{"text": "[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} fun x y => \u2200 (b : \u03b2), f x (f y b) = f y (f x b)\nb : \u03b2\n\u22a2 noncommFoldr f (\u2191l) comm b = List.foldr f b l\n[PROOFSTEP]\nsimp only [noncommFoldr, coe_foldr, coe_attach, List.attach, Function.comp]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} fun x y => \u2200 (b : \u03b2), f x (f y b) = f y (f x b)\nb : \u03b2\n\u22a2 List.foldr (fun x => f \u2191x) b (List.pmap Subtype.mk l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l)) = List.foldr f b l\n[PROOFSTEP]\nrw [\u2190 List.foldr_map]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} fun x y => \u2200 (b : \u03b2), f x (f y b) = f y (f x b)\nb : \u03b2\n\u22a2 List.foldr f b (List.map (fun x => \u2191x) (List.pmap Subtype.mk l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l))) = List.foldr f b l\n[PROOFSTEP]\nsimp [List.map_pmap, List.pmap_eq_map]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ns : Multiset \u03b1\na : \u03b1\nh : Set.Pairwise {x | x \u2208 a ::\u2098 s} fun x y => \u2200 (b : \u03b2), f x (f y b) = f y (f x b)\nh' : Set.Pairwise {x | x \u2208 s} fun x y => \u2200 (b : \u03b2), f x (f y b) = f y (f x b)\nb : \u03b2\n\u22a2 noncommFoldr f (a ::\u2098 s) h b = f a (noncommFoldr f s h' b)\n[PROOFSTEP]\ninduction s using Quotient.inductionOn\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\na : \u03b1\nb : \u03b2\na\u271d : List \u03b1\nh : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) a\u271d} fun x y => \u2200 (b : \u03b2), f x (f y b) = f y (f x b)\nh' : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} fun x y => \u2200 (b : \u03b2), f x (f y b) = f y (f x b)\n\u22a2 noncommFoldr f (a ::\u2098 Quotient.mk (List.isSetoid \u03b1) a\u271d) h b =\n f a (noncommFoldr f (Quotient.mk (List.isSetoid \u03b1) a\u271d) h' b)\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ns : Multiset \u03b1\nh : LeftCommutative f\nb : \u03b2\n\u22a2 noncommFoldr f s\n (_ : \u2200 (x : \u03b1), x \u2208 {x | x \u2208 s} \u2192 \u2200 (y : \u03b1), y \u2208 {x | x \u2208 s} \u2192 x \u2260 y \u2192 \u2200 (b : \u03b2), f x (f y b) = f y (f x b)) b =\n foldr f h b s\n[PROOFSTEP]\ninduction s using Quotient.inductionOn\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nh : LeftCommutative f\nb : \u03b2\na\u271d : List \u03b1\n\u22a2 noncommFoldr f (Quotient.mk (List.isSetoid \u03b1) a\u271d)\n (_ :\n \u2200 (x : \u03b1),\n x \u2208 {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} \u2192\n \u2200 (y : \u03b1), y \u2208 {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} \u2192 x \u2260 y \u2192 \u2200 (b : \u03b2), f x (f y b) = f y (f x b))\n b =\n foldr f h b (Quotient.mk (List.isSetoid \u03b1) a\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nassoc : IsAssociative \u03b1 op\ns : Multiset \u03b1\ncomm : Set.Pairwise {x | x \u2208 s} fun x y => op x y = op y x\nx : \u03b1\nhx : x \u2208 {x | x \u2208 s}\ny : \u03b1\nhy : y \u2208 {x | x \u2208 s}\nh : x \u2260 y\nb : \u03b1\n\u22a2 op x (op y b) = op y (op x b)\n[PROOFSTEP]\nrw [\u2190 assoc.assoc, comm hx hy h, assoc.assoc]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nassoc : IsAssociative \u03b1 op\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} fun x y => op x y = op y x\na : \u03b1\n\u22a2 noncommFold op (\u2191l) comm a = List.foldr op a l\n[PROOFSTEP]\nsimp [noncommFold]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nassoc : IsAssociative \u03b1 op\ns : Multiset \u03b1\na : \u03b1\nh : Set.Pairwise {x | x \u2208 a ::\u2098 s} fun x y => op x y = op y x\nh' : Set.Pairwise {x | x \u2208 s} fun x y => op x y = op y x\nx : \u03b1\n\u22a2 noncommFold op (a ::\u2098 s) h x = op a (noncommFold op s h' x)\n[PROOFSTEP]\ninduction s using Quotient.inductionOn\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nassoc : IsAssociative \u03b1 op\na x : \u03b1\na\u271d : List \u03b1\nh : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) a\u271d} fun x y => op x y = op y x\nh' : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} fun x y => op x y = op y x\n\u22a2 noncommFold op (a ::\u2098 Quotient.mk (List.isSetoid \u03b1) a\u271d) h x =\n op a (noncommFold op (Quotient.mk (List.isSetoid \u03b1) a\u271d) h' x)\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nassoc : IsAssociative \u03b1 op\ns : Multiset \u03b1\ninst\u271d : IsCommutative \u03b1 op\na : \u03b1\n\u22a2 noncommFold op s (_ : \u2200 (x : \u03b1), x \u2208 {x | x \u2208 s} \u2192 \u2200 (y : \u03b1), y \u2208 {x | x \u2208 s} \u2192 x \u2260 y \u2192 op x y = op y x) a =\n fold op a s\n[PROOFSTEP]\ninduction s using Quotient.inductionOn\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nassoc : IsAssociative \u03b1 op\ninst\u271d : IsCommutative \u03b1 op\na : \u03b1\na\u271d : List \u03b1\n\u22a2 noncommFold op (Quotient.mk (List.isSetoid \u03b1) a\u271d)\n (_ :\n \u2200 (x : \u03b1),\n x \u2208 {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} \u2192\n \u2200 (y : \u03b1), y \u2208 {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} \u2192 x \u2260 y \u2192 op x y = op y x)\n a =\n fold op a (Quotient.mk (List.isSetoid \u03b1) a\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} Commute\n\u22a2 noncommProd (\u2191l) comm = List.prod l\n[PROOFSTEP]\nrw [noncommProd]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} Commute\n\u22a2 noncommFold (fun x x_1 => x * x_1) (\u2191l) comm 1 = List.prod l\n[PROOFSTEP]\nsimp only [noncommFold_coe]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nl : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 \u2191l} Commute\n\u22a2 List.foldr (fun x x_1 => x * x_1) 1 l = List.prod l\n[PROOFSTEP]\ninduction' l with hd tl hl\n[GOAL]\ncase nil\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nl : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute\ncomm : Set.Pairwise {x | x \u2208 \u2191[]} Commute\n\u22a2 List.foldr (fun x x_1 => x * x_1) 1 [] = List.prod []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nl : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute\nhd : \u03b1\ntl : List \u03b1\nhl : Set.Pairwise {x | x \u2208 \u2191tl} Commute \u2192 List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl\ncomm : Set.Pairwise {x | x \u2208 \u2191(hd :: tl)} Commute\n\u22a2 List.foldr (fun x x_1 => x * x_1) 1 (hd :: tl) = List.prod (hd :: tl)\n[PROOFSTEP]\nrw [List.prod_cons, List.foldr, hl]\n[GOAL]\ncase cons\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nl : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute\nhd : \u03b1\ntl : List \u03b1\nhl : Set.Pairwise {x | x \u2208 \u2191tl} Commute \u2192 List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl\ncomm : Set.Pairwise {x | x \u2208 \u2191(hd :: tl)} Commute\n\u22a2 Set.Pairwise {x | x \u2208 \u2191tl} Commute\n[PROOFSTEP]\nintro x hx y hy\n[GOAL]\ncase cons\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nl : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 \u2191l} Commute\nhd : \u03b1\ntl : List \u03b1\nhl : Set.Pairwise {x | x \u2208 \u2191tl} Commute \u2192 List.foldr (fun x x_1 => x * x_1) 1 tl = List.prod tl\ncomm : Set.Pairwise {x | x \u2208 \u2191(hd :: tl)} Commute\nx : \u03b1\nhx : x \u2208 {x | x \u2208 \u2191tl}\ny : \u03b1\nhy : y \u2208 {x | x \u2208 \u2191tl}\n\u22a2 x \u2260 y \u2192 Commute x y\n[PROOFSTEP]\nexact comm (List.mem_cons_of_mem _ hx) (List.mem_cons_of_mem _ hy)\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns : Multiset \u03b1\na : \u03b1\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 s} Commute\n\u22a2 noncommProd (a ::\u2098 s) comm = a * noncommProd s (_ : Set.Pairwise {x | x \u2208 s} Commute)\n[PROOFSTEP]\ninduction s using Quotient.inductionOn\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\na : \u03b1\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\n\u22a2 noncommProd (a ::\u2098 Quotient.mk (List.isSetoid \u03b1) a\u271d) comm =\n a *\n noncommProd (Quotient.mk (List.isSetoid \u03b1) a\u271d)\n (_ : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute)\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns : Multiset \u03b1\na : \u03b1\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 s} Commute\n\u22a2 noncommProd (a ::\u2098 s) comm = noncommProd s (_ : Set.Pairwise {x | x \u2208 s} Commute) * a\n[PROOFSTEP]\ninduction' s using Quotient.inductionOn with s\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\n\u22a2 noncommProd (a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s) comm =\n noncommProd (Quotient.mk (List.isSetoid \u03b1) s) (_ : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) s} Commute) *\n a\n[PROOFSTEP]\nsimp only [quot_mk_to_coe, cons_coe, noncommProd_coe, List.prod_cons]\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\n\u22a2 a * List.prod s = List.prod s * a\n[PROOFSTEP]\ninduction' s with hd tl IH\n[GOAL]\ncase h.nil\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d\u00b9 : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) []} Commute\n\u22a2 a * List.prod [] = List.prod [] * a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.cons\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d\u00b9 : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\nhd : \u03b1\ntl : List \u03b1\nIH : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl} Commute \u2192 a * List.prod tl = List.prod tl * a\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)} Commute\n\u22a2 a * List.prod (hd :: tl) = List.prod (hd :: tl) * a\n[PROOFSTEP]\nrw [List.prod_cons, mul_assoc, \u2190 IH, \u2190 mul_assoc, \u2190 mul_assoc]\n[GOAL]\ncase h.cons\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d\u00b9 : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\nhd : \u03b1\ntl : List \u03b1\nIH : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl} Commute \u2192 a * List.prod tl = List.prod tl * a\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)} Commute\n\u22a2 a * hd * List.prod tl = hd * a * List.prod tl\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.cons.e_a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d\u00b9 : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\nhd : \u03b1\ntl : List \u03b1\nIH : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl} Commute \u2192 a * List.prod tl = List.prod tl * a\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)} Commute\n\u22a2 a * hd = hd * a\n[PROOFSTEP]\napply comm.of_refl\n[GOAL]\ncase h.cons.e_a.a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d\u00b9 : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\nhd : \u03b1\ntl : List \u03b1\nIH : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl} Commute \u2192 a * List.prod tl = List.prod tl * a\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)} Commute\n\u22a2 a \u2208 {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.cons.e_a.a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d\u00b9 : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\nhd : \u03b1\ntl : List \u03b1\nIH : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl} Commute \u2192 a * List.prod tl = List.prod tl * a\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)} Commute\n\u22a2 hd \u2208 {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.cons\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d\u00b9 : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\nhd : \u03b1\ntl : List \u03b1\nIH : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl} Commute \u2192 a * List.prod tl = List.prod tl * a\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)} Commute\n\u22a2 Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl} Commute\n[PROOFSTEP]\nintro x hx y hy\n[GOAL]\ncase h.cons\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d\u00b9 : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\nhd : \u03b1\ntl : List \u03b1\nIH : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl} Commute \u2192 a * List.prod tl = List.prod tl * a\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)} Commute\nx : \u03b1\nhx : x \u2208 {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl}\ny : \u03b1\nhy : y \u2208 {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl}\n\u22a2 x \u2260 y \u2192 Commute x y\n[PROOFSTEP]\nsimp only [quot_mk_to_coe, List.mem_cons, mem_coe, cons_coe] at hx hy \n[GOAL]\ncase h.cons\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d\u00b9 : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\nhd : \u03b1\ntl : List \u03b1\nIH : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl} Commute \u2192 a * List.prod tl = List.prod tl * a\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)} Commute\nx y : \u03b1\nhx : x \u2208 {x | x = a \u2228 x \u2208 tl}\nhy : y \u2208 {x | x = a \u2228 x \u2208 tl}\n\u22a2 x \u2260 y \u2192 Commute x y\n[PROOFSTEP]\napply comm\n[GOAL]\ncase h.cons.a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d\u00b9 : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\nhd : \u03b1\ntl : List \u03b1\nIH : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl} Commute \u2192 a * List.prod tl = List.prod tl * a\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)} Commute\nx y : \u03b1\nhx : x \u2208 {x | x = a \u2228 x \u2208 tl}\nhy : y \u2208 {x | x = a \u2228 x \u2208 tl}\n\u22a2 x \u2208 {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)}\n[PROOFSTEP]\ncases hx\n[GOAL]\ncase h.cons.a.inl\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d\u00b9 : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\nhd : \u03b1\ntl : List \u03b1\nIH : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl} Commute \u2192 a * List.prod tl = List.prod tl * a\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)} Commute\nx y : \u03b1\nhy : y \u2208 {x | x = a \u2228 x \u2208 tl}\nh\u271d : x = a\n\u22a2 x \u2208 {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)}\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase h.cons.a.inr\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d\u00b9 : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\nhd : \u03b1\ntl : List \u03b1\nIH : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl} Commute \u2192 a * List.prod tl = List.prod tl * a\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)} Commute\nx y : \u03b1\nhy : y \u2208 {x | x = a \u2228 x \u2208 tl}\nh\u271d : x \u2208 tl\n\u22a2 x \u2208 {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)}\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase h.cons.a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d\u00b9 : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\nhd : \u03b1\ntl : List \u03b1\nIH : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl} Commute \u2192 a * List.prod tl = List.prod tl * a\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)} Commute\nx y : \u03b1\nhx : x \u2208 {x | x = a \u2228 x \u2208 tl}\nhy : y \u2208 {x | x = a \u2228 x \u2208 tl}\n\u22a2 y \u2208 {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)}\n[PROOFSTEP]\ncases hy\n[GOAL]\ncase h.cons.a.inl\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d\u00b9 : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\nhd : \u03b1\ntl : List \u03b1\nIH : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl} Commute \u2192 a * List.prod tl = List.prod tl * a\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)} Commute\nx y : \u03b1\nhx : x \u2208 {x | x = a \u2228 x \u2208 tl}\nh\u271d : y = a\n\u22a2 y \u2208 {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)}\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase h.cons.a.inr\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns\u271d : Multiset \u03b1\na : \u03b1\ncomm\u271d\u00b9 : Set.Pairwise {x | x \u2208 a ::\u2098 s\u271d} Commute\ns : List \u03b1\ncomm\u271d : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) s} Commute\nhd : \u03b1\ntl : List \u03b1\nIH : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) tl} Commute \u2192 a * List.prod tl = List.prod tl * a\ncomm : Set.Pairwise {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)} Commute\nx y : \u03b1\nhx : x \u2208 {x | x = a \u2228 x \u2208 tl}\nh\u271d : y \u2208 tl\n\u22a2 y \u2208 {x | x \u2208 a ::\u2098 Quotient.mk (List.isSetoid \u03b1) (hd :: tl)}\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns t : Multiset \u03b1\ncomm : Set.Pairwise {x | x \u2208 s + t} Commute\n\u22a2 noncommProd (s + t) comm =\n noncommProd s (_ : Set.Pairwise {x | x \u2208 s} Commute) * noncommProd t (_ : Set.Pairwise {x | x \u2208 t} Commute)\n[PROOFSTEP]\nrcases s with \u27e8\u27e9\n[GOAL]\ncase mk\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns t : Multiset \u03b1\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quot.mk Setoid.r a\u271d + t} Commute\n\u22a2 noncommProd (Quot.mk Setoid.r a\u271d + t) comm =\n noncommProd (Quot.mk Setoid.r a\u271d) (_ : Set.Pairwise {x | x \u2208 Quot.mk Setoid.r a\u271d} Commute) *\n noncommProd t (_ : Set.Pairwise {x | x \u2208 t} Commute)\n[PROOFSTEP]\nrcases t with \u27e8\u27e9\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns t : Multiset \u03b1\na\u271d\u00b9 a\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quot.mk Setoid.r a\u271d\u00b9 + Quot.mk Setoid.r a\u271d} Commute\n\u22a2 noncommProd (Quot.mk Setoid.r a\u271d\u00b9 + Quot.mk Setoid.r a\u271d) comm =\n noncommProd (Quot.mk Setoid.r a\u271d\u00b9) (_ : Set.Pairwise {x | x \u2208 Quot.mk Setoid.r a\u271d\u00b9} Commute) *\n noncommProd (Quot.mk Setoid.r a\u271d) (_ : Set.Pairwise {x | x \u2208 Quot.mk Setoid.r a\u271d} Commute)\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\ns : Multiset \u03b1\ncomm : Set.Pairwise {x | x \u2208 s} Commute\nf : F\n\u22a2 Set.Pairwise {x | x \u2208 map (\u2191f) s} Commute\n[PROOFSTEP]\nsimp only [Multiset.mem_map]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\ns : Multiset \u03b1\ncomm : Set.Pairwise {x | x \u2208 s} Commute\nf : F\n\u22a2 Set.Pairwise {x | \u2203 a, a \u2208 s \u2227 \u2191f a = x} Commute\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9 _ \u27e8y, hy, rfl\u27e9 _\n[GOAL]\ncase intro.intro.intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\ns : Multiset \u03b1\ncomm : Set.Pairwise {x | x \u2208 s} Commute\nf : F\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\na\u271d : \u2191f x \u2260 \u2191f y\n\u22a2 Commute (\u2191f x) (\u2191f y)\n[PROOFSTEP]\nexact (comm.of_refl hx hy).map f\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\ns : Multiset \u03b1\ncomm : Set.Pairwise {x | x \u2208 s} Commute\nf : F\n\u22a2 \u2191f (noncommProd s comm) = noncommProd (map (\u2191f) s) (_ : Set.Pairwise {x | x \u2208 map (\u2191f) s} Commute)\n[PROOFSTEP]\ninduction s using Quotient.inductionOn\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : F\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\n\u22a2 \u2191f (noncommProd (Quotient.mk (List.isSetoid \u03b1) a\u271d) comm) =\n noncommProd (map (\u2191f) (Quotient.mk (List.isSetoid \u03b1) a\u271d))\n (_ : Set.Pairwise {x | x \u2208 map (\u2191f) (Quotient.mk (List.isSetoid \u03b1) a\u271d)} Commute)\n[PROOFSTEP]\nsimpa using map_list_prod f _\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns : Multiset \u03b1\ncomm : Set.Pairwise {x | x \u2208 s} Commute\nm : \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 x = m\n\u22a2 noncommProd s comm = m ^ \u2191card s\n[PROOFSTEP]\ninduction s using Quotient.inductionOn\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nm : \u03b1\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\nh : \u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d \u2192 x = m\n\u22a2 noncommProd (Quotient.mk (List.isSetoid \u03b1) a\u271d) comm = m ^ \u2191card (Quotient.mk (List.isSetoid \u03b1) a\u271d)\n[PROOFSTEP]\nsimp only [quot_mk_to_coe, noncommProd_coe, coe_card, mem_coe] at *\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nm : \u03b1\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\nh : \u2200 (x : \u03b1), x \u2208 a\u271d \u2192 x = m\n\u22a2 List.prod a\u271d = m ^ List.length a\u271d\n[PROOFSTEP]\nexact List.prod_eq_pow_card _ m h\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1\u271d \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1\u271d \u2192 \u03b1\u271d \u2192 \u03b1\u271d\ninst\u271d\u00b2 : Monoid \u03b1\u271d\ninst\u271d\u00b9 : Monoid \u03b2\n\u03b1 : Type u_6\ninst\u271d : CommMonoid \u03b1\ns : Multiset \u03b1\n\u22a2 noncommProd s (_ : \u2200 (x : \u03b1), x \u2208 {x | x \u2208 s} \u2192 \u2200 (x_2 : \u03b1), x_2 \u2208 {x | x \u2208 s} \u2192 x \u2260 x_2 \u2192 Commute x x_2) = prod s\n[PROOFSTEP]\ninduction s using Quotient.inductionOn\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1\u271d : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1\u271d \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1\u271d \u2192 \u03b1\u271d \u2192 \u03b1\u271d\ninst\u271d\u00b2 : Monoid \u03b1\u271d\ninst\u271d\u00b9 : Monoid \u03b2\n\u03b1 : Type u_6\ninst\u271d : CommMonoid \u03b1\na\u271d : List \u03b1\n\u22a2 noncommProd (Quotient.mk (List.isSetoid \u03b1) a\u271d)\n (_ :\n \u2200 (x : \u03b1),\n x \u2208 {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} \u2192\n \u2200 (x_2 : \u03b1), x_2 \u2208 {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} \u2192 x \u2260 x_2 \u2192 Commute x x_2) =\n prod (Quotient.mk (List.isSetoid \u03b1) a\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns : Multiset \u03b1\ncomm : Set.Pairwise {x | x \u2208 s} Commute\ny : \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y x\n\u22a2 Commute y (noncommProd s comm)\n[PROOFSTEP]\ninduction s using Quotient.inductionOn\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ny : \u03b1\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\nh : \u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d \u2192 Commute y x\n\u22a2 Commute y (noncommProd (Quotient.mk (List.isSetoid \u03b1) a\u271d) comm)\n[PROOFSTEP]\nsimp only [quot_mk_to_coe, noncommProd_coe]\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ny : \u03b1\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\nh : \u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d \u2192 Commute y x\n\u22a2 Commute y (List.prod a\u271d)\n[PROOFSTEP]\nexact Commute.list_prod_right _ _ h\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\n\u22a2 Set.Pairwise {x | x \u2208 Multiset.map f s.val} Commute\n[PROOFSTEP]\nsimp_rw [Multiset.mem_map]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\n\u22a2 Set.Pairwise {x | \u2203 a, a \u2208 s.val \u2227 f a = x} Commute\n[PROOFSTEP]\nrintro _ \u27e8a, ha, rfl\u27e9 _ \u27e8b, hb, rfl\u27e9 _\n[GOAL]\ncase intro.intro.intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\na : \u03b1\nha : a \u2208 s.val\nb : \u03b1\nhb : b \u2208 s.val\na\u271d : f a \u2260 f b\n\u22a2 Commute (f a) (f b)\n[PROOFSTEP]\nexact comm.of_refl ha hb\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u2081 s\u2082 : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\nh\u2081 : s\u2081 = s\u2082\nh\u2082 : \u2200 (x : \u03b1), x \u2208 s\u2082 \u2192 f x = g x\ncomm : Set.Pairwise \u2191s\u2081 fun a b => Commute (f a) (f b)\nx : \u03b1\nhx : x \u2208 \u2191s\u2082\ny : \u03b1\nhy : y \u2208 \u2191s\u2082\nh : x \u2260 y\n\u22a2 (fun a b => Commute (g a) (g b)) x y\n[PROOFSTEP]\ndsimp only\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u2081 s\u2082 : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\nh\u2081 : s\u2081 = s\u2082\nh\u2082 : \u2200 (x : \u03b1), x \u2208 s\u2082 \u2192 f x = g x\ncomm : Set.Pairwise \u2191s\u2081 fun a b => Commute (f a) (f b)\nx : \u03b1\nhx : x \u2208 \u2191s\u2082\ny : \u03b1\nhy : y \u2208 \u2191s\u2082\nh : x \u2260 y\n\u22a2 Commute (g x) (g y)\n[PROOFSTEP]\nrw [\u2190 h\u2082 _ hx, \u2190 h\u2082 _ hy]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u2081 s\u2082 : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\nh\u2081 : s\u2081 = s\u2082\nh\u2082 : \u2200 (x : \u03b1), x \u2208 s\u2082 \u2192 f x = g x\ncomm : Set.Pairwise \u2191s\u2081 fun a b => Commute (f a) (f b)\nx : \u03b1\nhx : x \u2208 \u2191s\u2082\ny : \u03b1\nhy : y \u2208 \u2191s\u2082\nh : x \u2260 y\n\u22a2 Commute (f x) (f y)\n[PROOFSTEP]\nsubst h\u2081\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u2081 : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s\u2081 fun a b => Commute (f a) (f b)\nx y : \u03b1\nh : x \u2260 y\nh\u2082 : \u2200 (x : \u03b1), x \u2208 s\u2081 \u2192 f x = g x\nhx : x \u2208 \u2191s\u2081\nhy : y \u2208 \u2191s\u2081\n\u22a2 Commute (f x) (f y)\n[PROOFSTEP]\nexact comm hx hy h\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u2081 s\u2082 : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\nh\u2081 : s\u2081 = s\u2082\nh\u2082 : \u2200 (x : \u03b1), x \u2208 s\u2082 \u2192 f x = g x\ncomm : Set.Pairwise \u2191s\u2081 fun a b => Commute (f a) (f b)\n\u22a2 noncommProd s\u2081 f comm =\n noncommProd s\u2082 g (_ : \u2200 (x : \u03b1), x \u2208 \u2191s\u2082 \u2192 \u2200 (y : \u03b1), y \u2208 \u2191s\u2082 \u2192 x \u2260 y \u2192 (fun a b => Commute (g a) (g b)) x y)\n[PROOFSTEP]\nsimp_rw [noncommProd, Multiset.map_congr (congr_arg _ h\u2081) h\u2082]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(List.toFinset l) fun a b => Commute (f a) (f b)\nhl : List.Nodup l\n\u22a2 noncommProd (List.toFinset l) f comm = List.prod (List.map f l)\n[PROOFSTEP]\nrw [\u2190 List.dedup_eq_self] at hl \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(List.toFinset l) fun a b => Commute (f a) (f b)\nhl\u271d : List.Nodup l\nhl : List.dedup l = l\n\u22a2 noncommProd (List.toFinset l) f comm = List.prod (List.map f l)\n[PROOFSTEP]\nsimp [noncommProd, hl]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\n\u22a2 Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute\n[PROOFSTEP]\nconvert noncommProd_lemma _ f comm using 3\n[GOAL]\ncase h.e'_2.h.e'_2.h.a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 f a ::\u2098 Multiset.map f s.val \u2194 x\u271d \u2208 Multiset.map f (insert a s).val\n[PROOFSTEP]\nsimp [@eq_comm _ (f a)]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\n\u22a2 Multiset.noncommProd (Multiset.map f (insert a s).val)\n (_ : Set.Pairwise {x | x \u2208 Multiset.map f (insert a s).val} Commute) =\n Multiset.noncommProd (f a ::\u2098 Multiset.map f s.val)\n (_ : Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute)\n[PROOFSTEP]\n{ congr\n rw [insert_val_of_not_mem ha, Multiset.map_cons]\n}\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\n\u22a2 Multiset.noncommProd (Multiset.map f (insert a s).val)\n (_ : Set.Pairwise {x | x \u2208 Multiset.map f (insert a s).val} Commute) =\n Multiset.noncommProd (f a ::\u2098 Multiset.map f s.val)\n (_ : Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\n\u22a2 Multiset.map f (insert a s).val = f a ::\u2098 Multiset.map f s.val\n[PROOFSTEP]\nrw [insert_val_of_not_mem ha, Multiset.map_cons]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\n\u22a2 Multiset.noncommProd (f a ::\u2098 Multiset.map f s.val)\n (_ : Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute) =\n f a * noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b))\n[PROOFSTEP]\nrw [Multiset.noncommProd_cons, noncommProd]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\n\u22a2 Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute\n[PROOFSTEP]\nconvert noncommProd_lemma _ f comm using 3\n[GOAL]\ncase h.e'_2.h.e'_2.h.a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 f a ::\u2098 Multiset.map f s.val \u2194 x\u271d \u2208 Multiset.map f (insert a s).val\n[PROOFSTEP]\nsimp [@eq_comm _ (f a)]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\n\u22a2 Multiset.noncommProd (Multiset.map f (insert a s).val)\n (_ : Set.Pairwise {x | x \u2208 Multiset.map f (insert a s).val} Commute) =\n Multiset.noncommProd (f a ::\u2098 Multiset.map f s.val)\n (_ : Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute)\n[PROOFSTEP]\n{ congr\n rw [insert_val_of_not_mem ha, Multiset.map_cons]\n}\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\n\u22a2 Multiset.noncommProd (Multiset.map f (insert a s).val)\n (_ : Set.Pairwise {x | x \u2208 Multiset.map f (insert a s).val} Commute) =\n Multiset.noncommProd (f a ::\u2098 Multiset.map f s.val)\n (_ : Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\n\u22a2 Multiset.map f (insert a s).val = f a ::\u2098 Multiset.map f s.val\n[PROOFSTEP]\nrw [insert_val_of_not_mem ha, Multiset.map_cons]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b)\nha : \u00aca \u2208 s\n\u22a2 Multiset.noncommProd (f a ::\u2098 Multiset.map f s.val)\n (_ : Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute) =\n noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) * f a\n[PROOFSTEP]\nrw [Multiset.noncommProd_cons', noncommProd]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\na : \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 Set.Pairwise \u2191{a} fun a b => Commute (f a) (f b)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\na : \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 Set.Pairwise {a} fun a b => Commute (f a) (f b)\n[PROOFSTEP]\nexact Set.pairwise_singleton _ _\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : MonoidHomClass F \u03b2 \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ng : F\n\u22a2 \u2191g (noncommProd s f comm) =\n noncommProd s (fun i => \u2191g (f i))\n (_ : \u2200 (x : \u03b1), x \u2208 \u2191s \u2192 \u2200 (y : \u03b1), y \u2208 \u2191s \u2192 x \u2260 y \u2192 Commute (\u2191g (f x)) (\u2191g (f y)))\n[PROOFSTEP]\nsimp [noncommProd, Multiset.noncommProd_map]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\nm : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x = m\n\u22a2 noncommProd s f comm = m ^ card s\n[PROOFSTEP]\nrw [noncommProd, Multiset.noncommProd_eq_pow_card _ _ m]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\nm : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x = m\n\u22a2 m ^ \u2191Multiset.card (Multiset.map f s.val) = m ^ card s\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\nm : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x = m\n\u22a2 \u2200 (x : \u03b2), x \u2208 Multiset.map f s.val \u2192 x = m\n[PROOFSTEP]\nsimp only [Finset.card_def, Multiset.card_map]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\nm : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x = m\n\u22a2 \u2200 (x : \u03b2), x \u2208 Multiset.map f s.val \u2192 x = m\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ny : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y (f x)\n\u22a2 Commute y (noncommProd s f comm)\n[PROOFSTEP]\napply Multiset.noncommProd_commute\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ny : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y (f x)\n\u22a2 \u2200 (x : \u03b2), x \u2208 Multiset.map f s.val \u2192 Commute y x\n[PROOFSTEP]\nintro y\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ny\u271d : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y\u271d (f x)\ny : \u03b2\n\u22a2 y \u2208 Multiset.map f s.val \u2192 Commute y\u271d y\n[PROOFSTEP]\nrw [Multiset.mem_map]\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ny\u271d : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y\u271d (f x)\ny : \u03b2\n\u22a2 (\u2203 a, a \u2208 s.val \u2227 f a = y) \u2192 Commute y\u271d y\n[PROOFSTEP]\nrintro \u27e8x, \u27e8hx, rfl\u27e9\u27e9\n[GOAL]\ncase h.intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)\ny : \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y (f x)\nx : \u03b1\nhx : x \u2208 s.val\n\u22a2 Commute y (f x)\n[PROOFSTEP]\nexact h x hx\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2\u271d \u2192 \u03b2\u271d\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\u271d\ninst\u271d\u00b9 : Monoid \u03b3\n\u03b2 : Type u_6\ninst\u271d : CommMonoid \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 noncommProd s f (_ : \u2200 (x : \u03b1), x \u2208 \u2191s \u2192 \u2200 (x_2 : \u03b1), x_2 \u2208 \u2191s \u2192 x \u2260 x_2 \u2192 Commute (f x) (f x_2)) = Finset.prod s f\n[PROOFSTEP]\nclassical\ninduction' s using Finset.induction_on with a s ha IH\n\u00b7 simp\n\u00b7 simp [ha, IH]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2\u271d \u2192 \u03b2\u271d\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\u271d\ninst\u271d\u00b9 : Monoid \u03b3\n\u03b2 : Type u_6\ninst\u271d : CommMonoid \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 noncommProd s f (_ : \u2200 (x : \u03b1), x \u2208 \u2191s \u2192 \u2200 (x_2 : \u03b1), x_2 \u2208 \u2191s \u2192 x \u2260 x_2 \u2192 Commute (f x) (f x_2)) = Finset.prod s f\n[PROOFSTEP]\ninduction' s using Finset.induction_on with a s ha IH\n[GOAL]\ncase empty\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2\u271d \u2192 \u03b2\u271d\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\u271d\ninst\u271d\u00b9 : Monoid \u03b3\n\u03b2 : Type u_6\ninst\u271d : CommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 noncommProd \u2205 f (_ : \u2200 (x : \u03b1), x \u2208 \u2191\u2205 \u2192 \u2200 (x_2 : \u03b1), x_2 \u2208 \u2191\u2205 \u2192 x \u2260 x_2 \u2192 Commute (f x) (f x_2)) = Finset.prod \u2205 f\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2\u271d \u2192 \u03b2\u271d\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\u271d\ninst\u271d\u00b9 : Monoid \u03b3\n\u03b2 : Type u_6\ninst\u271d : CommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nIH : noncommProd s f (_ : \u2200 (x : \u03b1), x \u2208 \u2191s \u2192 \u2200 (x_2 : \u03b1), x_2 \u2208 \u2191s \u2192 x \u2260 x_2 \u2192 Commute (f x) (f x_2)) = Finset.prod s f\n\u22a2 noncommProd (insert a s) f\n (_ : \u2200 (x : \u03b1), x \u2208 \u2191(insert a s) \u2192 \u2200 (x_2 : \u03b1), x_2 \u2208 \u2191(insert a s) \u2192 x \u2260 x_2 \u2192 Commute (f x) (f x_2)) =\n Finset.prod (insert a s) f\n[PROOFSTEP]\nsimp [ha, IH]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : Disjoint s t\nf : \u03b1 \u2192 \u03b2\ncomm : Set.Pairwise {x | x \u2208 s \u222a t} fun a b => Commute (f a) (f b)\n\u22a2 noncommProd (s \u222a t) f comm =\n noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) *\n noncommProd t f (_ : Set.Pairwise \u2191t fun a b => Commute (f a) (f b))\n[PROOFSTEP]\nobtain \u27e8sl, sl', rfl\u27e9 := exists_list_nodup_eq s\n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nt : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\nh : Disjoint (List.toFinset sl) t\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a t} fun a b => Commute (f a) (f b)\n\u22a2 noncommProd (List.toFinset sl \u222a t) f comm =\n noncommProd (List.toFinset sl) f (_ : Set.Pairwise \u2191(List.toFinset sl) fun a b => Commute (f a) (f b)) *\n noncommProd t f (_ : Set.Pairwise \u2191t fun a b => Commute (f a) (f b))\n[PROOFSTEP]\nobtain \u27e8tl, tl', rfl\u27e9 := exists_list_nodup_eq t\n[GOAL]\ncase intro.intro.intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : Disjoint (List.toFinset sl) (List.toFinset tl)\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 noncommProd (List.toFinset sl \u222a List.toFinset tl) f comm =\n noncommProd (List.toFinset sl) f (_ : Set.Pairwise \u2191(List.toFinset sl) fun a b => Commute (f a) (f b)) *\n noncommProd (List.toFinset tl) f (_ : Set.Pairwise \u2191(List.toFinset tl) fun a b => Commute (f a) (f b))\n[PROOFSTEP]\nrw [List.disjoint_toFinset_iff_disjoint] at h \n[GOAL]\ncase intro.intro.intro.intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 noncommProd (List.toFinset sl \u222a List.toFinset tl) f comm =\n noncommProd (List.toFinset sl) f (_ : Set.Pairwise \u2191(List.toFinset sl) fun a b => Commute (f a) (f b)) *\n noncommProd (List.toFinset tl) f (_ : Set.Pairwise \u2191(List.toFinset tl) fun a b => Commute (f a) (f b))\n[PROOFSTEP]\ncalc\n noncommProd (List.toFinset sl \u222a List.toFinset tl) f comm =\n noncommProd \u27e8\u2191(sl ++ tl), Multiset.coe_nodup.2 (sl'.append tl' h)\u27e9 f (by convert comm; simp [Set.ext_iff]) :=\n noncommProd_congr (by ext; simp) (by simp) _\n _ =\n noncommProd (List.toFinset sl) f (comm.mono <| coe_subset.2 <| subset_union_left _ _) *\n noncommProd (List.toFinset tl) f (comm.mono <| coe_subset.2 <| subset_union_right _ _) :=\n by simp [noncommProd, List.dedup_eq_self.2 sl', List.dedup_eq_self.2 tl', h]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 Set.Pairwise \u2191{ val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } fun a b => Commute (f a) (f b)\n[PROOFSTEP]\nconvert comm\n[GOAL]\ncase h.e'_2\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 \u2191{ val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } = {x | x \u2208 List.toFinset sl \u222a List.toFinset tl}\n[PROOFSTEP]\nsimp [Set.ext_iff]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 List.toFinset sl \u222a List.toFinset tl = { val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) }\n[PROOFSTEP]\next\n[GOAL]\ncase a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 List.toFinset sl \u222a List.toFinset tl \u2194 a\u271d \u2208 { val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) }\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 \u2200 (x : \u03b1), x \u2208 { val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } \u2192 f x = f x\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : Monoid \u03b3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nsl : List \u03b1\nsl' : List.Nodup sl\ntl : List \u03b1\ntl' : List.Nodup tl\nh : List.Disjoint sl tl\ncomm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b)\n\u22a2 noncommProd { val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } f\n (_ :\n Set.Pairwise \u2191{ val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } fun a b =>\n Commute (f a) (f b)) =\n noncommProd (List.toFinset sl) f (_ : Set.Pairwise \u2191(List.toFinset sl) fun a b => Commute (f a) (f b)) *\n noncommProd (List.toFinset tl) f (_ : Set.Pairwise \u2191(List.toFinset tl) fun a b => Commute (f a) (f b))\n[PROOFSTEP]\nsimp [noncommProd, List.dedup_eq_self.2 sl', List.dedup_eq_self.2 tl', h]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)\n\u22a2 Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)\n[PROOFSTEP]\nintro x hx y hy h\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)\nx : \u03b1\nhx : x \u2208 \u2191s\ny : \u03b1\nhy : y \u2208 \u2191s\nh : x \u2260 y\n\u22a2 Commute ((f * g) x) ((f * g) y)\n[PROOFSTEP]\napply Commute.mul_left\n[GOAL]\ncase hac\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)\nx : \u03b1\nhx : x \u2208 \u2191s\ny : \u03b1\nhy : y \u2208 \u2191s\nh : x \u2260 y\n\u22a2 Commute (f x) ((f * g) y)\n[PROOFSTEP]\napply Commute.mul_right\n[GOAL]\ncase hbc\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)\nx : \u03b1\nhx : x \u2208 \u2191s\ny : \u03b1\nhy : y \u2208 \u2191s\nh : x \u2260 y\n\u22a2 Commute (g x) ((f * g) y)\n[PROOFSTEP]\napply Commute.mul_right\n[GOAL]\ncase hac.hab\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)\nx : \u03b1\nhx : x \u2208 \u2191s\ny : \u03b1\nhy : y \u2208 \u2191s\nh : x \u2260 y\n\u22a2 Commute (f x) (f y)\n[PROOFSTEP]\nexact comm_ff.of_refl hx hy\n[GOAL]\ncase hac.hac\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)\nx : \u03b1\nhx : x \u2208 \u2191s\ny : \u03b1\nhy : y \u2208 \u2191s\nh : x \u2260 y\n\u22a2 Commute (f x) (g y)\n[PROOFSTEP]\nexact (comm_gf hy hx h.symm).symm\n[GOAL]\ncase hbc.hab\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)\nx : \u03b1\nhx : x \u2208 \u2191s\ny : \u03b1\nhy : y \u2208 \u2191s\nh : x \u2260 y\n\u22a2 Commute (g x) (f y)\n[PROOFSTEP]\nexact comm_gf hx hy h\n[GOAL]\ncase hbc.hac\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)\nx : \u03b1\nhx : x \u2208 \u2191s\ny : \u03b1\nhy : y \u2208 \u2191s\nh : x \u2260 y\n\u22a2 Commute (g x) (g y)\n[PROOFSTEP]\nexact comm_gg.of_refl hx hy\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)\n\u22a2 noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n noncommProd s f comm_ff * noncommProd s g comm_gg\n[PROOFSTEP]\nclassical\ninduction' s using Finset.induction_on with x s hnmem ih\n\u00b7 simp\nsimp only [Finset.noncommProd_insert_of_not_mem _ _ _ _ hnmem]\nspecialize\n ih (comm_ff.mono fun _ => mem_insert_of_mem) (comm_gg.mono fun _ => mem_insert_of_mem)\n (comm_gf.mono fun _ => mem_insert_of_mem)\nrw [ih, Pi.mul_apply]\nsimp only [mul_assoc]\ncongr 1\nsimp only [\u2190 mul_assoc]\ncongr 1\nrefine' noncommProd_commute _ _ _ _ fun y hy => _\nexact comm_gf (mem_insert_self x s) (mem_insert_of_mem hy) (ne_of_mem_of_not_mem hy hnmem).symm\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)\n\u22a2 noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n noncommProd s f comm_ff * noncommProd s g comm_gg\n[PROOFSTEP]\ninduction' s using Finset.induction_on with x s hnmem ih\n[GOAL]\ncase empty\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)\ncomm_ff : Set.Pairwise \u2191\u2205 fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191\u2205 fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191\u2205 fun x y => Commute (g x) (f y)\n\u22a2 noncommProd \u2205 (f * g) (_ : Set.Pairwise \u2191\u2205 fun x y => Commute ((f * g) x) ((f * g) y)) =\n noncommProd \u2205 f comm_ff * noncommProd \u2205 g comm_gg\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\nih :\n \u2200 (comm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y))\n (comm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n (comm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)),\n noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n noncommProd s f comm_ff * noncommProd s g comm_gg\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\n\u22a2 noncommProd (insert x s) (f * g) (_ : Set.Pairwise \u2191(insert x s) fun x y => Commute ((f * g) x) ((f * g) y)) =\n noncommProd (insert x s) f comm_ff * noncommProd (insert x s) g comm_gg\n[PROOFSTEP]\nsimp only [Finset.noncommProd_insert_of_not_mem _ _ _ _ hnmem]\n[GOAL]\ncase insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\nih :\n \u2200 (comm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y))\n (comm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n (comm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)),\n noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n noncommProd s f comm_ff * noncommProd s g comm_gg\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\n\u22a2 (f * g) x * noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun a b => Commute ((f * g) a) ((f * g) b)) =\n f x * noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) *\n (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun a b => Commute (g a) (g b)))\n[PROOFSTEP]\nspecialize\n ih (comm_ff.mono fun _ => mem_insert_of_mem) (comm_gg.mono fun _ => mem_insert_of_mem)\n (comm_gf.mono fun _ => mem_insert_of_mem)\n[GOAL]\ncase insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 (f * g) x * noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun a b => Commute ((f * g) a) ((f * g) b)) =\n f x * noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) *\n (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun a b => Commute (g a) (g b)))\n[PROOFSTEP]\nrw [ih, Pi.mul_apply]\n[GOAL]\ncase insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 f x * g x *\n (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))) =\n f x * noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) *\n (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun a b => Commute (g a) (g b)))\n[PROOFSTEP]\nsimp only [mul_assoc]\n[GOAL]\ncase insert\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 f x *\n (g x *\n (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)))) =\n f x *\n (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase insert.e_a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 g x *\n (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))) =\n noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)))\n[PROOFSTEP]\nsimp only [\u2190 mul_assoc]\n[GOAL]\ncase insert.e_a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 g x * noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)) =\n noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) * g x *\n noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase insert.e_a.e_a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\n\u22a2 g x * noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) =\n noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) * g x\n[PROOFSTEP]\nrefine' noncommProd_commute _ _ _ _ fun y hy => _\n[GOAL]\ncase insert.e_a.e_a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : Monoid \u03b3\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u03b2\ncomm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y)\ncomm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y)\ncomm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y)\nx : \u03b1\ns : Finset \u03b1\nhnmem : \u00acx \u2208 s\ncomm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y)\ncomm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y)\ncomm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y)\nih :\n noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =\n noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *\n noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))\ny : \u03b1\nhy : y \u2208 s\n\u22a2 Commute (g x) (f y)\n[PROOFSTEP]\nexact comm_gf (mem_insert_self x s) (mem_insert_of_mem hy) (ne_of_mem_of_not_mem hy hnmem).symm\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\n\u22a2 noncommProd univ (fun i => Pi.mulSingle i (x i))\n (_ :\n \u2200 (i : \u03b9), i \u2208 \u2191univ \u2192 \u2200 (j : \u03b9), j \u2208 \u2191univ \u2192 i \u2260 j \u2192 Commute (Pi.mulSingle i (x i)) (Pi.mulSingle j (x j))) =\n x\n[PROOFSTEP]\next i\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni : \u03b9\n\u22a2 noncommProd univ (fun i => Pi.mulSingle i (x i))\n (_ : \u2200 (i : \u03b9), i \u2208 \u2191univ \u2192 \u2200 (j : \u03b9), j \u2208 \u2191univ \u2192 i \u2260 j \u2192 Commute (Pi.mulSingle i (x i)) (Pi.mulSingle j (x j)))\n i =\n x i\n[PROOFSTEP]\napply (univ.noncommProd_map (fun i => MonoidHom.single M i (x i)) _ (Pi.evalMonoidHom M i)).trans\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni : \u03b9\n\u22a2 noncommProd univ (fun i_1 => \u2191(Pi.evalMonoidHom M i) (\u2191(MonoidHom.single M i_1) (x i_1)))\n (_ :\n \u2200 (x_1 : \u03b9),\n x_1 \u2208 \u2191univ \u2192\n \u2200 (y : \u03b9),\n y \u2208 \u2191univ \u2192\n x_1 \u2260 y \u2192\n Commute (\u2191(Pi.evalMonoidHom M i) (\u2191(MonoidHom.single M x_1) (x x_1)))\n (\u2191(Pi.evalMonoidHom M i) (\u2191(MonoidHom.single M y) (x y)))) =\n x i\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni : \u03b9\n\u22a2 Set.Pairwise \u2191univ fun a b =>\n Commute ((fun i => \u2191(MonoidHom.single M i) (x i)) a) ((fun i => \u2191(MonoidHom.single M i) (x i)) b)\n[PROOFSTEP]\nrefine' (noncommProd_congr (insert_erase (mem_univ i)).symm _ _).trans _\n[GOAL]\ncase h.refine'_1\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni : \u03b9\n\u22a2 Set.Pairwise \u2191univ fun a b =>\n Commute ((fun i => \u2191(MonoidHom.single M i) (x i)) a) ((fun i => \u2191(MonoidHom.single M i) (x i)) b)\n[PROOFSTEP]\nintro i _ j _ _\n[GOAL]\ncase h.refine'_1\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni\u271d i : \u03b9\na\u271d\u00b2 : i \u2208 \u2191univ\nj : \u03b9\na\u271d\u00b9 : j \u2208 \u2191univ\na\u271d : i \u2260 j\n\u22a2 Commute ((fun i => \u2191(MonoidHom.single M i) (x i)) i) ((fun i => \u2191(MonoidHom.single M i) (x i)) j)\n[PROOFSTEP]\nexact Pi.mulSingle_apply_commute x i j\n[GOAL]\ncase h.refine'_2\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni : \u03b9\n\u22a2 \u03b9 \u2192\n (fun x => M i)\n (noncommProd univ (fun i => \u2191(MonoidHom.single M i) (x i))\n (_ :\n \u2200 \u2983i : \u03b9\u2984, i \u2208 \u2191univ \u2192 \u2200 \u2983j : \u03b9\u2984, j \u2208 \u2191univ \u2192 i \u2260 j \u2192 Commute (Pi.mulSingle i (x i)) (Pi.mulSingle j (x j))))\n[PROOFSTEP]\nintro j\n[GOAL]\ncase h.refine'_2\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni j : \u03b9\n\u22a2 (fun x => M i)\n (noncommProd univ (fun i => \u2191(MonoidHom.single M i) (x i))\n (_ : \u2200 \u2983i : \u03b9\u2984, i \u2208 \u2191univ \u2192 \u2200 \u2983j : \u03b9\u2984, j \u2208 \u2191univ \u2192 i \u2260 j \u2192 Commute (Pi.mulSingle i (x i)) (Pi.mulSingle j (x j))))\n[PROOFSTEP]\nexact Pi.mulSingle j (x j) i\n[GOAL]\ncase h.refine'_3\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni : \u03b9\n\u22a2 \u2200 (x_1 : \u03b9),\n x_1 \u2208 insert i (erase univ i) \u2192\n \u2191(Pi.evalMonoidHom M i) (\u2191(MonoidHom.single M x_1) (x x_1)) = Pi.mulSingle x_1 (x x_1) i\n[PROOFSTEP]\nintro j _\n[GOAL]\ncase h.refine'_3\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni j : \u03b9\na\u271d : j \u2208 insert i (erase univ i)\n\u22a2 \u2191(Pi.evalMonoidHom M i) (\u2191(MonoidHom.single M j) (x j)) = Pi.mulSingle j (x j) i\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.refine'_4\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni : \u03b9\n\u22a2 noncommProd (insert i (erase univ i)) (fun j => Pi.mulSingle j (x j) i)\n (_ :\n \u2200 (x_1 : \u03b9),\n x_1 \u2208 \u2191(insert i (erase univ i)) \u2192\n \u2200 (y : \u03b9),\n y \u2208 \u2191(insert i (erase univ i)) \u2192\n x_1 \u2260 y \u2192 (fun a b => Commute (Pi.mulSingle a (x a) i) (Pi.mulSingle b (x b) i)) x_1 y) =\n x i\n[PROOFSTEP]\nrw [noncommProd_insert_of_not_mem _ _ _ _ (not_mem_erase _ _), noncommProd_eq_pow_card (univ.erase i), one_pow, mul_one]\n[GOAL]\ncase h.refine'_4\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni : \u03b9\n\u22a2 Pi.mulSingle i (x i) i = x i\ncase h.refine'_4.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni : \u03b9\n\u22a2 \u2200 (x_1 : \u03b9), x_1 \u2208 erase univ i \u2192 Pi.mulSingle x_1 (x x_1) i = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.refine'_4.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni : \u03b9\n\u22a2 \u2200 (x_1 : \u03b9), x_1 \u2208 erase univ i \u2192 Pi.mulSingle x_1 (x x_1) i = 1\n[PROOFSTEP]\nintro j hj\n[GOAL]\ncase h.refine'_4.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni j : \u03b9\nhj : j \u2208 erase univ i\n\u22a2 Pi.mulSingle j (x j) i = 1\n[PROOFSTEP]\nsimp at hj \n[GOAL]\ncase h.refine'_4.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni j : \u03b9\nhj : \u00acj = i\n\u22a2 Pi.mulSingle j (x j) i = 1\n[PROOFSTEP]\nsimp [Pi.mulSingle, Function.update]\n[GOAL]\ncase h.refine'_4.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni j : \u03b9\nhj : \u00acj = i\n\u22a2 \u2200 (h : i = j), (_ : j = i) \u25b8 x j = 1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.refine'_4.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nx : (i : \u03b9) \u2192 M i\ni j : \u03b9\nhj : \u00acj = i\nh : i = j\n\u22a2 (_ : j = i) \u25b8 x j = 1\n[PROOFSTEP]\nsimp [*] at *\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3\nh : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x)\n\u22a2 f = g\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3\nh : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x)\nval\u271d : Fintype \u03b9\n\u22a2 f = g\n[PROOFSTEP]\next x\n[GOAL]\ncase intro.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3\nh : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x)\nval\u271d : Fintype \u03b9\nx : (i : \u03b9) \u2192 M i\n\u22a2 \u2191f x = \u2191g x\n[PROOFSTEP]\nrw [\u2190 noncommProd_mul_single x, univ.noncommProd_map, univ.noncommProd_map]\n[GOAL]\ncase intro.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3\nh : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x)\nval\u271d : Fintype \u03b9\nx : (i : \u03b9) \u2192 M i\n\u22a2 noncommProd univ (fun i => \u2191f (Pi.mulSingle i (x i)))\n (_ :\n \u2200 (x_1 : \u03b9),\n x_1 \u2208 \u2191univ \u2192\n \u2200 (y : \u03b9), y \u2208 \u2191univ \u2192 x_1 \u2260 y \u2192 Commute (\u2191f (Pi.mulSingle x_1 (x x_1))) (\u2191f (Pi.mulSingle y (x y)))) =\n noncommProd univ (fun i => \u2191g (Pi.mulSingle i (x i)))\n (_ :\n \u2200 (x_1 : \u03b9),\n x_1 \u2208 \u2191univ \u2192\n \u2200 (y : \u03b9), y \u2208 \u2191univ \u2192 x_1 \u2260 y \u2192 Commute (\u2191g (Pi.mulSingle x_1 (x x_1))) (\u2191g (Pi.mulSingle y (x y))))\n[PROOFSTEP]\ncongr 1 with i\n[GOAL]\ncase intro.h.e_f.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : Monoid \u03b3\nM : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3\nh : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x)\nval\u271d : Fintype \u03b9\nx : (i : \u03b9) \u2192 M i\ni : \u03b9\n\u22a2 \u2191f (Pi.mulSingle i (x i)) = \u2191g (Pi.mulSingle i (x i))\n[PROOFSTEP]\nexact h i (x i)\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.NoncommProd", "llama_tokens": 38871, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707283, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.44998973438497775}} {"text": "[GOAL]\nV : Type (u + 1)\ninst\u271d : LargeCategory V\nG : MonCat\nA : Action V G\n\u22a2 \u2191A.\u03c1 1 = \ud835\udfd9 A.V\n[PROOFSTEP]\nrw [MonoidHom.map_one]\n[GOAL]\nV : Type (u + 1)\ninst\u271d : LargeCategory V\nG : MonCat\nA : Action V G\n\u22a2 1 = \ud835\udfd9 A.V\n[PROOFSTEP]\nrfl\n[GOAL]\nV : Type (u + 1)\ninst\u271d : LargeCategory V\nG : GroupCat\nA : Action V (MonCat.of \u2191G)\ng : \u2191G\n\u22a2 \u2191A.\u03c1 (g\u207b\u00b9 * g) = \ud835\udfd9 A.V\n[PROOFSTEP]\nrw [inv_mul_self, \u03c1_one]\n[GOAL]\nV : Type (u + 1)\ninst\u271d : LargeCategory V\nG : GroupCat\nA : Action V (MonCat.of \u2191G)\ng : \u2191G\n\u22a2 \u2191A.\u03c1 (g * g\u207b\u00b9) = \ud835\udfd9 A.V\n[PROOFSTEP]\nrw [mul_inv_self, \u03c1_one]\n[GOAL]\nV : Type (u + 1)\ninst\u271d : LargeCategory V\nG : MonCat\nM N : Action V G\nf : M.V \u2245 N.V\ncomm : autoParam (\u2200 (g : \u2191G), \u2191M.\u03c1 g \u226b f.hom = f.hom \u226b \u2191N.\u03c1 g) _auto\u271d\ng : \u2191G\n\u22a2 \u2191N.\u03c1 g \u226b f.inv = f.inv \u226b \u2191M.\u03c1 g\n[PROOFSTEP]\nhave w := comm g =\u226b f.inv\n[GOAL]\nV : Type (u + 1)\ninst\u271d : LargeCategory V\nG : MonCat\nM N : Action V G\nf : M.V \u2245 N.V\ncomm : autoParam (\u2200 (g : \u2191G), \u2191M.\u03c1 g \u226b f.hom = f.hom \u226b \u2191N.\u03c1 g) _auto\u271d\ng : \u2191G\nw : (\u2191M.\u03c1 g \u226b f.hom) \u226b f.inv = (f.hom \u226b \u2191N.\u03c1 g) \u226b f.inv\n\u22a2 \u2191N.\u03c1 g \u226b f.inv = f.inv \u226b \u2191M.\u03c1 g\n[PROOFSTEP]\nsimp at w \n[GOAL]\nV : Type (u + 1)\ninst\u271d : LargeCategory V\nG : MonCat\nM N : Action V G\nf : M.V \u2245 N.V\ncomm : autoParam (\u2200 (g : \u2191G), \u2191M.\u03c1 g \u226b f.hom = f.hom \u226b \u2191N.\u03c1 g) _auto\u271d\ng : \u2191G\nw : \u2191M.\u03c1 g = f.hom \u226b \u2191N.\u03c1 g \u226b f.inv\n\u22a2 \u2191N.\u03c1 g \u226b f.inv = f.inv \u226b \u2191M.\u03c1 g\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nV : Type (u + 1)\ninst\u271d : LargeCategory V\nG : MonCat\nM N : Action V G\nf : M \u2245 N\ng : \u2191G\n\u22a2 \u2191N.\u03c1 g = \u2191(Iso.conj ((forget V G).mapIso f)) (\u2191M.\u03c1 g)\n[PROOFSTEP]\nrw [Iso.conj_apply, Iso.eq_inv_comp]\n[GOAL]\nV : Type (u + 1)\ninst\u271d : LargeCategory V\nG : MonCat\nM N : Action V G\nf : M \u2245 N\ng : \u2191G\n\u22a2 ((forget V G).mapIso f).hom \u226b \u2191N.\u03c1 g = \u2191M.\u03c1 g \u226b ((forget V G).mapIso f).hom\n[PROOFSTEP]\nsimp [f.hom.comm]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : HasZeroMorphisms V\nX Y : Action V G\n\u22a2 \u2200 (g : \u2191G), \u2191X.\u03c1 g \u226b 0 = 0 \u226b \u2191Y.\u03c1 g\n[PROOFSTEP]\naesop_cat\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\nf g : X \u27f6 Y\n\u22a2 \u2200 (g_1 : \u2191G), \u2191X.\u03c1 g_1 \u226b (f.hom + g.hom) = (f.hom + g.hom) \u226b \u2191Y.\u03c1 g_1\n[PROOFSTEP]\nsimp [f.comm, g.comm]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\n\u22a2 \u2200 (a b c : X \u27f6 Y), a + b + c = a + (b + c)\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\na\u271d b\u271d c\u271d : X \u27f6 Y\n\u22a2 a\u271d + b\u271d + c\u271d = a\u271d + (b\u271d + c\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\na\u271d b\u271d c\u271d : X \u27f6 Y\n\u22a2 (a\u271d + b\u271d + c\u271d).hom = (a\u271d + (b\u271d + c\u271d)).hom\n[PROOFSTEP]\nexact add_assoc _ _ _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\n\u22a2 \u2200 (a : X \u27f6 Y), 0 + a = a\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\na\u271d : X \u27f6 Y\n\u22a2 0 + a\u271d = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\na\u271d : X \u27f6 Y\n\u22a2 (0 + a\u271d).hom = a\u271d.hom\n[PROOFSTEP]\nexact zero_add _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\n\u22a2 \u2200 (a : X \u27f6 Y), a + 0 = a\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\na\u271d : X \u27f6 Y\n\u22a2 a\u271d + 0 = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\na\u271d : X \u27f6 Y\n\u22a2 (a\u271d + 0).hom = a\u271d.hom\n[PROOFSTEP]\nexact add_zero _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\nf : X \u27f6 Y\n\u22a2 \u2200 (g : \u2191G), \u2191X.\u03c1 g \u226b (-f.hom) = (-f.hom) \u226b \u2191Y.\u03c1 g\n[PROOFSTEP]\nsimp [f.comm]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\n\u22a2 \u2200 (a : X \u27f6 Y), -a + a = 0\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\na\u271d : X \u27f6 Y\n\u22a2 -a\u271d + a\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\na\u271d : X \u27f6 Y\n\u22a2 (-a\u271d + a\u271d).hom = 0.hom\n[PROOFSTEP]\nexact add_left_neg _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\n\u22a2 \u2200 (a b : X \u27f6 Y), a + b = b + a\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\na\u271d b\u271d : X \u27f6 Y\n\u22a2 a\u271d + b\u271d = b\u271d + a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nX Y : Action V G\na\u271d b\u271d : X \u27f6 Y\n\u22a2 (a\u271d + b\u271d).hom = (b\u271d + a\u271d).hom\n[PROOFSTEP]\nexact add_comm _ _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\n\u22a2 \u2200 (P Q R : Action V G) (f f' : P \u27f6 Q) (g : Q \u27f6 R), (f + f') \u226b g = f \u226b g + f' \u226b g\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nP\u271d Q\u271d R\u271d : Action V G\nf\u271d f'\u271d : P\u271d \u27f6 Q\u271d\ng\u271d : Q\u271d \u27f6 R\u271d\n\u22a2 (f\u271d + f'\u271d) \u226b g\u271d = f\u271d \u226b g\u271d + f'\u271d \u226b g\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nP\u271d Q\u271d R\u271d : Action V G\nf\u271d f'\u271d : P\u271d \u27f6 Q\u271d\ng\u271d : Q\u271d \u27f6 R\u271d\n\u22a2 ((f\u271d + f'\u271d) \u226b g\u271d).hom = (f\u271d \u226b g\u271d + f'\u271d \u226b g\u271d).hom\n[PROOFSTEP]\nexact Preadditive.add_comp _ _ _ _ _ _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\n\u22a2 \u2200 (P Q R : Action V G) (f : P \u27f6 Q) (g g' : Q \u27f6 R), f \u226b (g + g') = f \u226b g + f \u226b g'\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nP\u271d Q\u271d R\u271d : Action V G\nf\u271d : P\u271d \u27f6 Q\u271d\ng\u271d g'\u271d : Q\u271d \u27f6 R\u271d\n\u22a2 f\u271d \u226b (g\u271d + g'\u271d) = f\u271d \u226b g\u271d + f\u271d \u226b g'\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : Preadditive V\nP\u271d Q\u271d R\u271d : Action V G\nf\u271d : P\u271d \u27f6 Q\u271d\ng\u271d g'\u271d : Q\u271d \u27f6 R\u271d\n\u22a2 (f\u271d \u226b (g\u271d + g'\u271d)).hom = (f\u271d \u226b g\u271d + f\u271d \u226b g'\u271d).hom\n[PROOFSTEP]\nexact Preadditive.comp_add _ _ _ _ _ _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\nr : R\nf : X \u27f6 Y\n\u22a2 \u2200 (g : \u2191G), \u2191X.\u03c1 g \u226b (r \u2022 f.hom) = (r \u2022 f.hom) \u226b \u2191Y.\u03c1 g\n[PROOFSTEP]\nsimp [f.comm]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\n\u22a2 \u2200 (b : X \u27f6 Y), 1 \u2022 b = b\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\nb\u271d : X \u27f6 Y\n\u22a2 1 \u2022 b\u271d = b\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\nb\u271d : X \u27f6 Y\n\u22a2 (1 \u2022 b\u271d).hom = b\u271d.hom\n[PROOFSTEP]\nexact one_smul _ _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\n\u22a2 \u2200 (x y : R) (b : X \u27f6 Y), (x * y) \u2022 b = x \u2022 y \u2022 b\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\nx\u271d y\u271d : R\nb\u271d : X \u27f6 Y\n\u22a2 (x\u271d * y\u271d) \u2022 b\u271d = x\u271d \u2022 y\u271d \u2022 b\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\nx\u271d y\u271d : R\nb\u271d : X \u27f6 Y\n\u22a2 ((x\u271d * y\u271d) \u2022 b\u271d).hom = (x\u271d \u2022 y\u271d \u2022 b\u271d).hom\n[PROOFSTEP]\nexact mul_smul _ _ _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\n\u22a2 \u2200 (a : R), a \u2022 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\na\u271d : R\n\u22a2 a\u271d \u2022 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\na\u271d : R\n\u22a2 (a\u271d \u2022 0).hom = 0.hom\n[PROOFSTEP]\nexact smul_zero _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\n\u22a2 \u2200 (a : R) (x y : X \u27f6 Y), a \u2022 (x + y) = a \u2022 x + a \u2022 y\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\na\u271d : R\nx\u271d y\u271d : X \u27f6 Y\n\u22a2 a\u271d \u2022 (x\u271d + y\u271d) = a\u271d \u2022 x\u271d + a\u271d \u2022 y\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\na\u271d : R\nx\u271d y\u271d : X \u27f6 Y\n\u22a2 (a\u271d \u2022 (x\u271d + y\u271d)).hom = (a\u271d \u2022 x\u271d + a\u271d \u2022 y\u271d).hom\n[PROOFSTEP]\nexact smul_add _ _ _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\n\u22a2 \u2200 (r s : R) (x : X \u27f6 Y), (r + s) \u2022 x = r \u2022 x + s \u2022 x\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\nr\u271d s\u271d : R\nx\u271d : X \u27f6 Y\n\u22a2 (r\u271d + s\u271d) \u2022 x\u271d = r\u271d \u2022 x\u271d + s\u271d \u2022 x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\nr\u271d s\u271d : R\nx\u271d : X \u27f6 Y\n\u22a2 ((r\u271d + s\u271d) \u2022 x\u271d).hom = (r\u271d \u2022 x\u271d + s\u271d \u2022 x\u271d).hom\n[PROOFSTEP]\nexact add_smul _ _ _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\n\u22a2 \u2200 (x : X \u27f6 Y), 0 \u2022 x = 0\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\nx\u271d : X \u27f6 Y\n\u22a2 0 \u2022 x\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX Y : Action V G\nx\u271d : X \u27f6 Y\n\u22a2 (0 \u2022 x\u271d).hom = 0.hom\n[PROOFSTEP]\nexact zero_smul _ _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\n\u22a2 \u2200 (X Y Z : Action V G) (r : R) (f : X \u27f6 Y) (g : Y \u27f6 Z), (r \u2022 f) \u226b g = r \u2022 f \u226b g\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX\u271d Y\u271d Z\u271d : Action V G\nr\u271d : R\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 (r\u271d \u2022 f\u271d) \u226b g\u271d = r\u271d \u2022 f\u271d \u226b g\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX\u271d Y\u271d Z\u271d : Action V G\nr\u271d : R\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 ((r\u271d \u2022 f\u271d) \u226b g\u271d).hom = (r\u271d \u2022 f\u271d \u226b g\u271d).hom\n[PROOFSTEP]\nexact Linear.smul_comp _ _ _ _ _ _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\n\u22a2 \u2200 (X Y Z : Action V G) (f : X \u27f6 Y) (r : R) (g : Y \u27f6 Z), f \u226b (r \u2022 g) = r \u2022 f \u226b g\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX\u271d Y\u271d Z\u271d : Action V G\nf\u271d : X\u271d \u27f6 Y\u271d\nr\u271d : R\ng\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 f\u271d \u226b (r\u271d \u2022 g\u271d) = r\u271d \u2022 f\u271d \u226b g\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG : MonCat\ninst\u271d\u00b2 : Preadditive V\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R V\nX\u271d Y\u271d Z\u271d : Action V G\nf\u271d : X\u271d \u27f6 Y\u271d\nr\u271d : R\ng\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 (f\u271d \u226b (r\u271d \u2022 g\u271d)).hom = (r\u271d \u2022 f\u271d \u226b g\u271d).hom\n[PROOFSTEP]\nexact Linear.comp_smul _ _ _ _ _ _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX Y Z : Action V G\n\u22a2 (\u03b1_ X Y Z).hom.hom = (\u03b1_ X.V Y.V Z.V).hom\n[PROOFSTEP]\ndsimp [Monoidal.transport_associator]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX Y Z : Action V G\n\u22a2 (\ud835\udfd9 (X.V \u2297 Y.V) \u2297 \ud835\udfd9 Z.V) \u226b (\u03b1_ X.V Y.V Z.V).hom \u226b (\ud835\udfd9 X.V \u2297 \ud835\udfd9 (Y.V \u2297 Z.V)) = (\u03b1_ X.V Y.V Z.V).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX Y Z : Action V G\n\u22a2 (\u03b1_ X Y Z).inv.hom = (\u03b1_ X.V Y.V Z.V).inv\n[PROOFSTEP]\ndsimp [Monoidal.transport_associator]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX Y Z : Action V G\n\u22a2 ((\ud835\udfd9 X.V \u2297 \ud835\udfd9 (Y.V \u2297 Z.V)) \u226b (\u03b1_ X.V Y.V Z.V).inv) \u226b (\ud835\udfd9 (X.V \u2297 Y.V) \u2297 \ud835\udfd9 Z.V) = (\u03b1_ X.V Y.V Z.V).inv\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 (\u03bb_ X).hom.hom = (\u03bb_ X.V).hom\n[PROOFSTEP]\ndsimp [Monoidal.transport_leftUnitor]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 ((\ud835\udfd9 (\ud835\udfd9_ V) \u2297 \ud835\udfd9 X.V) \u226b (\u03bb_ X.V).hom) \u226b \ud835\udfd9 X.V = (\u03bb_ X.V).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 (\u03bb_ X).inv.hom = (\u03bb_ X.V).inv\n[PROOFSTEP]\ndsimp [Monoidal.transport_leftUnitor]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 \ud835\udfd9 X.V \u226b (\u03bb_ X.V).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ V) \u2297 \ud835\udfd9 X.V) = (\u03bb_ X.V).inv\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 (\u03c1_ X).hom.hom = (\u03c1_ X.V).hom\n[PROOFSTEP]\ndsimp [Monoidal.transport_rightUnitor]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 ((\ud835\udfd9 X.V \u2297 \ud835\udfd9 (\ud835\udfd9_ V)) \u226b (\u03c1_ X.V).hom) \u226b \ud835\udfd9 X.V = (\u03c1_ X.V).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 (\u03c1_ X).inv.hom = (\u03c1_ X.V).inv\n[PROOFSTEP]\ndsimp [Monoidal.transport_rightUnitor]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 \ud835\udfd9 X.V \u226b (\u03c1_ X.V).inv \u226b (\ud835\udfd9 X.V \u2297 \ud835\udfd9 (\ud835\udfd9_ V)) = (\u03c1_ X.V).inv\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : V\nf : \ud835\udfd9_ V \u2245 X\nx\u271d : \u2191G\n\u22a2 \u2191(\ud835\udfd9_ (Action V G)).\u03c1 x\u271d \u226b f.hom = f.hom \u226b \u2191{ V := X, \u03c1 := 1 }.\u03c1 x\u271d\n[PROOFSTEP]\nsimp only [MonoidHom.one_apply, End.one_def, Category.id_comp f.hom, tensorUnit_rho, MonCat.oneHom_apply, MonCat.one_of,\n Category.comp_id]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\n\u22a2 Faithful (forgetMonoidal V G).toLaxMonoidalFunctor.toFunctor\n[PROOFSTEP]\nchange Faithful (forget V G)\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\n\u22a2 Faithful (forget V G)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : BraidedCategory V\nX Y : Action V G\ng : \u2191G\n\u22a2 \u2191(X \u2297 Y).\u03c1 g \u226b\n (\u03b2_ (((CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).inverse.obj X).obj PUnit.unit)\n (((CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).inverse.obj Y).obj PUnit.unit)).hom =\n (\u03b2_ (((CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).inverse.obj X).obj PUnit.unit)\n (((CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).inverse.obj Y).obj PUnit.unit)).hom \u226b\n \u2191(Y \u2297 X).\u03c1 g\n[PROOFSTEP]\nsimp [FunctorCategoryEquivalence.inverse]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : BraidedCategory V\n\u22a2 \u2200 (X Y : Action V G),\n LaxMonoidalFunctor.\u03bc (forgetMonoidal V G).toLaxMonoidalFunctor X Y \u226b\n (forgetMonoidal V G).toLaxMonoidalFunctor.toFunctor.map\n ((fun X Y =>\n mkIso\n (\u03b2_\n (((CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).inverse.obj X).obj PUnit.unit)\n (((CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).inverse.obj Y).obj\n PUnit.unit)))\n X Y).hom =\n (\u03b2_ ((forgetMonoidal V G).toLaxMonoidalFunctor.toFunctor.obj X)\n ((forgetMonoidal V G).toLaxMonoidalFunctor.toFunctor.obj Y)).hom \u226b\n LaxMonoidalFunctor.\u03bc (forgetMonoidal V G).toLaxMonoidalFunctor Y X\n[PROOFSTEP]\naesop_cat\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : BraidedCategory V\n\u22a2 Faithful (forgetBraided V G).toMonoidalFunctor.toLaxMonoidalFunctor.toFunctor\n[PROOFSTEP]\nchange Faithful (forget V G)\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : BraidedCategory V\n\u22a2 Faithful (forget V G)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\n\u22a2 IsEquivalence (functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor\n[PROOFSTEP]\nchange IsEquivalence (Action.functorCategoryEquivalence _ _).functor\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\n\u22a2 IsEquivalence (functorCategoryEquivalence V G).functor\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 NatTrans.app (LaxMonoidalFunctor.\u03bc (functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor A B) PUnit.unit =\n \ud835\udfd9\n (((functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj A \u2297\n (functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj B).obj\n PUnit.unit)\n[PROOFSTEP]\ndsimp only [functorCategoryMonoidalEquivalence]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 NatTrans.app\n (LaxMonoidalFunctor.\u03bc\n (Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G))).toLaxMonoidalFunctor\n A B)\n PUnit.unit =\n \ud835\udfd9\n (((Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm\n (functorCategoryEquivalence V G))).toLaxMonoidalFunctor.toFunctor.obj\n A \u2297\n (Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm\n (functorCategoryEquivalence V G))).toLaxMonoidalFunctor.toFunctor.obj\n B).obj\n PUnit.unit)\n[PROOFSTEP]\nsimp only [Monoidal.fromTransported_toLaxMonoidalFunctor_\u03bc, NatTrans.comp_app]\n -- porting note: Lean3 was able to see through some defeq, as the mathlib3 proof was\n -- show (\ud835\udfd9 A.V \u2297 \ud835\udfd9 B.V) \u226b \ud835\udfd9 (A.V \u2297 B.V) \u226b (\ud835\udfd9 A.V \u2297 \ud835\udfd9 B.V) = \ud835\udfd9 (A.V \u2297 B.V)\n -- simp only [monoidal_category.tensor_id, category.comp_id]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 NatTrans.app\n (NatTrans.app (Equivalence.unit (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n ((Functor.asEquivalence\n (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).functor).inverse.obj\n A \u2297\n (Functor.asEquivalence\n (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).functor).inverse.obj\n B))\n PUnit.unit \u226b\n NatTrans.app\n (inv\n ((Functor.asEquivalence\n (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).functor).inverse.map\n ((CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).functor.map\n (NatTrans.app (Equivalence.unitInv (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n ((Functor.asEquivalence\n (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).functor).inverse.obj\n A) \u2297\n NatTrans.app (Equivalence.unitInv (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n ((Functor.asEquivalence\n (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).functor).inverse.obj\n B)))))\n PUnit.unit \u226b\n NatTrans.app\n ((Functor.asEquivalence\n (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).functor).inverse.map\n ((CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).functor.map\n ((CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).inverse.map\n (NatTrans.app (Equivalence.counit (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n A) \u2297\n (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)).inverse.map\n (NatTrans.app (Equivalence.counit (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n B))))\n PUnit.unit =\n \ud835\udfd9\n (((Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm\n (functorCategoryEquivalence V G))).toLaxMonoidalFunctor.toFunctor.obj\n A \u2297\n (Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm\n (functorCategoryEquivalence V G))).toLaxMonoidalFunctor.toFunctor.obj\n B).obj\n PUnit.unit)\n[PROOFSTEP]\ndsimp [Equivalence.unit]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 \ud835\udfd9\n (((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.obj A).obj PUnit.unit \u2297\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.obj B).obj PUnit.unit) \u226b\n NatTrans.app\n (inv\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.map\n (FunctorCategoryEquivalence.inverse.map\n (NatTrans.app (Equivalence.unitInv (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.obj A) \u2297\n NatTrans.app (Equivalence.unitInv (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.obj B)))))\n PUnit.unit \u226b\n NatTrans.app\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.map\n (FunctorCategoryEquivalence.inverse.map\n (FunctorCategoryEquivalence.functor.map\n (NatTrans.app (Equivalence.counit (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n A) \u2297\n FunctorCategoryEquivalence.functor.map\n (NatTrans.app (Equivalence.counit (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n B))))\n PUnit.unit =\n \ud835\udfd9\n (((Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm\n (functorCategoryEquivalence V G))).toLaxMonoidalFunctor.toFunctor.obj\n A).obj\n PUnit.unit \u2297\n ((Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm\n (functorCategoryEquivalence V G))).toLaxMonoidalFunctor.toFunctor.obj\n B).obj\n PUnit.unit)\n[PROOFSTEP]\nerw [Category.id_comp]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 NatTrans.app\n (inv\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.map\n (FunctorCategoryEquivalence.inverse.map\n (NatTrans.app (Equivalence.unitInv (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.obj A) \u2297\n NatTrans.app (Equivalence.unitInv (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.obj B)))))\n PUnit.unit \u226b\n NatTrans.app\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.map\n (FunctorCategoryEquivalence.inverse.map\n (FunctorCategoryEquivalence.functor.map\n (NatTrans.app (Equivalence.counit (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n A) \u2297\n FunctorCategoryEquivalence.functor.map\n (NatTrans.app (Equivalence.counit (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n B))))\n PUnit.unit =\n \ud835\udfd9\n (((Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm\n (functorCategoryEquivalence V G))).toLaxMonoidalFunctor.toFunctor.obj\n A).obj\n PUnit.unit \u2297\n ((Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm\n (functorCategoryEquivalence V G))).toLaxMonoidalFunctor.toFunctor.obj\n B).obj\n PUnit.unit)\n[PROOFSTEP]\nrw [NatIso.isIso_inv_app, IsIso.inv_comp_eq]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 NatTrans.app\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.map\n (FunctorCategoryEquivalence.inverse.map\n (FunctorCategoryEquivalence.functor.map\n (NatTrans.app (Equivalence.counit (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G))) A) \u2297\n FunctorCategoryEquivalence.functor.map\n (NatTrans.app (Equivalence.counit (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n B))))\n PUnit.unit =\n NatTrans.app\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.map\n (FunctorCategoryEquivalence.inverse.map\n (NatTrans.app (Equivalence.unitInv (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.obj A) \u2297\n NatTrans.app (Equivalence.unitInv (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.obj B))))\n PUnit.unit \u226b\n \ud835\udfd9\n (((Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm\n (functorCategoryEquivalence V G))).toLaxMonoidalFunctor.toFunctor.obj\n A).obj\n PUnit.unit \u2297\n ((Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm\n (functorCategoryEquivalence V G))).toLaxMonoidalFunctor.toFunctor.obj\n B).obj\n PUnit.unit)\n[PROOFSTEP]\nerw [MonoidalCategory.tensor_id]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 NatTrans.app\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.map\n (FunctorCategoryEquivalence.inverse.map\n (\ud835\udfd9\n (FunctorCategoryEquivalence.functor.obj\n (FunctorCategoryEquivalence.inverse.obj\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.obj A)) \u2297\n FunctorCategoryEquivalence.functor.obj\n (FunctorCategoryEquivalence.inverse.obj\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.obj B))))))\n PUnit.unit =\n NatTrans.app\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.map\n (FunctorCategoryEquivalence.inverse.map\n (\ud835\udfd9\n (FunctorCategoryEquivalence.functor.obj\n (FunctorCategoryEquivalence.inverse.obj\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.obj A)) \u2297\n FunctorCategoryEquivalence.functor.obj\n (FunctorCategoryEquivalence.inverse.obj\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.obj B))))))\n PUnit.unit \u226b\n \ud835\udfd9\n (((Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm\n (functorCategoryEquivalence V G))).toLaxMonoidalFunctor.toFunctor.obj\n A).obj\n PUnit.unit \u2297\n ((Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm\n (functorCategoryEquivalence V G))).toLaxMonoidalFunctor.toFunctor.obj\n B).obj\n PUnit.unit)\n[PROOFSTEP]\nerw [(functorCategoryEquivalence V G).inverse.map_id, (functorCategoryEquivalence V G).functor.map_id, Category.id_comp]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 NatTrans.app\n (\ud835\udfd9\n ((functorCategoryEquivalence V G).functor.obj\n (FunctorCategoryEquivalence.inverse.obj\n (FunctorCategoryEquivalence.functor.obj\n (FunctorCategoryEquivalence.inverse.obj\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.obj A)) \u2297\n FunctorCategoryEquivalence.functor.obj\n (FunctorCategoryEquivalence.inverse.obj\n ((Functor.asEquivalence FunctorCategoryEquivalence.inverse).inverse.obj B))))))\n PUnit.unit =\n \ud835\udfd9\n (((Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm\n (functorCategoryEquivalence V G))).toLaxMonoidalFunctor.toFunctor.obj\n A).obj\n PUnit.unit \u2297\n ((Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm\n (functorCategoryEquivalence V G))).toLaxMonoidalFunctor.toFunctor.obj\n B).obj\n PUnit.unit)\n[PROOFSTEP]\nrfl\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 NatTrans.app (MonoidalFunctor.\u03bcIso (functorCategoryMonoidalEquivalence V G) A B).inv PUnit.unit =\n \ud835\udfd9 (((functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj (A \u2297 B)).obj PUnit.unit)\n[PROOFSTEP]\nrw [\u2190 NatIso.app_inv, \u2190 IsIso.Iso.inv_hom]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 inv ((MonoidalFunctor.\u03bcIso (functorCategoryMonoidalEquivalence V G) A B).app PUnit.unit).hom =\n \ud835\udfd9 (((functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj (A \u2297 B)).obj PUnit.unit)\n[PROOFSTEP]\nrefine' IsIso.inv_eq_of_hom_inv_id _\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 ((MonoidalFunctor.\u03bcIso (functorCategoryMonoidalEquivalence V G) A B).app PUnit.unit).hom \u226b\n \ud835\udfd9 (((functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj (A \u2297 B)).obj PUnit.unit) =\n \ud835\udfd9\n (((functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj A \u2297\n (functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj B).obj\n PUnit.unit)\n[PROOFSTEP]\nrw [Category.comp_id, NatIso.app_hom, MonoidalFunctor.\u03bcIso_hom, functorCategoryMonoidalEquivalence.\u03bc_app]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\n\u22a2 NatTrans.app (functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.\u03b5 PUnit.unit =\n \ud835\udfd9 ((\ud835\udfd9_ (SingleObj \u2191G \u2964 V)).obj PUnit.unit)\n[PROOFSTEP]\ndsimp only [functorCategoryMonoidalEquivalence]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\n\u22a2 NatTrans.app\n (Monoidal.fromTransported\n (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G))).toLaxMonoidalFunctor.\u03b5\n PUnit.unit =\n \ud835\udfd9 ((\ud835\udfd9_ (SingleObj \u2191G \u2964 V)).obj PUnit.unit)\n[PROOFSTEP]\nsimp only [Monoidal.fromTransported_toLaxMonoidalFunctor_\u03b5]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\n\u22a2 NatTrans.app\n (NatTrans.app (Equivalence.unit (CategoryTheory.Equivalence.symm (functorCategoryEquivalence V G)))\n (\ud835\udfd9_ (SingleObj \u2191G \u2964 V)))\n PUnit.unit =\n \ud835\udfd9 ((\ud835\udfd9_ (SingleObj \u2191G \u2964 V)).obj PUnit.unit)\n[PROOFSTEP]\nrfl\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : GroupCat\ninst\u271d : RightRigidCategory V\n\u22a2 RightRigidCategory (SingleObj \u2191((forget\u2082 GroupCat MonCat).obj H) \u2964 V)\n[PROOFSTEP]\nchange RightRigidCategory (SingleObj H \u2964 V)\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : GroupCat\ninst\u271d : RightRigidCategory V\n\u22a2 RightRigidCategory (SingleObj \u2191H \u2964 V)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : GroupCat\ninst\u271d : LeftRigidCategory V\n\u22a2 LeftRigidCategory (SingleObj \u2191((forget\u2082 GroupCat MonCat).obj H) \u2964 V)\n[PROOFSTEP]\nchange LeftRigidCategory (SingleObj H \u2964 V)\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : GroupCat\ninst\u271d : LeftRigidCategory V\n\u22a2 LeftRigidCategory (SingleObj \u2191H \u2964 V)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : GroupCat\ninst\u271d : RigidCategory V\n\u22a2 RigidCategory (SingleObj \u2191((forget\u2082 GroupCat MonCat).obj H) \u2964 V)\n[PROOFSTEP]\nchange RigidCategory (SingleObj H \u2964 V)\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : GroupCat\ninst\u271d : RigidCategory V\n\u22a2 RigidCategory (SingleObj \u2191H \u2964 V)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget\u2082 GroupCat MonCat).obj H)\ninst\u271d : RightRigidCategory V\nh : \u2191H\n\u22a2 \u2191X\u1601.\u03c1 h = \u2191X.\u03c1 h\u207b\u00b9\u1601\n[PROOFSTEP]\nrw [\u2190 SingleObj.inv_as_inv]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget\u2082 GroupCat MonCat).obj H)\ninst\u271d : RightRigidCategory V\nh : \u2191H\n\u22a2 \u2191X\u1601.\u03c1 h = \u2191X.\u03c1 (inv h)\u1601\ncase x\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget\u2082 GroupCat MonCat).obj H)\ninst\u271d : RightRigidCategory V\nh : \u2191H\n\u22a2 SingleObj H.1\ncase y\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget\u2082 GroupCat MonCat).obj H)\ninst\u271d : RightRigidCategory V\nh : \u2191H\n\u22a2 SingleObj H.1\n[PROOFSTEP]\nrfl\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget\u2082 GroupCat MonCat).obj H)\ninst\u271d : LeftRigidCategory V\nh : \u2191H\n\u22a2 \u2191(\u1601X).\u03c1 h = \u1601\u2191X.\u03c1 h\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 SingleObj.inv_as_inv]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget\u2082 GroupCat MonCat).obj H)\ninst\u271d : LeftRigidCategory V\nh : \u2191H\n\u22a2 \u2191(\u1601X).\u03c1 h = \u1601\u2191X.\u03c1 (inv h)\ncase x\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget\u2082 GroupCat MonCat).obj H)\ninst\u271d : LeftRigidCategory V\nh : \u2191H\n\u22a2 SingleObj H.1\ncase y\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget\u2082 GroupCat MonCat).obj H)\ninst\u271d : LeftRigidCategory V\nh : \u2191H\n\u22a2 SingleObj H.1\n[PROOFSTEP]\nrfl\n[GOAL]\nV : Type (u + 1)\ninst\u271d : LargeCategory V\nG : MonCat\nX\u271d Y\u271d : V\nf : X\u271d \u27f6 Y\u271d\nx\u271d : \u2191(MonCat.of PUnit)\n\u22a2 \u2191((fun X => { V := X, \u03c1 := 1 }) X\u271d).\u03c1 PUnit.unit \u226b f = f \u226b \u2191((fun X => { V := X, \u03c1 := 1 }) Y\u271d).\u03c1 PUnit.unit\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type (u + 1)\ninst\u271d : LargeCategory V\nG : MonCat\nX : Action V (MonCat.of PUnit)\nx\u271d : \u2191(MonCat.of PUnit)\n\u22a2 \u2191((\ud835\udfed (Action V (MonCat.of PUnit))).obj X).\u03c1 PUnit.unit \u226b (Iso.refl ((\ud835\udfed (Action V (MonCat.of PUnit))).obj X).V).hom =\n (Iso.refl ((\ud835\udfed (Action V (MonCat.of PUnit))).obj X).V).hom \u226b\n \u2191((forget V (MonCat.of PUnit) \u22d9\n CategoryTheory.Functor.mk { obj := fun X => { V := X, \u03c1 := 1 }, map := fun {X Y} f => Hom.mk f }).obj\n X).\u03c1\n PUnit.unit\n[PROOFSTEP]\nsimp only [MonCat.oneHom_apply, MonCat.one_of, End.one_def, id_eq, Functor.comp_obj, forget_obj, Iso.refl_hom,\n Category.comp_id]\n[GOAL]\nV : Type (u + 1)\ninst\u271d : LargeCategory V\nG : MonCat\nX : Action V (MonCat.of PUnit)\nx\u271d : \u2191(MonCat.of PUnit)\n\u22a2 \u2191((\ud835\udfed (Action V (MonCat.of PUnit))).obj X).\u03c1 PUnit.unit = \ud835\udfd9 ((\ud835\udfed (Action V (MonCat.of PUnit))).obj X).V\n[PROOFSTEP]\nexact \u03c1_one X\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG\u271d H\u271d : MonCat\nf : G\u271d \u27f6 H\u271d\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nG H : Type u\ninst\u271d\u00b9 : Monoid G\ninst\u271d : MulAction G H\n\u22a2 MulAction (\u2191(MonCat.of G)) H\n[PROOFSTEP]\nassumption\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\n\u03b9 : Type v\nG : Type (max v u)\ninst\u271d\u00b9 : Monoid G\nF : \u03b9 \u2192 Type (max v u)\ninst\u271d : (i : \u03b9) \u2192 MulAction G (F i)\ns : Cone (Discrete.functor fun i => ofMulAction G (F i))\ng : \u2191(MonCat.of G)\n\u22a2 (\u2191s.pt.\u03c1 g \u226b fun x i => Hom.hom (NatTrans.app s.\u03c0 { as := i }) x) =\n (fun x i => Hom.hom (NatTrans.app s.\u03c0 { as := i }) x) \u226b\n \u2191{ pt := ofMulAction G ((i : \u03b9) \u2192 F i), \u03c0 := Discrete.natTrans fun i => Hom.mk fun x => x i.as }.pt.\u03c1 g\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\n\u03b9 : Type v\nG : Type (max v u)\ninst\u271d\u00b9 : Monoid G\nF : \u03b9 \u2192 Type (max v u)\ninst\u271d : (i : \u03b9) \u2192 MulAction G (F i)\ns : Cone (Discrete.functor fun i => ofMulAction G (F i))\ng : \u2191(MonCat.of G)\nx : s.pt.V\n\u22a2 (\u2191s.pt.\u03c1 g \u226b fun x i => Hom.hom (NatTrans.app s.\u03c0 { as := i }) x) x =\n ((fun x i => Hom.hom (NatTrans.app s.\u03c0 { as := i }) x) \u226b\n \u2191{ pt := ofMulAction G ((i : \u03b9) \u2192 F i), \u03c0 := Discrete.natTrans fun i => Hom.mk fun x => x i.as }.pt.\u03c1 g)\n x\n[PROOFSTEP]\nfunext j\n[GOAL]\ncase h.h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\n\u03b9 : Type v\nG : Type (max v u)\ninst\u271d\u00b9 : Monoid G\nF : \u03b9 \u2192 Type (max v u)\ninst\u271d : (i : \u03b9) \u2192 MulAction G (F i)\ns : Cone (Discrete.functor fun i => ofMulAction G (F i))\ng : \u2191(MonCat.of G)\nx : s.pt.V\nj : \u03b9\n\u22a2 (\u2191s.pt.\u03c1 g \u226b fun x i => Hom.hom (NatTrans.app s.\u03c0 { as := i }) x) x j =\n ((fun x i => Hom.hom (NatTrans.app s.\u03c0 { as := i }) x) \u226b\n \u2191{ pt := ofMulAction G ((i : \u03b9) \u2192 F i), \u03c0 := Discrete.natTrans fun i => Hom.mk fun x => x i.as }.pt.\u03c1 g)\n x j\n[PROOFSTEP]\nexact congr_fun ((s.\u03c0.app \u27e8j\u27e9).comm g) x\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG\u271d H : MonCat\nf\u271d : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\n\u03b9 : Type v\nG : Type (max v u)\ninst\u271d\u00b9 : Monoid G\nF : \u03b9 \u2192 Type (max v u)\ninst\u271d : (i : \u03b9) \u2192 MulAction G (F i)\ns : Cone (Discrete.functor fun i => ofMulAction G (F i))\nf : s.pt \u27f6 { pt := ofMulAction G ((i : \u03b9) \u2192 F i), \u03c0 := Discrete.natTrans fun i => Hom.mk fun x => x i.as }.pt\nh :\n \u2200 (j : Discrete \u03b9),\n f \u226b\n NatTrans.app { pt := ofMulAction G ((i : \u03b9) \u2192 F i), \u03c0 := Discrete.natTrans fun i => Hom.mk fun x => x i.as }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\n\u22a2 f = (fun s => Hom.mk fun x i => Hom.hom (NatTrans.app s.\u03c0 { as := i }) x) s\n[PROOFSTEP]\next x\n[GOAL]\ncase h.h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG\u271d H : MonCat\nf\u271d : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\n\u03b9 : Type v\nG : Type (max v u)\ninst\u271d\u00b9 : Monoid G\nF : \u03b9 \u2192 Type (max v u)\ninst\u271d : (i : \u03b9) \u2192 MulAction G (F i)\ns : Cone (Discrete.functor fun i => ofMulAction G (F i))\nf : s.pt \u27f6 { pt := ofMulAction G ((i : \u03b9) \u2192 F i), \u03c0 := Discrete.natTrans fun i => Hom.mk fun x => x i.as }.pt\nh :\n \u2200 (j : Discrete \u03b9),\n f \u226b\n NatTrans.app { pt := ofMulAction G ((i : \u03b9) \u2192 F i), \u03c0 := Discrete.natTrans fun i => Hom.mk fun x => x i.as }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\nx : s.pt.V\n\u22a2 Hom.hom f x = Hom.hom ((fun s => Hom.mk fun x i => Hom.hom (NatTrans.app s.\u03c0 { as := i }) x) s) x\n[PROOFSTEP]\nfunext j\n[GOAL]\ncase h.h.h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG\u271d H : MonCat\nf\u271d : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\n\u03b9 : Type v\nG : Type (max v u)\ninst\u271d\u00b9 : Monoid G\nF : \u03b9 \u2192 Type (max v u)\ninst\u271d : (i : \u03b9) \u2192 MulAction G (F i)\ns : Cone (Discrete.functor fun i => ofMulAction G (F i))\nf : s.pt \u27f6 { pt := ofMulAction G ((i : \u03b9) \u2192 F i), \u03c0 := Discrete.natTrans fun i => Hom.mk fun x => x i.as }.pt\nh :\n \u2200 (j : Discrete \u03b9),\n f \u226b\n NatTrans.app { pt := ofMulAction G ((i : \u03b9) \u2192 F i), \u03c0 := Discrete.natTrans fun i => Hom.mk fun x => x i.as }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\nx : s.pt.V\nj : \u03b9\n\u22a2 Hom.hom f x j = Hom.hom ((fun s => Hom.mk fun x i => Hom.hom (NatTrans.app s.\u03c0 { as := i }) x) s) x j\n[PROOFSTEP]\ndsimp at *\n[GOAL]\ncase h.h.h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG\u271d H : MonCat\nf\u271d : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\n\u03b9 : Type v\nG : Type (max v u)\ninst\u271d\u00b9 : Monoid G\nF : \u03b9 \u2192 Type (max v u)\ninst\u271d : (i : \u03b9) \u2192 MulAction G (F i)\ns : Cone (Discrete.functor fun i => ofMulAction G (F i))\nf : s.pt \u27f6 { pt := ofMulAction G ((i : \u03b9) \u2192 F i), \u03c0 := Discrete.natTrans fun i => Hom.mk fun x => x i.as }.pt\nh : \u2200 (j : Discrete \u03b9), (f \u226b Hom.mk fun x => x j.as) = NatTrans.app s.\u03c0 j\nx : s.pt.V\nj : \u03b9\n\u22a2 Hom.hom f x j = Hom.hom (NatTrans.app s.\u03c0 { as := j }) x\n[PROOFSTEP]\nrw [\u2190 h \u27e8j\u27e9]\n[GOAL]\ncase h.h.h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG\u271d H : MonCat\nf\u271d : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\n\u03b9 : Type v\nG : Type (max v u)\ninst\u271d\u00b9 : Monoid G\nF : \u03b9 \u2192 Type (max v u)\ninst\u271d : (i : \u03b9) \u2192 MulAction G (F i)\ns : Cone (Discrete.functor fun i => ofMulAction G (F i))\nf : s.pt \u27f6 { pt := ofMulAction G ((i : \u03b9) \u2192 F i), \u03c0 := Discrete.natTrans fun i => Hom.mk fun x => x i.as }.pt\nh : \u2200 (j : Discrete \u03b9), (f \u226b Hom.mk fun x => x j.as) = NatTrans.app s.\u03c0 j\nx : s.pt.V\nj : \u03b9\n\u22a2 Hom.hom f x j = Hom.hom (f \u226b Hom.mk fun x => x { as := j }.as) x\n[PROOFSTEP]\nrfl\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\ng : G\n\u22a2 (\u2191(leftRegular G \u2297 X).\u03c1 g \u226b fun g => (g.fst, \u2191X.\u03c1 g.fst\u207b\u00b9 g.snd)) =\n (fun g => (g.fst, \u2191X.\u03c1 g.fst\u207b\u00b9 g.snd)) \u226b \u2191(leftRegular G \u2297 { V := X.V, \u03c1 := 1 }).\u03c1 g\n[PROOFSTEP]\nfunext \u27e8(x\u2081 : G), (x\u2082 : X.V)\u27e9\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\ng x\u2081 : G\nx\u2082 : X.V\n\u22a2 (\u2191(leftRegular G \u2297 X).\u03c1 g \u226b fun g => (g.fst, \u2191X.\u03c1 g.fst\u207b\u00b9 g.snd)) (x\u2081, x\u2082) =\n ((fun g => (g.fst, \u2191X.\u03c1 g.fst\u207b\u00b9 g.snd)) \u226b \u2191(leftRegular G \u2297 { V := X.V, \u03c1 := 1 }).\u03c1 g) (x\u2081, x\u2082)\n[PROOFSTEP]\nrefine' Prod.ext rfl _\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\ng x\u2081 : G\nx\u2082 : X.V\n\u22a2 ((\u2191(leftRegular G \u2297 X).\u03c1 g \u226b fun g => (g.fst, \u2191X.\u03c1 g.fst\u207b\u00b9 g.snd)) (x\u2081, x\u2082)).snd =\n (((fun g => (g.fst, \u2191X.\u03c1 g.fst\u207b\u00b9 g.snd)) \u226b \u2191(leftRegular G \u2297 { V := X.V, \u03c1 := 1 }).\u03c1 g) (x\u2081, x\u2082)).snd\n[PROOFSTEP]\nchange (X.\u03c1 ((g * x\u2081)\u207b\u00b9 : G) * X.\u03c1 g) x\u2082 = X.\u03c1 _ _\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\ng x\u2081 : G\nx\u2082 : X.V\n\u22a2 (\u2191X.\u03c1 (g * x\u2081)\u207b\u00b9 * \u2191X.\u03c1 g) x\u2082 = \u2191X.\u03c1 (x\u2081, x\u2082).fst\u207b\u00b9 (x\u2081, x\u2082).snd\n[PROOFSTEP]\nrw [mul_inv_rev, \u2190 X.\u03c1.map_mul, inv_mul_cancel_right]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\ng : G\n\u22a2 (\u2191(leftRegular G \u2297 { V := X.V, \u03c1 := 1 }).\u03c1 g \u226b fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)) =\n (fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)) \u226b \u2191(leftRegular G \u2297 X).\u03c1 g\n[PROOFSTEP]\nfunext \u27e8(x\u2081 : G), (x\u2082 : X.V)\u27e9\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\ng x\u2081 : G\nx\u2082 : X.V\n\u22a2 (\u2191(leftRegular G \u2297 { V := X.V, \u03c1 := 1 }).\u03c1 g \u226b fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)) (x\u2081, x\u2082) =\n ((fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)) \u226b \u2191(leftRegular G \u2297 X).\u03c1 g) (x\u2081, x\u2082)\n[PROOFSTEP]\nrefine' Prod.ext rfl _\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\ng x\u2081 : G\nx\u2082 : X.V\n\u22a2 ((\u2191(leftRegular G \u2297 { V := X.V, \u03c1 := 1 }).\u03c1 g \u226b fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)) (x\u2081, x\u2082)).snd =\n (((fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)) \u226b \u2191(leftRegular G \u2297 X).\u03c1 g) (x\u2081, x\u2082)).snd\n[PROOFSTEP]\nerw [tensor_rho, tensor_rho]\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\ng x\u2081 : G\nx\u2082 : X.V\n\u22a2 (((\u2191(leftRegular G).\u03c1 g \u2297 \u2191{ V := X.V, \u03c1 := 1 }.\u03c1 g) \u226b fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)) (x\u2081, x\u2082)).snd =\n (((fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)) \u226b (\u2191(leftRegular G).\u03c1 g \u2297 \u2191X.\u03c1 g)) (x\u2081, x\u2082)).snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\ng x\u2081 : G\nx\u2082 : X.V\n\u22a2 \u2191X.\u03c1 (\u2191(leftRegular G).\u03c1 g x\u2081) x\u2082 = \u2191X.\u03c1 g (\u2191X.\u03c1 x\u2081 x\u2082)\n[PROOFSTEP]\nrw [leftRegular_\u03c1_apply]\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\ng x\u2081 : G\nx\u2082 : X.V\n\u22a2 \u2191X.\u03c1 (g \u2022 x\u2081) x\u2082 = \u2191X.\u03c1 g (\u2191X.\u03c1 x\u2081 x\u2082)\n[PROOFSTEP]\nerw [map_mul]\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\ng x\u2081 : G\nx\u2082 : X.V\n\u22a2 (\u2191X.\u03c1 g * \u2191X.\u03c1 x\u2081) x\u2082 = \u2191X.\u03c1 g (\u2191X.\u03c1 x\u2081 x\u2082)\n[PROOFSTEP]\nrfl\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\n\u22a2 ((Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst\u207b\u00b9 g.snd)) \u226b Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)) = \ud835\udfd9 (leftRegular G \u2297 X)\n[PROOFSTEP]\napply Hom.ext\n[GOAL]\ncase hom\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\n\u22a2 ((Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst\u207b\u00b9 g.snd)) \u226b Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)).hom =\n (\ud835\udfd9 (leftRegular G \u2297 X)).hom\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase hom.h\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\nx : (leftRegular G \u2297 X).V\n\u22a2 Hom.hom ((Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst\u207b\u00b9 g.snd)) \u226b Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)) x =\n Hom.hom (\ud835\udfd9 (leftRegular G \u2297 X)) x\n[PROOFSTEP]\nrefine' Prod.ext rfl _\n[GOAL]\ncase hom.h\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\nx : (leftRegular G \u2297 X).V\n\u22a2 (Hom.hom ((Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst\u207b\u00b9 g.snd)) \u226b Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)) x).snd =\n (Hom.hom (\ud835\udfd9 (leftRegular G \u2297 X)) x).snd\n[PROOFSTEP]\nchange (X.\u03c1 x.1 * X.\u03c1 (x.1\u207b\u00b9 : G)) x.2 = x.2\n[GOAL]\ncase hom.h\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\nx : (leftRegular G \u2297 X).V\n\u22a2 (\u2191X.\u03c1 x.fst * \u2191X.\u03c1 x.fst\u207b\u00b9) x.snd = x.snd\n[PROOFSTEP]\nrw [\u2190 X.\u03c1.map_mul, mul_inv_self, X.\u03c1.map_one, MonCat.one_of, End.one_def, types_id_apply]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\n\u22a2 ((Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)) \u226b Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst\u207b\u00b9 g.snd)) =\n \ud835\udfd9 (leftRegular G \u2297 { V := X.V, \u03c1 := 1 })\n[PROOFSTEP]\napply Hom.ext\n[GOAL]\ncase hom\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\n\u22a2 ((Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)) \u226b Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst\u207b\u00b9 g.snd)).hom =\n (\ud835\udfd9 (leftRegular G \u2297 { V := X.V, \u03c1 := 1 })).hom\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase hom.h\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\nx : (leftRegular G \u2297 { V := X.V, \u03c1 := 1 }).V\n\u22a2 Hom.hom ((Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)) \u226b Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst\u207b\u00b9 g.snd)) x =\n Hom.hom (\ud835\udfd9 (leftRegular G \u2297 { V := X.V, \u03c1 := 1 })) x\n[PROOFSTEP]\nrefine' Prod.ext rfl _\n[GOAL]\ncase hom.h\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\nx : (leftRegular G \u2297 { V := X.V, \u03c1 := 1 }).V\n\u22a2 (Hom.hom ((Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst g.snd)) \u226b Hom.mk fun g => (g.fst, \u2191X.\u03c1 g.fst\u207b\u00b9 g.snd)) x).snd =\n (Hom.hom (\ud835\udfd9 (leftRegular G \u2297 { V := X.V, \u03c1 := 1 })) x).snd\n[PROOFSTEP]\nchange (X.\u03c1 (x.1\u207b\u00b9 : G) * X.\u03c1 x.1) x.2 = x.2\n[GOAL]\ncase hom.h\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG\u271d H : MonCat\nf : G\u271d \u27f6 H\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nG : Type u\ninst\u271d : Group G\nX : Action (Type u) (MonCat.of G)\nx : (leftRegular G \u2297 { V := X.V, \u03c1 := 1 }).V\n\u22a2 (\u2191X.\u03c1 x.fst\u207b\u00b9 * \u2191X.\u03c1 x.fst) x.snd = x.snd\n[PROOFSTEP]\nrw [\u2190 X.\u03c1.map_mul, inv_mul_self, X.\u03c1.map_one, MonCat.one_of, End.one_def, types_id_apply]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nW : Type (u + 1)\ninst\u271d : LargeCategory W\nF : V \u2964 W\nG : MonCat\nM : Action V G\n\u22a2 (fun g => F.map (\u2191M.\u03c1 g)) 1 = 1\n[PROOFSTEP]\nsimp only [End.one_def, Action.\u03c1_one, F.map_id, MonCat.one_of]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nW : Type (u + 1)\ninst\u271d : LargeCategory W\nF : V \u2964 W\nG : MonCat\nM : Action V G\ng h : \u2191G\n\u22a2 OneHom.toFun { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) } (g * h) =\n OneHom.toFun { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) } g *\n OneHom.toFun { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) } h\n[PROOFSTEP]\ndsimp\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nW : Type (u + 1)\ninst\u271d : LargeCategory W\nF : V \u2964 W\nG : MonCat\nM : Action V G\ng h : \u2191G\n\u22a2 F.map (\u2191M.\u03c1 (g * h)) = F.map (\u2191M.\u03c1 g) * F.map (\u2191M.\u03c1 h)\n[PROOFSTEP]\nrw [map_mul, MonCat.mul_of, End.mul_def, End.mul_def, F.map_comp]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nW : Type (u + 1)\ninst\u271d : LargeCategory W\nF : V \u2964 W\nG : MonCat\nX\u271d Y\u271d : Action V G\nf : X\u271d \u27f6 Y\u271d\ng : \u2191G\n\u22a2 \u2191((fun M =>\n { V := F.obj M.V,\n \u03c1 :=\n {\n toOneHom :=\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) },\n map_mul' :=\n (_ :\n \u2200 (g h : \u2191G),\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n (g * h) =\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n g *\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n h) } })\n X\u271d).\u03c1\n g \u226b\n F.map f.hom =\n F.map f.hom \u226b\n \u2191((fun M =>\n { V := F.obj M.V,\n \u03c1 :=\n {\n toOneHom :=\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) },\n map_mul' :=\n (_ :\n \u2200 (g h : \u2191G),\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n (g * h) =\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n g *\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n h) } })\n Y\u271d).\u03c1\n g\n[PROOFSTEP]\ndsimp\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nW : Type (u + 1)\ninst\u271d : LargeCategory W\nF : V \u2964 W\nG : MonCat\nX\u271d Y\u271d : Action V G\nf : X\u271d \u27f6 Y\u271d\ng : \u2191G\n\u22a2 F.map (\u2191X\u271d.\u03c1 g) \u226b F.map f.hom = F.map f.hom \u226b F.map (\u2191Y\u271d.\u03c1 g)\n[PROOFSTEP]\nrw [\u2190 F.map_comp, f.comm, F.map_comp]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nW : Type (u + 1)\ninst\u271d : LargeCategory W\nF : V \u2964 W\nG : MonCat\nM : Action V G\n\u22a2 {\n obj := fun M =>\n { V := F.obj M.V,\n \u03c1 :=\n { toOneHom := { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) },\n map_mul' :=\n (_ :\n \u2200 (g h : \u2191G),\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n (g * h) =\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) } g *\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n h) } },\n map := fun {X Y} f => Action.Hom.mk (F.map f.hom) }.map\n (\ud835\udfd9 M) =\n \ud835\udfd9\n ({\n obj := fun M =>\n { V := F.obj M.V,\n \u03c1 :=\n { toOneHom := { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) },\n map_mul' :=\n (_ :\n \u2200 (g h : \u2191G),\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n (g * h) =\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n g *\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n h) } },\n map := fun {X Y} f => Action.Hom.mk (F.map f.hom) }.obj\n M)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nW : Type (u + 1)\ninst\u271d : LargeCategory W\nF : V \u2964 W\nG : MonCat\nM : Action V G\n\u22a2 ({\n obj := fun M =>\n { V := F.obj M.V,\n \u03c1 :=\n { toOneHom := { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) },\n map_mul' :=\n (_ :\n \u2200 (g h : \u2191G),\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n (g * h) =\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n g *\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n h) } },\n map := fun {X Y} f => Action.Hom.mk (F.map f.hom) }.map\n (\ud835\udfd9 M)).hom =\n (\ud835\udfd9\n ({\n obj := fun M =>\n { V := F.obj M.V,\n \u03c1 :=\n {\n toOneHom :=\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) },\n map_mul' :=\n (_ :\n \u2200 (g h : \u2191G),\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n (g * h) =\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n g *\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n h) } },\n map := fun {X Y} f => Action.Hom.mk (F.map f.hom) }.obj\n M)).hom\n[PROOFSTEP]\nsimp only [Action.id_hom, F.map_id]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nW : Type (u + 1)\ninst\u271d : LargeCategory W\nF : V \u2964 W\nG : MonCat\nX\u271d Y\u271d Z\u271d : Action V G\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 {\n obj := fun M =>\n { V := F.obj M.V,\n \u03c1 :=\n { toOneHom := { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) },\n map_mul' :=\n (_ :\n \u2200 (g h : \u2191G),\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n (g * h) =\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) } g *\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n h) } },\n map := fun {X Y} f => Action.Hom.mk (F.map f.hom) }.map\n (f \u226b g) =\n {\n obj := fun M =>\n { V := F.obj M.V,\n \u03c1 :=\n { toOneHom := { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) },\n map_mul' :=\n (_ :\n \u2200 (g h : \u2191G),\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n (g * h) =\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n g *\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n h) } },\n map := fun {X Y} f => Action.Hom.mk (F.map f.hom) }.map\n f \u226b\n {\n obj := fun M =>\n { V := F.obj M.V,\n \u03c1 :=\n { toOneHom := { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) },\n map_mul' :=\n (_ :\n \u2200 (g h : \u2191G),\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n (g * h) =\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n g *\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n h) } },\n map := fun {X Y} f => Action.Hom.mk (F.map f.hom) }.map\n g\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nW : Type (u + 1)\ninst\u271d : LargeCategory W\nF : V \u2964 W\nG : MonCat\nX\u271d Y\u271d Z\u271d : Action V G\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 ({\n obj := fun M =>\n { V := F.obj M.V,\n \u03c1 :=\n { toOneHom := { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) },\n map_mul' :=\n (_ :\n \u2200 (g h : \u2191G),\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n (g * h) =\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n g *\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n h) } },\n map := fun {X Y} f => Action.Hom.mk (F.map f.hom) }.map\n (f \u226b g)).hom =\n ({\n obj := fun M =>\n { V := F.obj M.V,\n \u03c1 :=\n {\n toOneHom :=\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) },\n map_mul' :=\n (_ :\n \u2200 (g h : \u2191G),\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n (g * h) =\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n g *\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n h) } },\n map := fun {X Y} f => Action.Hom.mk (F.map f.hom) }.map\n f \u226b\n {\n obj := fun M =>\n { V := F.obj M.V,\n \u03c1 :=\n {\n toOneHom :=\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) },\n map_mul' :=\n (_ :\n \u2200 (g h : \u2191G),\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n (g * h) =\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n g *\n OneHom.toFun\n { toFun := fun g => F.map (\u2191M.\u03c1 g), map_one' := (_ : F.map (\u2191M.\u03c1 1) = \ud835\udfd9 (F.obj M.V)) }\n h) } },\n map := fun {X Y} f => Action.Hom.mk (F.map f.hom) }.map\n g).hom\n[PROOFSTEP]\nsimp only [Action.comp_hom, F.map_comp]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\ng : \u2191G\n\u22a2 \u2191(MonoidalCategory.tensorUnit (Action W G)).\u03c1 g \u226b F.\u03b5 =\n F.\u03b5 \u226b \u2191((Functor.mk src\u271d.toPrefunctor).obj (MonoidalCategory.tensorUnit (Action V G))).\u03c1 g\n[PROOFSTEP]\ndsimp [FunctorCategoryEquivalence.inverse, Functor.mapAction]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\ng : \u2191G\n\u22a2 \ud835\udfd9 (MonoidalCategory.tensorUnit W) \u226b F.\u03b5 = F.\u03b5 \u226b F.map (\ud835\udfd9 (MonoidalCategory.tensorUnit V))\n[PROOFSTEP]\nrw [Category.id_comp, F.map_id, Category.comp_id]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\n\u22a2 \u2200 {X Y X' Y' : Action V G} (f : X \u27f6 Y) (g : X' \u27f6 Y'),\n MonoidalCategory.tensorHom ((Functor.mk src\u271d.toPrefunctor).map f) ((Functor.mk src\u271d.toPrefunctor).map g) \u226b\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) Y Y' =\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) X X' \u226b\n (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.tensorHom f g)\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\nX\u271d Y\u271d X'\u271d Y'\u271d : Action V G\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : X'\u271d \u27f6 Y'\u271d\n\u22a2 MonoidalCategory.tensorHom ((Functor.mk src\u271d.toPrefunctor).map f\u271d) ((Functor.mk src\u271d.toPrefunctor).map g\u271d) \u226b\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) Y\u271d Y'\u271d =\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) X\u271d X'\u271d \u226b\n (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.tensorHom f\u271d g\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\nX\u271d Y\u271d X'\u271d Y'\u271d : Action V G\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : X'\u271d \u27f6 Y'\u271d\n\u22a2 (MonoidalCategory.tensorHom ((Functor.mk src\u271d.toPrefunctor).map f\u271d) ((Functor.mk src\u271d.toPrefunctor).map g\u271d) \u226b\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) Y\u271d Y'\u271d).hom =\n ((fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) X\u271d X'\u271d \u226b\n (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.tensorHom f\u271d g\u271d)).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\n\u22a2 \u2200 (X Y Z : Action V G),\n MonoidalCategory.tensorHom ((fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) X Y)\n (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj Z)) \u226b\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) (MonoidalCategory.tensorObj X Y) Z \u226b\n (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.associator X Y Z).hom =\n (MonoidalCategory.associator ((Functor.mk src\u271d.toPrefunctor).obj X) ((Functor.mk src\u271d.toPrefunctor).obj Y)\n ((Functor.mk src\u271d.toPrefunctor).obj Z)).hom \u226b\n MonoidalCategory.tensorHom (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj X))\n ((fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) Y Z) \u226b\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) X (MonoidalCategory.tensorObj Y Z)\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\nX\u271d Y\u271d Z\u271d : Action V G\n\u22a2 MonoidalCategory.tensorHom ((fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) X\u271d Y\u271d)\n (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj Z\u271d)) \u226b\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) (MonoidalCategory.tensorObj X\u271d Y\u271d) Z\u271d \u226b\n (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.associator X\u271d Y\u271d Z\u271d).hom =\n (MonoidalCategory.associator ((Functor.mk src\u271d.toPrefunctor).obj X\u271d) ((Functor.mk src\u271d.toPrefunctor).obj Y\u271d)\n ((Functor.mk src\u271d.toPrefunctor).obj Z\u271d)).hom \u226b\n MonoidalCategory.tensorHom (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj X\u271d))\n ((fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) Y\u271d Z\u271d) \u226b\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) X\u271d (MonoidalCategory.tensorObj Y\u271d Z\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\nX\u271d Y\u271d Z\u271d : Action V G\n\u22a2 (MonoidalCategory.tensorHom ((fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) X\u271d Y\u271d)\n (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj Z\u271d)) \u226b\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) (MonoidalCategory.tensorObj X\u271d Y\u271d)\n Z\u271d \u226b\n (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.associator X\u271d Y\u271d Z\u271d).hom).hom =\n ((MonoidalCategory.associator ((Functor.mk src\u271d.toPrefunctor).obj X\u271d) ((Functor.mk src\u271d.toPrefunctor).obj Y\u271d)\n ((Functor.mk src\u271d.toPrefunctor).obj Z\u271d)).hom \u226b\n MonoidalCategory.tensorHom (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj X\u271d))\n ((fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) Y\u271d Z\u271d) \u226b\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) X\u271d\n (MonoidalCategory.tensorObj Y\u271d Z\u271d)).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\n\u22a2 \u2200 (X : Action V G),\n (MonoidalCategory.leftUnitor ((Functor.mk src\u271d.toPrefunctor).obj X)).hom =\n MonoidalCategory.tensorHom (Hom.mk F.\u03b5) (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj X)) \u226b\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V))\n (MonoidalCategory.tensorUnit (Action V G)) X \u226b\n (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.leftUnitor X).hom\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\nX\u271d : Action V G\n\u22a2 (MonoidalCategory.leftUnitor ((Functor.mk src\u271d.toPrefunctor).obj X\u271d)).hom =\n MonoidalCategory.tensorHom (Hom.mk F.\u03b5) (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj X\u271d)) \u226b\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V))\n (MonoidalCategory.tensorUnit (Action V G)) X\u271d \u226b\n (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.leftUnitor X\u271d).hom\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\nX\u271d : Action V G\n\u22a2 (MonoidalCategory.leftUnitor ((Functor.mk src\u271d.toPrefunctor).obj X\u271d)).hom.hom =\n (MonoidalCategory.tensorHom (Hom.mk F.\u03b5) (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj X\u271d)) \u226b\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V))\n (MonoidalCategory.tensorUnit (Action V G)) X\u271d \u226b\n (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.leftUnitor X\u271d).hom).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\n\u22a2 \u2200 (X : Action V G),\n (MonoidalCategory.rightUnitor ((Functor.mk src\u271d.toPrefunctor).obj X)).hom =\n MonoidalCategory.tensorHom (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj X)) (Hom.mk F.\u03b5) \u226b\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) X\n (MonoidalCategory.tensorUnit (Action V G)) \u226b\n (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.rightUnitor X).hom\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\nX\u271d : Action V G\n\u22a2 (MonoidalCategory.rightUnitor ((Functor.mk src\u271d.toPrefunctor).obj X\u271d)).hom =\n MonoidalCategory.tensorHom (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj X\u271d)) (Hom.mk F.\u03b5) \u226b\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) X\u271d\n (MonoidalCategory.tensorUnit (Action V G)) \u226b\n (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.rightUnitor X\u271d).hom\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\nX\u271d : Action V G\n\u22a2 (MonoidalCategory.rightUnitor ((Functor.mk src\u271d.toPrefunctor).obj X\u271d)).hom.hom =\n (MonoidalCategory.tensorHom (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj X\u271d)) (Hom.mk F.\u03b5) \u226b\n (fun X Y => Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)) X\u271d\n (MonoidalCategory.tensorUnit (Action V G)) \u226b\n (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.rightUnitor X\u271d).hom).hom\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\nX\u271d : Action V G\n\u22a2 (MonoidalCategory.tensorHom (\ud835\udfd9 (F.obj X\u271d.V)) (\ud835\udfd9 (MonoidalCategory.tensorUnit W)) \u226b\n (MonoidalCategory.rightUnitor (F.obj X\u271d.V)).hom) \u226b\n \ud835\udfd9 (F.obj X\u271d.V) =\n MonoidalCategory.tensorHom (\ud835\udfd9 (F.obj X\u271d.V)) F.\u03b5 \u226b\n LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X\u271d.V (MonoidalCategory.tensorUnit V) \u226b\n F.map\n ((MonoidalCategory.tensorHom (\ud835\udfd9 X\u271d.V) (\ud835\udfd9 (MonoidalCategory.tensorUnit V)) \u226b\n (MonoidalCategory.rightUnitor X\u271d.V).hom) \u226b\n \ud835\udfd9 X\u271d.V)\n[PROOFSTEP]\nsimp only [MonoidalCategory.rightUnitor_conjugation, LaxMonoidalFunctor.right_unitality, Category.id_comp,\n Category.assoc, LaxMonoidalFunctor.right_unitality_inv_assoc, Category.comp_id, Iso.hom_inv_id]\n[GOAL]\ncase h\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\nX\u271d : Action V G\n\u22a2 MonoidalCategory.tensorHom (\ud835\udfd9 (F.obj X\u271d.V)) F.\u03b5 \u226b\n LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X\u271d.V (MonoidalCategory.tensorUnit V) \u226b\n F.map (MonoidalCategory.rightUnitor X\u271d.V).hom \u226b\n F.map (MonoidalCategory.rightUnitor X\u271d.V).inv \u226b F.map (MonoidalCategory.rightUnitor X\u271d.V).hom =\n MonoidalCategory.tensorHom (\ud835\udfd9 (F.obj X\u271d.V)) F.\u03b5 \u226b\n LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X\u271d.V (MonoidalCategory.tensorUnit V) \u226b\n F.map (MonoidalCategory.rightUnitor X\u271d.V).hom\n[PROOFSTEP]\nrw [\u2190 F.map_comp, Iso.inv_hom_id, F.map_id, Category.comp_id]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\n\u22a2 IsIso\n (LaxMonoidalFunctor.mk (Functor.mk src\u271d.toPrefunctor) (Hom.mk F.\u03b5) fun X Y =>\n Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)).\u03b5\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nsrc\u271d : Action V G \u2964 Action W G := Functor.mapAction F.toFunctor G\n\u22a2 \u2200 (X Y : Action V G),\n IsIso\n (LaxMonoidalFunctor.\u03bc\n (LaxMonoidalFunctor.mk (Functor.mk src\u271d.toPrefunctor) (Hom.mk F.\u03b5) fun X Y =>\n Hom.mk (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V))\n X Y)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\n\u22a2 (inv (mapAction F G).toLaxMonoidalFunctor.\u03b5).hom = inv F.\u03b5\n[PROOFSTEP]\nrw [\u2190 cancel_mono F.\u03b5, IsIso.inv_hom_id, \u2190 F.mapAction_toLaxMonoidalFunctor_\u03b5_hom G, \u2190 Action.comp_hom,\n IsIso.inv_hom_id, Action.id_hom]\n[GOAL]\nV : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nX Y : Action V G\n\u22a2 (inv (LaxMonoidalFunctor.\u03bc (mapAction F G).toLaxMonoidalFunctor X Y)).hom =\n inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X.V Y.V)\n[PROOFSTEP]\nrw [\u2190 cancel_mono (F.\u03bc X.V Y.V), IsIso.inv_hom_id, \u2190 F.mapAction_toLaxMonoidalFunctor_\u03bc_hom G, \u2190 Action.comp_hom,\n IsIso.inv_hom_id, Action.id_hom]\n", "meta": {"mathlib_filename": "Mathlib.RepresentationTheory.Action", "llama_tokens": 35489, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7634837635542925, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4496072898748403}} {"text": "[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u2074 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b9 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nf : F\ns : Set M\nN : Submodule R\u2082 M\u2082\n\u22a2 map f (span R s) \u2264 N \u2194 \u2200 (m : M), m \u2208 s \u2192 \u2191f m \u2208 N\n[PROOFSTEP]\nrw [map_span, span_le, Set.image_subset_iff]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u2074 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b9 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nf : F\ns : Set M\nN : Submodule R\u2082 M\u2082\n\u22a2 s \u2286 \u2191f \u207b\u00b9' \u2191N \u2194 \u2200 (m : M), m \u2208 s \u2192 \u2191f m \u2208 N\n[PROOFSTEP]\nexact Iff.rfl\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u22a2 span R (insert 0 s) = span R s\n[PROOFSTEP]\nrefine' le_antisymm _ (Submodule.span_mono (Set.subset_insert 0 s))\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u22a2 span R (insert 0 s) \u2264 span R s\n[PROOFSTEP]\nrw [span_le, Set.insert_subset_iff]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u22a2 0 \u2208 \u2191(span R s) \u2227 s \u2286 \u2191(span R s)\n[PROOFSTEP]\nexact \u27e8by simp only [SetLike.mem_coe, Submodule.zero_mem], Submodule.subset_span\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u22a2 0 \u2208 \u2191(span R s)\n[PROOFSTEP]\nsimp only [SetLike.mem_coe, Submodule.zero_mem]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\nf : F\ns : Set M\u2082\n\u22a2 span R (\u2191f \u207b\u00b9' s) \u2264 comap f (span R\u2082 s)\n[PROOFSTEP]\nrw [span_le, comap_coe]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\nf : F\ns : Set M\u2082\n\u22a2 \u2191f \u207b\u00b9' s \u2286 \u2191f \u207b\u00b9' \u2191(span R\u2082 s)\n[PROOFSTEP]\nexact preimage_mono subset_span\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\np : M \u2192 Prop\nh : x \u2208 span R s\u271d\nHs : \u2200 (x : M), x \u2208 s\u271d \u2192 p x\nH0 : p 0\nH1 : \u2200 (x y : M), p x \u2192 p y \u2192 p (x + y)\nH2 : \u2200 (a : R) (x : M), p x \u2192 p (a \u2022 x)\ns : Set M\n\u22a2 \u2200 {a b : M}, a \u2208 p \u2192 b \u2208 p \u2192 a + b \u2208 p\n[PROOFSTEP]\nintros x y\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\np : M \u2192 Prop\nh : x\u271d \u2208 span R s\u271d\nHs : \u2200 (x : M), x \u2208 s\u271d \u2192 p x\nH0 : p 0\nH1 : \u2200 (x y : M), p x \u2192 p y \u2192 p (x + y)\nH2 : \u2200 (a : R) (x : M), p x \u2192 p (a \u2022 x)\ns : Set M\nx y : M\n\u22a2 x \u2208 p \u2192 y \u2208 p \u2192 x + y \u2208 p\n[PROOFSTEP]\nexact H1 x y\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\np : (x : M) \u2192 x \u2208 span R s \u2192 Prop\nHs : \u2200 (x : M) (h : x \u2208 s), p x (_ : x \u2208 \u2191(span R s))\nH0 : p 0 (_ : 0 \u2208 span R s)\nH1 : \u2200 (x : M) (hx : x \u2208 span R s) (y : M) (hy : y \u2208 span R s), p x hx \u2192 p y hy \u2192 p (x + y) (_ : x + y \u2208 span R s)\nH2 : \u2200 (a : R) (x : M) (hx : x \u2208 span R s), p x hx \u2192 p (a \u2022 x) (_ : a \u2022 x \u2208 span R s)\nx : M\nhx : x \u2208 span R s\n\u22a2 p x hx\n[PROOFSTEP]\nrefine' Exists.elim _ fun (hx : x \u2208 span R s) (hc : p x hx) => hc\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\np : (x : M) \u2192 x \u2208 span R s \u2192 Prop\nHs : \u2200 (x : M) (h : x \u2208 s), p x (_ : x \u2208 \u2191(span R s))\nH0 : p 0 (_ : 0 \u2208 span R s)\nH1 : \u2200 (x : M) (hx : x \u2208 span R s) (y : M) (hy : y \u2208 span R s), p x hx \u2192 p y hy \u2192 p (x + y) (_ : x + y \u2208 span R s)\nH2 : \u2200 (a : R) (x : M) (hx : x \u2208 span R s), p x hx \u2192 p (a \u2022 x) (_ : a \u2022 x \u2208 span R s)\nx : M\nhx : x \u2208 span R s\n\u22a2 \u2203 x_1, p x x_1\n[PROOFSTEP]\nrefine'\n span_induction hx (fun m hm => \u27e8subset_span hm, Hs m hm\u27e9) \u27e8zero_mem _, H0\u27e9\n (fun x y hx hy => Exists.elim hx fun hx' hx => Exists.elim hy fun hy' hy => \u27e8add_mem hx' hy', H1 _ _ _ _ hx hy\u27e9)\n fun r x hx => Exists.elim hx fun hx' hx => \u27e8smul_mem _ _ hx', H2 r _ _ hx\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b2 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx\u271d\u00b9 : { x // x \u2208 \u2191(span R s) }\nx : M\nhx : x \u2208 \u2191(span R s)\nx\u271d : { val := x, property := hx } \u2208 \u22a4\n\u22a2 { val := x, property := hx } \u2208 span R (Subtype.val \u207b\u00b9' s)\n[PROOFSTEP]\nrefine'\n span_induction' (p := fun x hx \u21a6 (\u27e8x, hx\u27e9 : span R s) \u2208 span R (Subtype.val \u207b\u00b9' s)) (fun x' hx' \u21a6 subset_span hx') _\n (fun x _ y _ \u21a6 _) (fun r x _ \u21a6 _) hx\n[GOAL]\ncase refine'_1\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b2 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx\u271d\u00b9 : { x // x \u2208 \u2191(span R s) }\nx : M\nhx : x \u2208 \u2191(span R s)\nx\u271d : { val := x, property := hx } \u2208 \u22a4\n\u22a2 (fun x hx => { val := x, property := hx } \u2208 span R (Subtype.val \u207b\u00b9' s)) 0 (_ : 0 \u2208 span R s)\n[PROOFSTEP]\nexact zero_mem _\n[GOAL]\ncase refine'_2\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u2075 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx\u271d\u2074 : { x // x \u2208 \u2191(span R s) }\nx\u271d\u00b3 : M\nhx : x\u271d\u00b3 \u2208 \u2191(span R s)\nx\u271d\u00b2 : { val := x\u271d\u00b3, property := hx } \u2208 \u22a4\nx : M\nx\u271d\u00b9 : x \u2208 span R s\ny : M\nx\u271d : y \u2208 span R s\n\u22a2 (fun x hx => { val := x, property := hx } \u2208 span R (Subtype.val \u207b\u00b9' s)) x x\u271d\u00b9 \u2192\n (fun x hx => { val := x, property := hx } \u2208 span R (Subtype.val \u207b\u00b9' s)) y x\u271d \u2192\n (fun x hx => { val := x, property := hx } \u2208 span R (Subtype.val \u207b\u00b9' s)) (x + y) (_ : x + y \u2208 span R s)\n[PROOFSTEP]\nexact add_mem\n[GOAL]\ncase refine'_3\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u2074 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx\u271d\u00b3 : { x // x \u2208 \u2191(span R s) }\nx\u271d\u00b2 : M\nhx : x\u271d\u00b2 \u2208 \u2191(span R s)\nx\u271d\u00b9 : { val := x\u271d\u00b2, property := hx } \u2208 \u22a4\nr : R\nx : M\nx\u271d : x \u2208 span R s\n\u22a2 (fun x hx => { val := x, property := hx } \u2208 span R (Subtype.val \u207b\u00b9' s)) x x\u271d \u2192\n (fun x hx => { val := x, property := hx } \u2208 span R (Subtype.val \u207b\u00b9' s)) (r \u2022 x) (_ : r \u2022 x \u2208 span R s)\n[PROOFSTEP]\nexact smul_mem _ _\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t s : Set M\n\u22a2 (span \u2115 s).toAddSubmonoid = AddSubmonoid.closure s\n[PROOFSTEP]\nrefine' Eq.symm (AddSubmonoid.closure_eq_of_le subset_span _)\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t s : Set M\n\u22a2 (span \u2115 s).toAddSubmonoid \u2264 AddSubmonoid.closure s\n[PROOFSTEP]\napply\n (OrderIso.to_galoisConnection (AddSubmonoid.toNatSubmodule (M := M)).symm).l_le (a := span \u2115 s) (b :=\n AddSubmonoid.closure s)\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t s : Set M\n\u22a2 span \u2115 s \u2264 \u2191(OrderIso.symm (OrderIso.symm AddSubmonoid.toNatSubmodule)) (AddSubmonoid.closure s)\n[PROOFSTEP]\nrw [span_le]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t s : Set M\n\u22a2 s \u2286 \u2191(\u2191(OrderIso.symm (OrderIso.symm AddSubmonoid.toNatSubmodule)) (AddSubmonoid.closure s))\n[PROOFSTEP]\nexact AddSubmonoid.subset_closure\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\ns : AddSubmonoid M\n\u22a2 (span \u2115 \u2191s).toAddSubmonoid = s\n[PROOFSTEP]\nrw [span_nat_eq_addSubmonoid_closure, s.closure_eq]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\u271d\ninst\u271d\u2075 : Module R M\u271d\nx : M\u271d\np p' : Submodule R M\u271d\ninst\u271d\u2074 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b9 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns\u271d t : Set M\u271d\nM : Type u_9\ninst\u271d : AddCommGroup M\ns : AddSubgroup M\n\u22a2 toAddSubgroup (span \u2124 \u2191s) = s\n[PROOFSTEP]\nrw [span_int_eq_addSubgroup_closure, s.closure_eq]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u2074 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b9 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\ninst\u271d : DecidableEq M\n\u03b1 : Type u_9\ns : Finset \u03b1\nf : { x // x \u2208 s } \u2192 Finset M\n\u22a2 span R \u2191(Finset.biUnion (Finset.attach s) f) = \u2a06 (x : { x // x \u2208 s }), span R \u2191(f x)\n[PROOFSTEP]\nsimp [span_iUnion]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u22a2 p \u2294 span R s = span R (\u2191p \u222a s)\n[PROOFSTEP]\nrw [Submodule.span_union, p.span_eq]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u22a2 span R s \u2294 p = span R (s \u222a \u2191p)\n[PROOFSTEP]\nrw [Submodule.span_union, p.span_eq]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t s : Set M\n\u22a2 span R s = \u2a06 (x : M) (_ : x \u2208 s), span R {x}\n[PROOFSTEP]\nsimp only [\u2190 span_iUnion, Set.biUnion_of_singleton s]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Type u_9\nv : \u03b9 \u2192 M\n\u22a2 span R (range v) = \u2a06 (i : \u03b9), span R {v i}\n[PROOFSTEP]\nrw [span_eq_iSup_of_singleton_spans, iSup_range]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t s : Set M\nr : R\n\u22a2 span R (r \u2022 s) \u2264 span R s\n[PROOFSTEP]\nrw [span_le]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t s : Set M\nr : R\n\u22a2 r \u2022 s \u2286 \u2191(span R s)\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t s : Set M\nr : R\nx : M\nhx : x \u2208 s\n\u22a2 (fun x => r \u2022 x) x \u2208 \u2191(span R s)\n[PROOFSTEP]\nexact smul_mem (span R s) r (subset_span hx)\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t s : Set M\nr : R\nhr : IsUnit r\n\u22a2 span R (r \u2022 s) = span R s\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t s : Set M\nr : R\nhr : IsUnit r\n\u22a2 span R (r \u2022 s) \u2264 span R s\n[PROOFSTEP]\napply span_smul_le\n[GOAL]\ncase a\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t s : Set M\nr : R\nhr : IsUnit r\n\u22a2 span R s \u2264 span R (r \u2022 s)\n[PROOFSTEP]\nconvert span_smul_le (r \u2022 s) ((hr.unit\u207b\u00b9 : _) : R)\n[GOAL]\ncase h.e'_3.h.e'_6\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t s : Set M\nr : R\nhr : IsUnit r\n\u22a2 s = \u2191(IsUnit.unit hr)\u207b\u00b9 \u2022 r \u2022 s\n[PROOFSTEP]\nrw [smul_smul]\n[GOAL]\ncase h.e'_3.h.e'_6\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t s : Set M\nr : R\nhr : IsUnit r\n\u22a2 s = (\u2191(IsUnit.unit hr)\u207b\u00b9 * r) \u2022 s\n[PROOFSTEP]\nerw [hr.unit.inv_val]\n[GOAL]\ncase h.e'_3.h.e'_6\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t s : Set M\nr : R\nhr : IsUnit r\n\u22a2 s = 1 \u2022 s\n[PROOFSTEP]\nrw [one_smul]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submodule R M\nH : Directed (fun x x_1 => x \u2264 x_1) S\n\u22a2 \u2191(iSup S) = \u22c3 (i : \u03b9), \u2191(S i)\n[PROOFSTEP]\nrefine' Subset.antisymm _ (iUnion_subset <| le_iSup S)\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submodule R M\nH : Directed (fun x x_1 => x \u2264 x_1) S\n\u22a2 \u2191(iSup S) \u2286 \u22c3 (i : \u03b9), \u2191(S i)\n[PROOFSTEP]\nsuffices (span R (\u22c3 i, (S i : Set M)) : Set M) \u2286 \u22c3 i : \u03b9, \u2191(S i) by simpa only [span_iUnion, span_eq] using this\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submodule R M\nH : Directed (fun x x_1 => x \u2264 x_1) S\nthis : \u2191(span R (\u22c3 (i : \u03b9), \u2191(S i))) \u2286 \u22c3 (i : \u03b9), \u2191(S i)\n\u22a2 \u2191(iSup S) \u2286 \u22c3 (i : \u03b9), \u2191(S i)\n[PROOFSTEP]\nsimpa only [span_iUnion, span_eq] using this\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submodule R M\nH : Directed (fun x x_1 => x \u2264 x_1) S\n\u22a2 \u2191(span R (\u22c3 (i : \u03b9), \u2191(S i))) \u2286 \u22c3 (i : \u03b9), \u2191(S i)\n[PROOFSTEP]\nrefine' fun x hx => span_induction hx (fun _ => id) _ _ _\n[GOAL]\ncase refine'_1\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submodule R M\nH : Directed (fun x x_1 => x \u2264 x_1) S\nx : M\nhx : x \u2208 \u2191(span R (\u22c3 (i : \u03b9), \u2191(S i)))\n\u22a2 0 \u2208 \u22c3 (i : \u03b9), \u2191(S i)\n[PROOFSTEP]\nsimp only [mem_iUnion, exists_imp]\n[GOAL]\ncase refine'_2\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submodule R M\nH : Directed (fun x x_1 => x \u2264 x_1) S\nx : M\nhx : x \u2208 \u2191(span R (\u22c3 (i : \u03b9), \u2191(S i)))\n\u22a2 \u2200 (x y : M), x \u2208 \u22c3 (i : \u03b9), \u2191(S i) \u2192 y \u2208 \u22c3 (i : \u03b9), \u2191(S i) \u2192 x + y \u2208 \u22c3 (i : \u03b9), \u2191(S i)\n[PROOFSTEP]\nsimp only [mem_iUnion, exists_imp]\n[GOAL]\ncase refine'_3\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submodule R M\nH : Directed (fun x x_1 => x \u2264 x_1) S\nx : M\nhx : x \u2208 \u2191(span R (\u22c3 (i : \u03b9), \u2191(S i)))\n\u22a2 \u2200 (a : R) (x : M), x \u2208 \u22c3 (i : \u03b9), \u2191(S i) \u2192 a \u2022 x \u2208 \u22c3 (i : \u03b9), \u2191(S i)\n[PROOFSTEP]\nsimp only [mem_iUnion, exists_imp]\n[GOAL]\ncase refine'_1\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submodule R M\nH : Directed (fun x x_1 => x \u2264 x_1) S\nx : M\nhx : x \u2208 \u2191(span R (\u22c3 (i : \u03b9), \u2191(S i)))\n\u22a2 \u2203 i, 0 \u2208 \u2191(S i)\n[PROOFSTEP]\nexact h\u03b9.elim fun i => \u27e8i, (S i).zero_mem\u27e9\n[GOAL]\ncase refine'_2\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submodule R M\nH : Directed (fun x x_1 => x \u2264 x_1) S\nx : M\nhx : x \u2208 \u2191(span R (\u22c3 (i : \u03b9), \u2191(S i)))\n\u22a2 \u2200 (x y : M) (x_1 : \u03b9), x \u2208 \u2191(S x_1) \u2192 \u2200 (x_2 : \u03b9), y \u2208 \u2191(S x_2) \u2192 \u2203 i, x + y \u2208 \u2191(S i)\n[PROOFSTEP]\nintro x y i hi j hj\n[GOAL]\ncase refine'_2\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submodule R M\nH : Directed (fun x x_1 => x \u2264 x_1) S\nx\u271d : M\nhx : x\u271d \u2208 \u2191(span R (\u22c3 (i : \u03b9), \u2191(S i)))\nx y : M\ni : \u03b9\nhi : x \u2208 \u2191(S i)\nj : \u03b9\nhj : y \u2208 \u2191(S j)\n\u22a2 \u2203 i, x + y \u2208 \u2191(S i)\n[PROOFSTEP]\nrcases H i j with \u27e8k, ik, jk\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submodule R M\nH : Directed (fun x x_1 => x \u2264 x_1) S\nx\u271d : M\nhx : x\u271d \u2208 \u2191(span R (\u22c3 (i : \u03b9), \u2191(S i)))\nx y : M\ni : \u03b9\nhi : x \u2208 \u2191(S i)\nj : \u03b9\nhj : y \u2208 \u2191(S j)\nk : \u03b9\nik : S i \u2264 S k\njk : S j \u2264 S k\n\u22a2 \u2203 i, x + y \u2208 \u2191(S i)\n[PROOFSTEP]\nexact \u27e8k, add_mem (ik hi) (jk hj)\u27e9\n[GOAL]\ncase refine'_3\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submodule R M\nH : Directed (fun x x_1 => x \u2264 x_1) S\nx : M\nhx : x \u2208 \u2191(span R (\u22c3 (i : \u03b9), \u2191(S i)))\n\u22a2 \u2200 (a : R) (x : M) (x_1 : \u03b9), x \u2208 \u2191(S x_1) \u2192 \u2203 i, a \u2022 x \u2208 \u2191(S i)\n[PROOFSTEP]\nexact fun a x i hi => \u27e8i, smul_mem _ a hi\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2074 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b9 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\ninst\u271d : Nonempty \u03b9\nS : \u03b9 \u2192 Submodule R M\nH : Directed (fun x x_1 => x \u2264 x_1) S\nx : M\n\u22a2 x \u2208 iSup S \u2194 \u2203 i, x \u2208 S i\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, coe_iSup_of_directed S H, mem_iUnion]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2074 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b9 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\ninst\u271d : Nonempty \u03b9\nS : \u03b9 \u2192 Submodule R M\nH : Directed (fun x x_1 => x \u2264 x_1) S\nx : M\n\u22a2 (\u2203 i, x \u2208 \u2191(S i)) \u2194 \u2203 i, x \u2208 S i\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\ns : Set (Submodule R M)\nz : M\nhs : Set.Nonempty s\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\n\u22a2 z \u2208 sSup s \u2194 \u2203 y, y \u2208 s \u2227 z \u2208 y\n[PROOFSTEP]\nhave : Nonempty s := hs.to_subtype\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\ns : Set (Submodule R M)\nz : M\nhs : Set.Nonempty s\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\nthis : Nonempty \u2191s\n\u22a2 z \u2208 sSup s \u2194 \u2203 y, y \u2208 s \u2227 z \u2208 y\n[PROOFSTEP]\nsimp only [sSup_eq_iSup', mem_iSup_of_directed _ hdir.directed_val, SetCoe.exists, Subtype.coe_mk, exists_prop]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh : x \u2208 p \u2294 p'\n\u22a2 \u2203 y, y \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y + z = x\n[PROOFSTEP]\nrw [\u2190 span_eq p, \u2190 span_eq p', \u2190 span_union] at h \n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh : x \u2208 span R (\u2191p \u222a \u2191p')\n\u22a2 \u2203 y, y \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y + z = x\n[PROOFSTEP]\nrefine span_induction h ?_ ?_ ?_ ?_\n[GOAL]\ncase refine_1\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh : x \u2208 span R (\u2191p \u222a \u2191p')\n\u22a2 \u2200 (x : M), x \u2208 \u2191p \u222a \u2191p' \u2192 \u2203 y, y \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y + z = x\n[PROOFSTEP]\nrintro y (h | h)\n[GOAL]\ncase refine_1.inl\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh\u271d : x \u2208 span R (\u2191p \u222a \u2191p')\ny : M\nh : y \u2208 \u2191p\n\u22a2 \u2203 y_1, y_1 \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y_1 + z = y\n[PROOFSTEP]\nexact \u27e8y, h, 0, by simp, by simp\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh\u271d : x \u2208 span R (\u2191p \u222a \u2191p')\ny : M\nh : y \u2208 \u2191p\n\u22a2 0 \u2208 p'\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh\u271d : x \u2208 span R (\u2191p \u222a \u2191p')\ny : M\nh : y \u2208 \u2191p\n\u22a2 y + 0 = y\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_1.inr\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh\u271d : x \u2208 span R (\u2191p \u222a \u2191p')\ny : M\nh : y \u2208 \u2191p'\n\u22a2 \u2203 y_1, y_1 \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y_1 + z = y\n[PROOFSTEP]\nexact \u27e80, by simp, y, h, by simp\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh\u271d : x \u2208 span R (\u2191p \u222a \u2191p')\ny : M\nh : y \u2208 \u2191p'\n\u22a2 0 \u2208 p\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh\u271d : x \u2208 span R (\u2191p \u222a \u2191p')\ny : M\nh : y \u2208 \u2191p'\n\u22a2 0 + y = y\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_2\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh : x \u2208 span R (\u2191p \u222a \u2191p')\n\u22a2 \u2203 y, y \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y + z = 0\n[PROOFSTEP]\nexact \u27e80, by simp, 0, by simp\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh : x \u2208 span R (\u2191p \u222a \u2191p')\n\u22a2 0 \u2208 p\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh : x \u2208 span R (\u2191p \u222a \u2191p')\n\u22a2 0 \u2208 p' \u2227 0 + 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_3\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh : x \u2208 span R (\u2191p \u222a \u2191p')\n\u22a2 \u2200 (x y : M),\n (\u2203 y, y \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y + z = x) \u2192\n (\u2203 y_1, y_1 \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y_1 + z = y) \u2192 \u2203 y_1, y_1 \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y_1 + z = x + y\n[PROOFSTEP]\nrintro _ _ \u27e8y\u2081, hy\u2081, z\u2081, hz\u2081, rfl\u27e9 \u27e8y\u2082, hy\u2082, z\u2082, hz\u2082, rfl\u27e9\n[GOAL]\ncase refine_3.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh : x \u2208 span R (\u2191p \u222a \u2191p')\ny\u2081 : M\nhy\u2081 : y\u2081 \u2208 p\nz\u2081 : M\nhz\u2081 : z\u2081 \u2208 p'\ny\u2082 : M\nhy\u2082 : y\u2082 \u2208 p\nz\u2082 : M\nhz\u2082 : z\u2082 \u2208 p'\n\u22a2 \u2203 y, y \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y + z = y\u2081 + z\u2081 + (y\u2082 + z\u2082)\n[PROOFSTEP]\nexact\n \u27e8_, add_mem hy\u2081 hy\u2082, _, add_mem hz\u2081 hz\u2082, by rw [add_assoc, add_assoc, \u2190 add_assoc y\u2082, \u2190 add_assoc z\u2081, add_comm y\u2082]\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh : x \u2208 span R (\u2191p \u222a \u2191p')\ny\u2081 : M\nhy\u2081 : y\u2081 \u2208 p\nz\u2081 : M\nhz\u2081 : z\u2081 \u2208 p'\ny\u2082 : M\nhy\u2082 : y\u2082 \u2208 p\nz\u2082 : M\nhz\u2082 : z\u2082 \u2208 p'\n\u22a2 y\u2081 + y\u2082 + (z\u2081 + z\u2082) = y\u2081 + z\u2081 + (y\u2082 + z\u2082)\n[PROOFSTEP]\nrw [add_assoc, add_assoc, \u2190 add_assoc y\u2082, \u2190 add_assoc z\u2081, add_comm y\u2082]\n[GOAL]\ncase refine_4\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh : x \u2208 span R (\u2191p \u222a \u2191p')\n\u22a2 \u2200 (a : R) (x : M), (\u2203 y, y \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y + z = x) \u2192 \u2203 y, y \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y + z = a \u2022 x\n[PROOFSTEP]\nrintro a _ \u27e8y, hy, z, hz, rfl\u27e9\n[GOAL]\ncase refine_4.intro.intro.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh : x \u2208 span R (\u2191p \u222a \u2191p')\na : R\ny : M\nhy : y \u2208 p\nz : M\nhz : z \u2208 p'\n\u22a2 \u2203 y_1, y_1 \u2208 p \u2227 \u2203 z_1, z_1 \u2208 p' \u2227 y_1 + z_1 = a \u2022 (y + z)\n[PROOFSTEP]\nexact \u27e8_, smul_mem _ a hy, _, smul_mem _ a hz, by simp [smul_add]\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nh : x \u2208 span R (\u2191p \u222a \u2191p')\na : R\ny : M\nhy : y \u2208 p\nz : M\nhz : z \u2208 p'\n\u22a2 a \u2022 y + a \u2022 z = a \u2022 (y + z)\n[PROOFSTEP]\nsimp [smul_add]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u22a2 (\u2203 y, y \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y + z = x) \u2192 x \u2208 p \u2294 p'\n[PROOFSTEP]\nrintro \u27e8y, hy, z, hz, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\nhy : y \u2208 p\nz : M\nhz : z \u2208 p'\n\u22a2 y + z \u2208 p \u2294 p'\n[PROOFSTEP]\nexact add_mem ((le_sup_left : p \u2264 p \u2294 p') hy) ((le_sup_right : p' \u2264 p \u2294 p') hz)\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u22a2 (\u2203 y, y \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y + z = x) \u2194 \u2203 y z, \u2191y + \u2191z = x\n[PROOFSTEP]\nsimp only [Subtype.exists, exists_prop]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u22a2 \u2191(p \u2294 p') = \u2191p + \u2191p'\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx\u271d : M\n\u22a2 x\u271d \u2208 \u2191(p \u2294 p') \u2194 x\u271d \u2208 \u2191p + \u2191p'\n[PROOFSTEP]\nrw [SetLike.mem_coe, mem_sup, Set.mem_add]\n[GOAL]\ncase h\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx\u271d : M\n\u22a2 (\u2203 y, y \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y + z = x\u271d) \u2194 \u2203 x y, x \u2208 \u2191p \u2227 y \u2208 \u2191p' \u2227 x + y = x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u22a2 (p \u2294 p').toAddSubmonoid = p.toAddSubmonoid \u2294 p'.toAddSubmonoid\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\n\u22a2 x \u2208 (p \u2294 p').toAddSubmonoid \u2194 x \u2208 p.toAddSubmonoid \u2294 p'.toAddSubmonoid\n[PROOFSTEP]\nrw [mem_toAddSubmonoid, mem_sup, AddSubmonoid.mem_sup]\n[GOAL]\ncase h\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\n\u22a2 (\u2203 y, y \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y + z = x) \u2194 \u2203 y, y \u2208 p.toAddSubmonoid \u2227 \u2203 z, z \u2208 p'.toAddSubmonoid \u2227 y + z = x\n[PROOFSTEP]\nrfl\n[GOAL]\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2079 : Semiring R\u271d\ninst\u271d\u2078 : AddCommMonoid M\u271d\ninst\u271d\u2077 : Module R\u271d M\u271d\nx : M\u271d\np\u271d p'\u271d : Submodule R\u271d M\u271d\ninst\u271d\u2076 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b3 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\np p' : Submodule R M\n\u22a2 toAddSubgroup (p \u2294 p') = toAddSubgroup p \u2294 toAddSubgroup p'\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2079 : Semiring R\u271d\ninst\u271d\u2078 : AddCommMonoid M\u271d\ninst\u271d\u2077 : Module R\u271d M\u271d\nx\u271d : M\u271d\np\u271d p'\u271d : Submodule R\u271d M\u271d\ninst\u271d\u2076 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b3 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\np p' : Submodule R M\nx : M\n\u22a2 x \u2208 toAddSubgroup (p \u2294 p') \u2194 x \u2208 toAddSubgroup p \u2294 toAddSubgroup p'\n[PROOFSTEP]\nrw [mem_toAddSubgroup, mem_sup, AddSubgroup.mem_sup]\n[GOAL]\ncase h\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2079 : Semiring R\u271d\ninst\u271d\u2078 : AddCommMonoid M\u271d\ninst\u271d\u2077 : Module R\u271d M\u271d\nx\u271d : M\u271d\np\u271d p'\u271d : Submodule R\u271d M\u271d\ninst\u271d\u2076 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b3 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\np p' : Submodule R M\nx : M\n\u22a2 (\u2203 y, y \u2208 p \u2227 \u2203 z, z \u2208 p' \u2227 y + z = x) \u2194 \u2203 y, y \u2208 toAddSubgroup p \u2227 \u2203 z, z \u2208 toAddSubgroup p' \u2227 y + z = x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\nh : x \u2260 0\n\u22a2 \u2203 x_1 y, x_1 \u2260 y\n[PROOFSTEP]\nuse 0, \u27e8x, Submodule.mem_span_singleton_self x\u27e9\n[GOAL]\ncase h\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\nh : x \u2260 0\n\u22a2 0 \u2260 { val := x, property := (_ : x \u2208 span R {x}) }\n[PROOFSTEP]\nintro H\n[GOAL]\ncase h\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\nh : x \u2260 0\nH : 0 = { val := x, property := (_ : x \u2208 span R {x}) }\n\u22a2 False\n[PROOFSTEP]\nrw [eq_comm, Submodule.mk_eq_zero] at H \n[GOAL]\ncase h\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\nh : x \u2260 0\nH : x = 0\n\u22a2 False\n[PROOFSTEP]\nexact h H\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\nh : x \u2208 span R {y}\n\u22a2 \u2203 a, a \u2022 y = x\n[PROOFSTEP]\nrefine span_induction h ?_ ?_ ?_ ?_\n[GOAL]\ncase refine_1\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\nh : x \u2208 span R {y}\n\u22a2 \u2200 (x : M), x \u2208 {y} \u2192 \u2203 a, a \u2022 y = x\n[PROOFSTEP]\nrintro y (rfl | \u27e8\u27e8_\u27e9\u27e9)\n[GOAL]\ncase refine_1.refl\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\nh : x \u2208 span R {y}\n\u22a2 \u2203 a, a \u2022 y = y\n[PROOFSTEP]\nexact \u27e81, by simp\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\nh : x \u2208 span R {y}\n\u22a2 1 \u2022 y = y\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_2\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\nh : x \u2208 span R {y}\n\u22a2 \u2203 a, a \u2022 y = 0\n[PROOFSTEP]\nexact \u27e80, by simp\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\nh : x \u2208 span R {y}\n\u22a2 0 \u2022 y = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_3\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\nh : x \u2208 span R {y}\n\u22a2 \u2200 (x y_1 : M), (\u2203 a, a \u2022 y = x) \u2192 (\u2203 a, a \u2022 y = y_1) \u2192 \u2203 a, a \u2022 y = x + y_1\n[PROOFSTEP]\nrintro _ _ \u27e8a, rfl\u27e9 \u27e8b, rfl\u27e9\n[GOAL]\ncase refine_3.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\nh : x \u2208 span R {y}\na b : R\n\u22a2 \u2203 a_1, a_1 \u2022 y = a \u2022 y + b \u2022 y\n[PROOFSTEP]\nexact \u27e8a + b, by simp [add_smul]\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\nh : x \u2208 span R {y}\na b : R\n\u22a2 (a + b) \u2022 y = a \u2022 y + b \u2022 y\n[PROOFSTEP]\nsimp [add_smul]\n[GOAL]\ncase refine_4\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\nh : x \u2208 span R {y}\n\u22a2 \u2200 (a : R) (x : M), (\u2203 a, a \u2022 y = x) \u2192 \u2203 a_2, a_2 \u2022 y = a \u2022 x\n[PROOFSTEP]\nrintro a _ \u27e8b, rfl\u27e9\n[GOAL]\ncase refine_4.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\nh : x \u2208 span R {y}\na b : R\n\u22a2 \u2203 a_1, a_1 \u2022 y = a \u2022 b \u2022 y\n[PROOFSTEP]\nexact \u27e8a * b, by simp [smul_smul]\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\nh : x \u2208 span R {y}\na b : R\n\u22a2 (a * b) \u2022 y = a \u2022 b \u2022 y\n[PROOFSTEP]\nsimp [smul_smul]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\n\u22a2 (\u2203 a, a \u2022 y = x) \u2192 x \u2208 span R {y}\n[PROOFSTEP]\nrintro \u27e8a, y, rfl\u27e9\n[GOAL]\ncase intro.refl\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\na : R\n\u22a2 a \u2022 y \u2208 span R {y}\n[PROOFSTEP]\nexact smul_mem _ _ (subset_span <| by simp)\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\na : R\n\u22a2 y \u2208 {y}\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\ns : Submodule R M\nv\u2080 : M\n\u22a2 s \u2264 span R {v\u2080} \u2194 \u2200 (v : M), v \u2208 s \u2192 \u2203 r, r \u2022 v\u2080 = v\n[PROOFSTEP]\nsimp_rw [SetLike.le_def, mem_span_singleton]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\n\u22a2 span R {x} = \u22a4 \u2194 \u2200 (v : M), \u2203 r, r \u2022 x = v\n[PROOFSTEP]\nrw [eq_top_iff, le_span_singleton_iff]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\n\u22a2 (\u2200 (v : M), v \u2208 \u22a4 \u2192 \u2203 r, r \u2022 x = v) \u2194 \u2200 (v : M), \u2203 r, r \u2022 x = v\n[PROOFSTEP]\ntauto\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u22a2 span R {0} = \u22a5\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx\u271d : M\n\u22a2 x\u271d \u2208 span R {0} \u2194 x\u271d \u2208 \u22a5\n[PROOFSTEP]\nsimp [mem_span_singleton, eq_comm]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nS : Type u_9\ninst\u271d\u00b3 : Monoid S\ninst\u271d\u00b2 : SMul S R\ninst\u271d\u00b9 : MulAction S M\ninst\u271d : IsScalarTower S R M\nr : S\nx : M\n\u22a2 span R {r \u2022 x} \u2264 span R {x}\n[PROOFSTEP]\nrw [span_le, Set.singleton_subset_iff, SetLike.mem_coe]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nS : Type u_9\ninst\u271d\u00b3 : Monoid S\ninst\u271d\u00b2 : SMul S R\ninst\u271d\u00b9 : MulAction S M\ninst\u271d : IsScalarTower S R M\nr : S\nx : M\n\u22a2 r \u2022 x \u2208 span R {x}\n[PROOFSTEP]\nexact smul_of_tower_mem _ _ (mem_span_singleton_self _)\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nG : Type u_9\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : SMul G R\ninst\u271d\u00b9 : MulAction G M\ninst\u271d : IsScalarTower G R M\ng : G\nx : M\n\u22a2 span R {g \u2022 x} = span R {x}\n[PROOFSTEP]\nrefine' le_antisymm (span_singleton_smul_le R g x) _\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nG : Type u_9\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : SMul G R\ninst\u271d\u00b9 : MulAction G M\ninst\u271d : IsScalarTower G R M\ng : G\nx : M\n\u22a2 span R {x} \u2264 span R {g \u2022 x}\n[PROOFSTEP]\nconvert span_singleton_smul_le R g\u207b\u00b9 (g \u2022 x)\n[GOAL]\ncase h.e'_3.h.e'_6.h.e'_4\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nG : Type u_9\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : SMul G R\ninst\u271d\u00b9 : MulAction G M\ninst\u271d : IsScalarTower G R M\ng : G\nx : M\n\u22a2 x = g\u207b\u00b9 \u2022 g \u2022 x\n[PROOFSTEP]\nexact (inv_smul_smul g x).symm\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nr : R\nhr : IsUnit r\nx : M\n\u22a2 span R {r \u2022 x} = span R {x}\n[PROOFSTEP]\nlift r to R\u02e3 using hr\n[GOAL]\ncase intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\nr : R\u02e3\n\u22a2 span R {\u2191r \u2022 x} = span R {x}\n[PROOFSTEP]\nrw [\u2190 Units.smul_def]\n[GOAL]\ncase intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\nr : R\u02e3\n\u22a2 span R {r \u2022 x} = span R {x}\n[PROOFSTEP]\nexact span_singleton_group_smul_eq R r x\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK\u271d : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2076 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b3 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\nK : Type u_9\nE : Type u_10\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module K E\ns : Submodule K E\nx : E\n\u22a2 Disjoint s (span K {x}) \u2194 x \u2208 s \u2192 x = 0\n[PROOFSTEP]\nrefine' disjoint_def.trans \u27e8fun H hx => H x hx <| subset_span <| mem_singleton x, _\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK\u271d : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2076 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b3 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\nK : Type u_9\nE : Type u_10\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module K E\ns : Submodule K E\nx : E\n\u22a2 (x \u2208 s \u2192 x = 0) \u2192 \u2200 (x_1 : E), x_1 \u2208 s \u2192 x_1 \u2208 span K {x} \u2192 x_1 = 0\n[PROOFSTEP]\nintro H y hy hyx\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK\u271d : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2076 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b3 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\nK : Type u_9\nE : Type u_10\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module K E\ns : Submodule K E\nx : E\nH : x \u2208 s \u2192 x = 0\ny : E\nhy : y \u2208 s\nhyx : y \u2208 span K {x}\n\u22a2 y = 0\n[PROOFSTEP]\nobtain \u27e8c, rfl\u27e9 := mem_span_singleton.1 hyx\n[GOAL]\ncase intro\nR : Type u_1\nR\u2082 : Type u_2\nK\u271d : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2076 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b3 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\nK : Type u_9\nE : Type u_10\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module K E\ns : Submodule K E\nx : E\nH : x \u2208 s \u2192 x = 0\nc : K\nhy : c \u2022 x \u2208 s\nhyx : c \u2022 x \u2208 span K {x}\n\u22a2 c \u2022 x = 0\n[PROOFSTEP]\nby_cases hc : c = 0\n[GOAL]\ncase pos\nR : Type u_1\nR\u2082 : Type u_2\nK\u271d : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2076 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b3 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\nK : Type u_9\nE : Type u_10\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module K E\ns : Submodule K E\nx : E\nH : x \u2208 s \u2192 x = 0\nc : K\nhy : c \u2022 x \u2208 s\nhyx : c \u2022 x \u2208 span K {x}\nhc : c = 0\n\u22a2 c \u2022 x = 0\n[PROOFSTEP]\nrw [hc, zero_smul]\n[GOAL]\ncase neg\nR : Type u_1\nR\u2082 : Type u_2\nK\u271d : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2076 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b3 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\nK : Type u_9\nE : Type u_10\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module K E\ns : Submodule K E\nx : E\nH : x \u2208 s \u2192 x = 0\nc : K\nhy : c \u2022 x \u2208 s\nhyx : c \u2022 x \u2208 span K {x}\nhc : \u00acc = 0\n\u22a2 c \u2022 x = 0\n[PROOFSTEP]\nrw [s.smul_mem_iff hc] at hy \n[GOAL]\ncase neg\nR : Type u_1\nR\u2082 : Type u_2\nK\u271d : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2076 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b3 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\nK : Type u_9\nE : Type u_10\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module K E\ns : Submodule K E\nx : E\nH : x \u2208 s \u2192 x = 0\nc : K\nhy : x \u2208 s\nhyx : c \u2022 x \u2208 span K {x}\nhc : \u00acc = 0\n\u22a2 c \u2022 x = 0\n[PROOFSTEP]\nrw [H hy, smul_zero]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx y z : M\nhxy : x \u2208 span R {y}\nhyz : y \u2208 span R {z}\n\u22a2 x \u2208 span R {z}\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, \u2190 singleton_subset_iff] at *\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx y z : M\nhxy : {x} \u2286 \u2191(span R {y})\nhyz : {y} \u2286 \u2191(span R {z})\n\u22a2 {x} \u2286 \u2191(span R {z})\n[PROOFSTEP]\nexact Submodule.subset_span_trans hxy hyz\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\n\u22a2 x \u2208 span R (insert y s) \u2194 \u2203 a z, z \u2208 span R s \u2227 x = a \u2022 y + z\n[PROOFSTEP]\nsimp only [\u2190 union_singleton, span_union, mem_sup, mem_span_singleton, exists_prop, exists_exists_eq_and]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\n\u22a2 (\u2203 y_1, y_1 \u2208 span R s \u2227 \u2203 a, y_1 + a \u2022 y = x) \u2194 \u2203 a z, z \u2208 span R s \u2227 x = a \u2022 y + z\n[PROOFSTEP]\nrw [exists_comm]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\ny : M\n\u22a2 (\u2203 y_1, y_1 \u2208 span R s \u2227 \u2203 a, y_1 + a \u2022 y = x) \u2194 \u2203 b a, b \u2208 span R s \u2227 x = a \u2022 y + b\n[PROOFSTEP]\nsimp only [eq_comm, add_comm, exists_and_left]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx y z : M\n\u22a2 z \u2208 span R {x, y} \u2194 \u2203 a b, a \u2022 x + b \u2022 y = z\n[PROOFSTEP]\nsimp_rw [mem_span_insert, mem_span_singleton, exists_exists_eq_and, eq_comm]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\nx : M\ns : Set M\n\u22a2 span R (insert x s) = span R {x} \u2294 span R s\n[PROOFSTEP]\nrw [insert_eq, span_union]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u22a2 (\u2200 (x_1 : M), x_1 \u2208 {x} \u2192 x_1 = 0) \u2194 x = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u22a2 span R 0 = \u22a5\n[PROOFSTEP]\nrw [\u2190 singleton_zero, span_singleton_eq_bot]\n[GOAL]\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\n\u22a2 span R {x} = span R {y} \u2194 \u2203 z, z \u2022 x = y\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : x = 0\n\u22a2 span R {x} = span R {y} \u2194 \u2203 z, z \u2022 x = y\n[PROOFSTEP]\nrw [hx, span_zero_singleton, eq_comm, span_singleton_eq_bot]\n[GOAL]\ncase pos\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : x = 0\n\u22a2 y = 0 \u2194 \u2203 z, z \u2022 0 = y\n[PROOFSTEP]\nexact \u27e8fun hy => \u27e81, by rw [hy, smul_zero]\u27e9, fun \u27e8_, hz\u27e9 => by rw [\u2190 hz, smul_zero]\u27e9\n[GOAL]\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : x = 0\nhy : y = 0\n\u22a2 1 \u2022 0 = y\n[PROOFSTEP]\nrw [hy, smul_zero]\n[GOAL]\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d\u00b9 : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : x = 0\nx\u271d : \u2203 z, z \u2022 0 = y\nw\u271d : R\u02e3\nhz : w\u271d \u2022 0 = y\n\u22a2 y = 0\n[PROOFSTEP]\nrw [\u2190 hz, smul_zero]\n[GOAL]\ncase neg\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\n\u22a2 span R {x} = span R {y} \u2194 \u2203 z, z \u2022 x = y\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 span R {x} = span R {y} \u2194 \u2203 z, z \u2022 x = y\n[PROOFSTEP]\nrw [hy, span_zero_singleton, span_singleton_eq_bot]\n[GOAL]\ncase pos\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 x = 0 \u2194 \u2203 z, z \u2022 x = 0\n[PROOFSTEP]\nexact \u27e8fun hx => \u27e81, by rw [hx, smul_zero]\u27e9, fun \u27e8z, hz\u27e9 => (smul_eq_zero_iff_eq z).mp hz\u27e9\n[GOAL]\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx\u271d : \u00acx = 0\nhy : y = 0\nhx : x = 0\n\u22a2 1 \u2022 x = 0\n[PROOFSTEP]\nrw [hx, smul_zero]\n[GOAL]\ncase neg\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 span R {x} = span R {y} \u2194 \u2203 z, z \u2022 x = y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase neg.mp\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 span R {x} = span R {y} \u2192 \u2203 z, z \u2022 x = y\n[PROOFSTEP]\nintro hxy\n[GOAL]\ncase neg.mp\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : span R {x} = span R {y}\n\u22a2 \u2203 z, z \u2022 x = y\n[PROOFSTEP]\ncases'\n mem_span_singleton.mp\n (by\n rw [hxy]\n apply mem_span_singleton_self) with\n v hv\n[GOAL]\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : span R {x} = span R {y}\n\u22a2 ?m.246152 \u2208 span ?m.246042 {?m.246153}\n[PROOFSTEP]\nrw [hxy]\n[GOAL]\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : span R {x} = span R {y}\n\u22a2 ?m.246152 \u2208 span R {y}\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : span R {x} = span R {y}\n\u22a2 M\n[PROOFSTEP]\napply mem_span_singleton_self\n[GOAL]\ncase neg.mp.intro\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : span R {x} = span R {y}\nv : R\nhv : v \u2022 x = y\n\u22a2 \u2203 z, z \u2022 x = y\n[PROOFSTEP]\ncases'\n mem_span_singleton.mp\n (by\n rw [\u2190 hxy]\n apply mem_span_singleton_self) with\n i hi\n[GOAL]\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : span R {x} = span R {y}\nv : R\nhv : v \u2022 x = y\n\u22a2 ?m.246933 \u2208 span ?m.246823 {?m.246934}\n[PROOFSTEP]\nrw [\u2190 hxy]\n[GOAL]\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : span R {x} = span R {y}\nv : R\nhv : v \u2022 x = y\n\u22a2 ?m.246933 \u2208 span R {x}\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : span R {x} = span R {y}\nv : R\nhv : v \u2022 x = y\n\u22a2 M\n[PROOFSTEP]\napply mem_span_singleton_self\n[GOAL]\ncase neg.mp.intro.intro\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : span R {x} = span R {y}\nv : R\nhv : v \u2022 x = y\ni : R\nhi : i \u2022 y = x\n\u22a2 \u2203 z, z \u2022 x = y\n[PROOFSTEP]\nhave vi : v * i = 1 := by\n rw [\u2190 one_smul R y, \u2190 hi, smul_smul] at hv \n exact smul_left_injective R hy hv\n[GOAL]\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : span R {x} = span R {y}\nv : R\nhv : v \u2022 x = y\ni : R\nhi : i \u2022 y = x\n\u22a2 v * i = 1\n[PROOFSTEP]\nrw [\u2190 one_smul R y, \u2190 hi, smul_smul] at hv \n[GOAL]\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : span R {x} = span R {y}\nv i : R\nhv : (v * i) \u2022 y = 1 \u2022 y\nhi : i \u2022 y = x\n\u22a2 v * i = 1\n[PROOFSTEP]\nexact smul_left_injective R hy hv\n[GOAL]\ncase neg.mp.intro.intro\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : span R {x} = span R {y}\nv : R\nhv : v \u2022 x = y\ni : R\nhi : i \u2022 y = x\nvi : v * i = 1\n\u22a2 \u2203 z, z \u2022 x = y\n[PROOFSTEP]\nhave iv : i * v = 1 := by\n rw [\u2190 one_smul R x, \u2190 hv, smul_smul] at hi \n exact smul_left_injective R hx hi\n[GOAL]\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : span R {x} = span R {y}\nv : R\nhv : v \u2022 x = y\ni : R\nhi : i \u2022 y = x\nvi : v * i = 1\n\u22a2 i * v = 1\n[PROOFSTEP]\nrw [\u2190 one_smul R x, \u2190 hv, smul_smul] at hi \n[GOAL]\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : span R {x} = span R {y}\nv : R\nhv : v \u2022 x = y\ni : R\nhi : (i * v) \u2022 x = 1 \u2022 x\nvi : v * i = 1\n\u22a2 i * v = 1\n[PROOFSTEP]\nexact smul_left_injective R hx hi\n[GOAL]\ncase neg.mp.intro.intro\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : span R {x} = span R {y}\nv : R\nhv : v \u2022 x = y\ni : R\nhi : i \u2022 y = x\nvi : v * i = 1\niv : i * v = 1\n\u22a2 \u2203 z, z \u2022 x = y\n[PROOFSTEP]\nexact \u27e8\u27e8v, i, vi, iv\u27e9, hv\u27e9\n[GOAL]\ncase neg.mpr\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 (\u2203 z, z \u2022 x = y) \u2192 span R {x} = span R {y}\n[PROOFSTEP]\nrintro \u27e8v, rfl\u27e9\n[GOAL]\ncase neg.mpr.intro\nR\u271d : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b9\u2070 : Semiring R\u271d\ninst\u271d\u2079 : AddCommMonoid M\u271d\ninst\u271d\u2078 : Module R\u271d M\u271d\nx\u271d : M\u271d\np p' : Submodule R\u271d M\u271d\ninst\u271d\u2077 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M\u271d M\u2082\ns t : Set M\u271d\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx : M\nhx : \u00acx = 0\nv : R\u02e3\nhy : \u00acv \u2022 x = 0\n\u22a2 span R {x} = span R {v \u2022 x}\n[PROOFSTEP]\nrw [span_singleton_group_smul_eq]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2074 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b9 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nf : F\nx : M\ns : Set M\nh : x \u2208 span R s\n\u22a2 \u2191f x \u2208 span R\u2082 (\u2191f '' s)\n[PROOFSTEP]\nrw [Submodule.span_image]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2074 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b9 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nf : F\nx : M\ns : Set M\nh : x \u2208 span R s\n\u22a2 \u2191f x \u2208 map f (span R s)\n[PROOFSTEP]\nexact Submodule.mem_map_of_mem h\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2074 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b9 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nf : F\nx : M\ns : Set M\nhf : Injective \u2191f\n\u22a2 \u2191f x \u2208 span R\u2082 (\u2191f '' s) \u2194 x \u2208 span R s\n[PROOFSTEP]\nrw [\u2190 Submodule.mem_comap, \u2190 Submodule.map_span, Submodule.comap_map_eq_of_injective hf]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\np : Submodule R M\nx : { x // x \u2208 p }\n\u22a2 map (Submodule.subtype p) (span R {x}) = span R {\u2191x}\n[PROOFSTEP]\nsimp [\u2190 span_image]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\n\u22a2 \u2a06 (i : \u03b9), p i = span R (\u22c3 (i : \u03b9), \u2191(p i))\n[PROOFSTEP]\nsimp_rw [\u2190 iSup_span, span_eq]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\n\u22a2 (\u2a06 (i : \u03b9), p i).toAddSubmonoid = \u2a06 (i : \u03b9), (p i).toAddSubmonoid\n[PROOFSTEP]\nrefine' le_antisymm (fun x => _) (iSup_le fun i => toAddSubmonoid_mono <| le_iSup _ i)\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nx : M\n\u22a2 x \u2208 (\u2a06 (i : \u03b9), p i).toAddSubmonoid \u2192 x \u2208 \u2a06 (i : \u03b9), (p i).toAddSubmonoid\n[PROOFSTEP]\nsimp_rw [iSup_eq_span, AddSubmonoid.iSup_eq_closure, mem_toAddSubmonoid, coe_toAddSubmonoid]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nx : M\n\u22a2 x \u2208 span R (\u22c3 (i : \u03b9), \u2191(p i)) \u2192 x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n[PROOFSTEP]\nintro hx\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nx : M\nhx : x \u2208 span R (\u22c3 (i : \u03b9), \u2191(p i))\n\u22a2 x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n[PROOFSTEP]\nrefine' Submodule.span_induction hx (fun x hx => _) _ (fun x y hx hy => _) fun r x hx => _\n[GOAL]\ncase refine'_1\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nx\u271d : M\nhx\u271d : x\u271d \u2208 span R (\u22c3 (i : \u03b9), \u2191(p i))\nx : M\nhx : x \u2208 \u22c3 (i : \u03b9), \u2191(p i)\n\u22a2 x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n[PROOFSTEP]\nexact AddSubmonoid.subset_closure hx\n[GOAL]\ncase refine'_2\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nx : M\nhx : x \u2208 span R (\u22c3 (i : \u03b9), \u2191(p i))\n\u22a2 0 \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n[PROOFSTEP]\nexact AddSubmonoid.zero_mem _\n[GOAL]\ncase refine'_3\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nx\u271d : M\nhx\u271d : x\u271d \u2208 span R (\u22c3 (i : \u03b9), \u2191(p i))\nx y : M\nhx : x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\nhy : y \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n\u22a2 x + y \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n[PROOFSTEP]\nexact AddSubmonoid.add_mem _ hx hy\n[GOAL]\ncase refine'_4\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nx\u271d : M\nhx\u271d : x\u271d \u2208 span R (\u22c3 (i : \u03b9), \u2191(p i))\nr : R\nx : M\nhx : x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n\u22a2 r \u2022 x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n[PROOFSTEP]\nrefine AddSubmonoid.closure_induction hx ?_ ?_ ?_\n[GOAL]\ncase refine'_4.refine_1\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nx\u271d : M\nhx\u271d : x\u271d \u2208 span R (\u22c3 (i : \u03b9), \u2191(p i))\nr : R\nx : M\nhx : x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n\u22a2 \u2200 (x : M), x \u2208 \u22c3 (i : \u03b9), \u2191(p i) \u2192 r \u2022 x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n[PROOFSTEP]\nrintro x \u27e8_, \u27e8i, rfl\u27e9, hix : x \u2208 p i\u27e9\n[GOAL]\ncase refine'_4.refine_1.intro.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b2 : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nx\u271d\u00b9 : M\nhx\u271d : x\u271d\u00b9 \u2208 span R (\u22c3 (i : \u03b9), \u2191(p i))\nr : R\nx\u271d : M\nhx : x\u271d \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\nx : M\ni : \u03b9\nhix : x \u2208 p i\n\u22a2 r \u2022 x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n[PROOFSTEP]\napply AddSubmonoid.subset_closure (Set.mem_iUnion.mpr \u27e8i, _\u27e9)\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b2 : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nx\u271d\u00b9 : M\nhx\u271d : x\u271d\u00b9 \u2208 span R (\u22c3 (i : \u03b9), \u2191(p i))\nr : R\nx\u271d : M\nhx : x\u271d \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\nx : M\ni : \u03b9\nhix : x \u2208 p i\n\u22a2 r \u2022 x \u2208 \u2191(p i)\n[PROOFSTEP]\nexact smul_mem _ r hix\n[GOAL]\ncase refine'_4.refine_2\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nx\u271d : M\nhx\u271d : x\u271d \u2208 span R (\u22c3 (i : \u03b9), \u2191(p i))\nr : R\nx : M\nhx : x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n\u22a2 r \u2022 0 \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n[PROOFSTEP]\nrw [smul_zero]\n[GOAL]\ncase refine'_4.refine_2\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nx\u271d : M\nhx\u271d : x\u271d \u2208 span R (\u22c3 (i : \u03b9), \u2191(p i))\nr : R\nx : M\nhx : x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n\u22a2 0 \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n[PROOFSTEP]\nexact AddSubmonoid.zero_mem _\n[GOAL]\ncase refine'_4.refine_3\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nx\u271d : M\nhx\u271d : x\u271d \u2208 span R (\u22c3 (i : \u03b9), \u2191(p i))\nr : R\nx : M\nhx : x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n\u22a2 \u2200 (x y : M),\n r \u2022 x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i)) \u2192\n r \u2022 y \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i)) \u2192 r \u2022 (x + y) \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n[PROOFSTEP]\nintro x y hx hy\n[GOAL]\ncase refine'_4.refine_3\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b2 : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nx\u271d\u00b9 : M\nhx\u271d\u00b9 : x\u271d\u00b9 \u2208 span R (\u22c3 (i : \u03b9), \u2191(p i))\nr : R\nx\u271d : M\nhx\u271d : x\u271d \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\nx y : M\nhx : r \u2022 x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\nhy : r \u2022 y \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n\u22a2 r \u2022 (x + y) \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n[PROOFSTEP]\nrw [smul_add]\n[GOAL]\ncase refine'_4.refine_3\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b2 : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nx\u271d\u00b9 : M\nhx\u271d\u00b9 : x\u271d\u00b9 \u2208 span R (\u22c3 (i : \u03b9), \u2191(p i))\nr : R\nx\u271d : M\nhx\u271d : x\u271d \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\nx y : M\nhx : r \u2022 x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\nhy : r \u2022 y \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n\u22a2 r \u2022 x + r \u2022 y \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), \u2191(p i))\n[PROOFSTEP]\nexact AddSubmonoid.add_mem _ hx hy\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nC : M \u2192 Prop\nx : M\nhx : x \u2208 \u2a06 (i : \u03b9), p i\nhp : \u2200 (i : \u03b9) (x : M), x \u2208 p i \u2192 C x\nh0 : C 0\nhadd : \u2200 (x y : M), C x \u2192 C y \u2192 C (x + y)\n\u22a2 C x\n[PROOFSTEP]\nrw [\u2190 mem_toAddSubmonoid, iSup_toAddSubmonoid] at hx \n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nC : M \u2192 Prop\nx : M\nhx : x \u2208 \u2a06 (i : \u03b9), (p i).toAddSubmonoid\nhp : \u2200 (i : \u03b9) (x : M), x \u2208 p i \u2192 C x\nh0 : C 0\nhadd : \u2200 (x y : M), C x \u2192 C y \u2192 C (x + y)\n\u22a2 C x\n[PROOFSTEP]\nexact AddSubmonoid.iSup_induction (x := x) _ hx hp h0 hadd\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nC : (x : M) \u2192 x \u2208 \u2a06 (i : \u03b9), p i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : M) (hx : x \u2208 p i), C x (_ : x \u2208 \u2a06 (i : \u03b9), p i)\nh0 : C 0 (_ : 0 \u2208 \u2a06 (i : \u03b9), p i)\nhadd :\n \u2200 (x y : M) (hx : x \u2208 \u2a06 (i : \u03b9), p i) (hy : y \u2208 \u2a06 (i : \u03b9), p i),\n C x hx \u2192 C y hy \u2192 C (x + y) (_ : x + y \u2208 \u2a06 (i : \u03b9), p i)\nx : M\nhx : x \u2208 \u2a06 (i : \u03b9), p i\n\u22a2 C x hx\n[PROOFSTEP]\nrefine' Exists.elim _ fun (hx : x \u2208 \u2a06 i, p i) (hc : C x hx) => hc\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nC : (x : M) \u2192 x \u2208 \u2a06 (i : \u03b9), p i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : M) (hx : x \u2208 p i), C x (_ : x \u2208 \u2a06 (i : \u03b9), p i)\nh0 : C 0 (_ : 0 \u2208 \u2a06 (i : \u03b9), p i)\nhadd :\n \u2200 (x y : M) (hx : x \u2208 \u2a06 (i : \u03b9), p i) (hy : y \u2208 \u2a06 (i : \u03b9), p i),\n C x hx \u2192 C y hy \u2192 C (x + y) (_ : x + y \u2208 \u2a06 (i : \u03b9), p i)\nx : M\nhx : x \u2208 \u2a06 (i : \u03b9), p i\n\u22a2 \u2203 x_1, C x x_1\n[PROOFSTEP]\nrefine' iSup_induction p (C := fun x : M \u21a6 \u2203 (hx : x \u2208 \u2a06 i, p i), C x hx) hx (fun i x hx => _) _ fun x y => _\n[GOAL]\ncase refine'_1\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nC : (x : M) \u2192 x \u2208 \u2a06 (i : \u03b9), p i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : M) (hx : x \u2208 p i), C x (_ : x \u2208 \u2a06 (i : \u03b9), p i)\nh0 : C 0 (_ : 0 \u2208 \u2a06 (i : \u03b9), p i)\nhadd :\n \u2200 (x y : M) (hx : x \u2208 \u2a06 (i : \u03b9), p i) (hy : y \u2208 \u2a06 (i : \u03b9), p i),\n C x hx \u2192 C y hy \u2192 C (x + y) (_ : x + y \u2208 \u2a06 (i : \u03b9), p i)\nx\u271d : M\nhx\u271d : x\u271d \u2208 \u2a06 (i : \u03b9), p i\ni : \u03b9\nx : M\nhx : x \u2208 p i\n\u22a2 (fun x => \u2203 hx, C x hx) x\n[PROOFSTEP]\nexact \u27e8_, hp _ _ hx\u27e9\n[GOAL]\ncase refine'_2\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nC : (x : M) \u2192 x \u2208 \u2a06 (i : \u03b9), p i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : M) (hx : x \u2208 p i), C x (_ : x \u2208 \u2a06 (i : \u03b9), p i)\nh0 : C 0 (_ : 0 \u2208 \u2a06 (i : \u03b9), p i)\nhadd :\n \u2200 (x y : M) (hx : x \u2208 \u2a06 (i : \u03b9), p i) (hy : y \u2208 \u2a06 (i : \u03b9), p i),\n C x hx \u2192 C y hy \u2192 C (x + y) (_ : x + y \u2208 \u2a06 (i : \u03b9), p i)\nx : M\nhx : x \u2208 \u2a06 (i : \u03b9), p i\n\u22a2 (fun x => \u2203 hx, C x hx) 0\n[PROOFSTEP]\nexact \u27e8_, h0\u27e9\n[GOAL]\ncase refine'_3\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nC : (x : M) \u2192 x \u2208 \u2a06 (i : \u03b9), p i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : M) (hx : x \u2208 p i), C x (_ : x \u2208 \u2a06 (i : \u03b9), p i)\nh0 : C 0 (_ : 0 \u2208 \u2a06 (i : \u03b9), p i)\nhadd :\n \u2200 (x y : M) (hx : x \u2208 \u2a06 (i : \u03b9), p i) (hy : y \u2208 \u2a06 (i : \u03b9), p i),\n C x hx \u2192 C y hy \u2192 C (x + y) (_ : x + y \u2208 \u2a06 (i : \u03b9), p i)\nx\u271d : M\nhx : x\u271d \u2208 \u2a06 (i : \u03b9), p i\nx y : M\n\u22a2 (fun x => \u2203 hx, C x hx) x \u2192 (fun x => \u2203 hx, C x hx) y \u2192 (fun x => \u2203 hx, C x hx) (x + y)\n[PROOFSTEP]\nrintro \u27e8_, Cx\u27e9 \u27e8_, Cy\u27e9\n[GOAL]\ncase refine'_3.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nC : (x : M) \u2192 x \u2208 \u2a06 (i : \u03b9), p i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : M) (hx : x \u2208 p i), C x (_ : x \u2208 \u2a06 (i : \u03b9), p i)\nh0 : C 0 (_ : 0 \u2208 \u2a06 (i : \u03b9), p i)\nhadd :\n \u2200 (x y : M) (hx : x \u2208 \u2a06 (i : \u03b9), p i) (hy : y \u2208 \u2a06 (i : \u03b9), p i),\n C x hx \u2192 C y hy \u2192 C (x + y) (_ : x + y \u2208 \u2a06 (i : \u03b9), p i)\nx\u271d : M\nhx : x\u271d \u2208 \u2a06 (i : \u03b9), p i\nx y : M\nw\u271d\u00b9 : x \u2208 \u2a06 (i : \u03b9), p i\nCx : C x w\u271d\u00b9\nw\u271d : y \u2208 \u2a06 (i : \u03b9), p i\nCy : C y w\u271d\n\u22a2 \u2203 hx, C (x + y) hx\n[PROOFSTEP]\nrefine' \u27e8_, hadd _ _ _ _ Cx Cy\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nm : M\np : Submodule R M\n\u22a2 span R {m} \u2264 p \u2194 m \u2208 p\n[PROOFSTEP]\nrw [span_le, singleton_subset_iff, SetLike.mem_coe]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\n\u22a2 CompleteLattice.IsCompactElement (span R {x})\n[PROOFSTEP]\nrw [CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\n\u22a2 \u2200 (s : Set (Submodule R M)),\n Set.Nonempty s \u2192 DirectedOn (fun x x_1 => x \u2264 x_1) s \u2192 span R {x} \u2264 sSup s \u2192 \u2203 x_1, x_1 \u2208 s \u2227 span R {x} \u2264 x_1\n[PROOFSTEP]\nintro d hemp hdir hsup\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\nd : Set (Submodule R M)\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) d\nhsup : span R {x} \u2264 sSup d\n\u22a2 \u2203 x_1, x_1 \u2208 d \u2227 span R {x} \u2264 x_1\n[PROOFSTEP]\nhave : x \u2208 (sSup d) := (SetLike.le_def.mp hsup) (mem_span_singleton_self x)\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\nd : Set (Submodule R M)\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) d\nhsup : span R {x} \u2264 sSup d\nthis : x \u2208 sSup d\n\u22a2 \u2203 x_1, x_1 \u2208 d \u2227 span R {x} \u2264 x_1\n[PROOFSTEP]\nobtain \u27e8y, \u27e8hyd, hxy\u27e9\u27e9 := (mem_sSup_of_directed hemp hdir).mp this\n[GOAL]\ncase intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\nd : Set (Submodule R M)\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) d\nhsup : span R {x} \u2264 sSup d\nthis : x \u2208 sSup d\ny : Submodule R M\nhyd : y \u2208 d\nhxy : x \u2208 y\n\u22a2 \u2203 x_1, x_1 \u2208 d \u2227 span R {x} \u2264 x_1\n[PROOFSTEP]\nexact \u27e8y, \u27e8hyd, by simpa only [span_le, singleton_subset_iff]\u27e9\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nx : M\nd : Set (Submodule R M)\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) d\nhsup : span R {x} \u2264 sSup d\nthis : x \u2208 sSup d\ny : Submodule R M\nhyd : y \u2208 d\nhxy : x \u2208 y\n\u22a2 span R {x} \u2264 y\n[PROOFSTEP]\nsimpa only [span_le, singleton_subset_iff]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nS : Finset M\n\u22a2 CompleteLattice.IsCompactElement (span R \u2191S)\n[PROOFSTEP]\nrw [span_eq_iSup_of_singleton_spans]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nS : Finset M\n\u22a2 CompleteLattice.IsCompactElement (\u2a06 (x : M) (_ : x \u2208 \u2191S), span R {x})\n[PROOFSTEP]\nsimp only [Finset.mem_coe]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nS : Finset M\n\u22a2 CompleteLattice.IsCompactElement (\u2a06 (x : M) (_ : x \u2208 S), span R {x})\n[PROOFSTEP]\nrw [\u2190 Finset.sup_eq_iSup]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nS : Finset M\n\u22a2 CompleteLattice.IsCompactElement (Finset.sup S fun x => span R {x})\n[PROOFSTEP]\nexact CompleteLattice.finset_sup_compact_of_compact S fun x _ => singleton_span_isCompactElement x\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t\u271d : Set M\ns t : Submodule R M\nht : t \u2208 (fun x => span R {x}) '' \u2191s\n\u22a2 CompleteLattice.IsCompactElement t\n[PROOFSTEP]\nrcases(Set.mem_image _ _ _).1 ht with \u27e8x, _, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\ns : Submodule R M\nx : M\nleft\u271d : x \u2208 \u2191s\nht : span R {x} \u2208 (fun x => span R {x}) '' \u2191s\n\u22a2 CompleteLattice.IsCompactElement (span R {x})\n[PROOFSTEP]\napply singleton_span_isCompactElement\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns\u271d t : Set M\ns : Submodule R M\n\u22a2 sSup ((fun x => span R {x}) '' \u2191s) = s\n[PROOFSTEP]\nrw [sSup_eq_iSup, iSup_image, \u2190 span_eq_iSup_of_singleton_spans, span_eq]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\np : Submodule R M\n\u22a2 p = sSup {T | \u2203 m x x, T = span R {m}}\n[PROOFSTEP]\nlet S := {T : Submodule R M | \u2203 (m : M) (_ : m \u2208 p) (_ : m \u2260 0), T = span R { m }}\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | \u2203 m x x, T = span R {m}}\n\u22a2 p = sSup {T | \u2203 m x x, T = span R {m}}\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | \u2203 m x x, T = span R {m}}\n\u22a2 p \u2264 sSup {T | \u2203 m x x, T = span R {m}}\n[PROOFSTEP]\nintro m hm\n[GOAL]\ncase a\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | \u2203 m x x, T = span R {m}}\nm : M\nhm : m \u2208 p\n\u22a2 m \u2208 sSup {T | \u2203 m x x, T = span R {m}}\n[PROOFSTEP]\nby_cases h : m = 0\n[GOAL]\ncase pos\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | \u2203 m x x, T = span R {m}}\nm : M\nhm : m \u2208 p\nh : m = 0\n\u22a2 m \u2208 sSup {T | \u2203 m x x, T = span R {m}}\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase pos\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | \u2203 m x x, T = span R {m}}\nm : M\nhm : m \u2208 p\nh : m = 0\n\u22a2 0 \u2208 sSup {T | \u2203 m x x, T = span R {m}}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | \u2203 m x x, T = span R {m}}\nm : M\nhm : m \u2208 p\nh : \u00acm = 0\n\u22a2 m \u2208 sSup {T | \u2203 m x x, T = span R {m}}\n[PROOFSTEP]\nexact @le_sSup _ _ S _ \u27e8m, \u27e8hm, \u27e8h, rfl\u27e9\u27e9\u27e9 m (mem_span_singleton_self m)\n[GOAL]\ncase a\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | \u2203 m x x, T = span R {m}}\n\u22a2 sSup {T | \u2203 m x x, T = span R {m}} \u2264 p\n[PROOFSTEP]\nrw [sSup_le_iff]\n[GOAL]\ncase a\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | \u2203 m x x, T = span R {m}}\n\u22a2 \u2200 (b : Submodule R M), b \u2208 {T | \u2203 m x x, T = span R {m}} \u2192 b \u2264 p\n[PROOFSTEP]\nrintro S \u27e8_, \u27e8_, \u27e8_, rfl\u27e9\u27e9\u27e9\n[GOAL]\ncase a.intro.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\np : Submodule R M\nS : Set (Submodule R M) := {T | \u2203 m x x, T = span R {m}}\nw\u271d\u00b2 : M\nw\u271d\u00b9 : w\u271d\u00b2 \u2208 p\nw\u271d : w\u271d\u00b2 \u2260 0\n\u22a2 span R {w\u271d\u00b2} \u2264 p\n[PROOFSTEP]\nrwa [span_singleton_le_iff_mem]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\n\u22a2 I < I \u2294 span R {a} \u2194 \u00aca \u2208 I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\n\u22a2 I < I \u2294 span R {a} \u2192 \u00aca \u2208 I\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : I < I \u2294 span R {a}\n\u22a2 \u00aca \u2208 I\n[PROOFSTEP]\nby_contra akey\n[GOAL]\ncase mp\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : I < I \u2294 span R {a}\nakey : a \u2208 I\n\u22a2 False\n[PROOFSTEP]\nhave h1 : (I \u2294 R \u2219 a) \u2264 I := by\n simp only [sup_le_iff]\n constructor\n \u00b7 exact le_refl I\n \u00b7 exact (span_singleton_le_iff_mem a I).mpr akey\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : I < I \u2294 span R {a}\nakey : a \u2208 I\n\u22a2 I \u2294 span R {a} \u2264 I\n[PROOFSTEP]\nsimp only [sup_le_iff]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : I < I \u2294 span R {a}\nakey : a \u2208 I\n\u22a2 I \u2264 I \u2227 span R {a} \u2264 I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : I < I \u2294 span R {a}\nakey : a \u2208 I\n\u22a2 I \u2264 I\n[PROOFSTEP]\nexact le_refl I\n[GOAL]\ncase right\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : I < I \u2294 span R {a}\nakey : a \u2208 I\n\u22a2 span R {a} \u2264 I\n[PROOFSTEP]\nexact (span_singleton_le_iff_mem a I).mpr akey\n[GOAL]\ncase mp\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : I < I \u2294 span R {a}\nakey : a \u2208 I\nh1 : I \u2294 span R {a} \u2264 I\n\u22a2 False\n[PROOFSTEP]\nhave h2 := gt_of_ge_of_gt h1 h\n[GOAL]\ncase mp\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : I < I \u2294 span R {a}\nakey : a \u2208 I\nh1 : I \u2294 span R {a} \u2264 I\nh2 : I > I\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl I h2\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\n\u22a2 \u00aca \u2208 I \u2192 I < I \u2294 span R {a}\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : \u00aca \u2208 I\n\u22a2 I < I \u2294 span R {a}\n[PROOFSTEP]\napply SetLike.lt_iff_le_and_exists.mpr\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : \u00aca \u2208 I\n\u22a2 I \u2264 I \u2294 span R {a} \u2227 \u2203 x, x \u2208 I \u2294 span R {a} \u2227 \u00acx \u2208 I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.left\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : \u00aca \u2208 I\n\u22a2 I \u2264 I \u2294 span R {a}\ncase mpr.right\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : \u00aca \u2208 I\n\u22a2 \u2203 x, x \u2208 I \u2294 span R {a} \u2227 \u00acx \u2208 I\n[PROOFSTEP]\nsimp only [le_sup_left]\n[GOAL]\ncase mpr.right\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : \u00aca \u2208 I\n\u22a2 \u2203 x, x \u2208 I \u2294 span R {a} \u2227 \u00acx \u2208 I\n[PROOFSTEP]\nuse a\n[GOAL]\ncase h\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : \u00aca \u2208 I\n\u22a2 a \u2208 I \u2294 span R {a} \u2227 \u00aca \u2208 I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : \u00aca \u2208 I\n\u22a2 a \u2208 I \u2294 span R {a}\ncase h.right\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : \u00aca \u2208 I\n\u22a2 \u00aca \u2208 I\n[PROOFSTEP]\nswap\n[GOAL]\ncase h.right\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : \u00aca \u2208 I\n\u22a2 \u00aca \u2208 I\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h.left\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : \u00aca \u2208 I\n\u22a2 a \u2208 I \u2294 span R {a}\n[PROOFSTEP]\nhave : (R \u2219 a) \u2264 I \u2294 R \u2219 a := le_sup_right\n[GOAL]\ncase h.left\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\nh : \u00aca \u2208 I\nthis : span R {a} \u2264 I \u2294 span R {a}\n\u22a2 a \u2208 I \u2294 span R {a}\n[PROOFSTEP]\nexact this (mem_span_singleton_self a)\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nm : M\n\u22a2 m \u2208 \u2a06 (i : \u03b9), p i \u2194 \u2200 (N : Submodule R M), (\u2200 (i : \u03b9), p i \u2264 N) \u2192 m \u2208 N\n[PROOFSTEP]\nrw [\u2190 span_singleton_le_iff_mem, le_iSup_iff]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx : M\np\u271d p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\n\u03b9 : Sort u_9\np : \u03b9 \u2192 Submodule R M\nm : M\n\u22a2 (\u2200 (b : Submodule R M), (\u2200 (i : \u03b9), p i \u2264 b) \u2192 span R {m} \u2264 b) \u2194 \u2200 (N : Submodule R M), (\u2200 (i : \u03b9), p i \u2264 N) \u2192 m \u2208 N\n[PROOFSTEP]\nsimp only [span_singleton_le_iff_mem]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx : M\nhx : x \u2208 span R S\n\u22a2 \u2203 T, \u2191T \u2286 S \u2227 x \u2208 span R \u2191T\n[PROOFSTEP]\nclassical\nrefine' span_induction hx (fun x hx => _) _ _ _\n\u00b7 refine' \u27e8{ x }, _, _\u27e9\n \u00b7 rwa [Finset.coe_singleton, Set.singleton_subset_iff]\n \u00b7 rw [Finset.coe_singleton]\n exact Submodule.mem_span_singleton_self x\n\u00b7 use\u2205\n simp\n\u00b7 rintro x y \u27e8X, hX, hxX\u27e9 \u27e8Y, hY, hyY\u27e9\n refine' \u27e8X \u222a Y, _, _\u27e9\n \u00b7 rw [Finset.coe_union]\n exact Set.union_subset hX hY\n rw [Finset.coe_union, span_union, mem_sup]\n exact \u27e8x, hxX, y, hyY, rfl\u27e9\n\u00b7 rintro a x \u27e8T, hT, h2\u27e9\n exact \u27e8T, hT, smul_mem _ _ h2\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx : M\nhx : x \u2208 span R S\n\u22a2 \u2203 T, \u2191T \u2286 S \u2227 x \u2208 span R \u2191T\n[PROOFSTEP]\nrefine' span_induction hx (fun x hx => _) _ _ _\n[GOAL]\ncase refine'_1\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx\u271d : M\nhx\u271d : x\u271d \u2208 span R S\nx : M\nhx : x \u2208 S\n\u22a2 \u2203 T, \u2191T \u2286 S \u2227 x \u2208 span R \u2191T\n[PROOFSTEP]\nrefine' \u27e8{ x }, _, _\u27e9\n[GOAL]\ncase refine'_1.refine'_1\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx\u271d : M\nhx\u271d : x\u271d \u2208 span R S\nx : M\nhx : x \u2208 S\n\u22a2 \u2191{x} \u2286 S\n[PROOFSTEP]\nrwa [Finset.coe_singleton, Set.singleton_subset_iff]\n[GOAL]\ncase refine'_1.refine'_2\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx\u271d : M\nhx\u271d : x\u271d \u2208 span R S\nx : M\nhx : x \u2208 S\n\u22a2 x \u2208 span R \u2191{x}\n[PROOFSTEP]\nrw [Finset.coe_singleton]\n[GOAL]\ncase refine'_1.refine'_2\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx\u271d : M\nhx\u271d : x\u271d \u2208 span R S\nx : M\nhx : x \u2208 S\n\u22a2 x \u2208 span R {x}\n[PROOFSTEP]\nexact Submodule.mem_span_singleton_self x\n[GOAL]\ncase refine'_2\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx : M\nhx : x \u2208 span R S\n\u22a2 \u2203 T, \u2191T \u2286 S \u2227 0 \u2208 span R \u2191T\n[PROOFSTEP]\nuse\u2205\n[GOAL]\ncase h\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx : M\nhx : x \u2208 span R S\n\u22a2 \u2191\u2205 \u2286 S \u2227 0 \u2208 span R \u2191\u2205\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx : M\nhx : x \u2208 span R S\n\u22a2 \u2200 (x y : M), (\u2203 T, \u2191T \u2286 S \u2227 x \u2208 span R \u2191T) \u2192 (\u2203 T, \u2191T \u2286 S \u2227 y \u2208 span R \u2191T) \u2192 \u2203 T, \u2191T \u2286 S \u2227 x + y \u2208 span R \u2191T\n[PROOFSTEP]\nrintro x y \u27e8X, hX, hxX\u27e9 \u27e8Y, hY, hyY\u27e9\n[GOAL]\ncase refine'_3.intro.intro.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx\u271d : M\nhx : x\u271d \u2208 span R S\nx y : M\nX : Finset M\nhX : \u2191X \u2286 S\nhxX : x \u2208 span R \u2191X\nY : Finset M\nhY : \u2191Y \u2286 S\nhyY : y \u2208 span R \u2191Y\n\u22a2 \u2203 T, \u2191T \u2286 S \u2227 x + y \u2208 span R \u2191T\n[PROOFSTEP]\nrefine' \u27e8X \u222a Y, _, _\u27e9\n[GOAL]\ncase refine'_3.intro.intro.intro.intro.refine'_1\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx\u271d : M\nhx : x\u271d \u2208 span R S\nx y : M\nX : Finset M\nhX : \u2191X \u2286 S\nhxX : x \u2208 span R \u2191X\nY : Finset M\nhY : \u2191Y \u2286 S\nhyY : y \u2208 span R \u2191Y\n\u22a2 \u2191(X \u222a Y) \u2286 S\n[PROOFSTEP]\nrw [Finset.coe_union]\n[GOAL]\ncase refine'_3.intro.intro.intro.intro.refine'_1\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx\u271d : M\nhx : x\u271d \u2208 span R S\nx y : M\nX : Finset M\nhX : \u2191X \u2286 S\nhxX : x \u2208 span R \u2191X\nY : Finset M\nhY : \u2191Y \u2286 S\nhyY : y \u2208 span R \u2191Y\n\u22a2 \u2191X \u222a \u2191Y \u2286 S\n[PROOFSTEP]\nexact Set.union_subset hX hY\n[GOAL]\ncase refine'_3.intro.intro.intro.intro.refine'_2\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx\u271d : M\nhx : x\u271d \u2208 span R S\nx y : M\nX : Finset M\nhX : \u2191X \u2286 S\nhxX : x \u2208 span R \u2191X\nY : Finset M\nhY : \u2191Y \u2286 S\nhyY : y \u2208 span R \u2191Y\n\u22a2 x + y \u2208 span R \u2191(X \u222a Y)\n[PROOFSTEP]\nrw [Finset.coe_union, span_union, mem_sup]\n[GOAL]\ncase refine'_3.intro.intro.intro.intro.refine'_2\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx\u271d : M\nhx : x\u271d \u2208 span R S\nx y : M\nX : Finset M\nhX : \u2191X \u2286 S\nhxX : x \u2208 span R \u2191X\nY : Finset M\nhY : \u2191Y \u2286 S\nhyY : y \u2208 span R \u2191Y\n\u22a2 \u2203 y_1, y_1 \u2208 span R \u2191X \u2227 \u2203 z, z \u2208 span R \u2191Y \u2227 y_1 + z = x + y\n[PROOFSTEP]\nexact \u27e8x, hxX, y, hyY, rfl\u27e9\n[GOAL]\ncase refine'_4\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx : M\nhx : x \u2208 span R S\n\u22a2 \u2200 (a : R) (x : M), (\u2203 T, \u2191T \u2286 S \u2227 x \u2208 span R \u2191T) \u2192 \u2203 T, \u2191T \u2286 S \u2227 a \u2022 x \u2208 span R \u2191T\n[PROOFSTEP]\nrintro a x \u27e8T, hT, h2\u27e9\n[GOAL]\ncase refine'_4.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS\u271d : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nx\u271d\u00b9 : M\np p' : Submodule R M\ninst\u271d\u00b3 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t S : Set M\nx\u271d : M\nhx : x\u271d \u2208 span R S\na : R\nx : M\nT : Finset M\nhT : \u2191T \u2286 S\nh2 : x \u2208 span R \u2191T\n\u22a2 \u2203 T, \u2191T \u2286 S \u2227 a \u2022 x \u2208 span R \u2191T\n[PROOFSTEP]\nexact \u27e8T, hT, smul_mem _ _ h2\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u2075 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b2 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nM' : Type u_9\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nq\u2081 q\u2081' : Submodule R M'\nsrc\u271d : AddSubmonoid (M \u00d7 M') := AddSubmonoid.prod p.toAddSubmonoid q\u2081.toAddSubmonoid\n\u22a2 \u2200 (c : R) {x : M \u00d7 M'},\n x \u2208\n {\n toAddSubsemigroup :=\n { carrier := \u2191p \u00d7\u02e2 \u2191q\u2081,\n add_mem' := (_ : \u2200 {a b : M \u00d7 M'}, a \u2208 src\u271d.carrier \u2192 b \u2208 src\u271d.carrier \u2192 a + b \u2208 src\u271d.carrier) },\n zero_mem' := (_ : 0 \u2208 src\u271d.carrier) }.toAddSubsemigroup.carrier \u2192\n c \u2022 x \u2208\n {\n toAddSubsemigroup :=\n { carrier := \u2191p \u00d7\u02e2 \u2191q\u2081,\n add_mem' := (_ : \u2200 {a b : M \u00d7 M'}, a \u2208 src\u271d.carrier \u2192 b \u2208 src\u271d.carrier \u2192 a + b \u2208 src\u271d.carrier) },\n zero_mem' := (_ : 0 \u2208 src\u271d.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrintro a \u27e8x, y\u27e9 \u27e8hx, hy\u27e9\n[GOAL]\ncase mk.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2075 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b2 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nM' : Type u_9\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nq\u2081 q\u2081' : Submodule R M'\nsrc\u271d : AddSubmonoid (M \u00d7 M') := AddSubmonoid.prod p.toAddSubmonoid q\u2081.toAddSubmonoid\na : R\nx : M\ny : M'\nhx : (x, y).fst \u2208 \u2191p\nhy : (x, y).snd \u2208 \u2191q\u2081\n\u22a2 a \u2022 (x, y) \u2208\n {\n toAddSubsemigroup :=\n { carrier := \u2191p \u00d7\u02e2 \u2191q\u2081,\n add_mem' := (_ : \u2200 {a b : M \u00d7 M'}, a \u2208 src\u271d.carrier \u2192 b \u2208 src\u271d.carrier \u2192 a + b \u2208 src\u271d.carrier) },\n zero_mem' := (_ : 0 \u2208 src\u271d.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nexact \u27e8smul_mem _ a hx, smul_mem _ a hy\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u2075 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b2 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nM' : Type u_9\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nq\u2081 q\u2081' : Submodule R M'\n\u22a2 prod \u22a4 \u22a4 = \u22a4\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u2075 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b2 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nM' : Type u_9\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nq\u2081 q\u2081' : Submodule R M'\nx\u271d : M \u00d7 M'\n\u22a2 x\u271d \u2208 prod \u22a4 \u22a4 \u2194 x\u271d \u2208 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u2075 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b2 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nM' : Type u_9\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nq\u2081 q\u2081' : Submodule R M'\n\u22a2 prod \u22a5 \u22a5 = \u22a5\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2075 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b2 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nM' : Type u_9\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nq\u2081 q\u2081' : Submodule R M'\nx : M\ny : M'\n\u22a2 (x, y) \u2208 prod \u22a5 \u22a5 \u2194 (x, y) \u2208 \u22a5\n[PROOFSTEP]\nsimp [Prod.zero_eq_mk]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u2075 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b2 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nM' : Type u_9\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nq\u2081 q\u2081' : Submodule R M'\n\u22a2 prod p q\u2081 \u2294 prod p' q\u2081' = prod (p \u2294 p') (q\u2081 \u2294 q\u2081')\n[PROOFSTEP]\nrefine' le_antisymm (sup_le (prod_mono le_sup_left le_sup_left) (prod_mono le_sup_right le_sup_right)) _\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u2075 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b2 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nM' : Type u_9\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nq\u2081 q\u2081' : Submodule R M'\n\u22a2 prod (p \u2294 p') (q\u2081 \u2294 q\u2081') \u2264 prod p q\u2081 \u2294 prod p' q\u2081'\n[PROOFSTEP]\nsimp [SetLike.le_def]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u2075 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b2 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nM' : Type u_9\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nq\u2081 q\u2081' : Submodule R M'\n\u22a2 \u2200 (a : M) (b : M'), a \u2208 p \u2294 p' \u2192 b \u2208 q\u2081 \u2294 q\u2081' \u2192 (a, b) \u2208 prod p q\u2081 \u2294 prod p' q\u2081'\n[PROOFSTEP]\nintro xx yy hxx hyy\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u2075 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b2 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nM' : Type u_9\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nq\u2081 q\u2081' : Submodule R M'\nxx : M\nyy : M'\nhxx : xx \u2208 p \u2294 p'\nhyy : yy \u2208 q\u2081 \u2294 q\u2081'\n\u22a2 (xx, yy) \u2208 prod p q\u2081 \u2294 prod p' q\u2081'\n[PROOFSTEP]\nrcases mem_sup.1 hxx with \u27e8x, hx, x', hx', rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2075 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b2 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nM' : Type u_9\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nq\u2081 q\u2081' : Submodule R M'\nyy : M'\nhyy : yy \u2208 q\u2081 \u2294 q\u2081'\nx : M\nhx : x \u2208 p\nx' : M\nhx' : x' \u2208 p'\nhxx : x + x' \u2208 p \u2294 p'\n\u22a2 (x + x', yy) \u2208 prod p q\u2081 \u2294 prod p' q\u2081'\n[PROOFSTEP]\nrcases mem_sup.1 hyy with \u27e8y, hy, y', hy', rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2075 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b2 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nM' : Type u_9\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nq\u2081 q\u2081' : Submodule R M'\nx : M\nhx : x \u2208 p\nx' : M\nhx' : x' \u2208 p'\nhxx : x + x' \u2208 p \u2294 p'\ny : M'\nhy : y \u2208 q\u2081\ny' : M'\nhy' : y' \u2208 q\u2081'\nhyy : y + y' \u2208 q\u2081 \u2294 q\u2081'\n\u22a2 (x + x', y + y') \u2208 prod p q\u2081 \u2294 prod p' q\u2081'\n[PROOFSTEP]\nrefine' mem_sup.2 \u27e8(x, y), \u27e8hx, hy\u27e9, (x', y'), \u27e8hx', hy'\u27e9, rfl\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ns : Set M\n\u22a2 span R (-s) = span R (\u2191(-LinearMap.id) '' s)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ns : Set M\n\u22a2 map (-LinearMap.id) (span R s) = span R s\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx y : M\ns : Set M\n\u22a2 x \u2208 span R (insert y s) \u2194 \u2203 a, x + a \u2022 y \u2208 span R s\n[PROOFSTEP]\nrw [mem_span_insert]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx y : M\ns : Set M\n\u22a2 (\u2203 a z, z \u2208 span R s \u2227 x = a \u2022 y + z) \u2194 \u2203 a, x + a \u2022 y \u2208 span R s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx y : M\ns : Set M\n\u22a2 (\u2203 a z, z \u2208 span R s \u2227 x = a \u2022 y + z) \u2192 \u2203 a, x + a \u2022 y \u2208 span R s\n[PROOFSTEP]\nrintro \u27e8a, z, hz, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ny : M\ns : Set M\na : R\nz : M\nhz : z \u2208 span R s\n\u22a2 \u2203 a_1, a \u2022 y + z + a_1 \u2022 y \u2208 span R s\n[PROOFSTEP]\nexact \u27e8-a, by simp [hz, add_assoc]\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ny : M\ns : Set M\na : R\nz : M\nhz : z \u2208 span R s\n\u22a2 a \u2022 y + z + -a \u2022 y \u2208 span R s\n[PROOFSTEP]\nsimp [hz, add_assoc]\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx y : M\ns : Set M\n\u22a2 (\u2203 a, x + a \u2022 y \u2208 span R s) \u2192 \u2203 a z, z \u2208 span R s \u2227 x = a \u2022 y + z\n[PROOFSTEP]\nrintro \u27e8a, h\u27e9\n[GOAL]\ncase mpr.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx y : M\ns : Set M\na : R\nh : x + a \u2022 y \u2208 span R s\n\u22a2 \u2203 a z, z \u2208 span R s \u2227 x = a \u2022 y + z\n[PROOFSTEP]\nexact \u27e8-a, _, h, by simp [add_comm, add_left_comm]\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx y : M\ns : Set M\na : R\nh : x + a \u2022 y \u2208 span R s\n\u22a2 x = -a \u2022 y + (x + a \u2022 y)\n[PROOFSTEP]\nsimp [add_comm, add_left_comm]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx\u271d y z : Submodule R M\nxz : x\u271d \u2264 z\na : M\nha : a \u2208 (x\u271d \u2294 y) \u2293 z\n\u22a2 a \u2208 x\u271d \u2294 y \u2293 z\n[PROOFSTEP]\nrw [mem_inf, mem_sup] at ha \n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx\u271d y z : Submodule R M\nxz : x\u271d \u2264 z\na : M\nha : (\u2203 y_1, y_1 \u2208 x\u271d \u2227 \u2203 z, z \u2208 y \u2227 y_1 + z = a) \u2227 a \u2208 z\n\u22a2 a \u2208 x\u271d \u2294 y \u2293 z\n[PROOFSTEP]\nrcases ha with \u27e8\u27e8b, hb, c, hc, rfl\u27e9, haz\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx\u271d y z : Submodule R M\nxz : x\u271d \u2264 z\nb : M\nhb : b \u2208 x\u271d\nc : M\nhc : c \u2208 y\nhaz : b + c \u2208 z\n\u22a2 b + c \u2208 x\u271d \u2294 y \u2293 z\n[PROOFSTEP]\nrw [mem_sup]\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx\u271d y z : Submodule R M\nxz : x\u271d \u2264 z\nb : M\nhb : b \u2208 x\u271d\nc : M\nhc : c \u2208 y\nhaz : b + c \u2208 z\n\u22a2 \u2203 y_1, y_1 \u2208 x\u271d \u2227 \u2203 z_1, z_1 \u2208 y \u2293 z \u2227 y_1 + z_1 = b + c\n[PROOFSTEP]\nrefine' \u27e8b, hb, c, mem_inf.2 \u27e8hc, _\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx\u271d y z : Submodule R M\nxz : x\u271d \u2264 z\nb : M\nhb : b \u2208 x\u271d\nc : M\nhc : c \u2208 y\nhaz : b + c \u2208 z\n\u22a2 c \u2208 z\n[PROOFSTEP]\nrw [\u2190 add_sub_cancel c b, add_comm]\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx\u271d y z : Submodule R M\nxz : x\u271d \u2264 z\nb : M\nhb : b \u2208 x\u271d\nc : M\nhc : c \u2208 y\nhaz : b + c \u2208 z\n\u22a2 b + c - b \u2208 z\n[PROOFSTEP]\napply z.sub_mem haz (xz hb)\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring R\u2082\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nF : Type u_8\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\np : Submodule R M\n\u22a2 comap f (map f p) = p \u2294 LinearMap.ker f\n[PROOFSTEP]\nrefine' le_antisymm _ (sup_le (le_comap_map _ _) (comap_mono bot_le))\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring R\u2082\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nF : Type u_8\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\np : Submodule R M\n\u22a2 comap f (map f p) \u2264 p \u2294 LinearMap.ker f\n[PROOFSTEP]\nrintro x \u27e8y, hy, e\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring R\u2082\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nF : Type u_8\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\np : Submodule R M\nx y : M\nhy : y \u2208 \u2191p\ne : \u2191f y = \u2191f x\n\u22a2 x \u2208 p \u2294 LinearMap.ker f\n[PROOFSTEP]\nexact mem_sup.2 \u27e8y, hy, x - y, by simpa using sub_eq_zero.2 e.symm, by simp\u27e9\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring R\u2082\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nF : Type u_8\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\np : Submodule R M\nx y : M\nhy : y \u2208 \u2191p\ne : \u2191f y = \u2191f x\n\u22a2 x - y \u2208 LinearMap.ker f\n[PROOFSTEP]\nsimpa using sub_eq_zero.2 e.symm\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring R\u2082\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nF : Type u_8\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\np : Submodule R M\nx y : M\nhy : y \u2208 \u2191p\ne : \u2191f y = \u2191f x\n\u22a2 y + (x - y) = x\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring R\u2082\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nF : Type u_8\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\np : Submodule R M\nh : LinearMap.ker f \u2264 p\n\u22a2 comap f (map f p) = p\n[PROOFSTEP]\nrw [Submodule.comap_map_eq, sup_of_le_left h]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring R\u2082\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nF : Type u_8\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\np p' : Submodule R M\n\u22a2 map f p \u2264 map f p' \u2194 p \u2264 p' \u2294 ker f\n[PROOFSTEP]\nrw [map_le_iff_le_comap, Submodule.comap_map_eq]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring R\u2082\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nF : Type u_8\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\nhf : ker f = \u22a5\np p' : Submodule R M\n\u22a2 map f p \u2264 map f p' \u2194 p \u2264 p'\n[PROOFSTEP]\nrw [LinearMap.map_le_map_iff, hf, sup_bot_eq]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring R\u2082\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nF : Type u_8\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\nhf : range f = \u22a4\np : Submodule R M\n\u22a2 map f p = \u22a4 \u2194 p \u2294 ker f = \u22a4\n[PROOFSTEP]\nsimp_rw [\u2190 top_le_iff, \u2190 hf, range_eq_map, LinearMap.map_le_map_iff]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nx y : M\n\u22a2 y \u2208 span R {x} \u2194 y \u2208 range (toSpanSingleton R M x)\n[PROOFSTEP]\nrefine' Iff.trans _ LinearMap.mem_range.symm\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nx y : M\n\u22a2 y \u2208 span R {x} \u2194 \u2203 y_1, \u2191(toSpanSingleton R M x) y_1 = y\n[PROOFSTEP]\nexact mem_span_singleton\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u22a2 toSpanSingleton R M 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u22a2 \u2191(toSpanSingleton R M 0) 1 = \u21910 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Semiring R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns : Set M\nf g : F\n\u22a2 Set.EqOn \u2191f \u2191g \u2191(span R s) \u2194 Set.EqOn (\u2191f) (\u2191g) s\n[PROOFSTEP]\nrw [\u2190 le_eqLocus, span_le]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Semiring R\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\nF : Type u_8\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns : Set M\nf g : F\n\u22a2 s \u2286 \u2191(eqLocus f g) \u2194 Set.EqOn (\u2191f) (\u2191g) s\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : V \u2192\u2097[K] K\nx : V\nhx : \u2191f x \u2260 0\ny : V\nx\u271d : y \u2208 \u22a4\n\u22a2 y - (\u2191f y * (\u2191f x)\u207b\u00b9) \u2022 x \u2208 ker f\n[PROOFSTEP]\nrw [LinearMap.mem_ker, f.map_sub, f.map_smul, smul_eq_mul, mul_assoc, inv_mul_cancel hx, mul_one, sub_self]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : V \u2192\u2097[K] K\nx : V\nhx : \u2191f x \u2260 0\ny : V\nx\u271d : y \u2208 \u22a4\n\u22a2 (\u2191f y * (\u2191f x)\u207b\u00b9) \u2022 x + (y - (\u2191f y * (\u2191f x)\u207b\u00b9) \u2022 x) = y\n[PROOFSTEP]\nsimp only [add_sub_cancel'_right]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx : M\nh : x \u2260 0\n\u22a2 \u2191(toSpanNonzeroSingleton R M x h) 1 = { val := x, property := (_ : x \u2208 Submodule.span R {x}) }\n[PROOFSTEP]\napply SetLike.coe_eq_coe.mp\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx : M\nh : x \u2260 0\n\u22a2 \u2191(\u2191(toSpanNonzeroSingleton R M x h) 1) = \u2191{ val := x, property := (_ : x \u2208 Submodule.span R {x}) }\n[PROOFSTEP]\nhave : \u2191(toSpanNonzeroSingleton R M x h 1) = toSpanSingleton R M x 1 := rfl\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx : M\nh : x \u2260 0\nthis : \u2191(\u2191(toSpanNonzeroSingleton R M x h) 1) = \u2191(toSpanSingleton R M x) 1\n\u22a2 \u2191(\u2191(toSpanNonzeroSingleton R M x h) 1) = \u2191{ val := x, property := (_ : x \u2208 Submodule.span R {x}) }\n[PROOFSTEP]\nrw [this, toSpanSingleton_one, Submodule.coe_mk]\n[GOAL]\nR : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx : M\nh : x \u2260 0\n\u22a2 \u2191(coord R M x h) { val := x, property := (_ : x \u2208 Submodule.span R {x}) } = 1\n[PROOFSTEP]\nrw [\u2190 toSpanNonzeroSingleton_one R M x h, LinearEquiv.symm_apply_apply]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Span", "llama_tokens": 85273, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7634837635542925, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4496072898748403}} {"text": "[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2081 : \u2203 k, k \u2260 1 \u2227 x + y = k \u2022 (x - y)\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\n\u22a2 \u2016x - y\u2016 * \u2016x + y\u2016 = |\u2016z + y\u2016 ^ 2 - \u2016z - x\u2016 ^ 2|\n[PROOFSTEP]\nobtain \u27e8k, hk_ne_one, hk\u27e9 := h\u2081\n[GOAL]\ncase intro.intro\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\n\u22a2 \u2016x - y\u2016 * \u2016x + y\u2016 = |\u2016z + y\u2016 ^ 2 - \u2016z - x\u2016 ^ 2|\n[PROOFSTEP]\nlet r := (k - 1)\u207b\u00b9 * (k + 1)\n[GOAL]\ncase intro.intro\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\n\u22a2 \u2016x - y\u2016 * \u2016x + y\u2016 = |\u2016z + y\u2016 ^ 2 - \u2016z - x\u2016 ^ 2|\n[PROOFSTEP]\nhave hxy : x = r \u2022 y :=\n by\n rw [\u2190 smul_smul, eq_inv_smul_iff\u2080 (sub_ne_zero.mpr hk_ne_one), \u2190 sub_eq_zero]\n calc\n (k - 1) \u2022 x - (k + 1) \u2022 y = k \u2022 x - x - (k \u2022 y + y) := by simp_rw [sub_smul, add_smul, one_smul]\n _ = k \u2022 x - k \u2022 y - (x + y) := by simp_rw [\u2190 sub_sub, sub_right_comm]\n _ = k \u2022 (x - y) - (x + y) := by rw [\u2190 smul_sub k x y]\n _ = 0 := sub_eq_zero.mpr hk.symm\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\n\u22a2 x = r \u2022 y\n[PROOFSTEP]\nrw [\u2190 smul_smul, eq_inv_smul_iff\u2080 (sub_ne_zero.mpr hk_ne_one), \u2190 sub_eq_zero]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\n\u22a2 (k - 1) \u2022 x - (k + 1) \u2022 y = 0\n[PROOFSTEP]\ncalc\n (k - 1) \u2022 x - (k + 1) \u2022 y = k \u2022 x - x - (k \u2022 y + y) := by simp_rw [sub_smul, add_smul, one_smul]\n _ = k \u2022 x - k \u2022 y - (x + y) := by simp_rw [\u2190 sub_sub, sub_right_comm]\n _ = k \u2022 (x - y) - (x + y) := by rw [\u2190 smul_sub k x y]\n _ = 0 := sub_eq_zero.mpr hk.symm\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\n\u22a2 (k - 1) \u2022 x - (k + 1) \u2022 y = k \u2022 x - x - (k \u2022 y + y)\n[PROOFSTEP]\nsimp_rw [sub_smul, add_smul, one_smul]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\n\u22a2 k \u2022 x - x - (k \u2022 y + y) = k \u2022 x - k \u2022 y - (x + y)\n[PROOFSTEP]\nsimp_rw [\u2190 sub_sub, sub_right_comm]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\n\u22a2 k \u2022 x - k \u2022 y - (x + y) = k \u2022 (x - y) - (x + y)\n[PROOFSTEP]\nrw [\u2190 smul_sub k x y]\n[GOAL]\ncase intro.intro\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\nhxy : x = r \u2022 y\n\u22a2 \u2016x - y\u2016 * \u2016x + y\u2016 = |\u2016z + y\u2016 ^ 2 - \u2016z - x\u2016 ^ 2|\n[PROOFSTEP]\nhave hzy : \u27eaz, y\u27eb = 0 := by\n rwa [inner_eq_zero_iff_angle_eq_pi_div_two, \u2190 norm_add_eq_norm_sub_iff_angle_eq_pi_div_two, eq_comm]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\nhxy : x = r \u2022 y\n\u22a2 inner z y = 0\n[PROOFSTEP]\nrwa [inner_eq_zero_iff_angle_eq_pi_div_two, \u2190 norm_add_eq_norm_sub_iff_angle_eq_pi_div_two, eq_comm]\n[GOAL]\ncase intro.intro\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\nhxy : x = r \u2022 y\nhzy : inner z y = 0\n\u22a2 \u2016x - y\u2016 * \u2016x + y\u2016 = |\u2016z + y\u2016 ^ 2 - \u2016z - x\u2016 ^ 2|\n[PROOFSTEP]\nhave hzx : \u27eaz, x\u27eb = 0 := by rw [hxy, inner_smul_right, hzy, mul_zero]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\nhxy : x = r \u2022 y\nhzy : inner z y = 0\n\u22a2 inner z x = 0\n[PROOFSTEP]\nrw [hxy, inner_smul_right, hzy, mul_zero]\n[GOAL]\ncase intro.intro\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\nhxy : x = r \u2022 y\nhzy : inner z y = 0\nhzx : inner z x = 0\n\u22a2 \u2016x - y\u2016 * \u2016x + y\u2016 = |\u2016z + y\u2016 ^ 2 - \u2016z - x\u2016 ^ 2|\n[PROOFSTEP]\ncalc\n \u2016x - y\u2016 * \u2016x + y\u2016 = \u2016(r - 1) \u2022 y\u2016 * \u2016(r + 1) \u2022 y\u2016 := by simp [sub_smul, add_smul, hxy]\n _ = \u2016r - 1\u2016 * \u2016y\u2016 * (\u2016r + 1\u2016 * \u2016y\u2016) := by simp_rw [norm_smul]\n _ = \u2016r - 1\u2016 * \u2016r + 1\u2016 * \u2016y\u2016 ^ 2 := by ring\n _ = |(r - 1) * (r + 1) * \u2016y\u2016 ^ 2| := by simp [abs_mul]\n _ = |r ^ 2 * \u2016y\u2016 ^ 2 - \u2016y\u2016 ^ 2| := by ring_nf\n _ = |\u2016x\u2016 ^ 2 - \u2016y\u2016 ^ 2| := by simp [hxy, norm_smul, mul_pow, sq_abs]\n _ = |\u2016z + y\u2016 ^ 2 - \u2016z - x\u2016 ^ 2| := by simp [norm_add_sq_real, norm_sub_sq_real, hzy, hzx, abs_sub_comm]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\nhxy : x = r \u2022 y\nhzy : inner z y = 0\nhzx : inner z x = 0\n\u22a2 \u2016x - y\u2016 * \u2016x + y\u2016 = \u2016(r - 1) \u2022 y\u2016 * \u2016(r + 1) \u2022 y\u2016\n[PROOFSTEP]\nsimp [sub_smul, add_smul, hxy]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\nhxy : x = r \u2022 y\nhzy : inner z y = 0\nhzx : inner z x = 0\n\u22a2 \u2016(r - 1) \u2022 y\u2016 * \u2016(r + 1) \u2022 y\u2016 = \u2016r - 1\u2016 * \u2016y\u2016 * (\u2016r + 1\u2016 * \u2016y\u2016)\n[PROOFSTEP]\nsimp_rw [norm_smul]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\nhxy : x = r \u2022 y\nhzy : inner z y = 0\nhzx : inner z x = 0\n\u22a2 \u2016r - 1\u2016 * \u2016y\u2016 * (\u2016r + 1\u2016 * \u2016y\u2016) = \u2016r - 1\u2016 * \u2016r + 1\u2016 * \u2016y\u2016 ^ 2\n[PROOFSTEP]\nring\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\nhxy : x = r \u2022 y\nhzy : inner z y = 0\nhzx : inner z x = 0\n\u22a2 \u2016r - 1\u2016 * \u2016r + 1\u2016 * \u2016y\u2016 ^ 2 = |(r - 1) * (r + 1) * \u2016y\u2016 ^ 2|\n[PROOFSTEP]\nsimp [abs_mul]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\nhxy : x = r \u2022 y\nhzy : inner z y = 0\nhzx : inner z x = 0\n\u22a2 |(r - 1) * (r + 1) * \u2016y\u2016 ^ 2| = |r ^ 2 * \u2016y\u2016 ^ 2 - \u2016y\u2016 ^ 2|\n[PROOFSTEP]\nring_nf\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\nhxy : x = r \u2022 y\nhzy : inner z y = 0\nhzx : inner z x = 0\n\u22a2 |r ^ 2 * \u2016y\u2016 ^ 2 - \u2016y\u2016 ^ 2| = |\u2016x\u2016 ^ 2 - \u2016y\u2016 ^ 2|\n[PROOFSTEP]\nsimp [hxy, norm_smul, mul_pow, sq_abs]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y z : V\nh\u2082 : \u2016z - y\u2016 = \u2016z + y\u2016\nk : \u211d\nhk_ne_one : k \u2260 1\nhk : x + y = k \u2022 (x - y)\nr : \u211d := (k - 1)\u207b\u00b9 * (k + 1)\nhxy : x = r \u2022 y\nhzy : inner z y = 0\nhzx : inner z x = 0\n\u22a2 |\u2016x\u2016 ^ 2 - \u2016y\u2016 ^ 2| = |\u2016z + y\u2016 ^ 2 - \u2016z - x\u2016 ^ 2|\n[PROOFSTEP]\nsimp [norm_add_sq_real, norm_sub_sq_real, hzy, hzx, abs_sub_comm]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b p q : P\nhp : \u2203 k, k \u2260 1 \u2227 b -\u1d65 p = k \u2022 (a -\u1d65 p)\nhq : dist a q = dist b q\n\u22a2 dist a p * dist b p = |dist b q ^ 2 - dist p q ^ 2|\n[PROOFSTEP]\nlet m : P := midpoint \u211d a b\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b p q : P\nhp : \u2203 k, k \u2260 1 \u2227 b -\u1d65 p = k \u2022 (a -\u1d65 p)\nhq : dist a q = dist b q\nm : P := midpoint \u211d a b\n\u22a2 dist a p * dist b p = |dist b q ^ 2 - dist p q ^ 2|\n[PROOFSTEP]\nhave h1 := vsub_sub_vsub_cancel_left a p m\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b p q : P\nhp : \u2203 k, k \u2260 1 \u2227 b -\u1d65 p = k \u2022 (a -\u1d65 p)\nhq : dist a q = dist b q\nm : P := midpoint \u211d a b\nh1 : m -\u1d65 p - (m -\u1d65 a) = a -\u1d65 p\n\u22a2 dist a p * dist b p = |dist b q ^ 2 - dist p q ^ 2|\n[PROOFSTEP]\nhave h2 := vsub_sub_vsub_cancel_left p q m\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b p q : P\nhp : \u2203 k, k \u2260 1 \u2227 b -\u1d65 p = k \u2022 (a -\u1d65 p)\nhq : dist a q = dist b q\nm : P := midpoint \u211d a b\nh1 : m -\u1d65 p - (m -\u1d65 a) = a -\u1d65 p\nh2 : m -\u1d65 q - (m -\u1d65 p) = p -\u1d65 q\n\u22a2 dist a p * dist b p = |dist b q ^ 2 - dist p q ^ 2|\n[PROOFSTEP]\nhave h3 := vsub_sub_vsub_cancel_left a q m\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b p q : P\nhp : \u2203 k, k \u2260 1 \u2227 b -\u1d65 p = k \u2022 (a -\u1d65 p)\nhq : dist a q = dist b q\nm : P := midpoint \u211d a b\nh1 : m -\u1d65 p - (m -\u1d65 a) = a -\u1d65 p\nh2 : m -\u1d65 q - (m -\u1d65 p) = p -\u1d65 q\nh3 : m -\u1d65 q - (m -\u1d65 a) = a -\u1d65 q\n\u22a2 dist a p * dist b p = |dist b q ^ 2 - dist p q ^ 2|\n[PROOFSTEP]\nhave h : \u2200 r, b -\u1d65 r = m -\u1d65 r + (m -\u1d65 a) := fun r => by\n rw [midpoint_vsub_left, \u2190 right_vsub_midpoint, add_comm, vsub_add_vsub_cancel]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b p q : P\nhp : \u2203 k, k \u2260 1 \u2227 b -\u1d65 p = k \u2022 (a -\u1d65 p)\nhq : dist a q = dist b q\nm : P := midpoint \u211d a b\nh1 : m -\u1d65 p - (m -\u1d65 a) = a -\u1d65 p\nh2 : m -\u1d65 q - (m -\u1d65 p) = p -\u1d65 q\nh3 : m -\u1d65 q - (m -\u1d65 a) = a -\u1d65 q\nr : P\n\u22a2 b -\u1d65 r = m -\u1d65 r + (m -\u1d65 a)\n[PROOFSTEP]\nrw [midpoint_vsub_left, \u2190 right_vsub_midpoint, add_comm, vsub_add_vsub_cancel]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b p q : P\nhp : \u2203 k, k \u2260 1 \u2227 b -\u1d65 p = k \u2022 (a -\u1d65 p)\nhq : dist a q = dist b q\nm : P := midpoint \u211d a b\nh1 : m -\u1d65 p - (m -\u1d65 a) = a -\u1d65 p\nh2 : m -\u1d65 q - (m -\u1d65 p) = p -\u1d65 q\nh3 : m -\u1d65 q - (m -\u1d65 a) = a -\u1d65 q\nh : \u2200 (r : P), b -\u1d65 r = m -\u1d65 r + (m -\u1d65 a)\n\u22a2 dist a p * dist b p = |dist b q ^ 2 - dist p q ^ 2|\n[PROOFSTEP]\niterate 4 rw [dist_eq_norm_vsub V]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b p q : P\nhp : \u2203 k, k \u2260 1 \u2227 b -\u1d65 p = k \u2022 (a -\u1d65 p)\nhq : dist a q = dist b q\nm : P := midpoint \u211d a b\nh1 : m -\u1d65 p - (m -\u1d65 a) = a -\u1d65 p\nh2 : m -\u1d65 q - (m -\u1d65 p) = p -\u1d65 q\nh3 : m -\u1d65 q - (m -\u1d65 a) = a -\u1d65 q\nh : \u2200 (r : P), b -\u1d65 r = m -\u1d65 r + (m -\u1d65 a)\n\u22a2 dist a p * dist b p = |dist b q ^ 2 - dist p q ^ 2|\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b p q : P\nhp : \u2203 k, k \u2260 1 \u2227 b -\u1d65 p = k \u2022 (a -\u1d65 p)\nhq : dist a q = dist b q\nm : P := midpoint \u211d a b\nh1 : m -\u1d65 p - (m -\u1d65 a) = a -\u1d65 p\nh2 : m -\u1d65 q - (m -\u1d65 p) = p -\u1d65 q\nh3 : m -\u1d65 q - (m -\u1d65 a) = a -\u1d65 q\nh : \u2200 (r : P), b -\u1d65 r = m -\u1d65 r + (m -\u1d65 a)\n\u22a2 \u2016a -\u1d65 p\u2016 * dist b p = |dist b q ^ 2 - dist p q ^ 2|\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b p q : P\nhp : \u2203 k, k \u2260 1 \u2227 b -\u1d65 p = k \u2022 (a -\u1d65 p)\nhq : dist a q = dist b q\nm : P := midpoint \u211d a b\nh1 : m -\u1d65 p - (m -\u1d65 a) = a -\u1d65 p\nh2 : m -\u1d65 q - (m -\u1d65 p) = p -\u1d65 q\nh3 : m -\u1d65 q - (m -\u1d65 a) = a -\u1d65 q\nh : \u2200 (r : P), b -\u1d65 r = m -\u1d65 r + (m -\u1d65 a)\n\u22a2 \u2016a -\u1d65 p\u2016 * \u2016b -\u1d65 p\u2016 = |dist b q ^ 2 - dist p q ^ 2|\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b p q : P\nhp : \u2203 k, k \u2260 1 \u2227 b -\u1d65 p = k \u2022 (a -\u1d65 p)\nhq : dist a q = dist b q\nm : P := midpoint \u211d a b\nh1 : m -\u1d65 p - (m -\u1d65 a) = a -\u1d65 p\nh2 : m -\u1d65 q - (m -\u1d65 p) = p -\u1d65 q\nh3 : m -\u1d65 q - (m -\u1d65 a) = a -\u1d65 q\nh : \u2200 (r : P), b -\u1d65 r = m -\u1d65 r + (m -\u1d65 a)\n\u22a2 \u2016a -\u1d65 p\u2016 * \u2016b -\u1d65 p\u2016 = |\u2016b -\u1d65 q\u2016 ^ 2 - dist p q ^ 2|\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b p q : P\nhp : \u2203 k, k \u2260 1 \u2227 b -\u1d65 p = k \u2022 (a -\u1d65 p)\nhq : dist a q = dist b q\nm : P := midpoint \u211d a b\nh1 : m -\u1d65 p - (m -\u1d65 a) = a -\u1d65 p\nh2 : m -\u1d65 q - (m -\u1d65 p) = p -\u1d65 q\nh3 : m -\u1d65 q - (m -\u1d65 a) = a -\u1d65 q\nh : \u2200 (r : P), b -\u1d65 r = m -\u1d65 r + (m -\u1d65 a)\n\u22a2 \u2016a -\u1d65 p\u2016 * \u2016b -\u1d65 p\u2016 = |\u2016b -\u1d65 q\u2016 ^ 2 - \u2016p -\u1d65 q\u2016 ^ 2|\n[PROOFSTEP]\nrw [\u2190 h1, \u2190 h2, h, h]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b p q : P\nhp : \u2203 k, k \u2260 1 \u2227 b -\u1d65 p = k \u2022 (a -\u1d65 p)\nhq : dist a q = dist b q\nm : P := midpoint \u211d a b\nh1 : m -\u1d65 p - (m -\u1d65 a) = a -\u1d65 p\nh2 : m -\u1d65 q - (m -\u1d65 p) = p -\u1d65 q\nh3 : m -\u1d65 q - (m -\u1d65 a) = a -\u1d65 q\nh : \u2200 (r : P), b -\u1d65 r = m -\u1d65 r + (m -\u1d65 a)\n\u22a2 \u2016m -\u1d65 p - (m -\u1d65 a)\u2016 * \u2016m -\u1d65 p + (m -\u1d65 a)\u2016 = |\u2016m -\u1d65 q + (m -\u1d65 a)\u2016 ^ 2 - \u2016m -\u1d65 q - (m -\u1d65 p)\u2016 ^ 2|\n[PROOFSTEP]\nrw [\u2190 h1, h] at hp \n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b p q : P\nhq : dist a q = dist b q\nm : P := midpoint \u211d a b\nhp : \u2203 k, k \u2260 1 \u2227 m -\u1d65 p + (m -\u1d65 a) = k \u2022 (m -\u1d65 p - (m -\u1d65 a))\nh1 : m -\u1d65 p - (m -\u1d65 a) = a -\u1d65 p\nh2 : m -\u1d65 q - (m -\u1d65 p) = p -\u1d65 q\nh3 : m -\u1d65 q - (m -\u1d65 a) = a -\u1d65 q\nh : \u2200 (r : P), b -\u1d65 r = m -\u1d65 r + (m -\u1d65 a)\n\u22a2 \u2016m -\u1d65 p - (m -\u1d65 a)\u2016 * \u2016m -\u1d65 p + (m -\u1d65 a)\u2016 = |\u2016m -\u1d65 q + (m -\u1d65 a)\u2016 ^ 2 - \u2016m -\u1d65 q - (m -\u1d65 p)\u2016 ^ 2|\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V a q, dist_eq_norm_vsub V b q, \u2190 h3, h] at hq \n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b p q : P\nm : P := midpoint \u211d a b\nhq : \u2016m -\u1d65 q - (m -\u1d65 a)\u2016 = \u2016m -\u1d65 q + (m -\u1d65 a)\u2016\nhp : \u2203 k, k \u2260 1 \u2227 m -\u1d65 p + (m -\u1d65 a) = k \u2022 (m -\u1d65 p - (m -\u1d65 a))\nh1 : m -\u1d65 p - (m -\u1d65 a) = a -\u1d65 p\nh2 : m -\u1d65 q - (m -\u1d65 p) = p -\u1d65 q\nh3 : m -\u1d65 q - (m -\u1d65 a) = a -\u1d65 q\nh : \u2200 (r : P), b -\u1d65 r = m -\u1d65 r + (m -\u1d65 a)\n\u22a2 \u2016m -\u1d65 p - (m -\u1d65 a)\u2016 * \u2016m -\u1d65 p + (m -\u1d65 a)\u2016 = |\u2016m -\u1d65 q + (m -\u1d65 a)\u2016 ^ 2 - \u2016m -\u1d65 q - (m -\u1d65 p)\u2016 ^ 2|\n[PROOFSTEP]\nexact mul_norm_eq_abs_sub_sq_norm hp hq\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2203 k\u2081, k\u2081 \u2260 1 \u2227 b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nhcpd : \u2203 k\u2082, k\u2082 \u2260 1 \u2227 d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\n\u22a2 dist a p * dist b p = dist c p * dist d p\n[PROOFSTEP]\nobtain \u27e8q, r, h'\u27e9 := (cospherical_def { a, b, c, d }).mp h\n[GOAL]\ncase intro.intro\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2203 k\u2081, k\u2081 \u2260 1 \u2227 b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nhcpd : \u2203 k\u2082, k\u2082 \u2260 1 \u2227 d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\nq : P\nr : \u211d\nh' : \u2200 (p : P), p \u2208 {a, b, c, d} \u2192 dist p q = r\n\u22a2 dist a p * dist b p = dist c p * dist d p\n[PROOFSTEP]\nobtain \u27e8ha, hb, hc, hd\u27e9 := h' a (by simp), h' b (by simp), h' c (by simp), h' d (by simp)\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2203 k\u2081, k\u2081 \u2260 1 \u2227 b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nhcpd : \u2203 k\u2082, k\u2082 \u2260 1 \u2227 d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\nq : P\nr : \u211d\nh' : \u2200 (p : P), p \u2208 {a, b, c, d} \u2192 dist p q = r\n\u22a2 a \u2208 {a, b, c, d}\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2203 k\u2081, k\u2081 \u2260 1 \u2227 b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nhcpd : \u2203 k\u2082, k\u2082 \u2260 1 \u2227 d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\nq : P\nr : \u211d\nh' : \u2200 (p : P), p \u2208 {a, b, c, d} \u2192 dist p q = r\n\u22a2 b \u2208 {a, b, c, d}\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2203 k\u2081, k\u2081 \u2260 1 \u2227 b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nhcpd : \u2203 k\u2082, k\u2082 \u2260 1 \u2227 d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\nq : P\nr : \u211d\nh' : \u2200 (p : P), p \u2208 {a, b, c, d} \u2192 dist p q = r\n\u22a2 c \u2208 {a, b, c, d}\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2203 k\u2081, k\u2081 \u2260 1 \u2227 b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nhcpd : \u2203 k\u2082, k\u2082 \u2260 1 \u2227 d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\nq : P\nr : \u211d\nh' : \u2200 (p : P), p \u2208 {a, b, c, d} \u2192 dist p q = r\n\u22a2 d \u2208 {a, b, c, d}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2203 k\u2081, k\u2081 \u2260 1 \u2227 b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nhcpd : \u2203 k\u2082, k\u2082 \u2260 1 \u2227 d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\nq : P\nr : \u211d\nh' : \u2200 (p : P), p \u2208 {a, b, c, d} \u2192 dist p q = r\nha : dist a q = r\nhb : dist b q = r\nhc : dist c q = r\nhd : dist d q = r\n\u22a2 dist a p * dist b p = dist c p * dist d p\n[PROOFSTEP]\nrw [\u2190 hd] at hc \n[GOAL]\ncase intro.intro\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2203 k\u2081, k\u2081 \u2260 1 \u2227 b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nhcpd : \u2203 k\u2082, k\u2082 \u2260 1 \u2227 d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\nq : P\nr : \u211d\nh' : \u2200 (p : P), p \u2208 {a, b, c, d} \u2192 dist p q = r\nha : dist a q = r\nhb : dist b q = r\nhc : dist c q = dist d q\nhd : dist d q = r\n\u22a2 dist a p * dist b p = dist c p * dist d p\n[PROOFSTEP]\nrw [\u2190 hb] at ha \n[GOAL]\ncase intro.intro\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2203 k\u2081, k\u2081 \u2260 1 \u2227 b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nhcpd : \u2203 k\u2082, k\u2082 \u2260 1 \u2227 d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\nq : P\nr : \u211d\nh' : \u2200 (p : P), p \u2208 {a, b, c, d} \u2192 dist p q = r\nha : dist a q = dist b q\nhb : dist b q = r\nhc : dist c q = dist d q\nhd : dist d q = r\n\u22a2 dist a p * dist b p = dist c p * dist d p\n[PROOFSTEP]\nrw [mul_dist_eq_abs_sub_sq_dist hapb ha, hb, mul_dist_eq_abs_sub_sq_dist hcpd hc, hd]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2220 a p b = \u03c0\nhcpd : \u2220 c p d = \u03c0\n\u22a2 dist a p * dist b p = dist c p * dist d p\n[PROOFSTEP]\nobtain \u27e8-, k\u2081, _, hab\u27e9 := angle_eq_pi_iff.mp hapb\n[GOAL]\ncase intro.intro.intro\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2220 a p b = \u03c0\nhcpd : \u2220 c p d = \u03c0\nk\u2081 : \u211d\nleft\u271d : k\u2081 < 0\nhab : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\n\u22a2 dist a p * dist b p = dist c p * dist d p\n[PROOFSTEP]\nobtain \u27e8-, k\u2082, _, hcd\u27e9 := angle_eq_pi_iff.mp hcpd\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2220 a p b = \u03c0\nhcpd : \u2220 c p d = \u03c0\nk\u2081 : \u211d\nleft\u271d\u00b9 : k\u2081 < 0\nhab : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nk\u2082 : \u211d\nleft\u271d : k\u2082 < 0\nhcd : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\n\u22a2 dist a p * dist b p = dist c p * dist d p\n[PROOFSTEP]\nexact mul_dist_eq_mul_dist_of_cospherical h \u27e8k\u2081, by linarith, hab\u27e9 \u27e8k\u2082, by linarith, hcd\u27e9\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2220 a p b = \u03c0\nhcpd : \u2220 c p d = \u03c0\nk\u2081 : \u211d\nleft\u271d\u00b9 : k\u2081 < 0\nhab : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nk\u2082 : \u211d\nleft\u271d : k\u2082 < 0\nhcd : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\n\u22a2 k\u2081 \u2260 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2220 a p b = \u03c0\nhcpd : \u2220 c p d = \u03c0\nk\u2081 : \u211d\nleft\u271d\u00b9 : k\u2081 < 0\nhab : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nk\u2082 : \u211d\nleft\u271d : k\u2082 < 0\nhcd : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\n\u22a2 k\u2082 \u2260 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhab : a \u2260 b\nhcd : c \u2260 d\nhapb : \u2220 a p b = 0\nhcpd : \u2220 c p d = 0\n\u22a2 dist a p * dist b p = dist c p * dist d p\n[PROOFSTEP]\nobtain \u27e8-, k\u2081, -, hab\u2081\u27e9 := angle_eq_zero_iff.mp hapb\n[GOAL]\ncase intro.intro.intro\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhab : a \u2260 b\nhcd : c \u2260 d\nhapb : \u2220 a p b = 0\nhcpd : \u2220 c p d = 0\nk\u2081 : \u211d\nhab\u2081 : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\n\u22a2 dist a p * dist b p = dist c p * dist d p\n[PROOFSTEP]\nobtain \u27e8-, k\u2082, -, hcd\u2081\u27e9 := angle_eq_zero_iff.mp hcpd\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhab : a \u2260 b\nhcd : c \u2260 d\nhapb : \u2220 a p b = 0\nhcpd : \u2220 c p d = 0\nk\u2081 : \u211d\nhab\u2081 : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nk\u2082 : \u211d\nhcd\u2081 : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\n\u22a2 dist a p * dist b p = dist c p * dist d p\n[PROOFSTEP]\nrefine' mul_dist_eq_mul_dist_of_cospherical h \u27e8k\u2081, _, hab\u2081\u27e9 \u27e8k\u2082, _, hcd\u2081\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhab : a \u2260 b\nhcd : c \u2260 d\nhapb : \u2220 a p b = 0\nhcpd : \u2220 c p d = 0\nk\u2081 : \u211d\nhab\u2081 : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nk\u2082 : \u211d\nhcd\u2081 : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\n\u22a2 k\u2081 \u2260 1\n[PROOFSTEP]\nby_contra hnot\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhab : a \u2260 b\nhcd : c \u2260 d\nhapb : \u2220 a p b = 0\nhcpd : \u2220 c p d = 0\nk\u2081 : \u211d\nhab\u2081 : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nk\u2082 : \u211d\nhcd\u2081 : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\n\u22a2 k\u2082 \u2260 1\n[PROOFSTEP]\nby_contra hnot\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhab : a \u2260 b\nhcd : c \u2260 d\nhapb : \u2220 a p b = 0\nhcpd : \u2220 c p d = 0\nk\u2081 : \u211d\nhab\u2081 : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nk\u2082 : \u211d\nhcd\u2081 : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\nhnot : k\u2081 = 1\n\u22a2 False\n[PROOFSTEP]\nsimp_all only [Classical.not_not, one_smul]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhab : a \u2260 b\nhcd : c \u2260 d\nhapb : \u2220 a p b = 0\nhcpd : \u2220 c p d = 0\nk\u2081 : \u211d\nhab\u2081 : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nk\u2082 : \u211d\nhcd\u2081 : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\nhnot : k\u2082 = 1\n\u22a2 False\n[PROOFSTEP]\nsimp_all only [Classical.not_not, one_smul]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhab : a \u2260 b\nhcd : c \u2260 d\nhapb : \u2220 a p b = 0\nhcpd : \u2220 c p d = 0\nk\u2081 k\u2082 : \u211d\nhab\u2081 : b -\u1d65 p = a -\u1d65 p\nhcd\u2081 : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\nhnot : k\u2081 = 1\n\u22a2 False\ncase intro.intro.intro.intro.intro.intro.refine'_2\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhab : a \u2260 b\nhcd : c \u2260 d\nhapb : \u2220 a p b = 0\nhcpd : \u2220 c p d = 0\nk\u2081 : \u211d\nhab\u2081 : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nk\u2082 : \u211d\nhcd\u2081 : d -\u1d65 p = c -\u1d65 p\nhnot : k\u2082 = 1\n\u22a2 False\n[PROOFSTEP]\nexacts [hab (vsub_left_cancel hab\u2081).symm, hcd (vsub_left_cancel hcd\u2081).symm]\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Euclidean.Sphere.Power", "llama_tokens": 14181, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7634837635542924, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4496072898748402}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.28, u_1} C\ninst\u271d : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\n\u22a2 \u2200 [inst : IsIso (N\u2081.map f)], IsIso f\n[PROOFSTEP]\nintro\n -- restating the result in a way that allows induction on the degree n\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nsuffices \u2200 n : \u2115, IsIso (f.app (op [n]))\n by\n haveI : \u2200 \u0394 : SimplexCategory\u1d52\u1d56, IsIso (f.app \u0394) := fun \u0394 => this \u0394.unop.len\n apply NatIso.isIso_of_isIso_app\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nthis : \u2200 (n : \u2115), IsIso (NatTrans.app f (op [n]))\n\u22a2 IsIso f\n[PROOFSTEP]\nhaveI : \u2200 \u0394 : SimplexCategory\u1d52\u1d56, IsIso (f.app \u0394) := fun \u0394 => this \u0394.unop.len\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nthis\u271d : \u2200 (n : \u2115), IsIso (NatTrans.app f (op [n]))\nthis : \u2200 (\u0394 : SimplexCategory\u1d52\u1d56), IsIso (NatTrans.app f \u0394)\n\u22a2 IsIso f\n[PROOFSTEP]\napply NatIso.isIso_of_isIso_app\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\n\u22a2 \u2200 (n : \u2115), IsIso (NatTrans.app f (op [n]))\n[PROOFSTEP]\nhave h\u2081 := HomologicalComplex.congr_hom (Karoubi.hom_ext_iff.mp (IsIso.hom_inv_id (N\u2081.map f)))\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nh\u2081 : \u2200 (i : \u2115), HomologicalComplex.Hom.f (N\u2081.map f \u226b inv (N\u2081.map f)).f i = HomologicalComplex.Hom.f (\ud835\udfd9 (N\u2081.obj X)).f i\n\u22a2 \u2200 (n : \u2115), IsIso (NatTrans.app f (op [n]))\n[PROOFSTEP]\nhave h\u2082 := HomologicalComplex.congr_hom (Karoubi.hom_ext_iff.mp (IsIso.inv_hom_id (N\u2081.map f)))\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nh\u2081 : \u2200 (i : \u2115), HomologicalComplex.Hom.f (N\u2081.map f \u226b inv (N\u2081.map f)).f i = HomologicalComplex.Hom.f (\ud835\udfd9 (N\u2081.obj X)).f i\nh\u2082 : \u2200 (i : \u2115), HomologicalComplex.Hom.f (inv (N\u2081.map f) \u226b N\u2081.map f).f i = HomologicalComplex.Hom.f (\ud835\udfd9 (N\u2081.obj Y)).f i\n\u22a2 \u2200 (n : \u2115), IsIso (NatTrans.app f (op [n]))\n[PROOFSTEP]\nhave h\u2083 := fun n => Karoubi.HomologicalComplex.p_comm_f_assoc (inv (N\u2081.map f)) n (f.app (op [n]))\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nh\u2081 : \u2200 (i : \u2115), HomologicalComplex.Hom.f (N\u2081.map f \u226b inv (N\u2081.map f)).f i = HomologicalComplex.Hom.f (\ud835\udfd9 (N\u2081.obj X)).f i\nh\u2082 : \u2200 (i : \u2115), HomologicalComplex.Hom.f (inv (N\u2081.map f) \u226b N\u2081.map f).f i = HomologicalComplex.Hom.f (\ud835\udfd9 (N\u2081.obj Y)).f i\nh\u2083 :\n \u2200 (n : \u2115),\n HomologicalComplex.Hom.f (N\u2081.obj Y).p n \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b NatTrans.app f (op [n]) =\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b HomologicalComplex.Hom.f (N\u2081.obj X).p n \u226b NatTrans.app f (op [n])\n\u22a2 \u2200 (n : \u2115), IsIso (NatTrans.app f (op [n]))\n[PROOFSTEP]\nsimp only [N\u2081_map_f, Karoubi.comp_f, HomologicalComplex.comp_f, AlternatingFaceMapComplex.map_f, N\u2081_obj_p,\n Karoubi.id_eq, assoc] at h\u2081 h\u2082 h\u2083 \n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nh\u2082 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i \u226b HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) =\n HomologicalComplex.Hom.f PInfty i\nh\u2083 :\n \u2200 (n : \u2115),\n HomologicalComplex.Hom.f PInfty n \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b NatTrans.app f (op [n]) =\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app f (op [n])\nh\u2081 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i =\n HomologicalComplex.Hom.f PInfty i\n\u22a2 \u2200 (n : \u2115), IsIso (NatTrans.app f (op [n]))\n[PROOFSTEP]\nintro n\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nh\u2082 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i \u226b HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) =\n HomologicalComplex.Hom.f PInfty i\nh\u2083 :\n \u2200 (n : \u2115),\n HomologicalComplex.Hom.f PInfty n \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b NatTrans.app f (op [n]) =\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app f (op [n])\nh\u2081 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i =\n HomologicalComplex.Hom.f PInfty i\nn : \u2115\n\u22a2 IsIso (NatTrans.app f (op [n]))\n[PROOFSTEP]\ninduction' n with n hn\n[GOAL]\ncase zero\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nh\u2082 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i \u226b HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) =\n HomologicalComplex.Hom.f PInfty i\nh\u2083 :\n \u2200 (n : \u2115),\n HomologicalComplex.Hom.f PInfty n \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b NatTrans.app f (op [n]) =\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app f (op [n])\nh\u2081 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i =\n HomologicalComplex.Hom.f PInfty i\n\u22a2 IsIso (NatTrans.app f (op [Nat.zero]))\n[PROOFSTEP]\nuse(inv (N\u2081.map f)).f.f 0\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nh\u2082 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i \u226b HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) =\n HomologicalComplex.Hom.f PInfty i\nh\u2083 :\n \u2200 (n : \u2115),\n HomologicalComplex.Hom.f PInfty n \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b NatTrans.app f (op [n]) =\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app f (op [n])\nh\u2081 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i =\n HomologicalComplex.Hom.f PInfty i\n\u22a2 NatTrans.app f (op [Nat.zero]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 = \ud835\udfd9 (X.obj (op [Nat.zero])) \u2227\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 \u226b NatTrans.app f (op [Nat.zero]) = \ud835\udfd9 (Y.obj (op [Nat.zero]))\n[PROOFSTEP]\nhave h\u2081\u2080 := h\u2081 0\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nh\u2082 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i \u226b HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) =\n HomologicalComplex.Hom.f PInfty i\nh\u2083 :\n \u2200 (n : \u2115),\n HomologicalComplex.Hom.f PInfty n \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b NatTrans.app f (op [n]) =\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app f (op [n])\nh\u2081 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i =\n HomologicalComplex.Hom.f PInfty i\nh\u2081\u2080 :\n HomologicalComplex.Hom.f PInfty 0 \u226b NatTrans.app f (op [0]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 =\n HomologicalComplex.Hom.f PInfty 0\n\u22a2 NatTrans.app f (op [Nat.zero]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 = \ud835\udfd9 (X.obj (op [Nat.zero])) \u2227\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 \u226b NatTrans.app f (op [Nat.zero]) = \ud835\udfd9 (Y.obj (op [Nat.zero]))\n[PROOFSTEP]\nhave h\u2082\u2080 := h\u2082 0\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nh\u2082 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i \u226b HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) =\n HomologicalComplex.Hom.f PInfty i\nh\u2083 :\n \u2200 (n : \u2115),\n HomologicalComplex.Hom.f PInfty n \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b NatTrans.app f (op [n]) =\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app f (op [n])\nh\u2081 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i =\n HomologicalComplex.Hom.f PInfty i\nh\u2081\u2080 :\n HomologicalComplex.Hom.f PInfty 0 \u226b NatTrans.app f (op [0]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 =\n HomologicalComplex.Hom.f PInfty 0\nh\u2082\u2080 :\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 \u226b HomologicalComplex.Hom.f PInfty 0 \u226b NatTrans.app f (op [0]) =\n HomologicalComplex.Hom.f PInfty 0\n\u22a2 NatTrans.app f (op [Nat.zero]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 = \ud835\udfd9 (X.obj (op [Nat.zero])) \u2227\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 \u226b NatTrans.app f (op [Nat.zero]) = \ud835\udfd9 (Y.obj (op [Nat.zero]))\n[PROOFSTEP]\ndsimp at h\u2081\u2080 h\u2082\u2080 \n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nh\u2082 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i \u226b HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) =\n HomologicalComplex.Hom.f PInfty i\nh\u2083 :\n \u2200 (n : \u2115),\n HomologicalComplex.Hom.f PInfty n \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b NatTrans.app f (op [n]) =\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app f (op [n])\nh\u2081 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i =\n HomologicalComplex.Hom.f PInfty i\nh\u2081\u2080 : \ud835\udfd9 (X.obj (op [0])) \u226b NatTrans.app f (op [0]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 = \ud835\udfd9 (X.obj (op [0]))\nh\u2082\u2080 : HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 \u226b \ud835\udfd9 (X.obj (op [0])) \u226b NatTrans.app f (op [0]) = \ud835\udfd9 (Y.obj (op [0]))\n\u22a2 NatTrans.app f (op [Nat.zero]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 = \ud835\udfd9 (X.obj (op [Nat.zero])) \u2227\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 \u226b NatTrans.app f (op [Nat.zero]) = \ud835\udfd9 (Y.obj (op [Nat.zero]))\n[PROOFSTEP]\nsimp only [id_comp, comp_id] at h\u2081\u2080 h\u2082\u2080 \n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nh\u2082 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i \u226b HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) =\n HomologicalComplex.Hom.f PInfty i\nh\u2083 :\n \u2200 (n : \u2115),\n HomologicalComplex.Hom.f PInfty n \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b NatTrans.app f (op [n]) =\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app f (op [n])\nh\u2081 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i =\n HomologicalComplex.Hom.f PInfty i\nh\u2081\u2080 : NatTrans.app f (op [0]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 = \ud835\udfd9 (X.obj (op [0]))\nh\u2082\u2080 : HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 \u226b NatTrans.app f (op [0]) = \ud835\udfd9 (Y.obj (op [0]))\n\u22a2 NatTrans.app f (op [Nat.zero]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 = \ud835\udfd9 (X.obj (op [Nat.zero])) \u2227\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f 0 \u226b NatTrans.app f (op [Nat.zero]) = \ud835\udfd9 (Y.obj (op [Nat.zero]))\n[PROOFSTEP]\ntauto\n[GOAL]\ncase succ\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nh\u2082 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i \u226b HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) =\n HomologicalComplex.Hom.f PInfty i\nh\u2083 :\n \u2200 (n : \u2115),\n HomologicalComplex.Hom.f PInfty n \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b NatTrans.app f (op [n]) =\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app f (op [n])\nh\u2081 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i =\n HomologicalComplex.Hom.f PInfty i\nn : \u2115\nhn : IsIso (NatTrans.app f (op [n]))\n\u22a2 IsIso (NatTrans.app f (op [Nat.succ n]))\n[PROOFSTEP]\nhaveI := hn\n[GOAL]\ncase succ\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nh\u2082 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i \u226b HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) =\n HomologicalComplex.Hom.f PInfty i\nh\u2083 :\n \u2200 (n : \u2115),\n HomologicalComplex.Hom.f PInfty n \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b NatTrans.app f (op [n]) =\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app f (op [n])\nh\u2081 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i =\n HomologicalComplex.Hom.f PInfty i\nn : \u2115\nhn this : IsIso (NatTrans.app f (op [n]))\n\u22a2 IsIso (NatTrans.app f (op [Nat.succ n]))\n[PROOFSTEP]\nuse \u03c6 {\n a := PInfty.f (n + 1) \u226b (inv (N\u2081.map f)).f.f (n + 1)\n b := fun i => inv (f.app (op [n])) \u226b X.\u03c3 i }\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.28, u_1} C\ninst\u271d\u00b9 : Preadditive C\nX Y : SimplicialObject C\nf : X \u27f6 Y\ninst\u271d : IsIso (N\u2081.map f)\nh\u2082 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i \u226b HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) =\n HomologicalComplex.Hom.f PInfty i\nh\u2083 :\n \u2200 (n : \u2115),\n HomologicalComplex.Hom.f PInfty n \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b NatTrans.app f (op [n]) =\n HomologicalComplex.Hom.f (inv (N\u2081.map f)).f n \u226b HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app f (op [n])\nh\u2081 :\n \u2200 (i : \u2115),\n HomologicalComplex.Hom.f PInfty i \u226b NatTrans.app f (op [i]) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f i =\n HomologicalComplex.Hom.f PInfty i\nn : \u2115\nhn this : IsIso (NatTrans.app f (op [n]))\n\u22a2 NatTrans.app f (op [Nat.succ n]) \u226b\n \u03c6\n { a := HomologicalComplex.Hom.f PInfty (n + 1) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f (n + 1),\n b := fun i => inv (NatTrans.app f (op [n])) \u226b SimplicialObject.\u03c3 X i } =\n \ud835\udfd9 (X.obj (op [Nat.succ n])) \u2227\n \u03c6\n { a := HomologicalComplex.Hom.f PInfty (n + 1) \u226b HomologicalComplex.Hom.f (inv (N\u2081.map f)).f (n + 1),\n b := fun i => inv (NatTrans.app f (op [n])) \u226b SimplicialObject.\u03c3 X i } \u226b\n NatTrans.app f (op [Nat.succ n]) =\n \ud835\udfd9 (Y.obj (op [Nat.succ n]))\n[PROOFSTEP]\nsimp only [MorphComponents.id, \u2190 id_\u03c6, \u2190 preComp_\u03c6, preComp, \u2190 postComp_\u03c6, postComp, PInfty_f_naturality_assoc,\n IsIso.hom_inv_id_assoc, assoc, IsIso.inv_hom_id_assoc, SimplicialObject.\u03c3_naturality, h\u2081, h\u2082, h\u2083]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\n\u22a2 N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor =\n karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)\n[PROOFSTEP]\nrefine' CategoryTheory.Functor.ext (fun P => _) fun P Q f => _\n[GOAL]\ncase refine'_1\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\n\u22a2 (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P =\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n P\n[PROOFSTEP]\nrefine' HomologicalComplex.ext _ _\n[GOAL]\ncase refine'_1.refine'_1\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\n\u22a2 ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P).X =\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n P).X\n[PROOFSTEP]\next n\n[GOAL]\ncase refine'_1.refine'_1.h.h_X\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 (HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n).X =\n (HomologicalComplex.X\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n P)\n n).X\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.refine'_1.h.h_p\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 (HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n).p \u226b\n eqToHom\n (_ :\n (HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n).X =\n (HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n).X) =\n eqToHom\n (_ :\n (HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n).X =\n (HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n).X) \u226b\n (HomologicalComplex.X\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n P)\n n).p\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_1.refine'_1.h.h_p\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 (HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app P.p (op [n])) \u226b \ud835\udfd9 (P.X.obj (op [n])) =\n \ud835\udfd9 (P.X.obj (op [n])) \u226b (HomologicalComplex.Hom.f PInfty n).f\n[PROOFSTEP]\nsimp only [karoubi_PInfty_f, comp_id, PInfty_f_naturality, id_comp, eqToHom_refl]\n[GOAL]\ncase refine'_1.refine'_2\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\n\u22a2 \u2200 (i j : \u2115),\n ComplexShape.Rel (ComplexShape.down \u2115) i j \u2192\n HomologicalComplex.d ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) i j \u226b\n eqToHom\n (_ :\n HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) j =\n HomologicalComplex.X\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n (ComplexShape.down \u2115)).obj\n P)\n j) =\n eqToHom\n (_ :\n HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) i =\n HomologicalComplex.X\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n (ComplexShape.down \u2115)).obj\n P)\n i) \u226b\n HomologicalComplex.d\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n P)\n i j\n[PROOFSTEP]\nrintro _ n (rfl : n + 1 = _)\n[GOAL]\ncase refine'_1.refine'_2\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 HomologicalComplex.d ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) n \u226b\n eqToHom\n (_ :\n HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n =\n HomologicalComplex.X\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n P)\n n) =\n eqToHom\n (_ :\n HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) =\n HomologicalComplex.X\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n P)\n (n + 1)) \u226b\n HomologicalComplex.d\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n P)\n (n + 1) n\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_1.refine'_2.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 (HomologicalComplex.d ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) n \u226b\n eqToHom\n (_ :\n HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n =\n HomologicalComplex.X\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n (ComplexShape.down \u2115)).obj\n P)\n n)).f =\n (eqToHom\n (_ :\n HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) =\n HomologicalComplex.X\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n (ComplexShape.down \u2115)).obj\n P)\n (n + 1)) \u226b\n HomologicalComplex.d\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n P)\n (n + 1) n).f\n[PROOFSTEP]\nhave h := (AlternatingFaceMapComplex.map P.p).comm (n + 1) n\n[GOAL]\ncase refine'_1.refine'_2.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\nh :\n HomologicalComplex.Hom.f (AlternatingFaceMapComplex.map P.p) (n + 1) \u226b\n HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n =\n HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n \u226b\n HomologicalComplex.Hom.f (AlternatingFaceMapComplex.map P.p) n\n\u22a2 (HomologicalComplex.d ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) n \u226b\n eqToHom\n (_ :\n HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n =\n HomologicalComplex.X\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n (ComplexShape.down \u2115)).obj\n P)\n n)).f =\n (eqToHom\n (_ :\n HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) =\n HomologicalComplex.X\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n (ComplexShape.down \u2115)).obj\n P)\n (n + 1)) \u226b\n HomologicalComplex.d\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n P)\n (n + 1) n).f\n[PROOFSTEP]\ndsimp [N\u2082, karoubiChainComplexEquivalence, KaroubiHomologicalComplexEquivalence.Functor.obj] at h \u22a2\n[GOAL]\ncase refine'_1.refine'_2.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\nh :\n NatTrans.app P.p (op [n + 1]) \u226b HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n =\n HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n \u226b NatTrans.app P.p (op [n])\n\u22a2 ((HomologicalComplex.Hom.f PInfty (n + 1) \u226b NatTrans.app P.p (op [n + 1])) \u226b\n HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n) \u226b\n (eqToHom\n (_ :\n Karoubi.mk (P.X.obj (op [n])) (HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app P.p (op [n])) =\n HomologicalComplex.X\n ((Functor.mapHomologicalComplex (KaroubiKaroubi.inverse C) (ComplexShape.down \u2115)).obj\n (HomologicalComplex.mk\n (fun n =>\n Karoubi.mk ((KaroubiFunctorCategoryEmbedding.obj P).obj (op [n]))\n (HomologicalComplex.Hom.f PInfty n))\n fun i j =>\n Karoubi.Hom.mk\n (HomologicalComplex.Hom.f PInfty i \u226b\n HomologicalComplex.d (AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)) i\n j)))\n n)).f =\n (eqToHom\n (_ :\n Karoubi.mk (P.X.obj (op [n + 1]))\n (HomologicalComplex.Hom.f PInfty (n + 1) \u226b NatTrans.app P.p (op [n + 1])) =\n HomologicalComplex.X\n ((Functor.mapHomologicalComplex (KaroubiKaroubi.inverse C) (ComplexShape.down \u2115)).obj\n (HomologicalComplex.mk\n (fun n =>\n Karoubi.mk ((KaroubiFunctorCategoryEmbedding.obj P).obj (op [n]))\n (HomologicalComplex.Hom.f PInfty n))\n fun i j =>\n Karoubi.Hom.mk\n (HomologicalComplex.Hom.f PInfty i \u226b\n HomologicalComplex.d (AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)) i\n j)))\n (n + 1))).f \u226b\n (HomologicalComplex.Hom.f PInfty (n + 1)).f \u226b\n (HomologicalComplex.d (AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)) (n + 1) n).f\n[PROOFSTEP]\nsimp only [assoc, Karoubi.eqToHom_f, eqToHom_refl, comp_id, karoubi_alternatingFaceMapComplex_d, karoubi_PInfty_f, \u2190\n HomologicalComplex.Hom.comm_assoc, \u2190 h, app_idem_assoc]\n[GOAL]\ncase refine'_2\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nP Q : Karoubi (SimplicialObject C)\nf : P \u27f6 Q\n\u22a2 (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).map f =\n eqToHom\n (_ :\n (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P =\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n P) \u226b\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).map\n f \u226b\n eqToHom\n (_ :\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n Q =\n (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj Q)\n[PROOFSTEP]\next n\n[GOAL]\ncase refine'_2.h.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nP Q : Karoubi (SimplicialObject C)\nf : P \u27f6 Q\nn : \u2115\n\u22a2 (HomologicalComplex.Hom.f ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).map f) n).f =\n (HomologicalComplex.Hom.f\n (eqToHom\n (_ :\n (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P =\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n (ComplexShape.down \u2115)).obj\n P) \u226b\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).map\n f \u226b\n eqToHom\n (_ :\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n (ComplexShape.down \u2115)).obj\n Q =\n (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj Q))\n n).f\n[PROOFSTEP]\ndsimp [KaroubiKaroubi.inverse, Functor.mapHomologicalComplex]\n[GOAL]\ncase refine'_2.h.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nP Q : Karoubi (SimplicialObject C)\nf : P \u27f6 Q\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app f.f (op [n]) =\n (HomologicalComplex.Hom.f\n (eqToHom\n (_ :\n (karoubiChainComplexEquivalence C \u2115).functor.obj (N\u2082.obj P) =\n HomologicalComplex.mk (fun i => Karoubi.mk (P.X.obj (op [i])) (HomologicalComplex.Hom.f PInfty i).f)\n fun i j =>\n Karoubi.Hom.mk\n (HomologicalComplex.d\n ((karoubiChainComplexEquivalence (Karoubi C) \u2115).functor.obj\n (N\u2081.obj (KaroubiFunctorCategoryEmbedding.obj P)))\n i j).f.f))\n n).f \u226b\n ((HomologicalComplex.Hom.f PInfty n).f \u226b NatTrans.app f.f (op [n])) \u226b\n (HomologicalComplex.Hom.f\n (eqToHom\n (_ :\n (HomologicalComplex.mk (fun i => Karoubi.mk (Q.X.obj (op [i])) (HomologicalComplex.Hom.f PInfty i).f)\n fun i j =>\n Karoubi.Hom.mk\n (HomologicalComplex.d\n ((karoubiChainComplexEquivalence (Karoubi C) \u2115).functor.obj\n (N\u2081.obj (KaroubiFunctorCategoryEmbedding.obj Q)))\n i j).f.f) =\n (karoubiChainComplexEquivalence C \u2115).functor.obj (N\u2082.obj Q)))\n n).f\n[PROOFSTEP]\nsimp only [karoubi_PInfty_f, HomologicalComplex.eqToHom_f, Karoubi.eqToHom_f, assoc, comp_id, PInfty_f_naturality,\n app_p_comp, karoubiChainComplexEquivalence_functor_obj_X_p, N\u2082_obj_p_f, eqToHom_refl, PInfty_f_naturality_assoc,\n app_comp_p, PInfty_f_idem_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.74447, u_1} C\ninst\u271d : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\n\u22a2 \u2200 [inst : IsIso (N\u2082.map f)], IsIso f\n[PROOFSTEP]\nintro\n -- The following functor `F` reflects isomorphism because it is\n -- a composition of four functors which reflects isomorphisms.\n -- Then, it suffices to show that `F.map f` is an isomorphism.\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.74447, u_1} C\ninst\u271d\u00b9 : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\ninst\u271d : IsIso (N\u2082.map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nlet F\u2081 := karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.74447, u_1} C\ninst\u271d\u00b9 : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\ninst\u271d : IsIso (N\u2082.map f)\nF\u2081 : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C\n\u22a2 IsIso f\n[PROOFSTEP]\nlet F\u2082 : SimplicialObject (Karoubi C) \u2964 _ := N\u2081\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.74447, u_1} C\ninst\u271d\u00b9 : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\ninst\u271d : IsIso (N\u2082.map f)\nF\u2081 : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C\nF\u2082 : SimplicialObject (Karoubi C) \u2964 Karoubi (ChainComplex (Karoubi C) \u2115) := N\u2081\n\u22a2 IsIso f\n[PROOFSTEP]\nlet F\u2083 := (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.74447, u_1} C\ninst\u271d\u00b9 : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\ninst\u271d : IsIso (N\u2082.map f)\nF\u2081 : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C\nF\u2082 : SimplicialObject (Karoubi C) \u2964 Karoubi (ChainComplex (Karoubi C) \u2115) := N\u2081\nF\u2083 : Karoubi (ChainComplex (Karoubi C) \u2115) \u2964 ChainComplex (Karoubi (Karoubi C)) \u2115 :=\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor\n\u22a2 IsIso f\n[PROOFSTEP]\nlet F\u2084 := Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.74447, u_1} C\ninst\u271d\u00b9 : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\ninst\u271d : IsIso (N\u2082.map f)\nF\u2081 : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C\nF\u2082 : SimplicialObject (Karoubi C) \u2964 Karoubi (ChainComplex (Karoubi C) \u2115) := N\u2081\nF\u2083 : Karoubi (ChainComplex (Karoubi C) \u2115) \u2964 ChainComplex (Karoubi (Karoubi C)) \u2115 :=\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor\nF\u2084 : HomologicalComplex (Karoubi (Karoubi C)) (ComplexShape.down \u2115) \u2964\n HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) :=\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)\n\u22a2 IsIso f\n[PROOFSTEP]\nlet F := F\u2081 \u22d9 F\u2082 \u22d9 F\u2083 \u22d9 F\u2084\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.74447, u_1} C\ninst\u271d\u00b9 : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\ninst\u271d : IsIso (N\u2082.map f)\nF\u2081 : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C\nF\u2082 : SimplicialObject (Karoubi C) \u2964 Karoubi (ChainComplex (Karoubi C) \u2115) := N\u2081\nF\u2083 : Karoubi (ChainComplex (Karoubi C) \u2115) \u2964 ChainComplex (Karoubi (Karoubi C)) \u2115 :=\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor\nF\u2084 : HomologicalComplex (Karoubi (Karoubi C)) (ComplexShape.down \u2115) \u2964\n HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) :=\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)\nF : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) := F\u2081 \u22d9 F\u2082 \u22d9 F\u2083 \u22d9 F\u2084\n\u22a2 IsIso f\n[PROOFSTEP]\nhaveI : ReflectsIsomorphisms F\u2081 := reflectsIsomorphisms_of_full_and_faithful _\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.74447, u_1} C\ninst\u271d\u00b9 : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\ninst\u271d : IsIso (N\u2082.map f)\nF\u2081 : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C\nF\u2082 : SimplicialObject (Karoubi C) \u2964 Karoubi (ChainComplex (Karoubi C) \u2115) := N\u2081\nF\u2083 : Karoubi (ChainComplex (Karoubi C) \u2115) \u2964 ChainComplex (Karoubi (Karoubi C)) \u2115 :=\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor\nF\u2084 : HomologicalComplex (Karoubi (Karoubi C)) (ComplexShape.down \u2115) \u2964\n HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) :=\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)\nF : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) := F\u2081 \u22d9 F\u2082 \u22d9 F\u2083 \u22d9 F\u2084\nthis : ReflectsIsomorphisms F\u2081\n\u22a2 IsIso f\n[PROOFSTEP]\nhaveI : ReflectsIsomorphisms F\u2082 := by infer_instance\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.74447, u_1} C\ninst\u271d\u00b9 : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\ninst\u271d : IsIso (N\u2082.map f)\nF\u2081 : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C\nF\u2082 : SimplicialObject (Karoubi C) \u2964 Karoubi (ChainComplex (Karoubi C) \u2115) := N\u2081\nF\u2083 : Karoubi (ChainComplex (Karoubi C) \u2115) \u2964 ChainComplex (Karoubi (Karoubi C)) \u2115 :=\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor\nF\u2084 : HomologicalComplex (Karoubi (Karoubi C)) (ComplexShape.down \u2115) \u2964\n HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) :=\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)\nF : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) := F\u2081 \u22d9 F\u2082 \u22d9 F\u2083 \u22d9 F\u2084\nthis : ReflectsIsomorphisms F\u2081\n\u22a2 ReflectsIsomorphisms F\u2082\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.74447, u_1} C\ninst\u271d\u00b9 : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\ninst\u271d : IsIso (N\u2082.map f)\nF\u2081 : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C\nF\u2082 : SimplicialObject (Karoubi C) \u2964 Karoubi (ChainComplex (Karoubi C) \u2115) := N\u2081\nF\u2083 : Karoubi (ChainComplex (Karoubi C) \u2115) \u2964 ChainComplex (Karoubi (Karoubi C)) \u2115 :=\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor\nF\u2084 : HomologicalComplex (Karoubi (Karoubi C)) (ComplexShape.down \u2115) \u2964\n HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) :=\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)\nF : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) := F\u2081 \u22d9 F\u2082 \u22d9 F\u2083 \u22d9 F\u2084\nthis\u271d : ReflectsIsomorphisms F\u2081\nthis : ReflectsIsomorphisms F\u2082\n\u22a2 IsIso f\n[PROOFSTEP]\nhaveI : ReflectsIsomorphisms F\u2083 := reflectsIsomorphisms_of_full_and_faithful _\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.74447, u_1} C\ninst\u271d\u00b9 : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\ninst\u271d : IsIso (N\u2082.map f)\nF\u2081 : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C\nF\u2082 : SimplicialObject (Karoubi C) \u2964 Karoubi (ChainComplex (Karoubi C) \u2115) := N\u2081\nF\u2083 : Karoubi (ChainComplex (Karoubi C) \u2115) \u2964 ChainComplex (Karoubi (Karoubi C)) \u2115 :=\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor\nF\u2084 : HomologicalComplex (Karoubi (Karoubi C)) (ComplexShape.down \u2115) \u2964\n HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) :=\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)\nF : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) := F\u2081 \u22d9 F\u2082 \u22d9 F\u2083 \u22d9 F\u2084\nthis\u271d\u00b9 : ReflectsIsomorphisms F\u2081\nthis\u271d : ReflectsIsomorphisms F\u2082\nthis : ReflectsIsomorphisms F\u2083\n\u22a2 IsIso f\n[PROOFSTEP]\nhaveI : ReflectsIsomorphisms ((KaroubiKaroubi.equivalence C).inverse) := reflectsIsomorphisms_of_full_and_faithful _\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.74447, u_1} C\ninst\u271d\u00b9 : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\ninst\u271d : IsIso (N\u2082.map f)\nF\u2081 : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C\nF\u2082 : SimplicialObject (Karoubi C) \u2964 Karoubi (ChainComplex (Karoubi C) \u2115) := N\u2081\nF\u2083 : Karoubi (ChainComplex (Karoubi C) \u2115) \u2964 ChainComplex (Karoubi (Karoubi C)) \u2115 :=\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor\nF\u2084 : HomologicalComplex (Karoubi (Karoubi C)) (ComplexShape.down \u2115) \u2964\n HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) :=\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)\nF : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) := F\u2081 \u22d9 F\u2082 \u22d9 F\u2083 \u22d9 F\u2084\nthis\u271d\u00b2 : ReflectsIsomorphisms F\u2081\nthis\u271d\u00b9 : ReflectsIsomorphisms F\u2082\nthis\u271d : ReflectsIsomorphisms F\u2083\nthis : ReflectsIsomorphisms (KaroubiKaroubi.equivalence C).inverse\n\u22a2 IsIso f\n[PROOFSTEP]\nhave : IsIso (F.map f) := by\n simp only\n rw [\u2190 compatibility_N\u2082_N\u2081_karoubi, Functor.comp_map]\n apply Functor.map_isIso\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.74447, u_1} C\ninst\u271d\u00b9 : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\ninst\u271d : IsIso (N\u2082.map f)\nF\u2081 : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C\nF\u2082 : SimplicialObject (Karoubi C) \u2964 Karoubi (ChainComplex (Karoubi C) \u2115) := N\u2081\nF\u2083 : Karoubi (ChainComplex (Karoubi C) \u2115) \u2964 ChainComplex (Karoubi (Karoubi C)) \u2115 :=\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor\nF\u2084 : HomologicalComplex (Karoubi (Karoubi C)) (ComplexShape.down \u2115) \u2964\n HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) :=\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)\nF : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) := F\u2081 \u22d9 F\u2082 \u22d9 F\u2083 \u22d9 F\u2084\nthis\u271d\u00b2 : ReflectsIsomorphisms F\u2081\nthis\u271d\u00b9 : ReflectsIsomorphisms F\u2082\nthis\u271d : ReflectsIsomorphisms F\u2083\nthis : ReflectsIsomorphisms (KaroubiKaroubi.equivalence C).inverse\n\u22a2 IsIso (F.map f)\n[PROOFSTEP]\nsimp only\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.74447, u_1} C\ninst\u271d\u00b9 : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\ninst\u271d : IsIso (N\u2082.map f)\nF\u2081 : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C\nF\u2082 : SimplicialObject (Karoubi C) \u2964 Karoubi (ChainComplex (Karoubi C) \u2115) := N\u2081\nF\u2083 : Karoubi (ChainComplex (Karoubi C) \u2115) \u2964 ChainComplex (Karoubi (Karoubi C)) \u2115 :=\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor\nF\u2084 : HomologicalComplex (Karoubi (Karoubi C)) (ComplexShape.down \u2115) \u2964\n HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) :=\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)\nF : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) := F\u2081 \u22d9 F\u2082 \u22d9 F\u2083 \u22d9 F\u2084\nthis\u271d\u00b2 : ReflectsIsomorphisms F\u2081\nthis\u271d\u00b9 : ReflectsIsomorphisms F\u2082\nthis\u271d : ReflectsIsomorphisms F\u2083\nthis : ReflectsIsomorphisms (KaroubiKaroubi.equivalence C).inverse\n\u22a2 IsIso\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n N\u2081 \u22d9\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).map\n f)\n[PROOFSTEP]\nrw [\u2190 compatibility_N\u2082_N\u2081_karoubi, Functor.comp_map]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.74447, u_1} C\ninst\u271d\u00b9 : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\ninst\u271d : IsIso (N\u2082.map f)\nF\u2081 : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C\nF\u2082 : SimplicialObject (Karoubi C) \u2964 Karoubi (ChainComplex (Karoubi C) \u2115) := N\u2081\nF\u2083 : Karoubi (ChainComplex (Karoubi C) \u2115) \u2964 ChainComplex (Karoubi (Karoubi C)) \u2115 :=\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor\nF\u2084 : HomologicalComplex (Karoubi (Karoubi C)) (ComplexShape.down \u2115) \u2964\n HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) :=\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)\nF : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) := F\u2081 \u22d9 F\u2082 \u22d9 F\u2083 \u22d9 F\u2084\nthis\u271d\u00b2 : ReflectsIsomorphisms F\u2081\nthis\u271d\u00b9 : ReflectsIsomorphisms F\u2082\nthis\u271d : ReflectsIsomorphisms F\u2083\nthis : ReflectsIsomorphisms (KaroubiKaroubi.equivalence C).inverse\n\u22a2 IsIso ((karoubiChainComplexEquivalence C \u2115).functor.map (N\u2082.map f))\n[PROOFSTEP]\napply Functor.map_isIso\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.74447, u_1} C\ninst\u271d\u00b9 : Preadditive C\nA\u271d B\u271d : Karoubi (SimplicialObject C)\nf : A\u271d \u27f6 B\u271d\ninst\u271d : IsIso (N\u2082.map f)\nF\u2081 : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C\nF\u2082 : SimplicialObject (Karoubi C) \u2964 Karoubi (ChainComplex (Karoubi C) \u2115) := N\u2081\nF\u2083 : Karoubi (ChainComplex (Karoubi C) \u2115) \u2964 ChainComplex (Karoubi (Karoubi C)) \u2115 :=\n (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor\nF\u2084 : HomologicalComplex (Karoubi (Karoubi C)) (ComplexShape.down \u2115) \u2964\n HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) :=\n Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)\nF : Karoubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u2964 HomologicalComplex (Karoubi C) (ComplexShape.down \u2115) := F\u2081 \u22d9 F\u2082 \u22d9 F\u2083 \u22d9 F\u2084\nthis\u271d\u00b3 : ReflectsIsomorphisms F\u2081\nthis\u271d\u00b2 : ReflectsIsomorphisms F\u2082\nthis\u271d\u00b9 : ReflectsIsomorphisms F\u2083\nthis\u271d : ReflectsIsomorphisms (KaroubiKaroubi.equivalence C).inverse\nthis : IsIso (F.map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nexact isIso_of_reflects_iso f F\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicTopology.DoldKan.NReflectsIso", "llama_tokens": 20161, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850402140659, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.4496068146270216}} {"text": "[GOAL]\nR : Type u\ninst\u271d : LinearOrder R\nsrc\u271d : PartialOrder (Tropical R) := instPartialOrderTropical\na b : Tropical R\n\u22a2 untrop (min a b) = untrop (if a \u2264 b then a else b)\n[PROOFSTEP]\nsimp [min_def]\n[GOAL]\nR : Type u\ninst\u271d : LinearOrder R\nsrc\u271d : PartialOrder (Tropical R) := instPartialOrderTropical\na b : Tropical R\n\u22a2 (if a \u2264 b then untrop a else untrop b) = untrop (if a \u2264 b then a else b)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u\ninst\u271d : LinearOrder R\nsrc\u271d : PartialOrder (Tropical R) := instPartialOrderTropical\na b : Tropical R\nh\u271d : a \u2264 b\n\u22a2 untrop a = untrop a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u\ninst\u271d : LinearOrder R\nsrc\u271d : PartialOrder (Tropical R) := instPartialOrderTropical\na b : Tropical R\nh\u271d : \u00aca \u2264 b\n\u22a2 untrop b = untrop b\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d : LinearOrder R\nsrc\u271d : PartialOrder (Tropical R) := instPartialOrderTropical\na b : Tropical R\n\u22a2 untrop (max a b) = untrop (if a \u2264 b then b else a)\n[PROOFSTEP]\nsimp [max_def]\n[GOAL]\nR : Type u\ninst\u271d : LinearOrder R\nsrc\u271d : PartialOrder (Tropical R) := instPartialOrderTropical\na b : Tropical R\n\u22a2 (if a \u2264 b then untrop b else untrop a) = untrop (if a \u2264 b then b else a)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u\ninst\u271d : LinearOrder R\nsrc\u271d : PartialOrder (Tropical R) := instPartialOrderTropical\na b : Tropical R\nh\u271d : a \u2264 b\n\u22a2 untrop b = untrop b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u\ninst\u271d : LinearOrder R\nsrc\u271d : PartialOrder (Tropical R) := instPartialOrderTropical\na b : Tropical R\nh\u271d : \u00aca \u2264 b\n\u22a2 untrop a = untrop a\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d : LinearOrder R\nx y : Tropical R\nh : x \u2264 y\n\u22a2 untrop (x + y) = untrop x\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nR : Type u\ninst\u271d : LinearOrder R\nx y : Tropical R\nh : y \u2264 x\n\u22a2 untrop (x + y) = untrop y\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nR : Type u\ninst\u271d : LinearOrder R\nx y : Tropical R\n\u22a2 x + y = x \u2194 x \u2264 y\n[PROOFSTEP]\nrw [trop_add_def, trop_eq_iff_eq_untrop, \u2190 untrop_le_iff, min_eq_left_iff]\n[GOAL]\nR : Type u\ninst\u271d : LinearOrder R\nx y : Tropical R\n\u22a2 x + y = y \u2194 y \u2264 x\n[PROOFSTEP]\nrw [trop_add_def, trop_eq_iff_eq_untrop, \u2190 untrop_le_iff, min_eq_right_iff]\n[GOAL]\nR : Type u\ninst\u271d : LinearOrder R\nx y z : Tropical R\n\u22a2 x + y = z \u2194 x = z \u2227 x \u2264 y \u2228 y = z \u2227 y \u2264 x\n[PROOFSTEP]\nrw [trop_add_def, trop_eq_iff_eq_untrop]\n[GOAL]\nR : Type u\ninst\u271d : LinearOrder R\nx y z : Tropical R\n\u22a2 min (untrop x) (untrop y) = untrop z \u2194 x = z \u2227 x \u2264 y \u2228 y = z \u2227 y \u2264 x\n[PROOFSTEP]\nsimp [min_eq_iff]\n[GOAL]\nR : Type u\ninst\u271d : LinearOrder R\na b : Tropical (WithTop R)\n\u22a2 a + b = 0 \u2194 a = 0 \u2227 b = 0\n[PROOFSTEP]\nrw [add_eq_iff]\n[GOAL]\nR : Type u\ninst\u271d : LinearOrder R\na b : Tropical (WithTop R)\n\u22a2 a = 0 \u2227 a \u2264 b \u2228 b = 0 \u2227 b \u2264 a \u2194 a = 0 \u2227 b = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\ninst\u271d : LinearOrder R\na b : Tropical (WithTop R)\n\u22a2 a = 0 \u2227 a \u2264 b \u2228 b = 0 \u2227 b \u2264 a \u2192 a = 0 \u2227 b = 0\n[PROOFSTEP]\nrintro (\u27e8rfl, h\u27e9 | \u27e8rfl, h\u27e9)\n[GOAL]\ncase mp.inl.intro\nR : Type u\ninst\u271d : LinearOrder R\nb : Tropical (WithTop R)\nh : 0 \u2264 b\n\u22a2 0 = 0 \u2227 b = 0\n[PROOFSTEP]\nexact \u27e8rfl, le_antisymm (le_zero _) h\u27e9\n[GOAL]\ncase mp.inr.intro\nR : Type u\ninst\u271d : LinearOrder R\na : Tropical (WithTop R)\nh : 0 \u2264 a\n\u22a2 a = 0 \u2227 0 = 0\n[PROOFSTEP]\nexact \u27e8le_antisymm (le_zero _) h, rfl\u27e9\n[GOAL]\ncase mpr\nR : Type u\ninst\u271d : LinearOrder R\na b : Tropical (WithTop R)\n\u22a2 a = 0 \u2227 b = 0 \u2192 a = 0 \u2227 a \u2264 b \u2228 b = 0 \u2227 b \u2264 a\n[PROOFSTEP]\nrintro \u27e8rfl, rfl\u27e9\n[GOAL]\ncase mpr.intro\nR : Type u\ninst\u271d : LinearOrder R\n\u22a2 0 = 0 \u2227 0 \u2264 0 \u2228 0 = 0 \u2227 0 \u2264 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : OrderTop R\ninst\u271d : Zero R\nsrc\u271d\u00b9 : One (Tropical R) := instOneTropical\nsrc\u271d : AddCommMonoid (Tropical R) := instAddCommMonoidTropical\nn : \u2115\n\u22a2 untrop (NatCast.natCast (n + 1)) = untrop (NatCast.natCast n + 1)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nR : Type u\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : OrderTop R\ninst\u271d : Zero R\nsrc\u271d\u00b9 : One (Tropical R) := instOneTropical\nsrc\u271d : AddCommMonoid (Tropical R) := instAddCommMonoidTropical\n\u22a2 untrop (NatCast.natCast (Nat.zero + 1)) = untrop (NatCast.natCast Nat.zero + 1)\n[PROOFSTEP]\nsimp [Nat.cast]\n[GOAL]\ncase succ\nR : Type u\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : OrderTop R\ninst\u271d : Zero R\nsrc\u271d\u00b9 : One (Tropical R) := instOneTropical\nsrc\u271d : AddCommMonoid (Tropical R) := instAddCommMonoidTropical\nn\u271d : \u2115\n\u22a2 untrop (NatCast.natCast (Nat.succ n\u271d + 1)) = untrop (NatCast.natCast (Nat.succ n\u271d) + 1)\n[PROOFSTEP]\nsimp [Nat.cast]\n[GOAL]\nR : Type u\ninst\u271d : AddMonoid R\nsrc\u271d\u00b9 : MulOneClass (Tropical R) := instMulOneClassTropical\nsrc\u271d : Semigroup (Tropical R) := instSemigroupTropical\nx\u271d : Tropical R\n\u22a2 untrop ((fun n x => x ^ n) 0 x\u271d) = untrop 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d : AddGroup R\nsrc\u271d : Monoid (Tropical R) := instMonoidTropical\nx\u271d\u00b9 x\u271d : Tropical R\n\u22a2 untrop (x\u271d\u00b9 / x\u271d) = untrop (x\u271d\u00b9 * x\u271d\u207b\u00b9)\n[PROOFSTEP]\nsimp [sub_eq_add_neg]\n[GOAL]\nR : Type u\ninst\u271d : LinearOrder R\nx y z : Tropical R\nh : y \u2264 z\n\u22a2 x + y \u2264 x + z\n[PROOFSTEP]\ncases' le_total x y with hx hy\n[GOAL]\ncase inl\nR : Type u\ninst\u271d : LinearOrder R\nx y z : Tropical R\nh : y \u2264 z\nhx : x \u2264 y\n\u22a2 x + y \u2264 x + z\n[PROOFSTEP]\nrw [add_eq_left hx, add_eq_left (hx.trans h)]\n[GOAL]\ncase inr\nR : Type u\ninst\u271d : LinearOrder R\nx y z : Tropical R\nh : y \u2264 z\nhy : y \u2264 x\n\u22a2 x + y \u2264 x + z\n[PROOFSTEP]\nrw [add_eq_right hy]\n[GOAL]\ncase inr\nR : Type u\ninst\u271d : LinearOrder R\nx y z : Tropical R\nh : y \u2264 z\nhy : y \u2264 x\n\u22a2 y \u2264 x + z\n[PROOFSTEP]\ncases' le_total x z with hx hx\n[GOAL]\ncase inr.inl\nR : Type u\ninst\u271d : LinearOrder R\nx y z : Tropical R\nh : y \u2264 z\nhy : y \u2264 x\nhx : x \u2264 z\n\u22a2 y \u2264 x + z\n[PROOFSTEP]\nrwa [add_eq_left hx]\n[GOAL]\ncase inr.inr\nR : Type u\ninst\u271d : LinearOrder R\nx y z : Tropical R\nh : y \u2264 z\nhy : y \u2264 x\nhx : z \u2264 x\n\u22a2 y \u2264 x + z\n[PROOFSTEP]\nrwa [add_eq_right hx]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : LinearOrder R\ninst\u271d\u00b2 : AddMonoid R\ninst\u271d\u00b9 : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass R R (Function.swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nx y : Tropical R\nn : \u2115\n\u22a2 (x + y) ^ n = x ^ n + y ^ n\n[PROOFSTEP]\ncases' le_total x y with h h\n[GOAL]\ncase inl\nR : Type u\ninst\u271d\u00b3 : LinearOrder R\ninst\u271d\u00b2 : AddMonoid R\ninst\u271d\u00b9 : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass R R (Function.swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nx y : Tropical R\nn : \u2115\nh : x \u2264 y\n\u22a2 (x + y) ^ n = x ^ n + y ^ n\n[PROOFSTEP]\nrw [add_eq_left h, add_eq_left (pow_le_pow_of_le_left' h _)]\n[GOAL]\ncase inr\nR : Type u\ninst\u271d\u00b3 : LinearOrder R\ninst\u271d\u00b2 : AddMonoid R\ninst\u271d\u00b9 : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass R R (Function.swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nx y : Tropical R\nn : \u2115\nh : y \u2264 x\n\u22a2 (x + y) ^ n = x ^ n + y ^ n\n[PROOFSTEP]\nrw [add_eq_right h, add_eq_right (pow_le_pow_of_le_left' h _)]\n[GOAL]\nR : Type u\ninst\u271d : LinearOrderedAddCommMonoidWithTop R\nsrc\u271d\u00b3 : AddMonoidWithOne (Tropical R) := instAddMonoidWithOneTropical\nsrc\u271d\u00b2 : Distrib (Tropical R) := instDistribTropical\nsrc\u271d\u00b9 : AddCommMonoid (Tropical R) := instAddCommMonoidTropical\nsrc\u271d : CommMonoid (Tropical R) := instCommMonoidTropical\nx\u271d : Tropical R\n\u22a2 untrop (0 * x\u271d) = untrop 0\n[PROOFSTEP]\nsimp [top_add]\n[GOAL]\nR : Type u\ninst\u271d : LinearOrderedAddCommMonoidWithTop R\nsrc\u271d\u00b3 : AddMonoidWithOne (Tropical R) := instAddMonoidWithOneTropical\nsrc\u271d\u00b2 : Distrib (Tropical R) := instDistribTropical\nsrc\u271d\u00b9 : AddCommMonoid (Tropical R) := instAddCommMonoidTropical\nsrc\u271d : CommMonoid (Tropical R) := instCommMonoidTropical\nx\u271d : Tropical R\n\u22a2 untrop (x\u271d * 0) = untrop 0\n[PROOFSTEP]\nsimp [add_top]\n[GOAL]\nR\u271d : Type u\ninst\u271d\u00b2 : LinearOrderedAddCommMonoidWithTop R\u271d\nR : Type u_1\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\nx : Tropical R\nn : \u2115\n\u22a2 (n + 1) \u2022 x = x\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\nR\u271d : Type u\ninst\u271d\u00b2 : LinearOrderedAddCommMonoidWithTop R\u271d\nR : Type u_1\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\nx : Tropical R\n\u22a2 (Nat.zero + 1) \u2022 x = x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR\u271d : Type u\ninst\u271d\u00b2 : LinearOrderedAddCommMonoidWithTop R\u271d\nR : Type u_1\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\nx : Tropical R\nn : \u2115\nIH : (n + 1) \u2022 x = x\n\u22a2 (Nat.succ n + 1) \u2022 x = x\n[PROOFSTEP]\nrw [add_nsmul, IH, one_nsmul, add_self]\n[GOAL]\nR\u271d : Type u\ninst\u271d\u00b9 : LinearOrderedAddCommMonoidWithTop R\u271d\nR : Type u_1\ninst\u271d : LinearOrderedAddCommMonoid R\na b : Tropical (WithTop R)\n\u22a2 a * b = 0 \u2194 a = 0 \u2228 b = 0\n[PROOFSTEP]\nsimp [\u2190 untrop_inj_iff, WithTop.add_eq_top]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Tropical.Basic", "llama_tokens": 4154, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.6370308013713525, "lm_q1q2_score": 0.4496068018789136}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : Norm E\na : \ud835\udd5c\nf : \ud835\udd5c \u2192 E\nh : IsBoundedUnder (fun x x_1 => x \u2264 x_1) (\ud835\udcdd[{a}\u1d9c] a) (norm \u2218 f)\n\u22a2 f =o[\ud835\udcdd[{a}\u1d9c] a] fun x => (x - a)\u207b\u00b9\n[PROOFSTEP]\nrefine' (h.isBigO_const (one_ne_zero' \u211d)).trans_isLittleO (isLittleO_const_left.2 <| Or.inr _)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : Norm E\na : \ud835\udd5c\nf : \ud835\udd5c \u2192 E\nh : IsBoundedUnder (fun x x_1 => x \u2264 x_1) (\ud835\udcdd[{a}\u1d9c] a) (norm \u2218 f)\n\u22a2 Tendsto (norm \u2218 fun x => (x - a)\u207b\u00b9) (\ud835\udcdd[{a}\u1d9c] a) atTop\n[PROOFSTEP]\nsimp only [(\u00b7 \u2218 \u00b7), norm_inv]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : Norm E\na : \ud835\udd5c\nf : \ud835\udd5c \u2192 E\nh : IsBoundedUnder (fun x x_1 => x \u2264 x_1) (\ud835\udcdd[{a}\u1d9c] a) (norm \u2218 f)\n\u22a2 Tendsto (fun x => \u2016x - a\u2016\u207b\u00b9) (\ud835\udcdd[{a}\u1d9c] a) atTop\n[PROOFSTEP]\nexact (tendsto_norm_sub_self_punctured_nhds a).inv_tendsto_zero\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\np q : \u2115\n\u22a2 (fun x => x ^ p / x ^ q) =\u1da0[atTop] fun x => x ^ (\u2191p - \u2191q)\n[PROOFSTEP]\napply (eventually_gt_atTop (0 : \ud835\udd5c)).mono fun x hx => _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\np q : \u2115\n\u22a2 \u2200 (x : \ud835\udd5c), 0 < x \u2192 (fun x => x ^ p / x ^ q) x = (fun x => x ^ (\u2191p - \u2191q)) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\np q : \u2115\nx : \ud835\udd5c\nhx : 0 < x\n\u22a2 (fun x => x ^ p / x ^ q) x = (fun x => x ^ (\u2191p - \u2191q)) x\n[PROOFSTEP]\nsimp [zpow_sub\u2080 hx.ne']\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\np q : \u2115\n\u22a2 (fun x => x ^ p / x ^ q) =\u1da0[atBot] fun x => x ^ (\u2191p - \u2191q)\n[PROOFSTEP]\napply (eventually_lt_atBot (0 : \ud835\udd5c)).mono fun x hx => _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\np q : \u2115\n\u22a2 \u2200 (x : \ud835\udd5c), x < 0 \u2192 (fun x => x ^ p / x ^ q) x = (fun x => x ^ (\u2191p - \u2191q)) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\np q : \u2115\nx : \ud835\udd5c\nhx : x < 0\n\u22a2 (fun x => x ^ p / x ^ q) x = (fun x => x ^ (\u2191p - \u2191q)) x\n[PROOFSTEP]\nsimp [zpow_sub\u2080 hx.ne]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\nn : \u2124\nhn : 0 < n\n\u22a2 Tendsto (fun x => x ^ n) atTop atTop\n[PROOFSTEP]\nlift n to \u2115 using hn.le\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\nn : \u2115\nhn : 0 < \u2191n\n\u22a2 Tendsto (fun x => x ^ \u2191n) atTop atTop\n[PROOFSTEP]\nsimp only [zpow_ofNat]\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\nn : \u2115\nhn : 0 < \u2191n\n\u22a2 Tendsto (fun x => x ^ n) atTop atTop\n[PROOFSTEP]\nexact tendsto_pow_atTop (Nat.cast_pos.mp hn).ne'\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\np q : \u2115\nhpq : q < p\n\u22a2 Tendsto (fun x => x ^ p / x ^ q) atTop atTop\n[PROOFSTEP]\nrw [tendsto_congr' pow_div_pow_eventuallyEq_atTop]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\np q : \u2115\nhpq : q < p\n\u22a2 Tendsto (fun x => x ^ (\u2191p - \u2191q)) atTop atTop\n[PROOFSTEP]\napply tendsto_zpow_atTop_atTop\n[GOAL]\ncase hn\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\np q : \u2115\nhpq : q < p\n\u22a2 0 < \u2191p - \u2191q\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\np q : \u2115\nhpq : p < q\n\u22a2 Tendsto (fun x => x ^ p / x ^ q) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [tendsto_congr' pow_div_pow_eventuallyEq_atTop]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\np q : \u2115\nhpq : p < q\n\u22a2 Tendsto (fun x => x ^ (\u2191p - \u2191q)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\napply tendsto_zpow_atTop_zero\n[GOAL]\ncase hn\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\np q : \u2115\nhpq : p < q\n\u22a2 \u2191p - \u2191q < 0\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\np q : \u2115\nhpq : p < q\n\u22a2 (fun x => x ^ p) =o[atTop] fun x => x ^ q\n[PROOFSTEP]\nrefine' (isLittleO_iff_tendsto' _).mpr (tendsto_pow_div_pow_atTop_zero hpq)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\np q : \u2115\nhpq : p < q\n\u22a2 \u2200\u1da0 (x : \ud835\udd5c) in atTop, x ^ q = 0 \u2192 x ^ p = 0\n[PROOFSTEP]\nexact (eventually_gt_atTop 0).mono fun x hx hxq => (pow_ne_zero q hx.ne' hxq).elim\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : NormedLinearOrderedField \ud835\udd5c\n\u03b1 : Type u_2\nu v : \u03b1 \u2192 \ud835\udd5c\nl : Filter \u03b1\nhuv : u =O[l] v\nhu : Tendsto (fun x => \u2016u x\u2016) l atTop\n\u22a2 Tendsto (fun x => \u2016v x\u2016) l atTop\n[PROOFSTEP]\nrcases huv.exists_pos with \u27e8c, hc, hcuv\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d : NormedLinearOrderedField \ud835\udd5c\n\u03b1 : Type u_2\nu v : \u03b1 \u2192 \ud835\udd5c\nl : Filter \u03b1\nhuv : u =O[l] v\nhu : Tendsto (fun x => \u2016u x\u2016) l atTop\nc : \u211d\nhc : 0 < c\nhcuv : IsBigOWith c l u v\n\u22a2 Tendsto (fun x => \u2016v x\u2016) l atTop\n[PROOFSTEP]\nrw [IsBigOWith] at hcuv \n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d : NormedLinearOrderedField \ud835\udd5c\n\u03b1 : Type u_2\nu v : \u03b1 \u2192 \ud835\udd5c\nl : Filter \u03b1\nhuv : u =O[l] v\nhu : Tendsto (fun x => \u2016u x\u2016) l atTop\nc : \u211d\nhc : 0 < c\nhcuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 c * \u2016v x\u2016\n\u22a2 Tendsto (fun x => \u2016v x\u2016) l atTop\n[PROOFSTEP]\nconvert Tendsto.atTop_div_const hc (tendsto_atTop_mono' l hcuv hu)\n[GOAL]\ncase h.e'_3.h\n\ud835\udd5c : Type u_1\ninst\u271d : NormedLinearOrderedField \ud835\udd5c\n\u03b1 : Type u_2\nu v : \u03b1 \u2192 \ud835\udd5c\nl : Filter \u03b1\nhuv : u =O[l] v\nhu : Tendsto (fun x => \u2016u x\u2016) l atTop\nc : \u211d\nhc : 0 < c\nhcuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 c * \u2016v x\u2016\nx\u271d : \u03b1\n\u22a2 \u2016v x\u271d\u2016 = c * \u2016v x\u271d\u2016 / c\n[PROOFSTEP]\nrw [mul_div_cancel_left _ hc.ne.symm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\n\u22a2 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2211 i in range n, g i\n[PROOFSTEP]\nhave A : \u2200 i, \u2016g i\u2016 = g i := fun i => Real.norm_of_nonneg (hg i)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\n\u22a2 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2211 i in range n, g i\n[PROOFSTEP]\nhave B : \u2200 n, \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i := fun n => by rwa [Real.norm_eq_abs, abs_sum_of_nonneg']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nn : \u2115\n\u22a2 \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n[PROOFSTEP]\nrwa [Real.norm_eq_abs, abs_sum_of_nonneg']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u22a2 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2211 i in range n, g i\n[PROOFSTEP]\napply isLittleO_iff.2 fun \u03b5 \u03b5pos => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u22a2 \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (x : \u2115) in atTop, \u2016\u2211 i in range x, f i\u2016 \u2264 \u03b5 * \u2016\u2211 i in range x, g i\u2016\n[PROOFSTEP]\nintro \u03b5 \u03b5pos\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop, \u2016\u2211 i in range x, f i\u2016 \u2264 \u03b5 * \u2016\u2211 i in range x, g i\u2016\n[PROOFSTEP]\nobtain \u27e8N, hN\u27e9 : \u2203 N : \u2115, \u2200 b : \u2115, N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b := by\n simpa only [A, eventually_atTop] using isLittleO_iff.mp h (half_pos \u03b5pos)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2203 N, \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\n[PROOFSTEP]\nsimpa only [A, eventually_atTop] using isLittleO_iff.mp h (half_pos \u03b5pos)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop, \u2016\u2211 i in range x, f i\u2016 \u2264 \u03b5 * \u2016\u2211 i in range x, g i\u2016\n[PROOFSTEP]\nhave : (fun _ : \u2115 => \u2211 i in range N, f i) =o[atTop] fun n : \u2115 => \u2211 i in range n, g i :=\n by\n apply isLittleO_const_left.2\n exact Or.inr (h'g.congr fun n => (B n).symm)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\n\u22a2 (fun x => \u2211 i in range N, f i) =o[atTop] fun n => \u2211 i in range n, g i\n[PROOFSTEP]\napply isLittleO_const_left.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\n\u22a2 \u2211 i in range N, f i = 0 \u2228 Tendsto (norm \u2218 fun n => \u2211 i in range n, g i) atTop atTop\n[PROOFSTEP]\nexact Or.inr (h'g.congr fun n => (B n).symm)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\nthis : (fun x => \u2211 i in range N, f i) =o[atTop] fun n => \u2211 i in range n, g i\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop, \u2016\u2211 i in range x, f i\u2016 \u2264 \u03b5 * \u2016\u2211 i in range x, g i\u2016\n[PROOFSTEP]\nfilter_upwards [isLittleO_iff.1 this (half_pos \u03b5pos), Ici_mem_atTop N] with n hn Nn\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\nthis : (fun x => \u2211 i in range N, f i) =o[atTop] fun n => \u2211 i in range n, g i\nn : \u2115\nhn : \u2016\u2211 i in range N, f i\u2016 \u2264 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016\nNn : n \u2208 Set.Ici N\n\u22a2 \u2016\u2211 i in range n, f i\u2016 \u2264 \u03b5 * \u2016\u2211 i in range n, g i\u2016\n[PROOFSTEP]\ncalc\n \u2016\u2211 i in range n, f i\u2016 = \u2016(\u2211 i in range N, f i) + \u2211 i in Ico N n, f i\u2016 := by rw [sum_range_add_sum_Ico _ Nn]\n _ \u2264 \u2016\u2211 i in range N, f i\u2016 + \u2016\u2211 i in Ico N n, f i\u2016 := (norm_add_le _ _)\n _ \u2264 \u2016\u2211 i in range N, f i\u2016 + \u2211 i in Ico N n, \u03b5 / 2 * g i :=\n (add_le_add le_rfl (norm_sum_le_of_le _ fun i hi => hN _ (mem_Ico.1 hi).1))\n _ \u2264 \u2016\u2211 i in range N, f i\u2016 + \u2211 i in range n, \u03b5 / 2 * g i :=\n by\n gcongr\n apply sum_le_sum_of_subset_of_nonneg\n \u00b7 rw [range_eq_Ico]\n exact Ico_subset_Ico (zero_le _) le_rfl\n \u00b7 intro i _ _\n exact mul_nonneg (half_pos \u03b5pos).le (hg i)\n _ \u2264 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016 + \u03b5 / 2 * \u2211 i in range n, g i := by rw [\u2190 mul_sum]; gcongr\n _ = \u03b5 * \u2016\u2211 i in range n, g i\u2016 := by\n simp only [B]\n ring\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\nthis : (fun x => \u2211 i in range N, f i) =o[atTop] fun n => \u2211 i in range n, g i\nn : \u2115\nhn : \u2016\u2211 i in range N, f i\u2016 \u2264 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016\nNn : n \u2208 Set.Ici N\n\u22a2 \u2016\u2211 i in range n, f i\u2016 = \u2016\u2211 i in range N, f i + \u2211 i in Ico N n, f i\u2016\n[PROOFSTEP]\nrw [sum_range_add_sum_Ico _ Nn]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\nthis : (fun x => \u2211 i in range N, f i) =o[atTop] fun n => \u2211 i in range n, g i\nn : \u2115\nhn : \u2016\u2211 i in range N, f i\u2016 \u2264 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016\nNn : n \u2208 Set.Ici N\n\u22a2 \u2016\u2211 i in range N, f i\u2016 + \u2211 i in Ico N n, \u03b5 / 2 * g i \u2264 \u2016\u2211 i in range N, f i\u2016 + \u2211 i in range n, \u03b5 / 2 * g i\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase bc\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\nthis : (fun x => \u2211 i in range N, f i) =o[atTop] fun n => \u2211 i in range n, g i\nn : \u2115\nhn : \u2016\u2211 i in range N, f i\u2016 \u2264 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016\nNn : n \u2208 Set.Ici N\n\u22a2 \u2211 i in Ico N n, \u03b5 / 2 * g i \u2264 \u2211 i in range n, \u03b5 / 2 * g i\n[PROOFSTEP]\napply sum_le_sum_of_subset_of_nonneg\n[GOAL]\ncase bc.h\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\nthis : (fun x => \u2211 i in range N, f i) =o[atTop] fun n => \u2211 i in range n, g i\nn : \u2115\nhn : \u2016\u2211 i in range N, f i\u2016 \u2264 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016\nNn : n \u2208 Set.Ici N\n\u22a2 Ico N n \u2286 range n\n[PROOFSTEP]\nrw [range_eq_Ico]\n[GOAL]\ncase bc.h\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\nthis : (fun x => \u2211 i in range N, f i) =o[atTop] fun n => \u2211 i in range n, g i\nn : \u2115\nhn : \u2016\u2211 i in range N, f i\u2016 \u2264 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016\nNn : n \u2208 Set.Ici N\n\u22a2 Ico N n \u2286 Ico 0 n\n[PROOFSTEP]\nexact Ico_subset_Ico (zero_le _) le_rfl\n[GOAL]\ncase bc.hf\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\nthis : (fun x => \u2211 i in range N, f i) =o[atTop] fun n => \u2211 i in range n, g i\nn : \u2115\nhn : \u2016\u2211 i in range N, f i\u2016 \u2264 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016\nNn : n \u2208 Set.Ici N\n\u22a2 \u2200 (i : \u2115), i \u2208 range n \u2192 \u00aci \u2208 Ico N n \u2192 0 \u2264 \u03b5 / 2 * g i\n[PROOFSTEP]\nintro i _ _\n[GOAL]\ncase bc.hf\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\nthis : (fun x => \u2211 i in range N, f i) =o[atTop] fun n => \u2211 i in range n, g i\nn : \u2115\nhn : \u2016\u2211 i in range N, f i\u2016 \u2264 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016\nNn : n \u2208 Set.Ici N\ni : \u2115\na\u271d\u00b9 : i \u2208 range n\na\u271d : \u00aci \u2208 Ico N n\n\u22a2 0 \u2264 \u03b5 / 2 * g i\n[PROOFSTEP]\nexact mul_nonneg (half_pos \u03b5pos).le (hg i)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\nthis : (fun x => \u2211 i in range N, f i) =o[atTop] fun n => \u2211 i in range n, g i\nn : \u2115\nhn : \u2016\u2211 i in range N, f i\u2016 \u2264 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016\nNn : n \u2208 Set.Ici N\n\u22a2 \u2016\u2211 i in range N, f i\u2016 + \u2211 i in range n, \u03b5 / 2 * g i \u2264 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016 + \u03b5 / 2 * \u2211 i in range n, g i\n[PROOFSTEP]\nrw [\u2190 mul_sum]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\nthis : (fun x => \u2211 i in range N, f i) =o[atTop] fun n => \u2211 i in range n, g i\nn : \u2115\nhn : \u2016\u2211 i in range N, f i\u2016 \u2264 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016\nNn : n \u2208 Set.Ici N\n\u22a2 \u2016\u2211 i in range N, f i\u2016 + \u03b5 / 2 * \u2211 x in range n, g x \u2264 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016 + \u03b5 / 2 * \u2211 i in range n, g i\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\nthis : (fun x => \u2211 i in range N, f i) =o[atTop] fun n => \u2211 i in range n, g i\nn : \u2115\nhn : \u2016\u2211 i in range N, f i\u2016 \u2264 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016\nNn : n \u2208 Set.Ici N\n\u22a2 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016 + \u03b5 / 2 * \u2211 i in range n, g i = \u03b5 * \u2016\u2211 i in range n, g i\u2016\n[PROOFSTEP]\nsimp only [B]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\ng : \u2115 \u2192 \u211d\nh : f =o[atTop] g\nhg : 0 \u2264 g\nh'g : Tendsto (fun n => \u2211 i in range n, g i) atTop atTop\nA : \u2200 (i : \u2115), \u2016g i\u2016 = g i\nB : \u2200 (n : \u2115), \u2016\u2211 i in range n, g i\u2016 = \u2211 i in range n, g i\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016f b\u2016 \u2264 \u03b5 / 2 * g b\nthis : (fun x => \u2211 i in range N, f i) =o[atTop] fun n => \u2211 i in range n, g i\nn : \u2115\nhn : \u2016\u2211 i in range N, f i\u2016 \u2264 \u03b5 / 2 * \u2016\u2211 i in range n, g i\u2016\nNn : n \u2208 Set.Ici N\n\u22a2 \u03b5 / 2 * \u2211 i in range n, g i + \u03b5 / 2 * \u2211 i in range n, g i = \u03b5 * \u2211 i in range n, g i\n[PROOFSTEP]\nring\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nh : Tendsto f atTop (\ud835\udcdd 0)\n\u22a2 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2191n\n[PROOFSTEP]\nhave := ((isLittleO_one_iff \u211d).2 h).sum_range fun i => zero_le_one\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nh : Tendsto f atTop (\ud835\udcdd 0)\nthis :\n Tendsto (fun n => \u2211 i in range n, 1) atTop atTop \u2192 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2211 i in range n, 1\n\u22a2 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2191n\n[PROOFSTEP]\nsimp only [sum_const, card_range, Nat.smul_one_eq_coe] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nh : Tendsto f atTop (\ud835\udcdd 0)\nthis : Tendsto (fun n => \u2191n) atTop atTop \u2192 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2191n\n\u22a2 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2191n\n[PROOFSTEP]\nexact this tendsto_nat_cast_atTop_atTop\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nu : \u2115 \u2192 E\nl : E\nh : Tendsto u atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, u i) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nrw [\u2190 tendsto_sub_nhds_zero_iff, \u2190 isLittleO_one_iff \u211d]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nu : \u2115 \u2192 E\nl : E\nh : Tendsto u atTop (\ud835\udcdd l)\n\u22a2 (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, u i - l) =o[atTop] fun _x => 1\n[PROOFSTEP]\nhave := Asymptotics.isLittleO_sum_range_of_tendsto_zero (tendsto_sub_nhds_zero_iff.2 h)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nu : \u2115 \u2192 E\nl : E\nh : Tendsto u atTop (\ud835\udcdd l)\nthis : (fun n => \u2211 i in range n, (u i - l)) =o[atTop] fun n => \u2191n\n\u22a2 (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, u i - l) =o[atTop] fun _x => 1\n[PROOFSTEP]\napply ((isBigO_refl (fun n : \u2115 => (n : \u211d)\u207b\u00b9) atTop).smul_isLittleO this).congr' _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nu : \u2115 \u2192 E\nl : E\nh : Tendsto u atTop (\ud835\udcdd l)\nthis : (fun n => \u2211 i in range n, (u i - l)) =o[atTop] fun n => \u2191n\n\u22a2 (fun x => (\u2191x)\u207b\u00b9 \u2022 \u2211 i in range x, (u i - l)) =\u1da0[atTop] fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, u i - l\n[PROOFSTEP]\nfilter_upwards [Ici_mem_atTop 1] with n npos\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nu : \u2115 \u2192 E\nl : E\nh : Tendsto u atTop (\ud835\udcdd l)\nthis : (fun n => \u2211 i in range n, (u i - l)) =o[atTop] fun n => \u2191n\nn : \u2115\nnpos : n \u2208 Set.Ici 1\n\u22a2 (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, (u i - l) = (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, u i - l\n[PROOFSTEP]\nhave npos\u211d : (0 : \u211d) < n := Nat.cast_pos.2 npos\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nu : \u2115 \u2192 E\nl : E\nh : Tendsto u atTop (\ud835\udcdd l)\nthis : (fun n => \u2211 i in range n, (u i - l)) =o[atTop] fun n => \u2191n\nn : \u2115\nnpos : n \u2208 Set.Ici 1\nnpos\u211d : 0 < \u2191n\n\u22a2 (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, (u i - l) = (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, u i - l\n[PROOFSTEP]\nsimp only [smul_sub, sum_sub_distrib, sum_const, card_range, sub_right_inj]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nu : \u2115 \u2192 E\nl : E\nh : Tendsto u atTop (\ud835\udcdd l)\nthis : (fun n => \u2211 i in range n, (u i - l)) =o[atTop] fun n => \u2191n\nn : \u2115\nnpos : n \u2208 Set.Ici 1\nnpos\u211d : 0 < \u2191n\n\u22a2 (\u2191n)\u207b\u00b9 \u2022 n \u2022 l = l\n[PROOFSTEP]\nrw [nsmul_eq_smul_cast \u211d, smul_smul, inv_mul_cancel npos\u211d.ne', one_smul]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nu : \u2115 \u2192 E\nl : E\nh : Tendsto u atTop (\ud835\udcdd l)\nthis : (fun n => \u2211 i in range n, (u i - l)) =o[atTop] fun n => \u2191n\n\u22a2 (fun x => (\u2191x)\u207b\u00b9 \u2022 \u2191x) =\u1da0[atTop] fun _x => 1\n[PROOFSTEP]\nfilter_upwards [Ici_mem_atTop 1] with n npos\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nu : \u2115 \u2192 E\nl : E\nh : Tendsto u atTop (\ud835\udcdd l)\nthis : (fun n => \u2211 i in range n, (u i - l)) =o[atTop] fun n => \u2191n\nn : \u2115\nnpos : n \u2208 Set.Ici 1\n\u22a2 (\u2191n)\u207b\u00b9 \u2022 \u2191n = 1\n[PROOFSTEP]\nhave npos\u211d : (0 : \u211d) < n := Nat.cast_pos.2 npos\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nu : \u2115 \u2192 E\nl : E\nh : Tendsto u atTop (\ud835\udcdd l)\nthis : (fun n => \u2211 i in range n, (u i - l)) =o[atTop] fun n => \u2191n\nn : \u2115\nnpos : n \u2208 Set.Ici 1\nnpos\u211d : 0 < \u2191n\n\u22a2 (\u2191n)\u207b\u00b9 \u2022 \u2191n = 1\n[PROOFSTEP]\nrw [Algebra.id.smul_eq_mul, inv_mul_cancel npos\u211d.ne']\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "llama_tokens": 11710, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147276, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.4495934253821279}} {"text": "[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nsrc\u271d\u00b9 : Module R (A \u00d7 B) := instModule\nsrc\u271d : R \u2192+* A \u00d7 B := RingHom.prod (algebraMap R A) (algebraMap R B)\n\u22a2 \u2200 (r : R) (x : (fun x => A \u00d7 B) r),\n \u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n map_add' :=\n (_ : \u2200 (x y : R), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) }\n r *\n x =\n x *\n \u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n map_add' :=\n (_ : \u2200 (x y : R), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) }\n r\n[PROOFSTEP]\nrintro r \u27e8a, b\u27e9\n[GOAL]\ncase mk\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nsrc\u271d\u00b9 : Module R (A \u00d7 B) := instModule\nsrc\u271d : R \u2192+* A \u00d7 B := RingHom.prod (algebraMap R A) (algebraMap R B)\nr : R\na : A\nb : B\n\u22a2 \u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n map_add' :=\n (_ : \u2200 (x y : R), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) }\n r *\n (a, b) =\n (a, b) *\n \u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n map_add' :=\n (_ : \u2200 (x y : R), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) }\n r\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nsrc\u271d\u00b9 : Module R (A \u00d7 B) := instModule\nsrc\u271d : R \u2192+* A \u00d7 B := RingHom.prod (algebraMap R A) (algebraMap R B)\nr : R\na : A\nb : B\n\u22a2 (\u2191(algebraMap R A) r * a, \u2191(algebraMap R B) r * b) = (a * \u2191(algebraMap R A) r, b * \u2191(algebraMap R B) r)\n[PROOFSTEP]\nrw [commutes r a, commutes r b]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nsrc\u271d\u00b9 : Module R (A \u00d7 B) := instModule\nsrc\u271d : R \u2192+* A \u00d7 B := RingHom.prod (algebraMap R A) (algebraMap R B)\n\u22a2 \u2200 (r : R) (x : (fun x => A \u00d7 B) r),\n r \u2022 x =\n \u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n map_add' :=\n (_ : \u2200 (x y : R), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) }\n r *\n x\n[PROOFSTEP]\nrintro r \u27e8a, b\u27e9\n[GOAL]\ncase mk\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nsrc\u271d\u00b9 : Module R (A \u00d7 B) := instModule\nsrc\u271d : R \u2192+* A \u00d7 B := RingHom.prod (algebraMap R A) (algebraMap R B)\nr : R\na : A\nb : B\n\u22a2 r \u2022 (a, b) =\n \u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n map_add' :=\n (_ : \u2200 (x y : R), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) }\n r *\n (a, b)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nsrc\u271d\u00b9 : Module R (A \u00d7 B) := instModule\nsrc\u271d : R \u2192+* A \u00d7 B := RingHom.prod (algebraMap R A) (algebraMap R B)\nr : R\na : A\nb : B\n\u22a2 (r \u2022 a, r \u2022 b) = (\u2191(algebraMap R A) r * a, \u2191(algebraMap R B) r * b)\n[PROOFSTEP]\nrw [Algebra.smul_def r a, Algebra.smul_def r b]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2090[R] B\ng : A \u2192\u2090[R] C\nsrc\u271d : A \u2192+* B \u00d7 C := RingHom.prod \u2191f \u2191g\nr : R\n\u22a2 OneHom.toFun\n (\u2191\u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n map_add' :=\n (_ : \u2200 (x y : A), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) })\n (\u2191(algebraMap R A) r) =\n \u2191(algebraMap R (B \u00d7 C)) r\n[PROOFSTEP]\nsimp only [toRingHom_eq_coe, RingHom.toFun_eq_coe, RingHom.prod_apply, coe_toRingHom, commutes, Prod.algebraMap_apply]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2090[R] B\ng : A \u2192\u2090[R] C\n\u22a2 comp (fst R B C) (prod f g) = f\n[PROOFSTEP]\next\n[GOAL]\ncase H\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2090[R] B\ng : A \u2192\u2090[R] C\nx\u271d : A\n\u22a2 \u2191(comp (fst R B C) (prod f g)) x\u271d = \u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2090[R] B\ng : A \u2192\u2090[R] C\n\u22a2 comp (snd R B C) (prod f g) = g\n[PROOFSTEP]\next\n[GOAL]\ncase H\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2090[R] B\ng : A \u2192\u2090[R] C\nx\u271d : A\n\u22a2 \u2191(comp (snd R B C) (prod f g)) x\u271d = \u2191g x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : (A \u2192\u2090[R] B) \u00d7 (A \u2192\u2090[R] C)\n\u22a2 (fun f => (comp (fst R B C) f, comp (snd R B C) f)) ((fun f => prod f.fst f.snd) f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081.H\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : (A \u2192\u2090[R] B) \u00d7 (A \u2192\u2090[R] C)\nx\u271d : A\n\u22a2 \u2191((fun f => (comp (fst R B C) f, comp (snd R B C) f)) ((fun f => prod f.fst f.snd) f)).fst x\u271d = \u2191f.fst x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2082.H\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : (A \u2192\u2090[R] B) \u00d7 (A \u2192\u2090[R] C)\nx\u271d : A\n\u22a2 \u2191((fun f => (comp (fst R B C) f, comp (snd R B C) f)) ((fun f => prod f.fst f.snd) f)).snd x\u271d = \u2191f.snd x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2090[R] B \u00d7 C\n\u22a2 (fun f => prod f.fst f.snd) ((fun f => (comp (fst R B C) f, comp (snd R B C) f)) f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase H.h\u2081\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2090[R] B \u00d7 C\nx\u271d : A\n\u22a2 (\u2191((fun f => prod f.fst f.snd) ((fun f => (comp (fst R B C) f, comp (snd R B C) f)) f)) x\u271d).fst = (\u2191f x\u271d).fst\n[PROOFSTEP]\nrfl\n[GOAL]\ncase H.h\u2082\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2090[R] B \u00d7 C\nx\u271d : A\n\u22a2 (\u2191((fun f => prod f.fst f.snd) ((fun f => (comp (fst R B C) f, comp (snd R B C) f)) f)) x\u271d).snd = (\u2191f x\u271d).snd\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Algebra.Prod", "llama_tokens": 4200, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891435927269, "lm_q2_score": 0.546738151984614, "lm_q1q2_score": 0.44952217294969987}} {"text": "[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na\u271d b c : B\nf : a\u271d \u27f6 b\ng : b \u27f6 a\u271d\na : B\n\u22a2 leftZigzag (\u03c1_ (\ud835\udfd9 a)).inv (\u03c1_ (\ud835\udfd9 a)).hom = (\u03bb_ (\ud835\udfd9 a)).hom \u226b (\u03c1_ (\ud835\udfd9 a)).inv\n[PROOFSTEP]\ndsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na\u271d b c : B\nf : a\u271d \u27f6 b\ng : b \u27f6 a\u271d\na : B\n\u22a2 (\u03c1_ (\ud835\udfd9 a)).inv \u25b7 \ud835\udfd9 a \u2297\u226b \ud835\udfd9 a \u25c1 (\u03c1_ (\ud835\udfd9 a)).hom = (\u03bb_ (\ud835\udfd9 a)).hom \u226b (\u03c1_ (\ud835\udfd9 a)).inv\n[PROOFSTEP]\ncoherence\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na\u271d b c : B\nf : a\u271d \u27f6 b\ng : b \u27f6 a\u271d\na : B\n\u22a2 rightZigzag (\u03c1_ (\ud835\udfd9 a)).inv (\u03c1_ (\ud835\udfd9 a)).hom = (\u03c1_ (\ud835\udfd9 a)).hom \u226b (\u03bb_ (\ud835\udfd9 a)).inv\n[PROOFSTEP]\ndsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na\u271d b c : B\nf : a\u271d \u27f6 b\ng : b \u27f6 a\u271d\na : B\n\u22a2 \ud835\udfd9 a \u25c1 (\u03c1_ (\ud835\udfd9 a)).inv \u2297\u226b (\u03c1_ (\ud835\udfd9 a)).hom \u25b7 \ud835\udfd9 a = (\u03c1_ (\ud835\udfd9 a)).hom \u226b (\u03bb_ (\ud835\udfd9 a)).inv\n[PROOFSTEP]\ncoherence\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 (leftZigzagIso \u03b7 \u03b5).inv = rightZigzag \u03b5.inv \u03b7.inv\n[PROOFSTEP]\nsimp [bicategoricalComp, bicategoricalIsoComp]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 (rightZigzagIso \u03b7 \u03b5).inv = leftZigzag \u03b5.inv \u03b7.inv\n[PROOFSTEP]\nsimp [bicategoricalComp, bicategoricalIsoComp]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7\u271d : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5\u271d : g \u226b f \u2245 \ud835\udfd9 b\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 leftZigzagIso \u03b7 (adjointifyCounit \u03b7 \u03b5) = \u03bb_ f \u226a\u226b (\u03c1_ f).symm\n[PROOFSTEP]\napply Iso.ext\n[GOAL]\ncase w\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7\u271d : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5\u271d : g \u226b f \u2245 \ud835\udfd9 b\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 (leftZigzagIso \u03b7 (adjointifyCounit \u03b7 \u03b5)).hom = (\u03bb_ f \u226a\u226b (\u03c1_ f).symm).hom\n[PROOFSTEP]\ndsimp [adjointifyCounit, bicategoricalIsoComp]\n[GOAL]\ncase w\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7\u271d : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5\u271d : g \u226b f \u2245 \ud835\udfd9 b\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 \u03b7.hom \u25b7 f \u226b\n ((\u03b1_ f g f).hom \u226b \ud835\udfd9 f \u25b7 (g \u226b f)) \u226b\n f \u25c1 (g \u25c1 ((\u03c1_ f).inv \u226b (f \u25c1 \u03b5.inv \u226b (\ud835\udfd9 f \u25b7 (g \u226b f) \u226b (\u03b1_ f g f).inv) \u226b \u03b7.inv \u25b7 f) \u226b (\u03bb_ f).hom) \u226b \u03b5.hom) =\n (\u03bb_ f).hom \u226b (\u03c1_ f).inv\n[PROOFSTEP]\ncalc\n _ = \ud835\udfd9 _ \u2297\u226b (\u03b7.hom \u25b7 (f \u226b \ud835\udfd9 b) \u226b (f \u226b g) \u25c1 f \u25c1 \u03b5.inv) \u2297\u226b f \u25c1 g \u25c1 \u03b7.inv \u25b7 f \u2297\u226b f \u25c1 \u03b5.hom := by simp [bicategoricalComp];\n coherence\n _ = \ud835\udfd9 _ \u2297\u226b f \u25c1 \u03b5.inv \u2297\u226b (\u03b7.hom \u25b7 (f \u226b g) \u226b (f \u226b g) \u25c1 \u03b7.inv) \u25b7 f \u2297\u226b f \u25c1 \u03b5.hom := by\n rw [\u2190 whisker_exchange \u03b7.hom (f \u25c1 \u03b5.inv)]; simp [bicategoricalComp]; coherence\n _ = \ud835\udfd9 _ \u2297\u226b f \u25c1 \u03b5.inv \u2297\u226b (\u03b7.inv \u226b \u03b7.hom) \u25b7 f \u2297\u226b f \u25c1 \u03b5.hom := by rw [\u2190 whisker_exchange \u03b7.hom \u03b7.inv]; coherence\n _ = \ud835\udfd9 _ \u2297\u226b f \u25c1 (\u03b5.inv \u226b \u03b5.hom) := by rw [Iso.inv_hom_id]; simp [bicategoricalComp]\n _ = _ := by rw [Iso.inv_hom_id]; simp [bicategoricalComp]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7\u271d : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5\u271d : g \u226b f \u2245 \ud835\udfd9 b\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 \u03b7.hom \u25b7 f \u226b\n ((\u03b1_ f g f).hom \u226b \ud835\udfd9 f \u25b7 (g \u226b f)) \u226b\n f \u25c1 (g \u25c1 ((\u03c1_ f).inv \u226b (f \u25c1 \u03b5.inv \u226b (\ud835\udfd9 f \u25b7 (g \u226b f) \u226b (\u03b1_ f g f).inv) \u226b \u03b7.inv \u25b7 f) \u226b (\u03bb_ f).hom) \u226b \u03b5.hom) =\n \ud835\udfd9 (\ud835\udfd9 a \u226b f) \u2297\u226b (\u03b7.hom \u25b7 (f \u226b \ud835\udfd9 b) \u226b (f \u226b g) \u25c1 f \u25c1 \u03b5.inv) \u2297\u226b f \u25c1 g \u25c1 \u03b7.inv \u25b7 f \u2297\u226b f \u25c1 \u03b5.hom\n[PROOFSTEP]\nsimp [bicategoricalComp]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7\u271d : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5\u271d : g \u226b f \u2245 \ud835\udfd9 b\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 \u03b7.hom \u25b7 f \u226b\n (\u03b1_ f g f).hom \u226b\n (\u03c1_ (f \u226b g \u226b f)).inv \u226b\n (\u03b1_ f (g \u226b f) (\ud835\udfd9 b)).hom \u226b\n f \u25c1 (\u03b1_ g f (\ud835\udfd9 b)).hom \u226b\n f \u25c1 g \u25c1 f \u25c1 \u03b5.inv \u226b f \u25c1 g \u25c1 (\u03b1_ f g f).inv \u226b f \u25c1 g \u25c1 \u03b7.inv \u25b7 f \u226b f \u25c1 g \u25c1 (\u03bb_ f).hom \u226b f \u25c1 \u03b5.hom =\n (\u03bb_ f).hom \u226b\n (\u03c1_ f).inv \u226b\n (\u03bb_ (f \u226b \ud835\udfd9 b)).inv \u226b\n (\u03b1_ (\ud835\udfd9 a) f (\ud835\udfd9 b)).inv \u226b\n (\u03c1_ (\ud835\udfd9 a \u226b f)).hom \u226b\n \u03b7.hom \u25b7 f \u226b\n (\u03c1_ ((f \u226b g) \u226b f)).inv \u226b\n (\u03b1_ (f \u226b g) f (\ud835\udfd9 b)).hom \u226b\n (\u03b1_ f g (f \u226b \ud835\udfd9 b)).hom \u226b\n f \u25c1 g \u25c1 f \u25c1 \u03b5.inv \u226b f \u25c1 g \u25c1 (\u03b1_ f g f).inv \u226b f \u25c1 g \u25c1 \u03b7.inv \u25b7 f \u226b f \u25c1 g \u25c1 (\u03bb_ f).hom \u226b f \u25c1 \u03b5.hom\n[PROOFSTEP]\ncoherence\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7\u271d : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5\u271d : g \u226b f \u2245 \ud835\udfd9 b\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 \ud835\udfd9 (\ud835\udfd9 a \u226b f) \u2297\u226b (\u03b7.hom \u25b7 (f \u226b \ud835\udfd9 b) \u226b (f \u226b g) \u25c1 f \u25c1 \u03b5.inv) \u2297\u226b f \u25c1 g \u25c1 \u03b7.inv \u25b7 f \u2297\u226b f \u25c1 \u03b5.hom =\n \ud835\udfd9 (\ud835\udfd9 a \u226b f) \u2297\u226b f \u25c1 \u03b5.inv \u2297\u226b (\u03b7.hom \u25b7 (f \u226b g) \u226b (f \u226b g) \u25c1 \u03b7.inv) \u25b7 f \u2297\u226b f \u25c1 \u03b5.hom\n[PROOFSTEP]\nrw [\u2190 whisker_exchange \u03b7.hom (f \u25c1 \u03b5.inv)]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7\u271d : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5\u271d : g \u226b f \u2245 \ud835\udfd9 b\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 \ud835\udfd9 (\ud835\udfd9 a \u226b f) \u2297\u226b (\ud835\udfd9 a \u25c1 f \u25c1 \u03b5.inv \u226b \u03b7.hom \u25b7 (f \u226b g \u226b f)) \u2297\u226b f \u25c1 g \u25c1 \u03b7.inv \u25b7 f \u2297\u226b f \u25c1 \u03b5.hom =\n \ud835\udfd9 (\ud835\udfd9 a \u226b f) \u2297\u226b f \u25c1 \u03b5.inv \u2297\u226b (\u03b7.hom \u25b7 (f \u226b g) \u226b (f \u226b g) \u25c1 \u03b7.inv) \u25b7 f \u2297\u226b f \u25c1 \u03b5.hom\n[PROOFSTEP]\nsimp [bicategoricalComp]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7\u271d : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5\u271d : g \u226b f \u2245 \ud835\udfd9 b\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 f \u25c1 \u03b5.inv \u226b\n (\u03bb_ (f \u226b g \u226b f)).inv \u226b\n (\u03b1_ (\ud835\udfd9 a) f (g \u226b f)).inv \u226b\n (\u03b1_ (\ud835\udfd9 a \u226b f) g f).inv \u226b\n \u03b7.hom \u25b7 f \u25b7 g \u25b7 f \u226b\n (\u03b1_ ((f \u226b g) \u226b f) g f).hom \u226b\n (\u03b1_ (f \u226b g) f (g \u226b f)).hom \u226b\n (\u03b1_ f g (f \u226b g \u226b f)).hom \u226b\n f \u25c1 g \u25c1 (\u03b1_ f g f).inv \u226b f \u25c1 g \u25c1 \u03b7.inv \u25b7 f \u226b f \u25c1 g \u25c1 (\u03bb_ f).hom \u226b f \u25c1 \u03b5.hom =\n f \u25c1 \u03b5.inv \u226b\n (\u03b1_ f g f).inv \u226b\n (\u03bb_ ((f \u226b g) \u226b f)).inv \u226b\n (\u03b1_ (\ud835\udfd9 a) (f \u226b g) f).inv \u226b\n (\u03b1_ (\ud835\udfd9 a) f g).inv \u25b7 f \u226b\n \u03b7.hom \u25b7 f \u25b7 g \u25b7 f \u226b\n (\u03b1_ (f \u226b g) f g).hom \u25b7 f \u226b\n (\u03b1_ (f \u226b g) (f \u226b g) f).hom \u226b\n (\u03b1_ f g ((f \u226b g) \u226b f)).hom \u226b f \u25c1 g \u25c1 \u03b7.inv \u25b7 f \u226b f \u25c1 g \u25c1 (\u03bb_ f).hom \u226b f \u25c1 \u03b5.hom\n[PROOFSTEP]\ncoherence\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7\u271d : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5\u271d : g \u226b f \u2245 \ud835\udfd9 b\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 \ud835\udfd9 (\ud835\udfd9 a \u226b f) \u2297\u226b f \u25c1 \u03b5.inv \u2297\u226b (\u03b7.hom \u25b7 (f \u226b g) \u226b (f \u226b g) \u25c1 \u03b7.inv) \u25b7 f \u2297\u226b f \u25c1 \u03b5.hom =\n \ud835\udfd9 (\ud835\udfd9 a \u226b f) \u2297\u226b f \u25c1 \u03b5.inv \u2297\u226b (\u03b7.inv \u226b \u03b7.hom) \u25b7 f \u2297\u226b f \u25c1 \u03b5.hom\n[PROOFSTEP]\nrw [\u2190 whisker_exchange \u03b7.hom \u03b7.inv]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7\u271d : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5\u271d : g \u226b f \u2245 \ud835\udfd9 b\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 \ud835\udfd9 (\ud835\udfd9 a \u226b f) \u2297\u226b f \u25c1 \u03b5.inv \u2297\u226b (\ud835\udfd9 a \u25c1 \u03b7.inv \u226b \u03b7.hom \u25b7 \ud835\udfd9 a) \u25b7 f \u2297\u226b f \u25c1 \u03b5.hom =\n \ud835\udfd9 (\ud835\udfd9 a \u226b f) \u2297\u226b f \u25c1 \u03b5.inv \u2297\u226b (\u03b7.inv \u226b \u03b7.hom) \u25b7 f \u2297\u226b f \u25c1 \u03b5.hom\n[PROOFSTEP]\ncoherence\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7\u271d : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5\u271d : g \u226b f \u2245 \ud835\udfd9 b\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 \ud835\udfd9 (\ud835\udfd9 a \u226b f) \u2297\u226b f \u25c1 \u03b5.inv \u2297\u226b (\u03b7.inv \u226b \u03b7.hom) \u25b7 f \u2297\u226b f \u25c1 \u03b5.hom = \ud835\udfd9 (\ud835\udfd9 a \u226b f) \u2297\u226b f \u25c1 (\u03b5.inv \u226b \u03b5.hom)\n[PROOFSTEP]\nrw [Iso.inv_hom_id]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7\u271d : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5\u271d : g \u226b f \u2245 \ud835\udfd9 b\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 \ud835\udfd9 (\ud835\udfd9 a \u226b f) \u2297\u226b f \u25c1 \u03b5.inv \u2297\u226b \ud835\udfd9 (f \u226b g) \u25b7 f \u2297\u226b f \u25c1 \u03b5.hom = \ud835\udfd9 (\ud835\udfd9 a \u226b f) \u2297\u226b f \u25c1 (\u03b5.inv \u226b \u03b5.hom)\n[PROOFSTEP]\nsimp [bicategoricalComp]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7\u271d : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5\u271d : g \u226b f \u2245 \ud835\udfd9 b\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 \ud835\udfd9 (\ud835\udfd9 a \u226b f) \u2297\u226b f \u25c1 (\u03b5.inv \u226b \u03b5.hom) = (\u03bb_ f).hom \u226b (\u03c1_ f).inv\n[PROOFSTEP]\nrw [Iso.inv_hom_id]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 a\n\u03b7\u271d : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5\u271d : g \u226b f \u2245 \ud835\udfd9 b\n\u03b7 : \ud835\udfd9 a \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\n\u22a2 \ud835\udfd9 (\ud835\udfd9 a \u226b f) \u2297\u226b f \u25c1 \ud835\udfd9 (\ud835\udfd9 b) = (\u03bb_ f).hom \u226b (\u03c1_ f).inv\n[PROOFSTEP]\nsimp [bicategoricalComp]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na\u271d b c : B\nf : a\u271d \u27f6 b\ng : b \u27f6 a\u271d\n\u03b7 : \ud835\udfd9 a\u271d \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\na : B\n\u22a2 leftZigzagIso (\u03c1_ (\ud835\udfd9 a)).symm (\u03c1_ (\ud835\udfd9 a)) = \u03bb_ (\ud835\udfd9 a) \u226a\u226b (\u03c1_ (\ud835\udfd9 a)).symm\n[PROOFSTEP]\next\n[GOAL]\ncase w\nB : Type u\ninst\u271d : Bicategory B\na\u271d b c : B\nf : a\u271d \u27f6 b\ng : b \u27f6 a\u271d\n\u03b7 : \ud835\udfd9 a\u271d \u2245 f \u226b g\n\u03b5 : g \u226b f \u2245 \ud835\udfd9 b\na : B\n\u22a2 (leftZigzagIso (\u03c1_ (\ud835\udfd9 a)).symm (\u03c1_ (\ud835\udfd9 a))).hom = (\u03bb_ (\ud835\udfd9 a) \u226a\u226b (\u03c1_ (\ud835\udfd9 a)).symm).hom\n[PROOFSTEP]\nsimp [bicategoricalIsoComp]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Bicategory.Adjunction", "llama_tokens": 5562, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.822189121808099, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.4495221610392126}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nI : Finset \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\n\u22a2 \u2191(\u2211 i in I, \u03ba i) a = \u2211 i in I, \u2191(\u03ba i) a\n[PROOFSTEP]\nrw [coe_finset_sum, Finset.sum_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nI : Finset \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\ns : Set \u03b2\n\u22a2 \u2191\u2191(\u2191(\u2211 i in I, \u03ba i) a) s = \u2211 i in I, \u2191\u2191(\u2191(\u03ba i) a) s\n[PROOFSTEP]\nrw [finset_sum_apply, Measure.finset_sum_apply]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b9 : Type u_3\nm\u03b1\u271d : MeasurableSpace \u03b1\u271d\nm\u03b2\u271d : MeasurableSpace \u03b2\u271d\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nx\u271d : \u03b1\n\u22a2 \u2191\u2191(\u21910 x\u271d) Set.univ \u2264 0\n[PROOFSTEP]\nsimp only [kernel.zero_apply, Measure.coe_zero, Pi.zero_apply, le_zero_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsFiniteKernel \u03ba\ninst\u271d : IsFiniteKernel \u03b7\n\u22a2 IsFiniteKernel (\u03ba + \u03b7)\n[PROOFSTEP]\nrefine\n \u27e8\u27e8IsFiniteKernel.bound \u03ba + IsFiniteKernel.bound \u03b7,\n ENNReal.add_lt_top.mpr \u27e8IsFiniteKernel.bound_lt_top \u03ba, IsFiniteKernel.bound_lt_top \u03b7\u27e9, fun a => ?_\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsFiniteKernel \u03ba\ninst\u271d : IsFiniteKernel \u03b7\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(\u03ba + \u03b7) a) Set.univ \u2264 bound \u03ba + bound \u03b7\n[PROOFSTEP]\nexact add_le_add (kernel.measure_le_bound _ _ _) (kernel.measure_le_bound _ _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u22a2 \u03ba = \u03b7 \u2194 \u2200 (a : \u03b1) (s : Set \u03b2), MeasurableSet s \u2192 \u2191\u2191(\u2191\u03ba a) s = \u2191\u2191(\u2191\u03b7 a) s\n[PROOFSTEP]\nsimp_rw [ext_iff, Measure.ext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : \u2200 (a : \u03b1) (f : \u03b2 \u2192 \u211d\u22650\u221e), Measurable f \u2192 \u222b\u207b (b : \u03b2), f b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2), f b \u2202\u2191\u03b7 a\n\u22a2 \u03ba = \u03b7\n[PROOFSTEP]\next a s hs\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : \u2200 (a : \u03b1) (f : \u03b2 \u2192 \u211d\u22650\u221e), Measurable f \u2192 \u222b\u207b (b : \u03b2), f b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2), f b \u2202\u2191\u03b7 a\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191\u03ba a) s = \u2191\u2191(\u2191\u03b7 a) s\n[PROOFSTEP]\nspecialize h a (s.indicator fun _ => 1) (Measurable.indicator measurable_const hs)\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nh : \u222b\u207b (b : \u03b2), Set.indicator s (fun x => 1) b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2), Set.indicator s (fun x => 1) b \u2202\u2191\u03b7 a\n\u22a2 \u2191\u2191(\u2191\u03ba a) s = \u2191\u2191(\u2191\u03b7 a) s\n[PROOFSTEP]\nsimp_rw [lintegral_indicator_const hs, one_mul] at h \n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nh : \u2191\u2191(\u2191\u03ba a) s = \u2191\u2191(\u2191\u03b7 a) s\n\u22a2 \u2191\u2191(\u2191\u03ba a) s = \u2191\u2191(\u2191\u03b7 a) s\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : \u03ba = \u03b7\na : \u03b1\nf : \u03b2 \u2192 \u211d\u22650\u221e\nx\u271d : Measurable f\n\u22a2 \u222b\u207b (b : \u03b2), f b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2), f b \u2202\u2191\u03b7 a\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Integrable fun x => ENNReal.toReal (\u2191\u2191(\u2191\u03ba x) s)\n[PROOFSTEP]\nrefine'\n Integrable.mono' (integrable_const (IsFiniteKernel.bound \u03ba).toReal)\n ((kernel.measurable_coe \u03ba hs).ennreal_toReal.aestronglyMeasurable) (ae_of_all \u03bc <| fun x => _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\ns : Set \u03b2\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2016ENNReal.toReal (\u2191\u2191(\u2191\u03ba x) s)\u2016 \u2264 ENNReal.toReal (IsFiniteKernel.bound \u03ba)\n[PROOFSTEP]\nrw [Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg,\n ENNReal.toReal_le_toReal (measure_ne_top _ _) (IsFiniteKernel.bound_ne_top _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\ns : Set \u03b2\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191(\u2191\u03ba x) s \u2264 IsFiniteKernel.bound \u03ba\n[PROOFSTEP]\nexact kernel.measure_le_bound _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\n\u22a2 (fun a => Measure.sum fun n => \u2191(\u03ba n) a) \u2208 kernel \u03b1 \u03b2\n[PROOFSTEP]\nrefine' Measure.measurable_of_measurable_coe _ fun s hs => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Measurable fun b => \u2191\u2191(Measure.sum fun n => \u2191(\u03ba n) b) s\n[PROOFSTEP]\nsimp_rw [Measure.sum_apply _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Measurable fun b => \u2211' (i : \u03b9), \u2191\u2191(\u2191(\u03ba i) b) s\n[PROOFSTEP]\nexact Measurable.ennreal_tsum fun n => kernel.measurable_coe (\u03ba n) hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.sum \u03ba) a) s = \u2211' (n : \u03b9), \u2191\u2191(\u2191(\u03ba n) a) s\n[PROOFSTEP]\nrw [sum_apply \u03ba a, Measure.sum_apply _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u22a2 (kernel.sum fun x => 0) = 0\n[PROOFSTEP]\next a s hs\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.sum fun x => 0) a) s = \u2191\u2191(\u21910 a) s\n[PROOFSTEP]\nrw [sum_apply' _ a hs]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2211' (n : \u03b9), \u2191\u2191(\u21910 a) s = \u2191\u2191(\u21910 a) s\n[PROOFSTEP]\nsimp only [zero_apply, Measure.coe_zero, Pi.zero_apply, tsum_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\n\u22a2 (kernel.sum fun n => kernel.sum (\u03ba n)) = kernel.sum fun m => kernel.sum fun n => \u03ba n m\n[PROOFSTEP]\next a s\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\ns : Set \u03b2\na\u271d : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.sum fun n => kernel.sum (\u03ba n)) a) s = \u2191\u2191(\u2191(kernel.sum fun m => kernel.sum fun n => \u03ba n m) a) s\n[PROOFSTEP]\nsimp_rw [sum_apply]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03ba : \u03b9 \u2192 \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\ns : Set \u03b2\na\u271d : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.sum fun n => Measure.sum fun n_1 => \u2191(\u03ba n n_1) a) s =\n \u2191\u2191(Measure.sum fun n => Measure.sum fun n_1 => \u2191(\u03ba n_1 n) a) s\n[PROOFSTEP]\nrw [Measure.sum_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Fintype \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\n\u22a2 kernel.sum \u03ba = \u2211 i : \u03b9, \u03ba i\n[PROOFSTEP]\next a s hs\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Fintype \u03b9\n\u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.sum \u03ba) a) s = \u2191\u2191(\u2191(\u2211 i : \u03b9, \u03ba i) a) s\n[PROOFSTEP]\nsimp only [sum_apply' \u03ba a hs, finset_sum_apply' _ \u03ba a s, tsum_fintype]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03ba \u03b7 : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\n\u22a2 (kernel.sum fun n => \u03ba n + \u03b7 n) = kernel.sum \u03ba + kernel.sum \u03b7\n[PROOFSTEP]\next a s hs\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03ba \u03b7 : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.sum fun n => \u03ba n + \u03b7 n) a) s = \u2191\u2191(\u2191(kernel.sum \u03ba + kernel.sum \u03b7) a) s\n[PROOFSTEP]\nsimp only [coeFn_add, Pi.add_apply, sum_apply, Measure.sum_apply _ hs, Pi.add_apply, Measure.coe_add,\n tsum_add ENNReal.summable ENNReal.summable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : IsFiniteKernel \u03ba\nn : \u2115\n\u22a2 IsFiniteKernel ((fun n => if n = 0 then \u03ba else 0) n)\n[PROOFSTEP]\nsimp only\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : IsFiniteKernel \u03ba\nn : \u2115\n\u22a2 IsFiniteKernel (if n = 0 then \u03ba else 0)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : IsFiniteKernel \u03ba\nn : \u2115\nh\u271d : n = 0\n\u22a2 IsFiniteKernel \u03ba\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : IsFiniteKernel \u03ba\nn : \u2115\nh\u271d : \u00acn = 0\n\u22a2 IsFiniteKernel 0\n[PROOFSTEP]\nexact h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : IsFiniteKernel \u03ba\nn : \u2115\nh\u271d : \u00acn = 0\n\u22a2 IsFiniteKernel 0\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : IsFiniteKernel \u03ba\n\u22a2 \u03ba = kernel.sum fun n => if n = 0 then \u03ba else 0\n[PROOFSTEP]\next a s hs\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : IsFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191\u03ba a) s = \u2191\u2191(\u2191(kernel.sum fun n => if n = 0 then \u03ba else 0) a) s\n[PROOFSTEP]\nrw [kernel.sum_apply' _ _ hs]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : IsFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191\u03ba a) s = \u2211' (n : \u2115), \u2191\u2191(\u2191(if n = 0 then \u03ba else 0) a) s\n[PROOFSTEP]\nhave : (fun i => ((ite (i = 0) \u03ba 0) a) s) = fun i => ite (i = 0) (\u03ba a s) 0 := by ext1 i; split_ifs <;> rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : IsFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 (fun i => \u2191\u2191(\u2191(if i = 0 then \u03ba else 0) a) s) = fun i => if i = 0 then \u2191\u2191(\u2191\u03ba a) s else 0\n[PROOFSTEP]\next1 i\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : IsFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\ni : \u2115\n\u22a2 \u2191\u2191(\u2191(if i = 0 then \u03ba else 0) a) s = if i = 0 then \u2191\u2191(\u2191\u03ba a) s else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : IsFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\ni : \u2115\nh\u271d : i = 0\n\u22a2 \u2191\u2191(\u2191\u03ba a) s = \u2191\u2191(\u2191\u03ba a) s\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : IsFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\ni : \u2115\nh\u271d : \u00aci = 0\n\u22a2 \u2191\u2191(\u21910 a) s = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : IsFiniteKernel \u03ba\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\nthis : (fun i => \u2191\u2191(\u2191(if i = 0 then \u03ba else 0) a) s) = fun i => if i = 0 then \u2191\u2191(\u2191\u03ba a) s else 0\n\u22a2 \u2191\u2191(\u2191\u03ba a) s = \u2211' (n : \u2115), \u2191\u2191(\u2191(if n = 0 then \u03ba else 0) a) s\n[PROOFSTEP]\nrw [this, tsum_ite_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nh : IsSFiniteKernel \u03ba\na : \u03b1\n\u22a2 (Measure.sum fun n => \u2191(seq \u03ba n) a) = \u2191\u03ba a\n[PROOFSTEP]\nrw [\u2190 kernel.sum_apply, kernel_sum_seq \u03ba]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d \u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\n\u22a2 IsSFiniteKernel (\u03ba + \u03b7)\n[PROOFSTEP]\nrefine' \u27e8\u27e8fun n => seq \u03ba n + seq \u03b7 n, fun n => inferInstance, _\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d \u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\n\u22a2 \u03ba + \u03b7 = kernel.sum fun n => seq \u03ba n + seq \u03b7 n\n[PROOFSTEP]\nrw [sum_add, kernel_sum_seq \u03ba, kernel_sum_seq \u03b7]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI : Finset \u03b9\nh : \u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)\n\u22a2 IsSFiniteKernel (\u2211 i in I, \u03bas i)\n[PROOFSTEP]\nclassical\ninduction' I using Finset.induction with i I hi_nmem_I h_ind h\n\u00b7 rw [Finset.sum_empty]; infer_instance\n\u00b7 rw [Finset.sum_insert hi_nmem_I]\n haveI : IsSFiniteKernel (\u03bas i) := h i (Finset.mem_insert_self _ _)\n have : IsSFiniteKernel (\u2211 x : \u03b9 in I, \u03bas x) := h_ind fun i hiI => h i (Finset.mem_insert_of_mem hiI)\n exact IsSFiniteKernel.add _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI : Finset \u03b9\nh : \u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)\n\u22a2 IsSFiniteKernel (\u2211 i in I, \u03bas i)\n[PROOFSTEP]\ninduction' I using Finset.induction with i I hi_nmem_I h_ind h\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)\nh : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 IsSFiniteKernel (\u03bas i)\n\u22a2 IsSFiniteKernel (\u2211 i in \u2205, \u03bas i)\n[PROOFSTEP]\nrw [Finset.sum_empty]\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)\nh : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 IsSFiniteKernel (\u03bas i)\n\u22a2 IsSFiniteKernel 0\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i)\ni : \u03b9\nI : Finset \u03b9\nhi_nmem_I : \u00aci \u2208 I\nh_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1)\n\u22a2 IsSFiniteKernel (\u2211 i in insert i I, \u03bas i)\n[PROOFSTEP]\nrw [Finset.sum_insert hi_nmem_I]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i)\ni : \u03b9\nI : Finset \u03b9\nhi_nmem_I : \u00aci \u2208 I\nh_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1)\n\u22a2 IsSFiniteKernel (\u03bas i + \u2211 x in I, \u03bas x)\n[PROOFSTEP]\nhaveI : IsSFiniteKernel (\u03bas i) := h i (Finset.mem_insert_self _ _)\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i)\ni : \u03b9\nI : Finset \u03b9\nhi_nmem_I : \u00aci \u2208 I\nh_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1)\nthis : IsSFiniteKernel (\u03bas i)\n\u22a2 IsSFiniteKernel (\u03bas i + \u2211 x in I, \u03bas x)\n[PROOFSTEP]\nhave : IsSFiniteKernel (\u2211 x : \u03b9 in I, \u03bas x) := h_ind fun i hiI => h i (Finset.mem_insert_of_mem hiI)\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nI\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 IsSFiniteKernel (\u03bas i)\ni : \u03b9\nI : Finset \u03b9\nhi_nmem_I : \u00aci \u2208 I\nh_ind : (\u2200 (i : \u03b9), i \u2208 I \u2192 IsSFiniteKernel (\u03bas i)) \u2192 IsSFiniteKernel (\u2211 i in I, \u03bas i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsSFiniteKernel (\u03bas i_1)\nthis\u271d : IsSFiniteKernel (\u03bas i)\nthis : IsSFiniteKernel (\u2211 x in I, \u03bas x)\n\u22a2 IsSFiniteKernel (\u03bas i + \u2211 x in I, \u03bas x)\n[PROOFSTEP]\nexact IsSFiniteKernel.add _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Denumerable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\n\u22a2 IsSFiniteKernel (kernel.sum \u03bas)\n[PROOFSTEP]\nlet e : \u2115 \u2243 \u03b9 \u00d7 \u2115 := (Denumerable.eqv (\u03b9 \u00d7 \u2115)).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Denumerable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\ne : \u2115 \u2243 \u03b9 \u00d7 \u2115 := (Denumerable.eqv (\u03b9 \u00d7 \u2115)).symm\n\u22a2 IsSFiniteKernel (kernel.sum \u03bas)\n[PROOFSTEP]\nrefine' \u27e8\u27e8fun n => seq (\u03bas (e n).1) (e n).2, inferInstance, _\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Denumerable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\ne : \u2115 \u2243 \u03b9 \u00d7 \u2115 := (Denumerable.eqv (\u03b9 \u00d7 \u2115)).symm\n\u22a2 kernel.sum \u03bas = kernel.sum fun n => seq (\u03bas (\u2191e n).fst) (\u2191e n).snd\n[PROOFSTEP]\nhave h\u03ba_eq : kernel.sum \u03bas = kernel.sum fun n => kernel.sum (seq (\u03bas n)) := by simp_rw [kernel_sum_seq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Denumerable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\ne : \u2115 \u2243 \u03b9 \u00d7 \u2115 := (Denumerable.eqv (\u03b9 \u00d7 \u2115)).symm\n\u22a2 kernel.sum \u03bas = kernel.sum fun n => kernel.sum (seq (\u03bas n))\n[PROOFSTEP]\nsimp_rw [kernel_sum_seq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Denumerable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\ne : \u2115 \u2243 \u03b9 \u00d7 \u2115 := (Denumerable.eqv (\u03b9 \u00d7 \u2115)).symm\nh\u03ba_eq : kernel.sum \u03bas = kernel.sum fun n => kernel.sum (seq (\u03bas n))\n\u22a2 kernel.sum \u03bas = kernel.sum fun n => seq (\u03bas (\u2191e n).fst) (\u2191e n).snd\n[PROOFSTEP]\next a s hs\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Denumerable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\ne : \u2115 \u2243 \u03b9 \u00d7 \u2115 := (Denumerable.eqv (\u03b9 \u00d7 \u2115)).symm\nh\u03ba_eq : kernel.sum \u03bas = kernel.sum fun n => kernel.sum (seq (\u03bas n))\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.sum \u03bas) a) s = \u2191\u2191(\u2191(kernel.sum fun n => seq (\u03bas (\u2191e n).fst) (\u2191e n).snd) a) s\n[PROOFSTEP]\nrw [h\u03ba_eq]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Denumerable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\ne : \u2115 \u2243 \u03b9 \u00d7 \u2115 := (Denumerable.eqv (\u03b9 \u00d7 \u2115)).symm\nh\u03ba_eq : kernel.sum \u03bas = kernel.sum fun n => kernel.sum (seq (\u03bas n))\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(kernel.sum fun n => kernel.sum (seq (\u03bas n))) a) s = \u2191\u2191(\u2191(kernel.sum fun n => seq (\u03bas (\u2191e n).fst) (\u2191e n).snd) a) s\n[PROOFSTEP]\nsimp_rw [kernel.sum_apply' _ _ hs]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Denumerable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\ne : \u2115 \u2243 \u03b9 \u00d7 \u2115 := (Denumerable.eqv (\u03b9 \u00d7 \u2115)).symm\nh\u03ba_eq : kernel.sum \u03bas = kernel.sum fun n => kernel.sum (seq (\u03bas n))\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2211' (n : \u03b9) (n_1 : \u2115), \u2191\u2191(\u2191(seq (\u03bas n) n_1) a) s =\n \u2211' (n : \u2115), \u2191\u2191(\u2191(seq (\u03bas (\u2191(Denumerable.eqv (\u03b9 \u00d7 \u2115)).symm n).fst) (\u2191(Denumerable.eqv (\u03b9 \u00d7 \u2115)).symm n).snd) a) s\n[PROOFSTEP]\nchange (\u2211' i, \u2211' m, seq (\u03bas i) m a s) = \u2211' n, (fun im : \u03b9 \u00d7 \u2115 => seq (\u03bas im.fst) im.snd a s) (e n)\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Denumerable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\ne : \u2115 \u2243 \u03b9 \u00d7 \u2115 := (Denumerable.eqv (\u03b9 \u00d7 \u2115)).symm\nh\u03ba_eq : kernel.sum \u03bas = kernel.sum fun n => kernel.sum (seq (\u03bas n))\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2211' (i : \u03b9) (m : \u2115), \u2191\u2191(\u2191(seq (\u03bas i) m) a) s = \u2211' (n : \u2115), (fun im => \u2191\u2191(\u2191(seq (\u03bas im.fst) im.snd) a) s) (\u2191e n)\n[PROOFSTEP]\nrw [e.tsum_eq (fun im : \u03b9 \u00d7 \u2115 => seq (\u03bas im.fst) im.snd a s), tsum_prod' ENNReal.summable fun _ => ENNReal.summable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\n\u22a2 IsSFiniteKernel (kernel.sum \u03bas)\n[PROOFSTEP]\ncases fintypeOrInfinite \u03b9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\nval\u271d : Fintype \u03b9\n\u22a2 IsSFiniteKernel (kernel.sum \u03bas)\n[PROOFSTEP]\nrw [sum_fintype]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\nval\u271d : Fintype \u03b9\n\u22a2 IsSFiniteKernel (\u2211 i : \u03b9, \u03bas i)\n[PROOFSTEP]\nexact IsSFiniteKernel.finset_sum Finset.univ fun i _ => h\u03bas i\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\nval\u271d : Infinite \u03b9\n\u22a2 IsSFiniteKernel (kernel.sum \u03bas)\n[PROOFSTEP]\ncases nonempty_denumerable \u03b9\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : Countable \u03b9\n\u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 }\nh\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n)\nval\u271d\u00b9 : Infinite \u03b9\nval\u271d : Denumerable \u03b9\n\u22a2 IsSFiniteKernel (kernel.sum \u03bas)\n[PROOFSTEP]\nexact isSFiniteKernel_sum_of_denumerable h\u03bas\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\n\u22a2 (fun a => Measure.dirac (f a)) \u2208 kernel \u03b1 \u03b2\n[PROOFSTEP]\nrefine' Measure.measurable_of_measurable_coe _ fun s hs => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Measurable fun b => \u2191\u2191(Measure.dirac (f b)) s\n[PROOFSTEP]\nsimp_rw [Measure.dirac_apply' _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Measurable fun b => Set.indicator s 1 (f b)\n[PROOFSTEP]\nexact measurable_one.indicator (hf hs)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(deterministic f hf) a) s = Set.indicator s (fun x => 1) (f a)\n[PROOFSTEP]\nrw [deterministic]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191{ val := fun a => Measure.dirac (f a), property := (_ : Measurable fun a => Measure.dirac (f a)) } a) s =\n Set.indicator s (fun x => 1) (f a)\n[PROOFSTEP]\nchange Measure.dirac (f a) s = s.indicator 1 (f a)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.dirac (f a)) s = Set.indicator s 1 (f a)\n[PROOFSTEP]\nsimp_rw [Measure.dirac_apply' _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\na : \u03b1\n\u22a2 IsProbabilityMeasure (\u2191(deterministic f hf) a)\n[PROOFSTEP]\nrw [deterministic_apply hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\na : \u03b1\n\u22a2 IsProbabilityMeasure (Measure.dirac (f a))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\na : \u03b1\nhg : Measurable g\nhf : Measurable f\n\u22a2 \u222b\u207b (x : \u03b2), f x \u2202\u2191(deterministic g hg) a = f (g a)\n[PROOFSTEP]\nrw [kernel.deterministic_apply, lintegral_dirac' _ hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\na : \u03b1\nhg : Measurable g\ninst\u271d : MeasurableSingletonClass \u03b2\n\u22a2 \u222b\u207b (x : \u03b2), f x \u2202\u2191(deterministic g hg) a = f (g a)\n[PROOFSTEP]\nrw [kernel.deterministic_apply, lintegral_dirac (g a) f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\na : \u03b1\nhg : Measurable g\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\ninst\u271d : Decidable (g a \u2208 s)\n\u22a2 \u222b\u207b (x : \u03b2) in s, f x \u2202\u2191(deterministic g hg) a = if g a \u2208 s then f (g a) else 0\n[PROOFSTEP]\nrw [kernel.deterministic_apply, set_lintegral_dirac' hf hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\na : \u03b1\nhg : Measurable g\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\ns : Set \u03b2\ninst\u271d : Decidable (g a \u2208 s)\n\u22a2 \u222b\u207b (x : \u03b2) in s, f x \u2202\u2191(deterministic g hg) a = if g a \u2208 s then f (g a) else 0\n[PROOFSTEP]\nrw [kernel.deterministic_apply, set_lintegral_dirac f s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nE : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b2 \u2192 E\ng : \u03b1 \u2192 \u03b2\na : \u03b1\nhg : Measurable g\nhf : StronglyMeasurable f\n\u22a2 \u222b (x : \u03b2), f x \u2202\u2191(deterministic g hg) a = f (g a)\n[PROOFSTEP]\nrw [kernel.deterministic_apply, integral_dirac' _ _ hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nE : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u03b2 \u2192 E\ng : \u03b1 \u2192 \u03b2\na : \u03b1\nhg : Measurable g\ninst\u271d : MeasurableSingletonClass \u03b2\n\u22a2 \u222b (x : \u03b2), f x \u2202\u2191(deterministic g hg) a = f (g a)\n[PROOFSTEP]\nrw [kernel.deterministic_apply, integral_dirac _ (g a)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nE : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u03b2 \u2192 E\ng : \u03b1 \u2192 \u03b2\na : \u03b1\nhg : Measurable g\nhf : StronglyMeasurable f\ns : Set \u03b2\nhs : MeasurableSet s\ninst\u271d : Decidable (g a \u2208 s)\n\u22a2 \u222b (x : \u03b2) in s, f x \u2202\u2191(deterministic g hg) a = if g a \u2208 s then f (g a) else 0\n[PROOFSTEP]\nrw [kernel.deterministic_apply, set_integral_dirac' hf _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nE : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf : \u03b2 \u2192 E\ng : \u03b1 \u2192 \u03b2\na : \u03b1\nhg : Measurable g\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\ns : Set \u03b2\ninst\u271d : Decidable (g a \u2208 s)\n\u22a2 \u222b (x : \u03b2) in s, f x \u2202\u2191(deterministic g hg) a = if g a \u2208 s then f (g a) else 0\n[PROOFSTEP]\nrw [kernel.deterministic_apply, set_integral_dirac f _ s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b2 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b2\na : \u03b1\n\u22a2 \u222b\u207b (x : \u03b2), f x \u2202\u2191(const \u03b1 \u03bc) a = \u222b\u207b (x : \u03b2), f x \u2202\u03bc\n[PROOFSTEP]\nrw [kernel.const_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nf : \u03b2 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b2\na : \u03b1\ns : Set \u03b2\n\u22a2 \u222b\u207b (x : \u03b2) in s, f x \u2202\u2191(const \u03b1 \u03bc) a = \u222b\u207b (x : \u03b2) in s, f x \u2202\u03bc\n[PROOFSTEP]\nrw [kernel.const_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\na : \u03b1\n\u22a2 \u222b (x : \u03b2), f x \u2202\u2191(const \u03b1 \u03bc) a = \u222b (x : \u03b2), f x \u2202\u03bc\n[PROOFSTEP]\nrw [kernel.const_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\na : \u03b1\ns : Set \u03b2\n\u22a2 \u222b (x : \u03b2) in s, f x \u2202\u2191(const \u03b1 \u03bc) a = \u222b (x : \u03b2) in s, f x \u2202\u03bc\n[PROOFSTEP]\nrw [kernel.const_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ns t : Set \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhs : MeasurableSet s\n\u22a2 (fun a => Measure.restrict (\u2191\u03ba a) s) \u2208 kernel \u03b1 \u03b2\n[PROOFSTEP]\nrefine' Measure.measurable_of_measurable_coe _ fun t ht => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ns t\u271d : Set \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhs : MeasurableSet s\nt : Set \u03b2\nht : MeasurableSet t\n\u22a2 Measurable fun b => \u2191\u2191(Measure.restrict (\u2191\u03ba b) s) t\n[PROOFSTEP]\nsimp_rw [Measure.restrict_apply ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ns t\u271d : Set \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhs : MeasurableSet s\nt : Set \u03b2\nht : MeasurableSet t\n\u22a2 Measurable fun b => \u2191\u2191(\u2191\u03ba b) (t \u2229 s)\n[PROOFSTEP]\nexact kernel.measurable_coe \u03ba (ht.inter hs)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ns t : Set \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhs : MeasurableSet s\na : \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(\u2191(kernel.restrict \u03ba hs) a) t = \u2191\u2191(\u2191\u03ba a) (t \u2229 s)\n[PROOFSTEP]\nrw [restrict_apply \u03ba hs a, Measure.restrict_apply ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ns t : Set \u03b2\n\u22a2 kernel.restrict \u03ba (_ : MeasurableSet Set.univ) = \u03ba\n[PROOFSTEP]\next1 a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ns t : Set \u03b2\na : \u03b1\n\u22a2 \u2191(kernel.restrict \u03ba (_ : MeasurableSet Set.univ)) a = \u2191\u03ba a\n[PROOFSTEP]\nrw [kernel.restrict_apply, Measure.restrict_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ns t : Set \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhs : MeasurableSet s\na : \u03b1\nf : \u03b2 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (b : \u03b2), f b \u2202\u2191(kernel.restrict \u03ba hs) a = \u222b\u207b (b : \u03b2) in s, f b \u2202\u2191\u03ba a\n[PROOFSTEP]\nrw [restrict_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ns t\u271d : Set \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhs : MeasurableSet s\na : \u03b1\nf : \u03b2 \u2192 \u211d\u22650\u221e\nt : Set \u03b2\n\u22a2 \u222b\u207b (b : \u03b2) in t, f b \u2202\u2191(kernel.restrict \u03ba hs) a = \u222b\u207b (b : \u03b2) in t \u2229 s, f b \u2202\u2191\u03ba a\n[PROOFSTEP]\nrw [restrict_apply, Measure.restrict_restrict' hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ns t\u271d : Set \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b2 \u2192 E\na : \u03b1\nhs : MeasurableSet s\nt : Set \u03b2\n\u22a2 \u222b (x : \u03b2) in t, f x \u2202\u2191(kernel.restrict \u03ba hs) a = \u222b (x : \u03b2) in t \u2229 s, f x \u2202\u2191\u03ba a\n[PROOFSTEP]\nrw [restrict_apply, Measure.restrict_restrict' hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ns t : Set \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhs : MeasurableSet s\n\u22a2 IsFiniteKernel (kernel.restrict \u03ba hs)\n[PROOFSTEP]\nrefine' \u27e8\u27e8IsFiniteKernel.bound \u03ba, IsFiniteKernel.bound_lt_top \u03ba, fun a => _\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ns t : Set \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhs : MeasurableSet s\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(kernel.restrict \u03ba hs) a) Set.univ \u2264 IsFiniteKernel.bound \u03ba\n[PROOFSTEP]\nrw [restrict_apply' \u03ba hs a MeasurableSet.univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ns t : Set \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhs : MeasurableSet s\na : \u03b1\n\u22a2 \u2191\u2191(\u2191\u03ba a) (Set.univ \u2229 s) \u2264 IsFiniteKernel.bound \u03ba\n[PROOFSTEP]\nexact measure_le_bound \u03ba a _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ns t : Set \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhs : MeasurableSet s\n\u22a2 IsSFiniteKernel (kernel.restrict \u03ba hs)\n[PROOFSTEP]\nrefine' \u27e8\u27e8fun n => kernel.restrict (seq \u03ba n) hs, inferInstance, _\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ns t : Set \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhs : MeasurableSet s\n\u22a2 kernel.restrict \u03ba hs = kernel.sum fun n => kernel.restrict (seq \u03ba n) hs\n[PROOFSTEP]\next1 a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\ns t : Set \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhs : MeasurableSet s\na : \u03b1\n\u22a2 \u2191(kernel.restrict \u03ba hs) a = \u2191(kernel.sum fun n => kernel.restrict (seq \u03ba n) hs) a\n[PROOFSTEP]\nsimp_rw [sum_apply, restrict_apply, \u2190 Measure.restrict_sum _ hs, \u2190 sum_apply, kernel_sum_seq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhf : MeasurableEmbedding f\n\u22a2 (fun a => Measure.comap f (\u2191\u03ba a)) \u2208 kernel \u03b1 \u03b3\n[PROOFSTEP]\nrefine' Measure.measurable_measure.mpr fun t ht => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhf : MeasurableEmbedding f\nt : Set \u03b3\nht : MeasurableSet t\n\u22a2 Measurable fun b => \u2191\u2191(Measure.comap f (\u2191\u03ba b)) t\n[PROOFSTEP]\nhave : (fun a => Measure.comap f (\u03ba a) t) = fun a => \u03ba a (f '' t) :=\n by\n ext1 a\n rw [Measure.comap_apply _ hf.injective _ _ ht]\n exact fun s' hs' \u21a6 hf.measurableSet_image.mpr hs'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhf : MeasurableEmbedding f\nt : Set \u03b3\nht : MeasurableSet t\n\u22a2 (fun a => \u2191\u2191(Measure.comap f (\u2191\u03ba a)) t) = fun a => \u2191\u2191(\u2191\u03ba a) (f '' t)\n[PROOFSTEP]\next1 a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhf : MeasurableEmbedding f\nt : Set \u03b3\nht : MeasurableSet t\na : \u03b1\n\u22a2 \u2191\u2191(Measure.comap f (\u2191\u03ba a)) t = \u2191\u2191(\u2191\u03ba a) (f '' t)\n[PROOFSTEP]\nrw [Measure.comap_apply _ hf.injective _ _ ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhf : MeasurableEmbedding f\nt : Set \u03b3\nht : MeasurableSet t\na : \u03b1\n\u22a2 \u2200 (s : Set \u03b3), MeasurableSet s \u2192 MeasurableSet (f '' s)\n[PROOFSTEP]\nexact fun s' hs' \u21a6 hf.measurableSet_image.mpr hs'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhf : MeasurableEmbedding f\nt : Set \u03b3\nht : MeasurableSet t\nthis : (fun a => \u2191\u2191(Measure.comap f (\u2191\u03ba a)) t) = fun a => \u2191\u2191(\u2191\u03ba a) (f '' t)\n\u22a2 Measurable fun b => \u2191\u2191(Measure.comap f (\u2191\u03ba b)) t\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhf : MeasurableEmbedding f\nt : Set \u03b3\nht : MeasurableSet t\nthis : (fun a => \u2191\u2191(Measure.comap f (\u2191\u03ba a)) t) = fun a => \u2191\u2191(\u2191\u03ba a) (f '' t)\n\u22a2 Measurable fun a => \u2191\u2191(\u2191\u03ba a) (f '' t)\n[PROOFSTEP]\nexact kernel.measurable_coe _ (hf.measurableSet_image.mpr ht)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhf : MeasurableEmbedding f\na : \u03b1\nt : Set \u03b3\nht : MeasurableSet t\n\u22a2 \u2191\u2191(\u2191(comapRight \u03ba hf) a) t = \u2191\u2191(\u2191\u03ba a) (f '' t)\n[PROOFSTEP]\nrw [comapRight_apply, Measure.comap_apply _ hf.injective (fun s => hf.measurableSet_image.mpr) _ ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhf : MeasurableEmbedding f\nh\u03ba : \u2200 (a : \u03b1), \u2191\u2191(\u2191\u03ba a) (Set.range f) = 1\n\u22a2 IsMarkovKernel (kernel.comapRight \u03ba hf)\n[PROOFSTEP]\nrefine' \u27e8fun a => \u27e8_\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhf : MeasurableEmbedding f\nh\u03ba : \u2200 (a : \u03b1), \u2191\u2191(\u2191\u03ba a) (Set.range f) = 1\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(kernel.comapRight \u03ba hf) a) Set.univ = 1\n[PROOFSTEP]\nrw [comapRight_apply' \u03ba hf a MeasurableSet.univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhf : MeasurableEmbedding f\nh\u03ba : \u2200 (a : \u03b1), \u2191\u2191(\u2191\u03ba a) (Set.range f) = 1\na : \u03b1\n\u22a2 \u2191\u2191(\u2191\u03ba a) (f '' Set.univ) = 1\n[PROOFSTEP]\nsimp only [Set.image_univ, Subtype.range_coe_subtype, Set.setOf_mem_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nhf : MeasurableEmbedding f\nh\u03ba : \u2200 (a : \u03b1), \u2191\u2191(\u2191\u03ba a) (Set.range f) = 1\na : \u03b1\n\u22a2 \u2191\u2191(\u2191\u03ba a) (Set.range f) = 1\n[PROOFSTEP]\nexact h\u03ba a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf : MeasurableEmbedding f\n\u22a2 IsFiniteKernel (kernel.comapRight \u03ba hf)\n[PROOFSTEP]\nrefine' \u27e8\u27e8IsFiniteKernel.bound \u03ba, IsFiniteKernel.bound_lt_top \u03ba, fun a => _\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf : MeasurableEmbedding f\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(kernel.comapRight \u03ba hf) a) Set.univ \u2264 IsFiniteKernel.bound \u03ba\n[PROOFSTEP]\nrw [comapRight_apply' \u03ba hf a .univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsFiniteKernel \u03ba\nhf : MeasurableEmbedding f\na : \u03b1\n\u22a2 \u2191\u2191(\u2191\u03ba a) (f '' Set.univ) \u2264 IsFiniteKernel.bound \u03ba\n[PROOFSTEP]\nexact measure_le_bound \u03ba a _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf : MeasurableEmbedding f\n\u22a2 IsSFiniteKernel (comapRight \u03ba hf)\n[PROOFSTEP]\nrefine' \u27e8\u27e8fun n => comapRight (seq \u03ba n) hf, inferInstance, _\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf : MeasurableEmbedding f\n\u22a2 comapRight \u03ba hf = kernel.sum fun n => comapRight (seq \u03ba n) hf\n[PROOFSTEP]\next1 a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf : MeasurableEmbedding f\na : \u03b1\n\u22a2 \u2191(comapRight \u03ba hf) a = \u2191(kernel.sum fun n => comapRight (seq \u03ba n) hf) a\n[PROOFSTEP]\nrw [sum_apply]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf : MeasurableEmbedding f\na : \u03b1\n\u22a2 \u2191(comapRight \u03ba hf) a = Measure.sum fun n => \u2191(comapRight (seq \u03ba n) hf) a\n[PROOFSTEP]\nsimp_rw [comapRight_apply _ hf]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf : MeasurableEmbedding f\na : \u03b1\n\u22a2 Measure.comap f (\u2191\u03ba a) = Measure.sum fun n => Measure.comap f (\u2191(seq \u03ba n) a)\n[PROOFSTEP]\nhave : (Measure.sum fun n => Measure.comap f (seq \u03ba n a)) = Measure.comap f (Measure.sum fun n => seq \u03ba n a) :=\n by\n ext1 t ht\n rw [Measure.comap_apply _ hf.injective (fun s' => hf.measurableSet_image.mpr) _ ht, Measure.sum_apply _ ht,\n Measure.sum_apply _ (hf.measurableSet_image.mpr ht)]\n congr with n : 1\n rw [Measure.comap_apply _ hf.injective (fun s' => hf.measurableSet_image.mpr) _ ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf : MeasurableEmbedding f\na : \u03b1\n\u22a2 (Measure.sum fun n => Measure.comap f (\u2191(seq \u03ba n) a)) = Measure.comap f (Measure.sum fun n => \u2191(seq \u03ba n) a)\n[PROOFSTEP]\next1 t ht\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf : MeasurableEmbedding f\na : \u03b1\nt : Set \u03b3\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.sum fun n => Measure.comap f (\u2191(seq \u03ba n) a)) t = \u2191\u2191(Measure.comap f (Measure.sum fun n => \u2191(seq \u03ba n) a)) t\n[PROOFSTEP]\nrw [Measure.comap_apply _ hf.injective (fun s' => hf.measurableSet_image.mpr) _ ht, Measure.sum_apply _ ht,\n Measure.sum_apply _ (hf.measurableSet_image.mpr ht)]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf : MeasurableEmbedding f\na : \u03b1\nt : Set \u03b3\nht : MeasurableSet t\n\u22a2 \u2211' (i : \u2115), \u2191\u2191(Measure.comap f (\u2191(seq \u03ba i) a)) t = \u2211' (i : \u2115), \u2191\u2191(\u2191(seq \u03ba i) a) (f '' t)\n[PROOFSTEP]\ncongr with n : 1\n[GOAL]\ncase h.e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf : MeasurableEmbedding f\na : \u03b1\nt : Set \u03b3\nht : MeasurableSet t\nn : \u2115\n\u22a2 \u2191\u2191(Measure.comap f (\u2191(seq \u03ba n) a)) t = \u2191\u2191(\u2191(seq \u03ba n) a) (f '' t)\n[PROOFSTEP]\nrw [Measure.comap_apply _ hf.injective (fun s' => hf.measurableSet_image.mpr) _ ht]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d : IsSFiniteKernel \u03ba\nhf : MeasurableEmbedding f\na : \u03b1\nthis : (Measure.sum fun n => Measure.comap f (\u2191(seq \u03ba n) a)) = Measure.comap f (Measure.sum fun n => \u2191(seq \u03ba n) a)\n\u22a2 Measure.comap f (\u2191\u03ba a) = Measure.sum fun n => Measure.comap f (\u2191(seq \u03ba n) a)\n[PROOFSTEP]\nrw [this, measure_sum_seq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\nt : Set \u03b2\n\u22a2 \u2191\u2191(\u2191(piecewise hs \u03ba \u03b7) a) t = if a \u2208 s then \u2191\u2191(\u2191\u03ba a) t else \u2191\u2191(\u2191\u03b7 a) t\n[PROOFSTEP]\nrw [piecewise_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\nt : Set \u03b2\n\u22a2 \u2191\u2191(if a \u2208 s then \u2191\u03ba a else \u2191\u03b7 a) t = if a \u2208 s then \u2191\u2191(\u2191\u03ba a) t else \u2191\u2191(\u2191\u03b7 a) t\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\nt : Set \u03b2\nh\u271d : a \u2208 s\n\u22a2 \u2191\u2191(\u2191\u03ba a) t = \u2191\u2191(\u2191\u03ba a) t\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\nt : Set \u03b2\nh\u271d : \u00aca \u2208 s\n\u22a2 \u2191\u2191(\u2191\u03b7 a) t = \u2191\u2191(\u2191\u03b7 a) t\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\ninst\u271d\u00b9 : IsMarkovKernel \u03ba\ninst\u271d : IsMarkovKernel \u03b7\n\u22a2 IsMarkovKernel (kernel.piecewise hs \u03ba \u03b7)\n[PROOFSTEP]\nrefine' \u27e8fun a => \u27e8_\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\ninst\u271d\u00b9 : IsMarkovKernel \u03ba\ninst\u271d : IsMarkovKernel \u03b7\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(kernel.piecewise hs \u03ba \u03b7) a) Set.univ = 1\n[PROOFSTEP]\nrw [piecewise_apply', measure_univ, measure_univ, ite_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\ninst\u271d\u00b9 : IsFiniteKernel \u03ba\ninst\u271d : IsFiniteKernel \u03b7\n\u22a2 IsFiniteKernel (kernel.piecewise hs \u03ba \u03b7)\n[PROOFSTEP]\nrefine' \u27e8\u27e8max (IsFiniteKernel.bound \u03ba) (IsFiniteKernel.bound \u03b7), _, fun a => _\u27e9\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\ninst\u271d\u00b9 : IsFiniteKernel \u03ba\ninst\u271d : IsFiniteKernel \u03b7\n\u22a2 max (IsFiniteKernel.bound \u03ba) (IsFiniteKernel.bound \u03b7) < \u22a4\n[PROOFSTEP]\nexact max_lt (IsFiniteKernel.bound_lt_top \u03ba) (IsFiniteKernel.bound_lt_top \u03b7)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\ninst\u271d\u00b9 : IsFiniteKernel \u03ba\ninst\u271d : IsFiniteKernel \u03b7\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(kernel.piecewise hs \u03ba \u03b7) a) Set.univ \u2264 max (IsFiniteKernel.bound \u03ba) (IsFiniteKernel.bound \u03b7)\n[PROOFSTEP]\nrw [piecewise_apply']\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\ninst\u271d\u00b9 : IsFiniteKernel \u03ba\ninst\u271d : IsFiniteKernel \u03b7\na : \u03b1\n\u22a2 (if a \u2208 s then \u2191\u2191(\u2191\u03ba a) Set.univ else \u2191\u2191(\u2191\u03b7 a) Set.univ) \u2264 max (IsFiniteKernel.bound \u03ba) (IsFiniteKernel.bound \u03b7)\n[PROOFSTEP]\nexact (ite_le_sup _ _ _).trans (sup_le_sup (measure_le_bound _ _ _) (measure_le_bound _ _ _))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\n\u22a2 IsSFiniteKernel (piecewise hs \u03ba \u03b7)\n[PROOFSTEP]\nrefine' \u27e8\u27e8fun n => piecewise hs (seq \u03ba n) (seq \u03b7 n), inferInstance, _\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\n\u22a2 piecewise hs \u03ba \u03b7 = kernel.sum fun n => piecewise hs (seq \u03ba n) (seq \u03b7 n)\n[PROOFSTEP]\next1 a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\n\u22a2 \u2191(piecewise hs \u03ba \u03b7) a = \u2191(kernel.sum fun n => piecewise hs (seq \u03ba n) (seq \u03b7 n)) a\n[PROOFSTEP]\nsimp_rw [sum_apply, kernel.piecewise_apply]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\n\u22a2 (if a \u2208 s then \u2191\u03ba a else \u2191\u03b7 a) = Measure.sum fun n => if a \u2208 s then \u2191(seq \u03ba n) a else \u2191(seq \u03b7 n) a\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nh\u271d : a \u2208 s\n\u22a2 \u2191\u03ba a = Measure.sum fun n => \u2191(seq \u03ba n) a\n[PROOFSTEP]\nexact (measure_sum_seq _ a).symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nh\u271d : \u00aca \u2208 s\n\u22a2 \u2191\u03b7 a = Measure.sum fun n => \u2191(seq \u03b7 n) a\n[PROOFSTEP]\nexact (measure_sum_seq _ a).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\ng : \u03b2 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (b : \u03b2), g b \u2202\u2191(piecewise hs \u03ba \u03b7) a = if a \u2208 s then \u222b\u207b (b : \u03b2), g b \u2202\u2191\u03ba a else \u222b\u207b (b : \u03b2), g b \u2202\u2191\u03b7 a\n[PROOFSTEP]\nsimp_rw [piecewise_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\ng : \u03b2 \u2192 \u211d\u22650\u221e\n\u22a2 (\u222b\u207b (b : \u03b2), g b \u2202if a \u2208 s then \u2191\u03ba a else \u2191\u03b7 a) = if a \u2208 s then \u222b\u207b (b : \u03b2), g b \u2202\u2191\u03ba a else \u222b\u207b (b : \u03b2), g b \u2202\u2191\u03b7 a\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\ng : \u03b2 \u2192 \u211d\u22650\u221e\nh\u271d : a \u2208 s\n\u22a2 \u222b\u207b (b : \u03b2), g b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2), g b \u2202\u2191\u03ba a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\ng : \u03b2 \u2192 \u211d\u22650\u221e\nh\u271d : \u00aca \u2208 s\n\u22a2 \u222b\u207b (b : \u03b2), g b \u2202\u2191\u03b7 a = \u222b\u207b (b : \u03b2), g b \u2202\u2191\u03b7 a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\ng : \u03b2 \u2192 \u211d\u22650\u221e\nt : Set \u03b2\n\u22a2 \u222b\u207b (b : \u03b2) in t, g b \u2202\u2191(piecewise hs \u03ba \u03b7) a = if a \u2208 s then \u222b\u207b (b : \u03b2) in t, g b \u2202\u2191\u03ba a else \u222b\u207b (b : \u03b2) in t, g b \u2202\u2191\u03b7 a\n[PROOFSTEP]\nsimp_rw [piecewise_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\ng : \u03b2 \u2192 \u211d\u22650\u221e\nt : Set \u03b2\n\u22a2 (\u222b\u207b (b : \u03b2) in t, g b \u2202if a \u2208 s then \u2191\u03ba a else \u2191\u03b7 a) =\n if a \u2208 s then \u222b\u207b (b : \u03b2) in t, g b \u2202\u2191\u03ba a else \u222b\u207b (b : \u03b2) in t, g b \u2202\u2191\u03b7 a\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\ng : \u03b2 \u2192 \u211d\u22650\u221e\nt : Set \u03b2\nh\u271d : a \u2208 s\n\u22a2 \u222b\u207b (b : \u03b2) in t, g b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2) in t, g b \u2202\u2191\u03ba a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\ng : \u03b2 \u2192 \u211d\u22650\u221e\nt : Set \u03b2\nh\u271d : \u00aca \u2208 s\n\u22a2 \u222b\u207b (b : \u03b2) in t, g b \u2202\u2191\u03b7 a = \u222b\u207b (b : \u03b2) in t, g b \u2202\u2191\u03b7 a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\na : \u03b1\ng : \u03b2 \u2192 E\n\u22a2 \u222b (b : \u03b2), g b \u2202\u2191(piecewise hs \u03ba \u03b7) a = if a \u2208 s then \u222b (b : \u03b2), g b \u2202\u2191\u03ba a else \u222b (b : \u03b2), g b \u2202\u2191\u03b7 a\n[PROOFSTEP]\nsimp_rw [piecewise_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\na : \u03b1\ng : \u03b2 \u2192 E\n\u22a2 (\u222b (b : \u03b2), g b \u2202if a \u2208 s then \u2191\u03ba a else \u2191\u03b7 a) = if a \u2208 s then \u222b (b : \u03b2), g b \u2202\u2191\u03ba a else \u222b (b : \u03b2), g b \u2202\u2191\u03b7 a\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\na : \u03b1\ng : \u03b2 \u2192 E\nh\u271d : a \u2208 s\n\u22a2 \u222b (b : \u03b2), g b \u2202\u2191\u03ba a = \u222b (b : \u03b2), g b \u2202\u2191\u03ba a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\na : \u03b1\ng : \u03b2 \u2192 E\nh\u271d : \u00aca \u2208 s\n\u22a2 \u222b (b : \u03b2), g b \u2202\u2191\u03b7 a = \u222b (b : \u03b2), g b \u2202\u2191\u03b7 a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\na : \u03b1\ng : \u03b2 \u2192 E\nt : Set \u03b2\n\u22a2 \u222b (b : \u03b2) in t, g b \u2202\u2191(piecewise hs \u03ba \u03b7) a = if a \u2208 s then \u222b (b : \u03b2) in t, g b \u2202\u2191\u03ba a else \u222b (b : \u03b2) in t, g b \u2202\u2191\u03b7 a\n[PROOFSTEP]\nsimp_rw [piecewise_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\na : \u03b1\ng : \u03b2 \u2192 E\nt : Set \u03b2\n\u22a2 (\u222b (b : \u03b2) in t, g b \u2202if a \u2208 s then \u2191\u03ba a else \u2191\u03b7 a) =\n if a \u2208 s then \u222b (b : \u03b2) in t, g b \u2202\u2191\u03ba a else \u222b (b : \u03b2) in t, g b \u2202\u2191\u03b7 a\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\na : \u03b1\ng : \u03b2 \u2192 E\nt : Set \u03b2\nh\u271d : a \u2208 s\n\u22a2 \u222b (b : \u03b2) in t, g b \u2202\u2191\u03ba a = \u222b (b : \u03b2) in t, g b \u2202\u2191\u03ba a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d\u00b2 : DecidablePred fun x => x \u2208 s\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\na : \u03b1\ng : \u03b2 \u2192 E\nt : Set \u03b2\nh\u271d : \u00aca \u2208 s\n\u22a2 \u222b (b : \u03b2) in t, g b \u2202\u2191\u03b7 a = \u222b (b : \u03b2) in t, g b \u2202\u2191\u03b7 a\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Probability.Kernel.Basic", "llama_tokens": 28761, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.449255793491485}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : LipschitzMul \u03b2\n\u22a2 \u2200 (p q : \u03b2 \u00d7 \u03b2), dist (p.fst * p.snd) (q.fst * q.snd) \u2264 \u2191(LipschitzMul.C \u03b2) * dist p q\n[PROOFSTEP]\nrw [\u2190 lipschitzWith_iff_dist_le_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : LipschitzMul \u03b2\n\u22a2 LipschitzWith (LipschitzMul.C \u03b2) fun p => p.fst * p.snd\n[PROOFSTEP]\nexact lipschitzWith_lipschitz_const_mul_edist\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : LipschitzMul \u03b2\ns : Submonoid \u03b2\n\u22a2 LipschitzWith (LipschitzMul.C \u03b2) fun p => p.fst * p.snd\n[PROOFSTEP]\nrintro \u27e8x\u2081, x\u2082\u27e9 \u27e8y\u2081, y\u2082\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : LipschitzMul \u03b2\ns : Submonoid \u03b2\nx\u2081 x\u2082 y\u2081 y\u2082 : { x // x \u2208 s }\n\u22a2 edist ((fun p => p.fst * p.snd) (x\u2081, x\u2082)) ((fun p => p.fst * p.snd) (y\u2081, y\u2082)) \u2264\n \u2191(LipschitzMul.C \u03b2) * edist (x\u2081, x\u2082) (y\u2081, y\u2082)\n[PROOFSTEP]\nconvert lipschitzWith_lipschitz_const_mul_edist \u27e8(x\u2081 : \u03b2), x\u2082\u27e9 \u27e8y\u2081, y\u2082\u27e9 using 1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : LipschitzMul \u03b2\np q : \u211d \u00d7 \u211d\n\u22a2 dist (p.fst + p.snd) (q.fst + q.snd) \u2264 \u21912 * dist p q\n[PROOFSTEP]\nsimp only [Real.dist_eq, Prod.dist_eq, Prod.fst_sub, Prod.snd_sub, NNReal.coe_ofNat, add_sub_add_comm, two_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : LipschitzMul \u03b2\np q : \u211d \u00d7 \u211d\n\u22a2 |p.fst - q.fst + (p.snd - q.snd)| \u2264 max |p.fst - q.fst| |p.snd - q.snd| + max |p.fst - q.fst| |p.snd - q.snd|\n[PROOFSTEP]\nrefine le_trans (abs_add (p.1 - q.1) (p.2 - q.2)) ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : LipschitzMul \u03b2\np q : \u211d \u00d7 \u211d\n\u22a2 |p.fst - q.fst| + |p.snd - q.snd| \u2264 max |p.fst - q.fst| |p.snd - q.snd| + max |p.fst - q.fst| |p.snd - q.snd|\n[PROOFSTEP]\nexact add_le_add (le_max_left _ _) (le_max_right _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : LipschitzMul \u03b2\n\u22a2 LipschitzWith (LipschitzAdd.C \u211d) fun p => p.fst + p.snd\n[PROOFSTEP]\nrintro \u27e8x\u2081, x\u2082\u27e9 \u27e8y\u2081, y\u2082\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : LipschitzMul \u03b2\nx\u2081 x\u2082 y\u2081 y\u2082 : \u211d\u22650\n\u22a2 edist ((fun p => p.fst + p.snd) (x\u2081, x\u2082)) ((fun p => p.fst + p.snd) (y\u2081, y\u2082)) \u2264\n \u2191(LipschitzAdd.C \u211d) * edist (x\u2081, x\u2082) (y\u2081, y\u2082)\n[PROOFSTEP]\nexact lipschitzWith_lipschitz_const_add_edist \u27e8(x\u2081 : \u211d), x\u2082\u27e9 \u27e8y\u2081, y\u2082\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\n\u22a2 Continuous fun p => p.fst \u2022 p.snd\n[PROOFSTEP]\nrw [Metric.continuous_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\n\u22a2 \u2200 (b : \u03b1 \u00d7 \u03b2) (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (a : \u03b1 \u00d7 \u03b2), dist a b < \u03b4 \u2192 dist (a.fst \u2022 a.snd) (b.fst \u2022 b.snd) < \u03b5\n[PROOFSTEP]\nrintro \u27e8a, b\u27e9 \u03b5 \u03b50\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (a_1 : \u03b1 \u00d7 \u03b2), dist a_1 (a, b) < \u03b4 \u2192 dist (a_1.fst \u2022 a_1.snd) ((a, b).fst \u2022 (a, b).snd) < \u03b5\n[PROOFSTEP]\nobtain \u27e8\u03b4, \u03b40, h\u03b4\u03b5\u27e9 : \u2203 \u03b4 > 0, \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 < \u03b5\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 < \u03b5\n[PROOFSTEP]\nhave : Continuous fun \u03b4 \u21a6 \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 := by continuity\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u22a2 Continuous fun \u03b4 => \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4\n[PROOFSTEP]\ncontinuity\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nthis : Continuous fun \u03b4 => \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 < \u03b5\n[PROOFSTEP]\nrefine ((this.tendsto' _ _ ?_).eventually (gt_mem_nhds \u03b50)).exists_gt\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nthis : Continuous fun \u03b4 => \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4\n\u22a2 0 * (0 + dist b 0) + dist a 0 * 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4\u03b5 : \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 < \u03b5\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (a_1 : \u03b1 \u00d7 \u03b2), dist a_1 (a, b) < \u03b4 \u2192 dist (a_1.fst \u2022 a_1.snd) ((a, b).fst \u2022 (a, b).snd) < \u03b5\n[PROOFSTEP]\nrefine \u27e8\u03b4, \u03b40, fun (a', b') hab' => ?_\u27e9\n[GOAL]\ncase mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4\u03b5 : \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 < \u03b5\nx\u271d : \u03b1 \u00d7 \u03b2\na' : \u03b1\nb' : \u03b2\nhab' : dist (a', b') (a, b) < \u03b4\n\u22a2 dist ((a', b').fst \u2022 (a', b').snd) ((a, b).fst \u2022 (a, b).snd) < \u03b5\n[PROOFSTEP]\nobtain \u27e8ha, hb\u27e9 := max_lt_iff.1 hab'\n[GOAL]\ncase mk.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4\u03b5 : \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 < \u03b5\nx\u271d : \u03b1 \u00d7 \u03b2\na' : \u03b1\nb' : \u03b2\nhab' : dist (a', b') (a, b) < \u03b4\nha : dist (a', b').fst (a, b).fst < \u03b4\nhb : dist (a', b').snd (a, b).snd < \u03b4\n\u22a2 dist ((a', b').fst \u2022 (a', b').snd) ((a, b).fst \u2022 (a, b).snd) < \u03b5\n[PROOFSTEP]\ncalc\n dist (a' \u2022 b') (a \u2022 b) \u2264 dist (a' \u2022 b') (a \u2022 b') + dist (a \u2022 b') (a \u2022 b) := dist_triangle ..\n _ \u2264 dist a' a * dist b' 0 + dist a 0 * dist b' b := (add_le_add (dist_pair_smul _ _ _) (dist_smul_pair _ _ _))\n _ \u2264 \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 :=\n by\n have : dist b' 0 \u2264 \u03b4 + dist b 0 := (dist_triangle _ _ _).trans <| add_le_add_right hb.le _\n mono* <;> apply_rules [dist_nonneg, le_of_lt]\n _ < \u03b5 := h\u03b4\u03b5\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4\u03b5 : \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 < \u03b5\nx\u271d : \u03b1 \u00d7 \u03b2\na' : \u03b1\nb' : \u03b2\nhab' : dist (a', b') (a, b) < \u03b4\nha : dist (a', b').fst (a, b).fst < \u03b4\nhb : dist (a', b').snd (a, b).snd < \u03b4\n\u22a2 dist a' a * dist b' 0 + dist a 0 * dist b' b \u2264 \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4\n[PROOFSTEP]\nhave : dist b' 0 \u2264 \u03b4 + dist b 0 := (dist_triangle _ _ _).trans <| add_le_add_right hb.le _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4\u03b5 : \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 < \u03b5\nx\u271d : \u03b1 \u00d7 \u03b2\na' : \u03b1\nb' : \u03b2\nhab' : dist (a', b') (a, b) < \u03b4\nha : dist (a', b').fst (a, b).fst < \u03b4\nhb : dist (a', b').snd (a, b).snd < \u03b4\nthis : dist b' 0 \u2264 \u03b4 + dist b 0\n\u22a2 dist a' a * dist b' 0 + dist a 0 * dist b' b \u2264 \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4\n[PROOFSTEP]\nmono*\n[GOAL]\ncase h\u2081.h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4\u03b5 : \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 < \u03b5\nx\u271d : \u03b1 \u00d7 \u03b2\na' : \u03b1\nb' : \u03b2\nhab' : dist (a', b') (a, b) < \u03b4\nha : dist (a', b').fst (a, b).fst < \u03b4\nhb : dist (a', b').snd (a, b).snd < \u03b4\nthis : dist b' 0 \u2264 \u03b4 + dist b 0\n\u22a2 dist a' a \u2264 \u03b4\n[PROOFSTEP]\napply_rules [dist_nonneg, le_of_lt]\n[GOAL]\ncase h\u2081.c0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4\u03b5 : \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 < \u03b5\nx\u271d : \u03b1 \u00d7 \u03b2\na' : \u03b1\nb' : \u03b2\nhab' : dist (a', b') (a, b) < \u03b4\nha : dist (a', b').fst (a, b).fst < \u03b4\nhb : dist (a', b').snd (a, b).snd < \u03b4\nthis : dist b' 0 \u2264 \u03b4 + dist b 0\n\u22a2 0 \u2264 dist b' 0\n[PROOFSTEP]\napply_rules [dist_nonneg, le_of_lt]\n[GOAL]\ncase h\u2081.b0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4\u03b5 : \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 < \u03b5\nx\u271d : \u03b1 \u00d7 \u03b2\na' : \u03b1\nb' : \u03b2\nhab' : dist (a', b') (a, b) < \u03b4\nha : dist (a', b').fst (a, b).fst < \u03b4\nhb : dist (a', b').snd (a, b).snd < \u03b4\nthis : dist b' 0 \u2264 \u03b4 + dist b 0\n\u22a2 0 \u2264 \u03b4\n[PROOFSTEP]\napply_rules [dist_nonneg, le_of_lt]\n[GOAL]\ncase h\u2082.h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4\u03b5 : \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 < \u03b5\nx\u271d : \u03b1 \u00d7 \u03b2\na' : \u03b1\nb' : \u03b2\nhab' : dist (a', b') (a, b) < \u03b4\nha : dist (a', b').fst (a, b).fst < \u03b4\nhb : dist (a', b').snd (a, b).snd < \u03b4\nthis : dist b' 0 \u2264 \u03b4 + dist b 0\n\u22a2 dist b' b \u2264 \u03b4\n[PROOFSTEP]\napply_rules [dist_nonneg, le_of_lt]\n[GOAL]\ncase h\u2082.c0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4\u03b5 : \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 < \u03b5\nx\u271d : \u03b1 \u00d7 \u03b2\na' : \u03b1\nb' : \u03b2\nhab' : dist (a', b') (a, b) < \u03b4\nha : dist (a', b').fst (a, b).fst < \u03b4\nhb : dist (a', b').snd (a, b).snd < \u03b4\nthis : dist b' 0 \u2264 \u03b4 + dist b 0\n\u22a2 0 \u2264 dist b' b\n[PROOFSTEP]\napply_rules [dist_nonneg, le_of_lt]\n[GOAL]\ncase h\u2082.b0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4\u03b5 : \u03b4 * (\u03b4 + dist b 0) + dist a 0 * \u03b4 < \u03b5\nx\u271d : \u03b1 \u00d7 \u03b2\na' : \u03b1\nb' : \u03b2\nhab' : dist (a', b') (a, b) < \u03b4\nha : dist (a', b').fst (a, b).fst < \u03b4\nhb : dist (a', b').snd (a, b).snd < \u03b4\nthis : dist b' 0 \u2264 \u03b4 + dist b 0\n\u22a2 0 \u2264 dist a 0\n[PROOFSTEP]\napply_rules [dist_nonneg, le_of_lt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\nx y\u2081 y\u2082 : \u211d\n\u22a2 dist (x \u2022 y\u2081) (x \u2022 y\u2082) \u2264 dist x 0 * dist y\u2081 y\u2082\n[PROOFSTEP]\nsimpa [Real.dist_eq, mul_sub] using (abs_mul x (y\u2081 - y\u2082)).le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\nx\u2081 x\u2082 y : \u211d\n\u22a2 dist (x\u2081 \u2022 y) (x\u2082 \u2022 y) \u2264 dist x\u2081 x\u2082 * dist y 0\n[PROOFSTEP]\nsimpa [Real.dist_eq, sub_mul] using (abs_mul (x\u2081 - x\u2082) y).le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\nx y\u2081 y\u2082 : \u211d\u22650\n\u22a2 dist (x \u2022 y\u2081) (x \u2022 y\u2082) \u2264 dist x 0 * dist y\u2081 y\u2082\n[PROOFSTEP]\nconvert dist_smul_pair (x : \u211d) (y\u2081 : \u211d) y\u2082 using 1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PseudoMetricSpace \u03b1\ninst\u271d\u2074 : PseudoMetricSpace \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : BoundedSMul \u03b1 \u03b2\nx\u2081 x\u2082 y : \u211d\u22650\n\u22a2 dist (x\u2081 \u2022 y) (x\u2082 \u2022 y) \u2264 dist x\u2081 x\u2082 * dist y 0\n[PROOFSTEP]\nconvert dist_pair_smul (x\u2081 : \u211d) x\u2082 (y : \u211d) using 1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : PseudoMetricSpace \u03b1\ninst\u271d\u2076 : PseudoMetricSpace \u03b2\ninst\u271d\u2075 : Zero \u03b1\ninst\u271d\u2074 : Zero \u03b2\ninst\u271d\u00b3 : SMul \u03b1 \u03b2\ninst\u271d\u00b2 : BoundedSMul \u03b1 \u03b2\ninst\u271d\u00b9 : SMul \u03b1\u1d50\u1d52\u1d56 \u03b2\ninst\u271d : IsCentralScalar \u03b1 \u03b2\nx : \u03b1\ny\u2081 y\u2082 : \u03b2\n\u22a2 dist (MulOpposite.op x \u2022 y\u2081) (MulOpposite.op x \u2022 y\u2082) \u2264 dist (MulOpposite.op x) 0 * dist y\u2081 y\u2082\n[PROOFSTEP]\nsimpa only [op_smul_eq_smul] using dist_smul_pair x y\u2081 y\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : PseudoMetricSpace \u03b1\ninst\u271d\u2076 : PseudoMetricSpace \u03b2\ninst\u271d\u2075 : Zero \u03b1\ninst\u271d\u2074 : Zero \u03b2\ninst\u271d\u00b3 : SMul \u03b1 \u03b2\ninst\u271d\u00b2 : BoundedSMul \u03b1 \u03b2\ninst\u271d\u00b9 : SMul \u03b1\u1d50\u1d52\u1d56 \u03b2\ninst\u271d : IsCentralScalar \u03b1 \u03b2\nx\u2081 x\u2082 : \u03b1\ny : \u03b2\n\u22a2 dist (MulOpposite.op x\u2081 \u2022 y) (MulOpposite.op x\u2082 \u2022 y) \u2264 dist (MulOpposite.op x\u2081) (MulOpposite.op x\u2082) * dist y 0\n[PROOFSTEP]\nsimpa only [op_smul_eq_smul] using dist_pair_smul x\u2081 x\u2082 y\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.Algebra", "llama_tokens": 7361, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.4491194134876732}} {"text": "[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 \u2191(bind\u2081 (frobeniusPolyRat p)) (wittPolynomial p \u211a n) = wittPolynomial p \u211a (n + 1)\n[PROOFSTEP]\ndelta frobeniusPolyRat\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 \u2191(bind\u2081 fun n => \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a n)) (wittPolynomial p \u211a n) =\n wittPolynomial p \u211a (n + 1)\n[PROOFSTEP]\nrw [\u2190 bind\u2081_bind\u2081, bind\u2081_xInTermsOfW_wittPolynomial, bind\u2081_X_right, Function.comp_apply]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 frobeniusPolyAux p n =\n X (n + 1) -\n \u2211 i in range n,\n \u2211 j in range (p ^ (n - i)),\n (X i ^ p) ^ (p ^ (n - i) - (j + 1)) * frobeniusPolyAux p i ^ (j + 1) *\n \u2191C\n \u2191(Nat.choose (p ^ (n - i)) (j + 1) /\n p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }))\n[PROOFSTEP]\nrw [frobeniusPolyAux, \u2190 Fin.sum_univ_eq_sum_range]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn j : \u2115\nhj : j < p ^ n\n\u22a2 p ^ (n - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) \u2223 Nat.choose (p ^ n) (j + 1)\n[PROOFSTEP]\napply multiplicity.pow_dvd_of_le_multiplicity\n[GOAL]\ncase a\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn j : \u2115\nhj : j < p ^ n\n\u22a2 \u2191(n - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) \u2264 multiplicity p (Nat.choose (p ^ n) (j + 1))\n[PROOFSTEP]\nrw [hp.out.multiplicity_choose_prime_pow hj j.succ_ne_zero]\n[GOAL]\ncase a\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn j : \u2115\nhj : j < p ^ n\n\u22a2 \u2191(n - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) \u2264\n \u2191(n - Part.get (multiplicity p (Nat.succ j)) (_ : multiplicity.Finite p (Nat.succ j)))\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn i j : \u2115\nhi : i \u2264 n\nhj : j < p ^ (n - i)\n\u22a2 j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) } + n =\n i + j + (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })\n[PROOFSTEP]\ngeneralize h : v p \u27e8j + 1, j.succ_pos\u27e9 = m\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn i j : \u2115\nhi : i \u2264 n\nhj : j < p ^ (n - i)\nm : \u2115\nh : v p { val := j + 1, property := (_ : 0 < Nat.succ j) } = m\n\u22a2 j - m + n = i + j + (n - i - m)\n[PROOFSTEP]\nrsuffices \u27e8h\u2081, h\u2082\u27e9 : m \u2264 n - i \u2227 m \u2264 j\n[GOAL]\ncase intro\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn i j : \u2115\nhi : i \u2264 n\nhj : j < p ^ (n - i)\nm : \u2115\nh : v p { val := j + 1, property := (_ : 0 < Nat.succ j) } = m\nh\u2081 : m \u2264 n - i\nh\u2082 : m \u2264 j\n\u22a2 j - m + n = i + j + (n - i - m)\n[PROOFSTEP]\nrw [tsub_add_eq_add_tsub h\u2082, add_comm i j, add_tsub_assoc_of_le (h\u2081.trans (Nat.sub_le n i)), add_assoc, tsub_right_comm,\n add_comm i, tsub_add_cancel_of_le (le_tsub_of_add_le_right ((le_tsub_iff_left hi).mp h\u2081))]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn i j : \u2115\nhi : i \u2264 n\nhj : j < p ^ (n - i)\nm : \u2115\nh : v p { val := j + 1, property := (_ : 0 < Nat.succ j) } = m\n\u22a2 m \u2264 n - i \u2227 m \u2264 j\n[PROOFSTEP]\nhave hle : p ^ m \u2264 j + 1 := h \u25b8 Nat.le_of_dvd j.succ_pos (multiplicity.pow_multiplicity_dvd _)\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn i j : \u2115\nhi : i \u2264 n\nhj : j < p ^ (n - i)\nm : \u2115\nh : v p { val := j + 1, property := (_ : 0 < Nat.succ j) } = m\nhle : p ^ m \u2264 j + 1\n\u22a2 m \u2264 n - i \u2227 m \u2264 j\n[PROOFSTEP]\nexact \u27e8(pow_le_pow_iff hp.1.one_lt).1 (hle.trans hj), Nat.le_of_lt_succ ((Nat.lt_pow_self hp.1.one_lt m).trans_le hle)\u27e9\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPoly p n) = frobeniusPolyRat p n\n[PROOFSTEP]\nrw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast, Int.cast_ofNat,\n frobeniusPolyRat]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 X n ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p n) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a n)\n[PROOFSTEP]\nrefine Nat.strong_induction_on n ?_\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 \u2200 (n : \u2115),\n (\u2200 (m : \u2115),\n m < n \u2192\n X m ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p m) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a m)) \u2192\n X n ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p n) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a n)\n[PROOFSTEP]\nclear n\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2200 (n : \u2115),\n (\u2200 (m : \u2115),\n m < n \u2192\n X m ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p m) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a m)) \u2192\n X n ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p n) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a n)\n[PROOFSTEP]\nintro n IH\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nIH :\n \u2200 (m : \u2115),\n m < n \u2192\n X m ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p m) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a m)\n\u22a2 X n ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p n) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a n)\n[PROOFSTEP]\nrw [xInTermsOfW_eq]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nIH :\n \u2200 (m : \u2115),\n m < n \u2192\n X m ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p m) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a m)\n\u22a2 X n ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p n) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1))\n ((X n - \u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)) * \u2191C (\u215f\u2191p ^ n))\n[PROOFSTEP]\nsimp only [AlgHom.map_sum, AlgHom.map_sub, AlgHom.map_mul, AlgHom.map_pow, bind\u2081_C_right]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nIH :\n \u2200 (m : \u2115),\n m < n \u2192\n X m ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p m) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a m)\n\u22a2 X n ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p n) =\n (\u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (X n) -\n \u2211 x in range n,\n \u2191C (\u2191p ^ x) * \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a x) ^ p ^ (n - x)) *\n \u2191C (\u215f\u2191p ^ n)\n[PROOFSTEP]\nhave h1 : (p : \u211a) ^ n * \u215f(p : \u211a) ^ n = 1 := by rw [\u2190 mul_pow, mul_invOf_self, one_pow]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nIH :\n \u2200 (m : \u2115),\n m < n \u2192\n X m ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p m) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a m)\n\u22a2 \u2191p ^ n * \u215f\u2191p ^ n = 1\n[PROOFSTEP]\nrw [\u2190 mul_pow, mul_invOf_self, one_pow]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nIH :\n \u2200 (m : \u2115),\n m < n \u2192\n X m ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p m) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a m)\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\n\u22a2 X n ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p n) =\n (\u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (X n) -\n \u2211 x in range n,\n \u2191C (\u2191p ^ x) * \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a x) ^ p ^ (n - x)) *\n \u2191C (\u215f\u2191p ^ n)\n[PROOFSTEP]\nrw [bind\u2081_X_right, Function.comp_apply, wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ, sum_range_succ, tsub_self,\n add_tsub_cancel_left, pow_zero, pow_one, pow_one, sub_mul, add_mul, add_mul, mul_right_comm,\n mul_right_comm (C ((p : \u211a) ^ (n + 1))), \u2190 C_mul, \u2190 C_mul, pow_succ, mul_assoc (p : \u211a) ((p : \u211a) ^ n), h1, mul_one, C_1,\n one_mul, add_comm _ (X n ^ p), add_assoc, \u2190 add_sub, add_right_inj, frobeniusPolyAux_eq, RingHom.map_sub, map_X,\n mul_sub, sub_eq_add_neg, add_comm _ (C (p : \u211a) * X (n + 1)), \u2190 add_sub,\n show (Int.castRingHom \u211a) \u2191p = (p : \u211a) from rfl, add_right_inj, neg_eq_iff_eq_neg, neg_sub, eq_comm]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nIH :\n \u2200 (m : \u2115),\n m < n \u2192\n X m ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p m) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a m)\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\n\u22a2 (\u2211 x in range n, \u2191C (\u2191p ^ x) * \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a x) ^ p ^ (n - x)) *\n \u2191C (\u215f\u2191p ^ n) -\n (\u2211 x in range n, \u2191C (\u2191p ^ x) * X x ^ p ^ (n + 1 - x)) * \u2191C (\u215f\u2191p ^ n) =\n \u2191C \u2191p *\n \u2191(MvPolynomial.map (Int.castRingHom \u211a))\n (\u2211 i in range n,\n \u2211 j in range (p ^ (n - i)),\n (X i ^ p) ^ (p ^ (n - i) - (j + 1)) * frobeniusPolyAux p i ^ (j + 1) *\n \u2191C\n \u2191(Nat.choose (p ^ (n - i)) (j + 1) /\n p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })))\n[PROOFSTEP]\nsimp only [map_sum, mul_sum, sum_mul, \u2190 sum_sub_distrib]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nIH :\n \u2200 (m : \u2115),\n m < n \u2192\n X m ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p m) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a m)\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\n\u22a2 \u2211 x in range n,\n (\u2191C (\u2191p ^ x) * \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a x) ^ p ^ (n - x) * \u2191C (\u215f\u2191p ^ n) -\n \u2191C (\u2191p ^ x) * X x ^ p ^ (n + 1 - x) * \u2191C (\u215f\u2191p ^ n)) =\n \u2211 x in range n,\n \u2211 x_1 in range (p ^ (n - x)),\n \u2191C \u2191p *\n \u2191(MvPolynomial.map (Int.castRingHom \u211a))\n ((X x ^ p) ^ (p ^ (n - x) - (x_1 + 1)) * frobeniusPolyAux p x ^ (x_1 + 1) *\n \u2191C\n \u2191(Nat.choose (p ^ (n - x)) (x_1 + 1) /\n p ^ (n - x - v p { val := x_1 + 1, property := (_ : 0 < Nat.succ x_1) }) *\n p ^ (x_1 - v p { val := x_1 + 1, property := (_ : 0 < Nat.succ x_1) })))\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nIH :\n \u2200 (m : \u2115),\n m < n \u2192\n X m ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p m) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a m)\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\n\u22a2 \u2200 (x : \u2115),\n x \u2208 range n \u2192\n \u2191C (\u2191p ^ x) * \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a x) ^ p ^ (n - x) * \u2191C (\u215f\u2191p ^ n) -\n \u2191C (\u2191p ^ x) * X x ^ p ^ (n + 1 - x) * \u2191C (\u215f\u2191p ^ n) =\n \u2211 x_1 in range (p ^ (n - x)),\n \u2191C \u2191p *\n \u2191(MvPolynomial.map (Int.castRingHom \u211a))\n ((X x ^ p) ^ (p ^ (n - x) - (x_1 + 1)) * frobeniusPolyAux p x ^ (x_1 + 1) *\n \u2191C\n \u2191(Nat.choose (p ^ (n - x)) (x_1 + 1) /\n p ^ (n - x - v p { val := x_1 + 1, property := (_ : 0 < Nat.succ x_1) }) *\n p ^ (x_1 - v p { val := x_1 + 1, property := (_ : 0 < Nat.succ x_1) })))\n[PROOFSTEP]\nintro i hi\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nIH :\n \u2200 (m : \u2115),\n m < n \u2192\n X m ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p m) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a m)\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i \u2208 range n\n\u22a2 \u2191C (\u2191p ^ i) * \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a i) ^ p ^ (n - i) * \u2191C (\u215f\u2191p ^ n) -\n \u2191C (\u2191p ^ i) * X i ^ p ^ (n + 1 - i) * \u2191C (\u215f\u2191p ^ n) =\n \u2211 x in range (p ^ (n - i)),\n \u2191C \u2191p *\n \u2191(MvPolynomial.map (Int.castRingHom \u211a))\n ((X i ^ p) ^ (p ^ (n - i) - (x + 1)) * frobeniusPolyAux p i ^ (x + 1) *\n \u2191C\n \u2191(Nat.choose (p ^ (n - i)) (x + 1) /\n p ^ (n - i - v p { val := x + 1, property := (_ : 0 < Nat.succ x) }) *\n p ^ (x - v p { val := x + 1, property := (_ : 0 < Nat.succ x) })))\n[PROOFSTEP]\nrw [mem_range] at hi \n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nIH :\n \u2200 (m : \u2115),\n m < n \u2192\n X m ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p m) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a m)\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\n\u22a2 \u2191C (\u2191p ^ i) * \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a i) ^ p ^ (n - i) * \u2191C (\u215f\u2191p ^ n) -\n \u2191C (\u2191p ^ i) * X i ^ p ^ (n + 1 - i) * \u2191C (\u215f\u2191p ^ n) =\n \u2211 x in range (p ^ (n - i)),\n \u2191C \u2191p *\n \u2191(MvPolynomial.map (Int.castRingHom \u211a))\n ((X i ^ p) ^ (p ^ (n - i) - (x + 1)) * frobeniusPolyAux p i ^ (x + 1) *\n \u2191C\n \u2191(Nat.choose (p ^ (n - i)) (x + 1) /\n p ^ (n - i - v p { val := x + 1, property := (_ : 0 < Nat.succ x) }) *\n p ^ (x - v p { val := x + 1, property := (_ : 0 < Nat.succ x) })))\n[PROOFSTEP]\nrw [\u2190 IH i hi]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nIH :\n \u2200 (m : \u2115),\n m < n \u2192\n X m ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p m) =\n \u2191(bind\u2081 (wittPolynomial p \u211a \u2218 fun n => n + 1)) (xInTermsOfW p \u211a m)\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\n\u22a2 \u2191C (\u2191p ^ i) *\n (X i ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p i)) ^\n p ^ (n - i) *\n \u2191C (\u215f\u2191p ^ n) -\n \u2191C (\u2191p ^ i) * X i ^ p ^ (n + 1 - i) * \u2191C (\u215f\u2191p ^ n) =\n \u2211 x in range (p ^ (n - i)),\n \u2191C \u2191p *\n \u2191(MvPolynomial.map (Int.castRingHom \u211a))\n ((X i ^ p) ^ (p ^ (n - i) - (x + 1)) * frobeniusPolyAux p i ^ (x + 1) *\n \u2191C\n \u2191(Nat.choose (p ^ (n - i)) (x + 1) /\n p ^ (n - i - v p { val := x + 1, property := (_ : 0 < Nat.succ x) }) *\n p ^ (x - v p { val := x + 1, property := (_ : 0 < Nat.succ x) })))\n[PROOFSTEP]\nclear IH\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\n\u22a2 \u2191C (\u2191p ^ i) *\n (X i ^ p + \u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p i)) ^\n p ^ (n - i) *\n \u2191C (\u215f\u2191p ^ n) -\n \u2191C (\u2191p ^ i) * X i ^ p ^ (n + 1 - i) * \u2191C (\u215f\u2191p ^ n) =\n \u2211 x in range (p ^ (n - i)),\n \u2191C \u2191p *\n \u2191(MvPolynomial.map (Int.castRingHom \u211a))\n ((X i ^ p) ^ (p ^ (n - i) - (x + 1)) * frobeniusPolyAux p i ^ (x + 1) *\n \u2191C\n \u2191(Nat.choose (p ^ (n - i)) (x + 1) /\n p ^ (n - i - v p { val := x + 1, property := (_ : 0 < Nat.succ x) }) *\n p ^ (x - v p { val := x + 1, property := (_ : 0 < Nat.succ x) })))\n[PROOFSTEP]\nrw [add_comm (X i ^ p), add_pow, sum_range_succ', pow_zero, tsub_zero, Nat.choose_zero_right, one_mul, Nat.cast_one,\n mul_one, mul_add, add_mul, Nat.succ_sub (le_of_lt hi), Nat.succ_eq_add_one (n - i), pow_succ, pow_mul, add_sub_cancel,\n mul_sum, sum_mul]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\n\u22a2 \u2211 x in range (p ^ (n - i)),\n \u2191C (\u2191p ^ i) *\n ((\u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p i)) ^ (x + 1) *\n (X i ^ p) ^ (p ^ (n - i) - (x + 1)) *\n \u2191(Nat.choose (p ^ (n - i)) (x + 1))) *\n \u2191C (\u215f\u2191p ^ n) =\n \u2211 x in range (p ^ (n - i)),\n \u2191C \u2191p *\n \u2191(MvPolynomial.map (Int.castRingHom \u211a))\n ((X i ^ p) ^ (p ^ (n - i) - (x + 1)) * frobeniusPolyAux p i ^ (x + 1) *\n \u2191C\n \u2191(Nat.choose (p ^ (n - i)) (x + 1) /\n p ^ (n - i - v p { val := x + 1, property := (_ : 0 < Nat.succ x) }) *\n p ^ (x - v p { val := x + 1, property := (_ : 0 < Nat.succ x) })))\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\n\u22a2 \u2200 (x : \u2115),\n x \u2208 range (p ^ (n - i)) \u2192\n \u2191C (\u2191p ^ i) *\n ((\u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p i)) ^ (x + 1) *\n (X i ^ p) ^ (p ^ (n - i) - (x + 1)) *\n \u2191(Nat.choose (p ^ (n - i)) (x + 1))) *\n \u2191C (\u215f\u2191p ^ n) =\n \u2191C \u2191p *\n \u2191(MvPolynomial.map (Int.castRingHom \u211a))\n ((X i ^ p) ^ (p ^ (n - i) - (x + 1)) * frobeniusPolyAux p i ^ (x + 1) *\n \u2191C\n \u2191(Nat.choose (p ^ (n - i)) (x + 1) /\n p ^ (n - i - v p { val := x + 1, property := (_ : 0 < Nat.succ x) }) *\n p ^ (x - v p { val := x + 1, property := (_ : 0 < Nat.succ x) })))\n[PROOFSTEP]\nintro j hj\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j \u2208 range (p ^ (n - i))\n\u22a2 \u2191C (\u2191p ^ i) *\n ((\u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p i)) ^ (j + 1) *\n (X i ^ p) ^ (p ^ (n - i) - (j + 1)) *\n \u2191(Nat.choose (p ^ (n - i)) (j + 1))) *\n \u2191C (\u215f\u2191p ^ n) =\n \u2191C \u2191p *\n \u2191(MvPolynomial.map (Int.castRingHom \u211a))\n ((X i ^ p) ^ (p ^ (n - i) - (j + 1)) * frobeniusPolyAux p i ^ (j + 1) *\n \u2191C\n \u2191(Nat.choose (p ^ (n - i)) (j + 1) / p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })))\n[PROOFSTEP]\nrw [mem_range] at hj \n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\n\u22a2 \u2191C (\u2191p ^ i) *\n ((\u2191C (\u2191(Int.castRingHom \u211a) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p i)) ^ (j + 1) *\n (X i ^ p) ^ (p ^ (n - i) - (j + 1)) *\n \u2191(Nat.choose (p ^ (n - i)) (j + 1))) *\n \u2191C (\u215f\u2191p ^ n) =\n \u2191C \u2191p *\n \u2191(MvPolynomial.map (Int.castRingHom \u211a))\n ((X i ^ p) ^ (p ^ (n - i) - (j + 1)) * frobeniusPolyAux p i ^ (j + 1) *\n \u2191C\n \u2191(Nat.choose (p ^ (n - i)) (j + 1) / p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })))\n[PROOFSTEP]\nrw [RingHom.map_mul, RingHom.map_mul, RingHom.map_pow, RingHom.map_pow, RingHom.map_pow, RingHom.map_pow,\n RingHom.map_pow, map_C, map_X, mul_pow]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\n\u22a2 \u2191C \u2191p ^ i *\n (\u2191C (\u2191(Int.castRingHom \u211a) \u2191p) ^ (j + 1) *\n \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p i) ^ (j + 1) *\n (X i ^ p) ^ (p ^ (n - i) - (j + 1)) *\n \u2191(Nat.choose (p ^ (n - i)) (j + 1))) *\n \u2191C \u215f\u2191p ^ n =\n \u2191C \u2191p *\n ((X i ^ p) ^ (p ^ (n - i) - (j + 1)) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p i) ^ (j + 1) *\n \u2191C\n (\u2191(Int.castRingHom \u211a)\n \u2191(Nat.choose (p ^ (n - i)) (j + 1) / p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }))))\n[PROOFSTEP]\nrw [mul_comm (C (p : \u211a) ^ i), mul_comm _ ((X i ^ p) ^ _), show (Int.castRingHom \u211a) \u2191p = (p : \u211a) from rfl,\n mul_comm (C (p : \u211a) ^ (j + 1)), mul_comm (C (p : \u211a))]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\n\u22a2 (X i ^ p) ^ (p ^ (n - i) - (j + 1)) *\n (\u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p i) ^ (j + 1) * \u2191C \u2191p ^ (j + 1)) *\n \u2191(Nat.choose (p ^ (n - i)) (j + 1)) *\n \u2191C \u2191p ^ i *\n \u2191C \u215f\u2191p ^ n =\n (X i ^ p) ^ (p ^ (n - i) - (j + 1)) * \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p i) ^ (j + 1) *\n \u2191C\n (\u2191(Int.castRingHom \u211a)\n \u2191(Nat.choose (p ^ (n - i)) (j + 1) / p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }))) *\n \u2191C \u2191p\n[PROOFSTEP]\nsimp only [mul_assoc]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\n\u22a2 (X i ^ p) ^ (p ^ (n - i) - (j + 1)) *\n (\u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p i) ^ (j + 1) *\n (\u2191C \u2191p ^ (j + 1) * (\u2191(Nat.choose (p ^ (n - i)) (j + 1)) * (\u2191C \u2191p ^ i * \u2191C \u215f\u2191p ^ n)))) =\n (X i ^ p) ^ (p ^ (n - i) - (j + 1)) *\n (\u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p i) ^ (j + 1) *\n (\u2191C\n (\u2191(Int.castRingHom \u211a)\n \u2191(Nat.choose (p ^ (n - i)) (j + 1) /\n p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }))) *\n \u2191C \u2191p))\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\n\u22a2 \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p i) ^ (j + 1) *\n (\u2191C \u2191p ^ (j + 1) * (\u2191(Nat.choose (p ^ (n - i)) (j + 1)) * (\u2191C \u2191p ^ i * \u2191C \u215f\u2191p ^ n))) =\n \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (frobeniusPolyAux p i) ^ (j + 1) *\n (\u2191C\n (\u2191(Int.castRingHom \u211a)\n \u2191(Nat.choose (p ^ (n - i)) (j + 1) / p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }))) *\n \u2191C \u2191p)\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h.h\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\n\u22a2 \u2191C \u2191p ^ (j + 1) * (\u2191(Nat.choose (p ^ (n - i)) (j + 1)) * (\u2191C \u2191p ^ i * \u2191C \u215f\u2191p ^ n)) =\n \u2191C\n (\u2191(Int.castRingHom \u211a)\n \u2191(Nat.choose (p ^ (n - i)) (j + 1) / p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }))) *\n \u2191C \u2191p\n[PROOFSTEP]\nrw [\u2190 C_eq_coe_nat]\n[GOAL]\ncase h.h\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\n\u22a2 \u2191C \u2191p ^ (j + 1) * (\u2191C \u2191(Nat.choose (p ^ (n - i)) (j + 1)) * (\u2191C \u2191p ^ i * \u2191C \u215f\u2191p ^ n)) =\n \u2191C\n (\u2191(Int.castRingHom \u211a)\n \u2191(Nat.choose (p ^ (n - i)) (j + 1) / p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }))) *\n \u2191C \u2191p\n[PROOFSTEP]\nsimp only [\u2190 RingHom.map_pow, \u2190 C_mul]\n[GOAL]\ncase h.h\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\n\u22a2 \u2191C (\u2191p ^ (j + 1) * (\u2191(Nat.choose (p ^ (n - i)) (j + 1)) * (\u2191p ^ i * \u215f\u2191p ^ n))) =\n \u2191C\n (\u2191(Int.castRingHom \u211a)\n \u2191(Nat.choose (p ^ (n - i)) (j + 1) / p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })) *\n \u2191p)\n[PROOFSTEP]\nrw [C_inj]\n[GOAL]\ncase h.h\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\n\u22a2 \u2191p ^ (j + 1) * (\u2191(Nat.choose (p ^ (n - i)) (j + 1)) * (\u2191p ^ i * \u215f\u2191p ^ n)) =\n \u2191(Int.castRingHom \u211a)\n \u2191(Nat.choose (p ^ (n - i)) (j + 1) / p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })) *\n \u2191p\n[PROOFSTEP]\nsimp only [invOf_eq_inv, eq_intCast, inv_pow, Int.cast_ofNat, Nat.cast_mul, Int.cast_mul]\n[GOAL]\ncase h.h\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\n\u22a2 \u2191p ^ (j + 1) * (\u2191(Nat.choose (p ^ (n - i)) (j + 1)) * (\u2191p ^ i * (\u2191p ^ n)\u207b\u00b9)) =\n \u2191(Nat.choose (p ^ (n - i)) (j + 1) / p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })) *\n \u2191(p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })) *\n \u2191p\n[PROOFSTEP]\nrw [Rat.coe_nat_div _ _ (map_frobeniusPoly.key\u2081 p (n - i) j hj)]\n[GOAL]\ncase h.h\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\n\u22a2 \u2191p ^ (j + 1) * (\u2191(Nat.choose (p ^ (n - i)) (j + 1)) * (\u2191p ^ i * (\u2191p ^ n)\u207b\u00b9)) =\n \u2191(Nat.choose (p ^ (n - i)) (j + 1)) / \u2191(p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })) *\n \u2191(p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })) *\n \u2191p\n[PROOFSTEP]\nsimp only [Nat.cast_pow, pow_add, pow_one]\n[GOAL]\ncase h.h\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\n\u22a2 \u2191p ^ j * \u2191p * (\u2191(Nat.choose (p ^ (n - i)) (j + 1)) * (\u2191p ^ i * (\u2191p ^ n)\u207b\u00b9)) =\n \u2191(Nat.choose (p ^ (n - i)) (j + 1)) / \u2191p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n \u2191p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n \u2191p\n[PROOFSTEP]\nsuffices\n (((p ^ (n - i)).choose (j + 1) : \u211a) * (p : \u211a) ^ (j - v p \u27e8j + 1, j.succ_pos\u27e9) * \u2191p * (p ^ n : \u211a)) =\n (p : \u211a) ^ j * p * \u2191((p ^ (n - i)).choose (j + 1) * p ^ i) * (p : \u211a) ^ (n - i - v p \u27e8j + 1, j.succ_pos\u27e9)\n by\n have aux : \u2200 k : \u2115, (p : \u211a) ^ k \u2260 0 := by intro; apply pow_ne_zero; exact_mod_cast hp.1.ne_zero\n simpa [aux, -one_div, field_simps] using this.symm\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\nthis :\n \u2191(Nat.choose (p ^ (n - i)) (j + 1)) * \u2191p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) * \u2191p *\n \u2191(p ^ n) =\n \u2191p ^ j * \u2191p * \u2191(Nat.choose (p ^ (n - i)) (j + 1) * p ^ i) *\n \u2191p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })\n\u22a2 \u2191p ^ j * \u2191p * (\u2191(Nat.choose (p ^ (n - i)) (j + 1)) * (\u2191p ^ i * (\u2191p ^ n)\u207b\u00b9)) =\n \u2191(Nat.choose (p ^ (n - i)) (j + 1)) / \u2191p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n \u2191p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n \u2191p\n[PROOFSTEP]\nhave aux : \u2200 k : \u2115, (p : \u211a) ^ k \u2260 0 := by intro; apply pow_ne_zero; exact_mod_cast hp.1.ne_zero\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\nthis :\n \u2191(Nat.choose (p ^ (n - i)) (j + 1)) * \u2191p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) * \u2191p *\n \u2191(p ^ n) =\n \u2191p ^ j * \u2191p * \u2191(Nat.choose (p ^ (n - i)) (j + 1) * p ^ i) *\n \u2191p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })\n\u22a2 \u2200 (k : \u2115), \u2191p ^ k \u2260 0\n[PROOFSTEP]\nintro\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\nthis :\n \u2191(Nat.choose (p ^ (n - i)) (j + 1)) * \u2191p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) * \u2191p *\n \u2191(p ^ n) =\n \u2191p ^ j * \u2191p * \u2191(Nat.choose (p ^ (n - i)) (j + 1) * p ^ i) *\n \u2191p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })\nk\u271d : \u2115\n\u22a2 \u2191p ^ k\u271d \u2260 0\n[PROOFSTEP]\napply pow_ne_zero\n[GOAL]\ncase h\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\nthis :\n \u2191(Nat.choose (p ^ (n - i)) (j + 1)) * \u2191p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) * \u2191p *\n \u2191(p ^ n) =\n \u2191p ^ j * \u2191p * \u2191(Nat.choose (p ^ (n - i)) (j + 1) * p ^ i) *\n \u2191p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })\nk\u271d : \u2115\n\u22a2 \u2191p \u2260 0\n[PROOFSTEP]\nexact_mod_cast hp.1.ne_zero\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\nthis :\n \u2191(Nat.choose (p ^ (n - i)) (j + 1)) * \u2191p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) * \u2191p *\n \u2191(p ^ n) =\n \u2191p ^ j * \u2191p * \u2191(Nat.choose (p ^ (n - i)) (j + 1) * p ^ i) *\n \u2191p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })\naux : \u2200 (k : \u2115), \u2191p ^ k \u2260 0\n\u22a2 \u2191p ^ j * \u2191p * (\u2191(Nat.choose (p ^ (n - i)) (j + 1)) * (\u2191p ^ i * (\u2191p ^ n)\u207b\u00b9)) =\n \u2191(Nat.choose (p ^ (n - i)) (j + 1)) / \u2191p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n \u2191p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) *\n \u2191p\n[PROOFSTEP]\nsimpa [aux, -one_div, field_simps] using this.symm\n[GOAL]\ncase h.h\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\n\u22a2 \u2191(Nat.choose (p ^ (n - i)) (j + 1)) * \u2191p ^ (j - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }) * \u2191p *\n \u2191(p ^ n) =\n \u2191p ^ j * \u2191p * \u2191(Nat.choose (p ^ (n - i)) (j + 1) * p ^ i) *\n \u2191p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })\n[PROOFSTEP]\nrw [mul_comm _ (p : \u211a), mul_assoc, Nat.cast_pow, mul_assoc, \u2190 pow_add, map_frobeniusPoly.key\u2082 p hi.le hj, Nat.cast_mul,\n Nat.cast_pow]\n[GOAL]\ncase h.h\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh1 : \u2191p ^ n * \u215f\u2191p ^ n = 1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < p ^ (n - i)\n\u22a2 \u2191p *\n (\u2191(Nat.choose (p ^ (n - i)) (j + 1)) *\n \u2191p ^ (i + j + (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) }))) =\n \u2191p ^ j * \u2191p * (\u2191(Nat.choose (p ^ (n - i)) (j + 1)) * \u2191p ^ i) *\n \u2191p ^ (n - i - v p { val := j + 1, property := (_ : 0 < Nat.succ j) })\n[PROOFSTEP]\nring\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 \u2191(MvPolynomial.map (Int.castRingHom (ZMod p))) (frobeniusPoly p n) = X n ^ p\n[PROOFSTEP]\nrw [frobeniusPoly, RingHom.map_add, RingHom.map_pow, RingHom.map_mul, map_X, map_C]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 X n ^ p +\n \u2191C (\u2191(Int.castRingHom (ZMod p)) \u2191p) * \u2191(MvPolynomial.map (Int.castRingHom (ZMod p))) (frobeniusPolyAux p n) =\n X n ^ p\n[PROOFSTEP]\nsimp only [Int.cast_ofNat, add_zero, eq_intCast, ZMod.nat_cast_self, zero_mul, C_0]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 \u2191(bind\u2081 (frobeniusPoly p)) (wittPolynomial p \u2124 n) = wittPolynomial p \u2124 (n + 1)\n[PROOFSTEP]\napply MvPolynomial.map_injective (Int.castRingHom \u211a) Int.cast_injective\n[GOAL]\ncase a\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (\u2191(bind\u2081 (frobeniusPoly p)) (wittPolynomial p \u2124 n)) =\n \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (wittPolynomial p \u2124 (n + 1))\n[PROOFSTEP]\nsimp only [map_bind\u2081, map_frobeniusPoly, bind\u2081_frobeniusPolyRat_wittPolynomial, map_wittPolynomial]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 coeff (frobeniusFun x) n = \u2191(aeval x.coeff) (frobeniusPoly p n)\n[PROOFSTEP]\nrw [frobeniusFun, coeff_mk]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2200 \u2983R : Type ?u.420485\u2984 [inst : CommRing R] (x : \ud835\udd4e R),\n (frobeniusFun x).coeff = fun n => \u2191(aeval x.coeff) (frobeniusPoly p n)\n[PROOFSTEP]\nintros\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nR\u271d : Type ?u.420485\ninst\u271d : CommRing R\u271d\nx\u271d : \ud835\udd4e R\u271d\n\u22a2 (frobeniusFun x\u271d).coeff = fun n => \u2191(aeval x\u271d.coeff) (frobeniusPoly p n)\n[PROOFSTEP]\nfunext n\n[GOAL]\ncase h\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nR\u271d : Type ?u.420485\ninst\u271d : CommRing R\u271d\nx\u271d : \ud835\udd4e R\u271d\nn : \u2115\n\u22a2 coeff (frobeniusFun x\u271d) n = \u2191(aeval x\u271d.coeff) (frobeniusPoly p n)\n[PROOFSTEP]\napply coeff_frobeniusFun\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nx : \ud835\udd4e R\n\u22a2 \u2191(ghostComponent n) (frobeniusFun x) = \u2191(ghostComponent (n + 1)) x\n[PROOFSTEP]\nsimp only [ghostComponent_apply, frobeniusFun, coeff_mk, \u2190 bind\u2081_frobeniusPoly_wittPolynomial, aeval_bind\u2081]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 frobeniusFun 1 = 1\n[PROOFSTEP]\nrefine\n -- Porting note: removing the placeholders give an errorIsPoly.ext\n (@IsPoly.comp p _ _ (frobeniusFun_isPoly p) WittVector.oneIsPoly)\n (@IsPoly.comp p _ _ WittVector.oneIsPoly (frobeniusFun_isPoly p)) ?_ _ 0\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2200 (R : Type u_1) [_Rcr : CommRing R] (x : \ud835\udd4e R) (n : \u2115),\n \u2191(ghostComponent n) ((frobeniusFun \u2218 fun x => 1) x) = \u2191(ghostComponent n) (((fun x => 1) \u2218 frobeniusFun) x)\n[PROOFSTEP]\nsimp only [Function.comp_apply, map_one, forall_const]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2200 (R : Type u_1) [_Rcr : CommRing R] (n : \u2115), \u2191(ghostComponent n) (frobeniusFun 1) = 1\n[PROOFSTEP]\nghost_simp\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2200 (x y : \ud835\udd4e R),\n OneHom.toFun\n { toFun := frobeniusFun, map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n (x * y) =\n OneHom.toFun\n { toFun := frobeniusFun, map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n x *\n OneHom.toFun\n { toFun := frobeniusFun, map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) } y\n[PROOFSTEP]\nghost_calc _ _\n[GOAL]\ncase refine_3\np : \u2115\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing S\nR : Type u_1\nR._inst : CommRing R\nx\u271d y\u271d : \ud835\udd4e R\n\u22a2 \u2200 (n : \u2115),\n \u2191(ghostComponent n)\n (OneHom.toFun\n { toFun := frobeniusFun, map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n (x\u271d * y\u271d)) =\n \u2191(ghostComponent n)\n (OneHom.toFun\n { toFun := frobeniusFun, map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n x\u271d *\n OneHom.toFun\n { toFun := frobeniusFun, map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n y\u271d)\n[PROOFSTEP]\nghost_simp\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) },\n map_mul' :=\n (_ :\n \u2200 (x y : \ud835\udd4e R),\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n (x * y) =\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n x *\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n y) })\n 0 =\n 0\n[PROOFSTEP]\nrefine\n IsPoly.ext (@IsPoly.comp p _ _ (frobeniusFun_isPoly p) WittVector.zeroIsPoly)\n (@IsPoly.comp p _ _ WittVector.zeroIsPoly (frobeniusFun_isPoly p)) ?_ _ 0\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2200 (R : Type u_1) [_Rcr : CommRing R] (x : \ud835\udd4e R) (n : \u2115),\n \u2191(ghostComponent n) ((frobeniusFun \u2218 fun x => 0) x) = \u2191(ghostComponent n) (((fun x => 0) \u2218 frobeniusFun) x)\n[PROOFSTEP]\nsimp only [Function.comp_apply, map_zero, forall_const]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2200 (R : Type u_1) [_Rcr : CommRing R] (n : \u2115), \u2191(ghostComponent n) (frobeniusFun 0) = 0\n[PROOFSTEP]\nghost_simp\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2200 (x y : \ud835\udd4e R),\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) },\n map_mul' :=\n (_ :\n \u2200 (x y : \ud835\udd4e R),\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n (x * y) =\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n x *\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n y) })\n (x + y) =\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) },\n map_mul' :=\n (_ :\n \u2200 (x y : \ud835\udd4e R),\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n (x * y) =\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n x *\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n y) })\n x +\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) },\n map_mul' :=\n (_ :\n \u2200 (x y : \ud835\udd4e R),\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n (x * y) =\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n x *\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n y) })\n y\n[PROOFSTEP]\nghost_calc _ _\n[GOAL]\ncase refine_3\np : \u2115\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing S\nR : Type u_1\nR._inst : CommRing R\nx\u271d y\u271d : \ud835\udd4e R\n\u22a2 \u2200 (n : \u2115),\n \u2191(ghostComponent n)\n (OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) },\n map_mul' :=\n (_ :\n \u2200 (x y : \ud835\udd4e R),\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n (x * y) =\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n x *\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n y) })\n (x\u271d + y\u271d)) =\n \u2191(ghostComponent n)\n (OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) },\n map_mul' :=\n (_ :\n \u2200 (x y : \ud835\udd4e R),\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n (x * y) =\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n x *\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n y) })\n x\u271d +\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) },\n map_mul' :=\n (_ :\n \u2200 (x y : \ud835\udd4e R),\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n (x * y) =\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n x *\n OneHom.toFun\n { toFun := frobeniusFun,\n map_one' := (_ : (frobeniusFun \u2218 fun x => 1) 0 = ((fun x => 1) \u2218 frobeniusFun) 0) }\n y) })\n y\u271d)\n[PROOFSTEP]\nghost_simp\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CharP R p\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 coeff (\u2191frobenius x) n = coeff x n ^ p\n[PROOFSTEP]\nrw [coeff_frobenius]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CharP R p\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 \u2191(aeval x.coeff) (frobeniusPoly p n) = coeff x n ^ p\n[PROOFSTEP]\nletI : Algebra (ZMod p) R :=\n ZMod.algebra _\n _\n -- outline of the calculation, proofs follow below\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CharP R p\nx : \ud835\udd4e R\nn : \u2115\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n\u22a2 \u2191(aeval x.coeff) (frobeniusPoly p n) = coeff x n ^ p\n[PROOFSTEP]\ncalc\n aeval (fun k => x.coeff k) (frobeniusPoly p n) =\n aeval (fun k => x.coeff k) (MvPolynomial.map (Int.castRingHom (ZMod p)) (frobeniusPoly p n)) :=\n ?_\n _ = aeval (fun k => x.coeff k) (X n ^ p : MvPolynomial \u2115 (ZMod p)) := ?_\n _ = x.coeff n ^ p := ?_\n[GOAL]\ncase calc_1\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CharP R p\nx : \ud835\udd4e R\nn : \u2115\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n\u22a2 \u2191(aeval fun k => coeff x k) (frobeniusPoly p n) =\n \u2191(aeval fun k => coeff x k) (\u2191(MvPolynomial.map (Int.castRingHom (ZMod p))) (frobeniusPoly p n))\n[PROOFSTEP]\nconv_rhs => rw [aeval_eq_eval\u2082Hom, eval\u2082Hom_map_hom]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CharP R p\nx : \ud835\udd4e R\nn : \u2115\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n| \u2191(aeval fun k => coeff x k) (\u2191(MvPolynomial.map (Int.castRingHom (ZMod p))) (frobeniusPoly p n))\n[PROOFSTEP]\nrw [aeval_eq_eval\u2082Hom, eval\u2082Hom_map_hom]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CharP R p\nx : \ud835\udd4e R\nn : \u2115\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n| \u2191(aeval fun k => coeff x k) (\u2191(MvPolynomial.map (Int.castRingHom (ZMod p))) (frobeniusPoly p n))\n[PROOFSTEP]\nrw [aeval_eq_eval\u2082Hom, eval\u2082Hom_map_hom]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CharP R p\nx : \ud835\udd4e R\nn : \u2115\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n| \u2191(aeval fun k => coeff x k) (\u2191(MvPolynomial.map (Int.castRingHom (ZMod p))) (frobeniusPoly p n))\n[PROOFSTEP]\nrw [aeval_eq_eval\u2082Hom, eval\u2082Hom_map_hom]\n[GOAL]\ncase calc_1\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CharP R p\nx : \ud835\udd4e R\nn : \u2115\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n\u22a2 \u2191(aeval fun k => coeff x k) (frobeniusPoly p n) =\n \u2191(eval\u2082Hom (RingHom.comp (algebraMap (ZMod p) R) (Int.castRingHom (ZMod p))) fun k => coeff x k) (frobeniusPoly p n)\n[PROOFSTEP]\napply eval\u2082Hom_congr (RingHom.ext_int _ _) rfl rfl\n[GOAL]\ncase calc_2\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CharP R p\nx : \ud835\udd4e R\nn : \u2115\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n\u22a2 \u2191(aeval fun k => coeff x k) (\u2191(MvPolynomial.map (Int.castRingHom (ZMod p))) (frobeniusPoly p n)) =\n \u2191(aeval fun k => coeff x k) (X n ^ p)\n[PROOFSTEP]\nrw [frobeniusPoly_zmod]\n[GOAL]\ncase calc_3\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CharP R p\nx : \ud835\udd4e R\nn : \u2115\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n\u22a2 \u2191(aeval fun k => coeff x k) (X n ^ p) = coeff x n ^ p\n[PROOFSTEP]\nrw [map_pow, aeval_X]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CharP R p\n\u22a2 frobenius = map (_root_.frobenius R p)\n[PROOFSTEP]\next (x n)\n[GOAL]\ncase a.h\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CharP R p\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 coeff (\u2191frobenius x) n = coeff (\u2191(map (_root_.frobenius R p)) x) n\n[PROOFSTEP]\nsimp only [coeff_frobenius_charP, map_coeff, frobenius_def]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CharP R p\nx : \ud835\udd4e (ZMod p)\n\u22a2 \u2191frobenius x = x\n[PROOFSTEP]\nsimp only [ext_iff, coeff_frobenius_charP, ZMod.pow_card, eq_self_iff_true, forall_const]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : CharP R p\ninst\u271d : PerfectRing R p\nsrc\u271d : \ud835\udd4e R \u2192+* \ud835\udd4e R := frobenius\nf : \ud835\udd4e R\nn : \u2115\n\u22a2 coeff (\u2191(map \u2191(RingEquiv.symm (_root_.frobeniusEquiv R p))) (\u2191frobenius f)) n = coeff f n\n[PROOFSTEP]\nrw [frobenius_eq_map_frobenius]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : CharP R p\ninst\u271d : PerfectRing R p\nsrc\u271d : \ud835\udd4e R \u2192+* \ud835\udd4e R := frobenius\nf : \ud835\udd4e R\nn : \u2115\n\u22a2 coeff (\u2191(map \u2191(RingEquiv.symm (_root_.frobeniusEquiv R p))) (\u2191(map (_root_.frobenius R p)) f)) n = coeff f n\n[PROOFSTEP]\nexact frobeniusEquiv_symm_apply_frobenius R p _\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : CharP R p\ninst\u271d : PerfectRing R p\nsrc\u271d : \ud835\udd4e R \u2192+* \ud835\udd4e R := frobenius\nf : \ud835\udd4e R\nn : \u2115\n\u22a2 coeff (\u2191frobenius (\u2191(map \u2191(RingEquiv.symm (_root_.frobeniusEquiv R p))) f)) n = coeff f n\n[PROOFSTEP]\nrw [frobenius_eq_map_frobenius]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : CharP R p\ninst\u271d : PerfectRing R p\nsrc\u271d : \ud835\udd4e R \u2192+* \ud835\udd4e R := frobenius\nf : \ud835\udd4e R\nn : \u2115\n\u22a2 coeff (\u2191(map (_root_.frobenius R p)) (\u2191(map \u2191(RingEquiv.symm (_root_.frobeniusEquiv R p))) f)) n = coeff f n\n[PROOFSTEP]\nexact frobenius_apply_frobeniusEquiv_symm R p _\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.WittVector.Frobenius", "llama_tokens": 25176, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191214879991, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.4490999886943441}} {"text": "[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : HeytingAlgebra \u03b1\nx\u271d : HeytingAlgebra \u03b2\ninst\u271d : HeytingHomClass F \u03b1 \u03b2\nsrc\u271d : HeytingHomClass F \u03b1 \u03b2 := inst\u271d\nf : F\n\u22a2 \u2191f \u22a4 = \u22a4\n[PROOFSTEP]\nrw [\u2190 @himp_self \u03b1 _ \u22a5, \u2190 himp_self, map_himp]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : CoheytingAlgebra \u03b1\nx\u271d : CoheytingAlgebra \u03b2\ninst\u271d : CoheytingHomClass F \u03b1 \u03b2\nsrc\u271d : CoheytingHomClass F \u03b1 \u03b2 := inst\u271d\nf : F\n\u22a2 \u2191f \u22a5 = \u22a5\n[PROOFSTEP]\nrw [\u2190 @sdiff_self \u03b1 _ \u22a4, \u2190 sdiff_self, map_sdiff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : BiheytingAlgebra \u03b1\nx\u271d : BiheytingAlgebra \u03b2\ninst\u271d : BiheytingHomClass F \u03b1 \u03b2\nsrc\u271d : BiheytingHomClass F \u03b1 \u03b2 := inst\u271d\nf : F\n\u22a2 \u2191f \u22a5 = \u22a5\n[PROOFSTEP]\nrw [\u2190 @sdiff_self \u03b1 _ \u22a4, \u2190 sdiff_self, BiheytingHomClass.map_sdiff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : BiheytingAlgebra \u03b1\nx\u271d : BiheytingAlgebra \u03b2\ninst\u271d : BiheytingHomClass F \u03b1 \u03b2\nsrc\u271d : BiheytingHomClass F \u03b1 \u03b2 := inst\u271d\nf : F\n\u22a2 \u2191f \u22a4 = \u22a4\n[PROOFSTEP]\nrw [\u2190 @himp_self \u03b1 _ \u22a5, \u2190 himp_self, map_himp]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : HeytingAlgebra \u03b1\nx\u271d : HeytingAlgebra \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nsrc\u271d : BoundedLatticeHomClass F \u03b1 \u03b2 := toBoundedLatticeHomClass\nf : F\na b : \u03b1\nc : (fun x => \u03b2) (a \u21e8 b)\n\u22a2 c \u2264 \u2191f (a \u21e8 b) \u2194 c \u2264 \u2191f a \u21e8 \u2191f b\n[PROOFSTEP]\nsimp only [\u2190 map_inv_le_iff, le_himp_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : HeytingAlgebra \u03b1\nx\u271d : HeytingAlgebra \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nsrc\u271d : BoundedLatticeHomClass F \u03b1 \u03b2 := toBoundedLatticeHomClass\nf : F\na b : \u03b1\nc : (fun x => \u03b2) (a \u21e8 b)\n\u22a2 EquivLike.inv f c \u2293 a \u2264 b \u2194 EquivLike.inv f (c \u2293 \u2191f a) \u2264 b\n[PROOFSTEP]\nrw [\u2190 OrderIsoClass.map_le_map_iff f]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : HeytingAlgebra \u03b1\nx\u271d : HeytingAlgebra \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nsrc\u271d : BoundedLatticeHomClass F \u03b1 \u03b2 := toBoundedLatticeHomClass\nf : F\na b : \u03b1\nc : (fun x => \u03b2) (a \u21e8 b)\n\u22a2 \u2191f (EquivLike.inv f c \u2293 a) \u2264 \u2191f b \u2194 EquivLike.inv f (c \u2293 \u2191f a) \u2264 b\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : CoheytingAlgebra \u03b1\nx\u271d : CoheytingAlgebra \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nsrc\u271d : BoundedLatticeHomClass F \u03b1 \u03b2 := toBoundedLatticeHomClass\nf : F\na b : \u03b1\nc : (fun x => \u03b2) (a \\ b)\n\u22a2 \u2191f (a \\ b) \u2264 c \u2194 \u2191f a \\ \u2191f b \u2264 c\n[PROOFSTEP]\nsimp only [\u2190 le_map_inv_iff, sdiff_le_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : CoheytingAlgebra \u03b1\nx\u271d : CoheytingAlgebra \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nsrc\u271d : BoundedLatticeHomClass F \u03b1 \u03b2 := toBoundedLatticeHomClass\nf : F\na b : \u03b1\nc : (fun x => \u03b2) (a \\ b)\n\u22a2 a \u2264 b \u2294 EquivLike.inv f c \u2194 a \u2264 EquivLike.inv f (\u2191f b \u2294 c)\n[PROOFSTEP]\nrw [\u2190 OrderIsoClass.map_le_map_iff f]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : CoheytingAlgebra \u03b1\nx\u271d : CoheytingAlgebra \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nsrc\u271d : BoundedLatticeHomClass F \u03b1 \u03b2 := toBoundedLatticeHomClass\nf : F\na b : \u03b1\nc : (fun x => \u03b2) (a \\ b)\n\u22a2 \u2191f a \u2264 \u2191f (b \u2294 EquivLike.inv f c) \u2194 a \u2264 EquivLike.inv f (\u2191f b \u2294 c)\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : BiheytingAlgebra \u03b1\nx\u271d : BiheytingAlgebra \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nsrc\u271d : LatticeHomClass F \u03b1 \u03b2 := toLatticeHomClass\nf : F\na b : \u03b1\nc : (fun x => \u03b2) (a \u21e8 b)\n\u22a2 c \u2264 \u2191f (a \u21e8 b) \u2194 c \u2264 \u2191f a \u21e8 \u2191f b\n[PROOFSTEP]\nsimp only [\u2190 map_inv_le_iff, le_himp_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : BiheytingAlgebra \u03b1\nx\u271d : BiheytingAlgebra \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nsrc\u271d : LatticeHomClass F \u03b1 \u03b2 := toLatticeHomClass\nf : F\na b : \u03b1\nc : (fun x => \u03b2) (a \u21e8 b)\n\u22a2 EquivLike.inv f c \u2293 a \u2264 b \u2194 EquivLike.inv f (c \u2293 \u2191f a) \u2264 b\n[PROOFSTEP]\nrw [\u2190 OrderIsoClass.map_le_map_iff f]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : BiheytingAlgebra \u03b1\nx\u271d : BiheytingAlgebra \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nsrc\u271d : LatticeHomClass F \u03b1 \u03b2 := toLatticeHomClass\nf : F\na b : \u03b1\nc : (fun x => \u03b2) (a \u21e8 b)\n\u22a2 \u2191f (EquivLike.inv f c \u2293 a) \u2264 \u2191f b \u2194 EquivLike.inv f (c \u2293 \u2191f a) \u2264 b\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : BiheytingAlgebra \u03b1\nx\u271d : BiheytingAlgebra \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nsrc\u271d : LatticeHomClass F \u03b1 \u03b2 := toLatticeHomClass\nf : F\na b : \u03b1\nc : (fun x => \u03b2) (a \\ b)\n\u22a2 \u2191f (a \\ b) \u2264 c \u2194 \u2191f a \\ \u2191f b \u2264 c\n[PROOFSTEP]\nsimp only [\u2190 le_map_inv_iff, sdiff_le_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : BiheytingAlgebra \u03b1\nx\u271d : BiheytingAlgebra \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nsrc\u271d : LatticeHomClass F \u03b1 \u03b2 := toLatticeHomClass\nf : F\na b : \u03b1\nc : (fun x => \u03b2) (a \\ b)\n\u22a2 a \u2264 b \u2294 EquivLike.inv f c \u2194 a \u2264 EquivLike.inv f (\u2191f b \u2294 c)\n[PROOFSTEP]\nrw [\u2190 OrderIsoClass.map_le_map_iff f]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b9 : BiheytingAlgebra \u03b1\nx\u271d : BiheytingAlgebra \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nsrc\u271d : LatticeHomClass F \u03b1 \u03b2 := toLatticeHomClass\nf : F\na b : \u03b1\nc : (fun x => \u03b2) (a \\ b)\n\u22a2 \u2191f a \u2264 \u2191f (b \u2294 EquivLike.inv f c) \u2194 a \u2264 EquivLike.inv f (\u2191f b \u2294 c)\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : BooleanAlgebra \u03b2\ninst\u271d : BoundedLatticeHomClass F \u03b1 \u03b2\nsrc\u271d : BoundedLatticeHomClass F \u03b1 \u03b2 := inst\u271d\nf : F\na b : \u03b1\n\u22a2 \u2191f (a \u21e8 b) = \u2191f a \u21e8 \u2191f b\n[PROOFSTEP]\nrw [himp_eq, himp_eq, map_sup, (isCompl_compl.map _).compl_eq]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : BooleanAlgebra \u03b2\ninst\u271d : BoundedLatticeHomClass F \u03b1 \u03b2\nsrc\u271d : BoundedLatticeHomClass F \u03b1 \u03b2 := inst\u271d\nf : F\na b : \u03b1\n\u22a2 \u2191f (a \\ b) = \u2191f a \\ \u2191f b\n[PROOFSTEP]\nrw [sdiff_eq, sdiff_eq, map_inf, (isCompl_compl.map _).compl_eq]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b2 : HeytingAlgebra \u03b1\ninst\u271d\u00b9 : HeytingAlgebra \u03b2\ninst\u271d : HeytingHomClass F \u03b1 \u03b2\nf : F\na : \u03b1\n\u22a2 \u2191f a\u1d9c = (\u2191f a)\u1d9c\n[PROOFSTEP]\nrw [\u2190 himp_bot, \u2190 himp_bot, map_himp, map_bot]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b2 : HeytingAlgebra \u03b1\ninst\u271d\u00b9 : HeytingAlgebra \u03b2\ninst\u271d : HeytingHomClass F \u03b1 \u03b2\nf : F\na b : \u03b1\n\u22a2 \u2191f (a \u21d4 b) = \u2191f a \u21d4 \u2191f b\n[PROOFSTEP]\nsimp_rw [bihimp, map_inf, map_himp]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b2 : CoheytingAlgebra \u03b1\ninst\u271d\u00b9 : CoheytingAlgebra \u03b2\ninst\u271d : CoheytingHomClass F \u03b1 \u03b2\nf : F\na : \u03b1\n\u22a2 \u2191f (\uffe2a) = \uffe2\u2191f a\n[PROOFSTEP]\nrw [\u2190 top_sdiff', \u2190 top_sdiff', map_sdiff, map_top]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b2 : CoheytingAlgebra \u03b1\ninst\u271d\u00b9 : CoheytingAlgebra \u03b2\ninst\u271d : CoheytingHomClass F \u03b1 \u03b2\nf : F\na b : \u03b1\n\u22a2 \u2191f (a \u2206 b) = \u2191f a \u2206 \u2191f b\n[PROOFSTEP]\nsimp_rw [symmDiff, map_sup, map_sdiff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : HeytingAlgebra \u03b1\ninst\u271d\u00b2 : HeytingAlgebra \u03b2\ninst\u271d\u00b9 : HeytingAlgebra \u03b3\ninst\u271d : HeytingAlgebra \u03b4\nf g : HeytingHom \u03b1 \u03b2\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8_, _\u27e9, _\u27e9, _\u27e9 := f\n[GOAL]\ncase mk.mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : HeytingAlgebra \u03b1\ninst\u271d\u00b2 : HeytingAlgebra \u03b2\ninst\u271d\u00b9 : HeytingAlgebra \u03b3\ninst\u271d : HeytingAlgebra \u03b4\ng : HeytingHom \u03b1 \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_sup'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2294 b) = toFun\u271d a \u2294 toFun\u271d b\nmap_inf'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } (a \u2293 b) =\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } a \u2293\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } b\nmap_bot'\u271d :\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom \u22a5 = \u22a5\nmap_himp'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom (a \u21e8 b) =\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom a \u21e8\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom b\nh :\n (fun f => f.toFun)\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_bot' := map_bot'\u271d, map_himp' := map_himp'\u271d } =\n (fun f => f.toFun) g\n\u22a2 { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_bot' := map_bot'\u271d, map_himp' := map_himp'\u271d } =\n g\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8_, _\u27e9, _\u27e9, _\u27e9 := g\n[GOAL]\ncase mk.mk.mk.mk.mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : HeytingAlgebra \u03b1\ninst\u271d\u00b2 : HeytingAlgebra \u03b2\ninst\u271d\u00b9 : HeytingAlgebra \u03b3\ninst\u271d : HeytingAlgebra \u03b4\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmap_sup'\u271d\u00b9 : \u2200 (a b : \u03b1), toFun\u271d\u00b9 (a \u2294 b) = toFun\u271d\u00b9 a \u2294 toFun\u271d\u00b9 b\nmap_inf'\u271d\u00b9 :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } (a \u2293 b) =\n SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } a \u2293\n SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } b\nmap_bot'\u271d\u00b9 :\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom \u22a5 = \u22a5\nmap_himp'\u271d\u00b9 :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom (a \u21e8 b) =\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom a \u21e8\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom b\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_sup'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2294 b) = toFun\u271d a \u2294 toFun\u271d b\nmap_inf'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } (a \u2293 b) =\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } a \u2293\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } b\nmap_bot'\u271d :\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom \u22a5 = \u22a5\nmap_himp'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom (a \u21e8 b) =\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom a \u21e8\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom b\nh :\n (fun f => f.toFun)\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 },\n map_bot' := map_bot'\u271d\u00b9, map_himp' := map_himp'\u271d\u00b9 } =\n (fun f => f.toFun)\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_bot' := map_bot'\u271d, map_himp' := map_himp'\u271d }\n\u22a2 { toLatticeHom := { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 },\n map_bot' := map_bot'\u271d\u00b9, map_himp' := map_himp'\u271d\u00b9 } =\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_bot' := map_bot'\u271d, map_himp' := map_himp'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : HeytingAlgebra \u03b1\ninst\u271d\u00b2 : HeytingAlgebra \u03b2\ninst\u271d\u00b9 : HeytingAlgebra \u03b3\ninst\u271d : HeytingAlgebra \u03b4\nf : HeytingHom \u03b1 \u03b2\nf' : \u03b1 \u2192 \u03b2\nh : f' = \u2191f\n\u22a2 \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b\n[PROOFSTEP]\nsimpa only [h] using map_sup f\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : HeytingAlgebra \u03b1\ninst\u271d\u00b2 : HeytingAlgebra \u03b2\ninst\u271d\u00b9 : HeytingAlgebra \u03b3\ninst\u271d : HeytingAlgebra \u03b4\nf : HeytingHom \u03b1 \u03b2\nf' : \u03b1 \u2192 \u03b2\nh : f' = \u2191f\n\u22a2 \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } b\n[PROOFSTEP]\nsimpa only [h] using map_inf f\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : HeytingAlgebra \u03b1\ninst\u271d\u00b2 : HeytingAlgebra \u03b2\ninst\u271d\u00b9 : HeytingAlgebra \u03b3\ninst\u271d : HeytingAlgebra \u03b4\nf : HeytingHom \u03b1 \u03b2\nf' : \u03b1 \u2192 \u03b2\nh : f' = \u2191f\n\u22a2 SupHom.toFun\n { toSupHom := { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } b) }.toSupHom\n \u22a5 =\n \u22a5\n[PROOFSTEP]\nsimpa only [h] using map_bot f\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : HeytingAlgebra \u03b1\ninst\u271d\u00b2 : HeytingAlgebra \u03b2\ninst\u271d\u00b9 : HeytingAlgebra \u03b3\ninst\u271d : HeytingAlgebra \u03b4\nf : HeytingHom \u03b1 \u03b2\nf' : \u03b1 \u2192 \u03b2\nh : f' = \u2191f\n\u22a2 \u2200 (a b : \u03b1),\n SupHom.toFun\n { toSupHom := { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) }\n b) }.toSupHom\n (a \u21e8 b) =\n SupHom.toFun\n { toSupHom := { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) }\n b) }.toSupHom\n a \u21e8\n SupHom.toFun\n { toSupHom := { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) }\n b) }.toSupHom\n b\n[PROOFSTEP]\nsimpa only [h] using map_himp f\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : HeytingAlgebra \u03b1\ninst\u271d\u00b2 : HeytingAlgebra \u03b2\ninst\u271d\u00b9 : HeytingAlgebra \u03b3\ninst\u271d : HeytingAlgebra \u03b4\nf : HeytingHom \u03b2 \u03b3\ng : HeytingHom \u03b1 \u03b2\nsrc\u271d : LatticeHom \u03b1 \u03b3 := LatticeHom.comp f.toLatticeHom g.toLatticeHom\n\u22a2 SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191f \u2218 \u2191g,\n map_sup' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2294 b) = SupHom.toFun src\u271d.toSupHom a \u2294 SupHom.toFun src\u271d.toSupHom b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2293 b) =\n SupHom.toFun src\u271d.toSupHom a \u2293 SupHom.toFun src\u271d.toSupHom b) }.toSupHom\n \u22a5 =\n \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : HeytingAlgebra \u03b1\ninst\u271d\u00b2 : HeytingAlgebra \u03b2\ninst\u271d\u00b9 : HeytingAlgebra \u03b3\ninst\u271d : HeytingAlgebra \u03b4\nf : HeytingHom \u03b2 \u03b3\ng : HeytingHom \u03b1 \u03b2\nsrc\u271d : LatticeHom \u03b1 \u03b3 := LatticeHom.comp f.toLatticeHom g.toLatticeHom\na b : \u03b1\n\u22a2 SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191f \u2218 \u2191g,\n map_sup' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2294 b) = SupHom.toFun src\u271d.toSupHom a \u2294 SupHom.toFun src\u271d.toSupHom b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2293 b) =\n SupHom.toFun src\u271d.toSupHom a \u2293 SupHom.toFun src\u271d.toSupHom b) }.toSupHom\n (a \u21e8 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191f \u2218 \u2191g,\n map_sup' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2294 b) =\n SupHom.toFun src\u271d.toSupHom a \u2294 SupHom.toFun src\u271d.toSupHom b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2293 b) =\n SupHom.toFun src\u271d.toSupHom a \u2293 SupHom.toFun src\u271d.toSupHom b) }.toSupHom\n a \u21e8\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191f \u2218 \u2191g,\n map_sup' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2294 b) =\n SupHom.toFun src\u271d.toSupHom a \u2294 SupHom.toFun src\u271d.toSupHom b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2293 b) =\n SupHom.toFun src\u271d.toSupHom a \u2293 SupHom.toFun src\u271d.toSupHom b) }.toSupHom\n b\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : HeytingAlgebra \u03b1\ninst\u271d\u00b2 : HeytingAlgebra \u03b2\ninst\u271d\u00b9 : HeytingAlgebra \u03b3\ninst\u271d : HeytingAlgebra \u03b4\nf f\u2081 f\u2082 : HeytingHom \u03b1 \u03b2\ng g\u2081 g\u2082 : HeytingHom \u03b2 \u03b3\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\na : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a)\n[PROOFSTEP]\nrw [\u2190 comp_apply, h, comp_apply]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : CoheytingAlgebra \u03b1\ninst\u271d\u00b2 : CoheytingAlgebra \u03b2\ninst\u271d\u00b9 : CoheytingAlgebra \u03b3\ninst\u271d : CoheytingAlgebra \u03b4\nf g : CoheytingHom \u03b1 \u03b2\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8_, _\u27e9, _\u27e9, _\u27e9 := f\n[GOAL]\ncase mk.mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : CoheytingAlgebra \u03b1\ninst\u271d\u00b2 : CoheytingAlgebra \u03b2\ninst\u271d\u00b9 : CoheytingAlgebra \u03b3\ninst\u271d : CoheytingAlgebra \u03b4\ng : CoheytingHom \u03b1 \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_sup'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2294 b) = toFun\u271d a \u2294 toFun\u271d b\nmap_inf'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } (a \u2293 b) =\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } a \u2293\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } b\nmap_top'\u271d :\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom \u22a4 = \u22a4\nmap_sdiff'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom (a \\ b) =\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom a \\\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom b\nh :\n (fun f => f.toFun)\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_top' := map_top'\u271d, map_sdiff' := map_sdiff'\u271d } =\n (fun f => f.toFun) g\n\u22a2 { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_top' := map_top'\u271d, map_sdiff' := map_sdiff'\u271d } =\n g\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8_, _\u27e9, _\u27e9, _\u27e9 := g\n[GOAL]\ncase mk.mk.mk.mk.mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : CoheytingAlgebra \u03b1\ninst\u271d\u00b2 : CoheytingAlgebra \u03b2\ninst\u271d\u00b9 : CoheytingAlgebra \u03b3\ninst\u271d : CoheytingAlgebra \u03b4\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmap_sup'\u271d\u00b9 : \u2200 (a b : \u03b1), toFun\u271d\u00b9 (a \u2294 b) = toFun\u271d\u00b9 a \u2294 toFun\u271d\u00b9 b\nmap_inf'\u271d\u00b9 :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } (a \u2293 b) =\n SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } a \u2293\n SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } b\nmap_top'\u271d\u00b9 :\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom \u22a4 = \u22a4\nmap_sdiff'\u271d\u00b9 :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom (a \\ b) =\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom a \\\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom b\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_sup'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2294 b) = toFun\u271d a \u2294 toFun\u271d b\nmap_inf'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } (a \u2293 b) =\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } a \u2293\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } b\nmap_top'\u271d :\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom \u22a4 = \u22a4\nmap_sdiff'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom (a \\ b) =\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom a \\\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom b\nh :\n (fun f => f.toFun)\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 },\n map_top' := map_top'\u271d\u00b9, map_sdiff' := map_sdiff'\u271d\u00b9 } =\n (fun f => f.toFun)\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_top' := map_top'\u271d, map_sdiff' := map_sdiff'\u271d }\n\u22a2 { toLatticeHom := { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 },\n map_top' := map_top'\u271d\u00b9, map_sdiff' := map_sdiff'\u271d\u00b9 } =\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_top' := map_top'\u271d, map_sdiff' := map_sdiff'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : CoheytingAlgebra \u03b1\ninst\u271d\u00b2 : CoheytingAlgebra \u03b2\ninst\u271d\u00b9 : CoheytingAlgebra \u03b3\ninst\u271d : CoheytingAlgebra \u03b4\nf : CoheytingHom \u03b1 \u03b2\nf' : \u03b1 \u2192 \u03b2\nh : f' = \u2191f\n\u22a2 \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b\n[PROOFSTEP]\nsimpa only [h] using map_sup f\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : CoheytingAlgebra \u03b1\ninst\u271d\u00b2 : CoheytingAlgebra \u03b2\ninst\u271d\u00b9 : CoheytingAlgebra \u03b3\ninst\u271d : CoheytingAlgebra \u03b4\nf : CoheytingHom \u03b1 \u03b2\nf' : \u03b1 \u2192 \u03b2\nh : f' = \u2191f\n\u22a2 \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } b\n[PROOFSTEP]\nsimpa only [h] using map_inf f\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : CoheytingAlgebra \u03b1\ninst\u271d\u00b2 : CoheytingAlgebra \u03b2\ninst\u271d\u00b9 : CoheytingAlgebra \u03b3\ninst\u271d : CoheytingAlgebra \u03b4\nf : CoheytingHom \u03b1 \u03b2\nf' : \u03b1 \u2192 \u03b2\nh : f' = \u2191f\n\u22a2 SupHom.toFun\n { toSupHom := { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } b) }.toSupHom\n \u22a4 =\n \u22a4\n[PROOFSTEP]\nsimpa only [h] using map_top f\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : CoheytingAlgebra \u03b1\ninst\u271d\u00b2 : CoheytingAlgebra \u03b2\ninst\u271d\u00b9 : CoheytingAlgebra \u03b3\ninst\u271d : CoheytingAlgebra \u03b4\nf : CoheytingHom \u03b1 \u03b2\nf' : \u03b1 \u2192 \u03b2\nh : f' = \u2191f\n\u22a2 \u2200 (a b : \u03b1),\n SupHom.toFun\n { toSupHom := { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) }\n b) }.toSupHom\n (a \\ b) =\n SupHom.toFun\n { toSupHom := { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) }\n b) }.toSupHom\n a \\\n SupHom.toFun\n { toSupHom := { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) }\n b) }.toSupHom\n b\n[PROOFSTEP]\nsimpa only [h] using map_sdiff f\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : CoheytingAlgebra \u03b1\ninst\u271d\u00b2 : CoheytingAlgebra \u03b2\ninst\u271d\u00b9 : CoheytingAlgebra \u03b3\ninst\u271d : CoheytingAlgebra \u03b4\nf : CoheytingHom \u03b2 \u03b3\ng : CoheytingHom \u03b1 \u03b2\nsrc\u271d : LatticeHom \u03b1 \u03b3 := LatticeHom.comp f.toLatticeHom g.toLatticeHom\n\u22a2 SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191f \u2218 \u2191g,\n map_sup' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2294 b) = SupHom.toFun src\u271d.toSupHom a \u2294 SupHom.toFun src\u271d.toSupHom b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2293 b) =\n SupHom.toFun src\u271d.toSupHom a \u2293 SupHom.toFun src\u271d.toSupHom b) }.toSupHom\n \u22a4 =\n \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : CoheytingAlgebra \u03b1\ninst\u271d\u00b2 : CoheytingAlgebra \u03b2\ninst\u271d\u00b9 : CoheytingAlgebra \u03b3\ninst\u271d : CoheytingAlgebra \u03b4\nf : CoheytingHom \u03b2 \u03b3\ng : CoheytingHom \u03b1 \u03b2\nsrc\u271d : LatticeHom \u03b1 \u03b3 := LatticeHom.comp f.toLatticeHom g.toLatticeHom\na b : \u03b1\n\u22a2 SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191f \u2218 \u2191g,\n map_sup' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2294 b) = SupHom.toFun src\u271d.toSupHom a \u2294 SupHom.toFun src\u271d.toSupHom b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2293 b) =\n SupHom.toFun src\u271d.toSupHom a \u2293 SupHom.toFun src\u271d.toSupHom b) }.toSupHom\n (a \\ b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191f \u2218 \u2191g,\n map_sup' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2294 b) =\n SupHom.toFun src\u271d.toSupHom a \u2294 SupHom.toFun src\u271d.toSupHom b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2293 b) =\n SupHom.toFun src\u271d.toSupHom a \u2293 SupHom.toFun src\u271d.toSupHom b) }.toSupHom\n a \\\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191f \u2218 \u2191g,\n map_sup' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2294 b) =\n SupHom.toFun src\u271d.toSupHom a \u2294 SupHom.toFun src\u271d.toSupHom b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2293 b) =\n SupHom.toFun src\u271d.toSupHom a \u2293 SupHom.toFun src\u271d.toSupHom b) }.toSupHom\n b\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : CoheytingAlgebra \u03b1\ninst\u271d\u00b2 : CoheytingAlgebra \u03b2\ninst\u271d\u00b9 : CoheytingAlgebra \u03b3\ninst\u271d : CoheytingAlgebra \u03b4\nf f\u2081 f\u2082 : CoheytingHom \u03b1 \u03b2\ng g\u2081 g\u2082 : CoheytingHom \u03b2 \u03b3\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\na : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a)\n[PROOFSTEP]\nrw [\u2190 comp_apply, h, comp_apply]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : BiheytingAlgebra \u03b1\ninst\u271d\u00b2 : BiheytingAlgebra \u03b2\ninst\u271d\u00b9 : BiheytingAlgebra \u03b3\ninst\u271d : BiheytingAlgebra \u03b4\nf g : BiheytingHom \u03b1 \u03b2\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8_, _\u27e9, _\u27e9, _\u27e9 := f\n[GOAL]\ncase mk.mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : BiheytingAlgebra \u03b1\ninst\u271d\u00b2 : BiheytingAlgebra \u03b2\ninst\u271d\u00b9 : BiheytingAlgebra \u03b3\ninst\u271d : BiheytingAlgebra \u03b4\ng : BiheytingHom \u03b1 \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_sup'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2294 b) = toFun\u271d a \u2294 toFun\u271d b\nmap_inf'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } (a \u2293 b) =\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } a \u2293\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } b\nmap_himp'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom (a \u21e8 b) =\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom a \u21e8\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom b\nmap_sdiff'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom (a \\ b) =\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom a \\\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom b\nh :\n (fun f => f.toFun)\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_himp' := map_himp'\u271d, map_sdiff' := map_sdiff'\u271d } =\n (fun f => f.toFun) g\n\u22a2 { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_himp' := map_himp'\u271d, map_sdiff' := map_sdiff'\u271d } =\n g\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8_, _\u27e9, _\u27e9, _\u27e9 := g\n[GOAL]\ncase mk.mk.mk.mk.mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : BiheytingAlgebra \u03b1\ninst\u271d\u00b2 : BiheytingAlgebra \u03b2\ninst\u271d\u00b9 : BiheytingAlgebra \u03b3\ninst\u271d : BiheytingAlgebra \u03b4\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmap_sup'\u271d\u00b9 : \u2200 (a b : \u03b1), toFun\u271d\u00b9 (a \u2294 b) = toFun\u271d\u00b9 a \u2294 toFun\u271d\u00b9 b\nmap_inf'\u271d\u00b9 :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } (a \u2293 b) =\n SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } a \u2293\n SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } b\nmap_himp'\u271d\u00b9 :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom (a \u21e8 b) =\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom a \u21e8\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom b\nmap_sdiff'\u271d\u00b9 :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom (a \\ b) =\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom a \\\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom b\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_sup'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2294 b) = toFun\u271d a \u2294 toFun\u271d b\nmap_inf'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } (a \u2293 b) =\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } a \u2293\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } b\nmap_himp'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom (a \u21e8 b) =\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom a \u21e8\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom b\nmap_sdiff'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom (a \\ b) =\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom a \\\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom b\nh :\n (fun f => f.toFun)\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 },\n map_himp' := map_himp'\u271d\u00b9, map_sdiff' := map_sdiff'\u271d\u00b9 } =\n (fun f => f.toFun)\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_himp' := map_himp'\u271d, map_sdiff' := map_sdiff'\u271d }\n\u22a2 { toLatticeHom := { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 },\n map_himp' := map_himp'\u271d\u00b9, map_sdiff' := map_sdiff'\u271d\u00b9 } =\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_himp' := map_himp'\u271d, map_sdiff' := map_sdiff'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : BiheytingAlgebra \u03b1\ninst\u271d\u00b2 : BiheytingAlgebra \u03b2\ninst\u271d\u00b9 : BiheytingAlgebra \u03b3\ninst\u271d : BiheytingAlgebra \u03b4\nf : BiheytingHom \u03b1 \u03b2\nf' : \u03b1 \u2192 \u03b2\nh : f' = \u2191f\n\u22a2 \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b\n[PROOFSTEP]\nsimpa only [h] using map_sup f\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : BiheytingAlgebra \u03b1\ninst\u271d\u00b2 : BiheytingAlgebra \u03b2\ninst\u271d\u00b9 : BiheytingAlgebra \u03b3\ninst\u271d : BiheytingAlgebra \u03b4\nf : BiheytingHom \u03b1 \u03b2\nf' : \u03b1 \u2192 \u03b2\nh : f' = \u2191f\n\u22a2 \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } b\n[PROOFSTEP]\nsimpa only [h] using map_inf f\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : BiheytingAlgebra \u03b1\ninst\u271d\u00b2 : BiheytingAlgebra \u03b2\ninst\u271d\u00b9 : BiheytingAlgebra \u03b3\ninst\u271d : BiheytingAlgebra \u03b4\nf : BiheytingHom \u03b1 \u03b2\nf' : \u03b1 \u2192 \u03b2\nh : f' = \u2191f\n\u22a2 \u2200 (a b : \u03b1),\n SupHom.toFun\n { toSupHom := { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) }\n b) }.toSupHom\n (a \u21e8 b) =\n SupHom.toFun\n { toSupHom := { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) }\n b) }.toSupHom\n a \u21e8\n SupHom.toFun\n { toSupHom := { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) }\n b) }.toSupHom\n b\n[PROOFSTEP]\nsimpa only [h] using map_himp f\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : BiheytingAlgebra \u03b1\ninst\u271d\u00b2 : BiheytingAlgebra \u03b2\ninst\u271d\u00b9 : BiheytingAlgebra \u03b3\ninst\u271d : BiheytingAlgebra \u03b4\nf : BiheytingHom \u03b1 \u03b2\nf' : \u03b1 \u2192 \u03b2\nh : f' = \u2191f\n\u22a2 \u2200 (a b : \u03b1),\n SupHom.toFun\n { toSupHom := { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) }\n b) }.toSupHom\n (a \\ b) =\n SupHom.toFun\n { toSupHom := { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) }\n b) }.toSupHom\n a \\\n SupHom.toFun\n { toSupHom := { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } (a \u2293 b) =\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) } a \u2293\n SupHom.toFun { toFun := f', map_sup' := (_ : \u2200 (a b : \u03b1), f' (a \u2294 b) = f' a \u2294 f' b) }\n b) }.toSupHom\n b\n[PROOFSTEP]\nsimpa only [h] using map_sdiff f\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : BiheytingAlgebra \u03b1\ninst\u271d\u00b2 : BiheytingAlgebra \u03b2\ninst\u271d\u00b9 : BiheytingAlgebra \u03b3\ninst\u271d : BiheytingAlgebra \u03b4\nf : BiheytingHom \u03b2 \u03b3\ng : BiheytingHom \u03b1 \u03b2\nsrc\u271d : LatticeHom \u03b1 \u03b3 := LatticeHom.comp f.toLatticeHom g.toLatticeHom\na b : \u03b1\n\u22a2 SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191f \u2218 \u2191g,\n map_sup' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2294 b) = SupHom.toFun src\u271d.toSupHom a \u2294 SupHom.toFun src\u271d.toSupHom b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2293 b) =\n SupHom.toFun src\u271d.toSupHom a \u2293 SupHom.toFun src\u271d.toSupHom b) }.toSupHom\n (a \u21e8 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191f \u2218 \u2191g,\n map_sup' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2294 b) =\n SupHom.toFun src\u271d.toSupHom a \u2294 SupHom.toFun src\u271d.toSupHom b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2293 b) =\n SupHom.toFun src\u271d.toSupHom a \u2293 SupHom.toFun src\u271d.toSupHom b) }.toSupHom\n a \u21e8\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191f \u2218 \u2191g,\n map_sup' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2294 b) =\n SupHom.toFun src\u271d.toSupHom a \u2294 SupHom.toFun src\u271d.toSupHom b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2293 b) =\n SupHom.toFun src\u271d.toSupHom a \u2293 SupHom.toFun src\u271d.toSupHom b) }.toSupHom\n b\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : BiheytingAlgebra \u03b1\ninst\u271d\u00b2 : BiheytingAlgebra \u03b2\ninst\u271d\u00b9 : BiheytingAlgebra \u03b3\ninst\u271d : BiheytingAlgebra \u03b4\nf : BiheytingHom \u03b2 \u03b3\ng : BiheytingHom \u03b1 \u03b2\nsrc\u271d : LatticeHom \u03b1 \u03b3 := LatticeHom.comp f.toLatticeHom g.toLatticeHom\na b : \u03b1\n\u22a2 SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191f \u2218 \u2191g,\n map_sup' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2294 b) = SupHom.toFun src\u271d.toSupHom a \u2294 SupHom.toFun src\u271d.toSupHom b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2293 b) =\n SupHom.toFun src\u271d.toSupHom a \u2293 SupHom.toFun src\u271d.toSupHom b) }.toSupHom\n (a \\ b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191f \u2218 \u2191g,\n map_sup' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2294 b) =\n SupHom.toFun src\u271d.toSupHom a \u2294 SupHom.toFun src\u271d.toSupHom b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2293 b) =\n SupHom.toFun src\u271d.toSupHom a \u2293 SupHom.toFun src\u271d.toSupHom b) }.toSupHom\n a \\\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191f \u2218 \u2191g,\n map_sup' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2294 b) =\n SupHom.toFun src\u271d.toSupHom a \u2294 SupHom.toFun src\u271d.toSupHom b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u03b1),\n SupHom.toFun src\u271d.toSupHom (a \u2293 b) =\n SupHom.toFun src\u271d.toSupHom a \u2293 SupHom.toFun src\u271d.toSupHom b) }.toSupHom\n b\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : BiheytingAlgebra \u03b1\ninst\u271d\u00b2 : BiheytingAlgebra \u03b2\ninst\u271d\u00b9 : BiheytingAlgebra \u03b3\ninst\u271d : BiheytingAlgebra \u03b4\nf f\u2081 f\u2082 : BiheytingHom \u03b1 \u03b2\ng g\u2081 g\u2082 : BiheytingHom \u03b2 \u03b3\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\na : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a)\n[PROOFSTEP]\nrw [\u2190 comp_apply, h, comp_apply]\n", "meta": {"mathlib_filename": "Mathlib.Order.Heyting.Hom", "llama_tokens": 21708, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7879311956428946, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4490046809690304}} {"text": "[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : DecidableEq m\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : DecidableEq n\ninst\u271d\u2077 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u2076 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u2075 : Field \ud835\udd42\ninst\u271d\u2074 : Ring \ud835\udd38\ninst\u271d\u00b3 : TopologicalSpace \ud835\udd38\ninst\u271d\u00b2 : TopologicalRing \ud835\udd38\ninst\u271d\u00b9 : Algebra \ud835\udd42 \ud835\udd38\ninst\u271d : T2Space \ud835\udd38\nv : m \u2192 \ud835\udd38\n\u22a2 exp \ud835\udd42 (diagonal v) = diagonal (exp \ud835\udd42 v)\n[PROOFSTEP]\nsimp_rw [exp_eq_tsum, diagonal_pow, \u2190 diagonal_smul, \u2190 diagonal_tsum]\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : DecidableEq m\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : DecidableEq n\ninst\u271d\u2077 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u2076 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u2075 : Field \ud835\udd42\ninst\u271d\u2074 : Ring \ud835\udd38\ninst\u271d\u00b3 : TopologicalSpace \ud835\udd38\ninst\u271d\u00b2 : TopologicalRing \ud835\udd38\ninst\u271d\u00b9 : Algebra \ud835\udd42 \ud835\udd38\ninst\u271d : T2Space \ud835\udd38\nv : m \u2192 Matrix n n \ud835\udd38\n\u22a2 exp \ud835\udd42 (blockDiagonal v) = blockDiagonal (exp \ud835\udd42 v)\n[PROOFSTEP]\nsimp_rw [exp_eq_tsum, \u2190 blockDiagonal_pow, \u2190 blockDiagonal_smul, \u2190 blockDiagonal_tsum]\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : DecidableEq m\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : DecidableEq n\ninst\u271d\u2077 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u2076 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u2075 : Field \ud835\udd42\ninst\u271d\u2074 : Ring \ud835\udd38\ninst\u271d\u00b3 : TopologicalSpace \ud835\udd38\ninst\u271d\u00b2 : TopologicalRing \ud835\udd38\ninst\u271d\u00b9 : Algebra \ud835\udd42 \ud835\udd38\ninst\u271d : T2Space \ud835\udd38\nv : (i : m) \u2192 Matrix (n' i) (n' i) \ud835\udd38\n\u22a2 exp \ud835\udd42 (blockDiagonal' v) = blockDiagonal' (exp \ud835\udd42 v)\n[PROOFSTEP]\nsimp_rw [exp_eq_tsum, \u2190 blockDiagonal'_pow, \u2190 blockDiagonal'_smul, \u2190 blockDiagonal'_tsum]\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : Field \ud835\udd42\ninst\u271d\u2074 : CommRing \ud835\udd38\ninst\u271d\u00b3 : TopologicalSpace \ud835\udd38\ninst\u271d\u00b2 : TopologicalRing \ud835\udd38\ninst\u271d\u00b9 : Algebra \ud835\udd42 \ud835\udd38\ninst\u271d : T2Space \ud835\udd38\nA : Matrix m m \ud835\udd38\n\u22a2 exp \ud835\udd42 A\u1d40 = (exp \ud835\udd42 A)\u1d40\n[PROOFSTEP]\nsimp_rw [exp_eq_tsum, transpose_tsum, transpose_smul, transpose_pow]\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA B : Matrix m m \ud835\udd38\nh : Commute A B\n\u22a2 exp \ud835\udd42 (A + B) = exp \ud835\udd42 A * exp \ud835\udd42 B\n[PROOFSTEP]\nletI : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA B : Matrix m m \ud835\udd38\nh : Commute A B\nthis : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\n\u22a2 exp \ud835\udd42 (A + B) = exp \ud835\udd42 A * exp \ud835\udd42 B\n[PROOFSTEP]\nletI : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA B : Matrix m m \ud835\udd38\nh : Commute A B\nthis\u271d : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\n\u22a2 exp \ud835\udd42 (A + B) = exp \ud835\udd42 A * exp \ud835\udd42 B\n[PROOFSTEP]\nletI : NormedAlgebra \ud835\udd42 (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedAlgebra\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA B : Matrix m m \ud835\udd38\nh : Commute A B\nthis\u271d\u00b9 : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis\u271d : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\nthis : NormedAlgebra \ud835\udd42 (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedAlgebra\n\u22a2 exp \ud835\udd42 (A + B) = exp \ud835\udd42 A * exp \ud835\udd42 B\n[PROOFSTEP]\nexact exp_add_of_commute h\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\n\u03b9 : Type u_7\ns : Finset \u03b9\nf : \u03b9 \u2192 Matrix m m \ud835\udd38\nh : Set.Pairwise \u2191s fun i j => Commute (f i) (f j)\n\u22a2 exp \ud835\udd42 (\u2211 i in s, f i) =\n Finset.noncommProd s (fun i => exp \ud835\udd42 (f i))\n (_ : \u2200 (i : \u03b9), i \u2208 \u2191s \u2192 \u2200 (j : \u03b9), j \u2208 \u2191s \u2192 i \u2260 j \u2192 Commute (exp \ud835\udd42 (f i)) (exp \ud835\udd42 (f j)))\n[PROOFSTEP]\nletI : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\n\u03b9 : Type u_7\ns : Finset \u03b9\nf : \u03b9 \u2192 Matrix m m \ud835\udd38\nh : Set.Pairwise \u2191s fun i j => Commute (f i) (f j)\nthis : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\n\u22a2 exp \ud835\udd42 (\u2211 i in s, f i) =\n Finset.noncommProd s (fun i => exp \ud835\udd42 (f i))\n (_ : \u2200 (i : \u03b9), i \u2208 \u2191s \u2192 \u2200 (j : \u03b9), j \u2208 \u2191s \u2192 i \u2260 j \u2192 Commute (exp \ud835\udd42 (f i)) (exp \ud835\udd42 (f j)))\n[PROOFSTEP]\nletI : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\n\u03b9 : Type u_7\ns : Finset \u03b9\nf : \u03b9 \u2192 Matrix m m \ud835\udd38\nh : Set.Pairwise \u2191s fun i j => Commute (f i) (f j)\nthis\u271d : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\n\u22a2 exp \ud835\udd42 (\u2211 i in s, f i) =\n Finset.noncommProd s (fun i => exp \ud835\udd42 (f i))\n (_ : \u2200 (i : \u03b9), i \u2208 \u2191s \u2192 \u2200 (j : \u03b9), j \u2208 \u2191s \u2192 i \u2260 j \u2192 Commute (exp \ud835\udd42 (f i)) (exp \ud835\udd42 (f j)))\n[PROOFSTEP]\nletI : NormedAlgebra \ud835\udd42 (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedAlgebra\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\n\u03b9 : Type u_7\ns : Finset \u03b9\nf : \u03b9 \u2192 Matrix m m \ud835\udd38\nh : Set.Pairwise \u2191s fun i j => Commute (f i) (f j)\nthis\u271d\u00b9 : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis\u271d : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\nthis : NormedAlgebra \ud835\udd42 (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedAlgebra\n\u22a2 exp \ud835\udd42 (\u2211 i in s, f i) =\n Finset.noncommProd s (fun i => exp \ud835\udd42 (f i))\n (_ : \u2200 (i : \u03b9), i \u2208 \u2191s \u2192 \u2200 (j : \u03b9), j \u2208 \u2191s \u2192 i \u2260 j \u2192 Commute (exp \ud835\udd42 (f i)) (exp \ud835\udd42 (f j)))\n[PROOFSTEP]\nexact exp_sum_of_commute s f h\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn\u271d : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\u271d\ninst\u271d\u2075 : DecidableEq n\u271d\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nn : \u2115\nA : Matrix m m \ud835\udd38\n\u22a2 exp \ud835\udd42 (n \u2022 A) = exp \ud835\udd42 A ^ n\n[PROOFSTEP]\nletI : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn\u271d : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\u271d\ninst\u271d\u2075 : DecidableEq n\u271d\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nn : \u2115\nA : Matrix m m \ud835\udd38\nthis : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\n\u22a2 exp \ud835\udd42 (n \u2022 A) = exp \ud835\udd42 A ^ n\n[PROOFSTEP]\nletI : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn\u271d : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\u271d\ninst\u271d\u2075 : DecidableEq n\u271d\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nn : \u2115\nA : Matrix m m \ud835\udd38\nthis\u271d : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\n\u22a2 exp \ud835\udd42 (n \u2022 A) = exp \ud835\udd42 A ^ n\n[PROOFSTEP]\nletI : NormedAlgebra \ud835\udd42 (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedAlgebra\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn\u271d : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\u271d\ninst\u271d\u2075 : DecidableEq n\u271d\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nn : \u2115\nA : Matrix m m \ud835\udd38\nthis\u271d\u00b9 : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis\u271d : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\nthis : NormedAlgebra \ud835\udd42 (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedAlgebra\n\u22a2 exp \ud835\udd42 (n \u2022 A) = exp \ud835\udd42 A ^ n\n[PROOFSTEP]\nexact exp_nsmul n A\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\n\u22a2 IsUnit (exp \ud835\udd42 A)\n[PROOFSTEP]\nletI : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nthis : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\n\u22a2 IsUnit (exp \ud835\udd42 A)\n[PROOFSTEP]\nletI : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nthis\u271d : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\n\u22a2 IsUnit (exp \ud835\udd42 A)\n[PROOFSTEP]\nletI : NormedAlgebra \ud835\udd42 (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedAlgebra\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nthis\u271d\u00b9 : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis\u271d : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\nthis : NormedAlgebra \ud835\udd42 (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedAlgebra\n\u22a2 IsUnit (exp \ud835\udd42 A)\n[PROOFSTEP]\nexact isUnit_exp _ A\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nU : (Matrix m m \ud835\udd38)\u02e3\nA : Matrix m m \ud835\udd38\n\u22a2 exp \ud835\udd42 (\u2191U * A * \u2191U\u207b\u00b9) = \u2191U * exp \ud835\udd42 A * \u2191U\u207b\u00b9\n[PROOFSTEP]\nletI : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nU : (Matrix m m \ud835\udd38)\u02e3\nA : Matrix m m \ud835\udd38\nthis : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\n\u22a2 exp \ud835\udd42 (\u2191U * A * \u2191U\u207b\u00b9) = \u2191U * exp \ud835\udd42 A * \u2191U\u207b\u00b9\n[PROOFSTEP]\nletI : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nU : (Matrix m m \ud835\udd38)\u02e3\nA : Matrix m m \ud835\udd38\nthis\u271d : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\n\u22a2 exp \ud835\udd42 (\u2191U * A * \u2191U\u207b\u00b9) = \u2191U * exp \ud835\udd42 A * \u2191U\u207b\u00b9\n[PROOFSTEP]\nletI : NormedAlgebra \ud835\udd42 (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedAlgebra\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nU : (Matrix m m \ud835\udd38)\u02e3\nA : Matrix m m \ud835\udd38\nthis\u271d\u00b9 : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis\u271d : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\nthis : NormedAlgebra \ud835\udd42 (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedAlgebra\n\u22a2 exp \ud835\udd42 (\u2191U * A * \u2191U\u207b\u00b9) = \u2191U * exp \ud835\udd42 A * \u2191U\u207b\u00b9\n[PROOFSTEP]\nexact exp_units_conj _ U A\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\n\u22a2 exp \ud835\udd42 (-A) = (exp \ud835\udd42 A)\u207b\u00b9\n[PROOFSTEP]\nrw [nonsing_inv_eq_ring_inverse]\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\n\u22a2 exp \ud835\udd42 (-A) = Ring.inverse (exp \ud835\udd42 A)\n[PROOFSTEP]\nletI : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nthis : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\n\u22a2 exp \ud835\udd42 (-A) = Ring.inverse (exp \ud835\udd42 A)\n[PROOFSTEP]\nletI : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nthis\u271d : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\n\u22a2 exp \ud835\udd42 (-A) = Ring.inverse (exp \ud835\udd42 A)\n[PROOFSTEP]\nletI : NormedAlgebra \ud835\udd42 (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedAlgebra\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nthis\u271d\u00b9 : SeminormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpSemiNormedRing\nthis\u271d : NormedRing (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedRing\nthis : NormedAlgebra \ud835\udd42 (Matrix m m \ud835\udd38) := Matrix.linftyOpNormedAlgebra\n\u22a2 exp \ud835\udd42 (-A) = Ring.inverse (exp \ud835\udd42 A)\n[PROOFSTEP]\nexact (Ring.inverse_exp _ A).symm\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nz : \u2124\nA : Matrix m m \ud835\udd38\n\u22a2 exp \ud835\udd42 (z \u2022 A) = exp \ud835\udd42 A ^ z\n[PROOFSTEP]\nobtain \u27e8n, rfl | rfl\u27e9 := z.eq_nat_or_neg\n[GOAL]\ncase intro.inl\n\ud835\udd42 : Type u_1\nm : Type u_2\nn\u271d : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\u271d\ninst\u271d\u2075 : DecidableEq n\u271d\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nn : \u2115\n\u22a2 exp \ud835\udd42 (\u2191n \u2022 A) = exp \ud835\udd42 A ^ \u2191n\n[PROOFSTEP]\nrw [zpow_ofNat, coe_nat_zsmul, exp_nsmul]\n[GOAL]\ncase intro.inr\n\ud835\udd42 : Type u_1\nm : Type u_2\nn\u271d : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\u271d\ninst\u271d\u2075 : DecidableEq n\u271d\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nn : \u2115\n\u22a2 exp \ud835\udd42 (-\u2191n \u2022 A) = exp \ud835\udd42 A ^ (-\u2191n)\n[PROOFSTEP]\nhave : IsUnit (exp \ud835\udd42 A).det := (Matrix.isUnit_iff_isUnit_det _).mp (isUnit_exp _ _)\n[GOAL]\ncase intro.inr\n\ud835\udd42 : Type u_1\nm : Type u_2\nn\u271d : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\u271d\ninst\u271d\u2075 : DecidableEq n\u271d\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nn : \u2115\nthis : IsUnit (det (exp \ud835\udd42 A))\n\u22a2 exp \ud835\udd42 (-\u2191n \u2022 A) = exp \ud835\udd42 A ^ (-\u2191n)\n[PROOFSTEP]\nrw [Matrix.zpow_neg this, zpow_ofNat, neg_smul, exp_neg, coe_nat_zsmul, exp_nsmul]\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nU A : Matrix m m \ud835\udd38\nhy : IsUnit U\nu : (Matrix m m \ud835\udd38)\u02e3\nhu : \u2191u = U\n\u22a2 exp \ud835\udd42 (\u2191u * A * (\u2191u)\u207b\u00b9) = \u2191u * exp \ud835\udd42 A * (\u2191u)\u207b\u00b9\n[PROOFSTEP]\nsimpa only [Matrix.coe_units_inv] using exp_units_conj \ud835\udd42 u A\n[GOAL]\n\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : IsROrC \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nU A : Matrix m m \ud835\udd38\nhy : IsUnit U\nu : (Matrix m m \ud835\udd38)\u02e3\nhu : \u2191u = U\n\u22a2 exp \ud835\udd42 ((\u2191u)\u207b\u00b9 * A * \u2191u) = (\u2191u)\u207b\u00b9 * exp \ud835\udd42 A * \u2191u\n[PROOFSTEP]\nsimpa only [Matrix.coe_units_inv] using exp_units_conj' \ud835\udd42 u A\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.MatrixExponential", "llama_tokens": 11943, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7879311956428946, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4490046809690304}} {"text": "[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u2074 : LE \u03b1\ninst\u271d\u00b3 : OrderTop \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderTop \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nf : F\na : \u03b1\n\u22a2 \u2191f a = \u22a4 \u2194 a = \u22a4\n[PROOFSTEP]\nletI : TopHomClass F \u03b1 \u03b2 := OrderIsoClass.toTopHomClass\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u2074 : LE \u03b1\ninst\u271d\u00b3 : OrderTop \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderTop \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nf : F\na : \u03b1\nthis : TopHomClass F \u03b1 \u03b2 := OrderIsoClass.toTopHomClass\n\u22a2 \u2191f a = \u22a4 \u2194 a = \u22a4\n[PROOFSTEP]\nrw [\u2190 map_top f, (EquivLike.injective f).eq_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u2074 : LE \u03b1\ninst\u271d\u00b3 : OrderBot \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderBot \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nf : F\na : \u03b1\n\u22a2 \u2191f a = \u22a5 \u2194 a = \u22a5\n[PROOFSTEP]\nletI : BotHomClass F \u03b1 \u03b2 := OrderIsoClass.toBotHomClass\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u2074 : LE \u03b1\ninst\u271d\u00b3 : OrderBot \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderBot \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nf : F\na : \u03b1\nthis : BotHomClass F \u03b1 \u03b2 := OrderIsoClass.toBotHomClass\n\u22a2 \u2191f a = \u22a5 \u2194 a = \u22a5\n[PROOFSTEP]\nrw [\u2190 map_bot f, (EquivLike.injective f).eq_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : Top \u03b1\ninst\u271d\u00b2 : Top \u03b2\ninst\u271d\u00b9 : Top \u03b3\ninst\u271d : Top \u03b4\nf g : TopHom \u03b1 \u03b2\nh : f.toFun = g.toFun\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : Top \u03b1\ninst\u271d\u00b2 : Top \u03b2\ninst\u271d\u00b9 : Top \u03b3\ninst\u271d : Top \u03b4\ng : TopHom \u03b1 \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_top'\u271d : toFun\u271d \u22a4 = \u22a4\nh : { toFun := toFun\u271d, map_top' := map_top'\u271d }.toFun = g.toFun\n\u22a2 { toFun := toFun\u271d, map_top' := map_top'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : Top \u03b1\ninst\u271d\u00b2 : Top \u03b2\ninst\u271d\u00b9 : Top \u03b3\ninst\u271d : Top \u03b4\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmap_top'\u271d\u00b9 : toFun\u271d\u00b9 \u22a4 = \u22a4\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_top'\u271d : toFun\u271d \u22a4 = \u22a4\nh : { toFun := toFun\u271d\u00b9, map_top' := map_top'\u271d\u00b9 }.toFun = { toFun := toFun\u271d, map_top' := map_top'\u271d }.toFun\n\u22a2 { toFun := toFun\u271d\u00b9, map_top' := map_top'\u271d\u00b9 } = { toFun := toFun\u271d, map_top' := map_top'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : Top \u03b1\ninst\u271d\u00b2 : Top \u03b2\ninst\u271d\u00b9 : Top \u03b3\ninst\u271d : Top \u03b4\nf : TopHom \u03b2 \u03b3\ng : TopHom \u03b1 \u03b2\n\u22a2 (\u2191f \u2218 \u2191g) \u22a4 = \u22a4\n[PROOFSTEP]\nrw [comp_apply, map_top, map_top]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : Top \u03b1\ninst\u271d\u00b2 : Top \u03b2\ninst\u271d\u00b9 : Top \u03b3\ninst\u271d : Top \u03b4\ng : TopHom \u03b2 \u03b3\nf\u2081 f\u2082 : TopHom \u03b1 \u03b2\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\na : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a)\n[PROOFSTEP]\nrw [\u2190 TopHom.comp_apply, h, TopHom.comp_apply]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b2 : Top \u03b1\ninst\u271d\u00b9 : SemilatticeInf \u03b2\ninst\u271d : OrderTop \u03b2\nf\u271d g\u271d f g : TopHom \u03b1 \u03b2\n\u22a2 (\u2191f \u2293 \u2191g) \u22a4 = \u22a4\n[PROOFSTEP]\nrw [Pi.inf_apply, map_top, map_top, inf_top_eq]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b2 : Top \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : OrderTop \u03b2\nf\u271d g\u271d f g : TopHom \u03b1 \u03b2\n\u22a2 (\u2191f \u2294 \u2191g) \u22a4 = \u22a4\n[PROOFSTEP]\nrw [Pi.sup_apply, map_top, map_top, sup_top_eq]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : Bot \u03b1\ninst\u271d\u00b2 : Bot \u03b2\ninst\u271d\u00b9 : Bot \u03b3\ninst\u271d : Bot \u03b4\nf g : BotHom \u03b1 \u03b2\nh : f.toFun = g.toFun\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : Bot \u03b1\ninst\u271d\u00b2 : Bot \u03b2\ninst\u271d\u00b9 : Bot \u03b3\ninst\u271d : Bot \u03b4\ng : BotHom \u03b1 \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_bot'\u271d : toFun\u271d \u22a5 = \u22a5\nh : { toFun := toFun\u271d, map_bot' := map_bot'\u271d }.toFun = g.toFun\n\u22a2 { toFun := toFun\u271d, map_bot' := map_bot'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : Bot \u03b1\ninst\u271d\u00b2 : Bot \u03b2\ninst\u271d\u00b9 : Bot \u03b3\ninst\u271d : Bot \u03b4\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmap_bot'\u271d\u00b9 : toFun\u271d\u00b9 \u22a5 = \u22a5\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_bot'\u271d : toFun\u271d \u22a5 = \u22a5\nh : { toFun := toFun\u271d\u00b9, map_bot' := map_bot'\u271d\u00b9 }.toFun = { toFun := toFun\u271d, map_bot' := map_bot'\u271d }.toFun\n\u22a2 { toFun := toFun\u271d\u00b9, map_bot' := map_bot'\u271d\u00b9 } = { toFun := toFun\u271d, map_bot' := map_bot'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : Bot \u03b1\ninst\u271d\u00b2 : Bot \u03b2\ninst\u271d\u00b9 : Bot \u03b3\ninst\u271d : Bot \u03b4\nf : BotHom \u03b2 \u03b3\ng : BotHom \u03b1 \u03b2\n\u22a2 (\u2191f \u2218 \u2191g) \u22a5 = \u22a5\n[PROOFSTEP]\nrw [comp_apply, map_bot, map_bot]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : Bot \u03b1\ninst\u271d\u00b2 : Bot \u03b2\ninst\u271d\u00b9 : Bot \u03b3\ninst\u271d : Bot \u03b4\ng : BotHom \u03b2 \u03b3\nf\u2081 f\u2082 : BotHom \u03b1 \u03b2\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\na : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a)\n[PROOFSTEP]\nrw [\u2190 BotHom.comp_apply, h, BotHom.comp_apply]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b2 : Bot \u03b1\ninst\u271d\u00b9 : SemilatticeInf \u03b2\ninst\u271d : OrderBot \u03b2\nf\u271d g\u271d f g : BotHom \u03b1 \u03b2\n\u22a2 (\u2191f \u2293 \u2191g) \u22a5 = \u22a5\n[PROOFSTEP]\nrw [Pi.inf_apply, map_bot, map_bot, inf_bot_eq]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b2 : Bot \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : OrderBot \u03b2\nf\u271d g\u271d f g : BotHom \u03b1 \u03b2\n\u22a2 (\u2191f \u2294 \u2191g) \u22a5 = \u22a5\n[PROOFSTEP]\nrw [Pi.sup_apply, map_bot, map_bot, sup_bot_eq]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : BoundedOrder \u03b2\ninst\u271d\u00b9 : BoundedOrder \u03b3\ninst\u271d : BoundedOrder \u03b4\nf g : BoundedOrderHom \u03b1 \u03b2\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\nobtain \u27e8\u27e8_, _\u27e9, _\u27e9 := f\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : BoundedOrder \u03b2\ninst\u271d\u00b9 : BoundedOrder \u03b3\ninst\u271d : BoundedOrder \u03b4\ng : BoundedOrderHom \u03b1 \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmonotone'\u271d : Monotone toFun\u271d\nmap_top'\u271d : OrderHom.toFun { toFun := toFun\u271d, monotone' := monotone'\u271d } \u22a4 = \u22a4\nmap_bot'\u271d : OrderHom.toFun { toFun := toFun\u271d, monotone' := monotone'\u271d } \u22a5 = \u22a5\nh :\n (fun f => f.toFun)\n { toOrderHom := { toFun := toFun\u271d, monotone' := monotone'\u271d }, map_top' := map_top'\u271d, map_bot' := map_bot'\u271d } =\n (fun f => f.toFun) g\n\u22a2 { toOrderHom := { toFun := toFun\u271d, monotone' := monotone'\u271d }, map_top' := map_top'\u271d, map_bot' := map_bot'\u271d } = g\n[PROOFSTEP]\nobtain \u27e8\u27e8_, _\u27e9, _\u27e9 := g\n[GOAL]\ncase mk.mk.mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : BoundedOrder \u03b2\ninst\u271d\u00b9 : BoundedOrder \u03b3\ninst\u271d : BoundedOrder \u03b4\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmonotone'\u271d\u00b9 : Monotone toFun\u271d\u00b9\nmap_top'\u271d\u00b9 : OrderHom.toFun { toFun := toFun\u271d\u00b9, monotone' := monotone'\u271d\u00b9 } \u22a4 = \u22a4\nmap_bot'\u271d\u00b9 : OrderHom.toFun { toFun := toFun\u271d\u00b9, monotone' := monotone'\u271d\u00b9 } \u22a5 = \u22a5\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmonotone'\u271d : Monotone toFun\u271d\nmap_top'\u271d : OrderHom.toFun { toFun := toFun\u271d, monotone' := monotone'\u271d } \u22a4 = \u22a4\nmap_bot'\u271d : OrderHom.toFun { toFun := toFun\u271d, monotone' := monotone'\u271d } \u22a5 = \u22a5\nh :\n (fun f => f.toFun)\n { toOrderHom := { toFun := toFun\u271d\u00b9, monotone' := monotone'\u271d\u00b9 }, map_top' := map_top'\u271d\u00b9, map_bot' := map_bot'\u271d\u00b9 } =\n (fun f => f.toFun)\n { toOrderHom := { toFun := toFun\u271d, monotone' := monotone'\u271d }, map_top' := map_top'\u271d, map_bot' := map_bot'\u271d }\n\u22a2 { toOrderHom := { toFun := toFun\u271d\u00b9, monotone' := monotone'\u271d\u00b9 }, map_top' := map_top'\u271d\u00b9, map_bot' := map_bot'\u271d\u00b9 } =\n { toOrderHom := { toFun := toFun\u271d, monotone' := monotone'\u271d }, map_top' := map_top'\u271d, map_bot' := map_bot'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : BoundedOrder \u03b2\ninst\u271d\u00b9 : BoundedOrder \u03b3\ninst\u271d : BoundedOrder \u03b4\ng : BoundedOrderHom \u03b2 \u03b3\nf\u2081 f\u2082 : BoundedOrderHom \u03b1 \u03b2\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\na : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a)\n[PROOFSTEP]\nrw [\u2190 BoundedOrderHom.comp_apply, h, BoundedOrderHom.comp_apply]\n", "meta": {"mathlib_filename": "Mathlib.Order.Hom.Bounded", "llama_tokens": 4388, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7826624789529376, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.4489965644985547}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nx y : E\n\u22a2 x \u2208 \u2191(convexHull \ud835\udd5c) s \u2194 \u2200 (t : Set E), s \u2286 t \u2192 Convex \ud835\udd5c t \u2192 x \u2208 t\n[PROOFSTEP]\nsimp_rw [convexHull_eq_iInter, mem_iInter]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nx y : E\n\u22a2 \u2191(convexHull \ud835\udd5c) s = \u2205 \u2194 s = \u2205\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nx y : E\n\u22a2 \u2191(convexHull \ud835\udd5c) s = \u2205 \u2192 s = \u2205\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nx y : E\nh : \u2191(convexHull \ud835\udd5c) s = \u2205\n\u22a2 s = \u2205\n[PROOFSTEP]\nrw [\u2190 Set.subset_empty_iff, \u2190 h]\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nx y : E\nh : \u2191(convexHull \ud835\udd5c) s = \u2205\n\u22a2 s \u2286 \u2191(convexHull \ud835\udd5c) s\n[PROOFSTEP]\nexact subset_convexHull \ud835\udd5c _\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nx y : E\n\u22a2 s = \u2205 \u2192 \u2191(convexHull \ud835\udd5c) s = \u2205\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nt : Set E\nx y : E\n\u22a2 \u2191(convexHull \ud835\udd5c) \u2205 = \u2205\n[PROOFSTEP]\nexact convexHull_empty\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nx y : E\n\u22a2 Set.Nonempty (\u2191(convexHull \ud835\udd5c) s) \u2194 Set.Nonempty s\n[PROOFSTEP]\nrw [nonempty_iff_ne_empty, nonempty_iff_ne_empty, Ne.def, Ne.def]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nx y : E\n\u22a2 \u00ac\u2191(convexHull \ud835\udd5c) s = \u2205 \u2194 \u00acs = \u2205\n[PROOFSTEP]\nexact not_congr convexHull_empty_iff\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nx\u271d y\u271d x y : E\n\u22a2 \u2191(convexHull \ud835\udd5c) {x, y} = segment \ud835\udd5c x y\n[PROOFSTEP]\nrefine\n (convexHull_min ?_ <| convex_segment _ _).antisymm\n (segment_subset_convexHull (mem_insert _ _) <| subset_insert _ _ <| mem_singleton _)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nx\u271d y\u271d x y : E\n\u22a2 {x, y} \u2286 segment \ud835\udd5c x y\n[PROOFSTEP]\nrw [insert_subset_iff, singleton_subset_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nx\u271d y\u271d x y : E\n\u22a2 x \u2208 segment \ud835\udd5c x y \u2227 y \u2208 segment \ud835\udd5c x y\n[PROOFSTEP]\nexact \u27e8left_mem_segment _ _ _, right_mem_segment _ _ _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\n\u22a2 Convex \ud835\udd5c (s \\ {x}) \u2194 \u00acx \u2208 \u2191(convexHull \ud835\udd5c) (s \\ {x})\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\n\u22a2 Convex \ud835\udd5c (s \\ {x}) \u2192 \u00acx \u2208 \u2191(convexHull \ud835\udd5c) (s \\ {x})\n[PROOFSTEP]\nrintro hsx hx\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhsx : Convex \ud835\udd5c (s \\ {x})\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) (s \\ {x})\n\u22a2 False\n[PROOFSTEP]\nrw [hsx.convexHull_eq] at hx \n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhsx : Convex \ud835\udd5c (s \\ {x})\nhx : x \u2208 s \\ {x}\n\u22a2 False\n[PROOFSTEP]\nexact hx.2 (mem_singleton _)\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\n\u22a2 \u00acx \u2208 \u2191(convexHull \ud835\udd5c) (s \\ {x}) \u2192 Convex \ud835\udd5c (s \\ {x})\n[PROOFSTEP]\nrintro hx\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhx : \u00acx \u2208 \u2191(convexHull \ud835\udd5c) (s \\ {x})\n\u22a2 Convex \ud835\udd5c (s \\ {x})\n[PROOFSTEP]\nsuffices h : s \\ { x } = convexHull \ud835\udd5c (s \\ { x })\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhx : \u00acx \u2208 \u2191(convexHull \ud835\udd5c) (s \\ {x})\nh : s \\ {x} = \u2191(convexHull \ud835\udd5c) (s \\ {x})\n\u22a2 Convex \ud835\udd5c (s \\ {x})\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhx : \u00acx \u2208 \u2191(convexHull \ud835\udd5c) (s \\ {x})\nh : s \\ {x} = \u2191(convexHull \ud835\udd5c) (s \\ {x})\n\u22a2 Convex \ud835\udd5c (\u2191(convexHull \ud835\udd5c) (s \\ {x}))\n[PROOFSTEP]\nexact convex_convexHull \ud835\udd5c _\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhx : \u00acx \u2208 \u2191(convexHull \ud835\udd5c) (s \\ {x})\n\u22a2 s \\ {x} = \u2191(convexHull \ud835\udd5c) (s \\ {x})\n[PROOFSTEP]\nexact\n Subset.antisymm (subset_convexHull \ud835\udd5c _) fun y hy =>\n \u27e8convexHull_min (diff_subset _ _) hs hy, by\n rintro (rfl : y = x)\n exact hx hy\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y\u271d : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhx : \u00acx \u2208 \u2191(convexHull \ud835\udd5c) (s \\ {x})\ny : E\nhy : y \u2208 \u2191(convexHull \ud835\udd5c) (s \\ {x})\n\u22a2 \u00acy \u2208 {x}\n[PROOFSTEP]\nrintro (rfl : y = x)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx y\u271d : E\ns : Set E\nhs : Convex \ud835\udd5c s\ny : E\nhx : \u00acy \u2208 \u2191(convexHull \ud835\udd5c) (s \\ {y})\nhy : y \u2208 \u2191(convexHull \ud835\udd5c) (s \\ {y})\n\u22a2 False\n[PROOFSTEP]\nexact hx hy\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns : Set E\nf : E \u2192\u1d43[\ud835\udd5c] F\n\u22a2 \u2191f '' \u2191(convexHull \ud835\udd5c) s = \u2191(convexHull \ud835\udd5c) (\u2191f '' s)\n[PROOFSTEP]\napply Set.Subset.antisymm\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns : Set E\nf : E \u2192\u1d43[\ud835\udd5c] F\n\u22a2 \u2191f '' \u2191(convexHull \ud835\udd5c) s \u2286 \u2191(convexHull \ud835\udd5c) (\u2191f '' s)\n[PROOFSTEP]\nrw [Set.image_subset_iff]\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns : Set E\nf : E \u2192\u1d43[\ud835\udd5c] F\n\u22a2 \u2191(convexHull \ud835\udd5c) s \u2286 \u2191f \u207b\u00b9' \u2191(convexHull \ud835\udd5c) (\u2191f '' s)\n[PROOFSTEP]\nrefine' convexHull_min _ ((convex_convexHull \ud835\udd5c (f '' s)).affine_preimage f)\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns : Set E\nf : E \u2192\u1d43[\ud835\udd5c] F\n\u22a2 s \u2286 \u2191f \u207b\u00b9' \u2191(convexHull \ud835\udd5c) (\u2191f '' s)\n[PROOFSTEP]\nrw [\u2190 Set.image_subset_iff]\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns : Set E\nf : E \u2192\u1d43[\ud835\udd5c] F\n\u22a2 \u2191f '' s \u2286 \u2191(convexHull \ud835\udd5c) (\u2191f '' s)\n[PROOFSTEP]\nexact subset_convexHull \ud835\udd5c (f '' s)\n[GOAL]\ncase h\u2082\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns : Set E\nf : E \u2192\u1d43[\ud835\udd5c] F\n\u22a2 \u2191(convexHull \ud835\udd5c) (\u2191f '' s) \u2286 \u2191f '' \u2191(convexHull \ud835\udd5c) s\n[PROOFSTEP]\nexact convexHull_min (Set.image_subset _ (subset_convexHull \ud835\udd5c s)) ((convex_convexHull \ud835\udd5c s).affine_image f)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns : Set E\n\u22a2 affineSpan \ud835\udd5c (\u2191(convexHull \ud835\udd5c) s) = affineSpan \ud835\udd5c s\n[PROOFSTEP]\nrefine' le_antisymm _ (affineSpan_mono \ud835\udd5c (subset_convexHull \ud835\udd5c s))\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns : Set E\n\u22a2 affineSpan \ud835\udd5c (\u2191(convexHull \ud835\udd5c) s) \u2264 affineSpan \ud835\udd5c s\n[PROOFSTEP]\nrw [affineSpan_le]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns : Set E\n\u22a2 \u2191(convexHull \ud835\udd5c) s \u2286 \u2191(affineSpan \ud835\udd5c s)\n[PROOFSTEP]\nexact convexHull_subset_affineSpan s\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d s : Set E\n\u22a2 \u2191(convexHull \ud835\udd5c) (-s) = -\u2191(convexHull \ud835\udd5c) s\n[PROOFSTEP]\nsimp_rw [\u2190 image_neg]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d s : Set E\n\u22a2 \u2191(convexHull \ud835\udd5c) (Neg.neg '' s) = Neg.neg '' \u2191(convexHull \ud835\udd5c) s\n[PROOFSTEP]\nexact (AffineMap.image_convexHull _ <| -1).symm\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.Hull", "llama_tokens": 5555, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7772998611746912, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.4488870671879591}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nr : \u211d\u22650\n\u22a2 \u2191r \u2022 j = r \u2022 j\n[PROOFSTEP]\nrw [real_smul_def, if_pos (NNReal.coe_nonneg r), Real.toNNReal_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 (r \u2022 j).posPart = Real.toNNReal r \u2022 j.posPart\n[PROOFSTEP]\nrw [real_smul_def, \u2190 smul_posPart, if_pos hr]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 (r \u2022 j).negPart = Real.toNNReal r \u2022 j.negPart\n[PROOFSTEP]\nrw [real_smul_def, \u2190 smul_negPart, if_pos hr]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nr : \u211d\nhr : r < 0\n\u22a2 (r \u2022 j).posPart = Real.toNNReal (-r) \u2022 j.negPart\n[PROOFSTEP]\nrw [real_smul_def, \u2190 smul_negPart, if_neg (not_le.2 hr), neg_posPart]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nr : \u211d\nhr : r < 0\n\u22a2 (r \u2022 j).negPart = Real.toNNReal (-r) \u2022 j.posPart\n[PROOFSTEP]\nrw [real_smul_def, \u2190 smul_posPart, if_neg (not_le.2 hr), neg_negPart]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\n\u22a2 toSignedMeasure 0 = 0\n[PROOFSTEP]\next1 i hi\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(toSignedMeasure 0) i = \u21910 i\n[PROOFSTEP]\nrw [toSignedMeasure, toSignedMeasure_sub_apply hi, zero_posPart, zero_negPart, sub_self, VectorMeasure.coe_zero,\n Pi.zero_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\n\u22a2 toSignedMeasure (-j) = -toSignedMeasure j\n[PROOFSTEP]\next1 i hi\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(toSignedMeasure (-j)) i = \u2191(-toSignedMeasure j) i\n[PROOFSTEP]\nrw [neg_apply, toSignedMeasure, toSignedMeasure, toSignedMeasure_sub_apply hi, toSignedMeasure_sub_apply hi, neg_sub,\n neg_posPart, neg_negPart]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nr : \u211d\u22650\n\u22a2 toSignedMeasure (r \u2022 j) = r \u2022 toSignedMeasure j\n[PROOFSTEP]\next1 i hi\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nr : \u211d\u22650\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(toSignedMeasure (r \u2022 j)) i = \u2191(r \u2022 toSignedMeasure j) i\n[PROOFSTEP]\nrw [VectorMeasure.smul_apply, toSignedMeasure, toSignedMeasure, toSignedMeasure_sub_apply hi,\n toSignedMeasure_sub_apply hi, smul_sub, smul_posPart, smul_negPart, \u2190 ENNReal.toReal_smul, \u2190 ENNReal.toReal_smul,\n smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\n\u22a2 \u2203 S,\n MeasurableSet S \u2227\n VectorMeasure.restrict (toSignedMeasure j) S \u2264 VectorMeasure.restrict 0 S \u2227\n VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j) S\u1d9c \u2227\n \u2191\u2191j.posPart S = 0 \u2227 \u2191\u2191j.negPart S\u1d9c = 0\n[PROOFSTEP]\nobtain \u27e8S, hS\u2081, hS\u2082, hS\u2083\u27e9 := j.mutuallySingular\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191j.posPart S = 0\nhS\u2083 : \u2191\u2191j.negPart S\u1d9c = 0\n\u22a2 \u2203 S,\n MeasurableSet S \u2227\n VectorMeasure.restrict (toSignedMeasure j) S \u2264 VectorMeasure.restrict 0 S \u2227\n VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j) S\u1d9c \u2227\n \u2191\u2191j.posPart S = 0 \u2227 \u2191\u2191j.negPart S\u1d9c = 0\n[PROOFSTEP]\nrefine' \u27e8S, hS\u2081, _, _, hS\u2082, hS\u2083\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191j.posPart S = 0\nhS\u2083 : \u2191\u2191j.negPart S\u1d9c = 0\n\u22a2 VectorMeasure.restrict (toSignedMeasure j) S \u2264 VectorMeasure.restrict 0 S\n[PROOFSTEP]\nrefine' restrict_le_restrict_of_subset_le _ _ fun A hA hA\u2081 => _\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191j.posPart S = 0\nhS\u2083 : \u2191\u2191j.negPart S\u1d9c = 0\nA : Set \u03b1\nhA : MeasurableSet A\nhA\u2081 : A \u2286 S\n\u22a2 \u2191(toSignedMeasure j) A \u2264 \u21910 A\n[PROOFSTEP]\nrw [toSignedMeasure, toSignedMeasure_sub_apply hA,\n show j.posPart A = 0 from nonpos_iff_eq_zero.1 (hS\u2082 \u25b8 measure_mono hA\u2081), ENNReal.zero_toReal, zero_sub, neg_le,\n zero_apply, neg_zero]\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191j.posPart S = 0\nhS\u2083 : \u2191\u2191j.negPart S\u1d9c = 0\nA : Set \u03b1\nhA : MeasurableSet A\nhA\u2081 : A \u2286 S\n\u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191j.negPart A)\n[PROOFSTEP]\nexact ENNReal.toReal_nonneg\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191j.posPart S = 0\nhS\u2083 : \u2191\u2191j.negPart S\u1d9c = 0\n\u22a2 VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j) S\u1d9c\n[PROOFSTEP]\nrefine' restrict_le_restrict_of_subset_le _ _ fun A hA hA\u2081 => _\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191j.posPart S = 0\nhS\u2083 : \u2191\u2191j.negPart S\u1d9c = 0\nA : Set \u03b1\nhA : MeasurableSet A\nhA\u2081 : A \u2286 S\u1d9c\n\u22a2 \u21910 A \u2264 \u2191(toSignedMeasure j) A\n[PROOFSTEP]\nrw [toSignedMeasure, toSignedMeasure_sub_apply hA,\n show j.negPart A = 0 from nonpos_iff_eq_zero.1 (hS\u2083 \u25b8 measure_mono hA\u2081), ENNReal.zero_toReal, sub_zero]\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191j.posPart S = 0\nhS\u2083 : \u2191\u2191j.negPart S\u1d9c = 0\nA : Set \u03b1\nhA : MeasurableSet A\nhA\u2081 : A \u2286 S\u1d9c\n\u22a2 \u21910 A \u2264 ENNReal.toReal (\u2191\u2191j.posPart A)\n[PROOFSTEP]\nexact ENNReal.toReal_nonneg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns\u271d : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ns : SignedMeasure \u03b1\ni : Set \u03b1 :=\n choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)\nhi : MeasurableSet\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2227\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2264\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2227\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c \u2264\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c :=\n choose_spec (exists_compl_positive_negative s)\n\u22a2 toMeasureOfZeroLE s i\n (_ :\n MeasurableSet\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)))\n (_ :\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2264\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))) \u27c2\u2098\n toMeasureOfLEZero s i\u1d9c\n (_ :\n MeasurableSet\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c)\n (_ :\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c \u2264\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c)\n[PROOFSTEP]\nrefine'\n \u27e8i\u1d9c, hi.1.compl, _, _\u27e9\n -- Porting note: added `\u2190 NNReal.eq_iff`\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns\u271d : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ns : SignedMeasure \u03b1\ni : Set \u03b1 :=\n choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)\nhi : MeasurableSet\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2227\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2264\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2227\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c \u2264\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c :=\n choose_spec (exists_compl_positive_negative s)\n\u22a2 \u2191\u2191(toMeasureOfZeroLE s i\n (_ :\n MeasurableSet\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)))\n (_ :\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2264\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))))\n i\u1d9c =\n 0\n[PROOFSTEP]\nrw [toMeasureOfZeroLE_apply _ _ hi.1 hi.1.compl]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns\u271d : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ns : SignedMeasure \u03b1\ni : Set \u03b1 :=\n choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)\nhi : MeasurableSet\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2227\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2264\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2227\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c \u2264\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c :=\n choose_spec (exists_compl_positive_negative s)\n\u22a2 \u2191{\n val :=\n \u2191s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c) \u2229\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c),\n property :=\n (_ :\n 0 \u2264\n \u2191s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c) \u2229\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c)) } =\n 0\n[PROOFSTEP]\nsimp [\u2190 NNReal.eq_iff]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns\u271d : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ns : SignedMeasure \u03b1\ni : Set \u03b1 :=\n choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)\nhi : MeasurableSet\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2227\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2264\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2227\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c \u2264\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c :=\n choose_spec (exists_compl_positive_negative s)\n\u22a2 \u2191\u2191(toMeasureOfLEZero s i\u1d9c\n (_ :\n MeasurableSet\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c)\n (_ :\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c \u2264\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c))\n i\u1d9c\u1d9c =\n 0\n[PROOFSTEP]\nrw [toMeasureOfLEZero_apply _ _ hi.1.compl hi.1.compl.compl]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns\u271d : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ns : SignedMeasure \u03b1\ni : Set \u03b1 :=\n choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)\nhi : MeasurableSet\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2227\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2264\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2227\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c \u2264\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c :=\n choose_spec (exists_compl_positive_negative s)\n\u22a2 \u2191{\n val :=\n -\u2191s\n ((choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c \u2229\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c\u1d9c),\n property :=\n (_ :\n 0 \u2264\n -\u2191s\n ((choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c \u2229\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c\u1d9c)) } =\n 0\n[PROOFSTEP]\nsimp [\u2190 NNReal.eq_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns\u271d : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ns : SignedMeasure \u03b1\n\u22a2 \u2203 i hi\u2081 hi\u2082 hi\u2083,\n (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082 \u2227\n (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n[PROOFSTEP]\nset i := choose s.exists_compl_positive_negative\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns\u271d : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ns : SignedMeasure \u03b1\ni : Set \u03b1 :=\n choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)\n\u22a2 \u2203 i hi\u2081 hi\u2082 hi\u2083,\n (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082 \u2227\n (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n[PROOFSTEP]\nobtain \u27e8hi\u2081, hi\u2082, hi\u2083\u27e9 := choose_spec s.exists_compl_positive_negative\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns\u271d : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ns : SignedMeasure \u03b1\ni : Set \u03b1 :=\n choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)\nhi\u2081 :\n MeasurableSet\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\nhi\u2082 :\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c)) \u2264\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\nhi\u2083 :\n VectorMeasure.restrict s\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c \u2264\n VectorMeasure.restrict 0\n (choose\n (_ :\n \u2203 i,\n MeasurableSet i \u2227\n VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i \u2227\n VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c))\u1d9c\n\u22a2 \u2203 i hi\u2081 hi\u2082 hi\u2083,\n (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082 \u2227\n (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n[PROOFSTEP]\nexact \u27e8i, hi\u2081, hi\u2082, hi\u2083, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns\u271d : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ns : SignedMeasure \u03b1\n\u22a2 JordanDecomposition.toSignedMeasure (toJordanDecomposition s) = s\n[PROOFSTEP]\nobtain \u27e8i, hi\u2081, hi\u2082, hi\u2083, h\u03bc, h\u03bd\u27e9 := s.toJordanDecomposition_spec\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns\u271d : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nh\u03bc : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nh\u03bd : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 JordanDecomposition.toSignedMeasure (toJordanDecomposition s) = s\n[PROOFSTEP]\nsimp only [JordanDecomposition.toSignedMeasure, h\u03bc, h\u03bd]\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns\u271d : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nh\u03bc : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nh\u03bd : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 Measure.toSignedMeasure (toMeasureOfZeroLE s i hi\u2081 hi\u2082) -\n Measure.toSignedMeasure (toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083) =\n s\n[PROOFSTEP]\next k hk\n[GOAL]\ncase intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns\u271d : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nh\u03bc : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nh\u03bd : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\nk : Set \u03b1\nhk : MeasurableSet k\n\u22a2 \u2191(Measure.toSignedMeasure (toMeasureOfZeroLE s i hi\u2081 hi\u2082) -\n Measure.toSignedMeasure (toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083))\n k =\n \u2191s k\n[PROOFSTEP]\nrw [toSignedMeasure_sub_apply hk, toMeasureOfZeroLE_apply _ hi\u2082 hi\u2081 hk, toMeasureOfLEZero_apply _ hi\u2083 hi\u2081.compl hk]\n[GOAL]\ncase intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns\u271d : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nh\u03bc : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nh\u03bd : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\nk : Set \u03b1\nhk : MeasurableSet k\n\u22a2 ENNReal.toReal \u2191{ val := \u2191s (i \u2229 k), property := (_ : 0 \u2264 \u2191s (i \u2229 k)) } -\n ENNReal.toReal \u2191{ val := -\u2191s (i\u1d9c \u2229 k), property := (_ : 0 \u2264 -\u2191s (i\u1d9c \u2229 k)) } =\n \u2191s k\n[PROOFSTEP]\nsimp only [ENNReal.coe_toReal, NNReal.coe_mk, ENNReal.some_eq_coe, sub_neg_eq_add]\n[GOAL]\ncase intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns\u271d : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nh\u03bc : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nh\u03bd : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\nk : Set \u03b1\nhk : MeasurableSet k\n\u22a2 \u2191s (i \u2229 k) + \u2191s (i\u1d9c \u2229 k) = \u2191s k\n[PROOFSTEP]\nrw [\u2190 of_union _ (MeasurableSet.inter hi\u2081 hk) (MeasurableSet.inter hi\u2081.compl hk), Set.inter_comm i, Set.inter_comm i\u1d9c,\n Set.inter_union_compl _ _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns\u271d : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nh\u03bc : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nh\u03bd : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\nk : Set \u03b1\nhk : MeasurableSet k\n\u22a2 Disjoint (i \u2229 k) (i\u1d9c \u2229 k)\n[PROOFSTEP]\nexact (disjoint_compl_right.inf_left _).inf_right _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhw\u2081 : \u2191s w = 0\nhw\u2082 : w \u2286 u\nhwt : v \u2286 w\n\u22a2 \u2191s v = 0\n[PROOFSTEP]\nhave : s v + s (w \\ v) = 0 := by\n rw [\u2190 hw\u2081, \u2190 of_union Set.disjoint_sdiff_right hv (hw.diff hv), Set.union_diff_self,\n Set.union_eq_self_of_subset_left hwt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhw\u2081 : \u2191s w = 0\nhw\u2082 : w \u2286 u\nhwt : v \u2286 w\n\u22a2 \u2191s v + \u2191s (w \\ v) = 0\n[PROOFSTEP]\nrw [\u2190 hw\u2081, \u2190 of_union Set.disjoint_sdiff_right hv (hw.diff hv), Set.union_diff_self,\n Set.union_eq_self_of_subset_left hwt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhw\u2081 : \u2191s w = 0\nhw\u2082 : w \u2286 u\nhwt : v \u2286 w\nthis : \u2191s v + \u2191s (w \\ v) = 0\n\u22a2 \u2191s v = 0\n[PROOFSTEP]\nhave h\u2081 := nonneg_of_zero_le_restrict _ (restrict_le_restrict_subset _ _ hu hsu (hwt.trans hw\u2082))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhw\u2081 : \u2191s w = 0\nhw\u2082 : w \u2286 u\nhwt : v \u2286 w\nthis : \u2191s v + \u2191s (w \\ v) = 0\nh\u2081 : 0 \u2264 \u2191s v\n\u22a2 \u2191s v = 0\n[PROOFSTEP]\nhave h\u2082 := nonneg_of_zero_le_restrict _ (restrict_le_restrict_subset _ _ hu hsu ((w.diff_subset v).trans hw\u2082))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhw\u2081 : \u2191s w = 0\nhw\u2082 : w \u2286 u\nhwt : v \u2286 w\nthis : \u2191s v + \u2191s (w \\ v) = 0\nh\u2081 : 0 \u2264 \u2191s v\nh\u2082 : 0 \u2264 \u2191s (w \\ v)\n\u22a2 \u2191s v = 0\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict s u \u2264 VectorMeasure.restrict 0 u\nhw\u2081 : \u2191s w = 0\nhw\u2082 : w \u2286 u\nhwt : v \u2286 w\n\u22a2 \u2191s v = 0\n[PROOFSTEP]\nrw [\u2190 s.neg_le_neg_iff _ hu, neg_zero] at hsu \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhw\u2081 : \u2191s w = 0\nhw\u2082 : w \u2286 u\nhwt : v \u2286 w\n\u22a2 \u2191s v = 0\n[PROOFSTEP]\nhave := subset_positive_null_set hu hv hw hsu\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhw\u2081 : \u2191s w = 0\nhw\u2082 : w \u2286 u\nhwt : v \u2286 w\nthis : \u2191(-s) w = 0 \u2192 w \u2286 u \u2192 v \u2286 w \u2192 \u2191(-s) v = 0\n\u22a2 \u2191s v = 0\n[PROOFSTEP]\nsimp only [Pi.neg_apply, neg_eq_zero, coe_neg] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhw\u2081 : \u2191s w = 0\nhw\u2082 : w \u2286 u\nhwt : v \u2286 w\nthis : \u2191s w = 0 \u2192 w \u2286 u \u2192 v \u2286 w \u2192 \u2191s v = 0\n\u22a2 \u2191s v = 0\n[PROOFSTEP]\nexact this hw\u2081 hw\u2082 hwt\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0\n[PROOFSTEP]\nrw [restrict_le_restrict_iff] at hsu hsv \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet v\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet u\n[PROOFSTEP]\nhave a := hsu (hu.diff hv) (u.diff_subset v)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j\nhs : \u2191s (u \u2206 v) = 0\na : \u21910 (u \\ v) \u2264 \u2191s (u \\ v)\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet v\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet u\n[PROOFSTEP]\nhave b := hsv (hv.diff hu) (v.diff_subset u)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j\nhs : \u2191s (u \u2206 v) = 0\na : \u21910 (u \\ v) \u2264 \u2191s (u \\ v)\nb : \u21910 (v \\ u) \u2264 \u2191s (v \\ u)\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet v\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet u\n[PROOFSTEP]\nerw [of_union (Set.disjoint_of_subset_left (u.diff_subset v) disjoint_sdiff_self_right) (hu.diff hv) (hv.diff hu)] at hs \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j\nhs : \u2191s (u \\ v) + \u2191s (v \\ u) = 0\na : \u21910 (u \\ v) \u2264 \u2191s (u \\ v)\nb : \u21910 (v \\ u) \u2264 \u2191s (v \\ u)\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet v\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet u\n[PROOFSTEP]\nrw [zero_apply] at a b \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j\nhs : \u2191s (u \\ v) + \u2191s (v \\ u) = 0\na : 0 \u2264 \u2191s (u \\ v)\nb : 0 \u2264 \u2191s (v \\ u)\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet v\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet u\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j\nhs : \u2191s (u \\ v) + \u2191s (v \\ u) = 0\na : 0 \u2264 \u2191s (u \\ v)\nb : 0 \u2264 \u2191s (v \\ u)\n\u22a2 \u2191s (u \\ v) = 0\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j\nhs : \u2191s (u \\ v) + \u2191s (v \\ u) = 0\na : 0 \u2264 \u2191s (u \\ v)\nb : 0 \u2264 \u2191s (v \\ u)\n\u22a2 \u2191s (v \\ u) = 0\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet v\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet u\n[PROOFSTEP]\nall_goals\n first\n | linarith\n | infer_instance\n | assumption\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j\nhs : \u2191s (u \\ v) + \u2191s (v \\ u) = 0\na : 0 \u2264 \u2191s (u \\ v)\nb : 0 \u2264 \u2191s (v \\ u)\n\u22a2 \u2191s (u \\ v) = 0\n[PROOFSTEP]\nfirst\n| linarith\n| infer_instance\n| assumption\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j\nhs : \u2191s (u \\ v) + \u2191s (v \\ u) = 0\na : 0 \u2264 \u2191s (u \\ v)\nb : 0 \u2264 \u2191s (v \\ u)\n\u22a2 \u2191s (u \\ v) = 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j\nhs : \u2191s (u \\ v) + \u2191s (v \\ u) = 0\na : 0 \u2264 \u2191s (u \\ v)\nb : 0 \u2264 \u2191s (v \\ u)\n\u22a2 \u2191s (v \\ u) = 0\n[PROOFSTEP]\nfirst\n| linarith\n| infer_instance\n| assumption\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j\nhs : \u2191s (u \\ v) + \u2191s (v \\ u) = 0\na : 0 \u2264 \u2191s (u \\ v)\nb : 0 \u2264 \u2191s (v \\ u)\n\u22a2 \u2191s (v \\ u) = 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet v\n[PROOFSTEP]\nfirst\n| linarith\n| infer_instance\n| assumption\n[GOAL]\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet v\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet v\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet v\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet u\n[PROOFSTEP]\nfirst\n| linarith\n| infer_instance\n| assumption\n[GOAL]\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet u\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet u\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 MeasurableSet u\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict s u \u2264 VectorMeasure.restrict 0 u\nhsv : VectorMeasure.restrict s v \u2264 VectorMeasure.restrict 0 v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0\n[PROOFSTEP]\nrw [\u2190 s.neg_le_neg_iff _ hu, neg_zero] at hsu \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict s v \u2264 VectorMeasure.restrict 0 v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0\n[PROOFSTEP]\nrw [\u2190 s.neg_le_neg_iff _ hv, neg_zero] at hsv \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict (-s) v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0\n[PROOFSTEP]\nhave := of_diff_eq_zero_of_symmDiff_eq_zero_positive hu hv hsu hsv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict (-s) v\nhs : \u2191s (u \u2206 v) = 0\nthis : \u2191(-s) (u \u2206 v) = 0 \u2192 \u2191(-s) (u \\ v) = 0 \u2227 \u2191(-s) (v \\ u) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0\n[PROOFSTEP]\nsimp only [Pi.neg_apply, neg_eq_zero, coe_neg] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict (-s) v\nhs : \u2191s (u \u2206 v) = 0\nthis : \u2191s (u \u2206 v) = 0 \u2192 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0\n[PROOFSTEP]\nexact this hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v)\n[PROOFSTEP]\nhave hwuv : s ((w \u2229 u) \u2206 (w \u2229 v)) = 0 :=\n by\n refine'\n subset_positive_null_set (hu.union hv) ((hw.inter hu).symmDiff (hw.inter hv)) (hu.symmDiff hv)\n (restrict_le_restrict_union _ _ hu hsu hv hsv) hs Set.symmDiff_subset_union _\n rw [\u2190 Set.inter_symmDiff_distrib_left]\n exact Set.inter_subset_right _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s ((w \u2229 u) \u2206 (w \u2229 v)) = 0\n[PROOFSTEP]\nrefine'\n subset_positive_null_set (hu.union hv) ((hw.inter hu).symmDiff (hw.inter hv)) (hu.symmDiff hv)\n (restrict_le_restrict_union _ _ hu hsu hv hsv) hs Set.symmDiff_subset_union _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 (w \u2229 u) \u2206 (w \u2229 v) \u2286 u \u2206 v\n[PROOFSTEP]\nrw [\u2190 Set.inter_symmDiff_distrib_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 w \u2229 u \u2206 v \u2286 u \u2206 v\n[PROOFSTEP]\nexact Set.inter_subset_right _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\nhwuv : \u2191s ((w \u2229 u) \u2206 (w \u2229 v)) = 0\n\u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v)\n[PROOFSTEP]\nobtain \u27e8huv, hvu\u27e9 :=\n of_diff_eq_zero_of_symmDiff_eq_zero_positive (hw.inter hu) (hw.inter hv)\n (restrict_le_restrict_subset _ _ hu hsu (w.inter_subset_right u))\n (restrict_le_restrict_subset _ _ hv hsv (w.inter_subset_right v)) hwuv\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v\nhs : \u2191s (u \u2206 v) = 0\nhwuv : \u2191s ((w \u2229 u) \u2206 (w \u2229 v)) = 0\nhuv : \u2191s ((w \u2229 u) \\ (w \u2229 v)) = 0\nhvu : \u2191s ((w \u2229 v) \\ (w \u2229 u)) = 0\n\u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v)\n[PROOFSTEP]\nrw [\u2190 of_diff_of_diff_eq_zero (hw.inter hu) (hw.inter hv) hvu, huv, zero_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict s u \u2264 VectorMeasure.restrict 0 u\nhsv : VectorMeasure.restrict s v \u2264 VectorMeasure.restrict 0 v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v)\n[PROOFSTEP]\nrw [\u2190 s.neg_le_neg_iff _ hu, neg_zero] at hsu \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict s v \u2264 VectorMeasure.restrict 0 v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v)\n[PROOFSTEP]\nrw [\u2190 s.neg_le_neg_iff _ hv, neg_zero] at hsv \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict (-s) v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v)\n[PROOFSTEP]\nhave := of_inter_eq_of_symmDiff_eq_zero_positive hu hv hw hsu hsv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict (-s) v\nhs : \u2191s (u \u2206 v) = 0\nthis : \u2191(-s) (u \u2206 v) = 0 \u2192 \u2191(-s) (w \u2229 u) = \u2191(-s) (w \u2229 v)\n\u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v)\n[PROOFSTEP]\nsimp only [Pi.neg_apply, neg_inj, neg_eq_zero, coe_neg] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict (-s) v\nhs : \u2191s (u \u2206 v) = 0\nthis : \u2191s (u \u2206 v) = 0 \u2192 \u2191s (w \u2229 u) = \u2191s (w \u2229 v)\n\u22a2 \u2191s (w \u2229 u) = \u2191s (w \u2229 v)\n[PROOFSTEP]\nexact this hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\n\u22a2 j\u2081 = j\u2082\n[PROOFSTEP]\next1\n[GOAL]\ncase posPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\n\u22a2 j\u2081.posPart = j\u2082.posPart\n[PROOFSTEP]\nexact hj\n[GOAL]\ncase negPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\n\u22a2 j\u2081.negPart = j\u2082.negPart\n[PROOFSTEP]\nrw [\u2190 toSignedMeasure_eq_toSignedMeasure_iff]\n -- Porting note: golfed\n[GOAL]\ncase negPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\n\u22a2 Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart\n[PROOFSTEP]\nunfold toSignedMeasure at hj' \n[GOAL]\ncase negPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' :\n Measure.toSignedMeasure j\u2081.posPart - Measure.toSignedMeasure j\u2081.negPart =\n Measure.toSignedMeasure j\u2082.posPart - Measure.toSignedMeasure j\u2082.negPart\n\u22a2 Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart\n[PROOFSTEP]\nsimp_rw [hj, sub_right_inj] at hj' \n[GOAL]\ncase negPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.posPart = j\u2082.posPart\nhj' : Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart\n\u22a2 Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart\n[PROOFSTEP]\nexact hj'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 Injective toSignedMeasure\n[PROOFSTEP]\nintro j\u2081 j\u2082 hj\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\n\u22a2 j\u2081 = j\u2082\n[PROOFSTEP]\nobtain \u27e8S, hS\u2081, hS\u2082, hS\u2083, hS\u2084, hS\u2085\u27e9 := j\u2081.exists_compl_positive_negative\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\n\u22a2 j\u2081 = j\u2082\n[PROOFSTEP]\nobtain \u27e8T, hT\u2081, hT\u2082, hT\u2083, hT\u2084, hT\u2085\u27e9 := j\u2082.exists_compl_positive_negative\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2082) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2082) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\n\u22a2 j\u2081 = j\u2082\n[PROOFSTEP]\nrw [\u2190 hj] at hT\u2082 hT\u2083 \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\n\u22a2 j\u2081 = j\u2082\n[PROOFSTEP]\nobtain \u27e8hST\u2081, -\u27e9 :=\n of_symmDiff_compl_positive_negative hS\u2081.compl hT\u2081.compl \u27e8hS\u2083, (compl_compl S).symm \u25b8 hS\u2082\u27e9\n \u27e8hT\u2083, (compl_compl T).symm \u25b8 hT\u2082\u27e9\n -- it suffices to show the Jordan decompositions have the same positive parts\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\n\u22a2 j\u2081 = j\u2082\n[PROOFSTEP]\nrefine' eq_of_posPart_eq_posPart _ hj\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\n\u22a2 j\u2081.posPart = j\u2082.posPart\n[PROOFSTEP]\next1 i hi\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191\u2191j\u2081.posPart i = \u2191\u2191j\u2082.posPart i\n[PROOFSTEP]\nhave h\u03bc\u2081 : (j\u2081.posPart i).toReal = j\u2081.toSignedMeasure (i \u2229 S\u1d9c) :=\n by\n rw [toSignedMeasure, toSignedMeasure_sub_apply (hi.inter hS\u2081.compl),\n show j\u2081.negPart (i \u2229 S\u1d9c) = 0 from nonpos_iff_eq_zero.1 (hS\u2085 \u25b8 measure_mono (Set.inter_subset_right _ _)),\n ENNReal.zero_toReal, sub_zero]\n conv_lhs => rw [\u2190 Set.inter_union_compl i S]\n rw [measure_union,\n show j\u2081.posPart (i \u2229 S) = 0 from nonpos_iff_eq_zero.1 (hS\u2084 \u25b8 measure_mono (Set.inter_subset_right _ _)), zero_add]\n \u00b7\n refine'\n Set.disjoint_of_subset_left (Set.inter_subset_right _ _)\n (Set.disjoint_of_subset_right (Set.inter_subset_right _ _) disjoint_compl_right)\n \u00b7 exact hi.inter hS\u2081.compl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n[PROOFSTEP]\nrw [toSignedMeasure, toSignedMeasure_sub_apply (hi.inter hS\u2081.compl),\n show j\u2081.negPart (i \u2229 S\u1d9c) = 0 from nonpos_iff_eq_zero.1 (hS\u2085 \u25b8 measure_mono (Set.inter_subset_right _ _)),\n ENNReal.zero_toReal, sub_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = ENNReal.toReal (\u2191\u2191j\u2081.posPart (i \u2229 S\u1d9c))\n[PROOFSTEP]\nconv_lhs => rw [\u2190 Set.inter_union_compl i S]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n| ENNReal.toReal (\u2191\u2191j\u2081.posPart i)\n[PROOFSTEP]\nrw [\u2190 Set.inter_union_compl i S]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n| ENNReal.toReal (\u2191\u2191j\u2081.posPart i)\n[PROOFSTEP]\nrw [\u2190 Set.inter_union_compl i S]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n| ENNReal.toReal (\u2191\u2191j\u2081.posPart i)\n[PROOFSTEP]\nrw [\u2190 Set.inter_union_compl i S]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2081.posPart (i \u2229 S \u222a i \u2229 S\u1d9c)) = ENNReal.toReal (\u2191\u2191j\u2081.posPart (i \u2229 S\u1d9c))\n[PROOFSTEP]\nrw [measure_union,\n show j\u2081.posPart (i \u2229 S) = 0 from nonpos_iff_eq_zero.1 (hS\u2084 \u25b8 measure_mono (Set.inter_subset_right _ _)), zero_add]\n[GOAL]\ncase hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Disjoint (i \u2229 S) (i \u2229 S\u1d9c)\n[PROOFSTEP]\nrefine'\n Set.disjoint_of_subset_left (Set.inter_subset_right _ _)\n (Set.disjoint_of_subset_right (Set.inter_subset_right _ _) disjoint_compl_right)\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 MeasurableSet (i \u2229 S\u1d9c)\n[PROOFSTEP]\nexact hi.inter hS\u2081.compl\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n\u22a2 \u2191\u2191j\u2081.posPart i = \u2191\u2191j\u2082.posPart i\n[PROOFSTEP]\nhave h\u03bc\u2082 : (j\u2082.posPart i).toReal = j\u2082.toSignedMeasure (i \u2229 T\u1d9c) :=\n by\n rw [toSignedMeasure, toSignedMeasure_sub_apply (hi.inter hT\u2081.compl),\n show j\u2082.negPart (i \u2229 T\u1d9c) = 0 from nonpos_iff_eq_zero.1 (hT\u2085 \u25b8 measure_mono (Set.inter_subset_right _ _)),\n ENNReal.zero_toReal, sub_zero]\n conv_lhs => rw [\u2190 Set.inter_union_compl i T]\n rw [measure_union,\n show j\u2082.posPart (i \u2229 T) = 0 from nonpos_iff_eq_zero.1 (hT\u2084 \u25b8 measure_mono (Set.inter_subset_right _ _)), zero_add]\n \u00b7\n exact\n Set.disjoint_of_subset_left (Set.inter_subset_right _ _)\n (Set.disjoint_of_subset_right (Set.inter_subset_right _ _) disjoint_compl_right)\n \u00b7 exact hi.inter hT\u2081.compl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2082.posPart i) = \u2191(toSignedMeasure j\u2082) (i \u2229 T\u1d9c)\n[PROOFSTEP]\nrw [toSignedMeasure, toSignedMeasure_sub_apply (hi.inter hT\u2081.compl),\n show j\u2082.negPart (i \u2229 T\u1d9c) = 0 from nonpos_iff_eq_zero.1 (hT\u2085 \u25b8 measure_mono (Set.inter_subset_right _ _)),\n ENNReal.zero_toReal, sub_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2082.posPart i) = ENNReal.toReal (\u2191\u2191j\u2082.posPart (i \u2229 T\u1d9c))\n[PROOFSTEP]\nconv_lhs => rw [\u2190 Set.inter_union_compl i T]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n| ENNReal.toReal (\u2191\u2191j\u2082.posPart i)\n[PROOFSTEP]\nrw [\u2190 Set.inter_union_compl i T]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n| ENNReal.toReal (\u2191\u2191j\u2082.posPart i)\n[PROOFSTEP]\nrw [\u2190 Set.inter_union_compl i T]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n| ENNReal.toReal (\u2191\u2191j\u2082.posPart i)\n[PROOFSTEP]\nrw [\u2190 Set.inter_union_compl i T]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n\u22a2 ENNReal.toReal (\u2191\u2191j\u2082.posPart (i \u2229 T \u222a i \u2229 T\u1d9c)) = ENNReal.toReal (\u2191\u2191j\u2082.posPart (i \u2229 T\u1d9c))\n[PROOFSTEP]\nrw [measure_union,\n show j\u2082.posPart (i \u2229 T) = 0 from nonpos_iff_eq_zero.1 (hT\u2084 \u25b8 measure_mono (Set.inter_subset_right _ _)), zero_add]\n[GOAL]\ncase hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n\u22a2 Disjoint (i \u2229 T) (i \u2229 T\u1d9c)\n[PROOFSTEP]\nexact\n Set.disjoint_of_subset_left (Set.inter_subset_right _ _)\n (Set.disjoint_of_subset_right (Set.inter_subset_right _ _) disjoint_compl_right)\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\n\u22a2 MeasurableSet (i \u2229 T\u1d9c)\n[PROOFSTEP]\nexact hi.inter hT\u2081.compl\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\nh\u03bc\u2082 : ENNReal.toReal (\u2191\u2191j\u2082.posPart i) = \u2191(toSignedMeasure j\u2082) (i \u2229 T\u1d9c)\n\u22a2 \u2191\u2191j\u2081.posPart i = \u2191\u2191j\u2082.posPart i\n[PROOFSTEP]\nrw [\u2190 ENNReal.toReal_eq_toReal (measure_ne_top _ _) (measure_ne_top _ _), h\u03bc\u2081, h\u03bc\u2082, \u2190 hj]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c\nhS\u2084 : \u2191\u2191j\u2081.posPart S = 0\nhS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c\nhT\u2084 : \u2191\u2191j\u2082.posPart T = 0\nhT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c)\nh\u03bc\u2082 : ENNReal.toReal (\u2191\u2191j\u2082.posPart i) = \u2191(toSignedMeasure j\u2082) (i \u2229 T\u1d9c)\n\u22a2 \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c) = \u2191(toSignedMeasure j\u2081) (i \u2229 T\u1d9c)\n[PROOFSTEP]\nexact of_inter_eq_of_symmDiff_eq_zero_positive hS\u2081.compl hT\u2081.compl hi hS\u2083 hT\u2083 hST\u2081\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\n\u22a2 toSignedMeasure j = toSignedMeasure (toJordanDecomposition (toSignedMeasure j))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 toJordanDecomposition 0 = 0\n[PROOFSTEP]\napply toSignedMeasure_injective\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 toSignedMeasure (toJordanDecomposition 0) = toSignedMeasure 0\n[PROOFSTEP]\nsimp [toSignedMeasure_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u22a2 toJordanDecomposition (-s) = -toJordanDecomposition s\n[PROOFSTEP]\napply toSignedMeasure_injective\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u22a2 toSignedMeasure (toJordanDecomposition (-s)) = toSignedMeasure (-toJordanDecomposition s)\n[PROOFSTEP]\nsimp [toSignedMeasure_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\u22650\n\u22a2 toJordanDecomposition (r \u2022 s) = r \u2022 toJordanDecomposition s\n[PROOFSTEP]\napply toSignedMeasure_injective\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\u22650\n\u22a2 toSignedMeasure (toJordanDecomposition (r \u2022 s)) = toSignedMeasure (r \u2022 toJordanDecomposition s)\n[PROOFSTEP]\nsimp [toSignedMeasure_smul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 toJordanDecomposition (r \u2022 s) = r \u2022 toJordanDecomposition s\n[PROOFSTEP]\nlift r to \u211d\u22650 using hr\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\u22650\n\u22a2 toJordanDecomposition (\u2191r \u2022 s) = \u2191r \u2022 toJordanDecomposition s\n[PROOFSTEP]\nrw [JordanDecomposition.coe_smul, \u2190 toJordanDecomposition_smul]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\u22650\n\u22a2 toJordanDecomposition (\u2191r \u2022 s) = toJordanDecomposition (r \u2022 s)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\n\u22a2 toJordanDecomposition (r \u2022 s) = r \u2022 toJordanDecomposition s\n[PROOFSTEP]\nby_cases hr : 0 \u2264 r\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 toJordanDecomposition (r \u2022 s) = r \u2022 toJordanDecomposition s\n[PROOFSTEP]\nexact toJordanDecomposition_smul_real_nonneg s r hr\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 toJordanDecomposition (r \u2022 s) = r \u2022 toJordanDecomposition s\n[PROOFSTEP]\next1\n[GOAL]\ncase neg.posPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 (toJordanDecomposition (r \u2022 s)).posPart = (r \u2022 toJordanDecomposition s).posPart\n[PROOFSTEP]\nrw [real_smul_posPart_neg _ _ (not_le.1 hr), show r \u2022 s = -(-r \u2022 s) by rw [neg_smul, neg_neg],\n toJordanDecomposition_neg, neg_posPart, toJordanDecomposition_smul_real_nonneg, \u2190 smul_negPart, real_smul_nonneg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 r \u2022 s = -(-r \u2022 s)\n[PROOFSTEP]\nrw [neg_smul, neg_neg]\n[GOAL]\ncase neg.posPart.hr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 0 \u2264 -r\ncase neg.posPart.hr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 r : \u211d hr : \u00ac0 \u2264 r \u22a2 0 \u2264 -r\n[PROOFSTEP]\nall_goals exact Left.nonneg_neg_iff.2 (le_of_lt (not_le.1 hr))\n[GOAL]\ncase neg.posPart.hr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 0 \u2264 -r\n[PROOFSTEP]\nexact Left.nonneg_neg_iff.2 (le_of_lt (not_le.1 hr))\n[GOAL]\ncase neg.posPart.hr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 0 \u2264 -r\n[PROOFSTEP]\nexact Left.nonneg_neg_iff.2 (le_of_lt (not_le.1 hr))\n[GOAL]\ncase neg.negPart\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 (toJordanDecomposition (r \u2022 s)).negPart = (r \u2022 toJordanDecomposition s).negPart\n[PROOFSTEP]\nrw [real_smul_negPart_neg _ _ (not_le.1 hr), show r \u2022 s = -(-r \u2022 s) by rw [neg_smul, neg_neg],\n toJordanDecomposition_neg, neg_negPart, toJordanDecomposition_smul_real_nonneg, \u2190 smul_posPart, real_smul_nonneg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 r \u2022 s = -(-r \u2022 s)\n[PROOFSTEP]\nrw [neg_smul, neg_neg]\n[GOAL]\ncase neg.negPart.hr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 0 \u2264 -r\ncase neg.negPart.hr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 r : \u211d hr : \u00ac0 \u2264 r \u22a2 0 \u2264 -r\n[PROOFSTEP]\nall_goals exact Left.nonneg_neg_iff.2 (le_of_lt (not_le.1 hr))\n[GOAL]\ncase neg.negPart.hr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 0 \u2264 -r\n[PROOFSTEP]\nexact Left.nonneg_neg_iff.2 (le_of_lt (not_le.1 hr))\n[GOAL]\ncase neg.negPart.hr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 0 \u2264 -r\n[PROOFSTEP]\nexact Left.nonneg_neg_iff.2 (le_of_lt (not_le.1 hr))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nj : JordanDecomposition \u03b1\nh : s = toSignedMeasure j\n\u22a2 toJordanDecomposition s = j\n[PROOFSTEP]\nrw [h, toJordanDecomposition_toSignedMeasure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 totalVariation 0 = 0\n[PROOFSTEP]\nsimp [totalVariation, toJordanDecomposition_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u22a2 totalVariation (-s) = totalVariation s\n[PROOFSTEP]\nsimp [totalVariation, toJordanDecomposition_neg, add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(totalVariation s) i = 0\n\u22a2 \u2191s i = 0\n[PROOFSTEP]\nrw [totalVariation, Measure.coe_add, Pi.add_apply, add_eq_zero_iff] at hs \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\n\u22a2 \u2191s i = 0\n[PROOFSTEP]\nrw [\u2190 toSignedMeasure_toJordanDecomposition s, toSignedMeasure, VectorMeasure.coe_sub, Pi.sub_apply,\n Measure.toSignedMeasure_apply, Measure.toSignedMeasure_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\n\u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -\n if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =\n 0\n[PROOFSTEP]\nby_cases hi : MeasurableSet i\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\nhi : MeasurableSet i\n\u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -\n if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =\n 0\n[PROOFSTEP]\nrw [if_pos hi, if_pos hi]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) - ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) = 0\n[PROOFSTEP]\nsimp [hs.1, hs.2]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\nhi : \u00acMeasurableSet i\n\u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -\n if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =\n 0\n[PROOFSTEP]\nsimp [if_neg hi]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\n\u22a2 s \u226a\u1d65 \u03bc \u2194 totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\n\u22a2 s \u226a\u1d65 \u03bc \u2192 totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\n\u22a2 totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc \u2192 s \u226a\u1d65 \u03bc\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : s \u226a\u1d65 \u03bc\n\u22a2 totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc\n[PROOFSTEP]\nrefine' Measure.AbsolutelyContinuous.mk fun S hS\u2081 hS\u2082 => _\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : s \u226a\u1d65 \u03bc\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0\n\u22a2 \u2191\u2191(totalVariation s) S = 0\n[PROOFSTEP]\nobtain \u27e8i, hi\u2081, hi\u2082, hi\u2083, hpos, hneg\u27e9 := s.toJordanDecomposition_spec\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : s \u226a\u1d65 \u03bc\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 \u2191\u2191(totalVariation s) S = 0\n[PROOFSTEP]\nrw [totalVariation, Measure.add_apply, hpos, hneg, toMeasureOfZeroLE_apply _ _ _ hS\u2081, toMeasureOfLEZero_apply _ _ _ hS\u2081]\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : s \u226a\u1d65 \u03bc\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 \u2191{ val := \u2191s (i \u2229 S), property := (_ : 0 \u2264 \u2191s (i \u2229 S)) } +\n \u2191{ val := -\u2191s (i\u1d9c \u2229 S), property := (_ : 0 \u2264 -\u2191s (i\u1d9c \u2229 S)) } =\n 0\n[PROOFSTEP]\nrw [\u2190 VectorMeasure.AbsolutelyContinuous.ennrealToMeasure] at h \n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : \u2200 \u2983s_1 : Set \u03b1\u2984, \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s_1 = 0 \u2192 \u2191s s_1 = 0\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 \u2191{ val := \u2191s (i \u2229 S), property := (_ : 0 \u2264 \u2191s (i \u2229 S)) } +\n \u2191{ val := -\u2191s (i\u1d9c \u2229 S), property := (_ : 0 \u2264 -\u2191s (i\u1d9c \u2229 S)) } =\n 0\n[PROOFSTEP]\nsimp [h (measure_mono_null (i.inter_subset_right S) hS\u2082), h (measure_mono_null (i\u1d9c.inter_subset_right S) hS\u2082), \u2190\n NNReal.eq_iff]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc\n\u22a2 s \u226a\u1d65 \u03bc\n[PROOFSTEP]\nrefine' VectorMeasure.AbsolutelyContinuous.mk fun S hS\u2081 hS\u2082 => _\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u03bc S = 0\n\u22a2 \u2191s S = 0\n[PROOFSTEP]\nrw [\u2190 VectorMeasure.ennrealToMeasure_apply hS\u2081] at hS\u2082 \n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0\n\u22a2 \u2191s S = 0\n[PROOFSTEP]\nexact null_of_totalVariation_zero s (h hS\u2082)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\n\u22a2 totalVariation s \u226a \u03bc \u2194 (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\n\u22a2 totalVariation s \u226a \u03bc \u2192 (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\n\u22a2 (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc \u2192 totalVariation s \u226a \u03bc\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nh : totalVariation s \u226a \u03bc\n\u22a2 (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nh : totalVariation s \u226a \u03bc\n\u22a2 (toJordanDecomposition s).posPart \u226a \u03bc\ncase mp.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nh : totalVariation s \u226a \u03bc\n\u22a2 (toJordanDecomposition s).negPart \u226a \u03bc\n[PROOFSTEP]\nall_goals\n refine' Measure.AbsolutelyContinuous.mk fun S _ hS\u2082 => _\n have := h hS\u2082\n rw [totalVariation, Measure.add_apply, add_eq_zero_iff] at this \n[GOAL]\ncase mp.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nh : totalVariation s \u226a \u03bc\n\u22a2 (toJordanDecomposition s).posPart \u226a \u03bc\n[PROOFSTEP]\nrefine' Measure.AbsolutelyContinuous.mk fun S _ hS\u2082 => _\n[GOAL]\ncase mp.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nh : totalVariation s \u226a \u03bc\nS : Set \u03b1\nx\u271d : MeasurableSet S\nhS\u2082 : \u2191\u2191\u03bc S = 0\n\u22a2 \u2191\u2191(toJordanDecomposition s).posPart S = 0\n[PROOFSTEP]\nhave := h hS\u2082\n[GOAL]\ncase mp.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nh : totalVariation s \u226a \u03bc\nS : Set \u03b1\nx\u271d : MeasurableSet S\nhS\u2082 : \u2191\u2191\u03bc S = 0\nthis : \u2191\u2191(totalVariation s) S = 0\n\u22a2 \u2191\u2191(toJordanDecomposition s).posPart S = 0\n[PROOFSTEP]\nrw [totalVariation, Measure.add_apply, add_eq_zero_iff] at this \n[GOAL]\ncase mp.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nh : totalVariation s \u226a \u03bc\n\u22a2 (toJordanDecomposition s).negPart \u226a \u03bc\n[PROOFSTEP]\nrefine' Measure.AbsolutelyContinuous.mk fun S _ hS\u2082 => _\n[GOAL]\ncase mp.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nh : totalVariation s \u226a \u03bc\nS : Set \u03b1\nx\u271d : MeasurableSet S\nhS\u2082 : \u2191\u2191\u03bc S = 0\n\u22a2 \u2191\u2191(toJordanDecomposition s).negPart S = 0\n[PROOFSTEP]\nhave := h hS\u2082\n[GOAL]\ncase mp.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nh : totalVariation s \u226a \u03bc\nS : Set \u03b1\nx\u271d : MeasurableSet S\nhS\u2082 : \u2191\u2191\u03bc S = 0\nthis : \u2191\u2191(totalVariation s) S = 0\n\u22a2 \u2191\u2191(toJordanDecomposition s).negPart S = 0\n[PROOFSTEP]\nrw [totalVariation, Measure.add_apply, add_eq_zero_iff] at this \n[GOAL]\ncase mp.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nh : totalVariation s \u226a \u03bc\nS : Set \u03b1\nx\u271d : MeasurableSet S\nhS\u2082 : \u2191\u2191\u03bc S = 0\nthis : \u2191\u2191(toJordanDecomposition s).posPart S = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart S = 0\n\u22a2 \u2191\u2191(toJordanDecomposition s).posPart S = 0\ncase mp.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nh : totalVariation s \u226a \u03bc\nS : Set \u03b1\nx\u271d : MeasurableSet S\nhS\u2082 : \u2191\u2191\u03bc S = 0\nthis : \u2191\u2191(toJordanDecomposition s).posPart S = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart S = 0\n\u22a2 \u2191\u2191(toJordanDecomposition s).negPart S = 0\n[PROOFSTEP]\nexacts [this.1, this.2]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\n\u22a2 totalVariation s \u226a \u03bc\n[PROOFSTEP]\nrefine' Measure.AbsolutelyContinuous.mk fun S _ hS\u2082 => _\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nh : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc\nS : Set \u03b1\nx\u271d : MeasurableSet S\nhS\u2082 : \u2191\u2191\u03bc S = 0\n\u22a2 \u2191\u2191(totalVariation s) S = 0\n[PROOFSTEP]\nrw [totalVariation, Measure.add_apply, h.1 hS\u2082, h.2 hS\u2082, add_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\n\u22a2 s \u27c2\u1d65 t \u2194 totalVariation s \u27c2\u2098 totalVariation t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\n\u22a2 s \u27c2\u1d65 t \u2192 totalVariation s \u27c2\u2098 totalVariation t\n[PROOFSTEP]\nrintro \u27e8u, hmeas, hu\u2081, hu\u2082\u27e9\n[GOAL]\ncase mp.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 (t : Set \u03b1), t \u2286 u \u2192 \u2191s t = 0\nhu\u2082 : \u2200 (t_1 : Set \u03b1), t_1 \u2286 u\u1d9c \u2192 \u2191t t_1 = 0\n\u22a2 totalVariation s \u27c2\u2098 totalVariation t\n[PROOFSTEP]\nobtain \u27e8i, hi\u2081, hi\u2082, hi\u2083, hipos, hineg\u27e9 := s.toJordanDecomposition_spec\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 (t : Set \u03b1), t \u2286 u \u2192 \u2191s t = 0\nhu\u2082 : \u2200 (t_1 : Set \u03b1), t_1 \u2286 u\u1d9c \u2192 \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhineg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 totalVariation s \u27c2\u2098 totalVariation t\n[PROOFSTEP]\nobtain \u27e8j, hj\u2081, hj\u2082, hj\u2083, hjpos, hjneg\u27e9 := t.toJordanDecomposition_spec\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 (t : Set \u03b1), t \u2286 u \u2192 \u2191s t = 0\nhu\u2082 : \u2200 (t_1 : Set \u03b1), t_1 \u2286 u\u1d9c \u2192 \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhineg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : VectorMeasure.restrict 0 j \u2264 VectorMeasure.restrict t j\nhj\u2083 : VectorMeasure.restrict t j\u1d9c \u2264 VectorMeasure.restrict 0 j\u1d9c\nhjpos : (toJordanDecomposition t).posPart = toMeasureOfZeroLE t j hj\u2081 hj\u2082\nhjneg : (toJordanDecomposition t).negPart = toMeasureOfLEZero t j\u1d9c (_ : MeasurableSet j\u1d9c) hj\u2083\n\u22a2 totalVariation s \u27c2\u2098 totalVariation t\n[PROOFSTEP]\nrefine' \u27e8u, hmeas, _, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 (t : Set \u03b1), t \u2286 u \u2192 \u2191s t = 0\nhu\u2082 : \u2200 (t_1 : Set \u03b1), t_1 \u2286 u\u1d9c \u2192 \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhineg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : VectorMeasure.restrict 0 j \u2264 VectorMeasure.restrict t j\nhj\u2083 : VectorMeasure.restrict t j\u1d9c \u2264 VectorMeasure.restrict 0 j\u1d9c\nhjpos : (toJordanDecomposition t).posPart = toMeasureOfZeroLE t j hj\u2081 hj\u2082\nhjneg : (toJordanDecomposition t).negPart = toMeasureOfLEZero t j\u1d9c (_ : MeasurableSet j\u1d9c) hj\u2083\n\u22a2 \u2191\u2191(totalVariation s) u = 0\n[PROOFSTEP]\nrw [totalVariation, Measure.add_apply, hipos, hineg, toMeasureOfZeroLE_apply _ _ _ hmeas,\n toMeasureOfLEZero_apply _ _ _ hmeas]\n -- Porting note: added `\u2190 NNReal.eq_iff`\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 (t : Set \u03b1), t \u2286 u \u2192 \u2191s t = 0\nhu\u2082 : \u2200 (t_1 : Set \u03b1), t_1 \u2286 u\u1d9c \u2192 \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhineg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : VectorMeasure.restrict 0 j \u2264 VectorMeasure.restrict t j\nhj\u2083 : VectorMeasure.restrict t j\u1d9c \u2264 VectorMeasure.restrict 0 j\u1d9c\nhjpos : (toJordanDecomposition t).posPart = toMeasureOfZeroLE t j hj\u2081 hj\u2082\nhjneg : (toJordanDecomposition t).negPart = toMeasureOfLEZero t j\u1d9c (_ : MeasurableSet j\u1d9c) hj\u2083\n\u22a2 \u2191{ val := \u2191s (i \u2229 u), property := (_ : 0 \u2264 \u2191s (i \u2229 u)) } +\n \u2191{ val := -\u2191s (i\u1d9c \u2229 u), property := (_ : 0 \u2264 -\u2191s (i\u1d9c \u2229 u)) } =\n 0\n[PROOFSTEP]\nsimp [hu\u2081 _ (Set.inter_subset_right _ _), \u2190 NNReal.eq_iff]\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 (t : Set \u03b1), t \u2286 u \u2192 \u2191s t = 0\nhu\u2082 : \u2200 (t_1 : Set \u03b1), t_1 \u2286 u\u1d9c \u2192 \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhineg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : VectorMeasure.restrict 0 j \u2264 VectorMeasure.restrict t j\nhj\u2083 : VectorMeasure.restrict t j\u1d9c \u2264 VectorMeasure.restrict 0 j\u1d9c\nhjpos : (toJordanDecomposition t).posPart = toMeasureOfZeroLE t j hj\u2081 hj\u2082\nhjneg : (toJordanDecomposition t).negPart = toMeasureOfLEZero t j\u1d9c (_ : MeasurableSet j\u1d9c) hj\u2083\n\u22a2 \u2191\u2191(totalVariation t) u\u1d9c = 0\n[PROOFSTEP]\nrw [totalVariation, Measure.add_apply, hjpos, hjneg, toMeasureOfZeroLE_apply _ _ _ hmeas.compl,\n toMeasureOfLEZero_apply _ _ _ hmeas.compl]\n -- Porting note: added `\u2190 NNReal.eq_iff`\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 (t : Set \u03b1), t \u2286 u \u2192 \u2191s t = 0\nhu\u2082 : \u2200 (t_1 : Set \u03b1), t_1 \u2286 u\u1d9c \u2192 \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhineg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : VectorMeasure.restrict 0 j \u2264 VectorMeasure.restrict t j\nhj\u2083 : VectorMeasure.restrict t j\u1d9c \u2264 VectorMeasure.restrict 0 j\u1d9c\nhjpos : (toJordanDecomposition t).posPart = toMeasureOfZeroLE t j hj\u2081 hj\u2082\nhjneg : (toJordanDecomposition t).negPart = toMeasureOfLEZero t j\u1d9c (_ : MeasurableSet j\u1d9c) hj\u2083\n\u22a2 \u2191{ val := \u2191t (j \u2229 u\u1d9c), property := (_ : 0 \u2264 \u2191t (j \u2229 u\u1d9c)) } +\n \u2191{ val := -\u2191t (j\u1d9c \u2229 u\u1d9c), property := (_ : 0 \u2264 -\u2191t (j\u1d9c \u2229 u\u1d9c)) } =\n 0\n[PROOFSTEP]\nsimp [hu\u2082 _ (Set.inter_subset_right _ _), \u2190 NNReal.eq_iff]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\n\u22a2 totalVariation s \u27c2\u2098 totalVariation t \u2192 s \u27c2\u1d65 t\n[PROOFSTEP]\nrintro \u27e8u, hmeas, hu\u2081, hu\u2082\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2191\u2191(totalVariation s) u = 0\nhu\u2082 : \u2191\u2191(totalVariation t) u\u1d9c = 0\n\u22a2 s \u27c2\u1d65 t\n[PROOFSTEP]\nexact\n \u27e8u, hmeas, fun t htu => null_of_totalVariation_zero _ (measure_mono_null htu hu\u2081), fun t htv =>\n null_of_totalVariation_zero _ (measure_mono_null htv hu\u2082)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\n\u22a2 s \u27c2\u1d65 \u03bc \u2194 totalVariation s \u27c2\u2098 VectorMeasure.ennrealToMeasure \u03bc\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\n\u22a2 s \u27c2\u1d65 \u03bc \u2192 totalVariation s \u27c2\u2098 VectorMeasure.ennrealToMeasure \u03bc\n[PROOFSTEP]\nrintro \u27e8u, hmeas, hu\u2081, hu\u2082\u27e9\n[GOAL]\ncase mp.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 (t : Set \u03b1), t \u2286 u \u2192 \u2191s t = 0\nhu\u2082 : \u2200 (t : Set \u03b1), t \u2286 u\u1d9c \u2192 \u2191\u03bc t = 0\n\u22a2 totalVariation s \u27c2\u2098 VectorMeasure.ennrealToMeasure \u03bc\n[PROOFSTEP]\nobtain \u27e8i, hi\u2081, hi\u2082, hi\u2083, hpos, hneg\u27e9 := s.toJordanDecomposition_spec\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 (t : Set \u03b1), t \u2286 u \u2192 \u2191s t = 0\nhu\u2082 : \u2200 (t : Set \u03b1), t \u2286 u\u1d9c \u2192 \u2191\u03bc t = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 totalVariation s \u27c2\u2098 VectorMeasure.ennrealToMeasure \u03bc\n[PROOFSTEP]\nrefine' \u27e8u, hmeas, _, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 (t : Set \u03b1), t \u2286 u \u2192 \u2191s t = 0\nhu\u2082 : \u2200 (t : Set \u03b1), t \u2286 u\u1d9c \u2192 \u2191\u03bc t = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 \u2191\u2191(totalVariation s) u = 0\n[PROOFSTEP]\nrw [totalVariation, Measure.add_apply, hpos, hneg, toMeasureOfZeroLE_apply _ _ _ hmeas,\n toMeasureOfLEZero_apply _ _ _ hmeas]\n -- Porting note: added `\u2190 NNReal.eq_iff`\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 (t : Set \u03b1), t \u2286 u \u2192 \u2191s t = 0\nhu\u2082 : \u2200 (t : Set \u03b1), t \u2286 u\u1d9c \u2192 \u2191\u03bc t = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 \u2191{ val := \u2191s (i \u2229 u), property := (_ : 0 \u2264 \u2191s (i \u2229 u)) } +\n \u2191{ val := -\u2191s (i\u1d9c \u2229 u), property := (_ : 0 \u2264 -\u2191s (i\u1d9c \u2229 u)) } =\n 0\n[PROOFSTEP]\nsimp [hu\u2081 _ (Set.inter_subset_right _ _), \u2190 NNReal.eq_iff]\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 (t : Set \u03b1), t \u2286 u \u2192 \u2191s t = 0\nhu\u2082 : \u2200 (t : Set \u03b1), t \u2286 u\u1d9c \u2192 \u2191\u03bc t = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) u\u1d9c = 0\n[PROOFSTEP]\nrw [VectorMeasure.ennrealToMeasure_apply hmeas.compl]\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 (t : Set \u03b1), t \u2286 u \u2192 \u2191s t = 0\nhu\u2082 : \u2200 (t : Set \u03b1), t \u2286 u\u1d9c \u2192 \u2191\u03bc t = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082\nhneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083\n\u22a2 \u2191\u03bc u\u1d9c = 0\n[PROOFSTEP]\nexact hu\u2082 _ (Set.Subset.refl _)\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\n\u22a2 totalVariation s \u27c2\u2098 VectorMeasure.ennrealToMeasure \u03bc \u2192 s \u27c2\u1d65 \u03bc\n[PROOFSTEP]\nrintro \u27e8u, hmeas, hu\u2081, hu\u2082\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2191\u2191(totalVariation s) u = 0\nhu\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) u\u1d9c = 0\n\u22a2 s \u27c2\u1d65 \u03bc\n[PROOFSTEP]\nrefine'\n VectorMeasure.MutuallySingular.mk u hmeas (fun t htu _ => null_of_totalVariation_zero _ (measure_mono_null htu hu\u2081))\n fun t htv hmt => _\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2191\u2191(totalVariation s) u = 0\nhu\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) u\u1d9c = 0\nt : Set \u03b1\nhtv : t \u2286 u\u1d9c\nhmt : MeasurableSet t\n\u22a2 \u2191\u03bc t = 0\n[PROOFSTEP]\nrw [\u2190 VectorMeasure.ennrealToMeasure_apply hmt]\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2191\u2191(totalVariation s) u = 0\nhu\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) u\u1d9c = 0\nt : Set \u03b1\nhtv : t \u2286 u\u1d9c\nhmt : MeasurableSet t\n\u22a2 \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) t = 0\n[PROOFSTEP]\nexact measure_mono_null htv hu\u2082\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Decomposition.Jordan", "llama_tokens": 50540, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124811, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.44882281820780445}} {"text": "[GOAL]\nG : Type u_1\ninst\u271d : Group G\nV : Set G\n\u22a2 index \u2205 V = 0\n[PROOFSTEP]\nsimp only [index, Nat.sInf_eq_zero]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nV : Set G\n\u22a2 0 \u2208 Finset.card '' {t | \u2205 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V} \u2228\n Finset.card '' {t | \u2205 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V} = \u2205\n[PROOFSTEP]\nleft\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d : Group G\nV : Set G\n\u22a2 0 \u2208 Finset.card '' {t | \u2205 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n[PROOFSTEP]\nuse\u2205\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d : Group G\nV : Set G\n\u22a2 \u2205 \u2208 {t | \u2205 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V} \u2227 Finset.card \u2205 = 0\n[PROOFSTEP]\nsimp only [Finset.card_empty, empty_subset, mem_setOf_eq, eq_self_iff_true, and_self_iff]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nK\u2080 : PositiveCompacts G\nU : Set G\n\u22a2 prehaar (\u2191K\u2080) U \u22a5 = 0\n[PROOFSTEP]\nrw [prehaar, Compacts.coe_bot, index_empty, Nat.cast_zero, zero_div]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\n\u22a2 0 \u2264 prehaar (\u2191K\u2080) U K\n[PROOFSTEP]\napply div_nonneg\n[GOAL]\ncase ha\nG : Type u_1\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\n\u22a2 0 \u2264 \u2191(index (\u2191K) U)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase hb\nG : Type u_1\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\n\u22a2 0 \u2264 \u2191(index (\u2191K\u2080) U)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase ha\nG : Type u_1\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\n\u22a2 0 \u2264 index (\u2191K) U\n[PROOFSTEP]\napply zero_le\n[GOAL]\ncase hb\nG : Type u_1\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\n\u22a2 0 \u2264 index (\u2191K\u2080) U\n[PROOFSTEP]\napply zero_le\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nK\u2080 : Set G\nf : Compacts G \u2192 \u211d\n\u22a2 f \u2208 haarProduct K\u2080 \u2194 \u2200 (K : Compacts G), f K \u2208 Icc 0 \u2191(index (\u2191K) K\u2080)\n[PROOFSTEP]\nsimp only [haarProduct, Set.pi, forall_prop_of_true, mem_univ, mem_setOf_eq]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK V : Set G\nhK : IsCompact K\nhV : Set.Nonempty (interior V)\n\u22a2 \u2203 n, n \u2208 Finset.card '' {t | K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n[PROOFSTEP]\nrcases compact_covered_by_mul_left_translates hK hV with \u27e8t, ht\u27e9\n[GOAL]\ncase intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK V : Set G\nhK : IsCompact K\nhV : Set.Nonempty (interior V)\nt : Finset G\nht : K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\n\u22a2 \u2203 n, n \u2208 Finset.card '' {t | K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n[PROOFSTEP]\nexact \u27e8t.card, t, ht, rfl\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK V : Set G\nhK : IsCompact K\nhV : Set.Nonempty (interior V)\n\u22a2 \u2203 t, K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V \u2227 Finset.card t = index K V\n[PROOFSTEP]\nhave := Nat.sInf_mem (index_defined hK hV)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK V : Set G\nhK : IsCompact K\nhV : Set.Nonempty (interior V)\nthis :\n sInf (Finset.card '' {t | K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}) \u2208\n Finset.card '' {t | K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n\u22a2 \u2203 t, K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V \u2227 Finset.card t = index K V\n[PROOFSTEP]\nrwa [mem_image] at this \n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 index (\u2191K) V \u2264 index \u2191K \u2191K\u2080 * index (\u2191K\u2080) V\n[PROOFSTEP]\nobtain \u27e8s, h1s, h2s\u27e9 := index_elim K.isCompact K\u2080.interior_nonempty\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\n\u22a2 index (\u2191K) V \u2264 index \u2191K \u2191K\u2080 * index (\u2191K\u2080) V\n[PROOFSTEP]\nobtain \u27e8t, h1t, h2t\u27e9 := index_elim K\u2080.isCompact hV\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 index (\u2191K) V \u2264 index \u2191K \u2191K\u2080 * index (\u2191K\u2080) V\n[PROOFSTEP]\nrw [\u2190 h2s, \u2190 h2t, mul_comm]\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 index (\u2191K) V \u2264 Finset.card t * Finset.card s\n[PROOFSTEP]\nrefine' le_trans _ Finset.card_mul_le\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 index (\u2191K) V \u2264 Finset.card (t * s)\n[PROOFSTEP]\napply Nat.sInf_le\n[GOAL]\ncase intro.intro.intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 Finset.card (t * s) \u2208 Finset.card '' {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n[PROOFSTEP]\nrefine' \u27e8_, _, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 t * s \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n[PROOFSTEP]\nrw [mem_setOf_eq]\n[GOAL]\ncase intro.intro.intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t * s), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\nrefine' Subset.trans h1s _\n[GOAL]\ncase intro.intro.intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t * s), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\napply iUnion\u2082_subset\n[GOAL]\ncase intro.intro.intro.intro.hm.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\n\u22a2 \u2200 (i : G), i \u2208 s \u2192 (fun h => i * h) \u207b\u00b9' \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t * s), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\nintro g\u2081 hg\u2081\n[GOAL]\ncase intro.intro.intro.intro.hm.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\n\u22a2 (fun h => g\u2081 * h) \u207b\u00b9' \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t * s), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\nrw [preimage_subset_iff]\n[GOAL]\ncase intro.intro.intro.intro.hm.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\n\u22a2 \u2200 (a : G), g\u2081 * a \u2208 \u2191K\u2080 \u2192 a \u2208 \u22c3 (g : G) (_ : g \u2208 t * s), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\nintro g\u2082 hg\u2082\n[GOAL]\ncase intro.intro.intro.intro.hm.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\ng\u2082 : G\nhg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080\n\u22a2 g\u2082 \u2208 \u22c3 (g : G) (_ : g \u2208 t * s), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\nhave := h1t hg\u2082\n[GOAL]\ncase intro.intro.intro.intro.hm.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\ng\u2082 : G\nhg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080\nthis : g\u2081 * g\u2082 \u2208 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\n\u22a2 g\u2082 \u2208 \u22c3 (g : G) (_ : g \u2208 t * s), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\nrcases this with \u27e8_, \u27e8g\u2083, rfl\u27e9, A, \u27e8hg\u2083, rfl\u27e9, h2V\u27e9\n[GOAL]\ncase intro.intro.intro.intro.hm.h.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\ng\u2082 : G\nhg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080\ng\u2083 : G\nhg\u2083 : g\u2083 \u2208 t\nh2V : g\u2081 * g\u2082 \u2208 (fun h => (fun h => g\u2083 * h) \u207b\u00b9' V) hg\u2083\n\u22a2 g\u2082 \u2208 \u22c3 (g : G) (_ : g \u2208 t * s), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\nrw [mem_preimage, \u2190 mul_assoc] at h2V \n[GOAL]\ncase intro.intro.intro.intro.hm.h.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' \u2191K\u2080\nh2s : Finset.card s = index \u2191K \u2191K\u2080\nt : Finset G\nh1t : \u2191K\u2080 \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index (\u2191K\u2080) V\ng\u2081 : G\nhg\u2081 : g\u2081 \u2208 s\ng\u2082 : G\nhg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080\ng\u2083 : G\nhg\u2083 : g\u2083 \u2208 t\nh2V : g\u2083 * g\u2081 * g\u2082 \u2208 V\n\u22a2 g\u2082 \u2208 \u22c3 (g : G) (_ : g \u2208 t * s), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\nexact mem_biUnion (Finset.mul_mem_mul hg\u2083 hg\u2081) h2V\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 0 < index (\u2191K) V\n[PROOFSTEP]\nunfold index\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 0 < sInf (Finset.card '' {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V})\n[PROOFSTEP]\nrw [Nat.sInf_def, Nat.find_pos, mem_image]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 \u00ac\u2203 x, x \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V} \u2227 Finset.card x = 0\n[PROOFSTEP]\nrintro \u27e8t, h1t, h2t\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nt : Finset G\nh1t : t \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\nh2t : Finset.card t = 0\n\u22a2 False\n[PROOFSTEP]\nrw [Finset.card_eq_zero] at h2t \n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nt : Finset G\nh1t : t \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\nh2t : t = \u2205\n\u22a2 False\n[PROOFSTEP]\nsubst h2t\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8g, hg\u27e9 := K.interior_nonempty\n[GOAL]\ncase intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 False\n[PROOFSTEP]\nshow g \u2208 (\u2205 : Set G)\n[GOAL]\ncase intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 g \u2208 \u2205\n[PROOFSTEP]\nconvert h1t (interior_subset hg)\n[GOAL]\ncase h.e'_5\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 \u2205 = \u22c3 (g : G) (_ : g \u2208 \u2205), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_5\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 \u22c3 (g : G) (_ : g \u2208 \u2205), (fun h => g * h) \u207b\u00b9' V = \u2205\n[PROOFSTEP]\nsimp only [Finset.not_mem_empty, iUnion_of_empty, iUnion_empty]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 \u2203 n, n \u2208 Finset.card '' {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n[PROOFSTEP]\nexact index_defined K.isCompact hV\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK K' V : Set G\nhK' : IsCompact K'\nh : K \u2286 K'\nhV : Set.Nonempty (interior V)\n\u22a2 index K V \u2264 index K' V\n[PROOFSTEP]\nrcases index_elim hK' hV with \u27e8s, h1s, h2s\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK K' V : Set G\nhK' : IsCompact K'\nh : K \u2286 K'\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K' \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K' V\n\u22a2 index K V \u2264 index K' V\n[PROOFSTEP]\napply Nat.sInf_le\n[GOAL]\ncase intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK K' V : Set G\nhK' : IsCompact K'\nh : K \u2286 K'\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K' \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K' V\n\u22a2 index K' V \u2208 Finset.card '' {t | K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n[PROOFSTEP]\nrw [mem_image]\n[GOAL]\ncase intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK K' V : Set G\nhK' : IsCompact K'\nh : K \u2286 K'\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K' \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K' V\n\u22a2 \u2203 x, x \u2208 {t | K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V} \u2227 Finset.card x = index K' V\n[PROOFSTEP]\nrefine' \u27e8s, Subset.trans h h1s, h2s\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 index (K\u2081.carrier \u222a K\u2082.carrier) V \u2264 index K\u2081.carrier V + index K\u2082.carrier V\n[PROOFSTEP]\nrcases index_elim K\u2081.2 hV with \u27e8s, h1s, h2s\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\n\u22a2 index (K\u2081.carrier \u222a K\u2082.carrier) V \u2264 index K\u2081.carrier V + index K\u2082.carrier V\n[PROOFSTEP]\nrcases index_elim K\u2082.2 hV with \u27e8t, h1t, h2t\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\n\u22a2 index (K\u2081.carrier \u222a K\u2082.carrier) V \u2264 index K\u2081.carrier V + index K\u2082.carrier V\n[PROOFSTEP]\nrw [\u2190 h2s, \u2190 h2t]\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\n\u22a2 index (K\u2081.carrier \u222a K\u2082.carrier) V \u2264 Finset.card s + Finset.card t\n[PROOFSTEP]\nrefine' le_trans _ (Finset.card_union_le _ _)\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\n\u22a2 index (K\u2081.carrier \u222a K\u2082.carrier) V \u2264 Finset.card (s \u222a t)\n[PROOFSTEP]\napply Nat.sInf_le\n[GOAL]\ncase intro.intro.intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\n\u22a2 Finset.card (s \u222a t) \u2208 Finset.card '' {t | K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n[PROOFSTEP]\nrefine' \u27e8_, _, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\n\u22a2 s \u222a t \u2208 {t | K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n[PROOFSTEP]\nrw [mem_setOf_eq]\n[GOAL]\ncase intro.intro.intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\n\u22a2 K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s \u222a t), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\napply union_subset\n[GOAL]\ncase intro.intro.intro.intro.hm.sr\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\n\u22a2 K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s \u222a t), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\nrefine' Subset.trans (by assumption) _\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\n\u22a2 K\u2081.carrier \u2286 ?m.46641\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.intro.hm.tr\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\n\u22a2 K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s \u222a t), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\nrefine' Subset.trans (by assumption) _\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\n\u22a2 K\u2082.carrier \u2286 ?m.46676\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.intro.hm.sr\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\n\u22a2 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2286 \u22c3 (g : G) (_ : g \u2208 s \u222a t), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\napply biUnion_subset_biUnion_left\n[GOAL]\ncase intro.intro.intro.intro.hm.tr\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\n\u22a2 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V \u2286 \u22c3 (g : G) (_ : g \u2208 s \u222a t), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\napply biUnion_subset_biUnion_left\n[GOAL]\ncase intro.intro.intro.intro.hm.sr.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\n\u22a2 (fun x => x \u2208 s.val) \u2286 fun x => x \u2208 (s \u222a t).val\n[PROOFSTEP]\nintro g hg\n[GOAL]\ncase intro.intro.intro.intro.hm.tr.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\n\u22a2 (fun x => x \u2208 t.val) \u2286 fun x => x \u2208 (s \u222a t).val\n[PROOFSTEP]\nintro g hg\n[GOAL]\ncase intro.intro.intro.intro.hm.sr.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\ng : G\nhg : g \u2208 fun x => x \u2208 s.val\n\u22a2 g \u2208 fun x => x \u2208 (s \u222a t).val\n[PROOFSTEP]\nsimp only [mem_def] at hg \n[GOAL]\ncase intro.intro.intro.intro.hm.tr.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\ng : G\nhg : g \u2208 fun x => x \u2208 t.val\n\u22a2 g \u2208 fun x => x \u2208 (s \u222a t).val\n[PROOFSTEP]\nsimp only [mem_def] at hg \n[GOAL]\ncase intro.intro.intro.intro.hm.sr.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\ng : G\nhg : g \u2208 s.val\n\u22a2 g \u2208 fun x => x \u2208 (s \u222a t).val\n[PROOFSTEP]\nsimp only [mem_def, Multiset.mem_union, Finset.union_val, hg, or_true_iff, true_or_iff]\n[GOAL]\ncase intro.intro.intro.intro.hm.tr.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\ns : Finset G\nh1s : K\u2081.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K\u2081.carrier V\nt : Finset G\nh1t : K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V\nh2t : Finset.card t = index K\u2082.carrier V\ng : G\nhg : g \u2208 t.val\n\u22a2 g \u2208 fun x => x \u2208 (s \u222a t).val\n[PROOFSTEP]\nsimp only [mem_def, Multiset.mem_union, Finset.union_val, hg, or_true_iff, true_or_iff]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\n\u22a2 index (K\u2081.carrier \u222a K\u2082.carrier) V = index K\u2081.carrier V + index K\u2082.carrier V\n[PROOFSTEP]\napply le_antisymm (index_union_le K\u2081 K\u2082 hV)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\n\u22a2 index K\u2081.carrier V + index K\u2082.carrier V \u2264 index (K\u2081.carrier \u222a K\u2082.carrier) V\n[PROOFSTEP]\nrcases index_elim (K\u2081.2.union K\u2082.2) hV with \u27e8s, h1s, h2s\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\n\u22a2 index K\u2081.carrier V + index K\u2082.carrier V \u2264 index (K\u2081.carrier \u222a K\u2082.carrier) V\n[PROOFSTEP]\nrw [\u2190 h2s]\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\n\u22a2 index K\u2081.carrier V + index K\u2082.carrier V \u2264 Finset.card s\n[PROOFSTEP]\nhave :\n \u2200 K : Set G,\n (K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' V) \u2192\n index K V \u2264 (s.filter fun g => ((fun h : G => g * h) \u207b\u00b9' V \u2229 K).Nonempty).card :=\n by\n intro K hK; apply Nat.sInf_le; refine' \u27e8_, _, rfl\u27e9; rw [mem_setOf_eq]\n intro g hg; rcases hK hg with \u27e8_, \u27e8g\u2080, rfl\u27e9, _, \u27e8h1g\u2080, rfl\u27e9, h2g\u2080\u27e9\n simp only [mem_preimage] at h2g\u2080 \n simp only [mem_iUnion]; use g\u2080; constructor; swap\n \u00b7 simp only [Finset.mem_filter, h1g\u2080, true_and_iff]; use g\n simp only [hg, h2g\u2080, mem_inter_iff, mem_preimage, and_self_iff]\n exact h2g\u2080\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\n\u22a2 \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\n[PROOFSTEP]\nintro K hK\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nK : Set G\nhK : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\n\u22a2 index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\n[PROOFSTEP]\napply Nat.sInf_le\n[GOAL]\ncase hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nK : Set G\nhK : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\n\u22a2 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s) \u2208\n Finset.card '' {t | K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n[PROOFSTEP]\nrefine' \u27e8_, _, rfl\u27e9\n[GOAL]\ncase hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nK : Set G\nhK : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\n\u22a2 Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s \u2208\n {t | K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n[PROOFSTEP]\nrw [mem_setOf_eq]\n[GOAL]\ncase hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nK : Set G\nhK : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\n\u22a2 K \u2286 \u22c3 (g : G) (_ : g \u2208 Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\nintro g hg\n[GOAL]\ncase hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nK : Set G\nhK : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\ng : G\nhg : g \u2208 K\n\u22a2 g \u2208 \u22c3 (g : G) (_ : g \u2208 Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\nrcases hK hg with \u27e8_, \u27e8g\u2080, rfl\u27e9, _, \u27e8h1g\u2080, rfl\u27e9, h2g\u2080\u27e9\n[GOAL]\ncase hm.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nK : Set G\nhK : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\ng : G\nhg : g \u2208 K\ng\u2080 : G\nh1g\u2080 : g\u2080 \u2208 s\nh2g\u2080 : g \u2208 (fun h => (fun h => g\u2080 * h) \u207b\u00b9' V) h1g\u2080\n\u22a2 g \u2208 \u22c3 (g : G) (_ : g \u2208 Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\nsimp only [mem_preimage] at h2g\u2080 \n[GOAL]\ncase hm.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nK : Set G\nhK : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\ng : G\nhg : g \u2208 K\ng\u2080 : G\nh1g\u2080 : g\u2080 \u2208 s\nh2g\u2080 : g\u2080 * g \u2208 V\n\u22a2 g \u2208 \u22c3 (g : G) (_ : g \u2208 Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s), (fun h => g * h) \u207b\u00b9' V\n[PROOFSTEP]\nsimp only [mem_iUnion]\n[GOAL]\ncase hm.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nK : Set G\nhK : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\ng : G\nhg : g \u2208 K\ng\u2080 : G\nh1g\u2080 : g\u2080 \u2208 s\nh2g\u2080 : g\u2080 * g \u2208 V\n\u22a2 \u2203 i i_1, g \u2208 (fun h => i * h) \u207b\u00b9' V\n[PROOFSTEP]\nuse g\u2080\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nK : Set G\nhK : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\ng : G\nhg : g \u2208 K\ng\u2080 : G\nh1g\u2080 : g\u2080 \u2208 s\nh2g\u2080 : g\u2080 * g \u2208 V\n\u22a2 \u2203 i, g \u2208 (fun h => g\u2080 * h) \u207b\u00b9' V\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nK : Set G\nhK : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\ng : G\nhg : g \u2208 K\ng\u2080 : G\nh1g\u2080 : g\u2080 \u2208 s\nh2g\u2080 : g\u2080 * g \u2208 V\n\u22a2 g \u2208 (fun h => g\u2080 * h) \u207b\u00b9' V\ncase h.w\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nK : Set G\nhK : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\ng : G\nhg : g \u2208 K\ng\u2080 : G\nh1g\u2080 : g\u2080 \u2208 s\nh2g\u2080 : g\u2080 * g \u2208 V\n\u22a2 g\u2080 \u2208 Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s\n[PROOFSTEP]\nswap\n[GOAL]\ncase h.w\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nK : Set G\nhK : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\ng : G\nhg : g \u2208 K\ng\u2080 : G\nh1g\u2080 : g\u2080 \u2208 s\nh2g\u2080 : g\u2080 * g \u2208 V\n\u22a2 g\u2080 \u2208 Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s\n[PROOFSTEP]\nsimp only [Finset.mem_filter, h1g\u2080, true_and_iff]\n[GOAL]\ncase h.w\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nK : Set G\nhK : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\ng : G\nhg : g \u2208 K\ng\u2080 : G\nh1g\u2080 : g\u2080 \u2208 s\nh2g\u2080 : g\u2080 * g \u2208 V\n\u22a2 Set.Nonempty ((fun h => g\u2080 * h) \u207b\u00b9' V \u2229 K)\n[PROOFSTEP]\nuse g\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nK : Set G\nhK : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\ng : G\nhg : g \u2208 K\ng\u2080 : G\nh1g\u2080 : g\u2080 \u2208 s\nh2g\u2080 : g\u2080 * g \u2208 V\n\u22a2 g \u2208 (fun h => g\u2080 * h) \u207b\u00b9' V \u2229 K\n[PROOFSTEP]\nsimp only [hg, h2g\u2080, mem_inter_iff, mem_preimage, and_self_iff]\n[GOAL]\ncase h.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nK : Set G\nhK : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\ng : G\nhg : g \u2208 K\ng\u2080 : G\nh1g\u2080 : g\u2080 \u2208 s\nh2g\u2080 : g\u2080 * g \u2208 V\n\u22a2 g \u2208 (fun h => g\u2080 * h) \u207b\u00b9' V\n[PROOFSTEP]\nexact h2g\u2080\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\n\u22a2 index K\u2081.carrier V + index K\u2082.carrier V \u2264 Finset.card s\n[PROOFSTEP]\nrefine'\n le_trans\n (add_le_add (this K\u2081.1 <| Subset.trans (subset_union_left _ _) h1s)\n (this K\u2082.1 <| Subset.trans (subset_union_right _ _) h1s))\n _\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\n\u22a2 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K\u2081.carrier)) s) +\n Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K\u2082.carrier)) s) \u2264\n Finset.card s\n[PROOFSTEP]\nrw [\u2190 Finset.card_union_eq, Finset.filter_union_right]\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\n\u22a2 Finset.card\n (Finset.filter\n (fun x =>\n Set.Nonempty ((fun h => x * h) \u207b\u00b9' V \u2229 K\u2081.carrier) \u2228 Set.Nonempty ((fun h => x * h) \u207b\u00b9' V \u2229 K\u2082.carrier))\n s) \u2264\n Finset.card s\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\n\u22a2 Disjoint (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K\u2081.carrier)) s)\n (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K\u2082.carrier)) s)\n[PROOFSTEP]\nexact s.card_filter_le _\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\n\u22a2 Disjoint (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K\u2081.carrier)) s)\n (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K\u2082.carrier)) s)\n[PROOFSTEP]\napply Finset.disjoint_filter.mpr\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\n\u22a2 \u2200 (x : G),\n x \u2208 s \u2192 Set.Nonempty ((fun h => x * h) \u207b\u00b9' V \u2229 K\u2081.carrier) \u2192 \u00acSet.Nonempty ((fun h => x * h) \u207b\u00b9' V \u2229 K\u2082.carrier)\n[PROOFSTEP]\nrintro g\u2081 _ \u27e8g\u2082, h1g\u2082, h2g\u2082\u27e9 \u27e8g\u2083, h1g\u2083, h2g\u2083\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\ng\u2081 : G\na\u271d : g\u2081 \u2208 s\ng\u2082 : G\nh1g\u2082 : g\u2082 \u2208 (fun h => g\u2081 * h) \u207b\u00b9' V\nh2g\u2082 : g\u2082 \u2208 K\u2081.carrier\ng\u2083 : G\nh1g\u2083 : g\u2083 \u2208 (fun h => g\u2081 * h) \u207b\u00b9' V\nh2g\u2083 : g\u2083 \u2208 K\u2082.carrier\n\u22a2 False\n[PROOFSTEP]\nsimp only [mem_preimage] at h1g\u2083 h1g\u2082 \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\ng\u2081 : G\na\u271d : g\u2081 \u2208 s\ng\u2082 : G\nh2g\u2082 : g\u2082 \u2208 K\u2081.carrier\ng\u2083 : G\nh2g\u2083 : g\u2083 \u2208 K\u2082.carrier\nh1g\u2083 : g\u2081 * g\u2083 \u2208 V\nh1g\u2082 : g\u2081 * g\u2082 \u2208 V\n\u22a2 False\n[PROOFSTEP]\nrefine' h.le_bot (_ : g\u2081\u207b\u00b9 \u2208 _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\ng\u2081 : G\na\u271d : g\u2081 \u2208 s\ng\u2082 : G\nh2g\u2082 : g\u2082 \u2208 K\u2081.carrier\ng\u2083 : G\nh2g\u2083 : g\u2083 \u2208 K\u2082.carrier\nh1g\u2083 : g\u2081 * g\u2083 \u2208 V\nh1g\u2082 : g\u2081 * g\u2082 \u2208 V\n\u22a2 g\u2081\u207b\u00b9 \u2208 K\u2081.carrier * V\u207b\u00b9 \u2293 K\u2082.carrier * V\u207b\u00b9\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.left\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\ng\u2081 : G\na\u271d : g\u2081 \u2208 s\ng\u2082 : G\nh2g\u2082 : g\u2082 \u2208 K\u2081.carrier\ng\u2083 : G\nh2g\u2083 : g\u2083 \u2208 K\u2082.carrier\nh1g\u2083 : g\u2081 * g\u2083 \u2208 V\nh1g\u2082 : g\u2081 * g\u2082 \u2208 V\n\u22a2 g\u2081\u207b\u00b9 \u2208 K\u2081.carrier * V\u207b\u00b9\n[PROOFSTEP]\nsimp only [Set.mem_inv, Set.mem_mul, exists_exists_and_eq_and, exists_and_left]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.right\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\ng\u2081 : G\na\u271d : g\u2081 \u2208 s\ng\u2082 : G\nh2g\u2082 : g\u2082 \u2208 K\u2081.carrier\ng\u2083 : G\nh2g\u2083 : g\u2083 \u2208 K\u2082.carrier\nh1g\u2083 : g\u2081 * g\u2083 \u2208 V\nh1g\u2082 : g\u2081 * g\u2082 \u2208 V\n\u22a2 g\u2081\u207b\u00b9 \u2208 K\u2082.carrier * V\u207b\u00b9\n[PROOFSTEP]\nsimp only [Set.mem_inv, Set.mem_mul, exists_exists_and_eq_and, exists_and_left]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.left\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\ng\u2081 : G\na\u271d : g\u2081 \u2208 s\ng\u2082 : G\nh2g\u2082 : g\u2082 \u2208 K\u2081.carrier\ng\u2083 : G\nh2g\u2083 : g\u2083 \u2208 K\u2082.carrier\nh1g\u2083 : g\u2081 * g\u2083 \u2208 V\nh1g\u2082 : g\u2081 * g\u2082 \u2208 V\n\u22a2 \u2203 x, x \u2208 K\u2081.carrier \u2227 \u2203 x_1, x_1\u207b\u00b9 \u2208 V \u2227 x * x_1 = g\u2081\u207b\u00b9\n[PROOFSTEP]\nrefine' \u27e8_, h2g\u2082, (g\u2081 * g\u2082)\u207b\u00b9, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.left.refine'_1\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\ng\u2081 : G\na\u271d : g\u2081 \u2208 s\ng\u2082 : G\nh2g\u2082 : g\u2082 \u2208 K\u2081.carrier\ng\u2083 : G\nh2g\u2083 : g\u2083 \u2208 K\u2082.carrier\nh1g\u2083 : g\u2081 * g\u2083 \u2208 V\nh1g\u2082 : g\u2081 * g\u2082 \u2208 V\n\u22a2 (g\u2081 * g\u2082)\u207b\u00b9\u207b\u00b9 \u2208 V\ncase intro.intro.intro.intro.intro.intro.left.refine'_2\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\ng\u2081 : G\na\u271d : g\u2081 \u2208 s\ng\u2082 : G\nh2g\u2082 : g\u2082 \u2208 K\u2081.carrier\ng\u2083 : G\nh2g\u2083 : g\u2083 \u2208 K\u2082.carrier\nh1g\u2083 : g\u2081 * g\u2083 \u2208 V\nh1g\u2082 : g\u2081 * g\u2082 \u2208 V\n\u22a2 g\u2082 * (g\u2081 * g\u2082)\u207b\u00b9 = g\u2081\u207b\u00b9\n[PROOFSTEP]\nsimp only [inv_inv, h1g\u2082]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.left.refine'_2\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\ng\u2081 : G\na\u271d : g\u2081 \u2208 s\ng\u2082 : G\nh2g\u2082 : g\u2082 \u2208 K\u2081.carrier\ng\u2083 : G\nh2g\u2083 : g\u2083 \u2208 K\u2082.carrier\nh1g\u2083 : g\u2081 * g\u2083 \u2208 V\nh1g\u2082 : g\u2081 * g\u2082 \u2208 V\n\u22a2 g\u2082 * (g\u2081 * g\u2082)\u207b\u00b9 = g\u2081\u207b\u00b9\n[PROOFSTEP]\nsimp only [mul_inv_rev, mul_inv_cancel_left]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.right\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\ng\u2081 : G\na\u271d : g\u2081 \u2208 s\ng\u2082 : G\nh2g\u2082 : g\u2082 \u2208 K\u2081.carrier\ng\u2083 : G\nh2g\u2083 : g\u2083 \u2208 K\u2082.carrier\nh1g\u2083 : g\u2081 * g\u2083 \u2208 V\nh1g\u2082 : g\u2081 * g\u2082 \u2208 V\n\u22a2 \u2203 x, x \u2208 K\u2082.carrier \u2227 \u2203 x_1, x_1\u207b\u00b9 \u2208 V \u2227 x * x_1 = g\u2081\u207b\u00b9\n[PROOFSTEP]\nrefine' \u27e8_, h2g\u2083, (g\u2081 * g\u2083)\u207b\u00b9, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.right.refine'_1\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\ng\u2081 : G\na\u271d : g\u2081 \u2208 s\ng\u2082 : G\nh2g\u2082 : g\u2082 \u2208 K\u2081.carrier\ng\u2083 : G\nh2g\u2083 : g\u2083 \u2208 K\u2082.carrier\nh1g\u2083 : g\u2081 * g\u2083 \u2208 V\nh1g\u2082 : g\u2081 * g\u2082 \u2208 V\n\u22a2 (g\u2081 * g\u2083)\u207b\u00b9\u207b\u00b9 \u2208 V\ncase intro.intro.intro.intro.intro.intro.right.refine'_2\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\ng\u2081 : G\na\u271d : g\u2081 \u2208 s\ng\u2082 : G\nh2g\u2082 : g\u2082 \u2208 K\u2081.carrier\ng\u2083 : G\nh2g\u2083 : g\u2083 \u2208 K\u2082.carrier\nh1g\u2083 : g\u2081 * g\u2083 \u2208 V\nh1g\u2082 : g\u2081 * g\u2082 \u2208 V\n\u22a2 g\u2083 * (g\u2081 * g\u2083)\u207b\u00b9 = g\u2081\u207b\u00b9\n[PROOFSTEP]\nsimp only [inv_inv, h1g\u2083]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.right.refine'_2\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2081 K\u2082 : Compacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : Disjoint (K\u2081.carrier * V\u207b\u00b9) (K\u2082.carrier * V\u207b\u00b9)\ns : Finset G\nh1s : K\u2081.carrier \u222a K\u2082.carrier \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index (K\u2081.carrier \u222a K\u2082.carrier) V\nthis :\n \u2200 (K : Set G),\n K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2192\n index K V \u2264 Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) \u207b\u00b9' V \u2229 K)) s)\ng\u2081 : G\na\u271d : g\u2081 \u2208 s\ng\u2082 : G\nh2g\u2082 : g\u2082 \u2208 K\u2081.carrier\ng\u2083 : G\nh2g\u2083 : g\u2083 \u2208 K\u2082.carrier\nh1g\u2083 : g\u2081 * g\u2083 \u2208 V\nh1g\u2082 : g\u2081 * g\u2082 \u2208 V\n\u22a2 g\u2083 * (g\u2081 * g\u2083)\u207b\u00b9 = g\u2081\u207b\u00b9\n[PROOFSTEP]\nsimp only [mul_inv_rev, mul_inv_cancel_left]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : Set.Nonempty (interior V)\ng : G\n\u22a2 index ((fun h => g * h) '' K) V \u2264 index K V\n[PROOFSTEP]\nrcases index_elim hK hV with \u27e8s, h1s, h2s\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : Set.Nonempty (interior V)\ng : G\ns : Finset G\nh1s : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K V\n\u22a2 index ((fun h => g * h) '' K) V \u2264 index K V\n[PROOFSTEP]\nrw [\u2190 h2s]\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : Set.Nonempty (interior V)\ng : G\ns : Finset G\nh1s : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K V\n\u22a2 index ((fun h => g * h) '' K) V \u2264 Finset.card s\n[PROOFSTEP]\napply Nat.sInf_le\n[GOAL]\ncase intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : Set.Nonempty (interior V)\ng : G\ns : Finset G\nh1s : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K V\n\u22a2 Finset.card s \u2208 Finset.card '' {t | (fun h => g * h) '' K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n[PROOFSTEP]\nrw [mem_image]\n[GOAL]\ncase intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : Set.Nonempty (interior V)\ng : G\ns : Finset G\nh1s : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K V\n\u22a2 \u2203 x, x \u2208 {t | (fun h => g * h) '' K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V} \u2227 Finset.card x = Finset.card s\n[PROOFSTEP]\nrefine' \u27e8s.map (Equiv.mulRight g\u207b\u00b9).toEmbedding, _, Finset.card_map _\u27e9\n[GOAL]\ncase intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : Set.Nonempty (interior V)\ng : G\ns : Finset G\nh1s : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K V\n\u22a2 Finset.map (Equiv.toEmbedding (Equiv.mulRight g\u207b\u00b9)) s \u2208\n {t | (fun h => g * h) '' K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n[PROOFSTEP]\nsimp only [mem_setOf_eq]\n[GOAL]\ncase intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : Set.Nonempty (interior V)\ng : G\ns : Finset G\nh1s : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K V\n\u22a2 (fun h => g * h) '' K \u2286\n \u22c3 (g_1 : G) (_ : g_1 \u2208 Finset.map (Equiv.toEmbedding (Equiv.mulRight g\u207b\u00b9)) s), (fun h => g_1 * h) \u207b\u00b9' V\n[PROOFSTEP]\nrefine' Subset.trans (image_subset _ h1s) _\n[GOAL]\ncase intro.intro.hm\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : Set.Nonempty (interior V)\ng : G\ns : Finset G\nh1s : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K V\n\u22a2 (fun h => g * h) '' \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V \u2286\n \u22c3 (g_1 : G) (_ : g_1 \u2208 Finset.map (Equiv.toEmbedding (Equiv.mulRight g\u207b\u00b9)) s), (fun h => g_1 * h) \u207b\u00b9' V\n[PROOFSTEP]\nrintro _ \u27e8g\u2081, \u27e8_, \u27e8g\u2082, rfl\u27e9, \u27e8_, \u27e8hg\u2082, rfl\u27e9, hg\u2081\u27e9\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.hm.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : Set.Nonempty (interior V)\ng : G\ns : Finset G\nh1s : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K V\ng\u2081 g\u2082 : G\nhg\u2082 : g\u2082 \u2208 s\nhg\u2081 : g\u2081 \u2208 (fun h => (fun h => g\u2082 * h) \u207b\u00b9' V) hg\u2082\n\u22a2 (fun h => g * h) g\u2081 \u2208\n \u22c3 (g_1 : G) (_ : g_1 \u2208 Finset.map (Equiv.toEmbedding (Equiv.mulRight g\u207b\u00b9)) s), (fun h => g_1 * h) \u207b\u00b9' V\n[PROOFSTEP]\nsimp only [mem_preimage] at hg\u2081 \n[GOAL]\ncase intro.intro.hm.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : Set.Nonempty (interior V)\ng : G\ns : Finset G\nh1s : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K V\ng\u2081 g\u2082 : G\nhg\u2082 : g\u2082 \u2208 s\nhg\u2081 : g\u2082 * g\u2081 \u2208 V\n\u22a2 (fun h => g * h) g\u2081 \u2208\n \u22c3 (g_1 : G) (_ : g_1 \u2208 Finset.map (Equiv.toEmbedding (Equiv.mulRight g\u207b\u00b9)) s), (fun h => g_1 * h) \u207b\u00b9' V\n[PROOFSTEP]\nsimp only [exists_prop, mem_iUnion, Finset.mem_map, Equiv.coe_mulRight, exists_exists_and_eq_and, mem_preimage,\n Equiv.toEmbedding_apply]\n[GOAL]\ncase intro.intro.hm.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : Set.Nonempty (interior V)\ng : G\ns : Finset G\nh1s : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K V\ng\u2081 g\u2082 : G\nhg\u2082 : g\u2082 \u2208 s\nhg\u2081 : g\u2082 * g\u2081 \u2208 V\n\u22a2 \u2203 a, a \u2208 s \u2227 a * g\u207b\u00b9 * (g * g\u2081) \u2208 V\n[PROOFSTEP]\nrefine' \u27e8_, hg\u2082, _\u27e9\n[GOAL]\ncase intro.intro.hm.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : Set.Nonempty (interior V)\ng : G\ns : Finset G\nh1s : K \u2286 \u22c3 (g : G) (_ : g \u2208 s), (fun h => g * h) \u207b\u00b9' V\nh2s : Finset.card s = index K V\ng\u2081 g\u2082 : G\nhg\u2082 : g\u2082 \u2208 s\nhg\u2081 : g\u2082 * g\u2081 \u2208 V\n\u22a2 g\u2082 * g\u207b\u00b9 * (g * g\u2081) \u2208 V\n[PROOFSTEP]\nsimp only [mul_assoc, hg\u2081, inv_mul_cancel_left]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\ng : G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 index ((fun h => g * h) '' K) V = index K V\n[PROOFSTEP]\nrefine' le_antisymm (mul_left_index_le hK hV g) _\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\ng : G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 index K V \u2264 index ((fun h => g * h) '' K) V\n[PROOFSTEP]\nconvert mul_left_index_le (hK.image <| continuous_mul_left g) hV g\u207b\u00b9\n[GOAL]\ncase h.e'_3.h.e'_3\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\ng : G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 K = (fun h => g\u207b\u00b9 * h) '' ((fun b => g * b) '' K)\n[PROOFSTEP]\nrw [image_image]\n[GOAL]\ncase h.e'_3.h.e'_3\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\ng : G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 K = (fun x => g\u207b\u00b9 * (g * x)) '' K\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_3.h.e'_3\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\ng : G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 (fun x => g\u207b\u00b9 * (g * x)) '' K = K\n[PROOFSTEP]\nconvert image_id' _ with h\n[GOAL]\ncase h.e'_2.h.e'_3.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : Set G\nhK : IsCompact K\ng : G\nV : Set G\nhV : Set.Nonempty (interior V)\nh : G\n\u22a2 g\u207b\u00b9 * (g * h) = h\n[PROOFSTEP]\napply inv_mul_cancel_left\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 prehaar (\u2191K\u2080) U K \u2264 \u2191(index \u2191K \u2191K\u2080)\n[PROOFSTEP]\nunfold prehaar\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 \u2191(index (\u2191K) U) / \u2191(index (\u2191K\u2080) U) \u2264 \u2191(index \u2191K \u2191K\u2080)\n[PROOFSTEP]\nrw [div_le_iff]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 \u2191(index (\u2191K) U) \u2264 \u2191(index \u2191K \u2191K\u2080) * \u2191(index (\u2191K\u2080) U)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 0 < \u2191(index (\u2191K\u2080) U)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 index (\u2191K) U \u2264 index \u2191K \u2191K\u2080 * index (\u2191K\u2080) U\n[PROOFSTEP]\napply le_index_mul K\u2080 K hU\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 0 < index (\u2191K\u2080) U\n[PROOFSTEP]\nexact index_pos K\u2080 hU\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK : Set G\nh1K : IsCompact K\nh2K : Set.Nonempty (interior K)\n\u22a2 0 < prehaar (\u2191K\u2080) U { carrier := K, isCompact' := h1K }\n[PROOFSTEP]\napply div_pos\n[GOAL]\ncase ha\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK : Set G\nh1K : IsCompact K\nh2K : Set.Nonempty (interior K)\n\u22a2 0 < \u2191(index (\u2191{ carrier := K, isCompact' := h1K }) U)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase hb\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK : Set G\nh1K : IsCompact K\nh2K : Set.Nonempty (interior K)\n\u22a2 0 < \u2191(index (\u2191K\u2080) U)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase ha\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK : Set G\nh1K : IsCompact K\nh2K : Set.Nonempty (interior K)\n\u22a2 0 < index (\u2191{ carrier := K, isCompact' := h1K }) U\ncase hb\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK : Set G\nh1K : IsCompact K\nh2K : Set.Nonempty (interior K)\n\u22a2 0 < index (\u2191K\u2080) U\n[PROOFSTEP]\napply index_pos \u27e8\u27e8K, h1K\u27e9, h2K\u27e9 hU\n[GOAL]\ncase hb\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK : Set G\nh1K : IsCompact K\nh2K : Set.Nonempty (interior K)\n\u22a2 0 < index (\u2191K\u2080) U\n[PROOFSTEP]\nexact index_pos K\u2080 hU\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 K\u2082.carrier\n\u22a2 prehaar (\u2191K\u2080) U K\u2081 \u2264 prehaar (\u2191K\u2080) U K\u2082\n[PROOFSTEP]\nsimp only [prehaar]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 K\u2082.carrier\n\u22a2 \u2191(index (\u2191K\u2081) U) / \u2191(index (\u2191K\u2080) U) \u2264 \u2191(index (\u2191K\u2082) U) / \u2191(index (\u2191K\u2080) U)\n[PROOFSTEP]\nrw [div_le_div_right]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 K\u2082.carrier\n\u22a2 \u2191(index (\u2191K\u2081) U) \u2264 \u2191(index (\u2191K\u2082) U)\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 K\u2082.carrier\n\u22a2 0 < \u2191(index (\u2191K\u2080) U)\n[PROOFSTEP]\nexact_mod_cast index_mono K\u2082.2 h hU\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 K\u2082.carrier\n\u22a2 0 < \u2191(index (\u2191K\u2080) U)\n[PROOFSTEP]\nexact_mod_cast index_pos K\u2080 hU\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\n\u22a2 0 < \u2191(index (\u2191K\u2080.toCompacts) U)\n[PROOFSTEP]\nexact_mod_cast index_pos K\u2080 hU\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082) \u2264 prehaar (\u2191K\u2080) U K\u2081 + prehaar (\u2191K\u2080) U K\u2082\n[PROOFSTEP]\nsimp only [prehaar]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 \u2191(index (\u2191(K\u2081 \u2294 K\u2082)) U) / \u2191(index (\u2191K\u2080) U) \u2264 \u2191(index (\u2191K\u2081) U) / \u2191(index (\u2191K\u2080) U) + \u2191(index (\u2191K\u2082) U) / \u2191(index (\u2191K\u2080) U)\n[PROOFSTEP]\nrw [div_add_div_same, div_le_div_right]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 \u2191(index (\u2191(K\u2081 \u2294 K\u2082)) U) \u2264 \u2191(index (\u2191K\u2081) U) + \u2191(index (\u2191K\u2082) U)\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 0 < \u2191(index (\u2191K\u2080) U)\n[PROOFSTEP]\nexact_mod_cast index_union_le K\u2081 K\u2082 hU\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\n\u22a2 0 < \u2191(index (\u2191K\u2080) U)\n[PROOFSTEP]\nexact_mod_cast index_pos K\u2080 hU\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\nh : Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9)\n\u22a2 prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082) = prehaar (\u2191K\u2080) U K\u2081 + prehaar (\u2191K\u2080) U K\u2082\n[PROOFSTEP]\nsimp only [prehaar]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\nh : Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9)\n\u22a2 \u2191(index (\u2191(K\u2081 \u2294 K\u2082)) U) / \u2191(index (\u2191K\u2080) U) = \u2191(index (\u2191K\u2081) U) / \u2191(index (\u2191K\u2080) U) + \u2191(index (\u2191K\u2082) U) / \u2191(index (\u2191K\u2080) U)\n[PROOFSTEP]\nrw [div_add_div_same]\n -- Porting note: Here was `congr`, but `to_additive` failed to generate a theorem.\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\nh : Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9)\n\u22a2 \u2191(index (\u2191(K\u2081 \u2294 K\u2082)) U) / \u2191(index (\u2191K\u2080) U) = (\u2191(index (\u2191K\u2081) U) + \u2191(index (\u2191K\u2082) U)) / \u2191(index (\u2191K\u2080) U)\n[PROOFSTEP]\nrefine congr_arg (fun x : \u211d => x / index K\u2080 U) ?_\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nK\u2081 K\u2082 : Compacts G\nhU : Set.Nonempty (interior U)\nh : Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9)\n\u22a2 \u2191(index (\u2191(K\u2081 \u2294 K\u2082)) U) = \u2191(index (\u2191K\u2081) U) + \u2191(index (\u2191K\u2082) U)\n[PROOFSTEP]\nexact_mod_cast index_union_eq K\u2081 K\u2082 hU h\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\ng : G\nK : Compacts G\n\u22a2 prehaar (\u2191K\u2080) U (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = prehaar (\u2191K\u2080) U K\n[PROOFSTEP]\nsimp only [prehaar, Compacts.coe_map, is_left_invariant_index K.isCompact _ hU]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\n\u22a2 prehaar (\u2191K\u2080) U \u2208 haarProduct \u2191K\u2080\n[PROOFSTEP]\nrintro \u27e8K, hK\u27e9 _\n[GOAL]\ncase mk\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK : Set G\nhK : IsCompact K\na\u271d : { carrier := K, isCompact' := hK } \u2208 univ\n\u22a2 prehaar (\u2191K\u2080) U { carrier := K, isCompact' := hK } \u2208\n (fun K => Icc 0 \u2191(index \u2191K \u2191K\u2080)) { carrier := K, isCompact' := hK }\n[PROOFSTEP]\nrw [mem_Icc]\n[GOAL]\ncase mk\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\nK : Set G\nhK : IsCompact K\na\u271d : { carrier := K, isCompact' := hK } \u2208 univ\n\u22a2 0 \u2264 prehaar (\u2191K\u2080) U { carrier := K, isCompact' := hK } \u2227\n prehaar (\u2191K\u2080) U { carrier := K, isCompact' := hK } \u2264 \u2191(index \u2191{ carrier := K, isCompact' := hK } \u2191K\u2080)\n[PROOFSTEP]\nexact \u27e8prehaar_nonneg K\u2080 _, prehaar_le_index K\u2080 _ hU\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 (V : OpenNhdsOf 1), clPrehaar (\u2191K\u2080) V)\n[PROOFSTEP]\nhave : IsCompact (haarProduct (K\u2080 : Set G)) := by apply isCompact_univ_pi; intro K; apply isCompact_Icc\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\n\u22a2 IsCompact (haarProduct \u2191K\u2080)\n[PROOFSTEP]\napply isCompact_univ_pi\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\n\u22a2 \u2200 (i : Compacts G), IsCompact (Icc 0 \u2191(index \u2191i \u2191K\u2080))\n[PROOFSTEP]\nintro K\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\n\u22a2 IsCompact (Icc 0 \u2191(index \u2191K \u2191K\u2080))\n[PROOFSTEP]\napply isCompact_Icc\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 (V : OpenNhdsOf 1), clPrehaar (\u2191K\u2080) V)\n[PROOFSTEP]\nrefine' this.inter_iInter_nonempty (clPrehaar K\u2080) (fun s => isClosed_closure) fun t => _\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 (i : OpenNhdsOf 1) (_ : i \u2208 t), clPrehaar (\u2191K\u2080) i)\n[PROOFSTEP]\nlet V\u2080 := \u22c2 V \u2208 t, (V : OpenNhdsOf (1 : G)).carrier\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 (i : OpenNhdsOf 1) (_ : i \u2208 t), clPrehaar (\u2191K\u2080) i)\n[PROOFSTEP]\nhave h1V\u2080 : IsOpen V\u2080 := by apply isOpen_biInter; apply Finset.finite_toSet; rintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\u27e9 _; exact hV\u2081\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\n\u22a2 IsOpen V\u2080\n[PROOFSTEP]\napply isOpen_biInter\n[GOAL]\ncase hs\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\n\u22a2 Set.Finite fun i => i \u2208 t.val\ncase h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\n\u22a2 \u2200 (i : OpenNhdsOf 1), (i \u2208 fun i => i \u2208 t.val) \u2192 IsOpen i.carrier\n[PROOFSTEP]\napply Finset.finite_toSet\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\n\u22a2 \u2200 (i : OpenNhdsOf 1), (i \u2208 fun i => i \u2208 t.val) \u2192 IsOpen i.carrier\n[PROOFSTEP]\nrintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\u27e9 _\n[GOAL]\ncase h.mk.mk\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nV : Set G\nhV\u2081 : IsOpen V\nhV\u2082 : 1 \u2208 { carrier := V, is_open' := hV\u2081 }.carrier\na\u271d : { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 } \u2208 fun i => i \u2208 t.val\n\u22a2 IsOpen { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 }.toOpens.carrier\n[PROOFSTEP]\nexact hV\u2081\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nh1V\u2080 : IsOpen V\u2080\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 (i : OpenNhdsOf 1) (_ : i \u2208 t), clPrehaar (\u2191K\u2080) i)\n[PROOFSTEP]\nhave h2V\u2080 : (1 : G) \u2208 V\u2080 := by simp only [mem_iInter]; rintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\u27e9 _; exact hV\u2082\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nh1V\u2080 : IsOpen V\u2080\n\u22a2 1 \u2208 V\u2080\n[PROOFSTEP]\nsimp only [mem_iInter]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nh1V\u2080 : IsOpen V\u2080\n\u22a2 \u2200 (i : OpenNhdsOf 1), i \u2208 t \u2192 1 \u2208 i.carrier\n[PROOFSTEP]\nrintro \u27e8\u27e8V, hV\u2081\u27e9, hV\u2082\u27e9 _\n[GOAL]\ncase mk.mk\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nh1V\u2080 : IsOpen V\u2080\nV : Set G\nhV\u2081 : IsOpen V\nhV\u2082 : 1 \u2208 { carrier := V, is_open' := hV\u2081 }.carrier\ni\u271d : { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 } \u2208 t\n\u22a2 1 \u2208 { toOpens := { carrier := V, is_open' := hV\u2081 }, mem' := hV\u2082 }.toOpens.carrier\n[PROOFSTEP]\nexact hV\u2082\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 (i : OpenNhdsOf 1) (_ : i \u2208 t), clPrehaar (\u2191K\u2080) i)\n[PROOFSTEP]\nrefine' \u27e8prehaar K\u2080 V\u2080, _\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 prehaar (\u2191K\u2080) V\u2080 \u2208 haarProduct \u2191K\u2080 \u2229 \u22c2 (i : OpenNhdsOf 1) (_ : i \u2208 t), clPrehaar (\u2191K\u2080) i\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 prehaar (\u2191K\u2080) V\u2080 \u2208 haarProduct \u2191K\u2080\n[PROOFSTEP]\napply prehaar_mem_haarProduct K\u2080\n[GOAL]\ncase left\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 Set.Nonempty (interior V\u2080)\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 1 \u2208 interior V\u2080\n[PROOFSTEP]\nrwa [h1V\u2080.interior_eq]\n[GOAL]\ncase right\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 prehaar (\u2191K\u2080) V\u2080 \u2208 \u22c2 (i : OpenNhdsOf 1) (_ : i \u2208 t), clPrehaar (\u2191K\u2080) i\n[PROOFSTEP]\nsimp only [mem_iInter]\n[GOAL]\ncase right\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\n\u22a2 \u2200 (i : OpenNhdsOf 1), i \u2208 t \u2192 prehaar (\u2191K\u2080) (\u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier) \u2208 clPrehaar (\u2191K\u2080) i\n[PROOFSTEP]\nrintro \u27e8V, hV\u27e9 h2V\n[GOAL]\ncase right.mk\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\nV : Opens G\nhV : 1 \u2208 V.carrier\nh2V : { toOpens := V, mem' := hV } \u2208 t\n\u22a2 prehaar (\u2191K\u2080) (\u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier) \u2208 clPrehaar \u2191K\u2080 { toOpens := V, mem' := hV }\n[PROOFSTEP]\napply subset_closure\n[GOAL]\ncase right.mk.a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\nV : Opens G\nhV : 1 \u2208 V.carrier\nh2V : { toOpens := V, mem' := hV } \u2208 t\n\u22a2 prehaar (\u2191K\u2080) (\u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier) \u2208\n prehaar \u2191K\u2080 '' {U | U \u2286 \u2191{ toOpens := V, mem' := hV }.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}\n[PROOFSTEP]\napply mem_image_of_mem\n[GOAL]\ncase right.mk.a.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\nV : Opens G\nhV : 1 \u2208 V.carrier\nh2V : { toOpens := V, mem' := hV } \u2208 t\n\u22a2 \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier \u2208 {U | U \u2286 \u2191{ toOpens := V, mem' := hV }.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}\n[PROOFSTEP]\nrw [mem_setOf_eq]\n[GOAL]\ncase right.mk.a.h\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nthis : IsCompact (haarProduct \u2191K\u2080)\nt : Finset (OpenNhdsOf 1)\nV\u2080 : Set G := \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\nh1V\u2080 : IsOpen V\u2080\nh2V\u2080 : 1 \u2208 V\u2080\nV : Opens G\nhV : 1 \u2208 V.carrier\nh2V : { toOpens := V, mem' := hV } \u2208 t\n\u22a2 \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier \u2286 \u2191{ toOpens := V, mem' := hV }.toOpens \u2227\n IsOpen (\u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier) \u2227 1 \u2208 \u22c2 (V : OpenNhdsOf 1) (_ : V \u2208 t), V.carrier\n[PROOFSTEP]\nexact \u27e8Subset.trans (iInter_subset _ \u27e8V, hV\u27e9) (iInter_subset _ h2V), h1V\u2080, h2V\u2080\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nV : OpenNhdsOf 1\n\u22a2 chaar K\u2080 \u2208 clPrehaar (\u2191K\u2080) V\n[PROOFSTEP]\nhave := (Classical.choose_spec (nonempty_iInter_clPrehaar K\u2080)).2\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nV : OpenNhdsOf 1\nthis :\n Classical.choose (_ : Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 (V : OpenNhdsOf 1), clPrehaar (\u2191K\u2080) V)) \u2208\n \u22c2 (V : OpenNhdsOf 1), clPrehaar (\u2191K\u2080) V\n\u22a2 chaar K\u2080 \u2208 clPrehaar (\u2191K\u2080) V\n[PROOFSTEP]\nrw [mem_iInter] at this \n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nV : OpenNhdsOf 1\nthis :\n \u2200 (i : OpenNhdsOf 1),\n Classical.choose (_ : Set.Nonempty (haarProduct \u2191K\u2080 \u2229 \u22c2 (V : OpenNhdsOf 1), clPrehaar (\u2191K\u2080) V)) \u2208 clPrehaar (\u2191K\u2080) i\n\u22a2 chaar K\u2080 \u2208 clPrehaar (\u2191K\u2080) V\n[PROOFSTEP]\nexact this V\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\n\u22a2 0 \u2264 chaar K\u2080 K\n[PROOFSTEP]\nhave := chaar_mem_haarProduct K\u2080 K (mem_univ _)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nthis : chaar K\u2080 K \u2208 (fun K => Icc 0 \u2191(index \u2191K \u2191K\u2080)) K\n\u22a2 0 \u2264 chaar K\u2080 K\n[PROOFSTEP]\nrw [mem_Icc] at this \n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK : Compacts G\nthis : 0 \u2264 chaar K\u2080 K \u2227 chaar K\u2080 K \u2264 \u2191(index \u2191K \u2191K\u2080)\n\u22a2 0 \u2264 chaar K\u2080 K\n[PROOFSTEP]\nexact this.1\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\n\u22a2 chaar K\u2080 \u22a5 = 0\n[PROOFSTEP]\nlet eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\n\u22a2 chaar K\u2080 \u22a5 = 0\n[PROOFSTEP]\nhave : Continuous eval := continuous_apply \u22a5\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u22a5 = 0\n[PROOFSTEP]\nshow chaar K\u2080 \u2208 eval \u207b\u00b9' {(0 : \u211d)}\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' {0}\n[PROOFSTEP]\napply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K\u2080 \u22a4)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\nthis : Continuous eval\n\u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' {0}\n[PROOFSTEP]\nunfold clPrehaar\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\nthis : Continuous eval\n\u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' {0}\n[PROOFSTEP]\nrw [IsClosed.closure_subset_iff]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\nthis : Continuous eval\n\u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' {0}\n[PROOFSTEP]\nrintro _ \u27e8U, _, rfl\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2208 {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' {0}\n[PROOFSTEP]\napply prehaar_empty\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\nthis : Continuous eval\n\u22a2 IsClosed (eval \u207b\u00b9' {0})\n[PROOFSTEP]\napply continuous_iff_isClosed.mp this\n[GOAL]\ncase a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f \u22a5\nthis : Continuous eval\n\u22a2 IsClosed {0}\n[PROOFSTEP]\nexact isClosed_singleton\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\n\u22a2 chaar K\u2080 K\u2080.toCompacts = 1\n[PROOFSTEP]\nlet eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\n\u22a2 chaar K\u2080 K\u2080.toCompacts = 1\n[PROOFSTEP]\nhave : Continuous eval := continuous_apply _\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 chaar K\u2080 K\u2080.toCompacts = 1\n[PROOFSTEP]\nshow chaar K\u2080 \u2208 eval \u207b\u00b9' {(1 : \u211d)}\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' {1}\n[PROOFSTEP]\napply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K\u2080 \u22a4)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' {1}\n[PROOFSTEP]\nunfold clPrehaar\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' {1}\n[PROOFSTEP]\nrw [IsClosed.closure_subset_iff]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' {1}\n[PROOFSTEP]\nrintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' {1}\n[PROOFSTEP]\napply prehaar_self\n[GOAL]\ncase intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty (interior U)\n[PROOFSTEP]\nrw [h2U.interior_eq]\n[GOAL]\ncase intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty U\n[PROOFSTEP]\nexact \u27e81, h3U\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 IsClosed (eval \u207b\u00b9' {1})\n[PROOFSTEP]\napply continuous_iff_isClosed.mp this\n[GOAL]\ncase a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts\nthis : Continuous eval\n\u22a2 IsClosed {1}\n[PROOFSTEP]\nexact isClosed_singleton\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\n\u22a2 chaar K\u2080 K\u2081 \u2264 chaar K\u2080 K\u2082\n[PROOFSTEP]\nlet eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\n\u22a2 chaar K\u2080 K\u2081 \u2264 chaar K\u2080 K\u2082\n[PROOFSTEP]\nhave : Continuous eval := (continuous_apply K\u2082).sub (continuous_apply K\u2081)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 chaar K\u2080 K\u2081 \u2264 chaar K\u2080 K\u2082\n[PROOFSTEP]\nrw [\u2190 sub_nonneg]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 0 \u2264 chaar K\u2080 K\u2082 - chaar K\u2080 K\u2081\n[PROOFSTEP]\nshow chaar K\u2080 \u2208 eval \u207b\u00b9' Ici (0 : \u211d)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' Ici 0\n[PROOFSTEP]\napply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K\u2080 \u22a4)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' Ici 0\n[PROOFSTEP]\nunfold clPrehaar\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' Ici 0\n[PROOFSTEP]\nrw [IsClosed.closure_subset_iff]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' Ici 0\n[PROOFSTEP]\nrintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' Ici 0\n[PROOFSTEP]\nsimp only [mem_preimage, mem_Ici, sub_nonneg]\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U K\u2081 \u2264 prehaar (\u2191K\u2080) U K\u2082\n[PROOFSTEP]\napply prehaar_mono _ h\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty (interior U)\n[PROOFSTEP]\nrw [h2U.interior_eq]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty U\n[PROOFSTEP]\nexact \u27e81, h3U\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 IsClosed (eval \u207b\u00b9' Ici 0)\n[PROOFSTEP]\napply continuous_iff_isClosed.mp this\n[GOAL]\ncase a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081\nthis : Continuous eval\n\u22a2 IsClosed (Ici 0)\n[PROOFSTEP]\nexact isClosed_Ici\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) \u2264 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082\n[PROOFSTEP]\nlet eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) \u2264 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082\n[PROOFSTEP]\nhave : Continuous eval := by exact ((continuous_apply K\u2081).add (continuous_apply K\u2082)).sub (continuous_apply (K\u2081 \u2294 K\u2082))\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\n\u22a2 Continuous eval\n[PROOFSTEP]\nexact ((continuous_apply K\u2081).add (continuous_apply K\u2082)).sub (continuous_apply (K\u2081 \u2294 K\u2082))\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) \u2264 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082\n[PROOFSTEP]\nrw [\u2190 sub_nonneg]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 0 \u2264 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 - chaar K\u2080 (K\u2081 \u2294 K\u2082)\n[PROOFSTEP]\nshow chaar K\u2080 \u2208 eval \u207b\u00b9' Ici (0 : \u211d)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' Ici 0\n[PROOFSTEP]\napply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K\u2080 \u22a4)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' Ici 0\n[PROOFSTEP]\nunfold clPrehaar\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' Ici 0\n[PROOFSTEP]\nrw [IsClosed.closure_subset_iff]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' Ici 0\n[PROOFSTEP]\nrintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' Ici 0\n[PROOFSTEP]\nsimp only [mem_preimage, mem_Ici, sub_nonneg]\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082) \u2264 prehaar (\u2191K\u2080) U K\u2081 + prehaar (\u2191K\u2080) U K\u2082\n[PROOFSTEP]\napply prehaar_sup_le\n[GOAL]\ncase intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty (interior U)\n[PROOFSTEP]\nrw [h2U.interior_eq]\n[GOAL]\ncase intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty U\n[PROOFSTEP]\nexact \u27e81, h3U\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 IsClosed (eval \u207b\u00b9' Ici 0)\n[PROOFSTEP]\napply continuous_iff_isClosed.mp this\n[GOAL]\ncase a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 IsClosed (Ici 0)\n[PROOFSTEP]\nexact isClosed_Ici\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082\n[PROOFSTEP]\nrcases isCompact_isCompact_separated K\u2081.2 K\u2082.2 h with \u27e8U\u2081, U\u2082, h1U\u2081, h1U\u2082, h2U\u2081, h2U\u2082, hU\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082\n[PROOFSTEP]\nrcases compact_open_separated_mul_right K\u2081.2 h1U\u2081 h2U\u2081 with \u27e8L\u2081, h1L\u2081, h2L\u2081\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nh2L\u2081 : K\u2081.carrier * L\u2081 \u2286 U\u2081\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082\n[PROOFSTEP]\nrcases mem_nhds_iff.mp h1L\u2081 with \u27e8V\u2081, h1V\u2081, h2V\u2081, h3V\u2081\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nh2L\u2081 : K\u2081.carrier * L\u2081 \u2286 U\u2081\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082\n[PROOFSTEP]\nreplace h2L\u2081 := Subset.trans (mul_subset_mul_left h1V\u2081) h2L\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082\n[PROOFSTEP]\nrcases compact_open_separated_mul_right K\u2082.2 h1U\u2082 h2U\u2082 with \u27e8L\u2082, h1L\u2082, h2L\u2082\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nh2L\u2082 : K\u2082.carrier * L\u2082 \u2286 U\u2082\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082\n[PROOFSTEP]\nrcases mem_nhds_iff.mp h1L\u2082 with \u27e8V\u2082, h1V\u2082, h2V\u2082, h3V\u2082\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nh2L\u2082 : K\u2082.carrier * L\u2082 \u2286 U\u2082\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082\n[PROOFSTEP]\nreplace h2L\u2082 := Subset.trans (mul_subset_mul_left h1V\u2082) h2L\u2082\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082\n[PROOFSTEP]\nlet eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082\n[PROOFSTEP]\nhave : Continuous eval := ((continuous_apply K\u2081).add (continuous_apply K\u2082)).sub (continuous_apply (K\u2081 \u2294 K\u2082))\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) = chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082\n[PROOFSTEP]\nrw [eq_comm, \u2190 sub_eq_zero]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 - chaar K\u2080 (K\u2081 \u2294 K\u2082) = 0\n[PROOFSTEP]\nshow chaar K\u2080 \u2208 eval \u207b\u00b9' {(0 : \u211d)}\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' {0}\n[PROOFSTEP]\nlet V := V\u2081 \u2229 V\u2082\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' {0}\n[PROOFSTEP]\napply\n mem_of_subset_of_mem _\n (chaar_mem_clPrehaar K\u2080\n \u27e8\u27e8V\u207b\u00b9, (h2V\u2081.inter h2V\u2082).preimage continuous_inv\u27e9, by\n simp only [mem_inv, inv_one, h3V\u2081, h3V\u2082, mem_inter_iff, true_and_iff]\u27e9)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 1 \u2208 { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) }.carrier\n[PROOFSTEP]\nsimp only [mem_inv, inv_one, h3V\u2081, h3V\u2082, mem_inter_iff, true_and_iff]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 clPrehaar \u2191K\u2080\n { toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) } \u2286\n eval \u207b\u00b9' {0}\n[PROOFSTEP]\nunfold clPrehaar\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 closure\n (prehaar \u2191K\u2080 ''\n {U |\n U \u2286\n \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens \u2227\n IsOpen U \u2227 1 \u2208 U}) \u2286\n eval \u207b\u00b9' {0}\n[PROOFSTEP]\nrw [IsClosed.closure_subset_iff]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 prehaar \u2191K\u2080 ''\n {U |\n U \u2286\n \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens \u2227\n IsOpen U \u2227 1 \u2208 U} \u2286\n eval \u207b\u00b9' {0}\n[PROOFSTEP]\nrintro _ \u27e8U, \u27e8h1U, h2U, h3U\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n U \u2286\n \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' {0}\n[PROOFSTEP]\nsimp only [mem_preimage, sub_eq_zero, mem_singleton_iff]\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n U \u2286\n \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U K\u2081 + prehaar (\u2191K\u2080) U K\u2082 = prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082)\n[PROOFSTEP]\nrw [eq_comm]\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n U \u2286\n \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082) = prehaar (\u2191K\u2080) U K\u2081 + prehaar (\u2191K\u2080) U K\u2082\n[PROOFSTEP]\napply prehaar_sup_eq\n[GOAL]\ncase intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n U \u2286\n \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty (interior U)\n[PROOFSTEP]\nrw [h2U.interior_eq]\n[GOAL]\ncase intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n U \u2286\n \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty U\n[PROOFSTEP]\nexact \u27e81, h3U\u27e9\n[GOAL]\ncase intro.intro.intro.intro.h\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n U \u2286\n \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Disjoint (K\u2081.carrier * U\u207b\u00b9) (K\u2082.carrier * U\u207b\u00b9)\n[PROOFSTEP]\nrefine' disjoint_of_subset _ _ hU\n[GOAL]\ncase intro.intro.intro.intro.h.refine'_1\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n U \u2286\n \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 K\u2081.carrier * U\u207b\u00b9 \u2286 U\u2081\n[PROOFSTEP]\nrefine' Subset.trans (mul_subset_mul Subset.rfl _) h2L\u2081\n[GOAL]\ncase intro.intro.intro.intro.h.refine'_1\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n U \u2286\n \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 U\u207b\u00b9 \u2286 V\u2081\n[PROOFSTEP]\nexact Subset.trans (inv_subset.mpr h1U) (inter_subset_left _ _)\n[GOAL]\ncase intro.intro.intro.intro.h.refine'_2\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n U \u2286\n \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 K\u2082.carrier * U\u207b\u00b9 \u2286 U\u2082\n[PROOFSTEP]\nrefine' Subset.trans (mul_subset_mul Subset.rfl _) h2L\u2082\n[GOAL]\ncase intro.intro.intro.intro.h.refine'_2\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\nU : Set G\nh1U :\n U \u2286\n \u2191{ toOpens := { carrier := V\u207b\u00b9, is_open' := (_ : IsOpen ((fun a => a\u207b\u00b9) \u207b\u00b9' (V\u2081 \u2229 V\u2082))) },\n mem' := (_ : 1 \u2208 (V\u2081 \u2229 V\u2082)\u207b\u00b9) }.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 U\u207b\u00b9 \u2286 V\u2082\n[PROOFSTEP]\nexact Subset.trans (inv_subset.mpr h1U) (inter_subset_right _ _)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 IsClosed (eval \u207b\u00b9' {0})\n[PROOFSTEP]\napply continuous_iff_isClosed.mp this\n[GOAL]\ncase a\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint K\u2081.carrier K\u2082.carrier\nU\u2081 U\u2082 : Set G\nh1U\u2081 : IsOpen U\u2081\nh1U\u2082 : IsOpen U\u2082\nh2U\u2081 : K\u2081.carrier \u2286 U\u2081\nh2U\u2082 : K\u2082.carrier \u2286 U\u2082\nhU : Disjoint U\u2081 U\u2082\nL\u2081 : Set G\nh1L\u2081 : L\u2081 \u2208 \ud835\udcdd 1\nV\u2081 : Set G\nh1V\u2081 : V\u2081 \u2286 L\u2081\nh2V\u2081 : IsOpen V\u2081\nh3V\u2081 : 1 \u2208 V\u2081\nh2L\u2081 : K\u2081.carrier * V\u2081 \u2286 U\u2081\nL\u2082 : Set G\nh1L\u2082 : L\u2082 \u2208 \ud835\udcdd 1\nV\u2082 : Set G\nh1V\u2082 : V\u2082 \u2286 L\u2082\nh2V\u2082 : IsOpen V\u2082\nh3V\u2082 : 1 \u2208 V\u2082\nh2L\u2082 : K\u2082.carrier * V\u2082 \u2286 U\u2082\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082)\nthis : Continuous eval\nV : Set G := V\u2081 \u2229 V\u2082\n\u22a2 IsClosed {0}\n[PROOFSTEP]\nexact isClosed_singleton\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\n\u22a2 chaar K\u2080 (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = chaar K\u2080 K\n[PROOFSTEP]\nlet eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (K.map _ <| continuous_mul_left g) - f K\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\n\u22a2 chaar K\u2080 (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = chaar K\u2080 K\n[PROOFSTEP]\nhave : Continuous eval := (continuous_apply (K.map _ _)).sub (continuous_apply K)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 chaar K\u2080 (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = chaar K\u2080 K\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 chaar K\u2080 (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - chaar K\u2080 K = 0\n[PROOFSTEP]\nshow chaar K\u2080 \u2208 eval \u207b\u00b9' {(0 : \u211d)}\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' {0}\n[PROOFSTEP]\napply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K\u2080 \u22a4)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' {0}\n[PROOFSTEP]\nunfold clPrehaar\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' {0}\n[PROOFSTEP]\nrw [IsClosed.closure_subset_iff]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' {0}\n[PROOFSTEP]\nrintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' {0}\n[PROOFSTEP]\nsimp only [mem_singleton_iff, mem_preimage, sub_eq_zero]\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 prehaar (\u2191K\u2080) U (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = prehaar (\u2191K\u2080) U K\n[PROOFSTEP]\napply is_left_invariant_prehaar\n[GOAL]\ncase intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty (interior U)\n[PROOFSTEP]\nrw [h2U.interior_eq]\n[GOAL]\ncase intro.intro.intro.intro.hU\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\nU : Set G\nleft\u271d : U \u2286 \u2191\u22a4.toOpens\nh2U : IsOpen U\nh3U : 1 \u2208 U\n\u22a2 Set.Nonempty U\n[PROOFSTEP]\nexact \u27e81, h3U\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 IsClosed (eval \u207b\u00b9' {0})\n[PROOFSTEP]\napply continuous_iff_isClosed.mp this\n[GOAL]\ncase a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\neval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K\nthis : Continuous eval\n\u22a2 IsClosed {0}\n[PROOFSTEP]\nexact isClosed_singleton\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : \u2191K\u2081 \u2286 \u2191K\u2082\n\u22a2 (fun K => { val := chaar K\u2080 K, property := (_ : 0 \u2264 chaar K\u2080 K) }) K\u2081 \u2264\n (fun K => { val := chaar K\u2080 K, property := (_ : 0 \u2264 chaar K\u2080 K) }) K\u2082\n[PROOFSTEP]\nsimp only [\u2190 NNReal.coe_le_coe, NNReal.toReal, chaar_mono, h]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint \u2191K\u2081 \u2191K\u2082\n\u22a2 (fun K => { val := chaar K\u2080 K, property := (_ : 0 \u2264 chaar K\u2080 K) }) (K\u2081 \u2294 K\u2082) =\n (fun K => { val := chaar K\u2080 K, property := (_ : 0 \u2264 chaar K\u2080 K) }) K\u2081 +\n (fun K => { val := chaar K\u2080 K, property := (_ : 0 \u2264 chaar K\u2080 K) }) K\u2082\n[PROOFSTEP]\nsimp only [chaar_sup_eq h]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\nh : Disjoint \u2191K\u2081 \u2191K\u2082\n\u22a2 { val := chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082, property := (_ : (fun r => 0 \u2264 r) (chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082)) } =\n { val := chaar K\u2080 K\u2081, property := (_ : 0 \u2264 chaar K\u2080 K\u2081) } +\n { val := chaar K\u2080 K\u2082, property := (_ : 0 \u2264 chaar K\u2080 K\u2082) }\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\n\u22a2 (fun K => { val := chaar K\u2080 K, property := (_ : 0 \u2264 chaar K\u2080 K) }) (K\u2081 \u2294 K\u2082) \u2264\n (fun K => { val := chaar K\u2080 K, property := (_ : 0 \u2264 chaar K\u2080 K) }) K\u2081 +\n (fun K => { val := chaar K\u2080 K, property := (_ : 0 \u2264 chaar K\u2080 K) }) K\u2082\n[PROOFSTEP]\nsimp only [\u2190 NNReal.coe_le_coe, NNReal.coe_add]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nK\u2081 K\u2082 : Compacts G\n\u22a2 \u2191{ val := chaar K\u2080 (K\u2081 \u2294 K\u2082), property := (_ : 0 \u2264 chaar K\u2080 (K\u2081 \u2294 K\u2082)) } \u2264\n \u2191{ val := chaar K\u2080 K\u2081, property := (_ : 0 \u2264 chaar K\u2080 K\u2081) } +\n \u2191{ val := chaar K\u2080 K\u2082, property := (_ : 0 \u2264 chaar K\u2080 K\u2082) }\n[PROOFSTEP]\nsimp only [NNReal.toReal, chaar_sup_le]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\n\u22a2 (fun s => \u2191(Content.toFun (haarContent K\u2080) s)) K\u2080.toCompacts = 1\n[PROOFSTEP]\nsimp_rw [\u2190 ENNReal.coe_one, haarContent_apply, ENNReal.coe_eq_coe, chaar_self]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\n\u22a2 { val := 1, property := (_ : (fun r => 0 \u2264 r) 1) } = 1\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\ng : G\nK : Compacts G\n\u22a2 (fun s => \u2191(Content.toFun (haarContent K\u2080) s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n (fun s => \u2191(Content.toFun (haarContent K\u2080) s)) K\n[PROOFSTEP]\nsimpa only [ENNReal.coe_eq_coe, \u2190 NNReal.coe_eq, haarContent_apply] using is_left_invariant_chaar g K\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\n\u22a2 0 < \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080\n[PROOFSTEP]\nrefine' zero_lt_one.trans_le _\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\n\u22a2 1 \u2264 \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080\n[PROOFSTEP]\nrw [Content.outerMeasure_eq_iInf]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\n\u22a2 1 \u2264\n \u2a05 (U : Set G) (hU : IsOpen U) (_ : \u2191K\u2080 \u2286 U), Content.innerContent (haarContent K\u2080) { carrier := U, is_open' := hU }\n[PROOFSTEP]\nrefine' le_iInf\u2082 fun U hU => le_iInf fun hK\u2080 => le_trans _ <| le_iSup\u2082 K\u2080.toCompacts hK\u2080\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : T2Space G\nK\u2080 : PositiveCompacts G\nU : Set G\nhU : IsOpen U\nhK\u2080 : \u2191K\u2080 \u2286 U\n\u22a2 1 \u2264 (fun s => \u2191(Content.toFun (haarContent K\u2080) s)) K\u2080.toCompacts\n[PROOFSTEP]\nexact haarContent_self.ge\n[GOAL]\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\ns : Set G\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(haarMeasure K\u2080) s = \u2191(Content.outerMeasure (haarContent K\u2080)) s / \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080\n[PROOFSTEP]\nchange ((haarContent K\u2080).outerMeasure K\u2080)\u207b\u00b9 * (haarContent K\u2080).measure s = _\n[GOAL]\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\ns : Set G\nhs : MeasurableSet s\n\u22a2 (\u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080)\u207b\u00b9 * \u2191\u2191(Content.measure (haarContent K\u2080)) s =\n \u2191(Content.outerMeasure (haarContent K\u2080)) s / \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080\n[PROOFSTEP]\nsimp only [hs, div_eq_mul_inv, mul_comm, Content.measure_apply]\n[GOAL]\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\n\u22a2 IsMulLeftInvariant (haarMeasure K\u2080)\n[PROOFSTEP]\nrw [\u2190 forall_measure_preimage_mul_iff]\n[GOAL]\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\n\u22a2 \u2200 (g : G) (A : Set G), MeasurableSet A \u2192 \u2191\u2191(haarMeasure K\u2080) ((fun h => g * h) \u207b\u00b9' A) = \u2191\u2191(haarMeasure K\u2080) A\n[PROOFSTEP]\nintro g A hA\n[GOAL]\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\ng : G\nA : Set G\nhA : MeasurableSet A\n\u22a2 \u2191\u2191(haarMeasure K\u2080) ((fun h => g * h) \u207b\u00b9' A) = \u2191\u2191(haarMeasure K\u2080) A\n[PROOFSTEP]\nrw [haarMeasure_apply hA, haarMeasure_apply (measurable_const_mul g hA)]\n -- Porting note: Here was `congr 1`, but `to_additive` failed to generate a theorem.\n[GOAL]\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\ng : G\nA : Set G\nhA : MeasurableSet A\n\u22a2 \u2191(Content.outerMeasure (haarContent K\u2080)) ((fun x => g * x) \u207b\u00b9' A) / \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080 =\n \u2191(Content.outerMeasure (haarContent K\u2080)) A / \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080\n[PROOFSTEP]\nrefine congr_arg (fun x : \u211d\u22650\u221e => x / (haarContent K\u2080).outerMeasure K\u2080) ?_\n[GOAL]\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\ng : G\nA : Set G\nhA : MeasurableSet A\n\u22a2 \u2191(Content.outerMeasure (haarContent K\u2080)) ((fun x => g * x) \u207b\u00b9' A) = \u2191(Content.outerMeasure (haarContent K\u2080)) A\n[PROOFSTEP]\napply Content.is_mul_left_invariant_outerMeasure\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\ng : G\nA : Set G\nhA : MeasurableSet A\n\u22a2 \u2200 (g : G) {K : Compacts G},\n (fun s => \u2191(Content.toFun (haarContent K\u2080) s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =\n (fun s => \u2191(Content.toFun (haarContent K\u2080) s)) K\n[PROOFSTEP]\napply is_left_invariant_haarContent\n[GOAL]\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\n\u22a2 \u2191\u2191(haarMeasure K\u2080) \u2191K\u2080 = 1\n[PROOFSTEP]\nhaveI : LocallyCompactSpace G := K\u2080.locallyCompactSpace_of_group\n[GOAL]\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 \u2191\u2191(haarMeasure K\u2080) \u2191K\u2080 = 1\n[PROOFSTEP]\nrw [haarMeasure_apply K\u2080.isCompact.measurableSet, ENNReal.div_self]\n[GOAL]\ncase h0\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080 \u2260 0\n[PROOFSTEP]\nrw [\u2190 pos_iff_ne_zero]\n[GOAL]\ncase h0\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 0 < \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080\n[PROOFSTEP]\nexact haarContent_outerMeasure_self_pos\n[GOAL]\ncase hI\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080 \u2260 \u22a4\n[PROOFSTEP]\nexact (Content.outerMeasure_lt_top_of_isCompact _ K\u2080.isCompact).ne\n[GOAL]\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\n\u22a2 Regular (haarMeasure K\u2080)\n[PROOFSTEP]\nhaveI : LocallyCompactSpace G := K\u2080.locallyCompactSpace_of_group\n[GOAL]\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 Regular (haarMeasure K\u2080)\n[PROOFSTEP]\napply Regular.smul\n[GOAL]\ncase hx\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 (\u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080)\u207b\u00b9 \u2260 \u22a4\n[PROOFSTEP]\nrw [ENNReal.inv_ne_top]\n[GOAL]\ncase hx\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 \u2191(Content.outerMeasure (haarContent K\u2080)) \u2191K\u2080 \u2260 0\n[PROOFSTEP]\nexact haarContent_outerMeasure_self_pos.ne'\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : T2Space G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : SecondCountableTopology G\nK\u2080 : PositiveCompacts G\n\u22a2 SigmaFinite (haarMeasure K\u2080)\n[PROOFSTEP]\nhaveI : LocallyCompactSpace G := K\u2080.locallyCompactSpace_of_group\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : T2Space G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : SecondCountableTopology G\nK\u2080 : PositiveCompacts G\nthis : LocallyCompactSpace G\n\u22a2 SigmaFinite (haarMeasure K\u2080)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\n\u22a2 IsHaarMeasure (haarMeasure K\u2080)\n[PROOFSTEP]\napply isHaarMeasure_of_isCompact_nonempty_interior (haarMeasure K\u2080) K\u2080 K\u2080.isCompact K\u2080.interior_nonempty\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\n\u22a2 \u2191\u2191(haarMeasure K\u2080) \u2191K\u2080 \u2260 0\n[PROOFSTEP]\nsimp only [haarMeasure_self]\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nexact one_ne_zero\n[GOAL]\ncase h'\nG : Type u_1\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : T2Space G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nK\u2080 : PositiveCompacts G\n\u22a2 \u2191\u2191(haarMeasure K\u2080) \u2191K\u2080 \u2260 \u22a4\n[PROOFSTEP]\nsimp only [haarMeasure_self]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK\u2080 : PositiveCompacts G\n\u22a2 (\u2191\u2191\u03bc \u2191K\u2080 / \u2191\u2191(haarMeasure K\u2080) \u2191K\u2080) \u2022 haarMeasure K\u2080 = \u2191\u2191\u03bc \u2191K\u2080 \u2022 haarMeasure K\u2080\n[PROOFSTEP]\nrw [haarMeasure_self, div_one]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nh2K : Set.Nonempty (interior K)\nh\u03bcK : \u2191\u2191\u03bc K \u2260 \u22a4\n\u22a2 Regular \u03bc\n[PROOFSTEP]\nrw [haarMeasure_unique \u03bc \u27e8\u27e8K, hK\u27e9, h2K\u27e9]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsMulLeftInvariant \u03bc\nK : Set G\nhK : IsCompact K\nh2K : Set.Nonempty (interior K)\nh\u03bcK : \u2191\u2191\u03bc K \u2260 \u22a4\n\u22a2 Regular\n (\u2191\u2191\u03bc \u2191{ toCompacts := { carrier := K, isCompact' := hK }, interior_nonempty' := h2K } \u2022\n haarMeasure { toCompacts := { carrier := K, isCompact' := hK }, interior_nonempty' := h2K })\n[PROOFSTEP]\nexact Regular.smul h\u03bcK\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : T2Space G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\ninst\u271d\u00b2 : LocallyCompactSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsHaarMeasure \u03bd\n\u22a2 \u2203 c, c \u2260 0 \u2227 c \u2260 \u22a4 \u2227 \u03bc = c \u2022 \u03bd\n[PROOFSTEP]\nhave K : PositiveCompacts G := Classical.arbitrary _\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : T2Space G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\ninst\u271d\u00b2 : LocallyCompactSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsHaarMeasure \u03bd\nK : PositiveCompacts G\n\u22a2 \u2203 c, c \u2260 0 \u2227 c \u2260 \u22a4 \u2227 \u03bc = c \u2022 \u03bd\n[PROOFSTEP]\nhave \u03bdpos : 0 < \u03bd K := measure_pos_of_nonempty_interior _ K.interior_nonempty\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : T2Space G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\ninst\u271d\u00b2 : LocallyCompactSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsHaarMeasure \u03bd\nK : PositiveCompacts G\n\u03bdpos : 0 < \u2191\u2191\u03bd \u2191K\n\u22a2 \u2203 c, c \u2260 0 \u2227 c \u2260 \u22a4 \u2227 \u03bc = c \u2022 \u03bd\n[PROOFSTEP]\nhave \u03bdne : \u03bd K \u2260 \u221e := K.isCompact.measure_lt_top.ne\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : T2Space G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\ninst\u271d\u00b2 : LocallyCompactSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsHaarMeasure \u03bd\nK : PositiveCompacts G\n\u03bdpos : 0 < \u2191\u2191\u03bd \u2191K\n\u03bdne : \u2191\u2191\u03bd \u2191K \u2260 \u22a4\n\u22a2 \u2203 c, c \u2260 0 \u2227 c \u2260 \u22a4 \u2227 \u03bc = c \u2022 \u03bd\n[PROOFSTEP]\nrefine' \u27e8\u03bc K / \u03bd K, _, _, _\u27e9\n[GOAL]\ncase refine'_1\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : T2Space G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\ninst\u271d\u00b2 : LocallyCompactSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsHaarMeasure \u03bd\nK : PositiveCompacts G\n\u03bdpos : 0 < \u2191\u2191\u03bd \u2191K\n\u03bdne : \u2191\u2191\u03bd \u2191K \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc \u2191K / \u2191\u2191\u03bd \u2191K \u2260 0\n[PROOFSTEP]\nsimp only [\u03bdne, (\u03bc.measure_pos_of_nonempty_interior K.interior_nonempty).ne', Ne.def, ENNReal.div_eq_zero_iff,\n not_false_iff, or_self_iff]\n[GOAL]\ncase refine'_2\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : T2Space G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\ninst\u271d\u00b2 : LocallyCompactSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsHaarMeasure \u03bd\nK : PositiveCompacts G\n\u03bdpos : 0 < \u2191\u2191\u03bd \u2191K\n\u03bdne : \u2191\u2191\u03bd \u2191K \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc \u2191K / \u2191\u2191\u03bd \u2191K \u2260 \u22a4\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, \u03bdpos.ne', (K.isCompact.measure_lt_top (\u03bc := \u03bc)).ne, or_self_iff, ENNReal.inv_eq_top,\n ENNReal.mul_eq_top, Ne.def, not_false_iff, and_false_iff, false_and_iff]\n[GOAL]\ncase refine'_3\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : T2Space G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\ninst\u271d\u00b2 : LocallyCompactSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsHaarMeasure \u03bd\nK : PositiveCompacts G\n\u03bdpos : 0 < \u2191\u2191\u03bd \u2191K\n\u03bdne : \u2191\u2191\u03bd \u2191K \u2260 \u22a4\n\u22a2 \u03bc = (\u2191\u2191\u03bc \u2191K / \u2191\u2191\u03bd \u2191K) \u2022 \u03bd\n[PROOFSTEP]\ncalc\n \u03bc = \u03bc K \u2022 haarMeasure K := haarMeasure_unique \u03bc K\n _ = (\u03bc K / \u03bd K) \u2022 \u03bd K \u2022 haarMeasure K := by\n rw [smul_smul, div_eq_mul_inv, mul_assoc, ENNReal.inv_mul_cancel \u03bdpos.ne' \u03bdne, mul_one]\n _ = (\u03bc K / \u03bd K) \u2022 \u03bd := by rw [\u2190 haarMeasure_unique \u03bd K]\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : T2Space G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\ninst\u271d\u00b2 : LocallyCompactSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsHaarMeasure \u03bd\nK : PositiveCompacts G\n\u03bdpos : 0 < \u2191\u2191\u03bd \u2191K\n\u03bdne : \u2191\u2191\u03bd \u2191K \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc \u2191K \u2022 haarMeasure K = (\u2191\u2191\u03bc \u2191K / \u2191\u2191\u03bd \u2191K) \u2022 \u2191\u2191\u03bd \u2191K \u2022 haarMeasure K\n[PROOFSTEP]\nrw [smul_smul, div_eq_mul_inv, mul_assoc, ENNReal.inv_mul_cancel \u03bdpos.ne' \u03bdne, mul_one]\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : T2Space G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\ninst\u271d\u00b2 : LocallyCompactSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsHaarMeasure \u03bd\nK : PositiveCompacts G\n\u03bdpos : 0 < \u2191\u2191\u03bd \u2191K\n\u03bdne : \u2191\u2191\u03bd \u2191K \u2260 \u22a4\n\u22a2 (\u2191\u2191\u03bc \u2191K / \u2191\u2191\u03bd \u2191K) \u2022 \u2191\u2191\u03bd \u2191K \u2022 haarMeasure K = (\u2191\u2191\u03bc \u2191K / \u2191\u2191\u03bd \u2191K) \u2022 \u03bd\n[PROOFSTEP]\nrw [\u2190 haarMeasure_unique \u03bd K]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : LocallyCompactSpace G\n\u03bc : Measure G\ninst\u271d : IsHaarMeasure \u03bc\n\u22a2 Regular \u03bc\n[PROOFSTEP]\nhave K : PositiveCompacts G := Classical.arbitrary _\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : LocallyCompactSpace G\n\u03bc : Measure G\ninst\u271d : IsHaarMeasure \u03bc\nK : PositiveCompacts G\n\u22a2 Regular \u03bc\n[PROOFSTEP]\nobtain \u27e8c, _, ctop, h\u03bc\u27e9 : \u2203 c : \u211d\u22650\u221e, c \u2260 0 \u2227 c \u2260 \u221e \u2227 \u03bc = c \u2022 haarMeasure K := isHaarMeasure_eq_smul_isHaarMeasure \u03bc _\n[GOAL]\ncase intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : LocallyCompactSpace G\n\u03bc : Measure G\ninst\u271d : IsHaarMeasure \u03bc\nK : PositiveCompacts G\nc : \u211d\u22650\u221e\nleft\u271d : c \u2260 0\nctop : c \u2260 \u22a4\nh\u03bc : \u03bc = c \u2022 haarMeasure K\n\u22a2 Regular \u03bc\n[PROOFSTEP]\nrw [h\u03bc]\n[GOAL]\ncase intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : LocallyCompactSpace G\n\u03bc : Measure G\ninst\u271d : IsHaarMeasure \u03bc\nK : PositiveCompacts G\nc : \u211d\u22650\u221e\nleft\u271d : c \u2260 0\nctop : c \u2260 \u22a4\nh\u03bc : \u03bc = c \u2022 haarMeasure K\n\u22a2 Regular (c \u2022 haarMeasure K)\n[PROOFSTEP]\nexact Regular.smul ctop\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\n\u22a2 E / E \u2208 \ud835\udcdd 1\n[PROOFSTEP]\nobtain \u27e8L, hL, hLE, hLpos, hLtop\u27e9 : \u2203 L : Set G, MeasurableSet L \u2227 L \u2286 E \u2227 0 < \u03bc L \u2227 \u03bc L < \u221e :=\n exists_subset_measure_lt_top hE hEpos\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\n\u22a2 E / E \u2208 \ud835\udcdd 1\n[PROOFSTEP]\nobtain \u27e8K, hKL, hK, hKpos\u27e9 : \u2203 (K : Set G), K \u2286 L \u2227 IsCompact K \u2227 0 < \u03bc K :=\n MeasurableSet.exists_lt_isCompact_of_ne_top hL (ne_of_lt hLtop) hLpos\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\n\u22a2 E / E \u2208 \ud835\udcdd 1\n[PROOFSTEP]\nhave hKtop : \u03bc K \u2260 \u221e := by\n apply ne_top_of_le_ne_top (ne_of_lt hLtop)\n apply measure_mono hKL\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc K \u2260 \u22a4\n[PROOFSTEP]\napply ne_top_of_le_ne_top (ne_of_lt hLtop)\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc K \u2264 \u2191\u2191\u03bc L\n[PROOFSTEP]\napply measure_mono hKL\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\n\u22a2 E / E \u2208 \ud835\udcdd 1\n[PROOFSTEP]\nobtain \u27e8U, hUK, hU, h\u03bcUK\u27e9 : \u2203 (U : Set G), U \u2287 K \u2227 IsOpen U \u2227 \u03bc U < \u03bc K + \u03bc K :=\n Set.exists_isOpen_lt_add K hKtop hKpos.ne'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\n\u22a2 E / E \u2208 \ud835\udcdd 1\n[PROOFSTEP]\nobtain \u27e8V, hV1, hVKU\u27e9 : \u2203 V \u2208 \ud835\udcdd (1 : G), V * K \u2286 U := compact_open_separated_mul_left hK hU hUK\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\n\u22a2 E / E \u2208 \ud835\udcdd 1\n[PROOFSTEP]\nhave hv : \u2200 v : G, v \u2208 V \u2192 \u00acDisjoint ({ v } * K) K :=\n by\n intro v hv hKv\n have hKvsub : { v } * K \u222a K \u2286 U := by\n apply Set.union_subset _ hUK\n apply _root_.subset_trans _ hVKU\n apply Set.mul_subset_mul _ (Set.Subset.refl K)\n simp only [Set.singleton_subset_iff, hv]\n replace hKvsub := @measure_mono _ _ \u03bc _ _ hKvsub\n have hcontr := lt_of_le_of_lt hKvsub h\u03bcUK\n rw [measure_union hKv (IsCompact.measurableSet hK)] at hcontr \n have hKtranslate : \u03bc ({ v } * K) = \u03bc K := by simp only [singleton_mul, image_mul_left, measure_preimage_mul]\n rw [hKtranslate, lt_self_iff_false] at hcontr \n assumption\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\n\u22a2 \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\n[PROOFSTEP]\nintro v hv hKv\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\n\u22a2 False\n[PROOFSTEP]\nhave hKvsub : { v } * K \u222a K \u2286 U := by\n apply Set.union_subset _ hUK\n apply _root_.subset_trans _ hVKU\n apply Set.mul_subset_mul _ (Set.Subset.refl K)\n simp only [Set.singleton_subset_iff, hv]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\n\u22a2 {v} * K \u222a K \u2286 U\n[PROOFSTEP]\napply Set.union_subset _ hUK\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\n\u22a2 {v} * K \u2286 U\n[PROOFSTEP]\napply _root_.subset_trans _ hVKU\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\n\u22a2 {v} * K \u2286 V * K\n[PROOFSTEP]\napply Set.mul_subset_mul _ (Set.Subset.refl K)\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\n\u22a2 {v} \u2286 V\n[PROOFSTEP]\nsimp only [Set.singleton_subset_iff, hv]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : {v} * K \u222a K \u2286 U\n\u22a2 False\n[PROOFSTEP]\nreplace hKvsub := @measure_mono _ _ \u03bc _ _ hKvsub\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\n\u22a2 False\n[PROOFSTEP]\nhave hcontr := lt_of_le_of_lt hKvsub h\u03bcUK\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\nhcontr : \u2191\u2191\u03bc ({v} * K \u222a K) < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\n\u22a2 False\n[PROOFSTEP]\nrw [measure_union hKv (IsCompact.measurableSet hK)] at hcontr \n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\nhcontr : \u2191\u2191\u03bc ({v} * K) + \u2191\u2191\u03bc K < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\n\u22a2 False\n[PROOFSTEP]\nhave hKtranslate : \u03bc ({ v } * K) = \u03bc K := by simp only [singleton_mul, image_mul_left, measure_preimage_mul]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\nhcontr : \u2191\u2191\u03bc ({v} * K) + \u2191\u2191\u03bc K < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\n\u22a2 \u2191\u2191\u03bc ({v} * K) = \u2191\u2191\u03bc K\n[PROOFSTEP]\nsimp only [singleton_mul, image_mul_left, measure_preimage_mul]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\nhcontr : \u2191\u2191\u03bc ({v} * K) + \u2191\u2191\u03bc K < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nhKtranslate : \u2191\u2191\u03bc ({v} * K) = \u2191\u2191\u03bc K\n\u22a2 False\n[PROOFSTEP]\nrw [hKtranslate, lt_self_iff_false] at hcontr \n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nv : G\nhv : v \u2208 V\nhKv : Disjoint ({v} * K) K\nhKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U\nhcontr : False\nhKtranslate : \u2191\u2191\u03bc ({v} * K) = \u2191\u2191\u03bc K\n\u22a2 False\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\n\u22a2 E / E \u2208 \ud835\udcdd 1\n[PROOFSTEP]\nsuffices V \u2286 E / E from Filter.mem_of_superset hV1 this\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\n\u22a2 V \u2286 E / E\n[PROOFSTEP]\nintro v hvV\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\nv : G\nhvV : v \u2208 V\n\u22a2 v \u2208 E / E\n[PROOFSTEP]\nobtain \u27e8x, hxK, hxvK\u27e9 : \u2203 x : G, x \u2208 { v } * K \u2227 x \u2208 K := Set.not_disjoint_iff.1 (hv v hvV)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\nv : G\nhvV : v \u2208 V\nx : G\nhxK : x \u2208 {v} * K\nhxvK : x \u2208 K\n\u22a2 v \u2208 E / E\n[PROOFSTEP]\nrefine' \u27e8x, v\u207b\u00b9 * x, hLE (hKL hxvK), _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\nv : G\nhvV : v \u2208 V\nx : G\nhxK : x \u2208 {v} * K\nhxvK : x \u2208 K\n\u22a2 v\u207b\u00b9 * x \u2208 E\n[PROOFSTEP]\napply hKL.trans hLE\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.a\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\nv : G\nhvV : v \u2208 V\nx : G\nhxK : x \u2208 {v} * K\nhxvK : x \u2208 K\n\u22a2 v\u207b\u00b9 * x \u2208 K\n[PROOFSTEP]\nsimpa only [singleton_mul, image_mul_left, mem_preimage] using hxK\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : TopologicalGroup G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEpos : 0 < \u2191\u2191\u03bc E\nL : Set G\nhL : MeasurableSet L\nhLE : L \u2286 E\nhLpos : 0 < \u2191\u2191\u03bc L\nhLtop : \u2191\u2191\u03bc L < \u22a4\nK : Set G\nhKL : K \u2286 L\nhK : IsCompact K\nhKpos : 0 < \u2191\u2191\u03bc K\nhKtop : \u2191\u2191\u03bc K \u2260 \u22a4\nU : Set G\nhUK : U \u2287 K\nhU : IsOpen U\nh\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K\nV : Set G\nhV1 : V \u2208 \ud835\udcdd 1\nhVKU : V * K \u2286 U\nhv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K\nv : G\nhvV : v \u2208 V\nx : G\nhxK : x \u2208 {v} * K\nhxvK : x \u2208 K\n\u22a2 (fun x x_1 => x / x_1) x (v\u207b\u00b9 * x) = v\n[PROOFSTEP]\nsimp only [div_eq_iff_eq_mul, \u2190 mul_assoc, mul_right_inv, one_mul]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\n\u22a2 IsInvInvariant \u03bc\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase inv_eq_self\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\n\u22a2 Measure.inv \u03bc = \u03bc\n[PROOFSTEP]\nhaveI : IsHaarMeasure (Measure.map Inv.inv \u03bc) := (MulEquiv.inv G).isHaarMeasure_map \u03bc continuous_inv continuous_inv\n[GOAL]\ncase inv_eq_self\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis : IsHaarMeasure (map inv \u03bc)\n\u22a2 Measure.inv \u03bc = \u03bc\n[PROOFSTEP]\nobtain \u27e8c, _, _, hc\u27e9 : \u2203 c : \u211d\u22650\u221e, c \u2260 0 \u2227 c \u2260 \u221e \u2227 Measure.map Inv.inv \u03bc = c \u2022 \u03bc :=\n isHaarMeasure_eq_smul_isHaarMeasure _ _\n[GOAL]\ncase inv_eq_self.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis : IsHaarMeasure (map inv \u03bc)\nc : \u211d\u22650\u221e\nleft\u271d\u00b9 : c \u2260 0\nleft\u271d : c \u2260 \u22a4\nhc : map inv \u03bc = c \u2022 \u03bc\n\u22a2 Measure.inv \u03bc = \u03bc\n[PROOFSTEP]\nhave : map Inv.inv (map Inv.inv \u03bc) = c ^ 2 \u2022 \u03bc := by simp only [hc, smul_smul, pow_two, Measure.map_smul]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis : IsHaarMeasure (map inv \u03bc)\nc : \u211d\u22650\u221e\nleft\u271d\u00b9 : c \u2260 0\nleft\u271d : c \u2260 \u22a4\nhc : map inv \u03bc = c \u2022 \u03bc\n\u22a2 map inv (map inv \u03bc) = c ^ 2 \u2022 \u03bc\n[PROOFSTEP]\nsimp only [hc, smul_smul, pow_two, Measure.map_smul]\n[GOAL]\ncase inv_eq_self.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis\u271d : IsHaarMeasure (map inv \u03bc)\nc : \u211d\u22650\u221e\nleft\u271d\u00b9 : c \u2260 0\nleft\u271d : c \u2260 \u22a4\nhc : map inv \u03bc = c \u2022 \u03bc\nthis : map inv (map inv \u03bc) = c ^ 2 \u2022 \u03bc\n\u22a2 Measure.inv \u03bc = \u03bc\n[PROOFSTEP]\nhave \u03bceq : \u03bc = c ^ 2 \u2022 \u03bc :=\n by\n rw [map_map continuous_inv.measurable continuous_inv.measurable] at this \n simpa only [inv_involutive, Involutive.comp_self, map_id]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis\u271d : IsHaarMeasure (map inv \u03bc)\nc : \u211d\u22650\u221e\nleft\u271d\u00b9 : c \u2260 0\nleft\u271d : c \u2260 \u22a4\nhc : map inv \u03bc = c \u2022 \u03bc\nthis : map inv (map inv \u03bc) = c ^ 2 \u2022 \u03bc\n\u22a2 \u03bc = c ^ 2 \u2022 \u03bc\n[PROOFSTEP]\nrw [map_map continuous_inv.measurable continuous_inv.measurable] at this \n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis\u271d : IsHaarMeasure (map inv \u03bc)\nc : \u211d\u22650\u221e\nleft\u271d\u00b9 : c \u2260 0\nleft\u271d : c \u2260 \u22a4\nhc : map inv \u03bc = c \u2022 \u03bc\nthis : map ((fun a => a\u207b\u00b9) \u2218 fun a => a\u207b\u00b9) \u03bc = c ^ 2 \u2022 \u03bc\n\u22a2 \u03bc = c ^ 2 \u2022 \u03bc\n[PROOFSTEP]\nsimpa only [inv_involutive, Involutive.comp_self, map_id]\n[GOAL]\ncase inv_eq_self.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis\u271d : IsHaarMeasure (map inv \u03bc)\nc : \u211d\u22650\u221e\nleft\u271d\u00b9 : c \u2260 0\nleft\u271d : c \u2260 \u22a4\nhc : map inv \u03bc = c \u2022 \u03bc\nthis : map inv (map inv \u03bc) = c ^ 2 \u2022 \u03bc\n\u03bceq : \u03bc = c ^ 2 \u2022 \u03bc\n\u22a2 Measure.inv \u03bc = \u03bc\n[PROOFSTEP]\nhave K : PositiveCompacts G := Classical.arbitrary _\n[GOAL]\ncase inv_eq_self.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis\u271d : IsHaarMeasure (map inv \u03bc)\nc : \u211d\u22650\u221e\nleft\u271d\u00b9 : c \u2260 0\nleft\u271d : c \u2260 \u22a4\nhc : map inv \u03bc = c \u2022 \u03bc\nthis : map inv (map inv \u03bc) = c ^ 2 \u2022 \u03bc\n\u03bceq : \u03bc = c ^ 2 \u2022 \u03bc\nK : PositiveCompacts G\n\u22a2 Measure.inv \u03bc = \u03bc\n[PROOFSTEP]\nhave : c ^ 2 * \u03bc K = 1 ^ 2 * \u03bc K := by\n conv_rhs => rw [\u03bceq]\n simp\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis\u271d : IsHaarMeasure (map inv \u03bc)\nc : \u211d\u22650\u221e\nleft\u271d\u00b9 : c \u2260 0\nleft\u271d : c \u2260 \u22a4\nhc : map inv \u03bc = c \u2022 \u03bc\nthis : map inv (map inv \u03bc) = c ^ 2 \u2022 \u03bc\n\u03bceq : \u03bc = c ^ 2 \u2022 \u03bc\nK : PositiveCompacts G\n\u22a2 c ^ 2 * \u2191\u2191\u03bc \u2191K = 1 ^ 2 * \u2191\u2191\u03bc \u2191K\n[PROOFSTEP]\nconv_rhs => rw [\u03bceq]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis\u271d : IsHaarMeasure (map inv \u03bc)\nc : \u211d\u22650\u221e\nleft\u271d\u00b9 : c \u2260 0\nleft\u271d : c \u2260 \u22a4\nhc : map inv \u03bc = c \u2022 \u03bc\nthis : map inv (map inv \u03bc) = c ^ 2 \u2022 \u03bc\n\u03bceq : \u03bc = c ^ 2 \u2022 \u03bc\nK : PositiveCompacts G\n| 1 ^ 2 * \u2191\u2191\u03bc \u2191K\n[PROOFSTEP]\nrw [\u03bceq]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis\u271d : IsHaarMeasure (map inv \u03bc)\nc : \u211d\u22650\u221e\nleft\u271d\u00b9 : c \u2260 0\nleft\u271d : c \u2260 \u22a4\nhc : map inv \u03bc = c \u2022 \u03bc\nthis : map inv (map inv \u03bc) = c ^ 2 \u2022 \u03bc\n\u03bceq : \u03bc = c ^ 2 \u2022 \u03bc\nK : PositiveCompacts G\n| 1 ^ 2 * \u2191\u2191\u03bc \u2191K\n[PROOFSTEP]\nrw [\u03bceq]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis\u271d : IsHaarMeasure (map inv \u03bc)\nc : \u211d\u22650\u221e\nleft\u271d\u00b9 : c \u2260 0\nleft\u271d : c \u2260 \u22a4\nhc : map inv \u03bc = c \u2022 \u03bc\nthis : map inv (map inv \u03bc) = c ^ 2 \u2022 \u03bc\n\u03bceq : \u03bc = c ^ 2 \u2022 \u03bc\nK : PositiveCompacts G\n| 1 ^ 2 * \u2191\u2191\u03bc \u2191K\n[PROOFSTEP]\nrw [\u03bceq]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis\u271d : IsHaarMeasure (map inv \u03bc)\nc : \u211d\u22650\u221e\nleft\u271d\u00b9 : c \u2260 0\nleft\u271d : c \u2260 \u22a4\nhc : map inv \u03bc = c \u2022 \u03bc\nthis : map inv (map inv \u03bc) = c ^ 2 \u2022 \u03bc\n\u03bceq : \u03bc = c ^ 2 \u2022 \u03bc\nK : PositiveCompacts G\n\u22a2 c ^ 2 * \u2191\u2191\u03bc \u2191K = 1 ^ 2 * \u2191\u2191(c ^ 2 \u2022 \u03bc) \u2191K\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inv_eq_self.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis\u271d\u00b9 : IsHaarMeasure (map inv \u03bc)\nc : \u211d\u22650\u221e\nleft\u271d\u00b9 : c \u2260 0\nleft\u271d : c \u2260 \u22a4\nhc : map inv \u03bc = c \u2022 \u03bc\nthis\u271d : map inv (map inv \u03bc) = c ^ 2 \u2022 \u03bc\n\u03bceq : \u03bc = c ^ 2 \u2022 \u03bc\nK : PositiveCompacts G\nthis : c ^ 2 * \u2191\u2191\u03bc \u2191K = 1 ^ 2 * \u2191\u2191\u03bc \u2191K\n\u22a2 Measure.inv \u03bc = \u03bc\n[PROOFSTEP]\nhave : c ^ 2 = 1 ^ 2 :=\n (ENNReal.mul_eq_mul_right (measure_pos_of_nonempty_interior _ K.interior_nonempty).ne'\n K.isCompact.measure_lt_top.ne).1\n this\n[GOAL]\ncase inv_eq_self.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis\u271d\u00b2 : IsHaarMeasure (map inv \u03bc)\nc : \u211d\u22650\u221e\nleft\u271d\u00b9 : c \u2260 0\nleft\u271d : c \u2260 \u22a4\nhc : map inv \u03bc = c \u2022 \u03bc\nthis\u271d\u00b9 : map inv (map inv \u03bc) = c ^ 2 \u2022 \u03bc\n\u03bceq : \u03bc = c ^ 2 \u2022 \u03bc\nK : PositiveCompacts G\nthis\u271d : c ^ 2 * \u2191\u2191\u03bc \u2191K = 1 ^ 2 * \u2191\u2191\u03bc \u2191K\nthis : c ^ 2 = 1 ^ 2\n\u22a2 Measure.inv \u03bc = \u03bc\n[PROOFSTEP]\nhave : c = 1 := (ENNReal.pow_strictMono two_ne_zero).injective this\n[GOAL]\ncase inv_eq_self.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : CommGroup G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : T2Space G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : LocallyCompactSpace G\nthis\u271d\u00b3 : IsHaarMeasure (map inv \u03bc)\nc : \u211d\u22650\u221e\nleft\u271d\u00b9 : c \u2260 0\nleft\u271d : c \u2260 \u22a4\nhc : map inv \u03bc = c \u2022 \u03bc\nthis\u271d\u00b2 : map inv (map inv \u03bc) = c ^ 2 \u2022 \u03bc\n\u03bceq : \u03bc = c ^ 2 \u2022 \u03bc\nK : PositiveCompacts G\nthis\u271d\u00b9 : c ^ 2 * \u2191\u2191\u03bc \u2191K = 1 ^ 2 * \u2191\u2191\u03bc \u2191K\nthis\u271d : c ^ 2 = 1 ^ 2\nthis : c = 1\n\u22a2 Measure.inv \u03bc = \u03bc\n[PROOFSTEP]\nrw [Measure.inv, hc, this, one_smul]\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc\n[PROOFSTEP]\nlet f := @zpowGroupHom G _ n\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc\n[PROOFSTEP]\nhave hf : Continuous f := continuous_zpow n\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc\n[PROOFSTEP]\nhaveI : (\u03bc.map f).IsHaarMeasure := isHaarMeasure_map \u03bc f hf (RootableBy.surjective_pow G \u2124 hn) (by simp)\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\n\u22a2 Filter.Tendsto (\u2191f) (Filter.cocompact G) (Filter.cocompact G)\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc\n[PROOFSTEP]\nobtain \u27e8C, -, -, hC\u27e9 := isHaarMeasure_eq_smul_isHaarMeasure (\u03bc.map f) \u03bc\n[GOAL]\ncase intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc\n[PROOFSTEP]\nsuffices C = 1 by rwa [this, one_smul] at hC \n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis\u271d : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\nthis : C = 1\n\u22a2 map (fun g => g ^ n) \u03bc = \u03bc\n[PROOFSTEP]\nrwa [this, one_smul] at hC \n[GOAL]\ncase intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\n\u22a2 C = 1\n[PROOFSTEP]\nhave h_univ : (\u03bc.map f) univ = \u03bc univ := by\n rw [map_apply_of_aemeasurable hf.measurable.aemeasurable MeasurableSet.univ, preimage_univ]\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\n\u22a2 \u2191\u2191(map (\u2191f) \u03bc) univ = \u2191\u2191\u03bc univ\n[PROOFSTEP]\nrw [map_apply_of_aemeasurable hf.measurable.aemeasurable MeasurableSet.univ, preimage_univ]\n[GOAL]\ncase intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\nh_univ : \u2191\u2191(map (\u2191f) \u03bc) univ = \u2191\u2191\u03bc univ\n\u22a2 C = 1\n[PROOFSTEP]\nhave h\u03bc\u2080 : \u03bc univ \u2260 0 := IsOpenPosMeasure.open_pos univ isOpen_univ univ_nonempty\n[GOAL]\ncase intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\nh_univ : \u2191\u2191(map (\u2191f) \u03bc) univ = \u2191\u2191\u03bc univ\nh\u03bc\u2080 : \u2191\u2191\u03bc univ \u2260 0\n\u22a2 C = 1\n[PROOFSTEP]\nhave h\u03bc\u2081 : \u03bc univ \u2260 \u221e := CompactSpace.isFiniteMeasure.measure_univ_lt_top.ne\n[GOAL]\ncase intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : CommGroup G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : T2Space G\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : SecondCountableTopology G\n\u03bc : Measure G\ninst\u271d\u00b2 : IsHaarMeasure \u03bc\ninst\u271d\u00b9 : CompactSpace G\ninst\u271d : RootableBy G \u2124\nn : \u2124\nhn : n \u2260 0\nf : G \u2192* G := zpowGroupHom n\nhf : Continuous \u2191f\nthis : IsHaarMeasure (map (\u2191f) \u03bc)\nC : \u211d\u22650\u221e\nhC : map (\u2191f) \u03bc = C \u2022 \u03bc\nh_univ : \u2191\u2191(map (\u2191f) \u03bc) univ = \u2191\u2191\u03bc univ\nh\u03bc\u2080 : \u2191\u2191\u03bc univ \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc univ \u2260 \u22a4\n\u22a2 C = 1\n[PROOFSTEP]\nrwa [hC, smul_apply, Algebra.id.smul_eq_mul, mul_comm, \u2190 ENNReal.eq_div_iff h\u03bc\u2080 h\u03bc\u2081, ENNReal.div_self h\u03bc\u2080 h\u03bc\u2081] at h_univ \n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.Haar.Basic", "llama_tokens": 93171, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.4485457442098706}} {"text": "[GOAL]\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_2\n\u03c3\u2081 : Type\n\u03b3 : Type u_1\n\u03c3\u2082 : Type\nxs : Vector \u03b1 n\nf\u2081 : \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr f\u2081 (mapAccumr f\u2082 xs s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\n (fun x s =>\n let r\u2082 := f\u2082 x s.snd;\n let r\u2081 := f\u2081 r\u2082.snd s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\ninduction xs using Vector.revInductionOn generalizing s\u2081 s\u2082\n[GOAL]\ncase nil\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_2\n\u03c3\u2081 : Type\n\u03b3 : Type u_1\n\u03c3\u2082 : Type\nxs : Vector \u03b1 n\nf\u2081 : \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr f\u2081 (mapAccumr f\u2082 nil s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\n (fun x s =>\n let r\u2082 := f\u2082 x s.snd;\n let r\u2081 := f\u2081 r\u2082.snd s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n nil (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_2\n\u03c3\u2081 : Type\n\u03b3 : Type u_1\n\u03c3\u2082 : Type\nxs : Vector \u03b1 n\nf\u2081 : \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nx\u271d : \u03b1\na\u271d :\n \u2200 {s\u2082 : \u03c3\u2082} {s\u2081 : \u03c3\u2081},\n mapAccumr f\u2081 (mapAccumr f\u2082 xs\u271d s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\n (fun x s =>\n let r\u2082 := f\u2082 x s.snd;\n let r\u2081 := f\u2081 r\u2082.snd s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs\u271d (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr f\u2081 (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\n (fun x s =>\n let r\u2082 := f\u2082 x s.snd;\n let r\u2081 := f\u2081 r\u2082.snd s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n (snoc xs\u271d x\u271d) (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_2\n\u03c3\u2081 : Type\n\u03b3 : Type u_1\n\u03c3\u2082 : Type ?u.3222\nxs : Vector \u03b1 n\nf\u2081 : \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082\u271d : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\ns : \u03c3\u2081\nf\u2082 : \u03b1 \u2192 \u03b2\n\u22a2 mapAccumr f\u2081 (map f\u2082 xs) s = mapAccumr (fun x s => f\u2081 (f\u2082 x) s) xs s\n[PROOFSTEP]\ninduction xs using Vector.revInductionOn generalizing s\n[GOAL]\ncase nil\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_2\n\u03c3\u2081 : Type\n\u03b3 : Type u_1\n\u03c3\u2082 : Type ?u.3222\nxs : Vector \u03b1 n\nf\u2081 : \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082\u271d : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\nf\u2082 : \u03b1 \u2192 \u03b2\ns : \u03c3\u2081\n\u22a2 mapAccumr f\u2081 (map f\u2082 nil) s = mapAccumr (fun x s => f\u2081 (f\u2082 x) s) nil s\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_2\n\u03c3\u2081 : Type\n\u03b3 : Type u_1\n\u03c3\u2082 : Type ?u.3222\nxs : Vector \u03b1 n\nf\u2081 : \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082\u271d : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\nf\u2082 : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nx\u271d : \u03b1\na\u271d : \u2200 {s : \u03c3\u2081}, mapAccumr f\u2081 (map f\u2082 xs\u271d) s = mapAccumr (fun x s => f\u2081 (f\u2082 x) s) xs\u271d s\ns : \u03c3\u2081\n\u22a2 mapAccumr f\u2081 (map f\u2082 (snoc xs\u271d x\u271d)) s = mapAccumr (fun x s => f\u2081 (f\u2082 x) s) (snoc xs\u271d x\u271d) s\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_2\n\u03c3\u2081 : Type ?u.4482\n\u03b3 : Type u_1\n\u03c3\u2082 : Type\nxs : Vector \u03b1 n\nf\u2081\u271d : \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\ns : \u03c3\u2082\nf\u2081 : \u03b2 \u2192 \u03b3\n\u22a2 map f\u2081 (mapAccumr f\u2082 xs s).snd =\n (mapAccumr\n (fun x s =>\n let r := f\u2082 x s;\n (r.fst, f\u2081 r.snd))\n xs s).snd\n[PROOFSTEP]\ninduction xs using Vector.revInductionOn generalizing s\n[GOAL]\ncase nil\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_2\n\u03c3\u2081 : Type ?u.4482\n\u03b3 : Type u_1\n\u03c3\u2082 : Type\nxs : Vector \u03b1 n\nf\u2081\u271d : \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\nf\u2081 : \u03b2 \u2192 \u03b3\ns : \u03c3\u2082\n\u22a2 map f\u2081 (mapAccumr f\u2082 nil s).snd =\n (mapAccumr\n (fun x s =>\n let r := f\u2082 x s;\n (r.fst, f\u2081 r.snd))\n nil s).snd\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_2\n\u03c3\u2081 : Type ?u.4482\n\u03b3 : Type u_1\n\u03c3\u2082 : Type\nxs : Vector \u03b1 n\nf\u2081\u271d : \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\nf\u2081 : \u03b2 \u2192 \u03b3\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nx\u271d : \u03b1\na\u271d :\n \u2200 {s : \u03c3\u2082},\n map f\u2081 (mapAccumr f\u2082 xs\u271d s).snd =\n (mapAccumr\n (fun x s =>\n let r := f\u2082 x s;\n (r.fst, f\u2081 r.snd))\n xs\u271d s).snd\ns : \u03c3\u2082\n\u22a2 map f\u2081 (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s).snd =\n (mapAccumr\n (fun x s =>\n let r := f\u2082 x s;\n (r.fst, f\u2081 r.snd))\n (snoc xs\u271d x\u271d) s).snd\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_2\n\u03c3\u2081 : Type ?u.6822\n\u03b3 : Type u_1\n\u03c3\u2082 : Type ?u.6830\nxs : Vector \u03b1 n\nf\u2081\u271d : \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082\u271d : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\nf\u2081 : \u03b2 \u2192 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2\n\u22a2 map f\u2081 (map f\u2082 xs) = map (fun x => f\u2081 (f\u2082 x)) xs\n[PROOFSTEP]\ninduction xs using Vector.inductionOn\n[GOAL]\ncase h_nil\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_2\n\u03c3\u2081 : Type ?u.6822\n\u03b3 : Type u_1\n\u03c3\u2082 : Type ?u.6830\nxs : Vector \u03b1 n\nf\u2081\u271d : \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082\u271d : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\nf\u2081 : \u03b2 \u2192 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2\n\u22a2 map f\u2081 (map f\u2082 nil) = map (fun x => f\u2081 (f\u2082 x)) nil\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase h_cons\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_2\n\u03c3\u2081 : Type ?u.6822\n\u03b3 : Type u_1\n\u03c3\u2082 : Type ?u.6830\nxs : Vector \u03b1 n\nf\u2081\u271d : \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082\u271d : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\nf\u2081 : \u03b2 \u2192 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\nx\u271d : \u03b1\nw\u271d : Vector \u03b1 n\u271d\na\u271d : map f\u2081 (map f\u2082 w\u271d) = map (fun x => f\u2081 (f\u2082 x)) w\u271d\n\u22a2 map f\u2081 (map f\u2082 (x\u271d ::\u1d65 w\u271d)) = map (fun x => f\u2081 (f\u2082 x)) (x\u271d ::\u1d65 w\u271d)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03b6 \u03c3\u2082 : Type\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\nf\u2081 : \u03b3 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b6\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\n\u22a2 mapAccumr\u2082 f\u2081 (mapAccumr f\u2082 xs s\u2082).snd ys s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y s =>\n let r\u2082 := f\u2082 x s.snd;\n let r\u2081 := f\u2081 r\u2082.snd y s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs ys (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\ninduction xs, ys using Vector.revInductionOn\u2082 generalizing s\u2081 s\u2082\n[GOAL]\ncase nil\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03b6 \u03c3\u2082 : Type\nf\u2081 : \u03b3 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b6\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr\u2082 f\u2081 (mapAccumr f\u2082 nil s\u2082).snd nil s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y s =>\n let r\u2082 := f\u2082 x s.snd;\n let r\u2081 := f\u2081 r\u2082.snd y s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n nil nil (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03b6 \u03c3\u2082 : Type\nf\u2081 : \u03b3 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b6\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d :\n \u2200 {s\u2082 : \u03c3\u2082} {s\u2081 : \u03c3\u2081},\n mapAccumr\u2082 f\u2081 (mapAccumr f\u2082 xs\u271d s\u2082).snd ys\u271d s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y s =>\n let r\u2082 := f\u2082 x s.snd;\n let r\u2081 := f\u2081 r\u2082.snd y s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs\u271d ys\u271d (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr\u2082 f\u2081 (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).snd (snoc ys\u271d y\u271d) s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y s =>\n let r\u2082 := f\u2082 x s.snd;\n let r\u2081 := f\u2081 r\u2082.snd y s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_4\nn : \u2115\n\u03b2 : Type u_3\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 : Type u_1\n\u03b6 : Type u_2\nf\u2081 : \u03b3 \u2192 \u03b2 \u2192 \u03b6\nf\u2082 : \u03b1 \u2192 \u03b3\n\u22a2 map\u2082 f\u2081 (map f\u2082 xs) ys = map\u2082 (fun x y => f\u2081 (f\u2082 x) y) xs ys\n[PROOFSTEP]\ninduction xs, ys using Vector.revInductionOn\u2082\n[GOAL]\ncase nil\n\u03b1 : Type u_4\nn : \u2115\n\u03b2 : Type u_3\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 : Type u_1\n\u03b6 : Type u_2\nf\u2081 : \u03b3 \u2192 \u03b2 \u2192 \u03b6\nf\u2082 : \u03b1 \u2192 \u03b3\n\u22a2 map\u2082 f\u2081 (map f\u2082 nil) nil = map\u2082 (fun x y => f\u2081 (f\u2082 x) y) nil nil\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type u_4\nn : \u2115\n\u03b2 : Type u_3\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 : Type u_1\n\u03b6 : Type u_2\nf\u2081 : \u03b3 \u2192 \u03b2 \u2192 \u03b6\nf\u2082 : \u03b1 \u2192 \u03b3\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d : map\u2082 f\u2081 (map f\u2082 xs\u271d) ys\u271d = map\u2082 (fun x y => f\u2081 (f\u2082 x) y) xs\u271d ys\u271d\n\u22a2 map\u2082 f\u2081 (map f\u2082 (snoc xs\u271d x\u271d)) (snoc ys\u271d y\u271d) = map\u2082 (fun x y => f\u2081 (f\u2082 x) y) (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03b6 \u03c3\u2082 : Type\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\nf\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b6\nf\u2082 : \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\n\u22a2 mapAccumr\u2082 f\u2081 xs (mapAccumr f\u2082 ys s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y s =>\n let r\u2082 := f\u2082 y s.snd;\n let r\u2081 := f\u2081 x r\u2082.snd s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs ys (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\ninduction xs, ys using Vector.revInductionOn\u2082 generalizing s\u2081 s\u2082\n[GOAL]\ncase nil\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03b6 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b6\nf\u2082 : \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr\u2082 f\u2081 nil (mapAccumr f\u2082 nil s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y s =>\n let r\u2082 := f\u2082 y s.snd;\n let r\u2081 := f\u2081 x r\u2082.snd s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n nil nil (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03b6 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b6\nf\u2082 : \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d :\n \u2200 {s\u2082 : \u03c3\u2082} {s\u2081 : \u03c3\u2081},\n mapAccumr\u2082 f\u2081 xs\u271d (mapAccumr f\u2082 ys\u271d s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y s =>\n let r\u2082 := f\u2082 y s.snd;\n let r\u2081 := f\u2081 x r\u2082.snd s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs\u271d ys\u271d (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (mapAccumr f\u2082 (snoc ys\u271d y\u271d) s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y s =>\n let r\u2082 := f\u2082 y s.snd;\n let r\u2081 := f\u2081 x r\u2082.snd s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_4\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 : Type u_1\n\u03b6 : Type u_2\nf\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b6\nf\u2082 : \u03b2 \u2192 \u03b3\n\u22a2 map\u2082 f\u2081 xs (map f\u2082 ys) = map\u2082 (fun x y => f\u2081 x (f\u2082 y)) xs ys\n[PROOFSTEP]\ninduction xs, ys using Vector.revInductionOn\u2082\n[GOAL]\ncase nil\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_4\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 : Type u_1\n\u03b6 : Type u_2\nf\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b6\nf\u2082 : \u03b2 \u2192 \u03b3\n\u22a2 map\u2082 f\u2081 nil (map f\u2082 nil) = map\u2082 (fun x y => f\u2081 x (f\u2082 y)) nil nil\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_4\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 : Type u_1\n\u03b6 : Type u_2\nf\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b6\nf\u2082 : \u03b2 \u2192 \u03b3\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d : map\u2082 f\u2081 xs\u271d (map f\u2082 ys\u271d) = map\u2082 (fun x y => f\u2081 x (f\u2082 y)) xs\u271d ys\u271d\n\u22a2 map\u2082 f\u2081 (snoc xs\u271d x\u271d) (map f\u2082 (snoc ys\u271d y\u271d)) = map\u2082 (fun x y => f\u2081 x (f\u2082 y)) (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03b6 \u03c3\u2082 : Type\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\nf\u2081 : \u03b3 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\n\u22a2 mapAccumr f\u2081 (mapAccumr\u2082 f\u2082 xs ys s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y s =>\n let r\u2082 := f\u2082 x y s.snd;\n let r\u2081 := f\u2081 r\u2082.snd s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs ys (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\ninduction xs, ys using Vector.revInductionOn\u2082 generalizing s\u2081 s\u2082\n[GOAL]\ncase nil\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03b6 \u03c3\u2082 : Type\nf\u2081 : \u03b3 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr f\u2081 (mapAccumr\u2082 f\u2082 nil nil s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y s =>\n let r\u2082 := f\u2082 x y s.snd;\n let r\u2081 := f\u2081 r\u2082.snd s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n nil nil (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03b6 \u03c3\u2082 : Type\nf\u2081 : \u03b3 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d :\n \u2200 {s\u2082 : \u03c3\u2082} {s\u2081 : \u03c3\u2081},\n mapAccumr f\u2081 (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y s =>\n let r\u2082 := f\u2082 x y s.snd;\n let r\u2081 := f\u2081 r\u2082.snd s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs\u271d ys\u271d (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr f\u2081 (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y s =>\n let r\u2082 := f\u2082 x y s.snd;\n let r\u2081 := f\u2081 r\u2082.snd s.fst;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_4\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 : Type u_1\n\u03b6 : Type u_2\nf\u2081 : \u03b3 \u2192 \u03b6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u22a2 map f\u2081 (map\u2082 f\u2082 xs ys) = map\u2082 (fun x y => f\u2081 (f\u2082 x y)) xs ys\n[PROOFSTEP]\ninduction xs, ys using Vector.revInductionOn\u2082\n[GOAL]\ncase nil\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_4\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 : Type u_1\n\u03b6 : Type u_2\nf\u2081 : \u03b3 \u2192 \u03b6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u22a2 map f\u2081 (map\u2082 f\u2082 nil nil) = map\u2082 (fun x y => f\u2081 (f\u2082 x y)) nil nil\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type u_3\nn : \u2115\n\u03b2 : Type u_4\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 : Type u_1\n\u03b6 : Type u_2\nf\u2081 : \u03b3 \u2192 \u03b6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d : map f\u2081 (map\u2082 f\u2082 xs\u271d ys\u271d) = map\u2082 (fun x y => f\u2081 (f\u2082 x y)) xs\u271d ys\u271d\n\u22a2 map f\u2081 (map\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d)) = map\u2082 (fun x y => f\u2081 (f\u2082 x y)) (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03c6 \u03c3\u2082 : Type\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\nf\u2081 : \u03b3 \u2192 \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03c6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\n\u22a2 mapAccumr\u2082 f\u2081 (mapAccumr\u2082 f\u2082 xs ys s\u2082).snd xs s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 r\u2082.snd x s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs ys (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\ninduction xs, ys using Vector.revInductionOn\u2082 generalizing s\u2081 s\u2082\n[GOAL]\ncase nil\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03c6 \u03c3\u2082 : Type\nf\u2081 : \u03b3 \u2192 \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03c6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr\u2082 f\u2081 (mapAccumr\u2082 f\u2082 nil nil s\u2082).snd nil s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 r\u2082.snd x s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n nil nil (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03c6 \u03c3\u2082 : Type\nf\u2081 : \u03b3 \u2192 \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03c6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d :\n \u2200 {s\u2082 : \u03c3\u2082} {s\u2081 : \u03c3\u2081},\n mapAccumr\u2082 f\u2081 (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).snd xs\u271d s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 r\u2082.snd x s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs\u271d ys\u271d (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr\u2082 f\u2081 (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).snd (snoc xs\u271d x\u271d) s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 r\u2082.snd x s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03c6 \u03c3\u2082 : Type\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\nf\u2081 : \u03b3 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03c6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\n\u22a2 mapAccumr\u2082 f\u2081 (mapAccumr\u2082 f\u2082 xs ys s\u2082).snd ys s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 r\u2082.snd y s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs ys (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\ninduction xs, ys using Vector.revInductionOn\u2082 generalizing s\u2081 s\u2082\n[GOAL]\ncase nil\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03c6 \u03c3\u2082 : Type\nf\u2081 : \u03b3 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03c6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr\u2082 f\u2081 (mapAccumr\u2082 f\u2082 nil nil s\u2082).snd nil s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 r\u2082.snd y s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n nil nil (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03c6 \u03c3\u2082 : Type\nf\u2081 : \u03b3 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03c6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d :\n \u2200 {s\u2082 : \u03c3\u2082} {s\u2081 : \u03c3\u2081},\n mapAccumr\u2082 f\u2081 (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).snd ys\u271d s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 r\u2082.snd y s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs\u271d ys\u271d (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr\u2082 f\u2081 (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).snd (snoc ys\u271d y\u271d) s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 r\u2082.snd y s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03c6 \u03c3\u2082 : Type\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\nf\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03c6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\n\u22a2 mapAccumr\u2082 f\u2081 xs (mapAccumr\u2082 f\u2082 xs ys s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 x r\u2082.snd s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs ys (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\ninduction xs, ys using Vector.revInductionOn\u2082 generalizing s\u2081 s\u2082\n[GOAL]\ncase nil\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03c6 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03c6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr\u2082 f\u2081 nil (mapAccumr\u2082 f\u2082 nil nil s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 x r\u2082.snd s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n nil nil (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03c6 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03c6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d :\n \u2200 {s\u2082 : \u03c3\u2082} {s\u2081 : \u03c3\u2081},\n mapAccumr\u2082 f\u2081 xs\u271d (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 x r\u2082.snd s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs\u271d ys\u271d (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 x r\u2082.snd s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03c6 \u03c3\u2082 : Type\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\nf\u2081 : \u03b2 \u2192 \u03b3 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03c6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\n\u22a2 mapAccumr\u2082 f\u2081 ys (mapAccumr\u2082 f\u2082 xs ys s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 y r\u2082.snd s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs ys (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\ninduction xs, ys using Vector.revInductionOn\u2082 generalizing s\u2081 s\u2082\n[GOAL]\ncase nil\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03c6 \u03c3\u2082 : Type\nf\u2081 : \u03b2 \u2192 \u03b3 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03c6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr\u2082 f\u2081 nil (mapAccumr\u2082 f\u2082 nil nil s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 y r\u2082.snd s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n nil nil (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03c6 \u03c3\u2082 : Type\nf\u2081 : \u03b2 \u2192 \u03b3 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03c6\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d :\n \u2200 {s\u2082 : \u03c3\u2082} {s\u2081 : \u03c3\u2081},\n mapAccumr\u2082 f\u2081 ys\u271d (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 y r\u2082.snd s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n xs\u271d ys\u271d (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\n\u22a2 mapAccumr\u2082 f\u2081 (snoc ys\u271d y\u271d) (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).snd s\u2081 =\n let m :=\n mapAccumr\u2082\n (fun x y x_1 =>\n match x_1 with\n | (s\u2081, s\u2082) =>\n let r\u2082 := f\u2082 x y s\u2082;\n let r\u2081 := f\u2081 y r\u2082.snd s\u2081;\n ((r\u2081.fst, r\u2082.fst), r\u2081.snd))\n (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) (s\u2081, s\u2082);\n (m.fst.fst, m.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03c3\u2081 : Type\n\u03b2 : Type u_1\n\u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b2\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nh\u2080 : R s\u2081 s\u2082\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1), R s q \u2192 R (f\u2081 a s).fst (f\u2082 a q).fst \u2227 (f\u2081 a s).snd = (f\u2082 a q).snd\n\u22a2 R (mapAccumr f\u2081 xs s\u2081).fst (mapAccumr f\u2082 xs s\u2082).fst \u2227 (mapAccumr f\u2081 xs s\u2081).snd = (mapAccumr f\u2082 xs s\u2082).snd\n[PROOFSTEP]\ninduction xs using Vector.revInductionOn generalizing s\u2081 s\u2082\n[GOAL]\ncase nil\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03c3\u2081 : Type\n\u03b2 : Type u_1\n\u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b2\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1), R s q \u2192 R (f\u2081 a s).fst (f\u2082 a q).fst \u2227 (f\u2081 a s).snd = (f\u2082 a q).snd\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr f\u2081 nil s\u2081).fst (mapAccumr f\u2082 nil s\u2082).fst \u2227 (mapAccumr f\u2081 nil s\u2081).snd = (mapAccumr f\u2082 nil s\u2082).snd\ncase snoc\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03c3\u2081 : Type\n\u03b2 : Type u_1\n\u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b2\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1), R s q \u2192 R (f\u2081 a s).fst (f\u2082 a q).fst \u2227 (f\u2081 a s).snd = (f\u2082 a q).snd\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nx\u271d : \u03b1\na\u271d :\n \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n R s\u2081 s\u2082 \u2192\n R (mapAccumr f\u2081 xs\u271d s\u2081).fst (mapAccumr f\u2082 xs\u271d s\u2082).fst \u2227 (mapAccumr f\u2081 xs\u271d s\u2081).snd = (mapAccumr f\u2082 xs\u271d s\u2082).snd\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).fst (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).fst \u2227\n (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).snd = (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).snd\n[PROOFSTEP]\nnext => exact \u27e8h\u2080, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03c3\u2081 : Type\n\u03b2 : Type u_1\n\u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b2\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1), R s q \u2192 R (f\u2081 a s).fst (f\u2082 a q).fst \u2227 (f\u2081 a s).snd = (f\u2082 a q).snd\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr f\u2081 nil s\u2081).fst (mapAccumr f\u2082 nil s\u2082).fst \u2227 (mapAccumr f\u2081 nil s\u2081).snd = (mapAccumr f\u2082 nil s\u2082).snd\n[PROOFSTEP]\nexact \u27e8h\u2080, rfl\u27e9\n[GOAL]\ncase snoc\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03c3\u2081 : Type\n\u03b2 : Type u_1\n\u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b2\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1), R s q \u2192 R (f\u2081 a s).fst (f\u2082 a q).fst \u2227 (f\u2081 a s).snd = (f\u2082 a q).snd\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nx\u271d : \u03b1\na\u271d :\n \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n R s\u2081 s\u2082 \u2192\n R (mapAccumr f\u2081 xs\u271d s\u2081).fst (mapAccumr f\u2082 xs\u271d s\u2082).fst \u2227 (mapAccumr f\u2081 xs\u271d s\u2081).snd = (mapAccumr f\u2082 xs\u271d s\u2082).snd\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).fst (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).fst \u2227\n (mapAccumr f\u2081 (snoc xs\u271d x\u271d) s\u2081).snd = (mapAccumr f\u2082 (snoc xs\u271d x\u271d) s\u2082).snd\n[PROOFSTEP]\nnext xs x ih =>\n rcases(hR x h\u2080) with \u27e8hR, _\u27e9\n simp only [mapAccumr_snoc, ih hR, true_and]\n congr 1\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\nxs\u271d : Vector \u03b1 n\n\u03c3\u2081 : Type\n\u03b2 : Type u_1\n\u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b2\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1), R s q \u2192 R (f\u2081 a s).fst (f\u2082 a q).fst \u2227 (f\u2081 a s).snd = (f\u2082 a q).snd\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nx : \u03b1\nih :\n \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs s\u2081).fst (mapAccumr f\u2082 xs s\u2082).fst \u2227 (mapAccumr f\u2081 xs s\u2081).snd = (mapAccumr f\u2082 xs s\u2082).snd\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr f\u2081 (snoc xs x) s\u2081).fst (mapAccumr f\u2082 (snoc xs x) s\u2082).fst \u2227\n (mapAccumr f\u2081 (snoc xs x) s\u2081).snd = (mapAccumr f\u2082 (snoc xs x) s\u2082).snd\n[PROOFSTEP]\nrcases(hR x h\u2080) with \u27e8hR, _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_2\nn : \u2115\nxs\u271d : Vector \u03b1 n\n\u03c3\u2081 : Type\n\u03b2 : Type u_1\n\u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b2\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR\u271d : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1), R s q \u2192 R (f\u2081 a s).fst (f\u2082 a q).fst \u2227 (f\u2081 a s).snd = (f\u2082 a q).snd\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nx : \u03b1\nih :\n \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs s\u2081).fst (mapAccumr f\u2082 xs s\u2082).fst \u2227 (mapAccumr f\u2081 xs s\u2081).snd = (mapAccumr f\u2082 xs s\u2082).snd\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\nhR : R (f\u2081 x s\u2081).fst (f\u2082 x s\u2082).fst\nright\u271d : (f\u2081 x s\u2081).snd = (f\u2082 x s\u2082).snd\n\u22a2 R (mapAccumr f\u2081 (snoc xs x) s\u2081).fst (mapAccumr f\u2082 (snoc xs x) s\u2082).fst \u2227\n (mapAccumr f\u2081 (snoc xs x) s\u2081).snd = (mapAccumr f\u2082 (snoc xs x) s\u2082).snd\n[PROOFSTEP]\nsimp only [mapAccumr_snoc, ih hR, true_and]\n[GOAL]\ncase intro\n\u03b1 : Type u_2\nn : \u2115\nxs\u271d : Vector \u03b1 n\n\u03c3\u2081 : Type\n\u03b2 : Type u_1\n\u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b2\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR\u271d : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1), R s q \u2192 R (f\u2081 a s).fst (f\u2082 a q).fst \u2227 (f\u2081 a s).snd = (f\u2082 a q).snd\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nx : \u03b1\nih :\n \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n R s\u2081 s\u2082 \u2192 R (mapAccumr f\u2081 xs s\u2081).fst (mapAccumr f\u2082 xs s\u2082).fst \u2227 (mapAccumr f\u2081 xs s\u2081).snd = (mapAccumr f\u2082 xs s\u2082).snd\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\nhR : R (f\u2081 x s\u2081).fst (f\u2082 x s\u2082).fst\nright\u271d : (f\u2081 x s\u2081).snd = (f\u2082 x s\u2082).snd\n\u22a2 snoc (mapAccumr f\u2082 xs (f\u2082 x s\u2082).fst).snd (f\u2081 x s\u2081).snd = snoc (mapAccumr f\u2082 xs (f\u2082 x s\u2082).fst).snd (f\u2082 x s\u2082).snd\n[PROOFSTEP]\ncongr 1\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03c3\u2081 : Type\n\u03b2 : Type u_1\n\u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b2\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh : \u2203 R, R s\u2081 s\u2082 \u2227 \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1), R s q \u2192 R (f\u2081 a s).fst (f\u2082 a q).fst \u2227 (f\u2081 a s).snd = (f\u2082 a q).snd\n\u22a2 (mapAccumr f\u2081 xs s\u2081).snd = (mapAccumr f\u2082 xs s\u2082).snd\n[PROOFSTEP]\nrcases h with \u27e8R, h\u2080, hR\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03c3\u2081 : Type\n\u03b2 : Type u_1\n\u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b2\nf\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nh\u2080 : R s\u2081 s\u2082\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1), R s q \u2192 R (f\u2081 a s).fst (f\u2082 a q).fst \u2227 (f\u2081 a s).snd = (f\u2082 a q).snd\n\u22a2 (mapAccumr f\u2081 xs s\u2081).snd = (mapAccumr f\u2082 xs s\u2082).snd\n[PROOFSTEP]\nexact (mapAccumr_bisim R h\u2080 hR).2\n[GOAL]\n\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nys : Vector \u03b2 n\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nh\u2080 : R s\u2081 s\u2082\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).fst (f\u2082 a b q).fst \u2227 (f\u2081 a b s).snd = (f\u2082 a b q).snd\n\u22a2 R (mapAccumr\u2082 f\u2081 xs ys s\u2081).fst (mapAccumr\u2082 f\u2082 xs ys s\u2082).fst \u2227\n (mapAccumr\u2082 f\u2081 xs ys s\u2081).snd = (mapAccumr\u2082 f\u2082 xs ys s\u2082).snd\n[PROOFSTEP]\ninduction xs, ys using Vector.revInductionOn\u2082 generalizing s\u2081 s\u2082\n[GOAL]\ncase nil\n\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).fst (f\u2082 a b q).fst \u2227 (f\u2081 a b s).snd = (f\u2082 a b q).snd\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr\u2082 f\u2081 nil nil s\u2081).fst (mapAccumr\u2082 f\u2082 nil nil s\u2082).fst \u2227\n (mapAccumr\u2082 f\u2081 nil nil s\u2081).snd = (mapAccumr\u2082 f\u2082 nil nil s\u2082).snd\ncase snoc\n\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).fst (f\u2082 a b q).fst \u2227 (f\u2081 a b s).snd = (f\u2082 a b q).snd\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d :\n \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n R s\u2081 s\u2082 \u2192\n R (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).fst (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).fst \u2227\n (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).snd = (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).snd\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).fst (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).fst \u2227\n (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).snd = (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).snd\n[PROOFSTEP]\nnext => exact \u27e8h\u2080, rfl\u27e9\n[GOAL]\n\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).fst (f\u2082 a b q).fst \u2227 (f\u2081 a b s).snd = (f\u2082 a b q).snd\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr\u2082 f\u2081 nil nil s\u2081).fst (mapAccumr\u2082 f\u2082 nil nil s\u2082).fst \u2227\n (mapAccumr\u2082 f\u2081 nil nil s\u2081).snd = (mapAccumr\u2082 f\u2082 nil nil s\u2082).snd\n[PROOFSTEP]\nexact \u27e8h\u2080, rfl\u27e9\n[GOAL]\ncase snoc\n\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).fst (f\u2082 a b q).fst \u2227 (f\u2081 a b s).snd = (f\u2082 a b q).snd\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d :\n \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n R s\u2081 s\u2082 \u2192\n R (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).fst (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).fst \u2227\n (mapAccumr\u2082 f\u2081 xs\u271d ys\u271d s\u2081).snd = (mapAccumr\u2082 f\u2082 xs\u271d ys\u271d s\u2082).snd\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).fst (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).fst \u2227\n (mapAccumr\u2082 f\u2081 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2081).snd = (mapAccumr\u2082 f\u2082 (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s\u2082).snd\n[PROOFSTEP]\nnext xs ys x y ih =>\n rcases(hR x y h\u2080) with \u27e8hR, _\u27e9\n simp only [mapAccumr\u2082_snoc, ih hR, true_and]\n congr 1\n[GOAL]\n\u03b1 : Type\nn : \u2115\nxs\u271d : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).fst (f\u2082 a b q).fst \u2227 (f\u2081 a b s).snd = (f\u2082 a b q).snd\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nx : \u03b1\ny : \u03b2\nih :\n \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n R s\u2081 s\u2082 \u2192\n R (mapAccumr\u2082 f\u2081 xs ys s\u2081).fst (mapAccumr\u2082 f\u2082 xs ys s\u2082).fst \u2227\n (mapAccumr\u2082 f\u2081 xs ys s\u2081).snd = (mapAccumr\u2082 f\u2082 xs ys s\u2082).snd\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\n\u22a2 R (mapAccumr\u2082 f\u2081 (snoc xs x) (snoc ys y) s\u2081).fst (mapAccumr\u2082 f\u2082 (snoc xs x) (snoc ys y) s\u2082).fst \u2227\n (mapAccumr\u2082 f\u2081 (snoc xs x) (snoc ys y) s\u2081).snd = (mapAccumr\u2082 f\u2082 (snoc xs x) (snoc ys y) s\u2082).snd\n[PROOFSTEP]\nrcases(hR x y h\u2080) with \u27e8hR, _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type\nn : \u2115\nxs\u271d : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR\u271d : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).fst (f\u2082 a b q).fst \u2227 (f\u2081 a b s).snd = (f\u2082 a b q).snd\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nx : \u03b1\ny : \u03b2\nih :\n \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n R s\u2081 s\u2082 \u2192\n R (mapAccumr\u2082 f\u2081 xs ys s\u2081).fst (mapAccumr\u2082 f\u2082 xs ys s\u2082).fst \u2227\n (mapAccumr\u2082 f\u2081 xs ys s\u2081).snd = (mapAccumr\u2082 f\u2082 xs ys s\u2082).snd\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\nhR : R (f\u2081 x y s\u2081).fst (f\u2082 x y s\u2082).fst\nright\u271d : (f\u2081 x y s\u2081).snd = (f\u2082 x y s\u2082).snd\n\u22a2 R (mapAccumr\u2082 f\u2081 (snoc xs x) (snoc ys y) s\u2081).fst (mapAccumr\u2082 f\u2082 (snoc xs x) (snoc ys y) s\u2082).fst \u2227\n (mapAccumr\u2082 f\u2081 (snoc xs x) (snoc ys y) s\u2081).snd = (mapAccumr\u2082 f\u2082 (snoc xs x) (snoc ys y) s\u2082).snd\n[PROOFSTEP]\nsimp only [mapAccumr\u2082_snoc, ih hR, true_and]\n[GOAL]\ncase intro\n\u03b1 : Type\nn : \u2115\nxs\u271d : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nhR\u271d : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).fst (f\u2082 a b q).fst \u2227 (f\u2081 a b s).snd = (f\u2082 a b q).snd\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nx : \u03b1\ny : \u03b2\nih :\n \u2200 {s\u2081 : \u03c3\u2081} {s\u2082 : \u03c3\u2082},\n R s\u2081 s\u2082 \u2192\n R (mapAccumr\u2082 f\u2081 xs ys s\u2081).fst (mapAccumr\u2082 f\u2082 xs ys s\u2082).fst \u2227\n (mapAccumr\u2082 f\u2081 xs ys s\u2081).snd = (mapAccumr\u2082 f\u2082 xs ys s\u2082).snd\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh\u2080 : R s\u2081 s\u2082\nhR : R (f\u2081 x y s\u2081).fst (f\u2082 x y s\u2082).fst\nright\u271d : (f\u2081 x y s\u2081).snd = (f\u2082 x y s\u2082).snd\n\u22a2 snoc (mapAccumr\u2082 f\u2082 xs ys (f\u2082 x y s\u2082).fst).snd (f\u2081 x y s\u2081).snd =\n snoc (mapAccumr\u2082 f\u2082 xs ys (f\u2082 x y s\u2082).fst).snd (f\u2082 x y s\u2082).snd\n[PROOFSTEP]\ncongr 1\n[GOAL]\n\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nys : Vector \u03b2 n\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nh :\n \u2203 R,\n R s\u2081 s\u2082 \u2227\n \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).fst (f\u2082 a b q).fst \u2227 (f\u2081 a b s).snd = (f\u2082 a b q).snd\n\u22a2 (mapAccumr\u2082 f\u2081 xs ys s\u2081).snd = (mapAccumr\u2082 f\u2082 xs ys s\u2082).snd\n[PROOFSTEP]\nrcases h with \u27e8R, h\u2080, hR\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type\nn : \u2115\nxs : Vector \u03b1 n\n\u03b2 \u03c3\u2081 \u03b3 \u03c3\u2082 : Type\nys : Vector \u03b2 n\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b3\nf\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\ns\u2081 : \u03c3\u2081\ns\u2082 : \u03c3\u2082\nR : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nh\u2080 : R s\u2081 s\u2082\nhR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1) (b : \u03b2), R s q \u2192 R (f\u2081 a b s).fst (f\u2082 a b q).fst \u2227 (f\u2081 a b s).snd = (f\u2082 a b q).snd\n\u22a2 (mapAccumr\u2082 f\u2081 xs ys s\u2081).snd = (mapAccumr\u2082 f\u2082 xs ys s\u2082).snd\n[PROOFSTEP]\nexact (mapAccumr\u2082_bisim R h\u2080 hR).2\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type ?u.34261\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b1\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d\n\u22a2 map f xs = (mapAccumr (fun x x_1 => ((), f x)) xs ()).snd\n[PROOFSTEP]\nclear ys\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type ?u.34261\nxs : Vector \u03b1 n\n\u03b1\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d\n\u22a2 map f xs = (mapAccumr (fun x x_1 => ((), f x)) xs ()).snd\n[PROOFSTEP]\ninduction xs using Vector.revInductionOn\n[GOAL]\ncase nil\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type ?u.34261\nxs : Vector \u03b1 n\n\u03b1\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d\n\u22a2 map f nil = (mapAccumr (fun x x_1 => ((), f x)) nil ()).snd\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type ?u.34261\nxs : Vector \u03b1 n\n\u03b1\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nx\u271d : \u03b1\na\u271d : map f xs\u271d = (mapAccumr (fun x x_1 => ((), f x)) xs\u271d ()).snd\n\u22a2 map f (snoc xs\u271d x\u271d) = (mapAccumr (fun x x_1 => ((), f x)) (snoc xs\u271d x\u271d) ()).snd\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 : Type\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (s : \u03c3), s \u2208 S \u2192 (f a s).fst \u2208 S\nout : \u2200 (a : \u03b1) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a s).snd = (f a s').snd\n\u22a2 (mapAccumr f xs s\u2080).snd = map (fun x => (f x s\u2080).snd) xs\n[PROOFSTEP]\nrw [Vector.map_eq_mapAccumr]\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 : Type\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (s : \u03c3), s \u2208 S \u2192 (f a s).fst \u2208 S\nout : \u2200 (a : \u03b1) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a s).snd = (f a s').snd\n\u22a2 (mapAccumr f xs s\u2080).snd = (mapAccumr (fun x x_1 => ((), (f x s\u2080).snd)) xs ()).snd\n[PROOFSTEP]\napply mapAccumr_bisim_tail\n[GOAL]\ncase h\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 : Type\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (s : \u03c3), s \u2208 S \u2192 (f a s).fst \u2208 S\nout : \u2200 (a : \u03b1) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a s).snd = (f a s').snd\n\u22a2 \u2203 R,\n R s\u2080 () \u2227\n \u2200 {s : \u03c3} {q : Unit} (a : \u03b1), R s q \u2192 R (f a s).fst ((), (f a s\u2080).snd).fst \u2227 (f a s).snd = ((), (f a s\u2080).snd).snd\n[PROOFSTEP]\nuse fun s _ => s \u2208 S, h\u2080\n[GOAL]\ncase right\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 : Type\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (s : \u03c3), s \u2208 S \u2192 (f a s).fst \u2208 S\nout : \u2200 (a : \u03b1) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a s).snd = (f a s').snd\n\u22a2 \u2200 {s : \u03c3} {q : Unit} (a : \u03b1), s \u2208 S \u2192 (f a s).fst \u2208 S \u2227 (f a s).snd = ((), (f a s\u2080).snd).snd\n[PROOFSTEP]\nexact @fun s _q a h => \u27e8closure a s h, out a s s\u2080 h h\u2080\u27e9\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b1\u271d : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\u271d\n\u22a2 map\u2082 f xs ys = (mapAccumr\u2082 (fun x y x_1 => ((), f x y)) xs ys ()).snd\n[PROOFSTEP]\ninduction xs, ys using Vector.revInductionOn\u2082\n[GOAL]\ncase nil\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b1\u271d : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\u271d\n\u22a2 map\u2082 f nil nil = (mapAccumr\u2082 (fun x y x_1 => ((), f x y)) nil nil ()).snd\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b1\u271d : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\u271d\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d : map\u2082 f xs\u271d ys\u271d = (mapAccumr\u2082 (fun x y x_1 => ((), f x y)) xs\u271d ys\u271d ()).snd\n\u22a2 map\u2082 f (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) = (mapAccumr\u2082 (fun x y x_1 => ((), f x y)) (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) ()).snd\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), s \u2208 S \u2192 (f a b s).fst \u2208 S\nout : \u2200 (a : \u03b1) (b : \u03b2) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a b s).snd = (f a b s').snd\n\u22a2 (mapAccumr\u2082 f xs ys s\u2080).snd = map\u2082 (fun x x_1 => (f x x_1 s\u2080).snd) xs ys\n[PROOFSTEP]\nrw [Vector.map\u2082_eq_mapAccumr\u2082]\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), s \u2208 S \u2192 (f a b s).fst \u2208 S\nout : \u2200 (a : \u03b1) (b : \u03b2) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a b s).snd = (f a b s').snd\n\u22a2 (mapAccumr\u2082 f xs ys s\u2080).snd = (mapAccumr\u2082 (fun x y x_1 => ((), (f x y s\u2080).snd)) xs ys ()).snd\n[PROOFSTEP]\napply mapAccumr\u2082_bisim_tail\n[GOAL]\ncase h\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), s \u2208 S \u2192 (f a b s).fst \u2208 S\nout : \u2200 (a : \u03b1) (b : \u03b2) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a b s).snd = (f a b s').snd\n\u22a2 \u2203 R,\n R s\u2080 () \u2227\n \u2200 {s : \u03c3} {q : Unit} (a : \u03b1) (b : \u03b2),\n R s q \u2192 R (f a b s).fst ((), (f a b s\u2080).snd).fst \u2227 (f a b s).snd = ((), (f a b s\u2080).snd).snd\n[PROOFSTEP]\nuse fun s _ => s \u2208 S, h\u2080\n[GOAL]\ncase right\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ns\u2080 : \u03c3\nS : Set \u03c3\nh\u2080 : s\u2080 \u2208 S\nclosure : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), s \u2208 S \u2192 (f a b s).fst \u2208 S\nout : \u2200 (a : \u03b1) (b : \u03b2) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a b s).snd = (f a b s').snd\n\u22a2 \u2200 {s : \u03c3} {q : Unit} (a : \u03b1) (b : \u03b2), s \u2208 S \u2192 (f a b s).fst \u2208 S \u2227 (f a b s).snd = ((), (f a b s\u2080).snd).snd\n[PROOFSTEP]\nexact @fun s _q a b h => \u27e8closure a b s h, out a b s s\u2080 h h\u2080\u27e9\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 : Type\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\ns : \u03c3\nh : \u2200 (a : \u03b1), (f a s).fst = s\n\u22a2 mapAccumr f xs s = (s, map (fun x => (f x s).snd) xs)\n[PROOFSTEP]\nclear ys\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\n\u03c3 : Type\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\ns : \u03c3\nh : \u2200 (a : \u03b1), (f a s).fst = s\n\u22a2 mapAccumr f xs s = (s, map (fun x => (f x s).snd) xs)\n[PROOFSTEP]\ninduction xs using revInductionOn\n[GOAL]\ncase nil\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\n\u03c3 : Type\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\ns : \u03c3\nh : \u2200 (a : \u03b1), (f a s).fst = s\n\u22a2 mapAccumr f nil s = (s, map (fun x => (f x s).snd) nil)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\n\u03c3 : Type\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\ns : \u03c3\nh : \u2200 (a : \u03b1), (f a s).fst = s\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nx\u271d : \u03b1\na\u271d : mapAccumr f xs\u271d s = (s, map (fun x => (f x s).snd) xs\u271d)\n\u22a2 mapAccumr f (snoc xs\u271d x\u271d) s = (s, map (fun x => (f x s).snd) (snoc xs\u271d x\u271d))\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ns : \u03c3\nh : \u2200 (a : \u03b1) (b : \u03b2), (f a b s).fst = s\n\u22a2 mapAccumr\u2082 f xs ys s = (s, map\u2082 (fun x y => (f x y s).snd) xs ys)\n[PROOFSTEP]\ninduction xs, ys using revInductionOn\u2082\n[GOAL]\ncase nil\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ns : \u03c3\nh : \u2200 (a : \u03b1) (b : \u03b2), (f a b s).fst = s\n\u22a2 mapAccumr\u2082 f nil nil s = (s, map\u2082 (fun x y => (f x y s).snd) nil nil)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase snoc\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ns : \u03c3\nh : \u2200 (a : \u03b1) (b : \u03b2), (f a b s).fst = s\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d : mapAccumr\u2082 f xs\u271d ys\u271d s = (s, map\u2082 (fun x y => (f x y s).snd) xs\u271d ys\u271d)\n\u22a2 mapAccumr\u2082 f (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s = (s, map\u2082 (fun x y => (f x y s).snd) (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d))\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 : Type\ns : \u03c3\nf : \u03b1 \u2192 \u03c3 \u00d7 \u03c3 \u2192 (\u03c3 \u00d7 \u03c3) \u00d7 \u03b2\nh : \u2200 (x : \u03b1) (s : \u03c3), (f x (s, s)).fst.fst = (f x (s, s)).fst.snd\n\u22a2 \u2203 R,\n R (s, s) s \u2227\n \u2200 {s : \u03c3 \u00d7 \u03c3} {q : \u03c3} (a : \u03b1),\n R s q \u2192\n R (f a s).fst ((f a (q, q)).fst.fst, (f a (q, q)).snd).fst \u2227\n (f a s).snd = ((f a (q, q)).fst.fst, (f a (q, q)).snd).snd\n[PROOFSTEP]\nuse fun (s\u2081, s\u2082) s => s\u2082 = s \u2227 s\u2081 = s\n[GOAL]\ncase h\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_1\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 : Type\ns : \u03c3\nf : \u03b1 \u2192 \u03c3 \u00d7 \u03c3 \u2192 (\u03c3 \u00d7 \u03c3) \u00d7 \u03b2\nh : \u2200 (x : \u03b1) (s : \u03c3), (f x (s, s)).fst.fst = (f x (s, s)).fst.snd\n\u22a2 (match (s, s) with\n | (s\u2081, s\u2082) => s\u2082 = s \u2227 s\u2081 = s) \u2227\n \u2200 {s : \u03c3 \u00d7 \u03c3} {q : \u03c3} (a : \u03b1),\n (match s with\n | (s\u2081, s\u2082) => s\u2082 = q \u2227 s\u2081 = q) \u2192\n (match (f a s).fst with\n | (s\u2081, s\u2082) =>\n s\u2082 = ((f a (q, q)).fst.fst, (f a (q, q)).snd).fst \u2227 s\u2081 = ((f a (q, q)).fst.fst, (f a (q, q)).snd).fst) \u2227\n (f a s).snd = ((f a (q, q)).fst.fst, (f a (q, q)).snd).snd\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ns : \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u00d7 \u03c3 \u2192 (\u03c3 \u00d7 \u03c3) \u00d7 \u03b3\nh :\n \u2200 (x : \u03b1) (y : \u03b2) (s : \u03c3),\n let s' := (f x y (s, s)).fst;\n s'.fst = s'.snd\n\u22a2 \u2203 R,\n R (s, s) s \u2227\n \u2200 {s : \u03c3 \u00d7 \u03c3} {q : \u03c3} (a : \u03b1) (b : \u03b2),\n R s q \u2192\n R (f a b s).fst ((f a b (q, q)).fst.fst, (f a b (q, q)).snd).fst \u2227\n (f a b s).snd = ((f a b (q, q)).fst.fst, (f a b (q, q)).snd).snd\n[PROOFSTEP]\nuse fun (s\u2081, s\u2082) s => s\u2082 = s \u2227 s\u2081 = s\n[GOAL]\ncase h\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ns : \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u00d7 \u03c3 \u2192 (\u03c3 \u00d7 \u03c3) \u00d7 \u03b3\nh :\n \u2200 (x : \u03b1) (y : \u03b2) (s : \u03c3),\n let s' := (f x y (s, s)).fst;\n s'.fst = s'.snd\n\u22a2 (match (s, s) with\n | (s\u2081, s\u2082) => s\u2082 = s \u2227 s\u2081 = s) \u2227\n \u2200 {s : \u03c3 \u00d7 \u03c3} {q : \u03c3} (a : \u03b1) (b : \u03b2),\n (match s with\n | (s\u2081, s\u2082) => s\u2082 = q \u2227 s\u2081 = q) \u2192\n (match (f a b s).fst with\n | (s\u2081, s\u2082) =>\n s\u2082 = ((f a b (q, q)).fst.fst, (f a b (q, q)).snd).fst \u2227\n s\u2081 = ((f a b (q, q)).fst.fst, (f a b (q, q)).snd).fst) \u2227\n (f a b s).snd = ((f a b (q, q)).fst.fst, (f a b (q, q)).snd).snd\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ns : \u03c3\ninst\u271d : Inhabited \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f default b s = f a b s\n\u22a2 mapAccumr\u2082 f xs ys s = mapAccumr (fun b s => f default b s) ys s\n[PROOFSTEP]\ninduction xs, ys using Vector.revInductionOn\u2082 generalizing s\n[GOAL]\ncase nil\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ninst\u271d : Inhabited \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f default b s = f a b s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f nil nil s = mapAccumr (fun b s => f default b s) nil s\ncase snoc\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ninst\u271d : Inhabited \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f default b s = f a b s\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d : \u2200 {s : \u03c3}, mapAccumr\u2082 f xs\u271d ys\u271d s = mapAccumr (fun b s => f default b s) ys\u271d s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s = mapAccumr (fun b s => f default b s) (snoc ys\u271d y\u271d) s\n[PROOFSTEP]\ncase nil => rfl\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ninst\u271d : Inhabited \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f default b s = f a b s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f nil nil s = mapAccumr (fun b s => f default b s) nil s\n[PROOFSTEP]\ncase nil => rfl\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ninst\u271d : Inhabited \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f default b s = f a b s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f nil nil s = mapAccumr (fun b s => f default b s) nil s\n[PROOFSTEP]\nrfl\n[GOAL]\ncase snoc\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ninst\u271d : Inhabited \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f default b s = f a b s\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d : \u2200 {s : \u03c3}, mapAccumr\u2082 f xs\u271d ys\u271d s = mapAccumr (fun b s => f default b s) ys\u271d s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s = mapAccumr (fun b s => f default b s) (snoc ys\u271d y\u271d) s\n[PROOFSTEP]\ncase snoc xs ys x y ih => simp [h x y s, ih]\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs\u271d : Vector \u03b1 n\nys\u271d : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ninst\u271d : Inhabited \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f default b s = f a b s\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nx : \u03b1\ny : \u03b2\nih : \u2200 {s : \u03c3}, mapAccumr\u2082 f xs ys s = mapAccumr (fun b s => f default b s) ys s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f (snoc xs x) (snoc ys y) s = mapAccumr (fun b s => f default b s) (snoc ys y) s\n[PROOFSTEP]\ncase snoc xs ys x y ih => simp [h x y s, ih]\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs\u271d : Vector \u03b1 n\nys\u271d : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ninst\u271d : Inhabited \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f default b s = f a b s\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nx : \u03b1\ny : \u03b2\nih : \u2200 {s : \u03c3}, mapAccumr\u2082 f xs ys s = mapAccumr (fun b s => f default b s) ys s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f (snoc xs x) (snoc ys y) s = mapAccumr (fun b s => f default b s) (snoc ys y) s\n[PROOFSTEP]\nsimp [h x y s, ih]\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ns : \u03c3\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f a default s = f a b s\n\u22a2 mapAccumr\u2082 f xs ys s = mapAccumr (fun a s => f a default s) xs s\n[PROOFSTEP]\ninduction xs, ys using Vector.revInductionOn\u2082 generalizing s\n[GOAL]\ncase nil\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f a default s = f a b s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f nil nil s = mapAccumr (fun a s => f a default s) nil s\ncase snoc\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f a default s = f a b s\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d : \u2200 {s : \u03c3}, mapAccumr\u2082 f xs\u271d ys\u271d s = mapAccumr (fun a s => f a default s) xs\u271d s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s = mapAccumr (fun a s => f a default s) (snoc xs\u271d x\u271d) s\n[PROOFSTEP]\ncase nil => rfl\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f a default s = f a b s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f nil nil s = mapAccumr (fun a s => f a default s) nil s\n[PROOFSTEP]\ncase nil => rfl\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f a default s = f a b s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f nil nil s = mapAccumr (fun a s => f a default s) nil s\n[PROOFSTEP]\nrfl\n[GOAL]\ncase snoc\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f a default s = f a b s\nn\u271d : \u2115\nxs\u271d : Vector \u03b1 n\u271d\nys\u271d : Vector \u03b2 n\u271d\nx\u271d : \u03b1\ny\u271d : \u03b2\na\u271d : \u2200 {s : \u03c3}, mapAccumr\u2082 f xs\u271d ys\u271d s = mapAccumr (fun a s => f a default s) xs\u271d s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f (snoc xs\u271d x\u271d) (snoc ys\u271d y\u271d) s = mapAccumr (fun a s => f a default s) (snoc xs\u271d x\u271d) s\n[PROOFSTEP]\ncase snoc xs ys x y ih => simp [h x y s, ih]\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs\u271d : Vector \u03b1 n\nys\u271d : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f a default s = f a b s\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nx : \u03b1\ny : \u03b2\nih : \u2200 {s : \u03c3}, mapAccumr\u2082 f xs ys s = mapAccumr (fun a s => f a default s) xs s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f (snoc xs x) (snoc ys y) s = mapAccumr (fun a s => f a default s) (snoc xs x) s\n[PROOFSTEP]\ncase snoc xs ys x y ih => simp [h x y s, ih]\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs\u271d : Vector \u03b1 n\nys\u271d : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), f a default s = f a b s\nn\u271d : \u2115\nxs : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nx : \u03b1\ny : \u03b2\nih : \u2200 {s : \u03c3}, mapAccumr\u2082 f xs ys s = mapAccumr (fun a s => f a default s) xs s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f (snoc xs x) (snoc ys y) s = mapAccumr (fun a s => f a default s) (snoc xs x) s\n[PROOFSTEP]\nsimp [h x y s, ih]\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\nxs ys : Vector \u03b1 n\n\u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b2\ncomm : \u2200 (a\u2081 a\u2082 : \u03b1), f a\u2081 a\u2082 = f a\u2082 a\u2081\n\u22a2 map\u2082 f xs ys = map\u2082 f ys xs\n[PROOFSTEP]\ninduction xs, ys using Vector.inductionOn\u2082\n[GOAL]\ncase nil\n\u03b1 : Type u_2\nn : \u2115\nxs ys : Vector \u03b1 n\n\u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b2\ncomm : \u2200 (a\u2081 a\u2082 : \u03b1), f a\u2081 a\u2082 = f a\u2082 a\u2081\n\u22a2 map\u2082 f nil nil = map\u2082 f nil nil\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase cons\n\u03b1 : Type u_2\nn : \u2115\nxs ys : Vector \u03b1 n\n\u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b2\ncomm : \u2200 (a\u2081 a\u2082 : \u03b1), f a\u2081 a\u2082 = f a\u2082 a\u2081\nn\u271d : \u2115\na\u271d\u00b9 b\u271d : \u03b1\nx\u271d y\u271d : Vector \u03b1 n\u271d\na\u271d : map\u2082 f x\u271d y\u271d = map\u2082 f y\u271d x\u271d\n\u22a2 map\u2082 f (a\u271d\u00b9 ::\u1d65 x\u271d) (b\u271d ::\u1d65 y\u271d) = map\u2082 f (b\u271d ::\u1d65 y\u271d) (a\u271d\u00b9 ::\u1d65 x\u271d)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type\nn : \u2115\nxs ys : Vector \u03b1 n\n\u03c3 \u03b3 : Type\ns : \u03c3\nf : \u03b1 \u2192 \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ncomm : \u2200 (a\u2081 a\u2082 : \u03b1) (s : \u03c3), f a\u2081 a\u2082 s = f a\u2082 a\u2081 s\n\u22a2 mapAccumr\u2082 f xs ys s = mapAccumr\u2082 f ys xs s\n[PROOFSTEP]\ninduction xs, ys using Vector.inductionOn\u2082 generalizing s\n[GOAL]\ncase nil\n\u03b1 : Type\nn : \u2115\nxs ys : Vector \u03b1 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ncomm : \u2200 (a\u2081 a\u2082 : \u03b1) (s : \u03c3), f a\u2081 a\u2082 s = f a\u2082 a\u2081 s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f nil nil s = mapAccumr\u2082 f nil nil s\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase cons\n\u03b1 : Type\nn : \u2115\nxs ys : Vector \u03b1 n\n\u03c3 \u03b3 : Type\nf : \u03b1 \u2192 \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\ncomm : \u2200 (a\u2081 a\u2082 : \u03b1) (s : \u03c3), f a\u2081 a\u2082 s = f a\u2082 a\u2081 s\nn\u271d : \u2115\na\u271d\u00b9 b\u271d : \u03b1\nx\u271d y\u271d : Vector \u03b1 n\u271d\na\u271d : \u2200 {s : \u03c3}, mapAccumr\u2082 f x\u271d y\u271d s = mapAccumr\u2082 f y\u271d x\u271d s\ns : \u03c3\n\u22a2 mapAccumr\u2082 f (a\u271d\u00b9 ::\u1d65 x\u271d) (b\u271d ::\u1d65 y\u271d) s = mapAccumr\u2082 f (b\u271d ::\u1d65 y\u271d) (a\u271d\u00b9 ::\u1d65 x\u271d) s\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_3\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u22a2 map\u2082 f xs ys = map\u2082 (flip f) ys xs\n[PROOFSTEP]\ninduction xs, ys using Vector.inductionOn\u2082\n[GOAL]\ncase nil\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_3\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u22a2 map\u2082 f nil nil = map\u2082 (flip f) nil nil\n[PROOFSTEP]\nsimp_all [flip]\n[GOAL]\ncase cons\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_3\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\na\u271d\u00b9 : \u03b1\nb\u271d : \u03b2\nx\u271d : Vector \u03b1 n\u271d\ny\u271d : Vector \u03b2 n\u271d\na\u271d : map\u2082 f x\u271d y\u271d = map\u2082 (flip f) y\u271d x\u271d\n\u22a2 map\u2082 f (a\u271d\u00b9 ::\u1d65 x\u271d) (b\u271d ::\u1d65 y\u271d) = map\u2082 (flip f) (b\u271d ::\u1d65 y\u271d) (a\u271d\u00b9 ::\u1d65 x\u271d)\n[PROOFSTEP]\nsimp_all [flip]\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ns : \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\n\u22a2 mapAccumr\u2082 f xs ys s = mapAccumr\u2082 (flip f) ys xs s\n[PROOFSTEP]\ninduction xs, ys using Vector.inductionOn\u2082\n[GOAL]\ncase nil\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ns : \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\n\u22a2 mapAccumr\u2082 f nil nil s = mapAccumr\u2082 (flip f) nil nil s\n[PROOFSTEP]\nsimp_all [flip]\n[GOAL]\ncase cons\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03b3 : Type\ns : \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3\nn\u271d : \u2115\na\u271d\u00b9 : \u03b1\nb\u271d : \u03b2\nx\u271d : Vector \u03b1 n\u271d\ny\u271d : Vector \u03b2 n\u271d\na\u271d : mapAccumr\u2082 f x\u271d y\u271d s = mapAccumr\u2082 (flip f) y\u271d x\u271d s\n\u22a2 mapAccumr\u2082 f (a\u271d\u00b9 ::\u1d65 x\u271d) (b\u271d ::\u1d65 y\u271d) s = mapAccumr\u2082 (flip f) (b\u271d ::\u1d65 y\u271d) (a\u271d\u00b9 ::\u1d65 x\u271d) s\n[PROOFSTEP]\nsimp_all [flip]\n", "meta": {"mathlib_filename": "Mathlib.Data.Vector.MapLemmas", "llama_tokens": 28859, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.4485252253981146}} {"text": "[GOAL]\n\u22a2 \u2200 (a : PrimeMultiset), \u22a5 \u2264 a\n[PROOFSTEP]\nsimp only [bot_le, forall_const]\n[GOAL]\n\u22a2 Repr PrimeMultiset\n[PROOFSTEP]\ndelta PrimeMultiset\n[GOAL]\n\u22a2 Repr (Multiset Nat.Primes)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nv : PrimeMultiset\np : \u2115\nh : p \u2208 toNatMultiset v\n\u22a2 Nat.Prime p\n[PROOFSTEP]\nrcases Multiset.mem_map.mp h with \u27e8\u27e8_, hp'\u27e9, \u27e8_, h_eq\u27e9\u27e9\n[GOAL]\ncase intro.mk.intro\nv : PrimeMultiset\np : \u2115\nh : p \u2208 toNatMultiset v\nval\u271d : \u2115\nhp' : Nat.Prime val\u271d\nleft\u271d : { val := val\u271d, property := hp' } \u2208 v\nh_eq : Coe.coe { val := val\u271d, property := hp' } = p\n\u22a2 Nat.Prime p\n[PROOFSTEP]\nexact h_eq \u25b8 hp'\n[GOAL]\nv : PrimeMultiset\np : \u2115+\nh : p \u2208 toPNatMultiset v\n\u22a2 PNat.Prime p\n[PROOFSTEP]\nrcases Multiset.mem_map.mp h with \u27e8\u27e8_, hp'\u27e9, \u27e8_, h_eq\u27e9\u27e9\n[GOAL]\ncase intro.mk.intro\nv : PrimeMultiset\np : \u2115+\nh : p \u2208 toPNatMultiset v\nval\u271d : \u2115\nhp' : Nat.Prime val\u271d\nleft\u271d : { val := val\u271d, property := hp' } \u2208 v\nh_eq : Coe.coe { val := val\u271d, property := hp' } = p\n\u22a2 PNat.Prime p\n[PROOFSTEP]\nexact h_eq \u25b8 hp'\n[GOAL]\nv : PrimeMultiset\n\u22a2 Multiset.map PNat.val (toPNatMultiset v) = toNatMultiset v\n[PROOFSTEP]\nchange (v.map (Coe.coe : Nat.Primes \u2192 \u2115+)).map Subtype.val = v.map Subtype.val\n[GOAL]\nv : PrimeMultiset\n\u22a2 Multiset.map Subtype.val (Multiset.map Coe.coe v) = Multiset.map Subtype.val v\n[PROOFSTEP]\nrw [Multiset.map_map]\n[GOAL]\nv : PrimeMultiset\n\u22a2 Multiset.map (Subtype.val \u2218 Coe.coe) v = Multiset.map Subtype.val v\n[PROOFSTEP]\ncongr\n[GOAL]\nv : PrimeMultiset\n\u22a2 \u2191(prod v) = Multiset.prod (toNatMultiset v)\n[PROOFSTEP]\nlet h : (v.prod : \u2115) = ((v.map Coe.coe).map Coe.coe).prod := PNat.coeMonoidHom.map_multiset_prod v.toPNatMultiset\n[GOAL]\nv : PrimeMultiset\nh : \u2191(prod v) = Multiset.prod (Multiset.map Coe.coe (Multiset.map Coe.coe v)) :=\n MonoidHom.map_multiset_prod PNat.coeMonoidHom (toPNatMultiset v)\n\u22a2 \u2191(prod v) = Multiset.prod (toNatMultiset v)\n[PROOFSTEP]\nrw [Multiset.map_map] at h \n[GOAL]\nv : PrimeMultiset\nh : \u2191(prod v) = Multiset.prod (Multiset.map (Coe.coe \u2218 Coe.coe) v)\n\u22a2 \u2191(prod v) = Multiset.prod (toNatMultiset v)\n[PROOFSTEP]\nhave : (Coe.coe : \u2115+ \u2192 \u2115) \u2218 (Coe.coe : Nat.Primes \u2192 \u2115+) = Coe.coe := funext fun p => rfl\n[GOAL]\nv : PrimeMultiset\nh : \u2191(prod v) = Multiset.prod (Multiset.map (Coe.coe \u2218 Coe.coe) v)\nthis : Coe.coe \u2218 Coe.coe = Coe.coe\n\u22a2 \u2191(prod v) = Multiset.prod (toNatMultiset v)\n[PROOFSTEP]\nrw [this] at h \n[GOAL]\nv : PrimeMultiset\nh : \u2191(prod v) = Multiset.prod (Multiset.map Coe.coe v)\nthis : Coe.coe \u2218 Coe.coe = Coe.coe\n\u22a2 \u2191(prod v) = Multiset.prod (toNatMultiset v)\n[PROOFSTEP]\nexact h\n[GOAL]\nv : Multiset \u2115\nh : \u2200 (p : \u2115), p \u2208 v \u2192 Nat.Prime p\n\u22a2 toNatMultiset (ofNatMultiset v h) = v\n[PROOFSTEP]\ndsimp [ofNatMultiset, toNatMultiset]\n[GOAL]\nv : Multiset \u2115\nh : \u2200 (p : \u2115), p \u2208 v \u2192 Nat.Prime p\n\u22a2 Multiset.map Coe.coe (Multiset.pmap (fun p hp => { val := p, property := hp }) v h) = v\n[PROOFSTEP]\nhave : (fun p h => (Coe.coe : Nat.Primes \u2192 \u2115) \u27e8p, h\u27e9) = fun p _ => id p :=\n by\n funext p h\n rfl\n[GOAL]\nv : Multiset \u2115\nh : \u2200 (p : \u2115), p \u2208 v \u2192 Nat.Prime p\n\u22a2 (fun p h => Coe.coe { val := p, property := h }) = fun p x => id p\n[PROOFSTEP]\nfunext p h\n[GOAL]\ncase h.h\nv : Multiset \u2115\nh\u271d : \u2200 (p : \u2115), p \u2208 v \u2192 Nat.Prime p\np : \u2115\nh : Nat.Prime p\n\u22a2 Coe.coe { val := p, property := h } = id p\n[PROOFSTEP]\nrfl\n[GOAL]\nv : Multiset \u2115\nh : \u2200 (p : \u2115), p \u2208 v \u2192 Nat.Prime p\nthis : (fun p h => Coe.coe { val := p, property := h }) = fun p x => id p\n\u22a2 Multiset.map Coe.coe (Multiset.pmap (fun p hp => { val := p, property := hp }) v h) = v\n[PROOFSTEP]\nrw [Multiset.map_pmap, this, Multiset.pmap_eq_map, Multiset.map_id]\n[GOAL]\nv : Multiset \u2115\nh : \u2200 (p : \u2115), p \u2208 v \u2192 Nat.Prime p\n\u22a2 \u2191(prod (ofNatMultiset v h)) = Multiset.prod v\n[PROOFSTEP]\nrw [coe_prod, to_ofNatMultiset]\n[GOAL]\nv : Multiset \u2115+\nh : \u2200 (p : \u2115+), p \u2208 v \u2192 PNat.Prime p\n\u22a2 toPNatMultiset (ofPNatMultiset v h) = v\n[PROOFSTEP]\ndsimp [ofPNatMultiset, toPNatMultiset]\n[GOAL]\nv : Multiset \u2115+\nh : \u2200 (p : \u2115+), p \u2208 v \u2192 PNat.Prime p\n\u22a2 Multiset.map Coe.coe (Multiset.pmap (fun p hp => { val := \u2191p, property := hp }) v h) = v\n[PROOFSTEP]\nhave : (fun (p : \u2115+) (h : p.Prime) => (Coe.coe : Nat.Primes \u2192 \u2115+) \u27e8p, h\u27e9) = fun p _ => id p :=\n by\n funext p h\n apply Subtype.eq\n rfl\n[GOAL]\nv : Multiset \u2115+\nh : \u2200 (p : \u2115+), p \u2208 v \u2192 PNat.Prime p\n\u22a2 (fun p h => Coe.coe { val := \u2191p, property := h }) = fun p x => id p\n[PROOFSTEP]\nfunext p h\n[GOAL]\ncase h.h\nv : Multiset \u2115+\nh\u271d : \u2200 (p : \u2115+), p \u2208 v \u2192 PNat.Prime p\np : \u2115+\nh : PNat.Prime p\n\u22a2 Coe.coe { val := \u2191p, property := h } = id p\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase h.h.a\nv : Multiset \u2115+\nh\u271d : \u2200 (p : \u2115+), p \u2208 v \u2192 PNat.Prime p\np : \u2115+\nh : PNat.Prime p\n\u22a2 \u2191(Coe.coe { val := \u2191p, property := h }) = \u2191(id p)\n[PROOFSTEP]\nrfl\n[GOAL]\nv : Multiset \u2115+\nh : \u2200 (p : \u2115+), p \u2208 v \u2192 PNat.Prime p\nthis : (fun p h => Coe.coe { val := \u2191p, property := h }) = fun p x => id p\n\u22a2 Multiset.map Coe.coe (Multiset.pmap (fun p hp => { val := \u2191p, property := hp }) v h) = v\n[PROOFSTEP]\nrw [Multiset.map_pmap, this, Multiset.pmap_eq_map, Multiset.map_id]\n[GOAL]\nv : Multiset \u2115+\nh : \u2200 (p : \u2115+), p \u2208 v \u2192 PNat.Prime p\n\u22a2 prod (ofPNatMultiset v h) = Multiset.prod v\n[PROOFSTEP]\ndsimp [prod]\n[GOAL]\nv : Multiset \u2115+\nh : \u2200 (p : \u2115+), p \u2208 v \u2192 PNat.Prime p\n\u22a2 Multiset.prod (toPNatMultiset (ofPNatMultiset v h)) = Multiset.prod v\n[PROOFSTEP]\nrw [to_ofPNatMultiset]\n[GOAL]\nl : List \u2115\nh : \u2200 (p : \u2115), p \u2208 l \u2192 Nat.Prime p\n\u22a2 \u2191(prod (ofNatList l h)) = List.prod l\n[PROOFSTEP]\nhave := prod_ofNatMultiset (l : Multiset \u2115) h\n[GOAL]\nl : List \u2115\nh : \u2200 (p : \u2115), p \u2208 l \u2192 Nat.Prime p\nthis : \u2191(prod (ofNatMultiset (\u2191l) h)) = Multiset.prod \u2191l\n\u22a2 \u2191(prod (ofNatList l h)) = List.prod l\n[PROOFSTEP]\nrw [Multiset.coe_prod] at this \n[GOAL]\nl : List \u2115\nh : \u2200 (p : \u2115), p \u2208 l \u2192 Nat.Prime p\nthis : \u2191(prod (ofNatMultiset (\u2191l) h)) = List.prod l\n\u22a2 \u2191(prod (ofNatList l h)) = List.prod l\n[PROOFSTEP]\nexact this\n[GOAL]\nl : List \u2115+\nh : \u2200 (p : \u2115+), p \u2208 l \u2192 PNat.Prime p\n\u22a2 prod (ofPNatList l h) = List.prod l\n[PROOFSTEP]\nhave := prod_ofPNatMultiset (l : Multiset \u2115+) h\n[GOAL]\nl : List \u2115+\nh : \u2200 (p : \u2115+), p \u2208 l \u2192 PNat.Prime p\nthis : prod (ofPNatMultiset (\u2191l) h) = Multiset.prod \u2191l\n\u22a2 prod (ofPNatList l h) = List.prod l\n[PROOFSTEP]\nrw [Multiset.coe_prod] at this \n[GOAL]\nl : List \u2115+\nh : \u2200 (p : \u2115+), p \u2208 l \u2192 PNat.Prime p\nthis : prod (ofPNatMultiset (\u2191l) h) = List.prod l\n\u22a2 prod (ofPNatList l h) = List.prod l\n[PROOFSTEP]\nexact this\n[GOAL]\n\u22a2 prod 0 = 1\n[PROOFSTEP]\nexact Multiset.prod_zero\n[GOAL]\nu v : PrimeMultiset\n\u22a2 prod (u + v) = prod u * prod v\n[PROOFSTEP]\nchange (coePNatMonoidHom (u + v)).prod = _\n[GOAL]\nu v : PrimeMultiset\n\u22a2 Multiset.prod (\u2191coePNatMonoidHom (u + v)) = prod u * prod v\n[PROOFSTEP]\nrw [coePNatMonoidHom.map_add]\n[GOAL]\nu v : PrimeMultiset\n\u22a2 Multiset.prod (\u2191coePNatMonoidHom u + \u2191coePNatMonoidHom v) = prod u * prod v\n[PROOFSTEP]\nexact Multiset.prod_add _ _\n[GOAL]\nd : \u2115\nu : PrimeMultiset\n\u22a2 prod (d \u2022 u) = Pow.pow (prod u) d\n[PROOFSTEP]\ninduction' d with n ih\n[GOAL]\ncase zero\nu : PrimeMultiset\n\u22a2 prod (Nat.zero \u2022 u) = Pow.pow (prod u) Nat.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nu : PrimeMultiset\nn : \u2115\nih : prod (n \u2022 u) = Pow.pow (prod u) n\n\u22a2 prod (Nat.succ n \u2022 u) = Pow.pow (prod u) (Nat.succ n)\n[PROOFSTEP]\nhave : \u2200 n' : \u2115, Pow.pow (prod u) n' = Monoid.npow n' (prod u) := fun _ \u21a6 rfl\n[GOAL]\ncase succ\nu : PrimeMultiset\nn : \u2115\nih : prod (n \u2022 u) = Pow.pow (prod u) n\nthis : \u2200 (n' : \u2115), Pow.pow (prod u) n' = Monoid.npow n' (prod u)\n\u22a2 prod (Nat.succ n \u2022 u) = Pow.pow (prod u) (Nat.succ n)\n[PROOFSTEP]\nrw [succ_nsmul, prod_add, ih, this, this, Monoid.npow_succ, mul_comm]\n[GOAL]\nn : \u2115+\n\u22a2 \u2191(PrimeMultiset.prod (factorMultiset n)) = \u2191n\n[PROOFSTEP]\ndsimp [factorMultiset]\n[GOAL]\nn : \u2115+\n\u22a2 \u2191(PrimeMultiset.prod (PrimeMultiset.ofNatList (Nat.factors \u2191n) (_ : \u2200 {p : \u2115}, p \u2208 Nat.factors \u2191n \u2192 Nat.Prime p))) =\n \u2191n\n[PROOFSTEP]\nrw [PrimeMultiset.prod_ofNatList]\n[GOAL]\nn : \u2115+\n\u22a2 List.prod (Nat.factors \u2191n) = \u2191n\n[PROOFSTEP]\nexact Nat.prod_factors n.ne_zero\n[GOAL]\nv : PrimeMultiset\n\u22a2 PNat.factorMultiset (prod v) = v\n[PROOFSTEP]\napply PrimeMultiset.coeNat_injective\n[GOAL]\ncase a\nv : PrimeMultiset\n\u22a2 Coe.coe (PNat.factorMultiset (prod v)) = Coe.coe v\n[PROOFSTEP]\nsuffices toNatMultiset (PNat.factorMultiset (prod v)) = toNatMultiset v by exact this\n[GOAL]\nv : PrimeMultiset\nthis : toNatMultiset (PNat.factorMultiset (prod v)) = toNatMultiset v\n\u22a2 Coe.coe (PNat.factorMultiset (prod v)) = Coe.coe v\n[PROOFSTEP]\nexact this\n[GOAL]\ncase a\nv : PrimeMultiset\n\u22a2 toNatMultiset (PNat.factorMultiset (prod v)) = toNatMultiset v\n[PROOFSTEP]\nrw [v.prod.coeNat_factorMultiset, PrimeMultiset.coe_prod]\n[GOAL]\ncase a\nv : PrimeMultiset\n\u22a2 \u2191(Nat.factors (Multiset.prod (toNatMultiset v))) = toNatMultiset v\n[PROOFSTEP]\nrcases v with\n \u27e8l\u27e9\n --unfold_coes\n[GOAL]\ncase a.mk\nv : PrimeMultiset\nl : List Nat.Primes\n\u22a2 \u2191(Nat.factors (Multiset.prod (toNatMultiset (Quot.mk Setoid.r l)))) = toNatMultiset (Quot.mk Setoid.r l)\n[PROOFSTEP]\ndsimp [PrimeMultiset.toNatMultiset]\n[GOAL]\ncase a.mk\nv : PrimeMultiset\nl : List Nat.Primes\n\u22a2 \u2191(Nat.factors (Multiset.prod \u2191(List.map Coe.coe l))) = \u2191(List.map Coe.coe l)\n[PROOFSTEP]\nrw [Multiset.coe_prod]\n[GOAL]\ncase a.mk\nv : PrimeMultiset\nl : List Nat.Primes\n\u22a2 \u2191(Nat.factors (List.prod (List.map Coe.coe l))) = \u2191(List.map Coe.coe l)\n[PROOFSTEP]\nlet l' := l.map (Coe.coe : Nat.Primes \u2192 \u2115)\n[GOAL]\ncase a.mk\nv : PrimeMultiset\nl : List Nat.Primes\nl' : List \u2115 := List.map Coe.coe l\n\u22a2 \u2191(Nat.factors (List.prod (List.map Coe.coe l))) = \u2191(List.map Coe.coe l)\n[PROOFSTEP]\nhave : \u2200 p : \u2115, p \u2208 l' \u2192 p.Prime := fun p hp =>\n by\n rcases List.mem_map.mp hp with \u27e8\u27e8_, hp'\u27e9, \u27e8_, h_eq\u27e9\u27e9\n exact h_eq \u25b8 hp'\n[GOAL]\nv : PrimeMultiset\nl : List Nat.Primes\nl' : List \u2115 := List.map Coe.coe l\np : \u2115\nhp : p \u2208 l'\n\u22a2 Nat.Prime p\n[PROOFSTEP]\nrcases List.mem_map.mp hp with \u27e8\u27e8_, hp'\u27e9, \u27e8_, h_eq\u27e9\u27e9\n[GOAL]\ncase intro.mk.intro\nv : PrimeMultiset\nl : List Nat.Primes\nl' : List \u2115 := List.map Coe.coe l\np : \u2115\nhp : p \u2208 l'\nval\u271d : \u2115\nhp' : Nat.Prime val\u271d\nleft\u271d : { val := val\u271d, property := hp' } \u2208 l\nh_eq : Coe.coe { val := val\u271d, property := hp' } = p\n\u22a2 Nat.Prime p\n[PROOFSTEP]\nexact h_eq \u25b8 hp'\n[GOAL]\ncase a.mk\nv : PrimeMultiset\nl : List Nat.Primes\nl' : List \u2115 := List.map Coe.coe l\nthis : \u2200 (p : \u2115), p \u2208 l' \u2192 Nat.Prime p\n\u22a2 \u2191(Nat.factors (List.prod (List.map Coe.coe l))) = \u2191(List.map Coe.coe l)\n[PROOFSTEP]\nexact Multiset.coe_eq_coe.mpr (@Nat.factors_unique _ l' rfl this).symm\n[GOAL]\n\u22a2 factorMultiset 1 = 0\n[PROOFSTEP]\nsimp [factorMultiset, PrimeMultiset.ofNatList, PrimeMultiset.ofNatMultiset]\n[GOAL]\nn m : \u2115+\n\u22a2 factorMultiset (n * m) = factorMultiset n + factorMultiset m\n[PROOFSTEP]\nlet u := factorMultiset n\n[GOAL]\nn m : \u2115+\nu : PrimeMultiset := factorMultiset n\n\u22a2 factorMultiset (n * m) = factorMultiset n + factorMultiset m\n[PROOFSTEP]\nlet v := factorMultiset m\n[GOAL]\nn m : \u2115+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\n\u22a2 factorMultiset (n * m) = factorMultiset n + factorMultiset m\n[PROOFSTEP]\nhave : n = u.prod := (prod_factorMultiset n).symm\n[GOAL]\nn m : \u2115+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis : n = PrimeMultiset.prod u\n\u22a2 factorMultiset (n * m) = factorMultiset n + factorMultiset m\n[PROOFSTEP]\nrw [this]\n[GOAL]\nn m : \u2115+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis : n = PrimeMultiset.prod u\n\u22a2 factorMultiset (PrimeMultiset.prod u * m) = factorMultiset (PrimeMultiset.prod u) + factorMultiset m\n[PROOFSTEP]\nhave : m = v.prod := (prod_factorMultiset m).symm\n[GOAL]\nn m : \u2115+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis\u271d : n = PrimeMultiset.prod u\nthis : m = PrimeMultiset.prod v\n\u22a2 factorMultiset (PrimeMultiset.prod u * m) = factorMultiset (PrimeMultiset.prod u) + factorMultiset m\n[PROOFSTEP]\nrw [this]\n[GOAL]\nn m : \u2115+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis\u271d : n = PrimeMultiset.prod u\nthis : m = PrimeMultiset.prod v\n\u22a2 factorMultiset (PrimeMultiset.prod u * PrimeMultiset.prod v) =\n factorMultiset (PrimeMultiset.prod u) + factorMultiset (PrimeMultiset.prod v)\n[PROOFSTEP]\nrw [\u2190 PrimeMultiset.prod_add]\n[GOAL]\nn m : \u2115+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis\u271d : n = PrimeMultiset.prod u\nthis : m = PrimeMultiset.prod v\n\u22a2 factorMultiset (PrimeMultiset.prod (u + v)) =\n factorMultiset (PrimeMultiset.prod u) + factorMultiset (PrimeMultiset.prod v)\n[PROOFSTEP]\nrepeat' rw [PrimeMultiset.factorMultiset_prod]\n[GOAL]\nn m : \u2115+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis\u271d : n = PrimeMultiset.prod u\nthis : m = PrimeMultiset.prod v\n\u22a2 factorMultiset (PrimeMultiset.prod (u + v)) =\n factorMultiset (PrimeMultiset.prod u) + factorMultiset (PrimeMultiset.prod v)\n[PROOFSTEP]\nrw [PrimeMultiset.factorMultiset_prod]\n[GOAL]\nn m : \u2115+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis\u271d : n = PrimeMultiset.prod u\nthis : m = PrimeMultiset.prod v\n\u22a2 u + v = factorMultiset (PrimeMultiset.prod u) + factorMultiset (PrimeMultiset.prod v)\n[PROOFSTEP]\nrw [PrimeMultiset.factorMultiset_prod]\n[GOAL]\nn m : \u2115+\nu : PrimeMultiset := factorMultiset n\nv : PrimeMultiset := factorMultiset m\nthis\u271d : n = PrimeMultiset.prod u\nthis : m = PrimeMultiset.prod v\n\u22a2 u + v = u + factorMultiset (PrimeMultiset.prod v)\n[PROOFSTEP]\nrw [PrimeMultiset.factorMultiset_prod]\n[GOAL]\nn : \u2115+\nm : \u2115\n\u22a2 factorMultiset (Pow.pow n m) = m \u2022 factorMultiset n\n[PROOFSTEP]\nlet u := factorMultiset n\n[GOAL]\nn : \u2115+\nm : \u2115\nu : PrimeMultiset := factorMultiset n\n\u22a2 factorMultiset (Pow.pow n m) = m \u2022 factorMultiset n\n[PROOFSTEP]\nhave : n = u.prod := (prod_factorMultiset n).symm\n[GOAL]\nn : \u2115+\nm : \u2115\nu : PrimeMultiset := factorMultiset n\nthis : n = PrimeMultiset.prod u\n\u22a2 factorMultiset (Pow.pow n m) = m \u2022 factorMultiset n\n[PROOFSTEP]\nrw [this, \u2190 PrimeMultiset.prod_smul]\n[GOAL]\nn : \u2115+\nm : \u2115\nu : PrimeMultiset := factorMultiset n\nthis : n = PrimeMultiset.prod u\n\u22a2 factorMultiset (PrimeMultiset.prod (m \u2022 u)) = m \u2022 factorMultiset (PrimeMultiset.prod u)\n[PROOFSTEP]\nrepeat' rw [PrimeMultiset.factorMultiset_prod]\n[GOAL]\nn : \u2115+\nm : \u2115\nu : PrimeMultiset := factorMultiset n\nthis : n = PrimeMultiset.prod u\n\u22a2 factorMultiset (PrimeMultiset.prod (m \u2022 u)) = m \u2022 factorMultiset (PrimeMultiset.prod u)\n[PROOFSTEP]\nrw [PrimeMultiset.factorMultiset_prod]\n[GOAL]\nn : \u2115+\nm : \u2115\nu : PrimeMultiset := factorMultiset n\nthis : n = PrimeMultiset.prod u\n\u22a2 m \u2022 u = m \u2022 factorMultiset (PrimeMultiset.prod u)\n[PROOFSTEP]\nrw [PrimeMultiset.factorMultiset_prod]\n[GOAL]\np : Nat.Primes\n\u22a2 factorMultiset \u2191p = PrimeMultiset.ofPrime p\n[PROOFSTEP]\napply factorMultisetEquiv.symm.injective\n[GOAL]\ncase a\np : Nat.Primes\n\u22a2 \u2191factorMultisetEquiv.symm (factorMultiset \u2191p) = \u2191factorMultisetEquiv.symm (PrimeMultiset.ofPrime p)\n[PROOFSTEP]\nchange (p : \u2115+).factorMultiset.prod = (PrimeMultiset.ofPrime p).prod\n[GOAL]\ncase a\np : Nat.Primes\n\u22a2 PrimeMultiset.prod (factorMultiset \u2191p) = PrimeMultiset.prod (PrimeMultiset.ofPrime p)\n[PROOFSTEP]\nrw [(p : \u2115+).prod_factorMultiset, PrimeMultiset.prod_ofPrime]\n[GOAL]\nm n : \u2115+\n\u22a2 factorMultiset m \u2264 factorMultiset n \u2194 m \u2223 n\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nm n : \u2115+\n\u22a2 factorMultiset m \u2264 factorMultiset n \u2192 m \u2223 n\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nm n : \u2115+\nh : factorMultiset m \u2264 factorMultiset n\n\u22a2 m \u2223 n\n[PROOFSTEP]\nrw [\u2190 prod_factorMultiset m, \u2190 prod_factorMultiset m]\n[GOAL]\ncase mp\nm n : \u2115+\nh : factorMultiset m \u2264 factorMultiset n\n\u22a2 PrimeMultiset.prod (factorMultiset (PrimeMultiset.prod (factorMultiset m))) \u2223 n\n[PROOFSTEP]\napply Dvd.intro (n.factorMultiset - m.factorMultiset).prod\n[GOAL]\ncase mp\nm n : \u2115+\nh : factorMultiset m \u2264 factorMultiset n\n\u22a2 PrimeMultiset.prod (factorMultiset (PrimeMultiset.prod (factorMultiset m))) *\n PrimeMultiset.prod (factorMultiset n - factorMultiset m) =\n n\n[PROOFSTEP]\nrw [\u2190 PrimeMultiset.prod_add, PrimeMultiset.factorMultiset_prod, add_tsub_cancel_of_le h, prod_factorMultiset]\n[GOAL]\ncase mpr\nm n : \u2115+\n\u22a2 m \u2223 n \u2192 factorMultiset m \u2264 factorMultiset n\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nm n : \u2115+\nh : m \u2223 n\n\u22a2 factorMultiset m \u2264 factorMultiset n\n[PROOFSTEP]\nrw [\u2190 mul_div_exact h, factorMultiset_mul]\n[GOAL]\ncase mpr\nm n : \u2115+\nh : m \u2223 n\n\u22a2 factorMultiset m \u2264 factorMultiset m + factorMultiset (divExact n m)\n[PROOFSTEP]\nexact le_self_add\n[GOAL]\nm : \u2115+\nv : PrimeMultiset\n\u22a2 factorMultiset m \u2264 v \u2194 m \u2223 PrimeMultiset.prod v\n[PROOFSTEP]\nlet h := @factorMultiset_le_iff m v.prod\n[GOAL]\nm : \u2115+\nv : PrimeMultiset\nh : factorMultiset m \u2264 factorMultiset (PrimeMultiset.prod v) \u2194 m \u2223 PrimeMultiset.prod v := factorMultiset_le_iff\n\u22a2 factorMultiset m \u2264 v \u2194 m \u2223 PrimeMultiset.prod v\n[PROOFSTEP]\nrw [v.factorMultiset_prod] at h \n[GOAL]\nm : \u2115+\nv : PrimeMultiset\nh : factorMultiset m \u2264 v \u2194 m \u2223 PrimeMultiset.prod v\n\u22a2 factorMultiset m \u2264 v \u2194 m \u2223 PrimeMultiset.prod v\n[PROOFSTEP]\nexact h\n[GOAL]\nu v : PrimeMultiset\n\u22a2 prod u \u2223 prod v \u2194 u \u2264 v\n[PROOFSTEP]\nlet h := @PNat.factorMultiset_le_iff' u.prod v\n[GOAL]\nu v : PrimeMultiset\nh : PNat.factorMultiset (prod u) \u2264 v \u2194 prod u \u2223 prod v := PNat.factorMultiset_le_iff'\n\u22a2 prod u \u2223 prod v \u2194 u \u2264 v\n[PROOFSTEP]\nrw [u.factorMultiset_prod] at h \n[GOAL]\nu v : PrimeMultiset\nh : u \u2264 v \u2194 prod u \u2223 prod v\n\u22a2 prod u \u2223 prod v \u2194 u \u2264 v\n[PROOFSTEP]\nexact h.symm\n[GOAL]\nu : PrimeMultiset\nn : \u2115+\n\u22a2 prod u \u2223 n \u2194 u \u2264 PNat.factorMultiset n\n[PROOFSTEP]\nlet h := @prod_dvd_iff u n.factorMultiset\n[GOAL]\nu : PrimeMultiset\nn : \u2115+\nh : prod u \u2223 prod (PNat.factorMultiset n) \u2194 u \u2264 PNat.factorMultiset n := prod_dvd_iff\n\u22a2 prod u \u2223 n \u2194 u \u2264 PNat.factorMultiset n\n[PROOFSTEP]\nrw [n.prod_factorMultiset] at h \n[GOAL]\nu : PrimeMultiset\nn : \u2115+\nh : prod u \u2223 n \u2194 u \u2264 PNat.factorMultiset n\n\u22a2 prod u \u2223 n \u2194 u \u2264 PNat.factorMultiset n\n[PROOFSTEP]\nexact h\n[GOAL]\nm n : \u2115+\n\u22a2 factorMultiset (gcd m n) = factorMultiset m \u2293 factorMultiset n\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nm n : \u2115+\n\u22a2 factorMultiset (gcd m n) \u2264 factorMultiset m \u2293 factorMultiset n\n[PROOFSTEP]\napply le_inf_iff.mpr\n[GOAL]\ncase a\nm n : \u2115+\n\u22a2 factorMultiset (gcd m n) \u2264 factorMultiset m \u2227 factorMultiset (gcd m n) \u2264 factorMultiset n\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.left\nm n : \u2115+\n\u22a2 factorMultiset (gcd m n) \u2264 factorMultiset m\n[PROOFSTEP]\napply factorMultiset_le_iff.mpr\n[GOAL]\ncase a.right\nm n : \u2115+\n\u22a2 factorMultiset (gcd m n) \u2264 factorMultiset n\n[PROOFSTEP]\napply factorMultiset_le_iff.mpr\n[GOAL]\ncase a.left\nm n : \u2115+\n\u22a2 gcd m n \u2223 m\ncase a.right m n : \u2115+ \u22a2 gcd m n \u2223 n\n[PROOFSTEP]\nexact gcd_dvd_left m n\n[GOAL]\ncase a.right\nm n : \u2115+\n\u22a2 gcd m n \u2223 n\n[PROOFSTEP]\nexact gcd_dvd_right m n\n[GOAL]\ncase a\nm n : \u2115+\n\u22a2 factorMultiset m \u2293 factorMultiset n \u2264 factorMultiset (gcd m n)\n[PROOFSTEP]\nrw [\u2190 PrimeMultiset.prod_dvd_iff, prod_factorMultiset]\n[GOAL]\ncase a\nm n : \u2115+\n\u22a2 PrimeMultiset.prod (factorMultiset m \u2293 factorMultiset n) \u2223 gcd m n\n[PROOFSTEP]\napply dvd_gcd\n[GOAL]\ncase a.hm\nm n : \u2115+\n\u22a2 PrimeMultiset.prod (factorMultiset m \u2293 factorMultiset n) \u2223 m\n[PROOFSTEP]\nrw [PrimeMultiset.prod_dvd_iff']\n[GOAL]\ncase a.hn\nm n : \u2115+\n\u22a2 PrimeMultiset.prod (factorMultiset m \u2293 factorMultiset n) \u2223 n\n[PROOFSTEP]\nrw [PrimeMultiset.prod_dvd_iff']\n[GOAL]\ncase a.hm\nm n : \u2115+\n\u22a2 factorMultiset m \u2293 factorMultiset n \u2264 factorMultiset m\ncase a.hn m n : \u2115+ \u22a2 factorMultiset m \u2293 factorMultiset n \u2264 factorMultiset n\n[PROOFSTEP]\nexact inf_le_left\n[GOAL]\ncase a.hn\nm n : \u2115+\n\u22a2 factorMultiset m \u2293 factorMultiset n \u2264 factorMultiset n\n[PROOFSTEP]\nexact inf_le_right\n[GOAL]\nm n : \u2115+\n\u22a2 factorMultiset (lcm m n) = factorMultiset m \u2294 factorMultiset n\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nm n : \u2115+\n\u22a2 factorMultiset (lcm m n) \u2264 factorMultiset m \u2294 factorMultiset n\n[PROOFSTEP]\nrw [\u2190 PrimeMultiset.prod_dvd_iff, prod_factorMultiset]\n[GOAL]\ncase a\nm n : \u2115+\n\u22a2 lcm m n \u2223 PrimeMultiset.prod (factorMultiset m \u2294 factorMultiset n)\n[PROOFSTEP]\napply lcm_dvd\n[GOAL]\ncase a.hm\nm n : \u2115+\n\u22a2 m \u2223 PrimeMultiset.prod (factorMultiset m \u2294 factorMultiset n)\n[PROOFSTEP]\nrw [\u2190 factorMultiset_le_iff']\n[GOAL]\ncase a.hn\nm n : \u2115+\n\u22a2 n \u2223 PrimeMultiset.prod (factorMultiset m \u2294 factorMultiset n)\n[PROOFSTEP]\nrw [\u2190 factorMultiset_le_iff']\n[GOAL]\ncase a.hm\nm n : \u2115+\n\u22a2 factorMultiset m \u2264 factorMultiset m \u2294 factorMultiset n\ncase a.hn m n : \u2115+ \u22a2 factorMultiset n \u2264 factorMultiset m \u2294 factorMultiset n\n[PROOFSTEP]\nexact le_sup_left\n[GOAL]\ncase a.hn\nm n : \u2115+\n\u22a2 factorMultiset n \u2264 factorMultiset m \u2294 factorMultiset n\n[PROOFSTEP]\nexact le_sup_right\n[GOAL]\ncase a\nm n : \u2115+\n\u22a2 factorMultiset m \u2294 factorMultiset n \u2264 factorMultiset (lcm m n)\n[PROOFSTEP]\napply sup_le_iff.mpr\n[GOAL]\ncase a\nm n : \u2115+\n\u22a2 factorMultiset m \u2264 factorMultiset (lcm m n) \u2227 factorMultiset n \u2264 factorMultiset (lcm m n)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.left\nm n : \u2115+\n\u22a2 factorMultiset m \u2264 factorMultiset (lcm m n)\n[PROOFSTEP]\napply factorMultiset_le_iff.mpr\n[GOAL]\ncase a.right\nm n : \u2115+\n\u22a2 factorMultiset n \u2264 factorMultiset (lcm m n)\n[PROOFSTEP]\napply factorMultiset_le_iff.mpr\n[GOAL]\ncase a.left\nm n : \u2115+\n\u22a2 m \u2223 lcm m n\ncase a.right m n : \u2115+ \u22a2 n \u2223 lcm m n\n[PROOFSTEP]\nexact dvd_lcm_left m n\n[GOAL]\ncase a.right\nm n : \u2115+\n\u22a2 n \u2223 lcm m n\n[PROOFSTEP]\nexact dvd_lcm_right m n\n[GOAL]\nm : \u2115+\np : Nat.Primes\nk : \u2115\n\u22a2 Pow.pow (\u2191p) k \u2223 m \u2194 k \u2264 Multiset.count p (factorMultiset m)\n[PROOFSTEP]\nintros\n[GOAL]\nm : \u2115+\np : Nat.Primes\nk : \u2115\n\u22a2 Pow.pow (\u2191p) k \u2223 m \u2194 k \u2264 Multiset.count p (factorMultiset m)\n[PROOFSTEP]\nrw [Multiset.le_count_iff_replicate_le, \u2190 factorMultiset_le_iff, factorMultiset_pow, factorMultiset_ofPrime]\n[GOAL]\nm : \u2115+\np : Nat.Primes\nk : \u2115\n\u22a2 k \u2022 PrimeMultiset.ofPrime p \u2264 factorMultiset m \u2194 Multiset.replicate k p \u2264 factorMultiset m\n[PROOFSTEP]\ncongr! 2\n[GOAL]\ncase a.h.e'_3.h\nm : \u2115+\np : Nat.Primes\nk : \u2115\ne_1\u271d : PrimeMultiset = Multiset Nat.Primes\n\u22a2 k \u2022 PrimeMultiset.ofPrime p = Multiset.replicate k p\n[PROOFSTEP]\napply Multiset.eq_replicate.mpr\n[GOAL]\ncase a.h.e'_3.h\nm : \u2115+\np : Nat.Primes\nk : \u2115\ne_1\u271d : PrimeMultiset = Multiset Nat.Primes\n\u22a2 \u2191Multiset.card (k \u2022 PrimeMultiset.ofPrime p) = k \u2227 \u2200 (b : Nat.Primes), b \u2208 k \u2022 PrimeMultiset.ofPrime p \u2192 b = p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.h.e'_3.h.left\nm : \u2115+\np : Nat.Primes\nk : \u2115\ne_1\u271d : PrimeMultiset = Multiset Nat.Primes\n\u22a2 \u2191Multiset.card (k \u2022 PrimeMultiset.ofPrime p) = k\n[PROOFSTEP]\nrw [Multiset.card_nsmul, PrimeMultiset.card_ofPrime, mul_one]\n[GOAL]\ncase a.h.e'_3.h.right\nm : \u2115+\np : Nat.Primes\nk : \u2115\ne_1\u271d : PrimeMultiset = Multiset Nat.Primes\n\u22a2 \u2200 (b : Nat.Primes), b \u2208 k \u2022 PrimeMultiset.ofPrime p \u2192 b = p\n[PROOFSTEP]\nintro q h\n[GOAL]\ncase a.h.e'_3.h.right\nm : \u2115+\np : Nat.Primes\nk : \u2115\ne_1\u271d : PrimeMultiset = Multiset Nat.Primes\nq : Nat.Primes\nh : q \u2208 k \u2022 PrimeMultiset.ofPrime p\n\u22a2 q = p\n[PROOFSTEP]\nrw [PrimeMultiset.ofPrime, Multiset.nsmul_singleton _ k] at h \n[GOAL]\ncase a.h.e'_3.h.right\nm : \u2115+\np : Nat.Primes\nk : \u2115\ne_1\u271d : PrimeMultiset = Multiset Nat.Primes\nq : Nat.Primes\nh : q \u2208 Multiset.replicate k p\n\u22a2 q = p\n[PROOFSTEP]\nexact Multiset.eq_of_mem_replicate h\n[GOAL]\nu v : PrimeMultiset\n\u22a2 prod (u \u2293 v) = PNat.gcd (prod u) (prod v)\n[PROOFSTEP]\nlet n := u.prod\n[GOAL]\nu v : PrimeMultiset\nn : \u2115+ := prod u\n\u22a2 prod (u \u2293 v) = PNat.gcd (prod u) (prod v)\n[PROOFSTEP]\nlet m := v.prod\n[GOAL]\nu v : PrimeMultiset\nn : \u2115+ := prod u\nm : \u2115+ := prod v\n\u22a2 prod (u \u2293 v) = PNat.gcd (prod u) (prod v)\n[PROOFSTEP]\nchange (u \u2293 v).prod = PNat.gcd n m\n[GOAL]\nu v : PrimeMultiset\nn : \u2115+ := prod u\nm : \u2115+ := prod v\n\u22a2 prod (u \u2293 v) = PNat.gcd n m\n[PROOFSTEP]\nhave : u = n.factorMultiset := u.factorMultiset_prod.symm\n[GOAL]\nu v : PrimeMultiset\nn : \u2115+ := prod u\nm : \u2115+ := prod v\nthis : u = PNat.factorMultiset n\n\u22a2 prod (u \u2293 v) = PNat.gcd n m\n[PROOFSTEP]\nrw [this]\n[GOAL]\nu v : PrimeMultiset\nn : \u2115+ := prod u\nm : \u2115+ := prod v\nthis : u = PNat.factorMultiset n\n\u22a2 prod (PNat.factorMultiset n \u2293 v) = PNat.gcd n m\n[PROOFSTEP]\nhave : v = m.factorMultiset := v.factorMultiset_prod.symm\n[GOAL]\nu v : PrimeMultiset\nn : \u2115+ := prod u\nm : \u2115+ := prod v\nthis\u271d : u = PNat.factorMultiset n\nthis : v = PNat.factorMultiset m\n\u22a2 prod (PNat.factorMultiset n \u2293 v) = PNat.gcd n m\n[PROOFSTEP]\nrw [this]\n[GOAL]\nu v : PrimeMultiset\nn : \u2115+ := prod u\nm : \u2115+ := prod v\nthis\u271d : u = PNat.factorMultiset n\nthis : v = PNat.factorMultiset m\n\u22a2 prod (PNat.factorMultiset n \u2293 PNat.factorMultiset m) = PNat.gcd n m\n[PROOFSTEP]\nrw [\u2190 PNat.factorMultiset_gcd n m, PNat.prod_factorMultiset]\n[GOAL]\nu v : PrimeMultiset\n\u22a2 prod (u \u2294 v) = PNat.lcm (prod u) (prod v)\n[PROOFSTEP]\nlet n := u.prod\n[GOAL]\nu v : PrimeMultiset\nn : \u2115+ := prod u\n\u22a2 prod (u \u2294 v) = PNat.lcm (prod u) (prod v)\n[PROOFSTEP]\nlet m := v.prod\n[GOAL]\nu v : PrimeMultiset\nn : \u2115+ := prod u\nm : \u2115+ := prod v\n\u22a2 prod (u \u2294 v) = PNat.lcm (prod u) (prod v)\n[PROOFSTEP]\nchange (u \u2294 v).prod = PNat.lcm n m\n[GOAL]\nu v : PrimeMultiset\nn : \u2115+ := prod u\nm : \u2115+ := prod v\n\u22a2 prod (u \u2294 v) = PNat.lcm n m\n[PROOFSTEP]\nhave : u = n.factorMultiset := u.factorMultiset_prod.symm\n[GOAL]\nu v : PrimeMultiset\nn : \u2115+ := prod u\nm : \u2115+ := prod v\nthis : u = PNat.factorMultiset n\n\u22a2 prod (u \u2294 v) = PNat.lcm n m\n[PROOFSTEP]\nrw [this]\n[GOAL]\nu v : PrimeMultiset\nn : \u2115+ := prod u\nm : \u2115+ := prod v\nthis : u = PNat.factorMultiset n\n\u22a2 prod (PNat.factorMultiset n \u2294 v) = PNat.lcm n m\n[PROOFSTEP]\nhave : v = m.factorMultiset := v.factorMultiset_prod.symm\n[GOAL]\nu v : PrimeMultiset\nn : \u2115+ := prod u\nm : \u2115+ := prod v\nthis\u271d : u = PNat.factorMultiset n\nthis : v = PNat.factorMultiset m\n\u22a2 prod (PNat.factorMultiset n \u2294 v) = PNat.lcm n m\n[PROOFSTEP]\nrw [this]\n[GOAL]\nu v : PrimeMultiset\nn : \u2115+ := prod u\nm : \u2115+ := prod v\nthis\u271d : u = PNat.factorMultiset n\nthis : v = PNat.factorMultiset m\n\u22a2 prod (PNat.factorMultiset n \u2294 PNat.factorMultiset m) = PNat.lcm n m\n[PROOFSTEP]\nrw [\u2190 PNat.factorMultiset_lcm n m, PNat.prod_factorMultiset]\n", "meta": {"mathlib_filename": "Mathlib.Data.PNat.Factors", "llama_tokens": 12269, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.640635868562172, "lm_q1q2_score": 0.4479674633809165}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\n\u22a2 AntilipschitzWith K f \u2194 \u2200 (x y : \u03b1), nndist x y \u2264 K * nndist (f x) (f y)\n[PROOFSTEP]\nsimp only [AntilipschitzWith, edist_nndist]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 (x y : \u03b1), \u2191(nndist x y) \u2264 \u2191K * \u2191(nndist (f x) (f y))) \u2194 \u2200 (x y : \u03b1), nndist x y \u2264 K * nndist (f x) (f y)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\n\u22a2 AntilipschitzWith K f \u2194 \u2200 (x y : \u03b1), dist x y \u2264 \u2191K * dist (f x) (f y)\n[PROOFSTEP]\nsimp only [antilipschitzWith_iff_le_mul_nndist, dist_nndist]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 (x y : \u03b1), nndist x y \u2264 K * nndist (f x) (f y)) \u2194 \u2200 (x y : \u03b1), \u2191(nndist x y) \u2264 \u2191K * \u2191(nndist (f x) (f y))\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\nx y : \u03b1\n\u22a2 K\u207b\u00b9 * nndist x y \u2264 nndist (f x) (f y)\n[PROOFSTEP]\nsimpa only [div_eq_inv_mul] using NNReal.div_le_of_le_mul' (hf.le_mul_nndist x y)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\nx y : \u03b1\n\u22a2 \u2191K\u207b\u00b9 * dist x y \u2264 dist (f x) (f y)\n[PROOFSTEP]\nexact_mod_cast hf.mul_le_nndist x y\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : PseudoEMetricSpace \u03b1\u271d\ninst\u271d\u00b3 : PseudoEMetricSpace \u03b2\u271d\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf\u271d : \u03b1\u271d \u2192 \u03b2\u271d\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : EMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\nx y : \u03b1\nh : f x = f y\n\u22a2 x = y\n[PROOFSTEP]\nsimpa only [h, edist_self, mul_zero, edist_le_zero] using hf x y\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\nx y : \u03b1\n\u22a2 (\u2191K)\u207b\u00b9 * edist x y \u2264 edist (f x) (f y)\n[PROOFSTEP]\nrw [mul_comm, \u2190 div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\nx y : \u03b1\n\u22a2 edist x y / \u2191K \u2264 edist (f x) (f y)\n[PROOFSTEP]\nexact ENNReal.div_le_of_le_mul' (hf x y)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\n\u22a2 edist x y \u2264 \u21911 * edist (id x) (id y)\n[PROOFSTEP]\nsimp only [ENNReal.coe_one, one_mul, id, le_refl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nKg : \u211d\u22650\ng : \u03b2 \u2192 \u03b3\nhg : AntilipschitzWith Kg g\nKf : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith Kf f\nx y : \u03b1\n\u22a2 \u2191Kf * (\u2191Kg * edist (g (f x)) (g (f y))) = \u2191(Kf * Kg) * edist ((g \u2218 f) x) ((g \u2218 f) y)\n[PROOFSTEP]\nrw [ENNReal.coe_mul, mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nKg : \u211d\u22650\ng : \u03b2 \u2192 \u03b3\nhg : AntilipschitzWith Kg g\nKf : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith Kf f\nx y : \u03b1\n\u22a2 \u2191Kf * (\u2191Kg * edist (g (f x)) (g (f y))) = \u2191Kf * (\u2191Kg * edist ((g \u2218 f) x) ((g \u2218 f) y))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhf : AntilipschitzWith K (Set.restrict s f)\ng : \u03b2 \u2192 \u03b1\nt : Set \u03b2\ng_maps : MapsTo g t s\ng_inv : RightInvOn g f t\nx y : \u2191t\n\u22a2 edist (Set.restrict t g x) (Set.restrict t g y) \u2264 \u2191K * edist x y\n[PROOFSTEP]\nsimpa only [restrict_apply, g_inv x.mem, g_inv y.mem, Subtype.edist_eq, Subtype.coe_mk] using\n hf \u27e8g x, g_maps x.mem\u27e9 \u27e8g y, g_maps y.mem\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\ng : \u03b2 \u2192 \u03b1\nhg : Function.RightInverse g f\n\u22a2 LipschitzWith K g\n[PROOFSTEP]\nintro x y\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\ng : \u03b2 \u2192 \u03b1\nhg : Function.RightInverse g f\nx y : \u03b2\n\u22a2 edist (g x) (g y) \u2264 \u2191K * edist x y\n[PROOFSTEP]\nhave := hf (g x) (g y)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\ng : \u03b2 \u2192 \u03b1\nhg : Function.RightInverse g f\nx y : \u03b2\nthis : edist (g x) (g y) \u2264 \u2191K * edist (f (g x)) (f (g y))\n\u22a2 edist (g x) (g y) \u2264 \u2191K * edist x y\n[PROOFSTEP]\nrwa [hg x, hg y] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\n\u22a2 comap (Prod.map f f) (\ud835\udce4 \u03b2) \u2264 \ud835\udce4 \u03b1\n[PROOFSTEP]\nrefine ((uniformity_basis_edist.comap _).le_basis_iff uniformity_basis_edist).2 fun \u03b5 h\u2080 => ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\n\u03b5 : \u211d\u22650\u221e\nh\u2080 : 0 < \u03b5\n\u22a2 \u2203 i, 0 < i \u2227 Prod.map f f \u207b\u00b9' {p | edist p.fst p.snd < i} \u2286 {p | edist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nrefine \u27e8(\u2191K)\u207b\u00b9 * \u03b5, ENNReal.mul_pos (ENNReal.inv_ne_zero.2 ENNReal.coe_ne_top) h\u2080.ne', ?_\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\n\u03b5 : \u211d\u22650\u221e\nh\u2080 : 0 < \u03b5\n\u22a2 Prod.map f f \u207b\u00b9' {p | edist p.fst p.snd < (\u2191K)\u207b\u00b9 * \u03b5} \u2286 {p | edist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nrefine' fun x hx => (hf x.1 x.2).trans_lt _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\n\u03b5 : \u211d\u22650\u221e\nh\u2080 : 0 < \u03b5\nx : \u03b1 \u00d7 \u03b1\nhx : x \u2208 Prod.map f f \u207b\u00b9' {p | edist p.fst p.snd < (\u2191K)\u207b\u00b9 * \u03b5}\n\u22a2 \u2191K * edist (f x.fst) (f x.snd) < \u03b5\n[PROOFSTEP]\nrw [mul_comm, \u2190 div_eq_mul_inv] at hx \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\n\u03b5 : \u211d\u22650\u221e\nh\u2080 : 0 < \u03b5\nx : \u03b1 \u00d7 \u03b1\nhx : x \u2208 Prod.map f f \u207b\u00b9' {p | edist p.fst p.snd < \u03b5 / \u2191K}\n\u22a2 \u2191K * edist (f x.fst) (f x.snd) < \u03b5\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\n\u03b5 : \u211d\u22650\u221e\nh\u2080 : 0 < \u03b5\nx : \u03b1 \u00d7 \u03b1\nhx : x \u2208 Prod.map f f \u207b\u00b9' {p | edist p.fst p.snd < \u03b5 / \u2191K}\n\u22a2 edist (f x.fst) (f x.snd) * \u2191K < \u03b5\n[PROOFSTEP]\nexact ENNReal.mul_lt_of_lt_div hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b2\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Subsingleton \u03b1\nK : \u211d\u22650\nx y : \u03b1\n\u22a2 edist x y \u2264 \u2191K * edist (f x) (f y)\n[PROOFSTEP]\nsimp only [Subsingleton.elim x y, edist_self, zero_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\ns : Set \u03b2\nhs : Metric.Bounded s\nx : \u03b1\nhx : x \u2208 f \u207b\u00b9' s\ny : \u03b1\nhy : y \u2208 f \u207b\u00b9' s\n\u22a2 \u2191K * dist (f x) (f y) \u2264 \u2191K * diam s\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith K f\ns : Set \u03b2\nhs : Metric.Bounded s\nx : \u03b1\nhx : x \u2208 f \u207b\u00b9' s\ny : \u03b1\nhy : y \u2208 f \u207b\u00b9' s\n\u22a2 dist (f x) (f y) \u2264 diam s\n[PROOFSTEP]\nexact dist_le_diam_of_mem hs hx hy\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\u271d\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\nK\u271d : \u211d\u22650\nf\u271d : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_4\ninst\u271d\u00b9 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\ninst\u271d : ProperSpace \u03b1\nhK : AntilipschitzWith K f\nf_cont : Continuous f\nhf : Function.Surjective f\n\u22a2 ProperSpace \u03b2\n[PROOFSTEP]\nrefine \u27e8fun x\u2080 r => ?_\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\u271d\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\nK\u271d : \u211d\u22650\nf\u271d : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_4\ninst\u271d\u00b9 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\ninst\u271d : ProperSpace \u03b1\nhK : AntilipschitzWith K f\nf_cont : Continuous f\nhf : Function.Surjective f\nx\u2080 : \u03b2\nr : \u211d\n\u22a2 IsCompact (closedBall x\u2080 r)\n[PROOFSTEP]\nlet K := f \u207b\u00b9' closedBall x\u2080 r\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\u271d\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\nK\u271d\u00b9 : \u211d\u22650\nf\u271d : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_4\ninst\u271d\u00b9 : MetricSpace \u03b1\nK\u271d : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\ninst\u271d : ProperSpace \u03b1\nhK : AntilipschitzWith K\u271d f\nf_cont : Continuous f\nhf : Function.Surjective f\nx\u2080 : \u03b2\nr : \u211d\nK : Set \u03b1 := f \u207b\u00b9' closedBall x\u2080 r\n\u22a2 IsCompact (closedBall x\u2080 r)\n[PROOFSTEP]\nhave A : IsClosed K := isClosed_ball.preimage f_cont\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\u271d\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\nK\u271d\u00b9 : \u211d\u22650\nf\u271d : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_4\ninst\u271d\u00b9 : MetricSpace \u03b1\nK\u271d : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\ninst\u271d : ProperSpace \u03b1\nhK : AntilipschitzWith K\u271d f\nf_cont : Continuous f\nhf : Function.Surjective f\nx\u2080 : \u03b2\nr : \u211d\nK : Set \u03b1 := f \u207b\u00b9' closedBall x\u2080 r\nA : IsClosed K\n\u22a2 IsCompact (closedBall x\u2080 r)\n[PROOFSTEP]\nhave B : Bounded K := hK.bounded_preimage bounded_closedBall\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\u271d\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\nK\u271d\u00b9 : \u211d\u22650\nf\u271d : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_4\ninst\u271d\u00b9 : MetricSpace \u03b1\nK\u271d : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\ninst\u271d : ProperSpace \u03b1\nhK : AntilipschitzWith K\u271d f\nf_cont : Continuous f\nhf : Function.Surjective f\nx\u2080 : \u03b2\nr : \u211d\nK : Set \u03b1 := f \u207b\u00b9' closedBall x\u2080 r\nA : IsClosed K\nB : Metric.Bounded K\n\u22a2 IsCompact (closedBall x\u2080 r)\n[PROOFSTEP]\nhave : IsCompact K := isCompact_iff_isClosed_bounded.2 \u27e8A, B\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\u271d\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\nK\u271d\u00b9 : \u211d\u22650\nf\u271d : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_4\ninst\u271d\u00b9 : MetricSpace \u03b1\nK\u271d : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\ninst\u271d : ProperSpace \u03b1\nhK : AntilipschitzWith K\u271d f\nf_cont : Continuous f\nhf : Function.Surjective f\nx\u2080 : \u03b2\nr : \u211d\nK : Set \u03b1 := f \u207b\u00b9' closedBall x\u2080 r\nA : IsClosed K\nB : Metric.Bounded K\nthis : IsCompact K\n\u22a2 IsCompact (closedBall x\u2080 r)\n[PROOFSTEP]\nconvert this.image f_cont\n[GOAL]\ncase h.e'_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\u271d\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\nK\u271d\u00b9 : \u211d\u22650\nf\u271d : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_4\ninst\u271d\u00b9 : MetricSpace \u03b1\nK\u271d : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\ninst\u271d : ProperSpace \u03b1\nhK : AntilipschitzWith K\u271d f\nf_cont : Continuous f\nhf : Function.Surjective f\nx\u2080 : \u03b2\nr : \u211d\nK : Set \u03b1 := f \u207b\u00b9' closedBall x\u2080 r\nA : IsClosed K\nB : Metric.Bounded K\nthis : IsCompact K\n\u22a2 closedBall x\u2080 r = f '' K\n[PROOFSTEP]\nexact (hf.image_preimage _).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : LipschitzWith K f\ng : \u03b2 \u2192 \u03b1\nhg : Function.RightInverse g f\nx y : \u03b2\n\u22a2 edist x y \u2264 \u2191K * edist (g x) (g y)\n[PROOFSTEP]\nsimpa only [hg _] using hf (g x) (g y)\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.Antilipschitz", "llama_tokens": 6144, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581741774411, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.4479438727888907}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na : \u03b1\n\u22a2 \u2191p a = 0 \u2194 \u00aca \u2208 support p\n[PROOFSTEP]\nrw [mem_support_iff, Classical.not_not]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na : \u03b1\n\u22a2 \u2191p a = 1 \u2194 support p = {a}\n[PROOFSTEP]\nrefine'\n \u27e8fun h =>\n Set.Subset.antisymm (fun a' ha' => by_contra fun ha => _) fun a' ha' =>\n ha'.symm \u25b8 (p.mem_support_iff a).2 fun ha => zero_ne_one <| ha.symm.trans h,\n fun h => _root_.trans (symm <| tsum_eq_single a fun a' ha' => (p.apply_eq_zero_iff a').2 (h.symm \u25b8 ha')) p.tsum_coe\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\n\u22a2 False\n[PROOFSTEP]\nsuffices : 1 < \u2211' a, p a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 1 < \u2211' (a : \u03b1), \u2191p a\n\u22a2 False\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\n\u22a2 1 < \u2211' (a : \u03b1), \u2191p a\n[PROOFSTEP]\nexact ne_of_lt this p.tsum_coe.symm\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\n\u22a2 1 < \u2211' (a : \u03b1), \u2191p a\n[PROOFSTEP]\nhave : 0 < \u2211' b, ite (b = a) 0 (p b) :=\n lt_of_le_of_ne' zero_le'\n ((tsum_ne_zero_iff ENNReal.summable).2 \u27e8a', ite_ne_left_iff.2 \u27e8ha, Ne.symm <| (p.mem_support_iff a').2 ha'\u27e9\u27e9)\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\n\u22a2 1 < \u2211' (a : \u03b1), \u2191p a\n[PROOFSTEP]\ncalc\n 1 = 1 + 0 := (add_zero 1).symm\n _ < p a + \u2211' b, ite (b = a) 0 (p b) := (ENNReal.add_lt_add_of_le_of_lt ENNReal.one_ne_top (le_of_eq h.symm) this)\n _ = ite (a = a) (p a) 0 + \u2211' b, ite (b = a) 0 (p b) := by rw [eq_self_iff_true, if_true]\n _ = (\u2211' b, ite (b = a) (p b) 0) + \u2211' b, ite (b = a) 0 (p b) :=\n by\n congr\n exact symm (tsum_eq_single a fun b hb => if_neg hb)\n _ = \u2211' b, (ite (b = a) (p b) 0 + ite (b = a) 0 (p b)) := ENNReal.tsum_add.symm\n _ = \u2211' b, p b := tsum_congr fun b => by split_ifs <;> simp only [zero_add, add_zero, le_rfl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\n\u22a2 (\u2191p a + \u2211' (b : \u03b1), if b = a then 0 else \u2191p b) = (if a = a then \u2191p a else 0) + \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\n[PROOFSTEP]\nrw [eq_self_iff_true, if_true]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\n\u22a2 ((if a = a then \u2191p a else 0) + \u2211' (b : \u03b1), if b = a then 0 else \u2191p b) =\n (\u2211' (b : \u03b1), if b = a then \u2191p b else 0) + \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\n\u22a2 (if a = a then \u2191p a else 0) = \u2211' (b : \u03b1), if b = a then \u2191p b else 0\n[PROOFSTEP]\nexact symm (tsum_eq_single a fun b hb => if_neg hb)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\nb : \u03b1\n\u22a2 ((if b = a then \u2191p b else 0) + if b = a then 0 else \u2191p b) = \u2191p b\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\nb : \u03b1\nh\u271d : b = a\n\u22a2 \u2191p b + 0 = \u2191p b\n[PROOFSTEP]\nsimp only [zero_add, add_zero, le_rfl]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na : \u03b1\nh : \u2191p a = 1\na' : \u03b1\nha' : a' \u2208 support p\nha : \u00aca' \u2208 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b\nb : \u03b1\nh\u271d : \u00acb = a\n\u22a2 0 + \u2191p b = \u2191p b\n[PROOFSTEP]\nsimp only [zero_add, add_zero, le_rfl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na : \u03b1\n\u22a2 \u2191p a \u2264 1\n[PROOFSTEP]\nrefine' hasSum_le (fun b => _) (hasSum_ite_eq a (p a)) (hasSum_coe_one p)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na b : \u03b1\n\u22a2 (if b = a then \u2191p a else 0) \u2264 \u2191p b\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na b : \u03b1\nh : b = a\n\u22a2 \u2191p a \u2264 \u2191p b\n[PROOFSTEP]\nsimp only [h, zero_le', le_rfl]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\na b : \u03b1\nh : \u00acb = a\n\u22a2 0 \u2264 \u2191p b\n[PROOFSTEP]\nsimp only [h, zero_le', le_rfl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\n\u22a2 OuterMeasure.caratheodory (toOuterMeasure p) = \u22a4\n[PROOFSTEP]\nrefine' eq_top_iff.2 <| le_trans (le_sInf fun x hx => _) (le_sum_caratheodory _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\nx : MeasurableSpace \u03b1\nhx : x \u2208 Set.range fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)\n\u22a2 \u22a4 \u2264 x\n[PROOFSTEP]\nhave \u27e8y, hy\u27e9 := hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\nx : MeasurableSpace \u03b1\nhx : x \u2208 Set.range fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)\ny : \u03b1\nhy : (fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)) y = x\n\u22a2 \u22a4 \u2264 x\n[PROOFSTEP]\nexact ((le_of_eq (dirac_caratheodory y).symm).trans (le_smul_caratheodory _ _)).trans (le_of_eq hy)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns\u271d t : Set \u03b1\ns : Finset \u03b1\n\u22a2 \u2191(toOuterMeasure p) \u2191s = \u2211 x in s, \u2191p x\n[PROOFSTEP]\nrefine' (toOuterMeasure_apply p s).trans ((tsum_eq_sum (s := s) _).trans _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns\u271d t : Set \u03b1\ns : Finset \u03b1\n\u22a2 \u2200 (b : \u03b1), \u00acb \u2208 s \u2192 Set.indicator (\u2191s) (\u2191p) b = 0\n[PROOFSTEP]\nexact fun x hx => Set.indicator_of_not_mem (Finset.mem_coe.not.2 hx) _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns\u271d t : Set \u03b1\ns : Finset \u03b1\n\u22a2 \u2211 b in s, Set.indicator (\u2191s) (\u2191p) b = \u2211 x in s, \u2191p x\n[PROOFSTEP]\nexact Finset.sum_congr rfl fun x hx => Set.indicator_of_mem (Finset.mem_coe.2 hx) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 \u2191(toOuterMeasure p) {a} = \u2191p a\n[PROOFSTEP]\nrefine' (p.toOuterMeasure_apply { a }).trans ((tsum_eq_single a fun b hb => _).trans _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2260 a\n\u22a2 Set.indicator {a} (\u2191p) b = 0\n[PROOFSTEP]\nexact ite_eq_right_iff.2 fun hb' => False.elim <| hb hb'\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 Set.indicator {a} (\u2191p) a = \u2191p a\n[PROOFSTEP]\nexact ite_eq_left_iff.2 fun ha' => False.elim <| ha' rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\n\u22a2 \u2191(toOuterMeasure p) s = 0 \u2194 Disjoint (support p) s\n[PROOFSTEP]\nrw [toOuterMeasure_apply, ENNReal.tsum_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\n\u22a2 (\u2200 (i : \u03b1), Set.indicator s (\u2191p) i = 0) \u2194 Disjoint (support p) s\n[PROOFSTEP]\nexact Function.funext_iff.symm.trans Set.indicator_eq_zero'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\n\u22a2 \u2191(toOuterMeasure p) s = 1 \u2194 support p \u2286 s\n[PROOFSTEP]\nrefine' (p.toOuterMeasure_apply s).symm \u25b8 \u27e8fun h a hap => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\na : \u03b1\nhap : a \u2208 support p\n\u22a2 a \u2208 s\n[PROOFSTEP]\nrefine' by_contra fun hs => ne_of_lt _ (h.trans p.tsum_coe.symm)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\na : \u03b1\nhap : a \u2208 support p\nhs : \u00aca \u2208 s\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x < \u2211' (a : \u03b1), \u2191p a\n[PROOFSTEP]\nhave hs' : s.indicator p a = 0 := Set.indicator_apply_eq_zero.2 fun hs' => False.elim <| hs hs'\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\na : \u03b1\nhap : a \u2208 support p\nhs : \u00aca \u2208 s\nhs' : Set.indicator s (\u2191p) a = 0\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x < \u2211' (a : \u03b1), \u2191p a\n[PROOFSTEP]\nhave hsa : s.indicator p a < p a := hs'.symm \u25b8 (p.apply_pos_iff a).2 hap\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\na : \u03b1\nhap : a \u2208 support p\nhs : \u00aca \u2208 s\nhs' : Set.indicator s (\u2191p) a = 0\nhsa : Set.indicator s (\u2191p) a < \u2191p a\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x < \u2211' (a : \u03b1), \u2191p a\n[PROOFSTEP]\nexact ENNReal.tsum_lt_tsum (p.tsum_coe_indicator_ne_top s) (fun x => Set.indicator_apply_le fun _ => le_rfl) hsa\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\nh : support p \u2286 s\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\n[PROOFSTEP]\nsuffices : \u2200 (x) (_ : x \u2209 s), p x = 0\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\nh : support p \u2286 s\nthis : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2191p x = 0\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1\ncase this \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : Pmf \u03b1 s t : Set \u03b1 h : support p \u2286 s \u22a2 \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2191p x = 0\n[PROOFSTEP]\nexact\n _root_.trans (tsum_congr fun a => (Set.indicator_apply s p a).trans (ite_eq_left_iff.2 <| symm \u2218 this a)) p.tsum_coe\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\nh : support p \u2286 s\n\u22a2 \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2191p x = 0\n[PROOFSTEP]\nexact fun a ha => (p.apply_eq_zero_iff a).2 <| Set.not_mem_subset h ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns t : Set \u03b1\n\u22a2 \u2191(toOuterMeasure p) (s \u2229 support p) = \u2191(toOuterMeasure p) s\n[PROOFSTEP]\nsimp only [toOuterMeasure_apply, Pmf.support, Set.indicator_inter_support]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\np : Pmf \u03b1\ns t : Set \u03b1\na : \u03b1\nh : MeasurableSet {a}\n\u22a2 \u2191\u2191(toMeasure p) {a} = \u2191p a\n[PROOFSTEP]\nsimp [toMeasure_apply_eq_toOuterMeasure_apply _ _ h, toOuterMeasure_apply_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\np : Pmf \u03b1\ns t : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(toMeasure p) s = 0 \u2194 Disjoint (support p) s\n[PROOFSTEP]\nrw [toMeasure_apply_eq_toOuterMeasure_apply p s hs, toOuterMeasure_apply_eq_zero_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\np : Pmf \u03b1\ns t : Set \u03b1\nhs : MeasurableSet s\nhp : MeasurableSet (support p)\n\u22a2 \u2191\u2191(toMeasure p) (s \u2229 support p) = \u2191\u2191(toMeasure p) s\n[PROOFSTEP]\nsimp [p.toMeasure_apply_eq_toOuterMeasure_apply s hs, p.toMeasure_apply_eq_toOuterMeasure_apply _ (hs.inter hp)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\np : Pmf \u03b1\ns\u271d t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nh : s \u2229 support p \u2286 t\n\u22a2 \u2191\u2191(toMeasure p) s \u2264 \u2191\u2191(toMeasure p) t\n[PROOFSTEP]\nsimpa only [p.toMeasure_apply_eq_toOuterMeasure_apply, hs, ht] using toOuterMeasure_mono p h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\np : Pmf \u03b1\ns\u271d t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nh : s \u2229 support p = t \u2229 support p\n\u22a2 \u2191\u2191(toMeasure p) s = \u2191\u2191(toMeasure p) t\n[PROOFSTEP]\nsimpa only [p.toMeasure_apply_eq_toOuterMeasure_apply, hs, ht] using toOuterMeasure_apply_eq_of_inter_support_eq p h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : MeasurableSpace \u03b1\np : Pmf \u03b1\ns t : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\n\u22a2 Function.Injective toMeasure\n[PROOFSTEP]\nintro p q h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : MeasurableSpace \u03b1\np\u271d : Pmf \u03b1\ns t : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\np q : Pmf \u03b1\nh : toMeasure p = toMeasure q\n\u22a2 p = q\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : MeasurableSpace \u03b1\np\u271d : Pmf \u03b1\ns t : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\np q : Pmf \u03b1\nh : toMeasure p = toMeasure q\nx : \u03b1\n\u22a2 \u2191p x = \u2191q x\n[PROOFSTEP]\nrw [\u2190 p.toMeasure_apply_singleton x <| measurableSet_singleton x, \u2190\n q.toMeasure_apply_singleton x <| measurableSet_singleton x, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : Countable \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(toMeasure (toPmf \u03bc)) s = \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [\u03bc.toPmf.toMeasure_apply s hs, \u2190 \u03bc.tsum_indicator_apply_singleton s hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : Countable \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191(toPmf \u03bc)) x = \u2211' (x : \u03b1), Set.indicator s (fun x => \u2191\u2191\u03bc {x}) x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\np : Pmf \u03b1\n\u22a2 \u2191\u2191(toMeasure p) Set.univ = 1\n[PROOFSTEP]\nsimpa only [MeasurableSet.univ, toMeasure_apply_eq_toOuterMeasure_apply, Set.indicator_univ, toOuterMeasure_apply,\n ENNReal.coe_eq_one] using tsum_coe p\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : Countable \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\np : Pmf \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\nx : \u03b1\n\u22a2 \u2191(Measure.toPmf (toMeasure p)) x = \u2191p x\n[PROOFSTEP]\nrw [\u2190 p.toMeasure_apply_singleton x (measurableSet_singleton x), p.toMeasure.toPmf_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : Countable \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSingletonClass \u03b1\np : Pmf \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\u271d\n\u03bc : Measure \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\n\u22a2 toMeasure p = \u03bc \u2194 p = Measure.toPmf \u03bc\n[PROOFSTEP]\nrw [\u2190 toMeasure_inj, Measure.toPmf_toMeasure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : Countable \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSingletonClass \u03b1\np : Pmf \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\u271d\n\u03bc : Measure \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\n\u22a2 Measure.toPmf \u03bc = p \u2194 \u03bc = toMeasure p\n[PROOFSTEP]\nrw [\u2190 toMeasure_inj, Measure.toPmf_toMeasure]\n", "meta": {"mathlib_filename": "Mathlib.Probability.ProbabilityMassFunction.Basic", "llama_tokens": 7207, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581626286834, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.44794386577113593}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : ULift C\nf : X \u27f6 Y\n\u22a2 (downFunctor \u22d9 upFunctor).map f \u226b (fun X => \ud835\udfd9 ((downFunctor \u22d9 upFunctor).obj X)) Y =\n (fun X => \ud835\udfd9 ((downFunctor \u22d9 upFunctor).obj X)) X \u226b (\ud835\udfed (ULift C)).map f\n[PROOFSTEP]\nchange f \u226b \ud835\udfd9 _ = \ud835\udfd9 _ \u226b f\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : ULift C\nf : X \u27f6 Y\n\u22a2 f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : ULift C\nf : X \u27f6 Y\n\u22a2 (\ud835\udfed (ULift C)).map f \u226b (fun X => \ud835\udfd9 ((\ud835\udfed (ULift C)).obj X)) Y =\n (fun X => \ud835\udfd9 ((\ud835\udfed (ULift C)).obj X)) X \u226b (downFunctor \u22d9 upFunctor).map f\n[PROOFSTEP]\nchange f \u226b \ud835\udfd9 _ = \ud835\udfd9 _ \u226b f\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : ULift C\nf : X \u27f6 Y\n\u22a2 f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u22a2 ((NatTrans.mk fun X => \ud835\udfd9 ((downFunctor \u22d9 upFunctor).obj X)) \u226b NatTrans.mk fun X => \ud835\udfd9 ((\ud835\udfed (ULift C)).obj X)) =\n \ud835\udfd9 (downFunctor \u22d9 upFunctor)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nx\u271d : ULift C\n\u22a2 NatTrans.app\n ((NatTrans.mk fun X => \ud835\udfd9 ((downFunctor \u22d9 upFunctor).obj X)) \u226b NatTrans.mk fun X => \ud835\udfd9 ((\ud835\udfed (ULift C)).obj X)) x\u271d =\n NatTrans.app (\ud835\udfd9 (downFunctor \u22d9 upFunctor)) x\u271d\n[PROOFSTEP]\nchange \ud835\udfd9 _ \u226b \ud835\udfd9 _ = \ud835\udfd9 _\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nx\u271d : ULift C\n\u22a2 \ud835\udfd9 ((downFunctor \u22d9 upFunctor).obj x\u271d) \u226b \ud835\udfd9 ((downFunctor \u22d9 upFunctor).obj x\u271d) = \ud835\udfd9 ((downFunctor \u22d9 upFunctor).obj x\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u22a2 ((NatTrans.mk fun X => \ud835\udfd9 ((\ud835\udfed (ULift C)).obj X)) \u226b NatTrans.mk fun X => \ud835\udfd9 ((downFunctor \u22d9 upFunctor).obj X)) =\n \ud835\udfd9 (\ud835\udfed (ULift C))\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nx\u271d : ULift C\n\u22a2 NatTrans.app\n ((NatTrans.mk fun X => \ud835\udfd9 ((\ud835\udfed (ULift C)).obj X)) \u226b NatTrans.mk fun X => \ud835\udfd9 ((downFunctor \u22d9 upFunctor).obj X)) x\u271d =\n NatTrans.app (\ud835\udfd9 (\ud835\udfed (ULift C))) x\u271d\n[PROOFSTEP]\nchange \ud835\udfd9 _ \u226b \ud835\udfd9 _ = \ud835\udfd9 _\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nx\u271d : ULift C\n\u22a2 \ud835\udfd9 ((\ud835\udfed (ULift C)).obj x\u271d) \u226b \ud835\udfd9 ((\ud835\udfed (ULift C)).obj x\u271d) = \ud835\udfd9 ((\ud835\udfed (ULift C)).obj x\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\n\u22a2 upFunctor.map (NatTrans.app (Iso.mk (\ud835\udfd9 (\ud835\udfed C)) (\ud835\udfd9 (upFunctor \u22d9 downFunctor))).hom X) \u226b\n NatTrans.app\n (Iso.mk (NatTrans.mk fun X => \ud835\udfd9 ((downFunctor \u22d9 upFunctor).obj X))\n (NatTrans.mk fun X => \ud835\udfd9 ((\ud835\udfed (ULift C)).obj X))).hom\n (upFunctor.obj X) =\n \ud835\udfd9 (upFunctor.obj X)\n[PROOFSTEP]\nchange \ud835\udfd9 X \u226b \ud835\udfd9 X = \ud835\udfd9 X\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\n\u22a2 \ud835\udfd9 X \u226b \ud835\udfd9 X = \ud835\udfd9 X\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Category.ULift", "llama_tokens": 1527, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506526772884, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.44793890167311246}} {"text": "[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u00b3 : Semiring R\u2081\ninst\u271d\u00b2 : Semiring R\u2082\ninst\u271d\u00b9 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3 : R\u2081 \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\u2081\ninst\u271d : RingHomInvPair \u03c3 \u03c3'\nx : R\u2081\n\u22a2 \u2191\u03c3' (\u2191\u03c3 x) = x\n[PROOFSTEP]\nrw [\u2190 RingHom.comp_apply, comp_eq]\n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u00b3 : Semiring R\u2081\ninst\u271d\u00b2 : Semiring R\u2082\ninst\u271d\u00b9 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3 : R\u2081 \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\u2081\ninst\u271d : RingHomInvPair \u03c3 \u03c3'\nx : R\u2081\n\u22a2 \u2191(RingHom.id R\u2081) x = x\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u00b3 : Semiring R\u2081\ninst\u271d\u00b2 : Semiring R\u2082\ninst\u271d\u00b9 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3 : R\u2081 \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\u2081\ninst\u271d : RingHomInvPair \u03c3 \u03c3'\nx : R\u2082\n\u22a2 \u2191\u03c3 (\u2191\u03c3' x) = x\n[PROOFSTEP]\nrw [\u2190 RingHom.comp_apply, comp_eq\u2082]\n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u00b3 : Semiring R\u2081\ninst\u271d\u00b2 : Semiring R\u2082\ninst\u271d\u00b9 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3 : R\u2081 \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\u2081\ninst\u271d : RingHomInvPair \u03c3 \u03c3'\nx : R\u2082\n\u22a2 \u2191(RingHom.id R\u2082) x = x\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u2074 : Semiring R\u2081\ninst\u271d\u00b3 : Semiring R\u2082\ninst\u271d\u00b2 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3 : R\u2081 \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3'\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\ninst\u271d : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\n\u22a2 RingHom.comp \u03c3\u2082\u2081 \u03c3\u2081\u2082 = RingHom.id R\u2081\n[PROOFSTEP]\nsimp only [comp_eq]\n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u2074 : Semiring R\u2081\ninst\u271d\u00b3 : Semiring R\u2082\ninst\u271d\u00b2 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3 : R\u2081 \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3'\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\ninst\u271d : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\n\u22a2 RingHom.comp \u03c3\u2081\u2082 \u03c3\u2082\u2081 = RingHom.id R\u2082\n[PROOFSTEP]\nsimp only [comp_eq\u2082]\n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u00b2 : Semiring R\u2081\ninst\u271d\u00b9 : Semiring R\u2082\ninst\u271d : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3 : R\u2081 \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\u2081\n\u22a2 RingHom.comp \u03c3\u2081\u2082 (RingHom.id R\u2081) = \u03c3\u2081\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u00b2 : Semiring R\u2081\ninst\u271d\u00b9 : Semiring R\u2082\ninst\u271d : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3 : R\u2081 \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\u2081\nx\u271d : R\u2081\n\u22a2 \u2191(RingHom.comp \u03c3\u2081\u2082 (RingHom.id R\u2081)) x\u271d = \u2191\u03c3\u2081\u2082 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u00b2 : Semiring R\u2081\ninst\u271d\u00b9 : Semiring R\u2082\ninst\u271d : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3 : R\u2081 \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\u2081\n\u22a2 RingHom.comp (RingHom.id R\u2082) \u03c3\u2081\u2082 = \u03c3\u2081\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u00b2 : Semiring R\u2081\ninst\u271d\u00b9 : Semiring R\u2082\ninst\u271d : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3 : R\u2081 \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\u2081\nx\u271d : R\u2081\n\u22a2 \u2191(RingHom.comp (RingHom.id R\u2082) \u03c3\u2081\u2082) x\u271d = \u2191\u03c3\u2081\u2082 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u2075 : Semiring R\u2081\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3 : R\u2081 \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\u2081\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b9 : RingHomSurjective \u03c3\u2081\u2082\ninst\u271d : RingHomSurjective \u03c3\u2082\u2083\n\u22a2 Function.Surjective \u2191\u03c3\u2081\u2083\n[PROOFSTEP]\nhave := \u03c3\u2082\u2083.surjective.comp \u03c3\u2081\u2082.surjective\n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u2075 : Semiring R\u2081\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3 : R\u2081 \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\u2081\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b9 : RingHomSurjective \u03c3\u2081\u2082\ninst\u271d : RingHomSurjective \u03c3\u2082\u2083\nthis : Function.Surjective (\u2191\u03c3\u2082\u2083 \u2218 \u2191\u03c3\u2081\u2082)\n\u22a2 Function.Surjective \u2191\u03c3\u2081\u2083\n[PROOFSTEP]\nrwa [\u2190 RingHom.coe_comp, RingHomCompTriple.comp_eq] at this \n", "meta": {"mathlib_filename": "Mathlib.Algebra.Ring.CompTypeclasses", "llama_tokens": 2178, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506526772884, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.44793890167311246}} {"text": "[GOAL]\nx y : \u211d\na b : Cauchy abs\n\u22a2 { cauchy := a } = { cauchy := b } \u2194 { cauchy := a }.cauchy = { cauchy := b }.cauchy\n[PROOFSTEP]\nrw [ofCauchy.injEq]\n[GOAL]\nx y : \u211d\na b : Cauchy abs\n\u22a2 { cauchy := a - b } = { cauchy := a } - { cauchy := b }\n[PROOFSTEP]\nrw [sub_eq_add_neg, ofCauchy_add, ofCauchy_neg]\n[GOAL]\nx y : \u211d\na b : Cauchy abs\n\u22a2 { cauchy := a } + -{ cauchy := b } = { cauchy := a } - { cauchy := b }\n[PROOFSTEP]\nrfl\n[GOAL]\nx y : \u211d\nf : Cauchy abs\n\u22a2 { cauchy := f\u207b\u00b9 } = Real.inv' { cauchy := f }\n[PROOFSTEP]\nrw [inv']\n[GOAL]\nx y : \u211d\n\u22a2 Real.zero.cauchy = 0\n[PROOFSTEP]\nrw [zero_def]\n[GOAL]\nx y : \u211d\n\u22a2 Real.one.cauchy = 1\n[PROOFSTEP]\nrw [one_def]\n[GOAL]\nx y : \u211d\na b : Cauchy abs\n\u22a2 (Real.add { cauchy := a } { cauchy := b }).cauchy = { cauchy := a }.cauchy + { cauchy := b }.cauchy\n[PROOFSTEP]\nrw [add_def]\n[GOAL]\nx y : \u211d\na : Cauchy abs\n\u22a2 (Real.neg { cauchy := a }).cauchy = -{ cauchy := a }.cauchy\n[PROOFSTEP]\nrw [neg_def]\n[GOAL]\nx y : \u211d\na b : Cauchy abs\n\u22a2 (Real.mul { cauchy := a } { cauchy := b }).cauchy = { cauchy := a }.cauchy * { cauchy := b }.cauchy\n[PROOFSTEP]\nrw [mul_def]\n[GOAL]\nx y : \u211d\na b : Cauchy abs\n\u22a2 ({ cauchy := a } - { cauchy := b }).cauchy = { cauchy := a }.cauchy - { cauchy := b }.cauchy\n[PROOFSTEP]\nrw [sub_eq_add_neg, \u2190 cauchy_neg, \u2190 cauchy_add]\n[GOAL]\nx y : \u211d\na b : Cauchy abs\n\u22a2 ({ cauchy := a } - { cauchy := b }).cauchy = ({ cauchy := a } + -{ cauchy := b }).cauchy\n[PROOFSTEP]\nrfl\n[GOAL]\nx y : \u211d\nf : Cauchy abs\n\u22a2 (Real.inv' { cauchy := f }).cauchy = { cauchy := f }.cauchy\u207b\u00b9\n[PROOFSTEP]\nrw [inv']\n[GOAL]\nx y : \u211d\n\u22a2 CommRing \u211d\n[PROOFSTEP]\nrefine'\n { natCast := fun n => \u27e8n\u27e9\n intCast := fun z => \u27e8z\u27e9\n zero := (0 : \u211d)\n one := (1 : \u211d)\n mul := (\u00b7 * \u00b7)\n add := (\u00b7 + \u00b7)\n neg := @Neg.neg \u211d _\n sub := @Sub.sub \u211d _\n npow := @npowRec \u211d \u27e81\u27e9 \u27e8(\u00b7 * \u00b7)\u27e9\n nsmul := @nsmulRec \u211d \u27e80\u27e9 \u27e8(\u00b7 + \u00b7)\u27e9\n zsmul := @zsmulRec \u211d \u27e80\u27e9 \u27e8(\u00b7 + \u00b7)\u27e9 \u27e8@Neg.neg \u211d _\u27e9, .. }\n[GOAL]\ncase refine'_1\nx y : \u211d\n\u22a2 \u2200 (a b c : \u211d), a + b + c = a + (b + c)\ncase refine'_2\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), 0 + a = a\ncase refine'_3\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), a + 0 = a\ncase refine'_4\nx y : \u211d\n\u22a2 \u2200 (x : \u211d), nsmulRec 0 x = 0\ncase refine'_5\nx y : \u211d\n\u22a2 \u2200 (n : \u2115) (x : \u211d), nsmulRec (n + 1) x = x + nsmulRec n x\ncase refine'_6\nx y : \u211d\n\u22a2 \u2200 (a b : \u211d), a + b = b + a\ncase refine'_7\nx y : \u211d\n\u22a2 \u2200 (a b c : \u211d), a * (b + c) = a * b + a * c\ncase refine'_8\nx y : \u211d\n\u22a2 \u2200 (a b c : \u211d), (a + b) * c = a * c + b * c\ncase refine'_9\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), 0 * a = 0\ncase refine'_10\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), a * 0 = 0\ncase refine'_11\nx y : \u211d\n\u22a2 \u2200 (a b c : \u211d), a * b * c = a * (b * c)\ncase refine'_12\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), 1 * a = a\ncase refine'_13\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), a * 1 = a\ncase refine'_14\nx y : \u211d\n\u22a2 NatCast.natCast 0 = 0\ncase refine'_15\nx y : \u211d\n\u22a2 \u2200 (n : \u2115), NatCast.natCast (n + 1) = NatCast.natCast n + 1\ncase refine'_16\nx y : \u211d\n\u22a2 \u2200 (x : \u211d), npowRec 0 x = 1\ncase refine'_17\nx y : \u211d\n\u22a2 \u2200 (n : \u2115) (x : \u211d), npowRec (n + 1) x = x * npowRec n x\ncase refine'_18\nx y : \u211d\n\u22a2 \u2200 (a b : \u211d), a - b = a + -b\ncase refine'_19\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), zsmulRec 0 a = 0\ncase refine'_20\nx y : \u211d\n\u22a2 \u2200 (n : \u2115) (a : \u211d), zsmulRec (Int.ofNat (Nat.succ n)) a = a + zsmulRec (Int.ofNat n) a\ncase refine'_21\nx y : \u211d\n\u22a2 \u2200 (n : \u2115) (a : \u211d), zsmulRec (Int.negSucc n) a = -zsmulRec (\u2191(Nat.succ n)) a\ncase refine'_22\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), -a + a = 0\ncase refine'_23\nx y : \u211d\n\u22a2 \u2200 (n : \u2115), IntCast.intCast \u2191n = \u2191n\ncase refine'_24\nx y : \u211d\n\u22a2 \u2200 (n : \u2115), IntCast.intCast (Int.negSucc n) = -\u2191(n + 1)\ncase refine'_25 x y : \u211d \u22a2 \u2200 (a b : \u211d), a * b = b * a\n[PROOFSTEP]\nall_goals\n intros\n first\n | rfl\n | apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_1\nx y : \u211d\n\u22a2 \u2200 (a b c : \u211d), a + b + c = a + (b + c)\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d + b\u271d + c\u271d = a\u271d + (b\u271d + c\u271d)\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_1\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d + b\u271d + c\u271d = a\u271d + (b\u271d + c\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d + b\u271d + c\u271d = a\u271d + (b\u271d + c\u271d)\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_1.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 (a\u271d + b\u271d + c\u271d).cauchy = (a\u271d + (b\u271d + c\u271d)).cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\ncase refine'_1.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d.cauchy + b\u271d.cauchy + c\u271d.cauchy = a\u271d.cauchy + (b\u271d.cauchy + c\u271d.cauchy)\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\ncase refine'_1.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d.cauchy + b\u271d.cauchy + c\u271d.cauchy = a\u271d.cauchy + (b\u271d.cauchy + c\u271d.cauchy)\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_1.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d.cauchy + b\u271d.cauchy + c\u271d.cauchy = a\u271d.cauchy + (b\u271d.cauchy + c\u271d.cauchy)\n[PROOFSTEP]\napply add_assoc\n[GOAL]\ncase refine'_2\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), 0 + a = a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\nx y a\u271d : \u211d\n\u22a2 0 + a\u271d = a\u271d\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_2\nx y a\u271d : \u211d\n\u22a2 0 + a\u271d = a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\nx y a\u271d : \u211d\n\u22a2 0 + a\u271d = a\u271d\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_2.a\nx y a\u271d : \u211d\n\u22a2 (0 + a\u271d).cauchy = a\u271d.cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\n\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_3\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), a + 0 = a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\nx y a\u271d : \u211d\n\u22a2 a\u271d + 0 = a\u271d\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_3\nx y a\u271d : \u211d\n\u22a2 a\u271d + 0 = a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_3\nx y a\u271d : \u211d\n\u22a2 a\u271d + 0 = a\u271d\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_3.a\nx y a\u271d : \u211d\n\u22a2 (a\u271d + 0).cauchy = a\u271d.cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\n\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_4\nx y : \u211d\n\u22a2 \u2200 (x : \u211d), nsmulRec 0 x = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_4\nx y x\u271d : \u211d\n\u22a2 nsmulRec 0 x\u271d = 0\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_4\nx y x\u271d : \u211d\n\u22a2 nsmulRec 0 x\u271d = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_5\nx y : \u211d\n\u22a2 \u2200 (n : \u2115) (x : \u211d), nsmulRec (n + 1) x = x + nsmulRec n x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_5\nx y : \u211d\nn\u271d : \u2115\nx\u271d : \u211d\n\u22a2 nsmulRec (n\u271d + 1) x\u271d = x\u271d + nsmulRec n\u271d x\u271d\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_5\nx y : \u211d\nn\u271d : \u2115\nx\u271d : \u211d\n\u22a2 nsmulRec (n\u271d + 1) x\u271d = x\u271d + nsmulRec n\u271d x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_6\nx y : \u211d\n\u22a2 \u2200 (a b : \u211d), a + b = b + a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_6\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d + b\u271d = b\u271d + a\u271d\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_6\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d + b\u271d = b\u271d + a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_6\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d + b\u271d = b\u271d + a\u271d\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_6.a\nx y a\u271d b\u271d : \u211d\n\u22a2 (a\u271d + b\u271d).cauchy = (b\u271d + a\u271d).cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\ncase refine'_6.a\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d.cauchy + b\u271d.cauchy = b\u271d.cauchy + a\u271d.cauchy\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\ncase refine'_6.a\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d.cauchy + b\u271d.cauchy = b\u271d.cauchy + a\u271d.cauchy\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_6.a\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d.cauchy + b\u271d.cauchy = b\u271d.cauchy + a\u271d.cauchy\n[PROOFSTEP]\napply add_assoc\n[GOAL]\ncase refine'_6.a\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d.cauchy + b\u271d.cauchy = b\u271d.cauchy + a\u271d.cauchy\n[PROOFSTEP]\napply add_comm\n[GOAL]\ncase refine'_7\nx y : \u211d\n\u22a2 \u2200 (a b c : \u211d), a * (b + c) = a * b + a * c\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_7\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d * (b\u271d + c\u271d) = a\u271d * b\u271d + a\u271d * c\u271d\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_7\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d * (b\u271d + c\u271d) = a\u271d * b\u271d + a\u271d * c\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_7\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d * (b\u271d + c\u271d) = a\u271d * b\u271d + a\u271d * c\u271d\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_7.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 (a\u271d * (b\u271d + c\u271d)).cauchy = (a\u271d * b\u271d + a\u271d * c\u271d).cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\ncase refine'_7.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d.cauchy * (b\u271d.cauchy + c\u271d.cauchy) = a\u271d.cauchy * b\u271d.cauchy + a\u271d.cauchy * c\u271d.cauchy\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\ncase refine'_7.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d.cauchy * (b\u271d.cauchy + c\u271d.cauchy) = a\u271d.cauchy * b\u271d.cauchy + a\u271d.cauchy * c\u271d.cauchy\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_7.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d.cauchy * (b\u271d.cauchy + c\u271d.cauchy) = a\u271d.cauchy * b\u271d.cauchy + a\u271d.cauchy * c\u271d.cauchy\n[PROOFSTEP]\napply add_assoc\n[GOAL]\ncase refine'_7.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d.cauchy * (b\u271d.cauchy + c\u271d.cauchy) = a\u271d.cauchy * b\u271d.cauchy + a\u271d.cauchy * c\u271d.cauchy\n[PROOFSTEP]\napply add_comm\n[GOAL]\ncase refine'_7.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d.cauchy * (b\u271d.cauchy + c\u271d.cauchy) = a\u271d.cauchy * b\u271d.cauchy + a\u271d.cauchy * c\u271d.cauchy\n[PROOFSTEP]\napply left_distrib\n[GOAL]\ncase refine'_8\nx y : \u211d\n\u22a2 \u2200 (a b c : \u211d), (a + b) * c = a * c + b * c\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_8\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 (a\u271d + b\u271d) * c\u271d = a\u271d * c\u271d + b\u271d * c\u271d\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_8\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 (a\u271d + b\u271d) * c\u271d = a\u271d * c\u271d + b\u271d * c\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_8\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 (a\u271d + b\u271d) * c\u271d = a\u271d * c\u271d + b\u271d * c\u271d\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_8.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 ((a\u271d + b\u271d) * c\u271d).cauchy = (a\u271d * c\u271d + b\u271d * c\u271d).cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\ncase refine'_8.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 (a\u271d.cauchy + b\u271d.cauchy) * c\u271d.cauchy = a\u271d.cauchy * c\u271d.cauchy + b\u271d.cauchy * c\u271d.cauchy\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\ncase refine'_8.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 (a\u271d.cauchy + b\u271d.cauchy) * c\u271d.cauchy = a\u271d.cauchy * c\u271d.cauchy + b\u271d.cauchy * c\u271d.cauchy\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_8.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 (a\u271d.cauchy + b\u271d.cauchy) * c\u271d.cauchy = a\u271d.cauchy * c\u271d.cauchy + b\u271d.cauchy * c\u271d.cauchy\n[PROOFSTEP]\napply add_assoc\n[GOAL]\ncase refine'_8.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 (a\u271d.cauchy + b\u271d.cauchy) * c\u271d.cauchy = a\u271d.cauchy * c\u271d.cauchy + b\u271d.cauchy * c\u271d.cauchy\n[PROOFSTEP]\napply add_comm\n[GOAL]\ncase refine'_8.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 (a\u271d.cauchy + b\u271d.cauchy) * c\u271d.cauchy = a\u271d.cauchy * c\u271d.cauchy + b\u271d.cauchy * c\u271d.cauchy\n[PROOFSTEP]\napply left_distrib\n[GOAL]\ncase refine'_8.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 (a\u271d.cauchy + b\u271d.cauchy) * c\u271d.cauchy = a\u271d.cauchy * c\u271d.cauchy + b\u271d.cauchy * c\u271d.cauchy\n[PROOFSTEP]\napply right_distrib\n[GOAL]\ncase refine'_9\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), 0 * a = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_9\nx y a\u271d : \u211d\n\u22a2 0 * a\u271d = 0\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_9\nx y a\u271d : \u211d\n\u22a2 0 * a\u271d = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_9\nx y a\u271d : \u211d\n\u22a2 0 * a\u271d = 0\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_9.a\nx y a\u271d : \u211d\n\u22a2 (0 * a\u271d).cauchy = 0.cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\n\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_10\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), a * 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_10\nx y a\u271d : \u211d\n\u22a2 a\u271d * 0 = 0\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_10\nx y a\u271d : \u211d\n\u22a2 a\u271d * 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_10\nx y a\u271d : \u211d\n\u22a2 a\u271d * 0 = 0\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_10.a\nx y a\u271d : \u211d\n\u22a2 (a\u271d * 0).cauchy = 0.cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\n\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_11\nx y : \u211d\n\u22a2 \u2200 (a b c : \u211d), a * b * c = a * (b * c)\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_11\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d * b\u271d * c\u271d = a\u271d * (b\u271d * c\u271d)\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_11\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d * b\u271d * c\u271d = a\u271d * (b\u271d * c\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_11\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d * b\u271d * c\u271d = a\u271d * (b\u271d * c\u271d)\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_11.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 (a\u271d * b\u271d * c\u271d).cauchy = (a\u271d * (b\u271d * c\u271d)).cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\ncase refine'_11.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d.cauchy * b\u271d.cauchy * c\u271d.cauchy = a\u271d.cauchy * (b\u271d.cauchy * c\u271d.cauchy)\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\ncase refine'_11.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d.cauchy * b\u271d.cauchy * c\u271d.cauchy = a\u271d.cauchy * (b\u271d.cauchy * c\u271d.cauchy)\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_11.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d.cauchy * b\u271d.cauchy * c\u271d.cauchy = a\u271d.cauchy * (b\u271d.cauchy * c\u271d.cauchy)\n[PROOFSTEP]\napply add_assoc\n[GOAL]\ncase refine'_11.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d.cauchy * b\u271d.cauchy * c\u271d.cauchy = a\u271d.cauchy * (b\u271d.cauchy * c\u271d.cauchy)\n[PROOFSTEP]\napply add_comm\n[GOAL]\ncase refine'_11.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d.cauchy * b\u271d.cauchy * c\u271d.cauchy = a\u271d.cauchy * (b\u271d.cauchy * c\u271d.cauchy)\n[PROOFSTEP]\napply left_distrib\n[GOAL]\ncase refine'_11.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d.cauchy * b\u271d.cauchy * c\u271d.cauchy = a\u271d.cauchy * (b\u271d.cauchy * c\u271d.cauchy)\n[PROOFSTEP]\napply right_distrib\n[GOAL]\ncase refine'_11.a\nx y a\u271d b\u271d c\u271d : \u211d\n\u22a2 a\u271d.cauchy * b\u271d.cauchy * c\u271d.cauchy = a\u271d.cauchy * (b\u271d.cauchy * c\u271d.cauchy)\n[PROOFSTEP]\napply mul_assoc\n[GOAL]\ncase refine'_12\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), 1 * a = a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_12\nx y a\u271d : \u211d\n\u22a2 1 * a\u271d = a\u271d\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_12\nx y a\u271d : \u211d\n\u22a2 1 * a\u271d = a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_12\nx y a\u271d : \u211d\n\u22a2 1 * a\u271d = a\u271d\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_12.a\nx y a\u271d : \u211d\n\u22a2 (1 * a\u271d).cauchy = a\u271d.cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\n\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_13\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), a * 1 = a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_13\nx y a\u271d : \u211d\n\u22a2 a\u271d * 1 = a\u271d\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_13\nx y a\u271d : \u211d\n\u22a2 a\u271d * 1 = a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_13\nx y a\u271d : \u211d\n\u22a2 a\u271d * 1 = a\u271d\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_13.a\nx y a\u271d : \u211d\n\u22a2 (a\u271d * 1).cauchy = a\u271d.cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\n\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_14\nx y : \u211d\n\u22a2 NatCast.natCast 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_14\nx y : \u211d\n\u22a2 NatCast.natCast 0 = 0\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_14\nx y : \u211d\n\u22a2 NatCast.natCast 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_14\nx y : \u211d\n\u22a2 NatCast.natCast 0 = 0\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_14.a\nx y : \u211d\n\u22a2 (NatCast.natCast 0).cauchy = 0.cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\n\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_15\nx y : \u211d\n\u22a2 \u2200 (n : \u2115), NatCast.natCast (n + 1) = NatCast.natCast n + 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_15\nx y : \u211d\nn\u271d : \u2115\n\u22a2 NatCast.natCast (n\u271d + 1) = NatCast.natCast n\u271d + 1\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_15\nx y : \u211d\nn\u271d : \u2115\n\u22a2 NatCast.natCast (n\u271d + 1) = NatCast.natCast n\u271d + 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_15\nx y : \u211d\nn\u271d : \u2115\n\u22a2 NatCast.natCast (n\u271d + 1) = NatCast.natCast n\u271d + 1\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_15.a\nx y : \u211d\nn\u271d : \u2115\n\u22a2 (NatCast.natCast (n\u271d + 1)).cauchy = (NatCast.natCast n\u271d + 1).cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\n\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_16\nx y : \u211d\n\u22a2 \u2200 (x : \u211d), npowRec 0 x = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_16\nx y x\u271d : \u211d\n\u22a2 npowRec 0 x\u271d = 1\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_16\nx y x\u271d : \u211d\n\u22a2 npowRec 0 x\u271d = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_17\nx y : \u211d\n\u22a2 \u2200 (n : \u2115) (x : \u211d), npowRec (n + 1) x = x * npowRec n x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_17\nx y : \u211d\nn\u271d : \u2115\nx\u271d : \u211d\n\u22a2 npowRec (n\u271d + 1) x\u271d = x\u271d * npowRec n\u271d x\u271d\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_17\nx y : \u211d\nn\u271d : \u2115\nx\u271d : \u211d\n\u22a2 npowRec (n\u271d + 1) x\u271d = x\u271d * npowRec n\u271d x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_18\nx y : \u211d\n\u22a2 \u2200 (a b : \u211d), a - b = a + -b\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_18\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d - b\u271d = a\u271d + -b\u271d\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_18\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d - b\u271d = a\u271d + -b\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_19\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), zsmulRec 0 a = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_19\nx y a\u271d : \u211d\n\u22a2 zsmulRec 0 a\u271d = 0\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_19\nx y a\u271d : \u211d\n\u22a2 zsmulRec 0 a\u271d = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_20\nx y : \u211d\n\u22a2 \u2200 (n : \u2115) (a : \u211d), zsmulRec (Int.ofNat (Nat.succ n)) a = a + zsmulRec (Int.ofNat n) a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_20\nx y : \u211d\nn\u271d : \u2115\na\u271d : \u211d\n\u22a2 zsmulRec (Int.ofNat (Nat.succ n\u271d)) a\u271d = a\u271d + zsmulRec (Int.ofNat n\u271d) a\u271d\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_20\nx y : \u211d\nn\u271d : \u2115\na\u271d : \u211d\n\u22a2 zsmulRec (Int.ofNat (Nat.succ n\u271d)) a\u271d = a\u271d + zsmulRec (Int.ofNat n\u271d) a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_21\nx y : \u211d\n\u22a2 \u2200 (n : \u2115) (a : \u211d), zsmulRec (Int.negSucc n) a = -zsmulRec (\u2191(Nat.succ n)) a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_21\nx y : \u211d\nn\u271d : \u2115\na\u271d : \u211d\n\u22a2 zsmulRec (Int.negSucc n\u271d) a\u271d = -zsmulRec (\u2191(Nat.succ n\u271d)) a\u271d\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_21\nx y : \u211d\nn\u271d : \u2115\na\u271d : \u211d\n\u22a2 zsmulRec (Int.negSucc n\u271d) a\u271d = -zsmulRec (\u2191(Nat.succ n\u271d)) a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_22\nx y : \u211d\n\u22a2 \u2200 (a : \u211d), -a + a = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_22\nx y a\u271d : \u211d\n\u22a2 -a\u271d + a\u271d = 0\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_22\nx y a\u271d : \u211d\n\u22a2 -a\u271d + a\u271d = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_22\nx y a\u271d : \u211d\n\u22a2 -a\u271d + a\u271d = 0\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_22.a\nx y a\u271d : \u211d\n\u22a2 (-a\u271d + a\u271d).cauchy = 0.cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\n\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_23\nx y : \u211d\n\u22a2 \u2200 (n : \u2115), IntCast.intCast \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_23\nx y : \u211d\nn\u271d : \u2115\n\u22a2 IntCast.intCast \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_23\nx y : \u211d\nn\u271d : \u2115\n\u22a2 IntCast.intCast \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_24\nx y : \u211d\n\u22a2 \u2200 (n : \u2115), IntCast.intCast (Int.negSucc n) = -\u2191(n + 1)\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_24\nx y : \u211d\nn\u271d : \u2115\n\u22a2 IntCast.intCast (Int.negSucc n\u271d) = -\u2191(n\u271d + 1)\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_24\nx y : \u211d\nn\u271d : \u2115\n\u22a2 IntCast.intCast (Int.negSucc n\u271d) = -\u2191(n\u271d + 1)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_24\nx y : \u211d\nn\u271d : \u2115\n\u22a2 IntCast.intCast (Int.negSucc n\u271d) = -\u2191(n\u271d + 1)\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_24.a\nx y : \u211d\nn\u271d : \u2115\n\u22a2 (IntCast.intCast (Int.negSucc n\u271d)).cauchy = (-\u2191(n\u271d + 1)).cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\n\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_25\nx y : \u211d\n\u22a2 \u2200 (a b : \u211d), a * b = b * a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_25\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d * b\u271d = b\u271d * a\u271d\n[PROOFSTEP]\nfirst\n| rfl\n| apply ext_cauchy\n simp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n first\n | done\n | apply add_assoc\n | apply add_comm\n | apply left_distrib\n | apply right_distrib\n | apply mul_assoc\n | apply mul_comm\n[GOAL]\ncase refine'_25\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d * b\u271d = b\u271d * a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_25\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d * b\u271d = b\u271d * a\u271d\n[PROOFSTEP]\napply ext_cauchy\n[GOAL]\ncase refine'_25.a\nx y a\u271d b\u271d : \u211d\n\u22a2 (a\u271d * b\u271d).cauchy = (b\u271d * a\u271d).cauchy\n[PROOFSTEP]\nsimp [cauchy_add, cauchy_zero, cauchy_one, cauchy_neg, cauchy_mul, cauchy_natCast, cauchy_intCast]\n[GOAL]\ncase refine'_25.a\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d.cauchy * b\u271d.cauchy = b\u271d.cauchy * a\u271d.cauchy\n[PROOFSTEP]\nfirst\n| done\n| apply add_assoc\n| apply add_comm\n| apply left_distrib\n| apply right_distrib\n| apply mul_assoc\n| apply mul_comm\n[GOAL]\ncase refine'_25.a\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d.cauchy * b\u271d.cauchy = b\u271d.cauchy * a\u271d.cauchy\n[PROOFSTEP]\ndone\n[GOAL]\ncase refine'_25.a\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d.cauchy * b\u271d.cauchy = b\u271d.cauchy * a\u271d.cauchy\n[PROOFSTEP]\napply add_assoc\n[GOAL]\ncase refine'_25.a\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d.cauchy * b\u271d.cauchy = b\u271d.cauchy * a\u271d.cauchy\n[PROOFSTEP]\napply add_comm\n[GOAL]\ncase refine'_25.a\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d.cauchy * b\u271d.cauchy = b\u271d.cauchy * a\u271d.cauchy\n[PROOFSTEP]\napply left_distrib\n[GOAL]\ncase refine'_25.a\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d.cauchy * b\u271d.cauchy = b\u271d.cauchy * a\u271d.cauchy\n[PROOFSTEP]\napply right_distrib\n[GOAL]\ncase refine'_25.a\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d.cauchy * b\u271d.cauchy = b\u271d.cauchy * a\u271d.cauchy\n[PROOFSTEP]\napply mul_assoc\n[GOAL]\ncase refine'_25.a\nx y a\u271d b\u271d : \u211d\n\u22a2 a\u271d.cauchy * b\u271d.cauchy = b\u271d.cauchy * a\u271d.cauchy\n[PROOFSTEP]\napply mul_comm\n[GOAL]\nx y : \u211d\n\u22a2 Ring \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 CommSemiring \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 Semiring \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 CommMonoidWithZero \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 MonoidWithZero \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 AddCommGroup \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 AddGroup \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 AddCommMonoid \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 AddMonoid \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 AddLeftCancelSemigroup \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 AddRightCancelSemigroup \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 AddCommSemigroup \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 AddSemigroup \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 CommMonoid \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 Monoid \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 CommSemigroup \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 Semigroup \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\nf g : CauSeq \u211a abs\n\u22a2 Real.lt { cauchy := Quotient.mk equiv f } { cauchy := Quotient.mk equiv g } \u2194 f < g\n[PROOFSTEP]\nrw [lt_def]\n[GOAL]\nx y : \u211d\nf g : CauSeq \u211a abs\n\u22a2 (match { cauchy := Quotient.mk equiv f }, { cauchy := Quotient.mk equiv g } with\n | { cauchy := x }, { cauchy := y } => Quotient.liftOn\u2082 x y (fun x x_1 => x < x_1) Real.definition.proof_1\u271d) \u2194\n f < g\n[PROOFSTEP]\nrfl\n[GOAL]\nx y : \u211d\n\u22a2 mk 0 = 0\n[PROOFSTEP]\nrw [\u2190 ofCauchy_zero]\n[GOAL]\nx y : \u211d\n\u22a2 mk 0 = { cauchy := 0 }\n[PROOFSTEP]\nrfl\n[GOAL]\nx y : \u211d\n\u22a2 mk 1 = 1\n[PROOFSTEP]\nrw [\u2190 ofCauchy_one]\n[GOAL]\nx y : \u211d\n\u22a2 mk 1 = { cauchy := 1 }\n[PROOFSTEP]\nrfl\n[GOAL]\nx y : \u211d\nf g : CauSeq \u211a abs\n\u22a2 mk (f + g) = mk f + mk g\n[PROOFSTEP]\nsimp [mk, \u2190 ofCauchy_add]\n[GOAL]\nx y : \u211d\nf g : CauSeq \u211a abs\n\u22a2 mk (f * g) = mk f * mk g\n[PROOFSTEP]\nsimp [mk, \u2190 ofCauchy_mul]\n[GOAL]\nx y : \u211d\nf : CauSeq \u211a abs\n\u22a2 mk (-f) = -mk f\n[PROOFSTEP]\nsimp [mk, \u2190 ofCauchy_neg]\n[GOAL]\nx y : \u211d\nf : CauSeq \u211a abs\n\u22a2 0 < mk f \u2194 Pos f\n[PROOFSTEP]\nrw [\u2190 mk_zero, mk_lt]\n[GOAL]\nx y : \u211d\nf : CauSeq \u211a abs\n\u22a2 0 < f \u2194 Pos f\n[PROOFSTEP]\nexact iff_of_eq (congr_arg Pos (sub_zero f))\n[GOAL]\nx\u271d y\u271d x y : \u211d\n\u22a2 Real.le x y \u2194 x < y \u2228 x = y\n[PROOFSTEP]\nrw [le_def]\n[GOAL]\nx y : \u211d\nf g : CauSeq \u211a abs\n\u22a2 mk f \u2264 mk g \u2194 f \u2264 g\n[PROOFSTEP]\nsimp [le_def', mk_eq]\n[GOAL]\nx y : \u211d\nf g : CauSeq \u211a abs\n\u22a2 f < g \u2228 f \u2248 g \u2194 f \u2264 g\n[PROOFSTEP]\nrfl\n[GOAL]\nx\u271d y : \u211d\nC : \u211d \u2192 Prop\nx : \u211d\nh : \u2200 (y : CauSeq \u211a abs), C (mk y)\n\u22a2 C x\n[PROOFSTEP]\ncases' x with x\n[GOAL]\ncase ofCauchy\nx\u271d y : \u211d\nC : \u211d \u2192 Prop\nh : \u2200 (y : CauSeq \u211a abs), C (mk y)\nx : Cauchy abs\n\u22a2 C { cauchy := x }\n[PROOFSTEP]\ninduction' x using Quot.induction_on with x\n[GOAL]\ncase ofCauchy.h\nx\u271d y : \u211d\nC : \u211d \u2192 Prop\nh : \u2200 (y : CauSeq \u211a abs), C (mk y)\nx : CauSeq \u211a abs\n\u22a2 C { cauchy := Quot.mk Setoid.r x }\n[PROOFSTEP]\nexact h x\n[GOAL]\nx y a b c : \u211d\n\u22a2 c + a < c + b \u2194 a < b\n[PROOFSTEP]\ninduction a using Real.ind_mk\n[GOAL]\ncase h\nx y b c : \u211d\ny\u271d : CauSeq \u211a abs\n\u22a2 c + mk y\u271d < c + b \u2194 mk y\u271d < b\n[PROOFSTEP]\ninduction b using Real.ind_mk\n[GOAL]\ncase h.h\nx y c : \u211d\ny\u271d\u00b9 y\u271d : CauSeq \u211a abs\n\u22a2 c + mk y\u271d\u00b9 < c + mk y\u271d \u2194 mk y\u271d\u00b9 < mk y\u271d\n[PROOFSTEP]\ninduction c using Real.ind_mk\n[GOAL]\ncase h.h.h\nx y : \u211d\ny\u271d\u00b2 y\u271d\u00b9 y\u271d : CauSeq \u211a abs\n\u22a2 mk y\u271d + mk y\u271d\u00b2 < mk y\u271d + mk y\u271d\u00b9 \u2194 mk y\u271d\u00b2 < mk y\u271d\u00b9\n[PROOFSTEP]\nsimp only [mk_lt, \u2190 mk_add]\n[GOAL]\ncase h.h.h\nx y : \u211d\ny\u271d\u00b2 y\u271d\u00b9 y\u271d : CauSeq \u211a abs\n\u22a2 y\u271d + y\u271d\u00b2 < y\u271d + y\u271d\u00b9 \u2194 y\u271d\u00b2 < y\u271d\u00b9\n[PROOFSTEP]\nshow Pos _ \u2194 Pos _\n[GOAL]\ncase h.h.h\nx y : \u211d\ny\u271d\u00b2 y\u271d\u00b9 y\u271d : CauSeq \u211a abs\n\u22a2 Pos (y\u271d + y\u271d\u00b9 - (y\u271d + y\u271d\u00b2)) \u2194 Pos (y\u271d\u00b9 - y\u271d\u00b2)\n[PROOFSTEP]\nrw [add_sub_add_left_eq_sub]\n[GOAL]\nx y a : \u211d\n\u22a2 a \u2264 a\n[PROOFSTEP]\ninduction' a using Real.ind_mk with a\n[GOAL]\ncase h\nx y : \u211d\na : CauSeq \u211a abs\n\u22a2 mk a \u2264 mk a\n[PROOFSTEP]\nrw [mk_le]\n[GOAL]\nx y a b c : \u211d\n\u22a2 a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n[PROOFSTEP]\ninduction' a using Real.ind_mk with a\n[GOAL]\ncase h\nx y b c : \u211d\na : CauSeq \u211a abs\n\u22a2 mk a \u2264 b \u2192 b \u2264 c \u2192 mk a \u2264 c\n[PROOFSTEP]\ninduction' b using Real.ind_mk with b\n[GOAL]\ncase h.h\nx y c : \u211d\na b : CauSeq \u211a abs\n\u22a2 mk a \u2264 mk b \u2192 mk b \u2264 c \u2192 mk a \u2264 c\n[PROOFSTEP]\ninduction' c using Real.ind_mk with c\n[GOAL]\ncase h.h.h\nx y : \u211d\na b c : CauSeq \u211a abs\n\u22a2 mk a \u2264 mk b \u2192 mk b \u2264 mk c \u2192 mk a \u2264 mk c\n[PROOFSTEP]\nsimpa using le_trans\n[GOAL]\nx y a b : \u211d\n\u22a2 a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n[PROOFSTEP]\ninduction' a using Real.ind_mk with a\n[GOAL]\ncase h\nx y b : \u211d\na : CauSeq \u211a abs\n\u22a2 mk a < b \u2194 mk a \u2264 b \u2227 \u00acb \u2264 mk a\n[PROOFSTEP]\ninduction' b using Real.ind_mk with b\n[GOAL]\ncase h.h\nx y : \u211d\na b : CauSeq \u211a abs\n\u22a2 mk a < mk b \u2194 mk a \u2264 mk b \u2227 \u00acmk b \u2264 mk a\n[PROOFSTEP]\nsimpa using lt_iff_le_not_le\n[GOAL]\nx y a b : \u211d\n\u22a2 a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\ninduction' a using Real.ind_mk with a\n[GOAL]\ncase h\nx y b : \u211d\na : CauSeq \u211a abs\n\u22a2 mk a \u2264 b \u2192 b \u2264 mk a \u2192 mk a = b\n[PROOFSTEP]\ninduction' b using Real.ind_mk with b\n[GOAL]\ncase h.h\nx y : \u211d\na b : CauSeq \u211a abs\n\u22a2 mk a \u2264 mk b \u2192 mk b \u2264 mk a \u2192 mk a = mk b\n[PROOFSTEP]\nsimpa [mk_eq] using @CauSeq.le_antisymm _ _ a b\n[GOAL]\nx y : \u211d\n\u22a2 Preorder \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx\u271d y\u271d : \u211d\nx y : \u211a\n\u22a2 \u2191x < \u2191y \u2194 x < y\n[PROOFSTEP]\nerw [mk_lt]\n[GOAL]\nx\u271d y\u271d : \u211d\nx y : \u211a\n\u22a2 const abs \u2191x < const abs \u2191y \u2194 x < y\n[PROOFSTEP]\nexact const_lt\n[GOAL]\nx y : \u211d\n\u22a2 0 < 1\n[PROOFSTEP]\nconvert ratCast_lt.2 zero_lt_one\n[GOAL]\ncase h.e'_3\nx y : \u211d\n\u22a2 0 = \u21910\n[PROOFSTEP]\nsimp [\u2190 ofCauchy_ratCast, ofCauchy_one, ofCauchy_zero]\n[GOAL]\ncase h.e'_4\nx y : \u211d\n\u22a2 1 = \u21911\n[PROOFSTEP]\nsimp [\u2190 ofCauchy_ratCast, ofCauchy_one, ofCauchy_zero]\n[GOAL]\nx y a b : \u211d\n\u22a2 0 < a \u2192 0 < b \u2192 0 < a * b\n[PROOFSTEP]\ninduction' a using Real.ind_mk with a\n[GOAL]\ncase h\nx y b : \u211d\na : CauSeq \u211a abs\n\u22a2 0 < mk a \u2192 0 < b \u2192 0 < mk a * b\n[PROOFSTEP]\ninduction' b using Real.ind_mk with b\n[GOAL]\ncase h.h\nx y : \u211d\na b : CauSeq \u211a abs\n\u22a2 0 < mk a \u2192 0 < mk b \u2192 0 < mk a * mk b\n[PROOFSTEP]\nsimpa only [mk_lt, mk_pos, \u2190 mk_mul] using CauSeq.mul_pos\n[GOAL]\nx y : \u211d\nsrc\u271d\u00b2 : CommRing \u211d := commRing\nsrc\u271d\u00b9 : PartialOrder \u211d := partialOrder\nsrc\u271d : Semiring \u211d := semiring\n\u22a2 \u2200 (a b : \u211d), a \u2264 b \u2192 \u2200 (c : \u211d), c + a \u2264 c + b\n[PROOFSTEP]\nsimp only [le_iff_eq_or_lt]\n[GOAL]\nx y : \u211d\nsrc\u271d\u00b2 : CommRing \u211d := commRing\nsrc\u271d\u00b9 : PartialOrder \u211d := partialOrder\nsrc\u271d : Semiring \u211d := semiring\n\u22a2 \u2200 (a b : \u211d), a = b \u2228 a < b \u2192 \u2200 (c : \u211d), c + a = c + b \u2228 c + a < c + b\n[PROOFSTEP]\nrintro a b \u27e8rfl, h\u27e9\n[GOAL]\ncase inl.refl\nx y : \u211d\nsrc\u271d\u00b2 : CommRing \u211d := commRing\nsrc\u271d\u00b9 : PartialOrder \u211d := partialOrder\nsrc\u271d : Semiring \u211d := semiring\na : \u211d\n\u22a2 \u2200 (c : \u211d), c + a = c + a \u2228 c + a < c + a\n[PROOFSTEP]\nsimp only [lt_self_iff_false, or_false, forall_const]\n[GOAL]\ncase inr\nx y : \u211d\nsrc\u271d\u00b2 : CommRing \u211d := commRing\nsrc\u271d\u00b9 : PartialOrder \u211d := partialOrder\nsrc\u271d : Semiring \u211d := semiring\na b : \u211d\nh\u271d : a < b\n\u22a2 \u2200 (c : \u211d), c + a = c + b \u2228 c + a < c + b\n[PROOFSTEP]\nexact fun c => Or.inr ((add_lt_add_iff_left c).2 \u2039_\u203a)\n[GOAL]\nx y : \u211d\na b : CauSeq \u211a abs\n\u22a2 { cauchy := Quotient.mk equiv (a \u2294 b) } = Real.sup { cauchy := Quotient.mk equiv a } { cauchy := Quotient.mk equiv b }\n[PROOFSTEP]\nrw [sup_def]\n[GOAL]\nx y : \u211d\na b : CauSeq \u211a abs\n\u22a2 { cauchy := Quotient.mk equiv (a \u2294 b) } =\n match { cauchy := Quotient.mk equiv a }, { cauchy := Quotient.mk equiv b } with\n | { cauchy := x }, { cauchy := y } =>\n {\n cauchy :=\n Quotient.map\u2082 (fun x x_1 => x \u2294 x_1)\n (_ : \u2200 (x x_1 : CauSeq \u211a abs), x \u2248 x_1 \u2192 \u2200 (x_2 x_3 : CauSeq \u211a abs), x_2 \u2248 x_3 \u2192 x \u2294 x_2 \u2248 x_1 \u2294 x_3) x y }\n[PROOFSTEP]\nrfl\n[GOAL]\nx y : \u211d\na b : CauSeq \u211a abs\n\u22a2 { cauchy := Quotient.mk equiv (a \u2293 b) } = Real.inf { cauchy := Quotient.mk equiv a } { cauchy := Quotient.mk equiv b }\n[PROOFSTEP]\nrw [inf_def]\n[GOAL]\nx y : \u211d\na b : CauSeq \u211a abs\n\u22a2 { cauchy := Quotient.mk equiv (a \u2293 b) } =\n match { cauchy := Quotient.mk equiv a }, { cauchy := Quotient.mk equiv b } with\n | { cauchy := x }, { cauchy := y } =>\n {\n cauchy :=\n Quotient.map\u2082 (fun x x_1 => x \u2293 x_1)\n (_ : \u2200 (x x_1 : CauSeq \u211a abs), x \u2248 x_1 \u2192 \u2200 (x_2 x_3 : CauSeq \u211a abs), x_2 \u2248 x_3 \u2192 x \u2293 x_2 \u2248 x_1 \u2293 x_3) x y }\n[PROOFSTEP]\nrfl\n[GOAL]\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\n\u22a2 \u2200 (a b : \u211d), a \u2264 a \u2294 b\n[PROOFSTEP]\nintros a b\n[GOAL]\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b : \u211d\n\u22a2 a \u2264 a \u2294 b\n[PROOFSTEP]\ninduction' a using Real.ind_mk with a\n[GOAL]\ncase h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\nb : \u211d\na : CauSeq \u211a abs\n\u22a2 mk a \u2264 mk a \u2294 b\n[PROOFSTEP]\ninduction' b using Real.ind_mk with b\n[GOAL]\ncase h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b : CauSeq \u211a abs\n\u22a2 mk a \u2264 mk a \u2294 mk b\n[PROOFSTEP]\nrw [\u2190 mk_sup, mk_le]\n[GOAL]\ncase h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b : CauSeq \u211a abs\n\u22a2 a \u2264 a \u2294 b\n[PROOFSTEP]\nexact CauSeq.le_sup_left\n[GOAL]\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\n\u22a2 \u2200 (a b : \u211d), b \u2264 a \u2294 b\n[PROOFSTEP]\nintros a b\n[GOAL]\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b : \u211d\n\u22a2 b \u2264 a \u2294 b\n[PROOFSTEP]\ninduction' a using Real.ind_mk with a\n[GOAL]\ncase h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\nb : \u211d\na : CauSeq \u211a abs\n\u22a2 b \u2264 mk a \u2294 b\n[PROOFSTEP]\ninduction' b using Real.ind_mk with b\n[GOAL]\ncase h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b : CauSeq \u211a abs\n\u22a2 mk b \u2264 mk a \u2294 mk b\n[PROOFSTEP]\nrw [\u2190 mk_sup, mk_le]\n[GOAL]\ncase h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b : CauSeq \u211a abs\n\u22a2 b \u2264 a \u2294 b\n[PROOFSTEP]\nexact CauSeq.le_sup_right\n[GOAL]\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\n\u22a2 \u2200 (a b c : \u211d), a \u2264 c \u2192 b \u2264 c \u2192 a \u2294 b \u2264 c\n[PROOFSTEP]\nintros a b c\n[GOAL]\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b c : \u211d\n\u22a2 a \u2264 c \u2192 b \u2264 c \u2192 a \u2294 b \u2264 c\n[PROOFSTEP]\ninduction' a using Real.ind_mk with a\n[GOAL]\ncase h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\nb c : \u211d\na : CauSeq \u211a abs\n\u22a2 mk a \u2264 c \u2192 b \u2264 c \u2192 mk a \u2294 b \u2264 c\n[PROOFSTEP]\ninduction' b using Real.ind_mk with b\n[GOAL]\ncase h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\nc : \u211d\na b : CauSeq \u211a abs\n\u22a2 mk a \u2264 c \u2192 mk b \u2264 c \u2192 mk a \u2294 mk b \u2264 c\n[PROOFSTEP]\ninduction' c using Real.ind_mk with c\n[GOAL]\ncase h.h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b c : CauSeq \u211a abs\n\u22a2 mk a \u2264 mk c \u2192 mk b \u2264 mk c \u2192 mk a \u2294 mk b \u2264 mk c\n[PROOFSTEP]\nsimp_rw [\u2190 mk_sup, mk_le]\n[GOAL]\ncase h.h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b c : CauSeq \u211a abs\n\u22a2 a \u2264 c \u2192 b \u2264 c \u2192 a \u2294 b \u2264 c\n[PROOFSTEP]\nexact CauSeq.sup_le\n[GOAL]\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\n\u22a2 \u2200 (a b : \u211d), a \u2293 b \u2264 a\n[PROOFSTEP]\nintros a b\n[GOAL]\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b : \u211d\n\u22a2 a \u2293 b \u2264 a\n[PROOFSTEP]\ninduction' a using Real.ind_mk with a\n[GOAL]\ncase h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\nb : \u211d\na : CauSeq \u211a abs\n\u22a2 mk a \u2293 b \u2264 mk a\n[PROOFSTEP]\ninduction' b using Real.ind_mk with b\n[GOAL]\ncase h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b : CauSeq \u211a abs\n\u22a2 mk a \u2293 mk b \u2264 mk a\n[PROOFSTEP]\nrw [\u2190 mk_inf, mk_le]\n[GOAL]\ncase h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b : CauSeq \u211a abs\n\u22a2 a \u2293 b \u2264 a\n[PROOFSTEP]\nexact CauSeq.inf_le_left\n[GOAL]\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\n\u22a2 \u2200 (a b : \u211d), a \u2293 b \u2264 b\n[PROOFSTEP]\nintros a b\n[GOAL]\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b : \u211d\n\u22a2 a \u2293 b \u2264 b\n[PROOFSTEP]\ninduction' a using Real.ind_mk with a\n[GOAL]\ncase h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\nb : \u211d\na : CauSeq \u211a abs\n\u22a2 mk a \u2293 b \u2264 b\n[PROOFSTEP]\ninduction' b using Real.ind_mk with b\n[GOAL]\ncase h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b : CauSeq \u211a abs\n\u22a2 mk a \u2293 mk b \u2264 mk b\n[PROOFSTEP]\nrw [\u2190 mk_inf, mk_le]\n[GOAL]\ncase h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b : CauSeq \u211a abs\n\u22a2 a \u2293 b \u2264 b\n[PROOFSTEP]\nexact CauSeq.inf_le_right\n[GOAL]\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\n\u22a2 \u2200 (a b c : \u211d), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\n[PROOFSTEP]\nintros a b c\n[GOAL]\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b c : \u211d\n\u22a2 a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\n[PROOFSTEP]\ninduction' a using Real.ind_mk with a\n[GOAL]\ncase h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\nb c : \u211d\na : CauSeq \u211a abs\n\u22a2 mk a \u2264 b \u2192 mk a \u2264 c \u2192 mk a \u2264 b \u2293 c\n[PROOFSTEP]\ninduction' b using Real.ind_mk with b\n[GOAL]\ncase h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\nc : \u211d\na b : CauSeq \u211a abs\n\u22a2 mk a \u2264 mk b \u2192 mk a \u2264 c \u2192 mk a \u2264 mk b \u2293 c\n[PROOFSTEP]\ninduction' c using Real.ind_mk with c\n[GOAL]\ncase h.h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b c : CauSeq \u211a abs\n\u22a2 mk a \u2264 mk b \u2192 mk a \u2264 mk c \u2192 mk a \u2264 mk b \u2293 mk c\n[PROOFSTEP]\nsimp_rw [\u2190 mk_inf, mk_le]\n[GOAL]\ncase h.h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b c : CauSeq \u211a abs\n\u22a2 a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\n[PROOFSTEP]\nexact CauSeq.le_inf\n[GOAL]\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\n\u22a2 \u2200 (x y z : \u211d), (x \u2294 y) \u2293 (x \u2294 z) \u2264 x \u2294 y \u2293 z\n[PROOFSTEP]\nintros a b c\n[GOAL]\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b c : \u211d\n\u22a2 (a \u2294 b) \u2293 (a \u2294 c) \u2264 a \u2294 b \u2293 c\n[PROOFSTEP]\ninduction' a using Real.ind_mk with a\n[GOAL]\ncase h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\nb c : \u211d\na : CauSeq \u211a abs\n\u22a2 (mk a \u2294 b) \u2293 (mk a \u2294 c) \u2264 mk a \u2294 b \u2293 c\n[PROOFSTEP]\ninduction' b using Real.ind_mk with b\n[GOAL]\ncase h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\nc : \u211d\na b : CauSeq \u211a abs\n\u22a2 (mk a \u2294 mk b) \u2293 (mk a \u2294 c) \u2264 mk a \u2294 mk b \u2293 c\n[PROOFSTEP]\ninduction' c using Real.ind_mk with c\n[GOAL]\ncase h.h.h\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b c : CauSeq \u211a abs\n\u22a2 (mk a \u2294 mk b) \u2293 (mk a \u2294 mk c) \u2264 mk a \u2294 mk b \u2293 mk c\n[PROOFSTEP]\napply Eq.le\n[GOAL]\ncase h.h.h.a\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b c : CauSeq \u211a abs\n\u22a2 (mk a \u2294 mk b) \u2293 (mk a \u2294 mk c) = mk a \u2294 mk b \u2293 mk c\n[PROOFSTEP]\nsimp only [\u2190 mk_sup, \u2190 mk_inf]\n[GOAL]\ncase h.h.h.a\nx y : \u211d\nsrc\u271d : PartialOrder \u211d := partialOrder\na b c : CauSeq \u211a abs\n\u22a2 mk ((a \u2294 b) \u2293 (a \u2294 c)) = mk (a \u2294 b \u2293 c)\n[PROOFSTEP]\nexact congr_arg mk (CauSeq.sup_inf_distrib_left _ _ _).symm\n[GOAL]\nx y : \u211d\n\u22a2 \u2200 (a b : \u211d), a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\nintros a b\n[GOAL]\nx y a b : \u211d\n\u22a2 a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\ninduction' a using Real.ind_mk with a\n[GOAL]\ncase h\nx y b : \u211d\na : CauSeq \u211a abs\n\u22a2 mk a \u2264 b \u2228 b \u2264 mk a\n[PROOFSTEP]\ninduction' b using Real.ind_mk with b\n[GOAL]\ncase h.h\nx y : \u211d\na b : CauSeq \u211a abs\n\u22a2 mk a \u2264 mk b \u2228 mk b \u2264 mk a\n[PROOFSTEP]\nsimpa using le_total a b\n[GOAL]\nx y : \u211d\n\u22a2 LinearOrderedRing \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 LinearOrderedSemiring \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\nsrc\u271d : LinearOrderedCommRing \u211d := linearOrderedCommRing\n\u22a2 \u2200 (a : \u211d), a \u2260 0 \u2192 a * a\u207b\u00b9 = 1\n[PROOFSTEP]\nrintro \u27e8a\u27e9 h\n[GOAL]\ncase ofCauchy\nx y : \u211d\nsrc\u271d : LinearOrderedCommRing \u211d := linearOrderedCommRing\na : Cauchy abs\nh : { cauchy := a } \u2260 0\n\u22a2 { cauchy := a } * { cauchy := a }\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase ofCauchy\nx y : \u211d\nsrc\u271d : LinearOrderedCommRing \u211d := linearOrderedCommRing\na : Cauchy abs\nh : { cauchy := a } \u2260 0\n\u22a2 { cauchy := a }\u207b\u00b9 * { cauchy := a } = 1\n[PROOFSTEP]\nsimp only [\u2190 ofCauchy_inv, \u2190 ofCauchy_mul, \u2190 ofCauchy_one, \u2190 ofCauchy_zero, Ne.def, ofCauchy.injEq] at *\n[GOAL]\ncase ofCauchy\nx y : \u211d\nsrc\u271d : LinearOrderedCommRing \u211d := linearOrderedCommRing\na : Cauchy abs\nh : \u00aca = 0\n\u22a2 a\u207b\u00b9 * a = 1\n[PROOFSTEP]\nexact CauSeq.Completion.inv_mul_cancel h\n[GOAL]\nx y : \u211d\nsrc\u271d : LinearOrderedCommRing \u211d := linearOrderedCommRing\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\nsimp [\u2190 ofCauchy_zero, \u2190 ofCauchy_inv]\n[GOAL]\nx y : \u211d\nsrc\u271d : LinearOrderedCommRing \u211d := linearOrderedCommRing\nn : \u2124\nd : \u2115\nhd : d \u2260 0\nh2 : Nat.coprime (Int.natAbs n) d\n\u22a2 \u2191(Rat.mk' n d) = \u2191n * (\u2191d)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 ofCauchy_ratCast, Rat.cast_mk', ofCauchy_mul, ofCauchy_inv, ofCauchy_natCast, ofCauchy_intCast]\n[GOAL]\nx y : \u211d\n\u22a2 LinearOrderedAddCommGroup \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 Field \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u22a2 DivisionRing \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y a b : \u211d\n\u22a2 Decidable (a < b)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y a b : \u211d\n\u22a2 Decidable (a \u2264 b)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y a b : \u211d\n\u22a2 Decidable (a = b)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\nf : CauSeq \u211a abs\n\u22a2 (\u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 x \u2264 \u2191(\u2191f j)) \u2192 x \u2264 mk f\n[PROOFSTEP]\nintro h\n[GOAL]\nx y : \u211d\nf : CauSeq \u211a abs\nh : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 x \u2264 \u2191(\u2191f j)\n\u22a2 x \u2264 mk f\n[PROOFSTEP]\ninduction' x using Real.ind_mk with x\n[GOAL]\ncase h\nx\u271d y : \u211d\nf : CauSeq \u211a abs\nh\u271d : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 x\u271d \u2264 \u2191(\u2191f j)\nx : CauSeq \u211a abs\nh : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 mk x \u2264 \u2191(\u2191f j)\n\u22a2 mk x \u2264 mk f\n[PROOFSTEP]\napply le_of_not_lt\n[GOAL]\ncase h.h\nx\u271d y : \u211d\nf : CauSeq \u211a abs\nh\u271d : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 x\u271d \u2264 \u2191(\u2191f j)\nx : CauSeq \u211a abs\nh : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 mk x \u2264 \u2191(\u2191f j)\n\u22a2 \u00acmk f < mk x\n[PROOFSTEP]\nrw [mk_lt]\n[GOAL]\ncase h.h\nx\u271d y : \u211d\nf : CauSeq \u211a abs\nh\u271d : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 x\u271d \u2264 \u2191(\u2191f j)\nx : CauSeq \u211a abs\nh : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 mk x \u2264 \u2191(\u2191f j)\n\u22a2 \u00acf < x\n[PROOFSTEP]\nrintro \u27e8K, K0, hK\u27e9\n[GOAL]\ncase h.h.intro.intro\nx\u271d y : \u211d\nf : CauSeq \u211a abs\nh\u271d : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 x\u271d \u2264 \u2191(\u2191f j)\nx : CauSeq \u211a abs\nh : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 mk x \u2264 \u2191(\u2191f j)\nK : \u211a\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 \u2191(x - f) j\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8i, H\u27e9 := exists_forall_ge_and h (exists_forall_ge_and hK (f.cauchy\u2083 <| half_pos K0))\n[GOAL]\ncase h.h.intro.intro.intro\nx\u271d y : \u211d\nf : CauSeq \u211a abs\nh\u271d : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 x\u271d \u2264 \u2191(\u2191f j)\nx : CauSeq \u211a abs\nh : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 mk x \u2264 \u2191(\u2191f j)\nK : \u211a\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 \u2191(x - f) j\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 mk x \u2264 \u2191(\u2191f j) \u2227 K \u2264 \u2191(x - f) j \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K / 2\n\u22a2 False\n[PROOFSTEP]\napply not_lt_of_le (H _ le_rfl).1\n[GOAL]\ncase h.h.intro.intro.intro\nx\u271d y : \u211d\nf : CauSeq \u211a abs\nh\u271d : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 x\u271d \u2264 \u2191(\u2191f j)\nx : CauSeq \u211a abs\nh : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 mk x \u2264 \u2191(\u2191f j)\nK : \u211a\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 \u2191(x - f) j\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 mk x \u2264 \u2191(\u2191f j) \u2227 K \u2264 \u2191(x - f) j \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K / 2\n\u22a2 \u2191(\u2191f i) < mk x\n[PROOFSTEP]\nerw [mk_lt]\n[GOAL]\ncase h.h.intro.intro.intro\nx\u271d y : \u211d\nf : CauSeq \u211a abs\nh\u271d : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 x\u271d \u2264 \u2191(\u2191f j)\nx : CauSeq \u211a abs\nh : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 mk x \u2264 \u2191(\u2191f j)\nK : \u211a\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 \u2191(x - f) j\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 mk x \u2264 \u2191(\u2191f j) \u2227 K \u2264 \u2191(x - f) j \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K / 2\n\u22a2 const abs \u2191(\u2191f i) < x\n[PROOFSTEP]\nrefine' \u27e8_, half_pos K0, i, fun j ij => _\u27e9\n[GOAL]\ncase h.h.intro.intro.intro\nx\u271d y : \u211d\nf : CauSeq \u211a abs\nh\u271d : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 x\u271d \u2264 \u2191(\u2191f j)\nx : CauSeq \u211a abs\nh : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 mk x \u2264 \u2191(\u2191f j)\nK : \u211a\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 \u2191(x - f) j\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 mk x \u2264 \u2191(\u2191f j) \u2227 K \u2264 \u2191(x - f) j \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K / 2\nj : \u2115\nij : j \u2265 i\n\u22a2 K / 2 \u2264 \u2191(x - const abs \u2191(\u2191f i)) j\n[PROOFSTEP]\nhave := add_le_add (H _ ij).2.1 (le_of_lt (abs_lt.1 <| (H _ le_rfl).2.2 _ ij).1)\n[GOAL]\ncase h.h.intro.intro.intro\nx\u271d y : \u211d\nf : CauSeq \u211a abs\nh\u271d : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 x\u271d \u2264 \u2191(\u2191f j)\nx : CauSeq \u211a abs\nh : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 mk x \u2264 \u2191(\u2191f j)\nK : \u211a\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 \u2191(x - f) j\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 mk x \u2264 \u2191(\u2191f j) \u2227 K \u2264 \u2191(x - f) j \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K / 2\nj : \u2115\nij : j \u2265 i\nthis : K + -(K / 2) \u2264 \u2191(x - f) j + (\u2191f j - \u2191f i)\n\u22a2 K / 2 \u2264 \u2191(x - const abs \u2191(\u2191f i)) j\n[PROOFSTEP]\nrwa [\u2190 sub_eq_add_neg, sub_self_div_two, sub_apply, sub_add_sub_cancel] at this \n[GOAL]\nx\u271d y : \u211d\nf : CauSeq \u211a abs\nx : \u211d\nh : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2191(\u2191f j) \u2264 x\n\u22a2 mk f \u2264 x\n[PROOFSTEP]\ncases' h with i H\n[GOAL]\ncase intro\nx\u271d y : \u211d\nf : CauSeq \u211a abs\nx : \u211d\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 \u2191(\u2191f j) \u2264 x\n\u22a2 mk f \u2264 x\n[PROOFSTEP]\nrw [\u2190 neg_le_neg_iff, \u2190 mk_neg]\n[GOAL]\ncase intro\nx\u271d y : \u211d\nf : CauSeq \u211a abs\nx : \u211d\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 \u2191(\u2191f j) \u2264 x\n\u22a2 -x \u2264 mk (-f)\n[PROOFSTEP]\nexact le_mk_of_forall_le \u27e8i, fun j ij => by simp [H _ ij]\u27e9\n[GOAL]\nx\u271d y : \u211d\nf : CauSeq \u211a abs\nx : \u211d\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 \u2191(\u2191f j) \u2264 x\nj : \u2115\nij : j \u2265 i\n\u22a2 -x \u2264 \u2191(\u2191(-f) j)\n[PROOFSTEP]\nsimp [H _ ij]\n[GOAL]\nx y : \u211d\nf : \u2115 \u2192 \u211a\nH : IsCauSeq abs f\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211a\n\u03b40 : 0 < \u03b4\n\u03b4\u03b5 : \u2191\u03b4 < \u03b5\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 |f j - f i| < \u03b4\nj : \u2115\nij : j \u2265 i\n\u22a2 |(fun i => \u2191(f i)) j - (fun i => \u2191(f i)) i| < \u2191\u03b4\n[PROOFSTEP]\nsimpa using (@Rat.cast_lt \u211d _ _ _).2 (hi _ ij)\n[GOAL]\nx y : \u211d\nf : \u2115 \u2192 \u211a\nH : IsCauSeq abs fun i => \u2191(f i)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 |(fun i => \u2191(f i)) j - (fun i => \u2191(f i)) i| < \u2191\u03b5\nj : \u2115\nij : j \u2265 i\n\u22a2 \u2191|f j - f i| < \u2191\u03b5\n[PROOFSTEP]\nsimpa using hi _ ij\n[GOAL]\nx y : \u211d\nS : Set \u211d\nhne : Set.Nonempty S\nhbdd : BddAbove S\n\u22a2 \u2203 x, IsLUB S x\n[PROOFSTEP]\nrcases hne, hbdd with \u27e8\u27e8L, hL\u27e9, \u27e8U, hU\u27e9\u27e9\n[GOAL]\ncase intro.intro\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\n\u22a2 \u2203 x, IsLUB S x\n[PROOFSTEP]\nhave : \u2200 d : \u2115, BddAbove {m : \u2124 | \u2203 y \u2208 S, (m : \u211d) \u2264 y * d} :=\n by\n cases' exists_int_gt U with k hk\n refine' fun d => \u27e8k * d, fun z h => _\u27e9\n rcases h with \u27e8y, yS, hy\u27e9\n refine' Int.cast_le.1 (hy.trans _)\n push_cast\n exact mul_le_mul_of_nonneg_right ((hU yS).trans hk.le) d.cast_nonneg\n[GOAL]\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\n\u22a2 \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\n[PROOFSTEP]\ncases' exists_int_gt U with k hk\n[GOAL]\ncase intro\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nk : \u2124\nhk : U < \u2191k\n\u22a2 \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\n[PROOFSTEP]\nrefine' fun d => \u27e8k * d, fun z h => _\u27e9\n[GOAL]\ncase intro\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nk : \u2124\nhk : U < \u2191k\nd : \u2115\nz : \u2124\nh : z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\n\u22a2 z \u2264 k * \u2191d\n[PROOFSTEP]\nrcases h with \u27e8y, yS, hy\u27e9\n[GOAL]\ncase intro.intro.intro\nx y\u271d : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nk : \u2124\nhk : U < \u2191k\nd : \u2115\nz : \u2124\ny : \u211d\nyS : y \u2208 S\nhy : \u2191z \u2264 y * \u2191d\n\u22a2 z \u2264 k * \u2191d\n[PROOFSTEP]\nrefine' Int.cast_le.1 (hy.trans _)\n[GOAL]\ncase intro.intro.intro\nx y\u271d : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nk : \u2124\nhk : U < \u2191k\nd : \u2115\nz : \u2124\ny : \u211d\nyS : y \u2208 S\nhy : \u2191z \u2264 y * \u2191d\n\u22a2 y * \u2191d \u2264 \u2191(k * \u2191d)\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase intro.intro.intro\nx y\u271d : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nk : \u2124\nhk : U < \u2191k\nd : \u2115\nz : \u2124\ny : \u211d\nyS : y \u2208 S\nhy : \u2191z \u2264 y * \u2191d\n\u22a2 y * \u2191d \u2264 \u2191k * \u2191d\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_right ((hU yS).trans hk.le) d.cast_nonneg\n[GOAL]\ncase intro.intro\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\n\u22a2 \u2203 x, IsLUB S x\n[PROOFSTEP]\nchoose f hf using fun d : \u2115 => Int.exists_greatest_of_bdd (this d) \u27e8\u230aL * d\u230b, L, hL, Int.floor_le _\u27e9\n[GOAL]\ncase intro.intro\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\n\u22a2 \u2203 x, IsLUB S x\n[PROOFSTEP]\nhave hf\u2081 : \u2200 n > 0, \u2203 y \u2208 S, ((f n / n : \u211a) : \u211d) \u2264 y := fun n n0 =>\n let \u27e8y, yS, hy\u27e9 := (hf n).1\n \u27e8y, yS, by simpa using (div_le_iff (Nat.cast_pos.2 n0 : (_ : \u211d) < _)).2 hy\u27e9\n[GOAL]\nx y\u271d : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nn : \u2115\nn0 : n > 0\ny : \u211d\nyS : y \u2208 S\nhy : \u2191(f n) \u2264 y * \u2191n\n\u22a2 \u2191(\u2191(f n) / \u2191n) \u2264 y\n[PROOFSTEP]\nsimpa using (div_le_iff (Nat.cast_pos.2 n0 : (_ : \u211d) < _)).2 hy\n[GOAL]\ncase intro.intro\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\n\u22a2 \u2203 x, IsLUB S x\n[PROOFSTEP]\nhave hf\u2082 : \u2200 n > 0, \u2200 y \u2208 S, (y - ((n : \u2115) : \u211d)\u207b\u00b9) < (f n / n : \u211a) :=\n by\n intro n n0 y yS\n have := (Int.sub_one_lt_floor _).trans_le (Int.cast_le.2 <| (hf n).2 _ \u27e8y, yS, Int.floor_le _\u27e9)\n simp only [Rat.cast_div, Rat.cast_coe_int, Rat.cast_coe_nat, gt_iff_lt]\n rwa [lt_div_iff (Nat.cast_pos.2 n0 : (_ : \u211d) < _), sub_mul, _root_.inv_mul_cancel]\n exact ne_of_gt (Nat.cast_pos.2 n0)\n[GOAL]\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\n\u22a2 \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n[PROOFSTEP]\nintro n n0 y yS\n[GOAL]\nx y\u271d : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nn : \u2115\nn0 : n > 0\ny : \u211d\nyS : y \u2208 S\n\u22a2 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n[PROOFSTEP]\nhave := (Int.sub_one_lt_floor _).trans_le (Int.cast_le.2 <| (hf n).2 _ \u27e8y, yS, Int.floor_le _\u27e9)\n[GOAL]\nx y\u271d : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis\u271d : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nn : \u2115\nn0 : n > 0\ny : \u211d\nyS : y \u2208 S\nthis : y * \u2191n - 1 < \u2191(f n)\n\u22a2 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n[PROOFSTEP]\nsimp only [Rat.cast_div, Rat.cast_coe_int, Rat.cast_coe_nat, gt_iff_lt]\n[GOAL]\nx y\u271d : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis\u271d : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nn : \u2115\nn0 : n > 0\ny : \u211d\nyS : y \u2208 S\nthis : y * \u2191n - 1 < \u2191(f n)\n\u22a2 y - (\u2191n)\u207b\u00b9 < \u2191(f n) / \u2191n\n[PROOFSTEP]\nrwa [lt_div_iff (Nat.cast_pos.2 n0 : (_ : \u211d) < _), sub_mul, _root_.inv_mul_cancel]\n[GOAL]\nx y\u271d : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis\u271d : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nn : \u2115\nn0 : n > 0\ny : \u211d\nyS : y \u2208 S\nthis : y * \u2191n - 1 < \u2191(f n)\n\u22a2 \u2191n \u2260 0\n[PROOFSTEP]\nexact ne_of_gt (Nat.cast_pos.2 n0)\n[GOAL]\ncase intro.intro\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n\u22a2 \u2203 x, IsLUB S x\n[PROOFSTEP]\nhave hg : IsCauSeq abs (fun n => f n / n : \u2115 \u2192 \u211a) := by\n intro \u03b5 \u03b50\n suffices \u2200 j \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a, \u2200 k \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a, (f j / j - f k / k : \u211a) < \u03b5\n by\n refine' \u27e8_, fun j ij => abs_lt.2 \u27e8_, this _ ij _ le_rfl\u27e9\u27e9\n rw [neg_lt, neg_sub]\n exact this _ le_rfl _ ij\n intro j ij k ik\n replace ij := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ij)\n replace ik := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ik)\n have j0 := Nat.cast_pos.1 ((inv_pos.2 \u03b50).trans_le ij)\n have k0 := Nat.cast_pos.1 ((inv_pos.2 \u03b50).trans_le ik)\n rcases hf\u2081 _ j0 with \u27e8y, yS, hy\u27e9\n refine' lt_of_lt_of_le ((@Rat.cast_lt \u211d _ _ _).1 _) ((inv_le \u03b50 (Nat.cast_pos.2 k0)).1 ik)\n simpa using sub_lt_iff_lt_add'.2 (lt_of_le_of_lt hy <| sub_lt_iff_lt_add.1 <| hf\u2082 _ k0 _ yS)\n[GOAL]\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n\u22a2 IsCauSeq abs fun n => \u2191(f n) / \u2191n\n[PROOFSTEP]\nintro \u03b5 \u03b50\n[GOAL]\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 |(fun n => \u2191(f n) / \u2191n) j - (fun n => \u2191(f n) / \u2191n) i| < \u03b5\n[PROOFSTEP]\nsuffices \u2200 j \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a, \u2200 k \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a, (f j / j - f k / k : \u211a) < \u03b5\n by\n refine' \u27e8_, fun j ij => abs_lt.2 \u27e8_, this _ ij _ le_rfl\u27e9\u27e9\n rw [neg_lt, neg_sub]\n exact this _ le_rfl _ ij\n[GOAL]\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis\u271d : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\nthis : \u2200 (j : \u2115), j \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a \u2192 \u2200 (k : \u2115), k \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a \u2192 \u2191(f j) / \u2191j - \u2191(f k) / \u2191k < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 |(fun n => \u2191(f n) / \u2191n) j - (fun n => \u2191(f n) / \u2191n) i| < \u03b5\n[PROOFSTEP]\nrefine' \u27e8_, fun j ij => abs_lt.2 \u27e8_, this _ ij _ le_rfl\u27e9\u27e9\n[GOAL]\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis\u271d : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\nthis : \u2200 (j : \u2115), j \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a \u2192 \u2200 (k : \u2115), k \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a \u2192 \u2191(f j) / \u2191j - \u2191(f k) / \u2191k < \u03b5\nj : \u2115\nij : j \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a\n\u22a2 -\u03b5 < (fun n => \u2191(f n) / \u2191n) j - (fun n => \u2191(f n) / \u2191n) \u2308\u03b5\u207b\u00b9\u2309\u208a\n[PROOFSTEP]\nrw [neg_lt, neg_sub]\n[GOAL]\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis\u271d : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\nthis : \u2200 (j : \u2115), j \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a \u2192 \u2200 (k : \u2115), k \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a \u2192 \u2191(f j) / \u2191j - \u2191(f k) / \u2191k < \u03b5\nj : \u2115\nij : j \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a\n\u22a2 (fun n => \u2191(f n) / \u2191n) \u2308\u03b5\u207b\u00b9\u2309\u208a - (fun n => \u2191(f n) / \u2191n) j < \u03b5\n[PROOFSTEP]\nexact this _ le_rfl _ ij\n[GOAL]\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\n\u22a2 \u2200 (j : \u2115), j \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a \u2192 \u2200 (k : \u2115), k \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a \u2192 \u2191(f j) / \u2191j - \u2191(f k) / \u2191k < \u03b5\n[PROOFSTEP]\nintro j ij k ik\n[GOAL]\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\nj : \u2115\nij : j \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a\nk : \u2115\nik : k \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a\n\u22a2 \u2191(f j) / \u2191j - \u2191(f k) / \u2191k < \u03b5\n[PROOFSTEP]\nreplace ij := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ij)\n[GOAL]\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\nj k : \u2115\nik : k \u2265 \u2308\u03b5\u207b\u00b9\u2309\u208a\nij : \u03b5\u207b\u00b9 \u2264 \u2191j\n\u22a2 \u2191(f j) / \u2191j - \u2191(f k) / \u2191k < \u03b5\n[PROOFSTEP]\nreplace ik := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ik)\n[GOAL]\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\nj k : \u2115\nij : \u03b5\u207b\u00b9 \u2264 \u2191j\nik : \u03b5\u207b\u00b9 \u2264 \u2191k\n\u22a2 \u2191(f j) / \u2191j - \u2191(f k) / \u2191k < \u03b5\n[PROOFSTEP]\nhave j0 := Nat.cast_pos.1 ((inv_pos.2 \u03b50).trans_le ij)\n[GOAL]\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\nj k : \u2115\nij : \u03b5\u207b\u00b9 \u2264 \u2191j\nik : \u03b5\u207b\u00b9 \u2264 \u2191k\nj0 : 0 < j\n\u22a2 \u2191(f j) / \u2191j - \u2191(f k) / \u2191k < \u03b5\n[PROOFSTEP]\nhave k0 := Nat.cast_pos.1 ((inv_pos.2 \u03b50).trans_le ik)\n[GOAL]\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\nj k : \u2115\nij : \u03b5\u207b\u00b9 \u2264 \u2191j\nik : \u03b5\u207b\u00b9 \u2264 \u2191k\nj0 : 0 < j\nk0 : 0 < k\n\u22a2 \u2191(f j) / \u2191j - \u2191(f k) / \u2191k < \u03b5\n[PROOFSTEP]\nrcases hf\u2081 _ j0 with \u27e8y, yS, hy\u27e9\n[GOAL]\ncase intro.intro\nx y\u271d : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\nj k : \u2115\nij : \u03b5\u207b\u00b9 \u2264 \u2191j\nik : \u03b5\u207b\u00b9 \u2264 \u2191k\nj0 : 0 < j\nk0 : 0 < k\ny : \u211d\nyS : y \u2208 S\nhy : \u2191(\u2191(f j) / \u2191j) \u2264 y\n\u22a2 \u2191(f j) / \u2191j - \u2191(f k) / \u2191k < \u03b5\n[PROOFSTEP]\nrefine' lt_of_lt_of_le ((@Rat.cast_lt \u211d _ _ _).1 _) ((inv_le \u03b50 (Nat.cast_pos.2 k0)).1 ik)\n[GOAL]\ncase intro.intro\nx y\u271d : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\nj k : \u2115\nij : \u03b5\u207b\u00b9 \u2264 \u2191j\nik : \u03b5\u207b\u00b9 \u2264 \u2191k\nj0 : 0 < j\nk0 : 0 < k\ny : \u211d\nyS : y \u2208 S\nhy : \u2191(\u2191(f j) / \u2191j) \u2264 y\n\u22a2 \u2191(\u2191(f j) / \u2191j - \u2191(f k) / \u2191k) < \u2191(\u2191k)\u207b\u00b9\n[PROOFSTEP]\nsimpa using sub_lt_iff_lt_add'.2 (lt_of_le_of_lt hy <| sub_lt_iff_lt_add.1 <| hf\u2082 _ k0 _ yS)\n[GOAL]\ncase intro.intro\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\nhg : IsCauSeq abs fun n => \u2191(f n) / \u2191n\n\u22a2 \u2203 x, IsLUB S x\n[PROOFSTEP]\nlet g : CauSeq \u211a abs := \u27e8fun n => f n / n, hg\u27e9\n[GOAL]\ncase intro.intro\nx y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\nhg : IsCauSeq abs fun n => \u2191(f n) / \u2191n\ng : CauSeq \u211a abs := { val := fun n => \u2191(f n) / \u2191n, property := hg }\n\u22a2 \u2203 x, IsLUB S x\n[PROOFSTEP]\nrefine' \u27e8mk g, \u27e8fun x xS => _, fun y h => _\u27e9\u27e9\n[GOAL]\ncase intro.intro.refine'_1\nx\u271d y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\nhg : IsCauSeq abs fun n => \u2191(f n) / \u2191n\ng : CauSeq \u211a abs := { val := fun n => \u2191(f n) / \u2191n, property := hg }\nx : \u211d\nxS : x \u2208 S\n\u22a2 x \u2264 mk g\n[PROOFSTEP]\nrefine' le_of_forall_ge_of_dense fun z xz => _\n[GOAL]\ncase intro.intro.refine'_1\nx\u271d y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\nhg : IsCauSeq abs fun n => \u2191(f n) / \u2191n\ng : CauSeq \u211a abs := { val := fun n => \u2191(f n) / \u2191n, property := hg }\nx : \u211d\nxS : x \u2208 S\nz : \u211d\nxz : z < x\n\u22a2 z \u2264 mk g\n[PROOFSTEP]\ncases' exists_nat_gt (x - z)\u207b\u00b9 with K hK\n[GOAL]\ncase intro.intro.refine'_1.intro\nx\u271d y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\nhg : IsCauSeq abs fun n => \u2191(f n) / \u2191n\ng : CauSeq \u211a abs := { val := fun n => \u2191(f n) / \u2191n, property := hg }\nx : \u211d\nxS : x \u2208 S\nz : \u211d\nxz : z < x\nK : \u2115\nhK : (x - z)\u207b\u00b9 < \u2191K\n\u22a2 z \u2264 mk g\n[PROOFSTEP]\nrefine' le_mk_of_forall_le \u27e8K, fun n nK => _\u27e9\n[GOAL]\ncase intro.intro.refine'_1.intro\nx\u271d y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\nhg : IsCauSeq abs fun n => \u2191(f n) / \u2191n\ng : CauSeq \u211a abs := { val := fun n => \u2191(f n) / \u2191n, property := hg }\nx : \u211d\nxS : x \u2208 S\nz : \u211d\nxz : z < x\nK : \u2115\nhK : (x - z)\u207b\u00b9 < \u2191K\nn : \u2115\nnK : n \u2265 K\n\u22a2 z \u2264 \u2191(\u2191g n)\n[PROOFSTEP]\nreplace xz := sub_pos.2 xz\n[GOAL]\ncase intro.intro.refine'_1.intro\nx\u271d y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\nhg : IsCauSeq abs fun n => \u2191(f n) / \u2191n\ng : CauSeq \u211a abs := { val := fun n => \u2191(f n) / \u2191n, property := hg }\nx : \u211d\nxS : x \u2208 S\nz : \u211d\nK : \u2115\nhK : (x - z)\u207b\u00b9 < \u2191K\nn : \u2115\nnK : n \u2265 K\nxz : 0 < x - z\n\u22a2 z \u2264 \u2191(\u2191g n)\n[PROOFSTEP]\nreplace hK := hK.le.trans (Nat.cast_le.2 nK)\n[GOAL]\ncase intro.intro.refine'_1.intro\nx\u271d y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\nhg : IsCauSeq abs fun n => \u2191(f n) / \u2191n\ng : CauSeq \u211a abs := { val := fun n => \u2191(f n) / \u2191n, property := hg }\nx : \u211d\nxS : x \u2208 S\nz : \u211d\nK n : \u2115\nnK : n \u2265 K\nxz : 0 < x - z\nhK : (x - z)\u207b\u00b9 \u2264 \u2191n\n\u22a2 z \u2264 \u2191(\u2191g n)\n[PROOFSTEP]\nhave n0 : 0 < n := Nat.cast_pos.1 ((inv_pos.2 xz).trans_le hK)\n[GOAL]\ncase intro.intro.refine'_1.intro\nx\u271d y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\nhg : IsCauSeq abs fun n => \u2191(f n) / \u2191n\ng : CauSeq \u211a abs := { val := fun n => \u2191(f n) / \u2191n, property := hg }\nx : \u211d\nxS : x \u2208 S\nz : \u211d\nK n : \u2115\nnK : n \u2265 K\nxz : 0 < x - z\nhK : (x - z)\u207b\u00b9 \u2264 \u2191n\nn0 : 0 < n\n\u22a2 z \u2264 \u2191(\u2191g n)\n[PROOFSTEP]\nrefine' le_trans _ (hf\u2082 _ n0 _ xS).le\n[GOAL]\ncase intro.intro.refine'_1.intro\nx\u271d y : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\nhg : IsCauSeq abs fun n => \u2191(f n) / \u2191n\ng : CauSeq \u211a abs := { val := fun n => \u2191(f n) / \u2191n, property := hg }\nx : \u211d\nxS : x \u2208 S\nz : \u211d\nK n : \u2115\nnK : n \u2265 K\nxz : 0 < x - z\nhK : (x - z)\u207b\u00b9 \u2264 \u2191n\nn0 : 0 < n\n\u22a2 z \u2264 x - (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nrwa [le_sub_comm, inv_le (Nat.cast_pos.2 n0 : (_ : \u211d) < _) xz]\n[GOAL]\ncase intro.intro.refine'_2\nx y\u271d : \u211d\nS : Set \u211d\nL : \u211d\nhL : L \u2208 S\nU : \u211d\nhU : U \u2208 upperBounds S\nthis : \u2200 (d : \u2115), BddAbove {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d}\nf : \u2115 \u2192 \u2124\nhf : \u2200 (d : \u2115), f d \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2227 \u2200 (z : \u2124), z \u2208 {m | \u2203 y, y \u2208 S \u2227 \u2191m \u2264 y * \u2191d} \u2192 z \u2264 f d\nhf\u2081 : \u2200 (n : \u2115), n > 0 \u2192 \u2203 y, y \u2208 S \u2227 \u2191(\u2191(f n) / \u2191n) \u2264 y\nhf\u2082 : \u2200 (n : \u2115), n > 0 \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y - (\u2191n)\u207b\u00b9 < \u2191(\u2191(f n) / \u2191n)\nhg : IsCauSeq abs fun n => \u2191(f n) / \u2191n\ng : CauSeq \u211a abs := { val := fun n => \u2191(f n) / \u2191n, property := hg }\ny : \u211d\nh : y \u2208 upperBounds S\n\u22a2 mk g \u2264 y\n[PROOFSTEP]\nexact\n mk_le_of_forall_le\n \u27e81, fun n n1 =>\n let \u27e8x, xS, hx\u27e9 := hf\u2081 _ n1\n le_trans hx (h xS)\u27e9\n[GOAL]\nx y : \u211d\nS : Set \u211d\nh\u2081 : Set.Nonempty S\nh\u2082 : BddAbove S\n\u22a2 IsLUB S (sSup S)\n[PROOFSTEP]\nsimp only [sSup_def, dif_pos (And.intro h\u2081 h\u2082)]\n[GOAL]\nx y : \u211d\nS : Set \u211d\nh\u2081 : Set.Nonempty S\nh\u2082 : BddAbove S\n\u22a2 IsLUB S (choose (_ : \u2203 x, IsLUB S x))\n[PROOFSTEP]\napply Classical.choose_spec\n[GOAL]\nx y : \u211d\nS : Set \u211d\nh\u2081 : Set.Nonempty S\nh\u2082 : BddBelow S\n\u22a2 IsGLB S (sInf S)\n[PROOFSTEP]\nrw [sInf_def, \u2190 isLUB_neg', neg_neg]\n[GOAL]\nx y : \u211d\nS : Set \u211d\nh\u2081 : Set.Nonempty S\nh\u2082 : BddBelow S\n\u22a2 IsLUB (-S) (sSup (-S))\n[PROOFSTEP]\nexact Real.isLUB_sSup _ h\u2081.neg h\u2082.neg\n[GOAL]\nx y : \u211d\nsrc\u271d\u00b9 : LinearOrder \u211d := linearOrder\nsrc\u271d : Lattice \u211d := lattice\ns : Set \u211d\nhs : \u00acBddAbove s\n\u22a2 sSup s = sSup Set.univ\n[PROOFSTEP]\nsimp [hs, sSup_def]\n[GOAL]\nx y : \u211d\nsrc\u271d\u00b9 : LinearOrder \u211d := linearOrder\nsrc\u271d : Lattice \u211d := lattice\ns : Set \u211d\nhs : \u00acBddBelow s\n\u22a2 sInf s = sInf Set.univ\n[PROOFSTEP]\nsimp [hs, sInf_def, sSup_def]\n[GOAL]\nx y : \u211d\ns : Set \u211d\nh : BddBelow s\nh' : Set.Nonempty s\na : \u211d\n\u22a2 sInf s \u2264 a \u2194 \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 x, x \u2208 s \u2227 x < a + \u03b5\n[PROOFSTEP]\nrw [le_iff_forall_pos_lt_add]\n[GOAL]\nx y : \u211d\ns : Set \u211d\nh : BddBelow s\nh' : Set.Nonempty s\na : \u211d\n\u22a2 (\u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 sInf s < a + \u03b5) \u2194 \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 x, x \u2208 s \u2227 x < a + \u03b5\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nx y : \u211d\ns : Set \u211d\nh : BddBelow s\nh' : Set.Nonempty s\na : \u211d\n\u22a2 (\u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 sInf s < a + \u03b5) \u2192 \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 x, x \u2208 s \u2227 x < a + \u03b5\n[PROOFSTEP]\nintro H \u03b5 \u03b5_pos\n[GOAL]\ncase mpr\nx y : \u211d\ns : Set \u211d\nh : BddBelow s\nh' : Set.Nonempty s\na : \u211d\n\u22a2 (\u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 x, x \u2208 s \u2227 x < a + \u03b5) \u2192 \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 sInf s < a + \u03b5\n[PROOFSTEP]\nintro H \u03b5 \u03b5_pos\n[GOAL]\ncase mp\nx y : \u211d\ns : Set \u211d\nh : BddBelow s\nh' : Set.Nonempty s\na : \u211d\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 sInf s < a + \u03b5\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\n\u22a2 \u2203 x, x \u2208 s \u2227 x < a + \u03b5\n[PROOFSTEP]\nexact exists_lt_of_csInf_lt h' (H \u03b5 \u03b5_pos)\n[GOAL]\ncase mpr\nx y : \u211d\ns : Set \u211d\nh : BddBelow s\nh' : Set.Nonempty s\na : \u211d\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 x, x \u2208 s \u2227 x < a + \u03b5\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\n\u22a2 sInf s < a + \u03b5\n[PROOFSTEP]\nrcases H \u03b5 \u03b5_pos with \u27e8x, x_in, hx\u27e9\n[GOAL]\ncase mpr.intro.intro\nx\u271d y : \u211d\ns : Set \u211d\nh : BddBelow s\nh' : Set.Nonempty s\na : \u211d\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 x, x \u2208 s \u2227 x < a + \u03b5\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nx : \u211d\nx_in : x \u2208 s\nhx : x < a + \u03b5\n\u22a2 sInf s < a + \u03b5\n[PROOFSTEP]\nexact csInf_lt_of_lt h x_in hx\n[GOAL]\nx y : \u211d\ns : Set \u211d\nh : BddAbove s\nh' : Set.Nonempty s\na : \u211d\n\u22a2 a \u2264 sSup s \u2194 \u2200 (\u03b5 : \u211d), \u03b5 < 0 \u2192 \u2203 x, x \u2208 s \u2227 a + \u03b5 < x\n[PROOFSTEP]\nrw [le_iff_forall_pos_lt_add]\n[GOAL]\nx y : \u211d\ns : Set \u211d\nh : BddAbove s\nh' : Set.Nonempty s\na : \u211d\n\u22a2 (\u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 a < sSup s + \u03b5) \u2194 \u2200 (\u03b5 : \u211d), \u03b5 < 0 \u2192 \u2203 x, x \u2208 s \u2227 a + \u03b5 < x\n[PROOFSTEP]\nrefine' \u27e8fun H \u03b5 \u03b5_neg => _, fun H \u03b5 \u03b5_pos => _\u27e9\n[GOAL]\ncase refine'_1\nx y : \u211d\ns : Set \u211d\nh : BddAbove s\nh' : Set.Nonempty s\na : \u211d\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 a < sSup s + \u03b5\n\u03b5 : \u211d\n\u03b5_neg : \u03b5 < 0\n\u22a2 \u2203 x, x \u2208 s \u2227 a + \u03b5 < x\n[PROOFSTEP]\nexact exists_lt_of_lt_csSup h' (lt_sub_iff_add_lt.mp (H _ (neg_pos.mpr \u03b5_neg)))\n[GOAL]\ncase refine'_2\nx y : \u211d\ns : Set \u211d\nh : BddAbove s\nh' : Set.Nonempty s\na : \u211d\nH : \u2200 (\u03b5 : \u211d), \u03b5 < 0 \u2192 \u2203 x, x \u2208 s \u2227 a + \u03b5 < x\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\n\u22a2 a < sSup s + \u03b5\n[PROOFSTEP]\nrcases H _ (neg_lt_zero.mpr \u03b5_pos) with \u27e8x, x_in, hx\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nx\u271d y : \u211d\ns : Set \u211d\nh : BddAbove s\nh' : Set.Nonempty s\na : \u211d\nH : \u2200 (\u03b5 : \u211d), \u03b5 < 0 \u2192 \u2203 x, x \u2208 s \u2227 a + \u03b5 < x\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nx : \u211d\nx_in : x \u2208 s\nhx : a + -\u03b5 < x\n\u22a2 a < sSup s + \u03b5\n[PROOFSTEP]\nexact sub_lt_iff_lt_add.mp (lt_csSup_of_lt h x_in hx)\n[GOAL]\nx y : \u211d\n\u22a2 \u00ac(Set.Nonempty \u2205 \u2227 BddAbove \u2205)\n[PROOFSTEP]\nsimp\n[GOAL]\nx y : \u211d\n\u03b1 : Sort u_1\ninst\u271d : IsEmpty \u03b1\nf : \u03b1 \u2192 \u211d\n\u22a2 \u2a06 (i : \u03b1), f i = 0\n[PROOFSTEP]\ndsimp [iSup]\n[GOAL]\nx y : \u211d\n\u03b1 : Sort u_1\ninst\u271d : IsEmpty \u03b1\nf : \u03b1 \u2192 \u211d\n\u22a2 sSup (Set.range fun i => f i) = 0\n[PROOFSTEP]\nconvert Real.sSup_empty\n[GOAL]\ncase h.e'_2.h.e'_3\nx y : \u211d\n\u03b1 : Sort u_1\ninst\u271d : IsEmpty \u03b1\nf : \u03b1 \u2192 \u211d\n\u22a2 (Set.range fun i => f i) = \u2205\n[PROOFSTEP]\nrw [Set.range_eq_empty_iff]\n[GOAL]\ncase h.e'_2.h.e'_3\nx y : \u211d\n\u03b1 : Sort u_1\ninst\u271d : IsEmpty \u03b1\nf : \u03b1 \u2192 \u211d\n\u22a2 IsEmpty \u03b1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx y : \u211d\n\u03b1 : Sort u_1\n\u22a2 \u2a06 (x : \u03b1), 0 = 0\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b1\n[GOAL]\ncase inl\nx y : \u211d\n\u03b1 : Sort u_1\nh\u271d : IsEmpty \u03b1\n\u22a2 \u2a06 (x : \u03b1), 0 = 0\n[PROOFSTEP]\nexact Real.ciSup_empty _\n[GOAL]\ncase inr\nx y : \u211d\n\u03b1 : Sort u_1\nh\u271d : Nonempty \u03b1\n\u22a2 \u2a06 (x : \u03b1), 0 = 0\n[PROOFSTEP]\nexact ciSup_const\n[GOAL]\nx y : \u211d\n\u22a2 sInf \u2205 = 0\n[PROOFSTEP]\nsimp [sInf_def, sSup_empty]\n[GOAL]\nx y : \u211d\n\u03b1 : Sort u_1\ninst\u271d : IsEmpty \u03b1\nf : \u03b1 \u2192 \u211d\n\u22a2 \u2a05 (i : \u03b1), f i = 0\n[PROOFSTEP]\nrw [iInf_of_empty', sInf_empty]\n[GOAL]\nx y : \u211d\n\u03b1 : Sort u_1\n\u22a2 \u2a05 (x : \u03b1), 0 = 0\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b1\n[GOAL]\ncase inl\nx y : \u211d\n\u03b1 : Sort u_1\nh\u271d : IsEmpty \u03b1\n\u22a2 \u2a05 (x : \u03b1), 0 = 0\n[PROOFSTEP]\nexact Real.ciInf_empty _\n[GOAL]\ncase inr\nx y : \u211d\n\u03b1 : Sort u_1\nh\u271d : Nonempty \u03b1\n\u22a2 \u2a05 (x : \u03b1), 0 = 0\n[PROOFSTEP]\nexact ciInf_const\n[GOAL]\nx y : \u211d\nS : Set \u211d\nhS : \u2200 (x : \u211d), x \u2208 S \u2192 0 \u2264 x\n\u22a2 0 \u2264 sSup S\n[PROOFSTEP]\nrcases S.eq_empty_or_nonempty with (rfl | \u27e8y, hy\u27e9)\n[GOAL]\ncase inl\nx y : \u211d\nhS : \u2200 (x : \u211d), x \u2208 \u2205 \u2192 0 \u2264 x\n\u22a2 0 \u2264 sSup \u2205\n[PROOFSTEP]\nexact sSup_empty.ge\n[GOAL]\ncase inr.intro\nx y\u271d : \u211d\nS : Set \u211d\nhS : \u2200 (x : \u211d), x \u2208 S \u2192 0 \u2264 x\ny : \u211d\nhy : y \u2208 S\n\u22a2 0 \u2264 sSup S\n[PROOFSTEP]\napply dite _ (fun h => le_csSup_of_le h hy <| hS y hy) fun h => (sSup_of_not_bddAbove h).ge\n[GOAL]\nx y : \u211d\nS : Set \u211d\na : \u211d\nhS : \u2200 (x : \u211d), x \u2208 S \u2192 x \u2264 a\nha : 0 \u2264 a\n\u22a2 sSup S \u2264 a\n[PROOFSTEP]\nrcases S.eq_empty_or_nonempty with (rfl | hS\u2082)\n[GOAL]\ncase inl\nx y a : \u211d\nha : 0 \u2264 a\nhS : \u2200 (x : \u211d), x \u2208 \u2205 \u2192 x \u2264 a\n\u22a2 sSup \u2205 \u2264 a\ncase inr x y : \u211d S : Set \u211d a : \u211d hS : \u2200 (x : \u211d), x \u2208 S \u2192 x \u2264 a ha : 0 \u2264 a hS\u2082 : Set.Nonempty S \u22a2 sSup S \u2264 a\n[PROOFSTEP]\nexacts [sSup_empty.trans_le ha, csSup_le hS\u2082 hS]\n[GOAL]\nx y : \u211d\nS : Set \u211d\nhS : \u2200 (x : \u211d), x \u2208 S \u2192 0 \u2264 x\n\u22a2 0 \u2264 sInf S\n[PROOFSTEP]\nrcases S.eq_empty_or_nonempty with (rfl | hS\u2082)\n[GOAL]\ncase inl\nx y : \u211d\nhS : \u2200 (x : \u211d), x \u2208 \u2205 \u2192 0 \u2264 x\n\u22a2 0 \u2264 sInf \u2205\ncase inr x y : \u211d S : Set \u211d hS : \u2200 (x : \u211d), x \u2208 S \u2192 0 \u2264 x hS\u2082 : Set.Nonempty S \u22a2 0 \u2264 sInf S\n[PROOFSTEP]\nexacts [sInf_empty.ge, le_csInf hS\u2082 hS]\n[GOAL]\nx y : \u211d\nS : Set \u211d\nhS : \u2200 (x : \u211d), x \u2208 S \u2192 x \u2264 0\n\u22a2 sInf S \u2264 0\n[PROOFSTEP]\nrcases S.eq_empty_or_nonempty with (rfl | \u27e8y, hy\u27e9)\n[GOAL]\ncase inl\nx y : \u211d\nhS : \u2200 (x : \u211d), x \u2208 \u2205 \u2192 x \u2264 0\n\u22a2 sInf \u2205 \u2264 0\n[PROOFSTEP]\nexact sInf_empty.le\n[GOAL]\ncase inr.intro\nx y\u271d : \u211d\nS : Set \u211d\nhS : \u2200 (x : \u211d), x \u2208 S \u2192 x \u2264 0\ny : \u211d\nhy : y \u2208 S\n\u22a2 sInf S \u2264 0\n[PROOFSTEP]\napply dite _ (fun h => csInf_le_of_le h hy <| hS y hy) fun h => (sInf_of_not_bddBelow h).le\n[GOAL]\nx y : \u211d\ns : Set \u211d\nh\u2081 : BddBelow s\nh\u2082 : BddAbove s\n\u22a2 sInf s \u2264 sSup s\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | hne)\n[GOAL]\ncase inl\nx y : \u211d\nh\u2081 : BddBelow \u2205\nh\u2082 : BddAbove \u2205\n\u22a2 sInf \u2205 \u2264 sSup \u2205\n[PROOFSTEP]\nrw [sInf_empty, sSup_empty]\n[GOAL]\ncase inr\nx y : \u211d\ns : Set \u211d\nh\u2081 : BddBelow s\nh\u2082 : BddAbove s\nhne : Set.Nonempty s\n\u22a2 sInf s \u2264 sSup s\n[PROOFSTEP]\nexact csInf_le_csSup h\u2081 h\u2082 hne\n[GOAL]\nx y : \u211d\nf : CauSeq \u211d abs\n\u22a2 \u2203 x, f \u2248 const abs x\n[PROOFSTEP]\nlet S := {x : \u211d | const abs x < f}\n[GOAL]\nx y : \u211d\nf : CauSeq \u211d abs\nS : Set \u211d := {x | const abs x < f}\n\u22a2 \u2203 x, f \u2248 const abs x\n[PROOFSTEP]\nhave lb : \u2203 x, x \u2208 S := exists_lt f\n[GOAL]\nx y : \u211d\nf : CauSeq \u211d abs\nS : Set \u211d := {x | const abs x < f}\nlb : \u2203 x, x \u2208 S\n\u22a2 \u2203 x, f \u2248 const abs x\n[PROOFSTEP]\nhave ub' : \u2200 x, f < const abs x \u2192 \u2200 y \u2208 S, y \u2264 x := fun x h y yS => le_of_lt <| const_lt.1 <| CauSeq.lt_trans yS h\n[GOAL]\nx y : \u211d\nf : CauSeq \u211d abs\nS : Set \u211d := {x | const abs x < f}\nlb : \u2203 x, x \u2208 S\nub' : \u2200 (x : \u211d), f < const abs x \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\n\u22a2 \u2203 x, f \u2248 const abs x\n[PROOFSTEP]\nhave ub : \u2203 x, \u2200 y \u2208 S, y \u2264 x := (exists_gt f).imp ub'\n[GOAL]\nx y : \u211d\nf : CauSeq \u211d abs\nS : Set \u211d := {x | const abs x < f}\nlb : \u2203 x, x \u2208 S\nub' : \u2200 (x : \u211d), f < const abs x \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\nub : \u2203 x, \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\n\u22a2 \u2203 x, f \u2248 const abs x\n[PROOFSTEP]\nrefine' \u27e8sSup S, ((lt_total _ _).resolve_left fun h => _).resolve_right fun h => _\u27e9\n[GOAL]\ncase refine'_1\nx y : \u211d\nf : CauSeq \u211d abs\nS : Set \u211d := {x | const abs x < f}\nlb : \u2203 x, x \u2208 S\nub' : \u2200 (x : \u211d), f < const abs x \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\nub : \u2203 x, \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\nh : f < const abs (sSup S)\n\u22a2 False\n[PROOFSTEP]\nrcases h with \u27e8\u03b5, \u03b50, i, ih\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\nx y : \u211d\nf : CauSeq \u211d abs\nS : Set \u211d := {x | const abs x < f}\nlb : \u2203 x, x \u2208 S\nub' : \u2200 (x : \u211d), f < const abs x \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\nub : \u2203 x, \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nih : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(const abs (sSup S) - f) j\n\u22a2 False\n[PROOFSTEP]\nrefine' (csSup_le lb (ub' _ _)).not_lt (sub_lt_self _ (half_pos \u03b50))\n[GOAL]\ncase refine'_1.intro.intro.intro\nx y : \u211d\nf : CauSeq \u211d abs\nS : Set \u211d := {x | const abs x < f}\nlb : \u2203 x, x \u2208 S\nub' : \u2200 (x : \u211d), f < const abs x \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\nub : \u2203 x, \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nih : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(const abs (sSup S) - f) j\n\u22a2 f < const abs (sSup S - \u03b5 / 2)\n[PROOFSTEP]\nrefine' \u27e8_, half_pos \u03b50, i, fun j ij => _\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\nx y : \u211d\nf : CauSeq \u211d abs\nS : Set \u211d := {x | const abs x < f}\nlb : \u2203 x, x \u2208 S\nub' : \u2200 (x : \u211d), f < const abs x \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\nub : \u2203 x, \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nih : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(const abs (sSup S) - f) j\nj : \u2115\nij : j \u2265 i\n\u22a2 \u03b5 / 2 \u2264 \u2191(const abs (sSup S - \u03b5 / 2) - f) j\n[PROOFSTEP]\nrw [sub_apply, const_apply, sub_right_comm, le_sub_iff_add_le, add_halves]\n[GOAL]\ncase refine'_1.intro.intro.intro\nx y : \u211d\nf : CauSeq \u211d abs\nS : Set \u211d := {x | const abs x < f}\nlb : \u2203 x, x \u2208 S\nub' : \u2200 (x : \u211d), f < const abs x \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\nub : \u2203 x, \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nih : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(const abs (sSup S) - f) j\nj : \u2115\nij : j \u2265 i\n\u22a2 \u03b5 \u2264 sSup S - \u2191f j\n[PROOFSTEP]\nexact ih _ ij\n[GOAL]\ncase refine'_2\nx y : \u211d\nf : CauSeq \u211d abs\nS : Set \u211d := {x | const abs x < f}\nlb : \u2203 x, x \u2208 S\nub' : \u2200 (x : \u211d), f < const abs x \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\nub : \u2203 x, \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\nh : const abs (sSup S) < f\n\u22a2 False\n[PROOFSTEP]\nrcases h with \u27e8\u03b5, \u03b50, i, ih\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\nx y : \u211d\nf : CauSeq \u211d abs\nS : Set \u211d := {x | const abs x < f}\nlb : \u2203 x, x \u2208 S\nub' : \u2200 (x : \u211d), f < const abs x \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\nub : \u2203 x, \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nih : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(f - const abs (sSup S)) j\n\u22a2 False\n[PROOFSTEP]\nrefine' (le_csSup ub _).not_lt ((lt_add_iff_pos_left _).2 (half_pos \u03b50))\n[GOAL]\ncase refine'_2.intro.intro.intro\nx y : \u211d\nf : CauSeq \u211d abs\nS : Set \u211d := {x | const abs x < f}\nlb : \u2203 x, x \u2208 S\nub' : \u2200 (x : \u211d), f < const abs x \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\nub : \u2203 x, \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nih : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(f - const abs (sSup S)) j\n\u22a2 \u03b5 / 2 + sSup S \u2208 S\n[PROOFSTEP]\nrefine' \u27e8_, half_pos \u03b50, i, fun j ij => _\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\nx y : \u211d\nf : CauSeq \u211d abs\nS : Set \u211d := {x | const abs x < f}\nlb : \u2203 x, x \u2208 S\nub' : \u2200 (x : \u211d), f < const abs x \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\nub : \u2203 x, \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nih : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(f - const abs (sSup S)) j\nj : \u2115\nij : j \u2265 i\n\u22a2 \u03b5 / 2 \u2264 \u2191(f - const abs (\u03b5 / 2 + sSup S)) j\n[PROOFSTEP]\nrw [sub_apply, const_apply, add_comm, \u2190 sub_sub, le_sub_iff_add_le, add_halves]\n[GOAL]\ncase refine'_2.intro.intro.intro\nx y : \u211d\nf : CauSeq \u211d abs\nS : Set \u211d := {x | const abs x < f}\nlb : \u2203 x, x \u2208 S\nub' : \u2200 (x : \u211d), f < const abs x \u2192 \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\nub : \u2203 x, \u2200 (y : \u211d), y \u2208 S \u2192 y \u2264 x\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nih : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(f - const abs (sSup S)) j\nj : \u2115\nij : j \u2265 i\n\u22a2 \u03b5 \u2264 \u2191f j - sSup S\n[PROOFSTEP]\nexact ih _ ij\n[GOAL]\nx\u271d y : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\n\u22a2 \u2a05 (r : \u2191(Ioi x)), f \u2191r = \u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\nx\u271d y : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\n\u22a2 \u2a05 (r : \u2191(Ioi x)), f \u2191r \u2264 \u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q\n[PROOFSTEP]\nhave : Nonempty { r' : \u211a // x < \u2191r' } := by\n obtain \u27e8r, hrx\u27e9 := exists_rat_gt x\n exact \u27e8\u27e8r, hrx\u27e9\u27e9\n[GOAL]\nx\u271d y : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\n\u22a2 Nonempty { r' // x < \u2191r' }\n[PROOFSTEP]\nobtain \u27e8r, hrx\u27e9 := exists_rat_gt x\n[GOAL]\ncase intro\nx\u271d y : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nr : \u211a\nhrx : x < \u2191r\n\u22a2 Nonempty { r' // x < \u2191r' }\n[PROOFSTEP]\nexact \u27e8\u27e8r, hrx\u27e9\u27e9\n[GOAL]\ncase refine'_1\nx\u271d y : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nthis : Nonempty { r' // x < \u2191r' }\n\u22a2 \u2a05 (r : \u2191(Ioi x)), f \u2191r \u2264 \u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q\n[PROOFSTEP]\nrefine' le_ciInf fun r => _\n[GOAL]\ncase refine'_1\nx\u271d y : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nthis : Nonempty { r' // x < \u2191r' }\nr : { q' // x < \u2191q' }\n\u22a2 \u2a05 (r : \u2191(Ioi x)), f \u2191r \u2264 f \u2191\u2191r\n[PROOFSTEP]\nobtain \u27e8y, hxy, hyr\u27e9 := exists_rat_btwn r.prop\n[GOAL]\ncase refine'_1.intro.intro\nx\u271d y\u271d : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nthis : Nonempty { r' // x < \u2191r' }\nr : { q' // x < \u2191q' }\ny : \u211a\nhxy : x < \u2191y\nhyr : \u2191y < \u2191\u2191r\n\u22a2 \u2a05 (r : \u2191(Ioi x)), f \u2191r \u2264 f \u2191\u2191r\n[PROOFSTEP]\nrefine' ciInf_set_le hf (hxy.trans _)\n[GOAL]\ncase refine'_1.intro.intro\nx\u271d y\u271d : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nthis : Nonempty { r' // x < \u2191r' }\nr : { q' // x < \u2191q' }\ny : \u211a\nhxy : x < \u2191y\nhyr : \u2191y < \u2191\u2191r\n\u22a2 \u2191y < \u2191\u2191r\n[PROOFSTEP]\nexact_mod_cast hyr\n[GOAL]\ncase refine'_2\nx\u271d y : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\n\u22a2 \u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q \u2264 \u2a05 (r : \u2191(Ioi x)), f \u2191r\n[PROOFSTEP]\nrefine' le_ciInf fun q => _\n[GOAL]\ncase refine'_2\nx\u271d y : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\n\u22a2 \u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q \u2264 f \u2191q\n[PROOFSTEP]\nhave hq := q.prop\n[GOAL]\ncase refine'_2\nx\u271d y : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : \u2191q \u2208 Ioi x\n\u22a2 \u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q \u2264 f \u2191q\n[PROOFSTEP]\nrw [mem_Ioi] at hq \n[GOAL]\ncase refine'_2\nx\u271d y : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\n\u22a2 \u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q \u2264 f \u2191q\n[PROOFSTEP]\nobtain \u27e8y, hxy, hyq\u27e9 := exists_rat_btwn hq\n[GOAL]\ncase refine'_2.intro.intro\nx\u271d y\u271d : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\n\u22a2 \u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q \u2264 f \u2191q\n[PROOFSTEP]\nrefine' (ciInf_le _ _).trans _\n[GOAL]\ncase refine'_2.intro.intro.refine'_1\nx\u271d y\u271d : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\n\u22a2 BddBelow (range fun q => f \u2191\u2191q)\n[PROOFSTEP]\nrefine' \u27e8hf.some, fun z => _\u27e9\n[GOAL]\ncase refine'_2.intro.intro.refine'_1\nx\u271d y\u271d : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\nz : \u211d\n\u22a2 (z \u2208 range fun q => f \u2191\u2191q) \u2192 Set.Nonempty.some hf \u2264 z\n[PROOFSTEP]\nrintro \u27e8u, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.refine'_1.intro\nx\u271d y\u271d : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\nu : { q' // x < \u2191q' }\n\u22a2 Set.Nonempty.some hf \u2264 (fun q => f \u2191\u2191q) u\n[PROOFSTEP]\nsuffices hfu : f u \u2208 f '' Ioi x\n[GOAL]\ncase refine'_2.intro.intro.refine'_1.intro\nx\u271d y\u271d : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\nu : { q' // x < \u2191q' }\nhfu : f \u2191\u2191u \u2208 f '' Ioi x\n\u22a2 Set.Nonempty.some hf \u2264 (fun q => f \u2191\u2191q) u\ncase hfu\nx\u271d y\u271d : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\nu : { q' // x < \u2191q' }\n\u22a2 f \u2191\u2191u \u2208 f '' Ioi x\n[PROOFSTEP]\nexact hf.choose_spec hfu\n[GOAL]\ncase hfu\nx\u271d y\u271d : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\nu : { q' // x < \u2191q' }\n\u22a2 f \u2191\u2191u \u2208 f '' Ioi x\n[PROOFSTEP]\nexact \u27e8u, u.prop, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.refine'_2\nx\u271d y\u271d : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\n\u22a2 { q' // x < \u2191q' }\n[PROOFSTEP]\nexact \u27e8y, hxy\u27e9\n[GOAL]\ncase refine'_2.intro.intro.refine'_3\nx\u271d y\u271d : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\n\u22a2 f \u2191\u2191{ val := y, property := hxy } \u2264 f \u2191q\n[PROOFSTEP]\nrefine' hf_mono (le_trans _ hyq.le)\n[GOAL]\ncase refine'_2.intro.intro.refine'_3\nx\u271d y\u271d : \u211d\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\n\u22a2 \u2191\u2191{ val := y, property := hxy } \u2264 \u2191y\n[PROOFSTEP]\nnorm_cast\n", "meta": {"mathlib_filename": "Mathlib.Data.Real.Basic", "llama_tokens": 51509, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754371026367, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.4477688504758319}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : Decidable (a < x \u2227 x < b)\n\u22a2 Decidable (x \u2208 Ioo a b)\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : Decidable (a \u2264 x \u2227 x < b)\n\u22a2 Decidable (x \u2208 Ico a b)\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : Decidable (x < b)\n\u22a2 Decidable (x \u2208 Iio b)\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : Decidable (a \u2264 x \u2227 x \u2264 b)\n\u22a2 Decidable (x \u2208 Icc a b)\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : Decidable (x \u2264 b)\n\u22a2 Decidable (x \u2208 Iic b)\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : Decidable (a < x \u2227 x \u2264 b)\n\u22a2 Decidable (x \u2208 Ioc a b)\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : Decidable (a \u2264 x)\n\u22a2 Decidable (x \u2208 Ici a)\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : Decidable (a < x)\n\u22a2 Decidable (x \u2208 Ioi a)\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 a \u2208 Ioo a b \u2194 False\n[PROOFSTEP]\nsimp [lt_irrefl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 a \u2208 Ico a b \u2194 a < b\n[PROOFSTEP]\nsimp [le_refl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 a \u2208 Icc a b \u2194 a \u2264 b\n[PROOFSTEP]\nsimp [le_refl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 a \u2208 Ioc a b \u2194 False\n[PROOFSTEP]\nsimp [lt_irrefl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 a \u2208 Ici a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 b \u2208 Ioo a b \u2194 False\n[PROOFSTEP]\nsimp [lt_irrefl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 b \u2208 Ico a b \u2194 False\n[PROOFSTEP]\nsimp [lt_irrefl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 b \u2208 Icc a b \u2194 a \u2264 b\n[PROOFSTEP]\nsimp [le_refl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 b \u2208 Ioc a b \u2194 a < b\n[PROOFSTEP]\nsimp [le_refl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 a \u2208 Iic a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Icc a b = \u2205 \u2194 \u00aca \u2264 b\n[PROOFSTEP]\nrw [\u2190 not_nonempty_iff_eq_empty, not_iff_not, nonempty_Icc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Ico a b = \u2205 \u2194 \u00aca < b\n[PROOFSTEP]\nrw [\u2190 not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ico]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Ioc a b = \u2205 \u2194 \u00aca < b\n[PROOFSTEP]\nrw [\u2190 not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ioc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : DenselyOrdered \u03b1\n\u22a2 Ioo a b = \u2205 \u2194 \u00aca < b\n[PROOFSTEP]\nrw [\u2190 not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ioo]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na\u271d b c a : \u03b1\n\u22a2 \u2200 (x : \u03b1), x \u2208 Icc a a \u2194 x \u2208 {a}\n[PROOFSTEP]\nsimp [Icc, le_antisymm_iff, and_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\n\u22a2 Icc a b = {c} \u2194 a = c \u2227 b = c\n[PROOFSTEP]\nrefine' \u27e8fun h => _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : Icc a b = {c}\n\u22a2 a = c \u2227 b = c\n[PROOFSTEP]\nhave hab : a \u2264 b := nonempty_Icc.1 (h.symm.subst <| singleton_nonempty c)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : Icc a b = {c}\nhab : a \u2264 b\n\u22a2 a = c \u2227 b = c\n[PROOFSTEP]\nexact \u27e8eq_of_mem_singleton <| h.subst <| left_mem_Icc.2 hab, eq_of_mem_singleton <| h.subst <| right_mem_Icc.2 hab\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\n\u22a2 a = c \u2227 b = c \u2192 Icc a b = {c}\n[PROOFSTEP]\nrintro \u27e8rfl, rfl\u27e9\n[GOAL]\ncase refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\nb : \u03b1\n\u22a2 Icc b b = {b}\n[PROOFSTEP]\nexact Icc_self _\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\u271d\na\u271d b\u271d c : \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 Set.Subsingleton (Icc a b) \u2194 b \u2264 a\n[PROOFSTEP]\nrefine' \u27e8fun h \u21a6 _, subsingleton_Icc_of_ge\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\u271d\na\u271d b\u271d c : \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : Set.Subsingleton (Icc a b)\n\u22a2 b \u2264 a\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\u271d\na\u271d b\u271d c : \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a < b\n\u22a2 \u00acSet.Subsingleton (Icc a b)\n[PROOFSTEP]\nsimp only [ge_iff_le, gt_iff_lt, not_subsingleton_iff]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\u271d\na\u271d b\u271d c : \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a < b\n\u22a2 Set.Nontrivial (Icc a b)\n[PROOFSTEP]\nexact \u27e8a, \u27e8le_refl _, h.le\u27e9, b, \u27e8h.le, le_refl _\u27e9, h.ne\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c x : \u03b1\n\u22a2 x \u2208 Icc a b \\ {a} \u2194 x \u2208 Ioc a b\n[PROOFSTEP]\nsimp [lt_iff_le_and_ne, eq_comm, and_right_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c x : \u03b1\n\u22a2 x \u2208 Icc a b \\ {b} \u2194 x \u2208 Ico a b\n[PROOFSTEP]\nsimp [lt_iff_le_and_ne, and_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c x : \u03b1\n\u22a2 x \u2208 Ico a b \\ {a} \u2194 x \u2208 Ioo a b\n[PROOFSTEP]\nsimp [and_right_comm, \u2190 lt_iff_le_and_ne, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c x : \u03b1\n\u22a2 x \u2208 Ioc a b \\ {b} \u2194 x \u2208 Ioo a b\n[PROOFSTEP]\nsimp [and_assoc, \u2190 lt_iff_le_and_ne]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\n\u22a2 Icc a b \\ {a, b} = Ioo a b\n[PROOFSTEP]\nrw [insert_eq, \u2190 diff_diff, Icc_diff_left, Ioc_diff_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c x : \u03b1\n\u22a2 x \u2208 Ici a \\ {a} \u2194 x \u2208 Ioi a\n[PROOFSTEP]\nsimp [lt_iff_le_and_ne, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c x : \u03b1\n\u22a2 x \u2208 Iic a \\ {a} \u2194 x \u2208 Iio a\n[PROOFSTEP]\nsimp [lt_iff_le_and_ne]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : a < b\n\u22a2 Ico a b \\ Ioo a b = {a}\n[PROOFSTEP]\nrw [\u2190 Ico_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 <| left_mem_Ico.2 h)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : a < b\n\u22a2 Ioc a b \\ Ioo a b = {b}\n[PROOFSTEP]\nrw [\u2190 Ioc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <| right_mem_Ioc.2 h)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : a \u2264 b\n\u22a2 Icc a b \\ Ico a b = {b}\n[PROOFSTEP]\nrw [\u2190 Icc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <| right_mem_Icc.2 h)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : a \u2264 b\n\u22a2 Icc a b \\ Ioc a b = {a}\n[PROOFSTEP]\nrw [\u2190 Icc_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 <| left_mem_Icc.2 h)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : a \u2264 b\n\u22a2 Icc a b \\ Ioo a b = {a, b}\n[PROOFSTEP]\nrw [\u2190 Icc_diff_both, diff_diff_cancel_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : a \u2264 b\n\u22a2 {a, b} \u2286 Icc a b\n[PROOFSTEP]\nsimp [insert_subset_iff, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\n\u22a2 Ici a \\ Ioi a = {a}\n[PROOFSTEP]\nrw [\u2190 Ici_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 left_mem_Ici)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\n\u22a2 Iic a \\ Iio a = {a}\n[PROOFSTEP]\nrw [\u2190 Iic_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 right_mem_Iic)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c x : \u03b1\n\u22a2 x \u2208 Ioi a \u222a {a} \u2194 x \u2208 Ici a\n[PROOFSTEP]\nsimp [eq_comm, le_iff_eq_or_lt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nhab : a < b\n\u22a2 Ioo a b \u222a {a} = Ico a b\n[PROOFSTEP]\nrw [\u2190 Ico_diff_left, diff_union_self, union_eq_self_of_subset_right (singleton_subset_iff.2 <| left_mem_Ico.2 hab)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nhab : a < b\n\u22a2 Ioo a b \u222a {b} = Ioc a b\n[PROOFSTEP]\nsimpa only [dual_Ioo, dual_Ico] using Ioo_union_left hab.dual\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nhab : a \u2264 b\n\u22a2 Ioc a b \u222a {a} = Icc a b\n[PROOFSTEP]\nrw [\u2190 Icc_diff_left, diff_union_self, union_eq_self_of_subset_right (singleton_subset_iff.2 <| left_mem_Icc.2 hab)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nhab : a \u2264 b\n\u22a2 Ico a b \u222a {b} = Icc a b\n[PROOFSTEP]\nsimpa only [dual_Ioc, dual_Icc] using Ioc_union_left hab.dual\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : a \u2264 b\n\u22a2 insert b (Ico a b) = Icc a b\n[PROOFSTEP]\nrw [insert_eq, union_comm, Ico_union_right h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : a \u2264 b\n\u22a2 insert a (Ioc a b) = Icc a b\n[PROOFSTEP]\nrw [insert_eq, union_comm, Ioc_union_left h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : a < b\n\u22a2 insert a (Ioo a b) = Ico a b\n[PROOFSTEP]\nrw [insert_eq, union_comm, Ioo_union_left h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\nh : a < b\n\u22a2 insert b (Ioo a b) = Ioc a b\n[PROOFSTEP]\nrw [insert_eq, union_comm, Ioo_union_right h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioi a \u2286 s\nhc : s \u2286 Ici a\nh : a \u2208 s\n\u22a2 Ici a \u2286 s\n[PROOFSTEP]\nrw [\u2190 Ioi_union_left, union_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioi a \u2286 s\nhc : s \u2286 Ici a\nh : a \u2208 s\n\u22a2 Ioi a \u2286 s \u2227 {a} \u2286 s\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\n\u22a2 s \u2208 {Icc a b, Ico a b, Ioc a b, Ioo a b}\n[PROOFSTEP]\nclassical\nby_cases ha : a \u2208 s <;> by_cases hb : b \u2208 s\n\u00b7 refine' Or.inl (Subset.antisymm hc _)\n rwa [\u2190 Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha, \u2190 Icc_diff_right, diff_singleton_subset_iff,\n insert_eq_of_mem hb] at ho \n\u00b7 refine' Or.inr <| Or.inl <| Subset.antisymm _ _\n \u00b7 rw [\u2190 Icc_diff_right]\n exact subset_diff_singleton hc hb\n \u00b7 rwa [\u2190 Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha] at ho \n\u00b7 refine' Or.inr <| Or.inr <| Or.inl <| Subset.antisymm _ _\n \u00b7 rw [\u2190 Icc_diff_left]\n exact subset_diff_singleton hc ha\n \u00b7 rwa [\u2190 Ioc_diff_right, diff_singleton_subset_iff, insert_eq_of_mem hb] at ho \n\u00b7 refine' Or.inr <| Or.inr <| Or.inr <| Subset.antisymm _ ho\n rw [\u2190 Ico_diff_left, \u2190 Icc_diff_right]\n apply_rules [subset_diff_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\n\u22a2 s \u2208 {Icc a b, Ico a b, Ioc a b, Ioo a b}\n[PROOFSTEP]\nby_cases ha : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\nha : a \u2208 s\n\u22a2 s \u2208 {Icc a b, Ico a b, Ioc a b, Ioo a b}\n[PROOFSTEP]\nby_cases hb : b \u2208 s\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\nha : \u00aca \u2208 s\n\u22a2 s \u2208 {Icc a b, Ico a b, Ioc a b, Ioo a b}\n[PROOFSTEP]\nby_cases hb : b \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 s \u2208 {Icc a b, Ico a b, Ioc a b, Ioo a b}\n[PROOFSTEP]\nrefine' Or.inl (Subset.antisymm hc _)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 Icc a b \u2286 s\n[PROOFSTEP]\nrwa [\u2190 Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha, \u2190 Icc_diff_right, diff_singleton_subset_iff,\n insert_eq_of_mem hb] at ho \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\nha : a \u2208 s\nhb : \u00acb \u2208 s\n\u22a2 s \u2208 {Icc a b, Ico a b, Ioc a b, Ioo a b}\n[PROOFSTEP]\nrefine' Or.inr <| Or.inl <| Subset.antisymm _ _\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\nha : a \u2208 s\nhb : \u00acb \u2208 s\n\u22a2 s \u2286 Ico a b\n[PROOFSTEP]\nrw [\u2190 Icc_diff_right]\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\nha : a \u2208 s\nhb : \u00acb \u2208 s\n\u22a2 s \u2286 Icc a b \\ {b}\n[PROOFSTEP]\nexact subset_diff_singleton hc hb\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\nha : a \u2208 s\nhb : \u00acb \u2208 s\n\u22a2 Ico a b \u2286 s\n[PROOFSTEP]\nrwa [\u2190 Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha] at ho \n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\nha : \u00aca \u2208 s\nhb : b \u2208 s\n\u22a2 s \u2208 {Icc a b, Ico a b, Ioc a b, Ioo a b}\n[PROOFSTEP]\nrefine' Or.inr <| Or.inr <| Or.inl <| Subset.antisymm _ _\n[GOAL]\ncase pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\nha : \u00aca \u2208 s\nhb : b \u2208 s\n\u22a2 s \u2286 Ioc a b\n[PROOFSTEP]\nrw [\u2190 Icc_diff_left]\n[GOAL]\ncase pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\nha : \u00aca \u2208 s\nhb : b \u2208 s\n\u22a2 s \u2286 Icc a b \\ {a}\n[PROOFSTEP]\nexact subset_diff_singleton hc ha\n[GOAL]\ncase pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\nha : \u00aca \u2208 s\nhb : b \u2208 s\n\u22a2 Ioc a b \u2286 s\n[PROOFSTEP]\nrwa [\u2190 Ioc_diff_right, diff_singleton_subset_iff, insert_eq_of_mem hb] at ho \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\nha : \u00aca \u2208 s\nhb : \u00acb \u2208 s\n\u22a2 s \u2208 {Icc a b, Ico a b, Ioc a b, Ioo a b}\n[PROOFSTEP]\nrefine' Or.inr <| Or.inr <| Or.inr <| Subset.antisymm _ ho\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\nha : \u00aca \u2208 s\nhb : \u00acb \u2208 s\n\u22a2 s \u2286 Ioo a b\n[PROOFSTEP]\nrw [\u2190 Ico_diff_left, \u2190 Icc_diff_right]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\ns : Set \u03b1\nho : Ioo a b \u2286 s\nhc : s \u2286 Icc a b\nha : \u00aca \u2208 s\nhb : \u00acb \u2208 s\n\u22a2 s \u2286 (Icc a b \\ {b}) \\ {a}\n[PROOFSTEP]\napply_rules [subset_diff_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderTop \u03b1\na : \u03b1\n\u22a2 Icc a \u22a4 = Ici a\n[PROOFSTEP]\nsimp [\u2190 Ici_inter_Iic]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderTop \u03b1\na : \u03b1\n\u22a2 Ioc a \u22a4 = Ioi a\n[PROOFSTEP]\nsimp [\u2190 Ioi_inter_Iic]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\na : \u03b1\n\u22a2 Icc \u22a5 a = Iic a\n[PROOFSTEP]\nsimp [\u2190 Ici_inter_Iic]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\na : \u03b1\n\u22a2 Ico \u22a5 a = Iio a\n[PROOFSTEP]\nsimp [\u2190 Ici_inter_Iio]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : BoundedOrder \u03b1\n\u22a2 Icc \u22a5 \u22a4 = univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ici a \\ Ici b = Ico a b\n[PROOFSTEP]\nrw [diff_eq, compl_Ici, Ici_inter_Iio]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ici a \\ Ioi b = Icc a b\n[PROOFSTEP]\nrw [diff_eq, compl_Ioi, Ici_inter_Iic]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioi a \\ Ioi b = Ioc a b\n[PROOFSTEP]\nrw [diff_eq, compl_Ioi, Ioi_inter_Iic]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioi a \\ Ici b = Ioo a b\n[PROOFSTEP]\nrw [diff_eq, compl_Ici, Ioi_inter_Iio]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Iic b \\ Iic a = Ioc a b\n[PROOFSTEP]\nrw [diff_eq, compl_Iic, inter_comm, Ioi_inter_Iic]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Iio b \\ Iic a = Ioo a b\n[PROOFSTEP]\nrw [diff_eq, compl_Iic, inter_comm, Ioi_inter_Iio]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Iic b \\ Iio a = Icc a b\n[PROOFSTEP]\nrw [diff_eq, compl_Iio, inter_comm, Ici_inter_Iic]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Iio b \\ Iio a = Ico a b\n[PROOFSTEP]\nrw [diff_eq, compl_Iio, inter_comm, Ici_inter_Iio]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : a\u2081 < b\u2081\n\u22a2 Ioc a\u2081 b\u2081 \u2286 Ioc a\u2082 b\u2082 \u2194 b\u2081 \u2264 b\u2082 \u2227 a\u2082 \u2264 a\u2081\n[PROOFSTEP]\nconvert @Ico_subset_Ico_iff \u03b1\u1d52\u1d48 _ b\u2081 b\u2082 a\u2081 a\u2082 h\u2081 using 2\n[GOAL]\ncase h.e'_1.h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : a\u2081 < b\u2081\ne_1\u271d : Set \u03b1 = Set \u03b1\u1d52\u1d48\n\u22a2 Ioc a\u2081 b\u2081 = Ico b\u2081 a\u2081\n[PROOFSTEP]\nexact (@dual_Ico \u03b1 _ _ _).symm\n[GOAL]\ncase h.e'_1.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : a\u2081 < b\u2081\ne_1\u271d : Set \u03b1 = Set \u03b1\u1d52\u1d48\n\u22a2 Ioc a\u2082 b\u2082 = Ico b\u2082 a\u2082\n[PROOFSTEP]\nexact (@dual_Ico \u03b1 _ _ _).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\ninst\u271d : DenselyOrdered \u03b1\nh\u2081 : a\u2081 < b\u2081\nh : Ioo a\u2081 b\u2081 \u2286 Ioo a\u2082 b\u2082\n\u22a2 a\u2082 \u2264 a\u2081 \u2227 b\u2081 \u2264 b\u2082\n[PROOFSTEP]\nrcases exists_between h\u2081 with \u27e8x, xa, xb\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\ninst\u271d : DenselyOrdered \u03b1\nh\u2081 : a\u2081 < b\u2081\nh : Ioo a\u2081 b\u2081 \u2286 Ioo a\u2082 b\u2082\nx : \u03b1\nxa : a\u2081 < x\nxb : x < b\u2081\n\u22a2 a\u2082 \u2264 a\u2081 \u2227 b\u2081 \u2264 b\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\ninst\u271d : DenselyOrdered \u03b1\nh\u2081 : a\u2081 < b\u2081\nh : Ioo a\u2081 b\u2081 \u2286 Ioo a\u2082 b\u2082\nx : \u03b1\nxa : a\u2081 < x\nxb : x < b\u2081\n\u22a2 a\u2082 \u2264 a\u2081\n[PROOFSTEP]\nrefine' le_of_not_lt fun h' => _\n[GOAL]\ncase intro.intro.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\ninst\u271d : DenselyOrdered \u03b1\nh\u2081 : a\u2081 < b\u2081\nh : Ioo a\u2081 b\u2081 \u2286 Ioo a\u2082 b\u2082\nx : \u03b1\nxa : a\u2081 < x\nxb : x < b\u2081\n\u22a2 b\u2081 \u2264 b\u2082\n[PROOFSTEP]\nrefine' le_of_not_lt fun h' => _\n[GOAL]\ncase intro.intro.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\ninst\u271d : DenselyOrdered \u03b1\nh\u2081 : a\u2081 < b\u2081\nh : Ioo a\u2081 b\u2081 \u2286 Ioo a\u2082 b\u2082\nx : \u03b1\nxa : a\u2081 < x\nxb : x < b\u2081\nh' : a\u2081 < a\u2082\n\u22a2 False\n[PROOFSTEP]\nhave ab := (h \u27e8xa, xb\u27e9).1.trans xb\n[GOAL]\ncase intro.intro.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\ninst\u271d : DenselyOrdered \u03b1\nh\u2081 : a\u2081 < b\u2081\nh : Ioo a\u2081 b\u2081 \u2286 Ioo a\u2082 b\u2082\nx : \u03b1\nxa : a\u2081 < x\nxb : x < b\u2081\nh' : a\u2081 < a\u2082\nab : a\u2082 < b\u2081\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ (h \u27e8h', ab\u27e9).1\n[GOAL]\ncase intro.intro.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\ninst\u271d : DenselyOrdered \u03b1\nh\u2081 : a\u2081 < b\u2081\nh : Ioo a\u2081 b\u2081 \u2286 Ioo a\u2082 b\u2082\nx : \u03b1\nxa : a\u2081 < x\nxb : x < b\u2081\nh' : b\u2082 < b\u2081\n\u22a2 False\n[PROOFSTEP]\nhave ab := xa.trans (h \u27e8xa, xb\u27e9).2\n[GOAL]\ncase intro.intro.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\ninst\u271d : DenselyOrdered \u03b1\nh\u2081 : a\u2081 < b\u2081\nh : Ioo a\u2081 b\u2081 \u2286 Ioo a\u2082 b\u2082\nx : \u03b1\nxa : a\u2081 < x\nxb : x < b\u2081\nh' : b\u2082 < b\u2081\nab : a\u2081 < b\u2082\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ (h \u27e8ab, h'\u27e9).2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2081 < b\u2081 \u2228 a\u2082 < b\u2082\ne : Ico a\u2081 b\u2081 = Ico a\u2082 b\u2082\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082\n[PROOFSTEP]\nsimp [Subset.antisymm_iff] at e \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2081 < b\u2081 \u2228 a\u2082 < b\u2082\ne : Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2227 Ico a\u2082 b\u2082 \u2286 Ico a\u2081 b\u2081\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082\n[PROOFSTEP]\nsimp [le_antisymm_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2081 < b\u2081 \u2228 a\u2082 < b\u2082\ne : Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2227 Ico a\u2082 b\u2082 \u2286 Ico a\u2081 b\u2081\n\u22a2 (a\u2081 \u2264 a\u2082 \u2227 a\u2082 \u2264 a\u2081) \u2227 b\u2081 \u2264 b\u2082 \u2227 b\u2082 \u2264 b\u2081\n[PROOFSTEP]\ncases' h with h h <;> simp [Ico_subset_Ico_iff h] at e <;> [rcases e with \u27e8\u27e8h\u2081, h\u2082\u27e9, e'\u27e9;\n rcases e with \u27e8e', \u27e8h\u2081, h\u2082\u27e9\u27e9] <;>\n -- Porting note: restore `tauto`have hab := (Ico_subset_Ico_iff <| h\u2081.trans_lt <| h.trans_le h\u2082).1 e' <;>\n [exact \u27e8\u27e8hab.left, h\u2081\u27e9, \u27e8h\u2082, hab.right\u27e9\u27e9; exact \u27e8\u27e8h\u2081, hab.left\u27e9, \u27e8hab.right, h\u2082\u27e9\u27e9]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2081 < b\u2081 \u2228 a\u2082 < b\u2082\ne : Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2227 Ico a\u2082 b\u2082 \u2286 Ico a\u2081 b\u2081\n\u22a2 (a\u2081 \u2264 a\u2082 \u2227 a\u2082 \u2264 a\u2081) \u2227 b\u2081 \u2264 b\u2082 \u2227 b\u2082 \u2264 b\u2081\n[PROOFSTEP]\ncases' h with h h <;> simp [Ico_subset_Ico_iff h] at e <;> [rcases e with \u27e8\u27e8h\u2081, h\u2082\u27e9, e'\u27e9; rcases e with \u27e8e', \u27e8h\u2081, h\u2082\u27e9\u27e9]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2081 < b\u2081 \u2228 a\u2082 < b\u2082\ne : Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2227 Ico a\u2082 b\u2082 \u2286 Ico a\u2081 b\u2081\n\u22a2 (a\u2081 \u2264 a\u2082 \u2227 a\u2082 \u2264 a\u2081) \u2227 b\u2081 \u2264 b\u2082 \u2227 b\u2082 \u2264 b\u2081\n[PROOFSTEP]\ncases' h with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\ne : Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2227 Ico a\u2082 b\u2082 \u2286 Ico a\u2081 b\u2081\nh : a\u2081 < b\u2081\n\u22a2 (a\u2081 \u2264 a\u2082 \u2227 a\u2082 \u2264 a\u2081) \u2227 b\u2081 \u2264 b\u2082 \u2227 b\u2082 \u2264 b\u2081\n[PROOFSTEP]\nsimp [Ico_subset_Ico_iff h] at e \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\ne : Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2227 Ico a\u2082 b\u2082 \u2286 Ico a\u2081 b\u2081\nh : a\u2082 < b\u2082\n\u22a2 (a\u2081 \u2264 a\u2082 \u2227 a\u2082 \u2264 a\u2081) \u2227 b\u2081 \u2264 b\u2082 \u2227 b\u2082 \u2264 b\u2081\n[PROOFSTEP]\nsimp [Ico_subset_Ico_iff h] at e \n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2081 < b\u2081\ne : (a\u2082 \u2264 a\u2081 \u2227 b\u2081 \u2264 b\u2082) \u2227 Ico a\u2082 b\u2082 \u2286 Ico a\u2081 b\u2081\n\u22a2 (a\u2081 \u2264 a\u2082 \u2227 a\u2082 \u2264 a\u2081) \u2227 b\u2081 \u2264 b\u2082 \u2227 b\u2082 \u2264 b\u2081\n[PROOFSTEP]\nrcases e with \u27e8\u27e8h\u2081, h\u2082\u27e9, e'\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2082 < b\u2082\ne : Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2227 a\u2081 \u2264 a\u2082 \u2227 b\u2082 \u2264 b\u2081\n\u22a2 (a\u2081 \u2264 a\u2082 \u2227 a\u2082 \u2264 a\u2081) \u2227 b\u2081 \u2264 b\u2082 \u2227 b\u2082 \u2264 b\u2081\n[PROOFSTEP]\nrcases e with \u27e8e', \u27e8h\u2081, h\u2082\u27e9\u27e9\n[GOAL]\ncase inl.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2081 < b\u2081\ne' : Ico a\u2082 b\u2082 \u2286 Ico a\u2081 b\u2081\nh\u2081 : a\u2082 \u2264 a\u2081\nh\u2082 : b\u2081 \u2264 b\u2082\n\u22a2 (a\u2081 \u2264 a\u2082 \u2227 a\u2082 \u2264 a\u2081) \u2227 b\u2081 \u2264 b\u2082 \u2227 b\u2082 \u2264 b\u2081\n[PROOFSTEP]\nhave hab := (Ico_subset_Ico_iff <| h\u2081.trans_lt <| h.trans_le h\u2082).1 e'\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2082 < b\u2082\ne' : Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082\nh\u2081 : a\u2081 \u2264 a\u2082\nh\u2082 : b\u2082 \u2264 b\u2081\n\u22a2 (a\u2081 \u2264 a\u2082 \u2227 a\u2082 \u2264 a\u2081) \u2227 b\u2081 \u2264 b\u2082 \u2227 b\u2082 \u2264 b\u2081\n[PROOFSTEP]\nhave hab := (Ico_subset_Ico_iff <| h\u2081.trans_lt <| h.trans_le h\u2082).1 e'\n[GOAL]\ncase inl.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2081 < b\u2081\ne' : Ico a\u2082 b\u2082 \u2286 Ico a\u2081 b\u2081\nh\u2081 : a\u2082 \u2264 a\u2081\nh\u2082 : b\u2081 \u2264 b\u2082\nhab : a\u2081 \u2264 a\u2082 \u2227 b\u2082 \u2264 b\u2081\n\u22a2 (a\u2081 \u2264 a\u2082 \u2227 a\u2082 \u2264 a\u2081) \u2227 b\u2081 \u2264 b\u2082 \u2227 b\u2082 \u2264 b\u2081\n[PROOFSTEP]\nexact \u27e8\u27e8hab.left, h\u2081\u27e9, \u27e8h\u2082, hab.right\u27e9\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2082 < b\u2082\ne' : Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082\nh\u2081 : a\u2081 \u2264 a\u2082\nh\u2082 : b\u2082 \u2264 b\u2081\nhab : a\u2082 \u2264 a\u2081 \u2227 b\u2081 \u2264 b\u2082\n\u22a2 (a\u2081 \u2264 a\u2082 \u2227 a\u2082 \u2264 a\u2081) \u2227 b\u2081 \u2264 b\u2082 \u2227 b\u2082 \u2264 b\u2081\n[PROOFSTEP]\nexact \u27e8\u27e8h\u2081, hab.left\u27e9, \u27e8hab.right, h\u2082\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : a\u2081 < b\u2081 \u2228 a\u2082 < b\u2082\nx\u271d : a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082\nh\u2081 : a\u2081 = a\u2082\nh\u2082 : b\u2081 = b\u2082\n\u22a2 Ico a\u2081 b\u2081 = Ico a\u2082 b\u2082\n[PROOFSTEP]\nrw [h\u2081, h\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioi b \u2286 Ioi a \u2194 a \u2264 b\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => Ioi_subset_Ioi h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : Ioi b \u2286 Ioi a\n\u22a2 a \u2264 b\n[PROOFSTEP]\nby_contra ba\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : Ioi b \u2286 Ioi a\nba : \u00aca \u2264 b\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ (h (not_le.mp ba))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\ninst\u271d : DenselyOrdered \u03b1\n\u22a2 Ioi b \u2286 Ici a \u2194 a \u2264 b\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => Ioi_subset_Ici h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\ninst\u271d : DenselyOrdered \u03b1\nh : Ioi b \u2286 Ici a\n\u22a2 a \u2264 b\n[PROOFSTEP]\nby_contra ba\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\ninst\u271d : DenselyOrdered \u03b1\nh : Ioi b \u2286 Ici a\nba : \u00aca \u2264 b\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8c, bc, ca\u27e9 : \u2203 c, b < c \u2227 c < a := exists_between (not_le.mp ba)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c\u271d d : \u03b1\ninst\u271d : DenselyOrdered \u03b1\nh : Ioi b \u2286 Ici a\nba : \u00aca \u2264 b\nc : \u03b1\nbc : b < c\nca : c < a\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ (ca.trans_le (h bc))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Iio a \u2286 Iio b \u2194 a \u2264 b\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => Iio_subset_Iio h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : Iio a \u2286 Iio b\n\u22a2 a \u2264 b\n[PROOFSTEP]\nby_contra ab\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : Iio a \u2286 Iio b\nab : \u00aca \u2264 b\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ (h (not_le.mp ab))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\ninst\u271d : DenselyOrdered \u03b1\n\u22a2 Iio a \u2286 Iic b \u2194 a \u2264 b\n[PROOFSTEP]\nrw [\u2190 diff_eq_empty, Iio_diff_Iic, Ioo_eq_empty_iff, not_lt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\n\u22a2 Ioo a b \u222a Ioi c = Ioi (min a c)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nx : \u03b1\n\u22a2 x \u2208 Ioo a b \u222a Ioi c \u2194 x \u2208 Ioi (min a c)\n[PROOFSTEP]\nsimp_rw [mem_union, mem_Ioo, mem_Ioi, min_lt_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nx : \u03b1\n\u22a2 a < x \u2227 x < b \u2228 c < x \u2194 a < x \u2228 c < x\n[PROOFSTEP]\nby_cases hc : c < x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nx : \u03b1\nhc : c < x\n\u22a2 a < x \u2227 x < b \u2228 c < x \u2194 a < x \u2228 c < x\n[PROOFSTEP]\nsimp only [hc, or_true]\n -- Porting note: restore `tauto`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nx : \u03b1\nhc : \u00acc < x\n\u22a2 a < x \u2227 x < b \u2228 c < x \u2194 a < x \u2228 c < x\n[PROOFSTEP]\nhave hxb : x < b := (le_of_not_gt hc).trans_lt h\u2081\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nx : \u03b1\nhc : \u00acc < x\nhxb : x < b\n\u22a2 a < x \u2227 x < b \u2228 c < x \u2194 a < x \u2228 c < x\n[PROOFSTEP]\nsimp only [hxb, and_true]\n -- Porting note: restore `tauto`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : c < max a b\n\u22a2 Ioo a b \u222a Ioi c = Ioi (min a c)\n[PROOFSTEP]\ncases' le_total a b with hab hab\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : c < max a b\nhab : a \u2264 b\n\u22a2 Ioo a b \u222a Ioi c = Ioi (min a c)\n[PROOFSTEP]\nsimp [hab] at h \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : c < max a b\nhab : b \u2264 a\n\u22a2 Ioo a b \u222a Ioi c = Ioi (min a c)\n[PROOFSTEP]\nsimp [hab] at h \n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nh : c < b\n\u22a2 Ioo a b \u222a Ioi c = Ioi (min a c)\n[PROOFSTEP]\nexact Ioo_union_Ioi' h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nh : c < a\n\u22a2 Ioo a b \u222a Ioi c = Ioi (min a c)\n[PROOFSTEP]\nrw [min_comm]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nh : c < a\n\u22a2 Ioo a b \u222a Ioi c = Ioi (min c a)\n[PROOFSTEP]\nsimp [*, min_eq_left_of_lt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\n\u22a2 Ico a b \u222a Ici c = Ici (min a c)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 x \u2208 Ico a b \u222a Ici c \u2194 x \u2208 Ici (min a c)\n[PROOFSTEP]\nsimp_rw [mem_union, mem_Ico, mem_Ici, min_le_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 a \u2264 x \u2227 x < b \u2228 c \u2264 x \u2194 a \u2264 x \u2228 c \u2264 x\n[PROOFSTEP]\nby_cases hc : c \u2264 x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : c \u2264 x\n\u22a2 a \u2264 x \u2227 x < b \u2228 c \u2264 x \u2194 a \u2264 x \u2228 c \u2264 x\n[PROOFSTEP]\nsimp only [hc, or_true]\n -- Porting note: restore `tauto`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc \u2264 x\n\u22a2 a \u2264 x \u2227 x < b \u2228 c \u2264 x \u2194 a \u2264 x \u2228 c \u2264 x\n[PROOFSTEP]\nhave hxb : x < b := (lt_of_not_ge hc).trans_le h\u2081\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc \u2264 x\nhxb : x < b\n\u22a2 a \u2264 x \u2227 x < b \u2228 c \u2264 x \u2194 a \u2264 x \u2228 c \u2264 x\n[PROOFSTEP]\nsimp only [hxb, and_true]\n -- Porting note: restore `tauto`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : c \u2264 max a b\n\u22a2 Ico a b \u222a Ici c = Ici (min a c)\n[PROOFSTEP]\ncases' le_total a b with hab hab\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : c \u2264 max a b\nhab : a \u2264 b\n\u22a2 Ico a b \u222a Ici c = Ici (min a c)\n[PROOFSTEP]\nsimp [hab] at h \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : c \u2264 max a b\nhab : b \u2264 a\n\u22a2 Ico a b \u222a Ici c = Ici (min a c)\n[PROOFSTEP]\nsimp [hab] at h \n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nh : c \u2264 b\n\u22a2 Ico a b \u222a Ici c = Ici (min a c)\n[PROOFSTEP]\nexact Ico_union_Ici' h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nh : c \u2264 a\n\u22a2 Ico a b \u222a Ici c = Ici (min a c)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\n\u22a2 Ioc a b \u222a Ioi c = Ioi (min a c)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 x \u2208 Ioc a b \u222a Ioi c \u2194 x \u2208 Ioi (min a c)\n[PROOFSTEP]\nsimp_rw [mem_union, mem_Ioc, mem_Ioi, min_lt_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2194 a < x \u2228 c < x\n[PROOFSTEP]\nby_cases hc : c < x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : c < x\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2194 a < x \u2228 c < x\n[PROOFSTEP]\nsimp only [hc, or_true]\n -- Porting note: restore `tauto`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc < x\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2194 a < x \u2228 c < x\n[PROOFSTEP]\nhave hxb : x \u2264 b := (le_of_not_gt hc).trans h\u2081\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc < x\nhxb : x \u2264 b\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2194 a < x \u2228 c < x\n[PROOFSTEP]\nsimp only [hxb, and_true]\n -- Porting note: restore `tauto`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : c \u2264 max a b\n\u22a2 Ioc a b \u222a Ioi c = Ioi (min a c)\n[PROOFSTEP]\ncases' le_total a b with hab hab\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : c \u2264 max a b\nhab : a \u2264 b\n\u22a2 Ioc a b \u222a Ioi c = Ioi (min a c)\n[PROOFSTEP]\nsimp [hab] at h \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : c \u2264 max a b\nhab : b \u2264 a\n\u22a2 Ioc a b \u222a Ioi c = Ioi (min a c)\n[PROOFSTEP]\nsimp [hab] at h \n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nh : c \u2264 b\n\u22a2 Ioc a b \u222a Ioi c = Ioi (min a c)\n[PROOFSTEP]\nexact Ioc_union_Ioi' h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nh : c \u2264 a\n\u22a2 Ioc a b \u222a Ioi c = Ioi (min a c)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\n\u22a2 Icc a b \u222a Ici c = Ici (min a c)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 x \u2208 Icc a b \u222a Ici c \u2194 x \u2208 Ici (min a c)\n[PROOFSTEP]\nsimp_rw [mem_union, mem_Icc, mem_Ici, min_le_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 a \u2264 x \u2227 x \u2264 b \u2228 c \u2264 x \u2194 a \u2264 x \u2228 c \u2264 x\n[PROOFSTEP]\nby_cases hc : c \u2264 x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : c \u2264 x\n\u22a2 a \u2264 x \u2227 x \u2264 b \u2228 c \u2264 x \u2194 a \u2264 x \u2228 c \u2264 x\n[PROOFSTEP]\nsimp only [hc, or_true]\n -- Porting note: restore `tauto`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc \u2264 x\n\u22a2 a \u2264 x \u2227 x \u2264 b \u2228 c \u2264 x \u2194 a \u2264 x \u2228 c \u2264 x\n[PROOFSTEP]\nhave hxb : x \u2264 b := (le_of_not_ge hc).trans h\u2081\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc \u2264 x\nhxb : x \u2264 b\n\u22a2 a \u2264 x \u2227 x \u2264 b \u2228 c \u2264 x \u2194 a \u2264 x \u2228 c \u2264 x\n[PROOFSTEP]\nsimp only [hxb, and_true]\n -- Porting note: restore `tauto`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : c \u2264 max a b\n\u22a2 Icc a b \u222a Ici c = Ici (min a c)\n[PROOFSTEP]\ncases' le_or_lt a b with hab hab\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : c \u2264 max a b\nhab : a \u2264 b\n\u22a2 Icc a b \u222a Ici c = Ici (min a c)\n[PROOFSTEP]\nsimp [hab] at h \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : c \u2264 max a b\nhab : b < a\n\u22a2 Icc a b \u222a Ici c = Ici (min a c)\n[PROOFSTEP]\nsimp [hab] at h \n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nh : c \u2264 b\n\u22a2 Icc a b \u222a Ici c = Ici (min a c)\n[PROOFSTEP]\nexact Icc_union_Ici' h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b < a\nh : c \u2264 a \u2228 c \u2264 b\n\u22a2 Icc a b \u222a Ici c = Ici (min a c)\n[PROOFSTEP]\ncases' h with h h\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b < a\nh : c \u2264 a\n\u22a2 Icc a b \u222a Ici c = Ici (min a c)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b < a\nh : c \u2264 b\n\u22a2 Icc a b \u222a Ici c = Ici (min a c)\n[PROOFSTEP]\nhave hca : c \u2264 a := h.trans hab.le\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b < a\nh : c \u2264 b\nhca : c \u2264 a\n\u22a2 Icc a b \u222a Ici c = Ici (min a c)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\n\u22a2 Iio b \u222a Ico c d = Iio (max b d)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 x \u2208 Iio b \u222a Ico c d \u2194 x \u2208 Iio (max b d)\n[PROOFSTEP]\nsimp_rw [mem_union, mem_Iio, mem_Ico, lt_max_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 x < b \u2228 c \u2264 x \u2227 x < d \u2194 x < b \u2228 x < d\n[PROOFSTEP]\nby_cases hc : c \u2264 x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : c \u2264 x\n\u22a2 x < b \u2228 c \u2264 x \u2227 x < d \u2194 x < b \u2228 x < d\n[PROOFSTEP]\nsimp only [hc, true_and]\n -- Porting note: restore `tauto`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc \u2264 x\n\u22a2 x < b \u2228 c \u2264 x \u2227 x < d \u2194 x < b \u2228 x < d\n[PROOFSTEP]\nhave hxb : x < b := (lt_of_not_ge hc).trans_le h\u2081\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc \u2264 x\nhxb : x < b\n\u22a2 x < b \u2228 c \u2264 x \u2227 x < d \u2194 x < b \u2228 x < d\n[PROOFSTEP]\nsimp only [hxb, true_or]\n -- Porting note: restore `tauto`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : min c d \u2264 b\n\u22a2 Iio b \u222a Ico c d = Iio (max b d)\n[PROOFSTEP]\ncases' le_total c d with hcd hcd\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : min c d \u2264 b\nhcd : c \u2264 d\n\u22a2 Iio b \u222a Ico c d = Iio (max b d)\n[PROOFSTEP]\nsimp [hcd] at h \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : min c d \u2264 b\nhcd : d \u2264 c\n\u22a2 Iio b \u222a Ico c d = Iio (max b d)\n[PROOFSTEP]\nsimp [hcd] at h \n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhcd : c \u2264 d\nh : c \u2264 b\n\u22a2 Iio b \u222a Ico c d = Iio (max b d)\n[PROOFSTEP]\nexact Iio_union_Ico' h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhcd : d \u2264 c\nh : d \u2264 b\n\u22a2 Iio b \u222a Ico c d = Iio (max b d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\n\u22a2 Iic b \u222a Ioc c d = Iic (max b d)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nx : \u03b1\n\u22a2 x \u2208 Iic b \u222a Ioc c d \u2194 x \u2208 Iic (max b d)\n[PROOFSTEP]\nsimp_rw [mem_union, mem_Iic, mem_Ioc, le_max_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nx : \u03b1\n\u22a2 x \u2264 b \u2228 c < x \u2227 x \u2264 d \u2194 x \u2264 b \u2228 x \u2264 d\n[PROOFSTEP]\nby_cases hc : c < x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nx : \u03b1\nhc : c < x\n\u22a2 x \u2264 b \u2228 c < x \u2227 x \u2264 d \u2194 x \u2264 b \u2228 x \u2264 d\n[PROOFSTEP]\nsimp only [hc, true_and]\n -- Porting note: restore `tauto`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nx : \u03b1\nhc : \u00acc < x\n\u22a2 x \u2264 b \u2228 c < x \u2227 x \u2264 d \u2194 x \u2264 b \u2228 x \u2264 d\n[PROOFSTEP]\nhave hxb : x \u2264 b := (le_of_not_gt hc).trans h\u2081.le\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nx : \u03b1\nhc : \u00acc < x\nhxb : x \u2264 b\n\u22a2 x \u2264 b \u2228 c < x \u2227 x \u2264 d \u2194 x \u2264 b \u2228 x \u2264 d\n[PROOFSTEP]\nsimp only [hxb, true_or]\n -- Porting note: restore `tauto`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : min c d < b\n\u22a2 Iic b \u222a Ioc c d = Iic (max b d)\n[PROOFSTEP]\ncases' le_total c d with hcd hcd\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : min c d < b\nhcd : c \u2264 d\n\u22a2 Iic b \u222a Ioc c d = Iic (max b d)\n[PROOFSTEP]\nsimp [hcd] at h \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : min c d < b\nhcd : d \u2264 c\n\u22a2 Iic b \u222a Ioc c d = Iic (max b d)\n[PROOFSTEP]\nsimp [hcd] at h \n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhcd : c \u2264 d\nh : c < b\n\u22a2 Iic b \u222a Ioc c d = Iic (max b d)\n[PROOFSTEP]\nexact Iic_union_Ioc' h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhcd : d \u2264 c\nh : d < b\n\u22a2 Iic b \u222a Ioc c d = Iic (max b d)\n[PROOFSTEP]\nrw [max_comm]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhcd : d \u2264 c\nh : d < b\n\u22a2 Iic b \u222a Ioc c d = Iic (max d b)\n[PROOFSTEP]\nsimp [*, max_eq_right_of_lt h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\n\u22a2 Iio b \u222a Ioo c d = Iio (max b d)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nx : \u03b1\n\u22a2 x \u2208 Iio b \u222a Ioo c d \u2194 x \u2208 Iio (max b d)\n[PROOFSTEP]\ncases' lt_or_le x b with hba hba\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nx : \u03b1\nhba : x < b\n\u22a2 x \u2208 Iio b \u222a Ioo c d \u2194 x \u2208 Iio (max b d)\n[PROOFSTEP]\nsimp [hba, h\u2081]\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nx : \u03b1\nhba : b \u2264 x\n\u22a2 x \u2208 Iio b \u222a Ioo c d \u2194 x \u2208 Iio (max b d)\n[PROOFSTEP]\nsimp only [mem_Iio, mem_union, mem_Ioo, lt_max_iff]\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nx : \u03b1\nhba : b \u2264 x\n\u22a2 x < b \u2228 c < x \u2227 x < d \u2194 x < b \u2228 x < d\n[PROOFSTEP]\nrefine' or_congr Iff.rfl \u27e8And.right, _\u27e9\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nx : \u03b1\nhba : b \u2264 x\n\u22a2 x < d \u2192 c < x \u2227 x < d\n[PROOFSTEP]\nexact fun h\u2082 => \u27e8h\u2081.trans_le hba, h\u2082\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : min c d < b\n\u22a2 Iio b \u222a Ioo c d = Iio (max b d)\n[PROOFSTEP]\ncases' le_total c d with hcd hcd\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : min c d < b\nhcd : c \u2264 d\n\u22a2 Iio b \u222a Ioo c d = Iio (max b d)\n[PROOFSTEP]\nsimp [hcd] at h \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : min c d < b\nhcd : d \u2264 c\n\u22a2 Iio b \u222a Ioo c d = Iio (max b d)\n[PROOFSTEP]\nsimp [hcd] at h \n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhcd : c \u2264 d\nh : c < b\n\u22a2 Iio b \u222a Ioo c d = Iio (max b d)\n[PROOFSTEP]\nexact Iio_union_Ioo' h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhcd : d \u2264 c\nh : d < b\n\u22a2 Iio b \u222a Ioo c d = Iio (max b d)\n[PROOFSTEP]\nrw [max_comm]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhcd : d \u2264 c\nh : d < b\n\u22a2 Iio b \u222a Ioo c d = Iio (max d b)\n[PROOFSTEP]\nsimp [*, max_eq_right_of_lt h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\n\u22a2 Iic b \u222a Icc c d = Iic (max b d)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 x \u2208 Iic b \u222a Icc c d \u2194 x \u2208 Iic (max b d)\n[PROOFSTEP]\nsimp_rw [mem_union, mem_Iic, mem_Icc, le_max_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 x \u2264 b \u2228 c \u2264 x \u2227 x \u2264 d \u2194 x \u2264 b \u2228 x \u2264 d\n[PROOFSTEP]\nby_cases hc : c \u2264 x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : c \u2264 x\n\u22a2 x \u2264 b \u2228 c \u2264 x \u2227 x \u2264 d \u2194 x \u2264 b \u2228 x \u2264 d\n[PROOFSTEP]\nsimp only [hc, true_and]\n -- Porting note: restore `tauto`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc \u2264 x\n\u22a2 x \u2264 b \u2228 c \u2264 x \u2227 x \u2264 d \u2194 x \u2264 b \u2228 x \u2264 d\n[PROOFSTEP]\nhave hxb : x \u2264 b := (le_of_not_ge hc).trans h\u2081\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc \u2264 x\nhxb : x \u2264 b\n\u22a2 x \u2264 b \u2228 c \u2264 x \u2227 x \u2264 d \u2194 x \u2264 b \u2228 x \u2264 d\n[PROOFSTEP]\nsimp only [hxb, true_or]\n -- Porting note: restore `tauto`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : min c d \u2264 b\n\u22a2 Iic b \u222a Icc c d = Iic (max b d)\n[PROOFSTEP]\ncases' le_or_lt c d with hcd hcd\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : min c d \u2264 b\nhcd : c \u2264 d\n\u22a2 Iic b \u222a Icc c d = Iic (max b d)\n[PROOFSTEP]\nsimp [hcd] at h \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : min c d \u2264 b\nhcd : d < c\n\u22a2 Iic b \u222a Icc c d = Iic (max b d)\n[PROOFSTEP]\nsimp [hcd] at h \n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhcd : c \u2264 d\nh : c \u2264 b\n\u22a2 Iic b \u222a Icc c d = Iic (max b d)\n[PROOFSTEP]\nexact Iic_union_Icc' h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhcd : d < c\nh : c \u2264 b \u2228 d \u2264 b\n\u22a2 Iic b \u222a Icc c d = Iic (max b d)\n[PROOFSTEP]\ncases' h with h h\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhcd : d < c\nh : c \u2264 b\n\u22a2 Iic b \u222a Icc c d = Iic (max b d)\n[PROOFSTEP]\nhave hdb : d \u2264 b := hcd.le.trans h\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhcd : d < c\nh : c \u2264 b\nhdb : d \u2264 b\n\u22a2 Iic b \u222a Icc c d = Iic (max b d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhcd : d < c\nh : d \u2264 b\n\u22a2 Iic b \u222a Icc c d = Iic (max b d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\n\u22a2 x \u2208 Ico a b \u222a Ico c d \u2194 x \u2208 Ico (min a c) (max b d)\n[PROOFSTEP]\nsimp_rw [mem_union, mem_Ico, min_le_iff, lt_max_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\n\u22a2 a \u2264 x \u2227 x < b \u2228 c \u2264 x \u2227 x < d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nby_cases hc : c \u2264 x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : c \u2264 x\n\u22a2 a \u2264 x \u2227 x < b \u2228 c \u2264 x \u2227 x < d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nby_cases hd : x < d\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : \u00acc \u2264 x\n\u22a2 a \u2264 x \u2227 x < b \u2228 c \u2264 x \u2227 x < d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nby_cases hd : x < d\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : c \u2264 x\nhd : x < d\n\u22a2 a \u2264 x \u2227 x < b \u2228 c \u2264 x \u2227 x < d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nsimp only [hc, hd, and_self, or_true]\n -- Porting note: restore `tauto`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : c \u2264 x\nhd : \u00acx < d\n\u22a2 a \u2264 x \u2227 x < b \u2228 c \u2264 x \u2227 x < d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nhave hax : a \u2264 x := h\u2082.trans (le_of_not_gt hd)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : c \u2264 x\nhd : \u00acx < d\nhax : a \u2264 x\n\u22a2 a \u2264 x \u2227 x < b \u2228 c \u2264 x \u2227 x < d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nsimp only [hax, true_and, hc, or_self]\n -- Porting note: restore `tauto`\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : \u00acc \u2264 x\nhd : x < d\n\u22a2 a \u2264 x \u2227 x < b \u2228 c \u2264 x \u2227 x < d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nhave hxb : x < b := (lt_of_not_ge hc).trans_le h\u2081\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : \u00acc \u2264 x\nhd : x < d\nhxb : x < b\n\u22a2 a \u2264 x \u2227 x < b \u2228 c \u2264 x \u2227 x < d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nsimp only [hxb, and_true, hc, false_and, or_false, true_or]\n -- Porting note: restore `tauto`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : \u00acc \u2264 x\nhd : \u00acx < d\n\u22a2 a \u2264 x \u2227 x < b \u2228 c \u2264 x \u2227 x < d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nsimp only [hc, hd, and_self, or_false]\n -- Porting note: restore `tauto`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\n[PROOFSTEP]\ncases' le_total a b with hab hab\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\nhab : a \u2264 b\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\n[PROOFSTEP]\ncases' le_total c d with hcd hcd\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\nhab : b \u2264 a\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\n[PROOFSTEP]\ncases' le_total c d with hcd hcd\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\nhab : a \u2264 b\nhcd : c \u2264 d\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\n[PROOFSTEP]\nsimp [hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\nhab : a \u2264 b\nhcd : d \u2264 c\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\n[PROOFSTEP]\nsimp [hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\nhab : b \u2264 a\nhcd : c \u2264 d\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\n[PROOFSTEP]\nsimp [hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\nhab : b \u2264 a\nhcd : d \u2264 c\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\n[PROOFSTEP]\nsimp [hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nhcd : c \u2264 d\nh\u2081 : a \u2264 d\nh\u2082 : c \u2264 b\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\n[PROOFSTEP]\nexact Ico_union_Ico' h\u2082 h\u2081\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nhcd : d \u2264 c\nh\u2081 : a \u2264 c\nh\u2082 : d \u2264 b\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nhcd : c \u2264 d\nh\u2081 : b \u2264 d\nh\u2082 : c \u2264 a\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nhcd : d \u2264 c\nh\u2081 : b \u2264 c\nh\u2082 : d \u2264 a\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\n[PROOFSTEP]\nall_goals simp [*]\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nhcd : d \u2264 c\nh\u2081 : a \u2264 c\nh\u2082 : d \u2264 b\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nhcd : c \u2264 d\nh\u2081 : b \u2264 d\nh\u2082 : c \u2264 a\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nhcd : d \u2264 c\nh\u2081 : b \u2264 c\nh\u2082 : d \u2264 a\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\n\u22a2 Ioc a b \u222a Ioc c d = Ioc (min a c) (max b d)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\n\u22a2 x \u2208 Ioc a b \u222a Ioc c d \u2194 x \u2208 Ioc (min a c) (max b d)\n[PROOFSTEP]\nsimp_rw [mem_union, mem_Ioc, min_lt_iff, le_max_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2227 x \u2264 d \u2194 (a < x \u2228 c < x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nby_cases hc : c < x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : c < x\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2227 x \u2264 d \u2194 (a < x \u2228 c < x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nby_cases hd : x \u2264 d\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : \u00acc < x\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2227 x \u2264 d \u2194 (a < x \u2228 c < x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nby_cases hd : x \u2264 d\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : c < x\nhd : x \u2264 d\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2227 x \u2264 d \u2194 (a < x \u2228 c < x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nsimp only [hc, hd, and_self, or_true]\n -- Porting note: restore `tauto`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : c < x\nhd : \u00acx \u2264 d\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2227 x \u2264 d \u2194 (a < x \u2228 c < x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nhave hax : a < x := h\u2082.trans_lt (lt_of_not_ge hd)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : c < x\nhd : \u00acx \u2264 d\nhax : a < x\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2227 x \u2264 d \u2194 (a < x \u2228 c < x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nsimp only [hax, true_and, hc, or_self]\n -- Porting note: restore `tauto`\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : \u00acc < x\nhd : x \u2264 d\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2227 x \u2264 d \u2194 (a < x \u2228 c < x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nhave hxb : x \u2264 b := (le_of_not_gt hc).trans h\u2081\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : \u00acc < x\nhd : x \u2264 d\nhxb : x \u2264 b\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2227 x \u2264 d \u2194 (a < x \u2228 c < x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nsimp only [hxb, and_true, hc, false_and, or_false, true_or]\n -- Porting note: restore `tauto`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : \u00acc < x\nhd : \u00acx \u2264 d\n\u22a2 a < x \u2227 x \u2264 b \u2228 c < x \u2227 x \u2264 d \u2194 (a < x \u2228 c < x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nsimp only [hc, hd, and_self, or_false]\n -- Porting note: restore `tauto`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\n\u22a2 Ioc a b \u222a Ioc c d = Ioc (min a c) (max b d)\n[PROOFSTEP]\ncases' le_total a b with hab hab\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\nhab : a \u2264 b\n\u22a2 Ioc a b \u222a Ioc c d = Ioc (min a c) (max b d)\n[PROOFSTEP]\ncases' le_total c d with hcd hcd\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\nhab : b \u2264 a\n\u22a2 Ioc a b \u222a Ioc c d = Ioc (min a c) (max b d)\n[PROOFSTEP]\ncases' le_total c d with hcd hcd\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\nhab : a \u2264 b\nhcd : c \u2264 d\n\u22a2 Ioc a b \u222a Ioc c d = Ioc (min a c) (max b d)\n[PROOFSTEP]\nsimp [hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\nhab : a \u2264 b\nhcd : d \u2264 c\n\u22a2 Ioc a b \u222a Ioc c d = Ioc (min a c) (max b d)\n[PROOFSTEP]\nsimp [hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\nhab : b \u2264 a\nhcd : c \u2264 d\n\u22a2 Ioc a b \u222a Ioc c d = Ioc (min a c) (max b d)\n[PROOFSTEP]\nsimp [hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\nhab : b \u2264 a\nhcd : d \u2264 c\n\u22a2 Ioc a b \u222a Ioc c d = Ioc (min a c) (max b d)\n[PROOFSTEP]\nsimp [hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nhcd : c \u2264 d\nh\u2081 : a \u2264 d\nh\u2082 : c \u2264 b\n\u22a2 Ioc a b \u222a Ioc c d = Ioc (min a c) (max b d)\n[PROOFSTEP]\nexact Ioc_union_Ioc' h\u2082 h\u2081\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nhcd : d \u2264 c\nh\u2081 : a \u2264 c\nh\u2082 : d \u2264 b\n\u22a2 Ioc a b \u222a Ioc c d = Ioc (min a c) (max b d)\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nhcd : c \u2264 d\nh\u2081 : b \u2264 d\nh\u2082 : c \u2264 a\n\u22a2 Ioc a b \u222a Ioc c d = Ioc (min a c) (max b d)\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nhcd : d \u2264 c\nh\u2081 : b \u2264 c\nh\u2082 : d \u2264 a\n\u22a2 Ioc a b \u222a Ioc c d = Ioc (min a c) (max b d)\n[PROOFSTEP]\nall_goals simp [*]\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nhcd : d \u2264 c\nh\u2081 : a \u2264 c\nh\u2082 : d \u2264 b\n\u22a2 Ioc a b \u222a Ioc c d = Ioc (min a c) (max b d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nhcd : c \u2264 d\nh\u2081 : b \u2264 d\nh\u2082 : c \u2264 a\n\u22a2 Ioc a b \u222a Ioc c d = Ioc (min a c) (max b d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nhcd : d \u2264 c\nh\u2081 : b \u2264 c\nh\u2082 : d \u2264 a\n\u22a2 Ioc a b \u222a Ioc c d = Ioc (min a c) (max b d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\n\u22a2 Icc a b \u222a Icc c d = Icc (min a c) (max b d)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\n\u22a2 x \u2208 Icc a b \u222a Icc c d \u2194 x \u2208 Icc (min a c) (max b d)\n[PROOFSTEP]\nsimp_rw [mem_union, mem_Icc, min_le_iff, le_max_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\n\u22a2 a \u2264 x \u2227 x \u2264 b \u2228 c \u2264 x \u2227 x \u2264 d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nby_cases hc : c \u2264 x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : c \u2264 x\n\u22a2 a \u2264 x \u2227 x \u2264 b \u2228 c \u2264 x \u2227 x \u2264 d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nby_cases hd : x \u2264 d\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : \u00acc \u2264 x\n\u22a2 a \u2264 x \u2227 x \u2264 b \u2228 c \u2264 x \u2227 x \u2264 d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nby_cases hd : x \u2264 d\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : c \u2264 x\nhd : x \u2264 d\n\u22a2 a \u2264 x \u2227 x \u2264 b \u2228 c \u2264 x \u2227 x \u2264 d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nsimp only [hc, hd, and_self, or_true]\n -- Porting note: restore `tauto`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : c \u2264 x\nhd : \u00acx \u2264 d\n\u22a2 a \u2264 x \u2227 x \u2264 b \u2228 c \u2264 x \u2227 x \u2264 d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nhave hax : a \u2264 x := h\u2082.trans (le_of_not_ge hd)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : c \u2264 x\nhd : \u00acx \u2264 d\nhax : a \u2264 x\n\u22a2 a \u2264 x \u2227 x \u2264 b \u2228 c \u2264 x \u2227 x \u2264 d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nsimp only [hax, true_and, hc, or_self]\n -- Porting note: restore `tauto`\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : \u00acc \u2264 x\nhd : x \u2264 d\n\u22a2 a \u2264 x \u2227 x \u2264 b \u2228 c \u2264 x \u2227 x \u2264 d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nhave hxb : x \u2264 b := (le_of_not_ge hc).trans h\u2081\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : \u00acc \u2264 x\nhd : x \u2264 d\nhxb : x \u2264 b\n\u22a2 a \u2264 x \u2227 x \u2264 b \u2228 c \u2264 x \u2227 x \u2264 d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nsimp only [hxb, and_true, hc, false_and, or_false, true_or]\n -- Porting note: restore `tauto`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nh\u2082 : a \u2264 d\nx : \u03b1\nhc : \u00acc \u2264 x\nhd : \u00acx \u2264 d\n\u22a2 a \u2264 x \u2227 x \u2264 b \u2228 c \u2264 x \u2227 x \u2264 d \u2194 (a \u2264 x \u2228 c \u2264 x) \u2227 (x \u2264 b \u2228 x \u2264 d)\n[PROOFSTEP]\nsimp only [hc, hd, and_self, or_false]\n -- Porting note: restore `tauto`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b < max c d\nh\u2082 : min c d < max a b\n\u22a2 Icc a b \u222a Icc c d = Icc (min a c) (max b d)\n[PROOFSTEP]\ncases' le_or_lt a b with hab hab\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b < max c d\nh\u2082 : min c d < max a b\nhab : a \u2264 b\n\u22a2 Icc a b \u222a Icc c d = Icc (min a c) (max b d)\n[PROOFSTEP]\ncases' le_or_lt c d with hcd hcd\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b < max c d\nh\u2082 : min c d < max a b\nhab : b < a\n\u22a2 Icc a b \u222a Icc c d = Icc (min a c) (max b d)\n[PROOFSTEP]\ncases' le_or_lt c d with hcd hcd\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b < max c d\nh\u2082 : min c d < max a b\nhab : a \u2264 b\nhcd : c \u2264 d\n\u22a2 Icc a b \u222a Icc c d = Icc (min a c) (max b d)\n[PROOFSTEP]\nsimp only [min_eq_left, min_eq_right, max_eq_left, max_eq_right, min_eq_left_of_lt, min_eq_right_of_lt,\n max_eq_left_of_lt, max_eq_right_of_lt, hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b < max c d\nh\u2082 : min c d < max a b\nhab : a \u2264 b\nhcd : d < c\n\u22a2 Icc a b \u222a Icc c d = Icc (min a c) (max b d)\n[PROOFSTEP]\nsimp only [min_eq_left, min_eq_right, max_eq_left, max_eq_right, min_eq_left_of_lt, min_eq_right_of_lt,\n max_eq_left_of_lt, max_eq_right_of_lt, hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b < max c d\nh\u2082 : min c d < max a b\nhab : b < a\nhcd : c \u2264 d\n\u22a2 Icc a b \u222a Icc c d = Icc (min a c) (max b d)\n[PROOFSTEP]\nsimp only [min_eq_left, min_eq_right, max_eq_left, max_eq_right, min_eq_left_of_lt, min_eq_right_of_lt,\n max_eq_left_of_lt, max_eq_right_of_lt, hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b < max c d\nh\u2082 : min c d < max a b\nhab : b < a\nhcd : d < c\n\u22a2 Icc a b \u222a Icc c d = Icc (min a c) (max b d)\n[PROOFSTEP]\nsimp only [min_eq_left, min_eq_right, max_eq_left, max_eq_right, min_eq_left_of_lt, min_eq_right_of_lt,\n max_eq_left_of_lt, max_eq_right_of_lt, hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nhcd : c \u2264 d\nh\u2081 : a < d\nh\u2082 : c < b\n\u22a2 Icc a b \u222a Icc c d = Icc (min a c) (max b d)\n[PROOFSTEP]\nexact Icc_union_Icc' h\u2082.le h\u2081.le\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nhcd : d < c\nh\u2081 : a < c\nh\u2082 : d < b\n\u22a2 Icc a b \u222a Icc c d = Icc (min a c) (max b d)\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b < a\nhcd : c \u2264 d\nh\u2081 : b < d\nh\u2082 : c < a\n\u22a2 Icc a b \u222a Icc c d = Icc (min a c) (max b d)\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b < a\nhcd : d < c\nh\u2081 : b < c\nh\u2082 : d < a\n\u22a2 Icc a b \u222a Icc c d = Icc (min a c) (max b d)\n[PROOFSTEP]\nall_goals simp [*, min_eq_left_of_lt, max_eq_left_of_lt, min_eq_right_of_lt, max_eq_right_of_lt]\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nhcd : d < c\nh\u2081 : a < c\nh\u2082 : d < b\n\u22a2 Icc a b \u222a Icc c d = Icc (min a c) (max b d)\n[PROOFSTEP]\nsimp [*, min_eq_left_of_lt, max_eq_left_of_lt, min_eq_right_of_lt, max_eq_right_of_lt]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b < a\nhcd : c \u2264 d\nh\u2081 : b < d\nh\u2082 : c < a\n\u22a2 Icc a b \u222a Icc c d = Icc (min a c) (max b d)\n[PROOFSTEP]\nsimp [*, min_eq_left_of_lt, max_eq_left_of_lt, min_eq_right_of_lt, max_eq_right_of_lt]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b < a\nhcd : d < c\nh\u2081 : b < c\nh\u2082 : d < a\n\u22a2 Icc a b \u222a Icc c d = Icc (min a c) (max b d)\n[PROOFSTEP]\nsimp [*, min_eq_left_of_lt, max_eq_left_of_lt, min_eq_right_of_lt, max_eq_right_of_lt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nh\u2082 : a < d\n\u22a2 Ioo a b \u222a Ioo c d = Ioo (min a c) (max b d)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nh\u2082 : a < d\nx : \u03b1\n\u22a2 x \u2208 Ioo a b \u222a Ioo c d \u2194 x \u2208 Ioo (min a c) (max b d)\n[PROOFSTEP]\nsimp_rw [mem_union, mem_Ioo, min_lt_iff, lt_max_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nh\u2082 : a < d\nx : \u03b1\n\u22a2 a < x \u2227 x < b \u2228 c < x \u2227 x < d \u2194 (a < x \u2228 c < x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nby_cases hc : c < x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nh\u2082 : a < d\nx : \u03b1\nhc : c < x\n\u22a2 a < x \u2227 x < b \u2228 c < x \u2227 x < d \u2194 (a < x \u2228 c < x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nby_cases hd : x < d\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nh\u2082 : a < d\nx : \u03b1\nhc : \u00acc < x\n\u22a2 a < x \u2227 x < b \u2228 c < x \u2227 x < d \u2194 (a < x \u2228 c < x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nby_cases hd : x < d\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nh\u2082 : a < d\nx : \u03b1\nhc : c < x\nhd : x < d\n\u22a2 a < x \u2227 x < b \u2228 c < x \u2227 x < d \u2194 (a < x \u2228 c < x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nsimp only [hc, hd, and_self, or_true]\n -- Porting note: restore `tauto`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nh\u2082 : a < d\nx : \u03b1\nhc : c < x\nhd : \u00acx < d\n\u22a2 a < x \u2227 x < b \u2228 c < x \u2227 x < d \u2194 (a < x \u2228 c < x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nhave hax : a < x := h\u2082.trans_le (le_of_not_lt hd)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nh\u2082 : a < d\nx : \u03b1\nhc : c < x\nhd : \u00acx < d\nhax : a < x\n\u22a2 a < x \u2227 x < b \u2228 c < x \u2227 x < d \u2194 (a < x \u2228 c < x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nsimp only [hax, true_and, hc, or_self]\n -- Porting note: restore `tauto`\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nh\u2082 : a < d\nx : \u03b1\nhc : \u00acc < x\nhd : x < d\n\u22a2 a < x \u2227 x < b \u2228 c < x \u2227 x < d \u2194 (a < x \u2228 c < x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nhave hxb : x < b := (le_of_not_lt hc).trans_lt h\u2081\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nh\u2082 : a < d\nx : \u03b1\nhc : \u00acc < x\nhd : x < d\nhxb : x < b\n\u22a2 a < x \u2227 x < b \u2228 c < x \u2227 x < d \u2194 (a < x \u2228 c < x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nsimp only [hxb, and_true, hc, false_and, or_false, true_or]\n -- Porting note: restore `tauto`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c < b\nh\u2082 : a < d\nx : \u03b1\nhc : \u00acc < x\nhd : \u00acx < d\n\u22a2 a < x \u2227 x < b \u2228 c < x \u2227 x < d \u2194 (a < x \u2228 c < x) \u2227 (x < b \u2228 x < d)\n[PROOFSTEP]\nsimp only [hc, hd, and_self, or_false]\n -- Porting note: restore `tauto`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b < max c d\nh\u2082 : min c d < max a b\n\u22a2 Ioo a b \u222a Ioo c d = Ioo (min a c) (max b d)\n[PROOFSTEP]\ncases' le_total a b with hab hab\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b < max c d\nh\u2082 : min c d < max a b\nhab : a \u2264 b\n\u22a2 Ioo a b \u222a Ioo c d = Ioo (min a c) (max b d)\n[PROOFSTEP]\ncases' le_total c d with hcd hcd\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b < max c d\nh\u2082 : min c d < max a b\nhab : b \u2264 a\n\u22a2 Ioo a b \u222a Ioo c d = Ioo (min a c) (max b d)\n[PROOFSTEP]\ncases' le_total c d with hcd hcd\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b < max c d\nh\u2082 : min c d < max a b\nhab : a \u2264 b\nhcd : c \u2264 d\n\u22a2 Ioo a b \u222a Ioo c d = Ioo (min a c) (max b d)\n[PROOFSTEP]\nsimp only [min_eq_left, min_eq_right, max_eq_left, max_eq_right, hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b < max c d\nh\u2082 : min c d < max a b\nhab : a \u2264 b\nhcd : d \u2264 c\n\u22a2 Ioo a b \u222a Ioo c d = Ioo (min a c) (max b d)\n[PROOFSTEP]\nsimp only [min_eq_left, min_eq_right, max_eq_left, max_eq_right, hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b < max c d\nh\u2082 : min c d < max a b\nhab : b \u2264 a\nhcd : c \u2264 d\n\u22a2 Ioo a b \u222a Ioo c d = Ioo (min a c) (max b d)\n[PROOFSTEP]\nsimp only [min_eq_left, min_eq_right, max_eq_left, max_eq_right, hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : min a b < max c d\nh\u2082 : min c d < max a b\nhab : b \u2264 a\nhcd : d \u2264 c\n\u22a2 Ioo a b \u222a Ioo c d = Ioo (min a c) (max b d)\n[PROOFSTEP]\nsimp only [min_eq_left, min_eq_right, max_eq_left, max_eq_right, hab, hcd] at h\u2081 h\u2082 \n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nhcd : c \u2264 d\nh\u2081 : a < d\nh\u2082 : c < b\n\u22a2 Ioo a b \u222a Ioo c d = Ioo (min a c) (max b d)\n[PROOFSTEP]\nexact Ioo_union_Ioo' h\u2082 h\u2081\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nhcd : d \u2264 c\nh\u2081 : a < c\nh\u2082 : d < b\n\u22a2 Ioo a b \u222a Ioo c d = Ioo (min a c) (max b d)\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nhcd : c \u2264 d\nh\u2081 : b < d\nh\u2082 : c < a\n\u22a2 Ioo a b \u222a Ioo c d = Ioo (min a c) (max b d)\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nhcd : d \u2264 c\nh\u2081 : b < c\nh\u2082 : d < a\n\u22a2 Ioo a b \u222a Ioo c d = Ioo (min a c) (max b d)\n[PROOFSTEP]\nall_goals\n simp [*, min_eq_left_of_lt, min_eq_right_of_lt, max_eq_left_of_lt, max_eq_right_of_lt, le_of_lt h\u2082, le_of_lt h\u2081]\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : a \u2264 b\nhcd : d \u2264 c\nh\u2081 : a < c\nh\u2082 : d < b\n\u22a2 Ioo a b \u222a Ioo c d = Ioo (min a c) (max b d)\n[PROOFSTEP]\nsimp [*, min_eq_left_of_lt, min_eq_right_of_lt, max_eq_left_of_lt, max_eq_right_of_lt, le_of_lt h\u2082, le_of_lt h\u2081]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nhcd : c \u2264 d\nh\u2081 : b < d\nh\u2082 : c < a\n\u22a2 Ioo a b \u222a Ioo c d = Ioo (min a c) (max b d)\n[PROOFSTEP]\nsimp [*, min_eq_left_of_lt, min_eq_right_of_lt, max_eq_left_of_lt, max_eq_right_of_lt, le_of_lt h\u2082, le_of_lt h\u2081]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nhab : b \u2264 a\nhcd : d \u2264 c\nh\u2081 : b < c\nh\u2082 : d < a\n\u22a2 Ioo a b \u222a Ioo c d = Ioo (min a c) (max b d)\n[PROOFSTEP]\nsimp [*, min_eq_left_of_lt, min_eq_right_of_lt, max_eq_left_of_lt, max_eq_right_of_lt, le_of_lt h\u2082, le_of_lt h\u2081]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : SemilatticeInf \u03b1\na b : \u03b1\n\u22a2 Iic a \u2229 Iic b = Iic (a \u2293 b)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : SemilatticeInf \u03b1\na b x : \u03b1\n\u22a2 x \u2208 Iic a \u2229 Iic b \u2194 x \u2208 Iic (a \u2293 b)\n[PROOFSTEP]\nsimp [Iic]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : SemilatticeInf \u03b1\na b c : \u03b1\n\u22a2 Ioc a b \u2229 Iic c = Ioc a (b \u2293 c)\n[PROOFSTEP]\nrw [\u2190 Ioi_inter_Iic, \u2190 Ioi_inter_Iic, inter_assoc, Iic_inter_Iic]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : SemilatticeSup \u03b1\na b : \u03b1\n\u22a2 Ici a \u2229 Ici b = Ici (a \u2294 b)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : SemilatticeSup \u03b1\na b x : \u03b1\n\u22a2 x \u2208 Ici a \u2229 Ici b \u2194 x \u2208 Ici (a \u2294 b)\n[PROOFSTEP]\nsimp [Ici]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : SemilatticeSup \u03b1\na b c : \u03b1\n\u22a2 Ico a b \u2229 Ici c = Ico (a \u2294 c) b\n[PROOFSTEP]\nrw [\u2190 Ici_inter_Iio, \u2190 Ici_inter_Iio, \u2190 Ici_inter_Ici, inter_right_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Lattice \u03b1\na b c a\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\n\u22a2 Icc a\u2081 b\u2081 \u2229 Icc a\u2082 b\u2082 = Icc (a\u2081 \u2294 a\u2082) (b\u2081 \u2293 b\u2082)\n[PROOFSTEP]\nsimp only [Ici_inter_Iic.symm, Ici_inter_Ici.symm, Iic_inter_Iic.symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Lattice \u03b1\na b c a\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\n\u22a2 Ici a\u2081 \u2229 Iic b\u2081 \u2229 (Ici a\u2082 \u2229 Iic b\u2082) = Ici a\u2081 \u2229 Ici a\u2082 \u2229 (Iic b\u2081 \u2229 Iic b\u2082)\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Lattice \u03b1\na b c a\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\n\u22a2 Icc a b \u2229 Icc b c = {b}\n[PROOFSTEP]\nrw [Icc_inter_Icc, sup_of_le_right hab, inf_of_le_left hbc, Icc_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ico a\u2081 b\u2081 \u2229 Ico a\u2082 b\u2082 = Ico (a\u2081 \u2294 a\u2082) (b\u2081 \u2293 b\u2082)\n[PROOFSTEP]\nsimp only [Ici_inter_Iio.symm, Ici_inter_Ici.symm, Iio_inter_Iio.symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ici a\u2081 \u2229 Iio b\u2081 \u2229 (Ici a\u2082 \u2229 Iio b\u2082) = Ici a\u2081 \u2229 Ici a\u2082 \u2229 (Iio b\u2081 \u2229 Iio b\u2082)\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioc a\u2081 b\u2081 \u2229 Ioc a\u2082 b\u2082 = Ioc (a\u2081 \u2294 a\u2082) (b\u2081 \u2293 b\u2082)\n[PROOFSTEP]\nsimp only [Ioi_inter_Iic.symm, Ioi_inter_Ioi.symm, Iic_inter_Iic.symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioi a\u2081 \u2229 Iic b\u2081 \u2229 (Ioi a\u2082 \u2229 Iic b\u2082) = Ioi a\u2081 \u2229 Ioi a\u2082 \u2229 (Iic b\u2081 \u2229 Iic b\u2082)\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioo a\u2081 b\u2081 \u2229 Ioo a\u2082 b\u2082 = Ioo (a\u2081 \u2294 a\u2082) (b\u2081 \u2293 b\u2082)\n[PROOFSTEP]\nsimp only [Ioi_inter_Iio.symm, Ioi_inter_Ioi.symm, Iio_inter_Iio.symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioi a\u2081 \u2229 Iio b\u2081 \u2229 (Ioi a\u2082 \u2229 Iio b\u2082) = Ioi a\u2081 \u2229 Ioi a\u2082 \u2229 (Iio b\u2081 \u2229 Iio b\u2082)\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : b\u2081 < b\u2082\nx : \u03b1\n\u22a2 x \u2208 Ioc a\u2081 b\u2081 \u2229 Ioo a\u2082 b\u2082 \u2194 x \u2208 Ioc (max a\u2081 a\u2082) b\u2081\n[PROOFSTEP]\nsimp [and_assoc, @and_left_comm (x \u2264 _), and_iff_left_iff_imp.2 fun h' => lt_of_le_of_lt h' h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : b\u2082 \u2264 b\u2081\nx : \u03b1\n\u22a2 x \u2208 Ioc a\u2081 b\u2081 \u2229 Ioo a\u2082 b\u2082 \u2194 x \u2208 Ioo (max a\u2081 a\u2082) b\u2082\n[PROOFSTEP]\nsimp [and_assoc, @and_left_comm (x \u2264 _), and_iff_right_iff_imp.2 fun h' => (le_of_lt h').trans h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : b\u2081 \u2264 b\u2082\n\u22a2 Ioo a\u2081 b\u2081 \u2229 Ioc a\u2082 b\u2082 = Ioo (max a\u2081 a\u2082) b\u2081\n[PROOFSTEP]\nrw [inter_comm, Ioc_inter_Ioo_of_right_le h, max_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh : b\u2082 < b\u2081\n\u22a2 Ioo a\u2081 b\u2081 \u2229 Ioc a\u2082 b\u2082 = Ioc (max a\u2081 a\u2082) b\u2082\n[PROOFSTEP]\nrw [inter_comm, Ioc_inter_Ioo_of_left_lt h, max_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ico a b \\ Iio c = Ico (max a c) b\n[PROOFSTEP]\nrw [diff_eq, compl_Iio, Ico_inter_Ici, sup_eq_max]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 \u2200 (x : \u03b1), x \u2208 Ioc a b \\ Ioi c \u2194 x \u2208 Ioc a (min b c)\n[PROOFSTEP]\nsimp (config := { contextual := true }) [iff_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioc a b \u2229 Ioi c = Ioc (a \u2294 c) b\n[PROOFSTEP]\nrw [\u2190 Ioi_inter_Iic, inter_assoc, inter_comm, inter_assoc, Ioi_inter_Ioi, inter_comm, Ioi_inter_Iic, sup_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 \u2200 (x : \u03b1), x \u2208 Ico a b \u2229 Iio c \u2194 x \u2208 Ico a (min b c)\n[PROOFSTEP]\nsimp (config := { contextual := true }) [iff_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioc a b \\ Iic c = Ioc (max a c) b\n[PROOFSTEP]\nrw [diff_eq, compl_Iic, Ioc_inter_Ioi, sup_eq_max]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioc a b \u222a Ioc a c = Ioc a (max b c)\n[PROOFSTEP]\nrw [Ioc_union_Ioc, min_self]\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 min a b \u2264 max a c\n[PROOFSTEP]\nexact (min_le_left _ _).trans (le_max_left _ _)\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 min a c \u2264 max a b\n[PROOFSTEP]\nexact (min_le_left _ _).trans (le_max_left _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioc a c \u222a Ioc b c = Ioc (min a b) c\n[PROOFSTEP]\nrw [Ioc_union_Ioc, max_self]\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 min a c \u2264 max b c\n[PROOFSTEP]\nexact (min_le_right _ _).trans (le_max_right _ _)\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 min b c \u2264 max a c\n[PROOFSTEP]\nexact (min_le_right _ _).trans (le_max_right _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioc a b \u222a Ioc b a = Ioc (min a b) (max a b)\n[PROOFSTEP]\nrw [max_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioc a b \u222a Ioc b a = Ioc (min a b) (max b a)\n[PROOFSTEP]\napply Ioc_union_Ioc\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 min a b \u2264 max b a\n[PROOFSTEP]\nrw [max_comm]\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 min b a \u2264 max a b\n[PROOFSTEP]\nrw [max_comm]\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 min a b \u2264 max a b\n[PROOFSTEP]\nexact min_le_max\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 min b a \u2264 max b a\n[PROOFSTEP]\nexact min_le_max\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioc a b \u222a Ioc b c \u222a Ioc c a = Ioc (min a (min b c)) (max a (max b c))\n[PROOFSTEP]\nrw [Ioc_union_Ioc, Ioc_union_Ioc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioc (min (min a b) c) (max (max b c) a) = Ioc (min a (min b c)) (max a (max b c))\n[PROOFSTEP]\nsimp [min_le_of_left_le, min_le_of_right_le, le_max_of_le_left, le_max_of_le_right, le_refl, min_assoc, max_comm]\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 min (min a b) (max b c) \u2264 max c a\n[PROOFSTEP]\nsimp [min_le_of_left_le, min_le_of_right_le, le_max_of_le_left, le_max_of_le_right, le_refl, min_assoc, max_comm]\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 min c a \u2264 max (min a b) (max b c)\n[PROOFSTEP]\nsimp [min_le_of_left_le, min_le_of_right_le, le_max_of_le_left, le_max_of_le_right, le_refl, min_assoc, max_comm]\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 min a b \u2264 max b c\n[PROOFSTEP]\nsimp [min_le_of_left_le, min_le_of_right_le, le_max_of_le_left, le_max_of_le_right, le_refl, min_assoc, max_comm]\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 min b c \u2264 max a b\n[PROOFSTEP]\nsimp [min_le_of_left_le, min_le_of_right_le, le_max_of_le_left, le_max_of_le_right, le_refl, min_assoc, max_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : \u03b2\n\u22a2 Icc a\u2081 a\u2082 \u00d7\u02e2 Icc b\u2081 b\u2082 = Icc (a\u2081, b\u2081) (a\u2082, b\u2082)\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 (x, y) \u2208 Icc a\u2081 a\u2082 \u00d7\u02e2 Icc b\u2081 b\u2082 \u2194 (x, y) \u2208 Icc (a\u2081, b\u2081) (a\u2082, b\u2082)\n[PROOFSTEP]\nsimp [and_assoc, and_comm, and_left_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na b : \u03b1 \u00d7 \u03b2\n\u22a2 Icc a b = Icc a.fst b.fst \u00d7\u02e2 Icc a.snd b.snd\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DenselyOrdered \u03b1\nx y : \u03b1\nx\u271d : \u2191(Set.Ioo x y)\na : \u03b1\nha\u2081 : x < a\nha\u2082 : a < y\n\u22a2 \u2203 b, b < { val := a, property := (_ : x < a \u2227 a < y) }\n[PROOFSTEP]\nrcases exists_between ha\u2081 with \u27e8b, hb\u2081, hb\u2082\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DenselyOrdered \u03b1\nx y : \u03b1\nx\u271d : \u2191(Set.Ioo x y)\na : \u03b1\nha\u2081 : x < a\nha\u2082 : a < y\nb : \u03b1\nhb\u2081 : x < b\nhb\u2082 : b < a\n\u22a2 \u2203 b, b < { val := a, property := (_ : x < a \u2227 a < y) }\n[PROOFSTEP]\nexact \u27e8\u27e8b, hb\u2081, hb\u2082.trans ha\u2082\u27e9, hb\u2082\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DenselyOrdered \u03b1\nx y : \u03b1\nx\u271d : \u2191(Set.Ioc x y)\na : \u03b1\nha\u2081 : x < a\nha\u2082 : a \u2264 y\n\u22a2 \u2203 b, b < { val := a, property := (_ : x < a \u2227 a \u2264 y) }\n[PROOFSTEP]\nrcases exists_between ha\u2081 with \u27e8b, hb\u2081, hb\u2082\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DenselyOrdered \u03b1\nx y : \u03b1\nx\u271d : \u2191(Set.Ioc x y)\na : \u03b1\nha\u2081 : x < a\nha\u2082 : a \u2264 y\nb : \u03b1\nhb\u2081 : x < b\nhb\u2082 : b < a\n\u22a2 \u2203 b, b < { val := a, property := (_ : x < a \u2227 a \u2264 y) }\n[PROOFSTEP]\nexact \u27e8\u27e8b, hb\u2081, hb\u2082.le.trans ha\u2082\u27e9, hb\u2082\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DenselyOrdered \u03b1\nx y : \u03b1\nx\u271d : \u2191(Set.Ioi x)\na : \u03b1\nha : a \u2208 Set.Ioi x\n\u22a2 \u2203 b, b < { val := a, property := ha }\n[PROOFSTEP]\nrcases exists_between ha with \u27e8b, hb\u2081, hb\u2082\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DenselyOrdered \u03b1\nx y : \u03b1\nx\u271d : \u2191(Set.Ioi x)\na : \u03b1\nha : a \u2208 Set.Ioi x\nb : \u03b1\nhb\u2081 : x < b\nhb\u2082 : b < a\n\u22a2 \u2203 b, b < { val := a, property := ha }\n[PROOFSTEP]\nexact \u27e8\u27e8b, hb\u2081\u27e9, hb\u2082\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DenselyOrdered \u03b1\nx y : \u03b1\nx\u271d : \u2191(Set.Ioo x y)\na : \u03b1\nha\u2081 : x < a\nha\u2082 : a < y\n\u22a2 \u2203 b, { val := a, property := (_ : x < a \u2227 a < y) } < b\n[PROOFSTEP]\nrcases exists_between ha\u2082 with \u27e8b, hb\u2081, hb\u2082\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DenselyOrdered \u03b1\nx y : \u03b1\nx\u271d : \u2191(Set.Ioo x y)\na : \u03b1\nha\u2081 : x < a\nha\u2082 : a < y\nb : \u03b1\nhb\u2081 : a < b\nhb\u2082 : b < y\n\u22a2 \u2203 b, { val := a, property := (_ : x < a \u2227 a < y) } < b\n[PROOFSTEP]\nexact \u27e8\u27e8b, ha\u2081.trans hb\u2081, hb\u2082\u27e9, hb\u2081\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DenselyOrdered \u03b1\nx y : \u03b1\nx\u271d : \u2191(Set.Ico x y)\na : \u03b1\nha\u2081 : x \u2264 a\nha\u2082 : a < y\n\u22a2 \u2203 b, { val := a, property := (_ : x \u2264 a \u2227 a < y) } < b\n[PROOFSTEP]\nrcases exists_between ha\u2082 with \u27e8b, hb\u2081, hb\u2082\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DenselyOrdered \u03b1\nx y : \u03b1\nx\u271d : \u2191(Set.Ico x y)\na : \u03b1\nha\u2081 : x \u2264 a\nha\u2082 : a < y\nb : \u03b1\nhb\u2081 : a < b\nhb\u2082 : b < y\n\u22a2 \u2203 b, { val := a, property := (_ : x \u2264 a \u2227 a < y) } < b\n[PROOFSTEP]\nexact \u27e8\u27e8b, ha\u2081.trans hb\u2081.le, hb\u2082\u27e9, hb\u2081\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DenselyOrdered \u03b1\nx y : \u03b1\nx\u271d : \u2191(Set.Iio x)\na : \u03b1\nha : a \u2208 Set.Iio x\n\u22a2 \u2203 b, { val := a, property := ha } < b\n[PROOFSTEP]\nrcases exists_between ha with \u27e8b, hb\u2081, hb\u2082\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DenselyOrdered \u03b1\nx y : \u03b1\nx\u271d : \u2191(Set.Iio x)\na : \u03b1\nha : a \u2208 Set.Iio x\nb : \u03b1\nhb\u2081 : a < b\nhb\u2082 : b < x\n\u22a2 \u2203 b, { val := a, property := ha } < b\n[PROOFSTEP]\nexact \u27e8\u27e8b, hb\u2082\u27e9, hb\u2081\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Intervals.Basic", "llama_tokens": 44725, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952052, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.44753327092659057}} {"text": "[GOAL]\na b c d : Prop\np : ?m.25026\nh : Decidable True\n\u22a2 decide True = true\n[PROOFSTEP]\nsimp\n[GOAL]\na b c d : Prop\np : ?m.25099\nh : Decidable False\n\u22a2 decide False = false\n[PROOFSTEP]\nsimp\n[GOAL]\na b c d p : Prop\nh : Decidable p\nh\u2081 : p \u2192 Sort u\nh\u2082 : \u00acp \u2192 Sort u\nh\u2083 : p\nh\u2084 : h\u2081 h\u2083\n\u22a2 h\u2081 h\u2083 = Decidable.recOn h h\u2082 h\u2081\n[PROOFSTEP]\nmatch h with\n| .isTrue _ => rfl\n[GOAL]\na b c d p : Prop\nh : Decidable p\nh\u2081 : p \u2192 Sort u\nh\u2082 : \u00acp \u2192 Sort u\nh\u2083 : p\nh\u2084 : h\u2081 h\u2083\nh\u271d : p\n\u22a2 h\u2081 h\u2083 = Decidable.recOn (isTrue h\u271d) h\u2082 h\u2081\n[PROOFSTEP]\nrfl\n[GOAL]\na b c d p : Prop\nh : Decidable p\nh\u2081 : p \u2192 Sort u\nh\u2082 : \u00acp \u2192 Sort u\nh\u2083 : \u00acp\nh\u2084 : h\u2082 h\u2083\n\u22a2 h\u2082 h\u2083 = Decidable.recOn h h\u2082 h\u2081\n[PROOFSTEP]\nmatch h with\n| .isFalse _ => rfl\n[GOAL]\na b c d p : Prop\nh : Decidable p\nh\u2081 : p \u2192 Sort u\nh\u2082 : \u00acp \u2192 Sort u\nh\u2083 : \u00acp\nh\u2084 : h\u2082 h\u2083\nh\u271d : \u00acp\n\u22a2 h\u2082 h\u2083 = Decidable.recOn (isFalse h\u271d) h\u2082 h\u2081\n[PROOFSTEP]\nrfl\n[GOAL]\na b c d : Prop\np\u271d : ?m.26639\n\u03b1 : Sort u\np : \u03b1 \u2192 \u03b1 \u2192 Bool\nh\u2081 : IsDecEq p\nh\u2082 : IsDecRefl p\nx y : \u03b1\nhp : \u00acp x y = true\nhxy : x = y\n\u22a2 \u00acp y y = true\n[PROOFSTEP]\nrwa [hxy] at hp \n[GOAL]\na b c\u271d d : Prop\np : ?m.27547\nc t e : Prop\ninst\u271d : Decidable c\nh : if c then t else e\nhc : c\n\u22a2 t\n[PROOFSTEP]\nhave := if_pos hc \u25b8 h\n[GOAL]\na b c\u271d d : Prop\np : ?m.27547\nc t e : Prop\ninst\u271d : Decidable c\nh : if c then t else e\nhc : c\nthis : t\n\u22a2 t\n[PROOFSTEP]\nexact this\n[GOAL]\na b c\u271d d : Prop\np : ?m.27609\nc t e : Prop\ninst\u271d : Decidable c\nh : if c then t else e\nhnc : \u00acc\n\u22a2 e\n[PROOFSTEP]\nhave := if_neg hnc \u25b8 h\n[GOAL]\na b c\u271d d : Prop\np : ?m.27609\nc t e : Prop\ninst\u271d : Decidable c\nh : if c then t else e\nhnc : \u00acc\nthis : e\n\u22a2 e\n[PROOFSTEP]\nexact this\n[GOAL]\na b\u271d c d : Prop\np : ?m.29900\n\u03b1 : Sort v\n\u03b2 : \u03b1 \u2192 Sort u\na\u2081 a\u2082 : \u03b1\nb : (x : \u03b1) \u2192 \u03b2 x\nh : a\u2081 = a\u2082\n\u22a2 HEq\n (let x := a\u2081;\n b x)\n (let x := a\u2082;\n b x)\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\na b\u271d c d : Prop\np : ?m.29900\n\u03b1 : Sort v\n\u03b2 : \u03b1 \u2192 Sort u\na\u2081 : \u03b1\nb : (x : \u03b1) \u2192 \u03b2 x\n\u22a2 HEq\n (let x := a\u2081;\n b x)\n (let x := a\u2081;\n b x)\n[PROOFSTEP]\nrfl\n[GOAL]\na\u271d b c d : Prop\np : ?m.30034\n\u03b1 : Sort v\n\u03b2 : \u03b1 \u2192 Sort u\na : \u03b1\nb\u2081 b\u2082 : (x : \u03b1) \u2192 \u03b2 x\nh : \u2200 (x : \u03b1), b\u2081 x = b\u2082 x\n\u22a2 (let x := a;\n b\u2081 x) =\n let x := a;\n b\u2082 x\n[PROOFSTEP]\nexact h _ \u25b8 rfl\n[GOAL]\na b c d : Prop\np : ?m.30108\n\u03b1 : Sort v\n\u03b2 : Sort u\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : \u03b1 \u2192 \u03b2\nh\u2081 : a\u2081 = a\u2082\nh\u2082 : \u2200 (x : \u03b1), b\u2081 x = b\u2082 x\n\u22a2 (let x := a\u2081;\n b\u2081 x) =\n let x := a\u2082;\n b\u2082 x\n[PROOFSTEP]\nsimp [h\u2081, h\u2082]\n", "meta": {"mathlib_filename": "Mathlib.Init.Logic", "llama_tokens": 1318, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984444, "lm_q2_score": 0.7025300449389326, "lm_q1q2_score": 0.4475332629915615}} {"text": "[GOAL]\nm n b : \u2115\nhm : m \u2260 1\nhn : 0 < n\nhb : log m n < b\n\u22a2 multiplicity m n = \u2191(card (Ico 1 (Part.get (multiplicity m n) (_ : multiplicity.Finite m n) + 1)))\n[PROOFSTEP]\nsimp\n[GOAL]\nm n b : \u2115\nhm : m \u2260 1\nhn : 0 < n\nhb : log m n < b\ni : \u2115\n\u22a2 i \u2208 Ico 1 (Part.get (multiplicity m n) (_ : multiplicity.Finite m n) + 1) \u2194 i \u2208 filter (fun i => m ^ i \u2223 n) (Ico 1 b)\n[PROOFSTEP]\nrw [mem_filter, mem_Ico, mem_Ico, lt_succ_iff, \u2190 @PartENat.coe_le_coe i, PartENat.natCast_get, \u2190\n pow_dvd_iff_le_multiplicity, and_right_comm]\n[GOAL]\nm n b : \u2115\nhm : m \u2260 1\nhn : 0 < n\nhb : log m n < b\ni : \u2115\n\u22a2 1 \u2264 i \u2227 m ^ i \u2223 n \u2194 (1 \u2264 i \u2227 m ^ i \u2223 n) \u2227 i < b\n[PROOFSTEP]\nrefine' (and_iff_left_of_imp fun h => lt_of_le_of_lt _ hb).symm\n[GOAL]\nm n b : \u2115\nhm : m \u2260 1\nhn : 0 < n\nhb : log m n < b\ni : \u2115\nh : 1 \u2264 i \u2227 m ^ i \u2223 n\n\u22a2 i \u2264 log m n\n[PROOFSTEP]\ncases' m with m\n[GOAL]\ncase zero\nn b : \u2115\nhn : 0 < n\ni : \u2115\nhm : zero \u2260 1\nhb : log zero n < b\nh : 1 \u2264 i \u2227 zero ^ i \u2223 n\n\u22a2 i \u2264 log zero n\n[PROOFSTEP]\nrw [zero_eq, zero_pow, zero_dvd_iff] at h \n[GOAL]\ncase zero\nn b : \u2115\nhn : 0 < n\ni : \u2115\nhm : zero \u2260 1\nhb : log zero n < b\nh : 1 \u2264 i \u2227 n = 0\n\u22a2 i \u2264 log zero n\ncase zero n b : \u2115 hn : 0 < n i : \u2115 hm : zero \u2260 1 hb : log zero n < b h : 1 \u2264 i \u2227 0 ^ i \u2223 n \u22a2 0 < i\n[PROOFSTEP]\nexacts [(hn.ne' h.2).elim, h.1]\n[GOAL]\ncase succ\nn b : \u2115\nhn : 0 < n\ni m : \u2115\nhm : succ m \u2260 1\nhb : log (succ m) n < b\nh : 1 \u2264 i \u2227 succ m ^ i \u2223 n\n\u22a2 i \u2264 log (succ m) n\n[PROOFSTEP]\nexact le_log_of_pow_le (one_lt_iff_ne_zero_and_ne_one.2 \u27e8m.succ_ne_zero, hm\u27e9) (le_of_dvd hn h.2)\n[GOAL]\np : \u2115\nhp : Prime p\nb : \u2115\nx\u271d : log p 0 < b\n\u22a2 multiplicity p 0! = \u2191(\u2211 i in Ico 1 b, 0 / p ^ i)\n[PROOFSTEP]\nsimp [Ico, hp.multiplicity_one]\n[GOAL]\np : \u2115\nhp : Prime p\nn b : \u2115\nhb : log p (n + 1) < b\n\u22a2 multiplicity p (n + 1)! = multiplicity p n ! + multiplicity p (n + 1)\n[PROOFSTEP]\nrw [factorial_succ, hp.multiplicity_mul, add_comm]\n[GOAL]\np : \u2115\nhp : Prime p\nn b : \u2115\nhb : log p (n + 1) < b\n\u22a2 multiplicity p n ! + multiplicity p (n + 1) =\n \u2191(\u2211 i in Ico 1 b, n / p ^ i) + \u2191(card (filter (fun i => p ^ i \u2223 n + 1) (Ico 1 b)))\n[PROOFSTEP]\nrw [multiplicity_factorial hp ((log_mono_right <| le_succ _).trans_lt hb), \u2190\n multiplicity_eq_card_pow_dvd hp.ne_one (succ_pos _) hb]\n[GOAL]\np : \u2115\nhp : Prime p\nn b : \u2115\nhb : log p (n + 1) < b\n\u22a2 \u2191(\u2211 i in Ico 1 b, n / p ^ i) + \u2191(card (filter (fun i => p ^ i \u2223 n + 1) (Ico 1 b))) =\n \u2191(\u2211 i in Ico 1 b, (n / p ^ i + if p ^ i \u2223 n + 1 then 1 else 0))\n[PROOFSTEP]\nrw [sum_add_distrib, sum_boole]\n[GOAL]\np : \u2115\nhp : Prime p\nn b : \u2115\nhb : log p (n + 1) < b\n\u22a2 \u2191(\u2211 i in Ico 1 b, n / p ^ i) + \u2191(card (filter (fun i => p ^ i \u2223 n + 1) (Ico 1 b))) =\n \u2191(\u2211 x in Ico 1 b, n / p ^ x + \u2191(card (filter (fun x => p ^ x \u2223 n + 1) (Ico 1 b))))\n[PROOFSTEP]\nsimp\n[GOAL]\nn p : \u2115\nhp : Prime p\n\u22a2 multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1\n[PROOFSTEP]\nhave hp' := hp.prime\n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\n\u22a2 multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1\n[PROOFSTEP]\nhave h0 : 2 \u2264 p := hp.two_le\n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\n\u22a2 multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1\n[PROOFSTEP]\nhave h1 : 1 \u2264 p * n + 1 := Nat.le_add_left _ _\n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\n\u22a2 multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1\n[PROOFSTEP]\nhave h2 : p * n + 1 \u2264 p * (n + 1)\n[GOAL]\ncase h2\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\n\u22a2 p * n + 1 \u2264 p * (n + 1)\n[PROOFSTEP]\nlinarith\n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\nh2 : p * n + 1 \u2264 p * (n + 1)\n\u22a2 multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1\n[PROOFSTEP]\nhave h3 : p * n + 1 \u2264 p * (n + 1) + 1\n[GOAL]\ncase h3\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\nh2 : p * n + 1 \u2264 p * (n + 1)\n\u22a2 p * n + 1 \u2264 p * (n + 1) + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\nh2 : p * n + 1 \u2264 p * (n + 1)\nh3 : p * n + 1 \u2264 p * (n + 1) + 1\n\u22a2 multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1\n[PROOFSTEP]\nhave hm : multiplicity p (p * n)! \u2260 \u22a4 :=\n by\n rw [Ne.def, eq_top_iff_not_finite, Classical.not_not, finite_nat_iff]\n exact \u27e8hp.ne_one, factorial_pos _\u27e9\n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\nh2 : p * n + 1 \u2264 p * (n + 1)\nh3 : p * n + 1 \u2264 p * (n + 1) + 1\n\u22a2 multiplicity p (p * n)! \u2260 \u22a4\n[PROOFSTEP]\nrw [Ne.def, eq_top_iff_not_finite, Classical.not_not, finite_nat_iff]\n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\nh2 : p * n + 1 \u2264 p * (n + 1)\nh3 : p * n + 1 \u2264 p * (n + 1) + 1\n\u22a2 p \u2260 1 \u2227 0 < (p * n)!\n[PROOFSTEP]\nexact \u27e8hp.ne_one, factorial_pos _\u27e9\n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\nh2 : p * n + 1 \u2264 p * (n + 1)\nh3 : p * n + 1 \u2264 p * (n + 1) + 1\nhm : multiplicity p (p * n)! \u2260 \u22a4\n\u22a2 multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1\n[PROOFSTEP]\nrevert hm\n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\nh2 : p * n + 1 \u2264 p * (n + 1)\nh3 : p * n + 1 \u2264 p * (n + 1) + 1\n\u22a2 multiplicity p (p * n)! \u2260 \u22a4 \u2192 multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1\n[PROOFSTEP]\nhave h4 : \u2200 m \u2208 Ico (p * n + 1) (p * (n + 1)), multiplicity p m = 0 :=\n by\n intro m hm\n rw [multiplicity_eq_zero, \u2190 not_dvd_iff_between_consec_multiples _ hp.pos]\n rw [mem_Ico] at hm \n exact \u27e8n, lt_of_succ_le hm.1, hm.2\u27e9\n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\nh2 : p * n + 1 \u2264 p * (n + 1)\nh3 : p * n + 1 \u2264 p * (n + 1) + 1\n\u22a2 \u2200 (m : \u2115), m \u2208 Ico (p * n + 1) (p * (n + 1)) \u2192 multiplicity p m = 0\n[PROOFSTEP]\nintro m hm\n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\nh2 : p * n + 1 \u2264 p * (n + 1)\nh3 : p * n + 1 \u2264 p * (n + 1) + 1\nm : \u2115\nhm : m \u2208 Ico (p * n + 1) (p * (n + 1))\n\u22a2 multiplicity p m = 0\n[PROOFSTEP]\nrw [multiplicity_eq_zero, \u2190 not_dvd_iff_between_consec_multiples _ hp.pos]\n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\nh2 : p * n + 1 \u2264 p * (n + 1)\nh3 : p * n + 1 \u2264 p * (n + 1) + 1\nm : \u2115\nhm : m \u2208 Ico (p * n + 1) (p * (n + 1))\n\u22a2 \u2203 k, p * k < m \u2227 m < p * (k + 1)\n[PROOFSTEP]\nrw [mem_Ico] at hm \n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\nh2 : p * n + 1 \u2264 p * (n + 1)\nh3 : p * n + 1 \u2264 p * (n + 1) + 1\nm : \u2115\nhm : p * n + 1 \u2264 m \u2227 m < p * (n + 1)\n\u22a2 \u2203 k, p * k < m \u2227 m < p * (k + 1)\n[PROOFSTEP]\nexact \u27e8n, lt_of_succ_le hm.1, hm.2\u27e9\n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\nh2 : p * n + 1 \u2264 p * (n + 1)\nh3 : p * n + 1 \u2264 p * (n + 1) + 1\nh4 : \u2200 (m : \u2115), m \u2208 Ico (p * n + 1) (p * (n + 1)) \u2192 multiplicity p m = 0\n\u22a2 multiplicity p (p * n)! \u2260 \u22a4 \u2192 multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1\n[PROOFSTEP]\nsimp_rw [\u2190 prod_Ico_id_eq_factorial, multiplicity.Finset.prod hp', \u2190 sum_Ico_consecutive _ h1 h3, add_assoc]\n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\nh2 : p * n + 1 \u2264 p * (n + 1)\nh3 : p * n + 1 \u2264 p * (n + 1) + 1\nh4 : \u2200 (m : \u2115), m \u2208 Ico (p * n + 1) (p * (n + 1)) \u2192 multiplicity p m = 0\n\u22a2 \u2211 x in Ico 1 (p * n + 1), multiplicity p x \u2260 \u22a4 \u2192\n \u2211 x in Ico 1 (p * n + 1), multiplicity p x + \u2211 x in Ico (p * n + 1) (p * (n + 1) + 1), multiplicity p x =\n \u2211 x in Ico 1 (p * n + 1), multiplicity p x + (multiplicity p (n + 1) + 1)\n[PROOFSTEP]\nintro h\n[GOAL]\nn p : \u2115\nhp : Prime p\nhp' : _root_.Prime p\nh0 : 2 \u2264 p\nh1 : 1 \u2264 p * n + 1\nh2 : p * n + 1 \u2264 p * (n + 1)\nh3 : p * n + 1 \u2264 p * (n + 1) + 1\nh4 : \u2200 (m : \u2115), m \u2208 Ico (p * n + 1) (p * (n + 1)) \u2192 multiplicity p m = 0\nh : \u2211 x in Ico 1 (p * n + 1), multiplicity p x \u2260 \u22a4\n\u22a2 \u2211 x in Ico 1 (p * n + 1), multiplicity p x + \u2211 x in Ico (p * n + 1) (p * (n + 1) + 1), multiplicity p x =\n \u2211 x in Ico 1 (p * n + 1), multiplicity p x + (multiplicity p (n + 1) + 1)\n[PROOFSTEP]\nrw [PartENat.add_left_cancel_iff h, sum_Ico_succ_top h2, multiplicity.mul hp', hp.multiplicity_self, sum_congr rfl h4,\n sum_const_zero, zero_add, add_comm (1 : PartENat)]\n[GOAL]\nn p : \u2115\nhp : Prime p\n\u22a2 multiplicity p (p * n)! = multiplicity p n ! + \u2191n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\np : \u2115\nhp : Prime p\n\u22a2 multiplicity p (p * zero)! = multiplicity p zero ! + \u2191zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\np : \u2115\nhp : Prime p\nn : \u2115\nih : multiplicity p (p * n)! = multiplicity p n ! + \u2191n\n\u22a2 multiplicity p (p * succ n)! = multiplicity p (succ n)! + \u2191(succ n)\n[PROOFSTEP]\nsimp only [succ_eq_add_one, multiplicity.mul, hp, hp.prime, ih, multiplicity_factorial_mul_succ, \u2190 add_assoc,\n Nat.cast_one, Nat.cast_add, factorial_succ]\n[GOAL]\ncase succ\np : \u2115\nhp : Prime p\nn : \u2115\nih : multiplicity p (p * n)! = multiplicity p n ! + \u2191n\n\u22a2 multiplicity p n ! + \u2191n + multiplicity p (n + 1) + 1 = multiplicity p (n + 1) + multiplicity p n ! + \u2191n + 1\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase succ.e_a\np : \u2115\nhp : Prime p\nn : \u2115\nih : multiplicity p (p * n)! = multiplicity p n ! + \u2191n\n\u22a2 multiplicity p n ! + \u2191n + multiplicity p (n + 1) = multiplicity p (n + 1) + multiplicity p n ! + \u2191n\n[PROOFSTEP]\nrw [add_comm, add_assoc]\n[GOAL]\np n r b : \u2115\nhp : Prime p\nhbn : log p n < b\n\u22a2 p ^ r \u2223 n ! \u2194 r \u2264 \u2211 i in Ico 1 b, n / p ^ i\n[PROOFSTEP]\nrw [\u2190 PartENat.coe_le_coe, \u2190 hp.multiplicity_factorial hbn, \u2190 pow_dvd_iff_le_multiplicity]\n[GOAL]\np : \u2115\nhp : Prime p\nn : \u2115\n\u22a2 multiplicity p n ! \u2264 \u2191(n / (p - 1))\n[PROOFSTEP]\nrw [hp.multiplicity_factorial (lt_succ_self _), PartENat.coe_le_coe]\n[GOAL]\np : \u2115\nhp : Prime p\nn : \u2115\n\u22a2 \u2211 i in Ico 1 (succ (log p n)), n / p ^ i \u2264 n / (p - 1)\n[PROOFSTEP]\nexact Nat.geom_sum_Ico_le hp.two_le _ _\n[GOAL]\np n b k : \u2115\nhp : Prime p\nhkn : k \u2264 n\n\u22a2 \u2211 i in Ico 1 b, n / p ^ i = \u2211 i in Ico 1 b, (k + (n - k)) / p ^ i\n[PROOFSTEP]\nsimp only [add_tsub_cancel_of_le hkn]\n[GOAL]\np n b k : \u2115\nhp : Prime p\nhkn : k \u2264 n\n\u22a2 \u2211 i in Ico 1 b, (k + (n - k)) / p ^ i =\n \u2211 i in Ico 1 b, (k / p ^ i + (n - k) / p ^ i + if p ^ i \u2264 k % p ^ i + (n - k) % p ^ i then 1 else 0)\n[PROOFSTEP]\nsimp only [Nat.add_div (pow_pos hp.pos _)]\n[GOAL]\np n b k : \u2115\nhp : Prime p\nhkn : k \u2264 n\n\u22a2 \u2211 i in Ico 1 b, (k / p ^ i + (n - k) / p ^ i + if p ^ i \u2264 k % p ^ i + (n - k) % p ^ i then 1 else 0) =\n \u2211 i in Ico 1 b, k / p ^ i + \u2211 i in Ico 1 b, (n - k) / p ^ i +\n card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))\n[PROOFSTEP]\nsimp [sum_add_distrib, sum_boole]\n[GOAL]\np n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\n\u22a2 multiplicity p (choose n k) = \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b)))\n[PROOFSTEP]\nhave h\u2081 :\n multiplicity p (choose n k) + multiplicity p (k ! * (n - k)!) =\n ((Finset.Ico 1 b).filter fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i).card + multiplicity p (k ! * (n - k)!) :=\n by\n rw [\u2190 hp.multiplicity_mul, \u2190 mul_assoc, choose_mul_factorial_mul_factorial hkn, hp.multiplicity_factorial hnb,\n hp.multiplicity_mul, hp.multiplicity_factorial ((log_mono_right hkn).trans_lt hnb),\n hp.multiplicity_factorial (lt_of_le_of_lt (log_mono_right tsub_le_self) hnb), multiplicity_choose_aux hp hkn]\n simp [add_comm]\n[GOAL]\np n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\n\u22a2 multiplicity p (choose n k) + multiplicity p (k ! * (n - k)!) =\n \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) + multiplicity p (k ! * (n - k)!)\n[PROOFSTEP]\nrw [\u2190 hp.multiplicity_mul, \u2190 mul_assoc, choose_mul_factorial_mul_factorial hkn, hp.multiplicity_factorial hnb,\n hp.multiplicity_mul, hp.multiplicity_factorial ((log_mono_right hkn).trans_lt hnb),\n hp.multiplicity_factorial (lt_of_le_of_lt (log_mono_right tsub_le_self) hnb), multiplicity_choose_aux hp hkn]\n[GOAL]\np n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\n\u22a2 \u2191(\u2211 i in Ico 1 b, k / p ^ i + \u2211 i in Ico 1 b, (n - k) / p ^ i +\n card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) =\n \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) +\n (\u2191(\u2211 i in Ico 1 b, k / p ^ i) + \u2191(\u2211 i in Ico 1 b, (n - k) / p ^ i))\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\np n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\nh\u2081 :\n multiplicity p (choose n k) + multiplicity p (k ! * (n - k)!) =\n \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) + multiplicity p (k ! * (n - k)!)\n\u22a2 multiplicity p (choose n k) = \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b)))\n[PROOFSTEP]\nrefine (PartENat.add_right_cancel_iff ?_).1 h\u2081\n[GOAL]\np n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\nh\u2081 :\n multiplicity p (choose n k) + multiplicity p (k ! * (n - k)!) =\n \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) + multiplicity p (k ! * (n - k)!)\n\u22a2 multiplicity p (k ! * (n - k)!) \u2260 \u22a4\n[PROOFSTEP]\napply PartENat.ne_top_iff_dom.2\n[GOAL]\np n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\nh\u2081 :\n multiplicity p (choose n k) + multiplicity p (k ! * (n - k)!) =\n \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) + multiplicity p (k ! * (n - k)!)\n\u22a2 (multiplicity p (k ! * (n - k)!)).Dom\n[PROOFSTEP]\nexact finite_nat_iff.2 \u27e8hp.ne_one, mul_pos (factorial_pos k) (factorial_pos (n - k))\u27e9\n[GOAL]\np : \u2115\nhp : Prime p\nx\u271d : \u2115\n\u22a2 multiplicity p x\u271d \u2264 multiplicity p (choose x\u271d 0) + multiplicity p 0\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\nhp : Prime p\nn\u271d : \u2115\n\u22a2 multiplicity p 0 \u2264 multiplicity p (choose 0 (n\u271d + 1)) + multiplicity p (n\u271d + 1)\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\nhp : Prime p\nn k : \u2115\n\u22a2 multiplicity p (n + 1) \u2264 multiplicity p (choose (n + 1) (k + 1)) + multiplicity p (k + 1)\n[PROOFSTEP]\nrw [\u2190 hp.multiplicity_mul]\n[GOAL]\np : \u2115\nhp : Prime p\nn k : \u2115\n\u22a2 multiplicity p (n + 1) \u2264 multiplicity p (choose (n + 1) (k + 1) * (k + 1))\n[PROOFSTEP]\nrefine' multiplicity_le_multiplicity_of_dvd_right _\n[GOAL]\np : \u2115\nhp : Prime p\nn k : \u2115\n\u22a2 n + 1 \u2223 choose (n + 1) (k + 1) * (k + 1)\n[PROOFSTEP]\nrw [\u2190 succ_mul_choose_eq]\n[GOAL]\np : \u2115\nhp : Prime p\nn k : \u2115\n\u22a2 n + 1 \u2223 succ n * choose n k\n[PROOFSTEP]\nexact dvd_mul_right _ _\n[GOAL]\np n k : \u2115\nhp : Prime p\nhkn : k \u2264 p ^ n\nhk0 : k \u2260 0\n\u22a2 multiplicity p (choose (p ^ n) k) + multiplicity p k \u2264 \u2191n\n[PROOFSTEP]\nhave hdisj :\n Disjoint ((Ico 1 n.succ).filter fun i => p ^ i \u2264 k % p ^ i + (p ^ n - k) % p ^ i)\n ((Ico 1 n.succ).filter fun i => p ^ i \u2223 k) :=\n by simp (config := { contextual := true }) [disjoint_right, *, dvd_iff_mod_eq_zero, Nat.mod_lt _ (pow_pos hp.pos _)]\n[GOAL]\np n k : \u2115\nhp : Prime p\nhkn : k \u2264 p ^ n\nhk0 : k \u2260 0\n\u22a2 Disjoint (filter (fun i => p ^ i \u2264 k % p ^ i + (p ^ n - k) % p ^ i) (Ico 1 (succ n)))\n (filter (fun i => p ^ i \u2223 k) (Ico 1 (succ n)))\n[PROOFSTEP]\nsimp (config := { contextual := true }) [disjoint_right, *, dvd_iff_mod_eq_zero, Nat.mod_lt _ (pow_pos hp.pos _)]\n[GOAL]\np n k : \u2115\nhp : Prime p\nhkn : k \u2264 p ^ n\nhk0 : k \u2260 0\nhdisj :\n Disjoint (filter (fun i => p ^ i \u2264 k % p ^ i + (p ^ n - k) % p ^ i) (Ico 1 (succ n)))\n (filter (fun i => p ^ i \u2223 k) (Ico 1 (succ n)))\n\u22a2 multiplicity p (choose (p ^ n) k) + multiplicity p k \u2264 \u2191n\n[PROOFSTEP]\nrw [multiplicity_choose hp hkn (lt_succ_self _),\n multiplicity_eq_card_pow_dvd (ne_of_gt hp.one_lt) hk0.bot_lt (lt_succ_of_le (log_mono_right hkn)), \u2190 Nat.cast_add,\n PartENat.coe_le_coe, log_pow hp.one_lt, \u2190 card_disjoint_union hdisj, filter_union_right]\n[GOAL]\np n k : \u2115\nhp : Prime p\nhkn : k \u2264 p ^ n\nhk0 : k \u2260 0\nhdisj :\n Disjoint (filter (fun i => p ^ i \u2264 k % p ^ i + (p ^ n - k) % p ^ i) (Ico 1 (succ n)))\n (filter (fun i => p ^ i \u2223 k) (Ico 1 (succ n)))\n\u22a2 card (filter (fun x => p ^ x \u2264 k % p ^ x + (p ^ n - k) % p ^ x \u2228 p ^ x \u2223 k) (Ico 1 (succ n))) \u2264 n\n[PROOFSTEP]\nhave filter_le_Ico := (Ico 1 n.succ).card_filter_le fun x => p ^ x \u2264 k % p ^ x + (p ^ n - k) % p ^ x \u2228 p ^ x \u2223 k\n[GOAL]\np n k : \u2115\nhp : Prime p\nhkn : k \u2264 p ^ n\nhk0 : k \u2260 0\nhdisj :\n Disjoint (filter (fun i => p ^ i \u2264 k % p ^ i + (p ^ n - k) % p ^ i) (Ico 1 (succ n)))\n (filter (fun i => p ^ i \u2223 k) (Ico 1 (succ n)))\nfilter_le_Ico :\n card (filter (fun x => p ^ x \u2264 k % p ^ x + (p ^ n - k) % p ^ x \u2228 p ^ x \u2223 k) (Ico 1 (succ n))) \u2264 card (Ico 1 (succ n))\n\u22a2 card (filter (fun x => p ^ x \u2264 k % p ^ x + (p ^ n - k) % p ^ x \u2228 p ^ x \u2223 k) (Ico 1 (succ n))) \u2264 n\n[PROOFSTEP]\nrwa [card_Ico 1 n.succ] at filter_le_Ico \n[GOAL]\np n k : \u2115\nhp : Prime p\nhkn : k \u2264 p ^ n\nhk0 : k \u2260 0\n\u22a2 \u2191n \u2264 multiplicity p (choose (p ^ n) k) + multiplicity p k\n[PROOFSTEP]\nrw [\u2190 hp.multiplicity_pow_self]\n[GOAL]\np n k : \u2115\nhp : Prime p\nhkn : k \u2264 p ^ n\nhk0 : k \u2260 0\n\u22a2 multiplicity p (p ^ n) \u2264 multiplicity p (choose (p ^ n) k) + multiplicity p k\n[PROOFSTEP]\nexact multiplicity_le_multiplicity_choose_add hp _ _\n[GOAL]\np n k : \u2115\nhp : Prime p\nhk : k \u2260 0\nhkp : k \u2260 p ^ n\n\u22a2 p \u2223 choose (p ^ n) k\n[PROOFSTEP]\nobtain hkp | hkp := hkp.symm.lt_or_lt\n[GOAL]\ncase inl\np n k : \u2115\nhp : Prime p\nhk : k \u2260 0\nhkp\u271d : k \u2260 p ^ n\nhkp : p ^ n < k\n\u22a2 p \u2223 choose (p ^ n) k\n[PROOFSTEP]\nsimp [choose_eq_zero_of_lt hkp]\n[GOAL]\ncase inr\np n k : \u2115\nhp : Prime p\nhk : k \u2260 0\nhkp\u271d : k \u2260 p ^ n\nhkp : k < p ^ n\n\u22a2 p \u2223 choose (p ^ n) k\n[PROOFSTEP]\nrefine' multiplicity_ne_zero.1 fun h => hkp.not_le <| Nat.le_of_dvd hk.bot_lt _\n[GOAL]\ncase inr\np n k : \u2115\nhp : Prime p\nhk : k \u2260 0\nhkp\u271d : k \u2260 p ^ n\nhkp : k < p ^ n\nh : multiplicity p (choose (p ^ n) k) = 0\n\u22a2 p ^ n \u2223 k\n[PROOFSTEP]\nhave H := hp.multiplicity_choose_prime_pow_add_multiplicity hkp.le hk\n[GOAL]\ncase inr\np n k : \u2115\nhp : Prime p\nhk : k \u2260 0\nhkp\u271d : k \u2260 p ^ n\nhkp : k < p ^ n\nh : multiplicity p (choose (p ^ n) k) = 0\nH : multiplicity p (choose (p ^ n) k) + multiplicity p k = \u2191n\n\u22a2 p ^ n \u2223 k\n[PROOFSTEP]\nrw [h, zero_add, eq_coe_iff] at H \n[GOAL]\ncase inr\np n k : \u2115\nhp : Prime p\nhk : k \u2260 0\nhkp\u271d : k \u2260 p ^ n\nhkp : k < p ^ n\nh : multiplicity p (choose (p ^ n) k) = 0\nH : p ^ n \u2223 k \u2227 \u00acp ^ (n + 1) \u2223 k\n\u22a2 p ^ n \u2223 k\n[PROOFSTEP]\nexact H.1\n[GOAL]\np n k : \u2115\nhp : Prime p\n\u22a2 p \u2223 choose (p ^ n) k \u2194 k \u2260 0 \u2227 k \u2260 p ^ n\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8_, _\u27e9, fun h => dvd_choose_pow hp h.1 h.2\u27e9\n[GOAL]\ncase refine'_1\np n k : \u2115\nhp : Prime p\nh : p \u2223 choose (p ^ n) k\n\u22a2 k \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_2\np n k : \u2115\nhp : Prime p\nh : p \u2223 choose (p ^ n) k\n\u22a2 k \u2260 p ^ n\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_1\np n : \u2115\nhp : Prime p\nh : p \u2223 choose (p ^ n) 0\n\u22a2 False\n[PROOFSTEP]\nsimp [hp.ne_one] at h \n[GOAL]\ncase refine'_2\np n : \u2115\nhp : Prime p\nh : p \u2223 choose (p ^ n) (p ^ n)\n\u22a2 False\n[PROOFSTEP]\nsimp [hp.ne_one] at h \n[GOAL]\n\u22a2 \u2200 {n : \u2115}, n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\n[PROOFSTEP]\nhave h2 := prime_two.prime\n[GOAL]\nh2 : _root_.Prime 2\n\u22a2 \u2200 {n : \u2115}, n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\n[PROOFSTEP]\nrefine' binaryRec _ _\n[GOAL]\ncase refine'_1\nh2 : _root_.Prime 2\n\u22a2 0 \u2260 0 \u2192 multiplicity 2 0! < \u21910\n[PROOFSTEP]\nexact fun h => False.elim <| h rfl\n[GOAL]\ncase refine'_2\nh2 : _root_.Prime 2\n\u22a2 \u2200 (b : Bool) (n : \u2115), (n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n) \u2192 bit b n \u2260 0 \u2192 multiplicity 2 (bit b n)! < \u2191(bit b n)\n[PROOFSTEP]\nintro b n ih h\n[GOAL]\ncase refine'_2\nh2 : _root_.Prime 2\nb : Bool\nn : \u2115\nih : n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\nh : bit b n \u2260 0\n\u22a2 multiplicity 2 (bit b n)! < \u2191(bit b n)\n[PROOFSTEP]\nby_cases hn : n = 0\n[GOAL]\ncase pos\nh2 : _root_.Prime 2\nb : Bool\nn : \u2115\nih : n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\nh : bit b n \u2260 0\nhn : n = 0\n\u22a2 multiplicity 2 (bit b n)! < \u2191(bit b n)\n[PROOFSTEP]\nsubst hn\n[GOAL]\ncase pos\nh2 : _root_.Prime 2\nb : Bool\nih : 0 \u2260 0 \u2192 multiplicity 2 0! < \u21910\nh : bit b 0 \u2260 0\n\u22a2 multiplicity 2 (bit b 0)! < \u2191(bit b 0)\n[PROOFSTEP]\nsimp only [ne_eq, bit_eq_zero, true_and, Bool.not_eq_false] at h \n[GOAL]\ncase pos\nh2 : _root_.Prime 2\nb : Bool\nih : 0 \u2260 0 \u2192 multiplicity 2 0! < \u21910\nh : b = true\n\u22a2 multiplicity 2 (bit b 0)! < \u2191(bit b 0)\n[PROOFSTEP]\nsimp only [h, bit_true, bit1_zero, factorial, mul_one, Nat.isUnit_iff, cast_one]\n[GOAL]\ncase pos\nh2 : _root_.Prime 2\nb : Bool\nih : 0 \u2260 0 \u2192 multiplicity 2 0! < \u21910\nh : b = true\n\u22a2 multiplicity 2 (succ 0) < 1\n[PROOFSTEP]\nrw [Prime.multiplicity_one]\n[GOAL]\ncase pos\nh2 : _root_.Prime 2\nb : Bool\nih : 0 \u2260 0 \u2192 multiplicity 2 0! < \u21910\nh : b = true\n\u22a2 0 < 1\ncase pos h2 : _root_.Prime 2 b : Bool ih : 0 \u2260 0 \u2192 multiplicity 2 0! < \u21910 h : b = true \u22a2 Prime 2\n[PROOFSTEP]\nsimp only [zero_lt_one]\n[GOAL]\ncase pos\nh2 : _root_.Prime 2\nb : Bool\nih : 0 \u2260 0 \u2192 multiplicity 2 0! < \u21910\nh : b = true\n\u22a2 Prime 2\n[PROOFSTEP]\ndecide\n[GOAL]\ncase neg\nh2 : _root_.Prime 2\nb : Bool\nn : \u2115\nih : n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\nh : bit b n \u2260 0\nhn : \u00acn = 0\n\u22a2 multiplicity 2 (bit b n)! < \u2191(bit b n)\n[PROOFSTEP]\nhave : multiplicity 2 (2 * n)! < (2 * n : \u2115) :=\n by\n rw [prime_two.multiplicity_factorial_mul]\n refine' (PartENat.add_lt_add_right (ih hn) (PartENat.natCast_ne_top _)).trans_le _\n rw [two_mul]\n norm_cast\n[GOAL]\nh2 : _root_.Prime 2\nb : Bool\nn : \u2115\nih : n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\nh : bit b n \u2260 0\nhn : \u00acn = 0\n\u22a2 multiplicity 2 (2 * n)! < \u2191(2 * n)\n[PROOFSTEP]\nrw [prime_two.multiplicity_factorial_mul]\n[GOAL]\nh2 : _root_.Prime 2\nb : Bool\nn : \u2115\nih : n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\nh : bit b n \u2260 0\nhn : \u00acn = 0\n\u22a2 multiplicity 2 n ! + \u2191n < \u2191(2 * n)\n[PROOFSTEP]\nrefine' (PartENat.add_lt_add_right (ih hn) (PartENat.natCast_ne_top _)).trans_le _\n[GOAL]\nh2 : _root_.Prime 2\nb : Bool\nn : \u2115\nih : n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\nh : bit b n \u2260 0\nhn : \u00acn = 0\n\u22a2 \u2191n + \u2191n \u2264 \u2191(2 * n)\n[PROOFSTEP]\nrw [two_mul]\n[GOAL]\nh2 : _root_.Prime 2\nb : Bool\nn : \u2115\nih : n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\nh : bit b n \u2260 0\nhn : \u00acn = 0\n\u22a2 \u2191n + \u2191n \u2264 \u2191(n + n)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase neg\nh2 : _root_.Prime 2\nb : Bool\nn : \u2115\nih : n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\nh : bit b n \u2260 0\nhn : \u00acn = 0\nthis : multiplicity 2 (2 * n)! < \u2191(2 * n)\n\u22a2 multiplicity 2 (bit b n)! < \u2191(bit b n)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase neg.false\nh2 : _root_.Prime 2\nn : \u2115\nih : n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\nhn : \u00acn = 0\nthis : multiplicity 2 (2 * n)! < \u2191(2 * n)\nh : bit false n \u2260 0\n\u22a2 multiplicity 2 (bit false n)! < \u2191(bit false n)\n[PROOFSTEP]\nsimpa [bit0_eq_two_mul n]\n[GOAL]\ncase neg.true\nh2 : _root_.Prime 2\nn : \u2115\nih : n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\nhn : \u00acn = 0\nthis : multiplicity 2 (2 * n)! < \u2191(2 * n)\nh : bit true n \u2260 0\n\u22a2 multiplicity 2 (bit true n)! < \u2191(bit true n)\n[PROOFSTEP]\nsuffices multiplicity 2 (2 * n + 1) + multiplicity 2 (2 * n)! < \u2191(2 * n) + 1 by\n simpa [succ_eq_add_one, multiplicity.mul, h2, prime_two, Nat.bit1_eq_succ_bit0, bit0_eq_two_mul n]\n[GOAL]\nh2 : _root_.Prime 2\nn : \u2115\nih : n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\nhn : \u00acn = 0\nthis\u271d : multiplicity 2 (2 * n)! < \u2191(2 * n)\nh : bit true n \u2260 0\nthis : multiplicity 2 (2 * n + 1) + multiplicity 2 (2 * n)! < \u2191(2 * n) + 1\n\u22a2 multiplicity 2 (bit true n)! < \u2191(bit true n)\n[PROOFSTEP]\nsimpa [succ_eq_add_one, multiplicity.mul, h2, prime_two, Nat.bit1_eq_succ_bit0, bit0_eq_two_mul n]\n[GOAL]\ncase neg.true\nh2 : _root_.Prime 2\nn : \u2115\nih : n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\nhn : \u00acn = 0\nthis : multiplicity 2 (2 * n)! < \u2191(2 * n)\nh : bit true n \u2260 0\n\u22a2 multiplicity 2 (2 * n + 1) + multiplicity 2 (2 * n)! < \u2191(2 * n) + 1\n[PROOFSTEP]\nrw [multiplicity_eq_zero.2 (two_not_dvd_two_mul_add_one n), zero_add]\n[GOAL]\ncase neg.true\nh2 : _root_.Prime 2\nn : \u2115\nih : n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\nhn : \u00acn = 0\nthis : multiplicity 2 (2 * n)! < \u2191(2 * n)\nh : bit true n \u2260 0\n\u22a2 multiplicity 2 (2 * n)! < \u2191(2 * n) + 1\n[PROOFSTEP]\nrefine' this.trans _\n[GOAL]\ncase neg.true\nh2 : _root_.Prime 2\nn : \u2115\nih : n \u2260 0 \u2192 multiplicity 2 n ! < \u2191n\nhn : \u00acn = 0\nthis : multiplicity 2 (2 * n)! < \u2191(2 * n)\nh : bit true n \u2260 0\n\u22a2 \u2191(2 * n) < \u2191(2 * n) + 1\n[PROOFSTEP]\nexact_mod_cast lt_succ_self _\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Multiplicity", "llama_tokens": 12205, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085909370422, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4475101827237892}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Z \u27f6 X\nw : h \u226b f = h \u226b g\n\u22a2 G.map h \u226b G.map f = G.map h \u226b G.map g\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp, w]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Z \u27f6 X\nw : h \u226b f = h \u226b g\n\u22a2 (Iso.refl\n ((Cones.postcompose (diagramIsoParallelPair (parallelPair f g \u22d9 G)).hom).obj\n (G.mapCone (Fork.of\u03b9 h w))).pt).hom \u226b\n Fork.\u03b9 (Fork.of\u03b9 (G.map h) (_ : G.map h \u226b G.map f = G.map h \u226b G.map g)) =\n Fork.\u03b9 ((Cones.postcompose (diagramIsoParallelPair (parallelPair f g \u22d9 G)).hom).obj (G.mapCone (Fork.of\u03b9 h w)))\n[PROOFSTEP]\nsimp [Fork.\u03b9]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Z \u27f6 X\nw : h \u226b f = h \u226b g\ninst\u271d : PreservesLimit (parallelPair f g) G\nl : IsLimit (Fork.of\u03b9 h w)\n\u22a2 G.map h \u226b G.map f = G.map h \u226b G.map g\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp, w]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Z \u27f6 X\nw : h \u226b f = h \u226b g\ninst\u271d : ReflectsLimit (parallelPair f g) G\n\u22a2 G.map h \u226b G.map f = G.map h \u226b G.map g\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp, w]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Z \u27f6 X\nw : h \u226b f = h \u226b g\ninst\u271d\u00b9 : HasEqualizer f g\ninst\u271d : PreservesLimit (parallelPair f g) G\n\u22a2 G.map (equalizer.\u03b9 f g) \u226b G.map f = G.map (equalizer.\u03b9 f g) \u226b G.map g\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Z \u27f6 X\nw : h \u226b f = h \u226b g\ninst\u271d\u00b9 : HasEqualizer f g\ninst\u271d : PreservesLimit (parallelPair f g) G\n\u22a2 G.map (equalizer.\u03b9 f g \u226b f) = G.map (equalizer.\u03b9 f g \u226b g)\n[PROOFSTEP]\nrw [equalizer.condition]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Z \u27f6 X\nw : h \u226b f = h \u226b g\ninst\u271d\u00b9 : HasEqualizer f g\ninst\u271d : HasEqualizer (G.map f) (G.map g)\ni : IsIso (equalizerComparison f g G)\n\u22a2 PreservesLimit (parallelPair f g) G\n[PROOFSTEP]\napply preservesLimitOfPreservesLimitCone (equalizerIsEqualizer f g)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Z \u27f6 X\nw : h \u226b f = h \u226b g\ninst\u271d\u00b9 : HasEqualizer f g\ninst\u271d : HasEqualizer (G.map f) (G.map g)\ni : IsIso (equalizerComparison f g G)\n\u22a2 IsLimit (G.mapCone (Fork.of\u03b9 (equalizer.\u03b9 f g) (_ : equalizer.\u03b9 f g \u226b f = equalizer.\u03b9 f g \u226b g)))\n[PROOFSTEP]\napply (isLimitMapConeForkEquiv _ _).symm _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Z \u27f6 X\nw : h \u226b f = h \u226b g\ninst\u271d\u00b9 : HasEqualizer f g\ninst\u271d : HasEqualizer (G.map f) (G.map g)\ni : IsIso (equalizerComparison f g G)\n\u22a2 IsLimit\n (Fork.of\u03b9 (G.map (equalizer.\u03b9 f g)) (_ : G.map (equalizer.\u03b9 f g) \u226b G.map f = G.map (equalizer.\u03b9 f g) \u226b G.map g))\n[PROOFSTEP]\nrefine @IsLimit.ofPointIso _ _ _ _ _ _ _ (limit.isLimit (parallelPair (G.map f) (G.map g))) ?_\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Z \u27f6 X\nw : h \u226b f = h \u226b g\ninst\u271d\u00b9 : HasEqualizer f g\ninst\u271d : HasEqualizer (G.map f) (G.map g)\ni : IsIso (equalizerComparison f g G)\n\u22a2 IsIso\n (IsLimit.lift (limit.isLimit (parallelPair (G.map f) (G.map g)))\n (Fork.of\u03b9 (G.map (equalizer.\u03b9 f g)) (_ : G.map (equalizer.\u03b9 f g) \u226b G.map f = G.map (equalizer.\u03b9 f g) \u226b G.map g)))\n[PROOFSTEP]\napply i\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Z \u27f6 X\nw : h \u226b f = h \u226b g\ninst\u271d\u00b2 : HasEqualizer f g\ninst\u271d\u00b9 : HasEqualizer (G.map f) (G.map g)\ninst\u271d : PreservesLimit (parallelPair f g) G\n\u22a2 IsIso (equalizerComparison f g G)\n[PROOFSTEP]\nrw [\u2190 PreservesEqualizer.iso_hom]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Z \u27f6 X\nw : h \u226b f = h \u226b g\ninst\u271d\u00b2 : HasEqualizer f g\ninst\u271d\u00b9 : HasEqualizer (G.map f) (G.map g)\ninst\u271d : PreservesLimit (parallelPair f g) G\n\u22a2 IsIso (PreservesEqualizer.iso G f g).hom\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\n\u22a2 G.map f \u226b G.map h = G.map g \u226b G.map h\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp, w]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\n\u22a2 Cofork.\u03c0\n ((Cocones.precompose (diagramIsoParallelPair (parallelPair f g \u22d9 G)).inv).obj (G.mapCocone (Cofork.of\u03c0 h w))) \u226b\n (Iso.refl\n ((Cocones.precompose (diagramIsoParallelPair (parallelPair f g \u22d9 G)).inv).obj\n (G.mapCocone (Cofork.of\u03c0 h w))).pt).hom =\n Cofork.\u03c0 (Cofork.of\u03c0 (G.map h) (_ : G.map f \u226b G.map h = G.map g \u226b G.map h))\n[PROOFSTEP]\ndsimp only [Cofork.\u03c0, Cofork.of\u03c0_\u03b9_app]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\n\u22a2 NatTrans.app\n ((Cocones.precompose (diagramIsoParallelPair (parallelPair f g \u22d9 G)).inv).obj (G.mapCocone (Cofork.of\u03c0 h w))).\u03b9\n WalkingParallelPair.one \u226b\n (Iso.refl\n ((Cocones.precompose (diagramIsoParallelPair (parallelPair f g \u22d9 G)).inv).obj\n (G.mapCocone (Cofork.of\u03c0 h w))).pt).hom =\n G.map h\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\n\u22a2 (\ud835\udfd9 (G.obj Y) \u226b G.map h) \u226b \ud835\udfd9 (G.obj Z) = G.map h\n[PROOFSTEP]\nrw [Category.comp_id, Category.id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\ninst\u271d : PreservesColimit (parallelPair f g) G\nl : IsColimit (Cofork.of\u03c0 h w)\n\u22a2 G.map f \u226b G.map h = G.map g \u226b G.map h\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp, w]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\ninst\u271d : ReflectsColimit (parallelPair f g) G\n\u22a2 G.map f \u226b G.map h = G.map g \u226b G.map h\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp, w]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\ninst\u271d\u00b9 : HasCoequalizer f g\ninst\u271d : PreservesColimit (parallelPair f g) G\n\u22a2 G.map f \u226b G.map (coequalizer.\u03c0 f g) = G.map g \u226b G.map (coequalizer.\u03c0 f g)\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\ninst\u271d\u00b9 : HasCoequalizer f g\ninst\u271d : PreservesColimit (parallelPair f g) G\n\u22a2 G.map (f \u226b coequalizer.\u03c0 f g) = G.map (g \u226b coequalizer.\u03c0 f g)\n[PROOFSTEP]\nrw [coequalizer.condition]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\ninst\u271d\u00b9 : HasCoequalizer f g\ninst\u271d : HasCoequalizer (G.map f) (G.map g)\ni : IsIso (coequalizerComparison f g G)\n\u22a2 PreservesColimit (parallelPair f g) G\n[PROOFSTEP]\napply preservesColimitOfPreservesColimitCocone (coequalizerIsCoequalizer f g)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\ninst\u271d\u00b9 : HasCoequalizer f g\ninst\u271d : HasCoequalizer (G.map f) (G.map g)\ni : IsIso (coequalizerComparison f g G)\n\u22a2 IsColimit (G.mapCocone (Cofork.of\u03c0 (coequalizer.\u03c0 f g) (_ : f \u226b coequalizer.\u03c0 f g = g \u226b coequalizer.\u03c0 f g)))\n[PROOFSTEP]\napply (isColimitMapCoconeCoforkEquiv _ _).symm _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\ninst\u271d\u00b9 : HasCoequalizer f g\ninst\u271d : HasCoequalizer (G.map f) (G.map g)\ni : IsIso (coequalizerComparison f g G)\n\u22a2 IsColimit\n (Cofork.of\u03c0 (G.map (coequalizer.\u03c0 f g))\n (_ : G.map f \u226b G.map (coequalizer.\u03c0 f g) = G.map g \u226b G.map (coequalizer.\u03c0 f g)))\n[PROOFSTEP]\nrefine @IsColimit.ofPointIso _ _ _ _ _ _ _ (colimit.isColimit (parallelPair (G.map f) (G.map g))) ?_\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\ninst\u271d\u00b9 : HasCoequalizer f g\ninst\u271d : HasCoequalizer (G.map f) (G.map g)\ni : IsIso (coequalizerComparison f g G)\n\u22a2 IsIso\n (IsColimit.desc (colimit.isColimit (parallelPair (G.map f) (G.map g)))\n (Cofork.of\u03c0 (G.map (coequalizer.\u03c0 f g))\n (_ : G.map f \u226b G.map (coequalizer.\u03c0 f g) = G.map g \u226b G.map (coequalizer.\u03c0 f g))))\n[PROOFSTEP]\napply i\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\ninst\u271d\u00b2 : HasCoequalizer f g\ninst\u271d\u00b9 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d : PreservesColimit (parallelPair f g) G\n\u22a2 IsIso (coequalizerComparison f g G)\n[PROOFSTEP]\nrw [\u2190 PreservesCoequalizer.iso_hom]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\ninst\u271d\u00b2 : HasCoequalizer f g\ninst\u271d\u00b9 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d : PreservesColimit (parallelPair f g) G\n\u22a2 IsIso (PreservesCoequalizer.iso G f g).hom\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b2 : HasCoequalizer f g\ninst\u271d\u00b9 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d : PreservesColimit (parallelPair f g) G\nW : D\nh k : G.obj (coequalizer f g) \u27f6 W\n\u22a2 G.map (coequalizer.\u03c0 f g) \u226b h = G.map (coequalizer.\u03c0 f g) \u226b k \u2192 h = k\n[PROOFSTEP]\nrw [\u2190 \u03b9_comp_coequalizerComparison]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b2 : HasCoequalizer f g\ninst\u271d\u00b9 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d : PreservesColimit (parallelPair f g) G\nW : D\nh k : G.obj (coequalizer f g) \u27f6 W\n\u22a2 (coequalizer.\u03c0 (G.map f) (G.map g) \u226b coequalizerComparison f g G) \u226b h =\n (coequalizer.\u03c0 (G.map f) (G.map g) \u226b coequalizerComparison f g G) \u226b k \u2192\n h = k\n[PROOFSTEP]\nhaveI : Epi (coequalizer.\u03c0 (G.map f) (G.map g) \u226b coequalizerComparison f g G) := by apply epi_comp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b2 : HasCoequalizer f g\ninst\u271d\u00b9 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d : PreservesColimit (parallelPair f g) G\nW : D\nh k : G.obj (coequalizer f g) \u27f6 W\n\u22a2 Epi (coequalizer.\u03c0 (G.map f) (G.map g) \u226b coequalizerComparison f g G)\n[PROOFSTEP]\napply epi_comp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b2 : HasCoequalizer f g\ninst\u271d\u00b9 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d : PreservesColimit (parallelPair f g) G\nW : D\nh k : G.obj (coequalizer f g) \u27f6 W\nthis : Epi (coequalizer.\u03c0 (G.map f) (G.map g) \u226b coequalizerComparison f g G)\n\u22a2 (coequalizer.\u03c0 (G.map f) (G.map g) \u226b coequalizerComparison f g G) \u226b h =\n (coequalizer.\u03c0 (G.map f) (G.map g) \u226b coequalizerComparison f g G) \u226b k \u2192\n h = k\n[PROOFSTEP]\napply (cancel_epi _).1\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\ninst\u271d\u00b2 : HasCoequalizer f g\ninst\u271d\u00b9 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d : PreservesColimit (parallelPair f g) G\n\u22a2 G.map (coequalizer.\u03c0 f g) \u226b (PreservesCoequalizer.iso G f g).inv = coequalizer.\u03c0 (G.map f) (G.map g)\n[PROOFSTEP]\nrw [\u2190 \u03b9_comp_coequalizerComparison_assoc, \u2190 PreservesCoequalizer.iso_hom, Iso.hom_inv_id, comp_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\ninst\u271d\u00b2 : HasCoequalizer f g\ninst\u271d\u00b9 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d : PreservesColimit (parallelPair f g) G\nW : D\nk : G.obj Y \u27f6 W\nwk : G.map f \u226b k = G.map g \u226b k\n\u22a2 G.map (coequalizer.\u03c0 f g) \u226b (PreservesCoequalizer.iso G f g).inv \u226b coequalizer.desc k wk = k\n[PROOFSTEP]\nrw [\u2190 Category.assoc, map_\u03c0_preserves_coequalizer_inv, coequalizer.\u03c0_desc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\ninst\u271d\u00b3 : HasCoequalizer f g\ninst\u271d\u00b2 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' \u27f6 Y'\ninst\u271d : HasCoequalizer f' g'\np : G.obj X \u27f6 X'\nq : G.obj Y \u27f6 Y'\nwf : G.map f \u226b q = p \u226b f'\nwg : G.map g \u226b q = p \u226b g'\n\u22a2 G.map (coequalizer.\u03c0 f g) \u226b\n (PreservesCoequalizer.iso G f g).inv \u226b colimMap (parallelPairHom (G.map f) (G.map g) f' g' p q wf wg) =\n q \u226b coequalizer.\u03c0 f' g'\n[PROOFSTEP]\nrw [\u2190 Category.assoc, map_\u03c0_preserves_coequalizer_inv, \u03b9_colimMap, parallelPairHom_app_one]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b3 : HasCoequalizer f g\ninst\u271d\u00b2 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' \u27f6 Y'\ninst\u271d : HasCoequalizer f' g'\np : G.obj X \u27f6 X'\nq : G.obj Y \u27f6 Y'\nwf : G.map f \u226b q = p \u226b f'\nwg : G.map g \u226b q = p \u226b g'\nZ' : D\nh : Y' \u27f6 Z'\nwh : f' \u226b h = g' \u226b h\n\u22a2 G.map (coequalizer.\u03c0 f g) \u226b\n (PreservesCoequalizer.iso G f g).inv \u226b\n colimMap (parallelPairHom (G.map f) (G.map g) f' g' p q wf wg) \u226b coequalizer.desc h wh =\n q \u226b h\n[PROOFSTEP]\nslice_lhs 1 3 => rw [map_\u03c0_preserves_coequalizer_inv_colimMap]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b3 : HasCoequalizer f g\ninst\u271d\u00b2 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' \u27f6 Y'\ninst\u271d : HasCoequalizer f' g'\np : G.obj X \u27f6 X'\nq : G.obj Y \u27f6 Y'\nwf : G.map f \u226b q = p \u226b f'\nwg : G.map g \u226b q = p \u226b g'\nZ' : D\nh : Y' \u27f6 Z'\nwh : f' \u226b h = g' \u226b h\n| G.map (coequalizer.\u03c0 f g) \u226b\n (PreservesCoequalizer.iso G f g).inv \u226b colimMap (parallelPairHom (G.map f) (G.map g) f' g' p q wf wg)\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b3 : HasCoequalizer f g\ninst\u271d\u00b2 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' \u27f6 Y'\ninst\u271d : HasCoequalizer f' g'\np : G.obj X \u27f6 X'\nq : G.obj Y \u27f6 Y'\nwf : G.map f \u226b q = p \u226b f'\nwg : G.map g \u226b q = p \u226b g'\nZ' : D\nh : Y' \u27f6 Z'\nwh : f' \u226b h = g' \u226b h\n| coequalizer.desc h wh\n[PROOFSTEP]\nrw [map_\u03c0_preserves_coequalizer_inv_colimMap]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b3 : HasCoequalizer f g\ninst\u271d\u00b2 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' \u27f6 Y'\ninst\u271d : HasCoequalizer f' g'\np : G.obj X \u27f6 X'\nq : G.obj Y \u27f6 Y'\nwf : G.map f \u226b q = p \u226b f'\nwg : G.map g \u226b q = p \u226b g'\nZ' : D\nh : Y' \u27f6 Z'\nwh : f' \u226b h = g' \u226b h\n| G.map (coequalizer.\u03c0 f g) \u226b\n (PreservesCoequalizer.iso G f g).inv \u226b colimMap (parallelPairHom (G.map f) (G.map g) f' g' p q wf wg)\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b3 : HasCoequalizer f g\ninst\u271d\u00b2 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' \u27f6 Y'\ninst\u271d : HasCoequalizer f' g'\np : G.obj X \u27f6 X'\nq : G.obj Y \u27f6 Y'\nwf : G.map f \u226b q = p \u226b f'\nwg : G.map g \u226b q = p \u226b g'\nZ' : D\nh : Y' \u27f6 Z'\nwh : f' \u226b h = g' \u226b h\n| coequalizer.desc h wh\n[PROOFSTEP]\nrw [map_\u03c0_preserves_coequalizer_inv_colimMap]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b3 : HasCoequalizer f g\ninst\u271d\u00b2 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' \u27f6 Y'\ninst\u271d : HasCoequalizer f' g'\np : G.obj X \u27f6 X'\nq : G.obj Y \u27f6 Y'\nwf : G.map f \u226b q = p \u226b f'\nwg : G.map g \u226b q = p \u226b g'\nZ' : D\nh : Y' \u27f6 Z'\nwh : f' \u226b h = g' \u226b h\n| G.map (coequalizer.\u03c0 f g) \u226b\n (PreservesCoequalizer.iso G f g).inv \u226b colimMap (parallelPairHom (G.map f) (G.map g) f' g' p q wf wg)\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b3 : HasCoequalizer f g\ninst\u271d\u00b2 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' \u27f6 Y'\ninst\u271d : HasCoequalizer f' g'\np : G.obj X \u27f6 X'\nq : G.obj Y \u27f6 Y'\nwf : G.map f \u226b q = p \u226b f'\nwg : G.map g \u226b q = p \u226b g'\nZ' : D\nh : Y' \u27f6 Z'\nwh : f' \u226b h = g' \u226b h\n| coequalizer.desc h wh\n[PROOFSTEP]\nrw [map_\u03c0_preserves_coequalizer_inv_colimMap]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b3 : HasCoequalizer f g\ninst\u271d\u00b2 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' \u27f6 Y'\ninst\u271d : HasCoequalizer f' g'\np : G.obj X \u27f6 X'\nq : G.obj Y \u27f6 Y'\nwf : G.map f \u226b q = p \u226b f'\nwg : G.map g \u226b q = p \u226b g'\nZ' : D\nh : Y' \u27f6 Z'\nwh : f' \u226b h = g' \u226b h\n\u22a2 (q \u226b coequalizer.\u03c0 f' g') \u226b coequalizer.desc h wh = q \u226b h\n[PROOFSTEP]\nslice_lhs 2 3 => rw [coequalizer.\u03c0_desc]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b3 : HasCoequalizer f g\ninst\u271d\u00b2 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' \u27f6 Y'\ninst\u271d : HasCoequalizer f' g'\np : G.obj X \u27f6 X'\nq : G.obj Y \u27f6 Y'\nwf : G.map f \u226b q = p \u226b f'\nwg : G.map g \u226b q = p \u226b g'\nZ' : D\nh : Y' \u27f6 Z'\nwh : f' \u226b h = g' \u226b h\n| coequalizer.\u03c0 f' g' \u226b coequalizer.desc h wh\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b3 : HasCoequalizer f g\ninst\u271d\u00b2 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' \u27f6 Y'\ninst\u271d : HasCoequalizer f' g'\np : G.obj X \u27f6 X'\nq : G.obj Y \u27f6 Y'\nwf : G.map f \u226b q = p \u226b f'\nwg : G.map g \u226b q = p \u226b g'\nZ' : D\nh : Y' \u27f6 Z'\nwh : f' \u226b h = g' \u226b h\n| q\n[PROOFSTEP]\nrw [coequalizer.\u03c0_desc]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b3 : HasCoequalizer f g\ninst\u271d\u00b2 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' \u27f6 Y'\ninst\u271d : HasCoequalizer f' g'\np : G.obj X \u27f6 X'\nq : G.obj Y \u27f6 Y'\nwf : G.map f \u226b q = p \u226b f'\nwg : G.map g \u226b q = p \u226b g'\nZ' : D\nh : Y' \u27f6 Z'\nwh : f' \u226b h = g' \u226b h\n| coequalizer.\u03c0 f' g' \u226b coequalizer.desc h wh\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b3 : HasCoequalizer f g\ninst\u271d\u00b2 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' \u27f6 Y'\ninst\u271d : HasCoequalizer f' g'\np : G.obj X \u27f6 X'\nq : G.obj Y \u27f6 Y'\nwf : G.map f \u226b q = p \u226b f'\nwg : G.map g \u226b q = p \u226b g'\nZ' : D\nh : Y' \u27f6 Z'\nwh : f' \u226b h = g' \u226b h\n| q\n[PROOFSTEP]\nrw [coequalizer.\u03c0_desc]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b3 : HasCoequalizer f g\ninst\u271d\u00b2 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' \u27f6 Y'\ninst\u271d : HasCoequalizer f' g'\np : G.obj X \u27f6 X'\nq : G.obj Y \u27f6 Y'\nwf : G.map f \u226b q = p \u226b f'\nwg : G.map g \u226b q = p \u226b g'\nZ' : D\nh : Y' \u27f6 Z'\nwh : f' \u226b h = g' \u226b h\n| coequalizer.\u03c0 f' g' \u226b coequalizer.desc h wh\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh\u271d : Y \u27f6 Z\nw : f \u226b h\u271d = g \u226b h\u271d\ninst\u271d\u00b3 : HasCoequalizer f g\ninst\u271d\u00b2 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f g) G\nX' Y' : D\nf' g' : X' \u27f6 Y'\ninst\u271d : HasCoequalizer f' g'\np : G.obj X \u27f6 X'\nq : G.obj Y \u27f6 Y'\nwf : G.map f \u226b q = p \u226b f'\nwg : G.map g \u226b q = p \u226b g'\nZ' : D\nh : Y' \u27f6 Z'\nwh : f' \u226b h = g' \u226b h\n| q\n[PROOFSTEP]\nrw [coequalizer.\u03c0_desc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf\u271d g\u271d : X \u27f6 Y\nh : Y \u27f6 Z\nw : f\u271d \u226b h = g\u271d \u226b h\ninst\u271d\u00b3 : HasCoequalizer f\u271d g\u271d\ninst\u271d\u00b2 : HasCoequalizer (G.map f\u271d) (G.map g\u271d)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f\u271d g\u271d) G\nf g : X \u27f6 Y\ninst\u271d : HasSplitCoequalizer f g\n\u22a2 PreservesColimit (parallelPair f g) G\n[PROOFSTEP]\napply preservesColimitOfPreservesColimitCocone (HasSplitCoequalizer.isSplitCoequalizer f g).isCoequalizer\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf\u271d g\u271d : X \u27f6 Y\nh : Y \u27f6 Z\nw : f\u271d \u226b h = g\u271d \u226b h\ninst\u271d\u00b3 : HasCoequalizer f\u271d g\u271d\ninst\u271d\u00b2 : HasCoequalizer (G.map f\u271d) (G.map g\u271d)\ninst\u271d\u00b9 : PreservesColimit (parallelPair f\u271d g\u271d) G\nf g : X \u27f6 Y\ninst\u271d : HasSplitCoequalizer f g\n\u22a2 IsColimit (G.mapCocone (IsSplitCoequalizer.asCofork (HasSplitCoequalizer.isSplitCoequalizer f g)))\n[PROOFSTEP]\napply (isColimitMapCoconeCoforkEquiv G _).symm ((HasSplitCoequalizer.isSplitCoequalizer f g).map G).isCoequalizer\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers", "llama_tokens": 11835, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149758396752, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.44741289198530904}} {"text": "[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b3 : OrderedCancelCommMonoid \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b2 : One \u03b2\ninst\u271d\u00b9 : Mul \u03b2\ninst\u271d : Pow \u03b2 \u2115\nf : \u03b2 \u2192 \u03b1\nhf : Injective f\none : f 1 = 1\nmul : \u2200 (x y : \u03b2), f (x * y) = f x * f y\nnpow : \u2200 (x : \u03b2) (n : \u2115), f (x ^ n) = f x ^ n\nsrc\u271d : OrderedCommMonoid \u03b2 := orderedCommMonoid f hf one mul npow\na b c : \u03b2\nbc : f (a * b) \u2264 f (a * c)\n\u22a2 f a * f b \u2264 f a * f c\n[PROOFSTEP]\nrwa [\u2190 mul, \u2190 mul]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Monoid.Cancel.Basic", "llama_tokens": 222, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7745833841649233, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.4473183143051844}} {"text": "[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nu : R\u02e3\n\u22a2 \u2191(constantCoeff R) (invUnitsSub u) = 1 /\u209a u\n[PROOFSTEP]\nrw [\u2190 coeff_zero_eq_constantCoeff_apply, coeff_invUnitsSub, zero_add, pow_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nu : R\u02e3\n\u22a2 invUnitsSub u * X = invUnitsSub u * \u2191(C R) \u2191u - 1\n[PROOFSTEP]\next (_ | n)\n[GOAL]\ncase h.zero\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nu : R\u02e3\n\u22a2 \u2191(coeff R Nat.zero) (invUnitsSub u * X) = \u2191(coeff R Nat.zero) (invUnitsSub u * \u2191(C R) \u2191u - 1)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.succ\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nu : R\u02e3\nn : \u2115\n\u22a2 \u2191(coeff R (Nat.succ n)) (invUnitsSub u * X) = \u2191(coeff R (Nat.succ n)) (invUnitsSub u * \u2191(C R) \u2191u - 1)\n[PROOFSTEP]\nsimp [n.succ_ne_zero, pow_succ]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nu : R\u02e3\n\u22a2 invUnitsSub u * (\u2191(C R) \u2191u - X) = 1\n[PROOFSTEP]\nsimp [mul_sub, sub_sub_cancel]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nu : R\u02e3\n\u22a2 \u2191(map f) (invUnitsSub u) = invUnitsSub (\u2191(Units.map \u2191f) u)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nu : R\u02e3\nn\u271d : \u2115\n\u22a2 \u2191(coeff S n\u271d) (\u2191(map f) (invUnitsSub u)) = \u2191(coeff S n\u271d) (invUnitsSub (\u2191(Units.map \u2191f) u))\n[PROOFSTEP]\nsimp only [\u2190 map_pow, coeff_map, coeff_invUnitsSub, one_divp]\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nu : R\u02e3\nn\u271d : \u2115\n\u22a2 \u2191f \u2191(u ^ (n\u271d + 1))\u207b\u00b9 = \u2191(\u2191(Units.map \u2191f) (u ^ (n\u271d + 1)))\u207b\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\nA : Type u_1\nA' : Type u_2\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : Ring A'\ninst\u271d\u2075 : Algebra \u211a A\ninst\u271d\u2074 : Algebra \u211a A'\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring A'\ninst\u271d\u00b9 : Algebra \u211a A\ninst\u271d : Algebra \u211a A'\nn : \u2115\nf : A \u2192+* A'\n\u22a2 \u2191(constantCoeff A) (exp A) = 1\n[PROOFSTEP]\nrw [\u2190 coeff_zero_eq_constantCoeff_apply, coeff_exp]\n[GOAL]\nA : Type u_1\nA' : Type u_2\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : Ring A'\ninst\u271d\u2075 : Algebra \u211a A\ninst\u271d\u2074 : Algebra \u211a A'\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring A'\ninst\u271d\u00b9 : Algebra \u211a A\ninst\u271d : Algebra \u211a A'\nn : \u2115\nf : A \u2192+* A'\n\u22a2 \u2191(algebraMap \u211a A) (1 / \u21910!) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u_1\nA' : Type u_2\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : Ring A'\ninst\u271d\u2075 : Algebra \u211a A\ninst\u271d\u2074 : Algebra \u211a A'\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring A'\ninst\u271d\u00b9 : Algebra \u211a A\ninst\u271d : Algebra \u211a A'\nn : \u2115\nf : A \u2192+* A'\n\u22a2 \u2191(coeff A (bit0 n)) (sin A) = 0\n[PROOFSTEP]\nrw [sin, coeff_mk, if_pos (even_bit0 n)]\n[GOAL]\nA : Type u_1\nA' : Type u_2\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : Ring A'\ninst\u271d\u2075 : Algebra \u211a A\ninst\u271d\u2074 : Algebra \u211a A'\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring A'\ninst\u271d\u00b9 : Algebra \u211a A\ninst\u271d : Algebra \u211a A'\nn : \u2115\nf : A \u2192+* A'\n\u22a2 \u2191(coeff A (bit1 n)) (sin A) = (-1) ^ n * \u2191(coeff A (bit1 n)) (exp A)\n[PROOFSTEP]\nrw [sin, coeff_mk, if_neg n.not_even_bit1, Nat.bit1_div_two, \u2190 mul_one_div, map_mul, map_pow, map_neg, map_one,\n coeff_exp]\n[GOAL]\nA : Type u_1\nA' : Type u_2\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : Ring A'\ninst\u271d\u2075 : Algebra \u211a A\ninst\u271d\u2074 : Algebra \u211a A'\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring A'\ninst\u271d\u00b9 : Algebra \u211a A\ninst\u271d : Algebra \u211a A'\nn : \u2115\nf : A \u2192+* A'\n\u22a2 \u2191(coeff A (bit0 n)) (cos A) = (-1) ^ n * \u2191(coeff A (bit0 n)) (exp A)\n[PROOFSTEP]\nrw [cos, coeff_mk, if_pos (even_bit0 n), Nat.bit0_div_two, \u2190 mul_one_div, map_mul, map_pow, map_neg, map_one, coeff_exp]\n[GOAL]\nA : Type u_1\nA' : Type u_2\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : Ring A'\ninst\u271d\u2075 : Algebra \u211a A\ninst\u271d\u2074 : Algebra \u211a A'\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring A'\ninst\u271d\u00b9 : Algebra \u211a A\ninst\u271d : Algebra \u211a A'\nn : \u2115\nf : A \u2192+* A'\n\u22a2 \u2191(coeff A (bit1 n)) (cos A) = 0\n[PROOFSTEP]\nrw [cos, coeff_mk, if_neg n.not_even_bit1]\n[GOAL]\nA : Type u_1\nA' : Type u_2\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : Ring A'\ninst\u271d\u2075 : Algebra \u211a A\ninst\u271d\u2074 : Algebra \u211a A'\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring A'\ninst\u271d\u00b9 : Algebra \u211a A\ninst\u271d : Algebra \u211a A'\nn : \u2115\nf : A \u2192+* A'\n\u22a2 \u2191(map f) (exp A) = exp A'\n[PROOFSTEP]\next\n[GOAL]\ncase h\nA : Type u_1\nA' : Type u_2\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : Ring A'\ninst\u271d\u2075 : Algebra \u211a A\ninst\u271d\u2074 : Algebra \u211a A'\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring A'\ninst\u271d\u00b9 : Algebra \u211a A\ninst\u271d : Algebra \u211a A'\nn : \u2115\nf : A \u2192+* A'\nn\u271d : \u2115\n\u22a2 \u2191(coeff A' n\u271d) (\u2191(map f) (exp A)) = \u2191(coeff A' n\u271d) (exp A')\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u_1\nA' : Type u_2\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : Ring A'\ninst\u271d\u2075 : Algebra \u211a A\ninst\u271d\u2074 : Algebra \u211a A'\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring A'\ninst\u271d\u00b9 : Algebra \u211a A\ninst\u271d : Algebra \u211a A'\nn : \u2115\nf : A \u2192+* A'\n\u22a2 \u2191(map f) (sin A) = sin A'\n[PROOFSTEP]\next\n[GOAL]\ncase h\nA : Type u_1\nA' : Type u_2\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : Ring A'\ninst\u271d\u2075 : Algebra \u211a A\ninst\u271d\u2074 : Algebra \u211a A'\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring A'\ninst\u271d\u00b9 : Algebra \u211a A\ninst\u271d : Algebra \u211a A'\nn : \u2115\nf : A \u2192+* A'\nn\u271d : \u2115\n\u22a2 \u2191(coeff A' n\u271d) (\u2191(map f) (sin A)) = \u2191(coeff A' n\u271d) (sin A')\n[PROOFSTEP]\nsimp [sin, apply_ite f]\n[GOAL]\nA : Type u_1\nA' : Type u_2\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : Ring A'\ninst\u271d\u2075 : Algebra \u211a A\ninst\u271d\u2074 : Algebra \u211a A'\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring A'\ninst\u271d\u00b9 : Algebra \u211a A\ninst\u271d : Algebra \u211a A'\nn : \u2115\nf : A \u2192+* A'\n\u22a2 \u2191(map f) (cos A) = cos A'\n[PROOFSTEP]\next\n[GOAL]\ncase h\nA : Type u_1\nA' : Type u_2\ninst\u271d\u2077 : Ring A\ninst\u271d\u2076 : Ring A'\ninst\u271d\u2075 : Algebra \u211a A\ninst\u271d\u2074 : Algebra \u211a A'\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring A'\ninst\u271d\u00b9 : Algebra \u211a A\ninst\u271d : Algebra \u211a A'\nn : \u2115\nf : A \u2192+* A'\nn\u271d : \u2115\n\u22a2 \u2191(coeff A' n\u271d) (\u2191(map f) (cos A)) = \u2191(coeff A' n\u271d) (cos A')\n[PROOFSTEP]\nsimp [cos, apply_ite f]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\n\u22a2 \u2191(rescale a) (exp A) * \u2191(rescale b) (exp A) = \u2191(rescale (a + b)) (exp A)\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn : \u2115\n\u22a2 \u2191(coeff A n) (\u2191(rescale a) (exp A) * \u2191(rescale b) (exp A)) = \u2191(coeff A n) (\u2191(rescale (a + b)) (exp A))\n[PROOFSTEP]\nsimp only [coeff_mul, exp, rescale, coeff_mk, MonoidHom.coe_mk, OneHom.coe_mk, coe_mk, factorial,\n Nat.sum_antidiagonal_eq_sum_range_succ_mk, add_pow, sum_mul]\n[GOAL]\ncase h\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn : \u2115\n\u22a2 (Finset.sum (Finset.range (succ n)) fun x =>\n a ^ x * \u2191(algebraMap \u211a A) (1 / \u2191x !) * (b ^ (n - x) * \u2191(algebraMap \u211a A) (1 / \u2191(n - x)!))) =\n Finset.sum (Finset.range (n + 1)) fun x => a ^ x * b ^ (n - x) * \u2191(Nat.choose n x) * \u2191(algebraMap \u211a A) (1 / \u2191n !)\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase h\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn : \u2115\n\u22a2 \u2200 (x : \u2115),\n x \u2208 Finset.range (succ n) \u2192\n a ^ x * \u2191(algebraMap \u211a A) (1 / \u2191x !) * (b ^ (n - x) * \u2191(algebraMap \u211a A) (1 / \u2191(n - x)!)) =\n a ^ x * b ^ (n - x) * \u2191(Nat.choose n x) * \u2191(algebraMap \u211a A) (1 / \u2191n !)\n[PROOFSTEP]\nrintro x hx\n[GOAL]\ncase h\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 a ^ x * \u2191(algebraMap \u211a A) (1 / \u2191x !) * (b ^ (n - x) * \u2191(algebraMap \u211a A) (1 / \u2191(n - x)!)) =\n a ^ x * b ^ (n - x) * \u2191(Nat.choose n x) * \u2191(algebraMap \u211a A) (1 / \u2191n !)\n[PROOFSTEP]\nsuffices\n a ^ x * b ^ (n - x) * (algebraMap \u211a A (1 / \u2191x.factorial) * algebraMap \u211a A (1 / \u2191(n - x).factorial)) =\n a ^ x * b ^ (n - x) * (\u2191(n.choose x) * (algebraMap \u211a A) (1 / \u2191n.factorial))\n by convert this using 1 <;> ring\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\nthis :\n a ^ x * b ^ (n - x) * (\u2191(algebraMap \u211a A) (1 / \u2191x !) * \u2191(algebraMap \u211a A) (1 / \u2191(n - x)!)) =\n a ^ x * b ^ (n - x) * (\u2191(Nat.choose n x) * \u2191(algebraMap \u211a A) (1 / \u2191n !))\n\u22a2 a ^ x * \u2191(algebraMap \u211a A) (1 / \u2191x !) * (b ^ (n - x) * \u2191(algebraMap \u211a A) (1 / \u2191(n - x)!)) =\n a ^ x * b ^ (n - x) * \u2191(Nat.choose n x) * \u2191(algebraMap \u211a A) (1 / \u2191n !)\n[PROOFSTEP]\nconvert this using 1\n[GOAL]\ncase h.e'_2\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\nthis :\n a ^ x * b ^ (n - x) * (\u2191(algebraMap \u211a A) (1 / \u2191x !) * \u2191(algebraMap \u211a A) (1 / \u2191(n - x)!)) =\n a ^ x * b ^ (n - x) * (\u2191(Nat.choose n x) * \u2191(algebraMap \u211a A) (1 / \u2191n !))\n\u22a2 a ^ x * \u2191(algebraMap \u211a A) (1 / \u2191x !) * (b ^ (n - x) * \u2191(algebraMap \u211a A) (1 / \u2191(n - x)!)) =\n a ^ x * b ^ (n - x) * (\u2191(algebraMap \u211a A) (1 / \u2191x !) * \u2191(algebraMap \u211a A) (1 / \u2191(n - x)!))\n[PROOFSTEP]\nring\n[GOAL]\ncase h.e'_3\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\nthis :\n a ^ x * b ^ (n - x) * (\u2191(algebraMap \u211a A) (1 / \u2191x !) * \u2191(algebraMap \u211a A) (1 / \u2191(n - x)!)) =\n a ^ x * b ^ (n - x) * (\u2191(Nat.choose n x) * \u2191(algebraMap \u211a A) (1 / \u2191n !))\n\u22a2 a ^ x * b ^ (n - x) * \u2191(Nat.choose n x) * \u2191(algebraMap \u211a A) (1 / \u2191n !) =\n a ^ x * b ^ (n - x) * (\u2191(Nat.choose n x) * \u2191(algebraMap \u211a A) (1 / \u2191n !))\n[PROOFSTEP]\nring\n[GOAL]\ncase h\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 a ^ x * b ^ (n - x) * (\u2191(algebraMap \u211a A) (1 / \u2191x !) * \u2191(algebraMap \u211a A) (1 / \u2191(n - x)!)) =\n a ^ x * b ^ (n - x) * (\u2191(Nat.choose n x) * \u2191(algebraMap \u211a A) (1 / \u2191n !))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 \u2191(algebraMap \u211a A) (1 / \u2191x !) * \u2191(algebraMap \u211a A) (1 / \u2191(n - x)!) = \u2191(Nat.choose n x) * \u2191(algebraMap \u211a A) (1 / \u2191n !)\n[PROOFSTEP]\nrw [\u2190 map_natCast (algebraMap \u211a A) (n.choose x), \u2190 map_mul, \u2190 map_mul]\n[GOAL]\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 \u2191(algebraMap \u211a A) (1 / \u2191x ! * (1 / \u2191(n - x)!)) = \u2191(algebraMap \u211a A) (\u2191(Nat.choose n x) * (1 / \u2191n !))\n[PROOFSTEP]\nrefine' RingHom.congr_arg _ _\n[GOAL]\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 1 / \u2191x ! * (1 / \u2191(n - x)!) = \u2191(Nat.choose n x) * (1 / \u2191n !)\n[PROOFSTEP]\nrw [mul_one_div (\u2191(n.choose x) : \u211a), one_div_mul_one_div]\n[GOAL]\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 1 / (\u2191x ! * \u2191(n - x)!) = \u2191(Nat.choose n x) / \u2191n !\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 \u2191(Nat.choose n x) / \u2191n ! = 1 / (\u2191x ! * \u2191(n - x)!)\n[PROOFSTEP]\nrw [div_eq_iff, div_mul_eq_mul_div, one_mul, choose_eq_factorial_div_factorial]\n[GOAL]\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 \u2191(n ! / (x ! * (n - x)!)) = \u2191n ! / (\u2191x ! * \u2191(n - x)!)\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 x \u2264 n\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 \u2191n ! \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 \u2191(n ! / (x ! * (n - x)!)) = \u2191n ! / \u2191(x ! * (n - x)!)\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 x \u2264 n\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 \u2191n ! \u2260 0\n[PROOFSTEP]\nrw [cast_div_charZero]\n[GOAL]\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 x ! * (n - x)! \u2223 n !\n[PROOFSTEP]\napply factorial_mul_factorial_dvd_factorial (mem_range_succ_iff.1 hx)\n[GOAL]\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 x \u2264 n\n[PROOFSTEP]\napply mem_range_succ_iff.1 hx\n[GOAL]\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\n\u22a2 \u2191n ! \u2260 0\n[PROOFSTEP]\nrintro h\n[GOAL]\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\nh : \u2191n ! = 0\n\u22a2 False\n[PROOFSTEP]\napply factorial_ne_zero n\n[GOAL]\ncase h.e_a\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\na b : A\nn x : \u2115\nhx : x \u2208 Finset.range (succ n)\nh : \u2191n ! = 0\n\u22a2 n ! = 0\n[PROOFSTEP]\nrw [cast_eq_zero.1 h]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\n\u22a2 exp A * \u2191evalNegHom (exp A) = 1\n[PROOFSTEP]\nconvert exp_mul_exp_eq_exp_add (1 : A) (-1)\n[GOAL]\ncase h.e'_2.h.e'_5\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\n\u22a2 exp A = \u2191(rescale 1) (exp A)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\n\u22a2 1 = \u2191(rescale (1 + -1)) (exp A)\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\nk : \u2115\n\u22a2 exp A ^ k = \u2191(rescale \u2191k) (exp A)\n[PROOFSTEP]\ninduction' k with k h\n[GOAL]\ncase zero\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\n\u22a2 exp A ^ zero = \u2191(rescale \u2191zero) (exp A)\n[PROOFSTEP]\nsimp only [rescale_zero, constantCoeff_exp, Function.comp_apply, map_one, cast_zero, zero_eq, pow_zero (exp A),\n coe_comp]\n[GOAL]\ncase succ\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\nk : \u2115\nh : exp A ^ k = \u2191(rescale \u2191k) (exp A)\n\u22a2 exp A ^ succ k = \u2191(rescale \u2191(succ k)) (exp A)\n[PROOFSTEP]\nsimpa only [succ_eq_add_one, cast_add, \u2190 exp_mul_exp_eq_exp_add (k : A), \u2190 h, cast_one, id_apply, rescale_one] using\n pow_succ' (exp A) k\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\nn : \u2115\n\u22a2 (Finset.sum (Finset.range n) fun k => exp A ^ k) =\n mk fun p => Finset.sum (Finset.range n) fun k => \u2191(k ^ p) * \u2191(algebraMap \u211a A) (\u2191p !)\u207b\u00b9\n[PROOFSTEP]\nsimp only [exp_pow_eq_rescale_exp, rescale]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\nn : \u2115\n\u22a2 (Finset.sum (Finset.range n) fun x =>\n \u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n g) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n g) })\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x_1 y : PowerSeries A),\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n g) })\n (x_1 + y) =\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n g) })\n x_1 +\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n g) })\n y) }\n (exp A)) =\n mk fun p => Finset.sum (Finset.range n) fun x => \u2191(x ^ p) * \u2191(algebraMap \u211a A) (\u2191p !)\u207b\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase h\nA : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\nn n\u271d : \u2115\n\u22a2 \u2191(coeff A n\u271d)\n (Finset.sum (Finset.range n) fun x =>\n \u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n g) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n g) })\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x_1 y : PowerSeries A),\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n g) })\n (x_1 + y) =\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n g) })\n x_1 +\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (f g : PowerSeries A),\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f,\n map_one' := (_ : (fun f => mk fun n => \u2191x ^ n * \u2191(coeff A n) f) 1 = 1) }\n g) })\n y) }\n (exp A)) =\n \u2191(coeff A n\u271d) (mk fun p => Finset.sum (Finset.range n) fun x => \u2191(x ^ p) * \u2191(algebraMap \u211a A) (\u2191p !)\u207b\u00b9)\n[PROOFSTEP]\nsimp only [one_div, coeff_mk, cast_pow, coe_mk, MonoidHom.coe_mk, OneHom.coe_mk, coeff_exp, factorial,\n LinearMap.map_sum]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.PowerSeries.WellKnown", "llama_tokens": 11659, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833841649233, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.4473183143051844}} {"text": "[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\np : R[X]\n\u22a2 \u2191(\u2191(toFunBilinear R A) a) p = sum p fun n r => \u2191(monomial n) (a * \u2191(algebraMap R A) r)\n[PROOFSTEP]\nsimp only [toFunBilinear_apply_apply, aeval_def, eval\u2082_eq_sum, Polynomial.sum, Finset.smul_sum]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\np : R[X]\n\u22a2 (Finset.sum (support p) fun x => a \u2022 (\u2191(algebraMap R A[X]) (coeff p x) * X ^ x)) =\n Finset.sum (support p) fun x => \u2191(monomial x) (a * \u2191(algebraMap R A) (coeff p x))\n[PROOFSTEP]\ncongr with i : 1\n[GOAL]\ncase e_f.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\np : R[X]\ni : \u2115\n\u22a2 a \u2022 (\u2191(algebraMap R A[X]) (coeff p i) * X ^ i) = \u2191(monomial i) (a * \u2191(algebraMap R A) (coeff p i))\n[PROOFSTEP]\nrw [\u2190 Algebra.smul_def, \u2190 C_mul', mul_smul_comm, C_mul_X_pow_eq_monomial, \u2190 Algebra.commutes, \u2190 Algebra.smul_def,\n smul_monomial]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np : R[X]\nk : \u2115\nh : Decidable \u00accoeff p k = 0\na : A\n\u22a2 (if \u00accoeff p k = 0 then a * \u2191(algebraMap R A) (coeff p k) else 0) = a * \u2191(algebraMap R A) (coeff p k)\n[PROOFSTEP]\nclassical split_ifs <;> simp [*]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np : R[X]\nk : \u2115\nh : Decidable \u00accoeff p k = 0\na : A\n\u22a2 (if \u00accoeff p k = 0 then a * \u2191(algebraMap R A) (coeff p k) else 0) = a * \u2191(algebraMap R A) (coeff p k)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np : R[X]\nk : \u2115\nh : Decidable \u00accoeff p k = 0\na : A\nh\u271d : coeff p k = 0\n\u22a2 0 = a * \u2191(algebraMap R A) (coeff p k)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np : R[X]\nk : \u2115\nh : Decidable \u00accoeff p k = 0\na : A\nh\u271d : \u00accoeff p k = 0\n\u22a2 a * \u2191(algebraMap R A) (coeff p k) = a * \u2191(algebraMap R A) (coeff p k)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nk : \u2115\na\u2081 a\u2082 : A\np\u2081 p\u2082 : R[X]\n\u22a2 a\u2081 * a\u2082 * \u2191(algebraMap R A) (coeff (p\u2081 * p\u2082) k) =\n Finset.sum (Finset.Nat.antidiagonal k) fun x =>\n a\u2081 * \u2191(algebraMap R A) (coeff p\u2081 x.fst) * (a\u2082 * \u2191(algebraMap R A) (coeff p\u2082 x.snd))\n[PROOFSTEP]\nsimp_rw [mul_assoc, Algebra.commutes, \u2190 Finset.mul_sum, mul_assoc, \u2190 Finset.mul_sum]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nk : \u2115\na\u2081 a\u2082 : A\np\u2081 p\u2082 : R[X]\n\u22a2 a\u2081 * (a\u2082 * \u2191(algebraMap R A) (coeff (p\u2081 * p\u2082) k)) =\n a\u2081 *\n (a\u2082 *\n Finset.sum (Finset.Nat.antidiagonal k) fun x =>\n \u2191(algebraMap R A) (coeff p\u2082 x.snd) * \u2191(algebraMap R A) (coeff p\u2081 x.fst))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nk : \u2115\na\u2081 a\u2082 : A\np\u2081 p\u2082 : R[X]\n\u22a2 \u2191(algebraMap R A) (coeff (p\u2081 * p\u2082) k) =\n Finset.sum (Finset.Nat.antidiagonal k) fun x =>\n \u2191(algebraMap R A) (coeff p\u2082 x.snd) * \u2191(algebraMap R A) (coeff p\u2081 x.fst)\n[PROOFSTEP]\nsimp_rw [Algebra.commutes (coeff p\u2082 _), coeff_mul, map_sum, RingHom.map_mul]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na\u2081 a\u2082 : A\np\u2081 p\u2082 : R[X]\n\u22a2 \u2191(toFunLinear R A) ((a\u2081 * a\u2082) \u2297\u209c[R] (p\u2081 * p\u2082)) = \u2191(toFunLinear R A) (a\u2081 \u2297\u209c[R] p\u2081) * \u2191(toFunLinear R A) (a\u2082 \u2297\u209c[R] p\u2082)\n[PROOFSTEP]\nclassical\nsimp only [toFunLinear_tmul_apply, toFunBilinear_apply_eq_sum]\next k\nsimp_rw [coeff_sum, coeff_monomial, sum_def, Finset.sum_ite_eq', mem_support_iff, Ne.def]\nconv_rhs => rw [coeff_mul]\nsimp_rw [finset_sum_coeff, coeff_monomial, Finset.sum_ite_eq', mem_support_iff, Ne.def, mul_ite, mul_zero, ite_mul,\n zero_mul]\nsimp_rw [ite_mul_zero_left (\u00accoeff p\u2081 _ = 0) (a\u2081 * (algebraMap R A) (coeff p\u2081 _))]\nsimp_rw [ite_mul_zero_right (\u00accoeff p\u2082 _ = 0) _ (_ * _)]\nsimp_rw [toFunLinear_mul_tmul_mul_aux_1, toFunLinear_mul_tmul_mul_aux_2]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na\u2081 a\u2082 : A\np\u2081 p\u2082 : R[X]\n\u22a2 \u2191(toFunLinear R A) ((a\u2081 * a\u2082) \u2297\u209c[R] (p\u2081 * p\u2082)) = \u2191(toFunLinear R A) (a\u2081 \u2297\u209c[R] p\u2081) * \u2191(toFunLinear R A) (a\u2082 \u2297\u209c[R] p\u2082)\n[PROOFSTEP]\nsimp only [toFunLinear_tmul_apply, toFunBilinear_apply_eq_sum]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na\u2081 a\u2082 : A\np\u2081 p\u2082 : R[X]\n\u22a2 (sum (p\u2081 * p\u2082) fun n r => \u2191(monomial n) (a\u2081 * a\u2082 * \u2191(algebraMap R A) r)) =\n (sum p\u2081 fun n r => \u2191(monomial n) (a\u2081 * \u2191(algebraMap R A) r)) *\n sum p\u2082 fun n r => \u2191(monomial n) (a\u2082 * \u2191(algebraMap R A) r)\n[PROOFSTEP]\next k\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na\u2081 a\u2082 : A\np\u2081 p\u2082 : R[X]\nk : \u2115\n\u22a2 coeff (sum (p\u2081 * p\u2082) fun n r => \u2191(monomial n) (a\u2081 * a\u2082 * \u2191(algebraMap R A) r)) k =\n coeff\n ((sum p\u2081 fun n r => \u2191(monomial n) (a\u2081 * \u2191(algebraMap R A) r)) *\n sum p\u2082 fun n r => \u2191(monomial n) (a\u2082 * \u2191(algebraMap R A) r))\n k\n[PROOFSTEP]\nsimp_rw [coeff_sum, coeff_monomial, sum_def, Finset.sum_ite_eq', mem_support_iff, Ne.def]\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na\u2081 a\u2082 : A\np\u2081 p\u2082 : R[X]\nk : \u2115\n\u22a2 (if \u00accoeff (p\u2081 * p\u2082) k = 0 then a\u2081 * a\u2082 * \u2191(algebraMap R A) (coeff (p\u2081 * p\u2082) k) else 0) =\n coeff\n ((Finset.sum (support p\u2081) fun n => \u2191(monomial n) (a\u2081 * \u2191(algebraMap R A) (coeff p\u2081 n))) *\n Finset.sum (support p\u2082) fun n => \u2191(monomial n) (a\u2082 * \u2191(algebraMap R A) (coeff p\u2082 n)))\n k\n[PROOFSTEP]\nconv_rhs => rw [coeff_mul]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na\u2081 a\u2082 : A\np\u2081 p\u2082 : R[X]\nk : \u2115\n| coeff\n ((Finset.sum (support p\u2081) fun n => \u2191(monomial n) (a\u2081 * \u2191(algebraMap R A) (coeff p\u2081 n))) *\n Finset.sum (support p\u2082) fun n => \u2191(monomial n) (a\u2082 * \u2191(algebraMap R A) (coeff p\u2082 n)))\n k\n[PROOFSTEP]\nrw [coeff_mul]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na\u2081 a\u2082 : A\np\u2081 p\u2082 : R[X]\nk : \u2115\n| coeff\n ((Finset.sum (support p\u2081) fun n => \u2191(monomial n) (a\u2081 * \u2191(algebraMap R A) (coeff p\u2081 n))) *\n Finset.sum (support p\u2082) fun n => \u2191(monomial n) (a\u2082 * \u2191(algebraMap R A) (coeff p\u2082 n)))\n k\n[PROOFSTEP]\nrw [coeff_mul]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na\u2081 a\u2082 : A\np\u2081 p\u2082 : R[X]\nk : \u2115\n| coeff\n ((Finset.sum (support p\u2081) fun n => \u2191(monomial n) (a\u2081 * \u2191(algebraMap R A) (coeff p\u2081 n))) *\n Finset.sum (support p\u2082) fun n => \u2191(monomial n) (a\u2082 * \u2191(algebraMap R A) (coeff p\u2082 n)))\n k\n[PROOFSTEP]\nrw [coeff_mul]\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na\u2081 a\u2082 : A\np\u2081 p\u2082 : R[X]\nk : \u2115\n\u22a2 (if \u00accoeff (p\u2081 * p\u2082) k = 0 then a\u2081 * a\u2082 * \u2191(algebraMap R A) (coeff (p\u2081 * p\u2082) k) else 0) =\n Finset.sum (Finset.Nat.antidiagonal k) fun x =>\n coeff (Finset.sum (support p\u2081) fun n => \u2191(monomial n) (a\u2081 * \u2191(algebraMap R A) (coeff p\u2081 n))) x.fst *\n coeff (Finset.sum (support p\u2082) fun n => \u2191(monomial n) (a\u2082 * \u2191(algebraMap R A) (coeff p\u2082 n))) x.snd\n[PROOFSTEP]\nsimp_rw [finset_sum_coeff, coeff_monomial, Finset.sum_ite_eq', mem_support_iff, Ne.def, mul_ite, mul_zero, ite_mul,\n zero_mul]\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na\u2081 a\u2082 : A\np\u2081 p\u2082 : R[X]\nk : \u2115\n\u22a2 (if \u00accoeff (p\u2081 * p\u2082) k = 0 then a\u2081 * a\u2082 * \u2191(algebraMap R A) (coeff (p\u2081 * p\u2082) k) else 0) =\n Finset.sum (Finset.Nat.antidiagonal k) fun x =>\n if \u00accoeff p\u2082 x.snd = 0 then\n if \u00accoeff p\u2081 x.fst = 0 then a\u2081 * \u2191(algebraMap R A) (coeff p\u2081 x.fst) * (a\u2082 * \u2191(algebraMap R A) (coeff p\u2082 x.snd))\n else 0\n else 0\n[PROOFSTEP]\nsimp_rw [ite_mul_zero_left (\u00accoeff p\u2081 _ = 0) (a\u2081 * (algebraMap R A) (coeff p\u2081 _))]\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na\u2081 a\u2082 : A\np\u2081 p\u2082 : R[X]\nk : \u2115\n\u22a2 (if \u00accoeff (p\u2081 * p\u2082) k = 0 then a\u2081 * a\u2082 * \u2191(algebraMap R A) (coeff (p\u2081 * p\u2082) k) else 0) =\n Finset.sum (Finset.Nat.antidiagonal k) fun x =>\n if \u00accoeff p\u2082 x.snd = 0 then\n (if \u00accoeff p\u2081 x.fst = 0 then a\u2081 * \u2191(algebraMap R A) (coeff p\u2081 x.fst) else 0) *\n (a\u2082 * \u2191(algebraMap R A) (coeff p\u2082 x.snd))\n else 0\n[PROOFSTEP]\nsimp_rw [ite_mul_zero_right (\u00accoeff p\u2082 _ = 0) _ (_ * _)]\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na\u2081 a\u2082 : A\np\u2081 p\u2082 : R[X]\nk : \u2115\n\u22a2 (if \u00accoeff (p\u2081 * p\u2082) k = 0 then a\u2081 * a\u2082 * \u2191(algebraMap R A) (coeff (p\u2081 * p\u2082) k) else 0) =\n Finset.sum (Finset.Nat.antidiagonal k) fun x =>\n (if \u00accoeff p\u2081 x.fst = 0 then a\u2081 * \u2191(algebraMap R A) (coeff p\u2081 x.fst) else 0) *\n if \u00accoeff p\u2082 x.snd = 0 then a\u2082 * \u2191(algebraMap R A) (coeff p\u2082 x.snd) else 0\n[PROOFSTEP]\nsimp_rw [toFunLinear_mul_tmul_mul_aux_1, toFunLinear_mul_tmul_mul_aux_2]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nr : R\n\u22a2 \u2191(toFunLinear R A) (\u2191(algebraMap R A) r \u2297\u209c[R] 1) = \u2191(algebraMap R A[X]) r\n[PROOFSTEP]\nrw [toFunLinear_tmul_apply, toFunBilinear_apply_apply, Polynomial.aeval_one, algebraMap_smul,\n Algebra.algebraMap_eq_smul_one, Algebra.algebraMap_eq_smul_one]\n -- porting note: had to\n -- add extra `Algebra.algebraMap_eq_smul_one`\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np q : A[X]\n\u22a2 invFun R A (p + q) = invFun R A p + invFun R A q\n[PROOFSTEP]\nsimp only [invFun, eval\u2082_add]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A \u2297[R] R[X]\n\u22a2 invFun R A (\u2191(toFunAlgHom R A) x) = x\n[PROOFSTEP]\nrefine TensorProduct.induction_on x ?_ ?_ ?_\n[GOAL]\ncase refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A \u2297[R] R[X]\n\u22a2 invFun R A (\u2191(toFunAlgHom R A) 0) = 0\n[PROOFSTEP]\nsimp [invFun]\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A \u2297[R] R[X]\n\u22a2 \u2200 (x : A) (y : R[X]), invFun R A (\u2191(toFunAlgHom R A) (x \u2297\u209c[R] y)) = x \u2297\u209c[R] y\n[PROOFSTEP]\nintro a p\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A \u2297[R] R[X]\na : A\np : R[X]\n\u22a2 invFun R A (\u2191(toFunAlgHom R A) (a \u2297\u209c[R] p)) = a \u2297\u209c[R] p\n[PROOFSTEP]\ndsimp only [invFun]\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A \u2297[R] R[X]\na : A\np : R[X]\n\u22a2 eval\u2082 (\u2191includeLeft) (1 \u2297\u209c[R] X) (\u2191(toFunAlgHom R A) (a \u2297\u209c[R] p)) = a \u2297\u209c[R] p\n[PROOFSTEP]\nrw [toFunAlgHom_apply_tmul, eval\u2082_sum]\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A \u2297[R] R[X]\na : A\np : R[X]\n\u22a2 (sum p fun n a_1 => eval\u2082 (\u2191includeLeft) (1 \u2297\u209c[R] X) (\u2191(monomial n) (a * \u2191(algebraMap R A) a_1))) = a \u2297\u209c[R] p\n[PROOFSTEP]\nsimp_rw [eval\u2082_monomial, AlgHom.coe_toRingHom, Algebra.TensorProduct.tmul_pow, one_pow,\n Algebra.TensorProduct.includeLeft_apply, Algebra.TensorProduct.tmul_mul_tmul, mul_one, one_mul, \u2190 Algebra.commutes, \u2190\n Algebra.smul_def, smul_tmul, sum_def, \u2190 tmul_sum]\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A \u2297[R] R[X]\na : A\np : R[X]\n\u22a2 (a \u2297\u209c[R] Finset.sum (support p) fun a => coeff p a \u2022 X ^ a) = a \u2297\u209c[R] p\n[PROOFSTEP]\nconv_rhs => rw [\u2190 sum_C_mul_X_pow_eq p]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A \u2297[R] R[X]\na : A\np : R[X]\n| a \u2297\u209c[R] p\n[PROOFSTEP]\nrw [\u2190 sum_C_mul_X_pow_eq p]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A \u2297[R] R[X]\na : A\np : R[X]\n| a \u2297\u209c[R] p\n[PROOFSTEP]\nrw [\u2190 sum_C_mul_X_pow_eq p]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A \u2297[R] R[X]\na : A\np : R[X]\n| a \u2297\u209c[R] p\n[PROOFSTEP]\nrw [\u2190 sum_C_mul_X_pow_eq p]\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A \u2297[R] R[X]\na : A\np : R[X]\n\u22a2 (a \u2297\u209c[R] Finset.sum (support p) fun a => coeff p a \u2022 X ^ a) = a \u2297\u209c[R] sum p fun n a => \u2191C a * X ^ n\n[PROOFSTEP]\nsimp only [Algebra.smul_def]\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A \u2297[R] R[X]\na : A\np : R[X]\n\u22a2 (a \u2297\u209c[R] Finset.sum (support p) fun x => \u2191(algebraMap R R[X]) (coeff p x) * X ^ x) =\n a \u2297\u209c[R] sum p fun n a => \u2191C a * X ^ n\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A \u2297[R] R[X]\n\u22a2 \u2200 (x y : A \u2297[R] R[X]),\n invFun R A (\u2191(toFunAlgHom R A) x) = x \u2192\n invFun R A (\u2191(toFunAlgHom R A) y) = y \u2192 invFun R A (\u2191(toFunAlgHom R A) (x + y)) = x + y\n[PROOFSTEP]\nintro p q hp hq\n[GOAL]\ncase refine_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx p q : A \u2297[R] R[X]\nhp : invFun R A (\u2191(toFunAlgHom R A) p) = p\nhq : invFun R A (\u2191(toFunAlgHom R A) q) = q\n\u22a2 invFun R A (\u2191(toFunAlgHom R A) (p + q)) = p + q\n[PROOFSTEP]\nsimp only [AlgHom.map_add, invFun_add, hp, hq]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A[X]\n\u22a2 \u2191(toFunAlgHom R A) (invFun R A x) = x\n[PROOFSTEP]\nrefine Polynomial.induction_on' x ?_ ?_\n[GOAL]\ncase refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A[X]\n\u22a2 \u2200 (p q : A[X]),\n \u2191(toFunAlgHom R A) (invFun R A p) = p \u2192\n \u2191(toFunAlgHom R A) (invFun R A q) = q \u2192 \u2191(toFunAlgHom R A) (invFun R A (p + q)) = p + q\n[PROOFSTEP]\nintro p q hp hq\n[GOAL]\ncase refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx p q : A[X]\nhp : \u2191(toFunAlgHom R A) (invFun R A p) = p\nhq : \u2191(toFunAlgHom R A) (invFun R A q) = q\n\u22a2 \u2191(toFunAlgHom R A) (invFun R A (p + q)) = p + q\n[PROOFSTEP]\nsimp only [invFun_add, AlgHom.map_add, hp, hq]\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A[X]\n\u22a2 \u2200 (n : \u2115) (a : A), \u2191(toFunAlgHom R A) (invFun R A (\u2191(monomial n) a)) = \u2191(monomial n) a\n[PROOFSTEP]\nintro n a\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A[X]\nn : \u2115\na : A\n\u22a2 \u2191(toFunAlgHom R A) (invFun R A (\u2191(monomial n) a)) = \u2191(monomial n) a\n[PROOFSTEP]\nrw [invFun_monomial, Algebra.TensorProduct.tmul_pow, one_pow, Algebra.TensorProduct.tmul_mul_tmul, mul_one, one_mul,\n toFunAlgHom_apply_tmul, X_pow_eq_monomial, sum_monomial_index]\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A[X]\nn : \u2115\na : A\n\u22a2 \u2191(monomial n) (a * \u2191(algebraMap R A) 1) = \u2191(monomial n) a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_2.hf\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx : A[X]\nn : \u2115\na : A\n\u22a2 \u2191(monomial n) (a * \u2191(algebraMap R A) 0) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\nk : \u2115\nx : R\n\u22a2 \u2191matPolyEquiv (stdBasisMatrix i j (\u2191(monomial k) x)) = \u2191(monomial k) (stdBasisMatrix i j x)\n[PROOFSTEP]\nsimp only [matPolyEquiv, AlgEquiv.trans_apply, matrixEquivTensor_apply_std_basis]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\nk : \u2115\nx : R\n\u22a2 \u2191(AlgEquiv.symm (polyEquivTensor R (Matrix n n R)))\n (\u2191(Algebra.TensorProduct.comm R R[X] (Matrix n n R)) (\u2191(monomial k) x \u2297\u209c[R] stdBasisMatrix i j 1)) =\n \u2191(monomial k) (stdBasisMatrix i j x)\n[PROOFSTEP]\napply (polyEquivTensor R (Matrix n n R)).injective\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\nk : \u2115\nx : R\n\u22a2 \u2191(polyEquivTensor R (Matrix n n R))\n (\u2191(AlgEquiv.symm (polyEquivTensor R (Matrix n n R)))\n (\u2191(Algebra.TensorProduct.comm R R[X] (Matrix n n R)) (\u2191(monomial k) x \u2297\u209c[R] stdBasisMatrix i j 1))) =\n \u2191(polyEquivTensor R (Matrix n n R)) (\u2191(monomial k) (stdBasisMatrix i j x))\n[PROOFSTEP]\nsimp only [AlgEquiv.apply_symm_apply, Algebra.TensorProduct.comm_tmul, polyEquivTensor_apply, eval\u2082_monomial]\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\nk : \u2115\nx : R\n\u22a2 stdBasisMatrix i j 1 \u2297\u209c[R] \u2191(monomial k) x = \u2191\u2191includeLeft (stdBasisMatrix i j x) * 1 \u2297\u209c[R] X ^ k\n[PROOFSTEP]\nsimp only [Algebra.TensorProduct.tmul_mul_tmul, one_pow, one_mul, Matrix.mul_one, Algebra.TensorProduct.tmul_pow,\n Algebra.TensorProduct.includeLeft_apply]\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\nk : \u2115\nx : R\n\u22a2 stdBasisMatrix i j 1 \u2297\u209c[R] \u2191(monomial k) x = \u2191\u2191includeLeft (stdBasisMatrix i j x) * 1 \u2297\u209c[R] (X ^ k)\n[PROOFSTEP]\nrw [\u2190 smul_X_eq_monomial, \u2190 TensorProduct.smul_tmul]\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\nk : \u2115\nx : R\n\u22a2 (x \u2022 stdBasisMatrix i j 1) \u2297\u209c[R] (X ^ k) = \u2191\u2191includeLeft (stdBasisMatrix i j x) * 1 \u2297\u209c[R] (X ^ k)\n[PROOFSTEP]\ncongr with i'\n[GOAL]\ncase a.e_m.a.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\nk : \u2115\nx : R\ni' x\u271d : n\n\u22a2 (x \u2022 stdBasisMatrix i j 1) i' x\u271d = \u2191(LinearMap.mulLeft R (stdBasisMatrix i j x, 1).fst) (1, X ^ k).fst i' x\u271d\n[PROOFSTEP]\nsimp [stdBasisMatrix]\n[GOAL]\ncase a.e_n.a\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\nk : \u2115\nx : R\ni' : \u2115\n\u22a2 coeff (X ^ k) i' = coeff (\u2191(LinearMap.mulLeft R (stdBasisMatrix i j x, 1).snd) (1, X ^ k).snd) i'\n[PROOFSTEP]\nsimp [stdBasisMatrix]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\np : R[X]\nk : \u2115\n\u22a2 coeff (\u2191matPolyEquiv (stdBasisMatrix i j p)) k = stdBasisMatrix i j (coeff p k)\n[PROOFSTEP]\nrefine Polynomial.induction_on' p ?_ ?_\n[GOAL]\ncase refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\np : R[X]\nk : \u2115\n\u22a2 \u2200 (p q : R[X]),\n coeff (\u2191matPolyEquiv (stdBasisMatrix i j p)) k = stdBasisMatrix i j (coeff p k) \u2192\n coeff (\u2191matPolyEquiv (stdBasisMatrix i j q)) k = stdBasisMatrix i j (coeff q k) \u2192\n coeff (\u2191matPolyEquiv (stdBasisMatrix i j (p + q))) k = stdBasisMatrix i j (coeff (p + q) k)\n[PROOFSTEP]\nintro p q hp hq\n[GOAL]\ncase refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\np\u271d : R[X]\nk : \u2115\np q : R[X]\nhp : coeff (\u2191matPolyEquiv (stdBasisMatrix i j p)) k = stdBasisMatrix i j (coeff p k)\nhq : coeff (\u2191matPolyEquiv (stdBasisMatrix i j q)) k = stdBasisMatrix i j (coeff q k)\n\u22a2 coeff (\u2191matPolyEquiv (stdBasisMatrix i j (p + q))) k = stdBasisMatrix i j (coeff (p + q) k)\n[PROOFSTEP]\next\n[GOAL]\ncase refine_1.a.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\np\u271d : R[X]\nk : \u2115\np q : R[X]\nhp : coeff (\u2191matPolyEquiv (stdBasisMatrix i j p)) k = stdBasisMatrix i j (coeff p k)\nhq : coeff (\u2191matPolyEquiv (stdBasisMatrix i j q)) k = stdBasisMatrix i j (coeff q k)\ni\u271d x\u271d : n\n\u22a2 coeff (\u2191matPolyEquiv (stdBasisMatrix i j (p + q))) k i\u271d x\u271d = stdBasisMatrix i j (coeff (p + q) k) i\u271d x\u271d\n[PROOFSTEP]\nsimp [hp, hq, coeff_add, DMatrix.add_apply, stdBasisMatrix_add]\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\np : R[X]\nk : \u2115\n\u22a2 \u2200 (n_1 : \u2115) (a : R),\n coeff (\u2191matPolyEquiv (stdBasisMatrix i j (\u2191(monomial n_1) a))) k = stdBasisMatrix i j (coeff (\u2191(monomial n_1) a) k)\n[PROOFSTEP]\nintro k x\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\np : R[X]\nk\u271d k : \u2115\nx : R\n\u22a2 coeff (\u2191matPolyEquiv (stdBasisMatrix i j (\u2191(monomial k) x))) k\u271d = stdBasisMatrix i j (coeff (\u2191(monomial k) x) k\u271d)\n[PROOFSTEP]\nsimp only [matPolyEquiv_coeff_apply_aux_1, coeff_monomial]\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\np : R[X]\nk\u271d k : \u2115\nx : R\n\u22a2 (if k = k\u271d then stdBasisMatrix i j x else 0) = stdBasisMatrix i j (if k = k\u271d then x else 0)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\np : R[X]\nk\u271d k : \u2115\nx : R\nh\u271d : k = k\u271d\n\u22a2 stdBasisMatrix i j x = stdBasisMatrix i j x\n[PROOFSTEP]\nfunext\n[GOAL]\ncase pos.h.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\np : R[X]\nk\u271d k : \u2115\nx : R\nh\u271d : k = k\u271d\nx\u271d\u00b9 x\u271d : n\n\u22a2 stdBasisMatrix i j x x\u271d\u00b9 x\u271d = stdBasisMatrix i j x x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\np : R[X]\nk\u271d k : \u2115\nx : R\nh\u271d : \u00ack = k\u271d\n\u22a2 0 = stdBasisMatrix i j 0\n[PROOFSTEP]\nfunext\n[GOAL]\ncase neg.h.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\ni j : n\np : R[X]\nk\u271d k : \u2115\nx : R\nh\u271d : \u00ack = k\u271d\nx\u271d\u00b9 x\u271d : n\n\u22a2 OfNat.ofNat 0 x\u271d\u00b9 x\u271d = stdBasisMatrix i j 0 x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\nk : \u2115\ni j : n\n\u22a2 coeff (\u2191matPolyEquiv m) k i j = coeff (m i j) k\n[PROOFSTEP]\nrefine Matrix.induction_on' m ?_ ?_ ?_\n[GOAL]\ncase refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\nk : \u2115\ni j : n\n\u22a2 coeff (\u2191matPolyEquiv 0) k i j = coeff (OfNat.ofNat 0 i j) k\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\nk : \u2115\ni j : n\n\u22a2 \u2200 (p q : Matrix n n R[X]),\n coeff (\u2191matPolyEquiv p) k i j = coeff (p i j) k \u2192\n coeff (\u2191matPolyEquiv q) k i j = coeff (q i j) k \u2192 coeff (\u2191matPolyEquiv (p + q)) k i j = coeff ((p + q) i j) k\n[PROOFSTEP]\nintro p q hp hq\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\nk : \u2115\ni j : n\np q : Matrix n n R[X]\nhp : coeff (\u2191matPolyEquiv p) k i j = coeff (p i j) k\nhq : coeff (\u2191matPolyEquiv q) k i j = coeff (q i j) k\n\u22a2 coeff (\u2191matPolyEquiv (p + q)) k i j = coeff ((p + q) i j) k\n[PROOFSTEP]\nsimp [hp, hq]\n[GOAL]\ncase refine_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\nk : \u2115\ni j : n\n\u22a2 \u2200 (i_1 j_1 : n) (x : R[X]),\n coeff (\u2191matPolyEquiv (stdBasisMatrix i_1 j_1 x)) k i j = coeff (stdBasisMatrix i_1 j_1 x i j) k\n[PROOFSTEP]\nintro i' j' x\n[GOAL]\ncase refine_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\nk : \u2115\ni j i' j' : n\nx : R[X]\n\u22a2 coeff (\u2191matPolyEquiv (stdBasisMatrix i' j' x)) k i j = coeff (stdBasisMatrix i' j' x i j) k\n[PROOFSTEP]\nerw [matPolyEquiv_coeff_apply_aux_2]\n[GOAL]\ncase refine_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\nk : \u2115\ni j i' j' : n\nx : R[X]\n\u22a2 stdBasisMatrix i' j' (coeff x k) i j = coeff (stdBasisMatrix i' j' x i j) k\n[PROOFSTEP]\ndsimp [stdBasisMatrix]\n[GOAL]\ncase refine_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\nk : \u2115\ni j i' j' : n\nx : R[X]\n\u22a2 (if i' = i \u2227 j' = j then coeff x k else 0) = coeff (if i' = i \u2227 j' = j then x else 0) k\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\nk : \u2115\ni j i' j' : n\nx : R[X]\nh\u271d : i' = i \u2227 j' = j\n\u22a2 coeff x k = coeff x k\n[PROOFSTEP]\nrename_i h\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\nk : \u2115\ni j i' j' : n\nx : R[X]\nh\u271d : \u00ac(i' = i \u2227 j' = j)\n\u22a2 0 = coeff 0 k\n[PROOFSTEP]\nrename_i h\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\nk : \u2115\ni j i' j' : n\nx : R[X]\nh : i' = i \u2227 j' = j\n\u22a2 coeff x k = coeff x k\n[PROOFSTEP]\nrcases h with \u27e8rfl, rfl\u27e9\n[GOAL]\ncase pos.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\nk : \u2115\ni' j' : n\nx : R[X]\n\u22a2 coeff x k = coeff x k\n[PROOFSTEP]\nsimp [stdBasisMatrix]\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\nk : \u2115\ni j i' j' : n\nx : R[X]\nh : \u00ac(i' = i \u2227 j' = j)\n\u22a2 0 = coeff 0 k\n[PROOFSTEP]\nsimp [stdBasisMatrix, h]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : (Matrix n n R)[X]\ni j : n\nk : \u2115\n\u22a2 coeff (\u2191(AlgEquiv.symm matPolyEquiv) p i j) k = coeff p k i j\n[PROOFSTEP]\nhave t : p = matPolyEquiv (matPolyEquiv.symm p) := by simp\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : (Matrix n n R)[X]\ni j : n\nk : \u2115\n\u22a2 p = \u2191matPolyEquiv (\u2191(AlgEquiv.symm matPolyEquiv) p)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : (Matrix n n R)[X]\ni j : n\nk : \u2115\nt : p = \u2191matPolyEquiv (\u2191(AlgEquiv.symm matPolyEquiv) p)\n\u22a2 coeff (\u2191(AlgEquiv.symm matPolyEquiv) p i j) k = coeff p k i j\n[PROOFSTEP]\nconv_rhs => rw [t]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : (Matrix n n R)[X]\ni j : n\nk : \u2115\nt : p = \u2191matPolyEquiv (\u2191(AlgEquiv.symm matPolyEquiv) p)\n| coeff p k i j\n[PROOFSTEP]\nrw [t]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : (Matrix n n R)[X]\ni j : n\nk : \u2115\nt : p = \u2191matPolyEquiv (\u2191(AlgEquiv.symm matPolyEquiv) p)\n| coeff p k i j\n[PROOFSTEP]\nrw [t]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : (Matrix n n R)[X]\ni j : n\nk : \u2115\nt : p = \u2191matPolyEquiv (\u2191(AlgEquiv.symm matPolyEquiv) p)\n| coeff p k i j\n[PROOFSTEP]\nrw [t]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : (Matrix n n R)[X]\ni j : n\nk : \u2115\nt : p = \u2191matPolyEquiv (\u2191(AlgEquiv.symm matPolyEquiv) p)\n\u22a2 coeff (\u2191(AlgEquiv.symm matPolyEquiv) p i j) k = coeff (\u2191matPolyEquiv (\u2191(AlgEquiv.symm matPolyEquiv) p)) k i j\n[PROOFSTEP]\nsimp only [matPolyEquiv_coeff_apply]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : R[X]\n\u22a2 \u2191matPolyEquiv (p \u2022 1) = Polynomial.map (algebraMap R (Matrix n n R)) p\n[PROOFSTEP]\next m i j\n[GOAL]\ncase a.a.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : R[X]\nm : \u2115\ni j : n\n\u22a2 coeff (\u2191matPolyEquiv (p \u2022 1)) m i j = coeff (Polynomial.map (algebraMap R (Matrix n n R)) p) m i j\n[PROOFSTEP]\nsimp only [coeff_map, one_apply, algebraMap_matrix_apply, mul_boole, Pi.smul_apply, matPolyEquiv_coeff_apply]\n[GOAL]\ncase a.a.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : R[X]\nm : \u2115\ni j : n\n\u22a2 coeff ((p \u2022 1) i j) m = if i = j then \u2191(algebraMap R R) (coeff p m) else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : R[X]\nm : \u2115\ni j : n\nh\u271d : i = j\n\u22a2 coeff ((p \u2022 1) i j) m = \u2191(algebraMap R R) (coeff p m)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : R[X]\nm : \u2115\ni j : n\nh\u271d : \u00aci = j\n\u22a2 coeff ((p \u2022 1) i j) m = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : R[X]\nm : \u2115\ni j : n\nh\u271d : i = j\n\u22a2 coeff (p * OfNat.ofNat 1 i j) m = coeff p m\n[PROOFSTEP]\nrename_i h\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : R[X]\nm : \u2115\ni j : n\nh\u271d : \u00aci = j\n\u22a2 coeff (p * OfNat.ofNat 1 i j) m = 0\n[PROOFSTEP]\nrename_i h\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : R[X]\nm : \u2115\ni j : n\nh : i = j\n\u22a2 coeff (p * OfNat.ofNat 1 i j) m = coeff p m\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\np : R[X]\nm : \u2115\ni j : n\nh : \u00aci = j\n\u22a2 coeff (p * OfNat.ofNat 1 i j) m = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\ni j : n\n\u22a2 support (m i j) \u2286 support (\u2191matPolyEquiv m)\n[PROOFSTEP]\nintro k\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\ni j : n\nk : \u2115\n\u22a2 k \u2208 support (m i j) \u2192 k \u2208 support (\u2191matPolyEquiv m)\n[PROOFSTEP]\ncontrapose\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\ni j : n\nk : \u2115\n\u22a2 \u00ack \u2208 support (\u2191matPolyEquiv m) \u2192 \u00ack \u2208 support (m i j)\n[PROOFSTEP]\nsimp only [not_mem_support_iff]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\ni j : n\nk : \u2115\n\u22a2 coeff (\u2191matPolyEquiv m) k = 0 \u2192 coeff (m i j) k = 0\n[PROOFSTEP]\nintro hk\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\ni j : n\nk : \u2115\nhk : coeff (\u2191matPolyEquiv m) k = 0\n\u22a2 coeff (m i j) k = 0\n[PROOFSTEP]\nrw [\u2190 matPolyEquiv_coeff_apply, hk]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nm : Matrix n n R[X]\ni j : n\nk : \u2115\nhk : coeff (\u2191matPolyEquiv m) k = 0\n\u22a2 OfNat.ofNat 0 i j = 0\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.PolynomialAlgebra", "llama_tokens": 16998, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833841649233, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.44731831430518426}} {"text": "[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nh : x \u2208+ l\n\u22a2 x \u2208 l\n[PROOFSTEP]\ninduction' h with l' _ y l' _ hm\n[GOAL]\ncase cons_mem\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nl' : List \u03b1\na\u271d : x \u2208 l'\n\u22a2 x \u2208 x :: l'\n[PROOFSTEP]\nexact mem_cons_self _ _\n[GOAL]\ncase cons_duplicate\n\u03b1 : Type u_1\nl : List \u03b1\nx y : \u03b1\nl' : List \u03b1\na\u271d : x \u2208+ l'\nhm : x \u2208 l'\n\u22a2 x \u2208 y :: l'\n[PROOFSTEP]\nexact mem_cons_of_mem _ hm\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nh : x \u2208+ x :: l\n\u22a2 x \u2208 l\n[PROOFSTEP]\ncases' h with _ h _ _ h\n[GOAL]\ncase cons_mem\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nh : x \u2208 l\n\u22a2 x \u2208 l\n[PROOFSTEP]\nexact h\n[GOAL]\ncase cons_duplicate\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nh : x \u2208+ l\n\u22a2 x \u2208 l\n[PROOFSTEP]\nexact h.mem\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nh : x \u2208+ l\ny : \u03b1\n\u22a2 l \u2260 [y]\n[PROOFSTEP]\ninduction' h with l' h z l' h _\n[GOAL]\ncase cons_mem\n\u03b1 : Type u_1\nl : List \u03b1\nx y : \u03b1\nl' : List \u03b1\nh : x \u2208 l'\n\u22a2 x :: l' \u2260 [y]\n[PROOFSTEP]\nsimp [ne_nil_of_mem h]\n[GOAL]\ncase cons_duplicate\n\u03b1 : Type u_1\nl : List \u03b1\nx y z : \u03b1\nl' : List \u03b1\nh : x \u2208+ l'\na_ih\u271d : l' \u2260 [y]\n\u22a2 z :: l' \u2260 [y]\n[PROOFSTEP]\nsimp [ne_nil_of_mem h.mem]\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx y : \u03b1\n\u22a2 x \u2208+ y :: l \u2194 y = x \u2227 x \u2208 l \u2228 x \u2208+ l\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nl : List \u03b1\nx y : \u03b1\nh : x \u2208+ y :: l\n\u22a2 y = x \u2227 x \u2208 l \u2228 x \u2208+ l\n[PROOFSTEP]\ncases' h with _ hm _ _ hm\n[GOAL]\ncase refine'_1.cons_mem\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nhm : x \u2208 l\n\u22a2 x = x \u2227 x \u2208 l \u2228 x \u2208+ l\n[PROOFSTEP]\nexact Or.inl \u27e8rfl, hm\u27e9\n[GOAL]\ncase refine'_1.cons_duplicate\n\u03b1 : Type u_1\nl : List \u03b1\nx y : \u03b1\nhm : x \u2208+ l\n\u22a2 y = x \u2227 x \u2208 l \u2228 x \u2208+ l\n[PROOFSTEP]\nexact Or.inr hm\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nl : List \u03b1\nx y : \u03b1\nh : y = x \u2227 x \u2208 l \u2228 x \u2208+ l\n\u22a2 x \u2208+ y :: l\n[PROOFSTEP]\nrcases h with (\u27e8rfl | h\u27e9 | h)\n[GOAL]\ncase refine'_2.inl.intro.refl\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nright\u271d : x \u2208 l\n\u22a2 x \u2208+ x :: l\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase refine'_2.inr\n\u03b1 : Type u_1\nl : List \u03b1\nx y : \u03b1\nh : x \u2208+ l\n\u22a2 x \u2208+ y :: l\n[PROOFSTEP]\nexact h.cons_duplicate\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx y : \u03b1\nh : x \u2208+ y :: l\nhx : x \u2260 y\n\u22a2 x \u2208+ l\n[PROOFSTEP]\nsimpa [duplicate_cons_iff, hx.symm] using h\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx y : \u03b1\nhne : x \u2260 y\n\u22a2 x \u2208+ y :: l \u2194 x \u2208+ l\n[PROOFSTEP]\nsimp [duplicate_cons_iff, hne.symm]\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nl' : List \u03b1\nhx : x \u2208+ l\nh : l <+ l'\n\u22a2 x \u2208+ l'\n[PROOFSTEP]\ninduction' h with l\u2081 l\u2082 y _ IH l\u2081 l\u2082 y h IH\n[GOAL]\ncase slnil\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nl' : List \u03b1\nhx\u271d : x \u2208+ l\nhx : x \u2208+ []\n\u22a2 x \u2208+ []\n[PROOFSTEP]\nexact hx\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nl' : List \u03b1\nhx\u271d : x \u2208+ l\nl\u2081 l\u2082 : List \u03b1\ny : \u03b1\na\u271d : l\u2081 <+ l\u2082\nIH : x \u2208+ l\u2081 \u2192 x \u2208+ l\u2082\nhx : x \u2208+ l\u2081\n\u22a2 x \u2208+ y :: l\u2082\n[PROOFSTEP]\nexact (IH hx).duplicate_cons _\n[GOAL]\ncase cons\u2082\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nl' : List \u03b1\nhx\u271d : x \u2208+ l\nl\u2081 l\u2082 : List \u03b1\ny : \u03b1\nh : l\u2081 <+ l\u2082\nIH : x \u2208+ l\u2081 \u2192 x \u2208+ l\u2082\nhx : x \u2208+ y :: l\u2081\n\u22a2 x \u2208+ y :: l\u2082\n[PROOFSTEP]\nrw [duplicate_cons_iff] at hx \u22a2\n[GOAL]\ncase cons\u2082\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nl' : List \u03b1\nhx\u271d : x \u2208+ l\nl\u2081 l\u2082 : List \u03b1\ny : \u03b1\nh : l\u2081 <+ l\u2082\nIH : x \u2208+ l\u2081 \u2192 x \u2208+ l\u2082\nhx : y = x \u2227 x \u2208 l\u2081 \u2228 x \u2208+ l\u2081\n\u22a2 y = x \u2227 x \u2208 l\u2082 \u2228 x \u2208+ l\u2082\n[PROOFSTEP]\nrcases hx with (\u27e8rfl, hx\u27e9 | hx)\n[GOAL]\ncase cons\u2082.inl.intro\n\u03b1 : Type u_1\nl l' l\u2081 l\u2082 : List \u03b1\ny : \u03b1\nh : l\u2081 <+ l\u2082\nhx\u271d : y \u2208+ l\nIH : y \u2208+ l\u2081 \u2192 y \u2208+ l\u2082\nhx : y \u2208 l\u2081\n\u22a2 y = y \u2227 y \u2208 l\u2082 \u2228 y \u2208+ l\u2082\n[PROOFSTEP]\nsimp [h.subset hx]\n[GOAL]\ncase cons\u2082.inr\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nl' : List \u03b1\nhx\u271d : x \u2208+ l\nl\u2081 l\u2082 : List \u03b1\ny : \u03b1\nh : l\u2081 <+ l\u2082\nIH : x \u2208+ l\u2081 \u2192 x \u2208+ l\u2082\nhx : x \u2208+ l\u2081\n\u22a2 y = x \u2227 x \u2208 l\u2082 \u2228 x \u2208+ l\u2082\n[PROOFSTEP]\nsimp [IH hx]\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\n\u22a2 x \u2208+ l \u2194 [x, x] <+ l\n[PROOFSTEP]\ninduction' l with y l IH\n[GOAL]\ncase nil\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\n\u22a2 x \u2208+ [] \u2194 [x, x] <+ []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nl\u271d : List \u03b1\nx y : \u03b1\nl : List \u03b1\nIH : x \u2208+ l \u2194 [x, x] <+ l\n\u22a2 x \u2208+ y :: l \u2194 [x, x] <+ y :: l\n[PROOFSTEP]\nby_cases hx : x = y\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nl\u271d : List \u03b1\nx y : \u03b1\nl : List \u03b1\nIH : x \u2208+ l \u2194 [x, x] <+ l\nhx : x = y\n\u22a2 x \u2208+ y :: l \u2194 [x, x] <+ y :: l\n[PROOFSTEP]\nsimp [hx, cons_sublist_cons_iff, singleton_sublist]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nl\u271d : List \u03b1\nx y : \u03b1\nl : List \u03b1\nIH : x \u2208+ l \u2194 [x, x] <+ l\nhx : \u00acx = y\n\u22a2 x \u2208+ y :: l \u2194 [x, x] <+ y :: l\n[PROOFSTEP]\nrw [duplicate_cons_iff_of_ne hx, IH]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nl\u271d : List \u03b1\nx y : \u03b1\nl : List \u03b1\nIH : x \u2208+ l \u2194 [x, x] <+ l\nhx : \u00acx = y\n\u22a2 [x, x] <+ l \u2194 [x, x] <+ y :: l\n[PROOFSTEP]\nrefine' \u27e8sublist_cons_of_sublist y, fun h => _\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nl\u271d : List \u03b1\nx y : \u03b1\nl : List \u03b1\nIH : x \u2208+ l \u2194 [x, x] <+ l\nhx : \u00acx = y\nh : [x, x] <+ y :: l\n\u22a2 [x, x] <+ l\n[PROOFSTEP]\ncases h\n[GOAL]\ncase neg.cons\n\u03b1 : Type u_1\nl\u271d : List \u03b1\nx y : \u03b1\nl : List \u03b1\nIH : x \u2208+ l \u2194 [x, x] <+ l\nhx : \u00acx = y\na\u271d : [x, x] <+ l\n\u22a2 [x, x] <+ l\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg.cons\u2082\n\u03b1 : Type u_1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nIH : x \u2208+ l \u2194 [x, x] <+ l\nhx : \u00acx = x\na\u271d : [x] <+ l\n\u22a2 [x, x] <+ l\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\n\u22a2 Nodup l \u2194 \u2200 (x : \u03b1), \u00acx \u2208+ l\n[PROOFSTEP]\nsimp_rw [nodup_iff_sublist, duplicate_iff_sublist]\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\n\u22a2 (\u2203 x, x \u2208+ l) \u2194 \u00acNodup l\n[PROOFSTEP]\nsimp [nodup_iff_forall_not_duplicate]\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 x \u2208+ l \u2194 2 \u2264 count x l\n[PROOFSTEP]\nsimp [duplicate_iff_sublist, le_count_iff_replicate_sublist]\n[GOAL]\n\u03b1 : Type u_1\nl\u271d : List \u03b1\nx\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nl : List \u03b1\nh : \u00acx \u2208+ l\nhx : \u00ac(y = x \u2227 x \u2208 l)\n\u22a2 \u00acx \u2208+ y :: l\n[PROOFSTEP]\nsimpa [duplicate_cons_iff, h] using hx\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Duplicate", "llama_tokens": 3166, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.6619228691808011, "lm_q1q2_score": 0.44730404359734727}} {"text": "[GOAL]\nl : List Bool\nb : Bool\n\u22a2 count (!b) l + count b l = length l\n[PROOFSTEP]\nsimp only [length_eq_countp_add_countp (\u00b7 == !b), count, add_right_inj]\n[GOAL]\nl : List Bool\nb : Bool\n\u22a2 countp (fun x => x == b) l = countp (fun a => decide \u00ac(a == !b) = true) l\n[PROOFSTEP]\nsuffices : (fun x => x == b) = (fun a => decide \u00ac(a == !b) = true)\n[GOAL]\nl : List Bool\nb : Bool\nthis : (fun x => x == b) = fun a => decide \u00ac(a == !b) = true\n\u22a2 countp (fun x => x == b) l = countp (fun a => decide \u00ac(a == !b) = true) l\ncase this l : List Bool b : Bool \u22a2 (fun x => x == b) = fun a => decide \u00ac(a == !b) = true\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase this\nl : List Bool\nb : Bool\n\u22a2 (fun x => x == b) = fun a => decide \u00ac(a == !b) = true\n[PROOFSTEP]\next x\n[GOAL]\ncase this.h\nl : List Bool\nb x : Bool\n\u22a2 (x == b) = decide \u00ac(x == !b) = true\n[PROOFSTEP]\ncases x\n[GOAL]\ncase this.h.false\nl : List Bool\nb : Bool\n\u22a2 (false == b) = decide \u00ac(false == !b) = true\n[PROOFSTEP]\ncases b\n[GOAL]\ncase this.h.true\nl : List Bool\nb : Bool\n\u22a2 (true == b) = decide \u00ac(true == !b) = true\n[PROOFSTEP]\ncases b\n[GOAL]\ncase this.h.false.false\nl : List Bool\n\u22a2 (false == false) = decide \u00ac(false == !false) = true\n[PROOFSTEP]\nrfl\n[GOAL]\ncase this.h.false.true\nl : List Bool\n\u22a2 (false == true) = decide \u00ac(false == !true) = true\n[PROOFSTEP]\nrfl\n[GOAL]\ncase this.h.true.false\nl : List Bool\n\u22a2 (true == false) = decide \u00ac(true == !false) = true\n[PROOFSTEP]\nrfl\n[GOAL]\ncase this.h.true.true\nl : List Bool\n\u22a2 (true == true) = decide \u00ac(true == !true) = true\n[PROOFSTEP]\nrfl\n[GOAL]\nl : List Bool\nb : Bool\n\u22a2 count b l + count (!b) l = length l\n[PROOFSTEP]\nrw [add_comm, count_not_add_count]\n[GOAL]\nb x : Bool\nl : List Bool\nh : Chain (fun x x_1 => x \u2260 x_1) b (x :: l)\n\u22a2 count (!b) (x :: l) = count b (x :: l) + length (x :: l) % 2\n[PROOFSTEP]\nobtain rfl : b = !x := Bool.eq_not_iff.2 (rel_of_chain_cons h)\n[GOAL]\nx : Bool\nl : List Bool\nh : Chain (fun x x_1 => x \u2260 x_1) (!x) (x :: l)\n\u22a2 count (!!x) (x :: l) = count (!x) (x :: l) + length (x :: l) % 2\n[PROOFSTEP]\nrw [Bool.not_not, count_cons_self, count_cons_of_ne x.not_ne_self, Chain.count_not (chain_of_chain_cons h), length,\n add_assoc, Nat.mod_two_add_succ_mod_two]\n[GOAL]\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh2 : Even (length l)\nb : Bool\n\u22a2 count (!b) l = count b l\n[PROOFSTEP]\ncases' l with x l\n[GOAL]\ncase nil\nb : Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) []\nh2 : Even (length [])\n\u22a2 count (!b) [] = count b []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\nb x : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) (x :: l)\nh2 : Even (length (x :: l))\n\u22a2 count (!b) (x :: l) = count b (x :: l)\n[PROOFSTEP]\nrw [length_cons, Nat.even_add_one, Nat.not_even_iff] at h2 \n[GOAL]\ncase cons\nb x : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) (x :: l)\nh2 : length l % 2 = 1\n\u22a2 count (!b) (x :: l) = count b (x :: l)\n[PROOFSTEP]\nsuffices count (!x) (x :: l) = count x (x :: l) by\n -- Porting note: old proof is\n -- cases b <;> cases x <;> try exact this;cases b <;> cases x <;> revert this <;>\n simp only [Bool.not_false, Bool.not_true] <;>\n intro this <;>\n (try exact this) <;>\n exact this.symm\n[GOAL]\nb x : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) (x :: l)\nh2 : length l % 2 = 1\nthis : count (!x) (x :: l) = count x (x :: l)\n\u22a2 count (!b) (x :: l) = count b (x :: l)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\nx : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) (x :: l)\nh2 : length l % 2 = 1\nthis : count (!x) (x :: l) = count x (x :: l)\n\u22a2 count (!false) (x :: l) = count false (x :: l)\n[PROOFSTEP]\ncases x\n[GOAL]\ncase true\nx : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) (x :: l)\nh2 : length l % 2 = 1\nthis : count (!x) (x :: l) = count x (x :: l)\n\u22a2 count (!true) (x :: l) = count true (x :: l)\n[PROOFSTEP]\ncases x\n[GOAL]\ncase false.false\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (false :: l)\nthis : count (!false) (false :: l) = count false (false :: l)\n\u22a2 count (!false) (false :: l) = count false (false :: l)\n[PROOFSTEP]\nrevert this\n[GOAL]\ncase false.true\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (true :: l)\nthis : count (!true) (true :: l) = count true (true :: l)\n\u22a2 count (!false) (true :: l) = count false (true :: l)\n[PROOFSTEP]\nrevert this\n[GOAL]\ncase true.false\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (false :: l)\nthis : count (!false) (false :: l) = count false (false :: l)\n\u22a2 count (!true) (false :: l) = count true (false :: l)\n[PROOFSTEP]\nrevert this\n[GOAL]\ncase true.true\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (true :: l)\nthis : count (!true) (true :: l) = count true (true :: l)\n\u22a2 count (!true) (true :: l) = count true (true :: l)\n[PROOFSTEP]\nrevert this\n[GOAL]\ncase false.false\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (false :: l)\n\u22a2 count (!false) (false :: l) = count false (false :: l) \u2192 count (!false) (false :: l) = count false (false :: l)\n[PROOFSTEP]\nsimp only [Bool.not_false, Bool.not_true]\n[GOAL]\ncase false.true\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (true :: l)\n\u22a2 count (!true) (true :: l) = count true (true :: l) \u2192 count (!false) (true :: l) = count false (true :: l)\n[PROOFSTEP]\nsimp only [Bool.not_false, Bool.not_true]\n[GOAL]\ncase true.false\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (false :: l)\n\u22a2 count (!false) (false :: l) = count false (false :: l) \u2192 count (!true) (false :: l) = count true (false :: l)\n[PROOFSTEP]\nsimp only [Bool.not_false, Bool.not_true]\n[GOAL]\ncase true.true\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (true :: l)\n\u22a2 count (!true) (true :: l) = count true (true :: l) \u2192 count (!true) (true :: l) = count true (true :: l)\n[PROOFSTEP]\nsimp only [Bool.not_false, Bool.not_true]\n[GOAL]\ncase false.false\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (false :: l)\n\u22a2 count true (false :: l) = count false (false :: l) \u2192 count true (false :: l) = count false (false :: l)\n[PROOFSTEP]\nintro this\n[GOAL]\ncase false.true\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (true :: l)\n\u22a2 count false (true :: l) = count true (true :: l) \u2192 count true (true :: l) = count false (true :: l)\n[PROOFSTEP]\nintro this\n[GOAL]\ncase true.false\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (false :: l)\n\u22a2 count true (false :: l) = count false (false :: l) \u2192 count false (false :: l) = count true (false :: l)\n[PROOFSTEP]\nintro this\n[GOAL]\ncase true.true\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (true :: l)\n\u22a2 count false (true :: l) = count true (true :: l) \u2192 count false (true :: l) = count true (true :: l)\n[PROOFSTEP]\nintro this\n[GOAL]\ncase false.false\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (false :: l)\nthis : count true (false :: l) = count false (false :: l)\n\u22a2 count true (false :: l) = count false (false :: l)\n[PROOFSTEP]\ntry exact this\n[GOAL]\ncase false.false\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (false :: l)\nthis : count true (false :: l) = count false (false :: l)\n\u22a2 count true (false :: l) = count false (false :: l)\n[PROOFSTEP]\nexact this\n[GOAL]\ncase false.true\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (true :: l)\nthis : count false (true :: l) = count true (true :: l)\n\u22a2 count true (true :: l) = count false (true :: l)\n[PROOFSTEP]\ntry exact this\n[GOAL]\ncase false.true\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (true :: l)\nthis : count false (true :: l) = count true (true :: l)\n\u22a2 count true (true :: l) = count false (true :: l)\n[PROOFSTEP]\nexact this\n[GOAL]\ncase true.false\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (false :: l)\nthis : count true (false :: l) = count false (false :: l)\n\u22a2 count false (false :: l) = count true (false :: l)\n[PROOFSTEP]\ntry exact this\n[GOAL]\ncase true.false\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (false :: l)\nthis : count true (false :: l) = count false (false :: l)\n\u22a2 count false (false :: l) = count true (false :: l)\n[PROOFSTEP]\nexact this\n[GOAL]\ncase true.true\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (true :: l)\nthis : count false (true :: l) = count true (true :: l)\n\u22a2 count false (true :: l) = count true (true :: l)\n[PROOFSTEP]\ntry exact this\n[GOAL]\ncase true.true\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (true :: l)\nthis : count false (true :: l) = count true (true :: l)\n\u22a2 count false (true :: l) = count true (true :: l)\n[PROOFSTEP]\nexact this\n[GOAL]\ncase false.true\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (true :: l)\nthis : count false (true :: l) = count true (true :: l)\n\u22a2 count true (true :: l) = count false (true :: l)\n[PROOFSTEP]\nexact this.symm\n[GOAL]\ncase true.false\nl : List Bool\nh2 : length l % 2 = 1\nhl : Chain' (fun x x_1 => x \u2260 x_1) (false :: l)\nthis : count true (false :: l) = count false (false :: l)\n\u22a2 count false (false :: l) = count true (false :: l)\n[PROOFSTEP]\nexact this.symm\n[GOAL]\ncase cons\nb x : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) (x :: l)\nh2 : length l % 2 = 1\n\u22a2 count (!x) (x :: l) = count x (x :: l)\n[PROOFSTEP]\nrw [count_cons_of_ne x.not_ne_self, hl.count_not, h2, count_cons_self]\n[GOAL]\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\n\u22a2 count (!b) l \u2264 count b l + 1\n[PROOFSTEP]\ncases' l with x l\n[GOAL]\ncase nil\nb : Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) []\n\u22a2 count (!b) [] \u2264 count b [] + 1\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase cons\nb x : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) (x :: l)\n\u22a2 count (!b) (x :: l) \u2264 count b (x :: l) + 1\n[PROOFSTEP]\nobtain rfl | rfl : b = x \u2228 b = !x := by simp only [Bool.eq_not_iff, em]\n[GOAL]\nb x : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) (x :: l)\n\u22a2 b = x \u2228 b = !x\n[PROOFSTEP]\nsimp only [Bool.eq_not_iff, em]\n[GOAL]\ncase cons.inl\nb : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) (b :: l)\n\u22a2 count (!b) (b :: l) \u2264 count b (b :: l) + 1\n[PROOFSTEP]\nrw [count_cons_of_ne b.not_ne_self, count_cons_self, hl.count_not, add_assoc]\n[GOAL]\ncase cons.inl\nb : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) (b :: l)\n\u22a2 count b l + length l % 2 \u2264 count b l + (1 + 1)\n[PROOFSTEP]\nexact add_le_add_left (Nat.mod_lt _ two_pos).le _\n[GOAL]\ncase cons.inr\nx : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) (x :: l)\n\u22a2 count (!!x) (x :: l) \u2264 count (!x) (x :: l) + 1\n[PROOFSTEP]\nrw [Bool.not_not, count_cons_self, count_cons_of_ne x.not_ne_self, hl.count_not]\n[GOAL]\ncase cons.inr\nx : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) (x :: l)\n\u22a2 count x l + 1 \u2264 count x l + length l % 2 + 1\n[PROOFSTEP]\nexact add_le_add_right (le_add_right le_rfl) _\n[GOAL]\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh2 : Even (length l)\nb : Bool\n\u22a2 2 * count b l = length l\n[PROOFSTEP]\nrw [\u2190 count_not_add_count l b, hl.count_not_eq_count h2, two_mul]\n[GOAL]\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\n\u22a2 2 * count b l =\n if Even (length l) then length l else if (some b == head? l) = true then length l + 1 else length l - 1\n[PROOFSTEP]\nby_cases h2 : Even (length l)\n[GOAL]\ncase pos\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\nh2 : Even (length l)\n\u22a2 2 * count b l =\n if Even (length l) then length l else if (some b == head? l) = true then length l + 1 else length l - 1\n[PROOFSTEP]\nrw [if_pos h2, hl.two_mul_count_bool_of_even h2]\n[GOAL]\ncase neg\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\nh2 : \u00acEven (length l)\n\u22a2 2 * count b l =\n if Even (length l) then length l else if (some b == head? l) = true then length l + 1 else length l - 1\n[PROOFSTEP]\ncases' l with x l\n[GOAL]\ncase neg.nil\nb : Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) []\nh2 : \u00acEven (length [])\n\u22a2 2 * count b [] =\n if Even (length []) then length [] else if (some b == head? []) = true then length [] + 1 else length [] - 1\n[PROOFSTEP]\nexact (h2 even_zero).elim\n[GOAL]\ncase neg.cons\nb x : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) (x :: l)\nh2 : \u00acEven (length (x :: l))\n\u22a2 2 * count b (x :: l) =\n if Even (length (x :: l)) then length (x :: l)\n else if (some b == head? (x :: l)) = true then length (x :: l) + 1 else length (x :: l) - 1\n[PROOFSTEP]\nsimp only [if_neg h2, count_cons', mul_add, head?, Option.mem_some_iff, @eq_comm _ x]\n[GOAL]\ncase neg.cons\nb x : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) (x :: l)\nh2 : \u00acEven (length (x :: l))\n\u22a2 (2 * count b l + 2 * if b = x then 1 else 0) =\n if (some b == some x) = true then length (x :: l) + 1 else length (x :: l) - 1\n[PROOFSTEP]\nrw [length_cons, Nat.even_add_one, not_not] at h2 \n[GOAL]\ncase neg.cons\nb x : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) (x :: l)\nh2 : Even (length l)\n\u22a2 (2 * count b l + 2 * if b = x then 1 else 0) =\n if (some b == some x) = true then length (x :: l) + 1 else length (x :: l) - 1\n[PROOFSTEP]\nreplace hl : l.Chain' (\u00b7 \u2260 \u00b7) := hl.tail\n[GOAL]\ncase neg.cons\nb x : Bool\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\n\u22a2 (2 * count b l + 2 * if b = x then 1 else 0) =\n if (some b == some x) = true then length (x :: l) + 1 else length (x :: l) - 1\n[PROOFSTEP]\nrw [hl.two_mul_count_bool_of_even h2]\n[GOAL]\ncase neg.cons\nb x : Bool\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\n\u22a2 (length l + 2 * if b = x then 1 else 0) =\n if (some b == some x) = true then length (x :: l) + 1 else length (x :: l) - 1\n[PROOFSTEP]\ncases b\n[GOAL]\ncase neg.cons.false\nx : Bool\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\n\u22a2 (length l + 2 * if false = x then 1 else 0) =\n if (some false == some x) = true then length (x :: l) + 1 else length (x :: l) - 1\n[PROOFSTEP]\ncases x\n[GOAL]\ncase neg.cons.true\nx : Bool\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\n\u22a2 (length l + 2 * if true = x then 1 else 0) =\n if (some true == some x) = true then length (x :: l) + 1 else length (x :: l) - 1\n[PROOFSTEP]\ncases x\n[GOAL]\ncase neg.cons.false.false\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\n\u22a2 (length l + 2 * if false = false then 1 else 0) =\n if (some false == some false) = true then length (false :: l) + 1 else length (false :: l) - 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase neg.cons.false.true\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\n\u22a2 (length l + 2 * if false = true then 1 else 0) =\n if (some false == some true) = true then length (true :: l) + 1 else length (true :: l) - 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase neg.cons.true.false\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\n\u22a2 (length l + 2 * if true = false then 1 else 0) =\n if (some true == some false) = true then length (false :: l) + 1 else length (false :: l) - 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase neg.cons.true.true\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\n\u22a2 (length l + 2 * if true = true then 1 else 0) =\n if (some true == some true) = true then length (true :: l) + 1 else length (true :: l) - 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b9 : false = false\nh\u271d : (some false == some false) = true\n\u22a2 length l + 2 * 1 = length (false :: l) + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b9 : false = false\nh\u271d : \u00ac(some false == some false) = true\n\u22a2 length l + 2 * 1 = length (false :: l) - 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b9 : \u00acfalse = false\nh\u271d : (some false == some false) = true\n\u22a2 length l + 2 * 0 = length (false :: l) + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b9 : \u00acfalse = false\nh\u271d : \u00ac(some false == some false) = true\n\u22a2 length l + 2 * 0 = length (false :: l) - 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : false = true\nh\u271d\u00b9 : False\nh\u271d : (some false == some true) = true\n\u22a2 length l + 2 * 1 = length (true :: l) + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : false = true\nh\u271d\u00b9 : False\nh\u271d : \u00ac(some false == some true) = true\n\u22a2 length l + 2 * 1 = length (true :: l) - 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : false = true\nh\u271d\u00b9 : \u00acFalse\nh\u271d : (some false == some true) = true\n\u22a2 length l + 2 * 0 = length (true :: l) + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : false = true\nh\u271d\u00b9 : \u00acFalse\nh\u271d : \u00ac(some false == some true) = true\n\u22a2 length l + 2 * 0 = length (true :: l) - 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : \u00acfalse = true\nh\u271d\u00b9 : False\nh\u271d : (some false == some true) = true\n\u22a2 length l + 2 * 1 = length (true :: l) + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : \u00acfalse = true\nh\u271d\u00b9 : False\nh\u271d : \u00ac(some false == some true) = true\n\u22a2 length l + 2 * 1 = length (true :: l) - 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : \u00acfalse = true\nh\u271d\u00b9 : \u00acFalse\nh\u271d : (some false == some true) = true\n\u22a2 length l + 2 * 0 = length (true :: l) + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : \u00acfalse = true\nh\u271d\u00b9 : \u00acFalse\nh\u271d : \u00ac(some false == some true) = true\n\u22a2 length l + 2 * 0 = length (true :: l) - 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : true = false\nh\u271d\u00b9 : False\nh\u271d : (some true == some false) = true\n\u22a2 length l + 2 * 1 = length (false :: l) + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : true = false\nh\u271d\u00b9 : False\nh\u271d : \u00ac(some true == some false) = true\n\u22a2 length l + 2 * 1 = length (false :: l) - 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : true = false\nh\u271d\u00b9 : \u00acFalse\nh\u271d : (some true == some false) = true\n\u22a2 length l + 2 * 0 = length (false :: l) + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : true = false\nh\u271d\u00b9 : \u00acFalse\nh\u271d : \u00ac(some true == some false) = true\n\u22a2 length l + 2 * 0 = length (false :: l) - 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : \u00actrue = false\nh\u271d\u00b9 : False\nh\u271d : (some true == some false) = true\n\u22a2 length l + 2 * 1 = length (false :: l) + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : \u00actrue = false\nh\u271d\u00b9 : False\nh\u271d : \u00ac(some true == some false) = true\n\u22a2 length l + 2 * 1 = length (false :: l) - 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : \u00actrue = false\nh\u271d\u00b9 : \u00acFalse\nh\u271d : (some true == some false) = true\n\u22a2 length l + 2 * 0 = length (false :: l) + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : \u00actrue = false\nh\u271d\u00b9 : \u00acFalse\nh\u271d : \u00ac(some true == some false) = true\n\u22a2 length l + 2 * 0 = length (false :: l) - 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b9 : true = true\nh\u271d : (some true == some true) = true\n\u22a2 length l + 2 * 1 = length (true :: l) + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b9 : true = true\nh\u271d : \u00ac(some true == some true) = true\n\u22a2 length l + 2 * 1 = length (true :: l) - 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b9 : \u00actrue = true\nh\u271d : (some true == some true) = true\n\u22a2 length l + 2 * 0 = length (true :: l) + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b9 : \u00actrue = true\nh\u271d : \u00ac(some true == some true) = true\n\u22a2 length l + 2 * 0 = length (true :: l) - 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b9 : false = false\nh\u271d : \u00ac(some false == some false) = true\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b9 : \u00acfalse = false\nh\u271d : (some false == some false) = true\n\u22a2 length l = Nat.succ (length l) + 1\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : false = true\nh\u271d\u00b9 : False\nh\u271d : \u00ac(some false == some true) = true\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : false = true\nh\u271d\u00b9 : \u00acFalse\nh\u271d : (some false == some true) = true\n\u22a2 length l = Nat.succ (length l) + 1\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : \u00acfalse = true\nh\u271d\u00b9 : False\nh\u271d : \u00ac(some false == some true) = true\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : \u00acfalse = true\nh\u271d\u00b9 : \u00acFalse\nh\u271d : (some false == some true) = true\n\u22a2 length l = Nat.succ (length l) + 1\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : true = false\nh\u271d\u00b9 : False\nh\u271d : \u00ac(some true == some false) = true\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : true = false\nh\u271d\u00b9 : \u00acFalse\nh\u271d : (some true == some false) = true\n\u22a2 length l = Nat.succ (length l) + 1\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : \u00actrue = false\nh\u271d\u00b9 : False\nh\u271d : \u00ac(some true == some false) = true\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b2 : \u00actrue = false\nh\u271d\u00b9 : \u00acFalse\nh\u271d : (some true == some false) = true\n\u22a2 length l = Nat.succ (length l) + 1\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b9 : true = true\nh\u271d : \u00ac(some true == some true) = true\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nh\u271d\u00b9 : \u00actrue = true\nh\u271d : (some true == some true) = true\n\u22a2 length l = Nat.succ (length l) + 1\n[PROOFSTEP]\ncontradiction\n[GOAL]\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\n\u22a2 length l - 1 \u2264 2 * count b l\n[PROOFSTEP]\nrw [hl.two_mul_count_bool_eq_ite]\n[GOAL]\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\n\u22a2 length l - 1 \u2264 if Even (length l) then length l else if (some b == head? l) = true then length l + 1 else length l - 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\nh\u271d : Even (length l)\n\u22a2 length l - 1 \u2264 length l\n[PROOFSTEP]\nsimp [le_tsub_add, Nat.le_succ_of_le]\n[GOAL]\ncase pos\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\nh\u271d\u00b9 : \u00acEven (length l)\nh\u271d : (some b == head? l) = true\n\u22a2 length l - 1 \u2264 length l + 1\n[PROOFSTEP]\nsimp [le_tsub_add, Nat.le_succ_of_le]\n[GOAL]\ncase neg\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\nh\u271d\u00b9 : \u00acEven (length l)\nh\u271d : \u00ac(some b == head? l) = true\n\u22a2 length l - 1 \u2264 length l - 1\n[PROOFSTEP]\nsimp [le_tsub_add, Nat.le_succ_of_le]\n[GOAL]\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\n\u22a2 length l / 2 \u2264 count b l\n[PROOFSTEP]\nrw [Nat.div_le_iff_le_mul_add_pred two_pos, \u2190 tsub_le_iff_right]\n[GOAL]\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\n\u22a2 length l - (2 - 1) \u2264 2 * count b l\n[PROOFSTEP]\nexact length_sub_one_le_two_mul_count_bool hl b\n[GOAL]\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\n\u22a2 2 * count b l \u2264 length l + 1\n[PROOFSTEP]\nrw [hl.two_mul_count_bool_eq_ite]\n[GOAL]\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\n\u22a2 (if Even (length l) then length l else if (some b == head? l) = true then length l + 1 else length l - 1) \u2264\n length l + 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\nh\u271d : Even (length l)\n\u22a2 length l \u2264 length l + 1\n[PROOFSTEP]\nsimp [Nat.le_succ_of_le]\n[GOAL]\ncase pos\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\nh\u271d\u00b9 : \u00acEven (length l)\nh\u271d : (some b == head? l) = true\n\u22a2 length l + 1 \u2264 length l + 1\n[PROOFSTEP]\nsimp [Nat.le_succ_of_le]\n[GOAL]\ncase neg\nl : List Bool\nhl : Chain' (fun x x_1 => x \u2260 x_1) l\nb : Bool\nh\u271d\u00b9 : \u00acEven (length l)\nh\u271d : \u00ac(some b == head? l) = true\n\u22a2 length l - 1 \u2264 length l + 1\n[PROOFSTEP]\nsimp [Nat.le_succ_of_le]\n", "meta": {"mathlib_filename": "Mathlib.Data.Bool.Count", "llama_tokens": 11457, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707283, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4470687187626755}} {"text": "[GOAL]\nX Y : WalkingCospan\n\u22a2 Subsingleton (X \u27f6 Y)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase allEq\nX Y : WalkingCospan\n\u22a2 \u2200 (a b : X \u27f6 Y), a = b\n[PROOFSTEP]\nintros\n[GOAL]\ncase allEq\nX Y : WalkingCospan\na\u271d b\u271d : X \u27f6 Y\n\u22a2 a\u271d = b\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nX Y : WalkingSpan\n\u22a2 Subsingleton (X \u27f6 Y)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase allEq\nX Y : WalkingSpan\n\u22a2 \u2200 (a b : X \u27f6 Y), a = b\n[PROOFSTEP]\nintros a b\n[GOAL]\ncase allEq\nX Y : WalkingSpan\na b : X \u27f6 Y\n\u22a2 a = b\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\ns t : Cone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03c0 left = i.hom \u226b NatTrans.app t.\u03c0 left\nw\u2082 : NatTrans.app s.\u03c0 right = i.hom \u226b NatTrans.app t.\u03c0 right\n\u22a2 s \u2245 t\n[PROOFSTEP]\napply Cones.ext i _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\ns t : Cone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03c0 left = i.hom \u226b NatTrans.app t.\u03c0 left\nw\u2082 : NatTrans.app s.\u03c0 right = i.hom \u226b NatTrans.app t.\u03c0 right\n\u22a2 \u2200 (j : WalkingCospan), NatTrans.app s.\u03c0 j = i.hom \u226b NatTrans.app t.\u03c0 j\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9)\n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\ns t : Cone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03c0 left = i.hom \u226b NatTrans.app t.\u03c0 left\nw\u2082 : NatTrans.app s.\u03c0 right = i.hom \u226b NatTrans.app t.\u03c0 right\n\u22a2 NatTrans.app s.\u03c0 none = i.hom \u226b NatTrans.app t.\u03c0 none\n[PROOFSTEP]\nhave h\u2081 := s.\u03c0.naturality WalkingCospan.Hom.inl\n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\ns t : Cone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03c0 left = i.hom \u226b NatTrans.app t.\u03c0 left\nw\u2082 : NatTrans.app s.\u03c0 right = i.hom \u226b NatTrans.app t.\u03c0 right\nh\u2081 : ((Functor.const WalkingCospan).obj s.pt).map inl \u226b NatTrans.app s.\u03c0 one = NatTrans.app s.\u03c0 left \u226b F.map inl\n\u22a2 NatTrans.app s.\u03c0 none = i.hom \u226b NatTrans.app t.\u03c0 none\n[PROOFSTEP]\ndsimp at h\u2081 \n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\ns t : Cone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03c0 left = i.hom \u226b NatTrans.app t.\u03c0 left\nw\u2082 : NatTrans.app s.\u03c0 right = i.hom \u226b NatTrans.app t.\u03c0 right\nh\u2081 : \ud835\udfd9 s.pt \u226b NatTrans.app s.\u03c0 one = NatTrans.app s.\u03c0 left \u226b F.map inl\n\u22a2 NatTrans.app s.\u03c0 none = i.hom \u226b NatTrans.app t.\u03c0 none\n[PROOFSTEP]\nsimp only [Category.id_comp] at h\u2081 \n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\ns t : Cone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03c0 left = i.hom \u226b NatTrans.app t.\u03c0 left\nw\u2082 : NatTrans.app s.\u03c0 right = i.hom \u226b NatTrans.app t.\u03c0 right\nh\u2081 : NatTrans.app s.\u03c0 one = NatTrans.app s.\u03c0 left \u226b F.map inl\n\u22a2 NatTrans.app s.\u03c0 none = i.hom \u226b NatTrans.app t.\u03c0 none\n[PROOFSTEP]\nhave h\u2082 := t.\u03c0.naturality WalkingCospan.Hom.inl\n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\ns t : Cone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03c0 left = i.hom \u226b NatTrans.app t.\u03c0 left\nw\u2082 : NatTrans.app s.\u03c0 right = i.hom \u226b NatTrans.app t.\u03c0 right\nh\u2081 : NatTrans.app s.\u03c0 one = NatTrans.app s.\u03c0 left \u226b F.map inl\nh\u2082 : ((Functor.const WalkingCospan).obj t.pt).map inl \u226b NatTrans.app t.\u03c0 one = NatTrans.app t.\u03c0 left \u226b F.map inl\n\u22a2 NatTrans.app s.\u03c0 none = i.hom \u226b NatTrans.app t.\u03c0 none\n[PROOFSTEP]\ndsimp at h\u2082 \n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\ns t : Cone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03c0 left = i.hom \u226b NatTrans.app t.\u03c0 left\nw\u2082 : NatTrans.app s.\u03c0 right = i.hom \u226b NatTrans.app t.\u03c0 right\nh\u2081 : NatTrans.app s.\u03c0 one = NatTrans.app s.\u03c0 left \u226b F.map inl\nh\u2082 : \ud835\udfd9 t.pt \u226b NatTrans.app t.\u03c0 one = NatTrans.app t.\u03c0 left \u226b F.map inl\n\u22a2 NatTrans.app s.\u03c0 none = i.hom \u226b NatTrans.app t.\u03c0 none\n[PROOFSTEP]\nsimp only [Category.id_comp] at h\u2082 \n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\ns t : Cone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03c0 left = i.hom \u226b NatTrans.app t.\u03c0 left\nw\u2082 : NatTrans.app s.\u03c0 right = i.hom \u226b NatTrans.app t.\u03c0 right\nh\u2081 : NatTrans.app s.\u03c0 one = NatTrans.app s.\u03c0 left \u226b F.map inl\nh\u2082 : NatTrans.app t.\u03c0 one = NatTrans.app t.\u03c0 left \u226b F.map inl\n\u22a2 NatTrans.app s.\u03c0 none = i.hom \u226b NatTrans.app t.\u03c0 none\n[PROOFSTEP]\nsimp_rw [h\u2082, \u2190 Category.assoc, \u2190 w\u2081, \u2190 h\u2081]\n[GOAL]\ncase some.left\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\ns t : Cone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03c0 left = i.hom \u226b NatTrans.app t.\u03c0 left\nw\u2082 : NatTrans.app s.\u03c0 right = i.hom \u226b NatTrans.app t.\u03c0 right\n\u22a2 NatTrans.app s.\u03c0 (some WalkingPair.left) = i.hom \u226b NatTrans.app t.\u03c0 (some WalkingPair.left)\n[PROOFSTEP]\nexact w\u2081\n[GOAL]\ncase some.right\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\ns t : Cone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03c0 left = i.hom \u226b NatTrans.app t.\u03c0 left\nw\u2082 : NatTrans.app s.\u03c0 right = i.hom \u226b NatTrans.app t.\u03c0 right\n\u22a2 NatTrans.app s.\u03c0 (some WalkingPair.right) = i.hom \u226b NatTrans.app t.\u03c0 (some WalkingPair.right)\n[PROOFSTEP]\nexact w\u2082\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\ns t : Cocone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03b9 WalkingCospan.left \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.left\nw\u2082 : NatTrans.app s.\u03b9 WalkingCospan.right \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.right\n\u22a2 s \u2245 t\n[PROOFSTEP]\napply Cocones.ext i _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\ns t : Cocone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03b9 WalkingCospan.left \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.left\nw\u2082 : NatTrans.app s.\u03b9 WalkingCospan.right \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.right\n\u22a2 \u2200 (j : WalkingSpan), NatTrans.app s.\u03b9 j \u226b i.hom = NatTrans.app t.\u03b9 j\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9)\n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\ns t : Cocone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03b9 WalkingCospan.left \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.left\nw\u2082 : NatTrans.app s.\u03b9 WalkingCospan.right \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.right\n\u22a2 NatTrans.app s.\u03b9 none \u226b i.hom = NatTrans.app t.\u03b9 none\n[PROOFSTEP]\nhave h\u2081 := s.\u03b9.naturality WalkingSpan.Hom.fst\n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\ns t : Cocone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03b9 WalkingCospan.left \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.left\nw\u2082 : NatTrans.app s.\u03b9 WalkingCospan.right \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.right\nh\u2081 : F.map fst \u226b NatTrans.app s.\u03b9 left = NatTrans.app s.\u03b9 zero \u226b ((Functor.const WalkingSpan).obj s.pt).map fst\n\u22a2 NatTrans.app s.\u03b9 none \u226b i.hom = NatTrans.app t.\u03b9 none\n[PROOFSTEP]\ndsimp at h\u2081 \n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\ns t : Cocone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03b9 WalkingCospan.left \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.left\nw\u2082 : NatTrans.app s.\u03b9 WalkingCospan.right \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.right\nh\u2081 : F.map fst \u226b NatTrans.app s.\u03b9 left = NatTrans.app s.\u03b9 zero \u226b \ud835\udfd9 s.pt\n\u22a2 NatTrans.app s.\u03b9 none \u226b i.hom = NatTrans.app t.\u03b9 none\n[PROOFSTEP]\nsimp only [Category.comp_id] at h\u2081 \n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\ns t : Cocone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03b9 WalkingCospan.left \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.left\nw\u2082 : NatTrans.app s.\u03b9 WalkingCospan.right \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.right\nh\u2081 : F.map fst \u226b NatTrans.app s.\u03b9 left = NatTrans.app s.\u03b9 zero\n\u22a2 NatTrans.app s.\u03b9 none \u226b i.hom = NatTrans.app t.\u03b9 none\n[PROOFSTEP]\nhave h\u2082 := t.\u03b9.naturality WalkingSpan.Hom.fst\n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\ns t : Cocone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03b9 WalkingCospan.left \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.left\nw\u2082 : NatTrans.app s.\u03b9 WalkingCospan.right \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.right\nh\u2081 : F.map fst \u226b NatTrans.app s.\u03b9 left = NatTrans.app s.\u03b9 zero\nh\u2082 : F.map fst \u226b NatTrans.app t.\u03b9 left = NatTrans.app t.\u03b9 zero \u226b ((Functor.const WalkingSpan).obj t.pt).map fst\n\u22a2 NatTrans.app s.\u03b9 none \u226b i.hom = NatTrans.app t.\u03b9 none\n[PROOFSTEP]\ndsimp at h\u2082 \n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\ns t : Cocone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03b9 WalkingCospan.left \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.left\nw\u2082 : NatTrans.app s.\u03b9 WalkingCospan.right \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.right\nh\u2081 : F.map fst \u226b NatTrans.app s.\u03b9 left = NatTrans.app s.\u03b9 zero\nh\u2082 : F.map fst \u226b NatTrans.app t.\u03b9 left = NatTrans.app t.\u03b9 zero \u226b \ud835\udfd9 t.pt\n\u22a2 NatTrans.app s.\u03b9 none \u226b i.hom = NatTrans.app t.\u03b9 none\n[PROOFSTEP]\nsimp only [Category.comp_id] at h\u2082 \n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\ns t : Cocone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03b9 WalkingCospan.left \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.left\nw\u2082 : NatTrans.app s.\u03b9 WalkingCospan.right \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.right\nh\u2081 : F.map fst \u226b NatTrans.app s.\u03b9 left = NatTrans.app s.\u03b9 zero\nh\u2082 : F.map fst \u226b NatTrans.app t.\u03b9 left = NatTrans.app t.\u03b9 zero\n\u22a2 NatTrans.app s.\u03b9 none \u226b i.hom = NatTrans.app t.\u03b9 none\n[PROOFSTEP]\nsimp_rw [\u2190 h\u2081, Category.assoc, w\u2081, h\u2082]\n[GOAL]\ncase some.left\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\ns t : Cocone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03b9 WalkingCospan.left \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.left\nw\u2082 : NatTrans.app s.\u03b9 WalkingCospan.right \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.right\n\u22a2 NatTrans.app s.\u03b9 (some WalkingPair.left) \u226b i.hom = NatTrans.app t.\u03b9 (some WalkingPair.left)\n[PROOFSTEP]\nexact w\u2081\n[GOAL]\ncase some.right\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\ns t : Cocone F\ni : s.pt \u2245 t.pt\nw\u2081 : NatTrans.app s.\u03b9 WalkingCospan.left \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.left\nw\u2082 : NatTrans.app s.\u03b9 WalkingCospan.right \u226b i.hom = NatTrans.app t.\u03b9 WalkingCospan.right\n\u22a2 NatTrans.app s.\u03b9 (some WalkingPair.right) \u226b i.hom = NatTrans.app t.\u03b9 (some WalkingPair.right)\n[PROOFSTEP]\nexact w\u2082\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\nj : WalkingCospan\n\u22a2 F.obj j = (cospan (F.map inl) (F.map inr)).obj j\n[PROOFSTEP]\nrcases j with (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9)\n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\n\u22a2 F.obj none = (cospan (F.map inl) (F.map inr)).obj none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.left\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\n\u22a2 F.obj (some WalkingPair.left) = (cospan (F.map inl) (F.map inr)).obj (some WalkingPair.left)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.right\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\n\u22a2 F.obj (some WalkingPair.right) = (cospan (F.map inl) (F.map inr)).obj (some WalkingPair.right)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\n\u22a2 \u2200 {X Y : WalkingCospan} (f : X \u27f6 Y),\n F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) Y).hom =\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) X).hom \u226b\n (cospan (F.map inl) (F.map inr)).map f\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9) (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9) f\n[GOAL]\ncase none.none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\nf : none \u27f6 none\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) none).hom =\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) none).hom \u226b\n (cospan (F.map inl) (F.map inr)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some.left\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\nf : none \u27f6 some WalkingPair.left\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.left)).hom =\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) none).hom \u226b\n (cospan (F.map inl) (F.map inr)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some.right\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\nf : none \u27f6 some WalkingPair.right\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.right)).hom =\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) none).hom \u226b\n (cospan (F.map inl) (F.map inr)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\nf : some WalkingPair.left \u27f6 none\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) none).hom =\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.left)).hom \u226b\n (cospan (F.map inl) (F.map inr)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.some.left\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\nf : some WalkingPair.left \u27f6 some WalkingPair.left\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.left)).hom =\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.left)).hom \u226b\n (cospan (F.map inl) (F.map inr)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.some.right\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\nf : some WalkingPair.left \u27f6 some WalkingPair.right\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.right)).hom =\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.left)).hom \u226b\n (cospan (F.map inl) (F.map inr)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\nf : some WalkingPair.right \u27f6 none\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) none).hom =\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.right)).hom \u226b\n (cospan (F.map inl) (F.map inr)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.some.left\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\nf : some WalkingPair.right \u27f6 some WalkingPair.left\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.left)).hom =\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.right)).hom \u226b\n (cospan (F.map inl) (F.map inr)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.some.right\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\nf : some WalkingPair.right \u27f6 some WalkingPair.right\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.right)).hom =\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.right)).hom \u226b\n (cospan (F.map inl) (F.map inr)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\n\u22a2 F.map (WidePullbackShape.Hom.id none) \u226b\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) none).hom =\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) none).hom \u226b\n (cospan (F.map inl) (F.map inr)).map (WidePullbackShape.Hom.id none)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.left.none.term\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\n\u22a2 F.map (term WalkingPair.left) \u226b ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) none).hom =\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.left)).hom \u226b\n (cospan (F.map inl) (F.map inr)).map (term WalkingPair.left)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\n\u22a2 F.map (WidePullbackShape.Hom.id (some WalkingPair.left)) \u226b\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.left)).hom =\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.left)).hom \u226b\n (cospan (F.map inl) (F.map inr)).map (WidePullbackShape.Hom.id (some WalkingPair.left))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.right.none.term\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\n\u22a2 F.map (term WalkingPair.right) \u226b\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) none).hom =\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.right)).hom \u226b\n (cospan (F.map inl) (F.map inr)).map (term WalkingPair.right)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\n\u22a2 F.map (WidePullbackShape.Hom.id (some WalkingPair.right)) \u226b\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.right)).hom =\n ((fun j => eqToIso (_ : F.obj j = (cospan (F.map inl) (F.map inr)).obj j)) (some WalkingPair.right)).hom \u226b\n (cospan (F.map inl) (F.map inr)).map (WidePullbackShape.Hom.id (some WalkingPair.right))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\n\u22a2 F.map (\ud835\udfd9 none) \u226b \ud835\udfd9 (F.obj none) = \ud835\udfd9 (F.obj none) \u226b (cospan (F.map inl) (F.map inr)).map (\ud835\udfd9 none)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.left.none.term\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\n\u22a2 F.map (term WalkingPair.left) \u226b \ud835\udfd9 (F.obj none) = \ud835\udfd9 (F.obj (some WalkingPair.left)) \u226b F.map inl\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\n\u22a2 F.map (\ud835\udfd9 (some WalkingPair.left)) \u226b \ud835\udfd9 (F.obj (some WalkingPair.left)) =\n \ud835\udfd9 (F.obj (some WalkingPair.left)) \u226b (cospan (F.map inl) (F.map inr)).map (\ud835\udfd9 (some WalkingPair.left))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.right.none.term\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\n\u22a2 F.map (term WalkingPair.right) \u226b \ud835\udfd9 (F.obj none) = \ud835\udfd9 (F.obj (some WalkingPair.right)) \u226b F.map inr\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingCospan \u2964 C\n\u22a2 F.map (\ud835\udfd9 (some WalkingPair.right)) \u226b \ud835\udfd9 (F.obj (some WalkingPair.right)) =\n \ud835\udfd9 (F.obj (some WalkingPair.right)) \u226b (cospan (F.map inl) (F.map inr)).map (\ud835\udfd9 (some WalkingPair.right))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\nj : WalkingSpan\n\u22a2 F.obj j = (span (F.map fst) (F.map snd)).obj j\n[PROOFSTEP]\nrcases j with (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9)\n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\n\u22a2 F.obj none = (span (F.map fst) (F.map snd)).obj none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.left\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\n\u22a2 F.obj (some WalkingPair.left) = (span (F.map fst) (F.map snd)).obj (some WalkingPair.left)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.right\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\n\u22a2 F.obj (some WalkingPair.right) = (span (F.map fst) (F.map snd)).obj (some WalkingPair.right)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\n\u22a2 \u2200 {X Y : WalkingSpan} (f : X \u27f6 Y),\n F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) Y).hom =\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) X).hom \u226b\n (span (F.map fst) (F.map snd)).map f\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9) (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9) f\n[GOAL]\ncase none.none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\nf : none \u27f6 none\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) none).hom =\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) none).hom \u226b\n (span (F.map fst) (F.map snd)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some.left\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\nf : none \u27f6 some WalkingPair.left\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.left)).hom =\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) none).hom \u226b\n (span (F.map fst) (F.map snd)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some.right\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\nf : none \u27f6 some WalkingPair.right\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.right)).hom =\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) none).hom \u226b\n (span (F.map fst) (F.map snd)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\nf : some WalkingPair.left \u27f6 none\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) none).hom =\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.left)).hom \u226b\n (span (F.map fst) (F.map snd)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.some.left\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\nf : some WalkingPair.left \u27f6 some WalkingPair.left\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.left)).hom =\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.left)).hom \u226b\n (span (F.map fst) (F.map snd)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.some.right\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\nf : some WalkingPair.left \u27f6 some WalkingPair.right\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.right)).hom =\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.left)).hom \u226b\n (span (F.map fst) (F.map snd)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.none\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\nf : some WalkingPair.right \u27f6 none\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) none).hom =\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.right)).hom \u226b\n (span (F.map fst) (F.map snd)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.some.left\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\nf : some WalkingPair.right \u27f6 some WalkingPair.left\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.left)).hom =\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.right)).hom \u226b\n (span (F.map fst) (F.map snd)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.some.right\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\nf : some WalkingPair.right \u27f6 some WalkingPair.right\n\u22a2 F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.right)).hom =\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.right)).hom \u226b\n (span (F.map fst) (F.map snd)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\n\u22a2 F.map (WidePushoutShape.Hom.id none) \u226b\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) none).hom =\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) none).hom \u226b\n (span (F.map fst) (F.map snd)).map (WidePushoutShape.Hom.id none)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.some.left.init\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\n\u22a2 F.map (init WalkingPair.left) \u226b\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.left)).hom =\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) none).hom \u226b\n (span (F.map fst) (F.map snd)).map (init WalkingPair.left)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.some.right.init\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\n\u22a2 F.map (init WalkingPair.right) \u226b\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.right)).hom =\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) none).hom \u226b\n (span (F.map fst) (F.map snd)).map (init WalkingPair.right)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\n\u22a2 F.map (WidePushoutShape.Hom.id (some WalkingPair.left)) \u226b\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.left)).hom =\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.left)).hom \u226b\n (span (F.map fst) (F.map snd)).map (WidePushoutShape.Hom.id (some WalkingPair.left))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\n\u22a2 F.map (WidePushoutShape.Hom.id (some WalkingPair.right)) \u226b\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.right)).hom =\n ((fun j => eqToIso (_ : F.obj j = (span (F.map fst) (F.map snd)).obj j)) (some WalkingPair.right)).hom \u226b\n (span (F.map fst) (F.map snd)).map (WidePushoutShape.Hom.id (some WalkingPair.right))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\n\u22a2 F.map (\ud835\udfd9 none) \u226b \ud835\udfd9 (F.obj none) = \ud835\udfd9 (F.obj none) \u226b (span (F.map fst) (F.map snd)).map (\ud835\udfd9 none)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase none.some.left.init\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\n\u22a2 F.map (init WalkingPair.left) \u226b \ud835\udfd9 (F.obj (some WalkingPair.left)) = \ud835\udfd9 (F.obj none) \u226b F.map fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase none.some.right.init\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\n\u22a2 F.map (init WalkingPair.right) \u226b \ud835\udfd9 (F.obj (some WalkingPair.right)) = \ud835\udfd9 (F.obj none) \u226b F.map snd\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\n\u22a2 F.map (\ud835\udfd9 (some WalkingPair.left)) \u226b \ud835\udfd9 (F.obj (some WalkingPair.left)) =\n \ud835\udfd9 (F.obj (some WalkingPair.left)) \u226b (span (F.map fst) (F.map snd)).map (\ud835\udfd9 (some WalkingPair.left))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WalkingSpan \u2964 C\n\u22a2 F.map (\ud835\udfd9 (some WalkingPair.right)) \u226b \ud835\udfd9 (F.obj (some WalkingPair.right)) =\n \ud835\udfd9 (F.obj (some WalkingPair.right)) \u226b (span (F.map fst) (F.map snd)).map (\ud835\udfd9 (some WalkingPair.right))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 (X_1 : WalkingCospan) \u2192 (cospan f g \u22d9 F).obj X_1 \u2245 (cospan (F.map f) (F.map g)).obj X_1\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9)\n[GOAL]\ncase none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 (cospan f g \u22d9 F).obj none \u2245 (cospan (F.map f) (F.map g)).obj none\n[PROOFSTEP]\nexact Iso.refl _\n[GOAL]\ncase some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 (cospan f g \u22d9 F).obj (some WalkingPair.left) \u2245 (cospan (F.map f) (F.map g)).obj (some WalkingPair.left)\n[PROOFSTEP]\nexact Iso.refl _\n[GOAL]\ncase some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 (cospan f g \u22d9 F).obj (some WalkingPair.right) \u2245 (cospan (F.map f) (F.map g)).obj (some WalkingPair.right)\n[PROOFSTEP]\nexact Iso.refl _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 \u2200 {X_1 Y_1 : WalkingCospan} (f_1 : X_1 \u27f6 Y_1),\n (cospan f g \u22d9 F).map f_1 \u226b\n (Option.casesOn Y_1 (Iso.refl ((cospan f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn X_1 (Iso.refl ((cospan f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (cospan (F.map f) (F.map g)).map f_1\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9) (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9) f\n[GOAL]\ncase none.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf : none \u27f6 none\n\u22a2 (cospan f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn none (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn none (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (cospan (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf : none \u27f6 some WalkingPair.left\n\u22a2 (cospan f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn none (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (cospan (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf : none \u27f6 some WalkingPair.right\n\u22a2 (cospan f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn none (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (cospan (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf : some WalkingPair.left \u27f6 none\n\u22a2 (cospan f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn none (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (cospan (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf : some WalkingPair.left \u27f6 some WalkingPair.left\n\u22a2 (cospan f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (cospan (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf : some WalkingPair.left \u27f6 some WalkingPair.right\n\u22a2 (cospan f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (cospan (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf : some WalkingPair.right \u27f6 none\n\u22a2 (cospan f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn none (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (cospan (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf : some WalkingPair.right \u27f6 some WalkingPair.left\n\u22a2 (cospan f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (cospan (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf : some WalkingPair.right \u27f6 some WalkingPair.right\n\u22a2 (cospan f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((cospan f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (cospan (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 (cospan f g \u22d9 F).map (WidePullbackShape.Hom.id none) \u226b\n (Option.casesOn none (Iso.refl ((cospan f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn none (Iso.refl ((cospan f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (cospan (F.map f) (F.map g)).map (WidePullbackShape.Hom.id none)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.left.none.term\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 (cospan f g \u22d9 F).map (term WalkingPair.left) \u226b\n (Option.casesOn none (Iso.refl ((cospan f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((cospan f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (cospan (F.map f) (F.map g)).map (term WalkingPair.left)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 (cospan f g \u22d9 F).map (WidePullbackShape.Hom.id (some WalkingPair.left)) \u226b\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((cospan f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((cospan f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (cospan (F.map f) (F.map g)).map (WidePullbackShape.Hom.id (some WalkingPair.left))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.right.none.term\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 (cospan f g \u22d9 F).map (term WalkingPair.right) \u226b\n (Option.casesOn none (Iso.refl ((cospan f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((cospan f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (cospan (F.map f) (F.map g)).map (term WalkingPair.right)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 (cospan f g \u22d9 F).map (WidePullbackShape.Hom.id (some WalkingPair.right)) \u226b\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((cospan f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((cospan f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((cospan f g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (cospan (F.map f) (F.map g)).map (WidePullbackShape.Hom.id (some WalkingPair.right))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 F.map ((cospan f g).map (\ud835\udfd9 none)) \u226b \ud835\udfd9 (F.obj Z) = \ud835\udfd9 (F.obj Z) \u226b (cospan (F.map f) (F.map g)).map (\ud835\udfd9 none)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.left.none.term\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 F.map f \u226b \ud835\udfd9 (F.obj Z) = \ud835\udfd9 (F.obj X) \u226b F.map f\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 F.map ((cospan f g).map (\ud835\udfd9 (some WalkingPair.left))) \u226b \ud835\udfd9 (F.obj X) =\n \ud835\udfd9 (F.obj X) \u226b (cospan (F.map f) (F.map g)).map (\ud835\udfd9 (some WalkingPair.left))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.right.none.term\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 F.map g \u226b \ud835\udfd9 (F.obj Z) = \ud835\udfd9 (F.obj Y) \u226b F.map g\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 F.map ((cospan f g).map (\ud835\udfd9 (some WalkingPair.right))) \u226b \ud835\udfd9 (F.obj Y) =\n \ud835\udfd9 (F.obj Y) \u226b (cospan (F.map f) (F.map g)).map (\ud835\udfd9 (some WalkingPair.right))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 (X_1 : WalkingSpan) \u2192 (span f g \u22d9 F).obj X_1 \u2245 (span (F.map f) (F.map g)).obj X_1\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9)\n[GOAL]\ncase none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 (span f g \u22d9 F).obj none \u2245 (span (F.map f) (F.map g)).obj none\n[PROOFSTEP]\nexact Iso.refl _\n[GOAL]\ncase some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 (span f g \u22d9 F).obj (some WalkingPair.left) \u2245 (span (F.map f) (F.map g)).obj (some WalkingPair.left)\n[PROOFSTEP]\nexact Iso.refl _\n[GOAL]\ncase some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 (span f g \u22d9 F).obj (some WalkingPair.right) \u2245 (span (F.map f) (F.map g)).obj (some WalkingPair.right)\n[PROOFSTEP]\nexact Iso.refl _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 \u2200 {X_1 Y_1 : WalkingSpan} (f_1 : X_1 \u27f6 Y_1),\n (span f g \u22d9 F).map f_1 \u226b\n (Option.casesOn Y_1 (Iso.refl ((span f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn X_1 (Iso.refl ((span f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (span (F.map f) (F.map g)).map f_1\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9) (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9) f\n[GOAL]\ncase none.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf : none \u27f6 none\n\u22a2 (span f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn none (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn none (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (span (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf : none \u27f6 some WalkingPair.left\n\u22a2 (span f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn none (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (span (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf : none \u27f6 some WalkingPair.right\n\u22a2 (span f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn none (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (span (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf : some WalkingPair.left \u27f6 none\n\u22a2 (span f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn none (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (span (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf : some WalkingPair.left \u27f6 some WalkingPair.left\n\u22a2 (span f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (span (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf : some WalkingPair.left \u27f6 some WalkingPair.right\n\u22a2 (span f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (span (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf : some WalkingPair.right \u27f6 none\n\u22a2 (span f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn none (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (span (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf : some WalkingPair.right \u27f6 some WalkingPair.left\n\u22a2 (span f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (span (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf : some WalkingPair.right \u27f6 some WalkingPair.right\n\u22a2 (span f\u271d g \u22d9 F).map f \u226b\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((span f\u271d g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f\u271d g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (span (F.map f\u271d) (F.map g)).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 (span f g \u22d9 F).map (WidePushoutShape.Hom.id none) \u226b\n (Option.casesOn none (Iso.refl ((span f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn none (Iso.refl ((span f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (span (F.map f) (F.map g)).map (WidePushoutShape.Hom.id none)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.some.left.init\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 (span f g \u22d9 F).map (init WalkingPair.left) \u226b\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((span f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn none (Iso.refl ((span f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (span (F.map f) (F.map g)).map (init WalkingPair.left)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.some.right.init\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 (span f g \u22d9 F).map (init WalkingPair.right) \u226b\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((span f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn none (Iso.refl ((span f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (span (F.map f) (F.map g)).map (init WalkingPair.right)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 (span f g \u22d9 F).map (WidePushoutShape.Hom.id (some WalkingPair.left)) \u226b\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((span f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.left) (Iso.refl ((span f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (span (F.map f) (F.map g)).map (WidePushoutShape.Hom.id (some WalkingPair.left))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 (span f g \u22d9 F).map (WidePushoutShape.Hom.id (some WalkingPair.right)) \u226b\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((span f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.right)))).hom =\n (Option.casesOn (some WalkingPair.right) (Iso.refl ((span f g \u22d9 F).obj none)) fun val =>\n WalkingPair.casesOn val (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.left)))\n (Iso.refl ((span f g \u22d9 F).obj (some WalkingPair.right)))).hom \u226b\n (span (F.map f) (F.map g)).map (WidePushoutShape.Hom.id (some WalkingPair.right))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 F.map ((span f g).map (\ud835\udfd9 none)) \u226b \ud835\udfd9 (F.obj X) = \ud835\udfd9 (F.obj X) \u226b (span (F.map f) (F.map g)).map (\ud835\udfd9 none)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase none.some.left.init\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 F.map f \u226b \ud835\udfd9 (F.obj Y) = \ud835\udfd9 (F.obj X) \u226b F.map f\n[PROOFSTEP]\nsimp\n[GOAL]\ncase none.some.right.init\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 F.map g \u226b \ud835\udfd9 (F.obj Z) = \ud835\udfd9 (F.obj X) \u226b F.map g\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 F.map ((span f g).map (\ud835\udfd9 (some WalkingPair.left))) \u226b \ud835\udfd9 (F.obj Y) =\n \ud835\udfd9 (F.obj Y) \u226b (span (F.map f) (F.map g)).map (\ud835\udfd9 (some WalkingPair.left))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 F.map ((span f g).map (\ud835\udfd9 (some WalkingPair.right))) \u226b \ud835\udfd9 (F.obj Z) =\n \ud835\udfd9 (F.obj Z) \u226b (span (F.map f) (F.map g)).map (\ud835\udfd9 (some WalkingPair.right))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (X_1 : WalkingCospan) \u2192 (cospan f g).obj X_1 \u2245 (cospan f' g').obj X_1\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9)\n[GOAL]\ncase none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (cospan f g).obj none \u2245 (cospan f' g').obj none\ncase some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (cospan f g).obj (some WalkingPair.left) \u2245 (cospan f' g').obj (some WalkingPair.left)\ncase some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (cospan f g).obj (some WalkingPair.right) \u2245 (cospan f' g').obj (some WalkingPair.right)\n[PROOFSTEP]\nexacts [iZ, iX, iY]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 \u2200 {X_1 Y_1 : WalkingCospan} (f_1 : X_1 \u27f6 Y_1),\n (cospan f g).map f_1 \u226b (Option.casesOn Y_1 iZ fun val => WalkingPair.casesOn val iX iY).hom =\n (Option.casesOn X_1 iZ fun val => WalkingPair.casesOn val iX iY).hom \u226b (cospan f' g').map f_1\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9) (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9) f\n[GOAL]\ncase none.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\nf : none \u27f6 none\n\u22a2 (cospan f\u271d g).map f \u226b (Option.casesOn none iZ fun val => WalkingPair.casesOn val iX iY).hom =\n (Option.casesOn none iZ fun val => WalkingPair.casesOn val iX iY).hom \u226b (cospan f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\nf : none \u27f6 some WalkingPair.left\n\u22a2 (cospan f\u271d g).map f \u226b (Option.casesOn (some WalkingPair.left) iZ fun val => WalkingPair.casesOn val iX iY).hom =\n (Option.casesOn none iZ fun val => WalkingPair.casesOn val iX iY).hom \u226b (cospan f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\nf : none \u27f6 some WalkingPair.right\n\u22a2 (cospan f\u271d g).map f \u226b (Option.casesOn (some WalkingPair.right) iZ fun val => WalkingPair.casesOn val iX iY).hom =\n (Option.casesOn none iZ fun val => WalkingPair.casesOn val iX iY).hom \u226b (cospan f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\nf : some WalkingPair.left \u27f6 none\n\u22a2 (cospan f\u271d g).map f \u226b (Option.casesOn none iZ fun val => WalkingPair.casesOn val iX iY).hom =\n (Option.casesOn (some WalkingPair.left) iZ fun val => WalkingPair.casesOn val iX iY).hom \u226b (cospan f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\nf : some WalkingPair.left \u27f6 some WalkingPair.left\n\u22a2 (cospan f\u271d g).map f \u226b (Option.casesOn (some WalkingPair.left) iZ fun val => WalkingPair.casesOn val iX iY).hom =\n (Option.casesOn (some WalkingPair.left) iZ fun val => WalkingPair.casesOn val iX iY).hom \u226b (cospan f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\nf : some WalkingPair.left \u27f6 some WalkingPair.right\n\u22a2 (cospan f\u271d g).map f \u226b (Option.casesOn (some WalkingPair.right) iZ fun val => WalkingPair.casesOn val iX iY).hom =\n (Option.casesOn (some WalkingPair.left) iZ fun val => WalkingPair.casesOn val iX iY).hom \u226b (cospan f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\nf : some WalkingPair.right \u27f6 none\n\u22a2 (cospan f\u271d g).map f \u226b (Option.casesOn none iZ fun val => WalkingPair.casesOn val iX iY).hom =\n (Option.casesOn (some WalkingPair.right) iZ fun val => WalkingPair.casesOn val iX iY).hom \u226b (cospan f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\nf : some WalkingPair.right \u27f6 some WalkingPair.left\n\u22a2 (cospan f\u271d g).map f \u226b (Option.casesOn (some WalkingPair.left) iZ fun val => WalkingPair.casesOn val iX iY).hom =\n (Option.casesOn (some WalkingPair.right) iZ fun val => WalkingPair.casesOn val iX iY).hom \u226b (cospan f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\nf : some WalkingPair.right \u27f6 some WalkingPair.right\n\u22a2 (cospan f\u271d g).map f \u226b (Option.casesOn (some WalkingPair.right) iZ fun val => WalkingPair.casesOn val iX iY).hom =\n (Option.casesOn (some WalkingPair.right) iZ fun val => WalkingPair.casesOn val iX iY).hom \u226b (cospan f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (cospan f g).map (WidePullbackShape.Hom.id none) \u226b\n (Option.casesOn none iZ fun val => WalkingPair.casesOn val iX iY).hom =\n (Option.casesOn none iZ fun val => WalkingPair.casesOn val iX iY).hom \u226b\n (cospan f' g').map (WidePullbackShape.Hom.id none)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.left.none.term\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (cospan f g).map (term WalkingPair.left) \u226b (Option.casesOn none iZ fun val => WalkingPair.casesOn val iX iY).hom =\n (Option.casesOn (some WalkingPair.left) iZ fun val => WalkingPair.casesOn val iX iY).hom \u226b\n (cospan f' g').map (term WalkingPair.left)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (cospan f g).map (WidePullbackShape.Hom.id (some WalkingPair.left)) \u226b\n (Option.casesOn (some WalkingPair.left) iZ fun val => WalkingPair.casesOn val iX iY).hom =\n (Option.casesOn (some WalkingPair.left) iZ fun val => WalkingPair.casesOn val iX iY).hom \u226b\n (cospan f' g').map (WidePullbackShape.Hom.id (some WalkingPair.left))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.right.none.term\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (cospan f g).map (term WalkingPair.right) \u226b (Option.casesOn none iZ fun val => WalkingPair.casesOn val iX iY).hom =\n (Option.casesOn (some WalkingPair.right) iZ fun val => WalkingPair.casesOn val iX iY).hom \u226b\n (cospan f' g').map (term WalkingPair.right)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (cospan f g).map (WidePullbackShape.Hom.id (some WalkingPair.right)) \u226b\n (Option.casesOn (some WalkingPair.right) iZ fun val => WalkingPair.casesOn val iX iY).hom =\n (Option.casesOn (some WalkingPair.right) iZ fun val => WalkingPair.casesOn val iX iY).hom \u226b\n (cospan f' g').map (WidePullbackShape.Hom.id (some WalkingPair.right))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (cospan f g).map (\ud835\udfd9 none) \u226b iZ.hom = iZ.hom \u226b (cospan f' g').map (\ud835\udfd9 none)\n[PROOFSTEP]\nsimp [wf, wg]\n[GOAL]\ncase some.left.none.term\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 f \u226b iZ.hom = iX.hom \u226b f'\n[PROOFSTEP]\nsimp [wf, wg]\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (cospan f g).map (\ud835\udfd9 (some WalkingPair.left)) \u226b iX.hom = iX.hom \u226b (cospan f' g').map (\ud835\udfd9 (some WalkingPair.left))\n[PROOFSTEP]\nsimp [wf, wg]\n[GOAL]\ncase some.right.none.term\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 g \u226b iZ.hom = iY.hom \u226b g'\n[PROOFSTEP]\nsimp [wf, wg]\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (cospan f g).map (\ud835\udfd9 (some WalkingPair.right)) \u226b iY.hom = iY.hom \u226b (cospan f' g').map (\ud835\udfd9 (some WalkingPair.right))\n[PROOFSTEP]\nsimp [wf, wg]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (cospanExt iX iY iZ wf wg).app WalkingCospan.left = iX\n[PROOFSTEP]\ndsimp [cospanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (cospanExt iX iY iZ wf wg).app WalkingCospan.right = iY\n[PROOFSTEP]\ndsimp [cospanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (cospanExt iX iY iZ wf wg).app WalkingCospan.one = iZ\n[PROOFSTEP]\ndsimp [cospanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 NatTrans.app (cospanExt iX iY iZ wf wg).hom WalkingCospan.left = iX.hom\n[PROOFSTEP]\ndsimp [cospanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 NatTrans.app (cospanExt iX iY iZ wf wg).hom WalkingCospan.right = iY.hom\n[PROOFSTEP]\ndsimp [cospanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 NatTrans.app (cospanExt iX iY iZ wf wg).hom WalkingCospan.one = iZ.hom\n[PROOFSTEP]\ndsimp [cospanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 NatTrans.app (cospanExt iX iY iZ wf wg).inv WalkingCospan.left = iX.inv\n[PROOFSTEP]\ndsimp [cospanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 NatTrans.app (cospanExt iX iY iZ wf wg).inv WalkingCospan.right = iY.inv\n[PROOFSTEP]\ndsimp [cospanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 NatTrans.app (cospanExt iX iY iZ wf wg).inv WalkingCospan.one = iZ.inv\n[PROOFSTEP]\ndsimp [cospanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (X_1 : WalkingSpan) \u2192 (span f g).obj X_1 \u2245 (span f' g').obj X_1\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9)\n[GOAL]\ncase none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (span f g).obj none \u2245 (span f' g').obj none\ncase some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (span f g).obj (some WalkingPair.left) \u2245 (span f' g').obj (some WalkingPair.left)\ncase some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (span f g).obj (some WalkingPair.right) \u2245 (span f' g').obj (some WalkingPair.right)\n[PROOFSTEP]\nexacts [iX, iY, iZ]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 \u2200 {X_1 Y_1 : WalkingSpan} (f_1 : X_1 \u27f6 Y_1),\n (span f g).map f_1 \u226b (Option.casesOn Y_1 iX fun val => WalkingPair.casesOn val iY iZ).hom =\n (Option.casesOn X_1 iX fun val => WalkingPair.casesOn val iY iZ).hom \u226b (span f' g').map f_1\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9) (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9) f\n[GOAL]\ncase none.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\nf : none \u27f6 none\n\u22a2 (span f\u271d g).map f \u226b (Option.casesOn none iX fun val => WalkingPair.casesOn val iY iZ).hom =\n (Option.casesOn none iX fun val => WalkingPair.casesOn val iY iZ).hom \u226b (span f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\nf : none \u27f6 some WalkingPair.left\n\u22a2 (span f\u271d g).map f \u226b (Option.casesOn (some WalkingPair.left) iX fun val => WalkingPair.casesOn val iY iZ).hom =\n (Option.casesOn none iX fun val => WalkingPair.casesOn val iY iZ).hom \u226b (span f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\nf : none \u27f6 some WalkingPair.right\n\u22a2 (span f\u271d g).map f \u226b (Option.casesOn (some WalkingPair.right) iX fun val => WalkingPair.casesOn val iY iZ).hom =\n (Option.casesOn none iX fun val => WalkingPair.casesOn val iY iZ).hom \u226b (span f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\nf : some WalkingPair.left \u27f6 none\n\u22a2 (span f\u271d g).map f \u226b (Option.casesOn none iX fun val => WalkingPair.casesOn val iY iZ).hom =\n (Option.casesOn (some WalkingPair.left) iX fun val => WalkingPair.casesOn val iY iZ).hom \u226b (span f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\nf : some WalkingPair.left \u27f6 some WalkingPair.left\n\u22a2 (span f\u271d g).map f \u226b (Option.casesOn (some WalkingPair.left) iX fun val => WalkingPair.casesOn val iY iZ).hom =\n (Option.casesOn (some WalkingPair.left) iX fun val => WalkingPair.casesOn val iY iZ).hom \u226b (span f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\nf : some WalkingPair.left \u27f6 some WalkingPair.right\n\u22a2 (span f\u271d g).map f \u226b (Option.casesOn (some WalkingPair.right) iX fun val => WalkingPair.casesOn val iY iZ).hom =\n (Option.casesOn (some WalkingPair.left) iX fun val => WalkingPair.casesOn val iY iZ).hom \u226b (span f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\nf : some WalkingPair.right \u27f6 none\n\u22a2 (span f\u271d g).map f \u226b (Option.casesOn none iX fun val => WalkingPair.casesOn val iY iZ).hom =\n (Option.casesOn (some WalkingPair.right) iX fun val => WalkingPair.casesOn val iY iZ).hom \u226b (span f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\nf : some WalkingPair.right \u27f6 some WalkingPair.left\n\u22a2 (span f\u271d g).map f \u226b (Option.casesOn (some WalkingPair.left) iX fun val => WalkingPair.casesOn val iY iZ).hom =\n (Option.casesOn (some WalkingPair.right) iX fun val => WalkingPair.casesOn val iY iZ).hom \u226b (span f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f\u271d \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\nf : some WalkingPair.right \u27f6 some WalkingPair.right\n\u22a2 (span f\u271d g).map f \u226b (Option.casesOn (some WalkingPair.right) iX fun val => WalkingPair.casesOn val iY iZ).hom =\n (Option.casesOn (some WalkingPair.right) iX fun val => WalkingPair.casesOn val iY iZ).hom \u226b (span f' g').map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (span f g).map (WidePushoutShape.Hom.id none) \u226b\n (Option.casesOn none iX fun val => WalkingPair.casesOn val iY iZ).hom =\n (Option.casesOn none iX fun val => WalkingPair.casesOn val iY iZ).hom \u226b\n (span f' g').map (WidePushoutShape.Hom.id none)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.some.left.init\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (span f g).map (init WalkingPair.left) \u226b\n (Option.casesOn (some WalkingPair.left) iX fun val => WalkingPair.casesOn val iY iZ).hom =\n (Option.casesOn none iX fun val => WalkingPair.casesOn val iY iZ).hom \u226b (span f' g').map (init WalkingPair.left)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.some.right.init\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (span f g).map (init WalkingPair.right) \u226b\n (Option.casesOn (some WalkingPair.right) iX fun val => WalkingPair.casesOn val iY iZ).hom =\n (Option.casesOn none iX fun val => WalkingPair.casesOn val iY iZ).hom \u226b (span f' g').map (init WalkingPair.right)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (span f g).map (WidePushoutShape.Hom.id (some WalkingPair.left)) \u226b\n (Option.casesOn (some WalkingPair.left) iX fun val => WalkingPair.casesOn val iY iZ).hom =\n (Option.casesOn (some WalkingPair.left) iX fun val => WalkingPair.casesOn val iY iZ).hom \u226b\n (span f' g').map (WidePushoutShape.Hom.id (some WalkingPair.left))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (span f g).map (WidePushoutShape.Hom.id (some WalkingPair.right)) \u226b\n (Option.casesOn (some WalkingPair.right) iX fun val => WalkingPair.casesOn val iY iZ).hom =\n (Option.casesOn (some WalkingPair.right) iX fun val => WalkingPair.casesOn val iY iZ).hom \u226b\n (span f' g').map (WidePushoutShape.Hom.id (some WalkingPair.right))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (span f g).map (\ud835\udfd9 none) \u226b iX.hom = iX.hom \u226b (span f' g').map (\ud835\udfd9 none)\n[PROOFSTEP]\nsimp [wf, wg]\n[GOAL]\ncase none.some.left.init\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 f \u226b iY.hom = iX.hom \u226b f'\n[PROOFSTEP]\nsimp [wf, wg]\n[GOAL]\ncase none.some.right.init\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 g \u226b iZ.hom = iX.hom \u226b g'\n[PROOFSTEP]\nsimp [wf, wg]\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (span f g).map (\ud835\udfd9 (some WalkingPair.left)) \u226b iY.hom = iY.hom \u226b (span f' g').map (\ud835\udfd9 (some WalkingPair.left))\n[PROOFSTEP]\nsimp [wf, wg]\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (span f g).map (\ud835\udfd9 (some WalkingPair.right)) \u226b iZ.hom = iZ.hom \u226b (span f' g').map (\ud835\udfd9 (some WalkingPair.right))\n[PROOFSTEP]\nsimp [wf, wg]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (spanExt iX iY iZ wf wg).app WalkingSpan.left = iY\n[PROOFSTEP]\ndsimp [spanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (spanExt iX iY iZ wf wg).app WalkingSpan.right = iZ\n[PROOFSTEP]\ndsimp [spanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (spanExt iX iY iZ wf wg).app WalkingSpan.zero = iX\n[PROOFSTEP]\ndsimp [spanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 NatTrans.app (spanExt iX iY iZ wf wg).hom WalkingSpan.left = iY.hom\n[PROOFSTEP]\ndsimp [spanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 NatTrans.app (spanExt iX iY iZ wf wg).hom WalkingSpan.right = iZ.hom\n[PROOFSTEP]\ndsimp [spanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 NatTrans.app (spanExt iX iY iZ wf wg).hom WalkingSpan.zero = iX.hom\n[PROOFSTEP]\ndsimp [spanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 NatTrans.app (spanExt iX iY iZ wf wg).inv WalkingSpan.left = iY.inv\n[PROOFSTEP]\ndsimp [spanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 NatTrans.app (spanExt iX iY iZ wf wg).inv WalkingSpan.right = iZ.inv\n[PROOFSTEP]\ndsimp [spanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Y\ng : X \u27f6 Z\nf' : X' \u27f6 Y'\ng' : X' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iY.hom\nwg : iX.hom \u226b g' = g \u226b iZ.hom\n\u22a2 NatTrans.app (spanExt iX iY iZ wf wg).inv WalkingSpan.zero = iX.inv\n[PROOFSTEP]\ndsimp [spanExt]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\n\u22a2 NatTrans.app t.\u03c0 WalkingCospan.one = fst t \u226b f\n[PROOFSTEP]\nhave w := t.\u03c0.naturality WalkingCospan.Hom.inl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nw :\n ((Functor.const WalkingCospan).obj t.pt).map inl \u226b NatTrans.app t.\u03c0 WalkingCospan.one =\n NatTrans.app t.\u03c0 WalkingCospan.left \u226b (cospan f g).map inl\n\u22a2 NatTrans.app t.\u03c0 WalkingCospan.one = fst t \u226b f\n[PROOFSTEP]\ndsimp at w \n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nw : \ud835\udfd9 t.pt \u226b NatTrans.app t.\u03c0 WalkingCospan.one = fst t \u226b f\n\u22a2 NatTrans.app t.\u03c0 WalkingCospan.one = fst t \u226b f\n[PROOFSTEP]\nsimpa using w\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nlift : (s : PullbackCone f g) \u2192 s.pt \u27f6 t.pt\nfac_left : \u2200 (s : PullbackCone f g), lift s \u226b fst t = fst s\nfac_right : \u2200 (s : PullbackCone f g), lift s \u226b snd t = snd s\nuniq :\n \u2200 (s : PullbackCone f g) (m : s.pt \u27f6 t.pt),\n (\u2200 (j : WalkingCospan), m \u226b NatTrans.app t.\u03c0 j = NatTrans.app s.\u03c0 j) \u2192 m = lift s\ns : Cone (cospan f g)\nj : WalkingCospan\n\u22a2 lift s \u226b NatTrans.app t.\u03c0 none = NatTrans.app s.\u03c0 none\n[PROOFSTEP]\nrw [\u2190 s.w inl, \u2190 t.w inl, \u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nlift : (s : PullbackCone f g) \u2192 s.pt \u27f6 t.pt\nfac_left : \u2200 (s : PullbackCone f g), lift s \u226b fst t = fst s\nfac_right : \u2200 (s : PullbackCone f g), lift s \u226b snd t = snd s\nuniq :\n \u2200 (s : PullbackCone f g) (m : s.pt \u27f6 t.pt),\n (\u2200 (j : WalkingCospan), m \u226b NatTrans.app t.\u03c0 j = NatTrans.app s.\u03c0 j) \u2192 m = lift s\ns : Cone (cospan f g)\nj : WalkingCospan\n\u22a2 (lift s \u226b NatTrans.app t.\u03c0 WalkingCospan.left) \u226b (cospan f g).map inl =\n NatTrans.app s.\u03c0 WalkingCospan.left \u226b (cospan f g).map inl\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nlift : (s : PullbackCone f g) \u2192 s.pt \u27f6 t.pt\nfac_left : \u2200 (s : PullbackCone f g), lift s \u226b fst t = fst s\nfac_right : \u2200 (s : PullbackCone f g), lift s \u226b snd t = snd s\nuniq :\n \u2200 (s : PullbackCone f g) (m : s.pt \u27f6 t.pt),\n (\u2200 (j : WalkingCospan), m \u226b NatTrans.app t.\u03c0 j = NatTrans.app s.\u03c0 j) \u2192 m = lift s\ns : Cone (cospan f g)\nj : WalkingCospan\n\u22a2 lift s \u226b NatTrans.app t.\u03c0 WalkingCospan.left = NatTrans.app s.\u03c0 WalkingCospan.left\n[PROOFSTEP]\nexact fac_left s\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\n\u22a2 \u2200 \u2983X_1 Y_1 : WalkingCospan\u2984 (f_1 : X_1 \u27f6 Y_1),\n ((Functor.const WalkingCospan).obj W).map f_1 \u226b\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) Y_1 =\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) X_1 \u226b (cospan f g).map f_1\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9) (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9) j\n[GOAL]\ncase none.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\nj : none \u27f6 none\n\u22a2 ((Functor.const WalkingCospan).obj W).map j \u226b\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) none =\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) none \u226b (cospan f g).map j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase none.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\nj : none \u27f6 some WalkingPair.left\n\u22a2 ((Functor.const WalkingCospan).obj W).map j \u226b\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.left) =\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) none \u226b (cospan f g).map j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase none.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\nj : none \u27f6 some WalkingPair.right\n\u22a2 ((Functor.const WalkingCospan).obj W).map j \u226b\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.right) =\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) none \u226b (cospan f g).map j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase some.left.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\nj : some WalkingPair.left \u27f6 none\n\u22a2 ((Functor.const WalkingCospan).obj W).map j \u226b\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) none =\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.left) \u226b\n (cospan f g).map j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase some.left.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\nj : some WalkingPair.left \u27f6 some WalkingPair.left\n\u22a2 ((Functor.const WalkingCospan).obj W).map j \u226b\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.left) =\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.left) \u226b\n (cospan f g).map j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase some.left.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\nj : some WalkingPair.left \u27f6 some WalkingPair.right\n\u22a2 ((Functor.const WalkingCospan).obj W).map j \u226b\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.right) =\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.left) \u226b\n (cospan f g).map j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase some.right.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\nj : some WalkingPair.right \u27f6 none\n\u22a2 ((Functor.const WalkingCospan).obj W).map j \u226b\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) none =\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.right) \u226b\n (cospan f g).map j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase some.right.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\nj : some WalkingPair.right \u27f6 some WalkingPair.left\n\u22a2 ((Functor.const WalkingCospan).obj W).map j \u226b\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.left) =\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.right) \u226b\n (cospan f g).map j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase some.right.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\nj : some WalkingPair.right \u27f6 some WalkingPair.right\n\u22a2 ((Functor.const WalkingCospan).obj W).map j \u226b\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.right) =\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.right) \u226b\n (cospan f g).map j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\n\u22a2 ((Functor.const WalkingCospan).obj W).map (WidePullbackShape.Hom.id none) \u226b\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) none =\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) none \u226b\n (cospan f g).map (WidePullbackShape.Hom.id none)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.left.none.term\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\n\u22a2 ((Functor.const WalkingCospan).obj W).map (term WalkingPair.left) \u226b\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) none =\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.left) \u226b\n (cospan f g).map (term WalkingPair.left)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\n\u22a2 ((Functor.const WalkingCospan).obj W).map (WidePullbackShape.Hom.id (some WalkingPair.left)) \u226b\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.left) =\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.left) \u226b\n (cospan f g).map (WidePullbackShape.Hom.id (some WalkingPair.left))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.right.none.term\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\n\u22a2 ((Functor.const WalkingCospan).obj W).map (term WalkingPair.right) \u226b\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) none =\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.right) \u226b\n (cospan f g).map (term WalkingPair.right)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\n\u22a2 ((Functor.const WalkingCospan).obj W).map (WidePullbackShape.Hom.id (some WalkingPair.right)) \u226b\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.right) =\n (fun j => Option.casesOn j (fst \u226b f) fun j' => WalkingPair.casesOn j' fst snd) (some WalkingPair.right) \u226b\n (cospan f g).map (WidePullbackShape.Hom.id (some WalkingPair.right))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\n\u22a2 \ud835\udfd9 W \u226b fst \u226b f = (fst \u226b f) \u226b (cospan f g).map (\ud835\udfd9 none)\n[PROOFSTEP]\nsimp [eq]\n[GOAL]\ncase some.left.none.term\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\n\u22a2 \ud835\udfd9 W \u226b fst \u226b f = fst \u226b f\n[PROOFSTEP]\nsimp [eq]\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\n\u22a2 \ud835\udfd9 W \u226b fst = fst \u226b (cospan f g).map (\ud835\udfd9 (some WalkingPair.left))\n[PROOFSTEP]\nsimp [eq]\n[GOAL]\ncase some.right.none.term\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\n\u22a2 \ud835\udfd9 W \u226b fst \u226b f = snd \u226b g\n[PROOFSTEP]\nsimp [eq]\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nfst : W \u27f6 X\nsnd : W \u27f6 Y\neq : fst \u226b f = snd \u226b g\n\u22a2 \ud835\udfd9 W \u226b snd = snd \u226b (cospan f g).map (\ud835\udfd9 (some WalkingPair.right))\n[PROOFSTEP]\nsimp [eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nW : C\nk l : W \u27f6 t.pt\nh\u2080 : k \u226b fst t = l \u226b fst t\nh\u2081 : k \u226b snd t = l \u226b snd t\n\u22a2 k \u226b NatTrans.app t.\u03c0 none = l \u226b NatTrans.app t.\u03c0 none\n[PROOFSTEP]\nrw [\u2190 t.w inl]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nW : C\nk l : W \u27f6 t.pt\nh\u2080 : k \u226b fst t = l \u226b fst t\nh\u2081 : k \u226b snd t = l \u226b snd t\n\u22a2 k \u226b NatTrans.app t.\u03c0 WalkingCospan.left \u226b (cospan f g).map inl =\n l \u226b NatTrans.app t.\u03c0 WalkingCospan.left \u226b (cospan f g).map inl\n[PROOFSTEP]\ndsimp [h\u2080]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nW : C\nk l : W \u27f6 t.pt\nh\u2080 : k \u226b fst t = l \u226b fst t\nh\u2081 : k \u226b snd t = l \u226b snd t\n\u22a2 k \u226b fst t \u226b f = l \u226b fst t \u226b f\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc, congrArg (\u00b7 \u226b f) h\u2080]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nht : IsLimit t\ninst\u271d : Mono f\n\u22a2 Mono (snd t)\n[PROOFSTEP]\nrefine \u27e8fun {W} h k i => IsLimit.hom_ext ht ?_ i\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nht : IsLimit t\ninst\u271d : Mono f\nW : C\nh k : W \u27f6 t.pt\ni : h \u226b snd t = k \u226b snd t\n\u22a2 h \u226b fst t = k \u226b fst t\n[PROOFSTEP]\nrw [\u2190 cancel_mono f, Category.assoc, Category.assoc, condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nht : IsLimit t\ninst\u271d : Mono f\nW : C\nh k : W \u27f6 t.pt\ni : h \u226b snd t = k \u226b snd t\n\u22a2 h \u226b snd t \u226b g = k \u226b snd t \u226b g\n[PROOFSTEP]\nhave := congrArg (\u00b7 \u226b g) i\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nht : IsLimit t\ninst\u271d : Mono f\nW : C\nh k : W \u27f6 t.pt\ni : h \u226b snd t = k \u226b snd t\nthis : (fun x => x \u226b g) (h \u226b snd t) = (fun x => x \u226b g) (k \u226b snd t)\n\u22a2 h \u226b snd t \u226b g = k \u226b snd t \u226b g\n[PROOFSTEP]\ndsimp at this \n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nht : IsLimit t\ninst\u271d : Mono f\nW : C\nh k : W \u27f6 t.pt\ni : h \u226b snd t = k \u226b snd t\nthis : (h \u226b snd t) \u226b g = (k \u226b snd t) \u226b g\n\u22a2 h \u226b snd t \u226b g = k \u226b snd t \u226b g\n[PROOFSTEP]\nrwa [Category.assoc, Category.assoc] at this \n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nht : IsLimit t\ninst\u271d : Mono g\n\u22a2 Mono (fst t)\n[PROOFSTEP]\nrefine \u27e8fun {W} h k i => IsLimit.hom_ext ht i ?_\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nht : IsLimit t\ninst\u271d : Mono g\nW : C\nh k : W \u27f6 t.pt\ni : h \u226b fst t = k \u226b fst t\n\u22a2 h \u226b snd t = k \u226b snd t\n[PROOFSTEP]\nrw [\u2190 cancel_mono g, Category.assoc, Category.assoc, \u2190 condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nht : IsLimit t\ninst\u271d : Mono g\nW : C\nh k : W \u27f6 t.pt\ni : h \u226b fst t = k \u226b fst t\n\u22a2 h \u226b fst t \u226b f = k \u226b fst t \u226b f\n[PROOFSTEP]\nhave := congrArg (\u00b7 \u226b f) i\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nht : IsLimit t\ninst\u271d : Mono g\nW : C\nh k : W \u27f6 t.pt\ni : h \u226b fst t = k \u226b fst t\nthis : (fun x => x \u226b f) (h \u226b fst t) = (fun x => x \u226b f) (k \u226b fst t)\n\u22a2 h \u226b fst t \u226b f = k \u226b fst t \u226b f\n[PROOFSTEP]\ndsimp at this \n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nht : IsLimit t\ninst\u271d : Mono g\nW : C\nh k : W \u27f6 t.pt\ni : h \u226b fst t = k \u226b fst t\nthis : (h \u226b fst t) \u226b f = (k \u226b fst t) \u226b f\n\u22a2 h \u226b fst t \u226b f = k \u226b fst t \u226b f\n[PROOFSTEP]\nrwa [Category.assoc, Category.assoc] at this \n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nt : PullbackCone f g\nht : IsLimit t\nW : C\nh : W \u27f6 X\nk : W \u27f6 Y\nw : h \u226b f = k \u226b g\n\u22a2 lift ht h k w \u226b fst t = h \u2227 lift ht h k w \u226b snd t = k\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nh : W \u27f6 X\nk : W \u27f6 Y\ncomm : h \u226b f = k \u226b g\nt : IsLimit (mk k h (_ : k \u226b g = h \u226b f))\ns : PullbackCone f g\n\u22a2 { l //\n l \u226b fst (mk h k comm) = fst s \u2227\n l \u226b snd (mk h k comm) = snd s \u2227\n \u2200 {m : s.pt \u27f6 (mk h k comm).pt}, m \u226b fst (mk h k comm) = fst s \u2192 m \u226b snd (mk h k comm) = snd s \u2192 m = l }\n[PROOFSTEP]\nrefine'\n \u27e8(IsLimit.lift' t _ _ s.condition.symm).1, (IsLimit.lift' t _ _ _).2.2, (IsLimit.lift' t _ _ _).2.1, fun m\u2081 m\u2082 =>\n t.hom_ext _\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nh : W \u27f6 X\nk : W \u27f6 Y\ncomm : h \u226b f = k \u226b g\nt : IsLimit (mk k h (_ : k \u226b g = h \u226b f))\ns : PullbackCone f g\nm\u271d : s.pt \u27f6 (mk h k comm).pt\nm\u2081 : m\u271d \u226b fst (mk h k comm) = fst s\nm\u2082 : m\u271d \u226b snd (mk h k comm) = snd s\n\u22a2 \u2200 (j : WalkingCospan),\n m\u271d \u226b NatTrans.app (mk k h (_ : k \u226b g = h \u226b f)).\u03c0 j =\n \u2191(IsLimit.lift' t (snd s) (fst s) (_ : snd s \u226b g = fst s \u226b f)) \u226b NatTrans.app (mk k h (_ : k \u226b g = h \u226b f)).\u03c0 j\n[PROOFSTEP]\napply (mk k h _).equalizer_ext\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nh : W \u27f6 X\nk : W \u27f6 Y\ncomm : h \u226b f = k \u226b g\nt : IsLimit (mk k h (_ : k \u226b g = h \u226b f))\ns : PullbackCone f g\nm\u271d : s.pt \u27f6 (mk h k comm).pt\nm\u2081 : m\u271d \u226b fst (mk h k comm) = fst s\nm\u2082 : m\u271d \u226b snd (mk h k comm) = snd s\n\u22a2 m\u271d \u226b fst (mk k h (_ : k \u226b g = h \u226b f)) =\n \u2191(IsLimit.lift' t (snd s) (fst s) (_ : snd s \u226b g = fst s \u226b f)) \u226b fst (mk k h (_ : k \u226b g = h \u226b f))\n[PROOFSTEP]\nrwa [(IsLimit.lift' t _ _ _).2.1]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nW : C\nh : W \u27f6 X\nk : W \u27f6 Y\ncomm : h \u226b f = k \u226b g\nt : IsLimit (mk k h (_ : k \u226b g = h \u226b f))\ns : PullbackCone f g\nm\u271d : s.pt \u27f6 (mk h k comm).pt\nm\u2081 : m\u271d \u226b fst (mk h k comm) = fst s\nm\u2082 : m\u271d \u226b snd (mk h k comm) = snd s\n\u22a2 m\u271d \u226b snd (mk k h (_ : k \u226b g = h \u226b f)) =\n \u2191(IsLimit.lift' t (snd s) (fst s) (_ : snd s \u226b g = fst s \u226b f)) \u226b snd (mk k h (_ : k \u226b g = h \u226b f))\n[PROOFSTEP]\nrwa [(IsLimit.lift' t _ _ _).2.2]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf : X \u27f6 Y\ninst\u271d : Mono f\ns : PullbackCone f f\n\u22a2 (fun s => fst s) s \u226b \ud835\udfd9 X = snd s\n[PROOFSTEP]\nrw [\u2190 cancel_mono f, Category.comp_id, s.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\nf : X \u27f6 Y\ninst\u271d : Mono f\ns : PullbackCone f f\nm : s.pt \u27f6 X\nm\u2081 : m \u226b \ud835\udfd9 X = fst s\nx\u271d : m \u226b \ud835\udfd9 X = snd s\n\u22a2 m = (fun s => fst s) s\n[PROOFSTEP]\nsimpa using m\u2081\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z\u271d : C\nf\u271d : X \u27f6 Z\u271d\ng\u271d : Y \u27f6 Z\u271d\nf : X \u27f6 Y\nt : IsLimit (mk (\ud835\udfd9 X) (\ud835\udfd9 X) (_ : \ud835\udfd9 X \u226b f = \ud835\udfd9 X \u226b f))\nZ : C\ng h : Z \u27f6 X\neq : g \u226b f = h \u226b f\n\u22a2 g = h\n[PROOFSTEP]\nrcases PullbackCone.IsLimit.lift' t _ _ eq with \u27e8_, rfl, rfl\u27e9\n[GOAL]\ncase mk.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z\u271d : C\nf\u271d : X \u27f6 Z\u271d\ng : Y \u27f6 Z\u271d\nf : X \u27f6 Y\nt : IsLimit (mk (\ud835\udfd9 X) (\ud835\udfd9 X) (_ : \ud835\udfd9 X \u226b f = \ud835\udfd9 X \u226b f))\nZ : C\nval\u271d : Z \u27f6 (mk (\ud835\udfd9 X) (\ud835\udfd9 X) (_ : \ud835\udfd9 X \u226b f = \ud835\udfd9 X \u226b f)).pt\neq :\n (val\u271d \u226b fst (mk (\ud835\udfd9 X) (\ud835\udfd9 X) (_ : \ud835\udfd9 X \u226b f = \ud835\udfd9 X \u226b f))) \u226b f = (val\u271d \u226b snd (mk (\ud835\udfd9 X) (\ud835\udfd9 X) (_ : \ud835\udfd9 X \u226b f = \ud835\udfd9 X \u226b f))) \u226b f\n\u22a2 val\u271d \u226b fst (mk (\ud835\udfd9 X) (\ud835\udfd9 X) (_ : \ud835\udfd9 X \u226b f = \ud835\udfd9 X \u226b f)) = val\u271d \u226b snd (mk (\ud835\udfd9 X) (\ud835\udfd9 X) (_ : \ud835\udfd9 X \u226b f = \ud835\udfd9 X \u226b f))\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : W \u27f6 Z\ninst\u271d : Mono h\nx : X \u27f6 W\ny : Y \u27f6 W\nhxh : x \u226b h = f\nhyh : y \u226b h = g\ns : PullbackCone f g\nhs : IsLimit s\n\u22a2 (fst s \u226b x) \u226b h = (snd s \u226b y) \u226b h\n[PROOFSTEP]\nsimp only [Category.assoc, hxh, hyh, s.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : W \u27f6 Z\ninst\u271d : Mono h\nx : X \u27f6 W\ny : Y \u27f6 W\nhxh : x \u226b h = f\nhyh : y \u226b h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\n\u22a2 fst t \u226b x \u226b h = snd t \u226b y \u226b h\n[PROOFSTEP]\nrw [\u2190 Category.assoc, \u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : W \u27f6 Z\ninst\u271d : Mono h\nx : X \u27f6 W\ny : Y \u27f6 W\nhxh : x \u226b h = f\nhyh : y \u226b h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\n\u22a2 (fst t \u226b x) \u226b h = (snd t \u226b y) \u226b h\n[PROOFSTEP]\napply congrArg (\u00b7 \u226b h) t.condition\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : W \u27f6 Z\ninst\u271d : Mono h\nx : X \u27f6 W\ny : Y \u27f6 W\nhxh : x \u226b h = f\nhyh : y \u226b h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\nthis : fst t \u226b x \u226b h = snd t \u226b y \u226b h\n\u22a2 fst t \u226b f = snd t \u226b g\n[PROOFSTEP]\nrw [\u2190 hxh, \u2190 hyh, this]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : W \u27f6 Z\ninst\u271d : Mono h\nx : X \u27f6 W\ny : Y \u27f6 W\nhxh : x \u226b h = f\nhyh : y \u226b h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\nthis : fst t \u226b x \u226b h = snd t \u226b y \u226b h\nm\u271d : t.pt \u27f6 (mk (fst s) (snd s) (_ : fst s \u226b x = snd s \u226b y)).pt\nhr : m\u271d \u226b fst (mk (fst s) (snd s) (_ : fst s \u226b x = snd s \u226b y)) = fst t\nhr' : m\u271d \u226b snd (mk (fst s) (snd s) (_ : fst s \u226b x = snd s \u226b y)) = snd t\n\u22a2 m\u271d = Limits.IsLimit.lift hs (mk (fst t) (snd t) (_ : fst t \u226b f = snd t \u226b g))\n[PROOFSTEP]\napply PullbackCone.IsLimit.hom_ext hs\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : W \u27f6 Z\ninst\u271d : Mono h\nx : X \u27f6 W\ny : Y \u27f6 W\nhxh : x \u226b h = f\nhyh : y \u226b h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\nthis : fst t \u226b x \u226b h = snd t \u226b y \u226b h\nm\u271d : t.pt \u27f6 (mk (fst s) (snd s) (_ : fst s \u226b x = snd s \u226b y)).pt\nhr : m\u271d \u226b fst (mk (fst s) (snd s) (_ : fst s \u226b x = snd s \u226b y)) = fst t\nhr' : m\u271d \u226b snd (mk (fst s) (snd s) (_ : fst s \u226b x = snd s \u226b y)) = snd t\n\u22a2 m\u271d \u226b fst s = Limits.IsLimit.lift hs (mk (fst t) (snd t) (_ : fst t \u226b f = snd t \u226b g)) \u226b fst s\n[PROOFSTEP]\nsimp only [PullbackCone.mk_fst, PullbackCone.mk_snd] at hr hr' \u22a2\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : W \u27f6 Z\ninst\u271d : Mono h\nx : X \u27f6 W\ny : Y \u27f6 W\nhxh : x \u226b h = f\nhyh : y \u226b h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\nthis : fst t \u226b x \u226b h = snd t \u226b y \u226b h\nm\u271d : t.pt \u27f6 (mk (fst s) (snd s) (_ : fst s \u226b x = snd s \u226b y)).pt\nhr : m\u271d \u226b fst (mk (fst s) (snd s) (_ : fst s \u226b x = snd s \u226b y)) = fst t\nhr' : m\u271d \u226b snd (mk (fst s) (snd s) (_ : fst s \u226b x = snd s \u226b y)) = snd t\n\u22a2 m\u271d \u226b snd s = Limits.IsLimit.lift hs (mk (fst t) (snd t) (_ : fst t \u226b f = snd t \u226b g)) \u226b snd s\n[PROOFSTEP]\nsimp only [PullbackCone.mk_fst, PullbackCone.mk_snd] at hr hr' \u22a2\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : W \u27f6 Z\ninst\u271d : Mono h\nx : X \u27f6 W\ny : Y \u27f6 W\nhxh : x \u226b h = f\nhyh : y \u226b h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\nthis : fst t \u226b x \u226b h = snd t \u226b y \u226b h\nm\u271d : t.pt \u27f6 (mk (fst s) (snd s) (_ : fst s \u226b x = snd s \u226b y)).pt\nhr : m\u271d \u226b fst s = fst t\nhr' : m\u271d \u226b snd s = snd t\n\u22a2 m\u271d \u226b fst s = Limits.IsLimit.lift hs (mk (fst t) (snd t) (_ : fst t \u226b f = snd t \u226b g)) \u226b fst s\n[PROOFSTEP]\nsimp only [hr, hr']\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : W \u27f6 Z\ninst\u271d : Mono h\nx : X \u27f6 W\ny : Y \u27f6 W\nhxh : x \u226b h = f\nhyh : y \u226b h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\nthis : fst t \u226b x \u226b h = snd t \u226b y \u226b h\nm\u271d : t.pt \u27f6 (mk (fst s) (snd s) (_ : fst s \u226b x = snd s \u226b y)).pt\nhr : m\u271d \u226b fst s = fst t\nhr' : m\u271d \u226b snd s = snd t\n\u22a2 m\u271d \u226b snd s = Limits.IsLimit.lift hs (mk (fst t) (snd t) (_ : fst t \u226b f = snd t \u226b g)) \u226b snd s\n[PROOFSTEP]\nsimp only [hr, hr']\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : W \u27f6 Z\ninst\u271d : Mono h\nx : X \u27f6 W\ny : Y \u27f6 W\nhxh : x \u226b h = f\nhyh : y \u226b h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\nthis : fst t \u226b x \u226b h = snd t \u226b y \u226b h\nm\u271d : t.pt \u27f6 (mk (fst s) (snd s) (_ : fst s \u226b x = snd s \u226b y)).pt\nhr : m\u271d \u226b fst s = fst t\nhr' : m\u271d \u226b snd s = snd t\n\u22a2 fst t = Limits.IsLimit.lift hs (mk (fst t) (snd t) (_ : fst t \u226b f = snd t \u226b g)) \u226b fst s\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : W \u27f6 Z\ninst\u271d : Mono h\nx : X \u27f6 W\ny : Y \u27f6 W\nhxh : x \u226b h = f\nhyh : y \u226b h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\nthis : fst t \u226b x \u226b h = snd t \u226b y \u226b h\nm\u271d : t.pt \u27f6 (mk (fst s) (snd s) (_ : fst s \u226b x = snd s \u226b y)).pt\nhr : m\u271d \u226b fst s = fst t\nhr' : m\u271d \u226b snd s = snd t\n\u22a2 snd t = Limits.IsLimit.lift hs (mk (fst t) (snd t) (_ : fst t \u226b f = snd t \u226b g)) \u226b snd s\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : W \u27f6 Z\ninst\u271d : Mono h\nx : X \u27f6 W\ny : Y \u27f6 W\nhxh : x \u226b h = f\nhyh : y \u226b h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\nthis : fst t \u226b x \u226b h = snd t \u226b y \u226b h\nm\u271d : t.pt \u27f6 (mk (fst s) (snd s) (_ : fst s \u226b x = snd s \u226b y)).pt\nhr : m\u271d \u226b fst s = fst t\nhr' : m\u271d \u226b snd s = snd t\n\u22a2 Limits.IsLimit.lift hs (mk (fst t) (snd t) (_ : fst t \u226b f = snd t \u226b g)) \u226b fst s = fst t\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : W \u27f6 Z\ninst\u271d : Mono h\nx : X \u27f6 W\ny : Y \u27f6 W\nhxh : x \u226b h = f\nhyh : y \u226b h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\nthis : fst t \u226b x \u226b h = snd t \u226b y \u226b h\nm\u271d : t.pt \u27f6 (mk (fst s) (snd s) (_ : fst s \u226b x = snd s \u226b y)).pt\nhr : m\u271d \u226b fst s = fst t\nhr' : m\u271d \u226b snd s = snd t\n\u22a2 Limits.IsLimit.lift hs (mk (fst t) (snd t) (_ : fst t \u226b f = snd t \u226b g)) \u226b snd s = snd t\n[PROOFSTEP]\nexacts [hs.fac _ WalkingCospan.left, hs.fac _ WalkingCospan.right]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 W\ng : Y \u27f6 W\ni : W \u27f6 Z\ninst\u271d : Mono i\ns : PullbackCone f g\nH : IsLimit s\n\u22a2 fst s \u226b f \u226b i = snd s \u226b g \u226b i\n[PROOFSTEP]\nrw [\u2190 Category.assoc, \u2190 Category.assoc, s.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 W\ng : Y \u27f6 W\ni : W \u27f6 Z\ninst\u271d : Mono i\ns : PullbackCone f g\nH : IsLimit s\n\u22a2 IsLimit (mk (fst s) (snd s) (_ : fst s \u226b f \u226b i = snd s \u226b g \u226b i))\n[PROOFSTEP]\napply PullbackCone.isLimitAux'\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 W\ng : Y \u27f6 W\ni : W \u27f6 Z\ninst\u271d : Mono i\ns : PullbackCone f g\nH : IsLimit s\n\u22a2 (s_1 : PullbackCone (f \u226b i) (g \u226b i)) \u2192\n { l //\n l \u226b fst (mk (fst s) (snd s) (_ : fst s \u226b f \u226b i = snd s \u226b g \u226b i)) = fst s_1 \u2227\n l \u226b snd (mk (fst s) (snd s) (_ : fst s \u226b f \u226b i = snd s \u226b g \u226b i)) = snd s_1 \u2227\n \u2200 {m : s_1.pt \u27f6 (mk (fst s) (snd s) (_ : fst s \u226b f \u226b i = snd s \u226b g \u226b i)).pt},\n m \u226b fst (mk (fst s) (snd s) (_ : fst s \u226b f \u226b i = snd s \u226b g \u226b i)) = fst s_1 \u2192\n m \u226b snd (mk (fst s) (snd s) (_ : fst s \u226b f \u226b i = snd s \u226b g \u226b i)) = snd s_1 \u2192 m = l }\n[PROOFSTEP]\nintro s\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 W\ng : Y \u27f6 W\ni : W \u27f6 Z\ninst\u271d : Mono i\ns\u271d : PullbackCone f g\nH : IsLimit s\u271d\ns : PullbackCone (f \u226b i) (g \u226b i)\n\u22a2 { l //\n l \u226b fst (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)) = fst s \u2227\n l \u226b snd (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)) = snd s \u2227\n \u2200 {m : s.pt \u27f6 (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)).pt},\n m \u226b fst (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)) = fst s \u2192\n m \u226b snd (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)) = snd s \u2192 m = l }\n[PROOFSTEP]\nrcases PullbackCone.IsLimit.lift' H s.fst s.snd ((cancel_mono i).mp (by simpa using s.condition)) with \u27e8l, h\u2081, h\u2082\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 W\ng : Y \u27f6 W\ni : W \u27f6 Z\ninst\u271d : Mono i\ns\u271d : PullbackCone f g\nH : IsLimit s\u271d\ns : PullbackCone (f \u226b i) (g \u226b i)\n\u22a2 (fst s \u226b f) \u226b i = (snd s \u226b g) \u226b i\n[PROOFSTEP]\nsimpa using s.condition\n[GOAL]\ncase create.mk.intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 W\ng : Y \u27f6 W\ni : W \u27f6 Z\ninst\u271d : Mono i\ns\u271d : PullbackCone f g\nH : IsLimit s\u271d\ns : PullbackCone (f \u226b i) (g \u226b i)\nl : s.pt \u27f6 s\u271d.pt\nh\u2081 : l \u226b fst s\u271d = fst s\nh\u2082 : l \u226b snd s\u271d = snd s\n\u22a2 { l //\n l \u226b fst (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)) = fst s \u2227\n l \u226b snd (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)) = snd s \u2227\n \u2200 {m : s.pt \u27f6 (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)).pt},\n m \u226b fst (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)) = fst s \u2192\n m \u226b snd (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)) = snd s \u2192 m = l }\n[PROOFSTEP]\nrefine' \u27e8l, h\u2081, h\u2082, _\u27e9\n[GOAL]\ncase create.mk.intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 W\ng : Y \u27f6 W\ni : W \u27f6 Z\ninst\u271d : Mono i\ns\u271d : PullbackCone f g\nH : IsLimit s\u271d\ns : PullbackCone (f \u226b i) (g \u226b i)\nl : s.pt \u27f6 s\u271d.pt\nh\u2081 : l \u226b fst s\u271d = fst s\nh\u2082 : l \u226b snd s\u271d = snd s\n\u22a2 \u2200 {m : s.pt \u27f6 (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)).pt},\n m \u226b fst (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)) = fst s \u2192\n m \u226b snd (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)) = snd s \u2192 m = l\n[PROOFSTEP]\nintro m hm\u2081 hm\u2082\n[GOAL]\ncase create.mk.intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 W\ng : Y \u27f6 W\ni : W \u27f6 Z\ninst\u271d : Mono i\ns\u271d : PullbackCone f g\nH : IsLimit s\u271d\ns : PullbackCone (f \u226b i) (g \u226b i)\nl : s.pt \u27f6 s\u271d.pt\nh\u2081 : l \u226b fst s\u271d = fst s\nh\u2082 : l \u226b snd s\u271d = snd s\nm : s.pt \u27f6 (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)).pt\nhm\u2081 : m \u226b fst (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)) = fst s\nhm\u2082 : m \u226b snd (mk (fst s\u271d) (snd s\u271d) (_ : fst s\u271d \u226b f \u226b i = snd s\u271d \u226b g \u226b i)) = snd s\n\u22a2 m = l\n[PROOFSTEP]\nexact (PullbackCone.IsLimit.hom_ext H (hm\u2081.trans h\u2081.symm) (hm\u2082.trans h\u2082.symm) : _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nt : PushoutCocone f g\n\u22a2 NatTrans.app t.\u03b9 WalkingSpan.zero = f \u226b inl t\n[PROOFSTEP]\nhave w := t.\u03b9.naturality WalkingSpan.Hom.fst\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nt : PushoutCocone f g\nw :\n (span f g).map fst \u226b NatTrans.app t.\u03b9 WalkingSpan.left =\n NatTrans.app t.\u03b9 WalkingSpan.zero \u226b ((Functor.const WalkingSpan).obj t.pt).map fst\n\u22a2 NatTrans.app t.\u03b9 WalkingSpan.zero = f \u226b inl t\n[PROOFSTEP]\ndsimp at w \n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nt : PushoutCocone f g\nw : f \u226b inl t = NatTrans.app t.\u03b9 WalkingSpan.zero \u226b \ud835\udfd9 t.pt\n\u22a2 NatTrans.app t.\u03b9 WalkingSpan.zero = f \u226b inl t\n[PROOFSTEP]\nsimpa using w.symm\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nt : PushoutCocone f g\ndesc : (s : PushoutCocone f g) \u2192 t.pt \u27f6 s.pt\nfac_left : \u2200 (s : PushoutCocone f g), inl t \u226b desc s = inl s\nfac_right : \u2200 (s : PushoutCocone f g), inr t \u226b desc s = inr s\nuniq :\n \u2200 (s : PushoutCocone f g) (m : t.pt \u27f6 s.pt),\n (\u2200 (j : WalkingSpan), NatTrans.app t.\u03b9 j \u226b m = NatTrans.app s.\u03b9 j) \u2192 m = desc s\ns : Cocone (span f g)\nj : WalkingSpan\n\u22a2 NatTrans.app t.\u03b9 none \u226b desc s = NatTrans.app s.\u03b9 none\n[PROOFSTEP]\nsimp [\u2190 s.w fst, \u2190 t.w fst, fac_left s]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 \u2200 \u2983X_1 Y_1 : WalkingSpan\u2984 (f_1 : X_1 \u27f6 Y_1),\n (span f g).map f_1 \u226b (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) Y_1 =\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) X_1 \u226b\n ((Functor.const WalkingSpan).obj W).map f_1\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9) (\u27e8\u27e9 | \u27e8\u27e8\u27e9\u27e9)\n[GOAL]\ncase none.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 \u2200 (f_1 : none \u27f6 none),\n (span f g).map f_1 \u226b (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none =\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none \u226b\n ((Functor.const WalkingSpan).obj W).map f_1\n[PROOFSTEP]\nintro f\n[GOAL]\ncase none.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 \u2200 (f_1 : none \u27f6 some WalkingPair.left),\n (span f g).map f_1 \u226b\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.left) =\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none \u226b\n ((Functor.const WalkingSpan).obj W).map f_1\n[PROOFSTEP]\nintro f\n[GOAL]\ncase none.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 \u2200 (f_1 : none \u27f6 some WalkingPair.right),\n (span f g).map f_1 \u226b\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.right) =\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none \u226b\n ((Functor.const WalkingSpan).obj W).map f_1\n[PROOFSTEP]\nintro f\n[GOAL]\ncase some.left.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 \u2200 (f_1 : some WalkingPair.left \u27f6 none),\n (span f g).map f_1 \u226b (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none =\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.left) \u226b\n ((Functor.const WalkingSpan).obj W).map f_1\n[PROOFSTEP]\nintro f\n[GOAL]\ncase some.left.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 \u2200 (f_1 : some WalkingPair.left \u27f6 some WalkingPair.left),\n (span f g).map f_1 \u226b\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.left) =\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.left) \u226b\n ((Functor.const WalkingSpan).obj W).map f_1\n[PROOFSTEP]\nintro f\n[GOAL]\ncase some.left.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 \u2200 (f_1 : some WalkingPair.left \u27f6 some WalkingPair.right),\n (span f g).map f_1 \u226b\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.right) =\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.left) \u226b\n ((Functor.const WalkingSpan).obj W).map f_1\n[PROOFSTEP]\nintro f\n[GOAL]\ncase some.right.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 \u2200 (f_1 : some WalkingPair.right \u27f6 none),\n (span f g).map f_1 \u226b (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none =\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.right) \u226b\n ((Functor.const WalkingSpan).obj W).map f_1\n[PROOFSTEP]\nintro f\n[GOAL]\ncase some.right.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 \u2200 (f_1 : some WalkingPair.right \u27f6 some WalkingPair.left),\n (span f g).map f_1 \u226b\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.left) =\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.right) \u226b\n ((Functor.const WalkingSpan).obj W).map f_1\n[PROOFSTEP]\nintro f\n[GOAL]\ncase some.right.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 \u2200 (f_1 : some WalkingPair.right \u27f6 some WalkingPair.right),\n (span f g).map f_1 \u226b\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.right) =\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.right) \u226b\n ((Functor.const WalkingSpan).obj W).map f_1\n[PROOFSTEP]\nintro f\n[GOAL]\ncase none.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f\u271d \u226b inl = g \u226b inr\nf : none \u27f6 none\n\u22a2 (span f\u271d g).map f \u226b (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none =\n (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none \u226b\n ((Functor.const WalkingSpan).obj W).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f\u271d \u226b inl = g \u226b inr\nf : none \u27f6 some WalkingPair.left\n\u22a2 (span f\u271d g).map f \u226b\n (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.left) =\n (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none \u226b\n ((Functor.const WalkingSpan).obj W).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f\u271d \u226b inl = g \u226b inr\nf : none \u27f6 some WalkingPair.right\n\u22a2 (span f\u271d g).map f \u226b\n (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.right) =\n (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none \u226b\n ((Functor.const WalkingSpan).obj W).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f\u271d \u226b inl = g \u226b inr\nf : some WalkingPair.left \u27f6 none\n\u22a2 (span f\u271d g).map f \u226b (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none =\n (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.left) \u226b\n ((Functor.const WalkingSpan).obj W).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f\u271d \u226b inl = g \u226b inr\nf : some WalkingPair.left \u27f6 some WalkingPair.left\n\u22a2 (span f\u271d g).map f \u226b\n (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.left) =\n (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.left) \u226b\n ((Functor.const WalkingSpan).obj W).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.left.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f\u271d \u226b inl = g \u226b inr\nf : some WalkingPair.left \u27f6 some WalkingPair.right\n\u22a2 (span f\u271d g).map f \u226b\n (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.right) =\n (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.left) \u226b\n ((Functor.const WalkingSpan).obj W).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f\u271d \u226b inl = g \u226b inr\nf : some WalkingPair.right \u27f6 none\n\u22a2 (span f\u271d g).map f \u226b (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none =\n (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.right) \u226b\n ((Functor.const WalkingSpan).obj W).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f\u271d \u226b inl = g \u226b inr\nf : some WalkingPair.right \u27f6 some WalkingPair.left\n\u22a2 (span f\u271d g).map f \u226b\n (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.left) =\n (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.right) \u226b\n ((Functor.const WalkingSpan).obj W).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.right.some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f\u271d \u226b inl = g \u226b inr\nf : some WalkingPair.right \u27f6 some WalkingPair.right\n\u22a2 (span f\u271d g).map f \u226b\n (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.right) =\n (fun j => Option.casesOn j (f\u271d \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.right) \u226b\n ((Functor.const WalkingSpan).obj W).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 (span f g).map (WidePushoutShape.Hom.id none) \u226b\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none =\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none \u226b\n ((Functor.const WalkingSpan).obj W).map (WidePushoutShape.Hom.id none)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.some.left.init\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 (span f g).map (init WalkingPair.left) \u226b\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.left) =\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none \u226b\n ((Functor.const WalkingSpan).obj W).map (init WalkingPair.left)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.some.right.init\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 (span f g).map (init WalkingPair.right) \u226b\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.right) =\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) none \u226b\n ((Functor.const WalkingSpan).obj W).map (init WalkingPair.right)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 (span f g).map (WidePushoutShape.Hom.id (some WalkingPair.left)) \u226b\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.left) =\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.left) \u226b\n ((Functor.const WalkingSpan).obj W).map (WidePushoutShape.Hom.id (some WalkingPair.left))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 (span f g).map (WidePushoutShape.Hom.id (some WalkingPair.right)) \u226b\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.right) =\n (fun j => Option.casesOn j (f \u226b inl) fun j' => WalkingPair.casesOn j' inl inr) (some WalkingPair.right) \u226b\n ((Functor.const WalkingSpan).obj W).map (WidePushoutShape.Hom.id (some WalkingPair.right))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.none.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 (span f g).map (\ud835\udfd9 none) \u226b f \u226b inl = (f \u226b inl) \u226b \ud835\udfd9 W\n[PROOFSTEP]\naesop\n[GOAL]\ncase none.some.left.init\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 f \u226b inl = (f \u226b inl) \u226b \ud835\udfd9 W\n[PROOFSTEP]\naesop\n[GOAL]\ncase none.some.right.init\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 g \u226b inr = (f \u226b inl) \u226b \ud835\udfd9 W\n[PROOFSTEP]\naesop\n[GOAL]\ncase some.left.some.left.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 (span f g).map (\ud835\udfd9 (some WalkingPair.left)) \u226b inl = inl \u226b \ud835\udfd9 W\n[PROOFSTEP]\naesop\n[GOAL]\ncase some.right.some.right.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\ninl : Y \u27f6 W\ninr : Z \u27f6 W\neq : f \u226b inl = g \u226b inr\n\u22a2 (span f g).map (\ud835\udfd9 (some WalkingPair.right)) \u226b inr = inr \u226b \ud835\udfd9 W\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nt : PushoutCocone f g\nW : C\nk l : t.pt \u27f6 W\nh\u2080 : inl t \u226b k = inl t \u226b l\nh\u2081 : inr t \u226b k = inr t \u226b l\n\u22a2 NatTrans.app t.\u03b9 none \u226b k = NatTrans.app t.\u03b9 none \u226b l\n[PROOFSTEP]\nrw [\u2190 t.w fst, Category.assoc, Category.assoc, h\u2080]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nt : PushoutCocone f g\nht : IsColimit t\nW : C\nh : Y \u27f6 W\nk : Z \u27f6 W\nw : f \u226b h = g \u226b k\n\u22a2 inl t \u226b desc ht h k w = h \u2227 inr t \u226b desc ht h k w = k\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nt : PushoutCocone f g\nht : IsColimit t\ninst\u271d : Epi f\nW : C\nh k : t.pt \u27f6 W\ni : inr t \u226b h = inr t \u226b k\n\u22a2 inl t \u226b h = inl t \u226b k\n[PROOFSTEP]\nsimp [\u2190 cancel_epi f, t.condition_assoc, i]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nt : PushoutCocone f g\nht : IsColimit t\ninst\u271d : Epi g\nW : C\nh k : t.pt \u27f6 W\ni : inl t \u226b h = inl t \u226b k\n\u22a2 inr t \u226b h = inr t \u226b k\n[PROOFSTEP]\nsimp [\u2190 cancel_epi g, \u2190 t.condition_assoc, i]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\nh : Y \u27f6 W\nk : Z \u27f6 W\ncomm : f \u226b h = g \u226b k\nt : IsColimit (mk k h (_ : g \u226b k = f \u226b h))\ns : PushoutCocone f g\n\u22a2 { l //\n inl (mk h k comm) \u226b l = inl s \u2227\n inr (mk h k comm) \u226b l = inr s \u2227\n \u2200 {m : (mk h k comm).pt \u27f6 s.pt}, inl (mk h k comm) \u226b m = inl s \u2192 inr (mk h k comm) \u226b m = inr s \u2192 m = l }\n[PROOFSTEP]\nrefine'\n \u27e8(IsColimit.desc' t _ _ s.condition.symm).1, (IsColimit.desc' t _ _ _).2.2, (IsColimit.desc' t _ _ _).2.1,\n fun m\u2081 m\u2082 => t.hom_ext _\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\nh : Y \u27f6 W\nk : Z \u27f6 W\ncomm : f \u226b h = g \u226b k\nt : IsColimit (mk k h (_ : g \u226b k = f \u226b h))\ns : PushoutCocone f g\nm\u271d : (mk h k comm).pt \u27f6 s.pt\nm\u2081 : inl (mk h k comm) \u226b m\u271d = inl s\nm\u2082 : inr (mk h k comm) \u226b m\u271d = inr s\n\u22a2 \u2200 (j : WalkingSpan),\n NatTrans.app (mk k h (_ : g \u226b k = f \u226b h)).\u03b9 j \u226b m\u271d =\n NatTrans.app (mk k h (_ : g \u226b k = f \u226b h)).\u03b9 j \u226b \u2191(IsColimit.desc' t (inr s) (inl s) (_ : g \u226b inr s = f \u226b inl s))\n[PROOFSTEP]\napply (mk k h _).coequalizer_ext\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\nh : Y \u27f6 W\nk : Z \u27f6 W\ncomm : f \u226b h = g \u226b k\nt : IsColimit (mk k h (_ : g \u226b k = f \u226b h))\ns : PushoutCocone f g\nm\u271d : (mk h k comm).pt \u27f6 s.pt\nm\u2081 : inl (mk h k comm) \u226b m\u271d = inl s\nm\u2082 : inr (mk h k comm) \u226b m\u271d = inr s\n\u22a2 inl (mk k h (_ : g \u226b k = f \u226b h)) \u226b m\u271d =\n inl (mk k h (_ : g \u226b k = f \u226b h)) \u226b \u2191(IsColimit.desc' t (inr s) (inl s) (_ : g \u226b inr s = f \u226b inl s))\n[PROOFSTEP]\nrwa [(IsColimit.desc' t _ _ _).2.1]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nW : C\nh : Y \u27f6 W\nk : Z \u27f6 W\ncomm : f \u226b h = g \u226b k\nt : IsColimit (mk k h (_ : g \u226b k = f \u226b h))\ns : PushoutCocone f g\nm\u271d : (mk h k comm).pt \u27f6 s.pt\nm\u2081 : inl (mk h k comm) \u226b m\u271d = inl s\nm\u2082 : inr (mk h k comm) \u226b m\u271d = inr s\n\u22a2 inr (mk k h (_ : g \u226b k = f \u226b h)) \u226b m\u271d =\n inr (mk k h (_ : g \u226b k = f \u226b h)) \u226b \u2191(IsColimit.desc' t (inr s) (inl s) (_ : g \u226b inr s = f \u226b inl s))\n[PROOFSTEP]\nrwa [(IsColimit.desc' t _ _ _).2.2]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf : X \u27f6 Y\ninst\u271d : Epi f\ns : PushoutCocone f f\n\u22a2 \ud835\udfd9 Y \u226b (fun s => inl s) s = inr s\n[PROOFSTEP]\nrw [\u2190 cancel_epi f, Category.id_comp, s.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\nf : X \u27f6 Y\ninst\u271d : Epi f\ns : PushoutCocone f f\nm : Y \u27f6 s.pt\nm\u2081 : \ud835\udfd9 Y \u226b m = inl s\nx\u271d : \ud835\udfd9 Y \u226b m = inr s\n\u22a2 m = (fun s => inl s) s\n[PROOFSTEP]\nsimpa using m\u2081\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z\u271d : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\u271d\nf : X \u27f6 Y\nt : IsColimit (mk (\ud835\udfd9 Y) (\ud835\udfd9 Y) (_ : f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y))\nZ : C\ng h : Y \u27f6 Z\neq : f \u226b g = f \u226b h\n\u22a2 g = h\n[PROOFSTEP]\nrcases PushoutCocone.IsColimit.desc' t _ _ eq with \u27e8_, rfl, rfl\u27e9\n[GOAL]\ncase mk.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z\u271d : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\u271d\nf : X \u27f6 Y\nt : IsColimit (mk (\ud835\udfd9 Y) (\ud835\udfd9 Y) (_ : f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y))\nZ : C\nval\u271d : (mk (\ud835\udfd9 Y) (\ud835\udfd9 Y) (_ : f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y)).pt \u27f6 Z\neq : f \u226b inl (mk (\ud835\udfd9 Y) (\ud835\udfd9 Y) (_ : f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y)) \u226b val\u271d = f \u226b inr (mk (\ud835\udfd9 Y) (\ud835\udfd9 Y) (_ : f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y)) \u226b val\u271d\n\u22a2 inl (mk (\ud835\udfd9 Y) (\ud835\udfd9 Y) (_ : f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y)) \u226b val\u271d = inr (mk (\ud835\udfd9 Y) (\ud835\udfd9 Y) (_ : f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y)) \u226b val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : X \u27f6 W\ninst\u271d : Epi h\nx : W \u27f6 Y\ny : W \u27f6 Z\nhhx : h \u226b x = f\nhhy : h \u226b y = g\ns : PushoutCocone f g\nhs : IsColimit s\n\u22a2 h \u226b x \u226b inl s = f \u226b inl s\n[PROOFSTEP]\nrw [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : X \u27f6 W\ninst\u271d : Epi h\nx : W \u27f6 Y\ny : W \u27f6 Z\nhhx : h \u226b x = f\nhhy : h \u226b y = g\ns : PushoutCocone f g\nhs : IsColimit s\n\u22a2 (h \u226b x) \u226b inl s = f \u226b inl s\n[PROOFSTEP]\napply congrArg (\u00b7 \u226b inl s) hhx\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : X \u27f6 W\ninst\u271d : Epi h\nx : W \u27f6 Y\ny : W \u27f6 Z\nhhx : h \u226b x = f\nhhy : h \u226b y = g\ns : PushoutCocone f g\nhs : IsColimit s\nreassoc\u2081 : h \u226b x \u226b inl s = f \u226b inl s\n\u22a2 h \u226b y \u226b inr s = g \u226b inr s\n[PROOFSTEP]\nrw [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : X \u27f6 W\ninst\u271d : Epi h\nx : W \u27f6 Y\ny : W \u27f6 Z\nhhx : h \u226b x = f\nhhy : h \u226b y = g\ns : PushoutCocone f g\nhs : IsColimit s\nreassoc\u2081 : h \u226b x \u226b inl s = f \u226b inl s\n\u22a2 (h \u226b y) \u226b inr s = g \u226b inr s\n[PROOFSTEP]\napply congrArg (\u00b7 \u226b inr s) hhy\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : X \u27f6 W\ninst\u271d : Epi h\nx : W \u27f6 Y\ny : W \u27f6 Z\nhhx : h \u226b x = f\nhhy : h \u226b y = g\ns : PushoutCocone f g\nhs : IsColimit s\nreassoc\u2081 : h \u226b x \u226b inl s = f \u226b inl s\nreassoc\u2082 : h \u226b y \u226b inr s = g \u226b inr s\n\u22a2 h \u226b x \u226b inl s = h \u226b y \u226b inr s\n[PROOFSTEP]\nrw [reassoc\u2081, reassoc\u2082, s.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : X \u27f6 W\ninst\u271d : Epi h\nx : W \u27f6 Y\ny : W \u27f6 Z\nhhx : h \u226b x = f\nhhy : h \u226b y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\n\u22a2 f \u226b inl t = g \u226b inr t\n[PROOFSTEP]\nrw [\u2190 hhx, \u2190 hhy, Category.assoc, Category.assoc, t.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : X \u27f6 W\ninst\u271d : Epi h\nx : W \u27f6 Y\ny : W \u27f6 Z\nhhx : h \u226b x = f\nhhy : h \u226b y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\nm\u271d : (mk (inl s) (inr s) (_ : x \u226b inl s = y \u226b inr s)).pt \u27f6 t.pt\nhr : inl (mk (inl s) (inr s) (_ : x \u226b inl s = y \u226b inr s)) \u226b m\u271d = inl t\nhr' : inr (mk (inl s) (inr s) (_ : x \u226b inl s = y \u226b inr s)) \u226b m\u271d = inr t\n\u22a2 m\u271d = Limits.IsColimit.desc hs (mk (inl t) (inr t) (_ : f \u226b inl t = g \u226b inr t))\n[PROOFSTEP]\napply PushoutCocone.IsColimit.hom_ext hs\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : X \u27f6 W\ninst\u271d : Epi h\nx : W \u27f6 Y\ny : W \u27f6 Z\nhhx : h \u226b x = f\nhhy : h \u226b y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\nm\u271d : (mk (inl s) (inr s) (_ : x \u226b inl s = y \u226b inr s)).pt \u27f6 t.pt\nhr : inl (mk (inl s) (inr s) (_ : x \u226b inl s = y \u226b inr s)) \u226b m\u271d = inl t\nhr' : inr (mk (inl s) (inr s) (_ : x \u226b inl s = y \u226b inr s)) \u226b m\u271d = inr t\n\u22a2 inl s \u226b m\u271d = inl s \u226b Limits.IsColimit.desc hs (mk (inl t) (inr t) (_ : f \u226b inl t = g \u226b inr t))\n[PROOFSTEP]\nsimp only [PushoutCocone.mk_inl, PushoutCocone.mk_inr] at hr hr' \u22a2\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : X \u27f6 W\ninst\u271d : Epi h\nx : W \u27f6 Y\ny : W \u27f6 Z\nhhx : h \u226b x = f\nhhy : h \u226b y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\nm\u271d : (mk (inl s) (inr s) (_ : x \u226b inl s = y \u226b inr s)).pt \u27f6 t.pt\nhr : inl s \u226b m\u271d = inl t\nhr' : inr s \u226b m\u271d = inr t\n\u22a2 inl s \u226b m\u271d = inl s \u226b Limits.IsColimit.desc hs (mk (inl t) (inr t) (_ : f \u226b inl t = g \u226b inr t))\n[PROOFSTEP]\nsimp only [hr, hr']\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : X \u27f6 W\ninst\u271d : Epi h\nx : W \u27f6 Y\ny : W \u27f6 Z\nhhx : h \u226b x = f\nhhy : h \u226b y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\nm\u271d : (mk (inl s) (inr s) (_ : x \u226b inl s = y \u226b inr s)).pt \u27f6 t.pt\nhr : inl s \u226b m\u271d = inl t\nhr' : inr s \u226b m\u271d = inr t\n\u22a2 inl t = inl s \u226b Limits.IsColimit.desc hs (mk (inl t) (inr t) (_ : f \u226b inl t = g \u226b inr t))\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : X \u27f6 W\ninst\u271d : Epi h\nx : W \u27f6 Y\ny : W \u27f6 Z\nhhx : h \u226b x = f\nhhy : h \u226b y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\nm\u271d : (mk (inl s) (inr s) (_ : x \u226b inl s = y \u226b inr s)).pt \u27f6 t.pt\nhr : inl s \u226b m\u271d = inl t\nhr' : inr s \u226b m\u271d = inr t\n\u22a2 inl s \u226b Limits.IsColimit.desc hs (mk (inl t) (inr t) (_ : f \u226b inl t = g \u226b inr t)) = inl t\n[PROOFSTEP]\nexact hs.fac _ WalkingSpan.left\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : X \u27f6 W\ninst\u271d : Epi h\nx : W \u27f6 Y\ny : W \u27f6 Z\nhhx : h \u226b x = f\nhhy : h \u226b y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\nm\u271d : (mk (inl s) (inr s) (_ : x \u226b inl s = y \u226b inr s)).pt \u27f6 t.pt\nhr : inl (mk (inl s) (inr s) (_ : x \u226b inl s = y \u226b inr s)) \u226b m\u271d = inl t\nhr' : inr (mk (inl s) (inr s) (_ : x \u226b inl s = y \u226b inr s)) \u226b m\u271d = inr t\n\u22a2 inr s \u226b m\u271d = inr s \u226b Limits.IsColimit.desc hs (mk (inl t) (inr t) (_ : f \u226b inl t = g \u226b inr t))\n[PROOFSTEP]\nsimp only [PushoutCocone.mk_inl, PushoutCocone.mk_inr] at hr hr' \u22a2\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : X \u27f6 W\ninst\u271d : Epi h\nx : W \u27f6 Y\ny : W \u27f6 Z\nhhx : h \u226b x = f\nhhy : h \u226b y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\nm\u271d : (mk (inl s) (inr s) (_ : x \u226b inl s = y \u226b inr s)).pt \u27f6 t.pt\nhr : inl s \u226b m\u271d = inl t\nhr' : inr s \u226b m\u271d = inr t\n\u22a2 inr s \u226b m\u271d = inr s \u226b Limits.IsColimit.desc hs (mk (inl t) (inr t) (_ : f \u226b inl t = g \u226b inr t))\n[PROOFSTEP]\nsimp only [hr, hr']\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : X \u27f6 W\ninst\u271d : Epi h\nx : W \u27f6 Y\ny : W \u27f6 Z\nhhx : h \u226b x = f\nhhy : h \u226b y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\nm\u271d : (mk (inl s) (inr s) (_ : x \u226b inl s = y \u226b inr s)).pt \u27f6 t.pt\nhr : inl s \u226b m\u271d = inl t\nhr' : inr s \u226b m\u271d = inr t\n\u22a2 inr t = inr s \u226b Limits.IsColimit.desc hs (mk (inl t) (inr t) (_ : f \u226b inl t = g \u226b inr t))\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : X \u27f6 W\ninst\u271d : Epi h\nx : W \u27f6 Y\ny : W \u27f6 Z\nhhx : h \u226b x = f\nhhy : h \u226b y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\nm\u271d : (mk (inl s) (inr s) (_ : x \u226b inl s = y \u226b inr s)).pt \u27f6 t.pt\nhr : inl s \u226b m\u271d = inl t\nhr' : inr s \u226b m\u271d = inr t\n\u22a2 inr s \u226b Limits.IsColimit.desc hs (mk (inl t) (inr t) (_ : f \u226b inl t = g \u226b inr t)) = inr t\n[PROOFSTEP]\nexact hs.fac _ WalkingSpan.right\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : W \u27f6 X\ninst\u271d : Epi h\ns : PushoutCocone f g\nH : IsColimit s\n\u22a2 (h \u226b f) \u226b inl s = (h \u226b g) \u226b inr s\n[PROOFSTEP]\nrw [Category.assoc, Category.assoc, s.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : W \u27f6 X\ninst\u271d : Epi h\ns : PushoutCocone f g\nH : IsColimit s\n\u22a2 IsColimit (mk (inl s) (inr s) (_ : (h \u226b f) \u226b inl s = (h \u226b g) \u226b inr s))\n[PROOFSTEP]\napply PushoutCocone.isColimitAux'\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : W \u27f6 X\ninst\u271d : Epi h\ns : PushoutCocone f g\nH : IsColimit s\n\u22a2 (s_1 : PushoutCocone (h \u226b f) (h \u226b g)) \u2192\n { l //\n inl (mk (inl s) (inr s) (_ : (h \u226b f) \u226b inl s = (h \u226b g) \u226b inr s)) \u226b l = inl s_1 \u2227\n inr (mk (inl s) (inr s) (_ : (h \u226b f) \u226b inl s = (h \u226b g) \u226b inr s)) \u226b l = inr s_1 \u2227\n \u2200 {m : (mk (inl s) (inr s) (_ : (h \u226b f) \u226b inl s = (h \u226b g) \u226b inr s)).pt \u27f6 s_1.pt},\n inl (mk (inl s) (inr s) (_ : (h \u226b f) \u226b inl s = (h \u226b g) \u226b inr s)) \u226b m = inl s_1 \u2192\n inr (mk (inl s) (inr s) (_ : (h \u226b f) \u226b inl s = (h \u226b g) \u226b inr s)) \u226b m = inr s_1 \u2192 m = l }\n[PROOFSTEP]\nintro s\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : W \u27f6 X\ninst\u271d : Epi h\ns\u271d : PushoutCocone f g\nH : IsColimit s\u271d\ns : PushoutCocone (h \u226b f) (h \u226b g)\n\u22a2 { l //\n inl (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)) \u226b l = inl s \u2227\n inr (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)) \u226b l = inr s \u2227\n \u2200 {m : (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)).pt \u27f6 s.pt},\n inl (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)) \u226b m = inl s \u2192\n inr (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)) \u226b m = inr s \u2192 m = l }\n[PROOFSTEP]\nrcases PushoutCocone.IsColimit.desc' H s.inl s.inr ((cancel_epi h).mp (by simpa using s.condition)) with \u27e8l, h\u2081, h\u2082\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : W \u27f6 X\ninst\u271d : Epi h\ns\u271d : PushoutCocone f g\nH : IsColimit s\u271d\ns : PushoutCocone (h \u226b f) (h \u226b g)\n\u22a2 h \u226b f \u226b inl s = h \u226b g \u226b inr s\n[PROOFSTEP]\nsimpa using s.condition\n[GOAL]\ncase create.mk.intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : W \u27f6 X\ninst\u271d : Epi h\ns\u271d : PushoutCocone f g\nH : IsColimit s\u271d\ns : PushoutCocone (h \u226b f) (h \u226b g)\nl : s\u271d.pt \u27f6 s.pt\nh\u2081 : inl s\u271d \u226b l = inl s\nh\u2082 : inr s\u271d \u226b l = inr s\n\u22a2 { l //\n inl (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)) \u226b l = inl s \u2227\n inr (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)) \u226b l = inr s \u2227\n \u2200 {m : (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)).pt \u27f6 s.pt},\n inl (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)) \u226b m = inl s \u2192\n inr (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)) \u226b m = inr s \u2192 m = l }\n[PROOFSTEP]\nrefine' \u27e8l, h\u2081, h\u2082, _\u27e9\n[GOAL]\ncase create.mk.intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : W \u27f6 X\ninst\u271d : Epi h\ns\u271d : PushoutCocone f g\nH : IsColimit s\u271d\ns : PushoutCocone (h \u226b f) (h \u226b g)\nl : s\u271d.pt \u27f6 s.pt\nh\u2081 : inl s\u271d \u226b l = inl s\nh\u2082 : inr s\u271d \u226b l = inr s\n\u22a2 \u2200 {m : (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)).pt \u27f6 s.pt},\n inl (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)) \u226b m = inl s \u2192\n inr (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)) \u226b m = inr s \u2192 m = l\n[PROOFSTEP]\nintro m hm\u2081 hm\u2082\n[GOAL]\ncase create.mk.intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : W \u27f6 X\ninst\u271d : Epi h\ns\u271d : PushoutCocone f g\nH : IsColimit s\u271d\ns : PushoutCocone (h \u226b f) (h \u226b g)\nl : s\u271d.pt \u27f6 s.pt\nh\u2081 : inl s\u271d \u226b l = inl s\nh\u2082 : inr s\u271d \u226b l = inr s\nm : (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)).pt \u27f6 s.pt\nhm\u2081 : inl (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)) \u226b m = inl s\nhm\u2082 : inr (mk (inl s\u271d) (inr s\u271d) (_ : (h \u226b f) \u226b inl s\u271d = (h \u226b g) \u226b inr s\u271d)) \u226b m = inr s\n\u22a2 m = l\n[PROOFSTEP]\nexact (PushoutCocone.IsColimit.hom_ext H (hm\u2081.trans h\u2081.symm) (hm\u2082.trans h\u2082.symm) : _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nF : WalkingCospan \u2964 C\nt : Cone F\n\u22a2 \u2200 (j : WalkingCospan),\n NatTrans.app ((Cones.postcompose (diagramIsoCospan F).hom).obj t).\u03c0 j =\n (Iso.refl ((Cones.postcompose (diagramIsoCospan F).hom).obj t).pt).hom \u226b\n NatTrans.app\n (mk (NatTrans.app t.\u03c0 WalkingCospan.left) (NatTrans.app t.\u03c0 WalkingCospan.right)\n (_ :\n NatTrans.app t.\u03c0 WalkingCospan.left \u226b F.map inl = NatTrans.app t.\u03c0 WalkingCospan.right \u226b F.map inr)).\u03c0\n j\n[PROOFSTEP]\nrintro (_ | (_ | _))\n[GOAL]\ncase none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nF : WalkingCospan \u2964 C\nt : Cone F\n\u22a2 NatTrans.app ((Cones.postcompose (diagramIsoCospan F).hom).obj t).\u03c0 none =\n (Iso.refl ((Cones.postcompose (diagramIsoCospan F).hom).obj t).pt).hom \u226b\n NatTrans.app\n (mk (NatTrans.app t.\u03c0 WalkingCospan.left) (NatTrans.app t.\u03c0 WalkingCospan.right)\n (_ : NatTrans.app t.\u03c0 WalkingCospan.left \u226b F.map inl = NatTrans.app t.\u03c0 WalkingCospan.right \u226b F.map inr)).\u03c0\n none\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nF : WalkingCospan \u2964 C\nt : Cone F\n\u22a2 NatTrans.app t.\u03c0 none \u226b \ud835\udfd9 (F.obj none) = \ud835\udfd9 t.pt \u226b NatTrans.app t.\u03c0 WalkingCospan.left \u226b F.map inl\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nF : WalkingCospan \u2964 C\nt : Cone F\n\u22a2 NatTrans.app ((Cones.postcompose (diagramIsoCospan F).hom).obj t).\u03c0 (some WalkingPair.left) =\n (Iso.refl ((Cones.postcompose (diagramIsoCospan F).hom).obj t).pt).hom \u226b\n NatTrans.app\n (mk (NatTrans.app t.\u03c0 WalkingCospan.left) (NatTrans.app t.\u03c0 WalkingCospan.right)\n (_ : NatTrans.app t.\u03c0 WalkingCospan.left \u226b F.map inl = NatTrans.app t.\u03c0 WalkingCospan.right \u226b F.map inr)).\u03c0\n (some WalkingPair.left)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nF : WalkingCospan \u2964 C\nt : Cone F\n\u22a2 NatTrans.app t.\u03c0 (some WalkingPair.left) \u226b \ud835\udfd9 (F.obj (some WalkingPair.left)) =\n \ud835\udfd9 t.pt \u226b NatTrans.app t.\u03c0 WalkingCospan.left\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nF : WalkingCospan \u2964 C\nt : Cone F\n\u22a2 NatTrans.app ((Cones.postcompose (diagramIsoCospan F).hom).obj t).\u03c0 (some WalkingPair.right) =\n (Iso.refl ((Cones.postcompose (diagramIsoCospan F).hom).obj t).pt).hom \u226b\n NatTrans.app\n (mk (NatTrans.app t.\u03c0 WalkingCospan.left) (NatTrans.app t.\u03c0 WalkingCospan.right)\n (_ : NatTrans.app t.\u03c0 WalkingCospan.left \u226b F.map inl = NatTrans.app t.\u03c0 WalkingCospan.right \u226b F.map inr)).\u03c0\n (some WalkingPair.right)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nF : WalkingCospan \u2964 C\nt : Cone F\n\u22a2 NatTrans.app t.\u03c0 (some WalkingPair.right) \u226b \ud835\udfd9 (F.obj (some WalkingPair.right)) =\n \ud835\udfd9 t.pt \u226b NatTrans.app t.\u03c0 WalkingCospan.right\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nF : WalkingSpan \u2964 C\nt : Cocone F\n\u22a2 \u2200 (j : WalkingSpan),\n NatTrans.app ((Cocones.precompose (diagramIsoSpan F).inv).obj t).\u03b9 j \u226b\n (Iso.refl ((Cocones.precompose (diagramIsoSpan F).inv).obj t).pt).hom =\n NatTrans.app\n (mk (NatTrans.app t.\u03b9 WalkingSpan.left) (NatTrans.app t.\u03b9 WalkingSpan.right)\n (_ : F.map fst \u226b NatTrans.app t.\u03b9 WalkingSpan.left = F.map snd \u226b NatTrans.app t.\u03b9 WalkingSpan.right)).\u03b9\n j\n[PROOFSTEP]\nrintro (_ | (_ | _))\n[GOAL]\ncase none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nF : WalkingSpan \u2964 C\nt : Cocone F\n\u22a2 NatTrans.app ((Cocones.precompose (diagramIsoSpan F).inv).obj t).\u03b9 none \u226b\n (Iso.refl ((Cocones.precompose (diagramIsoSpan F).inv).obj t).pt).hom =\n NatTrans.app\n (mk (NatTrans.app t.\u03b9 WalkingSpan.left) (NatTrans.app t.\u03b9 WalkingSpan.right)\n (_ : F.map fst \u226b NatTrans.app t.\u03b9 WalkingSpan.left = F.map snd \u226b NatTrans.app t.\u03b9 WalkingSpan.right)).\u03b9\n none\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nF : WalkingSpan \u2964 C\nt : Cocone F\n\u22a2 (\ud835\udfd9 (F.obj WalkingSpan.zero) \u226b NatTrans.app t.\u03b9 none) \u226b \ud835\udfd9 t.pt = F.map fst \u226b NatTrans.app t.\u03b9 WalkingSpan.left\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nF : WalkingSpan \u2964 C\nt : Cocone F\n\u22a2 NatTrans.app ((Cocones.precompose (diagramIsoSpan F).inv).obj t).\u03b9 (some WalkingPair.left) \u226b\n (Iso.refl ((Cocones.precompose (diagramIsoSpan F).inv).obj t).pt).hom =\n NatTrans.app\n (mk (NatTrans.app t.\u03b9 WalkingSpan.left) (NatTrans.app t.\u03b9 WalkingSpan.right)\n (_ : F.map fst \u226b NatTrans.app t.\u03b9 WalkingSpan.left = F.map snd \u226b NatTrans.app t.\u03b9 WalkingSpan.right)).\u03b9\n (some WalkingPair.left)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nF : WalkingSpan \u2964 C\nt : Cocone F\n\u22a2 (\ud835\udfd9 (F.obj WalkingSpan.left) \u226b NatTrans.app t.\u03b9 (some WalkingPair.left)) \u226b \ud835\udfd9 t.pt = NatTrans.app t.\u03b9 WalkingSpan.left\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nF : WalkingSpan \u2964 C\nt : Cocone F\n\u22a2 NatTrans.app ((Cocones.precompose (diagramIsoSpan F).inv).obj t).\u03b9 (some WalkingPair.right) \u226b\n (Iso.refl ((Cocones.precompose (diagramIsoSpan F).inv).obj t).pt).hom =\n NatTrans.app\n (mk (NatTrans.app t.\u03b9 WalkingSpan.left) (NatTrans.app t.\u03b9 WalkingSpan.right)\n (_ : F.map fst \u226b NatTrans.app t.\u03b9 WalkingSpan.left = F.map snd \u226b NatTrans.app t.\u03b9 WalkingSpan.right)).\u03b9\n (some WalkingPair.right)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nF : WalkingSpan \u2964 C\nt : Cocone F\n\u22a2 (\ud835\udfd9 (F.obj WalkingSpan.right) \u226b NatTrans.app t.\u03b9 (some WalkingPair.right)) \u226b \ud835\udfd9 t.pt =\n NatTrans.app t.\u03b9 WalkingSpan.right\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d W X Y Z S T : C\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ninst\u271d\u00b9 : HasPullback f\u2081 f\u2082\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ninst\u271d : HasPullback g\u2081 g\u2082\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\n\u22a2 (fst \u226b i\u2081) \u226b g\u2081 = (snd \u226b i\u2082) \u226b g\u2082\n[PROOFSTEP]\nsimp [\u2190 eq\u2081, \u2190 eq\u2082, pullback.condition_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d W X Y Z S T : C\nf\u2081 : S \u27f6 W\nf\u2082 : S \u27f6 X\ninst\u271d\u00b9 : HasPushout f\u2081 f\u2082\ng\u2081 : T \u27f6 Y\ng\u2082 : T \u27f6 Z\ninst\u271d : HasPushout g\u2081 g\u2082\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\neq\u2081 : f\u2081 \u226b i\u2081 = i\u2083 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2082 = i\u2083 \u226b g\u2082\n\u22a2 f\u2081 \u226b i\u2081 \u226b inl = f\u2082 \u226b i\u2082 \u226b inr\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc, eq\u2081, eq\u2082]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d W X Y Z S T : C\nf\u2081 : S \u27f6 W\nf\u2082 : S \u27f6 X\ninst\u271d\u00b9 : HasPushout f\u2081 f\u2082\ng\u2081 : T \u27f6 Y\ng\u2082 : T \u27f6 Z\ninst\u271d : HasPushout g\u2081 g\u2082\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\neq\u2081 : f\u2081 \u226b i\u2081 = i\u2083 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2082 = i\u2083 \u226b g\u2082\n\u22a2 (i\u2083 \u226b g\u2081) \u226b inl = (i\u2083 \u226b g\u2082) \u226b inr\n[PROOFSTEP]\nsimp [pushout.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : HasPullback f g\n\u22a2 \u2200 (s : PullbackCone f g),\n (fun s =>\n pullback.lift (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g))\n s \u226b\n pullback.fst =\n PullbackCone.fst s\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : HasPullback f g\n\u22a2 \u2200 (s : PullbackCone f g),\n (fun s =>\n pullback.lift (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g))\n s \u226b\n pullback.snd =\n PullbackCone.snd s\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : HasPullback f g\n\u22a2 \u2200 (s : PullbackCone f g) (m : s.pt \u27f6 pullback f g),\n m \u226b pullback.fst = PullbackCone.fst s \u2192\n m \u226b pullback.snd = PullbackCone.snd s \u2192\n m =\n (fun s =>\n pullback.lift (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g))\n s\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC\u271d : Type u\ninst\u271d\u2074 : Category.{v, u} C\u271d\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d : C\u271d\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.224485, u_1} C\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasBinaryProduct X Y\nW : C\ni\u2081 i\u2082 : W \u27f6 pullback f g\nh : i\u2081 \u226b prod.lift pullback.fst pullback.snd = i\u2082 \u226b prod.lift pullback.fst pullback.snd\n\u22a2 i\u2081 = i\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC\u271d : Type u\ninst\u271d\u2074 : Category.{v, u} C\u271d\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d : C\u271d\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.224485, u_1} C\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasBinaryProduct X Y\nW : C\ni\u2081 i\u2082 : W \u27f6 pullback f g\nh : i\u2081 \u226b prod.lift pullback.fst pullback.snd = i\u2082 \u226b prod.lift pullback.fst pullback.snd\n\u22a2 i\u2081 \u226b pullback.fst = i\u2082 \u226b pullback.fst\n[PROOFSTEP]\nsimpa using congrArg (fun f => f \u226b prod.fst) h\n[GOAL]\ncase h\u2081\nC\u271d : Type u\ninst\u271d\u2074 : Category.{v, u} C\u271d\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d : C\u271d\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.224485, u_1} C\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasBinaryProduct X Y\nW : C\ni\u2081 i\u2082 : W \u27f6 pullback f g\nh : i\u2081 \u226b prod.lift pullback.fst pullback.snd = i\u2082 \u226b prod.lift pullback.fst pullback.snd\n\u22a2 i\u2081 \u226b pullback.snd = i\u2082 \u226b pullback.snd\n[PROOFSTEP]\nsimpa using congrArg (fun f => f \u226b prod.snd) h\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : HasPushout f g\n\u22a2 \u2200 (s : PushoutCocone f g),\n pushout.inl \u226b\n (fun s =>\n pushout.desc (PushoutCocone.inl s) (PushoutCocone.inr s)\n (_ : f \u226b PushoutCocone.inl s = g \u226b PushoutCocone.inr s))\n s =\n PushoutCocone.inl s\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : HasPushout f g\n\u22a2 \u2200 (s : PushoutCocone f g),\n pushout.inr \u226b\n (fun s =>\n pushout.desc (PushoutCocone.inl s) (PushoutCocone.inr s)\n (_ : f \u226b PushoutCocone.inl s = g \u226b PushoutCocone.inr s))\n s =\n PushoutCocone.inr s\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : HasPushout f g\n\u22a2 \u2200 (s : PushoutCocone f g) (m : pushout f g \u27f6 s.pt),\n pushout.inl \u226b m = PushoutCocone.inl s \u2192\n pushout.inr \u226b m = PushoutCocone.inr s \u2192\n m =\n (fun s =>\n pushout.desc (PushoutCocone.inl s) (PushoutCocone.inr s)\n (_ : f \u226b PushoutCocone.inl s = g \u226b PushoutCocone.inr s))\n s\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC\u271d : Type u\ninst\u271d\u2074 : Category.{v, u} C\u271d\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d : C\u271d\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.241525, u_1} C\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : HasPushout f g\ninst\u271d : HasBinaryCoproduct Y Z\nW : C\ni\u2081 i\u2082 : pushout f g \u27f6 W\nh : coprod.desc pushout.inl pushout.inr \u226b i\u2081 = coprod.desc pushout.inl pushout.inr \u226b i\u2082\n\u22a2 i\u2081 = i\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC\u271d : Type u\ninst\u271d\u2074 : Category.{v, u} C\u271d\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d : C\u271d\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.241525, u_1} C\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : HasPushout f g\ninst\u271d : HasBinaryCoproduct Y Z\nW : C\ni\u2081 i\u2082 : pushout f g \u27f6 W\nh : coprod.desc pushout.inl pushout.inr \u226b i\u2081 = coprod.desc pushout.inl pushout.inr \u226b i\u2082\n\u22a2 pushout.inl \u226b i\u2081 = pushout.inl \u226b i\u2082\n[PROOFSTEP]\nsimpa using congrArg (fun f => coprod.inl \u226b f) h\n[GOAL]\ncase h\u2081\nC\u271d : Type u\ninst\u271d\u2074 : Category.{v, u} C\u271d\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d : C\u271d\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.241525, u_1} C\nX Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : HasPushout f g\ninst\u271d : HasBinaryCoproduct Y Z\nW : C\ni\u2081 i\u2082 : pushout f g \u27f6 W\nh : coprod.desc pushout.inl pushout.inr \u226b i\u2081 = coprod.desc pushout.inl pushout.inr \u226b i\u2082\n\u22a2 pushout.inr \u226b i\u2081 = pushout.inr \u226b i\u2082\n[PROOFSTEP]\nsimpa using congrArg (fun f => coprod.inr \u226b f) h\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d W X Y Z S T : C\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ninst\u271d\u2074 : HasPullback f\u2081 f\u2082\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ninst\u271d\u00b3 : HasPullback g\u2081 g\u2082\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\ninst\u271d\u00b2 : IsIso i\u2081\ninst\u271d\u00b9 : IsIso i\u2082\ninst\u271d : IsIso i\u2083\n\u22a2 IsIso (map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)\n[PROOFSTEP]\nrefine' \u27e8\u27e8pullback.map _ _ _ _ (inv i\u2081) (inv i\u2082) (inv i\u2083) _ _, _, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d W X Y Z S T : C\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ninst\u271d\u2074 : HasPullback f\u2081 f\u2082\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ninst\u271d\u00b3 : HasPullback g\u2081 g\u2082\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\ninst\u271d\u00b2 : IsIso i\u2081\ninst\u271d\u00b9 : IsIso i\u2082\ninst\u271d : IsIso i\u2083\n\u22a2 g\u2081 \u226b inv i\u2083 = inv i\u2081 \u226b f\u2081\n[PROOFSTEP]\nrw [IsIso.comp_inv_eq, Category.assoc, eq\u2081, IsIso.inv_hom_id_assoc]\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d W X Y Z S T : C\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ninst\u271d\u2074 : HasPullback f\u2081 f\u2082\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ninst\u271d\u00b3 : HasPullback g\u2081 g\u2082\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\ninst\u271d\u00b2 : IsIso i\u2081\ninst\u271d\u00b9 : IsIso i\u2082\ninst\u271d : IsIso i\u2083\n\u22a2 g\u2082 \u226b inv i\u2083 = inv i\u2082 \u226b f\u2082\n[PROOFSTEP]\nrw [IsIso.comp_inv_eq, Category.assoc, eq\u2082, IsIso.inv_hom_id_assoc]\n[GOAL]\ncase refine'_3\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d W X Y Z S T : C\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ninst\u271d\u2074 : HasPullback f\u2081 f\u2082\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ninst\u271d\u00b3 : HasPullback g\u2081 g\u2082\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\ninst\u271d\u00b2 : IsIso i\u2081\ninst\u271d\u00b9 : IsIso i\u2082\ninst\u271d : IsIso i\u2083\n\u22a2 map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082 \u226b\n map g\u2081 g\u2082 f\u2081 f\u2082 (inv i\u2081) (inv i\u2082) (inv i\u2083) (_ : g\u2081 \u226b inv i\u2083 = inv i\u2081 \u226b f\u2081) (_ : g\u2082 \u226b inv i\u2083 = inv i\u2082 \u226b f\u2082) =\n \ud835\udfd9 (pullback f\u2081 f\u2082)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase refine'_4\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d W X Y Z S T : C\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ninst\u271d\u2074 : HasPullback f\u2081 f\u2082\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ninst\u271d\u00b3 : HasPullback g\u2081 g\u2082\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\ninst\u271d\u00b2 : IsIso i\u2081\ninst\u271d\u00b9 : IsIso i\u2082\ninst\u271d : IsIso i\u2083\n\u22a2 map g\u2081 g\u2082 f\u2081 f\u2082 (inv i\u2081) (inv i\u2082) (inv i\u2083) (_ : g\u2081 \u226b inv i\u2083 = inv i\u2081 \u226b f\u2081) (_ : g\u2082 \u226b inv i\u2083 = inv i\u2082 \u226b f\u2082) \u226b\n map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082 =\n \ud835\udfd9 (pullback g\u2081 g\u2082)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Z\ng\u2081 g\u2082 : Y \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPullback f\u2081 g\u2081\ninst\u271d : HasPullback f\u2082 g\u2082\n\u22a2 f\u2081 \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f\u2082\n[PROOFSTEP]\nsimp [h\u2081]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Z\ng\u2081 g\u2082 : Y \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPullback f\u2081 g\u2081\ninst\u271d : HasPullback f\u2082 g\u2082\n\u22a2 g\u2081 \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g\u2082\n[PROOFSTEP]\nsimp [h\u2082]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Z\ng\u2081 g\u2082 : Y \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPullback f\u2081 g\u2081\ninst\u271d : HasPullback f\u2082 g\u2082\n\u22a2 f\u2082 \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f\u2081\n[PROOFSTEP]\nsimp [h\u2081]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Z\ng\u2081 g\u2082 : Y \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPullback f\u2081 g\u2081\ninst\u271d : HasPullback f\u2082 g\u2082\n\u22a2 g\u2082 \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g\u2081\n[PROOFSTEP]\nsimp [h\u2082]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Z\ng\u2081 g\u2082 : Y \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPullback f\u2081 g\u2081\ninst\u271d : HasPullback f\u2082 g\u2082\n\u22a2 (congrHom h\u2081 h\u2082).inv = map f\u2082 g\u2082 f\u2081 g\u2081 (\ud835\udfd9 X) (\ud835\udfd9 Y) (\ud835\udfd9 Z) (_ : f\u2082 \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f\u2081) (_ : g\u2082 \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g\u2081)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Z\ng\u2081 g\u2082 : Y \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPullback f\u2081 g\u2081\ninst\u271d : HasPullback f\u2082 g\u2082\n\u22a2 (congrHom h\u2081 h\u2082).inv \u226b fst =\n map f\u2082 g\u2082 f\u2081 g\u2081 (\ud835\udfd9 X) (\ud835\udfd9 Y) (\ud835\udfd9 Z) (_ : f\u2082 \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f\u2081) (_ : g\u2082 \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g\u2081) \u226b fst\n[PROOFSTEP]\nerw [pullback.lift_fst]\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Z\ng\u2081 g\u2082 : Y \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPullback f\u2081 g\u2081\ninst\u271d : HasPullback f\u2082 g\u2082\n\u22a2 (congrHom h\u2081 h\u2082).inv \u226b fst = fst \u226b \ud835\udfd9 X\n[PROOFSTEP]\nrw [Iso.inv_comp_eq]\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Z\ng\u2081 g\u2082 : Y \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPullback f\u2081 g\u2081\ninst\u271d : HasPullback f\u2082 g\u2082\n\u22a2 fst = (congrHom h\u2081 h\u2082).hom \u226b fst \u226b \ud835\udfd9 X\n[PROOFSTEP]\nerw [pullback.lift_fst_assoc]\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Z\ng\u2081 g\u2082 : Y \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPullback f\u2081 g\u2081\ninst\u271d : HasPullback f\u2082 g\u2082\n\u22a2 fst = (fst \u226b \ud835\udfd9 X) \u226b \ud835\udfd9 X\n[PROOFSTEP]\nrw [Category.comp_id, Category.comp_id]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Z\ng\u2081 g\u2082 : Y \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPullback f\u2081 g\u2081\ninst\u271d : HasPullback f\u2082 g\u2082\n\u22a2 (congrHom h\u2081 h\u2082).inv \u226b snd =\n map f\u2082 g\u2082 f\u2081 g\u2081 (\ud835\udfd9 X) (\ud835\udfd9 Y) (\ud835\udfd9 Z) (_ : f\u2082 \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f\u2081) (_ : g\u2082 \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g\u2081) \u226b snd\n[PROOFSTEP]\nerw [pullback.lift_snd]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Z\ng\u2081 g\u2082 : Y \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPullback f\u2081 g\u2081\ninst\u271d : HasPullback f\u2082 g\u2082\n\u22a2 (congrHom h\u2081 h\u2082).inv \u226b snd = snd \u226b \ud835\udfd9 Y\n[PROOFSTEP]\nrw [Iso.inv_comp_eq]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Z\ng\u2081 g\u2082 : Y \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPullback f\u2081 g\u2081\ninst\u271d : HasPullback f\u2082 g\u2082\n\u22a2 snd = (congrHom h\u2081 h\u2082).hom \u226b snd \u226b \ud835\udfd9 Y\n[PROOFSTEP]\nerw [pullback.lift_snd_assoc]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Z\ng\u2081 g\u2082 : Y \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPullback f\u2081 g\u2081\ninst\u271d : HasPullback f\u2082 g\u2082\n\u22a2 snd = (snd \u226b \ud835\udfd9 Y) \u226b \ud835\udfd9 Y\n[PROOFSTEP]\nrw [Category.comp_id, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d W X Y Z S T : C\nf\u2081 : S \u27f6 W\nf\u2082 : S \u27f6 X\ninst\u271d\u2074 : HasPushout f\u2081 f\u2082\ng\u2081 : T \u27f6 Y\ng\u2082 : T \u27f6 Z\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\neq\u2081 : f\u2081 \u226b i\u2081 = i\u2083 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2082 = i\u2083 \u226b g\u2082\ninst\u271d\u00b2 : IsIso i\u2081\ninst\u271d\u00b9 : IsIso i\u2082\ninst\u271d : IsIso i\u2083\n\u22a2 IsIso (map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)\n[PROOFSTEP]\nrefine' \u27e8\u27e8pushout.map _ _ _ _ (inv i\u2081) (inv i\u2082) (inv i\u2083) _ _, _, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d W X Y Z S T : C\nf\u2081 : S \u27f6 W\nf\u2082 : S \u27f6 X\ninst\u271d\u2074 : HasPushout f\u2081 f\u2082\ng\u2081 : T \u27f6 Y\ng\u2082 : T \u27f6 Z\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\neq\u2081 : f\u2081 \u226b i\u2081 = i\u2083 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2082 = i\u2083 \u226b g\u2082\ninst\u271d\u00b2 : IsIso i\u2081\ninst\u271d\u00b9 : IsIso i\u2082\ninst\u271d : IsIso i\u2083\n\u22a2 g\u2081 \u226b inv i\u2081 = inv i\u2083 \u226b f\u2081\n[PROOFSTEP]\nrw [IsIso.comp_inv_eq, Category.assoc, eq\u2081, IsIso.inv_hom_id_assoc]\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d W X Y Z S T : C\nf\u2081 : S \u27f6 W\nf\u2082 : S \u27f6 X\ninst\u271d\u2074 : HasPushout f\u2081 f\u2082\ng\u2081 : T \u27f6 Y\ng\u2082 : T \u27f6 Z\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\neq\u2081 : f\u2081 \u226b i\u2081 = i\u2083 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2082 = i\u2083 \u226b g\u2082\ninst\u271d\u00b2 : IsIso i\u2081\ninst\u271d\u00b9 : IsIso i\u2082\ninst\u271d : IsIso i\u2083\n\u22a2 g\u2082 \u226b inv i\u2082 = inv i\u2083 \u226b f\u2082\n[PROOFSTEP]\nrw [IsIso.comp_inv_eq, Category.assoc, eq\u2082, IsIso.inv_hom_id_assoc]\n[GOAL]\ncase refine'_3\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d W X Y Z S T : C\nf\u2081 : S \u27f6 W\nf\u2082 : S \u27f6 X\ninst\u271d\u2074 : HasPushout f\u2081 f\u2082\ng\u2081 : T \u27f6 Y\ng\u2082 : T \u27f6 Z\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\neq\u2081 : f\u2081 \u226b i\u2081 = i\u2083 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2082 = i\u2083 \u226b g\u2082\ninst\u271d\u00b2 : IsIso i\u2081\ninst\u271d\u00b9 : IsIso i\u2082\ninst\u271d : IsIso i\u2083\n\u22a2 map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082 \u226b\n map g\u2081 g\u2082 f\u2081 f\u2082 (inv i\u2081) (inv i\u2082) (inv i\u2083) (_ : g\u2081 \u226b inv i\u2081 = inv i\u2083 \u226b f\u2081) (_ : g\u2082 \u226b inv i\u2082 = inv i\u2083 \u226b f\u2082) =\n \ud835\udfd9 (pushout f\u2081 f\u2082)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase refine'_4\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\nW\u271d X\u271d Y\u271d Z\u271d W X Y Z S T : C\nf\u2081 : S \u27f6 W\nf\u2082 : S \u27f6 X\ninst\u271d\u2074 : HasPushout f\u2081 f\u2082\ng\u2081 : T \u27f6 Y\ng\u2082 : T \u27f6 Z\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\neq\u2081 : f\u2081 \u226b i\u2081 = i\u2083 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2082 = i\u2083 \u226b g\u2082\ninst\u271d\u00b2 : IsIso i\u2081\ninst\u271d\u00b9 : IsIso i\u2082\ninst\u271d : IsIso i\u2083\n\u22a2 map g\u2081 g\u2082 f\u2081 f\u2082 (inv i\u2081) (inv i\u2082) (inv i\u2083) (_ : g\u2081 \u226b inv i\u2081 = inv i\u2083 \u226b f\u2081) (_ : g\u2082 \u226b inv i\u2082 = inv i\u2083 \u226b f\u2082) \u226b\n map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082 =\n \ud835\udfd9 (pushout g\u2081 g\u2082)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z X Y S T S' : C\nf : X \u27f6 T\ng : Y \u27f6 T\ni : T \u27f6 S\ni' : S \u27f6 S'\ninst\u271d\u00b3 : HasPullback f g\ninst\u271d\u00b2 : HasPullback (f \u226b i) (g \u226b i)\ninst\u271d\u00b9 : HasPullback (f \u226b i \u226b i') (g \u226b i \u226b i')\ninst\u271d : HasPullback ((f \u226b i) \u226b i') ((g \u226b i) \u226b i')\n\u22a2 mapDesc f g (i \u226b i') =\n mapDesc f g i \u226b\n mapDesc (f \u226b i) (g \u226b i) i' \u226b (congrHom (_ : (f \u226b i) \u226b i' = f \u226b i \u226b i') (_ : (g \u226b i) \u226b i' = g \u226b i \u226b i')).hom\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Y\ng\u2081 g\u2082 : X \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPushout f\u2081 g\u2081\ninst\u271d : HasPushout f\u2082 g\u2082\n\u22a2 f\u2081 \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u2082\n[PROOFSTEP]\nsimp [h\u2081]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Y\ng\u2081 g\u2082 : X \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPushout f\u2081 g\u2081\ninst\u271d : HasPushout f\u2082 g\u2082\n\u22a2 g\u2081 \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b g\u2082\n[PROOFSTEP]\nsimp [h\u2082]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Y\ng\u2081 g\u2082 : X \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPushout f\u2081 g\u2081\ninst\u271d : HasPushout f\u2082 g\u2082\n\u22a2 f\u2082 \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u2081\n[PROOFSTEP]\nsimp [h\u2081]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Y\ng\u2081 g\u2082 : X \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPushout f\u2081 g\u2081\ninst\u271d : HasPushout f\u2082 g\u2082\n\u22a2 g\u2082 \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b g\u2081\n[PROOFSTEP]\nsimp [h\u2082]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Y\ng\u2081 g\u2082 : X \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPushout f\u2081 g\u2081\ninst\u271d : HasPushout f\u2082 g\u2082\n\u22a2 (congrHom h\u2081 h\u2082).inv = map f\u2082 g\u2082 f\u2081 g\u2081 (\ud835\udfd9 Y) (\ud835\udfd9 Z) (\ud835\udfd9 X) (_ : f\u2082 \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u2081) (_ : g\u2082 \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b g\u2081)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Y\ng\u2081 g\u2082 : X \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPushout f\u2081 g\u2081\ninst\u271d : HasPushout f\u2082 g\u2082\n\u22a2 inl \u226b (congrHom h\u2081 h\u2082).inv =\n inl \u226b map f\u2082 g\u2082 f\u2081 g\u2081 (\ud835\udfd9 Y) (\ud835\udfd9 Z) (\ud835\udfd9 X) (_ : f\u2082 \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u2081) (_ : g\u2082 \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b g\u2081)\n[PROOFSTEP]\nerw [pushout.inl_desc]\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Y\ng\u2081 g\u2082 : X \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPushout f\u2081 g\u2081\ninst\u271d : HasPushout f\u2082 g\u2082\n\u22a2 inl \u226b (congrHom h\u2081 h\u2082).inv = \ud835\udfd9 Y \u226b inl\n[PROOFSTEP]\nrw [Iso.comp_inv_eq, Category.id_comp]\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Y\ng\u2081 g\u2082 : X \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPushout f\u2081 g\u2081\ninst\u271d : HasPushout f\u2082 g\u2082\n\u22a2 inl = inl \u226b (congrHom h\u2081 h\u2082).hom\n[PROOFSTEP]\nerw [pushout.inl_desc]\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Y\ng\u2081 g\u2082 : X \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPushout f\u2081 g\u2081\ninst\u271d : HasPushout f\u2082 g\u2082\n\u22a2 inl = \ud835\udfd9 Y \u226b inl\n[PROOFSTEP]\nrw [Category.id_comp]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Y\ng\u2081 g\u2082 : X \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPushout f\u2081 g\u2081\ninst\u271d : HasPushout f\u2082 g\u2082\n\u22a2 inr \u226b (congrHom h\u2081 h\u2082).inv =\n inr \u226b map f\u2082 g\u2082 f\u2081 g\u2081 (\ud835\udfd9 Y) (\ud835\udfd9 Z) (\ud835\udfd9 X) (_ : f\u2082 \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u2081) (_ : g\u2082 \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b g\u2081)\n[PROOFSTEP]\nerw [pushout.inr_desc]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Y\ng\u2081 g\u2082 : X \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPushout f\u2081 g\u2081\ninst\u271d : HasPushout f\u2082 g\u2082\n\u22a2 inr \u226b (congrHom h\u2081 h\u2082).inv = \ud835\udfd9 Z \u226b inr\n[PROOFSTEP]\nrw [Iso.comp_inv_eq, Category.id_comp]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Y\ng\u2081 g\u2082 : X \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPushout f\u2081 g\u2081\ninst\u271d : HasPushout f\u2082 g\u2082\n\u22a2 inr = inr \u226b (congrHom h\u2081 h\u2082).hom\n[PROOFSTEP]\nerw [pushout.inr_desc]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d X Y Z : C\nf\u2081 f\u2082 : X \u27f6 Y\ng\u2081 g\u2082 : X \u27f6 Z\nh\u2081 : f\u2081 = f\u2082\nh\u2082 : g\u2081 = g\u2082\ninst\u271d\u00b9 : HasPushout f\u2081 g\u2081\ninst\u271d : HasPushout f\u2082 g\u2082\n\u22a2 inr = \ud835\udfd9 Z \u226b inr\n[PROOFSTEP]\nrw [Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z X Y S T S' : C\nf : T \u27f6 X\ng : T \u27f6 Y\ni : S \u27f6 T\ni' : S' \u27f6 S\ninst\u271d\u00b3 : HasPushout f g\ninst\u271d\u00b2 : HasPushout (i \u226b f) (i \u226b g)\ninst\u271d\u00b9 : HasPushout (i' \u226b i \u226b f) (i' \u226b i \u226b g)\ninst\u271d : HasPushout ((i' \u226b i) \u226b f) ((i' \u226b i) \u226b g)\n\u22a2 mapLift f g (i' \u226b i) =\n (congrHom (_ : (i' \u226b i) \u226b f = i' \u226b i \u226b f) (_ : (i' \u226b i) \u226b g = i' \u226b i \u226b g)).hom \u226b\n mapLift (i \u226b f) (i \u226b g) i' \u226b mapLift f g i\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nG : C \u2964 D\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback (G.map f) (G.map g)\n\u22a2 G.map pullback.fst \u226b G.map f = G.map pullback.snd \u226b G.map g\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp, pullback.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nG : C \u2964 D\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback (G.map f) (G.map g)\nW : C\nh : W \u27f6 X\nk : W \u27f6 Y\nw : h \u226b f = k \u226b g\n\u22a2 G.map h \u226b G.map f = G.map k \u226b G.map g\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp, w]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nG : C \u2964 D\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback (G.map f) (G.map g)\nW : C\nh : W \u27f6 X\nk : W \u27f6 Y\nw : h \u226b f = k \u226b g\n\u22a2 G.map (pullback.lift h k w) \u226b pullbackComparison G f g =\n pullback.lift (G.map h) (G.map k) (_ : G.map h \u226b G.map f = G.map k \u226b G.map g)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nG : C \u2964 D\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback (G.map f) (G.map g)\nW : C\nh : W \u27f6 X\nk : W \u27f6 Y\nw : h \u226b f = k \u226b g\n\u22a2 (G.map (pullback.lift h k w) \u226b pullbackComparison G f g) \u226b pullback.fst =\n pullback.lift (G.map h) (G.map k) (_ : G.map h \u226b G.map f = G.map k \u226b G.map g) \u226b pullback.fst\n[PROOFSTEP]\nsimp [\u2190 G.map_comp]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nG : C \u2964 D\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback (G.map f) (G.map g)\nW : C\nh : W \u27f6 X\nk : W \u27f6 Y\nw : h \u226b f = k \u226b g\n\u22a2 (G.map (pullback.lift h k w) \u226b pullbackComparison G f g) \u226b pullback.snd =\n pullback.lift (G.map h) (G.map k) (_ : G.map h \u226b G.map f = G.map k \u226b G.map g) \u226b pullback.snd\n[PROOFSTEP]\nsimp [\u2190 G.map_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nG : C \u2964 D\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : HasPushout f g\ninst\u271d : HasPushout (G.map f) (G.map g)\n\u22a2 G.map f \u226b G.map pushout.inl = G.map g \u226b G.map pushout.inr\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp, pushout.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nG : C \u2964 D\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : HasPushout f g\ninst\u271d : HasPushout (G.map f) (G.map g)\nW : C\nh : Y \u27f6 W\nk : Z \u27f6 W\nw : f \u226b h = g \u226b k\n\u22a2 G.map f \u226b G.map h = G.map g \u226b G.map k\n[PROOFSTEP]\nsimp only [\u2190 G.map_comp, w]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nG : C \u2964 D\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : HasPushout f g\ninst\u271d : HasPushout (G.map f) (G.map g)\nW : C\nh : Y \u27f6 W\nk : Z \u27f6 W\nw : f \u226b h = g \u226b k\n\u22a2 pushoutComparison G f g \u226b G.map (pushout.desc h k w) =\n pushout.desc (G.map h) (G.map k) (_ : G.map f \u226b G.map h = G.map g \u226b G.map k)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nG : C \u2964 D\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : HasPushout f g\ninst\u271d : HasPushout (G.map f) (G.map g)\nW : C\nh : Y \u27f6 W\nk : Z \u27f6 W\nw : f \u226b h = g \u226b k\n\u22a2 pushout.inl \u226b pushoutComparison G f g \u226b G.map (pushout.desc h k w) =\n pushout.inl \u226b pushout.desc (G.map h) (G.map k) (_ : G.map f \u226b G.map h = G.map g \u226b G.map k)\n[PROOFSTEP]\nsimp [\u2190 G.map_comp]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW\u271d X Y Z : C\nG : C \u2964 D\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : HasPushout f g\ninst\u271d : HasPushout (G.map f) (G.map g)\nW : C\nh : Y \u27f6 W\nk : Z \u27f6 W\nw : f \u226b h = g \u226b k\n\u22a2 pushout.inr \u226b pushoutComparison G f g \u226b G.map (pushout.desc h k w) =\n pushout.inr \u226b pushout.desc (G.map h) (G.map k) (_ : G.map f \u226b G.map h = G.map g \u226b G.map k)\n[PROOFSTEP]\nsimp [\u2190 G.map_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : HasPullback f g\n\u22a2 (pullbackSymmetry f g).hom \u226b pullback.fst = pullback.snd\n[PROOFSTEP]\nsimp [pullbackSymmetry]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : HasPullback f g\n\u22a2 (pullbackSymmetry f g).hom \u226b pullback.snd = pullback.fst\n[PROOFSTEP]\nsimp [pullbackSymmetry]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : HasPullback f g\n\u22a2 (pullbackSymmetry f g).inv \u226b pullback.fst = pullback.snd\n[PROOFSTEP]\nsimp [Iso.inv_comp_eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : HasPullback f g\n\u22a2 (pullbackSymmetry f g).inv \u226b pullback.snd = pullback.fst\n[PROOFSTEP]\nsimp [Iso.inv_comp_eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : HasPushout f g\n\u22a2 pushout.inl \u226b (pushoutSymmetry f g).inv = pushout.inr\n[PROOFSTEP]\nsimp [Iso.comp_inv_eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : HasPushout f g\n\u22a2 pushout.inr \u226b (pushoutSymmetry f g).inv = pushout.inl\n[PROOFSTEP]\nsimp [Iso.comp_inv_eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 W\ng : Y \u27f6 W\ni : W \u27f6 Z\ninst\u271d\u00b9 : Mono i\ninst\u271d : HasPullback f g\n\u22a2 pullback.fst \u226b f \u226b i = pullback.snd \u226b g \u226b i\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 W\ng : Y \u27f6 W\ni : W \u27f6 Z\ninst\u271d\u00b9 : Mono i\ninst\u271d : HasPullback f g\n\u22a2 (pullback.fst \u226b f) \u226b i = (pullback.snd \u226b g) \u226b i\n[PROOFSTEP]\nrw [cancel_mono]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 W\ng : Y \u27f6 W\ni : W \u27f6 Z\ninst\u271d\u00b9 : Mono i\ninst\u271d : HasPullback f g\n\u22a2 pullback.fst \u226b f = pullback.snd \u226b g\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 W\ng : Y \u27f6 W\ni : W \u27f6 Z\ninst\u271d\u00b9 : Mono i\ninst\u271d : HasPullback f g\n\u22a2 C\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 W\ng : Y \u27f6 W\ni : W \u27f6 Z\ninst\u271d\u00b9 : Mono i\ninst\u271d : HasPullback f g\n\u22a2 X \u27f6 ?m.358762\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 W\ng : Y \u27f6 W\ni : W \u27f6 Z\ninst\u271d\u00b9 : Mono i\ninst\u271d : HasPullback f g\n\u22a2 Y \u27f6 ?m.358762\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 W\ng : Y \u27f6 W\ni : W \u27f6 Z\ninst\u271d\u00b9 : Mono i\ninst\u271d : HasPullback f g\n\u22a2 HasPullback ?m.358763 ?m.358764\n[PROOFSTEP]\napply pullback.condition\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 (g \u226b inv f) \u226b f = \ud835\udfd9 Y \u226b g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 NatTrans.app (pullbackConeOfLeftIso f g).\u03c0 none = g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso f\ns : PullbackCone f g\n\u22a2 PullbackCone.snd s \u226b PullbackCone.fst (pullbackConeOfLeftIso f g) = PullbackCone.fst s \u2227\n PullbackCone.snd s \u226b PullbackCone.snd (pullbackConeOfLeftIso f g) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (pullbackConeOfLeftIso f g).pt},\n m \u226b PullbackCone.fst (pullbackConeOfLeftIso f g) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (pullbackConeOfLeftIso f g) = PullbackCone.snd s \u2192 m = PullbackCone.snd s\n[PROOFSTEP]\nsimp [\u2190 s.condition_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 IsIso pullback.snd\n[PROOFSTEP]\nrefine' \u27e8\u27e8pullback.lift (g \u226b inv f) (\ud835\udfd9 _) (by simp), _, by simp\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 (g \u226b inv f) \u226b f = \ud835\udfd9 Y \u226b g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 pullback.lift (g \u226b inv f) (\ud835\udfd9 Y) (_ : (g \u226b inv f) \u226b f = \ud835\udfd9 Y \u226b g) \u226b pullback.snd = \ud835\udfd9 Y\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 pullback.snd \u226b pullback.lift (g \u226b inv f) (\ud835\udfd9 Y) (_ : (g \u226b inv f) \u226b f = \ud835\udfd9 Y \u226b g) = \ud835\udfd9 (pullback f g)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 (pullback.snd \u226b pullback.lift (g \u226b inv f) (\ud835\udfd9 Y) (_ : (g \u226b inv f) \u226b f = \ud835\udfd9 Y \u226b g)) \u226b pullback.fst =\n \ud835\udfd9 (pullback f g) \u226b pullback.fst\n[PROOFSTEP]\nsimp [\u2190 pullback.condition_assoc]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 (pullback.snd \u226b pullback.lift (g \u226b inv f) (\ud835\udfd9 Y) (_ : (g \u226b inv f) \u226b f = \ud835\udfd9 Y \u226b g)) \u226b pullback.snd =\n \ud835\udfd9 (pullback f g) \u226b pullback.snd\n[PROOFSTEP]\nsimp [pullback.condition_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n\u22a2 HasPullback i (f \u226b i)\n[PROOFSTEP]\nconv =>\n congr\n rw [\u2190 Category.id_comp i]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n| HasPullback i (f \u226b i)\n[PROOFSTEP]\n congr\n rw [\u2190 Category.id_comp i]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n| HasPullback i (f \u226b i)\n[PROOFSTEP]\n congr\n rw [\u2190 Category.id_comp i]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n| HasPullback i (f \u226b i)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase f\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n| i\ncase g\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n| f \u226b i\n[PROOFSTEP]\nrw [\u2190 Category.id_comp i]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n\u22a2 HasPullback (\ud835\udfd9 Z \u226b i) (f \u226b i)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n\u22a2 IsIso pullback.snd\n[PROOFSTEP]\nconvert\n (congrArg IsIso\n (show _ \u226b pullback.snd = _ from\n limit.isoLimitCone_hom_\u03c0 \u27e8_, pullbackIsPullbackOfCompMono (\ud835\udfd9 _) f i\u27e9 WalkingCospan.right)).mp\n inferInstance\n[GOAL]\ncase h.e'_3.h.e'_6.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n\u22a2 i = \ud835\udfd9 Z \u226b i\n[PROOFSTEP]\nexact (Category.id_comp _).symm\n[GOAL]\ncase h.e'_5.e'_6.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\ne_3\u271d : pullback i (f \u226b i) = limit (cospan (\ud835\udfd9 Z \u226b i) (f \u226b i))\ne_4\u271d : X = (cospan (\ud835\udfd9 Z \u226b i) (f \u226b i)).obj right\n\u22a2 i = \ud835\udfd9 Z \u226b i\n[PROOFSTEP]\nexact (Category.id_comp _).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 \ud835\udfd9 X \u226b f = (f \u226b inv g) \u226b g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 NatTrans.app (pullbackConeOfRightIso f g).\u03c0 none = f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso g\ns : PullbackCone f g\n\u22a2 PullbackCone.fst s \u226b PullbackCone.fst (pullbackConeOfRightIso f g) = PullbackCone.fst s \u2227\n PullbackCone.fst s \u226b PullbackCone.snd (pullbackConeOfRightIso f g) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (pullbackConeOfRightIso f g).pt},\n m \u226b PullbackCone.fst (pullbackConeOfRightIso f g) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (pullbackConeOfRightIso f g) = PullbackCone.snd s \u2192 m = PullbackCone.fst s\n[PROOFSTEP]\nsimp [s.condition_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 IsIso pullback.fst\n[PROOFSTEP]\nrefine' \u27e8\u27e8pullback.lift (\ud835\udfd9 _) (f \u226b inv g) (by simp), _, by simp\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 \ud835\udfd9 X \u226b f = (f \u226b inv g) \u226b g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 pullback.lift (\ud835\udfd9 X) (f \u226b inv g) (_ : \ud835\udfd9 X \u226b f = (f \u226b inv g) \u226b g) \u226b pullback.fst = \ud835\udfd9 X\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 pullback.fst \u226b pullback.lift (\ud835\udfd9 X) (f \u226b inv g) (_ : \ud835\udfd9 X \u226b f = (f \u226b inv g) \u226b g) = \ud835\udfd9 (pullback f g)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 (pullback.fst \u226b pullback.lift (\ud835\udfd9 X) (f \u226b inv g) (_ : \ud835\udfd9 X \u226b f = (f \u226b inv g) \u226b g)) \u226b pullback.fst =\n \ud835\udfd9 (pullback f g) \u226b pullback.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 (pullback.fst \u226b pullback.lift (\ud835\udfd9 X) (f \u226b inv g) (_ : \ud835\udfd9 X \u226b f = (f \u226b inv g) \u226b g)) \u226b pullback.snd =\n \ud835\udfd9 (pullback f g) \u226b pullback.snd\n[PROOFSTEP]\nsimp [pullback.condition_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso g\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n\u22a2 HasPullback (f \u226b i) i\n[PROOFSTEP]\nconv =>\n congr\n case g => rw [\u2190 Category.id_comp i]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso g\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n| HasPullback (f \u226b i) i\n[PROOFSTEP]\n congr\n case g => rw [\u2190 Category.id_comp i]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso g\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n| HasPullback (f \u226b i) i\n[PROOFSTEP]\n congr\n case g => rw [\u2190 Category.id_comp i]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso g\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n| HasPullback (f \u226b i) i\n[PROOFSTEP]\ncongr\n[GOAL]\ncase f\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso g\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n| f \u226b i\ncase g\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso g\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n| i\n[PROOFSTEP]\ncase g => rw [\u2190 Category.id_comp i]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso g\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n| i\n[PROOFSTEP]\nrw [\u2190 Category.id_comp i]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso g\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n| i\n[PROOFSTEP]\nrw [\u2190 Category.id_comp i]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso g\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n| i\n[PROOFSTEP]\nrw [\u2190 Category.id_comp i]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso g\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n\u22a2 HasPullback (f \u226b i) (\ud835\udfd9 Z \u226b i)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso g\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n\u22a2 IsIso pullback.fst\n[PROOFSTEP]\nconvert\n (congrArg IsIso\n (show _ \u226b pullback.fst = _ from\n limit.isoLimitCone_hom_\u03c0 \u27e8_, pullbackIsPullbackOfCompMono f (\ud835\udfd9 _) i\u27e9 WalkingCospan.left)).mp\n inferInstance\n[GOAL]\ncase h.e'_3.h.e'_7.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso g\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\n\u22a2 i = \ud835\udfd9 Z \u226b i\n[PROOFSTEP]\nexact (Category.id_comp _).symm\n[GOAL]\ncase h.e'_5.e'_7.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso g\ni : Z \u27f6 W\ninst\u271d : Mono i\nf : X \u27f6 Z\ne_3\u271d : pullback (f \u226b i) i = limit (cospan (f \u226b i) (\ud835\udfd9 Z \u226b i))\ne_4\u271d : X = (cospan (f \u226b i) (\ud835\udfd9 Z \u226b i)).obj left\n\u22a2 i = \ud835\udfd9 Z \u226b i\n[PROOFSTEP]\nexact (Category.id_comp _).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : W \u27f6 X\ninst\u271d\u00b9 : Epi h\ninst\u271d : HasPushout f g\n\u22a2 (h \u226b f) \u226b pushout.inl = (h \u226b g) \u226b pushout.inr\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : W \u27f6 X\ninst\u271d\u00b9 : Epi h\ninst\u271d : HasPushout f g\n\u22a2 h \u226b f \u226b pushout.inl = h \u226b g \u226b pushout.inr\n[PROOFSTEP]\nrw [cancel_epi]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : W \u27f6 X\ninst\u271d\u00b9 : Epi h\ninst\u271d : HasPushout f g\n\u22a2 f \u226b pushout.inl = g \u226b pushout.inr\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : W \u27f6 X\ninst\u271d\u00b9 : Epi h\ninst\u271d : HasPushout f g\n\u22a2 C\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : W \u27f6 X\ninst\u271d\u00b9 : Epi h\ninst\u271d : HasPushout f g\n\u22a2 ?m.421520 \u27f6 Y\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : W \u27f6 X\ninst\u271d\u00b9 : Epi h\ninst\u271d : HasPushout f g\n\u22a2 ?m.421520 \u27f6 Z\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : W \u27f6 X\ninst\u271d\u00b9 : Epi h\ninst\u271d : HasPushout f g\n\u22a2 HasPushout ?m.421523 ?m.421524\n[PROOFSTEP]\nexact pushout.condition\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 f \u226b inv f \u226b g = g \u226b \ud835\udfd9 Z\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 NatTrans.app (pushoutCoconeOfLeftIso f g).\u03b9 none = g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso f\ns : PushoutCocone f g\n\u22a2 PushoutCocone.inl (pushoutCoconeOfLeftIso f g) \u226b PushoutCocone.inr s = PushoutCocone.inl s \u2227\n PushoutCocone.inr (pushoutCoconeOfLeftIso f g) \u226b PushoutCocone.inr s = PushoutCocone.inr s \u2227\n \u2200 {m : (pushoutCoconeOfLeftIso f g).pt \u27f6 s.pt},\n PushoutCocone.inl (pushoutCoconeOfLeftIso f g) \u226b m = PushoutCocone.inl s \u2192\n PushoutCocone.inr (pushoutCoconeOfLeftIso f g) \u226b m = PushoutCocone.inr s \u2192 m = PushoutCocone.inr s\n[PROOFSTEP]\nsimp [\u2190 s.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 IsIso pushout.inr\n[PROOFSTEP]\nrefine' \u27e8\u27e8pushout.desc (inv f \u226b g) (\ud835\udfd9 _) (by simp), by simp, _\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 f \u226b inv f \u226b g = g \u226b \ud835\udfd9 Z\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 pushout.inr \u226b pushout.desc (inv f \u226b g) (\ud835\udfd9 Z) (_ : f \u226b inv f \u226b g = g \u226b \ud835\udfd9 Z) = \ud835\udfd9 Z\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 pushout.desc (inv f \u226b g) (\ud835\udfd9 Z) (_ : f \u226b inv f \u226b g = g \u226b \ud835\udfd9 Z) \u226b pushout.inr = \ud835\udfd9 (pushout f g)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 pushout.inl \u226b pushout.desc (inv f \u226b g) (\ud835\udfd9 Z) (_ : f \u226b inv f \u226b g = g \u226b \ud835\udfd9 Z) \u226b pushout.inr =\n pushout.inl \u226b \ud835\udfd9 (pushout f g)\n[PROOFSTEP]\nsimp [\u2190 pushout.condition]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso f\n\u22a2 pushout.inr \u226b pushout.desc (inv f \u226b g) (\ud835\udfd9 Z) (_ : f \u226b inv f \u226b g = g \u226b \ud835\udfd9 Z) \u226b pushout.inr =\n pushout.inr \u226b \ud835\udfd9 (pushout f g)\n[PROOFSTEP]\nsimp [pushout.condition_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n\u22a2 HasPushout h (h \u226b f)\n[PROOFSTEP]\nconv =>\n congr\n rw [\u2190 Category.comp_id h]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n| HasPushout h (h \u226b f)\n[PROOFSTEP]\n congr\n rw [\u2190 Category.comp_id h]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n| HasPushout h (h \u226b f)\n[PROOFSTEP]\n congr\n rw [\u2190 Category.comp_id h]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n| HasPushout h (h \u226b f)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase f\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n| h\ncase g\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n| h \u226b f\n[PROOFSTEP]\nrw [\u2190 Category.comp_id h]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n\u22a2 HasPushout (h \u226b \ud835\udfd9 X) (h \u226b f)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n\u22a2 IsIso pushout.inr\n[PROOFSTEP]\nconvert\n (congrArg IsIso\n (show pushout.inr \u226b _ = _ from\n colimit.isoColimitCocone_\u03b9_inv \u27e8_, pushoutIsPushoutOfEpiComp (\ud835\udfd9 _) f h\u27e9 WalkingSpan.right)).mp\n inferInstance\n[GOAL]\ncase h.e'_4.h.e'_6.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n\u22a2 h = h \u226b \ud835\udfd9 X\n[PROOFSTEP]\napply (Category.comp_id _).symm\n[GOAL]\ncase h.e'_5.e'_6.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso f\u271d\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\ne_3\u271d : Y = (span (h \u226b \ud835\udfd9 X) (h \u226b f)).obj right\ne_4\u271d : pushout h (h \u226b f) = colimit (span (h \u226b \ud835\udfd9 X) (h \u226b f))\n\u22a2 h = h \u226b \ud835\udfd9 X\n[PROOFSTEP]\napply (Category.comp_id _).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 f \u226b \ud835\udfd9 Y = g \u226b inv g \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 NatTrans.app (pushoutCoconeOfRightIso f g).\u03b9 none = f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso g\ns : PushoutCocone f g\n\u22a2 PushoutCocone.inl (pushoutCoconeOfRightIso f g) \u226b PushoutCocone.inl s = PushoutCocone.inl s \u2227\n PushoutCocone.inr (pushoutCoconeOfRightIso f g) \u226b PushoutCocone.inl s = PushoutCocone.inr s \u2227\n \u2200 {m : (pushoutCoconeOfRightIso f g).pt \u27f6 s.pt},\n PushoutCocone.inl (pushoutCoconeOfRightIso f g) \u226b m = PushoutCocone.inl s \u2192\n PushoutCocone.inr (pushoutCoconeOfRightIso f g) \u226b m = PushoutCocone.inr s \u2192 m = PushoutCocone.inl s\n[PROOFSTEP]\nsimp [\u2190 s.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 IsIso pushout.inl\n[PROOFSTEP]\nrefine' \u27e8\u27e8pushout.desc (\ud835\udfd9 _) (inv g \u226b f) (by simp), by simp, _\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 f \u226b \ud835\udfd9 Y = g \u226b inv g \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 pushout.inl \u226b pushout.desc (\ud835\udfd9 Y) (inv g \u226b f) (_ : f \u226b \ud835\udfd9 Y = g \u226b inv g \u226b f) = \ud835\udfd9 Y\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 pushout.desc (\ud835\udfd9 Y) (inv g \u226b f) (_ : f \u226b \ud835\udfd9 Y = g \u226b inv g \u226b f) \u226b pushout.inl = \ud835\udfd9 (pushout f g)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 pushout.inl \u226b pushout.desc (\ud835\udfd9 Y) (inv g \u226b f) (_ : f \u226b \ud835\udfd9 Y = g \u226b inv g \u226b f) \u226b pushout.inl =\n pushout.inl \u226b \ud835\udfd9 (pushout f g)\n[PROOFSTEP]\nsimp [\u2190 pushout.condition]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 pushout.inr \u226b pushout.desc (\ud835\udfd9 Y) (inv g \u226b f) (_ : f \u226b \ud835\udfd9 Y = g \u226b inv g \u226b f) \u226b pushout.inl =\n pushout.inr \u226b \ud835\udfd9 (pushout f g)\n[PROOFSTEP]\nsimp [pushout.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso g\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n\u22a2 HasPushout (h \u226b f) h\n[PROOFSTEP]\nconv =>\n congr\n case g => rw [\u2190 Category.comp_id h]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso g\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n| HasPushout (h \u226b f) h\n[PROOFSTEP]\n congr\n case g => rw [\u2190 Category.comp_id h]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso g\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n| HasPushout (h \u226b f) h\n[PROOFSTEP]\n congr\n case g => rw [\u2190 Category.comp_id h]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso g\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n| HasPushout (h \u226b f) h\n[PROOFSTEP]\ncongr\n[GOAL]\ncase f\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso g\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n| h \u226b f\ncase g\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso g\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n| h\n[PROOFSTEP]\ncase g => rw [\u2190 Category.comp_id h]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso g\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n| h\n[PROOFSTEP]\nrw [\u2190 Category.comp_id h]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso g\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n| h\n[PROOFSTEP]\nrw [\u2190 Category.comp_id h]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso g\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n| h\n[PROOFSTEP]\nrw [\u2190 Category.comp_id h]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso g\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n\u22a2 HasPushout (h \u226b f) (h \u226b \ud835\udfd9 X)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso g\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n\u22a2 IsIso pushout.inl\n[PROOFSTEP]\nconvert\n (congrArg IsIso\n (show pushout.inl \u226b _ = _ from\n colimit.isoColimitCocone_\u03b9_inv \u27e8_, pushoutIsPushoutOfEpiComp f (\ud835\udfd9 _) h\u27e9 WalkingSpan.left)).mp\n inferInstance\n[GOAL]\ncase h.e'_4.h.e'_7.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso g\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\n\u22a2 h = h \u226b \ud835\udfd9 X\n[PROOFSTEP]\nexact (Category.comp_id _).symm\n[GOAL]\ncase h.e'_5.e'_7.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d\u00b9 : IsIso g\nh : W \u27f6 X\ninst\u271d : Epi h\nf : X \u27f6 Y\ne_3\u271d : Y = (span (h \u226b f) (h \u226b \ud835\udfd9 X)).obj left\ne_4\u271d : pushout (h \u226b f) h = colimit (span (h \u226b f) (h \u226b \ud835\udfd9 X))\n\u22a2 h = h \u226b \ud835\udfd9 X\n[PROOFSTEP]\nexact (Category.comp_id _).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 (pullbackSymmetry f f).hom = \ud835\udfd9 (pullback f f)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 (pullbackSymmetry f f).hom \u226b pullback.fst = \ud835\udfd9 (pullback f f) \u226b pullback.fst\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 (pullbackSymmetry f f).hom \u226b pullback.snd = \ud835\udfd9 (pullback f f) \u226b pullback.snd\n[PROOFSTEP]\nsimp [fst_eq_snd_of_mono_eq]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 (pullbackSymmetry f f).hom \u226b pullback.snd = \ud835\udfd9 (pullback f f) \u226b pullback.snd\n[PROOFSTEP]\nsimp [fst_eq_snd_of_mono_eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 IsIso pullback.fst\n[PROOFSTEP]\nrefine' \u27e8\u27e8pullback.lift (\ud835\udfd9 _) (\ud835\udfd9 _) (by simp), _, by simp\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 \ud835\udfd9 X \u226b f = \ud835\udfd9 X \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 pullback.lift (\ud835\udfd9 X) (\ud835\udfd9 X) (_ : \ud835\udfd9 X \u226b f = \ud835\udfd9 X \u226b f) \u226b pullback.fst = \ud835\udfd9 X\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 pullback.fst \u226b pullback.lift (\ud835\udfd9 X) (\ud835\udfd9 X) (_ : \ud835\udfd9 X \u226b f = \ud835\udfd9 X \u226b f) = \ud835\udfd9 (pullback f f)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 (pullback.fst \u226b pullback.lift (\ud835\udfd9 X) (\ud835\udfd9 X) (_ : \ud835\udfd9 X \u226b f = \ud835\udfd9 X \u226b f)) \u226b pullback.fst = \ud835\udfd9 (pullback f f) \u226b pullback.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 (pullback.fst \u226b pullback.lift (\ud835\udfd9 X) (\ud835\udfd9 X) (_ : \ud835\udfd9 X \u226b f = \ud835\udfd9 X \u226b f)) \u226b pullback.snd = \ud835\udfd9 (pullback f f) \u226b pullback.snd\n[PROOFSTEP]\nsimp [fst_eq_snd_of_mono_eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 IsIso pullback.snd\n[PROOFSTEP]\nrw [\u2190 fst_eq_snd_of_mono_eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 IsIso pullback.fst\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Epi f\n\u22a2 (pushoutSymmetry f f).hom = \ud835\udfd9 (pushout f f)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Epi f\n\u22a2 pushout.inl \u226b (pushoutSymmetry f f).hom = pushout.inl \u226b \ud835\udfd9 (pushout f f)\n[PROOFSTEP]\nsimp [inl_eq_inr_of_epi_eq]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Epi f\n\u22a2 pushout.inr \u226b (pushoutSymmetry f f).hom = pushout.inr \u226b \ud835\udfd9 (pushout f f)\n[PROOFSTEP]\nsimp [inl_eq_inr_of_epi_eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Epi f\n\u22a2 IsIso pushout.inl\n[PROOFSTEP]\nrefine' \u27e8\u27e8pushout.desc (\ud835\udfd9 _) (\ud835\udfd9 _) (by simp), by simp, _\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Epi f\n\u22a2 f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Epi f\n\u22a2 pushout.inl \u226b pushout.desc (\ud835\udfd9 Y) (\ud835\udfd9 Y) (_ : f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y) = \ud835\udfd9 Y\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Epi f\n\u22a2 pushout.desc (\ud835\udfd9 Y) (\ud835\udfd9 Y) (_ : f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y) \u226b pushout.inl = \ud835\udfd9 (pushout f f)\n[PROOFSTEP]\napply pushout.hom_ext\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Epi f\n\u22a2 pushout.inl \u226b pushout.desc (\ud835\udfd9 Y) (\ud835\udfd9 Y) (_ : f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y) \u226b pushout.inl = pushout.inl \u226b \ud835\udfd9 (pushout f f)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Epi f\n\u22a2 pushout.inr \u226b pushout.desc (\ud835\udfd9 Y) (\ud835\udfd9 Y) (_ : f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y) \u226b pushout.inl = pushout.inr \u226b \ud835\udfd9 (pushout f f)\n[PROOFSTEP]\nsimp [inl_eq_inr_of_epi_eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Epi f\n\u22a2 IsIso pushout.inr\n[PROOFSTEP]\nrw [\u2190 inl_eq_inr_of_epi_eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ninst\u271d : Epi f\n\u22a2 IsIso pushout.inl\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\n\u22a2 i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083\n[PROOFSTEP]\nrw [\u2190 Category.assoc, h\u2081, Category.assoc, h\u2082, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\n\u22a2 IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\n[PROOFSTEP]\nfapply PullbackCone.isLimitAux'\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\n\u22a2 (s : PullbackCone (g\u2081 \u226b g\u2082) i\u2083) \u2192\n { l //\n l \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) =\n PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) =\n PullbackCone.snd s \u2192\n m = l }\n[PROOFSTEP]\nintro s\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\n\u22a2 { l //\n l \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) =\n PullbackCone.snd s \u2192\n m = l }\n[PROOFSTEP]\nhave : (s.fst \u226b g\u2081) \u226b g\u2082 = s.snd \u226b i\u2083 := by rw [\u2190 s.condition, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\n\u22a2 (PullbackCone.fst s \u226b g\u2081) \u226b g\u2082 = PullbackCone.snd s \u226b i\u2083\n[PROOFSTEP]\nrw [\u2190 s.condition, Category.assoc]\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\nthis : (PullbackCone.fst s \u226b g\u2081) \u226b g\u2082 = PullbackCone.snd s \u226b i\u2083\n\u22a2 { l //\n l \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) =\n PullbackCone.snd s \u2192\n m = l }\n[PROOFSTEP]\nrcases PullbackCone.IsLimit.lift' H (s.fst \u226b g\u2081) s.snd this with \u27e8l\u2081, hl\u2081, hl\u2081'\u27e9\n[GOAL]\ncase create.mk.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\nthis : (PullbackCone.fst s \u226b g\u2081) \u226b g\u2082 = PullbackCone.snd s \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2082 f\u2082 h\u2082).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.fst s \u226b g\u2081\nhl\u2081' : l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.snd s\n\u22a2 { l //\n l \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) =\n PullbackCone.snd s \u2192\n m = l }\n[PROOFSTEP]\nrcases PullbackCone.IsLimit.lift' H' s.fst l\u2081 hl\u2081.symm with \u27e8l\u2082, hl\u2082, hl\u2082'\u27e9\n[GOAL]\ncase create.mk.intro.mk.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\nthis : (PullbackCone.fst s \u226b g\u2081) \u226b g\u2082 = PullbackCone.snd s \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2082 f\u2082 h\u2082).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.fst s \u226b g\u2081\nhl\u2081' : l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.snd s\nl\u2082 : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhl\u2082 : l\u2082 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhl\u2082' : l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = l\u2081\n\u22a2 { l //\n l \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) =\n PullbackCone.snd s \u2192\n m = l }\n[PROOFSTEP]\nuse l\u2082\n[GOAL]\ncase property\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\nthis : (PullbackCone.fst s \u226b g\u2081) \u226b g\u2082 = PullbackCone.snd s \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2082 f\u2082 h\u2082).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.fst s \u226b g\u2081\nhl\u2081' : l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.snd s\nl\u2082 : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhl\u2082 : l\u2082 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhl\u2082' : l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = l\u2081\n\u22a2 l\u2082 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s \u2227\n l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u2192\n m = l\u2082\n[PROOFSTEP]\nuse hl\u2082\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\nthis : (PullbackCone.fst s \u226b g\u2081) \u226b g\u2082 = PullbackCone.snd s \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2082 f\u2082 h\u2082).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.fst s \u226b g\u2081\nhl\u2081' : l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.snd s\nl\u2082 : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhl\u2082 : l\u2082 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhl\u2082' : l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = l\u2081\n\u22a2 l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u2192\n m = l\u2082\n[PROOFSTEP]\nuse show l\u2082 \u226b f\u2081 \u226b f\u2082 = s.snd by\n rw [\u2190 hl\u2081', \u2190 hl\u2082', Category.assoc]\n rfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\nthis : (PullbackCone.fst s \u226b g\u2081) \u226b g\u2082 = PullbackCone.snd s \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2082 f\u2082 h\u2082).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.fst s \u226b g\u2081\nhl\u2081' : l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.snd s\nl\u2082 : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhl\u2082 : l\u2082 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhl\u2082' : l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = l\u2081\n\u22a2 l\u2082 \u226b f\u2081 \u226b f\u2082 = PullbackCone.snd s\n[PROOFSTEP]\nrw [\u2190 hl\u2081', \u2190 hl\u2082', Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\nthis : (PullbackCone.fst s \u226b g\u2081) \u226b g\u2082 = PullbackCone.snd s \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2082 f\u2082 h\u2082).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.fst s \u226b g\u2081\nhl\u2081' : l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.snd s\nl\u2082 : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhl\u2082 : l\u2082 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhl\u2082' : l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = l\u2081\n\u22a2 l\u2082 \u226b f\u2081 \u226b f\u2082 = l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\nthis : (PullbackCone.fst s \u226b g\u2081) \u226b g\u2082 = PullbackCone.snd s \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2082 f\u2082 h\u2082).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.fst s \u226b g\u2081\nhl\u2081' : l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.snd s\nl\u2082 : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhl\u2082 : l\u2082 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhl\u2082' : l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = l\u2081\n\u22a2 \u2200 {m : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u2192\n m = l\u2082\n[PROOFSTEP]\nintro m hm\u2081 hm\u2082\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\nthis : (PullbackCone.fst s \u226b g\u2081) \u226b g\u2082 = PullbackCone.snd s \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2082 f\u2082 h\u2082).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.fst s \u226b g\u2081\nhl\u2081' : l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.snd s\nl\u2082 : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhl\u2082 : l\u2082 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhl\u2082' : l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = l\u2081\nm : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhm\u2081 : m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhm\u2082 : m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s\n\u22a2 m = l\u2082\n[PROOFSTEP]\napply PullbackCone.IsLimit.hom_ext H'\n[GOAL]\ncase right.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\nthis : (PullbackCone.fst s \u226b g\u2081) \u226b g\u2082 = PullbackCone.snd s \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2082 f\u2082 h\u2082).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.fst s \u226b g\u2081\nhl\u2081' : l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.snd s\nl\u2082 : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhl\u2082 : l\u2082 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhl\u2082' : l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = l\u2081\nm : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhm\u2081 : m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhm\u2082 : m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s\n\u22a2 m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = l\u2082 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081)\n[PROOFSTEP]\nerw [hm\u2081, hl\u2082]\n[GOAL]\ncase right.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\nthis : (PullbackCone.fst s \u226b g\u2081) \u226b g\u2082 = PullbackCone.snd s \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2082 f\u2082 h\u2082).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.fst s \u226b g\u2081\nhl\u2081' : l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.snd s\nl\u2082 : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhl\u2082 : l\u2082 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhl\u2082' : l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = l\u2081\nm : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhm\u2081 : m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhm\u2082 : m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s\n\u22a2 m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081)\n[PROOFSTEP]\napply PullbackCone.IsLimit.hom_ext H\n[GOAL]\ncase right.h\u2081.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\nthis : (PullbackCone.fst s \u226b g\u2081) \u226b g\u2082 = PullbackCone.snd s \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2082 f\u2082 h\u2082).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.fst s \u226b g\u2081\nhl\u2081' : l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.snd s\nl\u2082 : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhl\u2082 : l\u2082 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhl\u2082' : l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = l\u2081\nm : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhm\u2081 : m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhm\u2082 : m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s\n\u22a2 (m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081)) \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082) =\n (l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081)) \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082)\n[PROOFSTEP]\nerw [Category.assoc, \u2190 h\u2081, \u2190 Category.assoc, hm\u2081, \u2190 hl\u2082, Category.assoc, Category.assoc, h\u2081]\n[GOAL]\ncase right.h\u2081.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\nthis : (PullbackCone.fst s \u226b g\u2081) \u226b g\u2082 = PullbackCone.snd s \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2082 f\u2082 h\u2082).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.fst s \u226b g\u2081\nhl\u2081' : l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.snd s\nl\u2082 : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhl\u2082 : l\u2082 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhl\u2082' : l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = l\u2081\nm : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhm\u2081 : m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhm\u2082 : m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s\n\u22a2 l\u2082 \u226b f\u2081 \u226b i\u2082 = l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right.h\u2081.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\ns : PullbackCone (g\u2081 \u226b g\u2082) i\u2083\nthis : (PullbackCone.fst s \u226b g\u2081) \u226b g\u2082 = PullbackCone.snd s \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2082 f\u2082 h\u2082).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.fst s \u226b g\u2081\nhl\u2081' : l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082) = PullbackCone.snd s\nl\u2082 : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhl\u2082 : l\u2082 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhl\u2082' : l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = l\u2081\nm : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhm\u2081 : m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhm\u2082 : m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s\n\u22a2 (m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081)) \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082) =\n (l\u2082 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081)) \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082)\n[PROOFSTEP]\nerw [Category.assoc, hm\u2082, \u2190 hl\u2081', \u2190 hl\u2082']\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\n\u22a2 i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083\n[PROOFSTEP]\nrw [\u2190 Category.assoc, h\u2081, Category.assoc, h\u2082, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\n\u22a2 IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\n[PROOFSTEP]\nfapply PushoutCocone.isColimitAux'\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\n\u22a2 (s : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)) \u2192\n { l //\n PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l = PushoutCocone.inl s \u2227\n PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l =\n PushoutCocone.inr s \u2227\n \u2200 {m : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt},\n PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m =\n PushoutCocone.inl s \u2192\n PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m =\n PushoutCocone.inr s \u2192\n m = l }\n[PROOFSTEP]\nintro s\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\n\u22a2 { l //\n PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l = PushoutCocone.inl s \u2227\n PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l = PushoutCocone.inr s \u2227\n \u2200 {m : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt},\n PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m =\n PushoutCocone.inl s \u2192\n PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m =\n PushoutCocone.inr s \u2192\n m = l }\n[PROOFSTEP]\nhave : i\u2081 \u226b s.inl = f\u2081 \u226b f\u2082 \u226b s.inr := by rw [s.condition, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\n\u22a2 i\u2081 \u226b PushoutCocone.inl s = f\u2081 \u226b f\u2082 \u226b PushoutCocone.inr s\n[PROOFSTEP]\nrw [s.condition, Category.assoc]\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\nthis : i\u2081 \u226b PushoutCocone.inl s = f\u2081 \u226b f\u2082 \u226b PushoutCocone.inr s\n\u22a2 { l //\n PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l = PushoutCocone.inl s \u2227\n PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l = PushoutCocone.inr s \u2227\n \u2200 {m : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt},\n PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m =\n PushoutCocone.inl s \u2192\n PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m =\n PushoutCocone.inr s \u2192\n m = l }\n[PROOFSTEP]\nrcases PushoutCocone.IsColimit.desc' H' s.inl (f\u2082 \u226b s.inr) this with \u27e8l\u2081, hl\u2081, hl\u2081'\u27e9\n[GOAL]\ncase create.mk.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\nthis : i\u2081 \u226b PushoutCocone.inl s = f\u2081 \u226b f\u2082 \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk g\u2081 i\u2082 h\u2081).pt \u27f6 s.pt\nhl\u2081 : PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = PushoutCocone.inl s\nhl\u2081' : PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = f\u2082 \u226b PushoutCocone.inr s\n\u22a2 { l //\n PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l = PushoutCocone.inl s \u2227\n PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l = PushoutCocone.inr s \u2227\n \u2200 {m : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt},\n PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m =\n PushoutCocone.inl s \u2192\n PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m =\n PushoutCocone.inr s \u2192\n m = l }\n[PROOFSTEP]\nrcases PushoutCocone.IsColimit.desc' H l\u2081 s.inr hl\u2081' with \u27e8l\u2082, hl\u2082, hl\u2082'\u27e9\n[GOAL]\ncase create.mk.intro.mk.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\nthis : i\u2081 \u226b PushoutCocone.inl s = f\u2081 \u226b f\u2082 \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk g\u2081 i\u2082 h\u2081).pt \u27f6 s.pt\nhl\u2081 : PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = PushoutCocone.inl s\nhl\u2081' : PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = f\u2082 \u226b PushoutCocone.inr s\nl\u2082 : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt\nhl\u2082 : PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = l\u2081\nhl\u2082' : PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = PushoutCocone.inr s\n\u22a2 { l //\n PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l = PushoutCocone.inl s \u2227\n PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l = PushoutCocone.inr s \u2227\n \u2200 {m : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt},\n PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m =\n PushoutCocone.inl s \u2192\n PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m =\n PushoutCocone.inr s \u2192\n m = l }\n[PROOFSTEP]\nuse l\u2082\n[GOAL]\ncase property\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\nthis : i\u2081 \u226b PushoutCocone.inl s = f\u2081 \u226b f\u2082 \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk g\u2081 i\u2082 h\u2081).pt \u27f6 s.pt\nhl\u2081 : PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = PushoutCocone.inl s\nhl\u2081' : PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = f\u2082 \u226b PushoutCocone.inr s\nl\u2082 : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt\nhl\u2082 : PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = l\u2081\nhl\u2082' : PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = PushoutCocone.inr s\n\u22a2 PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l\u2082 = PushoutCocone.inl s \u2227\n PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l\u2082 = PushoutCocone.inr s \u2227\n \u2200 {m : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt},\n PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m =\n PushoutCocone.inl s \u2192\n PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m =\n PushoutCocone.inr s \u2192\n m = l\u2082\n[PROOFSTEP]\nuse show (g\u2081 \u226b g\u2082) \u226b l\u2082 = s.inl by\n rw [\u2190 hl\u2081, \u2190 hl\u2082, Category.assoc]\n rfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\nthis : i\u2081 \u226b PushoutCocone.inl s = f\u2081 \u226b f\u2082 \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk g\u2081 i\u2082 h\u2081).pt \u27f6 s.pt\nhl\u2081 : PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = PushoutCocone.inl s\nhl\u2081' : PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = f\u2082 \u226b PushoutCocone.inr s\nl\u2082 : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt\nhl\u2082 : PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = l\u2081\nhl\u2082' : PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = PushoutCocone.inr s\n\u22a2 (g\u2081 \u226b g\u2082) \u226b l\u2082 = PushoutCocone.inl s\n[PROOFSTEP]\nrw [\u2190 hl\u2081, \u2190 hl\u2082, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\nthis : i\u2081 \u226b PushoutCocone.inl s = f\u2081 \u226b f\u2082 \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk g\u2081 i\u2082 h\u2081).pt \u27f6 s.pt\nhl\u2081 : PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = PushoutCocone.inl s\nhl\u2081' : PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = f\u2082 \u226b PushoutCocone.inr s\nl\u2082 : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt\nhl\u2082 : PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = l\u2081\nhl\u2082' : PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = PushoutCocone.inr s\n\u22a2 g\u2081 \u226b g\u2082 \u226b l\u2082 = PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\nthis : i\u2081 \u226b PushoutCocone.inl s = f\u2081 \u226b f\u2082 \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk g\u2081 i\u2082 h\u2081).pt \u27f6 s.pt\nhl\u2081 : PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = PushoutCocone.inl s\nhl\u2081' : PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = f\u2082 \u226b PushoutCocone.inr s\nl\u2082 : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt\nhl\u2082 : PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = l\u2081\nhl\u2082' : PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = PushoutCocone.inr s\n\u22a2 PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l\u2082 = PushoutCocone.inr s \u2227\n \u2200 {m : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt},\n PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m = PushoutCocone.inl s \u2192\n PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m =\n PushoutCocone.inr s \u2192\n m = l\u2082\n[PROOFSTEP]\nuse hl\u2082'\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\nthis : i\u2081 \u226b PushoutCocone.inl s = f\u2081 \u226b f\u2082 \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk g\u2081 i\u2082 h\u2081).pt \u27f6 s.pt\nhl\u2081 : PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = PushoutCocone.inl s\nhl\u2081' : PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = f\u2082 \u226b PushoutCocone.inr s\nl\u2082 : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt\nhl\u2082 : PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = l\u2081\nhl\u2082' : PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = PushoutCocone.inr s\n\u22a2 \u2200 {m : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt},\n PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m = PushoutCocone.inl s \u2192\n PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m = PushoutCocone.inr s \u2192\n m = l\u2082\n[PROOFSTEP]\nintro m hm\u2081 hm\u2082\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\nthis : i\u2081 \u226b PushoutCocone.inl s = f\u2081 \u226b f\u2082 \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk g\u2081 i\u2082 h\u2081).pt \u27f6 s.pt\nhl\u2081 : PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = PushoutCocone.inl s\nhl\u2081' : PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = f\u2082 \u226b PushoutCocone.inr s\nl\u2082 : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt\nhl\u2082 : PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = l\u2081\nhl\u2082' : PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = PushoutCocone.inr s\nm : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhm\u2081 : PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m = PushoutCocone.inl s\nhm\u2082 : PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m = PushoutCocone.inr s\n\u22a2 m = l\u2082\n[PROOFSTEP]\napply PushoutCocone.IsColimit.hom_ext H\n[GOAL]\ncase right.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\nthis : i\u2081 \u226b PushoutCocone.inl s = f\u2081 \u226b f\u2082 \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk g\u2081 i\u2082 h\u2081).pt \u27f6 s.pt\nhl\u2081 : PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = PushoutCocone.inl s\nhl\u2081' : PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = f\u2082 \u226b PushoutCocone.inr s\nl\u2082 : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt\nhl\u2082 : PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = l\u2081\nhl\u2082' : PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = PushoutCocone.inr s\nm : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhm\u2081 : PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m = PushoutCocone.inl s\nhm\u2082 : PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m = PushoutCocone.inr s\n\u22a2 PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b m = PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082\n[PROOFSTEP]\napply PushoutCocone.IsColimit.hom_ext H'\n[GOAL]\ncase right.h\u2080.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\nthis : i\u2081 \u226b PushoutCocone.inl s = f\u2081 \u226b f\u2082 \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk g\u2081 i\u2082 h\u2081).pt \u27f6 s.pt\nhl\u2081 : PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = PushoutCocone.inl s\nhl\u2081' : PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = f\u2082 \u226b PushoutCocone.inr s\nl\u2082 : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt\nhl\u2082 : PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = l\u2081\nhl\u2082' : PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = PushoutCocone.inr s\nm : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhm\u2081 : PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m = PushoutCocone.inl s\nhm\u2082 : PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m = PushoutCocone.inr s\n\u22a2 PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b m =\n PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082\n[PROOFSTEP]\nerw [\u2190 Category.assoc, hm\u2081, hl\u2082, hl\u2081]\n[GOAL]\ncase right.h\u2080.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\nthis : i\u2081 \u226b PushoutCocone.inl s = f\u2081 \u226b f\u2082 \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk g\u2081 i\u2082 h\u2081).pt \u27f6 s.pt\nhl\u2081 : PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = PushoutCocone.inl s\nhl\u2081' : PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = f\u2082 \u226b PushoutCocone.inr s\nl\u2082 : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt\nhl\u2082 : PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = l\u2081\nhl\u2082' : PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = PushoutCocone.inr s\nm : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhm\u2081 : PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m = PushoutCocone.inl s\nhm\u2082 : PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m = PushoutCocone.inr s\n\u22a2 PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b m =\n PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082\n[PROOFSTEP]\nerw [\u2190 Category.assoc, h\u2082, Category.assoc, hm\u2082, \u2190 hl\u2082', \u2190 Category.assoc, \u2190 Category.assoc, \u2190 h\u2082]\n[GOAL]\ncase right.h\u2080.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\nthis : i\u2081 \u226b PushoutCocone.inl s = f\u2081 \u226b f\u2082 \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk g\u2081 i\u2082 h\u2081).pt \u27f6 s.pt\nhl\u2081 : PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = PushoutCocone.inl s\nhl\u2081' : PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = f\u2082 \u226b PushoutCocone.inr s\nl\u2082 : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt\nhl\u2082 : PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = l\u2081\nhl\u2082' : PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = PushoutCocone.inr s\nm : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhm\u2081 : PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m = PushoutCocone.inl s\nhm\u2082 : PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m = PushoutCocone.inr s\n\u22a2 (i\u2082 \u226b g\u2082) \u226b l\u2082 = (PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082)) \u226b l\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\nH' : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\ns : PushoutCocone i\u2081 (f\u2081 \u226b f\u2082)\nthis : i\u2081 \u226b PushoutCocone.inl s = f\u2081 \u226b f\u2082 \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk g\u2081 i\u2082 h\u2081).pt \u27f6 s.pt\nhl\u2081 : PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = PushoutCocone.inl s\nhl\u2081' : PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b l\u2081 = f\u2082 \u226b PushoutCocone.inr s\nl\u2082 : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt\nhl\u2082 : PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = l\u2081\nhl\u2082' : PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082 = PushoutCocone.inr s\nm : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhm\u2081 : PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m = PushoutCocone.inl s\nhm\u2082 : PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m = PushoutCocone.inr s\n\u22a2 PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b m = PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l\u2082\n[PROOFSTEP]\nerw [hm\u2082, hl\u2082']\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\n\u22a2 i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083\n[PROOFSTEP]\nrw [\u2190 Category.assoc, h\u2081, Category.assoc, h\u2082, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\n\u22a2 IsLimit (PullbackCone.mk i\u2081 f\u2081 h\u2081)\n[PROOFSTEP]\nfapply PullbackCone.isLimitAux'\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\n\u22a2 (s : PullbackCone g\u2081 i\u2082) \u2192\n { l //\n l \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s \u2192 m = l }\n[PROOFSTEP]\nintro s\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\n\u22a2 { l //\n l \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s \u2192 m = l }\n[PROOFSTEP]\nhave : s.fst \u226b g\u2081 \u226b g\u2082 = (s.snd \u226b f\u2082) \u226b i\u2083 := by rw [\u2190 Category.assoc, s.condition, Category.assoc, Category.assoc, h\u2082]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\n\u22a2 PullbackCone.fst s \u226b g\u2081 \u226b g\u2082 = (PullbackCone.snd s \u226b f\u2082) \u226b i\u2083\n[PROOFSTEP]\nrw [\u2190 Category.assoc, s.condition, Category.assoc, Category.assoc, h\u2082]\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\nthis : PullbackCone.fst s \u226b g\u2081 \u226b g\u2082 = (PullbackCone.snd s \u226b f\u2082) \u226b i\u2083\n\u22a2 { l //\n l \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s \u2192 m = l }\n[PROOFSTEP]\nrcases PullbackCone.IsLimit.lift' H' s.fst (s.snd \u226b f\u2082) this with \u27e8l\u2081, hl\u2081, hl\u2081'\u27e9\n[GOAL]\ncase create.mk.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\nthis : PullbackCone.fst s \u226b g\u2081 \u226b g\u2082 = (PullbackCone.snd s \u226b f\u2082) \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhl\u2081' :\n l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u226b f\u2082\n\u22a2 { l //\n l \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s \u2192 m = l }\n[PROOFSTEP]\nuse l\u2081\n[GOAL]\ncase property\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\nthis : PullbackCone.fst s \u226b g\u2081 \u226b g\u2082 = (PullbackCone.snd s \u226b f\u2082) \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhl\u2081' :\n l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u226b f\u2082\n\u22a2 l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s \u2227\n l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s \u2192 m = l\u2081\n[PROOFSTEP]\nuse hl\u2081\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\nthis : PullbackCone.fst s \u226b g\u2081 \u226b g\u2082 = (PullbackCone.snd s \u226b f\u2082) \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhl\u2081' :\n l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u226b f\u2082\n\u22a2 l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s \u2192 m = l\u2081\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase right.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\nthis : PullbackCone.fst s \u226b g\u2081 \u226b g\u2082 = (PullbackCone.snd s \u226b f\u2082) \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhl\u2081' :\n l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u226b f\u2082\n\u22a2 l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s\n[PROOFSTEP]\napply PullbackCone.IsLimit.hom_ext H\n[GOAL]\ncase right.left.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\nthis : PullbackCone.fst s \u226b g\u2081 \u226b g\u2082 = (PullbackCone.snd s \u226b f\u2082) \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhl\u2081' :\n l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u226b f\u2082\n\u22a2 (l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081)) \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082) =\n PullbackCone.snd s \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082)\n[PROOFSTEP]\nerw [Category.assoc, \u2190 h\u2081, \u2190 Category.assoc, hl\u2081, s.condition]\n[GOAL]\ncase right.left.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\nthis : PullbackCone.fst s \u226b g\u2081 \u226b g\u2082 = (PullbackCone.snd s \u226b f\u2082) \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhl\u2081' :\n l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u226b f\u2082\n\u22a2 PullbackCone.snd s \u226b i\u2082 = PullbackCone.snd s \u226b PullbackCone.fst (PullbackCone.mk i\u2082 f\u2082 h\u2082)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right.left.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\nthis : PullbackCone.fst s \u226b g\u2081 \u226b g\u2082 = (PullbackCone.snd s \u226b f\u2082) \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhl\u2081' :\n l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u226b f\u2082\n\u22a2 (l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081)) \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082) =\n PullbackCone.snd s \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082)\n[PROOFSTEP]\nerw [Category.assoc, hl\u2081']\n[GOAL]\ncase right.left.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\nthis : PullbackCone.fst s \u226b g\u2081 \u226b g\u2082 = (PullbackCone.snd s \u226b f\u2082) \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhl\u2081' :\n l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u226b f\u2082\n\u22a2 PullbackCone.snd s \u226b f\u2082 = PullbackCone.snd s \u226b PullbackCone.snd (PullbackCone.mk i\u2082 f\u2082 h\u2082)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\nthis : PullbackCone.fst s \u226b g\u2081 \u226b g\u2082 = (PullbackCone.snd s \u226b f\u2082) \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhl\u2081' :\n l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u226b f\u2082\n\u22a2 \u2200 {m : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s \u2192 m = l\u2081\n[PROOFSTEP]\nintro m hm\u2081 hm\u2082\n[GOAL]\ncase right.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\nthis : PullbackCone.fst s \u226b g\u2081 \u226b g\u2082 = (PullbackCone.snd s \u226b f\u2082) \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhl\u2081' :\n l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u226b f\u2082\nm : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhm\u2081 : m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhm\u2082 : m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s\n\u22a2 m = l\u2081\n[PROOFSTEP]\napply PullbackCone.IsLimit.hom_ext H'\n[GOAL]\ncase right.right.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\nthis : PullbackCone.fst s \u226b g\u2081 \u226b g\u2082 = (PullbackCone.snd s \u226b f\u2082) \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhl\u2081' :\n l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u226b f\u2082\nm : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhm\u2081 : m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhm\u2082 : m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s\n\u22a2 m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) =\n l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\n[PROOFSTEP]\nerw [hm\u2081, hl\u2081]\n[GOAL]\ncase right.right.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\nthis : PullbackCone.fst s \u226b g\u2081 \u226b g\u2082 = (PullbackCone.snd s \u226b f\u2082) \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhl\u2081' :\n l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u226b f\u2082\nm : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhm\u2081 : m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhm\u2082 : m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s\n\u22a2 m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) =\n l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\n[PROOFSTEP]\nerw [hl\u2081', \u2190 hm\u2082]\n[GOAL]\ncase right.right.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsLimit (PullbackCone.mk i\u2082 f\u2082 h\u2082)\nH' : IsLimit (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PullbackCone g\u2081 i\u2082\nthis : PullbackCone.fst s \u226b g\u2081 \u226b g\u2082 = (PullbackCone.snd s \u226b f\u2082) \u226b i\u2083\nl\u2081 : s.pt \u27f6 (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt\nhl\u2081 : l\u2081 \u226b PullbackCone.fst (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.fst s\nhl\u2081' :\n l\u2081 \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) = PullbackCone.snd s \u226b f\u2082\nm : s.pt \u27f6 (PullbackCone.mk i\u2081 f\u2081 h\u2081).pt\nhm\u2081 : m \u226b PullbackCone.fst (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.fst s\nhm\u2082 : m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081) = PullbackCone.snd s\n\u22a2 m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 (f\u2081 \u226b f\u2082) (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) =\n (m \u226b PullbackCone.snd (PullbackCone.mk i\u2081 f\u2081 h\u2081)) \u226b f\u2082\n[PROOFSTEP]\nexact (Category.assoc _ _ _).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\n\u22a2 i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083\n[PROOFSTEP]\nrw [\u2190 Category.assoc, h\u2081, Category.assoc, h\u2082, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\n\u22a2 IsColimit (PushoutCocone.mk g\u2082 i\u2083 h\u2082)\n[PROOFSTEP]\nfapply PushoutCocone.isColimitAux'\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\n\u22a2 (s : PushoutCocone i\u2082 f\u2082) \u2192\n { l //\n PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l = PushoutCocone.inl s \u2227\n PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l = PushoutCocone.inr s \u2227\n \u2200 {m : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt},\n PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b m = PushoutCocone.inl s \u2192\n PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b m = PushoutCocone.inr s \u2192 m = l }\n[PROOFSTEP]\nintro s\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PushoutCocone i\u2082 f\u2082\n\u22a2 { l //\n PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l = PushoutCocone.inl s \u2227\n PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l = PushoutCocone.inr s \u2227\n \u2200 {m : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt},\n PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b m = PushoutCocone.inl s \u2192\n PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b m = PushoutCocone.inr s \u2192 m = l }\n[PROOFSTEP]\nhave : i\u2081 \u226b g\u2081 \u226b s.inl = (f\u2081 \u226b f\u2082) \u226b s.inr := by\n rw [Category.assoc, \u2190 s.condition, \u2190 Category.assoc, \u2190 Category.assoc, h\u2081]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PushoutCocone i\u2082 f\u2082\n\u22a2 i\u2081 \u226b g\u2081 \u226b PushoutCocone.inl s = (f\u2081 \u226b f\u2082) \u226b PushoutCocone.inr s\n[PROOFSTEP]\nrw [Category.assoc, \u2190 s.condition, \u2190 Category.assoc, \u2190 Category.assoc, h\u2081]\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PushoutCocone i\u2082 f\u2082\nthis : i\u2081 \u226b g\u2081 \u226b PushoutCocone.inl s = (f\u2081 \u226b f\u2082) \u226b PushoutCocone.inr s\n\u22a2 { l //\n PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l = PushoutCocone.inl s \u2227\n PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l = PushoutCocone.inr s \u2227\n \u2200 {m : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt},\n PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b m = PushoutCocone.inl s \u2192\n PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b m = PushoutCocone.inr s \u2192 m = l }\n[PROOFSTEP]\nrcases PushoutCocone.IsColimit.desc' H' (g\u2081 \u226b s.inl) s.inr this with \u27e8l\u2081, hl\u2081, hl\u2081'\u27e9\n[GOAL]\ncase create.mk.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PushoutCocone i\u2082 f\u2082\nthis : i\u2081 \u226b g\u2081 \u226b PushoutCocone.inl s = (f\u2081 \u226b f\u2082) \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhl\u2081 :\n PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l\u2081 = g\u2081 \u226b PushoutCocone.inl s\nhl\u2081' : PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l\u2081 = PushoutCocone.inr s\n\u22a2 { l //\n PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l = PushoutCocone.inl s \u2227\n PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b l = PushoutCocone.inr s \u2227\n \u2200 {m : (PushoutCocone.mk g\u2082 i\u2083 h\u2082).pt \u27f6 s.pt},\n PushoutCocone.inl (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b m = PushoutCocone.inl s \u2192\n PushoutCocone.inr (PushoutCocone.mk g\u2082 i\u2083 h\u2082) \u226b m = PushoutCocone.inr s \u2192 m = l }\n[PROOFSTEP]\ndsimp at *\n[GOAL]\ncase create.mk.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PushoutCocone i\u2082 f\u2082\nthis : i\u2081 \u226b g\u2081 \u226b PushoutCocone.inl s = (f\u2081 \u226b f\u2082) \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhl\u2081 : (g\u2081 \u226b g\u2082) \u226b l\u2081 = g\u2081 \u226b PushoutCocone.inl s\nhl\u2081' : i\u2083 \u226b l\u2081 = PushoutCocone.inr s\n\u22a2 { l //\n g\u2082 \u226b l = PushoutCocone.inl s \u2227\n i\u2083 \u226b l = PushoutCocone.inr s \u2227\n \u2200 {m : Y\u2083 \u27f6 s.pt}, g\u2082 \u226b m = PushoutCocone.inl s \u2192 i\u2083 \u226b m = PushoutCocone.inr s \u2192 m = l }\n[PROOFSTEP]\nuse l\u2081\n[GOAL]\ncase property\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PushoutCocone i\u2082 f\u2082\nthis : i\u2081 \u226b g\u2081 \u226b PushoutCocone.inl s = (f\u2081 \u226b f\u2082) \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhl\u2081 : (g\u2081 \u226b g\u2082) \u226b l\u2081 = g\u2081 \u226b PushoutCocone.inl s\nhl\u2081' : i\u2083 \u226b l\u2081 = PushoutCocone.inr s\n\u22a2 g\u2082 \u226b l\u2081 = PushoutCocone.inl s \u2227\n i\u2083 \u226b l\u2081 = PushoutCocone.inr s \u2227\n \u2200 {m : Y\u2083 \u27f6 s.pt}, g\u2082 \u226b m = PushoutCocone.inl s \u2192 i\u2083 \u226b m = PushoutCocone.inr s \u2192 m = l\u2081\n[PROOFSTEP]\nrefine' \u27e8_, _, _\u27e9\n[GOAL]\ncase property.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PushoutCocone i\u2082 f\u2082\nthis : i\u2081 \u226b g\u2081 \u226b PushoutCocone.inl s = (f\u2081 \u226b f\u2082) \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhl\u2081 : (g\u2081 \u226b g\u2082) \u226b l\u2081 = g\u2081 \u226b PushoutCocone.inl s\nhl\u2081' : i\u2083 \u226b l\u2081 = PushoutCocone.inr s\n\u22a2 g\u2082 \u226b l\u2081 = PushoutCocone.inl s\n[PROOFSTEP]\napply PushoutCocone.IsColimit.hom_ext H\n[GOAL]\ncase property.refine'_1.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PushoutCocone i\u2082 f\u2082\nthis : i\u2081 \u226b g\u2081 \u226b PushoutCocone.inl s = (f\u2081 \u226b f\u2082) \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhl\u2081 : (g\u2081 \u226b g\u2082) \u226b l\u2081 = g\u2081 \u226b PushoutCocone.inl s\nhl\u2081' : i\u2083 \u226b l\u2081 = PushoutCocone.inr s\n\u22a2 PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b g\u2082 \u226b l\u2081 =\n PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b PushoutCocone.inl s\n[PROOFSTEP]\nerw [\u2190 Category.assoc, hl\u2081]\n[GOAL]\ncase property.refine'_1.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PushoutCocone i\u2082 f\u2082\nthis : i\u2081 \u226b g\u2081 \u226b PushoutCocone.inl s = (f\u2081 \u226b f\u2082) \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhl\u2081 : (g\u2081 \u226b g\u2082) \u226b l\u2081 = g\u2081 \u226b PushoutCocone.inl s\nhl\u2081' : i\u2083 \u226b l\u2081 = PushoutCocone.inr s\n\u22a2 g\u2081 \u226b PushoutCocone.inl s = PushoutCocone.inl (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b PushoutCocone.inl s\n[PROOFSTEP]\nrfl\n[GOAL]\ncase property.refine'_1.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PushoutCocone i\u2082 f\u2082\nthis : i\u2081 \u226b g\u2081 \u226b PushoutCocone.inl s = (f\u2081 \u226b f\u2082) \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhl\u2081 : (g\u2081 \u226b g\u2082) \u226b l\u2081 = g\u2081 \u226b PushoutCocone.inl s\nhl\u2081' : i\u2083 \u226b l\u2081 = PushoutCocone.inr s\n\u22a2 PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b g\u2082 \u226b l\u2081 =\n PushoutCocone.inr (PushoutCocone.mk g\u2081 i\u2082 h\u2081) \u226b PushoutCocone.inl s\n[PROOFSTEP]\nerw [\u2190 Category.assoc, h\u2082, Category.assoc, hl\u2081', s.condition]\n[GOAL]\ncase property.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PushoutCocone i\u2082 f\u2082\nthis : i\u2081 \u226b g\u2081 \u226b PushoutCocone.inl s = (f\u2081 \u226b f\u2082) \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhl\u2081 : (g\u2081 \u226b g\u2082) \u226b l\u2081 = g\u2081 \u226b PushoutCocone.inl s\nhl\u2081' : i\u2083 \u226b l\u2081 = PushoutCocone.inr s\n\u22a2 i\u2083 \u226b l\u2081 = PushoutCocone.inr s\n[PROOFSTEP]\nexact hl\u2081'\n[GOAL]\ncase property.refine'_3\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PushoutCocone i\u2082 f\u2082\nthis : i\u2081 \u226b g\u2081 \u226b PushoutCocone.inl s = (f\u2081 \u226b f\u2082) \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhl\u2081 : (g\u2081 \u226b g\u2082) \u226b l\u2081 = g\u2081 \u226b PushoutCocone.inl s\nhl\u2081' : i\u2083 \u226b l\u2081 = PushoutCocone.inr s\n\u22a2 \u2200 {m : Y\u2083 \u27f6 s.pt}, g\u2082 \u226b m = PushoutCocone.inl s \u2192 i\u2083 \u226b m = PushoutCocone.inr s \u2192 m = l\u2081\n[PROOFSTEP]\nintro m hm\u2081 hm\u2082\n[GOAL]\ncase property.refine'_3\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PushoutCocone i\u2082 f\u2082\nthis : i\u2081 \u226b g\u2081 \u226b PushoutCocone.inl s = (f\u2081 \u226b f\u2082) \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhl\u2081 : (g\u2081 \u226b g\u2082) \u226b l\u2081 = g\u2081 \u226b PushoutCocone.inl s\nhl\u2081' : i\u2083 \u226b l\u2081 = PushoutCocone.inr s\nm : Y\u2083 \u27f6 s.pt\nhm\u2081 : g\u2082 \u226b m = PushoutCocone.inl s\nhm\u2082 : i\u2083 \u226b m = PushoutCocone.inr s\n\u22a2 m = l\u2081\n[PROOFSTEP]\napply PushoutCocone.IsColimit.hom_ext H'\n[GOAL]\ncase property.refine'_3.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PushoutCocone i\u2082 f\u2082\nthis : i\u2081 \u226b g\u2081 \u226b PushoutCocone.inl s = (f\u2081 \u226b f\u2082) \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhl\u2081 : (g\u2081 \u226b g\u2082) \u226b l\u2081 = g\u2081 \u226b PushoutCocone.inl s\nhl\u2081' : i\u2083 \u226b l\u2081 = PushoutCocone.inr s\nm : Y\u2083 \u27f6 s.pt\nhm\u2081 : g\u2082 \u226b m = PushoutCocone.inl s\nhm\u2082 : i\u2083 \u226b m = PushoutCocone.inr s\n\u22a2 PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m =\n PushoutCocone.inl (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l\u2081\n[PROOFSTEP]\nerw [hl\u2081, Category.assoc, hm\u2081]\n[GOAL]\ncase property.refine'_3.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 X\u2082\nf\u2082 : X\u2082 \u27f6 X\u2083\ng\u2081 : Y\u2081 \u27f6 Y\u2082\ng\u2082 : Y\u2082 \u27f6 Y\u2083\ni\u2081 : X\u2081 \u27f6 Y\u2081\ni\u2082 : X\u2082 \u27f6 Y\u2082\ni\u2083 : X\u2083 \u27f6 Y\u2083\nh\u2081 : i\u2081 \u226b g\u2081 = f\u2081 \u226b i\u2082\nh\u2082 : i\u2082 \u226b g\u2082 = f\u2082 \u226b i\u2083\nH : IsColimit (PushoutCocone.mk g\u2081 i\u2082 h\u2081)\nH' : IsColimit (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083))\ns : PushoutCocone i\u2082 f\u2082\nthis : i\u2081 \u226b g\u2081 \u226b PushoutCocone.inl s = (f\u2081 \u226b f\u2082) \u226b PushoutCocone.inr s\nl\u2081 : (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)).pt \u27f6 s.pt\nhl\u2081 : (g\u2081 \u226b g\u2082) \u226b l\u2081 = g\u2081 \u226b PushoutCocone.inl s\nhl\u2081' : i\u2083 \u226b l\u2081 = PushoutCocone.inr s\nm : Y\u2083 \u27f6 s.pt\nhm\u2081 : g\u2082 \u226b m = PushoutCocone.inl s\nhm\u2082 : i\u2083 \u226b m = PushoutCocone.inr s\n\u22a2 PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b m =\n PushoutCocone.inr (PushoutCocone.mk (g\u2081 \u226b g\u2082) i\u2083 (_ : i\u2081 \u226b g\u2081 \u226b g\u2082 = (f\u2081 \u226b f\u2082) \u226b i\u2083)) \u226b l\u2081\n[PROOFSTEP]\nerw [hm\u2082, hl\u2081']\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : W \u27f6 X\ninst\u271d\u00b2 : HasPullback f g\ninst\u271d\u00b9 : HasPullback f' pullback.fst\ninst\u271d : HasPullback (f' \u226b f) g\n\u22a2 pullback f' pullback.fst \u2245 pullback (f' \u226b f) g\n[PROOFSTEP]\nlet this :=\n bigSquareIsPullback (pullback.snd : pullback f' (pullback.fst : pullback f g \u27f6 _) \u27f6 _) pullback.snd f' f pullback.fst\n pullback.fst g pullback.condition pullback.condition (pullbackIsPullback _ _) (pullbackIsPullback _ _)\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : W \u27f6 X\ninst\u271d\u00b2 : HasPullback f g\ninst\u271d\u00b9 : HasPullback f' pullback.fst\ninst\u271d : HasPullback (f' \u226b f) g\nthis : IsLimit\n (PullbackCone.mk pullback.fst (pullback.snd \u226b pullback.snd)\n (_ : pullback.fst \u226b f' \u226b f = (pullback.snd \u226b pullback.snd) \u226b g)) :=\n bigSquareIsPullback pullback.snd pullback.snd f' f pullback.fst pullback.fst g\n (_ : pullback.fst \u226b f' = pullback.snd \u226b pullback.fst) (_ : pullback.fst \u226b f = pullback.snd \u226b g)\n (pullbackIsPullback f g) (pullbackIsPullback f' pullback.fst)\n\u22a2 pullback f' pullback.fst \u2245 pullback (f' \u226b f) g\n[PROOFSTEP]\nexact (this.conePointUniqueUpToIso (pullbackIsPullback _ _) : _)\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : W \u27f6 X\ninst\u271d\u00b2 : HasPullback f g\ninst\u271d\u00b9 : HasPullback f' pullback.fst\ninst\u271d : HasPullback (f' \u226b f) g\n\u22a2 (pullbackRightPullbackFstIso f g f').inv \u226b pullback.snd \u226b pullback.fst = pullback.fst \u226b f'\n[PROOFSTEP]\nrw [\u2190 pullback.condition]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : W \u27f6 X\ninst\u271d\u00b2 : HasPullback f g\ninst\u271d\u00b9 : HasPullback f' pullback.fst\ninst\u271d : HasPullback (f' \u226b f) g\n\u22a2 (pullbackRightPullbackFstIso f g f').inv \u226b pullback.fst \u226b f' = pullback.fst \u226b f'\n[PROOFSTEP]\nexact pullbackRightPullbackFstIso_inv_fst_assoc _ _ _ _\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ng' : Z \u27f6 W\ninst\u271d\u00b2 : HasPushout f g\ninst\u271d\u00b9 : HasPushout pushout.inr g'\ninst\u271d : HasPushout f (g \u226b g')\n\u22a2 pushout.inr \u226b (pushoutLeftPushoutInrIso f g g').inv = pushout.inr\n[PROOFSTEP]\nrw [Iso.comp_inv_eq, inr_pushoutLeftPushoutInrIso_hom]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ng' : Z \u27f6 W\ninst\u271d\u00b2 : HasPushout f g\ninst\u271d\u00b9 : HasPushout pushout.inr g'\ninst\u271d : HasPushout f (g \u226b g')\n\u22a2 pushout.inl \u226b pushout.inl \u226b (pushoutLeftPushoutInrIso f g g').hom = pushout.inl\n[PROOFSTEP]\nrw [\u2190 Category.assoc, \u2190 Iso.eq_comp_inv, inl_pushoutLeftPushoutInrIso_inv]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ng' : Z \u27f6 W\ninst\u271d\u00b2 : HasPushout f g\ninst\u271d\u00b9 : HasPushout pushout.inr g'\ninst\u271d : HasPushout f (g \u226b g')\n\u22a2 pushout.inr \u226b pushout.inl \u226b (pushoutLeftPushoutInrIso f g g').hom = g' \u226b pushout.inr\n[PROOFSTEP]\nrw [\u2190 Category.assoc, \u2190 Iso.eq_comp_inv, Category.assoc, inr_pushoutLeftPushoutInrIso_inv, pushout.condition]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 IsLimit\n (PullbackCone.mk pullback.fst\n (pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084))\n (_ :\n pullback.fst \u226b pullback.snd =\n pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.fst))\n[PROOFSTEP]\napply leftSquareIsPullback\n[GOAL]\ncase H\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 IsLimit (PullbackCone.mk pullback.fst ?f\u2082 ?h\u2082)\ncase H'\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 IsLimit\n (PullbackCone.mk pullback.fst\n (pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n ?f\u2082)\n (_ :\n pullback.fst \u226b pullback.snd \u226b ?g\u2082 =\n (pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n ?f\u2082) \u226b\n ?i\u2083))\ncase X\u2083\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 C\ncase Y\u2083\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 C\ncase f\u2082\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 Z\u2082 \u27f6 ?X\u2083\ncase g\u2082\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 X\u2082 \u27f6 ?Y\u2083\ncase i\u2083\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 ?X\u2083 \u27f6 ?Y\u2083\ncase h\u2082\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 pullback.fst \u226b ?g\u2082 = ?f\u2082 \u226b ?i\u2083\n[PROOFSTEP]\nexact pullbackIsPullback f\u2083 f\u2084\n[GOAL]\ncase H'\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 IsLimit\n (PullbackCone.mk pullback.fst\n (pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.snd)\n (_ :\n pullback.fst \u226b pullback.snd \u226b f\u2083 =\n (pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.snd) \u226b\n f\u2084))\n[PROOFSTEP]\nconvert pullbackIsPullback (g\u2082 \u226b f\u2083) f\u2084\n[GOAL]\ncase h.e'_6.h.e'_10\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.snd =\n pullback.snd\n[PROOFSTEP]\nrw [pullback.lift_snd]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 (pullback.fst \u226b pullback.fst) \u226b f\u2081 =\n pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.fst \u226b f\u2082\n[PROOFSTEP]\nrw [pullback.lift_fst_assoc, Category.assoc, Category.assoc, pullback.condition]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 IsLimit\n (PullbackCone.mk (pullback.fst \u226b pullback.fst)\n (pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084))\n (_ :\n (pullback.fst \u226b pullback.fst) \u226b f\u2081 =\n pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.fst \u226b f\u2082))\n[PROOFSTEP]\napply PullbackCone.flipIsLimit\n[GOAL]\ncase t\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 IsLimit\n (PullbackCone.mk\n (pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084))\n (pullback.fst \u226b pullback.fst)\n (_ :\n pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.fst \u226b f\u2082 =\n (pullback.fst \u226b pullback.fst) \u226b f\u2081))\n[PROOFSTEP]\napply bigSquareIsPullback\n[GOAL]\ncase t.H\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 IsLimit (PullbackCone.mk ?t.i\u2082 pullback.fst ?t.h\u2082)\n[PROOFSTEP]\napply PullbackCone.flipIsLimit\n[GOAL]\ncase t.H.t\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 IsLimit (PullbackCone.mk pullback.fst ?t.i\u2082 (_ : pullback.fst \u226b f\u2081 = ?t.i\u2082 \u226b f\u2082))\n[PROOFSTEP]\nexact pullbackIsPullback f\u2081 f\u2082\n[GOAL]\ncase t.H'\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 IsLimit\n (PullbackCone.mk\n (pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084))\n pullback.fst ?t.h\u2081)\n[PROOFSTEP]\napply PullbackCone.flipIsLimit\n[GOAL]\ncase t.H'.t\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 IsLimit\n (PullbackCone.mk pullback.fst\n (pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084))\n (_ :\n pullback.fst \u226b pullback.snd =\n pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.fst))\n[PROOFSTEP]\napply pullbackPullbackLeftIsPullback\n[GOAL]\ncase t.h\u2081\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.fst =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nexact pullback.lift_fst _ _ _\n[GOAL]\ncase t.h\u2082\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback (pullback.snd \u226b f\u2083) f\u2084\n\u22a2 pullback.snd \u226b f\u2082 = pullback.fst \u226b f\u2081\n[PROOFSTEP]\nexact pullback.condition.symm\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 IsLimit\n (PullbackCone.mk\n (pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082))\n pullback.snd\n (_ :\n pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.snd =\n pullback.snd \u226b pullback.fst))\n[PROOFSTEP]\napply PullbackCone.flipIsLimit\n[GOAL]\ncase t\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 IsLimit\n (PullbackCone.mk pullback.snd\n (pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082))\n (_ :\n pullback.snd \u226b pullback.fst =\n pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.snd))\n[PROOFSTEP]\napply leftSquareIsPullback\n[GOAL]\ncase t.H\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 IsLimit (PullbackCone.mk pullback.snd ?t.f\u2082 ?t.h\u2082)\n[PROOFSTEP]\napply PullbackCone.flipIsLimit\n[GOAL]\ncase t.H.t\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 IsLimit (PullbackCone.mk ?t.f\u2082 pullback.snd (_ : ?t.f\u2082 \u226b ?t.i\u2083 = pullback.snd \u226b ?t.g\u2082))\n[PROOFSTEP]\nexact pullbackIsPullback f\u2081 f\u2082\n[GOAL]\ncase t.H'\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 IsLimit\n (PullbackCone.mk pullback.snd\n (pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.fst)\n (_ :\n pullback.snd \u226b pullback.fst \u226b f\u2082 =\n (pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.fst) \u226b\n f\u2081))\n[PROOFSTEP]\napply PullbackCone.flipIsLimit\n[GOAL]\ncase t.H'.t\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 IsLimit\n (PullbackCone.mk\n (pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.fst)\n pullback.snd\n (_ :\n (pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.fst) \u226b\n f\u2081 =\n pullback.snd \u226b pullback.fst \u226b f\u2082))\n[PROOFSTEP]\nconvert pullbackIsPullback f\u2081 (g\u2083 \u226b f\u2082)\n[GOAL]\ncase h.e'_6.h.e'_9\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.fst =\n pullback.fst\n[PROOFSTEP]\nrw [pullback.lift_fst]\n[GOAL]\ncase t.h\u2082\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.snd \u226b f\u2082 = pullback.fst \u226b f\u2081\n[PROOFSTEP]\nexact pullback.condition.symm\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.snd \u226b f\u2083 =\n (pullback.snd \u226b pullback.snd) \u226b f\u2084\n[PROOFSTEP]\nrw [pullback.lift_snd_assoc, Category.assoc, Category.assoc, pullback.condition]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 IsLimit\n (PullbackCone.mk\n (pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082))\n (pullback.snd \u226b pullback.snd)\n (_ :\n pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.snd \u226b f\u2083 =\n (pullback.snd \u226b pullback.snd) \u226b f\u2084))\n[PROOFSTEP]\napply bigSquareIsPullback\n[GOAL]\ncase H\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 IsLimit (PullbackCone.mk ?i\u2082 pullback.snd ?h\u2082)\ncase H'\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 IsLimit\n (PullbackCone.mk\n (pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082))\n pullback.snd ?h\u2081)\ncase i\u2082\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 Z\u2082 \u27f6 X\u2082\ncase h\u2081\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.snd =\n pullback.snd \u226b ?i\u2082\ncase h\u2082\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 ?i\u2082 \u226b f\u2083 = pullback.snd \u226b f\u2084\n[PROOFSTEP]\nexact pullbackIsPullback f\u2083 f\u2084\n[GOAL]\ncase H'\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 IsLimit\n (PullbackCone.mk\n (pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082))\n pullback.snd ?h\u2081)\ncase h\u2081\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u00b2 : HasPullback f\u2081 f\u2082\ninst\u271d\u00b9 : HasPullback f\u2083 f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.snd =\n pullback.snd \u226b pullback.fst\n[PROOFSTEP]\napply pullbackPullbackRightIsPullback\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 (pullbackAssoc f\u2081 f\u2082 f\u2083 f\u2084).inv \u226b pullback.fst \u226b pullback.fst = pullback.fst\n[PROOFSTEP]\ntrans l\u2081' \u226b pullback.fst\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 (pullbackAssoc f\u2081 f\u2082 f\u2083 f\u2084).inv \u226b pullback.fst \u226b pullback.fst =\n pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.fst\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.fst =\n pullback.fst\n[PROOFSTEP]\nrw [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 ((pullbackAssoc f\u2081 f\u2082 f\u2083 f\u2084).inv \u226b pullback.fst) \u226b pullback.fst =\n pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.fst\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.fst =\n pullback.fst\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 (pullbackAssoc f\u2081 f\u2082 f\u2083 f\u2084).inv \u226b pullback.fst =\n pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082)\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.fst =\n pullback.fst\n[PROOFSTEP]\nexact IsLimit.conePointUniqueUpToIso_inv_comp _ _ WalkingCospan.left\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift pullback.fst (pullback.snd \u226b pullback.fst)\n (_ : pullback.fst \u226b f\u2081 = (pullback.snd \u226b pullback.fst) \u226b f\u2082) \u226b\n pullback.fst =\n pullback.fst\n[PROOFSTEP]\nexact pullback.lift_fst _ _ _\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 (pullbackAssoc f\u2081 f\u2082 f\u2083 f\u2084).hom \u226b pullback.fst = pullback.fst \u226b pullback.fst\n[PROOFSTEP]\nrw [\u2190 Iso.eq_inv_comp, pullbackAssoc_inv_fst_fst]\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 (pullbackAssoc f\u2081 f\u2082 f\u2083 f\u2084).hom \u226b pullback.snd \u226b pullback.fst = pullback.fst \u226b pullback.snd\n[PROOFSTEP]\ntrans l\u2082 \u226b pullback.fst\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 (pullbackAssoc f\u2081 f\u2082 f\u2083 f\u2084).hom \u226b pullback.snd \u226b pullback.fst =\n pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.fst\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.fst =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nrw [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 ((pullbackAssoc f\u2081 f\u2082 f\u2083 f\u2084).hom \u226b pullback.snd) \u226b pullback.fst =\n pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.fst\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.fst =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 (pullbackAssoc f\u2081 f\u2082 f\u2083 f\u2084).hom \u226b pullback.snd =\n pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084)\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.fst =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nexact IsLimit.conePointUniqueUpToIso_hom_comp _ _ WalkingCospan.right\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.fst =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nexact pullback.lift_fst _ _ _\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 (pullbackAssoc f\u2081 f\u2082 f\u2083 f\u2084).hom \u226b pullback.snd \u226b pullback.snd = pullback.snd\n[PROOFSTEP]\ntrans l\u2082 \u226b pullback.snd\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 (pullbackAssoc f\u2081 f\u2082 f\u2083 f\u2084).hom \u226b pullback.snd \u226b pullback.snd =\n pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.snd\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.snd =\n pullback.snd\n[PROOFSTEP]\nrw [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 ((pullbackAssoc f\u2081 f\u2082 f\u2083 f\u2084).hom \u226b pullback.snd) \u226b pullback.snd =\n pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.snd\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.snd =\n pullback.snd\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 (pullbackAssoc f\u2081 f\u2082 f\u2083 f\u2084).hom \u226b pullback.snd =\n pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084)\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.snd =\n pullback.snd\n[PROOFSTEP]\nexact IsLimit.conePointUniqueUpToIso_hom_comp _ _ WalkingCospan.right\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 pullback.lift (pullback.fst \u226b pullback.snd) pullback.snd\n (_ : (pullback.fst \u226b pullback.snd) \u226b f\u2083 = pullback.snd \u226b f\u2084) \u226b\n pullback.snd =\n pullback.snd\n[PROOFSTEP]\nexact pullback.lift_snd _ _ _\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 (pullbackAssoc f\u2081 f\u2082 f\u2083 f\u2084).inv \u226b pullback.fst \u226b pullback.snd = pullback.snd \u226b pullback.fst\n[PROOFSTEP]\nrw [Iso.inv_comp_eq, pullbackAssoc_hom_snd_fst]\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2081\nf\u2083 : X\u2082 \u27f6 Y\u2082\nf\u2084 : X\u2083 \u27f6 Y\u2082\ninst\u271d\u2075 : HasPullback f\u2081 f\u2082\ninst\u271d\u2074 : HasPullback f\u2083 f\u2084\ninst\u271d\u00b3 : HasPullback (pullback.snd \u226b sorryAx (sorryAx C true \u27f6 Y\u2082) true) f\u2084\ninst\u271d\u00b2 : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\ninst\u271d\u00b9 : HasPullback (pullback.snd \u226b f\u2083) f\u2084\ninst\u271d : HasPullback f\u2081 (pullback.fst \u226b f\u2082)\n\u22a2 (pullbackAssoc f\u2081 f\u2082 f\u2083 f\u2084).inv \u226b pullback.snd = pullback.snd \u226b pullback.snd\n[PROOFSTEP]\nrw [Iso.inv_comp_eq, pullbackAssoc_hom_snd_snd]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\n\u22a2 IsColimit\n (PushoutCocone.mk pushout.inl\n (pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr))\n (_ :\n pushout.inr \u226b pushout.inl =\n pushout.inl \u226b\n pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr\n (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr)))\n[PROOFSTEP]\napply PushoutCocone.flipIsColimit\n[GOAL]\ncase t\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\n\u22a2 IsColimit\n (PushoutCocone.mk\n (pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr))\n pushout.inl\n (_ :\n pushout.inl \u226b\n pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr\n (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr) =\n pushout.inr \u226b pushout.inl))\n[PROOFSTEP]\napply rightSquareIsPushout\n[GOAL]\ncase t.H\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\n\u22a2 IsColimit (PushoutCocone.mk ?t.g\u2081 pushout.inl ?t.h\u2081)\n[PROOFSTEP]\napply PushoutCocone.flipIsColimit\n[GOAL]\ncase t.H.t\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\n\u22a2 IsColimit (PushoutCocone.mk pushout.inl ?t.g\u2081 (_ : ?t.f\u2081 \u226b pushout.inl = ?t.i\u2081 \u226b ?t.g\u2081))\n[PROOFSTEP]\nexact pushoutIsPushout _ _\n[GOAL]\ncase t.H'\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\n\u22a2 IsColimit\n (PushoutCocone.mk\n (pushout.inr \u226b\n pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr))\n pushout.inl\n (_ :\n g\u2084 \u226b\n pushout.inr \u226b\n pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr\n (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr) =\n (g\u2083 \u226b pushout.inr) \u226b pushout.inl))\n[PROOFSTEP]\napply PushoutCocone.flipIsColimit\n[GOAL]\ncase t.H'.t\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\n\u22a2 IsColimit\n (PushoutCocone.mk pushout.inl\n (pushout.inr \u226b\n pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr))\n (_ :\n (g\u2083 \u226b pushout.inr) \u226b pushout.inl =\n g\u2084 \u226b\n pushout.inr \u226b\n pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr\n (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr)))\n[PROOFSTEP]\nconvert pushoutIsPushout (g\u2083 \u226b f\u2082) g\u2084\n[GOAL]\ncase h.e'_6.h.e'_10\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\n\u22a2 pushout.inr \u226b\n pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr) =\n pushout.inr\n[PROOFSTEP]\nexact pushout.inr_desc _ _ _\n[GOAL]\ncase t.h\u2081\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\n\u22a2 g\u2084 \u226b pushout.inr = g\u2083 \u226b pushout.inl\n[PROOFSTEP]\nexact pushout.condition.symm\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\n\u22a2 g\u2081 \u226b pushout.inl \u226b pushout.inl =\n (g\u2082 \u226b pushout.inl) \u226b\n pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr)\n[PROOFSTEP]\nrw [Category.assoc, pushout.inl_desc, pushout.condition_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\n\u22a2 IsColimit\n (PushoutCocone.mk (pushout.inl \u226b pushout.inl)\n (pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr))\n (_ :\n g\u2081 \u226b pushout.inl \u226b pushout.inl =\n (g\u2082 \u226b pushout.inl) \u226b\n pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr\n (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr)))\n[PROOFSTEP]\napply bigSquareIsPushout\n[GOAL]\ncase H\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\n\u22a2 IsColimit\n (PushoutCocone.mk pushout.inl\n (pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr))\n ?h\u2082)\n[PROOFSTEP]\napply pushoutPushoutLeftIsPushout\n[GOAL]\ncase H'\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\n\u22a2 IsColimit (PushoutCocone.mk pushout.inl pushout.inr ?h\u2081)\n[PROOFSTEP]\nexact pushoutIsPushout _ _\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 IsColimit\n (PushoutCocone.mk\n (pushout.desc pushout.inl (pushout.inl \u226b pushout.inr) (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr))\n pushout.inr\n (_ :\n pushout.inr \u226b\n pushout.desc pushout.inl (pushout.inl \u226b pushout.inr)\n (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr) =\n pushout.inl \u226b pushout.inr))\n[PROOFSTEP]\napply rightSquareIsPushout\n[GOAL]\ncase H\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 IsColimit (PushoutCocone.mk ?g\u2081 pushout.inr ?h\u2081)\n[PROOFSTEP]\nexact pushoutIsPushout _ _\n[GOAL]\ncase H'\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 IsColimit\n (PushoutCocone.mk\n (pushout.inl \u226b\n pushout.desc pushout.inl (pushout.inl \u226b pushout.inr) (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr))\n pushout.inr\n (_ :\n g\u2081 \u226b\n pushout.inl \u226b\n pushout.desc pushout.inl (pushout.inl \u226b pushout.inr)\n (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr) =\n (g\u2082 \u226b pushout.inl) \u226b pushout.inr))\n[PROOFSTEP]\nconvert pushoutIsPushout g\u2081 (g\u2082 \u226b f\u2083)\n[GOAL]\ncase h.e'_6.h.e'_9\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inl \u226b\n pushout.desc pushout.inl (pushout.inl \u226b pushout.inr) (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr) =\n pushout.inl\n[PROOFSTEP]\nrw [pushout.inl_desc]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 (g\u2083 \u226b pushout.inr) \u226b\n pushout.desc pushout.inl (pushout.inl \u226b pushout.inr) (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr) =\n g\u2084 \u226b pushout.inr \u226b pushout.inr\n[PROOFSTEP]\nrw [Category.assoc, pushout.inr_desc, pushout.condition_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 IsColimit\n (PushoutCocone.mk\n (pushout.desc pushout.inl (pushout.inl \u226b pushout.inr) (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr))\n (pushout.inr \u226b pushout.inr)\n (_ :\n (g\u2083 \u226b pushout.inr) \u226b\n pushout.desc pushout.inl (pushout.inl \u226b pushout.inr)\n (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr) =\n g\u2084 \u226b pushout.inr \u226b pushout.inr))\n[PROOFSTEP]\napply PushoutCocone.flipIsColimit\n[GOAL]\ncase t\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 IsColimit\n (PushoutCocone.mk (pushout.inr \u226b pushout.inr)\n (pushout.desc pushout.inl (pushout.inl \u226b pushout.inr) (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr))\n (_ :\n g\u2084 \u226b pushout.inr \u226b pushout.inr =\n (g\u2083 \u226b pushout.inr) \u226b\n pushout.desc pushout.inl (pushout.inl \u226b pushout.inr)\n (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr)))\n[PROOFSTEP]\napply bigSquareIsPushout\n[GOAL]\ncase t.H\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 IsColimit\n (PushoutCocone.mk pushout.inr\n (pushout.desc pushout.inl (pushout.inl \u226b pushout.inr) (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr))\n ?t.h\u2082)\n[PROOFSTEP]\napply PushoutCocone.flipIsColimit\n[GOAL]\ncase t.H.t\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 IsColimit\n (PushoutCocone.mk\n (pushout.desc pushout.inl (pushout.inl \u226b pushout.inr) (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr))\n pushout.inr\n (_ :\n pushout.inr \u226b\n pushout.desc pushout.inl (pushout.inl \u226b pushout.inr)\n (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr) =\n ?t.i\u2082 \u226b pushout.inr))\n[PROOFSTEP]\napply pushoutPushoutRightIsPushout\n[GOAL]\ncase t.H'\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 IsColimit (PushoutCocone.mk pushout.inr pushout.inl ?t.h\u2081)\n[PROOFSTEP]\napply PushoutCocone.flipIsColimit\n[GOAL]\ncase t.H'.t\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 IsColimit (PushoutCocone.mk pushout.inl pushout.inr (_ : g\u2083 \u226b pushout.inl = g\u2084 \u226b pushout.inr))\n[PROOFSTEP]\nexact pushoutIsPushout _ _\n[GOAL]\ncase t.h\u2081\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 g\u2084 \u226b pushout.inr = g\u2083 \u226b pushout.inl\n[PROOFSTEP]\nexact pushout.condition.symm\n[GOAL]\ncase t.h\u2082\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b2 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b9 : HasPushout g\u2083 g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inl \u226b pushout.inr =\n pushout.inr \u226b\n pushout.desc pushout.inl (pushout.inl \u226b pushout.inr) (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr)\n[PROOFSTEP]\nexact (pushout.inr_desc _ _ _).symm\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b2 : HasPushout g\u2083 g\u2084\ninst\u271d\u00b9 : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inl \u226b pushout.inl \u226b (pushoutAssoc g\u2081 g\u2082 g\u2083 g\u2084).hom = pushout.inl\n[PROOFSTEP]\ntrans f\u2081 \u226b l\u2081\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b2 : HasPushout g\u2083 g\u2084\ninst\u271d\u00b9 : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inl \u226b pushout.inl \u226b (pushoutAssoc g\u2081 g\u2082 g\u2083 g\u2084).hom =\n pushout.inl \u226b\n pushout.desc pushout.inl (pushout.inl \u226b pushout.inr) (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b2 : HasPushout g\u2083 g\u2084\ninst\u271d\u00b9 : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inl \u226b (pushoutAssoc g\u2081 g\u2082 g\u2083 g\u2084).hom =\n pushout.desc pushout.inl (pushout.inl \u226b pushout.inr) (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr)\n[PROOFSTEP]\nexact (pushoutPushoutLeftIsPushout g\u2081 g\u2082 g\u2083 g\u2084).comp_coconePointUniqueUpToIso_hom _ WalkingCospan.left\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b2 : HasPushout g\u2083 g\u2084\ninst\u271d\u00b9 : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inl \u226b\n pushout.desc pushout.inl (pushout.inl \u226b pushout.inr) (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr) =\n pushout.inl\n[PROOFSTEP]\nexact pushout.inl_desc _ _ _\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b2 : HasPushout g\u2083 g\u2084\ninst\u271d\u00b9 : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inr \u226b pushout.inl \u226b (pushoutAssoc g\u2081 g\u2082 g\u2083 g\u2084).hom = pushout.inl \u226b pushout.inr\n[PROOFSTEP]\ntrans f\u2082 \u226b l\u2081\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b2 : HasPushout g\u2083 g\u2084\ninst\u271d\u00b9 : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inr \u226b pushout.inl \u226b (pushoutAssoc g\u2081 g\u2082 g\u2083 g\u2084).hom =\n pushout.inr \u226b\n pushout.desc pushout.inl (pushout.inl \u226b pushout.inr) (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b2 : HasPushout g\u2083 g\u2084\ninst\u271d\u00b9 : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inl \u226b (pushoutAssoc g\u2081 g\u2082 g\u2083 g\u2084).hom =\n pushout.desc pushout.inl (pushout.inl \u226b pushout.inr) (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr)\n[PROOFSTEP]\nexact (pushoutPushoutLeftIsPushout g\u2081 g\u2082 g\u2083 g\u2084).comp_coconePointUniqueUpToIso_hom _ WalkingCospan.left\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b2 : HasPushout g\u2083 g\u2084\ninst\u271d\u00b9 : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inr \u226b\n pushout.desc pushout.inl (pushout.inl \u226b pushout.inr) (_ : g\u2081 \u226b pushout.inl = g\u2082 \u226b pushout.inl \u226b pushout.inr) =\n pushout.inl \u226b pushout.inr\n[PROOFSTEP]\nexact pushout.inr_desc _ _ _\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b2 : HasPushout g\u2083 g\u2084\ninst\u271d\u00b9 : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inr \u226b pushout.inr \u226b (pushoutAssoc g\u2081 g\u2082 g\u2083 g\u2084).inv = pushout.inr\n[PROOFSTEP]\ntrans f\u2084 \u226b l\u2082'\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b2 : HasPushout g\u2083 g\u2084\ninst\u271d\u00b9 : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inr \u226b pushout.inr \u226b (pushoutAssoc g\u2081 g\u2082 g\u2083 g\u2084).inv =\n pushout.inr \u226b\n pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b2 : HasPushout g\u2083 g\u2084\ninst\u271d\u00b9 : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inr \u226b (pushoutAssoc g\u2081 g\u2082 g\u2083 g\u2084).inv =\n pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr)\n[PROOFSTEP]\nexact\n (pushoutPushoutLeftIsPushout g\u2081 g\u2082 g\u2083 g\u2084).comp_coconePointUniqueUpToIso_inv (pushoutPushoutRightIsPushout g\u2081 g\u2082 g\u2083 g\u2084)\n WalkingCospan.right\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b2 : HasPushout g\u2083 g\u2084\ninst\u271d\u00b9 : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inr \u226b\n pushout.desc (pushout.inr \u226b pushout.inl) pushout.inr (_ : g\u2083 \u226b pushout.inr \u226b pushout.inl = g\u2084 \u226b pushout.inr) =\n pushout.inr\n[PROOFSTEP]\nexact pushout.inr_desc _ _ _\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b2 : HasPushout g\u2083 g\u2084\ninst\u271d\u00b9 : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inl \u226b (pushoutAssoc g\u2081 g\u2082 g\u2083 g\u2084).inv = pushout.inl \u226b pushout.inl\n[PROOFSTEP]\nrw [Iso.comp_inv_eq, Category.assoc, inl_inl_pushoutAssoc_hom]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b2 : HasPushout g\u2083 g\u2084\ninst\u271d\u00b9 : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inl \u226b pushout.inr \u226b (pushoutAssoc g\u2081 g\u2082 g\u2083 g\u2084).inv = pushout.inr \u226b pushout.inl\n[PROOFSTEP]\nrw [\u2190 Category.assoc, Iso.comp_inv_eq, Category.assoc, inr_inl_pushoutAssoc_hom]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\nW X Y Z X\u2081 X\u2082 X\u2083 Z\u2081 Z\u2082 : C\ng\u2081 : Z\u2081 \u27f6 X\u2081\ng\u2082 : Z\u2081 \u27f6 X\u2082\ng\u2083 : Z\u2082 \u27f6 X\u2082\ng\u2084 : Z\u2082 \u27f6 X\u2083\ninst\u271d\u00b3 : HasPushout g\u2081 g\u2082\ninst\u271d\u00b2 : HasPushout g\u2083 g\u2084\ninst\u271d\u00b9 : HasPushout (g\u2083 \u226b pushout.inr) g\u2084\ninst\u271d : HasPushout g\u2081 (g\u2082 \u226b pushout.inl)\n\u22a2 pushout.inr \u226b (pushoutAssoc g\u2081 g\u2082 g\u2083 g\u2084).hom = pushout.inr \u226b pushout.inr\n[PROOFSTEP]\nrw [\u2190 Iso.eq_comp_inv, Category.assoc, inr_inr_pushoutAssoc_inv]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD\u271d : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\u271d\nW X Y Z : C\nD : Type u\nh : Category.{v, u} D\nh' : HasWidePullbacks D\n\u22a2 HasPullbacks D\n[PROOFSTEP]\nhaveI I := @hasWidePullbacks_shrink.{0, w} D h h'\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD\u271d : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\u271d\nW X Y Z : C\nD : Type u\nh : Category.{v, u} D\nh' : HasWidePullbacks D\nI : HasWidePullbacks D\n\u22a2 HasPullbacks D\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD\u271d : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\u271d\nW X Y Z : C\nD : Type u\nh : Category.{v, u} D\nh' : HasWidePushouts D\n\u22a2 HasPushouts D\n[PROOFSTEP]\nhaveI I := @hasWidePushouts_shrink.{0, w} D h h'\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD\u271d : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\u271d\nW X Y Z : C\nD : Type u\nh : Category.{v, u} D\nh' : HasWidePushouts D\nI : HasWidePushouts D\n\u22a2 HasPushouts D\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d\u00b9 Y\u271d\u00b9 Z : C\ninst\u271d : HasPullbacks C\nX Y : C\nf : X \u27f6 Y\nX\u271d Y\u271d : Over Y\ni : X\u271d \u27f6 Y\u271d\n\u22a2 X\u271d.hom \u226b \ud835\udfd9 ((Functor.fromPUnit Y).obj X\u271d.right) = i.left \u226b Y\u271d.hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d\u00b9 Y\u271d\u00b9 Z : C\ninst\u271d : HasPullbacks C\nX Y : C\nf : X \u27f6 Y\nX\u271d Y\u271d : Over Y\ni : X\u271d \u27f6 Y\u271d\n\u22a2 f \u226b \ud835\udfd9 ((Functor.fromPUnit Y).obj X\u271d.right) = \ud835\udfd9 X \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d : C\ninst\u271d : HasPullbacks C\nX Y : C\nf : X \u27f6 Y\nZ : Over Y\n\u22a2 { obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f Y_1.hom f i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f)) }.map\n (\ud835\udfd9 Z) =\n \ud835\udfd9\n ({ obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f Y_1.hom f i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f)) }.obj\n Z)\n[PROOFSTEP]\napply Over.OverMorphism.ext\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d : C\ninst\u271d : HasPullbacks C\nX Y : C\nf : X \u27f6 Y\nZ : Over Y\n\u22a2 ({ obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f Y_1.hom f i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f)) }.map\n (\ud835\udfd9 Z)).left =\n (\ud835\udfd9\n ({ obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f Y_1.hom f i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f)) }.obj\n Z)).left\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h.h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d : C\ninst\u271d : HasPullbacks C\nX Y : C\nf : X \u27f6 Y\nZ : Over Y\n\u22a2 ({ obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f Y_1.hom f i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f)) }.map\n (\ud835\udfd9 Z)).left \u226b\n pullback.fst =\n (\ud835\udfd9\n ({ obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f Y_1.hom f i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f)) }.obj\n Z)).left \u226b\n pullback.fst\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d : C\ninst\u271d : HasPullbacks C\nX Y : C\nf : X \u27f6 Y\nZ : Over Y\n\u22a2 pullback.map Z.hom f Z.hom f (\ud835\udfd9 Z.left) (\ud835\udfd9 X) (\ud835\udfd9 Y) (_ : Z.hom \u226b \ud835\udfd9 Y = \ud835\udfd9 Z.left \u226b Z.hom) (_ : f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f) \u226b\n pullback.fst =\n \ud835\udfd9 (pullback Z.hom f) \u226b pullback.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d : C\ninst\u271d : HasPullbacks C\nX Y : C\nf : X \u27f6 Y\nZ : Over Y\n\u22a2 ({ obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f Y_1.hom f i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f)) }.map\n (\ud835\udfd9 Z)).left \u226b\n pullback.snd =\n (\ud835\udfd9\n ({ obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f Y_1.hom f i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f)) }.obj\n Z)).left \u226b\n pullback.snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d Y\u271d Z\u271d : C\ninst\u271d : HasPullbacks C\nX Y : C\nf : X \u27f6 Y\nZ : Over Y\n\u22a2 pullback.map Z.hom f Z.hom f (\ud835\udfd9 Z.left) (\ud835\udfd9 X) (\ud835\udfd9 Y) (_ : Z.hom \u226b \ud835\udfd9 Y = \ud835\udfd9 Z.left \u226b Z.hom) (_ : f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f) \u226b\n pullback.snd =\n \ud835\udfd9 (pullback Z.hom f) \u226b pullback.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d\u00b9 Y\u271d\u00b9 Z : C\ninst\u271d : HasPullbacks C\nX Y : C\nf\u271d : X \u27f6 Y\nX\u271d Y\u271d Z\u271d : Over Y\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f\u271d Y_1.hom f\u271d i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d)) }.map\n (f \u226b g) =\n { obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f\u271d Y_1.hom f\u271d i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d)) }.map\n f \u226b\n { obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f\u271d Y_1.hom f\u271d i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d)) }.map\n g\n[PROOFSTEP]\napply Over.OverMorphism.ext\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d\u00b9 Y\u271d\u00b9 Z : C\ninst\u271d : HasPullbacks C\nX Y : C\nf\u271d : X \u27f6 Y\nX\u271d Y\u271d Z\u271d : Over Y\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 ({ obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f\u271d Y_1.hom f\u271d i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d)) }.map\n (f \u226b g)).left =\n ({ obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f\u271d Y_1.hom f\u271d i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d)) }.map\n f \u226b\n { obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f\u271d Y_1.hom f\u271d i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d)) }.map\n g).left\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h.h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d\u00b9 Y\u271d\u00b9 Z : C\ninst\u271d : HasPullbacks C\nX Y : C\nf\u271d : X \u27f6 Y\nX\u271d Y\u271d Z\u271d : Over Y\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 ({ obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f\u271d Y_1.hom f\u271d i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d)) }.map\n (f \u226b g)).left \u226b\n pullback.fst =\n ({ obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f\u271d Y_1.hom f\u271d i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d)) }.map\n f \u226b\n { obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f\u271d Y_1.hom f\u271d i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d)) }.map\n g).left \u226b\n pullback.fst\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d\u00b9 Y\u271d\u00b9 Z : C\ninst\u271d : HasPullbacks C\nX Y : C\nf\u271d : X \u27f6 Y\nX\u271d Y\u271d Z\u271d : Over Y\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 pullback.map X\u271d.hom f\u271d Z\u271d.hom f\u271d (f.left \u226b g.left) (\ud835\udfd9 X) (\ud835\udfd9 Y) (_ : X\u271d.hom \u226b \ud835\udfd9 Y = (f.left \u226b g.left) \u226b Z\u271d.hom)\n (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d) \u226b\n pullback.fst =\n (pullback.map X\u271d.hom f\u271d Y\u271d.hom f\u271d f.left (\ud835\udfd9 X) (\ud835\udfd9 Y) (_ : X\u271d.hom \u226b \ud835\udfd9 Y = f.left \u226b Y\u271d.hom)\n (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d) \u226b\n pullback.map Y\u271d.hom f\u271d Z\u271d.hom f\u271d g.left (\ud835\udfd9 X) (\ud835\udfd9 Y) (_ : Y\u271d.hom \u226b \ud835\udfd9 Y = g.left \u226b Z\u271d.hom)\n (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d)) \u226b\n pullback.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d\u00b9 Y\u271d\u00b9 Z : C\ninst\u271d : HasPullbacks C\nX Y : C\nf\u271d : X \u27f6 Y\nX\u271d Y\u271d Z\u271d : Over Y\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 ({ obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f\u271d Y_1.hom f\u271d i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d)) }.map\n (f \u226b g)).left \u226b\n pullback.snd =\n ({ obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f\u271d Y_1.hom f\u271d i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d)) }.map\n f \u226b\n { obj := fun g => Over.mk pullback.snd,\n map := fun {X_1 Y_1} i =>\n Over.homMk\n (pullback.map X_1.hom f\u271d Y_1.hom f\u271d i.left (\ud835\udfd9 X) (\ud835\udfd9 ((Functor.fromPUnit Y).obj X_1.right))\n (_ : X_1.hom \u226b \ud835\udfd9 Y = i.left \u226b Y_1.hom) (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d)) }.map\n g).left \u226b\n pullback.snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nW X\u271d\u00b9 Y\u271d\u00b9 Z : C\ninst\u271d : HasPullbacks C\nX Y : C\nf\u271d : X \u27f6 Y\nX\u271d Y\u271d Z\u271d : Over Y\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 pullback.map X\u271d.hom f\u271d Z\u271d.hom f\u271d (f.left \u226b g.left) (\ud835\udfd9 X) (\ud835\udfd9 Y) (_ : X\u271d.hom \u226b \ud835\udfd9 Y = (f.left \u226b g.left) \u226b Z\u271d.hom)\n (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d) \u226b\n pullback.snd =\n (pullback.map X\u271d.hom f\u271d Y\u271d.hom f\u271d f.left (\ud835\udfd9 X) (\ud835\udfd9 Y) (_ : X\u271d.hom \u226b \ud835\udfd9 Y = f.left \u226b Y\u271d.hom)\n (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d) \u226b\n pullback.map Y\u271d.hom f\u271d Z\u271d.hom f\u271d g.left (\ud835\udfd9 X) (\ud835\udfd9 Y) (_ : Y\u271d.hom \u226b \ud835\udfd9 Y = g.left \u226b Z\u271d.hom)\n (_ : f\u271d \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\u271d)) \u226b\n pullback.snd\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "llama_tokens": 194951, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707281, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.44706871876267545}} {"text": "[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b2 : MulZeroClass \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PosMulStrictMono \u03b1\nha : 0 < a\nhb : 0 < b\n\u22a2 0 < a * b\n[PROOFSTEP]\nsimpa only [mul_zero] using mul_lt_mul_of_pos_left hb ha\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b2 : MulZeroClass \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PosMulStrictMono \u03b1\nha : 0 < a\nhb : b < 0\n\u22a2 a * b < 0\n[PROOFSTEP]\nsimpa only [mul_zero] using mul_lt_mul_of_pos_left hb ha\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulZeroClass \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : PosMulStrictMono \u03b1\ninst\u271d : PosMulReflectLT \u03b1\nh : 0 < c\n\u22a2 0 < c * b \u2194 0 < b\n[PROOFSTEP]\nrw [\u2190 mul_zero c, mul_lt_mul_left h]\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulZeroClass \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : PosMulStrictMono \u03b1\ninst\u271d : PosMulReflectLT \u03b1\nh : 0 < c\n\u22a2 0 < b \u2194 c * 0 < b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b2 : MulZeroClass \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : MulPosStrictMono \u03b1\nha : 0 < a\nhb : 0 < b\n\u22a2 0 < a * b\n[PROOFSTEP]\nsimpa only [zero_mul] using mul_lt_mul_of_pos_right ha hb\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b2 : MulZeroClass \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : MulPosStrictMono \u03b1\nha : a < 0\nhb : 0 < b\n\u22a2 a * b < 0\n[PROOFSTEP]\nsimpa only [zero_mul] using mul_lt_mul_of_pos_right ha hb\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulZeroClass \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : MulPosStrictMono \u03b1\ninst\u271d : MulPosReflectLT \u03b1\nh : 0 < c\n\u22a2 0 < b * c \u2194 0 < b\n[PROOFSTEP]\nrw [\u2190 zero_mul c, mul_lt_mul_right h]\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulZeroClass \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : MulPosStrictMono \u03b1\ninst\u271d : MulPosReflectLT \u03b1\nh : 0 < c\n\u22a2 0 < b \u2194 0 * c < b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b2 : MulZeroClass \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PosMulMono \u03b1\nha : 0 \u2264 a\nhb : 0 \u2264 b\n\u22a2 0 \u2264 a * b\n[PROOFSTEP]\nsimpa only [mul_zero] using mul_le_mul_of_nonneg_left hb ha\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b2 : MulZeroClass \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PosMulMono \u03b1\nha : 0 \u2264 a\nhb : b \u2264 0\n\u22a2 a * b \u2264 0\n[PROOFSTEP]\nsimpa only [mul_zero] using mul_le_mul_of_nonneg_left hb ha\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b2 : MulZeroClass \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : MulPosMono \u03b1\nha : 0 \u2264 a\nhb : 0 \u2264 b\n\u22a2 0 \u2264 a * b\n[PROOFSTEP]\nsimpa only [zero_mul] using mul_le_mul_of_nonneg_right ha hb\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b2 : MulZeroClass \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : MulPosMono \u03b1\nha : a \u2264 0\nhb : 0 \u2264 b\n\u22a2 a * b \u2264 0\n[PROOFSTEP]\nsimpa only [zero_mul] using mul_le_mul_of_nonneg_right ha hb\n[GOAL]\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : MulZeroClass \u03b1\ninst\u271d : PartialOrder \u03b1\nh\u271d : CovariantClass { x // 0 < x } \u03b1 (fun x y => \u2191x * y) fun x x_1 => x \u2264 x_1\na : { x // 0 \u2264 x }\nb c : \u03b1\nh : b \u2264 c\n\u22a2 \u2191a * b \u2264 \u2191a * c\n[PROOFSTEP]\nobtain ha | ha := a.prop.eq_or_lt\n[GOAL]\ncase inl\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : MulZeroClass \u03b1\ninst\u271d : PartialOrder \u03b1\nh\u271d : CovariantClass { x // 0 < x } \u03b1 (fun x y => \u2191x * y) fun x x_1 => x \u2264 x_1\na : { x // 0 \u2264 x }\nb c : \u03b1\nh : b \u2264 c\nha : 0 = \u2191a\n\u22a2 \u2191a * b \u2264 \u2191a * c\n[PROOFSTEP]\nsimp [\u2190 ha]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : MulZeroClass \u03b1\ninst\u271d : PartialOrder \u03b1\nh\u271d : CovariantClass { x // 0 < x } \u03b1 (fun x y => \u2191x * y) fun x x_1 => x \u2264 x_1\na : { x // 0 \u2264 x }\nb c : \u03b1\nh : b \u2264 c\nha : 0 < \u2191a\n\u22a2 \u2191a * b \u2264 \u2191a * c\n[PROOFSTEP]\nexact @CovariantClass.elim \u03b1>0 \u03b1 (fun x y => x * y) (\u00b7 \u2264 \u00b7) _ \u27e8_, ha\u27e9 _ _ h\n[GOAL]\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : MulZeroClass \u03b1\ninst\u271d : PartialOrder \u03b1\nh\u271d : CovariantClass { x // 0 < x } \u03b1 (fun x y => y * \u2191x) fun x x_1 => x \u2264 x_1\na : { x // 0 \u2264 x }\nb c : \u03b1\nh : b \u2264 c\n\u22a2 b * \u2191a \u2264 c * \u2191a\n[PROOFSTEP]\nobtain ha | ha := a.prop.eq_or_lt\n[GOAL]\ncase inl\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : MulZeroClass \u03b1\ninst\u271d : PartialOrder \u03b1\nh\u271d : CovariantClass { x // 0 < x } \u03b1 (fun x y => y * \u2191x) fun x x_1 => x \u2264 x_1\na : { x // 0 \u2264 x }\nb c : \u03b1\nh : b \u2264 c\nha : 0 = \u2191a\n\u22a2 b * \u2191a \u2264 c * \u2191a\n[PROOFSTEP]\nsimp [\u2190 ha]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : MulZeroClass \u03b1\ninst\u271d : PartialOrder \u03b1\nh\u271d : CovariantClass { x // 0 < x } \u03b1 (fun x y => y * \u2191x) fun x x_1 => x \u2264 x_1\na : { x // 0 \u2264 x }\nb c : \u03b1\nh : b \u2264 c\nha : 0 < \u2191a\n\u22a2 b * \u2191a \u2264 c * \u2191a\n[PROOFSTEP]\nexact @CovariantClass.elim \u03b1>0 \u03b1 (fun x y => y * x) (\u00b7 \u2264 \u00b7) _ \u27e8_, ha\u27e9 _ _ h\n[GOAL]\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : MulZeroClass \u03b1\ninst\u271d : PartialOrder \u03b1\nh\u271d : ContravariantClass { x // 0 < x } \u03b1 (fun x y => \u2191x * y) fun x x_1 => x < x_1\na : { x // 0 \u2264 x }\nb c : \u03b1\nh : \u2191a * b < \u2191a * c\n\u22a2 b < c\n[PROOFSTEP]\nobtain ha | ha := a.prop.eq_or_lt\n[GOAL]\ncase inl\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : MulZeroClass \u03b1\ninst\u271d : PartialOrder \u03b1\nh\u271d : ContravariantClass { x // 0 < x } \u03b1 (fun x y => \u2191x * y) fun x x_1 => x < x_1\na : { x // 0 \u2264 x }\nb c : \u03b1\nh : \u2191a * b < \u2191a * c\nha : 0 = \u2191a\n\u22a2 b < c\n[PROOFSTEP]\nsimp [\u2190 ha] at h \n[GOAL]\ncase inr\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : MulZeroClass \u03b1\ninst\u271d : PartialOrder \u03b1\nh\u271d : ContravariantClass { x // 0 < x } \u03b1 (fun x y => \u2191x * y) fun x x_1 => x < x_1\na : { x // 0 \u2264 x }\nb c : \u03b1\nh : \u2191a * b < \u2191a * c\nha : 0 < \u2191a\n\u22a2 b < c\n[PROOFSTEP]\nexact @ContravariantClass.elim \u03b1>0 \u03b1 (fun x y => x * y) (\u00b7 < \u00b7) _ \u27e8_, ha\u27e9 _ _ h\n[GOAL]\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : MulZeroClass \u03b1\ninst\u271d : PartialOrder \u03b1\nh\u271d : ContravariantClass { x // 0 < x } \u03b1 (fun x y => y * \u2191x) fun x x_1 => x < x_1\na : { x // 0 \u2264 x }\nb c : \u03b1\nh : b * \u2191a < c * \u2191a\n\u22a2 b < c\n[PROOFSTEP]\nobtain ha | ha := a.prop.eq_or_lt\n[GOAL]\ncase inl\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : MulZeroClass \u03b1\ninst\u271d : PartialOrder \u03b1\nh\u271d : ContravariantClass { x // 0 < x } \u03b1 (fun x y => y * \u2191x) fun x x_1 => x < x_1\na : { x // 0 \u2264 x }\nb c : \u03b1\nh : b * \u2191a < c * \u2191a\nha : 0 = \u2191a\n\u22a2 b < c\n[PROOFSTEP]\nsimp [\u2190 ha] at h \n[GOAL]\ncase inr\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : MulZeroClass \u03b1\ninst\u271d : PartialOrder \u03b1\nh\u271d : ContravariantClass { x // 0 < x } \u03b1 (fun x y => y * \u2191x) fun x x_1 => x < x_1\na : { x // 0 \u2264 x }\nb c : \u03b1\nh : b * \u2191a < c * \u2191a\nha : 0 < \u2191a\n\u22a2 b < c\n[PROOFSTEP]\nexact @ContravariantClass.elim \u03b1>0 \u03b1 (fun x y => y * x) (\u00b7 < \u00b7) _ \u27e8_, ha\u27e9 _ _ h\n[GOAL]\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b2 : MulZeroClass \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PosMulMonoRev \u03b1\na : { x // 0 < x }\nb c : \u03b1\nh : \u2191a * b < \u2191a * c\n\u22a2 b \u2260 c\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\na\u271d b\u271d c d : \u03b1\ninst\u271d\u00b2 : MulZeroClass \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PosMulMonoRev \u03b1\na : { x // 0 < x }\nb : \u03b1\nh : \u2191a * b < \u2191a * b\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b2 : MulZeroClass \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : MulPosMonoRev \u03b1\na : { x // 0 < x }\nb c : \u03b1\nh : b * \u2191a < c * \u2191a\n\u22a2 b \u2260 c\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\na\u271d b\u271d c d : \u03b1\ninst\u271d\u00b2 : MulZeroClass \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : MulPosMonoRev \u03b1\na : { x // 0 < x }\nb : \u03b1\nh : b * \u2191a < b * \u2191a\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2075 : MulZeroClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : PosMulStrictMono \u03b1\ninst\u271d\u00b2 : MulPosStrictMono \u03b1\ninst\u271d\u00b9 : PosMulMonoRev \u03b1\ninst\u271d : MulPosMonoRev \u03b1\nhac : a \u2264 b\nhbd : c \u2264 d\na0 : 0 < a\nd0 : 0 < d\n\u22a2 a * c = b * d \u2194 a = b \u2227 c = d\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => congr_arg\u2082 (\u00b7 * \u00b7) h.1 h.2\u27e9\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2075 : MulZeroClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : PosMulStrictMono \u03b1\ninst\u271d\u00b2 : MulPosStrictMono \u03b1\ninst\u271d\u00b9 : PosMulMonoRev \u03b1\ninst\u271d : MulPosMonoRev \u03b1\nhac : a \u2264 b\nhbd : c \u2264 d\na0 : 0 < a\nd0 : 0 < d\nh : a * c = b * d\n\u22a2 a = b \u2227 c = d\n[PROOFSTEP]\nrcases hac.eq_or_lt with (rfl | hac)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\na c d : \u03b1\ninst\u271d\u2075 : MulZeroClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : PosMulStrictMono \u03b1\ninst\u271d\u00b2 : MulPosStrictMono \u03b1\ninst\u271d\u00b9 : PosMulMonoRev \u03b1\ninst\u271d : MulPosMonoRev \u03b1\nhbd : c \u2264 d\na0 : 0 < a\nd0 : 0 < d\nhac : a \u2264 a\nh : a * c = a * d\n\u22a2 a = a \u2227 c = d\n[PROOFSTEP]\nexact \u27e8rfl, (mul_left_cancel_iff_of_pos a0).mp h\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2075 : MulZeroClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : PosMulStrictMono \u03b1\ninst\u271d\u00b2 : MulPosStrictMono \u03b1\ninst\u271d\u00b9 : PosMulMonoRev \u03b1\ninst\u271d : MulPosMonoRev \u03b1\nhac\u271d : a \u2264 b\nhbd : c \u2264 d\na0 : 0 < a\nd0 : 0 < d\nh : a * c = b * d\nhac : a < b\n\u22a2 a = b \u2227 c = d\n[PROOFSTEP]\nrcases eq_or_lt_of_le hbd with (rfl | hbd)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\na b c : \u03b1\ninst\u271d\u2075 : MulZeroClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : PosMulStrictMono \u03b1\ninst\u271d\u00b2 : MulPosStrictMono \u03b1\ninst\u271d\u00b9 : PosMulMonoRev \u03b1\ninst\u271d : MulPosMonoRev \u03b1\nhac\u271d : a \u2264 b\na0 : 0 < a\nhac : a < b\nhbd : c \u2264 c\nd0 : 0 < c\nh : a * c = b * c\n\u22a2 a = b \u2227 c = c\n[PROOFSTEP]\nexact \u27e8(mul_right_cancel_iff_of_pos d0).mp h, rfl\u27e9\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2075 : MulZeroClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : PosMulStrictMono \u03b1\ninst\u271d\u00b2 : MulPosStrictMono \u03b1\ninst\u271d\u00b9 : PosMulMonoRev \u03b1\ninst\u271d : MulPosMonoRev \u03b1\nhac\u271d : a \u2264 b\nhbd\u271d : c \u2264 d\na0 : 0 < a\nd0 : 0 < d\nh : a * c = b * d\nhac : a < b\nhbd : c < d\n\u22a2 a = b \u2227 c = d\n[PROOFSTEP]\nexact ((mul_lt_mul_of_pos_of_pos hac hbd a0 d0).ne h).elim\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2075 : MulZeroClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : PosMulStrictMono \u03b1\ninst\u271d\u00b2 : MulPosStrictMono \u03b1\ninst\u271d\u00b9 : PosMulMonoRev \u03b1\ninst\u271d : MulPosMonoRev \u03b1\nhac : a \u2264 b\nhbd : c \u2264 d\nb0 : 0 < b\nc0 : 0 < c\n\u22a2 a * c = b * d \u2194 a = b \u2227 c = d\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => congr_arg\u2082 (\u00b7 * \u00b7) h.1 h.2\u27e9\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2075 : MulZeroClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : PosMulStrictMono \u03b1\ninst\u271d\u00b2 : MulPosStrictMono \u03b1\ninst\u271d\u00b9 : PosMulMonoRev \u03b1\ninst\u271d : MulPosMonoRev \u03b1\nhac : a \u2264 b\nhbd : c \u2264 d\nb0 : 0 < b\nc0 : 0 < c\nh : a * c = b * d\n\u22a2 a = b \u2227 c = d\n[PROOFSTEP]\nrcases hac.eq_or_lt with (rfl | hac)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\na c d : \u03b1\ninst\u271d\u2075 : MulZeroClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : PosMulStrictMono \u03b1\ninst\u271d\u00b2 : MulPosStrictMono \u03b1\ninst\u271d\u00b9 : PosMulMonoRev \u03b1\ninst\u271d : MulPosMonoRev \u03b1\nhbd : c \u2264 d\nc0 : 0 < c\nhac : a \u2264 a\nb0 : 0 < a\nh : a * c = a * d\n\u22a2 a = a \u2227 c = d\n[PROOFSTEP]\nexact \u27e8rfl, (mul_left_cancel_iff_of_pos b0).mp h\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2075 : MulZeroClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : PosMulStrictMono \u03b1\ninst\u271d\u00b2 : MulPosStrictMono \u03b1\ninst\u271d\u00b9 : PosMulMonoRev \u03b1\ninst\u271d : MulPosMonoRev \u03b1\nhac\u271d : a \u2264 b\nhbd : c \u2264 d\nb0 : 0 < b\nc0 : 0 < c\nh : a * c = b * d\nhac : a < b\n\u22a2 a = b \u2227 c = d\n[PROOFSTEP]\nrcases eq_or_lt_of_le hbd with (rfl | hbd)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\na b c : \u03b1\ninst\u271d\u2075 : MulZeroClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : PosMulStrictMono \u03b1\ninst\u271d\u00b2 : MulPosStrictMono \u03b1\ninst\u271d\u00b9 : PosMulMonoRev \u03b1\ninst\u271d : MulPosMonoRev \u03b1\nhac\u271d : a \u2264 b\nb0 : 0 < b\nc0 : 0 < c\nhac : a < b\nhbd : c \u2264 c\nh : a * c = b * c\n\u22a2 a = b \u2227 c = c\n[PROOFSTEP]\nexact \u27e8(mul_right_cancel_iff_of_pos c0).mp h, rfl\u27e9\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2075 : MulZeroClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : PosMulStrictMono \u03b1\ninst\u271d\u00b2 : MulPosStrictMono \u03b1\ninst\u271d\u00b9 : PosMulMonoRev \u03b1\ninst\u271d : MulPosMonoRev \u03b1\nhac\u271d : a \u2264 b\nhbd\u271d : c \u2264 d\nb0 : 0 < b\nc0 : 0 < c\nh : a * c = b * d\nhac : a < b\nhbd : c < d\n\u22a2 a = b \u2227 c = d\n[PROOFSTEP]\nexact ((mul_lt_mul_of_lt_of_lt' hac hbd b0 c0).ne h).elim\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulZeroClass \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : PosMulMono \u03b1\ninst\u271d : MulPosMono \u03b1\nhab : 0 < a * b\n\u22a2 0 < a \u2227 0 < b \u2228 a < 0 \u2227 b < 0\n[PROOFSTEP]\nrcases lt_trichotomy a 0 with (ha | rfl | ha)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulZeroClass \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : PosMulMono \u03b1\ninst\u271d : MulPosMono \u03b1\nhab : 0 < a * b\nha : a < 0\n\u22a2 0 < a \u2227 0 < b \u2228 a < 0 \u2227 b < 0\n[PROOFSTEP]\nrefine' Or.inr \u27e8ha, lt_imp_lt_of_le_imp_le (fun hb => _) hab\u27e9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulZeroClass \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : PosMulMono \u03b1\ninst\u271d : MulPosMono \u03b1\nhab : 0 < a * b\nha : a < 0\nhb : 0 \u2264 b\n\u22a2 a * b \u2264 0\n[PROOFSTEP]\nexact mul_nonpos_of_nonpos_of_nonneg ha.le hb\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nb c d : \u03b1\ninst\u271d\u00b3 : MulZeroClass \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : PosMulMono \u03b1\ninst\u271d : MulPosMono \u03b1\nhab : 0 < 0 * b\n\u22a2 0 < 0 \u2227 0 < b \u2228 0 < 0 \u2227 b < 0\n[PROOFSTEP]\nrw [zero_mul] at hab \n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nb c d : \u03b1\ninst\u271d\u00b3 : MulZeroClass \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : PosMulMono \u03b1\ninst\u271d : MulPosMono \u03b1\nhab : 0 < 0\n\u22a2 0 < 0 \u2227 0 < b \u2228 0 < 0 \u2227 b < 0\n[PROOFSTEP]\nexact hab.false.elim\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulZeroClass \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : PosMulMono \u03b1\ninst\u271d : MulPosMono \u03b1\nhab : 0 < a * b\nha : 0 < a\n\u22a2 0 < a \u2227 0 < b \u2228 a < 0 \u2227 b < 0\n[PROOFSTEP]\nrefine' Or.inl \u27e8ha, lt_imp_lt_of_le_imp_le (fun hb => _) hab\u27e9\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulZeroClass \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : PosMulMono \u03b1\ninst\u271d : MulPosMono \u03b1\nhab : 0 < a * b\nha : 0 < a\nhb : b \u2264 0\n\u22a2 a * b \u2264 0\n[PROOFSTEP]\nexact mul_nonpos_of_nonneg_of_nonpos ha.le hb\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2074 : MulOneClass \u03b1\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : PosMulMono \u03b1\ninst\u271d : PosMulMonoRev \u03b1\na0 : 0 < a\n\u22a2 a \u2264 a * b \u2194 a * 1 \u2264 a * b\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2074 : MulOneClass \u03b1\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : PosMulStrictMono \u03b1\ninst\u271d : PosMulReflectLT \u03b1\na0 : 0 < a\n\u22a2 a < a * b \u2194 a * 1 < a * b\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2074 : MulOneClass \u03b1\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : PosMulMono \u03b1\ninst\u271d : PosMulMonoRev \u03b1\na0 : 0 < a\n\u22a2 a * b \u2264 a \u2194 a * b \u2264 a * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2074 : MulOneClass \u03b1\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : PosMulStrictMono \u03b1\ninst\u271d : PosMulReflectLT \u03b1\na0 : 0 < a\n\u22a2 a * b < a \u2194 a * b < a * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2074 : MulOneClass \u03b1\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : MulPosMono \u03b1\ninst\u271d : MulPosMonoRev \u03b1\na0 : 0 < a\n\u22a2 a \u2264 b * a \u2194 1 * a \u2264 b * a\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2074 : MulOneClass \u03b1\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : MulPosStrictMono \u03b1\ninst\u271d : MulPosReflectLT \u03b1\na0 : 0 < a\n\u22a2 a < b * a \u2194 1 * a < b * a\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2074 : MulOneClass \u03b1\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : MulPosMono \u03b1\ninst\u271d : MulPosMonoRev \u03b1\nb0 : 0 < b\n\u22a2 a * b \u2264 b \u2194 a * b \u2264 1 * b\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u2074 : MulOneClass \u03b1\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : MulPosStrictMono \u03b1\ninst\u271d : MulPosReflectLT \u03b1\nb0 : 0 < b\n\u22a2 a * b < b \u2194 a * b < 1 * b\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : MulPosMono \u03b1\nhb : 0 \u2264 b\nh : a \u2264 1\n\u22a2 a * b \u2264 b\n[PROOFSTEP]\nsimpa only [one_mul] using mul_le_mul_of_nonneg_right h hb\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : MulPosMono \u03b1\nhb : 0 \u2264 b\nh : 1 \u2264 a\n\u22a2 b \u2264 a * b\n[PROOFSTEP]\nsimpa only [one_mul] using mul_le_mul_of_nonneg_right h hb\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PosMulMono \u03b1\nha : 0 \u2264 a\nh : b \u2264 1\n\u22a2 a * b \u2264 a\n[PROOFSTEP]\nsimpa only [mul_one] using mul_le_mul_of_nonneg_left h ha\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PosMulMono \u03b1\nha : 0 \u2264 a\nh : 1 \u2264 b\n\u22a2 a \u2264 a * b\n[PROOFSTEP]\nsimpa only [mul_one] using mul_le_mul_of_nonneg_left h ha\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : MulPosStrictMono \u03b1\nhb : 0 < b\nh : a < 1\n\u22a2 a * b < b\n[PROOFSTEP]\nsimpa only [one_mul] using mul_lt_mul_of_pos_right h hb\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : MulPosStrictMono \u03b1\nhb : 0 < b\nh : 1 < a\n\u22a2 b < a * b\n[PROOFSTEP]\nsimpa only [one_mul] using mul_lt_mul_of_pos_right h hb\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PosMulStrictMono \u03b1\nha : 0 < a\nh : b < 1\n\u22a2 a * b < a\n[PROOFSTEP]\nsimpa only [mul_one] using mul_lt_mul_of_pos_left h ha\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PosMulStrictMono \u03b1\nha : 0 < a\nh : 1 < b\n\u22a2 a < a * b\n[PROOFSTEP]\nsimpa only [mul_one] using mul_lt_mul_of_pos_left h ha\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : PosMulStrictMono \u03b1\na0 : 0 < a\n\u22a2 \u2203 b, b * b \u2264 a\n[PROOFSTEP]\nobtain ha | ha := lt_or_le a 1\n[GOAL]\ncase inl\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : PosMulStrictMono \u03b1\na0 : 0 < a\nha : a < 1\n\u22a2 \u2203 b, b * b \u2264 a\n[PROOFSTEP]\nexact \u27e8a, (mul_lt_of_lt_one_right a0 ha).le\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : PosMulStrictMono \u03b1\na0 : 0 < a\nha : 1 \u2264 a\n\u22a2 \u2203 b, b * b \u2264 a\n[PROOFSTEP]\nexact \u27e81, by rwa [mul_one]\u27e9\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : PosMulStrictMono \u03b1\na0 : 0 < a\nha : 1 \u2264 a\n\u22a2 1 * 1 \u2264 a\n[PROOFSTEP]\nrwa [mul_one]\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b2 : CommSemigroup \u03b1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : Preorder \u03b1\n\u22a2 PosMulStrictMono \u03b1 \u2194 MulPosStrictMono \u03b1\n[PROOFSTEP]\nsimp only [PosMulStrictMono, MulPosStrictMono, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b2 : CommSemigroup \u03b1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : Preorder \u03b1\n\u22a2 PosMulReflectLT \u03b1 \u2194 MulPosReflectLT \u03b1\n[PROOFSTEP]\nsimp only [PosMulReflectLT, MulPosReflectLT, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b2 : CommSemigroup \u03b1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : Preorder \u03b1\n\u22a2 PosMulMono \u03b1 \u2194 MulPosMono \u03b1\n[PROOFSTEP]\nsimp only [PosMulMono, MulPosMono, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b2 : CommSemigroup \u03b1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : Preorder \u03b1\n\u22a2 PosMulMonoRev \u03b1 \u2194 MulPosMonoRev \u03b1\n[PROOFSTEP]\nsimp only [PosMulMonoRev, MulPosMonoRev, mul_comm]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Ring.Lemmas", "llama_tokens": 9634, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.6334102498375401, "lm_q1q2_score": 0.4470514708138365}} {"text": "[GOAL]\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\n\u22a2 \u2200 (a : \u2115), Acc (Nat.lbp p) a\n[PROOFSTEP]\nlet \u27e8n, pn\u27e9 := H\n[GOAL]\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nn : \u2115\npn : true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\n\u22a2 \u2200 (a : \u2115), Acc (Nat.lbp p) a\n[PROOFSTEP]\nsuffices \u2200 m k, n \u2264 k + m \u2192 Acc (lbp p) k by exact fun a => this _ _ (Nat.le_add_left _ _)\n[GOAL]\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nn : \u2115\npn : true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nthis : \u2200 (m k : \u2115), n \u2264 k + m \u2192 Acc (Nat.lbp p) k\n\u22a2 \u2200 (a : \u2115), Acc (Nat.lbp p) a\n[PROOFSTEP]\nexact fun a => this _ _ (Nat.le_add_left _ _)\n[GOAL]\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nn : \u2115\npn : true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\n\u22a2 \u2200 (m k : \u2115), n \u2264 k + m \u2192 Acc (Nat.lbp p) k\n[PROOFSTEP]\nintro m k kn\n[GOAL]\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nn : \u2115\npn : true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm k : \u2115\nkn : n \u2264 k + m\n\u22a2 Acc (Nat.lbp p) k\n[PROOFSTEP]\ninduction' m with m IH generalizing k\n[GOAL]\ncase zero\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nn : \u2115\npn : true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm k\u271d : \u2115\nkn\u271d : n \u2264 k\u271d + m\nk : \u2115\nkn : n \u2264 k + zero\n\u22a2 Acc (Nat.lbp p) k\n[PROOFSTEP]\nrefine' \u27e8_, fun y r => _\u27e9\n[GOAL]\ncase succ\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nn : \u2115\npn : true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm\u271d k\u271d : \u2115\nkn\u271d : n \u2264 k\u271d + m\u271d\nm : \u2115\nIH : \u2200 (k : \u2115), n \u2264 k + m \u2192 Acc (Nat.lbp p) k\nk : \u2115\nkn : n \u2264 k + succ m\n\u22a2 Acc (Nat.lbp p) k\n[PROOFSTEP]\nrefine' \u27e8_, fun y r => _\u27e9\n[GOAL]\ncase zero\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nn : \u2115\npn : true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm k\u271d : \u2115\nkn\u271d : n \u2264 k\u271d + m\nk : \u2115\nkn : n \u2264 k + zero\ny : \u2115\nr : Nat.lbp p y k\n\u22a2 Acc (Nat.lbp p) y\n[PROOFSTEP]\nrcases r with \u27e8rfl, a\u27e9\n[GOAL]\ncase succ\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nn : \u2115\npn : true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm\u271d k\u271d : \u2115\nkn\u271d : n \u2264 k\u271d + m\u271d\nm : \u2115\nIH : \u2200 (k : \u2115), n \u2264 k + m \u2192 Acc (Nat.lbp p) k\nk : \u2115\nkn : n \u2264 k + succ m\ny : \u2115\nr : Nat.lbp p y k\n\u22a2 Acc (Nat.lbp p) y\n[PROOFSTEP]\nrcases r with \u27e8rfl, a\u27e9\n[GOAL]\ncase zero.intro\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nn : \u2115\npn : true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm k\u271d : \u2115\nkn\u271d : n \u2264 k\u271d + m\nk : \u2115\nkn : n \u2264 k + zero\na : \u2200 (k_1 : \u2115), k_1 \u2264 k \u2192 false \u2208 p k_1\n\u22a2 Acc (Nat.lbp p) (k + 1)\n[PROOFSTEP]\ninjection mem_unique pn.1 (a _ kn)\n[GOAL]\ncase succ.intro\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nn : \u2115\npn : true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm\u271d k\u271d : \u2115\nkn\u271d : n \u2264 k\u271d + m\u271d\nm : \u2115\nIH : \u2200 (k : \u2115), n \u2264 k + m \u2192 Acc (Nat.lbp p) k\nk : \u2115\nkn : n \u2264 k + succ m\na : \u2200 (k_1 : \u2115), k_1 \u2264 k \u2192 false \u2208 p k_1\n\u22a2 Acc (Nat.lbp p) (k + 1)\n[PROOFSTEP]\nexact IH _ (by rw [Nat.add_right_comm]; exact kn)\n[GOAL]\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nn : \u2115\npn : true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm\u271d k\u271d : \u2115\nkn\u271d : n \u2264 k\u271d + m\u271d\nm : \u2115\nIH : \u2200 (k : \u2115), n \u2264 k + m \u2192 Acc (Nat.lbp p) k\nk : \u2115\nkn : n \u2264 k + succ m\na : \u2200 (k_1 : \u2115), k_1 \u2264 k \u2192 false \u2208 p k_1\n\u22a2 n \u2264 k + 1 + m\n[PROOFSTEP]\nrw [Nat.add_right_comm]\n[GOAL]\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nn : \u2115\npn : true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm\u271d k\u271d : \u2115\nkn\u271d : n \u2264 k\u271d + m\u271d\nm : \u2115\nIH : \u2200 (k : \u2115), n \u2264 k + m \u2192 Acc (Nat.lbp p) k\nk : \u2115\nkn : n \u2264 k + succ m\na : \u2200 (k_1 : \u2115), k_1 \u2264 k \u2192 false \u2208 p k_1\n\u22a2 n \u2264 k + m + 1\n[PROOFSTEP]\nexact kn\n[GOAL]\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\n\u22a2 (x : \u2115) \u2192\n ((y : \u2115) \u2192\n Nat.lbp p y x \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }) \u2192\n (\u2200 (n : \u2115), n < x \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\n[PROOFSTEP]\nintro m IH al\n[GOAL]\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp p y m \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\nal : \u2200 (n : \u2115), n < m \u2192 false \u2208 p n\n\u22a2 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\n[PROOFSTEP]\nhave pm : (p m).Dom := by\n rcases H with \u27e8n, h\u2081, h\u2082\u27e9\n rcases lt_trichotomy m n with (h\u2083 | h\u2083 | h\u2083)\n \u00b7 exact h\u2082 _ h\u2083\n \u00b7 rw [h\u2083]\n exact h\u2081.fst\n \u00b7 injection mem_unique h\u2081 (al _ h\u2083)\n[GOAL]\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp p y m \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\nal : \u2200 (n : \u2115), n < m \u2192 false \u2208 p n\n\u22a2 (p m).Dom\n[PROOFSTEP]\nrcases H with \u27e8n, h\u2081, h\u2082\u27e9\n[GOAL]\ncase intro.intro\np : \u2115 \u2192. Bool\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp p y m \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\nal : \u2200 (n : \u2115), n < m \u2192 false \u2208 p n\nn : \u2115\nh\u2081 : true \u2208 p n\nh\u2082 : \u2200 (k : \u2115), k < n \u2192 (p k).Dom\n\u22a2 (p m).Dom\n[PROOFSTEP]\nrcases lt_trichotomy m n with (h\u2083 | h\u2083 | h\u2083)\n[GOAL]\ncase intro.intro.inl\np : \u2115 \u2192. Bool\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp p y m \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\nal : \u2200 (n : \u2115), n < m \u2192 false \u2208 p n\nn : \u2115\nh\u2081 : true \u2208 p n\nh\u2082 : \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nh\u2083 : m < n\n\u22a2 (p m).Dom\n[PROOFSTEP]\nexact h\u2082 _ h\u2083\n[GOAL]\ncase intro.intro.inr.inl\np : \u2115 \u2192. Bool\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp p y m \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\nal : \u2200 (n : \u2115), n < m \u2192 false \u2208 p n\nn : \u2115\nh\u2081 : true \u2208 p n\nh\u2082 : \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nh\u2083 : m = n\n\u22a2 (p m).Dom\n[PROOFSTEP]\nrw [h\u2083]\n[GOAL]\ncase intro.intro.inr.inl\np : \u2115 \u2192. Bool\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp p y m \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\nal : \u2200 (n : \u2115), n < m \u2192 false \u2208 p n\nn : \u2115\nh\u2081 : true \u2208 p n\nh\u2082 : \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nh\u2083 : m = n\n\u22a2 (p n).Dom\n[PROOFSTEP]\nexact h\u2081.fst\n[GOAL]\ncase intro.intro.inr.inr\np : \u2115 \u2192. Bool\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp p y m \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\nal : \u2200 (n : \u2115), n < m \u2192 false \u2208 p n\nn : \u2115\nh\u2081 : true \u2208 p n\nh\u2082 : \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nh\u2083 : n < m\n\u22a2 (p m).Dom\n[PROOFSTEP]\ninjection mem_unique h\u2081 (al _ h\u2083)\n[GOAL]\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp p y m \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\nal : \u2200 (n : \u2115), n < m \u2192 false \u2208 p n\npm : (p m).Dom\n\u22a2 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\n[PROOFSTEP]\ncases e : (p m).get pm\n[GOAL]\ncase false\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp p y m \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\nal : \u2200 (n : \u2115), n < m \u2192 false \u2208 p n\npm : (p m).Dom\ne : Part.get (p m) pm = false\n\u22a2 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\n[PROOFSTEP]\nsuffices \u2200\u1d49 k \u2264 m, false \u2208 p k from IH _ \u27e8rfl, this\u27e9 fun n h => this _ (le_of_lt_succ h)\n[GOAL]\ncase false\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp p y m \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\nal : \u2200 (n : \u2115), n < m \u2192 false \u2208 p n\npm : (p m).Dom\ne : Part.get (p m) pm = false\n\u22a2 \u2200 (k : \u2115), k \u2264 m \u2192 false \u2208 p k\n[PROOFSTEP]\nintro n h\n[GOAL]\ncase false\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp p y m \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\nal : \u2200 (n : \u2115), n < m \u2192 false \u2208 p n\npm : (p m).Dom\ne : Part.get (p m) pm = false\nn : \u2115\nh : n \u2264 m\n\u22a2 false \u2208 p n\n[PROOFSTEP]\ncases' h.lt_or_eq_dec with h h\n[GOAL]\ncase false.inl\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp p y m \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\nal : \u2200 (n : \u2115), n < m \u2192 false \u2208 p n\npm : (p m).Dom\ne : Part.get (p m) pm = false\nn : \u2115\nh\u271d : n \u2264 m\nh : n < m\n\u22a2 false \u2208 p n\n[PROOFSTEP]\nexact al _ h\n[GOAL]\ncase false.inr\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp p y m \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\nal : \u2200 (n : \u2115), n < m \u2192 false \u2208 p n\npm : (p m).Dom\ne : Part.get (p m) pm = false\nn : \u2115\nh\u271d : n \u2264 m\nh : n = m\n\u22a2 false \u2208 p n\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase false.inr\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp p y m \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\nal : \u2200 (n : \u2115), n < m \u2192 false \u2208 p n\npm : (p m).Dom\ne : Part.get (p m) pm = false\nn : \u2115\nh\u271d : n \u2264 m\nh : n = m\n\u22a2 false \u2208 p m\n[PROOFSTEP]\nexact \u27e8_, e\u27e9\n[GOAL]\ncase true\np : \u2115 \u2192. Bool\nH : \u2203 n, true \u2208 p n \u2227 \u2200 (k : \u2115), k < n \u2192 (p k).Dom\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp p y m \u2192 (\u2200 (n : \u2115), n < y \u2192 false \u2208 p n) \u2192 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\nal : \u2200 (n : \u2115), n < m \u2192 false \u2208 p n\npm : (p m).Dom\ne : Part.get (p m) pm = true\n\u22a2 { n // true \u2208 p n \u2227 \u2200 (m : \u2115), m < n \u2192 false \u2208 p m }\n[PROOFSTEP]\nexact \u27e8m, \u27e8_, e\u27e9, al\u27e9\n[GOAL]\np : \u2115 \u2192. Bool\nn : \u2115\nx\u271d : true \u2208 p n \u2227 \u2200 {m : \u2115}, m < n \u2192 false \u2208 p m\nh\u2081 : true \u2208 p n\nh\u2082 : \u2200 {m : \u2115}, m < n \u2192 false \u2208 p m\n\u22a2 n \u2208 rfind p\n[PROOFSTEP]\nlet \u27e8m, hm\u27e9 := dom_iff_mem.1 <| (@rfind_dom p).2 \u27e8_, h\u2081, fun {m} mn => (h\u2082 mn).fst\u27e9\n[GOAL]\np : \u2115 \u2192. Bool\nn : \u2115\nx\u271d : true \u2208 p n \u2227 \u2200 {m : \u2115}, m < n \u2192 false \u2208 p m\nh\u2081 : true \u2208 p n\nh\u2082 : \u2200 {m : \u2115}, m < n \u2192 false \u2208 p m\nm : \u2115\nhm : m \u2208 rfind p\n\u22a2 n \u2208 rfind p\n[PROOFSTEP]\nrcases lt_trichotomy m n with (h | h | h)\n[GOAL]\ncase inl\np : \u2115 \u2192. Bool\nn : \u2115\nx\u271d : true \u2208 p n \u2227 \u2200 {m : \u2115}, m < n \u2192 false \u2208 p m\nh\u2081 : true \u2208 p n\nh\u2082 : \u2200 {m : \u2115}, m < n \u2192 false \u2208 p m\nm : \u2115\nhm : m \u2208 rfind p\nh : m < n\n\u22a2 n \u2208 rfind p\n[PROOFSTEP]\ninjection mem_unique (h\u2082 h) (rfind_spec hm)\n[GOAL]\ncase inr.inl\np : \u2115 \u2192. Bool\nn : \u2115\nx\u271d : true \u2208 p n \u2227 \u2200 {m : \u2115}, m < n \u2192 false \u2208 p m\nh\u2081 : true \u2208 p n\nh\u2082 : \u2200 {m : \u2115}, m < n \u2192 false \u2208 p m\nm : \u2115\nhm : m \u2208 rfind p\nh : m = n\n\u22a2 n \u2208 rfind p\n[PROOFSTEP]\nrwa [\u2190 h]\n[GOAL]\ncase inr.inr\np : \u2115 \u2192. Bool\nn : \u2115\nx\u271d : true \u2208 p n \u2227 \u2200 {m : \u2115}, m < n \u2192 false \u2208 p m\nh\u2081 : true \u2208 p n\nh\u2082 : \u2200 {m : \u2115}, m < n \u2192 false \u2208 p m\nm : \u2115\nhm : m \u2208 rfind p\nh : n < m\n\u22a2 n \u2208 rfind p\n[PROOFSTEP]\ninjection mem_unique h\u2081 (rfind_min hm h)\n[GOAL]\np : \u2115 \u2192 Bool\nm : \u2115\npm : p m = true\nthis : true \u2208 \u2191p m\nn : \u2115\nhn : n \u2208 rfind \u2191p\nh : m < n\n\u22a2 False\n[PROOFSTEP]\ninjection mem_unique this (rfind_min hn h)\n[GOAL]\n\u03b1 : Type u_1\nf : \u2115 \u2192 Option \u03b1\nh : \u2203 n a, a \u2208 f n\n\u22a2 (rfindOpt f).Dom\n[PROOFSTEP]\nhave h' : \u2203 n, (f n).isSome := h.imp fun n => Option.isSome_iff_exists.2\n[GOAL]\n\u03b1 : Type u_1\nf : \u2115 \u2192 Option \u03b1\nh : \u2203 n a, a \u2208 f n\nh' : \u2203 n, Option.isSome (f n) = true\n\u22a2 (rfindOpt f).Dom\n[PROOFSTEP]\nhave s := Nat.find_spec h'\n[GOAL]\n\u03b1 : Type u_1\nf : \u2115 \u2192 Option \u03b1\nh : \u2203 n a, a \u2208 f n\nh' : \u2203 n, Option.isSome (f n) = true\ns : Option.isSome (f (Nat.find h')) = true\n\u22a2 (rfindOpt f).Dom\n[PROOFSTEP]\nhave fd : (rfind fun n => (f n).isSome).Dom := \u27e8Nat.find h', by simpa using s.symm, fun _ _ => trivial\u27e9\n[GOAL]\n\u03b1 : Type u_1\nf : \u2115 \u2192 Option \u03b1\nh : \u2203 n a, a \u2208 f n\nh' : \u2203 n, Option.isSome (f n) = true\ns : Option.isSome (f (Nat.find h')) = true\n\u22a2 true \u2208 (fun n => \u2191(Option.some (Option.isSome (f n)))) (Nat.find h')\n[PROOFSTEP]\nsimpa using s.symm\n[GOAL]\n\u03b1 : Type u_1\nf : \u2115 \u2192 Option \u03b1\nh : \u2203 n a, a \u2208 f n\nh' : \u2203 n, Option.isSome (f n) = true\ns : Option.isSome (f (Nat.find h')) = true\nfd : (rfind fun n => \u2191(Option.some (Option.isSome (f n)))).Dom\n\u22a2 (rfindOpt f).Dom\n[PROOFSTEP]\nrefine' \u27e8fd, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\nf : \u2115 \u2192 Option \u03b1\nh : \u2203 n a, a \u2208 f n\nh' : \u2203 n, Option.isSome (f n) = true\ns : Option.isSome (f (Nat.find h')) = true\nfd : (rfind fun n => \u2191(Option.some (Option.isSome (f n)))).Dom\n\u22a2 ((fun b => (fun n => \u2191(f n)) (Part.get (rfind fun n => \u2191(Option.some (Option.isSome (f n)))) b)) fd).Dom\n[PROOFSTEP]\nhave := rfind_spec (get_mem fd)\n[GOAL]\n\u03b1 : Type u_1\nf : \u2115 \u2192 Option \u03b1\nh : \u2203 n a, a \u2208 f n\nh' : \u2203 n, Option.isSome (f n) = true\ns : Option.isSome (f (Nat.find h')) = true\nfd : (rfind fun n => \u2191(Option.some (Option.isSome (f n)))).Dom\nthis : true \u2208 \u2191(Option.some (Option.isSome (f (Part.get (rfind fun n => \u2191(Option.some (Option.isSome (f n)))) fd))))\n\u22a2 ((fun b => (fun n => \u2191(f n)) (Part.get (rfind fun n => \u2191(Option.some (Option.isSome (f n)))) b)) fd).Dom\n[PROOFSTEP]\nsimp at this \u22a2\n[GOAL]\n\u03b1 : Type u_1\nf : \u2115 \u2192 Option \u03b1\nh : \u2203 n a, a \u2208 f n\nh' : \u2203 n, Option.isSome (f n) = true\ns : Option.isSome (f (Nat.find h')) = true\nfd : (rfind fun n => \u2191(Option.some (Option.isSome (f n)))).Dom\nthis :\n true =\n Option.isSome\n (f\n (Part.get (rfind fun n => Part.some (Option.isSome (f n)))\n (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom)))\n\u22a2 Option.isSome\n (f\n (Part.get (rfind fun n => Part.some (Option.isSome (f n)))\n (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom))) =\n true\n[PROOFSTEP]\ncases' Option.isSome_iff_exists.1 this.symm with a e\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nf : \u2115 \u2192 Option \u03b1\nh : \u2203 n a, a \u2208 f n\nh' : \u2203 n, Option.isSome (f n) = true\ns : Option.isSome (f (Nat.find h')) = true\nfd : (rfind fun n => \u2191(Option.some (Option.isSome (f n)))).Dom\nthis :\n true =\n Option.isSome\n (f\n (Part.get (rfind fun n => Part.some (Option.isSome (f n)))\n (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom)))\na : \u03b1\ne :\n f\n (Part.get (rfind fun n => Part.some (Option.isSome (f n)))\n (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom)) =\n Option.some a\n\u22a2 Option.isSome\n (f\n (Part.get (rfind fun n => Part.some (Option.isSome (f n)))\n (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom))) =\n true\n[PROOFSTEP]\nrw [e]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nf : \u2115 \u2192 Option \u03b1\nh : \u2203 n a, a \u2208 f n\nh' : \u2203 n, Option.isSome (f n) = true\ns : Option.isSome (f (Nat.find h')) = true\nfd : (rfind fun n => \u2191(Option.some (Option.isSome (f n)))).Dom\nthis :\n true =\n Option.isSome\n (f\n (Part.get (rfind fun n => Part.some (Option.isSome (f n)))\n (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom)))\na : \u03b1\ne :\n f\n (Part.get (rfind fun n => Part.some (Option.isSome (f n)))\n (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom)) =\n Option.some a\n\u22a2 Option.isSome (Option.some a) = true\n[PROOFSTEP]\ntrivial\n[GOAL]\n\u03b1 : Type u_1\nf : \u2115 \u2192 Option \u03b1\nH : \u2200 {a : \u03b1} {m n : \u2115}, m \u2264 n \u2192 a \u2208 f m \u2192 a \u2208 f n\na : \u03b1\nx\u271d : \u2203 n, a \u2208 f n\nn : \u2115\nh : a \u2208 f n\n\u22a2 a \u2208 rfindOpt f\n[PROOFSTEP]\nhave h' := rfindOpt_dom.2 \u27e8_, _, h\u27e9\n[GOAL]\n\u03b1 : Type u_1\nf : \u2115 \u2192 Option \u03b1\nH : \u2200 {a : \u03b1} {m n : \u2115}, m \u2264 n \u2192 a \u2208 f m \u2192 a \u2208 f n\na : \u03b1\nx\u271d : \u2203 n, a \u2208 f n\nn : \u2115\nh : a \u2208 f n\nh' : (rfindOpt f).Dom\n\u22a2 a \u2208 rfindOpt f\n[PROOFSTEP]\ncases' rfindOpt_spec \u27e8h', rfl\u27e9 with k hk\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nf : \u2115 \u2192 Option \u03b1\nH : \u2200 {a : \u03b1} {m n : \u2115}, m \u2264 n \u2192 a \u2208 f m \u2192 a \u2208 f n\na : \u03b1\nx\u271d : \u2203 n, a \u2208 f n\nn : \u2115\nh : a \u2208 f n\nh' : (rfindOpt f).Dom\nk : \u2115\nhk : Part.get (rfindOpt f) h' \u2208 f k\n\u22a2 a \u2208 rfindOpt f\n[PROOFSTEP]\nhave := (H (le_max_left _ _) h).symm.trans (H (le_max_right _ _) hk)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nf : \u2115 \u2192 Option \u03b1\nH : \u2200 {a : \u03b1} {m n : \u2115}, m \u2264 n \u2192 a \u2208 f m \u2192 a \u2208 f n\na : \u03b1\nx\u271d : \u2203 n, a \u2208 f n\nn : \u2115\nh : a \u2208 f n\nh' : (rfindOpt f).Dom\nk : \u2115\nhk : Part.get (rfindOpt f) h' \u2208 f k\nthis : Option.some a = Option.some (Part.get (rfindOpt f) h')\n\u22a2 a \u2208 rfindOpt f\n[PROOFSTEP]\nsimp at this \n[GOAL]\ncase intro\n\u03b1 : Type u_1\nf : \u2115 \u2192 Option \u03b1\nH : \u2200 {a : \u03b1} {m n : \u2115}, m \u2264 n \u2192 a \u2208 f m \u2192 a \u2208 f n\na : \u03b1\nx\u271d : \u2203 n, a \u2208 f n\nn : \u2115\nh : a \u2208 f n\nh' : (rfindOpt f).Dom\nk : \u2115\nhk : Part.get (rfindOpt f) h' \u2208 f k\nthis : a = Part.get (rfindOpt f) h'\n\u22a2 a \u2208 rfindOpt f\n[PROOFSTEP]\nsimp [this, get_mem]\n[GOAL]\nf : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\n\u22a2 Partrec \u2191f\n[PROOFSTEP]\ninduction hf with\n| zero => exact zero\n| succ => exact succ\n| left => exact left\n| right => exact right\n| pair _ _ pf pg =>\n refine' (pf.pair pg).of_eq_tot fun n => _\n simp [Seq.seq]\n| comp _ _ pf pg =>\n refine' (pf.comp pg).of_eq_tot fun n => _\n simp\n| prec _ _ pf pg =>\n refine' (pf.prec pg).of_eq_tot fun n => _\n simp only [unpaired, PFun.coe_val, bind_eq_bind]\n induction n.unpair.2 with\n | zero => simp\n | succ m IH =>\n simp only [mem_bind_iff, mem_some_iff]\n exact \u27e8_, IH, rfl\u27e9\n[GOAL]\nf : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\n\u22a2 Partrec \u2191f\n[PROOFSTEP]\ninduction hf with\n| zero => exact zero\n| succ => exact succ\n| left => exact left\n| right => exact right\n| pair _ _ pf pg =>\n refine' (pf.pair pg).of_eq_tot fun n => _\n simp [Seq.seq]\n| comp _ _ pf pg =>\n refine' (pf.comp pg).of_eq_tot fun n => _\n simp\n| prec _ _ pf pg =>\n refine' (pf.prec pg).of_eq_tot fun n => _\n simp only [unpaired, PFun.coe_val, bind_eq_bind]\n induction n.unpair.2 with\n | zero => simp\n | succ m IH =>\n simp only [mem_bind_iff, mem_some_iff]\n exact \u27e8_, IH, rfl\u27e9\n[GOAL]\ncase zero\nf : \u2115 \u2192 \u2115\n\u22a2 Partrec \u2191fun x => 0\n[PROOFSTEP]\n\n| zero => exact zero\n[GOAL]\ncase zero\nf : \u2115 \u2192 \u2115\n\u22a2 Partrec \u2191fun x => 0\n[PROOFSTEP]\nexact zero\n[GOAL]\ncase succ\nf : \u2115 \u2192 \u2115\n\u22a2 Partrec \u2191Nat.succ\n[PROOFSTEP]\n\n| succ => exact succ\n[GOAL]\ncase succ\nf : \u2115 \u2192 \u2115\n\u22a2 Partrec \u2191Nat.succ\n[PROOFSTEP]\nexact succ\n[GOAL]\ncase left\nf : \u2115 \u2192 \u2115\n\u22a2 Partrec \u2191fun n => (unpair n).fst\n[PROOFSTEP]\n\n| left => exact left\n[GOAL]\ncase left\nf : \u2115 \u2192 \u2115\n\u22a2 Partrec \u2191fun n => (unpair n).fst\n[PROOFSTEP]\nexact left\n[GOAL]\ncase right\nf : \u2115 \u2192 \u2115\n\u22a2 Partrec \u2191fun n => (unpair n).snd\n[PROOFSTEP]\n\n| right => exact right\n[GOAL]\ncase right\nf : \u2115 \u2192 \u2115\n\u22a2 Partrec \u2191fun n => (unpair n).snd\n[PROOFSTEP]\nexact right\n[GOAL]\ncase pair\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\n\u22a2 Partrec \u2191fun n => Nat.pair (f\u271d n) (g\u271d n)\n[PROOFSTEP]\n\n| pair _ _ pf pg =>\n refine' (pf.pair pg).of_eq_tot fun n => _\n simp [Seq.seq]\n[GOAL]\ncase pair\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\n\u22a2 Partrec \u2191fun n => Nat.pair (f\u271d n) (g\u271d n)\n[PROOFSTEP]\nrefine' (pf.pair pg).of_eq_tot fun n => _\n[GOAL]\ncase pair\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\nn : \u2115\n\u22a2 Nat.pair (f\u271d n) (g\u271d n) \u2208 Seq.seq (Nat.pair <$> \u2191f\u271d n) fun x => \u2191g\u271d n\n[PROOFSTEP]\nsimp [Seq.seq]\n[GOAL]\ncase comp\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\n\u22a2 Partrec \u2191fun n => f\u271d (g\u271d n)\n[PROOFSTEP]\n\n| comp _ _ pf pg =>\n refine' (pf.comp pg).of_eq_tot fun n => _\n simp\n[GOAL]\ncase comp\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\n\u22a2 Partrec \u2191fun n => f\u271d (g\u271d n)\n[PROOFSTEP]\nrefine' (pf.comp pg).of_eq_tot fun n => _\n[GOAL]\ncase comp\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\nn : \u2115\n\u22a2 f\u271d (g\u271d n) \u2208 \u2191g\u271d n >>= \u2191f\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase prec\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\n\u22a2 Partrec \u2191(unpaired fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (Nat.pair z (Nat.pair y IH))) n)\n[PROOFSTEP]\n\n| prec _ _ pf pg =>\n refine' (pf.prec pg).of_eq_tot fun n => _\n simp only [unpaired, PFun.coe_val, bind_eq_bind]\n induction n.unpair.2 with\n | zero => simp\n | succ m IH =>\n simp only [mem_bind_iff, mem_some_iff]\n exact \u27e8_, IH, rfl\u27e9\n[GOAL]\ncase prec\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\n\u22a2 Partrec \u2191(unpaired fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (Nat.pair z (Nat.pair y IH))) n)\n[PROOFSTEP]\nrefine' (pf.prec pg).of_eq_tot fun n => _\n[GOAL]\ncase prec\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\nn : \u2115\n\u22a2 unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (Nat.pair z (Nat.pair y IH))) n) n \u2208\n unpaired\n (fun a n =>\n Nat.rec (\u2191f\u271d a)\n (fun y IH => do\n let i \u2190 IH\n \u2191g\u271d (Nat.pair a (Nat.pair y i)))\n n)\n n\n[PROOFSTEP]\nsimp only [unpaired, PFun.coe_val, bind_eq_bind]\n[GOAL]\ncase prec\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\nn : \u2115\n\u22a2 Nat.rec (f\u271d (unpair n).fst) (fun y IH => g\u271d (Nat.pair (unpair n).fst (Nat.pair y IH))) (unpair n).snd \u2208\n Nat.rec (Part.some (f\u271d (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g\u271d (Nat.pair (unpair n).fst (Nat.pair y i)))) (unpair n).snd\n[PROOFSTEP]\ninduction n.unpair.2 with\n| zero => simp\n| succ m IH =>\n simp only [mem_bind_iff, mem_some_iff]\n exact \u27e8_, IH, rfl\u27e9\n[GOAL]\ncase prec\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\nn x\u271d : \u2115\n\u22a2 Nat.rec (f\u271d (unpair n).fst) (fun y IH => g\u271d (Nat.pair (unpair n).fst (Nat.pair y IH))) x\u271d \u2208\n Nat.rec (Part.some (f\u271d (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g\u271d (Nat.pair (unpair n).fst (Nat.pair y i)))) x\u271d\n[PROOFSTEP]\ninduction n.unpair.2 with\n| zero => simp\n| succ m IH =>\n simp only [mem_bind_iff, mem_some_iff]\n exact \u27e8_, IH, rfl\u27e9\n[GOAL]\ncase prec.zero\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\nn : \u2115\n\u22a2 Nat.rec (f\u271d (unpair n).fst) (fun y IH => g\u271d (Nat.pair (unpair n).fst (Nat.pair y IH))) Nat.zero \u2208\n Nat.rec (Part.some (f\u271d (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g\u271d (Nat.pair (unpair n).fst (Nat.pair y i)))) Nat.zero\n[PROOFSTEP]\n\n| zero => simp\n[GOAL]\ncase prec.zero\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\nn : \u2115\n\u22a2 Nat.rec (f\u271d (unpair n).fst) (fun y IH => g\u271d (Nat.pair (unpair n).fst (Nat.pair y IH))) Nat.zero \u2208\n Nat.rec (Part.some (f\u271d (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g\u271d (Nat.pair (unpair n).fst (Nat.pair y i)))) Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase prec.succ\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\nn m : \u2115\nIH :\n Nat.rec (f\u271d (unpair n).fst) (fun y IH => g\u271d (Nat.pair (unpair n).fst (Nat.pair y IH))) m \u2208\n Nat.rec (Part.some (f\u271d (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g\u271d (Nat.pair (unpair n).fst (Nat.pair y i)))) m\n\u22a2 Nat.rec (f\u271d (unpair n).fst) (fun y IH => g\u271d (Nat.pair (unpair n).fst (Nat.pair y IH))) (Nat.succ m) \u2208\n Nat.rec (Part.some (f\u271d (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g\u271d (Nat.pair (unpair n).fst (Nat.pair y i)))) (Nat.succ m)\n[PROOFSTEP]\n\n| succ m IH =>\n simp only [mem_bind_iff, mem_some_iff]\n exact \u27e8_, IH, rfl\u27e9\n[GOAL]\ncase prec.succ\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\nn m : \u2115\nIH :\n Nat.rec (f\u271d (unpair n).fst) (fun y IH => g\u271d (Nat.pair (unpair n).fst (Nat.pair y IH))) m \u2208\n Nat.rec (Part.some (f\u271d (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g\u271d (Nat.pair (unpair n).fst (Nat.pair y i)))) m\n\u22a2 Nat.rec (f\u271d (unpair n).fst) (fun y IH => g\u271d (Nat.pair (unpair n).fst (Nat.pair y IH))) (Nat.succ m) \u2208\n Nat.rec (Part.some (f\u271d (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g\u271d (Nat.pair (unpair n).fst (Nat.pair y i)))) (Nat.succ m)\n[PROOFSTEP]\nsimp only [mem_bind_iff, mem_some_iff]\n[GOAL]\ncase prec.succ\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\npf : Partrec \u2191f\u271d\npg : Partrec \u2191g\u271d\nn m : \u2115\nIH :\n Nat.rec (f\u271d (unpair n).fst) (fun y IH => g\u271d (Nat.pair (unpair n).fst (Nat.pair y IH))) m \u2208\n Nat.rec (Part.some (f\u271d (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g\u271d (Nat.pair (unpair n).fst (Nat.pair y i)))) m\n\u22a2 \u2203 a,\n a \u2208\n Nat.rec (Part.some (f\u271d (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g\u271d (Nat.pair (unpair n).fst (Nat.pair y i)))) m \u2227\n g\u271d\n (Nat.pair (unpair n).fst\n (Nat.pair m (Nat.rec (f\u271d (unpair n).fst) (fun y IH => g\u271d (Nat.pair (unpair n).fst (Nat.pair y IH))) m))) =\n g\u271d (Nat.pair (unpair n).fst (Nat.pair m a))\n[PROOFSTEP]\nexact \u27e8_, IH, rfl\u27e9\n[GOAL]\nn a : \u2115\nx\u271d : a \u2208 Nat.rfind fun n_1 => (fun m => decide (m = 0)) <$> (\u2191fun x => 1) (Nat.pair n n_1)\nh : (Nat.rfind fun n_1 => (fun m => decide (m = 0)) <$> (\u2191fun x => 1) (Nat.pair n n_1)).Dom\nh\u271d : Part.get (Nat.rfind fun n_1 => (fun m => decide (m = 0)) <$> (\u2191fun x => 1) (Nat.pair n n_1)) h = a\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\nf g h : \u2115 \u2192. \u2115\nhf : Partrec f\nhg : Partrec g\nhh : Partrec h\na s : \u2115\n\u22a2 (s \u2208\n (Seq.seq (Nat.pair <$> Part.some a) fun x => f a) >>=\n unpaired fun a n =>\n Nat.rec (g a)\n (fun y IH => do\n let i \u2190 IH\n h (Nat.pair a (Nat.pair y i)))\n n) \u2194\n s \u2208\n Part.bind (f a) fun n =>\n Nat.rec (g a)\n (fun y IH => do\n let i \u2190 IH\n h (Nat.pair a (Nat.pair y i)))\n n\n[PROOFSTEP]\nsimp [Seq.seq]\n[GOAL]\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\nn : \u2115\n\u22a2 (Nat.rfind fun n_1 =>\n (fun m => decide (m = 0)) <$> \u2191(unpaired fun n m => if n = Nat.succ m then 0 else 1) (Nat.pair n n_1)) =\n \u2191(Nat.ppred n)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\n\u22a2 (Nat.rfind fun n =>\n (fun m => decide (m = 0)) <$> \u2191(unpaired fun n m => if n = Nat.succ m then 0 else 1) (Nat.pair Nat.zero n)) =\n \u2191(Nat.ppred Nat.zero)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\nn\u271d : \u2115\n\u22a2 (Nat.rfind fun n =>\n (fun m => decide (m = 0)) <$> \u2191(unpaired fun n m => if n = Nat.succ m then 0 else 1) (Nat.pair (Nat.succ n\u271d) n)) =\n \u2191(Nat.ppred (Nat.succ n\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase zero\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\n\u22a2 (Nat.rfind fun n => Part.some false) = Part.none\n[PROOFSTEP]\nexact eq_none_iff.2 fun a \u27e8\u27e8m, h, _\u27e9, _\u27e9 => by simp [show 0 \u2260 m.succ by intro h; injection h] at h \n[GOAL]\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\na : \u2115\nx\u271d : a \u2208 Nat.rfind fun n => Part.some false\nm : \u2115\nh : true \u2208 (fun n => Part.some false) m\nright\u271d : \u2200 (k : \u2115), k < m \u2192 ((fun n => Part.some false) k).Dom\nh\u271d :\n Part.get (Nat.rfind fun n => Part.some false)\n (_ : \u2203 n, true \u2208 (fun n => Part.some false) n \u2227 \u2200 (k : \u2115), k < n \u2192 ((fun n => Part.some false) k).Dom) =\n a\n\u22a2 False\n[PROOFSTEP]\nsimp [show 0 \u2260 m.succ by intro h; injection h] at h \n[GOAL]\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\na : \u2115\nx\u271d : a \u2208 Nat.rfind fun n => Part.some false\nm : \u2115\nh : true \u2208 (fun n => Part.some false) m\nright\u271d : \u2200 (k : \u2115), k < m \u2192 ((fun n => Part.some false) k).Dom\nh\u271d :\n Part.get (Nat.rfind fun n => Part.some false)\n (_ : \u2203 n, true \u2208 (fun n => Part.some false) n \u2227 \u2200 (k : \u2115), k < n \u2192 ((fun n => Part.some false) k).Dom) =\n a\n\u22a2 0 \u2260 Nat.succ m\n[PROOFSTEP]\nintro h\n[GOAL]\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\na : \u2115\nx\u271d : a \u2208 Nat.rfind fun n => Part.some false\nm : \u2115\nh\u271d\u00b9 : true \u2208 (fun n => Part.some false) m\nright\u271d : \u2200 (k : \u2115), k < m \u2192 ((fun n => Part.some false) k).Dom\nh\u271d :\n Part.get (Nat.rfind fun n => Part.some false)\n (_ : \u2203 n, true \u2208 (fun n => Part.some false) n \u2227 \u2200 (k : \u2115), k < n \u2192 ((fun n => Part.some false) k).Dom) =\n a\nh : 0 = Nat.succ m\n\u22a2 False\n[PROOFSTEP]\ninjection h\n[GOAL]\ncase succ\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\nn\u271d : \u2115\n\u22a2 (Nat.rfind fun n => Part.some (decide (\u00acn\u271d = n \u2192 False))) = Part.some n\u271d\n[PROOFSTEP]\nrefine' eq_some_iff.2 _\n[GOAL]\ncase succ\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\nn\u271d : \u2115\n\u22a2 n\u271d \u2208 Nat.rfind fun n => Part.some (decide (\u00acn\u271d = n \u2192 False))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\nn\u271d : \u2115\n\u22a2 \u2200 {m : \u2115}, m < n\u271d \u2192 \u00ac(\u00acn\u271d = m \u2192 False)\n[PROOFSTEP]\nintro m h\n[GOAL]\ncase succ\nthis : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1\nn\u271d m : \u2115\nh : m < n\u271d\n\u22a2 \u00ac(\u00acn\u271d = m \u2192 False)\n[PROOFSTEP]\nsimp [ne_of_gt h]\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03c3\nhf : Primrec f\nn : \u2115\n\u22a2 (do\n let n \u2190 (\u2191fun n => encode (Option.map f (decode n))) n\n \u2191(Nat.ppred n)) =\n Part.bind \u2191(decode n) fun a => map encode (\u2191f a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03c3\nhf : Primrec f\nn : \u2115\n\u22a2 \u2191(Nat.ppred (encode (Option.map f (decode n)))) = Part.bind \u2191(decode n) fun a => Part.some (encode (f a))\n[PROOFSTEP]\ncases decode (\u03b1 := \u03b1) n\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03c3\nhf : Primrec f\nn : \u2115\n\u22a2 \u2191(Nat.ppred (encode (Option.map f Option.none))) = Part.bind \u2191Option.none fun a => Part.some (encode (f a))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03c3\nhf : Primrec f\nn : \u2115\nval\u271d : \u03b1\n\u22a2 \u2191(Nat.ppred (encode (Option.map f (Option.some val\u271d)))) =\n Part.bind \u2191(Option.some val\u271d) fun a => Part.some (encode (f a))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Option \u03b2\nhf : Computable f\nn : \u2115\n\u22a2 (do\n let n \u2190 Part.bind \u2191(decode n) fun a => map encode (\u2191f a)\n \u2191(Nat.ppred n)) =\n Part.bind \u2191(decode n) fun a => map encode ((fun a => \u2191(f a)) a)\n[PROOFSTEP]\ncases' decode (\u03b1 := \u03b1) n with a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Option \u03b2\nhf : Computable f\nn : \u2115\n\u22a2 (do\n let n \u2190 Part.bind \u2191Option.none fun a => map encode (\u2191f a)\n \u2191(Nat.ppred n)) =\n Part.bind \u2191Option.none fun a => map encode ((fun a => \u2191(f a)) a)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Option \u03b2\nhf : Computable f\nn : \u2115\na : \u03b1\n\u22a2 (do\n let n \u2190 Part.bind \u2191(Option.some a) fun a => map encode (\u2191f a)\n \u2191(Nat.ppred n)) =\n Part.bind \u2191(Option.some a) fun a => map encode ((fun a => \u2191(f a)) a)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Option \u03b2\nhf : Computable f\nn : \u2115\na : \u03b1\n\u22a2 \u2191(Nat.ppred (encode (f a))) = map encode \u2191(f a)\n[PROOFSTEP]\ncases' f a with b\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Option \u03b2\nhf : Computable f\nn : \u2115\na : \u03b1\n\u22a2 \u2191(Nat.ppred (encode Option.none)) = map encode \u2191Option.none\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Option \u03b2\nhf : Computable f\nn : \u2115\na : \u03b1\nb : \u03b2\n\u22a2 \u2191(Nat.ppred (encode (Option.some b))) = map encode \u2191(Option.some b)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\nhf : Computable f\nhg : Computable g\nn : \u2115\n\u22a2 (Seq.seq (Nat.pair <$> Part.bind \u2191(decode n) fun a => map encode (\u2191f a)) fun x =>\n Part.bind \u2191(decode n) fun a => map encode (\u2191g a)) =\n Part.bind \u2191(decode n) fun a => map encode ((\u2191fun a => (f a, g a)) a)\n[PROOFSTEP]\ncases decode (\u03b1 := \u03b1) n\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\nhf : Computable f\nhg : Computable g\nn : \u2115\n\u22a2 (Seq.seq (Nat.pair <$> Part.bind \u2191Option.none fun a => map encode (\u2191f a)) fun x =>\n Part.bind \u2191Option.none fun a => map encode (\u2191g a)) =\n Part.bind \u2191Option.none fun a => map encode ((\u2191fun a => (f a, g a)) a)\n[PROOFSTEP]\nsimp [Seq.seq]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\nhf : Computable f\nhg : Computable g\nn : \u2115\nval\u271d : \u03b1\n\u22a2 (Seq.seq (Nat.pair <$> Part.bind \u2191(Option.some val\u271d) fun a => map encode (\u2191f a)) fun x =>\n Part.bind \u2191(Option.some val\u271d) fun a => map encode (\u2191g a)) =\n Part.bind \u2191(Option.some val\u271d) fun a => map encode ((\u2191fun a => (f a, g a)) a)\n[PROOFSTEP]\nsimp [Seq.seq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nn : \u2115\n\u22a2 Part.none = Part.bind \u2191(decode n) fun a => map encode ((fun x => Part.none) a)\n[PROOFSTEP]\ncases decode (\u03b1 := \u03b1) n\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nn : \u2115\n\u22a2 Part.none = Part.bind \u2191Option.none fun a => map encode ((fun x => Part.none) a)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nn : \u2115\nval\u271d : \u03b1\n\u22a2 Part.none = Part.bind \u2191(Option.some val\u271d) fun a => map encode ((fun x => Part.none) a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03b2\ng : \u03b1 \u2192 \u03b2 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec\u2082 g\nn : \u2115\n\u22a2 (do\n let n \u2190 Seq.seq (Nat.pair <$> Part.some n) fun x => Part.bind \u2191(decode n) fun a => map encode (f a)\n Part.bind \u2191(decode n) fun a => map encode ((fun p => g p.fst p.snd) a)) =\n Part.bind \u2191(decode n) fun a => map encode ((fun a => Part.bind (f a) (g a)) a)\n[PROOFSTEP]\nsimp [Seq.seq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03b2\ng : \u03b1 \u2192 \u03b2 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec\u2082 g\nn : \u2115\n\u22a2 (Part.bind (Part.bind \u2191(decode n) fun y => map (Nat.pair n) (map encode (f y))) fun n =>\n Part.bind \u2191(Option.bind (decode (Nat.unpair n).fst) fun a => Option.map (Prod.mk a) (decode (Nat.unpair n).snd))\n fun a => map encode (g a.fst a.snd)) =\n Part.bind \u2191(decode n) fun a => Part.bind (f a) fun y => map encode (g a y)\n[PROOFSTEP]\ncases' e : decode (\u03b1 := \u03b1) n with a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03b2\ng : \u03b1 \u2192 \u03b2 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec\u2082 g\nn : \u2115\ne : decode n = Option.none\n\u22a2 (Part.bind (Part.bind \u2191Option.none fun y => map (Nat.pair n) (map encode (f y))) fun n =>\n Part.bind \u2191(Option.bind (decode (Nat.unpair n).fst) fun a => Option.map (Prod.mk a) (decode (Nat.unpair n).snd))\n fun a => map encode (g a.fst a.snd)) =\n Part.bind \u2191Option.none fun a => Part.bind (f a) fun y => map encode (g a y)\n[PROOFSTEP]\nsimp [e, encodek]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03b2\ng : \u03b1 \u2192 \u03b2 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec\u2082 g\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\n\u22a2 (Part.bind (Part.bind \u2191(Option.some a) fun y => map (Nat.pair n) (map encode (f y))) fun n =>\n Part.bind \u2191(Option.bind (decode (Nat.unpair n).fst) fun a => Option.map (Prod.mk a) (decode (Nat.unpair n).snd))\n fun a => map encode (g a.fst a.snd)) =\n Part.bind \u2191(Option.some a) fun a => Part.bind (f a) fun y => map encode (g a y)\n[PROOFSTEP]\nsimp [e, encodek]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03b2\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nhf : Partrec f\nhg : Computable\u2082 g\n\u22a2 Partrec fun a => Part.map (g a) (f a)\n[PROOFSTEP]\nsimpa [bind_some_eq_map] using @Partrec.bind _ _ _ _ _ _ _ (fun a => Part.some \u2218 (g a)) hf hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\n\u22a2 (Part.bind (Part.bind \u2191(decode n) fun a => Part.map encode (\u2191f a)) fun n_1 =>\n Nat.rec (Part.bind \u2191(decode n) fun a => Part.map encode (g a))\n (fun y IH => do\n let i \u2190 IH\n Part.bind \u2191(decode (Nat.pair n (Nat.pair y i))) fun a => Part.map encode ((fun p => h p.fst p.snd) a))\n n_1) =\n Part.bind \u2191(decode n) fun a =>\n Part.map encode ((fun a => Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) (f a)) a)\n[PROOFSTEP]\ncases' e : decode (\u03b1 := \u03b1) n with a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\ne : decode n = Option.none\n\u22a2 (Part.bind (Part.bind \u2191Option.none fun a => Part.map encode (\u2191f a)) fun n_1 =>\n Nat.rec (Part.bind \u2191Option.none fun a => Part.map encode (g a))\n (fun y IH => do\n let i \u2190 IH\n Part.bind \u2191(decode (Nat.pair n (Nat.pair y i))) fun a => Part.map encode ((fun p => h p.fst p.snd) a))\n n_1) =\n Part.bind \u2191Option.none fun a =>\n Part.map encode ((fun a => Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) (f a)) a)\n[PROOFSTEP]\nsimp [e]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\n\u22a2 (Part.bind (Part.bind \u2191(Option.some a) fun a => Part.map encode (\u2191f a)) fun n_1 =>\n Nat.rec (Part.bind \u2191(Option.some a) fun a => Part.map encode (g a))\n (fun y IH => do\n let i \u2190 IH\n Part.bind \u2191(decode (Nat.pair n (Nat.pair y i))) fun a => Part.map encode ((fun p => h p.fst p.snd) a))\n n_1) =\n Part.bind \u2191(Option.some a) fun a =>\n Part.map encode ((fun a => Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) (f a)) a)\n[PROOFSTEP]\nsimp [e]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\n\u22a2 Nat.rec (Part.map encode (g a))\n (fun y IH =>\n Part.bind IH fun i =>\n Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.fst a.snd))\n (f a) =\n Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) (f a))\n[PROOFSTEP]\ninduction' f a with m IH\n[GOAL]\ncase some.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\n\u22a2 Nat.rec (Part.map encode (g a))\n (fun y IH =>\n Part.bind IH fun i =>\n Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.fst a.snd))\n Nat.zero =\n Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) Nat.zero)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\nm : \u2115\nIH :\n Nat.rec (Part.map encode (g a))\n (fun y IH =>\n Part.bind IH fun i =>\n Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.fst a.snd))\n m =\n Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m)\n\u22a2 Nat.rec (Part.map encode (g a))\n (fun y IH =>\n Part.bind IH fun i =>\n Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.fst a.snd))\n (Nat.succ m) =\n Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) (Nat.succ m))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\nm : \u2115\nIH :\n Nat.rec (Part.map encode (g a))\n (fun y IH =>\n Part.bind IH fun i =>\n Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.fst a.snd))\n m =\n Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m)\n\u22a2 (Part.bind\n (Nat.rec (Part.map encode (g a))\n (fun y IH =>\n Part.bind IH fun i =>\n Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.fst a.snd))\n m)\n fun i => Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode i)) fun a => Part.map encode (h a.fst a.snd)) =\n Part.bind (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) fun y => Part.map encode (h a (m, y))\n[PROOFSTEP]\nrw [IH, Part.bind_map]\n[GOAL]\ncase some.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\nm : \u2115\nIH :\n Nat.rec (Part.map encode (g a))\n (fun y IH =>\n Part.bind IH fun i =>\n Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.fst a.snd))\n m =\n Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m)\n\u22a2 (Part.bind (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) fun y =>\n Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode (encode y))) fun a => Part.map encode (h a.fst a.snd)) =\n Part.bind (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m) fun y => Part.map encode (h a (m, y))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase some.succ.e_g\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\nm : \u2115\nIH :\n Nat.rec (Part.map encode (g a))\n (fun y IH =>\n Part.bind IH fun i =>\n Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.fst a.snd))\n m =\n Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m)\n\u22a2 (fun y =>\n Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode (encode y))) fun a => Part.map encode (h a.fst a.snd)) =\n fun y => Part.map encode (h a (m, y))\n[PROOFSTEP]\nfunext s\n[GOAL]\ncase some.succ.e_g.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192. \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec g\nhh : Partrec\u2082 h\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\nm : \u2115\nIH :\n Nat.rec (Part.map encode (g a))\n (fun y IH =>\n Part.bind IH fun i =>\n Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk y) (decode i)) fun a => Part.map encode (h a.fst a.snd))\n m =\n Part.map encode (Nat.rec (g a) (fun y IH => Part.bind IH fun i => h a (y, i)) m)\ns : \u03c3\n\u22a2 (Part.bind \u2191(Option.map (Prod.mk a \u2218 Prod.mk m) (decode (encode s))) fun a => Part.map encode (h a.fst a.snd)) =\n Part.map encode (h a (m, s))\n[PROOFSTEP]\nsimp [encodek]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b2 \u2192. \u03c3\ng : \u03b1 \u2192 \u03b2\nhf : Partrec f\nhg : Computable g\nn : \u2115\n\u22a2 (do\n let n \u2190 Part.bind \u2191(decode n) fun a => Part.map encode (\u2191g a)\n Part.bind \u2191(decode n) fun a => Part.map encode (f a)) =\n Part.bind \u2191(decode n) fun a => Part.map encode ((fun a => f (g a)) a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b2 \u2192. \u03c3\ng : \u03b1 \u2192 \u03b2\nhf : Partrec f\nhg : Computable g\nn : \u2115\n\u22a2 (Part.bind (Part.bind \u2191(decode n) fun a => Part.some (encode (g a))) fun n =>\n Part.bind \u2191(decode n) fun a => Part.map encode (f a)) =\n Part.bind \u2191(decode n) fun a => Part.map encode (f (g a))\n[PROOFSTEP]\ncases' e : decode (\u03b1 := \u03b1) n with a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b2 \u2192. \u03c3\ng : \u03b1 \u2192 \u03b2\nhf : Partrec f\nhg : Computable g\nn : \u2115\ne : decode n = Option.none\n\u22a2 (Part.bind (Part.bind \u2191Option.none fun a => Part.some (encode (g a))) fun n =>\n Part.bind \u2191(decode n) fun a => Part.map encode (f a)) =\n Part.bind \u2191Option.none fun a => Part.map encode (f (g a))\n[PROOFSTEP]\nsimp [e, encodek]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b2 \u2192. \u03c3\ng : \u03b1 \u2192 \u03b2\nhf : Partrec f\nhg : Computable g\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\n\u22a2 (Part.bind (Part.bind \u2191(Option.some a) fun a => Part.some (encode (g a))) fun n =>\n Part.bind \u2191(decode n) fun a => Part.map encode (f a)) =\n Part.bind \u2191(Option.some a) fun a => Part.map encode (f (g a))\n[PROOFSTEP]\nsimp [e, encodek]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u2115 \u2192. \u2115\n\u22a2 Partrec f \u2194 Nat.Partrec f\n[PROOFSTEP]\nsimp [Partrec, map_id']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\nf : \u2115 \u2192 \u2115 \u2192. \u03b1\nh : Partrec (Nat.unpaired f)\n\u22a2 Partrec\u2082 f\n[PROOFSTEP]\nsimpa using Partrec.comp (g := fun p : \u2115 \u00d7 \u2115 => (p.1, p.2)) h Primrec\u2082.pair.to_comp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn : \u2115\n\u22a2 (Nat.rfind fun n_1 =>\n (fun m => decide (m = 0)) <$>\n Part.bind \u2191(decode (Nat.pair n n_1)) fun a =>\n Part.map encode ((fun a => Part.map (fun b => bif (a, b).snd then 0 else 1) (p a.fst a.snd)) a)) =\n Part.bind \u2191(decode n) fun a => Part.map encode ((fun a => Nat.rfind (p a)) a)\n[PROOFSTEP]\ncases' e : decode (\u03b1 := \u03b1) n with a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn : \u2115\ne : decode n = Option.none\n\u22a2 (Nat.rfind fun n_1 =>\n (fun m => decide (m = 0)) <$>\n Part.bind \u2191(decode (Nat.pair n n_1)) fun a =>\n Part.map encode ((fun a => Part.map (fun b => bif (a, b).snd then 0 else 1) (p a.fst a.snd)) a)) =\n Part.bind \u2191Option.none fun a => Part.map encode ((fun a => Nat.rfind (p a)) a)\n[PROOFSTEP]\nsimp [e, Nat.rfind_zero_none, map_id']\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\n\u22a2 (Nat.rfind fun n_1 =>\n (fun m => decide (m = 0)) <$>\n Part.bind \u2191(decode (Nat.pair n n_1)) fun a =>\n Part.map encode ((fun a => Part.map (fun b => bif (a, b).snd then 0 else 1) (p a.fst a.snd)) a)) =\n Part.bind \u2191(Option.some a) fun a => Part.map encode ((fun a => Nat.rfind (p a)) a)\n[PROOFSTEP]\nsimp [e, Nat.rfind_zero_none, map_id']\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\n\u22a2 (Nat.rfind fun n => Part.map (fun m => decide (m = 0)) (Part.map (fun b => bif b then 0 else 1) (p a n))) =\n Nat.rfind (p a)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase some.e_p\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn : \u2115\na : \u03b1\ne : decode n = Option.some a\n\u22a2 (fun n => Part.map (fun m => decide (m = 0)) (Part.map (fun b => bif b then 0 else 1) (p a n))) = p a\n[PROOFSTEP]\nfunext n\n[GOAL]\ncase some.e_p.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn\u271d : \u2115\na : \u03b1\ne : decode n\u271d = Option.some a\nn : \u2115\n\u22a2 Part.map (fun m => decide (m = 0)) (Part.map (fun b => bif b then 0 else 1) (p a n)) = p a n\n[PROOFSTEP]\nsimp [Part.map_map, (\u00b7 \u2218 \u00b7)]\n[GOAL]\ncase some.e_p.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn\u271d : \u2115\na : \u03b1\ne : decode n\u271d = Option.some a\nn : \u2115\n\u22a2 Part.map (fun x => decide ((bif x then 0 else 1) = 0)) (p a n) = p a n\n[PROOFSTEP]\nrefine map_id' (fun b => ?_) _\n[GOAL]\ncase some.e_p.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn\u271d : \u2115\na : \u03b1\ne : decode n\u271d = Option.some a\nn : \u2115\nb : Bool\n\u22a2 decide ((bif b then 0 else 1) = 0) = b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some.e_p.h.false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn\u271d : \u2115\na : \u03b1\ne : decode n\u271d = Option.some a\nn : \u2115\n\u22a2 decide ((bif false then 0 else 1) = 0) = false\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.e_p.h.true\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 \u2115 \u2192. Bool\nhp : Partrec\u2082 p\nn\u271d : \u2115\na : \u03b1\ne : decode n\u271d = Option.some a\nn : \u2115\n\u22a2 decide ((bif true then 0 else 1) = 0) = true\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u2192. \u03c3\nhf : Computable f\nhg : Computable g\nhh : Partrec\u2082 h\na : \u03b1\n\u22a2 Nat.rec (\u2191g a) (fun y IH => Part.bind IH fun i => h (a, y, i).fst (Nat.pred (f (a, y, i).fst))) (f a) =\n Nat.casesOn (f a) (Part.some (g a)) (h a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u2192. \u03c3\nhf : Computable f\nhg : Computable g\nhh : Partrec\u2082 h\na : \u03b1\n\u22a2 Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a (Nat.pred (f a))) (f a) =\n Nat.rec (Part.some (g a)) (fun n n_ih => h a n) (f a)\n[PROOFSTEP]\ncases' f a with n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u2192. \u03c3\nhf : Computable f\nhg : Computable g\nhh : Partrec\u2082 h\na : \u03b1\n\u22a2 Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a (Nat.pred Nat.zero)) Nat.zero =\n Nat.rec (Part.some (g a)) (fun n n_ih => h a n) Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u2192. \u03c3\nhf : Computable f\nhg : Computable g\nhh : Partrec\u2082 h\na : \u03b1\nn : \u2115\n\u22a2 Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a (Nat.pred (Nat.succ n))) (Nat.succ n) =\n Nat.rec (Part.some (g a)) (fun n n_ih => h a n) (Nat.succ n)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u2192. \u03c3\nhf : Computable f\nhg : Computable g\nhh : Partrec\u2082 h\na : \u03b1\nn : \u2115\n\u22a2 (Part.bind (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n) fun i => h a n) = h a n\n[PROOFSTEP]\nrefine' ext fun b => \u27e8fun H => _, fun H => _\u27e9\n[GOAL]\ncase succ.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u2192. \u03c3\nhf : Computable f\nhg : Computable g\nhh : Partrec\u2082 h\na : \u03b1\nn : \u2115\nb : \u03c3\nH : b \u2208 Part.bind (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n) fun i => h a n\n\u22a2 b \u2208 h a n\n[PROOFSTEP]\nrcases mem_bind_iff.1 H with \u27e8c, _, h\u2082\u27e9\n[GOAL]\ncase succ.refine'_1.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u2192. \u03c3\nhf : Computable f\nhg : Computable g\nhh : Partrec\u2082 h\na : \u03b1\nn : \u2115\nb : \u03c3\nH : b \u2208 Part.bind (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n) fun i => h a n\nc : \u03c3\nleft\u271d : c \u2208 Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n\nh\u2082 : b \u2208 h a n\n\u22a2 b \u2208 h a n\n[PROOFSTEP]\nexact h\u2082\n[GOAL]\ncase succ.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u2192. \u03c3\nhf : Computable f\nhg : Computable g\nhh : Partrec\u2082 h\na : \u03b1\nn : \u2115\nb : \u03c3\nH : b \u2208 h a n\n\u22a2 b \u2208 Part.bind (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n) fun i => h a n\n[PROOFSTEP]\nhave : \u2200 m, (Nat.rec (motive := fun _ => Part \u03c3) (Part.some (g a)) (fun y IH => IH.bind fun _ => h a n) m).Dom :=\n by\n intro m\n induction m <;> simp [*, H.fst]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u2192. \u03c3\nhf : Computable f\nhg : Computable g\nhh : Partrec\u2082 h\na : \u03b1\nn : \u2115\nb : \u03c3\nH : b \u2208 h a n\n\u22a2 \u2200 (m : \u2115), (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun x => h a n) m).Dom\n[PROOFSTEP]\nintro m\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u2192. \u03c3\nhf : Computable f\nhg : Computable g\nhh : Partrec\u2082 h\na : \u03b1\nn : \u2115\nb : \u03c3\nH : b \u2208 h a n\nm : \u2115\n\u22a2 (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun x => h a n) m).Dom\n[PROOFSTEP]\ninduction m\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u2192. \u03c3\nhf : Computable f\nhg : Computable g\nhh : Partrec\u2082 h\na : \u03b1\nn : \u2115\nb : \u03c3\nH : b \u2208 h a n\n\u22a2 (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun x => h a n) Nat.zero).Dom\n[PROOFSTEP]\nsimp [*, H.fst]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u2192. \u03c3\nhf : Computable f\nhg : Computable g\nhh : Partrec\u2082 h\na : \u03b1\nn : \u2115\nb : \u03c3\nH : b \u2208 h a n\nn\u271d : \u2115\nn_ih\u271d : (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun x => h a n) n\u271d).Dom\n\u22a2 (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun x => h a n) (Nat.succ n\u271d)).Dom\n[PROOFSTEP]\nsimp [*, H.fst]\n[GOAL]\ncase succ.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u2192. \u03c3\nhf : Computable f\nhg : Computable g\nhh : Partrec\u2082 h\na : \u03b1\nn : \u2115\nb : \u03c3\nH : b \u2208 h a n\nthis : \u2200 (m : \u2115), (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun x => h a n) m).Dom\n\u22a2 b \u2208 Part.bind (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n) fun i => h a n\n[PROOFSTEP]\nexact \u27e8\u27e8this n, H.fst\u27e9, H.snd\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03c3\nh : Nat.Partrec fun n => Part.bind \u2191(decode\u2082 \u03b1 n) fun a => Part.map encode (f a)\n\u22a2 Partrec fun a => Part.map encode (f a)\n[PROOFSTEP]\nsimpa [encodek\u2082] using (nat_iff.2 h).comp (@Computable.encode \u03b1 _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192. \u03c3\nhf : \u2200 (i : Fin (n + 1)), Partrec (f i)\n\u22a2 Partrec fun a => Vector.mOfFn fun i => f i a\n[PROOFSTEP]\nsimp [Vector.mOfFn]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192. \u03c3\nhf : \u2200 (i : Fin (n + 1)), Partrec (f i)\n\u22a2 Partrec fun a =>\n Part.bind (f 0 a) fun a_1 => Part.bind (Vector.mOfFn fun i => f (Fin.succ i) a) fun v => Part.some (a_1 ::\u1d65 v)\n[PROOFSTEP]\nexact\n (hf 0).bind\n (Partrec.bind ((vector_mOfFn fun i => hf i.succ).comp fst) (Primrec.vector_cons.to_comp.comp (snd.comp fst) snd))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable\u2082 fun a n => Option.bind (decode n) (f a)\nn : \u2115\n\u22a2 (Part.bind \u2191(decode n) fun a =>\n map encode\n ((fun a =>\n Part.bind\n (do\n let n \u2190 (\u2191fun n => encode (decode n)) a.snd\n \u2191(Nat.ppred n))\n \u2191fun b => (fun a n => Option.bind (decode n) (f a)) (a, b).fst.fst (a, b).fst.snd)\n a)) =\n Part.bind \u2191(decode n) fun a => map encode ((\u2191fun p => f p.fst p.snd) a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable\u2082 fun a n => Option.bind (decode n) (f a)\nn : \u2115\n\u22a2 (Part.bind \u2191(Option.bind (decode (Nat.unpair n).fst) fun a => Option.some (a, (Nat.unpair n).snd)) fun a =>\n Part.bind \u2191(Nat.ppred (encode (decode a.snd))) fun y =>\n Part.some (encode (Option.bind (decode a.snd) (f a.fst)))) =\n Part.bind \u2191(Option.bind (decode (Nat.unpair n).fst) fun a => Option.map (Prod.mk a) (decode (Nat.unpair n).snd))\n fun a => Part.some (encode (f a.fst a.snd))\n[PROOFSTEP]\ncases decode (\u03b1 := \u03b1) n.unpair.1\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable\u2082 fun a n => Option.bind (decode n) (f a)\nn : \u2115\n\u22a2 (Part.bind \u2191(Option.bind Option.none fun a => Option.some (a, (Nat.unpair n).snd)) fun a =>\n Part.bind \u2191(Nat.ppred (encode (decode a.snd))) fun y =>\n Part.some (encode (Option.bind (decode a.snd) (f a.fst)))) =\n Part.bind \u2191(Option.bind Option.none fun a => Option.map (Prod.mk a) (decode (Nat.unpair n).snd)) fun a =>\n Part.some (encode (f a.fst a.snd))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable\u2082 fun a n => Option.bind (decode n) (f a)\nn : \u2115\nval\u271d : \u03b1\n\u22a2 (Part.bind \u2191(Option.bind (Option.some val\u271d) fun a => Option.some (a, (Nat.unpair n).snd)) fun a =>\n Part.bind \u2191(Nat.ppred (encode (decode a.snd))) fun y =>\n Part.some (encode (Option.bind (decode a.snd) (f a.fst)))) =\n Part.bind \u2191(Option.bind (Option.some val\u271d) fun a => Option.map (Prod.mk a) (decode (Nat.unpair n).snd)) fun a =>\n Part.some (encode (f a.fst a.snd))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable\u2082 fun a n => Option.bind (decode n) (f a)\nn : \u2115\nval\u271d : \u03b1\n\u22a2 (Part.bind \u2191(Nat.ppred (encode (decode (Nat.unpair n).snd))) fun y =>\n Part.some (encode (Option.bind (decode (Nat.unpair n).snd) (f val\u271d)))) =\n Part.bind \u2191(Option.map (Prod.mk val\u271d) (decode (Nat.unpair n).snd)) fun a => Part.some (encode (f a.fst a.snd))\n[PROOFSTEP]\ncases decode (\u03b1 := \u03b2) n.unpair.2\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable\u2082 fun a n => Option.bind (decode n) (f a)\nn : \u2115\nval\u271d : \u03b1\n\u22a2 (Part.bind \u2191(Nat.ppred (encode Option.none)) fun y => Part.some (encode (Option.bind Option.none (f val\u271d)))) =\n Part.bind \u2191(Option.map (Prod.mk val\u271d) Option.none) fun a => Part.some (encode (f a.fst a.snd))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable\u2082 fun a n => Option.bind (decode n) (f a)\nn : \u2115\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 (Part.bind \u2191(Nat.ppred (encode (Option.some val\u271d))) fun y =>\n Part.some (encode (Option.bind (Option.some val\u271d) (f val\u271d\u00b9)))) =\n Part.bind \u2191(Option.map (Prod.mk val\u271d\u00b9) (Option.some val\u271d)) fun a => Part.some (encode (f a.fst a.snd))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable\u2082 f\n\u22a2 Computable\u2082 fun a n => Option.bind (decode n) (f a)\n[PROOFSTEP]\nhave :\n Partrec fun a : \u03b1 \u00d7 \u2115 =>\n (encode (decode (\u03b1 := \u03b2) a.2)).casesOn (some Option.none) fun n => Part.map (f a.1) (decode (\u03b1 := \u03b2) n) :=\n Partrec.nat_casesOn_right (h := fun (a : \u03b1 \u00d7 \u2115) (n : \u2115) \u21a6 map (fun b \u21a6 f a.1 b) (Part.ofOption (decode n)))\n (Primrec.encdec.to_comp.comp snd) (const Option.none)\n ((ofOption (Computable.decode.comp snd)).map (hf.comp (fst.comp <| fst.comp fst) snd).to\u2082)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable\u2082 f\nthis : Partrec fun a => Nat.casesOn (encode (decode a.snd)) (Part.some Option.none) fun n => map (f a.fst) \u2191(decode n)\n\u22a2 Computable\u2082 fun a n => Option.bind (decode n) (f a)\n[PROOFSTEP]\nrefine' this.of_eq fun a => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable\u2082 f\nthis : Partrec fun a => Nat.casesOn (encode (decode a.snd)) (Part.some Option.none) fun n => map (f a.fst) \u2191(decode n)\na : \u03b1 \u00d7 \u2115\n\u22a2 (Nat.casesOn (encode (decode a.snd)) (Part.some Option.none) fun n => map (f a.fst) \u2191(decode n)) =\n (\u2191fun p => (fun a n => Option.bind (decode n) (f a)) p.fst p.snd) a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable\u2082 f\nthis : Partrec fun a => Nat.casesOn (encode (decode a.snd)) (Part.some Option.none) fun n => map (f a.fst) \u2191(decode n)\na : \u03b1 \u00d7 \u2115\n\u22a2 Nat.rec (Part.some Option.none) (fun n n_ih => map (f a.fst) \u2191(decode n)) (encode (decode a.snd)) =\n Part.some (Option.bind (decode a.snd) (f a.fst))\n[PROOFSTEP]\ncases decode (\u03b1 := \u03b2) a.2\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable\u2082 f\nthis : Partrec fun a => Nat.casesOn (encode (decode a.snd)) (Part.some Option.none) fun n => map (f a.fst) \u2191(decode n)\na : \u03b1 \u00d7 \u2115\n\u22a2 Nat.rec (Part.some Option.none) (fun n n_ih => map (f a.fst) \u2191(decode n)) (encode Option.none) =\n Part.some (Option.bind Option.none (f a.fst))\n[PROOFSTEP]\nsimp [encodek]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable\u2082 f\nthis : Partrec fun a => Nat.casesOn (encode (decode a.snd)) (Part.some Option.none) fun n => map (f a.fst) \u2191(decode n)\na : \u03b1 \u00d7 \u2115\nval\u271d : \u03b2\n\u22a2 Nat.rec (Part.some Option.none) (fun n n_ih => map (f a.fst) \u2191(decode n)) (encode (Option.some val\u271d)) =\n Part.some (Option.bind (Option.some val\u271d) (f a.fst))\n[PROOFSTEP]\nsimp [encodek]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3\n\u22a2 (Computable\u2082 fun a n => Option.map (f a) (decode n)) \u2194 Computable\u2082 f\n[PROOFSTEP]\nconvert (bind_decode_iff (f := fun a => Option.some \u2218 f a)).trans option_some_iff\n[GOAL]\ncase h.e'_1.h.e'_7.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nx\u271d\u00b9 : \u03b1\nx\u271d : \u2115\n\u22a2 Option.map (f x\u271d\u00b9) (decode x\u271d) = Option.bind (decode x\u271d) (Option.some \u2218 f x\u271d\u00b9)\n[PROOFSTEP]\napply Option.map_eq_bind\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192 \u03c3\nhf : Computable f\nhg : Computable g\nhh : Computable\u2082 h\na : \u03b1\n\u22a2 Nat.rec (\u2191g a) (fun y IH => Part.bind IH fun i => \u2191(h a) (y, i)) (f a) =\n (\u2191fun a => Nat.rec (g a) (fun y IH => h a (y, IH)) (f a)) a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192 \u03c3\nhf : Computable f\nhg : Computable g\nhh : Computable\u2082 h\na : \u03b1\n\u22a2 Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => Part.some (h a (y, i))) (f a) =\n Part.some (Nat.rec (g a) (fun y IH => h a (y, IH)) (f a))\n[PROOFSTEP]\ninduction f a\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192 \u03c3\nhf : Computable f\nhg : Computable g\nhh : Computable\u2082 h\na : \u03b1\n\u22a2 Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => Part.some (h a (y, i))) Nat.zero =\n Part.some (Nat.rec (g a) (fun y IH => h a (y, IH)) Nat.zero)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115\ng : \u03b1 \u2192 \u03c3\nh : \u03b1 \u2192 \u2115 \u00d7 \u03c3 \u2192 \u03c3\nhf : Computable f\nhg : Computable g\nhh : Computable\u2082 h\na : \u03b1\nn\u271d : \u2115\nn_ih\u271d :\n Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => Part.some (h a (y, i))) n\u271d =\n Part.some (Nat.rec (g a) (fun y IH => h a (y, IH)) n\u271d)\n\u22a2 Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => Part.some (h a (y, i))) (Nat.succ n\u271d) =\n Part.some (Nat.rec (g a) (fun y IH => h a (y, IH)) (Nat.succ n\u271d))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Bool\nf g : \u03b1 \u2192 \u03c3\nhc : Computable c\nhf : Computable f\nhg : Computable g\na : \u03b1\n\u22a2 (Nat.casesOn (encode (c a)) (g a) fun b => f (a, b).fst) = bif c a then f a else g a\n[PROOFSTEP]\ncases c a\n[GOAL]\ncase false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Bool\nf g : \u03b1 \u2192 \u03c3\nhc : Computable c\nhf : Computable f\nhg : Computable g\na : \u03b1\n\u22a2 (Nat.casesOn (encode false) (g a) fun b => f (a, b).fst) = bif false then f a else g a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Bool\nf g : \u03b1 \u2192 \u03c3\nhc : Computable c\nhf : Computable f\nhg : Computable g\na : \u03b1\n\u22a2 (Nat.casesOn (encode true) (g a) fun b => f (a, b).fst) = bif true then f a else g a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\no : \u03b1 \u2192 Option \u03b2\nf : \u03b1 \u2192 \u03c3\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nho : Computable o\nhf : Computable f\nhg : Computable\u2082 g\na : \u03b1\n\u22a2 (Nat.casesOn (encode (o a)) (Option.some (f a)) fun n => Option.map (g a) (decode n)) =\n Option.some (Option.casesOn (o a) (f a) (g a))\n[PROOFSTEP]\ncases o a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\no : \u03b1 \u2192 Option \u03b2\nf : \u03b1 \u2192 \u03c3\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nho : Computable o\nhf : Computable f\nhg : Computable\u2082 g\na : \u03b1\n\u22a2 (Nat.casesOn (encode Option.none) (Option.some (f a)) fun n => Option.map (g a) (decode n)) =\n Option.some (Option.casesOn Option.none (f a) (g a))\n[PROOFSTEP]\nsimp [encodek]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\no : \u03b1 \u2192 Option \u03b2\nf : \u03b1 \u2192 \u03c3\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nho : Computable o\nhf : Computable f\nhg : Computable\u2082 g\na : \u03b1\nval\u271d : \u03b2\n\u22a2 (Nat.casesOn (encode (Option.some val\u271d)) (Option.some (f a)) fun n => Option.map (g a) (decode n)) =\n Option.some (Option.casesOn (Option.some val\u271d) (f a) (g a))\n[PROOFSTEP]\nsimp [encodek]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Option \u03b2\ng : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable f\nhg : Computable\u2082 g\na : \u03b1\n\u22a2 Option.casesOn (f a) Option.none (g a) = Option.bind (f a) (g a)\n[PROOFSTEP]\ncases f a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Option \u03b2\ng : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable f\nhg : Computable\u2082 g\na : \u03b1\n\u22a2 Option.casesOn Option.none Option.none (g a) = Option.bind Option.none (g a)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Option \u03b2\ng : \u03b1 \u2192 \u03b2 \u2192 Option \u03c3\nhf : Computable f\nhg : Computable\u2082 g\na : \u03b1\nval\u271d : \u03b2\n\u22a2 Option.casesOn (Option.some val\u271d) Option.none (g a) = Option.bind (Option.some val\u271d) (g a)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Option \u03b2\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nhf : Computable f\nhg : Computable\u2082 g\n\u22a2 Computable fun a => Option.map (g a) (f a)\n[PROOFSTEP]\nconvert option_bind hf (option_some.comp\u2082 hg)\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Option \u03b2\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nhf : Computable f\nhg : Computable\u2082 g\nx\u271d : \u03b1\n\u22a2 Option.map (g x\u271d) (f x\u271d) = Option.bind (f x\u271d) fun b => Option.some (g x\u271d b)\n[PROOFSTEP]\napply Option.map_eq_bind\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Option \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : Computable f\nhg : Computable g\na : \u03b1\n\u22a2 (Option.casesOn (f a) (g a) fun b => b) = Option.getD (f a) (g a)\n[PROOFSTEP]\ncases f a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Option \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : Computable f\nhg : Computable g\na : \u03b1\n\u22a2 (Option.casesOn Option.none (g a) fun b => b) = Option.getD Option.none (g a)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Option \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : Computable f\nhg : Computable g\na : \u03b1\nval\u271d : \u03b2\n\u22a2 (Option.casesOn (Option.some val\u271d) (g a) fun b => b) = Option.getD (Option.some val\u271d) (g a)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nh : \u03b1 \u2192 \u03b3 \u2192 \u03c3\nhf : Computable f\nhg : Computable\u2082 g\nhh : Computable\u2082 h\na : \u03b1\n\u22a2 (bif Nat.bodd (encode (f a)) then Option.map (h a) (decode (Nat.div2 (encode (f a))))\n else Option.map (g a) (decode (Nat.div2 (encode (f a))))) =\n Option.some (Sum.casesOn (f a) (g a) (h a))\n[PROOFSTEP]\ncases' f a with b c\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nh : \u03b1 \u2192 \u03b3 \u2192 \u03c3\nhf : Computable f\nhg : Computable\u2082 g\nhh : Computable\u2082 h\na : \u03b1\nb : \u03b2\n\u22a2 (bif Nat.bodd (encode (Sum.inl b)) then Option.map (h a) (decode (Nat.div2 (encode (Sum.inl b))))\n else Option.map (g a) (decode (Nat.div2 (encode (Sum.inl b))))) =\n Option.some (Sum.casesOn (Sum.inl b) (g a) (h a))\n[PROOFSTEP]\nsimp [Nat.div2_val]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nh : \u03b1 \u2192 \u03b3 \u2192 \u03c3\nhf : Computable f\nhg : Computable\u2082 g\nhh : Computable\u2082 h\na : \u03b1\nc : \u03b3\n\u22a2 (bif Nat.bodd (encode (Sum.inr c)) then Option.map (h a) (decode (Nat.div2 (encode (Sum.inr c))))\n else Option.map (g a) (decode (Nat.div2 (encode (Sum.inr c))))) =\n Option.some (Sum.casesOn (Sum.inr c) (g a) (h a))\n[PROOFSTEP]\nsimp [Nat.div2_val]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115 \u2192 \u03c3\ng : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3\nhg : Computable\u2082 g\nH : \u2200 (a : \u03b1) (n : \u2115), g a (List.map (f a) (List.range n)) = Option.some (f a n)\na : \u03b1 \u00d7 \u2115\n\u22a2 Nat.rec (Option.some [])\n (fun y IH =>\n Option.bind (a, y, IH).snd.snd fun b =>\n Option.map (fun b_1 => (((a, y, IH), b), b_1).fst.snd ++ [(((a, y, IH), b), b_1).snd])\n (g ((a, y, IH), b).fst.fst.fst ((a, y, IH), b).snd))\n a.snd =\n Option.some ((fun a n => List.map (f a) (List.range n)) a.fst a.snd)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115 \u2192 \u03c3\ng : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3\nhg : Computable\u2082 g\nH : \u2200 (a : \u03b1) (n : \u2115), g a (List.map (f a) (List.range n)) = Option.some (f a n)\na : \u03b1 \u00d7 \u2115\n\u22a2 Nat.rec (Option.some []) (fun y IH => Option.bind IH fun b => Option.map (fun b_1 => b ++ [b_1]) (g a.fst b)) a.snd =\n Option.some (List.map (f a.fst) (List.range a.snd))\n[PROOFSTEP]\ninduction' a.2 with n IH\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115 \u2192 \u03c3\ng : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3\nhg : Computable\u2082 g\nH : \u2200 (a : \u03b1) (n : \u2115), g a (List.map (f a) (List.range n)) = Option.some (f a n)\na : \u03b1 \u00d7 \u2115\n\u22a2 Nat.rec (Option.some []) (fun y IH => Option.bind IH fun b => Option.map (fun b_1 => b ++ [b_1]) (g a.fst b))\n Nat.zero =\n Option.some (List.map (f a.fst) (List.range Nat.zero))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115 \u2192 \u03c3\ng : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3\nhg : Computable\u2082 g\nH : \u2200 (a : \u03b1) (n : \u2115), g a (List.map (f a) (List.range n)) = Option.some (f a n)\na : \u03b1 \u00d7 \u2115\nn : \u2115\nIH :\n Nat.rec (Option.some []) (fun y IH => Option.bind IH fun b => Option.map (fun b_1 => b ++ [b_1]) (g a.fst b)) n =\n Option.some (List.map (f a.fst) (List.range n))\n\u22a2 Nat.rec (Option.some []) (fun y IH => Option.bind IH fun b => Option.map (fun b_1 => b ++ [b_1]) (g a.fst b))\n (Nat.succ n) =\n Option.some (List.map (f a.fst) (List.range (Nat.succ n)))\n[PROOFSTEP]\nsimp [IH, H, List.range_succ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u2115 \u2192 \u03c3\ng : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3\nhg : Computable\u2082 g\nH : \u2200 (a : \u03b1) (n : \u2115), g a (List.map (f a) (List.range n)) = Option.some (f a n)\nthis : Computable\u2082 fun a n => List.map (f a) (List.range n)\na : \u03b1 \u00d7 \u2115\n\u22a2 (fun a b => List.get? (List.map (f (a, b).fst) (List.range (Nat.succ (a, b).snd))) (a, b).snd) a.fst a.snd =\n Option.some (f a.fst a.snd)\n[PROOFSTEP]\nsimp [List.get?_range (Nat.lt_succ_self a.2)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03c3\nhf : \u2200 (i : Fin (n + 1)), Computable (f i)\n\u22a2 Computable fun a => List.ofFn fun i => f i a\n[PROOFSTEP]\nsimp [List.ofFn_succ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03c3\nhf : \u2200 (i : Fin (n + 1)), Computable (f i)\n\u22a2 Computable fun a => f 0 a :: List.ofFn fun i => f (Fin.succ i) a\n[PROOFSTEP]\nexact list_cons.comp (hf 0) (list_ofFn fun i => hf i.succ)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nn : \u2115\nf : Fin n \u2192 \u03b1 \u2192 \u03c3\nhf : \u2200 (i : Fin n), Computable (f i)\na : \u03b1\n\u22a2 (Vector.mOfFn fun i => \u2191(f i) a) = (\u2191fun a => Vector.ofFn fun i => f i a) a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03c3\nh : Partrec fun a => Part.map Option.some (f a)\nn : \u2115\n\u22a2 (do\n let n \u2190 Part.bind \u2191(decode n) fun a => Part.map encode ((fun a => Part.map Option.some (f a)) a)\n \u2191(Nat.ppred n)) =\n Part.bind \u2191(decode n) fun a => Part.map encode (f a)\n[PROOFSTEP]\nsimp [Part.bind_assoc, \u2190 Function.comp_apply (f := Part.some) (g := encode), bind_some_eq_map, -Function.comp_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\no : \u03b1 \u2192 Option \u03b2\nf : \u03b1 \u2192 \u03c3\ng : \u03b1 \u2192 \u03b2 \u2192. \u03c3\nho : Computable o\nhf : Computable f\nhg : Partrec\u2082 g\nthis : Partrec fun a => Nat.casesOn (encode (o a)) (Part.some (f a)) fun n => Part.bind (\u2191(decode n)) (g a)\na : \u03b1\n\u22a2 (Nat.casesOn (encode (o a)) (Part.some (f a)) fun n => Part.bind (\u2191(decode n)) (g a)) =\n Option.casesOn (o a) (Part.some (f a)) (g a)\n[PROOFSTEP]\ncases' o a with b\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\no : \u03b1 \u2192 Option \u03b2\nf : \u03b1 \u2192 \u03c3\ng : \u03b1 \u2192 \u03b2 \u2192. \u03c3\nho : Computable o\nhf : Computable f\nhg : Partrec\u2082 g\nthis : Partrec fun a => Nat.casesOn (encode (o a)) (Part.some (f a)) fun n => Part.bind (\u2191(decode n)) (g a)\na : \u03b1\n\u22a2 (Nat.casesOn (encode Option.none) (Part.some (f a)) fun n => Part.bind (\u2191(decode n)) (g a)) =\n Option.casesOn Option.none (Part.some (f a)) (g a)\n[PROOFSTEP]\nsimp [encodek]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\no : \u03b1 \u2192 Option \u03b2\nf : \u03b1 \u2192 \u03c3\ng : \u03b1 \u2192 \u03b2 \u2192. \u03c3\nho : Computable o\nhf : Computable f\nhg : Partrec\u2082 g\nthis : Partrec fun a => Nat.casesOn (encode (o a)) (Part.some (f a)) fun n => Part.bind (\u2191(decode n)) (g a)\na : \u03b1\nb : \u03b2\n\u22a2 (Nat.casesOn (encode (Option.some b)) (Part.some (f a)) fun n => Part.bind (\u2191(decode n)) (g a)) =\n Option.casesOn (Option.some b) (Part.some (f a)) (g a)\n[PROOFSTEP]\nsimp [encodek]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nh : \u03b1 \u2192 \u03b3 \u2192. \u03c3\nhf : Computable f\nhg : Computable\u2082 g\nhh : Partrec\u2082 h\nthis :\n Partrec fun a =>\n Option.casesOn (Sum.casesOn (f a) (fun x => Option.none) Option.some)\n (Part.some (Sum.casesOn (f a) (fun b => Option.some (g a b)) fun x => Option.none)) fun c =>\n Part.map Option.some (h a c)\na : \u03b1\n\u22a2 (Option.casesOn (Sum.casesOn (f a) (fun x => Option.none) Option.some)\n (Part.some (Sum.casesOn (f a) (fun b => Option.some (g a b)) fun x => Option.none)) fun c =>\n Part.map Option.some (h a c)) =\n Part.map Option.some (Sum.casesOn (f a) (fun b => Part.some (g a b)) (h a))\n[PROOFSTEP]\ncases f a\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nh : \u03b1 \u2192 \u03b3 \u2192. \u03c3\nhf : Computable f\nhg : Computable\u2082 g\nhh : Partrec\u2082 h\nthis :\n Partrec fun a =>\n Option.casesOn (Sum.casesOn (f a) (fun x => Option.none) Option.some)\n (Part.some (Sum.casesOn (f a) (fun b => Option.some (g a b)) fun x => Option.none)) fun c =>\n Part.map Option.some (h a c)\na : \u03b1\nval\u271d : \u03b2\n\u22a2 (Option.casesOn (Sum.casesOn (Sum.inl val\u271d) (fun x => Option.none) Option.some)\n (Part.some (Sum.casesOn (Sum.inl val\u271d) (fun b => Option.some (g a b)) fun x => Option.none)) fun c =>\n Part.map Option.some (h a c)) =\n Part.map Option.some (Sum.casesOn (Sum.inl val\u271d) (fun b => Part.some (g a b)) (h a))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nh : \u03b1 \u2192 \u03b3 \u2192. \u03c3\nhf : Computable f\nhg : Computable\u2082 g\nhh : Partrec\u2082 h\nthis :\n Partrec fun a =>\n Option.casesOn (Sum.casesOn (f a) (fun x => Option.none) Option.some)\n (Part.some (Sum.casesOn (f a) (fun b => Option.some (g a b)) fun x => Option.none)) fun c =>\n Part.map Option.some (h a c)\na : \u03b1\nval\u271d : \u03b3\n\u22a2 (Option.casesOn (Sum.casesOn (Sum.inr val\u271d) (fun x => Option.none) Option.some)\n (Part.some (Sum.casesOn (Sum.inr val\u271d) (fun b => Option.some (g a b)) fun x => Option.none)) fun c =>\n Part.map Option.some (h a c)) =\n Part.map Option.some (Sum.casesOn (Sum.inr val\u271d) (fun b => Part.some (g a b)) (h a))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\ng : \u03b1 \u2192 \u03b2 \u2192. \u03c3\nh : \u03b1 \u2192 \u03b3 \u2192 \u03c3\nhf : Computable f\nhg : Partrec\u2082 g\nhh : Computable\u2082 h\na : \u03b1\n\u22a2 Sum.casesOn (Sum.casesOn (f a) (fun b => Sum.inr (a, b).snd) fun b => Sum.inl (a, b).snd) (fun b => Part.some (h a b))\n (g a) =\n Sum.casesOn (f a) (g a) fun c => Part.some (h a c)\n[PROOFSTEP]\ncases f a\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\ng : \u03b1 \u2192 \u03b2 \u2192. \u03c3\nh : \u03b1 \u2192 \u03b3 \u2192 \u03c3\nhf : Computable f\nhg : Partrec\u2082 g\nhh : Computable\u2082 h\na : \u03b1\nval\u271d : \u03b2\n\u22a2 Sum.casesOn (Sum.casesOn (Sum.inl val\u271d) (fun b => Sum.inr (a, b).snd) fun b => Sum.inl (a, b).snd)\n (fun b => Part.some (h a b)) (g a) =\n Sum.casesOn (Sum.inl val\u271d) (g a) fun c => Part.some (h a c)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\ng : \u03b1 \u2192 \u03b2 \u2192. \u03c3\nh : \u03b1 \u2192 \u03b3 \u2192 \u03c3\nhf : Computable f\nhg : Partrec\u2082 g\nhh : Computable\u2082 h\na : \u03b1\nval\u271d : \u03b3\n\u22a2 Sum.casesOn (Sum.casesOn (Sum.inr val\u271d) (fun b => Sum.inr (a, b).snd) fun b => Sum.inl (a, b).snd)\n (fun b => Part.some (h a b)) (g a) =\n Sum.casesOn (Sum.inr val\u271d) (g a) fun c => Part.some (h a c)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\n\u22a2 let F := fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n;\n (\u2203 n, ((\u2203 b', Sum.inl b' \u2208 F a n) \u2227 \u2200 {m : \u2115}, m < n \u2192 \u2203 b, Sum.inr b \u2208 F a m) \u2227 Sum.inl b \u2208 F a n) \u2194 b \u2208 PFun.fix f a\n[PROOFSTEP]\nintro F\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\n\u22a2 (\u2203 n, ((\u2203 b', Sum.inl b' \u2208 F a n) \u2227 \u2200 {m : \u2115}, m < n \u2192 \u2203 b, Sum.inr b \u2208 F a m) \u2227 Sum.inl b \u2208 F a n) \u2194 b \u2208 PFun.fix f a\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : \u2203 n, ((\u2203 b', Sum.inl b' \u2208 F a n) \u2227 \u2200 {m : \u2115}, m < n \u2192 \u2203 b, Sum.inr b \u2208 F a m) \u2227 Sum.inl b \u2208 F a n\n\u22a2 b \u2208 PFun.fix f a\n[PROOFSTEP]\nrcases h with \u27e8n, \u27e8_x, h\u2081\u27e9, h\u2082\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nn : \u2115\nh\u2082 : Sum.inl b \u2208 F a n\n_x : \u2203 b', Sum.inl b' \u2208 F a n\nh\u2081 : \u2200 {m : \u2115}, m < n \u2192 \u2203 b, Sum.inr b \u2208 F a m\n\u22a2 b \u2208 PFun.fix f a\n[PROOFSTEP]\nhave : \u2200 (m a') (_ : Sum.inr a' \u2208 F a m) (_ : b \u2208 PFun.fix f a'), b \u2208 PFun.fix f a :=\n by\n intro m a' am ba\n induction' m with m IH generalizing a' <;> simp at am \n \u00b7 rwa [\u2190 am]\n rcases am with \u27e8a\u2082, am\u2082, fa\u2082\u27e9\n exact IH _ am\u2082 (PFun.mem_fix_iff.2 (Or.inr \u27e8_, fa\u2082, ba\u27e9))\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nn : \u2115\nh\u2082 : Sum.inl b \u2208 F a n\n_x : \u2203 b', Sum.inl b' \u2208 F a n\nh\u2081 : \u2200 {m : \u2115}, m < n \u2192 \u2203 b, Sum.inr b \u2208 F a m\n\u22a2 \u2200 (m : \u2115) (a' : \u03b1), Sum.inr a' \u2208 F a m \u2192 b \u2208 PFun.fix f a' \u2192 b \u2208 PFun.fix f a\n[PROOFSTEP]\nintro m a' am ba\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nn : \u2115\nh\u2082 : Sum.inl b \u2208 F a n\n_x : \u2203 b', Sum.inl b' \u2208 F a n\nh\u2081 : \u2200 {m : \u2115}, m < n \u2192 \u2203 b, Sum.inr b \u2208 F a m\nm : \u2115\na' : \u03b1\nam : Sum.inr a' \u2208 F a m\nba : b \u2208 PFun.fix f a'\n\u22a2 b \u2208 PFun.fix f a\n[PROOFSTEP]\ninduction' m with m IH generalizing a'\n[GOAL]\ncase zero\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nn : \u2115\nh\u2082 : Sum.inl b \u2208 F a n\n_x : \u2203 b', Sum.inl b' \u2208 F a n\nh\u2081 : \u2200 {m : \u2115}, m < n \u2192 \u2203 b, Sum.inr b \u2208 F a m\nm : \u2115\na'\u271d : \u03b1\nam\u271d : Sum.inr a'\u271d \u2208 F a m\nba\u271d : b \u2208 PFun.fix f a'\u271d\na' : \u03b1\nam : Sum.inr a' \u2208 F a Nat.zero\nba : b \u2208 PFun.fix f a'\n\u22a2 b \u2208 PFun.fix f a\n[PROOFSTEP]\nsimp at am \n[GOAL]\ncase succ\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nn : \u2115\nh\u2082 : Sum.inl b \u2208 F a n\n_x : \u2203 b', Sum.inl b' \u2208 F a n\nh\u2081 : \u2200 {m : \u2115}, m < n \u2192 \u2203 b, Sum.inr b \u2208 F a m\nm\u271d : \u2115\na'\u271d : \u03b1\nam\u271d : Sum.inr a'\u271d \u2208 F a m\u271d\nba\u271d : b \u2208 PFun.fix f a'\u271d\nm : \u2115\nIH : \u2200 (a' : \u03b1), Sum.inr a' \u2208 F a m \u2192 b \u2208 PFun.fix f a' \u2192 b \u2208 PFun.fix f a\na' : \u03b1\nam : Sum.inr a' \u2208 F a (Nat.succ m)\nba : b \u2208 PFun.fix f a'\n\u22a2 b \u2208 PFun.fix f a\n[PROOFSTEP]\nsimp at am \n[GOAL]\ncase zero\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nn : \u2115\nh\u2082 : Sum.inl b \u2208 F a n\n_x : \u2203 b', Sum.inl b' \u2208 F a n\nh\u2081 : \u2200 {m : \u2115}, m < n \u2192 \u2203 b, Sum.inr b \u2208 F a m\nm : \u2115\na'\u271d : \u03b1\nam\u271d : Sum.inr a'\u271d \u2208 F a m\nba\u271d : b \u2208 PFun.fix f a'\u271d\na' : \u03b1\nba : b \u2208 PFun.fix f a'\nam : a' = a\n\u22a2 b \u2208 PFun.fix f a\n[PROOFSTEP]\nrwa [\u2190 am]\n[GOAL]\ncase succ\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nn : \u2115\nh\u2082 : Sum.inl b \u2208 F a n\n_x : \u2203 b', Sum.inl b' \u2208 F a n\nh\u2081 : \u2200 {m : \u2115}, m < n \u2192 \u2203 b, Sum.inr b \u2208 F a m\nm\u271d : \u2115\na'\u271d : \u03b1\nam\u271d : Sum.inr a'\u271d \u2208 F a m\u271d\nba\u271d : b \u2208 PFun.fix f a'\u271d\nm : \u2115\nIH : \u2200 (a' : \u03b1), Sum.inr a' \u2208 F a m \u2192 b \u2208 PFun.fix f a' \u2192 b \u2208 PFun.fix f a\na' : \u03b1\nba : b \u2208 PFun.fix f a'\nam :\n \u2203 b,\n Sum.inr b \u2208\n Nat.rec (Part.some (Sum.inr a))\n (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) m \u2227\n Sum.inr a' \u2208 f b\n\u22a2 b \u2208 PFun.fix f a\n[PROOFSTEP]\nrcases am with \u27e8a\u2082, am\u2082, fa\u2082\u27e9\n[GOAL]\ncase succ.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nn : \u2115\nh\u2082 : Sum.inl b \u2208 F a n\n_x : \u2203 b', Sum.inl b' \u2208 F a n\nh\u2081 : \u2200 {m : \u2115}, m < n \u2192 \u2203 b, Sum.inr b \u2208 F a m\nm\u271d : \u2115\na'\u271d : \u03b1\nam : Sum.inr a'\u271d \u2208 F a m\u271d\nba\u271d : b \u2208 PFun.fix f a'\u271d\nm : \u2115\nIH : \u2200 (a' : \u03b1), Sum.inr a' \u2208 F a m \u2192 b \u2208 PFun.fix f a' \u2192 b \u2208 PFun.fix f a\na' : \u03b1\nba : b \u2208 PFun.fix f a'\na\u2082 : \u03b1\nam\u2082 :\n Sum.inr a\u2082 \u2208\n Nat.rec (Part.some (Sum.inr a))\n (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) m\nfa\u2082 : Sum.inr a' \u2208 f a\u2082\n\u22a2 b \u2208 PFun.fix f a\n[PROOFSTEP]\nexact IH _ am\u2082 (PFun.mem_fix_iff.2 (Or.inr \u27e8_, fa\u2082, ba\u27e9))\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nn : \u2115\nh\u2082 : Sum.inl b \u2208 F a n\n_x : \u2203 b', Sum.inl b' \u2208 F a n\nh\u2081 : \u2200 {m : \u2115}, m < n \u2192 \u2203 b, Sum.inr b \u2208 F a m\nthis : \u2200 (m : \u2115) (a' : \u03b1), Sum.inr a' \u2208 F a m \u2192 b \u2208 PFun.fix f a' \u2192 b \u2208 PFun.fix f a\n\u22a2 b \u2208 PFun.fix f a\n[PROOFSTEP]\ncases n\n[GOAL]\ncase refine'_1.intro.intro.intro.zero\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nthis : \u2200 (m : \u2115) (a' : \u03b1), Sum.inr a' \u2208 F a m \u2192 b \u2208 PFun.fix f a' \u2192 b \u2208 PFun.fix f a\nh\u2082 : Sum.inl b \u2208 F a Nat.zero\n_x : \u2203 b', Sum.inl b' \u2208 F a Nat.zero\nh\u2081 : \u2200 {m : \u2115}, m < Nat.zero \u2192 \u2203 b, Sum.inr b \u2208 F a m\n\u22a2 b \u2208 PFun.fix f a\n[PROOFSTEP]\nsimp at h\u2082 \n[GOAL]\ncase refine'_1.intro.intro.intro.succ\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nthis : \u2200 (m : \u2115) (a' : \u03b1), Sum.inr a' \u2208 F a m \u2192 b \u2208 PFun.fix f a' \u2192 b \u2208 PFun.fix f a\nn\u271d : \u2115\nh\u2082 : Sum.inl b \u2208 F a (Nat.succ n\u271d)\n_x : \u2203 b', Sum.inl b' \u2208 F a (Nat.succ n\u271d)\nh\u2081 : \u2200 {m : \u2115}, m < Nat.succ n\u271d \u2192 \u2203 b, Sum.inr b \u2208 F a m\n\u22a2 b \u2208 PFun.fix f a\n[PROOFSTEP]\nsimp at h\u2082 \n[GOAL]\ncase refine'_1.intro.intro.intro.succ\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nthis : \u2200 (m : \u2115) (a' : \u03b1), Sum.inr a' \u2208 F a m \u2192 b \u2208 PFun.fix f a' \u2192 b \u2208 PFun.fix f a\nn\u271d : \u2115\n_x : \u2203 b', Sum.inl b' \u2208 F a (Nat.succ n\u271d)\nh\u2081 : \u2200 {m : \u2115}, m < Nat.succ n\u271d \u2192 \u2203 b, Sum.inr b \u2208 F a m\nh\u2082 :\n Sum.inl b \u2208\n Nat.rec (Part.some (Sum.inr a))\n (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) n\u271d \u2228\n \u2203 b_1,\n Sum.inr b_1 \u2208\n Nat.rec (Part.some (Sum.inr a))\n (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) n\u271d \u2227\n Sum.inl b \u2208 f b_1\n\u22a2 b \u2208 PFun.fix f a\n[PROOFSTEP]\nrcases h\u2082 with (h\u2082 | \u27e8a', am', fa'\u27e9)\n[GOAL]\ncase refine'_1.intro.intro.intro.succ.inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nthis : \u2200 (m : \u2115) (a' : \u03b1), Sum.inr a' \u2208 F a m \u2192 b \u2208 PFun.fix f a' \u2192 b \u2208 PFun.fix f a\nn\u271d : \u2115\n_x : \u2203 b', Sum.inl b' \u2208 F a (Nat.succ n\u271d)\nh\u2081 : \u2200 {m : \u2115}, m < Nat.succ n\u271d \u2192 \u2203 b, Sum.inr b \u2208 F a m\nh\u2082 :\n Sum.inl b \u2208\n Nat.rec (Part.some (Sum.inr a))\n (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) n\u271d\n\u22a2 b \u2208 PFun.fix f a\n[PROOFSTEP]\ncases' h\u2081 (Nat.lt_succ_self _) with a' h\n[GOAL]\ncase refine'_1.intro.intro.intro.succ.inl.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nthis : \u2200 (m : \u2115) (a' : \u03b1), Sum.inr a' \u2208 F a m \u2192 b \u2208 PFun.fix f a' \u2192 b \u2208 PFun.fix f a\nn\u271d : \u2115\n_x : \u2203 b', Sum.inl b' \u2208 F a (Nat.succ n\u271d)\nh\u2081 : \u2200 {m : \u2115}, m < Nat.succ n\u271d \u2192 \u2203 b, Sum.inr b \u2208 F a m\nh\u2082 :\n Sum.inl b \u2208\n Nat.rec (Part.some (Sum.inr a))\n (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) n\u271d\na' : \u03b1\nh : Sum.inr a' \u2208 F a n\u271d\n\u22a2 b \u2208 PFun.fix f a\n[PROOFSTEP]\ninjection mem_unique h h\u2082\n[GOAL]\ncase refine'_1.intro.intro.intro.succ.inr.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nthis : \u2200 (m : \u2115) (a' : \u03b1), Sum.inr a' \u2208 F a m \u2192 b \u2208 PFun.fix f a' \u2192 b \u2208 PFun.fix f a\nn\u271d : \u2115\n_x : \u2203 b', Sum.inl b' \u2208 F a (Nat.succ n\u271d)\nh\u2081 : \u2200 {m : \u2115}, m < Nat.succ n\u271d \u2192 \u2203 b, Sum.inr b \u2208 F a m\na' : \u03b1\nam' :\n Sum.inr a' \u2208\n Nat.rec (Part.some (Sum.inr a))\n (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) n\u271d\nfa' : Sum.inl b \u2208 f a'\n\u22a2 b \u2208 PFun.fix f a\n[PROOFSTEP]\nexact this _ _ am' (PFun.mem_fix_iff.2 (Or.inl fa'))\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\n\u22a2 \u2203 n, ((\u2203 b', Sum.inl b' \u2208 F a n) \u2227 \u2200 {m : \u2115}, m < n \u2192 \u2203 b, Sum.inr b \u2208 F a m) \u2227 Sum.inl b \u2208 F a n\n[PROOFSTEP]\nsuffices\n \u2200 (a') (_ : b \u2208 PFun.fix f a') (k) (_ : Sum.inr a' \u2208 F a k),\n \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 m < n, \u2200 (_ : k \u2264 m), \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\n by\n rcases this _ h 0 (by simp) with \u27e8n, hn\u2081, hn\u2082\u27e9\n exact \u27e8_, \u27e8\u27e8_, hn\u2081\u27e9, fun {m} mn => hn\u2082 m mn (Nat.zero_le _)\u27e9, hn\u2081\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\nthis :\n \u2200 (a' : \u03b1),\n b \u2208 PFun.fix f a' \u2192\n \u2200 (k : \u2115), Sum.inr a' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\n\u22a2 \u2203 n, ((\u2203 b', Sum.inl b' \u2208 F a n) \u2227 \u2200 {m : \u2115}, m < n \u2192 \u2203 b, Sum.inr b \u2208 F a m) \u2227 Sum.inl b \u2208 F a n\n[PROOFSTEP]\nrcases this _ h 0 (by simp) with \u27e8n, hn\u2081, hn\u2082\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\nthis :\n \u2200 (a' : \u03b1),\n b \u2208 PFun.fix f a' \u2192\n \u2200 (k : \u2115), Sum.inr a' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\n\u22a2 Sum.inr a \u2208 F a 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\nthis :\n \u2200 (a' : \u03b1),\n b \u2208 PFun.fix f a' \u2192\n \u2200 (k : \u2115), Sum.inr a' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nn : \u2115\nhn\u2081 : Sum.inl b \u2208 F a n\nhn\u2082 : \u2200 (m : \u2115), m < n \u2192 0 \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\n\u22a2 \u2203 n, ((\u2203 b', Sum.inl b' \u2208 F a n) \u2227 \u2200 {m : \u2115}, m < n \u2192 \u2203 b, Sum.inr b \u2208 F a m) \u2227 Sum.inl b \u2208 F a n\n[PROOFSTEP]\nexact \u27e8_, \u27e8\u27e8_, hn\u2081\u27e9, fun {m} mn => hn\u2082 m mn (Nat.zero_le _)\u27e9, hn\u2081\u27e9\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\n\u22a2 \u2200 (a' : \u03b1),\n b \u2208 PFun.fix f a' \u2192\n \u2200 (k : \u2115), Sum.inr a' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\n[PROOFSTEP]\nintro a\u2081 h\u2081\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\na\u2081 : \u03b1\nh\u2081 : b \u2208 PFun.fix f a\u2081\n\u22a2 \u2200 (k : \u2115), Sum.inr a\u2081 \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\n[PROOFSTEP]\napply @PFun.fixInduction _ _ _ _ _ _ h\u2081\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\na\u2081 : \u03b1\nh\u2081 : b \u2208 PFun.fix f a\u2081\n\u22a2 \u2200 (a' : \u03b1),\n b \u2208 PFun.fix f a' \u2192\n (\u2200 (a'' : \u03b1),\n Sum.inr a'' \u2208 f a' \u2192\n \u2200 (k : \u2115),\n Sum.inr a'' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m) \u2192\n \u2200 (k : \u2115), Sum.inr a' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\n[PROOFSTEP]\nintro a\u2082 h\u2082 IH k hk\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\na\u2081 : \u03b1\nh\u2081 : b \u2208 PFun.fix f a\u2081\na\u2082 : \u03b1\nh\u2082 : b \u2208 PFun.fix f a\u2082\nIH :\n \u2200 (a'' : \u03b1),\n Sum.inr a'' \u2208 f a\u2082 \u2192\n \u2200 (k : \u2115), Sum.inr a'' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nk : \u2115\nhk : Sum.inr a\u2082 \u2208 F a k\n\u22a2 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\n[PROOFSTEP]\nrcases PFun.mem_fix_iff.1 h\u2082 with (h\u2082 | \u27e8a\u2083, am\u2083, _\u27e9)\n[GOAL]\ncase refine'_2.inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\na\u2081 : \u03b1\nh\u2081 : b \u2208 PFun.fix f a\u2081\na\u2082 : \u03b1\nh\u2082\u271d : b \u2208 PFun.fix f a\u2082\nIH :\n \u2200 (a'' : \u03b1),\n Sum.inr a'' \u2208 f a\u2082 \u2192\n \u2200 (k : \u2115), Sum.inr a'' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nk : \u2115\nhk : Sum.inr a\u2082 \u2208 F a k\nh\u2082 : Sum.inl b \u2208 f a\u2082\n\u22a2 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\n[PROOFSTEP]\nrefine' \u27e8k.succ, _, fun m mk km => \u27e8a\u2082, _\u27e9\u27e9\n[GOAL]\ncase refine'_2.inl.refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\na\u2081 : \u03b1\nh\u2081 : b \u2208 PFun.fix f a\u2081\na\u2082 : \u03b1\nh\u2082\u271d : b \u2208 PFun.fix f a\u2082\nIH :\n \u2200 (a'' : \u03b1),\n Sum.inr a'' \u2208 f a\u2082 \u2192\n \u2200 (k : \u2115), Sum.inr a'' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nk : \u2115\nhk : Sum.inr a\u2082 \u2208 F a k\nh\u2082 : Sum.inl b \u2208 f a\u2082\n\u22a2 Sum.inl b \u2208 F a (Nat.succ k)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2.inl.refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\na\u2081 : \u03b1\nh\u2081 : b \u2208 PFun.fix f a\u2081\na\u2082 : \u03b1\nh\u2082\u271d : b \u2208 PFun.fix f a\u2082\nIH :\n \u2200 (a'' : \u03b1),\n Sum.inr a'' \u2208 f a\u2082 \u2192\n \u2200 (k : \u2115), Sum.inr a'' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nk : \u2115\nhk : Sum.inr a\u2082 \u2208 F a k\nh\u2082 : Sum.inl b \u2208 f a\u2082\n\u22a2 Sum.inl b \u2208\n Nat.rec (Part.some (Sum.inr a))\n (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) k \u2228\n \u2203 b_1,\n Sum.inr b_1 \u2208\n Nat.rec (Part.some (Sum.inr a))\n (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) k \u2227\n Sum.inl b \u2208 f b_1\n[PROOFSTEP]\nexact Or.inr \u27e8_, hk, h\u2082\u27e9\n[GOAL]\ncase refine'_2.inl.refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\na\u2081 : \u03b1\nh\u2081 : b \u2208 PFun.fix f a\u2081\na\u2082 : \u03b1\nh\u2082\u271d : b \u2208 PFun.fix f a\u2082\nIH :\n \u2200 (a'' : \u03b1),\n Sum.inr a'' \u2208 f a\u2082 \u2192\n \u2200 (k : \u2115), Sum.inr a'' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nk : \u2115\nhk : Sum.inr a\u2082 \u2208 F a k\nh\u2082 : Sum.inl b \u2208 f a\u2082\nm : \u2115\nmk : m < Nat.succ k\nkm : k \u2264 m\n\u22a2 Sum.inr a\u2082 \u2208 F a m\n[PROOFSTEP]\nrwa [le_antisymm (Nat.le_of_lt_succ mk) km]\n[GOAL]\ncase refine'_2.inr.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\na\u2081 : \u03b1\nh\u2081 : b \u2208 PFun.fix f a\u2081\na\u2082 : \u03b1\nh\u2082 : b \u2208 PFun.fix f a\u2082\nIH :\n \u2200 (a'' : \u03b1),\n Sum.inr a'' \u2208 f a\u2082 \u2192\n \u2200 (k : \u2115), Sum.inr a'' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nk : \u2115\nhk : Sum.inr a\u2082 \u2208 F a k\na\u2083 : \u03b1\nam\u2083 : Sum.inr a\u2083 \u2208 f a\u2082\nright\u271d : b \u2208 PFun.fix f a\u2083\n\u22a2 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\n[PROOFSTEP]\nrcases IH _ am\u2083 k.succ (by simp; exact \u27e8_, hk, am\u2083\u27e9) with \u27e8n, hn\u2081, hn\u2082\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\na\u2081 : \u03b1\nh\u2081 : b \u2208 PFun.fix f a\u2081\na\u2082 : \u03b1\nh\u2082 : b \u2208 PFun.fix f a\u2082\nIH :\n \u2200 (a'' : \u03b1),\n Sum.inr a'' \u2208 f a\u2082 \u2192\n \u2200 (k : \u2115), Sum.inr a'' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nk : \u2115\nhk : Sum.inr a\u2082 \u2208 F a k\na\u2083 : \u03b1\nam\u2083 : Sum.inr a\u2083 \u2208 f a\u2082\nright\u271d : b \u2208 PFun.fix f a\u2083\n\u22a2 Sum.inr a\u2083 \u2208 F a (Nat.succ k)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\na\u2081 : \u03b1\nh\u2081 : b \u2208 PFun.fix f a\u2081\na\u2082 : \u03b1\nh\u2082 : b \u2208 PFun.fix f a\u2082\nIH :\n \u2200 (a'' : \u03b1),\n Sum.inr a'' \u2208 f a\u2082 \u2192\n \u2200 (k : \u2115), Sum.inr a'' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nk : \u2115\nhk : Sum.inr a\u2082 \u2208 F a k\na\u2083 : \u03b1\nam\u2083 : Sum.inr a\u2083 \u2208 f a\u2082\nright\u271d : b \u2208 PFun.fix f a\u2083\n\u22a2 \u2203 b,\n Sum.inr b \u2208\n Nat.rec (Part.some (Sum.inr a))\n (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) k \u2227\n Sum.inr a\u2083 \u2208 f b\n[PROOFSTEP]\nexact \u27e8_, hk, am\u2083\u27e9\n[GOAL]\ncase refine'_2.inr.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\na\u2081 : \u03b1\nh\u2081 : b \u2208 PFun.fix f a\u2081\na\u2082 : \u03b1\nh\u2082 : b \u2208 PFun.fix f a\u2082\nIH :\n \u2200 (a'' : \u03b1),\n Sum.inr a'' \u2208 f a\u2082 \u2192\n \u2200 (k : \u2115), Sum.inr a'' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nk : \u2115\nhk : Sum.inr a\u2082 \u2208 F a k\na\u2083 : \u03b1\nam\u2083 : Sum.inr a\u2083 \u2208 f a\u2082\nright\u271d : b \u2208 PFun.fix f a\u2083\nn : \u2115\nhn\u2081 : Sum.inl b \u2208 F a n\nhn\u2082 : \u2200 (m : \u2115), m < n \u2192 Nat.succ k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\n\u22a2 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\n[PROOFSTEP]\nrefine' \u27e8n, hn\u2081, fun m mn km => _\u27e9\n[GOAL]\ncase refine'_2.inr.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\na\u2081 : \u03b1\nh\u2081 : b \u2208 PFun.fix f a\u2081\na\u2082 : \u03b1\nh\u2082 : b \u2208 PFun.fix f a\u2082\nIH :\n \u2200 (a'' : \u03b1),\n Sum.inr a'' \u2208 f a\u2082 \u2192\n \u2200 (k : \u2115), Sum.inr a'' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nk : \u2115\nhk : Sum.inr a\u2082 \u2208 F a k\na\u2083 : \u03b1\nam\u2083 : Sum.inr a\u2083 \u2208 f a\u2082\nright\u271d : b \u2208 PFun.fix f a\u2083\nn : \u2115\nhn\u2081 : Sum.inl b \u2208 F a n\nhn\u2082 : \u2200 (m : \u2115), m < n \u2192 Nat.succ k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nm : \u2115\nmn : m < n\nkm : k \u2264 m\n\u22a2 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\n[PROOFSTEP]\ncases' km.lt_or_eq_dec with km km\n[GOAL]\ncase refine'_2.inr.intro.intro.intro.intro.inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\na\u2081 : \u03b1\nh\u2081 : b \u2208 PFun.fix f a\u2081\na\u2082 : \u03b1\nh\u2082 : b \u2208 PFun.fix f a\u2082\nIH :\n \u2200 (a'' : \u03b1),\n Sum.inr a'' \u2208 f a\u2082 \u2192\n \u2200 (k : \u2115), Sum.inr a'' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nk : \u2115\nhk : Sum.inr a\u2082 \u2208 F a k\na\u2083 : \u03b1\nam\u2083 : Sum.inr a\u2083 \u2208 f a\u2082\nright\u271d : b \u2208 PFun.fix f a\u2083\nn : \u2115\nhn\u2081 : Sum.inl b \u2208 F a n\nhn\u2082 : \u2200 (m : \u2115), m < n \u2192 Nat.succ k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nm : \u2115\nmn : m < n\nkm\u271d : k \u2264 m\nkm : k < m\n\u22a2 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\n[PROOFSTEP]\nexact hn\u2082 _ mn km\n[GOAL]\ncase refine'_2.inr.intro.intro.intro.intro.inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3\u271d : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\u271d\n\u03b1 : Type u_5\n\u03c3 : Type u_6\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\na : \u03b1\nb : \u03c3\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nh : b \u2208 PFun.fix f a\na\u2081 : \u03b1\nh\u2081 : b \u2208 PFun.fix f a\u2081\na\u2082 : \u03b1\nh\u2082 : b \u2208 PFun.fix f a\u2082\nIH :\n \u2200 (a'' : \u03b1),\n Sum.inr a'' \u2208 f a\u2082 \u2192\n \u2200 (k : \u2115), Sum.inr a'' \u2208 F a k \u2192 \u2203 n, Sum.inl b \u2208 F a n \u2227 \u2200 (m : \u2115), m < n \u2192 k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nk : \u2115\nhk : Sum.inr a\u2082 \u2208 F a k\na\u2083 : \u03b1\nam\u2083 : Sum.inr a\u2083 \u2208 f a\u2082\nright\u271d : b \u2208 PFun.fix f a\u2083\nn : \u2115\nhn\u2081 : Sum.inl b \u2208 F a n\nhn\u2082 : \u2200 (m : \u2115), m < n \u2192 Nat.succ k \u2264 m \u2192 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\nm : \u2115\nmn : m < n\nkm\u271d : k \u2264 m\nkm : k = m\n\u22a2 \u2203 a\u2082, Sum.inr a\u2082 \u2208 F a m\n[PROOFSTEP]\nexact km \u25b8 \u27e8_, hk\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\nhf : Partrec f\n\u22a2 Partrec (PFun.fix f)\n[PROOFSTEP]\nlet F : \u03b1 \u2192 \u2115 \u2192. Sum \u03c3 \u03b1 := fun a n =>\n n.rec (some (Sum.inr a)) fun _ IH => IH.bind fun s => Sum.casesOn s (fun _ => Part.some s) f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\n\u22a2 Partrec (PFun.fix f)\n[PROOFSTEP]\nhave hF : Partrec\u2082 F :=\n Partrec.nat_rec snd (sum_inr.comp fst).partrec\n (sum_casesOn_right (snd.comp snd) (snd.comp <| snd.comp fst).to\u2082 (hf.comp snd).to\u2082).to\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nhF : Partrec\u2082 F\n\u22a2 Partrec (PFun.fix f)\n[PROOFSTEP]\nlet p a n := @Part.map _ Bool (fun s => Sum.casesOn s (fun _ => true) fun _ => false) (F a n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nhF : Partrec\u2082 F\np : \u03b1 \u2192 \u2115 \u2192 Part Bool := fun a n => Part.map (fun s => Sum.casesOn s (fun x => true) fun x => false) (F a n)\n\u22a2 Partrec (PFun.fix f)\n[PROOFSTEP]\nhave hp : Partrec\u2082 p := hF.map ((sum_casesOn Computable.id (const true).to\u2082 (const false).to\u2082).comp snd).to\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nhF : Partrec\u2082 F\np : \u03b1 \u2192 \u2115 \u2192 Part Bool := fun a n => Part.map (fun s => Sum.casesOn s (fun x => true) fun x => false) (F a n)\nhp : Partrec\u2082 p\n\u22a2 Partrec (PFun.fix f)\n[PROOFSTEP]\nexact\n (hp.rfind.bind (hF.bind (sum_casesOn_right snd snd.to\u2082 none.to\u2082).to\u2082).to\u2082).of_eq fun a =>\n ext fun b => by simp; apply fix_aux f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nhF : Partrec\u2082 F\np : \u03b1 \u2192 \u2115 \u2192 Part Bool := fun a n => Part.map (fun s => Sum.casesOn s (fun x => true) fun x => false) (F a n)\nhp : Partrec\u2082 p\na : \u03b1\nb : \u03c3\n\u22a2 (b \u2208\n Part.bind (Nat.rfind (p a)) fun b =>\n Part.bind (F (a, b).fst (a, b).snd) fun b_1 =>\n Sum.casesOn ((a, b), b_1).snd (fun b_2 => Part.some (((a, b), b_1), b_2).snd) fun b => Part.none) \u2194\n b \u2208 PFun.fix f a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03c3 \u2295 \u03b1\nhf : Partrec f\nF : \u03b1 \u2192 \u2115 \u2192. \u03c3 \u2295 \u03b1 :=\n fun a n =>\n Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n\nhF : Partrec\u2082 F\np : \u03b1 \u2192 \u2115 \u2192 Part Bool := fun a n => Part.map (fun s => Sum.casesOn s (fun x => true) fun x => false) (F a n)\nhp : Partrec\u2082 p\na : \u03b1\nb : \u03c3\n\u22a2 (\u2203 a_1,\n ((\u2203 a_2,\n Sum.inl a_2 \u2208\n Nat.rec (Part.some (Sum.inr a))\n (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) a_1) \u2227\n \u2200 {m : \u2115},\n m < a_1 \u2192\n \u2203 b,\n Sum.inr b \u2208\n Nat.rec (Part.some (Sum.inr a))\n (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) m) \u2227\n Sum.inl b \u2208\n Nat.rec (Part.some (Sum.inr a))\n (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) a_1) \u2194\n b \u2208 PFun.fix f a\n[PROOFSTEP]\napply fix_aux f\n", "meta": {"mathlib_filename": "Mathlib.Computability.Partrec", "llama_tokens": 59602, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006920020959543, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.4470479589186075}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X\u271d : C\nY Y' : D\nX : C\n\u22a2 \u2191(homEquiv adj X (F.obj X)) (\ud835\udfd9 (F.obj X)) = NatTrans.app adj.unit X\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X\u271d : C\nY Y' X : D\n\u22a2 \u2191(homEquiv adj (G.obj X) X).symm (\ud835\udfd9 (G.obj X)) = NatTrans.app adj.counit X\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nf : X' \u27f6 X\ng : X \u27f6 G.obj Y\n\u22a2 \u2191(homEquiv adj X' Y).symm (f \u226b g) = F.map f \u226b \u2191(homEquiv adj X Y).symm g\n[PROOFSTEP]\nrw [homEquiv_counit, F.map_comp, assoc, adj.homEquiv_counit.symm]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nf : X' \u27f6 X\ng : F.obj X \u27f6 Y\n\u22a2 \u2191(homEquiv adj X' Y) (F.map f \u226b g) = f \u226b \u2191(homEquiv adj X Y) g\n[PROOFSTEP]\nrw [\u2190 Equiv.eq_symm_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nf : X' \u27f6 X\ng : F.obj X \u27f6 Y\n\u22a2 F.map f \u226b g = \u2191(homEquiv adj X' Y).symm (f \u226b \u2191(homEquiv adj X Y) g)\n[PROOFSTEP]\nsimp only [Equiv.symm_apply_apply, eq_self_iff_true, homEquiv_naturality_left_symm]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nf : F.obj X \u27f6 Y\ng : Y \u27f6 Y'\n\u22a2 \u2191(homEquiv adj X Y') (f \u226b g) = \u2191(homEquiv adj X Y) f \u226b G.map g\n[PROOFSTEP]\nrw [homEquiv_unit, G.map_comp, \u2190 assoc, \u2190 homEquiv_unit]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nf : X \u27f6 G.obj Y\ng : Y \u27f6 Y'\n\u22a2 \u2191(homEquiv adj X Y').symm (f \u226b G.map g) = \u2191(homEquiv adj X Y).symm f \u226b g\n[PROOFSTEP]\nrw [Equiv.symm_apply_eq]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nf : X \u27f6 G.obj Y\ng : Y \u27f6 Y'\n\u22a2 f \u226b G.map g = \u2191(homEquiv adj X Y') (\u2191(homEquiv adj X Y).symm f \u226b g)\n[PROOFSTEP]\nsimp only [homEquiv_naturality_right, eq_self_iff_true, Equiv.apply_symm_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\n\u22a2 whiskerRight adj.unit F \u226b whiskerLeft F adj.counit = \ud835\udfd9 (\ud835\udfed C \u22d9 F)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nx\u271d : C\n\u22a2 NatTrans.app (whiskerRight adj.unit F \u226b whiskerLeft F adj.counit) x\u271d = NatTrans.app (\ud835\udfd9 (\ud835\udfed C \u22d9 F)) x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nx\u271d : C\n\u22a2 F.map (NatTrans.app adj.unit x\u271d) \u226b NatTrans.app adj.counit (F.obj x\u271d) = \ud835\udfd9 (F.obj x\u271d)\n[PROOFSTEP]\nerw [\u2190 adj.homEquiv_counit, Equiv.symm_apply_eq, adj.homEquiv_unit]\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nx\u271d : C\n\u22a2 NatTrans.app adj.unit x\u271d = NatTrans.app adj.unit x\u271d \u226b G.map (\ud835\udfd9 (F.obj x\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\n\u22a2 whiskerLeft G adj.unit \u226b whiskerRight adj.counit G = \ud835\udfd9 (G \u22d9 \ud835\udfed C)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' x\u271d : D\n\u22a2 NatTrans.app (whiskerLeft G adj.unit \u226b whiskerRight adj.counit G) x\u271d = NatTrans.app (\ud835\udfd9 (G \u22d9 \ud835\udfed C)) x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' x\u271d : D\n\u22a2 NatTrans.app adj.unit (G.obj x\u271d) \u226b G.map (NatTrans.app adj.counit x\u271d) = \ud835\udfd9 (G.obj x\u271d)\n[PROOFSTEP]\nerw [\u2190 adj.homEquiv_unit, \u2190 Equiv.eq_symm_apply, adj.homEquiv_counit]\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' x\u271d : D\n\u22a2 NatTrans.app adj.counit x\u271d = F.map (\ud835\udfd9 (G.obj x\u271d)) \u226b NatTrans.app adj.counit x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nA : C\nB : D\nf : F.obj A \u27f6 B\ng : A \u27f6 G.obj B\nh : \u2191(homEquiv adj A B) f = g\n\u22a2 f = \u2191(homEquiv adj A B).symm g\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nA : C\nB : D\nf : F.obj A \u27f6 B\n\u22a2 f = \u2191(homEquiv adj A B).symm (\u2191(homEquiv adj A B) f)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nA : C\nB : D\nf : F.obj A \u27f6 B\ng : A \u27f6 G.obj B\nh : f = \u2191(homEquiv adj A B).symm g\n\u22a2 \u2191(homEquiv adj A B) f = g\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nA : C\nB : D\ng : A \u27f6 G.obj B\n\u22a2 \u2191(homEquiv adj A B) (\u2191(homEquiv adj A B).symm g) = g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nA : C\nB : D\nf : F.obj A \u27f6 B\ng : A \u27f6 G.obj B\nh : g = \u2191(homEquiv adj A B) f\n\u22a2 \u2191(homEquiv adj A B).symm g = f\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nA : C\nB : D\nf : F.obj A \u27f6 B\n\u22a2 \u2191(homEquiv adj A B).symm (\u2191(homEquiv adj A B) f) = f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nA : C\nB : D\nf : F.obj A \u27f6 B\ng : A \u27f6 G.obj B\nh : \u2191(homEquiv adj A B).symm g = f\n\u22a2 g = \u2191(homEquiv adj A B) f\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nX' X : C\nY Y' : D\nA : C\nB : D\ng : A \u27f6 G.obj B\n\u22a2 g = \u2191(homEquiv adj A B) (\u2191(homEquiv adj A B).symm g)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX' X : C\nY Y' : D\nf : X' \u27f6 X\ng : F.obj X \u27f6 Y\n\u22a2 \u2191(homEquiv adj X' (F.obj X)) (F.map f) \u226b G.map g = f \u226b \u2191(homEquiv adj X Y) g\n[PROOFSTEP]\nrw [\u2190 homEquiv_naturality_right, \u2190 Equiv.eq_symm_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX' X : C\nY Y' : D\nf : X' \u27f6 X\ng : F.obj X \u27f6 Y\n\u22a2 F.map f \u226b g = \u2191(homEquiv adj X' Y).symm (f \u226b \u2191(homEquiv adj X Y) g)\n[PROOFSTEP]\nsimp\n -- @[simp] -- Porting note: LHS simplifies, added aux lemma above\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX' X : C\nY Y' : D\nf : X' \u27f6 X\ng : F.obj X \u27f6 Y\n\u22a2 \u2191(homEquiv adj X' Y) (F.map f \u226b g) = f \u226b \u2191(homEquiv adj X Y) g\n[PROOFSTEP]\nrw [\u2190 Equiv.eq_symm_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX' X : C\nY Y' : D\nf : X' \u27f6 X\ng : F.obj X \u27f6 Y\n\u22a2 F.map f \u226b g = \u2191(homEquiv adj X' Y).symm (f \u226b \u2191(homEquiv adj X Y) g)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX' X : C\nY Y' : D\nf : X \u27f6 G.obj Y\ng : Y \u27f6 Y'\n\u22a2 F.map f \u226b \u2191(homEquiv adj (G.obj Y) Y').symm (G.map g) = \u2191(homEquiv adj X Y).symm f \u226b g\n[PROOFSTEP]\nrw [\u2190 homEquiv_naturality_left_symm, Equiv.symm_apply_eq]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX' X : C\nY Y' : D\nf : X \u27f6 G.obj Y\ng : Y \u27f6 Y'\n\u22a2 f \u226b G.map g = \u2191(homEquiv adj X Y') (\u2191(homEquiv adj X Y).symm f \u226b g)\n[PROOFSTEP]\nsimp\n -- @[simp] -- Porting note: LHS simplifies, added aux lemma above\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX' X : C\nY Y' : D\nf : X \u27f6 G.obj Y\ng : Y \u27f6 Y'\n\u22a2 \u2191(homEquiv adj X Y').symm (f \u226b G.map g) = \u2191(homEquiv adj X Y).symm f \u226b g\n[PROOFSTEP]\nrw [Equiv.symm_apply_eq]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX' X : C\nY Y' : D\nf : X \u27f6 G.obj Y\ng : Y \u27f6 Y'\n\u22a2 f \u226b G.map g = \u2191(homEquiv adj X Y') (\u2191(homEquiv adj X Y).symm f \u226b g)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\n\u22a2 \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y),\n (\ud835\udfed C).map f \u226b (fun X => \u2191(CoreHomEquiv.homEquiv adj X (F.obj X)) (\ud835\udfd9 (F.obj X))) Y =\n (fun X => \u2191(CoreHomEquiv.homEquiv adj X (F.obj X)) (\ud835\udfd9 (F.obj X))) X \u226b (F \u22d9 G).map f\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (\ud835\udfed C).map f\u271d \u226b (fun X => \u2191(CoreHomEquiv.homEquiv adj X (F.obj X)) (\ud835\udfd9 (F.obj X))) Y\u271d =\n (fun X => \u2191(CoreHomEquiv.homEquiv adj X (F.obj X)) (\ud835\udfd9 (F.obj X))) X\u271d \u226b (F \u22d9 G).map f\u271d\n[PROOFSTEP]\nerw [\u2190 adj.homEquiv_naturality_left, \u2190 adj.homEquiv_naturality_right]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 \u2191(CoreHomEquiv.homEquiv adj ((\ud835\udfed C).obj X\u271d) (F.obj Y\u271d)) (F.map ((\ud835\udfed C).map f\u271d) \u226b \ud835\udfd9 (F.obj Y\u271d)) =\n \u2191(CoreHomEquiv.homEquiv adj ((\ud835\udfed C).obj X\u271d) (F.obj Y\u271d)) (\ud835\udfd9 (F.obj X\u271d) \u226b F.map f\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 \u2191(CoreHomEquiv.homEquiv adj X\u271d (F.obj Y\u271d)) (F.map f\u271d \u226b \ud835\udfd9 (F.obj Y\u271d)) =\n \u2191(CoreHomEquiv.homEquiv adj X\u271d (F.obj Y\u271d)) (\ud835\udfd9 (F.obj X\u271d) \u226b F.map f\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\n\u22a2 \u2200 \u2983X Y : D\u2984 (f : X \u27f6 Y),\n (G \u22d9 F).map f \u226b (fun Y => Equiv.invFun (CoreHomEquiv.homEquiv adj (G.obj Y) ((\ud835\udfed D).obj Y)) (\ud835\udfd9 (G.obj Y))) Y =\n (fun Y => Equiv.invFun (CoreHomEquiv.homEquiv adj (G.obj Y) ((\ud835\udfed D).obj Y)) (\ud835\udfd9 (G.obj Y))) X \u226b (\ud835\udfed D).map f\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX\u271d Y\u271d : D\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (G \u22d9 F).map f\u271d \u226b (fun Y => Equiv.invFun (CoreHomEquiv.homEquiv adj (G.obj Y) ((\ud835\udfed D).obj Y)) (\ud835\udfd9 (G.obj Y))) Y\u271d =\n (fun Y => Equiv.invFun (CoreHomEquiv.homEquiv adj (G.obj Y) ((\ud835\udfed D).obj Y)) (\ud835\udfd9 (G.obj Y))) X\u271d \u226b (\ud835\udfed D).map f\u271d\n[PROOFSTEP]\nerw [\u2190 adj.homEquiv_naturality_left_symm, \u2190 adj.homEquiv_naturality_right_symm]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX\u271d Y\u271d : D\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 \u2191(CoreHomEquiv.homEquiv adj (G.obj X\u271d) ((\ud835\udfed D).obj Y\u271d)).symm (G.map f\u271d \u226b \ud835\udfd9 (G.obj Y\u271d)) =\n \u2191(CoreHomEquiv.homEquiv adj (G.obj X\u271d) ((\ud835\udfed D).obj Y\u271d)).symm (\ud835\udfd9 (G.obj X\u271d) \u226b G.map ((\ud835\udfed D).map f\u271d))\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX\u271d Y\u271d : D\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 \u2191(CoreHomEquiv.homEquiv adj (G.obj X\u271d) Y\u271d).symm (G.map f\u271d \u226b \ud835\udfd9 (G.obj Y\u271d)) =\n \u2191(CoreHomEquiv.homEquiv adj (G.obj X\u271d) Y\u271d).symm (\ud835\udfd9 (G.obj X\u271d) \u226b G.map f\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX : C\nY : D\nf : F.obj X \u27f6 Y\n\u22a2 \u2191(CoreHomEquiv.homEquiv adj X Y) f =\n NatTrans.app (NatTrans.mk fun X => \u2191(CoreHomEquiv.homEquiv adj X (F.obj X)) (\ud835\udfd9 (F.obj X))) X \u226b G.map f\n[PROOFSTEP]\nerw [\u2190 adj.homEquiv_naturality_right]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX : C\nY : D\nf : F.obj X \u27f6 Y\n\u22a2 \u2191(CoreHomEquiv.homEquiv adj X Y) f = \u2191(CoreHomEquiv.homEquiv adj ((\ud835\udfed C).obj X) Y) (\ud835\udfd9 (F.obj X) \u226b f)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX : C\nY : D\nf : X \u27f6 G.obj Y\n\u22a2 \u2191(CoreHomEquiv.homEquiv adj X Y).symm f =\n F.map f \u226b\n NatTrans.app (NatTrans.mk fun Y => Equiv.invFun (CoreHomEquiv.homEquiv adj (G.obj Y) ((\ud835\udfed D).obj Y)) (\ud835\udfd9 (G.obj Y)))\n Y\n[PROOFSTEP]\nerw [\u2190 adj.homEquiv_naturality_left_symm]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreHomEquiv F G\nX : C\nY : D\nf : X \u27f6 G.obj Y\n\u22a2 \u2191(CoreHomEquiv.homEquiv adj X Y).symm f = \u2191(CoreHomEquiv.homEquiv adj X ((\ud835\udfed D).obj Y)).symm (f \u226b \ud835\udfd9 (G.obj Y))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreUnitCounit F G\nX : C\nY : D\nf : F.obj X \u27f6 Y\n\u22a2 (fun g => F.map g \u226b NatTrans.app adj.counit Y) ((fun f => NatTrans.app adj.unit X \u226b G.map f) f) = f\n[PROOFSTEP]\nchange F.map (_ \u226b _) \u226b _ = _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreUnitCounit F G\nX : C\nY : D\nf : F.obj X \u27f6 Y\n\u22a2 F.map (NatTrans.app adj.unit X \u226b G.map f) \u226b NatTrans.app adj.counit Y = f\n[PROOFSTEP]\nrw [F.map_comp, assoc, \u2190 Functor.comp_map, adj.counit.naturality, \u2190 assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreUnitCounit F G\nX : C\nY : D\nf : F.obj X \u27f6 Y\n\u22a2 (F.map (NatTrans.app adj.unit X) \u226b NatTrans.app adj.counit (F.obj X)) \u226b (\ud835\udfed D).map f = f\n[PROOFSTEP]\nconvert id_comp f\n[GOAL]\ncase h.e'_2.h.e'_6.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreUnitCounit F G\nX : C\nY : D\nf : F.obj X \u27f6 Y\ne_4\u271d : (\ud835\udfed D).obj (F.obj X) = F.obj X\n\u22a2 F.map (NatTrans.app adj.unit X) \u226b NatTrans.app adj.counit (F.obj X) = \ud835\udfd9 (F.obj X)\n[PROOFSTEP]\nhave t := congrArg (fun (s : NatTrans (\ud835\udfed C \u22d9 F) (F \u22d9 \ud835\udfed D)) => s.app X) adj.left_triangle\n[GOAL]\ncase h.e'_2.h.e'_6.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreUnitCounit F G\nX : C\nY : D\nf : F.obj X \u27f6 Y\ne_4\u271d : (\ud835\udfed D).obj (F.obj X) = F.obj X\nt :\n (fun s => NatTrans.app s X) (whiskerRight adj.unit F \u226b (Functor.associator F G F).hom \u226b whiskerLeft F adj.counit) =\n (fun s => NatTrans.app s X) (NatTrans.id (\ud835\udfed C \u22d9 F))\n\u22a2 F.map (NatTrans.app adj.unit X) \u226b NatTrans.app adj.counit (F.obj X) = \ud835\udfd9 (F.obj X)\n[PROOFSTEP]\ndsimp at t \n[GOAL]\ncase h.e'_2.h.e'_6.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreUnitCounit F G\nX : C\nY : D\nf : F.obj X \u27f6 Y\ne_4\u271d : (\ud835\udfed D).obj (F.obj X) = F.obj X\nt : F.map (NatTrans.app adj.unit X) \u226b \ud835\udfd9 (F.obj (G.obj (F.obj X))) \u226b NatTrans.app adj.counit (F.obj X) = \ud835\udfd9 (F.obj X)\n\u22a2 F.map (NatTrans.app adj.unit X) \u226b NatTrans.app adj.counit (F.obj X) = \ud835\udfd9 (F.obj X)\n[PROOFSTEP]\nsimp only [id_comp] at t \n[GOAL]\ncase h.e'_2.h.e'_6.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreUnitCounit F G\nX : C\nY : D\nf : F.obj X \u27f6 Y\ne_4\u271d : (\ud835\udfed D).obj (F.obj X) = F.obj X\nt : F.map (NatTrans.app adj.unit X) \u226b NatTrans.app adj.counit (F.obj X) = \ud835\udfd9 (F.obj X)\n\u22a2 F.map (NatTrans.app adj.unit X) \u226b NatTrans.app adj.counit (F.obj X) = \ud835\udfd9 (F.obj X)\n[PROOFSTEP]\nexact t\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreUnitCounit F G\nX : C\nY : D\ng : X \u27f6 G.obj Y\n\u22a2 (fun f => NatTrans.app adj.unit X \u226b G.map f) ((fun g => F.map g \u226b NatTrans.app adj.counit Y) g) = g\n[PROOFSTEP]\nchange _ \u226b G.map (_ \u226b _) = _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreUnitCounit F G\nX : C\nY : D\ng : X \u27f6 G.obj Y\n\u22a2 NatTrans.app adj.unit X \u226b G.map (F.map g \u226b NatTrans.app adj.counit Y) = g\n[PROOFSTEP]\nrw [G.map_comp, \u2190 assoc, \u2190 Functor.comp_map, \u2190 adj.unit.naturality, assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreUnitCounit F G\nX : C\nY : D\ng : X \u27f6 G.obj Y\n\u22a2 (\ud835\udfed C).map g \u226b NatTrans.app adj.unit (G.obj Y) \u226b G.map (NatTrans.app adj.counit Y) = g\n[PROOFSTEP]\nconvert comp_id g\n[GOAL]\ncase h.e'_2.h.e'_7.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreUnitCounit F G\nX : C\nY : D\ng : X \u27f6 G.obj Y\ne_4\u271d : (\ud835\udfed C).obj (G.obj Y) = G.obj Y\n\u22a2 NatTrans.app adj.unit (G.obj Y) \u226b G.map (NatTrans.app adj.counit Y) = \ud835\udfd9 (G.obj Y)\n[PROOFSTEP]\nhave t := congrArg (fun t : NatTrans (G \u22d9 \ud835\udfed C) (\ud835\udfed D \u22d9 G) => t.app Y) adj.right_triangle\n[GOAL]\ncase h.e'_2.h.e'_7.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreUnitCounit F G\nX : C\nY : D\ng : X \u27f6 G.obj Y\ne_4\u271d : (\ud835\udfed C).obj (G.obj Y) = G.obj Y\nt :\n (fun t => NatTrans.app t Y) (whiskerLeft G adj.unit \u226b (Functor.associator G F G).inv \u226b whiskerRight adj.counit G) =\n (fun t => NatTrans.app t Y) (NatTrans.id (G \u22d9 \ud835\udfed C))\n\u22a2 NatTrans.app adj.unit (G.obj Y) \u226b G.map (NatTrans.app adj.counit Y) = \ud835\udfd9 (G.obj Y)\n[PROOFSTEP]\ndsimp at t \n[GOAL]\ncase h.e'_2.h.e'_7.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreUnitCounit F G\nX : C\nY : D\ng : X \u27f6 G.obj Y\ne_4\u271d : (\ud835\udfed C).obj (G.obj Y) = G.obj Y\nt : NatTrans.app adj.unit (G.obj Y) \u226b \ud835\udfd9 (G.obj (F.obj (G.obj Y))) \u226b G.map (NatTrans.app adj.counit Y) = \ud835\udfd9 (G.obj Y)\n\u22a2 NatTrans.app adj.unit (G.obj Y) \u226b G.map (NatTrans.app adj.counit Y) = \ud835\udfd9 (G.obj Y)\n[PROOFSTEP]\nsimp only [id_comp] at t \n[GOAL]\ncase h.e'_2.h.e'_7.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nadj : CoreUnitCounit F G\nX : C\nY : D\ng : X \u27f6 G.obj Y\ne_4\u271d : (\ud835\udfed C).obj (G.obj Y) = G.obj Y\nt : NatTrans.app adj.unit (G.obj Y) \u226b G.map (NatTrans.app adj.counit Y) = \ud835\udfd9 (G.obj Y)\n\u22a2 NatTrans.app adj.unit (G.obj Y) \u226b G.map (NatTrans.app adj.counit Y) = \ud835\udfd9 (G.obj Y)\n[PROOFSTEP]\nexact t\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF\u271d : C \u2964 D\nG : D \u2964 C\nF F' : C \u2964 D\niso : F \u2245 F'\nX : C\nY : D\nf : F.obj X \u27f6 Y\n\u22a2 (fun g => NatTrans.app iso.hom X \u226b g) ((fun f => NatTrans.app iso.inv X \u226b f) f) = f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF\u271d : C \u2964 D\nG : D \u2964 C\nF F' : C \u2964 D\niso : F \u2245 F'\nX : C\nY : D\ng : F'.obj X \u27f6 Y\n\u22a2 (fun f => NatTrans.app iso.inv X \u226b f) ((fun g => NatTrans.app iso.hom X \u226b g) g) = g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG\u271d G G' : D \u2964 C\niso : G \u2245 G'\nX : C\nY : D\nf : X \u27f6 G.obj Y\n\u22a2 (fun g => g \u226b NatTrans.app iso.inv Y) ((fun f => f \u226b NatTrans.app iso.hom Y) f) = f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG\u271d G G' : D \u2964 C\niso : G \u2245 G'\nX : C\nY : D\ng : X \u27f6 G'.obj Y\n\u22a2 (fun f => f \u226b NatTrans.app iso.hom Y) ((fun g => g \u226b NatTrans.app iso.inv Y) g) = g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX : C\nY Y' : D\nf : X \u27f6 G.obj Y\ng : Y \u27f6 Y'\n\u22a2 \u2191(e X Y').symm (f \u226b G.map g) = \u2191(e X Y).symm f \u226b g\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX : C\nY Y' : D\nf : X \u27f6 G.obj Y\ng : Y \u27f6 Y'\n\u22a2 \u2191(e X Y').symm (f \u226b G.map g) = \u2191(e X Y).symm f \u226b g\n[PROOFSTEP]\nrw [Equiv.symm_apply_eq, he]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX : C\nY Y' : D\nf : X \u27f6 G.obj Y\ng : Y \u27f6 Y'\n\u22a2 f \u226b G.map g = \u2191(e X Y) (\u2191(e X Y).symm f) \u226b G.map g\n[PROOFSTEP]\nsimp\n -- #align category_theory.adjunction.he' category_theory.adjunction.he'\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX\u271d Y\u271d Z\u271d : C\nf : X\u271d \u27f6 Y\u271d\nf' : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := F_obj, map := fun {X X'} f => \u2191(e X (F_obj X')).symm (f \u226b \u2191(e X' (F_obj X')) (\ud835\udfd9 (F_obj X'))) }.map (f \u226b f') =\n { obj := F_obj, map := fun {X X'} f => \u2191(e X (F_obj X')).symm (f \u226b \u2191(e X' (F_obj X')) (\ud835\udfd9 (F_obj X'))) }.map f \u226b\n { obj := F_obj, map := fun {X X'} f => \u2191(e X (F_obj X')).symm (f \u226b \u2191(e X' (F_obj X')) (\ud835\udfd9 (F_obj X'))) }.map f'\n[PROOFSTEP]\nrw [Equiv.symm_apply_eq, he, Equiv.apply_symm_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX\u271d Y\u271d Z\u271d : C\nf : X\u271d \u27f6 Y\u271d\nf' : Y\u271d \u27f6 Z\u271d\n\u22a2 (f \u226b f') \u226b \u2191(e Z\u271d (F_obj Z\u271d)) (\ud835\udfd9 (F_obj Z\u271d)) =\n (f \u226b \u2191(e Y\u271d (F_obj Y\u271d)) (\ud835\udfd9 (F_obj Y\u271d))) \u226b\n G.map\n ({ obj := F_obj, map := fun {X X'} f => \u2191(e X (F_obj X')).symm (f \u226b \u2191(e X' (F_obj X')) (\ud835\udfd9 (F_obj X'))) }.map f')\n[PROOFSTEP]\nconv =>\n rhs\n rw [assoc, \u2190 he, id_comp, Equiv.apply_symm_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX\u271d Y\u271d Z\u271d : C\nf : X\u271d \u27f6 Y\u271d\nf' : Y\u271d \u27f6 Z\u271d\n| (f \u226b f') \u226b \u2191(e Z\u271d (F_obj Z\u271d)) (\ud835\udfd9 (F_obj Z\u271d)) =\n (f \u226b \u2191(e Y\u271d (F_obj Y\u271d)) (\ud835\udfd9 (F_obj Y\u271d))) \u226b\n G.map\n ({ obj := F_obj, map := fun {X X'} f => \u2191(e X (F_obj X')).symm (f \u226b \u2191(e X' (F_obj X')) (\ud835\udfd9 (F_obj X'))) }.map f')\n[PROOFSTEP]\n rhs\n rw [assoc, \u2190 he, id_comp, Equiv.apply_symm_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX\u271d Y\u271d Z\u271d : C\nf : X\u271d \u27f6 Y\u271d\nf' : Y\u271d \u27f6 Z\u271d\n| (f \u226b f') \u226b \u2191(e Z\u271d (F_obj Z\u271d)) (\ud835\udfd9 (F_obj Z\u271d)) =\n (f \u226b \u2191(e Y\u271d (F_obj Y\u271d)) (\ud835\udfd9 (F_obj Y\u271d))) \u226b\n G.map\n ({ obj := F_obj, map := fun {X X'} f => \u2191(e X (F_obj X')).symm (f \u226b \u2191(e X' (F_obj X')) (\ud835\udfd9 (F_obj X'))) }.map f')\n[PROOFSTEP]\n rhs\n rw [assoc, \u2190 he, id_comp, Equiv.apply_symm_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX\u271d Y\u271d Z\u271d : C\nf : X\u271d \u27f6 Y\u271d\nf' : Y\u271d \u27f6 Z\u271d\n| (f \u226b f') \u226b \u2191(e Z\u271d (F_obj Z\u271d)) (\ud835\udfd9 (F_obj Z\u271d)) =\n (f \u226b \u2191(e Y\u271d (F_obj Y\u271d)) (\ud835\udfd9 (F_obj Y\u271d))) \u226b\n G.map\n ({ obj := F_obj, map := fun {X X'} f => \u2191(e X (F_obj X')).symm (f \u226b \u2191(e X' (F_obj X')) (\ud835\udfd9 (F_obj X'))) }.map f')\n[PROOFSTEP]\nrhs\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX\u271d Y\u271d Z\u271d : C\nf : X\u271d \u27f6 Y\u271d\nf' : Y\u271d \u27f6 Z\u271d\n| (f \u226b \u2191(e Y\u271d (F_obj Y\u271d)) (\ud835\udfd9 (F_obj Y\u271d))) \u226b\n G.map\n ({ obj := F_obj, map := fun {X X'} f => \u2191(e X (F_obj X')).symm (f \u226b \u2191(e X' (F_obj X')) (\ud835\udfd9 (F_obj X'))) }.map f')\n[PROOFSTEP]\nrw [assoc, \u2190 he, id_comp, Equiv.apply_symm_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX\u271d Y\u271d Z\u271d : C\nf : X\u271d \u27f6 Y\u271d\nf' : Y\u271d \u27f6 Z\u271d\n\u22a2 (f \u226b f') \u226b \u2191(e Z\u271d (F_obj Z\u271d)) (\ud835\udfd9 (F_obj Z\u271d)) = f \u226b f' \u226b \u2191(e Z\u271d (F_obj Z\u271d)) (\ud835\udfd9 (F_obj Z\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX' X : C\nY : D\nf : X' \u27f6 X\ng : X \u27f6 G.obj Y\n\u22a2 \u2191(e X' Y).symm (f \u226b g) = (leftAdjointOfEquiv e he).map f \u226b \u2191(e X Y).symm g\n[PROOFSTEP]\nhave := @he' C _ D _ G F_obj e he\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX' X : C\nY : D\nf : X' \u27f6 X\ng : X \u27f6 G.obj Y\nthis : \u2200 {X : C} {Y Y' : D} (f : X \u27f6 G.obj Y) (g : Y \u27f6 Y'), \u2191(e X Y').symm (f \u226b G.map g) = \u2191(e X Y).symm f \u226b g\n\u22a2 \u2191(e X' Y).symm (f \u226b g) = (leftAdjointOfEquiv e he).map f \u226b \u2191(e X Y).symm g\n[PROOFSTEP]\nerw [\u2190 this, \u2190 Equiv.apply_eq_iff_eq (e X' Y)]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX' X : C\nY : D\nf : X' \u27f6 X\ng : X \u27f6 G.obj Y\nthis : \u2200 {X : C} {Y Y' : D} (f : X \u27f6 G.obj Y) (g : Y \u27f6 Y'), \u2191(e X Y').symm (f \u226b G.map g) = \u2191(e X Y).symm f \u226b g\n\u22a2 \u2191(e X' Y) (\u2191(e X' Y).symm (f \u226b g)) =\n \u2191(e X' Y)\n (\u2191(e X' Y).symm ((f \u226b \u2191(e X ((fun X => F_obj X) X)) (\ud835\udfd9 ((fun X => F_obj X) X))) \u226b G.map (\u2191(e X Y).symm g)))\n[PROOFSTEP]\nsimp [(he X' (F_obj X) Y (e X Y |>.symm g) (leftAdjointOfEquiv e he |>.map f)).symm]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX' X : C\nY : D\nf : X' \u27f6 X\ng : X \u27f6 G.obj Y\nthis : \u2200 {X : C} {Y Y' : D} (f : X \u27f6 G.obj Y) (g : Y \u27f6 Y'), \u2191(e X Y').symm (f \u226b G.map g) = \u2191(e X Y).symm f \u226b g\n\u22a2 f \u226b g = f \u226b \u2191(e X (F_obj X)) (\ud835\udfd9 (F_obj X)) \u226b G.map (\u2191(e X Y).symm g)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX' X : C\nY : D\nf : X' \u27f6 X\ng : X \u27f6 G.obj Y\nthis : \u2200 {X : C} {Y Y' : D} (f : X \u27f6 G.obj Y) (g : Y \u27f6 Y'), \u2191(e X Y').symm (f \u226b G.map g) = \u2191(e X Y).symm f \u226b g\n\u22a2 g = \u2191(e X (F_obj X)) (\ud835\udfd9 (F_obj X)) \u226b G.map (\u2191(e X Y).symm g)\n[PROOFSTEP]\nrw [\u2190 he]\n[GOAL]\ncase e_a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nF_obj : C \u2192 D\ne : (X : C) \u2192 (Y : D) \u2192 (F_obj X \u27f6 Y) \u2243 (X \u27f6 G.obj Y)\nhe : \u2200 (X : C) (Y Y' : D) (g : Y \u27f6 Y') (h : F_obj X \u27f6 Y), \u2191(e X Y') (h \u226b g) = \u2191(e X Y) h \u226b G.map g\nX' X : C\nY : D\nf : X' \u27f6 X\ng : X \u27f6 G.obj Y\nthis : \u2200 {X : C} {Y Y' : D} (f : X \u27f6 G.obj Y) (g : Y \u27f6 Y'), \u2191(e X Y').symm (f \u226b G.map g) = \u2191(e X Y).symm f \u226b g\n\u22a2 g = \u2191(e X Y) (\ud835\udfd9 (F_obj X) \u226b \u2191(e X Y).symm g)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\nX' X : C\nY : D\nf : X' \u27f6 X\ng : X \u27f6 G_obj Y\n\u22a2 F.map f \u226b \u2191(e X Y).symm g = \u2191(e X' Y).symm (f \u226b g)\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\nX' X : C\nY : D\nf : X' \u27f6 X\ng : X \u27f6 G_obj Y\n\u22a2 F.map f \u226b \u2191(e X Y).symm g = \u2191(e X' Y).symm (f \u226b g)\n[PROOFSTEP]\nrw [Equiv.eq_symm_apply, he]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\nX' X : C\nY : D\nf : X' \u27f6 X\ng : X \u27f6 G_obj Y\n\u22a2 f \u226b \u2191(e X Y) (\u2191(e X Y).symm g) = f \u226b g\n[PROOFSTEP]\nsimp\n -- #align category_theory.adjunction.he' category_theory.adjunction.he'\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\nY Y' Y'' : D\ng : Y \u27f6 Y'\ng' : Y' \u27f6 Y''\n\u22a2 { obj := G_obj, map := fun {Y Y'} g => \u2191(e (G_obj Y) Y') (\u2191(e (G_obj Y) Y).symm (\ud835\udfd9 (G_obj Y)) \u226b g) }.map (g \u226b g') =\n { obj := G_obj, map := fun {Y Y'} g => \u2191(e (G_obj Y) Y') (\u2191(e (G_obj Y) Y).symm (\ud835\udfd9 (G_obj Y)) \u226b g) }.map g \u226b\n { obj := G_obj, map := fun {Y Y'} g => \u2191(e (G_obj Y) Y') (\u2191(e (G_obj Y) Y).symm (\ud835\udfd9 (G_obj Y)) \u226b g) }.map g'\n[PROOFSTEP]\nrw [\u2190 Equiv.eq_symm_apply, \u2190 he'' e he, Equiv.symm_apply_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\nY Y' Y'' : D\ng : Y \u27f6 Y'\ng' : Y' \u27f6 Y''\n\u22a2 \u2191(e (G_obj Y) Y).symm (\ud835\udfd9 (G_obj Y)) \u226b g \u226b g' =\n F.map ({ obj := G_obj, map := fun {Y Y'} g => \u2191(e (G_obj Y) Y') (\u2191(e (G_obj Y) Y).symm (\ud835\udfd9 (G_obj Y)) \u226b g) }.map g) \u226b\n \u2191(e (G_obj Y') Y').symm (\ud835\udfd9 (G_obj Y')) \u226b g'\n[PROOFSTEP]\nconv =>\n rhs\n rw [\u2190 assoc, he'' e he, comp_id, Equiv.symm_apply_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\nY Y' Y'' : D\ng : Y \u27f6 Y'\ng' : Y' \u27f6 Y''\n| \u2191(e (G_obj Y) Y).symm (\ud835\udfd9 (G_obj Y)) \u226b g \u226b g' =\n F.map ({ obj := G_obj, map := fun {Y Y'} g => \u2191(e (G_obj Y) Y') (\u2191(e (G_obj Y) Y).symm (\ud835\udfd9 (G_obj Y)) \u226b g) }.map g) \u226b\n \u2191(e (G_obj Y') Y').symm (\ud835\udfd9 (G_obj Y')) \u226b g'\n[PROOFSTEP]\n rhs\n rw [\u2190 assoc, he'' e he, comp_id, Equiv.symm_apply_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\nY Y' Y'' : D\ng : Y \u27f6 Y'\ng' : Y' \u27f6 Y''\n| \u2191(e (G_obj Y) Y).symm (\ud835\udfd9 (G_obj Y)) \u226b g \u226b g' =\n F.map ({ obj := G_obj, map := fun {Y Y'} g => \u2191(e (G_obj Y) Y') (\u2191(e (G_obj Y) Y).symm (\ud835\udfd9 (G_obj Y)) \u226b g) }.map g) \u226b\n \u2191(e (G_obj Y') Y').symm (\ud835\udfd9 (G_obj Y')) \u226b g'\n[PROOFSTEP]\n rhs\n rw [\u2190 assoc, he'' e he, comp_id, Equiv.symm_apply_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\nY Y' Y'' : D\ng : Y \u27f6 Y'\ng' : Y' \u27f6 Y''\n| \u2191(e (G_obj Y) Y).symm (\ud835\udfd9 (G_obj Y)) \u226b g \u226b g' =\n F.map ({ obj := G_obj, map := fun {Y Y'} g => \u2191(e (G_obj Y) Y') (\u2191(e (G_obj Y) Y).symm (\ud835\udfd9 (G_obj Y)) \u226b g) }.map g) \u226b\n \u2191(e (G_obj Y') Y').symm (\ud835\udfd9 (G_obj Y')) \u226b g'\n[PROOFSTEP]\nrhs\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\nY Y' Y'' : D\ng : Y \u27f6 Y'\ng' : Y' \u27f6 Y''\n| F.map ({ obj := G_obj, map := fun {Y Y'} g => \u2191(e (G_obj Y) Y') (\u2191(e (G_obj Y) Y).symm (\ud835\udfd9 (G_obj Y)) \u226b g) }.map g) \u226b\n \u2191(e (G_obj Y') Y').symm (\ud835\udfd9 (G_obj Y')) \u226b g'\n[PROOFSTEP]\nrw [\u2190 assoc, he'' e he, comp_id, Equiv.symm_apply_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\nY Y' Y'' : D\ng : Y \u27f6 Y'\ng' : Y' \u27f6 Y''\n\u22a2 \u2191(e (G_obj Y) Y).symm (\ud835\udfd9 (G_obj Y)) \u226b g \u226b g' = (\u2191(e (G_obj Y) Y).symm (\ud835\udfd9 (G_obj Y)) \u226b g) \u226b g'\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\n\u22a2 \u2200 {X' X : C} {Y : D} (f : X' \u27f6 X) (g : X \u27f6 (rightAdjointOfEquiv e he).obj Y),\n \u2191(e X' Y).symm (f \u226b g) = F.map f \u226b \u2191(e X Y).symm g\n[PROOFSTEP]\nintro X X' Y f g\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\nX X' : C\nY : D\nf : X \u27f6 X'\ng : X' \u27f6 (rightAdjointOfEquiv e he).obj Y\n\u22a2 \u2191(e X Y).symm (f \u226b g) = F.map f \u226b \u2191(e X' Y).symm g\n[PROOFSTEP]\nrw [Equiv.symm_apply_eq]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\nX X' : C\nY : D\nf : X \u27f6 X'\ng : X' \u27f6 (rightAdjointOfEquiv e he).obj Y\n\u22a2 f \u226b g = \u2191(e X Y) (F.map f \u226b \u2191(e X' Y).symm g)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\nX X' : C\nY : D\nf : X \u27f6 X'\ng : X' \u27f6 (rightAdjointOfEquiv e he).obj Y\n\u22a2 f \u226b g = \u2191(e X Y) (F.map f \u226b \u2191(e X' Y).symm g)\n[PROOFSTEP]\nrw [he]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\nX X' : C\nY : D\nf : X \u27f6 X'\ng : X' \u27f6 (rightAdjointOfEquiv e he).obj Y\n\u22a2 f \u226b g = f \u226b \u2191(e X' Y) (\u2191(e X' Y).symm g)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\n\u22a2 \u2200 {X : C} {Y Y' : D} (f : F.obj X \u27f6 Y) (g : Y \u27f6 Y'), \u2191(e X Y') (f \u226b g) = \u2191(e X Y) f \u226b (rightAdjointOfEquiv e he).map g\n[PROOFSTEP]\nintro X Y Y' g h\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nG : D \u2964 C\nG_obj : D \u2192 C\ne : (X : C) \u2192 (Y : D) \u2192 (F.obj X \u27f6 Y) \u2243 (X \u27f6 G_obj Y)\nhe : \u2200 (X' X : C) (Y : D) (f : X' \u27f6 X) (g : F.obj X \u27f6 Y), \u2191(e X' Y) (F.map f \u226b g) = f \u226b \u2191(e X Y) g\nX : C\nY Y' : D\ng : F.obj X \u27f6 Y\nh : Y \u27f6 Y'\n\u22a2 \u2191(e X Y') (g \u226b h) = \u2191(e X Y) g \u226b (rightAdjointOfEquiv e he).map h\n[PROOFSTEP]\nerw [\u2190 he, Equiv.apply_eq_iff_eq, \u2190 assoc, he'' e he, comp_id, Equiv.symm_apply_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : C \u224c D\n\u22a2 whiskerRight (unit e) e.functor \u226b\n (Functor.associator e.functor e.inverse e.functor).hom \u226b whiskerLeft e.functor (counit e) =\n NatTrans.id (\ud835\udfed C \u22d9 e.functor)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : C \u224c D\nx\u271d : C\n\u22a2 NatTrans.app\n (whiskerRight (unit e) e.functor \u226b\n (Functor.associator e.functor e.inverse e.functor).hom \u226b whiskerLeft e.functor (counit e))\n x\u271d =\n NatTrans.app (NatTrans.id (\ud835\udfed C \u22d9 e.functor)) x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : C \u224c D\nx\u271d : C\n\u22a2 e.functor.map (NatTrans.app (unit e) x\u271d) \u226b\n \ud835\udfd9 (e.functor.obj (e.inverse.obj (e.functor.obj x\u271d))) \u226b NatTrans.app (counit e) (e.functor.obj x\u271d) =\n \ud835\udfd9 (e.functor.obj x\u271d)\n[PROOFSTEP]\nsimp only [id_comp]\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : C \u224c D\nx\u271d : C\n\u22a2 e.functor.map (NatTrans.app (unit e) x\u271d) \u226b NatTrans.app (counit e) (e.functor.obj x\u271d) = \ud835\udfd9 (e.functor.obj x\u271d)\n[PROOFSTEP]\nexact e.functor_unit_comp _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : C \u224c D\n\u22a2 whiskerLeft e.inverse (unit e) \u226b\n (Functor.associator e.inverse e.functor e.inverse).inv \u226b whiskerRight (counit e) e.inverse =\n NatTrans.id (e.inverse \u22d9 \ud835\udfed C)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : C \u224c D\nx\u271d : D\n\u22a2 NatTrans.app\n (whiskerLeft e.inverse (unit e) \u226b\n (Functor.associator e.inverse e.functor e.inverse).inv \u226b whiskerRight (counit e) e.inverse)\n x\u271d =\n NatTrans.app (NatTrans.id (e.inverse \u22d9 \ud835\udfed C)) x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : C \u224c D\nx\u271d : D\n\u22a2 NatTrans.app (unit e) (e.inverse.obj x\u271d) \u226b\n \ud835\udfd9 (e.inverse.obj (e.functor.obj (e.inverse.obj x\u271d))) \u226b e.inverse.map (NatTrans.app (counit e) x\u271d) =\n \ud835\udfd9 (e.inverse.obj x\u271d)\n[PROOFSTEP]\nsimp only [id_comp]\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : C \u224c D\nx\u271d : D\n\u22a2 NatTrans.app (unit e) (e.inverse.obj x\u271d) \u226b e.inverse.map (NatTrans.app (counit e) x\u271d) = \ud835\udfd9 (e.inverse.obj x\u271d)\n[PROOFSTEP]\nexact e.unit_inverse_comp _\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Adjunction.Basic", "llama_tokens": 23154, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.44701485974316446}} {"text": "[GOAL]\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedRing \u03b2\nhn : IsAbsoluteValue norm\nf : CauSeq \u03b2 norm\ninst\u271d : IsComplete \u03b2 norm\n\u22a2 \u2200 (s : Set \u03b2), IsOpen s \u2192 lim f \u2208 s \u2192 \u2191f \u207b\u00b9' s \u2208 atTop\n[PROOFSTEP]\nintro s os lfs\n[GOAL]\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedRing \u03b2\nhn : IsAbsoluteValue norm\nf : CauSeq \u03b2 norm\ninst\u271d : IsComplete \u03b2 norm\ns : Set \u03b2\nos : IsOpen s\nlfs : lim f \u2208 s\n\u22a2 \u2191f \u207b\u00b9' s \u2208 atTop\n[PROOFSTEP]\nsuffices \u2203 a : \u2115, \u2200 b : \u2115, b \u2265 a \u2192 f b \u2208 s by simpa using this\n[GOAL]\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedRing \u03b2\nhn : IsAbsoluteValue norm\nf : CauSeq \u03b2 norm\ninst\u271d : IsComplete \u03b2 norm\ns : Set \u03b2\nos : IsOpen s\nlfs : lim f \u2208 s\nthis : \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 \u2191f b \u2208 s\n\u22a2 \u2191f \u207b\u00b9' s \u2208 atTop\n[PROOFSTEP]\nsimpa using this\n[GOAL]\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedRing \u03b2\nhn : IsAbsoluteValue norm\nf : CauSeq \u03b2 norm\ninst\u271d : IsComplete \u03b2 norm\ns : Set \u03b2\nos : IsOpen s\nlfs : lim f \u2208 s\n\u22a2 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 \u2191f b \u2208 s\n[PROOFSTEP]\nrcases Metric.isOpen_iff.1 os _ lfs with \u27e8\u03b5, \u27e8h\u03b5, h\u03b5s\u27e9\u27e9\n[GOAL]\ncase intro.intro\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedRing \u03b2\nhn : IsAbsoluteValue norm\nf : CauSeq \u03b2 norm\ninst\u271d : IsComplete \u03b2 norm\ns : Set \u03b2\nos : IsOpen s\nlfs : lim f \u2208 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : Metric.ball (lim f) \u03b5 \u2286 s\n\u22a2 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 \u2191f b \u2208 s\n[PROOFSTEP]\ncases' Setoid.symm (CauSeq.equiv_lim f) _ h\u03b5 with N hN\n[GOAL]\ncase intro.intro.intro\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedRing \u03b2\nhn : IsAbsoluteValue norm\nf : CauSeq \u03b2 norm\ninst\u271d : IsComplete \u03b2 norm\ns : Set \u03b2\nos : IsOpen s\nlfs : lim f \u2208 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : Metric.ball (lim f) \u03b5 \u2286 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2016\u2191(const norm (lim f) - f) j\u2016 < \u03b5\n\u22a2 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 \u2191f b \u2208 s\n[PROOFSTEP]\nexists N\n[GOAL]\ncase intro.intro.intro\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedRing \u03b2\nhn : IsAbsoluteValue norm\nf : CauSeq \u03b2 norm\ninst\u271d : IsComplete \u03b2 norm\ns : Set \u03b2\nos : IsOpen s\nlfs : lim f \u2208 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : Metric.ball (lim f) \u03b5 \u2286 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2016\u2191(const norm (lim f) - f) j\u2016 < \u03b5\n\u22a2 \u2200 (b : \u2115), b \u2265 N \u2192 \u2191f b \u2208 s\n[PROOFSTEP]\nintro b hb\n[GOAL]\ncase intro.intro.intro\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedRing \u03b2\nhn : IsAbsoluteValue norm\nf : CauSeq \u03b2 norm\ninst\u271d : IsComplete \u03b2 norm\ns : Set \u03b2\nos : IsOpen s\nlfs : lim f \u2208 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : Metric.ball (lim f) \u03b5 \u2286 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2016\u2191(const norm (lim f) - f) j\u2016 < \u03b5\nb : \u2115\nhb : b \u2265 N\n\u22a2 \u2191f b \u2208 s\n[PROOFSTEP]\napply h\u03b5s\n[GOAL]\ncase intro.intro.intro.a\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedRing \u03b2\nhn : IsAbsoluteValue norm\nf : CauSeq \u03b2 norm\ninst\u271d : IsComplete \u03b2 norm\ns : Set \u03b2\nos : IsOpen s\nlfs : lim f \u2208 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : Metric.ball (lim f) \u03b5 \u2286 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2016\u2191(const norm (lim f) - f) j\u2016 < \u03b5\nb : \u2115\nhb : b \u2265 N\n\u22a2 \u2191f b \u2208 Metric.ball (lim f) \u03b5\n[PROOFSTEP]\ndsimp [Metric.ball]\n[GOAL]\ncase intro.intro.intro.a\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedRing \u03b2\nhn : IsAbsoluteValue norm\nf : CauSeq \u03b2 norm\ninst\u271d : IsComplete \u03b2 norm\ns : Set \u03b2\nos : IsOpen s\nlfs : lim f \u2208 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : Metric.ball (lim f) \u03b5 \u2286 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2016\u2191(const norm (lim f) - f) j\u2016 < \u03b5\nb : \u2115\nhb : b \u2265 N\n\u22a2 dist (\u2191f b) (lim f) < \u03b5\n[PROOFSTEP]\nrw [dist_comm, dist_eq_norm]\n[GOAL]\ncase intro.intro.intro.a\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedRing \u03b2\nhn : IsAbsoluteValue norm\nf : CauSeq \u03b2 norm\ninst\u271d : IsComplete \u03b2 norm\ns : Set \u03b2\nos : IsOpen s\nlfs : lim f \u2208 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : Metric.ball (lim f) \u03b5 \u2286 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2016\u2191(const norm (lim f) - f) j\u2016 < \u03b5\nb : \u2115\nhb : b \u2265 N\n\u22a2 \u2016lim f - \u2191f b\u2016 < \u03b5\n[PROOFSTEP]\nsolve_by_elim\n[GOAL]\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : \u2115 \u2192 \u03b2\nhf : CauchySeq f\n\u22a2 IsCauSeq norm f\n[PROOFSTEP]\ncases' cauchy_iff.1 hf with hf1 hf2\n[GOAL]\ncase intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : \u2115 \u2192 \u03b2\nhf : CauchySeq f\nhf1 : NeBot (map f atTop)\nhf2 : \u2200 (s : Set (\u03b2 \u00d7 \u03b2)), s \u2208 uniformity \u03b2 \u2192 \u2203 t, t \u2208 map f atTop \u2227 t \u00d7\u02e2 t \u2286 s\n\u22a2 IsCauSeq norm f\n[PROOFSTEP]\nintro \u03b5 h\u03b5\n[GOAL]\ncase intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : \u2115 \u2192 \u03b2\nhf : CauchySeq f\nhf1 : NeBot (map f atTop)\nhf2 : \u2200 (s : Set (\u03b2 \u00d7 \u03b2)), s \u2208 uniformity \u03b2 \u2192 \u2203 t, t \u2208 map f atTop \u2227 t \u00d7\u02e2 t \u2286 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016f j - f i\u2016 < \u03b5\n[PROOFSTEP]\nrcases hf2 {x | dist x.1 x.2 < \u03b5} (dist_mem_uniformity h\u03b5) with \u27e8t, \u27e8ht, htsub\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : \u2115 \u2192 \u03b2\nhf : CauchySeq f\nhf1 : NeBot (map f atTop)\nhf2 : \u2200 (s : Set (\u03b2 \u00d7 \u03b2)), s \u2208 uniformity \u03b2 \u2192 \u2203 t, t \u2208 map f atTop \u2227 t \u00d7\u02e2 t \u2286 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nt : Set \u03b2\nht : t \u2208 map f atTop\nhtsub : t \u00d7\u02e2 t \u2286 {x | dist x.fst x.snd < \u03b5}\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016f j - f i\u2016 < \u03b5\n[PROOFSTEP]\nsimp at ht \n[GOAL]\ncase intro.intro.intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : \u2115 \u2192 \u03b2\nhf : CauchySeq f\nhf1 : NeBot (map f atTop)\nhf2 : \u2200 (s : Set (\u03b2 \u00d7 \u03b2)), s \u2208 uniformity \u03b2 \u2192 \u2203 t, t \u2208 map f atTop \u2227 t \u00d7\u02e2 t \u2286 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nt : Set \u03b2\nhtsub : t \u00d7\u02e2 t \u2286 {x | dist x.fst x.snd < \u03b5}\nht : \u2203 a, \u2200 (b : \u2115), a \u2264 b \u2192 f b \u2208 t\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016f j - f i\u2016 < \u03b5\n[PROOFSTEP]\ncases' ht with N hN\n[GOAL]\ncase intro.intro.intro.intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : \u2115 \u2192 \u03b2\nhf : CauchySeq f\nhf1 : NeBot (map f atTop)\nhf2 : \u2200 (s : Set (\u03b2 \u00d7 \u03b2)), s \u2208 uniformity \u03b2 \u2192 \u2203 t, t \u2208 map f atTop \u2227 t \u00d7\u02e2 t \u2286 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nt : Set \u03b2\nhtsub : t \u00d7\u02e2 t \u2286 {x | dist x.fst x.snd < \u03b5}\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 f b \u2208 t\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016f j - f i\u2016 < \u03b5\n[PROOFSTEP]\nexists N\n[GOAL]\ncase intro.intro.intro.intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : \u2115 \u2192 \u03b2\nhf : CauchySeq f\nhf1 : NeBot (map f atTop)\nhf2 : \u2200 (s : Set (\u03b2 \u00d7 \u03b2)), s \u2208 uniformity \u03b2 \u2192 \u2203 t, t \u2208 map f atTop \u2227 t \u00d7\u02e2 t \u2286 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nt : Set \u03b2\nhtsub : t \u00d7\u02e2 t \u2286 {x | dist x.fst x.snd < \u03b5}\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 f b \u2208 t\n\u22a2 \u2200 (j : \u2115), j \u2265 N \u2192 \u2016f j - f N\u2016 < \u03b5\n[PROOFSTEP]\nintro j hj\n[GOAL]\ncase intro.intro.intro.intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : \u2115 \u2192 \u03b2\nhf : CauchySeq f\nhf1 : NeBot (map f atTop)\nhf2 : \u2200 (s : Set (\u03b2 \u00d7 \u03b2)), s \u2208 uniformity \u03b2 \u2192 \u2203 t, t \u2208 map f atTop \u2227 t \u00d7\u02e2 t \u2286 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nt : Set \u03b2\nhtsub : t \u00d7\u02e2 t \u2286 {x | dist x.fst x.snd < \u03b5}\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 f b \u2208 t\nj : \u2115\nhj : j \u2265 N\n\u22a2 \u2016f j - f N\u2016 < \u03b5\n[PROOFSTEP]\nrw [\u2190 dist_eq_norm]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : \u2115 \u2192 \u03b2\nhf : CauchySeq f\nhf1 : NeBot (map f atTop)\nhf2 : \u2200 (s : Set (\u03b2 \u00d7 \u03b2)), s \u2208 uniformity \u03b2 \u2192 \u2203 t, t \u2208 map f atTop \u2227 t \u00d7\u02e2 t \u2286 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nt : Set \u03b2\nhtsub : t \u00d7\u02e2 t \u2286 {x | dist x.fst x.snd < \u03b5}\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 f b \u2208 t\nj : \u2115\nhj : j \u2265 N\n\u22a2 dist (f j) (f N) < \u03b5\n[PROOFSTEP]\napply @htsub (f j, f N)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : \u2115 \u2192 \u03b2\nhf : CauchySeq f\nhf1 : NeBot (map f atTop)\nhf2 : \u2200 (s : Set (\u03b2 \u00d7 \u03b2)), s \u2208 uniformity \u03b2 \u2192 \u2203 t, t \u2208 map f atTop \u2227 t \u00d7\u02e2 t \u2286 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nt : Set \u03b2\nhtsub : t \u00d7\u02e2 t \u2286 {x | dist x.fst x.snd < \u03b5}\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 f b \u2208 t\nj : \u2115\nhj : j \u2265 N\n\u22a2 (f j, f N) \u2208 t \u00d7\u02e2 t\n[PROOFSTEP]\napply Set.mk_mem_prod\n[GOAL]\ncase intro.intro.intro.intro.ha\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : \u2115 \u2192 \u03b2\nhf : CauchySeq f\nhf1 : NeBot (map f atTop)\nhf2 : \u2200 (s : Set (\u03b2 \u00d7 \u03b2)), s \u2208 uniformity \u03b2 \u2192 \u2203 t, t \u2208 map f atTop \u2227 t \u00d7\u02e2 t \u2286 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nt : Set \u03b2\nhtsub : t \u00d7\u02e2 t \u2286 {x | dist x.fst x.snd < \u03b5}\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 f b \u2208 t\nj : \u2115\nhj : j \u2265 N\n\u22a2 f j \u2208 t\n[PROOFSTEP]\nsolve_by_elim [le_refl]\n[GOAL]\ncase intro.intro.intro.intro.hb\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : \u2115 \u2192 \u03b2\nhf : CauchySeq f\nhf1 : NeBot (map f atTop)\nhf2 : \u2200 (s : Set (\u03b2 \u00d7 \u03b2)), s \u2208 uniformity \u03b2 \u2192 \u2203 t, t \u2208 map f atTop \u2227 t \u00d7\u02e2 t \u2286 s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nt : Set \u03b2\nhtsub : t \u00d7\u02e2 t \u2286 {x | dist x.fst x.snd < \u03b5}\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 f b \u2208 t\nj : \u2115\nhj : j \u2265 N\n\u22a2 f N \u2208 t\n[PROOFSTEP]\nsolve_by_elim [le_refl]\n[GOAL]\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\n\u22a2 CauchySeq \u2191f\n[PROOFSTEP]\nrefine' cauchy_iff.2 \u27e8by infer_instance, fun s hs => _\u27e9\n[GOAL]\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\n\u22a2 NeBot (map (\u2191f) atTop)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u22a2 \u2203 t, t \u2208 map (\u2191f) atTop \u2227 t \u00d7\u02e2 t \u2286 s\n[PROOFSTEP]\nrcases mem_uniformity_dist.1 hs with \u27e8\u03b5, \u27e8h\u03b5, h\u03b5s\u27e9\u27e9\n[GOAL]\ncase intro.intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : \u2200 {a b : \u03b2}, dist a b < \u03b5 \u2192 (a, b) \u2208 s\n\u22a2 \u2203 t, t \u2208 map (\u2191f) atTop \u2227 t \u00d7\u02e2 t \u2286 s\n[PROOFSTEP]\ncases' CauSeq.cauchy\u2082 f h\u03b5 with N hN\n[GOAL]\ncase intro.intro.intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : \u2200 {a b : \u03b2}, dist a b < \u03b5 \u2192 (a, b) \u2208 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 \u2016\u2191f j - \u2191f k\u2016 < \u03b5\n\u22a2 \u2203 t, t \u2208 map (\u2191f) atTop \u2227 t \u00d7\u02e2 t \u2286 s\n[PROOFSTEP]\nexists {n | n \u2265 N}.image f\n[GOAL]\ncase intro.intro.intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : \u2200 {a b : \u03b2}, dist a b < \u03b5 \u2192 (a, b) \u2208 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 \u2016\u2191f j - \u2191f k\u2016 < \u03b5\n\u22a2 \u2191f '' {n | n \u2265 N} \u2208 map (\u2191f) atTop \u2227 (\u2191f '' {n | n \u2265 N}) \u00d7\u02e2 (\u2191f '' {n | n \u2265 N}) \u2286 s\n[PROOFSTEP]\nsimp only [exists_prop, mem_atTop_sets, mem_map, mem_image, ge_iff_le, mem_setOf_eq]\n[GOAL]\ncase intro.intro.intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : \u2200 {a b : \u03b2}, dist a b < \u03b5 \u2192 (a, b) \u2208 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 \u2016\u2191f j - \u2191f k\u2016 < \u03b5\n\u22a2 (\u2203 a, \u2200 (b : \u2115), a \u2264 b \u2192 b \u2208 \u2191f \u207b\u00b9' (\u2191f '' {n | N \u2264 n})) \u2227 (\u2191f '' {n | N \u2264 n}) \u00d7\u02e2 (\u2191f '' {n | N \u2264 n}) \u2286 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.intro.left\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : \u2200 {a b : \u03b2}, dist a b < \u03b5 \u2192 (a, b) \u2208 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 \u2016\u2191f j - \u2191f k\u2016 < \u03b5\n\u22a2 \u2203 a, \u2200 (b : \u2115), a \u2264 b \u2192 b \u2208 \u2191f \u207b\u00b9' (\u2191f '' {n | N \u2264 n})\n[PROOFSTEP]\nexists N\n[GOAL]\ncase intro.intro.intro.left\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : \u2200 {a b : \u03b2}, dist a b < \u03b5 \u2192 (a, b) \u2208 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 \u2016\u2191f j - \u2191f k\u2016 < \u03b5\n\u22a2 \u2200 (b : \u2115), N \u2264 b \u2192 b \u2208 \u2191f \u207b\u00b9' (\u2191f '' {n | N \u2264 n})\n[PROOFSTEP]\nintro b hb\n[GOAL]\ncase intro.intro.intro.left\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : \u2200 {a b : \u03b2}, dist a b < \u03b5 \u2192 (a, b) \u2208 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 \u2016\u2191f j - \u2191f k\u2016 < \u03b5\nb : \u2115\nhb : N \u2264 b\n\u22a2 b \u2208 \u2191f \u207b\u00b9' (\u2191f '' {n | N \u2264 n})\n[PROOFSTEP]\nexists b\n[GOAL]\ncase intro.intro.intro.right\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : \u2200 {a b : \u03b2}, dist a b < \u03b5 \u2192 (a, b) \u2208 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 \u2016\u2191f j - \u2191f k\u2016 < \u03b5\n\u22a2 (\u2191f '' {n | N \u2264 n}) \u00d7\u02e2 (\u2191f '' {n | N \u2264 n}) \u2286 s\n[PROOFSTEP]\nrintro \u27e8a, b\u27e9 \u27e8\u27e8a', \u27e8ha'1, ha'2\u27e9\u27e9, \u27e8b', \u27e8hb'1, hb'2\u27e9\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.right.mk.intro.intro.intro.intro.intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : \u2200 {a b : \u03b2}, dist a b < \u03b5 \u2192 (a, b) \u2208 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 \u2016\u2191f j - \u2191f k\u2016 < \u03b5\na b : \u03b2\na' : \u2115\nha'1 : a' \u2208 {n | N \u2264 n}\nha'2 : \u2191f a' = (a, b).fst\nb' : \u2115\nhb'1 : b' \u2208 {n | N \u2264 n}\nhb'2 : \u2191f b' = (a, b).snd\n\u22a2 (a, b) \u2208 s\n[PROOFSTEP]\ndsimp at ha'1 ha'2 hb'1 hb'2 \n[GOAL]\ncase intro.intro.intro.right.mk.intro.intro.intro.intro.intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : \u2200 {a b : \u03b2}, dist a b < \u03b5 \u2192 (a, b) \u2208 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 \u2016\u2191f j - \u2191f k\u2016 < \u03b5\na b : \u03b2\na' : \u2115\nha'1 : N \u2264 a'\nha'2 : \u2191f a' = a\nb' : \u2115\nhb'1 : N \u2264 b'\nhb'2 : \u2191f b' = b\n\u22a2 (a, b) \u2208 s\n[PROOFSTEP]\nrw [\u2190 ha'2, \u2190 hb'2]\n[GOAL]\ncase intro.intro.intro.right.mk.intro.intro.intro.intro.intro\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : \u2200 {a b : \u03b2}, dist a b < \u03b5 \u2192 (a, b) \u2208 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 \u2016\u2191f j - \u2191f k\u2016 < \u03b5\na b : \u03b2\na' : \u2115\nha'1 : N \u2264 a'\nha'2 : \u2191f a' = a\nb' : \u2115\nhb'1 : N \u2264 b'\nhb'2 : \u2191f b' = b\n\u22a2 (\u2191f a', \u2191f b') \u2208 s\n[PROOFSTEP]\napply h\u03b5s\n[GOAL]\ncase intro.intro.intro.right.mk.intro.intro.intro.intro.intro.a\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : \u2200 {a b : \u03b2}, dist a b < \u03b5 \u2192 (a, b) \u2208 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 \u2016\u2191f j - \u2191f k\u2016 < \u03b5\na b : \u03b2\na' : \u2115\nha'1 : N \u2264 a'\nha'2 : \u2191f a' = a\nb' : \u2115\nhb'1 : N \u2264 b'\nhb'2 : \u2191f b' = b\n\u22a2 dist (\u2191f a') (\u2191f b') < \u03b5\n[PROOFSTEP]\nrw [dist_eq_norm]\n[GOAL]\ncase intro.intro.intro.right.mk.intro.intro.intro.intro.intro.a\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : \u2200 {a b : \u03b2}, dist a b < \u03b5 \u2192 (a, b) \u2208 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 \u2016\u2191f j - \u2191f k\u2016 < \u03b5\na b : \u03b2\na' : \u2115\nha'1 : N \u2264 a'\nha'2 : \u2191f a' = a\nb' : \u2115\nhb'1 : N \u2264 b'\nhb'2 : \u2191f b' = b\n\u22a2 \u2016\u2191f a' - \u2191f b'\u2016 < \u03b5\n[PROOFSTEP]\napply hN\n[GOAL]\ncase intro.intro.intro.right.mk.intro.intro.intro.intro.intro.a.x\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : \u2200 {a b : \u03b2}, dist a b < \u03b5 \u2192 (a, b) \u2208 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 \u2016\u2191f j - \u2191f k\u2016 < \u03b5\na b : \u03b2\na' : \u2115\nha'1 : N \u2264 a'\nha'2 : \u2191f a' = a\nb' : \u2115\nhb'1 : N \u2264 b'\nhb'2 : \u2191f b' = b\n\u22a2 a' \u2265 N\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.right.mk.intro.intro.intro.intro.intro.a.x\n\u03b2 : Type v\ninst\u271d : NormedField \u03b2\nf : CauSeq \u03b2 norm\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 uniformity \u03b2\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5s : \u2200 {a b : \u03b2}, dist a b < \u03b5 \u2192 (a, b) \u2208 s\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 \u2016\u2191f j - \u2191f k\u2016 < \u03b5\na b : \u03b2\na' : \u2115\nha'1 : N \u2264 a'\nha'2 : \u2191f a' = a\nb' : \u2115\nhb'1 : N \u2264 b'\nhb'2 : \u2191f b' = b\n\u22a2 b' \u2265 N\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedField \u03b2\ninst\u271d : CauSeq.IsComplete \u03b2 norm\n\u22a2 CompleteSpace \u03b2\n[PROOFSTEP]\napply complete_of_cauchySeq_tendsto\n[GOAL]\ncase a\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedField \u03b2\ninst\u271d : CauSeq.IsComplete \u03b2 norm\n\u22a2 \u2200 (u : \u2115 \u2192 \u03b2), CauchySeq u \u2192 \u2203 a, Tendsto u atTop (\ud835\udcdd a)\n[PROOFSTEP]\nintro u hu\n[GOAL]\ncase a\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedField \u03b2\ninst\u271d : CauSeq.IsComplete \u03b2 norm\nu : \u2115 \u2192 \u03b2\nhu : CauchySeq u\n\u22a2 \u2203 a, Tendsto u atTop (\ud835\udcdd a)\n[PROOFSTEP]\nhave C : IsCauSeq norm u := isCauSeq_iff_cauchySeq.2 hu\n[GOAL]\ncase a\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedField \u03b2\ninst\u271d : CauSeq.IsComplete \u03b2 norm\nu : \u2115 \u2192 \u03b2\nhu : CauchySeq u\nC : IsCauSeq norm u\n\u22a2 \u2203 a, Tendsto u atTop (\ud835\udcdd a)\n[PROOFSTEP]\nexists CauSeq.lim \u27e8u, C\u27e9\n[GOAL]\ncase a\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedField \u03b2\ninst\u271d : CauSeq.IsComplete \u03b2 norm\nu : \u2115 \u2192 \u03b2\nhu : CauchySeq u\nC : IsCauSeq norm u\n\u22a2 Tendsto u atTop (\ud835\udcdd (CauSeq.lim { val := u, property := C }))\n[PROOFSTEP]\nrw [Metric.tendsto_atTop]\n[GOAL]\ncase a\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedField \u03b2\ninst\u271d : CauSeq.IsComplete \u03b2 norm\nu : \u2115 \u2192 \u03b2\nhu : CauchySeq u\nC : IsCauSeq norm u\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (CauSeq.lim { val := u, property := C }) < \u03b5\n[PROOFSTEP]\nintro \u03b5 \u03b5pos\n[GOAL]\ncase a\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedField \u03b2\ninst\u271d : CauSeq.IsComplete \u03b2 norm\nu : \u2115 \u2192 \u03b2\nhu : CauchySeq u\nC : IsCauSeq norm u\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (CauSeq.lim { val := u, property := C }) < \u03b5\n[PROOFSTEP]\ncases' (CauSeq.equiv_lim \u27e8u, C\u27e9) _ \u03b5pos with N hN\n[GOAL]\ncase a.intro\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedField \u03b2\ninst\u271d : CauSeq.IsComplete \u03b2 norm\nu : \u2115 \u2192 \u03b2\nhu : CauchySeq u\nC : IsCauSeq norm u\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nN : \u2115\nhN :\n \u2200 (j : \u2115), j \u2265 N \u2192 \u2016\u2191({ val := u, property := C } - CauSeq.const norm (CauSeq.lim { val := u, property := C })) j\u2016 < \u03b5\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (CauSeq.lim { val := u, property := C }) < \u03b5\n[PROOFSTEP]\nexists N\n[GOAL]\ncase a.intro\n\u03b2 : Type v\ninst\u271d\u00b9 : NormedField \u03b2\ninst\u271d : CauSeq.IsComplete \u03b2 norm\nu : \u2115 \u2192 \u03b2\nhu : CauchySeq u\nC : IsCauSeq norm u\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nN : \u2115\nhN :\n \u2200 (j : \u2115), j \u2265 N \u2192 \u2016\u2191({ val := u, property := C } - CauSeq.const norm (CauSeq.lim { val := u, property := C })) j\u2016 < \u03b5\n\u22a2 \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (CauSeq.lim { val := u, property := C }) < \u03b5\n[PROOFSTEP]\nsimpa [dist_eq_norm] using hN\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.CauSeqFilter", "llama_tokens": 9050, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585669110203, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.44695602923165384}} {"text": "[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\n\u22a2 dist a b * dist c d + dist b c * dist d a = dist a c * dist b d\n[PROOFSTEP]\nhave h' : Cospherical ({ a, c, b, d } : Set P) := by rwa [Set.insert_comm c b { d }]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\n\u22a2 Cospherical {a, c, b, d}\n[PROOFSTEP]\nrwa [Set.insert_comm c b { d }]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\n\u22a2 dist a b * dist c d + dist b c * dist d a = dist a c * dist b d\n[PROOFSTEP]\nhave hmul := mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi h' hapc hbpd\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\n\u22a2 dist a b * dist c d + dist b c * dist d a = dist a c * dist b d\n[PROOFSTEP]\nhave hbp := left_dist_ne_zero_of_angle_eq_pi hbpd\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\n\u22a2 dist a b * dist c d + dist b c * dist d a = dist a c * dist b d\n[PROOFSTEP]\nhave h\u2081 : dist c d = dist c p / dist b p * dist a b :=\n by\n rw [dist_mul_of_eq_angle_of_dist_mul b p a c p d, dist_comm a b]\n \u00b7 rw [angle_eq_angle_of_angle_eq_pi_of_angle_eq_pi hbpd hapc, angle_comm]\n all_goals field_simp [mul_comm, hmul]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\n\u22a2 dist c d = dist c p / dist b p * dist a b\n[PROOFSTEP]\nrw [dist_mul_of_eq_angle_of_dist_mul b p a c p d, dist_comm a b]\n[GOAL]\ncase h\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\n\u22a2 \u2220 c p d = \u2220 b p a\n[PROOFSTEP]\nrw [angle_eq_angle_of_angle_eq_pi_of_angle_eq_pi hbpd hapc, angle_comm]\n[GOAL]\ncase hab\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\n\u22a2 dist c p = dist c p / dist b p * dist b p\ncase hcb\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\n\u22a2 dist d p = dist c p / dist b p * dist a p\n[PROOFSTEP]\nall_goals field_simp [mul_comm, hmul]\n[GOAL]\ncase hab\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\n\u22a2 dist c p = dist c p / dist b p * dist b p\n[PROOFSTEP]\nfield_simp [mul_comm, hmul]\n[GOAL]\ncase hcb\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\n\u22a2 dist d p = dist c p / dist b p * dist a p\n[PROOFSTEP]\nfield_simp [mul_comm, hmul]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\nh\u2081 : dist c d = dist c p / dist b p * dist a b\n\u22a2 dist a b * dist c d + dist b c * dist d a = dist a c * dist b d\n[PROOFSTEP]\nhave h\u2082 : dist d a = dist a p / dist b p * dist b c :=\n by\n rw [dist_mul_of_eq_angle_of_dist_mul c p b d p a, dist_comm c b]\n \u00b7 rwa [angle_comm, angle_eq_angle_of_angle_eq_pi_of_angle_eq_pi]; rwa [angle_comm]\n all_goals field_simp [mul_comm, hmul]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\nh\u2081 : dist c d = dist c p / dist b p * dist a b\n\u22a2 dist d a = dist a p / dist b p * dist b c\n[PROOFSTEP]\nrw [dist_mul_of_eq_angle_of_dist_mul c p b d p a, dist_comm c b]\n[GOAL]\ncase h\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\nh\u2081 : dist c d = dist c p / dist b p * dist a b\n\u22a2 \u2220 d p a = \u2220 c p b\n[PROOFSTEP]\nrwa [angle_comm, angle_eq_angle_of_angle_eq_pi_of_angle_eq_pi]\n[GOAL]\ncase h.hbpd\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\nh\u2081 : dist c d = dist c p / dist b p * dist a b\n\u22a2 \u2220 d p b = \u03c0\n[PROOFSTEP]\nrwa [angle_comm]\n[GOAL]\ncase hab\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\nh\u2081 : dist c d = dist c p / dist b p * dist a b\n\u22a2 dist d p = dist a p / dist b p * dist c p\ncase hcb\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\nh\u2081 : dist c d = dist c p / dist b p * dist a b\n\u22a2 dist a p = dist a p / dist b p * dist b p\n[PROOFSTEP]\nall_goals field_simp [mul_comm, hmul]\n[GOAL]\ncase hab\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\nh\u2081 : dist c d = dist c p / dist b p * dist a b\n\u22a2 dist d p = dist a p / dist b p * dist c p\n[PROOFSTEP]\nfield_simp [mul_comm, hmul]\n[GOAL]\ncase hcb\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\nh\u2081 : dist c d = dist c p / dist b p * dist a b\n\u22a2 dist a p = dist a p / dist b p * dist b p\n[PROOFSTEP]\nfield_simp [mul_comm, hmul]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\nh\u2081 : dist c d = dist c p / dist b p * dist a b\nh\u2082 : dist d a = dist a p / dist b p * dist b c\n\u22a2 dist a b * dist c d + dist b c * dist d a = dist a c * dist b d\n[PROOFSTEP]\nhave h\u2083 : dist d p = dist a p * dist c p / dist b p := by field_simp [mul_comm, hmul]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\nh\u2081 : dist c d = dist c p / dist b p * dist a b\nh\u2082 : dist d a = dist a p / dist b p * dist b c\n\u22a2 dist d p = dist a p * dist c p / dist b p\n[PROOFSTEP]\nfield_simp [mul_comm, hmul]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\nh\u2081 : dist c d = dist c p / dist b p * dist a b\nh\u2082 : dist d a = dist a p / dist b p * dist b c\nh\u2083 : dist d p = dist a p * dist c p / dist b p\n\u22a2 dist a b * dist c d + dist b c * dist d a = dist a c * dist b d\n[PROOFSTEP]\nhave h\u2084 : \u2200 x y : \u211d, x * (y * x) = x * x * y := fun x y => by rw [mul_left_comm, mul_comm]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\nh\u2081 : dist c d = dist c p / dist b p * dist a b\nh\u2082 : dist d a = dist a p / dist b p * dist b c\nh\u2083 : dist d p = dist a p * dist c p / dist b p\nx y : \u211d\n\u22a2 x * (y * x) = x * x * y\n[PROOFSTEP]\nrw [mul_left_comm, mul_comm]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_2\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapc : \u2220 a p c = \u03c0\nhbpd : \u2220 b p d = \u03c0\nh' : Cospherical {a, c, b, d}\nhmul : dist a p * dist c p = dist b p * dist d p\nhbp : dist b p \u2260 0\nh\u2081 : dist c d = dist c p / dist b p * dist a b\nh\u2082 : dist d a = dist a p / dist b p * dist b c\nh\u2083 : dist d p = dist a p * dist c p / dist b p\nh\u2084 : \u2200 (x y : \u211d), x * (y * x) = x * x * y\n\u22a2 dist a b * dist c d + dist b c * dist d a = dist a c * dist b d\n[PROOFSTEP]\nfield_simp [h\u2081, h\u2082, dist_eq_add_dist_of_angle_eq_pi hbpd, h\u2083, hbp, dist_comm a b, h\u2084, \u2190 sq,\n dist_sq_mul_dist_add_dist_sq_mul_dist b, hapc]\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Euclidean.Sphere.Ptolemy", "llama_tokens": 5725, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.6513548646660542, "lm_q1q2_score": 0.4467965144346148}} {"text": "[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nm n : \u2115\nh : m \u2264 n\n\u22a2 \u2191(n - m) + \u2191m = \u2191n\n[PROOFSTEP]\nrw [\u2190 cast_add, Nat.sub_add_cancel h]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nh : 0 < 0\n\u22a2 \u2191(0 - 1) = \u21910 - 1\n[PROOFSTEP]\ncases h\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nn : \u2115\nx\u271d : 0 < n + 1\n\u22a2 \u2191(n + 1 - 1) = \u2191(n + 1) - 1\n[PROOFSTEP]\nrw [cast_succ, add_sub_cancel]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nn : \u2115\nx\u271d : 0 < n + 1\n\u22a2 \u2191(n + 1 - 1) = \u2191n\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : AddGroupWithOne R\nn : \u2115\ninst\u271d : AtLeastTwo n\n\u22a2 \u2191(OfNat.ofNat n) = OfNat.ofNat n\n[PROOFSTEP]\nsimpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\n\u22a2 \u21911 = 1\n[PROOFSTEP]\nerw [cast_ofNat, Nat.cast_one]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\n\u22a2 \u2191(-\u21910) = -\u2191\u21910\n[PROOFSTEP]\nerw [cast_zero, neg_zero]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nn : \u2115\n\u22a2 \u2191(-\u2191(n + 1)) = -\u2191\u2191(n + 1)\n[PROOFSTEP]\nerw [cast_ofNat, cast_negSucc]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nn : \u2115\n\u22a2 \u2191(- -[n+1]) = -\u2191-[n+1]\n[PROOFSTEP]\nerw [cast_ofNat, cast_negSucc, neg_neg]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nm n : \u2115\n\u22a2 \u2191(subNatNat m n) = \u2191m - \u2191n\n[PROOFSTEP]\nunfold subNatNat\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nm n : \u2115\n\u22a2 \u2191(match n - m with\n | 0 => ofNat (m - n)\n | succ k => -[k+1]) =\n \u2191m - \u2191n\n[PROOFSTEP]\ncases e : n - m\n[GOAL]\ncase zero\nR : Type u\ninst\u271d : AddGroupWithOne R\nm n : \u2115\ne : n - m = zero\n\u22a2 \u2191(match zero with\n | 0 => ofNat (m - n)\n | succ k => -[k+1]) =\n \u2191m - \u2191n\n[PROOFSTEP]\nsimp only [ofNat_eq_coe]\n[GOAL]\ncase zero\nR : Type u\ninst\u271d : AddGroupWithOne R\nm n : \u2115\ne : n - m = zero\n\u22a2 \u2191\u2191(m - n) = \u2191m - \u2191n\n[PROOFSTEP]\nsimp [e, Nat.le_of_sub_eq_zero e]\n[GOAL]\ncase succ\nR : Type u\ninst\u271d : AddGroupWithOne R\nm n n\u271d : \u2115\ne : n - m = succ n\u271d\n\u22a2 \u2191(match succ n\u271d with\n | 0 => ofNat (m - n)\n | succ k => -[k+1]) =\n \u2191m - \u2191n\n[PROOFSTEP]\nrw [cast_negSucc, Nat.add_one, \u2190 e, Nat.cast_sub <| _root_.le_of_lt <| Nat.lt_of_sub_eq_succ e, neg_sub]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nn : \u2115\n\u22a2 \u2191(negOfNat n) = -\u2191n\n[PROOFSTEP]\nsimp [Int.cast_neg, negOfNat_eq]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nm n : \u2115\n\u22a2 \u2191(\u2191m + \u2191n) = \u2191\u2191m + \u2191\u2191n\n[PROOFSTEP]\nsimp [\u2190 Int.ofNat_add, Nat.cast_add]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nm n : \u2115\n\u22a2 \u2191(\u2191m + -[n+1]) = \u2191\u2191m + \u2191-[n+1]\n[PROOFSTEP]\nerw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nm n : \u2115\n\u22a2 \u2191(-[m+1] + \u2191n) = \u2191-[m+1] + \u2191\u2191n\n[PROOFSTEP]\nerw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_add, add_assoc, eq_neg_add_iff_add_eq, \u2190 Nat.cast_add, \u2190\n Nat.cast_add, Nat.add_comm]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nm n : \u2115\n\u22a2 \u2191-[m + n + 1+1] = \u2191-[m+1] + \u2191-[n+1]\n[PROOFSTEP]\nrw [cast_negSucc, cast_negSucc, cast_negSucc, \u2190 neg_add_rev, \u2190 Nat.cast_add, Nat.add_right_comm m n 1, Nat.add_assoc,\n Nat.add_comm]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nm n : \u2124\n\u22a2 \u2191(m - n) = \u2191m - \u2191n\n[PROOFSTEP]\nsimp [Int.sub_eq_add_neg, sub_eq_add_neg, Int.cast_neg, Int.cast_add]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nn : \u2124\n\u22a2 \u2191(bit1 n) = bit1 \u2191n\n[PROOFSTEP]\nrw [bit1, Int.cast_add, Int.cast_one, cast_bit0]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\nn : \u2124\n\u22a2 bit0 \u2191n + 1 = bit1 \u2191n\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\n\u22a2 \u2191\u21912 = \u21912\n[PROOFSTEP]\nrw [cast_ofNat, Nat.cast_ofNat]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\n\u22a2 \u2191\u21913 = \u21913\n[PROOFSTEP]\nrw [cast_ofNat, Nat.cast_ofNat]\n[GOAL]\nR : Type u\ninst\u271d : AddGroupWithOne R\n\u22a2 \u2191\u21914 = \u21914\n[PROOFSTEP]\nrw [cast_ofNat, Nat.cast_ofNat]\n", "meta": {"mathlib_filename": "Mathlib.Data.Int.Cast.Basic", "llama_tokens": 1986, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872243177518, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.4467804500264719}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\n\u22a2 (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03bb_ (\ud835\udfd9_ C)).hom = (\u03c1_ (\ud835\udfd9_ C)).hom\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\n\u22a2 ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03bb_ (\ud835\udfd9_ C)).hom =\n (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03bb_ (\ud835\udfd9_ C)).hom\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM : Mon_ C\nZ : C\nf : Z \u27f6 M.X\n\u22a2 (M.one \u2297 f) \u226b M.mul = (\u03bb_ Z).hom \u226b f\n[PROOFSTEP]\nrw [\u2190 id_tensor_comp_tensor_id, Category.assoc, M.one_mul, leftUnitor_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM : Mon_ C\nZ : C\nf : Z \u27f6 M.X\n\u22a2 (f \u2297 M.one) \u226b M.mul = (\u03c1_ Z).hom \u226b f\n[PROOFSTEP]\nrw [\u2190 tensor_id_comp_id_tensor, Category.assoc, M.mul_one, rightUnitor_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM : Mon_ C\n\u22a2 (\ud835\udfd9 M.X \u2297 M.mul) \u226b M.mul = (\u03b1_ M.X M.X M.X).inv \u226b (M.mul \u2297 \ud835\udfd9 M.X) \u226b M.mul\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM A\u271d B\u271d : Mon_ C\nf : A\u271d \u27f6 B\u271d\ne : IsIso ((forget C).map f)\n\u22a2 B\u271d.mul \u226b inv f.hom = (inv f.hom \u2297 inv f.hom) \u226b A\u271d.mul\n[PROOFSTEP]\nsimp only [IsIso.comp_inv_eq, Hom.mul_hom, Category.assoc, \u2190 tensor_comp_assoc, IsIso.inv_hom_id, tensor_id,\n Category.id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM A\u271d B\u271d : Mon_ C\nf : A\u271d \u27f6 B\u271d\ne : IsIso ((forget C).map f)\n\u22a2 f \u226b Hom.mk (inv f.hom) = \ud835\udfd9 A\u271d \u2227 Hom.mk (inv f.hom) \u226b f = \ud835\udfd9 B\u271d\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM\u271d M N : Mon_ C\nf : M.X \u2245 N.X\none_f : M.one \u226b f.hom = N.one\nmul_f : M.mul \u226b f.hom = (f.hom \u2297 f.hom) \u226b N.mul\n\u22a2 N.one \u226b f.inv = M.one\n[PROOFSTEP]\nrw [\u2190 one_f]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM\u271d M N : Mon_ C\nf : M.X \u2245 N.X\none_f : M.one \u226b f.hom = N.one\nmul_f : M.mul \u226b f.hom = (f.hom \u2297 f.hom) \u226b N.mul\n\u22a2 (M.one \u226b f.hom) \u226b f.inv = M.one\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM\u271d M N : Mon_ C\nf : M.X \u2245 N.X\none_f : M.one \u226b f.hom = N.one\nmul_f : M.mul \u226b f.hom = (f.hom \u2297 f.hom) \u226b N.mul\n\u22a2 N.mul \u226b f.inv = (f.inv \u2297 f.inv) \u226b M.mul\n[PROOFSTEP]\nrw [\u2190 cancel_mono f.hom]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM\u271d M N : Mon_ C\nf : M.X \u2245 N.X\none_f : M.one \u226b f.hom = N.one\nmul_f : M.mul \u226b f.hom = (f.hom \u2297 f.hom) \u226b N.mul\n\u22a2 (N.mul \u226b f.inv) \u226b f.hom = ((f.inv \u2297 f.inv) \u226b M.mul) \u226b f.hom\n[PROOFSTEP]\nslice_rhs 2 3 => rw [mul_f]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM\u271d M N : Mon_ C\nf : M.X \u2245 N.X\none_f : M.one \u226b f.hom = N.one\nmul_f : M.mul \u226b f.hom = (f.hom \u2297 f.hom) \u226b N.mul\n| M.mul \u226b f.hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM\u271d M N : Mon_ C\nf : M.X \u2245 N.X\none_f : M.one \u226b f.hom = N.one\nmul_f : M.mul \u226b f.hom = (f.hom \u2297 f.hom) \u226b N.mul\n| f.inv \u2297 f.inv\n[PROOFSTEP]\nrw [mul_f]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM\u271d M N : Mon_ C\nf : M.X \u2245 N.X\none_f : M.one \u226b f.hom = N.one\nmul_f : M.mul \u226b f.hom = (f.hom \u2297 f.hom) \u226b N.mul\n| M.mul \u226b f.hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM\u271d M N : Mon_ C\nf : M.X \u2245 N.X\none_f : M.one \u226b f.hom = N.one\nmul_f : M.mul \u226b f.hom = (f.hom \u2297 f.hom) \u226b N.mul\n| f.inv \u2297 f.inv\n[PROOFSTEP]\nrw [mul_f]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM\u271d M N : Mon_ C\nf : M.X \u2245 N.X\none_f : M.one \u226b f.hom = N.one\nmul_f : M.mul \u226b f.hom = (f.hom \u2297 f.hom) \u226b N.mul\n| M.mul \u226b f.hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM\u271d M N : Mon_ C\nf : M.X \u2245 N.X\none_f : M.one \u226b f.hom = N.one\nmul_f : M.mul \u226b f.hom = (f.hom \u2297 f.hom) \u226b N.mul\n| f.inv \u2297 f.inv\n[PROOFSTEP]\nrw [mul_f]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM\u271d M N : Mon_ C\nf : M.X \u2245 N.X\none_f : M.one \u226b f.hom = N.one\nmul_f : M.mul \u226b f.hom = (f.hom \u2297 f.hom) \u226b N.mul\n\u22a2 (N.mul \u226b f.inv) \u226b f.hom = (f.inv \u2297 f.inv) \u226b (f.hom \u2297 f.hom) \u226b N.mul\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM A : Mon_ C\n\u22a2 (trivial C).one \u226b A.one = A.one\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM A : Mon_ C\n\u22a2 \ud835\udfd9 (\ud835\udfd9_ C) \u226b A.one = A.one\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM A : Mon_ C\n\u22a2 (trivial C).mul \u226b A.one = (A.one \u2297 A.one) \u226b A.mul\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM A : Mon_ C\n\u22a2 (\u03bb_ (\ud835\udfd9_ C)).hom \u226b A.one = (A.one \u2297 A.one) \u226b A.mul\n[PROOFSTEP]\nsimp [A.one_mul, unitors_equal]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM A : Mon_ C\nf : trivial C \u27f6 A\n\u22a2 f = default\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM A : Mon_ C\nf : trivial C \u27f6 A\n\u22a2 f.hom = default.hom\n[PROOFSTEP]\nsimp\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM A : Mon_ C\nf : trivial C \u27f6 A\n\u22a2 f.hom = A.one\n[PROOFSTEP]\nrw [\u2190 Category.id_comp f.hom]\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM A : Mon_ C\nf : trivial C \u27f6 A\n\u22a2 \ud835\udfd9 (trivial C).X \u226b f.hom = A.one\n[PROOFSTEP]\nerw [f.one_hom]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (F.\u03b5 \u226b F.map A.one \u2297 \ud835\udfd9 (F.obj A.X)) \u226b \u03bc F A.X A.X \u226b F.map A.mul = (\u03bb_ (F.obj A.X)).hom\n[PROOFSTEP]\nconv_lhs => rw [comp_tensor_id, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (F.\u03b5 \u226b F.map A.one \u2297 \ud835\udfd9 (F.obj A.X)) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nrw [comp_tensor_id, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (F.\u03b5 \u226b F.map A.one \u2297 \ud835\udfd9 (F.obj A.X)) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nrw [comp_tensor_id, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (F.\u03b5 \u226b F.map A.one \u2297 \ud835\udfd9 (F.obj A.X)) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nrw [comp_tensor_id, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 ((F.\u03b5 \u2297 F.map (\ud835\udfd9 A.X)) \u226b (F.map A.one \u2297 F.map (\ud835\udfd9 A.X))) \u226b \u03bc F A.X A.X \u226b F.map A.mul = (\u03bb_ (F.obj A.X)).hom\n[PROOFSTEP]\nslice_lhs 2 3 => rw [F.\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (F.map A.one \u2297 F.map (\ud835\udfd9 A.X)) \u226b \u03bc F A.X A.X\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.\u03b5 \u2297 F.map (\ud835\udfd9 A.X)\n[PROOFSTEP]\nrw [F.\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (F.map A.one \u2297 F.map (\ud835\udfd9 A.X)) \u226b \u03bc F A.X A.X\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.\u03b5 \u2297 F.map (\ud835\udfd9 A.X)\n[PROOFSTEP]\nrw [F.\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (F.map A.one \u2297 F.map (\ud835\udfd9 A.X)) \u226b \u03bc F A.X A.X\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.\u03b5 \u2297 F.map (\ud835\udfd9 A.X)\n[PROOFSTEP]\nrw [F.\u03bc_natural]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (F.\u03b5 \u2297 F.map (\ud835\udfd9 A.X)) \u226b (\u03bc F (\ud835\udfd9_ C) A.X \u226b F.map (A.one \u2297 \ud835\udfd9 A.X)) \u226b F.map A.mul = (\u03bb_ (F.obj A.X)).hom\n[PROOFSTEP]\nslice_lhs 3 4 => rw [\u2190 F.toFunctor.map_comp, A.one_mul]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (A.one \u2297 \ud835\udfd9 A.X) \u226b F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.\u03b5 \u2297 F.map (\ud835\udfd9 A.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| \u03bc F (\ud835\udfd9_ C) A.X\n[PROOFSTEP]\nrw [\u2190 F.toFunctor.map_comp, A.one_mul]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (A.one \u2297 \ud835\udfd9 A.X) \u226b F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.\u03b5 \u2297 F.map (\ud835\udfd9 A.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| \u03bc F (\ud835\udfd9_ C) A.X\n[PROOFSTEP]\nrw [\u2190 F.toFunctor.map_comp, A.one_mul]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (A.one \u2297 \ud835\udfd9 A.X) \u226b F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.\u03b5 \u2297 F.map (\ud835\udfd9 A.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| \u03bc F (\ud835\udfd9_ C) A.X\n[PROOFSTEP]\nrw [\u2190 F.toFunctor.map_comp, A.one_mul]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (F.\u03b5 \u2297 F.map (\ud835\udfd9 A.X)) \u226b \u03bc F (\ud835\udfd9_ C) A.X \u226b F.map (\u03bb_ A.X).hom = (\u03bb_ (F.obj A.X)).hom\n[PROOFSTEP]\nrw [F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (F.\u03b5 \u2297 \ud835\udfd9 (F.obj A.X)) \u226b \u03bc F (\ud835\udfd9_ C) A.X \u226b F.map (\u03bb_ A.X).hom = (\u03bb_ (F.obj A.X)).hom\n[PROOFSTEP]\nrw [F.left_unitality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (\ud835\udfd9 (F.obj A.X) \u2297 F.\u03b5 \u226b F.map A.one) \u226b \u03bc F A.X A.X \u226b F.map A.mul = (\u03c1_ (F.obj A.X)).hom\n[PROOFSTEP]\nconv_lhs => rw [id_tensor_comp, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\ud835\udfd9 (F.obj A.X) \u2297 F.\u03b5 \u226b F.map A.one) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nrw [id_tensor_comp, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\ud835\udfd9 (F.obj A.X) \u2297 F.\u03b5 \u226b F.map A.one) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nrw [id_tensor_comp, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\ud835\udfd9 (F.obj A.X) \u2297 F.\u03b5 \u226b F.map A.one) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nrw [id_tensor_comp, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 ((F.map (\ud835\udfd9 A.X) \u2297 F.\u03b5) \u226b (F.map (\ud835\udfd9 A.X) \u2297 F.map A.one)) \u226b \u03bc F A.X A.X \u226b F.map A.mul = (\u03c1_ (F.obj A.X)).hom\n[PROOFSTEP]\nslice_lhs 2 3 => rw [F.\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (F.map (\ud835\udfd9 A.X) \u2297 F.map A.one) \u226b \u03bc F A.X A.X\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (\ud835\udfd9 A.X) \u2297 F.\u03b5\n[PROOFSTEP]\nrw [F.\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (F.map (\ud835\udfd9 A.X) \u2297 F.map A.one) \u226b \u03bc F A.X A.X\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (\ud835\udfd9 A.X) \u2297 F.\u03b5\n[PROOFSTEP]\nrw [F.\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (F.map (\ud835\udfd9 A.X) \u2297 F.map A.one) \u226b \u03bc F A.X A.X\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (\ud835\udfd9 A.X) \u2297 F.\u03b5\n[PROOFSTEP]\nrw [F.\u03bc_natural]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (F.map (\ud835\udfd9 A.X) \u2297 F.\u03b5) \u226b (\u03bc F A.X (\ud835\udfd9_ C) \u226b F.map (\ud835\udfd9 A.X \u2297 A.one)) \u226b F.map A.mul = (\u03c1_ (F.obj A.X)).hom\n[PROOFSTEP]\nslice_lhs 3 4 => rw [\u2190 F.toFunctor.map_comp, A.mul_one]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (\ud835\udfd9 A.X \u2297 A.one) \u226b F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (\ud835\udfd9 A.X) \u2297 F.\u03b5\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| \u03bc F A.X (\ud835\udfd9_ C)\n[PROOFSTEP]\nrw [\u2190 F.toFunctor.map_comp, A.mul_one]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (\ud835\udfd9 A.X \u2297 A.one) \u226b F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (\ud835\udfd9 A.X) \u2297 F.\u03b5\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| \u03bc F A.X (\ud835\udfd9_ C)\n[PROOFSTEP]\nrw [\u2190 F.toFunctor.map_comp, A.mul_one]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (\ud835\udfd9 A.X \u2297 A.one) \u226b F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (\ud835\udfd9 A.X) \u2297 F.\u03b5\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| \u03bc F A.X (\ud835\udfd9_ C)\n[PROOFSTEP]\nrw [\u2190 F.toFunctor.map_comp, A.mul_one]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (F.map (\ud835\udfd9 A.X) \u2297 F.\u03b5) \u226b \u03bc F A.X (\ud835\udfd9_ C) \u226b F.map (\u03c1_ A.X).hom = (\u03c1_ (F.obj A.X)).hom\n[PROOFSTEP]\nrw [F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (\ud835\udfd9 (F.obj A.X) \u2297 F.\u03b5) \u226b \u03bc F A.X (\ud835\udfd9_ C) \u226b F.map (\u03c1_ A.X).hom = (\u03c1_ (F.obj A.X)).hom\n[PROOFSTEP]\nrw [F.right_unitality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (\u03bc F A.X A.X \u226b F.map A.mul \u2297 \ud835\udfd9 (F.obj A.X)) \u226b \u03bc F A.X A.X \u226b F.map A.mul =\n (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b\n (\ud835\udfd9 (F.obj A.X) \u2297 \u03bc F A.X A.X \u226b F.map A.mul) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nconv_lhs => rw [comp_tensor_id, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03bc F A.X A.X \u226b F.map A.mul \u2297 \ud835\udfd9 (F.obj A.X)) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nrw [comp_tensor_id, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03bc F A.X A.X \u226b F.map A.mul \u2297 \ud835\udfd9 (F.obj A.X)) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nrw [comp_tensor_id, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03bc F A.X A.X \u226b F.map A.mul \u2297 \ud835\udfd9 (F.obj A.X)) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nrw [comp_tensor_id, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 ((\u03bc F A.X A.X \u2297 F.map (\ud835\udfd9 A.X)) \u226b (F.map A.mul \u2297 F.map (\ud835\udfd9 A.X))) \u226b \u03bc F A.X A.X \u226b F.map A.mul =\n (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b\n (\ud835\udfd9 (F.obj A.X) \u2297 \u03bc F A.X A.X \u226b F.map A.mul) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nslice_lhs 2 3 => rw [F.\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (F.map A.mul \u2297 F.map (\ud835\udfd9 A.X)) \u226b \u03bc F A.X A.X\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| \u03bc F A.X A.X \u2297 F.map (\ud835\udfd9 A.X)\n[PROOFSTEP]\nrw [F.\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (F.map A.mul \u2297 F.map (\ud835\udfd9 A.X)) \u226b \u03bc F A.X A.X\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| \u03bc F A.X A.X \u2297 F.map (\ud835\udfd9 A.X)\n[PROOFSTEP]\nrw [F.\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (F.map A.mul \u2297 F.map (\ud835\udfd9 A.X)) \u226b \u03bc F A.X A.X\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| \u03bc F A.X A.X \u2297 F.map (\ud835\udfd9 A.X)\n[PROOFSTEP]\nrw [F.\u03bc_natural]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (\u03bc F A.X A.X \u2297 F.map (\ud835\udfd9 A.X)) \u226b (\u03bc F (A.X \u2297 A.X) A.X \u226b F.map (A.mul \u2297 \ud835\udfd9 A.X)) \u226b F.map A.mul =\n (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b\n (\ud835\udfd9 (F.obj A.X) \u2297 \u03bc F A.X A.X \u226b F.map A.mul) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nslice_lhs 3 4 => rw [\u2190 F.toFunctor.map_comp, A.mul_assoc]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (A.mul \u2297 \ud835\udfd9 A.X) \u226b F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| \u03bc F A.X A.X \u2297 F.map (\ud835\udfd9 A.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| \u03bc F (A.X \u2297 A.X) A.X\n[PROOFSTEP]\nrw [\u2190 F.toFunctor.map_comp, A.mul_assoc]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (A.mul \u2297 \ud835\udfd9 A.X) \u226b F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| \u03bc F A.X A.X \u2297 F.map (\ud835\udfd9 A.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| \u03bc F (A.X \u2297 A.X) A.X\n[PROOFSTEP]\nrw [\u2190 F.toFunctor.map_comp, A.mul_assoc]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (A.mul \u2297 \ud835\udfd9 A.X) \u226b F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| \u03bc F A.X A.X \u2297 F.map (\ud835\udfd9 A.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| \u03bc F (A.X \u2297 A.X) A.X\n[PROOFSTEP]\nrw [\u2190 F.toFunctor.map_comp, A.mul_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (\u03bc F A.X A.X \u2297 F.map (\ud835\udfd9 A.X)) \u226b \u03bc F (A.X \u2297 A.X) A.X \u226b F.map ((\u03b1_ A.X A.X A.X).hom \u226b (\ud835\udfd9 A.X \u2297 A.mul) \u226b A.mul) =\n (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b\n (\ud835\udfd9 (F.obj A.X) \u2297 \u03bc F A.X A.X \u226b F.map A.mul) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nconv_lhs => rw [F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03bc F A.X A.X \u2297 F.map (\ud835\udfd9 A.X)) \u226b \u03bc F (A.X \u2297 A.X) A.X \u226b F.map ((\u03b1_ A.X A.X A.X).hom \u226b (\ud835\udfd9 A.X \u2297 A.mul) \u226b A.mul)\n[PROOFSTEP]\nrw [F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03bc F A.X A.X \u2297 F.map (\ud835\udfd9 A.X)) \u226b \u03bc F (A.X \u2297 A.X) A.X \u226b F.map ((\u03b1_ A.X A.X A.X).hom \u226b (\ud835\udfd9 A.X \u2297 A.mul) \u226b A.mul)\n[PROOFSTEP]\nrw [F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03bc F A.X A.X \u2297 F.map (\ud835\udfd9 A.X)) \u226b \u03bc F (A.X \u2297 A.X) A.X \u226b F.map ((\u03b1_ A.X A.X A.X).hom \u226b (\ud835\udfd9 A.X \u2297 A.mul) \u226b A.mul)\n[PROOFSTEP]\nrw [F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (\u03bc F A.X A.X \u2297 \ud835\udfd9 (F.obj A.X)) \u226b \u03bc F (A.X \u2297 A.X) A.X \u226b F.map ((\u03b1_ A.X A.X A.X).hom \u226b (\ud835\udfd9 A.X \u2297 A.mul) \u226b A.mul) =\n (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b\n (\ud835\udfd9 (F.obj A.X) \u2297 \u03bc F A.X A.X \u226b F.map A.mul) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nconv_lhs => rw [F.toFunctor.map_comp, F.toFunctor.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03bc F A.X A.X \u2297 \ud835\udfd9 (F.obj A.X)) \u226b \u03bc F (A.X \u2297 A.X) A.X \u226b F.map ((\u03b1_ A.X A.X A.X).hom \u226b (\ud835\udfd9 A.X \u2297 A.mul) \u226b A.mul)\n[PROOFSTEP]\nrw [F.toFunctor.map_comp, F.toFunctor.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03bc F A.X A.X \u2297 \ud835\udfd9 (F.obj A.X)) \u226b \u03bc F (A.X \u2297 A.X) A.X \u226b F.map ((\u03b1_ A.X A.X A.X).hom \u226b (\ud835\udfd9 A.X \u2297 A.mul) \u226b A.mul)\n[PROOFSTEP]\nrw [F.toFunctor.map_comp, F.toFunctor.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03bc F A.X A.X \u2297 \ud835\udfd9 (F.obj A.X)) \u226b \u03bc F (A.X \u2297 A.X) A.X \u226b F.map ((\u03b1_ A.X A.X A.X).hom \u226b (\ud835\udfd9 A.X \u2297 A.mul) \u226b A.mul)\n[PROOFSTEP]\nrw [F.toFunctor.map_comp, F.toFunctor.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (\u03bc F A.X A.X \u2297 \ud835\udfd9 (F.obj A.X)) \u226b\n \u03bc F (A.X \u2297 A.X) A.X \u226b F.map (\u03b1_ A.X A.X A.X).hom \u226b F.map (\ud835\udfd9 A.X \u2297 A.mul) \u226b F.map A.mul =\n (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b\n (\ud835\udfd9 (F.obj A.X) \u2297 \u03bc F A.X A.X \u226b F.map A.mul) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nconv_rhs => rw [id_tensor_comp, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b (\ud835\udfd9 (F.obj A.X) \u2297 \u03bc F A.X A.X \u226b F.map A.mul) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nrw [id_tensor_comp, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b (\ud835\udfd9 (F.obj A.X) \u2297 \u03bc F A.X A.X \u226b F.map A.mul) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nrw [id_tensor_comp, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b (\ud835\udfd9 (F.obj A.X) \u2297 \u03bc F A.X A.X \u226b F.map A.mul) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nrw [id_tensor_comp, \u2190 F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (\u03bc F A.X A.X \u2297 \ud835\udfd9 (F.obj A.X)) \u226b\n \u03bc F (A.X \u2297 A.X) A.X \u226b F.map (\u03b1_ A.X A.X A.X).hom \u226b F.map (\ud835\udfd9 A.X \u2297 A.mul) \u226b F.map A.mul =\n (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b\n ((F.map (\ud835\udfd9 A.X) \u2297 \u03bc F A.X A.X) \u226b (F.map (\ud835\udfd9 A.X) \u2297 F.map A.mul)) \u226b \u03bc F A.X A.X \u226b F.map A.mul\n[PROOFSTEP]\nslice_rhs 3 4 => rw [F.\u03bc_natural]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (F.map (\ud835\udfd9 A.X) \u2297 F.map A.mul) \u226b \u03bc F A.X A.X\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (\ud835\udfd9 A.X) \u2297 \u03bc F A.X A.X\n[PROOFSTEP]\nrw [F.\u03bc_natural]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (F.map (\ud835\udfd9 A.X) \u2297 F.map A.mul) \u226b \u03bc F A.X A.X\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (\ud835\udfd9 A.X) \u2297 \u03bc F A.X A.X\n[PROOFSTEP]\nrw [F.\u03bc_natural]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (F.map (\ud835\udfd9 A.X) \u2297 F.map A.mul) \u226b \u03bc F A.X A.X\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map A.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (\ud835\udfd9 A.X) \u2297 \u03bc F A.X A.X\n[PROOFSTEP]\nrw [F.\u03bc_natural]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (\u03bc F A.X A.X \u2297 \ud835\udfd9 (F.obj A.X)) \u226b\n \u03bc F (A.X \u2297 A.X) A.X \u226b F.map (\u03b1_ A.X A.X A.X).hom \u226b F.map (\ud835\udfd9 A.X \u2297 A.mul) \u226b F.map A.mul =\n (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b\n (F.map (\ud835\udfd9 A.X) \u2297 \u03bc F A.X A.X) \u226b (\u03bc F A.X (A.X \u2297 A.X) \u226b F.map (\ud835\udfd9 A.X \u2297 A.mul)) \u226b F.map A.mul\n[PROOFSTEP]\nconv_rhs => rw [F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b\n (F.map (\ud835\udfd9 A.X) \u2297 \u03bc F A.X A.X) \u226b (\u03bc F A.X (A.X \u2297 A.X) \u226b F.map (\ud835\udfd9 A.X \u2297 A.mul)) \u226b F.map A.mul\n[PROOFSTEP]\nrw [F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b\n (F.map (\ud835\udfd9 A.X) \u2297 \u03bc F A.X A.X) \u226b (\u03bc F A.X (A.X \u2297 A.X) \u226b F.map (\ud835\udfd9 A.X \u2297 A.mul)) \u226b F.map A.mul\n[PROOFSTEP]\nrw [F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b\n (F.map (\ud835\udfd9 A.X) \u2297 \u03bc F A.X A.X) \u226b (\u03bc F A.X (A.X \u2297 A.X) \u226b F.map (\ud835\udfd9 A.X \u2297 A.mul)) \u226b F.map A.mul\n[PROOFSTEP]\nrw [F.toFunctor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (\u03bc F A.X A.X \u2297 \ud835\udfd9 (F.obj A.X)) \u226b\n \u03bc F (A.X \u2297 A.X) A.X \u226b F.map (\u03b1_ A.X A.X A.X).hom \u226b F.map (\ud835\udfd9 A.X \u2297 A.mul) \u226b F.map A.mul =\n (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b\n (\ud835\udfd9 (F.obj A.X) \u2297 \u03bc F A.X A.X) \u226b (\u03bc F A.X (A.X \u2297 A.X) \u226b F.map (\ud835\udfd9 A.X \u2297 A.mul)) \u226b F.map A.mul\n[PROOFSTEP]\nslice_rhs 1 3 => rw [\u2190 F.associativity]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b (\ud835\udfd9 (F.obj A.X) \u2297 \u03bc F A.X A.X) \u226b \u03bc F A.X (A.X \u2297 A.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (\ud835\udfd9 A.X \u2297 A.mul)\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map A.mul\n[PROOFSTEP]\nrw [\u2190 F.associativity]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b (\ud835\udfd9 (F.obj A.X) \u2297 \u03bc F A.X A.X) \u226b \u03bc F A.X (A.X \u2297 A.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (\ud835\udfd9 A.X \u2297 A.mul)\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map A.mul\n[PROOFSTEP]\nrw [\u2190 F.associativity]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| (\u03b1_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom \u226b (\ud835\udfd9 (F.obj A.X) \u2297 \u03bc F A.X A.X) \u226b \u03bc F A.X (A.X \u2297 A.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map (\ud835\udfd9 A.X \u2297 A.mul)\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n| F.map A.mul\n[PROOFSTEP]\nrw [\u2190 F.associativity]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 (\u03bc F A.X A.X \u2297 \ud835\udfd9 (F.obj A.X)) \u226b\n \u03bc F (A.X \u2297 A.X) A.X \u226b F.map (\u03b1_ A.X A.X A.X).hom \u226b F.map (\ud835\udfd9 A.X \u2297 A.mul) \u226b F.map A.mul =\n (((\u03bc F A.X A.X \u2297 \ud835\udfd9 (F.obj A.X)) \u226b \u03bc F (A.X \u2297 A.X) A.X \u226b F.map (\u03b1_ A.X A.X A.X).hom) \u226b F.map (\ud835\udfd9 A.X \u2297 A.mul)) \u226b\n F.map A.mul\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nX\u271d Y\u271d : Mon_ C\nf : X\u271d \u27f6 Y\u271d\n\u22a2 ((fun A => Mon_.mk (F.obj A.X) (F.\u03b5 \u226b F.map A.one) (\u03bc F A.X A.X \u226b F.map A.mul)) X\u271d).one \u226b F.map f.hom =\n ((fun A => Mon_.mk (F.obj A.X) (F.\u03b5 \u226b F.map A.one) (\u03bc F A.X A.X \u226b F.map A.mul)) Y\u271d).one\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nX\u271d Y\u271d : Mon_ C\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (F.\u03b5 \u226b F.map X\u271d.one) \u226b F.map f.hom = F.\u03b5 \u226b F.map Y\u271d.one\n[PROOFSTEP]\nrw [Category.assoc, \u2190 F.toFunctor.map_comp, f.one_hom]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nX\u271d Y\u271d : Mon_ C\nf : X\u271d \u27f6 Y\u271d\n\u22a2 ((fun A => Mon_.mk (F.obj A.X) (F.\u03b5 \u226b F.map A.one) (\u03bc F A.X A.X \u226b F.map A.mul)) X\u271d).mul \u226b F.map f.hom =\n (F.map f.hom \u2297 F.map f.hom) \u226b\n ((fun A => Mon_.mk (F.obj A.X) (F.\u03b5 \u226b F.map A.one) (\u03bc F A.X A.X \u226b F.map A.mul)) Y\u271d).mul\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nX\u271d Y\u271d : Mon_ C\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (\u03bc F X\u271d.X X\u271d.X \u226b F.map X\u271d.mul) \u226b F.map f.hom = (F.map f.hom \u2297 F.map f.hom) \u226b \u03bc F Y\u271d.X Y\u271d.X \u226b F.map Y\u271d.mul\n[PROOFSTEP]\nrw [Category.assoc, F.\u03bc_natural_assoc, \u2190 F.toFunctor.map_comp, \u2190 F.toFunctor.map_comp, f.mul_hom]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 { obj := fun A => Mon_.mk (F.obj A.X) (F.\u03b5 \u226b F.map A.one) (\u03bc F A.X A.X \u226b F.map A.mul),\n map := fun {X Y} f => Mon_.Hom.mk (F.map f.hom) }.map\n (\ud835\udfd9 A) =\n \ud835\udfd9\n ({ obj := fun A => Mon_.mk (F.obj A.X) (F.\u03b5 \u226b F.map A.one) (\u03bc F A.X A.X \u226b F.map A.mul),\n map := fun {X Y} f => Mon_.Hom.mk (F.map f.hom) }.obj\n A)\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nA : Mon_ C\n\u22a2 ({ obj := fun A => Mon_.mk (F.obj A.X) (F.\u03b5 \u226b F.map A.one) (\u03bc F A.X A.X \u226b F.map A.mul),\n map := fun {X Y} f => Mon_.Hom.mk (F.map f.hom) }.map\n (\ud835\udfd9 A)).hom =\n (\ud835\udfd9\n ({ obj := fun A => Mon_.mk (F.obj A.X) (F.\u03b5 \u226b F.map A.one) (\u03bc F A.X A.X \u226b F.map A.mul),\n map := fun {X Y} f => Mon_.Hom.mk (F.map f.hom) }.obj\n A)).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nX\u271d Y\u271d Z\u271d : Mon_ C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun A => Mon_.mk (F.obj A.X) (F.\u03b5 \u226b F.map A.one) (\u03bc F A.X A.X \u226b F.map A.mul),\n map := fun {X Y} f => Mon_.Hom.mk (F.map f.hom) }.map\n (f \u226b g) =\n { obj := fun A => Mon_.mk (F.obj A.X) (F.\u03b5 \u226b F.map A.one) (\u03bc F A.X A.X \u226b F.map A.mul),\n map := fun {X Y} f => Mon_.Hom.mk (F.map f.hom) }.map\n f \u226b\n { obj := fun A => Mon_.mk (F.obj A.X) (F.\u03b5 \u226b F.map A.one) (\u03bc F A.X A.X \u226b F.map A.mul),\n map := fun {X Y} f => Mon_.Hom.mk (F.map f.hom) }.map\n g\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nX\u271d Y\u271d Z\u271d : Mon_ C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 ({ obj := fun A => Mon_.mk (F.obj A.X) (F.\u03b5 \u226b F.map A.one) (\u03bc F A.X A.X \u226b F.map A.mul),\n map := fun {X Y} f => Mon_.Hom.mk (F.map f.hom) }.map\n (f \u226b g)).hom =\n ({ obj := fun A => Mon_.mk (F.obj A.X) (F.\u03b5 \u226b F.map A.one) (\u03bc F A.X A.X \u226b F.map A.mul),\n map := fun {X Y} f => Mon_.Hom.mk (F.map f.hom) }.map\n f \u226b\n { obj := fun A => Mon_.mk (F.obj A.X) (F.\u03b5 \u226b F.map A.one) (\u03bc F A.X A.X \u226b F.map A.mul),\n map := fun {X Y} f => Mon_.Hom.mk (F.map f.hom) }.map\n g).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nX\u271d Y\u271d : Mon_ C\nf : X\u271d \u27f6 Y\u271d\nx\u271d\u00b2 x\u271d\u00b9 : Discrete PUnit\nx\u271d : x\u271d\u00b2 \u27f6 x\u271d\u00b9\n\u22a2 ((fun A =>\n LaxMonoidalFunctor.mk\n (CategoryTheory.Functor.mk { obj := fun x => A.X, map := fun {X Y} x => \ud835\udfd9 ((fun x => A.X) X) }) A.one\n fun x x => A.mul)\n X\u271d).map\n x\u271d \u226b\n (fun x => f.hom) x\u271d\u00b9 =\n (fun x => f.hom) x\u271d\u00b2 \u226b\n ((fun A =>\n LaxMonoidalFunctor.mk\n (CategoryTheory.Functor.mk { obj := fun x => A.X, map := fun {X Y} x => \ud835\udfd9 ((fun x => A.X) X) }) A.one\n fun x x => A.mul)\n Y\u271d).map\n x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nX\u271d Y\u271d : Mon_ C\nf : X\u271d \u27f6 Y\u271d\nx\u271d\u00b2 x\u271d\u00b9 : Discrete PUnit\nx\u271d : x\u271d\u00b2 \u27f6 x\u271d\u00b9\n\u22a2 \ud835\udfd9 X\u271d.X \u226b f.hom = f.hom \u226b \ud835\udfd9 Y\u271d.X\n[PROOFSTEP]\nrw [Category.id_comp, Category.comp_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nF : LaxMonoidalFunctor (Discrete PUnit) C\nx\u271d : Discrete PUnit\n\u22a2 x\u271d = (trivial (Discrete PUnit)).X\n[PROOFSTEP]\next\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nF : LaxMonoidalFunctor (Discrete PUnit) C\n\u22a2 \u2200 {X Y : Discrete PUnit} (f : X \u27f6 Y),\n ((\ud835\udfed (LaxMonoidalFunctor (Discrete PUnit) C)).obj F).map f \u226b\n ((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) Y).hom =\n ((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) X).hom \u226b\n ((laxMonoidalToMon C \u22d9 monToLaxMonoidal C).obj F).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nF : LaxMonoidalFunctor (Discrete PUnit) C\n\u22a2 ((\ud835\udfed (LaxMonoidalFunctor (Discrete PUnit) C)).obj F).\u03b5 \u226b\n ((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) (\ud835\udfd9_ (Discrete PUnit))).hom =\n ((laxMonoidalToMon C \u22d9 monToLaxMonoidal C).obj F).\u03b5\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nF : LaxMonoidalFunctor (Discrete PUnit) C\n\u22a2 \u2200 (X Y : Discrete PUnit),\n \u03bc ((\ud835\udfed (LaxMonoidalFunctor (Discrete PUnit) C)).obj F) X Y \u226b\n ((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) (X \u2297 Y)).hom =\n (((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) X).hom \u2297\n ((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) Y).hom) \u226b\n \u03bc ((laxMonoidalToMon C \u22d9 monToLaxMonoidal C).obj F) X Y\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\n\u22a2 \u2200 {X Y : LaxMonoidalFunctor (Discrete PUnit) C} (f : X \u27f6 Y),\n (\ud835\udfed (LaxMonoidalFunctor (Discrete PUnit) C)).map f \u226b\n ((fun F =>\n MonoidalNatIso.ofComponents (fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X)))\n (_ :\n \u2200 {X Y : Discrete PUnit} (f : X \u27f6 Y),\n ((\ud835\udfed (LaxMonoidalFunctor (Discrete PUnit) C)).obj F).map f \u226b\n ((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) Y).hom =\n ((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) X).hom \u226b\n ((laxMonoidalToMon C \u22d9 monToLaxMonoidal C).obj F).map f)\n (_ : F.\u03b5 \u226b (F.mapIso (Iso.refl (\ud835\udfd9_ (Discrete PUnit)))).hom = F.\u03b5 \u226b F.map (\ud835\udfd9 (\ud835\udfd9_ (Discrete PUnit))))\n (_ :\n \u2200 (X Y : Discrete PUnit),\n \u03bc ((\ud835\udfed (LaxMonoidalFunctor (Discrete PUnit) C)).obj F) X Y \u226b\n ((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) (X \u2297 Y)).hom =\n (((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) X).hom \u2297\n ((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) Y).hom) \u226b\n \u03bc ((laxMonoidalToMon C \u22d9 monToLaxMonoidal C).obj F) X Y))\n Y).hom =\n ((fun F =>\n MonoidalNatIso.ofComponents (fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X)))\n (_ :\n \u2200 {X Y : Discrete PUnit} (f : X \u27f6 Y),\n ((\ud835\udfed (LaxMonoidalFunctor (Discrete PUnit) C)).obj F).map f \u226b\n ((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) Y).hom =\n ((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) X).hom \u226b\n ((laxMonoidalToMon C \u22d9 monToLaxMonoidal C).obj F).map f)\n (_ : F.\u03b5 \u226b (F.mapIso (Iso.refl (\ud835\udfd9_ (Discrete PUnit)))).hom = F.\u03b5 \u226b F.map (\ud835\udfd9 (\ud835\udfd9_ (Discrete PUnit))))\n (_ :\n \u2200 (X Y : Discrete PUnit),\n \u03bc ((\ud835\udfed (LaxMonoidalFunctor (Discrete PUnit) C)).obj F) X Y \u226b\n ((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) (X \u2297 Y)).hom =\n (((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) X).hom \u2297\n ((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit)).X))) Y).hom) \u226b\n \u03bc ((laxMonoidalToMon C \u22d9 monToLaxMonoidal C).obj F) X Y))\n X).hom \u226b\n (laxMonoidalToMon C \u22d9 monToLaxMonoidal C).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\n\u22a2 \u2200 {X Y : Mon_ C} (f : X \u27f6 Y),\n (monToLaxMonoidal C \u22d9 laxMonoidalToMon C).map f \u226b\n ((fun F =>\n Iso.mk (Hom.mk (\ud835\udfd9 ((monToLaxMonoidal C \u22d9 laxMonoidalToMon C).obj F).X))\n (Hom.mk (\ud835\udfd9 ((\ud835\udfed (Mon_ C)).obj F).X)))\n Y).hom =\n ((fun F =>\n Iso.mk (Hom.mk (\ud835\udfd9 ((monToLaxMonoidal C \u22d9 laxMonoidalToMon C).obj F).X))\n (Hom.mk (\ud835\udfd9 ((\ud835\udfed (Mon_ C)).obj F).X)))\n X).hom \u226b\n (\ud835\udfed (Mon_ C)).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n\u22a2 ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n\u22a2 ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one) \u226b (\u03b1_ M.X N.X P.X).hom = M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)\n[PROOFSTEP]\nslice_lhs 1 3 => rw [\u2190 Category.id_comp P.one, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n| ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one) \u226b (\u03b1_ M.X N.X P.X).hom\n[PROOFSTEP]\nrw [\u2190 Category.id_comp P.one, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n| ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one) \u226b (\u03b1_ M.X N.X P.X).hom\n[PROOFSTEP]\nrw [\u2190 Category.id_comp P.one, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n| ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one) \u226b (\u03b1_ M.X N.X P.X).hom\n[PROOFSTEP]\nrw [\u2190 Category.id_comp P.one, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n\u22a2 (((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b ((M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)\n[PROOFSTEP]\nslice_lhs 2 3 => rw [associator_naturality]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n| ((M.one \u2297 N.one) \u2297 P.one) \u226b (\u03b1_ M.X N.X P.X).hom\ncase a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C M N P : Mon_ C | (\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n| ((M.one \u2297 N.one) \u2297 P.one) \u226b (\u03b1_ M.X N.X P.X).hom\ncase a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C M N P : Mon_ C | (\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n| ((M.one \u2297 N.one) \u2297 P.one) \u226b (\u03b1_ M.X N.X P.X).hom\ncase a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C M N P : Mon_ C | (\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n\u22a2 ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03b1_ tensorUnit' (\ud835\udfd9_ C) (\ud835\udfd9_ C)).hom \u226b (M.one \u2297 N.one \u2297 P.one) =\n M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)\n[PROOFSTEP]\nslice_rhs 1 2 => rw [\u2190 Category.id_comp M.one, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n| M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)\n[PROOFSTEP]\nrw [\u2190 Category.id_comp M.one, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n| M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)\n[PROOFSTEP]\nrw [\u2190 Category.id_comp M.one, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n| M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)\n[PROOFSTEP]\nrw [\u2190 Category.id_comp M.one, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n\u22a2 ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03b1_ tensorUnit' (\ud835\udfd9_ C) (\ud835\udfd9_ C)).hom \u226b (M.one \u2297 N.one \u2297 P.one) =\n (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (M.one \u2297 N.one \u2297 P.one)\n[PROOFSTEP]\nslice_lhs 1 2 => rw [\u2190 leftUnitor_tensor_inv]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n| ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03b1_ tensorUnit' (\ud835\udfd9_ C) (\ud835\udfd9_ C)).hom\ncase a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C M N P : Mon_ C | M.one \u2297 N.one \u2297 P.one\n[PROOFSTEP]\nrw [\u2190 leftUnitor_tensor_inv]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n| ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03b1_ tensorUnit' (\ud835\udfd9_ C) (\ud835\udfd9_ C)).hom\ncase a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C M N P : Mon_ C | M.one \u2297 N.one \u2297 P.one\n[PROOFSTEP]\nrw [\u2190 leftUnitor_tensor_inv]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n| ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03b1_ tensorUnit' (\ud835\udfd9_ C) (\ud835\udfd9_ C)).hom\ncase a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C M N P : Mon_ C | M.one \u2297 N.one \u2297 P.one\n[PROOFSTEP]\nrw [\u2190 leftUnitor_tensor_inv]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n\u22a2 (\u03bb_ (tensorUnit' \u2297 \ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one \u2297 P.one) = (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (M.one \u2297 N.one \u2297 P.one)\n[PROOFSTEP]\nrw [\u2190 cancel_epi (\u03bb_ (\ud835\udfd9_ C)).inv]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n\u22a2 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\u03bb_ (tensorUnit' \u2297 \ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one \u2297 P.one) =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (M.one \u2297 N.one \u2297 P.one)\n[PROOFSTEP]\nslice_lhs 1 2 => rw [leftUnitor_inv_naturality]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n| (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\u03bb_ (tensorUnit' \u2297 \ud835\udfd9_ C)).inv\ncase a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C M N P : Mon_ C | M.one \u2297 N.one \u2297 P.one\n[PROOFSTEP]\nrw [leftUnitor_inv_naturality]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n| (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\u03bb_ (tensorUnit' \u2297 \ud835\udfd9_ C)).inv\ncase a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C M N P : Mon_ C | M.one \u2297 N.one \u2297 P.one\n[PROOFSTEP]\nrw [leftUnitor_inv_naturality]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n| (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\u03bb_ (tensorUnit' \u2297 \ud835\udfd9_ C)).inv\ncase a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C M N P : Mon_ C | M.one \u2297 N.one \u2297 P.one\n[PROOFSTEP]\nrw [leftUnitor_inv_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM N P : Mon_ C\n\u22a2 ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 tensorUnit' \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv)) \u226b (M.one \u2297 N.one \u2297 P.one) =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (M.one \u2297 N.one \u2297 P.one)\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM : Mon_ C\n\u22a2 ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.one)) \u226b (\u03bb_ M.X).hom = M.one\n[PROOFSTEP]\nslice_lhs 2 3 => rw [leftUnitor_naturality]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM : Mon_ C\n| (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.one) \u226b (\u03bb_ M.X).hom\ncase a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C M : Mon_ C | (\u03bb_ (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [leftUnitor_naturality]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM : Mon_ C\n| (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.one) \u226b (\u03bb_ M.X).hom\ncase a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C M : Mon_ C | (\u03bb_ (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [leftUnitor_naturality]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM : Mon_ C\n| (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.one) \u226b (\u03bb_ M.X).hom\ncase a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C M : Mon_ C | (\u03bb_ (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [leftUnitor_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM : Mon_ C\n\u22a2 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\u03bb_ (\ud835\udfd9_ C)).hom \u226b M.one = M.one\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM : Mon_ C\n\u22a2 ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ M.X).hom = M.one\n[PROOFSTEP]\nslice_lhs 2 3 => rw [rightUnitor_naturality, \u2190 unitors_equal]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM : Mon_ C\n| (M.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03c1_ M.X).hom\ncase a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C M : Mon_ C | (\u03bb_ (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [rightUnitor_naturality, \u2190 unitors_equal]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM : Mon_ C\n| (M.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03c1_ M.X).hom\ncase a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C M : Mon_ C | (\u03bb_ (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [rightUnitor_naturality, \u2190 unitors_equal]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM : Mon_ C\n| (M.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03c1_ M.X).hom\ncase a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C M : Mon_ C | (\u03bb_ (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [rightUnitor_naturality, \u2190 unitors_equal]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nM : Mon_ C\n\u22a2 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\u03bb_ (\ud835\udfd9_ C)).hom \u226b M.one = M.one\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) =\n (\u03bb_ (M.X \u2297 N.X)).hom\n[PROOFSTEP]\nrw [\u2190 Category.id_comp (\ud835\udfd9 (M.X \u2297 N.X)), tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b ((M.one \u2297 N.one) \u2297 \ud835\udfd9 (M.X \u2297 N.X))) \u226b\n tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) =\n (\u03bb_ (M.X \u2297 N.X)).hom\n[PROOFSTEP]\nslice_lhs 2 3 => rw [\u2190 tensor_id, tensor_\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| ((M.one \u2297 N.one) \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b tensor_\u03bc C (M.X, N.X) (M.X, N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (M.X \u2297 N.X)\n[PROOFSTEP]\nrw [\u2190 tensor_id, tensor_\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| ((M.one \u2297 N.one) \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b tensor_\u03bc C (M.X, N.X) (M.X, N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (M.X \u2297 N.X)\n[PROOFSTEP]\nrw [\u2190 tensor_id, tensor_\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| ((M.one \u2297 N.one) \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b tensor_\u03bc C (M.X, N.X) (M.X, N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (M.X \u2297 N.X)\n[PROOFSTEP]\nrw [\u2190 tensor_id, tensor_\u03bc_natural]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b\n (tensor_\u03bc C (tensorUnit', \ud835\udfd9_ C) (M.X, N.X) \u226b ((M.one \u2297 \ud835\udfd9 M.X) \u2297 N.one \u2297 \ud835\udfd9 N.X)) \u226b (M.mul \u2297 N.mul) =\n (\u03bb_ (M.X \u2297 N.X)).hom\n[PROOFSTEP]\nslice_lhs 3 4 => rw [\u2190 tensor_comp, one_mul M, one_mul N]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| ((M.one \u2297 \ud835\udfd9 M.X) \u2297 N.one \u2297 \ud835\udfd9 N.X) \u226b (M.mul \u2297 N.mul)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (M.X \u2297 N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (tensorUnit', \ud835\udfd9_ C) (M.X, N.X)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, one_mul M, one_mul N]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| ((M.one \u2297 \ud835\udfd9 M.X) \u2297 N.one \u2297 \ud835\udfd9 N.X) \u226b (M.mul \u2297 N.mul)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (M.X \u2297 N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (tensorUnit', \ud835\udfd9_ C) (M.X, N.X)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, one_mul M, one_mul N]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| ((M.one \u2297 \ud835\udfd9 M.X) \u2297 N.one \u2297 \ud835\udfd9 N.X) \u226b (M.mul \u2297 N.mul)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (M.X \u2297 N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (tensorUnit', \ud835\udfd9_ C) (M.X, N.X)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, one_mul M, one_mul N]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b tensor_\u03bc C (tensorUnit', \ud835\udfd9_ C) (M.X, N.X) \u226b ((\u03bb_ M.X).hom \u2297 (\u03bb_ N.X).hom) =\n (\u03bb_ (M.X \u2297 N.X)).hom\n[PROOFSTEP]\nsymm\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (\u03bb_ (M.X \u2297 N.X)).hom =\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b tensor_\u03bc C (tensorUnit', \ud835\udfd9_ C) (M.X, N.X) \u226b ((\u03bb_ M.X).hom \u2297 (\u03bb_ N.X).hom)\n[PROOFSTEP]\nexact tensor_left_unitality C M.X N.X\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (\ud835\udfd9 (M.X \u2297 N.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) \u226b tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) =\n (\u03c1_ (M.X \u2297 N.X)).hom\n[PROOFSTEP]\nrw [\u2190 Category.id_comp (\ud835\udfd9 (M.X \u2297 N.X)), tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 ((\ud835\udfd9 (M.X \u2297 N.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\ud835\udfd9 (M.X \u2297 N.X) \u2297 M.one \u2297 N.one)) \u226b\n tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) =\n (\u03c1_ (M.X \u2297 N.X)).hom\n[PROOFSTEP]\nslice_lhs 2 3 => rw [\u2190 tensor_id, tensor_\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\ud835\udfd9 (M.X \u2297 N.X) \u2297 M.one \u2297 N.one) \u226b tensor_\u03bc C (M.X, N.X) (M.X, N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| \ud835\udfd9 (M.X \u2297 N.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 tensor_id, tensor_\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\ud835\udfd9 (M.X \u2297 N.X) \u2297 M.one \u2297 N.one) \u226b tensor_\u03bc C (M.X, N.X) (M.X, N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| \ud835\udfd9 (M.X \u2297 N.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 tensor_id, tensor_\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\ud835\udfd9 (M.X \u2297 N.X) \u2297 M.one \u2297 N.one) \u226b tensor_\u03bc C (M.X, N.X) (M.X, N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| \ud835\udfd9 (M.X \u2297 N.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 tensor_id, tensor_\u03bc_natural]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (\ud835\udfd9 (M.X \u2297 N.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b\n (tensor_\u03bc C (M.X, N.X) (tensorUnit', \ud835\udfd9_ C) \u226b ((\ud835\udfd9 M.X \u2297 M.one) \u2297 \ud835\udfd9 N.X \u2297 N.one)) \u226b (M.mul \u2297 N.mul) =\n (\u03c1_ (M.X \u2297 N.X)).hom\n[PROOFSTEP]\nslice_lhs 3 4 => rw [\u2190 tensor_comp, mul_one M, mul_one N]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| ((\ud835\udfd9 M.X \u2297 M.one) \u2297 \ud835\udfd9 N.X \u2297 N.one) \u226b (M.mul \u2297 N.mul)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| \ud835\udfd9 (M.X \u2297 N.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (tensorUnit', \ud835\udfd9_ C)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, mul_one M, mul_one N]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| ((\ud835\udfd9 M.X \u2297 M.one) \u2297 \ud835\udfd9 N.X \u2297 N.one) \u226b (M.mul \u2297 N.mul)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| \ud835\udfd9 (M.X \u2297 N.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (tensorUnit', \ud835\udfd9_ C)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, mul_one M, mul_one N]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| ((\ud835\udfd9 M.X \u2297 M.one) \u2297 \ud835\udfd9 N.X \u2297 N.one) \u226b (M.mul \u2297 N.mul)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| \ud835\udfd9 (M.X \u2297 N.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (tensorUnit', \ud835\udfd9_ C)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, mul_one M, mul_one N]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (\ud835\udfd9 (M.X \u2297 N.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b tensor_\u03bc C (M.X, N.X) (tensorUnit', \ud835\udfd9_ C) \u226b ((\u03c1_ M.X).hom \u2297 (\u03c1_ N.X).hom) =\n (\u03c1_ (M.X \u2297 N.X)).hom\n[PROOFSTEP]\nsymm\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (\u03c1_ (M.X \u2297 N.X)).hom =\n (\ud835\udfd9 (M.X \u2297 N.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b tensor_\u03bc C (M.X, N.X) (tensorUnit', \ud835\udfd9_ C) \u226b ((\u03c1_ M.X).hom \u2297 (\u03c1_ N.X).hom)\n[PROOFSTEP]\nexact tensor_right_unitality C M.X N.X\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b\n tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) =\n (\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom \u226b\n (\ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)) \u226b\n tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)\n[PROOFSTEP]\nrw [\u2190 Category.id_comp (\ud835\udfd9 (M.X \u2297 N.X)), tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 ((tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b ((M.mul \u2297 N.mul) \u2297 \ud835\udfd9 (M.X \u2297 N.X))) \u226b\n tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) =\n (\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom \u226b\n (\ud835\udfd9 (M.X \u2297 N.X) \u226b \ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)) \u226b\n tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)\n[PROOFSTEP]\nslice_lhs 2 3 => rw [\u2190 tensor_id, tensor_\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| ((M.mul \u2297 N.mul) \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b tensor_\u03bc C (M.X, N.X) (M.X, N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (M.X \u2297 N.X)\n[PROOFSTEP]\nrw [\u2190 tensor_id, tensor_\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| ((M.mul \u2297 N.mul) \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b tensor_\u03bc C (M.X, N.X) (M.X, N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (M.X \u2297 N.X)\n[PROOFSTEP]\nrw [\u2190 tensor_id, tensor_\u03bc_natural]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| ((M.mul \u2297 N.mul) \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b tensor_\u03bc C (M.X, N.X) (M.X, N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (M.X \u2297 N.X)\n[PROOFSTEP]\nrw [\u2190 tensor_id, tensor_\u03bc_natural]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b\n (tensor_\u03bc C (((M.X, N.X) \u2297 (M.X, N.X)).fst, ((M.X, N.X) \u2297 (M.X, N.X)).snd) (M.X, N.X) \u226b\n ((M.mul \u2297 \ud835\udfd9 M.X) \u2297 N.mul \u2297 \ud835\udfd9 N.X)) \u226b\n (M.mul \u2297 N.mul) =\n (\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom \u226b\n (\ud835\udfd9 (M.X \u2297 N.X) \u226b \ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)) \u226b\n tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)\n[PROOFSTEP]\nslice_lhs 3 4 =>\n rw [\u2190 tensor_comp, mul_assoc M, mul_assoc N, tensor_comp, tensor_comp]\n -- Porting note: needed to add `dsimp` here.\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| ((M.mul \u2297 \ud835\udfd9 M.X) \u2297 N.mul \u2297 \ud835\udfd9 N.X) \u226b (M.mul \u2297 N.mul)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (M.X \u2297 N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (((M.X, N.X) \u2297 (M.X, N.X)).fst, ((M.X, N.X) \u2297 (M.X, N.X)).snd) (M.X, N.X)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, mul_assoc M, mul_assoc N, tensor_comp, tensor_comp]\n -- Porting note: needed to add `dsimp` here.\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| ((M.mul \u2297 \ud835\udfd9 M.X) \u2297 N.mul \u2297 \ud835\udfd9 N.X) \u226b (M.mul \u2297 N.mul)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (M.X \u2297 N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (((M.X, N.X) \u2297 (M.X, N.X)).fst, ((M.X, N.X) \u2297 (M.X, N.X)).snd) (M.X, N.X)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, mul_assoc M, mul_assoc N, tensor_comp, tensor_comp]\n -- Porting note: needed to add `dsimp` here.\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| ((M.mul \u2297 \ud835\udfd9 M.X) \u2297 N.mul \u2297 \ud835\udfd9 N.X) \u226b (M.mul \u2297 N.mul)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (M.X \u2297 N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (((M.X, N.X) \u2297 (M.X, N.X)).fst, ((M.X, N.X) \u2297 (M.X, N.X)).snd) (M.X, N.X)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, mul_assoc M, mul_assoc N, tensor_comp, tensor_comp]\n -- Porting note: needed to add `dsimp` here.\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b\n tensor_\u03bc C (((M.X, N.X) \u2297 (M.X, N.X)).fst, ((M.X, N.X) \u2297 (M.X, N.X)).snd) (M.X, N.X) \u226b\n ((\u03b1_ M.X M.X M.X).hom \u2297 (\u03b1_ N.X N.X N.X).hom) \u226b ((\ud835\udfd9 M.X \u2297 M.mul) \u2297 \ud835\udfd9 N.X \u2297 N.mul) \u226b (M.mul \u2297 N.mul) =\n (\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom \u226b\n (\ud835\udfd9 (M.X \u2297 N.X) \u226b \ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)) \u226b\n tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)\n[PROOFSTEP]\nslice_lhs 1 3 => dsimp; rw [tensor_associativity]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b\n tensor_\u03bc C (((M.X, N.X) \u2297 (M.X, N.X)).fst, ((M.X, N.X) \u2297 (M.X, N.X)).snd) (M.X, N.X) \u226b\n ((\u03b1_ M.X M.X M.X).hom \u2297 (\u03b1_ N.X N.X N.X).hom)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\ud835\udfd9 M.X \u2297 M.mul) \u2297 \ud835\udfd9 N.X \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\n[PROOFSTEP]\ndsimp; rw [tensor_associativity]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b\n tensor_\u03bc C (((M.X, N.X) \u2297 (M.X, N.X)).fst, ((M.X, N.X) \u2297 (M.X, N.X)).snd) (M.X, N.X) \u226b\n ((\u03b1_ M.X M.X M.X).hom \u2297 (\u03b1_ N.X N.X N.X).hom)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\ud835\udfd9 M.X \u2297 M.mul) \u2297 \ud835\udfd9 N.X \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\n[PROOFSTEP]\ndsimp; rw [tensor_associativity]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b\n tensor_\u03bc C (((M.X, N.X) \u2297 (M.X, N.X)).fst, ((M.X, N.X) \u2297 (M.X, N.X)).snd) (M.X, N.X) \u226b\n ((\u03b1_ M.X M.X M.X).hom \u2297 (\u03b1_ N.X N.X N.X).hom)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\ud835\udfd9 M.X \u2297 M.mul) \u2297 \ud835\udfd9 N.X \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (M.X \u2297 N.X)) \u226b\n tensor_\u03bc C (M.X \u2297 M.X, N.X \u2297 N.X) (M.X, N.X) \u226b ((\u03b1_ M.X M.X M.X).hom \u2297 (\u03b1_ N.X N.X N.X).hom)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\ud835\udfd9 M.X \u2297 M.mul) \u2297 \ud835\udfd9 N.X \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\n[PROOFSTEP]\nrw [tensor_associativity]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (((\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom \u226b\n (\ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X)) \u226b tensor_\u03bc C (M.X, N.X) (M.X \u2297 M.X, N.X \u2297 N.X)) \u226b\n ((\ud835\udfd9 M.X \u2297 M.mul) \u2297 \ud835\udfd9 N.X \u2297 N.mul)) \u226b\n (M.mul \u2297 N.mul) =\n (\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom \u226b\n (\ud835\udfd9 (M.X \u2297 N.X) \u226b \ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)) \u226b\n tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)\n[PROOFSTEP]\nslice_lhs 3 4 => rw [\u2190 tensor_\u03bc_natural]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (M.X \u2297 M.X, N.X \u2297 N.X) \u226b ((\ud835\udfd9 M.X \u2297 M.mul) \u2297 \ud835\udfd9 N.X \u2297 N.mul)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| \ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X)\n[PROOFSTEP]\nrw [\u2190 tensor_\u03bc_natural]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (M.X \u2297 M.X, N.X \u2297 N.X) \u226b ((\ud835\udfd9 M.X \u2297 M.mul) \u2297 \ud835\udfd9 N.X \u2297 N.mul)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| \ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X)\n[PROOFSTEP]\nrw [\u2190 tensor_\u03bc_natural]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (M.X \u2297 M.X, N.X \u2297 N.X) \u226b ((\ud835\udfd9 M.X \u2297 M.mul) \u2297 \ud835\udfd9 N.X \u2297 N.mul)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| \ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X)\n[PROOFSTEP]\nrw [\u2190 tensor_\u03bc_natural]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom \u226b\n (\ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X)) \u226b\n (((\ud835\udfd9 M.X \u2297 \ud835\udfd9 N.X) \u2297 M.mul \u2297 N.mul) \u226b tensor_\u03bc C (M.X, N.X) (M.X, N.X)) \u226b (M.mul \u2297 N.mul) =\n (\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom \u226b\n (\ud835\udfd9 (M.X \u2297 N.X) \u226b \ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)) \u226b\n tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)\n[PROOFSTEP]\nslice_lhs 2 3 => rw [\u2190 tensor_comp, tensor_id]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X)) \u226b ((\ud835\udfd9 M.X \u2297 \ud835\udfd9 N.X) \u2297 M.mul \u2297 N.mul)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (M.X, N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, tensor_id]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X)) \u226b ((\ud835\udfd9 M.X \u2297 \ud835\udfd9 N.X) \u2297 M.mul \u2297 N.mul)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (M.X, N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, tensor_id]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X)) \u226b ((\ud835\udfd9 M.X \u2297 \ud835\udfd9 N.X) \u2297 M.mul \u2297 N.mul)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (M.X, N.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| M.mul \u2297 N.mul\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n| (\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, tensor_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N : Mon_ C\n\u22a2 (\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom \u226b\n ((\ud835\udfd9 (M.X \u2297 N.X) \u226b \ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)) \u226b\n tensor_\u03bc C (M.X, N.X) (M.X, N.X)) \u226b\n (M.mul \u2297 N.mul) =\n (\u03b1_ (M.X \u2297 N.X) (M.X \u2297 N.X) (M.X \u2297 N.X)).hom \u226b\n (\ud835\udfd9 (M.X \u2297 N.X) \u226b \ud835\udfd9 (M.X \u2297 N.X) \u2297 tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)) \u226b\n tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n\u22a2 (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n\u22a2 tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul) \u226b (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))\n[PROOFSTEP]\nslice_lhs 2 3 => rw [\u2190 Category.id_comp P.mul, tensor_comp]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul) \u226b (\u03b1_ M.X N.X P.X).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X)\n[PROOFSTEP]\nrw [\u2190 Category.id_comp P.mul, tensor_comp]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul) \u226b (\u03b1_ M.X N.X P.X).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X)\n[PROOFSTEP]\nrw [\u2190 Category.id_comp P.mul, tensor_comp]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul) \u226b (\u03b1_ M.X N.X P.X).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X)\n[PROOFSTEP]\nrw [\u2190 Category.id_comp P.mul, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n\u22a2 tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n ((tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (P.X \u2297 P.X)) \u226b ((M.mul \u2297 N.mul) \u2297 P.mul)) \u226b (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))\n[PROOFSTEP]\nslice_lhs 3 4 => rw [associator_naturality]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| ((M.mul \u2297 N.mul) \u2297 P.mul) \u226b (\u03b1_ M.X N.X P.X).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (P.X \u2297 P.X)\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| ((M.mul \u2297 N.mul) \u2297 P.mul) \u226b (\u03b1_ M.X N.X P.X).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (P.X \u2297 P.X)\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| ((M.mul \u2297 N.mul) \u2297 P.mul) \u226b (\u03b1_ M.X N.X P.X).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (P.X \u2297 P.X)\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n\u22a2 tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (P.X \u2297 P.X)) \u226b\n (\u03b1_ (M.X \u2297 M.X) (N.X \u2297 N.X) (P.X \u2297 P.X)).hom \u226b (M.mul \u2297 N.mul \u2297 P.mul) =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))\n[PROOFSTEP]\nslice_rhs 3 4 => rw [\u2190 Category.id_comp M.mul, tensor_comp]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| (\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X)\n[PROOFSTEP]\nrw [\u2190 Category.id_comp M.mul, tensor_comp]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| (\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X)\n[PROOFSTEP]\nrw [\u2190 Category.id_comp M.mul, tensor_comp]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| (\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X)\n[PROOFSTEP]\nrw [\u2190 Category.id_comp M.mul, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n\u22a2 tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (P.X \u2297 P.X)) \u226b\n (\u03b1_ (M.X \u2297 M.X) (N.X \u2297 N.X) (P.X \u2297 P.X)).hom \u226b (M.mul \u2297 N.mul \u2297 P.mul) =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (\ud835\udfd9 (M.X \u2297 M.X) \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X)) \u226b (M.mul \u2297 N.mul \u2297 P.mul)\n[PROOFSTEP]\nslice_lhs 1 3 => rw [associator_monoidal]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (P.X \u2297 P.X)) \u226b (\u03b1_ (M.X \u2297 M.X) (N.X \u2297 N.X) (P.X \u2297 P.X)).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| M.mul \u2297 N.mul \u2297 P.mul\n[PROOFSTEP]\nrw [associator_monoidal]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (P.X \u2297 P.X)) \u226b (\u03b1_ (M.X \u2297 M.X) (N.X \u2297 N.X) (P.X \u2297 P.X)).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| M.mul \u2297 N.mul \u2297 P.mul\n[PROOFSTEP]\nrw [associator_monoidal]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u2297 \ud835\udfd9 (P.X \u2297 P.X)) \u226b (\u03b1_ (M.X \u2297 M.X) (N.X \u2297 N.X) (P.X \u2297 P.X)).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n| M.mul \u2297 N.mul \u2297 P.mul\n[PROOFSTEP]\nrw [associator_monoidal]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM N P : Mon_ C\n\u22a2 (((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b (\ud835\udfd9 (M.X \u2297 M.X) \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X))) \u226b\n (M.mul \u2297 N.mul \u2297 P.mul) =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (\ud835\udfd9 (M.X \u2297 M.X) \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X)) \u226b (M.mul \u2297 N.mul \u2297 P.mul)\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 (tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 M.mul)) \u226b (\u03bb_ M.X).hom =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b M.mul\n[PROOFSTEP]\nrw [\u2190 Category.comp_id (\u03bb_ (\ud835\udfd9_ C)).hom, \u2190 Category.id_comp M.mul, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 (tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (M.X \u2297 M.X)) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.mul)) \u226b (\u03bb_ M.X).hom =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul\n[PROOFSTEP]\nslice_lhs 3 4 => rw [leftUnitor_naturality]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.mul) \u226b (\u03bb_ M.X).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| (\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (M.X \u2297 M.X)\n[PROOFSTEP]\nrw [leftUnitor_naturality]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.mul) \u226b (\u03bb_ M.X).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| (\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (M.X \u2297 M.X)\n[PROOFSTEP]\nrw [leftUnitor_naturality]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.mul) \u226b (\u03bb_ M.X).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| (\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (M.X \u2297 M.X)\n[PROOFSTEP]\nrw [leftUnitor_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (M.X \u2297 M.X)) \u226b (\u03bb_ (M.X \u2297 M.X)).hom \u226b M.mul =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul\n[PROOFSTEP]\nslice_lhs 1 3 => rw [\u2190 leftUnitor_monoidal]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (M.X \u2297 M.X)) \u226b (\u03bb_ (M.X \u2297 M.X)).hom\ncase a C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C inst\u271d\u00b9 : MonoidalCategory C inst\u271d : BraidedCategory C M : Mon_ C | M.mul\n[PROOFSTEP]\nrw [\u2190 leftUnitor_monoidal]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (M.X \u2297 M.X)) \u226b (\u03bb_ (M.X \u2297 M.X)).hom\ncase a C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C inst\u271d\u00b9 : MonoidalCategory C inst\u271d : BraidedCategory C M : Mon_ C | M.mul\n[PROOFSTEP]\nrw [\u2190 leftUnitor_monoidal]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (M.X \u2297 M.X)) \u226b (\u03bb_ (M.X \u2297 M.X)).hom\ncase a C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C inst\u271d\u00b9 : MonoidalCategory C inst\u271d : BraidedCategory C M : Mon_ C | M.mul\n[PROOFSTEP]\nrw [\u2190 leftUnitor_monoidal]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b M.mul = ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul\n[PROOFSTEP]\nsimp only [Category.assoc, Category.id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.mul \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)) \u226b (\u03c1_ M.X).hom =\n ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul\n[PROOFSTEP]\nrw [\u2190 Category.id_comp M.mul, \u2190 Category.comp_id (\u03bb_ (\ud835\udfd9_ C)).hom, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (\ud835\udfd9 (M.X \u2297 M.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (M.mul \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ M.X).hom =\n ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul\n[PROOFSTEP]\nslice_lhs 3 4 => rw [rightUnitor_naturality]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| (M.mul \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03c1_ M.X).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| \ud835\udfd9 (M.X \u2297 M.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom\n[PROOFSTEP]\nrw [rightUnitor_naturality]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| (M.mul \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03c1_ M.X).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| \ud835\udfd9 (M.X \u2297 M.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom\n[PROOFSTEP]\nrw [rightUnitor_naturality]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| (M.mul \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03c1_ M.X).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| \ud835\udfd9 (M.X \u2297 M.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom\n[PROOFSTEP]\nrw [rightUnitor_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (\ud835\udfd9 (M.X \u2297 M.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03c1_ (M.X \u2297 M.X)).hom \u226b M.mul =\n ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul\n[PROOFSTEP]\nslice_lhs 1 3 => rw [\u2190 rightUnitor_monoidal]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (\ud835\udfd9 (M.X \u2297 M.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03c1_ (M.X \u2297 M.X)).hom\ncase a C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C inst\u271d\u00b9 : MonoidalCategory C inst\u271d : BraidedCategory C M : Mon_ C | M.mul\n[PROOFSTEP]\nrw [\u2190 rightUnitor_monoidal]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (\ud835\udfd9 (M.X \u2297 M.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03c1_ (M.X \u2297 M.X)).hom\ncase a C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C inst\u271d\u00b9 : MonoidalCategory C inst\u271d : BraidedCategory C M : Mon_ C | M.mul\n[PROOFSTEP]\nrw [\u2190 rightUnitor_monoidal]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n| tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (\ud835\udfd9 (M.X \u2297 M.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03c1_ (M.X \u2297 M.X)).hom\ncase a C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C inst\u271d\u00b9 : MonoidalCategory C inst\u271d : BraidedCategory C M : Mon_ C | M.mul\n[PROOFSTEP]\nrw [\u2190 rightUnitor_monoidal]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul = ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul\n[PROOFSTEP]\nsimp only [Category.assoc, Category.id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n\u22a2 ((fun M N => mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n X\u2081\u271d X\u2082\u271d).one \u226b\n (f.hom \u2297 g.hom) =\n ((fun M N =>\n mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n Y\u2081\u271d Y\u2082\u271d).one\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n\u22a2 ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u2081\u271d.one \u2297 X\u2082\u271d.one)) \u226b (f.hom \u2297 g.hom) = (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u2081\u271d.one \u2297 Y\u2082\u271d.one)\n[PROOFSTEP]\nslice_lhs 2 3 => rw [\u2190 tensor_comp, Hom.one_hom f, Hom.one_hom g]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| (X\u2081\u271d.one \u2297 X\u2082\u271d.one) \u226b (f.hom \u2297 g.hom)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| (\u03bb_ (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, Hom.one_hom f, Hom.one_hom g]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| (X\u2081\u271d.one \u2297 X\u2082\u271d.one) \u226b (f.hom \u2297 g.hom)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| (\u03bb_ (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, Hom.one_hom f, Hom.one_hom g]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| (X\u2081\u271d.one \u2297 X\u2082\u271d.one) \u226b (f.hom \u2297 g.hom)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| (\u03bb_ (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, Hom.one_hom f, Hom.one_hom g]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n\u22a2 ((fun M N => mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n X\u2081\u271d X\u2082\u271d).mul \u226b\n (f.hom \u2297 g.hom) =\n ((f.hom \u2297 g.hom) \u2297 f.hom \u2297 g.hom) \u226b\n ((fun M N =>\n mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n Y\u2081\u271d Y\u2082\u271d).mul\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n\u22a2 (tensor_\u03bc C (X\u2081\u271d.X, X\u2082\u271d.X) (X\u2081\u271d.X, X\u2082\u271d.X) \u226b (X\u2081\u271d.mul \u2297 X\u2082\u271d.mul)) \u226b (f.hom \u2297 g.hom) =\n ((f.hom \u2297 g.hom) \u2297 f.hom \u2297 g.hom) \u226b tensor_\u03bc C (Y\u2081\u271d.X, Y\u2082\u271d.X) (Y\u2081\u271d.X, Y\u2082\u271d.X) \u226b (Y\u2081\u271d.mul \u2297 Y\u2082\u271d.mul)\n[PROOFSTEP]\nslice_rhs 1 2 => rw [tensor_\u03bc_natural]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| ((f.hom \u2297 g.hom) \u2297 f.hom \u2297 g.hom) \u226b tensor_\u03bc C (Y\u2081\u271d.X, Y\u2082\u271d.X) (Y\u2081\u271d.X, Y\u2082\u271d.X)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| Y\u2081\u271d.mul \u2297 Y\u2082\u271d.mul\n[PROOFSTEP]\nrw [tensor_\u03bc_natural]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| ((f.hom \u2297 g.hom) \u2297 f.hom \u2297 g.hom) \u226b tensor_\u03bc C (Y\u2081\u271d.X, Y\u2082\u271d.X) (Y\u2081\u271d.X, Y\u2082\u271d.X)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| Y\u2081\u271d.mul \u2297 Y\u2082\u271d.mul\n[PROOFSTEP]\nrw [tensor_\u03bc_natural]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| ((f.hom \u2297 g.hom) \u2297 f.hom \u2297 g.hom) \u226b tensor_\u03bc C (Y\u2081\u271d.X, Y\u2082\u271d.X) (Y\u2081\u271d.X, Y\u2082\u271d.X)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| Y\u2081\u271d.mul \u2297 Y\u2082\u271d.mul\n[PROOFSTEP]\nrw [tensor_\u03bc_natural]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n\u22a2 (tensor_\u03bc C (X\u2081\u271d.X, X\u2082\u271d.X) (X\u2081\u271d.X, X\u2082\u271d.X) \u226b (X\u2081\u271d.mul \u2297 X\u2082\u271d.mul)) \u226b (f.hom \u2297 g.hom) =\n (tensor_\u03bc C (X\u2081\u271d.X, X\u2082\u271d.X) (X\u2081\u271d.X, X\u2082\u271d.X) \u226b ((f.hom \u2297 f.hom) \u2297 g.hom \u2297 g.hom)) \u226b (Y\u2081\u271d.mul \u2297 Y\u2082\u271d.mul)\n[PROOFSTEP]\nslice_lhs 2 3 => rw [\u2190 tensor_comp, Hom.mul_hom f, Hom.mul_hom g, tensor_comp]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| (X\u2081\u271d.mul \u2297 X\u2082\u271d.mul) \u226b (f.hom \u2297 g.hom)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| tensor_\u03bc C (X\u2081\u271d.X, X\u2082\u271d.X) (X\u2081\u271d.X, X\u2082\u271d.X)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, Hom.mul_hom f, Hom.mul_hom g, tensor_comp]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| (X\u2081\u271d.mul \u2297 X\u2082\u271d.mul) \u226b (f.hom \u2297 g.hom)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| tensor_\u03bc C (X\u2081\u271d.X, X\u2082\u271d.X) (X\u2081\u271d.X, X\u2082\u271d.X)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, Hom.mul_hom f, Hom.mul_hom g, tensor_comp]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| (X\u2081\u271d.mul \u2297 X\u2082\u271d.mul) \u226b (f.hom \u2297 g.hom)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n| tensor_\u03bc C (X\u2081\u271d.X, X\u2082\u271d.X) (X\u2081\u271d.X, X\u2082\u271d.X)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, Hom.mul_hom f, Hom.mul_hom g, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Mon_ C\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n\u22a2 tensor_\u03bc C (X\u2081\u271d.X, X\u2082\u271d.X) (X\u2081\u271d.X, X\u2082\u271d.X) \u226b ((f.hom \u2297 f.hom) \u2297 g.hom \u2297 g.hom) \u226b (Y\u2081\u271d.mul \u2297 Y\u2082\u271d.mul) =\n (tensor_\u03bc C (X\u2081\u271d.X, X\u2082\u271d.X) (X\u2081\u271d.X, X\u2082\u271d.X) \u226b ((f.hom \u2297 f.hom) \u2297 g.hom \u2297 g.hom)) \u226b (Y\u2081\u271d.mul \u2297 Y\u2082\u271d.mul)\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\n\u22a2 \u2200 (X\u2081 X\u2082 : Mon_ C),\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) (\ud835\udfd9 X\u2081) (\ud835\udfd9 X\u2082) =\n \ud835\udfd9\n ((fun M N =>\n mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n X\u2081 X\u2082)\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d X\u2082\u271d : Mon_ C\n\u22a2 (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) (\ud835\udfd9 X\u2081\u271d) (\ud835\udfd9 X\u2082\u271d) =\n \ud835\udfd9\n ((fun M N =>\n mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n X\u2081\u271d X\u2082\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d X\u2082\u271d : Mon_ C\n\u22a2 ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) (\ud835\udfd9 X\u2081\u271d) (\ud835\udfd9 X\u2082\u271d)).hom =\n (\ud835\udfd9\n ((fun M N =>\n mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n X\u2081\u271d X\u2082\u271d)).hom\n[PROOFSTEP]\napply tensor_id\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\n\u22a2 \u2200 {X\u2081 Y\u2081 Z\u2081 X\u2082 Y\u2082 Z\u2082 : Mon_ C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (g\u2081 : Y\u2081 \u27f6 Z\u2081) (g\u2082 : Y\u2082 \u27f6 Z\u2082),\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) (f\u2081 \u226b g\u2081) (f\u2082 \u226b g\u2082) =\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) f\u2081 f\u2082 \u226b (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) g\u2081 g\u2082\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d Z\u2081\u271d X\u2082\u271d Y\u2082\u271d Z\u2082\u271d : Mon_ C\nf\u2081\u271d : X\u2081\u271d \u27f6 Y\u2081\u271d\nf\u2082\u271d : X\u2082\u271d \u27f6 Y\u2082\u271d\ng\u2081\u271d : Y\u2081\u271d \u27f6 Z\u2081\u271d\ng\u2082\u271d : Y\u2082\u271d \u27f6 Z\u2082\u271d\n\u22a2 (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) (f\u2081\u271d \u226b g\u2081\u271d) (f\u2082\u271d \u226b g\u2082\u271d) =\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) f\u2081\u271d f\u2082\u271d \u226b\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) g\u2081\u271d g\u2082\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d Y\u2081\u271d Z\u2081\u271d X\u2082\u271d Y\u2082\u271d Z\u2082\u271d : Mon_ C\nf\u2081\u271d : X\u2081\u271d \u27f6 Y\u2081\u271d\nf\u2082\u271d : X\u2082\u271d \u27f6 Y\u2082\u271d\ng\u2081\u271d : Y\u2081\u271d \u27f6 Z\u2081\u271d\ng\u2082\u271d : Y\u2082\u271d \u27f6 Z\u2082\u271d\n\u22a2 ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) (f\u2081\u271d \u226b g\u2081\u271d) (f\u2082\u271d \u226b g\u2082\u271d)).hom =\n ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) f\u2081\u271d f\u2082\u271d \u226b\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) g\u2081\u271d g\u2082\u271d).hom\n[PROOFSTEP]\napply tensor_comp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\n\u22a2 \u2200 {X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : Mon_ C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083),\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) f\u2081 f\u2082) f\u2083 \u226b\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n Y\u2081 Y\u2082 Y\u2083).hom =\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n X\u2081 X\u2082 X\u2083).hom \u226b\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) f\u2081 ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) f\u2082 f\u2083)\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d X\u2082\u271d X\u2083\u271d Y\u2081\u271d Y\u2082\u271d Y\u2083\u271d : Mon_ C\nf\u2081\u271d : X\u2081\u271d \u27f6 Y\u2081\u271d\nf\u2082\u271d : X\u2082\u271d \u27f6 Y\u2082\u271d\nf\u2083\u271d : X\u2083\u271d \u27f6 Y\u2083\u271d\n\u22a2 (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) f\u2081\u271d f\u2082\u271d) f\u2083\u271d \u226b\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n Y\u2081\u271d Y\u2082\u271d Y\u2083\u271d).hom =\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n X\u2081\u271d X\u2082\u271d X\u2083\u271d).hom \u226b\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) f\u2081\u271d ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) f\u2082\u271d f\u2083\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d X\u2082\u271d X\u2083\u271d Y\u2081\u271d Y\u2082\u271d Y\u2083\u271d : Mon_ C\nf\u2081\u271d : X\u2081\u271d \u27f6 Y\u2081\u271d\nf\u2082\u271d : X\u2082\u271d \u27f6 Y\u2082\u271d\nf\u2083\u271d : X\u2083\u271d \u27f6 Y\u2083\u271d\n\u22a2 ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) f\u2081\u271d f\u2082\u271d) f\u2083\u271d \u226b\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n Y\u2081\u271d Y\u2082\u271d Y\u2083\u271d).hom).hom =\n (((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n X\u2081\u271d X\u2082\u271d X\u2083\u271d).hom \u226b\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) f\u2081\u271d\n ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) f\u2082\u271d f\u2083\u271d)).hom\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u2081\u271d X\u2082\u271d X\u2083\u271d Y\u2081\u271d Y\u2082\u271d Y\u2083\u271d : Mon_ C\nf\u2081\u271d : X\u2081\u271d \u27f6 Y\u2081\u271d\nf\u2082\u271d : X\u2082\u271d \u27f6 Y\u2082\u271d\nf\u2083\u271d : X\u2083\u271d \u27f6 Y\u2083\u271d\n\u22a2 ((f\u2081\u271d.hom \u2297 f\u2082\u271d.hom) \u2297 f\u2083\u271d.hom) \u226b\n (isoOfIso (\u03b1_ Y\u2081\u271d.X Y\u2082\u271d.X Y\u2083\u271d.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u2081\u271d.one \u2297 Y\u2082\u271d.one) \u2297 Y\u2083\u271d.one)) \u226b (\u03b1_ Y\u2081\u271d.X Y\u2082\u271d.X Y\u2083\u271d.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u2081\u271d.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u2082\u271d.one \u2297 Y\u2083\u271d.one)))\n (_ :\n (tensor_\u03bc C (Y\u2081\u271d.X \u2297 Y\u2082\u271d.X, Y\u2083\u271d.X) (Y\u2081\u271d.X \u2297 Y\u2082\u271d.X, Y\u2083\u271d.X) \u226b\n (tensor_\u03bc C (Y\u2081\u271d.X, Y\u2082\u271d.X) (Y\u2081\u271d.X, Y\u2082\u271d.X) \u226b (Y\u2081\u271d.mul \u2297 Y\u2082\u271d.mul) \u2297 Y\u2083\u271d.mul)) \u226b\n (\u03b1_ Y\u2081\u271d.X Y\u2082\u271d.X Y\u2083\u271d.X).hom =\n ((\u03b1_ Y\u2081\u271d.X Y\u2082\u271d.X Y\u2083\u271d.X).hom \u2297 (\u03b1_ Y\u2081\u271d.X Y\u2082\u271d.X Y\u2083\u271d.X).hom) \u226b\n tensor_\u03bc C (Y\u2081\u271d.X, Y\u2082\u271d.X \u2297 Y\u2083\u271d.X) (Y\u2081\u271d.X, Y\u2082\u271d.X \u2297 Y\u2083\u271d.X) \u226b\n (Y\u2081\u271d.mul \u2297 tensor_\u03bc C (Y\u2082\u271d.X, Y\u2083\u271d.X) (Y\u2082\u271d.X, Y\u2083\u271d.X) \u226b (Y\u2082\u271d.mul \u2297 Y\u2083\u271d.mul)))).hom.hom =\n (isoOfIso (\u03b1_ X\u2081\u271d.X X\u2082\u271d.X X\u2083\u271d.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u2081\u271d.one \u2297 X\u2082\u271d.one) \u2297 X\u2083\u271d.one)) \u226b (\u03b1_ X\u2081\u271d.X X\u2082\u271d.X X\u2083\u271d.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u2081\u271d.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u2082\u271d.one \u2297 X\u2083\u271d.one)))\n (_ :\n (tensor_\u03bc C (X\u2081\u271d.X \u2297 X\u2082\u271d.X, X\u2083\u271d.X) (X\u2081\u271d.X \u2297 X\u2082\u271d.X, X\u2083\u271d.X) \u226b\n (tensor_\u03bc C (X\u2081\u271d.X, X\u2082\u271d.X) (X\u2081\u271d.X, X\u2082\u271d.X) \u226b (X\u2081\u271d.mul \u2297 X\u2082\u271d.mul) \u2297 X\u2083\u271d.mul)) \u226b\n (\u03b1_ X\u2081\u271d.X X\u2082\u271d.X X\u2083\u271d.X).hom =\n ((\u03b1_ X\u2081\u271d.X X\u2082\u271d.X X\u2083\u271d.X).hom \u2297 (\u03b1_ X\u2081\u271d.X X\u2082\u271d.X X\u2083\u271d.X).hom) \u226b\n tensor_\u03bc C (X\u2081\u271d.X, X\u2082\u271d.X \u2297 X\u2083\u271d.X) (X\u2081\u271d.X, X\u2082\u271d.X \u2297 X\u2083\u271d.X) \u226b\n (X\u2081\u271d.mul \u2297 tensor_\u03bc C (X\u2082\u271d.X, X\u2083\u271d.X) (X\u2082\u271d.X, X\u2083\u271d.X) \u226b (X\u2082\u271d.mul \u2297 X\u2083\u271d.mul)))).hom.hom \u226b\n (f\u2081\u271d.hom \u2297 f\u2082\u271d.hom \u2297 f\u2083\u271d.hom)\n[PROOFSTEP]\napply associator_naturality\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\n\u22a2 \u2200 {X Y : Mon_ C} (f : X \u27f6 Y),\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) (\ud835\udfd9 (trivial C)) f \u226b\n ((fun M =>\n isoOfIso (\u03bb_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.one)) \u226b (\u03bb_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 M.mul)) \u226b (\u03bb_ M.X).hom =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b M.mul))\n Y).hom =\n ((fun M =>\n isoOfIso (\u03bb_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.one)) \u226b (\u03bb_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 M.mul)) \u226b (\u03bb_ M.X).hom =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b M.mul))\n X).hom \u226b\n f\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u271d Y\u271d : Mon_ C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) (\ud835\udfd9 (trivial C)) f\u271d \u226b\n ((fun M =>\n isoOfIso (\u03bb_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.one)) \u226b (\u03bb_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 M.mul)) \u226b (\u03bb_ M.X).hom =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b M.mul))\n Y\u271d).hom =\n ((fun M =>\n isoOfIso (\u03bb_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.one)) \u226b (\u03bb_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 M.mul)) \u226b (\u03bb_ M.X).hom =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b M.mul))\n X\u271d).hom \u226b\n f\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u271d Y\u271d : Mon_ C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) (\ud835\udfd9 (trivial C)) f\u271d \u226b\n ((fun M =>\n isoOfIso (\u03bb_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.one)) \u226b (\u03bb_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 M.mul)) \u226b (\u03bb_ M.X).hom =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b M.mul))\n Y\u271d).hom).hom =\n (((fun M =>\n isoOfIso (\u03bb_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.one)) \u226b (\u03bb_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 M.mul)) \u226b (\u03bb_ M.X).hom =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b M.mul))\n X\u271d).hom \u226b\n f\u271d).hom\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u271d Y\u271d : Mon_ C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 f\u271d.hom) \u226b\n (isoOfIso (\u03bb_ Y\u271d.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 Y\u271d.one)) \u226b (\u03bb_ Y\u271d.X).hom = Y\u271d.one)\n (_ :\n (tensor_\u03bc C (\ud835\udfd9_ C, Y\u271d.X) (\ud835\udfd9_ C, Y\u271d.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 Y\u271d.mul)) \u226b (\u03bb_ Y\u271d.X).hom =\n ((\u03bb_ Y\u271d.X).hom \u2297 (\u03bb_ Y\u271d.X).hom) \u226b Y\u271d.mul)).hom.hom =\n (isoOfIso (\u03bb_ X\u271d.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 X\u271d.one)) \u226b (\u03bb_ X\u271d.X).hom = X\u271d.one)\n (_ :\n (tensor_\u03bc C (\ud835\udfd9_ C, X\u271d.X) (\ud835\udfd9_ C, X\u271d.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 X\u271d.mul)) \u226b (\u03bb_ X\u271d.X).hom =\n ((\u03bb_ X\u271d.X).hom \u2297 (\u03bb_ X\u271d.X).hom) \u226b X\u271d.mul)).hom.hom \u226b\n f\u271d.hom\n[PROOFSTEP]\napply leftUnitor_naturality\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\n\u22a2 \u2200 {X Y : Mon_ C} (f : X \u27f6 Y),\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) f (\ud835\udfd9 (trivial C)) \u226b\n ((fun M =>\n isoOfIso (\u03c1_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.mul \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)) \u226b (\u03c1_ M.X).hom =\n ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul))\n Y).hom =\n ((fun M =>\n isoOfIso (\u03c1_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.mul \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)) \u226b (\u03c1_ M.X).hom =\n ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul))\n X).hom \u226b\n f\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u271d Y\u271d : Mon_ C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) f\u271d (\ud835\udfd9 (trivial C)) \u226b\n ((fun M =>\n isoOfIso (\u03c1_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.mul \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)) \u226b (\u03c1_ M.X).hom =\n ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul))\n Y\u271d).hom =\n ((fun M =>\n isoOfIso (\u03c1_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.mul \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)) \u226b (\u03c1_ M.X).hom =\n ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul))\n X\u271d).hom \u226b\n f\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u271d Y\u271d : Mon_ C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) f\u271d (\ud835\udfd9 (trivial C)) \u226b\n ((fun M =>\n isoOfIso (\u03c1_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.mul \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)) \u226b (\u03c1_ M.X).hom =\n ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul))\n Y\u271d).hom).hom =\n (((fun M =>\n isoOfIso (\u03c1_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.mul \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)) \u226b (\u03c1_ M.X).hom =\n ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul))\n X\u271d).hom \u226b\n f\u271d).hom\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u271d Y\u271d : Mon_ C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (f\u271d.hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b\n (isoOfIso (\u03c1_ Y\u271d.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ Y\u271d.X).hom = Y\u271d.one)\n (_ :\n (tensor_\u03bc C (Y\u271d.X, \ud835\udfd9_ C) (Y\u271d.X, \ud835\udfd9_ C) \u226b (Y\u271d.mul \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)) \u226b (\u03c1_ Y\u271d.X).hom =\n ((\u03c1_ Y\u271d.X).hom \u2297 (\u03c1_ Y\u271d.X).hom) \u226b Y\u271d.mul)).hom.hom =\n (isoOfIso (\u03c1_ X\u271d.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ X\u271d.X).hom = X\u271d.one)\n (_ :\n (tensor_\u03bc C (X\u271d.X, \ud835\udfd9_ C) (X\u271d.X, \ud835\udfd9_ C) \u226b (X\u271d.mul \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)) \u226b (\u03c1_ X\u271d.X).hom =\n ((\u03c1_ X\u271d.X).hom \u2297 (\u03c1_ X\u271d.X).hom) \u226b X\u271d.mul)).hom.hom \u226b\n f\u271d.hom\n[PROOFSTEP]\napply rightUnitor_naturality\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\n\u22a2 \u2200 (W X Y Z : Mon_ C),\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom))\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n W X Y).hom\n (\ud835\udfd9 Z) \u226b\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n W\n ((fun M N =>\n mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one))\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n X Y)\n Z).hom \u226b\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) (\ud835\udfd9 W)\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n X Y Z).hom =\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n ((fun M N =>\n mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n W X)\n Y Z).hom \u226b\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n W X\n ((fun M N =>\n mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n Y Z)).hom\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nW\u271d X\u271d Y\u271d Z\u271d : Mon_ C\n\u22a2 (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom))\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n W\u271d X\u271d Y\u271d).hom\n (\ud835\udfd9 Z\u271d) \u226b\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n W\u271d\n ((fun M N =>\n mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n X\u271d Y\u271d)\n Z\u271d).hom \u226b\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) (\ud835\udfd9 W\u271d)\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n X\u271d Y\u271d Z\u271d).hom =\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n ((fun M N =>\n mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n W\u271d X\u271d)\n Y\u271d Z\u271d).hom \u226b\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n W\u271d X\u271d\n ((fun M N =>\n mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n Y\u271d Z\u271d)).hom\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nW\u271d X\u271d Y\u271d Z\u271d : Mon_ C\n\u22a2 ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom))\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n W\u271d X\u271d Y\u271d).hom\n (\ud835\udfd9 Z\u271d) \u226b\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n W\u271d\n ((fun M N =>\n mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one))\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n X\u271d Y\u271d)\n Z\u271d).hom \u226b\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) (\ud835\udfd9 W\u271d)\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n X\u271d Y\u271d Z\u271d).hom).hom =\n (((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n ((fun M N =>\n mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n W\u271d X\u271d)\n Y\u271d Z\u271d).hom \u226b\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n W\u271d X\u271d\n ((fun M N =>\n mk (M.X \u2297 N.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one)) (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul)))\n Y\u271d Z\u271d)).hom).hom\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nW\u271d X\u271d Y\u271d Z\u271d : Mon_ C\n\u22a2 ((isoOfIso (\u03b1_ W\u271d.X X\u271d.X Y\u271d.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one) \u2297 Y\u271d.one)) \u226b (\u03b1_ W\u271d.X X\u271d.X Y\u271d.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one)))\n (_ :\n (tensor_\u03bc C (W\u271d.X \u2297 X\u271d.X, Y\u271d.X) (W\u271d.X \u2297 X\u271d.X, Y\u271d.X) \u226b\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul) \u2297 Y\u271d.mul)) \u226b\n (\u03b1_ W\u271d.X X\u271d.X Y\u271d.X).hom =\n ((\u03b1_ W\u271d.X X\u271d.X Y\u271d.X).hom \u2297 (\u03b1_ W\u271d.X X\u271d.X Y\u271d.X).hom) \u226b\n tensor_\u03bc C (W\u271d.X, X\u271d.X \u2297 Y\u271d.X) (W\u271d.X, X\u271d.X \u2297 Y\u271d.X) \u226b\n (W\u271d.mul \u2297 tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul)))).hom.hom \u2297\n \ud835\udfd9 Z\u271d.X) \u226b\n (isoOfIso (\u03b1_ W\u271d.X (X\u271d.X \u2297 Y\u271d.X) Z\u271d.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b\n (W\u271d.one \u2297\n (mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).one) \u2297\n Z\u271d.one)) \u226b\n (\u03b1_ W\u271d.X\n (mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).X\n Z\u271d.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b\n (W\u271d.one \u2297\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b\n ((mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).one \u2297\n Z\u271d.one)))\n (_ :\n (tensor_\u03bc C\n (W\u271d.X \u2297\n (mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).X,\n Z\u271d.X)\n (W\u271d.X \u2297\n (mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).X,\n Z\u271d.X) \u226b\n (tensor_\u03bc C\n (W\u271d.X,\n (mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).X)\n (W\u271d.X,\n (mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).X) \u226b\n (W\u271d.mul \u2297\n (mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).mul) \u2297\n Z\u271d.mul)) \u226b\n (\u03b1_ W\u271d.X\n (mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).X\n Z\u271d.X).hom =\n ((\u03b1_ W\u271d.X\n (mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).X\n Z\u271d.X).hom \u2297\n (\u03b1_ W\u271d.X\n (mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).X\n Z\u271d.X).hom) \u226b\n tensor_\u03bc C\n (W\u271d.X,\n (mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).X \u2297\n Z\u271d.X)\n (W\u271d.X,\n (mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).X \u2297\n Z\u271d.X) \u226b\n (W\u271d.mul \u2297\n tensor_\u03bc C\n ((mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).X,\n Z\u271d.X)\n ((mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).X,\n Z\u271d.X) \u226b\n ((mk (X\u271d.X \u2297 Y\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one))\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul))).mul \u2297\n Z\u271d.mul)))).hom.hom \u226b\n (\ud835\udfd9 W\u271d.X \u2297\n (isoOfIso (\u03b1_ X\u271d.X Y\u271d.X Z\u271d.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 Y\u271d.one) \u2297 Z\u271d.one)) \u226b (\u03b1_ X\u271d.X Y\u271d.X Z\u271d.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one)))\n (_ :\n (tensor_\u03bc C (X\u271d.X \u2297 Y\u271d.X, Z\u271d.X) (X\u271d.X \u2297 Y\u271d.X, Z\u271d.X) \u226b\n (tensor_\u03bc C (X\u271d.X, Y\u271d.X) (X\u271d.X, Y\u271d.X) \u226b (X\u271d.mul \u2297 Y\u271d.mul) \u2297 Z\u271d.mul)) \u226b\n (\u03b1_ X\u271d.X Y\u271d.X Z\u271d.X).hom =\n ((\u03b1_ X\u271d.X Y\u271d.X Z\u271d.X).hom \u2297 (\u03b1_ X\u271d.X Y\u271d.X Z\u271d.X).hom) \u226b\n tensor_\u03bc C (X\u271d.X, Y\u271d.X \u2297 Z\u271d.X) (X\u271d.X, Y\u271d.X \u2297 Z\u271d.X) \u226b\n (X\u271d.mul \u2297 tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul)))).hom.hom) =\n (isoOfIso (\u03b1_ (W\u271d.X \u2297 X\u271d.X) Y\u271d.X Z\u271d.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b\n ((mk (W\u271d.X \u2297 X\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one))\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul))).one \u2297\n Y\u271d.one) \u2297\n Z\u271d.one)) \u226b\n (\u03b1_\n (mk (W\u271d.X \u2297 X\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one))\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul))).X\n Y\u271d.X Z\u271d.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b\n ((mk (W\u271d.X \u2297 X\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one))\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul))).one \u2297\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one)))\n (_ :\n (tensor_\u03bc C\n ((mk (W\u271d.X \u2297 X\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one))\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul))).X \u2297\n Y\u271d.X,\n Z\u271d.X)\n ((mk (W\u271d.X \u2297 X\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one))\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul))).X \u2297\n Y\u271d.X,\n Z\u271d.X) \u226b\n (tensor_\u03bc C\n ((mk (W\u271d.X \u2297 X\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one))\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul))).X,\n Y\u271d.X)\n ((mk (W\u271d.X \u2297 X\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one))\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul))).X,\n Y\u271d.X) \u226b\n ((mk (W\u271d.X \u2297 X\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one))\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul))).mul \u2297\n Y\u271d.mul) \u2297\n Z\u271d.mul)) \u226b\n (\u03b1_\n (mk (W\u271d.X \u2297 X\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one))\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul))).X\n Y\u271d.X Z\u271d.X).hom =\n ((\u03b1_\n (mk (W\u271d.X \u2297 X\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one))\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul))).X\n Y\u271d.X Z\u271d.X).hom \u2297\n (\u03b1_\n (mk (W\u271d.X \u2297 X\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one))\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul))).X\n Y\u271d.X Z\u271d.X).hom) \u226b\n tensor_\u03bc C\n ((mk (W\u271d.X \u2297 X\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one))\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul))).X,\n Y\u271d.X \u2297 Z\u271d.X)\n ((mk (W\u271d.X \u2297 X\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one))\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul))).X,\n Y\u271d.X \u2297 Z\u271d.X) \u226b\n ((mk (W\u271d.X \u2297 X\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one))\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul))).mul \u2297\n tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul)))).hom.hom \u226b\n (isoOfIso (\u03b1_ W\u271d.X X\u271d.X (Y\u271d.X \u2297 Z\u271d.X))\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (W\u271d.one \u2297 X\u271d.one) \u2297\n (mk (Y\u271d.X \u2297 Z\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one))\n (tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul))).one)) \u226b\n (\u03b1_ W\u271d.X X\u271d.X\n (mk (Y\u271d.X \u2297 Z\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one))\n (tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul))).X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b\n (W\u271d.one \u2297\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b\n (X\u271d.one \u2297\n (mk (Y\u271d.X \u2297 Z\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one))\n (tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul))).one)))\n (_ :\n (tensor_\u03bc C\n (W\u271d.X \u2297 X\u271d.X,\n (mk (Y\u271d.X \u2297 Z\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one))\n (tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul))).X)\n (W\u271d.X \u2297 X\u271d.X,\n (mk (Y\u271d.X \u2297 Z\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one))\n (tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul))).X) \u226b\n (tensor_\u03bc C (W\u271d.X, X\u271d.X) (W\u271d.X, X\u271d.X) \u226b (W\u271d.mul \u2297 X\u271d.mul) \u2297\n (mk (Y\u271d.X \u2297 Z\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one))\n (tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul))).mul)) \u226b\n (\u03b1_ W\u271d.X X\u271d.X\n (mk (Y\u271d.X \u2297 Z\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one))\n (tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul))).X).hom =\n ((\u03b1_ W\u271d.X X\u271d.X\n (mk (Y\u271d.X \u2297 Z\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one))\n (tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul))).X).hom \u2297\n (\u03b1_ W\u271d.X X\u271d.X\n (mk (Y\u271d.X \u2297 Z\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one))\n (tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul))).X).hom) \u226b\n tensor_\u03bc C\n (W\u271d.X,\n X\u271d.X \u2297\n (mk (Y\u271d.X \u2297 Z\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one))\n (tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul))).X)\n (W\u271d.X,\n X\u271d.X \u2297\n (mk (Y\u271d.X \u2297 Z\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one))\n (tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul))).X) \u226b\n (W\u271d.mul \u2297\n tensor_\u03bc C\n (X\u271d.X,\n (mk (Y\u271d.X \u2297 Z\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one))\n (tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul))).X)\n (X\u271d.X,\n (mk (Y\u271d.X \u2297 Z\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one))\n (tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul))).X) \u226b\n (X\u271d.mul \u2297\n (mk (Y\u271d.X \u2297 Z\u271d.X) ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (Y\u271d.one \u2297 Z\u271d.one))\n (tensor_\u03bc C (Y\u271d.X, Z\u271d.X) (Y\u271d.X, Z\u271d.X) \u226b (Y\u271d.mul \u2297 Z\u271d.mul))).mul)))).hom.hom\n[PROOFSTEP]\napply pentagon\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\n\u22a2 \u2200 (X Y : Mon_ C),\n ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n X (trivial C) Y).hom \u226b\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) (\ud835\udfd9 X)\n ((fun M =>\n isoOfIso (\u03bb_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.one)) \u226b (\u03bb_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 M.mul)) \u226b (\u03bb_ M.X).hom =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b M.mul))\n Y).hom =\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom))\n ((fun M =>\n isoOfIso (\u03c1_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.mul \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)) \u226b (\u03c1_ M.X).hom =\n ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul))\n X).hom\n (\ud835\udfd9 Y)\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u271d Y\u271d : Mon_ C\n\u22a2 ((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n X\u271d (trivial C) Y\u271d).hom \u226b\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) (\ud835\udfd9 X\u271d)\n ((fun M =>\n isoOfIso (\u03bb_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.one)) \u226b (\u03bb_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 M.mul)) \u226b (\u03bb_ M.X).hom =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b M.mul))\n Y\u271d).hom =\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom))\n ((fun M =>\n isoOfIso (\u03c1_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.mul \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)) \u226b (\u03c1_ M.X).hom =\n ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul))\n X\u271d).hom\n (\ud835\udfd9 Y\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u271d Y\u271d : Mon_ C\n\u22a2 (((fun M N P =>\n isoOfIso (\u03b1_ M.X N.X P.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 N.one) \u2297 P.one)) \u226b (\u03b1_ M.X N.X P.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (N.one \u2297 P.one)))\n (_ :\n (tensor_\u03bc C (M.X \u2297 N.X, P.X) (M.X \u2297 N.X, P.X) \u226b\n (tensor_\u03bc C (M.X, N.X) (M.X, N.X) \u226b (M.mul \u2297 N.mul) \u2297 P.mul)) \u226b\n (\u03b1_ M.X N.X P.X).hom =\n ((\u03b1_ M.X N.X P.X).hom \u2297 (\u03b1_ M.X N.X P.X).hom) \u226b\n tensor_\u03bc C (M.X, N.X \u2297 P.X) (M.X, N.X \u2297 P.X) \u226b\n (M.mul \u2297 tensor_\u03bc C (N.X, P.X) (N.X, P.X) \u226b (N.mul \u2297 P.mul))))\n X\u271d (trivial C) Y\u271d).hom \u226b\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom)) (\ud835\udfd9 X\u271d)\n ((fun M =>\n isoOfIso (\u03bb_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 M.one)) \u226b (\u03bb_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (\ud835\udfd9_ C, M.X) (\ud835\udfd9_ C, M.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 M.mul)) \u226b (\u03bb_ M.X).hom =\n ((\u03bb_ M.X).hom \u2297 (\u03bb_ M.X).hom) \u226b M.mul))\n Y\u271d).hom).hom =\n ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => Hom.mk (f.hom \u2297 g.hom))\n ((fun M =>\n isoOfIso (\u03c1_ M.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (M.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ M.X).hom = M.one)\n (_ :\n (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.mul \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)) \u226b (\u03c1_ M.X).hom =\n ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul))\n X\u271d).hom\n (\ud835\udfd9 Y\u271d)).hom\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX\u271d Y\u271d : Mon_ C\n\u22a2 (isoOfIso (\u03b1_ X\u271d.X (\ud835\udfd9_ C) Y\u271d.X)\n (_ :\n ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 (trivial C).one) \u2297 Y\u271d.one)) \u226b\n (\u03b1_ X\u271d.X (trivial C).X Y\u271d.X).hom =\n (\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv \u226b ((trivial C).one \u2297 Y\u271d.one)))\n (_ :\n (tensor_\u03bc C (X\u271d.X \u2297 (trivial C).X, Y\u271d.X) (X\u271d.X \u2297 (trivial C).X, Y\u271d.X) \u226b\n (tensor_\u03bc C (X\u271d.X, (trivial C).X) (X\u271d.X, (trivial C).X) \u226b (X\u271d.mul \u2297 (trivial C).mul) \u2297 Y\u271d.mul)) \u226b\n (\u03b1_ X\u271d.X (trivial C).X Y\u271d.X).hom =\n ((\u03b1_ X\u271d.X (trivial C).X Y\u271d.X).hom \u2297 (\u03b1_ X\u271d.X (trivial C).X Y\u271d.X).hom) \u226b\n tensor_\u03bc C (X\u271d.X, (trivial C).X \u2297 Y\u271d.X) (X\u271d.X, (trivial C).X \u2297 Y\u271d.X) \u226b\n (X\u271d.mul \u2297\n tensor_\u03bc C ((trivial C).X, Y\u271d.X) ((trivial C).X, Y\u271d.X) \u226b ((trivial C).mul \u2297 Y\u271d.mul)))).hom.hom \u226b\n (\ud835\udfd9 X\u271d.X \u2297\n (isoOfIso (\u03bb_ Y\u271d.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 Y\u271d.one)) \u226b (\u03bb_ Y\u271d.X).hom = Y\u271d.one)\n (_ :\n (tensor_\u03bc C (\ud835\udfd9_ C, Y\u271d.X) (\ud835\udfd9_ C, Y\u271d.X) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 Y\u271d.mul)) \u226b (\u03bb_ Y\u271d.X).hom =\n ((\u03bb_ Y\u271d.X).hom \u2297 (\u03bb_ Y\u271d.X).hom) \u226b Y\u271d.mul)).hom.hom) =\n (isoOfIso (\u03c1_ X\u271d.X) (_ : ((\u03bb_ (\ud835\udfd9_ C)).inv \u226b (X\u271d.one \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ X\u271d.X).hom = X\u271d.one)\n (_ :\n (tensor_\u03bc C (X\u271d.X, \ud835\udfd9_ C) (X\u271d.X, \ud835\udfd9_ C) \u226b (X\u271d.mul \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)) \u226b (\u03c1_ X\u271d.X).hom =\n ((\u03c1_ X\u271d.X).hom \u2297 (\u03c1_ X\u271d.X).hom) \u226b X\u271d.mul)).hom.hom \u2297\n \ud835\udfd9 Y\u271d.X\n[PROOFSTEP]\napply triangle\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monoidal.Mon_", "llama_tokens": 87044, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147275, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.446780443344617}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 (fun s => if MeasurableSet s then \u222b (x : \u03b1) in s, f x \u2202\u03bc else 0) \u2205 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f\ns : \u2115 \u2192 Set \u03b1\nhs\u2081 : \u2200 (i : \u2115), MeasurableSet (s i)\nhs\u2082 : Pairwise (Disjoint on s)\n\u22a2 HasSum (fun i => (fun s => if MeasurableSet s then \u222b (x : \u03b1) in s, f x \u2202\u03bc else 0) (s i))\n ((fun s => if MeasurableSet s then \u222b (x : \u03b1) in s, f x \u2202\u03bc else 0) (\u22c3 (i : \u2115), s i))\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f\ns : \u2115 \u2192 Set \u03b1\nhs\u2081 : \u2200 (i : \u2115), MeasurableSet (s i)\nhs\u2082 : Pairwise (Disjoint on s)\n\u22a2 HasSum (fun i => if MeasurableSet (s i) then \u222b (x : \u03b1) in s i, f x \u2202\u03bc else 0)\n (if MeasurableSet (\u22c3 (i : \u2115), s i) then \u222b (x : \u03b1) in \u22c3 (i : \u2115), s i, f x \u2202\u03bc else 0)\n[PROOFSTEP]\nconvert hasSum_integral_iUnion hs\u2081 hs\u2082 hf.integrableOn with n\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f\ns : \u2115 \u2192 Set \u03b1\nhs\u2081 : \u2200 (i : \u2115), MeasurableSet (s i)\nhs\u2082 : Pairwise (Disjoint on s)\nn : \u2115\n\u22a2 (if MeasurableSet (s n) then \u222b (x : \u03b1) in s n, f x \u2202\u03bc else 0) = \u222b (a : \u03b1) in s n, f a \u2202\u03bc\n[PROOFSTEP]\nrw [if_pos (hs\u2081 n)]\n[GOAL]\ncase h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f\ns : \u2115 \u2192 Set \u03b1\nhs\u2081 : \u2200 (i : \u2115), MeasurableSet (s i)\nhs\u2082 : Pairwise (Disjoint on s)\n\u22a2 (if MeasurableSet (\u22c3 (i : \u2115), s i) then \u222b (x : \u03b1) in \u22c3 (i : \u2115), s i, f x \u2202\u03bc else 0) =\n \u222b (a : \u03b1) in \u22c3 (n : \u2115), s n, f a \u2202\u03bc\n[PROOFSTEP]\nrw [if_pos (MeasurableSet.iUnion hs\u2081)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191(withDensity\u1d65 \u03bc f) s = \u222b (x : \u03b1) in s, f x \u2202\u03bc\n[PROOFSTEP]\nrw [withDensity\u1d65, dif_pos hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191{ measureOf' := fun s => if MeasurableSet s then \u222b (x : \u03b1) in s, f x \u2202\u03bc else 0,\n empty' := (_ : (if MeasurableSet \u2205 then \u222b (x : \u03b1) in \u2205, f x \u2202\u03bc else 0) = 0),\n not_measurable' :=\n (_ : \u2200 (s : Set \u03b1), \u00acMeasurableSet s \u2192 (if MeasurableSet s then \u222b (x : \u03b1) in s, f x \u2202\u03bc else 0) = 0),\n m_iUnion' :=\n (_ :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (i : \u2115), MeasurableSet (s i)) \u2192\n Pairwise (Disjoint on s) \u2192\n HasSum (fun i => (fun s => if MeasurableSet s then \u222b (x : \u03b1) in s, f x \u2202\u03bc else 0) (s i))\n ((fun s => if MeasurableSet s then \u222b (x : \u03b1) in s, f x \u2202\u03bc else 0) (\u22c3 (i : \u2115), s i))) }\n s =\n \u222b (x : \u03b1) in s, f x \u2202\u03bc\n[PROOFSTEP]\nexact dif_pos hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\n\u22a2 withDensity\u1d65 \u03bc 0 = 0\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191(withDensity\u1d65 \u03bc 0) s = \u21910 s\n[PROOFSTEP]\nerw [withDensity\u1d65_apply (integrable_zero \u03b1 E \u03bc) hs]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b (x : \u03b1) in s, 0 \u2202\u03bc = \u21910 s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\n\u22a2 withDensity\u1d65 \u03bc (-f) = -withDensity\u1d65 \u03bc f\n[PROOFSTEP]\nby_cases hf : Integrable f \u03bc\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 withDensity\u1d65 \u03bc (-f) = -withDensity\u1d65 \u03bc f\n[PROOFSTEP]\next1 i hi\n[GOAL]\ncase pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(withDensity\u1d65 \u03bc (-f)) i = \u2191(-withDensity\u1d65 \u03bc f) i\n[PROOFSTEP]\nrw [VectorMeasure.neg_apply, withDensity\u1d65_apply hf hi, \u2190 integral_neg, withDensity\u1d65_apply hf.neg hi]\n[GOAL]\ncase pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b (x : \u03b1) in i, (-f) x \u2202\u03bc = \u222b (a : \u03b1) in i, -f a \u2202\u03bc\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 withDensity\u1d65 \u03bc (-f) = -withDensity\u1d65 \u03bc f\n[PROOFSTEP]\nrw [withDensity\u1d65, withDensity\u1d65, dif_neg hf, dif_neg, neg_zero]\n[GOAL]\ncase neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 \u00acIntegrable (-f)\n[PROOFSTEP]\nrwa [integrable_neg_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\n\u22a2 withDensity\u1d65 \u03bc (f + g) = withDensity\u1d65 \u03bc f + withDensity\u1d65 \u03bc g\n[PROOFSTEP]\next1 i hi\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(withDensity\u1d65 \u03bc (f + g)) i = \u2191(withDensity\u1d65 \u03bc f + withDensity\u1d65 \u03bc g) i\n[PROOFSTEP]\nrw [withDensity\u1d65_apply (hf.add hg) hi, VectorMeasure.add_apply, withDensity\u1d65_apply hf hi, withDensity\u1d65_apply hg hi]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b (x : \u03b1) in i, (f + g) x \u2202\u03bc = \u222b (x : \u03b1) in i, f x \u2202\u03bc + \u222b (x : \u03b1) in i, g x \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [Pi.add_apply]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b (x : \u03b1) in i, f x + g x \u2202\u03bc = \u222b (x : \u03b1) in i, f x \u2202\u03bc + \u222b (x : \u03b1) in i, g x \u2202\u03bc\n[PROOFSTEP]\nrw [integral_add]\n[GOAL]\ncase h.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Integrable fun x => f x\n[PROOFSTEP]\nrw [\u2190 integrableOn_univ]\n[GOAL]\ncase h.hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Integrable fun x => g x\n[PROOFSTEP]\nrw [\u2190 integrableOn_univ]\n[GOAL]\ncase h.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 IntegrableOn (fun x => f x) Set.univ\n[PROOFSTEP]\nexact hf.integrableOn.restrict MeasurableSet.univ\n[GOAL]\ncase h.hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 IntegrableOn (fun x => g x) Set.univ\n[PROOFSTEP]\nexact hg.integrableOn.restrict MeasurableSet.univ\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\n\u22a2 withDensity\u1d65 \u03bc (f - g) = withDensity\u1d65 \u03bc f - withDensity\u1d65 \u03bc g\n[PROOFSTEP]\nrw [sub_eq_add_neg, sub_eq_add_neg, withDensity\u1d65_add hf hg.neg, withDensity\u1d65_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\n\u22a2 withDensity\u1d65 \u03bc (r \u2022 f) = r \u2022 withDensity\u1d65 \u03bc f\n[PROOFSTEP]\nby_cases hf : Integrable f \u03bc\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : Integrable f\n\u22a2 withDensity\u1d65 \u03bc (r \u2022 f) = r \u2022 withDensity\u1d65 \u03bc f\n[PROOFSTEP]\next1 i hi\n[GOAL]\ncase pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : Integrable f\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(withDensity\u1d65 \u03bc (r \u2022 f)) i = \u2191(r \u2022 withDensity\u1d65 \u03bc f) i\n[PROOFSTEP]\nrw [withDensity\u1d65_apply (hf.smul r) hi, VectorMeasure.smul_apply, withDensity\u1d65_apply hf hi, \u2190 integral_smul r f]\n[GOAL]\ncase pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : Integrable f\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b (x : \u03b1) in i, (r \u2022 f) x \u2202\u03bc = \u222b (a : \u03b1) in i, r \u2022 f a \u2202\u03bc\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : \u00acIntegrable f\n\u22a2 withDensity\u1d65 \u03bc (r \u2022 f) = r \u2022 withDensity\u1d65 \u03bc f\n[PROOFSTEP]\nby_cases hr : r = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : \u00acIntegrable f\nhr : r = 0\n\u22a2 withDensity\u1d65 \u03bc (r \u2022 f) = r \u2022 withDensity\u1d65 \u03bc f\n[PROOFSTEP]\nrw [hr, zero_smul, zero_smul, withDensity\u1d65_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : \u00acIntegrable f\nhr : \u00acr = 0\n\u22a2 withDensity\u1d65 \u03bc (r \u2022 f) = r \u2022 withDensity\u1d65 \u03bc f\n[PROOFSTEP]\nrw [withDensity\u1d65, withDensity\u1d65, dif_neg hf, dif_neg, smul_zero]\n[GOAL]\ncase neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : \u00acIntegrable f\nhr : \u00acr = 0\n\u22a2 \u00acIntegrable (r \u2022 f)\n[PROOFSTEP]\nrwa [integrable_smul_iff hr f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\n\u22a2 withDensity\u1d65 \u03bc f \u226a\u1d65 toENNRealVectorMeasure \u03bc\n[PROOFSTEP]\nby_cases hf : Integrable f \u03bc\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 withDensity\u1d65 \u03bc f \u226a\u1d65 toENNRealVectorMeasure \u03bc\n[PROOFSTEP]\nrefine' VectorMeasure.AbsolutelyContinuous.mk fun i hi\u2081 hi\u2082 => _\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191(toENNRealVectorMeasure \u03bc) i = 0\n\u22a2 \u2191(withDensity\u1d65 \u03bc f) i = 0\n[PROOFSTEP]\nrw [toENNRealVectorMeasure_apply_measurable hi\u2081] at hi\u2082 \n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191\u2191\u03bc i = 0\n\u22a2 \u2191(withDensity\u1d65 \u03bc f) i = 0\n[PROOFSTEP]\nrw [withDensity\u1d65_apply hf hi\u2081, Measure.restrict_zero_set hi\u2082, integral_zero_measure]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u00acIntegrable f\n\u22a2 withDensity\u1d65 \u03bc f \u226a\u1d65 toENNRealVectorMeasure \u03bc\n[PROOFSTEP]\nrw [withDensity\u1d65, dif_neg hf]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u00acIntegrable f\n\u22a2 0 \u226a\u1d65 toENNRealVectorMeasure \u03bc\n[PROOFSTEP]\nexact VectorMeasure.AbsolutelyContinuous.zero _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : withDensity\u1d65 \u03bc f = withDensity\u1d65 \u03bc g\n\u22a2 f =\u1da0[ae \u03bc] g\n[PROOFSTEP]\nrefine' hf.ae_eq_of_forall_set_integral_eq f g hg fun i hi _ => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : withDensity\u1d65 \u03bc f = withDensity\u1d65 \u03bc g\ni : Set \u03b1\nhi : MeasurableSet i\nx\u271d : \u2191\u2191\u03bc i < \u22a4\n\u22a2 \u222b (x : \u03b1) in i, f x \u2202\u03bc = \u222b (x : \u03b1) in i, g x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 withDensity\u1d65_apply hf hi, hfg, withDensity\u1d65_apply hg hi]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\n\u22a2 withDensity\u1d65 \u03bc f = withDensity\u1d65 \u03bc g\n[PROOFSTEP]\nby_cases hf : Integrable f \u03bc\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : Integrable f\n\u22a2 withDensity\u1d65 \u03bc f = withDensity\u1d65 \u03bc g\n[PROOFSTEP]\next i hi\n[GOAL]\ncase pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : Integrable f\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(withDensity\u1d65 \u03bc f) i = \u2191(withDensity\u1d65 \u03bc g) i\n[PROOFSTEP]\nrw [withDensity\u1d65_apply hf hi, withDensity\u1d65_apply (hf.congr h) hi]\n[GOAL]\ncase pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : Integrable f\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b (x : \u03b1) in i, f x \u2202\u03bc = \u222b (x : \u03b1) in i, g x \u2202\u03bc\n[PROOFSTEP]\nexact integral_congr_ae (ae_restrict_of_ae h)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : \u00acIntegrable f\n\u22a2 withDensity\u1d65 \u03bc f = withDensity\u1d65 \u03bc g\n[PROOFSTEP]\nhave hg : \u00acIntegrable g \u03bc := by intro hg; exact hf (hg.congr h.symm)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : \u00acIntegrable f\n\u22a2 \u00acIntegrable g\n[PROOFSTEP]\nintro hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : \u00acIntegrable f\nhg : Integrable g\n\u22a2 False\n[PROOFSTEP]\nexact hf (hg.congr h.symm)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : \u00acIntegrable f\nhg : \u00acIntegrable g\n\u22a2 withDensity\u1d65 \u03bc f = withDensity\u1d65 \u03bc g\n[PROOFSTEP]\nrw [withDensity\u1d65, withDensity\u1d65, dif_neg hf, dif_neg hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhfm : AEMeasurable f\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u22a2 (withDensity\u1d65 \u03bc fun x => ENNReal.toReal (f x)) = toSignedMeasure (withDensity \u03bc f)\n[PROOFSTEP]\nhave hfi := integrable_toReal_of_lintegral_ne_top hfm hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhfm : AEMeasurable f\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhfi : Integrable fun x => ENNReal.toReal (f x)\n\u22a2 (withDensity\u1d65 \u03bc fun x => ENNReal.toReal (f x)) = toSignedMeasure (withDensity \u03bc f)\n[PROOFSTEP]\nhaveI := isFiniteMeasure_withDensity hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhfm : AEMeasurable f\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity \u03bc fun a => f a)\n\u22a2 (withDensity\u1d65 \u03bc fun x => ENNReal.toReal (f x)) = toSignedMeasure (withDensity \u03bc f)\n[PROOFSTEP]\next i hi\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhfm : AEMeasurable f\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity \u03bc fun a => f a)\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(withDensity\u1d65 \u03bc fun x => ENNReal.toReal (f x)) i = \u2191(toSignedMeasure (withDensity \u03bc f)) i\n[PROOFSTEP]\nrw [withDensity\u1d65_apply hfi hi, toSignedMeasure_apply_measurable hi, withDensity_apply _ hi,\n integral_toReal hfm.restrict]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhfm : AEMeasurable f\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity \u03bc fun a => f a)\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc i, f x < \u22a4\n[PROOFSTEP]\nrefine' ae_lt_top' hfm.restrict (ne_top_of_le_ne_top hf _)\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhfm : AEMeasurable f\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity \u03bc fun a => f a)\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b\u207b (x : \u03b1) in i, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nconv_rhs => rw [\u2190 set_lintegral_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhfm : AEMeasurable f\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity \u03bc fun a => f a)\ni : Set \u03b1\nhi : MeasurableSet i\n| \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 set_lintegral_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhfm : AEMeasurable f\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity \u03bc fun a => f a)\ni : Set \u03b1\nhi : MeasurableSet i\n| \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 set_lintegral_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhfm : AEMeasurable f\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity \u03bc fun a => f a)\ni : Set \u03b1\nhi : MeasurableSet i\n| \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 set_lintegral_univ]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhfm : AEMeasurable f\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity \u03bc fun a => f a)\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b\u207b (x : \u03b1) in i, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in Set.univ, f x \u2202\u03bc\n[PROOFSTEP]\nexact lintegral_mono_set (Set.subset_univ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\n\u22a2 withDensity\u1d65 \u03bc f =\n toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) -\n toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (-f x))\n[PROOFSTEP]\nhaveI := isFiniteMeasure_withDensity_ofReal hfi.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x))\n\u22a2 withDensity\u1d65 \u03bc f =\n toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) -\n toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (-f x))\n[PROOFSTEP]\nhaveI := isFiniteMeasure_withDensity_ofReal hfi.neg.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nthis\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x))\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x))\n\u22a2 withDensity\u1d65 \u03bc f =\n toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) -\n toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (-f x))\n[PROOFSTEP]\next i hi\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\nf : \u03b1 \u2192 \u211d\nhfi : Integrable f\nthis\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x))\nthis : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x))\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(withDensity\u1d65 \u03bc f) i =\n \u2191(toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) -\n toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (-f x)))\n i\n[PROOFSTEP]\nrw [withDensity\u1d65_apply hfi hi, integral_eq_lintegral_pos_part_sub_lintegral_neg_part hfi.integrableOn,\n VectorMeasure.sub_apply, toSignedMeasure_apply_measurable hi, toSignedMeasure_apply_measurable hi,\n withDensity_apply _ hi, withDensity_apply _ hi]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(VectorMeasure.trim (withDensity\u1d65 \u03bc f) hm) i = \u222b (x : \u03b1) in i, f x \u2202\u03bc\n[PROOFSTEP]\nrw [VectorMeasure.trim_measurableSet_eq hm hi, withDensity\u1d65_apply hf (hm _ hi)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhfi : Integrable f\n\u22a2 VectorMeasure.trim (withDensity\u1d65 \u03bc f) hm \u226a\u1d65 toENNRealVectorMeasure (Measure.trim \u03bc hm)\n[PROOFSTEP]\nrefine' VectorMeasure.AbsolutelyContinuous.mk fun j hj\u2081 hj\u2082 => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhfi : Integrable f\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : \u2191(toENNRealVectorMeasure (Measure.trim \u03bc hm)) j = 0\n\u22a2 \u2191(VectorMeasure.trim (withDensity\u1d65 \u03bc f) hm) j = 0\n[PROOFSTEP]\nrw [Measure.toENNRealVectorMeasure_apply_measurable hj\u2081, trim_measurableSet_eq hm hj\u2081] at hj\u2082 \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhfi : Integrable f\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : \u2191\u2191\u03bc j = 0\n\u22a2 \u2191(VectorMeasure.trim (withDensity\u1d65 \u03bc f) hm) j = 0\n[PROOFSTEP]\nrw [VectorMeasure.trim_measurableSet_eq hm hj\u2081, withDensity\u1d65_apply hfi (hm _ hj\u2081)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhfi : Integrable f\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : \u2191\u2191\u03bc j = 0\n\u22a2 \u222b (x : \u03b1) in j, f x \u2202\u03bc = 0\n[PROOFSTEP]\nsimp only [Measure.restrict_eq_zero.mpr hj\u2082, integral_zero_measure]\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "llama_tokens": 13784, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147275, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.446780443344617}} {"text": "[GOAL]\nx : PGame\n\u22a2 birthday x = max (lsub fun i => birthday (moveLeft x i)) (lsub fun i => birthday (moveRight x i))\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1\u271d \u03b2\u271d : Type u\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\n\u22a2 birthday (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) =\n max (lsub fun i => birthday (moveLeft (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i)) (lsub fun i => birthday (moveRight (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i))\n[PROOFSTEP]\nrw [birthday]\n[GOAL]\ncase mk\n\u03b1\u271d \u03b2\u271d : Type u\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\n\u22a2 max (lsub fun i => birthday (a\u271d\u00b9 i)) (lsub fun i => birthday (a\u271d i)) =\n max (lsub fun i => birthday (moveLeft (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i)) (lsub fun i => birthday (moveRight (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i))\n[PROOFSTEP]\nrfl\n[GOAL]\nx : PGame\ni : LeftMoves x\n\u22a2 birthday (moveLeft x i) < birthday x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1\u271d \u03b2\u271d : Type u_1\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\ni : LeftMoves (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 birthday (moveLeft (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i) < birthday (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n[PROOFSTEP]\nrw [birthday]\n[GOAL]\ncase mk\n\u03b1\u271d \u03b2\u271d : Type u_1\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\ni : LeftMoves (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 birthday (moveLeft (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i) < max (lsub fun i => birthday (a\u271d\u00b9 i)) (lsub fun i => birthday (a\u271d i))\n[PROOFSTEP]\nexact lt_max_of_lt_left (lt_lsub _ i)\n[GOAL]\nx : PGame\ni : RightMoves x\n\u22a2 birthday (moveRight x i) < birthday x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1\u271d \u03b2\u271d : Type u_1\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\ni : RightMoves (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 birthday (moveRight (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i) < birthday (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n[PROOFSTEP]\nrw [birthday]\n[GOAL]\ncase mk\n\u03b1\u271d \u03b2\u271d : Type u_1\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\ni : RightMoves (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 birthday (moveRight (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i) < max (lsub fun i => birthday (a\u271d\u00b9 i)) (lsub fun i => birthday (a\u271d i))\n[PROOFSTEP]\nexact lt_max_of_lt_right (lt_lsub _ i)\n[GOAL]\nx : PGame\no : Ordinal.{u_1}\n\u22a2 o < birthday x \u2194 (\u2203 i, o \u2264 birthday (moveLeft x i)) \u2228 \u2203 i, o \u2264 birthday (moveRight x i)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nx : PGame\no : Ordinal.{u_1}\n\u22a2 o < birthday x \u2192 (\u2203 i, o \u2264 birthday (moveLeft x i)) \u2228 \u2203 i, o \u2264 birthday (moveRight x i)\n[PROOFSTEP]\nrw [birthday_def]\n[GOAL]\ncase mp\nx : PGame\no : Ordinal.{u_1}\n\u22a2 o < max (lsub fun i => birthday (moveLeft x i)) (lsub fun i => birthday (moveRight x i)) \u2192\n (\u2203 i, o \u2264 birthday (moveLeft x i)) \u2228 \u2203 i, o \u2264 birthday (moveRight x i)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nx : PGame\no : Ordinal.{u_1}\nh : o < max (lsub fun i => birthday (moveLeft x i)) (lsub fun i => birthday (moveRight x i))\n\u22a2 (\u2203 i, o \u2264 birthday (moveLeft x i)) \u2228 \u2203 i, o \u2264 birthday (moveRight x i)\n[PROOFSTEP]\ncases' lt_max_iff.1 h with h' h'\n[GOAL]\ncase mp.inl\nx : PGame\no : Ordinal.{u_1}\nh : o < max (lsub fun i => birthday (moveLeft x i)) (lsub fun i => birthday (moveRight x i))\nh' : o < lsub fun i => birthday (moveLeft x i)\n\u22a2 (\u2203 i, o \u2264 birthday (moveLeft x i)) \u2228 \u2203 i, o \u2264 birthday (moveRight x i)\n[PROOFSTEP]\nleft\n[GOAL]\ncase mp.inl.h\nx : PGame\no : Ordinal.{u_1}\nh : o < max (lsub fun i => birthday (moveLeft x i)) (lsub fun i => birthday (moveRight x i))\nh' : o < lsub fun i => birthday (moveLeft x i)\n\u22a2 \u2203 i, o \u2264 birthday (moveLeft x i)\n[PROOFSTEP]\nrwa [lt_lsub_iff] at h' \n[GOAL]\ncase mp.inr\nx : PGame\no : Ordinal.{u_1}\nh : o < max (lsub fun i => birthday (moveLeft x i)) (lsub fun i => birthday (moveRight x i))\nh' : o < lsub fun i => birthday (moveRight x i)\n\u22a2 (\u2203 i, o \u2264 birthday (moveLeft x i)) \u2228 \u2203 i, o \u2264 birthday (moveRight x i)\n[PROOFSTEP]\nright\n[GOAL]\ncase mp.inr.h\nx : PGame\no : Ordinal.{u_1}\nh : o < max (lsub fun i => birthday (moveLeft x i)) (lsub fun i => birthday (moveRight x i))\nh' : o < lsub fun i => birthday (moveRight x i)\n\u22a2 \u2203 i, o \u2264 birthday (moveRight x i)\n[PROOFSTEP]\nrwa [lt_lsub_iff] at h' \n[GOAL]\ncase mpr\nx : PGame\no : Ordinal.{u_1}\n\u22a2 ((\u2203 i, o \u2264 birthday (moveLeft x i)) \u2228 \u2203 i, o \u2264 birthday (moveRight x i)) \u2192 o < birthday x\n[PROOFSTEP]\nrintro (\u27e8i, hi\u27e9 | \u27e8i, hi\u27e9)\n[GOAL]\ncase mpr.inl.intro\nx : PGame\no : Ordinal.{u_1}\ni : LeftMoves x\nhi : o \u2264 birthday (moveLeft x i)\n\u22a2 o < birthday x\n[PROOFSTEP]\nexact hi.trans_lt (birthday_moveLeft_lt i)\n[GOAL]\ncase mpr.inr.intro\nx : PGame\no : Ordinal.{u_1}\ni : RightMoves x\nhi : o \u2264 birthday (moveRight x i)\n\u22a2 o < birthday x\n[PROOFSTEP]\nexact hi.trans_lt (birthday_moveRight_lt i)\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 birthday (PGame.mk xl xr xL xR) = birthday (PGame.mk yl yr yL yR)\n[PROOFSTEP]\nunfold birthday\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 max (lsub fun i => birthday (xL i)) (lsub fun i => birthday (xR i)) =\n max (lsub fun i => birthday (yL i)) (lsub fun i => birthday (yR i))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 (lsub fun i => birthday (xL i)) = lsub fun i => birthday (yL i)\ncase e_a\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 (lsub fun i => birthday (xR i)) = lsub fun i => birthday (yR i)\n[PROOFSTEP]\nall_goals\n apply lsub_eq_of_range_eq.{u, u, u}\n ext i; constructor\n[GOAL]\ncase e_a\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 (lsub fun i => birthday (xL i)) = lsub fun i => birthday (yL i)\n[PROOFSTEP]\napply lsub_eq_of_range_eq.{u, u, u}\n[GOAL]\ncase e_a\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 (Set.range fun i => birthday (xL i)) = Set.range fun i => birthday (yL i)\n[PROOFSTEP]\next i\n[GOAL]\ncase e_a.h\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => birthday (xL i)) \u2194 i \u2208 Set.range fun i => birthday (yL i)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase e_a\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 (lsub fun i => birthday (xR i)) = lsub fun i => birthday (yR i)\n[PROOFSTEP]\napply lsub_eq_of_range_eq.{u, u, u}\n[GOAL]\ncase e_a\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 (Set.range fun i => birthday (xR i)) = Set.range fun i => birthday (yR i)\n[PROOFSTEP]\next i\n[GOAL]\ncase e_a.h\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => birthday (xR i)) \u2194 i \u2208 Set.range fun i => birthday (yR i)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase e_a.h.mp\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => birthday (xL i)) \u2192 i \u2208 Set.range fun i => birthday (yL i)\ncase e_a.h.mpr\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => birthday (yL i)) \u2192 i \u2208 Set.range fun i => birthday (xL i)\ncase e_a.h.mp\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => birthday (xR i)) \u2192 i \u2208 Set.range fun i => birthday (yR i)\ncase e_a.h.mpr\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => birthday (yR i)) \u2192 i \u2208 Set.range fun i => birthday (xR i)\n[PROOFSTEP]\nall_goals rintro \u27e8j, rfl\u27e9\n[GOAL]\ncase e_a.h.mp\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => birthday (xL i)) \u2192 i \u2208 Set.range fun i => birthday (yL i)\n[PROOFSTEP]\nrintro \u27e8j, rfl\u27e9\n[GOAL]\ncase e_a.h.mpr\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => birthday (yL i)) \u2192 i \u2208 Set.range fun i => birthday (xL i)\n[PROOFSTEP]\nrintro \u27e8j, rfl\u27e9\n[GOAL]\ncase e_a.h.mp\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => birthday (xR i)) \u2192 i \u2208 Set.range fun i => birthday (yR i)\n[PROOFSTEP]\nrintro \u27e8j, rfl\u27e9\n[GOAL]\ncase e_a.h.mpr\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => birthday (yR i)) \u2192 i \u2208 Set.range fun i => birthday (xR i)\n[PROOFSTEP]\nrintro \u27e8j, rfl\u27e9\n[GOAL]\ncase e_a.h.mp.intro\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\nj : xl\n\u22a2 (fun i => birthday (xL i)) j \u2208 Set.range fun i => birthday (yL i)\n[PROOFSTEP]\nexact \u27e8_, (r.moveLeft j).birthday_congr.symm\u27e9\n[GOAL]\ncase e_a.h.mpr.intro\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\nj : yl\n\u22a2 (fun i => birthday (yL i)) j \u2208 Set.range fun i => birthday (xL i)\n[PROOFSTEP]\nexact \u27e8_, (r.moveLeftSymm j).birthday_congr\u27e9\n[GOAL]\ncase e_a.h.mp.intro\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\nj : xr\n\u22a2 (fun i => birthday (xR i)) j \u2208 Set.range fun i => birthday (yR i)\n[PROOFSTEP]\nexact \u27e8_, (r.moveRight j).birthday_congr.symm\u27e9\n[GOAL]\ncase e_a.h.mpr.intro\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\nj : yr\n\u22a2 (fun i => birthday (yR i)) j \u2208 Set.range fun i => birthday (xR i)\n[PROOFSTEP]\nexact \u27e8_, (r.moveRightSymm j).birthday_congr\u27e9\n[GOAL]\nx : PGame\n\u22a2 birthday x = 0 \u2194 IsEmpty (LeftMoves x) \u2227 IsEmpty (RightMoves x)\n[PROOFSTEP]\nrw [birthday_def, max_eq_zero, lsub_eq_zero_iff, lsub_eq_zero_iff]\n[GOAL]\n\u22a2 birthday 0 = 0\n[PROOFSTEP]\nsimp [inferInstanceAs (IsEmpty PEmpty)]\n[GOAL]\n\u22a2 birthday 1 = 1\n[PROOFSTEP]\nrw [birthday_def]\n[GOAL]\n\u22a2 max (lsub fun i => birthday (moveLeft 1 i)) (lsub fun i => birthday (moveRight 1 i)) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 birthday star = 1\n[PROOFSTEP]\nrw [birthday_def]\n[GOAL]\n\u22a2 max (lsub fun i => birthday (moveLeft star i)) (lsub fun i => birthday (moveRight star i)) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 birthday (-mk xl xr xL xR) = birthday (mk xl xr xL xR)\n[PROOFSTEP]\nrw [birthday_def, birthday_def, max_comm]\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 max (lsub fun i => birthday (moveRight (-mk xl xr xL xR) i)) (lsub fun i => birthday (moveLeft (-mk xl xr xL xR) i)) =\n max (lsub fun i => birthday (moveLeft (mk xl xr xL xR) i)) (lsub fun i => birthday (moveRight (mk xl xr xL xR) i))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_f\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 (fun i => birthday (moveRight (-mk xl xr xL xR) i)) = fun i => birthday (moveLeft (mk xl xr xL xR) i)\n[PROOFSTEP]\nfunext\n[GOAL]\ncase e_a.e_f\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 (fun i => birthday (moveLeft (-mk xl xr xL xR) i)) = fun i => birthday (moveRight (mk xl xr xL xR) i)\n[PROOFSTEP]\nfunext\n[GOAL]\ncase e_a.e_f.h\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nx\u271d : RightMoves (-mk xl xr xL xR)\n\u22a2 birthday (moveRight (-mk xl xr xL xR) x\u271d) = birthday (moveLeft (mk xl xr xL xR) x\u271d)\n[PROOFSTEP]\napply neg_birthday\n[GOAL]\ncase e_a.e_f.h\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nx\u271d : LeftMoves (-mk xl xr xL xR)\n\u22a2 birthday (moveLeft (-mk xl xr xL xR) x\u271d) = birthday (moveRight (mk xl xr xL xR) x\u271d)\n[PROOFSTEP]\napply neg_birthday\n[GOAL]\no : Ordinal.{u_1}\n\u22a2 birthday (toPGame o) = o\n[PROOFSTEP]\ninduction' o using Ordinal.induction with o IH\n[GOAL]\ncase h\no : Ordinal.{u_1}\nIH : \u2200 (k : Ordinal.{u_1}), k < o \u2192 birthday (toPGame k) = k\n\u22a2 birthday (toPGame o) = o\n[PROOFSTEP]\nrw [toPGame_def, PGame.birthday]\n[GOAL]\ncase h\no : Ordinal.{u_1}\nIH : \u2200 (k : Ordinal.{u_1}), k < o \u2192 birthday (toPGame k) = k\n\u22a2 max (lsub fun i => birthday (toPGame (typein (fun x x_1 => x < x_1) i))) (lsub fun i => birthday (PEmpty.elim i)) = o\n[PROOFSTEP]\nsimp only [lsub_empty, max_zero_right]\n -- Porting note: was `nth_rw 1 [\u2190 lsub_typein o]`\n[GOAL]\ncase h\no : Ordinal.{u_1}\nIH : \u2200 (k : Ordinal.{u_1}), k < o \u2192 birthday (toPGame k) = k\n\u22a2 (lsub fun i => birthday (toPGame (typein (fun x x_1 => x < x_1) i))) = o\n[PROOFSTEP]\nconv_rhs => rw [\u2190 lsub_typein o]\n[GOAL]\no : Ordinal.{u_1}\nIH : \u2200 (k : Ordinal.{u_1}), k < o \u2192 birthday (toPGame k) = k\n| o\n[PROOFSTEP]\nrw [\u2190 lsub_typein o]\n[GOAL]\no : Ordinal.{u_1}\nIH : \u2200 (k : Ordinal.{u_1}), k < o \u2192 birthday (toPGame k) = k\n| o\n[PROOFSTEP]\nrw [\u2190 lsub_typein o]\n[GOAL]\no : Ordinal.{u_1}\nIH : \u2200 (k : Ordinal.{u_1}), k < o \u2192 birthday (toPGame k) = k\n| o\n[PROOFSTEP]\nrw [\u2190 lsub_typein o]\n[GOAL]\ncase h\no : Ordinal.{u_1}\nIH : \u2200 (k : Ordinal.{u_1}), k < o \u2192 birthday (toPGame k) = k\n\u22a2 (lsub fun i => birthday (toPGame (typein (fun x x_1 => x < x_1) i))) = lsub (typein fun x x_1 => x < x_1)\n[PROOFSTEP]\ncongr with x\n[GOAL]\ncase h.e_f.h\no : Ordinal.{u_1}\nIH : \u2200 (k : Ordinal.{u_1}), k < o \u2192 birthday (toPGame k) = k\nx : (Quotient.out o).\u03b1\n\u22a2 birthday (toPGame (typein (fun x x_1 => x < x_1) x)) = typein (fun x x_1 => x < x_1) x\n[PROOFSTEP]\nexact IH _ (typein_lt_self x)\n[GOAL]\nxl \u03b2\u271d : Type u_1\nxL : xl \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\ni : LeftMoves (mk xl \u03b2\u271d xL a\u271d)\n\u22a2 moveLeft (mk xl \u03b2\u271d xL a\u271d) i \u2264\n moveLeft (toPGame (birthday (mk xl \u03b2\u271d xL a\u271d)))\n (\u2191toLeftMovesToPGame\n { val := birthday (moveLeft (mk xl \u03b2\u271d xL a\u271d) i),\n property := (_ : birthday (moveLeft (mk xl \u03b2\u271d xL a\u271d) i) < birthday (mk xl \u03b2\u271d xL a\u271d)) })\n[PROOFSTEP]\nsimp [le_birthday (xL i)]\n[GOAL]\na b x : PGame\n\u22a2 -toPGame (birthday x) \u2264 x\n[PROOFSTEP]\nsimpa only [neg_birthday, \u2190 neg_le_iff] using le_birthday (-x)\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 birthday (mk xl xr xL xR + mk yl yr yL yR) = birthday (mk xl xr xL xR) \u266f birthday (mk yl yr yL yR)\n[PROOFSTEP]\nrw [birthday_def, nadd_def]\n -- Porting note: `simp` doesn't apply\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (lsub fun i => birthday (moveLeft (mk xl xr xL xR + mk yl yr yL yR) i))\n (lsub fun i => birthday (moveRight (mk xl xr xL xR + mk yl yr yL yR) i)) =\n max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' \u266f birthday (mk yl yr yL yR))\n (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) \u266f b')\n[PROOFSTEP]\nerw [lsub_sum, lsub_sum]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max\n (max (lsub fun a => birthday (moveLeft (mk xl xr xL xR + mk yl yr yL yR) (Sum.inl a)))\n (lsub fun b => birthday (moveLeft (mk xl xr xL xR + mk yl yr yL yR) (Sum.inr b))))\n (max (lsub fun a => birthday (moveRight (mk xl xr xL xR + mk yl yr yL yR) (Sum.inl a)))\n (lsub fun b => birthday (moveRight (mk xl xr xL xR + mk yl yr yL yR) (Sum.inr b)))) =\n max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' \u266f birthday (mk yl yr yL yR))\n (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) \u266f b')\n[PROOFSTEP]\nsimp only [lsub_sum, mk_add_moveLeft_inl, moveLeft_mk, mk_add_moveLeft_inr, mk_add_moveRight_inl, moveRight_mk,\n mk_add_moveRight_inr]\n -- Porting note: Originally `simp only [birthday_add]`, but this causes an error in\n -- `termination_by`. Use a workaround.\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (max (lsub fun a => birthday (xL a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yL b)))\n (max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b))) =\n max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' \u266f birthday (mk yl yr yL yR))\n (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) \u266f b')\n[PROOFSTEP]\nconv_lhs => left; left; right; intro a; rw [birthday_add (xL a) \u27e8yl, yr, yL, yR\u27e9]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max (max (lsub fun a => birthday (xL a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yL b)))\n (max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b)))\n[PROOFSTEP]\nleft; left; right; intro a; rw [birthday_add (xL a) \u27e8yl, yr, yL, yR\u27e9]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max (max (lsub fun a => birthday (xL a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yL b)))\n (max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b)))\n[PROOFSTEP]\nleft; left; right; intro a; rw [birthday_add (xL a) \u27e8yl, yr, yL, yR\u27e9]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max (max (lsub fun a => birthday (xL a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yL b)))\n (max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b)))\n[PROOFSTEP]\nleft\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max (lsub fun a => birthday (xL a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yL b))\n[PROOFSTEP]\nleft\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| lsub fun a => birthday (xL a + mk yl yr yL yR)\n[PROOFSTEP]\nright\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| fun a => birthday (xL a + mk yl yr yL yR)\n[PROOFSTEP]\nintro a\n[GOAL]\ncase h\na\u271d b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\na : xl\n| birthday (xL a + mk yl yr yL yR)\n[PROOFSTEP]\nrw [birthday_add (xL a) \u27e8yl, yr, yL, yR\u27e9]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yL b)))\n (max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b))) =\n max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' \u266f birthday (mk yl yr yL yR))\n (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) \u266f b')\n[PROOFSTEP]\nconv_lhs => left; right; right; intro b; rw [birthday_add \u27e8xl, xr, xL, xR\u27e9 (yL b)]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max (max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yL b)))\n (max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b)))\n[PROOFSTEP]\nleft; right; right; intro b; rw [birthday_add \u27e8xl, xr, xL, xR\u27e9 (yL b)]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max (max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yL b)))\n (max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b)))\n[PROOFSTEP]\nleft; right; right; intro b; rw [birthday_add \u27e8xl, xr, xL, xR\u27e9 (yL b)]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max (max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yL b)))\n (max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b)))\n[PROOFSTEP]\nleft\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yL b))\n[PROOFSTEP]\nright\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| lsub fun b => birthday (mk xl xr xL xR + yL b)\n[PROOFSTEP]\nright\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| fun b => birthday (mk xl xr xL xR + yL b)\n[PROOFSTEP]\nintro b\n[GOAL]\ncase h\na b\u271d x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nb : yl\n| birthday (mk xl xr xL xR + yL b)\n[PROOFSTEP]\nrw [birthday_add \u27e8xl, xr, xL, xR\u27e9 (yL b)]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max\n (max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b)))\n (max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b))) =\n max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' \u266f birthday (mk yl yr yL yR))\n (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) \u266f b')\n[PROOFSTEP]\nconv_lhs => right; left; right; intro a; rw [birthday_add (xR a) \u27e8yl, yr, yL, yR\u27e9]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max\n (max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b)))\n (max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b)))\n[PROOFSTEP]\nright; left; right; intro a; rw [birthday_add (xR a) \u27e8yl, yr, yL, yR\u27e9]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max\n (max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b)))\n (max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b)))\n[PROOFSTEP]\nright; left; right; intro a; rw [birthday_add (xR a) \u27e8yl, yr, yL, yR\u27e9]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max\n (max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b)))\n (max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b)))\n[PROOFSTEP]\nright\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max (lsub fun a => birthday (xR a + mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b))\n[PROOFSTEP]\nleft\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| lsub fun a => birthday (xR a + mk yl yr yL yR)\n[PROOFSTEP]\nright\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| fun a => birthday (xR a + mk yl yr yL yR)\n[PROOFSTEP]\nintro a\n[GOAL]\ncase h\na\u271d b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\na : xr\n| birthday (xR a + mk yl yr yL yR)\n[PROOFSTEP]\nrw [birthday_add (xR a) \u27e8yl, yr, yL, yR\u27e9]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max\n (max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b)))\n (max (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR))\n (lsub fun b => birthday (mk xl xr xL xR + yR b))) =\n max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' \u266f birthday (mk yl yr yL yR))\n (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) \u266f b')\n[PROOFSTEP]\nconv_lhs => right; right; right; intro b; rw [birthday_add \u27e8xl, xr, xL, xR\u27e9 (yR b)]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max\n (max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b)))\n (max (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b)))\n[PROOFSTEP]\nright; right; right; intro b; rw [birthday_add \u27e8xl, xr, xL, xR\u27e9 (yR b)]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max\n (max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b)))\n (max (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b)))\n[PROOFSTEP]\nright; right; right; intro b; rw [birthday_add \u27e8xl, xr, xL, xR\u27e9 (yR b)]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max\n (max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b)))\n (max (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b)))\n[PROOFSTEP]\nright\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| max (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR)) (lsub fun b => birthday (mk xl xr xL xR + yR b))\n[PROOFSTEP]\nright\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| lsub fun b => birthday (mk xl xr xL xR + yR b)\n[PROOFSTEP]\nright\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n| fun b => birthday (mk xl xr xL xR + yR b)\n[PROOFSTEP]\nintro b\n[GOAL]\ncase h\na b\u271d x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nb : yr\n| birthday (mk xl xr xL xR + yR b)\n[PROOFSTEP]\nrw [birthday_add \u27e8xl, xr, xL, xR\u27e9 (yR b)]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max\n (max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b)))\n (max (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yR b))) =\n max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' \u266f birthday (mk yl yr yL yR))\n (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) \u266f b')\n[PROOFSTEP]\nrw [max_max_max_comm]\n[GOAL]\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max\n (max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR)))\n (max (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yR b))) =\n max (blsub (birthday (mk xl xr xL xR)) fun a' x => a' \u266f birthday (mk yl yr yL yR))\n (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) \u266f b')\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR)) =\n blsub (birthday (mk xl xr xL xR)) fun a' x => a' \u266f birthday (mk yl yr yL yR)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase e_a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yR b)) =\n blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) \u266f b'\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR)) \u2264\n blsub (birthday (mk xl xr xL xR)) fun a' x => a' \u266f birthday (mk yl yr yL yR)\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 (blsub (birthday (mk xl xr xL xR)) fun a' x => a' \u266f birthday (mk yl yr yL yR)) \u2264\n max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR))\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yR b)) \u2264\n blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) \u266f b'\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) \u266f b') \u2264\n max (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yR b))\n[PROOFSTEP]\nany_goals\n exact\n max_le_iff.2\n \u27e8lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveLeft_lt _),\n lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveRight_lt _)\u27e9\n[GOAL]\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR)) \u2264\n blsub (birthday (mk xl xr xL xR)) fun a' x => a' \u266f birthday (mk yl yr yL yR)\n[PROOFSTEP]\nexact\n max_le_iff.2\n \u27e8lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveLeft_lt _),\n lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveRight_lt _)\u27e9\n[GOAL]\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 (blsub (birthday (mk xl xr xL xR)) fun a' x => a' \u266f birthday (mk yl yr yL yR)) \u2264\n max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR))\n[PROOFSTEP]\nexact\n max_le_iff.2\n \u27e8lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveLeft_lt _),\n lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveRight_lt _)\u27e9\n[GOAL]\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yR b)) \u2264\n blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) \u266f b'\n[PROOFSTEP]\nexact\n max_le_iff.2\n \u27e8lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveLeft_lt _),\n lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveRight_lt _)\u27e9\n[GOAL]\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) \u266f b') \u2264\n max (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yR b))\n[PROOFSTEP]\nexact\n max_le_iff.2\n \u27e8lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveLeft_lt _),\n lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveRight_lt _)\u27e9\n[GOAL]\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 (blsub (birthday (mk xl xr xL xR)) fun a' x => a' \u266f birthday (mk yl yr yL yR)) \u2264\n max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR))\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) \u266f b') \u2264\n max (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yR b))\n[PROOFSTEP]\nall_goals\n refine blsub_le_iff.2 fun i hi => ?_\n rcases lt_birthday_iff.1 hi with (\u27e8j, hj\u27e9 | \u27e8j, hj\u27e9)\n[GOAL]\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 (blsub (birthday (mk xl xr xL xR)) fun a' x => a' \u266f birthday (mk yl yr yL yR)) \u2264\n max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR))\n[PROOFSTEP]\nrefine blsub_le_iff.2 fun i hi => ?_\n[GOAL]\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < birthday (mk xl xr xL xR)\n\u22a2 i \u266f birthday (mk yl yr yL yR) <\n max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR))\n[PROOFSTEP]\nrcases lt_birthday_iff.1 hi with (\u27e8j, hj\u27e9 | \u27e8j, hj\u27e9)\n[GOAL]\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 (blsub (birthday (mk yl yr yL yR)) fun b' x => birthday (mk xl xr xL xR) \u266f b') \u2264\n max (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yR b))\n[PROOFSTEP]\nrefine blsub_le_iff.2 fun i hi => ?_\n[GOAL]\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < birthday (mk yl yr yL yR)\n\u22a2 birthday (mk xl xr xL xR) \u266f i <\n max (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yR b))\n[PROOFSTEP]\nrcases lt_birthday_iff.1 hi with (\u27e8j, hj\u27e9 | \u27e8j, hj\u27e9)\n[GOAL]\ncase e_a.a.inl.intro\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < birthday (mk xl xr xL xR)\nj : LeftMoves (mk xl xr xL xR)\nhj : i \u2264 birthday (moveLeft (mk xl xr xL xR) j)\n\u22a2 i \u266f birthday (mk yl yr yL yR) <\n max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR))\n[PROOFSTEP]\nexact lt_max_of_lt_left ((nadd_le_nadd_right hj _).trans_lt (lt_lsub _ _))\n[GOAL]\ncase e_a.a.inr.intro\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < birthday (mk xl xr xL xR)\nj : RightMoves (mk xl xr xL xR)\nhj : i \u2264 birthday (moveRight (mk xl xr xL xR) j)\n\u22a2 i \u266f birthday (mk yl yr yL yR) <\n max (lsub fun a => birthday (xL a) \u266f birthday (mk yl yr yL yR))\n (lsub fun a => birthday (xR a) \u266f birthday (mk yl yr yL yR))\n[PROOFSTEP]\nexact lt_max_of_lt_right ((nadd_le_nadd_right hj _).trans_lt (lt_lsub _ _))\n[GOAL]\ncase e_a.a.inl.intro\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < birthday (mk yl yr yL yR)\nj : LeftMoves (mk yl yr yL yR)\nhj : i \u2264 birthday (moveLeft (mk yl yr yL yR) j)\n\u22a2 birthday (mk xl xr xL xR) \u266f i <\n max (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yR b))\n[PROOFSTEP]\nexact lt_max_of_lt_left ((nadd_le_nadd_left hj _).trans_lt (lt_lsub _ _))\n[GOAL]\ncase e_a.a.inr.intro\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < birthday (mk yl yr yL yR)\nj : RightMoves (mk yl yr yL yR)\nhj : i \u2264 birthday (moveRight (mk yl yr yL yR) j)\n\u22a2 birthday (mk xl xr xL xR) \u266f i <\n max (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yL b))\n (lsub fun b => birthday (mk xl xr xL xR) \u266f birthday (yR b))\n[PROOFSTEP]\nexact lt_max_of_lt_right ((nadd_le_nadd_left hj _).trans_lt (lt_lsub _ _))\n[GOAL]\na b x : PGame\n\u22a2 birthday (a + 0) = birthday a\n[PROOFSTEP]\nsimp\n[GOAL]\na b x : PGame\n\u22a2 birthday (0 + a) = birthday a\n[PROOFSTEP]\nsimp\n[GOAL]\na b x : PGame\n\u22a2 birthday (a + 1) = Order.succ (birthday a)\n[PROOFSTEP]\nsimp\n[GOAL]\na b x : PGame\n\u22a2 birthday (1 + a) = Order.succ (birthday a)\n[PROOFSTEP]\nsimp\n[GOAL]\na b x : PGame\nn : \u2115\n\u22a2 birthday \u2191(n + 1) = \u2191(n + 1)\n[PROOFSTEP]\nsimp [birthday_nat_cast]\n[GOAL]\na b x : PGame\nn : \u2115\n\u22a2 birthday (a + \u2191n) = birthday a + \u2191n\n[PROOFSTEP]\nsimp\n[GOAL]\na b x : PGame\nn : \u2115\n\u22a2 birthday (\u2191n + a) = birthday a + \u2191n\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.SetTheory.Game.Birthday", "llama_tokens": 19101, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147275, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.44678044334461686}} {"text": "[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y : L\nhx : x \u2208 0.carrier\n_hy : y \u2208 0.carrier\n\u22a2 \u2045x, y\u2046 \u2208 0.carrier\n[PROOFSTEP]\nrw [(Submodule.mem_bot R).1 hx, zero_lie]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y : L\nhx : x \u2208 0.carrier\n_hy : y \u2208 0.carrier\n\u22a2 0 \u2208 0.carrier\n[PROOFSTEP]\nexact Submodule.zero_mem 0\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' L'' : LieSubalgebra R L\nh : (fun L' => L'.carrier) L' = (fun L' => L'.carrier) L''\n\u22a2 L' = L''\n[PROOFSTEP]\nrcases L' with \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.mk\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL'' : LieSubalgebra R L\ntoAddSubmonoid\u271d : AddSubmonoid L\nsmul_mem'\u271d : \u2200 (c : R) {x : L}, x \u2208 toAddSubmonoid\u271d.carrier \u2192 c \u2022 x \u2208 toAddSubmonoid\u271d.carrier\nlie_mem'\u271d :\n \u2200 {x y : L},\n x \u2208 { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n y \u2208 { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n \u2045x, y\u2046 \u2208 { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier\nh :\n (fun L' => L'.carrier)\n { toSubmodule := { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }, lie_mem' := lie_mem'\u271d } =\n (fun L' => L'.carrier) L''\n\u22a2 { toSubmodule := { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }, lie_mem' := lie_mem'\u271d } = L''\n[PROOFSTEP]\nrcases L'' with \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.mk.mk.mk\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\ntoAddSubmonoid\u271d\u00b9 : AddSubmonoid L\nsmul_mem'\u271d\u00b9 : \u2200 (c : R) {x : L}, x \u2208 toAddSubmonoid\u271d\u00b9.carrier \u2192 c \u2022 x \u2208 toAddSubmonoid\u271d\u00b9.carrier\nlie_mem'\u271d\u00b9 :\n \u2200 {x y : L},\n x \u2208 { toAddSubmonoid := toAddSubmonoid\u271d\u00b9, smul_mem' := smul_mem'\u271d\u00b9 }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n y \u2208 { toAddSubmonoid := toAddSubmonoid\u271d\u00b9, smul_mem' := smul_mem'\u271d\u00b9 }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n \u2045x, y\u2046 \u2208\n { toAddSubmonoid := toAddSubmonoid\u271d\u00b9, smul_mem' := smul_mem'\u271d\u00b9 }.toAddSubmonoid.toAddSubsemigroup.carrier\ntoAddSubmonoid\u271d : AddSubmonoid L\nsmul_mem'\u271d : \u2200 (c : R) {x : L}, x \u2208 toAddSubmonoid\u271d.carrier \u2192 c \u2022 x \u2208 toAddSubmonoid\u271d.carrier\nlie_mem'\u271d :\n \u2200 {x y : L},\n x \u2208 { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n y \u2208 { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n \u2045x, y\u2046 \u2208 { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier\nh :\n (fun L' => L'.carrier)\n { toSubmodule := { toAddSubmonoid := toAddSubmonoid\u271d\u00b9, smul_mem' := smul_mem'\u271d\u00b9 }, lie_mem' := lie_mem'\u271d\u00b9 } =\n (fun L' => L'.carrier)\n { toSubmodule := { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }, lie_mem' := lie_mem'\u271d }\n\u22a2 { toSubmodule := { toAddSubmonoid := toAddSubmonoid\u271d\u00b9, smul_mem' := smul_mem'\u271d\u00b9 }, lie_mem' := lie_mem'\u271d\u00b9 } =\n { toSubmodule := { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }, lie_mem' := lie_mem'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.mk.mk.e_toSubmodule.e_toAddSubmonoid\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\ntoAddSubmonoid\u271d\u00b9 : AddSubmonoid L\nsmul_mem'\u271d\u00b9 : \u2200 (c : R) {x : L}, x \u2208 toAddSubmonoid\u271d\u00b9.carrier \u2192 c \u2022 x \u2208 toAddSubmonoid\u271d\u00b9.carrier\nlie_mem'\u271d\u00b9 :\n \u2200 {x y : L},\n x \u2208 { toAddSubmonoid := toAddSubmonoid\u271d\u00b9, smul_mem' := smul_mem'\u271d\u00b9 }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n y \u2208 { toAddSubmonoid := toAddSubmonoid\u271d\u00b9, smul_mem' := smul_mem'\u271d\u00b9 }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n \u2045x, y\u2046 \u2208\n { toAddSubmonoid := toAddSubmonoid\u271d\u00b9, smul_mem' := smul_mem'\u271d\u00b9 }.toAddSubmonoid.toAddSubsemigroup.carrier\ntoAddSubmonoid\u271d : AddSubmonoid L\nsmul_mem'\u271d : \u2200 (c : R) {x : L}, x \u2208 toAddSubmonoid\u271d.carrier \u2192 c \u2022 x \u2208 toAddSubmonoid\u271d.carrier\nlie_mem'\u271d :\n \u2200 {x y : L},\n x \u2208 { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n y \u2208 { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n \u2045x, y\u2046 \u2208 { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier\nh :\n (fun L' => L'.carrier)\n { toSubmodule := { toAddSubmonoid := toAddSubmonoid\u271d\u00b9, smul_mem' := smul_mem'\u271d\u00b9 }, lie_mem' := lie_mem'\u271d\u00b9 } =\n (fun L' => L'.carrier)\n { toSubmodule := { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }, lie_mem' := lie_mem'\u271d }\n\u22a2 toAddSubmonoid\u271d\u00b9 = toAddSubmonoid\u271d\n[PROOFSTEP]\nexact SetLike.coe_injective' h\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\n\u22a2 \u2200 (x y z : { x // x \u2208 L' }), \u2045x + y, z\u2046 = \u2045x, z\u2046 + \u2045y, z\u2046\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\nx\u271d y\u271d z\u271d : { x // x \u2208 L' }\n\u22a2 \u2045x\u271d + y\u271d, z\u271d\u2046 = \u2045x\u271d, z\u271d\u2046 + \u2045y\u271d, z\u271d\u2046\n[PROOFSTEP]\napply SetCoe.ext\n[GOAL]\ncase a\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\nx\u271d y\u271d z\u271d : { x // x \u2208 L' }\n\u22a2 \u2191\u2045x\u271d + y\u271d, z\u271d\u2046 = \u2191(\u2045x\u271d, z\u271d\u2046 + \u2045y\u271d, z\u271d\u2046)\n[PROOFSTEP]\napply add_lie\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\n\u22a2 \u2200 (x y z : { x // x \u2208 L' }), \u2045x, y + z\u2046 = \u2045x, y\u2046 + \u2045x, z\u2046\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\nx\u271d y\u271d z\u271d : { x // x \u2208 L' }\n\u22a2 \u2045x\u271d, y\u271d + z\u271d\u2046 = \u2045x\u271d, y\u271d\u2046 + \u2045x\u271d, z\u271d\u2046\n[PROOFSTEP]\napply SetCoe.ext\n[GOAL]\ncase a\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\nx\u271d y\u271d z\u271d : { x // x \u2208 L' }\n\u22a2 \u2191\u2045x\u271d, y\u271d + z\u271d\u2046 = \u2191(\u2045x\u271d, y\u271d\u2046 + \u2045x\u271d, z\u271d\u2046)\n[PROOFSTEP]\napply lie_add\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\n\u22a2 \u2200 (x : { x // x \u2208 L' }), \u2045x, x\u2046 = 0\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\nx\u271d : { x // x \u2208 L' }\n\u22a2 \u2045x\u271d, x\u271d\u2046 = 0\n[PROOFSTEP]\napply SetCoe.ext\n[GOAL]\ncase a\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\nx\u271d : { x // x \u2208 L' }\n\u22a2 \u2191\u2045x\u271d, x\u271d\u2046 = \u21910\n[PROOFSTEP]\napply lie_self\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\n\u22a2 \u2200 (x y z : { x // x \u2208 L' }), \u2045x, \u2045y, z\u2046\u2046 = \u2045\u2045x, y\u2046, z\u2046 + \u2045y, \u2045x, z\u2046\u2046\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\nx\u271d y\u271d z\u271d : { x // x \u2208 L' }\n\u22a2 \u2045x\u271d, \u2045y\u271d, z\u271d\u2046\u2046 = \u2045\u2045x\u271d, y\u271d\u2046, z\u271d\u2046 + \u2045y\u271d, \u2045x\u271d, z\u271d\u2046\u2046\n[PROOFSTEP]\napply SetCoe.ext\n[GOAL]\ncase a\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\nx\u271d y\u271d z\u271d : { x // x \u2208 L' }\n\u22a2 \u2191\u2045x\u271d, \u2045y\u271d, z\u271d\u2046\u2046 = \u2191(\u2045\u2045x\u271d, y\u271d\u2046, z\u271d\u2046 + \u2045y\u271d, \u2045x\u271d, z\u271d\u2046\u2046)\n[PROOFSTEP]\napply leibniz_lie\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\n\u22a2 \u2200 (t : R) (x y : { x // x \u2208 L' }), \u2045x, t \u2022 y\u2046 = t \u2022 \u2045x, y\u2046\n[PROOFSTEP]\n{ intros\n apply SetCoe.ext\n apply lie_smul\n}\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\n\u22a2 \u2200 (t : R) (x y : { x // x \u2208 L' }), \u2045x, t \u2022 y\u2046 = t \u2022 \u2045x, y\u2046\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\nt\u271d : R\nx\u271d y\u271d : { x // x \u2208 L' }\n\u22a2 \u2045x\u271d, t\u271d \u2022 y\u271d\u2046 = t\u271d \u2022 \u2045x\u271d, y\u271d\u2046\n[PROOFSTEP]\napply SetCoe.ext\n[GOAL]\ncase a\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\nt\u271d : R\nx\u271d y\u271d : { x // x \u2208 L' }\n\u22a2 \u2191\u2045x\u271d, t\u271d \u2022 y\u271d\u2046 = \u2191(t\u271d \u2022 \u2045x\u271d, y\u271d\u2046)\n[PROOFSTEP]\napply lie_smul\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\np : Submodule R L\nh :\n \u2200 {x y : L},\n x \u2208\n { toAddSubmonoid := p.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L}, x \u2208 p.carrier \u2192 c \u2022 x \u2208 p.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n y \u2208\n { toAddSubmonoid := p.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L}, x \u2208 p.carrier \u2192 c \u2022 x \u2208 p.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n \u2045x, y\u2046 \u2208\n { toAddSubmonoid := p.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L}, x \u2208 p.carrier \u2192 c \u2022 x \u2208 p.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 {\n toSubmodule :=\n { toAddSubmonoid := p.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 p.carrier \u2192 c \u2022 x \u2208 p.carrier) },\n lie_mem' := h }.toSubmodule =\n p\n[PROOFSTEP]\ncases p\n[GOAL]\ncase mk\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' : LieSubalgebra R L\ntoAddSubmonoid\u271d : AddSubmonoid L\nsmul_mem'\u271d : \u2200 (c : R) {x : L}, x \u2208 toAddSubmonoid\u271d.carrier \u2192 c \u2022 x \u2208 toAddSubmonoid\u271d.carrier\nh :\n \u2200 {x y : L},\n x \u2208\n { toAddSubmonoid := { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L},\n x \u2208\n { toAddSubmonoid := toAddSubmonoid\u271d,\n smul_mem' := smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n c \u2022 x \u2208\n { toAddSubmonoid := toAddSubmonoid\u271d,\n smul_mem' :=\n smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n y \u2208\n { toAddSubmonoid := { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L},\n x \u2208\n { toAddSubmonoid := toAddSubmonoid\u271d,\n smul_mem' := smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n c \u2022 x \u2208\n { toAddSubmonoid := toAddSubmonoid\u271d,\n smul_mem' :=\n smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n \u2045x, y\u2046 \u2208\n { toAddSubmonoid := { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L},\n x \u2208\n { toAddSubmonoid := toAddSubmonoid\u271d,\n smul_mem' := smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n c \u2022 x \u2208\n { toAddSubmonoid := toAddSubmonoid\u271d,\n smul_mem' :=\n smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 {\n toSubmodule :=\n { toAddSubmonoid := { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L},\n x \u2208\n { toAddSubmonoid := toAddSubmonoid\u271d,\n smul_mem' := smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n c \u2022 x \u2208\n { toAddSubmonoid := toAddSubmonoid\u271d,\n smul_mem' := smul_mem'\u271d }.toAddSubmonoid.toAddSubsemigroup.carrier) },\n lie_mem' := h }.toSubmodule =\n { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' L\u2081' L\u2082' : LieSubalgebra R L\nh : L\u2081'.toSubmodule = L\u2082'.toSubmodule\n\u22a2 L\u2081' = L\u2082'\n[PROOFSTEP]\nrw [SetLike.ext'_iff] at h \n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' L\u2081' L\u2082' : LieSubalgebra R L\nh\u271d : L\u2081'.toSubmodule = L\u2082'.toSubmodule\nh : \u2191L\u2081'.toSubmodule = \u2191L\u2082'.toSubmodule\n\u22a2 L\u2081' = L\u2082'\n[PROOFSTEP]\nrw [\u2190 coe_set_eq]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nL' L\u2081' L\u2082' : LieSubalgebra R L\nh\u271d : L\u2081'.toSubmodule = L\u2082'.toSubmodule\nh : \u2191L\u2081'.toSubmodule = \u2191L\u2082'.toSubmodule\n\u22a2 \u2191L\u2081' = \u2191L\u2082'\n[PROOFSTEP]\nexact h\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\nL' : LieSubalgebra R L\nM : Type w\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : LieRingModule L M\nN : Type w\u2081\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : LieModule R L N\ninst\u271d\u00b9 : Module R M\ninst\u271d : LieModule R L M\nt : R\nx : { x // x \u2208 L' }\nm : M\n\u22a2 \u2045t \u2022 x, m\u2046 = t \u2022 \u2045x, m\u2046\n[PROOFSTEP]\nsimp only [coe_bracket_of_module, smul_lie, Submodule.coe_smul_of_tower]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\nL' : LieSubalgebra R L\nM : Type w\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : LieRingModule L M\nN : Type w\u2081\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : LieModule R L N\ninst\u271d\u00b9 : Module R M\ninst\u271d : LieModule R L M\nt : R\nx : { x // x \u2208 L' }\nm : M\n\u22a2 \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046\n[PROOFSTEP]\nsimp only [coe_bracket_of_module, lie_smul]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nsrc\u271d : Submodule R L\u2082 := LinearMap.range \u2191f\n\u22a2 \u2200 {x y : L\u2082},\n x \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082},\n x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n y \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082},\n x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n \u2045x, y\u2046 \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082},\n x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrintro - - \u27e8x, rfl\u27e9 \u27e8y, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nsrc\u271d : Submodule R L\u2082 := LinearMap.range \u2191f\nx y : L\n\u22a2 \u2045\u2191\u2191f x, \u2191\u2191f y\u2046 \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nexact \u27e8\u2045x, y\u2046, f.map_lie x y\u27e9\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nsrc\u271d : L \u2192\u2097[R] { x // x \u2208 LinearMap.range \u2191f } := LinearMap.rangeRestrict \u2191f\nx y : L\n\u22a2 AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : L),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n \u2045x, y\u2046 =\n \u2045AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : L),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n x,\n AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : L),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n y\u2046\n[PROOFSTEP]\napply Subtype.ext\n[GOAL]\ncase a\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nsrc\u271d : L \u2192\u2097[R] { x // x \u2208 LinearMap.range \u2191f } := LinearMap.rangeRestrict \u2191f\nx y : L\n\u22a2 \u2191(AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : L),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n \u2045x, y\u2046) =\n \u2191\u2045AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : L),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n x,\n AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : L),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n y\u2046\n[PROOFSTEP]\nexact f.map_lie x y\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\n\u22a2 Function.Surjective \u2191(rangeRestrict f)\n[PROOFSTEP]\nrintro \u27e8y, hy\u27e9\n[GOAL]\ncase mk\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\ny : L\u2082\nhy : y \u2208 range f\n\u22a2 \u2203 a, \u2191(rangeRestrict f) a = { val := y, property := hy }\n[PROOFSTEP]\nerw [mem_range] at hy \n[GOAL]\ncase mk\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\ny : L\u2082\nhy\u271d : y \u2208 range f\nhy : \u2203 y_1, \u2191f y_1 = y\n\u22a2 \u2203 a, \u2191(rangeRestrict f) a = { val := y, property := hy\u271d }\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 := hy\n[GOAL]\ncase mk.intro\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nx : L\nhy : \u2191f x \u2208 range f\n\u22a2 \u2203 a, \u2191(rangeRestrict f) a = { val := \u2191f x, property := hy }\n[PROOFSTEP]\nuse x\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nx : L\nhy : \u2191f x \u2208 range f\n\u22a2 \u2191(rangeRestrict f) x = { val := \u2191f x, property := hy }\n[PROOFSTEP]\nsimp only [Subtype.mk_eq_mk, rangeRestrict_apply]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nh : Function.Injective \u2191f\nx y : L\nhxy : \u2191(rangeRestrict f) x = \u2191(rangeRestrict f) y\n\u22a2 x = y\n[PROOFSTEP]\nsimp only [Subtype.mk_eq_mk, rangeRestrict_apply] at hxy \n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nh : Function.Injective \u2191f\nx y : L\nhxy : \u2191f x = \u2191f y\n\u22a2 x = y\n[PROOFSTEP]\nexact h hxy\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\np : Submodule R L\n\u22a2 (\u2203 K, K.toSubmodule = p) \u2194 \u2200 (x y : L), x \u2208 p \u2192 y \u2208 p \u2192 \u2045x, y\u2046 \u2208 p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\np : Submodule R L\n\u22a2 (\u2203 K, K.toSubmodule = p) \u2192 \u2200 (x y : L), x \u2208 p \u2192 y \u2208 p \u2192 \u2045x, y\u2046 \u2208 p\n[PROOFSTEP]\nrintro \u27e8K, rfl\u27e9 _ _\n[GOAL]\ncase mp.intro\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK : LieSubalgebra R L\nx\u271d y\u271d : L\n\u22a2 x\u271d \u2208 K.toSubmodule \u2192 y\u271d \u2208 K.toSubmodule \u2192 \u2045x\u271d, y\u271d\u2046 \u2208 K.toSubmodule\n[PROOFSTEP]\nexact K.lie_mem'\n[GOAL]\ncase mpr\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\np : Submodule R L\n\u22a2 (\u2200 (x y : L), x \u2208 p \u2192 y \u2208 p \u2192 \u2045x, y\u2046 \u2208 p) \u2192 \u2203 K, K.toSubmodule = p\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\np : Submodule R L\nh : \u2200 (x y : L), x \u2208 p \u2192 y \u2208 p \u2192 \u2045x, y\u2046 \u2208 p\n\u22a2 \u2203 K, K.toSubmodule = p\n[PROOFSTEP]\nuse{ p with lie_mem' := h _ _ }\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\n\u22a2 LieHom.range (incl K) = K\n[PROOFSTEP]\nrw [\u2190 coe_to_submodule_eq_iff]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\n\u22a2 (LieHom.range (incl K)).toSubmodule = K.toSubmodule\n[PROOFSTEP]\nexact (K : Submodule R L).range_subtype\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nsrc\u271d : Submodule R L\u2082 := Submodule.map (\u2191f) K.toSubmodule\nx y : L\u2082\nhx :\n x \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\nhy :\n y \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 \u2045x, y\u2046 \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nerw [Submodule.mem_map] at hx \n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nsrc\u271d : Submodule R L\u2082 := Submodule.map (\u2191f) K.toSubmodule\nx y : L\u2082\nhx : \u2203 y, y \u2208 K.toSubmodule \u2227 \u2191\u2191f y = x\nhy :\n y \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 \u2045x, y\u2046 \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrcases hx with \u27e8x', hx', hx\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nsrc\u271d : Submodule R L\u2082 := Submodule.map (\u2191f) K.toSubmodule\nx y : L\u2082\nhy :\n y \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\nx' : L\nhx' : x' \u2208 K.toSubmodule\nhx : \u2191\u2191f x' = x\n\u22a2 \u2045x, y\u2046 \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrw [\u2190 hx]\n[GOAL]\ncase intro.intro\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nsrc\u271d : Submodule R L\u2082 := Submodule.map (\u2191f) K.toSubmodule\nx y : L\u2082\nhy :\n y \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\nx' : L\nhx' : x' \u2208 K.toSubmodule\nhx : \u2191\u2191f x' = x\n\u22a2 \u2045\u2191\u2191f x', y\u2046 \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nerw [Submodule.mem_map] at hy \n[GOAL]\ncase intro.intro\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nsrc\u271d : Submodule R L\u2082 := Submodule.map (\u2191f) K.toSubmodule\nx y : L\u2082\nhy : \u2203 y_1, y_1 \u2208 K.toSubmodule \u2227 \u2191\u2191f y_1 = y\nx' : L\nhx' : x' \u2208 K.toSubmodule\nhx : \u2191\u2191f x' = x\n\u22a2 \u2045\u2191\u2191f x', y\u2046 \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrcases hy with \u27e8y', hy', hy\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nsrc\u271d : Submodule R L\u2082 := Submodule.map (\u2191f) K.toSubmodule\nx y : L\u2082\nx' : L\nhx' : x' \u2208 K.toSubmodule\nhx : \u2191\u2191f x' = x\ny' : L\nhy' : y' \u2208 K.toSubmodule\nhy : \u2191\u2191f y' = y\n\u22a2 \u2045\u2191\u2191f x', y\u2046 \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrw [\u2190 hy]\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nsrc\u271d : Submodule R L\u2082 := Submodule.map (\u2191f) K.toSubmodule\nx y : L\u2082\nx' : L\nhx' : x' \u2208 K.toSubmodule\nhx : \u2191\u2191f x' = x\ny' : L\nhy' : y' \u2208 K.toSubmodule\nhy : \u2191\u2191f y' = y\n\u22a2 \u2045\u2191\u2191f x', \u2191\u2191f y'\u2046 \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L\u2082}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nerw [Submodule.mem_map]\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nsrc\u271d : Submodule R L\u2082 := Submodule.map (\u2191f) K.toSubmodule\nx y : L\u2082\nx' : L\nhx' : x' \u2208 K.toSubmodule\nhx : \u2191\u2191f x' = x\ny' : L\nhy' : y' \u2208 K.toSubmodule\nhy : \u2191\u2191f y' = y\n\u22a2 \u2203 y, y \u2208 K.toSubmodule \u2227 \u2191\u2191f y = \u2045\u2191\u2191f x', \u2191\u2191f y'\u2046\n[PROOFSTEP]\nexact \u27e8\u2045x', y'\u2046, K.lie_mem hx' hy', f.map_lie x' y'\u27e9\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nsrc\u271d : Submodule R L := Submodule.comap (\u2191f) K\u2082.toSubmodule\nx y : L\nhx :\n x \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\nhy :\n y \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 \u2045x, y\u2046 \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nsuffices \u2045f x, f y\u2046 \u2208 K\u2082 by simp [this]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nsrc\u271d : Submodule R L := Submodule.comap (\u2191f) K\u2082.toSubmodule\nx y : L\nhx :\n x \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\nhy :\n y \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\nthis : \u2045\u2191f x, \u2191f y\u2046 \u2208 K\u2082\n\u22a2 \u2045x, y\u2046 \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nsimp [this]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nsrc\u271d : Submodule R L := Submodule.comap (\u2191f) K\u2082.toSubmodule\nx y : L\nhx :\n x \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\nhy :\n y \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 \u2045\u2191f x, \u2191f y\u2046 \u2208 K\u2082\n[PROOFSTEP]\nexact K\u2082.lie_mem hx hy\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\n\u22a2 LieHom.range f = map f \u22a4\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nx\u271d : L\u2082\n\u22a2 x\u271d \u2208 LieHom.range f \u2194 x\u271d \u2208 map f \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nS : Set (LieSubalgebra R L)\nsrc\u271d : Submodule R L := sInf {x | \u2203 s, s \u2208 S \u2227 s.toSubmodule = x}\nx y : L\nhx :\n x \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\nhy :\n y \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 \u2045x, y\u2046 \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nsimp only [Submodule.mem_carrier, mem_iInter, Submodule.sInf_coe, mem_setOf_eq, forall_apply_eq_imp_iff\u2082, exists_imp,\n and_imp] at hx hy \u22a2\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nS : Set (LieSubalgebra R L)\nsrc\u271d : Submodule R L := sInf {x | \u2203 s, s \u2208 S \u2227 s.toSubmodule = x}\nx y : L\nhx : \u2200 (a : LieSubalgebra R L), a \u2208 S \u2192 x \u2208 \u2191a.toSubmodule\nhy : \u2200 (a : LieSubalgebra R L), a \u2208 S \u2192 y \u2208 \u2191a.toSubmodule\n\u22a2 \u2200 (a : LieSubalgebra R L), a \u2208 S \u2192 \u2045x, y\u2046 \u2208 \u2191a.toSubmodule\n[PROOFSTEP]\nintro K hK\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK\u271d K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nS : Set (LieSubalgebra R L)\nsrc\u271d : Submodule R L := sInf {x | \u2203 s, s \u2208 S \u2227 s.toSubmodule = x}\nx y : L\nhx : \u2200 (a : LieSubalgebra R L), a \u2208 S \u2192 x \u2208 \u2191a.toSubmodule\nhy : \u2200 (a : LieSubalgebra R L), a \u2208 S \u2192 y \u2208 \u2191a.toSubmodule\nK : LieSubalgebra R L\nhK : K \u2208 S\n\u22a2 \u2045x, y\u2046 \u2208 \u2191K.toSubmodule\n[PROOFSTEP]\nexact K.lie_mem (hx K hK) (hy K hK)\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nS : Set (LieSubalgebra R L)\n\u22a2 \u2191(sInf S) = \u22c2 (s : LieSubalgebra R L) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\nrw [\u2190 coe_to_submodule, sInf_coe_to_submodule, Submodule.sInf_coe]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nS : Set (LieSubalgebra R L)\n\u22a2 \u22c2 (p : Submodule R L) (_ : p \u2208 {x | \u2203 s, s \u2208 S \u2227 s.toSubmodule = x}), \u2191p = \u22c2 (s : LieSubalgebra R L) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nS : Set (LieSubalgebra R L)\nx : L\n\u22a2 x \u2208 \u22c2 (p : Submodule R L) (_ : p \u2208 {x | \u2203 s, s \u2208 S \u2227 s.toSubmodule = x}), \u2191p \u2194\n x \u2208 \u22c2 (s : LieSubalgebra R L) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nS : Set (LieSubalgebra R L)\n\u22a2 IsGLB S (sInf S)\n[PROOFSTEP]\nhave h : \u2200 K K' : LieSubalgebra R L, (K : Set L) \u2264 K' \u2194 K \u2264 K' :=\n by\n intros\n exact Iff.rfl\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nS : Set (LieSubalgebra R L)\n\u22a2 \u2200 (K K' : LieSubalgebra R L), \u2191K \u2264 \u2191K' \u2194 K \u2264 K'\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nS : Set (LieSubalgebra R L)\nK\u271d K'\u271d : LieSubalgebra R L\n\u22a2 \u2191K\u271d \u2264 \u2191K'\u271d \u2194 K\u271d \u2264 K'\u271d\n[PROOFSTEP]\nexact Iff.rfl\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nS : Set (LieSubalgebra R L)\nh : \u2200 (K K' : LieSubalgebra R L), \u2191K \u2264 \u2191K' \u2194 K \u2264 K'\n\u22a2 IsGLB S (sInf S)\n[PROOFSTEP]\napply IsGLB.of_image @h\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nS : Set (LieSubalgebra R L)\nh : \u2200 (K K' : LieSubalgebra R L), \u2191K \u2264 \u2191K' \u2194 K \u2264 K'\n\u22a2 IsGLB ((fun x => \u2191x) '' S) \u2191(sInf S)\n[PROOFSTEP]\nsimp only [sInf_coe]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nS : Set (LieSubalgebra R L)\nh : \u2200 (K K' : LieSubalgebra R L), \u2191K \u2264 \u2191K' \u2194 K \u2264 K'\n\u22a2 IsGLB ((fun x => \u2191x) '' S) (\u22c2 (s : LieSubalgebra R L) (_ : s \u2208 S), \u2191s)\n[PROOFSTEP]\nexact isGLB_biInf\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nsrc\u271d : CompleteLattice (LieSubalgebra R L) :=\n completeLatticeOfInf (LieSubalgebra R L) (_ : \u2200 (S : Set (LieSubalgebra R L)), IsGLB S (sInf S))\nN : LieSubalgebra R L\nx\u271d : L\nh : x\u271d \u2208 \u22a5\n\u22a2 x\u271d \u2208 N\n[PROOFSTEP]\nrw [mem_bot] at h \n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nsrc\u271d : CompleteLattice (LieSubalgebra R L) :=\n completeLatticeOfInf (LieSubalgebra R L) (_ : \u2200 (S : Set (LieSubalgebra R L)), IsGLB S (sInf S))\nN : LieSubalgebra R L\nx\u271d : L\nh : x\u271d = 0\n\u22a2 x\u271d \u2208 N\n[PROOFSTEP]\nrw [h]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nsrc\u271d : CompleteLattice (LieSubalgebra R L) :=\n completeLatticeOfInf (LieSubalgebra R L) (_ : \u2200 (S : Set (LieSubalgebra R L)), IsGLB S (sInf S))\nN : LieSubalgebra R L\nx\u271d : L\nh : x\u271d = 0\n\u22a2 0 \u2208 N\n[PROOFSTEP]\nexact N.zero_mem'\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nx : L\n\u22a2 x \u2208 K \u2293 K' \u2194 x \u2208 K \u2227 x \u2208 K'\n[PROOFSTEP]\nrw [\u2190 mem_coe_submodule, \u2190 mem_coe_submodule, \u2190 mem_coe_submodule, inf_coe_to_submodule, Submodule.mem_inf]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\n\u22a2 K = \u22a5 \u2194 \u2200 (x : L), x \u2208 K \u2192 x = 0\n[PROOFSTEP]\nrw [_root_.eq_bot_iff]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\n\u22a2 K \u2264 \u22a5 \u2194 \u2200 (x : L), x \u2208 K \u2192 x = 0\n[PROOFSTEP]\nexact Iff.rfl\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\n\u22a2 Subsingleton (LieSubalgebra R { x // x \u2208 \u22a5 })\n[PROOFSTEP]\napply subsingleton_of_bot_eq_top\n[GOAL]\ncase h\u03b1\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\n\u22a2 \u22a5 = \u22a4\n[PROOFSTEP]\next \u27e8x, hx\u27e9\n[GOAL]\ncase h\u03b1.h.mk\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nx : L\nhx : x \u2208 \u22a5\n\u22a2 { val := x, property := hx } \u2208 \u22a5 \u2194 { val := x, property := hx } \u2208 \u22a4\n[PROOFSTEP]\nchange x \u2208 \u22a5 at hx \n[GOAL]\ncase h\u03b1.h.mk\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nx : L\nhx : x \u2208 \u22a5\n\u22a2 { val := x, property := hx } \u2208 \u22a5 \u2194 { val := x, property := hx } \u2208 \u22a4\n[PROOFSTEP]\nrw [LieSubalgebra.mem_bot] at hx \n[GOAL]\ncase h\u03b1.h.mk\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nx : L\nhx\u271d : x \u2208 \u22a5\nhx : x = 0\n\u22a2 { val := x, property := hx\u271d } \u2208 \u22a5 \u2194 { val := x, property := hx\u271d } \u2208 \u22a4\n[PROOFSTEP]\nsubst hx\n[GOAL]\ncase h\u03b1.h.mk\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nhx : 0 \u2208 \u22a5\n\u22a2 { val := 0, property := hx } \u2208 \u22a5 \u2194 { val := 0, property := hx } \u2208 \u22a4\n[PROOFSTEP]\nsimp only [true_iff_iff, eq_self_iff_true, Submodule.mk_eq_zero, mem_bot, mem_top]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\n\u22a2 Subsingleton \u2191\u2191\u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nh : K \u2264 K'\nx y : { x // x \u2208 K }\n\u22a2 \u2191(homOfLe h) x = \u2191(homOfLe h) y \u2192 x = y\n[PROOFSTEP]\nsimp only [homOfLe_apply, imp_self, Subtype.mk_eq_mk, SetLike.coe_eq_coe]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nh : K \u2264 K'\nx : { x // x \u2208 K' }\n\u22a2 x \u2208 ofLe h \u2194 \u2191x \u2208 K\n[PROOFSTEP]\nsimp only [ofLe, homOfLe_apply, LieHom.mem_range]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nh : K \u2264 K'\nx : { x // x \u2208 K' }\n\u22a2 (\u2203 y, { val := \u2191y, property := (_ : \u2191y \u2208 K') } = x) \u2194 \u2191x \u2208 K\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nh : K \u2264 K'\nx : { x // x \u2208 K' }\n\u22a2 (\u2203 y, { val := \u2191y, property := (_ : \u2191y \u2208 K') } = x) \u2192 \u2191x \u2208 K\n[PROOFSTEP]\nrintro \u27e8y, rfl\u27e9\n[GOAL]\ncase mp.intro\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nh : K \u2264 K'\ny : { x // x \u2208 K }\n\u22a2 \u2191{ val := \u2191y, property := (_ : \u2191y \u2208 K') } \u2208 K\n[PROOFSTEP]\nexact y.property\n[GOAL]\ncase mpr\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nh : K \u2264 K'\nx : { x // x \u2208 K' }\n\u22a2 \u2191x \u2208 K \u2192 \u2203 y, { val := \u2191y, property := (_ : \u2191y \u2208 K') } = x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nh\u271d : K \u2264 K'\nx : { x // x \u2208 K' }\nh : \u2191x \u2208 K\n\u22a2 \u2203 y, { val := \u2191y, property := (_ : \u2191y \u2208 K') } = x\n[PROOFSTEP]\nuse\u27e8(x : L), h\u27e9\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nh : K \u2264 K'\n\u22a2 ofLe h = comap (incl K') K\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nh : K \u2264 K'\nx\u271d : { x // x \u2208 K' }\n\u22a2 x\u271d \u2208 ofLe h \u2194 x\u271d \u2208 comap (incl K') K\n[PROOFSTEP]\nrw [mem_ofLe]\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\nh : K \u2264 K'\nx\u271d : { x // x \u2208 K' }\n\u22a2 \u2191x\u271d \u2208 K \u2194 x\u271d \u2208 comap (incl K') K\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\nx : L\n\u22a2 x \u2208 lieSpan R L s \u2194 \u2200 (K : LieSubalgebra R L), s \u2286 \u2191K \u2192 x \u2208 K\n[PROOFSTEP]\nchange x \u2208 (lieSpan R L s : Set L) \u2194 _\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\nx : L\n\u22a2 x \u2208 \u2191(lieSpan R L s) \u2194 \u2200 (K : LieSubalgebra R L), s \u2286 \u2191K \u2192 x \u2208 K\n[PROOFSTEP]\nerw [sInf_coe]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\nx : L\n\u22a2 x \u2208 \u22c2 (s_1 : LieSubalgebra R L) (_ : s_1 \u2208 {N | s \u2286 \u2191N}), \u2191s_1 \u2194 \u2200 (K : LieSubalgebra R L), s \u2286 \u2191K \u2192 x \u2208 K\n[PROOFSTEP]\nexact Set.mem_iInter\u2082\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\n\u22a2 s \u2286 \u2191(lieSpan R L s)\n[PROOFSTEP]\nintro m hm\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\nm : L\nhm : m \u2208 s\n\u22a2 m \u2208 \u2191(lieSpan R L s)\n[PROOFSTEP]\nerw [mem_lieSpan]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\nm : L\nhm : m \u2208 s\n\u22a2 \u2200 (K : LieSubalgebra R L), s \u2286 \u2191K \u2192 m \u2208 K\n[PROOFSTEP]\nintro K hK\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK\u271d K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\nm : L\nhm : m \u2208 s\nK : LieSubalgebra R L\nhK : s \u2286 \u2191K\n\u22a2 m \u2208 K\n[PROOFSTEP]\nexact hK hm\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\n\u22a2 Submodule.span R s \u2264 (lieSpan R L s).toSubmodule\n[PROOFSTEP]\nrw [Submodule.span_le]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\n\u22a2 s \u2286 \u2191(lieSpan R L s).toSubmodule\n[PROOFSTEP]\napply subset_lieSpan\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK\u271d K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\nK : LieSubalgebra R L\n\u22a2 lieSpan R L s \u2264 K \u2194 s \u2286 \u2191K\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK\u271d K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\nK : LieSubalgebra R L\n\u22a2 lieSpan R L s \u2264 K \u2192 s \u2286 \u2191K\n[PROOFSTEP]\nexact Set.Subset.trans subset_lieSpan\n[GOAL]\ncase mpr\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK\u271d K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\nK : LieSubalgebra R L\n\u22a2 s \u2286 \u2191K \u2192 lieSpan R L s \u2264 K\n[PROOFSTEP]\nintro hs m hm\n[GOAL]\ncase mpr\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK\u271d K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\nK : LieSubalgebra R L\nhs : s \u2286 \u2191K\nm : L\nhm : m \u2208 lieSpan R L s\n\u22a2 m \u2208 K\n[PROOFSTEP]\nrw [mem_lieSpan] at hm \n[GOAL]\ncase mpr\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK\u271d K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\nK : LieSubalgebra R L\nhs : s \u2286 \u2191K\nm : L\nhm : \u2200 (K : LieSubalgebra R L), s \u2286 \u2191K \u2192 m \u2208 K\n\u22a2 m \u2208 K\n[PROOFSTEP]\nexact hm _ hs\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns t : Set L\nh : s \u2286 t\n\u22a2 lieSpan R L s \u2264 lieSpan R L t\n[PROOFSTEP]\nrw [lieSpan_le]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns t : Set L\nh : s \u2286 t\n\u22a2 s \u2286 \u2191(lieSpan R L t)\n[PROOFSTEP]\nexact Set.Subset.trans h subset_lieSpan\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\np : Submodule R L\n\u22a2 (lieSpan R L \u2191p).toSubmodule = p \u2194 \u2203 K, K.toSubmodule = p\n[PROOFSTEP]\nrw [p.exists_lieSubalgebra_coe_eq_iff]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\np : Submodule R L\n\u22a2 (lieSpan R L \u2191p).toSubmodule = p \u2194 \u2200 (x y : L), x \u2208 p \u2192 y \u2208 p \u2192 \u2045x, y\u2046 \u2208 p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\np : Submodule R L\n\u22a2 (lieSpan R L \u2191p).toSubmodule = p \u2192 \u2200 (x y : L), x \u2208 p \u2192 y \u2208 p \u2192 \u2045x, y\u2046 \u2208 p\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\np : Submodule R L\n\u22a2 (\u2200 (x y : L), x \u2208 p \u2192 y \u2208 p \u2192 \u2045x, y\u2046 \u2208 p) \u2192 (lieSpan R L \u2191p).toSubmodule = p\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\np : Submodule R L\nh : (lieSpan R L \u2191p).toSubmodule = p\n\u22a2 \u2200 (x y : L), x \u2208 p \u2192 y \u2208 p \u2192 \u2045x, y\u2046 \u2208 p\n[PROOFSTEP]\nintro x m hm\n[GOAL]\ncase mp\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\np : Submodule R L\nh : (lieSpan R L \u2191p).toSubmodule = p\nx m : L\nhm : x \u2208 p\n\u22a2 m \u2208 p \u2192 \u2045x, m\u2046 \u2208 p\n[PROOFSTEP]\nrw [\u2190 h, mem_coe_submodule]\n[GOAL]\ncase mp\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\np : Submodule R L\nh : (lieSpan R L \u2191p).toSubmodule = p\nx m : L\nhm : x \u2208 p\n\u22a2 m \u2208 lieSpan R L \u2191p \u2192 \u2045x, m\u2046 \u2208 (lieSpan R L \u2191p).toSubmodule\n[PROOFSTEP]\nexact lie_mem _ (subset_lieSpan hm)\n[GOAL]\ncase mpr\nR : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns : Set L\np : Submodule R L\nh : \u2200 (x y : L), x \u2208 p \u2192 y \u2208 p \u2192 \u2045x, y\u2046 \u2208 p\n\u22a2 (lieSpan R L \u2191p).toSubmodule = p\n[PROOFSTEP]\nrw [\u2190 coe_to_submodule_mk p @h, coe_to_submodule, coe_to_submodule_eq_iff, lieSpan_eq]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\u2081\ninst\u271d\u00b2 : LieRing L\u2082\ninst\u271d\u00b9 : LieAlgebra R L\u2081\ninst\u271d : LieAlgebra R L\u2082\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nh : Function.Injective \u2191f\n\u22a2 Function.Injective \u2191\u2191f\n[PROOFSTEP]\nrwa [LieHom.coe_toLinearMap]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\u2081\ninst\u271d\u00b2 : LieRing L\u2082\ninst\u271d\u00b9 : LieAlgebra R L\u2081\ninst\u271d : LieAlgebra R L\u2082\nL\u2081' L\u2081'' : LieSubalgebra R L\u2081\nL\u2082' : LieSubalgebra R L\u2082\nh : \u2191L\u2081' = \u2191L\u2081''\n\u22a2 L\u2081'.toSubmodule = L\u2081''.toSubmodule\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\u2081\ninst\u271d\u00b2 : LieRing L\u2082\ninst\u271d\u00b9 : LieAlgebra R L\u2081\ninst\u271d : LieAlgebra R L\u2082\nL\u2081' L\u2081'' : LieSubalgebra R L\u2081\nL\u2082' : LieSubalgebra R L\u2082\nh : \u2191L\u2081' = \u2191L\u2081''\nx : L\u2081\n\u22a2 x \u2208 L\u2081'.toSubmodule \u2194 x \u2208 L\u2081''.toSubmodule\n[PROOFSTEP]\nchange x \u2208 (L\u2081' : Set L\u2081) \u2194 x \u2208 (L\u2081'' : Set L\u2081)\n[GOAL]\ncase h\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\u2081\ninst\u271d\u00b2 : LieRing L\u2082\ninst\u271d\u00b9 : LieAlgebra R L\u2081\ninst\u271d : LieAlgebra R L\u2082\nL\u2081' L\u2081'' : LieSubalgebra R L\u2081\nL\u2082' : LieSubalgebra R L\u2082\nh : \u2191L\u2081' = \u2191L\u2081''\nx : L\u2081\n\u22a2 x \u2208 \u2191L\u2081' \u2194 x \u2208 \u2191L\u2081''\n[PROOFSTEP]\nrw [h]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\u2081\ninst\u271d\u00b2 : LieRing L\u2082\ninst\u271d\u00b9 : LieAlgebra R L\u2081\ninst\u271d : LieAlgebra R L\u2082\nL\u2081' L\u2081'' : LieSubalgebra R L\u2081\nL\u2082' : LieSubalgebra R L\u2082\ne : L\u2081 \u2243\u2097\u2045R\u2046 L\u2082\nsrc\u271d : { x // x \u2208 L\u2081''.toSubmodule } \u2243\u2097[R] { x // x \u2208 Submodule.map (\u2191(toLinearEquiv e)) L\u2081''.toSubmodule } :=\n LinearEquiv.submoduleMap (toLinearEquiv e) L\u2081''.toSubmodule\nx y : { x // x \u2208 L\u2081'' }\n\u22a2 AddHom.toFun src\u271d.toAddHom \u2045x, y\u2046 = \u2045AddHom.toFun src\u271d.toAddHom x, AddHom.toFun src\u271d.toAddHom y\u2046\n[PROOFSTEP]\napply SetCoe.ext\n[GOAL]\ncase a\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\u2081\ninst\u271d\u00b2 : LieRing L\u2082\ninst\u271d\u00b9 : LieAlgebra R L\u2081\ninst\u271d : LieAlgebra R L\u2082\nL\u2081' L\u2081'' : LieSubalgebra R L\u2081\nL\u2082' : LieSubalgebra R L\u2082\ne : L\u2081 \u2243\u2097\u2045R\u2046 L\u2082\nsrc\u271d : { x // x \u2208 L\u2081''.toSubmodule } \u2243\u2097[R] { x // x \u2208 Submodule.map (\u2191(toLinearEquiv e)) L\u2081''.toSubmodule } :=\n LinearEquiv.submoduleMap (toLinearEquiv e) L\u2081''.toSubmodule\nx y : { x // x \u2208 L\u2081'' }\n\u22a2 \u2191(AddHom.toFun src\u271d.toAddHom \u2045x, y\u2046) = \u2191\u2045AddHom.toFun src\u271d.toAddHom x, AddHom.toFun src\u271d.toAddHom y\u2046\n[PROOFSTEP]\nexact LieHom.map_lie (\u2191e : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082) \u2191x \u2191y\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\u2081\ninst\u271d\u00b2 : LieRing L\u2082\ninst\u271d\u00b9 : LieAlgebra R L\u2081\ninst\u271d : LieAlgebra R L\u2082\nL\u2081' L\u2081'' : LieSubalgebra R L\u2081\nL\u2082' : LieSubalgebra R L\u2082\ne : L\u2081 \u2243\u2097\u2045R\u2046 L\u2082\nh : LieSubalgebra.map e.toLieHom L\u2081' = L\u2082'\n\u22a2 Submodule.map (\u2191(toLinearEquiv e)) L\u2081'.toSubmodule = L\u2082'.toSubmodule\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\u2081\ninst\u271d\u00b2 : LieRing L\u2082\ninst\u271d\u00b9 : LieAlgebra R L\u2081\ninst\u271d : LieAlgebra R L\u2082\nL\u2081' L\u2081'' : LieSubalgebra R L\u2081\nL\u2082' : LieSubalgebra R L\u2082\ne : L\u2081 \u2243\u2097\u2045R\u2046 L\u2082\nh : LieSubalgebra.map e.toLieHom L\u2081' = L\u2082'\n\u22a2 Submodule.map (\u2191(toLinearEquiv e)) L\u2081'.toSubmodule = (LieSubalgebra.map e.toLieHom L\u2081').toSubmodule\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\u2081\ninst\u271d\u00b2 : LieRing L\u2082\ninst\u271d\u00b9 : LieAlgebra R L\u2081\ninst\u271d : LieAlgebra R L\u2082\nL\u2081' L\u2081'' : LieSubalgebra R L\u2081\nL\u2082' : LieSubalgebra R L\u2082\ne : L\u2081 \u2243\u2097\u2045R\u2046 L\u2082\nh : LieSubalgebra.map e.toLieHom L\u2081' = L\u2082'\nsrc\u271d : { x // x \u2208 L\u2081'.toSubmodule } \u2243\u2097[R] { x // x \u2208 L\u2082'.toSubmodule } :=\n LinearEquiv.ofSubmodules (toLinearEquiv e) L\u2081'.toSubmodule L\u2082'.toSubmodule\n (_ : Submodule.map (\u2191(toLinearEquiv e)) L\u2081'.toSubmodule = L\u2082'.toSubmodule)\nx y : { x // x \u2208 L\u2081' }\n\u22a2 AddHom.toFun src\u271d.toAddHom \u2045x, y\u2046 = \u2045AddHom.toFun src\u271d.toAddHom x, AddHom.toFun src\u271d.toAddHom y\u2046\n[PROOFSTEP]\napply SetCoe.ext\n[GOAL]\ncase a\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\u2081\ninst\u271d\u00b2 : LieRing L\u2082\ninst\u271d\u00b9 : LieAlgebra R L\u2081\ninst\u271d : LieAlgebra R L\u2082\nL\u2081' L\u2081'' : LieSubalgebra R L\u2081\nL\u2082' : LieSubalgebra R L\u2082\ne : L\u2081 \u2243\u2097\u2045R\u2046 L\u2082\nh : LieSubalgebra.map e.toLieHom L\u2081' = L\u2082'\nsrc\u271d : { x // x \u2208 L\u2081'.toSubmodule } \u2243\u2097[R] { x // x \u2208 L\u2082'.toSubmodule } :=\n LinearEquiv.ofSubmodules (toLinearEquiv e) L\u2081'.toSubmodule L\u2082'.toSubmodule\n (_ : Submodule.map (\u2191(toLinearEquiv e)) L\u2081'.toSubmodule = L\u2082'.toSubmodule)\nx y : { x // x \u2208 L\u2081' }\n\u22a2 \u2191(AddHom.toFun src\u271d.toAddHom \u2045x, y\u2046) = \u2191\u2045AddHom.toFun src\u271d.toAddHom x, AddHom.toFun src\u271d.toAddHom y\u2046\n[PROOFSTEP]\nexact LieHom.map_lie (\u2191e : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082) \u2191x \u2191y\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Lie.Subalgebra", "llama_tokens": 29294, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7634837527911057, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.44671522497349037}} {"text": "[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrcases em (\u2203 i, m i = 0) with (\u27e8i, hi\u27e9 | hm) <;> [skip; push_neg at hm ]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrcases em (\u2203 i, m i = 0) with (\u27e8i, hi\u27e9 | hm)\n[GOAL]\ncase inl.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : m i = 0\n\u22a2 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nskip\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\nhm : \u00ac\u2203 i, m i = 0\n\u22a2 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\npush_neg at hm \n[GOAL]\ncase inl.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : m i = 0\n\u22a2 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nsimp [f.map_coord_zero i hi, prod_eq_zero (mem_univ i), hi]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\nhm : \u2200 (i : \u03b9), m i \u2260 0\n\u22a2 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nchoose \u03b4 h\u03b40 h\u03b4m_lt hle_\u03b4m _ using fun i => rescale_to_shell (hc i) (h\u03b5 i) (hm i)\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\nhm : \u2200 (i : \u03b9), m i \u2260 0\n\u03b4 : \u03b9 \u2192 \ud835\udd5c\nh\u03b40 : \u2200 (i : \u03b9), \u03b4 i \u2260 0\nh\u03b4m_lt : \u2200 (i : \u03b9), \u2016\u03b4 i \u2022 m i\u2016 < \u03b5 i\nhle_\u03b4m : \u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016\u03b4 i \u2022 m i\u2016\n\u271d : \u2200 (i : \u03b9), \u2016\u03b4 i\u2016\u207b\u00b9 \u2264 (\u03b5 i)\u207b\u00b9 * \u2016c i\u2016 * \u2016m i\u2016\n\u22a2 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nhave h\u03b40 : 0 < \u220f i, \u2016\u03b4 i\u2016 := prod_pos fun i _ => norm_pos_iff.2 (h\u03b40 i)\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\n\u03b5 : \u03b9 \u2192 \u211d\nC : \u211d\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nc : \u03b9 \u2192 \ud835\udd5c\nhc : \u2200 (i : \u03b9), 1 < \u2016c i\u2016\nhf : \u2200 (m : (i : \u03b9) \u2192 E i), (\u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5 i) \u2192 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\nhm : \u2200 (i : \u03b9), m i \u2260 0\n\u03b4 : \u03b9 \u2192 \ud835\udd5c\nh\u03b40\u271d : \u2200 (i : \u03b9), \u03b4 i \u2260 0\nh\u03b4m_lt : \u2200 (i : \u03b9), \u2016\u03b4 i \u2022 m i\u2016 < \u03b5 i\nhle_\u03b4m : \u2200 (i : \u03b9), \u03b5 i / \u2016c i\u2016 \u2264 \u2016\u03b4 i \u2022 m i\u2016\n\u271d : \u2200 (i : \u03b9), \u2016\u03b4 i\u2016\u207b\u00b9 \u2264 (\u03b5 i)\u207b\u00b9 * \u2016c i\u2016 * \u2016m i\u2016\nh\u03b40 : 0 < \u220f i : \u03b9, \u2016\u03b4 i\u2016\n\u22a2 \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nsimpa [map_smul_univ, norm_smul, prod_mul_distrib, mul_left_comm C, mul_le_mul_left h\u03b40] using\n hf (fun i => \u03b4 i \u2022 m i) hle_\u03b4m h\u03b4m_lt\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf : Continuous \u2191f\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b9\n[GOAL]\ncase inl\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf : Continuous \u2191f\nh\u271d : IsEmpty \u03b9\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrefine' \u27e8\u2016f 0\u2016 + 1, add_pos_of_nonneg_of_pos (norm_nonneg _) zero_lt_one, fun m => _\u27e9\n[GOAL]\ncase inl\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf : Continuous \u2191f\nh\u271d : IsEmpty \u03b9\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191f m\u2016 \u2264 (\u2016\u2191f 0\u2016 + 1) * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nobtain rfl : m = 0\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf : Continuous \u2191f\nh\u271d : IsEmpty \u03b9\nm : (i : \u03b9) \u2192 E i\n\u22a2 m = 0\ncase inl\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf : Continuous \u2191f\nh\u271d : IsEmpty \u03b9\n\u22a2 \u2016\u2191f 0\u2016 \u2264 (\u2016\u2191f 0\u2016 + 1) * \u220f i : \u03b9, \u2016OfNat.ofNat 0 i\u2016\n[PROOFSTEP]\nexact funext (IsEmpty.elim \u2039_\u203a)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf : Continuous \u2191f\nh\u271d : IsEmpty \u03b9\n\u22a2 \u2016\u2191f 0\u2016 \u2264 (\u2016\u2191f 0\u2016 + 1) * \u220f i : \u03b9, \u2016OfNat.ofNat 0 i\u2016\n[PROOFSTEP]\nsimp [univ_eq_empty, zero_le_one]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf : Continuous \u2191f\nh\u271d : Nonempty \u03b9\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nobtain \u27e8\u03b5 : \u211d, \u03b50 : 0 < \u03b5, h\u03b5 : \u2200 m : \u2200 i, E i, \u2016m - 0\u2016 < \u03b5 \u2192 \u2016f m - f 0\u2016 < 1\u27e9 :=\n NormedAddCommGroup.tendsto_nhds_nhds.1 (hf.tendsto 0) 1 zero_lt_one\n[GOAL]\ncase inr.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf : Continuous \u2191f\nh\u271d : Nonempty \u03b9\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5 : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016m - 0\u2016 < \u03b5 \u2192 \u2016\u2191f m - \u2191f 0\u2016 < 1\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nsimp only [sub_zero, f.map_zero] at h\u03b5 \n[GOAL]\ncase inr.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf : Continuous \u2191f\nh\u271d : Nonempty \u03b9\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5 : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016m\u2016 < \u03b5 \u2192 \u2016\u2191f m\u2016 < 1\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrcases NormedField.exists_one_lt_norm \ud835\udd5c with \u27e8c, hc\u27e9\n[GOAL]\ncase inr.intro.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf : Continuous \u2191f\nh\u271d : Nonempty \u03b9\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5 : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016m\u2016 < \u03b5 \u2192 \u2016\u2191f m\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nhave : 0 < (\u2016c\u2016 / \u03b5) ^ Fintype.card \u03b9 := pow_pos (div_pos (zero_lt_one.trans hc) \u03b50) _\n[GOAL]\ncase inr.intro.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf : Continuous \u2191f\nh\u271d : Nonempty \u03b9\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5 : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016m\u2016 < \u03b5 \u2192 \u2016\u2191f m\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < (\u2016c\u2016 / \u03b5) ^ Fintype.card \u03b9\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrefine' \u27e8_, this, _\u27e9\n[GOAL]\ncase inr.intro.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf : Continuous \u2191f\nh\u271d : Nonempty \u03b9\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5 : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016m\u2016 < \u03b5 \u2192 \u2016\u2191f m\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < (\u2016c\u2016 / \u03b5) ^ Fintype.card \u03b9\n\u22a2 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 (\u2016c\u2016 / \u03b5) ^ Fintype.card \u03b9 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrefine' f.bound_of_shell (fun _ => \u03b50) (fun _ => hc) fun m hcm hm => _\n[GOAL]\ncase inr.intro.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf : Continuous \u2191f\nh\u271d : Nonempty \u03b9\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5 : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016m\u2016 < \u03b5 \u2192 \u2016\u2191f m\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < (\u2016c\u2016 / \u03b5) ^ Fintype.card \u03b9\nm : (i : \u03b9) \u2192 E i\nhcm : \u2200 (i : \u03b9), \u03b5 / \u2016c\u2016 \u2264 \u2016m i\u2016\nhm : \u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5\n\u22a2 \u2016\u2191f m\u2016 \u2264 (\u2016c\u2016 / \u03b5) ^ Fintype.card \u03b9 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrefine' (h\u03b5 m ((pi_norm_lt_iff \u03b50).2 hm)).le.trans _\n[GOAL]\ncase inr.intro.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf : Continuous \u2191f\nh\u271d : Nonempty \u03b9\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5 : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016m\u2016 < \u03b5 \u2192 \u2016\u2191f m\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < (\u2016c\u2016 / \u03b5) ^ Fintype.card \u03b9\nm : (i : \u03b9) \u2192 E i\nhcm : \u2200 (i : \u03b9), \u03b5 / \u2016c\u2016 \u2264 \u2016m i\u2016\nhm : \u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5\n\u22a2 1 \u2264 (\u2016c\u2016 / \u03b5) ^ Fintype.card \u03b9 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrw [\u2190 div_le_iff' this, one_div, \u2190 inv_pow, inv_div, Fintype.card, \u2190 prod_const]\n[GOAL]\ncase inr.intro.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nhf : Continuous \u2191f\nh\u271d : Nonempty \u03b9\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5 : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016m\u2016 < \u03b5 \u2192 \u2016\u2191f m\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < (\u2016c\u2016 / \u03b5) ^ Fintype.card \u03b9\nm : (i : \u03b9) \u2192 E i\nhcm : \u2200 (i : \u03b9), \u03b5 / \u2016c\u2016 \u2264 \u2016m i\u2016\nhm : \u2200 (i : \u03b9), \u2016m i\u2016 < \u03b5\n\u22a2 \u220f _x : \u03b9, \u03b5 / \u2016c\u2016 \u2264 \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nexact prod_le_prod (fun _ _ => div_nonneg \u03b50.le (norm_nonneg _)) fun i _ => hcm i\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191f m\u2081 - \u2191f m\u2082\u2016 \u2264 C * \u2211 i : \u03b9, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nhave A :\n \u2200 s : Finset \u03b9,\n \u2016f m\u2081 - f (s.piecewise m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016 :=\n by\n intro s\n induction' s using Finset.induction with i s his Hrec\n \u00b7 simp\n have I :\n \u2016f (s.piecewise m\u2082 m\u2081) - f ((insert i s).piecewise m\u2082 m\u2081)\u2016 \u2264\n C * \u220f j, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016 :=\n by\n have A : (insert i s).piecewise m\u2082 m\u2081 = Function.update (s.piecewise m\u2082 m\u2081) i (m\u2082 i) := s.piecewise_insert _ _ _\n have B : s.piecewise m\u2082 m\u2081 = Function.update (s.piecewise m\u2082 m\u2081) i (m\u2081 i) :=\n by\n ext j\n by_cases h : j = i\n \u00b7 rw [h]\n simp [his]\n \u00b7 simp [h]\n rw [B, A, \u2190 f.map_sub]\n apply le_trans (H _)\n gcongr with j\n \u00b7 exact fun j _ => norm_nonneg _\n by_cases h : j = i\n \u00b7 rw [h]\n simp\n \u00b7 by_cases h' : j \u2208 s <;> simp [h', h, le_refl]\n calc\n \u2016f m\u2081 - f ((insert i s).piecewise m\u2082 m\u2081)\u2016 \u2264\n \u2016f m\u2081 - f (s.piecewise m\u2082 m\u2081)\u2016 + \u2016f (s.piecewise m\u2082 m\u2081) - f ((insert i s).piecewise m\u2082 m\u2081)\u2016 :=\n by\n rw [\u2190 dist_eq_norm, \u2190 dist_eq_norm, \u2190 dist_eq_norm]\n exact dist_triangle _ _ _\n _ \u2264\n (C * \u2211 i in s, \u220f j, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) +\n C * \u220f j, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016 :=\n (add_le_add Hrec I)\n _ = C * \u2211 i in insert i s, \u220f j, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016 := by\n simp [his, add_comm, left_distrib]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\n\u22a2 \u2200 (s : Finset \u03b9),\n \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nintro s\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ns : Finset \u03b9\n\u22a2 \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\ninduction' s using Finset.induction with i s his Hrec\n[GOAL]\ncase empty\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191f m\u2081 - \u2191f (piecewise \u2205 m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in \u2205, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 \u2016\u2191f m\u2081 - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n C * \u2211 i in insert i s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nhave I :\n \u2016f (s.piecewise m\u2082 m\u2081) - f ((insert i s).piecewise m\u2082 m\u2081)\u2016 \u2264\n C * \u220f j, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016 :=\n by\n have A : (insert i s).piecewise m\u2082 m\u2081 = Function.update (s.piecewise m\u2082 m\u2081) i (m\u2082 i) := s.piecewise_insert _ _ _\n have B : s.piecewise m\u2082 m\u2081 = Function.update (s.piecewise m\u2082 m\u2081) i (m\u2081 i) :=\n by\n ext j\n by_cases h : j = i\n \u00b7 rw [h]\n simp [his]\n \u00b7 simp [h]\n rw [B, A, \u2190 f.map_sub]\n apply le_trans (H _)\n gcongr with j\n \u00b7 exact fun j _ => norm_nonneg _\n by_cases h : j = i\n \u00b7 rw [h]\n simp\n \u00b7 by_cases h' : j \u2208 s <;> simp [h', h, le_refl]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nhave A : (insert i s).piecewise m\u2082 m\u2081 = Function.update (s.piecewise m\u2082 m\u2081) i (m\u2082 i) := s.piecewise_insert _ _ _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\n\u22a2 \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nhave B : s.piecewise m\u2082 m\u2081 = Function.update (s.piecewise m\u2082 m\u2081) i (m\u2081 i) :=\n by\n ext j\n by_cases h : j = i\n \u00b7 rw [h]\n simp [his]\n \u00b7 simp [h]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\n\u22a2 piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nj : \u03b9\n\u22a2 piecewise s m\u2082 m\u2081 j = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i) j\n[PROOFSTEP]\nby_cases h : j = i\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nj : \u03b9\nh : j = i\n\u22a2 piecewise s m\u2082 m\u2081 j = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i) j\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nj : \u03b9\nh : j = i\n\u22a2 piecewise s m\u2082 m\u2081 i = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i) i\n[PROOFSTEP]\nsimp [his]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nj : \u03b9\nh : \u00acj = i\n\u22a2 piecewise s m\u2082 m\u2081 j = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i) j\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\n\u22a2 \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nrw [B, A, \u2190 f.map_sub]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\n\u22a2 \u2016\u2191f (Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i))\u2016 \u2264\n C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\napply le_trans (H _)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\n\u22a2 C * \u220f i_1 : \u03b9, \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) i_1\u2016 \u2264\n C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\ngcongr with j\n[GOAL]\ncase h.h0\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 univ \u2192 0 \u2264 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) i_1\u2016\n[PROOFSTEP]\nexact fun j _ => norm_nonneg _\n[GOAL]\ncase h.h1\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\nj : \u03b9\na\u271d : j \u2208 univ\n\u22a2 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) j\u2016 \u2264 if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nby_cases h : j = i\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\nj : \u03b9\na\u271d : j \u2208 univ\nh : j = i\n\u22a2 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) j\u2016 \u2264 if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\nj : \u03b9\na\u271d : j \u2208 univ\nh : j = i\n\u22a2 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) i\u2016 \u2264 if i = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 i\u2016 \u2016m\u2082 i\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\nj : \u03b9\na\u271d : j \u2208 univ\nh : \u00acj = i\n\u22a2 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) j\u2016 \u2264 if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nby_cases h' : j \u2208 s\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\nj : \u03b9\na\u271d : j \u2208 univ\nh : \u00acj = i\nh' : j \u2208 s\n\u22a2 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) j\u2016 \u2264 if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nsimp [h', h, le_refl]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\nj : \u03b9\na\u271d : j \u2208 univ\nh : \u00acj = i\nh' : \u00acj \u2208 s\n\u22a2 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) j\u2016 \u2264 if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nsimp [h', h, le_refl]\n[GOAL]\ncase insert\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nI :\n \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 \u2016\u2191f m\u2081 - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n C * \u2211 i in insert i s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\ncalc\n \u2016f m\u2081 - f ((insert i s).piecewise m\u2082 m\u2081)\u2016 \u2264\n \u2016f m\u2081 - f (s.piecewise m\u2082 m\u2081)\u2016 + \u2016f (s.piecewise m\u2082 m\u2081) - f ((insert i s).piecewise m\u2082 m\u2081)\u2016 :=\n by\n rw [\u2190 dist_eq_norm, \u2190 dist_eq_norm, \u2190 dist_eq_norm]\n exact dist_triangle _ _ _\n _ \u2264\n (C * \u2211 i in s, \u220f j, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) +\n C * \u220f j, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016 :=\n (add_le_add Hrec I)\n _ = C * \u2211 i in insert i s, \u220f j, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016 := by\n simp [his, add_comm, left_distrib]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nI :\n \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 \u2016\u2191f m\u2081 - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 + \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016\n[PROOFSTEP]\nrw [\u2190 dist_eq_norm, \u2190 dist_eq_norm, \u2190 dist_eq_norm]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nI :\n \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 dist (\u2191f m\u2081) (\u2191f (piecewise (insert i s) m\u2082 m\u2081)) \u2264\n dist (\u2191f m\u2081) (\u2191f (piecewise s m\u2082 m\u2081)) + dist (\u2191f (piecewise s m\u2082 m\u2081)) (\u2191f (piecewise (insert i s) m\u2082 m\u2081))\n[PROOFSTEP]\nexact dist_triangle _ _ _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nI :\n \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 ((C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) +\n C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) =\n C * \u2211 i in insert i s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nsimp [his, add_comm, left_distrib]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\nA :\n \u2200 (s : Finset \u03b9),\n \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 \u2016\u2191f m\u2081 - \u2191f m\u2082\u2016 \u2264 C * \u2211 i : \u03b9, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nconvert A univ\n[GOAL]\ncase h.e'_3.h.e'_3.h.e'_6.h.e'_6\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\nA :\n \u2200 (s : Finset \u03b9),\n \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 m\u2082 = piecewise univ m\u2082 m\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191f m\u2081 - \u2191f m\u2082\u2016 \u2264 C * \u2191(Fintype.card \u03b9) * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1) * \u2016m\u2081 - m\u2082\u2016\n[PROOFSTEP]\nclassical\nhave A :\n \u2200 i : \u03b9,\n \u220f j, (if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264 \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1) :=\n by\n intro i\n calc\n \u220f j, (if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264\n \u220f j : \u03b9, Function.update (fun _ => max \u2016m\u2081\u2016 \u2016m\u2082\u2016) i \u2016m\u2081 - m\u2082\u2016 j :=\n by\n apply prod_le_prod\n \u00b7 intro j _\n by_cases h : j = i <;> simp [h, norm_nonneg]\n \u00b7 intro j _\n by_cases h : j = i\n \u00b7 rw [h]\n simp\n exact norm_le_pi_norm (m\u2081 - m\u2082) i\n \u00b7 simp [h, -le_max_iff, -max_le_iff, max_le_max, norm_le_pi_norm (_ : \u2200 i, E i)]\n _ = \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1) :=\n by\n rw [prod_update_of_mem (Finset.mem_univ _)]\n simp [card_univ_diff]\ncalc\n \u2016f m\u2081 - f m\u2082\u2016 \u2264 C * \u2211 i, \u220f j, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016 :=\n f.norm_image_sub_le_of_bound' hC H m\u2081 m\u2082\n _ \u2264 C * \u2211 i, \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1) := by gcongr; apply A\n _ = C * Fintype.card \u03b9 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1) * \u2016m\u2081 - m\u2082\u2016 :=\n by\n rw [sum_const, card_univ, nsmul_eq_mul]\n ring\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191f m\u2081 - \u2191f m\u2082\u2016 \u2264 C * \u2191(Fintype.card \u03b9) * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1) * \u2016m\u2081 - m\u2082\u2016\n[PROOFSTEP]\nhave A :\n \u2200 i : \u03b9,\n \u220f j, (if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264 \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1) :=\n by\n intro i\n calc\n \u220f j, (if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264\n \u220f j : \u03b9, Function.update (fun _ => max \u2016m\u2081\u2016 \u2016m\u2082\u2016) i \u2016m\u2081 - m\u2082\u2016 j :=\n by\n apply prod_le_prod\n \u00b7 intro j _\n by_cases h : j = i <;> simp [h, norm_nonneg]\n \u00b7 intro j _\n by_cases h : j = i\n \u00b7 rw [h]\n simp\n exact norm_le_pi_norm (m\u2081 - m\u2082) i\n \u00b7 simp [h, -le_max_iff, -max_le_iff, max_le_max, norm_le_pi_norm (_ : \u2200 i, E i)]\n _ = \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1) :=\n by\n rw [prod_update_of_mem (Finset.mem_univ _)]\n simp [card_univ_diff]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\n\u22a2 \u2200 (i : \u03b9),\n (\u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264 \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1)\n[PROOFSTEP]\nintro i\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\n\u22a2 (\u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264 \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1)\n[PROOFSTEP]\ncalc\n \u220f j, (if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264\n \u220f j : \u03b9, Function.update (fun _ => max \u2016m\u2081\u2016 \u2016m\u2082\u2016) i \u2016m\u2081 - m\u2082\u2016 j :=\n by\n apply prod_le_prod\n \u00b7 intro j _\n by_cases h : j = i <;> simp [h, norm_nonneg]\n \u00b7 intro j _\n by_cases h : j = i\n \u00b7 rw [h]\n simp\n exact norm_le_pi_norm (m\u2081 - m\u2082) i\n \u00b7 simp [h, -le_max_iff, -max_le_iff, max_le_max, norm_le_pi_norm (_ : \u2200 i, E i)]\n _ = \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1) :=\n by\n rw [prod_update_of_mem (Finset.mem_univ _)]\n simp [card_univ_diff]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\n\u22a2 (\u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264\n \u220f j : \u03b9, Function.update (fun x => max \u2016m\u2081\u2016 \u2016m\u2082\u2016) i \u2016m\u2081 - m\u2082\u2016 j\n[PROOFSTEP]\napply prod_le_prod\n[GOAL]\ncase h0\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 univ \u2192 0 \u2264 if i_1 = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 i_1\u2016 \u2016m\u2082 i_1\u2016\n[PROOFSTEP]\nintro j _\n[GOAL]\ncase h0\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni j : \u03b9\na\u271d : j \u2208 univ\n\u22a2 0 \u2264 if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nby_cases h : j = i\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni j : \u03b9\na\u271d : j \u2208 univ\nh : j = i\n\u22a2 0 \u2264 if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nsimp [h, norm_nonneg]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni j : \u03b9\na\u271d : j \u2208 univ\nh : \u00acj = i\n\u22a2 0 \u2264 if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n[PROOFSTEP]\nsimp [h, norm_nonneg]\n[GOAL]\ncase h1\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\n\u22a2 \u2200 (i_1 : \u03b9),\n i_1 \u2208 univ \u2192\n (if i_1 = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 i_1\u2016 \u2016m\u2082 i_1\u2016) \u2264\n Function.update (fun x => max \u2016m\u2081\u2016 \u2016m\u2082\u2016) i \u2016m\u2081 - m\u2082\u2016 i_1\n[PROOFSTEP]\nintro j _\n[GOAL]\ncase h1\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni j : \u03b9\na\u271d : j \u2208 univ\n\u22a2 (if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264 Function.update (fun x => max \u2016m\u2081\u2016 \u2016m\u2082\u2016) i \u2016m\u2081 - m\u2082\u2016 j\n[PROOFSTEP]\nby_cases h : j = i\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni j : \u03b9\na\u271d : j \u2208 univ\nh : j = i\n\u22a2 (if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264 Function.update (fun x => max \u2016m\u2081\u2016 \u2016m\u2082\u2016) i \u2016m\u2081 - m\u2082\u2016 j\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni j : \u03b9\na\u271d : j \u2208 univ\nh : j = i\n\u22a2 (if i = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 i\u2016 \u2016m\u2082 i\u2016) \u2264 Function.update (fun x => max \u2016m\u2081\u2016 \u2016m\u2082\u2016) i \u2016m\u2081 - m\u2082\u2016 i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni j : \u03b9\na\u271d : j \u2208 univ\nh : j = i\n\u22a2 \u2016m\u2081 i - m\u2082 i\u2016 \u2264 \u2016m\u2081 - m\u2082\u2016\n[PROOFSTEP]\nexact norm_le_pi_norm (m\u2081 - m\u2082) i\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni j : \u03b9\na\u271d : j \u2208 univ\nh : \u00acj = i\n\u22a2 (if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264 Function.update (fun x => max \u2016m\u2081\u2016 \u2016m\u2082\u2016) i \u2016m\u2081 - m\u2082\u2016 j\n[PROOFSTEP]\nsimp [h, -le_max_iff, -max_le_iff, max_le_max, norm_le_pi_norm (_ : \u2200 i, E i)]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\n\u22a2 \u220f j : \u03b9, Function.update (fun x => max \u2016m\u2081\u2016 \u2016m\u2082\u2016) i \u2016m\u2081 - m\u2082\u2016 j = \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1)\n[PROOFSTEP]\nrw [prod_update_of_mem (Finset.mem_univ _)]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\n\u22a2 \u2016m\u2081 - m\u2082\u2016 * \u220f x in univ \\ {i}, max \u2016m\u2081\u2016 \u2016m\u2082\u2016 = \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1)\n[PROOFSTEP]\nsimp [card_univ_diff]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\nA :\n \u2200 (i : \u03b9),\n (\u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264 \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1)\n\u22a2 \u2016\u2191f m\u2081 - \u2191f m\u2082\u2016 \u2264 C * \u2191(Fintype.card \u03b9) * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1) * \u2016m\u2081 - m\u2082\u2016\n[PROOFSTEP]\ncalc\n \u2016f m\u2081 - f m\u2082\u2016 \u2264 C * \u2211 i, \u220f j, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016 :=\n f.norm_image_sub_le_of_bound' hC H m\u2081 m\u2082\n _ \u2264 C * \u2211 i, \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1) := by gcongr; apply A\n _ = C * Fintype.card \u03b9 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1) * \u2016m\u2081 - m\u2082\u2016 :=\n by\n rw [sum_const, card_univ, nsmul_eq_mul]\n ring\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\nA :\n \u2200 (i : \u03b9),\n (\u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264 \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1)\n\u22a2 (C * \u2211 i : \u03b9, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264\n C * \u2211 i : ?m.189282, \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h.h\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\nA :\n \u2200 (i : \u03b9),\n (\u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264 \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1)\ni\u271d : \u03b9\na\u271d : i\u271d \u2208 univ\n\u22a2 (\u220f j : \u03b9, if j = i\u271d then \u2016m\u2081 i\u271d - m\u2082 i\u271d\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264 \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1)\n[PROOFSTEP]\napply A\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\nA :\n \u2200 (i : \u03b9),\n (\u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264 \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1)\n\u22a2 C * \u2211 i : \u03b9, \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1) =\n C * \u2191(Fintype.card \u03b9) * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1) * \u2016m\u2081 - m\u2082\u2016\n[PROOFSTEP]\nrw [sum_const, card_univ, nsmul_eq_mul]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\nA :\n \u2200 (i : \u03b9),\n (\u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) \u2264 \u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1)\n\u22a2 C * (\u2191(Fintype.card \u03b9) * (\u2016m\u2081 - m\u2082\u2016 * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1))) =\n C * \u2191(Fintype.card \u03b9) * max \u2016m\u2081\u2016 \u2016m\u2082\u2016 ^ (Fintype.card \u03b9 - 1) * \u2016m\u2081 - m\u2082\u2016\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\n\u22a2 Continuous \u2191f\n[PROOFSTEP]\nlet D := max C 1\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nD : \u211d := max C 1\n\u22a2 Continuous \u2191f\n[PROOFSTEP]\nhave D_pos : 0 \u2264 D := le_trans zero_le_one (le_max_right _ _)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nD : \u211d := max C 1\nD_pos : 0 \u2264 D\n\u22a2 Continuous \u2191f\n[PROOFSTEP]\nreplace H : \u2200 m, \u2016f m\u2016 \u2264 D * \u220f i, \u2016m i\u2016\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nD : \u211d := max C 1\nD_pos : 0 \u2264 D\n\u22a2 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 D * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nintro m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nD : \u211d := max C 1\nD_pos : 0 \u2264 D\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191f m\u2016 \u2264 D * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\napply le_trans (H m) (mul_le_mul_of_nonneg_right (le_max_left _ _) _)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nD : \u211d := max C 1\nD_pos : 0 \u2264 D\nm : (i : \u03b9) \u2192 E i\n\u22a2 0 \u2264 \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nexact prod_nonneg fun (i : \u03b9) _ => norm_nonneg (m i)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nD : \u211d := max C 1\nD_pos : 0 \u2264 D\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 D * \u220f i : \u03b9, \u2016m i\u2016\n\u22a2 Continuous \u2191f\n[PROOFSTEP]\nrefine' continuous_iff_continuousAt.2 fun m => _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nD : \u211d := max C 1\nD_pos : 0 \u2264 D\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 D * \u220f i : \u03b9, \u2016m i\u2016\nm : (i : \u03b9) \u2192 E i\n\u22a2 ContinuousAt (\u2191f) m\n[PROOFSTEP]\nrefine'\n continuousAt_of_locally_lipschitz zero_lt_one (D * Fintype.card \u03b9 * (\u2016m\u2016 + 1) ^ (Fintype.card \u03b9 - 1)) fun m' h' => _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nD : \u211d := max C 1\nD_pos : 0 \u2264 D\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 D * \u220f i : \u03b9, \u2016m i\u2016\nm m' : (i : \u03b9) \u2192 E i\nh' : dist m' m < 1\n\u22a2 dist (\u2191f m') (\u2191f m) \u2264 D * \u2191(Fintype.card \u03b9) * (\u2016m\u2016 + 1) ^ (Fintype.card \u03b9 - 1) * dist m' m\n[PROOFSTEP]\nrw [dist_eq_norm, dist_eq_norm]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nD : \u211d := max C 1\nD_pos : 0 \u2264 D\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 D * \u220f i : \u03b9, \u2016m i\u2016\nm m' : (i : \u03b9) \u2192 E i\nh' : dist m' m < 1\n\u22a2 \u2016\u2191f m' - \u2191f m\u2016 \u2264 D * \u2191(Fintype.card \u03b9) * (\u2016m\u2016 + 1) ^ (Fintype.card \u03b9 - 1) * \u2016m' - m\u2016\n[PROOFSTEP]\nhave : max \u2016m'\u2016 \u2016m\u2016 \u2264 \u2016m\u2016 + 1 := by simp [zero_le_one, norm_le_of_mem_closedBall (le_of_lt h')]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nD : \u211d := max C 1\nD_pos : 0 \u2264 D\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 D * \u220f i : \u03b9, \u2016m i\u2016\nm m' : (i : \u03b9) \u2192 E i\nh' : dist m' m < 1\n\u22a2 max \u2016m'\u2016 \u2016m\u2016 \u2264 \u2016m\u2016 + 1\n[PROOFSTEP]\nsimp [zero_le_one, norm_le_of_mem_closedBall (le_of_lt h')]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nD : \u211d := max C 1\nD_pos : 0 \u2264 D\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 D * \u220f i : \u03b9, \u2016m i\u2016\nm m' : (i : \u03b9) \u2192 E i\nh' : dist m' m < 1\nthis : max \u2016m'\u2016 \u2016m\u2016 \u2264 \u2016m\u2016 + 1\n\u22a2 \u2016\u2191f m' - \u2191f m\u2016 \u2264 D * \u2191(Fintype.card \u03b9) * (\u2016m\u2016 + 1) ^ (Fintype.card \u03b9 - 1) * \u2016m' - m\u2016\n[PROOFSTEP]\ncalc\n \u2016f m' - f m\u2016 \u2264 D * Fintype.card \u03b9 * max \u2016m'\u2016 \u2016m\u2016 ^ (Fintype.card \u03b9 - 1) * \u2016m' - m\u2016 :=\n f.norm_image_sub_le_of_bound D_pos H m' m\n _ \u2264 D * Fintype.card \u03b9 * (\u2016m\u2016 + 1) ^ (Fintype.card \u03b9 - 1) * \u2016m' - m\u2016 := by gcongr\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\nC : \u211d\nD : \u211d := max C 1\nD_pos : 0 \u2264 D\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 D * \u220f i : \u03b9, \u2016m i\u2016\nm m' : (i : \u03b9) \u2192 E i\nh' : dist m' m < 1\nthis : max \u2016m'\u2016 \u2016m\u2016 \u2264 \u2016m\u2016 + 1\n\u22a2 D * \u2191(Fintype.card \u03b9) * max \u2016m'\u2016 \u2016m\u2016 ^ (Fintype.card \u03b9 - 1) * \u2016m' - m\u2016 \u2264\n D * \u2191(Fintype.card \u03b9) * (\u2016m\u2016 + 1) ^ (Fintype.card \u03b9 - 1) * \u2016m' - m\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d : MultilinearMap \ud835\udd5c E G\nk n : \u2115\nf : MultilinearMap \ud835\udd5c (fun x => G) G'\ns : Finset (Fin n)\nhk : card s = k\nz : G\nC : \u211d\nH : \u2200 (m : Fin n \u2192 G), \u2016\u2191f m\u2016 \u2264 C * \u220f i : Fin n, \u2016m i\u2016\nv : Fin k \u2192 G\n\u22a2 \u2016\u2191(restr f s hk z) v\u2016 \u2264 C * \u2016z\u2016 ^ (n - k) * \u220f i : Fin k, \u2016v i\u2016\n[PROOFSTEP]\nrw [mul_right_comm, mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d : MultilinearMap \ud835\udd5c E G\nk n : \u2115\nf : MultilinearMap \ud835\udd5c (fun x => G) G'\ns : Finset (Fin n)\nhk : card s = k\nz : G\nC : \u211d\nH : \u2200 (m : Fin n \u2192 G), \u2016\u2191f m\u2016 \u2264 C * \u220f i : Fin n, \u2016m i\u2016\nv : Fin k \u2192 G\n\u22a2 \u2016\u2191(restr f s hk z) v\u2016 \u2264 C * ((\u220f i : Fin k, \u2016v i\u2016) * \u2016z\u2016 ^ (n - k))\n[PROOFSTEP]\nconvert H _ using 2\n[GOAL]\ncase h.e'_4.h.e'_6\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d : MultilinearMap \ud835\udd5c E G\nk n : \u2115\nf : MultilinearMap \ud835\udd5c (fun x => G) G'\ns : Finset (Fin n)\nhk : card s = k\nz : G\nC : \u211d\nH : \u2200 (m : Fin n \u2192 G), \u2016\u2191f m\u2016 \u2264 C * \u220f i : Fin n, \u2016m i\u2016\nv : Fin k \u2192 G\n\u22a2 (\u220f i : Fin k, \u2016v i\u2016) * \u2016z\u2016 ^ (n - k) =\n \u220f i : Fin n, \u2016if h : i \u2208 s then v (\u2191(OrderIso.symm (orderIsoOfFin s hk)) { val := i, property := h }) else z\u2016\n[PROOFSTEP]\nsimp only [apply_dite norm, Fintype.prod_dite, prod_const \u2016z\u2016, Finset.card_univ,\n Fintype.card_of_subtype s\u1d9c fun _ => mem_compl, card_compl, Fintype.card_fin, hk, mk_coe, \u2190\n (s.orderIsoOfFin hk).symm.bijective.prod_comp fun x => \u2016v x\u2016]\n[GOAL]\ncase h.e'_4.h.e'_6\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d : MultilinearMap \ud835\udd5c E G\nk n : \u2115\nf : MultilinearMap \ud835\udd5c (fun x => G) G'\ns : Finset (Fin n)\nhk : card s = k\nz : G\nC : \u211d\nH : \u2200 (m : Fin n \u2192 G), \u2016\u2191f m\u2016 \u2264 C * \u220f i : Fin n, \u2016m i\u2016\nv : Fin k \u2192 G\n\u22a2 (\u220f i : { x // x \u2208 s }, \u2016v (\u2191(OrderIso.symm (orderIsoOfFin s hk)) i)\u2016) * \u2016z\u2016 ^ (n - k) =\n (\u220f a : { a // a \u2208 s }, \u2016v (\u2191(OrderIso.symm (orderIsoOfFin s hk)) { val := \u2191a, property := (_ : \u2191a \u2208 s) })\u2016) *\n \u2016z\u2016 ^ (n - k)\n[PROOFSTEP]\nconvert rfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191f m\u2016 \u2264 \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nhave A : 0 \u2264 \u220f i, \u2016m i\u2016 := prod_nonneg fun j _ => norm_nonneg _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\nA : 0 \u2264 \u220f i : \u03b9, \u2016m i\u2016\n\u22a2 \u2016\u2191f m\u2016 \u2264 \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\ncases' A.eq_or_lt with h hlt\n[GOAL]\ncase inl\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\nA : 0 \u2264 \u220f i : \u03b9, \u2016m i\u2016\nh : 0 = \u220f i : \u03b9, \u2016m i\u2016\n\u22a2 \u2016\u2191f m\u2016 \u2264 \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrcases prod_eq_zero_iff.1 h.symm with \u27e8i, _, hi\u27e9\n[GOAL]\ncase inl.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\nA : 0 \u2264 \u220f i : \u03b9, \u2016m i\u2016\nh : 0 = \u220f i : \u03b9, \u2016m i\u2016\ni : \u03b9\nleft\u271d : i \u2208 univ\nhi : \u2016m i\u2016 = 0\n\u22a2 \u2016\u2191f m\u2016 \u2264 \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrw [norm_eq_zero] at hi \n[GOAL]\ncase inl.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\nA : 0 \u2264 \u220f i : \u03b9, \u2016m i\u2016\nh : 0 = \u220f i : \u03b9, \u2016m i\u2016\ni : \u03b9\nleft\u271d : i \u2208 univ\nhi : m i = 0\n\u22a2 \u2016\u2191f m\u2016 \u2264 \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nhave : f m = 0 := f.map_coord_zero i hi\n[GOAL]\ncase inl.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\nA : 0 \u2264 \u220f i : \u03b9, \u2016m i\u2016\nh : 0 = \u220f i : \u03b9, \u2016m i\u2016\ni : \u03b9\nleft\u271d : i \u2208 univ\nhi : m i = 0\nthis : \u2191f m = 0\n\u22a2 \u2016\u2191f m\u2016 \u2264 \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrw [this, norm_zero]\n[GOAL]\ncase inl.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\nA : 0 \u2264 \u220f i : \u03b9, \u2016m i\u2016\nh : 0 = \u220f i : \u03b9, \u2016m i\u2016\ni : \u03b9\nleft\u271d : i \u2208 univ\nhi : m i = 0\nthis : \u2191f m = 0\n\u22a2 0 \u2264 \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nexact mul_nonneg (op_norm_nonneg f) A\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\nA : 0 \u2264 \u220f i : \u03b9, \u2016m i\u2016\nhlt : 0 < \u220f i : \u03b9, \u2016m i\u2016\n\u22a2 \u2016\u2191f m\u2016 \u2264 \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrw [\u2190 div_le_iff hlt]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\nA : 0 \u2264 \u220f i : \u03b9, \u2016m i\u2016\nhlt : 0 < \u220f i : \u03b9, \u2016m i\u2016\n\u22a2 \u2016\u2191f m\u2016 / \u220f i : \u03b9, \u2016m i\u2016 \u2264 \u2016f\u2016\n[PROOFSTEP]\napply le_csInf bounds_nonempty\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\nA : 0 \u2264 \u220f i : \u03b9, \u2016m i\u2016\nhlt : 0 < \u220f i : \u03b9, \u2016m i\u2016\n\u22a2 \u2200 (b : \u211d), b \u2208 {c | 0 \u2264 c \u2227 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 c * \u220f i : \u03b9, \u2016m i\u2016} \u2192 \u2016\u2191f m\u2016 / \u220f i : \u03b9, \u2016m i\u2016 \u2264 b\n[PROOFSTEP]\nrintro c \u27e8_, hc\u27e9\n[GOAL]\ncase inr.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc\u271d : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\nA : 0 \u2264 \u220f i : \u03b9, \u2016m i\u2016\nhlt : 0 < \u220f i : \u03b9, \u2016m i\u2016\nc : \u211d\nleft\u271d : 0 \u2264 c\nhc : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 c * \u220f i : \u03b9, \u2016m i\u2016\n\u22a2 \u2016\u2191f m\u2016 / \u220f i : \u03b9, \u2016m i\u2016 \u2264 c\n[PROOFSTEP]\nrw [div_le_iff hlt]\n[GOAL]\ncase inr.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc\u271d : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\nA : 0 \u2264 \u220f i : \u03b9, \u2016m i\u2016\nhlt : 0 < \u220f i : \u03b9, \u2016m i\u2016\nc : \u211d\nleft\u271d : 0 \u2264 c\nhc : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 c * \u220f i : \u03b9, \u2016m i\u2016\n\u22a2 \u2016\u2191f m\u2016 \u2264 c * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\napply hc\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\nh : \u2016m\u2016 \u2264 1\n\u22a2 \u2016f\u2016 * \u220f i : \u03b9, 1 = \u2016f\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm x : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191(f + g) x\u2016 \u2264 (\u2016f\u2016 + \u2016g\u2016) * \u220f i : \u03b9, \u2016x i\u2016\n[PROOFSTEP]\nrw [add_mul]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm x : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191(f + g) x\u2016 \u2264 \u2016f\u2016 * \u220f i : \u03b9, \u2016x i\u2016 + \u2016g\u2016 * \u220f i : \u03b9, \u2016x i\u2016\n[PROOFSTEP]\nexact norm_add_le_of_le (le_op_norm _ _) (le_op_norm _ _)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d m : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u21910 m\u2016 \u2264 0 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\nh : \u2016f\u2016 = 0\n\u22a2 f = 0\n[PROOFSTEP]\next m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\nh : \u2016f\u2016 = 0\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2191f m = \u21910 m\n[PROOFSTEP]\nsimpa [h] using f.le_op_norm m\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\u22a2 f = 0 \u2192 \u2016f\u2016 = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\ng : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u20160\u2016 = 0\n[PROOFSTEP]\nexact op_norm_zero\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc\u271d : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\nc : \ud835\udd5c'\n\u22a2 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(c \u2022 f) m\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nintro m\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc\u271d : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\nc : \ud835\udd5c'\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191(c \u2022 f) m\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nerw [norm_smul, mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc\u271d : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\nc : \ud835\udd5c'\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016c\u2016 * \u2016\u2191f.toMultilinearMap m\u2016 \u2264 \u2016c\u2016 * (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_left (le_op_norm _ _) (norm_nonneg _)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\n\u22a2 \u2016-f\u2016 = \u2016f\u2016\n[PROOFSTEP]\nrw [norm_def]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\n\u22a2 sInf {c | 0 \u2264 c \u2227 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(-f) m\u2016 \u2264 c * \u220f i : \u03b9, \u2016m i\u2016} = \u2016f\u2016\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\n\u22a2 {c | 0 \u2264 c \u2227 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(-f) m\u2016 \u2264 c * \u220f i : \u03b9, \u2016m i\u2016} =\n {c | 0 \u2264 c \u2227 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 c * \u220f i : \u03b9, \u2016m i\u2016}\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\nx\u271d : \u211d\n\u22a2 x\u271d \u2208 {c | 0 \u2264 c \u2227 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(-f) m\u2016 \u2264 c * \u220f i : \u03b9, \u2016m i\u2016} \u2194\n x\u271d \u2208 {c | 0 \u2264 c \u2227 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 c * \u220f i : \u03b9, \u2016m i\u2016}\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\nb : \u211d\nhm : \u2200 (i : \u03b9), \u2016m i\u2016 \u2264 b\n\u22a2 \u2016\u2191f m\u2016 \u2264 \u2016f\u2016 * b ^ Fintype.card \u03b9\n[PROOFSTEP]\nsimpa only [prod_const] using f.le_op_norm_mul_prod_of_le m hm\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi\u271d : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2079 : Fintype \u03b9\ninst\u271d\u00b9\u2078 : Fintype \u03b9'\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei\u271d i)\ninst\u271d\u2079 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei\u271d i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b2 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\nEi : Fin n \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin n) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin n) \u2192 NormedSpace \ud835\udd5c (Ei i)\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\nm : (i : Fin n) \u2192 Ei i\nb : \u211d\nhm : \u2016m\u2016 \u2264 b\n\u22a2 \u2016\u2191f m\u2016 \u2264 \u2016f\u2016 * b ^ n\n[PROOFSTEP]\nsimpa only [Fintype.card_fin] using f.le_op_norm_mul_pow_card_of_le m fun i => (norm_le_pi_norm m i).trans hm\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\n\u22a2 \u2191\u2016\u2191f m\u2016\u208a \u2264 \u2191(\u2016f\u2016\u208a * \u220f i : \u03b9, \u2016m i\u2016\u208a)\n[PROOFSTEP]\npush_cast\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\n\u22a2 \u2016\u2191f m\u2016 \u2264 \u2016f\u2016 * \u220f x : \u03b9, \u2016m x\u2016\n[PROOFSTEP]\nexact f.le_op_norm m\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g\u271d : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\nf : ContinuousMultilinearMap \ud835\udd5c E G\ng : ContinuousMultilinearMap \ud835\udd5c E G'\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191(prod f g) m\u2016 \u2264 max \u2016f\u2016 \u2016g\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nhave H : 0 \u2264 \u220f i, \u2016m i\u2016 := prod_nonneg fun _ _ => norm_nonneg _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g\u271d : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\nf : ContinuousMultilinearMap \ud835\udd5c E G\ng : ContinuousMultilinearMap \ud835\udd5c E G'\nm : (i : \u03b9) \u2192 E i\nH : 0 \u2264 \u220f i : \u03b9, \u2016m i\u2016\n\u22a2 \u2016\u2191(prod f g) m\u2016 \u2264 max \u2016f\u2016 \u2016g\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nsimpa only [prod_apply, Prod.norm_def, max_mul_of_nonneg, H] using max_le_max (f.le_op_norm m) (g.le_op_norm m)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9'\u271d : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE'\u271d : \u03b9'\u271d \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b2\u2070 : Fintype \u03b9\ninst\u271d\u00b9\u2079 : Fintype \u03b9'\u271d\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2077 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9'\u271d) \u2192 NormedAddCommGroup (E'\u271d i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9'\u271d) \u2192 NormedSpace \ud835\udd5c (E'\u271d i)\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\n\u03b9' : Type v'\ninst\u271d\u00b2 : Fintype \u03b9'\nE' : \u03b9' \u2192 Type wE'\ninst\u271d\u00b9 : (i' : \u03b9') \u2192 NormedAddCommGroup (E' i')\ninst\u271d : (i' : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i')\nf : (i' : \u03b9') \u2192 ContinuousMultilinearMap \ud835\udd5c E (E' i')\n\u22a2 \u2016pi f\u2016 = \u2016f\u2016\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9'\u271d : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE'\u271d : \u03b9'\u271d \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b2\u2070 : Fintype \u03b9\ninst\u271d\u00b9\u2079 : Fintype \u03b9'\u271d\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2077 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9'\u271d) \u2192 NormedAddCommGroup (E'\u271d i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9'\u271d) \u2192 NormedSpace \ud835\udd5c (E'\u271d i)\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\n\u03b9' : Type v'\ninst\u271d\u00b2 : Fintype \u03b9'\nE' : \u03b9' \u2192 Type wE'\ninst\u271d\u00b9 : (i' : \u03b9') \u2192 NormedAddCommGroup (E' i')\ninst\u271d : (i' : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i')\nf : (i' : \u03b9') \u2192 ContinuousMultilinearMap \ud835\udd5c E (E' i')\n\u22a2 \u2016pi f\u2016 \u2264 \u2016f\u2016\n[PROOFSTEP]\nrefine' op_norm_le_bound _ (norm_nonneg f) fun m => _\n[GOAL]\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9'\u271d : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE'\u271d : \u03b9'\u271d \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b2\u2070 : Fintype \u03b9\ninst\u271d\u00b9\u2079 : Fintype \u03b9'\u271d\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2077 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9'\u271d) \u2192 NormedAddCommGroup (E'\u271d i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9'\u271d) \u2192 NormedSpace \ud835\udd5c (E'\u271d i)\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\n\u03b9' : Type v'\ninst\u271d\u00b2 : Fintype \u03b9'\nE' : \u03b9' \u2192 Type wE'\ninst\u271d\u00b9 : (i' : \u03b9') \u2192 NormedAddCommGroup (E' i')\ninst\u271d : (i' : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i')\nf : (i' : \u03b9') \u2192 ContinuousMultilinearMap \ud835\udd5c E (E' i')\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191(pi f) m\u2016 \u2264 \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9'\u271d : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE'\u271d : \u03b9'\u271d \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b2\u2070 : Fintype \u03b9\ninst\u271d\u00b9\u2079 : Fintype \u03b9'\u271d\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2077 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9'\u271d) \u2192 NormedAddCommGroup (E'\u271d i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9'\u271d) \u2192 NormedSpace \ud835\udd5c (E'\u271d i)\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\n\u03b9' : Type v'\ninst\u271d\u00b2 : Fintype \u03b9'\nE' : \u03b9' \u2192 Type wE'\ninst\u271d\u00b9 : (i' : \u03b9') \u2192 NormedAddCommGroup (E' i')\ninst\u271d : (i' : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i')\nf : (i' : \u03b9') \u2192 ContinuousMultilinearMap \ud835\udd5c E (E' i')\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016fun j => \u2191(f j) m\u2016 \u2264 \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrw [pi_norm_le_iff_of_nonneg]\n[GOAL]\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9'\u271d : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE'\u271d : \u03b9'\u271d \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b2\u2070 : Fintype \u03b9\ninst\u271d\u00b9\u2079 : Fintype \u03b9'\u271d\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2077 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9'\u271d) \u2192 NormedAddCommGroup (E'\u271d i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9'\u271d) \u2192 NormedSpace \ud835\udd5c (E'\u271d i)\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\n\u03b9' : Type v'\ninst\u271d\u00b2 : Fintype \u03b9'\nE' : \u03b9' \u2192 Type wE'\ninst\u271d\u00b9 : (i' : \u03b9') \u2192 NormedAddCommGroup (E' i')\ninst\u271d : (i' : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i')\nf : (i' : \u03b9') \u2192 ContinuousMultilinearMap \ud835\udd5c E (E' i')\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2200 (i : \u03b9'), \u2016\u2191(f i) m\u2016 \u2264 \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9'\u271d : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE'\u271d : \u03b9'\u271d \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b2\u2070 : Fintype \u03b9\ninst\u271d\u00b9\u2079 : Fintype \u03b9'\u271d\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2077 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9'\u271d) \u2192 NormedAddCommGroup (E'\u271d i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9'\u271d) \u2192 NormedSpace \ud835\udd5c (E'\u271d i)\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\n\u03b9' : Type v'\ninst\u271d\u00b2 : Fintype \u03b9'\nE' : \u03b9' \u2192 Type wE'\ninst\u271d\u00b9 : (i' : \u03b9') \u2192 NormedAddCommGroup (E' i')\ninst\u271d : (i' : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i')\nf : (i' : \u03b9') \u2192 ContinuousMultilinearMap \ud835\udd5c E (E' i')\nm : (i : \u03b9) \u2192 E i\n\u22a2 0 \u2264 \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nexacts [fun i => (f i).le_of_op_norm_le m (norm_le_pi_norm f i),\n mul_nonneg (norm_nonneg f) (prod_nonneg fun _ _ => norm_nonneg _)]\n[GOAL]\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9'\u271d : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE'\u271d : \u03b9'\u271d \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b2\u2070 : Fintype \u03b9\ninst\u271d\u00b9\u2079 : Fintype \u03b9'\u271d\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2077 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9'\u271d) \u2192 NormedAddCommGroup (E'\u271d i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9'\u271d) \u2192 NormedSpace \ud835\udd5c (E'\u271d i)\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\n\u03b9' : Type v'\ninst\u271d\u00b2 : Fintype \u03b9'\nE' : \u03b9' \u2192 Type wE'\ninst\u271d\u00b9 : (i' : \u03b9') \u2192 NormedAddCommGroup (E' i')\ninst\u271d : (i' : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i')\nf : (i' : \u03b9') \u2192 ContinuousMultilinearMap \ud835\udd5c E (E' i')\n\u22a2 \u2016f\u2016 \u2264 \u2016pi f\u2016\n[PROOFSTEP]\nrefine' (pi_norm_le_iff_of_nonneg (norm_nonneg _)).2 fun i => _\n[GOAL]\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9'\u271d : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE'\u271d : \u03b9'\u271d \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b2\u2070 : Fintype \u03b9\ninst\u271d\u00b9\u2079 : Fintype \u03b9'\u271d\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2077 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9'\u271d) \u2192 NormedAddCommGroup (E'\u271d i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9'\u271d) \u2192 NormedSpace \ud835\udd5c (E'\u271d i)\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\n\u03b9' : Type v'\ninst\u271d\u00b2 : Fintype \u03b9'\nE' : \u03b9' \u2192 Type wE'\ninst\u271d\u00b9 : (i' : \u03b9') \u2192 NormedAddCommGroup (E' i')\ninst\u271d : (i' : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i')\nf : (i' : \u03b9') \u2192 ContinuousMultilinearMap \ud835\udd5c E (E' i')\ni : \u03b9'\n\u22a2 \u2016f i\u2016 \u2264 \u2016pi f\u2016\n[PROOFSTEP]\nrefine' op_norm_le_bound _ (norm_nonneg _) fun m => _\n[GOAL]\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9'\u271d : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE'\u271d : \u03b9'\u271d \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b2\u2070 : Fintype \u03b9\ninst\u271d\u00b9\u2079 : Fintype \u03b9'\u271d\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2077 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9'\u271d) \u2192 NormedAddCommGroup (E'\u271d i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9'\u271d) \u2192 NormedSpace \ud835\udd5c (E'\u271d i)\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\n\u03b9' : Type v'\ninst\u271d\u00b2 : Fintype \u03b9'\nE' : \u03b9' \u2192 Type wE'\ninst\u271d\u00b9 : (i' : \u03b9') \u2192 NormedAddCommGroup (E' i')\ninst\u271d : (i' : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i')\nf : (i' : \u03b9') \u2192 ContinuousMultilinearMap \ud835\udd5c E (E' i')\ni : \u03b9'\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191(f i) m\u2016 \u2264 \u2016pi f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrefine' le_trans _ ((pi f).le_op_norm m)\n[GOAL]\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9'\u271d : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE'\u271d : \u03b9'\u271d \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b2\u2070 : Fintype \u03b9\ninst\u271d\u00b9\u2079 : Fintype \u03b9'\u271d\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2077 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9'\u271d) \u2192 NormedAddCommGroup (E'\u271d i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9'\u271d) \u2192 NormedSpace \ud835\udd5c (E'\u271d i)\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\n\u03b9' : Type v'\ninst\u271d\u00b2 : Fintype \u03b9'\nE' : \u03b9' \u2192 Type wE'\ninst\u271d\u00b9 : (i' : \u03b9') \u2192 NormedAddCommGroup (E' i')\ninst\u271d : (i' : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i')\nf : (i' : \u03b9') \u2192 ContinuousMultilinearMap \ud835\udd5c E (E' i')\ni : \u03b9'\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191(f i) m\u2016 \u2264 \u2016\u2191(pi f) m\u2016\n[PROOFSTEP]\nconvert norm_le_pi_norm (fun j => f j m) i\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2078 : Fintype \u03b9\ninst\u271d\u00b9\u2077 : Fintype \u03b9'\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2079 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\ninst\u271d : Subsingleton \u03b9\ni' : \u03b9\nm : \u03b9 \u2192 G\n\u22a2 \u2016\u2191(ofSubsingleton \ud835\udd5c G i') m\u2016 \u2264 1 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrw [Fintype.prod_subsingleton _ i', one_mul, ofSubsingleton_apply]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2079 : Fintype \u03b9\ninst\u271d\u00b9\u2078 : Fintype \u03b9'\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2079 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b2 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\ninst\u271d\u00b9 : Subsingleton \u03b9\ninst\u271d : Nontrivial G\ni' : \u03b9\n\u22a2 \u2016ofSubsingleton \ud835\udd5c G i'\u2016 = 1\n[PROOFSTEP]\napply le_antisymm (norm_ofSubsingleton_le \ud835\udd5c G i')\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2079 : Fintype \u03b9\ninst\u271d\u00b9\u2078 : Fintype \u03b9'\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2079 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b2 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\ninst\u271d\u00b9 : Subsingleton \u03b9\ninst\u271d : Nontrivial G\ni' : \u03b9\n\u22a2 1 \u2264 \u2016ofSubsingleton \ud835\udd5c G i'\u2016\n[PROOFSTEP]\nobtain \u27e8g, hg\u27e9 := exists_ne (0 : G)\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2079 : Fintype \u03b9\ninst\u271d\u00b9\u2078 : Fintype \u03b9'\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2079 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g\u271d : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b2 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\ninst\u271d\u00b9 : Subsingleton \u03b9\ninst\u271d : Nontrivial G\ni' : \u03b9\ng : G\nhg : g \u2260 0\n\u22a2 1 \u2264 \u2016ofSubsingleton \ud835\udd5c G i'\u2016\n[PROOFSTEP]\nrw [\u2190 norm_ne_zero_iff] at hg \n[GOAL]\ncase intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2079 : Fintype \u03b9\ninst\u271d\u00b9\u2078 : Fintype \u03b9'\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2079 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g\u271d : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b2 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\ninst\u271d\u00b9 : Subsingleton \u03b9\ninst\u271d : Nontrivial G\ni' : \u03b9\ng : G\nhg : \u2016g\u2016 \u2260 0\n\u22a2 1 \u2264 \u2016ofSubsingleton \ud835\udd5c G i'\u2016\n[PROOFSTEP]\nhave := (ofSubsingleton \ud835\udd5c G i').ratio_le_op_norm fun _ => g\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2079 : Fintype \u03b9\ninst\u271d\u00b9\u2078 : Fintype \u03b9'\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u00b9\u2070 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2079 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g\u271d : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b2 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\ninst\u271d\u00b9 : Subsingleton \u03b9\ninst\u271d : Nontrivial G\ni' : \u03b9\ng : G\nhg : \u2016g\u2016 \u2260 0\nthis : \u2016\u2191(ofSubsingleton \ud835\udd5c G i') fun x => g\u2016 / \u220f i : \u03b9, \u2016g\u2016 \u2264 \u2016ofSubsingleton \ud835\udd5c G i'\u2016\n\u22a2 1 \u2264 \u2016ofSubsingleton \ud835\udd5c G i'\u2016\n[PROOFSTEP]\nrwa [Fintype.prod_subsingleton _ i', ofSubsingleton_apply, div_self hg] at this \n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2078 : Fintype \u03b9\ninst\u271d\u00b9\u2077 : Fintype \u03b9'\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2079 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\ninst\u271d : IsEmpty \u03b9\nx : G\n\u22a2 \u2016constOfIsEmpty \ud835\udd5c E x\u2016 = \u2016x\u2016\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2078 : Fintype \u03b9\ninst\u271d\u00b9\u2077 : Fintype \u03b9'\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2079 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\ninst\u271d : IsEmpty \u03b9\nx : G\n\u22a2 \u2016constOfIsEmpty \ud835\udd5c E x\u2016 \u2264 \u2016x\u2016\n[PROOFSTEP]\nrefine' op_norm_le_bound _ (norm_nonneg _) fun x => _\n[GOAL]\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2078 : Fintype \u03b9\ninst\u271d\u00b9\u2077 : Fintype \u03b9'\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2079 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\ninst\u271d : IsEmpty \u03b9\nx\u271d : G\nx : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191(constOfIsEmpty \ud835\udd5c E x\u271d) x\u2016 \u2264 \u2016x\u271d\u2016 * \u220f i : \u03b9, \u2016x i\u2016\n[PROOFSTEP]\nrw [Fintype.prod_empty, mul_one, constOfIsEmpty_apply]\n[GOAL]\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2078 : Fintype \u03b9\ninst\u271d\u00b9\u2077 : Fintype \u03b9'\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2079 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c \ud835\udd5c' G\ninst\u271d : IsEmpty \u03b9\nx : G\n\u22a2 \u2016x\u2016 \u2264 \u2016constOfIsEmpty \ud835\udd5c E x\u2016\n[PROOFSTEP]\nsimpa using (constOfIsEmpty \ud835\udd5c E x).le_op_norm 0\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\nf : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'\n\u22a2 (fun f =>\n (ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.fst \ud835\udd5c G G') f,\n ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.snd \ud835\udd5c G G') f))\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }\n (c \u2022 f) =\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }\n (c \u2022 f)) }.toAddHom\n f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081.H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\nf : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'\nx\u271d : (i : \u03b9) \u2192 E i\n\u22a2 \u2191((fun f =>\n (ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.fst \ud835\udd5c G G') f,\n ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.snd \ud835\udd5c G G') f))\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }\n (c \u2022 f) =\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }\n (c \u2022 f)) }.toAddHom\n f)).fst\n x\u271d =\n \u2191f.fst x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2082.H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\nf : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'\nx\u271d : (i : \u03b9) \u2192 E i\n\u22a2 \u2191((fun f =>\n (ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.fst \ud835\udd5c G G') f,\n ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.snd \ud835\udd5c G G') f))\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }\n (c \u2022 f) =\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }\n (c \u2022 f)) }.toAddHom\n f)).snd\n x\u271d =\n \u2191f.snd x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\nf : ContinuousMultilinearMap \ud835\udd5c E (G \u00d7 G')\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }\n (c \u2022 f) =\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }\n (c \u2022 f)) }.toAddHom\n ((fun f =>\n (ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.fst \ud835\udd5c G G') f,\n ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.snd \ud835\udd5c G G') f))\n f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase H.h\u2081\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\nf : ContinuousMultilinearMap \ud835\udd5c E (G \u00d7 G')\nx\u271d : (i : \u03b9) \u2192 E i\n\u22a2 (\u2191(AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }\n (c \u2022 f) =\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }\n (c \u2022 f)) }.toAddHom\n ((fun f =>\n (ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.fst \ud835\udd5c G G') f,\n ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.snd \ud835\udd5c G G') f))\n f))\n x\u271d).fst =\n (\u2191f x\u271d).fst\n[PROOFSTEP]\nrfl\n[GOAL]\ncase H.h\u2082\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' G\nf : ContinuousMultilinearMap \ud835\udd5c E (G \u00d7 G')\nx\u271d : (i : \u03b9) \u2192 E i\n\u22a2 (\u2191(AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }\n (c \u2022 f) =\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ContinuousMultilinearMap \ud835\udd5c E G'),\n (fun f => prod f.fst f.snd) (f + g) = (fun f => prod f.fst f.snd) (f + g)) }\n (c \u2022 f)) }.toAddHom\n ((fun f =>\n (ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.fst \ud835\udd5c G G') f,\n ContinuousLinearMap.compContinuousMultilinearMap (ContinuousLinearMap.snd \ud835\udd5c G G') f))\n f))\n x\u271d).snd =\n (\u2191f x\u271d).snd\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\u22a2 Continuous fun p => \u2191p.fst p.snd\n[PROOFSTEP]\napply continuous_iff_continuousAt.2 fun p => ?_\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\n\u22a2 ContinuousAt (fun p => \u2191p.fst p.snd) p\n[PROOFSTEP]\napply\n continuousAt_of_locally_lipschitz zero_lt_one\n ((\u2016p\u2016 + 1) * Fintype.card \u03b9 * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i, \u2016p.2 i\u2016) fun q hq => ?_\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p\n[PROOFSTEP]\nhave : 0 \u2264 max \u2016q.2\u2016 \u2016p.2\u2016 := by simp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\n\u22a2 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p\n[PROOFSTEP]\nhave : 0 \u2264 \u2016p\u2016 + 1 := zero_le_one.trans ((le_add_iff_nonneg_left 1).2 <| norm_nonneg p)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis : 0 \u2264 \u2016p\u2016 + 1\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p\n[PROOFSTEP]\nhave A : \u2016q\u2016 \u2264 \u2016p\u2016 + 1 := norm_le_of_mem_closedBall hq.le\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p\n[PROOFSTEP]\nhave : max \u2016q.2\u2016 \u2016p.2\u2016 \u2264 \u2016p\u2016 + 1 := (max_le_max (norm_snd_le q) (norm_snd_le p)).trans (by simp [A, zero_le_one])\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\n\u22a2 max \u2016q\u2016 \u2016p\u2016 \u2264 \u2016p\u2016 + 1\n[PROOFSTEP]\nsimp [A, zero_le_one]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d\u00b9 : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis\u271d : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\nthis : max \u2016q.snd\u2016 \u2016p.snd\u2016 \u2264 \u2016p\u2016 + 1\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p\n[PROOFSTEP]\nhave : \u2200 i : \u03b9, i \u2208 univ \u2192 0 \u2264 \u2016p.2 i\u2016 := fun i _ => norm_nonneg _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d\u00b2 : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis\u271d\u00b9 : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\nthis\u271d : max \u2016q.snd\u2016 \u2016p.snd\u2016 \u2264 \u2016p\u2016 + 1\nthis : \u2200 (i : \u03b9), i \u2208 univ \u2192 0 \u2264 \u2016Prod.snd p i\u2016\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p\n[PROOFSTEP]\ncalc\n dist (q.1 q.2) (p.1 p.2) \u2264 dist (q.1 q.2) (q.1 p.2) + dist (q.1 p.2) (p.1 p.2) := dist_triangle _ _ _\n _ = \u2016q.1 q.2 - q.1 p.2\u2016 + \u2016q.1 p.2 - p.1 p.2\u2016 := by rw [dist_eq_norm, dist_eq_norm]\n _ \u2264 \u2016q.1\u2016 * Fintype.card \u03b9 * max \u2016q.2\u2016 \u2016p.2\u2016 ^ (Fintype.card \u03b9 - 1) * \u2016q.2 - p.2\u2016 + \u2016q.1 - p.1\u2016 * \u220f i, \u2016p.2 i\u2016 :=\n (add_le_add (norm_image_sub_le _ _ _) ((q.1 - p.1).le_op_norm p.2))\n _ \u2264 (\u2016p\u2016 + 1) * Fintype.card \u03b9 * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) * \u2016q - p\u2016 + \u2016q - p\u2016 * \u220f i, \u2016p.2 i\u2016 := by\n apply_rules [add_le_add, mul_le_mul, le_refl, le_trans (norm_fst_le q) A, Nat.cast_nonneg, mul_nonneg,\n pow_le_pow_of_le_left, pow_nonneg, norm_snd_le (q - p), norm_nonneg, norm_fst_le (q - p), prod_nonneg]\n _ = ((\u2016p\u2016 + 1) * Fintype.card \u03b9 * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i, \u2016p.2 i\u2016) * dist q p :=\n by\n rw [dist_eq_norm]\n ring\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d\u00b2 : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis\u271d\u00b9 : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\nthis\u271d : max \u2016q.snd\u2016 \u2016p.snd\u2016 \u2264 \u2016p\u2016 + 1\nthis : \u2200 (i : \u03b9), i \u2208 univ \u2192 0 \u2264 \u2016Prod.snd p i\u2016\n\u22a2 dist (\u2191q.fst q.snd) (\u2191q.fst p.snd) + dist (\u2191q.fst p.snd) (\u2191p.fst p.snd) =\n \u2016\u2191q.fst q.snd - \u2191q.fst p.snd\u2016 + \u2016\u2191q.fst p.snd - \u2191p.fst p.snd\u2016\n[PROOFSTEP]\nrw [dist_eq_norm, dist_eq_norm]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d\u00b2 : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis\u271d\u00b9 : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\nthis\u271d : max \u2016q.snd\u2016 \u2016p.snd\u2016 \u2264 \u2016p\u2016 + 1\nthis : \u2200 (i : \u03b9), i \u2208 univ \u2192 0 \u2264 \u2016Prod.snd p i\u2016\n\u22a2 \u2016q.fst\u2016 * \u2191(Fintype.card \u03b9) * max \u2016q.snd\u2016 \u2016p.snd\u2016 ^ (Fintype.card \u03b9 - 1) * \u2016q.snd - p.snd\u2016 +\n \u2016q.fst - p.fst\u2016 * \u220f i : \u03b9, \u2016Prod.snd p i\u2016 \u2264\n (\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) * \u2016q - p\u2016 + \u2016q - p\u2016 * \u220f i : \u03b9, \u2016Prod.snd p i\u2016\n[PROOFSTEP]\napply_rules [add_le_add, mul_le_mul, le_refl, le_trans (norm_fst_le q) A, Nat.cast_nonneg, mul_nonneg,\n pow_le_pow_of_le_left, pow_nonneg, norm_snd_le (q - p), norm_nonneg, norm_fst_le (q - p), prod_nonneg]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d\u00b2 : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis\u271d\u00b9 : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\nthis\u271d : max \u2016q.snd\u2016 \u2016p.snd\u2016 \u2264 \u2016p\u2016 + 1\nthis : \u2200 (i : \u03b9), i \u2208 univ \u2192 0 \u2264 \u2016Prod.snd p i\u2016\n\u22a2 (\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) * \u2016q - p\u2016 + \u2016q - p\u2016 * \u220f i : \u03b9, \u2016Prod.snd p i\u2016 =\n ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p\n[PROOFSTEP]\nrw [dist_eq_norm]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d\u00b2 : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis\u271d\u00b9 : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\nthis\u271d : max \u2016q.snd\u2016 \u2016p.snd\u2016 \u2264 \u2016p\u2016 + 1\nthis : \u2200 (i : \u03b9), i \u2208 univ \u2192 0 \u2264 \u2016Prod.snd p i\u2016\n\u22a2 (\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) * \u2016q - p\u2016 + \u2016q - p\u2016 * \u220f i : \u03b9, \u2016Prod.snd p i\u2016 =\n ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * \u2016q - p\u2016\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\u03b1 : Type u_1\np : \u03b1 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nq : ContinuousMultilinearMap \ud835\udd5c E G\nh : HasSum p q\nm : (i : \u03b9) \u2192 E i\n\u22a2 HasSum (fun a => \u2191(p a) m) (\u2191q m)\n[PROOFSTEP]\ndsimp [HasSum] at h \u22a2\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\u03b1 : Type u_1\np : \u03b1 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nq : ContinuousMultilinearMap \ud835\udd5c E G\nh : Filter.Tendsto (fun s => \u2211 b in s, p b) Filter.atTop (nhds q)\nm : (i : \u03b9) \u2192 E i\n\u22a2 Filter.Tendsto (fun s => \u2211 b in s, \u2191(p b) m) Filter.atTop (nhds (\u2191q m))\n[PROOFSTEP]\nconvert ((continuous_eval_left m).tendsto _).comp h using 1\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\u03b1 : Type u_1\np : \u03b1 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nq : ContinuousMultilinearMap \ud835\udd5c E G\nh : Filter.Tendsto (fun s => \u2211 b in s, p b) Filter.atTop (nhds q)\nm : (i : \u03b9) \u2192 E i\n\u22a2 (fun s => \u2211 b in s, \u2191(p b) m) = (fun p => \u2191p m) \u2218 fun s => \u2211 b in s, p b\n[PROOFSTEP]\next s\n[GOAL]\ncase h.e'_3.h\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\n\u03b1 : Type u_1\np : \u03b1 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nq : ContinuousMultilinearMap \ud835\udd5c E G\nh : Filter.Tendsto (fun s => \u2211 b in s, p b) Filter.atTop (nhds q)\nm : (i : \u03b9) \u2192 E i\ns : Finset \u03b1\n\u22a2 \u2211 b in s, \u2191(p b) m = ((fun p => \u2191p m) \u2218 fun s => \u2211 b in s, p b) s\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\n\u22a2 CompleteSpace (ContinuousMultilinearMap \ud835\udd5c E G)\n[PROOFSTEP]\nhave nonneg : \u2200 v : \u2200 i, E i, 0 \u2264 \u220f i, \u2016v i\u2016 := fun v =>\n Finset.prod_nonneg fun i _ =>\n norm_nonneg\n _\n -- We show that every Cauchy sequence converges.\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\n\u22a2 CompleteSpace (ContinuousMultilinearMap \ud835\udd5c E G)\n[PROOFSTEP]\nrefine'\n Metric.complete_of_cauchySeq_tendsto fun f hf =>\n _\n -- We now expand out the definition of a Cauchy sequence,\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\n\u22a2 \u2203 a, Tendsto f atTop (\ud835\udcdd a)\n[PROOFSTEP]\nrcases cauchySeq_iff_le_tendsto_0.1 hf with\n \u27e8b, b0, b_bound, b_lim\u27e9\n -- and establish that the evaluation at any point `v : \u03a0 i, E i` is Cauchy.\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\n\u22a2 \u2203 a, Tendsto f atTop (\ud835\udcdd a)\n[PROOFSTEP]\nhave cau : \u2200 v, CauchySeq fun n => f n v := by\n intro v\n apply cauchySeq_iff_le_tendsto_0.2 \u27e8fun n => b n * \u220f i, \u2016v i\u2016, _, _, _\u27e9\n \u00b7 intro\n exact mul_nonneg (b0 _) (nonneg v)\n \u00b7 intro n m N hn hm\n rw [dist_eq_norm]\n apply le_trans ((f n - f m).le_op_norm v) _\n exact mul_le_mul_of_nonneg_right (b_bound n m N hn hm) (nonneg v)\n \u00b7 simpa using b_lim.mul tendsto_const_nhds\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\n\u22a2 \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\n[PROOFSTEP]\nintro v\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\nv : (i : \u03b9) \u2192 E i\n\u22a2 CauchySeq fun n => \u2191(f n) v\n[PROOFSTEP]\napply cauchySeq_iff_le_tendsto_0.2 \u27e8fun n => b n * \u220f i, \u2016v i\u2016, _, _, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\nv : (i : \u03b9) \u2192 E i\n\u22a2 \u2200 (n : \u2115), 0 \u2264 (fun n => b n * \u220f i : \u03b9, \u2016v i\u2016) n\n[PROOFSTEP]\nintro\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\nv : (i : \u03b9) \u2192 E i\nn\u271d : \u2115\n\u22a2 0 \u2264 (fun n => b n * \u220f i : \u03b9, \u2016v i\u2016) n\u271d\n[PROOFSTEP]\nexact mul_nonneg (b0 _) (nonneg v)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\nv : (i : \u03b9) \u2192 E i\n\u22a2 \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (\u2191(f n) v) (\u2191(f m) v) \u2264 (fun n => b n * \u220f i : \u03b9, \u2016v i\u2016) N\n[PROOFSTEP]\nintro n m N hn hm\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\nv : (i : \u03b9) \u2192 E i\nn m N : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\n\u22a2 dist (\u2191(f n) v) (\u2191(f m) v) \u2264 (fun n => b n * \u220f i : \u03b9, \u2016v i\u2016) N\n[PROOFSTEP]\nrw [dist_eq_norm]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\nv : (i : \u03b9) \u2192 E i\nn m N : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\n\u22a2 \u2016\u2191(f n) v - \u2191(f m) v\u2016 \u2264 (fun n => b n * \u220f i : \u03b9, \u2016v i\u2016) N\n[PROOFSTEP]\napply le_trans ((f n - f m).le_op_norm v) _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\nv : (i : \u03b9) \u2192 E i\nn m N : \u2115\nhn : N \u2264 n\nhm : N \u2264 m\n\u22a2 \u2016f n - f m\u2016 * \u220f i : \u03b9, \u2016v i\u2016 \u2264 (fun n => b n * \u220f i : \u03b9, \u2016v i\u2016) N\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_right (b_bound n m N hn hm) (nonneg v)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\nv : (i : \u03b9) \u2192 E i\n\u22a2 Tendsto (fun n => b n * \u220f i : \u03b9, \u2016v i\u2016) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa using b_lim.mul tendsto_const_nhds\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\n\u22a2 \u2203 a, Tendsto f atTop (\ud835\udcdd a)\n[PROOFSTEP]\nchoose F hF using fun v =>\n cauchySeq_tendsto_of_complete\n (cau v)\n -- Next, we show that this `F` is multilinear,\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\n\u22a2 \u2203 a, Tendsto f atTop (\ud835\udcdd a)\n[PROOFSTEP]\nlet Fmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F\n map_add' := fun v i x y => by\n skip\n have A := hF (Function.update v i (x + y))\n have B := (hF (Function.update v i x)).add (hF (Function.update v i y))\n simp at A B \n exact tendsto_nhds_unique A B\n map_smul' := fun v i c x => by\n skip\n have A := hF (Function.update v i (c \u2022 x))\n have B := Filter.Tendsto.smul (@tendsto_const_nhds _ \u2115 _ c _) (hF (Function.update v i x))\n simp at A B \n exact tendsto_nhds_unique A B }\n -- and that `F` has norm at most `(b 0 + \u2016f 0\u2016)`.\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d\u00b9 : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\ninst\u271d : DecidableEq \u03b9\nv : (i : \u03b9) \u2192 E i\ni : \u03b9\nx y : E i\n\u22a2 F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)\n[PROOFSTEP]\nskip\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d\u00b9 : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\ninst\u271d : DecidableEq \u03b9\nv : (i : \u03b9) \u2192 E i\ni : \u03b9\nx y : E i\n\u22a2 F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)\n[PROOFSTEP]\nhave A := hF (Function.update v i (x + y))\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d\u00b9 : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\ninst\u271d : DecidableEq \u03b9\nv : (i : \u03b9) \u2192 E i\ni : \u03b9\nx y : E i\nA : Tendsto (fun n => \u2191(f n) (Function.update v i (x + y))) atTop (\ud835\udcdd (F (Function.update v i (x + y))))\n\u22a2 F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)\n[PROOFSTEP]\nhave B := (hF (Function.update v i x)).add (hF (Function.update v i y))\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d\u00b9 : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\ninst\u271d : DecidableEq \u03b9\nv : (i : \u03b9) \u2192 E i\ni : \u03b9\nx y : E i\nA : Tendsto (fun n => \u2191(f n) (Function.update v i (x + y))) atTop (\ud835\udcdd (F (Function.update v i (x + y))))\nB :\n Tendsto (fun x_1 => \u2191(f x_1) (Function.update v i x) + \u2191(f x_1) (Function.update v i y)) atTop\n (\ud835\udcdd (F (Function.update v i x) + F (Function.update v i y)))\n\u22a2 F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)\n[PROOFSTEP]\nsimp at A B \n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d\u00b9 : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\ninst\u271d : DecidableEq \u03b9\nv : (i : \u03b9) \u2192 E i\ni : \u03b9\nx y : E i\nB :\n Tendsto (fun x_1 => \u2191(f x_1) (Function.update v i x) + \u2191(f x_1) (Function.update v i y)) atTop\n (\ud835\udcdd (F (Function.update v i x) + F (Function.update v i y)))\nA :\n Tendsto (fun n => \u2191(f n) (Function.update v i x) + \u2191(f n) (Function.update v i y)) atTop\n (\ud835\udcdd (F (Function.update v i (x + y))))\n\u22a2 F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)\n[PROOFSTEP]\nexact tendsto_nhds_unique A B\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nc\u271d : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d\u00b9 : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\ninst\u271d : DecidableEq \u03b9\nv : (i : \u03b9) \u2192 E i\ni : \u03b9\nc : \ud835\udd5c\nx : E i\n\u22a2 F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)\n[PROOFSTEP]\nskip\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nc\u271d : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d\u00b9 : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\ninst\u271d : DecidableEq \u03b9\nv : (i : \u03b9) \u2192 E i\ni : \u03b9\nc : \ud835\udd5c\nx : E i\n\u22a2 F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)\n[PROOFSTEP]\nhave A := hF (Function.update v i (c \u2022 x))\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nc\u271d : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d\u00b9 : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\ninst\u271d : DecidableEq \u03b9\nv : (i : \u03b9) \u2192 E i\ni : \u03b9\nc : \ud835\udd5c\nx : E i\nA : Tendsto (fun n => \u2191(f n) (Function.update v i (c \u2022 x))) atTop (\ud835\udcdd (F (Function.update v i (c \u2022 x))))\n\u22a2 F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)\n[PROOFSTEP]\nhave B := Filter.Tendsto.smul (@tendsto_const_nhds _ \u2115 _ c _) (hF (Function.update v i x))\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nc\u271d : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d\u00b9 : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\ninst\u271d : DecidableEq \u03b9\nv : (i : \u03b9) \u2192 E i\ni : \u03b9\nc : \ud835\udd5c\nx : E i\nA : Tendsto (fun n => \u2191(f n) (Function.update v i (c \u2022 x))) atTop (\ud835\udcdd (F (Function.update v i (c \u2022 x))))\nB : Tendsto (fun x_1 => c \u2022 \u2191(f x_1) (Function.update v i x)) atTop (\ud835\udcdd (c \u2022 F (Function.update v i x)))\n\u22a2 F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)\n[PROOFSTEP]\nsimp at A B \n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nc\u271d : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d\u00b9 : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\ninst\u271d : DecidableEq \u03b9\nv : (i : \u03b9) \u2192 E i\ni : \u03b9\nc : \ud835\udd5c\nx : E i\nB : Tendsto (fun x_1 => c \u2022 \u2191(f x_1) (Function.update v i x)) atTop (\ud835\udcdd (c \u2022 F (Function.update v i x)))\nA : Tendsto (fun n => c \u2022 \u2191(f n) (Function.update v i x)) atTop (\ud835\udcdd (F (Function.update v i (c \u2022 x))))\n\u22a2 F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)\n[PROOFSTEP]\nexact tendsto_nhds_unique A B\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\n\u22a2 \u2203 a, Tendsto f atTop (\ud835\udcdd a)\n[PROOFSTEP]\nhave Fnorm : \u2200 v, \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i, \u2016v i\u2016 := by\n intro v\n have A : \u2200 n, \u2016f n v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i, \u2016v i\u2016 := by\n intro n\n apply le_trans ((f n).le_op_norm _) _\n apply mul_le_mul_of_nonneg_right _ (nonneg v)\n calc\n \u2016f n\u2016 = \u2016f n - f 0 + f 0\u2016 := by\n congr 1\n abel\n _ \u2264 \u2016f n - f 0\u2016 + \u2016f 0\u2016 := (norm_add_le _ _)\n _ \u2264 b 0 + \u2016f 0\u2016 := by\n apply add_le_add_right\n simpa [dist_eq_norm] using b_bound n 0 0 (zero_le _) (zero_le _)\n exact\n le_of_tendsto (hF v).norm\n (eventually_of_forall A)\n -- Thus `F` is continuous, and we propose that as the limit point of our original Cauchy sequence.\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\n\u22a2 \u2200 (v : (i : \u03b9) \u2192 E i), \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\n[PROOFSTEP]\nintro v\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nv : (i : \u03b9) \u2192 E i\n\u22a2 \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\n[PROOFSTEP]\nhave A : \u2200 n, \u2016f n v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i, \u2016v i\u2016 := by\n intro n\n apply le_trans ((f n).le_op_norm _) _\n apply mul_le_mul_of_nonneg_right _ (nonneg v)\n calc\n \u2016f n\u2016 = \u2016f n - f 0 + f 0\u2016 := by\n congr 1\n abel\n _ \u2264 \u2016f n - f 0\u2016 + \u2016f 0\u2016 := (norm_add_le _ _)\n _ \u2264 b 0 + \u2016f 0\u2016 := by\n apply add_le_add_right\n simpa [dist_eq_norm] using b_bound n 0 0 (zero_le _) (zero_le _)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nv : (i : \u03b9) \u2192 E i\n\u22a2 \u2200 (n : \u2115), \u2016\u2191(f n) v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\n[PROOFSTEP]\nintro n\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nv : (i : \u03b9) \u2192 E i\nn : \u2115\n\u22a2 \u2016\u2191(f n) v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\n[PROOFSTEP]\napply le_trans ((f n).le_op_norm _) _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nv : (i : \u03b9) \u2192 E i\nn : \u2115\n\u22a2 \u2016f n\u2016 * \u220f i : \u03b9, \u2016v i\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_right _ (nonneg v)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nv : (i : \u03b9) \u2192 E i\nn : \u2115\n\u22a2 \u2016f n\u2016 \u2264 b 0 + \u2016f 0\u2016\n[PROOFSTEP]\ncalc\n \u2016f n\u2016 = \u2016f n - f 0 + f 0\u2016 := by\n congr 1\n abel\n _ \u2264 \u2016f n - f 0\u2016 + \u2016f 0\u2016 := (norm_add_le _ _)\n _ \u2264 b 0 + \u2016f 0\u2016 := by\n apply add_le_add_right\n simpa [dist_eq_norm] using b_bound n 0 0 (zero_le _) (zero_le _)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nv : (i : \u03b9) \u2192 E i\nn : \u2115\n\u22a2 \u2016f n\u2016 = \u2016f n - f 0 + f 0\u2016\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nv : (i : \u03b9) \u2192 E i\nn : \u2115\n\u22a2 f n = f n - f 0 + f 0\n[PROOFSTEP]\nabel\n[GOAL]\ncase e_a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nv : (i : \u03b9) \u2192 E i\nn : \u2115\n\u22a2 f n = f n - f 0 + f 0\n[PROOFSTEP]\nabel\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nv : (i : \u03b9) \u2192 E i\nn : \u2115\n\u22a2 \u2016f n - f 0\u2016 + \u2016f 0\u2016 \u2264 b 0 + \u2016f 0\u2016\n[PROOFSTEP]\napply add_le_add_right\n[GOAL]\ncase bc\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nv : (i : \u03b9) \u2192 E i\nn : \u2115\n\u22a2 \u2016f n - f 0\u2016 \u2264 b 0\n[PROOFSTEP]\nsimpa [dist_eq_norm] using b_bound n 0 0 (zero_le _) (zero_le _)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nv : (i : \u03b9) \u2192 E i\nA : \u2200 (n : \u2115), \u2016\u2191(f n) v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\n\u22a2 \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\n[PROOFSTEP]\nexact\n le_of_tendsto (hF v).norm\n (eventually_of_forall A)\n -- Thus `F` is continuous, and we propose that as the limit point of our original Cauchy sequence.\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nFnorm : \u2200 (v : (i : \u03b9) \u2192 E i), \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\n\u22a2 \u2203 a, Tendsto f atTop (\ud835\udcdd a)\n[PROOFSTEP]\nlet Fcont := Fmult.mkContinuous _ Fnorm\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nFnorm : \u2200 (v : (i : \u03b9) \u2192 E i), \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\nFcont : ContinuousMultilinearMap \ud835\udd5c E G := MultilinearMap.mkContinuous Fmult (b 0 + \u2016f 0\u2016) Fnorm\n\u22a2 \u2203 a, Tendsto f atTop (\ud835\udcdd a)\n[PROOFSTEP]\nuse Fcont\n[GOAL]\ncase h\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nFnorm : \u2200 (v : (i : \u03b9) \u2192 E i), \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\nFcont : ContinuousMultilinearMap \ud835\udd5c E G := MultilinearMap.mkContinuous Fmult (b 0 + \u2016f 0\u2016) Fnorm\n\u22a2 Tendsto f atTop (\ud835\udcdd Fcont)\n[PROOFSTEP]\nhave : \u2200 n, \u2016f n - Fcont\u2016 \u2264 b n := by\n intro n\n apply op_norm_le_bound _ (b0 n) fun v => ?_\n have A : \u2200\u1da0 m in atTop, \u2016(f n - f m) v\u2016 \u2264 b n * \u220f i, \u2016v i\u2016 :=\n by\n refine' eventually_atTop.2 \u27e8n, fun m hm => _\u27e9\n apply le_trans ((f n - f m).le_op_norm _) _\n exact mul_le_mul_of_nonneg_right (b_bound n m n le_rfl hm) (nonneg v)\n have B : Tendsto (fun m => \u2016(f n - f m) v\u2016) atTop (\ud835\udcdd \u2016(f n - Fcont) v\u2016) :=\n Tendsto.norm (tendsto_const_nhds.sub (hF v))\n exact le_of_tendsto B A\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nFnorm : \u2200 (v : (i : \u03b9) \u2192 E i), \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\nFcont : ContinuousMultilinearMap \ud835\udd5c E G := MultilinearMap.mkContinuous Fmult (b 0 + \u2016f 0\u2016) Fnorm\n\u22a2 \u2200 (n : \u2115), \u2016f n - Fcont\u2016 \u2264 b n\n[PROOFSTEP]\nintro n\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nFnorm : \u2200 (v : (i : \u03b9) \u2192 E i), \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\nFcont : ContinuousMultilinearMap \ud835\udd5c E G := MultilinearMap.mkContinuous Fmult (b 0 + \u2016f 0\u2016) Fnorm\nn : \u2115\n\u22a2 \u2016f n - Fcont\u2016 \u2264 b n\n[PROOFSTEP]\napply op_norm_le_bound _ (b0 n) fun v => ?_\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nFnorm : \u2200 (v : (i : \u03b9) \u2192 E i), \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\nFcont : ContinuousMultilinearMap \ud835\udd5c E G := MultilinearMap.mkContinuous Fmult (b 0 + \u2016f 0\u2016) Fnorm\nn : \u2115\nv : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191(f n - Fcont) v\u2016 \u2264 b n * \u220f i : \u03b9, \u2016v i\u2016\n[PROOFSTEP]\nhave A : \u2200\u1da0 m in atTop, \u2016(f n - f m) v\u2016 \u2264 b n * \u220f i, \u2016v i\u2016 :=\n by\n refine' eventually_atTop.2 \u27e8n, fun m hm => _\u27e9\n apply le_trans ((f n - f m).le_op_norm _) _\n exact mul_le_mul_of_nonneg_right (b_bound n m n le_rfl hm) (nonneg v)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nFnorm : \u2200 (v : (i : \u03b9) \u2192 E i), \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\nFcont : ContinuousMultilinearMap \ud835\udd5c E G := MultilinearMap.mkContinuous Fmult (b 0 + \u2016f 0\u2016) Fnorm\nn : \u2115\nv : (i : \u03b9) \u2192 E i\n\u22a2 \u2200\u1da0 (m : \u2115) in atTop, \u2016\u2191(f n - f m) v\u2016 \u2264 b n * \u220f i : \u03b9, \u2016v i\u2016\n[PROOFSTEP]\nrefine' eventually_atTop.2 \u27e8n, fun m hm => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nFnorm : \u2200 (v : (i : \u03b9) \u2192 E i), \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\nFcont : ContinuousMultilinearMap \ud835\udd5c E G := MultilinearMap.mkContinuous Fmult (b 0 + \u2016f 0\u2016) Fnorm\nn : \u2115\nv : (i : \u03b9) \u2192 E i\nm : \u2115\nhm : m \u2265 n\n\u22a2 \u2016\u2191(f n - f m) v\u2016 \u2264 b n * \u220f i : \u03b9, \u2016v i\u2016\n[PROOFSTEP]\napply le_trans ((f n - f m).le_op_norm _) _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm\u271d : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nFnorm : \u2200 (v : (i : \u03b9) \u2192 E i), \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\nFcont : ContinuousMultilinearMap \ud835\udd5c E G := MultilinearMap.mkContinuous Fmult (b 0 + \u2016f 0\u2016) Fnorm\nn : \u2115\nv : (i : \u03b9) \u2192 E i\nm : \u2115\nhm : m \u2265 n\n\u22a2 \u2016f n - f m\u2016 * \u220f i : \u03b9, \u2016v i\u2016 \u2264 b n * \u220f i : \u03b9, \u2016v i\u2016\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_right (b_bound n m n le_rfl hm) (nonneg v)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nFnorm : \u2200 (v : (i : \u03b9) \u2192 E i), \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\nFcont : ContinuousMultilinearMap \ud835\udd5c E G := MultilinearMap.mkContinuous Fmult (b 0 + \u2016f 0\u2016) Fnorm\nn : \u2115\nv : (i : \u03b9) \u2192 E i\nA : \u2200\u1da0 (m : \u2115) in atTop, \u2016\u2191(f n - f m) v\u2016 \u2264 b n * \u220f i : \u03b9, \u2016v i\u2016\n\u22a2 \u2016\u2191(f n - Fcont) v\u2016 \u2264 b n * \u220f i : \u03b9, \u2016v i\u2016\n[PROOFSTEP]\nhave B : Tendsto (fun m => \u2016(f n - f m) v\u2016) atTop (\ud835\udcdd \u2016(f n - Fcont) v\u2016) := Tendsto.norm (tendsto_const_nhds.sub (hF v))\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nFnorm : \u2200 (v : (i : \u03b9) \u2192 E i), \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\nFcont : ContinuousMultilinearMap \ud835\udd5c E G := MultilinearMap.mkContinuous Fmult (b 0 + \u2016f 0\u2016) Fnorm\nn : \u2115\nv : (i : \u03b9) \u2192 E i\nA : \u2200\u1da0 (m : \u2115) in atTop, \u2016\u2191(f n - f m) v\u2016 \u2264 b n * \u220f i : \u03b9, \u2016v i\u2016\nB : Tendsto (fun m => \u2016\u2191(f n - f m) v\u2016) atTop (\ud835\udcdd \u2016\u2191(f n - Fcont) v\u2016)\n\u22a2 \u2016\u2191(f n - Fcont) v\u2016 \u2264 b n * \u220f i : \u03b9, \u2016v i\u2016\n[PROOFSTEP]\nexact le_of_tendsto B A\n[GOAL]\ncase h\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nFnorm : \u2200 (v : (i : \u03b9) \u2192 E i), \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\nFcont : ContinuousMultilinearMap \ud835\udd5c E G := MultilinearMap.mkContinuous Fmult (b 0 + \u2016f 0\u2016) Fnorm\nthis : \u2200 (n : \u2115), \u2016f n - Fcont\u2016 \u2264 b n\n\u22a2 Tendsto f atTop (\ud835\udcdd Fcont)\n[PROOFSTEP]\nerw [tendsto_iff_norm_tendsto_zero]\n[GOAL]\ncase h\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf\u271d g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\ninst\u271d : CompleteSpace G\nnonneg : \u2200 (v : (i : \u03b9) \u2192 E i), 0 \u2264 \u220f i : \u03b9, \u2016v i\u2016\nf : \u2115 \u2192 ContinuousMultilinearMap \ud835\udd5c E G\nhf : CauchySeq f\nb : \u2115 \u2192 \u211d\nb0 : \u2200 (n : \u2115), 0 \u2264 b n\nb_bound : \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (f n) (f m) \u2264 b N\nb_lim : Tendsto b atTop (\ud835\udcdd 0)\ncau : \u2200 (v : (i : \u03b9) \u2192 E i), CauchySeq fun n => \u2191(f n) v\nF : (v : (i : \u03b9) \u2192 E i) \u2192 (fun x => G) v\nhF : \u2200 (v : (i : \u03b9) \u2192 E i), Tendsto (fun n => \u2191(f n) v) atTop (\ud835\udcdd (F v))\nFmult : MultilinearMap \ud835\udd5c E G :=\n { toFun := F,\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n F (Function.update v i (c \u2022 x)) = c \u2022 F (Function.update v i x)) }\nFnorm : \u2200 (v : (i : \u03b9) \u2192 E i), \u2016F v\u2016 \u2264 (b 0 + \u2016f 0\u2016) * \u220f i : \u03b9, \u2016v i\u2016\nFcont : ContinuousMultilinearMap \ud835\udd5c E G := MultilinearMap.mkContinuous Fmult (b 0 + \u2016f 0\u2016) Fnorm\nthis : \u2200 (n : \u2115), \u2016f n - Fcont\u2016 \u2264 b n\n\u22a2 Tendsto (fun e => \u2016f e - Fcont\u2016) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact squeeze_zero (fun n => norm_nonneg _) this b_lim\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nk n : \u2115\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => G) G'\ns : Finset (Fin n)\nhk : card s = k\nz : G\n\u22a2 \u2016restr f s hk z\u2016 \u2264 \u2016f\u2016 * \u2016z\u2016 ^ (n - k)\n[PROOFSTEP]\napply MultilinearMap.mkContinuous_norm_le\n[GOAL]\ncase hC\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn\u271d : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n\u271d) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n\u271d)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nk n : \u2115\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => G) G'\ns : Finset (Fin n)\nhk : card s = k\nz : G\n\u22a2 0 \u2264 \u2016f\u2016 * \u2016z\u2016 ^ (n - k)\n[PROOFSTEP]\nexact mul_nonneg (norm_nonneg _) (pow_nonneg (norm_nonneg _) _)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : Nonempty \u03b9\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A\u2016 \u2264 1\n[PROOFSTEP]\nhave := fun f => @op_norm_le_bound \ud835\udd5c \u03b9 (fun _ => A) A _ _ _ _ _ _ f _ zero_le_one\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : Nonempty \u03b9\nthis : \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A), (\u2200 (m : \u03b9 \u2192 A), \u2016\u2191f m\u2016 \u2264 1 * \u220f i : \u03b9, \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 1\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A\u2016 \u2264 1\n[PROOFSTEP]\nrefine' this _ _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : Nonempty \u03b9\nthis : \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A), (\u2200 (m : \u03b9 \u2192 A), \u2016\u2191f m\u2016 \u2264 1 * \u220f i : \u03b9, \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 1\n\u22a2 \u2200 (m : \u03b9 \u2192 A), \u2016\u2191(ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A) m\u2016 \u2264 1 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nintro m\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : Nonempty \u03b9\nthis : \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A), (\u2200 (m : \u03b9 \u2192 A), \u2016\u2191f m\u2016 \u2264 1 * \u220f i : \u03b9, \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 1\nm : \u03b9 \u2192 A\n\u22a2 \u2016\u2191(ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A) m\u2016 \u2264 1 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nsimp only [ContinuousMultilinearMap.mkPiAlgebra_apply, one_mul]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : Nonempty \u03b9\nthis : \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A), (\u2200 (m : \u03b9 \u2192 A), \u2016\u2191f m\u2016 \u2264 1 * \u220f i : \u03b9, \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 1\nm : \u03b9 \u2192 A\n\u22a2 \u2016\u220f i : \u03b9, m i\u2016 \u2264 \u220f x : \u03b9, \u2016m x\u2016\n[PROOFSTEP]\nexact norm_prod_le' _ univ_nonempty _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : IsEmpty \u03b9\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A\u2016 = \u20161\u2016\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : IsEmpty \u03b9\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A\u2016 \u2264 \u20161\u2016\n[PROOFSTEP]\nhave := fun f => @op_norm_le_bound \ud835\udd5c \u03b9 (fun _ => A) A _ _ _ _ _ _ f _ (norm_nonneg (1 : A))\n[GOAL]\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : IsEmpty \u03b9\nthis : \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A), (\u2200 (m : \u03b9 \u2192 A), \u2016\u2191f m\u2016 \u2264 \u20161\u2016 * \u220f i : \u03b9, \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 \u20161\u2016\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A\u2016 \u2264 \u20161\u2016\n[PROOFSTEP]\nrefine' this _ _\n[GOAL]\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : IsEmpty \u03b9\nthis : \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A), (\u2200 (m : \u03b9 \u2192 A), \u2016\u2191f m\u2016 \u2264 \u20161\u2016 * \u220f i : \u03b9, \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 \u20161\u2016\n\u22a2 \u2200 (m : \u03b9 \u2192 A), \u2016\u2191(ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A) m\u2016 \u2264 \u20161\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nsimp\n -- Porting note: have to annotate types to get mvars to unify\n[GOAL]\ncase a\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : IsEmpty \u03b9\n\u22a2 \u20161\u2016 \u2264 \u2016ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A\u2016\n[PROOFSTEP]\nconvert ratio_le_op_norm (ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A) fun _ => (1 : A)\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : IsEmpty \u03b9\n\u22a2 \u20161\u2016 = \u2016\u2191(ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A) fun x => 1\u2016 / \u220f i : \u03b9, \u20161\u2016\n[PROOFSTEP]\nsimp [eq_empty_of_isEmpty (univ : Finset \u03b9)]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : NormOneClass A\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A\u2016 = 1\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b9\n[GOAL]\ncase inl\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : NormOneClass A\nh\u271d : IsEmpty \u03b9\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A\u2016 = 1\n[PROOFSTEP]\nsimp [norm_mkPiAlgebra_of_empty]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : NormOneClass A\nh\u271d : Nonempty \u03b9\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A\u2016 = 1\n[PROOFSTEP]\nrefine' le_antisymm norm_mkPiAlgebra_le _\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : NormOneClass A\nh\u271d : Nonempty \u03b9\n\u22a2 1 \u2264 \u2016ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A\u2016\n[PROOFSTEP]\nconvert ratio_le_op_norm (ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A) fun _ => 1\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedCommRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : NormOneClass A\nh\u271d : Nonempty \u03b9\n\u22a2 1 = \u2016\u2191(ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A) fun x => 1\u2016 / \u220f i : \u03b9, \u20161\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n) A\u2016 \u2264 1\n[PROOFSTEP]\nhave := fun f => @op_norm_le_bound \ud835\udd5c (Fin n.succ) (fun _ => A) A _ _ _ _ _ _ f _ zero_le_one\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\nthis :\n \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A),\n (\u2200 (m : Fin (Nat.succ n) \u2192 A), \u2016\u2191f m\u2016 \u2264 1 * \u220f i : Fin (Nat.succ n), \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 1\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n) A\u2016 \u2264 1\n[PROOFSTEP]\nrefine' this _ _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\nthis :\n \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A),\n (\u2200 (m : Fin (Nat.succ n) \u2192 A), \u2016\u2191f m\u2016 \u2264 1 * \u220f i : Fin (Nat.succ n), \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 1\n\u22a2 \u2200 (m : Fin (Nat.succ n) \u2192 A),\n \u2016\u2191(ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n) A) m\u2016 \u2264 1 * \u220f i : Fin (Nat.succ n), \u2016m i\u2016\n[PROOFSTEP]\nintro m\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\nthis :\n \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A),\n (\u2200 (m : Fin (Nat.succ n) \u2192 A), \u2016\u2191f m\u2016 \u2264 1 * \u220f i : Fin (Nat.succ n), \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 1\nm : Fin (Nat.succ n) \u2192 A\n\u22a2 \u2016\u2191(ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n) A) m\u2016 \u2264 1 * \u220f i : Fin (Nat.succ n), \u2016m i\u2016\n[PROOFSTEP]\nsimp only [ContinuousMultilinearMap.mkPiAlgebraFin_apply, one_mul, List.ofFn_eq_map, Fin.prod_univ_def,\n Multiset.coe_map, Multiset.coe_prod]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\nthis :\n \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A),\n (\u2200 (m : Fin (Nat.succ n) \u2192 A), \u2016\u2191f m\u2016 \u2264 1 * \u220f i : Fin (Nat.succ n), \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 1\nm : Fin (Nat.succ n) \u2192 A\n\u22a2 \u2016List.prod (List.map m (List.finRange (Nat.succ n)))\u2016 \u2264\n List.prod (List.map (fun i => \u2016m i\u2016) (List.finRange (Nat.succ n)))\n[PROOFSTEP]\nrefine' (List.norm_prod_le' _).trans_eq _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\nthis :\n \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A),\n (\u2200 (m : Fin (Nat.succ n) \u2192 A), \u2016\u2191f m\u2016 \u2264 1 * \u220f i : Fin (Nat.succ n), \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 1\nm : Fin (Nat.succ n) \u2192 A\n\u22a2 List.map m (List.finRange (Nat.succ n)) \u2260 []\n[PROOFSTEP]\nrw [Ne.def, List.map_eq_nil, List.finRange_eq_nil]\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\nthis :\n \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A),\n (\u2200 (m : Fin (Nat.succ n) \u2192 A), \u2016\u2191f m\u2016 \u2264 1 * \u220f i : Fin (Nat.succ n), \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 1\nm : Fin (Nat.succ n) \u2192 A\n\u22a2 \u00acNat.succ n = 0\n[PROOFSTEP]\nexact Nat.succ_ne_zero _\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\nthis :\n \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A),\n (\u2200 (m : Fin (Nat.succ n) \u2192 A), \u2016\u2191f m\u2016 \u2264 1 * \u220f i : Fin (Nat.succ n), \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 1\nm : Fin (Nat.succ n) \u2192 A\n\u22a2 List.prod (List.map norm (List.map m (List.finRange (Nat.succ n)))) =\n List.prod (List.map (fun i => \u2016m i\u2016) (List.finRange (Nat.succ n)))\n[PROOFSTEP]\nrw [List.map_map]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\nthis :\n \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A),\n (\u2200 (m : Fin (Nat.succ n) \u2192 A), \u2016\u2191f m\u2016 \u2264 1 * \u220f i : Fin (Nat.succ n), \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 1\nm : Fin (Nat.succ n) \u2192 A\n\u22a2 List.prod (List.map (norm \u2218 m) (List.finRange (Nat.succ n))) =\n List.prod (List.map (fun i => \u2016m i\u2016) (List.finRange (Nat.succ n)))\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\nhn : 0 < n\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c n A\u2016 \u2264 1\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero hn.ne'\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b3 : NormedRing A\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c A\nn : \u2115\nEi : Fin (Nat.succ (Nat.succ n)) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ (Nat.succ n))) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ (Nat.succ n))) \u2192 NormedSpace \ud835\udd5c (Ei i)\nhn : 0 < Nat.succ n\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n) A\u2016 \u2264 1\n[PROOFSTEP]\nexact norm_mkPiAlgebraFin_succ_le\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c 0 A\u2016 = \u20161\u2016\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c 0 A\u2016 \u2264 \u20161\u2016\n[PROOFSTEP]\nhave := fun f => @op_norm_le_bound \ud835\udd5c (Fin 0) (fun _ => A) A _ _ _ _ _ _ f _ (norm_nonneg (1 : A))\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\nthis :\n \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A), (\u2200 (m : Fin 0 \u2192 A), \u2016\u2191f m\u2016 \u2264 \u20161\u2016 * \u220f i : Fin 0, \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 \u20161\u2016\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c 0 A\u2016 \u2264 \u20161\u2016\n[PROOFSTEP]\nrefine' this _ _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\nthis :\n \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A), (\u2200 (m : Fin 0 \u2192 A), \u2016\u2191f m\u2016 \u2264 \u20161\u2016 * \u220f i : Fin 0, \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 \u20161\u2016\n\u22a2 \u2200 (m : Fin 0 \u2192 A), \u2016\u2191(ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c 0 A) m\u2016 \u2264 \u20161\u2016 * \u220f i : Fin 0, \u2016m i\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\n\u22a2 \u20161\u2016 \u2264 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c 0 A\u2016\n[PROOFSTEP]\nconvert ratio_le_op_norm (ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c 0 A) fun _ => (1 : A)\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2076 : Fintype \u03b9\ninst\u271d\u00b9\u2075 : Fintype \u03b9'\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\n\u22a2 \u20161\u2016 = \u2016\u2191(ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c 0 A) fun x => 1\u2016 / \u220f i : Fin 0, \u20161\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : NormOneClass A\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c n A\u2016 = 1\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b2 : NormOneClass A\nEi : Fin (Nat.succ Nat.zero) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedSpace \ud835\udd5c (Ei i)\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c Nat.zero A\u2016 = 1\n[PROOFSTEP]\nrw [norm_mkPiAlgebraFin_zero]\n[GOAL]\ncase zero\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b2 : NormOneClass A\nEi : Fin (Nat.succ Nat.zero) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedSpace \ud835\udd5c (Ei i)\n\u22a2 \u20161\u2016 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b2 : NormOneClass A\nn\u271d : \u2115\nEi : Fin (Nat.succ (Nat.succ n\u271d)) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedSpace \ud835\udd5c (Ei i)\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n\u271d) A\u2016 = 1\n[PROOFSTEP]\nrefine' le_antisymm norm_mkPiAlgebraFin_succ_le _\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b2 : NormOneClass A\nn\u271d : \u2115\nEi : Fin (Nat.succ (Nat.succ n\u271d)) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedSpace \ud835\udd5c (Ei i)\n\u22a2 1 \u2264 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n\u271d) A\u2016\n[PROOFSTEP]\nconvert ratio_le_op_norm (ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ _) A) fun _ => 1\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b2 : NormOneClass A\nn\u271d : \u2115\nEi : Fin (Nat.succ (Nat.succ n\u271d)) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedSpace \ud835\udd5c (Ei i)\n\u22a2 1 = \u2016\u2191(ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n\u271d) A) fun x => 1\u2016 / \u220f i : Fin (Nat.succ n\u271d), \u20161\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nz : G\nm : \u03b9 \u2192 \ud835\udd5c\n\u22a2 \u2016\u2191(MultilinearMap.mkPiRing \ud835\udd5c \u03b9 z) m\u2016 \u2264 \u2016z\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nsimp only [MultilinearMap.mkPiRing_apply, norm_smul, norm_prod, mul_comm]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nz : G\nm : \u03b9 \u2192 \ud835\udd5c\n\u22a2 \u2016z\u2016 * \u220f b : \u03b9, \u2016m b\u2016 \u2264 \u2016z\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nz : G\n\u22a2 \u2016z\u2016 \u2264\n \u2016MultilinearMap.mkContinuous (MultilinearMap.mkPiRing \ud835\udd5c \u03b9 z) \u2016z\u2016\n (_ : \u2200 (m : \u03b9 \u2192 \ud835\udd5c), \u2016\u2191(MultilinearMap.mkPiRing \ud835\udd5c \u03b9 z) m\u2016 \u2264 \u2016z\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\u2016\n[PROOFSTEP]\nsimpa using (ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z).le_op_norm fun _ => 1\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nz\u2081 z\u2082 : G\n\u22a2 ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z\u2081 = ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z\u2082 \u2194 z\u2081 = z\u2082\n[PROOFSTEP]\nrw [\u2190 toMultilinearMap_injective.eq_iff]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nz\u2081 z\u2082 : G\n\u22a2 (ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z\u2081).toMultilinearMap =\n (ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z\u2082).toMultilinearMap \u2194\n z\u2081 = z\u2082\n[PROOFSTEP]\nexact MultilinearMap.mkPiRing_eq_iff\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\n\u22a2 ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nx\u271d : \u03b9 \u2192 \ud835\udd5c\n\u22a2 \u2191(ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 0) x\u271d = \u21910 x\u271d\n[PROOFSTEP]\nrw [mkPiField_apply, smul_zero, ContinuousMultilinearMap.zero_apply]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nz : G\n\u22a2 ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z = 0 \u2194 z = 0\n[PROOFSTEP]\nrw [\u2190 mkPiField_zero, mkPiField_eq_iff]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nz z' : G\n\u22a2 (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) (z + z') =\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z + (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z'\n[PROOFSTEP]\next m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nz z' : G\nm : \u03b9 \u2192 \ud835\udd5c\n\u22a2 \u2191((fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) (z + z')) m =\n \u2191((fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z + (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z') m\n[PROOFSTEP]\nsimp [smul_add]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nz : G\n\u22a2 AddHom.toFun\n { toFun := fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z,\n map_add' :=\n (_ :\n \u2200 (z z' : G),\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) (z + z') =\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z +\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z') }\n (c \u2022 z) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z,\n map_add' :=\n (_ :\n \u2200 (z z' : G),\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) (z + z') =\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z +\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z') }\n z\n[PROOFSTEP]\next m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nz : G\nm : \u03b9 \u2192 \ud835\udd5c\n\u22a2 \u2191(AddHom.toFun\n { toFun := fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z,\n map_add' :=\n (_ :\n \u2200 (z z' : G),\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) (z + z') =\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z +\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z') }\n (c \u2022 z))\n m =\n \u2191(\u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z,\n map_add' :=\n (_ :\n \u2200 (z z' : G),\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) (z + z') =\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z +\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z') }\n z)\n m\n[PROOFSTEP]\nsimp [smul_smul, mul_comm]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nz : G\n\u22a2 (fun f => \u2191f fun i => 1)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z,\n map_add' :=\n (_ :\n \u2200 (z z' : G),\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) (z + z') =\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z +\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z') },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (z : G),\n AddHom.toFun\n { toFun := fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z,\n map_add' :=\n (_ :\n \u2200 (z z' : G),\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) (z + z') =\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z +\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z') }\n (c \u2022 z) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z,\n map_add' :=\n (_ :\n \u2200 (z z' : G),\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) (z + z') =\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z +\n (fun z => ContinuousMultilinearMap.mkPiField \ud835\udd5c \u03b9 z) z') }\n z) }.toAddHom\n z) =\n z\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192L[\ud835\udd5c] G'\ng\u2081 g\u2082 : ContinuousMultilinearMap \ud835\udd5c E G\n\u22a2 compContinuousMultilinearMap f (g\u2081 + g\u2082) = compContinuousMultilinearMap f g\u2081 + compContinuousMultilinearMap f g\u2082\n[PROOFSTEP]\next1\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192L[\ud835\udd5c] G'\ng\u2081 g\u2082 : ContinuousMultilinearMap \ud835\udd5c E G\nx\u271d : (i : \u03b9) \u2192 E i\n\u22a2 \u2191(compContinuousMultilinearMap f (g\u2081 + g\u2082)) x\u271d =\n \u2191(compContinuousMultilinearMap f g\u2081 + compContinuousMultilinearMap f g\u2082) x\u271d\n[PROOFSTEP]\napply f.map_add\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf : G \u2192L[\ud835\udd5c] G'\ng : ContinuousMultilinearMap \ud835\udd5c E G\n\u22a2 compContinuousMultilinearMap f (c \u2022 g) = c \u2022 compContinuousMultilinearMap f g\n[PROOFSTEP]\next1\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf : G \u2192L[\ud835\udd5c] G'\ng : ContinuousMultilinearMap \ud835\udd5c E G\nx\u271d : (i : \u03b9) \u2192 E i\n\u22a2 \u2191(compContinuousMultilinearMap f (c \u2022 g)) x\u271d = \u2191(c \u2022 compContinuousMultilinearMap f g) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192L[\ud835\udd5c] G'\ng : ContinuousMultilinearMap \ud835\udd5c E G\n\u22a2 \u2016\u2191(\u2191(LinearMap.mk\u2082 \ud835\udd5c compContinuousMultilinearMap\n (_ :\n \u2200 (f\u2081 f\u2082 : G \u2192L[\ud835\udd5c] G') (g : ContinuousMultilinearMap \ud835\udd5c E G),\n compContinuousMultilinearMap (f\u2081 + f\u2082) g = compContinuousMultilinearMap (f\u2081 + f\u2082) g)\n (_ :\n \u2200 (c : \ud835\udd5c) (f : G \u2192L[\ud835\udd5c] G') (g : ContinuousMultilinearMap \ud835\udd5c E G),\n compContinuousMultilinearMap (c \u2022 f) g = compContinuousMultilinearMap (c \u2022 f) g)\n (_ :\n \u2200 (f : G \u2192L[\ud835\udd5c] G') (g\u2081 g\u2082 : ContinuousMultilinearMap \ud835\udd5c E G),\n compContinuousMultilinearMap f (g\u2081 + g\u2082) =\n compContinuousMultilinearMap f g\u2081 + compContinuousMultilinearMap f g\u2082)\n (_ :\n \u2200 (c : \ud835\udd5c) (f : G \u2192L[\ud835\udd5c] G') (g : ContinuousMultilinearMap \ud835\udd5c E G),\n compContinuousMultilinearMap f (c \u2022 g) = c \u2022 compContinuousMultilinearMap f g))\n f)\n g\u2016 \u2264\n 1 * \u2016f\u2016 * \u2016g\u2016\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192L[\ud835\udd5c] G'\ng : ContinuousMultilinearMap \ud835\udd5c E G\n\u22a2 \u2016\u2191(\u2191(LinearMap.mk\u2082 \ud835\udd5c compContinuousMultilinearMap\n (_ :\n \u2200 (f\u2081 f\u2082 : G \u2192L[\ud835\udd5c] G') (g : ContinuousMultilinearMap \ud835\udd5c E G),\n compContinuousMultilinearMap (f\u2081 + f\u2082) g = compContinuousMultilinearMap (f\u2081 + f\u2082) g)\n (_ :\n \u2200 (c : \ud835\udd5c) (f : G \u2192L[\ud835\udd5c] G') (g : ContinuousMultilinearMap \ud835\udd5c E G),\n compContinuousMultilinearMap (c \u2022 f) g = compContinuousMultilinearMap (c \u2022 f) g)\n (_ :\n \u2200 (f : G \u2192L[\ud835\udd5c] G') (g\u2081 g\u2082 : ContinuousMultilinearMap \ud835\udd5c E G),\n compContinuousMultilinearMap f (g\u2081 + g\u2082) =\n compContinuousMultilinearMap f g\u2081 + compContinuousMultilinearMap f g\u2082)\n (_ :\n \u2200 (c : \ud835\udd5c) (f : G \u2192L[\ud835\udd5c] G') (g : ContinuousMultilinearMap \ud835\udd5c E G),\n compContinuousMultilinearMap f (c \u2022 g) = c \u2022 compContinuousMultilinearMap f g))\n f)\n g\u2016 \u2264\n \u2016f\u2016 * \u2016g\u2016\n[PROOFSTEP]\nexact f.norm_compContinuousMultilinearMap_le g\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : G \u2243L[\ud835\udd5c] G'\nsrc\u271d : (fun x => ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G') \u2191g :=\n \u2191(compContinuousMultilinearMapL \ud835\udd5c E G G') \u2191g\n\u22a2 Function.LeftInverse (\u2191(\u2191(compContinuousMultilinearMapL \ud835\udd5c E G' G) \u2191(ContinuousLinearEquiv.symm g))) src\u271d.toFun\n[PROOFSTEP]\nintro f\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : G \u2243L[\ud835\udd5c] G'\nsrc\u271d : (fun x => ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G') \u2191g :=\n \u2191(compContinuousMultilinearMapL \ud835\udd5c E G G') \u2191g\nf : ContinuousMultilinearMap \ud835\udd5c E G\n\u22a2 \u2191(\u2191(compContinuousMultilinearMapL \ud835\udd5c E G' G) \u2191(ContinuousLinearEquiv.symm g)) (AddHom.toFun src\u271d.toAddHom f) = f\n[PROOFSTEP]\next1 m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : G \u2243L[\ud835\udd5c] G'\nsrc\u271d : (fun x => ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G') \u2191g :=\n \u2191(compContinuousMultilinearMapL \ud835\udd5c E G G') \u2191g\nf : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2191(\u2191(\u2191(compContinuousMultilinearMapL \ud835\udd5c E G' G) \u2191(ContinuousLinearEquiv.symm g)) (AddHom.toFun src\u271d.toAddHom f)) m =\n \u2191f m\n[PROOFSTEP]\nsimp [compContinuousMultilinearMapL]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : G \u2243L[\ud835\udd5c] G'\nsrc\u271d : (fun x => ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G') \u2191g :=\n \u2191(compContinuousMultilinearMapL \ud835\udd5c E G G') \u2191g\n\u22a2 Function.RightInverse (\u2191(\u2191(compContinuousMultilinearMapL \ud835\udd5c E G' G) \u2191(ContinuousLinearEquiv.symm g))) src\u271d.toFun\n[PROOFSTEP]\nintro f\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : G \u2243L[\ud835\udd5c] G'\nsrc\u271d : (fun x => ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G') \u2191g :=\n \u2191(compContinuousMultilinearMapL \ud835\udd5c E G G') \u2191g\nf : ContinuousMultilinearMap \ud835\udd5c E G'\n\u22a2 AddHom.toFun src\u271d.toAddHom (\u2191(\u2191(compContinuousMultilinearMapL \ud835\udd5c E G' G) \u2191(ContinuousLinearEquiv.symm g)) f) = f\n[PROOFSTEP]\next1 m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : G \u2243L[\ud835\udd5c] G'\nsrc\u271d : (fun x => ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G') \u2191g :=\n \u2191(compContinuousMultilinearMapL \ud835\udd5c E G G') \u2191g\nf : ContinuousMultilinearMap \ud835\udd5c E G'\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2191(AddHom.toFun src\u271d.toAddHom (\u2191(\u2191(compContinuousMultilinearMapL \ud835\udd5c E G' G) \u2191(ContinuousLinearEquiv.symm g)) f)) m =\n \u2191f m\n[PROOFSTEP]\nsimp [compContinuousMultilinearMapL]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G'\nm : (i : \u03b9) \u2192 E i\nx y : G\n\u22a2 (fun x => \u2191(\u2191f x) m) (x + y) = (fun x => \u2191(\u2191f x) m) x + (fun x => \u2191(\u2191f x) m) y\n[PROOFSTEP]\nsimp only [map_add, ContinuousMultilinearMap.add_apply]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G'\nm : (i : \u03b9) \u2192 E i\nc : \ud835\udd5c\nx : G\n\u22a2 AddHom.toFun { toFun := fun x => \u2191(\u2191f x) m, map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) }\n (c \u2022 x) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := fun x => \u2191(\u2191f x) m, map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) } x\n[PROOFSTEP]\nsimp only [ContinuousMultilinearMap.smul_apply, map_smul, RingHom.id_apply]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G'\nm : (i : \u03b9) \u2192 E i\nx : G\n\u22a2 \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m, map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016\n[PROOFSTEP]\nrw [mul_right_comm]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G'\nm : (i : \u03b9) \u2192 E i\nx : G\n\u22a2 \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m, map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n \u2016f\u2016 * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nexact (f x).le_of_op_norm_le _ (f.le_op_norm x)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G'\ninst\u271d : DecidableEq \u03b9\nm : (i : \u03b9) \u2192 E i\ni : \u03b9\nx y : E i\n\u22a2 (fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m, map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016))\n (Function.update m i (x + y)) =\n (fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m, map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016))\n (Function.update m i x) +\n (fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m, map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016))\n (Function.update m i y)\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G'\ninst\u271d : DecidableEq \u03b9\nm : (i : \u03b9) \u2192 E i\ni : \u03b9\nx y : E i\nx\u271d : G\n\u22a2 \u2191((fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016))\n (Function.update m i (x + y)))\n x\u271d =\n \u2191((fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016))\n (Function.update m i x) +\n (fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016))\n (Function.update m i y))\n x\u271d\n[PROOFSTEP]\nsimp only [add_apply, ContinuousMultilinearMap.map_add, LinearMap.coe_mk, LinearMap.mkContinuous_apply, AddHom.coe_mk]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G'\ninst\u271d : DecidableEq \u03b9\nm : (i : \u03b9) \u2192 E i\ni : \u03b9\nc : \ud835\udd5c\nx : E i\n\u22a2 (fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m, map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016))\n (Function.update m i (c \u2022 x)) =\n c \u2022\n (fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m, map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016))\n (Function.update m i x)\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G'\ninst\u271d : DecidableEq \u03b9\nm : (i : \u03b9) \u2192 E i\ni : \u03b9\nc : \ud835\udd5c\nx : E i\nx\u271d : G\n\u22a2 \u2191((fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016))\n (Function.update m i (c \u2022 x)))\n x\u271d =\n \u2191(c \u2022\n (fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016))\n (Function.update m i x))\n x\u271d\n[PROOFSTEP]\nsimp only [coe_smul', ContinuousMultilinearMap.map_smul, LinearMap.coe_mk, LinearMap.mkContinuous_apply, Pi.smul_apply,\n AddHom.coe_mk]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G'\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191{\n toFun := fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016),\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n (fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016))\n (Function.update m i (x + y)) =\n (fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016))\n (Function.update m i x) +\n (fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016))\n (Function.update m i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n (fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016))\n (Function.update m i (c \u2022 x)) =\n c \u2022\n (fun m =>\n LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016))\n (Function.update m i x)) }\n m\u2016 \u2264\n \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\ndsimp only [MultilinearMap.coe_mk]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c E G'\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016LinearMap.mkContinuous\n {\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m, map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n { toFun := fun x => \u2191(\u2191f x) m,\n map_add' := (_ : \u2200 (x y : G), \u2191(\u2191f (x + y)) m = \u2191(\u2191f x) m + \u2191(\u2191f y) m) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c) (x : G), \u2191(\u2191f (c \u2022 x)) m = c \u2022 \u2191(\u2191f x) m) }\n x\u2016 \u2264\n (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u2016x\u2016)\u2016 \u2264\n \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrefine LinearMap.mkContinuous_norm_le _ (mul_nonneg (norm_nonneg f) (prod_nonneg fun i _ => norm_nonneg (m i))) _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : G \u2192\u2097\u1d62[\ud835\udd5c] G'\nf : ContinuousMultilinearMap \ud835\udd5c E G\n\u22a2 \u2016ContinuousLinearMap.compContinuousMultilinearMap (toContinuousLinearMap g) f\u2016 = \u2016f\u2016\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.compContinuousMultilinearMap_coe, LinearIsometry.coe_toContinuousLinearMap,\n LinearIsometry.norm_map, ContinuousMultilinearMap.norm_def, Function.comp_apply]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192\u2097[\ud835\udd5c] MultilinearMap \ud835\udd5c E G'\nC : \u211d\nH : \u2200 (x : G) (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016\nx y : G\n\u22a2 (fun x => mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016)) (x + y) =\n (fun x => mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016)) x +\n (fun x => mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016)) y\n[PROOFSTEP]\next1\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192\u2097[\ud835\udd5c] MultilinearMap \ud835\udd5c E G'\nC : \u211d\nH : \u2200 (x : G) (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016\nx y : G\nx\u271d : (i : \u03b9) \u2192 E i\n\u22a2 \u2191((fun x => mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n (x + y))\n x\u271d =\n \u2191((fun x => mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016)) x +\n (fun x => mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n y)\n x\u271d\n[PROOFSTEP]\nsimp only [_root_.map_add]\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192\u2097[\ud835\udd5c] MultilinearMap \ud835\udd5c E G'\nC : \u211d\nH : \u2200 (x : G) (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016\nx y : G\nx\u271d : (i : \u03b9) \u2192 E i\n\u22a2 \u2191(mkContinuous (\u2191f x + \u2191f y) (C * \u2016x + y\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x + \u2191f y) m\u2016 \u2264 C * \u2016x + y\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x\u271d =\n \u2191(mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016) +\n mkContinuous (\u2191f y) (C * \u2016y\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f y) m\u2016 \u2264 C * \u2016y\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192\u2097[\ud835\udd5c] MultilinearMap \ud835\udd5c E G'\nC : \u211d\nH : \u2200 (x : G) (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016\nc : \ud835\udd5c\nx : G\n\u22a2 AddHom.toFun\n {\n toFun := fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : G),\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x +\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n y) }\n (c \u2022 x) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n {\n toFun := fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : G),\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x +\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n y) }\n x\n[PROOFSTEP]\next1\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192\u2097[\ud835\udd5c] MultilinearMap \ud835\udd5c E G'\nC : \u211d\nH : \u2200 (x : G) (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016\nc : \ud835\udd5c\nx : G\nx\u271d : (i : \u03b9) \u2192 E i\n\u22a2 \u2191(AddHom.toFun\n {\n toFun := fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : G),\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x +\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n y) }\n (c \u2022 x))\n x\u271d =\n \u2191(\u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n {\n toFun := fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : G),\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x +\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n y) }\n x)\n x\u271d\n[PROOFSTEP]\nsimp only [SMulHomClass.map_smul]\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192\u2097[\ud835\udd5c] MultilinearMap \ud835\udd5c E G'\nC : \u211d\nH : \u2200 (x : G) (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016\nc : \ud835\udd5c\nx : G\nx\u271d : (i : \u03b9) \u2192 E i\n\u22a2 \u2191(mkContinuous (c \u2022 \u2191f x) (C * \u2016c \u2022 x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(c \u2022 \u2191f x) m\u2016 \u2264 C * \u2016c \u2022 x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x\u271d =\n \u2191(\u2191(RingHom.id \ud835\udd5c) c \u2022\n mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192\u2097[\ud835\udd5c] MultilinearMap \ud835\udd5c E G'\nC : \u211d\nH : \u2200 (x : G) (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016\nx : G\n\u22a2 \u2016\u2191{\n toAddHom :=\n {\n toFun := fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : G),\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x +\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n y) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (x : G),\n AddHom.toFun\n {\n toFun := fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : G),\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x +\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n y) }\n (c \u2022 x) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n {\n toFun := fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : G),\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x +\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n y) }\n x) }\n x\u2016 \u2264\n max C 0 * \u2016x\u2016\n[PROOFSTEP]\nrw [LinearMap.coe_mk, AddHom.coe_mk]\n -- Porting note: added\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192\u2097[\ud835\udd5c] MultilinearMap \ud835\udd5c E G'\nC : \u211d\nH : \u2200 (x : G) (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016\nx : G\n\u22a2 \u2016mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016)\u2016 \u2264 max C 0 * \u2016x\u2016\n[PROOFSTEP]\nexact ((f x).mkContinuous_norm_le' _).trans_eq <| by rw [max_mul_of_nonneg _ _ (norm_nonneg x), zero_mul]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192\u2097[\ud835\udd5c] MultilinearMap \ud835\udd5c E G'\nC : \u211d\nH : \u2200 (x : G) (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016\nx : G\n\u22a2 max (C * \u2016x\u2016) 0 = max C 0 * \u2016x\u2016\n[PROOFSTEP]\nrw [max_mul_of_nonneg _ _ (norm_nonneg x), zero_mul]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192\u2097[\ud835\udd5c] MultilinearMap \ud835\udd5c E G'\nC : \u211d\nH : \u2200 (x : G) (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n\u22a2 \u2016mkContinuousLinear f C H\u2016 \u2264 max C 0\n[PROOFSTEP]\ndsimp only [mkContinuousLinear]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : G \u2192\u2097[\ud835\udd5c] MultilinearMap \ud835\udd5c E G'\nC : \u211d\nH : \u2200 (x : G) (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n\u22a2 \u2016LinearMap.mkContinuous\n {\n toAddHom :=\n {\n toFun := fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016) (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : G),\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x +\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n y) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (x : G),\n AddHom.toFun\n {\n toFun := fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : G),\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x +\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n y) }\n (c \u2022 x) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n {\n toFun := fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : G),\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x +\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n y) }\n x) }\n (max C 0)\n (_ :\n \u2200 (x : G),\n \u2016\u2191{\n toAddHom :=\n {\n toFun := fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : G),\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x +\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n y) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (x : G),\n AddHom.toFun\n {\n toFun := fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : G),\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x +\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n y) }\n (c \u2022 x) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n {\n toFun := fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : G),\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n x +\n (fun x =>\n mkContinuous (\u2191f x) (C * \u2016x\u2016)\n (_ : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(\u2191f x) m\u2016 \u2264 C * \u2016x\u2016 * \u220f i : \u03b9, \u2016m i\u2016))\n y) }\n x) }\n x\u2016 \u2264\n max C 0 * \u2016x\u2016)\u2016 \u2264\n max C 0\n[PROOFSTEP]\nexact LinearMap.mkContinuous_norm_le _ (le_max_right _ _) _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E (MultilinearMap \ud835\udd5c E' G)\nC : \u211d\nH : \u2200 (m\u2081 : (i : \u03b9) \u2192 E i) (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m\u2081) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m\u2081 i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016\ninst\u271d : DecidableEq \u03b9\nm : (i : \u03b9) \u2192 E i\ni : \u03b9\nx y : E i\n\u22a2 (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i (x + y)) =\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i x) +\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i y)\n[PROOFSTEP]\next1\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E (MultilinearMap \ud835\udd5c E' G)\nC : \u211d\nH : \u2200 (m\u2081 : (i : \u03b9) \u2192 E i) (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m\u2081) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m\u2081 i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016\ninst\u271d : DecidableEq \u03b9\nm : (i : \u03b9) \u2192 E i\ni : \u03b9\nx y : E i\nx\u271d : (i : \u03b9') \u2192 E' i\n\u22a2 \u2191((fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i (x + y)))\n x\u271d =\n \u2191((fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i x) +\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i y))\n x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E (MultilinearMap \ud835\udd5c E' G)\nC : \u211d\nH : \u2200 (m\u2081 : (i : \u03b9) \u2192 E i) (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m\u2081) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m\u2081 i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016\ninst\u271d : DecidableEq \u03b9\nm : (i : \u03b9) \u2192 E i\ni : \u03b9\nc : \ud835\udd5c\nx : E i\n\u22a2 (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i (c \u2022 x)) =\n c \u2022\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i x)\n[PROOFSTEP]\next1\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E (MultilinearMap \ud835\udd5c E' G)\nC : \u211d\nH : \u2200 (m\u2081 : (i : \u03b9) \u2192 E i) (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m\u2081) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m\u2081 i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016\ninst\u271d : DecidableEq \u03b9\nm : (i : \u03b9) \u2192 E i\ni : \u03b9\nc : \ud835\udd5c\nx : E i\nx\u271d : (i : \u03b9') \u2192 E' i\n\u22a2 \u2191((fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i (c \u2022 x)))\n x\u271d =\n \u2191(c \u2022\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i x))\n x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E (MultilinearMap \ud835\udd5c E' G)\nC : \u211d\nH : \u2200 (m\u2081 : (i : \u03b9) \u2192 E i) (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m\u2081) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m\u2081 i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191{\n toFun := fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016),\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i (x + y)) =\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i x) +\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i (c \u2022 x)) =\n c \u2022\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i x)) }\n m\u2016 \u2264\n max C 0 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nsimp only [coe_mk]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E (MultilinearMap \ud835\udd5c E' G)\nC : \u211d\nH : \u2200 (m\u2081 : (i : \u03b9) \u2192 E i) (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m\u2081) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m\u2081 i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016)\u2016 \u2264\n max C 0 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrefine ((f m).mkContinuous_norm_le' _).trans_eq ?_\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E (MultilinearMap \ud835\udd5c E' G)\nC : \u211d\nH : \u2200 (m\u2081 : (i : \u03b9) \u2192 E i) (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m\u2081) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m\u2081 i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016\nm : (i : \u03b9) \u2192 E i\n\u22a2 max (C * \u220f i : \u03b9, \u2016m i\u2016) 0 = max C 0 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrw [max_mul_of_nonneg, zero_mul]\n[GOAL]\ncase hc\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E (MultilinearMap \ud835\udd5c E' G)\nC : \u211d\nH : \u2200 (m\u2081 : (i : \u03b9) \u2192 E i) (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m\u2081) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m\u2081 i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016\nm : (i : \u03b9) \u2192 E i\n\u22a2 0 \u2264 \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nexact prod_nonneg fun _ _ => norm_nonneg _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E (MultilinearMap \ud835\udd5c E' G)\nC : \u211d\nH : \u2200 (m\u2081 : (i : \u03b9) \u2192 E i) (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m\u2081) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m\u2081 i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016\n\u22a2 \u2016mkContinuousMultilinear f C H\u2016 \u2264 max C 0\n[PROOFSTEP]\ndsimp only [mkContinuousMultilinear]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E (MultilinearMap \ud835\udd5c E' G)\nC : \u211d\nH : \u2200 (m\u2081 : (i : \u03b9) \u2192 E i) (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m\u2081) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m\u2081 i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016\n\u22a2 \u2016mkContinuous\n {\n toFun := fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016),\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i (x + y)) =\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i x) +\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i (c \u2022 x)) =\n c \u2022\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i x)) }\n (max C 0)\n (_ :\n \u2200 (m : (i : \u03b9) \u2192 E i),\n \u2016\u2191{\n toFun := fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ : \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016),\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i (x + y)) =\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i),\n \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i x) +\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i),\n \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i), \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i (c \u2022 x)) =\n c \u2022\n (fun m =>\n mkContinuous (\u2191f m) (C * \u220f i : \u03b9, \u2016m i\u2016)\n (_ :\n \u2200 (m\u2082 : (i : \u03b9') \u2192 E' i),\n \u2016\u2191(\u2191f m) m\u2082\u2016 \u2264 (C * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m\u2082 i\u2016))\n (Function.update m i x)) }\n m\u2016 \u2264\n max C 0 * \u220f i : \u03b9, \u2016m i\u2016)\u2016 \u2264\n max C 0\n[PROOFSTEP]\nexact mkContinuous_norm_le _ (le_max_right _ _) _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\nf : (i : \u03b9) \u2192 E i \u2192L[\ud835\udd5c] E\u2081 i\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016g\u2016 * \u220f i : \u03b9, \u2016f i\u2016 * \u2016m i\u2016 = (\u2016g\u2016 * \u220f i : \u03b9, \u2016f i\u2016) * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrw [prod_mul_distrib, mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\nf : (i : \u03b9) \u2192 E i \u2192\u2097\u1d62[\ud835\udd5c] E\u2081 i\n\u22a2 \u2016compContinuousLinearMap g fun i => LinearIsometry.toContinuousLinearMap (f i)\u2016 \u2264 \u2016g\u2016\n[PROOFSTEP]\nrefine op_norm_le_bound _ (norm_nonneg _) fun m => ?_\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\nf : (i : \u03b9) \u2192 E i \u2192\u2097\u1d62[\ud835\udd5c] E\u2081 i\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191(compContinuousLinearMap g fun i => LinearIsometry.toContinuousLinearMap (f i)) m\u2016 \u2264 \u2016g\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\napply (g.le_op_norm _).trans _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\nf : (i : \u03b9) \u2192 E i \u2192\u2097\u1d62[\ud835\udd5c] E\u2081 i\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016g\u2016 * \u220f i : \u03b9, \u2016\u2191((fun i => \u2191((fun i => LinearIsometry.toContinuousLinearMap (f i)) i)) i) (m i)\u2016 \u2264\n \u2016g\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.coe_coe, LinearIsometry.coe_toContinuousLinearMap, LinearIsometry.norm_map]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\nf : (i : \u03b9) \u2192 E i \u2192\u2097\u1d62[\ud835\udd5c] E\u2081 i\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016g\u2016 * \u220f x : \u03b9, \u2016m x\u2016 \u2264 \u2016g\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\nf : (i : \u03b9) \u2192 E i \u2243\u2097\u1d62[\ud835\udd5c] E\u2081 i\n\u22a2 \u2016compContinuousLinearMap g fun i => \u2191(ContinuousLinearEquiv.mk (f i).toLinearEquiv)\u2016 = \u2016g\u2016\n[PROOFSTEP]\napply le_antisymm (g.norm_compContinuous_linearIsometry_le fun i => (f i).toLinearIsometry)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\nf : (i : \u03b9) \u2192 E i \u2243\u2097\u1d62[\ud835\udd5c] E\u2081 i\n\u22a2 \u2016g\u2016 \u2264\n \u2016compContinuousLinearMap g fun i =>\n LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (f i))\u2016\n[PROOFSTEP]\nhave :\n g =\n (g.compContinuousLinearMap fun i => (f i : E i \u2192L[\ud835\udd5c] E\u2081 i)).compContinuousLinearMap fun i =>\n ((f i).symm : E\u2081 i \u2192L[\ud835\udd5c] E i) :=\n by\n ext1 m\n simp only [compContinuousLinearMap_apply, LinearIsometryEquiv.coe_coe'', LinearIsometryEquiv.apply_symm_apply]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\nf : (i : \u03b9) \u2192 E i \u2243\u2097\u1d62[\ud835\udd5c] E\u2081 i\n\u22a2 g =\n compContinuousLinearMap (compContinuousLinearMap g fun i => \u2191(ContinuousLinearEquiv.mk (f i).toLinearEquiv))\n fun i => \u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv)\n[PROOFSTEP]\next1 m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\nf : (i : \u03b9) \u2192 E i \u2243\u2097\u1d62[\ud835\udd5c] E\u2081 i\nm : (i : \u03b9) \u2192 E\u2081 i\n\u22a2 \u2191g m =\n \u2191(compContinuousLinearMap (compContinuousLinearMap g fun i => \u2191(ContinuousLinearEquiv.mk (f i).toLinearEquiv))\n fun i => \u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv))\n m\n[PROOFSTEP]\nsimp only [compContinuousLinearMap_apply, LinearIsometryEquiv.coe_coe'', LinearIsometryEquiv.apply_symm_apply]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\nf : (i : \u03b9) \u2192 E i \u2243\u2097\u1d62[\ud835\udd5c] E\u2081 i\nthis :\n g =\n compContinuousLinearMap (compContinuousLinearMap g fun i => \u2191(ContinuousLinearEquiv.mk (f i).toLinearEquiv))\n fun i => \u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv)\n\u22a2 \u2016g\u2016 \u2264\n \u2016compContinuousLinearMap g fun i =>\n LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (f i))\u2016\n[PROOFSTEP]\nconv_lhs => rw [this]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\nf : (i : \u03b9) \u2192 E i \u2243\u2097\u1d62[\ud835\udd5c] E\u2081 i\nthis :\n g =\n compContinuousLinearMap (compContinuousLinearMap g fun i => \u2191(ContinuousLinearEquiv.mk (f i).toLinearEquiv))\n fun i => \u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv)\n| \u2016g\u2016\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\nf : (i : \u03b9) \u2192 E i \u2243\u2097\u1d62[\ud835\udd5c] E\u2081 i\nthis :\n g =\n compContinuousLinearMap (compContinuousLinearMap g fun i => \u2191(ContinuousLinearEquiv.mk (f i).toLinearEquiv))\n fun i => \u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv)\n| \u2016g\u2016\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\nf : (i : \u03b9) \u2192 E i \u2243\u2097\u1d62[\ud835\udd5c] E\u2081 i\nthis :\n g =\n compContinuousLinearMap (compContinuousLinearMap g fun i => \u2191(ContinuousLinearEquiv.mk (f i).toLinearEquiv))\n fun i => \u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv)\n| \u2016g\u2016\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\nf : (i : \u03b9) \u2192 E i \u2243\u2097\u1d62[\ud835\udd5c] E\u2081 i\nthis :\n g =\n compContinuousLinearMap (compContinuousLinearMap g fun i => \u2191(ContinuousLinearEquiv.mk (f i).toLinearEquiv))\n fun i => \u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv)\n\u22a2 \u2016compContinuousLinearMap (compContinuousLinearMap g fun i => \u2191(ContinuousLinearEquiv.mk (f i).toLinearEquiv)) fun i =>\n \u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv)\u2016 \u2264\n \u2016compContinuousLinearMap g fun i =>\n LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (f i))\u2016\n[PROOFSTEP]\napply\n (g.compContinuousLinearMap fun i => (f i : E i \u2192L[\ud835\udd5c] E\u2081 i)).norm_compContinuous_linearIsometry_le fun i =>\n (f i).symm.toLinearIsometry\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : (i : \u03b9) \u2192 E i \u2243L[\ud835\udd5c] E\u2081 i\nsrc\u271d : ContinuousMultilinearMap \ud835\udd5c (fun i => E\u2081 i) G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E i) G :=\n compContinuousLinearMapL fun i => \u2191(f i)\n\u22a2 Function.LeftInverse (\u2191(compContinuousLinearMapL fun i => \u2191(ContinuousLinearEquiv.symm (f i)))) src\u271d.toFun\n[PROOFSTEP]\nintro g\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : (i : \u03b9) \u2192 E i \u2243L[\ud835\udd5c] E\u2081 i\nsrc\u271d : ContinuousMultilinearMap \ud835\udd5c (fun i => E\u2081 i) G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E i) G :=\n compContinuousLinearMapL fun i => \u2191(f i)\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\n\u22a2 \u2191(compContinuousLinearMapL fun i => \u2191(ContinuousLinearEquiv.symm (f i))) (AddHom.toFun src\u271d.toAddHom g) = g\n[PROOFSTEP]\next1 m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : (i : \u03b9) \u2192 E i \u2243L[\ud835\udd5c] E\u2081 i\nsrc\u271d : ContinuousMultilinearMap \ud835\udd5c (fun i => E\u2081 i) G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E i) G :=\n compContinuousLinearMapL fun i => \u2191(f i)\ng : ContinuousMultilinearMap \ud835\udd5c E\u2081 G\nm : (i : \u03b9) \u2192 E\u2081 i\n\u22a2 \u2191(\u2191(compContinuousLinearMapL fun i => \u2191(ContinuousLinearEquiv.symm (f i))) (AddHom.toFun src\u271d.toAddHom g)) m = \u2191g m\n[PROOFSTEP]\nsimp only [LinearMap.toFun_eq_coe, ContinuousLinearMap.coe_coe, compContinuousLinearMapL_apply,\n compContinuousLinearMap_apply, ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.apply_symm_apply]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : (i : \u03b9) \u2192 E i \u2243L[\ud835\udd5c] E\u2081 i\nsrc\u271d : ContinuousMultilinearMap \ud835\udd5c (fun i => E\u2081 i) G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E i) G :=\n compContinuousLinearMapL fun i => \u2191(f i)\n\u22a2 Function.RightInverse (\u2191(compContinuousLinearMapL fun i => \u2191(ContinuousLinearEquiv.symm (f i)))) src\u271d.toFun\n[PROOFSTEP]\nintro g\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : (i : \u03b9) \u2192 E i \u2243L[\ud835\udd5c] E\u2081 i\nsrc\u271d : ContinuousMultilinearMap \ud835\udd5c (fun i => E\u2081 i) G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E i) G :=\n compContinuousLinearMapL fun i => \u2191(f i)\ng : ContinuousMultilinearMap \ud835\udd5c E G\n\u22a2 AddHom.toFun src\u271d.toAddHom (\u2191(compContinuousLinearMapL fun i => \u2191(ContinuousLinearEquiv.symm (f i))) g) = g\n[PROOFSTEP]\next1 m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : (i : \u03b9) \u2192 E i \u2243L[\ud835\udd5c] E\u2081 i\nsrc\u271d : ContinuousMultilinearMap \ud835\udd5c (fun i => E\u2081 i) G \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E i) G :=\n compContinuousLinearMapL fun i => \u2191(f i)\ng : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2191(AddHom.toFun src\u271d.toAddHom (\u2191(compContinuousLinearMapL fun i => \u2191(ContinuousLinearEquiv.symm (f i))) g)) m = \u2191g m\n[PROOFSTEP]\nsimp only [compContinuousLinearMapL_apply, LinearMap.toFun_eq_coe, ContinuousLinearMap.coe_coe,\n compContinuousLinearMap_apply, ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.symm_apply_apply]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G\nm : (i : Fin (Nat.succ n)) \u2192 Ei i\n\u22a2 \u2016f\u2016 * \u2016m 0\u2016 * \u220f i : Fin n, \u2016tail m i\u2016 = \u2016f\u2016 * (\u2016m 0\u2016 * \u220f i : Fin n, \u2016tail m i\u2016)\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G\nm : (i : Fin (Nat.succ n)) \u2192 Ei i\n\u22a2 \u2016f\u2016 * (\u2016m 0\u2016 * \u220f i : Fin n, \u2016tail m i\u2016) = \u2016f\u2016 * \u220f i : Fin (Nat.succ n), \u2016m i\u2016\n[PROOFSTEP]\nrw [prod_univ_succ]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G\nm : (i : Fin (Nat.succ n)) \u2192 Ei i\n\u22a2 \u2016f\u2016 * (\u2016m 0\u2016 * \u220f i : Fin n, \u2016tail m i\u2016) = \u2016f\u2016 * (\u2016m 0\u2016 * \u220f i : Fin n, \u2016m (succ i)\u2016)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)\nm : (i : Fin (Nat.succ n)) \u2192 Ei i\n\u22a2 \u2016f\u2016 * ((\u220f i : Fin n, \u2016init m i\u2016) * \u2016m (last n)\u2016) = \u2016f\u2016 * \u220f i : Fin (Nat.succ n), \u2016m i\u2016\n[PROOFSTEP]\nrw [prod_univ_castSucc]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)\nm : (i : Fin (Nat.succ n)) \u2192 Ei i\n\u22a2 \u2016f\u2016 * ((\u220f i : Fin n, \u2016init m i\u2016) * \u2016m (last n)\u2016) = \u2016f\u2016 * ((\u220f i : Fin n, \u2016m (castSucc i)\u2016) * \u2016m (last n)\u2016)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\nx : Ei 0\nm : (i : Fin n) \u2192 Ei (succ i)\n\u22a2 \u2016f\u2016 * \u220f i : Fin (n + 1), \u2016Fin.cons x m i\u2016 = \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016\n[PROOFSTEP]\nrw [prod_univ_succ]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\nx : Ei 0\nm : (i : Fin n) \u2192 Ei (succ i)\n\u22a2 \u2016f\u2016 * (\u2016Fin.cons x m 0\u2016 * \u220f i : Fin n, \u2016Fin.cons x m (succ i)\u2016) = \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\nm : (i : Fin n) \u2192 Ei (castSucc i)\nx : Ei (last n)\n\u22a2 \u2016f\u2016 * \u220f i : Fin (n + 1), \u2016snoc m x i\u2016 = (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016\n[PROOFSTEP]\nrw [prod_univ_castSucc]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\nm : (i : Fin n) \u2192 Ei (castSucc i)\nx : Ei (last n)\n\u22a2 \u2016f\u2016 * ((\u220f i : Fin n, \u2016snoc m x (castSucc i)\u2016) * \u2016snoc m x (last n)\u2016) = (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G\nm : (i : Fin (Nat.succ n)) \u2192 Ei i\n\u22a2 \u2016\u2191(LinearMap.uncurryLeft (LinearMap.comp toMultilinearMapLinear \u2191f)) m\u2016 \u2264 \u2016f\u2016 * \u220f i : Fin (Nat.succ n), \u2016m i\u2016\n[PROOFSTEP]\nexact ContinuousLinearMap.norm_map_tail_le f m\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\nx y : Ei 0\n\u22a2 (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n x +\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n y\n[PROOFSTEP]\next m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\nx y : Ei 0\nm : (i : Fin n) \u2192 Ei (succ i)\n\u22a2 \u2191((fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n (x + y))\n m =\n \u2191((fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n x +\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n y)\n m\n[PROOFSTEP]\nexact f.cons_add m x y\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\nc : \ud835\udd5c\nx : Ei 0\n\u22a2 AddHom.toFun\n {\n toFun := fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : Ei 0),\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n x +\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n y) }\n (c \u2022 x) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n {\n toFun := fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : Ei 0),\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n x +\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n y) }\n x\n[PROOFSTEP]\next m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\nc : \ud835\udd5c\nx : Ei 0\nm : (i : Fin n) \u2192 Ei (succ i)\n\u22a2 \u2191(AddHom.toFun\n {\n toFun := fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : Ei 0),\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n x +\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n y) }\n (c \u2022 x))\n m =\n \u2191(\u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n {\n toFun := fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : Ei 0),\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)),\n \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n x +\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)),\n \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n y) }\n x)\n m\n[PROOFSTEP]\nexact f.cons_smul m c x\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\nx : Ei 0\n\u22a2 \u2016\u2191{\n toAddHom :=\n {\n toFun := fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : Ei 0),\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)),\n \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)),\n \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n x +\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)),\n \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n y) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (x : Ei 0),\n AddHom.toFun\n {\n toFun := fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : Ei 0),\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x)\n (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)),\n \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x)\n (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)),\n \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n x +\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x)\n (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)),\n \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n y) }\n (c \u2022 x) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n {\n toFun := fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)),\n \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016),\n map_add' :=\n (_ :\n \u2200 (x y : Ei 0),\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x)\n (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)),\n \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n (x + y) =\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x)\n (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)),\n \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n x +\n (fun x =>\n MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x)\n (\u2016f\u2016 * \u2016x\u2016)\n (_ :\n \u2200 (m : (i : Fin n) \u2192 Ei (succ i)),\n \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016))\n y) }\n x) }\n x\u2016 \u2264\n \u2016f\u2016 * \u2016x\u2016\n[PROOFSTEP]\nrw [LinearMap.coe_mk, AddHom.coe_mk]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\nx : Ei 0\n\u22a2 \u2016MultilinearMap.mkContinuous (\u2191(MultilinearMap.curryLeft f.toMultilinearMap) x) (\u2016f\u2016 * \u2016x\u2016)\n (_ : \u2200 (m : (i : Fin n) \u2192 Ei (succ i)), \u2016\u2191f (Fin.cons x m)\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 * \u220f i : Fin n, \u2016m i\u2016)\u2016 \u2264\n \u2016f\u2016 * \u2016x\u2016\n[PROOFSTEP]\nexact MultilinearMap.mkContinuous_norm_le _ (mul_nonneg (norm_nonneg _) (norm_nonneg _)) _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G\n\u22a2 curryLeft (uncurryLeft f) = f\n[PROOFSTEP]\next m x\n[GOAL]\ncase h.H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G\nm : Ei 0\nx : (i : Fin n) \u2192 Ei (succ i)\n\u22a2 \u2191(\u2191(curryLeft (uncurryLeft f)) m) x = \u2191(\u2191f m) x\n[PROOFSTEP]\nsimp only [tail_cons, ContinuousLinearMap.uncurryLeft_apply, ContinuousMultilinearMap.curryLeft_apply]\n[GOAL]\ncase h.H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G\nm : Ei 0\nx : (i : Fin n) \u2192 Ei (succ i)\n\u22a2 \u2191(\u2191f (Fin.cons m x 0)) x = \u2191(\u2191f m) x\n[PROOFSTEP]\nrw [cons_zero]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u2081 f\u2082 : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G\n\u22a2 ContinuousLinearMap.uncurryLeft (f\u2081 + f\u2082) = ContinuousLinearMap.uncurryLeft f\u2081 + ContinuousLinearMap.uncurryLeft f\u2082\n[PROOFSTEP]\next m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u2081 f\u2082 : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G\nm : (i : Fin (Nat.succ n)) \u2192 Ei i\n\u22a2 \u2191(ContinuousLinearMap.uncurryLeft (f\u2081 + f\u2082)) m =\n \u2191(ContinuousLinearMap.uncurryLeft f\u2081 + ContinuousLinearMap.uncurryLeft f\u2082) m\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G\n\u22a2 AddHom.toFun\n { toFun := ContinuousLinearMap.uncurryLeft,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G),\n ContinuousLinearMap.uncurryLeft (f\u2081 + f\u2082) =\n ContinuousLinearMap.uncurryLeft f\u2081 + ContinuousLinearMap.uncurryLeft f\u2082) }\n (c \u2022 f) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := ContinuousLinearMap.uncurryLeft,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G),\n ContinuousLinearMap.uncurryLeft (f\u2081 + f\u2082) =\n ContinuousLinearMap.uncurryLeft f\u2081 + ContinuousLinearMap.uncurryLeft f\u2082) }\n f\n[PROOFSTEP]\next m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G\nm : (i : Fin (Nat.succ n)) \u2192 Ei i\n\u22a2 \u2191(AddHom.toFun\n { toFun := ContinuousLinearMap.uncurryLeft,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G),\n ContinuousLinearMap.uncurryLeft (f\u2081 + f\u2082) =\n ContinuousLinearMap.uncurryLeft f\u2081 + ContinuousLinearMap.uncurryLeft f\u2082) }\n (c \u2022 f))\n m =\n \u2191(\u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := ContinuousLinearMap.uncurryLeft,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G),\n ContinuousLinearMap.uncurryLeft (f\u2081 + f\u2082) =\n ContinuousLinearMap.uncurryLeft f\u2081 + ContinuousLinearMap.uncurryLeft f\u2082) }\n f)\n m\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G\n\u22a2 \u2016\u2191{\n toLinearMap :=\n {\n toAddHom :=\n { toFun := ContinuousLinearMap.uncurryLeft,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G),\n ContinuousLinearMap.uncurryLeft (f\u2081 + f\u2082) =\n ContinuousLinearMap.uncurryLeft f\u2081 + ContinuousLinearMap.uncurryLeft f\u2082) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G),\n AddHom.toFun\n { toFun := ContinuousLinearMap.uncurryLeft,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G),\n ContinuousLinearMap.uncurryLeft (f\u2081 + f\u2082) =\n ContinuousLinearMap.uncurryLeft f\u2081 + ContinuousLinearMap.uncurryLeft f\u2082) }\n (c \u2022 f) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := ContinuousLinearMap.uncurryLeft,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G),\n ContinuousLinearMap.uncurryLeft (f\u2081 + f\u2082) =\n ContinuousLinearMap.uncurryLeft f\u2081 + ContinuousLinearMap.uncurryLeft f\u2082) }\n f) },\n invFun := curryLeft,\n left_inv :=\n (_ :\n \u2200 (f : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G),\n curryLeft (ContinuousLinearMap.uncurryLeft f) = f),\n right_inv :=\n (_ : \u2200 (f : ContinuousMultilinearMap \ud835\udd5c Ei G), ContinuousLinearMap.uncurryLeft (curryLeft f) = f) }\n f\u2016 \u2264\n \u2016f\u2016\n[PROOFSTEP]\nsimp only [LinearEquiv.coe_mk]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G\n\u22a2 \u2016ContinuousLinearMap.uncurryLeft f\u2016 \u2264 \u2016f\u2016\n[PROOFSTEP]\nexact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\n\u22a2 \u2016\u2191(LinearEquiv.symm\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := ContinuousLinearMap.uncurryLeft,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G),\n ContinuousLinearMap.uncurryLeft (f\u2081 + f\u2082) =\n ContinuousLinearMap.uncurryLeft f\u2081 + ContinuousLinearMap.uncurryLeft f\u2082) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G),\n AddHom.toFun\n { toFun := ContinuousLinearMap.uncurryLeft,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G),\n ContinuousLinearMap.uncurryLeft (f\u2081 + f\u2082) =\n ContinuousLinearMap.uncurryLeft f\u2081 + ContinuousLinearMap.uncurryLeft f\u2082) }\n (c \u2022 f) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := ContinuousLinearMap.uncurryLeft,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G),\n ContinuousLinearMap.uncurryLeft (f\u2081 + f\u2082) =\n ContinuousLinearMap.uncurryLeft f\u2081 + ContinuousLinearMap.uncurryLeft f\u2082) }\n f) },\n invFun := curryLeft,\n left_inv :=\n (_ :\n \u2200 (f : Ei 0 \u2192L[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (succ i)) G),\n curryLeft (ContinuousLinearMap.uncurryLeft f) = f),\n right_inv :=\n (_ : \u2200 (f : ContinuousMultilinearMap \ud835\udd5c Ei G), ContinuousLinearMap.uncurryLeft (curryLeft f) = f) })\n f\u2016 \u2264\n \u2016f\u2016\n[PROOFSTEP]\nsimp only [LinearEquiv.coe_symm_mk]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\n\u22a2 \u2016curryLeft f\u2016 \u2264 \u2016f\u2016\n[PROOFSTEP]\nexact LinearMap.mkContinuous_norm_le _ (norm_nonneg f) _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)\ninst\u271d : DecidableEq (Fin n)\nm : (i : Fin n) \u2192 Ei (castSucc i)\ni : Fin n\nx y : Ei (castSucc i)\n\u22a2 (fun m => \u2191(\u2191f m)) (update m i (x + y)) = (fun m => \u2191(\u2191f m)) (update m i x) + (fun m => \u2191(\u2191f m)) (update m i y)\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)\ninst\u271d : DecidableEq (Fin n)\nm : (i : Fin n) \u2192 Ei (castSucc i)\ni : Fin n\nc : \ud835\udd5c\nx : Ei (castSucc i)\n\u22a2 (fun m => \u2191(\u2191f m)) (update m i (c \u2022 x)) = c \u2022 (fun m => \u2191(\u2191f m)) (update m i x)\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\ninst\u271d : DecidableEq (Fin n)\nm : (i : Fin n) \u2192 Ei (castSucc i)\ni : Fin n\nx y : Ei (castSucc i)\n\u22a2 (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016))\n (update m i (x + y)) =\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016))\n (update m i x) +\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016))\n (update m i y)\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\ninst\u271d : DecidableEq (Fin n)\nm : (i : Fin n) \u2192 Ei (castSucc i)\ni : Fin n\nx y : Ei (castSucc i)\n\u22a2 LinearMap.mkContinuous\n (\u2191(MultilinearMap.curryRight f.toMultilinearMap) (update m i x) +\n \u2191(MultilinearMap.curryRight f.toMultilinearMap) (update m i y))\n (\u2016f\u2016 * \u220f x_1 : Fin n, \u2016update m i (x + y) x_1\u2016)\n (_ :\n \u2200 (x_1 : Ei (last n)),\n \u2016\u2191(\u2191(MultilinearMap.curryRight f.toMultilinearMap) (update m i x) +\n \u2191(MultilinearMap.curryRight f.toMultilinearMap) (update m i y))\n x_1\u2016 \u2264\n (\u2016f\u2016 * \u220f x_2 : Fin n, \u2016update m i (x + y) x_2\u2016) * \u2016x_1\u2016) =\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) (update m i x))\n (\u2016f\u2016 * \u220f i_1 : Fin n, \u2016update m i x i_1\u2016)\n (_ :\n \u2200 (x_1 : Ei (last n)), \u2016\u2191f (snoc (update m i x) x_1)\u2016 \u2264 (\u2016f\u2016 * \u220f i_1 : Fin n, \u2016update m i x i_1\u2016) * \u2016x_1\u2016) +\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) (update m i y))\n (\u2016f\u2016 * \u220f i_1 : Fin n, \u2016update m i y i_1\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc (update m i y) x)\u2016 \u2264 (\u2016f\u2016 * \u220f i_1 : Fin n, \u2016update m i y i_1\u2016) * \u2016x\u2016)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\ninst\u271d : DecidableEq (Fin n)\nm : (i : Fin n) \u2192 Ei (castSucc i)\ni : Fin n\nc : \ud835\udd5c\nx : Ei (castSucc i)\n\u22a2 (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016))\n (update m i (c \u2022 x)) =\n c \u2022\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016))\n (update m i x)\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\ninst\u271d : DecidableEq (Fin n)\nm : (i : Fin n) \u2192 Ei (castSucc i)\ni : Fin n\nc : \ud835\udd5c\nx : Ei (castSucc i)\n\u22a2 LinearMap.mkContinuous (c \u2022 \u2191(MultilinearMap.curryRight f.toMultilinearMap) (update m i x))\n (\u2016f\u2016 * \u220f x_1 : Fin n, \u2016update m i (c \u2022 x) x_1\u2016)\n (_ :\n \u2200 (x_1 : Ei (last n)),\n \u2016\u2191(c \u2022 \u2191(MultilinearMap.curryRight f.toMultilinearMap) (update m i x)) x_1\u2016 \u2264\n (\u2016f\u2016 * \u220f x_2 : Fin n, \u2016update m i (c \u2022 x) x_2\u2016) * \u2016x_1\u2016) =\n c \u2022\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) (update m i x))\n (\u2016f\u2016 * \u220f i_1 : Fin n, \u2016update m i x i_1\u2016)\n (_ : \u2200 (x_1 : Ei (last n)), \u2016\u2191f (snoc (update m i x) x_1)\u2016 \u2264 (\u2016f\u2016 * \u220f i_1 : Fin n, \u2016update m i x i_1\u2016) * \u2016x_1\u2016)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\nf' : MultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G) :=\n {\n toFun := fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016),\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq (Fin n)] (m : (i : Fin n) \u2192 Ei (castSucc i)) (i : Fin n) (x y : Ei (castSucc i)),\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016))\n (update m i (x + y)) =\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016))\n (update m i x) +\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016))\n (update m i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq (Fin n)] (m : (i : Fin n) \u2192 Ei (castSucc i)) (i : Fin n) (c : \ud835\udd5c) (x : Ei (castSucc i)),\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016))\n (update m i (c \u2022 x)) =\n c \u2022\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016))\n (update m i x)) }\nm : (i : Fin n) \u2192 Ei (castSucc i)\n\u22a2 \u2016\u2191f' m\u2016 \u2264 \u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016\n[PROOFSTEP]\nsimp only [MultilinearMap.coe_mk]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\nf' : MultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G) :=\n {\n toFun := fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016),\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq (Fin n)] (m : (i : Fin n) \u2192 Ei (castSucc i)) (i : Fin n) (x y : Ei (castSucc i)),\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016))\n (update m i (x + y)) =\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016))\n (update m i x) +\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016))\n (update m i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq (Fin n)] (m : (i : Fin n) \u2192 Ei (castSucc i)) (i : Fin n) (c : \ud835\udd5c) (x : Ei (castSucc i)),\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016))\n (update m i (c \u2022 x)) =\n c \u2022\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016))\n (update m i x)) }\nm : (i : Fin n) \u2192 Ei (castSucc i)\n\u22a2 \u2016LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\n (_ : \u2200 (x : Ei (last n)), \u2016\u2191f (snoc m x)\u2016 \u2264 (\u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016) * \u2016x\u2016)\u2016 \u2264\n \u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016\n[PROOFSTEP]\nexact LinearMap.mkContinuous_norm_le _ (mul_nonneg (norm_nonneg _) (prod_nonneg fun _ _ => norm_nonneg _)) _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)\n\u22a2 curryRight (uncurryRight f) = f\n[PROOFSTEP]\next m x\n[GOAL]\ncase H.h\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)\nm : (i : Fin n) \u2192 Ei (castSucc i)\nx : Ei (last n)\n\u22a2 \u2191(\u2191(curryRight (uncurryRight f)) m) x = \u2191(\u2191f m) x\n[PROOFSTEP]\nsimp only [snoc_last, ContinuousMultilinearMap.curryRight_apply, ContinuousMultilinearMap.uncurryRight_apply]\n[GOAL]\ncase H.h\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)\nm : (i : Fin n) \u2192 Ei (castSucc i)\nx : Ei (last n)\n\u22a2 \u2191(\u2191f (init (snoc m x))) x = \u2191(\u2191f m) x\n[PROOFSTEP]\nrw [init_snoc]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\n\u22a2 uncurryRight (curryRight f) = f\n[PROOFSTEP]\next m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\nm : (i : Fin (Nat.succ n)) \u2192 Ei i\n\u22a2 \u2191(uncurryRight (curryRight f)) m = \u2191f m\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u2081 f\u2082 : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)\n\u22a2 uncurryRight (f\u2081 + f\u2082) = uncurryRight f\u2081 + uncurryRight f\u2082\n[PROOFSTEP]\next m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u2081 f\u2082 : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)\nm : (i : Fin (Nat.succ n)) \u2192 Ei i\n\u22a2 \u2191(uncurryRight (f\u2081 + f\u2082)) m = \u2191(uncurryRight f\u2081 + uncurryRight f\u2082) m\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)\n\u22a2 AddHom.toFun\n { toFun := uncurryRight,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)),\n uncurryRight (f\u2081 + f\u2082) = uncurryRight f\u2081 + uncurryRight f\u2082) }\n (c \u2022 f) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := uncurryRight,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)),\n uncurryRight (f\u2081 + f\u2082) = uncurryRight f\u2081 + uncurryRight f\u2082) }\n f\n[PROOFSTEP]\next m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)\nm : (i : Fin (Nat.succ n)) \u2192 Ei i\n\u22a2 \u2191(AddHom.toFun\n { toFun := uncurryRight,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)),\n uncurryRight (f\u2081 + f\u2082) = uncurryRight f\u2081 + uncurryRight f\u2082) }\n (c \u2022 f))\n m =\n \u2191(\u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := uncurryRight,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)),\n uncurryRight (f\u2081 + f\u2082) = uncurryRight f\u2081 + uncurryRight f\u2082) }\n f)\n m\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)\n\u22a2 \u2016\u2191{\n toLinearMap :=\n {\n toAddHom :=\n { toFun := uncurryRight,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)),\n uncurryRight (f\u2081 + f\u2082) = uncurryRight f\u2081 + uncurryRight f\u2082) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)),\n AddHom.toFun\n { toFun := uncurryRight,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)),\n uncurryRight (f\u2081 + f\u2082) = uncurryRight f\u2081 + uncurryRight f\u2082) }\n (c \u2022 f) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := uncurryRight,\n map_add' :=\n (_ :\n \u2200\n (f\u2081 f\u2082 :\n ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)),\n uncurryRight (f\u2081 + f\u2082) = uncurryRight f\u2081 + uncurryRight f\u2082) }\n f) },\n invFun := curryRight,\n left_inv :=\n (_ :\n \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)),\n curryRight (uncurryRight f) = f),\n right_inv := (_ : \u2200 (f : ContinuousMultilinearMap \ud835\udd5c Ei G), uncurryRight (curryRight f) = f) }\n f\u2016 \u2264\n \u2016f\u2016\n[PROOFSTEP]\nsimp only [uncurryRight, LinearEquiv.coe_mk]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)\n\u22a2 \u2016MultilinearMap.mkContinuous\n (MultilinearMap.uncurryRight\n { toFun := fun m => \u2191(\u2191f m),\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq (Fin n)] (m : (i : Fin n) \u2192 Ei (castSucc i)) (i : Fin n) (x y : Ei (castSucc i)),\n \u2191(\u2191f (update m i (x + y))) = \u2191(\u2191f (update m i x)) + \u2191(\u2191f (update m i y))),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq (Fin n)] (m : (i : Fin n) \u2192 Ei (castSucc i)) (i : Fin n) (c : \ud835\udd5c)\n (x : Ei (castSucc i)), \u2191(\u2191f (update m i (c \u2022 x))) = c \u2022 \u2191(\u2191f (update m i x))) })\n \u2016f\u2016\n (_ :\n \u2200 (m : (i : Fin (Nat.succ n)) \u2192 Ei i), \u2016\u2191(\u2191f (init m)) (m (last n))\u2016 \u2264 \u2016f\u2016 * \u220f i : Fin (Nat.succ n), \u2016m i\u2016)\u2016 \u2264\n \u2016f\u2016\n[PROOFSTEP]\nexact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\n\u22a2 \u2016\u2191(LinearEquiv.symm\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := uncurryRight,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)),\n uncurryRight (f\u2081 + f\u2082) = uncurryRight f\u2081 + uncurryRight f\u2082) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)),\n AddHom.toFun\n { toFun := uncurryRight,\n map_add' :=\n (_ :\n \u2200\n (f\u2081 f\u2082 :\n ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)),\n uncurryRight (f\u2081 + f\u2082) = uncurryRight f\u2081 + uncurryRight f\u2082) }\n (c \u2022 f) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := uncurryRight,\n map_add' :=\n (_ :\n \u2200\n (f\u2081 f\u2082 :\n ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)),\n uncurryRight (f\u2081 + f\u2082) = uncurryRight f\u2081 + uncurryRight f\u2082) }\n f) },\n invFun := curryRight,\n left_inv :=\n (_ :\n \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun i => Ei (castSucc i)) (Ei (last n) \u2192L[\ud835\udd5c] G)),\n curryRight (uncurryRight f) = f),\n right_inv := (_ : \u2200 (f : ContinuousMultilinearMap \ud835\udd5c Ei G), uncurryRight (curryRight f) = f) })\n f\u2016 \u2264\n \u2016f\u2016\n[PROOFSTEP]\nsimp only [curryRight, LinearEquiv.coe_symm_mk]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c Ei G\n\u22a2 \u2016MultilinearMap.mkContinuous\n {\n toFun := fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016)\n (_ :\n \u2200 (x : Ei (last n)),\n \u2016\u2191(\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) x\u2016 \u2264 (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016) * \u2016x\u2016),\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq (Fin n)] (m : (i : Fin n) \u2192 (fun i => Ei (castSucc i)) i) (i : Fin n)\n (x y : (fun i => Ei (castSucc i)) i),\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m)\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016)\n (_ :\n \u2200 (x : Ei (last n)),\n \u2016\u2191(\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) x\u2016 \u2264\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016) * \u2016x\u2016))\n (update m i (x + y)) =\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m)\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016)\n (_ :\n \u2200 (x : Ei (last n)),\n \u2016\u2191(\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) x\u2016 \u2264\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016) * \u2016x\u2016))\n (update m i x) +\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m)\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016)\n (_ :\n \u2200 (x : Ei (last n)),\n \u2016\u2191(\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) x\u2016 \u2264\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016) * \u2016x\u2016))\n (update m i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq (Fin n)] (m : (i : Fin n) \u2192 (fun i => Ei (castSucc i)) i) (i : Fin n) (c : \ud835\udd5c)\n (x : (fun i => Ei (castSucc i)) i),\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m)\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016)\n (_ :\n \u2200 (x : Ei (last n)),\n \u2016\u2191(\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) x\u2016 \u2264\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016) * \u2016x\u2016))\n (update m i (c \u2022 x)) =\n c \u2022\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m)\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016)\n (_ :\n \u2200 (x : Ei (last n)),\n \u2016\u2191(\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) x\u2016 \u2264\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016) * \u2016x\u2016))\n (update m i x)) }\n \u2016f\u2016\n (_ :\n \u2200 (m : (i : Fin n) \u2192 (fun i => Ei (castSucc i)) i),\n \u2016\u2191{\n toFun := fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m)\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016)\n (_ :\n \u2200 (x : Ei (last n)),\n \u2016\u2191(\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) x\u2016 \u2264\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016) * \u2016x\u2016),\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq (Fin n)] (m : (i : Fin n) \u2192 (fun i => Ei (castSucc i)) i) (i : Fin n)\n (x y : (fun i => Ei (castSucc i)) i),\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m)\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016)\n (_ :\n \u2200 (x : Ei (last n)),\n \u2016\u2191(\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) x\u2016 \u2264\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016) * \u2016x\u2016))\n (update m i (x + y)) =\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m)\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016)\n (_ :\n \u2200 (x : Ei (last n)),\n \u2016\u2191(\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) x\u2016 \u2264\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016) * \u2016x\u2016))\n (update m i x) +\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m)\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016)\n (_ :\n \u2200 (x : Ei (last n)),\n \u2016\u2191(\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) x\u2016 \u2264\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016) * \u2016x\u2016))\n (update m i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq (Fin n)] (m : (i : Fin n) \u2192 (fun i => Ei (castSucc i)) i) (i : Fin n)\n (c : \ud835\udd5c) (x : (fun i => Ei (castSucc i)) i),\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m)\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016)\n (_ :\n \u2200 (x : Ei (last n)),\n \u2016\u2191(\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) x\u2016 \u2264\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016) * \u2016x\u2016))\n (update m i (c \u2022 x)) =\n c \u2022\n (fun m =>\n LinearMap.mkContinuous (\u2191(MultilinearMap.curryRight f.toMultilinearMap) m)\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016)\n (_ :\n \u2200 (x : Ei (last n)),\n \u2016\u2191(\u2191(MultilinearMap.curryRight f.toMultilinearMap) m) x\u2016 \u2264\n (\u2016f\u2016 * \u220f x : Fin n, \u2016m x\u2016) * \u2016x\u2016))\n (update m i x)) }\n m\u2016 \u2264\n \u2016f\u2016 * \u220f i : Fin n, \u2016m i\u2016)\u2016 \u2264\n \u2016f\u2016\n[PROOFSTEP]\nexact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => G) G'\nx : Fin 0 \u2192 G\n\u22a2 curry0 \ud835\udd5c G (\u2191f x) = f\n[PROOFSTEP]\next m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => G) G'\nx m : Fin 0 \u2192 G\n\u22a2 \u2191(curry0 \ud835\udd5c G (\u2191f x)) m = \u2191f m\n[PROOFSTEP]\nsimp [(Subsingleton.elim _ _ : x = m)]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => G) G'\n\u22a2 curry0 \ud835\udd5c G (uncurry0 f) = f\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => G) G'\nx : Fin 0 \u2192 G\n\u22a2 \u2016\u2191f x\u2016 = \u2016f\u2016\n[PROOFSTEP]\nobtain rfl : x = 0 := Subsingleton.elim _ _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => G) G'\n\u22a2 \u2016\u2191f 0\u2016 = \u2016f\u2016\n[PROOFSTEP]\nrefine' le_antisymm (by simpa using f.le_op_norm 0) _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => G) G'\n\u22a2 \u2016\u2191f 0\u2016 \u2264 \u2016f\u2016\n[PROOFSTEP]\nsimpa using f.le_op_norm 0\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => G) G'\n\u22a2 \u2016f\u2016 \u2264 \u2016\u2191f 0\u2016\n[PROOFSTEP]\nhave : \u2016ContinuousMultilinearMap.curry0 \ud835\udd5c G f.uncurry0\u2016 \u2264 \u2016f.uncurry0\u2016 :=\n ContinuousMultilinearMap.op_norm_le_bound _ (norm_nonneg _) fun m => by\n simp [-ContinuousMultilinearMap.apply_zero_curry0]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => G) G'\nm : Fin 0 \u2192 G\n\u22a2 \u2016\u2191(curry0 \ud835\udd5c G (uncurry0 f)) m\u2016 \u2264 \u2016uncurry0 f\u2016 * \u220f i : Fin 0, \u2016m i\u2016\n[PROOFSTEP]\nsimp [-ContinuousMultilinearMap.apply_zero_curry0]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => G) G'\nthis : \u2016curry0 \ud835\udd5c G (uncurry0 f)\u2016 \u2264 \u2016uncurry0 f\u2016\n\u22a2 \u2016f\u2016 \u2264 \u2016\u2191f 0\u2016\n[PROOFSTEP]\nsimpa [-Matrix.zero_empty] using this\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => G) G'\n\u22a2 \u2016uncurry0 f\u2016 = \u2016f\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\n\u03c3 : \u03b9 \u2243 \u03b9'\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'\n\u22a2 \u2016domDomCongr \u03c3 f\u2016 = \u2016f\u2016\n[PROOFSTEP]\nsimp only [norm_def, LinearEquiv.coe_mk, \u2190 \u03c3.prod_comp, (\u03c3.arrowCongr (Equiv.refl G)).surjective.forall,\n domDomCongr_apply, Equiv.arrowCongr_apply, Equiv.coe_refl, comp.left_id, comp_apply, Equiv.symm_apply_apply, id]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'\nm : \u03b9 \u2192 G\nm' : \u03b9' \u2192 G\n\u22a2 \u2016\u2191(\u2191(MultilinearMap.currySum f.toMultilinearMap) m) m'\u2016 \u2264 (\u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016) * \u220f i : \u03b9', \u2016m' i\u2016\n[PROOFSTEP]\nsimpa [Fintype.prod_sum_type, mul_assoc] using f.le_op_norm (Sum.elim m m')\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => G) (ContinuousMultilinearMap \ud835\udd5c (fun x => G) G')\nm : \u03b9 \u2295 \u03b9' \u2192 G\n\u22a2 \u2016\u2191(MultilinearMap.uncurrySum (LinearMap.compMultilinearMap toMultilinearMapLinear f.toMultilinearMap)) m\u2016 \u2264\n \u2016f\u2016 * \u220f i : \u03b9 \u2295 \u03b9', \u2016m i\u2016\n[PROOFSTEP]\nsimpa [Fintype.prod_sum_type, mul_assoc] using (f (m \u2218 Sum.inl)).le_of_op_norm_le (m \u2218 Sum.inr) (f.le_op_norm _)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'\n\u22a2 currySum (f + g) = currySum f + currySum g\n[PROOFSTEP]\next\n[GOAL]\ncase H.H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'\nx\u271d\u00b9 : \u03b9 \u2192 G\nx\u271d : \u03b9' \u2192 G\n\u22a2 \u2191(\u2191(currySum (f + g)) x\u271d\u00b9) x\u271d = \u2191(\u2191(currySum f + currySum g) x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'\n\u22a2 AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ : \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'), currySum (f + g) = currySum f + currySum g) }\n (c \u2022 f) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ : \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'), currySum (f + g) = currySum f + currySum g) }\n f\n[PROOFSTEP]\next\n[GOAL]\ncase H.H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'\nx\u271d\u00b9 : \u03b9 \u2192 G\nx\u271d : \u03b9' \u2192 G\n\u22a2 \u2191(\u2191(AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'), currySum (f + g) = currySum f + currySum g) }\n (c \u2022 f))\n x\u271d\u00b9)\n x\u271d =\n \u2191(\u2191(\u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) }\n f)\n x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'\n\u22a2 uncurrySum\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'), currySum (f + g) = currySum f + currySum g) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) }\n (c \u2022 f) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) }\n f) }.toAddHom\n f) =\n f\n[PROOFSTEP]\next m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'\nm : \u03b9 \u2295 \u03b9' \u2192 G\n\u22a2 \u2191(uncurrySum\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) }\n (c \u2022 f) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) }\n f) }.toAddHom\n f))\n m =\n \u2191f m\n[PROOFSTEP]\nexact congr_arg f (Sum.elim_comp_inl_inr m)\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => G) (ContinuousMultilinearMap \ud835\udd5c (fun x => G) G')\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'), currySum (f + g) = currySum f + currySum g) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) }\n (c \u2022 f) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) }\n f) }.toAddHom\n (uncurrySum f) =\n f\n[PROOFSTEP]\next m\u2081 m\u2082\n[GOAL]\ncase H.H\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => G) (ContinuousMultilinearMap \ud835\udd5c (fun x => G) G')\nm\u2081 : \u03b9 \u2192 G\nm\u2082 : \u03b9' \u2192 G\n\u22a2 \u2191(\u2191(AddHom.toFun\n {\n toAddHom :=\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) }\n (c \u2022 f) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) }\n f) }.toAddHom\n (uncurrySum f))\n m\u2081)\n m\u2082 =\n \u2191(\u2191f m\u2081) m\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => G) (ContinuousMultilinearMap \ud835\udd5c (fun x => G) G')\n\u22a2 \u2016\u2191(LinearEquiv.symm\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) }\n (c \u2022 f) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) }\n f) },\n invFun := uncurrySum,\n left_inv :=\n (_ :\n \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n uncurrySum\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) }\n (c \u2022 f) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) }\n f) }.toAddHom\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => G) (ContinuousMultilinearMap \ud835\udd5c (fun x => G) G')),\n AddHom.toFun\n {\n toAddHom :=\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) }\n (c \u2022 f) =\n \u2191(RingHom.id \ud835\udd5c) c \u2022\n AddHom.toFun\n { toFun := currySum,\n map_add' :=\n (_ :\n \u2200 (f g : ContinuousMultilinearMap \ud835\udd5c (fun x => G) G'),\n currySum (f + g) = currySum f + currySum g) }\n f) }.toAddHom\n (uncurrySum f) =\n f) })\n f\u2016 \u2264\n \u2016f\u2016\n[PROOFSTEP]\nsimp only [LinearEquiv.coe_symm_mk]\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => G) (ContinuousMultilinearMap \ud835\udd5c (fun x => G) G')\n\u22a2 \u2016uncurrySum f\u2016 \u2264 \u2016f\u2016\n[PROOFSTEP]\nexact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nk l : \u2115\ns : Finset (Fin n)\nhk : card s = k\nhl : card s\u1d9c = l\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => G) G'\nx y : G\n\u22a2 (\u2191(\u2191(\u2191(curryFinFinset \ud835\udd5c G G' hk hl) f) fun x_1 => x) fun x => y) = \u2191f (piecewise s (fun x_1 => x) fun x => y)\n[PROOFSTEP]\nrefine'\n (curryFinFinset_symm_apply_piecewise_const hk hl _ _ _).symm.trans\n _\n -- `rw` fails\n[GOAL]\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nk l : \u2115\ns : Finset (Fin n)\nhk : card s = k\nhl : card s\u1d9c = l\nf : ContinuousMultilinearMap \ud835\udd5c (fun i => G) G'\nx y : G\n\u22a2 \u2191(\u2191(LinearIsometryEquiv.symm (curryFinFinset \ud835\udd5c G G' hk hl)) (\u2191(curryFinFinset \ud835\udd5c G G' hk hl) f))\n (piecewise s (fun x_1 => x) fun x => y) =\n \u2191f (piecewise s (fun x_1 => x) fun x => y)\n[PROOFSTEP]\nrw [LinearIsometryEquiv.symm_apply_apply]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.Multilinear", "llama_tokens": 286233, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891392358015, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.44633629551432324}} {"text": "[GOAL]\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\n\u22a2 \u03b3 \u2208 Gamma N \u2194 \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\n[PROOFSTEP]\nrw [Gamma_mem']\n[GOAL]\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\n\u22a2 \u2191(SpecialLinearGroup.map (Int.castRingHom (ZMod N))) \u03b3 = 1 \u2194\n \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\n\u22a2 \u2191(SpecialLinearGroup.map (Int.castRingHom (ZMod N))) \u03b3 = 1 \u2192\n \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\nh : \u2191(SpecialLinearGroup.map (Int.castRingHom (ZMod N))) \u03b3 = 1\n\u22a2 \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\n[PROOFSTEP]\nsimp [\u2190 SL_reduction_mod_hom_val N \u03b3, h]\n[GOAL]\ncase mpr\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\n\u22a2 \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1 \u2192\n \u2191(SpecialLinearGroup.map (Int.castRingHom (ZMod N))) \u03b3 = 1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\nh : \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\n\u22a2 \u2191(SpecialLinearGroup.map (Int.castRingHom (ZMod N))) \u03b3 = 1\n[PROOFSTEP]\next i j\n[GOAL]\ncase mpr.a\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\nh : \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\ni j : Fin 2\n\u22a2 \u2191(\u2191(SpecialLinearGroup.map (Int.castRingHom (ZMod N))) \u03b3) i j = \u21911 i j\n[PROOFSTEP]\nrw [SL_reduction_mod_hom_val N \u03b3]\n[GOAL]\ncase mpr.a\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\nh : \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\ni j : Fin 2\n\u22a2 \u2191(\u2191\u03b3 i j) = \u21911 i j\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase mpr.a.head\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\nh : \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\nj : Fin 2\n\u22a2 \u2191(\u2191\u03b3 { val := 0, isLt := (_ : 0 < 2) } j) = \u21911 { val := 0, isLt := (_ : 0 < 2) } j\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase mpr.a.tail.head\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\nh : \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\nj : Fin 2\n\u22a2 \u2191(\u2191\u03b3 { val := 1, isLt := (_ : (fun a => a < 2) 1) } j) = \u21911 { val := 1, isLt := (_ : (fun a => a < 2) 1) } j\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase mpr.a.head.head\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\nh : \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\n\u22a2 \u2191(\u2191\u03b3 { val := 0, isLt := (_ : 0 < 2) } { val := 0, isLt := (_ : 0 < 2) }) =\n \u21911 { val := 0, isLt := (_ : 0 < 2) } { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase mpr.a.head.tail.head\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\nh : \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\n\u22a2 \u2191(\u2191\u03b3 { val := 0, isLt := (_ : 0 < 2) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n \u21911 { val := 0, isLt := (_ : 0 < 2) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase mpr.a.tail.head.head\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\nh : \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\n\u22a2 \u2191(\u2191\u03b3 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 0, isLt := (_ : 0 < 2) }) =\n \u21911 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase mpr.a.tail.head.tail.head\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\nh : \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\n\u22a2 \u2191(\u2191\u03b3 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n \u21911 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase mpr.a.head.head\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\nh : \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\n\u22a2 \u2191(\u2191\u03b3 { val := 0, isLt := (_ : 0 < 2) } { val := 0, isLt := (_ : 0 < 2) }) =\n \u21911 { val := 0, isLt := (_ : 0 < 2) } { val := 0, isLt := (_ : 0 < 2) }\ncase mpr.a.head.tail.head\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\nh : \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\n\u22a2 \u2191(\u2191\u03b3 { val := 0, isLt := (_ : 0 < 2) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n \u21911 { val := 0, isLt := (_ : 0 < 2) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }\ncase mpr.a.tail.head.head\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\nh : \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\n\u22a2 \u2191(\u2191\u03b3 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 0, isLt := (_ : 0 < 2) }) =\n \u21911 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 0, isLt := (_ : 0 < 2) }\ncase mpr.a.tail.head.tail.head\nN\u271d N : \u2115\n\u03b3 : SL(2, \u2124)\nh : \u2191(\u2191\u03b3 0 0) = 1 \u2227 \u2191(\u2191\u03b3 0 1) = 0 \u2227 \u2191(\u2191\u03b3 1 0) = 0 \u2227 \u2191(\u2191\u03b3 1 1) = 1\n\u22a2 \u2191(\u2191\u03b3 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n \u21911 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nexacts [h.1, h.2.1, h.2.2.1, h.2.2.2]\n[GOAL]\nN : \u2115\n\u22a2 Gamma 1 = \u22a4\n[PROOFSTEP]\next\n[GOAL]\ncase h\nN : \u2115\nx\u271d : SL(2, \u2124)\n\u22a2 x\u271d \u2208 Gamma 1 \u2194 x\u271d \u2208 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nN : \u2115\n\u22a2 Gamma 0 = \u22a5\n[PROOFSTEP]\next\n[GOAL]\ncase h\nN : \u2115\nx\u271d : SL(2, \u2124)\n\u22a2 x\u271d \u2208 Gamma 0 \u2194 x\u271d \u2208 \u22a5\n[PROOFSTEP]\nsimp only [Gamma_mem, coe_matrix_coe, Int.coe_castRingHom, map_apply, Int.cast_id, Subgroup.mem_bot]\n[GOAL]\ncase h\nN : \u2115\nx\u271d : SL(2, \u2124)\n\u22a2 \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1 \u2194 x\u271d = 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nN : \u2115\nx\u271d : SL(2, \u2124)\n\u22a2 \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1 \u2192 x\u271d = 1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mp\nN : \u2115\nx\u271d : SL(2, \u2124)\nh : \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1\n\u22a2 x\u271d = 1\n[PROOFSTEP]\next i j\n[GOAL]\ncase h.mp.a\nN : \u2115\nx\u271d : SL(2, \u2124)\nh : \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1\ni j : Fin 2\n\u22a2 \u2191x\u271d i j = \u21911 i j\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase h.mp.a.head\nN : \u2115\nx\u271d : SL(2, \u2124)\nh : \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1\nj : Fin 2\n\u22a2 \u2191x\u271d { val := 0, isLt := (_ : 0 < 2) } j = \u21911 { val := 0, isLt := (_ : 0 < 2) } j\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase h.mp.a.tail.head\nN : \u2115\nx\u271d : SL(2, \u2124)\nh : \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1\nj : Fin 2\n\u22a2 \u2191x\u271d { val := 1, isLt := (_ : (fun a => a < 2) 1) } j = \u21911 { val := 1, isLt := (_ : (fun a => a < 2) 1) } j\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase h.mp.a.head.head\nN : \u2115\nx\u271d : SL(2, \u2124)\nh : \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1\n\u22a2 \u2191x\u271d { val := 0, isLt := (_ : 0 < 2) } { val := 0, isLt := (_ : 0 < 2) } =\n \u21911 { val := 0, isLt := (_ : 0 < 2) } { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase h.mp.a.head.tail.head\nN : \u2115\nx\u271d : SL(2, \u2124)\nh : \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1\n\u22a2 \u2191x\u271d { val := 0, isLt := (_ : 0 < 2) } { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n \u21911 { val := 0, isLt := (_ : 0 < 2) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase h.mp.a.tail.head.head\nN : \u2115\nx\u271d : SL(2, \u2124)\nh : \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1\n\u22a2 \u2191x\u271d { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 0, isLt := (_ : 0 < 2) } =\n \u21911 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase h.mp.a.tail.head.tail.head\nN : \u2115\nx\u271d : SL(2, \u2124)\nh : \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1\n\u22a2 \u2191x\u271d { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n \u21911 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase h.mp.a.head.head\nN : \u2115\nx\u271d : SL(2, \u2124)\nh : \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1\n\u22a2 \u2191x\u271d { val := 0, isLt := (_ : 0 < 2) } { val := 0, isLt := (_ : 0 < 2) } =\n \u21911 { val := 0, isLt := (_ : 0 < 2) } { val := 0, isLt := (_ : 0 < 2) }\ncase h.mp.a.head.tail.head\nN : \u2115\nx\u271d : SL(2, \u2124)\nh : \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1\n\u22a2 \u2191x\u271d { val := 0, isLt := (_ : 0 < 2) } { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n \u21911 { val := 0, isLt := (_ : 0 < 2) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }\ncase h.mp.a.tail.head.head\nN : \u2115\nx\u271d : SL(2, \u2124)\nh : \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1\n\u22a2 \u2191x\u271d { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 0, isLt := (_ : 0 < 2) } =\n \u21911 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 0, isLt := (_ : 0 < 2) }\ncase h.mp.a.tail.head.tail.head\nN : \u2115\nx\u271d : SL(2, \u2124)\nh : \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1\n\u22a2 \u2191x\u271d { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n \u21911 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nexacts [h.1, h.2.1, h.2.2.1, h.2.2.2]\n[GOAL]\ncase h.mpr\nN : \u2115\nx\u271d : SL(2, \u2124)\n\u22a2 x\u271d = 1 \u2192 \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mpr\nN : \u2115\nx\u271d : SL(2, \u2124)\nh : x\u271d = 1\n\u22a2 \u2191(\u2191x\u271d 0 0) = 1 \u2227 \u2191(\u2191x\u271d 0 1) = 0 \u2227 \u2191(\u2191x\u271d 1 0) = 0 \u2227 \u2191(\u2191x\u271d 1 1) = 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nN\u271d N : \u2115\n\u22a2 \u2200 {a b : SL(2, \u2124)}, a \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 b \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 a * b \u2208 {g | \u2191(\u2191g 1 0) = 0}\n[PROOFSTEP]\nintro a b ha hb\n[GOAL]\nN\u271d N : \u2115\na b : SL(2, \u2124)\nha : a \u2208 {g | \u2191(\u2191g 1 0) = 0}\nhb : b \u2208 {g | \u2191(\u2191g 1 0) = 0}\n\u22a2 a * b \u2208 {g | \u2191(\u2191g 1 0) = 0}\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq]\n[GOAL]\nN\u271d N : \u2115\na b : SL(2, \u2124)\nha : a \u2208 {g | \u2191(\u2191g 1 0) = 0}\nhb : b \u2208 {g | \u2191(\u2191g 1 0) = 0}\n\u22a2 \u2191(\u2191(a * b) 1 0) = 0\n[PROOFSTEP]\nhave h := (Matrix.two_mul_expl a.1 b.1).2.2.1\n[GOAL]\nN\u271d N : \u2115\na b : SL(2, \u2124)\nha : a \u2208 {g | \u2191(\u2191g 1 0) = 0}\nhb : b \u2208 {g | \u2191(\u2191g 1 0) = 0}\nh : (\u2191a * \u2191b) 1 0 = \u2191a 1 0 * \u2191b 0 0 + \u2191a 1 1 * \u2191b 1 0\n\u22a2 \u2191(\u2191(a * b) 1 0) = 0\n[PROOFSTEP]\nsimp only [coe_matrix_coe, coe_mul, Int.coe_castRingHom, map_apply, Set.mem_setOf_eq] at *\n[GOAL]\nN\u271d N : \u2115\na b : SL(2, \u2124)\nha : \u2191(\u2191a 1 0) = 0\nhb : \u2191(\u2191b 1 0) = 0\nh : (\u2191a * \u2191b) 1 0 = \u2191a 1 0 * \u2191b 0 0 + \u2191a 1 1 * \u2191b 1 0\n\u22a2 \u2191((\u2191a * \u2191b) 1 0) = 0\n[PROOFSTEP]\nrw [h]\n[GOAL]\nN\u271d N : \u2115\na b : SL(2, \u2124)\nha : \u2191(\u2191a 1 0) = 0\nhb : \u2191(\u2191b 1 0) = 0\nh : (\u2191a * \u2191b) 1 0 = \u2191a 1 0 * \u2191b 0 0 + \u2191a 1 1 * \u2191b 1 0\n\u22a2 \u2191(\u2191a 1 0 * \u2191b 0 0 + \u2191a 1 1 * \u2191b 1 0) = 0\n[PROOFSTEP]\nsimp [ha, hb]\n[GOAL]\nN\u271d N : \u2115\n\u22a2 1 \u2208\n { carrier := {g | \u2191(\u2191g 1 0) = 0},\n mul_mem' :=\n (_ :\n \u2200 {a b : SL(2, \u2124)},\n a \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 b \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 a * b \u2208 {g | \u2191(\u2191g 1 0) = 0}) }.carrier\n[PROOFSTEP]\nsimp\n[GOAL]\nN\u271d N : \u2115\n\u22a2 \u2200 {x : SL(2, \u2124)},\n x \u2208\n {\n toSubsemigroup :=\n { carrier := {g | \u2191(\u2191g 1 0) = 0},\n mul_mem' :=\n (_ :\n \u2200 {a b : SL(2, \u2124)},\n a \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 b \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 a * b \u2208 {g | \u2191(\u2191g 1 0) = 0}) },\n one_mem' := (_ : \u2191(OfNat.ofNat 1 1 0) = 0) }.toSubsemigroup.carrier \u2192\n x\u207b\u00b9 \u2208\n {\n toSubsemigroup :=\n { carrier := {g | \u2191(\u2191g 1 0) = 0},\n mul_mem' :=\n (_ :\n \u2200 {a b : SL(2, \u2124)},\n a \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 b \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 a * b \u2208 {g | \u2191(\u2191g 1 0) = 0}) },\n one_mem' := (_ : \u2191(OfNat.ofNat 1 1 0) = 0) }.toSubsemigroup.carrier\n[PROOFSTEP]\nintro a ha\n[GOAL]\nN\u271d N : \u2115\na : SL(2, \u2124)\nha :\n a \u2208\n {\n toSubsemigroup :=\n { carrier := {g | \u2191(\u2191g 1 0) = 0},\n mul_mem' :=\n (_ :\n \u2200 {a b : SL(2, \u2124)},\n a \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 b \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 a * b \u2208 {g | \u2191(\u2191g 1 0) = 0}) },\n one_mem' := (_ : \u2191(OfNat.ofNat 1 1 0) = 0) }.toSubsemigroup.carrier\n\u22a2 a\u207b\u00b9 \u2208\n {\n toSubsemigroup :=\n { carrier := {g | \u2191(\u2191g 1 0) = 0},\n mul_mem' :=\n (_ :\n \u2200 {a b : SL(2, \u2124)},\n a \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 b \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 a * b \u2208 {g | \u2191(\u2191g 1 0) = 0}) },\n one_mem' := (_ : \u2191(OfNat.ofNat 1 1 0) = 0) }.toSubsemigroup.carrier\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq]\n[GOAL]\nN\u271d N : \u2115\na : SL(2, \u2124)\nha :\n a \u2208\n {\n toSubsemigroup :=\n { carrier := {g | \u2191(\u2191g 1 0) = 0},\n mul_mem' :=\n (_ :\n \u2200 {a b : SL(2, \u2124)},\n a \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 b \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 a * b \u2208 {g | \u2191(\u2191g 1 0) = 0}) },\n one_mem' := (_ : \u2191(OfNat.ofNat 1 1 0) = 0) }.toSubsemigroup.carrier\n\u22a2 \u2191(\u2191a\u207b\u00b9 1 0) = 0\n[PROOFSTEP]\nrw [SL2_inv_expl a]\n[GOAL]\nN\u271d N : \u2115\na : SL(2, \u2124)\nha :\n a \u2208\n {\n toSubsemigroup :=\n { carrier := {g | \u2191(\u2191g 1 0) = 0},\n mul_mem' :=\n (_ :\n \u2200 {a b : SL(2, \u2124)},\n a \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 b \u2208 {g | \u2191(\u2191g 1 0) = 0} \u2192 a * b \u2208 {g | \u2191(\u2191g 1 0) = 0}) },\n one_mem' := (_ : \u2191(OfNat.ofNat 1 1 0) = 0) }.toSubsemigroup.carrier\n\u22a2 \u2191(\u2191{ val := ![![\u2191a 1 1, -\u2191a 0 1], ![-\u2191a 1 0, \u2191a 0 0]],\n property := (_ : det ![![\u2191a 1 1, -\u2191a 0 1], ![-\u2191a 1 0, \u2191a 0 0]] = 1) }\n 1 0) =\n 0\n[PROOFSTEP]\nsimp only [cons_val_zero, cons_val_one, head_cons, coe_matrix_coe, coe_mk, Int.coe_castRingHom, map_apply, Int.cast_neg,\n neg_eq_zero, Set.mem_setOf_eq] at *\n[GOAL]\nN\u271d N : \u2115\na : SL(2, \u2124)\nha : \u2191(\u2191a 1 0) = 0\n\u22a2 \u2191(\u2191a 1 0) = 0\n[PROOFSTEP]\nexact ha\n[GOAL]\nN\u271d N : \u2115\nA : { x // x \u2208 Gamma0 N }\n\u22a2 \u2191(det \u2191\u2191A) = 1\n[PROOFSTEP]\nsimp [A.1.property]\n[GOAL]\nN\u271d N : \u2115\n\u22a2 (fun g => \u2191(\u2191\u2191g 1 1)) 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nN\u271d N : \u2115\n\u22a2 \u2200 (x y : { x // x \u2208 Gamma0 N }),\n OneHom.toFun { toFun := fun g => \u2191(\u2191\u2191g 1 1), map_one' := (_ : \u2191(OfNat.ofNat 1 1 1) = 1) } (x * y) =\n OneHom.toFun { toFun := fun g => \u2191(\u2191\u2191g 1 1), map_one' := (_ : \u2191(OfNat.ofNat 1 1 1) = 1) } x *\n OneHom.toFun { toFun := fun g => \u2191(\u2191\u2191g 1 1), map_one' := (_ : \u2191(OfNat.ofNat 1 1 1) = 1) } y\n[PROOFSTEP]\nintro A B\n[GOAL]\nN\u271d N : \u2115\nA B : { x // x \u2208 Gamma0 N }\n\u22a2 OneHom.toFun { toFun := fun g => \u2191(\u2191\u2191g 1 1), map_one' := (_ : \u2191(OfNat.ofNat 1 1 1) = 1) } (A * B) =\n OneHom.toFun { toFun := fun g => \u2191(\u2191\u2191g 1 1), map_one' := (_ : \u2191(OfNat.ofNat 1 1 1) = 1) } A *\n OneHom.toFun { toFun := fun g => \u2191(\u2191\u2191g 1 1), map_one' := (_ : \u2191(OfNat.ofNat 1 1 1) = 1) } B\n[PROOFSTEP]\nhave := (two_mul_expl A.1.1 B.1.1).2.2.2\n[GOAL]\nN\u271d N : \u2115\nA B : { x // x \u2208 Gamma0 N }\nthis : (\u2191\u2191A * \u2191\u2191B) 1 1 = \u2191\u2191A 1 0 * \u2191\u2191B 0 1 + \u2191\u2191A 1 1 * \u2191\u2191B 1 1\n\u22a2 OneHom.toFun { toFun := fun g => \u2191(\u2191\u2191g 1 1), map_one' := (_ : \u2191(OfNat.ofNat 1 1 1) = 1) } (A * B) =\n OneHom.toFun { toFun := fun g => \u2191(\u2191\u2191g 1 1), map_one' := (_ : \u2191(OfNat.ofNat 1 1 1) = 1) } A *\n OneHom.toFun { toFun := fun g => \u2191(\u2191\u2191g 1 1), map_one' := (_ : \u2191(OfNat.ofNat 1 1 1) = 1) } B\n[PROOFSTEP]\nsimp only [Subgroup.coe_mul, coe_matrix_coe, coe_mul, Int.coe_castRingHom, map_apply] at *\n[GOAL]\nN\u271d N : \u2115\nA B : { x // x \u2208 Gamma0 N }\nthis : (\u2191\u2191A * \u2191\u2191B) 1 1 = \u2191\u2191A 1 0 * \u2191\u2191B 0 1 + \u2191\u2191A 1 1 * \u2191\u2191B 1 1\n\u22a2 \u2191((\u2191\u2191A * \u2191\u2191B) 1 1) = \u2191(\u2191\u2191A 1 1) * \u2191(\u2191\u2191B 1 1)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nN\u271d N : \u2115\nA B : { x // x \u2208 Gamma0 N }\nthis : (\u2191\u2191A * \u2191\u2191B) 1 1 = \u2191\u2191A 1 0 * \u2191\u2191B 0 1 + \u2191\u2191A 1 1 * \u2191\u2191B 1 1\n\u22a2 \u2191(\u2191\u2191A 1 0 * \u2191\u2191B 0 1 + \u2191\u2191A 1 1 * \u2191\u2191B 1 1) = \u2191(\u2191\u2191A 1 1) * \u2191(\u2191\u2191B 1 1)\n[PROOFSTEP]\nhave ha := A.property\n[GOAL]\nN\u271d N : \u2115\nA B : { x // x \u2208 Gamma0 N }\nthis : (\u2191\u2191A * \u2191\u2191B) 1 1 = \u2191\u2191A 1 0 * \u2191\u2191B 0 1 + \u2191\u2191A 1 1 * \u2191\u2191B 1 1\nha : \u2191A \u2208 Gamma0 N\n\u22a2 \u2191(\u2191\u2191A 1 0 * \u2191\u2191B 0 1 + \u2191\u2191A 1 1 * \u2191\u2191B 1 1) = \u2191(\u2191\u2191A 1 1) * \u2191(\u2191\u2191B 1 1)\n[PROOFSTEP]\nsimp only [Int.cast_add, Int.cast_mul, add_left_eq_self, Gamma0_mem, coe_matrix_coe, Int.coe_castRingHom,\n map_apply] at *\n[GOAL]\nN\u271d N : \u2115\nA B : { x // x \u2208 Gamma0 N }\nthis : (\u2191\u2191A * \u2191\u2191B) 1 1 = \u2191\u2191A 1 0 * \u2191\u2191B 0 1 + \u2191\u2191A 1 1 * \u2191\u2191B 1 1\nha : \u2191(\u2191\u2191A 1 0) = 0\n\u22a2 \u2191(\u2191\u2191A 1 0) * \u2191(\u2191\u2191B 0 1) = 0\n[PROOFSTEP]\nrw [ha]\n[GOAL]\nN\u271d N : \u2115\nA B : { x // x \u2208 Gamma0 N }\nthis : (\u2191\u2191A * \u2191\u2191B) 1 1 = \u2191\u2191A 1 0 * \u2191\u2191B 0 1 + \u2191\u2191A 1 1 * \u2191\u2191B 1 1\nha : \u2191(\u2191\u2191A 1 0) = 0\n\u22a2 0 * \u2191(\u2191\u2191B 0 1) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nN\u271d N : \u2115\nA : { x // x \u2208 Gamma0 N }\n\u22a2 A \u2208 Gamma1' N \u2194 \u2191(\u2191\u2191A 0 0) = 1 \u2227 \u2191(\u2191\u2191A 1 1) = 1 \u2227 \u2191(\u2191\u2191A 1 0) = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nN\u271d N : \u2115\nA : { x // x \u2208 Gamma0 N }\n\u22a2 A \u2208 Gamma1' N \u2192 \u2191(\u2191\u2191A 0 0) = 1 \u2227 \u2191(\u2191\u2191A 1 1) = 1 \u2227 \u2191(\u2191\u2191A 1 0) = 0\n[PROOFSTEP]\nintro ha\n[GOAL]\ncase mp\nN\u271d N : \u2115\nA : { x // x \u2208 Gamma0 N }\nha : A \u2208 Gamma1' N\n\u22a2 \u2191(\u2191\u2191A 0 0) = 1 \u2227 \u2191(\u2191\u2191A 1 1) = 1 \u2227 \u2191(\u2191\u2191A 1 0) = 0\n[PROOFSTEP]\nhave hA := A.property\n[GOAL]\ncase mp\nN\u271d N : \u2115\nA : { x // x \u2208 Gamma0 N }\nha : A \u2208 Gamma1' N\nhA : \u2191A \u2208 Gamma0 N\n\u22a2 \u2191(\u2191\u2191A 0 0) = 1 \u2227 \u2191(\u2191\u2191A 1 1) = 1 \u2227 \u2191(\u2191\u2191A 1 0) = 0\n[PROOFSTEP]\nrw [Gamma0_mem] at hA \n[GOAL]\ncase mp\nN\u271d N : \u2115\nA : { x // x \u2208 Gamma0 N }\nha : A \u2208 Gamma1' N\nhA : \u2191(\u2191\u2191A 1 0) = 0\n\u22a2 \u2191(\u2191\u2191A 0 0) = 1 \u2227 \u2191(\u2191\u2191A 1 1) = 1 \u2227 \u2191(\u2191\u2191A 1 0) = 0\n[PROOFSTEP]\nhave adet := Gamma0_det N A\n[GOAL]\ncase mp\nN\u271d N : \u2115\nA : { x // x \u2208 Gamma0 N }\nha : A \u2208 Gamma1' N\nhA : \u2191(\u2191\u2191A 1 0) = 0\nadet : \u2191(det \u2191\u2191A) = 1\n\u22a2 \u2191(\u2191\u2191A 0 0) = 1 \u2227 \u2191(\u2191\u2191A 1 1) = 1 \u2227 \u2191(\u2191\u2191A 1 0) = 0\n[PROOFSTEP]\nrw [Matrix.det_fin_two] at adet \n[GOAL]\ncase mp\nN\u271d N : \u2115\nA : { x // x \u2208 Gamma0 N }\nha : A \u2208 Gamma1' N\nhA : \u2191(\u2191\u2191A 1 0) = 0\nadet : \u2191(\u2191\u2191A 0 0 * \u2191\u2191A 1 1 - \u2191\u2191A 0 1 * \u2191\u2191A 1 0) = 1\n\u22a2 \u2191(\u2191\u2191A 0 0) = 1 \u2227 \u2191(\u2191\u2191A 1 1) = 1 \u2227 \u2191(\u2191\u2191A 1 0) = 0\n[PROOFSTEP]\nsimp only [Gamma0Map, coe_matrix_coe, Int.coe_castRingHom, map_apply, Gamma1_mem', MonoidHom.coe_mk, OneHom.coe_mk,\n Int.cast_sub, Int.cast_mul] at *\n[GOAL]\ncase mp\nN\u271d N : \u2115\nA : { x // x \u2208 Gamma0 N }\nhA : \u2191(\u2191\u2191A 1 0) = 0\nha : \u2191(\u2191\u2191A 1 1) = 1\nadet : \u2191(\u2191\u2191A 0 0) * \u2191(\u2191\u2191A 1 1) - \u2191(\u2191\u2191A 0 1) * \u2191(\u2191\u2191A 1 0) = 1\n\u22a2 \u2191(\u2191\u2191A 0 0) = 1 \u2227 \u2191(\u2191\u2191A 1 1) = 1 \u2227 \u2191(\u2191\u2191A 1 0) = 0\n[PROOFSTEP]\nrw [hA, ha] at adet \n[GOAL]\ncase mp\nN\u271d N : \u2115\nA : { x // x \u2208 Gamma0 N }\nhA : \u2191(\u2191\u2191A 1 0) = 0\nha : \u2191(\u2191\u2191A 1 1) = 1\nadet : \u2191(\u2191\u2191A 0 0) * 1 - \u2191(\u2191\u2191A 0 1) * 0 = 1\n\u22a2 \u2191(\u2191\u2191A 0 0) = 1 \u2227 \u2191(\u2191\u2191A 1 1) = 1 \u2227 \u2191(\u2191\u2191A 1 0) = 0\n[PROOFSTEP]\nsimp only [mul_one, mul_zero, sub_zero] at adet \n[GOAL]\ncase mp\nN\u271d N : \u2115\nA : { x // x \u2208 Gamma0 N }\nhA : \u2191(\u2191\u2191A 1 0) = 0\nha : \u2191(\u2191\u2191A 1 1) = 1\nadet : \u2191(\u2191\u2191A 0 0) = 1\n\u22a2 \u2191(\u2191\u2191A 0 0) = 1 \u2227 \u2191(\u2191\u2191A 1 1) = 1 \u2227 \u2191(\u2191\u2191A 1 0) = 0\n[PROOFSTEP]\nsimp only [adet, hA, ha, eq_self_iff_true, and_self_iff]\n[GOAL]\ncase mpr\nN\u271d N : \u2115\nA : { x // x \u2208 Gamma0 N }\n\u22a2 \u2191(\u2191\u2191A 0 0) = 1 \u2227 \u2191(\u2191\u2191A 1 1) = 1 \u2227 \u2191(\u2191\u2191A 1 0) = 0 \u2192 A \u2208 Gamma1' N\n[PROOFSTEP]\nintro ha\n[GOAL]\ncase mpr\nN\u271d N : \u2115\nA : { x // x \u2208 Gamma0 N }\nha : \u2191(\u2191\u2191A 0 0) = 1 \u2227 \u2191(\u2191\u2191A 1 1) = 1 \u2227 \u2191(\u2191\u2191A 1 0) = 0\n\u22a2 A \u2208 Gamma1' N\n[PROOFSTEP]\nsimp only [Gamma1_mem', Gamma0Map, MonoidHom.coe_mk, coe_matrix_coe, Int.coe_castRingHom, map_apply]\n[GOAL]\ncase mpr\nN\u271d N : \u2115\nA : { x // x \u2208 Gamma0 N }\nha : \u2191(\u2191\u2191A 0 0) = 1 \u2227 \u2191(\u2191\u2191A 1 1) = 1 \u2227 \u2191(\u2191\u2191A 1 0) = 0\n\u22a2 \u2191{ toFun := fun g => \u2191(\u2191\u2191g 1 1), map_one' := (_ : \u2191(OfNat.ofNat 1 1 1) = 1) } A = 1\n[PROOFSTEP]\nexact ha.2.1\n[GOAL]\nN\u271d N : \u2115\nA : SL(2, \u2124)\n\u22a2 A \u2208 Gamma1 N \u2194 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nN\u271d N : \u2115\nA : SL(2, \u2124)\n\u22a2 A \u2208 Gamma1 N \u2192 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n[PROOFSTEP]\nintro ha\n[GOAL]\ncase mp\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : A \u2208 Gamma1 N\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n[PROOFSTEP]\nsimp_rw [Gamma1, Subgroup.mem_map] at ha \n[GOAL]\ncase mp\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2203 x, x \u2208 \u22a4 \u2227 \u2191(MonoidHom.comp (Subgroup.subtype (Gamma0 N)) (Subgroup.subtype (Gamma1' N))) x = A\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n[PROOFSTEP]\nobtain \u27e8\u27e8x, hx\u27e9, hxx\u27e9 := ha\n[GOAL]\ncase mp.intro.mk\nN\u271d N : \u2115\nA : SL(2, \u2124)\nx : { x // x \u2208 Gamma0 N }\nhx : x \u2208 Gamma1' N\nhxx :\n { val := x, property := hx } \u2208 \u22a4 \u2227\n \u2191(MonoidHom.comp (Subgroup.subtype (Gamma0 N)) (Subgroup.subtype (Gamma1' N))) { val := x, property := hx } = A\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n[PROOFSTEP]\nrw [Gamma1_to_Gamma0_mem] at hx \n[GOAL]\ncase mp.intro.mk\nN\u271d N : \u2115\nA : SL(2, \u2124)\nx : { x // x \u2208 Gamma0 N }\nhx\u271d : x \u2208 Gamma1' N\nhx : \u2191(\u2191\u2191x 0 0) = 1 \u2227 \u2191(\u2191\u2191x 1 1) = 1 \u2227 \u2191(\u2191\u2191x 1 0) = 0\nhxx :\n { val := x, property := hx\u271d } \u2208 \u22a4 \u2227\n \u2191(MonoidHom.comp (Subgroup.subtype (Gamma0 N)) (Subgroup.subtype (Gamma1' N))) { val := x, property := hx\u271d } = A\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n[PROOFSTEP]\nsimp only [Subgroup.mem_top, true_and] at hxx \n[GOAL]\ncase mp.intro.mk\nN\u271d N : \u2115\nA : SL(2, \u2124)\nx : { x // x \u2208 Gamma0 N }\nhx\u271d : x \u2208 Gamma1' N\nhx : \u2191(\u2191\u2191x 0 0) = 1 \u2227 \u2191(\u2191\u2191x 1 1) = 1 \u2227 \u2191(\u2191\u2191x 1 0) = 0\nhxx : \u2191(MonoidHom.comp (Subgroup.subtype (Gamma0 N)) (Subgroup.subtype (Gamma1' N))) { val := x, property := hx\u271d } = A\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n[PROOFSTEP]\nrw [\u2190 hxx]\n[GOAL]\ncase mp.intro.mk\nN\u271d N : \u2115\nA : SL(2, \u2124)\nx : { x // x \u2208 Gamma0 N }\nhx\u271d : x \u2208 Gamma1' N\nhx : \u2191(\u2191\u2191x 0 0) = 1 \u2227 \u2191(\u2191\u2191x 1 1) = 1 \u2227 \u2191(\u2191\u2191x 1 0) = 0\nhxx : \u2191(MonoidHom.comp (Subgroup.subtype (Gamma0 N)) (Subgroup.subtype (Gamma1' N))) { val := x, property := hx\u271d } = A\n\u22a2 \u2191(\u2191(\u2191(MonoidHom.comp (Subgroup.subtype (Gamma0 N)) (Subgroup.subtype (Gamma1' N))) { val := x, property := hx\u271d }) 0\n 0) =\n 1 \u2227\n \u2191(\u2191(\u2191(MonoidHom.comp (Subgroup.subtype (Gamma0 N)) (Subgroup.subtype (Gamma1' N))) { val := x, property := hx\u271d }) 1\n 1) =\n 1 \u2227\n \u2191(\u2191(\u2191(MonoidHom.comp (Subgroup.subtype (Gamma0 N)) (Subgroup.subtype (Gamma1' N))) { val := x, property := hx\u271d })\n 1 0) =\n 0\n[PROOFSTEP]\nconvert hx\n[GOAL]\ncase mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0 \u2192 A \u2208 Gamma1 N\n[PROOFSTEP]\nintro ha\n[GOAL]\ncase mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n\u22a2 A \u2208 Gamma1 N\n[PROOFSTEP]\nsimp_rw [Gamma1, Subgroup.mem_map]\n[GOAL]\ncase mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n\u22a2 \u2203 x, x \u2208 \u22a4 \u2227 \u2191(MonoidHom.comp (Subgroup.subtype (Gamma0 N)) (Subgroup.subtype (Gamma1' N))) x = A\n[PROOFSTEP]\nhave hA : A \u2208 Gamma0 N := by simp [ha.right.right, Gamma0_mem]\n[GOAL]\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n\u22a2 A \u2208 Gamma0 N\n[PROOFSTEP]\nsimp [ha.right.right, Gamma0_mem]\n[GOAL]\ncase mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\nhA : A \u2208 Gamma0 N\n\u22a2 \u2203 x, x \u2208 \u22a4 \u2227 \u2191(MonoidHom.comp (Subgroup.subtype (Gamma0 N)) (Subgroup.subtype (Gamma1' N))) x = A\n[PROOFSTEP]\nhave HA : (\u27e8A, hA\u27e9 : Gamma0 N) \u2208 Gamma1' N :=\n by\n simp only [Gamma1_to_Gamma0_mem, Subgroup.coe_mk, coe_matrix_coe, Int.coe_castRingHom, map_apply]\n exact ha\n[GOAL]\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\nhA : A \u2208 Gamma0 N\n\u22a2 { val := A, property := hA } \u2208 Gamma1' N\n[PROOFSTEP]\nsimp only [Gamma1_to_Gamma0_mem, Subgroup.coe_mk, coe_matrix_coe, Int.coe_castRingHom, map_apply]\n[GOAL]\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\nhA : A \u2208 Gamma0 N\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n[PROOFSTEP]\nexact ha\n[GOAL]\ncase mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\nhA : A \u2208 Gamma0 N\nHA : { val := A, property := hA } \u2208 Gamma1' N\n\u22a2 \u2203 x, x \u2208 \u22a4 \u2227 \u2191(MonoidHom.comp (Subgroup.subtype (Gamma0 N)) (Subgroup.subtype (Gamma1' N))) x = A\n[PROOFSTEP]\nrefine' \u27e8(\u27e8(\u27e8A, hA\u27e9 : Gamma0 N), HA\u27e9 : (Gamma1' N : Subgroup (Gamma0 N))), _\u27e9\n[GOAL]\ncase mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\nhA : A \u2208 Gamma0 N\nHA : { val := A, property := hA } \u2208 Gamma1' N\n\u22a2 { val := { val := A, property := hA }, property := HA } \u2208 \u22a4 \u2227\n \u2191(MonoidHom.comp (Subgroup.subtype (Gamma0 N)) (Subgroup.subtype (Gamma1' N)))\n { val := { val := A, property := hA }, property := HA } =\n A\n[PROOFSTEP]\nsimp\n[GOAL]\nN\u271d N : \u2115\n\u22a2 Gamma1 N \u2264 Gamma0 N\n[PROOFSTEP]\nintro x HA\n[GOAL]\nN\u271d N : \u2115\nx : SL(2, \u2124)\nHA : x \u2208 Gamma1 N\n\u22a2 x \u2208 Gamma0 N\n[PROOFSTEP]\nsimp only [Gamma0_mem, Gamma1_mem, coe_matrix_coe, Int.coe_castRingHom, map_apply] at *\n[GOAL]\nN\u271d N : \u2115\nx : SL(2, \u2124)\nHA : \u2191(\u2191x 0 0) = 1 \u2227 \u2191(\u2191x 1 1) = 1 \u2227 \u2191(\u2191x 1 0) = 0\n\u22a2 \u2191(\u2191x 1 0) = 0\n[PROOFSTEP]\nexact HA.2.2\n[GOAL]\nN : \u2115\nH K : Subgroup SL(2, \u2124)\nh : H \u2264 K\nh2 : IsCongruenceSubgroup H\n\u22a2 IsCongruenceSubgroup K\n[PROOFSTEP]\nobtain \u27e8N, hN\u27e9 := h2\n[GOAL]\ncase intro\nN\u271d : \u2115\nH K : Subgroup SL(2, \u2124)\nh : H \u2264 K\nN : \u2115+\nhN : Gamma \u2191N \u2264 H\n\u22a2 IsCongruenceSubgroup K\n[PROOFSTEP]\nrefine' \u27e8N, le_trans hN h\u27e9\n[GOAL]\nN\u271d : \u2115\nN : \u2115+\n\u22a2 Gamma \u2191N \u2264 Gamma \u2191N\n[PROOFSTEP]\nsimp only [le_refl]\n[GOAL]\nN\u271d : \u2115\nN : \u2115+\n\u22a2 IsCongruenceSubgroup (Gamma1 \u2191N)\n[PROOFSTEP]\nrefine' \u27e8N, _\u27e9\n[GOAL]\nN\u271d : \u2115\nN : \u2115+\n\u22a2 Gamma \u2191N \u2264 Gamma1 \u2191N\n[PROOFSTEP]\nintro A hA\n[GOAL]\nN\u271d : \u2115\nN : \u2115+\nA : SL(2, \u2124)\nhA : A \u2208 Gamma \u2191N\n\u22a2 A \u2208 Gamma1 \u2191N\n[PROOFSTEP]\nsimp only [Gamma1_mem, Gamma_mem] at *\n[GOAL]\nN\u271d : \u2115\nN : \u2115+\nA : SL(2, \u2124)\nhA : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 0 1) = 0 \u2227 \u2191(\u2191A 1 0) = 0 \u2227 \u2191(\u2191A 1 1) = 1\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n[PROOFSTEP]\nsimp only [hA, eq_self_iff_true, and_self_iff]\n[GOAL]\nN\u271d N : \u2115\ng : ConjAct SL(2, \u2124)\n\u22a2 g \u2022 Gamma N = Gamma N\n[PROOFSTEP]\napply Subgroup.Normal.conjAct (Gamma_normal N)\n[GOAL]\nN : \u2115\ng : ConjAct SL(2, \u2124)\n\u0393 : Subgroup SL(2, \u2124)\nh : IsCongruenceSubgroup \u0393\n\u22a2 IsCongruenceSubgroup (g \u2022 \u0393)\n[PROOFSTEP]\nobtain \u27e8N, HN\u27e9 := h\n[GOAL]\ncase intro\nN\u271d : \u2115\ng : ConjAct SL(2, \u2124)\n\u0393 : Subgroup SL(2, \u2124)\nN : \u2115+\nHN : Gamma \u2191N \u2264 \u0393\n\u22a2 IsCongruenceSubgroup (g \u2022 \u0393)\n[PROOFSTEP]\nrefine' \u27e8N, _\u27e9\n[GOAL]\ncase intro\nN\u271d : \u2115\ng : ConjAct SL(2, \u2124)\n\u0393 : Subgroup SL(2, \u2124)\nN : \u2115+\nHN : Gamma \u2191N \u2264 \u0393\n\u22a2 Gamma \u2191N \u2264 g \u2022 \u0393\n[PROOFSTEP]\nrw [\u2190 Gamma_cong_eq_self N g, Subgroup.pointwise_smul_le_pointwise_smul_iff]\n[GOAL]\ncase intro\nN\u271d : \u2115\ng : ConjAct SL(2, \u2124)\n\u0393 : Subgroup SL(2, \u2124)\nN : \u2115+\nHN : Gamma \u2191N \u2264 \u0393\n\u22a2 Gamma \u2191N \u2264 \u0393\n[PROOFSTEP]\nexact HN\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.ModularForms.CongruenceSubgroups", "llama_tokens": 16022, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891174511733, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.4463362836882484}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommMonoid \u03b1\ns : Interval \u03b1\n\u22a2 Option.map (fun b => NonemptyInterval.pure 1 * b) s = s\n[PROOFSTEP]\nsimp_rw [NonemptyInterval.pure_one, one_mul, \u2190 id_def, Option.map_id, id]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommMonoid \u03b1\ns : Interval \u03b1\n\u22a2 Option.map (fun a => a * NonemptyInterval.pure 1) s = s\n[PROOFSTEP]\nsimp_rw [NonemptyInterval.pure_one, mul_one, \u2190 id_def, Option.map_id, id]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : NonemptyInterval \u03b1\n\u22a2 s * t = 1 \u2194 \u2203 a b, s = pure a \u2227 t = pure b \u2227 a * b = 1\n[PROOFSTEP]\nrefine' \u27e8fun h => _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : NonemptyInterval \u03b1\nh : s * t = 1\n\u22a2 \u2203 a b, s = pure a \u2227 t = pure b \u2227 a * b = 1\n[PROOFSTEP]\nrw [ext_iff, Prod.ext_iff] at h \n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : NonemptyInterval \u03b1\nh : (s * t).fst = 1.fst \u2227 (s * t).snd = 1.snd\n\u22a2 \u2203 a b, s = pure a \u2227 t = pure b \u2227 a * b = 1\n[PROOFSTEP]\nhave := (mul_le_mul_iff_of_ge s.fst_le_snd t.fst_le_snd).1 (h.2.trans h.1.symm).le\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : NonemptyInterval \u03b1\nh : (s * t).fst = 1.fst \u2227 (s * t).snd = 1.snd\nthis : s.fst = s.snd \u2227 t.fst = t.snd\n\u22a2 \u2203 a b, s = pure a \u2227 t = pure b \u2227 a * b = 1\n[PROOFSTEP]\nrefine' \u27e8s.fst, t.fst, _, _, h.1\u27e9\n[GOAL]\ncase refine'_1.refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : NonemptyInterval \u03b1\nh : (s * t).fst = 1.fst \u2227 (s * t).snd = 1.snd\nthis : s.fst = s.snd \u2227 t.fst = t.snd\n\u22a2 s = pure s.fst\n[PROOFSTEP]\napply NonemptyInterval.ext\n[GOAL]\ncase refine'_1.refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : NonemptyInterval \u03b1\nh : (s * t).fst = 1.fst \u2227 (s * t).snd = 1.snd\nthis : s.fst = s.snd \u2227 t.fst = t.snd\n\u22a2 t = pure t.fst\n[PROOFSTEP]\napply NonemptyInterval.ext\n[GOAL]\ncase refine'_1.refine'_1.h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : NonemptyInterval \u03b1\nh : (s * t).fst = 1.fst \u2227 (s * t).snd = 1.snd\nthis : s.fst = s.snd \u2227 t.fst = t.snd\n\u22a2 s.toProd = (pure s.fst).toProd\n[PROOFSTEP]\ndsimp [pure]\n[GOAL]\ncase refine'_1.refine'_2.h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : NonemptyInterval \u03b1\nh : (s * t).fst = 1.fst \u2227 (s * t).snd = 1.snd\nthis : s.fst = s.snd \u2227 t.fst = t.snd\n\u22a2 t.toProd = (pure t.fst).toProd\n[PROOFSTEP]\ndsimp [pure]\n[GOAL]\ncase refine'_1.refine'_1.h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : NonemptyInterval \u03b1\nh : (s * t).fst = 1.fst \u2227 (s * t).snd = 1.snd\nthis : s.fst = s.snd \u2227 t.fst = t.snd\n\u22a2 s.toProd = (s.fst, s.fst)\n[PROOFSTEP]\nnth_rw 2 [this.1]\n[GOAL]\ncase refine'_1.refine'_2.h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : NonemptyInterval \u03b1\nh : (s * t).fst = 1.fst \u2227 (s * t).snd = 1.snd\nthis : s.fst = s.snd \u2227 t.fst = t.snd\n\u22a2 t.toProd = (t.fst, t.fst)\n[PROOFSTEP]\nnth_rw 2 [this.2]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : NonemptyInterval \u03b1\n\u22a2 (\u2203 a b, s = pure a \u2227 t = pure b \u2227 a * b = 1) \u2192 s * t = 1\n[PROOFSTEP]\nrintro \u27e8b, c, rfl, rfl, h\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\nb c : \u03b1\nh : b * c = 1\n\u22a2 pure b * pure c = 1\n[PROOFSTEP]\nrw [pure_mul_pure, h, pure_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : NonemptyInterval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (NonemptyInterval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 s - t = s + -t\n[PROOFSTEP]\nrefine NonemptyInterval.ext _ _ (Prod.ext ?_ ?_)\n[GOAL]\ncase refine_1\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : NonemptyInterval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (NonemptyInterval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 (s - t).fst = (s + -t).fst\n[PROOFSTEP]\nexact sub_eq_add_neg _ _\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : NonemptyInterval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (NonemptyInterval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 (s - t).snd = (s + -t).snd\n[PROOFSTEP]\nexact sub_eq_add_neg _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t : NonemptyInterval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (NonemptyInterval \u03b1) := addCommMonoid\ns : NonemptyInterval \u03b1\n\u22a2 - -s = s\n[PROOFSTEP]\napply NonemptyInterval.ext\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t : NonemptyInterval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (NonemptyInterval \u03b1) := addCommMonoid\ns : NonemptyInterval \u03b1\n\u22a2 (- -s).toProd = s.toProd\n[PROOFSTEP]\nexact neg_neg _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : NonemptyInterval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (NonemptyInterval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 -(s + t) = -t + -s\n[PROOFSTEP]\nrefine NonemptyInterval.ext _ _ (Prod.ext ?_ ?_)\n[GOAL]\ncase refine_1\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : NonemptyInterval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (NonemptyInterval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 (-(s + t)).fst = (-t + -s).fst\n[PROOFSTEP]\nexact neg_add_rev _ _\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : NonemptyInterval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (NonemptyInterval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 (-(s + t)).snd = (-t + -s).snd\n[PROOFSTEP]\nexact neg_add_rev _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : NonemptyInterval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (NonemptyInterval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\nh : s + t = 0\n\u22a2 -s = t\n[PROOFSTEP]\nobtain \u27e8a, b, rfl, rfl, hab\u27e9 := NonemptyInterval.add_eq_zero_iff.1 h\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t : NonemptyInterval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (NonemptyInterval \u03b1) := addCommMonoid\na b : \u03b1\nhab : a + b = 0\nh : pure a + pure b = 0\n\u22a2 -pure a = pure b\n[PROOFSTEP]\nrw [neg_pure, neg_eq_of_add_eq_zero_right hab]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : NonemptyInterval \u03b1\nsrc\u271d : CommMonoid (NonemptyInterval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 s / t = s * t\u207b\u00b9\n[PROOFSTEP]\nrefine NonemptyInterval.ext _ _ (Prod.ext ?_ ?_)\n[GOAL]\ncase refine_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : NonemptyInterval \u03b1\nsrc\u271d : CommMonoid (NonemptyInterval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 (s / t).fst = (s * t\u207b\u00b9).fst\n[PROOFSTEP]\nexact div_eq_mul_inv _ _\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : NonemptyInterval \u03b1\nsrc\u271d : CommMonoid (NonemptyInterval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 (s / t).snd = (s * t\u207b\u00b9).snd\n[PROOFSTEP]\nexact div_eq_mul_inv _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t : NonemptyInterval \u03b1\nsrc\u271d : CommMonoid (NonemptyInterval \u03b1) := commMonoid\ns : NonemptyInterval \u03b1\n\u22a2 s\u207b\u00b9\u207b\u00b9 = s\n[PROOFSTEP]\napply NonemptyInterval.ext\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t : NonemptyInterval \u03b1\nsrc\u271d : CommMonoid (NonemptyInterval \u03b1) := commMonoid\ns : NonemptyInterval \u03b1\n\u22a2 s\u207b\u00b9\u207b\u00b9.toProd = s.toProd\n[PROOFSTEP]\nexact inv_inv _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : NonemptyInterval \u03b1\nsrc\u271d : CommMonoid (NonemptyInterval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 (s * t)\u207b\u00b9 = t\u207b\u00b9 * s\u207b\u00b9\n[PROOFSTEP]\nrefine NonemptyInterval.ext _ _ (Prod.ext ?_ ?_)\n[GOAL]\ncase refine_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : NonemptyInterval \u03b1\nsrc\u271d : CommMonoid (NonemptyInterval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 (s * t)\u207b\u00b9.fst = (t\u207b\u00b9 * s\u207b\u00b9).fst\n[PROOFSTEP]\nexact mul_inv_rev _ _\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : NonemptyInterval \u03b1\nsrc\u271d : CommMonoid (NonemptyInterval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 (s * t)\u207b\u00b9.snd = (t\u207b\u00b9 * s\u207b\u00b9).snd\n[PROOFSTEP]\nexact mul_inv_rev _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : NonemptyInterval \u03b1\nsrc\u271d : CommMonoid (NonemptyInterval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\nh : s * t = 1\n\u22a2 s\u207b\u00b9 = t\n[PROOFSTEP]\nobtain \u27e8a, b, rfl, rfl, hab\u27e9 := NonemptyInterval.mul_eq_one_iff.1 h\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : NonemptyInterval \u03b1\nsrc\u271d : CommMonoid (NonemptyInterval \u03b1) := commMonoid\na b : \u03b1\nhab : a * b = 1\nh : pure a * pure b = 1\n\u22a2 (pure a)\u207b\u00b9 = pure b\n[PROOFSTEP]\nrw [inv_pure, inv_eq_of_mul_eq_one_right hab]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : Interval \u03b1\n\u22a2 s * t = 1 \u2194 \u2203 a b, s = pure a \u2227 t = pure b \u2227 a * b = 1\n[PROOFSTEP]\ncases s\n[GOAL]\ncase none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\nt : Interval \u03b1\n\u22a2 none * t = 1 \u2194 \u2203 a b, none = pure a \u2227 t = pure b \u2227 a * b = 1\n[PROOFSTEP]\nsimp [WithBot.none_eq_bot]\n[GOAL]\ncase some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\nt : Interval \u03b1\nval\u271d : NonemptyInterval \u03b1\n\u22a2 some val\u271d * t = 1 \u2194 \u2203 a b, some val\u271d = pure a \u2227 t = pure b \u2227 a * b = 1\n[PROOFSTEP]\ncases t\n[GOAL]\ncase some.none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\nval\u271d : NonemptyInterval \u03b1\n\u22a2 some val\u271d * none = 1 \u2194 \u2203 a b, some val\u271d = pure a \u2227 none = pure b \u2227 a * b = 1\n[PROOFSTEP]\nsimp [WithBot.none_eq_bot]\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\nval\u271d\u00b9 val\u271d : NonemptyInterval \u03b1\n\u22a2 some val\u271d\u00b9 * some val\u271d = 1 \u2194 \u2203 a b, some val\u271d\u00b9 = pure a \u2227 some val\u271d = pure b \u2227 a * b = 1\n[PROOFSTEP]\nsimp_rw [WithBot.some_eq_coe, \u2190 NonemptyInterval.coe_mul_interval, \u2190 NonemptyInterval.coe_one_interval, WithBot.coe_inj,\n NonemptyInterval.coe_eq_pure]\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\nval\u271d\u00b9 val\u271d : NonemptyInterval \u03b1\n\u22a2 val\u271d\u00b9 * val\u271d = 1 \u2194 \u2203 a b, val\u271d\u00b9 = NonemptyInterval.pure a \u2227 val\u271d = NonemptyInterval.pure b \u2227 a * b = 1\n[PROOFSTEP]\nexact NonemptyInterval.mul_eq_one_iff\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\n\u22a2 \u2200 (a b : Interval \u03b1), a - b = a + -b\n[PROOFSTEP]\nrintro (_ | s) (_ | t)\n[GOAL]\ncase none.none\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\n\u22a2 none - none = none + -none\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (sub_eq_add_neg _ _)\n[GOAL]\ncase none.none\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\n\u22a2 none - none = none + -none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase none.some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t\u271d : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\nt : NonemptyInterval \u03b1\n\u22a2 none - some t = none + -some t\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (sub_eq_add_neg _ _)\n[GOAL]\ncase none.some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t\u271d : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\nt : NonemptyInterval \u03b1\n\u22a2 none - some t = none + -some t\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.none\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns : NonemptyInterval \u03b1\n\u22a2 some s - none = some s + -none\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (sub_eq_add_neg _ _)\n[GOAL]\ncase some.none\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns : NonemptyInterval \u03b1\n\u22a2 some s - none = some s + -none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 some s - some t = some s + -some t\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (sub_eq_add_neg _ _)\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 some s - some t = some s + -some t\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 some s - some t = some s + -some t\n[PROOFSTEP]\nexact congr_arg some (sub_eq_add_neg _ _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\n\u22a2 \u2200 (x : Interval \u03b1), - -x = x\n[PROOFSTEP]\nrintro (_ | s)\n[GOAL]\ncase none\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\n\u22a2 - -none = none\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (neg_neg _)\n[GOAL]\ncase none\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\n\u22a2 - -none = none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns : NonemptyInterval \u03b1\n\u22a2 - -some s = some s\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (neg_neg _)\n[GOAL]\ncase some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns : NonemptyInterval \u03b1\n\u22a2 - -some s = some s\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns : NonemptyInterval \u03b1\n\u22a2 - -some s = some s\n[PROOFSTEP]\nexact congr_arg some (neg_neg _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\n\u22a2 \u2200 (a b : Interval \u03b1), -(a + b) = -b + -a\n[PROOFSTEP]\nrintro (_ | s) (_ | t)\n[GOAL]\ncase none.none\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\n\u22a2 -(none + none) = -none + -none\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (neg_add_rev _ _)\n[GOAL]\ncase none.none\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\n\u22a2 -(none + none) = -none + -none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase none.some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t\u271d : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\nt : NonemptyInterval \u03b1\n\u22a2 -(none + some t) = -some t + -none\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (neg_add_rev _ _)\n[GOAL]\ncase none.some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t\u271d : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\nt : NonemptyInterval \u03b1\n\u22a2 -(none + some t) = -some t + -none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.none\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns : NonemptyInterval \u03b1\n\u22a2 -(some s + none) = -none + -some s\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (neg_add_rev _ _)\n[GOAL]\ncase some.none\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns : NonemptyInterval \u03b1\n\u22a2 -(some s + none) = -none + -some s\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 -(some s + some t) = -some t + -some s\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (neg_add_rev _ _)\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 -(some s + some t) = -some t + -some s\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 -(some s + some t) = -some t + -some s\n[PROOFSTEP]\nexact congr_arg some (neg_add_rev _ _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\n\u22a2 \u2200 (a b : Interval \u03b1), a + b = 0 \u2192 -a = b\n[PROOFSTEP]\nrintro (_ | s) (_ | t) h\n[GOAL]\ncase none.none\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\nh : none + none = 0\n\u22a2 -none = none\n[PROOFSTEP]\nfirst\n| cases h\n| exact congr_arg some (neg_eq_of_add_eq_zero_right <| Option.some_injective _ h)\n[GOAL]\ncase none.none\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\nh : none + none = 0\n\u22a2 -none = none\n[PROOFSTEP]\ncases h\n[GOAL]\ncase none.some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t\u271d : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\nt : NonemptyInterval \u03b1\nh : none + some t = 0\n\u22a2 -none = some t\n[PROOFSTEP]\nfirst\n| cases h\n| exact congr_arg some (neg_eq_of_add_eq_zero_right <| Option.some_injective _ h)\n[GOAL]\ncase none.some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns t\u271d : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\nt : NonemptyInterval \u03b1\nh : none + some t = 0\n\u22a2 -none = some t\n[PROOFSTEP]\ncases h\n[GOAL]\ncase some.none\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns : NonemptyInterval \u03b1\nh : some s + none = 0\n\u22a2 -some s = none\n[PROOFSTEP]\nfirst\n| cases h\n| exact congr_arg some (neg_eq_of_add_eq_zero_right <| Option.some_injective _ h)\n[GOAL]\ncase some.none\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns : NonemptyInterval \u03b1\nh : some s + none = 0\n\u22a2 -some s = none\n[PROOFSTEP]\ncases h\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\nh : some s + some t = 0\n\u22a2 -some s = some t\n[PROOFSTEP]\nfirst\n| cases h\n| exact congr_arg some (neg_eq_of_add_eq_zero_right <| Option.some_injective _ h)\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\nh : some s + some t = 0\n\u22a2 -some s = some t\n[PROOFSTEP]\ncases h\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b9 : OrderedCommGroup \u03b1\u271d\ns\u271d t\u271d : Interval \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : OrderedAddCommGroup \u03b1\nsrc\u271d : AddCommMonoid (Interval \u03b1) := addCommMonoid\ns t : NonemptyInterval \u03b1\nh : some s + some t = 0\n\u22a2 -some s = some t\n[PROOFSTEP]\nexact congr_arg some (neg_eq_of_add_eq_zero_right <| Option.some_injective _ h)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\n\u22a2 \u2200 (a b : Interval \u03b1), a / b = a * b\u207b\u00b9\n[PROOFSTEP]\nrintro (_ | s) (_ | t)\n[GOAL]\ncase none.none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\n\u22a2 none / none = none * none\u207b\u00b9\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (div_eq_mul_inv _ _)\n[GOAL]\ncase none.none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\n\u22a2 none / none = none * none\u207b\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\ncase none.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t\u271d : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\nt : NonemptyInterval \u03b1\n\u22a2 none / some t = none * (some t)\u207b\u00b9\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (div_eq_mul_inv _ _)\n[GOAL]\ncase none.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t\u271d : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\nt : NonemptyInterval \u03b1\n\u22a2 none / some t = none * (some t)\u207b\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns : NonemptyInterval \u03b1\n\u22a2 some s / none = some s * none\u207b\u00b9\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (div_eq_mul_inv _ _)\n[GOAL]\ncase some.none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns : NonemptyInterval \u03b1\n\u22a2 some s / none = some s * none\u207b\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 some s / some t = some s * (some t)\u207b\u00b9\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (div_eq_mul_inv _ _)\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 some s / some t = some s * (some t)\u207b\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 some s / some t = some s * (some t)\u207b\u00b9\n[PROOFSTEP]\nexact congr_arg some (div_eq_mul_inv _ _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\n\u22a2 \u2200 (x : Interval \u03b1), x\u207b\u00b9\u207b\u00b9 = x\n[PROOFSTEP]\nrintro (_ | s)\n[GOAL]\ncase none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\n\u22a2 none\u207b\u00b9\u207b\u00b9 = none\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (inv_inv _)\n[GOAL]\ncase none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\n\u22a2 none\u207b\u00b9\u207b\u00b9 = none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns : NonemptyInterval \u03b1\n\u22a2 (some s)\u207b\u00b9\u207b\u00b9 = some s\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (inv_inv _)\n[GOAL]\ncase some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns : NonemptyInterval \u03b1\n\u22a2 (some s)\u207b\u00b9\u207b\u00b9 = some s\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns : NonemptyInterval \u03b1\n\u22a2 (some s)\u207b\u00b9\u207b\u00b9 = some s\n[PROOFSTEP]\nexact congr_arg some (inv_inv _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\n\u22a2 \u2200 (a b : Interval \u03b1), (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9\n[PROOFSTEP]\nrintro (_ | s) (_ | t)\n[GOAL]\ncase none.none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\n\u22a2 (none * none)\u207b\u00b9 = none\u207b\u00b9 * none\u207b\u00b9\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (mul_inv_rev _ _)\n[GOAL]\ncase none.none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\n\u22a2 (none * none)\u207b\u00b9 = none\u207b\u00b9 * none\u207b\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\ncase none.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t\u271d : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\nt : NonemptyInterval \u03b1\n\u22a2 (none * some t)\u207b\u00b9 = (some t)\u207b\u00b9 * none\u207b\u00b9\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (mul_inv_rev _ _)\n[GOAL]\ncase none.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t\u271d : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\nt : NonemptyInterval \u03b1\n\u22a2 (none * some t)\u207b\u00b9 = (some t)\u207b\u00b9 * none\u207b\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns : NonemptyInterval \u03b1\n\u22a2 (some s * none)\u207b\u00b9 = none\u207b\u00b9 * (some s)\u207b\u00b9\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (mul_inv_rev _ _)\n[GOAL]\ncase some.none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns : NonemptyInterval \u03b1\n\u22a2 (some s * none)\u207b\u00b9 = none\u207b\u00b9 * (some s)\u207b\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 (some s * some t)\u207b\u00b9 = (some t)\u207b\u00b9 * (some s)\u207b\u00b9\n[PROOFSTEP]\nfirst\n| rfl\n| exact congr_arg some (mul_inv_rev _ _)\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 (some s * some t)\u207b\u00b9 = (some t)\u207b\u00b9 * (some s)\u207b\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\n\u22a2 (some s * some t)\u207b\u00b9 = (some t)\u207b\u00b9 * (some s)\u207b\u00b9\n[PROOFSTEP]\nexact congr_arg some (mul_inv_rev _ _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\n\u22a2 \u2200 (a b : Interval \u03b1), a * b = 1 \u2192 a\u207b\u00b9 = b\n[PROOFSTEP]\nrintro (_ | s) (_ | t) h\n[GOAL]\ncase none.none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\nh : none * none = 1\n\u22a2 none\u207b\u00b9 = none\n[PROOFSTEP]\nfirst\n| cases h\n| exact congr_arg some (inv_eq_of_mul_eq_one_right <| Option.some_injective _ h)\n[GOAL]\ncase none.none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\nh : none * none = 1\n\u22a2 none\u207b\u00b9 = none\n[PROOFSTEP]\ncases h\n[GOAL]\ncase none.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t\u271d : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\nt : NonemptyInterval \u03b1\nh : none * some t = 1\n\u22a2 none\u207b\u00b9 = some t\n[PROOFSTEP]\nfirst\n| cases h\n| exact congr_arg some (inv_eq_of_mul_eq_one_right <| Option.some_injective _ h)\n[GOAL]\ncase none.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns t\u271d : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\nt : NonemptyInterval \u03b1\nh : none * some t = 1\n\u22a2 none\u207b\u00b9 = some t\n[PROOFSTEP]\ncases h\n[GOAL]\ncase some.none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns : NonemptyInterval \u03b1\nh : some s * none = 1\n\u22a2 (some s)\u207b\u00b9 = none\n[PROOFSTEP]\nfirst\n| cases h\n| exact congr_arg some (inv_eq_of_mul_eq_one_right <| Option.some_injective _ h)\n[GOAL]\ncase some.none\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns : NonemptyInterval \u03b1\nh : some s * none = 1\n\u22a2 (some s)\u207b\u00b9 = none\n[PROOFSTEP]\ncases h\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\nh : some s * some t = 1\n\u22a2 (some s)\u207b\u00b9 = some t\n[PROOFSTEP]\nfirst\n| cases h\n| exact congr_arg some (inv_eq_of_mul_eq_one_right <| Option.some_injective _ h)\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\nh : some s * some t = 1\n\u22a2 (some s)\u207b\u00b9 = some t\n[PROOFSTEP]\ncases h\n[GOAL]\ncase some.some\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedCommGroup \u03b1\ns\u271d t\u271d : Interval \u03b1\nsrc\u271d : CommMonoid (Interval \u03b1) := commMonoid\ns t : NonemptyInterval \u03b1\nh : some s * some t = 1\n\u22a2 (some s)\u207b\u00b9 = some t\n[PROOFSTEP]\nexact congr_arg some (inv_eq_of_mul_eq_one_right <| Option.some_injective _ h)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedAddCommGroup \u03b1\ns t : NonemptyInterval \u03b1\na : \u03b1\n\u22a2 length (s - t) = length s + length t\n[PROOFSTEP]\nsimp [sub_eq_add_neg]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedAddCommGroup \u03b1\ns t : Interval \u03b1\na : \u03b1\nx\u271d : Interval \u03b1\n\u22a2 length (\u22a5 + x\u271d) \u2264 length \u22a5 + length x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedAddCommGroup \u03b1\ns t : Interval \u03b1\na : \u03b1\nx\u271d : Interval \u03b1\n\u22a2 length (x\u271d + \u22a5) \u2264 length x\u271d + length \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedAddCommGroup \u03b1\ns t : Interval \u03b1\na : \u03b1\n\u22a2 length (s - t) \u2264 length s + length t\n[PROOFSTEP]\nsimpa [sub_eq_add_neg] using length_add_le s (-t)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedAddCommGroup \u03b1\ns\u271d t : Interval \u03b1\na : \u03b1\nf : \u03b9 \u2192 Interval \u03b1\ns : Finset \u03b9\n\u22a2 length (\u2211 i in s, f i) \u2264 \u2211 i in s, length (f i)\n[PROOFSTEP]\napply Finset.le_sum_of_subadditive\n[GOAL]\ncase h_one\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedAddCommGroup \u03b1\ns\u271d t : Interval \u03b1\na : \u03b1\nf : \u03b9 \u2192 Interval \u03b1\ns : Finset \u03b9\n\u22a2 length 0 = 0\ncase h_mul\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedAddCommGroup \u03b1\ns\u271d t : Interval \u03b1\na : \u03b1\nf : \u03b9 \u2192 Interval \u03b1\ns : Finset \u03b9\n\u22a2 \u2200 (x y : Interval \u03b1), length (x + y) \u2264 length x + length y\n[PROOFSTEP]\nexact length_zero\n[GOAL]\ncase h_mul\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : OrderedAddCommGroup \u03b1\ns\u271d t : Interval \u03b1\na : \u03b1\nf : \u03b9 \u2192 Interval \u03b1\ns : Finset \u03b9\n\u22a2 \u2200 (x y : Interval \u03b1), length (x + y) \u2264 length x + length y\n[PROOFSTEP]\nexact length_add_le\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Interval", "llama_tokens": 14360, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419704455588, "lm_q2_score": 0.6442251133170357, "lm_q1q2_score": 0.446217351898425}} {"text": "[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Add R\ninst\u271d : Mul R\nc x y : RingCon R\nh : (fun c => Setoid.r) x = (fun c => Setoid.r) y\n\u22a2 x = y\n[PROOFSTEP]\nrcases x with \u27e8\u27e8x, _\u27e9, _\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Add R\ninst\u271d : Mul R\nc y : RingCon R\nx : R \u2192 R \u2192 Prop\niseqv\u271d : Equivalence x\nadd'\u271d : \u2200 {w x_1 y z : R}, Setoid.r w x_1 \u2192 Setoid.r y z \u2192 Setoid.r (w + y) (x_1 + z)\nmul'\u271d : \u2200 {w x_1 y z : R}, Setoid.r w x_1 \u2192 Setoid.r y z \u2192 Setoid.r (w * y) (x_1 * z)\nh :\n (fun c => Setoid.r) { toSetoid := { r := x, iseqv := iseqv\u271d }, add' := add'\u271d, mul' := mul'\u271d } = (fun c => Setoid.r) y\n\u22a2 { toSetoid := { r := x, iseqv := iseqv\u271d }, add' := add'\u271d, mul' := mul'\u271d } = y\n[PROOFSTEP]\nrcases y with \u27e8\u27e8y, _\u27e9, _\u27e9\n[GOAL]\ncase mk.mk.mk.mk\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Add R\ninst\u271d : Mul R\nc : RingCon R\nx : R \u2192 R \u2192 Prop\niseqv\u271d\u00b9 : Equivalence x\nadd'\u271d\u00b9 : \u2200 {w x_1 y z : R}, Setoid.r w x_1 \u2192 Setoid.r y z \u2192 Setoid.r (w + y) (x_1 + z)\nmul'\u271d\u00b9 : \u2200 {w x_1 y z : R}, Setoid.r w x_1 \u2192 Setoid.r y z \u2192 Setoid.r (w * y) (x_1 * z)\ny : R \u2192 R \u2192 Prop\niseqv\u271d : Equivalence y\nadd'\u271d : \u2200 {w x y_1 z : R}, Setoid.r w x \u2192 Setoid.r y_1 z \u2192 Setoid.r (w + y_1) (x + z)\nmul'\u271d : \u2200 {w x y_1 z : R}, Setoid.r w x \u2192 Setoid.r y_1 z \u2192 Setoid.r (w * y_1) (x * z)\nh :\n (fun c => Setoid.r) { toSetoid := { r := x, iseqv := iseqv\u271d\u00b9 }, add' := add'\u271d\u00b9, mul' := mul'\u271d\u00b9 } =\n (fun c => Setoid.r) { toSetoid := { r := y, iseqv := iseqv\u271d }, add' := add'\u271d, mul' := mul'\u271d }\n\u22a2 { toSetoid := { r := x, iseqv := iseqv\u271d\u00b9 }, add' := add'\u271d\u00b9, mul' := mul'\u271d\u00b9 } =\n { toSetoid := { r := y, iseqv := iseqv\u271d }, add' := add'\u271d, mul' := mul'\u271d }\n[PROOFSTEP]\nhave : x = y := h\n[GOAL]\ncase mk.mk.mk.mk\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Add R\ninst\u271d : Mul R\nc : RingCon R\nx : R \u2192 R \u2192 Prop\niseqv\u271d\u00b9 : Equivalence x\nadd'\u271d\u00b9 : \u2200 {w x_1 y z : R}, Setoid.r w x_1 \u2192 Setoid.r y z \u2192 Setoid.r (w + y) (x_1 + z)\nmul'\u271d\u00b9 : \u2200 {w x_1 y z : R}, Setoid.r w x_1 \u2192 Setoid.r y z \u2192 Setoid.r (w * y) (x_1 * z)\ny : R \u2192 R \u2192 Prop\niseqv\u271d : Equivalence y\nadd'\u271d : \u2200 {w x y_1 z : R}, Setoid.r w x \u2192 Setoid.r y_1 z \u2192 Setoid.r (w + y_1) (x + z)\nmul'\u271d : \u2200 {w x y_1 z : R}, Setoid.r w x \u2192 Setoid.r y_1 z \u2192 Setoid.r (w * y_1) (x * z)\nh :\n (fun c => Setoid.r) { toSetoid := { r := x, iseqv := iseqv\u271d\u00b9 }, add' := add'\u271d\u00b9, mul' := mul'\u271d\u00b9 } =\n (fun c => Setoid.r) { toSetoid := { r := y, iseqv := iseqv\u271d }, add' := add'\u271d, mul' := mul'\u271d }\nthis : x = y\n\u22a2 { toSetoid := { r := x, iseqv := iseqv\u271d\u00b9 }, add' := add'\u271d\u00b9, mul' := mul'\u271d\u00b9 } =\n { toSetoid := { r := y, iseqv := iseqv\u271d }, add' := add'\u271d, mul' := mul'\u271d }\n[PROOFSTEP]\nsubst x\n[GOAL]\ncase mk.mk.mk.mk\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Add R\ninst\u271d : Mul R\nc : RingCon R\ny : R \u2192 R \u2192 Prop\niseqv\u271d\u00b9 : Equivalence y\nadd'\u271d\u00b9 : \u2200 {w x y_1 z : R}, Setoid.r w x \u2192 Setoid.r y_1 z \u2192 Setoid.r (w + y_1) (x + z)\nmul'\u271d\u00b9 : \u2200 {w x y_1 z : R}, Setoid.r w x \u2192 Setoid.r y_1 z \u2192 Setoid.r (w * y_1) (x * z)\niseqv\u271d : Equivalence y\nadd'\u271d : \u2200 {w x y_1 z : R}, Setoid.r w x \u2192 Setoid.r y_1 z \u2192 Setoid.r (w + y_1) (x + z)\nmul'\u271d : \u2200 {w x y_1 z : R}, Setoid.r w x \u2192 Setoid.r y_1 z \u2192 Setoid.r (w * y_1) (x * z)\nh :\n (fun c => Setoid.r) { toSetoid := { r := y, iseqv := iseqv\u271d }, add' := add'\u271d, mul' := mul'\u271d } =\n (fun c => Setoid.r) { toSetoid := { r := y, iseqv := iseqv\u271d\u00b9 }, add' := add'\u271d\u00b9, mul' := mul'\u271d\u00b9 }\n\u22a2 { toSetoid := { r := y, iseqv := iseqv\u271d }, add' := add'\u271d, mul' := mul'\u271d } =\n { toSetoid := { r := y, iseqv := iseqv\u271d\u00b9 }, add' := add'\u271d\u00b9, mul' := mul'\u271d\u00b9 }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Add R\ninst\u271d : Mul R\nc : RingCon R\nd : (a b : R) \u2192 Decidable (\u2191c a b)\n\u22a2 DecidableEq (RingCon.Quotient c)\n[PROOFSTEP]\ndelta RingCon.Quotient\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Add R\ninst\u271d : Mul R\nc : RingCon R\nd : (a b : R) \u2192 Decidable (\u2191c a b)\n\u22a2 DecidableEq (Quotient c.toSetoid)\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Congruence", "llama_tokens": 2081, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7826624789529375, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.44600228879370835}} {"text": "[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Monoid M\n\u22a2 \u2200 (b : M), 1 \u2022 b = b\n[PROOFSTEP]\nsimp only [units_smul_def, ofConjAct_one, Units.val_one, one_mul, inv_one, mul_one, forall_const]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Monoid M\n\u22a2 \u2200 (x y : ConjAct M\u02e3) (b : M), (x * y) \u2022 b = x \u2022 y \u2022 b\n[PROOFSTEP]\nsimp only [units_smul_def]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Monoid M\n\u22a2 \u2200 (x y : ConjAct M\u02e3) (b : M),\n \u2191(\u2191ofConjAct (x * y)) * b * \u2191(\u2191ofConjAct (x * y))\u207b\u00b9 =\n \u2191(\u2191ofConjAct x) * (\u2191(\u2191ofConjAct y) * b * \u2191(\u2191ofConjAct y)\u207b\u00b9) * \u2191(\u2191ofConjAct x)\u207b\u00b9\n[PROOFSTEP]\nsimp only [map_mul, Units.val_mul, mul_assoc, mul_inv_rev, forall_const, forall]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Monoid M\n\u22a2 \u2200 (r : ConjAct M\u02e3) (x y : M), r \u2022 (x * y) = r \u2022 x * r \u2022 y\n[PROOFSTEP]\nsimp only [units_smul_def]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Monoid M\n\u22a2 \u2200 (r : ConjAct M\u02e3) (x y : M),\n \u2191(\u2191ofConjAct r) * (x * y) * \u2191(\u2191ofConjAct r)\u207b\u00b9 =\n \u2191(\u2191ofConjAct r) * x * \u2191(\u2191ofConjAct r)\u207b\u00b9 * (\u2191(\u2191ofConjAct r) * y * \u2191(\u2191ofConjAct r)\u207b\u00b9)\n[PROOFSTEP]\nsimp only [mul_assoc, Units.inv_mul_cancel_left, forall_const, forall]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Monoid M\n\u22a2 \u2200 (r : ConjAct M\u02e3), r \u2022 1 = 1\n[PROOFSTEP]\nsimp [units_smul_def, mul_one, Units.mul_inv, forall, forall_const]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : SMul \u03b1 M\ninst\u271d\u00b9 : SMulCommClass \u03b1 M M\ninst\u271d : IsScalarTower \u03b1 M M\na : \u03b1\num : ConjAct M\u02e3\nm : M\n\u22a2 a \u2022 um \u2022 m = um \u2022 a \u2022 m\n[PROOFSTEP]\nrw [units_smul_def, units_smul_def, mul_smul_comm, smul_mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Semiring R\nsrc\u271d : MulDistribMulAction (ConjAct R\u02e3) R := unitsMulDistribMulAction\n\u22a2 \u2200 (a : ConjAct R\u02e3), a \u2022 0 = 0\n[PROOFSTEP]\nsimp only [units_smul_def, mul_zero, zero_mul, forall, forall_const]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Semiring R\nsrc\u271d : MulDistribMulAction (ConjAct R\u02e3) R := unitsMulDistribMulAction\n\u22a2 \u2200 (a : ConjAct R\u02e3) (x y : R), a \u2022 (x + y) = a \u2022 x + a \u2022 y\n[PROOFSTEP]\nsimp only [units_smul_def]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Semiring R\nsrc\u271d : MulDistribMulAction (ConjAct R\u02e3) R := unitsMulDistribMulAction\n\u22a2 \u2200 (a : ConjAct R\u02e3) (x y : R),\n \u2191(\u2191ofConjAct a) * (x + y) * \u2191(\u2191ofConjAct a)\u207b\u00b9 =\n \u2191(\u2191ofConjAct a) * x * \u2191(\u2191ofConjAct a)\u207b\u00b9 + \u2191(\u2191ofConjAct a) * y * \u2191(\u2191ofConjAct a)\u207b\u00b9\n[PROOFSTEP]\nsimp only [mul_add, add_mul, forall_const, forall]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : GroupWithZero G\u2080\n\u22a2 \u2200 (b : G\u2080), 1 \u2022 b = b\n[PROOFSTEP]\nsimp only [smul_def]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : GroupWithZero G\u2080\n\u22a2 \u2200 (b : G\u2080), \u2191ofConjAct 1 * b * (\u2191ofConjAct 1)\u207b\u00b9 = b\n[PROOFSTEP]\nsimp only [map_one, one_mul, inv_one, mul_one, forall_const]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : GroupWithZero G\u2080\n\u22a2 \u2200 (x y : ConjAct G\u2080) (b : G\u2080), (x * y) \u2022 b = x \u2022 y \u2022 b\n[PROOFSTEP]\nsimp only [smul_def]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : GroupWithZero G\u2080\n\u22a2 \u2200 (x y : ConjAct G\u2080) (b : G\u2080),\n \u2191ofConjAct (x * y) * b * (\u2191ofConjAct (x * y))\u207b\u00b9 =\n \u2191ofConjAct x * (\u2191ofConjAct y * b * (\u2191ofConjAct y)\u207b\u00b9) * (\u2191ofConjAct x)\u207b\u00b9\n[PROOFSTEP]\nsimp only [map_mul, mul_assoc, mul_inv_rev, forall_const, forall]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d\u00b3 : GroupWithZero G\u2080\ninst\u271d\u00b2 : SMul \u03b1 G\u2080\ninst\u271d\u00b9 : SMulCommClass \u03b1 G\u2080 G\u2080\ninst\u271d : IsScalarTower \u03b1 G\u2080 G\u2080\na : \u03b1\nug : ConjAct G\u2080\ng : G\u2080\n\u22a2 a \u2022 ug \u2022 g = ug \u2022 a \u2022 g\n[PROOFSTEP]\nrw [smul_def, smul_def, mul_smul_comm, smul_mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : DivisionRing K\nsrc\u271d : MulAction (ConjAct K) K := mulAction\u2080\n\u22a2 \u2200 (a : ConjAct K), a \u2022 0 = 0\n[PROOFSTEP]\nsimp only [smul_def]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : DivisionRing K\nsrc\u271d : MulAction (ConjAct K) K := mulAction\u2080\n\u22a2 \u2200 (a : ConjAct K), \u2191ofConjAct a * 0 * (\u2191ofConjAct a)\u207b\u00b9 = 0\n[PROOFSTEP]\nsimp only [mul_zero, zero_mul, forall, forall_const]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : DivisionRing K\nsrc\u271d : MulAction (ConjAct K) K := mulAction\u2080\n\u22a2 \u2200 (a : ConjAct K) (x y : K), a \u2022 (x + y) = a \u2022 x + a \u2022 y\n[PROOFSTEP]\nsimp only [smul_def]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : DivisionRing K\nsrc\u271d : MulAction (ConjAct K) K := mulAction\u2080\n\u22a2 \u2200 (a : ConjAct K) (x y : K),\n \u2191ofConjAct a * (x + y) * (\u2191ofConjAct a)\u207b\u00b9 =\n \u2191ofConjAct a * x * (\u2191ofConjAct a)\u207b\u00b9 + \u2191ofConjAct a * y * (\u2191ofConjAct a)\u207b\u00b9\n[PROOFSTEP]\nsimp only [mul_add, add_mul, forall_const, forall]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Group G\n\u22a2 \u2200 (b : G), 1 \u2022 b = b\n[PROOFSTEP]\nsimp only [smul_def, ofConjAct_one, one_mul, inv_one, mul_one, forall_const]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Group G\n\u22a2 \u2200 (x y : ConjAct G) (b : G), (x * y) \u2022 b = x \u2022 y \u2022 b\n[PROOFSTEP]\nsimp only [smul_def]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Group G\n\u22a2 \u2200 (x y : ConjAct G) (b : G),\n \u2191ofConjAct (x * y) * b * (\u2191ofConjAct (x * y))\u207b\u00b9 =\n \u2191ofConjAct x * (\u2191ofConjAct y * b * (\u2191ofConjAct y)\u207b\u00b9) * (\u2191ofConjAct x)\u207b\u00b9\n[PROOFSTEP]\nsimp only [map_mul, mul_assoc, mul_inv_rev, forall_const, forall]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Group G\n\u22a2 \u2200 (r : ConjAct G) (x y : G), r \u2022 (x * y) = r \u2022 x * r \u2022 y\n[PROOFSTEP]\nsimp only [smul_def]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Group G\n\u22a2 \u2200 (r : ConjAct G) (x y : G),\n \u2191ofConjAct r * (x * y) * (\u2191ofConjAct r)\u207b\u00b9 =\n \u2191ofConjAct r * x * (\u2191ofConjAct r)\u207b\u00b9 * (\u2191ofConjAct r * y * (\u2191ofConjAct r)\u207b\u00b9)\n[PROOFSTEP]\nsimp only [mul_assoc, inv_mul_cancel_left, forall_const, forall]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Group G\n\u22a2 \u2200 (r : ConjAct G), r \u2022 1 = 1\n[PROOFSTEP]\nsimp only [smul_def, mul_one, mul_right_inv, forall, forall_const]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : SMul \u03b1 G\ninst\u271d\u00b9 : SMulCommClass \u03b1 G G\ninst\u271d : IsScalarTower \u03b1 G G\na : \u03b1\nug : ConjAct G\ng : G\n\u22a2 a \u2022 ug \u2022 g = ug \u2022 a \u2022 g\n[PROOFSTEP]\nrw [smul_def, smul_def, mul_smul_comm, smul_mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Group G\n\u22a2 fixedPoints (ConjAct G) G = \u2191(center G)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Group G\nx : G\n\u22a2 x \u2208 fixedPoints (ConjAct G) G \u2194 x \u2208 \u2191(center G)\n[PROOFSTEP]\nsimp [mem_center_iff, smul_def, mul_inv_eq_iff_eq_mul]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Group G\ng h : G\n\u22a2 g \u2208 orbit (ConjAct G) h \u2194 IsConj g h\n[PROOFSTEP]\nrw [isConj_comm, isConj_iff, mem_orbit_iff]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Group G\ng h : G\n\u22a2 (\u2203 x, x \u2022 h = g) \u2194 \u2203 c, c * h * c\u207b\u00b9 = g\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d : Group G\ng h : G\n\u22a2 Setoid.Rel (orbitRel (ConjAct G) G) g h = IsConj g h\n[PROOFSTEP]\nrw [orbitRel_apply, mem_orbit_conjAct]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d\u00b9 inst\u271d : Group G\ng : G\n\u22a2 orbit (ConjAct G) g = ConjClasses.carrier (ConjClasses.mk g)\n[PROOFSTEP]\next h\n[GOAL]\ncase h\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d\u00b9 inst\u271d : Group G\ng h : G\n\u22a2 h \u2208 orbit (ConjAct G) g \u2194 h \u2208 ConjClasses.carrier (ConjClasses.mk g)\n[PROOFSTEP]\nrw [ConjClasses.mem_carrier_iff_mk_eq, ConjClasses.mk_eq_mk_iff_isConj, mem_orbit_conjAct]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\ng : G\nh : { x // x \u2208 H }\n\u22a2 \u2191(\u2191(MulEquiv.symm (\u2191MulAut.conjNormal g)) h) = g\u207b\u00b9 * \u2191h * g\n[PROOFSTEP]\nchange _ * _\u207b\u00b9\u207b\u00b9 = _\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\ng : G\nh : { x // x \u2208 H }\n\u22a2 \u2191ofConjAct (\u2191(MulEquiv.toMonoidHom toConjAct) g)\u207b\u00b9 * \u2191h * (\u2191(MulEquiv.toMonoidHom toConjAct) g)\u207b\u00b9\u207b\u00b9 = g\u207b\u00b9 * \u2191h * g\n[PROOFSTEP]\nrw [inv_inv]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK : Type u_6\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\ng : G\nh : { x // x \u2208 H }\n\u22a2 \u2191ofConjAct (\u2191(MulEquiv.toMonoidHom toConjAct) g)\u207b\u00b9 * \u2191h * \u2191(MulEquiv.toMonoidHom toConjAct) g = g\u207b\u00b9 * \u2191h * g\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK\u271d : Type u_6\ninst\u271d : Group G\nH : Subgroup G\nhH : Normal H\nK : Subgroup { x // x \u2208 H }\nh : Characteristic K\na : G\nha : a \u2208 map (Subgroup.subtype H) K\nb : G\n\u22a2 b * a * b\u207b\u00b9 \u2208 map (Subgroup.subtype H) K\n[PROOFSTEP]\nobtain \u27e8a, ha, rfl\u27e9 := ha\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nM : Type u_2\nG : Type u_3\nG\u2080 : Type u_4\nR : Type u_5\nK\u271d : Type u_6\ninst\u271d : Group G\nH : Subgroup G\nhH : Normal H\nK : Subgroup { x // x \u2208 H }\nh : Characteristic K\nb : G\na : { x // x \u2208 H }\nha : a \u2208 \u2191K\n\u22a2 b * \u2191(Subgroup.subtype H) a * b\u207b\u00b9 \u2208 map (Subgroup.subtype H) K\n[PROOFSTEP]\nexact K.apply_coe_mem_map H.subtype \u27e8_, (SetLike.ext_iff.mp (h.fixed (MulAut.conjNormal b)) a).mpr ha\u27e9\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.GroupAction.ConjAct", "llama_tokens": 5282, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390747, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.44573604265087075}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx : E\nhAB : IsExtreme \ud835\udd5c A B\nhBC : IsExtreme \ud835\udd5c B C\n\u22a2 IsExtreme \ud835\udd5c A C\n[PROOFSTEP]\nrefine' \u27e8Subset.trans hBC.1 hAB.1, fun x\u2081 hx\u2081A x\u2082 hx\u2082A x hxC hx \u21a6 _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx\u271d : E\nhAB : IsExtreme \ud835\udd5c A B\nhBC : IsExtreme \ud835\udd5c B C\nx\u2081 : E\nhx\u2081A : x\u2081 \u2208 A\nx\u2082 : E\nhx\u2082A : x\u2082 \u2208 A\nx : E\nhxC : x \u2208 C\nhx : x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x\u2081 \u2208 C \u2227 x\u2082 \u2208 C\n[PROOFSTEP]\nobtain \u27e8hx\u2081B, hx\u2082B\u27e9 := hAB.2 hx\u2081A hx\u2082A (hBC.1 hxC) hx\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx\u271d : E\nhAB : IsExtreme \ud835\udd5c A B\nhBC : IsExtreme \ud835\udd5c B C\nx\u2081 : E\nhx\u2081A : x\u2081 \u2208 A\nx\u2082 : E\nhx\u2082A : x\u2082 \u2208 A\nx : E\nhxC : x \u2208 C\nhx : x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\nhx\u2081B : x\u2081 \u2208 B\nhx\u2082B : x\u2082 \u2208 B\n\u22a2 x\u2081 \u2208 C \u2227 x\u2082 \u2208 C\n[PROOFSTEP]\nexact hBC.2 hx\u2081B hx\u2082B hxC hx\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx : E\nhAB : IsExtreme \ud835\udd5c A B\nhAC : IsExtreme \ud835\udd5c A C\n\u22a2 IsExtreme \ud835\udd5c A (B \u2229 C)\n[PROOFSTEP]\nuse Subset.trans (inter_subset_left _ _) hAB.1\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx : E\nhAB : IsExtreme \ud835\udd5c A B\nhAC : IsExtreme \ud835\udd5c A C\n\u22a2 \u2200 \u2983x\u2081 : E\u2984, x\u2081 \u2208 A \u2192 \u2200 \u2983x\u2082 : E\u2984, x\u2082 \u2208 A \u2192 \u2200 \u2983x : E\u2984, x \u2208 B \u2229 C \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 \u2208 B \u2229 C \u2227 x\u2082 \u2208 B \u2229 C\n[PROOFSTEP]\nrintro x\u2081 hx\u2081A x\u2082 hx\u2082A x \u27e8hxB, hxC\u27e9 hx\n[GOAL]\ncase right.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx\u271d : E\nhAB : IsExtreme \ud835\udd5c A B\nhAC : IsExtreme \ud835\udd5c A C\nx\u2081 : E\nhx\u2081A : x\u2081 \u2208 A\nx\u2082 : E\nhx\u2082A : x\u2082 \u2208 A\nx : E\nhxB : x \u2208 B\nhxC : x \u2208 C\nhx : x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x\u2081 \u2208 B \u2229 C \u2227 x\u2082 \u2208 B \u2229 C\n[PROOFSTEP]\nobtain \u27e8hx\u2081B, hx\u2082B\u27e9 := hAB.2 hx\u2081A hx\u2082A hxB hx\n[GOAL]\ncase right.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx\u271d : E\nhAB : IsExtreme \ud835\udd5c A B\nhAC : IsExtreme \ud835\udd5c A C\nx\u2081 : E\nhx\u2081A : x\u2081 \u2208 A\nx\u2082 : E\nhx\u2082A : x\u2082 \u2208 A\nx : E\nhxB : x \u2208 B\nhxC : x \u2208 C\nhx : x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\nhx\u2081B : x\u2081 \u2208 B\nhx\u2082B : x\u2082 \u2208 B\n\u22a2 x\u2081 \u2208 B \u2229 C \u2227 x\u2082 \u2208 B \u2229 C\n[PROOFSTEP]\nobtain \u27e8hx\u2081C, hx\u2082C\u27e9 := hAC.2 hx\u2081A hx\u2082A hxC hx\n[GOAL]\ncase right.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx\u271d : E\nhAB : IsExtreme \ud835\udd5c A B\nhAC : IsExtreme \ud835\udd5c A C\nx\u2081 : E\nhx\u2081A : x\u2081 \u2208 A\nx\u2082 : E\nhx\u2082A : x\u2082 \u2208 A\nx : E\nhxB : x \u2208 B\nhxC : x \u2208 C\nhx : x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\nhx\u2081B : x\u2081 \u2208 B\nhx\u2082B : x\u2082 \u2208 B\nhx\u2081C : x\u2081 \u2208 C\nhx\u2082C : x\u2082 \u2208 C\n\u22a2 x\u2081 \u2208 B \u2229 C \u2227 x\u2082 \u2208 B \u2229 C\n[PROOFSTEP]\nexact \u27e8\u27e8hx\u2081B, hx\u2081C\u27e9, hx\u2082B, hx\u2082C\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\u03b9\u271d : Type u_4\n\u03c0 : \u03b9\u271d \u2192 Type u_5\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : SMul \ud835\udd5c E\nA B C : Set E\nx : E\n\u03b9 : Sort u_6\ninst\u271d : Nonempty \u03b9\nF : \u03b9 \u2192 Set E\nhAF : \u2200 (i : \u03b9), IsExtreme \ud835\udd5c A (F i)\n\u22a2 IsExtreme \ud835\udd5c A (\u22c2 (i : \u03b9), F i)\n[PROOFSTEP]\nobtain i := Classical.arbitrary \u03b9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\u03b9\u271d : Type u_4\n\u03c0 : \u03b9\u271d \u2192 Type u_5\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : SMul \ud835\udd5c E\nA B C : Set E\nx : E\n\u03b9 : Sort u_6\ninst\u271d : Nonempty \u03b9\nF : \u03b9 \u2192 Set E\nhAF : \u2200 (i : \u03b9), IsExtreme \ud835\udd5c A (F i)\ni : \u03b9\n\u22a2 IsExtreme \ud835\udd5c A (\u22c2 (i : \u03b9), F i)\n[PROOFSTEP]\nrefine' \u27e8iInter_subset_of_subset i (hAF i).1, fun x\u2081 hx\u2081A x\u2082 hx\u2082A x hxF hx \u21a6 _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\u03b9\u271d : Type u_4\n\u03c0 : \u03b9\u271d \u2192 Type u_5\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : SMul \ud835\udd5c E\nA B C : Set E\nx\u271d : E\n\u03b9 : Sort u_6\ninst\u271d : Nonempty \u03b9\nF : \u03b9 \u2192 Set E\nhAF : \u2200 (i : \u03b9), IsExtreme \ud835\udd5c A (F i)\ni : \u03b9\nx\u2081 : E\nhx\u2081A : x\u2081 \u2208 A\nx\u2082 : E\nhx\u2082A : x\u2082 \u2208 A\nx : E\nhxF : x \u2208 \u22c2 (i : \u03b9), F i\nhx : x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x\u2081 \u2208 \u22c2 (i : \u03b9), F i \u2227 x\u2082 \u2208 \u22c2 (i : \u03b9), F i\n[PROOFSTEP]\nsimp_rw [mem_iInter] at hxF \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\u03b9\u271d : Type u_4\n\u03c0 : \u03b9\u271d \u2192 Type u_5\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : SMul \ud835\udd5c E\nA B C : Set E\nx\u271d : E\n\u03b9 : Sort u_6\ninst\u271d : Nonempty \u03b9\nF : \u03b9 \u2192 Set E\nhAF : \u2200 (i : \u03b9), IsExtreme \ud835\udd5c A (F i)\ni : \u03b9\nx\u2081 : E\nhx\u2081A : x\u2081 \u2208 A\nx\u2082 : E\nhx\u2082A : x\u2082 \u2208 A\nx : E\nhx : x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\nhxF : \u2200 (i : \u03b9), x \u2208 F i\n\u22a2 (\u2200 (i : \u03b9), x\u2081 \u2208 F i) \u2227 \u2200 (i : \u03b9), x\u2082 \u2208 F i\n[PROOFSTEP]\nhave h := fun i \u21a6 (hAF i).2 hx\u2081A hx\u2082A (hxF i) hx\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\u03b9\u271d : Type u_4\n\u03c0 : \u03b9\u271d \u2192 Type u_5\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : SMul \ud835\udd5c E\nA B C : Set E\nx\u271d : E\n\u03b9 : Sort u_6\ninst\u271d : Nonempty \u03b9\nF : \u03b9 \u2192 Set E\nhAF : \u2200 (i : \u03b9), IsExtreme \ud835\udd5c A (F i)\ni : \u03b9\nx\u2081 : E\nhx\u2081A : x\u2081 \u2208 A\nx\u2082 : E\nhx\u2082A : x\u2082 \u2208 A\nx : E\nhx : x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\nhxF : \u2200 (i : \u03b9), x \u2208 F i\nh : \u2200 (i : \u03b9), x\u2081 \u2208 F i \u2227 x\u2082 \u2208 F i\n\u22a2 (\u2200 (i : \u03b9), x\u2081 \u2208 F i) \u2227 \u2200 (i : \u03b9), x\u2082 \u2208 F i\n[PROOFSTEP]\nexact \u27e8fun i \u21a6 (h i).1, fun i \u21a6 (h i).2\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx : E\nF : Set (Set E)\nhF : Set.Nonempty F\nhA : \u2200 (B : Set E), B \u2208 F \u2192 IsExtreme \ud835\udd5c A B\n\u22a2 IsExtreme \ud835\udd5c A (\u22c2 (B : Set E) (_ : B \u2208 F), B)\n[PROOFSTEP]\nhaveI := hF.to_subtype\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx : E\nF : Set (Set E)\nhF : Set.Nonempty F\nhA : \u2200 (B : Set E), B \u2208 F \u2192 IsExtreme \ud835\udd5c A B\nthis : Nonempty \u2191F\n\u22a2 IsExtreme \ud835\udd5c A (\u22c2 (B : Set E) (_ : B \u2208 F), B)\n[PROOFSTEP]\nsimpa only [iInter_subtype] using isExtreme_iInter fun i : F \u21a6 hA _ i.2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx : E\nF : Set (Set E)\nhF : Set.Nonempty F\nhAF : \u2200 (B : Set E), B \u2208 F \u2192 IsExtreme \ud835\udd5c A B\n\u22a2 IsExtreme \ud835\udd5c A (\u22c2\u2080 F)\n[PROOFSTEP]\nobtain \u27e8B, hB\u27e9 := hF\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B\u271d C : Set E\nx : E\nF : Set (Set E)\nhAF : \u2200 (B : Set E), B \u2208 F \u2192 IsExtreme \ud835\udd5c A B\nB : Set E\nhB : B \u2208 F\n\u22a2 IsExtreme \ud835\udd5c A (\u22c2\u2080 F)\n[PROOFSTEP]\nrefine' \u27e8(sInter_subset_of_mem hB).trans (hAF B hB).1, fun x\u2081 hx\u2081A x\u2082 hx\u2082A x hxF hx \u21a6 _\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B\u271d C : Set E\nx\u271d : E\nF : Set (Set E)\nhAF : \u2200 (B : Set E), B \u2208 F \u2192 IsExtreme \ud835\udd5c A B\nB : Set E\nhB : B \u2208 F\nx\u2081 : E\nhx\u2081A : x\u2081 \u2208 A\nx\u2082 : E\nhx\u2082A : x\u2082 \u2208 A\nx : E\nhxF : x \u2208 \u22c2\u2080 F\nhx : x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x\u2081 \u2208 \u22c2\u2080 F \u2227 x\u2082 \u2208 \u22c2\u2080 F\n[PROOFSTEP]\nsimp_rw [mem_sInter] at hxF \u22a2\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B\u271d C : Set E\nx\u271d : E\nF : Set (Set E)\nhAF : \u2200 (B : Set E), B \u2208 F \u2192 IsExtreme \ud835\udd5c A B\nB : Set E\nhB : B \u2208 F\nx\u2081 : E\nhx\u2081A : x\u2081 \u2208 A\nx\u2082 : E\nhx\u2082A : x\u2082 \u2208 A\nx : E\nhx : x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\nhxF : \u2200 (t : Set E), t \u2208 F \u2192 x \u2208 t\n\u22a2 (\u2200 (t : Set E), t \u2208 F \u2192 x\u2081 \u2208 t) \u2227 \u2200 (t : Set E), t \u2208 F \u2192 x\u2082 \u2208 t\n[PROOFSTEP]\nhave h := fun B hB \u21a6 (hAF B hB).2 hx\u2081A hx\u2082A (hxF B hB) hx\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B\u271d C : Set E\nx\u271d : E\nF : Set (Set E)\nhAF : \u2200 (B : Set E), B \u2208 F \u2192 IsExtreme \ud835\udd5c A B\nB : Set E\nhB : B \u2208 F\nx\u2081 : E\nhx\u2081A : x\u2081 \u2208 A\nx\u2082 : E\nhx\u2082A : x\u2082 \u2208 A\nx : E\nhx : x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\nhxF : \u2200 (t : Set E), t \u2208 F \u2192 x \u2208 t\nh : \u2200 (B : Set E), B \u2208 F \u2192 x\u2081 \u2208 B \u2227 x\u2082 \u2208 B\n\u22a2 (\u2200 (t : Set E), t \u2208 F \u2192 x\u2081 \u2208 t) \u2227 \u2200 (t : Set E), t \u2208 F \u2192 x\u2082 \u2208 t\n[PROOFSTEP]\nexact \u27e8fun B hB \u21a6 (h B hB).1, fun B hB \u21a6 (h B hB).2\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx : E\n\u22a2 x \u2208 extremePoints \ud835\udd5c A \u2194 IsExtreme \ud835\udd5c A {x}\n[PROOFSTEP]\nrefine' \u27e8_, fun hx \u21a6 \u27e8singleton_subset_iff.1 hx.1, fun x\u2081 hx\u2081 x\u2082 hx\u2082 \u21a6 hx.2 hx\u2081 hx\u2082 rfl\u27e9\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx : E\n\u22a2 x \u2208 extremePoints \ud835\udd5c A \u2192 IsExtreme \ud835\udd5c A {x}\n[PROOFSTEP]\nrintro \u27e8hxA, hAx\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx : E\nhxA : x \u2208 A\nhAx : \u2200 \u2983x\u2081 : E\u2984, x\u2081 \u2208 A \u2192 \u2200 \u2983x\u2082 : E\u2984, x\u2082 \u2208 A \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\n\u22a2 IsExtreme \ud835\udd5c A {x}\n[PROOFSTEP]\nuse singleton_subset_iff.2 hxA\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx : E\nhxA : x \u2208 A\nhAx : \u2200 \u2983x\u2081 : E\u2984, x\u2081 \u2208 A \u2192 \u2200 \u2983x\u2082 : E\u2984, x\u2082 \u2208 A \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\n\u22a2 \u2200 \u2983x\u2081 : E\u2984, x\u2081 \u2208 A \u2192 \u2200 \u2983x\u2082 : E\u2984, x\u2082 \u2208 A \u2192 \u2200 \u2983x_1 : E\u2984, x_1 \u2208 {x} \u2192 x_1 \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 \u2208 {x} \u2227 x\u2082 \u2208 {x}\n[PROOFSTEP]\nrintro x\u2081 hx\u2081A x\u2082 hx\u2082A y (rfl : y = x)\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nA B C : Set E\nx\u2081 : E\nhx\u2081A : x\u2081 \u2208 A\nx\u2082 : E\nhx\u2082A : x\u2082 \u2208 A\ny : E\nhxA : y \u2208 A\nhAx : \u2200 \u2983x\u2081 : E\u2984, x\u2081 \u2208 A \u2192 \u2200 \u2983x\u2082 : E\u2984, x\u2082 \u2208 A \u2192 y \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = y \u2227 x\u2082 = y\n\u22a2 y \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 \u2208 {y} \u2227 x\u2082 \u2208 {y}\n[PROOFSTEP]\nexact hAx hx\u2081A hx\u2082A\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : Set E\nt : Set F\n\u22a2 extremePoints \ud835\udd5c (s \u00d7\u02e2 t) = extremePoints \ud835\udd5c s \u00d7\u02e2 extremePoints \ud835\udd5c t\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : Set E\nt : Set F\nx\u271d : E \u00d7 F\n\u22a2 x\u271d \u2208 extremePoints \ud835\udd5c (s \u00d7\u02e2 t) \u2194 x\u271d \u2208 extremePoints \ud835\udd5c s \u00d7\u02e2 extremePoints \ud835\udd5c t\n[PROOFSTEP]\nrefine' (and_congr_right fun hx \u21a6 \u27e8fun h \u21a6 _, fun h \u21a6 _\u27e9).trans and_and_and_comm\n[GOAL]\ncase h.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : Set E\nt : Set F\nx\u271d : E \u00d7 F\nhx : x\u271d \u2208 s \u00d7\u02e2 t\nh : \u2200 \u2983x\u2081 : E \u00d7 F\u2984, x\u2081 \u2208 s \u00d7\u02e2 t \u2192 \u2200 \u2983x\u2082 : E \u00d7 F\u2984, x\u2082 \u2208 s \u00d7\u02e2 t \u2192 x\u271d \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d \u2227 x\u2082 = x\u271d\n\u22a2 (\u2200 \u2983x\u2081 : E\u2984, x\u2081 \u2208 s \u2192 \u2200 \u2983x\u2082 : E\u2984, x\u2082 \u2208 s \u2192 x\u271d.fst \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d.fst \u2227 x\u2082 = x\u271d.fst) \u2227\n \u2200 \u2983x\u2081 : F\u2984, x\u2081 \u2208 t \u2192 \u2200 \u2983x\u2082 : F\u2984, x\u2082 \u2208 t \u2192 x\u271d.snd \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d.snd \u2227 x\u2082 = x\u271d.snd\ncase h.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : Set E\nt : Set F\nx\u271d : E \u00d7 F\nhx : x\u271d \u2208 s \u00d7\u02e2 t\nh :\n (\u2200 \u2983x\u2081 : E\u2984, x\u2081 \u2208 s \u2192 \u2200 \u2983x\u2082 : E\u2984, x\u2082 \u2208 s \u2192 x\u271d.fst \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d.fst \u2227 x\u2082 = x\u271d.fst) \u2227\n \u2200 \u2983x\u2081 : F\u2984, x\u2081 \u2208 t \u2192 \u2200 \u2983x\u2082 : F\u2984, x\u2082 \u2208 t \u2192 x\u271d.snd \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d.snd \u2227 x\u2082 = x\u271d.snd\n\u22a2 \u2200 \u2983x\u2081 : E \u00d7 F\u2984, x\u2081 \u2208 s \u00d7\u02e2 t \u2192 \u2200 \u2983x\u2082 : E \u00d7 F\u2984, x\u2082 \u2208 s \u00d7\u02e2 t \u2192 x\u271d \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d \u2227 x\u2082 = x\u271d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.refine'_1.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : Set E\nt : Set F\nx\u271d : E \u00d7 F\nhx : x\u271d \u2208 s \u00d7\u02e2 t\nh : \u2200 \u2983x\u2081 : E \u00d7 F\u2984, x\u2081 \u2208 s \u00d7\u02e2 t \u2192 \u2200 \u2983x\u2082 : E \u00d7 F\u2984, x\u2082 \u2208 s \u00d7\u02e2 t \u2192 x\u271d \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d \u2227 x\u2082 = x\u271d\n\u22a2 \u2200 \u2983x\u2081 : E\u2984, x\u2081 \u2208 s \u2192 \u2200 \u2983x\u2082 : E\u2984, x\u2082 \u2208 s \u2192 x\u271d.fst \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d.fst \u2227 x\u2082 = x\u271d.fst\n[PROOFSTEP]\nrintro x\u2081 hx\u2081 x\u2082 hx\u2082 hx_fst\n[GOAL]\ncase h.refine'_1.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : Set E\nt : Set F\nx\u271d : E \u00d7 F\nhx : x\u271d \u2208 s \u00d7\u02e2 t\nh : \u2200 \u2983x\u2081 : E \u00d7 F\u2984, x\u2081 \u2208 s \u00d7\u02e2 t \u2192 \u2200 \u2983x\u2082 : E \u00d7 F\u2984, x\u2082 \u2208 s \u00d7\u02e2 t \u2192 x\u271d \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d \u2227 x\u2082 = x\u271d\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\nhx_fst : x\u271d.fst \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x\u2081 = x\u271d.fst \u2227 x\u2082 = x\u271d.fst\n[PROOFSTEP]\nrefine' (h (mk_mem_prod hx\u2081 hx.2) (mk_mem_prod hx\u2082 hx.2) _).imp (congr_arg Prod.fst) (congr_arg Prod.fst)\n[GOAL]\ncase h.refine'_1.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : Set E\nt : Set F\nx\u271d : E \u00d7 F\nhx : x\u271d \u2208 s \u00d7\u02e2 t\nh : \u2200 \u2983x\u2081 : E \u00d7 F\u2984, x\u2081 \u2208 s \u00d7\u02e2 t \u2192 \u2200 \u2983x\u2082 : E \u00d7 F\u2984, x\u2082 \u2208 s \u00d7\u02e2 t \u2192 x\u271d \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d \u2227 x\u2082 = x\u271d\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\nhx_fst : x\u271d.fst \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x\u271d \u2208 openSegment \ud835\udd5c (x\u2081, x\u271d.snd) (x\u2082, x\u271d.snd)\n[PROOFSTEP]\nrw [\u2190 Prod.image_mk_openSegment_left]\n[GOAL]\ncase h.refine'_1.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : Set E\nt : Set F\nx\u271d : E \u00d7 F\nhx : x\u271d \u2208 s \u00d7\u02e2 t\nh : \u2200 \u2983x\u2081 : E \u00d7 F\u2984, x\u2081 \u2208 s \u00d7\u02e2 t \u2192 \u2200 \u2983x\u2082 : E \u00d7 F\u2984, x\u2082 \u2208 s \u00d7\u02e2 t \u2192 x\u271d \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d \u2227 x\u2082 = x\u271d\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\nhx_fst : x\u271d.fst \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x\u271d \u2208 (fun x => (x, x\u271d.snd)) '' openSegment \ud835\udd5c x\u2081 x\u2082\n[PROOFSTEP]\nexact \u27e8_, hx_fst, Prod.mk.eta\u27e9\n[GOAL]\ncase h.refine'_1.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : Set E\nt : Set F\nx\u271d : E \u00d7 F\nhx : x\u271d \u2208 s \u00d7\u02e2 t\nh : \u2200 \u2983x\u2081 : E \u00d7 F\u2984, x\u2081 \u2208 s \u00d7\u02e2 t \u2192 \u2200 \u2983x\u2082 : E \u00d7 F\u2984, x\u2082 \u2208 s \u00d7\u02e2 t \u2192 x\u271d \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d \u2227 x\u2082 = x\u271d\n\u22a2 \u2200 \u2983x\u2081 : F\u2984, x\u2081 \u2208 t \u2192 \u2200 \u2983x\u2082 : F\u2984, x\u2082 \u2208 t \u2192 x\u271d.snd \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d.snd \u2227 x\u2082 = x\u271d.snd\n[PROOFSTEP]\nrintro x\u2081 hx\u2081 x\u2082 hx\u2082 hx_snd\n[GOAL]\ncase h.refine'_1.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : Set E\nt : Set F\nx\u271d : E \u00d7 F\nhx : x\u271d \u2208 s \u00d7\u02e2 t\nh : \u2200 \u2983x\u2081 : E \u00d7 F\u2984, x\u2081 \u2208 s \u00d7\u02e2 t \u2192 \u2200 \u2983x\u2082 : E \u00d7 F\u2984, x\u2082 \u2208 s \u00d7\u02e2 t \u2192 x\u271d \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d \u2227 x\u2082 = x\u271d\nx\u2081 : F\nhx\u2081 : x\u2081 \u2208 t\nx\u2082 : F\nhx\u2082 : x\u2082 \u2208 t\nhx_snd : x\u271d.snd \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x\u2081 = x\u271d.snd \u2227 x\u2082 = x\u271d.snd\n[PROOFSTEP]\nrefine' (h (mk_mem_prod hx.1 hx\u2081) (mk_mem_prod hx.1 hx\u2082) _).imp (congr_arg Prod.snd) (congr_arg Prod.snd)\n[GOAL]\ncase h.refine'_1.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : Set E\nt : Set F\nx\u271d : E \u00d7 F\nhx : x\u271d \u2208 s \u00d7\u02e2 t\nh : \u2200 \u2983x\u2081 : E \u00d7 F\u2984, x\u2081 \u2208 s \u00d7\u02e2 t \u2192 \u2200 \u2983x\u2082 : E \u00d7 F\u2984, x\u2082 \u2208 s \u00d7\u02e2 t \u2192 x\u271d \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d \u2227 x\u2082 = x\u271d\nx\u2081 : F\nhx\u2081 : x\u2081 \u2208 t\nx\u2082 : F\nhx\u2082 : x\u2082 \u2208 t\nhx_snd : x\u271d.snd \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x\u271d \u2208 openSegment \ud835\udd5c (x\u271d.fst, x\u2081) (x\u271d.fst, x\u2082)\n[PROOFSTEP]\nrw [\u2190 Prod.image_mk_openSegment_right]\n[GOAL]\ncase h.refine'_1.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : Set E\nt : Set F\nx\u271d : E \u00d7 F\nhx : x\u271d \u2208 s \u00d7\u02e2 t\nh : \u2200 \u2983x\u2081 : E \u00d7 F\u2984, x\u2081 \u2208 s \u00d7\u02e2 t \u2192 \u2200 \u2983x\u2082 : E \u00d7 F\u2984, x\u2082 \u2208 s \u00d7\u02e2 t \u2192 x\u271d \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d \u2227 x\u2082 = x\u271d\nx\u2081 : F\nhx\u2081 : x\u2081 \u2208 t\nx\u2082 : F\nhx\u2082 : x\u2082 \u2208 t\nhx_snd : x\u271d.snd \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x\u271d \u2208 (fun y => (x\u271d.fst, y)) '' openSegment \ud835\udd5c x\u2081 x\u2082\n[PROOFSTEP]\nexact \u27e8_, hx_snd, Prod.mk.eta\u27e9\n[GOAL]\ncase h.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : Set E\nt : Set F\nx\u271d : E \u00d7 F\nhx : x\u271d \u2208 s \u00d7\u02e2 t\nh :\n (\u2200 \u2983x\u2081 : E\u2984, x\u2081 \u2208 s \u2192 \u2200 \u2983x\u2082 : E\u2984, x\u2082 \u2208 s \u2192 x\u271d.fst \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d.fst \u2227 x\u2082 = x\u271d.fst) \u2227\n \u2200 \u2983x\u2081 : F\u2984, x\u2081 \u2208 t \u2192 \u2200 \u2983x\u2082 : F\u2984, x\u2082 \u2208 t \u2192 x\u271d.snd \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d.snd \u2227 x\u2082 = x\u271d.snd\n\u22a2 \u2200 \u2983x\u2081 : E \u00d7 F\u2984, x\u2081 \u2208 s \u00d7\u02e2 t \u2192 \u2200 \u2983x\u2082 : E \u00d7 F\u2984, x\u2082 \u2208 s \u00d7\u02e2 t \u2192 x\u271d \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d \u2227 x\u2082 = x\u271d\n[PROOFSTEP]\nrintro x\u2081 hx\u2081 x\u2082 hx\u2082 \u27e8a, b, ha, hb, hab, hx'\u27e9\n[GOAL]\ncase h.refine'_2.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : Set E\nt : Set F\nx\u271d : E \u00d7 F\nhx : x\u271d \u2208 s \u00d7\u02e2 t\nh :\n (\u2200 \u2983x\u2081 : E\u2984, x\u2081 \u2208 s \u2192 \u2200 \u2983x\u2082 : E\u2984, x\u2082 \u2208 s \u2192 x\u271d.fst \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d.fst \u2227 x\u2082 = x\u271d.fst) \u2227\n \u2200 \u2983x\u2081 : F\u2984, x\u2081 \u2208 t \u2192 \u2200 \u2983x\u2082 : F\u2984, x\u2082 \u2208 t \u2192 x\u271d.snd \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d.snd \u2227 x\u2082 = x\u271d.snd\nx\u2081 : E \u00d7 F\nhx\u2081 : x\u2081 \u2208 s \u00d7\u02e2 t\nx\u2082 : E \u00d7 F\nhx\u2082 : x\u2082 \u2208 s \u00d7\u02e2 t\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : a \u2022 x\u2081 + b \u2022 x\u2082 = x\u271d\n\u22a2 x\u2081 = x\u271d \u2227 x\u2082 = x\u271d\n[PROOFSTEP]\nsimp_rw [Prod.ext_iff]\n[GOAL]\ncase h.refine'_2.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : Set E\nt : Set F\nx\u271d : E \u00d7 F\nhx : x\u271d \u2208 s \u00d7\u02e2 t\nh :\n (\u2200 \u2983x\u2081 : E\u2984, x\u2081 \u2208 s \u2192 \u2200 \u2983x\u2082 : E\u2984, x\u2082 \u2208 s \u2192 x\u271d.fst \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d.fst \u2227 x\u2082 = x\u271d.fst) \u2227\n \u2200 \u2983x\u2081 : F\u2984, x\u2081 \u2208 t \u2192 \u2200 \u2983x\u2082 : F\u2984, x\u2082 \u2208 t \u2192 x\u271d.snd \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u271d.snd \u2227 x\u2082 = x\u271d.snd\nx\u2081 : E \u00d7 F\nhx\u2081 : x\u2081 \u2208 s \u00d7\u02e2 t\nx\u2082 : E \u00d7 F\nhx\u2082 : x\u2082 \u2208 s \u00d7\u02e2 t\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : a \u2022 x\u2081 + b \u2022 x\u2082 = x\u271d\n\u22a2 (x\u2081.fst = x\u271d.fst \u2227 x\u2081.snd = x\u271d.snd) \u2227 x\u2082.fst = x\u271d.fst \u2227 x\u2082.snd = x\u271d.snd\n[PROOFSTEP]\nexact\n and_and_and_comm.1\n \u27e8h.1 hx\u2081.1 hx\u2082.1 \u27e8a, b, ha, hb, hab, congr_arg Prod.fst hx'\u27e9,\n h.2 hx\u2081.2 hx\u2082.2 \u27e8a, b, ha, hb, hab, congr_arg Prod.snd hx'\u27e9\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\n\u22a2 extremePoints \ud835\udd5c (pi univ s) = pi univ fun i => extremePoints \ud835\udd5c (s i)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\n\u22a2 x \u2208 extremePoints \ud835\udd5c (pi univ s) \u2194 x \u2208 pi univ fun i => extremePoints \ud835\udd5c (s i)\n[PROOFSTEP]\nsimp only [mem_extremePoints, mem_pi, mem_univ, true_imp_iff, @forall_and \u03b9]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\n\u22a2 ((\u2200 (i : \u03b9), x i \u2208 s i) \u2227\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x) \u2194\n (\u2200 (i : \u03b9), x i \u2208 s i) \u2227\n \u2200 (x_1 : \u03b9) (x\u2081 : \u03c0 x_1),\n x\u2081 \u2208 s x_1 \u2192 \u2200 (x\u2082 : \u03c0 x_1), x\u2082 \u2208 s x_1 \u2192 x x_1 \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x x_1 \u2227 x\u2082 = x x_1\n[PROOFSTEP]\nrefine' and_congr_right fun hx \u21a6 \u27e8fun h i \u21a6 _, fun h \u21a6 _\u27e9\n[GOAL]\ncase h.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\n\u22a2 \u2200 (x\u2081 : \u03c0 i), x\u2081 \u2208 s i \u2192 \u2200 (x\u2082 : \u03c0 i), x\u2082 \u2208 s i \u2192 x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x i \u2227 x\u2082 = x i\n[PROOFSTEP]\nrintro x\u2081 hx\u2081 x\u2082 hx\u2082 hi\n[GOAL]\ncase h.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x\u2081 = x i \u2227 x\u2082 = x i\n[PROOFSTEP]\nrefine'\n (h (update x i x\u2081) _ (update x i x\u2082) _ _).imp (fun h\u2081 \u21a6 by rw [\u2190 h\u2081, update_same]) fun h\u2082 \u21a6 by rw [\u2190 h\u2082, update_same]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\nh\u2081 : update x i x\u2081 = x\n\u22a2 x\u2081 = x i\n[PROOFSTEP]\nrw [\u2190 h\u2081, update_same]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\nh\u2082 : update x i x\u2082 = x\n\u22a2 x\u2082 = x i\n[PROOFSTEP]\nrw [\u2190 h\u2082, update_same]\n[GOAL]\ncase h.refine'_1.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 \u2200 (i_1 : \u03b9), update x i x\u2081 i_1 \u2208 s i_1\ncase h.refine'_1.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 \u2200 (i_1 : \u03b9), update x i x\u2082 i_1 \u2208 s i_1\ncase h.refine'_1.refine'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x \u2208 openSegment \ud835\udd5c (update x i x\u2081) (update x i x\u2082)\n[PROOFSTEP]\niterate 2 \n rintro j\n obtain rfl | hji := eq_or_ne j i\n \u00b7 rwa [update_same]\n \u00b7 rw [update_noteq hji]\n exact hx _\n[GOAL]\ncase h.refine'_1.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 \u2200 (i_1 : \u03b9), update x i x\u2081 i_1 \u2208 s i_1\ncase h.refine'_1.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 \u2200 (i_1 : \u03b9), update x i x\u2082 i_1 \u2208 s i_1\ncase h.refine'_1.refine'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x \u2208 openSegment \ud835\udd5c (update x i x\u2081) (update x i x\u2082)\n[PROOFSTEP]\nrintro j\n[GOAL]\ncase h.refine'_1.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\nj : \u03b9\n\u22a2 update x i x\u2081 j \u2208 s j\ncase h.refine'_1.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 \u2200 (i_1 : \u03b9), update x i x\u2082 i_1 \u2208 s i_1\ncase h.refine'_1.refine'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x \u2208 openSegment \ud835\udd5c (update x i x\u2081) (update x i x\u2082)\n[PROOFSTEP]\nobtain rfl | hji := eq_or_ne j i\n[GOAL]\ncase h.refine'_1.refine'_1.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\nj : \u03b9\nx\u2081 : \u03c0 j\nhx\u2081 : x\u2081 \u2208 s j\nx\u2082 : \u03c0 j\nhx\u2082 : x\u2082 \u2208 s j\nhi : x j \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 update x j x\u2081 j \u2208 s j\n[PROOFSTEP]\nrwa [update_same]\n[GOAL]\ncase h.refine'_1.refine'_1.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\nj : \u03b9\nhji : j \u2260 i\n\u22a2 update x i x\u2081 j \u2208 s j\n[PROOFSTEP]\nrw [update_noteq hji]\n[GOAL]\ncase h.refine'_1.refine'_1.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\nj : \u03b9\nhji : j \u2260 i\n\u22a2 x j \u2208 s j\n[PROOFSTEP]\nexact hx _\n[GOAL]\ncase h.refine'_1.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 \u2200 (i_1 : \u03b9), update x i x\u2082 i_1 \u2208 s i_1\ncase h.refine'_1.refine'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x \u2208 openSegment \ud835\udd5c (update x i x\u2081) (update x i x\u2082)\n[PROOFSTEP]\nrintro j\n[GOAL]\ncase h.refine'_1.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\nj : \u03b9\n\u22a2 update x i x\u2082 j \u2208 s j\ncase h.refine'_1.refine'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x \u2208 openSegment \ud835\udd5c (update x i x\u2081) (update x i x\u2082)\n[PROOFSTEP]\nobtain rfl | hji := eq_or_ne j i\n[GOAL]\ncase h.refine'_1.refine'_2.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\nj : \u03b9\nx\u2081 : \u03c0 j\nhx\u2081 : x\u2081 \u2208 s j\nx\u2082 : \u03c0 j\nhx\u2082 : x\u2082 \u2208 s j\nhi : x j \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 update x j x\u2082 j \u2208 s j\n[PROOFSTEP]\nrwa [update_same]\n[GOAL]\ncase h.refine'_1.refine'_2.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\nj : \u03b9\nhji : j \u2260 i\n\u22a2 update x i x\u2082 j \u2208 s j\n[PROOFSTEP]\nrw [update_noteq hji]\n[GOAL]\ncase h.refine'_1.refine'_2.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\nj : \u03b9\nhji : j \u2260 i\n\u22a2 x j \u2208 s j\n[PROOFSTEP]\nexact hx _\n[GOAL]\ncase h.refine'_1.refine'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x \u2208 openSegment \ud835\udd5c (update x i x\u2081) (update x i x\u2082)\n[PROOFSTEP]\nrw [\u2190 Pi.image_update_openSegment]\n[GOAL]\ncase h.refine'_1.refine'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\ni : \u03b9\nx\u2081 : \u03c0 i\nhx\u2081 : x\u2081 \u2208 s i\nx\u2082 : \u03c0 i\nhx\u2082 : x\u2082 \u2208 s i\nhi : x i \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x \u2208 update x i '' openSegment \ud835\udd5c x\u2081 x\u2082\n[PROOFSTEP]\nexact \u27e8_, hi, update_eq_self _ _\u27e9\n[GOAL]\ncase h.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x_1 : \u03b9) (x\u2081 : \u03c0 x_1),\n x\u2081 \u2208 s x_1 \u2192 \u2200 (x\u2082 : \u03c0 x_1), x\u2082 \u2208 s x_1 \u2192 x x_1 \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x x_1 \u2227 x\u2082 = x x_1\n\u22a2 \u2200 (x\u2081 : (i : \u03b9) \u2192 \u03c0 i),\n (\u2200 (i : \u03b9), x\u2081 i \u2208 s i) \u2192\n \u2200 (x\u2082 : (i : \u03b9) \u2192 \u03c0 i), (\u2200 (i : \u03b9), x\u2082 i \u2208 s i) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\n[PROOFSTEP]\nrintro x\u2081 hx\u2081 x\u2082 hx\u2082 \u27e8a, b, ha, hb, hab, hx'\u27e9\n[GOAL]\ncase h.refine'_2.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x_1 : \u03b9) (x\u2081 : \u03c0 x_1),\n x\u2081 \u2208 s x_1 \u2192 \u2200 (x\u2082 : \u03c0 x_1), x\u2082 \u2208 s x_1 \u2192 x x_1 \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x x_1 \u2227 x\u2082 = x x_1\nx\u2081 : (i : \u03b9) \u2192 \u03c0 i\nhx\u2081 : \u2200 (i : \u03b9), x\u2081 i \u2208 s i\nx\u2082 : (i : \u03b9) \u2192 \u03c0 i\nhx\u2082 : \u2200 (i : \u03b9), x\u2082 i \u2208 s i\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : a \u2022 x\u2081 + b \u2022 x\u2082 = x\n\u22a2 x\u2081 = x \u2227 x\u2082 = x\n[PROOFSTEP]\nsimp_rw [funext_iff, \u2190 forall_and]\n[GOAL]\ncase h.refine'_2.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (\u03c0 i)\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\nA B : Set E\nx\u271d : E\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nh :\n \u2200 (x_1 : \u03b9) (x\u2081 : \u03c0 x_1),\n x\u2081 \u2208 s x_1 \u2192 \u2200 (x\u2082 : \u03c0 x_1), x\u2082 \u2208 s x_1 \u2192 x x_1 \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x x_1 \u2227 x\u2082 = x x_1\nx\u2081 : (i : \u03b9) \u2192 \u03c0 i\nhx\u2081 : \u2200 (i : \u03b9), x\u2081 i \u2208 s i\nx\u2082 : (i : \u03b9) \u2192 \u03c0 i\nhx\u2082 : \u2200 (i : \u03b9), x\u2082 i \u2208 s i\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : a \u2022 x\u2081 + b \u2022 x\u2082 = x\n\u22a2 \u2200 (x_1 : \u03b9), x\u2081 x_1 = x x_1 \u2227 x\u2082 x_1 = x x_1\n[PROOFSTEP]\nexact fun i \u21a6 h _ _ (hx\u2081 _) _ (hx\u2082 _) \u27e8a, b, ha, hb, hab, congr_fun hx' _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\n\u22a2 x \u2208 extremePoints \ud835\udd5c A \u2194 x \u2208 A \u2227 \u2200 (x\u2081 : E), x\u2081 \u2208 A \u2192 \u2200 (x\u2082 : E), x\u2082 \u2208 A \u2192 x \u2208 segment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2228 x\u2082 = x\n[PROOFSTEP]\nrefine' and_congr_right fun hxA \u21a6 forall\u2084_congr fun x\u2081 h\u2081 x\u2082 h\u2082 \u21a6 _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhxA : x \u2208 A\nx\u2081 : E\nh\u2081 : x\u2081 \u2208 A\nx\u2082 : E\nh\u2082 : x\u2082 \u2208 A\n\u22a2 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x \u2194 x \u2208 segment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2228 x\u2082 = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhxA : x \u2208 A\nx\u2081 : E\nh\u2081 : x\u2081 \u2208 A\nx\u2082 : E\nh\u2082 : x\u2082 \u2208 A\n\u22a2 (x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x) \u2192 x \u2208 segment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2228 x\u2082 = x\n[PROOFSTEP]\nrw [\u2190 insert_endpoints_openSegment]\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhxA : x \u2208 A\nx\u2081 : E\nh\u2081 : x\u2081 \u2208 A\nx\u2082 : E\nh\u2082 : x\u2082 \u2208 A\n\u22a2 (x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x) \u2192 x \u2208 insert x\u2081 (insert x\u2082 (openSegment \ud835\udd5c x\u2081 x\u2082)) \u2192 x\u2081 = x \u2228 x\u2082 = x\n[PROOFSTEP]\nrintro H (rfl | rfl | hx)\n[GOAL]\ncase mp.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhxA : x \u2208 A\nx\u2082 : E\nh\u2082 : x\u2082 \u2208 A\nh\u2081 : x \u2208 A\nH : x \u2208 openSegment \ud835\udd5c x x\u2082 \u2192 x = x \u2227 x\u2082 = x\n\u22a2 x = x \u2228 x\u2082 = x\ncase mp.inr.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhxA : x \u2208 A\nx\u2081 : E\nh\u2081 : x\u2081 \u2208 A\nh\u2082 : x \u2208 A\nH : x \u2208 openSegment \ud835\udd5c x\u2081 x \u2192 x\u2081 = x \u2227 x = x\n\u22a2 x\u2081 = x \u2228 x = x\ncase mp.inr.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhxA : x \u2208 A\nx\u2081 : E\nh\u2081 : x\u2081 \u2208 A\nx\u2082 : E\nh\u2082 : x\u2082 \u2208 A\nH : x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\nhx : x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x\u2081 = x \u2228 x\u2082 = x\n[PROOFSTEP]\nexacts [Or.inl rfl, Or.inr rfl, Or.inl <| (H hx).1]\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhxA : x \u2208 A\nx\u2081 : E\nh\u2081 : x\u2081 \u2208 A\nx\u2082 : E\nh\u2082 : x\u2082 \u2208 A\n\u22a2 (x \u2208 segment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2228 x\u2082 = x) \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x\n[PROOFSTEP]\nintro H hx\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhxA : x \u2208 A\nx\u2081 : E\nh\u2081 : x\u2081 \u2208 A\nx\u2082 : E\nh\u2082 : x\u2082 \u2208 A\nH : x \u2208 segment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2228 x\u2082 = x\nhx : x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x\u2081 = x \u2227 x\u2082 = x\n[PROOFSTEP]\nrcases H (openSegment_subset_segment _ _ _ hx) with (rfl | rfl)\n[GOAL]\ncase mpr.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx\u2081 : E\nh\u2081 : x\u2081 \u2208 A\nx\u2082 : E\nh\u2082 : x\u2082 \u2208 A\nhxA : x\u2081 \u2208 A\nH : x\u2081 \u2208 segment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u2081 \u2228 x\u2082 = x\u2081\nhx : x\u2081 \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x\u2081 = x\u2081 \u2227 x\u2082 = x\u2081\ncase mpr.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx\u2081 : E\nh\u2081 : x\u2081 \u2208 A\nx\u2082 : E\nh\u2082 hxA : x\u2082 \u2208 A\nH : x\u2082 \u2208 segment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x\u2082 \u2228 x\u2082 = x\u2082\nhx : x\u2082 \u2208 openSegment \ud835\udd5c x\u2081 x\u2082\n\u22a2 x\u2081 = x\u2082 \u2227 x\u2082 = x\u2082\n[PROOFSTEP]\nexacts [\u27e8rfl, (left_mem_openSegment_iff.1 hx).symm\u27e9, \u27e8right_mem_openSegment_iff.1 hx, rfl\u27e9]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhA : Convex \ud835\udd5c A\n\u22a2 x \u2208 extremePoints \ud835\udd5c A \u2194 x \u2208 A \u2227 Convex \ud835\udd5c (A \\ {x})\n[PROOFSTEP]\nuse fun hx \u21a6 \u27e8hx.1, (mem_extremePoints_iff_extreme_singleton.1 hx).convex_diff hA\u27e9\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhA : Convex \ud835\udd5c A\n\u22a2 x \u2208 A \u2227 Convex \ud835\udd5c (A \\ {x}) \u2192 x \u2208 extremePoints \ud835\udd5c A\n[PROOFSTEP]\nrintro \u27e8hxA, hAx\u27e9\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhA : Convex \ud835\udd5c A\nhxA : x \u2208 A\nhAx : Convex \ud835\udd5c (A \\ {x})\n\u22a2 x \u2208 extremePoints \ud835\udd5c A\n[PROOFSTEP]\nrefine' mem_extremePoints_iff_forall_segment.2 \u27e8hxA, fun x\u2081 hx\u2081 x\u2082 hx\u2082 hx \u21a6 _\u27e9\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhA : Convex \ud835\udd5c A\nhxA : x \u2208 A\nhAx : Convex \ud835\udd5c (A \\ {x})\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 A\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 A\nhx : x \u2208 [x\u2081-[\ud835\udd5c]x\u2082]\n\u22a2 x\u2081 = x \u2228 x\u2082 = x\n[PROOFSTEP]\nrw [convex_iff_segment_subset] at hAx \n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhA : Convex \ud835\udd5c A\nhxA : x \u2208 A\nhAx : \u2200 \u2983x_1 : E\u2984, x_1 \u2208 A \\ {x} \u2192 \u2200 \u2983y : E\u2984, y \u2208 A \\ {x} \u2192 [x_1-[\ud835\udd5c]y] \u2286 A \\ {x}\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 A\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 A\nhx : x \u2208 [x\u2081-[\ud835\udd5c]x\u2082]\n\u22a2 x\u2081 = x \u2228 x\u2082 = x\n[PROOFSTEP]\nby_contra' h\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhA : Convex \ud835\udd5c A\nhxA : x \u2208 A\nhAx : \u2200 \u2983x_1 : E\u2984, x_1 \u2208 A \\ {x} \u2192 \u2200 \u2983y : E\u2984, y \u2208 A \\ {x} \u2192 [x_1-[\ud835\udd5c]y] \u2286 A \\ {x}\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 A\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 A\nhx : x \u2208 [x\u2081-[\ud835\udd5c]x\u2082]\nh : x\u2081 \u2260 x \u2227 x\u2082 \u2260 x\n\u22a2 False\n[PROOFSTEP]\nexact (hAx \u27e8hx\u2081, fun hx\u2081 \u21a6 h.1 (mem_singleton_iff.2 hx\u2081)\u27e9 \u27e8hx\u2082, fun hx\u2082 \u21a6 h.2 (mem_singleton_iff.2 hx\u2082)\u27e9 hx).2 rfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhA : Convex \ud835\udd5c A\n\u22a2 x \u2208 extremePoints \ud835\udd5c A \u2194 x \u2208 A \\ \u2191(convexHull \ud835\udd5c) (A \\ {x})\n[PROOFSTEP]\nrw [hA.mem_extremePoints_iff_convex_diff, hA.convex_remove_iff_not_mem_convexHull_remove, mem_diff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\n\u22a2 extremePoints \ud835\udd5c (\u2191(convexHull \ud835\udd5c) A) \u2286 A\n[PROOFSTEP]\nrintro x hx\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx\u271d x : E\nhx : x \u2208 extremePoints \ud835\udd5c (\u2191(convexHull \ud835\udd5c) A)\n\u22a2 x \u2208 A\n[PROOFSTEP]\nrw [(convex_convexHull \ud835\udd5c _).mem_extremePoints_iff_convex_diff] at hx \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx\u271d x : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) A \u2227 Convex \ud835\udd5c (\u2191(convexHull \ud835\udd5c) A \\ {x})\n\u22a2 x \u2208 A\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx\u271d x : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) A \u2227 Convex \ud835\udd5c (\u2191(convexHull \ud835\udd5c) A \\ {x})\nh : \u00acx \u2208 A\n\u22a2 False\n[PROOFSTEP]\nexact (convexHull_min (subset_diff.2 \u27e8subset_convexHull \ud835\udd5c _, disjoint_singleton_right.2 h\u27e9) hx.2 hx.1).2 rfl\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.Extreme", "llama_tokens": 29451, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125848754472, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.4456927418008026}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nlet d : Set \u03b1 \u2192 \u211d := fun s => ((\u03bc s).toNNReal : \u211d) - (\u03bd s).toNNReal\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nlet c : Set \u211d := d '' {s | MeasurableSet s}\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nlet \u03b3 : \u211d := sSup c\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave h\u03bc : \u2200 s, \u03bc s \u2260 \u221e := measure_ne_top \u03bc\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave h\u03bd : \u2200 s, \u03bd s \u2260 \u221e := measure_ne_top \u03bd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave to_nnreal_\u03bc : \u2200 s, ((\u03bc s).toNNReal : \u211d\u22650\u221e) = \u03bc s := fun s => ENNReal.coe_toNNReal <| h\u03bc _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave to_nnreal_\u03bd : \u2200 s, ((\u03bd s).toNNReal : \u211d\u22650\u221e) = \u03bd s := fun s => ENNReal.coe_toNNReal <| h\u03bd _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave d_split : \u2200 s t, MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t) :=\n by\n intro s t _hs ht\n dsimp only\n rw [\u2190 measure_inter_add_diff s ht, \u2190 measure_inter_add_diff s ht, ENNReal.toNNReal_add (h\u03bc _) (h\u03bc _),\n ENNReal.toNNReal_add (h\u03bd _) (h\u03bd _), NNReal.coe_add, NNReal.coe_add]\n simp only [sub_eq_add_neg, neg_add]\n abel\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\n\u22a2 \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\n[PROOFSTEP]\nintro s t _hs ht\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\ns t : Set \u03b1\n_hs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 d s = d (s \\ t) + d (s \u2229 t)\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\ns t : Set \u03b1\n_hs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) =\n \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s \\ t))) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s \\ t))) +\n (\u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s \u2229 t))) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s \u2229 t))))\n[PROOFSTEP]\nrw [\u2190 measure_inter_add_diff s ht, \u2190 measure_inter_add_diff s ht, ENNReal.toNNReal_add (h\u03bc _) (h\u03bc _),\n ENNReal.toNNReal_add (h\u03bd _) (h\u03bd _), NNReal.coe_add, NNReal.coe_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\ns t : Set \u03b1\n_hs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s \u2229 t))) + \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s \\ t))) -\n (\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s \u2229 t))) + \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s \\ t)))) =\n \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s \\ t))) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s \\ t))) +\n (\u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s \u2229 t))) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s \u2229 t))))\n[PROOFSTEP]\nsimp only [sub_eq_add_neg, neg_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\ns t : Set \u03b1\n_hs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s \u2229 t))) + \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s \\ t))) +\n (-\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s \u2229 t))) + -\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s \\ t)))) =\n \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s \\ t))) + -\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s \\ t))) +\n (\u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s \u2229 t))) + -\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s \u2229 t))))\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\ns t : Set \u03b1\n_hs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s \u2229 t))) + \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s \\ t))) +\n (-\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s \u2229 t))) + -\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s \\ t)))) =\n \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s \\ t))) + -\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s \\ t))) +\n (\u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s \u2229 t))) + -\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s \u2229 t))))\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave d_Union : \u2200 s : \u2115 \u2192 Set \u03b1, Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 n, s n))) :=\n by\n intro s hm\n refine' Tendsto.sub _ _ <;>\n refine' NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal _).comp <| tendsto_measure_iUnion hm\n exact h\u03bc _\n exact h\u03bd _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\n\u22a2 \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\n[PROOFSTEP]\nintro s hm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\ns : \u2115 \u2192 Set \u03b1\nhm : Monotone s\n\u22a2 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\n[PROOFSTEP]\nrefine' Tendsto.sub _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\ns : \u2115 \u2192 Set \u03b1\nhm : Monotone s\n\u22a2 Tendsto (fun n => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s n)))) atTop (\ud835\udcdd \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (\u22c3 (n : \u2115), s n))))\n[PROOFSTEP]\nrefine' NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal _).comp <| tendsto_measure_iUnion hm\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\ns : \u2115 \u2192 Set \u03b1\nhm : Monotone s\n\u22a2 Tendsto (fun n => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s n)))) atTop (\ud835\udcdd \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (\u22c3 (n : \u2115), s n))))\n[PROOFSTEP]\nrefine' NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal _).comp <| tendsto_measure_iUnion hm\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\ns : \u2115 \u2192 Set \u03b1\nhm : Monotone s\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) \u2260 \u22a4\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\ns : \u2115 \u2192 Set \u03b1\nhm : Monotone s\n\u22a2 \u2191\u2191\u03bd (\u22c3 (n : \u2115), s n) \u2260 \u22a4\n[PROOFSTEP]\nexact h\u03bc _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\ns : \u2115 \u2192 Set \u03b1\nhm : Monotone s\n\u22a2 \u2191\u2191\u03bd (\u22c3 (n : \u2115), s n) \u2260 \u22a4\n[PROOFSTEP]\nexact h\u03bd _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave d_Inter :\n \u2200 s : \u2115 \u2192 Set \u03b1,\n (\u2200 n, MeasurableSet (s n)) \u2192 (\u2200 n m, n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 n, s n))) :=\n by\n intro s hs hm\n refine' Tendsto.sub _ _ <;>\n refine' NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal <| _).comp <| tendsto_measure_iInter hs hm _\n exacts [h\u03bc _, \u27e80, h\u03bc _\u27e9, h\u03bd _, \u27e80, h\u03bd _\u27e9]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\n\u22a2 \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\n[PROOFSTEP]\nintro s hs hm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nhm : \u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n\n\u22a2 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\n[PROOFSTEP]\nrefine' Tendsto.sub _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nhm : \u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n\n\u22a2 Tendsto (fun n => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (s n)))) atTop (\ud835\udcdd \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (\u22c2 (n : \u2115), s n))))\n[PROOFSTEP]\nrefine' NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal <| _).comp <| tendsto_measure_iInter hs hm _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nhm : \u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n\n\u22a2 Tendsto (fun n => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (s n)))) atTop (\ud835\udcdd \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (\u22c2 (n : \u2115), s n))))\n[PROOFSTEP]\nrefine' NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal <| _).comp <| tendsto_measure_iInter hs hm _\n[GOAL]\ncase refine'_1.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nhm : \u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n\n\u22a2 \u2191\u2191\u03bc (\u22c2 (n : \u2115), s n) \u2260 \u22a4\ncase refine'_1.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nhm : \u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n\n\u22a2 \u2203 i, \u2191\u2191\u03bc (s i) \u2260 \u22a4\ncase refine'_2.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nhm : \u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n\n\u22a2 \u2191\u2191\u03bd (\u22c2 (n : \u2115), s n) \u2260 \u22a4\ncase refine'_2.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nhm : \u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n\n\u22a2 \u2203 i, \u2191\u2191\u03bd (s i) \u2260 \u22a4\n[PROOFSTEP]\nexacts [h\u03bc _, \u27e80, h\u03bc _\u27e9, h\u03bd _, \u27e80, h\u03bd _\u27e9]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave bdd_c : BddAbove c := by\n use(\u03bc univ).toNNReal\n rintro r \u27e8s, _hs, rfl\u27e9\n refine' le_trans (sub_le_self _ <| NNReal.coe_nonneg _) _\n rw [NNReal.coe_le_coe, \u2190 ENNReal.coe_le_coe, to_nnreal_\u03bc, to_nnreal_\u03bc]\n exact measure_mono (subset_univ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\n\u22a2 BddAbove c\n[PROOFSTEP]\nuse(\u03bc univ).toNNReal\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\n\u22a2 \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc univ)) \u2208 upperBounds c\n[PROOFSTEP]\nrintro r \u27e8s, _hs, rfl\u27e9\n[GOAL]\ncase h.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\ns : Set \u03b1\n_hs : s \u2208 {s | MeasurableSet s}\n\u22a2 d s \u2264 \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc univ))\n[PROOFSTEP]\nrefine' le_trans (sub_le_self _ <| NNReal.coe_nonneg _) _\n[GOAL]\ncase h.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\ns : Set \u03b1\n_hs : s \u2208 {s | MeasurableSet s}\n\u22a2 \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) \u2264 \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc univ))\n[PROOFSTEP]\nrw [NNReal.coe_le_coe, \u2190 ENNReal.coe_le_coe, to_nnreal_\u03bc, to_nnreal_\u03bc]\n[GOAL]\ncase h.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\ns : Set \u03b1\n_hs : s \u2208 {s | MeasurableSet s}\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc univ\n[PROOFSTEP]\nexact measure_mono (subset_univ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave c_nonempty : c.Nonempty := Nonempty.image _ \u27e8_, MeasurableSet.empty\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave d_le_\u03b3 : \u2200 s, MeasurableSet s \u2192 d s \u2264 \u03b3 := fun s hs => le_csSup bdd_c \u27e8s, hs, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave : \u2200 n : \u2115, \u2203 s : Set \u03b1, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s :=\n by\n intro n\n have : \u03b3 - (1 / 2) ^ n < \u03b3 := sub_lt_self \u03b3 (pow_pos (half_pos zero_lt_one) n)\n rcases exists_lt_of_lt_csSup c_nonempty this with \u27e8r, \u27e8s, hs, rfl\u27e9, hlt\u27e9\n exact \u27e8s, hs, hlt\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\n\u22a2 \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nn : \u2115\n\u22a2 \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\n[PROOFSTEP]\nhave : \u03b3 - (1 / 2) ^ n < \u03b3 := sub_lt_self \u03b3 (pow_pos (half_pos zero_lt_one) n)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nn : \u2115\nthis : \u03b3 - (1 / 2) ^ n < \u03b3\n\u22a2 \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\n[PROOFSTEP]\nrcases exists_lt_of_lt_csSup c_nonempty this with \u27e8r, \u27e8s, hs, rfl\u27e9, hlt\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nn : \u2115\nthis : \u03b3 - (1 / 2) ^ n < \u03b3\ns : Set \u03b1\nhs : s \u2208 {s | MeasurableSet s}\nhlt : \u03b3 - (1 / 2) ^ n < d s\n\u22a2 \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\n[PROOFSTEP]\nexact \u27e8s, hs, hlt\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nrcases Classical.axiom_of_choice this with \u27e8e, he\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nchange \u2115 \u2192 Set \u03b1 at e \n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave he\u2081 : \u2200 n, MeasurableSet (e n) := fun n => (he n).1\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave he\u2082 : \u2200 n, \u03b3 - (1 / 2) ^ n < d (e n) := fun n => (he n).2\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nlet f : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => (Finset.Ico n (m + 1)).inf e\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave hf : \u2200 n m, MeasurableSet (f n m) := by\n intro n m\n simp only [Finset.inf_eq_iInf]\n exact MeasurableSet.biInter (to_countable _) fun i _ => he\u2081 _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\n\u22a2 \u2200 (n m : \u2115), MeasurableSet (f n m)\n[PROOFSTEP]\nintro n m\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nn m : \u2115\n\u22a2 MeasurableSet (f n m)\n[PROOFSTEP]\nsimp only [Finset.inf_eq_iInf]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nn m : \u2115\n\u22a2 MeasurableSet (\u2a05 (a : \u2115) (_ : a \u2208 Finset.Ico n (m + 1)), e a)\n[PROOFSTEP]\nexact MeasurableSet.biInter (to_countable _) fun i _ => he\u2081 _\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave f_subset_f : \u2200 {a b c d}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c :=\n by\n intro a b c d hab hcd\n simp_rw [Finset.inf_eq_iInf]\n exact biInter_subset_biInter_left (Finset.Ico_subset_Ico hab <| Nat.succ_le_succ hcd)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\n\u22a2 \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\n[PROOFSTEP]\nintro a b c d hab hcd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd\u271d : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc\u271d : Set \u211d := d\u271d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\u271d\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d\u271d s = d\u271d (s \\ t) + d\u271d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d\u271d (s n)) atTop (\ud835\udcdd (d\u271d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d\u271d (s n)) atTop (\ud835\udcdd (d\u271d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\u271d\nc_nonempty : Set.Nonempty c\u271d\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d\u271d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d\u271d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d\u271d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d\u271d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\na b c d : \u2115\nhab : a \u2264 b\nhcd : c \u2264 d\n\u22a2 f a d \u2286 f b c\n[PROOFSTEP]\nsimp_rw [Finset.inf_eq_iInf]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd\u271d : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc\u271d : Set \u211d := d\u271d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\u271d\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d\u271d s = d\u271d (s \\ t) + d\u271d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d\u271d (s n)) atTop (\ud835\udcdd (d\u271d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d\u271d (s n)) atTop (\ud835\udcdd (d\u271d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\u271d\nc_nonempty : Set.Nonempty c\u271d\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d\u271d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d\u271d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d\u271d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d\u271d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\na b c d : \u2115\nhab : a \u2264 b\nhcd : c \u2264 d\n\u22a2 \u2a05 (a_1 : \u2115) (_ : a_1 \u2208 Finset.Ico a (d + 1)), e a_1 \u2286 \u2a05 (a : \u2115) (_ : a \u2208 Finset.Ico b (c + 1)), e a\n[PROOFSTEP]\nexact biInter_subset_biInter_left (Finset.Ico_subset_Ico hab <| Nat.succ_le_succ hcd)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave f_succ : \u2200 n m, n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1) :=\n by\n intro n m hnm\n have : n \u2264 m + 1 := le_of_lt (Nat.succ_le_succ hnm)\n simp_rw [Nat.Ico_succ_right_eq_insert_Ico this, Finset.inf_insert, Set.inter_comm]\n rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\n\u22a2 \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\n[PROOFSTEP]\nintro n m hnm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nn m : \u2115\nhnm : n \u2264 m\n\u22a2 f n (m + 1) = f n m \u2229 e (m + 1)\n[PROOFSTEP]\nhave : n \u2264 m + 1 := le_of_lt (Nat.succ_le_succ hnm)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis\u271d : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nn m : \u2115\nhnm : n \u2264 m\nthis : n \u2264 m + 1\n\u22a2 f n (m + 1) = f n m \u2229 e (m + 1)\n[PROOFSTEP]\nsimp_rw [Nat.Ico_succ_right_eq_insert_Ico this, Finset.inf_insert, Set.inter_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis\u271d : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nn m : \u2115\nhnm : n \u2264 m\nthis : n \u2264 m + 1\n\u22a2 e (m + 1) \u2293 Finset.inf (Finset.Ico n (m + 1)) e = e (m + 1) \u2229 Finset.inf (Finset.Ico n (m + 1)) e\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave le_d_f : \u2200 n m, m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n) :=\n by\n intro n m h\n refine' Nat.le_induction _ _ n h\n \u00b7 have := he\u2082 m\n simp_rw [Nat.Ico_succ_singleton, Finset.inf_singleton]\n linarith\n \u00b7 intro n (hmn : m \u2264 n) ih\n have : \u03b3 + (\u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) \u2264 \u03b3 + d (f m (n + 1)) := by\n calc\n \u03b3 + (\u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) \u2264 \u03b3 + (\u03b3 - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) :=\n by\n refine' add_le_add_left (add_le_add_left _ _) \u03b3\n simp only [pow_add, pow_one, le_sub_iff_add_le]\n linarith\n _ = \u03b3 - (1 / 2) ^ (n + 1) + (\u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n) := by simp only [sub_eq_add_neg]; abel\n _ \u2264 d (e (n + 1)) + d (f m n) := (add_le_add (le_of_lt <| he\u2082 _) ih)\n _ \u2264 d (e (n + 1)) + d (f m n \\ e (n + 1)) + d (f m (n + 1)) := by\n rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he\u2081 _), add_assoc]\n _ = d (e (n + 1) \u222a f m n) + d (f m (n + 1)) :=\n by\n rw [d_split (e (n + 1) \u222a f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left]\n abel\n exact (he\u2081 _).union (hf _ _)\n exact he\u2081 _\n _ \u2264 \u03b3 + d (f m (n + 1)) := add_le_add_right (d_le_\u03b3 _ <| (he\u2081 _).union (hf _ _)) _\n exact (add_le_add_iff_left \u03b3).1 this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\n\u22a2 \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n[PROOFSTEP]\nintro n m h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn m : \u2115\nh : m \u2264 n\n\u22a2 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n[PROOFSTEP]\nrefine' Nat.le_induction _ _ n h\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn m : \u2115\nh : m \u2264 n\n\u22a2 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ m \u2264 d (f m m)\n[PROOFSTEP]\nhave := he\u2082 m\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis\u271d : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn m : \u2115\nh : m \u2264 n\nthis : \u03b3 - (1 / 2) ^ m < d (e m)\n\u22a2 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ m \u2264 d (f m m)\n[PROOFSTEP]\nsimp_rw [Nat.Ico_succ_singleton, Finset.inf_singleton]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis\u271d : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn m : \u2115\nh : m \u2264 n\nthis : \u03b3 - (1 / 2) ^ m < d (e m)\n\u22a2 sSup ((fun a => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc a)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd a))) '' {s | MeasurableSet s}) -\n 2 * (1 / 2) ^ m +\n (1 / 2) ^ m \u2264\n \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc (e m))) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd (e m)))\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn m : \u2115\nh : m \u2264 n\n\u22a2 \u2200 (n : \u2115),\n m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n) \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) \u2264 d (f m (n + 1))\n[PROOFSTEP]\nintro n (hmn : m \u2264 n) ih\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) \u2264 d (f m (n + 1))\n[PROOFSTEP]\nhave : \u03b3 + (\u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) \u2264 \u03b3 + d (f m (n + 1)) := by\n calc\n \u03b3 + (\u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) \u2264 \u03b3 + (\u03b3 - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) :=\n by\n refine' add_le_add_left (add_le_add_left _ _) \u03b3\n simp only [pow_add, pow_one, le_sub_iff_add_le]\n linarith\n _ = \u03b3 - (1 / 2) ^ (n + 1) + (\u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n) := by simp only [sub_eq_add_neg]; abel\n _ \u2264 d (e (n + 1)) + d (f m n) := (add_le_add (le_of_lt <| he\u2082 _) ih)\n _ \u2264 d (e (n + 1)) + d (f m n \\ e (n + 1)) + d (f m (n + 1)) := by\n rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he\u2081 _), add_assoc]\n _ = d (e (n + 1) \u222a f m n) + d (f m (n + 1)) :=\n by\n rw [d_split (e (n + 1) \u222a f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left]\n abel\n exact (he\u2081 _).union (hf _ _)\n exact he\u2081 _\n _ \u2264 \u03b3 + d (f m (n + 1)) := add_le_add_right (d_le_\u03b3 _ <| (he\u2081 _).union (hf _ _)) _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 \u03b3 + (\u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) \u2264 \u03b3 + d (f m (n + 1))\n[PROOFSTEP]\ncalc\n \u03b3 + (\u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) \u2264 \u03b3 + (\u03b3 - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) :=\n by\n refine' add_le_add_left (add_le_add_left _ _) \u03b3\n simp only [pow_add, pow_one, le_sub_iff_add_le]\n linarith\n _ = \u03b3 - (1 / 2) ^ (n + 1) + (\u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n) := by simp only [sub_eq_add_neg]; abel\n _ \u2264 d (e (n + 1)) + d (f m n) := (add_le_add (le_of_lt <| he\u2082 _) ih)\n _ \u2264 d (e (n + 1)) + d (f m n \\ e (n + 1)) + d (f m (n + 1)) := by\n rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he\u2081 _), add_assoc]\n _ = d (e (n + 1) \u222a f m n) + d (f m (n + 1)) :=\n by\n rw [d_split (e (n + 1) \u222a f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left]\n abel\n exact (he\u2081 _).union (hf _ _)\n exact he\u2081 _\n _ \u2264 \u03b3 + d (f m (n + 1)) := add_le_add_right (d_le_\u03b3 _ <| (he\u2081 _).union (hf _ _)) _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 \u03b3 + (\u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) \u2264 \u03b3 + (\u03b3 - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1)))\n[PROOFSTEP]\nrefine' add_le_add_left (add_le_add_left _ _) \u03b3\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 (1 / 2) ^ (n + 1) \u2264 (1 / 2) ^ n - (1 / 2) ^ (n + 1)\n[PROOFSTEP]\nsimp only [pow_add, pow_one, le_sub_iff_add_le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 (1 / 2) ^ n * (1 / 2) + (1 / 2) ^ n * (1 / 2) \u2264 (1 / 2) ^ n\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 \u03b3 + (\u03b3 - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) =\n \u03b3 - (1 / 2) ^ (n + 1) + (\u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n)\n[PROOFSTEP]\nsimp only [sub_eq_add_neg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 sSup ((fun a => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc a)) + -\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd a))) '' {s | MeasurableSet s}) +\n (sSup ((fun a => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc a)) + -\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd a))) '' {s | MeasurableSet s}) +\n -(2 * (1 / 2) ^ m) +\n ((1 / 2) ^ n + -(1 / 2) ^ (n + 1))) =\n sSup ((fun a => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc a)) + -\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd a))) '' {s | MeasurableSet s}) +\n -(1 / 2) ^ (n + 1) +\n (sSup ((fun a => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc a)) + -\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd a))) '' {s | MeasurableSet s}) +\n -(2 * (1 / 2) ^ m) +\n (1 / 2) ^ n)\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 sSup ((fun a => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc a)) + -\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd a))) '' {s | MeasurableSet s}) +\n (sSup ((fun a => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc a)) + -\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd a))) '' {s | MeasurableSet s}) +\n -(2 * (1 / 2) ^ m) +\n ((1 / 2) ^ n + -(1 / 2) ^ (n + 1))) =\n sSup ((fun a => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc a)) + -\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd a))) '' {s | MeasurableSet s}) +\n -(1 / 2) ^ (n + 1) +\n (sSup ((fun a => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc a)) + -\u2191(ENNReal.toNNReal (\u2191\u2191\u03bd a))) '' {s | MeasurableSet s}) +\n -(2 * (1 / 2) ^ m) +\n (1 / 2) ^ n)\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 d (e (n + 1)) + d (f m n) \u2264 d (e (n + 1)) + d (f m n \\ e (n + 1)) + d (f m (n + 1))\n[PROOFSTEP]\nrw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he\u2081 _), add_assoc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 d (e (n + 1)) + d (f m n \\ e (n + 1)) + d (f m (n + 1)) = d (e (n + 1) \u222a f m n) + d (f m (n + 1))\n[PROOFSTEP]\nrw [d_split (e (n + 1) \u222a f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 d (e (n + 1)) + d (f m n \\ e (n + 1)) + d (f m (n + 1)) = d (f m n \\ e (n + 1)) + d (e (n + 1)) + d (f m (n + 1))\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 MeasurableSet (e (n + 1) \u222a f m n)\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 MeasurableSet (e (n + 1))\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 d (e (n + 1)) + d (f m n \\ e (n + 1)) + d (f m (n + 1)) = d (f m n \\ e (n + 1)) + d (e (n + 1)) + d (f m (n + 1))\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 MeasurableSet (e (n + 1) \u222a f m n)\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 MeasurableSet (e (n + 1))\n[PROOFSTEP]\nabel\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 MeasurableSet (e (n + 1) \u222a f m n)\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 MeasurableSet (e (n + 1))\n[PROOFSTEP]\nexact (he\u2081 _).union (hf _ _)\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 MeasurableSet (e (n + 1))\n[PROOFSTEP]\nexact he\u2081 _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis\u271d : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nn\u271d m : \u2115\nh : m \u2264 n\u271d\nn : \u2115\nhmn : m \u2264 n\nih : \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\nthis : \u03b3 + (\u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) \u2264 \u03b3 + d (f m (n + 1))\n\u22a2 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) \u2264 d (f m (n + 1))\n[PROOFSTEP]\nexact (add_le_add_iff_left \u03b3).1 this\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nlet s := \u22c3 m, \u22c2 n, f m n\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave \u03b3_le_d_s : \u03b3 \u2264 d s :=\n by\n have h\u03b3 : Tendsto (fun m : \u2115 => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3) :=\n by\n suffices Tendsto (fun m : \u2115 => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd (\u03b3 - 2 * 0)) by simpa only [mul_zero, tsub_zero]\n exact\n tendsto_const_nhds.sub <|\n tendsto_const_nhds.mul <|\n tendsto_pow_atTop_nhds_0_of_lt_1 (le_of_lt <| half_pos <| zero_lt_one) (half_lt_self zero_lt_one)\n have hd : Tendsto (fun m => d (\u22c2 n, f m n)) atTop (\ud835\udcdd (d (\u22c3 m, \u22c2 n, f m n))) :=\n by\n refine' d_Union _ _\n exact fun n m hnm => subset_iInter fun i => Subset.trans (iInter_subset (f n) i) <| f_subset_f hnm <| le_rfl\n refine' le_of_tendsto_of_tendsto' h\u03b3 hd fun m => _\n have : Tendsto (fun n => d (f m n)) atTop (\ud835\udcdd (d (\u22c2 n, f m n))) :=\n by\n refine' d_Inter _ _ _\n \u00b7 intro n\n exact hf _ _\n \u00b7 intro n m hnm\n exact f_subset_f le_rfl hnm\n refine' ge_of_tendsto this (eventually_atTop.2 \u27e8m, fun n hmn => _\u27e9)\n change \u03b3 - 2 * (1 / 2) ^ m \u2264 d (f m n)\n refine' le_trans _ (le_d_f _ _ hmn)\n exact le_add_of_le_of_nonneg le_rfl (pow_nonneg (le_of_lt <| half_pos <| zero_lt_one) _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u22a2 \u03b3 \u2264 d s\n[PROOFSTEP]\nhave h\u03b3 : Tendsto (fun m : \u2115 => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3) :=\n by\n suffices Tendsto (fun m : \u2115 => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd (\u03b3 - 2 * 0)) by simpa only [mul_zero, tsub_zero]\n exact\n tendsto_const_nhds.sub <|\n tendsto_const_nhds.mul <|\n tendsto_pow_atTop_nhds_0_of_lt_1 (le_of_lt <| half_pos <| zero_lt_one) (half_lt_self zero_lt_one)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u22a2 Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\n[PROOFSTEP]\nsuffices Tendsto (fun m : \u2115 => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd (\u03b3 - 2 * 0)) by simpa only [mul_zero, tsub_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis\u271d : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\nthis : Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd (\u03b3 - 2 * 0))\n\u22a2 Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\n[PROOFSTEP]\nsimpa only [mul_zero, tsub_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u22a2 Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd (\u03b3 - 2 * 0))\n[PROOFSTEP]\nexact\n tendsto_const_nhds.sub <|\n tendsto_const_nhds.mul <|\n tendsto_pow_atTop_nhds_0_of_lt_1 (le_of_lt <| half_pos <| zero_lt_one) (half_lt_self zero_lt_one)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\nh\u03b3 : Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\n\u22a2 \u03b3 \u2264 d s\n[PROOFSTEP]\nhave hd : Tendsto (fun m => d (\u22c2 n, f m n)) atTop (\ud835\udcdd (d (\u22c3 m, \u22c2 n, f m n))) :=\n by\n refine' d_Union _ _\n exact fun n m hnm => subset_iInter fun i => Subset.trans (iInter_subset (f n) i) <| f_subset_f hnm <| le_rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\nh\u03b3 : Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\n\u22a2 Tendsto (fun m => d (\u22c2 (n : \u2115), f m n)) atTop (\ud835\udcdd (d (\u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n)))\n[PROOFSTEP]\nrefine' d_Union _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\nh\u03b3 : Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\n\u22a2 Monotone fun m => \u22c2 (n : \u2115), f m n\n[PROOFSTEP]\nexact fun n m hnm => subset_iInter fun i => Subset.trans (iInter_subset (f n) i) <| f_subset_f hnm <| le_rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\nh\u03b3 : Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\nhd : Tendsto (fun m => d (\u22c2 (n : \u2115), f m n)) atTop (\ud835\udcdd (d (\u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n)))\n\u22a2 \u03b3 \u2264 d s\n[PROOFSTEP]\nrefine' le_of_tendsto_of_tendsto' h\u03b3 hd fun m => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\nh\u03b3 : Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\nhd : Tendsto (fun m => d (\u22c2 (n : \u2115), f m n)) atTop (\ud835\udcdd (d (\u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n)))\nm : \u2115\n\u22a2 \u03b3 - 2 * (1 / 2) ^ m \u2264 d (\u22c2 (n : \u2115), f m n)\n[PROOFSTEP]\nhave : Tendsto (fun n => d (f m n)) atTop (\ud835\udcdd (d (\u22c2 n, f m n))) :=\n by\n refine' d_Inter _ _ _\n \u00b7 intro n\n exact hf _ _\n \u00b7 intro n m hnm\n exact f_subset_f le_rfl hnm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\nh\u03b3 : Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\nhd : Tendsto (fun m => d (\u22c2 (n : \u2115), f m n)) atTop (\ud835\udcdd (d (\u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n)))\nm : \u2115\n\u22a2 Tendsto (fun n => d (f m n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), f m n)))\n[PROOFSTEP]\nrefine' d_Inter _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\nh\u03b3 : Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\nhd : Tendsto (fun m => d (\u22c2 (n : \u2115), f m n)) atTop (\ud835\udcdd (d (\u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n)))\nm : \u2115\n\u22a2 \u2200 (n : \u2115), MeasurableSet (f m n)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\nh\u03b3 : Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\nhd : Tendsto (fun m => d (\u22c2 (n : \u2115), f m n)) atTop (\ud835\udcdd (d (\u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n)))\nm n : \u2115\n\u22a2 MeasurableSet (f m n)\n[PROOFSTEP]\nexact hf _ _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\nh\u03b3 : Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\nhd : Tendsto (fun m => d (\u22c2 (n : \u2115), f m n)) atTop (\ud835\udcdd (d (\u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n)))\nm : \u2115\n\u22a2 \u2200 (n m_1 : \u2115), n \u2264 m_1 \u2192 f m m_1 \u2286 f m n\n[PROOFSTEP]\nintro n m hnm\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\nh\u03b3 : Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\nhd : Tendsto (fun m => d (\u22c2 (n : \u2115), f m n)) atTop (\ud835\udcdd (d (\u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n)))\nm\u271d n m : \u2115\nhnm : n \u2264 m\n\u22a2 f m\u271d m \u2286 f m\u271d n\n[PROOFSTEP]\nexact f_subset_f le_rfl hnm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis\u271d : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\nh\u03b3 : Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\nhd : Tendsto (fun m => d (\u22c2 (n : \u2115), f m n)) atTop (\ud835\udcdd (d (\u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n)))\nm : \u2115\nthis : Tendsto (fun n => d (f m n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), f m n)))\n\u22a2 \u03b3 - 2 * (1 / 2) ^ m \u2264 d (\u22c2 (n : \u2115), f m n)\n[PROOFSTEP]\nrefine' ge_of_tendsto this (eventually_atTop.2 \u27e8m, fun n hmn => _\u27e9)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis\u271d : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\nh\u03b3 : Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\nhd : Tendsto (fun m => d (\u22c2 (n : \u2115), f m n)) atTop (\ud835\udcdd (d (\u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n)))\nm : \u2115\nthis : Tendsto (fun n => d (f m n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), f m n)))\nn : \u2115\nhmn : n \u2265 m\n\u22a2 \u03b3 - 2 * (1 / 2) ^ m \u2264 d (f m n)\n[PROOFSTEP]\nchange \u03b3 - 2 * (1 / 2) ^ m \u2264 d (f m n)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis\u271d : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\nh\u03b3 : Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\nhd : Tendsto (fun m => d (\u22c2 (n : \u2115), f m n)) atTop (\ud835\udcdd (d (\u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n)))\nm : \u2115\nthis : Tendsto (fun n => d (f m n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), f m n)))\nn : \u2115\nhmn : n \u2265 m\n\u22a2 \u03b3 - 2 * (1 / 2) ^ m \u2264 d (f m n)\n[PROOFSTEP]\nrefine' le_trans _ (le_d_f _ _ hmn)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis\u271d : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\nh\u03b3 : Tendsto (fun m => \u03b3 - 2 * (1 / 2) ^ m) atTop (\ud835\udcdd \u03b3)\nhd : Tendsto (fun m => d (\u22c2 (n : \u2115), f m n)) atTop (\ud835\udcdd (d (\u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n)))\nm : \u2115\nthis : Tendsto (fun n => d (f m n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), f m n)))\nn : \u2115\nhmn : n \u2265 m\n\u22a2 \u03b3 - 2 * (1 / 2) ^ m \u2264 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n\n[PROOFSTEP]\nexact le_add_of_le_of_nonneg le_rfl (pow_nonneg (le_of_lt <| half_pos <| zero_lt_one) _)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave hs : MeasurableSet s := MeasurableSet.iUnion fun n => MeasurableSet.iInter fun m => hf _ _\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\nhs : MeasurableSet s\n\u22a2 \u2203 s,\n MeasurableSet s \u2227\n (\u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t) \u2227 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nrefine' \u27e8s, hs, _, _\u27e9\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s \u2192 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t\n[PROOFSTEP]\nintro t ht hts\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\nhts : t \u2286 s\n\u22a2 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t\n[PROOFSTEP]\nhave : 0 \u2264 d t :=\n (add_le_add_iff_left \u03b3).1 <|\n calc\n \u03b3 + 0 \u2264 d s := by rw [add_zero]; exact \u03b3_le_d_s\n _ = d (s \\ t) + d t := by rw [d_split _ _ hs ht, inter_eq_self_of_subset_right hts]\n _ \u2264 \u03b3 + d t := add_le_add (d_le_\u03b3 _ (hs.diff ht)) le_rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\nhts : t \u2286 s\n\u22a2 \u03b3 + 0 \u2264 d s\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\nhts : t \u2286 s\n\u22a2 \u03b3 \u2264 d s\n[PROOFSTEP]\nexact \u03b3_le_d_s\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\nhts : t \u2286 s\n\u22a2 d s = d (s \\ t) + d t\n[PROOFSTEP]\nrw [d_split _ _ hs ht, inter_eq_self_of_subset_right hts]\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis\u271d : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\nhts : t \u2286 s\nthis : 0 \u2264 d t\n\u22a2 \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc t\n[PROOFSTEP]\nrw [\u2190 to_nnreal_\u03bc, \u2190 to_nnreal_\u03bd, ENNReal.coe_le_coe, \u2190 NNReal.coe_le_coe]\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis\u271d : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\nhts : t \u2286 s\nthis : 0 \u2264 d t\n\u22a2 \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd t)) \u2264 \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc t))\n[PROOFSTEP]\nsimpa only [le_sub_iff_add_le, zero_add] using this\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nintro t ht hts\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\nhts : t \u2286 s\u1d9c\n\u22a2 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave : d t \u2264 0 :=\n (add_le_add_iff_left \u03b3).1 <|\n calc\n \u03b3 + d t \u2264 d s + d t := add_le_add \u03b3_le_d_s le_rfl\n _ = d (s \u222a t) := by\n rw [d_split _ _ (hs.union ht) ht, union_diff_right, union_inter_cancel_right,\n (subset_compl_iff_disjoint_left.1 hts).sdiff_eq_left]\n _ \u2264 \u03b3 + 0 := by rw [add_zero]; exact d_le_\u03b3 _ (hs.union ht)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\nhts : t \u2286 s\u1d9c\n\u22a2 d s + d t = d (s \u222a t)\n[PROOFSTEP]\nrw [d_split _ _ (hs.union ht) ht, union_diff_right, union_inter_cancel_right,\n (subset_compl_iff_disjoint_left.1 hts).sdiff_eq_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\nhts : t \u2286 s\u1d9c\n\u22a2 d (s \u222a t) \u2264 \u03b3 + 0\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\nhts : t \u2286 s\u1d9c\n\u22a2 d (s \u222a t) \u2264 \u03b3\n[PROOFSTEP]\nexact d_le_\u03b3 _ (hs.union ht)\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis\u271d : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\nhts : t \u2286 s\u1d9c\nthis : d t \u2264 0\n\u22a2 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bd t\n[PROOFSTEP]\nrw [\u2190 to_nnreal_\u03bc, \u2190 to_nnreal_\u03bd, ENNReal.coe_le_coe, \u2190 NNReal.coe_le_coe]\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nd : Set \u03b1 \u2192 \u211d := fun s => \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) - \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s))\nc : Set \u211d := d '' {s | MeasurableSet s}\n\u03b3 : \u211d := sSup c\nh\u03bc : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bd : \u2200 (s : Set \u03b1), \u2191\u2191\u03bd s \u2260 \u22a4\nto_nnreal_\u03bc : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc s)) = \u2191\u2191\u03bc s\nto_nnreal_\u03bd : \u2200 (s : Set \u03b1), \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd s)) = \u2191\u2191\u03bd s\nd_split : \u2200 (s t : Set \u03b1), MeasurableSet s \u2192 MeasurableSet t \u2192 d s = d (s \\ t) + d (s \u2229 t)\nd_Union : \u2200 (s : \u2115 \u2192 Set \u03b1), Monotone s \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c3 (n : \u2115), s n)))\nd_Inter :\n \u2200 (s : \u2115 \u2192 Set \u03b1),\n (\u2200 (n : \u2115), MeasurableSet (s n)) \u2192\n (\u2200 (n m : \u2115), n \u2264 m \u2192 s m \u2286 s n) \u2192 Tendsto (fun n => d (s n)) atTop (\ud835\udcdd (d (\u22c2 (n : \u2115), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_\u03b3 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 d s \u2264 \u03b3\nthis\u271d : \u2200 (n : \u2115), \u2203 s, MeasurableSet s \u2227 \u03b3 - (1 / 2) ^ n < d s\ne : \u2115 \u2192 Set \u03b1\nhe : \u2200 (x : \u2115), MeasurableSet (e x) \u2227 \u03b3 - (1 / 2) ^ x < d (e x)\nhe\u2081 : \u2200 (n : \u2115), MeasurableSet (e n)\nhe\u2082 : \u2200 (n : \u2115), \u03b3 - (1 / 2) ^ n < d (e n)\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1 := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : \u2200 (n m : \u2115), MeasurableSet (f n m)\nf_subset_f : \u2200 {a b c d : \u2115}, a \u2264 b \u2192 c \u2264 d \u2192 f a d \u2286 f b c\nf_succ : \u2200 (n m : \u2115), n \u2264 m \u2192 f n (m + 1) = f n m \u2229 e (m + 1)\nle_d_f : \u2200 (n m : \u2115), m \u2264 n \u2192 \u03b3 - 2 * (1 / 2) ^ m + (1 / 2) ^ n \u2264 d (f m n)\ns : Set \u03b1 := \u22c3 (m : \u2115), \u22c2 (n : \u2115), f m n\n\u03b3_le_d_s : \u03b3 \u2264 d s\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\nhts : t \u2286 s\u1d9c\nthis : d t \u2264 0\n\u22a2 \u2191(ENNReal.toNNReal (\u2191\u2191\u03bc t)) \u2264 \u2191(ENNReal.toNNReal (\u2191\u2191\u03bd t))\n[PROOFSTEP]\nsimpa only [sub_le_iff_le_add, zero_add] using this\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Decomposition.UnsignedHahn", "llama_tokens": 93982, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.798186787341014, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.4456492298943171}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1\nhg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 1\n[PROOFSTEP]\ncalc\n (\u222b\u207b a : \u03b1, (f * g) a \u2202\u03bc) \u2264 \u222b\u207b a : \u03b1, f a ^ p / ENNReal.ofReal p + g a ^ q / ENNReal.ofReal q \u2202\u03bc :=\n lintegral_mono fun a => young_inequality (f a) (g a) hpq\n _ = 1 := by\n simp only [div_eq_mul_inv]\n rw [lintegral_add_left']\n \u00b7 rw [lintegral_mul_const'' _ (hf.pow_const p), lintegral_mul_const', hf_norm, hg_norm, \u2190 div_eq_mul_inv, \u2190\n div_eq_mul_inv, hpq.inv_add_inv_conj_ennreal]\n simp [hpq.symm.pos]\n \u00b7 exact (hf.pow_const _).mul_const _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1\nhg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1\n\u22a2 \u222b\u207b (a : \u03b1), f a ^ p / ENNReal.ofReal p + g a ^ q / ENNReal.ofReal q \u2202\u03bc = 1\n[PROOFSTEP]\nsimp only [div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1\nhg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1\n\u22a2 \u222b\u207b (a : \u03b1), f a ^ p * (ENNReal.ofReal p)\u207b\u00b9 + g a ^ q * (ENNReal.ofReal q)\u207b\u00b9 \u2202\u03bc = 1\n[PROOFSTEP]\nrw [lintegral_add_left']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1\nhg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1\n\u22a2 \u222b\u207b (a : \u03b1), f a ^ p * (ENNReal.ofReal p)\u207b\u00b9 \u2202\u03bc + \u222b\u207b (a : \u03b1), g a ^ q * (ENNReal.ofReal q)\u207b\u00b9 \u2202\u03bc = 1\n[PROOFSTEP]\nrw [lintegral_mul_const'' _ (hf.pow_const p), lintegral_mul_const', hf_norm, hg_norm, \u2190 div_eq_mul_inv, \u2190\n div_eq_mul_inv, hpq.inv_add_inv_conj_ennreal]\n[GOAL]\ncase hr\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1\nhg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1\n\u22a2 (ENNReal.ofReal q)\u207b\u00b9 \u2260 \u22a4\n[PROOFSTEP]\nsimp [hpq.symm.pos]\n[GOAL]\ncase hf\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1\nhg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1\n\u22a2 AEMeasurable fun a => f a ^ p * (ENNReal.ofReal p)\u207b\u00b9\n[PROOFSTEP]\nexact (hf.pow_const _).mul_const _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\na : \u03b1\n\u22a2 f a = funMulInvSnorm f p \u03bc a * (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nsimp [funMulInvSnorm, mul_assoc, ENNReal.inv_mul_cancel, hf_nonzero, hf_top]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 < p\nf : \u03b1 \u2192 \u211d\u22650\u221e\na : \u03b1\n\u22a2 funMulInvSnorm f p \u03bc a ^ p = f a ^ p * (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc)\u207b\u00b9\n[PROOFSTEP]\nrw [funMulInvSnorm, mul_rpow_of_nonneg _ _ (le_of_lt hp0)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 < p\nf : \u03b1 \u2192 \u211d\u22650\u221e\na : \u03b1\n\u22a2 f a ^ p * ((\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p))\u207b\u00b9 ^ p = f a ^ p * (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc)\u207b\u00b9\n[PROOFSTEP]\nsuffices h_inv_rpow : ((\u222b\u207b c : \u03b1, f c ^ p \u2202\u03bc) ^ (1 / p))\u207b\u00b9 ^ p = (\u222b\u207b c : \u03b1, f c ^ p \u2202\u03bc)\u207b\u00b9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 < p\nf : \u03b1 \u2192 \u211d\u22650\u221e\na : \u03b1\nh_inv_rpow : ((\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p))\u207b\u00b9 ^ p = (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc)\u207b\u00b9\n\u22a2 f a ^ p * ((\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p))\u207b\u00b9 ^ p = f a ^ p * (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc)\u207b\u00b9\n[PROOFSTEP]\nrw [h_inv_rpow]\n[GOAL]\ncase h_inv_rpow\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 < p\nf : \u03b1 \u2192 \u211d\u22650\u221e\na : \u03b1\n\u22a2 ((\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p))\u207b\u00b9 ^ p = (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc)\u207b\u00b9\n[PROOFSTEP]\nrw [inv_rpow, \u2190 rpow_mul, one_div_mul_cancel hp0.ne', rpow_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0_lt : 0 < p\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (c : \u03b1), funMulInvSnorm f p \u03bc c ^ p \u2202\u03bc = 1\n[PROOFSTEP]\nsimp_rw [funMulInvSnorm_rpow hp0_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0_lt : 0 < p\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (c : \u03b1), f c ^ p * (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc)\u207b\u00b9 \u2202\u03bc = 1\n[PROOFSTEP]\nrw [lintegral_mul_const', ENNReal.mul_inv_cancel hf_nonzero hf_top]\n[GOAL]\ncase hr\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0_lt : 0 < p\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc)\u207b\u00b9 \u2260 \u22a4\n[PROOFSTEP]\nrwa [inv_ne_top]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nlet npf := (\u222b\u207b c : \u03b1, f c ^ p \u2202\u03bc) ^ (1 / p)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nlet nqg := (\u222b\u207b c : \u03b1, g c ^ q \u2202\u03bc) ^ (1 / q)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\ncalc\n (\u222b\u207b a : \u03b1, (f * g) a \u2202\u03bc) = \u222b\u207b a : \u03b1, (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a * (npf * nqg) \u2202\u03bc :=\n by\n refine' lintegral_congr fun a => _\n rw [Pi.mul_apply, fun_eq_funMulInvSnorm_mul_snorm f hf_nonzero hf_nontop,\n fun_eq_funMulInvSnorm_mul_snorm g hg_nonzero hg_nontop, Pi.mul_apply]\n ring\n _ \u2264 npf * nqg :=\n by\n rw [lintegral_mul_const' (npf * nqg) _ (by simp [hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top])]\n refine' mul_le_of_le_one_left' _\n have hf1 := lintegral_rpow_funMulInvSnorm_eq_one hpq.pos hf_nonzero hf_nontop\n have hg1 := lintegral_rpow_funMulInvSnorm_eq_one hpq.symm.pos hg_nonzero hg_nontop\n exact lintegral_mul_le_one_of_lintegral_rpow_eq_one hpq (hf.mul_const _) hf1 hg1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a * (npf * nqg) \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_congr fun a => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\na : \u03b1\n\u22a2 (f * g) a = (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a * (npf * nqg)\n[PROOFSTEP]\nrw [Pi.mul_apply, fun_eq_funMulInvSnorm_mul_snorm f hf_nonzero hf_nontop,\n fun_eq_funMulInvSnorm_mul_snorm g hg_nonzero hg_nontop, Pi.mul_apply]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\na : \u03b1\n\u22a2 funMulInvSnorm f p \u03bc a * (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p) *\n (funMulInvSnorm g q \u03bc a * (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)) =\n funMulInvSnorm f p \u03bc a * funMulInvSnorm g q \u03bc a * (npf * nqg)\n[PROOFSTEP]\nring\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\n\u22a2 \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a * (npf * nqg) \u2202\u03bc \u2264 npf * nqg\n[PROOFSTEP]\nrw [lintegral_mul_const' (npf * nqg) _ (by simp [hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top])]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\n\u22a2 npf * nqg \u2260 \u22a4\n[PROOFSTEP]\nsimp [hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\n\u22a2 (\u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc) * (npf * nqg) \u2264 npf * nqg\n[PROOFSTEP]\nrefine' mul_le_of_le_one_left' _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\n\u22a2 \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc \u2264 1\n[PROOFSTEP]\nhave hf1 := lintegral_rpow_funMulInvSnorm_eq_one hpq.pos hf_nonzero hf_nontop\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\nhf1 : \u222b\u207b (c : \u03b1), funMulInvSnorm (fun a => f a) p \u03bc c ^ p \u2202\u03bc = 1\n\u22a2 \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc \u2264 1\n[PROOFSTEP]\nhave hg1 := lintegral_rpow_funMulInvSnorm_eq_one hpq.symm.pos hg_nonzero hg_nontop\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_nontop : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_nontop : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 \u22a4\nhf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nnpf : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc) ^ (1 / p)\nnqg : \u211d\u22650\u221e := (\u222b\u207b (c : \u03b1), g c ^ q \u2202\u03bc) ^ (1 / q)\nhf1 : \u222b\u207b (c : \u03b1), funMulInvSnorm (fun a => f a) p \u03bc c ^ p \u2202\u03bc = 1\nhg1 : \u222b\u207b (c : \u03b1), funMulInvSnorm (fun a => g a) q \u03bc c ^ q \u2202\u03bc = 1\n\u22a2 \u222b\u207b (a : \u03b1), (funMulInvSnorm f p \u03bc * funMulInvSnorm g q \u03bc) a \u2202\u03bc \u2264 1\n[PROOFSTEP]\nexact lintegral_mul_le_one_of_lintegral_rpow_eq_one hpq (hf.mul_const _) hf1 hg1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nrw [lintegral_eq_zero_iff' (hf.pow_const p)] at hf_zero \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : (fun x => f x ^ p) =\u1d50[\u03bc] 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nrefine' Filter.Eventually.mp hf_zero (Filter.eventually_of_forall fun x => _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : (fun x => f x ^ p) =\u1d50[\u03bc] 0\nx : \u03b1\n\u22a2 (fun x => f x ^ p) x = OfNat.ofNat 0 x \u2192 f x = OfNat.ofNat 0 x\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : (fun x => f x ^ p) =\u1d50[\u03bc] 0\nx : \u03b1\n\u22a2 f x ^ p = OfNat.ofNat 0 x \u2192 f x = OfNat.ofNat 0 x\n[PROOFSTEP]\nrw [Pi.zero_apply, \u2190 not_imp_not]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : (fun x => f x ^ p) =\u1d50[\u03bc] 0\nx : \u03b1\n\u22a2 \u00acf x = 0 \u2192 \u00acf x ^ p = 0\n[PROOFSTEP]\nexact fun hx => (rpow_pos_of_nonneg (pos_iff_ne_zero.2 hx) hp0).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0\n[PROOFSTEP]\nrw [\u2190 @lintegral_zero_fun \u03b1 _ \u03bc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = lintegral \u03bc 0\n[PROOFSTEP]\nrefine' lintegral_congr_ae _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 (fun a => (f * g) a) =\u1d50[\u03bc] fun a => OfNat.ofNat 0 a\n[PROOFSTEP]\nsuffices h_mul_zero : f * g =\u1d50[\u03bc] 0 * g\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nh_mul_zero : f * g =\u1d50[\u03bc] 0 * g\n\u22a2 (fun a => (f * g) a) =\u1d50[\u03bc] fun a => OfNat.ofNat 0 a\n[PROOFSTEP]\nrwa [zero_mul] at h_mul_zero \n[GOAL]\ncase h_mul_zero\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 f * g =\u1d50[\u03bc] 0 * g\n[PROOFSTEP]\nhave hf_eq_zero : f =\u1d50[\u03bc] 0 := ae_eq_zero_of_lintegral_rpow_eq_zero hp0 hf hf_zero\n[GOAL]\ncase h_mul_zero\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nhp0 : 0 \u2264 p\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhf_eq_zero : f =\u1d50[\u03bc] 0\n\u22a2 f * g =\u1d50[\u03bc] 0 * g\n[PROOFSTEP]\nexact hf_eq_zero.mul (ae_eq_refl g)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhp0_lt : 0 < p\nhq0 : 0 \u2264 q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nrefine' le_trans le_top (le_of_eq _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhp0_lt : 0 < p\nhq0 : 0 \u2264 q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\n\u22a2 \u22a4 = (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nhave hp0_inv_lt : 0 < 1 / p := by simp [hp0_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhp0_lt : 0 < p\nhq0 : 0 \u2264 q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\n\u22a2 0 < 1 / p\n[PROOFSTEP]\nsimp [hp0_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhp0_lt : 0 < p\nhq0 : 0 \u2264 q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nhp0_inv_lt : 0 < 1 / p\n\u22a2 \u22a4 = (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nrw [hf_top, ENNReal.top_rpow_of_pos hp0_inv_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhp0_lt : 0 < p\nhq0 : 0 \u2264 q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_nonzero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc \u2260 0\nhp0_inv_lt : 0 < 1 / p\n\u22a2 \u22a4 = \u22a4 * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nsimp [hq0, hg_nonzero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nby_cases hf_zero : \u222b\u207b a, f a ^ p \u2202\u03bc = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nrefine' Eq.trans_le _ (zero_le _)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0\n[PROOFSTEP]\nexact lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero hpq.nonneg hf hf_zero\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nby_cases hg_zero : \u222b\u207b a, g a ^ q \u2202\u03bc = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nrefine' Eq.trans_le _ (zero_le _)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (g * f) a \u2202\u03bc = 0\n[PROOFSTEP]\nexact lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero hpq.symm.nonneg hg hg_zero\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nby_cases hf_top : \u222b\u207b a, f a ^ p \u2202\u03bc = \u22a4\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nexact lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top hpq.pos hpq.symm.nonneg hf_top hg_zero\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nby_cases hg_top : \u222b\u207b a, g a ^ q \u2202\u03bc = \u22a4\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nrw [mul_comm, mul_comm ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p))]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (g * f) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nexact lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top hpq.symm.pos hpq.nonneg hg_top hf_zero\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nexact ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top hpq hf hf_top hg_top hf_zero hg_zero\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc < \u22a4\n[PROOFSTEP]\nhave hp0_lt : 0 < p := lt_of_lt_of_le zero_lt_one hp1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc < \u22a4\n[PROOFSTEP]\nhave hp0 : 0 \u2264 p := le_of_lt hp0_lt\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc < \u22a4\n[PROOFSTEP]\ncalc\n (\u222b\u207b a : \u03b1, (f a + g a) ^ p \u2202\u03bc) \u2264 \u222b\u207b a, (2 : \u211d\u22650\u221e) ^ (p - 1) * f a ^ p + (2 : \u211d\u22650\u221e) ^ (p - 1) * g a ^ p \u2202\u03bc :=\n by\n refine' lintegral_mono fun a => _\n dsimp only\n have h_zero_lt_half_rpow : (0 : \u211d\u22650\u221e) < (1 / 2 : \u211d\u22650\u221e) ^ p :=\n by\n rw [\u2190 ENNReal.zero_rpow_of_pos hp0_lt]\n exact ENNReal.rpow_lt_rpow (by simp [zero_lt_one]) hp0_lt\n have h_rw : (1 / 2 : \u211d\u22650\u221e) ^ p * (2 : \u211d\u22650\u221e) ^ (p - 1) = 1 / 2 := by\n rw [sub_eq_add_neg, ENNReal.rpow_add _ _ two_ne_zero ENNReal.coe_ne_top, \u2190 mul_assoc, \u2190\n ENNReal.mul_rpow_of_nonneg _ _ hp0, one_div, ENNReal.inv_mul_cancel two_ne_zero ENNReal.coe_ne_top,\n ENNReal.one_rpow, one_mul, ENNReal.rpow_neg_one]\n rw [\u2190 ENNReal.mul_le_mul_left (ne_of_lt h_zero_lt_half_rpow).symm _]\n \u00b7 rw [mul_add, \u2190 mul_assoc, \u2190 mul_assoc, h_rw, \u2190 ENNReal.mul_rpow_of_nonneg _ _ hp0, mul_add]\n refine' ENNReal.rpow_arith_mean_le_arith_mean2_rpow (1 / 2 : \u211d\u22650\u221e) (1 / 2 : \u211d\u22650\u221e) (f a) (g a) _ hp1\n rw [ENNReal.div_add_div_same, one_add_one_eq_two, ENNReal.div_self two_ne_zero ENNReal.coe_ne_top]\n \u00b7 rw [\u2190 lt_top_iff_ne_top]\n refine' ENNReal.rpow_lt_top_of_nonneg hp0 _\n rw [one_div, ENNReal.inv_ne_top]\n exact two_ne_zero\n _ < \u22a4 :=\n by\n have h_two : (2 : \u211d\u22650\u221e) ^ (p - 1) \u2260 \u22a4 := ENNReal.rpow_ne_top_of_nonneg (by simp [hp1]) ENNReal.coe_ne_top\n rw [lintegral_add_left', lintegral_const_mul'' _ (hf.pow_const p), lintegral_const_mul' _ _ h_two,\n ENNReal.add_lt_top]\n \u00b7 exact \u27e8ENNReal.mul_lt_top h_two hf_top.ne, ENNReal.mul_lt_top h_two hg_top.ne\u27e9\n \u00b7 exact (hf.pow_const p).const_mul _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\n\u22a2 \u222b\u207b (a : \u03b1), (f a + g a) ^ p \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_mono fun a => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\n\u22a2 (f a + g a) ^ p \u2264 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\n\u22a2 (f a + g a) ^ p \u2264 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p\n[PROOFSTEP]\nhave h_zero_lt_half_rpow : (0 : \u211d\u22650\u221e) < (1 / 2 : \u211d\u22650\u221e) ^ p :=\n by\n rw [\u2190 ENNReal.zero_rpow_of_pos hp0_lt]\n exact ENNReal.rpow_lt_rpow (by simp [zero_lt_one]) hp0_lt\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\n\u22a2 0 < (1 / 2) ^ p\n[PROOFSTEP]\nrw [\u2190 ENNReal.zero_rpow_of_pos hp0_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\n\u22a2 0 ^ p < (1 / 2) ^ p\n[PROOFSTEP]\nexact ENNReal.rpow_lt_rpow (by simp [zero_lt_one]) hp0_lt\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\n\u22a2 0 < 1 / 2\n[PROOFSTEP]\nsimp [zero_lt_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\n\u22a2 (f a + g a) ^ p \u2264 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p\n[PROOFSTEP]\nhave h_rw : (1 / 2 : \u211d\u22650\u221e) ^ p * (2 : \u211d\u22650\u221e) ^ (p - 1) = 1 / 2 := by\n rw [sub_eq_add_neg, ENNReal.rpow_add _ _ two_ne_zero ENNReal.coe_ne_top, \u2190 mul_assoc, \u2190\n ENNReal.mul_rpow_of_nonneg _ _ hp0, one_div, ENNReal.inv_mul_cancel two_ne_zero ENNReal.coe_ne_top,\n ENNReal.one_rpow, one_mul, ENNReal.rpow_neg_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\n\u22a2 (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n[PROOFSTEP]\nrw [sub_eq_add_neg, ENNReal.rpow_add _ _ two_ne_zero ENNReal.coe_ne_top, \u2190 mul_assoc, \u2190\n ENNReal.mul_rpow_of_nonneg _ _ hp0, one_div, ENNReal.inv_mul_cancel two_ne_zero ENNReal.coe_ne_top, ENNReal.one_rpow,\n one_mul, ENNReal.rpow_neg_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 (f a + g a) ^ p \u2264 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p\n[PROOFSTEP]\nrw [\u2190 ENNReal.mul_le_mul_left (ne_of_lt h_zero_lt_half_rpow).symm _]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 (1 / 2) ^ p * (f a + g a) ^ p \u2264 (1 / 2) ^ p * (2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p)\n[PROOFSTEP]\nrw [mul_add, \u2190 mul_assoc, \u2190 mul_assoc, h_rw, \u2190 ENNReal.mul_rpow_of_nonneg _ _ hp0, mul_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 (1 / 2 * f a + 1 / 2 * g a) ^ p \u2264 1 / 2 * f a ^ p + 1 / 2 * g a ^ p\n[PROOFSTEP]\nrefine' ENNReal.rpow_arith_mean_le_arith_mean2_rpow (1 / 2 : \u211d\u22650\u221e) (1 / 2 : \u211d\u22650\u221e) (f a) (g a) _ hp1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 1 / 2 + 1 / 2 = 1\n[PROOFSTEP]\nrw [ENNReal.div_add_div_same, one_add_one_eq_two, ENNReal.div_self two_ne_zero ENNReal.coe_ne_top]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 (1 / 2) ^ p \u2260 \u22a4\n[PROOFSTEP]\nrw [\u2190 lt_top_iff_ne_top]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 (1 / 2) ^ p < \u22a4\n[PROOFSTEP]\nrefine' ENNReal.rpow_lt_top_of_nonneg hp0 _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 1 / 2 \u2260 \u22a4\n[PROOFSTEP]\nrw [one_div, ENNReal.inv_ne_top]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\na : \u03b1\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / 2\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nexact two_ne_zero\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\n\u22a2 \u222b\u207b (a : \u03b1), 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p \u2202\u03bc < \u22a4\n[PROOFSTEP]\nhave h_two : (2 : \u211d\u22650\u221e) ^ (p - 1) \u2260 \u22a4 := ENNReal.rpow_ne_top_of_nonneg (by simp [hp1]) ENNReal.coe_ne_top\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\n\u22a2 0 \u2264 p - 1\n[PROOFSTEP]\nsimp [hp1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\nh_two : 2 ^ (p - 1) \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p \u2202\u03bc < \u22a4\n[PROOFSTEP]\nrw [lintegral_add_left', lintegral_const_mul'' _ (hf.pow_const p), lintegral_const_mul' _ _ h_two, ENNReal.add_lt_top]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\nh_two : 2 ^ (p - 1) \u2260 \u22a4\n\u22a2 2 ^ (p - 1) * \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 \u2227 2 ^ (p - 1) * \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\n[PROOFSTEP]\nexact \u27e8ENNReal.mul_lt_top h_two hf_top.ne, ENNReal.mul_lt_top h_two hg_top.ne\u27e9\n[GOAL]\ncase hf\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nhp1 : 1 \u2264 p\nhp0_lt : 0 < p\nhp0 : 0 \u2264 p\nh_two : 2 ^ (p - 1) \u2260 \u22a4\n\u22a2 AEMeasurable fun a => 2 ^ (p - 1) * f a ^ p\n[PROOFSTEP]\nexact (hf.pow_const p).const_mul _\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\n\u22a2 (\u222b\u207b (a : \u03b1), (f * g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r)\n[PROOFSTEP]\nhave hp0_ne : p \u2260 0 := (ne_of_lt hp0_lt).symm\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f * g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r)\n[PROOFSTEP]\nhave hp0 : 0 \u2264 p := le_of_lt hp0_lt\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\n\u22a2 (\u222b\u207b (a : \u03b1), (f * g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r)\n[PROOFSTEP]\nhave hq0_lt : 0 < q := lt_of_le_of_lt hp0 hpq\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\n\u22a2 (\u222b\u207b (a : \u03b1), (f * g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r)\n[PROOFSTEP]\nhave hq0_ne : q \u2260 0 := (ne_of_lt hq0_lt).symm\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f * g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r)\n[PROOFSTEP]\nhave h_one_div_r : 1 / r = 1 / p - 1 / q := by rw [hpqr]; simp\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\n\u22a2 1 / r = 1 / p - 1 / q\n[PROOFSTEP]\nrw [hpqr]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\n\u22a2 1 / r = 1 / q + 1 / r - 1 / q\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\n\u22a2 (\u222b\u207b (a : \u03b1), (f * g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r)\n[PROOFSTEP]\nhave _ : r \u2260 0 :=\n by\n have hr_inv_pos : 0 < 1 / r := by rwa [h_one_div_r, sub_pos, one_div_lt_one_div hq0_lt hp0_lt]\n rw [one_div, _root_.inv_pos] at hr_inv_pos \n exact (ne_of_lt hr_inv_pos).symm\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\n\u22a2 r \u2260 0\n[PROOFSTEP]\nhave hr_inv_pos : 0 < 1 / r := by rwa [h_one_div_r, sub_pos, one_div_lt_one_div hq0_lt hp0_lt]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\n\u22a2 0 < 1 / r\n[PROOFSTEP]\nrwa [h_one_div_r, sub_pos, one_div_lt_one_div hq0_lt hp0_lt]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nhr_inv_pos : 0 < 1 / r\n\u22a2 r \u2260 0\n[PROOFSTEP]\nrw [one_div, _root_.inv_pos] at hr_inv_pos \n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nhr_inv_pos : 0 < r\n\u22a2 r \u2260 0\n[PROOFSTEP]\nexact (ne_of_lt hr_inv_pos).symm\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f * g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r)\n[PROOFSTEP]\nlet p2 := q / p\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\n\u22a2 (\u222b\u207b (a : \u03b1), (f * g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r)\n[PROOFSTEP]\nlet q2 := p2.conjugateExponent\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\n\u22a2 (\u222b\u207b (a : \u03b1), (f * g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r)\n[PROOFSTEP]\nhave hp2q2 : p2.IsConjugateExponent q2 :=\n Real.isConjugateExponent_conjugateExponent (by simp [_root_.lt_div_iff, hpq, hp0_lt])\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\n\u22a2 1 < p2\n[PROOFSTEP]\nsimp [_root_.lt_div_iff, hpq, hp0_lt]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n\u22a2 (\u222b\u207b (a : \u03b1), (f * g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r)\n[PROOFSTEP]\ncalc\n (\u222b\u207b a : \u03b1, (f * g) a ^ p \u2202\u03bc) ^ (1 / p) = (\u222b\u207b a : \u03b1, f a ^ p * g a ^ p \u2202\u03bc) ^ (1 / p) := by\n simp_rw [Pi.mul_apply, ENNReal.mul_rpow_of_nonneg _ _ hp0]\n _ \u2264 ((\u222b\u207b a, f a ^ (p * p2) \u2202\u03bc) ^ (1 / p2) * (\u222b\u207b a, g a ^ (p * q2) \u2202\u03bc) ^ (1 / q2)) ^ (1 / p) :=\n by\n refine' ENNReal.rpow_le_rpow _ (by simp [hp0])\n simp_rw [ENNReal.rpow_mul]\n exact ENNReal.lintegral_mul_le_Lp_mul_Lq \u03bc hp2q2 (hf.pow_const _) (hg.pow_const _)\n _ = (\u222b\u207b a : \u03b1, f a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b a : \u03b1, g a ^ r \u2202\u03bc) ^ (1 / r) :=\n by\n rw [@ENNReal.mul_rpow_of_nonneg _ _ (1 / p) (by simp [hp0]), \u2190 ENNReal.rpow_mul, \u2190 ENNReal.rpow_mul]\n have hpp2 : p * p2 = q := by\n symm\n rw [mul_comm, \u2190 div_eq_iff hp0_ne]\n have hpq2 : p * q2 = r := by\n rw [\u2190 inv_inv r, \u2190 one_div, \u2190 one_div, h_one_div_r]\n field_simp [Real.conjugateExponent, hp0_ne, hq0_ne]\n simp_rw [div_mul_div_comm, mul_one, mul_comm p2, mul_comm q2, hpp2, hpq2]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n\u22a2 (\u222b\u207b (a : \u03b1), (f * g) a ^ p \u2202\u03bc) ^ (1 / p) = (\u222b\u207b (a : \u03b1), f a ^ p * g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nsimp_rw [Pi.mul_apply, ENNReal.mul_rpow_of_nonneg _ _ hp0]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n\u22a2 (\u222b\u207b (a : \u03b1), f a ^ p * g a ^ p \u2202\u03bc) ^ (1 / p) \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ (p * p2) \u2202\u03bc) ^ (1 / p2) * (\u222b\u207b (a : \u03b1), g a ^ (p * q2) \u2202\u03bc) ^ (1 / q2)) ^ (1 / p)\n[PROOFSTEP]\nrefine' ENNReal.rpow_le_rpow _ (by simp [hp0])\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n\u22a2 0 \u2264 1 / p\n[PROOFSTEP]\nsimp [hp0]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n\u22a2 \u222b\u207b (a : \u03b1), f a ^ p * g a ^ p \u2202\u03bc \u2264\n (\u222b\u207b (a : \u03b1), f a ^ (p * p2) \u2202\u03bc) ^ (1 / p2) * (\u222b\u207b (a : \u03b1), g a ^ (p * q2) \u2202\u03bc) ^ (1 / q2)\n[PROOFSTEP]\nsimp_rw [ENNReal.rpow_mul]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n\u22a2 \u222b\u207b (a : \u03b1), f a ^ p * g a ^ p \u2202\u03bc \u2264\n (\u222b\u207b (a : \u03b1), (f a ^ p) ^ (q / p) \u2202\u03bc) ^ (1 / (q / p)) *\n (\u222b\u207b (a : \u03b1), (g a ^ p) ^ Real.conjugateExponent (q / p) \u2202\u03bc) ^ (1 / Real.conjugateExponent (q / p))\n[PROOFSTEP]\nexact ENNReal.lintegral_mul_le_Lp_mul_Lq \u03bc hp2q2 (hf.pow_const _) (hg.pow_const _)\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n\u22a2 ((\u222b\u207b (a : \u03b1), f a ^ (p * p2) \u2202\u03bc) ^ (1 / p2) * (\u222b\u207b (a : \u03b1), g a ^ (p * q2) \u2202\u03bc) ^ (1 / q2)) ^ (1 / p) =\n (\u222b\u207b (a : \u03b1), f a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r)\n[PROOFSTEP]\nrw [@ENNReal.mul_rpow_of_nonneg _ _ (1 / p) (by simp [hp0]), \u2190 ENNReal.rpow_mul, \u2190 ENNReal.rpow_mul]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n\u22a2 0 \u2264 1 / p\n[PROOFSTEP]\nsimp [hp0]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n\u22a2 (\u222b\u207b (a : \u03b1), f a ^ (p * p2) \u2202\u03bc) ^ (1 / p2 * (1 / p)) * (\u222b\u207b (a : \u03b1), g a ^ (p * q2) \u2202\u03bc) ^ (1 / q2 * (1 / p)) =\n (\u222b\u207b (a : \u03b1), f a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r)\n[PROOFSTEP]\nhave hpp2 : p * p2 = q := by\n symm\n rw [mul_comm, \u2190 div_eq_iff hp0_ne]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n\u22a2 p * p2 = q\n[PROOFSTEP]\nsymm\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\n\u22a2 q = p * p2\n[PROOFSTEP]\nrw [mul_comm, \u2190 div_eq_iff hp0_ne]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\nhpp2 : p * p2 = q\n\u22a2 (\u222b\u207b (a : \u03b1), f a ^ (p * p2) \u2202\u03bc) ^ (1 / p2 * (1 / p)) * (\u222b\u207b (a : \u03b1), g a ^ (p * q2) \u2202\u03bc) ^ (1 / q2 * (1 / p)) =\n (\u222b\u207b (a : \u03b1), f a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r)\n[PROOFSTEP]\nhave hpq2 : p * q2 = r := by\n rw [\u2190 inv_inv r, \u2190 one_div, \u2190 one_div, h_one_div_r]\n field_simp [Real.conjugateExponent, hp0_ne, hq0_ne]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\nhpp2 : p * p2 = q\n\u22a2 p * q2 = r\n[PROOFSTEP]\nrw [\u2190 inv_inv r, \u2190 one_div, \u2190 one_div, h_one_div_r]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\nhpp2 : p * p2 = q\n\u22a2 p * q2 = 1 / (1 / p - 1 / q)\n[PROOFSTEP]\nfield_simp [Real.conjugateExponent, hp0_ne, hq0_ne]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\np q r : \u211d\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp0_ne : p \u2260 0\nhp0 : 0 \u2264 p\nhq0_lt : 0 < q\nhq0_ne : q \u2260 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\nx\u271d : r \u2260 0\np2 : \u211d := q / p\nq2 : \u211d := Real.conjugateExponent p2\nhp2q2 : Real.IsConjugateExponent p2 q2\nhpp2 : p * p2 = q\nhpq2 : p * q2 = r\n\u22a2 (\u222b\u207b (a : \u03b1), f a ^ (p * p2) \u2202\u03bc) ^ (1 / p2 * (1 / p)) * (\u222b\u207b (a : \u03b1), g a ^ (p * q2) \u2202\u03bc) ^ (1 / q2 * (1 / p)) =\n (\u222b\u207b (a : \u03b1), f a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r)\n[PROOFSTEP]\nsimp_rw [div_mul_div_comm, mul_one, mul_comm p2, mul_comm q2, hpp2, hpq2]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), f a * g a ^ (p - 1) \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nrefine' le_trans (ENNReal.lintegral_mul_le_Lp_mul_Lq \u03bc hpq hf (hg.pow_const _)) _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), (g a ^ (p - 1)) ^ q \u2202\u03bc) ^ (1 / q) \u2264\n (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nby_cases hf_zero_rpow : (\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p) = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhf_zero_rpow : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = 0\n\u22a2 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), (g a ^ (p - 1)) ^ q \u2202\u03bc) ^ (1 / q) \u2264\n (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nrw [hf_zero_rpow, zero_mul]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhf_zero_rpow : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = 0\n\u22a2 0 \u2264 0 * (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhf_zero_rpow : \u00ac(\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = 0\n\u22a2 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), (g a ^ (p - 1)) ^ q \u2202\u03bc) ^ (1 / q) \u2264\n (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nhave hf_top_rpow : (\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 :=\n by\n by_contra h\n refine' hf_top _\n have hp_not_neg : \u00acp < 0 := by simp [hpq.nonneg]\n simpa [hpq.pos, hp_not_neg] using h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhf_zero_rpow : \u00ac(\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = 0\n\u22a2 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhf_zero_rpow : \u00ac(\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = 0\nh : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = \u22a4\n\u22a2 False\n[PROOFSTEP]\nrefine' hf_top _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhf_zero_rpow : \u00ac(\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = 0\nh : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n[PROOFSTEP]\nhave hp_not_neg : \u00acp < 0 := by simp [hpq.nonneg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhf_zero_rpow : \u00ac(\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = 0\nh : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = \u22a4\n\u22a2 \u00acp < 0\n[PROOFSTEP]\nsimp [hpq.nonneg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhf_zero_rpow : \u00ac(\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = 0\nh : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = \u22a4\nhp_not_neg : \u00acp < 0\n\u22a2 \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n[PROOFSTEP]\nsimpa [hpq.pos, hp_not_neg] using h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhf_zero_rpow : \u00ac(\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = 0\nhf_top_rpow : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), (g a ^ (p - 1)) ^ q \u2202\u03bc) ^ (1 / q) \u2264\n (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nrefine' (ENNReal.mul_le_mul_left hf_zero_rpow hf_top_rpow).mpr (le_of_eq _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhf_zero_rpow : \u00ac(\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = 0\nhf_top_rpow : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (g a ^ (p - 1)) ^ q \u2202\u03bc) ^ (1 / q) = (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase neg.e_a.e_f\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhf_zero_rpow : \u00ac(\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = 0\nhf_top_rpow : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\n\u22a2 (fun a => (g a ^ (p - 1)) ^ q) = fun a => g a ^ p\n[PROOFSTEP]\next1 a\n[GOAL]\ncase neg.e_a.e_f.h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhf_zero_rpow : \u00ac(\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) = 0\nhf_top_rpow : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\na : \u03b1\n\u22a2 (g a ^ (p - 1)) ^ q = g a ^ p\n[PROOFSTEP]\nrw [\u2190 ENNReal.rpow_mul, hpq.sub_one_mul_conj]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n (\u222b\u207b (a : \u03b1), (f a + g a) ^ p \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\ncalc\n (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) \u2264 \u222b\u207b a, (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc :=\n by\n refine' lintegral_mono fun a => _\n dsimp only\n by_cases h_zero : (f + g) a = 0\n \u00b7 rw [h_zero, ENNReal.zero_rpow_of_pos hpq.pos]\n exact zero_le _\n by_cases h_top : (f + g) a = \u22a4\n \u00b7 rw [h_top, ENNReal.top_rpow_of_pos hpq.sub_one_pos, ENNReal.top_mul_top]\n exact le_top\n refine' le_of_eq _\n nth_rw 2 [\u2190 ENNReal.rpow_one ((f + g) a)]\n rw [\u2190 ENNReal.rpow_add _ _ h_zero h_top, add_sub_cancel'_right]\n _ = (\u222b\u207b a : \u03b1, f a * (f + g) a ^ (p - 1) \u2202\u03bc) + \u222b\u207b a : \u03b1, g a * (f + g) a ^ (p - 1) \u2202\u03bc :=\n by\n have h_add_m : AEMeasurable (fun a : \u03b1 => (f + g) a ^ (p - 1 : \u211d)) \u03bc := (hf.add hg).pow_const _\n have h_add_apply :\n (\u222b\u207b a : \u03b1, (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc) = \u222b\u207b a : \u03b1, (f a + g a) * (f + g) a ^ (p - 1) \u2202\u03bc := rfl\n simp_rw [h_add_apply, add_mul]\n rw [lintegral_add_left' (hf.mul h_add_m)]\n _ \u2264 ((\u222b\u207b a, f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b a, g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b a, (f a + g a) ^ p \u2202\u03bc) ^ (1 / q) :=\n by\n rw [add_mul]\n exact\n add_le_add (lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow hpq hf (hf.add hg) hf_top)\n (lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow hpq hg (hf.add hg) hg_top)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_mono fun a => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\na : \u03b1\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\na : \u03b1\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)\n[PROOFSTEP]\nby_cases h_zero : (f + g) a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\na : \u03b1\nh_zero : (f + g) a = 0\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)\n[PROOFSTEP]\nrw [h_zero, ENNReal.zero_rpow_of_pos hpq.pos]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\na : \u03b1\nh_zero : (f + g) a = 0\n\u22a2 0 \u2264 0 * 0 ^ (p - 1)\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)\n[PROOFSTEP]\nby_cases h_top : (f + g) a = \u22a4\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\nh_top : (f + g) a = \u22a4\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)\n[PROOFSTEP]\nrw [h_top, ENNReal.top_rpow_of_pos hpq.sub_one_pos, ENNReal.top_mul_top]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\nh_top : (f + g) a = \u22a4\n\u22a2 \u22a4 ^ p \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\nh_top : \u00ac(f + g) a = \u22a4\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)\n[PROOFSTEP]\nrefine' le_of_eq _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\nh_top : \u00ac(f + g) a = \u22a4\n\u22a2 (f + g) a ^ p = (f + g) a * (f + g) a ^ (p - 1)\n[PROOFSTEP]\nnth_rw 2 [\u2190 ENNReal.rpow_one ((f + g) a)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\nh_top : \u00ac(f + g) a = \u22a4\n\u22a2 (f + g) a ^ p = (f + g) a ^ 1 * (f + g) a ^ (p - 1)\n[PROOFSTEP]\nrw [\u2190 ENNReal.rpow_add _ _ h_zero h_top, add_sub_cancel'_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc =\n \u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc\n[PROOFSTEP]\nhave h_add_m : AEMeasurable (fun a : \u03b1 => (f + g) a ^ (p - 1 : \u211d)) \u03bc := (hf.add hg).pow_const _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_m : AEMeasurable fun a => (f + g) a ^ (p - 1)\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc =\n \u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc\n[PROOFSTEP]\nhave h_add_apply : (\u222b\u207b a : \u03b1, (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc) = \u222b\u207b a : \u03b1, (f a + g a) * (f + g) a ^ (p - 1) \u2202\u03bc :=\n rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_m : AEMeasurable fun a => (f + g) a ^ (p - 1)\nh_add_apply : \u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc = \u222b\u207b (a : \u03b1), (f a + g a) * (f + g) a ^ (p - 1) \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc =\n \u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [h_add_apply, add_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_m : AEMeasurable fun a => (f + g) a ^ (p - 1)\nh_add_apply : \u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc = \u222b\u207b (a : \u03b1), (f a + g a) * (f + g) a ^ (p - 1) \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) + g a * (f + g) a ^ (p - 1) \u2202\u03bc =\n \u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_add_left' (hf.mul h_add_m)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n (\u222b\u207b (a : \u03b1), (f a + g a) ^ p \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nrw [add_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc \u2264\n (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), (f a + g a) ^ p \u2202\u03bc) ^ (1 / q) +\n (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), (f a + g a) ^ p \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nexact\n add_le_add (lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow hpq hf (hf.add hg) hf_top)\n (lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow hpq hg (hf.add hg) hg_top)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nhave hp_not_nonpos : \u00acp \u2264 0 := by simp [hpq.pos]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 \u00acp \u2264 0\n[PROOFSTEP]\nsimp [hpq.pos]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nhave htop_rpow : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 :=\n by\n by_contra h\n exact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by simp [hpq.nonneg]) h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nh : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) = \u22a4\n\u22a2 False\n[PROOFSTEP]\nexact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by simp [hpq.nonneg]) h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nh : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) = \u22a4\n\u22a2 0 \u2264 1 / p\n[PROOFSTEP]\nsimp [hpq.nonneg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nhave h0_rpow : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 := by\n simp [h_add_zero, h_add_top, hpq.nonneg, hp_not_nonpos, -Pi.add_apply]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\n[PROOFSTEP]\nsimp [h_add_zero, h_add_top, hpq.nonneg, hp_not_nonpos, -Pi.add_apply]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nsuffices h :\n 1 \u2264 (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b a : \u03b1, g a ^ p \u2202\u03bc) ^ (1 / p))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n 1 \u2264\n (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nrwa [\u2190 mul_le_mul_left h0_rpow htop_rpow, \u2190 mul_assoc, \u2190 rpow_add _ _ h_add_zero h_add_top, \u2190 sub_eq_add_neg,\n _root_.sub_self, rpow_zero, one_mul, mul_one] at h \n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\n\u22a2 1 \u2264\n (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))\n[PROOFSTEP]\nhave h :\n (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc) \u2264\n ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b a : \u03b1, g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc) ^ (1 / q) :=\n lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add hpq hf hf_top hg hg_top\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\n\u22a2 1 \u2264\n (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))\n[PROOFSTEP]\nhave h_one_div_q : 1 / q = 1 - 1 / p := by\n nth_rw 2 [\u2190 hpq.inv_add_inv_conj]\n ring\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\n\u22a2 1 / q = 1 - 1 / p\n[PROOFSTEP]\nnth_rw 2 [\u2190 hpq.inv_add_inv_conj]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\n\u22a2 1 / q = 1 / p + 1 / q - 1 / p\n[PROOFSTEP]\nring\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\nh_one_div_q : 1 / q = 1 - 1 / p\n\u22a2 1 \u2264\n (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))\n[PROOFSTEP]\nsimp_rw [h_one_div_q, sub_eq_add_neg 1 (1 / p), ENNReal.rpow_add _ _ h_add_zero h_add_top, rpow_one] at h \n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)))\n\u22a2 1 \u2264\n (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))\n[PROOFSTEP]\nconv_rhs at h => enter [2]; rw [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)))\n| ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)))\n[PROOFSTEP]\nenter [2]; rw [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)))\n| ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)))\n[PROOFSTEP]\nenter [2]; rw [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)))\n| ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)))\n[PROOFSTEP]\nenter [2]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)))\n| (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p))\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc)\n\u22a2 1 \u2264\n (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))\n[PROOFSTEP]\nconv_lhs at h => rw [\u2190 one_mul (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc)\n| \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 one_mul (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc)\n| \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 one_mul (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc)\n| \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 one_mul (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc)]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0\nh_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n 1 * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc)\n\u22a2 1 \u2264\n (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))\n[PROOFSTEP]\nrwa [\u2190 mul_assoc, ENNReal.mul_le_mul_right h_add_zero h_add_top, mul_comm] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nhave hp_pos : 0 < p := lt_of_lt_of_le zero_lt_one hp1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nby_cases hf_top : \u222b\u207b a, f a ^ p \u2202\u03bc = \u22a4\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nsimp [hf_top, hp_pos]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nby_cases hg_top : \u222b\u207b a, g a ^ p \u2202\u03bc = \u22a4\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nsimp [hg_top, hp_pos]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nby_cases h1 : p = 1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : p = 1\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nrefine' le_of_eq _\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : p = 1\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) = (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nsimp_rw [h1, one_div_one, ENNReal.rpow_one]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : p = 1\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a \u2202\u03bc = \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nexact lintegral_add_left' hf _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nhave hp1_lt : 1 < p := by\n refine' lt_of_le_of_ne hp1 _\n symm\n exact h1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\n\u22a2 1 < p\n[PROOFSTEP]\nrefine' lt_of_le_of_ne hp1 _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\n\u22a2 1 \u2260 p\n[PROOFSTEP]\nsymm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\n\u22a2 p \u2260 1\n[PROOFSTEP]\nexact h1\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nhave hpq := Real.isConjugateExponent_conjugateExponent hp1_lt\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nby_cases h0 : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nrw [h0, @ENNReal.zero_rpow_of_pos (1 / p) (by simp [lt_of_lt_of_le zero_lt_one hp1])]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 0 < 1 / p\n[PROOFSTEP]\nsimp [lt_of_lt_of_le zero_lt_one hp1]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 0 \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nhave htop : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4 := by\n rw [\u2190 Ne.def] at hf_top hg_top \n rw [\u2190 lt_top_iff_ne_top] at hf_top hg_top \u22a2\n exact lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top hf hf_top hg_top hp1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\n[PROOFSTEP]\nrw [\u2190 Ne.def] at hf_top hg_top \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\n[PROOFSTEP]\nrw [\u2190 lt_top_iff_ne_top] at hf_top hg_top \u22a2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc < \u22a4\n[PROOFSTEP]\nexact lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top hf hf_top hg_top hp1\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4\nh1 : \u00acp = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0\nhtop : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nexact lintegral_Lp_add_le_aux hpq hf hf_top hg hg_top h0 htop\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 p\nhp1 : p \u2264 1\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264\n 2 ^ (1 / p - 1) * ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))\n[PROOFSTEP]\nrcases eq_or_lt_of_le hp0 with (rfl | hp)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 0\nhp1 : 0 \u2264 1\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ 0 \u2202\u03bc) ^ (1 / 0) \u2264\n 2 ^ (1 / 0 - 1) * ((\u222b\u207b (a : \u03b1), f a ^ 0 \u2202\u03bc) ^ (1 / 0) + (\u222b\u207b (a : \u03b1), g a ^ 0 \u2202\u03bc) ^ (1 / 0))\n[PROOFSTEP]\nsimp only [Pi.add_apply, rpow_zero, lintegral_one, _root_.div_zero, zero_sub]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 0\nhp1 : 0 \u2264 1\n\u22a2 1 \u2264 2 ^ (-1) * (1 + 1)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 0\nhp1 : 0 \u2264 1\n\u22a2 1 \u2264 2 ^ (-1) * 2\n[PROOFSTEP]\nrw [rpow_neg, rpow_one, ENNReal.inv_mul_cancel two_ne_zero two_ne_top]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 p\nhp1 : p \u2264 1\nhp : 0 < p\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264\n 2 ^ (1 / p - 1) * ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))\n[PROOFSTEP]\ncalc\n (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 ((\u222b\u207b a, f a ^ p \u2202\u03bc) + \u222b\u207b a, g a ^ p \u2202\u03bc) ^ (1 / p) :=\n by\n apply rpow_le_rpow _ (div_nonneg zero_le_one hp0)\n rw [\u2190 lintegral_add_left' (hf.pow_const p)]\n exact lintegral_mono fun a => rpow_add_le_add_rpow _ _ hp0 hp1\n _ \u2264 (2 : \u211d\u22650\u221e) ^ (1 / p - 1) * ((\u222b\u207b a, f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b a, g a ^ p \u2202\u03bc) ^ (1 / p)) :=\n rpow_add_le_mul_rpow_add_rpow _ _ ((one_le_div hp).2 hp1)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 p\nhp1 : p \u2264 1\nhp : 0 < p\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc + \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\napply rpow_le_rpow _ (div_nonneg zero_le_one hp0)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 p\nhp1 : p \u2264 1\nhp : 0 < p\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc + \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lintegral_add_left' (hf.pow_const p)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhp0 : 0 \u2264 p\nhp1 : p \u2264 1\nhp : 0 < p\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a ^ p + g a ^ p \u2202\u03bc\n[PROOFSTEP]\nexact lintegral_mono fun a => rpow_add_le_add_rpow _ _ hp0 hp1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\nhf : AEMeasurable f\nhg : AEMeasurable g\n\u22a2 \u222b\u207b (a : \u03b1), \u2191((f * g) a) \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), \u2191(f a) ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), \u2191(g a) ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nsimp_rw [Pi.mul_apply, ENNReal.coe_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\nhf : AEMeasurable f\nhg : AEMeasurable g\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(f a) * \u2191(g a) \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), \u2191(f a) ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), \u2191(g a) ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nexact ENNReal.lintegral_mul_le_Lp_mul_Lq \u03bc hpq hf.coe_nnreal_ennreal hg.coe_nnreal_ennreal\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Integral.MeanInequalities", "llama_tokens": 56761, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.766293653760418, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.4454488692720544}} {"text": "[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nh : m < n\n\u22a2 hitting u s n m \u03c9 = m\n[PROOFSTEP]\nsimp_rw [hitting]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nh : m < n\n\u22a2 (if \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s then sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) else m) = m\n[PROOFSTEP]\nhave h_not : \u00ac\u2203 (j : \u03b9) (_ : j \u2208 Set.Icc n m), u j \u03c9 \u2208 s :=\n by\n push_neg\n intro j\n rw [Set.Icc_eq_empty_of_lt h]\n simp only [Set.mem_empty_iff_false, IsEmpty.forall_iff]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nh : m < n\n\u22a2 \u00ac\u2203 j x, u j \u03c9 \u2208 s\n[PROOFSTEP]\npush_neg\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nh : m < n\n\u22a2 \u2200 (j : \u03b9), j \u2208 Set.Icc n m \u2192 \u00acu j \u03c9 \u2208 s\n[PROOFSTEP]\nintro j\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nh : m < n\nj : \u03b9\n\u22a2 j \u2208 Set.Icc n m \u2192 \u00acu j \u03c9 \u2208 s\n[PROOFSTEP]\nrw [Set.Icc_eq_empty_of_lt h]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nh : m < n\nj : \u03b9\n\u22a2 j \u2208 \u2205 \u2192 \u00acu j \u03c9 \u2208 s\n[PROOFSTEP]\nsimp only [Set.mem_empty_iff_false, IsEmpty.forall_iff]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nh : m < n\nh_not : \u00ac\u2203 j x, u j \u03c9 \u2208 s\n\u22a2 (if \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s then sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) else m) = m\n[PROOFSTEP]\nsimp only [exists_prop] at h_not \n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nh : m < n\nh_not : \u00ac\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 (if \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s then sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) else m) = m\n[PROOFSTEP]\nsimp only [h_not, if_false]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\n\u03c9 : \u03a9\n\u22a2 hitting u s n m \u03c9 \u2264 m\n[PROOFSTEP]\ncases' le_or_lt n m with h_le h_lt\n[GOAL]\ncase inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\n\u03c9 : \u03a9\nh_le : n \u2264 m\n\u22a2 hitting u s n m \u03c9 \u2264 m\n[PROOFSTEP]\nsimp only [hitting]\n[GOAL]\ncase inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\n\u03c9 : \u03a9\nh_le : n \u2264 m\n\u22a2 (if \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s then sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) else m) \u2264 m\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\n\u03c9 : \u03a9\nh_le : n \u2264 m\nh : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) \u2264 m\n[PROOFSTEP]\nobtain \u27e8j, hj\u2081, hj\u2082\u27e9 := h\n[GOAL]\ncase pos.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\n\u03c9 : \u03a9\nh_le : n \u2264 m\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) \u2264 m\n[PROOFSTEP]\nchange j \u2208 {i | u i \u03c9 \u2208 s} at hj\u2082 \n[GOAL]\ncase pos.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\n\u03c9 : \u03a9\nh_le : n \u2264 m\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\nhj\u2082 : j \u2208 {i | u i \u03c9 \u2208 s}\n\u22a2 sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) \u2264 m\n[PROOFSTEP]\nexact (csInf_le (BddBelow.inter_of_left bddBelow_Icc) (Set.mem_inter hj\u2081 hj\u2082)).trans hj\u2081.2\n[GOAL]\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\n\u03c9 : \u03a9\nh_le : n \u2264 m\nh : \u00ac\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 m \u2264 m\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\n\u03c9 : \u03a9\nh_lt : m < n\n\u22a2 hitting u s n m \u03c9 \u2264 m\n[PROOFSTEP]\nrw [hitting_of_lt h_lt]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm k : \u03b9\nhk\u2081 : k < hitting u s n m \u03c9\nhk\u2082 : n \u2264 k\n\u22a2 \u00acu k \u03c9 \u2208 s\n[PROOFSTEP]\nclassical\nintro h\nhave hexists : \u2203 j \u2208 Set.Icc n m, u j \u03c9 \u2208 s\nrefine' \u27e8k, \u27e8hk\u2082, le_trans hk\u2081.le <| hitting_le _\u27e9, h\u27e9\nrefine' not_le.2 hk\u2081 _\nsimp_rw [hitting, if_pos hexists]\nexact csInf_le bddBelow_Icc.inter_of_left \u27e8\u27e8hk\u2082, le_trans hk\u2081.le <| hitting_le _\u27e9, h\u27e9\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm k : \u03b9\nhk\u2081 : k < hitting u s n m \u03c9\nhk\u2082 : n \u2264 k\n\u22a2 \u00acu k \u03c9 \u2208 s\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm k : \u03b9\nhk\u2081 : k < hitting u s n m \u03c9\nhk\u2082 : n \u2264 k\nh : u k \u03c9 \u2208 s\n\u22a2 False\n[PROOFSTEP]\nhave hexists : \u2203 j \u2208 Set.Icc n m, u j \u03c9 \u2208 s\n[GOAL]\ncase hexists\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm k : \u03b9\nhk\u2081 : k < hitting u s n m \u03c9\nhk\u2082 : n \u2264 k\nh : u k \u03c9 \u2208 s\n\u22a2 \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm k : \u03b9\nhk\u2081 : k < hitting u s n m \u03c9\nhk\u2082 : n \u2264 k\nh : u k \u03c9 \u2208 s\nhexists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 False\n[PROOFSTEP]\nrefine' \u27e8k, \u27e8hk\u2082, le_trans hk\u2081.le <| hitting_le _\u27e9, h\u27e9\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm k : \u03b9\nhk\u2081 : k < hitting u s n m \u03c9\nhk\u2082 : n \u2264 k\nh : u k \u03c9 \u2208 s\nhexists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 False\n[PROOFSTEP]\nrefine' not_le.2 hk\u2081 _\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm k : \u03b9\nhk\u2081 : k < hitting u s n m \u03c9\nhk\u2082 : n \u2264 k\nh : u k \u03c9 \u2208 s\nhexists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 k\n[PROOFSTEP]\nsimp_rw [hitting, if_pos hexists]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm k : \u03b9\nhk\u2081 : k < hitting u s n m \u03c9\nhk\u2082 : n \u2264 k\nh : u k \u03c9 \u2208 s\nhexists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) \u2264 k\n[PROOFSTEP]\nexact csInf_le bddBelow_Icc.inter_of_left \u27e8\u27e8hk\u2082, le_trans hk\u2081.le <| hitting_le _\u27e9, h\u27e9\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\n\u22a2 hitting u s n m \u03c9 = m \u2194 (\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s) \u2192 sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) = m\n[PROOFSTEP]\nrw [hitting, ite_eq_right_iff]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nhmn : m \u2264 n\n\u22a2 hitting u s n m \u03c9 = m\n[PROOFSTEP]\nobtain rfl | h := le_iff_eq_or_lt.1 hmn\n[GOAL]\ncase inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\ni : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nhmn : m \u2264 m\n\u22a2 hitting u s m m \u03c9 = m\n[PROOFSTEP]\nrw [hitting, ite_eq_right_iff, forall_exists_index]\n[GOAL]\ncase inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\ni : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nhmn : m \u2264 m\n\u22a2 \u2200 (x : \u03b9), x \u2208 Set.Icc m m \u2227 u x \u03c9 \u2208 s \u2192 sInf (Set.Icc m m \u2229 {i | u i \u03c9 \u2208 s}) = m\n[PROOFSTEP]\nconv => intro; rw [Set.mem_Icc, Set.Icc_self, and_imp, and_imp]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\ni : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nhmn : m \u2264 m\n| \u2200 (x : \u03b9), x \u2208 Set.Icc m m \u2227 u x \u03c9 \u2208 s \u2192 sInf (Set.Icc m m \u2229 {i | u i \u03c9 \u2208 s}) = m\n[PROOFSTEP]\nintro; rw [Set.mem_Icc, Set.Icc_self, and_imp, and_imp]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\ni : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nhmn : m \u2264 m\n| \u2200 (x : \u03b9), x \u2208 Set.Icc m m \u2227 u x \u03c9 \u2208 s \u2192 sInf (Set.Icc m m \u2229 {i | u i \u03c9 \u2208 s}) = m\n[PROOFSTEP]\nintro; rw [Set.mem_Icc, Set.Icc_self, and_imp, and_imp]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\ni : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nhmn : m \u2264 m\n| \u2200 (x : \u03b9), x \u2208 Set.Icc m m \u2227 u x \u03c9 \u2208 s \u2192 sInf (Set.Icc m m \u2229 {i | u i \u03c9 \u2208 s}) = m\n[PROOFSTEP]\nintro\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\ni : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nhmn : m \u2264 m\nx\u271d : \u03b9\n| x\u271d \u2208 Set.Icc m m \u2227 u x\u271d \u03c9 \u2208 s \u2192 sInf (Set.Icc m m \u2229 {i | u i \u03c9 \u2208 s}) = m\n[PROOFSTEP]\nrw [Set.mem_Icc, Set.Icc_self, and_imp, and_imp]\n[GOAL]\ncase inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\ni : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nhmn : m \u2264 m\n\u22a2 \u2200 (x : \u03b9), m \u2264 x \u2192 x \u2264 m \u2192 u x \u03c9 \u2208 s \u2192 sInf ({m} \u2229 {i | u i \u03c9 \u2208 s}) = m\n[PROOFSTEP]\nintro i hi\u2081 hi\u2082 hi\n[GOAL]\ncase inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\ni\u271d : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nhmn : m \u2264 m\ni : \u03b9\nhi\u2081 : m \u2264 i\nhi\u2082 : i \u2264 m\nhi : u i \u03c9 \u2208 s\n\u22a2 sInf ({m} \u2229 {i | u i \u03c9 \u2208 s}) = m\n[PROOFSTEP]\nrw [Set.inter_eq_left_iff_subset.2, csInf_singleton]\n[GOAL]\ncase inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\ni\u271d : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nhmn : m \u2264 m\ni : \u03b9\nhi\u2081 : m \u2264 i\nhi\u2082 : i \u2264 m\nhi : u i \u03c9 \u2208 s\n\u22a2 {m} \u2286 {i | u i \u03c9 \u2208 s}\n[PROOFSTEP]\nexact Set.singleton_subset_iff.2 (le_antisymm hi\u2082 hi\u2081 \u25b8 hi)\n[GOAL]\ncase inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nhmn : m \u2264 n\nh : m < n\n\u22a2 hitting u s n m \u03c9 = m\n[PROOFSTEP]\nexact hitting_of_lt h\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\nhnm : n \u2264 m\n\u03c9 : \u03a9\n\u22a2 n \u2264 hitting u s n m \u03c9\n[PROOFSTEP]\nsimp only [hitting]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\nhnm : n \u2264 m\n\u03c9 : \u03a9\n\u22a2 n \u2264 if \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s then sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) else m\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\nhnm : n \u2264 m\n\u03c9 : \u03a9\nh : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 n \u2264 sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s})\n[PROOFSTEP]\nrefine' le_csInf _ fun b hb => _\n[GOAL]\ncase pos.refine'_1\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\nhnm : n \u2264 m\n\u03c9 : \u03a9\nh : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 Set.Nonempty (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s})\n[PROOFSTEP]\nobtain \u27e8k, hk_Icc, hk_s\u27e9 := h\n[GOAL]\ncase pos.refine'_1.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\nhnm : n \u2264 m\n\u03c9 : \u03a9\nk : \u03b9\nhk_Icc : k \u2208 Set.Icc n m\nhk_s : u k \u03c9 \u2208 s\n\u22a2 Set.Nonempty (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s})\n[PROOFSTEP]\nexact \u27e8k, hk_Icc, hk_s\u27e9\n[GOAL]\ncase pos.refine'_2\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\nhnm : n \u2264 m\n\u03c9 : \u03a9\nh : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\nb : \u03b9\nhb : b \u2208 Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}\n\u22a2 n \u2264 b\n[PROOFSTEP]\nrw [Set.mem_inter_iff] at hb \n[GOAL]\ncase pos.refine'_2\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\nhnm : n \u2264 m\n\u03c9 : \u03a9\nh : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\nb : \u03b9\nhb : b \u2208 Set.Icc n m \u2227 b \u2208 {i | u i \u03c9 \u2208 s}\n\u22a2 n \u2264 b\n[PROOFSTEP]\nexact hb.1.1\n[GOAL]\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9\u271d : \u03a9\nm : \u03b9\nhnm : n \u2264 m\n\u03c9 : \u03a9\nh : \u00ac\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 n \u2264 m\n[PROOFSTEP]\nexact hnm\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 n \u2264 hitting u s n m \u03c9\n[PROOFSTEP]\nrefine' le_hitting _ \u03c9\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 n \u2264 m\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\nh : \u00acn \u2264 m\n\u22a2 False\n[PROOFSTEP]\nrw [Set.Icc_eq_empty_of_lt (not_le.mp h)] at h_exists \n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nh_exists : \u2203 j, j \u2208 \u2205 \u2227 u j \u03c9 \u2208 s\nh : \u00acn \u2264 m\n\u22a2 False\n[PROOFSTEP]\nsimp at h_exists \n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 u (hitting u s n m \u03c9) \u03c9 \u2208 s\n[PROOFSTEP]\nsimp_rw [hitting, if_pos h_exists]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 u (sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s})) \u03c9 \u2208 s\n[PROOFSTEP]\nhave h_nonempty : (Set.Icc n m \u2229 {i : \u03b9 | u i \u03c9 \u2208 s}).Nonempty :=\n by\n obtain \u27e8k, hk\u2081, hk\u2082\u27e9 := h_exists\n exact \u27e8k, Set.mem_inter hk\u2081 hk\u2082\u27e9\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 Set.Nonempty (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s})\n[PROOFSTEP]\nobtain \u27e8k, hk\u2081, hk\u2082\u27e9 := h_exists\n[GOAL]\ncase intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm k : \u03b9\nhk\u2081 : k \u2208 Set.Icc n m\nhk\u2082 : u k \u03c9 \u2208 s\n\u22a2 Set.Nonempty (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s})\n[PROOFSTEP]\nexact \u27e8k, Set.mem_inter hk\u2081 hk\u2082\u27e9\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\nh_nonempty : Set.Nonempty (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s})\n\u22a2 u (sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s})) \u03c9 \u2208 s\n[PROOFSTEP]\nhave h_mem := csInf_mem h_nonempty\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\nh_nonempty : Set.Nonempty (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s})\nh_mem : sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) \u2208 Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}\n\u22a2 u (sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s})) \u03c9 \u2208 s\n[PROOFSTEP]\nrw [Set.mem_inter_iff] at h_mem \n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\nh_nonempty : Set.Nonempty (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s})\nh_mem : sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) \u2208 Set.Icc n m \u2227 sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) \u2208 {i | u i \u03c9 \u2208 s}\n\u22a2 u (sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s})) \u03c9 \u2208 s\n[PROOFSTEP]\nexact h_mem.2\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nhl : hitting u s n m \u03c9 < m\n\u22a2 u (hitting u s n m \u03c9) \u03c9 \u2208 s\n[PROOFSTEP]\nby_cases h : \u2203 j \u2208 Set.Icc n m, u j \u03c9 \u2208 s\n[GOAL]\ncase pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nhl : hitting u s n m \u03c9 < m\nh : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 u (hitting u s n m \u03c9) \u03c9 \u2208 s\n[PROOFSTEP]\nexact hitting_mem_set h\n[GOAL]\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nhl : hitting u s n m \u03c9 < m\nh : \u00ac\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 u (hitting u s n m \u03c9) \u03c9 \u2208 s\n[PROOFSTEP]\nsimp_rw [hitting, if_neg h] at hl \n[GOAL]\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh : \u00ac\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\nhl : m < m\n\u22a2 u (hitting u s n m \u03c9) \u03c9 \u2208 s\n[PROOFSTEP]\nexact False.elim (hl.ne rfl)\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nhin : n \u2264 i\nhim : i \u2264 m\nhis : u i \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 i\n[PROOFSTEP]\nhave h_exists : \u2203 k \u2208 Set.Icc n m, u k \u03c9 \u2208 s := \u27e8i, \u27e8hin, him\u27e9, his\u27e9\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nhin : n \u2264 i\nhim : i \u2264 m\nhis : u i \u03c9 \u2208 s\nh_exists : \u2203 k, k \u2208 Set.Icc n m \u2227 u k \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 i\n[PROOFSTEP]\nsimp_rw [hitting, if_pos h_exists]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm : \u03b9\nhin : n \u2264 i\nhim : i \u2264 m\nhis : u i \u03c9 \u2208 s\nh_exists : \u2203 k, k \u2208 Set.Icc n m \u2227 u k \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i\n[PROOFSTEP]\nexact csInf_le (BddBelow.inter_of_left bddBelow_Icc) (Set.mem_inter \u27e8hin, him\u27e9 his)\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 i \u2192 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase mpr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 (\u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s) \u2192 hitting u s n m \u03c9 \u2264 i\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase mp\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\nh' : hitting u s n m \u03c9 \u2264 i\n\u22a2 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nexact \u27e8hitting u s n m \u03c9, \u27e8le_hitting_of_exists h_exists, h'\u27e9, hitting_mem_set h_exists\u27e9\n[GOAL]\ncase mpr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\nh' : \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 i\n[PROOFSTEP]\nhave h'' : \u2203 k \u2208 Set.Icc n (min m i), u k \u03c9 \u2208 s :=\n by\n obtain \u27e8k\u2081, hk\u2081_mem, hk\u2081_s\u27e9 := h_exists\n obtain \u27e8k\u2082, hk\u2082_mem, hk\u2082_s\u27e9 := h'\n refine' \u27e8min k\u2081 k\u2082, \u27e8le_min hk\u2081_mem.1 hk\u2082_mem.1, min_le_min hk\u2081_mem.2 hk\u2082_mem.2\u27e9, _\u27e9\n exact min_rec' (fun j => u j \u03c9 \u2208 s) hk\u2081_s hk\u2082_s\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\nh' : \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s\n\u22a2 \u2203 k, k \u2208 Set.Icc n (min m i) \u2227 u k \u03c9 \u2208 s\n[PROOFSTEP]\nobtain \u27e8k\u2081, hk\u2081_mem, hk\u2081_s\u27e9 := h_exists\n[GOAL]\ncase intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh' : \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s\nk\u2081 : \u03b9\nhk\u2081_mem : k\u2081 \u2208 Set.Icc n m\nhk\u2081_s : u k\u2081 \u03c9 \u2208 s\n\u22a2 \u2203 k, k \u2208 Set.Icc n (min m i) \u2227 u k \u03c9 \u2208 s\n[PROOFSTEP]\nobtain \u27e8k\u2082, hk\u2082_mem, hk\u2082_s\u27e9 := h'\n[GOAL]\ncase intro.intro.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm k\u2081 : \u03b9\nhk\u2081_mem : k\u2081 \u2208 Set.Icc n m\nhk\u2081_s : u k\u2081 \u03c9 \u2208 s\nk\u2082 : \u03b9\nhk\u2082_mem : k\u2082 \u2208 Set.Icc n i\nhk\u2082_s : u k\u2082 \u03c9 \u2208 s\n\u22a2 \u2203 k, k \u2208 Set.Icc n (min m i) \u2227 u k \u03c9 \u2208 s\n[PROOFSTEP]\nrefine' \u27e8min k\u2081 k\u2082, \u27e8le_min hk\u2081_mem.1 hk\u2082_mem.1, min_le_min hk\u2081_mem.2 hk\u2082_mem.2\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm k\u2081 : \u03b9\nhk\u2081_mem : k\u2081 \u2208 Set.Icc n m\nhk\u2081_s : u k\u2081 \u03c9 \u2208 s\nk\u2082 : \u03b9\nhk\u2082_mem : k\u2082 \u2208 Set.Icc n i\nhk\u2082_s : u k\u2082 \u03c9 \u2208 s\n\u22a2 u (min k\u2081 k\u2082) \u03c9 \u2208 s\n[PROOFSTEP]\nexact min_rec' (fun j => u j \u03c9 \u2208 s) hk\u2081_s hk\u2082_s\n[GOAL]\ncase mpr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\nh' : \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s\nh'' : \u2203 k, k \u2208 Set.Icc n (min m i) \u2227 u k \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 i\n[PROOFSTEP]\nobtain \u27e8k, hk\u2081, hk\u2082\u27e9 := h''\n[GOAL]\ncase mpr.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\nh' : \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s\nk : \u03b9\nhk\u2081 : k \u2208 Set.Icc n (min m i)\nhk\u2082 : u k \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 i\n[PROOFSTEP]\nrefine' le_trans _ (hk\u2081.2.trans (min_le_right _ _))\n[GOAL]\ncase mpr.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm : \u03b9\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\nh' : \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s\nk : \u03b9\nhk\u2081 : k \u2208 Set.Icc n (min m i)\nhk\u2082 : u k \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 k\n[PROOFSTEP]\nexact hitting_le_of_mem hk\u2081.1 (hk\u2081.2.trans (min_le_left _ _)) hk\u2082\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\n\u22a2 hitting u s n m \u03c9 \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nby_cases h_exists : \u2203 j \u2208 Set.Icc n m, u j \u03c9 \u2208 s\n[GOAL]\ncase pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\nh_exists : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nrw [hitting_le_iff_of_exists h_exists]\n[GOAL]\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\nh_exists : \u00ac\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nsimp_rw [hitting, if_neg h_exists]\n[GOAL]\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\nh_exists : \u00ac\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 m \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\npush_neg at h_exists \n[GOAL]\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\nh_exists : \u2200 (j : \u03b9), j \u2208 Set.Icc n m \u2192 \u00acu j \u03c9 \u2208 s\n\u22a2 m \u2264 i \u2194 \u2203 j, j \u2208 Set.Icc n i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nsimp only [not_le.mpr hi, Set.mem_Icc, false_iff_iff, not_exists, not_and, and_imp]\n[GOAL]\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i < m\nh_exists : \u2200 (j : \u03b9), j \u2208 Set.Icc n m \u2192 \u00acu j \u03c9 \u2208 s\n\u22a2 \u2200 (x : \u03b9), n \u2264 x \u2192 x \u2264 i \u2192 \u00acu x \u03c9 \u2208 s\n[PROOFSTEP]\nexact fun k hkn hki => h_exists k \u27e8hkn, hki.trans hi.le\u27e9\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i \u2264 m\n\u22a2 hitting u s n m \u03c9 < i \u2194 \u2203 j, j \u2208 Set.Ico n i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i \u2264 m\n\u22a2 hitting u s n m \u03c9 < i \u2192 \u2203 j, j \u2208 Set.Ico n i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase mpr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i \u2264 m\n\u22a2 (\u2203 j, j \u2208 Set.Ico n i \u2227 u j \u03c9 \u2208 s) \u2192 hitting u s n m \u03c9 < i\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase mp\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i \u2264 m\nh' : hitting u s n m \u03c9 < i\n\u22a2 \u2203 j, j \u2208 Set.Ico n i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nhave h : \u2203 j \u2208 Set.Icc n m, u j \u03c9 \u2208 s := by\n by_contra h\n simp_rw [hitting, if_neg h, \u2190 not_le] at h' \n exact h' hi\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i \u2264 m\nh' : hitting u s n m \u03c9 < i\n\u22a2 \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i \u2264 m\nh' : hitting u s n m \u03c9 < i\nh : \u00ac\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 False\n[PROOFSTEP]\nsimp_rw [hitting, if_neg h, \u2190 not_le] at h' \n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i \u2264 m\nh : \u00ac\u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\nh' : \u00aci \u2264 m\n\u22a2 False\n[PROOFSTEP]\nexact h' hi\n[GOAL]\ncase mp\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i \u2264 m\nh' : hitting u s n m \u03c9 < i\nh : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 \u2203 j, j \u2208 Set.Ico n i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nexact \u27e8hitting u s n m \u03c9, \u27e8le_hitting_of_exists h, h'\u27e9, hitting_mem_set h\u27e9\n[GOAL]\ncase mpr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i \u2264 m\nh' : \u2203 j, j \u2208 Set.Ico n i \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 < i\n[PROOFSTEP]\nobtain \u27e8k, hk\u2081, hk\u2082\u27e9 := h'\n[GOAL]\ncase mpr.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i \u2264 m\nk : \u03b9\nhk\u2081 : k \u2208 Set.Ico n i\nhk\u2082 : u k \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 < i\n[PROOFSTEP]\nrefine' lt_of_le_of_lt _ hk\u2081.2\n[GOAL]\ncase mpr.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nm i : \u03b9\nhi : i \u2264 m\nk : \u03b9\nhk\u2081 : k \u2208 Set.Ico n i\nhk\u2082 : u k \u03c9 \u2208 s\n\u22a2 hitting u s n m \u03c9 \u2264 k\n[PROOFSTEP]\nexact hitting_le_of_mem hk\u2081.1 (hk\u2081.2.le.trans hi) hk\u2082\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nh' : \u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s n m\u2081 \u03c9 = hitting u s n m\u2082 \u03c9\n[PROOFSTEP]\nsimp only [hitting, if_pos h']\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nh' : \u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) =\n if \u2203 j, j \u2208 Set.Icc n m\u2082 \u2227 u j \u03c9 \u2208 s then sInf (Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}) else m\u2082\n[PROOFSTEP]\nobtain \u27e8j, hj\u2081, hj\u2082\u27e9 := h'\n[GOAL]\ncase intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) =\n if \u2203 j, j \u2208 Set.Icc n m\u2082 \u2227 u j \u03c9 \u2208 s then sInf (Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}) else m\u2082\n[PROOFSTEP]\nrw [if_pos]\n[GOAL]\ncase intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) = sInf (Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s})\n[PROOFSTEP]\nrefine'\n le_antisymm _\n (csInf_le_csInf bddBelow_Icc.inter_of_left \u27e8j, hj\u2081, hj\u2082\u27e9\n (Set.inter_subset_inter_left _ (Set.Icc_subset_Icc_right h)))\n[GOAL]\ncase intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 sInf (Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s})\n[PROOFSTEP]\nrefine' le_csInf \u27e8j, Set.Icc_subset_Icc_right h hj\u2081, hj\u2082\u27e9 fun i hi => _\n[GOAL]\ncase intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\ni : \u03b9\nhi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i\n[PROOFSTEP]\nby_cases hi' : i \u2264 m\u2081\n[GOAL]\ncase pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\ni : \u03b9\nhi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}\nhi' : i \u2264 m\u2081\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i\n[PROOFSTEP]\nexact csInf_le bddBelow_Icc.inter_of_left \u27e8\u27e8hi.1.1, hi'\u27e9, hi.2\u27e9\n[GOAL]\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\ni : \u03b9\nhi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}\nhi' : \u00aci \u2264 m\u2081\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i\n[PROOFSTEP]\nchange j \u2208 {i | u i \u03c9 \u2208 s} at hj\u2082 \n[GOAL]\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\ni : \u03b9\nhi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}\nhi' : \u00aci \u2264 m\u2081\nhj\u2082 : j \u2208 {i | u i \u03c9 \u2208 s}\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i\n[PROOFSTEP]\nexact ((csInf_le bddBelow_Icc.inter_of_left \u27e8hj\u2081, hj\u2082\u27e9).trans (hj\u2081.2.trans le_rfl)).trans (le_of_lt (not_le.1 hi'))\n[GOAL]\ncase intro.intro.hc\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 \u2203 j, j \u2208 Set.Icc n m\u2082 \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nexact \u27e8j, \u27e8hj\u2081.1, hj\u2081.2.trans h\u27e9, hj\u2082\u27e9\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nhm : m\u2081 \u2264 m\u2082\n\u22a2 hitting u s n m\u2081 \u03c9 \u2264 hitting u s n m\u2082 \u03c9\n[PROOFSTEP]\nby_cases h : \u2203 j \u2208 Set.Icc n m\u2081, u j \u03c9 \u2208 s\n[GOAL]\ncase pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nhm : m\u2081 \u2264 m\u2082\nh : \u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s n m\u2081 \u03c9 \u2264 hitting u s n m\u2082 \u03c9\n[PROOFSTEP]\nexact (hitting_eq_hitting_of_exists hm h).le\n[GOAL]\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nhm : m\u2081 \u2264 m\u2082\nh : \u00ac\u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s n m\u2081 \u03c9 \u2264 hitting u s n m\u2082 \u03c9\n[PROOFSTEP]\nsimp_rw [hitting, if_neg h]\n[GOAL]\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nhm : m\u2081 \u2264 m\u2082\nh : \u00ac\u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s\n\u22a2 m\u2081 \u2264 if \u2203 j, j \u2208 Set.Icc n m\u2082 \u2227 u j \u03c9 \u2208 s then sInf (Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}) else m\u2082\n[PROOFSTEP]\nsplit_ifs with h'\n[GOAL]\ncase pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nhm : m\u2081 \u2264 m\u2082\nh : \u00ac\u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s\nh' : \u2203 j, j \u2208 Set.Icc n m\u2082 \u2227 u j \u03c9 \u2208 s\n\u22a2 m\u2081 \u2264 sInf (Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s})\n[PROOFSTEP]\nobtain \u27e8j, hj\u2081, hj\u2082\u27e9 := h'\n[GOAL]\ncase pos.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nhm : m\u2081 \u2264 m\u2082\nh : \u00ac\u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2082\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 m\u2081 \u2264 sInf (Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s})\n[PROOFSTEP]\nrefine' le_csInf \u27e8j, hj\u2081, hj\u2082\u27e9 _\n[GOAL]\ncase pos.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nhm : m\u2081 \u2264 m\u2082\nh : \u00ac\u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2082\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 \u2200 (b : \u03b9), b \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s} \u2192 m\u2081 \u2264 b\n[PROOFSTEP]\nby_contra hneg\n[GOAL]\ncase pos.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nhm : m\u2081 \u2264 m\u2082\nh : \u00ac\u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2082\nhj\u2082 : u j \u03c9 \u2208 s\nhneg : \u00ac\u2200 (b : \u03b9), b \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s} \u2192 m\u2081 \u2264 b\n\u22a2 False\n[PROOFSTEP]\npush_neg at hneg \n[GOAL]\ncase pos.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nhm : m\u2081 \u2264 m\u2082\nh : \u00ac\u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2082\nhj\u2082 : u j \u03c9 \u2208 s\nhneg : \u2203 b, b \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s} \u2227 b < m\u2081\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8i, hi\u2081, hi\u2082\u27e9 := hneg\n[GOAL]\ncase pos.intro.intro.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nhm : m\u2081 \u2264 m\u2082\nh : \u00ac\u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2082\nhj\u2082 : u j \u03c9 \u2208 s\ni : \u03b9\nhi\u2081 : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}\nhi\u2082 : i < m\u2081\n\u22a2 False\n[PROOFSTEP]\nexact h \u27e8i, \u27e8hi\u2081.1.1, hi\u2082.le\u27e9, hi\u2081.2\u27e9\n[GOAL]\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nhm : m\u2081 \u2264 m\u2082\nh : \u00ac\u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s\nh' : \u00ac\u2203 j, j \u2208 Set.Icc n m\u2082 \u2227 u j \u03c9 \u2208 s\n\u22a2 m\u2081 \u2264 m\u2082\n[PROOFSTEP]\nexact hm\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2075 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2074 : Countable \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn n' : \u03b9\nhu : Adapted f u\nhs : MeasurableSet s\n\u22a2 IsStoppingTime f (hitting u s n n')\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2075 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2074 : Countable \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn n' : \u03b9\nhu : Adapted f u\nhs : MeasurableSet s\ni : \u03b9\n\u22a2 MeasurableSet {\u03c9 | hitting u s n n' \u03c9 \u2264 i}\n[PROOFSTEP]\ncases' le_or_lt n' i with hi hi\n[GOAL]\ncase inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2075 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2074 : Countable \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn n' : \u03b9\nhu : Adapted f u\nhs : MeasurableSet s\ni : \u03b9\nhi : n' \u2264 i\n\u22a2 MeasurableSet {\u03c9 | hitting u s n n' \u03c9 \u2264 i}\n[PROOFSTEP]\nhave h_le : \u2200 \u03c9, hitting u s n n' \u03c9 \u2264 i := fun x => (hitting_le x).trans hi\n[GOAL]\ncase inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2075 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2074 : Countable \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn n' : \u03b9\nhu : Adapted f u\nhs : MeasurableSet s\ni : \u03b9\nhi : n' \u2264 i\nh_le : \u2200 (\u03c9 : \u03a9), hitting u s n n' \u03c9 \u2264 i\n\u22a2 MeasurableSet {\u03c9 | hitting u s n n' \u03c9 \u2264 i}\n[PROOFSTEP]\nsimp [h_le]\n[GOAL]\ncase inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2075 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2074 : Countable \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn n' : \u03b9\nhu : Adapted f u\nhs : MeasurableSet s\ni : \u03b9\nhi : i < n'\n\u22a2 MeasurableSet {\u03c9 | hitting u s n n' \u03c9 \u2264 i}\n[PROOFSTEP]\nhave h_set_eq_Union : {\u03c9 | hitting u s n n' \u03c9 \u2264 i} = \u22c3 j \u2208 Set.Icc n i, u j \u207b\u00b9' s :=\n by\n ext x\n rw [Set.mem_setOf_eq, hitting_le_iff_of_lt _ hi]\n simp only [Set.mem_Icc, exists_prop, Set.mem_iUnion, Set.mem_preimage]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2075 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2074 : Countable \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn n' : \u03b9\nhu : Adapted f u\nhs : MeasurableSet s\ni : \u03b9\nhi : i < n'\n\u22a2 {\u03c9 | hitting u s n n' \u03c9 \u2264 i} = \u22c3 (j : \u03b9) (_ : j \u2208 Set.Icc n i), u j \u207b\u00b9' s\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2075 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2074 : Countable \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn n' : \u03b9\nhu : Adapted f u\nhs : MeasurableSet s\ni : \u03b9\nhi : i < n'\nx : \u03a9\n\u22a2 x \u2208 {\u03c9 | hitting u s n n' \u03c9 \u2264 i} \u2194 x \u2208 \u22c3 (j : \u03b9) (_ : j \u2208 Set.Icc n i), u j \u207b\u00b9' s\n[PROOFSTEP]\nrw [Set.mem_setOf_eq, hitting_le_iff_of_lt _ hi]\n[GOAL]\ncase h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2075 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2074 : Countable \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn n' : \u03b9\nhu : Adapted f u\nhs : MeasurableSet s\ni : \u03b9\nhi : i < n'\nx : \u03a9\n\u22a2 (\u2203 j, j \u2208 Set.Icc n i \u2227 u j x \u2208 s) \u2194 x \u2208 \u22c3 (j : \u03b9) (_ : j \u2208 Set.Icc n i), u j \u207b\u00b9' s\n[PROOFSTEP]\nsimp only [Set.mem_Icc, exists_prop, Set.mem_iUnion, Set.mem_preimage]\n[GOAL]\ncase inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2075 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2074 : Countable \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn n' : \u03b9\nhu : Adapted f u\nhs : MeasurableSet s\ni : \u03b9\nhi : i < n'\nh_set_eq_Union : {\u03c9 | hitting u s n n' \u03c9 \u2264 i} = \u22c3 (j : \u03b9) (_ : j \u2208 Set.Icc n i), u j \u207b\u00b9' s\n\u22a2 MeasurableSet {\u03c9 | hitting u s n n' \u03c9 \u2264 i}\n[PROOFSTEP]\nrw [h_set_eq_Union]\n[GOAL]\ncase inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2075 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2074 : Countable \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn n' : \u03b9\nhu : Adapted f u\nhs : MeasurableSet s\ni : \u03b9\nhi : i < n'\nh_set_eq_Union : {\u03c9 | hitting u s n n' \u03c9 \u2264 i} = \u22c3 (j : \u03b9) (_ : j \u2208 Set.Icc n i), u j \u207b\u00b9' s\n\u22a2 MeasurableSet (\u22c3 (j : \u03b9) (_ : j \u2208 Set.Icc n i), u j \u207b\u00b9' s)\n[PROOFSTEP]\nexact MeasurableSet.iUnion fun j => MeasurableSet.iUnion fun hj => f.mono hj.2 _ ((hu j).measurable hs)\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn m : \u03b9\n\u03c9 : \u03a9\nh : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 stoppedValue u (hitting u s n m) \u03c9 \u2208 s\n[PROOFSTEP]\nsimp only [stoppedValue, hitting, if_pos h]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn m : \u03b9\n\u03c9 : \u03a9\nh : \u2203 j, j \u2208 Set.Icc n m \u2227 u j \u03c9 \u2208 s\n\u22a2 u (sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s})) \u03c9 \u2208 s\n[PROOFSTEP]\nobtain \u27e8j, hj\u2081, hj\u2082\u27e9 := h\n[GOAL]\ncase intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn m : \u03b9\n\u03c9 : \u03a9\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 u (sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s})) \u03c9 \u2208 s\n[PROOFSTEP]\nhave : sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) \u2208 Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s} :=\n csInf_mem (Set.nonempty_of_mem \u27e8hj\u2081, hj\u2082\u27e9)\n[GOAL]\ncase intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn m : \u03b9\n\u03c9 : \u03a9\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\nhj\u2082 : u j \u03c9 \u2208 s\nthis : sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}) \u2208 Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s}\n\u22a2 u (sInf (Set.Icc n m \u2229 {i | u i \u03c9 \u2208 s})) \u03c9 \u2208 s\n[PROOFSTEP]\nexact this.2\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2078 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2077 : Countable \u03b9\ninst\u271d\u2076 : TopologicalSpace \u03b9\ninst\u271d\u2075 : OrderTopology \u03b9\ninst\u271d\u2074 : FirstCountableTopology \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nN : \u03b9\nh\u03c4bdd : \u2200 (x : \u03a9), \u03c4 x \u2264 N\ns : Set \u03b2\nhs : MeasurableSet s\nhf : Adapted f u\n\u22a2 IsStoppingTime f fun x => hitting u s (\u03c4 x) N x\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2078 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2077 : Countable \u03b9\ninst\u271d\u2076 : TopologicalSpace \u03b9\ninst\u271d\u2075 : OrderTopology \u03b9\ninst\u271d\u2074 : FirstCountableTopology \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nN : \u03b9\nh\u03c4bdd : \u2200 (x : \u03a9), \u03c4 x \u2264 N\ns : Set \u03b2\nhs : MeasurableSet s\nhf : Adapted f u\nn : \u03b9\n\u22a2 MeasurableSet {\u03c9 | (fun x => hitting u s (\u03c4 x) N x) \u03c9 \u2264 n}\n[PROOFSTEP]\nhave h\u2081 :\n {x | hitting u s (\u03c4 x) N x \u2264 n} =\n (\u22c3 i \u2264 n, {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}) \u222a \u22c3 i > n, {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n} :=\n by\n ext x\n simp [\u2190 exists_or, \u2190 or_and_right, le_or_lt]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2078 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2077 : Countable \u03b9\ninst\u271d\u2076 : TopologicalSpace \u03b9\ninst\u271d\u2075 : OrderTopology \u03b9\ninst\u271d\u2074 : FirstCountableTopology \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nN : \u03b9\nh\u03c4bdd : \u2200 (x : \u03a9), \u03c4 x \u2264 N\ns : Set \u03b2\nhs : MeasurableSet s\nhf : Adapted f u\nn : \u03b9\n\u22a2 {x | hitting u s (\u03c4 x) N x \u2264 n} =\n (\u22c3 (i : \u03b9) (_ : i \u2264 n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}) \u222a\n \u22c3 (i : \u03b9) (_ : i > n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2078 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2077 : Countable \u03b9\ninst\u271d\u2076 : TopologicalSpace \u03b9\ninst\u271d\u2075 : OrderTopology \u03b9\ninst\u271d\u2074 : FirstCountableTopology \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nN : \u03b9\nh\u03c4bdd : \u2200 (x : \u03a9), \u03c4 x \u2264 N\ns : Set \u03b2\nhs : MeasurableSet s\nhf : Adapted f u\nn : \u03b9\nx : \u03a9\n\u22a2 x \u2208 {x | hitting u s (\u03c4 x) N x \u2264 n} \u2194\n x \u2208\n (\u22c3 (i : \u03b9) (_ : i \u2264 n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}) \u222a\n \u22c3 (i : \u03b9) (_ : i > n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}\n[PROOFSTEP]\nsimp [\u2190 exists_or, \u2190 or_and_right, le_or_lt]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2078 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2077 : Countable \u03b9\ninst\u271d\u2076 : TopologicalSpace \u03b9\ninst\u271d\u2075 : OrderTopology \u03b9\ninst\u271d\u2074 : FirstCountableTopology \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nN : \u03b9\nh\u03c4bdd : \u2200 (x : \u03a9), \u03c4 x \u2264 N\ns : Set \u03b2\nhs : MeasurableSet s\nhf : Adapted f u\nn : \u03b9\nh\u2081 :\n {x | hitting u s (\u03c4 x) N x \u2264 n} =\n (\u22c3 (i : \u03b9) (_ : i \u2264 n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}) \u222a\n \u22c3 (i : \u03b9) (_ : i > n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}\n\u22a2 MeasurableSet {\u03c9 | (fun x => hitting u s (\u03c4 x) N x) \u03c9 \u2264 n}\n[PROOFSTEP]\nhave h\u2082 : \u22c3 i > n, {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n} = \u2205 :=\n by\n ext x\n simp only [gt_iff_lt, Set.mem_iUnion, Set.mem_inter_iff, Set.mem_setOf_eq, exists_prop, Set.mem_empty_iff_false,\n iff_false_iff, not_exists, not_and, not_le]\n rintro m hm rfl\n exact lt_of_lt_of_le hm (le_hitting (h\u03c4bdd _) _)\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2078 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2077 : Countable \u03b9\ninst\u271d\u2076 : TopologicalSpace \u03b9\ninst\u271d\u2075 : OrderTopology \u03b9\ninst\u271d\u2074 : FirstCountableTopology \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nN : \u03b9\nh\u03c4bdd : \u2200 (x : \u03a9), \u03c4 x \u2264 N\ns : Set \u03b2\nhs : MeasurableSet s\nhf : Adapted f u\nn : \u03b9\nh\u2081 :\n {x | hitting u s (\u03c4 x) N x \u2264 n} =\n (\u22c3 (i : \u03b9) (_ : i \u2264 n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}) \u222a\n \u22c3 (i : \u03b9) (_ : i > n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}\n\u22a2 \u22c3 (i : \u03b9) (_ : i > n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n} = \u2205\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2078 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2077 : Countable \u03b9\ninst\u271d\u2076 : TopologicalSpace \u03b9\ninst\u271d\u2075 : OrderTopology \u03b9\ninst\u271d\u2074 : FirstCountableTopology \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nN : \u03b9\nh\u03c4bdd : \u2200 (x : \u03a9), \u03c4 x \u2264 N\ns : Set \u03b2\nhs : MeasurableSet s\nhf : Adapted f u\nn : \u03b9\nh\u2081 :\n {x | hitting u s (\u03c4 x) N x \u2264 n} =\n (\u22c3 (i : \u03b9) (_ : i \u2264 n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}) \u222a\n \u22c3 (i : \u03b9) (_ : i > n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}\nx : \u03a9\n\u22a2 x \u2208 \u22c3 (i : \u03b9) (_ : i > n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n} \u2194 x \u2208 \u2205\n[PROOFSTEP]\nsimp only [gt_iff_lt, Set.mem_iUnion, Set.mem_inter_iff, Set.mem_setOf_eq, exists_prop, Set.mem_empty_iff_false,\n iff_false_iff, not_exists, not_and, not_le]\n[GOAL]\ncase h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2078 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2077 : Countable \u03b9\ninst\u271d\u2076 : TopologicalSpace \u03b9\ninst\u271d\u2075 : OrderTopology \u03b9\ninst\u271d\u2074 : FirstCountableTopology \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nN : \u03b9\nh\u03c4bdd : \u2200 (x : \u03a9), \u03c4 x \u2264 N\ns : Set \u03b2\nhs : MeasurableSet s\nhf : Adapted f u\nn : \u03b9\nh\u2081 :\n {x | hitting u s (\u03c4 x) N x \u2264 n} =\n (\u22c3 (i : \u03b9) (_ : i \u2264 n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}) \u222a\n \u22c3 (i : \u03b9) (_ : i > n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}\nx : \u03a9\n\u22a2 \u2200 (x_1 : \u03b9), n < x_1 \u2192 \u03c4 x = x_1 \u2192 n < hitting u s x_1 N x\n[PROOFSTEP]\nrintro m hm rfl\n[GOAL]\ncase h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2078 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2077 : Countable \u03b9\ninst\u271d\u2076 : TopologicalSpace \u03b9\ninst\u271d\u2075 : OrderTopology \u03b9\ninst\u271d\u2074 : FirstCountableTopology \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nN : \u03b9\nh\u03c4bdd : \u2200 (x : \u03a9), \u03c4 x \u2264 N\ns : Set \u03b2\nhs : MeasurableSet s\nhf : Adapted f u\nn : \u03b9\nh\u2081 :\n {x | hitting u s (\u03c4 x) N x \u2264 n} =\n (\u22c3 (i : \u03b9) (_ : i \u2264 n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}) \u222a\n \u22c3 (i : \u03b9) (_ : i > n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}\nx : \u03a9\nhm : n < \u03c4 x\n\u22a2 n < hitting u s (\u03c4 x) N x\n[PROOFSTEP]\nexact lt_of_lt_of_le hm (le_hitting (h\u03c4bdd _) _)\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2078 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2077 : Countable \u03b9\ninst\u271d\u2076 : TopologicalSpace \u03b9\ninst\u271d\u2075 : OrderTopology \u03b9\ninst\u271d\u2074 : FirstCountableTopology \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nN : \u03b9\nh\u03c4bdd : \u2200 (x : \u03a9), \u03c4 x \u2264 N\ns : Set \u03b2\nhs : MeasurableSet s\nhf : Adapted f u\nn : \u03b9\nh\u2081 :\n {x | hitting u s (\u03c4 x) N x \u2264 n} =\n (\u22c3 (i : \u03b9) (_ : i \u2264 n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}) \u222a\n \u22c3 (i : \u03b9) (_ : i > n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}\nh\u2082 : \u22c3 (i : \u03b9) (_ : i > n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n} = \u2205\n\u22a2 MeasurableSet {\u03c9 | (fun x => hitting u s (\u03c4 x) N x) \u03c9 \u2264 n}\n[PROOFSTEP]\nrw [h\u2081, h\u2082, Set.union_empty]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b9\ninst\u271d\u2078 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d\u2077 : Countable \u03b9\ninst\u271d\u2076 : TopologicalSpace \u03b9\ninst\u271d\u2075 : OrderTopology \u03b9\ninst\u271d\u2074 : FirstCountableTopology \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : Filtration \u03b9 m\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\n\u03c4 : \u03a9 \u2192 \u03b9\nh\u03c4 : IsStoppingTime f \u03c4\nN : \u03b9\nh\u03c4bdd : \u2200 (x : \u03a9), \u03c4 x \u2264 N\ns : Set \u03b2\nhs : MeasurableSet s\nhf : Adapted f u\nn : \u03b9\nh\u2081 :\n {x | hitting u s (\u03c4 x) N x \u2264 n} =\n (\u22c3 (i : \u03b9) (_ : i \u2264 n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}) \u222a\n \u22c3 (i : \u03b9) (_ : i > n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n}\nh\u2082 : \u22c3 (i : \u03b9) (_ : i > n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n} = \u2205\n\u22a2 MeasurableSet (\u22c3 (i : \u03b9) (_ : i \u2264 n), {x | \u03c4 x = i} \u2229 {x | hitting u s i N x \u2264 n})\n[PROOFSTEP]\nexact\n MeasurableSet.iUnion fun i =>\n MeasurableSet.iUnion fun hi => (f.mono hi _ (h\u03c4.measurableSet_eq i)).inter (hitting_isStoppingTime hf hs n)\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : CompleteLattice \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nf : Filtration \u03b9 m\n\u03c9 : \u03a9\n\u22a2 hitting u s \u22a5 \u22a4 \u03c9 = sInf {i | u i \u03c9 \u2208 s}\n[PROOFSTEP]\nsimp only [hitting, Set.mem_Icc, bot_le, le_top, and_self_iff, exists_true_left, Set.Icc_bot, Set.Iic_top,\n Set.univ_inter, ite_eq_left_iff, not_exists]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : CompleteLattice \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nf : Filtration \u03b9 m\n\u03c9 : \u03a9\n\u22a2 (\u2200 (x : \u03b9), \u00ac(x \u2208 Set.univ \u2227 u x \u03c9 \u2208 s)) \u2192 \u22a4 = sInf {i | u i \u03c9 \u2208 s}\n[PROOFSTEP]\nintro h_nmem_s\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : CompleteLattice \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nf : Filtration \u03b9 m\n\u03c9 : \u03a9\nh_nmem_s : \u2200 (x : \u03b9), \u00ac(x \u2208 Set.univ \u2227 u x \u03c9 \u2208 s)\n\u22a2 \u22a4 = sInf {i | u i \u03c9 \u2208 s}\n[PROOFSTEP]\nsymm\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : CompleteLattice \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nf : Filtration \u03b9 m\n\u03c9 : \u03a9\nh_nmem_s : \u2200 (x : \u03b9), \u00ac(x \u2208 Set.univ \u2227 u x \u03c9 \u2208 s)\n\u22a2 sInf {i | u i \u03c9 \u2208 s} = \u22a4\n[PROOFSTEP]\nrw [sInf_eq_top]\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : CompleteLattice \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nf : Filtration \u03b9 m\n\u03c9 : \u03a9\nh_nmem_s : \u2200 (x : \u03b9), \u00ac(x \u2208 Set.univ \u2227 u x \u03c9 \u2208 s)\n\u22a2 \u2200 (a : \u03b9), a \u2208 {i | u i \u03c9 \u2208 s} \u2192 a = \u22a4\n[PROOFSTEP]\nsimp only [Set.mem_univ, true_and] at h_nmem_s \n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : CompleteLattice \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nf : Filtration \u03b9 m\n\u03c9 : \u03a9\nh_nmem_s : \u2200 (x : \u03b9), \u00acu x \u03c9 \u2208 s\n\u22a2 \u2200 (a : \u03b9), a \u2208 {i | u i \u03c9 \u2208 s} \u2192 a = \u22a4\n[PROOFSTEP]\nexact fun i hi_mem_s => absurd hi_mem_s (h_nmem_s i)\n[GOAL]\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nf : Filtration \u2115 m\ni n : \u03b9\n\u03c9 : \u03a9\nhx : \u2203 j, j \u2264 n \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s \u22a5 n \u03c9 \u2264 i \u2194 \u2203 j, j \u2264 i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\ncases' lt_or_le i n with hi hi\n[GOAL]\ncase inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nf : Filtration \u2115 m\ni n : \u03b9\n\u03c9 : \u03a9\nhx : \u2203 j, j \u2264 n \u2227 u j \u03c9 \u2208 s\nhi : i < n\n\u22a2 hitting u s \u22a5 n \u03c9 \u2264 i \u2194 \u2203 j, j \u2264 i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nrw [hitting_le_iff_of_lt _ hi]\n[GOAL]\ncase inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nf : Filtration \u2115 m\ni n : \u03b9\n\u03c9 : \u03a9\nhx : \u2203 j, j \u2264 n \u2227 u j \u03c9 \u2208 s\nhi : i < n\n\u22a2 (\u2203 j, j \u2208 Set.Icc \u22a5 i \u2227 u j \u03c9 \u2208 s) \u2194 \u2203 j, j \u2264 i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nf : Filtration \u2115 m\ni n : \u03b9\n\u03c9 : \u03a9\nhx : \u2203 j, j \u2264 n \u2227 u j \u03c9 \u2208 s\nhi : n \u2264 i\n\u22a2 hitting u s \u22a5 n \u03c9 \u2264 i \u2194 \u2203 j, j \u2264 i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nsimp only [(hitting_le \u03c9).trans hi, true_iff_iff]\n[GOAL]\ncase inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nf : Filtration \u2115 m\ni n : \u03b9\n\u03c9 : \u03a9\nhx : \u2203 j, j \u2264 n \u2227 u j \u03c9 \u2208 s\nhi : n \u2264 i\n\u22a2 \u2203 j, j \u2264 i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nobtain \u27e8j, hj\u2081, hj\u2082\u27e9 := hx\n[GOAL]\ncase inr.intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nf : Filtration \u2115 m\ni n : \u03b9\n\u03c9 : \u03a9\nhi : n \u2264 i\nj : \u03b9\nhj\u2081 : j \u2264 n\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 \u2203 j, j \u2264 i \u2227 u j \u03c9 \u2208 s\n[PROOFSTEP]\nexact \u27e8j, hj\u2081.trans hi, hj\u2082\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Probability.Process.HittingTime", "llama_tokens": 31825, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936430859597, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4454488630669587}} {"text": "[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\n\u22a2 1 = 2 * 0 + 1 + 2 * 0\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\n\u22a2 2 * 0 + 1 + 2 * 0 \u2264 2 * diam univ + 1 + 2 * diam univ\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081.bc.h\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\n\u22a2 0 \u2264 diam univ\n[PROOFSTEP]\npositivity\n[GOAL]\ncase h\u2082.h\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\n\u22a2 0 \u2264 diam univ\n[PROOFSTEP]\npositivity\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 diam univ = diam (range inl \u222a range inr)\n[PROOFSTEP]\nrw [range_inl_union_range_inr]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 diam (range inl) + dist (inl default) (inr default) + diam (range inr) =\n diam univ + (dist default default + 1 + dist default default) + diam univ\n[PROOFSTEP]\nrw [isometry_inl.diam_range, isometry_inr.diam_range]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 diam univ + dist (inl default) (inr default) + diam univ =\n diam univ + (dist default default + 1 + dist default default) + diam univ\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Type ?u.18344\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Dist ?m.19041\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Inhabited ?m.19041\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Inhabited ?m.19041\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Type ?u.19756\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Dist ?m.20453\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Inhabited ?m.20453\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Inhabited ?m.20453\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Type ?u.18344\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Dist ?m.19041\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Inhabited ?m.19041\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Inhabited ?m.19041\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Type ?u.19756\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Dist ?m.20453\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Inhabited ?m.20453\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Inhabited ?m.20453\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 diam univ + (dist default default + 1 + dist default default) + diam univ = 1 * diam univ + 1 + 1 * diam univ\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 1 * diam univ + 1 + 1 * diam univ \u2264 2 * diam univ + 1 + 2 * diam univ\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081.bc.h\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 1 \u2264 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h\u2082.h\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 1 \u2264 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 0 \u2264 f (x, y)\n[PROOFSTEP]\nhave : 0 \u2264 2 * f (x, y) :=\n calc\n 0 = f (x, x) := (candidates_refl fA).symm\n _ \u2264 f (x, y) + f (y, x) := (candidates_triangle fA)\n _ = f (x, y) + f (x, y) := by rw [candidates_symm fA]\n _ = 2 * f (x, y) := by ring\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 f (x, y) + f (y, x) = f (x, y) + f (x, y)\n[PROOFSTEP]\nrw [candidates_symm fA]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 f (x, y) + f (x, y) = 2 * f (x, y)\n[PROOFSTEP]\nring\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nthis : 0 \u2264 2 * f (x, y)\n\u22a2 0 \u2264 f (x, y)\n[PROOFSTEP]\nlinarith\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx y : X\n\u22a2 dist x y = dist (inl x) (inl y)\n[PROOFSTEP]\nrw [@Sum.dist_eq X Y]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx y : X\n\u22a2 dist x y = Sum.dist (inl x) (inl y)\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx y : X\n\u22a2 dist (inl x) (inl y) = 1 * dist (inl x) (inl y)\n[PROOFSTEP]\nring\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx y : X\n\u22a2 1 * dist (inl x) (inl y) \u2264 \u2191(GromovHausdorff.maxVar X Y) * dist (inl x) (inl y)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx y : X\n\u22a2 1 \u2264 \u2191(GromovHausdorff.maxVar X Y)\n[PROOFSTEP]\nexact one_le_maxVar X Y\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx : X\ny : Y\n\u22a2 \u2191(GromovHausdorff.maxVar X Y) = \u2191(GromovHausdorff.maxVar X Y) * 1\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx : X\ny : Y\n\u22a2 \u2191(GromovHausdorff.maxVar X Y) * 1 \u2264 \u2191(GromovHausdorff.maxVar X Y) * dist (inl x) (inr y)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx : X\ny : Y\n\u22a2 1 \u2264 dist (inl x) (inr y)\n[PROOFSTEP]\napply Sum.one_le_dist_inl_inr\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx : Y\ny : X\n\u22a2 \u2191(GromovHausdorff.maxVar X Y) = \u2191(GromovHausdorff.maxVar X Y) * 1\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx : Y\ny : X\n\u22a2 \u2191(GromovHausdorff.maxVar X Y) * 1 \u2264 \u2191(GromovHausdorff.maxVar X Y) * dist (inl x) (inr y)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx : Y\ny : X\n\u22a2 1 \u2264 dist (inl x) (inr y)\n[PROOFSTEP]\napply Sum.one_le_dist_inl_inr\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx y : Y\n\u22a2 dist x y = dist (inr x) (inr y)\n[PROOFSTEP]\nrw [@Sum.dist_eq X Y]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx y : Y\n\u22a2 dist x y = Sum.dist (inr x) (inr y)\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx y : Y\n\u22a2 dist (inr x) (inr y) = 1 * dist (inr x) (inr y)\n[PROOFSTEP]\nring\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx y : Y\n\u22a2 1 * dist (inr x) (inr y) \u2264 \u2191(GromovHausdorff.maxVar X Y) * dist (inr x) (inr y)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nfA : f \u2208 candidates X Y\nx y : Y\n\u22a2 1 \u2264 \u2191(GromovHausdorff.maxVar X Y)\n[PROOFSTEP]\nexact one_le_maxVar X Y\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 f (x, y) - f (z, t) \u2264 f (x, t) + f (t, y) - f (z, t)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 f (x, y) \u2264 f (x, t) + f (t, y)\n[PROOFSTEP]\nexact candidates_triangle fA\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 f (x, t) + f (t, y) - f (z, t) \u2264 f (x, z) + f (z, t) + f (t, y) - f (z, t)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h.bc\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 f (x, t) \u2264 f (x, z) + f (z, t)\n[PROOFSTEP]\nexact candidates_triangle fA\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 f (x, z) + f (z, t) + f (t, y) - f (z, t) = f (x, z) + f (t, y)\n[PROOFSTEP]\nsimp [sub_eq_add_neg, add_assoc]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 f (x, z) + f (t, y) \u2264 \u2191(GromovHausdorff.maxVar X Y) * dist x z + \u2191(GromovHausdorff.maxVar X Y) * dist t y\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 f (x, z) \u2264 \u2191(GromovHausdorff.maxVar X Y) * dist x z\n[PROOFSTEP]\napply candidates_dist_bound fA\n[GOAL]\ncase h\u2082\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 f (t, y) \u2264 \u2191(GromovHausdorff.maxVar X Y) * dist t y\n[PROOFSTEP]\napply candidates_dist_bound fA\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 \u2191(GromovHausdorff.maxVar X Y) * dist x z + \u2191(GromovHausdorff.maxVar X Y) * dist t y \u2264\n \u2191(GromovHausdorff.maxVar X Y) * max (dist x z) (dist t y) +\n \u2191(GromovHausdorff.maxVar X Y) * max (dist x z) (dist t y)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081.h\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 dist x z \u2264 max (dist x z) (dist t y)\n[PROOFSTEP]\napply le_max_left\n[GOAL]\ncase h\u2082.h\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 dist t y \u2264 max (dist x z) (dist t y)\n[PROOFSTEP]\napply le_max_right\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 \u2191(GromovHausdorff.maxVar X Y) * max (dist x z) (dist t y) +\n \u2191(GromovHausdorff.maxVar X Y) * max (dist x z) (dist t y) =\n 2 * \u2191(GromovHausdorff.maxVar X Y) * max (dist x z) (dist y t)\n[PROOFSTEP]\nrw [dist_comm t y]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 \u2191(GromovHausdorff.maxVar X Y) * max (dist x z) (dist y t) +\n \u2191(GromovHausdorff.maxVar X Y) * max (dist x z) (dist y t) =\n 2 * \u2191(GromovHausdorff.maxVar X Y) * max (dist x z) (dist y t)\n[PROOFSTEP]\nring\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 2 * \u2191(GromovHausdorff.maxVar X Y) * max (dist x z) (dist y t) = 2 * \u2191(GromovHausdorff.maxVar X Y) * dist (x, y) (z, t)\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 LipschitzWith (2 * GromovHausdorff.maxVar X Y) f\n[PROOFSTEP]\napply LipschitzWith.of_dist_le_mul\n[GOAL]\ncase a\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nfA : f \u2208 candidates X Y\n\u22a2 \u2200 (x y : (X \u2295 Y) \u00d7 (X \u2295 Y)), dist (f x) (f y) \u2264 \u2191(2 * GromovHausdorff.maxVar X Y) * dist x y\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9 \u27e8z, t\u27e9\n[GOAL]\ncase a.mk.mk\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z\u271d t\u271d : X \u2295 Y\nfA : f \u2208 candidates X Y\nx y z t : X \u2295 Y\n\u22a2 dist (f (x, y)) (f (z, t)) \u2264 \u2191(2 * GromovHausdorff.maxVar X Y) * dist (x, y) (z, t)\n[PROOFSTEP]\nrw [Real.dist_eq, abs_sub_le_iff]\n[GOAL]\ncase a.mk.mk\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z\u271d t\u271d : X \u2295 Y\nfA : f \u2208 candidates X Y\nx y z t : X \u2295 Y\n\u22a2 f (x, y) - f (z, t) \u2264 \u2191(2 * GromovHausdorff.maxVar X Y) * dist (x, y) (z, t) \u2227\n f (z, t) - f (x, y) \u2264 \u2191(2 * GromovHausdorff.maxVar X Y) * dist (x, y) (z, t)\n[PROOFSTEP]\nuse candidates_lipschitz_aux fA\n[GOAL]\ncase right\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z\u271d t\u271d : X \u2295 Y\nfA : f \u2208 candidates X Y\nx y z t : X \u2295 Y\n\u22a2 f (z, t) - f (x, y) \u2264 \u2191(2 * GromovHausdorff.maxVar X Y) * dist (x, y) (z, t)\n[PROOFSTEP]\nrw [dist_comm]\n[GOAL]\ncase right\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z\u271d t\u271d : X \u2295 Y\nfA : f \u2208 candidates X Y\nx y z t : X \u2295 Y\n\u22a2 f (z, t) - f (x, y) \u2264 \u2191(2 * GromovHausdorff.maxVar X Y) * dist (z, t) (x, y)\n[PROOFSTEP]\nexact candidates_lipschitz_aux fA\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 (fun p => dist p.fst p.snd) \u2208 candidates X Y\n[PROOFSTEP]\nsimp_rw [candidates, Set.mem_setOf_eq, dist_comm, dist_triangle, dist_self, maxVar_bound, forall_const, and_true]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 (\u2200 (x y : X), dist (inl x) (inl y) = dist x y) \u2227 \u2200 (x y : Y), dist (inr x) (inr y) = dist x y\n[PROOFSTEP]\nexact \u27e8fun x y => rfl, fun x y => rfl\u27e9\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 IsClosed (GromovHausdorff.candidatesB X Y)\n[PROOFSTEP]\nhave I1 : \u2200 x y, IsClosed {f : Cb X Y | f (inl x, inl y) = dist x y} := fun x y =>\n isClosed_eq continuous_eval_const continuous_const\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\n\u22a2 IsClosed (GromovHausdorff.candidatesB X Y)\n[PROOFSTEP]\nhave I2 : \u2200 x y, IsClosed {f : Cb X Y | f (inr x, inr y) = dist x y} := fun x y =>\n isClosed_eq continuous_eval_const continuous_const\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\n\u22a2 IsClosed (GromovHausdorff.candidatesB X Y)\n[PROOFSTEP]\nhave I3 : \u2200 x y, IsClosed {f : Cb X Y | f (x, y) = f (y, x)} := fun x y =>\n isClosed_eq continuous_eval_const continuous_eval_const\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\n\u22a2 IsClosed (GromovHausdorff.candidatesB X Y)\n[PROOFSTEP]\nhave I4 : \u2200 x y z, IsClosed {f : Cb X Y | f (x, z) \u2264 f (x, y) + f (y, z)} := fun x y z =>\n isClosed_le continuous_eval_const (continuous_eval_const.add continuous_eval_const)\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\n\u22a2 IsClosed (GromovHausdorff.candidatesB X Y)\n[PROOFSTEP]\nhave I5 : \u2200 x, IsClosed {f : Cb X Y | f (x, x) = 0} := fun x => isClosed_eq continuous_eval_const continuous_const\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\n\u22a2 IsClosed (GromovHausdorff.candidatesB X Y)\n[PROOFSTEP]\nhave I6 : \u2200 x y, IsClosed {f : Cb X Y | f (x, y) \u2264 maxVar X Y} := fun x y =>\n isClosed_le continuous_eval_const continuous_const\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (GromovHausdorff.candidatesB X Y)\n[PROOFSTEP]\nhave :\n candidatesB X Y =\n (((((\u22c2 (x) (y), {f : Cb X Y | f (@inl X Y x, @inl X Y y) = dist x y}) \u2229\n \u22c2 (x) (y), {f : Cb X Y | f (@inr X Y x, @inr X Y y) = dist x y}) \u2229\n \u22c2 (x) (y), {f : Cb X Y | f (x, y) = f (y, x)}) \u2229\n \u22c2 (x) (y) (z), {f : Cb X Y | f (x, z) \u2264 f (x, y) + f (y, z)}) \u2229\n \u22c2 x, {f : Cb X Y | f (x, x) = 0}) \u2229\n \u22c2 (x) (y), {f : Cb X Y | f (x, y) \u2264 maxVar X Y} :=\n by\n ext\n simp only [candidatesB, candidates, mem_inter_iff, mem_iInter, mem_setOf_eq]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d : GromovHausdorff.Cb X Y\n\u22a2 x\u271d \u2208 GromovHausdorff.candidatesB X Y \u2194\n x\u271d \u2208\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\nsimp only [candidatesB, candidates, mem_inter_iff, mem_iInter, mem_setOf_eq]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (GromovHausdorff.candidatesB X Y)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed\n ((((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\nrepeat'\n first\n | apply IsClosed.inter _ _\n | apply isClosed_iInter _\n | apply I1 _ _\n | apply I2 _ _\n | apply I3 _ _\n | apply I4 _ _ _\n | apply I5 _\n | apply I6 _ _\n | intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed\n ((((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed\n ((((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed\n ((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed\n ((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed\n (((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229 \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed\n (((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229 \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed\n ((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229 \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed\n ((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229 \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X), IsClosed (\u22c2 (y : X), {f | \u2191f (inl i, inl y) = dist i y})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X), IsClosed (\u22c2 (y : X), {f | \u2191f (inl i, inl y) = dist i y})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X), IsClosed (\u22c2 (y : X), {f | \u2191f (inl i, inl y) = dist i y})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X), IsClosed (\u22c2 (y : X), {f | \u2191f (inl i, inl y) = dist i y})\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X), IsClosed (\u22c2 (y : X), {f | \u2191f (inl i, inl y) = dist i y})\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X), IsClosed (\u22c2 (y : X), {f | \u2191f (inl i, inl y) = dist i y})\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X), IsClosed (\u22c2 (y : X), {f | \u2191f (inl i, inl y) = dist i y})\n[PROOFSTEP]\napply I4 _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X), IsClosed (\u22c2 (y : X), {f | \u2191f (inl i, inl y) = dist i y})\n[PROOFSTEP]\napply I5 _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X), IsClosed (\u22c2 (y : X), {f | \u2191f (inl i, inl y) = dist i y})\n[PROOFSTEP]\napply I6 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X), IsClosed (\u22c2 (y : X), {f | \u2191f (inl i, inl y) = dist i y})\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X\n\u22a2 IsClosed (\u22c2 (y : X), {f | \u2191f (inl x, inl y) = dist x y})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X\n\u22a2 IsClosed (\u22c2 (y : X), {f | \u2191f (inl x, inl y) = dist x y})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X\n\u22a2 IsClosed (\u22c2 (y : X), {f | \u2191f (inl x, inl y) = dist x y})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X\n\u22a2 \u2200 (i : X), IsClosed {f | \u2191f (inl x, inl i) = dist x i}\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X\n\u22a2 \u2200 (i : X), IsClosed {f | \u2191f (inl x, inl i) = dist x i}\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X\n\u22a2 \u2200 (i : X), IsClosed {f | \u2191f (inl x, inl i) = dist x i}\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X\n\u22a2 \u2200 (i : X), IsClosed {f | \u2191f (inl x, inl i) = dist x i}\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X\n\u22a2 \u2200 (i : X), IsClosed {f | \u2191f (inl x, inl i) = dist x i}\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X\n\u22a2 \u2200 (i : X), IsClosed {f | \u2191f (inl x, inl i) = dist x i}\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X\n\u22a2 \u2200 (i : X), IsClosed {f | \u2191f (inl x, inl i) = dist x i}\n[PROOFSTEP]\napply I4 _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X\n\u22a2 \u2200 (i : X), IsClosed {f | \u2191f (inl x, inl i) = dist x i}\n[PROOFSTEP]\napply I5 _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X\n\u22a2 \u2200 (i : X), IsClosed {f | \u2191f (inl x, inl i) = dist x i}\n[PROOFSTEP]\napply I6 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X\n\u22a2 \u2200 (i : X), IsClosed {f | \u2191f (inl x, inl i) = dist x i}\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X\n\u22a2 IsClosed {f | \u2191f (inl x\u271d, inl x) = dist x\u271d x}\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X\n\u22a2 IsClosed {f | \u2191f (inl x\u271d, inl x) = dist x\u271d x}\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X\n\u22a2 IsClosed {f | \u2191f (inl x\u271d, inl x) = dist x\u271d x}\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X\n\u22a2 IsClosed {f | \u2191f (inl x\u271d, inl x) = dist x\u271d x}\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : Y), IsClosed (\u22c2 (y : Y), {f | \u2191f (inr i, inr y) = dist i y})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : Y), IsClosed (\u22c2 (y : Y), {f | \u2191f (inr i, inr y) = dist i y})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : Y), IsClosed (\u22c2 (y : Y), {f | \u2191f (inr i, inr y) = dist i y})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : Y), IsClosed (\u22c2 (y : Y), {f | \u2191f (inr i, inr y) = dist i y})\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : Y), IsClosed (\u22c2 (y : Y), {f | \u2191f (inr i, inr y) = dist i y})\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : Y), IsClosed (\u22c2 (y : Y), {f | \u2191f (inr i, inr y) = dist i y})\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : Y), IsClosed (\u22c2 (y : Y), {f | \u2191f (inr i, inr y) = dist i y})\n[PROOFSTEP]\napply I4 _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : Y), IsClosed (\u22c2 (y : Y), {f | \u2191f (inr i, inr y) = dist i y})\n[PROOFSTEP]\napply I5 _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : Y), IsClosed (\u22c2 (y : Y), {f | \u2191f (inr i, inr y) = dist i y})\n[PROOFSTEP]\napply I6 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : Y), IsClosed (\u22c2 (y : Y), {f | \u2191f (inr i, inr y) = dist i y})\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : Y\n\u22a2 IsClosed (\u22c2 (y : Y), {f | \u2191f (inr x, inr y) = dist x y})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : Y\n\u22a2 IsClosed (\u22c2 (y : Y), {f | \u2191f (inr x, inr y) = dist x y})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : Y\n\u22a2 IsClosed (\u22c2 (y : Y), {f | \u2191f (inr x, inr y) = dist x y})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : Y\n\u22a2 \u2200 (i : Y), IsClosed {f | \u2191f (inr x, inr i) = dist x i}\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : Y\n\u22a2 \u2200 (i : Y), IsClosed {f | \u2191f (inr x, inr i) = dist x i}\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : Y\n\u22a2 \u2200 (i : Y), IsClosed {f | \u2191f (inr x, inr i) = dist x i}\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : Y\n\u22a2 \u2200 (i : Y), IsClosed {f | \u2191f (inr x, inr i) = dist x i}\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : Y\n\u22a2 \u2200 (i : Y), IsClosed {f | \u2191f (inr x, inr i) = dist x i}\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : Y\n\u22a2 \u2200 (i : Y), IsClosed {f | \u2191f (inr x, inr i) = dist x i}\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : Y\n\u22a2 \u2200 (i : Y), IsClosed {f | \u2191f (inr x, inr i) = dist x i}\n[PROOFSTEP]\napply I4 _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : Y\n\u22a2 \u2200 (i : Y), IsClosed {f | \u2191f (inr x, inr i) = dist x i}\n[PROOFSTEP]\napply I5 _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : Y\n\u22a2 \u2200 (i : Y), IsClosed {f | \u2191f (inr x, inr i) = dist x i}\n[PROOFSTEP]\napply I6 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : Y\n\u22a2 \u2200 (i : Y), IsClosed {f | \u2191f (inr x, inr i) = dist x i}\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : Y\n\u22a2 IsClosed {f | \u2191f (inr x\u271d, inr x) = dist x\u271d x}\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : Y\n\u22a2 IsClosed {f | \u2191f (inr x\u271d, inr x) = dist x\u271d x}\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : Y\n\u22a2 IsClosed {f | \u2191f (inr x\u271d, inr x) = dist x\u271d x}\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : Y\n\u22a2 IsClosed {f | \u2191f (inr x\u271d, inr x) = dist x\u271d x}\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : Y\n\u22a2 IsClosed {f | \u2191f (inr x\u271d, inr x) = dist x\u271d x}\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) = \u2191f (y, i)})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) = \u2191f (y, i)})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) = \u2191f (y, i)})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) = \u2191f (y, i)})\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) = \u2191f (y, i)})\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) = \u2191f (y, i)})\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) = \u2191f (y, i)})\n[PROOFSTEP]\napply I4 _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) = \u2191f (y, i)})\n[PROOFSTEP]\napply I5 _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) = \u2191f (y, i)})\n[PROOFSTEP]\napply I6 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) = \u2191f (y, i)})\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) = \u2191f (i, x)}\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) = \u2191f (i, x)}\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) = \u2191f (i, x)}\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) = \u2191f (i, x)}\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) = \u2191f (i, x)}\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) = \u2191f (i, x)}\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) = \u2191f (i, x)}\n[PROOFSTEP]\napply I4 _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) = \u2191f (i, x)}\n[PROOFSTEP]\napply I5 _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) = \u2191f (i, x)}\n[PROOFSTEP]\napply I6 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) = \u2191f (i, x)}\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d, x) = \u2191f (x, x\u271d)}\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d, x) = \u2191f (x, x\u271d)}\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d, x) = \u2191f (x, x\u271d)}\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d, x) = \u2191f (x, x\u271d)}\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d, x) = \u2191f (x, x\u271d)}\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d, x) = \u2191f (x, x\u271d)}\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (i, z) \u2264 \u2191f (i, y) + \u2191f (y, z)})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (i, z) \u2264 \u2191f (i, y) + \u2191f (y, z)})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (i, z) \u2264 \u2191f (i, y) + \u2191f (y, z)})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (i, z) \u2264 \u2191f (i, y) + \u2191f (y, z)})\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (i, z) \u2264 \u2191f (i, y) + \u2191f (y, z)})\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (i, z) \u2264 \u2191f (i, y) + \u2191f (y, z)})\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (i, z) \u2264 \u2191f (i, y) + \u2191f (y, z)})\n[PROOFSTEP]\napply I4 _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (i, z) \u2264 \u2191f (i, y) + \u2191f (y, z)})\n[PROOFSTEP]\napply I5 _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (i, z) \u2264 \u2191f (i, y) + \u2191f (y, z)})\n[PROOFSTEP]\napply I6 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (i, z) \u2264 \u2191f (i, y) + \u2191f (y, z)})\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed (\u22c2 (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed (\u22c2 (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed (\u22c2 (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, i) + \u2191f (i, z)})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, i) + \u2191f (i, z)})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, i) + \u2191f (i, z)})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, i) + \u2191f (i, z)})\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, i) + \u2191f (i, z)})\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, i) + \u2191f (i, z)})\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, i) + \u2191f (i, z)})\n[PROOFSTEP]\napply I4 _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, i) + \u2191f (i, z)})\n[PROOFSTEP]\napply I5 _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, i) + \u2191f (i, z)})\n[PROOFSTEP]\napply I6 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, i) + \u2191f (i, z)})\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed (\u22c2 (z : X \u2295 Y), {f | \u2191f (x\u271d, z) \u2264 \u2191f (x\u271d, x) + \u2191f (x, z)})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed (\u22c2 (z : X \u2295 Y), {f | \u2191f (x\u271d, z) \u2264 \u2191f (x\u271d, x) + \u2191f (x, z)})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed (\u22c2 (z : X \u2295 Y), {f | \u2191f (x\u271d, z) \u2264 \u2191f (x\u271d, x) + \u2191f (x, z)})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x\u271d, i) \u2264 \u2191f (x\u271d, x) + \u2191f (x, i)}\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x\u271d, i) \u2264 \u2191f (x\u271d, x) + \u2191f (x, i)}\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x\u271d, i) \u2264 \u2191f (x\u271d, x) + \u2191f (x, i)}\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x\u271d, i) \u2264 \u2191f (x\u271d, x) + \u2191f (x, i)}\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x\u271d, i) \u2264 \u2191f (x\u271d, x) + \u2191f (x, i)}\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x\u271d, i) \u2264 \u2191f (x\u271d, x) + \u2191f (x, i)}\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x\u271d, i) \u2264 \u2191f (x\u271d, x) + \u2191f (x, i)}\n[PROOFSTEP]\napply I4 _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x\u271d, i) \u2264 \u2191f (x\u271d, x) + \u2191f (x, i)}\n[PROOFSTEP]\napply I5 _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x\u271d, i) \u2264 \u2191f (x\u271d, x) + \u2191f (x, i)}\n[PROOFSTEP]\napply I6 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x\u271d, i) \u2264 \u2191f (x\u271d, x) + \u2191f (x, i)}\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b2 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d\u00b9 x\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d\u00b9, x) \u2264 \u2191f (x\u271d\u00b9, x\u271d) + \u2191f (x\u271d, x)}\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b2 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d\u00b9 x\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d\u00b9, x) \u2264 \u2191f (x\u271d\u00b9, x\u271d) + \u2191f (x\u271d, x)}\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b2 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d\u00b9 x\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d\u00b9, x) \u2264 \u2191f (x\u271d\u00b9, x\u271d) + \u2191f (x\u271d, x)}\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b2 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d\u00b9 x\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d\u00b9, x) \u2264 \u2191f (x\u271d\u00b9, x\u271d) + \u2191f (x\u271d, x)}\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b2 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d\u00b9 x\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d\u00b9, x) \u2264 \u2191f (x\u271d\u00b9, x\u271d) + \u2191f (x\u271d, x)}\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b2 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d\u00b9 x\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d\u00b9, x) \u2264 \u2191f (x\u271d\u00b9, x\u271d) + \u2191f (x\u271d, x)}\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b2 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d\u00b9 x\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d\u00b9, x) \u2264 \u2191f (x\u271d\u00b9, x\u271d) + \u2191f (x\u271d, x)}\n[PROOFSTEP]\napply I4 _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (i, i) = 0}\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (i, i) = 0}\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (i, i) = 0}\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (i, i) = 0}\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (i, i) = 0}\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (i, i) = 0}\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (i, i) = 0}\n[PROOFSTEP]\napply I4 _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (i, i) = 0}\n[PROOFSTEP]\napply I5 _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (i, i) = 0}\n[PROOFSTEP]\napply I6 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (i, i) = 0}\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x, x) = 0}\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x, x) = 0}\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x, x) = 0}\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x, x) = 0}\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x, x) = 0}\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x, x) = 0}\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x, x) = 0}\n[PROOFSTEP]\napply I4 _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x, x) = 0}\n[PROOFSTEP]\napply I5 _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 IsClosed (\u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\napply I4 _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\napply I5 _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\napply I6 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (i, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 IsClosed (\u22c2 (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)})\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply I4 _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply I5 _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply I6 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx : X \u2295 Y\n\u22a2 \u2200 (i : X \u2295 Y), IsClosed {f | \u2191f (x, i) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d, x) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\nfirst\n| apply IsClosed.inter _ _\n| apply isClosed_iInter _\n| apply I1 _ _\n| apply I2 _ _\n| apply I3 _ _\n| apply I4 _ _ _\n| apply I5 _\n| apply I6 _ _\n| intro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d, x) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply IsClosed.inter _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d, x) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply isClosed_iInter _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d, x) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply I1 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d, x) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply I2 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d, x) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply I3 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d, x) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply I4 _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d, x) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply I5 _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y z t : X \u2295 Y\nI1 : \u2200 (x y : X), IsClosed {f | \u2191f (inl x, inl y) = dist x y}\nI2 : \u2200 (x y : Y), IsClosed {f | \u2191f (inr x, inr y) = dist x y}\nI3 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) = \u2191f (y, x)}\nI4 : \u2200 (x y z : X \u2295 Y), IsClosed {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}\nI5 : \u2200 (x : X \u2295 Y), IsClosed {f | \u2191f (x, x) = 0}\nI6 : \u2200 (x y : X \u2295 Y), IsClosed {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nthis :\n GromovHausdorff.candidatesB X Y =\n (((((\u22c2 (x : X) (y : X), {f | \u2191f (inl x, inl y) = dist x y}) \u2229\n \u22c2 (x : Y) (y : Y), {f | \u2191f (inr x, inr y) = dist x y}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) = \u2191f (y, x)}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y) (z : X \u2295 Y), {f | \u2191f (x, z) \u2264 \u2191f (x, y) + \u2191f (y, z)}) \u2229\n \u22c2 (x : X \u2295 Y), {f | \u2191f (x, x) = 0}) \u2229\n \u22c2 (x : X \u2295 Y) (y : X \u2295 Y), {f | \u2191f (x, y) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\nx\u271d x : X \u2295 Y\n\u22a2 IsClosed {f | \u2191f (x\u271d, x) \u2264 \u2191(GromovHausdorff.maxVar X Y)}\n[PROOFSTEP]\napply I6 _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 IsCompact (GromovHausdorff.candidatesB X Y)\n[PROOFSTEP]\nrefine' arzela_ascoli\u2082 (Icc 0 (maxVar X Y) : Set \u211d) isCompact_Icc (candidatesB X Y) closed_candidatesB _ _\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 \u2200 (f : (X \u2295 Y) \u00d7 (X \u2295 Y) \u2192\u1d47 \u211d) (x : (X \u2295 Y) \u00d7 (X \u2295 Y)),\n f \u2208 GromovHausdorff.candidatesB X Y \u2192 \u2191f x \u2208 Icc 0 \u2191(GromovHausdorff.maxVar X Y)\n[PROOFSTEP]\nrintro f \u27e8x1, x2\u27e9 hf\n[GOAL]\ncase refine'_1.mk\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf : (X \u2295 Y) \u00d7 (X \u2295 Y) \u2192\u1d47 \u211d\nx1 x2 : X \u2295 Y\nhf : f \u2208 GromovHausdorff.candidatesB X Y\n\u22a2 \u2191f (x1, x2) \u2208 Icc 0 \u2191(GromovHausdorff.maxVar X Y)\n[PROOFSTEP]\nsimp only [Set.mem_Icc]\n[GOAL]\ncase refine'_1.mk\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf : (X \u2295 Y) \u00d7 (X \u2295 Y) \u2192\u1d47 \u211d\nx1 x2 : X \u2295 Y\nhf : f \u2208 GromovHausdorff.candidatesB X Y\n\u22a2 0 \u2264 \u2191f (x1, x2) \u2227 \u2191f (x1, x2) \u2264 \u2191(GromovHausdorff.maxVar X Y)\n[PROOFSTEP]\nexact \u27e8candidates_nonneg hf, candidates_le_maxVar hf\u27e9\n[GOAL]\ncase refine'_2\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Equicontinuous fun x => \u2191\u2191x\n[PROOFSTEP]\nrefine' equicontinuous_of_continuity_modulus (fun t => 2 * maxVar X Y * t) _ _ _\n[GOAL]\ncase refine'_2.refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 Tendsto (fun t => 2 * \u2191(GromovHausdorff.maxVar X Y) * t) (\ud835\udcdd 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : Tendsto (fun t : \u211d => 2 * (maxVar X Y : \u211d) * t) (\ud835\udcdd 0) (\ud835\udcdd (2 * maxVar X Y * 0)) :=\n tendsto_const_nhds.mul tendsto_id\n[GOAL]\ncase refine'_2.refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nthis : Tendsto (fun t => 2 * \u2191(GromovHausdorff.maxVar X Y) * t) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u2191(GromovHausdorff.maxVar X Y) * 0))\n\u22a2 Tendsto (fun t => 2 * \u2191(GromovHausdorff.maxVar X Y) * t) (\ud835\udcdd 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa using this\n[GOAL]\ncase refine'_2.refine'_2\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 \u2200 (x y : (X \u2295 Y) \u00d7 (X \u2295 Y)) (i : \u2191(GromovHausdorff.candidatesB X Y)),\n dist (\u2191\u2191i x) (\u2191\u2191i y) \u2264 (fun t => 2 * \u2191(GromovHausdorff.maxVar X Y) * t) (dist x y)\n[PROOFSTEP]\nrintro x y \u27e8f, hf\u27e9\n[GOAL]\ncase refine'_2.refine'_2.mk\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nx y : (X \u2295 Y) \u00d7 (X \u2295 Y)\nf : (X \u2295 Y) \u00d7 (X \u2295 Y) \u2192\u1d47 \u211d\nhf : f \u2208 GromovHausdorff.candidatesB X Y\n\u22a2 dist (\u2191\u2191{ val := f, property := hf } x) (\u2191\u2191{ val := f, property := hf } y) \u2264\n (fun t => 2 * \u2191(GromovHausdorff.maxVar X Y) * t) (dist x y)\n[PROOFSTEP]\nexact (candidates_lipschitz hf).dist_le_mul _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf : GromovHausdorff.Cb X Y\nC : \u211d\n\u22a2 BddAbove (range fun x => \u2a05 (y : Y), \u2191f (inl x, inr y) + C)\n[PROOFSTEP]\nrcases(Real.bounded_iff_bddBelow_bddAbove.1 f.bounded_range).2 with \u27e8Cf, hCf\u27e9\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf : GromovHausdorff.Cb X Y\nC Cf : \u211d\nhCf : Cf \u2208 upperBounds (range \u2191f)\n\u22a2 BddAbove (range fun x => \u2a05 (y : Y), \u2191f (inl x, inr y) + C)\n[PROOFSTEP]\nrefine' \u27e8Cf + C, forall_range_iff.2 fun x => _\u27e9\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nf : GromovHausdorff.Cb X Y\nC Cf : \u211d\nhCf : Cf \u2208 upperBounds (range \u2191f)\nx : X\n\u22a2 \u2a05 (y : Y), \u2191f (inl x, inr y) + C \u2264 Cf + C\n[PROOFSTEP]\ncalc\n \u2a05 y, f (inl x, inr y) + C \u2264 f (inl x, inr default) + C := ciInf_le (HD_below_aux1 C) default\n _ \u2264 Cf + C := add_le_add ((fun x => hCf (mem_range_self x)) _) le_rfl\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf : GromovHausdorff.Cb X Y\nC : \u211d\n\u22a2 BddAbove (range fun y => \u2a05 (x : X), \u2191f (inl x, inr y) + C)\n[PROOFSTEP]\nrcases(Real.bounded_iff_bddBelow_bddAbove.1 f.bounded_range).2 with \u27e8Cf, hCf\u27e9\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf : GromovHausdorff.Cb X Y\nC Cf : \u211d\nhCf : Cf \u2208 upperBounds (range \u2191f)\n\u22a2 BddAbove (range fun y => \u2a05 (x : X), \u2191f (inl x, inr y) + C)\n[PROOFSTEP]\nrefine' \u27e8Cf + C, forall_range_iff.2 fun y => _\u27e9\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y\u271d z t : X \u2295 Y\nf : GromovHausdorff.Cb X Y\nC Cf : \u211d\nhCf : Cf \u2208 upperBounds (range \u2191f)\ny : Y\n\u22a2 \u2a05 (x : X), \u2191f (inl x, inr y) + C \u2264 Cf + C\n[PROOFSTEP]\ncalc\n \u2a05 x, f (inl x, inr y) + C \u2264 f (inl default, inr y) + C := ciInf_le (HD_below_aux2 C) default\n _ \u2264 Cf + C := add_le_add ((fun x => hCf (mem_range_self x)) _) le_rfl\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\n\u22a2 HD (candidatesBDist X Y) \u2264 diam univ + 1 + diam univ\n[PROOFSTEP]\nrefine' max_le (ciSup_le fun x => _) (ciSup_le fun y => _)\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nx : X\n\u22a2 \u2a05 (y : Y), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 diam univ + 1 + diam univ\n[PROOFSTEP]\nhave A : \u2a05 y, candidatesBDist X Y (inl x, inr y) \u2264 candidatesBDist X Y (inl x, inr default) :=\n ciInf_le (by simpa using HD_below_aux1 0) default\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nx : X\n\u22a2 BddBelow (range fun y => \u2191(candidatesBDist X Y) (inl x, inr y))\n[PROOFSTEP]\nsimpa using HD_below_aux1 0\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nx : X\nA : \u2a05 (y : Y), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 \u2191(candidatesBDist X Y) (inl x, inr default)\n\u22a2 \u2a05 (y : Y), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 diam univ + 1 + diam univ\n[PROOFSTEP]\nhave B : dist (inl x) (inr default) \u2264 diam (univ : Set X) + 1 + diam (univ : Set Y) :=\n calc\n dist (inl x) (inr (default : Y)) = dist x (default : X) + 1 + dist default default := rfl\n _ \u2264 diam (univ : Set X) + 1 + diam (univ : Set Y) := by\n gcongr <;> exact dist_le_diam_of_mem bounded_of_compactSpace (mem_univ _) (mem_univ _)\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nx : X\nA : \u2a05 (y : Y), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 \u2191(candidatesBDist X Y) (inl x, inr default)\n\u22a2 dist x default + 1 + dist default default \u2264 diam univ + 1 + diam univ\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081.bc\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nx : X\nA : \u2a05 (y : Y), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 \u2191(candidatesBDist X Y) (inl x, inr default)\n\u22a2 dist x default \u2264 diam univ\n[PROOFSTEP]\nexact dist_le_diam_of_mem bounded_of_compactSpace (mem_univ _) (mem_univ _)\n[GOAL]\ncase h\u2082\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nx : X\nA : \u2a05 (y : Y), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 \u2191(candidatesBDist X Y) (inl x, inr default)\n\u22a2 dist default default \u2264 diam univ\n[PROOFSTEP]\nexact dist_le_diam_of_mem bounded_of_compactSpace (mem_univ _) (mem_univ _)\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nx : X\nA : \u2a05 (y : Y), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 \u2191(candidatesBDist X Y) (inl x, inr default)\nB : dist (inl x) (inr default) \u2264 diam univ + 1 + diam univ\n\u22a2 \u2a05 (y : Y), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 diam univ + 1 + diam univ\n[PROOFSTEP]\nexact le_trans A B\n[GOAL]\ncase refine'_2\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y\u271d z t : X \u2295 Y\ny : Y\n\u22a2 \u2a05 (x : X), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 diam univ + 1 + diam univ\n[PROOFSTEP]\nhave A : \u2a05 x, candidatesBDist X Y (inl x, inr y) \u2264 candidatesBDist X Y (inl default, inr y) :=\n ciInf_le (by simpa using HD_below_aux2 0) default\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y\u271d z t : X \u2295 Y\ny : Y\n\u22a2 BddBelow (range fun x => \u2191(candidatesBDist X Y) (inl x, inr y))\n[PROOFSTEP]\nsimpa using HD_below_aux2 0\n[GOAL]\ncase refine'_2\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y\u271d z t : X \u2295 Y\ny : Y\nA : \u2a05 (x : X), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 \u2191(candidatesBDist X Y) (inl default, inr y)\n\u22a2 \u2a05 (x : X), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 diam univ + 1 + diam univ\n[PROOFSTEP]\nhave B : dist (inl default) (inr y) \u2264 diam (univ : Set X) + 1 + diam (univ : Set Y) :=\n calc\n dist (inl (default : X)) (inr y) = dist default default + 1 + dist default y := rfl\n _ \u2264 diam (univ : Set X) + 1 + diam (univ : Set Y) := by\n gcongr <;> exact dist_le_diam_of_mem bounded_of_compactSpace (mem_univ _) (mem_univ _)\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y\u271d z t : X \u2295 Y\ny : Y\nA : \u2a05 (x : X), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 \u2191(candidatesBDist X Y) (inl default, inr y)\n\u22a2 dist default default + 1 + dist default y \u2264 diam univ + 1 + diam univ\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081.bc\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y\u271d z t : X \u2295 Y\ny : Y\nA : \u2a05 (x : X), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 \u2191(candidatesBDist X Y) (inl default, inr y)\n\u22a2 dist default default \u2264 diam univ\n[PROOFSTEP]\nexact dist_le_diam_of_mem bounded_of_compactSpace (mem_univ _) (mem_univ _)\n[GOAL]\ncase h\u2082\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y\u271d z t : X \u2295 Y\ny : Y\nA : \u2a05 (x : X), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 \u2191(candidatesBDist X Y) (inl default, inr y)\n\u22a2 dist default y \u2264 diam univ\n[PROOFSTEP]\nexact dist_le_diam_of_mem bounded_of_compactSpace (mem_univ _) (mem_univ _)\n[GOAL]\ncase refine'_2\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y\u271d z t : X \u2295 Y\ny : Y\nA : \u2a05 (x : X), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 \u2191(candidatesBDist X Y) (inl default, inr y)\nB : dist (inl default) (inr y) \u2264 diam univ + 1 + diam univ\n\u22a2 \u2a05 (x : X), \u2191(candidatesBDist X Y) (inl x, inr y) \u2264 diam univ + 1 + diam univ\n[PROOFSTEP]\nexact le_trans A B\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\n\u22a2 \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 (\u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nrcases(Real.bounded_iff_bddBelow_bddAbove.1 g.bounded_range).1 with \u27e8cg, hcg\u27e9\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\n\u22a2 \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 (\u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nhave Hcg : \u2200 x, cg \u2264 g x := fun x => hcg (mem_range_self x)\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\n\u22a2 \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 (\u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nrcases(Real.bounded_iff_bddBelow_bddAbove.1 f.bounded_range).1 with \u27e8cf, hcf\u27e9\n[GOAL]\ncase intro.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\n\u22a2 \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 (\u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nhave Hcf : \u2200 x, cf \u2264 f x := fun x =>\n hcf\n (mem_range_self x)\n -- prove the inequality but with `dist f g` inside, by using inequalities comparing\n -- iSup to iSup and iInf to iInf\n[GOAL]\ncase intro.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\n\u22a2 \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 (\u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nhave Z : (\u2a06 x, \u2a05 y, f (inl x, inr y)) \u2264 \u2a06 x, \u2a05 y, g (inl x, inr y) + dist f g :=\n ciSup_mono (HD_bound_aux1 _ (dist f g)) fun x =>\n ciInf_mono \u27e8cf, forall_range_iff.2 fun i => Hcf _\u27e9 fun y => coe_le_coe_add_dist\n[GOAL]\ncase intro.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 \u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\n\u22a2 \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 (\u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nhave E1 : \u2200 x, (\u2a05 y, g (inl x, inr y)) + dist f g = \u2a05 y, g (inl x, inr y) + dist f g :=\n by\n intro x\n refine' Monotone.map_ciInf_of_continuousAt (continuousAt_id.add continuousAt_const) _ _\n \u00b7 intro x y hx\n simpa\n \u00b7 show BddBelow (range fun y : Y => g (inl x, inr y))\n exact \u27e8cg, forall_range_iff.2 fun i => Hcg _\u27e9\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 \u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\n\u22a2 \u2200 (x : X), (\u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g = \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 \u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\nx : X\n\u22a2 (\u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g = \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\n[PROOFSTEP]\nrefine' Monotone.map_ciInf_of_continuousAt (continuousAt_id.add continuousAt_const) _ _\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 \u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\nx : X\n\u22a2 Monotone fun x => id x + dist f g\n[PROOFSTEP]\nintro x y hx\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx\u271d\u00b9 y\u271d z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 \u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\nx\u271d : X\nx y : \u211d\nhx : x \u2264 y\n\u22a2 (fun x => id x + dist f g) x \u2264 (fun x => id x + dist f g) y\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase refine'_2\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 \u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\nx : X\n\u22a2 BddBelow (range fun y => \u2191g (inl x, inr y))\n[PROOFSTEP]\nshow BddBelow (range fun y : Y => g (inl x, inr y))\n[GOAL]\ncase refine'_2\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 \u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\nx : X\n\u22a2 BddBelow (range fun y => \u2191g (inl x, inr y))\n[PROOFSTEP]\nexact \u27e8cg, forall_range_iff.2 fun i => Hcg _\u27e9\n[GOAL]\ncase intro.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 \u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\nE1 : \u2200 (x : X), (\u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g = \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\n\u22a2 \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 (\u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nhave E2 : (\u2a06 x, \u2a05 y, g (inl x, inr y)) + dist f g = \u2a06 x, (\u2a05 y, g (inl x, inr y)) + dist f g :=\n by\n refine' Monotone.map_ciSup_of_continuousAt (continuousAt_id.add continuousAt_const) _ _\n \u00b7 intro x y hx\n simpa\n \u00b7\n simpa using\n HD_bound_aux1 _\n 0\n -- deduce the result from the above two steps\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 \u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\nE1 : \u2200 (x : X), (\u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g = \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\n\u22a2 (\u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g = \u2a06 (x : X), (\u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nrefine' Monotone.map_ciSup_of_continuousAt (continuousAt_id.add continuousAt_const) _ _\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 \u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\nE1 : \u2200 (x : X), (\u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g = \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\n\u22a2 Monotone fun x => id x + dist f g\n[PROOFSTEP]\nintro x y hx\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 \u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\nE1 : \u2200 (x : X), (\u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g = \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\nx y : \u211d\nhx : x \u2264 y\n\u22a2 (fun x => id x + dist f g) x \u2264 (fun x => id x + dist f g) y\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase refine'_2\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 \u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\nE1 : \u2200 (x : X), (\u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g = \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\n\u22a2 BddAbove (range fun x => \u2a05 (y : Y), \u2191g (inl x, inr y))\n[PROOFSTEP]\nsimpa using\n HD_bound_aux1 _\n 0\n -- deduce the result from the above two steps\n[GOAL]\ncase intro.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 \u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\nE1 : \u2200 (x : X), (\u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g = \u2a05 (y : Y), \u2191g (inl x, inr y) + dist f g\nE2 : (\u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g = \u2a06 (x : X), (\u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g\n\u22a2 \u2a06 (x : X), \u2a05 (y : Y), \u2191f (inl x, inr y) \u2264 (\u2a06 (x : X), \u2a05 (y : Y), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nsimpa [E2, E1, Function.comp]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\n\u22a2 \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 (\u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nrcases(Real.bounded_iff_bddBelow_bddAbove.1 g.bounded_range).1 with \u27e8cg, hcg\u27e9\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\n\u22a2 \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 (\u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nhave Hcg : \u2200 x, cg \u2264 g x := fun x => hcg (mem_range_self x)\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\n\u22a2 \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 (\u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nrcases(Real.bounded_iff_bddBelow_bddAbove.1 f.bounded_range).1 with \u27e8cf, hcf\u27e9\n[GOAL]\ncase intro.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\n\u22a2 \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 (\u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nhave Hcf : \u2200 x, cf \u2264 f x := fun x =>\n hcf\n (mem_range_self x)\n -- prove the inequality but with `dist f g` inside, by using inequalities comparing\n -- iSup to iSup and iInf to iInf\n[GOAL]\ncase intro.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\n\u22a2 \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 (\u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nhave Z : (\u2a06 y, \u2a05 x, f (inl x, inr y)) \u2264 \u2a06 y, \u2a05 x, g (inl x, inr y) + dist f g :=\n ciSup_mono (HD_bound_aux2 _ (dist f g)) fun y =>\n ciInf_mono \u27e8cf, forall_range_iff.2 fun i => Hcf _\u27e9 fun y => coe_le_coe_add_dist\n[GOAL]\ncase intro.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 \u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\n\u22a2 \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 (\u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nhave E1 : \u2200 y, (\u2a05 x, g (inl x, inr y)) + dist f g = \u2a05 x, g (inl x, inr y) + dist f g :=\n by\n intro y\n refine' Monotone.map_ciInf_of_continuousAt (continuousAt_id.add continuousAt_const) _ _\n \u00b7 intro x y hx\n simpa\n \u00b7 show BddBelow (range fun x : X => g (inl x, inr y))\n exact \u27e8cg, forall_range_iff.2 fun i => Hcg _\u27e9\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 \u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\n\u22a2 \u2200 (y : Y), (\u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g = \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\n[PROOFSTEP]\nintro y\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y\u271d z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 \u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\ny : Y\n\u22a2 (\u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g = \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\n[PROOFSTEP]\nrefine' Monotone.map_ciInf_of_continuousAt (continuousAt_id.add continuousAt_const) _ _\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y\u271d z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 \u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\ny : Y\n\u22a2 Monotone fun x => id x + dist f g\n[PROOFSTEP]\nintro x y hx\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d\u00b9 z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 \u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\ny\u271d : Y\nx y : \u211d\nhx : x \u2264 y\n\u22a2 (fun x => id x + dist f g) x \u2264 (fun x => id x + dist f g) y\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase refine'_2\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y\u271d z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 \u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\ny : Y\n\u22a2 BddBelow (range fun x => \u2191g (inl x, inr y))\n[PROOFSTEP]\nshow BddBelow (range fun x : X => g (inl x, inr y))\n[GOAL]\ncase refine'_2\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y\u271d z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 \u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\ny : Y\n\u22a2 BddBelow (range fun x => \u2191g (inl x, inr y))\n[PROOFSTEP]\nexact \u27e8cg, forall_range_iff.2 fun i => Hcg _\u27e9\n[GOAL]\ncase intro.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 \u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\nE1 : \u2200 (y : Y), (\u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g = \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\n\u22a2 \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 (\u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nhave E2 : (\u2a06 y, \u2a05 x, g (inl x, inr y)) + dist f g = \u2a06 y, (\u2a05 x, g (inl x, inr y)) + dist f g :=\n by\n refine' Monotone.map_ciSup_of_continuousAt (continuousAt_id.add continuousAt_const) _ _\n \u00b7 intro x y hx\n simpa\n \u00b7\n simpa using\n HD_bound_aux2 _\n 0\n -- deduce the result from the above two steps\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 \u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\nE1 : \u2200 (y : Y), (\u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g = \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\n\u22a2 (\u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g = \u2a06 (y : Y), (\u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nrefine' Monotone.map_ciSup_of_continuousAt (continuousAt_id.add continuousAt_const) _ _\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 \u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\nE1 : \u2200 (y : Y), (\u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g = \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\n\u22a2 Monotone fun x => id x + dist f g\n[PROOFSTEP]\nintro x y hx\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx\u271d y\u271d z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 \u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\nE1 : \u2200 (y : Y), (\u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g = \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\nx y : \u211d\nhx : x \u2264 y\n\u22a2 (fun x => id x + dist f g) x \u2264 (fun x => id x + dist f g) y\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase refine'_2\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 \u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\nE1 : \u2200 (y : Y), (\u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g = \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\n\u22a2 BddAbove (range fun y => \u2a05 (x : X), \u2191g (inl x, inr y))\n[PROOFSTEP]\nsimpa using\n HD_bound_aux2 _\n 0\n -- deduce the result from the above two steps\n[GOAL]\ncase intro.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf\u271d : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X \u2295 Y\nf g : GromovHausdorff.Cb X Y\ncg : \u211d\nhcg : cg \u2208 lowerBounds (range \u2191g)\nHcg : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cg \u2264 \u2191g x\ncf : \u211d\nhcf : cf \u2208 lowerBounds (range \u2191f)\nHcf : \u2200 (x : (X \u2295 Y) \u00d7 (X \u2295 Y)), cf \u2264 \u2191f x\nZ : \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 \u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\nE1 : \u2200 (y : Y), (\u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g = \u2a05 (x : X), \u2191g (inl x, inr y) + dist f g\nE2 : (\u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g = \u2a06 (y : Y), (\u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g\n\u22a2 \u2a06 (y : Y), \u2a05 (x : X), \u2191f (inl x, inr y) \u2264 (\u2a06 (y : Y), \u2a05 (x : X), \u2191g (inl x, inr y)) + dist f g\n[PROOFSTEP]\nsimpa [E2, E1]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\n\u22a2 GromovHausdorff.optimalGHDist X Y \u2208 GromovHausdorff.candidatesB X Y\n[PROOFSTEP]\ncases Classical.choose_spec (exists_minimizer X Y)\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nleft\u271d :\n choose\n (_ :\n \u2203 f,\n f \u2208 GromovHausdorff.candidatesB X Y \u2227\n \u2200 (g : GromovHausdorff.Cb X Y), g \u2208 GromovHausdorff.candidatesB X Y \u2192 HD f \u2264 HD g) \u2208\n GromovHausdorff.candidatesB X Y\nright\u271d :\n \u2200 (g : GromovHausdorff.Cb X Y),\n g \u2208 GromovHausdorff.candidatesB X Y \u2192\n HD\n (choose\n (_ :\n \u2203 f,\n f \u2208 GromovHausdorff.candidatesB X Y \u2227\n \u2200 (g : GromovHausdorff.Cb X Y), g \u2208 GromovHausdorff.candidatesB X Y \u2192 HD f \u2264 HD g)) \u2264\n HD g\n\u22a2 GromovHausdorff.optimalGHDist X Y \u2208 GromovHausdorff.candidatesB X Y\n[PROOFSTEP]\nassumption\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nx y : X \u2295 Y\n\u22a2 (fun x y =>\n \u2191{ val := \u2191(GromovHausdorff.optimalGHDist X Y) (x, y),\n property := (_ : 0 \u2264 \u2191(GromovHausdorff.optimalGHDist X Y) (x, y)) })\n x y =\n ENNReal.ofReal (dist x y)\n[PROOFSTEP]\nsimp only\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nx y : X \u2295 Y\n\u22a2 \u2191{ val := \u2191(GromovHausdorff.optimalGHDist X Y) (x, y),\n property := (_ : 0 \u2264 \u2191(GromovHausdorff.optimalGHDist X Y) (x, y)) } =\n ENNReal.ofReal (\u2191(GromovHausdorff.optimalGHDist X Y) (x, y))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_val\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nx y : X \u2295 Y\n\u22a2 \u2191(GromovHausdorff.optimalGHDist X Y) (x, y) = max (\u2191(GromovHausdorff.optimalGHDist X Y) (x, y)) 0\n[PROOFSTEP]\nsimp only [max, left_eq_sup]\n[GOAL]\ncase e_a.e_val\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nx y : X \u2295 Y\n\u22a2 0 \u2264 \u2191(GromovHausdorff.optimalGHDist X Y) (x, y)\n[PROOFSTEP]\nexact candidates_nonneg (optimalGHDist_mem_candidatesB X Y)\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\n\u22a2 MetricSpace (OptimalGHCoupling X Y)\n[PROOFSTEP]\nunfold OptimalGHCoupling\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\n\u22a2 MetricSpace (UniformSpace.SeparationQuotient (X \u2295 Y))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\n\u22a2 IsCompact univ\n[PROOFSTEP]\nrw [\u2190 range_quotient_mk']\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\n\u22a2 IsCompact (range Quotient.mk')\n[PROOFSTEP]\nexact\n isCompact_range\n (continuous_sum_dom.2 \u27e8(isometry_optimalGHInjl X Y).continuous, (isometry_optimalGHInjr X Y).continuous\u27e9)\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\n\u22a2 hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) \u2264 HD f\n[PROOFSTEP]\nrefine' le_trans (le_of_forall_le_of_dense fun r hr => _) (HD_optimalGHDist_le X Y f h)\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\n\u22a2 hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) \u2264 r\n[PROOFSTEP]\nhave A : \u2200 x \u2208 range (optimalGHInjl X Y), \u2203 y \u2208 range (optimalGHInjr X Y), dist x y \u2264 r :=\n by\n rintro _ \u27e8z, rfl\u27e9\n have I1 : (\u2a06 x, \u2a05 y, optimalGHDist X Y (inl x, inr y)) < r := lt_of_le_of_lt (le_max_left _ _) hr\n have I2 : \u2a05 y, optimalGHDist X Y (inl z, inr y) \u2264 \u2a06 x, \u2a05 y, optimalGHDist X Y (inl x, inr y) :=\n le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)\n have I : \u2a05 y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1\n rcases exists_lt_of_csInf_lt (range_nonempty _) I with \u27e8r', \u27e8z', rfl\u27e9, hr'\u27e9\n exact \u27e8optimalGHInjr X Y z', mem_range_self _, le_of_lt hr'\u27e9\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\n\u22a2 \u2200 (x : OptimalGHCoupling X Y), x \u2208 range (optimalGHInjl X Y) \u2192 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist x y \u2264 r\n[PROOFSTEP]\nrintro _ \u27e8z, rfl\u27e9\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nz : X\n\u22a2 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist (optimalGHInjl X Y z) y \u2264 r\n[PROOFSTEP]\nhave I1 : (\u2a06 x, \u2a05 y, optimalGHDist X Y (inl x, inr y)) < r := lt_of_le_of_lt (le_max_left _ _) hr\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nz : X\nI1 : \u2a06 (x : X), \u2a05 (y : Y), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r\n\u22a2 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist (optimalGHInjl X Y z) y \u2264 r\n[PROOFSTEP]\nhave I2 : \u2a05 y, optimalGHDist X Y (inl z, inr y) \u2264 \u2a06 x, \u2a05 y, optimalGHDist X Y (inl x, inr y) :=\n le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nz : X\nI1 : \u2a06 (x : X), \u2a05 (y : Y), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r\n\u22a2 BddAbove (range fun x => \u2a05 (y : Y), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y))\n[PROOFSTEP]\nsimpa using HD_bound_aux1 _ 0\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nz : X\nI1 : \u2a06 (x : X), \u2a05 (y : Y), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r\nI2 :\n \u2a05 (y : Y), \u2191(GromovHausdorff.optimalGHDist X Y) (inl z, inr y) \u2264\n \u2a06 (x : X), \u2a05 (y : Y), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y)\n\u22a2 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist (optimalGHInjl X Y z) y \u2264 r\n[PROOFSTEP]\nhave I : \u2a05 y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nz : X\nI1 : \u2a06 (x : X), \u2a05 (y : Y), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r\nI2 :\n \u2a05 (y : Y), \u2191(GromovHausdorff.optimalGHDist X Y) (inl z, inr y) \u2264\n \u2a06 (x : X), \u2a05 (y : Y), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y)\nI : \u2a05 (y : Y), \u2191(GromovHausdorff.optimalGHDist X Y) (inl z, inr y) < r\n\u22a2 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist (optimalGHInjl X Y z) y \u2264 r\n[PROOFSTEP]\nrcases exists_lt_of_csInf_lt (range_nonempty _) I with \u27e8r', \u27e8z', rfl\u27e9, hr'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nz : X\nI1 : \u2a06 (x : X), \u2a05 (y : Y), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r\nI2 :\n \u2a05 (y : Y), \u2191(GromovHausdorff.optimalGHDist X Y) (inl z, inr y) \u2264\n \u2a06 (x : X), \u2a05 (y : Y), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y)\nI : \u2a05 (y : Y), \u2191(GromovHausdorff.optimalGHDist X Y) (inl z, inr y) < r\nz' : Y\nhr' : (fun y => \u2191(GromovHausdorff.optimalGHDist X Y) (inl z, inr y)) z' < r\n\u22a2 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist (optimalGHInjl X Y z) y \u2264 r\n[PROOFSTEP]\nexact \u27e8optimalGHInjr X Y z', mem_range_self _, le_of_lt hr'\u27e9\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nA : \u2200 (x : OptimalGHCoupling X Y), x \u2208 range (optimalGHInjl X Y) \u2192 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist x y \u2264 r\n\u22a2 hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) \u2264 r\n[PROOFSTEP]\nrefine' hausdorffDist_le_of_mem_dist _ A _\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nA : \u2200 (x : OptimalGHCoupling X Y), x \u2208 range (optimalGHInjl X Y) \u2192 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist x y \u2264 r\n\u22a2 0 \u2264 r\n[PROOFSTEP]\ninhabit X\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nA : \u2200 (x : OptimalGHCoupling X Y), x \u2208 range (optimalGHInjl X Y) \u2192 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist x y \u2264 r\ninhabited_h : Inhabited X\n\u22a2 0 \u2264 r\n[PROOFSTEP]\nrcases A _ (mem_range_self default) with \u27e8y, -, hy\u27e9\n[GOAL]\ncase refine'_1.intro.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nA : \u2200 (x : OptimalGHCoupling X Y), x \u2208 range (optimalGHInjl X Y) \u2192 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist x y \u2264 r\ninhabited_h : Inhabited X\ny : OptimalGHCoupling X Y\nhy : dist (optimalGHInjl X Y default) y \u2264 r\n\u22a2 0 \u2264 r\n[PROOFSTEP]\nexact le_trans dist_nonneg hy\n[GOAL]\ncase refine'_2\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nA : \u2200 (x : OptimalGHCoupling X Y), x \u2208 range (optimalGHInjl X Y) \u2192 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist x y \u2264 r\n\u22a2 \u2200 (x : OptimalGHCoupling X Y), x \u2208 range (optimalGHInjr X Y) \u2192 \u2203 y, y \u2208 range (optimalGHInjl X Y) \u2227 dist x y \u2264 r\n[PROOFSTEP]\nrintro _ \u27e8z, rfl\u27e9\n[GOAL]\ncase refine'_2.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nA : \u2200 (x : OptimalGHCoupling X Y), x \u2208 range (optimalGHInjl X Y) \u2192 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist x y \u2264 r\nz : Y\n\u22a2 \u2203 y, y \u2208 range (optimalGHInjl X Y) \u2227 dist (optimalGHInjr X Y z) y \u2264 r\n[PROOFSTEP]\nhave I1 : (\u2a06 y, \u2a05 x, optimalGHDist X Y (inl x, inr y)) < r := lt_of_le_of_lt (le_max_right _ _) hr\n[GOAL]\ncase refine'_2.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nA : \u2200 (x : OptimalGHCoupling X Y), x \u2208 range (optimalGHInjl X Y) \u2192 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist x y \u2264 r\nz : Y\nI1 : \u2a06 (y : Y), \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r\n\u22a2 \u2203 y, y \u2208 range (optimalGHInjl X Y) \u2227 dist (optimalGHInjr X Y z) y \u2264 r\n[PROOFSTEP]\nhave I2 : \u2a05 x, optimalGHDist X Y (inl x, inr z) \u2264 \u2a06 y, \u2a05 x, optimalGHDist X Y (inl x, inr y) :=\n le_csSup (by simpa using HD_bound_aux2 _ 0) (mem_range_self _)\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nA : \u2200 (x : OptimalGHCoupling X Y), x \u2208 range (optimalGHInjl X Y) \u2192 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist x y \u2264 r\nz : Y\nI1 : \u2a06 (y : Y), \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r\n\u22a2 BddAbove (range fun y => \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y))\n[PROOFSTEP]\nsimpa using HD_bound_aux2 _ 0\n[GOAL]\ncase refine'_2.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nA : \u2200 (x : OptimalGHCoupling X Y), x \u2208 range (optimalGHInjl X Y) \u2192 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist x y \u2264 r\nz : Y\nI1 : \u2a06 (y : Y), \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r\nI2 :\n \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr z) \u2264\n \u2a06 (y : Y), \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y)\n\u22a2 \u2203 y, y \u2208 range (optimalGHInjl X Y) \u2227 dist (optimalGHInjr X Y z) y \u2264 r\n[PROOFSTEP]\nhave I : \u2a05 x, optimalGHDist X Y (inl x, inr z) < r := lt_of_le_of_lt I2 I1\n[GOAL]\ncase refine'_2.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nA : \u2200 (x : OptimalGHCoupling X Y), x \u2208 range (optimalGHInjl X Y) \u2192 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist x y \u2264 r\nz : Y\nI1 : \u2a06 (y : Y), \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r\nI2 :\n \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr z) \u2264\n \u2a06 (y : Y), \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y)\nI : \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr z) < r\n\u22a2 \u2203 y, y \u2208 range (optimalGHInjl X Y) \u2227 dist (optimalGHInjr X Y z) y \u2264 r\n[PROOFSTEP]\nrcases exists_lt_of_csInf_lt (range_nonempty _) I with \u27e8r', \u27e8z', rfl\u27e9, hr'\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nA : \u2200 (x : OptimalGHCoupling X Y), x \u2208 range (optimalGHInjl X Y) \u2192 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist x y \u2264 r\nz : Y\nI1 : \u2a06 (y : Y), \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r\nI2 :\n \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr z) \u2264\n \u2a06 (y : Y), \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y)\nI : \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr z) < r\nz' : X\nhr' : (fun x => \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr z)) z' < r\n\u22a2 \u2203 y, y \u2208 range (optimalGHInjl X Y) \u2227 dist (optimalGHInjr X Y z) y \u2264 r\n[PROOFSTEP]\nrefine' \u27e8optimalGHInjl X Y z', mem_range_self _, le_of_lt _\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\nX : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nf : GromovHausdorff.Cb X Y\nh : f \u2208 GromovHausdorff.candidatesB X Y\nr : \u211d\nhr : HD (GromovHausdorff.optimalGHDist X Y) < r\nA : \u2200 (x : OptimalGHCoupling X Y), x \u2208 range (optimalGHInjl X Y) \u2192 \u2203 y, y \u2208 range (optimalGHInjr X Y) \u2227 dist x y \u2264 r\nz : Y\nI1 : \u2a06 (y : Y), \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y) < r\nI2 :\n \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr z) \u2264\n \u2a06 (y : Y), \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr y)\nI : \u2a05 (x : X), \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr z) < r\nz' : X\nhr' : (fun x => \u2191(GromovHausdorff.optimalGHDist X Y) (inl x, inr z)) z' < r\n\u22a2 dist (optimalGHInjr X Y z) (optimalGHInjl X Y z') < r\n[PROOFSTEP]\nrwa [dist_comm]\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.GromovHausdorffRealized", "llama_tokens": 189961, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737963569014, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.4453361483100427}} {"text": "[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne R\u271d\ninst\u271d\u00b2 : CharZero R\u271d\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : CharZero R\nq n : \u2115\nhn : n \u2260 0\n\u22a2 \u2191q ^ n = 1 \u2194 q = 1\n[PROOFSTEP]\nrw [\u2190 cast_pow, cast_eq_one]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne R\u271d\ninst\u271d\u00b2 : CharZero R\u271d\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : CharZero R\nq n : \u2115\nhn : n \u2260 0\n\u22a2 q ^ n = 1 \u2194 q = 1\n[PROOFSTEP]\nexact pow_eq_one_iff hn\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne R\ninst\u271d\u00b2 : CharZero R\nk : Type u_2\ninst\u271d\u00b9 : DivisionSemiring k\ninst\u271d : CharZero k\nm n : \u2115\nn_dvd : n \u2223 m\n\u22a2 \u2191(m / n) = \u2191m / \u2191n\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn)\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne R\ninst\u271d\u00b2 : CharZero R\nk : Type u_2\ninst\u271d\u00b9 : DivisionSemiring k\ninst\u271d : CharZero k\nm : \u2115\nn_dvd : 0 \u2223 m\n\u22a2 \u2191(m / 0) = \u2191m / \u21910\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne R\ninst\u271d\u00b2 : CharZero R\nk : Type u_2\ninst\u271d\u00b9 : DivisionSemiring k\ninst\u271d : CharZero k\nm n : \u2115\nn_dvd : n \u2223 m\nhn : n \u2260 0\n\u22a2 \u2191(m / n) = \u2191m / \u2191n\n[PROOFSTEP]\nexact cast_div n_dvd (cast_ne_zero.2 hn)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : AddMonoidWithOne M\ninst\u271d : CharZero M\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nhave : ((2 : \u2115) : M) \u2260 0 := Nat.cast_ne_zero.2 (by decide)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : AddMonoidWithOne M\ninst\u271d : CharZero M\n\u22a2 2 \u2260 0\n[PROOFSTEP]\ndecide\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : AddMonoidWithOne M\ninst\u271d : CharZero M\nthis : \u21912 \u2260 0\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nrwa [Nat.cast_two] at this \n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocSemiring R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na\u271d a : R\n\u22a2 a + a = 0 \u2194 a = 0\n[PROOFSTEP]\nsimp only [(two_mul a).symm, mul_eq_zero, two_ne_zero, false_or_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocSemiring R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na\u271d a : R\n\u22a2 0 = bit0 a \u2194 a = 0\n[PROOFSTEP]\nrw [eq_comm]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocSemiring R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na\u271d a : R\n\u22a2 bit0 a = 0 \u2194 a = 0\n[PROOFSTEP]\nexact bit0_eq_zero\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\nn : \u2115\na b : R\nh : \u2191n * a = \u2191n * b\n\u22a2 n = 0 \u2228 a = b\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 mul_sub, mul_eq_zero, sub_eq_zero] at h \n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\nn : \u2115\na b : R\nh\u271d : \u2191n * a = \u2191n * b\nh : \u2191n = 0 \u2228 a = b\n\u22a2 n = 0 \u2228 a = b\n[PROOFSTEP]\nexact_mod_cast h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\nn : \u2115\na b : R\nh : \u2191n * a = \u2191n * b\nw : n \u2260 0\n\u22a2 a = b\n[PROOFSTEP]\nsimpa [w] using nat_mul_inj h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na b : R\nh : bit0 a = bit0 b\n\u22a2 a = b\n[PROOFSTEP]\ndsimp [bit0] at h \n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na b : R\nh : a + a = b + b\n\u22a2 a = b\n[PROOFSTEP]\nsimp only [(two_mul a).symm, (two_mul b).symm] at h \n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na b : R\nh : 2 * a = 2 * b\n\u22a2 a = b\n[PROOFSTEP]\nrefine' nat_mul_inj' _ two_ne_zero\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na b : R\nh : 2 * a = 2 * b\n\u22a2 \u21912 * a = \u21912 * b\n[PROOFSTEP]\nexact_mod_cast h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na b : R\nh : bit1 a = bit1 b\n\u22a2 a = b\n[PROOFSTEP]\nsimp only [bit1, add_left_inj] at h \n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na b : R\nh : bit0 a = bit0 b\n\u22a2 a = b\n[PROOFSTEP]\nexact bit0_injective h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na : R\n\u22a2 bit1 a = 1 \u2194 a = 0\n[PROOFSTEP]\nrw [show (1 : R) = bit1 0 by simp, bit1_eq_bit1]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na : R\n\u22a2 1 = bit1 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na : R\n\u22a2 1 = bit1 a \u2194 a = 0\n[PROOFSTEP]\nrw [eq_comm]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na : R\n\u22a2 bit1 a = 1 \u2194 a = 0\n[PROOFSTEP]\nexact bit1_eq_one\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\na : R\n\u22a2 (a + a) / 2 = a\n[PROOFSTEP]\nrw [\u2190 mul_two, mul_div_cancel a two_ne_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\na : R\n\u22a2 a / 2 + a / 2 = a\n[PROOFSTEP]\nrw [\u2190 add_div, half_add_self]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\na : R\n\u22a2 a - a / 2 = a / 2\n[PROOFSTEP]\nrw [sub_eq_iff_eq_add, add_halves']\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\na : R\n\u22a2 a / 2 - a = -(a / 2)\n[PROOFSTEP]\nrw [\u2190 neg_sub, sub_half]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : AddMonoidWithOne R\ninst\u271d : CharZero R\nm n : \u2115\nh : \u2191m = \u2191n\n\u22a2 m = n\n[PROOFSTEP]\nrwa [\u2190 coe_nat, \u2190 coe_nat n, coe_eq_coe, Nat.cast_inj] at h \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : AddMonoidWithOne R\ninst\u271d : CharZero R\nm n : \u2115\nh : \u2191m = \u2191n\n\u22a2 m = n\n[PROOFSTEP]\nrwa [\u2190 coe_nat, \u2190 coe_nat n, coe_eq_coe, Nat.cast_inj] at h \n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : NonAssocSemiring S\n\u03d5 : R \u2192+* S\nhS : CharZero S\na b : \u2115\nh : \u2191a = \u2191b\n\u22a2 \u2191a = \u2191b\n[PROOFSTEP]\nrw [\u2190 map_natCast \u03d5, \u2190 map_natCast \u03d5, h]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : NonAssocSemiring S\n\u03d5 : R \u2192+* S\nh\u03d5 : Function.Injective \u2191\u03d5\nhR : CharZero R\n\u22a2 Function.Injective Nat.cast\n[PROOFSTEP]\nintro a b h\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : NonAssocSemiring S\n\u03d5 : R \u2192+* S\nh\u03d5 : Function.Injective \u2191\u03d5\nhR : CharZero R\na b : \u2115\nh : \u2191a = \u2191b\n\u22a2 a = b\n[PROOFSTEP]\nrwa [\u2190 @Nat.cast_inj R, \u2190 h\u03d5.eq_iff, map_natCast \u03d5, map_natCast \u03d5]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.CharZero.Lemmas", "llama_tokens": 3122, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154240079185319, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.44532234949471966}} {"text": "[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : SupHomClass F \u03b1 \u03b2\nsrc\u271d : SupHomClass F \u03b1 \u03b2 := inst\u271d\nf : F\na b : \u03b1\nh : a \u2264 b\n\u22a2 \u2191f a \u2264 \u2191f b\n[PROOFSTEP]\nrw [\u2190 sup_eq_right, \u2190 map_sup, sup_eq_right.2 h]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : SemilatticeInf \u03b2\ninst\u271d : InfHomClass F \u03b1 \u03b2\nsrc\u271d : InfHomClass F \u03b1 \u03b2 := inst\u271d\nf : F\na b : \u03b1\nh : a \u2264 b\n\u22a2 \u2191f a \u2264 \u2191f b\n[PROOFSTEP]\nrw [\u2190 inf_eq_left, \u2190 map_inf, inf_eq_left.2 h]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nsrc\u271d : OrderHomClass F \u03b1 \u03b2 :=\n let_fun this := inferInstance;\n this\nf : F\na b : \u03b1\nc : (fun x => \u03b2) (a \u2294 b)\n\u22a2 \u2191f (a \u2294 b) \u2264 c \u2194 \u2191f a \u2294 \u2191f b \u2264 c\n[PROOFSTEP]\nsimp only [\u2190 le_map_inv_iff, sup_le_iff]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : SemilatticeInf \u03b2\ninst\u271d : OrderIsoClass F \u03b1 \u03b2\nsrc\u271d : OrderHomClass F \u03b1 \u03b2 :=\n let_fun this := inferInstance;\n this\nf : F\na b : \u03b1\nc : (fun x => \u03b2) (a \u2293 b)\n\u22a2 c \u2264 \u2191f (a \u2293 b) \u2194 c \u2264 \u2191f a \u2293 \u2191f b\n[PROOFSTEP]\nsimp only [\u2190 map_inv_le_iff, le_inf_iff]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : Lattice \u03b2\ninst\u271d\u00b9 : BoundedOrder \u03b2\ninst\u271d : BoundedLatticeHomClass F \u03b1 \u03b2\nf : F\na b : \u03b1\nh : Disjoint a b\n\u22a2 Disjoint (\u2191f a) (\u2191f b)\n[PROOFSTEP]\nrw [disjoint_iff, \u2190 map_inf, h.eq_bot, map_bot]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : Lattice \u03b2\ninst\u271d\u00b9 : BoundedOrder \u03b2\ninst\u271d : BoundedLatticeHomClass F \u03b1 \u03b2\nf : F\na b : \u03b1\nh : Codisjoint a b\n\u22a2 Codisjoint (\u2191f a) (\u2191f b)\n[PROOFSTEP]\nrw [codisjoint_iff, \u2190 map_sup, h.eq_top, map_top]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : BooleanAlgebra \u03b2\ninst\u271d : BoundedLatticeHomClass F \u03b1 \u03b2\nf : F\na b : \u03b1\n\u22a2 \u2191f (a \\ b) = \u2191f a \\ \u2191f b\n[PROOFSTEP]\nrw [sdiff_eq, sdiff_eq, map_inf, map_compl']\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : BooleanAlgebra \u03b2\ninst\u271d : BoundedLatticeHomClass F \u03b1 \u03b2\nf : F\na b : \u03b1\n\u22a2 \u2191f (a \u2206 b) = \u2191f a \u2206 \u2191f b\n[PROOFSTEP]\nrw [symmDiff, symmDiff, map_sup, map_sdiff', map_sdiff']\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Sup \u03b1\ninst\u271d\u00b2 : Sup \u03b2\ninst\u271d\u00b9 : Sup \u03b3\ninst\u271d : Sup \u03b4\nf g : SupHom \u03b1 \u03b2\nh : f.toFun = g.toFun\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Sup \u03b1\ninst\u271d\u00b2 : Sup \u03b2\ninst\u271d\u00b9 : Sup \u03b3\ninst\u271d : Sup \u03b4\ng : SupHom \u03b1 \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_sup'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2294 b) = toFun\u271d a \u2294 toFun\u271d b\nh : { toFun := toFun\u271d, map_sup' := map_sup'\u271d }.toFun = g.toFun\n\u22a2 { toFun := toFun\u271d, map_sup' := map_sup'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Sup \u03b1\ninst\u271d\u00b2 : Sup \u03b2\ninst\u271d\u00b9 : Sup \u03b3\ninst\u271d : Sup \u03b4\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmap_sup'\u271d\u00b9 : \u2200 (a b : \u03b1), toFun\u271d\u00b9 (a \u2294 b) = toFun\u271d\u00b9 a \u2294 toFun\u271d\u00b9 b\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_sup'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2294 b) = toFun\u271d a \u2294 toFun\u271d b\nh : { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }.toFun = { toFun := toFun\u271d, map_sup' := map_sup'\u271d }.toFun\n\u22a2 { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } = { toFun := toFun\u271d, map_sup' := map_sup'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Sup \u03b1\ninst\u271d\u00b2 : Sup \u03b2\ninst\u271d\u00b9 : Sup \u03b3\ninst\u271d : Sup \u03b4\nf : SupHom \u03b2 \u03b3\ng : SupHom \u03b1 \u03b2\na b : \u03b1\n\u22a2 (\u2191f \u2218 \u2191g) (a \u2294 b) = (\u2191f \u2218 \u2191g) a \u2294 (\u2191f \u2218 \u2191g) b\n[PROOFSTEP]\nrw [comp_apply, map_sup, map_sup]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Sup \u03b1\ninst\u271d\u00b2 : Sup \u03b2\ninst\u271d\u00b9 : Sup \u03b3\ninst\u271d : Sup \u03b4\nf : SupHom \u03b2 \u03b3\ng : SupHom \u03b1 \u03b2\na b : \u03b1\n\u22a2 \u2191f (\u2191g a) \u2294 \u2191f (\u2191g b) = (\u2191f \u2218 \u2191g) a \u2294 (\u2191f \u2218 \u2191g) b\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Sup \u03b1\ninst\u271d\u00b2 : Sup \u03b2\ninst\u271d\u00b9 : Sup \u03b3\ninst\u271d : Sup \u03b4\ng : SupHom \u03b2 \u03b3\nf\u2081 f\u2082 : SupHom \u03b1 \u03b2\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\na : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a)\n[PROOFSTEP]\nrw [\u2190 SupHom.comp_apply, h, SupHom.comp_apply]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf g : SupHom \u03b1 \u03b2\na b : \u03b1\n\u22a2 (\u2191f \u2294 \u2191g) (a \u2294 b) = (\u2191f \u2294 \u2191g) a \u2294 (\u2191f \u2294 \u2191g) b\n[PROOFSTEP]\nrw [Pi.sup_apply, map_sup, map_sup]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf g : SupHom \u03b1 \u03b2\na b : \u03b1\n\u22a2 \u2191f a \u2294 \u2191f b \u2294 (\u2191g a \u2294 \u2191g b) = (\u2191f \u2294 \u2191g) a \u2294 (\u2191f \u2294 \u2191g) b\n[PROOFSTEP]\nexact sup_sup_sup_comm _ _ _ _\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Inf \u03b1\ninst\u271d\u00b2 : Inf \u03b2\ninst\u271d\u00b9 : Inf \u03b3\ninst\u271d : Inf \u03b4\nf g : InfHom \u03b1 \u03b2\nh : f.toFun = g.toFun\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Inf \u03b1\ninst\u271d\u00b2 : Inf \u03b2\ninst\u271d\u00b9 : Inf \u03b3\ninst\u271d : Inf \u03b4\ng : InfHom \u03b1 \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_inf'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2293 b) = toFun\u271d a \u2293 toFun\u271d b\nh : { toFun := toFun\u271d, map_inf' := map_inf'\u271d }.toFun = g.toFun\n\u22a2 { toFun := toFun\u271d, map_inf' := map_inf'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Inf \u03b1\ninst\u271d\u00b2 : Inf \u03b2\ninst\u271d\u00b9 : Inf \u03b3\ninst\u271d : Inf \u03b4\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmap_inf'\u271d\u00b9 : \u2200 (a b : \u03b1), toFun\u271d\u00b9 (a \u2293 b) = toFun\u271d\u00b9 a \u2293 toFun\u271d\u00b9 b\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_inf'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2293 b) = toFun\u271d a \u2293 toFun\u271d b\nh : { toFun := toFun\u271d\u00b9, map_inf' := map_inf'\u271d\u00b9 }.toFun = { toFun := toFun\u271d, map_inf' := map_inf'\u271d }.toFun\n\u22a2 { toFun := toFun\u271d\u00b9, map_inf' := map_inf'\u271d\u00b9 } = { toFun := toFun\u271d, map_inf' := map_inf'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Inf \u03b1\ninst\u271d\u00b2 : Inf \u03b2\ninst\u271d\u00b9 : Inf \u03b3\ninst\u271d : Inf \u03b4\nf : InfHom \u03b2 \u03b3\ng : InfHom \u03b1 \u03b2\na b : \u03b1\n\u22a2 (\u2191f \u2218 \u2191g) (a \u2293 b) = (\u2191f \u2218 \u2191g) a \u2293 (\u2191f \u2218 \u2191g) b\n[PROOFSTEP]\nrw [comp_apply, map_inf, map_inf]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Inf \u03b1\ninst\u271d\u00b2 : Inf \u03b2\ninst\u271d\u00b9 : Inf \u03b3\ninst\u271d : Inf \u03b4\nf : InfHom \u03b2 \u03b3\ng : InfHom \u03b1 \u03b2\na b : \u03b1\n\u22a2 \u2191f (\u2191g a) \u2293 \u2191f (\u2191g b) = (\u2191f \u2218 \u2191g) a \u2293 (\u2191f \u2218 \u2191g) b\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Inf \u03b1\ninst\u271d\u00b2 : Inf \u03b2\ninst\u271d\u00b9 : Inf \u03b3\ninst\u271d : Inf \u03b4\ng : InfHom \u03b2 \u03b3\nf\u2081 f\u2082 : InfHom \u03b1 \u03b2\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\na : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a)\n[PROOFSTEP]\nrw [\u2190 InfHom.comp_apply, h, InfHom.comp_apply]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : SemilatticeInf \u03b2\nf g : InfHom \u03b1 \u03b2\na b : \u03b1\n\u22a2 (\u2191f \u2293 \u2191g) (a \u2293 b) = (\u2191f \u2293 \u2191g) a \u2293 (\u2191f \u2293 \u2191g) b\n[PROOFSTEP]\nrw [Pi.inf_apply, map_inf, map_inf]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : SemilatticeInf \u03b2\nf g : InfHom \u03b1 \u03b2\na b : \u03b1\n\u22a2 \u2191f a \u2293 \u2191f b \u2293 (\u2191g a \u2293 \u2191g b) = (\u2191f \u2293 \u2191g) a \u2293 (\u2191f \u2293 \u2191g) b\n[PROOFSTEP]\nexact inf_inf_inf_comm _ _ _ _\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u2077 : Sup \u03b1\ninst\u271d\u2076 : Bot \u03b1\ninst\u271d\u2075 : Sup \u03b2\ninst\u271d\u2074 : Bot \u03b2\ninst\u271d\u00b3 : Sup \u03b3\ninst\u271d\u00b2 : Bot \u03b3\ninst\u271d\u00b9 : Sup \u03b4\ninst\u271d : Bot \u03b4\nf g : SupBotHom \u03b1 \u03b2\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\nobtain \u27e8\u27e8_, _\u27e9, _\u27e9 := f\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u2077 : Sup \u03b1\ninst\u271d\u2076 : Bot \u03b1\ninst\u271d\u2075 : Sup \u03b2\ninst\u271d\u2074 : Bot \u03b2\ninst\u271d\u00b3 : Sup \u03b3\ninst\u271d\u00b2 : Bot \u03b3\ninst\u271d\u00b9 : Sup \u03b4\ninst\u271d : Bot \u03b4\ng : SupBotHom \u03b1 \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_sup'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2294 b) = toFun\u271d a \u2294 toFun\u271d b\nmap_bot'\u271d : SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } \u22a5 = \u22a5\nh :\n (fun f => f.toFun) { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_bot' := map_bot'\u271d } =\n (fun f => f.toFun) g\n\u22a2 { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_bot' := map_bot'\u271d } = g\n[PROOFSTEP]\nobtain \u27e8\u27e8_, _\u27e9, _\u27e9 := g\n[GOAL]\ncase mk.mk.mk.mk\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u2077 : Sup \u03b1\ninst\u271d\u2076 : Bot \u03b1\ninst\u271d\u2075 : Sup \u03b2\ninst\u271d\u2074 : Bot \u03b2\ninst\u271d\u00b3 : Sup \u03b3\ninst\u271d\u00b2 : Bot \u03b3\ninst\u271d\u00b9 : Sup \u03b4\ninst\u271d : Bot \u03b4\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmap_sup'\u271d\u00b9 : \u2200 (a b : \u03b1), toFun\u271d\u00b9 (a \u2294 b) = toFun\u271d\u00b9 a \u2294 toFun\u271d\u00b9 b\nmap_bot'\u271d\u00b9 : SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } \u22a5 = \u22a5\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_sup'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2294 b) = toFun\u271d a \u2294 toFun\u271d b\nmap_bot'\u271d : SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } \u22a5 = \u22a5\nh :\n (fun f => f.toFun) { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_bot' := map_bot'\u271d\u00b9 } =\n (fun f => f.toFun) { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_bot' := map_bot'\u271d }\n\u22a2 { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_bot' := map_bot'\u271d\u00b9 } =\n { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_bot' := map_bot'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u2077 : Sup \u03b1\ninst\u271d\u2076 : Bot \u03b1\ninst\u271d\u2075 : Sup \u03b2\ninst\u271d\u2074 : Bot \u03b2\ninst\u271d\u00b3 : Sup \u03b3\ninst\u271d\u00b2 : Bot \u03b3\ninst\u271d\u00b9 : Sup \u03b4\ninst\u271d : Bot \u03b4\ng : SupBotHom \u03b2 \u03b3\nf\u2081 f\u2082 : SupBotHom \u03b1 \u03b2\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\na : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a)\n[PROOFSTEP]\nrw [\u2190 comp_apply, h, comp_apply]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u2077 : Inf \u03b1\ninst\u271d\u2076 : Top \u03b1\ninst\u271d\u2075 : Inf \u03b2\ninst\u271d\u2074 : Top \u03b2\ninst\u271d\u00b3 : Inf \u03b3\ninst\u271d\u00b2 : Top \u03b3\ninst\u271d\u00b9 : Inf \u03b4\ninst\u271d : Top \u03b4\nf g : InfTopHom \u03b1 \u03b2\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\nobtain \u27e8\u27e8_, _\u27e9, _\u27e9 := f\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u2077 : Inf \u03b1\ninst\u271d\u2076 : Top \u03b1\ninst\u271d\u2075 : Inf \u03b2\ninst\u271d\u2074 : Top \u03b2\ninst\u271d\u00b3 : Inf \u03b3\ninst\u271d\u00b2 : Top \u03b3\ninst\u271d\u00b9 : Inf \u03b4\ninst\u271d : Top \u03b4\ng : InfTopHom \u03b1 \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_inf'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2293 b) = toFun\u271d a \u2293 toFun\u271d b\nmap_top'\u271d : InfHom.toFun { toFun := toFun\u271d, map_inf' := map_inf'\u271d } \u22a4 = \u22a4\nh :\n (fun f => f.toFun) { toInfHom := { toFun := toFun\u271d, map_inf' := map_inf'\u271d }, map_top' := map_top'\u271d } =\n (fun f => f.toFun) g\n\u22a2 { toInfHom := { toFun := toFun\u271d, map_inf' := map_inf'\u271d }, map_top' := map_top'\u271d } = g\n[PROOFSTEP]\nobtain \u27e8\u27e8_, _\u27e9, _\u27e9 := g\n[GOAL]\ncase mk.mk.mk.mk\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u2077 : Inf \u03b1\ninst\u271d\u2076 : Top \u03b1\ninst\u271d\u2075 : Inf \u03b2\ninst\u271d\u2074 : Top \u03b2\ninst\u271d\u00b3 : Inf \u03b3\ninst\u271d\u00b2 : Top \u03b3\ninst\u271d\u00b9 : Inf \u03b4\ninst\u271d : Top \u03b4\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmap_inf'\u271d\u00b9 : \u2200 (a b : \u03b1), toFun\u271d\u00b9 (a \u2293 b) = toFun\u271d\u00b9 a \u2293 toFun\u271d\u00b9 b\nmap_top'\u271d\u00b9 : InfHom.toFun { toFun := toFun\u271d\u00b9, map_inf' := map_inf'\u271d\u00b9 } \u22a4 = \u22a4\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_inf'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2293 b) = toFun\u271d a \u2293 toFun\u271d b\nmap_top'\u271d : InfHom.toFun { toFun := toFun\u271d, map_inf' := map_inf'\u271d } \u22a4 = \u22a4\nh :\n (fun f => f.toFun) { toInfHom := { toFun := toFun\u271d\u00b9, map_inf' := map_inf'\u271d\u00b9 }, map_top' := map_top'\u271d\u00b9 } =\n (fun f => f.toFun) { toInfHom := { toFun := toFun\u271d, map_inf' := map_inf'\u271d }, map_top' := map_top'\u271d }\n\u22a2 { toInfHom := { toFun := toFun\u271d\u00b9, map_inf' := map_inf'\u271d\u00b9 }, map_top' := map_top'\u271d\u00b9 } =\n { toInfHom := { toFun := toFun\u271d, map_inf' := map_inf'\u271d }, map_top' := map_top'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u2077 : Inf \u03b1\ninst\u271d\u2076 : Top \u03b1\ninst\u271d\u2075 : Inf \u03b2\ninst\u271d\u2074 : Top \u03b2\ninst\u271d\u00b3 : Inf \u03b3\ninst\u271d\u00b2 : Top \u03b3\ninst\u271d\u00b9 : Inf \u03b4\ninst\u271d : Top \u03b4\ng : InfTopHom \u03b2 \u03b3\nf\u2081 f\u2082 : InfTopHom \u03b1 \u03b2\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\na : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a)\n[PROOFSTEP]\nrw [\u2190 comp_apply, h, comp_apply]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Lattice \u03b2\ninst\u271d\u00b9 : Lattice \u03b3\ninst\u271d : Lattice \u03b4\nf g : LatticeHom \u03b1 \u03b2\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\nobtain \u27e8\u27e8_, _\u27e9, _\u27e9 := f\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Lattice \u03b2\ninst\u271d\u00b9 : Lattice \u03b3\ninst\u271d : Lattice \u03b4\ng : LatticeHom \u03b1 \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_sup'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2294 b) = toFun\u271d a \u2294 toFun\u271d b\nmap_inf'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } (a \u2293 b) =\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } a \u2293\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } b\nh :\n (fun f => f.toFun) { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d } =\n (fun f => f.toFun) g\n\u22a2 { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d } = g\n[PROOFSTEP]\nobtain \u27e8\u27e8_, _\u27e9, _\u27e9 := g\n[GOAL]\ncase mk.mk.mk.mk\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Lattice \u03b2\ninst\u271d\u00b9 : Lattice \u03b3\ninst\u271d : Lattice \u03b4\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmap_sup'\u271d\u00b9 : \u2200 (a b : \u03b1), toFun\u271d\u00b9 (a \u2294 b) = toFun\u271d\u00b9 a \u2294 toFun\u271d\u00b9 b\nmap_inf'\u271d\u00b9 :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } (a \u2293 b) =\n SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } a \u2293\n SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } b\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_sup'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2294 b) = toFun\u271d a \u2294 toFun\u271d b\nmap_inf'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } (a \u2293 b) =\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } a \u2293\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } b\nh :\n (fun f => f.toFun) { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 } =\n (fun f => f.toFun) { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }\n\u22a2 { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 } =\n { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Lattice \u03b2\ninst\u271d\u00b9 : Lattice \u03b3\ninst\u271d : Lattice \u03b4\ng : LatticeHom \u03b2 \u03b3\nf\u2081 f\u2082 : LatticeHom \u03b1 \u03b2\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\na : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a)\n[PROOFSTEP]\nrw [\u2190 LatticeHom.comp_apply, h, LatticeHom.comp_apply]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : Lattice \u03b2\ninst\u271d : OrderHomClass F \u03b1 \u03b2\nsrc\u271d : OrderHomClass F \u03b1 \u03b2 := inst\u271d\nf : F\na b : \u03b1\n\u22a2 \u2191f (a \u2294 b) = \u2191f a \u2294 \u2191f b\n[PROOFSTEP]\nobtain h | h := le_total a b\n[GOAL]\ncase inl\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : Lattice \u03b2\ninst\u271d : OrderHomClass F \u03b1 \u03b2\nsrc\u271d : OrderHomClass F \u03b1 \u03b2 := inst\u271d\nf : F\na b : \u03b1\nh : a \u2264 b\n\u22a2 \u2191f (a \u2294 b) = \u2191f a \u2294 \u2191f b\n[PROOFSTEP]\nrw [sup_eq_right.2 h, sup_eq_right.2 (OrderHomClass.mono f h : f a \u2264 f b)]\n[GOAL]\ncase inr\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : Lattice \u03b2\ninst\u271d : OrderHomClass F \u03b1 \u03b2\nsrc\u271d : OrderHomClass F \u03b1 \u03b2 := inst\u271d\nf : F\na b : \u03b1\nh : b \u2264 a\n\u22a2 \u2191f (a \u2294 b) = \u2191f a \u2294 \u2191f b\n[PROOFSTEP]\nrw [sup_eq_left.2 h, sup_eq_left.2 (OrderHomClass.mono f h : f b \u2264 f a)]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : Lattice \u03b2\ninst\u271d : OrderHomClass F \u03b1 \u03b2\nsrc\u271d : OrderHomClass F \u03b1 \u03b2 := inst\u271d\nf : F\na b : \u03b1\n\u22a2 \u2191f (a \u2293 b) = \u2191f a \u2293 \u2191f b\n[PROOFSTEP]\nobtain h | h := le_total a b\n[GOAL]\ncase inl\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : Lattice \u03b2\ninst\u271d : OrderHomClass F \u03b1 \u03b2\nsrc\u271d : OrderHomClass F \u03b1 \u03b2 := inst\u271d\nf : F\na b : \u03b1\nh : a \u2264 b\n\u22a2 \u2191f (a \u2293 b) = \u2191f a \u2293 \u2191f b\n[PROOFSTEP]\nrw [inf_eq_left.2 h, inf_eq_left.2 (OrderHomClass.mono f h : f a \u2264 f b)]\n[GOAL]\ncase inr\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : Lattice \u03b2\ninst\u271d : OrderHomClass F \u03b1 \u03b2\nsrc\u271d : OrderHomClass F \u03b1 \u03b2 := inst\u271d\nf : F\na b : \u03b1\nh : b \u2264 a\n\u22a2 \u2191f (a \u2293 b) = \u2191f a \u2293 \u2191f b\n[PROOFSTEP]\nrw [inf_eq_right.2 h, inf_eq_right.2 (OrderHomClass.mono f h : f b \u2264 f a)]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u2077 : Lattice \u03b1\ninst\u271d\u2076 : Lattice \u03b2\ninst\u271d\u2075 : Lattice \u03b3\ninst\u271d\u2074 : Lattice \u03b4\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : BoundedOrder \u03b2\ninst\u271d\u00b9 : BoundedOrder \u03b3\ninst\u271d : BoundedOrder \u03b4\nf g : BoundedLatticeHom \u03b1 \u03b2\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8_, _\u27e9, _\u27e9, _\u27e9 := f\n[GOAL]\ncase mk.mk.mk\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u2077 : Lattice \u03b1\ninst\u271d\u2076 : Lattice \u03b2\ninst\u271d\u2075 : Lattice \u03b3\ninst\u271d\u2074 : Lattice \u03b4\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : BoundedOrder \u03b2\ninst\u271d\u00b9 : BoundedOrder \u03b3\ninst\u271d : BoundedOrder \u03b4\ng : BoundedLatticeHom \u03b1 \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_sup'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2294 b) = toFun\u271d a \u2294 toFun\u271d b\nmap_inf'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } (a \u2293 b) =\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } a \u2293\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } b\nmap_top'\u271d :\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom \u22a4 = \u22a4\nmap_bot'\u271d :\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom \u22a5 = \u22a5\nh :\n (fun f => f.toFun)\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_top' := map_top'\u271d, map_bot' := map_bot'\u271d } =\n (fun f => f.toFun) g\n\u22a2 { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_top' := map_top'\u271d, map_bot' := map_bot'\u271d } =\n g\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8_, _\u27e9, _\u27e9, _\u27e9 := g\n[GOAL]\ncase mk.mk.mk.mk.mk.mk\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u2077 : Lattice \u03b1\ninst\u271d\u2076 : Lattice \u03b2\ninst\u271d\u2075 : Lattice \u03b3\ninst\u271d\u2074 : Lattice \u03b4\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : BoundedOrder \u03b2\ninst\u271d\u00b9 : BoundedOrder \u03b3\ninst\u271d : BoundedOrder \u03b4\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmap_sup'\u271d\u00b9 : \u2200 (a b : \u03b1), toFun\u271d\u00b9 (a \u2294 b) = toFun\u271d\u00b9 a \u2294 toFun\u271d\u00b9 b\nmap_inf'\u271d\u00b9 :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } (a \u2293 b) =\n SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } a \u2293\n SupHom.toFun { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 } b\nmap_top'\u271d\u00b9 :\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom \u22a4 = \u22a4\nmap_bot'\u271d\u00b9 :\n SupHom.toFun { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 }.toSupHom \u22a5 = \u22a5\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_sup'\u271d : \u2200 (a b : \u03b1), toFun\u271d (a \u2294 b) = toFun\u271d a \u2294 toFun\u271d b\nmap_inf'\u271d :\n \u2200 (a b : \u03b1),\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } (a \u2293 b) =\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } a \u2293\n SupHom.toFun { toFun := toFun\u271d, map_sup' := map_sup'\u271d } b\nmap_top'\u271d :\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom \u22a4 = \u22a4\nmap_bot'\u271d :\n SupHom.toFun { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d }.toSupHom \u22a5 = \u22a5\nh :\n (fun f => f.toFun)\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 },\n map_top' := map_top'\u271d\u00b9, map_bot' := map_bot'\u271d\u00b9 } =\n (fun f => f.toFun)\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_top' := map_top'\u271d, map_bot' := map_bot'\u271d }\n\u22a2 { toLatticeHom := { toSupHom := { toFun := toFun\u271d\u00b9, map_sup' := map_sup'\u271d\u00b9 }, map_inf' := map_inf'\u271d\u00b9 },\n map_top' := map_top'\u271d\u00b9, map_bot' := map_bot'\u271d\u00b9 } =\n { toLatticeHom := { toSupHom := { toFun := toFun\u271d, map_sup' := map_sup'\u271d }, map_inf' := map_inf'\u271d },\n map_top' := map_top'\u271d, map_bot' := map_bot'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u2077 : Lattice \u03b1\ninst\u271d\u2076 : Lattice \u03b2\ninst\u271d\u2075 : Lattice \u03b3\ninst\u271d\u2074 : Lattice \u03b4\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : BoundedOrder \u03b2\ninst\u271d\u00b9 : BoundedOrder \u03b3\ninst\u271d : BoundedOrder \u03b4\ng : BoundedLatticeHom \u03b2 \u03b3\nf\u2081 f\u2082 : BoundedLatticeHom \u03b1 \u03b2\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\na : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a)\n[PROOFSTEP]\nrw [\u2190 comp_apply, h, comp_apply]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Lattice \u03b2\ninst\u271d : Lattice \u03b3\n\u22a2 (fun f => \u2191f) (withTopWithBot (LatticeHom.id \u03b1)) = (fun f => \u2191f) (BoundedLatticeHom.id (WithTop (WithBot \u03b1)))\n[PROOFSTEP]\nrefine' (congr_arg Option.map _).trans Option.map_id\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Lattice \u03b2\ninst\u271d : Lattice \u03b3\n\u22a2 (fun x => \u2191(LatticeHom.withBot (LatticeHom.id \u03b1)).toSupHom x) = id\n[PROOFSTEP]\nrw [withBot_id]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Lattice \u03b2\ninst\u271d : Lattice \u03b3\n\u22a2 (fun x => \u2191(LatticeHom.id (WithBot \u03b1)).toSupHom x) = id\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Lattice \u03b2\ninst\u271d : Lattice \u03b3\nf : LatticeHom \u03b2 \u03b3\ng : LatticeHom \u03b1 \u03b2\n\u22a2 withTopWithBot (comp f g) = BoundedLatticeHom.comp (withTopWithBot f) (withTopWithBot g)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Lattice \u03b2\ninst\u271d : Lattice \u03b3\nf : LatticeHom \u03b2 \u03b3\ng : LatticeHom \u03b1 \u03b2\na\u271d : WithTop (WithBot \u03b1)\n\u22a2 \u2191(withTopWithBot (comp f g)) a\u271d = \u2191(BoundedLatticeHom.comp (withTopWithBot f) (withTopWithBot g)) a\u271d\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Order.Hom.Lattice", "llama_tokens": 12524, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506526772883, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.4450575748269225}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : SemilatticeSup \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 u v : Set \u03b1\na b c : \u03b1\n\u22a2 c \u2208 s \u22bb t \u2194 \u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = c\n[PROOFSTEP]\nsimp [(\u00b7 \u22bb \u00b7)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : SemilatticeSup \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Set \u03b1\na b c : \u03b1\ns t : Set \u03b1\n\u22a2 image2 (fun x x_1 => x \u2294 x_1) s t = s \u22bb t\n[PROOFSTEP]\nhave : (s \u00d7\u02e2 t).image (uncurry (\u00b7 \u2294 \u00b7)) = Set.image2 (fun x x_1 => x \u2294 x_1) s t := by\n simp only [ge_iff_le, image_uncurry_prod]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : SemilatticeSup \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Set \u03b1\na b c : \u03b1\ns t : Set \u03b1\n\u22a2 (uncurry fun x x_1 => x \u2294 x_1) '' s \u00d7\u02e2 t = image2 (fun x x_1 => x \u2294 x_1) s t\n[PROOFSTEP]\nsimp only [ge_iff_le, image_uncurry_prod]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : SemilatticeSup \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Set \u03b1\na b c : \u03b1\ns t : Set \u03b1\nthis : (uncurry fun x x_1 => x \u2294 x_1) '' s \u00d7\u02e2 t = image2 (fun x x_1 => x \u2294 x_1) s t\n\u22a2 image2 (fun x x_1 => x \u2294 x_1) s t = s \u22bb t\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : SemilatticeSup \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Set \u03b1\na b c : \u03b1\ns t : Set \u03b1\nthis : (uncurry fun x x_1 => x \u2294 x_1) '' s \u00d7\u02e2 t = image2 (fun x x_1 => x \u2294 x_1) s t\n\u22a2 (uncurry fun x x_1 => x \u2294 x_1) '' s \u00d7\u02e2 t = s \u22bb t\n[PROOFSTEP]\nexact image_uncurry_prod _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : SemilatticeInf \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 u v : Set \u03b1\na b c : \u03b1\n\u22a2 c \u2208 s \u22bc t \u2194 \u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = c\n[PROOFSTEP]\nsimp [(\u00b7 \u22bc \u00b7)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : SemilatticeInf \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Set \u03b1\na b c : \u03b1\ns t : Set \u03b1\n\u22a2 image2 (fun x x_1 => x \u2293 x_1) s t = s \u22bc t\n[PROOFSTEP]\nhave : (s \u00d7\u02e2 t).image (uncurry (\u00b7 \u2293 \u00b7)) = Set.image2 (fun x x_1 => x \u2293 x_1) s t := by\n simp only [@ge_iff_le, @Set.image_uncurry_prod]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : SemilatticeInf \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Set \u03b1\na b c : \u03b1\ns t : Set \u03b1\n\u22a2 (uncurry fun x x_1 => x \u2293 x_1) '' s \u00d7\u02e2 t = image2 (fun x x_1 => x \u2293 x_1) s t\n[PROOFSTEP]\nsimp only [@ge_iff_le, @Set.image_uncurry_prod]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : SemilatticeInf \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Set \u03b1\na b c : \u03b1\ns t : Set \u03b1\nthis : (uncurry fun x x_1 => x \u2293 x_1) '' s \u00d7\u02e2 t = image2 (fun x x_1 => x \u2293 x_1) s t\n\u22a2 image2 (fun x x_1 => x \u2293 x_1) s t = s \u22bc t\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : SemilatticeInf \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Set \u03b1\na b c : \u03b1\ns t : Set \u03b1\nthis : (uncurry fun x x_1 => x \u2293 x_1) '' s \u00d7\u02e2 t = image2 (fun x x_1 => x \u2293 x_1) s t\n\u22a2 (uncurry fun x x_1 => x \u2293 x_1) '' s \u00d7\u02e2 t = s \u22bc t\n[PROOFSTEP]\nexact image_uncurry_prod _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\n\u22a2 upperClosure (s \u22bb t) = upperClosure s \u2294 upperClosure t\n[PROOFSTEP]\next a\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 a \u2208 \u2191(upperClosure (s \u22bb t)) \u2194 a \u2208 \u2191(upperClosure s \u2294 upperClosure t)\n[PROOFSTEP]\nsimp only [SetLike.mem_coe, mem_upperClosure, Set.mem_sups, exists_and_left, exists_prop, UpperSet.coe_sup,\n Set.mem_inter_iff]\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_1) \u2227 a_1 \u2264 a) \u2194 (\u2203 a_1, a_1 \u2208 s \u2227 a_1 \u2264 a) \u2227 \u2203 a_1, a_1 \u2208 t \u2227 a_1 \u2264 a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.h.mp\n\u03b1 : Type u_1\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_1) \u2227 a_1 \u2264 a) \u2192 (\u2203 a_2, a_2 \u2208 s \u2227 a_2 \u2264 a) \u2227 \u2203 a_2, a_2 \u2208 t \u2227 a_2 \u2264 a\n[PROOFSTEP]\nrintro \u27e8_, \u27e8b, hb, c, hc, rfl\u27e9, ha\u27e9\n[GOAL]\ncase a.h.mp.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 t\nha : b \u2294 c \u2264 a\n\u22a2 (\u2203 a_1, a_1 \u2208 s \u2227 a_1 \u2264 a) \u2227 \u2203 a_1, a_1 \u2208 t \u2227 a_1 \u2264 a\n[PROOFSTEP]\nexact \u27e8\u27e8b, hb, le_sup_left.trans ha\u27e9, c, hc, le_sup_right.trans ha\u27e9\n[GOAL]\ncase a.h.mpr\n\u03b1 : Type u_1\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 ((\u2203 a_1, a_1 \u2208 s \u2227 a_1 \u2264 a) \u2227 \u2203 a_1, a_1 \u2208 t \u2227 a_1 \u2264 a) \u2192 \u2203 a_2, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_2) \u2227 a_2 \u2264 a\n[PROOFSTEP]\nrintro \u27e8\u27e8b, hb, hab\u27e9, c, hc, hac\u27e9\n[GOAL]\ncase a.h.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : SemilatticeSup \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2208 s\nhab : b \u2264 a\nc : \u03b1\nhc : c \u2208 t\nhac : c \u2264 a\n\u22a2 \u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_1) \u2227 a_1 \u2264 a\n[PROOFSTEP]\nexact \u27e8_, \u27e8b, hb, c, hc, rfl\u27e9, sup_le hab hac\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\n\u22a2 lowerClosure (s \u22bc t) = lowerClosure s \u2293 lowerClosure t\n[PROOFSTEP]\next a\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 a \u2208 \u2191(lowerClosure (s \u22bc t)) \u2194 a \u2208 \u2191(lowerClosure s \u2293 lowerClosure t)\n[PROOFSTEP]\nsimp only [SetLike.mem_coe, mem_lowerClosure, Set.mem_infs, exists_and_left, exists_prop, LowerSet.coe_sup,\n Set.mem_inter_iff]\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_1) \u2227 a \u2264 a_1) \u2194 a \u2208 lowerClosure s \u2293 lowerClosure t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.h.mp\n\u03b1 : Type u_1\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_1) \u2227 a \u2264 a_1) \u2192 a \u2208 lowerClosure s \u2293 lowerClosure t\n[PROOFSTEP]\nrintro \u27e8_, \u27e8b, hb, c, hc, rfl\u27e9, ha\u27e9\n[GOAL]\ncase a.h.mp.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 t\nha : a \u2264 b \u2293 c\n\u22a2 a \u2208 lowerClosure s \u2293 lowerClosure t\n[PROOFSTEP]\nexact \u27e8\u27e8b, hb, ha.trans inf_le_left\u27e9, c, hc, ha.trans inf_le_right\u27e9\n[GOAL]\ncase a.h.mpr\n\u03b1 : Type u_1\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 a \u2208 lowerClosure s \u2293 lowerClosure t \u2192 \u2203 a_2, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_2) \u2227 a \u2264 a_2\n[PROOFSTEP]\nrintro \u27e8\u27e8b, hb, hab\u27e9, c, hc, hac\u27e9\n[GOAL]\ncase a.h.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : SemilatticeInf \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2208 s\nhab : a \u2264 b\nc : \u03b1\nhc : c \u2208 t\nhac : a \u2264 c\n\u22a2 \u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_1) \u2227 a \u2264 a_1\n[PROOFSTEP]\nexact \u27e8_, \u27e8b, hb, c, hc, rfl\u27e9, le_inf hab hac\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Sups", "llama_tokens": 3351, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584174871563662, "lm_q2_score": 0.6757645879592642, "lm_q1q2_score": 0.4449352219133959}} {"text": "[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\n\u22a2 \u2203 T,\n Set.Countable T \u2227\n (\u2200 (u : Set X), u \u2208 T \u2192 IsOpen u) \u2227\n s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u \u2227 \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\n[PROOFSTEP]\nhave : \u2200 x : s, \u2203 u, IsOpen u \u2227 x.1 \u2208 u \u2227 IntegrableOn f (u \u2229 s) \u03bc :=\n by\n rintro \u27e8x, hx\u27e9\n rcases hf x hx with \u27e8t, ht, h't\u27e9\n rcases mem_nhdsWithin.1 ht with \u27e8u, u_open, x_mem, u_sub\u27e9\n refine' \u27e8u, u_open, x_mem, h't.mono_set u_sub\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\n\u22a2 \u2200 (x : \u2191s), \u2203 u, IsOpen u \u2227 \u2191x \u2208 u \u2227 IntegrableOn f (u \u2229 s)\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase mk\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nx : X\nhx : x \u2208 s\n\u22a2 \u2203 u, IsOpen u \u2227 \u2191{ val := x, property := hx } \u2208 u \u2227 IntegrableOn f (u \u2229 s)\n[PROOFSTEP]\nrcases hf x hx with \u27e8t, ht, h't\u27e9\n[GOAL]\ncase mk.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nx : X\nhx : x \u2208 s\nt : Set X\nht : t \u2208 \ud835\udcdd[s] x\nh't : IntegrableOn f t\n\u22a2 \u2203 u, IsOpen u \u2227 \u2191{ val := x, property := hx } \u2208 u \u2227 IntegrableOn f (u \u2229 s)\n[PROOFSTEP]\nrcases mem_nhdsWithin.1 ht with \u27e8u, u_open, x_mem, u_sub\u27e9\n[GOAL]\ncase mk.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nx : X\nhx : x \u2208 s\nt : Set X\nht : t \u2208 \ud835\udcdd[s] x\nh't : IntegrableOn f t\nu : Set X\nu_open : IsOpen u\nx_mem : x \u2208 u\nu_sub : u \u2229 s \u2286 t\n\u22a2 \u2203 u, IsOpen u \u2227 \u2191{ val := x, property := hx } \u2208 u \u2227 IntegrableOn f (u \u2229 s)\n[PROOFSTEP]\nrefine' \u27e8u, u_open, x_mem, h't.mono_set u_sub\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nthis : \u2200 (x : \u2191s), \u2203 u, IsOpen u \u2227 \u2191x \u2208 u \u2227 IntegrableOn f (u \u2229 s)\n\u22a2 \u2203 T,\n Set.Countable T \u2227\n (\u2200 (u : Set X), u \u2208 T \u2192 IsOpen u) \u2227\n s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u \u2227 \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\n[PROOFSTEP]\nchoose u u_open xu hu using this\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\n\u22a2 \u2203 T,\n Set.Countable T \u2227\n (\u2200 (u : Set X), u \u2208 T \u2192 IsOpen u) \u2227\n s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u \u2227 \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\n[PROOFSTEP]\nobtain \u27e8T, T_count, hT\u27e9 : \u2203 T : Set s, T.Countable \u2227 s \u2286 \u22c3 i \u2208 T, u i :=\n by\n have : s \u2286 \u22c3 x : s, u x := fun y hy => mem_iUnion_of_mem \u27e8y, hy\u27e9 (xu \u27e8y, hy\u27e9)\n obtain \u27e8T, hT_count, hT_un\u27e9 := isOpen_iUnion_countable u u_open\n exact \u27e8T, hT_count, by rwa [hT_un]\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\n\u22a2 \u2203 T, Set.Countable T \u2227 s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 T), u i\n[PROOFSTEP]\nhave : s \u2286 \u22c3 x : s, u x := fun y hy => mem_iUnion_of_mem \u27e8y, hy\u27e9 (xu \u27e8y, hy\u27e9)\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nthis : s \u2286 \u22c3 (x : \u2191s), u x\n\u22a2 \u2203 T, Set.Countable T \u2227 s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 T), u i\n[PROOFSTEP]\nobtain \u27e8T, hT_count, hT_un\u27e9 := isOpen_iUnion_countable u u_open\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nthis : s \u2286 \u22c3 (x : \u2191s), u x\nT : Set \u2191s\nhT_count : Set.Countable T\nhT_un : \u22c3 (i : \u2191s) (_ : i \u2208 T), u i = \u22c3 (i : \u2191s), u i\n\u22a2 \u2203 T, Set.Countable T \u2227 s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 T), u i\n[PROOFSTEP]\nexact \u27e8T, hT_count, by rwa [hT_un]\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nthis : s \u2286 \u22c3 (x : \u2191s), u x\nT : Set \u2191s\nhT_count : Set.Countable T\nhT_un : \u22c3 (i : \u2191s) (_ : i \u2208 T), u i = \u22c3 (i : \u2191s), u i\n\u22a2 s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 T), u i\n[PROOFSTEP]\nrwa [hT_un]\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nT_count : Set.Countable T\nhT : s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 T), u i\n\u22a2 \u2203 T,\n Set.Countable T \u2227\n (\u2200 (u : Set X), u \u2208 T \u2192 IsOpen u) \u2227\n s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u \u2227 \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\n[PROOFSTEP]\nrefine' \u27e8u '' T, T_count.image _, _, by rwa [biUnion_image], _\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nT_count : Set.Countable T\nhT : s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 T), u i\n\u22a2 s \u2286 \u22c3 (u_1 : Set X) (_ : u_1 \u2208 u '' T), u_1\n[PROOFSTEP]\nrwa [biUnion_image]\n[GOAL]\ncase intro.intro.refine'_1\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nT_count : Set.Countable T\nhT : s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 T), u i\n\u22a2 \u2200 (u_1 : Set X), u_1 \u2208 u '' T \u2192 IsOpen u_1\n[PROOFSTEP]\nrintro v \u27e8w, -, rfl\u27e9\n[GOAL]\ncase intro.intro.refine'_1.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nT_count : Set.Countable T\nhT : s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 T), u i\nw : \u2191s\n\u22a2 IsOpen (u w)\n[PROOFSTEP]\nexact u_open _\n[GOAL]\ncase intro.intro.refine'_2\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nT_count : Set.Countable T\nhT : s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 T), u i\n\u22a2 \u2200 (u_1 : Set X), u_1 \u2208 u '' T \u2192 IntegrableOn f (u_1 \u2229 s)\n[PROOFSTEP]\nrintro v \u27e8w, -, rfl\u27e9\n[GOAL]\ncase intro.intro.refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nT_count : Set.Countable T\nhT : s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 T), u i\nw : \u2191s\n\u22a2 IntegrableOn f (u w \u2229 s)\n[PROOFSTEP]\nexact hu _\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\n\u22a2 \u2203 u, (\u2200 (n : \u2115), IsOpen (u n)) \u2227 s \u2286 \u22c3 (n : \u2115), u n \u2227 \u2200 (n : \u2115), IntegrableOn f (u n \u2229 s)\n[PROOFSTEP]\nrcases hf.exists_countable_integrableOn with \u27e8T, T_count, T_open, sT, hT\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\n\u22a2 \u2203 u, (\u2200 (n : \u2115), IsOpen (u n)) \u2227 s \u2286 \u22c3 (n : \u2115), u n \u2227 \u2200 (n : \u2115), IntegrableOn f (u n \u2229 s)\n[PROOFSTEP]\nlet T' : Set (Set X) := insert \u2205 T\n[GOAL]\ncase intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\n\u22a2 \u2203 u, (\u2200 (n : \u2115), IsOpen (u n)) \u2227 s \u2286 \u22c3 (n : \u2115), u n \u2227 \u2200 (n : \u2115), IntegrableOn f (u n \u2229 s)\n[PROOFSTEP]\nhave T'_count : T'.Countable := Countable.insert \u2205 T_count\n[GOAL]\ncase intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\n\u22a2 \u2203 u, (\u2200 (n : \u2115), IsOpen (u n)) \u2227 s \u2286 \u22c3 (n : \u2115), u n \u2227 \u2200 (n : \u2115), IntegrableOn f (u n \u2229 s)\n[PROOFSTEP]\nhave T'_ne : T'.Nonempty := by simp only [insert_nonempty]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\n\u22a2 Set.Nonempty T'\n[PROOFSTEP]\nsimp only [insert_nonempty]\n[GOAL]\ncase intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\n\u22a2 \u2203 u, (\u2200 (n : \u2115), IsOpen (u n)) \u2227 s \u2286 \u22c3 (n : \u2115), u n \u2227 \u2200 (n : \u2115), IntegrableOn f (u n \u2229 s)\n[PROOFSTEP]\nrcases T'_count.exists_eq_range T'_ne with \u27e8u, hu\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\n\u22a2 \u2203 u, (\u2200 (n : \u2115), IsOpen (u n)) \u2227 s \u2286 \u22c3 (n : \u2115), u n \u2227 \u2200 (n : \u2115), IntegrableOn f (u n \u2229 s)\n[PROOFSTEP]\nrefine' \u27e8u, _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\n\u22a2 \u2200 (n : \u2115), IsOpen (u n)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nn : \u2115\n\u22a2 IsOpen (u n)\n[PROOFSTEP]\nhave : u n \u2208 T' := by rw [hu]; exact mem_range_self n\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nn : \u2115\n\u22a2 u n \u2208 T'\n[PROOFSTEP]\nrw [hu]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nn : \u2115\n\u22a2 u n \u2208 range u\n[PROOFSTEP]\nexact mem_range_self n\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nn : \u2115\nthis : u n \u2208 T'\n\u22a2 IsOpen (u n)\n[PROOFSTEP]\nrcases mem_insert_iff.1 this with h | h\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.inl\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nn : \u2115\nthis : u n \u2208 T'\nh : u n = \u2205\n\u22a2 IsOpen (u n)\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.inl\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nn : \u2115\nthis : u n \u2208 T'\nh : u n = \u2205\n\u22a2 IsOpen \u2205\n[PROOFSTEP]\nexact isOpen_empty\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.inr\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nn : \u2115\nthis : u n \u2208 T'\nh : u n \u2208 T\n\u22a2 IsOpen (u n)\n[PROOFSTEP]\nexact T_open _ h\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\n\u22a2 s \u2286 \u22c3 (n : \u2115), u n\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nx : X\nhx : x \u2208 s\n\u22a2 x \u2208 \u22c3 (n : \u2115), u n\n[PROOFSTEP]\nobtain \u27e8v, hv, h'v\u27e9 : \u2203 v, v \u2208 T \u2227 x \u2208 v := by simpa only [mem_iUnion, exists_prop] using sT hx\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nx : X\nhx : x \u2208 s\n\u22a2 \u2203 v, v \u2208 T \u2227 x \u2208 v\n[PROOFSTEP]\nsimpa only [mem_iUnion, exists_prop] using sT hx\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nx : X\nhx : x \u2208 s\nv : Set X\nhv : v \u2208 T\nh'v : x \u2208 v\n\u22a2 x \u2208 \u22c3 (n : \u2115), u n\n[PROOFSTEP]\nhave : v \u2208 range u := by rw [\u2190 hu]; exact subset_insert \u2205 T hv\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nx : X\nhx : x \u2208 s\nv : Set X\nhv : v \u2208 T\nh'v : x \u2208 v\n\u22a2 v \u2208 range u\n[PROOFSTEP]\nrw [\u2190 hu]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nx : X\nhx : x \u2208 s\nv : Set X\nhv : v \u2208 T\nh'v : x \u2208 v\n\u22a2 v \u2208 T'\n[PROOFSTEP]\nexact subset_insert \u2205 T hv\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nx : X\nhx : x \u2208 s\nv : Set X\nhv : v \u2208 T\nh'v : x \u2208 v\nthis : v \u2208 range u\n\u22a2 x \u2208 \u22c3 (n : \u2115), u n\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 : \u2203 n, u n = v := by simpa only [mem_range] using this\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nx : X\nhx : x \u2208 s\nv : Set X\nhv : v \u2208 T\nh'v : x \u2208 v\nthis : v \u2208 range u\n\u22a2 \u2203 n, u n = v\n[PROOFSTEP]\nsimpa only [mem_range] using this\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nx : X\nhx : x \u2208 s\nn : \u2115\nhv : u n \u2208 T\nh'v : x \u2208 u n\nthis : u n \u2208 range u\n\u22a2 x \u2208 \u22c3 (n : \u2115), u n\n[PROOFSTEP]\nexact mem_iUnion_of_mem _ h'v\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\n\u22a2 \u2200 (n : \u2115), IntegrableOn f (u n \u2229 s)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nn : \u2115\n\u22a2 IntegrableOn f (u n \u2229 s)\n[PROOFSTEP]\nhave : u n \u2208 T' := by rw [hu]; exact mem_range_self n\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nn : \u2115\n\u22a2 u n \u2208 T'\n[PROOFSTEP]\nrw [hu]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nn : \u2115\n\u22a2 u n \u2208 range u\n[PROOFSTEP]\nexact mem_range_self n\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nn : \u2115\nthis : u n \u2208 T'\n\u22a2 IntegrableOn f (u n \u2229 s)\n[PROOFSTEP]\nrcases mem_insert_iff.1 this with h | h\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3.inl\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nn : \u2115\nthis : u n \u2208 T'\nh : u n = \u2205\n\u22a2 IntegrableOn f (u n \u2229 s)\n[PROOFSTEP]\nsimp only [h, empty_inter, integrableOn_empty]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3.inr\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nT : Set (Set X)\nT_count : Set.Countable T\nT_open : \u2200 (u : Set X), u \u2208 T \u2192 IsOpen u\nsT : s \u2286 \u22c3 (u : Set X) (_ : u \u2208 T), u\nhT : \u2200 (u : Set X), u \u2208 T \u2192 IntegrableOn f (u \u2229 s)\nT' : Set (Set X) := insert \u2205 T\nT'_count : Set.Countable T'\nT'_ne : Set.Nonempty T'\nu : \u2115 \u2192 Set X\nhu : T' = range u\nn : \u2115\nthis : u n \u2208 T'\nh : u n \u2208 T\n\u22a2 IntegrableOn f (u n \u2229 s)\n[PROOFSTEP]\nexact hT _ h\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc s)\n[PROOFSTEP]\nrcases hf.exists_nat_integrableOn with \u27e8u, -, su, hu\u27e9\n[GOAL]\ncase intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2115 \u2192 Set X\nsu : s \u2286 \u22c3 (n : \u2115), u n\nhu : \u2200 (n : \u2115), IntegrableOn f (u n \u2229 s)\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc s)\n[PROOFSTEP]\nhave : s = \u22c3 n, u n \u2229 s := by rw [\u2190 iUnion_inter]; exact (inter_eq_right_iff_subset.mpr su).symm\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2115 \u2192 Set X\nsu : s \u2286 \u22c3 (n : \u2115), u n\nhu : \u2200 (n : \u2115), IntegrableOn f (u n \u2229 s)\n\u22a2 s = \u22c3 (n : \u2115), u n \u2229 s\n[PROOFSTEP]\nrw [\u2190 iUnion_inter]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2115 \u2192 Set X\nsu : s \u2286 \u22c3 (n : \u2115), u n\nhu : \u2200 (n : \u2115), IntegrableOn f (u n \u2229 s)\n\u22a2 s = (\u22c3 (i : \u2115), u i) \u2229 s\n[PROOFSTEP]\nexact (inter_eq_right_iff_subset.mpr su).symm\n[GOAL]\ncase intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2115 \u2192 Set X\nsu : s \u2286 \u22c3 (n : \u2115), u n\nhu : \u2200 (n : \u2115), IntegrableOn f (u n \u2229 s)\nthis : s = \u22c3 (n : \u2115), u n \u2229 s\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc s)\n[PROOFSTEP]\nrw [this, aestronglyMeasurable_iUnion_iff]\n[GOAL]\ncase intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2115 \u2192 Set X\nsu : s \u2286 \u22c3 (n : \u2115), u n\nhu : \u2200 (n : \u2115), IntegrableOn f (u n \u2229 s)\nthis : s = \u22c3 (n : \u2115), u n \u2229 s\n\u22a2 \u2200 (i : \u2115), AEStronglyMeasurable f (Measure.restrict \u03bc (u i \u2229 s))\n[PROOFSTEP]\nexact fun i : \u2115 => (hu i).aestronglyMeasurable\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nhs : IsClosed s \u2228 IsOpen s\n\u22a2 LocallyIntegrableOn f s \u2194 \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k\n[PROOFSTEP]\nrefine' \u27e8fun hf k hk => hf.integrableOn_compact_subset hk, fun hf x hx => _\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nhs : IsClosed s \u2228 IsOpen s\nhf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k\nx : X\nhx : x \u2208 s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x)\n[PROOFSTEP]\ncases hs with\n| inl hs =>\n exact\n let \u27e8K, hK, h2K\u27e9 := exists_compact_mem_nhds x\n \u27e8_, inter_mem_nhdsWithin s h2K,\n hf _ (inter_subset_left _ _) (isCompact_of_isClosed_subset hK (hs.inter hK.isClosed) (inter_subset_right _ _))\u27e9\n| inr hs =>\n obtain \u27e8K, hK, h2K, h3K\u27e9 := exists_compact_subset hs hx\n refine' \u27e8K, _, hf K h3K hK\u27e9\n simpa only [IsOpen.nhdsWithin_eq hs hx, interior_eq_nhds'] using h2K\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nhs : IsClosed s \u2228 IsOpen s\nhf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k\nx : X\nhx : x \u2208 s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x)\n[PROOFSTEP]\ncases hs with\n| inl hs =>\n exact\n let \u27e8K, hK, h2K\u27e9 := exists_compact_mem_nhds x\n \u27e8_, inter_mem_nhdsWithin s h2K,\n hf _ (inter_subset_left _ _) (isCompact_of_isClosed_subset hK (hs.inter hK.isClosed) (inter_subset_right _ _))\u27e9\n| inr hs =>\n obtain \u27e8K, hK, h2K, h3K\u27e9 := exists_compact_subset hs hx\n refine' \u27e8K, _, hf K h3K hK\u27e9\n simpa only [IsOpen.nhdsWithin_eq hs hx, interior_eq_nhds'] using h2K\n[GOAL]\ncase inl\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nhf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k\nx : X\nhx : x \u2208 s\nhs : IsClosed s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x)\n[PROOFSTEP]\n\n| inl hs =>\n exact\n let \u27e8K, hK, h2K\u27e9 := exists_compact_mem_nhds x\n \u27e8_, inter_mem_nhdsWithin s h2K,\n hf _ (inter_subset_left _ _) (isCompact_of_isClosed_subset hK (hs.inter hK.isClosed) (inter_subset_right _ _))\u27e9\n[GOAL]\ncase inl\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nhf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k\nx : X\nhx : x \u2208 s\nhs : IsClosed s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x)\n[PROOFSTEP]\nexact\n let \u27e8K, hK, h2K\u27e9 := exists_compact_mem_nhds x\n \u27e8_, inter_mem_nhdsWithin s h2K,\n hf _ (inter_subset_left _ _) (isCompact_of_isClosed_subset hK (hs.inter hK.isClosed) (inter_subset_right _ _))\u27e9\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nhf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k\nx : X\nhx : x \u2208 s\nhs : IsOpen s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x)\n[PROOFSTEP]\n\n| inr hs =>\n obtain \u27e8K, hK, h2K, h3K\u27e9 := exists_compact_subset hs hx\n refine' \u27e8K, _, hf K h3K hK\u27e9\n simpa only [IsOpen.nhdsWithin_eq hs hx, interior_eq_nhds'] using h2K\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nhf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k\nx : X\nhx : x \u2208 s\nhs : IsOpen s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x)\n[PROOFSTEP]\nobtain \u27e8K, hK, h2K, h3K\u27e9 := exists_compact_subset hs hx\n[GOAL]\ncase inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nhf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k\nx : X\nhx : x \u2208 s\nhs : IsOpen s\nK : Set X\nhK : IsCompact K\nh2K : x \u2208 interior K\nh3K : K \u2286 s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x)\n[PROOFSTEP]\nrefine' \u27e8K, _, hf K h3K hK\u27e9\n[GOAL]\ncase inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nhf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k\nx : X\nhx : x \u2208 s\nhs : IsOpen s\nK : Set X\nhK : IsCompact K\nh2K : x \u2208 interior K\nh3K : K \u2286 s\n\u22a2 K \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\nsimpa only [IsOpen.nhdsWithin_eq hs hx, interior_eq_nhds'] using h2K\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\n\u22a2 LocallyIntegrableOn f univ \u2194 LocallyIntegrable f\n[PROOFSTEP]\nsimp only [LocallyIntegrableOn, nhdsWithin_univ, mem_univ, true_imp_iff]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\n\u22a2 (\u2200 (x : X), IntegrableAtFilter f (\ud835\udcdd x)) \u2194 LocallyIntegrable f\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhf : LocallyIntegrable f\n\u22a2 LocallyIntegrableOn f s\n[PROOFSTEP]\nintro x _\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhf : LocallyIntegrable f\nx : X\na\u271d : x \u2208 s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x)\n[PROOFSTEP]\nobtain \u27e8t, ht_mem, ht_int\u27e9 := hf x\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhf : LocallyIntegrable f\nx : X\na\u271d : x \u2208 s\nt : Set X\nht_mem : t \u2208 \ud835\udcdd x\nht_int : IntegrableOn f t\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x)\n[PROOFSTEP]\nobtain \u27e8u, hu_sub, hu_o, hu_mem\u27e9 := mem_nhds_iff.mp ht_mem\n[GOAL]\ncase intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhf : LocallyIntegrable f\nx : X\na\u271d : x \u2208 s\nt : Set X\nht_mem : t \u2208 \ud835\udcdd x\nht_int : IntegrableOn f t\nu : Set X\nhu_sub : u \u2286 t\nhu_o : IsOpen u\nhu_mem : x \u2208 u\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[s] x)\n[PROOFSTEP]\nrefine' \u27e8_, inter_mem_nhdsWithin s (hu_o.mem_nhds hu_mem), _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhf : LocallyIntegrable f\nx : X\na\u271d : x \u2208 s\nt : Set X\nht_mem : t \u2208 \ud835\udcdd x\nht_int : IntegrableOn f t\nu : Set X\nhu_sub : u \u2286 t\nhu_o : IsOpen u\nhu_mem : x \u2208 u\n\u22a2 IntegrableOn f (s \u2229 u)\n[PROOFSTEP]\nsimpa only [IntegrableOn, Measure.restrict_restrict hu_o.measurableSet, inter_comm] using ht_int.mono_set hu_sub\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhs : IsClosed s\n\u22a2 LocallyIntegrableOn f s \u2194 LocallyIntegrable f\n[PROOFSTEP]\nrefine' \u27e8fun hf x => _, locallyIntegrableOn_of_locallyIntegrable_restrict\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)\n[PROOFSTEP]\nby_cases h : x \u2208 s\n[GOAL]\ncase pos\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : x \u2208 s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)\n[PROOFSTEP]\nobtain \u27e8t, ht_nhds, ht_int\u27e9 := hf x h\n[GOAL]\ncase pos.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : x \u2208 s\nt : Set X\nht_nhds : t \u2208 \ud835\udcdd[s] x\nht_int : IntegrableOn f t\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)\n[PROOFSTEP]\nobtain \u27e8u, hu_o, hu_x, hu_sub\u27e9 := mem_nhdsWithin.mp ht_nhds\n[GOAL]\ncase pos.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : x \u2208 s\nt : Set X\nht_nhds : t \u2208 \ud835\udcdd[s] x\nht_int : IntegrableOn f t\nu : Set X\nhu_o : IsOpen u\nhu_x : x \u2208 u\nhu_sub : u \u2229 s \u2286 t\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)\n[PROOFSTEP]\nrefine' \u27e8u, hu_o.mem_nhds hu_x, _\u27e9\n[GOAL]\ncase pos.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : x \u2208 s\nt : Set X\nht_nhds : t \u2208 \ud835\udcdd[s] x\nht_int : IntegrableOn f t\nu : Set X\nhu_o : IsOpen u\nhu_x : x \u2208 u\nhu_sub : u \u2229 s \u2286 t\n\u22a2 IntegrableOn f u\n[PROOFSTEP]\nrw [IntegrableOn, restrict_restrict hu_o.measurableSet]\n[GOAL]\ncase pos.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : x \u2208 s\nt : Set X\nht_nhds : t \u2208 \ud835\udcdd[s] x\nht_int : IntegrableOn f t\nu : Set X\nhu_o : IsOpen u\nhu_x : x \u2208 u\nhu_sub : u \u2229 s \u2286 t\n\u22a2 Integrable f\n[PROOFSTEP]\nexact ht_int.mono_set hu_sub\n[GOAL]\ncase neg\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhs : IsClosed s\nhf : LocallyIntegrableOn f s\nx : X\nh : \u00acx \u2208 s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)\n[PROOFSTEP]\nrw [\u2190 isOpen_compl_iff] at hs \n[GOAL]\ncase neg\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhs : IsOpen s\u1d9c\nhf : LocallyIntegrableOn f s\nx : X\nh : \u00acx \u2208 s\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)\n[PROOFSTEP]\nrefine' \u27e8s\u1d9c, hs.mem_nhds h, _\u27e9\n[GOAL]\ncase neg\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhs : IsOpen s\u1d9c\nhf : LocallyIntegrableOn f s\nx : X\nh : \u00acx \u2208 s\n\u22a2 IntegrableOn f s\u1d9c\n[PROOFSTEP]\nrw [IntegrableOn, restrict_restrict, inter_comm, inter_compl_self, \u2190 IntegrableOn]\n[GOAL]\ncase neg\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhs : IsOpen s\u1d9c\nhf : LocallyIntegrableOn f s\nx : X\nh : \u00acx \u2208 s\n\u22a2 IntegrableOn f \u2205\ncase neg\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : OpensMeasurableSpace X\nhs : IsOpen s\u1d9c\nhf : LocallyIntegrableOn f s\nx : X\nh : \u00acx \u2208 s\n\u22a2 MeasurableSet s\u1d9c\n[PROOFSTEP]\nexacts [integrableOn_empty, hs.measurableSet]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\nhf : LocallyIntegrable f\nk : Set X\nhk : IsCompact k\n\u22a2 \u2203 u, IsOpen u \u2227 k \u2286 u \u2227 IntegrableOn f u\n[PROOFSTEP]\nrefine' IsCompact.induction_on hk _ _ _ _\n[GOAL]\ncase refine'_1\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\nhf : LocallyIntegrable f\nk : Set X\nhk : IsCompact k\n\u22a2 \u2203 u, IsOpen u \u2227 \u2205 \u2286 u \u2227 IntegrableOn f u\n[PROOFSTEP]\nrefine' \u27e8\u2205, isOpen_empty, Subset.rfl, integrableOn_empty\u27e9\n[GOAL]\ncase refine'_2\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\nhf : LocallyIntegrable f\nk : Set X\nhk : IsCompact k\n\u22a2 \u2200 \u2983s t : Set X\u2984, s \u2286 t \u2192 (\u2203 u, IsOpen u \u2227 t \u2286 u \u2227 IntegrableOn f u) \u2192 \u2203 u, IsOpen u \u2227 s \u2286 u \u2227 IntegrableOn f u\n[PROOFSTEP]\nrintro s t hst \u27e8u, u_open, tu, hu\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\nhf : LocallyIntegrable f\nk : Set X\nhk : IsCompact k\ns t : Set X\nhst : s \u2286 t\nu : Set X\nu_open : IsOpen u\ntu : t \u2286 u\nhu : IntegrableOn f u\n\u22a2 \u2203 u, IsOpen u \u2227 s \u2286 u \u2227 IntegrableOn f u\n[PROOFSTEP]\nexact \u27e8u, u_open, hst.trans tu, hu\u27e9\n[GOAL]\ncase refine'_3\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\nhf : LocallyIntegrable f\nk : Set X\nhk : IsCompact k\n\u22a2 \u2200 \u2983s t : Set X\u2984,\n (\u2203 u, IsOpen u \u2227 s \u2286 u \u2227 IntegrableOn f u) \u2192\n (\u2203 u, IsOpen u \u2227 t \u2286 u \u2227 IntegrableOn f u) \u2192 \u2203 u, IsOpen u \u2227 s \u222a t \u2286 u \u2227 IntegrableOn f u\n[PROOFSTEP]\nrintro s t \u27e8u, u_open, su, hu\u27e9 \u27e8v, v_open, tv, hv\u27e9\n[GOAL]\ncase refine'_3.intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\nhf : LocallyIntegrable f\nk : Set X\nhk : IsCompact k\ns t u : Set X\nu_open : IsOpen u\nsu : s \u2286 u\nhu : IntegrableOn f u\nv : Set X\nv_open : IsOpen v\ntv : t \u2286 v\nhv : IntegrableOn f v\n\u22a2 \u2203 u, IsOpen u \u2227 s \u222a t \u2286 u \u2227 IntegrableOn f u\n[PROOFSTEP]\nexact \u27e8u \u222a v, u_open.union v_open, union_subset_union su tv, hu.union hv\u27e9\n[GOAL]\ncase refine'_4\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\nhf : LocallyIntegrable f\nk : Set X\nhk : IsCompact k\n\u22a2 \u2200 (x : X), x \u2208 k \u2192 \u2203 t, t \u2208 \ud835\udcdd[k] x \u2227 \u2203 u, IsOpen u \u2227 t \u2286 u \u2227 IntegrableOn f u\n[PROOFSTEP]\nintro x _\n[GOAL]\ncase refine'_4\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\nhf : LocallyIntegrable f\nk : Set X\nhk : IsCompact k\nx : X\na\u271d : x \u2208 k\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[k] x \u2227 \u2203 u, IsOpen u \u2227 t \u2286 u \u2227 IntegrableOn f u\n[PROOFSTEP]\nrcases hf x with \u27e8u, ux, hu\u27e9\n[GOAL]\ncase refine'_4.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\nhf : LocallyIntegrable f\nk : Set X\nhk : IsCompact k\nx : X\na\u271d : x \u2208 k\nu : Set X\nux : u \u2208 \ud835\udcdd x\nhu : IntegrableOn f u\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[k] x \u2227 \u2203 u, IsOpen u \u2227 t \u2286 u \u2227 IntegrableOn f u\n[PROOFSTEP]\nrcases mem_nhds_iff.1 ux with \u27e8v, vu, v_open, xv\u27e9\n[GOAL]\ncase refine'_4.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\nhf : LocallyIntegrable f\nk : Set X\nhk : IsCompact k\nx : X\na\u271d : x \u2208 k\nu : Set X\nux : u \u2208 \ud835\udcdd x\nhu : IntegrableOn f u\nv : Set X\nvu : v \u2286 u\nv_open : IsOpen v\nxv : x \u2208 v\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[k] x \u2227 \u2203 u, IsOpen u \u2227 t \u2286 u \u2227 IntegrableOn f u\n[PROOFSTEP]\nexact \u27e8v, nhdsWithin_le_nhds (v_open.mem_nhds xv), v, v_open, Subset.rfl, hu.mono_set vu\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrable f\n\u22a2 AEStronglyMeasurable f \u03bc\n[PROOFSTEP]\nsimpa only [restrict_univ] using (locallyIntegrableOn_univ.mpr hf).aestronglyMeasurable\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrable f\n\u22a2 \u2203 u, (\u2200 (n : \u2115), IsOpen (u n)) \u2227 \u22c3 (n : \u2115), u n = univ \u2227 \u2200 (n : \u2115), IntegrableOn f (u n)\n[PROOFSTEP]\nrcases(hf.locallyIntegrableOn univ).exists_nat_integrableOn with \u27e8u, u_open, u_union, hu\u27e9\n[GOAL]\ncase intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrable f\nu : \u2115 \u2192 Set X\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_union : univ \u2286 \u22c3 (n : \u2115), u n\nhu : \u2200 (n : \u2115), IntegrableOn f (u n \u2229 univ)\n\u22a2 \u2203 u, (\u2200 (n : \u2115), IsOpen (u n)) \u2227 \u22c3 (n : \u2115), u n = univ \u2227 \u2200 (n : \u2115), IntegrableOn f (u n)\n[PROOFSTEP]\nrefine' \u27e8u, u_open, eq_univ_of_univ_subset u_union, fun n \u21a6 _\u27e9\n[GOAL]\ncase intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrable f\nu : \u2115 \u2192 Set X\nu_open : \u2200 (n : \u2115), IsOpen (u n)\nu_union : univ \u2286 \u22c3 (n : \u2115), u n\nhu : \u2200 (n : \u2115), IntegrableOn f (u n \u2229 univ)\nn : \u2115\n\u22a2 IntegrableOn f (u n)\n[PROOFSTEP]\nsimpa only [inter_univ] using hu n\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nc : E\n\u22a2 LocallyIntegrable fun x => c\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nc : E\nx : X\n\u22a2 IntegrableAtFilter (fun x => c) (\ud835\udcdd x)\n[PROOFSTEP]\nrcases \u03bc.finiteAt_nhds x with \u27e8U, hU, h'U\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nc : E\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\n\u22a2 IntegrableAtFilter (fun x => c) (\ud835\udcdd x)\n[PROOFSTEP]\nrefine' \u27e8U, hU, _\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nc : E\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\n\u22a2 IntegrableOn (fun x => c) U\n[PROOFSTEP]\nsimp only [h'U, integrableOn_const, or_true_iff]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\nhf : LocallyIntegrable f\ns : Set X\nhs : MeasurableSet s\n\u22a2 LocallyIntegrable (Set.indicator s f)\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\nhf : LocallyIntegrable f\ns : Set X\nhs : MeasurableSet s\nx : X\n\u22a2 IntegrableAtFilter (Set.indicator s f) (\ud835\udcdd x)\n[PROOFSTEP]\nrcases hf x with \u27e8U, hU, h'U\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\nhf : LocallyIntegrable f\ns : Set X\nhs : MeasurableSet s\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : IntegrableOn f U\n\u22a2 IntegrableAtFilter (Set.indicator s f) (\ud835\udcdd x)\n[PROOFSTEP]\nexact \u27e8U, hU, h'U.indicator hs\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\n\u22a2 LocallyIntegrable f \u2194 LocallyIntegrable (f \u2218 \u2191e)\n[PROOFSTEP]\nrefine' \u27e8fun h x => _, fun h x => _\u27e9\n[GOAL]\ncase refine'_1\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable f\nx : X\n\u22a2 IntegrableAtFilter (f \u2218 \u2191e) (\ud835\udcdd x)\n[PROOFSTEP]\nrcases h (e x) with \u27e8U, hU, h'U\u27e9\n[GOAL]\ncase refine'_1.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable f\nx : X\nU : Set Y\nhU : U \u2208 \ud835\udcdd (\u2191e x)\nh'U : IntegrableOn f U\n\u22a2 IntegrableAtFilter (f \u2218 \u2191e) (\ud835\udcdd x)\n[PROOFSTEP]\nrefine' \u27e8e \u207b\u00b9' U, e.continuous.continuousAt.preimage_mem_nhds hU, _\u27e9\n[GOAL]\ncase refine'_1.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable f\nx : X\nU : Set Y\nhU : U \u2208 \ud835\udcdd (\u2191e x)\nh'U : IntegrableOn f U\n\u22a2 IntegrableOn (f \u2218 \u2191e) (\u2191e \u207b\u00b9' U)\n[PROOFSTEP]\nexact (integrableOn_map_equiv e.toMeasurableEquiv).1 h'U\n[GOAL]\ncase refine'_2\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)\n[PROOFSTEP]\nrcases h (e.symm x) with \u27e8U, hU, h'U\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x)\nh'U : IntegrableOn (f \u2218 \u2191e) U\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)\n[PROOFSTEP]\nrefine' \u27e8e.symm \u207b\u00b9' U, e.symm.continuous.continuousAt.preimage_mem_nhds hU, _\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x)\nh'U : IntegrableOn (f \u2218 \u2191e) U\n\u22a2 IntegrableOn f (\u2191(Homeomorph.symm e) \u207b\u00b9' U)\n[PROOFSTEP]\napply (integrableOn_map_equiv e.toMeasurableEquiv).2\n[GOAL]\ncase refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x)\nh'U : IntegrableOn (f \u2218 \u2191e) U\n\u22a2 IntegrableOn (f \u2218 \u2191(Homeomorph.toMeasurableEquiv e))\n (\u2191(Homeomorph.toMeasurableEquiv e) \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U))\n[PROOFSTEP]\nsimp only [Homeomorph.toMeasurableEquiv_coe]\n[GOAL]\ncase refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x)\nh'U : IntegrableOn (f \u2218 \u2191e) U\n\u22a2 IntegrableOn (f \u2218 \u2191e) (\u2191e \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U))\n[PROOFSTEP]\nconvert h'U\n[GOAL]\ncase h.e'_6\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x)\nh'U : IntegrableOn (f \u2218 \u2191e) U\n\u22a2 \u2191e \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U) = U\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_6.h\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2076 : MeasurableSpace X\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : MeasurableSpace Y\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc\u271d : Measure X\ns : Set X\ninst\u271d\u00b9 : BorelSpace X\ninst\u271d : BorelSpace Y\ne : X \u2243\u209c Y\nf : Y \u2192 E\n\u03bc : Measure X\nh : LocallyIntegrable (f \u2218 \u2191e)\nx\u271d : Y\nU : Set X\nhU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x\u271d)\nh'U : IntegrableOn (f \u2218 \u2191e) U\nx : X\n\u22a2 x \u2208 \u2191e \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U) \u2194 x \u2208 U\n[PROOFSTEP]\nsimp only [mem_preimage, Homeomorph.symm_apply_apply]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nhf : LocallyIntegrable f\ng : X \u2192 \u211d\nhg : Continuous g\nh'g : HasCompactSupport g\n\u22a2 Integrable fun x => g x \u2022 f x\n[PROOFSTEP]\nlet K := tsupport g\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nhf : LocallyIntegrable f\ng : X \u2192 \u211d\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\n\u22a2 Integrable fun x => g x \u2022 f x\n[PROOFSTEP]\nhave hK : IsCompact K := h'g\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nhf : LocallyIntegrable f\ng : X \u2192 \u211d\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\n\u22a2 Integrable fun x => g x \u2022 f x\n[PROOFSTEP]\nhave : K.indicator (fun x \u21a6 g x \u2022 f x) = (fun x \u21a6 g x \u2022 f x) :=\n by\n apply indicator_eq_self.2\n apply support_subset_iff'.2\n intros x hx\n simp [image_eq_zero_of_nmem_tsupport hx]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nhf : LocallyIntegrable f\ng : X \u2192 \u211d\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\n\u22a2 (Set.indicator K fun x => g x \u2022 f x) = fun x => g x \u2022 f x\n[PROOFSTEP]\napply indicator_eq_self.2\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nhf : LocallyIntegrable f\ng : X \u2192 \u211d\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\n\u22a2 (support fun x => g x \u2022 f x) \u2286 K\n[PROOFSTEP]\napply support_subset_iff'.2\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nhf : LocallyIntegrable f\ng : X \u2192 \u211d\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\n\u22a2 \u2200 (x : X), \u00acx \u2208 K \u2192 g x \u2022 f x = 0\n[PROOFSTEP]\nintros x hx\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nhf : LocallyIntegrable f\ng : X \u2192 \u211d\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nx : X\nhx : \u00acx \u2208 K\n\u22a2 g x \u2022 f x = 0\n[PROOFSTEP]\nsimp [image_eq_zero_of_nmem_tsupport hx]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nhf : LocallyIntegrable f\ng : X \u2192 \u211d\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => g x \u2022 f x) = fun x => g x \u2022 f x\n\u22a2 Integrable fun x => g x \u2022 f x\n[PROOFSTEP]\nrw [\u2190 this, indicator_smul]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nhf : LocallyIntegrable f\ng : X \u2192 \u211d\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => g x \u2022 f x) = fun x => g x \u2022 f x\n\u22a2 Integrable fun x => g x \u2022 Set.indicator K (fun x => f x) x\n[PROOFSTEP]\napply Integrable.smul_of_top_right\n[GOAL]\ncase hf\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nhf : LocallyIntegrable f\ng : X \u2192 \u211d\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => g x \u2022 f x) = fun x => g x \u2022 f x\n\u22a2 Integrable fun x => Set.indicator K (fun x => f x) x\n[PROOFSTEP]\nrw [integrable_indicator_iff hK.measurableSet]\n[GOAL]\ncase hf\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nhf : LocallyIntegrable f\ng : X \u2192 \u211d\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => g x \u2022 f x) = fun x => g x \u2022 f x\n\u22a2 IntegrableOn (fun x => f x) K\n[PROOFSTEP]\nexact hf.integrableOn_isCompact hK\n[GOAL]\ncase h\u03c6\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nhf : LocallyIntegrable f\ng : X \u2192 \u211d\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => g x \u2022 f x) = fun x => g x \u2022 f x\n\u22a2 Mem\u2112p (fun x => g x) \u22a4\n[PROOFSTEP]\nexact hg.mem\u2112p_top_of_hasCompactSupport h'g \u03bc\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\n\u22a2 Integrable fun x => f x \u2022 g x\n[PROOFSTEP]\nlet K := tsupport g\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\n\u22a2 Integrable fun x => f x \u2022 g x\n[PROOFSTEP]\nhave hK : IsCompact K := h'g\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\n\u22a2 Integrable fun x => f x \u2022 g x\n[PROOFSTEP]\nhave : K.indicator (fun x \u21a6 f x \u2022 g x) = (fun x \u21a6 f x \u2022 g x) :=\n by\n apply indicator_eq_self.2\n apply support_subset_iff'.2\n intros x hx\n simp [image_eq_zero_of_nmem_tsupport hx]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\n\u22a2 (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n[PROOFSTEP]\napply indicator_eq_self.2\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\n\u22a2 (support fun x => f x \u2022 g x) \u2286 K\n[PROOFSTEP]\napply support_subset_iff'.2\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\n\u22a2 \u2200 (x : X), \u00acx \u2208 K \u2192 f x \u2022 g x = 0\n[PROOFSTEP]\nintros x hx\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nx : X\nhx : \u00acx \u2208 K\n\u22a2 f x \u2022 g x = 0\n[PROOFSTEP]\nsimp [image_eq_zero_of_nmem_tsupport hx]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n\u22a2 Integrable fun x => f x \u2022 g x\n[PROOFSTEP]\nrw [\u2190 this, indicator_smul_left]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n\u22a2 Integrable fun x => Set.indicator K (fun x => f x) x \u2022 g x\n[PROOFSTEP]\napply Integrable.smul_of_top_left\n[GOAL]\ncase h\u03c6\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n\u22a2 Integrable fun x => Set.indicator K (fun x => f x) x\n[PROOFSTEP]\nrw [integrable_indicator_iff hK.measurableSet]\n[GOAL]\ncase h\u03c6\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n\u22a2 IntegrableOn (fun x => f x) K\n[PROOFSTEP]\nexact hf.integrableOn_isCompact hK\n[GOAL]\ncase hf\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OpensMeasurableSpace X\ninst\u271d : T2Space X\nf : X \u2192 \u211d\nhf : LocallyIntegrable f\ng : X \u2192 E\nhg : Continuous g\nh'g : HasCompactSupport g\nK : Set X := tsupport g\nhK : IsCompact K\nthis : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x\n\u22a2 Mem\u2112p (fun x => g x) \u22a4\n[PROOFSTEP]\nexact hg.mem\u2112p_top_of_hasCompactSupport h'g \u03bc\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\nK : Set X\na b : X\ninst\u271d : MetrizableSpace X\nhK : IsCompact K\nhf : ContinuousOn f K\n\u22a2 IntegrableOn f K\n[PROOFSTEP]\nletI := metrizableSpaceMetric X\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\nK : Set X\na b : X\ninst\u271d : MetrizableSpace X\nhK : IsCompact K\nhf : ContinuousOn f K\nthis : MetricSpace X := metrizableSpaceMetric X\n\u22a2 IntegrableOn f K\n[PROOFSTEP]\nrefine' LocallyIntegrableOn.integrableOn_isCompact (fun x hx => _) hK\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\nK : Set X\na b : X\ninst\u271d : MetrizableSpace X\nhK : IsCompact K\nhf : ContinuousOn f K\nthis : MetricSpace X := metrizableSpaceMetric X\nx : X\nhx : x \u2208 K\n\u22a2 IntegrableAtFilter f (\ud835\udcdd[K] x)\n[PROOFSTEP]\nexact hf.integrableAt_nhdsWithin_of_isSeparable hK.measurableSet hK.isSeparable hx\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2075 : BorelSpace X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder X\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b2 : OrderTopology X\ninst\u271d\u00b9 : OrderTopology E\ninst\u271d : SecondCountableTopology E\nhmono : MonotoneOn f s\na b : X\nha : IsLeast s a\nhb : IsGreatest s b\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nh's : MeasurableSet s\n\u22a2 IntegrableOn f s\n[PROOFSTEP]\nborelize E\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2075 : BorelSpace X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder X\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b2 : OrderTopology X\ninst\u271d\u00b9 : OrderTopology E\ninst\u271d : SecondCountableTopology E\nhmono : MonotoneOn f s\na b : X\nha : IsLeast s a\nhb : IsGreatest s b\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nh's : MeasurableSet s\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 IntegrableOn f s\n[PROOFSTEP]\nobtain rfl | _ := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ninst\u271d\u2075 : BorelSpace X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder X\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b2 : OrderTopology X\ninst\u271d\u00b9 : OrderTopology E\ninst\u271d : SecondCountableTopology E\na b : X\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nhmono : MonotoneOn f \u2205\nha : IsLeast \u2205 a\nhb : IsGreatest \u2205 b\nhs : \u2191\u2191\u03bc \u2205 \u2260 \u22a4\nh's : MeasurableSet \u2205\n\u22a2 IntegrableOn f \u2205\n[PROOFSTEP]\nexact integrableOn_empty\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2075 : BorelSpace X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder X\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b2 : OrderTopology X\ninst\u271d\u00b9 : OrderTopology E\ninst\u271d : SecondCountableTopology E\nhmono : MonotoneOn f s\na b : X\nha : IsLeast s a\nhb : IsGreatest s b\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nh's : MeasurableSet s\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nh\u271d : Set.Nonempty s\n\u22a2 IntegrableOn f s\n[PROOFSTEP]\nhave hbelow : BddBelow (f '' s) := \u27e8f a, fun x \u27e8y, hy, hyx\u27e9 => hyx \u25b8 hmono ha.1 hy (ha.2 hy)\u27e9\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2075 : BorelSpace X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder X\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b2 : OrderTopology X\ninst\u271d\u00b9 : OrderTopology E\ninst\u271d : SecondCountableTopology E\nhmono : MonotoneOn f s\na b : X\nha : IsLeast s a\nhb : IsGreatest s b\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nh's : MeasurableSet s\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nh\u271d : Set.Nonempty s\nhbelow : BddBelow (f '' s)\n\u22a2 IntegrableOn f s\n[PROOFSTEP]\nhave habove : BddAbove (f '' s) := \u27e8f b, fun x \u27e8y, hy, hyx\u27e9 => hyx \u25b8 hmono hy hb.1 (hb.2 hy)\u27e9\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2075 : BorelSpace X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder X\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b2 : OrderTopology X\ninst\u271d\u00b9 : OrderTopology E\ninst\u271d : SecondCountableTopology E\nhmono : MonotoneOn f s\na b : X\nha : IsLeast s a\nhb : IsGreatest s b\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nh's : MeasurableSet s\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nh\u271d : Set.Nonempty s\nhbelow : BddBelow (f '' s)\nhabove : BddAbove (f '' s)\n\u22a2 IntegrableOn f s\n[PROOFSTEP]\nhave : Metric.Bounded (f '' s) := Metric.bounded_of_bddAbove_of_bddBelow habove hbelow\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2075 : BorelSpace X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder X\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b2 : OrderTopology X\ninst\u271d\u00b9 : OrderTopology E\ninst\u271d : SecondCountableTopology E\nhmono : MonotoneOn f s\na b : X\nha : IsLeast s a\nhb : IsGreatest s b\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nh's : MeasurableSet s\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nh\u271d : Set.Nonempty s\nhbelow : BddBelow (f '' s)\nhabove : BddAbove (f '' s)\nthis : Metric.Bounded (f '' s)\n\u22a2 IntegrableOn f s\n[PROOFSTEP]\nrcases bounded_iff_forall_norm_le.mp this with \u27e8C, hC\u27e9\n[GOAL]\ncase inr.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2075 : BorelSpace X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder X\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b2 : OrderTopology X\ninst\u271d\u00b9 : OrderTopology E\ninst\u271d : SecondCountableTopology E\nhmono : MonotoneOn f s\na b : X\nha : IsLeast s a\nhb : IsGreatest s b\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nh's : MeasurableSet s\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nh\u271d : Set.Nonempty s\nhbelow : BddBelow (f '' s)\nhabove : BddAbove (f '' s)\nthis : Metric.Bounded (f '' s)\nC : \u211d\nhC : \u2200 (x : E), x \u2208 f '' s \u2192 \u2016x\u2016 \u2264 C\n\u22a2 IntegrableOn f s\n[PROOFSTEP]\nhave A : IntegrableOn (fun _ => C) s \u03bc := by simp only [hs.lt_top, integrableOn_const, or_true_iff]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2075 : BorelSpace X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder X\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b2 : OrderTopology X\ninst\u271d\u00b9 : OrderTopology E\ninst\u271d : SecondCountableTopology E\nhmono : MonotoneOn f s\na b : X\nha : IsLeast s a\nhb : IsGreatest s b\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nh's : MeasurableSet s\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nh\u271d : Set.Nonempty s\nhbelow : BddBelow (f '' s)\nhabove : BddAbove (f '' s)\nthis : Metric.Bounded (f '' s)\nC : \u211d\nhC : \u2200 (x : E), x \u2208 f '' s \u2192 \u2016x\u2016 \u2264 C\n\u22a2 IntegrableOn (fun x => C) s\n[PROOFSTEP]\nsimp only [hs.lt_top, integrableOn_const, or_true_iff]\n[GOAL]\ncase inr.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2075 : BorelSpace X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder X\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b2 : OrderTopology X\ninst\u271d\u00b9 : OrderTopology E\ninst\u271d : SecondCountableTopology E\nhmono : MonotoneOn f s\na b : X\nha : IsLeast s a\nhb : IsGreatest s b\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nh's : MeasurableSet s\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nh\u271d : Set.Nonempty s\nhbelow : BddBelow (f '' s)\nhabove : BddAbove (f '' s)\nthis : Metric.Bounded (f '' s)\nC : \u211d\nhC : \u2200 (x : E), x \u2208 f '' s \u2192 \u2016x\u2016 \u2264 C\nA : IntegrableOn (fun x => C) s\n\u22a2 IntegrableOn f s\n[PROOFSTEP]\nrefine'\n Integrable.mono' A (aemeasurable_restrict_of_monotoneOn h's hmono).aestronglyMeasurable\n ((ae_restrict_iff' h's).mpr <| ae_of_all _ fun y hy => hC (f y) (mem_image_of_mem f hy))\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsFiniteMeasureOnCompacts \u03bc\nhs : IsCompact s\nhmono : MonotoneOn f s\n\u22a2 IntegrableOn f s\n[PROOFSTEP]\nobtain rfl | h := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsFiniteMeasureOnCompacts \u03bc\nhs : IsCompact \u2205\nhmono : MonotoneOn f \u2205\n\u22a2 IntegrableOn f \u2205\n[PROOFSTEP]\nexact integrableOn_empty\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsFiniteMeasureOnCompacts \u03bc\nhs : IsCompact s\nhmono : MonotoneOn f s\nh : Set.Nonempty s\n\u22a2 IntegrableOn f s\n[PROOFSTEP]\nexact\n hmono.integrableOn_of_measure_ne_top (hs.isLeast_sInf h) (hs.isGreatest_sSup h) hs.measure_lt_top.ne hs.measurableSet\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\n\u22a2 LocallyIntegrable f\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)\n[PROOFSTEP]\nrcases \u03bc.finiteAt_nhds x with \u27e8U, hU, h'U\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)\n[PROOFSTEP]\nobtain \u27e8a, b, xab, hab, abU\u27e9 : \u2203 a b : X, x \u2208 Icc a b \u2227 Icc a b \u2208 \ud835\udcdd x \u2227 Icc a b \u2286 U\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\n\u22a2 \u2203 a b, x \u2208 Icc a b \u2227 Icc a b \u2208 \ud835\udcdd x \u2227 Icc a b \u2286 U\ncase intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\na b : X\nxab : x \u2208 Icc a b\nhab : Icc a b \u2208 \ud835\udcdd x\nabU : Icc a b \u2286 U\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)\n[PROOFSTEP]\nexact exists_Icc_mem_subset_of_mem_nhds hU\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\na b : X\nxab : x \u2208 Icc a b\nhab : Icc a b \u2208 \ud835\udcdd x\nabU : Icc a b \u2286 U\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)\n[PROOFSTEP]\nhave ab : a \u2264 b := xab.1.trans xab.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\na b : X\nxab : x \u2208 Icc a b\nhab : Icc a b \u2208 \ud835\udcdd x\nabU : Icc a b \u2286 U\nab : a \u2264 b\n\u22a2 IntegrableAtFilter f (\ud835\udcdd x)\n[PROOFSTEP]\nrefine' \u27e8Icc a b, hab, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b9 : MeasurableSpace X\ninst\u271d\u00b9\u2070 : TopologicalSpace X\ninst\u271d\u2079 : MeasurableSpace Y\ninst\u271d\u2078 : TopologicalSpace Y\ninst\u271d\u2077 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder E\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : OrderTopology E\ninst\u271d\u00b9 : SecondCountableTopology E\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhmono : Monotone f\nx : X\nU : Set X\nhU : U \u2208 \ud835\udcdd x\nh'U : \u2191\u2191\u03bc U < \u22a4\na b : X\nxab : x \u2208 Icc a b\nhab : Icc a b \u2208 \ud835\udcdd x\nabU : Icc a b \u2286 U\nab : a \u2264 b\n\u22a2 IntegrableOn f (Icc a b)\n[PROOFSTEP]\nexact\n (hmono.monotoneOn _).integrableOn_of_measure_ne_top (isLeast_Icc ab) (isGreatest_Icc ab)\n ((measure_mono abU).trans_lt h'U).ne measurableSet_Icc\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\ng g' : X \u2192 R\nhg : IntegrableOn g A\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A \u2286 K\n\u22a2 IntegrableOn (fun x => g x * g' x) A\n[PROOFSTEP]\nrcases IsCompact.exists_bound_of_continuousOn hK hg' with \u27e8C, hC\u27e9\n[GOAL]\ncase intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\ng g' : X \u2192 R\nhg : IntegrableOn g A\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A \u2286 K\nC : \u211d\nhC : \u2200 (x : X), x \u2208 K \u2192 \u2016g' x\u2016 \u2264 C\n\u22a2 IntegrableOn (fun x => g x * g' x) A\n[PROOFSTEP]\nrw [IntegrableOn, \u2190 mem\u2112p_one_iff_integrable] at hg \u22a2\n[GOAL]\ncase intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\ng g' : X \u2192 R\nhg : Mem\u2112p g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A \u2286 K\nC : \u211d\nhC : \u2200 (x : X), x \u2208 K \u2192 \u2016g' x\u2016 \u2264 C\n\u22a2 Mem\u2112p (fun x => g x * g' x) 1\n[PROOFSTEP]\nhave : \u2200\u1d50 x \u2202\u03bc.restrict A, \u2016g x * g' x\u2016 \u2264 C * \u2016g x\u2016 :=\n by\n filter_upwards [ae_restrict_mem hA] with x hx\n refine' (norm_mul_le _ _).trans _\n rw [mul_comm]\n apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\ng g' : X \u2192 R\nhg : Mem\u2112p g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A \u2286 K\nC : \u211d\nhC : \u2200 (x : X), x \u2208 K \u2192 \u2016g' x\u2016 \u2264 C\n\u22a2 \u2200\u1d50 (x : X) \u2202Measure.restrict \u03bc A, \u2016g x * g' x\u2016 \u2264 C * \u2016g x\u2016\n[PROOFSTEP]\nfilter_upwards [ae_restrict_mem hA] with x hx\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\ng g' : X \u2192 R\nhg : Mem\u2112p g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A \u2286 K\nC : \u211d\nhC : \u2200 (x : X), x \u2208 K \u2192 \u2016g' x\u2016 \u2264 C\nx : X\nhx : x \u2208 A\n\u22a2 \u2016g x * g' x\u2016 \u2264 C * \u2016g x\u2016\n[PROOFSTEP]\nrefine' (norm_mul_le _ _).trans _\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\ng g' : X \u2192 R\nhg : Mem\u2112p g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A \u2286 K\nC : \u211d\nhC : \u2200 (x : X), x \u2208 K \u2192 \u2016g' x\u2016 \u2264 C\nx : X\nhx : x \u2208 A\n\u22a2 \u2016g x\u2016 * \u2016g' x\u2016 \u2264 C * \u2016g x\u2016\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\ng g' : X \u2192 R\nhg : Mem\u2112p g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A \u2286 K\nC : \u211d\nhC : \u2200 (x : X), x \u2208 K \u2192 \u2016g' x\u2016 \u2264 C\nx : X\nhx : x \u2208 A\n\u22a2 \u2016g' x\u2016 * \u2016g x\u2016 \u2264 C * \u2016g x\u2016\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)\n[GOAL]\ncase intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\ng g' : X \u2192 R\nhg : Mem\u2112p g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A \u2286 K\nC : \u211d\nhC : \u2200 (x : X), x \u2208 K \u2192 \u2016g' x\u2016 \u2264 C\nthis : \u2200\u1d50 (x : X) \u2202Measure.restrict \u03bc A, \u2016g x * g' x\u2016 \u2264 C * \u2016g x\u2016\n\u22a2 Mem\u2112p (fun x => g x * g' x) 1\n[PROOFSTEP]\nexact Mem\u2112p.of_le_mul hg (hg.aestronglyMeasurable.mul <| (hg'.mono hAK).aestronglyMeasurable hA) this\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\ng g' : X \u2192 R\nhg : ContinuousOn g K\nhg' : IntegrableOn g' A\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A \u2286 K\n\u22a2 IntegrableOn (fun x => g x * g' x) A\n[PROOFSTEP]\nrcases IsCompact.exists_bound_of_continuousOn hK hg with \u27e8C, hC\u27e9\n[GOAL]\ncase intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\ng g' : X \u2192 R\nhg : ContinuousOn g K\nhg' : IntegrableOn g' A\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A \u2286 K\nC : \u211d\nhC : \u2200 (x : X), x \u2208 K \u2192 \u2016g x\u2016 \u2264 C\n\u22a2 IntegrableOn (fun x => g x * g' x) A\n[PROOFSTEP]\nrw [IntegrableOn, \u2190 mem\u2112p_one_iff_integrable] at hg' \u22a2\n[GOAL]\ncase intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\ng g' : X \u2192 R\nhg : ContinuousOn g K\nhg' : Mem\u2112p g' 1\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A \u2286 K\nC : \u211d\nhC : \u2200 (x : X), x \u2208 K \u2192 \u2016g x\u2016 \u2264 C\n\u22a2 Mem\u2112p (fun x => g x * g' x) 1\n[PROOFSTEP]\nhave : \u2200\u1d50 x \u2202\u03bc.restrict A, \u2016g x * g' x\u2016 \u2264 C * \u2016g' x\u2016 :=\n by\n filter_upwards [ae_restrict_mem hA] with x hx\n refine' (norm_mul_le _ _).trans _\n apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\ng g' : X \u2192 R\nhg : ContinuousOn g K\nhg' : Mem\u2112p g' 1\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A \u2286 K\nC : \u211d\nhC : \u2200 (x : X), x \u2208 K \u2192 \u2016g x\u2016 \u2264 C\n\u22a2 \u2200\u1d50 (x : X) \u2202Measure.restrict \u03bc A, \u2016g x * g' x\u2016 \u2264 C * \u2016g' x\u2016\n[PROOFSTEP]\nfilter_upwards [ae_restrict_mem hA] with x hx\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\ng g' : X \u2192 R\nhg : ContinuousOn g K\nhg' : Mem\u2112p g' 1\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A \u2286 K\nC : \u211d\nhC : \u2200 (x : X), x \u2208 K \u2192 \u2016g x\u2016 \u2264 C\nx : X\nhx : x \u2208 A\n\u22a2 \u2016g x * g' x\u2016 \u2264 C * \u2016g' x\u2016\n[PROOFSTEP]\nrefine' (norm_mul_le _ _).trans _\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\ng g' : X \u2192 R\nhg : ContinuousOn g K\nhg' : Mem\u2112p g' 1\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A \u2286 K\nC : \u211d\nhC : \u2200 (x : X), x \u2208 K \u2192 \u2016g x\u2016 \u2264 C\nx : X\nhx : x \u2208 A\n\u22a2 \u2016g x\u2016 * \u2016g' x\u2016 \u2264 C * \u2016g' x\u2016\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)\n[GOAL]\ncase intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u00b2 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\ng g' : X \u2192 R\nhg : ContinuousOn g K\nhg' : Mem\u2112p g' 1\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A \u2286 K\nC : \u211d\nhC : \u2200 (x : X), x \u2208 K \u2192 \u2016g x\u2016 \u2264 C\nthis : \u2200\u1d50 (x : X) \u2202Measure.restrict \u03bc A, \u2016g x * g' x\u2016 \u2264 C * \u2016g' x\u2016\n\u22a2 Mem\u2112p (fun x => g x * g' x) 1\n[PROOFSTEP]\nexact Mem\u2112p.of_le_mul hg' (((hg.mono hAK).aestronglyMeasurable hA).mul hg'.aestronglyMeasurable) this\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2074 : OpensMeasurableSpace X\nA K : Set X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : T2Space X\ninst\u271d : SecondCountableTopologyEither X \ud835\udd5c\ng : X \u2192 E\nhg : IntegrableOn g K\nf : X \u2192 \ud835\udd5c\nhf : ContinuousOn f K\nhK : IsCompact K\n\u22a2 IntegrableOn (fun x => f x \u2022 g x) K\n[PROOFSTEP]\nrw [IntegrableOn, \u2190 integrable_norm_iff]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2074 : OpensMeasurableSpace X\nA K : Set X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : T2Space X\ninst\u271d : SecondCountableTopologyEither X \ud835\udd5c\ng : X \u2192 E\nhg : IntegrableOn g K\nf : X \u2192 \ud835\udd5c\nhf : ContinuousOn f K\nhK : IsCompact K\n\u22a2 Integrable fun a => \u2016f a \u2022 g a\u2016\n[PROOFSTEP]\nsimp_rw [norm_smul]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2074 : OpensMeasurableSpace X\nA K : Set X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : T2Space X\ninst\u271d : SecondCountableTopologyEither X \ud835\udd5c\ng : X \u2192 E\nhg : IntegrableOn g K\nf : X \u2192 \ud835\udd5c\nhf : ContinuousOn f K\nhK : IsCompact K\n\u22a2 Integrable fun a => \u2016f a\u2016 * \u2016g a\u2016\n[PROOFSTEP]\nrefine' IntegrableOn.continuousOn_mul _ hg.norm hK\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2074 : OpensMeasurableSpace X\nA K : Set X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : T2Space X\ninst\u271d : SecondCountableTopologyEither X \ud835\udd5c\ng : X \u2192 E\nhg : IntegrableOn g K\nf : X \u2192 \ud835\udd5c\nhf : ContinuousOn f K\nhK : IsCompact K\n\u22a2 ContinuousOn (fun a => \u2016f a\u2016) K\n[PROOFSTEP]\nexact continuous_norm.comp_continuousOn hf\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2074 : OpensMeasurableSpace X\nA K : Set X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : T2Space X\ninst\u271d : SecondCountableTopologyEither X \ud835\udd5c\ng : X \u2192 E\nhg : IntegrableOn g K\nf : X \u2192 \ud835\udd5c\nhf : ContinuousOn f K\nhK : IsCompact K\n\u22a2 AEStronglyMeasurable (fun x => f x \u2022 g x) (Measure.restrict \u03bc K)\n[PROOFSTEP]\nexact (hf.aestronglyMeasurable hK.measurableSet).smul hg.1\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2074 : OpensMeasurableSpace X\nA K : Set X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : T2Space X\ninst\u271d : SecondCountableTopologyEither X E\nf : X \u2192 \ud835\udd5c\nhf : IntegrableOn f K\ng : X \u2192 E\nhg : ContinuousOn g K\nhK : IsCompact K\n\u22a2 IntegrableOn (fun x => f x \u2022 g x) K\n[PROOFSTEP]\nrw [IntegrableOn, \u2190 integrable_norm_iff]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2074 : OpensMeasurableSpace X\nA K : Set X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : T2Space X\ninst\u271d : SecondCountableTopologyEither X E\nf : X \u2192 \ud835\udd5c\nhf : IntegrableOn f K\ng : X \u2192 E\nhg : ContinuousOn g K\nhK : IsCompact K\n\u22a2 Integrable fun a => \u2016f a \u2022 g a\u2016\n[PROOFSTEP]\nsimp_rw [norm_smul]\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2074 : OpensMeasurableSpace X\nA K : Set X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : T2Space X\ninst\u271d : SecondCountableTopologyEither X E\nf : X \u2192 \ud835\udd5c\nhf : IntegrableOn f K\ng : X \u2192 E\nhg : ContinuousOn g K\nhK : IsCompact K\n\u22a2 Integrable fun a => \u2016f a\u2016 * \u2016g a\u2016\n[PROOFSTEP]\nrefine' IntegrableOn.mul_continuousOn hf.norm _ hK\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2074 : OpensMeasurableSpace X\nA K : Set X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : T2Space X\ninst\u271d : SecondCountableTopologyEither X E\nf : X \u2192 \ud835\udd5c\nhf : IntegrableOn f K\ng : X \u2192 E\nhg : ContinuousOn g K\nhK : IsCompact K\n\u22a2 ContinuousOn (fun a => \u2016g a\u2016) K\n[PROOFSTEP]\nexact continuous_norm.comp_continuousOn hg\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d\u2074 : OpensMeasurableSpace X\nA K : Set X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : T2Space X\ninst\u271d : SecondCountableTopologyEither X E\nf : X \u2192 \ud835\udd5c\nhf : IntegrableOn f K\ng : X \u2192 E\nhg : ContinuousOn g K\nhK : IsCompact K\n\u22a2 AEStronglyMeasurable (fun x => f x \u2022 g x) (Measure.restrict \u03bc K)\n[PROOFSTEP]\nexact hf.1.smul (hg.aestronglyMeasurable hK.measurableSet)\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\ninst\u271d\u2074 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b3 : LocallyCompactSpace X\ninst\u271d\u00b2 : T2Space X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\nf g : X \u2192 R\ns : Set X\nhf : LocallyIntegrableOn f s\nhg : ContinuousOn g s\nhs : IsOpen s\n\u22a2 LocallyIntegrableOn (fun x => g x * f x) s\n[PROOFSTEP]\nrw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf \u22a2\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\ninst\u271d\u2074 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b3 : LocallyCompactSpace X\ninst\u271d\u00b2 : T2Space X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\nf g : X \u2192 R\ns : Set X\nhf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k\nhg : ContinuousOn g s\nhs : IsOpen s\n\u22a2 \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn (fun x => g x * f x) k\n[PROOFSTEP]\nexact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_mul (hg.mono hk_sub) hk_c\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\ninst\u271d\u2074 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b3 : LocallyCompactSpace X\ninst\u271d\u00b2 : T2Space X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\nf g : X \u2192 R\ns : Set X\nhf : LocallyIntegrableOn f s\nhg : ContinuousOn g s\nhs : IsOpen s\n\u22a2 LocallyIntegrableOn (fun x => f x * g x) s\n[PROOFSTEP]\nrw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf \u22a2\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : TopologicalSpace Y\ninst\u271d\u2075 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\ninst\u271d\u2074 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u00b3 : LocallyCompactSpace X\ninst\u271d\u00b2 : T2Space X\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : SecondCountableTopologyEither X R\nf g : X \u2192 R\ns : Set X\nhf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k\nhg : ContinuousOn g s\nhs : IsOpen s\n\u22a2 \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn (fun x => f x * g x) k\n[PROOFSTEP]\nexact fun k hk_sub hk_c => (hf k hk_sub hk_c).mul_continuousOn (hg.mono hk_sub) hk_c\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\ninst\u271d\u2075 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u2074 : LocallyCompactSpace X\ninst\u271d\u00b3 : T2Space X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SecondCountableTopologyEither X \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : X \u2192 E\ng : X \u2192 \ud835\udd5c\ns : Set X\nhs : IsOpen s\nhf : LocallyIntegrableOn f s\nhg : ContinuousOn g s\n\u22a2 LocallyIntegrableOn (fun x => g x \u2022 f x) s\n[PROOFSTEP]\nrw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf \u22a2\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\ninst\u271d\u2075 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u2074 : LocallyCompactSpace X\ninst\u271d\u00b3 : T2Space X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SecondCountableTopologyEither X \ud835\udd5c\ninst\u271d : NormedSpace \ud835\udd5c E\nf : X \u2192 E\ng : X \u2192 \ud835\udd5c\ns : Set X\nhs : IsOpen s\nhf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k\nhg : ContinuousOn g s\n\u22a2 \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn (fun x => g x \u2022 f x) k\n[PROOFSTEP]\nexact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_smul (hg.mono hk_sub) hk_c\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\ninst\u271d\u2075 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u2074 : LocallyCompactSpace X\ninst\u271d\u00b3 : T2Space X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SecondCountableTopologyEither X E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : X \u2192 \ud835\udd5c\ng : X \u2192 E\ns : Set X\nhs : IsOpen s\nhf : LocallyIntegrableOn f s\nhg : ContinuousOn g s\n\u22a2 LocallyIntegrableOn (fun x => f x \u2022 g x) s\n[PROOFSTEP]\nrw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf \u22a2\n[GOAL]\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : TopologicalSpace Y\ninst\u271d\u2076 : NormedAddCommGroup E\nf\u271d : X \u2192 E\n\u03bc : Measure X\ns\u271d : Set X\ninst\u271d\u2075 : OpensMeasurableSpace X\nA K : Set X\ninst\u271d\u2074 : LocallyCompactSpace X\ninst\u271d\u00b3 : T2Space X\n\ud835\udd5c : Type u_5\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SecondCountableTopologyEither X E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : X \u2192 \ud835\udd5c\ng : X \u2192 E\ns : Set X\nhs : IsOpen s\nhf : \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn f k\nhg : ContinuousOn g s\n\u22a2 \u2200 (k : Set X), k \u2286 s \u2192 IsCompact k \u2192 IntegrableOn (fun x => f x \u2022 g x) k\n[PROOFSTEP]\nexact fun k hk_sub hk_c => (hf k hk_sub hk_c).smul_continuousOn (hg.mono hk_sub) hk_c\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.LocallyIntegrable", "llama_tokens": 55305, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.6513548782017745, "lm_q1q2_score": 0.4445977316418698}} {"text": "[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhf : HasDerivAtFilter f f' x L\nhg : HasDerivAtFilter g g' x L\n\u22a2 HasDerivAtFilter (fun y => f y + g y) (f' + g') x L\n[PROOFSTEP]\nsimpa using (hf.add hg).hasDerivAtFilter\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhf : HasStrictDerivAt f f' x\nhg : HasStrictDerivAt g g' x\n\u22a2 HasStrictDerivAt (fun y => f y + g y) (f' + g') x\n[PROOFSTEP]\nsimpa using (hf.add hg).hasStrictDerivAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\nc : F\n\u22a2 derivWithin (fun y => f y + c) s x = derivWithin f s x\n[PROOFSTEP]\nsimp only [derivWithin, fderivWithin_add_const hxs]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nc : F\n\u22a2 deriv (fun y => f y + c) x = deriv f x\n[PROOFSTEP]\nsimp only [deriv, fderiv_add_const]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\nc : F\n\u22a2 derivWithin (fun y => c + f y) s x = derivWithin f s x\n[PROOFSTEP]\nsimp only [derivWithin, fderivWithin_const_add hxs]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nc : F\n\u22a2 deriv (fun y => c + f y) x = deriv f x\n[PROOFSTEP]\nsimp only [deriv, fderiv_const_add]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\u03b9 : Type u_1\nu : Finset \u03b9\nA : \u03b9 \u2192 \ud835\udd5c \u2192 F\nA' : \u03b9 \u2192 F\nh : \u2200 (i : \u03b9), i \u2208 u \u2192 HasDerivAtFilter (A i) (A' i) x L\n\u22a2 HasDerivAtFilter (fun y => \u2211 i in u, A i y) (\u2211 i in u, A' i) x L\n[PROOFSTEP]\nsimpa [ContinuousLinearMap.sum_apply] using (HasFDerivAtFilter.sum h).hasDerivAtFilter\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\u03b9 : Type u_1\nu : Finset \u03b9\nA : \u03b9 \u2192 \ud835\udd5c \u2192 F\nA' : \u03b9 \u2192 F\nh : \u2200 (i : \u03b9), i \u2208 u \u2192 HasStrictDerivAt (A i) (A' i) x\n\u22a2 HasStrictDerivAt (fun y => \u2211 i in u, A i y) (\u2211 i in u, A' i) x\n[PROOFSTEP]\nsimpa [ContinuousLinearMap.sum_apply] using (HasStrictFDerivAt.sum h).hasStrictDerivAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nh : HasDerivAtFilter f f' x L\n\u22a2 HasDerivAtFilter (fun x => -f x) (-f') x L\n[PROOFSTEP]\nsimpa using h.neg.hasDerivAtFilter\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nh : HasStrictDerivAt f f' x\n\u22a2 HasStrictDerivAt (fun x => -f x) (-f') x\n[PROOFSTEP]\nsimpa using h.neg.hasStrictDerivAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 derivWithin (fun y => -f y) s x = -derivWithin f s x\n[PROOFSTEP]\nsimp only [derivWithin, fderivWithin_neg hxs, ContinuousLinearMap.neg_apply]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\u22a2 deriv (fun y => -f y) x = -deriv f x\n[PROOFSTEP]\nsimp only [deriv, fderiv_neg, ContinuousLinearMap.neg_apply]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhf : HasDerivAtFilter f f' x L\nhg : HasDerivAtFilter g g' x L\n\u22a2 HasDerivAtFilter (fun x => f x - g x) (f' - g') x L\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using hf.add hg.neg\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhf : HasStrictDerivAt f f' x\nhg : HasStrictDerivAt g g' x\n\u22a2 HasStrictDerivAt (fun x => f x - g x) (f' - g') x\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using hf.add hg.neg\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhf : HasDerivAtFilter f f' x L\nc : F\n\u22a2 HasDerivAtFilter (fun x => f x - c) f' x L\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using hf.add_const (-c)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\nc : F\n\u22a2 derivWithin (fun y => f y - c) s x = derivWithin f s x\n[PROOFSTEP]\nsimp only [derivWithin, fderivWithin_sub_const hxs]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nc : F\n\u22a2 deriv (fun y => f y - c) x = deriv f x\n[PROOFSTEP]\nsimp only [deriv, fderiv_sub_const]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nc : F\nhf : HasDerivAtFilter f f' x L\n\u22a2 HasDerivAtFilter (fun x => c - f x) (-f') x L\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using hf.neg.const_add c\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nc : F\nhf : HasStrictDerivAt f f' x\n\u22a2 HasStrictDerivAt (fun x => c - f x) (-f') x\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using hf.neg.const_add c\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\nc : F\n\u22a2 derivWithin (fun y => c - f y) s x = -derivWithin f s x\n[PROOFSTEP]\nsimp [derivWithin, fderivWithin_const_sub hxs]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nc : F\n\u22a2 deriv (fun y => c - f y) x = -deriv f x\n[PROOFSTEP]\nsimp only [\u2190 derivWithin_univ, derivWithin_const_sub (uniqueDiffWithinAt_univ : UniqueDiffWithinAt \ud835\udd5c _ _)]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.Deriv.Add", "llama_tokens": 4253, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850402140659, "lm_q2_score": 0.6297746143530797, "lm_q1q2_score": 0.4444855015169862}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup \u03b1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : HasSolidNorm \u03b1\nx\u271d\u00b2 x\u271d\u00b9 : \u211a\nx\u271d : |x\u271d\u00b2| \u2264 |x\u271d\u00b9|\n\u22a2 \u2016x\u271d\u00b2\u2016 \u2264 \u2016x\u271d\u00b9\u2016\n[PROOFSTEP]\nsimpa only [norm, \u2190 Rat.cast_abs, Rat.cast_le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b : \u03b1\nh : b \u2293 -b \u2264 a \u2293 -a\n\u22a2 \u2016a\u2016 \u2264 \u2016b\u2016\n[PROOFSTEP]\napply solid\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b : \u03b1\nh : b \u2293 -b \u2264 a \u2293 -a\n\u22a2 |a| \u2264 |b|\n[PROOFSTEP]\nrw [abs_eq_sup_neg]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b : \u03b1\nh : b \u2293 -b \u2264 a \u2293 -a\n\u22a2 a \u2294 -a \u2264 |b|\n[PROOFSTEP]\nnth_rw 1 [\u2190 neg_neg a]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b : \u03b1\nh : b \u2293 -b \u2264 a \u2293 -a\n\u22a2 - -a \u2294 -a \u2264 |b|\n[PROOFSTEP]\nrw [\u2190 neg_inf_eq_sup_neg]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b : \u03b1\nh : b \u2293 -b \u2264 a \u2293 -a\n\u22a2 -(-a \u2293 a) \u2264 |b|\n[PROOFSTEP]\nrw [abs_eq_sup_neg]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b : \u03b1\nh : b \u2293 -b \u2264 a \u2293 -a\n\u22a2 -(-a \u2293 a) \u2264 b \u2294 -b\n[PROOFSTEP]\nnth_rw 1 [\u2190 neg_neg b]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b : \u03b1\nh : b \u2293 -b \u2264 a \u2293 -a\n\u22a2 -(-a \u2293 a) \u2264 - -b \u2294 -b\n[PROOFSTEP]\nrwa [\u2190 neg_inf_eq_sup_neg, neg_le_neg_iff, @inf_comm _ _ _ b, @inf_comm _ _ _ a]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 \u2016a \u2293 b - c \u2293 d\u2016 \u2264 \u2016a - c\u2016 + \u2016b - d\u2016\n[PROOFSTEP]\nrw [\u2190 norm_abs_eq_norm (a - c), \u2190 norm_abs_eq_norm (b - d)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 \u2016a \u2293 b - c \u2293 d\u2016 \u2264 \u2016|a - c|\u2016 + \u2016|b - d|\u2016\n[PROOFSTEP]\nrefine' le_trans (solid _) (norm_add_le |a - c| |b - d|)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2293 b - c \u2293 d| \u2264 ||a - c| + |b - d||\n[PROOFSTEP]\nrw [abs_of_nonneg (|a - c| + |b - d|) (add_nonneg (abs_nonneg (a - c)) (abs_nonneg (b - d)))]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2293 b - c \u2293 d| \u2264 |a - c| + |b - d|\n[PROOFSTEP]\ncalc\n |a \u2293 b - c \u2293 d| = |a \u2293 b - c \u2293 b + (c \u2293 b - c \u2293 d)| := by rw [sub_add_sub_cancel]\n _ \u2264 |a \u2293 b - c \u2293 b| + |c \u2293 b - c \u2293 d| := (abs_add_le _ _)\n _ \u2264 |a - c| + |b - d| := by\n apply add_le_add\n \u00b7 exact abs_inf_sub_inf_le_abs _ _ _\n \u00b7 rw [@inf_comm _ _ c, @inf_comm _ _ c]\n exact abs_inf_sub_inf_le_abs _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2293 b - c \u2293 d| = |a \u2293 b - c \u2293 b + (c \u2293 b - c \u2293 d)|\n[PROOFSTEP]\nrw [sub_add_sub_cancel]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2293 b - c \u2293 b| + |c \u2293 b - c \u2293 d| \u2264 |a - c| + |b - d|\n[PROOFSTEP]\napply add_le_add\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2293 b - c \u2293 b| \u2264 |a - c|\n[PROOFSTEP]\nexact abs_inf_sub_inf_le_abs _ _ _\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |c \u2293 b - c \u2293 d| \u2264 |b - d|\n[PROOFSTEP]\nrw [@inf_comm _ _ c, @inf_comm _ _ c]\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |b \u2293 c - d \u2293 c| \u2264 |b - d|\n[PROOFSTEP]\nexact abs_inf_sub_inf_le_abs _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 \u2016a \u2294 b - c \u2294 d\u2016 \u2264 \u2016a - c\u2016 + \u2016b - d\u2016\n[PROOFSTEP]\nrw [\u2190 norm_abs_eq_norm (a - c), \u2190 norm_abs_eq_norm (b - d)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 \u2016a \u2294 b - c \u2294 d\u2016 \u2264 \u2016|a - c|\u2016 + \u2016|b - d|\u2016\n[PROOFSTEP]\nrefine' le_trans (solid _) (norm_add_le |a - c| |b - d|)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2294 b - c \u2294 d| \u2264 ||a - c| + |b - d||\n[PROOFSTEP]\nrw [abs_of_nonneg (|a - c| + |b - d|) (add_nonneg (abs_nonneg (a - c)) (abs_nonneg (b - d)))]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2294 b - c \u2294 d| \u2264 |a - c| + |b - d|\n[PROOFSTEP]\ncalc\n |a \u2294 b - c \u2294 d| = |a \u2294 b - c \u2294 b + (c \u2294 b - c \u2294 d)| := by rw [sub_add_sub_cancel]\n _ \u2264 |a \u2294 b - c \u2294 b| + |c \u2294 b - c \u2294 d| := (abs_add_le _ _)\n _ \u2264 |a - c| + |b - d| := by\n apply add_le_add\n \u00b7 exact abs_sup_sub_sup_le_abs _ _ _\n \u00b7 rw [@sup_comm _ _ c, @sup_comm _ _ c]\n exact abs_sup_sub_sup_le_abs _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2294 b - c \u2294 d| = |a \u2294 b - c \u2294 b + (c \u2294 b - c \u2294 d)|\n[PROOFSTEP]\nrw [sub_add_sub_cancel]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2294 b - c \u2294 b| + |c \u2294 b - c \u2294 d| \u2264 |a - c| + |b - d|\n[PROOFSTEP]\napply add_le_add\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2294 b - c \u2294 b| \u2264 |a - c|\n[PROOFSTEP]\nexact abs_sup_sub_sup_le_abs _ _ _\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |c \u2294 b - c \u2294 d| \u2264 |b - d|\n[PROOFSTEP]\nrw [@sup_comm _ _ c, @sup_comm _ _ c]\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |b \u2294 c - d \u2294 c| \u2264 |b - d|\n[PROOFSTEP]\nexact abs_sup_sub_sup_le_abs _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\nx y : \u03b1\n\u22a2 \u2016x \u2293 y\u2016 \u2264 \u2016x\u2016 + \u2016y\u2016\n[PROOFSTEP]\nhave h : \u2016x \u2293 y - 0 \u2293 0\u2016 \u2264 \u2016x - 0\u2016 + \u2016y - 0\u2016 := norm_inf_sub_inf_le_add_norm x y 0 0\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\nx y : \u03b1\nh : \u2016x \u2293 y - 0 \u2293 0\u2016 \u2264 \u2016x - 0\u2016 + \u2016y - 0\u2016\n\u22a2 \u2016x \u2293 y\u2016 \u2264 \u2016x\u2016 + \u2016y\u2016\n[PROOFSTEP]\nsimpa only [inf_idem, sub_zero] using h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\nx y : \u03b1\n\u22a2 \u2016x \u2294 y\u2016 \u2264 \u2016x\u2016 + \u2016y\u2016\n[PROOFSTEP]\nhave h : \u2016x \u2294 y - 0 \u2294 0\u2016 \u2264 \u2016x - 0\u2016 + \u2016y - 0\u2016 := norm_sup_sub_sup_le_add_norm x y 0 0\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\nx y : \u03b1\nh : \u2016x \u2294 y - 0 \u2294 0\u2016 \u2264 \u2016x - 0\u2016 + \u2016y - 0\u2016\n\u22a2 \u2016x \u2294 y\u2016 \u2264 \u2016x\u2016 + \u2016y\u2016\n[PROOFSTEP]\nsimpa only [sup_idem, sub_zero] using h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\n\u22a2 ContinuousInf \u03b1\n[PROOFSTEP]\nrefine' \u27e8continuous_iff_continuousAt.2 fun q => tendsto_iff_norm_tendsto_zero.2 <| _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\nq : \u03b1 \u00d7 \u03b1\n\u22a2 Filter.Tendsto (fun e => \u2016e.fst \u2293 e.snd - (fun p => p.fst \u2293 p.snd) q\u2016) (nhds q) (nhds 0)\n[PROOFSTEP]\nhave : \u2200 p : \u03b1 \u00d7 \u03b1, \u2016p.1 \u2293 p.2 - q.1 \u2293 q.2\u2016 \u2264 \u2016p.1 - q.1\u2016 + \u2016p.2 - q.2\u2016 := fun _ => norm_inf_sub_inf_le_add_norm _ _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\nq : \u03b1 \u00d7 \u03b1\nthis : \u2200 (p : \u03b1 \u00d7 \u03b1), \u2016p.fst \u2293 p.snd - q.fst \u2293 q.snd\u2016 \u2264 \u2016p.fst - q.fst\u2016 + \u2016p.snd - q.snd\u2016\n\u22a2 Filter.Tendsto (fun e => \u2016e.fst \u2293 e.snd - (fun p => p.fst \u2293 p.snd) q\u2016) (nhds q) (nhds 0)\n[PROOFSTEP]\nrefine' squeeze_zero (fun e => norm_nonneg _) this _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\nq : \u03b1 \u00d7 \u03b1\nthis : \u2200 (p : \u03b1 \u00d7 \u03b1), \u2016p.fst \u2293 p.snd - q.fst \u2293 q.snd\u2016 \u2264 \u2016p.fst - q.fst\u2016 + \u2016p.snd - q.snd\u2016\n\u22a2 Filter.Tendsto (fun t => \u2016t.fst - q.fst\u2016 + \u2016t.snd - q.snd\u2016) (nhds q) (nhds 0)\n[PROOFSTEP]\nconvert\n ((continuous_fst.tendsto q).sub <| tendsto_const_nhds).norm.add\n ((continuous_snd.tendsto q).sub <| tendsto_const_nhds).norm\n[GOAL]\ncase h.e'_5.h.e'_3\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\nq : \u03b1 \u00d7 \u03b1\nthis : \u2200 (p : \u03b1 \u00d7 \u03b1), \u2016p.fst \u2293 p.snd - q.fst \u2293 q.snd\u2016 \u2264 \u2016p.fst - q.fst\u2016 + \u2016p.snd - q.snd\u2016\n\u22a2 0 = \u2016q.fst - q.fst\u2016 + \u2016q.snd - q.snd\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\nz x y : \u03b1\n\u22a2 dist (x \u2294 z) (y \u2294 z) \u2264 \u21911 * dist x y\n[PROOFSTEP]\nrw [NNReal.coe_one, one_mul, dist_eq_norm, dist_eq_norm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\nz x y : \u03b1\n\u22a2 \u2016x \u2294 z - y \u2294 z\u2016 \u2264 \u2016x - y\u2016\n[PROOFSTEP]\nexact norm_sup_sub_sup_le_norm x y z\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\n\u22a2 Continuous NegPart.neg\n[PROOFSTEP]\nrefine continuous_pos.comp <| @continuous_neg _ _ _ TopologicalAddGroup.toContinuousNeg\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : NormedLatticeAddCommGroup \u03b1\nE : Type u_2\ninst\u271d : NormedLatticeAddCommGroup E\n\u22a2 IsClosed {x | 0 \u2264 x}\n[PROOFSTEP]\nsuffices {x : E | 0 \u2264 x} = NegPart.neg \u207b\u00b9' {(0 : E)} by\n rw [this]\n exact IsClosed.preimage continuous_neg' isClosed_singleton\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : NormedLatticeAddCommGroup \u03b1\nE : Type u_2\ninst\u271d : NormedLatticeAddCommGroup E\nthis : {x | 0 \u2264 x} = NegPart.neg \u207b\u00b9' {0}\n\u22a2 IsClosed {x | 0 \u2264 x}\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : NormedLatticeAddCommGroup \u03b1\nE : Type u_2\ninst\u271d : NormedLatticeAddCommGroup E\nthis : {x | 0 \u2264 x} = NegPart.neg \u207b\u00b9' {0}\n\u22a2 IsClosed (NegPart.neg \u207b\u00b9' {0})\n[PROOFSTEP]\nexact IsClosed.preimage continuous_neg' isClosed_singleton\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : NormedLatticeAddCommGroup \u03b1\nE : Type u_2\ninst\u271d : NormedLatticeAddCommGroup E\n\u22a2 {x | 0 \u2264 x} = NegPart.neg \u207b\u00b9' {0}\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : NormedLatticeAddCommGroup \u03b1\nE : Type u_2\ninst\u271d : NormedLatticeAddCommGroup E\nx : E\n\u22a2 x \u2208 {x | 0 \u2264 x} \u2194 x \u2208 NegPart.neg \u207b\u00b9' {0}\n[PROOFSTEP]\nsimp only [Set.mem_preimage, Set.mem_singleton_iff, Set.mem_setOf_eq,\n @neg_eq_zero_iff E _ _ (OrderedAddCommGroup.to_covariantClass_left_le E)]\n -- porting note: I'm not sure why Lean couldn't synthesize this instance because it works with\n -- `have : CovariantClass E E (\u00b7 + \u00b7) (\u00b7 \u2264 \u00b7) := inferInstance`\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : NormedLatticeAddCommGroup \u03b1\nG : Type u_2\ninst\u271d\u00b2 : OrderedAddCommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : ContinuousSub G\nh : IsClosed {x | 0 \u2264 x}\n\u22a2 IsClosed {p | p.fst \u2264 p.snd}\n[PROOFSTEP]\nhave : {p : G \u00d7 G | p.fst \u2264 p.snd} = (fun p : G \u00d7 G => p.snd - p.fst) \u207b\u00b9' {x : G | 0 \u2264 x} := by ext1 p;\n simp only [sub_nonneg, Set.preimage_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : NormedLatticeAddCommGroup \u03b1\nG : Type u_2\ninst\u271d\u00b2 : OrderedAddCommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : ContinuousSub G\nh : IsClosed {x | 0 \u2264 x}\n\u22a2 {p | p.fst \u2264 p.snd} = (fun p => p.snd - p.fst) \u207b\u00b9' {x | 0 \u2264 x}\n[PROOFSTEP]\next1 p\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : NormedLatticeAddCommGroup \u03b1\nG : Type u_2\ninst\u271d\u00b2 : OrderedAddCommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : ContinuousSub G\nh : IsClosed {x | 0 \u2264 x}\np : G \u00d7 G\n\u22a2 p \u2208 {p | p.fst \u2264 p.snd} \u2194 p \u2208 (fun p => p.snd - p.fst) \u207b\u00b9' {x | 0 \u2264 x}\n[PROOFSTEP]\nsimp only [sub_nonneg, Set.preimage_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : NormedLatticeAddCommGroup \u03b1\nG : Type u_2\ninst\u271d\u00b2 : OrderedAddCommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : ContinuousSub G\nh : IsClosed {x | 0 \u2264 x}\nthis : {p | p.fst \u2264 p.snd} = (fun p => p.snd - p.fst) \u207b\u00b9' {x | 0 \u2264 x}\n\u22a2 IsClosed {p | p.fst \u2264 p.snd}\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : NormedLatticeAddCommGroup \u03b1\nG : Type u_2\ninst\u271d\u00b2 : OrderedAddCommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : ContinuousSub G\nh : IsClosed {x | 0 \u2264 x}\nthis : {p | p.fst \u2264 p.snd} = (fun p => p.snd - p.fst) \u207b\u00b9' {x | 0 \u2264 x}\n\u22a2 IsClosed ((fun p => p.snd - p.fst) \u207b\u00b9' {x | 0 \u2264 x})\n[PROOFSTEP]\nexact IsClosed.preimage (continuous_snd.sub continuous_fst) h\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Normed.Order.Lattice", "llama_tokens": 5845, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929002541068, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4444810222940209}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nh : IsOrderRightAdjoint f g\ne : \u03b2 \u2243o \u03b3\ny : \u03b3\n\u22a2 IsLUB {x | (\u2191e \u2218 f) x \u2264 y} ((g \u2218 \u2191(OrderIso.symm e)) y)\n[PROOFSTEP]\nsimpa [e.le_symm_apply] using h (e.symm y)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nh : IsOrderRightAdjoint f g\ne : \u03b3 \u2243o \u03b1\n\u22a2 IsOrderRightAdjoint (f \u2218 \u2191e) (\u2191(OrderIso.symm e) \u2218 g)\n[PROOFSTEP]\nintro y\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nh : IsOrderRightAdjoint f g\ne : \u03b3 \u2243o \u03b1\ny : \u03b2\n\u22a2 IsLUB {x | (f \u2218 \u2191e) x \u2264 y} ((\u2191(OrderIso.symm e) \u2218 g) y)\n[PROOFSTEP]\nchange IsLUB (e \u207b\u00b9' {x | f x \u2264 y}) (e.symm (g y))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nh : IsOrderRightAdjoint f g\ne : \u03b3 \u2243o \u03b1\ny : \u03b2\n\u22a2 IsLUB (\u2191e \u207b\u00b9' {x | f x \u2264 y}) (\u2191(OrderIso.symm e) (g y))\n[PROOFSTEP]\nrw [e.isLUB_preimage, e.apply_symm_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nh : IsOrderRightAdjoint f g\ne : \u03b3 \u2243o \u03b1\ny : \u03b2\n\u22a2 IsLUB {x | f x \u2264 y} (g y)\n[PROOFSTEP]\nexact h y\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\nfa : \u03b1 \u2243o \u03b1\nfb : \u03b2 \u21aao \u03b2\ng : \u03b1 \u2192 \u03b2\nh : Semiconj g \u2191fa \u2191fb\ng' : \u03b2 \u2192 \u03b1\nhg' : IsOrderRightAdjoint g g'\n\u22a2 Semiconj g' \u2191fb \u2191fa\n[PROOFSTEP]\nrefine' fun y => (hg' _).unique _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\nfa : \u03b1 \u2243o \u03b1\nfb : \u03b2 \u21aao \u03b2\ng : \u03b1 \u2192 \u03b2\nh : Semiconj g \u2191fa \u2191fb\ng' : \u03b2 \u2192 \u03b1\nhg' : IsOrderRightAdjoint g g'\ny : \u03b2\n\u22a2 IsLUB {x | g x \u2264 \u2191fb y} (\u2191fa (g' y))\n[PROOFSTEP]\nrw [\u2190 fa.surjective.image_preimage {x | g x \u2264 fb y}, preimage_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\nfa : \u03b1 \u2243o \u03b1\nfb : \u03b2 \u21aao \u03b2\ng : \u03b1 \u2192 \u03b2\nh : Semiconj g \u2191fa \u2191fb\ng' : \u03b2 \u2192 \u03b1\nhg' : IsOrderRightAdjoint g g'\ny : \u03b2\n\u22a2 IsLUB (\u2191fa '' {a | g (\u2191fa a) \u2264 \u2191fb y}) (\u2191fa (g' y))\n[PROOFSTEP]\nsimp only [h.eq, fb.le_iff_le, fa.leftOrdContinuous (hg' _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : Type u_4\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* \u03b1 \u2243o \u03b1\nh : \u03b1 \u2192 \u03b1\nH : \u2200 (x : \u03b1), IsLUB (range fun g' => \u2191(\u2191f\u2081 g')\u207b\u00b9 (\u2191(\u2191f\u2082 g') x)) (h x)\ng : G\n\u22a2 Semiconj h \u2191(\u2191f\u2082 g) \u2191(\u2191f\u2081 g)\n[PROOFSTEP]\nrefine' fun y => (H _).unique _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : Type u_4\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* \u03b1 \u2243o \u03b1\nh : \u03b1 \u2192 \u03b1\nH : \u2200 (x : \u03b1), IsLUB (range fun g' => \u2191(\u2191f\u2081 g')\u207b\u00b9 (\u2191(\u2191f\u2082 g') x)) (h x)\ng : G\ny : \u03b1\n\u22a2 IsLUB (range fun g' => \u2191(\u2191f\u2081 g')\u207b\u00b9 (\u2191(\u2191f\u2082 g') (\u2191(\u2191f\u2082 g) y))) (\u2191(\u2191f\u2081 g) (h y))\n[PROOFSTEP]\nhave := (f\u2081 g).leftOrdContinuous (H y)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : Type u_4\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* \u03b1 \u2243o \u03b1\nh : \u03b1 \u2192 \u03b1\nH : \u2200 (x : \u03b1), IsLUB (range fun g' => \u2191(\u2191f\u2081 g')\u207b\u00b9 (\u2191(\u2191f\u2082 g') x)) (h x)\ng : G\ny : \u03b1\nthis : IsLUB (\u2191(\u2191f\u2081 g) '' range fun g' => \u2191(\u2191f\u2081 g')\u207b\u00b9 (\u2191(\u2191f\u2082 g') y)) (\u2191(\u2191f\u2081 g) (h y))\n\u22a2 IsLUB (range fun g' => \u2191(\u2191f\u2081 g')\u207b\u00b9 (\u2191(\u2191f\u2082 g') (\u2191(\u2191f\u2082 g) y))) (\u2191(\u2191f\u2081 g) (h y))\n[PROOFSTEP]\nrw [\u2190 range_comp, \u2190 (Equiv.mulRight g).surjective.range_comp _] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : Type u_4\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* \u03b1 \u2243o \u03b1\nh : \u03b1 \u2192 \u03b1\nH : \u2200 (x : \u03b1), IsLUB (range fun g' => \u2191(\u2191f\u2081 g')\u207b\u00b9 (\u2191(\u2191f\u2082 g') x)) (h x)\ng : G\ny : \u03b1\nthis : IsLUB (range ((\u2191(\u2191f\u2081 g) \u2218 fun g' => \u2191(\u2191f\u2081 g')\u207b\u00b9 (\u2191(\u2191f\u2082 g') y)) \u2218 \u2191(Equiv.mulRight g))) (\u2191(\u2191f\u2081 g) (h y))\n\u22a2 IsLUB (range fun g' => \u2191(\u2191f\u2081 g')\u207b\u00b9 (\u2191(\u2191f\u2082 g') (\u2191(\u2191f\u2082 g) y))) (\u2191(\u2191f\u2081 g) (h y))\n[PROOFSTEP]\nsimpa [(\u00b7 \u2218 \u00b7)] using this\n", "meta": {"mathlib_filename": "Mathlib.Order.SemiconjSup", "llama_tokens": 2127, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743735019595, "lm_q2_score": 0.6001883592602049, "lm_q1q2_score": 0.4442440427985911}} {"text": "[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 EDist (\u03b2 i)\nf g : PiLp \u22a4 \u03b2\n\u22a2 edist f g = \u2a06 (i : \u03b9), edist (f i) (g i)\n[PROOFSTEP]\ndsimp [edist]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 EDist (\u03b2 i)\nf g : PiLp \u22a4 \u03b2\n\u22a2 (if \u22a4 = 0 then \u2191(Finset.card (Finite.toFinset (_ : Set.Finite {i | f i \u2260 g i})))\n else if \u22a4 = \u22a4 then \u2a06 (i : \u03b9), edist (f i) (g i) else (\u2211 i : \u03b9, edist (f i) (g i) ^ 0) ^ (1 / 0)) =\n \u2a06 (i : \u03b9), edist (f i) (g i)\n[PROOFSTEP]\nexact if_neg ENNReal.top_ne_zero\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\nf : PiLp p \u03b2\n\u22a2 edist f f = 0\n[PROOFSTEP]\nrcases p.trichotomy with (rfl | rfl | h)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\nf : PiLp 0 \u03b2\n\u22a2 edist f f = 0\n[PROOFSTEP]\nsimp [edist_eq_card]\n[GOAL]\ncase inr.inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\nf : PiLp \u22a4 \u03b2\n\u22a2 edist f f = 0\n[PROOFSTEP]\nsimp [edist_eq_iSup]\n[GOAL]\ncase inr.inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\nf : PiLp p \u03b2\nh : 0 < ENNReal.toReal p\n\u22a2 edist f f = 0\n[PROOFSTEP]\nsimp [edist_eq_sum h, ENNReal.zero_rpow_of_pos h, ENNReal.zero_rpow_of_pos (inv_pos.2 <| h)]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\nf g : PiLp p \u03b2\n\u22a2 edist f g = edist g f\n[PROOFSTEP]\nrcases p.trichotomy with (rfl | rfl | h)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\nf g : PiLp 0 \u03b2\n\u22a2 edist f g = edist g f\n[PROOFSTEP]\nsimp only [edist_eq_card, eq_comm, Ne.def]\n[GOAL]\ncase inr.inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\nf g : PiLp \u22a4 \u03b2\n\u22a2 edist f g = edist g f\n[PROOFSTEP]\nsimp only [edist_eq_iSup, edist_comm]\n[GOAL]\ncase inr.inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\nf g : PiLp p \u03b2\nh : 0 < ENNReal.toReal p\n\u22a2 edist f g = edist g f\n[PROOFSTEP]\nsimp only [edist_eq_sum h, edist_comm]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 Dist (\u03b1 i)\nf g : PiLp \u22a4 \u03b1\n\u22a2 dist f g = \u2a06 (i : \u03b9), dist (f i) (g i)\n[PROOFSTEP]\ndsimp [dist]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 Dist (\u03b1 i)\nf g : PiLp \u22a4 \u03b1\n\u22a2 (if \u22a4 = 0 then \u2191(Finset.card (Finite.toFinset (_ : Set.Finite {i | f i \u2260 g i})))\n else if \u22a4 = \u22a4 then \u2a06 (i : \u03b9), dist (f i) (g i) else (\u2211 i : \u03b9, dist (f i) (g i) ^ 0) ^ (1 / 0)) =\n \u2a06 (i : \u03b9), dist (f i) (g i)\n[PROOFSTEP]\nexact if_neg ENNReal.top_ne_zero\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Norm (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nf : PiLp \u22a4 \u03b2\n\u22a2 \u2016f\u2016 = \u2a06 (i : \u03b9), \u2016f i\u2016\n[PROOFSTEP]\ndsimp [Norm.norm]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Norm (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nf : PiLp \u22a4 \u03b2\n\u22a2 (if \u22a4 = 0 then \u2191(Finset.card (Finite.toFinset (_ : Set.Finite {i | f i \u2260 0})))\n else if \u22a4 = \u22a4 then \u2a06 (i : \u03b9), \u2016f i\u2016 else (\u2211 i : \u03b9, \u2016f i\u2016 ^ 0) ^ (1 / 0)) =\n \u2a06 (i : \u03b9), \u2016f i\u2016\n[PROOFSTEP]\nexact if_neg ENNReal.top_ne_zero\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nf g h : PiLp p \u03b2\n\u22a2 edist f h \u2264 edist f g + edist g h\n[PROOFSTEP]\nrcases p.dichotomy with (rfl | hp)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g h : PiLp \u22a4 \u03b2\n\u22a2 edist f h \u2264 edist f g + edist g h\n[PROOFSTEP]\nsimp only [edist_eq_iSup]\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g h : PiLp \u22a4 \u03b2\n\u22a2 \u2a06 (i : \u03b9), edist (f i) (h i) \u2264 (\u2a06 (i : \u03b9), edist (f i) (g i)) + \u2a06 (i : \u03b9), edist (g i) (h i)\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b9\n[GOAL]\ncase inl.inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g h : PiLp \u22a4 \u03b2\nh\u271d : IsEmpty \u03b9\n\u22a2 \u2a06 (i : \u03b9), edist (f i) (h i) \u2264 (\u2a06 (i : \u03b9), edist (f i) (g i)) + \u2a06 (i : \u03b9), edist (g i) (h i)\n[PROOFSTEP]\nsimp only [ciSup_of_empty, ENNReal.bot_eq_zero, add_zero, nonpos_iff_eq_zero]\n -- Porting note: `le_iSup` needed some help\n[GOAL]\ncase inl.inr\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g h : PiLp \u22a4 \u03b2\nh\u271d : Nonempty \u03b9\n\u22a2 \u2a06 (i : \u03b9), edist (f i) (h i) \u2264 (\u2a06 (i : \u03b9), edist (f i) (g i)) + \u2a06 (i : \u03b9), edist (g i) (h i)\n[PROOFSTEP]\nrefine\n iSup_le fun i =>\n (edist_triangle _ (g i) _).trans <|\n add_le_add (le_iSup (fun k => edist (f k) (g k)) i) (le_iSup (fun k => edist (g k) (h k)) i)\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nf g h : PiLp p \u03b2\nhp : 1 \u2264 ENNReal.toReal p\n\u22a2 edist f h \u2264 edist f g + edist g h\n[PROOFSTEP]\nsimp only [edist_eq_sum (zero_lt_one.trans_le hp)]\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nf g h : PiLp p \u03b2\nhp : 1 \u2264 ENNReal.toReal p\n\u22a2 (\u2211 i : \u03b9, edist (f i) (h i) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) \u2264\n (\u2211 i : \u03b9, edist (f i) (g i) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) +\n (\u2211 i : \u03b9, edist (g i) (h i) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\ncalc\n (\u2211 i, edist (f i) (h i) ^ p.toReal) ^ (1 / p.toReal) \u2264\n (\u2211 i, (edist (f i) (g i) + edist (g i) (h i)) ^ p.toReal) ^ (1 / p.toReal) :=\n by\n apply ENNReal.rpow_le_rpow _ (one_div_nonneg.2 <| zero_le_one.trans hp)\n refine' Finset.sum_le_sum fun i _ => _\n exact ENNReal.rpow_le_rpow (edist_triangle _ _ _) (zero_le_one.trans hp)\n _ \u2264 (\u2211 i, edist (f i) (g i) ^ p.toReal) ^ (1 / p.toReal) + (\u2211 i, edist (g i) (h i) ^ p.toReal) ^ (1 / p.toReal) :=\n ENNReal.Lp_add_le _ _ _ hp\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nf g h : PiLp p \u03b2\nhp : 1 \u2264 ENNReal.toReal p\n\u22a2 (\u2211 i : \u03b9, edist (f i) (h i) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) \u2264\n (\u2211 i : \u03b9, (edist (f i) (g i) + edist (g i) (h i)) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\napply ENNReal.rpow_le_rpow _ (one_div_nonneg.2 <| zero_le_one.trans hp)\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nf g h : PiLp p \u03b2\nhp : 1 \u2264 ENNReal.toReal p\n\u22a2 \u2211 i : \u03b9, edist (f i) (h i) ^ ENNReal.toReal p \u2264 \u2211 i : \u03b9, (edist (f i) (g i) + edist (g i) (h i)) ^ ENNReal.toReal p\n[PROOFSTEP]\nrefine' Finset.sum_le_sum fun i _ => _\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nf g h : PiLp p \u03b2\nhp : 1 \u2264 ENNReal.toReal p\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 edist (f i) (h i) ^ ENNReal.toReal p \u2264 (edist (f i) (g i) + edist (g i) (h i)) ^ ENNReal.toReal p\n[PROOFSTEP]\nexact ENNReal.rpow_le_rpow (edist_triangle _ _ _) (zero_le_one.trans hp)\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9\u271d : Type u_3\n\u03b1\u271d : \u03b9\u271d \u2192 Type u_4\n\u03b2 : \u03b9\u271d \u2192 Type u_5\ninst\u271d\u2075 : Fact (1 \u2264 p)\ninst\u271d\u2074 : (i : \u03b9\u271d) \u2192 PseudoMetricSpace (\u03b1\u271d i)\ninst\u271d\u00b3 : (i : \u03b9\u271d) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b2 : Fintype \u03b9\u271d\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Finite \u03b9\n\u03b1 : \u03b9 \u2192 Type u_7\ninst\u271d : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\nf g : PiLp \u22a4 \u03b1\n\u22a2 \u2a06 (i : \u03b9), edist (f i) (g i) \u2260 \u22a4\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9\u271d : Type u_3\n\u03b1\u271d : \u03b9\u271d \u2192 Type u_4\n\u03b2 : \u03b9\u271d \u2192 Type u_5\ninst\u271d\u2075 : Fact (1 \u2264 p)\ninst\u271d\u2074 : (i : \u03b9\u271d) \u2192 PseudoMetricSpace (\u03b1\u271d i)\ninst\u271d\u00b3 : (i : \u03b9\u271d) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b2 : Fintype \u03b9\u271d\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Finite \u03b9\n\u03b1 : \u03b9 \u2192 Type u_7\ninst\u271d : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\nf g : PiLp \u22a4 \u03b1\nval\u271d : Fintype \u03b9\n\u22a2 \u2a06 (i : \u03b9), edist (f i) (g i) \u2260 \u22a4\n[PROOFSTEP]\nobtain \u27e8M, hM\u27e9 := Fintype.exists_le fun i => (\u27e8dist (f i) (g i), dist_nonneg\u27e9 : \u211d\u22650)\n[GOAL]\ncase intro.intro\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9\u271d : Type u_3\n\u03b1\u271d : \u03b9\u271d \u2192 Type u_4\n\u03b2 : \u03b9\u271d \u2192 Type u_5\ninst\u271d\u2075 : Fact (1 \u2264 p)\ninst\u271d\u2074 : (i : \u03b9\u271d) \u2192 PseudoMetricSpace (\u03b1\u271d i)\ninst\u271d\u00b3 : (i : \u03b9\u271d) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b2 : Fintype \u03b9\u271d\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Finite \u03b9\n\u03b1 : \u03b9 \u2192 Type u_7\ninst\u271d : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\nf g : PiLp \u22a4 \u03b1\nval\u271d : Fintype \u03b9\nM : { r // 0 \u2264 r }\nhM : \u2200 (i : \u03b9), { val := dist (f i) (g i), property := (_ : 0 \u2264 dist (f i) (g i)) } \u2264 M\n\u22a2 \u2a06 (i : \u03b9), edist (f i) (g i) \u2260 \u22a4\n[PROOFSTEP]\nrefine' ne_of_lt ((iSup_le fun i => _).trans_lt (@ENNReal.coe_lt_top M))\n[GOAL]\ncase intro.intro\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9\u271d : Type u_3\n\u03b1\u271d : \u03b9\u271d \u2192 Type u_4\n\u03b2 : \u03b9\u271d \u2192 Type u_5\ninst\u271d\u2075 : Fact (1 \u2264 p)\ninst\u271d\u2074 : (i : \u03b9\u271d) \u2192 PseudoMetricSpace (\u03b1\u271d i)\ninst\u271d\u00b3 : (i : \u03b9\u271d) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b2 : Fintype \u03b9\u271d\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Finite \u03b9\n\u03b1 : \u03b9 \u2192 Type u_7\ninst\u271d : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\nf g : PiLp \u22a4 \u03b1\nval\u271d : Fintype \u03b9\nM : { r // 0 \u2264 r }\nhM : \u2200 (i : \u03b9), { val := dist (f i) (g i), property := (_ : 0 \u2264 dist (f i) (g i)) } \u2264 M\ni : \u03b9\n\u22a2 edist (f i) (g i) \u2264 \u2191M\n[PROOFSTEP]\nsimp only [edist, PseudoMetricSpace.edist_dist, ENNReal.ofReal_eq_coe_nnreal dist_nonneg]\n[GOAL]\ncase intro.intro\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9\u271d : Type u_3\n\u03b1\u271d : \u03b9\u271d \u2192 Type u_4\n\u03b2 : \u03b9\u271d \u2192 Type u_5\ninst\u271d\u2075 : Fact (1 \u2264 p)\ninst\u271d\u2074 : (i : \u03b9\u271d) \u2192 PseudoMetricSpace (\u03b1\u271d i)\ninst\u271d\u00b3 : (i : \u03b9\u271d) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b2 : Fintype \u03b9\u271d\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Finite \u03b9\n\u03b1 : \u03b9 \u2192 Type u_7\ninst\u271d : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\nf g : PiLp \u22a4 \u03b1\nval\u271d : Fintype \u03b9\nM : { r // 0 \u2264 r }\nhM : \u2200 (i : \u03b9), { val := dist (f i) (g i), property := (_ : 0 \u2264 dist (f i) (g i)) } \u2264 M\ni : \u03b9\n\u22a2 \u2191{ val := dist (f i) (g i), property := (_ : 0 \u2264 dist (f i) (g i)) } \u2264 \u2191M\n[PROOFSTEP]\nexact_mod_cast hM i\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nf g : PiLp p \u03b1\n\u22a2 edist f g \u2260 \u22a4\n[PROOFSTEP]\nrcases p.dichotomy with (rfl | h)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g : PiLp \u22a4 \u03b1\n\u22a2 edist f g \u2260 \u22a4\n[PROOFSTEP]\nexact iSup_edist_ne_top_aux f g\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nf g : PiLp p \u03b1\nh : 1 \u2264 ENNReal.toReal p\n\u22a2 edist f g \u2260 \u22a4\n[PROOFSTEP]\nrw [edist_eq_sum (zero_lt_one.trans_le h)]\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nf g : PiLp p \u03b1\nh : 1 \u2264 ENNReal.toReal p\n\u22a2 (\u2211 i : \u03b9, edist (f i) (g i) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) \u2260 \u22a4\n[PROOFSTEP]\nexact\n ENNReal.rpow_ne_top_of_nonneg (one_div_nonneg.2 (zero_le_one.trans h))\n (ne_of_lt <| ENNReal.sum_lt_top fun i hi => ENNReal.rpow_ne_top_of_nonneg (zero_le_one.trans h) (edist_ne_top _ _))\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nf g : PiLp p \u03b1\n\u22a2 dist f g = ENNReal.toReal (edist f g)\n[PROOFSTEP]\nrcases p.dichotomy with (rfl | h)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g : PiLp \u22a4 \u03b1\n\u22a2 dist f g = ENNReal.toReal (edist f g)\n[PROOFSTEP]\nrw [edist_eq_iSup, dist_eq_iSup]\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g : PiLp \u22a4 \u03b1\n\u22a2 \u2a06 (i : \u03b9), dist (f i) (g i) = ENNReal.toReal (\u2a06 (i : \u03b9), edist (f i) (g i))\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b9\n[GOAL]\ncase inl.inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g : PiLp \u22a4 \u03b1\nh\u271d : IsEmpty \u03b9\n\u22a2 \u2a06 (i : \u03b9), dist (f i) (g i) = ENNReal.toReal (\u2a06 (i : \u03b9), edist (f i) (g i))\n[PROOFSTEP]\nsimp only [Real.ciSup_empty, ciSup_of_empty, ENNReal.bot_eq_zero, ENNReal.zero_toReal]\n[GOAL]\ncase inl.inr\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g : PiLp \u22a4 \u03b1\nh\u271d : Nonempty \u03b9\n\u22a2 \u2a06 (i : \u03b9), dist (f i) (g i) = ENNReal.toReal (\u2a06 (i : \u03b9), edist (f i) (g i))\n[PROOFSTEP]\nrefine' le_antisymm (ciSup_le fun i => _) _\n[GOAL]\ncase inl.inr.refine'_1\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g : PiLp \u22a4 \u03b1\nh\u271d : Nonempty \u03b9\ni : \u03b9\n\u22a2 dist (f i) (g i) \u2264 ENNReal.toReal (\u2a06 (i : \u03b9), edist (f i) (g i))\n[PROOFSTEP]\nrw [\u2190 ENNReal.ofReal_le_iff_le_toReal (iSup_edist_ne_top_aux f g), \u2190 PseudoMetricSpace.edist_dist]\n -- Porting note: `le_iSup` needed some help\n[GOAL]\ncase inl.inr.refine'_1\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g : PiLp \u22a4 \u03b1\nh\u271d : Nonempty \u03b9\ni : \u03b9\n\u22a2 PseudoMetricSpace.edist (f i) (g i) \u2264 \u2a06 (i : \u03b9), edist (f i) (g i)\n[PROOFSTEP]\nexact le_iSup (fun k => edist (f k) (g k)) i\n[GOAL]\ncase inl.inr.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g : PiLp \u22a4 \u03b1\nh\u271d : Nonempty \u03b9\n\u22a2 ENNReal.toReal (\u2a06 (i : \u03b9), edist (f i) (g i)) \u2264 \u2a06 (i : \u03b9), dist (f i) (g i)\n[PROOFSTEP]\nrefine' ENNReal.toReal_le_of_le_ofReal (Real.sSup_nonneg _ _) (iSup_le fun i => _)\n[GOAL]\ncase inl.inr.refine'_2.refine'_1\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g : PiLp \u22a4 \u03b1\nh\u271d : Nonempty \u03b9\n\u22a2 \u2200 (x : \u211d), (x \u2208 range fun i => dist (f i) (g i)) \u2192 0 \u2264 x\n[PROOFSTEP]\nrintro - \u27e8i, rfl\u27e9\n[GOAL]\ncase inl.inr.refine'_2.refine'_1.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g : PiLp \u22a4 \u03b1\nh\u271d : Nonempty \u03b9\ni : \u03b9\n\u22a2 0 \u2264 (fun i => dist (f i) (g i)) i\n[PROOFSTEP]\nexact dist_nonneg\n[GOAL]\ncase inl.inr.refine'_2.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g : PiLp \u22a4 \u03b1\nh\u271d : Nonempty \u03b9\ni : \u03b9\n\u22a2 edist (f i) (g i) \u2264 ENNReal.ofReal (\u2a06 (i : \u03b9), dist (f i) (g i))\n[PROOFSTEP]\nchange PseudoMetricSpace.edist _ _ \u2264 _\n[GOAL]\ncase inl.inr.refine'_2.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g : PiLp \u22a4 \u03b1\nh\u271d : Nonempty \u03b9\ni : \u03b9\n\u22a2 PseudoMetricSpace.edist (f i) (g i) \u2264 ENNReal.ofReal (\u2a06 (i : \u03b9), dist (f i) (g i))\n[PROOFSTEP]\nrw [PseudoMetricSpace.edist_dist]\n -- Porting note: `le_ciSup` needed some help\n[GOAL]\ncase inl.inr.refine'_2.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nf g : PiLp \u22a4 \u03b1\nh\u271d : Nonempty \u03b9\ni : \u03b9\n\u22a2 ENNReal.ofReal (dist (f i) (g i)) \u2264 ENNReal.ofReal (\u2a06 (i : \u03b9), dist (f i) (g i))\n[PROOFSTEP]\nexact ENNReal.ofReal_le_ofReal (le_ciSup (Fintype.bddAbove_range (fun k => dist (f k) (g k))) i)\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nf g : PiLp p \u03b1\nh : 1 \u2264 ENNReal.toReal p\n\u22a2 dist f g = ENNReal.toReal (edist f g)\n[PROOFSTEP]\nhave A : \u2200 i, edist (f i) (g i) ^ p.toReal \u2260 \u22a4 := fun i =>\n ENNReal.rpow_ne_top_of_nonneg (zero_le_one.trans h) (edist_ne_top _ _)\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nf g : PiLp p \u03b1\nh : 1 \u2264 ENNReal.toReal p\nA : \u2200 (i : \u03b9), edist (f i) (g i) ^ ENNReal.toReal p \u2260 \u22a4\n\u22a2 dist f g = ENNReal.toReal (edist f g)\n[PROOFSTEP]\nsimp only [edist_eq_sum (zero_lt_one.trans_le h), dist_edist, ENNReal.toReal_rpow, dist_eq_sum (zero_lt_one.trans_le h),\n \u2190 ENNReal.toReal_sum fun i _ => A i]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\n\u22a2 LipschitzWith 1 \u2191(equiv p \u03b2)\n[PROOFSTEP]\nintro x y\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\n\u22a2 edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y) \u2264 \u21911 * edist x y\n[PROOFSTEP]\nrcases p.dichotomy with (rfl | h)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nx y : PiLp \u22a4 \u03b2\n\u22a2 edist (\u2191(equiv \u22a4 \u03b2) x) (\u2191(equiv \u22a4 \u03b2) y) \u2264 \u21911 * edist x y\n[PROOFSTEP]\nsimpa only [ENNReal.coe_one, one_mul, edist_eq_iSup, edist, Finset.sup_le_iff, Finset.mem_univ, forall_true_left] using\n le_iSup fun i => edist (x i) (y i)\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\n\u22a2 edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y) \u2264 \u21911 * edist x y\n[PROOFSTEP]\nhave cancel : p.toReal * (1 / p.toReal) = 1 := mul_div_cancel' 1 (zero_lt_one.trans_le h).ne'\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y) \u2264 \u21911 * edist x y\n[PROOFSTEP]\nrw [edist_eq_sum (zero_lt_one.trans_le h)]\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y) \u2264\n \u21911 * (\u2211 i : \u03b9, edist (x i) (y i) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nsimp only [edist, forall_prop_of_true, one_mul, Finset.mem_univ, Finset.sup_le_iff, ENNReal.coe_one]\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 \u2200 (b : \u03b9),\n edist (\u2191(equiv p \u03b2) x b) (\u2191(equiv p \u03b2) y b) \u2264\n (\u2211 x_1 : \u03b9, edist (x x_1) (y x_1) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\ni : \u03b9\n\u22a2 edist (\u2191(equiv p \u03b2) x i) (\u2191(equiv p \u03b2) y i) \u2264\n (\u2211 x_1 : \u03b9, edist (x x_1) (y x_1) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\ncalc\n edist (x i) (y i) = (edist (x i) (y i) ^ p.toReal) ^ (1 / p.toReal) := by simp [\u2190 ENNReal.rpow_mul, cancel, -one_div]\n _ \u2264 (\u2211 i, edist (x i) (y i) ^ p.toReal) ^ (1 / p.toReal) :=\n by\n apply ENNReal.rpow_le_rpow _ (one_div_nonneg.2 <| zero_le_one.trans h)\n exact Finset.single_le_sum (fun i _ => (bot_le : (0 : \u211d\u22650\u221e) \u2264 _)) (Finset.mem_univ i)\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\ni : \u03b9\n\u22a2 edist (x i) (y i) = (edist (x i) (y i) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nsimp [\u2190 ENNReal.rpow_mul, cancel, -one_div]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\ni : \u03b9\n\u22a2 (edist (x i) (y i) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) \u2264\n (\u2211 i : \u03b9, edist (x i) (y i) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\napply ENNReal.rpow_le_rpow _ (one_div_nonneg.2 <| zero_le_one.trans h)\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\ni : \u03b9\n\u22a2 edist (x i) (y i) ^ ENNReal.toReal p \u2264 \u2211 i : \u03b9, edist (x i) (y i) ^ ENNReal.toReal p\n[PROOFSTEP]\nexact Finset.single_le_sum (fun i _ => (bot_le : (0 : \u211d\u22650\u221e) \u2264 _)) (Finset.mem_univ i)\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\n\u22a2 AntilipschitzWith (\u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p)) \u2191(equiv p \u03b2)\n[PROOFSTEP]\nintro x y\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\n\u22a2 edist x y \u2264 \u2191(\u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p)) * edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y)\n[PROOFSTEP]\nrcases p.dichotomy with (rfl | h)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nx y : PiLp \u22a4 \u03b2\n\u22a2 edist x y \u2264 \u2191(\u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / \u22a4)) * edist (\u2191(equiv \u22a4 \u03b2) x) (\u2191(equiv \u22a4 \u03b2) y)\n[PROOFSTEP]\nsimp only [edist_eq_iSup, ENNReal.div_top, ENNReal.zero_toReal, NNReal.rpow_zero, ENNReal.coe_one, one_mul, iSup_le_iff]\n -- Porting note: `Finset.le_sup` needed some help\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : Fact (1 \u2264 \u22a4)\nx y : PiLp \u22a4 \u03b2\n\u22a2 \u2200 (i : \u03b9), edist (x i) (y i) \u2264 edist (\u2191(equiv \u22a4 \u03b2) x) (\u2191(equiv \u22a4 \u03b2) y)\n[PROOFSTEP]\nexact fun i => Finset.le_sup (f := fun i => edist (x i) (y i)) (Finset.mem_univ i)\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\n\u22a2 edist x y \u2264 \u2191(\u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p)) * edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y)\n[PROOFSTEP]\nhave pos : 0 < p.toReal := zero_lt_one.trans_le h\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\n\u22a2 edist x y \u2264 \u2191(\u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p)) * edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y)\n[PROOFSTEP]\nhave nonneg : 0 \u2264 1 / p.toReal := one_div_nonneg.2 (le_of_lt pos)\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\n\u22a2 edist x y \u2264 \u2191(\u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p)) * edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y)\n[PROOFSTEP]\nhave cancel : p.toReal * (1 / p.toReal) = 1 := mul_div_cancel' 1 (ne_of_gt pos)\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 edist x y \u2264 \u2191(\u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p)) * edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y)\n[PROOFSTEP]\nrw [edist_eq_sum pos, ENNReal.toReal_div 1 p]\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 (\u2211 i : \u03b9, edist (x i) (y i) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) \u2264\n \u2191(\u2191(Fintype.card \u03b9) ^ (ENNReal.toReal 1 / ENNReal.toReal p)) * edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y)\n[PROOFSTEP]\nsimp only [edist, \u2190 one_div, ENNReal.one_toReal]\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 (\u2211 x_1 : \u03b9, edist (x x_1) (y x_1) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) \u2264\n \u2191(\u2191(Fintype.card \u03b9) ^ (1 / ENNReal.toReal p)) *\n Finset.sup Finset.univ fun b => edist (\u2191(equiv p \u03b2) x b) (\u2191(equiv p \u03b2) y b)\n[PROOFSTEP]\ncalc\n (\u2211 i, edist (x i) (y i) ^ p.toReal) ^ (1 / p.toReal) \u2264\n (\u2211 _i, edist (PiLp.equiv p \u03b2 x) (PiLp.equiv p \u03b2 y) ^ p.toReal) ^ (1 / p.toReal) :=\n by\n refine ENNReal.rpow_le_rpow ?_ nonneg\n swap\n refine Finset.sum_le_sum fun i _ => ?_\n apply ENNReal.rpow_le_rpow _ (le_of_lt pos)\n exact Finset.le_sup (f := fun i => edist (x i) (y i)) (Finset.mem_univ i)\n _ = ((Fintype.card \u03b9 : \u211d\u22650) ^ (1 / p.toReal) : \u211d\u22650) * edist (PiLp.equiv p \u03b2 x) (PiLp.equiv p \u03b2 y) :=\n by\n simp only [nsmul_eq_mul, Finset.card_univ, ENNReal.rpow_one, Finset.sum_const,\n ENNReal.mul_rpow_of_nonneg _ _ nonneg, \u2190 ENNReal.rpow_mul, cancel]\n have : (Fintype.card \u03b9 : \u211d\u22650\u221e) = (Fintype.card \u03b9 : \u211d\u22650) := (ENNReal.coe_nat (Fintype.card \u03b9)).symm\n rw [this, ENNReal.coe_rpow_of_nonneg _ nonneg]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 (\u2211 i : \u03b9, edist (x i) (y i) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) \u2264\n (\u2211 _i : ?m.67030, edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nrefine ENNReal.rpow_le_rpow ?_ nonneg\n[GOAL]\ncase refine_1\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 Fintype ?m.67030\ncase refine_2\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 \u2211 i : \u03b9, edist (x i) (y i) ^ ENNReal.toReal p \u2264\n \u2211 _i : ?m.67030, edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y) ^ ENNReal.toReal p\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine_2\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 \u2211 i : \u03b9, edist (x i) (y i) ^ ENNReal.toReal p \u2264\n \u2211 _i : ?m.67030, edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y) ^ ENNReal.toReal p\ncase refine_1\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 Fintype ?m.67030\n[PROOFSTEP]\nrefine Finset.sum_le_sum fun i _ => ?_\n[GOAL]\ncase refine_2\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 edist (x i) (y i) ^ ENNReal.toReal p \u2264 edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y) ^ ENNReal.toReal p\n[PROOFSTEP]\napply ENNReal.rpow_le_rpow _ (le_of_lt pos)\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 edist (x i) (y i) \u2264 edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y)\n[PROOFSTEP]\nexact Finset.le_sup (f := fun i => edist (x i) (y i)) (Finset.mem_univ i)\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 (\u2211 _i : \u03b9, edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) =\n \u2191(\u2191(Fintype.card \u03b9) ^ (1 / ENNReal.toReal p)) * edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y)\n[PROOFSTEP]\nsimp only [nsmul_eq_mul, Finset.card_univ, ENNReal.rpow_one, Finset.sum_const, ENNReal.mul_rpow_of_nonneg _ _ nonneg, \u2190\n ENNReal.rpow_mul, cancel]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 \u2191(Fintype.card \u03b9) ^ (1 / ENNReal.toReal p) * edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y) =\n \u2191(\u2191(Fintype.card \u03b9) ^ (1 / ENNReal.toReal p)) * edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y)\n[PROOFSTEP]\nhave : (Fintype.card \u03b9 : \u211d\u22650\u221e) = (Fintype.card \u03b9 : \u211d\u22650) := (ENNReal.coe_nat (Fintype.card \u03b9)).symm\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\nthis : \u2191(Fintype.card \u03b9) = \u2191\u2191(Fintype.card \u03b9)\n\u22a2 \u2191(Fintype.card \u03b9) ^ (1 / ENNReal.toReal p) * edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y) =\n \u2191(\u2191(Fintype.card \u03b9) ^ (1 / ENNReal.toReal p)) * edist (\u2191(equiv p \u03b2) x) (\u2191(equiv p \u03b2) y)\n[PROOFSTEP]\nrw [this, ENNReal.coe_rpow_of_nonneg _ nonneg]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\n\u22a2 \ud835\udce4 (PiLp p \u03b2) = \ud835\udce4 ((i : \u03b9) \u2192 \u03b2 i)\n[PROOFSTEP]\nhave A : UniformInducing (PiLp.equiv p \u03b2) :=\n (antilipschitzWith_equiv_aux p \u03b2).uniformInducing (lipschitzWith_equiv_aux p \u03b2).uniformContinuous\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nA : UniformInducing \u2191(equiv p \u03b2)\n\u22a2 \ud835\udce4 (PiLp p \u03b2) = \ud835\udce4 ((i : \u03b9) \u2192 \u03b2 i)\n[PROOFSTEP]\nhave : (fun x : PiLp p \u03b2 \u00d7 PiLp p \u03b2 => ((PiLp.equiv p \u03b2) x.fst, (PiLp.equiv p \u03b2) x.snd)) = id := by ext i <;> rfl\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nA : UniformInducing \u2191(equiv p \u03b2)\n\u22a2 (fun x => (\u2191(equiv p \u03b2) x.fst, \u2191(equiv p \u03b2) x.snd)) = id\n[PROOFSTEP]\next i\n[GOAL]\ncase h.h\u2081.h\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nA : UniformInducing \u2191(equiv p \u03b2)\ni : PiLp p \u03b2 \u00d7 PiLp p \u03b2\nx\u271d : \u03b9\n\u22a2 Prod.fst (\u2191(equiv p \u03b2) i.fst, \u2191(equiv p \u03b2) i.snd) x\u271d = Prod.fst (id i) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.h\u2082.h\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nA : UniformInducing \u2191(equiv p \u03b2)\ni : PiLp p \u03b2 \u00d7 PiLp p \u03b2\nx\u271d : \u03b9\n\u22a2 Prod.snd (\u2191(equiv p \u03b2) i.fst, \u2191(equiv p \u03b2) i.snd) x\u271d = Prod.snd (id i) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fact (1 \u2264 p)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\ninst\u271d : Fintype \u03b9\nA : UniformInducing \u2191(equiv p \u03b2)\nthis : (fun x => (\u2191(equiv p \u03b2) x.fst, \u2191(equiv p \u03b2) x.snd)) = id\n\u22a2 \ud835\udce4 (PiLp p \u03b2) = \ud835\udce4 ((i : \u03b9) \u2192 \u03b2 i)\n[PROOFSTEP]\nrw [\u2190 A.comap_uniformity, this, comap_id]\n[GOAL]\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fact (1 \u2264 p\u271d)\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b2 i)\nhp : p \u2260 \u22a4\nx y : PiLp p \u03b2\n\u22a2 \u2191(nndist x y) = \u2191((\u2211 i : \u03b9, nndist (x i) (y i) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p))\n[PROOFSTEP]\npush_cast\n[GOAL]\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fact (1 \u2264 p\u271d)\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b2 i)\nhp : p \u2260 \u22a4\nx y : PiLp p \u03b2\n\u22a2 dist x y = (\u2211 x_1 : \u03b9, dist (x x_1) (y x_1) ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nexact dist_eq_sum (p.toReal_pos_iff_ne_top.mpr hp) _ _\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b2 i)\nx y : PiLp \u22a4 \u03b2\n\u22a2 \u2191(nndist x y) = \u2191(\u2a06 (i : \u03b9), nndist (x i) (y i))\n[PROOFSTEP]\npush_cast\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 PseudoMetricSpace (\u03b2 i)\nx y : PiLp \u22a4 \u03b2\n\u22a2 dist x y = \u2a06 (i : \u03b9), dist (x i) (y i)\n[PROOFSTEP]\nexact dist_eq_iSup _ _\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\nx y : PiLp \u22a4 \u03b2\n\u22a2 edist (\u2191(equiv \u22a4 \u03b2) x) (\u2191(equiv \u22a4 \u03b2) y) \u2264 edist x y\n[PROOFSTEP]\nsimpa only [ENNReal.coe_one, one_mul] using lipschitzWith_equiv \u221e \u03b2 x y\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b2 i)\nx y : PiLp \u22a4 \u03b2\n\u22a2 edist x y \u2264 edist (\u2191(equiv \u22a4 \u03b2) x) (\u2191(equiv \u22a4 \u03b2) y)\n[PROOFSTEP]\nsimpa only [ENNReal.div_top, ENNReal.zero_toReal, NNReal.rpow_zero, ENNReal.coe_one, one_mul] using\n antilipschitzWith_equiv \u221e \u03b2 x y\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nsrc\u271d : AddCommGroup ((i : \u03b9) \u2192 \u03b2 i) := Pi.addCommGroup\nx y : PiLp p \u03b2\n\u22a2 dist x y = \u2016x - y\u2016\n[PROOFSTEP]\nrcases p.dichotomy with (rfl | h)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nsrc\u271d : AddCommGroup ((i : \u03b9) \u2192 \u03b2 i) := Pi.addCommGroup\ninst\u271d : Fact (1 \u2264 \u22a4)\nx y : PiLp \u22a4 \u03b2\n\u22a2 dist x y = \u2016x - y\u2016\n[PROOFSTEP]\nsimp only [dist_eq_iSup, norm_eq_ciSup, dist_eq_norm]\n -- Porting note: added\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nsrc\u271d : AddCommGroup ((i : \u03b9) \u2192 \u03b2 i) := Pi.addCommGroup\ninst\u271d : Fact (1 \u2264 \u22a4)\nx y : PiLp \u22a4 \u03b2\n\u22a2 \u2a06 (i : \u03b9), \u2016x i - y i\u2016 = \u2a06 (i : \u03b9), \u2016(x - y) i\u2016\n[PROOFSTEP]\ncongr\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nsrc\u271d : AddCommGroup ((i : \u03b9) \u2192 \u03b2 i) := Pi.addCommGroup\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\n\u22a2 dist x y = \u2016x - y\u2016\n[PROOFSTEP]\nhave : p \u2260 \u221e := by\n intro hp\n rw [hp, ENNReal.top_toReal] at h \n linarith\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nsrc\u271d : AddCommGroup ((i : \u03b9) \u2192 \u03b2 i) := Pi.addCommGroup\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\n\u22a2 p \u2260 \u22a4\n[PROOFSTEP]\nintro hp\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nsrc\u271d : AddCommGroup ((i : \u03b9) \u2192 \u03b2 i) := Pi.addCommGroup\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\nhp : p = \u22a4\n\u22a2 False\n[PROOFSTEP]\nrw [hp, ENNReal.top_toReal] at h \n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nsrc\u271d : AddCommGroup ((i : \u03b9) \u2192 \u03b2 i) := Pi.addCommGroup\nx y : PiLp p \u03b2\nh : 1 \u2264 0\nhp : p = \u22a4\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nsrc\u271d : AddCommGroup ((i : \u03b9) \u2192 \u03b2 i) := Pi.addCommGroup\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\nthis : p \u2260 \u22a4\n\u22a2 dist x y = \u2016x - y\u2016\n[PROOFSTEP]\nsimp only [dist_eq_sum (zero_lt_one.trans_le h), norm_eq_sum (zero_lt_one.trans_le h), dist_eq_norm]\n -- Porting note: added\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nsrc\u271d : AddCommGroup ((i : \u03b9) \u2192 \u03b2 i) := Pi.addCommGroup\nx y : PiLp p \u03b2\nh : 1 \u2264 ENNReal.toReal p\nthis : p \u2260 \u22a4\n\u22a2 (\u2211 x_1 : \u03b9, \u2016x x_1 - y x_1\u2016 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) =\n (\u2211 i : \u03b9, \u2016(x - y) i\u2016 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\ncongr\n[GOAL]\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fact (1 \u2264 p\u271d)\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\nhp : p \u2260 \u22a4\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nf : PiLp p \u03b2\n\u22a2 \u2016f\u2016\u208a = (\u2211 i : \u03b9, \u2016f i\u2016\u208a ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\next\n[GOAL]\ncase a\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fact (1 \u2264 p\u271d)\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\nhp : p \u2260 \u22a4\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nf : PiLp p \u03b2\n\u22a2 \u2191\u2016f\u2016\u208a = \u2191((\u2211 i : \u03b9, \u2016f i\u2016\u208a ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p))\n[PROOFSTEP]\nsimp [NNReal.coe_sum, norm_eq_sum (p.toReal_pos_iff_ne_top.mpr hp)]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nf : PiLp \u22a4 \u03b2\n\u22a2 \u2016f\u2016\u208a = \u2a06 (i : \u03b9), \u2016f i\u2016\u208a\n[PROOFSTEP]\next\n[GOAL]\ncase a\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nf : PiLp \u22a4 \u03b2\n\u22a2 \u2191\u2016f\u2016\u208a = \u2191(\u2a06 (i : \u03b9), \u2016f i\u2016\u208a)\n[PROOFSTEP]\nsimp [NNReal.coe_iSup, norm_eq_ciSup]\n[GOAL]\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fact (1 \u2264 p\u271d)\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nn : \u2115\nh : p = \u2191n\nf : PiLp p \u03b2\n\u22a2 \u2016f\u2016 = (\u2211 i : \u03b9, \u2016f i\u2016 ^ n) ^ (1 / \u2191n)\n[PROOFSTEP]\nhave := p.toReal_pos_iff_ne_top.mpr (ne_of_eq_of_ne h <| ENNReal.nat_ne_top n)\n[GOAL]\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fact (1 \u2264 p\u271d)\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nn : \u2115\nh : p = \u2191n\nf : PiLp p \u03b2\nthis : 0 < ENNReal.toReal p\n\u22a2 \u2016f\u2016 = (\u2211 i : \u03b9, \u2016f i\u2016 ^ n) ^ (1 / \u2191n)\n[PROOFSTEP]\nsimp only [one_div, h, Real.rpow_nat_cast, ENNReal.toReal_nat, eq_self_iff_true, Finset.sum_congr, norm_eq_sum this]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nx : PiLp 2 \u03b2\n\u22a2 \u2016x\u2016 = Real.sqrt (\u2211 i : \u03b9, \u2016x i\u2016 ^ 2)\n[PROOFSTEP]\nrw [norm_eq_of_nat 2 (by norm_cast) _]\n -- Porting note: was `convert`\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nx : PiLp 2 \u03b2\n\u22a2 2 = \u21912\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nx : PiLp 2 \u03b2\n\u22a2 (\u2211 i : \u03b9, \u2016x i\u2016 ^ 2) ^ (1 / \u21912) = Real.sqrt (\u2211 i : \u03b9, \u2016x i\u2016 ^ 2)\n[PROOFSTEP]\nrw [Real.sqrt_eq_rpow]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nx : PiLp 2 \u03b2\n\u22a2 (\u2211 i : \u03b9, \u2016x i\u2016 ^ 2) ^ (1 / \u21912) = (\u2211 i : \u03b9, \u2016x i\u2016 ^ 2) ^ (1 / 2)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nx : PiLp 2 \u03b2\n\u22a2 \u2191\u2016x\u2016\u208a = \u2191(\u2191NNReal.sqrt (\u2211 i : \u03b9, \u2016x i\u2016\u208a ^ 2))\n[PROOFSTEP]\npush_cast\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nx : PiLp 2 \u03b2\n\u22a2 \u2016x\u2016 = Real.sqrt (\u2211 x_1 : \u03b9, \u2016x x_1\u2016 ^ 2)\n[PROOFSTEP]\nexact norm_eq_of_L2 x\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nx : PiLp 2 \u03b2\n\u22a2 \u2016x\u2016 ^ 2 = \u2211 i : \u03b9, \u2016x i\u2016 ^ 2\n[PROOFSTEP]\nsuffices \u2016x\u2016\u208a ^ 2 = \u2211 i : \u03b9, \u2016x i\u2016\u208a ^ 2 by simpa only [NNReal.coe_sum] using congr_arg ((\u2191) : \u211d\u22650 \u2192 \u211d) this\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nx : PiLp 2 \u03b2\nthis : \u2016x\u2016\u208a ^ 2 = \u2211 i : \u03b9, \u2016x i\u2016\u208a ^ 2\n\u22a2 \u2016x\u2016 ^ 2 = \u2211 i : \u03b9, \u2016x i\u2016 ^ 2\n[PROOFSTEP]\nsimpa only [NNReal.coe_sum] using congr_arg ((\u2191) : \u211d\u22650 \u2192 \u211d) this\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nx : PiLp 2 \u03b2\n\u22a2 \u2016x\u2016\u208a ^ 2 = \u2211 i : \u03b9, \u2016x i\u2016\u208a ^ 2\n[PROOFSTEP]\nrw [nnnorm_eq_of_L2, NNReal.sq_sqrt]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nx y : PiLp 2 \u03b2\n\u22a2 dist x y = Real.sqrt (\u2211 i : \u03b9, dist (x i) (y i) ^ 2)\n[PROOFSTEP]\nsimp_rw [dist_eq_norm, norm_eq_of_L2]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nx y : PiLp 2 \u03b2\n\u22a2 Real.sqrt (\u2211 i : \u03b9, \u2016(x - y) i\u2016 ^ 2) = Real.sqrt (\u2211 x_1 : \u03b9, \u2016x x_1 - y x_1\u2016 ^ 2)\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nx y : PiLp 2 \u03b2\n\u22a2 \u2191(nndist x y) = \u2191(\u2191NNReal.sqrt (\u2211 i : \u03b9, nndist (x i) (y i) ^ 2))\n[PROOFSTEP]\npush_cast\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nx y : PiLp 2 \u03b2\n\u22a2 dist x y = Real.sqrt (\u2211 x_1 : \u03b9, dist (x x_1) (y x_1) ^ 2)\n[PROOFSTEP]\nexact dist_eq_of_L2 _ _\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : Fact (1 \u2264 p)\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\nx y : PiLp 2 \u03b2\n\u22a2 edist x y = (\u2211 i : \u03b9, edist (x i) (y i) ^ 2) ^ (1 / 2)\n[PROOFSTEP]\nsimp [PiLp.edist_eq_sum]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fact (1 \u2264 p)\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nf : PiLp p \u03b2\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nrcases p.dichotomy with (rfl | hp)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : Fact (1 \u2264 \u22a4)\nf : PiLp \u22a4 \u03b2\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nletI : Module \ud835\udd5c (PiLp \u221e \u03b2) := Pi.module \u03b9 \u03b2 \ud835\udd5c\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : Fact (1 \u2264 \u22a4)\nf : PiLp \u22a4 \u03b2\nthis : Module \ud835\udd5c (PiLp \u22a4 \u03b2) := Pi.module \u03b9 \u03b2 \ud835\udd5c\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nsuffices \u2016c \u2022 f\u2016\u208a = \u2016c\u2016\u208a * \u2016f\u2016\u208a by exact_mod_cast NNReal.coe_mono this.le\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : Fact (1 \u2264 \u22a4)\nf : PiLp \u22a4 \u03b2\nthis\u271d : Module \ud835\udd5c (PiLp \u22a4 \u03b2) := Pi.module \u03b9 \u03b2 \ud835\udd5c\nthis : \u2016c \u2022 f\u2016\u208a = \u2016c\u2016\u208a * \u2016f\u2016\u208a\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nexact_mod_cast NNReal.coe_mono this.le\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : Fact (1 \u2264 \u22a4)\nf : PiLp \u22a4 \u03b2\nthis : Module \ud835\udd5c (PiLp \u22a4 \u03b2) := Pi.module \u03b9 \u03b2 \ud835\udd5c\n\u22a2 \u2016c \u2022 f\u2016\u208a = \u2016c\u2016\u208a * \u2016f\u2016\u208a\n[PROOFSTEP]\nsimp only [nnnorm_eq_ciSup, NNReal.mul_iSup, \u2190 nnnorm_smul]\n -- Porting note: added\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : Fact (1 \u2264 \u22a4)\nf : PiLp \u22a4 \u03b2\nthis : Module \ud835\udd5c (PiLp \u22a4 \u03b2) := Pi.module \u03b9 \u03b2 \ud835\udd5c\n\u22a2 \u2a06 (i : \u03b9), \u2016(c \u2022 f) i\u2016\u208a = \u2a06 (i : \u03b9), \u2016c \u2022 f i\u2016\u208a\n[PROOFSTEP]\ncongr\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fact (1 \u2264 p)\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nf : PiLp p \u03b2\nhp : 1 \u2264 ENNReal.toReal p\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nhave : p.toReal * (1 / p.toReal) = 1 := mul_div_cancel' 1 (zero_lt_one.trans_le hp).ne'\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fact (1 \u2264 p)\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nf : PiLp p \u03b2\nhp : 1 \u2264 ENNReal.toReal p\nthis : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nhave smul_apply : \u2200 i : \u03b9, (c \u2022 f) i = c \u2022 (f i) := fun i => rfl\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fact (1 \u2264 p)\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nf : PiLp p \u03b2\nhp : 1 \u2264 ENNReal.toReal p\nthis : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\nsmul_apply : \u2200 (i : \u03b9), (c \u2022 f) i = c \u2022 f i\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nsimp only [norm_eq_sum (zero_lt_one.trans_le hp), norm_smul, Real.mul_rpow, norm_nonneg, \u2190 Finset.mul_sum, smul_apply]\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fact (1 \u2264 p)\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nf : PiLp p \u03b2\nhp : 1 \u2264 ENNReal.toReal p\nthis : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\nsmul_apply : \u2200 (i : \u03b9), (c \u2022 f) i = c \u2022 f i\n\u22a2 (\u2016c\u2016 ^ ENNReal.toReal p * \u2211 x : \u03b9, \u2016f x\u2016 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) \u2264\n \u2016c\u2016 * (\u2211 x : \u03b9, \u2016f x\u2016 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nrw [mul_rpow (rpow_nonneg_of_nonneg (norm_nonneg _) _), \u2190 rpow_mul (norm_nonneg _), this, Real.rpow_one]\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fact (1 \u2264 p)\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nf : PiLp p \u03b2\nhp : 1 \u2264 ENNReal.toReal p\nthis : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\nsmul_apply : \u2200 (i : \u03b9), (c \u2022 f) i = c \u2022 f i\n\u22a2 0 \u2264 \u2211 x : \u03b9, \u2016f x\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nexact Finset.sum_nonneg fun i _ => rpow_nonneg_of_nonneg (norm_nonneg _) _\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fact (1 \u2264 p)\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\nsrc\u271d : PiLp \u22a4 \u03b2 \u2243 ((i : \u03b9) \u2192 \u03b2 i) := equiv \u22a4 \u03b2\nf : PiLp \u22a4 \u03b2\n\u22a2 \u2016\u2191{\n toLinearMap :=\n {\n toAddHom :=\n { toFun := src\u271d.toFun,\n map_add' := (_ : \u2200 (f g : PiLp \u22a4 \u03b2), Equiv.toFun src\u271d (f + g) = Equiv.toFun src\u271d (f + g)) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : PiLp \u22a4 \u03b2),\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' := (_ : \u2200 (f g : PiLp \u22a4 \u03b2), Equiv.toFun src\u271d (f + g) = Equiv.toFun src\u271d (f + g)) }\n (c \u2022 f) =\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' := (_ : \u2200 (f g : PiLp \u22a4 \u03b2), Equiv.toFun src\u271d (f + g) = Equiv.toFun src\u271d (f + g)) }\n (c \u2022 f)) },\n invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n f\u2016 =\n \u2016f\u2016\n[PROOFSTEP]\nsuffices (Finset.univ.sup fun i => \u2016f i\u2016\u208a) = \u2a06 i, \u2016f i\u2016\u208a by\n simpa only [NNReal.coe_iSup] using congr_arg ((\u2191) : \u211d\u22650 \u2192 \u211d) this\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fact (1 \u2264 p)\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\nsrc\u271d : PiLp \u22a4 \u03b2 \u2243 ((i : \u03b9) \u2192 \u03b2 i) := equiv \u22a4 \u03b2\nf : PiLp \u22a4 \u03b2\nthis : (Finset.sup Finset.univ fun i => \u2016f i\u2016\u208a) = \u2a06 (i : \u03b9), \u2016f i\u2016\u208a\n\u22a2 \u2016\u2191{\n toLinearMap :=\n {\n toAddHom :=\n { toFun := src\u271d.toFun,\n map_add' := (_ : \u2200 (f g : PiLp \u22a4 \u03b2), Equiv.toFun src\u271d (f + g) = Equiv.toFun src\u271d (f + g)) },\n map_smul' :=\n (_ :\n \u2200 (c : \ud835\udd5c) (f : PiLp \u22a4 \u03b2),\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' := (_ : \u2200 (f g : PiLp \u22a4 \u03b2), Equiv.toFun src\u271d (f + g) = Equiv.toFun src\u271d (f + g)) }\n (c \u2022 f) =\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' := (_ : \u2200 (f g : PiLp \u22a4 \u03b2), Equiv.toFun src\u271d (f + g) = Equiv.toFun src\u271d (f + g)) }\n (c \u2022 f)) },\n invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n f\u2016 =\n \u2016f\u2016\n[PROOFSTEP]\nsimpa only [NNReal.coe_iSup] using congr_arg ((\u2191) : \u211d\u22650 \u2192 \u211d) this\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fact (1 \u2264 p)\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\nsrc\u271d : PiLp \u22a4 \u03b2 \u2243 ((i : \u03b9) \u2192 \u03b2 i) := equiv \u22a4 \u03b2\nf : PiLp \u22a4 \u03b2\n\u22a2 (Finset.sup Finset.univ fun i => \u2016f i\u2016\u208a) = \u2a06 (i : \u03b9), \u2016f i\u2016\u208a\n[PROOFSTEP]\nrefine' antisymm (Finset.sup_le fun i _ => le_ciSup (Fintype.bddAbove_range fun i => \u2016f i\u2016\u208a) _) _\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fact (1 \u2264 p)\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\nsrc\u271d : PiLp \u22a4 \u03b2 \u2243 ((i : \u03b9) \u2192 \u03b2 i) := equiv \u22a4 \u03b2\nf : PiLp \u22a4 \u03b2\n\u22a2 \u2a06 (i : \u03b9), \u2016f i\u2016\u208a \u2264 Finset.sup Finset.univ fun i => \u2016f i\u2016\u208a\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b9\n[GOAL]\ncase inl\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fact (1 \u2264 p)\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\nsrc\u271d : PiLp \u22a4 \u03b2 \u2243 ((i : \u03b9) \u2192 \u03b2 i) := equiv \u22a4 \u03b2\nf : PiLp \u22a4 \u03b2\nh\u271d : IsEmpty \u03b9\n\u22a2 \u2a06 (i : \u03b9), \u2016f i\u2016\u208a \u2264 Finset.sup Finset.univ fun i => \u2016f i\u2016\u208a\n[PROOFSTEP]\nsimp only [ciSup_of_empty, Finset.univ_eq_empty, Finset.sup_empty, le_rfl]\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fact (1 \u2264 p)\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\nsrc\u271d : PiLp \u22a4 \u03b2 \u2243 ((i : \u03b9) \u2192 \u03b2 i) := equiv \u22a4 \u03b2\nf : PiLp \u22a4 \u03b2\nh\u271d : Nonempty \u03b9\n\u22a2 \u2a06 (i : \u03b9), \u2016f i\u2016\u208a \u2264 Finset.sup Finset.univ fun i => \u2016f i\u2016\u208a\n[PROOFSTEP]\nexact ciSup_le fun i => Finset.le_sup (f := fun k => \u2016f k\u2016\u208a) (Finset.mem_univ i)\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2078 : Fintype \u03b9\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx'\u271d y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\nx' : PiLp p fun x => E\n\u22a2 \u2016\u2191(LinearEquiv.piCongrLeft' \ud835\udd5c (fun x => E) e) x'\u2016 = \u2016x'\u2016\n[PROOFSTEP]\nclear x y\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2078 : Fintype \u03b9\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx'\u271d y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\nx' : PiLp p fun x => E\n\u22a2 \u2016\u2191(LinearEquiv.piCongrLeft' \ud835\udd5c (fun x => E) e) x'\u2016 = \u2016x'\u2016\n[PROOFSTEP]\nrcases p.dichotomy with (rfl | h)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2078 : Fintype \u03b9\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx'\u271d y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fact (1 \u2264 \u22a4)\nx' : PiLp \u22a4 fun x => E\n\u22a2 \u2016\u2191(LinearEquiv.piCongrLeft' \ud835\udd5c (fun x => E) e) x'\u2016 = \u2016x'\u2016\n[PROOFSTEP]\nsimp_rw [norm_eq_ciSup]\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2078 : Fintype \u03b9\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx'\u271d y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fact (1 \u2264 \u22a4)\nx' : PiLp \u22a4 fun x => E\n\u22a2 \u2a06 (i : \u03b9'), \u2016\u2191(LinearEquiv.piCongrLeft' \ud835\udd5c (fun x => E) e) x' i\u2016 = \u2a06 (i : \u03b9), \u2016x' i\u2016\n[PROOFSTEP]\nexact e.symm.iSup_congr fun _ => rfl\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2078 : Fintype \u03b9\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx'\u271d y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\nx' : PiLp p fun x => E\nh : 1 \u2264 ENNReal.toReal p\n\u22a2 \u2016\u2191(LinearEquiv.piCongrLeft' \ud835\udd5c (fun x => E) e) x'\u2016 = \u2016x'\u2016\n[PROOFSTEP]\nsimp only [norm_eq_sum (zero_lt_one.trans_le h)]\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2078 : Fintype \u03b9\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx'\u271d y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\nx' : PiLp p fun x => E\nh : 1 \u2264 ENNReal.toReal p\n\u22a2 (\u2211 x : \u03b9', \u2016\u2191(LinearEquiv.piCongrLeft' \ud835\udd5c (fun x => E) e) x' x\u2016 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) =\n (\u2211 x : \u03b9, \u2016x' x\u2016 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase inr.e_a\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2078 : Fintype \u03b9\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx'\u271d y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\nx' : PiLp p fun x => E\nh : 1 \u2264 ENNReal.toReal p\n\u22a2 \u2211 x : \u03b9', \u2016\u2191(LinearEquiv.piCongrLeft' \ud835\udd5c (fun x => E) e) x' x\u2016 ^ ENNReal.toReal p = \u2211 x : \u03b9, \u2016x' x\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nexact Fintype.sum_equiv e.symm _ _ fun _ => rfl\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2078 : Fintype \u03b9\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\nz : PiLp p fun x => E\n\u22a2 \u2191(LinearIsometryEquiv.symm (LinearIsometryEquiv.piLpCongrLeft p \ud835\udd5c E e)) z =\n \u2191(LinearIsometryEquiv.piLpCongrLeft p \ud835\udd5c E e.symm) z\n[PROOFSTEP]\nsimp only [LinearIsometryEquiv.piLpCongrLeft, LinearIsometryEquiv.symm, LinearIsometryEquiv.coe_mk]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2078 : Fintype \u03b9\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\nz : PiLp p fun x => E\n\u22a2 \u2191(LinearEquiv.symm (LinearEquiv.piCongrLeft' \ud835\udd5c (fun x => E) e)) z =\n \u2191(LinearEquiv.piCongrLeft' \ud835\udd5c (fun x => E) e.symm) z\n[PROOFSTEP]\nunfold PiLp WithLp\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2078 : Fintype \u03b9\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\nz : PiLp p fun x => E\n\u22a2 \u2191(LinearEquiv.symm (LinearEquiv.piCongrLeft' \ud835\udd5c (fun x => E) e)) z =\n \u2191(LinearEquiv.piCongrLeft' \ud835\udd5c (fun x => E) e.symm) z\n[PROOFSTEP]\next\n[GOAL]\ncase h\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2078 : Fintype \u03b9\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\nz : PiLp p fun x => E\nx\u271d : \u03b9\n\u22a2 \u2191(LinearEquiv.symm (LinearEquiv.piCongrLeft' \ud835\udd5c (fun x => E) e)) z x\u271d =\n \u2191(LinearEquiv.piCongrLeft' \ud835\udd5c (fun x => E) e.symm) z x\u271d\n[PROOFSTEP]\nsimp only [LinearEquiv.piCongrLeft'_symm_apply, eq_rec_constant, LinearEquiv.piCongrLeft'_apply, Equiv.symm_symm_apply]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9\u2070 : Fintype \u03b9\ninst\u271d\u2079 : Fact (1 \u2264 p)\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedField \ud835\udd5c'\ninst\u271d\u2076 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u2074 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : DecidableEq \u03b9'\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\nv : E\n\u22a2 \u2191(LinearIsometryEquiv.piLpCongrLeft p \ud835\udd5c E e) (\u2191(equiv p fun x => E).symm (Pi.single i v)) =\n \u2191(equiv p fun x => E).symm (Pi.single (\u2191e i) v)\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase h\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9\u2070 : Fintype \u03b9\ninst\u271d\u2079 : Fact (1 \u2264 p)\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedField \ud835\udd5c'\ninst\u271d\u2076 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx\u271d y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u2074 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : DecidableEq \u03b9'\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\nv : E\nx : \u03b9'\n\u22a2 \u2191(LinearIsometryEquiv.piLpCongrLeft p \ud835\udd5c E e) (\u2191(equiv p fun x => E).symm (Pi.single i v)) x =\n \u2191(equiv p fun x => E).symm (Pi.single (\u2191e i) v) x\n[PROOFSTEP]\nsimp [LinearIsometryEquiv.piLpCongrLeft_apply, LinearEquiv.piCongrLeft', Equiv.piCongrLeft', Pi.single, Function.update,\n Equiv.symm_apply_eq]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\nhp : Fact (1 \u2264 p)\ni : \u03b9\nb : \u03b2 i\n\u22a2 \u2016\u2191(equiv p \u03b2).symm (Pi.single i b)\u2016\u208a = \u2016b\u2016\u208a\n[PROOFSTEP]\nclear x y\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\nhp : Fact (1 \u2264 p)\ni : \u03b9\nb : \u03b2 i\n\u22a2 \u2016\u2191(equiv p \u03b2).symm (Pi.single i b)\u2016\u208a = \u2016b\u2016\u208a\n[PROOFSTEP]\nhaveI : Nonempty \u03b9 := \u27e8i\u27e9\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\nhp : Fact (1 \u2264 p)\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\n\u22a2 \u2016\u2191(equiv p \u03b2).symm (Pi.single i b)\u2016\u208a = \u2016b\u2016\u208a\n[PROOFSTEP]\ninduction p using ENNReal.recTopCoe generalizing hp with\n| top =>\n simp_rw [nnnorm_eq_ciSup, equiv_symm_apply]\n refine' ciSup_eq_of_forall_le_of_forall_lt_exists_gt (fun j => _) fun n hn => \u27e8i, hn.trans_eq _\u27e9\n \u00b7 obtain rfl | hij := Decidable.eq_or_ne i j\n \u00b7 rw [Pi.single_eq_same]\n \u00b7 rw [Pi.single_eq_of_ne' hij, nnnorm_zero]\n exact zero_le _\n \u00b7 rw [Pi.single_eq_same]\n|\n coe p =>\n have hp0 : (p : \u211d) \u2260 0 := by exact_mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 \u2264 (p : \u211d\u22650\u221e))).ne'\n rw [nnnorm_eq_sum ENNReal.coe_ne_top, ENNReal.coe_toReal, Fintype.sum_eq_single i, equiv_symm_apply,\n Pi.single_eq_same, \u2190 NNReal.rpow_mul, one_div, mul_inv_cancel hp0, NNReal.rpow_one]\n intro j hij\n rw [equiv_symm_apply, Pi.single_eq_of_ne hij, nnnorm_zero, NNReal.zero_rpow hp0]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\nhp : Fact (1 \u2264 p)\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\n\u22a2 \u2016\u2191(equiv p \u03b2).symm (Pi.single i b)\u2016\u208a = \u2016b\u2016\u208a\n[PROOFSTEP]\ninduction p using ENNReal.recTopCoe generalizing hp with\n| top =>\n simp_rw [nnnorm_eq_ciSup, equiv_symm_apply]\n refine' ciSup_eq_of_forall_le_of_forall_lt_exists_gt (fun j => _) fun n hn => \u27e8i, hn.trans_eq _\u27e9\n \u00b7 obtain rfl | hij := Decidable.eq_or_ne i j\n \u00b7 rw [Pi.single_eq_same]\n \u00b7 rw [Pi.single_eq_of_ne' hij, nnnorm_zero]\n exact zero_le _\n \u00b7 rw [Pi.single_eq_same]\n|\n coe p =>\n have hp0 : (p : \u211d) \u2260 0 := by exact_mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 \u2264 (p : \u211d\u22650\u221e))).ne'\n rw [nnnorm_eq_sum ENNReal.coe_ne_top, ENNReal.coe_toReal, Fintype.sum_eq_single i, equiv_symm_apply,\n Pi.single_eq_same, \u2190 NNReal.rpow_mul, one_div, mul_inv_cancel hp0, NNReal.rpow_one]\n intro j hij\n rw [equiv_symm_apply, Pi.single_eq_of_ne hij, nnnorm_zero, NNReal.zero_rpow hp0]\n[GOAL]\ncase top\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\n\u22a2 \u2016\u2191(equiv \u22a4 \u03b2).symm (Pi.single i b)\u2016\u208a = \u2016b\u2016\u208a\n[PROOFSTEP]\n\n| top =>\n simp_rw [nnnorm_eq_ciSup, equiv_symm_apply]\n refine' ciSup_eq_of_forall_le_of_forall_lt_exists_gt (fun j => _) fun n hn => \u27e8i, hn.trans_eq _\u27e9\n \u00b7 obtain rfl | hij := Decidable.eq_or_ne i j\n \u00b7 rw [Pi.single_eq_same]\n \u00b7 rw [Pi.single_eq_of_ne' hij, nnnorm_zero]\n exact zero_le _\n \u00b7 rw [Pi.single_eq_same]\n[GOAL]\ncase top\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\n\u22a2 \u2016\u2191(equiv \u22a4 \u03b2).symm (Pi.single i b)\u2016\u208a = \u2016b\u2016\u208a\n[PROOFSTEP]\nsimp_rw [nnnorm_eq_ciSup, equiv_symm_apply]\n[GOAL]\ncase top\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\n\u22a2 \u2a06 (i_1 : \u03b9), \u2016Pi.single i b i_1\u2016\u208a = \u2016b\u2016\u208a\n[PROOFSTEP]\nrefine' ciSup_eq_of_forall_le_of_forall_lt_exists_gt (fun j => _) fun n hn => \u27e8i, hn.trans_eq _\u27e9\n[GOAL]\ncase top.refine'_1\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\nj : \u03b9\n\u22a2 \u2016Pi.single i b j\u2016\u208a \u2264 \u2016b\u2016\u208a\n[PROOFSTEP]\nobtain rfl | hij := Decidable.eq_or_ne i j\n[GOAL]\ncase top.refine'_1.inl\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\n\u22a2 \u2016Pi.single i b i\u2016\u208a \u2264 \u2016b\u2016\u208a\n[PROOFSTEP]\nrw [Pi.single_eq_same]\n[GOAL]\ncase top.refine'_1.inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\nj : \u03b9\nhij : i \u2260 j\n\u22a2 \u2016Pi.single i b j\u2016\u208a \u2264 \u2016b\u2016\u208a\n[PROOFSTEP]\nrw [Pi.single_eq_of_ne' hij, nnnorm_zero]\n[GOAL]\ncase top.refine'_1.inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\nj : \u03b9\nhij : i \u2260 j\n\u22a2 0 \u2264 \u2016b\u2016\u208a\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase top.refine'_2\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\nn : \u211d\u22650\nhn : n < \u2016b\u2016\u208a\n\u22a2 \u2016b\u2016\u208a = \u2016Pi.single i b i\u2016\u208a\n[PROOFSTEP]\nrw [Pi.single_eq_same]\n[GOAL]\ncase coe\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p\u271d)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\np : \u211d\u22650\nhp : Fact (1 \u2264 \u2191p)\n\u22a2 \u2016\u2191(equiv (\u2191p) \u03b2).symm (Pi.single i b)\u2016\u208a = \u2016b\u2016\u208a\n[PROOFSTEP]\n\n|\n coe p =>\n have hp0 : (p : \u211d) \u2260 0 := by exact_mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 \u2264 (p : \u211d\u22650\u221e))).ne'\n rw [nnnorm_eq_sum ENNReal.coe_ne_top, ENNReal.coe_toReal, Fintype.sum_eq_single i, equiv_symm_apply,\n Pi.single_eq_same, \u2190 NNReal.rpow_mul, one_div, mul_inv_cancel hp0, NNReal.rpow_one]\n intro j hij\n rw [equiv_symm_apply, Pi.single_eq_of_ne hij, nnnorm_zero, NNReal.zero_rpow hp0]\n[GOAL]\ncase coe\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p\u271d)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\np : \u211d\u22650\nhp : Fact (1 \u2264 \u2191p)\n\u22a2 \u2016\u2191(equiv (\u2191p) \u03b2).symm (Pi.single i b)\u2016\u208a = \u2016b\u2016\u208a\n[PROOFSTEP]\nhave hp0 : (p : \u211d) \u2260 0 := by exact_mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 \u2264 (p : \u211d\u22650\u221e))).ne'\n[GOAL]\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p\u271d)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\np : \u211d\u22650\nhp : Fact (1 \u2264 \u2191p)\n\u22a2 \u2191p \u2260 0\n[PROOFSTEP]\nexact_mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 \u2264 (p : \u211d\u22650\u221e))).ne'\n[GOAL]\ncase coe\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p\u271d)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\np : \u211d\u22650\nhp : Fact (1 \u2264 \u2191p)\nhp0 : \u2191p \u2260 0\n\u22a2 \u2016\u2191(equiv (\u2191p) \u03b2).symm (Pi.single i b)\u2016\u208a = \u2016b\u2016\u208a\n[PROOFSTEP]\nrw [nnnorm_eq_sum ENNReal.coe_ne_top, ENNReal.coe_toReal, Fintype.sum_eq_single i, equiv_symm_apply, Pi.single_eq_same,\n \u2190 NNReal.rpow_mul, one_div, mul_inv_cancel hp0, NNReal.rpow_one]\n[GOAL]\ncase coe\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p\u271d)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\np : \u211d\u22650\nhp : Fact (1 \u2264 \u2191p)\nhp0 : \u2191p \u2260 0\n\u22a2 \u2200 (x : \u03b9), x \u2260 i \u2192 \u2016\u2191(equiv (\u2191p) \u03b2).symm (Pi.single i b) x\u2016\u208a ^ \u2191p = 0\n[PROOFSTEP]\nintro j hij\n[GOAL]\ncase coe\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p\u271d)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\np : \u211d\u22650\nhp : Fact (1 \u2264 \u2191p)\nhp0 : \u2191p \u2260 0\nj : \u03b9\nhij : j \u2260 i\n\u22a2 \u2016\u2191(equiv (\u2191p) \u03b2).symm (Pi.single i b) j\u2016\u208a ^ \u2191p = 0\n[PROOFSTEP]\nrw [equiv_symm_apply, Pi.single_eq_of_ne hij, nnnorm_zero, NNReal.zero_rpow hp0]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb\u2081 b\u2082 : \u03b2 i\n\u22a2 nndist (\u2191(equiv p \u03b2).symm (Pi.single i b\u2081)) (\u2191(equiv p \u03b2).symm (Pi.single i b\u2082)) = nndist b\u2081 b\u2082\n[PROOFSTEP]\nrw [nndist_eq_nnnorm, nndist_eq_nnnorm, \u2190 WithLp.equiv_symm_sub, \u2190 Pi.single_sub, nnnorm_equiv_symm_single]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb\u2081 b\u2082 : \u03b2 i\n\u22a2 edist (\u2191(equiv p \u03b2).symm (Pi.single i b\u2081)) (\u2191(equiv p \u03b2).symm (Pi.single i b\u2082)) = edist b\u2081 b\u2082\n[PROOFSTEP]\nsimp only [edist_nndist, nndist_equiv_symm_single_same p \u03b2 i b\u2081 b\u2082]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2\u271d i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2\u271d i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\u271d\nx' y' : (i : \u03b9) \u2192 \u03b2\u271d i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\n\u03b2 : Type u_8\ninst\u271d : SeminormedAddCommGroup \u03b2\nhp : p \u2260 \u22a4\nb : \u03b2\n\u22a2 \u2016\u2191(equiv p fun x => \u03b2).symm (Function.const \u03b9 b)\u2016\u208a = \u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p) * \u2016b\u2016\u208a\n[PROOFSTEP]\nrcases p.dichotomy with (h | h)\n[GOAL]\ncase inl\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2\u271d i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2\u271d i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\u271d\nx' y' : (i : \u03b9) \u2192 \u03b2\u271d i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\n\u03b2 : Type u_8\ninst\u271d : SeminormedAddCommGroup \u03b2\nhp : p \u2260 \u22a4\nb : \u03b2\nh : p = \u22a4\n\u22a2 \u2016\u2191(equiv p fun x => \u03b2).symm (Function.const \u03b9 b)\u2016\u208a = \u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p) * \u2016b\u2016\u208a\n[PROOFSTEP]\nexact False.elim (hp h)\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2\u271d i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2\u271d i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\u271d\nx' y' : (i : \u03b9) \u2192 \u03b2\u271d i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\n\u03b2 : Type u_8\ninst\u271d : SeminormedAddCommGroup \u03b2\nhp : p \u2260 \u22a4\nb : \u03b2\nh : 1 \u2264 ENNReal.toReal p\n\u22a2 \u2016\u2191(equiv p fun x => \u03b2).symm (Function.const \u03b9 b)\u2016\u208a = \u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p) * \u2016b\u2016\u208a\n[PROOFSTEP]\nhave ne_zero : p.toReal \u2260 0 := (zero_lt_one.trans_le h).ne'\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2\u271d i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2\u271d i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\u271d\nx' y' : (i : \u03b9) \u2192 \u03b2\u271d i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\n\u03b2 : Type u_8\ninst\u271d : SeminormedAddCommGroup \u03b2\nhp : p \u2260 \u22a4\nb : \u03b2\nh : 1 \u2264 ENNReal.toReal p\nne_zero : ENNReal.toReal p \u2260 0\n\u22a2 \u2016\u2191(equiv p fun x => \u03b2).symm (Function.const \u03b9 b)\u2016\u208a = \u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p) * \u2016b\u2016\u208a\n[PROOFSTEP]\nsimp_rw [nnnorm_eq_sum hp, equiv_symm_apply, Function.const_apply, Finset.sum_const, Finset.card_univ, nsmul_eq_mul,\n NNReal.mul_rpow, \u2190 NNReal.rpow_mul, mul_one_div_cancel ne_zero, NNReal.rpow_one, ENNReal.toReal_div,\n ENNReal.one_toReal]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9\u2070 : Fintype \u03b9\ninst\u271d\u2079 : Fact (1 \u2264 p)\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedField \ud835\udd5c'\ninst\u271d\u2076 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2\u271d i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2\u271d i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\u271d\nx' y' : (i : \u03b9) \u2192 \u03b2\u271d i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u2074 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\n\u03b2 : Type u_8\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : Nonempty \u03b9\nb : \u03b2\n\u22a2 \u2016\u2191(equiv p fun x => \u03b2).symm (Function.const \u03b9 b)\u2016\u208a = \u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p) * \u2016b\u2016\u208a\n[PROOFSTEP]\nclear x y\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9\u2070 : Fintype \u03b9\ninst\u271d\u2079 : Fact (1 \u2264 p)\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedField \ud835\udd5c'\ninst\u271d\u2076 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2\u271d i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2\u271d i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2\u271d i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u2074 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\n\u03b2 : Type u_8\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : Nonempty \u03b9\nb : \u03b2\n\u22a2 \u2016\u2191(equiv p fun x => \u03b2).symm (Function.const \u03b9 b)\u2016\u208a = \u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p) * \u2016b\u2016\u208a\n[PROOFSTEP]\nrcases em <| p = \u221e with (rfl | hp)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9\u2070 : Fintype \u03b9\ninst\u271d\u2079 : NormedField \ud835\udd5c\ninst\u271d\u2078 : NormedField \ud835\udd5c'\ninst\u271d\u2077 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2\u271d i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2\u271d i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2\u271d i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u2075 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\n\u03b2 : Type u_8\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b2\ninst\u271d\u00b9 : Nonempty \u03b9\nb : \u03b2\ninst\u271d : Fact (1 \u2264 \u22a4)\n\u22a2 \u2016\u2191(equiv \u22a4 fun x => \u03b2).symm (Function.const \u03b9 b)\u2016\u208a = \u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / \u22a4) * \u2016b\u2016\u208a\n[PROOFSTEP]\nsimp only [equiv_symm_apply, ENNReal.div_top, ENNReal.zero_toReal, NNReal.rpow_zero, one_mul, nnnorm_eq_ciSup,\n Function.const_apply, ciSup_const]\n[GOAL]\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9\u2070 : Fintype \u03b9\ninst\u271d\u2079 : Fact (1 \u2264 p)\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedField \ud835\udd5c'\ninst\u271d\u2076 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2\u271d i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2\u271d i)\nc : \ud835\udd5c\nx' y' : (i : \u03b9) \u2192 \u03b2\u271d i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u2074 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\n\u03b2 : Type u_8\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : Nonempty \u03b9\nb : \u03b2\nhp : \u00acp = \u22a4\n\u22a2 \u2016\u2191(equiv p fun x => \u03b2).symm (Function.const \u03b9 b)\u2016\u208a = \u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p) * \u2016b\u2016\u208a\n[PROOFSTEP]\nexact nnnorm_equiv_symm_const hp b\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2\u271d i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2\u271d i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\u271d\nx' y' : (i : \u03b9) \u2192 \u03b2\u271d i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\n\u03b2 : Type u_8\ninst\u271d : SeminormedAddCommGroup \u03b2\nhp : p \u2260 \u22a4\nb : \u03b2\n\u22a2 \u2191(\u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p) * \u2016b\u2016\u208a) = \u2191(\u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p)) * \u2016b\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2\u271d : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9\u2070 : Fintype \u03b9\ninst\u271d\u2079 : Fact (1 \u2264 p)\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : NormedField \ud835\udd5c'\ninst\u271d\u2076 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2\u271d i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2\u271d i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\u271d\nx' y' : (i : \u03b9) \u2192 \u03b2\u271d i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u2074 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\n\u03b2 : Type u_8\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : Nonempty \u03b9\nb : \u03b2\n\u22a2 \u2191(\u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p) * \u2016b\u2016\u208a) = \u2191(\u2191(Fintype.card \u03b9) ^ ENNReal.toReal (1 / p)) * \u2016b\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b3 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\n\u22a2 \u2191(basisFun p \ud835\udd5c \u03b9) i = \u2191(equiv p fun i => \ud835\udd5c).symm (Pi.single i 1)\n[PROOFSTEP]\nsimp_rw [basisFun, Basis.coe_ofEquivFun, WithLp.linearEquiv_symm_apply, Pi.single]\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2078 : Fintype \u03b9\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nb : Basis \u03b9 \ud835\udd5c (PiLp p fun x => \ud835\udd5c)\nA : Matrix \u03b9 \u03b9 \ud835\udd5c\n\u22a2 Basis.toMatrix b \u2191(basisFun p \ud835\udd5c \u03b9) * A = \u2191of fun i j => \u2191(\u2191b.repr (\u2191(equiv p fun i => \ud835\udd5c).symm (A\u1d40 j))) i\n[PROOFSTEP]\nhave := basis_toMatrix_basisFun_mul (b.map (PiLp.linearEquiv _ \ud835\udd5c _)) A\n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2078 : Fintype \u03b9\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nb : Basis \u03b9 \ud835\udd5c (PiLp p fun x => \ud835\udd5c)\nA : Matrix \u03b9 \u03b9 \ud835\udd5c\nthis :\n Basis.toMatrix (Basis.map b (linearEquiv p \ud835\udd5c fun x => \ud835\udd5c)) \u2191(Pi.basisFun \ud835\udd5c \u03b9) * A =\n \u2191of fun i j => \u2191(\u2191(Basis.map b (linearEquiv p \ud835\udd5c fun x => \ud835\udd5c)).repr (A\u1d40 j)) i\n\u22a2 Basis.toMatrix b \u2191(basisFun p \ud835\udd5c \u03b9) * A = \u2191of fun i j => \u2191(\u2191b.repr (\u2191(equiv p fun i => \ud835\udd5c).symm (A\u1d40 j))) i\n[PROOFSTEP]\nsimp_rw [\u2190 PiLp.basisFun_map p, Basis.map_repr, LinearEquiv.trans_apply, WithLp.linearEquiv_symm_apply,\n Basis.toMatrix_map, Function.comp, Basis.map_apply, LinearEquiv.symm_apply_apply] at this \n[GOAL]\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\n\u03b9 : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\n\u03b2 : \u03b9 \u2192 Type u_5\ninst\u271d\u2078 : Fintype \u03b9\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedField \ud835\udd5c'\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\n\u03b9' : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9'\nE : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nb : Basis \u03b9 \ud835\udd5c (PiLp p fun x => \ud835\udd5c)\nA : Matrix \u03b9 \u03b9 \ud835\udd5c\nthis :\n (Basis.toMatrix b fun x => \u2191(basisFun p \ud835\udd5c \u03b9) x) * A =\n \u2191of fun i j => \u2191(\u2191b.repr (\u2191(WithLp.equiv p (\u03b9 \u2192 \ud835\udd5c)).symm (A\u1d40 j))) i\n\u22a2 Basis.toMatrix b \u2191(basisFun p \ud835\udd5c \u03b9) * A = \u2191of fun i j => \u2191(\u2191b.repr (\u2191(equiv p fun i => \ud835\udd5c).symm (A\u1d40 j))) i\n[PROOFSTEP]\nexact this\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.PiLp", "llama_tokens": 52346, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956581000631542, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.4442373955268555}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nX\u271d Y\u271d : C\nF : C \u2964 D\ninst\u271d : Full F\nX Y : C\nf : F.obj X \u27f6 F.obj Y\n\u22a2 F.map (F.preimage f) = f\n[PROOFSTEP]\nunfold preimage\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nX\u271d Y\u271d : C\nF : C \u2964 D\ninst\u271d : Full F\nX Y : C\nf : F.obj X \u27f6 F.obj Y\n\u22a2 F.map (Full.preimage f) = f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y : C\nF : C \u2964 D\nh : \u2200 (X Y : C) (f : F.obj X \u27f6 F.obj Y), \u2203 p, F.map p = f\n\u22a2 Full F\n[PROOFSTEP]\nchoose p hp using h\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y : C\nF : C \u2964 D\np : (X Y : C) \u2192 (F.obj X \u27f6 F.obj Y) \u2192 (X \u27f6 Y)\nhp : \u2200 (X Y : C) (f : F.obj X \u27f6 F.obj Y), F.map (p X Y f) = f\n\u22a2 Full F\n[PROOFSTEP]\nexact \u27e8@p, @hp\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX Y Z : C\n\u22a2 F.map (F.preimage (\ud835\udfd9 (F.obj X))) = F.map (\ud835\udfd9 X)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX Y Z : C\nf : F.obj X \u27f6 F.obj Y\ng : F.obj Y \u27f6 F.obj Z\n\u22a2 F.map (F.preimage (f \u226b g)) = F.map (F.preimage f \u226b F.preimage g)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX Y Z : C\nf : X \u27f6 Y\n\u22a2 F.map (F.preimage (F.map f)) = F.map f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX Y Z : C\nf : F.obj X \u2245 F.obj Y\n\u22a2 F.map (F.preimage f.hom \u226b F.preimage f.inv) = F.map (\ud835\udfd9 X)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX Y Z : C\nf : F.obj X \u2245 F.obj Y\n\u22a2 F.map (F.preimage f.inv \u226b F.preimage f.hom) = F.map (\ud835\udfd9 Y)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX Y Z : C\nf : X \u2245 Y\n\u22a2 preimageIso F (F.mapIso f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX Y Z : C\nf : X \u2245 Y\n\u22a2 (preimageIso F (F.mapIso f)).hom = f.hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b2 : Full F\ninst\u271d\u00b9 : Faithful F\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : IsIso (F.map f)\n\u22a2 F.map (f \u226b F.preimage (inv (F.map f))) = F.map (\ud835\udfd9 X)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b2 : Full F\ninst\u271d\u00b9 : Faithful F\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : IsIso (F.map f)\n\u22a2 F.map (F.preimage (inv (F.map f)) \u226b f) = F.map (\ud835\udfd9 Y)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX\u271d Y\u271d Z X Y : C\nf : X \u27f6 Y\n\u22a2 (fun f => F.preimage f) ((fun f => F.map f) f) = f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX\u271d Y\u271d Z X Y : C\nf : F.obj X \u27f6 F.obj Y\n\u22a2 (fun f => F.map f) ((fun f => F.preimage f) f) = f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX\u271d Y\u271d Z X Y : C\nf : X \u2245 Y\n\u22a2 (fun f => Functor.preimageIso F f) ((fun f => F.mapIso f) f) = f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX\u271d Y\u271d Z X Y : C\nf : F.obj X \u2245 F.obj Y\n\u22a2 (fun f => F.mapIso f) ((fun f => Functor.preimageIso F f) f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX\u271d Y\u271d Z X Y : C\nf : F.obj X \u2245 F.obj Y\n\u22a2 ((fun f => F.mapIso f) ((fun f => Functor.preimageIso F f) f)).hom = f.hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u_1\ninst\u271d\u00b2 : Category.{?u.9765, u_1} E\nF G : C \u2964 D\nH : D \u2964 E\ninst\u271d\u00b9 : Full H\ninst\u271d : Faithful H\n\u03b1 : F \u22d9 H \u27f6 G \u22d9 H\nX Y : C\nf : X \u27f6 Y\n\u22a2 F.map f \u226b (fun X => \u2191(equivOfFullyFaithful H).symm (NatTrans.app \u03b1 X)) Y =\n (fun X => \u2191(equivOfFullyFaithful H).symm (NatTrans.app \u03b1 X)) X \u226b G.map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u_1\ninst\u271d\u00b2 : Category.{?u.9765, u_1} E\nF G : C \u2964 D\nH : D \u2964 E\ninst\u271d\u00b9 : Full H\ninst\u271d : Faithful H\n\u03b1 : F \u22d9 H \u27f6 G \u22d9 H\nX Y : C\nf : X \u27f6 Y\n\u22a2 F.map f \u226b H.preimage (NatTrans.app \u03b1 Y) = H.preimage (NatTrans.app \u03b1 X) \u226b G.map f\n[PROOFSTEP]\napply H.map_injective\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u_1\ninst\u271d\u00b2 : Category.{?u.9765, u_1} E\nF G : C \u2964 D\nH : D \u2964 E\ninst\u271d\u00b9 : Full H\ninst\u271d : Faithful H\n\u03b1 : F \u22d9 H \u27f6 G \u22d9 H\nX Y : C\nf : X \u27f6 Y\n\u22a2 H.map (F.map f \u226b H.preimage (NatTrans.app \u03b1 Y)) = H.map (H.preimage (NatTrans.app \u03b1 X) \u226b G.map f)\n[PROOFSTEP]\nsimpa using \u03b1.naturality f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u_1\ninst\u271d\u00b2 : Category.{?u.11855, u_1} E\nF G : C \u2964 D\nH : D \u2964 E\ninst\u271d\u00b9 : Full H\ninst\u271d : Faithful H\ni : F \u22d9 H \u2245 G \u22d9 H\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\n\u22a2 F.map f \u226b ((fun X => \u2191(isoEquivOfFullyFaithful H).symm (i.app X)) Y\u271d).hom =\n ((fun X => \u2191(isoEquivOfFullyFaithful H).symm (i.app X)) X\u271d).hom \u226b G.map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u_1\ninst\u271d\u00b2 : Category.{?u.11855, u_1} E\nF G : C \u2964 D\nH : D \u2964 E\ninst\u271d\u00b9 : Full H\ninst\u271d : Faithful H\ni : F \u22d9 H \u2245 G \u22d9 H\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\n\u22a2 F.map f \u226b H.preimage (NatTrans.app i.hom Y\u271d) = H.preimage (NatTrans.app i.hom X\u271d) \u226b G.map f\n[PROOFSTEP]\napply H.map_injective\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u_1\ninst\u271d\u00b2 : Category.{?u.11855, u_1} E\nF G : C \u2964 D\nH : D \u2964 E\ninst\u271d\u00b9 : Full H\ninst\u271d : Faithful H\ni : F \u22d9 H \u2245 G \u22d9 H\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\n\u22a2 H.map (F.map f \u226b H.preimage (NatTrans.app i.hom Y\u271d)) = H.map (H.preimage (NatTrans.app i.hom X\u271d) \u226b G.map f)\n[PROOFSTEP]\nsimpa using i.hom.naturality f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} E\nF G : C \u2964 D\nH : D \u2964 E\ninst\u271d\u00b9 : Full H\ninst\u271d : Faithful H\ni : F \u22d9 H \u2245 G \u22d9 H\n\u22a2 (natIsoOfCompFullyFaithful H i).hom = natTransOfCompFullyFaithful H i.hom\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} E\nF G : C \u2964 D\nH : D \u2964 E\ninst\u271d\u00b9 : Full H\ninst\u271d : Faithful H\ni : F \u22d9 H \u2245 G \u22d9 H\nx\u271d : C\n\u22a2 NatTrans.app (natIsoOfCompFullyFaithful H i).hom x\u271d = NatTrans.app (natTransOfCompFullyFaithful H i.hom) x\u271d\n[PROOFSTEP]\nsimp [natIsoOfCompFullyFaithful]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} E\nF G : C \u2964 D\nH : D \u2964 E\ninst\u271d\u00b9 : Full H\ninst\u271d : Faithful H\ni : F \u22d9 H \u2245 G \u22d9 H\n\u22a2 (natIsoOfCompFullyFaithful H i).inv = natTransOfCompFullyFaithful H i.inv\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} E\nF G : C \u2964 D\nH : D \u2964 E\ninst\u271d\u00b9 : Full H\ninst\u271d : Faithful H\ni : F \u22d9 H \u2245 G \u22d9 H\nx\u271d : C\n\u22a2 NatTrans.app (natIsoOfCompFullyFaithful H i).inv x\u271d = NatTrans.app (natTransOfCompFullyFaithful H i.inv) x\u271d\n[PROOFSTEP]\nsimp [\u2190 preimage_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u_1\ninst\u271d\u00b2 : Category.{?u.17323, u_1} E\nF G : C \u2964 D\nH : D \u2964 E\ninst\u271d\u00b9 : Full H\ninst\u271d : Faithful H\n\u22a2 Function.LeftInverse (natTransOfCompFullyFaithful H) fun \u03b1 => \u03b1 \u25eb \ud835\udfd9 H\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u_1\ninst\u271d\u00b2 : Category.{?u.17323, u_1} E\nF G : C \u2964 D\nH : D \u2964 E\ninst\u271d\u00b9 : Full H\ninst\u271d : Faithful H\n\u22a2 Function.RightInverse (natTransOfCompFullyFaithful H) fun \u03b1 => \u03b1 \u25eb \ud835\udfd9 H\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u_1\ninst\u271d\u00b2 : Category.{?u.20694, u_1} E\nF G : C \u2964 D\nH : D \u2964 E\ninst\u271d\u00b9 : Full H\ninst\u271d : Faithful H\n\u22a2 Function.LeftInverse (natIsoOfCompFullyFaithful H) fun e => hcomp e (Iso.refl H)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u_1\ninst\u271d\u00b2 : Category.{?u.20694, u_1} E\nF G : C \u2964 D\nH : D \u2964 E\ninst\u271d\u00b9 : Full H\ninst\u271d : Faithful H\n\u22a2 Function.RightInverse (natIsoOfCompFullyFaithful H) fun e => hcomp e (Iso.refl H)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF F' : C \u2964 D\nG : D \u2964 E\ninst\u271d : Full F\n\u03b1 : F \u2245 F'\nX\u271d Y\u271d : C\nf : F'.obj X\u271d \u27f6 F'.obj Y\u271d\n\u22a2 F'.map ((fun {X Y} f => F.preimage ((\u03b1.app X).hom \u226b f \u226b (\u03b1.app Y).inv)) f) = f\n[PROOFSTEP]\nsimp [\u2190 NatIso.naturality_1 \u03b1]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF F' : C \u2964 D\nG : D \u2964 E\ninst\u271d : Faithful F\n\u03b1 : F \u2245 F'\nX\u271d Y\u271d : C\na\u2081\u271d a\u2082\u271d : X\u271d \u27f6 Y\u271d\nh : F'.map a\u2081\u271d = F'.map a\u2082\u271d\n\u22a2 F.map a\u2081\u271d = F.map a\u2082\u271d\n[PROOFSTEP]\nrw [\u2190 NatIso.naturality_1 \u03b1.symm, h, NatIso.naturality_1 \u03b1.symm]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\n\u22a2 \u2200 (X : C), { obj := obj, map := map }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := obj, map := map }.obj X)\n[PROOFSTEP]\nintros X\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX : C\n\u22a2 { obj := obj, map := map }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := obj, map := map }.obj X)\n[PROOFSTEP]\napply G.map_injective\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX : C\n\u22a2 G.map ({ obj := obj, map := map }.map (\ud835\udfd9 X)) = G.map (\ud835\udfd9 ({ obj := obj, map := map }.obj X))\n[PROOFSTEP]\napply eq_of_heq\n[GOAL]\ncase a.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX : C\n\u22a2 HEq (G.map ({ obj := obj, map := map }.map (\ud835\udfd9 X))) (G.map (\ud835\udfd9 ({ obj := obj, map := map }.obj X)))\n[PROOFSTEP]\ntrans F.map (\ud835\udfd9 X)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX : C\n\u22a2 HEq (G.map ({ obj := obj, map := map }.map (\ud835\udfd9 X))) (F.map (\ud835\udfd9 X))\n[PROOFSTEP]\nexact h_map\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX : C\n\u22a2 HEq (F.map (\ud835\udfd9 X)) (G.map (\ud835\udfd9 ({ obj := obj, map := map }.obj X)))\n[PROOFSTEP]\nrw [F.map_id, G.map_id, h_obj X]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\n\u22a2 \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := obj, map := map }.map (f \u226b g) = { obj := obj, map := map }.map f \u226b { obj := obj, map := map }.map g\n[PROOFSTEP]\nintros X Y Z f g\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 { obj := obj, map := map }.map (f \u226b g) = { obj := obj, map := map }.map f \u226b { obj := obj, map := map }.map g\n[PROOFSTEP]\nrefine G.map_injective <| eq_of_heq <| h_map.trans ?_\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 HEq (F.map (f \u226b g)) (G.map ({ obj := obj, map := map }.map f \u226b { obj := obj, map := map }.map g))\n[PROOFSTEP]\nsimp only [Functor.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 HEq (F.map f \u226b F.map g) (G.map (map f) \u226b G.map (map g))\n[PROOFSTEP]\nconvert HEq.refl (F.map f \u226b F.map g)\n[GOAL]\ncase h.e'_3.h.e'_3\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 G.obj (obj X) = F.obj X\ncase h.e'_3.h.e'_4\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 G.obj (obj Z) = F.obj Z\ncase h.e'_4.e'_3\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\n\u22a2 G.obj (obj X) = F.obj X\ncase h.e'_4.e'_4\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\n\u22a2 G.obj (obj Y) = F.obj Y\ncase h.e'_4.e'_5\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\n\u22a2 G.obj (obj Z) = F.obj Z\ncase h.e'_4.e'_6\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d\u00b9 : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\ne_3\u271d : G.obj (obj X) = F.obj X\ne_4\u271d : G.obj (obj Y) = F.obj Y\n\u22a2 HEq (G.map (map f)) (F.map f)\ncase h.e'_4.e'_7\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\ne_4\u271d : G.obj (obj Y) = F.obj Y\ne_5\u271d : G.obj (obj Z) = F.obj Z\n\u22a2 HEq (G.map (map g)) (F.map g)\n[PROOFSTEP]\nall_goals\n {first\n | apply h_obj\n | apply h_map\n}\n[GOAL]\ncase h.e'_3.h.e'_3\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 G.obj (obj X) = F.obj X\n[PROOFSTEP]\n{first\n | apply h_obj\n | apply h_map\n}\n[GOAL]\ncase h.e'_3.h.e'_3\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 G.obj (obj X) = F.obj X\n[PROOFSTEP]\nfirst\n| apply h_obj\n| apply h_map\n[GOAL]\ncase h.e'_3.h.e'_3\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 G.obj (obj X) = F.obj X\n[PROOFSTEP]\napply h_obj\n[GOAL]\ncase h.e'_3.h.e'_4\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 G.obj (obj Z) = F.obj Z\n[PROOFSTEP]\n{first\n | apply h_obj\n | apply h_map\n}\n[GOAL]\ncase h.e'_3.h.e'_4\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 G.obj (obj Z) = F.obj Z\n[PROOFSTEP]\nfirst\n| apply h_obj\n| apply h_map\n[GOAL]\ncase h.e'_3.h.e'_4\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 G.obj (obj Z) = F.obj Z\n[PROOFSTEP]\napply h_obj\n[GOAL]\ncase h.e'_4.e'_3\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\n\u22a2 G.obj (obj X) = F.obj X\n[PROOFSTEP]\n{first\n | apply h_obj\n | apply h_map\n}\n[GOAL]\ncase h.e'_4.e'_3\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\n\u22a2 G.obj (obj X) = F.obj X\n[PROOFSTEP]\nfirst\n| apply h_obj\n| apply h_map\n[GOAL]\ncase h.e'_4.e'_3\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\n\u22a2 G.obj (obj X) = F.obj X\n[PROOFSTEP]\napply h_obj\n[GOAL]\ncase h.e'_4.e'_4\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\n\u22a2 G.obj (obj Y) = F.obj Y\n[PROOFSTEP]\n{first\n | apply h_obj\n | apply h_map\n}\n[GOAL]\ncase h.e'_4.e'_4\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\n\u22a2 G.obj (obj Y) = F.obj Y\n[PROOFSTEP]\nfirst\n| apply h_obj\n| apply h_map\n[GOAL]\ncase h.e'_4.e'_4\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\n\u22a2 G.obj (obj Y) = F.obj Y\n[PROOFSTEP]\napply h_obj\n[GOAL]\ncase h.e'_4.e'_5\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\n\u22a2 G.obj (obj Z) = F.obj Z\n[PROOFSTEP]\n{first\n | apply h_obj\n | apply h_map\n}\n[GOAL]\ncase h.e'_4.e'_5\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\n\u22a2 G.obj (obj Z) = F.obj Z\n[PROOFSTEP]\nfirst\n| apply h_obj\n| apply h_map\n[GOAL]\ncase h.e'_4.e'_5\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\n\u22a2 G.obj (obj Z) = F.obj Z\n[PROOFSTEP]\napply h_obj\n[GOAL]\ncase h.e'_4.e'_6\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d\u00b9 : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\ne_3\u271d : G.obj (obj X) = F.obj X\ne_4\u271d : G.obj (obj Y) = F.obj Y\n\u22a2 HEq (G.map (map f)) (F.map f)\n[PROOFSTEP]\n{first\n | apply h_obj\n | apply h_map\n}\n[GOAL]\ncase h.e'_4.e'_6\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d\u00b9 : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\ne_3\u271d : G.obj (obj X) = F.obj X\ne_4\u271d : G.obj (obj Y) = F.obj Y\n\u22a2 HEq (G.map (map f)) (F.map f)\n[PROOFSTEP]\nfirst\n| apply h_obj\n| apply h_map\n[GOAL]\ncase h.e'_4.e'_6\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d\u00b9 : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\ne_3\u271d : G.obj (obj X) = F.obj X\ne_4\u271d : G.obj (obj Y) = F.obj Y\n\u22a2 HEq (G.map (map f)) (F.map f)\n[PROOFSTEP]\napply h_obj\n[GOAL]\ncase h.e'_4.e'_6\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d\u00b9 : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\ne_3\u271d : G.obj (obj X) = F.obj X\ne_4\u271d : G.obj (obj Y) = F.obj Y\n\u22a2 HEq (G.map (map f)) (F.map f)\n[PROOFSTEP]\napply h_map\n[GOAL]\ncase h.e'_4.e'_7\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\ne_4\u271d : G.obj (obj Y) = F.obj Y\ne_5\u271d : G.obj (obj Z) = F.obj Z\n\u22a2 HEq (G.map (map g)) (F.map g)\n[PROOFSTEP]\n{first\n | apply h_obj\n | apply h_map\n}\n[GOAL]\ncase h.e'_4.e'_7\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\ne_4\u271d : G.obj (obj Y) = F.obj Y\ne_5\u271d : G.obj (obj Z) = F.obj Z\n\u22a2 HEq (G.map (map g)) (F.map g)\n[PROOFSTEP]\nfirst\n| apply h_obj\n| apply h_map\n[GOAL]\ncase h.e'_4.e'_7\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\ne_4\u271d : G.obj (obj Y) = F.obj Y\ne_5\u271d : G.obj (obj Z) = F.obj Z\n\u22a2 HEq (G.map (map g)) (F.map g)\n[PROOFSTEP]\napply h_obj\n[GOAL]\ncase h.e'_4.e'_7\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ne_3\u271d : (G.obj (obj X) \u27f6 G.obj (obj Z)) = (F.obj X \u27f6 F.obj Z)\ne_4\u271d : G.obj (obj Y) = F.obj Y\ne_5\u271d : G.obj (obj Z) = F.obj Z\n\u22a2 HEq (G.map (map g)) (F.map g)\n[PROOFSTEP]\napply h_map\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF : C \u2964 E\ninst\u271d\u00b9 : Faithful F\nG : D \u2964 E\ninst\u271d : Faithful G\nobj : C \u2192 D\nh_obj : \u2200 (X : C), G.obj (obj X) = F.obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) (F.map f)\n\u22a2 Faithful.div F G obj h_obj map h_map \u22d9 G = F\n[PROOFSTEP]\ncases' F with F_pre _ _\n[GOAL]\ncase mk\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nF F' : C \u2964 D\nG\u271d G : D \u2964 E\ninst\u271d\u00b9 : Faithful G\nobj : C \u2192 D\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nF_pre : C \u2964q E\nmap_id\u271d : \u2200 (X : C), F_pre.map (\ud835\udfd9 X) = \ud835\udfd9 (F_pre.obj X)\nmap_comp\u271d : \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z), F_pre.map (f \u226b g) = F_pre.map f \u226b F_pre.map g\ninst\u271d : Faithful (Functor.mk F_pre)\nh_obj : \u2200 (X : C), G.obj (obj X) = (Functor.mk F_pre).obj X\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq (G.map (map f)) ((Functor.mk F_pre).map f)\n\u22a2 Faithful.div (Functor.mk F_pre) G obj h_obj map h_map \u22d9 G = Functor.mk F_pre\n[PROOFSTEP]\ncases' G with G_pre _ _\n[GOAL]\ncase mk.mk\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nF F' : C \u2964 D\nG : D \u2964 E\nobj : C \u2192 D\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nF_pre : C \u2964q E\nmap_id\u271d\u00b9 : \u2200 (X : C), F_pre.map (\ud835\udfd9 X) = \ud835\udfd9 (F_pre.obj X)\nmap_comp\u271d\u00b9 : \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z), F_pre.map (f \u226b g) = F_pre.map f \u226b F_pre.map g\ninst\u271d\u00b9 : Faithful (Functor.mk F_pre)\nG_pre : D \u2964q E\nmap_id\u271d : \u2200 (X : D), G_pre.map (\ud835\udfd9 X) = \ud835\udfd9 (G_pre.obj X)\nmap_comp\u271d : \u2200 {X Y Z : D} (f : X \u27f6 Y) (g : Y \u27f6 Z), G_pre.map (f \u226b g) = G_pre.map f \u226b G_pre.map g\ninst\u271d : Faithful (Functor.mk G_pre)\nh_obj : \u2200 (X : C), (Functor.mk G_pre).obj (obj X) = (Functor.mk F_pre).obj X\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq ((Functor.mk G_pre).map (map f)) ((Functor.mk F_pre).map f)\n\u22a2 Faithful.div (Functor.mk F_pre) (Functor.mk G_pre) obj h_obj map h_map \u22d9 Functor.mk G_pre = Functor.mk F_pre\n[PROOFSTEP]\ncases' F_pre with F_obj _\n[GOAL]\ncase mk.mk.mk\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nF F' : C \u2964 D\nG : D \u2964 E\nobj : C \u2192 D\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nG_pre : D \u2964q E\nmap_id\u271d\u00b9 : \u2200 (X : D), G_pre.map (\ud835\udfd9 X) = \ud835\udfd9 (G_pre.obj X)\nmap_comp\u271d\u00b9 : \u2200 {X Y Z : D} (f : X \u27f6 Y) (g : Y \u27f6 Z), G_pre.map (f \u226b g) = G_pre.map f \u226b G_pre.map g\ninst\u271d\u00b9 : Faithful (Functor.mk G_pre)\nF_obj : C \u2192 E\nmap\u271d : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (F_obj X \u27f6 F_obj Y)\nmap_id\u271d : \u2200 (X : C), { obj := F_obj, map := map\u271d }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := F_obj, map := map\u271d }.obj X)\nmap_comp\u271d :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := F_obj, map := map\u271d }.map (f \u226b g) =\n { obj := F_obj, map := map\u271d }.map f \u226b { obj := F_obj, map := map\u271d }.map g\ninst\u271d : Faithful (Functor.mk { obj := F_obj, map := map\u271d })\nh_obj : \u2200 (X : C), (Functor.mk G_pre).obj (obj X) = (Functor.mk { obj := F_obj, map := map\u271d }).obj X\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq ((Functor.mk G_pre).map (map f)) ((Functor.mk { obj := F_obj, map := map\u271d }).map f)\n\u22a2 Faithful.div (Functor.mk { obj := F_obj, map := map\u271d }) (Functor.mk G_pre) obj h_obj map h_map \u22d9 Functor.mk G_pre =\n Functor.mk { obj := F_obj, map := map\u271d }\n[PROOFSTEP]\ncases' G_pre with G_obj _\n[GOAL]\ncase mk.mk.mk.mk\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nF F' : C \u2964 D\nG : D \u2964 E\nobj : C \u2192 D\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nF_obj : C \u2192 E\nmap\u271d\u00b9 : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (F_obj X \u27f6 F_obj Y)\nmap_id\u271d\u00b9 : \u2200 (X : C), { obj := F_obj, map := map\u271d\u00b9 }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := F_obj, map := map\u271d\u00b9 }.obj X)\nmap_comp\u271d\u00b9 :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := F_obj, map := map\u271d\u00b9 }.map (f \u226b g) =\n { obj := F_obj, map := map\u271d\u00b9 }.map f \u226b { obj := F_obj, map := map\u271d\u00b9 }.map g\ninst\u271d\u00b9 : Faithful (Functor.mk { obj := F_obj, map := map\u271d\u00b9 })\nG_obj : D \u2192 E\nmap\u271d : {X Y : D} \u2192 (X \u27f6 Y) \u2192 (G_obj X \u27f6 G_obj Y)\nmap_id\u271d : \u2200 (X : D), { obj := G_obj, map := map\u271d }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := G_obj, map := map\u271d }.obj X)\nmap_comp\u271d :\n \u2200 {X Y Z : D} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := G_obj, map := map\u271d }.map (f \u226b g) =\n { obj := G_obj, map := map\u271d }.map f \u226b { obj := G_obj, map := map\u271d }.map g\ninst\u271d : Faithful (Functor.mk { obj := G_obj, map := map\u271d })\nh_obj :\n \u2200 (X : C), (Functor.mk { obj := G_obj, map := map\u271d }).obj (obj X) = (Functor.mk { obj := F_obj, map := map\u271d\u00b9 }).obj X\nh_map :\n \u2200 {X Y : C} {f : X \u27f6 Y},\n HEq ((Functor.mk { obj := G_obj, map := map\u271d }).map (map f)) ((Functor.mk { obj := F_obj, map := map\u271d\u00b9 }).map f)\n\u22a2 Faithful.div (Functor.mk { obj := F_obj, map := map\u271d\u00b9 }) (Functor.mk { obj := G_obj, map := map\u271d }) obj h_obj map\n h_map \u22d9\n Functor.mk { obj := G_obj, map := map\u271d } =\n Functor.mk { obj := F_obj, map := map\u271d\u00b9 }\n[PROOFSTEP]\nunfold Faithful.div Functor.comp\n[GOAL]\ncase mk.mk.mk.mk\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nF F' : C \u2964 D\nG : D \u2964 E\nobj : C \u2192 D\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nF_obj : C \u2192 E\nmap\u271d\u00b9 : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (F_obj X \u27f6 F_obj Y)\nmap_id\u271d\u00b9 : \u2200 (X : C), { obj := F_obj, map := map\u271d\u00b9 }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := F_obj, map := map\u271d\u00b9 }.obj X)\nmap_comp\u271d\u00b9 :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := F_obj, map := map\u271d\u00b9 }.map (f \u226b g) =\n { obj := F_obj, map := map\u271d\u00b9 }.map f \u226b { obj := F_obj, map := map\u271d\u00b9 }.map g\ninst\u271d\u00b9 : Faithful (Functor.mk { obj := F_obj, map := map\u271d\u00b9 })\nG_obj : D \u2192 E\nmap\u271d : {X Y : D} \u2192 (X \u27f6 Y) \u2192 (G_obj X \u27f6 G_obj Y)\nmap_id\u271d : \u2200 (X : D), { obj := G_obj, map := map\u271d }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := G_obj, map := map\u271d }.obj X)\nmap_comp\u271d :\n \u2200 {X Y Z : D} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := G_obj, map := map\u271d }.map (f \u226b g) =\n { obj := G_obj, map := map\u271d }.map f \u226b { obj := G_obj, map := map\u271d }.map g\ninst\u271d : Faithful (Functor.mk { obj := G_obj, map := map\u271d })\nh_obj :\n \u2200 (X : C), (Functor.mk { obj := G_obj, map := map\u271d }).obj (obj X) = (Functor.mk { obj := F_obj, map := map\u271d\u00b9 }).obj X\nh_map :\n \u2200 {X Y : C} {f : X \u27f6 Y},\n HEq ((Functor.mk { obj := G_obj, map := map\u271d }).map (map f)) ((Functor.mk { obj := F_obj, map := map\u271d\u00b9 }).map f)\n\u22a2 Functor.mk\n { obj := fun X => (Functor.mk { obj := G_obj, map := map\u271d }).obj ((Functor.mk { obj := obj, map := map }).obj X),\n map := fun {X Y} f =>\n (Functor.mk { obj := G_obj, map := map\u271d }).map ((Functor.mk { obj := obj, map := map }).map f) } =\n Functor.mk { obj := F_obj, map := map\u271d\u00b9 }\n[PROOFSTEP]\nhave : F_obj = G_obj \u2218 obj := (funext h_obj).symm\n[GOAL]\ncase mk.mk.mk.mk\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nF F' : C \u2964 D\nG : D \u2964 E\nobj : C \u2192 D\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nF_obj : C \u2192 E\nmap\u271d\u00b9 : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (F_obj X \u27f6 F_obj Y)\nmap_id\u271d\u00b9 : \u2200 (X : C), { obj := F_obj, map := map\u271d\u00b9 }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := F_obj, map := map\u271d\u00b9 }.obj X)\nmap_comp\u271d\u00b9 :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := F_obj, map := map\u271d\u00b9 }.map (f \u226b g) =\n { obj := F_obj, map := map\u271d\u00b9 }.map f \u226b { obj := F_obj, map := map\u271d\u00b9 }.map g\ninst\u271d\u00b9 : Faithful (Functor.mk { obj := F_obj, map := map\u271d\u00b9 })\nG_obj : D \u2192 E\nmap\u271d : {X Y : D} \u2192 (X \u27f6 Y) \u2192 (G_obj X \u27f6 G_obj Y)\nmap_id\u271d : \u2200 (X : D), { obj := G_obj, map := map\u271d }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := G_obj, map := map\u271d }.obj X)\nmap_comp\u271d :\n \u2200 {X Y Z : D} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := G_obj, map := map\u271d }.map (f \u226b g) =\n { obj := G_obj, map := map\u271d }.map f \u226b { obj := G_obj, map := map\u271d }.map g\ninst\u271d : Faithful (Functor.mk { obj := G_obj, map := map\u271d })\nh_obj :\n \u2200 (X : C), (Functor.mk { obj := G_obj, map := map\u271d }).obj (obj X) = (Functor.mk { obj := F_obj, map := map\u271d\u00b9 }).obj X\nh_map :\n \u2200 {X Y : C} {f : X \u27f6 Y},\n HEq ((Functor.mk { obj := G_obj, map := map\u271d }).map (map f)) ((Functor.mk { obj := F_obj, map := map\u271d\u00b9 }).map f)\nthis : F_obj = G_obj \u2218 obj\n\u22a2 Functor.mk\n { obj := fun X => (Functor.mk { obj := G_obj, map := map\u271d }).obj ((Functor.mk { obj := obj, map := map }).obj X),\n map := fun {X Y} f =>\n (Functor.mk { obj := G_obj, map := map\u271d }).map ((Functor.mk { obj := obj, map := map }).map f) } =\n Functor.mk { obj := F_obj, map := map\u271d\u00b9 }\n[PROOFSTEP]\nsubst this\n[GOAL]\ncase mk.mk.mk.mk\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nF F' : C \u2964 D\nG : D \u2964 E\nobj : C \u2192 D\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nG_obj : D \u2192 E\nmap\u271d\u00b9 : {X Y : D} \u2192 (X \u27f6 Y) \u2192 (G_obj X \u27f6 G_obj Y)\nmap_id\u271d\u00b9 : \u2200 (X : D), { obj := G_obj, map := map\u271d\u00b9 }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := G_obj, map := map\u271d\u00b9 }.obj X)\nmap_comp\u271d\u00b9 :\n \u2200 {X Y Z : D} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := G_obj, map := map\u271d\u00b9 }.map (f \u226b g) =\n { obj := G_obj, map := map\u271d\u00b9 }.map f \u226b { obj := G_obj, map := map\u271d\u00b9 }.map g\ninst\u271d\u00b9 : Faithful (Functor.mk { obj := G_obj, map := map\u271d\u00b9 })\nmap\u271d : {X Y : C} \u2192 (X \u27f6 Y) \u2192 ((G_obj \u2218 obj) X \u27f6 (G_obj \u2218 obj) Y)\nmap_id\u271d : \u2200 (X : C), { obj := G_obj \u2218 obj, map := map\u271d }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := G_obj \u2218 obj, map := map\u271d }.obj X)\nmap_comp\u271d :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := G_obj \u2218 obj, map := map\u271d }.map (f \u226b g) =\n { obj := G_obj \u2218 obj, map := map\u271d }.map f \u226b { obj := G_obj \u2218 obj, map := map\u271d }.map g\ninst\u271d : Faithful (Functor.mk { obj := G_obj \u2218 obj, map := map\u271d })\nh_obj :\n \u2200 (X : C),\n (Functor.mk { obj := G_obj, map := map\u271d\u00b9 }).obj (obj X) = (Functor.mk { obj := G_obj \u2218 obj, map := map\u271d }).obj X\nh_map :\n \u2200 {X Y : C} {f : X \u27f6 Y},\n HEq ((Functor.mk { obj := G_obj, map := map\u271d\u00b9 }).map (map f))\n ((Functor.mk { obj := G_obj \u2218 obj, map := map\u271d }).map f)\n\u22a2 Functor.mk\n { obj := fun X => (Functor.mk { obj := G_obj, map := map\u271d\u00b9 }).obj ((Functor.mk { obj := obj, map := map }).obj X),\n map := fun {X Y} f =>\n (Functor.mk { obj := G_obj, map := map\u271d\u00b9 }).map ((Functor.mk { obj := obj, map := map }).map f) } =\n Functor.mk { obj := G_obj \u2218 obj, map := map\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.mk.mk.e_toPrefunctor.e_map\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nF F' : C \u2964 D\nG : D \u2964 E\nobj : C \u2192 D\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nG_obj : D \u2192 E\nmap\u271d\u00b9 : {X Y : D} \u2192 (X \u27f6 Y) \u2192 (G_obj X \u27f6 G_obj Y)\nmap_id\u271d\u00b9 : \u2200 (X : D), { obj := G_obj, map := map\u271d\u00b9 }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := G_obj, map := map\u271d\u00b9 }.obj X)\nmap_comp\u271d\u00b9 :\n \u2200 {X Y Z : D} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := G_obj, map := map\u271d\u00b9 }.map (f \u226b g) =\n { obj := G_obj, map := map\u271d\u00b9 }.map f \u226b { obj := G_obj, map := map\u271d\u00b9 }.map g\ninst\u271d\u00b9 : Faithful (Functor.mk { obj := G_obj, map := map\u271d\u00b9 })\nmap\u271d : {X Y : C} \u2192 (X \u27f6 Y) \u2192 ((G_obj \u2218 obj) X \u27f6 (G_obj \u2218 obj) Y)\nmap_id\u271d : \u2200 (X : C), { obj := G_obj \u2218 obj, map := map\u271d }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := G_obj \u2218 obj, map := map\u271d }.obj X)\nmap_comp\u271d :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := G_obj \u2218 obj, map := map\u271d }.map (f \u226b g) =\n { obj := G_obj \u2218 obj, map := map\u271d }.map f \u226b { obj := G_obj \u2218 obj, map := map\u271d }.map g\ninst\u271d : Faithful (Functor.mk { obj := G_obj \u2218 obj, map := map\u271d })\nh_obj :\n \u2200 (X : C),\n (Functor.mk { obj := G_obj, map := map\u271d\u00b9 }).obj (obj X) = (Functor.mk { obj := G_obj \u2218 obj, map := map\u271d }).obj X\nh_map :\n \u2200 {X Y : C} {f : X \u27f6 Y},\n HEq ((Functor.mk { obj := G_obj, map := map\u271d\u00b9 }).map (map f))\n ((Functor.mk { obj := G_obj \u2218 obj, map := map\u271d }).map f)\n\u22a2 (fun {X Y} f => (Functor.mk { obj := G_obj, map := map\u271d\u00b9 }).map ((Functor.mk { obj := obj, map := map }).map f)) =\n map\u271d\n[PROOFSTEP]\nsimp only [Function.comp_apply, heq_eq_eq] at h_map \n[GOAL]\ncase mk.mk.mk.mk.e_toPrefunctor.e_map\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nF F' : C \u2964 D\nG : D \u2964 E\nobj : C \u2192 D\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nG_obj : D \u2192 E\nmap\u271d\u00b9 : {X Y : D} \u2192 (X \u27f6 Y) \u2192 (G_obj X \u27f6 G_obj Y)\nmap_id\u271d\u00b9 : \u2200 (X : D), { obj := G_obj, map := map\u271d\u00b9 }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := G_obj, map := map\u271d\u00b9 }.obj X)\nmap_comp\u271d\u00b9 :\n \u2200 {X Y Z : D} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := G_obj, map := map\u271d\u00b9 }.map (f \u226b g) =\n { obj := G_obj, map := map\u271d\u00b9 }.map f \u226b { obj := G_obj, map := map\u271d\u00b9 }.map g\ninst\u271d\u00b9 : Faithful (Functor.mk { obj := G_obj, map := map\u271d\u00b9 })\nmap\u271d : {X Y : C} \u2192 (X \u27f6 Y) \u2192 ((G_obj \u2218 obj) X \u27f6 (G_obj \u2218 obj) Y)\nmap_id\u271d : \u2200 (X : C), { obj := G_obj \u2218 obj, map := map\u271d }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := G_obj \u2218 obj, map := map\u271d }.obj X)\nmap_comp\u271d :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := G_obj \u2218 obj, map := map\u271d }.map (f \u226b g) =\n { obj := G_obj \u2218 obj, map := map\u271d }.map f \u226b { obj := G_obj \u2218 obj, map := map\u271d }.map g\ninst\u271d : Faithful (Functor.mk { obj := G_obj \u2218 obj, map := map\u271d })\nh_obj :\n \u2200 (X : C),\n (Functor.mk { obj := G_obj, map := map\u271d\u00b9 }).obj (obj X) = (Functor.mk { obj := G_obj \u2218 obj, map := map\u271d }).obj X\nh_map\u271d :\n \u2200 {X Y : C} {f : X \u27f6 Y},\n HEq ((Functor.mk { obj := G_obj, map := map\u271d\u00b9 }).map (map f))\n ((Functor.mk { obj := G_obj \u2218 obj, map := map\u271d }).map f)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, map\u271d\u00b9 (map f) = map\u271d f\n\u22a2 (fun {X Y} f => (Functor.mk { obj := G_obj, map := map\u271d\u00b9 }).map ((Functor.mk { obj := obj, map := map }).map f)) =\n map\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase mk.mk.mk.mk.e_toPrefunctor.e_map.h.h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nF F' : C \u2964 D\nG : D \u2964 E\nobj : C \u2192 D\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nG_obj : D \u2192 E\nmap\u271d\u00b9 : {X Y : D} \u2192 (X \u27f6 Y) \u2192 (G_obj X \u27f6 G_obj Y)\nmap_id\u271d\u00b9 : \u2200 (X : D), { obj := G_obj, map := map\u271d\u00b9 }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := G_obj, map := map\u271d\u00b9 }.obj X)\nmap_comp\u271d\u00b9 :\n \u2200 {X Y Z : D} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := G_obj, map := map\u271d\u00b9 }.map (f \u226b g) =\n { obj := G_obj, map := map\u271d\u00b9 }.map f \u226b { obj := G_obj, map := map\u271d\u00b9 }.map g\ninst\u271d\u00b9 : Faithful (Functor.mk { obj := G_obj, map := map\u271d\u00b9 })\nmap\u271d : {X Y : C} \u2192 (X \u27f6 Y) \u2192 ((G_obj \u2218 obj) X \u27f6 (G_obj \u2218 obj) Y)\nmap_id\u271d : \u2200 (X : C), { obj := G_obj \u2218 obj, map := map\u271d }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := G_obj \u2218 obj, map := map\u271d }.obj X)\nmap_comp\u271d :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := G_obj \u2218 obj, map := map\u271d }.map (f \u226b g) =\n { obj := G_obj \u2218 obj, map := map\u271d }.map f \u226b { obj := G_obj \u2218 obj, map := map\u271d }.map g\ninst\u271d : Faithful (Functor.mk { obj := G_obj \u2218 obj, map := map\u271d })\nh_obj :\n \u2200 (X : C),\n (Functor.mk { obj := G_obj, map := map\u271d\u00b9 }).obj (obj X) = (Functor.mk { obj := G_obj \u2218 obj, map := map\u271d }).obj X\nh_map\u271d :\n \u2200 {X Y : C} {f : X \u27f6 Y},\n HEq ((Functor.mk { obj := G_obj, map := map\u271d\u00b9 }).map (map f))\n ((Functor.mk { obj := G_obj \u2218 obj, map := map\u271d }).map f)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, map\u271d\u00b9 (map f) = map\u271d f\nx\u271d\u00b2 x\u271d\u00b9 : C\nx\u271d : x\u271d\u00b2 \u27f6 x\u271d\u00b9\n\u22a2 (Functor.mk { obj := G_obj, map := map\u271d\u00b9 }).map ((Functor.mk { obj := obj, map := map }).map x\u271d) = map\u271d x\u271d\n[PROOFSTEP]\nexact h_map\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nF\u271d F' : C \u2964 D\nG\u271d : D \u2964 E\nF G : C \u2964 D\nH : D \u2964 E\ninst\u271d\u00b9 : Full H\ninst\u271d : Faithful H\ncomp_iso : F \u22d9 H \u2245 G \u22d9 H\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\n\u22a2 H.map (F.map f \u226b ((fun X => Functor.preimageIso H (comp_iso.app X)) Y\u271d).hom) =\n H.map (((fun X => Functor.preimageIso H (comp_iso.app X)) X\u271d).hom \u226b G.map f)\n[PROOFSTEP]\nsimpa using comp_iso.hom.naturality f\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Functor.FullyFaithful", "llama_tokens": 26220, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585903489891, "lm_q2_score": 0.6076631698328917, "lm_q1q2_score": 0.4442373803450322}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM\u271d : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b2 : Zero R\ninst\u271d\u00b9 : Zero M\ninst\u271d : ZeroHomClass F R M\nf : F\nr : R\nneZero : NeZero (\u2191f r)\nh : r = 0\n\u22a2 \u2191f r = 0\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM\u271d : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b2 : Zero R\ninst\u271d\u00b9 : Zero M\ninst\u271d : ZeroHomClass F R M\nf : F\nr : R\nneZero : NeZero (\u2191f r)\nh : r = 0\n\u22a2 \u2191f 0 = 0\n[PROOFSTEP]\nexact ZeroHomClass.map_zero f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM\u271d : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Zero R\nr : R\ninst\u271d\u00b2 : NeZero r\ninst\u271d\u00b9 : Zero M\ninst\u271d : ZeroHomClass F R M\nf : F\nhf : Function.Injective \u2191f\n\u22a2 \u2191f r \u2260 0\n[PROOFSTEP]\nrw [\u2190 ZeroHomClass.map_zero f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM\u271d : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\nR : Type u_9\nM : Type u_10\ninst\u271d\u00b3 : Zero R\nr : R\ninst\u271d\u00b2 : NeZero r\ninst\u271d\u00b9 : Zero M\ninst\u271d : ZeroHomClass F R M\nf : F\nhf : Function.Injective \u2191f\n\u22a2 \u2191f r \u2260 \u2191f 0\n[PROOFSTEP]\nexact hf.ne NeZero.out\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : One M\ninst\u271d : One N\nf g : OneHom M N\nh : f.toFun = g.toFun\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ng : OneHom M N\ntoFun\u271d : M \u2192 N\nmap_one'\u271d : toFun\u271d 1 = 1\nh : { toFun := toFun\u271d, map_one' := map_one'\u271d }.toFun = g.toFun\n\u22a2 { toFun := toFun\u271d, map_one' := map_one'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ntoFun\u271d\u00b9 : M \u2192 N\nmap_one'\u271d\u00b9 : toFun\u271d\u00b9 1 = 1\ntoFun\u271d : M \u2192 N\nmap_one'\u271d : toFun\u271d 1 = 1\nh : { toFun := toFun\u271d\u00b9, map_one' := map_one'\u271d\u00b9 }.toFun = { toFun := toFun\u271d, map_one' := map_one'\u271d }.toFun\n\u22a2 { toFun := toFun\u271d\u00b9, map_one' := map_one'\u271d\u00b9 } = { toFun := toFun\u271d, map_one' := map_one'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : Mul M\ninst\u271d : Mul N\nf g : M \u2192\u2099* N\nh : f.toFun = g.toFun\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : Mul M\ninst\u271d : Mul N\ng : M \u2192\u2099* N\ntoFun\u271d : M \u2192 N\nmap_mul'\u271d : \u2200 (x y : M), toFun\u271d (x * y) = toFun\u271d x * toFun\u271d y\nh : { toFun := toFun\u271d, map_mul' := map_mul'\u271d }.toFun = g.toFun\n\u22a2 { toFun := toFun\u271d, map_mul' := map_mul'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : Mul M\ninst\u271d : Mul N\ntoFun\u271d\u00b9 : M \u2192 N\nmap_mul'\u271d\u00b9 : \u2200 (x y : M), toFun\u271d\u00b9 (x * y) = toFun\u271d\u00b9 x * toFun\u271d\u00b9 y\ntoFun\u271d : M \u2192 N\nmap_mul'\u271d : \u2200 (x y : M), toFun\u271d (x * y) = toFun\u271d x * toFun\u271d y\nh : { toFun := toFun\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }.toFun = { toFun := toFun\u271d, map_mul' := map_mul'\u271d }.toFun\n\u22a2 { toFun := toFun\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } = { toFun := toFun\u271d, map_mul' := map_mul'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : MulOneClass M\ninst\u271d : MulOneClass N\nf g : M \u2192* N\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : MulOneClass M\ninst\u271d : MulOneClass N\ng : M \u2192* N\ntoOneHom\u271d : OneHom M N\nmap_mul'\u271d : \u2200 (x y : M), OneHom.toFun toOneHom\u271d (x * y) = OneHom.toFun toOneHom\u271d x * OneHom.toFun toOneHom\u271d y\nh : (fun f => f.toFun) { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d } = (fun f => f.toFun) g\n\u22a2 { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : MulOneClass M\ninst\u271d : MulOneClass N\ntoOneHom\u271d\u00b9 : OneHom M N\nmap_mul'\u271d\u00b9 : \u2200 (x y : M), OneHom.toFun toOneHom\u271d\u00b9 (x * y) = OneHom.toFun toOneHom\u271d\u00b9 x * OneHom.toFun toOneHom\u271d\u00b9 y\ntoOneHom\u271d : OneHom M N\nmap_mul'\u271d : \u2200 (x y : M), OneHom.toFun toOneHom\u271d (x * y) = OneHom.toFun toOneHom\u271d x * OneHom.toFun toOneHom\u271d y\nh :\n (fun f => f.toFun) { toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } =\n (fun f => f.toFun) { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }\n\u22a2 { toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } = { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toOneHom\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : MulOneClass M\ninst\u271d : MulOneClass N\ntoOneHom\u271d\u00b9 : OneHom M N\nmap_mul'\u271d\u00b9 : \u2200 (x y : M), OneHom.toFun toOneHom\u271d\u00b9 (x * y) = OneHom.toFun toOneHom\u271d\u00b9 x * OneHom.toFun toOneHom\u271d\u00b9 y\ntoOneHom\u271d : OneHom M N\nmap_mul'\u271d : \u2200 (x y : M), OneHom.toFun toOneHom\u271d (x * y) = OneHom.toFun toOneHom\u271d x * OneHom.toFun toOneHom\u271d y\nh :\n (fun f => f.toFun) { toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } =\n (fun f => f.toFun) { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }\n\u22a2 toOneHom\u271d\u00b9 = toOneHom\u271d\n[PROOFSTEP]\napply FunLike.coe_injective'\n[GOAL]\ncase mk.mk.e_toOneHom.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : MulOneClass M\ninst\u271d : MulOneClass N\ntoOneHom\u271d\u00b9 : OneHom M N\nmap_mul'\u271d\u00b9 : \u2200 (x y : M), OneHom.toFun toOneHom\u271d\u00b9 (x * y) = OneHom.toFun toOneHom\u271d\u00b9 x * OneHom.toFun toOneHom\u271d\u00b9 y\ntoOneHom\u271d : OneHom M N\nmap_mul'\u271d : \u2200 (x y : M), OneHom.toFun toOneHom\u271d (x * y) = OneHom.toFun toOneHom\u271d x * OneHom.toFun toOneHom\u271d y\nh :\n (fun f => f.toFun) { toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } =\n (fun f => f.toFun) { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }\n\u22a2 \u2191toOneHom\u271d\u00b9 = \u2191toOneHom\u271d\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MonoidHomClass F M N\nf : F\na b : M\nh : a * b = 1\n\u22a2 \u2191f a * \u2191f b = 1\n[PROOFSTEP]\nrw [\u2190 map_mul, h, map_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2074 : MulOneClass M\ninst\u271d\u00b3 : MulOneClass N\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : DivInvMonoid H\ninst\u271d : MonoidHomClass F G H\nf : F\nhf : \u2200 (a : G), \u2191f a\u207b\u00b9 = (\u2191f a)\u207b\u00b9\na b : G\n\u22a2 \u2191f (a / b) = \u2191f a / \u2191f b\n[PROOFSTEP]\nrw [div_eq_mul_inv, div_eq_mul_inv, map_mul, hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2074 : MulOneClass M\ninst\u271d\u00b3 : MulOneClass N\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : DivisionMonoid H\ninst\u271d : MonoidHomClass F G H\nf : F\na b : G\n\u22a2 \u2191f (a * b\u207b\u00b9) = \u2191f a * (\u2191f b)\u207b\u00b9\n[PROOFSTEP]\nrw [map_mul, map_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2074 : MulOneClass M\ninst\u271d\u00b3 : MulOneClass N\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : Monoid H\ninst\u271d : MonoidHomClass F G H\nf : F\na : G\n\u22a2 \u2191f (a ^ 0) = \u2191f a ^ 0\n[PROOFSTEP]\nrw [pow_zero, pow_zero, map_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2074 : MulOneClass M\ninst\u271d\u00b3 : MulOneClass N\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : Monoid H\ninst\u271d : MonoidHomClass F G H\nf : F\na : G\nn : \u2115\n\u22a2 \u2191f (a ^ (n + 1)) = \u2191f a ^ (n + 1)\n[PROOFSTEP]\nrw [pow_succ, pow_succ, map_mul, map_pow f a n]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2074 : MulOneClass M\ninst\u271d\u00b3 : MulOneClass N\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : DivInvMonoid H\ninst\u271d : MonoidHomClass F G H\nf : F\nhf : \u2200 (x : G), \u2191f x\u207b\u00b9 = (\u2191f x)\u207b\u00b9\na : G\nn : \u2115\n\u22a2 \u2191f (a ^ \u2191n) = \u2191f a ^ \u2191n\n[PROOFSTEP]\nrw [zpow_ofNat, map_pow, zpow_ofNat]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2074 : MulOneClass M\ninst\u271d\u00b3 : MulOneClass N\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : DivInvMonoid H\ninst\u271d : MonoidHomClass F G H\nf : F\nhf : \u2200 (x : G), \u2191f x\u207b\u00b9 = (\u2191f x)\u207b\u00b9\na : G\nn : \u2115\n\u22a2 \u2191f (a ^ Int.negSucc n) = \u2191f a ^ Int.negSucc n\n[PROOFSTEP]\nrw [zpow_negSucc, hf, map_pow, \u2190 zpow_negSucc, \u2190 zpow_negSucc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : MulZeroOneClass M\ninst\u271d : MulZeroOneClass N\nf g : M \u2192*\u2080 N\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : MulZeroOneClass M\ninst\u271d : MulZeroOneClass N\ng : M \u2192*\u2080 N\ntoZeroHom\u271d : ZeroHom M N\nmap_one'\u271d : ZeroHom.toFun toZeroHom\u271d 1 = 1\nmap_mul'\u271d : \u2200 (x y : M), ZeroHom.toFun toZeroHom\u271d (x * y) = ZeroHom.toFun toZeroHom\u271d x * ZeroHom.toFun toZeroHom\u271d y\nh : (fun f => f.toFun) { toZeroHom := toZeroHom\u271d, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d } = (fun f => f.toFun) g\n\u22a2 { toZeroHom := toZeroHom\u271d, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : MulZeroOneClass M\ninst\u271d : MulZeroOneClass N\ntoZeroHom\u271d\u00b9 : ZeroHom M N\nmap_one'\u271d\u00b9 : ZeroHom.toFun toZeroHom\u271d\u00b9 1 = 1\nmap_mul'\u271d\u00b9 : \u2200 (x y : M), ZeroHom.toFun toZeroHom\u271d\u00b9 (x * y) = ZeroHom.toFun toZeroHom\u271d\u00b9 x * ZeroHom.toFun toZeroHom\u271d\u00b9 y\ntoZeroHom\u271d : ZeroHom M N\nmap_one'\u271d : ZeroHom.toFun toZeroHom\u271d 1 = 1\nmap_mul'\u271d : \u2200 (x y : M), ZeroHom.toFun toZeroHom\u271d (x * y) = ZeroHom.toFun toZeroHom\u271d x * ZeroHom.toFun toZeroHom\u271d y\nh :\n (fun f => f.toFun) { toZeroHom := toZeroHom\u271d\u00b9, map_one' := map_one'\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } =\n (fun f => f.toFun) { toZeroHom := toZeroHom\u271d, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d }\n\u22a2 { toZeroHom := toZeroHom\u271d\u00b9, map_one' := map_one'\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } =\n { toZeroHom := toZeroHom\u271d, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toZeroHom\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : MulZeroOneClass M\ninst\u271d : MulZeroOneClass N\ntoZeroHom\u271d\u00b9 : ZeroHom M N\nmap_one'\u271d\u00b9 : ZeroHom.toFun toZeroHom\u271d\u00b9 1 = 1\nmap_mul'\u271d\u00b9 : \u2200 (x y : M), ZeroHom.toFun toZeroHom\u271d\u00b9 (x * y) = ZeroHom.toFun toZeroHom\u271d\u00b9 x * ZeroHom.toFun toZeroHom\u271d\u00b9 y\ntoZeroHom\u271d : ZeroHom M N\nmap_one'\u271d : ZeroHom.toFun toZeroHom\u271d 1 = 1\nmap_mul'\u271d : \u2200 (x y : M), ZeroHom.toFun toZeroHom\u271d (x * y) = ZeroHom.toFun toZeroHom\u271d x * ZeroHom.toFun toZeroHom\u271d y\nh :\n (fun f => f.toFun) { toZeroHom := toZeroHom\u271d\u00b9, map_one' := map_one'\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } =\n (fun f => f.toFun) { toZeroHom := toZeroHom\u271d, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d }\n\u22a2 toZeroHom\u271d\u00b9 = toZeroHom\u271d\n[PROOFSTEP]\napply FunLike.coe_injective'\n[GOAL]\ncase mk.mk.e_toZeroHom.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : MulZeroOneClass M\ninst\u271d : MulZeroOneClass N\ntoZeroHom\u271d\u00b9 : ZeroHom M N\nmap_one'\u271d\u00b9 : ZeroHom.toFun toZeroHom\u271d\u00b9 1 = 1\nmap_mul'\u271d\u00b9 : \u2200 (x y : M), ZeroHom.toFun toZeroHom\u271d\u00b9 (x * y) = ZeroHom.toFun toZeroHom\u271d\u00b9 x * ZeroHom.toFun toZeroHom\u271d\u00b9 y\ntoZeroHom\u271d : ZeroHom M N\nmap_one'\u271d : ZeroHom.toFun toZeroHom\u271d 1 = 1\nmap_mul'\u271d : \u2200 (x y : M), ZeroHom.toFun toZeroHom\u271d (x * y) = ZeroHom.toFun toZeroHom\u271d x * ZeroHom.toFun toZeroHom\u271d y\nh :\n (fun f => f.toFun) { toZeroHom := toZeroHom\u271d\u00b9, map_one' := map_one'\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } =\n (fun f => f.toFun) { toZeroHom := toZeroHom\u271d, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d }\n\u22a2 \u2191toZeroHom\u271d\u00b9 = \u2191toZeroHom\u271d\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : One M\ninst\u271d\u00b9 : One N\ninst\u271d : One P\nhnp : OneHom N P\nhmn : OneHom M N\n\u22a2 (\u2191hnp \u2218 \u2191hmn) 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : Mul M\ninst\u271d\u00b9 : Mul N\ninst\u271d : Mul P\nhnp : N \u2192\u2099* P\nhmn : M \u2192\u2099* N\nx y : M\n\u22a2 (\u2191hnp \u2218 \u2191hmn) (x * y) = (\u2191hnp \u2218 \u2191hmn) x * (\u2191hnp \u2218 \u2191hmn) y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nhnp : N \u2192* P\nhmn : M \u2192* N\n\u22a2 (\u2191hnp \u2218 \u2191hmn) 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nhnp : N \u2192* P\nhmn : M \u2192* N\n\u22a2 \u2200 (x y : M),\n OneHom.toFun { toFun := \u2191hnp \u2218 \u2191hmn, map_one' := (_ : \u2191hnp (\u2191hmn 1) = 1) } (x * y) =\n OneHom.toFun { toFun := \u2191hnp \u2218 \u2191hmn, map_one' := (_ : \u2191hnp (\u2191hmn 1) = 1) } x *\n OneHom.toFun { toFun := \u2191hnp \u2218 \u2191hmn, map_one' := (_ : \u2191hnp (\u2191hmn 1) = 1) } y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : MulZeroOneClass M\ninst\u271d\u00b9 : MulZeroOneClass N\ninst\u271d : MulZeroOneClass P\nhnp : N \u2192*\u2080 P\nhmn : M \u2192*\u2080 N\n\u22a2 (\u2191hnp \u2218 \u2191hmn) 0 = 0\n[PROOFSTEP]\nrw [Function.comp_apply, map_zero, map_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : MulZeroOneClass M\ninst\u271d\u00b9 : MulZeroOneClass N\ninst\u271d : MulZeroOneClass P\nhnp : N \u2192*\u2080 P\nhmn : M \u2192*\u2080 N\n\u22a2 ZeroHom.toFun { toFun := \u2191hnp \u2218 \u2191hmn, map_zero' := (_ : (\u2191hnp \u2218 \u2191hmn) 0 = 0) } 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : MulZeroOneClass M\ninst\u271d\u00b9 : MulZeroOneClass N\ninst\u271d : MulZeroOneClass P\nhnp : N \u2192*\u2080 P\nhmn : M \u2192*\u2080 N\n\u22a2 \u2200 (x y : M),\n ZeroHom.toFun { toFun := \u2191hnp \u2218 \u2191hmn, map_zero' := (_ : (\u2191hnp \u2218 \u2191hmn) 0 = 0) } (x * y) =\n ZeroHom.toFun { toFun := \u2191hnp \u2218 \u2191hmn, map_zero' := (_ : (\u2191hnp \u2218 \u2191hmn) 0 = 0) } x *\n ZeroHom.toFun { toFun := \u2191hnp \u2218 \u2191hmn, map_zero' := (_ : (\u2191hnp \u2218 \u2191hmn) 0 = 0) } y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : One M\ninst\u271d\u00b9 : One N\ninst\u271d : One P\ng : OneHom N P\nf\u2081 f\u2082 : OneHom M N\nhg : Function.Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\nx : M\n\u22a2 \u2191g (\u2191f\u2081 x) = \u2191g (\u2191f\u2082 x)\n[PROOFSTEP]\nrw [\u2190 OneHom.comp_apply, h, OneHom.comp_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : Mul M\ninst\u271d\u00b9 : Mul N\ninst\u271d : Mul P\ng : N \u2192\u2099* P\nf\u2081 f\u2082 : M \u2192\u2099* N\nhg : Function.Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\nx : M\n\u22a2 \u2191g (\u2191f\u2081 x) = \u2191g (\u2191f\u2082 x)\n[PROOFSTEP]\nrw [\u2190 MulHom.comp_apply, h, MulHom.comp_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\ng : N \u2192* P\nf\u2081 f\u2082 : M \u2192* N\nhg : Function.Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\nx : M\n\u22a2 \u2191g (\u2191f\u2081 x) = \u2191g (\u2191f\u2082 x)\n[PROOFSTEP]\nrw [\u2190 MonoidHom.comp_apply, h, MonoidHom.comp_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : MulZeroOneClass M\ninst\u271d\u00b9 : MulZeroOneClass N\ninst\u271d : MulZeroOneClass P\ng : N \u2192*\u2080 P\nf\u2081 f\u2082 : M \u2192*\u2080 N\nhg : Function.Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\nx : M\n\u22a2 \u2191g (\u2191f\u2081 x) = \u2191g (\u2191f\u2082 x)\n[PROOFSTEP]\nrw [\u2190 MonoidWithZeroHom.comp_apply, h, MonoidWithZeroHom.comp_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : One M\ninst\u271d\u00b9 : One N\ninst\u271d : One P\nf : OneHom N P\n\u22a2 comp f 1 = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : One M\ninst\u271d\u00b9 : One N\ninst\u271d : One P\nf : OneHom N P\nx\u271d : M\n\u22a2 \u2191(comp f 1) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\nsimp only [OneHom.map_one, OneHom.coe_comp, Function.comp_apply, OneHom.one_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : Mul M\ninst\u271d : CommSemigroup N\nf g : M \u2192\u2099* N\nx y : M\n\u22a2 (fun m => \u2191f m * \u2191g m) (x * y) = (fun m => \u2191f m * \u2191g m) x * (fun m => \u2191f m * \u2191g m) y\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : Mul M\ninst\u271d : CommSemigroup N\nf g : M \u2192\u2099* N\nx y : M\n\u22a2 (fun m => \u2191f m * \u2191g m) (x * y) = (fun m => \u2191f m * \u2191g m) x * (fun m => \u2191f m * \u2191g m) y\n[PROOFSTEP]\nshow f (x * y) * g (x * y) = f x * g x * (f y * g y)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : Mul M\ninst\u271d : CommSemigroup N\nf g : M \u2192\u2099* N\nx y : M\n\u22a2 \u2191f (x * y) * \u2191g (x * y) = \u2191f x * \u2191g x * (\u2191f y * \u2191g y)\n[PROOFSTEP]\nrw [f.map_mul, g.map_mul, \u2190 mul_assoc, \u2190 mul_assoc, mul_right_comm (f x)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : Mul M\ninst\u271d\u00b9 : CommSemigroup N\ninst\u271d : CommSemigroup P\ng : N \u2192\u2099* P\nf\u2081 f\u2082 : M \u2192\u2099* N\n\u22a2 comp g (f\u2081 * f\u2082) = comp g f\u2081 * comp g f\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : Mul M\ninst\u271d\u00b9 : CommSemigroup N\ninst\u271d : CommSemigroup P\ng : N \u2192\u2099* P\nf\u2081 f\u2082 : M \u2192\u2099* N\nx\u271d : M\n\u22a2 \u2191(comp g (f\u2081 * f\u2082)) x\u271d = \u2191(comp g f\u2081 * comp g f\u2082) x\u271d\n[PROOFSTEP]\nsimp only [mul_apply, Function.comp_apply, map_mul, coe_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM\u271d : Type u_3\nN\u271d : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : CommGroup H\nM : Type ?u.198672\nN : Type ?u.198675\ninst\u271d\u00b9 : MulOneClass M\ninst\u271d : CommMonoid N\nf g : M \u2192* N\n\u22a2 \u2191f 1 * \u2191g 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM\u271d : Type u_3\nN\u271d : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : CommGroup H\nM : Type ?u.198672\nN : Type ?u.198675\ninst\u271d\u00b9 : MulOneClass M\ninst\u271d : CommMonoid N\nf g : M \u2192* N\nx y : M\n\u22a2 OneHom.toFun { toFun := fun m => \u2191f m * \u2191g m, map_one' := (_ : \u2191f 1 * \u2191g 1 = 1) } (x * y) =\n OneHom.toFun { toFun := fun m => \u2191f m * \u2191g m, map_one' := (_ : \u2191f 1 * \u2191g 1 = 1) } x *\n OneHom.toFun { toFun := fun m => \u2191f m * \u2191g m, map_one' := (_ : \u2191f 1 * \u2191g 1 = 1) } y\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM\u271d : Type u_3\nN\u271d : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : CommGroup H\nM : Type ?u.198672\nN : Type ?u.198675\ninst\u271d\u00b9 : MulOneClass M\ninst\u271d : CommMonoid N\nf g : M \u2192* N\nx y : M\n\u22a2 OneHom.toFun { toFun := fun m => \u2191f m * \u2191g m, map_one' := (_ : \u2191f 1 * \u2191g 1 = 1) } (x * y) =\n OneHom.toFun { toFun := fun m => \u2191f m * \u2191g m, map_one' := (_ : \u2191f 1 * \u2191g 1 = 1) } x *\n OneHom.toFun { toFun := fun m => \u2191f m * \u2191g m, map_one' := (_ : \u2191f 1 * \u2191g 1 = 1) } y\n[PROOFSTEP]\nshow f (x * y) * g (x * y) = f x * g x * (f y * g y)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM\u271d : Type u_3\nN\u271d : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : CommGroup H\nM : Type ?u.198672\nN : Type ?u.198675\ninst\u271d\u00b9 : MulOneClass M\ninst\u271d : CommMonoid N\nf g : M \u2192* N\nx y : M\n\u22a2 \u2191f (x * y) * \u2191g (x * y) = \u2191f x * \u2191g x * (\u2191f y * \u2191g y)\n[PROOFSTEP]\nrw [f.map_mul, g.map_mul, \u2190 mul_assoc, \u2190 mul_assoc, mul_right_comm (f x)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : CommGroup H\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nf : N \u2192* P\n\u22a2 comp f 1 = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : CommGroup H\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nf : N \u2192* P\nx\u271d : M\n\u22a2 \u2191(comp f 1) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\nsimp only [map_one, coe_comp, Function.comp_apply, one_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : CommGroup H\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : CommMonoid P\ng : N \u2192* P\nf\u2081 f\u2082 : M \u2192* N\n\u22a2 comp g (f\u2081 * f\u2082) = comp g f\u2081 * comp g f\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : CommGroup H\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : CommMonoid P\ng : N \u2192* P\nf\u2081 f\u2082 : M \u2192* N\nx\u271d : M\n\u22a2 \u2191(comp g (f\u2081 * f\u2082)) x\u271d = \u2191(comp g f\u2081 * comp g f\u2082) x\u271d\n[PROOFSTEP]\nsimp only [mul_apply, Function.comp_apply, map_mul, coe_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : CommGroup H\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : DivisionMonoid \u03b2\nf : \u03b1 \u2192* \u03b2\ng h : \u03b1\n\u22a2 \u2191f (g * h\u207b\u00b9) = \u2191f g * (\u2191f h)\u207b\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG\u271d : Type u_6\nH\u271d : Type u_7\nF : Type u_8\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : CommGroup H\u271d\nG : Type u_9\nH : Type u_10\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulOneClass H\ninst\u271d : MonoidHomClass F G H\nf : F\nh : \u2200 (a : G), \u2191f a = 1 \u2192 a = 1\nx y : G\nhxy : \u2191f x = \u2191f y\n\u22a2 \u2191f (x * y\u207b\u00b9) = 1\n[PROOFSTEP]\nrw [map_mul, hxy, \u2190 map_mul, mul_inv_self, map_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : CommGroup H\ninst\u271d : MulOneClass M\nf : M \u2192 G\nmap_mul : \u2200 (a b : M), f (a * b) = f a * f b\n\u22a2 f 1 * ?m.224335 f map_mul = ?m.224335 f map_mul\n[PROOFSTEP]\nrw [\u2190 map_mul, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH\u271d : Type u_7\nF : Type u_8\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : CommGroup H\u271d\ninst\u271d\u00b9 : MulOneClass M\nH : Type u_9\ninst\u271d : Group H\nf : G \u2192 H\nmap_div : \u2200 (a b : G), f (a * b\u207b\u00b9) = f a * (f b)\u207b\u00b9\nx y : G\n\u22a2 f (x * y) = f x * (f (1 * 1\u207b\u00b9 * y\u207b\u00b9))\u207b\u00b9\n[PROOFSTEP]\n{simp only [one_mul, inv_one, \u2190 map_div, inv_inv]\n}\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH\u271d : Type u_7\nF : Type u_8\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : CommGroup H\u271d\ninst\u271d\u00b9 : MulOneClass M\nH : Type u_9\ninst\u271d : Group H\nf : G \u2192 H\nmap_div : \u2200 (a b : G), f (a * b\u207b\u00b9) = f a * (f b)\u207b\u00b9\nx y : G\n\u22a2 f (x * y) = f x * (f (1 * 1\u207b\u00b9 * y\u207b\u00b9))\u207b\u00b9\n[PROOFSTEP]\nsimp only [one_mul, inv_one, \u2190 map_div, inv_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH\u271d : Type u_7\nF : Type u_8\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : CommGroup H\u271d\ninst\u271d\u00b9 : MulOneClass M\nH : Type u_9\ninst\u271d : Group H\nf : G \u2192 H\nmap_div : \u2200 (a b : G), f (a * b\u207b\u00b9) = f a * (f b)\u207b\u00b9\nx y : G\n\u22a2 f x * (f (1 * 1\u207b\u00b9 * y\u207b\u00b9))\u207b\u00b9 = f x * f y\n[PROOFSTEP]\n{ simp only [map_div]\n simp only [mul_right_inv, one_mul, inv_inv]\n}\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH\u271d : Type u_7\nF : Type u_8\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : CommGroup H\u271d\ninst\u271d\u00b9 : MulOneClass M\nH : Type u_9\ninst\u271d : Group H\nf : G \u2192 H\nmap_div : \u2200 (a b : G), f (a * b\u207b\u00b9) = f a * (f b)\u207b\u00b9\nx y : G\n\u22a2 f x * (f (1 * 1\u207b\u00b9 * y\u207b\u00b9))\u207b\u00b9 = f x * f y\n[PROOFSTEP]\nsimp only [map_div]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH\u271d : Type u_7\nF : Type u_8\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : CommGroup H\u271d\ninst\u271d\u00b9 : MulOneClass M\nH : Type u_9\ninst\u271d : Group H\nf : G \u2192 H\nmap_div : \u2200 (a b : G), f (a * b\u207b\u00b9) = f a * (f b)\u207b\u00b9\nx y : G\n\u22a2 f x * (f 1 * (f 1)\u207b\u00b9 * (f y)\u207b\u00b9)\u207b\u00b9 = f x * f y\n[PROOFSTEP]\nsimp only [mul_right_inv, one_mul, inv_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH\u271d : Type u_7\nF : Type u_8\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : CommGroup H\u271d\ninst\u271d\u00b9 : MulOneClass M\nH : Type u_9\ninst\u271d : Group H\nf : G \u2192 H\nhf : \u2200 (x y : G), f (x / y) = f x / f y\n\u22a2 \u2200 (a b : G), f (a * b\u207b\u00b9) = f a * (f b)\u207b\u00b9\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM\u271d : Type u_3\nN : Type u_4\nP : Type u_5\nG\u271d : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : CommGroup H\ninst\u271d\u00b2 : MulOneClass M\u271d\nM : Type ?u.229390\nG : Type ?u.229393\ninst\u271d\u00b9 : MulOneClass M\ninst\u271d : CommGroup G\nf : M \u2192* G\na b : M\n\u22a2 (\u2191f (a * b))\u207b\u00b9 = (\u2191f a)\u207b\u00b9 * (\u2191f b)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 mul_inv, f.map_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM\u271d : Type u_3\nN\u271d : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : CommGroup H\ninst\u271d\u00b3 : MulOneClass M\u271d\nM : Type u_9\nN : Type u_10\nA : Type u_11\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : CommGroup A\n\u03c6 : N \u2192* A\n\u03c8 : M \u2192* N\n\u22a2 comp \u03c6\u207b\u00b9 \u03c8 = (comp \u03c6 \u03c8)\u207b\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM\u271d : Type u_3\nN\u271d : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : CommGroup H\ninst\u271d\u00b3 : MulOneClass M\u271d\nM : Type u_9\nN : Type u_10\nA : Type u_11\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : CommGroup A\n\u03c6 : N \u2192* A\n\u03c8 : M \u2192* N\nx\u271d : M\n\u22a2 \u2191(comp \u03c6\u207b\u00b9 \u03c8) x\u271d = \u2191(comp \u03c6 \u03c8)\u207b\u00b9 x\u271d\n[PROOFSTEP]\nsimp only [Function.comp_apply, inv_apply, coe_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM\u271d : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : CommGroup H\ninst\u271d\u00b3 : MulOneClass M\u271d\nM : Type u_9\nA : Type u_10\nB : Type u_11\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : CommGroup A\ninst\u271d : CommGroup B\n\u03c6 : A \u2192* B\n\u03c8 : M \u2192* A\n\u22a2 comp \u03c6 \u03c8\u207b\u00b9 = (comp \u03c6 \u03c8)\u207b\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM\u271d : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : CommGroup H\ninst\u271d\u00b3 : MulOneClass M\u271d\nM : Type u_9\nA : Type u_10\nB : Type u_11\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : CommGroup A\ninst\u271d : CommGroup B\n\u03c6 : A \u2192* B\n\u03c8 : M \u2192* A\nx\u271d : M\n\u22a2 \u2191(comp \u03c6 \u03c8\u207b\u00b9) x\u271d = \u2191(comp \u03c6 \u03c8)\u207b\u00b9 x\u271d\n[PROOFSTEP]\nsimp only [Function.comp_apply, inv_apply, map_inv, coe_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM\u271d : Type u_3\nN : Type u_4\nP : Type u_5\nG\u271d : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : CommGroup H\ninst\u271d\u00b2 : MulOneClass M\u271d\nM : Type ?u.233732\nG : Type ?u.233735\ninst\u271d\u00b9 : MulOneClass M\ninst\u271d : CommGroup G\nf g : M \u2192* G\na b : M\n\u22a2 \u2191f (a * b) / \u2191g (a * b) = \u2191f a / \u2191g a * (\u2191f b / \u2191g b)\n[PROOFSTEP]\nsimp [div_eq_mul_inv, mul_assoc, mul_left_comm, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : MulZeroOneClass M\ninst\u271d : CommMonoidWithZero N\nf g : M \u2192*\u2080 N\nsrc\u271d : M \u2192* N := \u2191f * \u2191g\n\u22a2 (fun a => \u2191f a * \u2191g a) 0 = 0\n[PROOFSTEP]\ndsimp only []\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nG : Type u_6\nH : Type u_7\nF : Type u_8\ninst\u271d\u00b9 : MulZeroOneClass M\ninst\u271d : CommMonoidWithZero N\nf g : M \u2192*\u2080 N\nsrc\u271d : M \u2192* N := \u2191f * \u2191g\n\u22a2 \u2191f 0 * \u2191g 0 = 0\n[PROOFSTEP]\nrw [map_zero, zero_mul]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Hom.Group", "llama_tokens": 14747, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707281, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.4441402723643508}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 radius p = liminf (fun n => 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))) atTop\n[PROOFSTEP]\nhave : \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 ((r : \u211d\u22650\u221e) \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / (n : \u211d))) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1) :=\n by\n intro r n hn\n have : 0 < (n : \u211d) := Nat.cast_pos.2 hn\n conv_lhs =>\n rw [one_div, ENNReal.le_inv_iff_mul_le, \u2190 ENNReal.coe_mul, ENNReal.coe_le_one_iff, one_div, \u2190 NNReal.rpow_one r, \u2190\n mul_inv_cancel this.ne', NNReal.rpow_mul, \u2190 NNReal.mul_rpow, \u2190 NNReal.one_rpow n\u207b\u00b9,\n NNReal.rpow_le_rpow_iff (inv_pos.2 this), mul_comm, NNReal.rpow_nat_cast]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 (\u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1)\n[PROOFSTEP]\nintro r n hn\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\nn : \u2115\nhn : 0 < n\n\u22a2 \u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1\n[PROOFSTEP]\nhave : 0 < (n : \u211d) := Nat.cast_pos.2 hn\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\nn : \u2115\nhn : 0 < n\nthis : 0 < \u2191n\n\u22a2 \u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1\n[PROOFSTEP]\nconv_lhs =>\n rw [one_div, ENNReal.le_inv_iff_mul_le, \u2190 ENNReal.coe_mul, ENNReal.coe_le_one_iff, one_div, \u2190 NNReal.rpow_one r, \u2190\n mul_inv_cancel this.ne', NNReal.rpow_mul, \u2190 NNReal.mul_rpow, \u2190 NNReal.one_rpow n\u207b\u00b9,\n NNReal.rpow_le_rpow_iff (inv_pos.2 this), mul_comm, NNReal.rpow_nat_cast]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\nn : \u2115\nhn : 0 < n\nthis : 0 < \u2191n\n| \u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))\n[PROOFSTEP]\nrw [one_div, ENNReal.le_inv_iff_mul_le, \u2190 ENNReal.coe_mul, ENNReal.coe_le_one_iff, one_div, \u2190 NNReal.rpow_one r, \u2190\n mul_inv_cancel this.ne', NNReal.rpow_mul, \u2190 NNReal.mul_rpow, \u2190 NNReal.one_rpow n\u207b\u00b9,\n NNReal.rpow_le_rpow_iff (inv_pos.2 this), mul_comm, NNReal.rpow_nat_cast]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\nn : \u2115\nhn : 0 < n\nthis : 0 < \u2191n\n| \u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))\n[PROOFSTEP]\nrw [one_div, ENNReal.le_inv_iff_mul_le, \u2190 ENNReal.coe_mul, ENNReal.coe_le_one_iff, one_div, \u2190 NNReal.rpow_one r, \u2190\n mul_inv_cancel this.ne', NNReal.rpow_mul, \u2190 NNReal.mul_rpow, \u2190 NNReal.one_rpow n\u207b\u00b9,\n NNReal.rpow_le_rpow_iff (inv_pos.2 this), mul_comm, NNReal.rpow_nat_cast]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\nn : \u2115\nhn : 0 < n\nthis : 0 < \u2191n\n| \u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))\n[PROOFSTEP]\nrw [one_div, ENNReal.le_inv_iff_mul_le, \u2190 ENNReal.coe_mul, ENNReal.coe_le_one_iff, one_div, \u2190 NNReal.rpow_one r, \u2190\n mul_inv_cancel this.ne', NNReal.rpow_mul, \u2190 NNReal.mul_rpow, \u2190 NNReal.one_rpow n\u207b\u00b9,\n NNReal.rpow_le_rpow_iff (inv_pos.2 this), mul_comm, NNReal.rpow_nat_cast]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nthis : \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 (\u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1)\n\u22a2 radius p = liminf (fun n => 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))) atTop\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nthis : \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 (\u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1)\n\u22a2 radius p \u2264 liminf (fun n => 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))) atTop\n[PROOFSTEP]\nrefine' ENNReal.le_of_forall_nnreal_lt fun r hr => _\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nthis : \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 (\u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1)\n\u22a2 liminf (fun n => 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))) atTop \u2264 radius p\n[PROOFSTEP]\nrefine' ENNReal.le_of_forall_nnreal_lt fun r hr => _\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nthis : \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 (\u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1)\nr : \u211d\u22650\nhr : \u2191r < radius p\n\u22a2 \u2191r \u2264 liminf (fun n => 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))) atTop\n[PROOFSTEP]\nhave := ((TFAE_exists_lt_isLittleO_pow (fun n => \u2016p n\u2016 * r ^ n) 1).out 1 7).1 (p.isLittleO_of_lt_radius hr)\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nthis\u271d : \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 (\u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1)\nr : \u211d\u22650\nhr : \u2191r < radius p\nthis : \u2203 a, a \u2208 Set.Ioo 0 1 \u2227 \u2200\u1da0 (n : \u2115) in atTop, |\u2016p n\u2016 * \u2191(r ^ n)| \u2264 a ^ n\n\u22a2 \u2191r \u2264 liminf (fun n => 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))) atTop\n[PROOFSTEP]\nobtain \u27e8a, ha, H\u27e9 := this\n[GOAL]\ncase a.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nthis : \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 (\u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1)\nr : \u211d\u22650\nhr : \u2191r < radius p\na : \u211d\nha : a \u2208 Set.Ioo 0 1\nH : \u2200\u1da0 (n : \u2115) in atTop, |\u2016p n\u2016 * \u2191(r ^ n)| \u2264 a ^ n\n\u22a2 \u2191r \u2264 liminf (fun n => 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))) atTop\n[PROOFSTEP]\napply le_liminf_of_le\n[GOAL]\ncase a.intro.intro.hf\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nthis : \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 (\u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1)\nr : \u211d\u22650\nhr : \u2191r < radius p\na : \u211d\nha : a \u2208 Set.Ioo 0 1\nH : \u2200\u1da0 (n : \u2115) in atTop, |\u2016p n\u2016 * \u2191(r ^ n)| \u2264 a ^ n\n\u22a2 autoParam (IsCoboundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))) _auto\u271d\n[PROOFSTEP]\ninfer_param\n[GOAL]\ncase a.intro.intro.h\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nthis : \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 (\u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1)\nr : \u211d\u22650\nhr : \u2191r < radius p\na : \u211d\nha : a \u2208 Set.Ioo 0 1\nH : \u2200\u1da0 (n : \u2115) in atTop, |\u2016p n\u2016 * \u2191(r ^ n)| \u2264 a ^ n\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))\n[PROOFSTEP]\nrw [\u2190 eventually_map]\n[GOAL]\ncase a.intro.intro.h\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nthis : \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 (\u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1)\nr : \u211d\u22650\nhr : \u2191r < radius p\na : \u211d\nha : a \u2208 Set.Ioo 0 1\nH : \u2200\u1da0 (n : \u2115) in atTop, |\u2016p n\u2016 * \u2191(r ^ n)| \u2264 a ^ n\n\u22a2 \u2200\u1da0 (b : \u211d\u22650\u221e) in map (fun n => 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))) atTop, \u2191r \u2264 b\n[PROOFSTEP]\nrefine' H.mp ((eventually_gt_atTop 0).mono fun n hn\u2080 hn => (this _ hn\u2080).2 (NNReal.coe_le_coe.1 _))\n[GOAL]\ncase a.intro.intro.h\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nthis : \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 (\u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1)\nr : \u211d\u22650\nhr : \u2191r < radius p\na : \u211d\nha : a \u2208 Set.Ioo 0 1\nH : \u2200\u1da0 (n : \u2115) in atTop, |\u2016p n\u2016 * \u2191(r ^ n)| \u2264 a ^ n\nn : \u2115\nhn\u2080 : 0 < n\nhn : |\u2016p n\u2016 * \u2191(r ^ n)| \u2264 a ^ n\n\u22a2 \u2191(\u2016p n\u2016\u208a * r ^ n) \u2264 \u21911\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase a.intro.intro.h\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nthis : \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 (\u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1)\nr : \u211d\u22650\nhr : \u2191r < radius p\na : \u211d\nha : a \u2208 Set.Ioo 0 1\nH : \u2200\u1da0 (n : \u2115) in atTop, |\u2016p n\u2016 * \u2191(r ^ n)| \u2264 a ^ n\nn : \u2115\nhn\u2080 : 0 < n\nhn : |\u2016p n\u2016 * \u2191(r ^ n)| \u2264 a ^ n\n\u22a2 \u2016p n\u2016 * \u2191r ^ n \u2264 1\n[PROOFSTEP]\nexact (le_abs_self _).trans (hn.trans (pow_le_one _ ha.1.le ha.2.le))\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nthis : \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 (\u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1)\nr : \u211d\u22650\nhr : \u2191r < liminf (fun n => 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))) atTop\n\u22a2 \u2191r \u2264 radius p\n[PROOFSTEP]\nrefine' p.le_radius_of_isBigO (IsBigO.of_bound 1 _)\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nthis : \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 (\u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1)\nr : \u211d\u22650\nhr : \u2191r < liminf (fun n => 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))) atTop\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop, \u2016\u2016p x\u2016 * \u2191r ^ x\u2016 \u2264 1 * \u20161\u2016\n[PROOFSTEP]\nrefine' (eventually_lt_of_lt_liminf hr).mp ((eventually_gt_atTop 0).mono fun n hn\u2080 hn => _)\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : FormalMultilinearSeries \ud835\udd5c E F\nthis : \u2200 (r : \u211d\u22650) {n : \u2115}, 0 < n \u2192 (\u2191r \u2264 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n)) \u2194 \u2016p n\u2016\u208a * r ^ n \u2264 1)\nr : \u211d\u22650\nhr : \u2191r < liminf (fun n => 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))) atTop\nn : \u2115\nhn\u2080 : 0 < n\nhn : \u2191r < 1 / \u2191(\u2016p n\u2016\u208a ^ (1 / \u2191n))\n\u22a2 \u2016\u2016p n\u2016 * \u2191r ^ n\u2016 \u2264 1 * \u20161\u2016\n[PROOFSTEP]\nsimpa using NNReal.coe_le_coe.2 ((this _ hn\u2080).1 hn.le)\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Analytic.RadiusLiminf", "llama_tokens": 6308, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.44393548099487945}} {"text": "[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u00b3 : CommRing A\nS : Submonoid A\nhS\u271d : S \u2264 A\u2070\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nhS : S \u2264 A\u2070\ns : { x // x \u2208 S }\n\u22a2 IsUnit (\u2191(algebraMap A K) \u2191s)\n[PROOFSTEP]\napply IsLocalization.map_units K (\u27e8s.1, hS s.2\u27e9 : A\u2070)\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS\u271d : S \u2264 A\u2070\ninst\u271d\u2075 : CommRing K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nhS : S \u2264 A\u2070\nsrc\u271d : B \u2192+* K := IsLocalization.lift (_ : \u2200 (s : { x // x \u2208 S }), IsUnit (\u2191(algebraMap A K) \u2191s))\na : A\n\u22a2 OneHom.toFun\n (\u2191\u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n map_add' :=\n (_ : \u2200 (x y : B), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) })\n (\u2191(algebraMap A B) a) =\n \u2191(algebraMap A K) a\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS\u271d : S \u2264 A\u2070\ninst\u271d\u2075 : CommRing K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nhS : S \u2264 A\u2070\nx : K\n\u22a2 x \u2208 AlgHom.range (mapToFractionRing K S B hS) \u2192\n \u2203 a s hs, x = IsLocalization.mk' K a { val := s, property := (_ : s \u2208 A\u2070) }\n[PROOFSTEP]\nrintro \u27e8x, rfl\u27e9\n[GOAL]\ncase intro\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS\u271d : S \u2264 A\u2070\ninst\u271d\u2075 : CommRing K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nhS : S \u2264 A\u2070\nx : B\n\u22a2 \u2203 a s hs, \u2191\u2191(mapToFractionRing K S B hS) x = IsLocalization.mk' K a { val := s, property := (_ : s \u2208 A\u2070) }\n[PROOFSTEP]\nobtain \u27e8a, s, rfl\u27e9 := IsLocalization.mk'_surjective S x\n[GOAL]\ncase intro.intro.intro\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS\u271d : S \u2264 A\u2070\ninst\u271d\u2075 : CommRing K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nhS : S \u2264 A\u2070\na : A\ns : { x // x \u2208 S }\n\u22a2 \u2203 a_1 s_1 hs,\n \u2191\u2191(mapToFractionRing K S B hS) (IsLocalization.mk' B a s) =\n IsLocalization.mk' K a_1 { val := s_1, property := (_ : s_1 \u2208 A\u2070) }\n[PROOFSTEP]\nuse a, s, s.2\n[GOAL]\ncase h\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS\u271d : S \u2264 A\u2070\ninst\u271d\u2075 : CommRing K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nhS : S \u2264 A\u2070\na : A\ns : { x // x \u2208 S }\n\u22a2 \u2191\u2191(mapToFractionRing K S B hS) (IsLocalization.mk' B a s) =\n IsLocalization.mk' K a { val := \u2191s, property := (_ : \u2191s \u2208 A\u2070) }\n[PROOFSTEP]\napply IsLocalization.lift_mk'\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS\u271d : S \u2264 A\u2070\ninst\u271d\u2075 : CommRing K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nhS : S \u2264 A\u2070\nx : K\n\u22a2 (\u2203 a s hs, x = IsLocalization.mk' K a { val := s, property := (_ : s \u2208 A\u2070) }) \u2192\n x \u2208 AlgHom.range (mapToFractionRing K S B hS)\n[PROOFSTEP]\nrintro \u27e8a, s, hs, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS\u271d : S \u2264 A\u2070\ninst\u271d\u2075 : CommRing K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nhS : S \u2264 A\u2070\na s : A\nhs : s \u2208 S\n\u22a2 IsLocalization.mk' K a { val := s, property := (_ : s \u2208 A\u2070) } \u2208 AlgHom.range (mapToFractionRing K S B hS)\n[PROOFSTEP]\nuse IsLocalization.mk' _ a \u27e8s, hs\u27e9\n[GOAL]\ncase h\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS\u271d : S \u2264 A\u2070\ninst\u271d\u2075 : CommRing K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nhS : S \u2264 A\u2070\na s : A\nhs : s \u2208 S\n\u22a2 \u2191\u2191(mapToFractionRing K S B hS) (IsLocalization.mk' B a { val := s, property := hs }) =\n IsLocalization.mk' K a { val := s, property := (_ : s \u2208 A\u2070) }\n[PROOFSTEP]\napply IsLocalization.lift_mk'\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS\u271d : S \u2264 A\u2070\ninst\u271d\u2075 : CommRing K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nhS : S \u2264 A\u2070\n\u22a2 Function.Bijective \u2191(AlgHom.rangeRestrict (mapToFractionRing K S B hS))\n[PROOFSTEP]\nrefine' \u27e8fun a b h => _, Set.surjective_onto_range\u27e9\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS\u271d : S \u2264 A\u2070\ninst\u271d\u2075 : CommRing K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nhS : S \u2264 A\u2070\na b : B\nh : \u2191(AlgHom.rangeRestrict (mapToFractionRing K S B hS)) a = \u2191(AlgHom.rangeRestrict (mapToFractionRing K S B hS)) b\n\u22a2 a = b\n[PROOFSTEP]\nrefine' (IsLocalization.lift_injective_iff _).2 (fun a b => _) (Subtype.ext_iff.1 h)\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS\u271d : S \u2264 A\u2070\ninst\u271d\u2075 : CommRing K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nhS : S \u2264 A\u2070\na\u271d b\u271d : B\nh : \u2191(AlgHom.rangeRestrict (mapToFractionRing K S B hS)) a\u271d = \u2191(AlgHom.rangeRestrict (mapToFractionRing K S B hS)) b\u271d\na b : A\n\u22a2 \u2191(algebraMap A B) a = \u2191(algebraMap A B) b \u2194 \u2191(algebraMap A K) a = \u2191(algebraMap A K) b\n[PROOFSTEP]\nexact \u27e8fun h => congr_arg _ (IsLocalization.injective _ hS h), fun h => congr_arg _ (IsFractionRing.injective A K h)\u27e9\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u00b3 : CommRing A\nS : Submonoid A\nhS\u271d : S \u2264 A\u2070\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nhS : S \u2264 A\u2070\n\u22a2 {x | \u2203 a s hs, x = IsLocalization.mk' K a { val := s, property := (_ : s \u2208 A\u2070) }} =\n \u2191(AlgHom.range (mapToFractionRing K S (Localization S) hS))\n[PROOFSTEP]\next\n[GOAL]\ncase h\nA : Type u_1\nK : Type u_2\ninst\u271d\u00b3 : CommRing A\nS : Submonoid A\nhS\u271d : S \u2264 A\u2070\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nhS : S \u2264 A\u2070\nx\u271d : K\n\u22a2 x\u271d \u2208 {x | \u2203 a s hs, x = IsLocalization.mk' K a { val := s, property := (_ : s \u2208 A\u2070) }} \u2194\n x\u271d \u2208 \u2191(AlgHom.range (mapToFractionRing K S (Localization S) hS))\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h\nA : Type u_1\nK : Type u_2\ninst\u271d\u00b3 : CommRing A\nS : Submonoid A\nhS\u271d : S \u2264 A\u2070\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nhS : S \u2264 A\u2070\nx\u271d : K\n\u22a2 x\u271d \u2208 \u2191(AlgHom.range (mapToFractionRing K S (Localization S) hS)) \u2194\n x\u271d \u2208 {x | \u2203 a s hs, x = IsLocalization.mk' K a { val := s, property := (_ : s \u2208 A\u2070) }}\n[PROOFSTEP]\napply mem_range_mapToFractionRing_iff\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u00b3 : CommRing A\nS : Submonoid A\nhS : S \u2264 A\u2070\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 IsLocalization S { x // x \u2208 subalgebra K S hS }\n[PROOFSTEP]\ndsimp only [Localization.subalgebra]\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u00b3 : CommRing A\nS : Submonoid A\nhS : S \u2264 A\u2070\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 IsLocalization S\n { x //\n x \u2208\n Subalgebra.copy (AlgHom.range (mapToFractionRing K S (Localization S) hS))\n {x | \u2203 a s hs, x = IsLocalization.mk' K a { val := s, property := (_ : s \u2208 A\u2070) }}\n (_ :\n {x | \u2203 a s hs, x = IsLocalization.mk' K a { val := s, property := (_ : s \u2208 A\u2070) }} =\n \u2191(AlgHom.range (mapToFractionRing K S (Localization S) hS))) }\n[PROOFSTEP]\nrw [Subalgebra.copy_eq]\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u00b3 : CommRing A\nS : Submonoid A\nhS : S \u2264 A\u2070\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 IsLocalization S { x // x \u2208 AlgHom.range (mapToFractionRing K S (Localization S) hS) }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS : S \u2264 A\u2070\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nx : K\n\u22a2 x \u2208 AlgHom.range (mapToFractionRing K S B hS) \u2194 \u2203 a s x_1, x = \u2191(algebraMap A K) a * (\u2191(algebraMap A K) s)\u207b\u00b9\n[PROOFSTEP]\nrw [mem_range_mapToFractionRing_iff]\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS : S \u2264 A\u2070\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nx : K\n\u22a2 (\u2203 a s hs, x = IsLocalization.mk' K a { val := s, property := (_ : s \u2208 A\u2070) }) \u2194\n \u2203 a s x_1, x = \u2191(algebraMap A K) a * (\u2191(algebraMap A K) s)\u207b\u00b9\n[PROOFSTEP]\nconvert Iff.rfl\n[GOAL]\ncase h.e'_2.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_3\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS : S \u2264 A\u2070\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nx : K\nx\u271d\u00b2 x\u271d\u00b9 : A\nx\u271d : x\u271d\u00b9 \u2208 S\n\u22a2 \u2191(algebraMap A K) x\u271d\u00b2 * (\u2191(algebraMap A K) x\u271d\u00b9)\u207b\u00b9 =\n IsLocalization.mk' K x\u271d\u00b2 { val := x\u271d\u00b9, property := (_ : x\u271d\u00b9 \u2208 A\u2070) }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e'_2.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_3.e_a\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS : S \u2264 A\u2070\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nx : K\nx\u271d\u00b2 x\u271d\u00b9 : A\nx\u271d : x\u271d\u00b9 \u2208 S\n\u22a2 (\u2191(algebraMap A K) x\u271d\u00b9)\u207b\u00b9 =\n \u2191(\u2191(IsUnit.liftRight\n (MonoidHom.restrict (Submonoid.LocalizationMap.toMap (IsLocalization.toLocalizationMap A\u2070 K)) A\u2070)\n (_ :\n \u2200 (y : { x // x \u2208 A\u2070 }),\n IsUnit (\u2191(Submonoid.LocalizationMap.toMap (IsLocalization.toLocalizationMap A\u2070 K)) \u2191y)))\n { val := x\u271d\u00b9, property := (_ : x\u271d\u00b9 \u2208 A\u2070) })\u207b\u00b9\n[PROOFSTEP]\nrw [Units.val_inv_eq_inv_val]\n[GOAL]\ncase h.e'_2.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_3.e_a\nA : Type u_1\nK : Type u_2\ninst\u271d\u2076 : CommRing A\nS : Submonoid A\nhS : S \u2264 A\u2070\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra A K\ninst\u271d\u00b3 : IsFractionRing A K\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsLocalization S B\nx : K\nx\u271d\u00b2 x\u271d\u00b9 : A\nx\u271d : x\u271d\u00b9 \u2208 S\n\u22a2 (\u2191(algebraMap A K) x\u271d\u00b9)\u207b\u00b9 =\n (\u2191(\u2191(IsUnit.liftRight\n (MonoidHom.restrict (Submonoid.LocalizationMap.toMap (IsLocalization.toLocalizationMap A\u2070 K)) A\u2070)\n (_ :\n \u2200 (y : { x // x \u2208 A\u2070 }),\n IsUnit (\u2191(Submonoid.LocalizationMap.toMap (IsLocalization.toLocalizationMap A\u2070 K)) \u2191y)))\n { val := x\u271d\u00b9, property := (_ : x\u271d\u00b9 \u2208 A\u2070) }))\u207b\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u00b3 : CommRing A\nS : Submonoid A\nhS : S \u2264 A\u2070\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 {x | \u2203 a s x_1, x = \u2191(algebraMap A K) a * (\u2191(algebraMap A K) s)\u207b\u00b9} =\n \u2191(AlgHom.range (mapToFractionRing K S (Localization S) hS))\n[PROOFSTEP]\next\n[GOAL]\ncase h\nA : Type u_1\nK : Type u_2\ninst\u271d\u00b3 : CommRing A\nS : Submonoid A\nhS : S \u2264 A\u2070\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx\u271d : K\n\u22a2 x\u271d \u2208 {x | \u2203 a s x_1, x = \u2191(algebraMap A K) a * (\u2191(algebraMap A K) s)\u207b\u00b9} \u2194\n x\u271d \u2208 \u2191(AlgHom.range (mapToFractionRing K S (Localization S) hS))\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h\nA : Type u_1\nK : Type u_2\ninst\u271d\u00b3 : CommRing A\nS : Submonoid A\nhS : S \u2264 A\u2070\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx\u271d : K\n\u22a2 x\u271d \u2208 \u2191(AlgHom.range (mapToFractionRing K S (Localization S) hS)) \u2194\n x\u271d \u2208 {x | \u2203 a s x_1, x = \u2191(algebraMap A K) a * (\u2191(algebraMap A K) s)\u207b\u00b9}\n[PROOFSTEP]\napply mem_range_mapToFractionRing_iff_ofField\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u00b3 : CommRing A\nS : Submonoid A\nhS : S \u2264 A\u2070\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 IsLocalization S { x // x \u2208 ofField K S hS }\n[PROOFSTEP]\ndsimp only [Localization.subalgebra.ofField]\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u00b3 : CommRing A\nS : Submonoid A\nhS : S \u2264 A\u2070\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 IsLocalization S\n { x //\n x \u2208\n Subalgebra.copy (AlgHom.range (mapToFractionRing K S (Localization S) hS))\n {x | \u2203 a s x_1, x = \u2191(algebraMap A K) a * (\u2191(algebraMap A K) s)\u207b\u00b9}\n (_ :\n {x | \u2203 a s x_1, x = \u2191(algebraMap A K) a * (\u2191(algebraMap A K) s)\u207b\u00b9} =\n \u2191(AlgHom.range (mapToFractionRing K S (Localization S) hS))) }\n[PROOFSTEP]\nrw [Subalgebra.copy_eq]\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u00b3 : CommRing A\nS : Submonoid A\nhS : S \u2264 A\u2070\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 IsLocalization S { x // x \u2208 AlgHom.range (mapToFractionRing K S (Localization S) hS) }\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Localization.AsSubring", "llama_tokens": 6301, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.44393548099487945}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : Filter \u03b1\ng : Filter \u03b2\n\u22a2 Set.univ \u2208 {s | \u2200\u1da0 (a : \u03b1) in f, \u2200\u1da0 (b : \u03b2) in g, (a, b) \u2208 s}\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq, Set.mem_univ, eventually_true]\n", "meta": {"mathlib_filename": "Mathlib.Order.Filter.Curry", "llama_tokens": 115, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.734119526900183, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.44335815205486057}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a : R\ns : Finset \u03b1\np : \u03b1 \u2192 R\nhp : \u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)\nhx : x * y = a * \u220f i in s, p i\n\u22a2 \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\n[PROOFSTEP]\ninduction' s using Finset.induction with i s his ih generalizing x y a\n[GOAL]\ncase empty\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx\u271d y\u271d a\u271d : R\ns : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)\nhx\u271d : x\u271d * y\u271d = a\u271d * \u220f i in s, p i\nx y a : R\nhp : \u2200 (i : \u03b1), i \u2208 \u2205 \u2192 Prime (p i)\nhx : x * y = a * \u220f i in \u2205, p i\n\u22a2 \u2203 t u b c, t \u222a u = \u2205 \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\n[PROOFSTEP]\nexact \u27e8\u2205, \u2205, x, y, by simp [hx]\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx\u271d y\u271d a\u271d : R\ns : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)\nhx\u271d : x\u271d * y\u271d = a\u271d * \u220f i in s, p i\nx y a : R\nhp : \u2200 (i : \u03b1), i \u2208 \u2205 \u2192 Prime (p i)\nhx : x * y = a * \u220f i in \u2205, p i\n\u22a2 \u2205 \u222a \u2205 = \u2205 \u2227 Disjoint \u2205 \u2205 \u2227 a = x * y \u2227 x = x * \u220f i in \u2205, p i \u2227 y = y * \u220f i in \u2205, p i\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase insert\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx\u271d y\u271d a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x\u271d * y\u271d = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\nx y a : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhx : x * y = a * \u220f i in insert i s, p i\n\u22a2 \u2203 t u b c, t \u222a u = insert i s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\n[PROOFSTEP]\nrw [prod_insert his, \u2190 mul_assoc] at hx \n[GOAL]\ncase insert\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx\u271d y\u271d a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x\u271d * y\u271d = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\nx y a : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhx : x * y = a * p i * \u220f x in s, p x\n\u22a2 \u2203 t u b c, t \u222a u = insert i s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\n[PROOFSTEP]\nhave hpi : Prime (p i) := hp i (mem_insert_self _ _)\n[GOAL]\ncase insert\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx\u271d y\u271d a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x\u271d * y\u271d = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\nx y a : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhx : x * y = a * p i * \u220f x in s, p x\nhpi : Prime (p i)\n\u22a2 \u2203 t u b c, t \u222a u = insert i s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\n[PROOFSTEP]\nrcases ih (fun i hi \u21a6 hp i (mem_insert_of_mem hi)) hx with \u27e8t, u, b, c, htus, htu, hbc, rfl, rfl\u27e9\n[GOAL]\ncase insert.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x * y = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\na : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhpi : Prime (p i)\nt u : Finset \u03b1\nb c : R\nhtus : t \u222a u = s\nhtu : Disjoint t u\nhbc : a * p i = b * c\nhx : (b * \u220f i in t, p i) * (c * \u220f i in u, p i) = a * p i * \u220f x in s, p x\n\u22a2 \u2203 t_1 u_1 b_1 c_1,\n t_1 \u222a u_1 = insert i s \u2227\n Disjoint t_1 u_1 \u2227\n a = b_1 * c_1 \u2227 b * \u220f i in t, p i = b_1 * \u220f i in t_1, p i \u2227 c * \u220f i in u, p i = c_1 * \u220f i in u_1, p i\n[PROOFSTEP]\nhave hit : i \u2209 t := fun hit \u21a6 his (htus \u25b8 mem_union_left _ hit)\n[GOAL]\ncase insert.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x * y = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\na : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhpi : Prime (p i)\nt u : Finset \u03b1\nb c : R\nhtus : t \u222a u = s\nhtu : Disjoint t u\nhbc : a * p i = b * c\nhx : (b * \u220f i in t, p i) * (c * \u220f i in u, p i) = a * p i * \u220f x in s, p x\nhit : \u00aci \u2208 t\n\u22a2 \u2203 t_1 u_1 b_1 c_1,\n t_1 \u222a u_1 = insert i s \u2227\n Disjoint t_1 u_1 \u2227\n a = b_1 * c_1 \u2227 b * \u220f i in t, p i = b_1 * \u220f i in t_1, p i \u2227 c * \u220f i in u, p i = c_1 * \u220f i in u_1, p i\n[PROOFSTEP]\nhave hiu : i \u2209 u := fun hiu \u21a6 his (htus \u25b8 mem_union_right _ hiu)\n[GOAL]\ncase insert.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x * y = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\na : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhpi : Prime (p i)\nt u : Finset \u03b1\nb c : R\nhtus : t \u222a u = s\nhtu : Disjoint t u\nhbc : a * p i = b * c\nhx : (b * \u220f i in t, p i) * (c * \u220f i in u, p i) = a * p i * \u220f x in s, p x\nhit : \u00aci \u2208 t\nhiu : \u00aci \u2208 u\n\u22a2 \u2203 t_1 u_1 b_1 c_1,\n t_1 \u222a u_1 = insert i s \u2227\n Disjoint t_1 u_1 \u2227\n a = b_1 * c_1 \u2227 b * \u220f i in t, p i = b_1 * \u220f i in t_1, p i \u2227 c * \u220f i in u, p i = c_1 * \u220f i in u_1, p i\n[PROOFSTEP]\nobtain \u27e8d, rfl\u27e9 | \u27e8d, rfl\u27e9 : p i \u2223 b \u2228 p i \u2223 c\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x * y = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\na : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhpi : Prime (p i)\nt u : Finset \u03b1\nb c : R\nhtus : t \u222a u = s\nhtu : Disjoint t u\nhbc : a * p i = b * c\nhx : (b * \u220f i in t, p i) * (c * \u220f i in u, p i) = a * p i * \u220f x in s, p x\nhit : \u00aci \u2208 t\nhiu : \u00aci \u2208 u\n\u22a2 p i \u2223 b \u2228 p i \u2223 c\ncase insert.intro.intro.intro.intro.intro.intro.intro.intro.inl.intro\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x * y = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\na : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhpi : Prime (p i)\nt u : Finset \u03b1\nc : R\nhtus : t \u222a u = s\nhtu : Disjoint t u\nhit : \u00aci \u2208 t\nhiu : \u00aci \u2208 u\nd : R\nhbc : a * p i = p i * d * c\nhx : (p i * d * \u220f i in t, p i) * (c * \u220f i in u, p i) = a * p i * \u220f x in s, p x\n\u22a2 \u2203 t_1 u_1 b c_1,\n t_1 \u222a u_1 = insert i s \u2227\n Disjoint t_1 u_1 \u2227\n a = b * c_1 \u2227 p i * d * \u220f i in t, p i = b * \u220f i in t_1, p i \u2227 c * \u220f i in u, p i = c_1 * \u220f i in u_1, p i\ncase insert.intro.intro.intro.intro.intro.intro.intro.intro.inr.intro\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x * y = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\na : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhpi : Prime (p i)\nt u : Finset \u03b1\nb : R\nhtus : t \u222a u = s\nhtu : Disjoint t u\nhit : \u00aci \u2208 t\nhiu : \u00aci \u2208 u\nd : R\nhbc : a * p i = b * (p i * d)\nhx : (b * \u220f i in t, p i) * (p i * d * \u220f i in u, p i) = a * p i * \u220f x in s, p x\n\u22a2 \u2203 t_1 u_1 b_1 c,\n t_1 \u222a u_1 = insert i s \u2227\n Disjoint t_1 u_1 \u2227\n a = b_1 * c \u2227 b * \u220f i in t, p i = b_1 * \u220f i in t_1, p i \u2227 p i * d * \u220f i in u, p i = c * \u220f i in u_1, p i\n[PROOFSTEP]\nexact hpi.dvd_or_dvd \u27e8a, by rw [\u2190 hbc, mul_comm]\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x * y = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\na : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhpi : Prime (p i)\nt u : Finset \u03b1\nb c : R\nhtus : t \u222a u = s\nhtu : Disjoint t u\nhbc : a * p i = b * c\nhx : (b * \u220f i in t, p i) * (c * \u220f i in u, p i) = a * p i * \u220f x in s, p x\nhit : \u00aci \u2208 t\nhiu : \u00aci \u2208 u\n\u22a2 b * c = p i * a\n[PROOFSTEP]\nrw [\u2190 hbc, mul_comm]\n[GOAL]\ncase insert.intro.intro.intro.intro.intro.intro.intro.intro.inl.intro\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x * y = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\na : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhpi : Prime (p i)\nt u : Finset \u03b1\nc : R\nhtus : t \u222a u = s\nhtu : Disjoint t u\nhit : \u00aci \u2208 t\nhiu : \u00aci \u2208 u\nd : R\nhbc : a * p i = p i * d * c\nhx : (p i * d * \u220f i in t, p i) * (c * \u220f i in u, p i) = a * p i * \u220f x in s, p x\n\u22a2 \u2203 t_1 u_1 b c_1,\n t_1 \u222a u_1 = insert i s \u2227\n Disjoint t_1 u_1 \u2227\n a = b * c_1 \u2227 p i * d * \u220f i in t, p i = b * \u220f i in t_1, p i \u2227 c * \u220f i in u, p i = c_1 * \u220f i in u_1, p i\n[PROOFSTEP]\nrw [mul_assoc, mul_comm a, mul_right_inj' hpi.ne_zero] at hbc \n[GOAL]\ncase insert.intro.intro.intro.intro.intro.intro.intro.intro.inl.intro\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x * y = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\na : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhpi : Prime (p i)\nt u : Finset \u03b1\nc : R\nhtus : t \u222a u = s\nhtu : Disjoint t u\nhit : \u00aci \u2208 t\nhiu : \u00aci \u2208 u\nd : R\nhbc : a = d * c\nhx : (p i * d * \u220f i in t, p i) * (c * \u220f i in u, p i) = a * p i * \u220f x in s, p x\n\u22a2 \u2203 t_1 u_1 b c_1,\n t_1 \u222a u_1 = insert i s \u2227\n Disjoint t_1 u_1 \u2227\n a = b * c_1 \u2227 p i * d * \u220f i in t, p i = b * \u220f i in t_1, p i \u2227 c * \u220f i in u, p i = c_1 * \u220f i in u_1, p i\n[PROOFSTEP]\nexact\n \u27e8insert i t, u, d, c, by rw [insert_union, htus], disjoint_insert_left.2 \u27e8hiu, htu\u27e9, by\n simp [hbc, prod_insert hit, mul_assoc, mul_comm, mul_left_comm]\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x * y = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\na : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhpi : Prime (p i)\nt u : Finset \u03b1\nc : R\nhtus : t \u222a u = s\nhtu : Disjoint t u\nhit : \u00aci \u2208 t\nhiu : \u00aci \u2208 u\nd : R\nhbc : a = d * c\nhx : (p i * d * \u220f i in t, p i) * (c * \u220f i in u, p i) = a * p i * \u220f x in s, p x\n\u22a2 insert i t \u222a u = insert i s\n[PROOFSTEP]\nrw [insert_union, htus]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x * y = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\na : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhpi : Prime (p i)\nt u : Finset \u03b1\nc : R\nhtus : t \u222a u = s\nhtu : Disjoint t u\nhit : \u00aci \u2208 t\nhiu : \u00aci \u2208 u\nd : R\nhbc : a = d * c\nhx : (p i * d * \u220f i in t, p i) * (c * \u220f i in u, p i) = a * p i * \u220f x in s, p x\n\u22a2 a = d * c \u2227 p i * d * \u220f i in t, p i = d * \u220f i in insert i t, p i \u2227 c * \u220f i in u, p i = c * \u220f i in u, p i\n[PROOFSTEP]\nsimp [hbc, prod_insert hit, mul_assoc, mul_comm, mul_left_comm]\n[GOAL]\ncase insert.intro.intro.intro.intro.intro.intro.intro.intro.inr.intro\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x * y = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\na : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhpi : Prime (p i)\nt u : Finset \u03b1\nb : R\nhtus : t \u222a u = s\nhtu : Disjoint t u\nhit : \u00aci \u2208 t\nhiu : \u00aci \u2208 u\nd : R\nhbc : a * p i = b * (p i * d)\nhx : (b * \u220f i in t, p i) * (p i * d * \u220f i in u, p i) = a * p i * \u220f x in s, p x\n\u22a2 \u2203 t_1 u_1 b_1 c,\n t_1 \u222a u_1 = insert i s \u2227\n Disjoint t_1 u_1 \u2227\n a = b_1 * c \u2227 b * \u220f i in t, p i = b_1 * \u220f i in t_1, p i \u2227 p i * d * \u220f i in u, p i = c * \u220f i in u_1, p i\n[PROOFSTEP]\nrw [\u2190 mul_assoc, mul_right_comm b, mul_left_inj' hpi.ne_zero] at hbc \n[GOAL]\ncase insert.intro.intro.intro.intro.intro.intro.intro.intro.inr.intro\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x * y = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\na : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhpi : Prime (p i)\nt u : Finset \u03b1\nb : R\nhtus : t \u222a u = s\nhtu : Disjoint t u\nhit : \u00aci \u2208 t\nhiu : \u00aci \u2208 u\nd : R\nhbc : a = b * d\nhx : (b * \u220f i in t, p i) * (p i * d * \u220f i in u, p i) = a * p i * \u220f x in s, p x\n\u22a2 \u2203 t_1 u_1 b_1 c,\n t_1 \u222a u_1 = insert i s \u2227\n Disjoint t_1 u_1 \u2227\n a = b_1 * c \u2227 b * \u220f i in t, p i = b_1 * \u220f i in t_1, p i \u2227 p i * d * \u220f i in u, p i = c * \u220f i in u_1, p i\n[PROOFSTEP]\nexact\n \u27e8t, insert i u, b, d, by rw [union_insert, htus], disjoint_insert_right.2 \u27e8hit, htu\u27e9, by\n simp [\u2190 hbc, prod_insert hiu, mul_assoc, mul_comm, mul_left_comm]\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x * y = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\na : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhpi : Prime (p i)\nt u : Finset \u03b1\nb : R\nhtus : t \u222a u = s\nhtu : Disjoint t u\nhit : \u00aci \u2208 t\nhiu : \u00aci \u2208 u\nd : R\nhbc : a = b * d\nhx : (b * \u220f i in t, p i) * (p i * d * \u220f i in u, p i) = a * p i * \u220f x in s, p x\n\u22a2 t \u222a insert i u = insert i s\n[PROOFSTEP]\nrw [union_insert, htus]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\n\u03b1 : Type u_2\ninst\u271d : DecidableEq \u03b1\nx y a\u271d : R\ns\u271d : Finset \u03b1\np : \u03b1 \u2192 R\nhp\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 Prime (p i)\nhx\u271d : x * y = a\u271d * \u220f i in s\u271d, p i\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih :\n \u2200 {x y a : R},\n (\u2200 (i : \u03b1), i \u2208 s \u2192 Prime (p i)) \u2192\n x * y = a * \u220f i in s, p i \u2192\n \u2203 t u b c, t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 x = b * \u220f i in t, p i \u2227 y = c * \u220f i in u, p i\na : R\nhp : \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 Prime (p i_1)\nhpi : Prime (p i)\nt u : Finset \u03b1\nb : R\nhtus : t \u222a u = s\nhtu : Disjoint t u\nhit : \u00aci \u2208 t\nhiu : \u00aci \u2208 u\nd : R\nhbc : a = b * d\nhx : (b * \u220f i in t, p i) * (p i * d * \u220f i in u, p i) = a * p i * \u220f x in s, p x\n\u22a2 a = b * d \u2227 b * \u220f i in t, p i = b * \u220f i in t, p i \u2227 p i * d * \u220f i in u, p i = d * \u220f i in insert i u, p i\n[PROOFSTEP]\nsimp [\u2190 hbc, prod_insert hiu, mul_assoc, mul_comm, mul_left_comm]\n[GOAL]\nR : Type u_1\ninst\u271d : CancelCommMonoidWithZero R\nx y a p : R\nn : \u2115\nhp : Prime p\nhx : x * y = a * p ^ n\n\u22a2 \u2203 i j b c, i + j = n \u2227 a = b * c \u2227 x = b * p ^ i \u2227 y = c * p ^ j\n[PROOFSTEP]\nrcases mul_eq_mul_prime_prod (fun _ _ \u21a6 hp) (show x * y = a * (range n).prod fun _ \u21a6 p by simpa) with\n \u27e8t, u, b, c, htus, htu, rfl, rfl, rfl\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d : CancelCommMonoidWithZero R\nx y a p : R\nn : \u2115\nhp : Prime p\nhx : x * y = a * p ^ n\n\u22a2 x * y = a * \u220f x in range n, p\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CancelCommMonoidWithZero R\np : R\nn : \u2115\nhp : Prime p\nt u : Finset \u2115\nb c : R\nhtus : t \u222a u = range n\nhtu : Disjoint t u\nhx : (b * \u220f i in t, p) * (c * \u220f i in u, p) = b * c * p ^ n\n\u22a2 \u2203 i j b_1 c_1, i + j = n \u2227 b * c = b_1 * c_1 \u2227 b * \u220f i in t, p = b_1 * p ^ i \u2227 c * \u220f i in u, p = c_1 * p ^ j\n[PROOFSTEP]\nexact \u27e8t.card, u.card, b, c, by rw [\u2190 card_disjoint_union htu, htus, card_range], by simp\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d : CancelCommMonoidWithZero R\np : R\nn : \u2115\nhp : Prime p\nt u : Finset \u2115\nb c : R\nhtus : t \u222a u = range n\nhtu : Disjoint t u\nhx : (b * \u220f i in t, p) * (c * \u220f i in u, p) = b * c * p ^ n\n\u22a2 card t + card u = n\n[PROOFSTEP]\nrw [\u2190 card_disjoint_union htu, htus, card_range]\n[GOAL]\nR : Type u_1\ninst\u271d : CancelCommMonoidWithZero R\np : R\nn : \u2115\nhp : Prime p\nt u : Finset \u2115\nb c : R\nhtus : t \u222a u = range n\nhtu : Disjoint t u\nhx : (b * \u220f i in t, p) * (c * \u220f i in u, p) = b * c * p ^ n\n\u22a2 b * c = b * c \u2227 b * \u220f i in t, p = b * p ^ card t \u2227 c * \u220f i in u, p = c * p ^ card u\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\np : \u03b1\nhp : Prime p\n\u22a2 Prime (-p)\n[PROOFSTEP]\nobtain \u27e8h1, h2, h3\u27e9 := hp\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\np : \u03b1\nh1 : p \u2260 0\nh2 : \u00acIsUnit p\nh3 : \u2200 (a b : \u03b1), p \u2223 a * b \u2192 p \u2223 a \u2228 p \u2223 b\n\u22a2 Prime (-p)\n[PROOFSTEP]\nexact \u27e8neg_ne_zero.mpr h1, by rwa [IsUnit.neg_iff], by simpa [neg_dvd] using h3\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\np : \u03b1\nh1 : p \u2260 0\nh2 : \u00acIsUnit p\nh3 : \u2200 (a b : \u03b1), p \u2223 a * b \u2192 p \u2223 a \u2228 p \u2223 b\n\u22a2 \u00acIsUnit (-p)\n[PROOFSTEP]\nrwa [IsUnit.neg_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\np : \u03b1\nh1 : p \u2260 0\nh2 : \u00acIsUnit p\nh3 : \u2200 (a b : \u03b1), p \u2223 a * b \u2192 p \u2223 a \u2228 p \u2223 b\n\u22a2 \u2200 (a b : \u03b1), -p \u2223 a * b \u2192 -p \u2223 a \u2228 -p \u2223 b\n[PROOFSTEP]\nsimpa [neg_dvd] using h3\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : LinearOrder \u03b1\np : \u03b1\nhp : Prime p\n\u22a2 Prime |p|\n[PROOFSTEP]\nobtain h | h := abs_choice p\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : LinearOrder \u03b1\np : \u03b1\nhp : Prime p\nh : |p| = p\n\u22a2 Prime |p|\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : LinearOrder \u03b1\np : \u03b1\nhp : Prime p\nh : |p| = -p\n\u22a2 Prime |p|\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : LinearOrder \u03b1\np : \u03b1\nhp : Prime p\nh : |p| = p\n\u22a2 Prime p\n[PROOFSTEP]\nexact hp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : LinearOrder \u03b1\np : \u03b1\nhp : Prime p\nh : |p| = -p\n\u22a2 Prime (-p)\n[PROOFSTEP]\nexact hp.neg\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Prime", "llama_tokens": 11639, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195269001831, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.4433581520548605}} {"text": "[GOAL]\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : X \u27f6 Y\nbij : Function.Bijective \u2191f\n\u22a2 IsIso f\n[PROOFSTEP]\nlet E := Equiv.ofBijective _ bij\n[GOAL]\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : X \u27f6 Y\nbij : Function.Bijective \u2191f\nE : (forget CompHaus).obj X \u2243 (forget CompHaus).obj Y := Equiv.ofBijective (\u2191f) bij\n\u22a2 IsIso f\n[PROOFSTEP]\nhave hE : Continuous E.symm := by\n rw [continuous_iff_isClosed]\n intro S hS\n rw [\u2190 E.image_eq_preimage]\n exact isClosedMap f S hS\n[GOAL]\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : X \u27f6 Y\nbij : Function.Bijective \u2191f\nE : (forget CompHaus).obj X \u2243 (forget CompHaus).obj Y := Equiv.ofBijective (\u2191f) bij\n\u22a2 Continuous \u2191E.symm\n[PROOFSTEP]\nrw [continuous_iff_isClosed]\n[GOAL]\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : X \u27f6 Y\nbij : Function.Bijective \u2191f\nE : (forget CompHaus).obj X \u2243 (forget CompHaus).obj Y := Equiv.ofBijective (\u2191f) bij\n\u22a2 \u2200 (s : Set ((forget CompHaus).obj X)), IsClosed s \u2192 IsClosed (\u2191E.symm \u207b\u00b9' s)\n[PROOFSTEP]\nintro S hS\n[GOAL]\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : X \u27f6 Y\nbij : Function.Bijective \u2191f\nE : (forget CompHaus).obj X \u2243 (forget CompHaus).obj Y := Equiv.ofBijective (\u2191f) bij\nS : Set ((forget CompHaus).obj X)\nhS : IsClosed S\n\u22a2 IsClosed (\u2191E.symm \u207b\u00b9' S)\n[PROOFSTEP]\nrw [\u2190 E.image_eq_preimage]\n[GOAL]\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : X \u27f6 Y\nbij : Function.Bijective \u2191f\nE : (forget CompHaus).obj X \u2243 (forget CompHaus).obj Y := Equiv.ofBijective (\u2191f) bij\nS : Set ((forget CompHaus).obj X)\nhS : IsClosed S\n\u22a2 IsClosed (\u2191E '' S)\n[PROOFSTEP]\nexact isClosedMap f S hS\n[GOAL]\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : X \u27f6 Y\nbij : Function.Bijective \u2191f\nE : (forget CompHaus).obj X \u2243 (forget CompHaus).obj Y := Equiv.ofBijective (\u2191f) bij\nhE : Continuous \u2191E.symm\n\u22a2 IsIso f\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u27e8E.symm, hE\u27e9, _, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : X \u27f6 Y\nbij : Function.Bijective \u2191f\nE : (forget CompHaus).obj X \u2243 (forget CompHaus).obj Y := Equiv.ofBijective (\u2191f) bij\nhE : Continuous \u2191E.symm\n\u22a2 f \u226b ContinuousMap.mk \u2191E.symm = \ud835\udfd9 X\n[PROOFSTEP]\next x\n[GOAL]\ncase refine'_1.w\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : X \u27f6 Y\nbij : Function.Bijective \u2191f\nE : (forget CompHaus).obj X \u2243 (forget CompHaus).obj Y := Equiv.ofBijective (\u2191f) bij\nhE : Continuous \u2191E.symm\nx : (forget CompHaus).obj X\n\u22a2 \u2191(f \u226b ContinuousMap.mk \u2191E.symm) x = \u2191(\ud835\udfd9 X) x\n[PROOFSTEP]\napply E.symm_apply_apply\n[GOAL]\ncase refine'_2\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : X \u27f6 Y\nbij : Function.Bijective \u2191f\nE : (forget CompHaus).obj X \u2243 (forget CompHaus).obj Y := Equiv.ofBijective (\u2191f) bij\nhE : Continuous \u2191E.symm\n\u22a2 ContinuousMap.mk \u2191E.symm \u226b f = \ud835\udfd9 Y\n[PROOFSTEP]\next x\n[GOAL]\ncase refine'_2.w\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : X \u27f6 Y\nbij : Function.Bijective \u2191f\nE : (forget CompHaus).obj X \u2243 (forget CompHaus).obj Y := Equiv.ofBijective (\u2191f) bij\nhE : Continuous \u2191E.symm\nx : (forget CompHaus).obj Y\n\u22a2 \u2191(ContinuousMap.mk \u2191E.symm \u226b f) x = \u2191(\ud835\udfd9 Y) x\n[PROOFSTEP]\napply E.apply_symm_apply\n[GOAL]\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : \u2191X.toTop \u2243\u209c \u2191Y.toTop\n\u22a2 ContinuousMap.mk \u2191f \u226b ContinuousMap.mk \u2191(Homeomorph.symm f) = \ud835\udfd9 X\n[PROOFSTEP]\next x\n[GOAL]\ncase w\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : \u2191X.toTop \u2243\u209c \u2191Y.toTop\nx : (forget CompHaus).obj X\n\u22a2 \u2191(ContinuousMap.mk \u2191f \u226b ContinuousMap.mk \u2191(Homeomorph.symm f)) x = \u2191(\ud835\udfd9 X) x\n[PROOFSTEP]\nexact f.symm_apply_apply x\n[GOAL]\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : \u2191X.toTop \u2243\u209c \u2191Y.toTop\n\u22a2 ContinuousMap.mk \u2191(Homeomorph.symm f) \u226b ContinuousMap.mk \u2191f = \ud835\udfd9 Y\n[PROOFSTEP]\next x\n[GOAL]\ncase w\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : \u2191X.toTop \u2243\u209c \u2191Y.toTop\nx : (forget CompHaus).obj Y\n\u22a2 \u2191(ContinuousMap.mk \u2191(Homeomorph.symm f) \u226b ContinuousMap.mk \u2191f) x = \u2191(\ud835\udfd9 Y) x\n[PROOFSTEP]\nexact f.apply_symm_apply x\n[GOAL]\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : X \u2245 Y\nx : \u2191X.toTop\n\u22a2 \u2191f.inv (\u2191f.hom x) = x\n[PROOFSTEP]\nsimp\n[GOAL]\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : X \u2245 Y\nx : \u2191Y.toTop\n\u22a2 \u2191f.hom (\u2191f.inv x) = x\n[PROOFSTEP]\nsimp\n[GOAL]\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : X \u2245 Y\n\u22a2 isoOfHomeo (homeoOfIso f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase w.w\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : X \u2245 Y\nx\u271d : (forget CompHaus).obj X\n\u22a2 \u2191(isoOfHomeo (homeoOfIso f)).hom x\u271d = \u2191f.hom x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : \u2191X.toTop \u2243\u209c \u2191Y.toTop\n\u22a2 homeoOfIso (isoOfHomeo f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase H\nX\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : CompactSpace X\u271d\ninst\u271d : T2Space X\u271d\nX Y : CompHaus\nf : \u2191X.toTop \u2243\u209c \u2191Y.toTop\nx\u271d : \u2191X.toTop\n\u22a2 \u2191(homeoOfIso (isoOfHomeo f)) x\u271d = \u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u2200 {A B : CompHaus} (f : A \u27f6 B) [inst : IsIso ((forget CompHaus).map f)], IsIso f\n[PROOFSTEP]\nintro A B f hf\n[GOAL]\nA B : CompHaus\nf : A \u27f6 B\nhf : IsIso ((forget CompHaus).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nexact CompHaus.isIso_of_bijective _ ((isIso_iff_bijective f).mp hf)\n[GOAL]\nX : TopCat\nY : CompHaus\n\u22a2 Function.LeftInverse (fun f => ContinuousMap.mk (stoneCechExtend (_ : Continuous f.toFun))) fun f =>\n ContinuousMap.mk (\u2191f \u2218 stoneCechUnit)\n[PROOFSTEP]\nrintro\n \u27e8f : StoneCech X \u27f6 Y, hf : Continuous f\u27e9\n -- Porting note: `ext` fails.\n[GOAL]\ncase mk\nX : TopCat\nY : CompHaus\nf : StoneCech \u2191X \u27f6 \u2191Y.toTop\nhf : Continuous f\n\u22a2 (fun f => ContinuousMap.mk (stoneCechExtend (_ : Continuous f.toFun)))\n ((fun f => ContinuousMap.mk (\u2191f \u2218 stoneCechUnit)) (ContinuousMap.mk f)) =\n ContinuousMap.mk f\n[PROOFSTEP]\napply ContinuousMap.ext\n[GOAL]\ncase mk.h\nX : TopCat\nY : CompHaus\nf : StoneCech \u2191X \u27f6 \u2191Y.toTop\nhf : Continuous f\n\u22a2 \u2200 (a : \u2191(stoneCechObj X).toTop),\n \u2191((fun f => ContinuousMap.mk (stoneCechExtend (_ : Continuous f.toFun)))\n ((fun f => ContinuousMap.mk (\u2191f \u2218 stoneCechUnit)) (ContinuousMap.mk f)))\n a =\n \u2191(ContinuousMap.mk f) a\n[PROOFSTEP]\nintro (x : StoneCech X)\n[GOAL]\ncase mk.h\nX : TopCat\nY : CompHaus\nf : StoneCech \u2191X \u27f6 \u2191Y.toTop\nhf : Continuous f\nx : StoneCech \u2191X\n\u22a2 \u2191((fun f => ContinuousMap.mk (stoneCechExtend (_ : Continuous f.toFun)))\n ((fun f => ContinuousMap.mk (\u2191f \u2218 stoneCechUnit)) (ContinuousMap.mk f)))\n x =\n \u2191(ContinuousMap.mk f) x\n[PROOFSTEP]\nrefine' congr_fun _ x\n[GOAL]\ncase mk.h\nX : TopCat\nY : CompHaus\nf : StoneCech \u2191X \u27f6 \u2191Y.toTop\nhf : Continuous f\nx : StoneCech \u2191X\n\u22a2 \u2191((fun f => ContinuousMap.mk (stoneCechExtend (_ : Continuous f.toFun)))\n ((fun f => ContinuousMap.mk (\u2191f \u2218 stoneCechUnit)) (ContinuousMap.mk f))) =\n \u2191(ContinuousMap.mk f)\n[PROOFSTEP]\napply Continuous.ext_on denseRange_stoneCechUnit (continuous_stoneCechExtend _) hf\n[GOAL]\ncase mk.h\nX : TopCat\nY : CompHaus\nf : StoneCech \u2191X \u27f6 \u2191Y.toTop\nhf : Continuous f\nx : StoneCech \u2191X\n\u22a2 Set.EqOn (stoneCechExtend ?m.48961) f (Set.range stoneCechUnit)\nX : TopCat\nY : CompHaus\nf : StoneCech \u2191X \u27f6 \u2191Y.toTop\nhf : Continuous f\nx : StoneCech \u2191X\n\u22a2 Continuous ((fun f => ContinuousMap.mk (\u2191f \u2218 stoneCechUnit)) (ContinuousMap.mk f)).toFun\n[PROOFSTEP]\nrintro _ \u27e8y, rfl\u27e9\n[GOAL]\ncase mk.h.intro\nX : TopCat\nY : CompHaus\nf : StoneCech \u2191X \u27f6 \u2191Y.toTop\nhf : Continuous f\nx : StoneCech \u2191X\ny : \u2191X\n\u22a2 stoneCechExtend ?m.48961 (stoneCechUnit y) = f (stoneCechUnit y)\nX : TopCat\nY : CompHaus\nf : StoneCech \u2191X \u27f6 \u2191Y.toTop\nhf : Continuous f\nx : StoneCech \u2191X\n\u22a2 Continuous ((fun f => ContinuousMap.mk (\u2191f \u2218 stoneCechUnit)) (ContinuousMap.mk f)).toFun\n[PROOFSTEP]\napply congr_fun (stoneCechExtend_extends (hf.comp _)) y\n[GOAL]\nX : TopCat\nY : CompHaus\nf : StoneCech \u2191X \u27f6 \u2191Y.toTop\nhf : Continuous f\nx : StoneCech \u2191X\n\u22a2 Continuous fun x => stoneCechUnit x\n[PROOFSTEP]\napply continuous_stoneCechUnit\n[GOAL]\nX : TopCat\nY : CompHaus\n\u22a2 Function.RightInverse (fun f => ContinuousMap.mk (stoneCechExtend (_ : Continuous f.toFun))) fun f =>\n ContinuousMap.mk (\u2191f \u2218 stoneCechUnit)\n[PROOFSTEP]\nrintro\n \u27e8f : (X : Type _) \u27f6 Y, hf : Continuous f\u27e9\n -- Porting note: `ext` fails.\n[GOAL]\ncase mk\nX : TopCat\nY : CompHaus\nf : \u2191X \u27f6 \u2191Y.toTop\nhf : Continuous f\n\u22a2 (fun f => ContinuousMap.mk (\u2191f \u2218 stoneCechUnit))\n ((fun f => ContinuousMap.mk (stoneCechExtend (_ : Continuous f.toFun))) (ContinuousMap.mk f)) =\n ContinuousMap.mk f\n[PROOFSTEP]\napply ContinuousMap.ext\n[GOAL]\ncase mk.h\nX : TopCat\nY : CompHaus\nf : \u2191X \u27f6 \u2191Y.toTop\nhf : Continuous f\n\u22a2 \u2200 (a : \u2191X),\n \u2191((fun f => ContinuousMap.mk (\u2191f \u2218 stoneCechUnit))\n ((fun f => ContinuousMap.mk (stoneCechExtend (_ : Continuous f.toFun))) (ContinuousMap.mk f)))\n a =\n \u2191(ContinuousMap.mk f) a\n[PROOFSTEP]\nintro\n[GOAL]\ncase mk.h\nX : TopCat\nY : CompHaus\nf : \u2191X \u27f6 \u2191Y.toTop\nhf : Continuous f\na\u271d : \u2191X\n\u22a2 \u2191((fun f => ContinuousMap.mk (\u2191f \u2218 stoneCechUnit))\n ((fun f => ContinuousMap.mk (stoneCechExtend (_ : Continuous f.toFun))) (ContinuousMap.mk f)))\n a\u271d =\n \u2191(ContinuousMap.mk f) a\u271d\n[PROOFSTEP]\nexact congr_fun (stoneCechExtend_extends hf) _\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\n\u22a2 CompactSpace \u2191(TopCat.limitCone FF).pt\n[PROOFSTEP]\nshow CompactSpace {u : \u2200 j, F.obj j | \u2200 {i j : J} (f : i \u27f6 j), (F.map f) (u i) = u j}\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\n\u22a2 CompactSpace \u2191{u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j}\n[PROOFSTEP]\nrw [\u2190 isCompact_iff_compactSpace]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\n\u22a2 IsCompact {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j}\n[PROOFSTEP]\napply IsClosed.isCompact\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\n\u22a2 IsClosed {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j}\n[PROOFSTEP]\nhave :\n {u : \u2200 j, F.obj j | \u2200 {i j : J} (f : i \u27f6 j), F.map f (u i) = u j} =\n \u22c2 (i : J) (j : J) (f : i \u27f6 j), {u | F.map f (u i) = u j} :=\n by\n ext1\n simp only [Set.mem_iInter, Set.mem_setOf_eq]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\n\u22a2 {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j} = \u22c2 (i : J) (j : J) (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j}\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\nx\u271d : (j : J) \u2192 \u2191(F.obj j).toTop\n\u22a2 x\u271d \u2208 {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j} \u2194\n x\u271d \u2208 \u22c2 (i : J) (j : J) (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j}\n[PROOFSTEP]\nsimp only [Set.mem_iInter, Set.mem_setOf_eq]\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\nthis :\n {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j} = \u22c2 (i : J) (j : J) (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j}\n\u22a2 IsClosed {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j}\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\nthis :\n {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j} = \u22c2 (i : J) (j : J) (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j}\n\u22a2 IsClosed (\u22c2 (i : J) (j : J) (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j})\n[PROOFSTEP]\napply isClosed_iInter\n[GOAL]\ncase h.h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\nthis :\n {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j} = \u22c2 (i : J) (j : J) (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j}\n\u22a2 \u2200 (i : J), IsClosed (\u22c2 (j : J) (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j})\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h.h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\nthis :\n {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j} = \u22c2 (i : J) (j : J) (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j}\ni : J\n\u22a2 IsClosed (\u22c2 (j : J) (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j})\n[PROOFSTEP]\napply isClosed_iInter\n[GOAL]\ncase h.h.h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\nthis :\n {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j} = \u22c2 (i : J) (j : J) (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j}\ni : J\n\u22a2 \u2200 (i_1 : J), IsClosed (\u22c2 (f : i \u27f6 i_1), {u | \u2191(F.map f) (u i) = u i_1})\n[PROOFSTEP]\nintro j\n[GOAL]\ncase h.h.h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\nthis :\n {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j} = \u22c2 (i : J) (j : J) (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j}\ni j : J\n\u22a2 IsClosed (\u22c2 (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j})\n[PROOFSTEP]\napply isClosed_iInter\n[GOAL]\ncase h.h.h.h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\nthis :\n {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j} = \u22c2 (i : J) (j : J) (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j}\ni j : J\n\u22a2 \u2200 (i_1 : i \u27f6 j), IsClosed {u | \u2191(F.map i_1) (u i) = u j}\n[PROOFSTEP]\nintro f\n[GOAL]\ncase h.h.h.h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\nthis :\n {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j} = \u22c2 (i : J) (j : J) (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j}\ni j : J\nf : i \u27f6 j\n\u22a2 IsClosed {u | \u2191(F.map f) (u i) = u j}\n[PROOFSTEP]\napply isClosed_eq\n[GOAL]\ncase h.h.h.h.hf\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\nthis :\n {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j} = \u22c2 (i : J) (j : J) (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j}\ni j : J\nf : i \u27f6 j\n\u22a2 Continuous fun x => \u2191(F.map f) (x i)\n[PROOFSTEP]\nexact (ContinuousMap.continuous (F.map f)).comp (continuous_apply i)\n[GOAL]\ncase h.h.h.h.hg\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\nthis :\n {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j} = \u22c2 (i : J) (j : J) (f : i \u27f6 j), {u | \u2191(F.map f) (u i) = u j}\ni j : J\nf : i \u27f6 j\n\u22a2 Continuous fun x => x j\n[PROOFSTEP]\nexact continuous_apply j\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\n\u22a2 \u2200 \u2983X Y : J\u2984 (f : X \u27f6 Y),\n ((Functor.const J).obj (mk (TopCat.limitCone FF).pt)).map f \u226b (fun j => NatTrans.app (TopCat.limitCone FF).\u03c0 j) Y =\n (fun j => NatTrans.app (TopCat.limitCone FF).\u03c0 j) X \u226b F.map f\n[PROOFSTEP]\nintro _ _ f\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\nX\u271d Y\u271d : J\nf : X\u271d \u27f6 Y\u271d\n\u22a2 ((Functor.const J).obj (mk (TopCat.limitCone FF).pt)).map f \u226b (fun j => NatTrans.app (TopCat.limitCone FF).\u03c0 j) Y\u271d =\n (fun j => NatTrans.app (TopCat.limitCone FF).\u03c0 j) X\u271d \u226b F.map f\n[PROOFSTEP]\next \u27e8x, hx\u27e9\n[GOAL]\ncase w.mk\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\nX\u271d Y\u271d : J\nf : X\u271d \u27f6 Y\u271d\nx : (j : J) \u2192 \u2191(FF.obj j)\nhx : x \u2208 {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(FF.map f) (u i) = u j}\n\u22a2 \u2191(((Functor.const J).obj (mk (TopCat.limitCone FF).pt)).map f \u226b (fun j => NatTrans.app (TopCat.limitCone FF).\u03c0 j) Y\u271d)\n { val := x, property := hx } =\n \u2191((fun j => NatTrans.app (TopCat.limitCone FF).\u03c0 j) X\u271d \u226b F.map f) { val := x, property := hx }\n[PROOFSTEP]\nsimp only [comp_apply, Functor.const_obj_map, id_apply]\n[GOAL]\ncase w.mk\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CompHaus\nFF : J \u2964 TopCatMax := F \u22d9 compHausToTop\nX\u271d Y\u271d : J\nf : X\u271d \u27f6 Y\u271d\nx : (j : J) \u2192 \u2191(FF.obj j)\nhx : x \u2208 {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(FF.map f) (u i) = u j}\n\u22a2 \u2191(NatTrans.app (TopCat.limitCone (F \u22d9 compHausToTop)).\u03c0 Y\u271d)\n (\u2191(\ud835\udfd9 (mk (TopCat.limitCone (F \u22d9 compHausToTop)).pt)) { val := x, property := hx }) =\n \u2191(F.map f) (\u2191(NatTrans.app (TopCat.limitCone (F \u22d9 compHausToTop)).\u03c0 X\u271d) { val := x, property := hx })\n[PROOFSTEP]\nexact (hx f).symm\n[GOAL]\nX Y : CompHaus\nf : X \u27f6 Y\n\u22a2 Epi f \u2194 Function.Surjective \u2191f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nX Y : CompHaus\nf : X \u27f6 Y\n\u22a2 Epi f \u2192 Function.Surjective \u2191f\n[PROOFSTEP]\ndsimp [Function.Surjective]\n[GOAL]\ncase mp\nX Y : CompHaus\nf : X \u27f6 Y\n\u22a2 Epi f \u2192 \u2200 (b : (forget CompHaus).obj Y), \u2203 a, \u2191f a = b\n[PROOFSTEP]\ncontrapose!\n[GOAL]\ncase mp\nX Y : CompHaus\nf : X \u27f6 Y\n\u22a2 (\u2203 b, \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 b) \u2192 \u00acEpi f\n[PROOFSTEP]\nrintro \u27e8y, hy\u27e9 hf\n[GOAL]\ncase mp.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\n\u22a2 False\n[PROOFSTEP]\nlet C := Set.range f\n[GOAL]\ncase mp.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\n\u22a2 False\n[PROOFSTEP]\nhave hC : IsClosed C := (isCompact_range f.continuous).isClosed\n[GOAL]\ncase mp.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\n\u22a2 False\n[PROOFSTEP]\nlet D := ({ y } : Set Y)\n[GOAL]\ncase mp.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\n\u22a2 False\n[PROOFSTEP]\nhave hD : IsClosed D := isClosed_singleton\n[GOAL]\ncase mp.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\n\u22a2 False\n[PROOFSTEP]\nhave hCD : Disjoint C D := by\n rw [Set.disjoint_singleton_right]\n rintro \u27e8y', hy'\u27e9\n exact hy y' hy'\n[GOAL]\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\n\u22a2 Disjoint C D\n[PROOFSTEP]\nrw [Set.disjoint_singleton_right]\n[GOAL]\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\n\u22a2 \u00acy \u2208 C\n[PROOFSTEP]\nrintro \u27e8y', hy'\u27e9\n[GOAL]\ncase intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\ny' : (forget CompHaus).obj X\nhy' : \u2191f y' = y\n\u22a2 False\n[PROOFSTEP]\nexact hy y' hy'\n[GOAL]\ncase mp.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\n\u22a2 False\n[PROOFSTEP]\nhaveI : NormalSpace ((forget CompHaus).obj Y) := normalOfCompactT2\n[GOAL]\ncase mp.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis : NormalSpace ((forget CompHaus).obj Y)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8\u03c6, h\u03c60, h\u03c61, h\u03c601\u27e9 := exists_continuous_zero_one_of_closed hC hD hCD\n[GOAL]\ncase mp.intro.intro.intro.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\n\u22a2 False\n[PROOFSTEP]\nhaveI : CompactSpace (ULift.{u} <| Set.Icc (0 : \u211d) 1) := Homeomorph.ulift.symm.compactSpace\n[GOAL]\ncase mp.intro.intro.intro.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis : CompactSpace (ULift \u2191(Set.Icc 0 1))\n\u22a2 False\n[PROOFSTEP]\nhaveI : T2Space (ULift.{u} <| Set.Icc (0 : \u211d) 1) := Homeomorph.ulift.symm.t2Space\n[GOAL]\ncase mp.intro.intro.intro.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\n\u22a2 False\n[PROOFSTEP]\nlet Z := of (ULift.{u} <| Set.Icc (0 : \u211d) 1)\n[GOAL]\ncase mp.intro.intro.intro.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\nZ : CompHaus := of (ULift \u2191(Set.Icc 0 1))\n\u22a2 False\n[PROOFSTEP]\nlet g : Y \u27f6 Z := \u27e8fun y' => \u27e8\u27e8\u03c6 y', h\u03c601 y'\u27e9\u27e9, continuous_uLift_up.comp (\u03c6.continuous.subtype_mk fun y' => h\u03c601 y')\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\nZ : CompHaus := of (ULift \u2191(Set.Icc 0 1))\ng : Y \u27f6 Z := ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }\n\u22a2 False\n[PROOFSTEP]\nlet h : Y \u27f6 Z := \u27e8fun _ => \u27e8\u27e80, Set.left_mem_Icc.mpr zero_le_one\u27e9\u27e9, continuous_const\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\nZ : CompHaus := of (ULift \u2191(Set.Icc 0 1))\ng : Y \u27f6 Z := ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }\nh : Y \u27f6 Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 \u2208 Set.Icc 0 1) } }\n\u22a2 False\n[PROOFSTEP]\nhave H : h = g := by\n rw [\u2190 cancel_epi f]\n ext x\n apply ULift.ext\n apply Subtype.ext\n dsimp\n -- Porting note: This `change` is not ideal.\n -- I think lean is having issues understanding when a `ContinuousMap` should be considered\n -- as a morphism.\n -- TODO(?): Make morphisms in `CompHaus` (and other topological categories)\n -- into a one-field-structure.\n change 0 = \u03c6 (f x)\n simp only [h\u03c60 (Set.mem_range_self x), Pi.zero_apply]\n[GOAL]\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\nZ : CompHaus := of (ULift \u2191(Set.Icc 0 1))\ng : Y \u27f6 Z := ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }\nh : Y \u27f6 Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 \u2208 Set.Icc 0 1) } }\n\u22a2 h = g\n[PROOFSTEP]\nrw [\u2190 cancel_epi f]\n[GOAL]\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\nZ : CompHaus := of (ULift \u2191(Set.Icc 0 1))\ng : Y \u27f6 Z := ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }\nh : Y \u27f6 Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 \u2208 Set.Icc 0 1) } }\n\u22a2 f \u226b h = f \u226b g\n[PROOFSTEP]\next x\n[GOAL]\ncase w\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\nZ : CompHaus := of (ULift \u2191(Set.Icc 0 1))\ng : Y \u27f6 Z := ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }\nh : Y \u27f6 Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 \u2208 Set.Icc 0 1) } }\nx : (forget CompHaus).obj X\n\u22a2 \u2191(f \u226b h) x = \u2191(f \u226b g) x\n[PROOFSTEP]\napply ULift.ext\n[GOAL]\ncase w.h\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\nZ : CompHaus := of (ULift \u2191(Set.Icc 0 1))\ng : Y \u27f6 Z := ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }\nh : Y \u27f6 Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 \u2208 Set.Icc 0 1) } }\nx : (forget CompHaus).obj X\n\u22a2 (\u2191(f \u226b h) x).down = (\u2191(f \u226b g) x).down\n[PROOFSTEP]\napply Subtype.ext\n[GOAL]\ncase w.h.a\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\nZ : CompHaus := of (ULift \u2191(Set.Icc 0 1))\ng : Y \u27f6 Z := ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }\nh : Y \u27f6 Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 \u2208 Set.Icc 0 1) } }\nx : (forget CompHaus).obj X\n\u22a2 \u2191(\u2191(f \u226b h) x).down = \u2191(\u2191(f \u226b g) x).down\n[PROOFSTEP]\ndsimp\n -- Porting note: This `change` is not ideal.\n -- I think lean is having issues understanding when a `ContinuousMap` should be considered\n -- as a morphism.\n -- TODO(?): Make morphisms in `CompHaus` (and other topological categories)\n -- into a one-field-structure.\n[GOAL]\ncase w.h.a\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\nZ : CompHaus := of (ULift \u2191(Set.Icc 0 1))\ng : Y \u27f6 Z := ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }\nh : Y \u27f6 Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 \u2208 Set.Icc 0 1) } }\nx : (forget CompHaus).obj X\n\u22a2 \u2191(\u2191(f \u226b ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 \u2208 Set.Icc 0 1) } }) x).down =\n \u2191(\u2191(f \u226b ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }) x).down\n[PROOFSTEP]\nchange 0 = \u03c6 (f x)\n[GOAL]\ncase w.h.a\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\nZ : CompHaus := of (ULift \u2191(Set.Icc 0 1))\ng : Y \u27f6 Z := ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }\nh : Y \u27f6 Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 \u2208 Set.Icc 0 1) } }\nx : (forget CompHaus).obj X\n\u22a2 0 = \u2191\u03c6 (\u2191f x)\n[PROOFSTEP]\nsimp only [h\u03c60 (Set.mem_range_self x), Pi.zero_apply]\n[GOAL]\ncase mp.intro.intro.intro.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\nZ : CompHaus := of (ULift \u2191(Set.Icc 0 1))\ng : Y \u27f6 Z := ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }\nh : Y \u27f6 Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 \u2208 Set.Icc 0 1) } }\nH : h = g\n\u22a2 False\n[PROOFSTEP]\napply_fun fun e => (e y).down.1 at H \n[GOAL]\ncase mp.intro.intro.intro.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\nZ : CompHaus := of (ULift \u2191(Set.Icc 0 1))\ng : Y \u27f6 Z := ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }\nh : Y \u27f6 Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 \u2208 Set.Icc 0 1) } }\nH : \u2191(\u2191h y).down = \u2191(\u2191g y).down\n\u22a2 False\n[PROOFSTEP]\ndsimp at H \n[GOAL]\ncase mp.intro.intro.intro.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\nZ : CompHaus := of (ULift \u2191(Set.Icc 0 1))\ng : Y \u27f6 Z := ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }\nh : Y \u27f6 Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 \u2208 Set.Icc 0 1) } }\nH :\n \u2191(\u2191(ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 \u2208 Set.Icc 0 1) } }) y).down =\n \u2191(\u2191(ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }) y).down\n\u22a2 False\n[PROOFSTEP]\nchange 0 = \u03c6 y at H \n[GOAL]\ncase mp.intro.intro.intro.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\nZ : CompHaus := of (ULift \u2191(Set.Icc 0 1))\ng : Y \u27f6 Z := ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }\nh : Y \u27f6 Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 \u2208 Set.Icc 0 1) } }\nH : 0 = \u2191\u03c6 y\n\u22a2 False\n[PROOFSTEP]\nsimp only [h\u03c61 (Set.mem_singleton y), Pi.one_apply] at H \n[GOAL]\ncase mp.intro.intro.intro.intro\nX Y : CompHaus\nf : X \u27f6 Y\ny : (forget CompHaus).obj Y\nhy : \u2200 (a : (forget CompHaus).obj X), \u2191f a \u2260 y\nhf : Epi f\nC : Set ((forget CompHaus).obj Y) := Set.range \u2191f\nhC : IsClosed C\nD : Set \u2191Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nthis\u271d\u00b9 : NormalSpace ((forget CompHaus).obj Y)\n\u03c6 : C((forget CompHaus).obj Y, \u211d)\nh\u03c60 : Set.EqOn (\u2191\u03c6) 0 C\nh\u03c61 : Set.EqOn (\u2191\u03c6) 1 D\nh\u03c601 : \u2200 (x : (forget CompHaus).obj Y), \u2191\u03c6 x \u2208 Set.Icc 0 1\nthis\u271d : CompactSpace (ULift \u2191(Set.Icc 0 1))\nthis : T2Space (ULift \u2191(Set.Icc 0 1))\nZ : CompHaus := of (ULift \u2191(Set.Icc 0 1))\ng : Y \u27f6 Z := ContinuousMap.mk fun y' => { down := { val := \u2191\u03c6 y', property := (_ : \u2191\u03c6 y' \u2208 Set.Icc 0 1) } }\nh : Y \u27f6 Z := ContinuousMap.mk fun x => { down := { val := 0, property := (_ : 0 \u2208 Set.Icc 0 1) } }\nH : 0 = 1\n\u22a2 False\n[PROOFSTEP]\nexact zero_ne_one H\n[GOAL]\ncase mpr\nX Y : CompHaus\nf : X \u27f6 Y\n\u22a2 Function.Surjective \u2191f \u2192 Epi f\n[PROOFSTEP]\nrw [\u2190 CategoryTheory.epi_iff_surjective]\n[GOAL]\ncase mpr\nX Y : CompHaus\nf : X \u27f6 Y\n\u22a2 Epi \u2191f \u2192 Epi f\n[PROOFSTEP]\napply (forget CompHaus).epi_of_epi_map\n[GOAL]\nX Y : CompHaus\nf : X \u27f6 Y\n\u22a2 Mono f \u2194 Function.Injective \u2191f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nX Y : CompHaus\nf : X \u27f6 Y\n\u22a2 Mono f \u2192 Function.Injective \u2191f\n[PROOFSTEP]\nintro hf x\u2081 x\u2082 h\n[GOAL]\ncase mp\nX Y : CompHaus\nf : X \u27f6 Y\nhf : Mono f\nx\u2081 x\u2082 : (forget CompHaus).obj X\nh : \u2191f x\u2081 = \u2191f x\u2082\n\u22a2 x\u2081 = x\u2082\n[PROOFSTEP]\nlet g\u2081 : of PUnit \u27f6 X := \u27e8fun _ => x\u2081, continuous_const\u27e9\n[GOAL]\ncase mp\nX Y : CompHaus\nf : X \u27f6 Y\nhf : Mono f\nx\u2081 x\u2082 : (forget CompHaus).obj X\nh : \u2191f x\u2081 = \u2191f x\u2082\ng\u2081 : of PUnit \u27f6 X := ContinuousMap.mk fun x => x\u2081\n\u22a2 x\u2081 = x\u2082\n[PROOFSTEP]\nlet g\u2082 : of PUnit \u27f6 X := \u27e8fun _ => x\u2082, continuous_const\u27e9\n[GOAL]\ncase mp\nX Y : CompHaus\nf : X \u27f6 Y\nhf : Mono f\nx\u2081 x\u2082 : (forget CompHaus).obj X\nh : \u2191f x\u2081 = \u2191f x\u2082\ng\u2081 : of PUnit \u27f6 X := ContinuousMap.mk fun x => x\u2081\ng\u2082 : of PUnit \u27f6 X := ContinuousMap.mk fun x => x\u2082\n\u22a2 x\u2081 = x\u2082\n[PROOFSTEP]\nhave : g\u2081 \u226b f = g\u2082 \u226b f := by\n ext\n exact h\n[GOAL]\nX Y : CompHaus\nf : X \u27f6 Y\nhf : Mono f\nx\u2081 x\u2082 : (forget CompHaus).obj X\nh : \u2191f x\u2081 = \u2191f x\u2082\ng\u2081 : of PUnit \u27f6 X := ContinuousMap.mk fun x => x\u2081\ng\u2082 : of PUnit \u27f6 X := ContinuousMap.mk fun x => x\u2082\n\u22a2 g\u2081 \u226b f = g\u2082 \u226b f\n[PROOFSTEP]\next\n[GOAL]\ncase w\nX Y : CompHaus\nf : X \u27f6 Y\nhf : Mono f\nx\u2081 x\u2082 : (forget CompHaus).obj X\nh : \u2191f x\u2081 = \u2191f x\u2082\ng\u2081 : of PUnit \u27f6 X := ContinuousMap.mk fun x => x\u2081\ng\u2082 : of PUnit \u27f6 X := ContinuousMap.mk fun x => x\u2082\nx\u271d : (forget CompHaus).obj (of PUnit)\n\u22a2 \u2191(g\u2081 \u226b f) x\u271d = \u2191(g\u2082 \u226b f) x\u271d\n[PROOFSTEP]\nexact h\n[GOAL]\ncase mp\nX Y : CompHaus\nf : X \u27f6 Y\nhf : Mono f\nx\u2081 x\u2082 : (forget CompHaus).obj X\nh : \u2191f x\u2081 = \u2191f x\u2082\ng\u2081 : of PUnit \u27f6 X := ContinuousMap.mk fun x => x\u2081\ng\u2082 : of PUnit \u27f6 X := ContinuousMap.mk fun x => x\u2082\nthis : g\u2081 \u226b f = g\u2082 \u226b f\n\u22a2 x\u2081 = x\u2082\n[PROOFSTEP]\nrw [cancel_mono] at this \n[GOAL]\ncase mp\nX Y : CompHaus\nf : X \u27f6 Y\nhf : Mono f\nx\u2081 x\u2082 : (forget CompHaus).obj X\nh : \u2191f x\u2081 = \u2191f x\u2082\ng\u2081 : of PUnit \u27f6 X := ContinuousMap.mk fun x => x\u2081\ng\u2082 : of PUnit \u27f6 X := ContinuousMap.mk fun x => x\u2082\nthis : g\u2081 = g\u2082\n\u22a2 x\u2081 = x\u2082\n[PROOFSTEP]\napply_fun fun e => e PUnit.unit at this \n[GOAL]\ncase mp\nX Y : CompHaus\nf : X \u27f6 Y\nhf : Mono f\nx\u2081 x\u2082 : (forget CompHaus).obj X\nh : \u2191f x\u2081 = \u2191f x\u2082\ng\u2081 : of PUnit \u27f6 X := ContinuousMap.mk fun x => x\u2081\ng\u2082 : of PUnit \u27f6 X := ContinuousMap.mk fun x => x\u2082\nthis : \u2191g\u2081 PUnit.unit = \u2191g\u2082 PUnit.unit\n\u22a2 x\u2081 = x\u2082\n[PROOFSTEP]\nexact this\n[GOAL]\ncase mpr\nX Y : CompHaus\nf : X \u27f6 Y\n\u22a2 Function.Injective \u2191f \u2192 Mono f\n[PROOFSTEP]\nrw [\u2190 CategoryTheory.mono_iff_injective]\n[GOAL]\ncase mpr\nX Y : CompHaus\nf : X \u27f6 Y\n\u22a2 Mono \u2191f \u2192 Mono f\n[PROOFSTEP]\napply (forget CompHaus).mono_of_mono_map\n", "meta": {"mathlib_filename": "Mathlib.Topology.Category.CompHaus.Basic", "llama_tokens": 19439, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.6224593312018545, "lm_q1q2_score": 0.4433355690618109}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n[PROOFSTEP]\nhave g_intble' : \u2200 t : \u211d, 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t :=\n by\n intro t ht\n cases' eq_or_lt_of_le ht with h h\n \u00b7 simp [\u2190 h]\n \u00b7 exact g_intble t h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\n\u22a2 \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n[PROOFSTEP]\nintro t ht\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nt : \u211d\nht : 0 \u2264 t\n\u22a2 IntervalIntegrable g volume 0 t\n[PROOFSTEP]\ncases' eq_or_lt_of_le ht with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nt : \u211d\nht : 0 \u2264 t\nh : 0 = t\n\u22a2 IntervalIntegrable g volume 0 t\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\nt : \u211d\nht : 0 \u2264 t\nh : 0 < t\n\u22a2 IntervalIntegrable g volume 0 t\n[PROOFSTEP]\nexact g_intble t h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n[PROOFSTEP]\nhave integrand_eq : \u2200 \u03c9, ENNReal.ofReal (\u222b t in (0)..f \u03c9, g t) = \u222b\u207b t in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) :=\n by\n intro \u03c9\n have g_ae_nn : 0 \u2264\u1d50[volume.restrict (Ioc 0 (f \u03c9))] g := by\n filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f \u03c9)))] with x hx using g_nn x hx.1\n rw [\u2190 ofReal_integral_eq_lintegral_ofReal (g_intble' (f \u03c9) (f_nn \u03c9)).1 g_ae_nn]\n congr\n exact intervalIntegral.integral_of_le (f_nn \u03c9)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u22a2 \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n[PROOFSTEP]\nintro \u03c9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u03c9 : \u03b1\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n[PROOFSTEP]\nhave g_ae_nn : 0 \u2264\u1d50[volume.restrict (Ioc 0 (f \u03c9))] g := by\n filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f \u03c9)))] with x hx using g_nn x hx.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u03c9 : \u03b1\n\u22a2 0 \u2264\u1d50[Measure.restrict volume (Ioc 0 (f \u03c9))] g\n[PROOFSTEP]\nfilter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f \u03c9)))] with x hx using g_nn x hx.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u03c9 : \u03b1\ng_ae_nn : 0 \u2264\u1d50[Measure.restrict volume (Ioc 0 (f \u03c9))] g\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n[PROOFSTEP]\nrw [\u2190 ofReal_integral_eq_lintegral_ofReal (g_intble' (f \u03c9) (f_nn \u03c9)).1 g_ae_nn]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u03c9 : \u03b1\ng_ae_nn : 0 \u2264\u1d50[Measure.restrict volume (Ioc 0 (f \u03c9))] g\n\u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = ENNReal.ofReal (\u222b (x : \u211d) in Ioc 0 (f \u03c9), g x)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_r\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\n\u03c9 : \u03b1\ng_ae_nn : 0 \u2264\u1d50[Measure.restrict volume (Ioc 0 (f \u03c9))] g\n\u22a2 \u222b (t : \u211d) in 0 ..f \u03c9, g t = \u222b (x : \u211d) in Ioc 0 (f \u03c9), g x\n[PROOFSTEP]\nexact intervalIntegral.integral_of_le (f_nn \u03c9)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n[PROOFSTEP]\nsimp_rw [integrand_eq, \u2190 lintegral_indicator (fun t => ENNReal.ofReal (g t)) measurableSet_Ioc]\n -- Porting note: was part of `simp_rw` on the previous line, but didn't trigger.\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), \u222b\u207b (a : \u211d), indicator (Ioc 0 (f \u03c9)) (fun t => ENNReal.ofReal (g t)) a \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n[PROOFSTEP]\nrw [\u2190 lintegral_indicator _ measurableSet_Ioi]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), \u222b\u207b (a : \u211d), indicator (Ioc 0 (f \u03c9)) (fun t => ENNReal.ofReal (g t)) a \u2202\u03bc =\n \u222b\u207b (a : \u211d), indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) a\n[PROOFSTEP]\nrw [lintegral_lintegral_swap]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (y : \u211d), \u222b\u207b (x : \u03b1), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y \u2202\u03bc =\n \u222b\u207b (a : \u211d), indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) a\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 (fun y => \u222b\u207b (x : \u03b1), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y \u2202\u03bc) = fun a =>\n indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) a\n[PROOFSTEP]\nfunext s\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\n\u22a2 \u222b\u207b (x : \u03b1), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s \u2202\u03bc =\n indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s\n[PROOFSTEP]\nhave aux\u2081 :\n (fun x => (Ioc 0 (f x)).indicator (fun t : \u211d => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * (Ioi (0 : \u211d)).indicator (fun _ => 1) s * (Ici s).indicator (fun _ : \u211d => (1 : \u211d\u22650\u221e)) (f x) :=\n by\n funext a\n by_cases s \u2208 Ioc (0 : \u211d) (f a)\n \u00b7 simp only [h, show s \u2208 Ioi (0 : \u211d) from h.1, show f a \u2208 Ici s from h.2, indicator_of_mem, mul_one]\n \u00b7 have h_copy := h\n simp only [mem_Ioc, not_and, not_le] at h \n by_cases h' : 0 < s\n \u00b7 simp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le, mul_zero]\n \u00b7 have : s \u2209 Ioi (0 : \u211d) := h'\n simp only [this, h', indicator_of_not_mem, not_false_iff, mul_zero, zero_mul, mem_Ioc, false_and_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\n\u22a2 (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n[PROOFSTEP]\nfunext a\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)\n[PROOFSTEP]\nby_cases s \u2208 Ioc (0 : \u211d) (f a)\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)\n[PROOFSTEP]\nby_cases s \u2208 Ioc (0 : \u211d) (f a)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh : s \u2208 Ioc 0 (f a)\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)\n[PROOFSTEP]\nsimp only [h, show s \u2208 Ioi (0 : \u211d) from h.1, show f a \u2208 Ici s from h.2, indicator_of_mem, mul_one]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh : \u00acs \u2208 Ioc 0 (f a)\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)\n[PROOFSTEP]\nhave h_copy := h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh h_copy : \u00acs \u2208 Ioc 0 (f a)\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)\n[PROOFSTEP]\nsimp only [mem_Ioc, not_and, not_le] at h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh_copy : \u00acs \u2208 Ioc 0 (f a)\nh : 0 < s \u2192 f a < s\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)\n[PROOFSTEP]\nby_cases h' : 0 < s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh_copy : \u00acs \u2208 Ioc 0 (f a)\nh : 0 < s \u2192 f a < s\nh' : 0 < s\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)\n[PROOFSTEP]\nsimp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le, mul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh_copy : \u00acs \u2208 Ioc 0 (f a)\nh : 0 < s \u2192 f a < s\nh' : \u00ac0 < s\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)\n[PROOFSTEP]\nhave : s \u2209 Ioi (0 : \u211d) := h'\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\na : \u03b1\nh_copy : \u00acs \u2208 Ioc 0 (f a)\nh : 0 < s \u2192 f a < s\nh' : \u00ac0 < s\nthis : \u00acs \u2208 Ioi 0\n\u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)\n[PROOFSTEP]\nsimp only [this, h', indicator_of_not_mem, not_false_iff, mul_zero, zero_mul, mem_Ioc, false_and_iff]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 \u222b\u207b (x : \u03b1), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s \u2202\u03bc =\n indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s\n[PROOFSTEP]\nsimp_rw [aux\u2081]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 \u222b\u207b (x : \u03b1), ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u2202\u03bc =\n indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s\n[PROOFSTEP]\nrw [lintegral_const_mul']\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1), indicator (Ici s) (fun x => 1) (f a) \u2202\u03bc =\n indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s\ncase h.h.hr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s \u2260 \u22a4\n[PROOFSTEP]\nswap\n[GOAL]\ncase h.h.hr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s \u2260 \u22a4\n[PROOFSTEP]\napply ENNReal.mul_ne_top ENNReal.ofReal_ne_top\n[GOAL]\ncase h.h.hr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 indicator (Ioi 0) (fun x => 1) s \u2260 \u22a4\n[PROOFSTEP]\nby_cases h : (0 : \u211d) < s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\nh : 0 < s\n\u22a2 indicator (Ioi 0) (fun x => 1) s \u2260 \u22a4\n[PROOFSTEP]\nsimp [indicator_apply, h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\nh : \u00ac0 < s\n\u22a2 indicator (Ioi 0) (fun x => 1) s \u2260 \u22a4\n[PROOFSTEP]\nsimp [indicator_apply, h]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1), indicator (Ici s) (fun x => 1) (f a) \u2202\u03bc =\n indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s\n[PROOFSTEP]\nsimp_rw [show\n (fun a => (Ici s).indicator (fun _ : \u211d => (1 : \u211d\u22650\u221e)) (f a)) = fun a => {a : \u03b1 | s \u2264 f a}.indicator (fun _ => 1) a\n by funext a; by_cases s \u2264 f a <;> simp [h]]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 (fun a => indicator (Ici s) (fun x => 1) (f a)) = fun a => indicator {a | s \u2264 f a} (fun x => 1) a\n[PROOFSTEP]\nfunext a\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\na : \u03b1\n\u22a2 indicator (Ici s) (fun x => 1) (f a) = indicator {a | s \u2264 f a} (fun x => 1) a\n[PROOFSTEP]\nby_cases s \u2264 f a\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\na : \u03b1\n\u22a2 indicator (Ici s) (fun x => 1) (f a) = indicator {a | s \u2264 f a} (fun x => 1) a\n[PROOFSTEP]\nby_cases s \u2264 f a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\na : \u03b1\nh : s \u2264 f a\n\u22a2 indicator (Ici s) (fun x => 1) (f a) = indicator {a | s \u2264 f a} (fun x => 1) a\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\na : \u03b1\nh : \u00acs \u2264 f a\n\u22a2 indicator (Ici s) (fun x => 1) (f a) = indicator {a | s \u2264 f a} (fun x => 1) a\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1), indicator {a | s \u2264 f a} (fun x => 1) a \u2202\u03bc =\n indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s\n[PROOFSTEP]\nrw [lintegral_indicator]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1) in {a | s \u2264 f a}, 1 \u2202\u03bc =\n indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s\ncase h.h.hs\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 MeasurableSet {a | s \u2264 f a}\n[PROOFSTEP]\nswap\n[GOAL]\ncase h.h.hs\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 MeasurableSet {a | s \u2264 f a}\n[PROOFSTEP]\nexact f_mble measurableSet_Ici\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1) in {a | s \u2264 f a}, 1 \u2202\u03bc =\n indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s\n[PROOFSTEP]\nrw [lintegral_one, Measure.restrict_apply MeasurableSet.univ, univ_inter, indicator_mul_left, mul_assoc,\n show\n (Ioi 0).indicator (fun _x : \u211d => (1 : \u211d\u22650\u221e)) s * \u03bc {a : \u03b1 | s \u2264 f a} =\n (Ioi 0).indicator (fun _x : \u211d => 1 * \u03bc {a : \u03b1 | s \u2264 f a}) s\n by by_cases 0 < s <;> simp [h]]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 indicator (Ioi 0) (fun _x => 1) s * \u2191\u2191\u03bc {a | s \u2264 f a} = indicator (Ioi 0) (fun _x => 1 * \u2191\u2191\u03bc {a | s \u2264 f a}) s\n[PROOFSTEP]\nby_cases 0 < s\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 indicator (Ioi 0) (fun _x => 1) s * \u2191\u2191\u03bc {a | s \u2264 f a} = indicator (Ioi 0) (fun _x => 1 * \u2191\u2191\u03bc {a | s \u2264 f a}) s\n[PROOFSTEP]\nby_cases 0 < s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\nh : 0 < s\n\u22a2 indicator (Ioi 0) (fun _x => 1) s * \u2191\u2191\u03bc {a | s \u2264 f a} = indicator (Ioi 0) (fun _x => 1 * \u2191\u2191\u03bc {a | s \u2264 f a}) s\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\nh : \u00ac0 < s\n\u22a2 indicator (Ioi 0) (fun _x => 1) s * \u2191\u2191\u03bc {a | s \u2264 f a} = indicator (Ioi 0) (fun _x => 1 * \u2191\u2191\u03bc {a | s \u2264 f a}) s\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun _x => 1 * \u2191\u2191\u03bc {a | s \u2264 f a}) s =\n indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a}) s * ENNReal.ofReal (g s)\n[PROOFSTEP]\nsimp_rw [mul_comm _ (ENNReal.ofReal _), one_mul]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\ns : \u211d\naux\u2081 :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n\u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun _x => \u2191\u2191\u03bc {a | s \u2264 f a}) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a}) s\n[PROOFSTEP]\nrfl\n[GOAL]\ncase hf\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 AEMeasurable (Function.uncurry fun \u03c9 a => indicator (Ioc 0 (f \u03c9)) (fun t => ENNReal.ofReal (g t)) a)\n[PROOFSTEP]\nhave aux\u2082 :\n (Function.uncurry fun (x : \u03b1) (y : \u211d) => (Ioc 0 (f x)).indicator (fun t : \u211d => ENNReal.ofReal (g t)) y) =\n {p : \u03b1 \u00d7 \u211d | p.2 \u2208 Ioc 0 (f p.1)}.indicator fun p => ENNReal.ofReal (g p.2) :=\n by\n funext p\n cases p with\n | mk p_fst p_snd => ?_\n rw [Function.uncurry_apply_pair]\n by_cases p_snd \u2208 Ioc 0 (f p_fst)\n \u00b7 have h' : (p_fst, p_snd) \u2208 {p : \u03b1 \u00d7 \u211d | p.snd \u2208 Ioc 0 (f p.fst)} := h\n rw [Set.indicator_of_mem h', Set.indicator_of_mem h]\n \u00b7 have h' : (p_fst, p_snd) \u2209 {p : \u03b1 \u00d7 \u211d | p.snd \u2208 Ioc 0 (f p.fst)} := h\n rw [Set.indicator_of_not_mem h', Set.indicator_of_not_mem h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\n\u22a2 (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd)\n[PROOFSTEP]\nfunext p\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np : \u03b1 \u00d7 \u211d\n\u22a2 Function.uncurry (fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) p =\n indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) p\n[PROOFSTEP]\ncases p with\n| mk p_fst p_snd => ?_\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np : \u03b1 \u00d7 \u211d\n\u22a2 Function.uncurry (fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) p =\n indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) p\n[PROOFSTEP]\ncases p with\n| mk p_fst p_snd => ?_\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\n\u22a2 Function.uncurry (fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) (p_fst, p_snd) =\n indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)\n[PROOFSTEP]\nrw [Function.uncurry_apply_pair]\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)\n[PROOFSTEP]\nby_cases p_snd \u2208 Ioc 0 (f p_fst)\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)\n[PROOFSTEP]\nby_cases p_snd \u2208 Ioc 0 (f p_fst)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\nh : p_snd \u2208 Ioc 0 (f p_fst)\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)\n[PROOFSTEP]\nhave h' : (p_fst, p_snd) \u2208 {p : \u03b1 \u00d7 \u211d | p.snd \u2208 Ioc 0 (f p.fst)} := h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\nh : p_snd \u2208 Ioc 0 (f p_fst)\nh' : (p_fst, p_snd) \u2208 {p | p.snd \u2208 Ioc 0 (f p.fst)}\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)\n[PROOFSTEP]\nrw [Set.indicator_of_mem h', Set.indicator_of_mem h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\nh : \u00acp_snd \u2208 Ioc 0 (f p_fst)\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)\n[PROOFSTEP]\nhave h' : (p_fst, p_snd) \u2209 {p : \u03b1 \u00d7 \u211d | p.snd \u2208 Ioc 0 (f p.fst)} := h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\np_fst : \u03b1\np_snd : \u211d\nh : \u00acp_snd \u2208 Ioc 0 (f p_fst)\nh' : \u00ac(p_fst, p_snd) \u2208 {p | p.snd \u2208 Ioc 0 (f p.fst)}\n\u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)\n[PROOFSTEP]\nrw [Set.indicator_of_not_mem h', Set.indicator_of_not_mem h]\n[GOAL]\ncase hf\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\naux\u2082 :\n (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd)\n\u22a2 AEMeasurable (Function.uncurry fun \u03c9 a => indicator (Ioc 0 (f \u03c9)) (fun t => ENNReal.ofReal (g t)) a)\n[PROOFSTEP]\nrw [aux\u2082]\n[GOAL]\ncase hf\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\naux\u2082 :\n (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd)\n\u22a2 AEMeasurable (indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd))\n[PROOFSTEP]\nhave mble := measurableSet_region_between_oc measurable_zero f_mble MeasurableSet.univ\n[GOAL]\ncase hf\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\naux\u2082 :\n (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd)\nmble : MeasurableSet {p | p.fst \u2208 univ \u2227 p.snd \u2208 Ioc (OfNat.ofNat 0 p.fst) (f p.fst)}\n\u22a2 AEMeasurable (indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd))\n[PROOFSTEP]\nsimp_rw [mem_univ, Pi.zero_apply, true_and_iff] at mble \n[GOAL]\ncase hf\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t\ng_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t\nintegrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)\naux\u2082 :\n (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd)\nmble : MeasurableSet {p | p.snd \u2208 Ioc 0 (f p.fst)}\n\u22a2 AEMeasurable (indicator {p | p.snd \u2208 Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd))\n[PROOFSTEP]\nexact (ENNReal.measurable_ofReal.comp (g_mble.comp measurable_snd)).aemeasurable.indicator mble\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n[PROOFSTEP]\nhave ex_G : \u2203 G : \u211d \u2192 \u211d, Measurable G \u2227 0 \u2264 G \u2227 g =\u1d50[volume.restrict (Ioi 0)] G :=\n by\n refine' AEMeasurable.exists_measurable_nonneg _ g_nn\n exact aemeasurable_Ioi_of_forall_Ioc fun t ht => (g_intble t ht).1.1.aemeasurable\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\n\u22a2 \u2203 G, Measurable G \u2227 0 \u2264 G \u2227 g =\u1d50[Measure.restrict volume (Ioi 0)] G\n[PROOFSTEP]\nrefine' AEMeasurable.exists_measurable_nonneg _ g_nn\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\n\u22a2 AEMeasurable g\n[PROOFSTEP]\nexact aemeasurable_Ioi_of_forall_Ioc fun t ht => (g_intble t ht).1.1.aemeasurable\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nex_G : \u2203 G, Measurable G \u2227 0 \u2264 G \u2227 g =\u1d50[Measure.restrict volume (Ioi 0)] G\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n[PROOFSTEP]\nrcases ex_G with \u27e8G, G_mble, G_nn, g_eq_G\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n[PROOFSTEP]\nhave g_eq_G_on : \u2200 t, g =\u1d50[volume.restrict (Ioc 0 t)] G := fun t =>\n ae_mono (Measure.restrict_mono Ioc_subset_Ioi_self le_rfl) g_eq_G\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1d50[Measure.restrict volume (Ioc 0 t)] G\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n[PROOFSTEP]\nhave G_intble : \u2200 t > 0, IntervalIntegrable G volume 0 t :=\n by\n refine' fun t t_pos => \u27e8(g_intble t t_pos).1.congr_fun_ae (g_eq_G_on t), _\u27e9\n rw [Ioc_eq_empty_of_le t_pos.lt.le]\n exact integrableOn_empty\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1d50[Measure.restrict volume (Ioc 0 t)] G\n\u22a2 \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\n[PROOFSTEP]\nrefine' fun t t_pos => \u27e8(g_intble t t_pos).1.congr_fun_ae (g_eq_G_on t), _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1d50[Measure.restrict volume (Ioc 0 t)] G\nt : \u211d\nt_pos : t > 0\n\u22a2 IntegrableOn G (Ioc t 0)\n[PROOFSTEP]\nrw [Ioc_eq_empty_of_le t_pos.lt.le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1d50[Measure.restrict volume (Ioc 0 t)] G\nt : \u211d\nt_pos : t > 0\n\u22a2 IntegrableOn G \u2205\n[PROOFSTEP]\nexact integrableOn_empty\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1d50[Measure.restrict volume (Ioc 0 t)] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n[PROOFSTEP]\nhave eq\u2081 :\n (\u222b\u207b t in Ioi 0, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t)) =\n \u222b\u207b t in Ioi 0, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (G t) :=\n by\n apply lintegral_congr_ae\n filter_upwards [g_eq_G] with a ha\n rw [ha]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1d50[Measure.restrict volume (Ioc 0 t)] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (G t)\n[PROOFSTEP]\napply lintegral_congr_ae\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1d50[Measure.restrict volume (Ioc 0 t)] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\n\u22a2 (fun a => \u2191\u2191\u03bc {a_1 | a \u2264 f a_1} * ENNReal.ofReal (g a)) =\u1d50[Measure.restrict volume (Ioi 0)] fun a =>\n \u2191\u2191\u03bc {a_1 | a \u2264 f a_1} * ENNReal.ofReal (G a)\n[PROOFSTEP]\nfilter_upwards [g_eq_G] with a ha\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1d50[Measure.restrict volume (Ioc 0 t)] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\na : \u211d\nha : g a = G a\n\u22a2 \u2191\u2191\u03bc {a_1 | a \u2264 f a_1} * ENNReal.ofReal (g a) = \u2191\u2191\u03bc {a_1 | a \u2264 f a_1} * ENNReal.ofReal (G a)\n[PROOFSTEP]\nrw [ha]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1d50[Measure.restrict volume (Ioc 0 t)] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\neq\u2081 :\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (G t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n[PROOFSTEP]\nhave eq\u2082 : \u2200 \u03c9, (\u222b t in (0)..f \u03c9, g t) = \u222b t in (0)..f \u03c9, G t :=\n by\n refine' fun \u03c9 => intervalIntegral.integral_congr_ae _\n have f\u03c9_nn : 0 \u2264 f \u03c9 := f_nn \u03c9\n rw [uIoc_of_le f\u03c9_nn, \u2190 ae_restrict_iff' (measurableSet_Ioc : MeasurableSet (Ioc (0 : \u211d) (f \u03c9)))]\n exact g_eq_G_on (f \u03c9)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1d50[Measure.restrict volume (Ioc 0 t)] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\neq\u2081 :\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (G t)\n\u22a2 \u2200 (\u03c9 : \u03b1), \u222b (t : \u211d) in 0 ..f \u03c9, g t = \u222b (t : \u211d) in 0 ..f \u03c9, G t\n[PROOFSTEP]\nrefine' fun \u03c9 => intervalIntegral.integral_congr_ae _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1d50[Measure.restrict volume (Ioc 0 t)] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\neq\u2081 :\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (G t)\n\u03c9 : \u03b1\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 \u0399 0 (f \u03c9) \u2192 g x = G x\n[PROOFSTEP]\nhave f\u03c9_nn : 0 \u2264 f \u03c9 := f_nn \u03c9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1d50[Measure.restrict volume (Ioc 0 t)] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\neq\u2081 :\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (G t)\n\u03c9 : \u03b1\nf\u03c9_nn : 0 \u2264 f \u03c9\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 \u0399 0 (f \u03c9) \u2192 g x = G x\n[PROOFSTEP]\nrw [uIoc_of_le f\u03c9_nn, \u2190 ae_restrict_iff' (measurableSet_Ioc : MeasurableSet (Ioc (0 : \u211d) (f \u03c9)))]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1d50[Measure.restrict volume (Ioc 0 t)] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\neq\u2081 :\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (G t)\n\u03c9 : \u03b1\nf\u03c9_nn : 0 \u2264 f \u03c9\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioc 0 (f \u03c9)), g x = G x\n[PROOFSTEP]\nexact g_eq_G_on (f \u03c9)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1d50[Measure.restrict volume (Ioc 0 t)] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\neq\u2081 :\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (G t)\neq\u2082 : \u2200 (\u03c9 : \u03b1), \u222b (t : \u211d) in 0 ..f \u03c9, g t = \u222b (t : \u211d) in 0 ..f \u03c9, G t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n[PROOFSTEP]\nsimp_rw [eq\u2081, eq\u2082]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nG : \u211d \u2192 \u211d\nG_mble : Measurable G\nG_nn : 0 \u2264 G\ng_eq_G : g =\u1d50[Measure.restrict volume (Ioi 0)] G\ng_eq_G_on : \u2200 (t : \u211d), g =\u1d50[Measure.restrict volume (Ioc 0 t)] G\nG_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t\neq\u2081 :\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (G t)\neq\u2082 : \u2200 (\u03c9 : \u03b1), \u222b (t : \u211d) in 0 ..f \u03c9, g t = \u222b (t : \u211d) in 0 ..f \u03c9, G t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, G t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (G t)\n[PROOFSTEP]\nexact lintegral_comp_eq_lintegral_meas_le_mul_of_measurable \u03bc f_nn f_mble G_intble G_mble fun t _ => G_nn t\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a}\n[PROOFSTEP]\nset cst := fun _ : \u211d => (1 : \u211d)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ncst : \u211d \u2192 \u211d := fun x => 1\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a}\n[PROOFSTEP]\nhave cst_intble : \u2200 t > 0, IntervalIntegrable cst volume 0 t := fun _ _ => intervalIntegrable_const\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ncst : \u211d \u2192 \u211d := fun x => 1\ncst_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable cst volume 0 t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a}\n[PROOFSTEP]\nhave key := lintegral_comp_eq_lintegral_meas_le_mul \u03bc f_nn f_mble cst_intble (eventually_of_forall fun _ => zero_le_one)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ncst : \u211d \u2192 \u211d := fun x => 1\ncst_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable cst volume 0 t\nkey :\n \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, cst t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (cst t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a}\n[PROOFSTEP]\nsimp_rw [ENNReal.ofReal_one, mul_one] at key \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ncst : \u211d \u2192 \u211d := fun x => 1\ncst_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable cst volume 0 t\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, 1) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a}\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a}\n[PROOFSTEP]\nrw [\u2190 key]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ncst : \u211d \u2192 \u211d := fun x => 1\ncst_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable cst volume 0 t\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, 1) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a}\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, 1) \u2202\u03bc\n[PROOFSTEP]\ncongr with \u03c9\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ncst : \u211d \u2192 \u211d := fun x => 1\ncst_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable cst volume 0 t\nkey : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, 1) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a}\n\u03c9 : \u03b1\n\u22a2 ENNReal.ofReal (f \u03c9) = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, 1)\n[PROOFSTEP]\nsimp only [intervalIntegral.integral_const, sub_zero, Algebra.id.smul_eq_mul, mul_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))\n[PROOFSTEP]\nhave one_lt_p : -1 < p - 1 := by linarith\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\n\u22a2 -1 < p - 1\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))\n[PROOFSTEP]\nhave obs : \u2200 x : \u211d, (\u222b t : \u211d in (0)..x, t ^ (p - 1)) = x ^ p / p :=\n by\n intro x\n rw [integral_rpow (Or.inl one_lt_p)]\n simp [Real.zero_rpow p_pos.ne.symm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\n\u22a2 \u2200 (x : \u211d), \u222b (t : \u211d) in 0 ..x, t ^ (p - 1) = x ^ p / p\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\nx : \u211d\n\u22a2 \u222b (t : \u211d) in 0 ..x, t ^ (p - 1) = x ^ p / p\n[PROOFSTEP]\nrw [integral_rpow (Or.inl one_lt_p)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\nx : \u211d\n\u22a2 (x ^ (p - 1 + 1) - 0 ^ (p - 1 + 1)) / (p - 1 + 1) = x ^ p / p\n[PROOFSTEP]\nsimp [Real.zero_rpow p_pos.ne.symm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\nobs : \u2200 (x : \u211d), \u222b (t : \u211d) in 0 ..x, t ^ (p - 1) = x ^ p / p\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))\n[PROOFSTEP]\nset g := fun t : \u211d => t ^ (p - 1)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))\n[PROOFSTEP]\nhave g_nn : \u2200\u1d50 t \u2202volume.restrict (Ioi (0 : \u211d)), 0 \u2264 g t :=\n by\n filter_upwards [self_mem_ae_restrict (measurableSet_Ioi : MeasurableSet (Ioi (0 : \u211d)))]\n intro t t_pos\n exact Real.rpow_nonneg_of_nonneg (mem_Ioi.mp t_pos).le (p - 1)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\n\u22a2 \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\n[PROOFSTEP]\nfilter_upwards [self_mem_ae_restrict (measurableSet_Ioi : MeasurableSet (Ioi (0 : \u211d)))]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 0 \u2264 a ^ (p - 1)\n[PROOFSTEP]\nintro t t_pos\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\nt : \u211d\nt_pos : t \u2208 Ioi 0\n\u22a2 0 \u2264 t ^ (p - 1)\n[PROOFSTEP]\nexact Real.rpow_nonneg_of_nonneg (mem_Ioi.mp t_pos).le (p - 1)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))\n[PROOFSTEP]\nhave g_intble : \u2200 t > 0, IntervalIntegrable g volume 0 t := fun _ _ =>\n intervalIntegral.intervalIntegrable_rpow' one_lt_p\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))\n[PROOFSTEP]\nhave key := lintegral_comp_eq_lintegral_meas_le_mul \u03bc f_nn f_mble g_intble g_nn\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1))\n[PROOFSTEP]\nrw [\u2190 key, \u2190 lintegral_const_mul (ENNReal.ofReal p)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc = \u222b\u207b (a : \u03b1), ENNReal.ofReal p * ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f a, g t) \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [obs]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 Measurable fun \u03c9 => ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t)\n[PROOFSTEP]\nsimp_rw [obs]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc = \u222b\u207b (a : \u03b1), ENNReal.ofReal p * ENNReal.ofReal (f a ^ p / p) \u2202\u03bc\n[PROOFSTEP]\ncongr with \u03c9\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u03c9 : \u03b1\n\u22a2 ENNReal.ofReal (f \u03c9 ^ p) = ENNReal.ofReal p * ENNReal.ofReal (f \u03c9 ^ p / p)\n[PROOFSTEP]\nrw [\u2190 ENNReal.ofReal_mul p_pos.le, mul_div_cancel' (f \u03c9 ^ p) p_pos.ne.symm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\ng\u271d : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\none_lt_p : -1 < p - 1\ng : \u211d \u2192 \u211d := fun t => t ^ (p - 1)\nobs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\nkey :\n \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)\n\u22a2 Measurable fun \u03c9 => ENNReal.ofReal (f \u03c9 ^ p / p)\n[PROOFSTEP]\nexact ((f_mble.pow measurable_const).div_const p).ennreal_ofReal\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\n\u22a2 0 < \u2191\u2191\u03bc {a | g a = t}\n[PROOFSTEP]\nhave uni : {a : \u03b1 | t \u2264 g a} = {a : \u03b1 | t < g a} \u222a {a : \u03b1 | t = g a} :=\n by\n ext a\n simp only [mem_setOf, mem_union]\n apply le_iff_lt_or_eq\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\n\u22a2 {a | t \u2264 g a} = {a | t < g a} \u222a {a | t = g a}\n[PROOFSTEP]\next a\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\na : \u03b1\n\u22a2 a \u2208 {a | t \u2264 g a} \u2194 a \u2208 {a | t < g a} \u222a {a | t = g a}\n[PROOFSTEP]\nsimp only [mem_setOf, mem_union]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\na : \u03b1\n\u22a2 t \u2264 g a \u2194 t < g a \u2228 t = g a\n[PROOFSTEP]\napply le_iff_lt_or_eq\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\nuni : {a | t \u2264 g a} = {a | t < g a} \u222a {a | t = g a}\n\u22a2 0 < \u2191\u2191\u03bc {a | g a = t}\n[PROOFSTEP]\nrw [show {a : \u03b1 | t = g a} = {a : \u03b1 | g a = t} by simp_rw [eq_comm]] at uni \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\nuni : {a | t \u2264 g a} = {a | t < g a} \u222a {a | t = g a}\n\u22a2 {a | t = g a} = {a | g a = t}\n[PROOFSTEP]\nsimp_rw [eq_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\nuni : {a | t \u2264 g a} = {a | t < g a} \u222a {a | g a = t}\n\u22a2 0 < \u2191\u2191\u03bc {a | g a = t}\n[PROOFSTEP]\nhave disj : {a : \u03b1 | t < g a} \u2229 {a : \u03b1 | g a = t} = \u2205 := by\n ext a\n simp only [mem_inter_iff, mem_setOf, mem_empty_iff_false, iff_false_iff, not_and]\n exact ne_of_gt\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\nuni : {a | t \u2264 g a} = {a | t < g a} \u222a {a | g a = t}\n\u22a2 {a | t < g a} \u2229 {a | g a = t} = \u2205\n[PROOFSTEP]\next a\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\nuni : {a | t \u2264 g a} = {a | t < g a} \u222a {a | g a = t}\na : \u03b1\n\u22a2 a \u2208 {a | t < g a} \u2229 {a | g a = t} \u2194 a \u2208 \u2205\n[PROOFSTEP]\nsimp only [mem_inter_iff, mem_setOf, mem_empty_iff_false, iff_false_iff, not_and]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\nuni : {a | t \u2264 g a} = {a | t < g a} \u222a {a | g a = t}\na : \u03b1\n\u22a2 t < g a \u2192 \u00acg a = t\n[PROOFSTEP]\nexact ne_of_gt\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\nuni : {a | t \u2264 g a} = {a | t < g a} \u222a {a | g a = t}\ndisj : {a | t < g a} \u2229 {a | g a = t} = \u2205\n\u22a2 0 < \u2191\u2191\u03bc {a | g a = t}\n[PROOFSTEP]\nhave \u03bc_add : \u03bc {a : \u03b1 | t \u2264 g a} = \u03bc {a : \u03b1 | t < g a} + \u03bc {a : \u03b1 | g a = t} := by\n rw [uni, measure_union (disjoint_iff_inter_eq_empty.mpr disj) (g_mble (Finite.measurableSet (finite_singleton t)))]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\nuni : {a | t \u2264 g a} = {a | t < g a} \u222a {a | g a = t}\ndisj : {a | t < g a} \u2229 {a | g a = t} = \u2205\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 g a} = \u2191\u2191\u03bc {a | t < g a} + \u2191\u2191\u03bc {a | g a = t}\n[PROOFSTEP]\nrw [uni, measure_union (disjoint_iff_inter_eq_empty.mpr disj) (g_mble (Finite.measurableSet (finite_singleton t)))]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\nuni : {a | t \u2264 g a} = {a | t < g a} \u222a {a | g a = t}\ndisj : {a | t < g a} \u2229 {a | g a = t} = \u2205\n\u03bc_add : \u2191\u2191\u03bc {a | t \u2264 g a} = \u2191\u2191\u03bc {a | t < g a} + \u2191\u2191\u03bc {a | g a = t}\n\u22a2 0 < \u2191\u2191\u03bc {a | g a = t}\n[PROOFSTEP]\nby_contra con\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\nuni : {a | t \u2264 g a} = {a | t < g a} \u222a {a | g a = t}\ndisj : {a | t < g a} \u2229 {a | g a = t} = \u2205\n\u03bc_add : \u2191\u2191\u03bc {a | t \u2264 g a} = \u2191\u2191\u03bc {a | t < g a} + \u2191\u2191\u03bc {a | g a = t}\ncon : \u00ac0 < \u2191\u2191\u03bc {a | g a = t}\n\u22a2 False\n[PROOFSTEP]\nrw [not_lt, nonpos_iff_eq_zero] at con \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\nuni : {a | t \u2264 g a} = {a | t < g a} \u222a {a | g a = t}\ndisj : {a | t < g a} \u2229 {a | g a = t} = \u2205\n\u03bc_add : \u2191\u2191\u03bc {a | t \u2264 g a} = \u2191\u2191\u03bc {a | t < g a} + \u2191\u2191\u03bc {a | g a = t}\ncon : \u2191\u2191\u03bc {a | g a = t} = 0\n\u22a2 False\n[PROOFSTEP]\nrw [con, add_zero] at \u03bc_add \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b2\nR : Type u_3\ninst\u271d\u00b2 : LinearOrder R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : MeasurableSingletonClass R\ng : \u03b1 \u2192 R\ng_mble : Measurable g\nt : R\nht : \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}\nuni : {a | t \u2264 g a} = {a | t < g a} \u222a {a | g a = t}\ndisj : {a | t < g a} \u2229 {a | g a = t} = \u2205\n\u03bc_add : \u2191\u2191\u03bc {a | t \u2264 g a} = \u2191\u2191\u03bc {a | t < g a}\ncon : \u2191\u2191\u03bc {a | g a = t} = 0\n\u22a2 False\n[PROOFSTEP]\nexact ht \u03bc_add\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (g t)\n[PROOFSTEP]\nrw [lintegral_comp_eq_lintegral_meas_le_mul \u03bc f_nn f_mble g_intble g_nn]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (g t)\n[PROOFSTEP]\napply lintegral_congr_ae\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\n\u22a2 (fun a => \u2191\u2191\u03bc {a_1 | a \u2264 f a_1} * ENNReal.ofReal (g a)) =\u1d50[Measure.restrict volume (Ioi 0)] fun a =>\n \u2191\u2191\u03bc {a_1 | a < f a_1} * ENNReal.ofReal (g a)\n[PROOFSTEP]\nfilter_upwards [Measure.meas_le_ae_eq_meas_lt \u03bc (volume.restrict (Ioi 0)) f_mble] with t ht\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\ng_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t\ng_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t\nt : \u211d\nht : \u2191\u2191\u03bc {a | t \u2264 f a} = \u2191\u2191\u03bc {a | t < f a}\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (g t)\n[PROOFSTEP]\nrw [ht]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\n[PROOFSTEP]\nrw [lintegral_eq_lintegral_meas_le \u03bc f_nn f_mble]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}\n[PROOFSTEP]\napply lintegral_congr_ae\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\n\u22a2 (fun a => \u2191\u2191\u03bc {a_1 | a \u2264 f a_1}) =\u1d50[Measure.restrict volume (Ioi 0)] fun a => \u2191\u2191\u03bc {a_1 | a < f a_1}\n[PROOFSTEP]\nfilter_upwards [Measure.meas_le_ae_eq_meas_lt \u03bc (volume.restrict (Ioi 0)) f_mble] with t ht\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\nt : \u211d\nht : \u2191\u2191\u03bc {a | t \u2264 f a} = \u2191\u2191\u03bc {a | t < f a}\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} = \u2191\u2191\u03bc {a | t < f a}\n[PROOFSTEP]\nrw [ht]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\n\u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =\n ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))\n[PROOFSTEP]\nrw [lintegral_rpow_eq_lintegral_meas_le_mul \u03bc f_nn f_mble p_pos]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\n\u22a2 ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) =\n ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))\n[PROOFSTEP]\napply congr_arg fun z => ENNReal.ofReal p * z\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) =\n \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))\n[PROOFSTEP]\napply lintegral_congr_ae\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\n\u22a2 (fun a => \u2191\u2191\u03bc {a_1 | a \u2264 f a_1} * ENNReal.ofReal (a ^ (p - 1))) =\u1d50[Measure.restrict volume (Ioi 0)] fun a =>\n \u2191\u2191\u03bc {a_1 | a < f a_1} * ENNReal.ofReal (a ^ (p - 1))\n[PROOFSTEP]\nfilter_upwards [Measure.meas_le_ae_eq_meas_lt \u03bc (volume.restrict (Ioi 0)) f_mble] with t ht\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\np : \u211d\np_pos : 0 < p\nt : \u211d\nht : \u2191\u2191\u03bc {a | t \u2264 f a} = \u2191\u2191\u03bc {a | t < f a}\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) = \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))\n[PROOFSTEP]\nrw [ht]\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Integral.Layercake", "llama_tokens": 48907, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124812, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.4432500335799584}} {"text": "[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\n\u22a2 IsTopologicalBasis (insert \u2205 s)\n[PROOFSTEP]\nrefine' \u27e8_, by rw [sUnion_insert, empty_union, h.sUnion_eq], _\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\n\u22a2 \u22c3\u2080 insert \u2205 s = univ\n[PROOFSTEP]\nrw [sUnion_insert, empty_union, h.sUnion_eq]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\n\u22a2 \u2200 (t\u2081 : Set \u03b1),\n t\u2081 \u2208 insert \u2205 s \u2192\n \u2200 (t\u2082 : Set \u03b1), t\u2082 \u2208 insert \u2205 s \u2192 \u2200 (x : \u03b1), x \u2208 t\u2081 \u2229 t\u2082 \u2192 \u2203 t\u2083, t\u2083 \u2208 insert \u2205 s \u2227 x \u2208 t\u2083 \u2227 t\u2083 \u2286 t\u2081 \u2229 t\u2082\n[PROOFSTEP]\nrintro t\u2081 (rfl | h\u2081) t\u2082 (rfl | h\u2082) x \u27e8hx\u2081, hx\u2082\u27e9\n[GOAL]\ncase refine'_1.inl.inl.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\nx : \u03b1\nhx\u2081 hx\u2082 : x \u2208 \u2205\n\u22a2 \u2203 t\u2083, t\u2083 \u2208 insert \u2205 s \u2227 x \u2208 t\u2083 \u2227 t\u2083 \u2286 \u2205 \u2229 \u2205\n[PROOFSTEP]\ncases hx\u2081\n[GOAL]\ncase refine'_1.inl.inr.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\nt\u2082 : Set \u03b1\nh\u2082 : t\u2082 \u2208 s\nx : \u03b1\nhx\u2081 : x \u2208 \u2205\nhx\u2082 : x \u2208 t\u2082\n\u22a2 \u2203 t\u2083, t\u2083 \u2208 insert \u2205 s \u2227 x \u2208 t\u2083 \u2227 t\u2083 \u2286 \u2205 \u2229 t\u2082\n[PROOFSTEP]\ncases hx\u2081\n[GOAL]\ncase refine'_1.inr.inl.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\nt\u2081 : Set \u03b1\nh\u2081 : t\u2081 \u2208 s\nx : \u03b1\nhx\u2081 : x \u2208 t\u2081\nhx\u2082 : x \u2208 \u2205\n\u22a2 \u2203 t\u2083, t\u2083 \u2208 insert \u2205 s \u2227 x \u2208 t\u2083 \u2227 t\u2083 \u2286 t\u2081 \u2229 \u2205\n[PROOFSTEP]\ncases hx\u2082\n[GOAL]\ncase refine'_1.inr.inr.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\nt\u2081 : Set \u03b1\nh\u2081 : t\u2081 \u2208 s\nt\u2082 : Set \u03b1\nh\u2082 : t\u2082 \u2208 s\nx : \u03b1\nhx\u2081 : x \u2208 t\u2081\nhx\u2082 : x \u2208 t\u2082\n\u22a2 \u2203 t\u2083, t\u2083 \u2208 insert \u2205 s \u2227 x \u2208 t\u2083 \u2227 t\u2083 \u2286 t\u2081 \u2229 t\u2082\n[PROOFSTEP]\nobtain \u27e8t\u2083, h\u2083, hs\u27e9 := h.exists_subset_inter _ h\u2081 _ h\u2082 x \u27e8hx\u2081, hx\u2082\u27e9\n[GOAL]\ncase refine'_1.inr.inr.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\nt\u2081 : Set \u03b1\nh\u2081 : t\u2081 \u2208 s\nt\u2082 : Set \u03b1\nh\u2082 : t\u2082 \u2208 s\nx : \u03b1\nhx\u2081 : x \u2208 t\u2081\nhx\u2082 : x \u2208 t\u2082\nt\u2083 : Set \u03b1\nh\u2083 : t\u2083 \u2208 s\nhs : x \u2208 t\u2083 \u2227 t\u2083 \u2286 t\u2081 \u2229 t\u2082\n\u22a2 \u2203 t\u2083, t\u2083 \u2208 insert \u2205 s \u2227 x \u2208 t\u2083 \u2227 t\u2083 \u2286 t\u2081 \u2229 t\u2082\n[PROOFSTEP]\nexact \u27e8t\u2083, .inr h\u2083, hs\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\n\u22a2 t = generateFrom (insert \u2205 s)\n[PROOFSTEP]\nrw [h.eq_generateFrom]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\n\u22a2 generateFrom s = generateFrom (insert \u2205 s)\n[PROOFSTEP]\nrefine' le_antisymm (le_generateFrom fun t => _) (generateFrom_anti <| subset_insert \u2205 s)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\nt : Set \u03b1\n\u22a2 t \u2208 insert \u2205 s \u2192 IsOpen t\n[PROOFSTEP]\nrintro (rfl | ht)\n[GOAL]\ncase refine'_2.inl\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\n\u22a2 IsOpen \u2205\n[PROOFSTEP]\nexact @isOpen_empty _ (generateFrom s)\n[GOAL]\ncase refine'_2.inr\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\nt : Set \u03b1\nht : t \u2208 s\n\u22a2 IsOpen t\n[PROOFSTEP]\nexact .basic t ht\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\n\u22a2 IsTopologicalBasis (s \\ {\u2205})\n[PROOFSTEP]\nrefine' \u27e8_, by rw [sUnion_diff_singleton_empty, h.sUnion_eq], _\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\n\u22a2 \u22c3\u2080 (s \\ {\u2205}) = univ\n[PROOFSTEP]\nrw [sUnion_diff_singleton_empty, h.sUnion_eq]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\n\u22a2 \u2200 (t\u2081 : Set \u03b1),\n t\u2081 \u2208 s \\ {\u2205} \u2192 \u2200 (t\u2082 : Set \u03b1), t\u2082 \u2208 s \\ {\u2205} \u2192 \u2200 (x : \u03b1), x \u2208 t\u2081 \u2229 t\u2082 \u2192 \u2203 t\u2083, t\u2083 \u2208 s \\ {\u2205} \u2227 x \u2208 t\u2083 \u2227 t\u2083 \u2286 t\u2081 \u2229 t\u2082\n[PROOFSTEP]\nrintro t\u2081 \u27e8h\u2081, -\u27e9 t\u2082 \u27e8h\u2082, -\u27e9 x hx\n[GOAL]\ncase refine'_1.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\nt\u2081 : Set \u03b1\nh\u2081 : t\u2081 \u2208 s\nt\u2082 : Set \u03b1\nh\u2082 : t\u2082 \u2208 s\nx : \u03b1\nhx : x \u2208 t\u2081 \u2229 t\u2082\n\u22a2 \u2203 t\u2083, t\u2083 \u2208 s \\ {\u2205} \u2227 x \u2208 t\u2083 \u2227 t\u2083 \u2286 t\u2081 \u2229 t\u2082\n[PROOFSTEP]\nobtain \u27e8t\u2083, h\u2083, hs\u27e9 := h.exists_subset_inter _ h\u2081 _ h\u2082 x hx\n[GOAL]\ncase refine'_1.intro.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\nt\u2081 : Set \u03b1\nh\u2081 : t\u2081 \u2208 s\nt\u2082 : Set \u03b1\nh\u2082 : t\u2082 \u2208 s\nx : \u03b1\nhx : x \u2208 t\u2081 \u2229 t\u2082\nt\u2083 : Set \u03b1\nh\u2083 : t\u2083 \u2208 s\nhs : x \u2208 t\u2083 \u2227 t\u2083 \u2286 t\u2081 \u2229 t\u2082\n\u22a2 \u2203 t\u2083, t\u2083 \u2208 s \\ {\u2205} \u2227 x \u2208 t\u2083 \u2227 t\u2083 \u2286 t\u2081 \u2229 t\u2082\n[PROOFSTEP]\nexact \u27e8t\u2083, \u27e8h\u2083, Nonempty.ne_empty \u27e8x, hs.1\u27e9\u27e9, hs\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\n\u22a2 t = generateFrom (s \\ {\u2205})\n[PROOFSTEP]\nrw [h.eq_generateFrom]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\n\u22a2 generateFrom s = generateFrom (s \\ {\u2205})\n[PROOFSTEP]\nrefine' le_antisymm (generateFrom_anti <| diff_subset s _) (le_generateFrom fun t ht => _)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\nt : Set \u03b1\nht : t \u2208 s\n\u22a2 IsOpen t\n[PROOFSTEP]\nobtain rfl | he := eq_or_ne t \u2205\n[GOAL]\ncase refine'_2.inl\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\nht : \u2205 \u2208 s\n\u22a2 IsOpen \u2205\n[PROOFSTEP]\nexact @isOpen_empty _ (generateFrom _)\n[GOAL]\ncase refine'_2.inr\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh : IsTopologicalBasis s\nt : Set \u03b1\nht : t \u2208 s\nhe : t \u2260 \u2205\n\u22a2 IsOpen t\n[PROOFSTEP]\nexact .basic t \u27e8ht, he\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nhs : t = generateFrom s\n\u22a2 IsTopologicalBasis ((fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 s})\n[PROOFSTEP]\nsubst t\n[GOAL]\n\u03b1 : Type u\ns : Set (Set \u03b1)\n\u22a2 IsTopologicalBasis ((fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 s})\n[PROOFSTEP]\nletI := generateFrom s\n[GOAL]\n\u03b1 : Type u\ns : Set (Set \u03b1)\nthis : TopologicalSpace \u03b1 := generateFrom s\n\u22a2 IsTopologicalBasis ((fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 s})\n[PROOFSTEP]\nrefine' \u27e8_, _, le_antisymm (le_generateFrom _) <| generateFrom_anti fun t ht => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\ns : Set (Set \u03b1)\nthis : TopologicalSpace \u03b1 := generateFrom s\n\u22a2 \u2200 (t\u2081 : Set \u03b1),\n t\u2081 \u2208 (fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 s} \u2192\n \u2200 (t\u2082 : Set \u03b1),\n t\u2082 \u2208 (fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 s} \u2192\n \u2200 (x : \u03b1), x \u2208 t\u2081 \u2229 t\u2082 \u2192 \u2203 t\u2083, t\u2083 \u2208 (fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 s} \u2227 x \u2208 t\u2083 \u2227 t\u2083 \u2286 t\u2081 \u2229 t\u2082\n[PROOFSTEP]\nrintro _ \u27e8t\u2081, \u27e8hft\u2081, ht\u2081b\u27e9, rfl\u27e9 _ \u27e8t\u2082, \u27e8hft\u2082, ht\u2082b\u27e9, rfl\u27e9 x h\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\nthis : TopologicalSpace \u03b1 := generateFrom s\nt\u2081 : Set (Set \u03b1)\nhft\u2081 : Set.Finite t\u2081\nht\u2081b : t\u2081 \u2286 s\nt\u2082 : Set (Set \u03b1)\nhft\u2082 : Set.Finite t\u2082\nht\u2082b : t\u2082 \u2286 s\nx : \u03b1\nh : x \u2208 (fun f => \u22c2\u2080 f) t\u2081 \u2229 (fun f => \u22c2\u2080 f) t\u2082\n\u22a2 \u2203 t\u2083, t\u2083 \u2208 (fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 s} \u2227 x \u2208 t\u2083 \u2227 t\u2083 \u2286 (fun f => \u22c2\u2080 f) t\u2081 \u2229 (fun f => \u22c2\u2080 f) t\u2082\n[PROOFSTEP]\nexact \u27e8_, \u27e8_, \u27e8hft\u2081.union hft\u2082, union_subset ht\u2081b ht\u2082b\u27e9, sInter_union t\u2081 t\u2082\u27e9, h, Subset.rfl\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\ns : Set (Set \u03b1)\nthis : TopologicalSpace \u03b1 := generateFrom s\n\u22a2 \u22c3\u2080 ((fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 s}) = univ\n[PROOFSTEP]\nrw [sUnion_image, iUnion\u2082_eq_univ_iff]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\ns : Set (Set \u03b1)\nthis : TopologicalSpace \u03b1 := generateFrom s\n\u22a2 \u2200 (a : \u03b1), \u2203 i j, a \u2208 \u22c2\u2080 i\n[PROOFSTEP]\nexact fun x => \u27e8\u2205, \u27e8finite_empty, empty_subset _\u27e9, sInter_empty.substr <| mem_univ x\u27e9\n[GOAL]\ncase refine'_3\n\u03b1 : Type u\ns : Set (Set \u03b1)\nthis : TopologicalSpace \u03b1 := generateFrom s\n\u22a2 \u2200 (s_1 : Set \u03b1), s_1 \u2208 (fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 s} \u2192 IsOpen s_1\n[PROOFSTEP]\nrintro _ \u27e8t, \u27e8hft, htb\u27e9, rfl\u27e9\n[GOAL]\ncase refine'_3.intro.intro.intro\n\u03b1 : Type u\ns : Set (Set \u03b1)\nthis : TopologicalSpace \u03b1 := generateFrom s\nt : Set (Set \u03b1)\nhft : Set.Finite t\nhtb : t \u2286 s\n\u22a2 IsOpen ((fun f => \u22c2\u2080 f) t)\n[PROOFSTEP]\napply isOpen_sInter\n[GOAL]\ncase refine'_3.intro.intro.intro.hs\n\u03b1 : Type u\ns : Set (Set \u03b1)\nthis : TopologicalSpace \u03b1 := generateFrom s\nt : Set (Set \u03b1)\nhft : Set.Finite t\nhtb : t \u2286 s\n\u22a2 Set.Finite t\ncase refine'_3.intro.intro.intro.a\n\u03b1 : Type u\ns : Set (Set \u03b1)\nthis : TopologicalSpace \u03b1 := generateFrom s\nt : Set (Set \u03b1)\nhft : Set.Finite t\nhtb : t \u2286 s\n\u22a2 \u2200 (t_1 : Set \u03b1), t_1 \u2208 t \u2192 IsOpen t_1\n[PROOFSTEP]\nexacts [hft, fun s hs => GenerateOpen.basic _ <| htb hs]\n[GOAL]\ncase refine'_4\n\u03b1 : Type u\ns : Set (Set \u03b1)\nthis : TopologicalSpace \u03b1 := generateFrom s\nt : Set \u03b1\nht : t \u2208 s\n\u22a2 t \u2208 (fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 s}\n[PROOFSTEP]\nrw [\u2190 sInter_singleton t]\n[GOAL]\ncase refine'_4\n\u03b1 : Type u\ns : Set (Set \u03b1)\nthis : TopologicalSpace \u03b1 := generateFrom s\nt : Set \u03b1\nht : t \u2208 s\n\u22a2 \u22c2\u2080 {t} \u2208 (fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 s}\n[PROOFSTEP]\nexact \u27e8{ t }, \u27e8finite_singleton t, singleton_subset_iff.2 ht\u27e9, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh_open : \u2200 (u : Set \u03b1), u \u2208 s \u2192 IsOpen u\nh_nhds : \u2200 (a : \u03b1) (u : Set \u03b1), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 s \u2227 a \u2208 v \u2227 v \u2286 u\n\u22a2 IsTopologicalBasis s\n[PROOFSTEP]\nrefine' \u27e8fun t\u2081 ht\u2081 t\u2082 ht\u2082 x hx => h_nhds _ _ hx (IsOpen.inter (h_open _ ht\u2081) (h_open _ ht\u2082)), _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh_open : \u2200 (u : Set \u03b1), u \u2208 s \u2192 IsOpen u\nh_nhds : \u2200 (a : \u03b1) (u : Set \u03b1), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 s \u2227 a \u2208 v \u2227 v \u2286 u\n\u22a2 \u22c3\u2080 s = univ\n[PROOFSTEP]\nrefine' sUnion_eq_univ_iff.2 fun a => _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh_open : \u2200 (u : Set \u03b1), u \u2208 s \u2192 IsOpen u\nh_nhds : \u2200 (a : \u03b1) (u : Set \u03b1), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 s \u2227 a \u2208 v \u2227 v \u2286 u\na : \u03b1\n\u22a2 \u2203 b, b \u2208 s \u2227 a \u2208 b\n[PROOFSTEP]\nrcases h_nhds a univ trivial isOpen_univ with \u27e8u, h\u2081, h\u2082, -\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh_open : \u2200 (u : Set \u03b1), u \u2208 s \u2192 IsOpen u\nh_nhds : \u2200 (a : \u03b1) (u : Set \u03b1), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 s \u2227 a \u2208 v \u2227 v \u2286 u\na : \u03b1\nu : Set \u03b1\nh\u2081 : u \u2208 s\nh\u2082 : a \u2208 u\n\u22a2 \u2203 b, b \u2208 s \u2227 a \u2208 b\n[PROOFSTEP]\nexact \u27e8u, h\u2081, h\u2082\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh_open : \u2200 (u : Set \u03b1), u \u2208 s \u2192 IsOpen u\nh_nhds : \u2200 (a : \u03b1) (u : Set \u03b1), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 s \u2227 a \u2208 v \u2227 v \u2286 u\n\u22a2 t = generateFrom s\n[PROOFSTEP]\nrefine' (le_generateFrom h_open).antisymm fun u hu => _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh_open : \u2200 (u : Set \u03b1), u \u2208 s \u2192 IsOpen u\nh_nhds : \u2200 (a : \u03b1) (u : Set \u03b1), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 s \u2227 a \u2208 v \u2227 v \u2286 u\nu : Set \u03b1\nhu : IsOpen u\n\u22a2 IsOpen u\n[PROOFSTEP]\nrefine' (@isOpen_iff_nhds \u03b1 (generateFrom s) u).mpr fun a ha => _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh_open : \u2200 (u : Set \u03b1), u \u2208 s \u2192 IsOpen u\nh_nhds : \u2200 (a : \u03b1) (u : Set \u03b1), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 s \u2227 a \u2208 v \u2227 v \u2286 u\nu : Set \u03b1\nhu : IsOpen u\na : \u03b1\nha : a \u2208 u\n\u22a2 \ud835\udcdd a \u2264 \ud835\udcdf u\n[PROOFSTEP]\nrcases h_nhds a u ha hu with \u27e8v, hvs, hav, hvu\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh_open : \u2200 (u : Set \u03b1), u \u2208 s \u2192 IsOpen u\nh_nhds : \u2200 (a : \u03b1) (u : Set \u03b1), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 s \u2227 a \u2208 v \u2227 v \u2286 u\nu : Set \u03b1\nhu : IsOpen u\na : \u03b1\nha : a \u2208 u\nv : Set \u03b1\nhvs : v \u2208 s\nhav : a \u2208 v\nhvu : v \u2286 u\n\u22a2 \ud835\udcdd a \u2264 \ud835\udcdf u\n[PROOFSTEP]\nrw [nhds_generateFrom]\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set (Set \u03b1)\nh_open : \u2200 (u : Set \u03b1), u \u2208 s \u2192 IsOpen u\nh_nhds : \u2200 (a : \u03b1) (u : Set \u03b1), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 s \u2227 a \u2208 v \u2227 v \u2286 u\nu : Set \u03b1\nhu : IsOpen u\na : \u03b1\nha : a \u2208 u\nv : Set \u03b1\nhvs : v \u2208 s\nhav : a \u2208 v\nhvu : v \u2286 u\n\u22a2 \u2a05 (s_1 : Set \u03b1) (_ : s_1 \u2208 {s_2 | a \u2208 s_2 \u2227 s_2 \u2208 s}), \ud835\udcdf s_1 \u2264 \ud835\udcdf u\n[PROOFSTEP]\nexact iInf\u2082_le_of_le v \u27e8hav, hvs\u27e9 (le_principal_iff.2 hvu)\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\na : \u03b1\ns : Set \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\n\u22a2 s \u2208 \ud835\udcdd a \u2194 \u2203 t, t \u2208 b \u2227 a \u2208 t \u2227 t \u2286 s\n[PROOFSTEP]\nchange s \u2208 (\ud835\udcdd a).sets \u2194 \u2203 t \u2208 b, a \u2208 t \u2227 t \u2286 s\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\na : \u03b1\ns : Set \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\n\u22a2 s \u2208 (\ud835\udcdd a).sets \u2194 \u2203 t, t \u2208 b \u2227 a \u2208 t \u2227 t \u2286 s\n[PROOFSTEP]\nrw [hb.eq_generateFrom, nhds_generateFrom, biInf_sets_eq]\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\na : \u03b1\ns : Set \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\n\u22a2 s \u2208 \u22c3 (i : Set \u03b1) (_ : i \u2208 {s | a \u2208 s \u2227 s \u2208 b}), (\ud835\udcdf i).sets \u2194 \u2203 t, t \u2208 b \u2227 a \u2208 t \u2227 t \u2286 s\n[PROOFSTEP]\nsimp [and_assoc, and_left_comm]\n[GOAL]\ncase h\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\na : \u03b1\ns : Set \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\n\u22a2 DirectedOn ((fun s => \ud835\udcdf s) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) {s | a \u2208 s \u2227 s \u2208 b}\n[PROOFSTEP]\nrintro s \u27e8hs\u2081, hs\u2082\u27e9 t \u27e8ht\u2081, ht\u2082\u27e9\n[GOAL]\ncase h.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\na : \u03b1\ns\u271d : Set \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\ns : Set \u03b1\nhs\u2081 : a \u2208 s\nhs\u2082 : s \u2208 b\nt : Set \u03b1\nht\u2081 : a \u2208 t\nht\u2082 : t \u2208 b\n\u22a2 \u2203 z,\n z \u2208 {s | a \u2208 s \u2227 s \u2208 b} \u2227\n ((fun s => \ud835\udcdf s) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) s z \u2227 ((fun s => \ud835\udcdf s) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) t z\n[PROOFSTEP]\nlet \u27e8u, hu\u2081, hu\u2082, hu\u2083\u27e9 := hb.1 _ hs\u2082 _ ht\u2082 _ \u27e8hs\u2081, ht\u2081\u27e9\n[GOAL]\ncase h.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\na : \u03b1\ns\u271d : Set \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\ns : Set \u03b1\nhs\u2081 : a \u2208 s\nhs\u2082 : s \u2208 b\nt : Set \u03b1\nht\u2081 : a \u2208 t\nht\u2082 : t \u2208 b\nu : Set \u03b1\nhu\u2081 : u \u2208 b\nhu\u2082 : a \u2208 u\nhu\u2083 : u \u2286 s \u2229 t\n\u22a2 \u2203 z,\n z \u2208 {s | a \u2208 s \u2227 s \u2208 b} \u2227\n ((fun s => \ud835\udcdf s) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) s z \u2227 ((fun s => \ud835\udcdf s) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) t z\n[PROOFSTEP]\nexact\n \u27e8u, \u27e8hu\u2082, hu\u2081\u27e9, le_principal_iff.2 (hu\u2083.trans (inter_subset_left _ _)),\n le_principal_iff.2 (hu\u2083.trans (inter_subset_right _ _))\u27e9\n[GOAL]\ncase ne\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\na : \u03b1\ns : Set \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\n\u22a2 Set.Nonempty {s | a \u2208 s \u2227 s \u2208 b}\n[PROOFSTEP]\nrcases eq_univ_iff_forall.1 hb.sUnion_eq a with \u27e8i, h1, h2\u27e9\n[GOAL]\ncase ne.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\na : \u03b1\ns : Set \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\ni : Set \u03b1\nh1 : i \u2208 b\nh2 : a \u2208 i\n\u22a2 Set.Nonempty {s | a \u2208 s \u2227 s \u2208 b}\n[PROOFSTEP]\nexact \u27e8i, h2, h1\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\n\u22a2 IsOpen s \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 t, t \u2208 b \u2227 a \u2208 t \u2227 t \u2286 s\n[PROOFSTEP]\nsimp [isOpen_iff_mem_nhds, hb.mem_nhds_iff]\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\na : \u03b1\ns : Set \u03b1\n\u22a2 (\u2203 t, t \u2208 b \u2227 a \u2208 t \u2227 t \u2286 s) \u2194 \u2203 i, (i \u2208 b \u2227 a \u2208 i) \u2227 i \u2286 s\n[PROOFSTEP]\nsimp only [and_assoc]\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\nhs : s \u2208 b\n\u22a2 IsOpen s\n[PROOFSTEP]\nrw [hb.eq_generateFrom]\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\nhs : s \u2208 b\n\u22a2 IsOpen s\n[PROOFSTEP]\nexact .basic s hs\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nB : Set (Set \u03b1)\nhB : IsTopologicalBasis B\nu : Set \u03b1\nou : IsOpen u\n\u22a2 u = \u22c3 (i : \u2191{s | s \u2208 B \u2227 s \u2286 u}), \u2191i\n[PROOFSTEP]\nrw [\u2190 sUnion_eq_iUnion]\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nB : Set (Set \u03b1)\nhB : IsTopologicalBasis B\nu : Set \u03b1\nou : IsOpen u\n\u22a2 u = \u22c3\u2080 {s | s \u2208 B \u2227 s \u2286 u}\n[PROOFSTEP]\napply hB.open_eq_sUnion' ou\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\ns : Set \u03b1\na : \u03b1\n\u22a2 (\u2200 (i : Set \u03b1), i \u2208 b \u2227 a \u2208 i \u2192 Set.Nonempty (i \u2229 s)) \u2194 \u2200 (o : Set \u03b1), o \u2208 b \u2192 a \u2208 o \u2192 Set.Nonempty (o \u2229 s)\n[PROOFSTEP]\nsimp only [and_imp]\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\ns : Set \u03b1\n\u22a2 Dense s \u2194 \u2200 (o : Set \u03b1), o \u2208 b \u2192 Set.Nonempty o \u2192 Set.Nonempty (o \u2229 s)\n[PROOFSTEP]\nsimp only [Dense, hb.mem_closure_iff]\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\ns : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1) (o : Set \u03b1), o \u2208 b \u2192 x \u2208 o \u2192 Set.Nonempty (o \u2229 s)) \u2194\n \u2200 (o : Set \u03b1), o \u2208 b \u2192 Set.Nonempty o \u2192 Set.Nonempty (o \u2229 s)\n[PROOFSTEP]\nexact \u27e8fun h o hb \u27e8a, ha\u27e9 => h a o hb ha, fun h a o hb ha => h o hb \u27e8a, ha\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nB : Set (Set \u03b1)\nhB : IsTopologicalBasis B\nf : \u03b1 \u2192 \u03b2\n\u22a2 IsOpenMap f \u2194 \u2200 (s : Set \u03b1), s \u2208 B \u2192 IsOpen (f '' s)\n[PROOFSTEP]\nrefine' \u27e8fun H o ho => H _ (hB.isOpen ho), fun hf o ho => _\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nB : Set (Set \u03b1)\nhB : IsTopologicalBasis B\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (s : Set \u03b1), s \u2208 B \u2192 IsOpen (f '' s)\no : Set \u03b1\nho : IsOpen o\n\u22a2 IsOpen (f '' o)\n[PROOFSTEP]\nrw [hB.open_eq_sUnion' ho, sUnion_eq_iUnion, image_iUnion]\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nB : Set (Set \u03b1)\nhB : IsTopologicalBasis B\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (s : Set \u03b1), s \u2208 B \u2192 IsOpen (f '' s)\no : Set \u03b1\nho : IsOpen o\n\u22a2 IsOpen (\u22c3 (i : \u2191{s | s \u2208 B \u2227 s \u2286 o}), f '' \u2191i)\n[PROOFSTEP]\nexact isOpen_iUnion fun s => hf s s.2.1\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u22a2 \u2200 (u : Set \u03b1), u \u2208 {U | IsOpen U} \u2192 IsOpen u\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u22a2 \u2200 (a : \u03b1) (u : Set \u03b1), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 {U | IsOpen U} \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nB\u2081 : Set (Set \u03b1)\nB\u2082 : Set (Set \u03b2)\nh\u2081 : IsTopologicalBasis B\u2081\nh\u2082 : IsTopologicalBasis B\u2082\n\u22a2 IsTopologicalBasis (image2 (fun x x_1 => x \u00d7\u02e2 x_1) B\u2081 B\u2082)\n[PROOFSTEP]\nrefine' isTopologicalBasis_of_open_of_nhds _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nB\u2081 : Set (Set \u03b1)\nB\u2082 : Set (Set \u03b2)\nh\u2081 : IsTopologicalBasis B\u2081\nh\u2082 : IsTopologicalBasis B\u2082\n\u22a2 \u2200 (u : Set (\u03b1 \u00d7 \u03b2)), u \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) B\u2081 B\u2082 \u2192 IsOpen u\n[PROOFSTEP]\nrintro _ \u27e8u\u2081, u\u2082, hu\u2081, hu\u2082, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nB\u2081 : Set (Set \u03b1)\nB\u2082 : Set (Set \u03b2)\nh\u2081 : IsTopologicalBasis B\u2081\nh\u2082 : IsTopologicalBasis B\u2082\nu\u2081 : Set \u03b1\nu\u2082 : Set \u03b2\nhu\u2081 : u\u2081 \u2208 B\u2081\nhu\u2082 : u\u2082 \u2208 B\u2082\n\u22a2 IsOpen ((fun x x_1 => x \u00d7\u02e2 x_1) u\u2081 u\u2082)\n[PROOFSTEP]\nexact (h\u2081.isOpen hu\u2081).prod (h\u2082.isOpen hu\u2082)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nB\u2081 : Set (Set \u03b1)\nB\u2082 : Set (Set \u03b2)\nh\u2081 : IsTopologicalBasis B\u2081\nh\u2082 : IsTopologicalBasis B\u2082\n\u22a2 \u2200 (a : \u03b1 \u00d7 \u03b2) (u : Set (\u03b1 \u00d7 \u03b2)), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) B\u2081 B\u2082 \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nrintro \u27e8a, b\u27e9 u hu uo\n[GOAL]\ncase refine'_2.mk\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nB\u2081 : Set (Set \u03b1)\nB\u2082 : Set (Set \u03b2)\nh\u2081 : IsTopologicalBasis B\u2081\nh\u2082 : IsTopologicalBasis B\u2082\na : \u03b1\nb : \u03b2\nu : Set (\u03b1 \u00d7 \u03b2)\nhu : (a, b) \u2208 u\nuo : IsOpen u\n\u22a2 \u2203 v, v \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) B\u2081 B\u2082 \u2227 (a, b) \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nrcases(h\u2081.nhds_hasBasis.prod_nhds h\u2082.nhds_hasBasis).mem_iff.1 (IsOpen.mem_nhds uo hu) with\n \u27e8\u27e8s, t\u27e9, \u27e8\u27e8hs, ha\u27e9, ht, hb\u27e9, hu\u27e9\n[GOAL]\ncase refine'_2.mk.intro.mk.intro.intro.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nB\u2081 : Set (Set \u03b1)\nB\u2082 : Set (Set \u03b2)\nh\u2081 : IsTopologicalBasis B\u2081\nh\u2082 : IsTopologicalBasis B\u2082\na : \u03b1\nb : \u03b2\nu : Set (\u03b1 \u00d7 \u03b2)\nhu\u271d : (a, b) \u2208 u\nuo : IsOpen u\ns : Set \u03b1\nt : Set \u03b2\nhu : (s, t).fst \u00d7\u02e2 (s, t).snd \u2286 u\nhs : (s, t).fst \u2208 B\u2081\nha : a \u2208 (s, t).fst\nht : (s, t).snd \u2208 B\u2082\nhb : b \u2208 (s, t).snd\n\u22a2 \u2203 v, v \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) B\u2081 B\u2082 \u2227 (a, b) \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nexact \u27e8s \u00d7\u02e2 t, mem_image2_of_mem hs ht, \u27e8ha, hb\u27e9, hu\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nT : Set (Set \u03b2)\nhf : Inducing f\nh : IsTopologicalBasis T\n\u22a2 IsTopologicalBasis (preimage f '' T)\n[PROOFSTEP]\nrefine' isTopologicalBasis_of_open_of_nhds _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nT : Set (Set \u03b2)\nhf : Inducing f\nh : IsTopologicalBasis T\n\u22a2 \u2200 (u : Set \u03b1), u \u2208 preimage f '' T \u2192 IsOpen u\n[PROOFSTEP]\nrintro _ \u27e8V, hV, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nT : Set (Set \u03b2)\nhf : Inducing f\nh : IsTopologicalBasis T\nV : Set \u03b2\nhV : V \u2208 T\n\u22a2 IsOpen (f \u207b\u00b9' V)\n[PROOFSTEP]\nrw [hf.isOpen_iff]\n[GOAL]\ncase refine'_1.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nT : Set (Set \u03b2)\nhf : Inducing f\nh : IsTopologicalBasis T\nV : Set \u03b2\nhV : V \u2208 T\n\u22a2 \u2203 t, IsOpen t \u2227 f \u207b\u00b9' t = f \u207b\u00b9' V\n[PROOFSTEP]\nrefine' \u27e8V, h.isOpen hV, rfl\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nT : Set (Set \u03b2)\nhf : Inducing f\nh : IsTopologicalBasis T\n\u22a2 \u2200 (a : \u03b1) (u : Set \u03b1), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 preimage f '' T \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nintro a U ha hU\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nT : Set (Set \u03b2)\nhf : Inducing f\nh : IsTopologicalBasis T\na : \u03b1\nU : Set \u03b1\nha : a \u2208 U\nhU : IsOpen U\n\u22a2 \u2203 v, v \u2208 preimage f '' T \u2227 a \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nrw [hf.isOpen_iff] at hU \n[GOAL]\ncase refine'_2\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nT : Set (Set \u03b2)\nhf : Inducing f\nh : IsTopologicalBasis T\na : \u03b1\nU : Set \u03b1\nha : a \u2208 U\nhU : \u2203 t, IsOpen t \u2227 f \u207b\u00b9' t = U\n\u22a2 \u2203 v, v \u2208 preimage f '' T \u2227 a \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nobtain \u27e8V, hV, rfl\u27e9 := hU\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nT : Set (Set \u03b2)\nhf : Inducing f\nh : IsTopologicalBasis T\na : \u03b1\nV : Set \u03b2\nhV : IsOpen V\nha : a \u2208 f \u207b\u00b9' V\n\u22a2 \u2203 v, v \u2208 preimage f '' T \u2227 a \u2208 v \u2227 v \u2286 f \u207b\u00b9' V\n[PROOFSTEP]\nobtain \u27e8S, hS, rfl\u27e9 := h.open_eq_sUnion hV\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nT : Set (Set \u03b2)\nhf : Inducing f\nh : IsTopologicalBasis T\na : \u03b1\nS : Set (Set \u03b2)\nhS : S \u2286 T\nhV : IsOpen (\u22c3\u2080 S)\nha : a \u2208 f \u207b\u00b9' \u22c3\u2080 S\n\u22a2 \u2203 v, v \u2208 preimage f '' T \u2227 a \u2208 v \u2227 v \u2286 f \u207b\u00b9' \u22c3\u2080 S\n[PROOFSTEP]\nobtain \u27e8W, hW, ha\u27e9 := ha\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nT : Set (Set \u03b2)\nhf : Inducing f\nh : IsTopologicalBasis T\na : \u03b1\nS : Set (Set \u03b2)\nhS : S \u2286 T\nhV : IsOpen (\u22c3\u2080 S)\nW : Set \u03b2\nhW : W \u2208 S\nha : f a \u2208 W\n\u22a2 \u2203 v, v \u2208 preimage f '' T \u2227 a \u2208 v \u2227 v \u2286 f \u207b\u00b9' \u22c3\u2080 S\n[PROOFSTEP]\nrefine' \u27e8f \u207b\u00b9' W, \u27e8_, hS hW, rfl\u27e9, ha, Set.preimage_mono <| Set.subset_sUnion_of_mem hW\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Sort u_1\nU : \u03b9 \u2192 Set \u03b1\nUo : \u2200 (i : \u03b9), IsOpen (U i)\nUc : \u22c3 (i : \u03b9), U i = univ\nb : (i : \u03b9) \u2192 Set (Set \u2191(U i))\nhb : \u2200 (i : \u03b9), IsTopologicalBasis (b i)\n\u22a2 IsTopologicalBasis (\u22c3 (i : \u03b9), image Subtype.val '' b i)\n[PROOFSTEP]\nrefine' isTopologicalBasis_of_open_of_nhds (fun u hu => _) _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Sort u_1\nU : \u03b9 \u2192 Set \u03b1\nUo : \u2200 (i : \u03b9), IsOpen (U i)\nUc : \u22c3 (i : \u03b9), U i = univ\nb : (i : \u03b9) \u2192 Set (Set \u2191(U i))\nhb : \u2200 (i : \u03b9), IsTopologicalBasis (b i)\nu : Set \u03b1\nhu : u \u2208 \u22c3 (i : \u03b9), image Subtype.val '' b i\n\u22a2 IsOpen u\n[PROOFSTEP]\nsimp only [mem_iUnion, mem_image] at hu \n[GOAL]\ncase refine'_1\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Sort u_1\nU : \u03b9 \u2192 Set \u03b1\nUo : \u2200 (i : \u03b9), IsOpen (U i)\nUc : \u22c3 (i : \u03b9), U i = univ\nb : (i : \u03b9) \u2192 Set (Set \u2191(U i))\nhb : \u2200 (i : \u03b9), IsTopologicalBasis (b i)\nu : Set \u03b1\nhu : \u2203 i x, x \u2208 b i \u2227 Subtype.val '' x = u\n\u22a2 IsOpen u\n[PROOFSTEP]\nrcases hu with \u27e8i, s, sb, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Sort u_1\nU : \u03b9 \u2192 Set \u03b1\nUo : \u2200 (i : \u03b9), IsOpen (U i)\nUc : \u22c3 (i : \u03b9), U i = univ\nb : (i : \u03b9) \u2192 Set (Set \u2191(U i))\nhb : \u2200 (i : \u03b9), IsTopologicalBasis (b i)\ni : \u03b9\ns : Set { x // x \u2208 U i }\nsb : s \u2208 b i\n\u22a2 IsOpen (Subtype.val '' s)\n[PROOFSTEP]\nexact (Uo i).isOpenMap_subtype_val _ ((hb i).isOpen sb)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Sort u_1\nU : \u03b9 \u2192 Set \u03b1\nUo : \u2200 (i : \u03b9), IsOpen (U i)\nUc : \u22c3 (i : \u03b9), U i = univ\nb : (i : \u03b9) \u2192 Set (Set \u2191(U i))\nhb : \u2200 (i : \u03b9), IsTopologicalBasis (b i)\n\u22a2 \u2200 (a : \u03b1) (u : Set \u03b1), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 \u22c3 (i : \u03b9), image Subtype.val '' b i \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nintro a u ha uo\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Sort u_1\nU : \u03b9 \u2192 Set \u03b1\nUo : \u2200 (i : \u03b9), IsOpen (U i)\nUc : \u22c3 (i : \u03b9), U i = univ\nb : (i : \u03b9) \u2192 Set (Set \u2191(U i))\nhb : \u2200 (i : \u03b9), IsTopologicalBasis (b i)\na : \u03b1\nu : Set \u03b1\nha : a \u2208 u\nuo : IsOpen u\n\u22a2 \u2203 v, v \u2208 \u22c3 (i : \u03b9), image Subtype.val '' b i \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nrcases iUnion_eq_univ_iff.1 Uc a with \u27e8i, hi\u27e9\n[GOAL]\ncase refine'_2.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Sort u_1\nU : \u03b9 \u2192 Set \u03b1\nUo : \u2200 (i : \u03b9), IsOpen (U i)\nUc : \u22c3 (i : \u03b9), U i = univ\nb : (i : \u03b9) \u2192 Set (Set \u2191(U i))\nhb : \u2200 (i : \u03b9), IsTopologicalBasis (b i)\na : \u03b1\nu : Set \u03b1\nha : a \u2208 u\nuo : IsOpen u\ni : \u03b9\nhi : a \u2208 U i\n\u22a2 \u2203 v, v \u2208 \u22c3 (i : \u03b9), image Subtype.val '' b i \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nlift a to \u21a5(U i) using hi\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Sort u_1\nU : \u03b9 \u2192 Set \u03b1\nUo : \u2200 (i : \u03b9), IsOpen (U i)\nUc : \u22c3 (i : \u03b9), U i = univ\nb : (i : \u03b9) \u2192 Set (Set \u2191(U i))\nhb : \u2200 (i : \u03b9), IsTopologicalBasis (b i)\nu : Set \u03b1\nuo : IsOpen u\ni : \u03b9\na : { x // x \u2208 U i }\nha : \u2191a \u2208 u\n\u22a2 \u2203 v, v \u2208 \u22c3 (i : \u03b9), image Subtype.val '' b i \u2227 \u2191a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nrcases(hb i).exists_subset_of_mem_open ha (uo.preimage continuous_subtype_val) with \u27e8v, hvb, hav, hvu\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Sort u_1\nU : \u03b9 \u2192 Set \u03b1\nUo : \u2200 (i : \u03b9), IsOpen (U i)\nUc : \u22c3 (i : \u03b9), U i = univ\nb : (i : \u03b9) \u2192 Set (Set \u2191(U i))\nhb : \u2200 (i : \u03b9), IsTopologicalBasis (b i)\nu : Set \u03b1\nuo : IsOpen u\ni : \u03b9\na : { x // x \u2208 U i }\nha : \u2191a \u2208 u\nv : Set \u2191(U i)\nhvb : v \u2208 b i\nhav : a \u2208 v\nhvu : v \u2286 Subtype.val \u207b\u00b9' u\n\u22a2 \u2203 v, v \u2208 \u22c3 (i : \u03b9), image Subtype.val '' b i \u2227 \u2191a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nexact \u27e8(\u2191) '' v, mem_iUnion.2 \u27e8i, mem_image_of_mem _ hvb\u27e9, mem_image_of_mem _ hav, image_subset_iff.2 hvu\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nB : Set (Set \u03b2)\nhB : IsTopologicalBasis B\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (s : Set \u03b2), s \u2208 B \u2192 IsOpen (f \u207b\u00b9' s)\n\u22a2 Continuous f\n[PROOFSTEP]\nrw [hB.eq_generateFrom]\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nB : Set (Set \u03b2)\nhB : IsTopologicalBasis B\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (s : Set \u03b2), s \u2208 B \u2192 IsOpen (f \u207b\u00b9' s)\n\u22a2 Continuous f\n[PROOFSTEP]\nexact continuous_generateFrom hf\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SeparableSpace \u03b1\ninst\u271d : Nonempty \u03b1\n\u22a2 \u2203 u, DenseRange u\n[PROOFSTEP]\nobtain \u27e8s : Set \u03b1, hs, s_dense\u27e9 := exists_countable_dense \u03b1\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SeparableSpace \u03b1\ninst\u271d : Nonempty \u03b1\ns : Set \u03b1\nhs : Set.Countable s\ns_dense : Dense s\n\u22a2 \u2203 u, DenseRange u\n[PROOFSTEP]\ncases' Set.countable_iff_exists_subset_range.mp hs with u hu\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SeparableSpace \u03b1\ninst\u271d : Nonempty \u03b1\ns : Set \u03b1\nhs : Set.Countable s\ns_dense : Dense s\nu : \u2115 \u2192 \u03b1\nhu : s \u2286 range u\n\u22a2 \u2203 u, DenseRange u\n[PROOFSTEP]\nexact \u27e8u, s_dense.mono hu\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type ?u.24188\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : SeparableSpace \u03b1\ninst\u271d : SeparableSpace \u03b2\n\u22a2 SeparableSpace (\u03b1 \u00d7 \u03b2)\n[PROOFSTEP]\nrcases exists_countable_dense \u03b1 with \u27e8s, hsc, hsd\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type ?u.24188\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : SeparableSpace \u03b1\ninst\u271d : SeparableSpace \u03b2\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\n\u22a2 SeparableSpace (\u03b1 \u00d7 \u03b2)\n[PROOFSTEP]\nrcases exists_countable_dense \u03b2 with \u27e8t, htc, htd\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type ?u.24188\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : SeparableSpace \u03b1\ninst\u271d : SeparableSpace \u03b2\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nt : Set \u03b2\nhtc : Set.Countable t\nhtd : Dense t\n\u22a2 SeparableSpace (\u03b1 \u00d7 \u03b2)\n[PROOFSTEP]\nexact \u27e8\u27e8s \u00d7\u02e2 t, hsc.prod htc, hsd.prod htd\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (X i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SeparableSpace (X i)\ninst\u271d : Countable \u03b9\n\u22a2 SeparableSpace ((i : \u03b9) \u2192 X i)\n[PROOFSTEP]\nchoose t htc htd using (exists_countable_dense <| X \u00b7)\n[GOAL]\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (X i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SeparableSpace (X i)\ninst\u271d : Countable \u03b9\nt : (x : \u03b9) \u2192 Set (X x)\nhtc : \u2200 (x : \u03b9), Set.Countable (t x)\nhtd : \u2200 (x : \u03b9), Dense (t x)\n\u22a2 SeparableSpace ((i : \u03b9) \u2192 X i)\n[PROOFSTEP]\nhaveI := fun i \u21a6 (htc i).to_subtype\n[GOAL]\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (X i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SeparableSpace (X i)\ninst\u271d : Countable \u03b9\nt : (x : \u03b9) \u2192 Set (X x)\nhtc : \u2200 (x : \u03b9), Set.Countable (t x)\nhtd : \u2200 (x : \u03b9), Dense (t x)\nthis : \u2200 (i : \u03b9), Countable \u2191(t i)\n\u22a2 SeparableSpace ((i : \u03b9) \u2192 X i)\n[PROOFSTEP]\nnontriviality \u2200 i, X i\n[GOAL]\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (X i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SeparableSpace (X i)\ninst\u271d : Countable \u03b9\nt : (x : \u03b9) \u2192 Set (X x)\nhtc : \u2200 (x : \u03b9), Set.Countable (t x)\nhtd : \u2200 (x : \u03b9), Dense (t x)\nthis : \u2200 (i : \u03b9), Countable \u2191(t i)\n\u271d : Nontrivial ((i : \u03b9) \u2192 X i)\n\u22a2 SeparableSpace ((i : \u03b9) \u2192 X i)\n[PROOFSTEP]\ninhabit \u2200 i, X i\n[GOAL]\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (X i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SeparableSpace (X i)\ninst\u271d : Countable \u03b9\nt : (x : \u03b9) \u2192 Set (X x)\nhtc : \u2200 (x : \u03b9), Set.Countable (t x)\nhtd : \u2200 (x : \u03b9), Dense (t x)\nthis : \u2200 (i : \u03b9), Countable \u2191(t i)\n\u271d : Nontrivial ((i : \u03b9) \u2192 X i)\ninhabited_h : Inhabited ((i : \u03b9) \u2192 X i)\n\u22a2 SeparableSpace ((i : \u03b9) \u2192 X i)\n[PROOFSTEP]\nclassical\nset f : (\u03a3 I : Finset \u03b9, \u2200 i : I, t i) \u2192 \u2200 i, X i := fun \u27e8I, g\u27e9 i \u21a6\n if hi : i \u2208 I then g \u27e8i, hi\u27e9 else (default : \u2200 i, X i) i\nrefine \u27e8\u27e8range f, countable_range f, dense_iff_inter_open.2 fun U hU \u27e8g, hg\u27e9 \u21a6 ?_\u27e9\u27e9\nrcases isOpen_pi_iff.1 hU g hg with \u27e8I, u, huo, huU\u27e9\nhave : \u2200 i : I, \u2203 y \u2208 t i, y \u2208 u i := fun i \u21a6 (htd i).exists_mem_open (huo i i.2).1 \u27e8_, (huo i i.2).2\u27e9\nchoose y hyt hyu using this\nlift y to \u2200 i : I, t i using hyt\nrefine \u27e8f \u27e8I, y\u27e9, huU fun i (hi : i \u2208 I) \u21a6 ?_, mem_range_self _\u27e9\nsimp only [dif_pos hi]\nexact hyu _\n[GOAL]\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (X i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SeparableSpace (X i)\ninst\u271d : Countable \u03b9\nt : (x : \u03b9) \u2192 Set (X x)\nhtc : \u2200 (x : \u03b9), Set.Countable (t x)\nhtd : \u2200 (x : \u03b9), Dense (t x)\nthis : \u2200 (i : \u03b9), Countable \u2191(t i)\n\u271d : Nontrivial ((i : \u03b9) \u2192 X i)\ninhabited_h : Inhabited ((i : \u03b9) \u2192 X i)\n\u22a2 SeparableSpace ((i : \u03b9) \u2192 X i)\n[PROOFSTEP]\nset f : (\u03a3 I : Finset \u03b9, \u2200 i : I, t i) \u2192 \u2200 i, X i := fun \u27e8I, g\u27e9 i \u21a6\n if hi : i \u2208 I then g \u27e8i, hi\u27e9 else (default : \u2200 i, X i) i\n[GOAL]\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (X i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SeparableSpace (X i)\ninst\u271d : Countable \u03b9\nt : (x : \u03b9) \u2192 Set (X x)\nhtc : \u2200 (x : \u03b9), Set.Countable (t x)\nhtd : \u2200 (x : \u03b9), Dense (t x)\nthis : \u2200 (i : \u03b9), Countable \u2191(t i)\n\u271d : Nontrivial ((i : \u03b9) \u2192 X i)\ninhabited_h : Inhabited ((i : \u03b9) \u2192 X i)\nf : (I : Finset \u03b9) \u00d7 ((i : { x // x \u2208 I }) \u2192 \u2191(t \u2191i)) \u2192 (i : \u03b9) \u2192 X i :=\n fun x i =>\n match x with\n | { fst := I, snd := g } => if hi : i \u2208 I then \u2191(g { val := i, property := hi }) else default i\n\u22a2 SeparableSpace ((i : \u03b9) \u2192 X i)\n[PROOFSTEP]\nrefine \u27e8\u27e8range f, countable_range f, dense_iff_inter_open.2 fun U hU \u27e8g, hg\u27e9 \u21a6 ?_\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (X i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SeparableSpace (X i)\ninst\u271d : Countable \u03b9\nt : (x : \u03b9) \u2192 Set (X x)\nhtc : \u2200 (x : \u03b9), Set.Countable (t x)\nhtd : \u2200 (x : \u03b9), Dense (t x)\nthis : \u2200 (i : \u03b9), Countable \u2191(t i)\n\u271d : Nontrivial ((i : \u03b9) \u2192 X i)\ninhabited_h : Inhabited ((i : \u03b9) \u2192 X i)\nf : (I : Finset \u03b9) \u00d7 ((i : { x // x \u2208 I }) \u2192 \u2191(t \u2191i)) \u2192 (i : \u03b9) \u2192 X i :=\n fun x i =>\n match x with\n | { fst := I, snd := g } => if hi : i \u2208 I then \u2191(g { val := i, property := hi }) else default i\nU : Set ((i : \u03b9) \u2192 X i)\nhU : IsOpen U\nx\u271d : Set.Nonempty U\ng : (i : \u03b9) \u2192 X i\nhg : g \u2208 U\n\u22a2 Set.Nonempty (U \u2229 range f)\n[PROOFSTEP]\nrcases isOpen_pi_iff.1 hU g hg with \u27e8I, u, huo, huU\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (X i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SeparableSpace (X i)\ninst\u271d : Countable \u03b9\nt : (x : \u03b9) \u2192 Set (X x)\nhtc : \u2200 (x : \u03b9), Set.Countable (t x)\nhtd : \u2200 (x : \u03b9), Dense (t x)\nthis : \u2200 (i : \u03b9), Countable \u2191(t i)\n\u271d : Nontrivial ((i : \u03b9) \u2192 X i)\ninhabited_h : Inhabited ((i : \u03b9) \u2192 X i)\nf : (I : Finset \u03b9) \u00d7 ((i : { x // x \u2208 I }) \u2192 \u2191(t \u2191i)) \u2192 (i : \u03b9) \u2192 X i :=\n fun x i =>\n match x with\n | { fst := I, snd := g } => if hi : i \u2208 I then \u2191(g { val := i, property := hi }) else default i\nU : Set ((i : \u03b9) \u2192 X i)\nhU : IsOpen U\nx\u271d : Set.Nonempty U\ng : (i : \u03b9) \u2192 X i\nhg : g \u2208 U\nI : Finset \u03b9\nu : (a : \u03b9) \u2192 Set (X a)\nhuo : \u2200 (a : \u03b9), a \u2208 I \u2192 IsOpen (u a) \u2227 g a \u2208 u a\nhuU : Set.pi (\u2191I) u \u2286 U\n\u22a2 Set.Nonempty (U \u2229 range f)\n[PROOFSTEP]\nhave : \u2200 i : I, \u2203 y \u2208 t i, y \u2208 u i := fun i \u21a6 (htd i).exists_mem_open (huo i i.2).1 \u27e8_, (huo i i.2).2\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (X i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SeparableSpace (X i)\ninst\u271d : Countable \u03b9\nt : (x : \u03b9) \u2192 Set (X x)\nhtc : \u2200 (x : \u03b9), Set.Countable (t x)\nhtd : \u2200 (x : \u03b9), Dense (t x)\nthis\u271d : \u2200 (i : \u03b9), Countable \u2191(t i)\n\u271d : Nontrivial ((i : \u03b9) \u2192 X i)\ninhabited_h : Inhabited ((i : \u03b9) \u2192 X i)\nf : (I : Finset \u03b9) \u00d7 ((i : { x // x \u2208 I }) \u2192 \u2191(t \u2191i)) \u2192 (i : \u03b9) \u2192 X i :=\n fun x i =>\n match x with\n | { fst := I, snd := g } => if hi : i \u2208 I then \u2191(g { val := i, property := hi }) else default i\nU : Set ((i : \u03b9) \u2192 X i)\nhU : IsOpen U\nx\u271d : Set.Nonempty U\ng : (i : \u03b9) \u2192 X i\nhg : g \u2208 U\nI : Finset \u03b9\nu : (a : \u03b9) \u2192 Set (X a)\nhuo : \u2200 (a : \u03b9), a \u2208 I \u2192 IsOpen (u a) \u2227 g a \u2208 u a\nhuU : Set.pi (\u2191I) u \u2286 U\nthis : \u2200 (i : { x // x \u2208 I }), \u2203 y, y \u2208 t \u2191i \u2227 y \u2208 u \u2191i\n\u22a2 Set.Nonempty (U \u2229 range f)\n[PROOFSTEP]\nchoose y hyt hyu using this\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (X i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SeparableSpace (X i)\ninst\u271d : Countable \u03b9\nt : (x : \u03b9) \u2192 Set (X x)\nhtc : \u2200 (x : \u03b9), Set.Countable (t x)\nhtd : \u2200 (x : \u03b9), Dense (t x)\nthis : \u2200 (i : \u03b9), Countable \u2191(t i)\n\u271d : Nontrivial ((i : \u03b9) \u2192 X i)\ninhabited_h : Inhabited ((i : \u03b9) \u2192 X i)\nf : (I : Finset \u03b9) \u00d7 ((i : { x // x \u2208 I }) \u2192 \u2191(t \u2191i)) \u2192 (i : \u03b9) \u2192 X i :=\n fun x i =>\n match x with\n | { fst := I, snd := g } => if hi : i \u2208 I then \u2191(g { val := i, property := hi }) else default i\nU : Set ((i : \u03b9) \u2192 X i)\nhU : IsOpen U\nx\u271d : Set.Nonempty U\ng : (i : \u03b9) \u2192 X i\nhg : g \u2208 U\nI : Finset \u03b9\nu : (a : \u03b9) \u2192 Set (X a)\nhuo : \u2200 (a : \u03b9), a \u2208 I \u2192 IsOpen (u a) \u2227 g a \u2208 u a\nhuU : Set.pi (\u2191I) u \u2286 U\ny : (i : { x // x \u2208 I }) \u2192 X \u2191i\nhyt : \u2200 (i : { x // x \u2208 I }), y i \u2208 t \u2191i\nhyu : \u2200 (i : { x // x \u2208 I }), y i \u2208 u \u2191i\n\u22a2 Set.Nonempty (U \u2229 range f)\n[PROOFSTEP]\nlift y to \u2200 i : I, t i using hyt\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (X i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SeparableSpace (X i)\ninst\u271d : Countable \u03b9\nt : (x : \u03b9) \u2192 Set (X x)\nhtc : \u2200 (x : \u03b9), Set.Countable (t x)\nhtd : \u2200 (x : \u03b9), Dense (t x)\nthis : \u2200 (i : \u03b9), Countable \u2191(t i)\n\u271d : Nontrivial ((i : \u03b9) \u2192 X i)\ninhabited_h : Inhabited ((i : \u03b9) \u2192 X i)\nf : (I : Finset \u03b9) \u00d7 ((i : { x // x \u2208 I }) \u2192 \u2191(t \u2191i)) \u2192 (i : \u03b9) \u2192 X i :=\n fun x i =>\n match x with\n | { fst := I, snd := g } => if hi : i \u2208 I then \u2191(g { val := i, property := hi }) else default i\nU : Set ((i : \u03b9) \u2192 X i)\nhU : IsOpen U\nx\u271d : Set.Nonempty U\ng : (i : \u03b9) \u2192 X i\nhg : g \u2208 U\nI : Finset \u03b9\nu : (a : \u03b9) \u2192 Set (X a)\nhuo : \u2200 (a : \u03b9), a \u2208 I \u2192 IsOpen (u a) \u2227 g a \u2208 u a\nhuU : Set.pi (\u2191I) u \u2286 U\ny : (i : { x // x \u2208 I }) \u2192 \u2191(t \u2191i)\nhyu : \u2200 (i : { x // x \u2208 I }), (fun i => \u2191(y i)) i \u2208 u \u2191i\n\u22a2 Set.Nonempty (U \u2229 range f)\n[PROOFSTEP]\nrefine \u27e8f \u27e8I, y\u27e9, huU fun i (hi : i \u2208 I) \u21a6 ?_, mem_range_self _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (X i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SeparableSpace (X i)\ninst\u271d : Countable \u03b9\nt : (x : \u03b9) \u2192 Set (X x)\nhtc : \u2200 (x : \u03b9), Set.Countable (t x)\nhtd : \u2200 (x : \u03b9), Dense (t x)\nthis : \u2200 (i : \u03b9), Countable \u2191(t i)\n\u271d : Nontrivial ((i : \u03b9) \u2192 X i)\ninhabited_h : Inhabited ((i : \u03b9) \u2192 X i)\nf : (I : Finset \u03b9) \u00d7 ((i : { x // x \u2208 I }) \u2192 \u2191(t \u2191i)) \u2192 (i : \u03b9) \u2192 X i :=\n fun x i =>\n match x with\n | { fst := I, snd := g } => if hi : i \u2208 I then \u2191(g { val := i, property := hi }) else default i\nU : Set ((i : \u03b9) \u2192 X i)\nhU : IsOpen U\nx\u271d : Set.Nonempty U\ng : (i : \u03b9) \u2192 X i\nhg : g \u2208 U\nI : Finset \u03b9\nu : (a : \u03b9) \u2192 Set (X a)\nhuo : \u2200 (a : \u03b9), a \u2208 I \u2192 IsOpen (u a) \u2227 g a \u2208 u a\nhuU : Set.pi (\u2191I) u \u2286 U\ny : (i : { x // x \u2208 I }) \u2192 \u2191(t \u2191i)\nhyu : \u2200 (i : { x // x \u2208 I }), (fun i => \u2191(y i)) i \u2208 u \u2191i\ni : \u03b9\nhi : i \u2208 I\n\u22a2 f { fst := I, snd := y } i \u2208 u i\n[PROOFSTEP]\nsimp only [dif_pos hi]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (X i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), SeparableSpace (X i)\ninst\u271d : Countable \u03b9\nt : (x : \u03b9) \u2192 Set (X x)\nhtc : \u2200 (x : \u03b9), Set.Countable (t x)\nhtd : \u2200 (x : \u03b9), Dense (t x)\nthis : \u2200 (i : \u03b9), Countable \u2191(t i)\n\u271d : Nontrivial ((i : \u03b9) \u2192 X i)\ninhabited_h : Inhabited ((i : \u03b9) \u2192 X i)\nf : (I : Finset \u03b9) \u00d7 ((i : { x // x \u2208 I }) \u2192 \u2191(t \u2191i)) \u2192 (i : \u03b9) \u2192 X i :=\n fun x i =>\n match x with\n | { fst := I, snd := g } => if hi : i \u2208 I then \u2191(g { val := i, property := hi }) else default i\nU : Set ((i : \u03b9) \u2192 X i)\nhU : IsOpen U\nx\u271d : Set.Nonempty U\ng : (i : \u03b9) \u2192 X i\nhg : g \u2208 U\nI : Finset \u03b9\nu : (a : \u03b9) \u2192 Set (X a)\nhuo : \u2200 (a : \u03b9), a \u2208 I \u2192 IsOpen (u a) \u2227 g a \u2208 u a\nhuU : Set.pi (\u2191I) u \u2286 U\ny : (i : { x // x \u2208 I }) \u2192 \u2191(t \u2191i)\nhyu : \u2200 (i : { x // x \u2208 I }), (fun i => \u2191(y i)) i \u2208 u \u2191i\ni : \u03b9\nhi : i \u2208 I\n\u22a2 \u2191(y { val := i, property := (_ : i \u2208 I) }) \u2208 u i\n[PROOFSTEP]\nexact hyu _\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : DiscreteTopology \u03b1\n\u22a2 SeparableSpace \u03b1 \u2194 Countable \u03b1\n[PROOFSTEP]\nsimp [SeparableSpace_iff, countable_univ_iff]\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SeparableSpace \u03b1\n\u03b9 : Type u_1\ns : \u03b9 \u2192 Set \u03b1\na : Set \u03b9\nh : PairwiseDisjoint a s\nha : \u2200 (i : \u03b9), i \u2208 a \u2192 IsOpen (s i)\nh'a : \u2200 (i : \u03b9), i \u2208 a \u2192 Set.Nonempty (s i)\n\u22a2 Set.Countable a\n[PROOFSTEP]\nrcases exists_countable_dense \u03b1 with \u27e8u, \u27e8u_encodable\u27e9, u_dense\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SeparableSpace \u03b1\n\u03b9 : Type u_1\ns : \u03b9 \u2192 Set \u03b1\na : Set \u03b9\nh : PairwiseDisjoint a s\nha : \u2200 (i : \u03b9), i \u2208 a \u2192 IsOpen (s i)\nh'a : \u2200 (i : \u03b9), i \u2208 a \u2192 Set.Nonempty (s i)\nu : Set \u03b1\nu_dense : Dense u\nu_encodable : Encodable \u2191u\n\u22a2 Set.Countable a\n[PROOFSTEP]\nhave : \u2200 i : a, \u2203 y, y \u2208 s i \u2229 u := fun i => dense_iff_inter_open.1 u_dense (s i) (ha i i.2) (h'a i i.2)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SeparableSpace \u03b1\n\u03b9 : Type u_1\ns : \u03b9 \u2192 Set \u03b1\na : Set \u03b9\nh : PairwiseDisjoint a s\nha : \u2200 (i : \u03b9), i \u2208 a \u2192 IsOpen (s i)\nh'a : \u2200 (i : \u03b9), i \u2208 a \u2192 Set.Nonempty (s i)\nu : Set \u03b1\nu_dense : Dense u\nu_encodable : Encodable \u2191u\nthis : \u2200 (i : \u2191a), \u2203 y, y \u2208 s \u2191i \u2229 u\n\u22a2 Set.Countable a\n[PROOFSTEP]\nchoose f hfs hfu using this\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SeparableSpace \u03b1\n\u03b9 : Type u_1\ns : \u03b9 \u2192 Set \u03b1\na : Set \u03b9\nh : PairwiseDisjoint a s\nha : \u2200 (i : \u03b9), i \u2208 a \u2192 IsOpen (s i)\nh'a : \u2200 (i : \u03b9), i \u2208 a \u2192 Set.Nonempty (s i)\nu : Set \u03b1\nu_dense : Dense u\nu_encodable : Encodable \u2191u\nf : \u2191a \u2192 \u03b1\nhfs : \u2200 (i : \u2191a), f i \u2208 s \u2191i\nhfu : \u2200 (i : \u2191a), f i \u2208 u\n\u22a2 Set.Countable a\n[PROOFSTEP]\nlift f to a \u2192 u using hfu\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SeparableSpace \u03b1\n\u03b9 : Type u_1\ns : \u03b9 \u2192 Set \u03b1\na : Set \u03b9\nh : PairwiseDisjoint a s\nha : \u2200 (i : \u03b9), i \u2208 a \u2192 IsOpen (s i)\nh'a : \u2200 (i : \u03b9), i \u2208 a \u2192 Set.Nonempty (s i)\nu : Set \u03b1\nu_dense : Dense u\nu_encodable : Encodable \u2191u\nf : \u2191a \u2192 \u2191u\nhfs : \u2200 (i : \u2191a), (fun i => \u2191(f i)) i \u2208 s \u2191i\n\u22a2 Set.Countable a\n[PROOFSTEP]\nhave f_inj : Injective f :=\n by\n refine' injective_iff_pairwise_ne.mpr ((h.subtype _ _).mono fun i j hij hfij => hij.le_bot \u27e8hfs i, _\u27e9)\n simp only [congr_arg Subtype.val hfij, hfs j]\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SeparableSpace \u03b1\n\u03b9 : Type u_1\ns : \u03b9 \u2192 Set \u03b1\na : Set \u03b9\nh : PairwiseDisjoint a s\nha : \u2200 (i : \u03b9), i \u2208 a \u2192 IsOpen (s i)\nh'a : \u2200 (i : \u03b9), i \u2208 a \u2192 Set.Nonempty (s i)\nu : Set \u03b1\nu_dense : Dense u\nu_encodable : Encodable \u2191u\nf : \u2191a \u2192 \u2191u\nhfs : \u2200 (i : \u2191a), (fun i => \u2191(f i)) i \u2208 s \u2191i\n\u22a2 Injective f\n[PROOFSTEP]\nrefine' injective_iff_pairwise_ne.mpr ((h.subtype _ _).mono fun i j hij hfij => hij.le_bot \u27e8hfs i, _\u27e9)\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SeparableSpace \u03b1\n\u03b9 : Type u_1\ns : \u03b9 \u2192 Set \u03b1\na : Set \u03b9\nh : PairwiseDisjoint a s\nha : \u2200 (i : \u03b9), i \u2208 a \u2192 IsOpen (s i)\nh'a : \u2200 (i : \u03b9), i \u2208 a \u2192 Set.Nonempty (s i)\nu : Set \u03b1\nu_dense : Dense u\nu_encodable : Encodable \u2191u\nf : \u2191a \u2192 \u2191u\nhfs : \u2200 (i : \u2191a), (fun i => \u2191(f i)) i \u2208 s \u2191i\ni j : \u2191a\nhij : (Disjoint on s) \u2191i \u2191j\nhfij : f i = f j\n\u22a2 (fun i => \u2191(f i)) i \u2208 s \u2191j\n[PROOFSTEP]\nsimp only [congr_arg Subtype.val hfij, hfs j]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SeparableSpace \u03b1\n\u03b9 : Type u_1\ns : \u03b9 \u2192 Set \u03b1\na : Set \u03b9\nh : PairwiseDisjoint a s\nha : \u2200 (i : \u03b9), i \u2208 a \u2192 IsOpen (s i)\nh'a : \u2200 (i : \u03b9), i \u2208 a \u2192 Set.Nonempty (s i)\nu : Set \u03b1\nu_dense : Dense u\nu_encodable : Encodable \u2191u\nf : \u2191a \u2192 \u2191u\nhfs : \u2200 (i : \u2191a), (fun i => \u2191(f i)) i \u2208 s \u2191i\nf_inj : Injective f\n\u22a2 Set.Countable a\n[PROOFSTEP]\nexact \u27e8@Encodable.ofInj _ _ u_encodable f f_inj\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns u : Set \u03b1\nhs : IsSeparable s\nhu : u \u2286 s\n\u22a2 IsSeparable u\n[PROOFSTEP]\nrcases hs with \u27e8c, c_count, hs\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns u : Set \u03b1\nhu : u \u2286 s\nc : Set \u03b1\nc_count : Set.Countable c\nhs : s \u2286 closure c\n\u22a2 IsSeparable u\n[PROOFSTEP]\nexact \u27e8c, c_count, hu.trans hs\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns u : Set \u03b1\nhs : IsSeparable s\nhu : IsSeparable u\n\u22a2 IsSeparable (s \u222a u)\n[PROOFSTEP]\nrcases hs with \u27e8cs, cs_count, hcs\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns u : Set \u03b1\nhu : IsSeparable u\ncs : Set \u03b1\ncs_count : Set.Countable cs\nhcs : s \u2286 closure cs\n\u22a2 IsSeparable (s \u222a u)\n[PROOFSTEP]\nrcases hu with \u27e8cu, cu_count, hcu\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns u cs : Set \u03b1\ncs_count : Set.Countable cs\nhcs : s \u2286 closure cs\ncu : Set \u03b1\ncu_count : Set.Countable cu\nhcu : u \u2286 closure cu\n\u22a2 IsSeparable (s \u222a u)\n[PROOFSTEP]\nrefine' \u27e8cs \u222a cu, cs_count.union cu_count, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns u cs : Set \u03b1\ncs_count : Set.Countable cs\nhcs : s \u2286 closure cs\ncu : Set \u03b1\ncu_count : Set.Countable cu\nhcu : u \u2286 closure cu\n\u22a2 s \u222a u \u2286 closure (cs \u222a cu)\n[PROOFSTEP]\nexact\n union_subset (hcs.trans (closure_mono (subset_union_left _ _))) (hcu.trans (closure_mono (subset_union_right _ _)))\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set \u03b1\nhs : IsSeparable s\n\u22a2 IsSeparable (_root_.closure s)\n[PROOFSTEP]\nrcases hs with \u27e8c, c_count, hs\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns c : Set \u03b1\nc_count : Set.Countable c\nhs : s \u2286 _root_.closure c\n\u22a2 IsSeparable (_root_.closure s)\n[PROOFSTEP]\nexact \u27e8c, c_count, by simpa using closure_mono hs\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns c : Set \u03b1\nc_count : Set.Countable c\nhs : s \u2286 _root_.closure c\n\u22a2 _root_.closure s \u2286 _root_.closure c\n[PROOFSTEP]\nsimpa using closure_mono hs\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsSeparable (s i)\n\u22a2 IsSeparable (\u22c3 (i : \u03b9), s i)\n[PROOFSTEP]\nchoose c hc h'c using hs\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\ninst\u271d : Countable \u03b9\ns c : \u03b9 \u2192 Set \u03b1\nhc : \u2200 (i : \u03b9), Set.Countable (c i)\nh'c : \u2200 (i : \u03b9), s i \u2286 closure (c i)\n\u22a2 IsSeparable (\u22c3 (i : \u03b9), s i)\n[PROOFSTEP]\nrefine' \u27e8\u22c3 i, c i, countable_iUnion hc, iUnion_subset_iff.2 fun i => _\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\ninst\u271d : Countable \u03b9\ns c : \u03b9 \u2192 Set \u03b1\nhc : \u2200 (i : \u03b9), Set.Countable (c i)\nh'c : \u2200 (i : \u03b9), s i \u2286 closure (c i)\ni : \u03b9\n\u22a2 s i \u2286 closure (\u22c3 (i : \u03b9), c i)\n[PROOFSTEP]\nexact (h'c i).trans (closure_mono (subset_iUnion _ i))\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u22a2 IsSeparable univ \u2194 SeparableSpace \u03b1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u22a2 IsSeparable univ \u2192 SeparableSpace \u03b1\n[PROOFSTEP]\nrintro \u27e8c, c_count, hc\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nc : Set \u03b1\nc_count : Set.Countable c\nhc : univ \u2286 closure c\n\u22a2 SeparableSpace \u03b1\n[PROOFSTEP]\nrefine' \u27e8\u27e8c, c_count, by rwa [dense_iff_closure_eq, \u2190 univ_subset_iff]\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nc : Set \u03b1\nc_count : Set.Countable c\nhc : univ \u2286 closure c\n\u22a2 Dense c\n[PROOFSTEP]\nrwa [dense_iff_closure_eq, \u2190 univ_subset_iff]\n[GOAL]\ncase mpr\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u22a2 SeparableSpace \u03b1 \u2192 IsSeparable univ\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nh : SeparableSpace \u03b1\n\u22a2 IsSeparable univ\n[PROOFSTEP]\nrcases exists_countable_dense \u03b1 with \u27e8c, c_count, hc\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nh : SeparableSpace \u03b1\nc : Set \u03b1\nc_count : Set.Countable c\nhc : Dense c\n\u22a2 IsSeparable univ\n[PROOFSTEP]\nexact \u27e8c, c_count, by rwa [univ_subset_iff, \u2190 dense_iff_closure_eq]\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nh : SeparableSpace \u03b1\nc : Set \u03b1\nc_count : Set.Countable c\nhc : Dense c\n\u22a2 univ \u2286 closure c\n[PROOFSTEP]\nrwa [univ_subset_iff, \u2190 dense_iff_closure_eq]\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nhs : IsSeparable s\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\n\u22a2 IsSeparable (f '' s)\n[PROOFSTEP]\nrcases hs with \u27e8c, c_count, hc\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nc : Set \u03b1\nc_count : Set.Countable c\nhc : s \u2286 _root_.closure c\n\u22a2 IsSeparable (f '' s)\n[PROOFSTEP]\nrefine' \u27e8f '' c, c_count.image _, _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nc : Set \u03b1\nc_count : Set.Countable c\nhc : s \u2286 _root_.closure c\n\u22a2 f '' s \u2286 _root_.closure (f '' c)\n[PROOFSTEP]\nrw [image_subset_iff]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nc : Set \u03b1\nc_count : Set.Countable c\nhc : s \u2286 _root_.closure c\n\u22a2 s \u2286 f \u207b\u00b9' _root_.closure (f '' c)\n[PROOFSTEP]\nexact hc.trans (closure_subset_preimage_closure_image hf)\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set \u03b1\ninst\u271d : SeparableSpace \u2191s\n\u22a2 IsSeparable s\n[PROOFSTEP]\nhave : IsSeparable (((\u2191) : s \u2192 \u03b1) '' (univ : Set s)) := (isSeparable_of_separableSpace _).image continuous_subtype_val\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns : Set \u03b1\ninst\u271d : SeparableSpace \u2191s\nthis : IsSeparable (Subtype.val '' univ)\n\u22a2 IsSeparable s\n[PROOFSTEP]\nsimpa only [image_univ, Subtype.range_val_subtype] using this\n[GOAL]\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\n\u22a2 IsTopologicalBasis {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = Set.pi (\u2191F) U}\n[PROOFSTEP]\nrefine' isTopologicalBasis_of_open_of_nhds _ _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\n\u22a2 \u2200 (u : Set ((i : \u03b9) \u2192 X i)), u \u2208 {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = Set.pi (\u2191F) U} \u2192 IsOpen u\n[PROOFSTEP]\nrintro _ \u27e8U, F, h1, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\nU : (i : \u03b9) \u2192 Set (X i)\nF : Finset \u03b9\nh1 : \u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i\n\u22a2 IsOpen (Set.pi (\u2191F) U)\n[PROOFSTEP]\napply isOpen_set_pi F.finite_toSet\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\nU : (i : \u03b9) \u2192 Set (X i)\nF : Finset \u03b9\nh1 : \u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i\n\u22a2 \u2200 (a : \u03b9), a \u2208 \u2191F \u2192 IsOpen (U a)\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\nU : (i : \u03b9) \u2192 Set (X i)\nF : Finset \u03b9\nh1 : \u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i\ni : \u03b9\nhi : i \u2208 \u2191F\n\u22a2 IsOpen (U i)\n[PROOFSTEP]\nexact (cond i).isOpen (h1 i hi)\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\n\u22a2 \u2200 (a : (i : \u03b9) \u2192 X i) (u : Set ((i : \u03b9) \u2192 X i)),\n a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = Set.pi (\u2191F) U} \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nintro a U ha hU\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\na : (i : \u03b9) \u2192 X i\nU : Set ((i : \u03b9) \u2192 X i)\nha : a \u2208 U\nhU : IsOpen U\n\u22a2 \u2203 v, v \u2208 {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = Set.pi (\u2191F) U} \u2227 a \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nobtain \u27e8I, t, hta, htU\u27e9 : \u2203 (I : Finset \u03b9) (t : \u2200 i : \u03b9, Set (X i)), (\u2200 i, t i \u2208 \ud835\udcdd (a i)) \u2227 Set.pi (\u2191I) t \u2286 U :=\n by\n rw [\u2190 Filter.mem_pi', \u2190 nhds_pi]\n exact hU.mem_nhds ha\n[GOAL]\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\na : (i : \u03b9) \u2192 X i\nU : Set ((i : \u03b9) \u2192 X i)\nha : a \u2208 U\nhU : IsOpen U\n\u22a2 \u2203 I t, (\u2200 (i : \u03b9), t i \u2208 \ud835\udcdd (a i)) \u2227 Set.pi (\u2191I) t \u2286 U\n[PROOFSTEP]\nrw [\u2190 Filter.mem_pi', \u2190 nhds_pi]\n[GOAL]\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\na : (i : \u03b9) \u2192 X i\nU : Set ((i : \u03b9) \u2192 X i)\nha : a \u2208 U\nhU : IsOpen U\n\u22a2 U \u2208 \ud835\udcdd fun i => a i\n[PROOFSTEP]\nexact hU.mem_nhds ha\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\na : (i : \u03b9) \u2192 X i\nU : Set ((i : \u03b9) \u2192 X i)\nha : a \u2208 U\nhU : IsOpen U\nI : Finset \u03b9\nt : (i : \u03b9) \u2192 Set (X i)\nhta : \u2200 (i : \u03b9), t i \u2208 \ud835\udcdd (a i)\nhtU : Set.pi (\u2191I) t \u2286 U\n\u22a2 \u2203 v, v \u2208 {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = Set.pi (\u2191F) U} \u2227 a \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nhave : \u2200 i, \u2203 V \u2208 T i, a i \u2208 V \u2227 V \u2286 t i := fun i => (cond i).mem_nhds_iff.1 (hta i)\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\na : (i : \u03b9) \u2192 X i\nU : Set ((i : \u03b9) \u2192 X i)\nha : a \u2208 U\nhU : IsOpen U\nI : Finset \u03b9\nt : (i : \u03b9) \u2192 Set (X i)\nhta : \u2200 (i : \u03b9), t i \u2208 \ud835\udcdd (a i)\nhtU : Set.pi (\u2191I) t \u2286 U\nthis : \u2200 (i : \u03b9), \u2203 V, V \u2208 T i \u2227 a i \u2208 V \u2227 V \u2286 t i\n\u22a2 \u2203 v, v \u2208 {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = Set.pi (\u2191F) U} \u2227 a \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nchoose V hVT haV hVt using this\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b9 : Type u_1\nX : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\na : (i : \u03b9) \u2192 X i\nU : Set ((i : \u03b9) \u2192 X i)\nha : a \u2208 U\nhU : IsOpen U\nI : Finset \u03b9\nt : (i : \u03b9) \u2192 Set (X i)\nhta : \u2200 (i : \u03b9), t i \u2208 \ud835\udcdd (a i)\nhtU : Set.pi (\u2191I) t \u2286 U\nV : (i : \u03b9) \u2192 Set (X i)\nhVT : \u2200 (i : \u03b9), V i \u2208 T i\nhaV : \u2200 (i : \u03b9), a i \u2208 V i\nhVt : \u2200 (i : \u03b9), V i \u2286 t i\n\u22a2 \u2203 v, v \u2208 {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = Set.pi (\u2191F) U} \u2227 a \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nexact \u27e8_, \u27e8V, I, fun i _ => hVT i, rfl\u27e9, fun i _ => haV i, (pi_mono fun i _ => hVt i).trans htU\u27e9\n[GOAL]\n\u03b2 : Type u_1\n\u03b9 : Type u_2\nX : \u03b9 \u2192 Type u_3\nt : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\nf : (i : \u03b9) \u2192 \u03b2 \u2192 X i\n\u22a2 IsTopologicalBasis {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = \u22c2 (i : \u03b9) (_ : i \u2208 F), f i \u207b\u00b9' U i}\n[PROOFSTEP]\nletI := \u2a05 i, induced (f i) (t i)\n[GOAL]\n\u03b2 : Type u_1\n\u03b9 : Type u_2\nX : \u03b9 \u2192 Type u_3\nt : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\nf : (i : \u03b9) \u2192 \u03b2 \u2192 X i\nthis : TopologicalSpace \u03b2 := \u2a05 (i : \u03b9), induced (f i) (t i)\n\u22a2 IsTopologicalBasis {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = \u22c2 (i : \u03b9) (_ : i \u2208 F), f i \u207b\u00b9' U i}\n[PROOFSTEP]\nconvert (isTopologicalBasis_pi cond).inducing (inducing_iInf_to_pi f)\n[GOAL]\ncase h.e'_3\n\u03b2 : Type u_1\n\u03b9 : Type u_2\nX : \u03b9 \u2192 Type u_3\nt : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\nf : (i : \u03b9) \u2192 \u03b2 \u2192 X i\nthis : TopologicalSpace \u03b2 := \u2a05 (i : \u03b9), induced (f i) (t i)\n\u22a2 {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = \u22c2 (i : \u03b9) (_ : i \u2208 F), f i \u207b\u00b9' U i} =\n (preimage fun x i => f i x) '' {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = Set.pi (\u2191F) U}\n[PROOFSTEP]\next V\n[GOAL]\ncase h.e'_3.h\n\u03b2 : Type u_1\n\u03b9 : Type u_2\nX : \u03b9 \u2192 Type u_3\nt : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\nf : (i : \u03b9) \u2192 \u03b2 \u2192 X i\nthis : TopologicalSpace \u03b2 := \u2a05 (i : \u03b9), induced (f i) (t i)\nV : Set \u03b2\n\u22a2 V \u2208 {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = \u22c2 (i : \u03b9) (_ : i \u2208 F), f i \u207b\u00b9' U i} \u2194\n V \u2208 (preimage fun x i => f i x) '' {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = Set.pi (\u2191F) U}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.e'_3.h.mp\n\u03b2 : Type u_1\n\u03b9 : Type u_2\nX : \u03b9 \u2192 Type u_3\nt : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\nf : (i : \u03b9) \u2192 \u03b2 \u2192 X i\nthis : TopologicalSpace \u03b2 := \u2a05 (i : \u03b9), induced (f i) (t i)\nV : Set \u03b2\n\u22a2 V \u2208 {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = \u22c2 (i : \u03b9) (_ : i \u2208 F), f i \u207b\u00b9' U i} \u2192\n V \u2208 (preimage fun x i => f i x) '' {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = Set.pi (\u2191F) U}\n[PROOFSTEP]\nrintro \u27e8U, F, h1, rfl\u27e9\n[GOAL]\ncase h.e'_3.h.mp.intro.intro.intro\n\u03b2 : Type u_1\n\u03b9 : Type u_2\nX : \u03b9 \u2192 Type u_3\nt : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\nf : (i : \u03b9) \u2192 \u03b2 \u2192 X i\nthis : TopologicalSpace \u03b2 := \u2a05 (i : \u03b9), induced (f i) (t i)\nU : (i : \u03b9) \u2192 Set (X i)\nF : Finset \u03b9\nh1 : \u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i\n\u22a2 \u22c2 (i : \u03b9) (_ : i \u2208 F), f i \u207b\u00b9' U i \u2208\n (preimage fun x i => f i x) '' {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = Set.pi (\u2191F) U}\n[PROOFSTEP]\nrefine' \u27e8(F : Set \u03b9).pi U, \u27e8U, F, h1, rfl\u27e9, _\u27e9\n[GOAL]\ncase h.e'_3.h.mp.intro.intro.intro\n\u03b2 : Type u_1\n\u03b9 : Type u_2\nX : \u03b9 \u2192 Type u_3\nt : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\nf : (i : \u03b9) \u2192 \u03b2 \u2192 X i\nthis : TopologicalSpace \u03b2 := \u2a05 (i : \u03b9), induced (f i) (t i)\nU : (i : \u03b9) \u2192 Set (X i)\nF : Finset \u03b9\nh1 : \u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i\n\u22a2 (fun x i => f i x) \u207b\u00b9' Set.pi (\u2191F) U = \u22c2 (i : \u03b9) (_ : i \u2208 F), f i \u207b\u00b9' U i\n[PROOFSTEP]\nsimp_rw [pi_def, Set.preimage_iInter]\n[GOAL]\ncase h.e'_3.h.mp.intro.intro.intro\n\u03b2 : Type u_1\n\u03b9 : Type u_2\nX : \u03b9 \u2192 Type u_3\nt : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\nf : (i : \u03b9) \u2192 \u03b2 \u2192 X i\nthis : TopologicalSpace \u03b2 := \u2a05 (i : \u03b9), induced (f i) (t i)\nU : (i : \u03b9) \u2192 Set (X i)\nF : Finset \u03b9\nh1 : \u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i\n\u22a2 \u22c2 (i : \u03b9) (_ : i \u2208 \u2191F), (fun x i => f i x) \u207b\u00b9' (eval i \u207b\u00b9' U i) = \u22c2 (i : \u03b9) (_ : i \u2208 F), f i \u207b\u00b9' U i\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3.h.mpr\n\u03b2 : Type u_1\n\u03b9 : Type u_2\nX : \u03b9 \u2192 Type u_3\nt : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\nf : (i : \u03b9) \u2192 \u03b2 \u2192 X i\nthis : TopologicalSpace \u03b2 := \u2a05 (i : \u03b9), induced (f i) (t i)\nV : Set \u03b2\n\u22a2 V \u2208 (preimage fun x i => f i x) '' {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = Set.pi (\u2191F) U} \u2192\n V \u2208 {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = \u22c2 (i : \u03b9) (_ : i \u2208 F), f i \u207b\u00b9' U i}\n[PROOFSTEP]\nrintro \u27e8U, \u27e8U, F, h1, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.intro\n\u03b2 : Type u_1\n\u03b9 : Type u_2\nX : \u03b9 \u2192 Type u_3\nt : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\nf : (i : \u03b9) \u2192 \u03b2 \u2192 X i\nthis : TopologicalSpace \u03b2 := \u2a05 (i : \u03b9), induced (f i) (t i)\nU : (i : \u03b9) \u2192 Set (X i)\nF : Finset \u03b9\nh1 : \u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i\n\u22a2 (fun x i => f i x) \u207b\u00b9' Set.pi (\u2191F) U \u2208\n {S | \u2203 U F, (\u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i) \u2227 S = \u22c2 (i : \u03b9) (_ : i \u2208 F), f i \u207b\u00b9' U i}\n[PROOFSTEP]\nrefine' \u27e8U, F, h1, _\u27e9\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.intro\n\u03b2 : Type u_1\n\u03b9 : Type u_2\nX : \u03b9 \u2192 Type u_3\nt : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\nf : (i : \u03b9) \u2192 \u03b2 \u2192 X i\nthis : TopologicalSpace \u03b2 := \u2a05 (i : \u03b9), induced (f i) (t i)\nU : (i : \u03b9) \u2192 Set (X i)\nF : Finset \u03b9\nh1 : \u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i\n\u22a2 (fun x i => f i x) \u207b\u00b9' Set.pi (\u2191F) U = \u22c2 (i : \u03b9) (_ : i \u2208 F), f i \u207b\u00b9' U i\n[PROOFSTEP]\nsimp_rw [pi_def, Set.preimage_iInter]\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.intro\n\u03b2 : Type u_1\n\u03b9 : Type u_2\nX : \u03b9 \u2192 Type u_3\nt : (i : \u03b9) \u2192 TopologicalSpace (X i)\nT : (i : \u03b9) \u2192 Set (Set (X i))\ncond : \u2200 (i : \u03b9), IsTopologicalBasis (T i)\nf : (i : \u03b9) \u2192 \u03b2 \u2192 X i\nthis : TopologicalSpace \u03b2 := \u2a05 (i : \u03b9), induced (f i) (t i)\nU : (i : \u03b9) \u2192 Set (X i)\nF : Finset \u03b9\nh1 : \u2200 (i : \u03b9), i \u2208 F \u2192 U i \u2208 T i\n\u22a2 \u22c2 (i : \u03b9) (_ : i \u2208 \u2191F), (fun x i => f i x) \u207b\u00b9' (eval i \u207b\u00b9' U i) = \u22c2 (i : \u03b9) (_ : i \u2208 F), f i \u207b\u00b9' U i\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ns : Set \u03b1\ninst\u271d : SeparableSpace \u2191s\nhs : Dense s\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Countable t \u2227 Dense t \u2227 (\u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t) \u2227 \u2200 (x : \u03b1), IsTop x \u2192 x \u2208 s \u2192 x \u2208 t\n[PROOFSTEP]\nrcases hs.exists_countable_dense_subset with \u27e8t, hts, htc, htd\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ns : Set \u03b1\ninst\u271d : SeparableSpace \u2191s\nhs : Dense s\nt : Set \u03b1\nhts : t \u2286 s\nhtc : Set.Countable t\nhtd : Dense t\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Countable t \u2227 Dense t \u2227 (\u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t) \u2227 \u2200 (x : \u03b1), IsTop x \u2192 x \u2208 s \u2192 x \u2208 t\n[PROOFSTEP]\nrefine' \u27e8(t \u222a ({x | IsBot x} \u222a {x | IsTop x})) \u2229 s, _, _, _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ns : Set \u03b1\ninst\u271d : SeparableSpace \u2191s\nhs : Dense s\nt : Set \u03b1\nhts : t \u2286 s\nhtc : Set.Countable t\nhtd : Dense t\n\u22a2 (t \u222a ({x | IsBot x} \u222a {x | IsTop x})) \u2229 s \u2286 s\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ns : Set \u03b1\ninst\u271d : SeparableSpace \u2191s\nhs : Dense s\nt : Set \u03b1\nhts : t \u2286 s\nhtc : Set.Countable t\nhtd : Dense t\n\u22a2 Set.Countable ((t \u222a ({x | IsBot x} \u222a {x | IsTop x})) \u2229 s)\ncase intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ns : Set \u03b1\ninst\u271d : SeparableSpace \u2191s\nhs : Dense s\nt : Set \u03b1\nhts : t \u2286 s\nhtc : Set.Countable t\nhtd : Dense t\n\u22a2 Dense ((t \u222a ({x | IsBot x} \u222a {x | IsTop x})) \u2229 s)\ncase intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ns : Set \u03b1\ninst\u271d : SeparableSpace \u2191s\nhs : Dense s\nt : Set \u03b1\nhts : t \u2286 s\nhtc : Set.Countable t\nhtd : Dense t\n\u22a2 \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 (t \u222a ({x | IsBot x} \u222a {x | IsTop x})) \u2229 s\ncase intro.intro.intro.refine'_5\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ns : Set \u03b1\ninst\u271d : SeparableSpace \u2191s\nhs : Dense s\nt : Set \u03b1\nhts : t \u2286 s\nhtc : Set.Countable t\nhtd : Dense t\n\u22a2 \u2200 (x : \u03b1), IsTop x \u2192 x \u2208 s \u2192 x \u2208 (t \u222a ({x | IsBot x} \u222a {x | IsTop x})) \u2229 s\n[PROOFSTEP]\nexacts [inter_subset_right _ _,\n (htc.union ((countable_isBot \u03b1).union (countable_isTop \u03b1))).mono (inter_subset_left _ _),\n htd.mono (subset_inter (subset_union_left _ _) hts), fun x hx hxs => \u27e8Or.inr <| Or.inl hx, hxs\u27e9, fun x hx hxs =>\n \u27e8Or.inr <| Or.inr hx, hxs\u27e9]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SeparableSpace \u03b1\ninst\u271d : PartialOrder \u03b1\n\u22a2 \u2203 s, Set.Countable s \u2227 Dense s \u2227 (\u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s) \u2227 \u2200 (x : \u03b1), IsTop x \u2192 x \u2208 s\n[PROOFSTEP]\nsimpa using dense_univ.exists_countable_dense_subset_bot_top\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type ?u.70125\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : FirstCountableTopology \u03b1\ninst\u271d : FirstCountableTopology \u03b2\nx\u271d : \u03b1 \u00d7 \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 IsCountablyGenerated (\ud835\udcdd (x, y))\n[PROOFSTEP]\nrw [nhds_prod_eq]\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type ?u.70125\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : FirstCountableTopology \u03b1\ninst\u271d : FirstCountableTopology \u03b2\nx\u271d : \u03b1 \u00d7 \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 IsCountablyGenerated (\ud835\udcdd x \u00d7\u02e2 \ud835\udcdd y)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), FirstCountableTopology (\u03c0 i)\nf : (i : \u03b9) \u2192 \u03c0 i\n\u22a2 IsCountablyGenerated (\ud835\udcdd f)\n[PROOFSTEP]\nrw [nhds_pi]\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), FirstCountableTopology (\u03c0 i)\nf : (i : \u03b9) \u2192 \u03c0 i\n\u22a2 IsCountablyGenerated (Filter.pi fun i => \ud835\udcdd (f i))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u22a2 \u2203 b, Set.Countable b \u2227 \u00ac\u2205 \u2208 b \u2227 IsTopologicalBasis b\n[PROOFSTEP]\nobtain \u27e8b, hb\u2081, hb\u2082\u27e9 := @SecondCountableTopology.is_open_generated_countable \u03b1 _ _\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nb : Set (Set \u03b1)\nhb\u2081 : Set.Countable b\nhb\u2082 : t = generateFrom b\n\u22a2 \u2203 b, Set.Countable b \u2227 \u00ac\u2205 \u2208 b \u2227 IsTopologicalBasis b\n[PROOFSTEP]\nrefine' \u27e8_, _, not_mem_diff_of_mem _, (isTopologicalBasis_of_subbasis hb\u2082).diff_empty\u27e9\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nb : Set (Set \u03b1)\nhb\u2081 : Set.Countable b\nhb\u2082 : t = generateFrom b\n\u22a2 Set.Countable ((fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 b} \\ {\u2205})\ncase intro.intro.refine'_2\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nb : Set (Set \u03b1)\nhb\u2081 : Set.Countable b\nhb\u2082 : t = generateFrom b\n\u22a2 \u2205 \u2208 {\u2205}\n[PROOFSTEP]\nexacts [((countable_setOf_finite_subset hb\u2081).image _).mono (diff_subset _ _), rfl]\n[GOAL]\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\nt : TopologicalSpace \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 SecondCountableTopology \u03b1\n[PROOFSTEP]\nrcases@SecondCountableTopology.is_open_generated_countable \u03b2 _ _ with \u27e8b, hb, eq\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\nt : TopologicalSpace \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nb : Set (Set \u03b2)\nhb : Set.Countable b\neq : t = generateFrom b\n\u22a2 SecondCountableTopology \u03b1\n[PROOFSTEP]\nletI := t.induced f\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\nt : TopologicalSpace \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nb : Set (Set \u03b2)\nhb : Set.Countable b\neq : t = generateFrom b\nthis : TopologicalSpace \u03b1 := induced f t\n\u22a2 SecondCountableTopology \u03b1\n[PROOFSTEP]\nrefine' { is_open_generated_countable := \u27e8preimage f '' b, hb.image _, _\u27e9 }\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\nt : TopologicalSpace \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nb : Set (Set \u03b2)\nhb : Set.Countable b\neq : t = generateFrom b\nthis : TopologicalSpace \u03b1 := induced f t\n\u22a2 induced f t = generateFrom (preimage f '' b)\n[PROOFSTEP]\nrw [eq, induced_generateFrom_eq]\n[GOAL]\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\nt : \u03b9 \u2192 TopologicalSpace \u03b1\nht : \u2200 (i : \u03b9), SecondCountableTopology \u03b1\n\u22a2 SecondCountableTopology \u03b1\n[PROOFSTEP]\nrw [funext fun i => @eq_generateFrom_countableBasis \u03b1 (t i) (ht i), \u2190 generateFrom_iUnion]\n[GOAL]\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\nt : \u03b9 \u2192 TopologicalSpace \u03b1\nht : \u2200 (i : \u03b9), SecondCountableTopology \u03b1\n\u22a2 SecondCountableTopology \u03b1\n[PROOFSTEP]\nexact\n SecondCountableTopology.mk' <|\n countable_iUnion fun i =>\n @countable_countableBasis _ (t i)\n (ht i)\n -- TODO: more fine grained instances for first_countable_topology, separable_space, t2_space, ...\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u22a2 SeparableSpace \u03b1\n[PROOFSTEP]\nchoose p hp using fun s : countableBasis \u03b1 => nonempty_of_mem_countableBasis s.2\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\np : \u2191(countableBasis \u03b1) \u2192 \u03b1\nhp : \u2200 (s : \u2191(countableBasis \u03b1)), p s \u2208 \u2191s\n\u22a2 SeparableSpace \u03b1\n[PROOFSTEP]\nexact\n \u27e8\u27e8range p, countable_range _,\n (isBasis_countableBasis \u03b1).dense_iff.2 fun o ho _ => \u27e8p \u27e8o, ho\u27e9, hp _, mem_range_self _\u27e9\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_1\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), IsOpen (s i)\n\u22a2 \u2203 T, Set.Countable T \u2227 \u22c3 (i : \u03b9) (_ : i \u2208 T), s i = \u22c3 (i : \u03b9), s i\n[PROOFSTEP]\nlet B := {b \u2208 countableBasis \u03b1 | \u2203 i, b \u2286 s i}\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_1\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), IsOpen (s i)\nB : Set (Set \u03b1) := {b | b \u2208 countableBasis \u03b1 \u2227 \u2203 i, b \u2286 s i}\n\u22a2 \u2203 T, Set.Countable T \u2227 \u22c3 (i : \u03b9) (_ : i \u2208 T), s i = \u22c3 (i : \u03b9), s i\n[PROOFSTEP]\nchoose f hf using fun b : B => b.2.2\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_1\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), IsOpen (s i)\nB : Set (Set \u03b1) := {b | b \u2208 countableBasis \u03b1 \u2227 \u2203 i, b \u2286 s i}\nf : \u2191B \u2192 \u03b9\nhf : \u2200 (b : \u2191B), \u2191b \u2286 s (f b)\n\u22a2 \u2203 T, Set.Countable T \u2227 \u22c3 (i : \u03b9) (_ : i \u2208 T), s i = \u22c3 (i : \u03b9), s i\n[PROOFSTEP]\nhaveI : Encodable B := ((countable_countableBasis \u03b1).mono (sep_subset _ _)).toEncodable\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_1\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), IsOpen (s i)\nB : Set (Set \u03b1) := {b | b \u2208 countableBasis \u03b1 \u2227 \u2203 i, b \u2286 s i}\nf : \u2191B \u2192 \u03b9\nhf : \u2200 (b : \u2191B), \u2191b \u2286 s (f b)\nthis : Encodable \u2191B\n\u22a2 \u2203 T, Set.Countable T \u2227 \u22c3 (i : \u03b9) (_ : i \u2208 T), s i = \u22c3 (i : \u03b9), s i\n[PROOFSTEP]\nrefine' \u27e8_, countable_range f, (iUnion\u2082_subset_iUnion _ _).antisymm (sUnion_subset _)\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_1\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), IsOpen (s i)\nB : Set (Set \u03b1) := {b | b \u2208 countableBasis \u03b1 \u2227 \u2203 i, b \u2286 s i}\nf : \u2191B \u2192 \u03b9\nhf : \u2200 (b : \u2191B), \u2191b \u2286 s (f b)\nthis : Encodable \u2191B\n\u22a2 \u2200 (t' : Set \u03b1), (t' \u2208 range fun i => s i) \u2192 t' \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 range f), s i\n[PROOFSTEP]\nrintro _ \u27e8i, rfl\u27e9 x xs\n[GOAL]\ncase intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_1\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), IsOpen (s i)\nB : Set (Set \u03b1) := {b | b \u2208 countableBasis \u03b1 \u2227 \u2203 i, b \u2286 s i}\nf : \u2191B \u2192 \u03b9\nhf : \u2200 (b : \u2191B), \u2191b \u2286 s (f b)\nthis : Encodable \u2191B\ni : \u03b9\nx : \u03b1\nxs : x \u2208 (fun i => s i) i\n\u22a2 x \u2208 \u22c3 (i : \u03b9) (_ : i \u2208 range f), s i\n[PROOFSTEP]\nrcases(isBasis_countableBasis \u03b1).exists_subset_of_mem_open xs (H _) with \u27e8b, hb, xb, bs\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_1\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), IsOpen (s i)\nB : Set (Set \u03b1) := {b | b \u2208 countableBasis \u03b1 \u2227 \u2203 i, b \u2286 s i}\nf : \u2191B \u2192 \u03b9\nhf : \u2200 (b : \u2191B), \u2191b \u2286 s (f b)\nthis : Encodable \u2191B\ni : \u03b9\nx : \u03b1\nxs : x \u2208 (fun i => s i) i\nb : Set \u03b1\nhb : b \u2208 countableBasis \u03b1\nxb : x \u2208 b\nbs : b \u2286 (fun i => s i) i\n\u22a2 x \u2208 \u22c3 (i : \u03b9) (_ : i \u2208 range f), s i\n[PROOFSTEP]\nexact \u27e8_, \u27e8_, rfl\u27e9, _, \u27e8\u27e8\u27e8_, hb, _, bs\u27e9, rfl\u27e9, rfl\u27e9, hf _ xb\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nS : Set (Set \u03b1)\nH : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsOpen s\nT : Set \u2191S\ncT : Set.Countable T\nhT : \u22c3 (i : \u2191S) (_ : i \u2208 T), \u2191i = \u22c3 (i : \u2191S), \u2191i\n\u22a2 \u22c3\u2080 (Subtype.val '' T) = \u22c3\u2080 S\n[PROOFSTEP]\nrwa [sUnion_image, sUnion_eq_iUnion]\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b1 \u2192 Set \u03b1\nhf : \u2200 (x : \u03b1), f x \u2208 \ud835\udcdd x\n\u22a2 \u2203 s, Set.Countable s \u2227 \u22c3 (x : \u03b1) (_ : x \u2208 s), f x = univ\n[PROOFSTEP]\nrcases isOpen_iUnion_countable (fun x => interior (f x)) fun x => isOpen_interior with \u27e8s, hsc, hsU\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b1 \u2192 Set \u03b1\nhf : \u2200 (x : \u03b1), f x \u2208 \ud835\udcdd x\ns : Set \u03b1\nhsc : Set.Countable s\nhsU : \u22c3 (i : \u03b1) (_ : i \u2208 s), interior (f i) = \u22c3 (i : \u03b1), interior (f i)\n\u22a2 \u2203 s, Set.Countable s \u2227 \u22c3 (x : \u03b1) (_ : x \u2208 s), f x = univ\n[PROOFSTEP]\nsuffices : \u22c3 x \u2208 s, interior (f x) = univ\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b1 \u2192 Set \u03b1\nhf : \u2200 (x : \u03b1), f x \u2208 \ud835\udcdd x\ns : Set \u03b1\nhsc : Set.Countable s\nhsU : \u22c3 (i : \u03b1) (_ : i \u2208 s), interior (f i) = \u22c3 (i : \u03b1), interior (f i)\nthis : \u22c3 (x : \u03b1) (_ : x \u2208 s), interior (f x) = univ\n\u22a2 \u2203 s, Set.Countable s \u2227 \u22c3 (x : \u03b1) (_ : x \u2208 s), f x = univ\ncase this\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b1 \u2192 Set \u03b1\nhf : \u2200 (x : \u03b1), f x \u2208 \ud835\udcdd x\ns : Set \u03b1\nhsc : Set.Countable s\nhsU : \u22c3 (i : \u03b1) (_ : i \u2208 s), interior (f i) = \u22c3 (i : \u03b1), interior (f i)\n\u22a2 \u22c3 (x : \u03b1) (_ : x \u2208 s), interior (f x) = univ\n[PROOFSTEP]\nexact \u27e8s, hsc, flip eq_univ_of_subset this <| iUnion\u2082_mono fun _ _ => interior_subset\u27e9\n[GOAL]\ncase this\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b1 \u2192 Set \u03b1\nhf : \u2200 (x : \u03b1), f x \u2208 \ud835\udcdd x\ns : Set \u03b1\nhsc : Set.Countable s\nhsU : \u22c3 (i : \u03b1) (_ : i \u2208 s), interior (f i) = \u22c3 (i : \u03b1), interior (f i)\n\u22a2 \u22c3 (x : \u03b1) (_ : x \u2208 s), interior (f x) = univ\n[PROOFSTEP]\nsimp only [hsU, eq_univ_iff_forall, mem_iUnion]\n[GOAL]\ncase this\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b1 \u2192 Set \u03b1\nhf : \u2200 (x : \u03b1), f x \u2208 \ud835\udcdd x\ns : Set \u03b1\nhsc : Set.Countable s\nhsU : \u22c3 (i : \u03b1) (_ : i \u2208 s), interior (f i) = \u22c3 (i : \u03b1), interior (f i)\n\u22a2 \u2200 (x : \u03b1), \u2203 i, x \u2208 interior (f i)\n[PROOFSTEP]\nexact fun x => \u27e8x, mem_interior_iff_mem_nhds.2 (hf x)\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2208 \ud835\udcdd[s] x\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Countable t \u2227 s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), f x\n[PROOFSTEP]\nhave : \u2200 x : s, (\u2191) \u207b\u00b9' f x \u2208 \ud835\udcdd x := fun x => preimage_coe_mem_nhds_subtype.2 (hf x x.2)\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2208 \ud835\udcdd[s] x\nthis : \u2200 (x : \u2191s), Subtype.val \u207b\u00b9' f \u2191x \u2208 \ud835\udcdd x\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Countable t \u2227 s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), f x\n[PROOFSTEP]\nrcases countable_cover_nhds this with \u27e8t, htc, htU\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2208 \ud835\udcdd[s] x\nthis : \u2200 (x : \u2191s), Subtype.val \u207b\u00b9' f \u2191x \u2208 \ud835\udcdd x\nt : Set \u2191s\nhtc : Set.Countable t\nhtU : \u22c3 (x : \u2191s) (_ : x \u2208 t), Subtype.val \u207b\u00b9' f \u2191x = univ\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Countable t \u2227 s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), f x\n[PROOFSTEP]\nrefine' \u27e8(\u2191) '' t, Subtype.coe_image_subset _ _, htc.image _, fun x hx => _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2208 \ud835\udcdd[s] x\nthis : \u2200 (x : \u2191s), Subtype.val \u207b\u00b9' f \u2191x \u2208 \ud835\udcdd x\nt : Set \u2191s\nhtc : Set.Countable t\nhtU : \u22c3 (x : \u2191s) (_ : x \u2208 t), Subtype.val \u207b\u00b9' f \u2191x = univ\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 Subtype.val '' t), f x\n[PROOFSTEP]\nsimp only [biUnion_image, eq_univ_iff_forall, \u2190 preimage_iUnion, mem_preimage] at htU \u22a2\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2208 \ud835\udcdd[s] x\nthis : \u2200 (x : \u2191s), Subtype.val \u207b\u00b9' f \u2191x \u2208 \ud835\udcdd x\nt : Set \u2191s\nhtc : Set.Countable t\nx : \u03b1\nhx : x \u2208 s\nhtU : \u2200 (x : \u2191s), \u2191x \u2208 \u22c3 (i : \u2191s) (_ : i \u2208 t), f \u2191i\n\u22a2 x \u2208 \u22c3 (y : \u2191s) (_ : y \u2208 t), f \u2191y\n[PROOFSTEP]\nexact htU \u27e8x, hx\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\n\u22a2 IsTopologicalBasis (\u22c3 (i : \u03b9), (fun u => Sigma.mk i '' u) '' s i)\n[PROOFSTEP]\napply isTopologicalBasis_of_open_of_nhds\n[GOAL]\ncase h_open\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\n\u22a2 \u2200 (u : Set ((i : \u03b9) \u00d7 E i)), u \u2208 \u22c3 (i : \u03b9), (fun u => Sigma.mk i '' u) '' s i \u2192 IsOpen u\n[PROOFSTEP]\nintro u hu\n[GOAL]\ncase h_open\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\nu : Set ((i : \u03b9) \u00d7 E i)\nhu : u \u2208 \u22c3 (i : \u03b9), (fun u => Sigma.mk i '' u) '' s i\n\u22a2 IsOpen u\n[PROOFSTEP]\nobtain \u27e8i, t, ts, rfl\u27e9 : \u2203 (i : \u03b9) (t : Set (E i)), t \u2208 s i \u2227 Sigma.mk i '' t = u := by\n simpa only [mem_iUnion, mem_image] using hu\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\nu : Set ((i : \u03b9) \u00d7 E i)\nhu : u \u2208 \u22c3 (i : \u03b9), (fun u => Sigma.mk i '' u) '' s i\n\u22a2 \u2203 i t, t \u2208 s i \u2227 Sigma.mk i '' t = u\n[PROOFSTEP]\nsimpa only [mem_iUnion, mem_image] using hu\n[GOAL]\ncase h_open.intro.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\ni : \u03b9\nt : Set (E i)\nts : t \u2208 s i\nhu : Sigma.mk i '' t \u2208 \u22c3 (i : \u03b9), (fun u => Sigma.mk i '' u) '' s i\n\u22a2 IsOpen (Sigma.mk i '' t)\n[PROOFSTEP]\nexact isOpenMap_sigmaMk _ ((hs i).isOpen ts)\n[GOAL]\ncase h_nhds\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\n\u22a2 \u2200 (a : (i : \u03b9) \u00d7 E i) (u : Set ((i : \u03b9) \u00d7 E i)),\n a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 \u22c3 (i : \u03b9), (fun u => Sigma.mk i '' u) '' s i \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nrintro \u27e8i, x\u27e9 u hxu u_open\n[GOAL]\ncase h_nhds.mk\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\ni : \u03b9\nx : E i\nu : Set ((i : \u03b9) \u00d7 E i)\nhxu : { fst := i, snd := x } \u2208 u\nu_open : IsOpen u\n\u22a2 \u2203 v, v \u2208 \u22c3 (i : \u03b9), (fun u => Sigma.mk i '' u) '' s i \u2227 { fst := i, snd := x } \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nhave hx : x \u2208 Sigma.mk i \u207b\u00b9' u := hxu\n[GOAL]\ncase h_nhds.mk\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\ni : \u03b9\nx : E i\nu : Set ((i : \u03b9) \u00d7 E i)\nhxu : { fst := i, snd := x } \u2208 u\nu_open : IsOpen u\nhx : x \u2208 Sigma.mk i \u207b\u00b9' u\n\u22a2 \u2203 v, v \u2208 \u22c3 (i : \u03b9), (fun u => Sigma.mk i '' u) '' s i \u2227 { fst := i, snd := x } \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nobtain \u27e8v, vs, xv, hv\u27e9 : \u2203 (v : Set (E i)), v \u2208 s i \u2227 x \u2208 v \u2227 v \u2286 Sigma.mk i \u207b\u00b9' u :=\n (hs i).exists_subset_of_mem_open hx (isOpen_sigma_iff.1 u_open i)\n[GOAL]\ncase h_nhds.mk.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\ni : \u03b9\nx : E i\nu : Set ((i : \u03b9) \u00d7 E i)\nhxu : { fst := i, snd := x } \u2208 u\nu_open : IsOpen u\nhx : x \u2208 Sigma.mk i \u207b\u00b9' u\nv : Set (E i)\nvs : v \u2208 s i\nxv : x \u2208 v\nhv : v \u2286 Sigma.mk i \u207b\u00b9' u\n\u22a2 \u2203 v, v \u2208 \u22c3 (i : \u03b9), (fun u => Sigma.mk i '' u) '' s i \u2227 { fst := i, snd := x } \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nexact \u27e8Sigma.mk i '' v, mem_iUnion.2 \u27e8i, mem_image_of_mem _ vs\u27e9, mem_image_of_mem _ xv, image_subset_iff.2 hv\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : \u2200 (i : \u03b9), SecondCountableTopology (E i)\n\u22a2 SecondCountableTopology ((i : \u03b9) \u00d7 E i)\n[PROOFSTEP]\nlet b := \u22c3 i : \u03b9, (fun u => (Sigma.mk i '' u : Set (\u03a3 i, E i))) '' countableBasis (E i)\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : \u2200 (i : \u03b9), SecondCountableTopology (E i)\nb : Set (Set ((i : \u03b9) \u00d7 E i)) := \u22c3 (i : \u03b9), (fun u => Sigma.mk i '' u) '' countableBasis (E i)\n\u22a2 SecondCountableTopology ((i : \u03b9) \u00d7 E i)\n[PROOFSTEP]\nhave A : IsTopologicalBasis b := IsTopologicalBasis.sigma fun i => isBasis_countableBasis _\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : \u2200 (i : \u03b9), SecondCountableTopology (E i)\nb : Set (Set ((i : \u03b9) \u00d7 E i)) := \u22c3 (i : \u03b9), (fun u => Sigma.mk i '' u) '' countableBasis (E i)\nA : IsTopologicalBasis b\n\u22a2 SecondCountableTopology ((i : \u03b9) \u00d7 E i)\n[PROOFSTEP]\nhave B : b.Countable := countable_iUnion fun i => (countable_countableBasis _).image _\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : \u2200 (i : \u03b9), SecondCountableTopology (E i)\nb : Set (Set ((i : \u03b9) \u00d7 E i)) := \u22c3 (i : \u03b9), (fun u => Sigma.mk i '' u) '' countableBasis (E i)\nA : IsTopologicalBasis b\nB : Set.Countable b\n\u22a2 SecondCountableTopology ((i : \u03b9) \u00d7 E i)\n[PROOFSTEP]\nexact A.secondCountableTopology B\n[GOAL]\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (Set \u03b1)\nhs : IsTopologicalBasis s\nt : Set (Set \u03b2)\nht : IsTopologicalBasis t\n\u22a2 IsTopologicalBasis ((fun u => Sum.inl '' u) '' s \u222a (fun u => Sum.inr '' u) '' t)\n[PROOFSTEP]\napply isTopologicalBasis_of_open_of_nhds\n[GOAL]\ncase h_open\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (Set \u03b1)\nhs : IsTopologicalBasis s\nt : Set (Set \u03b2)\nht : IsTopologicalBasis t\n\u22a2 \u2200 (u : Set (\u03b1 \u2295 \u03b2)), u \u2208 (fun u => Sum.inl '' u) '' s \u222a (fun u => Sum.inr '' u) '' t \u2192 IsOpen u\n[PROOFSTEP]\nrintro u (\u27e8w, hw, rfl\u27e9 | \u27e8w, hw, rfl\u27e9)\n[GOAL]\ncase h_open.inl.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (Set \u03b1)\nhs : IsTopologicalBasis s\nt : Set (Set \u03b2)\nht : IsTopologicalBasis t\nw : Set \u03b1\nhw : w \u2208 s\n\u22a2 IsOpen ((fun u => Sum.inl '' u) w)\n[PROOFSTEP]\nexact openEmbedding_inl.isOpenMap w (hs.isOpen hw)\n[GOAL]\ncase h_open.inr.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (Set \u03b1)\nhs : IsTopologicalBasis s\nt : Set (Set \u03b2)\nht : IsTopologicalBasis t\nw : Set \u03b2\nhw : w \u2208 t\n\u22a2 IsOpen ((fun u => Sum.inr '' u) w)\n[PROOFSTEP]\nexact openEmbedding_inr.isOpenMap w (ht.isOpen hw)\n[GOAL]\ncase h_nhds\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (Set \u03b1)\nhs : IsTopologicalBasis s\nt : Set (Set \u03b2)\nht : IsTopologicalBasis t\n\u22a2 \u2200 (a : \u03b1 \u2295 \u03b2) (u : Set (\u03b1 \u2295 \u03b2)),\n a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 (fun u => Sum.inl '' u) '' s \u222a (fun u => Sum.inr '' u) '' t \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nrintro (x | x) u hxu u_open\n[GOAL]\ncase h_nhds.inl\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (Set \u03b1)\nhs : IsTopologicalBasis s\nt : Set (Set \u03b2)\nht : IsTopologicalBasis t\nx : \u03b1\nu : Set (\u03b1 \u2295 \u03b2)\nhxu : Sum.inl x \u2208 u\nu_open : IsOpen u\n\u22a2 \u2203 v, v \u2208 (fun u => Sum.inl '' u) '' s \u222a (fun u => Sum.inr '' u) '' t \u2227 Sum.inl x \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nobtain \u27e8v, vs, xv, vu\u27e9 : \u2203 v \u2208 s, x \u2208 v \u2227 v \u2286 Sum.inl \u207b\u00b9' u :=\n hs.exists_subset_of_mem_open hxu (isOpen_sum_iff.1 u_open).1\n[GOAL]\ncase h_nhds.inl.intro.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (Set \u03b1)\nhs : IsTopologicalBasis s\nt : Set (Set \u03b2)\nht : IsTopologicalBasis t\nx : \u03b1\nu : Set (\u03b1 \u2295 \u03b2)\nhxu : Sum.inl x \u2208 u\nu_open : IsOpen u\nv : Set \u03b1\nvs : v \u2208 s\nxv : x \u2208 v\nvu : v \u2286 Sum.inl \u207b\u00b9' u\n\u22a2 \u2203 v, v \u2208 (fun u => Sum.inl '' u) '' s \u222a (fun u => Sum.inr '' u) '' t \u2227 Sum.inl x \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nexact \u27e8Sum.inl '' v, mem_union_left _ (mem_image_of_mem _ vs), mem_image_of_mem _ xv, image_subset_iff.2 vu\u27e9\n[GOAL]\ncase h_nhds.inr\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (Set \u03b1)\nhs : IsTopologicalBasis s\nt : Set (Set \u03b2)\nht : IsTopologicalBasis t\nx : \u03b2\nu : Set (\u03b1 \u2295 \u03b2)\nhxu : Sum.inr x \u2208 u\nu_open : IsOpen u\n\u22a2 \u2203 v, v \u2208 (fun u => Sum.inl '' u) '' s \u222a (fun u => Sum.inr '' u) '' t \u2227 Sum.inr x \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nobtain \u27e8v, vs, xv, vu\u27e9 : \u2203 v \u2208 t, x \u2208 v \u2227 v \u2286 Sum.inr \u207b\u00b9' u :=\n ht.exists_subset_of_mem_open hxu (isOpen_sum_iff.1 u_open).2\n[GOAL]\ncase h_nhds.inr.intro.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (Set \u03b1)\nhs : IsTopologicalBasis s\nt : Set (Set \u03b2)\nht : IsTopologicalBasis t\nx : \u03b2\nu : Set (\u03b1 \u2295 \u03b2)\nhxu : Sum.inr x \u2208 u\nu_open : IsOpen u\nv : Set \u03b2\nvs : v \u2208 t\nxv : x \u2208 v\nvu : v \u2286 Sum.inr \u207b\u00b9' u\n\u22a2 \u2203 v, v \u2208 (fun u => Sum.inl '' u) '' s \u222a (fun u => Sum.inr '' u) '' t \u2227 Sum.inr x \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nexact \u27e8Sum.inr '' v, mem_union_right _ (mem_image_of_mem _ vs), mem_image_of_mem _ xv, image_subset_iff.2 vu\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b2\n\u22a2 SecondCountableTopology (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\nlet b := (fun u => Sum.inl '' u) '' countableBasis \u03b1 \u222a (fun u => Sum.inr '' u) '' countableBasis \u03b2\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b2\nb : Set (Set (\u03b1 \u2295 \u03b2)) := (fun u => Sum.inl '' u) '' countableBasis \u03b1 \u222a (fun u => Sum.inr '' u) '' countableBasis \u03b2\n\u22a2 SecondCountableTopology (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\nhave A : IsTopologicalBasis b := (isBasis_countableBasis \u03b1).sum (isBasis_countableBasis \u03b2)\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b2\nb : Set (Set (\u03b1 \u2295 \u03b2)) := (fun u => Sum.inl '' u) '' countableBasis \u03b1 \u222a (fun u => Sum.inr '' u) '' countableBasis \u03b2\nA : IsTopologicalBasis b\n\u22a2 SecondCountableTopology (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\nhave B : b.Countable :=\n (Countable.image (countable_countableBasis _) _).union (Countable.image (countable_countableBasis _) _)\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b2\nb : Set (Set (\u03b1 \u2295 \u03b2)) := (fun u => Sum.inl '' u) '' countableBasis \u03b1 \u222a (fun u => Sum.inr '' u) '' countableBasis \u03b2\nA : IsTopologicalBasis b\nB : Set.Countable b\n\u22a2 SecondCountableTopology (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\nexact A.secondCountableTopology B\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\nY : Type u_2\ninst\u271d : TopologicalSpace Y\n\u03c0 : X \u2192 Y\nV : Set (Set X)\nhV : IsTopologicalBasis V\nh' : QuotientMap \u03c0\nh : IsOpenMap \u03c0\n\u22a2 IsTopologicalBasis (image \u03c0 '' V)\n[PROOFSTEP]\napply isTopologicalBasis_of_open_of_nhds\n[GOAL]\ncase h_open\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\nY : Type u_2\ninst\u271d : TopologicalSpace Y\n\u03c0 : X \u2192 Y\nV : Set (Set X)\nhV : IsTopologicalBasis V\nh' : QuotientMap \u03c0\nh : IsOpenMap \u03c0\n\u22a2 \u2200 (u : Set Y), u \u2208 image \u03c0 '' V \u2192 IsOpen u\n[PROOFSTEP]\nrintro - \u27e8U, U_in_V, rfl\u27e9\n[GOAL]\ncase h_open.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\nY : Type u_2\ninst\u271d : TopologicalSpace Y\n\u03c0 : X \u2192 Y\nV : Set (Set X)\nhV : IsTopologicalBasis V\nh' : QuotientMap \u03c0\nh : IsOpenMap \u03c0\nU : Set X\nU_in_V : U \u2208 V\n\u22a2 IsOpen (\u03c0 '' U)\n[PROOFSTEP]\napply h U (hV.isOpen U_in_V)\n[GOAL]\ncase h_nhds\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\nY : Type u_2\ninst\u271d : TopologicalSpace Y\n\u03c0 : X \u2192 Y\nV : Set (Set X)\nhV : IsTopologicalBasis V\nh' : QuotientMap \u03c0\nh : IsOpenMap \u03c0\n\u22a2 \u2200 (a : Y) (u : Set Y), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 image \u03c0 '' V \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nintro y U y_in_U U_open\n[GOAL]\ncase h_nhds\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\nY : Type u_2\ninst\u271d : TopologicalSpace Y\n\u03c0 : X \u2192 Y\nV : Set (Set X)\nhV : IsTopologicalBasis V\nh' : QuotientMap \u03c0\nh : IsOpenMap \u03c0\ny : Y\nU : Set Y\ny_in_U : y \u2208 U\nU_open : IsOpen U\n\u22a2 \u2203 v, v \u2208 image \u03c0 '' V \u2227 y \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 := h'.surjective y\n[GOAL]\ncase h_nhds.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\nY : Type u_2\ninst\u271d : TopologicalSpace Y\n\u03c0 : X \u2192 Y\nV : Set (Set X)\nhV : IsTopologicalBasis V\nh' : QuotientMap \u03c0\nh : IsOpenMap \u03c0\nU : Set Y\nU_open : IsOpen U\nx : X\ny_in_U : \u03c0 x \u2208 U\n\u22a2 \u2203 v, v \u2208 image \u03c0 '' V \u2227 \u03c0 x \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nlet W := \u03c0 \u207b\u00b9' U\n[GOAL]\ncase h_nhds.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\nY : Type u_2\ninst\u271d : TopologicalSpace Y\n\u03c0 : X \u2192 Y\nV : Set (Set X)\nhV : IsTopologicalBasis V\nh' : QuotientMap \u03c0\nh : IsOpenMap \u03c0\nU : Set Y\nU_open : IsOpen U\nx : X\ny_in_U : \u03c0 x \u2208 U\nW : Set X := \u03c0 \u207b\u00b9' U\n\u22a2 \u2203 v, v \u2208 image \u03c0 '' V \u2227 \u03c0 x \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nhave x_in_W : x \u2208 W := y_in_U\n[GOAL]\ncase h_nhds.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\nY : Type u_2\ninst\u271d : TopologicalSpace Y\n\u03c0 : X \u2192 Y\nV : Set (Set X)\nhV : IsTopologicalBasis V\nh' : QuotientMap \u03c0\nh : IsOpenMap \u03c0\nU : Set Y\nU_open : IsOpen U\nx : X\ny_in_U : \u03c0 x \u2208 U\nW : Set X := \u03c0 \u207b\u00b9' U\nx_in_W : x \u2208 W\n\u22a2 \u2203 v, v \u2208 image \u03c0 '' V \u2227 \u03c0 x \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nhave W_open : IsOpen W := U_open.preimage h'.continuous\n[GOAL]\ncase h_nhds.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\nY : Type u_2\ninst\u271d : TopologicalSpace Y\n\u03c0 : X \u2192 Y\nV : Set (Set X)\nhV : IsTopologicalBasis V\nh' : QuotientMap \u03c0\nh : IsOpenMap \u03c0\nU : Set Y\nU_open : IsOpen U\nx : X\ny_in_U : \u03c0 x \u2208 U\nW : Set X := \u03c0 \u207b\u00b9' U\nx_in_W : x \u2208 W\nW_open : IsOpen W\n\u22a2 \u2203 v, v \u2208 image \u03c0 '' V \u2227 \u03c0 x \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nobtain \u27e8Z, Z_in_V, x_in_Z, Z_in_W\u27e9 := hV.exists_subset_of_mem_open x_in_W W_open\n[GOAL]\ncase h_nhds.intro.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\nY : Type u_2\ninst\u271d : TopologicalSpace Y\n\u03c0 : X \u2192 Y\nV : Set (Set X)\nhV : IsTopologicalBasis V\nh' : QuotientMap \u03c0\nh : IsOpenMap \u03c0\nU : Set Y\nU_open : IsOpen U\nx : X\ny_in_U : \u03c0 x \u2208 U\nW : Set X := \u03c0 \u207b\u00b9' U\nx_in_W : x \u2208 W\nW_open : IsOpen W\nZ : Set X\nZ_in_V : Z \u2208 V\nx_in_Z : x \u2208 Z\nZ_in_W : Z \u2286 W\n\u22a2 \u2203 v, v \u2208 image \u03c0 '' V \u2227 \u03c0 x \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nhave \u03c0Z_in_U : \u03c0 '' Z \u2286 U := (Set.image_subset _ Z_in_W).trans (image_preimage_subset \u03c0 U)\n[GOAL]\ncase h_nhds.intro.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\nY : Type u_2\ninst\u271d : TopologicalSpace Y\n\u03c0 : X \u2192 Y\nV : Set (Set X)\nhV : IsTopologicalBasis V\nh' : QuotientMap \u03c0\nh : IsOpenMap \u03c0\nU : Set Y\nU_open : IsOpen U\nx : X\ny_in_U : \u03c0 x \u2208 U\nW : Set X := \u03c0 \u207b\u00b9' U\nx_in_W : x \u2208 W\nW_open : IsOpen W\nZ : Set X\nZ_in_V : Z \u2208 V\nx_in_Z : x \u2208 Z\nZ_in_W : Z \u2286 W\n\u03c0Z_in_U : \u03c0 '' Z \u2286 U\n\u22a2 \u2203 v, v \u2208 image \u03c0 '' V \u2227 \u03c0 x \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nexact \u27e8\u03c0 '' Z, \u27e8Z, Z_in_V, rfl\u27e9, \u27e8x, x_in_Z, rfl\u27e9, \u03c0Z_in_U\u27e9\n[GOAL]\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace Y\n\u03c0 : X \u2192 Y\ninst\u271d : SecondCountableTopology X\nh' : QuotientMap \u03c0\nh : IsOpenMap \u03c0\n\u22a2 \u2203 b, Set.Countable b \u2227 inst\u271d\u00b9 = generateFrom b\n[PROOFSTEP]\nobtain \u27e8V, V_countable, -, V_generates\u27e9 := exists_countable_basis X\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace Y\n\u03c0 : X \u2192 Y\ninst\u271d : SecondCountableTopology X\nh' : QuotientMap \u03c0\nh : IsOpenMap \u03c0\nV : Set (Set X)\nV_countable : Set.Countable V\nV_generates : IsTopologicalBasis V\n\u22a2 \u2203 b, Set.Countable b \u2227 inst\u271d\u00b9 = generateFrom b\n[PROOFSTEP]\nexact \u27e8Set.image \u03c0 '' V, V_countable.image (Set.image \u03c0), (V_generates.quotientMap h' h).eq_generateFrom\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nhf : Inducing f\n\u22a2 SecondCountableTopology \u03b1\n[PROOFSTEP]\nrw [hf.1]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nhf : Inducing f\n\u22a2 SecondCountableTopology \u03b1\n[PROOFSTEP]\nexact secondCountableTopology_induced \u03b1 \u03b2 f\n", "meta": {"mathlib_filename": "Mathlib.Topology.Bases", "llama_tokens": 44336, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.743167997235783, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.4432500267945186}} {"text": "[GOAL]\nw : EssSurj uliftFunctor.{u, v}\na : Type (max u v)\n\u22a2 Small.{v, max u v} a\n[PROOFSTEP]\nobtain \u27e8a', \u27e8m\u27e9\u27e9 := w.mem_essImage a\n[GOAL]\ncase intro.intro\nw : EssSurj uliftFunctor.{u, v}\na : Type (max u v)\na' : Type v\nm : uliftFunctor.{u, v}.obj a' \u2245 a\n\u22a2 Small.{v, max u v} a\n[PROOFSTEP]\nexact \u27e8a', \u27e8(Iso.toEquiv m).symm.trans Equiv.ulift\u27e9\u27e9\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.UnivLE", "llama_tokens": 179, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7248702880639791, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.4431726270988553}} {"text": "[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nv : M\nx fx : CliffordAlgebra Q\n\u22a2 \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx\n[PROOFSTEP]\nsimp only [contractLeftAux_apply_apply]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nv : M\nx fx : CliffordAlgebra Q\n\u22a2 \u2191d v \u2022 (\u2191(\u03b9 Q) v * x) - \u2191(\u03b9 Q) v * (\u2191d v \u2022 x - \u2191(\u03b9 Q) v * fx) = \u2191Q v \u2022 fx\n[PROOFSTEP]\nrw [mul_sub, \u2190 mul_assoc, \u03b9_sq_scalar, \u2190 Algebra.smul_def, \u2190 sub_add, mul_smul_comm, sub_self, zero_add]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' d\u2081 d\u2082 : Module.Dual R M\nx : CliffordAlgebra Q\n\u22a2 \u2191((fun d =>\n foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n (d\u2081 + d\u2082))\n x =\n \u2191((fun d =>\n foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n d\u2081 +\n (fun d =>\n foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n d\u2082)\n x\n[PROOFSTEP]\ndsimp only\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' d\u2081 d\u2082 : Module.Dual R M\nx : CliffordAlgebra Q\n\u22a2 \u2191(foldr' Q (contractLeftAux Q (d\u2081 + d\u2082))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n x =\n \u2191(foldr' Q (contractLeftAux Q d\u2081)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2081) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2081) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0 +\n foldr' Q (contractLeftAux Q d\u2082)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2082) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2082) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n x\n[PROOFSTEP]\nrw [LinearMap.add_apply]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' d\u2081 d\u2082 : Module.Dual R M\nx : CliffordAlgebra Q\n\u22a2 \u2191(foldr' Q (contractLeftAux Q (d\u2081 + d\u2082))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n x =\n \u2191(foldr' Q (contractLeftAux Q d\u2081)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2081) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2081) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n x +\n \u2191(foldr' Q (contractLeftAux Q d\u2082)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2082) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2082) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n x\n[PROOFSTEP]\ninduction' x using CliffordAlgebra.left_induction with r x y hx hy m x hx\n[GOAL]\ncase hr\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' d\u2081 d\u2082 : Module.Dual R M\nr : R\n\u22a2 \u2191(foldr' Q (contractLeftAux Q (d\u2081 + d\u2082))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n (\u2191(algebraMap R (CliffordAlgebra Q)) r) =\n \u2191(foldr' Q (contractLeftAux Q d\u2081)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2081) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2081) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n (\u2191(algebraMap R (CliffordAlgebra Q)) r) +\n \u2191(foldr' Q (contractLeftAux Q d\u2082)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2082) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2082) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n (\u2191(algebraMap R (CliffordAlgebra Q)) r)\n[PROOFSTEP]\nsimp_rw [foldr'_algebraMap, smul_zero, zero_add]\n[GOAL]\ncase h_add\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' d\u2081 d\u2082 : Module.Dual R M\nx y : CliffordAlgebra Q\nhx :\n \u2191(foldr' Q (contractLeftAux Q (d\u2081 + d\u2082))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n x =\n \u2191(foldr' Q (contractLeftAux Q d\u2081)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2081) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2081) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n x +\n \u2191(foldr' Q (contractLeftAux Q d\u2082)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2082) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2082) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n x\nhy :\n \u2191(foldr' Q (contractLeftAux Q (d\u2081 + d\u2082))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n y =\n \u2191(foldr' Q (contractLeftAux Q d\u2081)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2081) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2081) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n y +\n \u2191(foldr' Q (contractLeftAux Q d\u2082)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2082) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2082) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n y\n\u22a2 \u2191(foldr' Q (contractLeftAux Q (d\u2081 + d\u2082))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n (x + y) =\n \u2191(foldr' Q (contractLeftAux Q d\u2081)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2081) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2081) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n (x + y) +\n \u2191(foldr' Q (contractLeftAux Q d\u2082)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2082) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2082) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n (x + y)\n[PROOFSTEP]\nrw [map_add, map_add, map_add, add_add_add_comm, hx, hy]\n[GOAL]\ncase h_mul_\u03b9\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' d\u2081 d\u2082 : Module.Dual R M\nm : CliffordAlgebra Q\nx : M\nhx :\n \u2191(foldr' Q (contractLeftAux Q (d\u2081 + d\u2082))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n m =\n \u2191(foldr' Q (contractLeftAux Q d\u2081)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2081) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2081) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m +\n \u2191(foldr' Q (contractLeftAux Q d\u2082)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2082) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2082) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m\n\u22a2 \u2191(foldr' Q (contractLeftAux Q (d\u2081 + d\u2082))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n (\u2191(\u03b9 Q) x * m) =\n \u2191(foldr' Q (contractLeftAux Q d\u2081)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2081) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2081) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n (\u2191(\u03b9 Q) x * m) +\n \u2191(foldr' Q (contractLeftAux Q d\u2082)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2082) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2082) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n (\u2191(\u03b9 Q) x * m)\n[PROOFSTEP]\nrw [foldr'_\u03b9_mul, foldr'_\u03b9_mul, foldr'_\u03b9_mul, hx]\n[GOAL]\ncase h_mul_\u03b9\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' d\u2081 d\u2082 : Module.Dual R M\nm : CliffordAlgebra Q\nx : M\nhx :\n \u2191(foldr' Q (contractLeftAux Q (d\u2081 + d\u2082))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n m =\n \u2191(foldr' Q (contractLeftAux Q d\u2081)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2081) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2081) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m +\n \u2191(foldr' Q (contractLeftAux Q d\u2082)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2082) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2082) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m\n\u22a2 \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) x)\n (m,\n \u2191(foldr' Q (contractLeftAux Q d\u2081)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2081) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2081) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m +\n \u2191(foldr' Q (contractLeftAux Q d\u2082)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2082) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2082) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m) =\n \u2191(\u2191(contractLeftAux Q d\u2081) x)\n (m,\n \u2191(foldr' Q (contractLeftAux Q d\u2081)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2081) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2081) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m) +\n \u2191(\u2191(contractLeftAux Q d\u2082) x)\n (m,\n \u2191(foldr' Q (contractLeftAux Q d\u2082)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2082) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2082) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m)\n[PROOFSTEP]\ndsimp only [contractLeftAux_apply_apply]\n[GOAL]\ncase h_mul_\u03b9\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' d\u2081 d\u2082 : Module.Dual R M\nm : CliffordAlgebra Q\nx : M\nhx :\n \u2191(foldr' Q (contractLeftAux Q (d\u2081 + d\u2082))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (d\u2081 + d\u2082)) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n m =\n \u2191(foldr' Q (contractLeftAux Q d\u2081)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2081) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2081) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m +\n \u2191(foldr' Q (contractLeftAux Q d\u2082)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2082) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2082) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m\n\u22a2 \u2191(d\u2081 + d\u2082) x \u2022 m -\n \u2191(\u03b9 Q) x *\n (\u2191(foldr' Q (contractLeftAux Q d\u2081)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2081) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2081) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m +\n \u2191(foldr' Q (contractLeftAux Q d\u2082)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2082) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2082) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m) =\n \u2191d\u2081 x \u2022 m -\n \u2191(\u03b9 Q) x *\n \u2191(foldr' Q (contractLeftAux Q d\u2081)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2081) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2081) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m +\n (\u2191d\u2082 x \u2022 m -\n \u2191(\u03b9 Q) x *\n \u2191(foldr' Q (contractLeftAux Q d\u2082)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d\u2082) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d\u2082) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m)\n[PROOFSTEP]\nrw [sub_add_sub_comm, mul_add, LinearMap.add_apply, add_smul]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd\u271d d' : Module.Dual R M\nc : R\nd : Module.Dual R M\nx : CliffordAlgebra Q\n\u22a2 \u2191(AddHom.toFun\n {\n toFun := fun d =>\n foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0,\n map_add' :=\n (_ :\n \u2200 (d\u2081 d\u2082 : Module.Dual R M),\n (fun d =>\n foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n (d\u2081 + d\u2082) =\n (fun d =>\n foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n d\u2081 +\n (fun d =>\n foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n d\u2082) }\n (c \u2022 d))\n x =\n \u2191(\u2191(RingHom.id R) c \u2022\n AddHom.toFun\n {\n toFun := fun d =>\n foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0,\n map_add' :=\n (_ :\n \u2200 (d\u2081 d\u2082 : Module.Dual R M),\n (fun d =>\n foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n (d\u2081 + d\u2082) =\n (fun d =>\n foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n d\u2081 +\n (fun d =>\n foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) =\n \u2191Q v \u2022 fx)\n 0)\n d\u2082) }\n d)\n x\n[PROOFSTEP]\ndsimp only\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd\u271d d' : Module.Dual R M\nc : R\nd : Module.Dual R M\nx : CliffordAlgebra Q\n\u22a2 \u2191(foldr' Q (contractLeftAux Q (c \u2022 d))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n x =\n \u2191(\u2191(RingHom.id R) c \u2022\n foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n x\n[PROOFSTEP]\nrw [LinearMap.smul_apply, RingHom.id_apply]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd\u271d d' : Module.Dual R M\nc : R\nd : Module.Dual R M\nx : CliffordAlgebra Q\n\u22a2 \u2191(foldr' Q (contractLeftAux Q (c \u2022 d))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n x =\n c \u2022\n \u2191(foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n x\n[PROOFSTEP]\ninduction' x using CliffordAlgebra.left_induction with r x y hx hy m x hx\n[GOAL]\ncase hr\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd\u271d d' : Module.Dual R M\nc : R\nd : Module.Dual R M\nr : R\n\u22a2 \u2191(foldr' Q (contractLeftAux Q (c \u2022 d))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n (\u2191(algebraMap R (CliffordAlgebra Q)) r) =\n c \u2022\n \u2191(foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n (\u2191(algebraMap R (CliffordAlgebra Q)) r)\n[PROOFSTEP]\nsimp_rw [foldr'_algebraMap, smul_zero]\n[GOAL]\ncase h_add\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd\u271d d' : Module.Dual R M\nc : R\nd : Module.Dual R M\nx y : CliffordAlgebra Q\nhx :\n \u2191(foldr' Q (contractLeftAux Q (c \u2022 d))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n x =\n c \u2022\n \u2191(foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n x\nhy :\n \u2191(foldr' Q (contractLeftAux Q (c \u2022 d))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n y =\n c \u2022\n \u2191(foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n y\n\u22a2 \u2191(foldr' Q (contractLeftAux Q (c \u2022 d))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n (x + y) =\n c \u2022\n \u2191(foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n (x + y)\n[PROOFSTEP]\nrw [map_add, map_add, smul_add, hx, hy]\n[GOAL]\ncase h_mul_\u03b9\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd\u271d d' : Module.Dual R M\nc : R\nd : Module.Dual R M\nm : CliffordAlgebra Q\nx : M\nhx :\n \u2191(foldr' Q (contractLeftAux Q (c \u2022 d))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m =\n c \u2022\n \u2191(foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m\n\u22a2 \u2191(foldr' Q (contractLeftAux Q (c \u2022 d))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n (\u2191(\u03b9 Q) x * m) =\n c \u2022\n \u2191(foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n (\u2191(\u03b9 Q) x * m)\n[PROOFSTEP]\nrw [foldr'_\u03b9_mul, foldr'_\u03b9_mul, hx]\n[GOAL]\ncase h_mul_\u03b9\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd\u271d d' : Module.Dual R M\nc : R\nd : Module.Dual R M\nm : CliffordAlgebra Q\nx : M\nhx :\n \u2191(foldr' Q (contractLeftAux Q (c \u2022 d))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m =\n c \u2022\n \u2191(foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m\n\u22a2 \u2191(\u2191(contractLeftAux Q (c \u2022 d)) x)\n (m,\n c \u2022\n \u2191(foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m) =\n c \u2022\n \u2191(\u2191(contractLeftAux Q d) x)\n (m,\n \u2191(foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m)\n[PROOFSTEP]\ndsimp only [contractLeftAux_apply_apply]\n[GOAL]\ncase h_mul_\u03b9\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd\u271d d' : Module.Dual R M\nc : R\nd : Module.Dual R M\nm : CliffordAlgebra Q\nx : M\nhx :\n \u2191(foldr' Q (contractLeftAux Q (c \u2022 d))\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q (c \u2022 d)) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m =\n c \u2022\n \u2191(foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m\n\u22a2 \u2191(c \u2022 d) x \u2022 m -\n \u2191(\u03b9 Q) x *\n c \u2022\n \u2191(foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m =\n c \u2022\n (\u2191d x \u2022 m -\n \u2191(\u03b9 Q) x *\n \u2191(foldr' Q (contractLeftAux Q d)\n (_ :\n \u2200 (v : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx)\n 0)\n m)\n[PROOFSTEP]\nrw [LinearMap.smul_apply, smul_assoc, mul_smul_comm, smul_sub]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\na : M\nb : CliffordAlgebra Q\n\u22a2 \u2191(\u2191contractLeft d) (\u2191(\u03b9 Q) a * b) = \u2191d a \u2022 b - \u2191(\u03b9 Q) a * \u2191(\u2191contractLeft d) b\n[PROOFSTEP]\nrefine foldr'_\u03b9_mul _ _ ?_ _ _ _\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\na : M\nb : CliffordAlgebra Q\n\u22a2 \u2200 (m : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) m) (\u2191(\u03b9 Q) m * x, \u2191(\u2191(contractLeftAux Q d) m) (x, fx)) = \u2191Q m \u2022 fx\n[PROOFSTEP]\nexact fun m x fx \u21a6 contractLeftAux_contractLeftAux Q d m x fx\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\na : M\nb : CliffordAlgebra Q\n\u22a2 \u2191(\u2191contractRight (b * \u2191(\u03b9 Q) a)) d = \u2191d a \u2022 b - \u2191(\u2191contractRight b) d * \u2191(\u03b9 Q) a\n[PROOFSTEP]\nrw [contractRight_eq, reverse.map_mul, reverse_\u03b9, contractLeft_\u03b9_mul, map_sub, map_smul, reverse_reverse,\n reverse.map_mul, reverse_\u03b9, contractRight_eq]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nr : R\nb : CliffordAlgebra Q\n\u22a2 \u2191(\u2191contractLeft d) (\u2191(algebraMap R (CliffordAlgebra Q)) r * b) =\n \u2191(algebraMap R (CliffordAlgebra Q)) r * \u2191(\u2191contractLeft d) b\n[PROOFSTEP]\nrw [\u2190 Algebra.smul_def, map_smul, Algebra.smul_def, Algebra.smul_def]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\na : CliffordAlgebra Q\nr : R\n\u22a2 \u2191(\u2191contractLeft d) (a * \u2191(algebraMap R (CliffordAlgebra Q)) r) =\n \u2191(\u2191contractLeft d) a * \u2191(algebraMap R (CliffordAlgebra Q)) r\n[PROOFSTEP]\nrw [\u2190 Algebra.commutes, contractLeft_algebraMap_mul, Algebra.commutes, Algebra.commutes]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nr : R\nb : CliffordAlgebra Q\n\u22a2 \u2191(\u2191contractRight (\u2191(algebraMap R (CliffordAlgebra Q)) r * b)) d =\n \u2191(algebraMap R (CliffordAlgebra Q)) r * \u2191(\u2191contractRight b) d\n[PROOFSTEP]\nrw [\u2190 Algebra.smul_def, LinearMap.map_smul\u2082, Algebra.smul_def]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\na : CliffordAlgebra Q\nr : R\n\u22a2 \u2191(\u2191contractRight (a * \u2191(algebraMap R (CliffordAlgebra Q)) r)) d =\n \u2191(\u2191contractRight a) d * \u2191(algebraMap R (CliffordAlgebra Q)) r\n[PROOFSTEP]\nrw [\u2190 Algebra.commutes, contractRight_algebraMap_mul, Algebra.commutes]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : M\n\u22a2 \u2191(\u2191contractLeft d) (\u2191(\u03b9 Q) x) = \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191d x)\n[PROOFSTEP]\nrefine\n (foldr'_\u03b9 _ _ ?_ _ _).trans <| by\n simp_rw [contractLeftAux_apply_apply, mul_zero, sub_zero, Algebra.algebraMap_eq_smul_one]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : M\n\u22a2 \u2191(\u2191(contractLeftAux Q d) x) (1, 0) = \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191d x)\n[PROOFSTEP]\nsimp_rw [contractLeftAux_apply_apply, mul_zero, sub_zero, Algebra.algebraMap_eq_smul_one]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : M\n\u22a2 \u2200 (m : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) m) (\u2191(\u03b9 Q) m * x, \u2191(\u2191(contractLeftAux Q d) m) (x, fx)) = \u2191Q m \u2022 fx\n[PROOFSTEP]\nexact fun m x fx \u21a6 contractLeftAux_contractLeftAux Q d m x fx\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : M\n\u22a2 \u2191(\u2191contractRight (\u2191(\u03b9 Q) x)) d = \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191d x)\n[PROOFSTEP]\nrw [contractRight_eq, reverse_\u03b9, contractLeft_\u03b9, reverse.commutes]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nr : R\n\u22a2 \u2191(\u2191contractLeft d) (\u2191(algebraMap R (CliffordAlgebra Q)) r) = 0\n[PROOFSTEP]\nrefine (foldr'_algebraMap _ _ ?_ _ _).trans <| smul_zero _\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nr : R\n\u22a2 \u2200 (m : M) (x fx : CliffordAlgebra Q),\n \u2191(\u2191(contractLeftAux Q d) m) (\u2191(\u03b9 Q) m * x, \u2191(\u2191(contractLeftAux Q d) m) (x, fx)) = \u2191Q m \u2022 fx\n[PROOFSTEP]\nexact fun m x fx \u21a6 contractLeftAux_contractLeftAux Q d m x fx\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nr : R\n\u22a2 \u2191(\u2191contractRight (\u2191(algebraMap R (CliffordAlgebra Q)) r)) d = 0\n[PROOFSTEP]\nrw [contractRight_eq, reverse.commutes, contractLeft_algebraMap, map_zero]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\n\u22a2 \u2191(\u2191contractLeft d) 1 = 0\n[PROOFSTEP]\nsimpa only [map_one] using contractLeft_algebraMap Q d 1\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\n\u22a2 \u2191(\u2191contractRight 1) d = 0\n[PROOFSTEP]\nsimpa only [map_one] using contractRight_algebraMap Q d 1\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n\u22a2 \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d) x) = 0\n[PROOFSTEP]\ninduction' x using CliffordAlgebra.left_induction with r x y hx hy m x hx\n[GOAL]\ncase hr\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nr : R\n\u22a2 \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d) (\u2191(algebraMap R (CliffordAlgebra Q)) r)) = 0\n[PROOFSTEP]\nsimp_rw [contractLeft_algebraMap, map_zero]\n[GOAL]\ncase h_add\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx y : CliffordAlgebra Q\nhx : \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d) x) = 0\nhy : \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d) y) = 0\n\u22a2 \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d) (x + y)) = 0\n[PROOFSTEP]\nrw [map_add, map_add, hx, hy, add_zero]\n[GOAL]\ncase h_mul_\u03b9\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nm : CliffordAlgebra Q\nx : M\nhx : \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d) m) = 0\n\u22a2 \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d) (\u2191(\u03b9 Q) x * m)) = 0\n[PROOFSTEP]\nrw [contractLeft_\u03b9_mul, map_sub, contractLeft_\u03b9_mul, hx, LinearMap.map_smul, mul_zero, sub_zero, sub_self]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n\u22a2 \u2191(\u2191contractRight (\u2191(\u2191contractRight x) d)) d = 0\n[PROOFSTEP]\nrw [contractRight_eq, contractRight_eq, reverse_reverse, contractLeft_contractLeft, map_zero]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n\u22a2 \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d') x) = -\u2191(\u2191contractLeft d') (\u2191(\u2191contractLeft d) x)\n[PROOFSTEP]\ninduction' x using CliffordAlgebra.left_induction with r x y hx hy m x hx\n[GOAL]\ncase hr\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nr : R\n\u22a2 \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d') (\u2191(algebraMap R (CliffordAlgebra Q)) r)) =\n -\u2191(\u2191contractLeft d') (\u2191(\u2191contractLeft d) (\u2191(algebraMap R (CliffordAlgebra Q)) r))\n[PROOFSTEP]\nsimp_rw [contractLeft_algebraMap, map_zero, neg_zero]\n[GOAL]\ncase h_add\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx y : CliffordAlgebra Q\nhx : \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d') x) = -\u2191(\u2191contractLeft d') (\u2191(\u2191contractLeft d) x)\nhy : \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d') y) = -\u2191(\u2191contractLeft d') (\u2191(\u2191contractLeft d) y)\n\u22a2 \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d') (x + y)) = -\u2191(\u2191contractLeft d') (\u2191(\u2191contractLeft d) (x + y))\n[PROOFSTEP]\nrw [map_add, map_add, map_add, map_add, hx, hy, neg_add]\n[GOAL]\ncase h_mul_\u03b9\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nm : CliffordAlgebra Q\nx : M\nhx : \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d') m) = -\u2191(\u2191contractLeft d') (\u2191(\u2191contractLeft d) m)\n\u22a2 \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d') (\u2191(\u03b9 Q) x * m)) = -\u2191(\u2191contractLeft d') (\u2191(\u2191contractLeft d) (\u2191(\u03b9 Q) x * m))\n[PROOFSTEP]\nsimp only [contractLeft_\u03b9_mul, map_sub, LinearMap.map_smul]\n[GOAL]\ncase h_mul_\u03b9\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nm : CliffordAlgebra Q\nx : M\nhx : \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d') m) = -\u2191(\u2191contractLeft d') (\u2191(\u2191contractLeft d) m)\n\u22a2 \u2191d' x \u2022 \u2191(\u2191contractLeft d) m -\n (\u2191d x \u2022 \u2191(\u2191contractLeft d') m - \u2191(\u03b9 Q) x * \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft d') m)) =\n -(\u2191d x \u2022 \u2191(\u2191contractLeft d') m -\n (\u2191d' x \u2022 \u2191(\u2191contractLeft d) m - \u2191(\u03b9 Q) x * \u2191(\u2191contractLeft d') (\u2191(\u2191contractLeft d) m)))\n[PROOFSTEP]\nrw [neg_sub, sub_sub_eq_add_sub, hx, mul_neg, \u2190 sub_eq_add_neg]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx : CliffordAlgebra Q\n\u22a2 \u2191(\u2191contractRight (\u2191(\u2191contractRight x) d)) d' = -\u2191(\u2191contractRight (\u2191(\u2191contractRight x) d')) d\n[PROOFSTEP]\nrw [contractRight_eq, contractRight_eq, contractRight_eq, contractRight_eq, reverse_reverse, reverse_reverse,\n contractLeft_comm, map_neg]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nB : BilinForm R M\nv : M\nx : CliffordAlgebra Q\n\u22a2 \u2191(\u2191(changeFormAux Q B) v) (\u2191(\u2191(changeFormAux Q B) v) x) = (\u2191Q v - BilinForm.bilin B v v) \u2022 x\n[PROOFSTEP]\nsimp only [changeFormAux_apply_apply]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nB : BilinForm R M\nv : M\nx : CliffordAlgebra Q\n\u22a2 \u2191(\u03b9 Q) v * (\u2191(\u03b9 Q) v * x - \u2191(\u2191contractLeft (\u2191(\u2191BilinForm.toLin B) v)) x) -\n \u2191(\u2191contractLeft (\u2191(\u2191BilinForm.toLin B) v)) (\u2191(\u03b9 Q) v * x - \u2191(\u2191contractLeft (\u2191(\u2191BilinForm.toLin B) v)) x) =\n (\u2191Q v - BilinForm.bilin B v v) \u2022 x\n[PROOFSTEP]\nrw [mul_sub, \u2190 mul_assoc, \u03b9_sq_scalar, map_sub, contractLeft_\u03b9_mul, \u2190 sub_add, sub_sub_sub_comm, \u2190 Algebra.smul_def,\n BilinForm.toLin_apply, sub_self, sub_zero, contractLeft_contractLeft, add_zero, sub_smul]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh\u271d : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nh : BilinForm.toQuadraticForm B = Q' - Q\nm : M\nx : CliffordAlgebra Q'\n\u22a2 (\u2191Q' m - BilinForm.bilin B m m) \u2022 x = \u2191Q m \u2022 x\n[PROOFSTEP]\ndsimp only [\u2190 BilinForm.toQuadraticForm_apply]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh\u271d : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nh : BilinForm.toQuadraticForm B = Q' - Q\nm : M\nx : CliffordAlgebra Q'\n\u22a2 (\u2191Q' m - \u2191(BilinForm.toQuadraticForm B) m) \u2022 x = \u2191Q m \u2022 x\n[PROOFSTEP]\nrw [h, QuadraticForm.sub_apply, sub_sub_cancel]\n[GOAL]\nR : Type u1\ninst\u271d\u00b3 : CommRing R\nM : Type u2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\ninst\u271d : Invertible 2\n\u22a2 BilinForm.toQuadraticForm (\u2191QuadraticForm.associated (-Q)) = 0 - Q\n[PROOFSTEP]\nsimp [QuadraticForm.toQuadraticForm_associated]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\n\u22a2 \u2191(changeForm h) 1 = 1\n[PROOFSTEP]\nsimpa using changeForm_algebraMap h (1 : R)\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nm : M\n\u22a2 \u2191(\u03b9 Q') m = \u2191(\u2191(changeFormAux Q' B) m) 1\n[PROOFSTEP]\nrw [changeFormAux_apply_apply, mul_one, contractLeft_one, sub_zero]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nm : M\nx : CliffordAlgebra Q\n\u22a2 \u2191(\u2191(foldr Q (changeFormAux Q' B)\n (_ :\n \u2200 (m : M) (x : CliffordAlgebra Q'),\n \u2191(\u2191(changeFormAux Q' B) m) (\u2191(\u2191(changeFormAux Q' B) m) x) = \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (changeFormAux Q' B)\n (_ :\n \u2200 (m : M) (x : CliffordAlgebra Q'),\n \u2191(\u2191(changeFormAux Q' B) m) (\u2191(\u2191(changeFormAux Q' B) m) x) = \u2191Q m \u2022 x))\n 1)\n x))\n (\u2191(\u03b9 Q) m) =\n \u2191(\u03b9 Q') m * \u2191(changeForm h) x - \u2191(\u2191contractLeft (\u2191(\u2191BilinForm.toLin B) m)) (\u2191(changeForm h) x)\n[PROOFSTEP]\nrw [foldr_\u03b9]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nm : M\nx : CliffordAlgebra Q\n\u22a2 \u2191(\u2191(changeFormAux Q' B) m)\n (\u2191(\u2191(foldr Q (changeFormAux Q' B)\n (_ :\n \u2200 (m : M) (x : CliffordAlgebra Q'),\n \u2191(\u2191(changeFormAux Q' B) m) (\u2191(\u2191(changeFormAux Q' B) m) x) = \u2191Q m \u2022 x))\n 1)\n x) =\n \u2191(\u03b9 Q') m * \u2191(changeForm h) x - \u2191(\u2191contractLeft (\u2191(\u2191BilinForm.toLin B) m)) (\u2191(changeForm h) x)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nm\u2081 m\u2082 : M\n\u22a2 \u2191(changeForm h) (\u2191(\u03b9 Q) m\u2081 * \u2191(\u03b9 Q) m\u2082) =\n \u2191(\u03b9 Q') m\u2081 * \u2191(\u03b9 Q') m\u2082 - \u2191(algebraMap R (CliffordAlgebra Q')) (BilinForm.bilin B m\u2081 m\u2082)\n[PROOFSTEP]\nrw [changeForm_\u03b9_mul, changeForm_\u03b9, contractLeft_\u03b9, BilinForm.toLin_apply]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nd : Module.Dual R M\nx : CliffordAlgebra Q\n\u22a2 \u2191(changeForm h) (\u2191(\u2191contractLeft d) x) = \u2191(\u2191contractLeft d) (\u2191(changeForm h) x)\n[PROOFSTEP]\ninduction' x using CliffordAlgebra.left_induction with r x y hx hy m x hx\n[GOAL]\ncase hr\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nd : Module.Dual R M\nr : R\n\u22a2 \u2191(changeForm h) (\u2191(\u2191contractLeft d) (\u2191(algebraMap R (CliffordAlgebra Q)) r)) =\n \u2191(\u2191contractLeft d) (\u2191(changeForm h) (\u2191(algebraMap R (CliffordAlgebra Q)) r))\n[PROOFSTEP]\nsimp only [contractLeft_algebraMap, changeForm_algebraMap, map_zero]\n[GOAL]\ncase h_add\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nd : Module.Dual R M\nx y : CliffordAlgebra Q\nhx : \u2191(changeForm h) (\u2191(\u2191contractLeft d) x) = \u2191(\u2191contractLeft d) (\u2191(changeForm h) x)\nhy : \u2191(changeForm h) (\u2191(\u2191contractLeft d) y) = \u2191(\u2191contractLeft d) (\u2191(changeForm h) y)\n\u22a2 \u2191(changeForm h) (\u2191(\u2191contractLeft d) (x + y)) = \u2191(\u2191contractLeft d) (\u2191(changeForm h) (x + y))\n[PROOFSTEP]\nrw [map_add, map_add, map_add, map_add, hx, hy]\n[GOAL]\ncase h_mul_\u03b9\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nd : Module.Dual R M\nm : CliffordAlgebra Q\nx : M\nhx : \u2191(changeForm h) (\u2191(\u2191contractLeft d) m) = \u2191(\u2191contractLeft d) (\u2191(changeForm h) m)\n\u22a2 \u2191(changeForm h) (\u2191(\u2191contractLeft d) (\u2191(\u03b9 Q) x * m)) = \u2191(\u2191contractLeft d) (\u2191(changeForm h) (\u2191(\u03b9 Q) x * m))\n[PROOFSTEP]\nsimp only [contractLeft_\u03b9_mul, changeForm_\u03b9_mul, map_sub, LinearMap.map_smul]\n[GOAL]\ncase h_mul_\u03b9\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nd : Module.Dual R M\nm : CliffordAlgebra Q\nx : M\nhx : \u2191(changeForm h) (\u2191(\u2191contractLeft d) m) = \u2191(\u2191contractLeft d) (\u2191(changeForm h) m)\n\u22a2 \u2191d x \u2022 \u2191(changeForm h) m -\n (\u2191(\u03b9 Q') x * \u2191(changeForm h) (\u2191(\u2191contractLeft d) m) -\n \u2191(\u2191contractLeft (\u2191(\u2191BilinForm.toLin B) x)) (\u2191(changeForm h) (\u2191(\u2191contractLeft d) m))) =\n \u2191d x \u2022 \u2191(changeForm h) m - \u2191(\u03b9 Q') x * \u2191(\u2191contractLeft d) (\u2191(changeForm h) m) -\n \u2191(\u2191contractLeft d) (\u2191(\u2191contractLeft (\u2191(\u2191BilinForm.toLin B) x)) (\u2191(changeForm h) m))\n[PROOFSTEP]\nrw [\u2190 hx, contractLeft_comm, \u2190 sub_add, sub_neg_eq_add, \u2190 hx]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nx : CliffordAlgebra Q\n\u22a2 \u2191(changeForm (_ : BilinForm.toQuadraticForm 0 = Q - Q)) x = x\n[PROOFSTEP]\ninduction' x using CliffordAlgebra.left_induction with r x y hx hy m x hx\n[GOAL]\ncase hr\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nr : R\n\u22a2 \u2191(changeForm (_ : BilinForm.toQuadraticForm 0 = Q - Q)) (\u2191(algebraMap R (CliffordAlgebra Q)) r) =\n \u2191(algebraMap R (CliffordAlgebra Q)) r\n[PROOFSTEP]\nsimp_rw [changeForm_algebraMap]\n[GOAL]\ncase h_add\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nx y : CliffordAlgebra Q\nhx : \u2191(changeForm (_ : BilinForm.toQuadraticForm 0 = Q - Q)) x = x\nhy : \u2191(changeForm (_ : BilinForm.toQuadraticForm 0 = Q - Q)) y = y\n\u22a2 \u2191(changeForm (_ : BilinForm.toQuadraticForm 0 = Q - Q)) (x + y) = x + y\n[PROOFSTEP]\nrw [map_add, hx, hy]\n[GOAL]\ncase h_mul_\u03b9\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nm : CliffordAlgebra Q\nx : M\nhx : \u2191(changeForm (_ : BilinForm.toQuadraticForm 0 = Q - Q)) m = m\n\u22a2 \u2191(changeForm (_ : BilinForm.toQuadraticForm 0 = Q - Q)) (\u2191(\u03b9 Q) x * m) = \u2191(\u03b9 Q) x * m\n[PROOFSTEP]\nrw [changeForm_\u03b9_mul, hx, map_zero, LinearMap.zero_apply, map_zero, LinearMap.zero_apply, sub_zero]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nx : CliffordAlgebra Q\n\u22a2 \u2191(changeForm h') (\u2191(changeForm h) x) = \u2191(changeForm (_ : BilinForm.toQuadraticForm (B + B') = Q'' - Q)) x\n[PROOFSTEP]\ninduction' x using CliffordAlgebra.left_induction with r x y hx hy m x hx\n[GOAL]\ncase hr\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nr : R\n\u22a2 \u2191(changeForm h') (\u2191(changeForm h) (\u2191(algebraMap R (CliffordAlgebra Q)) r)) =\n \u2191(changeForm (_ : BilinForm.toQuadraticForm (B + B') = Q'' - Q)) (\u2191(algebraMap R (CliffordAlgebra Q)) r)\n[PROOFSTEP]\nsimp_rw [changeForm_algebraMap]\n[GOAL]\ncase h_add\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nx y : CliffordAlgebra Q\nhx : \u2191(changeForm h') (\u2191(changeForm h) x) = \u2191(changeForm (_ : BilinForm.toQuadraticForm (B + B') = Q'' - Q)) x\nhy : \u2191(changeForm h') (\u2191(changeForm h) y) = \u2191(changeForm (_ : BilinForm.toQuadraticForm (B + B') = Q'' - Q)) y\n\u22a2 \u2191(changeForm h') (\u2191(changeForm h) (x + y)) = \u2191(changeForm (_ : BilinForm.toQuadraticForm (B + B') = Q'' - Q)) (x + y)\n[PROOFSTEP]\nrw [map_add, map_add, map_add, hx, hy]\n[GOAL]\ncase h_mul_\u03b9\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nm : CliffordAlgebra Q\nx : M\nhx : \u2191(changeForm h') (\u2191(changeForm h) m) = \u2191(changeForm (_ : BilinForm.toQuadraticForm (B + B') = Q'' - Q)) m\n\u22a2 \u2191(changeForm h') (\u2191(changeForm h) (\u2191(\u03b9 Q) x * m)) =\n \u2191(changeForm (_ : BilinForm.toQuadraticForm (B + B') = Q'' - Q)) (\u2191(\u03b9 Q) x * m)\n[PROOFSTEP]\nrw [changeForm_\u03b9_mul, map_sub, changeForm_\u03b9_mul, changeForm_\u03b9_mul, hx, sub_sub, map_add, LinearMap.add_apply, map_add,\n LinearMap.add_apply, changeForm_contractLeft, hx, add_comm (_ : CliffordAlgebra Q'')]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nsrc\u271d : CliffordAlgebra Q \u2192\u2097[R] CliffordAlgebra Q' := changeForm h\nx : CliffordAlgebra Q\n\u22a2 \u2191(changeForm (_ : BilinForm.toQuadraticForm (-B) = Q - Q'))\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := \u2191(changeForm h),\n map_add' :=\n (_ :\n \u2200 (x y : CliffordAlgebra Q),\n AddHom.toFun src\u271d.toAddHom (x + y) =\n AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : CliffordAlgebra Q),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n x) =\n x\n[PROOFSTEP]\ndsimp only\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nsrc\u271d : CliffordAlgebra Q \u2192\u2097[R] CliffordAlgebra Q' := changeForm h\nx : CliffordAlgebra Q\n\u22a2 \u2191(changeForm (_ : BilinForm.toQuadraticForm (-B) = Q - Q')) (\u2191(changeForm h) x) = x\n[PROOFSTEP]\nexact (changeForm_changeForm _ _ x).trans <| by simp_rw [add_right_neg, changeForm_self_apply]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nsrc\u271d : CliffordAlgebra Q \u2192\u2097[R] CliffordAlgebra Q' := changeForm h\nx : CliffordAlgebra Q\n\u22a2 \u2191(changeForm (_ : BilinForm.toQuadraticForm (B + -B) = Q - Q)) x = x\n[PROOFSTEP]\nsimp_rw [add_right_neg, changeForm_self_apply]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nsrc\u271d : CliffordAlgebra Q \u2192\u2097[R] CliffordAlgebra Q' := changeForm h\nx : CliffordAlgebra Q'\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := \u2191(changeForm h),\n map_add' :=\n (_ :\n \u2200 (x y : CliffordAlgebra Q),\n AddHom.toFun src\u271d.toAddHom (x + y) = AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : CliffordAlgebra Q),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n (\u2191(changeForm (_ : BilinForm.toQuadraticForm (-B) = Q - Q')) x) =\n x\n[PROOFSTEP]\ndsimp only\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nsrc\u271d : CliffordAlgebra Q \u2192\u2097[R] CliffordAlgebra Q' := changeForm h\nx : CliffordAlgebra Q'\n\u22a2 \u2191(changeForm h) (\u2191(changeForm (_ : BilinForm.toQuadraticForm (-B) = Q - Q')) x) = x\n[PROOFSTEP]\nexact (changeForm_changeForm _ _ x).trans <| by simp_rw [add_left_neg, changeForm_self_apply]\n[GOAL]\nR : Type u1\ninst\u271d\u00b2 : CommRing R\nM : Type u2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinForm.toQuadraticForm B = Q' - Q\nh' : BilinForm.toQuadraticForm B' = Q'' - Q'\nsrc\u271d : CliffordAlgebra Q \u2192\u2097[R] CliffordAlgebra Q' := changeForm h\nx : CliffordAlgebra Q'\n\u22a2 \u2191(changeForm (_ : BilinForm.toQuadraticForm (-B + B) = Q' - Q')) x = x\n[PROOFSTEP]\nsimp_rw [add_left_neg, changeForm_self_apply]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.CliffordAlgebra.Contraction", "llama_tokens": 22761, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7826624789529375, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.44300146988989125}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\nx y z : \u03b1\n\u22a2 (fun x x_1 => x * x_1) x ((fun x x_1 => x * x_1) y z) = (fun x x_1 => x * x_1) y ((fun x x_1 => x * x_1) x z)\n[PROOFSTEP]\nsimp [mul_left_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ns : Multiset \u03b1\nx y z : \u03b1\n\u22a2 (fun x x_1 => x * x_1) x ((fun x x_1 => x * x_1) y z) = (fun x x_1 => x * x_1) y ((fun x x_1 => x * x_1) x z)\n[PROOFSTEP]\nsimp [mul_left_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ns : Multiset \u03b1\nx y z : \u03b1\n\u22a2 (fun x x_1 => x * x_1) ((fun x x_1 => x * x_1) x y) z = (fun x x_1 => x * x_1) ((fun x x_1 => x * x_1) x z) y\n[PROOFSTEP]\nsimp [mul_right_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ns : Multiset \u03b1\n\u22a2 foldl (fun x y => y * x) (_ : \u2200 (_x _y _z : \u03b1), _z * (_y * _x) = _y * (_z * _x)) 1 s =\n foldl (fun x x_1 => x * x_1) (_ : \u2200 (x y z : \u03b1), x * y * z = x * z * y) 1 s\n[PROOFSTEP]\nsimp [mul_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ns : Multiset \u03b1\n\u22a2 List.prod (toList s) = prod s\n[PROOFSTEP]\nconv_rhs => rw [\u2190 coe_toList s]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ns : Multiset \u03b1\n| prod s\n[PROOFSTEP]\nrw [\u2190 coe_toList s]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ns : Multiset \u03b1\n| prod s\n[PROOFSTEP]\nrw [\u2190 coe_toList s]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ns : Multiset \u03b1\n| prod s\n[PROOFSTEP]\nrw [\u2190 coe_toList s]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ns : Multiset \u03b1\n\u22a2 List.prod (toList s) = prod \u2191(toList s)\n[PROOFSTEP]\nrw [coe_prod]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a \u2208 s\n\u22a2 a * prod (erase s a) = prod s\n[PROOFSTEP]\nrw [\u2190 s.coe_toList, coe_erase, coe_prod, coe_prod, List.prod_erase (mem_toList.2 h)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ns t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b9\na : \u03b9\nh : a \u2208 m\n\u22a2 f a * prod (map f (erase m a)) = prod (map f m)\n[PROOFSTEP]\nrw [\u2190 m.coe_toList, coe_erase, coe_map, coe_map, coe_prod, coe_prod, List.prod_map_erase f (mem_toList.2 h)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\na : \u03b1\n\u22a2 prod {a} = a\n[PROOFSTEP]\nsimp only [mul_one, prod_cons, \u2190 cons_zero, eq_self_iff_true, prod_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\na b : \u03b1\n\u22a2 prod {a, b} = a * b\n[PROOFSTEP]\nrw [insert_eq_cons, prod_cons, prod_singleton]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t\u271d : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ns t : Multiset \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 prod (Quotient.mk (List.isSetoid \u03b1) l\u2081 + Quotient.mk (List.isSetoid \u03b1) l\u2082) =\n prod (Quotient.mk (List.isSetoid \u03b1) l\u2081) * prod (Quotient.mk (List.isSetoid \u03b1) l\u2082)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nm\u271d : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\nm : Multiset \u03b1\n\u22a2 prod (0 \u2022 m) = prod m ^ 0\n[PROOFSTEP]\nrw [zero_nsmul, pow_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nm\u271d : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\nm : Multiset \u03b1\n\u22a2 prod 0 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nm\u271d : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\nm : Multiset \u03b1\nn : \u2115\n\u22a2 prod ((n + 1) \u2022 m) = prod m ^ (n + 1)\n[PROOFSTEP]\nrw [add_nsmul, one_nsmul, pow_add, pow_one, prod_add, prod_nsmul m n]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\nn : \u2115\na : \u03b1\n\u22a2 prod (replicate n a) = a ^ n\n[PROOFSTEP]\nsimp [replicate, List.prod_replicate]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nhf : \u2200 (i' : \u03b9), i' \u2260 i \u2192 i' \u2208 m \u2192 f i' = 1\n\u22a2 prod (map f m) = f i ^ count i m\n[PROOFSTEP]\ninduction' m using Quotient.inductionOn with l\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nhf\u271d : \u2200 (i' : \u03b9), i' \u2260 i \u2192 i' \u2208 m \u2192 f i' = 1\nl : List \u03b9\nhf : \u2200 (i' : \u03b9), i' \u2260 i \u2192 i' \u2208 Quotient.mk (List.isSetoid \u03b9) l \u2192 f i' = 1\n\u22a2 prod (map f (Quotient.mk (List.isSetoid \u03b9) l)) = f i ^ count i (Quotient.mk (List.isSetoid \u03b9) l)\n[PROOFSTEP]\nsimp [List.prod_map_eq_pow_single i f hf]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ns t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nh : \u2200 (a' : \u03b1), a' \u2260 a \u2192 a' \u2208 s \u2192 a' = 1\n\u22a2 prod s = a ^ count a s\n[PROOFSTEP]\ninduction' s using Quotient.inductionOn with l\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ns t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nh\u271d : \u2200 (a' : \u03b1), a' \u2260 a \u2192 a' \u2208 s \u2192 a' = 1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2260 a \u2192 a' \u2208 Quotient.mk (List.isSetoid \u03b1) l \u2192 a' = 1\n\u22a2 prod (Quotient.mk (List.isSetoid \u03b1) l) = a ^ count a (Quotient.mk (List.isSetoid \u03b1) l)\n[PROOFSTEP]\nsimp [List.prod_eq_pow_single a h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ns t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 a ^ count a s = prod (filter (Eq a) s)\n[PROOFSTEP]\nrw [filter_eq, prod_replicate]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf\u271d g : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ns : Multiset \u03b1\nF : Type u_5\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : F\nl : List \u03b1\n\u22a2 prod (map (\u2191f) (Quotient.mk (List.isSetoid \u03b1) l)) = \u2191f (prod (Quotient.mk (List.isSetoid \u03b1) l))\n[PROOFSTEP]\nsimp only [l.prod_hom f, quot_mk_to_coe, coe_map, coe_prod]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf\u271d g\u271d : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ns : Multiset \u03b9\nF : Type u_5\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : F\ng : \u03b9 \u2192 \u03b1\n\u22a2 prod (map (fun i => \u2191f (g i)) s) = \u2191f (prod (map g s))\n[PROOFSTEP]\nconvert (s.map g).prod_hom f\n[GOAL]\ncase h.e'_2.h.e'_3\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf\u271d g\u271d : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ns : Multiset \u03b9\nF : Type u_5\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : F\ng : \u03b9 \u2192 \u03b1\n\u22a2 map (fun i => \u2191f (g i)) s = map (\u2191f) (map g s)\n[PROOFSTEP]\nexact (map_map _ _ _).symm\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf\u271d g : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : CommMonoid \u03b3\ns : Multiset \u03b9\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (a b : \u03b1) (c d : \u03b2), f (a * b) (c * d) = f a c * f b d\nhf' : f 1 1 = 1\nf\u2081 : \u03b9 \u2192 \u03b1\nf\u2082 : \u03b9 \u2192 \u03b2\nl : List \u03b9\n\u22a2 prod (map (fun i => f (f\u2081 i) (f\u2082 i)) (Quotient.mk (List.isSetoid \u03b9) l)) =\n f (prod (map f\u2081 (Quotient.mk (List.isSetoid \u03b9) l))) (prod (map f\u2082 (Quotient.mk (List.isSetoid \u03b9) l)))\n[PROOFSTEP]\nsimp only [l.prod_hom\u2082 f hf hf', quot_mk_to_coe, coe_map, coe_prod]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf\u271d g\u271d : \u03b9 \u2192 \u03b1\ninst\u271d : CommMonoid \u03b2\ns : Multiset \u03b9\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nh\u2081 : r 1 1\nh\u2082 : \u2200 \u2983a : \u03b9\u2984 \u2983b : \u03b1\u2984 \u2983c : \u03b2\u2984, r b c \u2192 r (f a * b) (g a * c)\nl : List \u03b9\n\u22a2 r (prod (map f (Quotient.mk (List.isSetoid \u03b9) l))) (prod (map g (Quotient.mk (List.isSetoid \u03b9) l)))\n[PROOFSTEP]\nsimp only [l.prod_hom_rel h\u2081 h\u2082, quot_mk_to_coe, coe_map, coe_prod]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\n\u22a2 prod (map (fun x => 1) m) = 1\n[PROOFSTEP]\nrw [map_const', prod_replicate, one_pow]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ninst\u271d : HasDistribNeg \u03b1\ns : Multiset \u03b1\n\u22a2 \u2200 (a : List \u03b1),\n prod (map Neg.neg (Quotient.mk (List.isSetoid \u03b1) a)) =\n (-1) ^ \u2191card (Quotient.mk (List.isSetoid \u03b1) a) * prod (Quotient.mk (List.isSetoid \u03b1) a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nm\u271d : Multiset \u03b9\nf\u271d g : \u03b9 \u2192 \u03b1\nm : Multiset \u03b2\nn : Multiset \u03b3\nf : \u03b2 \u2192 \u03b3 \u2192 \u03b1\n\u22a2 prod (map (fun a => prod (map (fun b => f a b) n)) 0) = prod (map (fun b => prod (map (fun a => f a b) 0)) n)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na\u271d : \u03b1\nm\u271d\u00b9 : Multiset \u03b9\nf\u271d g : \u03b9 \u2192 \u03b1\nm\u271d : Multiset \u03b2\nn : Multiset \u03b3\nf : \u03b2 \u2192 \u03b3 \u2192 \u03b1\na : \u03b2\nm : Multiset \u03b2\nih : prod (map (fun a => prod (map (fun b => f a b) n)) m) = prod (map (fun b => prod (map (fun a => f a b) m)) n)\n\u22a2 prod (map (fun a => prod (map (fun b => f a b) n)) (a ::\u2098 m)) =\n prod (map (fun b => prod (map (fun a => f a b) (a ::\u2098 m))) n)\n[PROOFSTEP]\nsimp [ih]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\ns : Multiset \u03b1\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\np_one : p 1\np_s : \u2200 (a : \u03b1), a \u2208 s \u2192 p a\n\u22a2 p (prod s)\n[PROOFSTEP]\nrw [prod_eq_foldr]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\ns : Multiset \u03b1\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\np_one : p 1\np_s : \u2200 (a : \u03b1), a \u2208 s \u2192 p a\n\u22a2 p (foldr (fun x x_1 => x * x_1) (_ : \u2200 (x y z : \u03b1), x * (y * z) = y * (x * z)) 1 s)\n[PROOFSTEP]\nexact foldr_induction (\u00b7 * \u00b7) (fun x y z => by simp [mul_left_comm]) 1 p s p_mul p_one p_s\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\ns : Multiset \u03b1\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\np_one : p 1\np_s : \u2200 (a : \u03b1), a \u2208 s \u2192 p a\nx y z : \u03b1\n\u22a2 (fun x x_1 => x * x_1) x ((fun x x_1 => x * x_1) y z) = (fun x x_1 => x * x_1) y ((fun x x_1 => x * x_1) x z)\n[PROOFSTEP]\nsimp [mul_left_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs : s \u2260 \u2205\np_s : \u2200 (a : \u03b1), a \u2208 s \u2192 p a\n\u22a2 p (prod s)\n[PROOFSTEP]\ninduction' s using Multiset.induction_on with a s hsa\n[GOAL]\ncase empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s \u2192 p a\nhs : 0 \u2260 \u2205\np_s : \u2200 (a : \u03b1), a \u2208 0 \u2192 p a\n\u22a2 p (prod 0)\n[PROOFSTEP]\nsimp at hs \n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s\u271d \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 p a\na : \u03b1\ns : Multiset \u03b1\nhsa : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 p (prod s)\nhs : a ::\u2098 s \u2260 \u2205\np_s : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\n\u22a2 p (prod (a ::\u2098 s))\n[PROOFSTEP]\nrw [prod_cons]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s\u271d \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 p a\na : \u03b1\ns : Multiset \u03b1\nhsa : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 p (prod s)\nhs : a ::\u2098 s \u2260 \u2205\np_s : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\n\u22a2 p (a * prod s)\n[PROOFSTEP]\nby_cases hs_empty : s = \u2205\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s\u271d \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 p a\na : \u03b1\ns : Multiset \u03b1\nhsa : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 p (prod s)\nhs : a ::\u2098 s \u2260 \u2205\np_s : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhs_empty : s = \u2205\n\u22a2 p (a * prod s)\n[PROOFSTEP]\nsimp [hs_empty, p_s a]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s\u271d \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 p a\na : \u03b1\ns : Multiset \u03b1\nhsa : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 p (prod s)\nhs : a ::\u2098 s \u2260 \u2205\np_s : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhs_empty : \u00acs = \u2205\n\u22a2 p (a * prod s)\n[PROOFSTEP]\nhave hps : \u2200 x, x \u2208 s \u2192 p x := fun x hxs => p_s x (mem_cons_of_mem hxs)\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\np : \u03b1 \u2192 Prop\np_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nhs\u271d : s\u271d \u2260 \u2205\np_s\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 p a\na : \u03b1\ns : Multiset \u03b1\nhsa : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 p (prod s)\nhs : a ::\u2098 s \u2260 \u2205\np_s : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhs_empty : \u00acs = \u2205\nhps : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\n\u22a2 p (a * prod s)\n[PROOFSTEP]\nexact p_mul a s.prod (p_s a (mem_cons_self a s)) (hsa hs_empty hps)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\nh : s \u2264 t\n\u22a2 prod s \u2223 prod t\n[PROOFSTEP]\nobtain \u27e8z, rfl\u27e9 := exists_add_of_le h\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\nz : Multiset \u03b1\nh : s \u2264 s + z\n\u22a2 prod s \u2223 prod (s + z)\n[PROOFSTEP]\nsimp only [prod_add, dvd_mul_right]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b2\nS : Multiset \u03b1\ng1 g2 : \u03b1 \u2192 \u03b2\nh : \u2200 (a : \u03b1), a \u2208 S \u2192 g1 a \u2223 g2 a\n\u22a2 prod (map g1 S) \u2223 prod (map g2 S)\n[PROOFSTEP]\napply Multiset.induction_on' S\n[GOAL]\ncase h\u2081\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b2\nS : Multiset \u03b1\ng1 g2 : \u03b1 \u2192 \u03b2\nh : \u2200 (a : \u03b1), a \u2208 S \u2192 g1 a \u2223 g2 a\n\u22a2 prod (map g1 0) \u2223 prod (map g2 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b2\nS : Multiset \u03b1\ng1 g2 : \u03b1 \u2192 \u03b2\nh : \u2200 (a : \u03b1), a \u2208 S \u2192 g1 a \u2223 g2 a\n\u22a2 \u2200 {a : \u03b1} {s : Multiset \u03b1},\n a \u2208 S \u2192 s \u2286 S \u2192 prod (map g1 s) \u2223 prod (map g2 s) \u2192 prod (map g1 (insert a s)) \u2223 prod (map g2 (insert a s))\n[PROOFSTEP]\nintro a T haS _ IH\n[GOAL]\ncase h\u2082\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b2\nS : Multiset \u03b1\ng1 g2 : \u03b1 \u2192 \u03b2\nh : \u2200 (a : \u03b1), a \u2208 S \u2192 g1 a \u2223 g2 a\na : \u03b1\nT : Multiset \u03b1\nhaS : a \u2208 S\na\u271d : T \u2286 S\nIH : prod (map g1 T) \u2223 prod (map g2 T)\n\u22a2 prod (map g1 (insert a T)) \u2223 prod (map g2 (insert a T))\n[PROOFSTEP]\nsimp [mul_dvd_mul (h a haS) IH]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\ns : Multiset \u03b1\nh : 0 \u2208 s\n\u22a2 prod s = 0\n[PROOFSTEP]\nrcases Multiset.exists_cons_of_mem h with \u27e8s', hs'\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoidWithZero \u03b1\ns : Multiset \u03b1\nh : 0 \u2208 s\ns' : Multiset \u03b1\nhs' : s = 0 ::\u2098 s'\n\u22a2 prod s = 0\n[PROOFSTEP]\nsimp [hs', Multiset.prod_cons]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : CommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NoZeroDivisors \u03b1\ninst\u271d : Nontrivial \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 prod (Quotient.mk (List.isSetoid \u03b1) l) = 0 \u2194 0 \u2208 Quotient.mk (List.isSetoid \u03b1) l\n[PROOFSTEP]\nrw [quot_mk_to_coe, coe_prod]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : CommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NoZeroDivisors \u03b1\ninst\u271d : Nontrivial \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 List.prod l = 0 \u2194 0 \u2208 \u2191l\n[PROOFSTEP]\nexact List.prod_eq_zero_iff\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionCommMonoid \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\n\u22a2 prod (map (fun i => (f i)\u207b\u00b9) m) = (prod (map f m))\u207b\u00b9\n[PROOFSTEP]\nsimp_rw [\u2190 (m.map f).prod_map_inv', map_map, Function.comp_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionCommMonoid \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\nn : \u2124\n\u22a2 prod (map (fun i => f i ^ n) m) = prod (map f m) ^ n\n[PROOFSTEP]\nconvert (m.map f).prod_hom (zpowGroupHom n : \u03b1 \u2192* \u03b1)\n[GOAL]\ncase h.e'_2.h.e'_3\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : DivisionCommMonoid \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\nn : \u2124\n\u22a2 map (fun i => f i ^ n) m = map (\u2191(zpowGroupHom n)) (map f m)\n[PROOFSTEP]\nsimp only [map_map, Function.comp_apply, zpowGroupHom_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\na : \u03b1\ns : Multiset \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 sum (map (fun i => a * f i) 0) = a * sum (map f 0)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\na : \u03b1\ns\u271d : Multiset \u03b9\nf : \u03b9 \u2192 \u03b1\ni : \u03b9\ns : Multiset \u03b9\nih : sum (map (fun i => a * f i) s) = a * sum (map f s)\n\u22a2 sum (map (fun i => a * f i) (i ::\u2098 s)) = a * sum (map f (i ::\u2098 s))\n[PROOFSTEP]\nsimp [ih, mul_add]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\na : \u03b1\ns : Multiset \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 sum (map (fun i => f i * a) 0) = sum (map f 0) * a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\na\u271d : \u03b1\ns\u271d : Multiset \u03b9\nf : \u03b9 \u2192 \u03b1\na : \u03b9\ns : Multiset \u03b9\nih : sum (map (fun i => f i * a\u271d) s) = sum (map f s) * a\u271d\n\u22a2 sum (map (fun i => f i * a\u271d) (a ::\u2098 s)) = sum (map f (a ::\u2098 s)) * a\u271d\n[PROOFSTEP]\nsimp [ih, add_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : Semiring \u03b1\na : \u03b1\ns\u271d : Multiset \u03b1\nx : \u03b1\ns : Multiset \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 s \u2192 a \u2223 x) \u2192 a \u2223 sum s\nh : \u2200 (x_1 : \u03b1), x_1 \u2208 x ::\u2098 s \u2192 a \u2223 x_1\n\u22a2 a \u2223 sum (x ::\u2098 s)\n[PROOFSTEP]\nrw [sum_cons]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : Semiring \u03b1\na : \u03b1\ns\u271d : Multiset \u03b1\nx : \u03b1\ns : Multiset \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 s \u2192 a \u2223 x) \u2192 a \u2223 sum s\nh : \u2200 (x_1 : \u03b1), x_1 \u2208 x ::\u2098 s \u2192 a \u2223 x_1\n\u22a2 a \u2223 x + sum s\n[PROOFSTEP]\nexact dvd_add (h _ (mem_cons_self _ _)) (ih fun y hy => h _ <| mem_cons.2 <| Or.inr hy)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nl : List \u03b1\nhl : \u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) l \u2192 1 \u2264 x\n\u22a2 1 \u2264 prod (Quotient.mk (List.isSetoid \u03b1) l)\n[PROOFSTEP]\nsimpa using List.one_le_prod_of_one_le hl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nl : List \u03b1\nhl : \u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) l \u2192 1 \u2264 x\nx : \u03b1\nhx : x \u2208 Quotient.mk (List.isSetoid \u03b1) l\n\u22a2 x \u2264 prod (Quotient.mk (List.isSetoid \u03b1) l)\n[PROOFSTEP]\nsimpa using List.single_le_prod hl x hx\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommMonoid \u03b1\ns\u271d t : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nn : \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2264 n\n\u22a2 prod s \u2264 n ^ \u2191card s\n[PROOFSTEP]\ninduction s using Quotient.inductionOn\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommMonoid \u03b1\ns t : Multiset \u03b1\na n : \u03b1\na\u271d : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d \u2192 x \u2264 n\n\u22a2 prod (Quotient.mk (List.isSetoid \u03b1) a\u271d) \u2264 n ^ \u2191card (Quotient.mk (List.isSetoid \u03b1) a\u271d)\n[PROOFSTEP]\nsimpa using List.prod_le_pow_card _ _ h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\n\u22a2 \u2200 (a : List \u03b1),\n (\u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) a \u2192 1 \u2264 x) \u2192\n prod (Quotient.mk (List.isSetoid \u03b1) a) = 1 \u2192 \u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) a \u2192 x = 1\n[PROOFSTEP]\nsimp only [quot_mk_to_coe, coe_prod, mem_coe]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\n\u22a2 \u2200 (a : List \u03b1), (\u2200 (x : \u03b1), x \u2208 a \u2192 1 \u2264 x) \u2192 List.prod a = 1 \u2192 \u2200 (x : \u03b1), x \u2208 a \u2192 x = 1\n[PROOFSTEP]\nexact fun l => List.all_one_of_le_one_le_of_prod_eq_one\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nh : Rel (fun x x_1 => x \u2264 x_1) s t\n\u22a2 prod s \u2264 prod t\n[PROOFSTEP]\ninduction' h with _ _ _ _ rh _ rt\n[GOAL]\ncase zero\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\n\u22a2 prod 0 \u2264 prod 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommMonoid \u03b1\ns t : Multiset \u03b1\na a\u271d\u00b9 b\u271d : \u03b1\nas\u271d bs\u271d : Multiset \u03b1\nrh : a\u271d\u00b9 \u2264 b\u271d\na\u271d : Rel (fun x x_1 => x \u2264 x_1) as\u271d bs\u271d\nrt : prod as\u271d \u2264 prod bs\u271d\n\u22a2 prod (a\u271d\u00b9 ::\u2098 as\u271d) \u2264 prod (b\u271d ::\u2098 bs\u271d)\n[PROOFSTEP]\nrw [prod_cons, prod_cons]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommMonoid \u03b1\ns t : Multiset \u03b1\na a\u271d\u00b9 b\u271d : \u03b1\nas\u271d bs\u271d : Multiset \u03b1\nrh : a\u271d\u00b9 \u2264 b\u271d\na\u271d : Rel (fun x x_1 => x \u2264 x_1) as\u271d bs\u271d\nrt : prod as\u271d \u2264 prod bs\u271d\n\u22a2 a\u271d\u00b9 * prod as\u271d \u2264 b\u271d * prod bs\u271d\n[PROOFSTEP]\nexact mul_le_mul' rh rt\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 a \u2264 x\n\u22a2 a ^ \u2191card s \u2264 prod s\n[PROOFSTEP]\nrw [\u2190 Multiset.prod_replicate, \u2190 Multiset.map_const]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommMonoid \u03b1\ns t : Multiset \u03b1\na : \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 a \u2264 x\n\u22a2 prod (map (Function.const \u03b1 a) s) \u2264 prod s\n[PROOFSTEP]\nexact prod_map_le_prod _ h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommSemiring \u03b1\nm : Multiset \u03b1\nh : \u2200 (a : \u03b1), a \u2208 m \u2192 0 \u2264 a\n\u22a2 0 \u2264 prod m\n[PROOFSTEP]\nrevert h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommSemiring \u03b1\nm : Multiset \u03b1\n\u22a2 (\u2200 (a : \u03b1), a \u2208 m \u2192 0 \u2264 a) \u2192 0 \u2264 prod m\n[PROOFSTEP]\nrefine' m.induction_on _ _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommSemiring \u03b1\nm : Multiset \u03b1\n\u22a2 (\u2200 (a : \u03b1), a \u2208 0 \u2192 0 \u2264 a) \u2192 0 \u2264 prod 0\n[PROOFSTEP]\nrintro -\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommSemiring \u03b1\nm : Multiset \u03b1\n\u22a2 0 \u2264 prod 0\n[PROOFSTEP]\nrw [prod_zero]\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommSemiring \u03b1\nm : Multiset \u03b1\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nexact zero_le_one\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommSemiring \u03b1\nm : Multiset \u03b1\n\u22a2 \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1},\n ((\u2200 (a : \u03b1), a \u2208 s \u2192 0 \u2264 a) \u2192 0 \u2264 prod s) \u2192 (\u2200 (a_2 : \u03b1), a_2 \u2208 a ::\u2098 s \u2192 0 \u2264 a_2) \u2192 0 \u2264 prod (a ::\u2098 s)\n[PROOFSTEP]\nintro a s hs ih\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommSemiring \u03b1\nm : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nhs : (\u2200 (a : \u03b1), a \u2208 s \u2192 0 \u2264 a) \u2192 0 \u2264 prod s\nih : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 0 \u2264 a_1\n\u22a2 0 \u2264 prod (a ::\u2098 s)\n[PROOFSTEP]\nrw [prod_cons]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : OrderedCommSemiring \u03b1\nm : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nhs : (\u2200 (a : \u03b1), a \u2208 s \u2192 0 \u2264 a) \u2192 0 \u2264 prod s\nih : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 0 \u2264 a_1\n\u22a2 0 \u2264 a * prod s\n[PROOFSTEP]\nexact mul_nonneg (ih _ <| mem_cons_self _ _) (hs fun a ha => ih _ <| mem_cons_of_mem ha)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\nm : Multiset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 m \u2192 x = 1\n\u22a2 prod m = 1\n[PROOFSTEP]\ninduction' m using Quotient.inductionOn with l\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\nm : Multiset \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 m \u2192 x = 1\nl : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) l \u2192 x = 1\n\u22a2 prod (Quotient.mk (List.isSetoid \u03b1) l) = 1\n[PROOFSTEP]\nsimp [List.prod_eq_one h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CanonicallyOrderedMonoid \u03b1\nm : Multiset \u03b1\na : \u03b1\nh : a \u2208 m\n\u22a2 a \u2264 prod m\n[PROOFSTEP]\nobtain \u27e8m', rfl\u27e9 := exists_cons_of_mem h\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CanonicallyOrderedMonoid \u03b1\na : \u03b1\nm' : Multiset \u03b1\nh : a \u2208 a ::\u2098 m'\n\u22a2 a \u2264 prod (a ::\u2098 m')\n[PROOFSTEP]\nrw [prod_cons]\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CanonicallyOrderedMonoid \u03b1\na : \u03b1\nm' : Multiset \u03b1\nh : a \u2208 a ::\u2098 m'\n\u22a2 a \u2264 a * prod m'\n[PROOFSTEP]\nexact _root_.le_mul_right (le_refl a)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_one : f 1 = 1\nhp_one : p 1\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\ns : Multiset \u03b1\nhps : \u2200 (a : \u03b1), a \u2208 s \u2192 p a\n\u22a2 f (prod s) \u2264 prod (map f s)\n[PROOFSTEP]\nrevert s\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_one : f 1 = 1\nhp_one : p 1\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\n\u22a2 \u2200 (s : Multiset \u03b1), (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 f (prod s) \u2264 prod (map f s)\n[PROOFSTEP]\nrefine' Multiset.induction _ _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_one : f 1 = 1\nhp_one : p 1\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\n\u22a2 (\u2200 (a : \u03b1), a \u2208 0 \u2192 p a) \u2192 f (prod 0) \u2264 prod (map f 0)\n[PROOFSTEP]\nsimp [le_of_eq h_one]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_one : f 1 = 1\nhp_one : p 1\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\n\u22a2 \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1},\n ((\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 f (prod s) \u2264 prod (map f s)) \u2192\n (\u2200 (a_2 : \u03b1), a_2 \u2208 a ::\u2098 s \u2192 p a_2) \u2192 f (prod (a ::\u2098 s)) \u2264 prod (map f (a ::\u2098 s))\n[PROOFSTEP]\nintro a s hs hpsa\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_one : f 1 = 1\nhp_one : p 1\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\na : \u03b1\ns : Multiset \u03b1\nhs : (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 f (prod s) \u2264 prod (map f s)\nhpsa : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\n\u22a2 f (prod (a ::\u2098 s)) \u2264 prod (map f (a ::\u2098 s))\n[PROOFSTEP]\nhave hps : \u2200 x, x \u2208 s \u2192 p x := fun x hx => hpsa x (mem_cons_of_mem hx)\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_one : f 1 = 1\nhp_one : p 1\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\na : \u03b1\ns : Multiset \u03b1\nhs : (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 f (prod s) \u2264 prod (map f s)\nhpsa : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhps : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\n\u22a2 f (prod (a ::\u2098 s)) \u2264 prod (map f (a ::\u2098 s))\n[PROOFSTEP]\nhave hp_prod : p s.prod := prod_induction p s hp_mul hp_one hps\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_one : f 1 = 1\nhp_one : p 1\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\na : \u03b1\ns : Multiset \u03b1\nhs : (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 f (prod s) \u2264 prod (map f s)\nhpsa : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhps : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\nhp_prod : p (prod s)\n\u22a2 f (prod (a ::\u2098 s)) \u2264 prod (map f (a ::\u2098 s))\n[PROOFSTEP]\nrw [prod_cons, map_cons, prod_cons]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_one : f 1 = 1\nhp_one : p 1\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\na : \u03b1\ns : Multiset \u03b1\nhs : (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 f (prod s) \u2264 prod (map f s)\nhpsa : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhps : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\nhp_prod : p (prod s)\n\u22a2 f (a * prod s) \u2264 f a * prod (map f s)\n[PROOFSTEP]\nexact (h_mul a s.prod (hpsa a (mem_cons_self a s)) hp_prod).trans (mul_le_mul_left' (hs hps) _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\nh_one : f 1 = 1\nh_mul : \u2200 (a b : \u03b1), f (a * b) \u2264 f a * f b\ns : Multiset \u03b1\n\u22a2 \u2200 (a b : \u03b1), (fun x => True) a \u2192 (fun x => True) b \u2192 (fun x => True) (a * b)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\nh_one : f 1 = 1\nh_mul : \u2200 (a b : \u03b1), f (a * b) \u2264 f a * f b\ns : Multiset \u03b1\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2192 (fun x => True) a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\ns : Multiset \u03b1\nhs_nonempty : s \u2260 \u2205\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 p a\n\u22a2 f (prod s) \u2264 prod (map f s)\n[PROOFSTEP]\nrevert s\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\n\u22a2 \u2200 (s : Multiset \u03b1), s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 f (prod s) \u2264 prod (map f s)\n[PROOFSTEP]\nrefine' Multiset.induction _ _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\n\u22a2 0 \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 0 \u2192 p a) \u2192 f (prod 0) \u2264 prod (map f 0)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nh : 0 \u2260 \u2205\n\u22a2 (\u2200 (a : \u03b1), a \u2208 0 \u2192 p a) \u2192 f (prod 0) \u2264 prod (map f 0)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase refine'_1.h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\nh : 0 \u2260 \u2205\n\u22a2 False\n[PROOFSTEP]\nexact h rfl\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\n\u22a2 \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1},\n (s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 f (prod s) \u2264 prod (map f s)) \u2192\n a ::\u2098 s \u2260 \u2205 \u2192 (\u2200 (a_2 : \u03b1), a_2 \u2208 a ::\u2098 s \u2192 p a_2) \u2192 f (prod (a ::\u2098 s)) \u2264 prod (map f (a ::\u2098 s))\n[PROOFSTEP]\nrintro a s hs - hsa_prop\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\na : \u03b1\ns : Multiset \u03b1\nhs : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 f (prod s) \u2264 prod (map f s)\nhsa_prop : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\n\u22a2 f (prod (a ::\u2098 s)) \u2264 prod (map f (a ::\u2098 s))\n[PROOFSTEP]\nrw [prod_cons, map_cons, prod_cons]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\na : \u03b1\ns : Multiset \u03b1\nhs : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 f (prod s) \u2264 prod (map f s)\nhsa_prop : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\n\u22a2 f (a * prod s) \u2264 f a * prod (map f s)\n[PROOFSTEP]\nby_cases hs_empty : s = \u2205\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\na : \u03b1\ns : Multiset \u03b1\nhs : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 f (prod s) \u2264 prod (map f s)\nhsa_prop : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhs_empty : s = \u2205\n\u22a2 f (a * prod s) \u2264 f a * prod (map f s)\n[PROOFSTEP]\nsimp [hs_empty]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\na : \u03b1\ns : Multiset \u03b1\nhs : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 f (prod s) \u2264 prod (map f s)\nhsa_prop : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhs_empty : \u00acs = \u2205\n\u22a2 f (a * prod s) \u2264 f a * prod (map f s)\n[PROOFSTEP]\nhave hsa_restrict : \u2200 x, x \u2208 s \u2192 p x := fun x hx => hsa_prop x (mem_cons_of_mem hx)\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\na : \u03b1\ns : Multiset \u03b1\nhs : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 f (prod s) \u2264 prod (map f s)\nhsa_prop : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhs_empty : \u00acs = \u2205\nhsa_restrict : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\n\u22a2 f (a * prod s) \u2264 f a * prod (map f s)\n[PROOFSTEP]\nhave hp_sup : p s.prod := prod_induction_nonempty p hp_mul hs_empty hsa_restrict\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\na : \u03b1\ns : Multiset \u03b1\nhs : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 f (prod s) \u2264 prod (map f s)\nhsa_prop : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhs_empty : \u00acs = \u2205\nhsa_restrict : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\nhp_sup : p (prod s)\n\u22a2 f (a * prod s) \u2264 f a * prod (map f s)\n[PROOFSTEP]\nhave hp_a : p a := hsa_prop a (mem_cons_self a s)\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\nh_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b\nhp_mul : \u2200 (a b : \u03b1), p a \u2192 p b \u2192 p (a * b)\na : \u03b1\ns : Multiset \u03b1\nhs : s \u2260 \u2205 \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 f (prod s) \u2264 prod (map f s)\nhsa_prop : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 p a_1\nhs_empty : \u00acs = \u2205\nhsa_restrict : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\nhp_sup : p (prod s)\nhp_a : p a\n\u22a2 f (a * prod s) \u2264 f a * prod (map f s)\n[PROOFSTEP]\nexact (h_mul a _ hp_a hp_sup).trans (mul_le_mul_left' (hs hs_empty hsa_restrict) _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\nh_mul : \u2200 (a b : \u03b1), f (a * b) \u2264 f a * f b\ns : Multiset \u03b1\nhs_nonempty : s \u2260 \u2205\n\u22a2 \u2200 (a b : \u03b1), (fun x => True) a \u2192 (fun x => True) b \u2192 f (a * b) \u2264 f a * f b\n[PROOFSTEP]\nsimp [h_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\nh_mul : \u2200 (a b : \u03b1), f (a * b) \u2264 f a * f b\ns : Multiset \u03b1\nhs_nonempty : s \u2260 \u2205\n\u22a2 \u2200 (a b : \u03b1), (fun x => True) a \u2192 (fun x => True) b \u2192 (fun x => True) (a * b)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : OrderedCommMonoid \u03b2\nf : \u03b1 \u2192 \u03b2\nh_mul : \u2200 (a b : \u03b1), f (a * b) \u2264 f a * f b\ns : Multiset \u03b1\nhs_nonempty : s \u2260 \u2205\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2192 (fun x => True) a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns : Multiset \u03b1\n\u22a2 sum (map (fun a => {a}) 0) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns : Multiset \u03b1\n\u22a2 \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1}, sum (map (fun a => {a}) s) = s \u2192 sum (map (fun a => {a}) (a ::\u2098 s)) = a ::\u2098 s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns : Multiset \u2115\nn : \u2115\n\u22a2 sum s % n = sum (map (fun x => x % n) s) % n\n[PROOFSTEP]\ninduction s using Multiset.induction\n[GOAL]\ncase empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nn : \u2115\n\u22a2 sum 0 % n = sum (map (fun x => x % n) 0) % n\n[PROOFSTEP]\nsimp [Nat.add_mod, *]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nn a\u271d\u00b9 : \u2115\ns\u271d : Multiset \u2115\na\u271d : sum s\u271d % n = sum (map (fun x => x % n) s\u271d) % n\n\u22a2 sum (a\u271d\u00b9 ::\u2098 s\u271d) % n = sum (map (fun x => x % n) (a\u271d\u00b9 ::\u2098 s\u271d)) % n\n[PROOFSTEP]\nsimp [Nat.add_mod, *]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns : Multiset \u2115\nn : \u2115\n\u22a2 prod s % n = prod (map (fun x => x % n) s) % n\n[PROOFSTEP]\ninduction s using Multiset.induction\n[GOAL]\ncase empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nn : \u2115\n\u22a2 prod 0 % n = prod (map (fun x => x % n) 0) % n\n[PROOFSTEP]\nsimp [Nat.mul_mod, *]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nn a\u271d\u00b9 : \u2115\ns\u271d : Multiset \u2115\na\u271d : prod s\u271d % n = prod (map (fun x => x % n) s\u271d) % n\n\u22a2 prod (a\u271d\u00b9 ::\u2098 s\u271d) % n = prod (map (fun x => x % n) (a\u271d\u00b9 ::\u2098 s\u271d)) % n\n[PROOFSTEP]\nsimp [Nat.mul_mod, *]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns : Multiset \u2124\nn : \u2124\n\u22a2 sum s % n = sum (map (fun x => x % n) s) % n\n[PROOFSTEP]\ninduction s using Multiset.induction\n[GOAL]\ncase empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nn : \u2124\n\u22a2 sum 0 % n = sum (map (fun x => x % n) 0) % n\n[PROOFSTEP]\nsimp [Int.add_emod, *]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nn a\u271d\u00b9 : \u2124\ns\u271d : Multiset \u2124\na\u271d : sum s\u271d % n = sum (map (fun x => x % n) s\u271d) % n\n\u22a2 sum (a\u271d\u00b9 ::\u2098 s\u271d) % n = sum (map (fun x => x % n) (a\u271d\u00b9 ::\u2098 s\u271d)) % n\n[PROOFSTEP]\nsimp [Int.add_emod, *]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns : Multiset \u2124\nn : \u2124\n\u22a2 prod s % n = prod (map (fun x => x % n) s) % n\n[PROOFSTEP]\ninduction s using Multiset.induction\n[GOAL]\ncase empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nn : \u2124\n\u22a2 prod 0 % n = prod (map (fun x => x % n) 0) % n\n[PROOFSTEP]\nsimp [Int.mul_emod, *]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nn a\u271d\u00b9 : \u2124\ns\u271d : Multiset \u2124\na\u271d : prod s\u271d % n = prod (map (fun x => x % n) s\u271d) % n\n\u22a2 prod (a\u271d\u00b9 ::\u2098 s\u271d) % n = prod (map (fun x => x % n) (a\u271d\u00b9 ::\u2098 s\u271d)) % n\n[PROOFSTEP]\nsimp [Int.mul_emod, *]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.BigOperators.Multiset.Basic", "llama_tokens": 21093, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7879311956428947, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.44295640250528573}} {"text": "[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\n\u22a2 AffineIndependent k p \u2194\n \u2200 (w : \u03b9 \u2192 k), \u2211 i : \u03b9, w i = 0 \u2192 \u2191(Finset.weightedVSub Finset.univ p) w = 0 \u2192 \u2200 (i : \u03b9), w i = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\n\u22a2 AffineIndependent k p \u2192\n \u2200 (w : \u03b9 \u2192 k), \u2211 i : \u03b9, w i = 0 \u2192 \u2191(Finset.weightedVSub Finset.univ p) w = 0 \u2192 \u2200 (i : \u03b9), w i = 0\n[PROOFSTEP]\nexact fun h w hw hs i => h Finset.univ w hw hs i (Finset.mem_univ _)\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\n\u22a2 (\u2200 (w : \u03b9 \u2192 k), \u2211 i : \u03b9, w i = 0 \u2192 \u2191(Finset.weightedVSub Finset.univ p) w = 0 \u2192 \u2200 (i : \u03b9), w i = 0) \u2192\n AffineIndependent k p\n[PROOFSTEP]\nintro h s w hw hs i hi\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nh : \u2200 (w : \u03b9 \u2192 k), \u2211 i : \u03b9, w i = 0 \u2192 \u2191(Finset.weightedVSub Finset.univ p) w = 0 \u2192 \u2200 (i : \u03b9), w i = 0\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 w i = 0\n[PROOFSTEP]\nrw [Finset.weightedVSub_indicator_subset _ _ (Finset.subset_univ s)] at hs \n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nh : \u2200 (w : \u03b9 \u2192 k), \u2211 i : \u03b9, w i = 0 \u2192 \u2191(Finset.weightedVSub Finset.univ p) w = 0 \u2192 \u2200 (i : \u03b9), w i = 0\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub Finset.univ p) (Set.indicator (\u2191s) w) = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 w i = 0\n[PROOFSTEP]\nrw [Set.sum_indicator_subset _ (Finset.subset_univ s)] at hw \n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nh : \u2200 (w : \u03b9 \u2192 k), \u2211 i : \u03b9, w i = 0 \u2192 \u2191(Finset.weightedVSub Finset.univ p) w = 0 \u2192 \u2200 (i : \u03b9), w i = 0\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i : \u03b9, Set.indicator (\u2191s) (fun i => w i) i = 0\nhs : \u2191(Finset.weightedVSub Finset.univ p) (Set.indicator (\u2191s) w) = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 w i = 0\n[PROOFSTEP]\nreplace h := h ((\u2191s : Set \u03b9).indicator w) hw hs i\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i : \u03b9, Set.indicator (\u2191s) (fun i => w i) i = 0\nhs : \u2191(Finset.weightedVSub Finset.univ p) (Set.indicator (\u2191s) w) = 0\ni : \u03b9\nhi : i \u2208 s\nh : Set.indicator (\u2191s) w i = 0\n\u22a2 w i = 0\n[PROOFSTEP]\nsimpa [hi] using h\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\n\u22a2 AffineIndependent k p \u2194 LinearIndependent k fun i => p \u2191i -\u1d65 p i1\n[PROOFSTEP]\nclassical\nconstructor\n\u00b7 intro h\n rw [linearIndependent_iff']\n intro s g hg i hi\n set f : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g \u27e8x, hx\u27e9 with hfdef\n let s2 : Finset \u03b9 := insert i1 (s.map (Embedding.subtype _))\n have hfg : \u2200 x : { x // x \u2260 i1 }, g x = f x := by\n intro x\n rw [hfdef]\n dsimp only\n erw [dif_neg x.property, Subtype.coe_eta]\n rw [hfg]\n have hf : \u2211 \u03b9 in s2, f \u03b9 = 0 :=\n by\n rw [Finset.sum_insert (Finset.not_mem_map_subtype_of_not_property s (Classical.not_not.2 rfl)),\n Finset.sum_subtype_map_embedding fun x _ => (hfg x).symm]\n rw [hfdef]\n dsimp only\n rw [dif_pos rfl]\n exact neg_add_self _\n have hs2 : s2.weightedVSub p f = (0 : V) :=\n by\n set f2 : \u03b9 \u2192 V := fun x => f x \u2022 (p x -\u1d65 p i1) with hf2def\n set g2 : { x // x \u2260 i1 } \u2192 V := fun x => g x \u2022 (p x -\u1d65 p i1)\n have hf2g2 : \u2200 x : { x // x \u2260 i1 }, f2 x = g2 x :=\n by\n simp only [hf2def]\n refine' fun x => _\n rw [hfg]\n rw [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero s2 f p hf (p i1), Finset.weightedVSubOfPoint_insert,\n Finset.weightedVSubOfPoint_apply, Finset.sum_subtype_map_embedding fun x _ => hf2g2 x]\n exact hg\n exact h s2 f hf hs2 i (Finset.mem_insert_of_mem (Finset.mem_map.2 \u27e8i, hi, rfl\u27e9))\n\u00b7 intro h\n rw [linearIndependent_iff'] at h \n intro s w hw hs i hi\n rw [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero s w p hw (p i1), \u2190 s.weightedVSubOfPoint_erase w p i1,\n Finset.weightedVSubOfPoint_apply] at hs \n let f : \u03b9 \u2192 V := fun i => w i \u2022 (p i -\u1d65 p i1)\n have hs2 : (\u2211 i in (s.erase i1).subtype fun i => i \u2260 i1, f i) = 0 :=\n by\n rw [\u2190 hs]\n convert Finset.sum_subtype_of_mem f fun x => Finset.ne_of_mem_erase\n have h2 := h ((s.erase i1).subtype fun i => i \u2260 i1) (fun x => w x) hs2\n simp_rw [Finset.mem_subtype] at h2 \n have h2b : \u2200 i \u2208 s, i \u2260 i1 \u2192 w i = 0 := fun i his hi => h2 \u27e8i, hi\u27e9 (Finset.mem_erase_of_ne_of_mem hi his)\n exact Finset.eq_zero_of_sum_eq_zero hw h2b i hi\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\n\u22a2 AffineIndependent k p \u2194 LinearIndependent k fun i => p \u2191i -\u1d65 p i1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\n\u22a2 AffineIndependent k p \u2192 LinearIndependent k fun i => p \u2191i -\u1d65 p i1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\n\u22a2 LinearIndependent k fun i => p \u2191i -\u1d65 p i1\n[PROOFSTEP]\nrw [linearIndependent_iff']\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\n\u22a2 \u2200 (s : Finset { x // x \u2260 i1 }) (g : { x // x \u2260 i1 } \u2192 k),\n \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0 \u2192 \u2200 (i : { x // x \u2260 i1 }), i \u2208 s \u2192 g i = 0\n[PROOFSTEP]\nintro s g hg i hi\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\n\u22a2 g i = 0\n[PROOFSTEP]\nset f : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g \u27e8x, hx\u27e9 with hfdef\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\n\u22a2 g i = 0\n[PROOFSTEP]\nlet s2 : Finset \u03b9 := insert i1 (s.map (Embedding.subtype _))\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\n\u22a2 g i = 0\n[PROOFSTEP]\nhave hfg : \u2200 x : { x // x \u2260 i1 }, g x = f x := by\n intro x\n rw [hfdef]\n dsimp only\n erw [dif_neg x.property, Subtype.coe_eta]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\n\u22a2 \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\n[PROOFSTEP]\nintro x\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nx : { x // x \u2260 i1 }\n\u22a2 g x = f \u2191x\n[PROOFSTEP]\nrw [hfdef]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nx : { x // x \u2260 i1 }\n\u22a2 g x = (fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }) \u2191x\n[PROOFSTEP]\ndsimp only\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nx : { x // x \u2260 i1 }\n\u22a2 g x = if hx : \u2191x = i1 then -\u2211 y in s, g y else g { val := \u2191x, property := hx }\n[PROOFSTEP]\nerw [dif_neg x.property, Subtype.coe_eta]\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\n\u22a2 g i = 0\n[PROOFSTEP]\nrw [hfg]\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\n\u22a2 f \u2191i = 0\n[PROOFSTEP]\nhave hf : \u2211 \u03b9 in s2, f \u03b9 = 0 :=\n by\n rw [Finset.sum_insert (Finset.not_mem_map_subtype_of_not_property s (Classical.not_not.2 rfl)),\n Finset.sum_subtype_map_embedding fun x _ => (hfg x).symm]\n rw [hfdef]\n dsimp only\n rw [dif_pos rfl]\n exact neg_add_self _\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\n\u22a2 \u2211 \u03b9 in s2, f \u03b9 = 0\n[PROOFSTEP]\nrw [Finset.sum_insert (Finset.not_mem_map_subtype_of_not_property s (Classical.not_not.2 rfl)),\n Finset.sum_subtype_map_embedding fun x _ => (hfg x).symm]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\n\u22a2 f i1 + \u2211 x in s, g x = 0\n[PROOFSTEP]\nrw [hfdef]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\n\u22a2 (fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }) i1 + \u2211 x in s, g x = 0\n[PROOFSTEP]\ndsimp only\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\n\u22a2 (if hx : i1 = i1 then -\u2211 y in s, g y else g { val := i1, property := hx }) + \u2211 x in s, g x = 0\n[PROOFSTEP]\nrw [dif_pos rfl]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\n\u22a2 -\u2211 y in s, g y + \u2211 x in s, g x = 0\n[PROOFSTEP]\nexact neg_add_self _\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\nhf : \u2211 \u03b9 in s2, f \u03b9 = 0\n\u22a2 f \u2191i = 0\n[PROOFSTEP]\nhave hs2 : s2.weightedVSub p f = (0 : V) :=\n by\n set f2 : \u03b9 \u2192 V := fun x => f x \u2022 (p x -\u1d65 p i1) with hf2def\n set g2 : { x // x \u2260 i1 } \u2192 V := fun x => g x \u2022 (p x -\u1d65 p i1)\n have hf2g2 : \u2200 x : { x // x \u2260 i1 }, f2 x = g2 x :=\n by\n simp only [hf2def]\n refine' fun x => _\n rw [hfg]\n rw [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero s2 f p hf (p i1), Finset.weightedVSubOfPoint_insert,\n Finset.weightedVSubOfPoint_apply, Finset.sum_subtype_map_embedding fun x _ => hf2g2 x]\n exact hg\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\nhf : \u2211 \u03b9 in s2, f \u03b9 = 0\n\u22a2 \u2191(Finset.weightedVSub s2 p) f = 0\n[PROOFSTEP]\nset f2 : \u03b9 \u2192 V := fun x => f x \u2022 (p x -\u1d65 p i1) with hf2def\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\nhf : \u2211 \u03b9 in s2, f \u03b9 = 0\nf2 : \u03b9 \u2192 V := fun x => f x \u2022 (p x -\u1d65 p i1)\nhf2def : f2 = fun x => f x \u2022 (p x -\u1d65 p i1)\n\u22a2 \u2191(Finset.weightedVSub s2 p) f = 0\n[PROOFSTEP]\nset g2 : { x // x \u2260 i1 } \u2192 V := fun x => g x \u2022 (p x -\u1d65 p i1)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\nhf : \u2211 \u03b9 in s2, f \u03b9 = 0\nf2 : \u03b9 \u2192 V := fun x => f x \u2022 (p x -\u1d65 p i1)\nhf2def : f2 = fun x => f x \u2022 (p x -\u1d65 p i1)\ng2 : { x // x \u2260 i1 } \u2192 V := fun x => g x \u2022 (p \u2191x -\u1d65 p i1)\nhg : Finset.sum s g2 = 0\n\u22a2 \u2191(Finset.weightedVSub s2 p) f = 0\n[PROOFSTEP]\nhave hf2g2 : \u2200 x : { x // x \u2260 i1 }, f2 x = g2 x :=\n by\n simp only [hf2def]\n refine' fun x => _\n rw [hfg]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\nhf : \u2211 \u03b9 in s2, f \u03b9 = 0\nf2 : \u03b9 \u2192 V := fun x => f x \u2022 (p x -\u1d65 p i1)\nhf2def : f2 = fun x => f x \u2022 (p x -\u1d65 p i1)\ng2 : { x // x \u2260 i1 } \u2192 V := fun x => g x \u2022 (p \u2191x -\u1d65 p i1)\nhg : Finset.sum s g2 = 0\n\u22a2 \u2200 (x : { x // x \u2260 i1 }), f2 \u2191x = g2 x\n[PROOFSTEP]\nsimp only [hf2def]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\nhf : \u2211 \u03b9 in s2, f \u03b9 = 0\nf2 : \u03b9 \u2192 V := fun x => f x \u2022 (p x -\u1d65 p i1)\nhf2def : f2 = fun x => f x \u2022 (p x -\u1d65 p i1)\ng2 : { x // x \u2260 i1 } \u2192 V := fun x => g x \u2022 (p \u2191x -\u1d65 p i1)\nhg : Finset.sum s g2 = 0\n\u22a2 \u2200 (x : { x // x \u2260 i1 }),\n (if hx : \u2191x = i1 then -\u2211 y in s, g y else g { val := \u2191x, property := hx }) \u2022 (p \u2191x -\u1d65 p i1) = g x \u2022 (p \u2191x -\u1d65 p i1)\n[PROOFSTEP]\nrefine' fun x => _\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\nhf : \u2211 \u03b9 in s2, f \u03b9 = 0\nf2 : \u03b9 \u2192 V := fun x => f x \u2022 (p x -\u1d65 p i1)\nhf2def : f2 = fun x => f x \u2022 (p x -\u1d65 p i1)\ng2 : { x // x \u2260 i1 } \u2192 V := fun x => g x \u2022 (p \u2191x -\u1d65 p i1)\nhg : Finset.sum s g2 = 0\nx : { x // x \u2260 i1 }\n\u22a2 (if hx : \u2191x = i1 then -\u2211 y in s, g y else g { val := \u2191x, property := hx }) \u2022 (p \u2191x -\u1d65 p i1) = g x \u2022 (p \u2191x -\u1d65 p i1)\n[PROOFSTEP]\nrw [hfg]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\nhf : \u2211 \u03b9 in s2, f \u03b9 = 0\nf2 : \u03b9 \u2192 V := fun x => f x \u2022 (p x -\u1d65 p i1)\nhf2def : f2 = fun x => f x \u2022 (p x -\u1d65 p i1)\ng2 : { x // x \u2260 i1 } \u2192 V := fun x => g x \u2022 (p \u2191x -\u1d65 p i1)\nhg : Finset.sum s g2 = 0\nhf2g2 : \u2200 (x : { x // x \u2260 i1 }), f2 \u2191x = g2 x\n\u22a2 \u2191(Finset.weightedVSub s2 p) f = 0\n[PROOFSTEP]\nrw [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero s2 f p hf (p i1), Finset.weightedVSubOfPoint_insert,\n Finset.weightedVSubOfPoint_apply, Finset.sum_subtype_map_embedding fun x _ => hf2g2 x]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\nhf : \u2211 \u03b9 in s2, f \u03b9 = 0\nf2 : \u03b9 \u2192 V := fun x => f x \u2022 (p x -\u1d65 p i1)\nhf2def : f2 = fun x => f x \u2022 (p x -\u1d65 p i1)\ng2 : { x // x \u2260 i1 } \u2192 V := fun x => g x \u2022 (p \u2191x -\u1d65 p i1)\nhg : Finset.sum s g2 = 0\nhf2g2 : \u2200 (x : { x // x \u2260 i1 }), f2 \u2191x = g2 x\n\u22a2 \u2211 x in s, g2 x = 0\n[PROOFSTEP]\nexact hg\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : AffineIndependent k p\ns : Finset { x // x \u2260 i1 }\ng : { x // x \u2260 i1 } \u2192 k\nhg : \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0\ni : { x // x \u2260 i1 }\nhi : i \u2208 s\nf : \u03b9 \u2192 k := fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\nhfdef : f = fun x => if hx : x = i1 then -\u2211 y in s, g y else g { val := x, property := hx }\ns2 : Finset \u03b9 := insert i1 (Finset.map (Embedding.subtype fun x => x \u2260 i1) s)\nhfg : \u2200 (x : { x // x \u2260 i1 }), g x = f \u2191x\nhf : \u2211 \u03b9 in s2, f \u03b9 = 0\nhs2 : \u2191(Finset.weightedVSub s2 p) f = 0\n\u22a2 f \u2191i = 0\n[PROOFSTEP]\nexact h s2 f hf hs2 i (Finset.mem_insert_of_mem (Finset.mem_map.2 \u27e8i, hi, rfl\u27e9))\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\n\u22a2 (LinearIndependent k fun i => p \u2191i -\u1d65 p i1) \u2192 AffineIndependent k p\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh : LinearIndependent k fun i => p \u2191i -\u1d65 p i1\n\u22a2 AffineIndependent k p\n[PROOFSTEP]\nrw [linearIndependent_iff'] at h \n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh :\n \u2200 (s : Finset { x // x \u2260 i1 }) (g : { x // x \u2260 i1 } \u2192 k),\n \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0 \u2192 \u2200 (i : { x // x \u2260 i1 }), i \u2208 s \u2192 g i = 0\n\u22a2 AffineIndependent k p\n[PROOFSTEP]\nintro s w hw hs i hi\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh :\n \u2200 (s : Finset { x // x \u2260 i1 }) (g : { x // x \u2260 i1 } \u2192 k),\n \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0 \u2192 \u2200 (i : { x // x \u2260 i1 }), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 w i = 0\n[PROOFSTEP]\nrw [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero s w p hw (p i1), \u2190 s.weightedVSubOfPoint_erase w p i1,\n Finset.weightedVSubOfPoint_apply] at hs \n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh :\n \u2200 (s : Finset { x // x \u2260 i1 }) (g : { x // x \u2260 i1 } \u2192 k),\n \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0 \u2192 \u2200 (i : { x // x \u2260 i1 }), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2211 i in Finset.erase s i1, w i \u2022 (p i -\u1d65 p i1) = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 w i = 0\n[PROOFSTEP]\nlet f : \u03b9 \u2192 V := fun i => w i \u2022 (p i -\u1d65 p i1)\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh :\n \u2200 (s : Finset { x // x \u2260 i1 }) (g : { x // x \u2260 i1 } \u2192 k),\n \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0 \u2192 \u2200 (i : { x // x \u2260 i1 }), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2211 i in Finset.erase s i1, w i \u2022 (p i -\u1d65 p i1) = 0\ni : \u03b9\nhi : i \u2208 s\nf : \u03b9 \u2192 V := fun i => w i \u2022 (p i -\u1d65 p i1)\n\u22a2 w i = 0\n[PROOFSTEP]\nhave hs2 : (\u2211 i in (s.erase i1).subtype fun i => i \u2260 i1, f i) = 0 :=\n by\n rw [\u2190 hs]\n convert Finset.sum_subtype_of_mem f fun x => Finset.ne_of_mem_erase\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh :\n \u2200 (s : Finset { x // x \u2260 i1 }) (g : { x // x \u2260 i1 } \u2192 k),\n \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0 \u2192 \u2200 (i : { x // x \u2260 i1 }), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2211 i in Finset.erase s i1, w i \u2022 (p i -\u1d65 p i1) = 0\ni : \u03b9\nhi : i \u2208 s\nf : \u03b9 \u2192 V := fun i => w i \u2022 (p i -\u1d65 p i1)\n\u22a2 \u2211 i in Finset.subtype (fun i => i \u2260 i1) (Finset.erase s i1), f \u2191i = 0\n[PROOFSTEP]\nrw [\u2190 hs]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh :\n \u2200 (s : Finset { x // x \u2260 i1 }) (g : { x // x \u2260 i1 } \u2192 k),\n \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0 \u2192 \u2200 (i : { x // x \u2260 i1 }), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2211 i in Finset.erase s i1, w i \u2022 (p i -\u1d65 p i1) = 0\ni : \u03b9\nhi : i \u2208 s\nf : \u03b9 \u2192 V := fun i => w i \u2022 (p i -\u1d65 p i1)\n\u22a2 \u2211 i in Finset.subtype (fun i => i \u2260 i1) (Finset.erase s i1), f \u2191i = \u2211 i in Finset.erase s i1, w i \u2022 (p i -\u1d65 p i1)\n[PROOFSTEP]\nconvert Finset.sum_subtype_of_mem f fun x => Finset.ne_of_mem_erase\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh :\n \u2200 (s : Finset { x // x \u2260 i1 }) (g : { x // x \u2260 i1 } \u2192 k),\n \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0 \u2192 \u2200 (i : { x // x \u2260 i1 }), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2211 i in Finset.erase s i1, w i \u2022 (p i -\u1d65 p i1) = 0\ni : \u03b9\nhi : i \u2208 s\nf : \u03b9 \u2192 V := fun i => w i \u2022 (p i -\u1d65 p i1)\nhs2 : \u2211 i in Finset.subtype (fun i => i \u2260 i1) (Finset.erase s i1), f \u2191i = 0\n\u22a2 w i = 0\n[PROOFSTEP]\nhave h2 := h ((s.erase i1).subtype fun i => i \u2260 i1) (fun x => w x) hs2\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh :\n \u2200 (s : Finset { x // x \u2260 i1 }) (g : { x // x \u2260 i1 } \u2192 k),\n \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0 \u2192 \u2200 (i : { x // x \u2260 i1 }), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2211 i in Finset.erase s i1, w i \u2022 (p i -\u1d65 p i1) = 0\ni : \u03b9\nhi : i \u2208 s\nf : \u03b9 \u2192 V := fun i => w i \u2022 (p i -\u1d65 p i1)\nhs2 : \u2211 i in Finset.subtype (fun i => i \u2260 i1) (Finset.erase s i1), f \u2191i = 0\nh2 : \u2200 (i : { x // x \u2260 i1 }), i \u2208 Finset.subtype (fun i => i \u2260 i1) (Finset.erase s i1) \u2192 (fun x => w \u2191x) i = 0\n\u22a2 w i = 0\n[PROOFSTEP]\nsimp_rw [Finset.mem_subtype] at h2 \n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh :\n \u2200 (s : Finset { x // x \u2260 i1 }) (g : { x // x \u2260 i1 } \u2192 k),\n \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0 \u2192 \u2200 (i : { x // x \u2260 i1 }), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2211 i in Finset.erase s i1, w i \u2022 (p i -\u1d65 p i1) = 0\ni : \u03b9\nhi : i \u2208 s\nf : \u03b9 \u2192 V := fun i => w i \u2022 (p i -\u1d65 p i1)\nhs2 : \u2211 i in Finset.subtype (fun i => i \u2260 i1) (Finset.erase s i1), f \u2191i = 0\nh2 : \u2200 (i : { x // x \u2260 i1 }), \u2191i \u2208 Finset.erase s i1 \u2192 w \u2191i = 0\n\u22a2 w i = 0\n[PROOFSTEP]\nhave h2b : \u2200 i \u2208 s, i \u2260 i1 \u2192 w i = 0 := fun i his hi => h2 \u27e8i, hi\u27e9 (Finset.mem_erase_of_ne_of_mem hi his)\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni1 : \u03b9\nh :\n \u2200 (s : Finset { x // x \u2260 i1 }) (g : { x // x \u2260 i1 } \u2192 k),\n \u2211 i in s, g i \u2022 (p \u2191i -\u1d65 p i1) = 0 \u2192 \u2200 (i : { x // x \u2260 i1 }), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2211 i in Finset.erase s i1, w i \u2022 (p i -\u1d65 p i1) = 0\ni : \u03b9\nhi : i \u2208 s\nf : \u03b9 \u2192 V := fun i => w i \u2022 (p i -\u1d65 p i1)\nhs2 : \u2211 i in Finset.subtype (fun i => i \u2260 i1) (Finset.erase s i1), f \u2191i = 0\nh2 : \u2200 (i : { x // x \u2260 i1 }), \u2191i \u2208 Finset.erase s i1 \u2192 w \u2191i = 0\nh2b : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 i1 \u2192 w i = 0\n\u22a2 w i = 0\n[PROOFSTEP]\nexact Finset.eq_zero_of_sum_eq_zero hw h2b i hi\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\n\u22a2 (AffineIndependent k fun p => \u2191p) \u2194 LinearIndependent k fun v => \u2191v\n[PROOFSTEP]\nrw [affineIndependent_iff_linearIndependent_vsub k (fun p => p : s \u2192 P) \u27e8p\u2081, hp\u2081\u27e9]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\n\u22a2 (LinearIndependent k fun i => \u2191\u2191i -\u1d65 \u2191{ val := p\u2081, property := hp\u2081 }) \u2194 LinearIndependent k fun v => \u2191v\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\n\u22a2 (LinearIndependent k fun i => \u2191\u2191i -\u1d65 \u2191{ val := p\u2081, property := hp\u2081 }) \u2192 LinearIndependent k fun v => \u2191v\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\nh : LinearIndependent k fun i => \u2191\u2191i -\u1d65 \u2191{ val := p\u2081, property := hp\u2081 }\n\u22a2 LinearIndependent k fun v => \u2191v\n[PROOFSTEP]\nhave hv : \u2200 v : (fun p => (p -\u1d65 p\u2081 : V)) '' (s \\ { p\u2081 }), (v : V) +\u1d65 p\u2081 \u2208 s \\ { p\u2081 } := fun v =>\n (vsub_left_injective p\u2081).mem_set_image.1 ((vadd_vsub (v : V) p\u2081).symm \u25b8 v.property)\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\nh : LinearIndependent k fun i => \u2191\u2191i -\u1d65 \u2191{ val := p\u2081, property := hp\u2081 }\nhv : \u2200 (v : \u2191((fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}))), \u2191v +\u1d65 p\u2081 \u2208 s \\ {p\u2081}\n\u22a2 LinearIndependent k fun v => \u2191v\n[PROOFSTEP]\nlet f : (fun p : P => (p -\u1d65 p\u2081 : V)) '' (s \\ { p\u2081 }) \u2192 { x : s // x \u2260 \u27e8p\u2081, hp\u2081\u27e9 } := fun x =>\n \u27e8\u27e8(x : V) +\u1d65 p\u2081, Set.mem_of_mem_diff (hv x)\u27e9, fun hx => Set.not_mem_of_mem_diff (hv x) (Subtype.ext_iff.1 hx)\u27e9\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\nh : LinearIndependent k fun i => \u2191\u2191i -\u1d65 \u2191{ val := p\u2081, property := hp\u2081 }\nhv : \u2200 (v : \u2191((fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}))), \u2191v +\u1d65 p\u2081 \u2208 s \\ {p\u2081}\nf : \u2191((fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081})) \u2192 { x // x \u2260 { val := p\u2081, property := hp\u2081 } } :=\n fun x =>\n { val := { val := \u2191x +\u1d65 p\u2081, property := (_ : \u2191x +\u1d65 p\u2081 \u2208 s) },\n property := (_ : { val := \u2191x +\u1d65 p\u2081, property := (_ : \u2191x +\u1d65 p\u2081 \u2208 s) } = { val := p\u2081, property := hp\u2081 } \u2192 False) }\n\u22a2 LinearIndependent k fun v => \u2191v\n[PROOFSTEP]\nconvert h.comp f fun x1 x2 hx => Subtype.ext (vadd_right_cancel p\u2081 (Subtype.ext_iff.1 (Subtype.ext_iff.1 hx)))\n[GOAL]\ncase h.e'_4.h.h.e\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\nh : LinearIndependent k fun i => \u2191\u2191i -\u1d65 \u2191{ val := p\u2081, property := hp\u2081 }\nhv : \u2200 (v : \u2191((fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}))), \u2191v +\u1d65 p\u2081 \u2208 s \\ {p\u2081}\nf : \u2191((fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081})) \u2192 { x // x \u2260 { val := p\u2081, property := hp\u2081 } } :=\n fun x =>\n { val := { val := \u2191x +\u1d65 p\u2081, property := (_ : \u2191x +\u1d65 p\u2081 \u2208 s) },\n property := (_ : { val := \u2191x +\u1d65 p\u2081, property := (_ : \u2191x +\u1d65 p\u2081 \u2208 s) } = { val := p\u2081, property := hp\u2081 } \u2192 False) }\nx\u271d : \u2191((fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}))\n\u22a2 Subtype.val = (fun i => \u2191\u2191i -\u1d65 \u2191{ val := p\u2081, property := hp\u2081 }) \u2218 f\n[PROOFSTEP]\next v\n[GOAL]\ncase h.e'_4.h.h.e.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\nh : LinearIndependent k fun i => \u2191\u2191i -\u1d65 \u2191{ val := p\u2081, property := hp\u2081 }\nhv : \u2200 (v : \u2191((fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}))), \u2191v +\u1d65 p\u2081 \u2208 s \\ {p\u2081}\nf : \u2191((fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081})) \u2192 { x // x \u2260 { val := p\u2081, property := hp\u2081 } } :=\n fun x =>\n { val := { val := \u2191x +\u1d65 p\u2081, property := (_ : \u2191x +\u1d65 p\u2081 \u2208 s) },\n property := (_ : { val := \u2191x +\u1d65 p\u2081, property := (_ : \u2191x +\u1d65 p\u2081 \u2208 s) } = { val := p\u2081, property := hp\u2081 } \u2192 False) }\nx\u271d : \u2191((fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}))\nv : { x // x \u2208 (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) }\n\u22a2 \u2191v = ((fun i => \u2191\u2191i -\u1d65 \u2191{ val := p\u2081, property := hp\u2081 }) \u2218 f) v\n[PROOFSTEP]\nexact (vadd_vsub (v : V) p\u2081).symm\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\n\u22a2 (LinearIndependent k fun v => \u2191v) \u2192 LinearIndependent k fun i => \u2191\u2191i -\u1d65 \u2191{ val := p\u2081, property := hp\u2081 }\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\nh : LinearIndependent k fun v => \u2191v\n\u22a2 LinearIndependent k fun i => \u2191\u2191i -\u1d65 \u2191{ val := p\u2081, property := hp\u2081 }\n[PROOFSTEP]\nlet f : { x : s // x \u2260 \u27e8p\u2081, hp\u2081\u27e9 } \u2192 (fun p : P => (p -\u1d65 p\u2081 : V)) '' (s \\ { p\u2081 }) := fun x =>\n \u27e8((x : s) : P) -\u1d65 p\u2081, \u27e8x, \u27e8\u27e8(x : s).property, fun hx => x.property (Subtype.ext hx)\u27e9, rfl\u27e9\u27e9\u27e9\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\nh : LinearIndependent k fun v => \u2191v\nf : { x // x \u2260 { val := p\u2081, property := hp\u2081 } } \u2192 \u2191((fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081})) :=\n fun x => { val := \u2191\u2191x -\u1d65 p\u2081, property := (_ : \u2203 a, a \u2208 s \\ {p\u2081} \u2227 (fun p => p -\u1d65 p\u2081) a = \u2191\u2191x -\u1d65 p\u2081) }\n\u22a2 LinearIndependent k fun i => \u2191\u2191i -\u1d65 \u2191{ val := p\u2081, property := hp\u2081 }\n[PROOFSTEP]\nconvert h.comp f fun x1 x2 hx => Subtype.ext (Subtype.ext (vsub_left_cancel (Subtype.ext_iff.1 hx)))\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\nhs : \u2200 (v : V), v \u2208 s \u2192 v \u2260 0\np\u2081 : P\n\u22a2 (LinearIndependent k fun v => \u2191v) \u2194 AffineIndependent k fun p => \u2191p\n[PROOFSTEP]\nrw [affineIndependent_set_iff_linearIndependent_vsub k (Set.mem_union_left _ (Set.mem_singleton p\u2081))]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\nhs : \u2200 (v : V), v \u2208 s \u2192 v \u2260 0\np\u2081 : P\n\u22a2 (LinearIndependent k fun v => \u2191v) \u2194 LinearIndependent k fun v => \u2191v\n[PROOFSTEP]\nhave h : (fun p => (p -\u1d65 p\u2081 : V)) '' (({ p\u2081 } \u222a (fun v => v +\u1d65 p\u2081) '' s) \\ { p\u2081 }) = s :=\n by\n simp_rw [Set.union_diff_left, Set.image_diff (vsub_left_injective p\u2081), Set.image_image, Set.image_singleton,\n vsub_self, vadd_vsub, Set.image_id']\n exact Set.diff_singleton_eq_self fun h => hs 0 h rfl\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\nhs : \u2200 (v : V), v \u2208 s \u2192 v \u2260 0\np\u2081 : P\n\u22a2 (fun p => p -\u1d65 p\u2081) '' (({p\u2081} \u222a (fun v => v +\u1d65 p\u2081) '' s) \\ {p\u2081}) = s\n[PROOFSTEP]\nsimp_rw [Set.union_diff_left, Set.image_diff (vsub_left_injective p\u2081), Set.image_image, Set.image_singleton, vsub_self,\n vadd_vsub, Set.image_id']\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\nhs : \u2200 (v : V), v \u2208 s \u2192 v \u2260 0\np\u2081 : P\n\u22a2 s \\ {0} = s\n[PROOFSTEP]\nexact Set.diff_singleton_eq_self fun h => hs 0 h rfl\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\nhs : \u2200 (v : V), v \u2208 s \u2192 v \u2260 0\np\u2081 : P\nh : (fun p => p -\u1d65 p\u2081) '' (({p\u2081} \u222a (fun v => v +\u1d65 p\u2081) '' s) \\ {p\u2081}) = s\n\u22a2 (LinearIndependent k fun v => \u2191v) \u2194 LinearIndependent k fun v => \u2191v\n[PROOFSTEP]\nrw [h]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\n\u22a2 AffineIndependent k p \u2194\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\n[PROOFSTEP]\nclassical\nconstructor\n\u00b7 intro ha s1 s2 w1 w2 hw1 hw2 heq\n ext i\n by_cases hi : i \u2208 s1 \u222a s2\n \u00b7 rw [\u2190 sub_eq_zero]\n rw [Set.sum_indicator_subset _ (Finset.subset_union_left s1 s2)] at hw1 \n rw [Set.sum_indicator_subset _ (Finset.subset_union_right s1 s2)] at hw2 \n have hws : (\u2211 i in s1 \u222a s2, (Set.indicator (\u2191s1) w1 - Set.indicator (\u2191s2) w2) i) = 0 := by simp [hw1, hw2]\n rw [Finset.affineCombination_indicator_subset _ _ (Finset.subset_union_left s1 s2),\n Finset.affineCombination_indicator_subset _ _ (Finset.subset_union_right s1 s2), \u2190 @vsub_eq_zero_iff_eq V,\n Finset.affineCombination_vsub] at heq \n exact ha (s1 \u222a s2) (Set.indicator (\u2191s1) w1 - Set.indicator (\u2191s2) w2) hws heq i hi\n \u00b7 rw [\u2190 Finset.mem_coe, Finset.coe_union] at hi \n have h\u2081 : Set.indicator (\u2191s1) w1 i = 0 :=\n by\n simp only [Set.indicator, Finset.mem_coe, ite_eq_right_iff]\n intro h\n by_contra\n exact (mt (@Set.mem_union_left _ i \u2191s1 \u2191s2) hi) h\n have h\u2082 : Set.indicator (\u2191s2) w2 i = 0 :=\n by\n simp only [Set.indicator, Finset.mem_coe, ite_eq_right_iff]\n intro h\n by_contra\n exact (mt (@Set.mem_union_right _ i \u2191s2 \u2191s1) hi) h\n simp [h\u2081, h\u2082]\n\u00b7 intro ha s w hw hs i0 hi0\n let w1 : \u03b9 \u2192 k := Function.update (Function.const \u03b9 0) i0 1\n have hw1 : \u2211 i in s, w1 i = 1 := by\n rw [Finset.sum_update_of_mem hi0]\n simp only [Finset.sum_const_zero, add_zero, const_apply]\n have hw1s : s.affineCombination k p w1 = p i0 :=\n s.affineCombination_of_eq_one_of_eq_zero w1 p hi0 (Function.update_same _ _ _) fun _ _ hne =>\n Function.update_noteq hne _ _\n let w2 := w + w1\n have hw2 : \u2211 i in s, w2 i = 1 := by simp_all only [Pi.add_apply, Finset.sum_add_distrib, zero_add]\n have hw2s : s.affineCombination k p w2 = p i0 := by\n simp_all only [\u2190 Finset.weightedVSub_vadd_affineCombination, zero_vadd]\n replace ha := ha s s w2 w1 hw2 hw1 (hw1s.symm \u25b8 hw2s)\n have hws : w2 i0 - w1 i0 = 0 := by\n rw [\u2190 Finset.mem_coe] at hi0 \n rw [\u2190 Set.indicator_of_mem hi0 w2, \u2190 Set.indicator_of_mem hi0 w1, ha, sub_self]\n simpa using hws\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\n\u22a2 AffineIndependent k p \u2194\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\n\u22a2 AffineIndependent k p \u2192\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\n[PROOFSTEP]\nintro ha s1 s2 w1 w2 hw1 hw2 heq\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\n\u22a2 Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\n[PROOFSTEP]\next i\n[GOAL]\ncase mp.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\n\u22a2 Set.indicator (\u2191s1) w1 i = Set.indicator (\u2191s2) w2 i\n[PROOFSTEP]\nby_cases hi : i \u2208 s1 \u222a s2\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : i \u2208 s1 \u222a s2\n\u22a2 Set.indicator (\u2191s1) w1 i = Set.indicator (\u2191s2) w2 i\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero]\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : i \u2208 s1 \u222a s2\n\u22a2 Set.indicator (\u2191s1) w1 i - Set.indicator (\u2191s2) w2 i = 0\n[PROOFSTEP]\nrw [Set.sum_indicator_subset _ (Finset.subset_union_left s1 s2)] at hw1 \n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1 \u222a s2, Set.indicator (\u2191s1) (fun i => w1 i) i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : i \u2208 s1 \u222a s2\n\u22a2 Set.indicator (\u2191s1) w1 i - Set.indicator (\u2191s2) w2 i = 0\n[PROOFSTEP]\nrw [Set.sum_indicator_subset _ (Finset.subset_union_right s1 s2)] at hw2 \n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1 \u222a s2, Set.indicator (\u2191s1) (fun i => w1 i) i = 1\nhw2 : \u2211 i in s1 \u222a s2, Set.indicator (\u2191s2) (fun i => w2 i) i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : i \u2208 s1 \u222a s2\n\u22a2 Set.indicator (\u2191s1) w1 i - Set.indicator (\u2191s2) w2 i = 0\n[PROOFSTEP]\nhave hws : (\u2211 i in s1 \u222a s2, (Set.indicator (\u2191s1) w1 - Set.indicator (\u2191s2) w2) i) = 0 := by simp [hw1, hw2]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1 \u222a s2, Set.indicator (\u2191s1) (fun i => w1 i) i = 1\nhw2 : \u2211 i in s1 \u222a s2, Set.indicator (\u2191s2) (fun i => w2 i) i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : i \u2208 s1 \u222a s2\n\u22a2 \u2211 i in s1 \u222a s2, (Set.indicator (\u2191s1) w1 - Set.indicator (\u2191s2) w2) i = 0\n[PROOFSTEP]\nsimp [hw1, hw2]\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1 \u222a s2, Set.indicator (\u2191s1) (fun i => w1 i) i = 1\nhw2 : \u2211 i in s1 \u222a s2, Set.indicator (\u2191s2) (fun i => w2 i) i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : i \u2208 s1 \u222a s2\nhws : \u2211 i in s1 \u222a s2, (Set.indicator (\u2191s1) w1 - Set.indicator (\u2191s2) w2) i = 0\n\u22a2 Set.indicator (\u2191s1) w1 i - Set.indicator (\u2191s2) w2 i = 0\n[PROOFSTEP]\nrw [Finset.affineCombination_indicator_subset _ _ (Finset.subset_union_left s1 s2),\n Finset.affineCombination_indicator_subset _ _ (Finset.subset_union_right s1 s2), \u2190 @vsub_eq_zero_iff_eq V,\n Finset.affineCombination_vsub] at heq \n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1 \u222a s2, Set.indicator (\u2191s1) (fun i => w1 i) i = 1\nhw2 : \u2211 i in s1 \u222a s2, Set.indicator (\u2191s2) (fun i => w2 i) i = 1\nheq : \u2191(Finset.weightedVSub (s1 \u222a s2) p) (Set.indicator (\u2191s1) w1 - Set.indicator (\u2191s2) w2) = 0\ni : \u03b9\nhi : i \u2208 s1 \u222a s2\nhws : \u2211 i in s1 \u222a s2, (Set.indicator (\u2191s1) w1 - Set.indicator (\u2191s2) w2) i = 0\n\u22a2 Set.indicator (\u2191s1) w1 i - Set.indicator (\u2191s2) w2 i = 0\n[PROOFSTEP]\nexact ha (s1 \u222a s2) (Set.indicator (\u2191s1) w1 - Set.indicator (\u2191s2) w2) hws heq i hi\n[GOAL]\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : \u00aci \u2208 s1 \u222a s2\n\u22a2 Set.indicator (\u2191s1) w1 i = Set.indicator (\u2191s2) w2 i\n[PROOFSTEP]\nrw [\u2190 Finset.mem_coe, Finset.coe_union] at hi \n[GOAL]\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : \u00aci \u2208 \u2191s1 \u222a \u2191s2\n\u22a2 Set.indicator (\u2191s1) w1 i = Set.indicator (\u2191s2) w2 i\n[PROOFSTEP]\nhave h\u2081 : Set.indicator (\u2191s1) w1 i = 0 :=\n by\n simp only [Set.indicator, Finset.mem_coe, ite_eq_right_iff]\n intro h\n by_contra\n exact (mt (@Set.mem_union_left _ i \u2191s1 \u2191s2) hi) h\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : \u00aci \u2208 \u2191s1 \u222a \u2191s2\n\u22a2 Set.indicator (\u2191s1) w1 i = 0\n[PROOFSTEP]\nsimp only [Set.indicator, Finset.mem_coe, ite_eq_right_iff]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : \u00aci \u2208 \u2191s1 \u222a \u2191s2\n\u22a2 i \u2208 s1 \u2192 w1 i = 0\n[PROOFSTEP]\nintro h\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : \u00aci \u2208 \u2191s1 \u222a \u2191s2\nh : i \u2208 s1\n\u22a2 w1 i = 0\n[PROOFSTEP]\nby_contra\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : \u00aci \u2208 \u2191s1 \u222a \u2191s2\nh : i \u2208 s1\nx\u271d : \u00acw1 i = 0\n\u22a2 False\n[PROOFSTEP]\nexact (mt (@Set.mem_union_left _ i \u2191s1 \u2191s2) hi) h\n[GOAL]\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : \u00aci \u2208 \u2191s1 \u222a \u2191s2\nh\u2081 : Set.indicator (\u2191s1) w1 i = 0\n\u22a2 Set.indicator (\u2191s1) w1 i = Set.indicator (\u2191s2) w2 i\n[PROOFSTEP]\nhave h\u2082 : Set.indicator (\u2191s2) w2 i = 0 :=\n by\n simp only [Set.indicator, Finset.mem_coe, ite_eq_right_iff]\n intro h\n by_contra\n exact (mt (@Set.mem_union_right _ i \u2191s2 \u2191s1) hi) h\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : \u00aci \u2208 \u2191s1 \u222a \u2191s2\nh\u2081 : Set.indicator (\u2191s1) w1 i = 0\n\u22a2 Set.indicator (\u2191s2) w2 i = 0\n[PROOFSTEP]\nsimp only [Set.indicator, Finset.mem_coe, ite_eq_right_iff]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : \u00aci \u2208 \u2191s1 \u222a \u2191s2\nh\u2081 : Set.indicator (\u2191s1) w1 i = 0\n\u22a2 i \u2208 s2 \u2192 w2 i = 0\n[PROOFSTEP]\nintro h\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : \u00aci \u2208 \u2191s1 \u222a \u2191s2\nh\u2081 : Set.indicator (\u2191s1) w1 i = 0\nh : i \u2208 s2\n\u22a2 w2 i = 0\n[PROOFSTEP]\nby_contra\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : \u00aci \u2208 \u2191s1 \u222a \u2191s2\nh\u2081 : Set.indicator (\u2191s1) w1 i = 0\nh : i \u2208 s2\nx\u271d : \u00acw2 i = 0\n\u22a2 False\n[PROOFSTEP]\nexact (mt (@Set.mem_union_right _ i \u2191s2 \u2191s1) hi) h\n[GOAL]\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nheq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\ni : \u03b9\nhi : \u00aci \u2208 \u2191s1 \u222a \u2191s2\nh\u2081 : Set.indicator (\u2191s1) w1 i = 0\nh\u2082 : Set.indicator (\u2191s2) w2 i = 0\n\u22a2 Set.indicator (\u2191s1) w1 i = Set.indicator (\u2191s2) w2 i\n[PROOFSTEP]\nsimp [h\u2081, h\u2082]\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\n\u22a2 (\u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2) \u2192\n AffineIndependent k p\n[PROOFSTEP]\nintro ha s w hw hs i0 hi0\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha :\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni0 : \u03b9\nhi0 : i0 \u2208 s\n\u22a2 w i0 = 0\n[PROOFSTEP]\nlet w1 : \u03b9 \u2192 k := Function.update (Function.const \u03b9 0) i0 1\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha :\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni0 : \u03b9\nhi0 : i0 \u2208 s\nw1 : \u03b9 \u2192 k := update (const \u03b9 0) i0 1\n\u22a2 w i0 = 0\n[PROOFSTEP]\nhave hw1 : \u2211 i in s, w1 i = 1 := by\n rw [Finset.sum_update_of_mem hi0]\n simp only [Finset.sum_const_zero, add_zero, const_apply]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha :\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni0 : \u03b9\nhi0 : i0 \u2208 s\nw1 : \u03b9 \u2192 k := update (const \u03b9 0) i0 1\n\u22a2 \u2211 i in s, w1 i = 1\n[PROOFSTEP]\nrw [Finset.sum_update_of_mem hi0]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha :\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni0 : \u03b9\nhi0 : i0 \u2208 s\nw1 : \u03b9 \u2192 k := update (const \u03b9 0) i0 1\n\u22a2 1 + \u2211 x in s \\ {i0}, const \u03b9 0 x = 1\n[PROOFSTEP]\nsimp only [Finset.sum_const_zero, add_zero, const_apply]\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha :\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni0 : \u03b9\nhi0 : i0 \u2208 s\nw1 : \u03b9 \u2192 k := update (const \u03b9 0) i0 1\nhw1 : \u2211 i in s, w1 i = 1\n\u22a2 w i0 = 0\n[PROOFSTEP]\nhave hw1s : s.affineCombination k p w1 = p i0 :=\n s.affineCombination_of_eq_one_of_eq_zero w1 p hi0 (Function.update_same _ _ _) fun _ _ hne =>\n Function.update_noteq hne _ _\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha :\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni0 : \u03b9\nhi0 : i0 \u2208 s\nw1 : \u03b9 \u2192 k := update (const \u03b9 0) i0 1\nhw1 : \u2211 i in s, w1 i = 1\nhw1s : \u2191(Finset.affineCombination k s p) w1 = p i0\n\u22a2 w i0 = 0\n[PROOFSTEP]\nlet w2 := w + w1\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha :\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni0 : \u03b9\nhi0 : i0 \u2208 s\nw1 : \u03b9 \u2192 k := update (const \u03b9 0) i0 1\nhw1 : \u2211 i in s, w1 i = 1\nhw1s : \u2191(Finset.affineCombination k s p) w1 = p i0\nw2 : \u03b9 \u2192 k := w + w1\n\u22a2 w i0 = 0\n[PROOFSTEP]\nhave hw2 : \u2211 i in s, w2 i = 1 := by simp_all only [Pi.add_apply, Finset.sum_add_distrib, zero_add]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha :\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni0 : \u03b9\nhi0 : i0 \u2208 s\nw1 : \u03b9 \u2192 k := update (const \u03b9 0) i0 1\nhw1 : \u2211 i in s, w1 i = 1\nhw1s : \u2191(Finset.affineCombination k s p) w1 = p i0\nw2 : \u03b9 \u2192 k := w + w1\n\u22a2 \u2211 i in s, w2 i = 1\n[PROOFSTEP]\nsimp_all only [Pi.add_apply, Finset.sum_add_distrib, zero_add]\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha :\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni0 : \u03b9\nhi0 : i0 \u2208 s\nw1 : \u03b9 \u2192 k := update (const \u03b9 0) i0 1\nhw1 : \u2211 i in s, w1 i = 1\nhw1s : \u2191(Finset.affineCombination k s p) w1 = p i0\nw2 : \u03b9 \u2192 k := w + w1\nhw2 : \u2211 i in s, w2 i = 1\n\u22a2 w i0 = 0\n[PROOFSTEP]\nhave hw2s : s.affineCombination k p w2 = p i0 := by\n simp_all only [\u2190 Finset.weightedVSub_vadd_affineCombination, zero_vadd]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha :\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni0 : \u03b9\nhi0 : i0 \u2208 s\nw1 : \u03b9 \u2192 k := update (const \u03b9 0) i0 1\nhw1 : \u2211 i in s, w1 i = 1\nhw1s : \u2191(Finset.affineCombination k s p) w1 = p i0\nw2 : \u03b9 \u2192 k := w + w1\nhw2 : \u2211 i in s, w2 i = 1\n\u22a2 \u2191(Finset.affineCombination k s p) w2 = p i0\n[PROOFSTEP]\nsimp_all only [\u2190 Finset.weightedVSub_vadd_affineCombination, zero_vadd]\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha :\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni0 : \u03b9\nhi0 : i0 \u2208 s\nw1 : \u03b9 \u2192 k := update (const \u03b9 0) i0 1\nhw1 : \u2211 i in s, w1 i = 1\nhw1s : \u2191(Finset.affineCombination k s p) w1 = p i0\nw2 : \u03b9 \u2192 k := w + w1\nhw2 : \u2211 i in s, w2 i = 1\nhw2s : \u2191(Finset.affineCombination k s p) w2 = p i0\n\u22a2 w i0 = 0\n[PROOFSTEP]\nreplace ha := ha s s w2 w1 hw2 hw1 (hw1s.symm \u25b8 hw2s)\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni0 : \u03b9\nhi0 : i0 \u2208 s\nw1 : \u03b9 \u2192 k := update (const \u03b9 0) i0 1\nhw1 : \u2211 i in s, w1 i = 1\nhw1s : \u2191(Finset.affineCombination k s p) w1 = p i0\nw2 : \u03b9 \u2192 k := w + w1\nhw2 : \u2211 i in s, w2 i = 1\nhw2s : \u2191(Finset.affineCombination k s p) w2 = p i0\nha : Set.indicator (\u2191s) w2 = Set.indicator (\u2191s) w1\n\u22a2 w i0 = 0\n[PROOFSTEP]\nhave hws : w2 i0 - w1 i0 = 0 := by\n rw [\u2190 Finset.mem_coe] at hi0 \n rw [\u2190 Set.indicator_of_mem hi0 w2, \u2190 Set.indicator_of_mem hi0 w1, ha, sub_self]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni0 : \u03b9\nhi0 : i0 \u2208 s\nw1 : \u03b9 \u2192 k := update (const \u03b9 0) i0 1\nhw1 : \u2211 i in s, w1 i = 1\nhw1s : \u2191(Finset.affineCombination k s p) w1 = p i0\nw2 : \u03b9 \u2192 k := w + w1\nhw2 : \u2211 i in s, w2 i = 1\nhw2s : \u2191(Finset.affineCombination k s p) w2 = p i0\nha : Set.indicator (\u2191s) w2 = Set.indicator (\u2191s) w1\n\u22a2 w2 i0 - w1 i0 = 0\n[PROOFSTEP]\nrw [\u2190 Finset.mem_coe] at hi0 \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni0 : \u03b9\nhi0 : i0 \u2208 \u2191s\nw1 : \u03b9 \u2192 k := update (const \u03b9 0) i0 1\nhw1 : \u2211 i in s, w1 i = 1\nhw1s : \u2191(Finset.affineCombination k s p) w1 = p i0\nw2 : \u03b9 \u2192 k := w + w1\nhw2 : \u2211 i in s, w2 i = 1\nhw2s : \u2191(Finset.affineCombination k s p) w2 = p i0\nha : Set.indicator (\u2191s) w2 = Set.indicator (\u2191s) w1\n\u22a2 w2 i0 - w1 i0 = 0\n[PROOFSTEP]\nrw [\u2190 Set.indicator_of_mem hi0 w2, \u2190 Set.indicator_of_mem hi0 w1, ha, sub_self]\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ni0 : \u03b9\nhi0 : i0 \u2208 s\nw1 : \u03b9 \u2192 k := update (const \u03b9 0) i0 1\nhw1 : \u2211 i in s, w1 i = 1\nhw1s : \u2191(Finset.affineCombination k s p) w1 = p i0\nw2 : \u03b9 \u2192 k := w + w1\nhw2 : \u2211 i in s, w2 i = 1\nhw2s : \u2191(Finset.affineCombination k s p) w2 = p i0\nha : Set.indicator (\u2191s) w2 = Set.indicator (\u2191s) w1\nhws : w2 i0 - w1 i0 = 0\n\u22a2 w i0 = 0\n[PROOFSTEP]\nsimpa using hws\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\n\u22a2 AffineIndependent k p \u2194\n \u2200 (w1 w2 : \u03b9 \u2192 k),\n \u2211 i : \u03b9, w1 i = 1 \u2192\n \u2211 i : \u03b9, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k Finset.univ p) w1 = \u2191(Finset.affineCombination k Finset.univ p) w2 \u2192 w1 = w2\n[PROOFSTEP]\nrw [affineIndependent_iff_indicator_eq_of_affineCombination_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\n\u22a2 (\u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2) \u2194\n \u2200 (w1 w2 : \u03b9 \u2192 k),\n \u2211 i : \u03b9, w1 i = 1 \u2192\n \u2211 i : \u03b9, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k Finset.univ p) w1 = \u2191(Finset.affineCombination k Finset.univ p) w2 \u2192 w1 = w2\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\n\u22a2 (\u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2) \u2192\n \u2200 (w1 w2 : \u03b9 \u2192 k),\n \u2211 i : \u03b9, w1 i = 1 \u2192\n \u2211 i : \u03b9, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k Finset.univ p) w1 = \u2191(Finset.affineCombination k Finset.univ p) w2 \u2192 w1 = w2\n[PROOFSTEP]\nintro h w1 w2 hw1 hw2 hweq\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nh :\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i : \u03b9, w1 i = 1\nhw2 : \u2211 i : \u03b9, w2 i = 1\nhweq : \u2191(Finset.affineCombination k Finset.univ p) w1 = \u2191(Finset.affineCombination k Finset.univ p) w2\n\u22a2 w1 = w2\n[PROOFSTEP]\nsimpa only [Set.indicator_univ, Finset.coe_univ] using h _ _ w1 w2 hw1 hw2 hweq\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\n\u22a2 (\u2200 (w1 w2 : \u03b9 \u2192 k),\n \u2211 i : \u03b9, w1 i = 1 \u2192\n \u2211 i : \u03b9, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k Finset.univ p) w1 = \u2191(Finset.affineCombination k Finset.univ p) w2 \u2192 w1 = w2) \u2192\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\n[PROOFSTEP]\nintro h s1 s2 w1 w2 hw1 hw2 hweq\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nh :\n \u2200 (w1 w2 : \u03b9 \u2192 k),\n \u2211 i : \u03b9, w1 i = 1 \u2192\n \u2211 i : \u03b9, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k Finset.univ p) w1 = \u2191(Finset.affineCombination k Finset.univ p) w2 \u2192 w1 = w2\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nhweq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\n\u22a2 Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\n[PROOFSTEP]\nhave hw1' : (\u2211 i, (s1 : Set \u03b9).indicator w1 i) = 1 := by\n rwa [Set.sum_indicator_subset _ (Finset.subset_univ s1)] at hw1 \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nh :\n \u2200 (w1 w2 : \u03b9 \u2192 k),\n \u2211 i : \u03b9, w1 i = 1 \u2192\n \u2211 i : \u03b9, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k Finset.univ p) w1 = \u2191(Finset.affineCombination k Finset.univ p) w2 \u2192 w1 = w2\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nhweq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\n\u22a2 \u2211 i : \u03b9, Set.indicator (\u2191s1) w1 i = 1\n[PROOFSTEP]\nrwa [Set.sum_indicator_subset _ (Finset.subset_univ s1)] at hw1 \n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nh :\n \u2200 (w1 w2 : \u03b9 \u2192 k),\n \u2211 i : \u03b9, w1 i = 1 \u2192\n \u2211 i : \u03b9, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k Finset.univ p) w1 = \u2191(Finset.affineCombination k Finset.univ p) w2 \u2192 w1 = w2\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nhweq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\nhw1' : \u2211 i : \u03b9, Set.indicator (\u2191s1) w1 i = 1\n\u22a2 Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\n[PROOFSTEP]\nhave hw2' : (\u2211 i, (s2 : Set \u03b9).indicator w2 i) = 1 := by\n rwa [Set.sum_indicator_subset _ (Finset.subset_univ s2)] at hw2 \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nh :\n \u2200 (w1 w2 : \u03b9 \u2192 k),\n \u2211 i : \u03b9, w1 i = 1 \u2192\n \u2211 i : \u03b9, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k Finset.univ p) w1 = \u2191(Finset.affineCombination k Finset.univ p) w2 \u2192 w1 = w2\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nhweq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\nhw1' : \u2211 i : \u03b9, Set.indicator (\u2191s1) w1 i = 1\n\u22a2 \u2211 i : \u03b9, Set.indicator (\u2191s2) w2 i = 1\n[PROOFSTEP]\nrwa [Set.sum_indicator_subset _ (Finset.subset_univ s2)] at hw2 \n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nh :\n \u2200 (w1 w2 : \u03b9 \u2192 k),\n \u2211 i : \u03b9, w1 i = 1 \u2192\n \u2211 i : \u03b9, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k Finset.univ p) w1 = \u2191(Finset.affineCombination k Finset.univ p) w2 \u2192 w1 = w2\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nhweq : \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2\nhw1' : \u2211 i : \u03b9, Set.indicator (\u2191s1) w1 i = 1\nhw2' : \u2211 i : \u03b9, Set.indicator (\u2191s2) w2 i = 1\n\u22a2 Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\n[PROOFSTEP]\nrw [Finset.affineCombination_indicator_subset w1 p (Finset.subset_univ s1),\n Finset.affineCombination_indicator_subset w2 p (Finset.subset_univ s2)] at hweq \n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nh :\n \u2200 (w1 w2 : \u03b9 \u2192 k),\n \u2211 i : \u03b9, w1 i = 1 \u2192\n \u2211 i : \u03b9, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k Finset.univ p) w1 = \u2191(Finset.affineCombination k Finset.univ p) w2 \u2192 w1 = w2\ns1 s2 : Finset \u03b9\nw1 w2 : \u03b9 \u2192 k\nhw1 : \u2211 i in s1, w1 i = 1\nhw2 : \u2211 i in s2, w2 i = 1\nhweq :\n \u2191(Finset.affineCombination k Finset.univ p) (Set.indicator (\u2191s1) w1) =\n \u2191(Finset.affineCombination k Finset.univ p) (Set.indicator (\u2191s2) w2)\nhw1' : \u2211 i : \u03b9, Set.indicator (\u2191s1) w1 i = 1\nhw2' : \u2211 i : \u03b9, Set.indicator (\u2191s2) w2 i = 1\n\u22a2 Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\n[PROOFSTEP]\nexact h _ _ hw1' hw2' hweq\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nhp : AffineIndependent k p\nj : \u03b9\nw : \u03b9 \u2192 k\u02e3\n\u22a2 AffineIndependent k fun i => \u2191(AffineMap.lineMap (p j) (p i)) \u2191(w i)\n[PROOFSTEP]\nrw [affineIndependent_iff_linearIndependent_vsub k _ j] at hp \u22a2\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nj : \u03b9\nhp : LinearIndependent k fun i => p \u2191i -\u1d65 p j\nw : \u03b9 \u2192 k\u02e3\n\u22a2 LinearIndependent k fun i => \u2191(AffineMap.lineMap (p j) (p \u2191i)) \u2191(w \u2191i) -\u1d65 \u2191(AffineMap.lineMap (p j) (p j)) \u2191(w j)\n[PROOFSTEP]\nsimp only [AffineMap.lineMap_vsub_left, AffineMap.coe_const, AffineMap.lineMap_same, const_apply]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nj : \u03b9\nhp : LinearIndependent k fun i => p \u2191i -\u1d65 p j\nw : \u03b9 \u2192 k\u02e3\n\u22a2 LinearIndependent k fun i => \u2191(w \u2191i) \u2022 (p \u2191i -\u1d65 p j)\n[PROOFSTEP]\nexact hp.units_smul fun i => w i\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\n\u22a2 Injective p\n[PROOFSTEP]\nintro i j hij\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ni j : \u03b9\nhij : p i = p j\n\u22a2 i = j\n[PROOFSTEP]\nrw [affineIndependent_iff_linearIndependent_vsub _ _ j] at ha \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\ni j : \u03b9\nha : LinearIndependent k fun i => p \u2191i -\u1d65 p j\nhij : p i = p j\n\u22a2 i = j\n[PROOFSTEP]\nby_contra hij'\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\ni j : \u03b9\nha : LinearIndependent k fun i => p \u2191i -\u1d65 p j\nhij : p i = p j\nhij' : \u00aci = j\n\u22a2 False\n[PROOFSTEP]\nrefine' ha.ne_zero \u27e8i, hij'\u27e9 (vsub_eq_zero_iff_eq.mpr _)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\ni j : \u03b9\nha : LinearIndependent k fun i => p \u2191i -\u1d65 p j\nhij : p i = p j\nhij' : \u00aci = j\n\u22a2 p \u2191{ val := i, property := hij' } = p j\n[PROOFSTEP]\nsimp_all only [ne_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\n\u22a2 AffineIndependent k (p \u2218 \u2191f)\n[PROOFSTEP]\nclassical\nintro fs w hw hs i0 hi0\nlet fs' := fs.map f\nlet w' i := if h : \u2203 i2, f i2 = i then w h.choose else 0\nhave hw' : \u2200 i2 : \u03b92, w' (f i2) = w i2 := by\n intro i2\n have h : \u2203 i : \u03b92, f i = f i2 := \u27e8i2, rfl\u27e9\n have hs : h.choose = i2 := f.injective h.choose_spec\n simp_rw [dif_pos h, hs]\nhave hw's : \u2211 i in fs', w' i = 0 := by\n rw [\u2190 hw, Finset.sum_map]\n simp [hw']\nhave hs' : fs'.weightedVSub p w' = (0 : V) :=\n by\n rw [\u2190 hs, Finset.weightedVSub_map]\n congr with i\n simp_all only [comp_apply, EmbeddingLike.apply_eq_iff_eq, exists_eq, dite_true]\nrw [\u2190 ha fs' w' hw's hs' (f i0) ((Finset.mem_map' _).2 hi0), hw']\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\n\u22a2 AffineIndependent k (p \u2218 \u2191f)\n[PROOFSTEP]\nintro fs w hw hs i0 hi0\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nfs : Finset \u03b92\nw : \u03b92 \u2192 k\nhw : \u2211 i in fs, w i = 0\nhs : \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w = 0\ni0 : \u03b92\nhi0 : i0 \u2208 fs\n\u22a2 w i0 = 0\n[PROOFSTEP]\nlet fs' := fs.map f\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nfs : Finset \u03b92\nw : \u03b92 \u2192 k\nhw : \u2211 i in fs, w i = 0\nhs : \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w = 0\ni0 : \u03b92\nhi0 : i0 \u2208 fs\nfs' : Finset \u03b9 := Finset.map f fs\n\u22a2 w i0 = 0\n[PROOFSTEP]\nlet w' i := if h : \u2203 i2, f i2 = i then w h.choose else 0\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nfs : Finset \u03b92\nw : \u03b92 \u2192 k\nhw : \u2211 i in fs, w i = 0\nhs : \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w = 0\ni0 : \u03b92\nhi0 : i0 \u2208 fs\nfs' : Finset \u03b9 := Finset.map f fs\nw' : \u03b9 \u2192 k := fun i => if h : \u2203 i2, \u2191f i2 = i then w (Exists.choose h) else 0\n\u22a2 w i0 = 0\n[PROOFSTEP]\nhave hw' : \u2200 i2 : \u03b92, w' (f i2) = w i2 := by\n intro i2\n have h : \u2203 i : \u03b92, f i = f i2 := \u27e8i2, rfl\u27e9\n have hs : h.choose = i2 := f.injective h.choose_spec\n simp_rw [dif_pos h, hs]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nfs : Finset \u03b92\nw : \u03b92 \u2192 k\nhw : \u2211 i in fs, w i = 0\nhs : \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w = 0\ni0 : \u03b92\nhi0 : i0 \u2208 fs\nfs' : Finset \u03b9 := Finset.map f fs\nw' : \u03b9 \u2192 k := fun i => if h : \u2203 i2, \u2191f i2 = i then w (Exists.choose h) else 0\n\u22a2 \u2200 (i2 : \u03b92), w' (\u2191f i2) = w i2\n[PROOFSTEP]\nintro i2\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nfs : Finset \u03b92\nw : \u03b92 \u2192 k\nhw : \u2211 i in fs, w i = 0\nhs : \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w = 0\ni0 : \u03b92\nhi0 : i0 \u2208 fs\nfs' : Finset \u03b9 := Finset.map f fs\nw' : \u03b9 \u2192 k := fun i => if h : \u2203 i2, \u2191f i2 = i then w (Exists.choose h) else 0\ni2 : \u03b92\n\u22a2 w' (\u2191f i2) = w i2\n[PROOFSTEP]\nhave h : \u2203 i : \u03b92, f i = f i2 := \u27e8i2, rfl\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nfs : Finset \u03b92\nw : \u03b92 \u2192 k\nhw : \u2211 i in fs, w i = 0\nhs : \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w = 0\ni0 : \u03b92\nhi0 : i0 \u2208 fs\nfs' : Finset \u03b9 := Finset.map f fs\nw' : \u03b9 \u2192 k := fun i => if h : \u2203 i2, \u2191f i2 = i then w (Exists.choose h) else 0\ni2 : \u03b92\nh : \u2203 i, \u2191f i = \u2191f i2\n\u22a2 w' (\u2191f i2) = w i2\n[PROOFSTEP]\nhave hs : h.choose = i2 := f.injective h.choose_spec\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nfs : Finset \u03b92\nw : \u03b92 \u2192 k\nhw : \u2211 i in fs, w i = 0\nhs\u271d : \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w = 0\ni0 : \u03b92\nhi0 : i0 \u2208 fs\nfs' : Finset \u03b9 := Finset.map f fs\nw' : \u03b9 \u2192 k := fun i => if h : \u2203 i2, \u2191f i2 = i then w (Exists.choose h) else 0\ni2 : \u03b92\nh : \u2203 i, \u2191f i = \u2191f i2\nhs : Exists.choose h = i2\n\u22a2 w' (\u2191f i2) = w i2\n[PROOFSTEP]\nsimp_rw [dif_pos h, hs]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nfs : Finset \u03b92\nw : \u03b92 \u2192 k\nhw : \u2211 i in fs, w i = 0\nhs : \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w = 0\ni0 : \u03b92\nhi0 : i0 \u2208 fs\nfs' : Finset \u03b9 := Finset.map f fs\nw' : \u03b9 \u2192 k := fun i => if h : \u2203 i2, \u2191f i2 = i then w (Exists.choose h) else 0\nhw' : \u2200 (i2 : \u03b92), w' (\u2191f i2) = w i2\n\u22a2 w i0 = 0\n[PROOFSTEP]\nhave hw's : \u2211 i in fs', w' i = 0 := by\n rw [\u2190 hw, Finset.sum_map]\n simp [hw']\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nfs : Finset \u03b92\nw : \u03b92 \u2192 k\nhw : \u2211 i in fs, w i = 0\nhs : \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w = 0\ni0 : \u03b92\nhi0 : i0 \u2208 fs\nfs' : Finset \u03b9 := Finset.map f fs\nw' : \u03b9 \u2192 k := fun i => if h : \u2203 i2, \u2191f i2 = i then w (Exists.choose h) else 0\nhw' : \u2200 (i2 : \u03b92), w' (\u2191f i2) = w i2\n\u22a2 \u2211 i in fs', w' i = 0\n[PROOFSTEP]\nrw [\u2190 hw, Finset.sum_map]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nfs : Finset \u03b92\nw : \u03b92 \u2192 k\nhw : \u2211 i in fs, w i = 0\nhs : \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w = 0\ni0 : \u03b92\nhi0 : i0 \u2208 fs\nfs' : Finset \u03b9 := Finset.map f fs\nw' : \u03b9 \u2192 k := fun i => if h : \u2203 i2, \u2191f i2 = i then w (Exists.choose h) else 0\nhw' : \u2200 (i2 : \u03b92), w' (\u2191f i2) = w i2\n\u22a2 \u2211 x in fs, w' (\u2191f x) = \u2211 i in fs, w i\n[PROOFSTEP]\nsimp [hw']\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nfs : Finset \u03b92\nw : \u03b92 \u2192 k\nhw : \u2211 i in fs, w i = 0\nhs : \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w = 0\ni0 : \u03b92\nhi0 : i0 \u2208 fs\nfs' : Finset \u03b9 := Finset.map f fs\nw' : \u03b9 \u2192 k := fun i => if h : \u2203 i2, \u2191f i2 = i then w (Exists.choose h) else 0\nhw' : \u2200 (i2 : \u03b92), w' (\u2191f i2) = w i2\nhw's : \u2211 i in fs', w' i = 0\n\u22a2 w i0 = 0\n[PROOFSTEP]\nhave hs' : fs'.weightedVSub p w' = (0 : V) :=\n by\n rw [\u2190 hs, Finset.weightedVSub_map]\n congr with i\n simp_all only [comp_apply, EmbeddingLike.apply_eq_iff_eq, exists_eq, dite_true]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nfs : Finset \u03b92\nw : \u03b92 \u2192 k\nhw : \u2211 i in fs, w i = 0\nhs : \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w = 0\ni0 : \u03b92\nhi0 : i0 \u2208 fs\nfs' : Finset \u03b9 := Finset.map f fs\nw' : \u03b9 \u2192 k := fun i => if h : \u2203 i2, \u2191f i2 = i then w (Exists.choose h) else 0\nhw' : \u2200 (i2 : \u03b92), w' (\u2191f i2) = w i2\nhw's : \u2211 i in fs', w' i = 0\n\u22a2 \u2191(Finset.weightedVSub fs' p) w' = 0\n[PROOFSTEP]\nrw [\u2190 hs, Finset.weightedVSub_map]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nfs : Finset \u03b92\nw : \u03b92 \u2192 k\nhw : \u2211 i in fs, w i = 0\nhs : \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w = 0\ni0 : \u03b92\nhi0 : i0 \u2208 fs\nfs' : Finset \u03b9 := Finset.map f fs\nw' : \u03b9 \u2192 k := fun i => if h : \u2203 i2, \u2191f i2 = i then w (Exists.choose h) else 0\nhw' : \u2200 (i2 : \u03b92), w' (\u2191f i2) = w i2\nhw's : \u2211 i in fs', w' i = 0\n\u22a2 \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) (w' \u2218 \u2191f) = \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w\n[PROOFSTEP]\ncongr with i\n[GOAL]\ncase h.e_6.h.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nfs : Finset \u03b92\nw : \u03b92 \u2192 k\nhw : \u2211 i in fs, w i = 0\nhs : \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w = 0\ni0 : \u03b92\nhi0 : i0 \u2208 fs\nfs' : Finset \u03b9 := Finset.map f fs\nw' : \u03b9 \u2192 k := fun i => if h : \u2203 i2, \u2191f i2 = i then w (Exists.choose h) else 0\nhw' : \u2200 (i2 : \u03b92), w' (\u2191f i2) = w i2\nhw's : \u2211 i in fs', w' i = 0\ni : \u03b92\n\u22a2 (w' \u2218 \u2191f) i = w i\n[PROOFSTEP]\nsimp_all only [comp_apply, EmbeddingLike.apply_eq_iff_eq, exists_eq, dite_true]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b92 : Type u_5\nf : \u03b92 \u21aa \u03b9\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nfs : Finset \u03b92\nw : \u03b92 \u2192 k\nhw : \u2211 i in fs, w i = 0\nhs : \u2191(Finset.weightedVSub fs (p \u2218 \u2191f)) w = 0\ni0 : \u03b92\nhi0 : i0 \u2208 fs\nfs' : Finset \u03b9 := Finset.map f fs\nw' : \u03b9 \u2192 k := fun i => if h : \u2203 i2, \u2191f i2 = i then w (Exists.choose h) else 0\nhw' : \u2200 (i2 : \u03b92), w' (\u2191f i2) = w i2\nhw's : \u2211 i in fs', w' i = 0\nhs' : \u2191(Finset.weightedVSub fs' p) w' = 0\n\u22a2 w i0 = 0\n[PROOFSTEP]\nrw [\u2190 ha fs' w' hw's hs' (f i0) ((Finset.mem_map' _).2 hi0), hw']\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\n\u22a2 AffineIndependent k fun x => \u2191x\n[PROOFSTEP]\nlet f : Set.range p \u2192 \u03b9 := fun x => x.property.choose\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nf : \u2191(Set.range p) \u2192 \u03b9 := fun x => Exists.choose (_ : \u2191x \u2208 Set.range p)\n\u22a2 AffineIndependent k fun x => \u2191x\n[PROOFSTEP]\nhave hf : \u2200 x, p (f x) = x := fun x => x.property.choose_spec\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nf : \u2191(Set.range p) \u2192 \u03b9 := fun x => Exists.choose (_ : \u2191x \u2208 Set.range p)\nhf : \u2200 (x : \u2191(Set.range p)), p (f x) = \u2191x\n\u22a2 AffineIndependent k fun x => \u2191x\n[PROOFSTEP]\nlet fe : Set.range p \u21aa \u03b9 := \u27e8f, fun x\u2081 x\u2082 he => Subtype.ext (hf x\u2081 \u25b8 hf x\u2082 \u25b8 he \u25b8 rfl)\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nf : \u2191(Set.range p) \u2192 \u03b9 := fun x => Exists.choose (_ : \u2191x \u2208 Set.range p)\nhf : \u2200 (x : \u2191(Set.range p)), p (f x) = \u2191x\nfe : \u2191(Set.range p) \u21aa \u03b9 := { toFun := f, inj' := (_ : \u2200 (x\u2081 x\u2082 : \u2191(Set.range p)), f x\u2081 = f x\u2082 \u2192 x\u2081 = x\u2082) }\n\u22a2 AffineIndependent k fun x => \u2191x\n[PROOFSTEP]\nconvert ha.comp_embedding fe\n[GOAL]\ncase h.e'_9.h.h.e\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nf : \u2191(Set.range p) \u2192 \u03b9 := fun x => Exists.choose (_ : \u2191x \u2208 Set.range p)\nhf : \u2200 (x : \u2191(Set.range p)), p (f x) = \u2191x\nfe : \u2191(Set.range p) \u21aa \u03b9 := { toFun := f, inj' := (_ : \u2200 (x\u2081 x\u2082 : \u2191(Set.range p)), f x\u2081 = f x\u2082 \u2192 x\u2081 = x\u2082) }\nx\u271d : \u2191(Set.range p)\n\u22a2 Subtype.val = p \u2218 \u2191fe\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_9.h.h.e.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nha : AffineIndependent k p\nf : \u2191(Set.range p) \u2192 \u03b9 := fun x => Exists.choose (_ : \u2191x \u2208 Set.range p)\nhf : \u2200 (x : \u2191(Set.range p)), p (f x) = \u2191x\nfe : \u2191(Set.range p) \u21aa \u03b9 := { toFun := f, inj' := (_ : \u2200 (x\u2081 x\u2082 : \u2191(Set.range p)), f x\u2081 = f x\u2082 \u2192 x\u2081 = x\u2082) }\nx\u271d\u00b9 : \u2191(Set.range p)\nx\u271d : { x // x \u2208 Set.range p }\n\u22a2 \u2191x\u271d = (p \u2218 \u2191fe) x\u271d\n[PROOFSTEP]\nsimp [hf]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b9' : Type u_5\ne : \u03b9 \u2243 \u03b9'\np : \u03b9' \u2192 P\n\u22a2 AffineIndependent k (p \u2218 \u2191e) \u2194 AffineIndependent k p\n[PROOFSTEP]\nrefine' \u27e8_, AffineIndependent.comp_embedding e.toEmbedding\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b9' : Type u_5\ne : \u03b9 \u2243 \u03b9'\np : \u03b9' \u2192 P\n\u22a2 AffineIndependent k (p \u2218 \u2191e) \u2192 AffineIndependent k p\n[PROOFSTEP]\nintro h\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b9' : Type u_5\ne : \u03b9 \u2243 \u03b9'\np : \u03b9' \u2192 P\nh : AffineIndependent k (p \u2218 \u2191e)\n\u22a2 AffineIndependent k p\n[PROOFSTEP]\nhave : p = p \u2218 e \u2218 e.symm.toEmbedding := by\n ext\n simp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b9' : Type u_5\ne : \u03b9 \u2243 \u03b9'\np : \u03b9' \u2192 P\nh : AffineIndependent k (p \u2218 \u2191e)\n\u22a2 p = p \u2218 \u2191e \u2218 \u2191(Equiv.toEmbedding e.symm)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b9' : Type u_5\ne : \u03b9 \u2243 \u03b9'\np : \u03b9' \u2192 P\nh : AffineIndependent k (p \u2218 \u2191e)\nx\u271d : \u03b9'\n\u22a2 p x\u271d = (p \u2218 \u2191e \u2218 \u2191(Equiv.toEmbedding e.symm)) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b9' : Type u_5\ne : \u03b9 \u2243 \u03b9'\np : \u03b9' \u2192 P\nh : AffineIndependent k (p \u2218 \u2191e)\nthis : p = p \u2218 \u2191e \u2218 \u2191(Equiv.toEmbedding e.symm)\n\u22a2 AffineIndependent k p\n[PROOFSTEP]\nrw [this]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u03b9' : Type u_5\ne : \u03b9 \u2243 \u03b9'\np : \u03b9' \u2192 P\nh : AffineIndependent k (p \u2218 \u2191e)\nthis : p = p \u2218 \u2191e \u2218 \u2191(Equiv.toEmbedding e.symm)\n\u22a2 AffineIndependent k (p \u2218 \u2191e \u2218 \u2191(Equiv.toEmbedding e.symm))\n[PROOFSTEP]\nexact h.comp_embedding e.symm.toEmbedding\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nf : P \u2192\u1d43[k] P\u2082\nhai : AffineIndependent k (\u2191f \u2218 p)\n\u22a2 AffineIndependent k p\n[PROOFSTEP]\ncases' isEmpty_or_nonempty \u03b9 with h h\n[GOAL]\ncase inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nf : P \u2192\u1d43[k] P\u2082\nhai : AffineIndependent k (\u2191f \u2218 p)\nh : IsEmpty \u03b9\n\u22a2 AffineIndependent k p\n[PROOFSTEP]\nhaveI := h\n[GOAL]\ncase inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nf : P \u2192\u1d43[k] P\u2082\nhai : AffineIndependent k (\u2191f \u2218 p)\nh this : IsEmpty \u03b9\n\u22a2 AffineIndependent k p\n[PROOFSTEP]\napply affineIndependent_of_subsingleton\n[GOAL]\ncase inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nf : P \u2192\u1d43[k] P\u2082\nhai : AffineIndependent k (\u2191f \u2218 p)\nh : Nonempty \u03b9\n\u22a2 AffineIndependent k p\n[PROOFSTEP]\nobtain \u27e8i\u27e9 := h\n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nf : P \u2192\u1d43[k] P\u2082\nhai : AffineIndependent k (\u2191f \u2218 p)\ni : \u03b9\n\u22a2 AffineIndependent k p\n[PROOFSTEP]\nrw [affineIndependent_iff_linearIndependent_vsub k p i]\n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nf : P \u2192\u1d43[k] P\u2082\nhai : AffineIndependent k (\u2191f \u2218 p)\ni : \u03b9\n\u22a2 LinearIndependent k fun i_1 => p \u2191i_1 -\u1d65 p i\n[PROOFSTEP]\nsimp_rw [affineIndependent_iff_linearIndependent_vsub k (f \u2218 p) i, Function.comp_apply, \u2190 f.linearMap_vsub] at hai \n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nf : P \u2192\u1d43[k] P\u2082\ni : \u03b9\nhai : LinearIndependent k fun i_1 => \u2191f.linear (p \u2191i_1 -\u1d65 p i)\n\u22a2 LinearIndependent k fun i_1 => p \u2191i_1 -\u1d65 p i\n[PROOFSTEP]\nexact LinearIndependent.of_comp f.linear hai\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nhai : AffineIndependent k p\nf : P \u2192\u1d43[k] P\u2082\nhf : Injective \u2191f\n\u22a2 AffineIndependent k (\u2191f \u2218 p)\n[PROOFSTEP]\ncases' isEmpty_or_nonempty \u03b9 with h h\n[GOAL]\ncase inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nhai : AffineIndependent k p\nf : P \u2192\u1d43[k] P\u2082\nhf : Injective \u2191f\nh : IsEmpty \u03b9\n\u22a2 AffineIndependent k (\u2191f \u2218 p)\n[PROOFSTEP]\nhaveI := h\n[GOAL]\ncase inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nhai : AffineIndependent k p\nf : P \u2192\u1d43[k] P\u2082\nhf : Injective \u2191f\nh this : IsEmpty \u03b9\n\u22a2 AffineIndependent k (\u2191f \u2218 p)\n[PROOFSTEP]\napply affineIndependent_of_subsingleton\n[GOAL]\ncase inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nhai : AffineIndependent k p\nf : P \u2192\u1d43[k] P\u2082\nhf : Injective \u2191f\nh : Nonempty \u03b9\n\u22a2 AffineIndependent k (\u2191f \u2218 p)\n[PROOFSTEP]\nobtain \u27e8i\u27e9 := h\n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nhai : AffineIndependent k p\nf : P \u2192\u1d43[k] P\u2082\nhf : Injective \u2191f\ni : \u03b9\n\u22a2 AffineIndependent k (\u2191f \u2218 p)\n[PROOFSTEP]\nrw [affineIndependent_iff_linearIndependent_vsub k p i] at hai \n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nf : P \u2192\u1d43[k] P\u2082\nhf : Injective \u2191f\ni : \u03b9\nhai : LinearIndependent k fun i_1 => p \u2191i_1 -\u1d65 p i\n\u22a2 AffineIndependent k (\u2191f \u2218 p)\n[PROOFSTEP]\nsimp_rw [affineIndependent_iff_linearIndependent_vsub k (f \u2218 p) i, Function.comp_apply, \u2190 f.linearMap_vsub]\n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nf : P \u2192\u1d43[k] P\u2082\nhf : Injective \u2191f\ni : \u03b9\nhai : LinearIndependent k fun i_1 => p \u2191i_1 -\u1d65 p i\n\u22a2 LinearIndependent k fun i_1 => \u2191f.linear (p \u2191i_1 -\u1d65 p i)\n[PROOFSTEP]\nhave hf' : LinearMap.ker f.linear = \u22a5 := by rwa [LinearMap.ker_eq_bot, f.linear_injective_iff]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nf : P \u2192\u1d43[k] P\u2082\nhf : Injective \u2191f\ni : \u03b9\nhai : LinearIndependent k fun i_1 => p \u2191i_1 -\u1d65 p i\n\u22a2 LinearMap.ker f.linear = \u22a5\n[PROOFSTEP]\nrwa [LinearMap.ker_eq_bot, f.linear_injective_iff]\n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\np : \u03b9 \u2192 P\nf : P \u2192\u1d43[k] P\u2082\nhf : Injective \u2191f\ni : \u03b9\nhai : LinearIndependent k fun i_1 => p \u2191i_1 -\u1d65 p i\nhf' : LinearMap.ker f.linear = \u22a5\n\u22a2 LinearIndependent k fun i_1 => \u2191f.linear (p \u2191i_1 -\u1d65 p i)\n[PROOFSTEP]\nexact LinearIndependent.map' hai f.linear hf'\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\ns : Set P\ne : P \u2243\u1d43[k] P\u2082\n\u22a2 AffineIndependent k Subtype.val \u2194 AffineIndependent k Subtype.val\n[PROOFSTEP]\nhave : e \u2218 ((\u2191) : s \u2192 P) = ((\u2191) : e '' s \u2192 P\u2082) \u2218 (e : P \u2243 P\u2082).image s := rfl\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module k V\ninst\u271d\u00b3 : AffineSpace V P\n\u03b9 : Type u_4\nV\u2082 : Type u_5\nP\u2082 : Type u_6\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module k V\u2082\ninst\u271d : AffineSpace V\u2082 P\u2082\ns : Set P\ne : P \u2243\u1d43[k] P\u2082\nthis : \u2191e \u2218 Subtype.val = Subtype.val \u2218 \u2191(Equiv.image (\u2191e) s)\n\u22a2 AffineIndependent k Subtype.val \u2194 AffineIndependent k Subtype.val\n[PROOFSTEP]\nrw [\u2190 e.affineIndependent_iff, this, affineIndependent_equiv]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Set \u03b9\np0 : P\nhp0s1 : p0 \u2208 affineSpan k (p '' s1)\nhp0s2 : p0 \u2208 affineSpan k (p '' s2)\n\u22a2 \u2203 i, i \u2208 s1 \u2229 s2\n[PROOFSTEP]\nrw [Set.image_eq_range] at hp0s1 hp0s2 \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Set \u03b9\np0 : P\nhp0s1 : p0 \u2208 affineSpan k (Set.range fun x => p \u2191x)\nhp0s2 : p0 \u2208 affineSpan k (Set.range fun x => p \u2191x)\n\u22a2 \u2203 i, i \u2208 s1 \u2229 s2\n[PROOFSTEP]\nrw [mem_affineSpan_iff_eq_affineCombination, \u2190 Finset.eq_affineCombination_subset_iff_eq_affineCombination_subtype] at\n hp0s1 hp0s2 \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Set \u03b9\np0 : P\nhp0s1 : \u2203 fs x w x, p0 = \u2191(Finset.affineCombination k fs p) w\nhp0s2 : \u2203 fs x w x, p0 = \u2191(Finset.affineCombination k fs p) w\n\u22a2 \u2203 i, i \u2208 s1 \u2229 s2\n[PROOFSTEP]\nrcases hp0s1 with \u27e8fs1, hfs1, w1, hw1, hp0s1\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Set \u03b9\np0 : P\nhp0s2 : \u2203 fs x w x, p0 = \u2191(Finset.affineCombination k fs p) w\nfs1 : Finset \u03b9\nhfs1 : \u2191fs1 \u2286 s1\nw1 : \u03b9 \u2192 k\nhw1 : \u2211 i in fs1, w1 i = 1\nhp0s1 : p0 = \u2191(Finset.affineCombination k fs1 p) w1\n\u22a2 \u2203 i, i \u2208 s1 \u2229 s2\n[PROOFSTEP]\nrcases hp0s2 with \u27e8fs2, hfs2, w2, hw2, hp0s2\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Set \u03b9\np0 : P\nfs1 : Finset \u03b9\nhfs1 : \u2191fs1 \u2286 s1\nw1 : \u03b9 \u2192 k\nhw1 : \u2211 i in fs1, w1 i = 1\nhp0s1 : p0 = \u2191(Finset.affineCombination k fs1 p) w1\nfs2 : Finset \u03b9\nhfs2 : \u2191fs2 \u2286 s2\nw2 : \u03b9 \u2192 k\nhw2 : \u2211 i in fs2, w2 i = 1\nhp0s2 : p0 = \u2191(Finset.affineCombination k fs2 p) w2\n\u22a2 \u2203 i, i \u2208 s1 \u2229 s2\n[PROOFSTEP]\nrw [affineIndependent_iff_indicator_eq_of_affineCombination_eq] at ha \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha :\n \u2200 (s1 s2 : Finset \u03b9) (w1 w2 : \u03b9 \u2192 k),\n \u2211 i in s1, w1 i = 1 \u2192\n \u2211 i in s2, w2 i = 1 \u2192\n \u2191(Finset.affineCombination k s1 p) w1 = \u2191(Finset.affineCombination k s2 p) w2 \u2192\n Set.indicator (\u2191s1) w1 = Set.indicator (\u2191s2) w2\ns1 s2 : Set \u03b9\np0 : P\nfs1 : Finset \u03b9\nhfs1 : \u2191fs1 \u2286 s1\nw1 : \u03b9 \u2192 k\nhw1 : \u2211 i in fs1, w1 i = 1\nhp0s1 : p0 = \u2191(Finset.affineCombination k fs1 p) w1\nfs2 : Finset \u03b9\nhfs2 : \u2191fs2 \u2286 s2\nw2 : \u03b9 \u2192 k\nhw2 : \u2211 i in fs2, w2 i = 1\nhp0s2 : p0 = \u2191(Finset.affineCombination k fs2 p) w2\n\u22a2 \u2203 i, i \u2208 s1 \u2229 s2\n[PROOFSTEP]\nreplace ha := ha fs1 fs2 w1 w2 hw1 hw2 (hp0s1 \u25b8 hp0s2)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\ns1 s2 : Set \u03b9\np0 : P\nfs1 : Finset \u03b9\nhfs1 : \u2191fs1 \u2286 s1\nw1 : \u03b9 \u2192 k\nhw1 : \u2211 i in fs1, w1 i = 1\nhp0s1 : p0 = \u2191(Finset.affineCombination k fs1 p) w1\nfs2 : Finset \u03b9\nhfs2 : \u2191fs2 \u2286 s2\nw2 : \u03b9 \u2192 k\nhw2 : \u2211 i in fs2, w2 i = 1\nhp0s2 : p0 = \u2191(Finset.affineCombination k fs2 p) w2\nha : Set.indicator (\u2191fs1) w1 = Set.indicator (\u2191fs2) w2\n\u22a2 \u2203 i, i \u2208 s1 \u2229 s2\n[PROOFSTEP]\nhave hnz : \u2211 i in fs1, w1 i \u2260 0 := hw1.symm \u25b8 one_ne_zero\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\ns1 s2 : Set \u03b9\np0 : P\nfs1 : Finset \u03b9\nhfs1 : \u2191fs1 \u2286 s1\nw1 : \u03b9 \u2192 k\nhw1 : \u2211 i in fs1, w1 i = 1\nhp0s1 : p0 = \u2191(Finset.affineCombination k fs1 p) w1\nfs2 : Finset \u03b9\nhfs2 : \u2191fs2 \u2286 s2\nw2 : \u03b9 \u2192 k\nhw2 : \u2211 i in fs2, w2 i = 1\nhp0s2 : p0 = \u2191(Finset.affineCombination k fs2 p) w2\nha : Set.indicator (\u2191fs1) w1 = Set.indicator (\u2191fs2) w2\nhnz : \u2211 i in fs1, w1 i \u2260 0\n\u22a2 \u2203 i, i \u2208 s1 \u2229 s2\n[PROOFSTEP]\nrcases Finset.exists_ne_zero_of_sum_ne_zero hnz with \u27e8i, hifs1, hinz\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\ns1 s2 : Set \u03b9\np0 : P\nfs1 : Finset \u03b9\nhfs1 : \u2191fs1 \u2286 s1\nw1 : \u03b9 \u2192 k\nhw1 : \u2211 i in fs1, w1 i = 1\nhp0s1 : p0 = \u2191(Finset.affineCombination k fs1 p) w1\nfs2 : Finset \u03b9\nhfs2 : \u2191fs2 \u2286 s2\nw2 : \u03b9 \u2192 k\nhw2 : \u2211 i in fs2, w2 i = 1\nhp0s2 : p0 = \u2191(Finset.affineCombination k fs2 p) w2\nha : Set.indicator (\u2191fs1) w1 = Set.indicator (\u2191fs2) w2\nhnz : \u2211 i in fs1, w1 i \u2260 0\ni : \u03b9\nhifs1 : i \u2208 fs1\nhinz : w1 i \u2260 0\n\u22a2 \u2203 i, i \u2208 s1 \u2229 s2\n[PROOFSTEP]\nsimp_rw [\u2190 Set.indicator_of_mem (Finset.mem_coe.2 hifs1) w1, ha] at hinz \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\ns1 s2 : Set \u03b9\np0 : P\nfs1 : Finset \u03b9\nhfs1 : \u2191fs1 \u2286 s1\nw1 : \u03b9 \u2192 k\nhw1 : \u2211 i in fs1, w1 i = 1\nhp0s1 : p0 = \u2191(Finset.affineCombination k fs1 p) w1\nfs2 : Finset \u03b9\nhfs2 : \u2191fs2 \u2286 s2\nw2 : \u03b9 \u2192 k\nhw2 : \u2211 i in fs2, w2 i = 1\nhp0s2 : p0 = \u2191(Finset.affineCombination k fs2 p) w2\nha : Set.indicator (\u2191fs1) w1 = Set.indicator (\u2191fs2) w2\nhnz : \u2211 i in fs1, w1 i \u2260 0\ni : \u03b9\nhifs1 : i \u2208 fs1\nhinz : Set.indicator (\u2191fs2) w2 i \u2260 0\n\u22a2 \u2203 i, i \u2208 s1 \u2229 s2\n[PROOFSTEP]\nuse i, hfs1 hifs1\n[GOAL]\ncase right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\ns1 s2 : Set \u03b9\np0 : P\nfs1 : Finset \u03b9\nhfs1 : \u2191fs1 \u2286 s1\nw1 : \u03b9 \u2192 k\nhw1 : \u2211 i in fs1, w1 i = 1\nhp0s1 : p0 = \u2191(Finset.affineCombination k fs1 p) w1\nfs2 : Finset \u03b9\nhfs2 : \u2191fs2 \u2286 s2\nw2 : \u03b9 \u2192 k\nhw2 : \u2211 i in fs2, w2 i = 1\nhp0s2 : p0 = \u2191(Finset.affineCombination k fs2 p) w2\nha : Set.indicator (\u2191fs1) w1 = Set.indicator (\u2191fs2) w2\nhnz : \u2211 i in fs1, w1 i \u2260 0\ni : \u03b9\nhifs1 : i \u2208 fs1\nhinz : Set.indicator (\u2191fs2) w2 i \u2260 0\n\u22a2 i \u2208 s2\n[PROOFSTEP]\nexact hfs2 (Set.mem_of_indicator_ne_zero hinz)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Set \u03b9\nhd : Disjoint s1 s2\n\u22a2 Disjoint \u2191(affineSpan k (p '' s1)) \u2191(affineSpan k (p '' s2))\n[PROOFSTEP]\nrefine' Set.disjoint_left.2 fun p0 hp0s1 hp0s2 => _\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Set \u03b9\nhd : Disjoint s1 s2\np0 : P\nhp0s1 : p0 \u2208 \u2191(affineSpan k (p '' s1))\nhp0s2 : p0 \u2208 \u2191(affineSpan k (p '' s2))\n\u22a2 False\n[PROOFSTEP]\ncases' ha.exists_mem_inter_of_exists_mem_inter_affineSpan hp0s1 hp0s2 with i hi\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ns1 s2 : Set \u03b9\nhd : Disjoint s1 s2\np0 : P\nhp0s1 : p0 \u2208 \u2191(affineSpan k (p '' s1))\nhp0s2 : p0 \u2208 \u2191(affineSpan k (p '' s2))\ni : \u03b9\nhi : i \u2208 s1 \u2229 s2\n\u22a2 False\n[PROOFSTEP]\nexact Set.disjoint_iff.1 hd hi\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ni : \u03b9\ns : Set \u03b9\n\u22a2 p i \u2208 affineSpan k (p '' s) \u2194 i \u2208 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ni : \u03b9\ns : Set \u03b9\n\u22a2 p i \u2208 affineSpan k (p '' s) \u2192 i \u2208 s\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ni : \u03b9\ns : Set \u03b9\nhs : p i \u2208 affineSpan k (p '' s)\n\u22a2 i \u2208 s\n[PROOFSTEP]\nhave h :=\n AffineIndependent.exists_mem_inter_of_exists_mem_inter_affineSpan ha hs\n (mem_affineSpan k (Set.mem_image_of_mem _ (Set.mem_singleton _)))\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ni : \u03b9\ns : Set \u03b9\nhs : p i \u2208 affineSpan k (p '' s)\nh : \u2203 i_1, i_1 \u2208 s \u2229 {i}\n\u22a2 i \u2208 s\n[PROOFSTEP]\nrwa [\u2190 Set.nonempty_def, Set.inter_singleton_nonempty] at h \n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ni : \u03b9\ns : Set \u03b9\n\u22a2 i \u2208 s \u2192 p i \u2208 affineSpan k (p '' s)\n[PROOFSTEP]\nexact fun h => mem_affineSpan k (Set.mem_image_of_mem p h)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ni : \u03b9\ns : Set \u03b9\n\u22a2 \u00acp i \u2208 affineSpan k (p '' (s \\ {i}))\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nh : \u00acAffineIndependent k Subtype.val\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nclassical\nrw [affineIndependent_iff_of_fintype] at h \nsimp only [exists_prop, not_forall] at h \nobtain \u27e8w, hw, hwt, i, hi\u27e9 := h\nsimp only [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero _ w ((\u2191) : t \u2192 V) hw 0, vsub_eq_sub,\n Finset.weightedVSubOfPoint_apply, sub_zero] at hwt \nlet f : \u2200 x : V, x \u2208 t \u2192 k := fun x hx => w \u27e8x, hx\u27e9\nrefine' \u27e8fun x => if hx : x \u2208 t then f x hx else (0 : k), _, _, by use i; simp [hi]\u27e9\nsuffices (\u2211 e : V in t, dite (e \u2208 t) (fun hx => f e hx \u2022 e) fun _ => 0) = 0\n by\n convert this\n rename V => x\n by_cases hx : x \u2208 t <;> simp [hx]\nall_goals simp only [Finset.sum_dite_of_true fun _ h => h, Finset.mk_coe, hwt, hw]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nh : \u00acAffineIndependent k Subtype.val\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nrw [affineIndependent_iff_of_fintype] at h \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nh :\n \u00ac\u2200 (w : { x // x \u2208 t } \u2192 k),\n \u2211 i : { x // x \u2208 t }, w i = 0 \u2192\n \u2191(Finset.weightedVSub Finset.univ Subtype.val) w = 0 \u2192 \u2200 (i : { x // x \u2208 t }), w i = 0\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nsimp only [exists_prop, not_forall] at h \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nh : \u2203 x, \u2211 i : { x // x \u2208 t }, x i = 0 \u2227 \u2191(Finset.weightedVSub Finset.univ Subtype.val) x = 0 \u2227 \u2203 x_1, \u00acx x_1 = 0\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nobtain \u27e8w, hw, hwt, i, hi\u27e9 := h\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\nhwt : \u2191(Finset.weightedVSub Finset.univ Subtype.val) w = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nsimp only [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero _ w ((\u2191) : t \u2192 V) hw 0, vsub_eq_sub,\n Finset.weightedVSubOfPoint_apply, sub_zero] at hwt \n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\nhwt : \u2211 x : { x // x \u2208 t }, w x \u2022 \u2191x = 0\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nlet f : \u2200 x : V, x \u2208 t \u2192 k := fun x hx => w \u27e8x, hx\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\nhwt : \u2211 x : { x // x \u2208 t }, w x \u2022 \u2191x = 0\nf : (x : V) \u2192 x \u2208 t \u2192 k := fun x hx => w { val := x, property := hx }\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nrefine' \u27e8fun x => if hx : x \u2208 t then f x hx else (0 : k), _, _, by use i; simp [hi]\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\nhwt : \u2211 x : { x // x \u2208 t }, w x \u2022 \u2191x = 0\nf : (x : V) \u2192 x \u2208 t \u2192 k := fun x hx => w { val := x, property := hx }\n\u22a2 \u2203 x, x \u2208 t \u2227 (fun x => if hx : x \u2208 t then f x hx else 0) x \u2260 0\n[PROOFSTEP]\nuse i\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\nhwt : \u2211 x : { x // x \u2208 t }, w x \u2022 \u2191x = 0\nf : (x : V) \u2192 x \u2208 t \u2192 k := fun x hx => w { val := x, property := hx }\n\u22a2 \u2191i \u2208 t \u2227 (fun x => if hx : x \u2208 t then f x hx else 0) \u2191i \u2260 0\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\nhwt : \u2211 x : { x // x \u2208 t }, w x \u2022 \u2191x = 0\nf : (x : V) \u2192 x \u2208 t \u2192 k := fun x hx => w { val := x, property := hx }\n\u22a2 \u2211 e in t, (fun x => if hx : x \u2208 t then f x hx else 0) e \u2022 e = 0\ncase intro.intro.intro.intro.refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\nhwt : \u2211 x : { x // x \u2208 t }, w x \u2022 \u2191x = 0\nf : (x : V) \u2192 x \u2208 t \u2192 k := fun x hx => w { val := x, property := hx }\n\u22a2 \u2211 e in t, (fun x => if hx : x \u2208 t then f x hx else 0) e = 0\n[PROOFSTEP]\nsuffices (\u2211 e : V in t, dite (e \u2208 t) (fun hx => f e hx \u2022 e) fun _ => 0) = 0\n by\n convert this\n rename V => x\n by_cases hx : x \u2208 t <;> simp [hx]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\nhwt : \u2211 x : { x // x \u2208 t }, w x \u2022 \u2191x = 0\nf : (x : V) \u2192 x \u2208 t \u2192 k := fun x hx => w { val := x, property := hx }\nthis : (\u2211 e in t, if hx : e \u2208 t then f e hx \u2022 e else 0) = 0\n\u22a2 \u2211 e in t, (fun x => if hx : x \u2208 t then f x hx else 0) e \u2022 e = 0\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_2.a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\nhwt : \u2211 x : { x // x \u2208 t }, w x \u2022 \u2191x = 0\nf : (x : V) \u2192 x \u2208 t \u2192 k := fun x hx => w { val := x, property := hx }\nthis : (\u2211 e in t, if hx : e \u2208 t then f e hx \u2022 e else 0) = 0\nx\u271d : V\na\u271d : x\u271d \u2208 t\n\u22a2 (fun x => if hx : x \u2208 t then f x hx else 0) x\u271d \u2022 x\u271d = if hx : x\u271d \u2208 t then f x\u271d hx \u2022 x\u271d else 0\n[PROOFSTEP]\nrename V => x\n[GOAL]\ncase h.e'_2.a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\nhwt : \u2211 x : { x // x \u2208 t }, w x \u2022 \u2191x = 0\nf : (x : V) \u2192 x \u2208 t \u2192 k := fun x hx => w { val := x, property := hx }\nthis : (\u2211 e in t, if hx : e \u2208 t then f e hx \u2022 e else 0) = 0\nx : V\na\u271d : x \u2208 t\n\u22a2 (fun x => if hx : x \u2208 t then f x hx else 0) x \u2022 x = if hx : x \u2208 t then f x hx \u2022 x else 0\n[PROOFSTEP]\nby_cases hx : x \u2208 t\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\nhwt : \u2211 x : { x // x \u2208 t }, w x \u2022 \u2191x = 0\nf : (x : V) \u2192 x \u2208 t \u2192 k := fun x hx => w { val := x, property := hx }\nthis : (\u2211 e in t, if hx : e \u2208 t then f e hx \u2022 e else 0) = 0\nx : V\na\u271d hx : x \u2208 t\n\u22a2 (fun x => if hx : x \u2208 t then f x hx else 0) x \u2022 x = if hx : x \u2208 t then f x hx \u2022 x else 0\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\nhwt : \u2211 x : { x // x \u2208 t }, w x \u2022 \u2191x = 0\nf : (x : V) \u2192 x \u2208 t \u2192 k := fun x hx => w { val := x, property := hx }\nthis : (\u2211 e in t, if hx : e \u2208 t then f e hx \u2022 e else 0) = 0\nx : V\na\u271d : x \u2208 t\nhx : \u00acx \u2208 t\n\u22a2 (fun x => if hx : x \u2208 t then f x hx else 0) x \u2022 x = if hx : x \u2208 t then f x hx \u2022 x else 0\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\nhwt : \u2211 x : { x // x \u2208 t }, w x \u2022 \u2191x = 0\nf : (x : V) \u2192 x \u2208 t \u2192 k := fun x hx => w { val := x, property := hx }\n\u22a2 (\u2211 e in t, if hx : e \u2208 t then f e hx \u2022 e else 0) = 0\ncase intro.intro.intro.intro.refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\nhwt : \u2211 x : { x // x \u2208 t }, w x \u2022 \u2191x = 0\nf : (x : V) \u2192 x \u2208 t \u2192 k := fun x hx => w { val := x, property := hx }\n\u22a2 \u2211 e in t, (fun x => if hx : x \u2208 t then f x hx else 0) e = 0\n[PROOFSTEP]\nall_goals simp only [Finset.sum_dite_of_true fun _ h => h, Finset.mk_coe, hwt, hw]\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\nhwt : \u2211 x : { x // x \u2208 t }, w x \u2022 \u2191x = 0\nf : (x : V) \u2192 x \u2208 t \u2192 k := fun x hx => w { val := x, property := hx }\n\u22a2 (\u2211 e in t, if hx : e \u2208 t then f e hx \u2022 e else 0) = 0\n[PROOFSTEP]\nsimp only [Finset.sum_dite_of_true fun _ h => h, Finset.mk_coe, hwt, hw]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nt : Finset V\nw : { x // x \u2208 t } \u2192 k\nhw : \u2211 i : { x // x \u2208 t }, w i = 0\ni : { x // x \u2208 t }\nhi : \u00acw i = 0\nhwt : \u2211 x : { x // x \u2208 t }, w x \u2022 \u2191x = 0\nf : (x : V) \u2192 x \u2208 t \u2192 k := fun x hx => w { val := x, property := hx }\n\u22a2 \u2211 e in t, (fun x => if hx : x \u2208 t then f x hx else 0) e = 0\n[PROOFSTEP]\nsimp only [Finset.sum_dite_of_true fun _ h => h, Finset.mk_coe, hwt, hw]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9\u271d : Type u_4\n\u03b9 : Type u_5\np : \u03b9 \u2192 V\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\n\u22a2 (\u2191(Finset.weightedVSub s p) w = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 w i = 0) \u2194 \u2211 e in s, w e \u2022 p e = 0 \u2192 \u2200 (e : \u03b9), e \u2208 s \u2192 w e = 0\n[PROOFSTEP]\nsimp [s.weightedVSub_eq_linear_combination hw]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\n\u22a2 \u2191(Finset.weightedVSub s p) w \u2208\n vectorSpan k {\u2191(Finset.affineCombination k s p) w\u2081, \u2191(Finset.affineCombination k s p) w\u2082} \u2194\n \u2203 r, \u2200 (i : \u03b9), i \u2208 s \u2192 w i = r * (w\u2081 i - w\u2082 i)\n[PROOFSTEP]\nrw [mem_vectorSpan_pair]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\n\u22a2 (\u2203 r,\n r \u2022 (\u2191(Finset.affineCombination k s p) w\u2081 -\u1d65 \u2191(Finset.affineCombination k s p) w\u2082) =\n \u2191(Finset.weightedVSub s p) w) \u2194\n \u2203 r, \u2200 (i : \u03b9), i \u2208 s \u2192 w i = r * (w\u2081 i - w\u2082 i)\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh\u271d : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nh :\n \u2203 r, r \u2022 (\u2191(Finset.affineCombination k s p) w\u2081 -\u1d65 \u2191(Finset.affineCombination k s p) w\u2082) = \u2191(Finset.weightedVSub s p) w\n\u22a2 \u2203 r, \u2200 (i : \u03b9), i \u2208 s \u2192 w i = r * (w\u2081 i - w\u2082 i)\n[PROOFSTEP]\nrcases h with \u27e8r, hr\u27e9\n[GOAL]\ncase refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nr : k\nhr : r \u2022 (\u2191(Finset.affineCombination k s p) w\u2081 -\u1d65 \u2191(Finset.affineCombination k s p) w\u2082) = \u2191(Finset.weightedVSub s p) w\n\u22a2 \u2203 r, \u2200 (i : \u03b9), i \u2208 s \u2192 w i = r * (w\u2081 i - w\u2082 i)\n[PROOFSTEP]\nrefine' \u27e8r, fun i hi => _\u27e9\n[GOAL]\ncase refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nr : k\nhr : r \u2022 (\u2191(Finset.affineCombination k s p) w\u2081 -\u1d65 \u2191(Finset.affineCombination k s p) w\u2082) = \u2191(Finset.weightedVSub s p) w\ni : \u03b9\nhi : i \u2208 s\n\u22a2 w i = r * (w\u2081 i - w\u2082 i)\n[PROOFSTEP]\nrw [s.affineCombination_vsub, \u2190 s.weightedVSub_const_smul, \u2190 sub_eq_zero, \u2190 map_sub] at hr \n[GOAL]\ncase refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nr : k\nhr\u271d : \u2191(Finset.weightedVSub s p) (r \u2022 (w\u2081 - w\u2082)) = \u2191(Finset.weightedVSub s p) w\nhr : \u2191(Finset.weightedVSub s p) (r \u2022 (w\u2081 - w\u2082) - w) = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 w i = r * (w\u2081 i - w\u2082 i)\n[PROOFSTEP]\nhave hw' : (\u2211 j in s, (r \u2022 (w\u2081 - w\u2082) - w) j) = 0 := by\n simp_rw [Pi.sub_apply, Pi.smul_apply, Pi.sub_apply, smul_sub, Finset.sum_sub_distrib, \u2190 Finset.smul_sum, hw, hw\u2081, hw\u2082,\n sub_self]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nr : k\nhr\u271d : \u2191(Finset.weightedVSub s p) (r \u2022 (w\u2081 - w\u2082)) = \u2191(Finset.weightedVSub s p) w\nhr : \u2191(Finset.weightedVSub s p) (r \u2022 (w\u2081 - w\u2082) - w) = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2211 j in s, (r \u2022 (w\u2081 - w\u2082) - w) j = 0\n[PROOFSTEP]\nsimp_rw [Pi.sub_apply, Pi.smul_apply, Pi.sub_apply, smul_sub, Finset.sum_sub_distrib, \u2190 Finset.smul_sum, hw, hw\u2081, hw\u2082,\n sub_self]\n[GOAL]\ncase refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nr : k\nhr\u271d : \u2191(Finset.weightedVSub s p) (r \u2022 (w\u2081 - w\u2082)) = \u2191(Finset.weightedVSub s p) w\nhr : \u2191(Finset.weightedVSub s p) (r \u2022 (w\u2081 - w\u2082) - w) = 0\ni : \u03b9\nhi : i \u2208 s\nhw' : \u2211 j in s, (r \u2022 (w\u2081 - w\u2082) - w) j = 0\n\u22a2 w i = r * (w\u2081 i - w\u2082 i)\n[PROOFSTEP]\nhave hr' := h s _ hw' hr i hi\n[GOAL]\ncase refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nr : k\nhr\u271d : \u2191(Finset.weightedVSub s p) (r \u2022 (w\u2081 - w\u2082)) = \u2191(Finset.weightedVSub s p) w\nhr : \u2191(Finset.weightedVSub s p) (r \u2022 (w\u2081 - w\u2082) - w) = 0\ni : \u03b9\nhi : i \u2208 s\nhw' : \u2211 j in s, (r \u2022 (w\u2081 - w\u2082) - w) j = 0\nhr' : (r \u2022 (w\u2081 - w\u2082) - w) i = 0\n\u22a2 w i = r * (w\u2081 i - w\u2082 i)\n[PROOFSTEP]\nrw [eq_comm, \u2190 sub_eq_zero, \u2190 smul_eq_mul]\n[GOAL]\ncase refine'_1.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nr : k\nhr\u271d : \u2191(Finset.weightedVSub s p) (r \u2022 (w\u2081 - w\u2082)) = \u2191(Finset.weightedVSub s p) w\nhr : \u2191(Finset.weightedVSub s p) (r \u2022 (w\u2081 - w\u2082) - w) = 0\ni : \u03b9\nhi : i \u2208 s\nhw' : \u2211 j in s, (r \u2022 (w\u2081 - w\u2082) - w) j = 0\nhr' : (r \u2022 (w\u2081 - w\u2082) - w) i = 0\n\u22a2 r \u2022 (w\u2081 i - w\u2082 i) - w i = 0\n[PROOFSTEP]\nexact hr'\n[GOAL]\ncase refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh\u271d : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nh : \u2203 r, \u2200 (i : \u03b9), i \u2208 s \u2192 w i = r * (w\u2081 i - w\u2082 i)\n\u22a2 \u2203 r, r \u2022 (\u2191(Finset.affineCombination k s p) w\u2081 -\u1d65 \u2191(Finset.affineCombination k s p) w\u2082) = \u2191(Finset.weightedVSub s p) w\n[PROOFSTEP]\nrcases h with \u27e8r, hr\u27e9\n[GOAL]\ncase refine'_2.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nr : k\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = r * (w\u2081 i - w\u2082 i)\n\u22a2 \u2203 r, r \u2022 (\u2191(Finset.affineCombination k s p) w\u2081 -\u1d65 \u2191(Finset.affineCombination k s p) w\u2082) = \u2191(Finset.weightedVSub s p) w\n[PROOFSTEP]\nrefine' \u27e8r, _\u27e9\n[GOAL]\ncase refine'_2.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nr : k\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = r * (w\u2081 i - w\u2082 i)\n\u22a2 r \u2022 (\u2191(Finset.affineCombination k s p) w\u2081 -\u1d65 \u2191(Finset.affineCombination k s p) w\u2082) = \u2191(Finset.weightedVSub s p) w\n[PROOFSTEP]\nlet w' i := r * (w\u2081 i - w\u2082 i)\n[GOAL]\ncase refine'_2.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nr : k\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = r * (w\u2081 i - w\u2082 i)\nw' : \u03b9 \u2192 k := fun i => r * (w\u2081 i - w\u2082 i)\n\u22a2 r \u2022 (\u2191(Finset.affineCombination k s p) w\u2081 -\u1d65 \u2191(Finset.affineCombination k s p) w\u2082) = \u2191(Finset.weightedVSub s p) w\n[PROOFSTEP]\nchange \u2200 i \u2208 s, w i = w' i at hr \n[GOAL]\ncase refine'_2.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nr : k\nw' : \u03b9 \u2192 k := fun i => r * (w\u2081 i - w\u2082 i)\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = w' i\n\u22a2 r \u2022 (\u2191(Finset.affineCombination k s p) w\u2081 -\u1d65 \u2191(Finset.affineCombination k s p) w\u2082) = \u2191(Finset.weightedVSub s p) w\n[PROOFSTEP]\nrw [s.weightedVSub_congr hr fun _ _ => rfl, s.affineCombination_vsub, \u2190 s.weightedVSub_const_smul]\n[GOAL]\ncase refine'_2.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 0\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nr : k\nw' : \u03b9 \u2192 k := fun i => r * (w\u2081 i - w\u2082 i)\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = w' i\n\u22a2 \u2191(Finset.weightedVSub s p) (r \u2022 (w\u2081 - w\u2082)) = \u2191(Finset.weightedVSub s fun x => p x) fun i => w' i\n[PROOFSTEP]\ncongr\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nx\u271d : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\n\u22a2 \u2191(Finset.affineCombination k s p) w \u2208\n affineSpan k {\u2191(Finset.affineCombination k s p) w\u2081, \u2191(Finset.affineCombination k s p) w\u2082} \u2194\n \u2203 r, \u2200 (i : \u03b9), i \u2208 s \u2192 w i = r * (w\u2082 i - w\u2081 i) + w\u2081 i\n[PROOFSTEP]\nrw [\u2190 vsub_vadd (s.affineCombination k p w) (s.affineCombination k p w\u2081),\n AffineSubspace.vadd_mem_iff_mem_direction _ (left_mem_affineSpan_pair _ _ _), direction_affineSpan,\n s.affineCombination_vsub, Set.pair_comm, weightedVSub_mem_vectorSpan_pair h _ hw\u2082 hw\u2081]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nx\u271d : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\n\u22a2 (\u2203 r, \u2200 (i : \u03b9), i \u2208 s \u2192 (w - w\u2081) i = r * (w\u2082 i - w\u2081 i)) \u2194 \u2203 r, \u2200 (i : \u03b9), i \u2208 s \u2192 w i = r * (w\u2082 i - w\u2081 i) + w\u2081 i\n[PROOFSTEP]\nsimp only [Pi.sub_apply, sub_eq_iff_eq_add]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nx\u271d : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\n\u22a2 \u2211 i in s, (w - w\u2081) i = 0\n[PROOFSTEP]\nsimp_all only [Pi.sub_apply, Finset.sum_sub_distrib, sub_self]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\nh : AffineIndependent k fun p => \u2191p\n\u22a2 \u2203 t, s \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | \u27e8p\u2081, hp\u2081\u27e9)\n[GOAL]\ncase inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nh : AffineIndependent k fun p => \u2191p\n\u22a2 \u2203 t, \u2205 \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nhave p\u2081 : P := AddTorsor.Nonempty.some\n[GOAL]\ncase inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nh : AffineIndependent k fun p => \u2191p\np\u2081 : P\n\u22a2 \u2203 t, \u2205 \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nlet hsv := Basis.ofVectorSpace k V\n[GOAL]\ncase inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nh : AffineIndependent k fun p => \u2191p\np\u2081 : P\nhsv : Basis (\u2191(Basis.ofVectorSpaceIndex k V)) k V := Basis.ofVectorSpace k V\n\u22a2 \u2203 t, \u2205 \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nhave hsvi := hsv.linearIndependent\n[GOAL]\ncase inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nh : AffineIndependent k fun p => \u2191p\np\u2081 : P\nhsv : Basis (\u2191(Basis.ofVectorSpaceIndex k V)) k V := Basis.ofVectorSpace k V\nhsvi : LinearIndependent k \u2191hsv\n\u22a2 \u2203 t, \u2205 \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nhave hsvt := hsv.span_eq\n[GOAL]\ncase inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nh : AffineIndependent k fun p => \u2191p\np\u2081 : P\nhsv : Basis (\u2191(Basis.ofVectorSpaceIndex k V)) k V := Basis.ofVectorSpace k V\nhsvi : LinearIndependent k \u2191hsv\nhsvt : Submodule.span k (Set.range \u2191hsv) = \u22a4\n\u22a2 \u2203 t, \u2205 \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nrw [Basis.coe_ofVectorSpace] at hsvi hsvt \n[GOAL]\ncase inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nh : AffineIndependent k fun p => \u2191p\np\u2081 : P\nhsv : Basis (\u2191(Basis.ofVectorSpaceIndex k V)) k V := Basis.ofVectorSpace k V\nhsvi : LinearIndependent k Subtype.val\nhsvt : Submodule.span k (Set.range Subtype.val) = \u22a4\n\u22a2 \u2203 t, \u2205 \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nhave h0 : \u2200 v : V, v \u2208 Basis.ofVectorSpaceIndex _ _ \u2192 v \u2260 0 :=\n by\n intro v hv\n simpa using hsv.ne_zero \u27e8v, hv\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nh : AffineIndependent k fun p => \u2191p\np\u2081 : P\nhsv : Basis (\u2191(Basis.ofVectorSpaceIndex k V)) k V := Basis.ofVectorSpace k V\nhsvi : LinearIndependent k Subtype.val\nhsvt : Submodule.span k (Set.range Subtype.val) = \u22a4\n\u22a2 \u2200 (v : V), v \u2208 Basis.ofVectorSpaceIndex (?m.389241 v) V \u2192 v \u2260 0\n[PROOFSTEP]\nintro v hv\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nh : AffineIndependent k fun p => \u2191p\np\u2081 : P\nhsv : Basis (\u2191(Basis.ofVectorSpaceIndex k V)) k V := Basis.ofVectorSpace k V\nhsvi : LinearIndependent k Subtype.val\nhsvt : Submodule.span k (Set.range Subtype.val) = \u22a4\nv : V\nhv : v \u2208 Basis.ofVectorSpaceIndex (?m.389241 v) V\n\u22a2 v \u2260 0\n[PROOFSTEP]\nsimpa using hsv.ne_zero \u27e8v, hv\u27e9\n[GOAL]\ncase inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nh : AffineIndependent k fun p => \u2191p\np\u2081 : P\nhsv : Basis (\u2191(Basis.ofVectorSpaceIndex k V)) k V := Basis.ofVectorSpace k V\nhsvi : LinearIndependent k Subtype.val\nhsvt : Submodule.span k (Set.range Subtype.val) = \u22a4\nh0 : \u2200 (v : V), v \u2208 Basis.ofVectorSpaceIndex k V \u2192 v \u2260 0\n\u22a2 \u2203 t, \u2205 \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nrw [linearIndependent_set_iff_affineIndependent_vadd_union_singleton k h0 p\u2081] at hsvi \n[GOAL]\ncase inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\nh : AffineIndependent k fun p => \u2191p\np\u2081 : P\nhsv : Basis (\u2191(Basis.ofVectorSpaceIndex k V)) k V := Basis.ofVectorSpace k V\nhsvi : AffineIndependent k fun p => \u2191p\nhsvt : Submodule.span k (Set.range Subtype.val) = \u22a4\nh0 : \u2200 (v : V), v \u2208 Basis.ofVectorSpaceIndex k V \u2192 v \u2260 0\n\u22a2 \u2203 t, \u2205 \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nexact\n \u27e8{ p\u2081 } \u222a (fun v => v +\u1d65 p\u2081) '' _, Set.empty_subset _, hsvi,\n affineSpan_singleton_union_vadd_eq_top_of_span_eq_top p\u2081 hsvt\u27e9\n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\nh : AffineIndependent k fun p => \u2191p\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\n\u22a2 \u2203 t, s \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nrw [affineIndependent_set_iff_linearIndependent_vsub k hp\u2081] at h \n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nh : LinearIndependent k fun v => \u2191v\nhp\u2081 : p\u2081 \u2208 s\n\u22a2 \u2203 t, s \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nlet bsv := Basis.extend h\n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nh : LinearIndependent k fun v => \u2191v\nhp\u2081 : p\u2081 \u2208 s\nbsv : Basis (\u2191(LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ))) k V := Basis.extend h\n\u22a2 \u2203 t, s \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nhave hsvi := bsv.linearIndependent\n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nh : LinearIndependent k fun v => \u2191v\nhp\u2081 : p\u2081 \u2208 s\nbsv : Basis (\u2191(LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ))) k V := Basis.extend h\nhsvi : LinearIndependent k \u2191bsv\n\u22a2 \u2203 t, s \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nhave hsvt := bsv.span_eq\n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nh : LinearIndependent k fun v => \u2191v\nhp\u2081 : p\u2081 \u2208 s\nbsv : Basis (\u2191(LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ))) k V := Basis.extend h\nhsvi : LinearIndependent k \u2191bsv\nhsvt : Submodule.span k (Set.range \u2191bsv) = \u22a4\n\u22a2 \u2203 t, s \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nrw [Basis.coe_extend] at hsvi hsvt \n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nh : LinearIndependent k fun v => \u2191v\nhp\u2081 : p\u2081 \u2208 s\nbsv : Basis (\u2191(LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ))) k V := Basis.extend h\nhsvi : LinearIndependent k Subtype.val\nhsvt : Submodule.span k (Set.range Subtype.val) = \u22a4\n\u22a2 \u2203 t, s \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nhave hsv := h.subset_extend (Set.subset_univ _)\n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nh : LinearIndependent k fun v => \u2191v\nhp\u2081 : p\u2081 \u2208 s\nbsv : Basis (\u2191(LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ))) k V := Basis.extend h\nhsvi : LinearIndependent k Subtype.val\nhsvt : Submodule.span k (Set.range Subtype.val) = \u22a4\nhsv : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ)\n\u22a2 \u2203 t, s \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nhave h0 : \u2200 v : V, v \u2208 h.extend _ \u2192 v \u2260 0 := by\n intro v hv\n simpa using bsv.ne_zero \u27e8v, hv\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nh : LinearIndependent k fun v => \u2191v\nhp\u2081 : p\u2081 \u2208 s\nbsv : Basis (\u2191(LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ))) k V := Basis.extend h\nhsvi : LinearIndependent k Subtype.val\nhsvt : Submodule.span k (Set.range Subtype.val) = \u22a4\nhsv : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ)\n\u22a2 \u2200 (v : V), v \u2208 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 ?m.393529 v) \u2192 v \u2260 0\n[PROOFSTEP]\nintro v hv\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nh : LinearIndependent k fun v => \u2191v\nhp\u2081 : p\u2081 \u2208 s\nbsv : Basis (\u2191(LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ))) k V := Basis.extend h\nhsvi : LinearIndependent k Subtype.val\nhsvt : Submodule.span k (Set.range Subtype.val) = \u22a4\nhsv : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ)\nv : V\nhv : v \u2208 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 ?m.393529 v)\n\u22a2 v \u2260 0\n[PROOFSTEP]\nsimpa using bsv.ne_zero \u27e8v, hv\u27e9\n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nh : LinearIndependent k fun v => \u2191v\nhp\u2081 : p\u2081 \u2208 s\nbsv : Basis (\u2191(LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ))) k V := Basis.extend h\nhsvi : LinearIndependent k Subtype.val\nhsvt : Submodule.span k (Set.range Subtype.val) = \u22a4\nhsv : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ)\nh0 : \u2200 (v : V), v \u2208 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ) \u2192 v \u2260 0\n\u22a2 \u2203 t, s \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nrw [linearIndependent_set_iff_affineIndependent_vadd_union_singleton k h0 p\u2081] at hsvi \n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nh : LinearIndependent k fun v => \u2191v\nhp\u2081 : p\u2081 \u2208 s\nbsv : Basis (\u2191(LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ))) k V := Basis.extend h\nhsvi : AffineIndependent k fun p => \u2191p\nhsvt : Submodule.span k (Set.range Subtype.val) = \u22a4\nhsv : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ)\nh0 : \u2200 (v : V), v \u2208 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ) \u2192 v \u2260 0\n\u22a2 \u2203 t, s \u2286 t \u2227 (AffineIndependent k fun p => \u2191p) \u2227 affineSpan k t = \u22a4\n[PROOFSTEP]\nrefine' \u27e8{ p\u2081 } \u222a (fun v => v +\u1d65 p\u2081) '' h.extend (Set.subset_univ _), _, _\u27e9\n[GOAL]\ncase inr.intro.refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nh : LinearIndependent k fun v => \u2191v\nhp\u2081 : p\u2081 \u2208 s\nbsv : Basis (\u2191(LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ))) k V := Basis.extend h\nhsvi : AffineIndependent k fun p => \u2191p\nhsvt : Submodule.span k (Set.range Subtype.val) = \u22a4\nhsv : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ)\nh0 : \u2200 (v : V), v \u2208 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ) \u2192 v \u2260 0\n\u22a2 s \u2286 {p\u2081} \u222a (fun v => v +\u1d65 p\u2081) '' LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ)\n[PROOFSTEP]\nrefine' Set.Subset.trans _ (Set.union_subset_union_right _ (Set.image_subset _ hsv))\n[GOAL]\ncase inr.intro.refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nh : LinearIndependent k fun v => \u2191v\nhp\u2081 : p\u2081 \u2208 s\nbsv : Basis (\u2191(LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ))) k V := Basis.extend h\nhsvi : AffineIndependent k fun p => \u2191p\nhsvt : Submodule.span k (Set.range Subtype.val) = \u22a4\nhsv : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ)\nh0 : \u2200 (v : V), v \u2208 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ) \u2192 v \u2260 0\n\u22a2 s \u2286 {p\u2081} \u222a (fun v => v +\u1d65 p\u2081) '' ((fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}))\n[PROOFSTEP]\nsimp [Set.image_image]\n[GOAL]\ncase inr.intro.refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nh : LinearIndependent k fun v => \u2191v\nhp\u2081 : p\u2081 \u2208 s\nbsv : Basis (\u2191(LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ))) k V := Basis.extend h\nhsvi : AffineIndependent k fun p => \u2191p\nhsvt : Submodule.span k (Set.range Subtype.val) = \u22a4\nhsv : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ)\nh0 : \u2200 (v : V), v \u2208 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ) \u2192 v \u2260 0\n\u22a2 (AffineIndependent k fun p => \u2191p) \u2227\n affineSpan k\n ({p\u2081} \u222a (fun v => v +\u1d65 p\u2081) '' LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ)) =\n \u22a4\n[PROOFSTEP]\nuse hsvi\n[GOAL]\ncase right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np\u2081 : P\nh : LinearIndependent k fun v => \u2191v\nhp\u2081 : p\u2081 \u2208 s\nbsv : Basis (\u2191(LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ))) k V := Basis.extend h\nhsvi : AffineIndependent k fun p => \u2191p\nhsvt : Submodule.span k (Set.range Subtype.val) = \u22a4\nhsv : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ)\nh0 : \u2200 (v : V), v \u2208 LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ) \u2192 v \u2260 0\n\u22a2 affineSpan k\n ({p\u2081} \u222a (fun v => v +\u1d65 p\u2081) '' LinearIndependent.extend h (_ : (fun p => p -\u1d65 p\u2081) '' (s \\ {p\u2081}) \u2286 Set.univ)) =\n \u22a4\n[PROOFSTEP]\nexact affineSpan_singleton_union_vadd_eq_top_of_span_eq_top p\u2081 hsvt\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k s \u2227 AffineIndependent k Subtype.val\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | \u27e8p, hp\u27e9)\n[GOAL]\ncase inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k \u2205 \u2227 AffineIndependent k Subtype.val\n[PROOFSTEP]\nexact \u27e8\u2205, Set.empty_subset \u2205, rfl, affineIndependent_of_subsingleton k _\u27e9\n[GOAL]\ncase inr.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k s \u2227 AffineIndependent k Subtype.val\n[PROOFSTEP]\nobtain \u27e8b, hb\u2081, hb\u2082, hb\u2083\u27e9 := exists_linearIndependent k ((Equiv.vaddConst p).symm '' s)\n[GOAL]\ncase inr.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : LinearIndependent k Subtype.val\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k s \u2227 AffineIndependent k Subtype.val\n[PROOFSTEP]\nhave hb\u2080 : \u2200 v : V, v \u2208 b \u2192 v \u2260 0 := fun v hv => hb\u2083.ne_zero (\u27e8v, hv\u27e9 : b)\n[GOAL]\ncase inr.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : LinearIndependent k Subtype.val\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k s \u2227 AffineIndependent k Subtype.val\n[PROOFSTEP]\nrw [linearIndependent_set_iff_affineIndependent_vadd_union_singleton k hb\u2080 p] at hb\u2083 \n[GOAL]\ncase inr.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k s \u2227 AffineIndependent k Subtype.val\n[PROOFSTEP]\nrefine' \u27e8{ p } \u222a Equiv.vaddConst p '' b, _, _, hb\u2083\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 {p} \u222a \u2191(Equiv.vaddConst p) '' b \u2286 s\n[PROOFSTEP]\napply Set.union_subset (Set.singleton_subset_iff.mpr hp)\n[GOAL]\ncase inr.intro.intro.intro.intro.refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 \u2191(Equiv.vaddConst p) '' b \u2286 s\n[PROOFSTEP]\nrwa [\u2190 (Equiv.vaddConst p).subset_image' b s]\n[GOAL]\ncase inr.intro.intro.intro.intro.refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b) = affineSpan k s\n[PROOFSTEP]\nrw [Equiv.coe_vaddConst_symm, \u2190 vectorSpan_eq_span_vsub_set_right k hp] at hb\u2082 \n[GOAL]\ncase inr.intro.intro.intro.intro.refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b) = affineSpan k s\n[PROOFSTEP]\napply AffineSubspace.ext_of_direction_eq\n[GOAL]\ncase inr.intro.intro.intro.intro.refine'_2.hd\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 AffineSubspace.direction (affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b)) = AffineSubspace.direction (affineSpan k s)\n[PROOFSTEP]\nhave : Submodule.span k b = Submodule.span k (insert 0 b) := by simp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 Submodule.span k b = Submodule.span k (insert 0 b)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.intro.intro.intro.intro.refine'_2.hd\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\nthis : Submodule.span k b = Submodule.span k (insert 0 b)\n\u22a2 AffineSubspace.direction (affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b)) = AffineSubspace.direction (affineSpan k s)\n[PROOFSTEP]\nsimp only [direction_affineSpan, \u2190 hb\u2082, Equiv.coe_vaddConst, Set.singleton_union,\n vectorSpan_eq_span_vsub_set_right k (Set.mem_insert p _), this]\n[GOAL]\ncase inr.intro.intro.intro.intro.refine'_2.hd\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\nthis : Submodule.span k b = Submodule.span k (insert 0 b)\n\u22a2 Submodule.span k ((fun x => x -\u1d65 p) '' insert p ((fun a => a +\u1d65 p) '' b)) = Submodule.span k (insert 0 b)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase inr.intro.intro.intro.intro.refine'_2.hd.e_s\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\nthis : Submodule.span k b = Submodule.span k (insert 0 b)\n\u22a2 (fun x => x -\u1d65 p) '' insert p ((fun a => a +\u1d65 p) '' b) = insert 0 b\n[PROOFSTEP]\nchange (Equiv.vaddConst p).symm '' insert p (Equiv.vaddConst p '' b) = _\n[GOAL]\ncase inr.intro.intro.intro.intro.refine'_2.hd.e_s\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\nthis : Submodule.span k b = Submodule.span k (insert 0 b)\n\u22a2 \u2191(Equiv.vaddConst p).symm '' insert p (\u2191(Equiv.vaddConst p) '' b) = insert 0 b\n[PROOFSTEP]\nrw [Set.image_insert_eq, \u2190 Set.image_comp]\n[GOAL]\ncase inr.intro.intro.intro.intro.refine'_2.hd.e_s\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\nthis : Submodule.span k b = Submodule.span k (insert 0 b)\n\u22a2 insert (\u2191(Equiv.vaddConst p).symm p) (\u2191(Equiv.vaddConst p).symm \u2218 \u2191(Equiv.vaddConst p) '' b) = insert 0 b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.intro.intro.intro.intro.refine'_2.hn\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 Set.Nonempty (\u2191(affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b)) \u2229 \u2191(affineSpan k s))\n[PROOFSTEP]\nuse p\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 p \u2208 \u2191(affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b)) \u2229 \u2191(affineSpan k s)\n[PROOFSTEP]\nsimp only [Equiv.coe_vaddConst, Set.singleton_union, Set.mem_inter_iff, coe_affineSpan]\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 p \u2208 spanPoints k (insert p ((fun a => a +\u1d65 p) '' b)) \u2227 p \u2208 spanPoints k s\n[PROOFSTEP]\nexact \u27e8mem_spanPoints k _ _ (Set.mem_insert p _), mem_spanPoints k _ _ hp\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nh : p\u2081 \u2260 p\u2082\n\u22a2 AffineIndependent k ![p\u2081, p\u2082]\n[PROOFSTEP]\nrw [affineIndependent_iff_linearIndependent_vsub k ![p\u2081, p\u2082] 0]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nh : p\u2081 \u2260 p\u2082\n\u22a2 LinearIndependent k fun i => Matrix.vecCons p\u2081 ![p\u2082] \u2191i -\u1d65 Matrix.vecCons p\u2081 ![p\u2082] 0\n[PROOFSTEP]\nlet i\u2081 : { x // x \u2260 (0 : Fin 2) } := \u27e81, by norm_num\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nh : p\u2081 \u2260 p\u2082\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nh : p\u2081 \u2260 p\u2082\ni\u2081 : { x // x \u2260 0 } := { val := 1, property := (_ : 1 \u2260 0) }\n\u22a2 LinearIndependent k fun i => Matrix.vecCons p\u2081 ![p\u2082] \u2191i -\u1d65 Matrix.vecCons p\u2081 ![p\u2082] 0\n[PROOFSTEP]\nhave he' : \u2200 i, i = i\u2081 := by\n rintro \u27e8i, hi\u27e9\n ext\n fin_cases i\n \u00b7 simp at hi \n \u00b7 simp only [Fin.val_one]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nh : p\u2081 \u2260 p\u2082\ni\u2081 : { x // x \u2260 0 } := { val := 1, property := (_ : 1 \u2260 0) }\n\u22a2 \u2200 (i : { x // x \u2260 0 }), i = i\u2081\n[PROOFSTEP]\nrintro \u27e8i, hi\u27e9\n[GOAL]\ncase mk\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nh : p\u2081 \u2260 p\u2082\ni\u2081 : { x // x \u2260 0 } := { val := 1, property := (_ : 1 \u2260 0) }\ni : Fin 2\nhi : i \u2260 0\n\u22a2 { val := i, property := hi } = i\u2081\n[PROOFSTEP]\next\n[GOAL]\ncase mk.a.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nh : p\u2081 \u2260 p\u2082\ni\u2081 : { x // x \u2260 0 } := { val := 1, property := (_ : 1 \u2260 0) }\ni : Fin 2\nhi : i \u2260 0\n\u22a2 \u2191\u2191{ val := i, property := hi } = \u2191\u2191i\u2081\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase mk.a.h.head\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nh : p\u2081 \u2260 p\u2082\ni\u2081 : { x // x \u2260 0 } := { val := 1, property := (_ : 1 \u2260 0) }\nhi : { val := 0, isLt := (_ : 0 < 2) } \u2260 0\n\u22a2 \u2191\u2191{ val := { val := 0, isLt := (_ : 0 < 2) }, property := hi } = \u2191\u2191i\u2081\n[PROOFSTEP]\nsimp at hi \n[GOAL]\ncase mk.a.h.tail.head\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nh : p\u2081 \u2260 p\u2082\ni\u2081 : { x // x \u2260 0 } := { val := 1, property := (_ : 1 \u2260 0) }\nhi : { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u2260 0\n\u22a2 \u2191\u2191{ val := { val := 1, isLt := (_ : (fun a => a < 2) 1) }, property := hi } = \u2191\u2191i\u2081\n[PROOFSTEP]\nsimp only [Fin.val_one]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nh : p\u2081 \u2260 p\u2082\ni\u2081 : { x // x \u2260 0 } := { val := 1, property := (_ : 1 \u2260 0) }\nhe' : \u2200 (i : { x // x \u2260 0 }), i = i\u2081\n\u22a2 LinearIndependent k fun i => Matrix.vecCons p\u2081 ![p\u2082] \u2191i -\u1d65 Matrix.vecCons p\u2081 ![p\u2082] 0\n[PROOFSTEP]\nhaveI : Unique { x // x \u2260 (0 : Fin 2) } := \u27e8\u27e8i\u2081\u27e9, he'\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nh : p\u2081 \u2260 p\u2082\ni\u2081 : { x // x \u2260 0 } := { val := 1, property := (_ : 1 \u2260 0) }\nhe' : \u2200 (i : { x // x \u2260 0 }), i = i\u2081\nthis : Unique { x // x \u2260 0 }\n\u22a2 LinearIndependent k fun i => Matrix.vecCons p\u2081 ![p\u2082] \u2191i -\u1d65 Matrix.vecCons p\u2081 ![p\u2082] 0\n[PROOFSTEP]\napply linearIndependent_unique\n[GOAL]\ncase a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nh : p\u2081 \u2260 p\u2082\ni\u2081 : { x // x \u2260 0 } := { val := 1, property := (_ : 1 \u2260 0) }\nhe' : \u2200 (i : { x // x \u2260 0 }), i = i\u2081\nthis : Unique { x // x \u2260 0 }\n\u22a2 Matrix.vecCons p\u2081 ![p\u2082] \u2191default -\u1d65 Matrix.vecCons p\u2081 ![p\u2082] 0 \u2260 0\n[PROOFSTEP]\nrw [he' default]\n[GOAL]\ncase a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nh : p\u2081 \u2260 p\u2082\ni\u2081 : { x // x \u2260 0 } := { val := 1, property := (_ : 1 \u2260 0) }\nhe' : \u2200 (i : { x // x \u2260 0 }), i = i\u2081\nthis : Unique { x // x \u2260 0 }\n\u22a2 Matrix.vecCons p\u2081 ![p\u2082] \u2191i\u2081 -\u1d65 Matrix.vecCons p\u2081 ![p\u2082] 0 \u2260 0\n[PROOFSTEP]\nsimpa using h.symm\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\n\u22a2 AffineIndependent k p\n[PROOFSTEP]\nclassical\nintro s w hw hs\nlet s' : Finset { y // y \u2260 i } := s.subtype (\u00b7 \u2260 i)\nlet p' : { y // y \u2260 i } \u2192 P := fun x => p x\nby_cases his : i \u2208 s \u2227 w i \u2260 0\n\u00b7 refine' False.elim (hi _)\n let wm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\n have hms : s.weightedVSub p wm = (0 : V) := by simp [hs]\n have hwm : \u2211 i in s, wm i = 0 := by simp [\u2190 Finset.mul_sum, hw]\n have hwmi : wm i = -1 := by simp [his.2]\n let w' : { y // y \u2260 i } \u2192 k := fun x => wm x\n have hw' : \u2211 x in s', w' x = 1 := by\n simp_rw [Finset.sum_subtype_eq_sum_filter]\n rw [\u2190 s.sum_filter_add_sum_filter_not (\u00b7 \u2260 i)] at hwm \n simp_rw [Classical.not_not] at hwm \n erw [Finset.filter_eq'] at hwm \n simp_rw [if_pos his.1, Finset.sum_singleton, hwmi, \u2190 sub_eq_add_neg, sub_eq_zero] at hwm \n exact hwm\n rw [\u2190 s.affineCombination_eq_of_weightedVSub_eq_zero_of_eq_neg_one hms his.1 hwmi, \u2190\n (Subtype.range_coe : _ = {x | x \u2260 i}), \u2190 Set.range_comp, \u2190 s.affineCombination_subtype_eq_filter]\n exact affineCombination_mem_affineSpan hw' p'\n\u00b7 rw [not_and_or, Classical.not_not] at his \n let w' : { y // y \u2260 i } \u2192 k := fun x => w x\n have hw' : \u2211 x in s', w' x = 0 := by\n simp_rw [Finset.sum_subtype_eq_sum_filter]\n rw [Finset.sum_filter_of_ne, hw]\n rintro x hxs hwx rfl\n exact hwx (his.neg_resolve_left hxs)\n have hs' : s'.weightedVSub p' w' = (0 : V) :=\n by\n simp_rw [Finset.weightedVSub_subtype_eq_filter]\n rw [Finset.weightedVSub_filter_of_ne, hs]\n rintro x hxs hwx rfl\n exact hwx (his.neg_resolve_left hxs)\n intro j hj\n by_cases hji : j = i\n \u00b7 rw [hji] at hj \n exact hji.symm \u25b8 his.neg_resolve_left hj\n \u00b7 exact ha s' w' hw' hs' \u27e8j, hji\u27e9 (Finset.mem_subtype.2 hj)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\n\u22a2 AffineIndependent k p\n[PROOFSTEP]\nintro s w hw hs\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 w i = 0\n[PROOFSTEP]\nlet s' : Finset { y // y \u2260 i } := s.subtype (\u00b7 \u2260 i)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 w i = 0\n[PROOFSTEP]\nlet p' : { y // y \u2260 i } \u2192 P := fun x => p x\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 w i = 0\n[PROOFSTEP]\nby_cases his : i \u2208 s \u2227 w i \u2260 0\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 w i = 0\n[PROOFSTEP]\nrefine' False.elim (hi _)\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\n\u22a2 p i \u2208 affineSpan k (p '' {x | x \u2260 i})\n[PROOFSTEP]\nlet wm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\n\u22a2 p i \u2208 affineSpan k (p '' {x | x \u2260 i})\n[PROOFSTEP]\nhave hms : s.weightedVSub p wm = (0 : V) := by simp [hs]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\n\u22a2 \u2191(Finset.weightedVSub s p) wm = 0\n[PROOFSTEP]\nsimp [hs]\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\nhms : \u2191(Finset.weightedVSub s p) wm = 0\n\u22a2 p i \u2208 affineSpan k (p '' {x | x \u2260 i})\n[PROOFSTEP]\nhave hwm : \u2211 i in s, wm i = 0 := by simp [\u2190 Finset.mul_sum, hw]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\nhms : \u2191(Finset.weightedVSub s p) wm = 0\n\u22a2 \u2211 i in s, wm i = 0\n[PROOFSTEP]\nsimp [\u2190 Finset.mul_sum, hw]\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\nhms : \u2191(Finset.weightedVSub s p) wm = 0\nhwm : \u2211 i in s, wm i = 0\n\u22a2 p i \u2208 affineSpan k (p '' {x | x \u2260 i})\n[PROOFSTEP]\nhave hwmi : wm i = -1 := by simp [his.2]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\nhms : \u2191(Finset.weightedVSub s p) wm = 0\nhwm : \u2211 i in s, wm i = 0\n\u22a2 wm i = -1\n[PROOFSTEP]\nsimp [his.2]\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\nhms : \u2191(Finset.weightedVSub s p) wm = 0\nhwm : \u2211 i in s, wm i = 0\nhwmi : wm i = -1\n\u22a2 p i \u2208 affineSpan k (p '' {x | x \u2260 i})\n[PROOFSTEP]\nlet w' : { y // y \u2260 i } \u2192 k := fun x => wm x\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\nhms : \u2191(Finset.weightedVSub s p) wm = 0\nhwm : \u2211 i in s, wm i = 0\nhwmi : wm i = -1\nw' : { y // y \u2260 i } \u2192 k := fun x => wm \u2191x\n\u22a2 p i \u2208 affineSpan k (p '' {x | x \u2260 i})\n[PROOFSTEP]\nhave hw' : \u2211 x in s', w' x = 1 := by\n simp_rw [Finset.sum_subtype_eq_sum_filter]\n rw [\u2190 s.sum_filter_add_sum_filter_not (\u00b7 \u2260 i)] at hwm \n simp_rw [Classical.not_not] at hwm \n erw [Finset.filter_eq'] at hwm \n simp_rw [if_pos his.1, Finset.sum_singleton, hwmi, \u2190 sub_eq_add_neg, sub_eq_zero] at hwm \n exact hwm\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\nhms : \u2191(Finset.weightedVSub s p) wm = 0\nhwm : \u2211 i in s, wm i = 0\nhwmi : wm i = -1\nw' : { y // y \u2260 i } \u2192 k := fun x => wm \u2191x\n\u22a2 \u2211 x in s', w' x = 1\n[PROOFSTEP]\nsimp_rw [Finset.sum_subtype_eq_sum_filter]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\nhms : \u2191(Finset.weightedVSub s p) wm = 0\nhwm : \u2211 i in s, wm i = 0\nhwmi : wm i = -1\nw' : { y // y \u2260 i } \u2192 k := fun x => wm \u2191x\n\u22a2 \u2211 x in Finset.filter (fun x => x \u2260 i) s, (-(w i)\u207b\u00b9 \u2022 w) x = 1\n[PROOFSTEP]\nrw [\u2190 s.sum_filter_add_sum_filter_not (\u00b7 \u2260 i)] at hwm \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\nhms : \u2191(Finset.weightedVSub s p) wm = 0\nhwm : \u2211 x in Finset.filter (fun x => x \u2260 i) s, wm x + \u2211 x in Finset.filter (fun x => \u00acx \u2260 i) s, wm x = 0\nhwmi : wm i = -1\nw' : { y // y \u2260 i } \u2192 k := fun x => wm \u2191x\n\u22a2 \u2211 x in Finset.filter (fun x => x \u2260 i) s, (-(w i)\u207b\u00b9 \u2022 w) x = 1\n[PROOFSTEP]\nsimp_rw [Classical.not_not] at hwm \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\nhms : \u2191(Finset.weightedVSub s p) wm = 0\nhwmi : wm i = -1\nw' : { y // y \u2260 i } \u2192 k := fun x => wm \u2191x\nhwm :\n \u2211 x in Finset.filter (fun x => x \u2260 i) s, (-(w i)\u207b\u00b9 \u2022 w) x +\n \u2211 x in Finset.filter (fun x => x = i) s, (-(w i)\u207b\u00b9 \u2022 w) x =\n 0\n\u22a2 \u2211 x in Finset.filter (fun x => x \u2260 i) s, (-(w i)\u207b\u00b9 \u2022 w) x = 1\n[PROOFSTEP]\nerw [Finset.filter_eq'] at hwm \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\nhms : \u2191(Finset.weightedVSub s p) wm = 0\nhwmi : wm i = -1\nw' : { y // y \u2260 i } \u2192 k := fun x => wm \u2191x\nhwm : \u2211 x in Finset.filter (fun x => x \u2260 i) s, (-(w i)\u207b\u00b9 \u2022 w) x + \u2211 x in if i \u2208 s then {i} else \u2205, (-(w i)\u207b\u00b9 \u2022 w) x = 0\n\u22a2 \u2211 x in Finset.filter (fun x => x \u2260 i) s, (-(w i)\u207b\u00b9 \u2022 w) x = 1\n[PROOFSTEP]\nsimp_rw [if_pos his.1, Finset.sum_singleton, hwmi, \u2190 sub_eq_add_neg, sub_eq_zero] at hwm \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\nhms : \u2191(Finset.weightedVSub s p) wm = 0\nhwmi : wm i = -1\nw' : { y // y \u2260 i } \u2192 k := fun x => wm \u2191x\nhwm : \u2211 x in Finset.filter (fun x => x \u2260 i) s, (-(w i)\u207b\u00b9 \u2022 w) x = 1\n\u22a2 \u2211 x in Finset.filter (fun x => x \u2260 i) s, (-(w i)\u207b\u00b9 \u2022 w) x = 1\n[PROOFSTEP]\nexact hwm\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\nhms : \u2191(Finset.weightedVSub s p) wm = 0\nhwm : \u2211 i in s, wm i = 0\nhwmi : wm i = -1\nw' : { y // y \u2260 i } \u2192 k := fun x => wm \u2191x\nhw' : \u2211 x in s', w' x = 1\n\u22a2 p i \u2208 affineSpan k (p '' {x | x \u2260 i})\n[PROOFSTEP]\nrw [\u2190 s.affineCombination_eq_of_weightedVSub_eq_zero_of_eq_neg_one hms his.1 hwmi, \u2190\n (Subtype.range_coe : _ = {x | x \u2260 i}), \u2190 Set.range_comp, \u2190 s.affineCombination_subtype_eq_filter]\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : i \u2208 s \u2227 w i \u2260 0\nwm : \u03b9 \u2192 k := -(w i)\u207b\u00b9 \u2022 w\nhms : \u2191(Finset.weightedVSub s p) wm = 0\nhwm : \u2211 i in s, wm i = 0\nhwmi : wm i = -1\nw' : { y // y \u2260 i } \u2192 k := fun x => wm \u2191x\nhw' : \u2211 x in s', w' x = 1\n\u22a2 (\u2191(Finset.affineCombination k (Finset.subtype (fun x => x \u2260 i) s) fun i_1 => p \u2191i_1) fun i_1 => wm \u2191i_1) \u2208\n affineSpan k (Set.range (p \u2218 Subtype.val))\n[PROOFSTEP]\nexact affineCombination_mem_affineSpan hw' p'\n[GOAL]\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : \u00ac(i \u2208 s \u2227 w i \u2260 0)\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 w i = 0\n[PROOFSTEP]\nrw [not_and_or, Classical.not_not] at his \n[GOAL]\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : \u00aci \u2208 s \u2228 w i = 0\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 w i = 0\n[PROOFSTEP]\nlet w' : { y // y \u2260 i } \u2192 k := fun x => w x\n[GOAL]\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : \u00aci \u2208 s \u2228 w i = 0\nw' : { y // y \u2260 i } \u2192 k := fun x => w \u2191x\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 w i = 0\n[PROOFSTEP]\nhave hw' : \u2211 x in s', w' x = 0 := by\n simp_rw [Finset.sum_subtype_eq_sum_filter]\n rw [Finset.sum_filter_of_ne, hw]\n rintro x hxs hwx rfl\n exact hwx (his.neg_resolve_left hxs)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : \u00aci \u2208 s \u2228 w i = 0\nw' : { y // y \u2260 i } \u2192 k := fun x => w \u2191x\n\u22a2 \u2211 x in s', w' x = 0\n[PROOFSTEP]\nsimp_rw [Finset.sum_subtype_eq_sum_filter]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : \u00aci \u2208 s \u2228 w i = 0\nw' : { y // y \u2260 i } \u2192 k := fun x => w \u2191x\n\u22a2 \u2211 x in Finset.filter (fun x => x \u2260 i) s, w x = 0\n[PROOFSTEP]\nrw [Finset.sum_filter_of_ne, hw]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : \u00aci \u2208 s \u2228 w i = 0\nw' : { y // y \u2260 i } \u2192 k := fun x => w \u2191x\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u2192 w x \u2260 0 \u2192 x \u2260 i\n[PROOFSTEP]\nrintro x hxs hwx rfl\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\nx : \u03b9\nhxs : x \u2208 s\nhwx : w x \u2260 0\nha : AffineIndependent k fun x_1 => p \u2191x_1\nhi : \u00acp x \u2208 affineSpan k (p '' {x_1 | x_1 \u2260 x})\ns' : Finset { y // y \u2260 x } := Finset.subtype (fun x_1 => x_1 \u2260 x) s\np' : { y // y \u2260 x } \u2192 P := fun x_1 => p \u2191x_1\nhis : \u00acx \u2208 s \u2228 w x = 0\nw' : { y // y \u2260 x } \u2192 k := fun x_1 => w \u2191x_1\n\u22a2 False\n[PROOFSTEP]\nexact hwx (his.neg_resolve_left hxs)\n[GOAL]\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : \u00aci \u2208 s \u2228 w i = 0\nw' : { y // y \u2260 i } \u2192 k := fun x => w \u2191x\nhw' : \u2211 x in s', w' x = 0\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 w i = 0\n[PROOFSTEP]\nhave hs' : s'.weightedVSub p' w' = (0 : V) :=\n by\n simp_rw [Finset.weightedVSub_subtype_eq_filter]\n rw [Finset.weightedVSub_filter_of_ne, hs]\n rintro x hxs hwx rfl\n exact hwx (his.neg_resolve_left hxs)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : \u00aci \u2208 s \u2228 w i = 0\nw' : { y // y \u2260 i } \u2192 k := fun x => w \u2191x\nhw' : \u2211 x in s', w' x = 0\n\u22a2 \u2191(Finset.weightedVSub s' p') w' = 0\n[PROOFSTEP]\nsimp_rw [Finset.weightedVSub_subtype_eq_filter]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : \u00aci \u2208 s \u2228 w i = 0\nw' : { y // y \u2260 i } \u2192 k := fun x => w \u2191x\nhw' : \u2211 x in s', w' x = 0\n\u22a2 \u2191(Finset.weightedVSub (Finset.filter (fun x => x \u2260 i) s) p) w = 0\n[PROOFSTEP]\nrw [Finset.weightedVSub_filter_of_ne, hs]\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : \u00aci \u2208 s \u2228 w i = 0\nw' : { y // y \u2260 i } \u2192 k := fun x => w \u2191x\nhw' : \u2211 x in s', w' x = 0\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 s \u2192 w i_1 \u2260 0 \u2192 i_1 \u2260 i\n[PROOFSTEP]\nrintro x hxs hwx rfl\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\nx : \u03b9\nhxs : x \u2208 s\nhwx : w x \u2260 0\nha : AffineIndependent k fun x_1 => p \u2191x_1\nhi : \u00acp x \u2208 affineSpan k (p '' {x_1 | x_1 \u2260 x})\ns' : Finset { y // y \u2260 x } := Finset.subtype (fun x_1 => x_1 \u2260 x) s\np' : { y // y \u2260 x } \u2192 P := fun x_1 => p \u2191x_1\nhis : \u00acx \u2208 s \u2228 w x = 0\nw' : { y // y \u2260 x } \u2192 k := fun x_1 => w \u2191x_1\nhw' : \u2211 x in s', w' x = 0\n\u22a2 False\n[PROOFSTEP]\nexact hwx (his.neg_resolve_left hxs)\n[GOAL]\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : \u00aci \u2208 s \u2228 w i = 0\nw' : { y // y \u2260 i } \u2192 k := fun x => w \u2191x\nhw' : \u2211 x in s', w' x = 0\nhs' : \u2191(Finset.weightedVSub s' p') w' = 0\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 w i = 0\n[PROOFSTEP]\nintro j hj\n[GOAL]\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : \u00aci \u2208 s \u2228 w i = 0\nw' : { y // y \u2260 i } \u2192 k := fun x => w \u2191x\nhw' : \u2211 x in s', w' x = 0\nhs' : \u2191(Finset.weightedVSub s' p') w' = 0\nj : \u03b9\nhj : j \u2208 s\n\u22a2 w j = 0\n[PROOFSTEP]\nby_cases hji : j = i\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : \u00aci \u2208 s \u2228 w i = 0\nw' : { y // y \u2260 i } \u2192 k := fun x => w \u2191x\nhw' : \u2211 x in s', w' x = 0\nhs' : \u2191(Finset.weightedVSub s' p') w' = 0\nj : \u03b9\nhj : j \u2208 s\nhji : j = i\n\u22a2 w j = 0\n[PROOFSTEP]\nrw [hji] at hj \n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : \u00aci \u2208 s \u2228 w i = 0\nw' : { y // y \u2260 i } \u2192 k := fun x => w \u2191x\nhw' : \u2211 x in s', w' x = 0\nhs' : \u2191(Finset.weightedVSub s' p') w' = 0\nj : \u03b9\nhj : i \u2208 s\nhji : j = i\n\u22a2 w j = 0\n[PROOFSTEP]\nexact hji.symm \u25b8 his.neg_resolve_left hj\n[GOAL]\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni : \u03b9\nha : AffineIndependent k fun x => p \u2191x\nhi : \u00acp i \u2208 affineSpan k (p '' {x | x \u2260 i})\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\nhs : \u2191(Finset.weightedVSub s p) w = 0\ns' : Finset { y // y \u2260 i } := Finset.subtype (fun x => x \u2260 i) s\np' : { y // y \u2260 i } \u2192 P := fun x => p \u2191x\nhis : \u00aci \u2208 s \u2228 w i = 0\nw' : { y // y \u2260 i } \u2192 k := fun x => w \u2191x\nhw' : \u2211 x in s', w' x = 0\nhs' : \u2191(Finset.weightedVSub s' p') w' = 0\nj : \u03b9\nhj : j \u2208 s\nhji : \u00acj = i\n\u22a2 w j = 0\n[PROOFSTEP]\nexact ha s' w' hw' hs' \u27e8j, hji\u27e9 (Finset.mem_subtype.2 hj)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\n\u22a2 AffineIndependent k ![p\u2081, p\u2082, p\u2083]\n[PROOFSTEP]\nhave ha : AffineIndependent k fun x : { x : Fin 3 // x \u2260 2 } => ![p\u2081, p\u2082, p\u2083] x :=\n by\n rw [\u2190 affineIndependent_equiv (finSuccAboveEquiv (2 : Fin 3)).toEquiv]\n convert affineIndependent_of_ne k hp\u2081p\u2082\n ext x\n fin_cases x <;> rfl\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\n\u22a2 AffineIndependent k fun x => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191x\n[PROOFSTEP]\nrw [\u2190 affineIndependent_equiv (finSuccAboveEquiv (2 : Fin 3)).toEquiv]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\n\u22a2 AffineIndependent k ((fun x => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191x) \u2218 \u2191(finSuccAboveEquiv 2).toEquiv)\n[PROOFSTEP]\nconvert affineIndependent_of_ne k hp\u2081p\u2082\n[GOAL]\ncase h.e'_9\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\n\u22a2 (fun x => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191x) \u2218 \u2191(finSuccAboveEquiv 2).toEquiv = ![p\u2081, p\u2082]\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_9.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\nx : Fin 2\n\u22a2 ((fun x => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191x) \u2218 \u2191(finSuccAboveEquiv 2).toEquiv) x = Matrix.vecCons p\u2081 ![p\u2082] x\n[PROOFSTEP]\nfin_cases x\n[GOAL]\ncase h.e'_9.h.head\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\n\u22a2 ((fun x => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191x) \u2218 \u2191(finSuccAboveEquiv 2).toEquiv) { val := 0, isLt := (_ : 0 < 2) } =\n Matrix.vecCons p\u2081 ![p\u2082] { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_9.h.tail.head\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\n\u22a2 ((fun x => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191x) \u2218 \u2191(finSuccAboveEquiv 2).toEquiv)\n { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n Matrix.vecCons p\u2081 ![p\u2082] { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nrfl\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\nha : AffineIndependent k fun x => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191x\n\u22a2 AffineIndependent k ![p\u2081, p\u2082, p\u2083]\n[PROOFSTEP]\nrefine' ha.affineIndependent_of_not_mem_span _\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\nha : AffineIndependent k fun x => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191x\n\u22a2 \u00acMatrix.vecCons p\u2081 ![p\u2082, p\u2083] 2 \u2208 affineSpan k (![p\u2081, p\u2082, p\u2083] '' {x | x \u2260 2})\n[PROOFSTEP]\nintro h\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\nha : AffineIndependent k fun x => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191x\nh : Matrix.vecCons p\u2081 ![p\u2082, p\u2083] 2 \u2208 affineSpan k (![p\u2081, p\u2082, p\u2083] '' {x | x \u2260 2})\n\u22a2 False\n[PROOFSTEP]\nrefine' hp\u2083 ((AffineSubspace.le_def' _ s).1 _ p\u2083 h)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\nha : AffineIndependent k fun x => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191x\nh : Matrix.vecCons p\u2081 ![p\u2082, p\u2083] 2 \u2208 affineSpan k (![p\u2081, p\u2082, p\u2083] '' {x | x \u2260 2})\n\u22a2 affineSpan k (![p\u2081, p\u2082, p\u2083] '' {x | x \u2260 2}) \u2264 s\n[PROOFSTEP]\nsimp_rw [affineSpan_le, Set.image_subset_iff, Set.subset_def, Set.mem_preimage]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\nha : AffineIndependent k fun x => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191x\nh : Matrix.vecCons p\u2081 ![p\u2082, p\u2083] 2 \u2208 affineSpan k (![p\u2081, p\u2082, p\u2083] '' {x | x \u2260 2})\n\u22a2 \u2200 (x : Fin 3), x \u2208 {x | x \u2260 2} \u2192 Matrix.vecCons p\u2081 ![p\u2082, p\u2083] x \u2208 \u2191s\n[PROOFSTEP]\nintro x\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\nha : AffineIndependent k fun x => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191x\nh : Matrix.vecCons p\u2081 ![p\u2082, p\u2083] 2 \u2208 affineSpan k (![p\u2081, p\u2082, p\u2083] '' {x | x \u2260 2})\nx : Fin 3\n\u22a2 x \u2208 {x | x \u2260 2} \u2192 Matrix.vecCons p\u2081 ![p\u2082, p\u2083] x \u2208 \u2191s\n[PROOFSTEP]\nfin_cases x\n[GOAL]\ncase head\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\nha : AffineIndependent k fun x => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191x\nh : Matrix.vecCons p\u2081 ![p\u2082, p\u2083] 2 \u2208 affineSpan k (![p\u2081, p\u2082, p\u2083] '' {x | x \u2260 2})\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } \u2208 {x | x \u2260 2} \u2192 Matrix.vecCons p\u2081 ![p\u2082, p\u2083] { val := 0, isLt := (_ : 0 < 3) } \u2208 \u2191s\n[PROOFSTEP]\nsimp [hp\u2081, hp\u2082]\n[GOAL]\ncase tail.head\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\nha : AffineIndependent k fun x => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191x\nh : Matrix.vecCons p\u2081 ![p\u2082, p\u2083] 2 \u2208 affineSpan k (![p\u2081, p\u2082, p\u2083] '' {x | x \u2260 2})\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2208 {x | x \u2260 2} \u2192\n Matrix.vecCons p\u2081 ![p\u2082, p\u2083] { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2208 \u2191s\n[PROOFSTEP]\nsimp [hp\u2081, hp\u2082]\n[GOAL]\ncase tail.tail.head\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : \u00acp\u2083 \u2208 s\nha : AffineIndependent k fun x => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191x\nh : Matrix.vecCons p\u2081 ![p\u2082, p\u2083] 2 \u2208 affineSpan k (![p\u2081, p\u2082, p\u2083] '' {x | x \u2260 2})\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2208 {x | x \u2260 2} \u2192\n Matrix.vecCons p\u2081 ![p\u2082, p\u2083] { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2208 \u2191s\n[PROOFSTEP]\nsimp [hp\u2081, hp\u2082]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : \u00acp\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\n\u22a2 AffineIndependent k ![p\u2081, p\u2082, p\u2083]\n[PROOFSTEP]\nrw [\u2190 affineIndependent_equiv (Equiv.swap (1 : Fin 3) 2)]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : \u00acp\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\n\u22a2 AffineIndependent k (![p\u2081, p\u2082, p\u2083] \u2218 \u2191(Equiv.swap 1 2))\n[PROOFSTEP]\nconvert affineIndependent_of_ne_of_mem_of_mem_of_not_mem hp\u2081p\u2083 hp\u2081 hp\u2083 hp\u2082 using 1\n[GOAL]\ncase h.e'_9\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : \u00acp\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\n\u22a2 ![p\u2081, p\u2082, p\u2083] \u2218 \u2191(Equiv.swap 1 2) = ![p\u2081, p\u2083, p\u2082]\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_9.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : \u00acp\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\nx : Fin 3\n\u22a2 (![p\u2081, p\u2082, p\u2083] \u2218 \u2191(Equiv.swap 1 2)) x = Matrix.vecCons p\u2081 ![p\u2083, p\u2082] x\n[PROOFSTEP]\nfin_cases x\n[GOAL]\ncase h.e'_9.h.head\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : \u00acp\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\n\u22a2 (![p\u2081, p\u2082, p\u2083] \u2218 \u2191(Equiv.swap 1 2)) { val := 0, isLt := (_ : 0 < 3) } =\n Matrix.vecCons p\u2081 ![p\u2083, p\u2082] { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_9.h.tail.head\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : \u00acp\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\n\u22a2 (![p\u2081, p\u2082, p\u2083] \u2218 \u2191(Equiv.swap 1 2)) { val := 1, isLt := (_ : (fun a => a < 3) 1) } =\n Matrix.vecCons p\u2081 ![p\u2083, p\u2082] { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_9.h.tail.tail.head\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : \u00acp\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\n\u22a2 (![p\u2081, p\u2082, p\u2083] \u2218 \u2191(Equiv.swap 1 2)) { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n Matrix.vecCons p\u2081 ![p\u2083, p\u2082] { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nrfl\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhp\u2081 : \u00acp\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\n\u22a2 AffineIndependent k ![p\u2081, p\u2082, p\u2083]\n[PROOFSTEP]\nrw [\u2190 affineIndependent_equiv (Equiv.swap (0 : Fin 3) 2)]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhp\u2081 : \u00acp\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\n\u22a2 AffineIndependent k (![p\u2081, p\u2082, p\u2083] \u2218 \u2191(Equiv.swap 0 2))\n[PROOFSTEP]\nconvert affineIndependent_of_ne_of_mem_of_mem_of_not_mem hp\u2082p\u2083.symm hp\u2083 hp\u2082 hp\u2081 using 1\n[GOAL]\ncase h.e'_9\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhp\u2081 : \u00acp\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\n\u22a2 ![p\u2081, p\u2082, p\u2083] \u2218 \u2191(Equiv.swap 0 2) = ![p\u2083, p\u2082, p\u2081]\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_9.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhp\u2081 : \u00acp\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\nx : Fin 3\n\u22a2 (![p\u2081, p\u2082, p\u2083] \u2218 \u2191(Equiv.swap 0 2)) x = Matrix.vecCons p\u2083 ![p\u2082, p\u2081] x\n[PROOFSTEP]\nfin_cases x\n[GOAL]\ncase h.e'_9.h.head\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhp\u2081 : \u00acp\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\n\u22a2 (![p\u2081, p\u2082, p\u2083] \u2218 \u2191(Equiv.swap 0 2)) { val := 0, isLt := (_ : 0 < 3) } =\n Matrix.vecCons p\u2083 ![p\u2082, p\u2081] { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_9.h.tail.head\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhp\u2081 : \u00acp\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\n\u22a2 (![p\u2081, p\u2082, p\u2083] \u2218 \u2191(Equiv.swap 0 2)) { val := 1, isLt := (_ : (fun a => a < 3) 1) } =\n Matrix.vecCons p\u2083 ![p\u2082, p\u2081] { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_9.h.tail.tail.head\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : AffineSubspace k P\np\u2081 p\u2082 p\u2083 : P\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhp\u2081 : \u00acp\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\n\u22a2 (![p\u2081, p\u2082, p\u2083] \u2218 \u2191(Equiv.swap 0 2)) { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n Matrix.vecCons p\u2083 ![p\u2082, p\u2081] { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nrfl\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nhs :\n \u2191(Finset.affineCombination k s p) w \u2208\n affineSpan k {\u2191(Finset.affineCombination k s p) w\u2081, \u2191(Finset.affineCombination k s p) w\u2082}\ni j : \u03b9\nhi : i \u2208 s\nhj : j \u2208 s\nhi0 : w\u2081 i = 0\nhj0 : w\u2081 j = 0\nhij : \u2191SignType.sign (w\u2082 i) = \u2191SignType.sign (w\u2082 j)\n\u22a2 \u2191SignType.sign (w i) = \u2191SignType.sign (w j)\n[PROOFSTEP]\nrw [affineCombination_mem_affineSpan_pair h hw hw\u2081 hw\u2082] at hs \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nhs : \u2203 r, \u2200 (i : \u03b9), i \u2208 s \u2192 w i = r * (w\u2082 i - w\u2081 i) + w\u2081 i\ni j : \u03b9\nhi : i \u2208 s\nhj : j \u2208 s\nhi0 : w\u2081 i = 0\nhj0 : w\u2081 j = 0\nhij : \u2191SignType.sign (w\u2082 i) = \u2191SignType.sign (w\u2082 j)\n\u22a2 \u2191SignType.sign (w i) = \u2191SignType.sign (w j)\n[PROOFSTEP]\nrcases hs with \u27e8r, hr\u27e9\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw w\u2081 w\u2082 : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\ni j : \u03b9\nhi : i \u2208 s\nhj : j \u2208 s\nhi0 : w\u2081 i = 0\nhj0 : w\u2081 j = 0\nhij : \u2191SignType.sign (w\u2082 i) = \u2191SignType.sign (w\u2082 j)\nr : k\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = r * (w\u2082 i - w\u2081 i) + w\u2081 i\n\u22a2 \u2191SignType.sign (w i) = \u2191SignType.sign (w j)\n[PROOFSTEP]\nrw [hr i hi, hr j hj, hi0, hj0, add_zero, add_zero, sub_zero, sub_zero, sign_mul, sign_mul, hij]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\ni\u2081 i\u2082 i\u2083 : \u03b9\nh\u2081 : i\u2081 \u2208 s\nh\u2082 : i\u2082 \u2208 s\nh\u2083 : i\u2083 \u2208 s\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nc : k\nhc0 : 0 < c\nhc1 : c < 1\nhs : \u2191(Finset.affineCombination k s p) w \u2208 affineSpan k {p i\u2081, \u2191(AffineMap.lineMap (p i\u2082) (p i\u2083)) c}\n\u22a2 \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\n[PROOFSTEP]\nclassical\nrw [\u2190 s.affineCombination_affineCombinationSingleWeights k p h\u2081, \u2190\n s.affineCombination_affineCombinationLineMapWeights p h\u2082 h\u2083 c] at hs \nrefine'\n sign_eq_of_affineCombination_mem_affineSpan_pair h hw (s.sum_affineCombinationSingleWeights k h\u2081)\n (s.sum_affineCombinationLineMapWeights h\u2082 h\u2083 c) hs h\u2082 h\u2083\n (Finset.affineCombinationSingleWeights_apply_of_ne k h\u2081\u2082.symm)\n (Finset.affineCombinationSingleWeights_apply_of_ne k h\u2081\u2083.symm) _\nrw [Finset.affineCombinationLineMapWeights_apply_left h\u2082\u2083, Finset.affineCombinationLineMapWeights_apply_right h\u2082\u2083]\nsimp_all only [sub_pos, sign_pos]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\ni\u2081 i\u2082 i\u2083 : \u03b9\nh\u2081 : i\u2081 \u2208 s\nh\u2082 : i\u2082 \u2208 s\nh\u2083 : i\u2083 \u2208 s\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nc : k\nhc0 : 0 < c\nhc1 : c < 1\nhs : \u2191(Finset.affineCombination k s p) w \u2208 affineSpan k {p i\u2081, \u2191(AffineMap.lineMap (p i\u2082) (p i\u2083)) c}\n\u22a2 \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\n[PROOFSTEP]\nrw [\u2190 s.affineCombination_affineCombinationSingleWeights k p h\u2081, \u2190\n s.affineCombination_affineCombinationLineMapWeights p h\u2082 h\u2083 c] at hs \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\ni\u2081 i\u2082 i\u2083 : \u03b9\nh\u2081 : i\u2081 \u2208 s\nh\u2082 : i\u2082 \u2208 s\nh\u2083 : i\u2083 \u2208 s\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nc : k\nhc0 : 0 < c\nhc1 : c < 1\nhs :\n \u2191(Finset.affineCombination k s p) w \u2208\n affineSpan k\n {\u2191(Finset.affineCombination k s p) (Finset.affineCombinationSingleWeights k i\u2081),\n \u2191(Finset.affineCombination k s p) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 c)}\n\u22a2 \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\n[PROOFSTEP]\nrefine'\n sign_eq_of_affineCombination_mem_affineSpan_pair h hw (s.sum_affineCombinationSingleWeights k h\u2081)\n (s.sum_affineCombinationLineMapWeights h\u2082 h\u2083 c) hs h\u2082 h\u2083\n (Finset.affineCombinationSingleWeights_apply_of_ne k h\u2081\u2082.symm)\n (Finset.affineCombinationSingleWeights_apply_of_ne k h\u2081\u2083.symm) _\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\ni\u2081 i\u2082 i\u2083 : \u03b9\nh\u2081 : i\u2081 \u2208 s\nh\u2082 : i\u2082 \u2208 s\nh\u2083 : i\u2083 \u2208 s\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nc : k\nhc0 : 0 < c\nhc1 : c < 1\nhs :\n \u2191(Finset.affineCombination k s p) w \u2208\n affineSpan k\n {\u2191(Finset.affineCombination k s p) (Finset.affineCombinationSingleWeights k i\u2081),\n \u2191(Finset.affineCombination k s p) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 c)}\n\u22a2 \u2191SignType.sign (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 c i\u2082) =\n \u2191SignType.sign (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 c i\u2083)\n[PROOFSTEP]\nrw [Finset.affineCombinationLineMapWeights_apply_left h\u2082\u2083, Finset.affineCombinationLineMapWeights_apply_right h\u2082\u2083]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\nh : AffineIndependent k p\nw : \u03b9 \u2192 k\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\ni\u2081 i\u2082 i\u2083 : \u03b9\nh\u2081 : i\u2081 \u2208 s\nh\u2082 : i\u2082 \u2208 s\nh\u2083 : i\u2083 \u2208 s\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nc : k\nhc0 : 0 < c\nhc1 : c < 1\nhs :\n \u2191(Finset.affineCombination k s p) w \u2208\n affineSpan k\n {\u2191(Finset.affineCombination k s p) (Finset.affineCombinationSingleWeights k i\u2081),\n \u2191(Finset.affineCombination k s p) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 c)}\n\u22a2 \u2191SignType.sign (1 - c) = \u2191SignType.sign c\n[PROOFSTEP]\nsimp_all only [sub_pos, sign_pos]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np : P\n\u22a2 Subsingleton (Fin (1 + 0))\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np : P\n\u22a2 Subsingleton (Fin 1)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nn : \u2115\ns1 s2 : Simplex k P n\nh : \u2200 (i : Fin (n + 1)), points s1 i = points s2 i\n\u22a2 s1 = s2\n[PROOFSTEP]\ncases s1\n[GOAL]\ncase mk\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nn : \u2115\ns2 : Simplex k P n\npoints\u271d : Fin (n + 1) \u2192 P\nIndependent\u271d : AffineIndependent k points\u271d\nh : \u2200 (i : Fin (n + 1)), points { points := points\u271d, Independent := Independent\u271d } i = points s2 i\n\u22a2 { points := points\u271d, Independent := Independent\u271d } = s2\n[PROOFSTEP]\ncases s2\n[GOAL]\ncase mk.mk\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nn : \u2115\npoints\u271d\u00b9 : Fin (n + 1) \u2192 P\nIndependent\u271d\u00b9 : AffineIndependent k points\u271d\u00b9\npoints\u271d : Fin (n + 1) \u2192 P\nIndependent\u271d : AffineIndependent k points\u271d\nh :\n \u2200 (i : Fin (n + 1)),\n points { points := points\u271d\u00b9, Independent := Independent\u271d\u00b9 } i =\n points { points := points\u271d, Independent := Independent\u271d } i\n\u22a2 { points := points\u271d\u00b9, Independent := Independent\u271d\u00b9 } = { points := points\u271d, Independent := Independent\u271d }\n[PROOFSTEP]\ncongr with i\n[GOAL]\ncase mk.mk.e_points.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nn : \u2115\npoints\u271d\u00b9 : Fin (n + 1) \u2192 P\nIndependent\u271d\u00b9 : AffineIndependent k points\u271d\u00b9\npoints\u271d : Fin (n + 1) \u2192 P\nIndependent\u271d : AffineIndependent k points\u271d\nh :\n \u2200 (i : Fin (n + 1)),\n points { points := points\u271d\u00b9, Independent := Independent\u271d\u00b9 } i =\n points { points := points\u271d, Independent := Independent\u271d } i\ni : Fin (n + 1)\n\u22a2 points\u271d\u00b9 i = points\u271d i\n[PROOFSTEP]\nexact h i\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nn : \u2115\ns : Simplex k P n\ni : Fin (n + 1)\n\u22a2 face s (_ : Finset.card {i} = 1) = mkOfPoint k (points s i)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nn : \u2115\ns : Simplex k P n\ni : Fin (n + 1)\ni\u271d : Fin (0 + 1)\n\u22a2 points (face s (_ : Finset.card {i} = 1)) i\u271d = points (mkOfPoint k (points s i)) i\u271d\n[PROOFSTEP]\nsimp only [Affine.Simplex.mkOfPoint_points, Affine.Simplex.face_points]\n -- Porting note: `simp` can't use the next lemma\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nn : \u2115\ns : Simplex k P n\ni : Fin (n + 1)\ni\u271d : Fin (0 + 1)\n\u22a2 points s (\u2191(Finset.orderEmbOfFin {i} (_ : Finset.card {i} = 1)) i\u271d) = points s i\n[PROOFSTEP]\nrw [Finset.orderEmbOfFin_singleton]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nn : \u2115\ns : Simplex k P n\nfs : Finset (Fin (n + 1))\nm : \u2115\nh : Finset.card fs = m + 1\n\u22a2 Set.range (face s h).points = s.points '' \u2191fs\n[PROOFSTEP]\nrw [face_points', Set.range_comp, Finset.range_orderEmbOfFin]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nm n : \u2115\ns : Simplex k P m\ne : Fin (m + 1) \u2243 Fin (n + 1)\n\u22a2 reindex (reindex s e) e.symm = s\n[PROOFSTEP]\nrw [\u2190 reindex_trans, Equiv.self_trans_symm, reindex_refl]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nm n : \u2115\ns : Simplex k P m\ne : Fin (n + 1) \u2243 Fin (m + 1)\n\u22a2 reindex (reindex s e.symm) e = s\n[PROOFSTEP]\nrw [\u2190 reindex_trans, Equiv.symm_trans_self, reindex_refl]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nm n : \u2115\ns : Simplex k P m\ne : Fin (m + 1) \u2243 Fin (n + 1)\n\u22a2 Set.range (reindex s e).points = Set.range s.points\n[PROOFSTEP]\nrw [reindex, Set.range_comp, Equiv.range_eq_univ, Set.image_univ]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nn : \u2115\ns : Simplex k P n\nfs : Finset (Fin (n + 1))\nm : \u2115\nh : Finset.card fs = m + 1\n\u22a2 Finset.centroid k Finset.univ (face s h).points = Finset.centroid k fs s.points\n[PROOFSTEP]\nconvert (Finset.univ.centroid_map k (fs.orderEmbOfFin h).toEmbedding s.points).symm\n[GOAL]\ncase h.e'_3.h.e'_9\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nn : \u2115\ns : Simplex k P n\nfs : Finset (Fin (n + 1))\nm : \u2115\nh : Finset.card fs = m + 1\n\u22a2 fs = Finset.map (Finset.orderEmbOfFin fs h).toEmbedding Finset.univ\n[PROOFSTEP]\nrw [\u2190 Finset.coe_inj, Finset.coe_map, Finset.coe_univ, Set.image_univ]\n[GOAL]\ncase h.e'_3.h.e'_9\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nn : \u2115\ns : Simplex k P n\nfs : Finset (Fin (n + 1))\nm : \u2115\nh : Finset.card fs = m + 1\n\u22a2 \u2191fs = Set.range \u2191(Finset.orderEmbOfFin fs h).toEmbedding\n[PROOFSTEP]\nsimp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\n\u22a2 Finset.centroid k fs\u2081 s.points = Finset.centroid k fs\u2082 s.points \u2194 fs\u2081 = fs\u2082\n[PROOFSTEP]\nrefine' \u27e8fun h => _, @congrArg _ _ fs\u2081 fs\u2082 (fun z => Finset.centroid k z s.points)\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\nh : Finset.centroid k fs\u2081 s.points = Finset.centroid k fs\u2082 s.points\n\u22a2 fs\u2081 = fs\u2082\n[PROOFSTEP]\nrw [Finset.centroid_eq_affineCombination_fintype, Finset.centroid_eq_affineCombination_fintype] at h \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\nh :\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2081) =\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2082)\n\u22a2 fs\u2081 = fs\u2082\n[PROOFSTEP]\nhave ha :=\n (affineIndependent_iff_indicator_eq_of_affineCombination_eq k s.points).1 s.Independent _ _ _ _\n (fs\u2081.sum_centroidWeightsIndicator_eq_one_of_card_eq_add_one k h\u2081)\n (fs\u2082.sum_centroidWeightsIndicator_eq_one_of_card_eq_add_one k h\u2082) h\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\nh :\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2081) =\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2082)\nha :\n (Set.indicator \u2191Finset.univ fun i => Finset.centroidWeightsIndicator k fs\u2081 i) =\n Set.indicator \u2191Finset.univ fun i => Finset.centroidWeightsIndicator k fs\u2082 i\n\u22a2 fs\u2081 = fs\u2082\n[PROOFSTEP]\nsimp_rw [Finset.coe_univ, Set.indicator_univ, Function.funext_iff, Finset.centroidWeightsIndicator_def,\n Finset.centroidWeights, h\u2081, h\u2082] at ha \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\nh :\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2081) =\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2082)\nha :\n \u2200 (a : Fin (n + 1)),\n Set.indicator (\u2191fs\u2081) (const (Fin (n + 1)) (\u2191(m\u2081 + 1))\u207b\u00b9) a =\n Set.indicator (\u2191fs\u2082) (const (Fin (n + 1)) (\u2191(m\u2082 + 1))\u207b\u00b9) a\n\u22a2 fs\u2081 = fs\u2082\n[PROOFSTEP]\next i\n[GOAL]\ncase a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\nh :\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2081) =\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2082)\nha :\n \u2200 (a : Fin (n + 1)),\n Set.indicator (\u2191fs\u2081) (const (Fin (n + 1)) (\u2191(m\u2081 + 1))\u207b\u00b9) a =\n Set.indicator (\u2191fs\u2082) (const (Fin (n + 1)) (\u2191(m\u2082 + 1))\u207b\u00b9) a\ni : Fin (n + 1)\n\u22a2 i \u2208 fs\u2081 \u2194 i \u2208 fs\u2082\n[PROOFSTEP]\nspecialize ha i\n[GOAL]\ncase a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\nh :\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2081) =\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2082)\ni : Fin (n + 1)\nha :\n Set.indicator (\u2191fs\u2081) (const (Fin (n + 1)) (\u2191(m\u2081 + 1))\u207b\u00b9) i =\n Set.indicator (\u2191fs\u2082) (const (Fin (n + 1)) (\u2191(m\u2082 + 1))\u207b\u00b9) i\n\u22a2 i \u2208 fs\u2081 \u2194 i \u2208 fs\u2082\n[PROOFSTEP]\nhave key : \u2200 n : \u2115, (n : k) + 1 \u2260 0 := fun n h => by norm_cast at h \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn\u271d : \u2115\ns : Simplex k P n\u271d\nfs\u2081 fs\u2082 : Finset (Fin (n\u271d + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\nh\u271d :\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2081) =\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2082)\ni : Fin (n\u271d + 1)\nha :\n Set.indicator (\u2191fs\u2081) (const (Fin (n\u271d + 1)) (\u2191(m\u2081 + 1))\u207b\u00b9) i =\n Set.indicator (\u2191fs\u2082) (const (Fin (n\u271d + 1)) (\u2191(m\u2082 + 1))\u207b\u00b9) i\nn : \u2115\nh : \u2191n + 1 = 0\n\u22a2 False\n[PROOFSTEP]\nnorm_cast at h \n[GOAL]\ncase a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\nh :\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2081) =\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2082)\ni : Fin (n + 1)\nha :\n Set.indicator (\u2191fs\u2081) (const (Fin (n + 1)) (\u2191(m\u2081 + 1))\u207b\u00b9) i =\n Set.indicator (\u2191fs\u2082) (const (Fin (n + 1)) (\u2191(m\u2082 + 1))\u207b\u00b9) i\nkey : \u2200 (n : \u2115), \u2191n + 1 \u2260 0\n\u22a2 i \u2208 fs\u2081 \u2194 i \u2208 fs\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\nh :\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2081) =\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2082)\ni : Fin (n + 1)\nha :\n Set.indicator (\u2191fs\u2081) (const (Fin (n + 1)) (\u2191(m\u2081 + 1))\u207b\u00b9) i =\n Set.indicator (\u2191fs\u2082) (const (Fin (n + 1)) (\u2191(m\u2082 + 1))\u207b\u00b9) i\nkey : \u2200 (n : \u2115), \u2191n + 1 \u2260 0\n\u22a2 i \u2208 fs\u2081 \u2192 i \u2208 fs\u2082\n[PROOFSTEP]\nintro hi\n[GOAL]\ncase a.mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\nh :\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2081) =\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2082)\ni : Fin (n + 1)\nha :\n Set.indicator (\u2191fs\u2081) (const (Fin (n + 1)) (\u2191(m\u2081 + 1))\u207b\u00b9) i =\n Set.indicator (\u2191fs\u2082) (const (Fin (n + 1)) (\u2191(m\u2082 + 1))\u207b\u00b9) i\nkey : \u2200 (n : \u2115), \u2191n + 1 \u2260 0\n\u22a2 i \u2208 fs\u2082 \u2192 i \u2208 fs\u2081\n[PROOFSTEP]\nintro hi\n[GOAL]\ncase a.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\nh :\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2081) =\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2082)\ni : Fin (n + 1)\nha :\n Set.indicator (\u2191fs\u2081) (const (Fin (n + 1)) (\u2191(m\u2081 + 1))\u207b\u00b9) i =\n Set.indicator (\u2191fs\u2082) (const (Fin (n + 1)) (\u2191(m\u2082 + 1))\u207b\u00b9) i\nkey : \u2200 (n : \u2115), \u2191n + 1 \u2260 0\nhi : i \u2208 fs\u2081\n\u22a2 i \u2208 fs\u2082\n[PROOFSTEP]\nby_contra hni\n[GOAL]\ncase a.mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\nh :\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2081) =\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2082)\ni : Fin (n + 1)\nha :\n Set.indicator (\u2191fs\u2081) (const (Fin (n + 1)) (\u2191(m\u2081 + 1))\u207b\u00b9) i =\n Set.indicator (\u2191fs\u2082) (const (Fin (n + 1)) (\u2191(m\u2082 + 1))\u207b\u00b9) i\nkey : \u2200 (n : \u2115), \u2191n + 1 \u2260 0\nhi : i \u2208 fs\u2082\n\u22a2 i \u2208 fs\u2081\n[PROOFSTEP]\nby_contra hni\n[GOAL]\ncase a.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\nh :\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2081) =\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2082)\ni : Fin (n + 1)\nha :\n Set.indicator (\u2191fs\u2081) (const (Fin (n + 1)) (\u2191(m\u2081 + 1))\u207b\u00b9) i =\n Set.indicator (\u2191fs\u2082) (const (Fin (n + 1)) (\u2191(m\u2082 + 1))\u207b\u00b9) i\nkey : \u2200 (n : \u2115), \u2191n + 1 \u2260 0\nhi : i \u2208 fs\u2081\nhni : \u00aci \u2208 fs\u2082\n\u22a2 False\n[PROOFSTEP]\nsimp [hni, hi, key] at ha \n[GOAL]\ncase a.mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\nh :\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2081) =\n \u2191(Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeightsIndicator k fs\u2082)\ni : Fin (n + 1)\nha :\n Set.indicator (\u2191fs\u2081) (const (Fin (n + 1)) (\u2191(m\u2081 + 1))\u207b\u00b9) i =\n Set.indicator (\u2191fs\u2082) (const (Fin (n + 1)) (\u2191(m\u2082 + 1))\u207b\u00b9) i\nkey : \u2200 (n : \u2115), \u2191n + 1 \u2260 0\nhi : i \u2208 fs\u2082\nhni : \u00aci \u2208 fs\u2081\n\u22a2 False\n[PROOFSTEP]\nsimpa [hni, hi, key] using ha.symm\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\n\u22a2 Finset.centroid k Finset.univ (face s h\u2081).points = Finset.centroid k Finset.univ (face s h\u2082).points \u2194 fs\u2081 = fs\u2082\n[PROOFSTEP]\nrw [face_centroid_eq_centroid, face_centroid_eq_centroid]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : CharZero k\nn : \u2115\ns : Simplex k P n\nfs\u2081 fs\u2082 : Finset (Fin (n + 1))\nm\u2081 m\u2082 : \u2115\nh\u2081 : Finset.card fs\u2081 = m\u2081 + 1\nh\u2082 : Finset.card fs\u2082 = m\u2082 + 1\n\u22a2 Finset.centroid k fs\u2081 s.points = Finset.centroid k fs\u2082 s.points \u2194 fs\u2081 = fs\u2082\n[PROOFSTEP]\nexact s.centroid_eq_iff h\u2081 h\u2082\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nn : \u2115\ns\u2081 s\u2082 : Simplex k P n\nh : Set.range s\u2081.points = Set.range s\u2082.points\n\u22a2 Finset.centroid k Finset.univ s\u2081.points = Finset.centroid k Finset.univ s\u2082.points\n[PROOFSTEP]\nrw [\u2190 Set.image_univ, \u2190 Set.image_univ, \u2190 Finset.coe_univ] at h \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nn : \u2115\ns\u2081 s\u2082 : Simplex k P n\nh\u271d : s\u2081.points '' Set.univ = s\u2082.points '' Set.univ\nh : s\u2081.points '' \u2191Finset.univ = s\u2082.points '' \u2191Finset.univ\n\u22a2 Finset.centroid k Finset.univ s\u2081.points = Finset.centroid k Finset.univ s\u2082.points\n[PROOFSTEP]\nexact\n Finset.univ.centroid_eq_of_inj_on_of_image_eq k _ (fun _ _ _ _ he => AffineIndependent.injective s\u2081.Independent he)\n (fun _ _ _ _ he => AffineIndependent.injective s\u2082.Independent he) h\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.AffineSpace.Independent", "llama_tokens": 115958, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7772998611746912, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.44294638683425397}} {"text": "[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d\u2076 : LinearOrderedCommMonoidWithZero \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Zero \u03b2\ninst\u271d\u2074 : One \u03b2\ninst\u271d\u00b3 : Mul \u03b2\ninst\u271d\u00b2 : Pow \u03b2 \u2115\ninst\u271d\u00b9 : Sup \u03b2\ninst\u271d : Inf \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Injective f\nzero : f 0 = 0\none : f 1 = 1\nmul : \u2200 (x y : \u03b2), f (x * y) = f x * f y\nnpow : \u2200 (x : \u03b2) (n : \u2115), f (x ^ n) = f x ^ n\nhsup : \u2200 (x y : \u03b2), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b2), f (x \u2293 y) = min (f x) (f y)\nsrc\u271d\u00b2 : LinearOrder \u03b2 := LinearOrder.lift f hf hsup hinf\nsrc\u271d\u00b9 : OrderedCommMonoid \u03b2 := orderedCommMonoid f hf one mul npow\nsrc\u271d : CommMonoidWithZero \u03b2 := Injective.commMonoidWithZero f hf zero one mul npow\n\u22a2 f 0 \u2264 f 1\n[PROOFSTEP]\nsimp only [zero, one, LinearOrderedCommMonoidWithZero.zero_le_one]\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommMonoidWithZero \u03b1\n\u22a2 0 \u2264 a\n[PROOFSTEP]\nsimpa only [mul_zero, mul_one] using mul_le_mul_left' zero_le_one a\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nh : c \u2260 0\nhab : a * c \u2264 b * c\n\u22a2 a \u2264 b\n[PROOFSTEP]\nsimpa only [mul_inv_cancel_right\u2080 h] using mul_le_mul_right' hab c\u207b\u00b9\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nh : c \u2260 0\nhab : a * c \u2264 b\n\u22a2 a * c \u2264 b * c\u207b\u00b9 * c\n[PROOFSTEP]\nsimpa [h] using hab\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nhab : a \u2264 b * c\n\u22a2 a * c\u207b\u00b9 \u2264 b\n[PROOFSTEP]\nby_cases h : c = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nhab : a \u2264 b * c\nh : c = 0\n\u22a2 a * c\u207b\u00b9 \u2264 b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nhab : a \u2264 b * c\nh : \u00acc = 0\n\u22a2 a * c\u207b\u00b9 \u2264 b\n[PROOFSTEP]\nexact le_of_le_mul_right h (by simpa [h] using hab)\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nhab : a \u2264 b * c\nh : \u00acc = 0\n\u22a2 a * c\u207b\u00b9 * c \u2264 b * c\n[PROOFSTEP]\nsimpa [h] using hab\n[GOAL]\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d d\u271d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\na b c d : \u03b1\nhb : b \u2260 0\nhd : d \u2260 0\n\u22a2 a * b\u207b\u00b9 \u2264 c * d\u207b\u00b9 \u2194 a * d \u2264 c * b\n[PROOFSTEP]\nrw [mul_inv_le_iff\u2080 hb, mul_right_comm, le_mul_inv_iff\u2080 hd]\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nhab : a \u2264 b\nhb : b \u2260 0\nhcd : c < d\nhd : d \u2260 0\nha : a = 0\n\u22a2 a * c < b * d\n[PROOFSTEP]\nrw [ha, zero_mul, zero_lt_iff]\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nhab : a \u2264 b\nhb : b \u2260 0\nhcd : c < d\nhd : d \u2260 0\nha : a = 0\n\u22a2 b * d \u2260 0\n[PROOFSTEP]\nexact mul_ne_zero hb hd\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nhab : a \u2264 b\nhb : b \u2260 0\nhcd : c < d\nhd : d \u2260 0\nha : \u00aca = 0\nhc : c = 0\n\u22a2 a * c < b * d\n[PROOFSTEP]\nrw [hc, mul_zero, zero_lt_iff]\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nhab : a \u2264 b\nhb : b \u2260 0\nhcd : c < d\nhd : d \u2260 0\nha : \u00aca = 0\nhc : c = 0\n\u22a2 b * d \u2260 0\n[PROOFSTEP]\nexact mul_ne_zero hb hd\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nh : x < y * z\n\u22a2 x * z\u207b\u00b9 < y\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nh : y \u2264 x * z\u207b\u00b9\n\u22a2 y * z \u2264 x\n[PROOFSTEP]\nsimpa only [inv_inv] using mul_inv_le_of_le_mul h\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nh : x < y * z\n\u22a2 y\u207b\u00b9 * x < z\n[PROOFSTEP]\nrw [mul_comm] at *\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nh : x < z * y\n\u22a2 x * y\u207b\u00b9 < z\n[PROOFSTEP]\nexact mul_inv_lt_of_lt_mul\u2080 h\n[GOAL]\n\u03b1 : Type u_1\na b c\u271d d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nc : \u03b1\nh : a < b\nhc : c \u2260 0\n\u22a2 a * c < b * c\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1 : Type u_1\na b c\u271d d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nc : \u03b1\nhc : c \u2260 0\nh : b * c \u2264 a * c\n\u22a2 b \u2264 a\n[PROOFSTEP]\nexact le_of_le_mul_right hc h\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nh : a * b < c * d\nhc : 0 < c\nhh : c \u2264 a\n\u22a2 b < d\n[PROOFSTEP]\nhave ha : a \u2260 0 := ne_of_gt (lt_of_lt_of_le hc hh)\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nh : a * b < c * d\nhc : 0 < c\nhh : c \u2264 a\nha : a \u2260 0\n\u22a2 b < d\n[PROOFSTEP]\nsimp_rw [\u2190 inv_le_inv\u2080 ha (ne_of_gt hc)] at hh \n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nh : a * b < c * d\nhc : 0 < c\nha : a \u2260 0\nhh : a\u207b\u00b9 \u2264 c\u207b\u00b9\n\u22a2 b < d\n[PROOFSTEP]\nhave := mul_lt_mul_of_lt_of_le\u2080 hh (inv_ne_zero (ne_of_gt hc)) h\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nh : a * b < c * d\nhc : 0 < c\nha : a \u2260 0\nhh : a\u207b\u00b9 \u2264 c\u207b\u00b9\nthis : a\u207b\u00b9 * (a * b) < c\u207b\u00b9 * (c * d)\n\u22a2 b < d\n[PROOFSTEP]\nsimpa [inv_mul_cancel_left\u2080 ha, inv_mul_cancel_left\u2080 (ne_of_gt hc)] using this\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nha : a \u2260 0\n\u22a2 a * b \u2264 a * c \u2194 b \u2264 c\n[PROOFSTEP]\nsimp only [mul_comm a]\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nha : a \u2260 0\n\u22a2 b * a \u2264 c * a \u2194 b \u2264 c\n[PROOFSTEP]\nexact mul_le_mul_right\u2080 ha\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nhc : c \u2260 0\n\u22a2 a / c \u2264 b / c \u2194 a \u2264 b\n[PROOFSTEP]\nrw [div_eq_mul_inv, div_eq_mul_inv, mul_le_mul_right\u2080 (inv_ne_zero hc)]\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhc : c \u2260 0\n\u22a2 a / b \u2264 a / c \u2194 c \u2264 b\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, mul_le_mul_left\u2080 ha, inv_le_inv\u2080 hb hc]\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nhc : c \u2260 0\n\u22a2 a \u2264 b / c \u2194 a * c \u2264 b\n[PROOFSTEP]\nrw [div_eq_mul_inv, le_mul_inv_iff\u2080 hc]\n[GOAL]\n\u03b1 : Type u_1\na b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\nhc : c \u2260 0\n\u22a2 a / c \u2264 b \u2194 a \u2264 b * c\n[PROOFSTEP]\nrw [div_eq_mul_inv, mul_inv_le_iff\u2080 hc]\n[GOAL]\n\u03b1 : Type u_1\na\u271d b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\na : \u03b1\nha : a \u2260 0\n\u22a2 symm (mulLeft\u2080' ha) = mulLeft\u2080' (_ : a\u207b\u00b9 \u2260 0)\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\na\u271d b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\na : \u03b1\nha : a \u2260 0\nx\u271d : \u03b1\n\u22a2 \u2191(symm (mulLeft\u2080' ha)) x\u271d = \u2191(mulLeft\u2080' (_ : a\u207b\u00b9 \u2260 0)) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\na\u271d b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\na : \u03b1\nha : a \u2260 0\n\u22a2 symm (mulRight\u2080' ha) = mulRight\u2080' (_ : a\u207b\u00b9 \u2260 0)\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\na\u271d b c d x y z : \u03b1\ninst\u271d : LinearOrderedCommGroupWithZero \u03b1\na : \u03b1\nha : a \u2260 0\nx\u271d : \u03b1\n\u22a2 \u2191(symm (mulRight\u2080' ha)) x\u271d = \u2191(mulRight\u2080' (_ : a\u207b\u00b9 \u2260 0)) x\u271d\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.WithZero", "llama_tokens": 3428, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.6334102498375401, "lm_q1q2_score": 0.4429149175428327}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nx y : ULift \u03b1\ninst\u271d : MulOneClass \u03b1\n\u22a2 \u2200 (x y : ULift \u03b1), \u2191Equiv.ulift (x * y) = \u2191Equiv.ulift x * \u2191Equiv.ulift y\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nx y : ULift \u03b1\ninst\u271d : MulOneClass \u03b1\nx\u271d y\u271d : ULift \u03b1\n\u22a2 \u2191Equiv.ulift (x\u271d * y\u271d) = \u2191Equiv.ulift x\u271d * \u2191Equiv.ulift y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nx y : ULift \u03b1\ninst\u271d : MulZeroOneClass \u03b1\n\u22a2 \u2200 (a b : ULift \u03b1), \u2191Equiv.ulift (a * b) = \u2191Equiv.ulift a * \u2191Equiv.ulift b\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nx y : ULift \u03b1\ninst\u271d : MulZeroOneClass \u03b1\na\u271d b\u271d : ULift \u03b1\n\u22a2 \u2191Equiv.ulift (a\u271d * b\u271d) = \u2191Equiv.ulift a\u271d * \u2191Equiv.ulift b\u271d\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Group.ULift", "llama_tokens": 384, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.44285203088788716}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d s t u : Finset \u03b1\n\u22a2 u \u2208 (fun s t => filter ((fun x x_1 => x \u2286 x_1) s) (powerset t)) s t \u2194 s \u2264 u \u2227 u \u2264 t\n[PROOFSTEP]\nrw [mem_filter, mem_powerset]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d s t u : Finset \u03b1\n\u22a2 u \u2286 t \u2227 (fun x x_1 => x \u2286 x_1) s u \u2194 s \u2264 u \u2227 u \u2264 t\n[PROOFSTEP]\nexact and_comm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d s t u : Finset \u03b1\n\u22a2 u \u2208 (fun s t => filter ((fun x x_1 => x \u2286 x_1) s) (ssubsets t)) s t \u2194 s \u2264 u \u2227 u < t\n[PROOFSTEP]\nrw [mem_filter, mem_ssubsets]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d s t u : Finset \u03b1\n\u22a2 u \u2282 t \u2227 (fun x x_1 => x \u2286 x_1) s u \u2194 s \u2264 u \u2227 u < t\n[PROOFSTEP]\nexact and_comm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d s t u : Finset \u03b1\n\u22a2 u \u2208 (fun s t => filter ((fun x x_1 => x \u2282 x_1) s) (powerset t)) s t \u2194 s < u \u2227 u \u2264 t\n[PROOFSTEP]\nrw [mem_filter, mem_powerset]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d s t u : Finset \u03b1\n\u22a2 u \u2286 t \u2227 (fun x x_1 => x \u2282 x_1) s u \u2194 s < u \u2227 u \u2264 t\n[PROOFSTEP]\nexact and_comm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d s t u : Finset \u03b1\n\u22a2 u \u2208 (fun s t => filter ((fun x x_1 => x \u2282 x_1) s) (ssubsets t)) s t \u2194 s < u \u2227 u < t\n[PROOFSTEP]\nrw [mem_filter, mem_ssubsets]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d s t u : Finset \u03b1\n\u22a2 u \u2282 t \u2227 (fun x x_1 => x \u2282 x_1) s u \u2194 s < u \u2227 u < t\n[PROOFSTEP]\nexact and_comm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\n\u22a2 Icc s t = image ((fun x x_1 => x \u222a x_1) s) (powerset (t \\ s))\n[PROOFSTEP]\next u\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 u \u2208 Icc s t \u2194 u \u2208 image ((fun x x_1 => x \u222a x_1) s) (powerset (t \\ s))\n[PROOFSTEP]\nsimp_rw [mem_Icc, mem_image, mem_powerset]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 s \u2264 u \u2227 u \u2264 t \u2194 \u2203 a, a \u2286 t \\ s \u2227 s \u222a a = u\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mp\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 s \u2264 u \u2227 u \u2264 t \u2192 \u2203 a, a \u2286 t \\ s \u2227 s \u222a a = u\n[PROOFSTEP]\nrintro \u27e8hs, ht\u27e9\n[GOAL]\ncase a.mp.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\nhs : s \u2264 u\nht : u \u2264 t\n\u22a2 \u2203 a, a \u2286 t \\ s \u2227 s \u222a a = u\n[PROOFSTEP]\nexact \u27e8u \\ s, sdiff_le_sdiff_right ht, sup_sdiff_cancel_right hs\u27e9\n[GOAL]\ncase a.mpr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 (\u2203 a, a \u2286 t \\ s \u2227 s \u222a a = u) \u2192 s \u2264 u \u2227 u \u2264 t\n[PROOFSTEP]\nrintro \u27e8v, hv, rfl\u27e9\n[GOAL]\ncase a.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nv : Finset \u03b1\nhv : v \u2286 t \\ s\n\u22a2 s \u2264 s \u222a v \u2227 s \u222a v \u2264 t\n[PROOFSTEP]\nexact \u27e8le_sup_left, union_subset h <| hv.trans <| sdiff_subset _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\n\u22a2 Ico s t = image ((fun x x_1 => x \u222a x_1) s) (ssubsets (t \\ s))\n[PROOFSTEP]\next u\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 u \u2208 Ico s t \u2194 u \u2208 image ((fun x x_1 => x \u222a x_1) s) (ssubsets (t \\ s))\n[PROOFSTEP]\nsimp_rw [mem_Ico, mem_image, mem_ssubsets]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 s \u2264 u \u2227 u < t \u2194 \u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mp\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 s \u2264 u \u2227 u < t \u2192 \u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u\n[PROOFSTEP]\nrintro \u27e8hs, ht\u27e9\n[GOAL]\ncase a.mp.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\nhs : s \u2264 u\nht : u < t\n\u22a2 \u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u\n[PROOFSTEP]\nexact \u27e8u \\ s, sdiff_lt_sdiff_right ht hs, sup_sdiff_cancel_right hs\u27e9\n[GOAL]\ncase a.mpr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\n\u22a2 (\u2203 a, a \u2282 t \\ s \u2227 s \u222a a = u) \u2192 s \u2264 u \u2227 u < t\n[PROOFSTEP]\nrintro \u27e8v, hv, rfl\u27e9\n[GOAL]\ncase a.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nv : Finset \u03b1\nhv : v \u2282 t \\ s\n\u22a2 s \u2264 s \u222a v \u2227 s \u222a v < t\n[PROOFSTEP]\nexact \u27e8le_sup_left, sup_lt_of_lt_sdiff_left hv h\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\n\u22a2 card (Icc s t) = 2 ^ (card t - card s)\n[PROOFSTEP]\nrw [\u2190 card_sdiff h, \u2190 card_powerset, Icc_eq_image_powerset h, Finset.card_image_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\n\u22a2 Set.InjOn ((fun x x_1 => x \u222a x_1) s) \u2191(powerset (t \\ s))\n[PROOFSTEP]\nrintro u hu v hv (huv : s \u2294 u = s \u2294 v)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\nhu : u \u2208 \u2191(powerset (t \\ s))\nv : Finset \u03b1\nhv : v \u2208 \u2191(powerset (t \\ s))\nhuv : s \u2294 u = s \u2294 v\n\u22a2 u = v\n[PROOFSTEP]\nrw [mem_coe, mem_powerset] at hu hv \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\nu : Finset \u03b1\nhu : u \u2286 t \\ s\nv : Finset \u03b1\nhv : v \u2286 t \\ s\nhuv : s \u2294 u = s \u2294 v\n\u22a2 u = v\n[PROOFSTEP]\nrw [\u2190 (disjoint_sdiff.mono_right hu : Disjoint s u).sup_sdiff_cancel_left, \u2190\n (disjoint_sdiff.mono_right hv : Disjoint s v).sup_sdiff_cancel_left, huv]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\n\u22a2 card (Ico s t) = 2 ^ (card t - card s) - 1\n[PROOFSTEP]\nrw [card_Ico_eq_card_Icc_sub_one, card_Icc_finset h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\n\u22a2 card (Ioc s t) = 2 ^ (card t - card s) - 1\n[PROOFSTEP]\nrw [card_Ioc_eq_card_Icc_sub_one, card_Icc_finset h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh : s \u2286 t\n\u22a2 card (Ioo s t) = 2 ^ (card t - card s) - 2\n[PROOFSTEP]\nrw [card_Ioo_eq_card_Icc_sub_two, card_Icc_finset h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 card (Iic s) = 2 ^ card s\n[PROOFSTEP]\nrw [Iic_eq_powerset, card_powerset]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 card (Iio s) = 2 ^ card s - 1\n[PROOFSTEP]\nrw [Iio_eq_ssubsets, ssubsets, card_erase_of_mem (mem_powerset_self _), card_powerset]\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.Interval", "llama_tokens": 3131, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.44285203088788716}} {"text": "[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : SemilatticeSup M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\n\u22a2 weightedTotalDegree' w p = \u22a5 \u2194 p = 0\n[PROOFSTEP]\nsimp only [weightedTotalDegree', Finset.sup_eq_bot_iff, mem_support_iff, WithBot.coe_ne_bot, MvPolynomial.eq_zero_iff]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : SemilatticeSup M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\n\u22a2 (\u2200 (s : \u03c3 \u2192\u2080 \u2115), coeff s p \u2260 0 \u2192 False) \u2194 \u2200 (d : \u03c3 \u2192\u2080 \u2115), coeff d p = 0\n[PROOFSTEP]\nexact forall_congr' fun _ => Classical.not_not\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : SemilatticeSup M\nw : \u03c3 \u2192 M\n\u22a2 weightedTotalDegree' w 0 = \u22a5\n[PROOFSTEP]\nsimp only [weightedTotalDegree', support_zero, Finset.sup_empty]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\nhp : p \u2260 0\n\u22a2 weightedTotalDegree' w p = \u2191(weightedTotalDegree w p)\n[PROOFSTEP]\nrw [Ne.def, \u2190 weightedTotalDegree'_eq_bot_iff w p, \u2190 Ne.def, WithBot.ne_bot_iff_exists] at hp \n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\nhp : \u2203 a, \u2191a = weightedTotalDegree' w p\n\u22a2 weightedTotalDegree' w p = \u2191(weightedTotalDegree w p)\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := hp\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\nm : M\nhm : \u2191m = weightedTotalDegree' w p\n\u22a2 weightedTotalDegree' w p = \u2191(weightedTotalDegree w p)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase intro.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\nm : M\nhm : \u2191m = weightedTotalDegree' w p\n\u22a2 weightedTotalDegree' w p \u2264 \u2191(weightedTotalDegree w p)\n[PROOFSTEP]\nsimp only [weightedTotalDegree, weightedTotalDegree', Finset.sup_le_iff, WithBot.coe_le_coe]\n[GOAL]\ncase intro.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\nm : M\nhm : \u2191m = weightedTotalDegree' w p\n\u22a2 \u2200 (b : \u03c3 \u2192\u2080 \u2115), b \u2208 support p \u2192 \u2191(weightedDegree' w) b \u2264 sup (support p) fun s => \u2191(weightedDegree' w) s\n[PROOFSTEP]\nintro b\n[GOAL]\ncase intro.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\nm : M\nhm : \u2191m = weightedTotalDegree' w p\nb : \u03c3 \u2192\u2080 \u2115\n\u22a2 b \u2208 support p \u2192 \u2191(weightedDegree' w) b \u2264 sup (support p) fun s => \u2191(weightedDegree' w) s\n[PROOFSTEP]\nexact Finset.le_sup\n[GOAL]\ncase intro.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\nm : M\nhm : \u2191m = weightedTotalDegree' w p\n\u22a2 \u2191(weightedTotalDegree w p) \u2264 weightedTotalDegree' w p\n[PROOFSTEP]\nsimp only [weightedTotalDegree]\n[GOAL]\ncase intro.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\nm : M\nhm : \u2191m = weightedTotalDegree' w p\n\u22a2 \u2191(sup (support p) fun s => \u2191(weightedDegree' w) s) \u2264 weightedTotalDegree' w p\n[PROOFSTEP]\nhave hm' : weightedTotalDegree' w p \u2264 m := le_of_eq hm.symm\n[GOAL]\ncase intro.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\nm : M\nhm : \u2191m = weightedTotalDegree' w p\nhm' : weightedTotalDegree' w p \u2264 \u2191m\n\u22a2 \u2191(sup (support p) fun s => \u2191(weightedDegree' w) s) \u2264 weightedTotalDegree' w p\n[PROOFSTEP]\nrw [\u2190 hm]\n[GOAL]\ncase intro.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\nm : M\nhm : \u2191m = weightedTotalDegree' w p\nhm' : weightedTotalDegree' w p \u2264 \u2191m\n\u22a2 \u2191(sup (support p) fun s => \u2191(weightedDegree' w) s) \u2264 \u2191m\n[PROOFSTEP]\nsimpa [weightedTotalDegree'] using hm'\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\n\u22a2 weightedTotalDegree w 0 = \u22a5\n[PROOFSTEP]\nsimp only [weightedTotalDegree, support_zero, Finset.sup_empty]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\na b : MvPolynomial \u03c3 R\nha : a \u2208 {x | IsWeightedHomogeneous w x m}\nhb : b \u2208 {x | IsWeightedHomogeneous w x m}\nc : \u03c3 \u2192\u2080 \u2115\nhc : coeff c (a + b) \u2260 0\n\u22a2 \u2191(weightedDegree' w) c = m\n[PROOFSTEP]\nrw [coeff_add] at hc \n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\na b : MvPolynomial \u03c3 R\nha : a \u2208 {x | IsWeightedHomogeneous w x m}\nhb : b \u2208 {x | IsWeightedHomogeneous w x m}\nc : \u03c3 \u2192\u2080 \u2115\nhc : coeff c a + coeff c b \u2260 0\n\u22a2 \u2191(weightedDegree' w) c = m\n[PROOFSTEP]\nobtain h | h : coeff c a \u2260 0 \u2228 coeff c b \u2260 0 := by\n contrapose! hc\n simp only [hc, add_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\na b : MvPolynomial \u03c3 R\nha : a \u2208 {x | IsWeightedHomogeneous w x m}\nhb : b \u2208 {x | IsWeightedHomogeneous w x m}\nc : \u03c3 \u2192\u2080 \u2115\nhc : coeff c a + coeff c b \u2260 0\n\u22a2 coeff c a \u2260 0 \u2228 coeff c b \u2260 0\n[PROOFSTEP]\ncontrapose! hc\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\na b : MvPolynomial \u03c3 R\nha : a \u2208 {x | IsWeightedHomogeneous w x m}\nhb : b \u2208 {x | IsWeightedHomogeneous w x m}\nc : \u03c3 \u2192\u2080 \u2115\nhc : coeff c a = 0 \u2227 coeff c b = 0\n\u22a2 coeff c a + coeff c b = 0\n[PROOFSTEP]\nsimp only [hc, add_zero]\n[GOAL]\ncase inl\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\na b : MvPolynomial \u03c3 R\nha : a \u2208 {x | IsWeightedHomogeneous w x m}\nhb : b \u2208 {x | IsWeightedHomogeneous w x m}\nc : \u03c3 \u2192\u2080 \u2115\nhc : coeff c a + coeff c b \u2260 0\nh : coeff c a \u2260 0\n\u22a2 \u2191(weightedDegree' w) c = m\n[PROOFSTEP]\nexact ha h\n[GOAL]\ncase inr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\na b : MvPolynomial \u03c3 R\nha : a \u2208 {x | IsWeightedHomogeneous w x m}\nhb : b \u2208 {x | IsWeightedHomogeneous w x m}\nc : \u03c3 \u2192\u2080 \u2115\nhc : coeff c a + coeff c b \u2260 0\nh : coeff c b \u2260 0\n\u22a2 \u2191(weightedDegree' w) c = m\n[PROOFSTEP]\nexact hb h\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\nr : R\na : MvPolynomial \u03c3 R\nha :\n a \u2208\n {\n toAddSubsemigroup :=\n { carrier := {x | IsWeightedHomogeneous w x m},\n add_mem' :=\n (_ :\n \u2200 {a b : MvPolynomial \u03c3 R},\n a \u2208 {x | IsWeightedHomogeneous w x m} \u2192\n b \u2208 {x | IsWeightedHomogeneous w x m} \u2192\n \u2200 (c : \u03c3 \u2192\u2080 \u2115), coeff c (a + b) \u2260 0 \u2192 \u2191(weightedDegree' w) c = m) },\n zero_mem' := (_ : \u2200 (d : \u03c3 \u2192\u2080 \u2115), coeff d 0 \u2260 0 \u2192 \u2191(weightedDegree' w) d = m) }.toAddSubsemigroup.carrier\nc : \u03c3 \u2192\u2080 \u2115\nhc : coeff c (r \u2022 a) \u2260 0\n\u22a2 \u2191(weightedDegree' w) c = m\n[PROOFSTEP]\nrw [coeff_smul] at hc \n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\nr : R\na : MvPolynomial \u03c3 R\nha :\n a \u2208\n {\n toAddSubsemigroup :=\n { carrier := {x | IsWeightedHomogeneous w x m},\n add_mem' :=\n (_ :\n \u2200 {a b : MvPolynomial \u03c3 R},\n a \u2208 {x | IsWeightedHomogeneous w x m} \u2192\n b \u2208 {x | IsWeightedHomogeneous w x m} \u2192\n \u2200 (c : \u03c3 \u2192\u2080 \u2115), coeff c (a + b) \u2260 0 \u2192 \u2191(weightedDegree' w) c = m) },\n zero_mem' := (_ : \u2200 (d : \u03c3 \u2192\u2080 \u2115), coeff d 0 \u2260 0 \u2192 \u2191(weightedDegree' w) d = m) }.toAddSubsemigroup.carrier\nc : \u03c3 \u2192\u2080 \u2115\nhc : r \u2022 coeff c a \u2260 0\n\u22a2 \u2191(weightedDegree' w) c = m\n[PROOFSTEP]\nexact ha (right_ne_zero_of_mul hc)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\n\u22a2 weightedHomogeneousSubmodule R w m = supported R R {d | \u2191(weightedDegree' w) d = m}\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\nx : MvPolynomial \u03c3 R\n\u22a2 x \u2208 weightedHomogeneousSubmodule R w m \u2194 x \u2208 supported R R {d | \u2191(weightedDegree' w) d = m}\n[PROOFSTEP]\nrw [mem_supported, Set.subset_def]\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\nx : MvPolynomial \u03c3 R\n\u22a2 x \u2208 weightedHomogeneousSubmodule R w m \u2194 \u2200 (x_1 : \u03c3 \u2192\u2080 \u2115), x_1 \u2208 \u2191x.support \u2192 x_1 \u2208 {d | \u2191(weightedDegree' w) d = m}\n[PROOFSTEP]\nsimp only [Finsupp.mem_support_iff, mem_coe]\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\nx : MvPolynomial \u03c3 R\n\u22a2 x \u2208 weightedHomogeneousSubmodule R w m \u2194 \u2200 (x_1 : \u03c3 \u2192\u2080 \u2115), \u2191x x_1 \u2260 0 \u2192 x_1 \u2208 {d | \u2191(weightedDegree' w) d = m}\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm n : M\n\u22a2 weightedHomogeneousSubmodule R w m * weightedHomogeneousSubmodule R w n \u2264 weightedHomogeneousSubmodule R w (m + n)\n[PROOFSTEP]\nclassical\nrw [Submodule.mul_le]\nintro \u03c6 h\u03c6 \u03c8 h\u03c8 c hc\nrw [coeff_mul] at hc \nobtain \u27e8\u27e8d, e\u27e9, hde, H\u27e9 := Finset.exists_ne_zero_of_sum_ne_zero hc\nhave aux : coeff d \u03c6 \u2260 0 \u2227 coeff e \u03c8 \u2260 0 := by\n contrapose! H\n by_cases h : coeff d \u03c6 = 0 <;> simp_all only [Ne.def, not_false_iff, zero_mul, mul_zero]\nrw [\u2190 Finsupp.mem_antidiagonal.mp hde, \u2190 h\u03c6 aux.1, \u2190 h\u03c8 aux.2, map_add]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm n : M\n\u22a2 weightedHomogeneousSubmodule R w m * weightedHomogeneousSubmodule R w n \u2264 weightedHomogeneousSubmodule R w (m + n)\n[PROOFSTEP]\nrw [Submodule.mul_le]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm n : M\n\u22a2 \u2200 (m_1 : MvPolynomial \u03c3 R),\n m_1 \u2208 weightedHomogeneousSubmodule R w m \u2192\n \u2200 (n_1 : MvPolynomial \u03c3 R),\n n_1 \u2208 weightedHomogeneousSubmodule R w n \u2192 m_1 * n_1 \u2208 weightedHomogeneousSubmodule R w (m + n)\n[PROOFSTEP]\nintro \u03c6 h\u03c6 \u03c8 h\u03c8 c hc\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm n : M\n\u03c6 : MvPolynomial \u03c3 R\nh\u03c6 : \u03c6 \u2208 weightedHomogeneousSubmodule R w m\n\u03c8 : MvPolynomial \u03c3 R\nh\u03c8 : \u03c8 \u2208 weightedHomogeneousSubmodule R w n\nc : \u03c3 \u2192\u2080 \u2115\nhc : coeff c (\u03c6 * \u03c8) \u2260 0\n\u22a2 \u2191(weightedDegree' w) c = m + n\n[PROOFSTEP]\nrw [coeff_mul] at hc \n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm n : M\n\u03c6 : MvPolynomial \u03c3 R\nh\u03c6 : \u03c6 \u2208 weightedHomogeneousSubmodule R w m\n\u03c8 : MvPolynomial \u03c3 R\nh\u03c8 : \u03c8 \u2208 weightedHomogeneousSubmodule R w n\nc : \u03c3 \u2192\u2080 \u2115\nhc\u271d : coeff c (\u03c6 * \u03c8) \u2260 0\nhc : \u2211 x in antidiagonal c, coeff x.fst \u03c6 * coeff x.snd \u03c8 \u2260 0\n\u22a2 \u2191(weightedDegree' w) c = m + n\n[PROOFSTEP]\nobtain \u27e8\u27e8d, e\u27e9, hde, H\u27e9 := Finset.exists_ne_zero_of_sum_ne_zero hc\n[GOAL]\ncase intro.mk.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm n : M\n\u03c6 : MvPolynomial \u03c3 R\nh\u03c6 : \u03c6 \u2208 weightedHomogeneousSubmodule R w m\n\u03c8 : MvPolynomial \u03c3 R\nh\u03c8 : \u03c8 \u2208 weightedHomogeneousSubmodule R w n\nc : \u03c3 \u2192\u2080 \u2115\nhc\u271d : coeff c (\u03c6 * \u03c8) \u2260 0\nhc : \u2211 x in antidiagonal c, coeff x.fst \u03c6 * coeff x.snd \u03c8 \u2260 0\nd e : \u03c3 \u2192\u2080 \u2115\nhde : (d, e) \u2208 antidiagonal c\nH : coeff (d, e).fst \u03c6 * coeff (d, e).snd \u03c8 \u2260 0\n\u22a2 \u2191(weightedDegree' w) c = m + n\n[PROOFSTEP]\nhave aux : coeff d \u03c6 \u2260 0 \u2227 coeff e \u03c8 \u2260 0 := by\n contrapose! H\n by_cases h : coeff d \u03c6 = 0 <;> simp_all only [Ne.def, not_false_iff, zero_mul, mul_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm n : M\n\u03c6 : MvPolynomial \u03c3 R\nh\u03c6 : \u03c6 \u2208 weightedHomogeneousSubmodule R w m\n\u03c8 : MvPolynomial \u03c3 R\nh\u03c8 : \u03c8 \u2208 weightedHomogeneousSubmodule R w n\nc : \u03c3 \u2192\u2080 \u2115\nhc\u271d : coeff c (\u03c6 * \u03c8) \u2260 0\nhc : \u2211 x in antidiagonal c, coeff x.fst \u03c6 * coeff x.snd \u03c8 \u2260 0\nd e : \u03c3 \u2192\u2080 \u2115\nhde : (d, e) \u2208 antidiagonal c\nH : coeff (d, e).fst \u03c6 * coeff (d, e).snd \u03c8 \u2260 0\n\u22a2 coeff d \u03c6 \u2260 0 \u2227 coeff e \u03c8 \u2260 0\n[PROOFSTEP]\ncontrapose! H\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm n : M\n\u03c6 : MvPolynomial \u03c3 R\nh\u03c6 : \u03c6 \u2208 weightedHomogeneousSubmodule R w m\n\u03c8 : MvPolynomial \u03c3 R\nh\u03c8 : \u03c8 \u2208 weightedHomogeneousSubmodule R w n\nc : \u03c3 \u2192\u2080 \u2115\nhc\u271d : coeff c (\u03c6 * \u03c8) \u2260 0\nhc : \u2211 x in antidiagonal c, coeff x.fst \u03c6 * coeff x.snd \u03c8 \u2260 0\nd e : \u03c3 \u2192\u2080 \u2115\nhde : (d, e) \u2208 antidiagonal c\nH : coeff d \u03c6 \u2260 0 \u2192 coeff e \u03c8 = 0\n\u22a2 coeff d \u03c6 * coeff e \u03c8 = 0\n[PROOFSTEP]\nby_cases h : coeff d \u03c6 = 0\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm n : M\n\u03c6 : MvPolynomial \u03c3 R\nh\u03c6 : \u03c6 \u2208 weightedHomogeneousSubmodule R w m\n\u03c8 : MvPolynomial \u03c3 R\nh\u03c8 : \u03c8 \u2208 weightedHomogeneousSubmodule R w n\nc : \u03c3 \u2192\u2080 \u2115\nhc\u271d : coeff c (\u03c6 * \u03c8) \u2260 0\nhc : \u2211 x in antidiagonal c, coeff x.fst \u03c6 * coeff x.snd \u03c8 \u2260 0\nd e : \u03c3 \u2192\u2080 \u2115\nhde : (d, e) \u2208 antidiagonal c\nH : coeff d \u03c6 \u2260 0 \u2192 coeff e \u03c8 = 0\nh : coeff d \u03c6 = 0\n\u22a2 coeff d \u03c6 * coeff e \u03c8 = 0\n[PROOFSTEP]\nsimp_all only [Ne.def, not_false_iff, zero_mul, mul_zero]\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm n : M\n\u03c6 : MvPolynomial \u03c3 R\nh\u03c6 : \u03c6 \u2208 weightedHomogeneousSubmodule R w m\n\u03c8 : MvPolynomial \u03c3 R\nh\u03c8 : \u03c8 \u2208 weightedHomogeneousSubmodule R w n\nc : \u03c3 \u2192\u2080 \u2115\nhc\u271d : coeff c (\u03c6 * \u03c8) \u2260 0\nhc : \u2211 x in antidiagonal c, coeff x.fst \u03c6 * coeff x.snd \u03c8 \u2260 0\nd e : \u03c3 \u2192\u2080 \u2115\nhde : (d, e) \u2208 antidiagonal c\nH : coeff d \u03c6 \u2260 0 \u2192 coeff e \u03c8 = 0\nh : \u00accoeff d \u03c6 = 0\n\u22a2 coeff d \u03c6 * coeff e \u03c8 = 0\n[PROOFSTEP]\nsimp_all only [Ne.def, not_false_iff, zero_mul, mul_zero]\n[GOAL]\ncase intro.mk.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm n : M\n\u03c6 : MvPolynomial \u03c3 R\nh\u03c6 : \u03c6 \u2208 weightedHomogeneousSubmodule R w m\n\u03c8 : MvPolynomial \u03c3 R\nh\u03c8 : \u03c8 \u2208 weightedHomogeneousSubmodule R w n\nc : \u03c3 \u2192\u2080 \u2115\nhc\u271d : coeff c (\u03c6 * \u03c8) \u2260 0\nhc : \u2211 x in antidiagonal c, coeff x.fst \u03c6 * coeff x.snd \u03c8 \u2260 0\nd e : \u03c3 \u2192\u2080 \u2115\nhde : (d, e) \u2208 antidiagonal c\nH : coeff (d, e).fst \u03c6 * coeff (d, e).snd \u03c8 \u2260 0\naux : coeff d \u03c6 \u2260 0 \u2227 coeff e \u03c8 \u2260 0\n\u22a2 \u2191(weightedDegree' w) c = m + n\n[PROOFSTEP]\nrw [\u2190 Finsupp.mem_antidiagonal.mp hde, \u2190 h\u03c6 aux.1, \u2190 h\u03c8 aux.2, map_add]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : \u2191(weightedDegree' w) d = m\n\u22a2 IsWeightedHomogeneous w (\u2191(monomial d) r) m\n[PROOFSTEP]\nclassical\nintro c hc\nrw [coeff_monomial] at hc \nsplit_ifs at hc with h\n\u00b7 subst c\n exact hm\n\u00b7 contradiction\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : \u2191(weightedDegree' w) d = m\n\u22a2 IsWeightedHomogeneous w (\u2191(monomial d) r) m\n[PROOFSTEP]\nintro c hc\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : \u2191(weightedDegree' w) d = m\nc : \u03c3 \u2192\u2080 \u2115\nhc : coeff c (\u2191(monomial d) r) \u2260 0\n\u22a2 \u2191(weightedDegree' w) c = m\n[PROOFSTEP]\nrw [coeff_monomial] at hc \n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : \u2191(weightedDegree' w) d = m\nc : \u03c3 \u2192\u2080 \u2115\nhc\u271d : coeff c (\u2191(monomial d) r) \u2260 0\nhc : (if d = c then r else 0) \u2260 0\n\u22a2 \u2191(weightedDegree' w) c = m\n[PROOFSTEP]\nsplit_ifs at hc with h\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : \u2191(weightedDegree' w) d = m\nc : \u03c3 \u2192\u2080 \u2115\nhc\u271d : coeff c (\u2191(monomial d) r) \u2260 0\nh : d = c\nhc : r \u2260 0\n\u22a2 \u2191(weightedDegree' w) c = m\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : \u2191(weightedDegree' w) d = m\nhc\u271d : r \u2260 0\nhc : coeff d (\u2191(monomial d) r) \u2260 0\n\u22a2 \u2191(weightedDegree' w) d = m\n[PROOFSTEP]\nexact hm\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : \u2191(weightedDegree' w) d = m\nc : \u03c3 \u2192\u2080 \u2115\nhc\u271d : coeff c (\u2191(monomial d) r) \u2260 0\nh : \u00acd = c\nhc : 0 \u2260 0\n\u22a2 \u2191(weightedDegree' w) c = m\n[PROOFSTEP]\ncontradiction\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\nhp : weightedTotalDegree w p = \u22a5\n\u22a2 IsWeightedHomogeneous w p \u22a5\n[PROOFSTEP]\nintro d hd\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\nhp : weightedTotalDegree w p = \u22a5\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d p \u2260 0\n\u22a2 \u2191(weightedDegree' w) d = \u22a5\n[PROOFSTEP]\nhave h := weightedTotalDegree_coe w p (MvPolynomial.ne_zero_iff.mpr \u27e8d, hd\u27e9)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\nhp : weightedTotalDegree w p = \u22a5\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d p \u2260 0\nh : weightedTotalDegree' w p = \u2191(weightedTotalDegree w p)\n\u22a2 \u2191(weightedDegree' w) d = \u22a5\n[PROOFSTEP]\nsimp only [weightedTotalDegree', hp] at h \n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\nhp : weightedTotalDegree w p = \u22a5\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d p \u2260 0\nh : (sup (support p) fun s => \u2191(\u2191(weightedDegree' w) s)) = \u2191\u22a5\n\u22a2 \u2191(weightedDegree' w) d = \u22a5\n[PROOFSTEP]\nrw [eq_bot_iff, \u2190 WithBot.coe_le_coe, \u2190 h]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nw : \u03c3 \u2192 M\np : MvPolynomial \u03c3 R\nhp : weightedTotalDegree w p = \u22a5\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d p \u2260 0\nh : (sup (support p) fun s => \u2191(\u2191(weightedDegree' w) s)) = \u2191\u22a5\n\u22a2 \u2191(\u2191(weightedDegree' w) d) \u2264 sup (support p) fun s => \u2191(\u2191(weightedDegree' w) s)\n[PROOFSTEP]\napply Finset.le_sup (mem_support_iff.mpr hd)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\ni : \u03c3\n\u22a2 IsWeightedHomogeneous w (X i) (w i)\n[PROOFSTEP]\napply isWeightedHomogeneous_monomial\n[GOAL]\ncase hm\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\ni : \u03c3\n\u22a2 \u2191(weightedDegree' w) (Finsupp.single i 1) = w i\n[PROOFSTEP]\nsimp only [weightedDegree', LinearMap.toAddMonoidHom_coe, total_single, one_nsmul]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : M\nw : \u03c3 \u2192 M\nh\u03c6 : IsWeightedHomogeneous w \u03c6 n\nd : \u03c3 \u2192\u2080 \u2115\nhd : \u2191(weightedDegree' w) d \u2260 n\n\u22a2 coeff d \u03c6 = 0\n[PROOFSTEP]\nhave aux := mt (@h\u03c6 d) hd\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : M\nw : \u03c3 \u2192 M\nh\u03c6 : IsWeightedHomogeneous w \u03c6 n\nd : \u03c3 \u2192\u2080 \u2115\nhd : \u2191(weightedDegree' w) d \u2260 n\naux : \u00accoeff d \u03c6 \u2260 0\n\u22a2 coeff d \u03c6 = 0\n[PROOFSTEP]\nrwa [Classical.not_not] at aux \n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : M\nw : \u03c3 \u2192 M\nh\u03c6 : \u03c6 \u2260 0\nhm : IsWeightedHomogeneous w \u03c6 m\nhn : IsWeightedHomogeneous w \u03c6 n\n\u22a2 m = n\n[PROOFSTEP]\nobtain \u27e8d, hd\u27e9 : \u2203 d, coeff d \u03c6 \u2260 0 := exists_coeff_ne_zero h\u03c6\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : M\nw : \u03c3 \u2192 M\nh\u03c6 : \u03c6 \u2260 0\nhm : IsWeightedHomogeneous w \u03c6 m\nhn : IsWeightedHomogeneous w \u03c6 n\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d \u03c6 \u2260 0\n\u22a2 m = n\n[PROOFSTEP]\nrw [\u2190 hm hd, \u2190 hn hd]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\n\u03c6\u271d \u03c8 : MvPolynomial \u03c3 R\nm n\u271d : M\n\u03b9 : Type u_4\ns : Finset \u03b9\n\u03c6 : \u03b9 \u2192 MvPolynomial \u03c3 R\nn : \u03b9 \u2192 M\nw : \u03c3 \u2192 M\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 IsWeightedHomogeneous w (\u03c6 i) (n i)) \u2192 IsWeightedHomogeneous w (\u220f i in s, \u03c6 i) (\u2211 i in s, n i)\n[PROOFSTEP]\nclassical\nrefine Finset.induction_on s ?_ ?_\n\u00b7 intro\n simp only [isWeightedHomogeneous_one, Finset.sum_empty, Finset.prod_empty]\n\u00b7 intro i s his IH h\n simp only [his, Finset.prod_insert, Finset.sum_insert, not_false_iff]\n apply (h i (Finset.mem_insert_self _ _)).mul (IH _)\n intro j hjs\n exact h j (Finset.mem_insert_of_mem hjs)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\n\u03c6\u271d \u03c8 : MvPolynomial \u03c3 R\nm n\u271d : M\n\u03b9 : Type u_4\ns : Finset \u03b9\n\u03c6 : \u03b9 \u2192 MvPolynomial \u03c3 R\nn : \u03b9 \u2192 M\nw : \u03c3 \u2192 M\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 IsWeightedHomogeneous w (\u03c6 i) (n i)) \u2192 IsWeightedHomogeneous w (\u220f i in s, \u03c6 i) (\u2211 i in s, n i)\n[PROOFSTEP]\nrefine Finset.induction_on s ?_ ?_\n[GOAL]\ncase refine_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\n\u03c6\u271d \u03c8 : MvPolynomial \u03c3 R\nm n\u271d : M\n\u03b9 : Type u_4\ns : Finset \u03b9\n\u03c6 : \u03b9 \u2192 MvPolynomial \u03c3 R\nn : \u03b9 \u2192 M\nw : \u03c3 \u2192 M\n\u22a2 (\u2200 (i : \u03b9), i \u2208 \u2205 \u2192 IsWeightedHomogeneous w (\u03c6 i) (n i)) \u2192 IsWeightedHomogeneous w (\u220f i in \u2205, \u03c6 i) (\u2211 i in \u2205, n i)\n[PROOFSTEP]\nintro\n[GOAL]\ncase refine_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\n\u03c6\u271d \u03c8 : MvPolynomial \u03c3 R\nm n\u271d : M\n\u03b9 : Type u_4\ns : Finset \u03b9\n\u03c6 : \u03b9 \u2192 MvPolynomial \u03c3 R\nn : \u03b9 \u2192 M\nw : \u03c3 \u2192 M\na\u271d : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 IsWeightedHomogeneous w (\u03c6 i) (n i)\n\u22a2 IsWeightedHomogeneous w (\u220f i in \u2205, \u03c6 i) (\u2211 i in \u2205, n i)\n[PROOFSTEP]\nsimp only [isWeightedHomogeneous_one, Finset.sum_empty, Finset.prod_empty]\n[GOAL]\ncase refine_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\n\u03c6\u271d \u03c8 : MvPolynomial \u03c3 R\nm n\u271d : M\n\u03b9 : Type u_4\ns : Finset \u03b9\n\u03c6 : \u03b9 \u2192 MvPolynomial \u03c3 R\nn : \u03b9 \u2192 M\nw : \u03c3 \u2192 M\n\u22a2 \u2200 \u2983a : \u03b9\u2984 {s : Finset \u03b9},\n \u00aca \u2208 s \u2192\n ((\u2200 (i : \u03b9), i \u2208 s \u2192 IsWeightedHomogeneous w (\u03c6 i) (n i)) \u2192\n IsWeightedHomogeneous w (\u220f i in s, \u03c6 i) (\u2211 i in s, n i)) \u2192\n (\u2200 (i : \u03b9), i \u2208 insert a s \u2192 IsWeightedHomogeneous w (\u03c6 i) (n i)) \u2192\n IsWeightedHomogeneous w (\u220f i in insert a s, \u03c6 i) (\u2211 i in insert a s, n i)\n[PROOFSTEP]\nintro i s his IH h\n[GOAL]\ncase refine_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\n\u03c6\u271d \u03c8 : MvPolynomial \u03c3 R\nm n\u271d : M\n\u03b9 : Type u_4\ns\u271d : Finset \u03b9\n\u03c6 : \u03b9 \u2192 MvPolynomial \u03c3 R\nn : \u03b9 \u2192 M\nw : \u03c3 \u2192 M\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nIH : (\u2200 (i : \u03b9), i \u2208 s \u2192 IsWeightedHomogeneous w (\u03c6 i) (n i)) \u2192 IsWeightedHomogeneous w (\u220f i in s, \u03c6 i) (\u2211 i in s, n i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 IsWeightedHomogeneous w (\u03c6 i_1) (n i_1)\n\u22a2 IsWeightedHomogeneous w (\u220f i in insert i s, \u03c6 i) (\u2211 i in insert i s, n i)\n[PROOFSTEP]\nsimp only [his, Finset.prod_insert, Finset.sum_insert, not_false_iff]\n[GOAL]\ncase refine_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\n\u03c6\u271d \u03c8 : MvPolynomial \u03c3 R\nm n\u271d : M\n\u03b9 : Type u_4\ns\u271d : Finset \u03b9\n\u03c6 : \u03b9 \u2192 MvPolynomial \u03c3 R\nn : \u03b9 \u2192 M\nw : \u03c3 \u2192 M\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nIH : (\u2200 (i : \u03b9), i \u2208 s \u2192 IsWeightedHomogeneous w (\u03c6 i) (n i)) \u2192 IsWeightedHomogeneous w (\u220f i in s, \u03c6 i) (\u2211 i in s, n i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 IsWeightedHomogeneous w (\u03c6 i_1) (n i_1)\n\u22a2 IsWeightedHomogeneous w (\u03c6 i * \u220f i in s, \u03c6 i) (n i + \u2211 i in s, n i)\n[PROOFSTEP]\napply (h i (Finset.mem_insert_self _ _)).mul (IH _)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\n\u03c6\u271d \u03c8 : MvPolynomial \u03c3 R\nm n\u271d : M\n\u03b9 : Type u_4\ns\u271d : Finset \u03b9\n\u03c6 : \u03b9 \u2192 MvPolynomial \u03c3 R\nn : \u03b9 \u2192 M\nw : \u03c3 \u2192 M\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nIH : (\u2200 (i : \u03b9), i \u2208 s \u2192 IsWeightedHomogeneous w (\u03c6 i) (n i)) \u2192 IsWeightedHomogeneous w (\u220f i in s, \u03c6 i) (\u2211 i in s, n i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 IsWeightedHomogeneous w (\u03c6 i_1) (n i_1)\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 IsWeightedHomogeneous w (\u03c6 i) (n i)\n[PROOFSTEP]\nintro j hjs\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\n\u03c6\u271d \u03c8 : MvPolynomial \u03c3 R\nm n\u271d : M\n\u03b9 : Type u_4\ns\u271d : Finset \u03b9\n\u03c6 : \u03b9 \u2192 MvPolynomial \u03c3 R\nn : \u03b9 \u2192 M\nw : \u03c3 \u2192 M\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nIH : (\u2200 (i : \u03b9), i \u2208 s \u2192 IsWeightedHomogeneous w (\u03c6 i) (n i)) \u2192 IsWeightedHomogeneous w (\u220f i in s, \u03c6 i) (\u2211 i in s, n i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 IsWeightedHomogeneous w (\u03c6 i_1) (n i_1)\nj : \u03b9\nhjs : j \u2208 s\n\u22a2 IsWeightedHomogeneous w (\u03c6 j) (n j)\n[PROOFSTEP]\nexact h j (Finset.mem_insert_of_mem hjs)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : M\ninst\u271d : SemilatticeSup M\nw : \u03c3 \u2192 M\nh\u03c6 : IsWeightedHomogeneous w \u03c6 n\nh : \u03c6 \u2260 0\n\u22a2 weightedTotalDegree' w \u03c6 = \u2191n\n[PROOFSTEP]\nsimp only [weightedTotalDegree']\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : M\ninst\u271d : SemilatticeSup M\nw : \u03c3 \u2192 M\nh\u03c6 : IsWeightedHomogeneous w \u03c6 n\nh : \u03c6 \u2260 0\n\u22a2 (sup (support \u03c6) fun s => \u2191(\u2191(weightedDegree' w) s)) = \u2191n\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : M\ninst\u271d : SemilatticeSup M\nw : \u03c3 \u2192 M\nh\u03c6 : IsWeightedHomogeneous w \u03c6 n\nh : \u03c6 \u2260 0\n\u22a2 (sup (support \u03c6) fun s => \u2191(\u2191(weightedDegree' w) s)) \u2264 \u2191n\n[PROOFSTEP]\nsimp only [Finset.sup_le_iff, mem_support_iff, WithBot.coe_le_coe]\n[GOAL]\ncase a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : M\ninst\u271d : SemilatticeSup M\nw : \u03c3 \u2192 M\nh\u03c6 : IsWeightedHomogeneous w \u03c6 n\nh : \u03c6 \u2260 0\n\u22a2 \u2200 (b : \u03c3 \u2192\u2080 \u2115), coeff b \u03c6 \u2260 0 \u2192 \u2191(weightedDegree' w) b \u2264 n\n[PROOFSTEP]\nexact fun d hd => le_of_eq (h\u03c6 hd)\n[GOAL]\ncase a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : M\ninst\u271d : SemilatticeSup M\nw : \u03c3 \u2192 M\nh\u03c6 : IsWeightedHomogeneous w \u03c6 n\nh : \u03c6 \u2260 0\n\u22a2 \u2191n \u2264 sup (support \u03c6) fun s => \u2191(\u2191(weightedDegree' w) s)\n[PROOFSTEP]\nobtain \u27e8d, hd\u27e9 : \u2203 d, coeff d \u03c6 \u2260 0 := exists_coeff_ne_zero h\n[GOAL]\ncase a.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : M\ninst\u271d : SemilatticeSup M\nw : \u03c3 \u2192 M\nh\u03c6 : IsWeightedHomogeneous w \u03c6 n\nh : \u03c6 \u2260 0\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d \u03c6 \u2260 0\n\u22a2 \u2191n \u2264 sup (support \u03c6) fun s => \u2191(\u2191(weightedDegree' w) s)\n[PROOFSTEP]\nsimp only [\u2190 h\u03c6 hd, Finsupp.sum]\n[GOAL]\ncase a.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : M\ninst\u271d : SemilatticeSup M\nw : \u03c3 \u2192 M\nh\u03c6 : IsWeightedHomogeneous w \u03c6 n\nh : \u03c6 \u2260 0\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d \u03c6 \u2260 0\n\u22a2 \u2191(\u2191(weightedDegree' w) d) \u2264 sup (support \u03c6) fun s => \u2191(\u2191(weightedDegree' w) s)\n[PROOFSTEP]\nreplace hd := Finsupp.mem_support_iff.mpr hd\n[GOAL]\ncase a.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : M\ninst\u271d : SemilatticeSup M\nw : \u03c3 \u2192 M\nh\u03c6 : IsWeightedHomogeneous w \u03c6 n\nh : \u03c6 \u2260 0\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 \u03c6.support\n\u22a2 \u2191(\u2191(weightedDegree' w) d) \u2264 sup (support \u03c6) fun s => \u2191(\u2191(weightedDegree' w) s)\n[PROOFSTEP]\napply Finset.le_sup hd\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\n\u22a2 IsWeightedHomogeneous w (\u2191(weightedHomogeneousComponent w n) \u03c6) n\n[PROOFSTEP]\nclassical\nintro d hd\ncontrapose! hd\nrw [coeff_weightedHomogeneousComponent, if_neg hd]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\n\u22a2 IsWeightedHomogeneous w (\u2191(weightedHomogeneousComponent w n) \u03c6) n\n[PROOFSTEP]\nintro d hd\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d (\u2191(weightedHomogeneousComponent w n) \u03c6) \u2260 0\n\u22a2 \u2191(weightedDegree' w) d = n\n[PROOFSTEP]\ncontrapose! hd\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhd : \u2191(weightedDegree' w) d \u2260 n\n\u22a2 coeff d (\u2191(weightedHomogeneousComponent w n) \u03c6) = 0\n[PROOFSTEP]\nrw [coeff_weightedHomogeneousComponent, if_neg hd]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nn : M\nr : R\n\u22a2 \u2191(weightedHomogeneousComponent w n) (\u2191C r * \u03c6) = \u2191C r * \u2191(weightedHomogeneousComponent w n) \u03c6\n[PROOFSTEP]\nsimp only [C_mul', LinearMap.map_smul]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nh : \u2200 (d : \u03c3 \u2192\u2080 \u2115), d \u2208 support \u03c6 \u2192 \u2191(weightedDegree' w) d \u2260 n\n\u22a2 \u2191(weightedHomogeneousComponent w n) \u03c6 = 0\n[PROOFSTEP]\nclassical\nrw [weightedHomogeneousComponent_apply, sum_eq_zero]\nintro d hd; rw [mem_filter] at hd \nexfalso; exact h _ hd.1 hd.2\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nh : \u2200 (d : \u03c3 \u2192\u2080 \u2115), d \u2208 support \u03c6 \u2192 \u2191(weightedDegree' w) d \u2260 n\n\u22a2 \u2191(weightedHomogeneousComponent w n) \u03c6 = 0\n[PROOFSTEP]\nrw [weightedHomogeneousComponent_apply, sum_eq_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nh : \u2200 (d : \u03c3 \u2192\u2080 \u2115), d \u2208 support \u03c6 \u2192 \u2191(weightedDegree' w) d \u2260 n\n\u22a2 \u2200 (x : \u03c3 \u2192\u2080 \u2115), x \u2208 Finset.filter (fun d => \u2191(weightedDegree' w) d = n) (support \u03c6) \u2192 \u2191(monomial x) (coeff x \u03c6) = 0\n[PROOFSTEP]\nintro d hd\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nh : \u2200 (d : \u03c3 \u2192\u2080 \u2115), d \u2208 support \u03c6 \u2192 \u2191(weightedDegree' w) d \u2260 n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 Finset.filter (fun d => \u2191(weightedDegree' w) d = n) (support \u03c6)\n\u22a2 \u2191(monomial d) (coeff d \u03c6) = 0\n[PROOFSTEP]\nrw [mem_filter] at hd \n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nh : \u2200 (d : \u03c3 \u2192\u2080 \u2115), d \u2208 support \u03c6 \u2192 \u2191(weightedDegree' w) d \u2260 n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support \u03c6 \u2227 \u2191(weightedDegree' w) d = n\n\u22a2 \u2191(monomial d) (coeff d \u03c6) = 0\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nh : \u2200 (d : \u03c3 \u2192\u2080 \u2115), d \u2208 support \u03c6 \u2192 \u2191(weightedDegree' w) d \u2260 n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support \u03c6 \u2227 \u2191(weightedDegree' w) d = n\n\u22a2 False\n[PROOFSTEP]\nexact h _ hd.1 hd.2\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\n\u22a2 \u2191(weightedHomogeneousComponent w n) \u03c6 = 0\n[PROOFSTEP]\nclassical\nrw [weightedHomogeneousComponent_apply, sum_eq_zero]\nintro d hd\nrw [Finset.mem_filter] at hd \nexfalso\napply lt_irrefl n\nnth_rw 1 [\u2190 hd.2]\nexact lt_of_le_of_lt (le_weightedTotalDegree w hd.1) h\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\n\u22a2 \u2191(weightedHomogeneousComponent w n) \u03c6 = 0\n[PROOFSTEP]\nrw [weightedHomogeneousComponent_apply, sum_eq_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\n\u22a2 \u2200 (x : \u03c3 \u2192\u2080 \u2115), x \u2208 Finset.filter (fun d => \u2191(weightedDegree' w) d = n) (support \u03c6) \u2192 \u2191(monomial x) (coeff x \u03c6) = 0\n[PROOFSTEP]\nintro d hd\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 Finset.filter (fun d => \u2191(weightedDegree' w) d = n) (support \u03c6)\n\u22a2 \u2191(monomial d) (coeff d \u03c6) = 0\n[PROOFSTEP]\nrw [Finset.mem_filter] at hd \n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support \u03c6 \u2227 \u2191(weightedDegree' w) d = n\n\u22a2 \u2191(monomial d) (coeff d \u03c6) = 0\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support \u03c6 \u2227 \u2191(weightedDegree' w) d = n\n\u22a2 False\n[PROOFSTEP]\napply lt_irrefl n\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support \u03c6 \u2227 \u2191(weightedDegree' w) d = n\n\u22a2 n < n\n[PROOFSTEP]\nnth_rw 1 [\u2190 hd.2]\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : SemilatticeSup M\ninst\u271d : OrderBot M\nh : weightedTotalDegree w \u03c6 < n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support \u03c6 \u2227 \u2191(weightedDegree' w) d = n\n\u22a2 \u2191(weightedDegree' w) d < n\n[PROOFSTEP]\nexact lt_of_le_of_lt (le_weightedTotalDegree w hd.1) h\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\n\u22a2 Set.Finite (Function.support fun m => \u2191(weightedHomogeneousComponent w m) \u03c6)\n[PROOFSTEP]\nsuffices (Function.support fun m => weightedHomogeneousComponent w m \u03c6) \u2286 (fun d => weightedDegree' w d) '' \u03c6.support by\n exact Finite.subset ((fun d : \u03c3 \u2192\u2080 \u2115 => (weightedDegree' w) d) '' \u2191(support \u03c6)).toFinite this\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nthis :\n (Function.support fun m => \u2191(weightedHomogeneousComponent w m) \u03c6) \u2286 (fun d => \u2191(weightedDegree' w) d) '' \u2191(support \u03c6)\n\u22a2 Set.Finite (Function.support fun m => \u2191(weightedHomogeneousComponent w m) \u03c6)\n[PROOFSTEP]\nexact Finite.subset ((fun d : \u03c3 \u2192\u2080 \u2115 => (weightedDegree' w) d) '' \u2191(support \u03c6)).toFinite this\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\n\u22a2 (Function.support fun m => \u2191(weightedHomogeneousComponent w m) \u03c6) \u2286 (fun d => \u2191(weightedDegree' w) d) '' \u2191(support \u03c6)\n[PROOFSTEP]\nintro m hm\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm : M\nhm : m \u2208 Function.support fun m => \u2191(weightedHomogeneousComponent w m) \u03c6\n\u22a2 m \u2208 (fun d => \u2191(weightedDegree' w) d) '' \u2191(support \u03c6)\n[PROOFSTEP]\nby_contra hm'\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm : M\nhm : m \u2208 Function.support fun m => \u2191(weightedHomogeneousComponent w m) \u03c6\nhm' : \u00acm \u2208 (fun d => \u2191(weightedDegree' w) d) '' \u2191(support \u03c6)\n\u22a2 False\n[PROOFSTEP]\napply hm\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm : M\nhm : m \u2208 Function.support fun m => \u2191(weightedHomogeneousComponent w m) \u03c6\nhm' : \u00acm \u2208 (fun d => \u2191(weightedDegree' w) d) '' \u2191(support \u03c6)\n\u22a2 (fun m => \u2191(weightedHomogeneousComponent w m) \u03c6) m = 0\n[PROOFSTEP]\nsimp only [mem_support, Ne.def] at hm \n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm : M\nhm' : \u00acm \u2208 (fun d => \u2191(weightedDegree' w) d) '' \u2191(support \u03c6)\nhm : \u00ac\u2191(weightedHomogeneousComponent w m) \u03c6 = 0\n\u22a2 (fun m => \u2191(weightedHomogeneousComponent w m) \u03c6) m = 0\n[PROOFSTEP]\nsimp only [Set.mem_image, not_exists, not_and] at hm' \n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm : M\nhm : \u00ac\u2191(weightedHomogeneousComponent w m) \u03c6 = 0\nhm' : \u2200 (x : \u03c3 \u2192\u2080 \u2115), x \u2208 \u2191(support \u03c6) \u2192 \u00ac\u2191(weightedDegree' w) x = m\n\u22a2 (fun m => \u2191(weightedHomogeneousComponent w m) \u03c6) m = 0\n[PROOFSTEP]\nexact weightedHomogeneousComponent_eq_zero' m \u03c6 hm'\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\n\u22a2 \u2211\u1da0 (m : M), \u2191(weightedHomogeneousComponent w m) \u03c6 = \u03c6\n[PROOFSTEP]\nclassical\nrw [finsum_eq_sum _ (weightedHomogeneousComponent_finsupp \u03c6)]\next1 d\nsimp only [coeff_sum, coeff_weightedHomogeneousComponent]\nrw [Finset.sum_eq_single (weightedDegree' w d)]\n\u00b7 rw [if_pos rfl]\n\u00b7 intro m _ hm'\n rw [if_neg hm'.symm]\n\u00b7 intro hm\n rw [if_pos rfl]\n simp only [Finite.mem_toFinset, mem_support, Ne.def, Classical.not_not] at hm \n have := coeff_weightedHomogeneousComponent (w := w) (weightedDegree' w d) \u03c6 d\n rw [hm, if_pos rfl, coeff_zero] at this \n exact this.symm\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\n\u22a2 \u2211\u1da0 (m : M), \u2191(weightedHomogeneousComponent w m) \u03c6 = \u03c6\n[PROOFSTEP]\nrw [finsum_eq_sum _ (weightedHomogeneousComponent_finsupp \u03c6)]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\n\u22a2 \u2211 i in Finite.toFinset (_ : Set.Finite (Function.support fun m => \u2191(weightedHomogeneousComponent w m) \u03c6)),\n \u2191(weightedHomogeneousComponent w i) \u03c6 =\n \u03c6\n[PROOFSTEP]\next1 d\n[GOAL]\ncase a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff d\n (\u2211 i in Finite.toFinset (_ : Set.Finite (Function.support fun m => \u2191(weightedHomogeneousComponent w m) \u03c6)),\n \u2191(weightedHomogeneousComponent w i) \u03c6) =\n coeff d \u03c6\n[PROOFSTEP]\nsimp only [coeff_sum, coeff_weightedHomogeneousComponent]\n[GOAL]\ncase a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n\u22a2 (\u2211 x in Finite.toFinset (_ : Set.Finite (Function.support fun m => \u2191(weightedHomogeneousComponent w m) \u03c6)),\n if \u2191(weightedDegree' w) d = x then coeff d \u03c6 else 0) =\n coeff d \u03c6\n[PROOFSTEP]\nrw [Finset.sum_eq_single (weightedDegree' w d)]\n[GOAL]\ncase a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n\u22a2 (if \u2191(weightedDegree' w) d = \u2191(weightedDegree' w) d then coeff d \u03c6 else 0) = coeff d \u03c6\n[PROOFSTEP]\nrw [if_pos rfl]\n[GOAL]\ncase a.h\u2080\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2200 (b : (fun x => M) d),\n b \u2208 Finite.toFinset (_ : Set.Finite (Function.support fun m => \u2191(weightedHomogeneousComponent w m) \u03c6)) \u2192\n b \u2260 \u2191(weightedDegree' w) d \u2192 (if \u2191(weightedDegree' w) d = b then coeff d \u03c6 else 0) = 0\n[PROOFSTEP]\nintro m _ hm'\n[GOAL]\ncase a.h\u2080\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nm : M\na\u271d : m \u2208 Finite.toFinset (_ : Set.Finite (Function.support fun m => \u2191(weightedHomogeneousComponent w m) \u03c6))\nhm' : m \u2260 \u2191(weightedDegree' w) d\n\u22a2 (if \u2191(weightedDegree' w) d = m then coeff d \u03c6 else 0) = 0\n[PROOFSTEP]\nrw [if_neg hm'.symm]\n[GOAL]\ncase a.h\u2081\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u00ac\u2191(weightedDegree' w) d \u2208\n Finite.toFinset (_ : Set.Finite (Function.support fun m => \u2191(weightedHomogeneousComponent w m) \u03c6)) \u2192\n (if \u2191(weightedDegree' w) d = \u2191(weightedDegree' w) d then coeff d \u03c6 else 0) = 0\n[PROOFSTEP]\nintro hm\n[GOAL]\ncase a.h\u2081\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhm :\n \u00ac\u2191(weightedDegree' w) d \u2208\n Finite.toFinset (_ : Set.Finite (Function.support fun m => \u2191(weightedHomogeneousComponent w m) \u03c6))\n\u22a2 (if \u2191(weightedDegree' w) d = \u2191(weightedDegree' w) d then coeff d \u03c6 else 0) = 0\n[PROOFSTEP]\nrw [if_pos rfl]\n[GOAL]\ncase a.h\u2081\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhm :\n \u00ac\u2191(weightedDegree' w) d \u2208\n Finite.toFinset (_ : Set.Finite (Function.support fun m => \u2191(weightedHomogeneousComponent w m) \u03c6))\n\u22a2 coeff d \u03c6 = 0\n[PROOFSTEP]\nsimp only [Finite.mem_toFinset, mem_support, Ne.def, Classical.not_not] at hm \n[GOAL]\ncase a.h\u2081\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhm : \u2191(weightedHomogeneousComponent w (\u2191(weightedDegree' w) d)) \u03c6 = 0\n\u22a2 coeff d \u03c6 = 0\n[PROOFSTEP]\nhave := coeff_weightedHomogeneousComponent (w := w) (weightedDegree' w d) \u03c6 d\n[GOAL]\ncase a.h\u2081\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhm : \u2191(weightedHomogeneousComponent w (\u2191(weightedDegree' w) d)) \u03c6 = 0\nthis :\n coeff d (\u2191(weightedHomogeneousComponent w (\u2191(weightedDegree' w) d)) \u03c6) =\n if \u2191(weightedDegree' w) d = \u2191(weightedDegree' w) d then coeff d \u03c6 else 0\n\u22a2 coeff d \u03c6 = 0\n[PROOFSTEP]\nrw [hm, if_pos rfl, coeff_zero] at this \n[GOAL]\ncase a.h\u2081\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nn : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nhm : \u2191(weightedHomogeneousComponent w (\u2191(weightedDegree' w) d)) \u03c6 = 0\nthis : 0 = coeff d \u03c6\n\u22a2 coeff d \u03c6 = 0\n[PROOFSTEP]\nexact this.symm\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : p \u2208 weightedHomogeneousSubmodule R w n\n\u22a2 \u2191(weightedHomogeneousComponent w m) p = if m = n then p else 0\n[PROOFSTEP]\nsimp only [mem_weightedHomogeneousSubmodule] at h \n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\n\u22a2 \u2191(weightedHomogeneousComponent w m) p = if m = n then p else 0\n[PROOFSTEP]\next x\n[GOAL]\ncase a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff x (\u2191(weightedHomogeneousComponent w m) p) = coeff x (if m = n then p else 0)\n[PROOFSTEP]\nrw [coeff_weightedHomogeneousComponent]\n[GOAL]\ncase a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\n\u22a2 (if \u2191(weightedDegree' w) x = m then coeff x p else 0) = coeff x (if m = n then p else 0)\n[PROOFSTEP]\nby_cases zero_coeff : coeff x p = 0\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\nzero_coeff : coeff x p = 0\n\u22a2 (if \u2191(weightedDegree' w) x = m then coeff x p else 0) = coeff x (if m = n then p else 0)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\nzero_coeff : coeff x p = 0\nh\u271d\u00b9 : \u2191(weightedDegree' w) x = m\nh\u271d : m = n\n\u22a2 coeff x p = coeff x p\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\nzero_coeff : coeff x p = 0\nh\u271d\u00b9 : \u2191(weightedDegree' w) x = m\nh\u271d : \u00acm = n\n\u22a2 coeff x p = coeff x 0\ncase pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\nzero_coeff : coeff x p = 0\nh\u271d\u00b9 : \u00ac\u2191(weightedDegree' w) x = m\nh\u271d : m = n\n\u22a2 0 = coeff x p\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\nzero_coeff : coeff x p = 0\nh\u271d\u00b9 : \u00ac\u2191(weightedDegree' w) x = m\nh\u271d : \u00acm = n\n\u22a2 0 = coeff x 0\n[PROOFSTEP]\nall_goals simp only [zero_coeff, coeff_zero]\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\nzero_coeff : coeff x p = 0\nh\u271d\u00b9 : \u2191(weightedDegree' w) x = m\nh\u271d : m = n\n\u22a2 coeff x p = coeff x p\n[PROOFSTEP]\nsimp only [zero_coeff, coeff_zero]\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\nzero_coeff : coeff x p = 0\nh\u271d\u00b9 : \u2191(weightedDegree' w) x = m\nh\u271d : \u00acm = n\n\u22a2 coeff x p = coeff x 0\n[PROOFSTEP]\nsimp only [zero_coeff, coeff_zero]\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\nzero_coeff : coeff x p = 0\nh\u271d\u00b9 : \u00ac\u2191(weightedDegree' w) x = m\nh\u271d : m = n\n\u22a2 0 = coeff x p\n[PROOFSTEP]\nsimp only [zero_coeff, coeff_zero]\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\nzero_coeff : coeff x p = 0\nh\u271d\u00b9 : \u00ac\u2191(weightedDegree' w) x = m\nh\u271d : \u00acm = n\n\u22a2 0 = coeff x 0\n[PROOFSTEP]\nsimp only [zero_coeff, coeff_zero]\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\nzero_coeff : \u00accoeff x p = 0\n\u22a2 (if \u2191(weightedDegree' w) x = m then coeff x p else 0) = coeff x (if m = n then p else 0)\n[PROOFSTEP]\nrw [h zero_coeff]\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\nzero_coeff : \u00accoeff x p = 0\n\u22a2 (if n = m then coeff x p else 0) = coeff x (if m = n then p else 0)\n[PROOFSTEP]\nsimp only [show n = m \u2194 m = n from eq_comm]\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\nzero_coeff : \u00accoeff x p = 0\n\u22a2 (if m = n then coeff x p else 0) = coeff x (if m = n then p else 0)\n[PROOFSTEP]\nsplit_ifs with h1\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\nzero_coeff : \u00accoeff x p = 0\nh1 : m = n\n\u22a2 coeff x p = coeff x p\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\nw : \u03c3 \u2192 M\nn\u271d : M\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq M\nm n : M\np : MvPolynomial \u03c3 R\nh : IsWeightedHomogeneous w p n\nx : \u03c3 \u2192\u2080 \u2115\nzero_coeff : \u00accoeff x p = 0\nh1 : \u00acm = n\n\u22a2 0 = coeff x 0\n[PROOFSTEP]\nsimp only [coeff_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid M\nw : \u03c3 \u2192 M\n\u03c6 : MvPolynomial \u03c3 R\ninst\u271d : NoZeroSMulDivisors \u2115 M\nhw : \u2200 (i : \u03c3), w i \u2260 0\n\u22a2 \u2191(weightedHomogeneousComponent w 0) \u03c6 = \u2191C (coeff 0 \u03c6)\n[PROOFSTEP]\nclassical\next1 d\nrcases Classical.em (d = 0) with (rfl | hd)\n\u00b7 simp only [coeff_weightedHomogeneousComponent, if_pos, map_zero, coeff_zero_C]\n\u00b7 rw [coeff_weightedHomogeneousComponent, if_neg, coeff_C, if_neg (Ne.symm hd)]\n simp only [weightedDegree', LinearMap.toAddMonoidHom_coe, Finsupp.total_apply, Finsupp.sum, sum_eq_zero_iff,\n Finsupp.mem_support_iff, Ne.def, smul_eq_zero, not_forall, not_or, and_self_left, exists_prop]\n simp only [FunLike.ext_iff, Finsupp.coe_zero, Pi.zero_apply, not_forall] at hd \n obtain \u27e8i, hi\u27e9 := hd\n exact \u27e8i, hi, hw i\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid M\nw : \u03c3 \u2192 M\n\u03c6 : MvPolynomial \u03c3 R\ninst\u271d : NoZeroSMulDivisors \u2115 M\nhw : \u2200 (i : \u03c3), w i \u2260 0\n\u22a2 \u2191(weightedHomogeneousComponent w 0) \u03c6 = \u2191C (coeff 0 \u03c6)\n[PROOFSTEP]\next1 d\n[GOAL]\ncase a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid M\nw : \u03c3 \u2192 M\n\u03c6 : MvPolynomial \u03c3 R\ninst\u271d : NoZeroSMulDivisors \u2115 M\nhw : \u2200 (i : \u03c3), w i \u2260 0\nd : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff d (\u2191(weightedHomogeneousComponent w 0) \u03c6) = coeff d (\u2191C (coeff 0 \u03c6))\n[PROOFSTEP]\nrcases Classical.em (d = 0) with (rfl | hd)\n[GOAL]\ncase a.inl\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid M\nw : \u03c3 \u2192 M\n\u03c6 : MvPolynomial \u03c3 R\ninst\u271d : NoZeroSMulDivisors \u2115 M\nhw : \u2200 (i : \u03c3), w i \u2260 0\n\u22a2 coeff 0 (\u2191(weightedHomogeneousComponent w 0) \u03c6) = coeff 0 (\u2191C (coeff 0 \u03c6))\n[PROOFSTEP]\nsimp only [coeff_weightedHomogeneousComponent, if_pos, map_zero, coeff_zero_C]\n[GOAL]\ncase a.inr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid M\nw : \u03c3 \u2192 M\n\u03c6 : MvPolynomial \u03c3 R\ninst\u271d : NoZeroSMulDivisors \u2115 M\nhw : \u2200 (i : \u03c3), w i \u2260 0\nd : \u03c3 \u2192\u2080 \u2115\nhd : \u00acd = 0\n\u22a2 coeff d (\u2191(weightedHomogeneousComponent w 0) \u03c6) = coeff d (\u2191C (coeff 0 \u03c6))\n[PROOFSTEP]\nrw [coeff_weightedHomogeneousComponent, if_neg, coeff_C, if_neg (Ne.symm hd)]\n[GOAL]\ncase a.inr.hnc\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid M\nw : \u03c3 \u2192 M\n\u03c6 : MvPolynomial \u03c3 R\ninst\u271d : NoZeroSMulDivisors \u2115 M\nhw : \u2200 (i : \u03c3), w i \u2260 0\nd : \u03c3 \u2192\u2080 \u2115\nhd : \u00acd = 0\n\u22a2 \u00ac\u2191(weightedDegree' w) d = 0\n[PROOFSTEP]\nsimp only [weightedDegree', LinearMap.toAddMonoidHom_coe, Finsupp.total_apply, Finsupp.sum, sum_eq_zero_iff,\n Finsupp.mem_support_iff, Ne.def, smul_eq_zero, not_forall, not_or, and_self_left, exists_prop]\n[GOAL]\ncase a.inr.hnc\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid M\nw : \u03c3 \u2192 M\n\u03c6 : MvPolynomial \u03c3 R\ninst\u271d : NoZeroSMulDivisors \u2115 M\nhw : \u2200 (i : \u03c3), w i \u2260 0\nd : \u03c3 \u2192\u2080 \u2115\nhd : \u00acd = 0\n\u22a2 \u2203 x, \u00ac\u2191d x = 0 \u2227 \u00acw x = 0\n[PROOFSTEP]\nsimp only [FunLike.ext_iff, Finsupp.coe_zero, Pi.zero_apply, not_forall] at hd \n[GOAL]\ncase a.inr.hnc\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid M\nw : \u03c3 \u2192 M\n\u03c6 : MvPolynomial \u03c3 R\ninst\u271d : NoZeroSMulDivisors \u2115 M\nhw : \u2200 (i : \u03c3), w i \u2260 0\nd : \u03c3 \u2192\u2080 \u2115\nhd : \u2203 x, \u00ac\u2191d x = 0\n\u22a2 \u2203 x, \u00ac\u2191d x = 0 \u2227 \u00acw x = 0\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := hd\n[GOAL]\ncase a.inr.hnc.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid M\nw : \u03c3 \u2192 M\n\u03c6 : MvPolynomial \u03c3 R\ninst\u271d : NoZeroSMulDivisors \u2115 M\nhw : \u2200 (i : \u03c3), w i \u2260 0\nd : \u03c3 \u2192\u2080 \u2115\ni : \u03c3\nhi : \u00ac\u2191d i = 0\n\u22a2 \u2203 x, \u00ac\u2191d x = 0 \u2227 \u00acw x = 0\n[PROOFSTEP]\nexact \u27e8i, hi, hw i\u27e9\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous", "llama_tokens": 26398, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.752012562644147, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4428520243420015}} {"text": "[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nhdeg : c1.deg = c2.deg\nhnum : \u2191c1.num = \u2191c2.num\nhden : \u2191c1.den = \u2191c2.den\n\u22a2 c1 = c2\n[PROOFSTEP]\nrcases c1 with \u27e8i1, \u27e8n1, hn1\u27e9, \u27e8d1, hd1\u27e9, h1\u27e9\n[GOAL]\ncase mk.mk.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc2 : NumDenSameDeg \ud835\udc9c x\ni1 : \u03b9\nn1 : A\nhn1 : n1 \u2208 \ud835\udc9c i1\nd1 : A\nhd1 : d1 \u2208 \ud835\udc9c i1\nh1 : \u2191{ val := d1, property := hd1 } \u2208 x\nhdeg :\n { deg := i1, num := { val := n1, property := hn1 }, den := { val := d1, property := hd1 }, den_mem := h1 }.deg =\n c2.deg\nhnum :\n \u2191{ deg := i1, num := { val := n1, property := hn1 }, den := { val := d1, property := hd1 }, den_mem := h1 }.num =\n \u2191c2.num\nhden :\n \u2191{ deg := i1, num := { val := n1, property := hn1 }, den := { val := d1, property := hd1 }, den_mem := h1 }.den =\n \u2191c2.den\n\u22a2 { deg := i1, num := { val := n1, property := hn1 }, den := { val := d1, property := hd1 }, den_mem := h1 } = c2\n[PROOFSTEP]\nrcases c2 with \u27e8i2, \u27e8n2, hn2\u27e9, \u27e8d2, hd2\u27e9, h2\u27e9\n[GOAL]\ncase mk.mk.mk.mk.mk.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\ni1 : \u03b9\nn1 : A\nhn1 : n1 \u2208 \ud835\udc9c i1\nd1 : A\nhd1 : d1 \u2208 \ud835\udc9c i1\nh1 : \u2191{ val := d1, property := hd1 } \u2208 x\ni2 : \u03b9\nn2 : A\nhn2 : n2 \u2208 \ud835\udc9c i2\nd2 : A\nhd2 : d2 \u2208 \ud835\udc9c i2\nh2 : \u2191{ val := d2, property := hd2 } \u2208 x\nhdeg :\n { deg := i1, num := { val := n1, property := hn1 }, den := { val := d1, property := hd1 }, den_mem := h1 }.deg =\n { deg := i2, num := { val := n2, property := hn2 }, den := { val := d2, property := hd2 }, den_mem := h2 }.deg\nhnum :\n \u2191{ deg := i1, num := { val := n1, property := hn1 }, den := { val := d1, property := hd1 }, den_mem := h1 }.num =\n \u2191{ deg := i2, num := { val := n2, property := hn2 }, den := { val := d2, property := hd2 }, den_mem := h2 }.num\nhden :\n \u2191{ deg := i1, num := { val := n1, property := hn1 }, den := { val := d1, property := hd1 }, den_mem := h1 }.den =\n \u2191{ deg := i2, num := { val := n2, property := hn2 }, den := { val := d2, property := hd2 }, den_mem := h2 }.den\n\u22a2 { deg := i1, num := { val := n1, property := hn1 }, den := { val := d1, property := hd1 }, den_mem := h1 } =\n { deg := i2, num := { val := n2, property := hn2 }, den := { val := d2, property := hd2 }, den_mem := h2 }\n[PROOFSTEP]\ndsimp only [Subtype.coe_mk] at *\n[GOAL]\ncase mk.mk.mk.mk.mk.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\ni1 : \u03b9\nn1 : A\nhn1 : n1 \u2208 \ud835\udc9c i1\nd1 : A\nhd1 : d1 \u2208 \ud835\udc9c i1\nh1 : \u2191{ val := d1, property := hd1 } \u2208 x\ni2 : \u03b9\nn2 : A\nhn2 : n2 \u2208 \ud835\udc9c i2\nd2 : A\nhd2 : d2 \u2208 \ud835\udc9c i2\nh2 : \u2191{ val := d2, property := hd2 } \u2208 x\nhdeg : i1 = i2\nhnum : n1 = n2\nhden : d1 = d2\n\u22a2 { deg := i1, num := { val := n1, property := hn1 }, den := { val := d1, property := hd1 }, den_mem := h1 } =\n { deg := i2, num := { val := n2, property := hn2 }, den := { val := d2, property := hd2 }, den_mem := h2 }\n[PROOFSTEP]\nsubst hdeg hnum hden\n[GOAL]\ncase mk.mk.mk.mk.mk.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\ni1 : \u03b9\nn1 : A\nhn1 : n1 \u2208 \ud835\udc9c i1\nd1 : A\nhd1 : d1 \u2208 \ud835\udc9c i1\nh1 : \u2191{ val := d1, property := hd1 } \u2208 x\nhn2 : n1 \u2208 \ud835\udc9c i1\nhd2 : d1 \u2208 \ud835\udc9c i1\nh2 : \u2191{ val := d1, property := hd2 } \u2208 x\n\u22a2 { deg := i1, num := { val := n1, property := hn1 }, den := { val := d1, property := hd1 }, den_mem := h1 } =\n { deg := i1, num := { val := n1, property := hn2 }, den := { val := d1, property := hd2 }, den_mem := h2 }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc : NumDenSameDeg \ud835\udc9c x\nn : \u2115\n\u22a2 \u2191(GradedMonoid.GMonoid.gnpow n c.den) \u2208 x\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc : NumDenSameDeg \ud835\udc9c x\n\u22a2 \u2191(GradedMonoid.GMonoid.gnpow Nat.zero c.den) \u2208 x\n[PROOFSTEP]\nsimpa only [Nat.zero_eq, coe_gnpow, pow_zero] using Submonoid.one_mem _\n[GOAL]\ncase succ\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc : NumDenSameDeg \ud835\udc9c x\nn : \u2115\nih : \u2191(GradedMonoid.GMonoid.gnpow n c.den) \u2208 x\n\u22a2 \u2191(GradedMonoid.GMonoid.gnpow (Nat.succ n) c.den) \u2208 x\n[PROOFSTEP]\nsimpa only [pow_succ', coe_gnpow] using x.mul_mem ih c.den_mem\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nz : HomogeneousLocalization \ud835\udc9c x\nn : \u2115\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Setoid.r ((fun x_1 => x_1 ^ n) c1) ((fun x_1 => x_1 ^ n) c2)\n[PROOFSTEP]\nchange Localization.mk _ _ = Localization.mk _ _\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nz : HomogeneousLocalization \ud835\udc9c x\nn : \u2115\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Localization.mk \u2191((fun x_1 => x_1 ^ n) c1).num\n { val := \u2191((fun x_1 => x_1 ^ n) c1).den, property := (_ : \u2191((fun x_1 => x_1 ^ n) c1).den \u2208 x) } =\n Localization.mk \u2191((fun x_1 => x_1 ^ n) c2).num\n { val := \u2191((fun x_1 => x_1 ^ n) c2).den, property := (_ : \u2191((fun x_1 => x_1 ^ n) c2).den \u2208 x) }\n[PROOFSTEP]\nsimp only [num_pow, den_pow]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nz : HomogeneousLocalization \ud835\udc9c x\nn : \u2115\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Localization.mk (\u2191c1.num ^ n) { val := \u2191c1.den ^ n, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c1.den ^ n)) } =\n Localization.mk (\u2191c2.num ^ n) { val := \u2191c2.den ^ n, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c2.den ^ n)) }\n[PROOFSTEP]\nconvert congr_arg (fun z : at x => z ^ n) h\n[GOAL]\ncase h.e'_2\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nz : HomogeneousLocalization \ud835\udc9c x\nn : \u2115\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Localization.mk (\u2191c1.num ^ n) { val := \u2191c1.den ^ n, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c1.den ^ n)) } =\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } ^ n\n[PROOFSTEP]\nerw [Localization.mk_pow]\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nz : HomogeneousLocalization \ud835\udc9c x\nn : \u2115\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Localization.mk (\u2191c2.num ^ n) { val := \u2191c2.den ^ n, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c2.den ^ n)) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) } ^ n\n[PROOFSTEP]\nerw [Localization.mk_pow]\n[GOAL]\ncase h.e'_2\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nz : HomogeneousLocalization \ud835\udc9c x\nn : \u2115\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Localization.mk (\u2191c1.num ^ n) { val := \u2191c1.den ^ n, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c1.den ^ n)) } =\n Localization.mk (\u2191c1.num ^ n) ({ val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } ^ n)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nz : HomogeneousLocalization \ud835\udc9c x\nn : \u2115\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Localization.mk (\u2191c2.num ^ n) { val := \u2191c2.den ^ n, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c2.den ^ n)) } =\n Localization.mk (\u2191c2.num ^ n) ({ val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) } ^ n)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2079 : AddCommMonoid \u03b9\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u2074 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\u03b1 : Type u_4\ninst\u271d\u00b3 : SMul \u03b1 R\ninst\u271d\u00b2 : SMul \u03b1 A\ninst\u271d\u00b9 : IsScalarTower \u03b1 R A\ninst\u271d : IsScalarTower \u03b1 A A\nm : \u03b1\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Setoid.r ((fun x_1 => m \u2022 x_1) c1) ((fun x_1 => m \u2022 x_1) c2)\n[PROOFSTEP]\nchange Localization.mk _ _ = Localization.mk _ _\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2079 : AddCommMonoid \u03b9\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u2074 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\u03b1 : Type u_4\ninst\u271d\u00b3 : SMul \u03b1 R\ninst\u271d\u00b2 : SMul \u03b1 A\ninst\u271d\u00b9 : IsScalarTower \u03b1 R A\ninst\u271d : IsScalarTower \u03b1 A A\nm : \u03b1\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Localization.mk \u2191((fun x_1 => m \u2022 x_1) c1).num\n { val := \u2191((fun x_1 => m \u2022 x_1) c1).den, property := (_ : \u2191((fun x_1 => m \u2022 x_1) c1).den \u2208 x) } =\n Localization.mk \u2191((fun x_1 => m \u2022 x_1) c2).num\n { val := \u2191((fun x_1 => m \u2022 x_1) c2).den, property := (_ : \u2191((fun x_1 => m \u2022 x_1) c2).den \u2208 x) }\n[PROOFSTEP]\nsimp only [num_smul, den_smul]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2079 : AddCommMonoid \u03b9\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u2074 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\u03b1 : Type u_4\ninst\u271d\u00b3 : SMul \u03b1 R\ninst\u271d\u00b2 : SMul \u03b1 A\ninst\u271d\u00b9 : IsScalarTower \u03b1 R A\ninst\u271d : IsScalarTower \u03b1 A A\nm : \u03b1\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Localization.mk \u2191(m \u2022 c1.num) { val := \u2191c1.den, property := (_ : (fun x_1 => x_1 \u2208 x) \u2191c1.den) } =\n Localization.mk \u2191(m \u2022 c2.num) { val := \u2191c2.den, property := (_ : (fun x_1 => x_1 \u2208 x) \u2191c2.den) }\n[PROOFSTEP]\nconvert congr_arg (fun z : at x => m \u2022 z) h\n[GOAL]\ncase h.e'_2\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2079 : AddCommMonoid \u03b9\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u2074 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\u03b1 : Type u_4\ninst\u271d\u00b3 : SMul \u03b1 R\ninst\u271d\u00b2 : SMul \u03b1 A\ninst\u271d\u00b9 : IsScalarTower \u03b1 R A\ninst\u271d : IsScalarTower \u03b1 A A\nm : \u03b1\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Localization.mk \u2191(m \u2022 c1.num) { val := \u2191c1.den, property := (_ : (fun x_1 => x_1 \u2208 x) \u2191c1.den) } =\n m \u2022 Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) }\n[PROOFSTEP]\nrw [Localization.smul_mk]\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2079 : AddCommMonoid \u03b9\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u2074 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\u03b1 : Type u_4\ninst\u271d\u00b3 : SMul \u03b1 R\ninst\u271d\u00b2 : SMul \u03b1 A\ninst\u271d\u00b9 : IsScalarTower \u03b1 R A\ninst\u271d : IsScalarTower \u03b1 A A\nm : \u03b1\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Localization.mk \u2191(m \u2022 c2.num) { val := \u2191c2.den, property := (_ : (fun x_1 => x_1 \u2208 x) \u2191c2.den) } =\n m \u2022 Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n[PROOFSTEP]\nrw [Localization.smul_mk]\n[GOAL]\ncase h.e'_2\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2079 : AddCommMonoid \u03b9\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u2074 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\u03b1 : Type u_4\ninst\u271d\u00b3 : SMul \u03b1 R\ninst\u271d\u00b2 : SMul \u03b1 A\ninst\u271d\u00b9 : IsScalarTower \u03b1 R A\ninst\u271d : IsScalarTower \u03b1 A A\nm : \u03b1\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Localization.mk \u2191(m \u2022 c1.num) { val := \u2191c1.den, property := (_ : (fun x_1 => x_1 \u2208 x) \u2191c1.den) } =\n Localization.mk (m \u2022 \u2191c1.num) { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2079 : AddCommMonoid \u03b9\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u2074 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\u03b1 : Type u_4\ninst\u271d\u00b3 : SMul \u03b1 R\ninst\u271d\u00b2 : SMul \u03b1 A\ninst\u271d\u00b9 : IsScalarTower \u03b1 R A\ninst\u271d : IsScalarTower \u03b1 A A\nm : \u03b1\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Localization.mk \u2191(m \u2022 c2.num) { val := \u2191c2.den, property := (_ : (fun x_1 => x_1 \u2208 x) \u2191c2.den) } =\n Localization.mk (m \u2022 \u2191c2.num) { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2079 : AddCommMonoid \u03b9\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u2074 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\u03b1 : Type u_4\ninst\u271d\u00b3 : SMul \u03b1 R\ninst\u271d\u00b2 : SMul \u03b1 A\ninst\u271d\u00b9 : IsScalarTower \u03b1 R A\ninst\u271d : IsScalarTower \u03b1 A A\ny : HomogeneousLocalization \ud835\udc9c x\nn : \u03b1\n\u22a2 val (n \u2022 y) = n \u2022 val y\n[PROOFSTEP]\ninduction y using Quotient.inductionOn\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2079 : AddCommMonoid \u03b9\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u2074 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\u03b1 : Type u_4\ninst\u271d\u00b3 : SMul \u03b1 R\ninst\u271d\u00b2 : SMul \u03b1 A\ninst\u271d\u00b9 : IsScalarTower \u03b1 R A\ninst\u271d : IsScalarTower \u03b1 A A\nn : \u03b1\na\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 val (n \u2022 Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d) = n \u2022 val (Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d)\n[PROOFSTEP]\nchange Localization.mk _ _ = n \u2022 Localization.mk _ _\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2079 : AddCommMonoid \u03b9\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u2074 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\u03b1 : Type u_4\ninst\u271d\u00b3 : SMul \u03b1 R\ninst\u271d\u00b2 : SMul \u03b1 A\ninst\u271d\u00b9 : IsScalarTower \u03b1 R A\ninst\u271d : IsScalarTower \u03b1 A A\nn : \u03b1\na\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 Localization.mk \u2191((fun x_1 => n \u2022 x_1) a\u271d).num\n { val := \u2191((fun x_1 => n \u2022 x_1) a\u271d).den, property := (_ : \u2191((fun x_1 => n \u2022 x_1) a\u271d).den \u2208 x) } =\n n \u2022 Localization.mk \u2191a\u271d.num { val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) }\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2079 : AddCommMonoid \u03b9\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u2074 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\u03b1 : Type u_4\ninst\u271d\u00b3 : SMul \u03b1 R\ninst\u271d\u00b2 : SMul \u03b1 A\ninst\u271d\u00b9 : IsScalarTower \u03b1 R A\ninst\u271d : IsScalarTower \u03b1 A A\nn : \u03b1\na\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 Localization.mk \u2191(n \u2022 a\u271d).num { val := \u2191(n \u2022 a\u271d).den, property := (_ : \u2191(n \u2022 a\u271d).den \u2208 x) } =\n n \u2022 Localization.mk \u2191a\u271d.num { val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) }\n[PROOFSTEP]\nrw [Localization.smul_mk]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2079 : AddCommMonoid \u03b9\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u2074 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\u03b1 : Type u_4\ninst\u271d\u00b3 : SMul \u03b1 R\ninst\u271d\u00b2 : SMul \u03b1 A\ninst\u271d\u00b9 : IsScalarTower \u03b1 R A\ninst\u271d : IsScalarTower \u03b1 A A\nn : \u03b1\na\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 Localization.mk \u2191(n \u2022 a\u271d).num { val := \u2191(n \u2022 a\u271d).den, property := (_ : \u2191(n \u2022 a\u271d).den \u2208 x) } =\n Localization.mk (n \u2022 \u2191a\u271d.num) { val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) }\n[PROOFSTEP]\ncongr 1\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Setoid.r (-c1) (-c2)\n[PROOFSTEP]\nchange Localization.mk _ _ = Localization.mk _ _\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 Localization.mk \u2191(-c1).num { val := \u2191(-c1).den, property := (_ : \u2191(-c1).den \u2208 x) } =\n Localization.mk \u2191(-c2).num { val := \u2191(-c2).den, property := (_ : \u2191(-c2).den \u2208 x) }\n[PROOFSTEP]\nsimp only [num_neg, den_neg, \u2190 Localization.neg_mk]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\n\u22a2 -Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : (fun x_1 => x_1 \u2208 x) \u2191c1.den) } =\n -Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : (fun x_1 => x_1 \u2208 x) \u2191c2.den) }\n[PROOFSTEP]\nexact congr_arg Neg.neg h\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\nc3 c4 : NumDenSameDeg \ud835\udc9c x\nh' :\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) } =\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n\u22a2 Setoid.r ((fun x_1 x_2 => x_1 + x_2) c1 c3) ((fun x_1 x_2 => x_1 + x_2) c2 c4)\n[PROOFSTEP]\nchange Localization.mk _ _ = Localization.mk _ _\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\nc3 c4 : NumDenSameDeg \ud835\udc9c x\nh' :\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) } =\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n\u22a2 Localization.mk \u2191((fun x_1 x_2 => x_1 + x_2) c1 c3).num\n { val := \u2191((fun x_1 x_2 => x_1 + x_2) c1 c3).den,\n property := (_ : \u2191((fun x_1 x_2 => x_1 + x_2) c1 c3).den \u2208 x) } =\n Localization.mk \u2191((fun x_1 x_2 => x_1 + x_2) c2 c4).num\n { val := \u2191((fun x_1 x_2 => x_1 + x_2) c2 c4).den, property := (_ : \u2191((fun x_1 x_2 => x_1 + x_2) c2 c4).den \u2208 x) }\n[PROOFSTEP]\nsimp only [num_add, den_add, \u2190 Localization.add_mk]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\nc3 c4 : NumDenSameDeg \ud835\udc9c x\nh' :\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) } =\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n\u22a2 Localization.mk (\u2191c1.den * \u2191c3.num + \u2191c3.den * \u2191c1.num)\n { val := \u2191c1.den * \u2191c3.den, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c1.den * \u2191c3.den)) } =\n Localization.mk (\u2191c2.den * \u2191c4.num + \u2191c4.den * \u2191c2.num)\n { val := \u2191c2.den * \u2191c4.den, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c2.den * \u2191c4.den)) }\n[PROOFSTEP]\nconvert congr_arg\u2082 (\u00b7 + \u00b7) h h'\n[GOAL]\ncase h.e'_2\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\nc3 c4 : NumDenSameDeg \ud835\udc9c x\nh' :\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) } =\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n\u22a2 Localization.mk (\u2191c1.den * \u2191c3.num + \u2191c3.den * \u2191c1.num)\n { val := \u2191c1.den * \u2191c3.den, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c1.den * \u2191c3.den)) } =\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } +\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) }\n[PROOFSTEP]\nerw [Localization.add_mk]\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\nc3 c4 : NumDenSameDeg \ud835\udc9c x\nh' :\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) } =\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n\u22a2 Localization.mk (\u2191c2.den * \u2191c4.num + \u2191c4.den * \u2191c2.num)\n { val := \u2191c2.den * \u2191c4.den, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c2.den * \u2191c4.den)) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) } +\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n[PROOFSTEP]\nerw [Localization.add_mk]\n[GOAL]\ncase h.e'_2\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\nc3 c4 : NumDenSameDeg \ud835\udc9c x\nh' :\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) } =\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n\u22a2 Localization.mk (\u2191c1.den * \u2191c3.num + \u2191c3.den * \u2191c1.num)\n { val := \u2191c1.den * \u2191c3.den, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c1.den * \u2191c3.den)) } =\n Localization.mk\n (\u2191{ val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } * \u2191c3.num +\n \u2191{ val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) } * \u2191c1.num)\n ({ val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } * { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) })\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\nc3 c4 : NumDenSameDeg \ud835\udc9c x\nh' :\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) } =\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n\u22a2 Localization.mk (\u2191c2.den * \u2191c4.num + \u2191c4.den * \u2191c2.num)\n { val := \u2191c2.den * \u2191c4.den, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c2.den * \u2191c4.den)) } =\n Localization.mk\n (\u2191{ val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) } * \u2191c4.num +\n \u2191{ val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) } * \u2191c2.num)\n ({ val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) } * { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) })\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\nc3 c4 : NumDenSameDeg \ud835\udc9c x\nh' :\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) } =\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n\u22a2 Setoid.r ((fun x_1 x_2 => x_1 * x_2) c1 c3) ((fun x_1 x_2 => x_1 * x_2) c2 c4)\n[PROOFSTEP]\nchange Localization.mk _ _ = Localization.mk _ _\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\nc3 c4 : NumDenSameDeg \ud835\udc9c x\nh' :\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) } =\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n\u22a2 Localization.mk \u2191((fun x_1 x_2 => x_1 * x_2) c1 c3).num\n { val := \u2191((fun x_1 x_2 => x_1 * x_2) c1 c3).den,\n property := (_ : \u2191((fun x_1 x_2 => x_1 * x_2) c1 c3).den \u2208 x) } =\n Localization.mk \u2191((fun x_1 x_2 => x_1 * x_2) c2 c4).num\n { val := \u2191((fun x_1 x_2 => x_1 * x_2) c2 c4).den, property := (_ : \u2191((fun x_1 x_2 => x_1 * x_2) c2 c4).den \u2208 x) }\n[PROOFSTEP]\nsimp only [num_mul, den_mul]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\nc3 c4 : NumDenSameDeg \ud835\udc9c x\nh' :\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) } =\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n\u22a2 Localization.mk (\u2191c1.num * \u2191c3.num)\n { val := \u2191c1.den * \u2191c3.den, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c1.den * \u2191c3.den)) } =\n Localization.mk (\u2191c2.num * \u2191c4.num)\n { val := \u2191c2.den * \u2191c4.den, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c2.den * \u2191c4.den)) }\n[PROOFSTEP]\nconvert congr_arg\u2082 (\u00b7 * \u00b7) h h'\n[GOAL]\ncase h.e'_2\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\nc3 c4 : NumDenSameDeg \ud835\udc9c x\nh' :\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) } =\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n\u22a2 Localization.mk (\u2191c1.num * \u2191c3.num)\n { val := \u2191c1.den * \u2191c3.den, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c1.den * \u2191c3.den)) } =\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } *\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) }\n[PROOFSTEP]\nerw [Localization.mk_mul]\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\nc3 c4 : NumDenSameDeg \ud835\udc9c x\nh' :\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) } =\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n\u22a2 Localization.mk (\u2191c2.num * \u2191c4.num)\n { val := \u2191c2.den * \u2191c4.den, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c2.den * \u2191c4.den)) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) } *\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n[PROOFSTEP]\nerw [Localization.mk_mul]\n[GOAL]\ncase h.e'_2\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\nc3 c4 : NumDenSameDeg \ud835\udc9c x\nh' :\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) } =\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n\u22a2 Localization.mk (\u2191c1.num * \u2191c3.num)\n { val := \u2191c1.den * \u2191c3.den, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c1.den * \u2191c3.den)) } =\n Localization.mk (\u2191c1.num * \u2191c3.num)\n ({ val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } * { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) })\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nc1 c2 : NumDenSameDeg \ud835\udc9c x\nh :\n Localization.mk \u2191c1.num { val := \u2191c1.den, property := (_ : \u2191c1.den \u2208 x) } =\n Localization.mk \u2191c2.num { val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) }\nc3 c4 : NumDenSameDeg \ud835\udc9c x\nh' :\n Localization.mk \u2191c3.num { val := \u2191c3.den, property := (_ : \u2191c3.den \u2208 x) } =\n Localization.mk \u2191c4.num { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) }\n\u22a2 Localization.mk (\u2191c2.num * \u2191c4.num)\n { val := \u2191c2.den * \u2191c4.den, property := (_ : (fun x_1 => x_1 \u2208 x) (\u2191c2.den * \u2191c4.den)) } =\n Localization.mk (\u2191c2.num * \u2191c4.num)\n ({ val := \u2191c2.den, property := (_ : \u2191c2.den \u2208 x) } * { val := \u2191c4.den, property := (_ : \u2191c4.den \u2208 x) })\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\ny1 y2 : HomogeneousLocalization \ud835\udc9c x\n\u22a2 val (y1 + y2) = val y1 + val y2\n[PROOFSTEP]\ninduction y1 using Quotient.inductionOn\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\ny2 : HomogeneousLocalization \ud835\udc9c x\na\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 val (Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d + y2) = val (Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d) + val y2\n[PROOFSTEP]\ninduction y2 using Quotient.inductionOn\n[GOAL]\ncase h.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d\u00b9 a\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 val (Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d\u00b9 + Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d) =\n val (Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d\u00b9) + val (Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d)\n[PROOFSTEP]\nchange Localization.mk _ _ = Localization.mk _ _ + Localization.mk _ _\n[GOAL]\ncase h.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d\u00b9 a\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 Localization.mk \u2191((fun x_1 x_2 => x_1 + x_2) a\u271d\u00b9 a\u271d).num\n { val := \u2191((fun x_1 x_2 => x_1 + x_2) a\u271d\u00b9 a\u271d).den,\n property := (_ : \u2191((fun x_1 x_2 => x_1 + x_2) a\u271d\u00b9 a\u271d).den \u2208 x) } =\n Localization.mk \u2191a\u271d\u00b9.num { val := \u2191a\u271d\u00b9.den, property := (_ : \u2191a\u271d\u00b9.den \u2208 x) } +\n Localization.mk \u2191a\u271d.num { val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) }\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d\u00b9 a\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 Localization.mk \u2191(a\u271d\u00b9 + a\u271d).num { val := \u2191(a\u271d\u00b9 + a\u271d).den, property := (_ : \u2191(a\u271d\u00b9 + a\u271d).den \u2208 x) } =\n Localization.mk \u2191a\u271d\u00b9.num { val := \u2191a\u271d\u00b9.den, property := (_ : \u2191a\u271d\u00b9.den \u2208 x) } +\n Localization.mk \u2191a\u271d.num { val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) }\n[PROOFSTEP]\nrw [Localization.add_mk]\n[GOAL]\ncase h.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d\u00b9 a\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 Localization.mk \u2191(a\u271d\u00b9 + a\u271d).num { val := \u2191(a\u271d\u00b9 + a\u271d).den, property := (_ : \u2191(a\u271d\u00b9 + a\u271d).den \u2208 x) } =\n Localization.mk\n (\u2191{ val := \u2191a\u271d\u00b9.den, property := (_ : \u2191a\u271d\u00b9.den \u2208 x) } * \u2191a\u271d.num +\n \u2191{ val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) } * \u2191a\u271d\u00b9.num)\n ({ val := \u2191a\u271d\u00b9.den, property := (_ : \u2191a\u271d\u00b9.den \u2208 x) } * { val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) })\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\ny1 y2 : HomogeneousLocalization \ud835\udc9c x\n\u22a2 val (y1 * y2) = val y1 * val y2\n[PROOFSTEP]\ninduction y1 using Quotient.inductionOn\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\ny2 : HomogeneousLocalization \ud835\udc9c x\na\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 val (Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d * y2) = val (Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d) * val y2\n[PROOFSTEP]\ninduction y2 using Quotient.inductionOn\n[GOAL]\ncase h.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d\u00b9 a\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 val (Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d\u00b9 * Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d) =\n val (Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d\u00b9) * val (Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d)\n[PROOFSTEP]\nchange Localization.mk _ _ = Localization.mk _ _ * Localization.mk _ _\n[GOAL]\ncase h.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d\u00b9 a\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 Localization.mk \u2191((fun x_1 x_2 => x_1 * x_2) a\u271d\u00b9 a\u271d).num\n { val := \u2191((fun x_1 x_2 => x_1 * x_2) a\u271d\u00b9 a\u271d).den,\n property := (_ : \u2191((fun x_1 x_2 => x_1 * x_2) a\u271d\u00b9 a\u271d).den \u2208 x) } =\n Localization.mk \u2191a\u271d\u00b9.num { val := \u2191a\u271d\u00b9.den, property := (_ : \u2191a\u271d\u00b9.den \u2208 x) } *\n Localization.mk \u2191a\u271d.num { val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) }\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d\u00b9 a\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 Localization.mk \u2191(a\u271d\u00b9 * a\u271d).num { val := \u2191(a\u271d\u00b9 * a\u271d).den, property := (_ : \u2191(a\u271d\u00b9 * a\u271d).den \u2208 x) } =\n Localization.mk \u2191a\u271d\u00b9.num { val := \u2191a\u271d\u00b9.den, property := (_ : \u2191a\u271d\u00b9.den \u2208 x) } *\n Localization.mk \u2191a\u271d.num { val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) }\n[PROOFSTEP]\nrw [Localization.mk_mul]\n[GOAL]\ncase h.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d\u00b9 a\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 Localization.mk \u2191(a\u271d\u00b9 * a\u271d).num { val := \u2191(a\u271d\u00b9 * a\u271d).den, property := (_ : \u2191(a\u271d\u00b9 * a\u271d).den \u2208 x) } =\n Localization.mk (\u2191a\u271d\u00b9.num * \u2191a\u271d.num)\n ({ val := \u2191a\u271d\u00b9.den, property := (_ : \u2191a\u271d\u00b9.den \u2208 x) } * { val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) })\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\ny : HomogeneousLocalization \ud835\udc9c x\n\u22a2 val (-y) = -val y\n[PROOFSTEP]\ninduction y using Quotient.inductionOn\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 val (-Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d) = -val (Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d)\n[PROOFSTEP]\nchange Localization.mk _ _ = -Localization.mk _ _\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 Localization.mk \u2191(-a\u271d).num { val := \u2191(-a\u271d).den, property := (_ : \u2191(-a\u271d).den \u2208 x) } =\n -Localization.mk \u2191a\u271d.num { val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) }\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 Localization.mk \u2191(-a\u271d).num { val := \u2191(-a\u271d).den, property := (_ : \u2191(-a\u271d).den \u2208 x) } =\n -Localization.mk \u2191a\u271d.num { val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) }\n[PROOFSTEP]\nrw [Localization.neg_mk]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 Localization.mk \u2191(-a\u271d).num { val := \u2191(-a\u271d).den, property := (_ : \u2191(-a\u271d).den \u2208 x) } =\n Localization.mk (-\u2191a\u271d.num) { val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\ny1 y2 : HomogeneousLocalization \ud835\udc9c x\n\u22a2 val (y1 - y2) = val y1 - val y2\n[PROOFSTEP]\nrw [show y1 - y2 = y1 + -y2 from rfl, add_val, neg_val]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\ny1 y2 : HomogeneousLocalization \ud835\udc9c x\n\u22a2 val y1 + -val y2 = val y1 - val y2\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\ny : HomogeneousLocalization \ud835\udc9c x\nn : \u2115\n\u22a2 val (y ^ n) = val y ^ n\n[PROOFSTEP]\ninduction y using Quotient.inductionOn\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nn : \u2115\na\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 val (Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d ^ n) = val (Quotient.mk (Setoid.ker (embedding \ud835\udc9c x)) a\u271d) ^ n\n[PROOFSTEP]\nchange Localization.mk _ _ = Localization.mk _ _ ^ n\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nn : \u2115\na\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 Localization.mk \u2191((fun x_1 => x_1 ^ n) a\u271d).num\n { val := \u2191((fun x_1 => x_1 ^ n) a\u271d).den, property := (_ : \u2191((fun x_1 => x_1 ^ n) a\u271d).den \u2208 x) } =\n Localization.mk \u2191a\u271d.num { val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) } ^ n\n[PROOFSTEP]\nrw [Localization.mk_pow]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nn : \u2115\na\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 Localization.mk \u2191((fun x_1 => x_1 ^ n) a\u271d).num\n { val := \u2191((fun x_1 => x_1 ^ n) a\u271d).den, property := (_ : \u2191((fun x_1 => x_1 ^ n) a\u271d).den \u2208 x) } =\n Localization.mk (\u2191a\u271d.num ^ n) ({ val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) } ^ n)\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nn : \u2115\na\u271d : NumDenSameDeg \ud835\udc9c x\n\u22a2 Localization.mk \u2191(a\u271d ^ n).num { val := \u2191(a\u271d ^ n).den, property := (_ : \u2191(a\u271d ^ n).den \u2208 x) } =\n Localization.mk (\u2191a\u271d.num ^ n) ({ val := \u2191a\u271d.den, property := (_ : \u2191a\u271d.den \u2208 x) } ^ n)\n[PROOFSTEP]\ncongr 1\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nn : \u2115\n\u22a2 val (Nat.unaryCast n) = \u2191n\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\u22a2 val (Nat.unaryCast Nat.zero) = \u2191Nat.zero\n[PROOFSTEP]\nsimp [Nat.unaryCast, zero_val, one_val, *]\n[GOAL]\ncase succ\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nn\u271d : \u2115\nn_ih\u271d : val (Nat.unaryCast n\u271d) = \u2191n\u271d\n\u22a2 val (Nat.unaryCast (Nat.succ n\u271d)) = \u2191(Nat.succ n\u271d)\n[PROOFSTEP]\nsimp [Nat.unaryCast, zero_val, one_val, *]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nn : \u2124\n\u22a2 val (Int.castDef n) = \u2191n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase ofNat\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d : \u2115\n\u22a2 val (Int.castDef (Int.ofNat a\u271d)) = \u2191(Int.ofNat a\u271d)\n[PROOFSTEP]\nsimp [Int.castDef, zero_val, one_val, *]\n[GOAL]\ncase negSucc\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d : \u2115\n\u22a2 val (Int.castDef (Int.negSucc a\u271d)) = \u2191(Int.negSucc a\u271d)\n[PROOFSTEP]\nsimp [Int.castDef, zero_val, one_val, *]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nf : HomogeneousLocalization \ud835\udc9c x\n\u22a2 val f = Localization.mk (num f) { val := den f, property := (_ : den f \u2208 x) }\n[PROOFSTEP]\nhave := Quotient.out_eq' f\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nf : HomogeneousLocalization \ud835\udc9c x\nthis : Quotient.mk'' (Quotient.out' f) = f\n\u22a2 val f = Localization.mk (num f) { val := den f, property := (_ : den f \u2208 x) }\n[PROOFSTEP]\napply_fun HomogeneousLocalization.val at this \n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nf : HomogeneousLocalization \ud835\udc9c x\nthis : val (Quotient.mk'' (Quotient.out' f)) = val f\n\u22a2 val f = Localization.mk (num f) { val := den f, property := (_ : den f \u2208 x) }\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nf : HomogeneousLocalization \ud835\udc9c x\nthis : val (Quotient.mk'' (Quotient.out' f)) = val f\n\u22a2 val (Quotient.mk'' (Quotient.out' f)) = Localization.mk (num f) { val := den f, property := (_ : den f \u2208 x) }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\nf g : HomogeneousLocalization \ud835\udc9c x\nh : val f = val g\n\u22a2 f = g\n[PROOFSTEP]\ninduction f using Quotient.inductionOn'\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\ng : HomogeneousLocalization \ud835\udc9c x\na\u271d : NumDenSameDeg \ud835\udc9c x\nh : val (Quotient.mk'' a\u271d) = val g\n\u22a2 Quotient.mk'' a\u271d = g\n[PROOFSTEP]\ninduction g using Quotient.inductionOn'\n[GOAL]\ncase h.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d\u00b9 a\u271d : NumDenSameDeg \ud835\udc9c x\nh : val (Quotient.mk'' a\u271d\u00b9) = val (Quotient.mk'' a\u271d)\n\u22a2 Quotient.mk'' a\u271d\u00b9 = Quotient.mk'' a\u271d\n[PROOFSTEP]\nrw [Quotient.eq'']\n[GOAL]\ncase h.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2075 : AddCommMonoid \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : Submonoid A\na\u271d\u00b9 a\u271d : NumDenSameDeg \ud835\udc9c x\nh : val (Quotient.mk'' a\u271d\u00b9) = val (Quotient.mk'' a\u271d)\n\u22a2 Setoid.r a\u271d\u00b9 a\u271d\n[PROOFSTEP]\nsimpa only [Quotient.liftOn'_mk] using h\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nh1 : IsUnit (val f)\n\u22a2 IsUnit f\n[PROOFSTEP]\nrcases h1 with \u27e8\u27e8a, b, eq0, eq1\u27e9, eq2 : a = f.val\u27e9\n[GOAL]\ncase intro.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\na b : at Ideal.primeCompl \ud835\udd2d\neq0 : a * b = 1\neq1 : b * a = 1\neq2 : a = val f\n\u22a2 IsUnit f\n[PROOFSTEP]\nrw [eq2] at eq0 eq1 \n[GOAL]\ncase intro.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\na b : at Ideal.primeCompl \ud835\udd2d\neq0 : val f * b = 1\neq1 : b * val f = 1\neq2 : a = val f\n\u22a2 IsUnit f\n[PROOFSTEP]\nclear a eq2\n[GOAL]\ncase intro.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb : at Ideal.primeCompl \ud835\udd2d\neq0 : val f * b = 1\neq1 : b * val f = 1\n\u22a2 IsUnit f\n[PROOFSTEP]\ninduction' b using Localization.induction_on with data\n[GOAL]\ncase intro.mk.H\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b = 1\neq1\u271d : b * val f = 1\ndata : A \u00d7 { x // x \u2208 Ideal.primeCompl \ud835\udd2d }\neq0 : val f * Localization.mk data.fst data.snd = 1\neq1 : Localization.mk data.fst data.snd * val f = 1\n\u22a2 IsUnit f\n[PROOFSTEP]\nrcases data with \u27e8a, \u27e8b, hb\u27e9\u27e9\n[GOAL]\ncase intro.mk.H.mk.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk (a, { val := b, property := hb }).fst (a, { val := b, property := hb }).snd = 1\neq1 : Localization.mk (a, { val := b, property := hb }).fst (a, { val := b, property := hb }).snd * val f = 1\n\u22a2 IsUnit f\n[PROOFSTEP]\ndsimp only at eq0 eq1 \n[GOAL]\ncase intro.mk.H.mk.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\neq1 : Localization.mk a { val := b, property := hb } * val f = 1\n\u22a2 IsUnit f\n[PROOFSTEP]\nhave b_f_den_not_mem : b * f.den \u2208 \ud835\udd2d.primeCompl := fun r =>\n Or.elim (Ideal.IsPrime.mem_or_mem inferInstance r) (hb \u00b7) (f.den_mem \u00b7)\n[GOAL]\ncase intro.mk.H.mk.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\neq1 : Localization.mk a { val := b, property := hb } * val f = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\n\u22a2 IsUnit f\n[PROOFSTEP]\nrw [f.eq_num_div_den, Localization.mk_mul, show (\u27e8b, hb\u27e9 : \ud835\udd2d.primeCompl) * \u27e8f.den, _\u27e9 = \u27e8b * f.den, _\u27e9 from rfl,\n show (1 : Localization.AtPrime \ud835\udd2d) = Localization.mk 1 1 by erw [Localization.mk_self 1], Localization.mk_eq_mk',\n IsLocalization.eq] at eq1 \n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\neq1 :\n Localization.mk (a * num f)\n { val := b * den f,\n property :=\n (_ :\n \u2191{ val := b, property := hb } * \u2191{ val := den f, property := (_ : den f \u2208 Ideal.primeCompl \ud835\udd2d) } \u2208\n Ideal.primeCompl \ud835\udd2d) } =\n 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\n\u22a2 1 = Localization.mk 1 1\n[PROOFSTEP]\nerw [Localization.mk_self 1]\n[GOAL]\ncase intro.mk.H.mk.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\neq1 :\n \u2203 c,\n \u2191c * (\u21911 * (a * num f)) =\n \u2191c *\n (\u2191{ val := b * den f,\n property :=\n (_ :\n \u2191{ val := b, property := hb } * \u2191{ val := den f, property := (_ : den f \u2208 Ideal.primeCompl \ud835\udd2d) } \u2208\n Ideal.primeCompl \ud835\udd2d) } *\n 1)\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\n\u22a2 IsUnit f\n[PROOFSTEP]\nrcases eq1 with \u27e8\u27e8c, hc\u27e9, eq1\u27e9\n[GOAL]\ncase intro.mk.H.mk.mk.intro.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 :\n \u2191{ val := c, property := hc } * (\u21911 * (a * num f)) =\n \u2191{ val := c, property := hc } *\n (\u2191{ val := b * den f,\n property :=\n (_ :\n \u2191{ val := b, property := hb } * \u2191{ val := den f, property := (_ : den f \u2208 Ideal.primeCompl \ud835\udd2d) } \u2208\n Ideal.primeCompl \ud835\udd2d) } *\n 1)\n\u22a2 IsUnit f\n[PROOFSTEP]\nchange c * (1 * (a * f.num)) = _ at eq1 \n[GOAL]\ncase intro.mk.H.mk.mk.intro.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 :\n c * (1 * (a * num f)) =\n \u2191{ val := c, property := hc } *\n (\u2191{ val := b * den f,\n property :=\n (_ :\n \u2191{ val := b, property := hb } * \u2191{ val := den f, property := (_ : den f \u2208 Ideal.primeCompl \ud835\udd2d) } \u2208\n Ideal.primeCompl \ud835\udd2d) } *\n 1)\n\u22a2 IsUnit f\n[PROOFSTEP]\nsimp only [one_mul, mul_one] at eq1 \n[GOAL]\ncase intro.mk.H.mk.mk.intro.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\n\u22a2 IsUnit f\n[PROOFSTEP]\nhave mem1 : c * (a * f.num) \u2208 \ud835\udd2d.primeCompl :=\n eq1.symm \u25b8 fun r => Or.elim (Ideal.IsPrime.mem_or_mem inferInstance r) (by tauto) (by tauto)\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\nr : c * (b * den f) \u2208 \u2191\ud835\udd2d\n\u22a2 c \u2208 \ud835\udd2d \u2192 False\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\nr : c * (b * den f) \u2208 \u2191\ud835\udd2d\n\u22a2 b * den f \u2208 \ud835\udd2d \u2192 False\n[PROOFSTEP]\ntauto\n[GOAL]\ncase intro.mk.H.mk.mk.intro.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\nmem1 : c * (a * num f) \u2208 Ideal.primeCompl \ud835\udd2d\n\u22a2 IsUnit f\n[PROOFSTEP]\nhave mem2 : f.num \u2209 \ud835\udd2d := by\n contrapose! mem1\n erw [Classical.not_not]\n exact Ideal.mul_mem_left _ _ (Ideal.mul_mem_left _ _ mem1)\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\nmem1 : c * (a * num f) \u2208 Ideal.primeCompl \ud835\udd2d\n\u22a2 \u00acnum f \u2208 \ud835\udd2d\n[PROOFSTEP]\ncontrapose! mem1\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\nmem1 : num f \u2208 \ud835\udd2d\n\u22a2 \u00acc * (a * num f) \u2208 Ideal.primeCompl \ud835\udd2d\n[PROOFSTEP]\nerw [Classical.not_not]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\nmem1 : num f \u2208 \ud835\udd2d\n\u22a2 c * (a * num f) \u2208 \u2191\ud835\udd2d\n[PROOFSTEP]\nexact Ideal.mul_mem_left _ _ (Ideal.mul_mem_left _ _ mem1)\n[GOAL]\ncase intro.mk.H.mk.mk.intro.mk\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\nmem1 : c * (a * num f) \u2208 Ideal.primeCompl \ud835\udd2d\nmem2 : \u00acnum f \u2208 \ud835\udd2d\n\u22a2 IsUnit f\n[PROOFSTEP]\nrefine' \u27e8\u27e8f, Quotient.mk'' \u27e8f.deg, \u27e8f.den, f.den_mem_deg\u27e9, \u27e8f.num, f.num_mem_deg\u27e9, mem2\u27e9, _, _\u27e9, rfl\u27e9\n[GOAL]\ncase intro.mk.H.mk.mk.intro.mk.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\nmem1 : c * (a * num f) \u2208 Ideal.primeCompl \ud835\udd2d\nmem2 : \u00acnum f \u2208 \ud835\udd2d\n\u22a2 f *\n Quotient.mk''\n { deg := deg f, num := { val := den f, property := (_ : den f \u2208 \ud835\udc9c (deg f)) },\n den := { val := num f, property := (_ : num f \u2208 \ud835\udc9c (deg f)) }, den_mem := mem2 } =\n 1\n[PROOFSTEP]\nsimp only [ext_iff_val, mul_val, val_mk'', f.eq_num_div_den, Localization.mk_mul, one_val]\n[GOAL]\ncase intro.mk.H.mk.mk.intro.mk.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\nmem1 : c * (a * num f) \u2208 Ideal.primeCompl \ud835\udd2d\nmem2 : \u00acnum f \u2208 \ud835\udd2d\n\u22a2 Quotient.mk''\n { deg := deg f, num := { val := den f, property := (_ : den f \u2208 \ud835\udc9c (deg f)) },\n den := { val := num f, property := (_ : num f \u2208 \ud835\udc9c (deg f)) }, den_mem := mem2 } *\n f =\n 1\n[PROOFSTEP]\nsimp only [ext_iff_val, mul_val, val_mk'', f.eq_num_div_den, Localization.mk_mul, one_val]\n[GOAL]\ncase intro.mk.H.mk.mk.intro.mk.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\nmem1 : c * (a * num f) \u2208 Ideal.primeCompl \ud835\udd2d\nmem2 : \u00acnum f \u2208 \ud835\udd2d\n\u22a2 Localization.mk (num f * den f)\n ({ val := den f, property := (_ : den f \u2208 Ideal.primeCompl \ud835\udd2d) } *\n { val := num f,\n property :=\n (_ :\n \u2191{ deg := deg f, num := { val := den f, property := (_ : den f \u2208 \ud835\udc9c (deg f)) },\n den := { val := num f, property := (_ : num f \u2208 \ud835\udc9c (deg f)) }, den_mem := mem2 }.den \u2208\n Ideal.primeCompl \ud835\udd2d) }) =\n 1\n[PROOFSTEP]\nconvert Localization.mk_self (M := A) _\n[GOAL]\ncase intro.mk.H.mk.mk.intro.mk.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\nmem1 : c * (a * num f) \u2208 Ideal.primeCompl \ud835\udd2d\nmem2 : \u00acnum f \u2208 \ud835\udd2d\n\u22a2 Localization.mk (den f * num f)\n ({ val := num f,\n property :=\n (_ :\n \u2191{ deg := deg f, num := { val := den f, property := (_ : den f \u2208 \ud835\udc9c (deg f)) },\n den := { val := num f, property := (_ : num f \u2208 \ud835\udc9c (deg f)) }, den_mem := mem2 }.den \u2208\n Ideal.primeCompl \ud835\udd2d) } *\n { val := den f, property := (_ : den f \u2208 Ideal.primeCompl \ud835\udd2d) }) =\n 1\n[PROOFSTEP]\nconvert Localization.mk_self (M := A) _\n[GOAL]\ncase h.e'_2.h.h.e'_4\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\nmem1 : c * (a * num f) \u2208 Ideal.primeCompl \ud835\udd2d\nmem2 : \u00acnum f \u2208 \ud835\udd2d\n\u22a2 num f * den f =\n \u2191({ val := den f, property := (_ : den f \u2208 Ideal.primeCompl \ud835\udd2d) } *\n { val := num f,\n property :=\n (_ :\n \u2191{ deg := deg f, num := { val := den f, property := (_ : den f \u2208 \ud835\udc9c (deg f)) },\n den := { val := num f, property := (_ : num f \u2208 \ud835\udc9c (deg f)) }, den_mem := mem2 }.den \u2208\n Ideal.primeCompl \ud835\udd2d) })\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase h.e'_2.h.h.e'_4\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\nmem1 : c * (a * num f) \u2208 Ideal.primeCompl \ud835\udd2d\nmem2 : \u00acnum f \u2208 \ud835\udd2d\n\u22a2 den f * num f =\n \u2191({ val := num f,\n property :=\n (_ :\n \u2191{ deg := deg f, num := { val := den f, property := (_ : den f \u2208 \ud835\udc9c (deg f)) },\n den := { val := num f, property := (_ : num f \u2208 \ud835\udc9c (deg f)) }, den_mem := mem2 }.den \u2208\n Ideal.primeCompl \ud835\udd2d) } *\n { val := den f, property := (_ : den f \u2208 Ideal.primeCompl \ud835\udd2d) })\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase h.e'_2.h.h.e'_4\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\nmem1 : c * (a * num f) \u2208 Ideal.primeCompl \ud835\udd2d\nmem2 : \u00acnum f \u2208 \ud835\udd2d\n\u22a2 den f * num f =\n \u2191({ val := den f, property := (_ : den f \u2208 Ideal.primeCompl \ud835\udd2d) } *\n { val := num f,\n property :=\n (_ :\n \u2191{ deg := deg f, num := { val := den f, property := (_ : den f \u2208 \ud835\udc9c (deg f)) },\n den := { val := num f, property := (_ : num f \u2208 \ud835\udc9c (deg f)) }, den_mem := mem2 }.den \u2208\n Ideal.primeCompl \ud835\udd2d) })\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_2.h.h.e'_4\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nb\u271d : at Ideal.primeCompl \ud835\udd2d\neq0\u271d : val f * b\u271d = 1\neq1\u271d : b\u271d * val f = 1\na b : A\nhb : b \u2208 Ideal.primeCompl \ud835\udd2d\neq0 : val f * Localization.mk a { val := b, property := hb } = 1\nb_f_den_not_mem : b * den f \u2208 Ideal.primeCompl \ud835\udd2d\nc : A\nhc : c \u2208 Ideal.primeCompl \ud835\udd2d\neq1 : c * (a * num f) = c * (b * den f)\nmem1 : c * (a * num f) \u2208 Ideal.primeCompl \ud835\udd2d\nmem2 : \u00acnum f \u2208 \ud835\udd2d\n\u22a2 num f * den f =\n \u2191({ val := num f,\n property :=\n (_ :\n \u2191{ deg := deg f, num := { val := den f, property := (_ : den f \u2208 \ud835\udc9c (deg f)) },\n den := { val := num f, property := (_ : num f \u2208 \ud835\udc9c (deg f)) }, den_mem := mem2 }.den \u2208\n Ideal.primeCompl \ud835\udd2d) } *\n { val := den f, property := (_ : den f \u2208 Ideal.primeCompl \ud835\udd2d) })\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nx\u271d : IsUnit f\nb : AtPrime \ud835\udc9c \ud835\udd2d\neq1 : f * b = 1\neq2 : b * f = 1\n\u22a2 IsUnit (val f)\n[PROOFSTEP]\nsimp only [ext_iff_val, mul_val, one_val] at eq1 eq2 \n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nf : AtPrime \ud835\udc9c \ud835\udd2d\nx\u271d : IsUnit f\nb : AtPrime \ud835\udc9c \ud835\udd2d\neq1 : val f * val b = 1\neq2 : val b * val f = 1\n\u22a2 IsUnit (val f)\n[PROOFSTEP]\nexact \u27e8\u27e8f.val, b.val, eq1, eq2\u27e9, rfl\u27e9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\nr : 0 = 1\n\u22a2 False\n[PROOFSTEP]\nsimp [ext_iff_val, zero_val, one_val, zero_ne_one] at r \n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : AtPrime \ud835\udc9c \ud835\udd2d\n\u22a2 IsUnit a \u2228 IsUnit (1 - a)\n[PROOFSTEP]\nsimp only [\u2190 isUnit_iff_isUnit_val, sub_val, one_val]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : AtPrime \ud835\udc9c \ud835\udd2d\n\u22a2 IsUnit (val a) \u2228 IsUnit (1 - val a)\n[PROOFSTEP]\ninduction' a using Quotient.inductionOn' with a\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\n\u22a2 IsUnit (val (Quotient.mk'' a)) \u2228 IsUnit (1 - val (Quotient.mk'' a))\n[PROOFSTEP]\nsimp only [HomogeneousLocalization.val_mk'']\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\n\u22a2 IsUnit (Localization.mk \u2191a.num { val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) }) \u2228\n IsUnit (1 - Localization.mk \u2191a.num { val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) })\n[PROOFSTEP]\nby_cases mem1 : a.num.1 \u2208 \ud835\udd2d\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\nmem1 : \u2191a.num \u2208 \ud835\udd2d\n\u22a2 IsUnit (Localization.mk \u2191a.num { val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) }) \u2228\n IsUnit (1 - Localization.mk \u2191a.num { val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) })\n[PROOFSTEP]\nright\n[GOAL]\ncase pos.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\nmem1 : \u2191a.num \u2208 \ud835\udd2d\n\u22a2 IsUnit (1 - Localization.mk \u2191a.num { val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) })\n[PROOFSTEP]\nhave : a.den.1 - a.num.1 \u2208 \ud835\udd2d.primeCompl := fun h =>\n a.den_mem (sub_add_cancel a.den.val a.num.val \u25b8 Ideal.add_mem _ h mem1 : a.den.1 \u2208 \ud835\udd2d)\n[GOAL]\ncase pos.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\nmem1 : \u2191a.num \u2208 \ud835\udd2d\nthis : \u2191a.den - \u2191a.num \u2208 Ideal.primeCompl \ud835\udd2d\n\u22a2 IsUnit (1 - Localization.mk \u2191a.num { val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) })\n[PROOFSTEP]\napply isUnit_of_mul_eq_one _ (Localization.mk a.den.1 \u27e8a.den.1 - a.num.1, this\u27e9)\n[GOAL]\ncase pos.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\nmem1 : \u2191a.num \u2208 \ud835\udd2d\nthis : \u2191a.den - \u2191a.num \u2208 Ideal.primeCompl \ud835\udd2d\n\u22a2 (1 - Localization.mk \u2191a.num { val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) }) *\n Localization.mk \u2191a.den { val := \u2191a.den - \u2191a.num, property := this } =\n 1\n[PROOFSTEP]\nsimp only [sub_mul, Localization.mk_mul, one_mul, Localization.sub_mk, Submonoid.coe_mul]\n[GOAL]\ncase pos.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\nmem1 : \u2191a.num \u2208 \ud835\udd2d\nthis : \u2191a.den - \u2191a.num \u2208 Ideal.primeCompl \ud835\udd2d\n\u22a2 Localization.mk (\u2191a.den * (\u2191a.den - \u2191a.num) * \u2191a.den - (\u2191a.den * (\u2191a.num * \u2191a.den) - \u2191a.num * (\u2191a.num * \u2191a.den)))\n ({ val := \u2191a.den - \u2191a.num, property := this } *\n ({ val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) } *\n { val := \u2191a.den - \u2191a.num, property := this })) =\n 1\n[PROOFSTEP]\nconvert Localization.mk_self (M := A) _\n[GOAL]\ncase h.e'_2.h.h.e'_4\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\nmem1 : \u2191a.num \u2208 \ud835\udd2d\nthis : \u2191a.den - \u2191a.num \u2208 Ideal.primeCompl \ud835\udd2d\n\u22a2 \u2191a.den * (\u2191a.den - \u2191a.num) * \u2191a.den - (\u2191a.den * (\u2191a.num * \u2191a.den) - \u2191a.num * (\u2191a.num * \u2191a.den)) =\n \u2191({ val := \u2191a.den - \u2191a.num, property := this } *\n ({ val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) } *\n { val := \u2191a.den - \u2191a.num, property := this }))\n[PROOFSTEP]\nsimp only [Submonoid.coe_mul]\n[GOAL]\ncase h.e'_2.h.h.e'_4\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\nmem1 : \u2191a.num \u2208 \ud835\udd2d\nthis : \u2191a.den - \u2191a.num \u2208 Ideal.primeCompl \ud835\udd2d\n\u22a2 \u2191a.den * (\u2191a.den - \u2191a.num) * \u2191a.den - (\u2191a.den * (\u2191a.num * \u2191a.den) - \u2191a.num * (\u2191a.num * \u2191a.den)) =\n (\u2191a.den - \u2191a.num) * (\u2191a.den * (\u2191a.den - \u2191a.num))\n[PROOFSTEP]\nring\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\nmem1 : \u00ac\u2191a.num \u2208 \ud835\udd2d\n\u22a2 IsUnit (Localization.mk \u2191a.num { val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) }) \u2228\n IsUnit (1 - Localization.mk \u2191a.num { val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) })\n[PROOFSTEP]\nleft\n[GOAL]\ncase neg.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\nmem1 : \u00ac\u2191a.num \u2208 \ud835\udd2d\n\u22a2 IsUnit (Localization.mk \u2191a.num { val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) })\n[PROOFSTEP]\nchange _ \u2208 \ud835\udd2d.primeCompl at mem1 \n[GOAL]\ncase neg.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\nmem1 : \u2191a.num \u2208 Ideal.primeCompl \ud835\udd2d\n\u22a2 IsUnit (Localization.mk \u2191a.num { val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) })\n[PROOFSTEP]\napply isUnit_of_mul_eq_one _ (Localization.mk a.den.1 \u27e8a.num.1, mem1\u27e9)\n[GOAL]\ncase neg.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\nmem1 : \u2191a.num \u2208 Ideal.primeCompl \ud835\udd2d\n\u22a2 Localization.mk \u2191a.num { val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) } *\n Localization.mk \u2191a.den { val := \u2191a.num, property := mem1 } =\n 1\n[PROOFSTEP]\nrw [Localization.mk_mul]\n[GOAL]\ncase neg.h\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\nmem1 : \u2191a.num \u2208 Ideal.primeCompl \ud835\udd2d\n\u22a2 Localization.mk (\u2191a.num * \u2191a.den)\n ({ val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) } * { val := \u2191a.num, property := mem1 }) =\n 1\n[PROOFSTEP]\nconvert Localization.mk_self (M := A) _\n[GOAL]\ncase h.e'_2.h.h.e'_4\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\nmem1 : \u2191a.num \u2208 Ideal.primeCompl \ud835\udd2d\n\u22a2 \u2191a.num * \u2191a.den =\n \u2191({ val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) } * { val := \u2191a.num, property := mem1 })\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase h.e'_2.h.h.e'_4\n\u03b9 : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2076 : AddCommMonoid \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u03b9 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nx : Submonoid A\n\ud835\udd2d : Ideal A\ninst\u271d : Ideal.IsPrime \ud835\udd2d\na : NumDenSameDeg \ud835\udc9c (Ideal.primeCompl \ud835\udd2d)\nmem1 : \u2191a.num \u2208 Ideal.primeCompl \ud835\udd2d\n\u22a2 \u2191a.den * \u2191a.num =\n \u2191({ val := \u2191a.den, property := (_ : \u2191a.den \u2208 Ideal.primeCompl \ud835\udd2d) } * { val := \u2191a.num, property := mem1 })\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization", "llama_tokens": 44312, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339676722393, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.44280559639665507}} {"text": "[GOAL]\n\u03b9 : Type u\u03b9\nR : Type uR\nA : \u03b9 \u2192 Type uA\nB : Type uB\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (A i)\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : GSemiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : GAlgebra R A\ninst\u271d : Algebra R B\na b : R\n\u22a2 OneHom.toFun\n { toFun := \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun),\n map_one' := (_ : \u2191(of A 0) (\u2191GAlgebra.toFun 1) = \u2191(of A 0) GradedMonoid.GOne.one) }\n (a * b) =\n OneHom.toFun\n { toFun := \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun),\n map_one' := (_ : \u2191(of A 0) (\u2191GAlgebra.toFun 1) = \u2191(of A 0) GradedMonoid.GOne.one) }\n a *\n OneHom.toFun\n { toFun := \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun),\n map_one' := (_ : \u2191(of A 0) (\u2191GAlgebra.toFun 1) = \u2191(of A 0) GradedMonoid.GOne.one) }\n b\n[PROOFSTEP]\nsimp only [AddMonoidHom.comp_apply]\n[GOAL]\n\u03b9 : Type u\u03b9\nR : Type uR\nA : \u03b9 \u2192 Type uA\nB : Type uB\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (A i)\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : GSemiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : GAlgebra R A\ninst\u271d : Algebra R B\na b : R\n\u22a2 \u2191(of A 0) (\u2191GAlgebra.toFun (a * b)) = \u2191(of A 0) (\u2191GAlgebra.toFun a) * \u2191(of A 0) (\u2191GAlgebra.toFun b)\n[PROOFSTEP]\nrw [of_mul_of]\n[GOAL]\n\u03b9 : Type u\u03b9\nR : Type uR\nA : \u03b9 \u2192 Type uA\nB : Type uB\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (A i)\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : GSemiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : GAlgebra R A\ninst\u271d : Algebra R B\na b : R\n\u22a2 \u2191(of A 0) (\u2191GAlgebra.toFun (a * b)) = \u2191(of A (0 + 0)) (GradedMonoid.GMul.mul (\u2191GAlgebra.toFun a) (\u2191GAlgebra.toFun b))\n[PROOFSTEP]\napply DFinsupp.single_eq_of_sigma_eq (GAlgebra.map_mul a b)\n[GOAL]\n\u03b9 : Type u\u03b9\nR : Type uR\nA : \u03b9 \u2192 Type uA\nB : Type uB\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (A i)\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : GSemiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : GAlgebra R A\ninst\u271d : Algebra R B\nr : R\nx : (fun x => \u2a01 (i : \u03b9), A i) r\n\u22a2 \u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun),\n map_one' := (_ : \u2191(of A 0) (\u2191GAlgebra.toFun 1) = \u2191(of A 0) GradedMonoid.GOne.one) },\n map_mul' :=\n (_ :\n \u2200 (a b : R),\n OneHom.toFun\n { toFun := \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun),\n map_one' := (_ : \u2191(of A 0) (\u2191GAlgebra.toFun 1) = \u2191(of A 0) GradedMonoid.GOne.one) }\n (a * b) =\n OneHom.toFun\n { toFun := \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun),\n map_one' := (_ : \u2191(of A 0) (\u2191GAlgebra.toFun 1) = \u2191(of A 0) GradedMonoid.GOne.one) }\n a *\n OneHom.toFun\n { toFun := \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun),\n map_one' := (_ : \u2191(of A 0) (\u2191GAlgebra.toFun 1) = \u2191(of A 0) GradedMonoid.GOne.one) }\n b) },\n map_zero' := (_ : \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun) 0 = 0),\n map_add' :=\n (_ :\n \u2200 (a b : R),\n \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun) (a + b) =\n \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun) a + \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun) b) }\n r *\n x =\n x *\n \u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun),\n map_one' := (_ : \u2191(of A 0) (\u2191GAlgebra.toFun 1) = \u2191(of A 0) GradedMonoid.GOne.one) },\n map_mul' :=\n (_ :\n \u2200 (a b : R),\n OneHom.toFun\n { toFun := \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun),\n map_one' := (_ : \u2191(of A 0) (\u2191GAlgebra.toFun 1) = \u2191(of A 0) GradedMonoid.GOne.one) }\n (a * b) =\n OneHom.toFun\n { toFun := \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun),\n map_one' := (_ : \u2191(of A 0) (\u2191GAlgebra.toFun 1) = \u2191(of A 0) GradedMonoid.GOne.one) }\n a *\n OneHom.toFun\n { toFun := \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun),\n map_one' := (_ : \u2191(of A 0) (\u2191GAlgebra.toFun 1) = \u2191(of A 0) GradedMonoid.GOne.one) }\n b) },\n map_zero' := (_ : \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun) 0 = 0),\n map_add' :=\n (_ :\n \u2200 (a b : R),\n \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun) (a + b) =\n \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun) a + \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun) b) }\n r\n[PROOFSTEP]\nchange AddMonoidHom.mul (DirectSum.of _ _ _) x = AddMonoidHom.mul.flip (DirectSum.of _ _ _) x\n[GOAL]\n\u03b9 : Type u\u03b9\nR : Type uR\nA : \u03b9 \u2192 Type uA\nB : Type uB\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (A i)\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : GSemiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : GAlgebra R A\ninst\u271d : Algebra R B\nr : R\nx : (fun x => \u2a01 (i : \u03b9), A i) r\n\u22a2 \u2191(\u2191AddMonoidHom.mul (\u2191(of (fun i => A i) 0) (\u2191GAlgebra.toFun r))) x =\n \u2191(\u2191(AddMonoidHom.flip AddMonoidHom.mul) (\u2191(of (fun i => A i) 0) (\u2191GAlgebra.toFun r))) x\n[PROOFSTEP]\napply FunLike.congr_fun _ x\n[GOAL]\n\u03b9 : Type u\u03b9\nR : Type uR\nA : \u03b9 \u2192 Type uA\nB : Type uB\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (A i)\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : GSemiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : GAlgebra R A\ninst\u271d : Algebra R B\nr : R\nx : (fun x => \u2a01 (i : \u03b9), A i) r\n\u22a2 \u2191AddMonoidHom.mul (\u2191(of (fun i => A i) 0) (\u2191GAlgebra.toFun r)) =\n \u2191(AddMonoidHom.flip AddMonoidHom.mul) (\u2191(of (fun i => A i) 0) (\u2191GAlgebra.toFun r))\n[PROOFSTEP]\next i xi : 2\n[GOAL]\ncase H.h\n\u03b9 : Type u\u03b9\nR : Type uR\nA : \u03b9 \u2192 Type uA\nB : Type uB\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (A i)\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : GSemiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : GAlgebra R A\ninst\u271d : Algebra R B\nr : R\nx : (fun x => \u2a01 (i : \u03b9), A i) r\ni : \u03b9\nxi : A i\n\u22a2 \u2191(AddMonoidHom.comp (\u2191AddMonoidHom.mul (\u2191(of (fun i => A i) 0) (\u2191GAlgebra.toFun r))) (of (fun i => A i) i)) xi =\n \u2191(AddMonoidHom.comp (\u2191(AddMonoidHom.flip AddMonoidHom.mul) (\u2191(of (fun i => A i) 0) (\u2191GAlgebra.toFun r)))\n (of (fun i => A i) i))\n xi\n[PROOFSTEP]\ndsimp only [AddMonoidHom.comp_apply, AddMonoidHom.mul_apply, AddMonoidHom.flip_apply]\n[GOAL]\ncase H.h\n\u03b9 : Type u\u03b9\nR : Type uR\nA : \u03b9 \u2192 Type uA\nB : Type uB\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (A i)\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : GSemiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : GAlgebra R A\ninst\u271d : Algebra R B\nr : R\nx : (fun x => \u2a01 (i : \u03b9), A i) r\ni : \u03b9\nxi : A i\n\u22a2 \u2191(of (fun i => A i) 0) (\u2191GAlgebra.toFun r) * \u2191(of (fun i => A i) i) xi =\n \u2191(of (fun i => A i) i) xi * \u2191(of (fun i => A i) 0) (\u2191GAlgebra.toFun r)\n[PROOFSTEP]\nrw [of_mul_of, of_mul_of]\n[GOAL]\ncase H.h\n\u03b9 : Type u\u03b9\nR : Type uR\nA : \u03b9 \u2192 Type uA\nB : Type uB\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (A i)\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : GSemiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : GAlgebra R A\ninst\u271d : Algebra R B\nr : R\nx : (fun x => \u2a01 (i : \u03b9), A i) r\ni : \u03b9\nxi : A i\n\u22a2 \u2191(of (fun i => A i) (0 + i)) (GradedMonoid.GMul.mul (\u2191GAlgebra.toFun r) xi) =\n \u2191(of (fun i => A i) (i + 0)) (GradedMonoid.GMul.mul xi (\u2191GAlgebra.toFun r))\n[PROOFSTEP]\napply DFinsupp.single_eq_of_sigma_eq (GAlgebra.commutes r \u27e8i, xi\u27e9)\n[GOAL]\n\u03b9 : Type u\u03b9\nR : Type uR\nA : \u03b9 \u2192 Type uA\nB : Type uB\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (A i)\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : GSemiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : GAlgebra R A\ninst\u271d : Algebra R B\nr : R\nx : (fun x => \u2a01 (i : \u03b9), A i) r\n\u22a2 r \u2022 x =\n \u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun),\n map_one' := (_ : \u2191(of A 0) (\u2191GAlgebra.toFun 1) = \u2191(of A 0) GradedMonoid.GOne.one) },\n map_mul' :=\n (_ :\n \u2200 (a b : R),\n OneHom.toFun\n { toFun := \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun),\n map_one' := (_ : \u2191(of A 0) (\u2191GAlgebra.toFun 1) = \u2191(of A 0) GradedMonoid.GOne.one) }\n (a * b) =\n OneHom.toFun\n { toFun := \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun),\n map_one' := (_ : \u2191(of A 0) (\u2191GAlgebra.toFun 1) = \u2191(of A 0) GradedMonoid.GOne.one) }\n a *\n OneHom.toFun\n { toFun := \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun),\n map_one' := (_ : \u2191(of A 0) (\u2191GAlgebra.toFun 1) = \u2191(of A 0) GradedMonoid.GOne.one) }\n b) },\n map_zero' := (_ : \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun) 0 = 0),\n map_add' :=\n (_ :\n \u2200 (a b : R),\n \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun) (a + b) =\n \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun) a + \u2191(AddMonoidHom.comp (of A 0) GAlgebra.toFun) b) }\n r *\n x\n[PROOFSTEP]\nchange DistribMulAction.toAddMonoidHom _ r x = AddMonoidHom.mul (DirectSum.of _ _ _) x\n[GOAL]\n\u03b9 : Type u\u03b9\nR : Type uR\nA : \u03b9 \u2192 Type uA\nB : Type uB\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (A i)\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : GSemiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : GAlgebra R A\ninst\u271d : Algebra R B\nr : R\nx : (fun x => \u2a01 (i : \u03b9), A i) r\n\u22a2 \u2191(DistribMulAction.toAddMonoidHom ((fun x => \u2a01 (i : \u03b9), A i) r) r) x =\n \u2191(\u2191AddMonoidHom.mul (\u2191(of (fun i => A i) 0) (\u2191GAlgebra.toFun r))) x\n[PROOFSTEP]\napply FunLike.congr_fun _ x\n[GOAL]\n\u03b9 : Type u\u03b9\nR : Type uR\nA : \u03b9 \u2192 Type uA\nB : Type uB\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (A i)\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : GSemiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : GAlgebra R A\ninst\u271d : Algebra R B\nr : R\nx : (fun x => \u2a01 (i : \u03b9), A i) r\n\u22a2 DistribMulAction.toAddMonoidHom ((fun x => \u2a01 (i : \u03b9), A i) r) r =\n \u2191AddMonoidHom.mul (\u2191(of (fun i => A i) 0) (\u2191GAlgebra.toFun r))\n[PROOFSTEP]\next i xi : 2\n[GOAL]\ncase H.h\n\u03b9 : Type u\u03b9\nR : Type uR\nA : \u03b9 \u2192 Type uA\nB : Type uB\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (A i)\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : GSemiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : GAlgebra R A\ninst\u271d : Algebra R B\nr : R\nx : (fun x => \u2a01 (i : \u03b9), A i) r\ni : \u03b9\nxi : A i\n\u22a2 \u2191(AddMonoidHom.comp (DistribMulAction.toAddMonoidHom ((fun x => \u2a01 (i : \u03b9), A i) r) r) (of (fun i => A i) i)) xi =\n \u2191(AddMonoidHom.comp (\u2191AddMonoidHom.mul (\u2191(of (fun i => A i) 0) (\u2191GAlgebra.toFun r))) (of (fun i => A i) i)) xi\n[PROOFSTEP]\ndsimp only [AddMonoidHom.comp_apply, DistribMulAction.toAddMonoidHom_apply, AddMonoidHom.mul_apply]\n[GOAL]\ncase H.h\n\u03b9 : Type u\u03b9\nR : Type uR\nA : \u03b9 \u2192 Type uA\nB : Type uB\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (A i)\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : GSemiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : GAlgebra R A\ninst\u271d : Algebra R B\nr : R\nx : (fun x => \u2a01 (i : \u03b9), A i) r\ni : \u03b9\nxi : A i\n\u22a2 r \u2022 \u2191(of (fun i => A i) i) xi = \u2191(of (fun i => A i) 0) (\u2191GAlgebra.toFun r) * \u2191(of (fun i => A i) i) xi\n[PROOFSTEP]\nrw [DirectSum.of_mul_of, \u2190 of_smul]\n[GOAL]\ncase H.h\n\u03b9 : Type u\u03b9\nR : Type uR\nA : \u03b9 \u2192 Type uA\nB : Type uB\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (A i)\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : GSemiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : GAlgebra R A\ninst\u271d : Algebra R B\nr : R\nx : (fun x => \u2a01 (i : \u03b9), A i) r\ni : \u03b9\nxi : A i\n\u22a2 \u2191(of (fun i => A i) i) (r \u2022 xi) = \u2191(of (fun i => A i) (0 + i)) (GradedMonoid.GMul.mul (\u2191GAlgebra.toFun r) xi)\n[PROOFSTEP]\napply DFinsupp.single_eq_of_sigma_eq (GAlgebra.smul_def r \u27e8i, xi\u27e9)\n", "meta": {"mathlib_filename": "Mathlib.Algebra.DirectSum.Algebra", "llama_tokens": 6093, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8333245994514084, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.44266983829151907}} {"text": "[GOAL]\nn k : \u2115\nh : MinFacHelper n k\n\u22a2 1 < n\n[PROOFSTEP]\nhave : 2 < minFac n := h.1.trans_le h.2.2\n[GOAL]\nn k : \u2115\nh : MinFacHelper n k\nthis : 2 < minFac n\n\u22a2 1 < n\n[PROOFSTEP]\nrcases eq_zero_or_pos n with rfl | h\n[GOAL]\ncase inl\nk : \u2115\nh : MinFacHelper 0 k\nthis : 2 < minFac 0\n\u22a2 1 < 0\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase inr\nn k : \u2115\nh\u271d : MinFacHelper n k\nthis : 2 < minFac n\nh : n > 0\n\u22a2 1 < n\n[PROOFSTEP]\nrcases(succ_le_of_lt h).eq_or_lt with rfl | h\n[GOAL]\ncase inr.inl\nk : \u2115\nh\u271d : MinFacHelper (succ 0) k\nthis : 2 < minFac (succ 0)\nh : succ 0 > 0\n\u22a2 1 < succ 0\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase inr.inr\nn k : \u2115\nh\u271d\u00b9 : MinFacHelper n k\nthis : 2 < minFac n\nh\u271d : n > 0\nh : succ 0 < n\n\u22a2 1 < n\n[PROOFSTEP]\nexact h\n[GOAL]\nn : \u2115\nh1 : ble 2 n = true\nh2 : 1 = n % 2\n\u22a2 MinFacHelper n 3\n[PROOFSTEP]\nrefine \u27e8by norm_num, by norm_num, ?_\u27e9\n[GOAL]\nn : \u2115\nh1 : ble 2 n = true\nh2 : 1 = n % 2\n\u22a2 2 < 3\n[PROOFSTEP]\nnorm_num\n[GOAL]\nn : \u2115\nh1 : ble 2 n = true\nh2 : 1 = n % 2\n\u22a2 3 % 2 = 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nn : \u2115\nh1 : ble 2 n = true\nh2 : 1 = n % 2\n\u22a2 3 \u2264 minFac n\n[PROOFSTEP]\nrefine (le_minFac'.mpr \u03bb p hp hpn \u21a6 ?_).resolve_left (Nat.ne_of_gt (Nat.le_of_ble_eq_true h1))\n[GOAL]\nn : \u2115\nh1 : ble 2 n = true\nh2 : 1 = n % 2\np : \u2115\nhp : 2 \u2264 p\nhpn : p \u2223 n\n\u22a2 3 \u2264 p\n[PROOFSTEP]\nrcases hp.eq_or_lt with rfl | h\n[GOAL]\ncase inl\nn : \u2115\nh1 : ble 2 n = true\nh2 : 1 = n % 2\nhp : 2 \u2264 2\nhpn : 2 \u2223 n\n\u22a2 3 \u2264 2\n[PROOFSTEP]\nsimp [(Nat.dvd_iff_mod_eq_zero ..).1 hpn] at h2 \n[GOAL]\ncase inr\nn : \u2115\nh1 : ble 2 n = true\nh2 : 1 = n % 2\np : \u2115\nhp : 2 \u2264 p\nhpn : p \u2223 n\nh : 2 < p\n\u22a2 3 \u2264 p\n[PROOFSTEP]\nexact h\n[GOAL]\nn k k' : \u2115\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : minFac n \u2260 k\n\u22a2 MinFacHelper n k'\n[PROOFSTEP]\nrw [\u2190 e]\n[GOAL]\nn k k' : \u2115\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : minFac n \u2260 k\n\u22a2 MinFacHelper n (k + 2)\n[PROOFSTEP]\nrefine \u27e8Nat.lt_add_right _ _ _ h.1, ?_, ?_\u27e9\n[GOAL]\ncase refine_1\nn k k' : \u2115\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : minFac n \u2260 k\n\u22a2 (k + 2) % 2 = 1\n[PROOFSTEP]\nrw [add_mod, mod_self, add_zero, mod_mod]\n[GOAL]\ncase refine_1\nn k k' : \u2115\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : minFac n \u2260 k\n\u22a2 k % 2 = 1\n[PROOFSTEP]\nexact h.2.1\n[GOAL]\ncase refine_2\nn k k' : \u2115\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : minFac n \u2260 k\n\u22a2 k + 2 \u2264 minFac n\n[PROOFSTEP]\nrcases h.2.2.eq_or_lt with rfl | h2\n[GOAL]\ncase refine_2.inl\nn k' : \u2115\ne : minFac n + 2 = k'\nh : MinFacHelper n (minFac n)\nnp : minFac n \u2260 minFac n\n\u22a2 minFac n + 2 \u2264 minFac n\n[PROOFSTEP]\nexact (np rfl).elim\n[GOAL]\ncase refine_2.inr\nn k k' : \u2115\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : minFac n \u2260 k\nh2 : k < minFac n\n\u22a2 k + 2 \u2264 minFac n\n[PROOFSTEP]\nrcases(succ_le_of_lt h2).eq_or_lt with h2 | h2\n[GOAL]\ncase refine_2.inr.inl\nn k k' : \u2115\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : minFac n \u2260 k\nh2\u271d : k < minFac n\nh2 : succ k = minFac n\n\u22a2 k + 2 \u2264 minFac n\n[PROOFSTEP]\nrefine ((h.1.trans_le h.2.2).ne ?_).elim\n[GOAL]\ncase refine_2.inr.inl\nn k k' : \u2115\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : minFac n \u2260 k\nh2\u271d : k < minFac n\nh2 : succ k = minFac n\n\u22a2 2 = minFac n\n[PROOFSTEP]\nhave h3 : 2 \u2223 minFac n\n[GOAL]\ncase h3\nn k k' : \u2115\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : minFac n \u2260 k\nh2\u271d : k < minFac n\nh2 : succ k = minFac n\n\u22a2 2 \u2223 minFac n\n[PROOFSTEP]\nrw [Nat.dvd_iff_mod_eq_zero, \u2190 h2, succ_eq_add_one, add_mod, h.2.1]\n[GOAL]\ncase h3\nn k k' : \u2115\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : minFac n \u2260 k\nh2\u271d : k < minFac n\nh2 : succ k = minFac n\n\u22a2 (1 + 1 % 2) % 2 = 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine_2.inr.inl\nn k k' : \u2115\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : minFac n \u2260 k\nh2\u271d : k < minFac n\nh2 : succ k = minFac n\nh3 : 2 \u2223 minFac n\n\u22a2 2 = minFac n\n[PROOFSTEP]\nrw [dvd_prime <| minFac_prime h.one_lt.ne'] at h3 \n[GOAL]\ncase refine_2.inr.inl\nn k k' : \u2115\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : minFac n \u2260 k\nh2\u271d : k < minFac n\nh2 : succ k = minFac n\nh3 : 2 = 1 \u2228 2 = minFac n\n\u22a2 2 = minFac n\n[PROOFSTEP]\nnorm_num at h3 \n[GOAL]\ncase refine_2.inr.inl\nn k k' : \u2115\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : minFac n \u2260 k\nh2\u271d : k < minFac n\nh2 : succ k = minFac n\nh3 : 2 = minFac n\n\u22a2 2 = minFac n\n[PROOFSTEP]\nexact h3\n[GOAL]\ncase refine_2.inr.inr\nn k k' : \u2115\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : minFac n \u2260 k\nh2\u271d : k < minFac n\nh2 : succ k < minFac n\n\u22a2 k + 2 \u2264 minFac n\n[PROOFSTEP]\nexact h2\n[GOAL]\nn k k' : \u2115\ne : k + 2 = k'\nnk : \u00acNat.Prime k\nh : MinFacHelper n k\n\u22a2 MinFacHelper n k'\n[PROOFSTEP]\nrefine minFacHelper_1 e h \u03bb h2 \u21a6 ?_\n[GOAL]\nn k k' : \u2115\ne : k + 2 = k'\nnk : \u00acNat.Prime k\nh : MinFacHelper n k\nh2 : minFac n = k\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 h2] at nk \n[GOAL]\nn k k' : \u2115\ne : k + 2 = k'\nnk : \u00acNat.Prime (minFac n)\nh : MinFacHelper n k\nh2 : minFac n = k\n\u22a2 False\n[PROOFSTEP]\nexact nk <| minFac_prime h.one_lt.ne'\n[GOAL]\nn k k' : \u2115\ne : k + 2 = k'\nnk : beq (n % k) 0 = false\nh : MinFacHelper n k\n\u22a2 MinFacHelper n k'\n[PROOFSTEP]\nrefine minFacHelper_1 e h \u03bb h2 \u21a6 ?_\n[GOAL]\nn k k' : \u2115\ne : k + 2 = k'\nnk : beq (n % k) 0 = false\nh : MinFacHelper n k\nh2 : minFac n = k\n\u22a2 False\n[PROOFSTEP]\nhave nk := Nat.ne_of_beq_eq_false nk\n[GOAL]\nn k k' : \u2115\ne : k + 2 = k'\nnk\u271d : beq (n % k) 0 = false\nh : MinFacHelper n k\nh2 : minFac n = k\nnk : \u00acn % k = 0\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 Nat.dvd_iff_mod_eq_zero, \u2190 h2] at nk \n[GOAL]\nn k k' : \u2115\ne : k + 2 = k'\nnk\u271d : beq (n % k) 0 = false\nh : MinFacHelper n k\nh2 : minFac n = k\nnk : \u00acminFac n \u2223 n\n\u22a2 False\n[PROOFSTEP]\nexact nk <| minFac_dvd n\n[GOAL]\na : \u2115\nh : a % 2 = 0\n\u22a2 minFac a = \u21912\n[PROOFSTEP]\nrw [cast_ofNat, minFac_eq_two_iff, Nat.dvd_iff_mod_eq_zero, h]\n[GOAL]\nn k : \u2115\nh1 : MinFacHelper n k\nh2 : 0 = n % k\n\u22a2 IsNat (minFac n) k\n[PROOFSTEP]\nrw [eq_comm, \u2190 Nat.dvd_iff_mod_eq_zero] at h2 \n[GOAL]\nn k : \u2115\nh1 : MinFacHelper n k\nh2 : k \u2223 n\n\u22a2 IsNat (minFac n) k\n[PROOFSTEP]\nexact \u27e8le_antisymm (minFac_le_of_dvd h1.1.le h2) h1.2.2\u27e9\n[GOAL]\nn k : \u2115\nh1 : MinFacHelper n k\nh2 : ble (k * k) n = false\n\u22a2 IsNat (minFac n) n\n[PROOFSTEP]\nrefine \u27e8(Nat.prime_def_minFac.mp ?_).2\u27e9\n[GOAL]\nn k : \u2115\nh1 : MinFacHelper n k\nh2 : ble (k * k) n = false\n\u22a2 Nat.Prime n\n[PROOFSTEP]\nrw [Nat.prime_def_le_sqrt]\n[GOAL]\nn k : \u2115\nh1 : MinFacHelper n k\nh2 : ble (k * k) n = false\n\u22a2 2 \u2264 n \u2227 \u2200 (m : \u2115), 2 \u2264 m \u2192 m \u2264 sqrt n \u2192 \u00acm \u2223 n\n[PROOFSTEP]\nrefine \u27e8h1.one_lt, \u03bb m hm hmn h2mn \u21a6 ?_\u27e9\n[GOAL]\nn k : \u2115\nh1 : MinFacHelper n k\nh2 : ble (k * k) n = false\nm : \u2115\nhm : 2 \u2264 m\nhmn : m \u2264 sqrt n\nh2mn : m \u2223 n\n\u22a2 False\n[PROOFSTEP]\nexact\n lt_irrefl m <|\n calc\n m \u2264 sqrt n := hmn\n _ < k := (sqrt_lt.mpr (ble_eq_false.mp h2))\n _ \u2264 n.minFac := h1.2.2\n _ \u2264 m := Nat.minFac_le_of_dvd hm h2mn\n", "meta": {"mathlib_filename": "Mathlib.Tactic.NormNum.Prime", "llama_tokens": 3706, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936537604181, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.4425310334179427}} {"text": "[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI I' : Ideal R\nJ J' : Ideal S\n\u22a2 \u2200 {a b : R \u00d7 S},\n a \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J} \u2192 b \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J} \u2192 a + b \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J}\n[PROOFSTEP]\nrintro \u27e8a\u2081, a\u2082\u27e9 \u27e8b\u2081, b\u2082\u27e9 \u27e8ha\u2081, ha\u2082\u27e9 \u27e8hb\u2081, hb\u2082\u27e9\n[GOAL]\ncase mk.mk.intro.intro\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI I' : Ideal R\nJ J' : Ideal S\na\u2081 : R\na\u2082 : S\nb\u2081 : R\nb\u2082 : S\nha\u2081 : (a\u2081, a\u2082).fst \u2208 I\nha\u2082 : (a\u2081, a\u2082).snd \u2208 J\nhb\u2081 : (b\u2081, b\u2082).fst \u2208 I\nhb\u2082 : (b\u2081, b\u2082).snd \u2208 J\n\u22a2 (a\u2081, a\u2082) + (b\u2081, b\u2082) \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J}\n[PROOFSTEP]\nexact \u27e8I.add_mem ha\u2081 hb\u2081, J.add_mem ha\u2082 hb\u2082\u27e9\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI I' : Ideal R\nJ J' : Ideal S\n\u22a2 0 \u2208\n { carrier := {x | x.fst \u2208 I \u2227 x.snd \u2208 J},\n add_mem' :=\n (_ :\n \u2200 {a b : R \u00d7 S},\n a \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J} \u2192\n b \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J} \u2192 a + b \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J}) }.carrier\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI I' : Ideal R\nJ J' : Ideal S\n\u22a2 \u2200 (c : R \u00d7 S) {x : R \u00d7 S},\n x \u2208\n {\n toAddSubsemigroup :=\n { carrier := {x | x.fst \u2208 I \u2227 x.snd \u2208 J},\n add_mem' :=\n (_ :\n \u2200 {a b : R \u00d7 S},\n a \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J} \u2192\n b \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J} \u2192 a + b \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J}) },\n zero_mem' := (_ : 0 \u2208 I \u2227 0 \u2208 J) }.toAddSubsemigroup.carrier \u2192\n c \u2022 x \u2208\n {\n toAddSubsemigroup :=\n { carrier := {x | x.fst \u2208 I \u2227 x.snd \u2208 J},\n add_mem' :=\n (_ :\n \u2200 {a b : R \u00d7 S},\n a \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J} \u2192\n b \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J} \u2192 a + b \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J}) },\n zero_mem' := (_ : 0 \u2208 I \u2227 0 \u2208 J) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrintro \u27e8a\u2081, a\u2082\u27e9 \u27e8b\u2081, b\u2082\u27e9 \u27e8hb\u2081, hb\u2082\u27e9\n[GOAL]\ncase mk.mk.intro\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI I' : Ideal R\nJ J' : Ideal S\na\u2081 : R\na\u2082 : S\nb\u2081 : R\nb\u2082 : S\nhb\u2081 : (b\u2081, b\u2082).fst \u2208 I\nhb\u2082 : (b\u2081, b\u2082).snd \u2208 J\n\u22a2 (a\u2081, a\u2082) \u2022 (b\u2081, b\u2082) \u2208\n {\n toAddSubsemigroup :=\n { carrier := {x | x.fst \u2208 I \u2227 x.snd \u2208 J},\n add_mem' :=\n (_ :\n \u2200 {a b : R \u00d7 S},\n a \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J} \u2192\n b \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J} \u2192 a + b \u2208 {x | x.fst \u2208 I \u2227 x.snd \u2208 J}) },\n zero_mem' := (_ : 0 \u2208 I \u2227 0 \u2208 J) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nexact \u27e8I.mul_mem_left _ hb\u2081, J.mul_mem_left _ hb\u2082\u27e9\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI I' : Ideal R\nJ J' : Ideal S\n\u22a2 \u2200 (x : R \u00d7 S), x \u2208 prod \u22a4 \u22a4 \u2194 x \u2208 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\n\u22a2 I = prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I)\n[PROOFSTEP]\napply Ideal.ext\n[GOAL]\ncase h\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\n\u22a2 \u2200 (x : R \u00d7 S), x \u2208 I \u2194 x \u2208 prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I)\n[PROOFSTEP]\nrintro \u27e8r, s\u27e9\n[GOAL]\ncase h.mk\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\nr : R\ns : S\n\u22a2 (r, s) \u2208 I \u2194 (r, s) \u2208 prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I)\n[PROOFSTEP]\nrw [mem_prod, mem_map_iff_of_surjective (RingHom.fst R S) Prod.fst_surjective,\n mem_map_iff_of_surjective (RingHom.snd R S) Prod.snd_surjective]\n[GOAL]\ncase h.mk\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\nr : R\ns : S\n\u22a2 (r, s) \u2208 I \u2194 (\u2203 x, x \u2208 I \u2227 \u2191(RingHom.fst R S) x = r) \u2227 \u2203 x, x \u2208 I \u2227 \u2191(RingHom.snd R S) x = s\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8\u27e8_, \u27e8h, rfl\u27e9\u27e9, \u27e8_, \u27e8h, rfl\u27e9\u27e9\u27e9, _\u27e9\n[GOAL]\ncase h.mk\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\nr : R\ns : S\n\u22a2 ((\u2203 x, x \u2208 I \u2227 \u2191(RingHom.fst R S) x = r) \u2227 \u2203 x, x \u2208 I \u2227 \u2191(RingHom.snd R S) x = s) \u2192 (r, s) \u2208 I\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e8r, s'\u27e9, \u27e8h\u2081, rfl\u27e9\u27e9, \u27e8\u27e8r', s\u27e9, \u27e8h\u2082, rfl\u27e9\u27e9\u27e9\n[GOAL]\ncase h.mk.intro.intro.mk.intro.intro.mk.intro\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\nr : R\ns' : S\nh\u2081 : (r, s') \u2208 I\nr' : R\ns : S\nh\u2082 : (r', s) \u2208 I\n\u22a2 (\u2191(RingHom.fst R S) (r, s'), \u2191(RingHom.snd R S) (r', s)) \u2208 I\n[PROOFSTEP]\nsimpa using I.add_mem (I.mul_mem_left (1, 0) h\u2081) (I.mul_mem_left (0, 1) h\u2082)\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\n\u22a2 map (RingHom.fst R S) (prod I J) = I\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx : R\n\u22a2 x \u2208 map (RingHom.fst R S) (prod I J) \u2194 x \u2208 I\n[PROOFSTEP]\nrw [mem_map_iff_of_surjective (RingHom.fst R S) Prod.fst_surjective]\n[GOAL]\ncase h\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx : R\n\u22a2 (\u2203 x_1, x_1 \u2208 prod I J \u2227 \u2191(RingHom.fst R S) x_1 = x) \u2194 x \u2208 I\n[PROOFSTEP]\nexact\n \u27e8by\n rintro \u27e8x, \u27e8h, rfl\u27e9\u27e9\n exact h.1, fun h => \u27e8\u27e8x, 0\u27e9, \u27e8\u27e8h, Ideal.zero_mem _\u27e9, rfl\u27e9\u27e9\u27e9\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx : R\n\u22a2 (\u2203 x_1, x_1 \u2208 prod I J \u2227 \u2191(RingHom.fst R S) x_1 = x) \u2192 x \u2208 I\n[PROOFSTEP]\nrintro \u27e8x, \u27e8h, rfl\u27e9\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx : R \u00d7 S\nh : x \u2208 prod I J\n\u22a2 \u2191(RingHom.fst R S) x \u2208 I\n[PROOFSTEP]\nexact h.1\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\n\u22a2 map (RingHom.snd R S) (prod I J) = J\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx : S\n\u22a2 x \u2208 map (RingHom.snd R S) (prod I J) \u2194 x \u2208 J\n[PROOFSTEP]\nrw [mem_map_iff_of_surjective (RingHom.snd R S) Prod.snd_surjective]\n[GOAL]\ncase h\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx : S\n\u22a2 (\u2203 x_1, x_1 \u2208 prod I J \u2227 \u2191(RingHom.snd R S) x_1 = x) \u2194 x \u2208 J\n[PROOFSTEP]\nexact\n \u27e8by\n rintro \u27e8x, \u27e8h, rfl\u27e9\u27e9\n exact h.2, fun h => \u27e8\u27e80, x\u27e9, \u27e8\u27e8Ideal.zero_mem _, h\u27e9, rfl\u27e9\u27e9\u27e9\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx : S\n\u22a2 (\u2203 x_1, x_1 \u2208 prod I J \u2227 \u2191(RingHom.snd R S) x_1 = x) \u2192 x \u2208 J\n[PROOFSTEP]\nrintro \u27e8x, \u27e8h, rfl\u27e9\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx : R \u00d7 S\nh : x \u2208 prod I J\n\u22a2 \u2191(RingHom.snd R S) x \u2208 J\n[PROOFSTEP]\nexact h.2\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI I' : Ideal R\nJ J' : Ideal S\n\u22a2 map (\u2191RingEquiv.prodComm) (prod I J) = prod J I\n[PROOFSTEP]\nrefine' Trans.trans (ideal_prod_eq _) _\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI I' : Ideal R\nJ J' : Ideal S\n\u22a2 prod (map (RingHom.fst S R) (map (\u2191RingEquiv.prodComm) (prod I J)))\n (map (RingHom.snd S R) (map (\u2191RingEquiv.prodComm) (prod I J))) =\n prod J I\n[PROOFSTEP]\nsimp [map_map]\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nx\u271d : Ideal R \u00d7 Ideal S\nI : Ideal R\nJ : Ideal S\n\u22a2 (fun I => (map (RingHom.fst R S) I, map (RingHom.snd R S) I)) ((fun I => prod I.fst I.snd) (I, J)) = (I, J)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I'\u271d : Ideal R\nJ\u271d J'\u271d : Ideal S\nI I' : Ideal R\nJ J' : Ideal S\n\u22a2 prod I J = prod I' J' \u2194 I = I' \u2227 J = J'\n[PROOFSTEP]\nsimp only [\u2190 idealProdEquiv_symm_apply, idealProdEquiv.symm.injective.eq_iff, Prod.mk.inj_iff]\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : IsPrime (prod I \u22a4)\n\u22a2 IsPrime I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase ne_top'\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : IsPrime (prod I \u22a4)\n\u22a2 I \u2260 \u22a4\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase ne_top'\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : I = \u22a4\n\u22a2 \u00acIsPrime (prod I \u22a4)\n[PROOFSTEP]\nrw [h, prod_top_top, isPrime_iff]\n[GOAL]\ncase ne_top'\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : I = \u22a4\n\u22a2 \u00ac(\u22a4 \u2260 \u22a4 \u2227 \u2200 {x y : R \u00d7 S}, x * y \u2208 \u22a4 \u2192 x \u2208 \u22a4 \u2228 y \u2208 \u22a4)\n[PROOFSTEP]\nsimp [isPrime_iff, h]\n[GOAL]\ncase mem_or_mem'\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : IsPrime (prod I \u22a4)\n\u22a2 \u2200 {x y : R}, x * y \u2208 I \u2192 x \u2208 I \u2228 y \u2208 I\n[PROOFSTEP]\nintro x y hxy\n[GOAL]\ncase mem_or_mem'\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : IsPrime (prod I \u22a4)\nx y : R\nhxy : x * y \u2208 I\n\u22a2 x \u2208 I \u2228 y \u2208 I\n[PROOFSTEP]\nhave : (\u27e8x, 1\u27e9 : R \u00d7 S) * \u27e8y, 1\u27e9 \u2208 prod I \u22a4 :=\n by\n rw [Prod.mk_mul_mk, mul_one, mem_prod]\n exact \u27e8hxy, trivial\u27e9\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : IsPrime (prod I \u22a4)\nx y : R\nhxy : x * y \u2208 I\n\u22a2 (x, 1) * (y, 1) \u2208 prod I \u22a4\n[PROOFSTEP]\nrw [Prod.mk_mul_mk, mul_one, mem_prod]\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : IsPrime (prod I \u22a4)\nx y : R\nhxy : x * y \u2208 I\n\u22a2 x * y \u2208 I \u2227 1 \u2208 \u22a4\n[PROOFSTEP]\nexact \u27e8hxy, trivial\u27e9\n[GOAL]\ncase mem_or_mem'\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : IsPrime (prod I \u22a4)\nx y : R\nhxy : x * y \u2208 I\nthis : (x, 1) * (y, 1) \u2208 prod I \u22a4\n\u22a2 x \u2208 I \u2228 y \u2208 I\n[PROOFSTEP]\nsimpa using h.mem_or_mem this\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' I : Ideal S\nh : IsPrime (prod \u22a4 I)\n\u22a2 IsPrime I\n[PROOFSTEP]\napply @isPrime_of_isPrime_prod_top _ R\n[GOAL]\ncase h\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' I : Ideal S\nh : IsPrime (prod \u22a4 I)\n\u22a2 IsPrime (prod I \u22a4)\n[PROOFSTEP]\nrw [\u2190 map_prodComm_prod]\n[GOAL]\ncase h\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' I : Ideal S\nh : IsPrime (prod \u22a4 I)\n\u22a2 IsPrime (map (\u2191RingEquiv.prodComm) (prod \u22a4 I))\n[PROOFSTEP]\nexact map_isPrime_of_equiv _\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : IsPrime I\n\u22a2 IsPrime (prod I \u22a4)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase ne_top'\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : IsPrime I\n\u22a2 prod I \u22a4 \u2260 \u22a4\n[PROOFSTEP]\nrcases h with \u27e8h, -\u27e9\n[GOAL]\ncase ne_top'.mk\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : I \u2260 \u22a4\n\u22a2 prod I \u22a4 \u2260 \u22a4\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase ne_top'.mk\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : prod I \u22a4 = \u22a4\n\u22a2 I = \u22a4\n[PROOFSTEP]\nrw [\u2190 prod_top_top, prod.ext_iff] at h \n[GOAL]\ncase ne_top'.mk\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : I = \u22a4 \u2227 \u22a4 = \u22a4\n\u22a2 I = \u22a4\n[PROOFSTEP]\nexact h.1\n[GOAL]\ncase mem_or_mem'\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : IsPrime I\n\u22a2 \u2200 {x y : R \u00d7 S}, x * y \u2208 prod I \u22a4 \u2192 x \u2208 prod I \u22a4 \u2228 y \u2208 prod I \u22a4\n[PROOFSTEP]\nrintro \u27e8r\u2081, s\u2081\u27e9 \u27e8r\u2082, s\u2082\u27e9 \u27e8h\u2081, _\u27e9\n[GOAL]\ncase mem_or_mem'.mk.mk.intro\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh : IsPrime I\nr\u2081 : R\ns\u2081 : S\nr\u2082 : R\ns\u2082 : S\nh\u2081 : ((r\u2081, s\u2081) * (r\u2082, s\u2082)).fst \u2208 I\nright\u271d : ((r\u2081, s\u2081) * (r\u2082, s\u2082)).snd \u2208 \u22a4\n\u22a2 (r\u2081, s\u2081) \u2208 prod I \u22a4 \u2228 (r\u2082, s\u2082) \u2208 prod I \u22a4\n[PROOFSTEP]\ncases' h.mem_or_mem h\u2081 with h h\n[GOAL]\ncase mem_or_mem'.mk.mk.intro.inl\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh\u271d : IsPrime I\nr\u2081 : R\ns\u2081 : S\nr\u2082 : R\ns\u2082 : S\nh\u2081 : ((r\u2081, s\u2081) * (r\u2082, s\u2082)).fst \u2208 I\nright\u271d : ((r\u2081, s\u2081) * (r\u2082, s\u2082)).snd \u2208 \u22a4\nh : (r\u2081, s\u2081).fst \u2208 I\n\u22a2 (r\u2081, s\u2081) \u2208 prod I \u22a4 \u2228 (r\u2082, s\u2082) \u2208 prod I \u22a4\n[PROOFSTEP]\nexact Or.inl \u27e8h, trivial\u27e9\n[GOAL]\ncase mem_or_mem'.mk.mk.intro.inr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal R\nh\u271d : IsPrime I\nr\u2081 : R\ns\u2081 : S\nr\u2082 : R\ns\u2082 : S\nh\u2081 : ((r\u2081, s\u2081) * (r\u2082, s\u2082)).fst \u2208 I\nright\u271d : ((r\u2081, s\u2081) * (r\u2082, s\u2082)).snd \u2208 \u22a4\nh : (r\u2082, s\u2082).fst \u2208 I\n\u22a2 (r\u2081, s\u2081) \u2208 prod I \u22a4 \u2228 (r\u2082, s\u2082) \u2208 prod I \u22a4\n[PROOFSTEP]\nexact Or.inr \u27e8h, trivial\u27e9\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' I : Ideal S\nh : IsPrime I\n\u22a2 IsPrime (prod \u22a4 I)\n[PROOFSTEP]\nletI : IsPrime (prod I (\u22a4 : Ideal R)) := isPrime_ideal_prod_top\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' I : Ideal S\nh : IsPrime I\nthis : IsPrime (prod I \u22a4) := isPrime_ideal_prod_top\n\u22a2 IsPrime (prod \u22a4 I)\n[PROOFSTEP]\nrw [\u2190 map_prodComm_prod]\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' I : Ideal S\nh : IsPrime I\nthis : IsPrime (prod I \u22a4) := isPrime_ideal_prod_top\n\u22a2 IsPrime (map (\u2191RingEquiv.prodComm) (prod I \u22a4))\n[PROOFSTEP]\napply map_isPrime_of_equiv _\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\n\u22a2 IsPrime (prod I J) \u2192 I = \u22a4 \u2228 J = \u22a4\n[PROOFSTEP]\ncontrapose!\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\n\u22a2 I \u2260 \u22a4 \u2227 J \u2260 \u22a4 \u2192 \u00acIsPrime (prod I J)\n[PROOFSTEP]\nsimp only [ne_top_iff_one, isPrime_iff, not_and, not_forall, not_or]\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\n\u22a2 \u00ac1 \u2208 I \u2227 \u00ac1 \u2208 J \u2192 \u00ac1 \u2208 prod I J \u2192 \u2203 x x_1 h, \u00acx \u2208 prod I J \u2227 \u00acx_1 \u2208 prod I J\n[PROOFSTEP]\nexact fun \u27e8hI, hJ\u27e9 _ => \u27e8\u27e80, 1\u27e9, \u27e81, 0\u27e9, by simp, by simp [hJ], by simp [hI]\u27e9\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx\u271d\u00b9 : \u00ac1 \u2208 I \u2227 \u00ac1 \u2208 J\nx\u271d : \u00ac1 \u2208 prod I J\nhI : \u00ac1 \u2208 I\nhJ : \u00ac1 \u2208 J\n\u22a2 (0, 1) * (1, 0) \u2208 prod I J\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx\u271d\u00b9 : \u00ac1 \u2208 I \u2227 \u00ac1 \u2208 J\nx\u271d : \u00ac1 \u2208 prod I J\nhI : \u00ac1 \u2208 I\nhJ : \u00ac1 \u2208 J\n\u22a2 \u00ac(0, 1) \u2208 prod I J\n[PROOFSTEP]\nsimp [hJ]\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ\u271d J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx\u271d\u00b9 : \u00ac1 \u2208 I \u2227 \u00ac1 \u2208 J\nx\u271d : \u00ac1 \u2208 prod I J\nhI : \u00ac1 \u2208 I\nhJ : \u00ac1 \u2208 J\n\u22a2 \u00ac(1, 0) \u2208 prod I J\n[PROOFSTEP]\nsimp [hI]\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\n\u22a2 IsPrime I \u2194 (\u2203 p, IsPrime p \u2227 I = prod p \u22a4) \u2228 \u2203 p, IsPrime p \u2227 I = prod \u22a4 p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\n\u22a2 IsPrime I \u2192 (\u2203 p, IsPrime p \u2227 I = prod p \u22a4) \u2228 \u2203 p, IsPrime p \u2227 I = prod \u22a4 p\n[PROOFSTEP]\nrw [ideal_prod_eq I]\n[GOAL]\ncase mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\n\u22a2 IsPrime (prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I)) \u2192\n (\u2203 p, IsPrime p \u2227 prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I) = prod p \u22a4) \u2228\n \u2203 p, IsPrime p \u2227 prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I) = prod \u22a4 p\n[PROOFSTEP]\nintro hI\n[GOAL]\ncase mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\nhI : IsPrime (prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I))\n\u22a2 (\u2203 p, IsPrime p \u2227 prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I) = prod p \u22a4) \u2228\n \u2203 p, IsPrime p \u2227 prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I) = prod \u22a4 p\n[PROOFSTEP]\nrcases ideal_prod_prime_aux hI with (h | h)\n[GOAL]\ncase mp.inl\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\nhI : IsPrime (prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I))\nh : map (RingHom.fst R S) I = \u22a4\n\u22a2 (\u2203 p, IsPrime p \u2227 prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I) = prod p \u22a4) \u2228\n \u2203 p, IsPrime p \u2227 prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I) = prod \u22a4 p\n[PROOFSTEP]\nright\n[GOAL]\ncase mp.inl.h\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\nhI : IsPrime (prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I))\nh : map (RingHom.fst R S) I = \u22a4\n\u22a2 \u2203 p, IsPrime p \u2227 prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I) = prod \u22a4 p\n[PROOFSTEP]\nrw [h] at hI \u22a2\n[GOAL]\ncase mp.inl.h\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\nhI : IsPrime (prod \u22a4 (map (RingHom.snd R S) I))\nh : map (RingHom.fst R S) I = \u22a4\n\u22a2 \u2203 p, IsPrime p \u2227 prod \u22a4 (map (RingHom.snd R S) I) = prod \u22a4 p\n[PROOFSTEP]\nexact \u27e8_, \u27e8isPrime_of_isPrime_prod_top' hI, rfl\u27e9\u27e9\n[GOAL]\ncase mp.inr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\nhI : IsPrime (prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I))\nh : map (RingHom.snd R S) I = \u22a4\n\u22a2 (\u2203 p, IsPrime p \u2227 prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I) = prod p \u22a4) \u2228\n \u2203 p, IsPrime p \u2227 prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I) = prod \u22a4 p\n[PROOFSTEP]\nleft\n[GOAL]\ncase mp.inr.h\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\nhI : IsPrime (prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I))\nh : map (RingHom.snd R S) I = \u22a4\n\u22a2 \u2203 p, IsPrime p \u2227 prod (map (RingHom.fst R S) I) (map (RingHom.snd R S) I) = prod p \u22a4\n[PROOFSTEP]\nrw [h] at hI \u22a2\n[GOAL]\ncase mp.inr.h\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\nhI : IsPrime (prod (map (RingHom.fst R S) I) \u22a4)\nh : map (RingHom.snd R S) I = \u22a4\n\u22a2 \u2203 p, IsPrime p \u2227 prod (map (RingHom.fst R S) I) \u22a4 = prod p \u22a4\n[PROOFSTEP]\nexact \u27e8_, \u27e8isPrime_of_isPrime_prod_top hI, rfl\u27e9\u27e9\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I' : Ideal R\nJ J' : Ideal S\nI : Ideal (R \u00d7 S)\n\u22a2 ((\u2203 p, IsPrime p \u2227 I = prod p \u22a4) \u2228 \u2203 p, IsPrime p \u2227 I = prod \u22a4 p) \u2192 IsPrime I\n[PROOFSTEP]\nrintro (\u27e8p, \u27e8h, rfl\u27e9\u27e9 | \u27e8p, \u27e8h, rfl\u27e9\u27e9)\n[GOAL]\ncase mpr.inl.intro.intro\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI I' : Ideal R\nJ J' : Ideal S\np : Ideal R\nh : IsPrime p\n\u22a2 IsPrime (prod p \u22a4)\n[PROOFSTEP]\nexact isPrime_ideal_prod_top\n[GOAL]\ncase mpr.inr.intro.intro\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI I' : Ideal R\nJ J' p : Ideal S\nh : IsPrime p\n\u22a2 IsPrime (prod \u22a4 p)\n[PROOFSTEP]\nexact isPrime_ideal_prod_top'\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Ideal.Prod", "llama_tokens": 10392, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.766293653760418, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.44253103341794264}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nx : \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 essSup f \u03bc = sInf {a | \u2191\u2191\u03bc {x | a < f x} = 0}\n[PROOFSTEP]\ndsimp [essSup, limsup, limsSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nx : \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 sInf {a | \u2200\u1da0 (n : \u03b2) in map f (Measure.ae \u03bc), n \u2264 a} = sInf {a | \u2191\u2191\u03bc {x | a < f x} = 0}\n[PROOFSTEP]\nsimp only [eventually_map, ae_iff, not_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nx : \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 essInf f \u03bc = sSup {a | \u2191\u2191\u03bc {x | f x < a} = 0}\n[PROOFSTEP]\ndsimp [essInf, liminf, limsInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nx : \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 sSup {a | \u2200\u1da0 (n : \u03b2) in map f (Measure.ae \u03bc), a \u2264 n} = sSup {a | \u2191\u2191\u03bc {x | f x < a} = 0}\n[PROOFSTEP]\nsimp only [eventually_map, ae_iff, not_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2\nx : \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : FirstCountableTopology \u03b2\ninst\u271d : OrderTopology \u03b2\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f) _auto\u271d\n\u22a2 \u2191\u2191\u03bc {y | essSup f \u03bc < f y} = 0\n[PROOFSTEP]\nsimp_rw [\u2190 not_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2\nx : \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : FirstCountableTopology \u03b2\ninst\u271d : OrderTopology \u03b2\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f) _auto\u271d\n\u22a2 \u2191\u2191\u03bc {y | \u00acf y \u2264 essSup f \u03bc} = 0\n[PROOFSTEP]\nexact ae_le_essSup hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2\nx : \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : FirstCountableTopology \u03b2\ninst\u271d : OrderTopology \u03b2\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bc) f) _auto\u271d\n\u22a2 \u2191\u2191\u03bc {y | f y < essInf f \u03bc} = 0\n[PROOFSTEP]\nsimp_rw [\u2190 not_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2\nx : \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : FirstCountableTopology \u03b2\ninst\u271d : OrderTopology \u03b2\nhf : autoParam (IsBoundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bc) f) _auto\u271d\n\u22a2 \u2191\u2191\u03bc {y | \u00acessInf f \u03bc \u2264 f y} = 0\n[PROOFSTEP]\nexact ae_essInf_le hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u22a5 \u2208 {a | \u2200\u1da0 (n : \u03b2) in map f (Measure.ae 0), n \u2264 a}\n[PROOFSTEP]\nsimp [Set.mem_setOf_eq, EventuallyLE, ae_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhf : f \u2264\u1d50[\u03bc] fun x => c\n\u22a2 essSup f \u03bc \u2264 c\n[PROOFSTEP]\nrefine' (essSup_mono_ae hf).trans _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhf : f \u2264\u1d50[\u03bc] fun x => c\n\u22a2 essSup (fun x => c) \u03bc \u2264 c\n[PROOFSTEP]\nby_cases h\u03bc : \u03bc = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhf : f \u2264\u1d50[\u03bc] fun x => c\nh\u03bc : \u03bc = 0\n\u22a2 essSup (fun x => c) \u03bc \u2264 c\n[PROOFSTEP]\nsimp [h\u03bc]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhf : f \u2264\u1d50[\u03bc] fun x => c\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 essSup (fun x => c) \u03bc \u2264 c\n[PROOFSTEP]\nrwa [essSup_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b2\nm : MeasurableSpace \u03b1\n\u03b3 : Type u_3\ninst\u271d : CompleteLattice \u03b3\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\ng : \u03b2 \u2243o \u03b3\n\u22a2 \u2191g (essSup f \u03bc) = essSup (fun x => \u2191g (f x)) \u03bc\n[PROOFSTEP]\nrefine' OrderIso.limsup_apply g _ _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b2\nm : MeasurableSpace \u03b1\n\u03b3 : Type u_3\ninst\u271d : CompleteLattice \u03b3\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\ng : \u03b2 \u2243o \u03b3\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b2\nm : MeasurableSpace \u03b1\n\u03b3 : Type u_3\ninst\u271d : CompleteLattice \u03b3\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\ng : \u03b2 \u2243o \u03b3\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b2\nm : MeasurableSpace \u03b1\n\u03b3 : Type u_3\ninst\u271d : CompleteLattice \u03b3\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\ng : \u03b2 \u2243o \u03b3\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) fun x => \u2191g (f x)\ncase refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b2\nm : MeasurableSpace \u03b1\n\u03b3 : Type u_3\ninst\u271d : CompleteLattice \u03b3\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\ng : \u03b2 \u2243o \u03b3\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) fun x => \u2191g (f x)\n[PROOFSTEP]\nall_goals isBoundedDefault\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b2\nm : MeasurableSpace \u03b1\n\u03b3 : Type u_3\ninst\u271d : CompleteLattice \u03b3\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\ng : \u03b2 \u2243o \u03b3\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b2\nm : MeasurableSpace \u03b1\n\u03b3 : Type u_3\ninst\u271d : CompleteLattice \u03b3\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\ng : \u03b2 \u2243o \u03b3\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b2\nm : MeasurableSpace \u03b1\n\u03b3 : Type u_3\ninst\u271d : CompleteLattice \u03b3\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\ng : \u03b2 \u2243o \u03b3\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) fun x => \u2191g (f x)\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b2\nm : MeasurableSpace \u03b1\n\u03b3 : Type u_3\ninst\u271d : CompleteLattice \u03b3\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\ng : \u03b2 \u2243o \u03b3\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) fun x => \u2191g (f x)\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bd \u226a \u03bc\n\u22a2 essSup f \u03bd \u2264 essSup f \u03bc\n[PROOFSTEP]\nrefine' limsup_le_limsup_of_le (Measure.ae_le_iff_absolutelyContinuous.mpr h\u03bc\u03bd) _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bd \u226a \u03bc\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bd) f\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bd \u226a \u03bc\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f\n[PROOFSTEP]\nall_goals isBoundedDefault\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bd \u226a \u03bc\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bd) f\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bd \u226a \u03bc\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bc \u226a \u03bd\n\u22a2 essInf f \u03bd \u2264 essInf f \u03bc\n[PROOFSTEP]\nrefine' liminf_le_liminf_of_le (Measure.ae_le_iff_absolutelyContinuous.mpr h\u03bc\u03bd) _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bc \u226a \u03bd\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bd) f\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bc \u226a \u03bd\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bc) f\n[PROOFSTEP]\nall_goals isBoundedDefault\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bc \u226a \u03bd\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bd) f\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u03bc\u03bd : \u03bc \u226a \u03bd\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bc) f\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u211d\u22650\u221e\nhc : c \u2260 0\n\u22a2 essSup f (c \u2022 \u03bc) = essSup f \u03bc\n[PROOFSTEP]\nsimp_rw [essSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u211d\u22650\u221e\nhc : c \u2260 0\n\u22a2 limsup f (Measure.ae (c \u2022 \u03bc)) = limsup f (Measure.ae \u03bc)\n[PROOFSTEP]\nsuffices h_smul : (c \u2022 \u03bc).ae = \u03bc.ae\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u211d\u22650\u221e\nhc : c \u2260 0\nh_smul : Measure.ae (c \u2022 \u03bc) = Measure.ae \u03bc\n\u22a2 limsup f (Measure.ae (c \u2022 \u03bc)) = limsup f (Measure.ae \u03bc)\n[PROOFSTEP]\nrw [h_smul]\n[GOAL]\ncase h_smul\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u211d\u22650\u221e\nhc : c \u2260 0\n\u22a2 Measure.ae (c \u2022 \u03bc) = Measure.ae \u03bc\n[PROOFSTEP]\next1\n[GOAL]\ncase h_smul.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u211d\u22650\u221e\nhc : c \u2260 0\ns\u271d : Set \u03b1\n\u22a2 s\u271d \u2208 Measure.ae (c \u2022 \u03bc) \u2194 s\u271d \u2208 Measure.ae \u03bc\n[PROOFSTEP]\nsimp_rw [mem_ae_iff]\n[GOAL]\ncase h_smul.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u211d\u22650\u221e\nhc : c \u2260 0\ns\u271d : Set \u03b1\n\u22a2 \u2191\u2191(c \u2022 \u03bc) s\u271d\u1d9c = 0 \u2194 \u2191\u2191\u03bc s\u271d\u1d9c = 0\n[PROOFSTEP]\nsimp [hc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 essSup (g \u2218 f) \u03bc \u2264 essSup g (Measure.map f \u03bc)\n[PROOFSTEP]\nrefine' limsSup_le_limsSup_of_le (fun t => _) (by isBoundedDefault) (by isBoundedDefault)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 IsCobounded (fun x x_1 => x \u2264 x_1) (map (g \u2218 f) (Measure.ae \u03bc))\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 IsBounded (fun x x_1 => x \u2264 x_1) (map g (Measure.ae (Measure.map f \u03bc)))\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\nhf : AEMeasurable f\nt : Set \u03b2\n\u22a2 t \u2208 map g (Measure.ae (Measure.map f \u03bc)) \u2192 t \u2208 map (g \u2218 f) (Measure.ae \u03bc)\n[PROOFSTEP]\nsimp_rw [Filter.mem_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\nhf : AEMeasurable f\nt : Set \u03b2\n\u22a2 g \u207b\u00b9' t \u2208 Measure.ae (Measure.map f \u03bc) \u2192 g \u2218 f \u207b\u00b9' t \u2208 Measure.ae \u03bc\n[PROOFSTEP]\nhave : g \u2218 f \u207b\u00b9' t = f \u207b\u00b9' (g \u207b\u00b9' t) := by\n ext1 x\n simp_rw [Set.mem_preimage, Function.comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\nhf : AEMeasurable f\nt : Set \u03b2\n\u22a2 g \u2218 f \u207b\u00b9' t = f \u207b\u00b9' (g \u207b\u00b9' t)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\nhf : AEMeasurable f\nt : Set \u03b2\nx : \u03b1\n\u22a2 x \u2208 g \u2218 f \u207b\u00b9' t \u2194 x \u2208 f \u207b\u00b9' (g \u207b\u00b9' t)\n[PROOFSTEP]\nsimp_rw [Set.mem_preimage, Function.comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\nhf : AEMeasurable f\nt : Set \u03b2\nthis : g \u2218 f \u207b\u00b9' t = f \u207b\u00b9' (g \u207b\u00b9' t)\n\u22a2 g \u207b\u00b9' t \u2208 Measure.ae (Measure.map f \u03bc) \u2192 g \u2218 f \u207b\u00b9' t \u2208 Measure.ae \u03bc\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\nhf : AEMeasurable f\nt : Set \u03b2\nthis : g \u2218 f \u207b\u00b9' t = f \u207b\u00b9' (g \u207b\u00b9' t)\n\u22a2 g \u207b\u00b9' t \u2208 Measure.ae (Measure.map f \u03bc) \u2192 f \u207b\u00b9' (g \u207b\u00b9' t) \u2208 Measure.ae \u03bc\n[PROOFSTEP]\nexact fun h => mem_ae_of_mem_ae_map hf h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u22a2 essSup g (Measure.map f \u03bc) = essSup (g \u2218 f) \u03bc\n[PROOFSTEP]\nrefine' le_antisymm _ (essSup_comp_le_essSup_map_measure hf.measurable.aemeasurable)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u22a2 essSup g (Measure.map f \u03bc) \u2264 essSup (g \u2218 f) \u03bc\n[PROOFSTEP]\nrefine' limsSup_le_limsSup (by isBoundedDefault) (by isBoundedDefault) (fun c h_le => _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u22a2 IsCobounded (fun x x_1 => x \u2264 x_1) (map g (Measure.ae (Measure.map f \u03bc)))\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u22a2 IsBounded (fun x x_1 => x \u2264 x_1) (map (g \u2218 f) (Measure.ae \u03bc))\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\nhf : MeasurableEmbedding f\nc : \u03b2\nh_le : \u2200\u1da0 (n : \u03b2) in map (g \u2218 f) (Measure.ae \u03bc), n \u2264 c\n\u22a2 \u2200\u1da0 (n : \u03b2) in map g (Measure.ae (Measure.map f \u03bc)), n \u2264 c\n[PROOFSTEP]\nrw [eventually_map] at h_le \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\nhf : MeasurableEmbedding f\nc : \u03b2\nh_le : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, (g \u2218 f) a \u2264 c\n\u22a2 \u2200\u1d50 (a : \u03b3) \u2202Measure.map f \u03bc, g a \u2264 c\n[PROOFSTEP]\nexact hf.ae_map_iff.mpr h_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : Measurable g\nhf : AEMeasurable f\n\u22a2 essSup g (Measure.map f \u03bc) = essSup (g \u2218 f) \u03bc\n[PROOFSTEP]\nrefine' le_antisymm _ (essSup_comp_le_essSup_map_measure hf)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : Measurable g\nhf : AEMeasurable f\n\u22a2 essSup g (Measure.map f \u03bc) \u2264 essSup (g \u2218 f) \u03bc\n[PROOFSTEP]\nrefine' limsSup_le_limsSup (by isBoundedDefault) (by isBoundedDefault) (fun c h_le => _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : Measurable g\nhf : AEMeasurable f\n\u22a2 IsCobounded (fun x x_1 => x \u2264 x_1) (map g (Measure.ae (Measure.map f \u03bc)))\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : Measurable g\nhf : AEMeasurable f\n\u22a2 IsBounded (fun x x_1 => x \u2264 x_1) (map (g \u2218 f) (Measure.ae \u03bc))\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : Measurable g\nhf : AEMeasurable f\nc : \u03b2\nh_le : \u2200\u1da0 (n : \u03b2) in map (g \u2218 f) (Measure.ae \u03bc), n \u2264 c\n\u22a2 \u2200\u1da0 (n : \u03b2) in map g (Measure.ae (Measure.map f \u03bc)), n \u2264 c\n[PROOFSTEP]\nrw [eventually_map] at h_le \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : Measurable g\nhf : AEMeasurable f\nc : \u03b2\nh_le : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, (g \u2218 f) a \u2264 c\n\u22a2 \u2200\u1d50 (a : \u03b3) \u2202Measure.map f \u03bc, g a \u2264 c\n[PROOFSTEP]\nrw [ae_map_iff hf (measurableSet_le hg measurable_const)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : Measurable g\nhf : AEMeasurable f\nc : \u03b2\nh_le : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, (g \u2218 f) a \u2264 c\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g (f x) \u2264 c\n[PROOFSTEP]\nexact h_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : AEMeasurable g\nhf : AEMeasurable f\n\u22a2 essSup g (Measure.map f \u03bc) = essSup (g \u2218 f) \u03bc\n[PROOFSTEP]\nrw [essSup_congr_ae hg.ae_eq_mk, essSup_map_measure_of_measurable hg.measurable_mk hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : AEMeasurable g\nhf : AEMeasurable f\n\u22a2 essSup (AEMeasurable.mk g hg \u2218 f) \u03bc = essSup (g \u2218 f) \u03bc\n[PROOFSTEP]\nrefine' essSup_congr_ae _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : AEMeasurable g\nhf : AEMeasurable f\n\u22a2 AEMeasurable.mk g hg \u2218 f =\u1d50[\u03bc] g \u2218 f\n[PROOFSTEP]\nhave h_eq := ae_of_ae_map hf hg.ae_eq_mk\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : AEMeasurable g\nhf : AEMeasurable f\nh_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g (f x) = AEMeasurable.mk g hg (f x)\n\u22a2 AEMeasurable.mk g hg \u2218 f =\u1d50[\u03bc] g \u2218 f\n[PROOFSTEP]\nrw [\u2190 EventuallyEq] at h_eq \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2075 : CompleteLattice \u03b2\n\u03b3 : Type u_3\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\nhg : AEMeasurable g\nhf : AEMeasurable f\nh_eq : (fun x => g (f x)) =\u1d50[\u03bc] fun x => AEMeasurable.mk g hg (f x)\n\u22a2 AEMeasurable.mk g hg \u2218 f =\u1d50[\u03bc] g \u2218 f\n[PROOFSTEP]\nexact h_eq.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\n\u22a2 essSup (indicator s f) \u03bc = essSup f (Measure.restrict \u03bc s)\n[PROOFSTEP]\nrefine'\n le_antisymm _\n (limsSup_le_limsSup_of_le (map_restrict_ae_le_map_indicator_ae hs) (by isBoundedDefault) (by isBoundedDefault))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\n\u22a2 IsCobounded (fun x x_1 => x \u2264 x_1) (map f (Measure.ae (Measure.restrict \u03bc s)))\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\n\u22a2 IsBounded (fun x x_1 => x \u2264 x_1) (map (indicator s f) (Measure.ae \u03bc))\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\n\u22a2 essSup (indicator s f) \u03bc \u2264 essSup f (Measure.restrict \u03bc s)\n[PROOFSTEP]\nrefine' limsSup_le_limsSup (by isBoundedDefault) (by isBoundedDefault) (fun c h_restrict_le => _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\n\u22a2 IsCobounded (fun x x_1 => x \u2264 x_1) (map (indicator s f) (Measure.ae \u03bc))\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\n\u22a2 IsBounded (fun x x_1 => x \u2264 x_1) (map f (Measure.ae (Measure.restrict \u03bc s)))\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1da0 (n : \u03b2) in map f (Measure.ae (Measure.restrict \u03bc s)), n \u2264 c\n\u22a2 \u2200\u1da0 (n : \u03b2) in map (indicator s f) (Measure.ae \u03bc), n \u2264 c\n[PROOFSTEP]\nrw [eventually_map] at h_restrict_le \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (a : \u03b1) \u2202Measure.restrict \u03bc s, f a \u2264 c\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, indicator s f a \u2264 c\n[PROOFSTEP]\nrw [ae_restrict_iff' hs] at h_restrict_le \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, indicator s f a \u2264 c\n[PROOFSTEP]\nhave hc : 0 \u2264 c := by\n rsuffices \u27e8x, hx\u27e9 : \u2203 x, 0 \u2264 f x \u2227 f x \u2264 c\n exact hx.1.trans hx.2\n refine' Frequently.exists _\n \u00b7 exact \u03bc.ae\n rw [EventuallyLE, ae_restrict_iff' hs] at hf \n have hs' : \u2203\u1d50 x \u2202\u03bc, x \u2208 s := by\n contrapose! hs_not_null\n rw [not_frequently, ae_iff] at hs_not_null \n suffices {a : \u03b1 | \u00aca \u2209 s} = s by rwa [\u2190 this]\n simp\n refine' hs'.mp (hf.mp (h_restrict_le.mono fun x hxs_imp_c hxf_nonneg hxs => _))\n rw [Pi.zero_apply] at hxf_nonneg \n exact \u27e8hxf_nonneg hxs, hxs_imp_c hxs\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 0 \u2264 c\n[PROOFSTEP]\nrsuffices \u27e8x, hx\u27e9 : \u2203 x, 0 \u2264 f x \u2227 f x \u2264 c\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nx : \u03b1\nhx : 0 \u2264 f x \u2227 f x \u2264 c\n\u22a2 0 \u2264 c\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2203 x, 0 \u2264 f x \u2227 f x \u2264 c\n[PROOFSTEP]\nexact hx.1.trans hx.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2203 x, 0 \u2264 f x \u2227 f x \u2264 c\n[PROOFSTEP]\nrefine' Frequently.exists _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 Filter \u03b1\n[PROOFSTEP]\nexact \u03bc.ae\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2203\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 f x \u2227 f x \u2264 c\n[PROOFSTEP]\nrw [EventuallyLE, ae_restrict_iff' hs] at hf \n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2203\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 f x \u2227 f x \u2264 c\n[PROOFSTEP]\nhave hs' : \u2203\u1d50 x \u2202\u03bc, x \u2208 s := by\n contrapose! hs_not_null\n rw [not_frequently, ae_iff] at hs_not_null \n suffices {a : \u03b1 | \u00aca \u2209 s} = s by rwa [\u2190 this]\n simp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\n\u22a2 \u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\n[PROOFSTEP]\ncontrapose! hs_not_null\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs_not_null : \u00ac\u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\n\u22a2 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nrw [not_frequently, ae_iff] at hs_not_null \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs_not_null : \u2191\u2191\u03bc {a | \u00ac\u00aca \u2208 s} = 0\n\u22a2 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nsuffices {a : \u03b1 | \u00aca \u2209 s} = s by rwa [\u2190 this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs_not_null : \u2191\u2191\u03bc {a | \u00ac\u00aca \u2208 s} = 0\nthis : {a | \u00ac\u00aca \u2208 s} = s\n\u22a2 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nrwa [\u2190 this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs_not_null : \u2191\u2191\u03bc {a | \u00ac\u00aca \u2208 s} = 0\n\u22a2 {a | \u00ac\u00aca \u2208 s} = s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs' : \u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\n\u22a2 \u2203\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 f x \u2227 f x \u2264 c\n[PROOFSTEP]\nrefine' hs'.mp (hf.mp (h_restrict_le.mono fun x hxs_imp_c hxf_nonneg hxs => _))\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs' : \u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\nx : \u03b1\nhxs_imp_c : x \u2208 s \u2192 f x \u2264 c\nhxf_nonneg : x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhxs : x \u2208 s\n\u22a2 0 \u2264 f x \u2227 f x \u2264 c\n[PROOFSTEP]\nrw [Pi.zero_apply] at hxf_nonneg \n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhs' : \u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\nx : \u03b1\nhxs_imp_c : x \u2208 s \u2192 f x \u2264 c\nhxf_nonneg : x \u2208 s \u2192 0 \u2264 f x\nhxs : x \u2208 s\n\u22a2 0 \u2264 f x \u2227 f x \u2264 c\n[PROOFSTEP]\nexact \u27e8hxf_nonneg hxs, hxs_imp_c hxs\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhc : 0 \u2264 c\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, indicator s f a \u2264 c\n[PROOFSTEP]\nrefine' h_restrict_le.mono fun x hxc => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhc : 0 \u2264 c\nx : \u03b1\nhxc : x \u2208 s \u2192 f x \u2264 c\n\u22a2 indicator s f x \u2264 c\n[PROOFSTEP]\nby_cases hxs : x \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhc : 0 \u2264 c\nx : \u03b1\nhxc : x \u2208 s \u2192 f x \u2264 c\nhxs : x \u2208 s\n\u22a2 indicator s f x \u2264 c\n[PROOFSTEP]\nsimpa [hxs] using hxc hxs\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : CompleteLinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f\nhs : MeasurableSet s\nhs_not_null : \u2191\u2191\u03bc s \u2260 0\nc : \u03b2\nh_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c\nhc : 0 \u2264 c\nx : \u03b1\nhxc : x \u2208 s \u2192 f x \u2264 c\nhxs : \u00acx \u2208 s\n\u22a2 indicator s f x \u2264 c\n[PROOFSTEP]\nsimpa [hxs] using hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\n\u03b9 : Type u_3\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 essSup (fun x => liminf (fun n => f n x) atTop) \u03bc \u2264 liminf (fun n => essSup (fun x => f n x) \u03bc) atTop\n[PROOFSTEP]\nsimp_rw [essSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\n\u03b9 : Type u_3\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 limsup (fun x => liminf (fun n => f n x) atTop) (Measure.ae \u03bc) \u2264\n liminf (fun n => limsup (fun x => f n x) (Measure.ae \u03bc)) atTop\n[PROOFSTEP]\nexact ENNReal.limsup_liminf_le_liminf_limsup fun a b => f b a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f\nr : \u211d\u22650\u221e\n\u22a2 r \u2264\n \u2a05 (a : \u211d\u22650) (_ : a \u2208 fun x => sets (map f (Measure.ae \u03bc)) {x_1 | (fun x_2 => (fun x x_3 => x \u2264 x_3) x_2 x) x_1}),\n \u2191a \u2194\n r \u2264 essSup (fun x => \u2191(f x)) \u03bc\n[PROOFSTEP]\nsimp [essSup, limsup, limsSup, eventually_map, ENNReal.forall_ennreal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f\nr : \u211d\u22650\u221e\n\u22a2 (\u2200 (i : \u211d\u22650), (i \u2208 fun x => sets (map f (Measure.ae \u03bc)) {x_1 | x_1 \u2264 x}) \u2192 r \u2264 \u2191i) \u2194\n \u2200 (r_1 : \u211d\u22650), (\u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 r_1) \u2192 r \u2264 \u2191r_1\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.EssSup", "llama_tokens": 17236, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300698514777, "lm_q2_score": 0.6297746004557471, "lm_q1q2_score": 0.44243559404886246}} {"text": "[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\nhx : IsIntegral A x\n\u22a2 Monic (minpoly A x)\n[PROOFSTEP]\ndelta minpoly\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\nhx : IsIntegral A x\n\u22a2 Monic\n (if hx : IsIntegral A x then\n WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 \u2227 eval\u2082 (algebraMap A B) x x_1 = 0) hx\n else 0)\n[PROOFSTEP]\nrw [dif_pos hx]\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\nhx : IsIntegral A x\n\u22a2 Monic\n (WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 \u2227 eval\u2082 (algebraMap A B) x x_1 = 0) hx)\n[PROOFSTEP]\nexact (degree_lt_wf.min_mem _ hx).1\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx\u271d : B\nf : B \u2192\u2090[A] B'\nhf : Injective \u2191f\nx : B\n\u22a2 minpoly A (\u2191f x) = minpoly A x\n[PROOFSTEP]\nrefine' dif_ctx_congr (isIntegral_algHom_iff _ hf) (fun _ => _) fun _ => rfl\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx\u271d\u00b9 : B\nf : B \u2192\u2090[A] B'\nhf : Injective \u2191f\nx : B\nx\u271d : IsIntegral A x\n\u22a2 WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 \u2227 eval\u2082 (algebraMap A ((fun x => B') x)) (\u2191f x) x_1 = 0) (_ : IsIntegral A (\u2191f x)) =\n WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 \u2227 eval\u2082 (algebraMap A B) x x_1 = 0) x\u271d\n[PROOFSTEP]\nsimp_rw [\u2190 Polynomial.aeval_def, aeval_algHom, AlgHom.comp_apply, _root_.map_eq_zero_iff f hf]\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\n\u22a2 \u2191(Polynomial.aeval x) (minpoly A x) = 0\n[PROOFSTEP]\ndelta minpoly\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\n\u22a2 \u2191(Polynomial.aeval x)\n (if hx : IsIntegral A x then\n WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 \u2227 eval\u2082 (algebraMap A B) x x_1 = 0) hx\n else 0) =\n 0\n[PROOFSTEP]\nsplit_ifs with hx\n[GOAL]\ncase pos\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\nhx : IsIntegral A x\n\u22a2 \u2191(Polynomial.aeval x)\n (if hx : IsIntegral A x then\n WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 \u2227 eval\u2082 (algebraMap A B) x x_1 = 0) hx\n else 0) =\n 0\n[PROOFSTEP]\nrw [dif_pos hx]\n[GOAL]\ncase pos\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\nhx : IsIntegral A x\n\u22a2 \u2191(Polynomial.aeval x)\n (WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 \u2227 eval\u2082 (algebraMap A B) x x_1 = 0) hx) =\n 0\n[PROOFSTEP]\nexact (degree_lt_wf.min_mem _ hx).2\n[GOAL]\ncase neg\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\nhx : \u00acIsIntegral A x\n\u22a2 \u2191(Polynomial.aeval x)\n (if hx : IsIntegral A x then\n WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 \u2227 eval\u2082 (algebraMap A B) x x_1 = 0) hx\n else 0) =\n 0\n[PROOFSTEP]\nrw [dif_neg hx]\n[GOAL]\ncase neg\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\nhx : \u00acIsIntegral A x\n\u22a2 \u2191(Polynomial.aeval x) 0 = 0\n[PROOFSTEP]\nexact aeval_zero _\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Ring B\ninst\u271d\u00b3 : Ring B'\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra A B'\nx : B\ninst\u271d : Nontrivial B\n\u22a2 minpoly A x \u2260 1\n[PROOFSTEP]\nintro h\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Ring B\ninst\u271d\u00b3 : Ring B'\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra A B'\nx : B\ninst\u271d : Nontrivial B\nh : minpoly A x = 1\n\u22a2 False\n[PROOFSTEP]\nrefine' (one_ne_zero : (1 : B) \u2260 0) _\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Ring B\ninst\u271d\u00b3 : Ring B'\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra A B'\nx : B\ninst\u271d : Nontrivial B\nh : minpoly A x = 1\n\u22a2 1 = 0\n[PROOFSTEP]\nsimpa using congr_arg (Polynomial.aeval x) h\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Ring B\ninst\u271d\u2075 : Ring B'\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Algebra A B'\nx : B\ninst\u271d\u00b2 : Nontrivial B\nR : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Nontrivial R\nf : A \u2192+* R\n\u22a2 map f (minpoly A x) \u2260 1\n[PROOFSTEP]\nby_cases hx : IsIntegral A x\n[GOAL]\ncase pos\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Ring B\ninst\u271d\u2075 : Ring B'\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Algebra A B'\nx : B\ninst\u271d\u00b2 : Nontrivial B\nR : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Nontrivial R\nf : A \u2192+* R\nhx : IsIntegral A x\n\u22a2 map f (minpoly A x) \u2260 1\n[PROOFSTEP]\nexact mt ((monic hx).eq_one_of_map_eq_one f) (ne_one A x)\n[GOAL]\ncase neg\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Ring B\ninst\u271d\u2075 : Ring B'\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Algebra A B'\nx : B\ninst\u271d\u00b2 : Nontrivial B\nR : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Nontrivial R\nf : A \u2192+* R\nhx : \u00acIsIntegral A x\n\u22a2 map f (minpoly A x) \u2260 1\n[PROOFSTEP]\nrw [eq_zero hx, Polynomial.map_zero]\n[GOAL]\ncase neg\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Ring B\ninst\u271d\u2075 : Ring B'\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Algebra A B'\nx : B\ninst\u271d\u00b2 : Nontrivial B\nR : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Nontrivial R\nf : A \u2192+* R\nhx : \u00acIsIntegral A x\n\u22a2 0 \u2260 1\n[PROOFSTEP]\nexact zero_ne_one\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Ring B\ninst\u271d\u00b3 : Ring B'\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra A B'\nx : B\ninst\u271d : Nontrivial B\n\u22a2 \u00acIsUnit (minpoly A x)\n[PROOFSTEP]\nhaveI : Nontrivial A := (algebraMap A B).domain_nontrivial\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Ring B\ninst\u271d\u00b3 : Ring B'\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra A B'\nx : B\ninst\u271d : Nontrivial B\nthis : Nontrivial A\n\u22a2 \u00acIsUnit (minpoly A x)\n[PROOFSTEP]\nby_cases hx : IsIntegral A x\n[GOAL]\ncase pos\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Ring B\ninst\u271d\u00b3 : Ring B'\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra A B'\nx : B\ninst\u271d : Nontrivial B\nthis : Nontrivial A\nhx : IsIntegral A x\n\u22a2 \u00acIsUnit (minpoly A x)\n[PROOFSTEP]\nexact mt (monic hx).eq_one_of_isUnit (ne_one A x)\n[GOAL]\ncase neg\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Ring B\ninst\u271d\u00b3 : Ring B'\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra A B'\nx : B\ninst\u271d : Nontrivial B\nthis : Nontrivial A\nhx : \u00acIsIntegral A x\n\u22a2 \u00acIsUnit (minpoly A x)\n[PROOFSTEP]\nrw [eq_zero hx]\n[GOAL]\ncase neg\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Ring B\ninst\u271d\u00b3 : Ring B'\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra A B'\nx : B\ninst\u271d : Nontrivial B\nthis : Nontrivial A\nhx : \u00acIsIntegral A x\n\u22a2 \u00acIsUnit 0\n[PROOFSTEP]\nexact not_isUnit_zero\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\nhx : degree (minpoly A x) = 1\n\u22a2 x \u2208 RingHom.range (algebraMap A B)\n[PROOFSTEP]\nhave h : IsIntegral A x := by\n by_contra h\n rw [eq_zero h, degree_zero, \u2190 WithBot.coe_one] at hx \n exact ne_of_lt (show \u22a5 < \u21911 from WithBot.bot_lt_coe 1) hx\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\nhx : degree (minpoly A x) = 1\n\u22a2 IsIntegral A x\n[PROOFSTEP]\nby_contra h\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\nhx : degree (minpoly A x) = 1\nh : \u00acIsIntegral A x\n\u22a2 False\n[PROOFSTEP]\nrw [eq_zero h, degree_zero, \u2190 WithBot.coe_one] at hx \n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\nhx : \u22a5 = \u21911\nh : \u00acIsIntegral A x\n\u22a2 False\n[PROOFSTEP]\nexact ne_of_lt (show \u22a5 < \u21911 from WithBot.bot_lt_coe 1) hx\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\nhx : degree (minpoly A x) = 1\nh : IsIntegral A x\n\u22a2 x \u2208 RingHom.range (algebraMap A B)\n[PROOFSTEP]\nhave key := minpoly.aeval A x\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\nhx : degree (minpoly A x) = 1\nh : IsIntegral A x\nkey : \u2191(Polynomial.aeval x) (minpoly A x) = 0\n\u22a2 x \u2208 RingHom.range (algebraMap A B)\n[PROOFSTEP]\nrw [eq_X_add_C_of_degree_eq_one hx, (minpoly.monic h).leadingCoeff, C_1, one_mul, aeval_add, aeval_C, aeval_X, \u2190\n eq_neg_iff_add_eq_zero, \u2190 RingHom.map_neg] at key \n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\nhx : degree (minpoly A x) = 1\nh : IsIntegral A x\nkey : x = \u2191(algebraMap A B) (-coeff (minpoly A x) 0)\n\u22a2 x \u2208 RingHom.range (algebraMap A B)\n[PROOFSTEP]\nexact \u27e8-(minpoly A x).coeff 0, key.symm\u27e9\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\n\u22a2 degree (minpoly A x) \u2264 degree p\n[PROOFSTEP]\ndelta minpoly\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\n\u22a2 degree\n (if hx : IsIntegral A x then\n WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 \u2227 eval\u2082 (algebraMap A B) x x_1 = 0) hx\n else 0) \u2264\n degree p\n[PROOFSTEP]\nsplit_ifs with hx\n[GOAL]\ncase pos\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhx : IsIntegral A x\n\u22a2 degree\n (WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 \u2227 eval\u2082 (algebraMap A B) x x_1 = 0) hx) \u2264\n degree p\n[PROOFSTEP]\nexact le_of_not_lt (degree_lt_wf.not_lt_min _ hx \u27e8pmonic, hp\u27e9)\n[GOAL]\ncase neg\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhx : \u00acIsIntegral A x\n\u22a2 degree 0 \u2264 degree p\n[PROOFSTEP]\nsimp only [degree_zero, bot_le]\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u22a2 p = minpoly A x\n[PROOFSTEP]\nnontriviality A\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\n\u22a2 p = minpoly A x\n[PROOFSTEP]\nhave hx : IsIntegral A x := \u27e8p, hm, hp\u27e9\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\n\u22a2 p = minpoly A x\n[PROOFSTEP]\nobtain h | h := hl _ ((minpoly A x).degree_modByMonic_lt hm)\n[GOAL]\ncase inl\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\n\u22a2 p = minpoly A x\ncase inr\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : \u2191(Polynomial.aeval x) (minpoly A x %\u2098 p) \u2260 0\n\u22a2 p = minpoly A x\n[PROOFSTEP]\nswap\n[GOAL]\ncase inr\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : \u2191(Polynomial.aeval x) (minpoly A x %\u2098 p) \u2260 0\n\u22a2 p = minpoly A x\n[PROOFSTEP]\nexact (h <| (aeval_modByMonic_eq_self_of_root hm hp).trans <| aeval A x).elim\n[GOAL]\ncase inl\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\n\u22a2 p = minpoly A x\n[PROOFSTEP]\nobtain \u27e8r, hr\u27e9 := (dvd_iff_modByMonic_eq_zero hm).1 h\n[GOAL]\ncase inl.intro\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\nr : A[X]\nhr : minpoly A x = p * r\n\u22a2 p = minpoly A x\n[PROOFSTEP]\nrw [hr]\n[GOAL]\ncase inl.intro\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\nr : A[X]\nhr : minpoly A x = p * r\n\u22a2 p = p * r\n[PROOFSTEP]\nhave hlead := congr_arg leadingCoeff hr\n[GOAL]\ncase inl.intro\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\nr : A[X]\nhr : minpoly A x = p * r\nhlead : leadingCoeff (minpoly A x) = leadingCoeff (p * r)\n\u22a2 p = p * r\n[PROOFSTEP]\nrw [mul_comm, leadingCoeff_mul_monic hm, (monic hx).leadingCoeff] at hlead \n[GOAL]\ncase inl.intro\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\nr : A[X]\nhr : minpoly A x = p * r\nhlead : 1 = leadingCoeff r\n\u22a2 p = p * r\n[PROOFSTEP]\nhave : natDegree r \u2264 0 :=\n by\n have hr0 : r \u2260 0 := by\n rintro rfl\n exact ne_zero hx (mul_zero p \u25b8 hr)\n apply_fun natDegree at hr \n rw [hm.natDegree_mul' hr0] at hr \n apply Nat.le_of_add_le_add_left\n rw [add_zero]\n exact hr.symm.trans_le (natDegree_le_natDegree <| min A x hm hp)\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\nr : A[X]\nhr : minpoly A x = p * r\nhlead : 1 = leadingCoeff r\n\u22a2 natDegree r \u2264 0\n[PROOFSTEP]\nhave hr0 : r \u2260 0 := by\n rintro rfl\n exact ne_zero hx (mul_zero p \u25b8 hr)\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\nr : A[X]\nhr : minpoly A x = p * r\nhlead : 1 = leadingCoeff r\n\u22a2 r \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\nhr : minpoly A x = p * 0\nhlead : 1 = leadingCoeff 0\n\u22a2 False\n[PROOFSTEP]\nexact ne_zero hx (mul_zero p \u25b8 hr)\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\nr : A[X]\nhr : minpoly A x = p * r\nhlead : 1 = leadingCoeff r\nhr0 : r \u2260 0\n\u22a2 natDegree r \u2264 0\n[PROOFSTEP]\napply_fun natDegree at hr \n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\nr : A[X]\nhlead : 1 = leadingCoeff r\nhr0 : r \u2260 0\nhr : natDegree (minpoly A x) = natDegree (p * r)\n\u22a2 natDegree r \u2264 0\n[PROOFSTEP]\nrw [hm.natDegree_mul' hr0] at hr \n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\nr : A[X]\nhlead : 1 = leadingCoeff r\nhr0 : r \u2260 0\nhr : natDegree (minpoly A x) = natDegree p + natDegree r\n\u22a2 natDegree r \u2264 0\n[PROOFSTEP]\napply Nat.le_of_add_le_add_left\n[GOAL]\ncase h\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\nr : A[X]\nhlead : 1 = leadingCoeff r\nhr0 : r \u2260 0\nhr : natDegree (minpoly A x) = natDegree p + natDegree r\n\u22a2 ?a + natDegree r \u2264 ?a + 0\ncase a\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\nr : A[X]\nhlead : 1 = leadingCoeff r\nhr0 : r \u2260 0\nhr : natDegree (minpoly A x) = natDegree p + natDegree r\n\u22a2 \u2115\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\ncase h\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\nr : A[X]\nhlead : 1 = leadingCoeff r\nhr0 : r \u2260 0\nhr : natDegree (minpoly A x) = natDegree p + natDegree r\n\u22a2 ?h + natDegree r \u2264 ?h\ncase h\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\nr : A[X]\nhlead : 1 = leadingCoeff r\nhr0 : r \u2260 0\nhr : natDegree (minpoly A x) = natDegree p + natDegree r\n\u22a2 \u2115\n[PROOFSTEP]\nexact hr.symm.trans_le (natDegree_le_natDegree <| min A x hm hp)\n[GOAL]\ncase inl.intro\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring B'\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra A B'\nx : B\np : A[X]\nhm : Monic p\nhp : \u2191(Polynomial.aeval x) p = 0\nhl : \u2200 (q : A[X]), degree q < degree p \u2192 q = 0 \u2228 \u2191(Polynomial.aeval x) q \u2260 0\n\u271d : Nontrivial A\nhx : IsIntegral A x\nh : minpoly A x %\u2098 p = 0\nr : A[X]\nhr : minpoly A x = p * r\nhlead : 1 = leadingCoeff r\nthis : natDegree r \u2264 0\n\u22a2 p = p * r\n[PROOFSTEP]\nrw [eq_C_of_natDegree_le_zero this, \u2190 Nat.eq_zero_of_le_zero this, \u2190 leadingCoeff, \u2190 hlead, C_1, mul_one]\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Ring B\ninst\u271d\u00b3 : Ring B'\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra A B'\nx : B\ninst\u271d : Subsingleton B\n\u22a2 minpoly A x = 1\n[PROOFSTEP]\nnontriviality A\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Ring B\ninst\u271d\u00b3 : Ring B'\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra A B'\nx : B\ninst\u271d : Subsingleton B\n\u271d : Nontrivial A\n\u22a2 minpoly A x = 1\n[PROOFSTEP]\nhave := minpoly.min A x monic_one (Subsingleton.elim _ _)\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Ring B\ninst\u271d\u00b3 : Ring B'\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra A B'\nx : B\ninst\u271d : Subsingleton B\n\u271d : Nontrivial A\nthis : degree (minpoly A x) \u2264 degree 1\n\u22a2 minpoly A x = 1\n[PROOFSTEP]\nrw [degree_one] at this \n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Ring B\ninst\u271d\u00b3 : Ring B'\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra A B'\nx : B\ninst\u271d : Subsingleton B\n\u271d : Nontrivial A\nthis : degree (minpoly A x) \u2264 0\n\u22a2 minpoly A x = 1\n[PROOFSTEP]\ncases' le_or_lt (minpoly A x).degree 0 with h h\n[GOAL]\ncase inl\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Ring B\ninst\u271d\u00b3 : Ring B'\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra A B'\nx : B\ninst\u271d : Subsingleton B\n\u271d : Nontrivial A\nthis : degree (minpoly A x) \u2264 0\nh : degree (minpoly A x) \u2264 0\n\u22a2 minpoly A x = 1\n[PROOFSTEP]\nrwa [(monic \u27e81, monic_one, by simp\u27e9 : (minpoly A x).Monic).degree_le_zero_iff_eq_one] at h \n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Ring B\ninst\u271d\u00b3 : Ring B'\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra A B'\nx : B\ninst\u271d : Subsingleton B\n\u271d : Nontrivial A\nthis : degree (minpoly A x) \u2264 0\nh : degree (minpoly A x) \u2264 0\n\u22a2 eval\u2082 (algebraMap A B) x 1 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Ring B\ninst\u271d\u00b3 : Ring B'\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra A B'\nx : B\ninst\u271d : Subsingleton B\n\u271d : Nontrivial A\nthis : degree (minpoly A x) \u2264 0\nh : 0 < degree (minpoly A x)\n\u22a2 minpoly A x = 1\n[PROOFSTEP]\nexact (this.not_lt h).elim\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra A B\nx : B\ninst\u271d : Nontrivial B\nhx : IsIntegral A x\n\u22a2 0 < natDegree (minpoly A x)\n[PROOFSTEP]\nrw [pos_iff_ne_zero]\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra A B\nx : B\ninst\u271d : Nontrivial B\nhx : IsIntegral A x\n\u22a2 natDegree (minpoly A x) \u2260 0\n[PROOFSTEP]\nintro ndeg_eq_zero\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra A B\nx : B\ninst\u271d : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : natDegree (minpoly A x) = 0\n\u22a2 False\n[PROOFSTEP]\nhave eq_one : minpoly A x = 1 := by\n rw [eq_C_of_natDegree_eq_zero ndeg_eq_zero]\n convert C_1 (R := A)\n simpa only [ndeg_eq_zero.symm] using (monic hx).leadingCoeff\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra A B\nx : B\ninst\u271d : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : natDegree (minpoly A x) = 0\n\u22a2 minpoly A x = 1\n[PROOFSTEP]\nrw [eq_C_of_natDegree_eq_zero ndeg_eq_zero]\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra A B\nx : B\ninst\u271d : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : natDegree (minpoly A x) = 0\n\u22a2 \u2191C (coeff (minpoly A x) 0) = 1\n[PROOFSTEP]\nconvert C_1 (R := A)\n[GOAL]\ncase h.e'_2.h.e'_6\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra A B\nx : B\ninst\u271d : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : natDegree (minpoly A x) = 0\n\u22a2 coeff (minpoly A x) 0 = 1\n[PROOFSTEP]\nsimpa only [ndeg_eq_zero.symm] using (monic hx).leadingCoeff\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra A B\nx : B\ninst\u271d : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : natDegree (minpoly A x) = 0\neq_one : minpoly A x = 1\n\u22a2 False\n[PROOFSTEP]\nsimpa only [eq_one, AlgHom.map_one, one_ne_zero] using aeval A x\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A\nhf : Injective \u2191(algebraMap A B)\n\u22a2 minpoly A (\u2191(algebraMap A B) a) = X - \u2191C a\n[PROOFSTEP]\nnontriviality A\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A\nhf : Injective \u2191(algebraMap A B)\n\u271d : Nontrivial A\n\u22a2 minpoly A (\u2191(algebraMap A B) a) = X - \u2191C a\n[PROOFSTEP]\nrefine' (unique' A _ (monic_X_sub_C a) _ _).symm\n[GOAL]\ncase refine'_1\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A\nhf : Injective \u2191(algebraMap A B)\n\u271d : Nontrivial A\n\u22a2 \u2191(Polynomial.aeval (\u2191(algebraMap A B) a)) (X - \u2191C a) = 0\n[PROOFSTEP]\nrw [map_sub, aeval_C, aeval_X, sub_self]\n[GOAL]\ncase refine'_2\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A\nhf : Injective \u2191(algebraMap A B)\n\u271d : Nontrivial A\n\u22a2 \u2200 (q : A[X]), degree q < degree (X - \u2191C a) \u2192 q = 0 \u2228 \u2191(Polynomial.aeval (\u2191(algebraMap A B) a)) q \u2260 0\n[PROOFSTEP]\nsimp_rw [or_iff_not_imp_left]\n[GOAL]\ncase refine'_2\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A\nhf : Injective \u2191(algebraMap A B)\n\u271d : Nontrivial A\n\u22a2 \u2200 (q : A[X]), degree q < degree (X - \u2191C a) \u2192 \u00acq = 0 \u2192 \u2191(Polynomial.aeval (\u2191(algebraMap A B) a)) q \u2260 0\n[PROOFSTEP]\nintro q hl h0\n[GOAL]\ncase refine'_2\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A\nhf : Injective \u2191(algebraMap A B)\n\u271d : Nontrivial A\nq : A[X]\nhl : degree q < degree (X - \u2191C a)\nh0 : \u00acq = 0\n\u22a2 \u2191(Polynomial.aeval (\u2191(algebraMap A B) a)) q \u2260 0\n[PROOFSTEP]\nrw [\u2190 natDegree_lt_natDegree_iff h0, natDegree_X_sub_C, Nat.lt_one_iff] at hl \n[GOAL]\ncase refine'_2\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A\nhf : Injective \u2191(algebraMap A B)\n\u271d : Nontrivial A\nq : A[X]\nhl : natDegree q = 0\nh0 : \u00acq = 0\n\u22a2 \u2191(Polynomial.aeval (\u2191(algebraMap A B) a)) q \u2260 0\n[PROOFSTEP]\nrw [eq_C_of_natDegree_eq_zero hl] at h0 \u22a2\n[GOAL]\ncase refine'_2\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A\nhf : Injective \u2191(algebraMap A B)\n\u271d : Nontrivial A\nq : A[X]\nhl : natDegree q = 0\nh0 : \u00ac\u2191C (coeff q 0) = 0\n\u22a2 \u2191(Polynomial.aeval (\u2191(algebraMap A B) a)) (\u2191C (coeff q 0)) \u2260 0\n[PROOFSTEP]\nrwa [aeval_C, map_ne_zero_iff _ hf, \u2190 C_ne_zero]\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A[X]\nhx : IsIntegral A x\nhamonic : Monic a\nhdvd : DvdNotUnit a (minpoly A x)\n\u22a2 \u2191(Polynomial.aeval x) a \u2260 0\n[PROOFSTEP]\nrefine' fun ha => (min A x hamonic ha).not_lt (degree_lt_degree _)\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A[X]\nhx : IsIntegral A x\nhamonic : Monic a\nhdvd : DvdNotUnit a (minpoly A x)\nha : \u2191(Polynomial.aeval x) a = 0\n\u22a2 natDegree a < natDegree (minpoly A x)\n[PROOFSTEP]\nobtain \u27e8_, c, hu, he\u27e9 := hdvd\n[GOAL]\ncase intro.intro.intro\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A[X]\nhx : IsIntegral A x\nhamonic : Monic a\nha : \u2191(Polynomial.aeval x) a = 0\nleft\u271d : a \u2260 0\nc : A[X]\nhu : \u00acIsUnit c\nhe : minpoly A x = a * c\n\u22a2 natDegree a < natDegree (minpoly A x)\n[PROOFSTEP]\nhave hcm := hamonic.of_mul_monic_left (he.subst <| monic hx)\n[GOAL]\ncase intro.intro.intro\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A[X]\nhx : IsIntegral A x\nhamonic : Monic a\nha : \u2191(Polynomial.aeval x) a = 0\nleft\u271d : a \u2260 0\nc : A[X]\nhu : \u00acIsUnit c\nhe : minpoly A x = a * c\nhcm : Monic c\n\u22a2 natDegree a < natDegree (minpoly A x)\n[PROOFSTEP]\nrw [he, hamonic.natDegree_mul hcm]\n -- TODO: port Nat.lt_add_of_zero_lt_left from lean3 core\n[GOAL]\ncase intro.intro.intro\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A[X]\nhx : IsIntegral A x\nhamonic : Monic a\nha : \u2191(Polynomial.aeval x) a = 0\nleft\u271d : a \u2260 0\nc : A[X]\nhu : \u00acIsUnit c\nhe : minpoly A x = a * c\nhcm : Monic c\n\u22a2 natDegree a < natDegree a + natDegree c\n[PROOFSTEP]\napply lt_add_of_pos_right\n[GOAL]\ncase intro.intro.intro.h\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A[X]\nhx : IsIntegral A x\nhamonic : Monic a\nha : \u2191(Polynomial.aeval x) a = 0\nleft\u271d : a \u2260 0\nc : A[X]\nhu : \u00acIsUnit c\nhe : minpoly A x = a * c\nhcm : Monic c\n\u22a2 0 < natDegree c\n[PROOFSTEP]\nrefine (lt_of_not_le fun h => hu ?_)\n[GOAL]\ncase intro.intro.intro.h\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A[X]\nhx : IsIntegral A x\nhamonic : Monic a\nha : \u2191(Polynomial.aeval x) a = 0\nleft\u271d : a \u2260 0\nc : A[X]\nhu : \u00acIsUnit c\nhe : minpoly A x = a * c\nhcm : Monic c\nh : natDegree c \u2264 0\n\u22a2 IsUnit c\n[PROOFSTEP]\nrw [eq_C_of_natDegree_le_zero h, \u2190 Nat.eq_zero_of_le_zero h, \u2190 leadingCoeff, hcm.leadingCoeff, C_1]\n[GOAL]\ncase intro.intro.intro.h\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Algebra A B\nx : B\na : A[X]\nhx : IsIntegral A x\nhamonic : Monic a\nha : \u2191(Polynomial.aeval x) a = 0\nleft\u271d : a \u2260 0\nc : A[X]\nhu : \u00acIsUnit c\nhe : minpoly A x = a * c\nhcm : Monic c\nh : natDegree c \u2264 0\n\u22a2 IsUnit 1\n[PROOFSTEP]\nexact isUnit_one\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Algebra A B\nx : B\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsDomain B\nhx : IsIntegral A x\n\u22a2 Irreducible (minpoly A x)\n[PROOFSTEP]\nrefine' (irreducible_of_monic (monic hx) <| ne_one A x).2 fun f g hf hg he => _\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Algebra A B\nx : B\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsDomain B\nhx : IsIntegral A x\nf g : A[X]\nhf : Monic f\nhg : Monic g\nhe : f * g = minpoly A x\n\u22a2 f = 1 \u2228 g = 1\n[PROOFSTEP]\nrw [\u2190 hf.isUnit_iff, \u2190 hg.isUnit_iff]\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Algebra A B\nx : B\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsDomain B\nhx : IsIntegral A x\nf g : A[X]\nhf : Monic f\nhg : Monic g\nhe : f * g = minpoly A x\n\u22a2 IsUnit f \u2228 IsUnit g\n[PROOFSTEP]\nby_contra' h\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Algebra A B\nx : B\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsDomain B\nhx : IsIntegral A x\nf g : A[X]\nhf : Monic f\nhg : Monic g\nhe : f * g = minpoly A x\nh : \u00acIsUnit f \u2227 \u00acIsUnit g\n\u22a2 False\n[PROOFSTEP]\nhave heval := congr_arg (Polynomial.aeval x) he\n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Algebra A B\nx : B\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsDomain B\nhx : IsIntegral A x\nf g : A[X]\nhf : Monic f\nhg : Monic g\nhe : f * g = minpoly A x\nh : \u00acIsUnit f \u2227 \u00acIsUnit g\nheval : \u2191(Polynomial.aeval x) (f * g) = \u2191(Polynomial.aeval x) (minpoly A x)\n\u22a2 False\n[PROOFSTEP]\nrw [aeval A x, aeval_mul, mul_eq_zero] at heval \n[GOAL]\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Algebra A B\nx : B\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsDomain B\nhx : IsIntegral A x\nf g : A[X]\nhf : Monic f\nhg : Monic g\nhe : f * g = minpoly A x\nh : \u00acIsUnit f \u2227 \u00acIsUnit g\nheval : \u2191(Polynomial.aeval x) f = 0 \u2228 \u2191(Polynomial.aeval x) g = 0\n\u22a2 False\n[PROOFSTEP]\ncases' heval with heval heval\n[GOAL]\ncase inl\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Algebra A B\nx : B\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsDomain B\nhx : IsIntegral A x\nf g : A[X]\nhf : Monic f\nhg : Monic g\nhe : f * g = minpoly A x\nh : \u00acIsUnit f \u2227 \u00acIsUnit g\nheval : \u2191(Polynomial.aeval x) f = 0\n\u22a2 False\n[PROOFSTEP]\nexact aeval_ne_zero_of_dvdNotUnit_minpoly hx hf \u27e8hf.ne_zero, g, h.2, he.symm\u27e9 heval\n[GOAL]\ncase inr\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Algebra A B\nx : B\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsDomain B\nhx : IsIntegral A x\nf g : A[X]\nhf : Monic f\nhg : Monic g\nhe : f * g = minpoly A x\nh : \u00acIsUnit f \u2227 \u00acIsUnit g\nheval : \u2191(Polynomial.aeval x) g = 0\n\u22a2 False\n[PROOFSTEP]\nrefine' aeval_ne_zero_of_dvdNotUnit_minpoly hx hg \u27e8hg.ne_zero, f, h.1, _\u27e9 heval\n[GOAL]\ncase inr\nA : Type u_1\nB : Type u_2\nB' : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Algebra A B\nx : B\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsDomain B\nhx : IsIntegral A x\nf g : A[X]\nhf : Monic f\nhg : Monic g\nhe : f * g = minpoly A x\nh : \u00acIsUnit f \u2227 \u00acIsUnit g\nheval : \u2191(Polynomial.aeval x) g = 0\n\u22a2 minpoly A x = g * f\n[PROOFSTEP]\nrw [mul_comm, he]\n", "meta": {"mathlib_filename": "Mathlib.FieldTheory.Minpoly.Basic", "llama_tokens": 17725, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581510799252, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.4424337412119466}} {"text": "[GOAL]\nR : Type u_1\nA : Type u\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalSemiring A\n\u22a2 Continuous \u2191(algebraMap R A) \u2194 Continuous fun p => p.fst \u2022 p.snd\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nA : Type u\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalSemiring A\nh : Continuous \u2191(algebraMap R A)\n\u22a2 Continuous fun p => p.fst \u2022 p.snd\n[PROOFSTEP]\nsimp only [Algebra.smul_def]\n[GOAL]\ncase refine'_1\nR : Type u_1\nA : Type u\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalSemiring A\nh : Continuous \u2191(algebraMap R A)\n\u22a2 Continuous fun p => \u2191(algebraMap R A) p.fst * p.snd\n[PROOFSTEP]\nexact (h.comp continuous_fst).mul continuous_snd\n[GOAL]\ncase refine'_2\nR : Type u_1\nA : Type u\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalSemiring A\nh : Continuous fun p => p.fst \u2022 p.snd\n\u22a2 Continuous \u2191(algebraMap R A)\n[PROOFSTEP]\nrw [algebraMap_eq_smul_one']\n[GOAL]\ncase refine'_2\nR : Type u_1\nA : Type u\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalSemiring A\nh : Continuous fun p => p.fst \u2022 p.snd\n\u22a2 Continuous fun r => r \u2022 1\n[PROOFSTEP]\nexact h.comp (continuous_id.prod_mk continuous_const)\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nA : Type u\ninst\u271d\u00b3 : TopologicalSpace A\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : TopologicalSemiring A\ns : Subalgebra R A\n\u22a2 IsClosed \u2191(topologicalClosure s)\n[PROOFSTEP]\nconvert @isClosed_closure A _ s\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommSemiring R\nA : Type u\ninst\u271d\u2077 : TopologicalSpace A\ninst\u271d\u2076 : Semiring A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : TopologicalSemiring A\ns : Subalgebra R A\nB : Type u_2\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : TopologicalRing B\ninst\u271d : Algebra R B\nf : B \u2192\u2090[R] A\nf' : B \u2243\u209c A\nw : \u2191f = \u2191f'\n\u22a2 comap f (topologicalClosure s) = topologicalClosure (comap f s)\n[PROOFSTEP]\napply SetLike.ext'\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2078 : CommSemiring R\nA : Type u\ninst\u271d\u2077 : TopologicalSpace A\ninst\u271d\u2076 : Semiring A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : TopologicalSemiring A\ns : Subalgebra R A\nB : Type u_2\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : TopologicalRing B\ninst\u271d : Algebra R B\nf : B \u2192\u2090[R] A\nf' : B \u2243\u209c A\nw : \u2191f = \u2191f'\n\u22a2 \u2191(comap f (topologicalClosure s)) = \u2191(topologicalClosure (comap f s))\n[PROOFSTEP]\nsimp only [Subalgebra.topologicalClosure_coe]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2078 : CommSemiring R\nA : Type u\ninst\u271d\u2077 : TopologicalSpace A\ninst\u271d\u2076 : Semiring A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : TopologicalSemiring A\ns : Subalgebra R A\nB : Type u_2\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : TopologicalRing B\ninst\u271d : Algebra R B\nf : B \u2192\u2090[R] A\nf' : B \u2243\u209c A\nw : \u2191f = \u2191f'\n\u22a2 \u2191(comap f (topologicalClosure s)) = closure \u2191(comap f s)\n[PROOFSTEP]\nsimp only [Subalgebra.coe_comap, Subsemiring.coe_comap, AlgHom.coe_toRingHom]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2078 : CommSemiring R\nA : Type u\ninst\u271d\u2077 : TopologicalSpace A\ninst\u271d\u2076 : Semiring A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : TopologicalSemiring A\ns : Subalgebra R A\nB : Type u_2\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : TopologicalRing B\ninst\u271d : Algebra R B\nf : B \u2192\u2090[R] A\nf' : B \u2243\u209c A\nw : \u2191f = \u2191f'\n\u22a2 \u2191f \u207b\u00b9' \u2191(topologicalClosure s) = closure (\u2191f \u207b\u00b9' \u2191s)\n[PROOFSTEP]\nrw [w]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2078 : CommSemiring R\nA : Type u\ninst\u271d\u2077 : TopologicalSpace A\ninst\u271d\u2076 : Semiring A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : TopologicalSemiring A\ns : Subalgebra R A\nB : Type u_2\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : TopologicalRing B\ninst\u271d : Algebra R B\nf : B \u2192\u2090[R] A\nf' : B \u2243\u209c A\nw : \u2191f = \u2191f'\n\u22a2 \u2191f' \u207b\u00b9' \u2191(topologicalClosure s) = closure (\u2191f' \u207b\u00b9' \u2191s)\n[PROOFSTEP]\nexact f'.preimage_closure _\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nA : Type u\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : TopologicalRing A\ninst\u271d : T2Space A\nx y : A\nhy : y \u2208 {x}\nz : A\nhz : z \u2208 {x}\n\u22a2 y * z = z * y\n[PROOFSTEP]\nrw [mem_singleton_iff] at hy hz \n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nA : Type u\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : TopologicalRing A\ninst\u271d : T2Space A\nx y : A\nhy : y = x\nz : A\nhz : z = x\n\u22a2 y * z = z * y\n[PROOFSTEP]\nrw [hy, hz]\n[GOAL]\nA : Type ?u.199010\ninst\u271d\u00b3 : DivisionRing A\ninst\u271d\u00b2 : TopologicalSpace A\ninst\u271d\u00b9 : ContinuousMul A\ninst\u271d : CharZero A\nr : \u211a\n\u22a2 Continuous fun x => r \u2022 x\n[PROOFSTEP]\nsimpa only [Algebra.smul_def] using continuous_const.mul continuous_id\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Algebra", "llama_tokens": 2267, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548511303338, "lm_q2_score": 0.6791786991753931, "lm_q1q2_score": 0.44238634049228187}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d : Mul \u03b2\n\u22a2 let mul := Equiv.mul e;\n \u03b1 \u2243* \u03b2\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d : Mul \u03b2\nmul\u271d : Mul \u03b1 := Equiv.mul e\n\u22a2 \u03b1 \u2243* \u03b2\n[PROOFSTEP]\nexact\n { e with\n map_mul' := fun x y => by\n apply e.symm.injective\n simp [mul_def] }\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d : Mul \u03b2\nmul\u271d : Mul \u03b1 := Equiv.mul e\nx y : \u03b1\n\u22a2 toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n (x * y) =\n toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n x *\n toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n y\n[PROOFSTEP]\napply e.symm.injective\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d : Mul \u03b2\nmul\u271d : Mul \u03b1 := Equiv.mul e\nx y : \u03b1\n\u22a2 \u2191e.symm\n (toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n (x * y)) =\n \u2191e.symm\n (toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n x *\n toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n y)\n[PROOFSTEP]\nsimp [mul_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d : Mul \u03b2\nb : \u03b2\n\u22a2 \u2191(MulEquiv.symm (mulEquiv e)) b = \u2191e.symm b\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d : Mul \u03b2\nb : \u03b2\n\u22a2 \u2191(MulEquiv.symm (mulEquiv e)) b = \u2191e.symm b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : Mul \u03b2\n\u22a2 Sort ?u.112825\n[PROOFSTEP]\nlet add := Equiv.add e\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : Mul \u03b2\nadd : Add \u03b1 := Equiv.add e\n\u22a2 Sort ?u.112825\n[PROOFSTEP]\nlet mul := Equiv.mul e\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : Mul \u03b2\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 Sort ?u.112825\n[PROOFSTEP]\nexact \u03b1 \u2243+* \u03b2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : Mul \u03b2\n\u22a2 let add := Equiv.add e;\n let mul := Equiv.mul e;\n \u03b1 \u2243+* \u03b2\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : Mul \u03b2\nadd\u271d : Add \u03b1 := Equiv.add e\nmul\u271d : Mul \u03b1 := Equiv.mul e\n\u22a2 \u03b1 \u2243+* \u03b2\n[PROOFSTEP]\nexact\n {\n e with\n map_add' := fun x y => by\n apply e.symm.injective\n simp [add_def]\n map_mul' := fun x y => by\n apply e.symm.injective\n simp [mul_def] }\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : Mul \u03b2\nadd\u271d : Add \u03b1 := Equiv.add e\nmul\u271d : Mul \u03b1 := Equiv.mul e\nx y : \u03b1\n\u22a2 toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n (x * y) =\n toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n x *\n toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n y\n[PROOFSTEP]\napply e.symm.injective\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : Mul \u03b2\nadd\u271d : Add \u03b1 := Equiv.add e\nmul\u271d : Mul \u03b1 := Equiv.mul e\nx y : \u03b1\n\u22a2 \u2191e.symm\n (toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n (x * y)) =\n \u2191e.symm\n (toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n x *\n toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n y)\n[PROOFSTEP]\nsimp [mul_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : Mul \u03b2\nadd\u271d : Add \u03b1 := Equiv.add e\nmul\u271d : Mul \u03b1 := Equiv.mul e\nx y : \u03b1\n\u22a2 toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n (x + y) =\n toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n x +\n toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n y\n[PROOFSTEP]\napply e.symm.injective\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : Mul \u03b2\nadd\u271d : Add \u03b1 := Equiv.add e\nmul\u271d : Mul \u03b1 := Equiv.mul e\nx y : \u03b1\n\u22a2 \u2191e.symm\n (toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n (x + y)) =\n \u2191e.symm\n (toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n x +\n toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n y)\n[PROOFSTEP]\nsimp [add_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : Mul \u03b2\nb : \u03b2\n\u22a2 Sort ?u.184035\n[PROOFSTEP]\nletI := Equiv.add e\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : Mul \u03b2\nb : \u03b2\nthis : Add \u03b1 := Equiv.add e\n\u22a2 Sort ?u.184035\n[PROOFSTEP]\nletI := Equiv.mul e\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : Mul \u03b2\nb : \u03b2\nthis\u271d : Add \u03b1 := Equiv.add e\nthis : Mul \u03b1 := Equiv.mul e\n\u22a2 Sort ?u.184035\n[PROOFSTEP]\nexact (ringEquiv e).symm b = e.symm b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : Mul \u03b2\nb : \u03b2\n\u22a2 \u2191(RingEquiv.symm (ringEquiv e)) b = \u2191e.symm b\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d e : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : Mul \u03b2\nb : \u03b2\n\u22a2 \u2191(RingEquiv.symm (ringEquiv e)) b = \u2191e.symm b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semigroup \u03b2\n\u22a2 Semigroup \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semigroup \u03b2\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 Semigroup \u03b1\n[PROOFSTEP]\napply e.injective.semigroup _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semigroup \u03b2\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semigroup \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : SemigroupWithZero \u03b2\n\u22a2 SemigroupWithZero \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : SemigroupWithZero \u03b2\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 SemigroupWithZero \u03b1\n[PROOFSTEP]\nlet zero := e.zero\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : SemigroupWithZero \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nzero : Zero \u03b1 := Equiv.zero e\n\u22a2 SemigroupWithZero \u03b1\n[PROOFSTEP]\napply e.injective.semigroupWithZero _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : SemigroupWithZero \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nzero : Zero \u03b1 := Equiv.zero e\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : SemigroupWithZero \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nzero : Zero \u03b1 := Equiv.zero e\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : SemigroupWithZero \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nzero : Zero \u03b1 := Equiv.zero e\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : SemigroupWithZero \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nzero : Zero \u03b1 := Equiv.zero e\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemigroup \u03b2\n\u22a2 CommSemigroup \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemigroup \u03b2\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 CommSemigroup \u03b1\n[PROOFSTEP]\napply e.injective.commSemigroup _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemigroup \u03b2\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemigroup \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroClass \u03b2\n\u22a2 MulZeroClass \u03b1\n[PROOFSTEP]\nlet zero := e.zero\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroClass \u03b2\nzero : Zero \u03b1 := Equiv.zero e\n\u22a2 MulZeroClass \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroClass \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 MulZeroClass \u03b1\n[PROOFSTEP]\napply e.injective.mulZeroClass _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroClass \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroClass \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2200 (a b : \u03b1), \u2191e (a * b) = \u2191e a * \u2191e b\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroClass \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroClass \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nmul : Mul \u03b1 := Equiv.mul e\na\u271d b\u271d : \u03b1\n\u22a2 \u2191e (a\u271d * b\u271d) = \u2191e a\u271d * \u2191e b\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulOneClass \u03b2\n\u22a2 MulOneClass \u03b1\n[PROOFSTEP]\nlet one := e.one\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulOneClass \u03b2\none : One \u03b1 := Equiv.one e\n\u22a2 MulOneClass \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulOneClass \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 MulOneClass \u03b1\n[PROOFSTEP]\napply e.injective.mulOneClass _\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulOneClass \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulOneClass \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulOneClass \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulOneClass \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroOneClass \u03b2\n\u22a2 MulZeroOneClass \u03b1\n[PROOFSTEP]\nlet zero := e.zero\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroOneClass \u03b2\nzero : Zero \u03b1 := Equiv.zero e\n\u22a2 MulZeroOneClass \u03b1\n[PROOFSTEP]\nlet one := e.one\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroOneClass \u03b2\nzero : Zero \u03b1 := Equiv.zero e\none : One \u03b1 := Equiv.one e\n\u22a2 MulZeroOneClass \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroOneClass \u03b2\nzero : Zero \u03b1 := Equiv.zero e\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 MulZeroOneClass \u03b1\n[PROOFSTEP]\napply e.injective.mulZeroOneClass _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroOneClass \u03b2\nzero : Zero \u03b1 := Equiv.zero e\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroOneClass \u03b2\nzero : Zero \u03b1 := Equiv.zero e\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroOneClass \u03b2\nzero : Zero \u03b1 := Equiv.zero e\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2200 (a b : \u03b1), \u2191e (a * b) = \u2191e a * \u2191e b\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroOneClass \u03b2\nzero : Zero \u03b1 := Equiv.zero e\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroOneClass \u03b2\nzero : Zero \u03b1 := Equiv.zero e\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulZeroOneClass \u03b2\nzero : Zero \u03b1 := Equiv.zero e\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\na\u271d b\u271d : \u03b1\n\u22a2 \u2191e (a\u271d * b\u271d) = \u2191e a\u271d * \u2191e b\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Monoid \u03b2\n\u22a2 Monoid \u03b1\n[PROOFSTEP]\nlet one := e.one\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Monoid \u03b2\none : One \u03b1 := Equiv.one e\n\u22a2 Monoid \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Monoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 Monoid \u03b1\n[PROOFSTEP]\nlet pow := e.pow \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Monoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\npow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 Monoid \u03b1\n[PROOFSTEP]\napply e.injective.monoid _\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Monoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\npow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Monoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\npow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Monoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\npow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (x ^ n) = \u2191e x ^ n\n[PROOFSTEP]\nintros\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Monoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\npow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Monoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\npow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Monoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\npow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (x\u271d ^ n\u271d) = \u2191e x\u271d ^ n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommMonoid \u03b2\n\u22a2 CommMonoid \u03b1\n[PROOFSTEP]\nlet one := e.one\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommMonoid \u03b2\none : One \u03b1 := Equiv.one e\n\u22a2 CommMonoid \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommMonoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 CommMonoid \u03b1\n[PROOFSTEP]\nlet pow := e.pow \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommMonoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\npow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 CommMonoid \u03b1\n[PROOFSTEP]\napply e.injective.commMonoid _\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommMonoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\npow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommMonoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\npow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommMonoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\npow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (x ^ n) = \u2191e x ^ n\n[PROOFSTEP]\nintros\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommMonoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\npow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommMonoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\npow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommMonoid \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\npow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (x\u271d ^ n\u271d) = \u2191e x\u271d ^ n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\n\u22a2 Group \u03b1\n[PROOFSTEP]\nlet one := e.one\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\n\u22a2 Group \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 Group \u03b1\n[PROOFSTEP]\nlet inv := e.Inv\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\n\u22a2 Group \u03b1\n[PROOFSTEP]\nlet div := e.div\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\n\u22a2 Group \u03b1\n[PROOFSTEP]\nlet npow := e.pow \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 Group \u03b1\n[PROOFSTEP]\nlet zpow := e.pow \u2124\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 Group \u03b1\n[PROOFSTEP]\napply e.injective.group _\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase inv\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 \u2200 (x : \u03b1), \u2191e x\u207b\u00b9 = (\u2191e x)\u207b\u00b9\n[PROOFSTEP]\nintros\n[GOAL]\ncase div\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x / y) = \u2191e x / \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (x ^ n) = \u2191e x ^ n\n[PROOFSTEP]\nintros\n[GOAL]\ncase zpow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 \u2200 (x : \u03b1) (n : \u2124), \u2191e (x ^ n) = \u2191e x ^ n\n[PROOFSTEP]\nintros\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase inv\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nx\u271d : \u03b1\n\u22a2 \u2191e x\u271d\u207b\u00b9 = (\u2191e x\u271d)\u207b\u00b9\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase div\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d / y\u271d) = \u2191e x\u271d / \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (x\u271d ^ n\u271d) = \u2191e x\u271d ^ n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase zpow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Group \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nx\u271d : \u03b1\nn\u271d : \u2124\n\u22a2 \u2191e (x\u271d ^ n\u271d) = \u2191e x\u271d ^ n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\n\u22a2 CommGroup \u03b1\n[PROOFSTEP]\nlet one := e.one\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\n\u22a2 CommGroup \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 CommGroup \u03b1\n[PROOFSTEP]\nlet inv := e.Inv\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\n\u22a2 CommGroup \u03b1\n[PROOFSTEP]\nlet div := e.div\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\n\u22a2 CommGroup \u03b1\n[PROOFSTEP]\nlet npow := e.pow \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 CommGroup \u03b1\n[PROOFSTEP]\nlet zpow := e.pow \u2124\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 CommGroup \u03b1\n[PROOFSTEP]\napply e.injective.commGroup _\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase inv\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 \u2200 (x : \u03b1), \u2191e x\u207b\u00b9 = (\u2191e x)\u207b\u00b9\n[PROOFSTEP]\nintros\n[GOAL]\ncase div\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x / y) = \u2191e x / \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (x ^ n) = \u2191e x ^ n\n[PROOFSTEP]\nintros\n[GOAL]\ncase zpow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 \u2200 (x : \u03b1) (n : \u2124), \u2191e (x ^ n) = \u2191e x ^ n\n[PROOFSTEP]\nintros\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase inv\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nx\u271d : \u03b1\n\u22a2 \u2191e x\u271d\u207b\u00b9 = (\u2191e x\u271d)\u207b\u00b9\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase div\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d / y\u271d) = \u2191e x\u271d / \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (x\u271d ^ n\u271d) = \u2191e x\u271d ^ n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase zpow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommGroup \u03b2\none : One \u03b1 := Equiv.one e\nmul : Mul \u03b1 := Equiv.mul e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nx\u271d : \u03b1\nn\u271d : \u2124\n\u22a2 \u2191e (x\u271d ^ n\u271d) = \u2191e x\u271d ^ n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\n\u22a2 NonUnitalNonAssocSemiring \u03b1\n[PROOFSTEP]\nlet zero := e.zero\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\n\u22a2 NonUnitalNonAssocSemiring \u03b1\n[PROOFSTEP]\nlet add := e.add\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\n\u22a2 NonUnitalNonAssocSemiring \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 NonUnitalNonAssocSemiring \u03b1\n[PROOFSTEP]\nlet nsmul := e.smul \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 NonUnitalNonAssocSemiring \u03b1\n[PROOFSTEP]\napply e.injective.nonUnitalNonAssocSemiring _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x + y) = \u2191e x + \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d + y\u271d) = \u2191e x\u271d + \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalSemiring \u03b2\n\u22a2 NonUnitalSemiring \u03b1\n[PROOFSTEP]\nlet zero := e.zero\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\n\u22a2 NonUnitalSemiring \u03b1\n[PROOFSTEP]\nlet add := e.add\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\n\u22a2 NonUnitalSemiring \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 NonUnitalSemiring \u03b1\n[PROOFSTEP]\nlet nsmul := e.smul \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 NonUnitalSemiring \u03b1\n[PROOFSTEP]\napply e.injective.nonUnitalSemiring _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x + y) = \u2191e x + \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d + y\u271d) = \u2191e x\u271d + \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : AddMonoidWithOne \u03b2\nsrc\u271d\u00b9 : AddMonoid \u03b1 := Equiv.addMonoid e\nsrc\u271d : One \u03b1 := Equiv.one e\n\u22a2 \u2191e (NatCast.natCast 0) = \u2191e 0\n[PROOFSTEP]\nsimp [zero_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : AddMonoidWithOne \u03b2\nsrc\u271d\u00b9 : AddMonoid \u03b1 := Equiv.addMonoid e\nsrc\u271d : One \u03b1 := Equiv.one e\nn : \u2115\n\u22a2 \u2191e (NatCast.natCast (n + 1)) = \u2191e (NatCast.natCast n + 1)\n[PROOFSTEP]\nsimp [add_def, one_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : AddGroupWithOne \u03b2\nsrc\u271d\u00b9 : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nsrc\u271d : AddGroup \u03b1 := Equiv.addGroup e\nn : \u2115\n\u22a2 IntCast.intCast \u2191n = \u2191n\n[PROOFSTEP]\nsimp only [Int.cast_ofNat]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : AddGroupWithOne \u03b2\nsrc\u271d\u00b9 : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nsrc\u271d : AddGroup \u03b1 := Equiv.addGroup e\nn : \u2115\n\u22a2 \u2191e.symm \u2191n = \u2191n\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocSemiring \u03b2\n\u22a2 NonAssocSemiring \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 NonAssocSemiring \u03b1\n[PROOFSTEP]\nlet add_monoid_with_one := e.addMonoidWithOne\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\n\u22a2 NonAssocSemiring \u03b1\n[PROOFSTEP]\napply e.injective.nonAssocSemiring _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x + y) = \u2191e x + \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\n\u22a2 \u2200 (n : \u2115), \u2191e \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d + y\u271d) = \u2191e x\u271d + \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nn\u271d : \u2115\n\u22a2 \u2191e \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\n\u22a2 Semiring \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 Semiring \u03b1\n[PROOFSTEP]\nlet add_monoid_with_one := e.addMonoidWithOne\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\n\u22a2 Semiring \u03b1\n[PROOFSTEP]\nlet npow := e.pow \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 Semiring \u03b1\n[PROOFSTEP]\napply e.injective.semiring _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x + y) = \u2191e x + \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (x ^ n) = \u2191e x ^ n\n[PROOFSTEP]\nintros\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (n : \u2115), \u2191e \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d + y\u271d) = \u2191e x\u271d + \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (x\u271d ^ n\u271d) = \u2191e x\u271d ^ n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nn\u271d : \u2115\n\u22a2 \u2191e \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommSemiring \u03b2\n\u22a2 NonUnitalCommSemiring \u03b1\n[PROOFSTEP]\nlet zero := e.zero\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\n\u22a2 NonUnitalCommSemiring \u03b1\n[PROOFSTEP]\nlet add := e.add\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\n\u22a2 NonUnitalCommSemiring \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 NonUnitalCommSemiring \u03b1\n[PROOFSTEP]\nlet nsmul := e.smul \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 NonUnitalCommSemiring \u03b1\n[PROOFSTEP]\napply e.injective.nonUnitalCommSemiring _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x + y) = \u2191e x + \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d + y\u271d) = \u2191e x\u271d + \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommSemiring \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\n\u22a2 CommSemiring \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 CommSemiring \u03b1\n[PROOFSTEP]\nlet add_monoid_with_one := e.addMonoidWithOne\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\n\u22a2 CommSemiring \u03b1\n[PROOFSTEP]\nlet npow := e.pow \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 CommSemiring \u03b1\n[PROOFSTEP]\napply e.injective.commSemiring _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x + y) = \u2191e x + \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (x ^ n) = \u2191e x ^ n\n[PROOFSTEP]\nintros\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (n : \u2115), \u2191e \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d + y\u271d) = \u2191e x\u271d + \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (x\u271d ^ n\u271d) = \u2191e x\u271d ^ n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommSemiring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_monoid_with_one : AddMonoidWithOne \u03b1 := Equiv.addMonoidWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nn\u271d : \u2115\n\u22a2 \u2191e \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\n\u22a2 NonUnitalNonAssocRing \u03b1\n[PROOFSTEP]\nlet zero := e.zero\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\n\u22a2 NonUnitalNonAssocRing \u03b1\n[PROOFSTEP]\nlet add := e.add\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\n\u22a2 NonUnitalNonAssocRing \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 NonUnitalNonAssocRing \u03b1\n[PROOFSTEP]\nlet neg := e.Neg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\n\u22a2 NonUnitalNonAssocRing \u03b1\n[PROOFSTEP]\nlet sub := e.sub\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\n\u22a2 NonUnitalNonAssocRing \u03b1\n[PROOFSTEP]\nlet nsmul := e.smul \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 NonUnitalNonAssocRing \u03b1\n[PROOFSTEP]\nlet zsmul := e.smul \u2124\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 NonUnitalNonAssocRing \u03b1\n[PROOFSTEP]\napply e.injective.nonUnitalNonAssocRing _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x + y) = \u2191e x + \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x : \u03b1), \u2191e (-x) = -\u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x - y) = \u2191e x - \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase zsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x : \u03b1) (n : \u2124), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d + y\u271d) = \u2191e x\u271d + \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d : \u03b1\n\u22a2 \u2191e (-x\u271d) = -\u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d - y\u271d) = \u2191e x\u271d - \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase zsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalNonAssocRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d : \u03b1\nn\u271d : \u2124\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\n\u22a2 NonUnitalRing \u03b1\n[PROOFSTEP]\nlet zero := e.zero\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\n\u22a2 NonUnitalRing \u03b1\n[PROOFSTEP]\nlet add := e.add\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\n\u22a2 NonUnitalRing \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 NonUnitalRing \u03b1\n[PROOFSTEP]\nlet neg := e.Neg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\n\u22a2 NonUnitalRing \u03b1\n[PROOFSTEP]\nlet sub := e.sub\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\n\u22a2 NonUnitalRing \u03b1\n[PROOFSTEP]\nlet nsmul := e.smul \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 NonUnitalRing \u03b1\n[PROOFSTEP]\nlet zsmul := e.smul \u2124\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 NonUnitalRing \u03b1\n[PROOFSTEP]\napply e.injective.nonUnitalRing _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x + y) = \u2191e x + \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x : \u03b1), \u2191e (-x) = -\u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x - y) = \u2191e x - \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase gsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x : \u03b1) (n : \u2124), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d + y\u271d) = \u2191e x\u271d + \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d : \u03b1\n\u22a2 \u2191e (-x\u271d) = -\u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d - y\u271d) = \u2191e x\u271d - \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase gsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d : \u03b1\nn\u271d : \u2124\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\n\u22a2 NonAssocRing \u03b1\n[PROOFSTEP]\nlet add_group_with_one := e.addGroupWithOne\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\n\u22a2 NonAssocRing \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 NonAssocRing \u03b1\n[PROOFSTEP]\napply e.injective.nonAssocRing _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x + y) = \u2191e x + \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2200 (x : \u03b1), \u2191e (-x) = -\u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x - y) = \u2191e x - \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase gsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2200 (x : \u03b1) (n : \u2124), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2200 (n : \u2115), \u2191e \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase int_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2200 (n : \u2124), \u2191e \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d + y\u271d) = \u2191e x\u271d + \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\nx\u271d : \u03b1\n\u22a2 \u2191e (-x\u271d) = -\u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d - y\u271d) = \u2191e x\u271d - \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase gsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\nx\u271d : \u03b1\nn\u271d : \u2124\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\nn\u271d : \u2115\n\u22a2 \u2191e \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase int_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonAssocRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nmul : Mul \u03b1 := Equiv.mul e\nn\u271d : \u2124\n\u22a2 \u2191e \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\n\u22a2 Ring \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 Ring \u03b1\n[PROOFSTEP]\nlet add_group_with_one := e.addGroupWithOne\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\n\u22a2 Ring \u03b1\n[PROOFSTEP]\nlet npow := e.pow \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 Ring \u03b1\n[PROOFSTEP]\napply e.injective.ring _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x + y) = \u2191e x + \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x : \u03b1), \u2191e (-x) = -\u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x - y) = \u2191e x - \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase zsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x : \u03b1) (n : \u2124), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (x ^ n) = \u2191e x ^ n\n[PROOFSTEP]\nintros\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (n : \u2115), \u2191e \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase int_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (n : \u2124), \u2191e \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d + y\u271d) = \u2191e x\u271d + \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d : \u03b1\n\u22a2 \u2191e (-x\u271d) = -\u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d - y\u271d) = \u2191e x\u271d - \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase zsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d : \u03b1\nn\u271d : \u2124\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (x\u271d ^ n\u271d) = \u2191e x\u271d ^ n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nn\u271d : \u2115\n\u22a2 \u2191e \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase int_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Ring \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nn\u271d : \u2124\n\u22a2 \u2191e \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\n\u22a2 NonUnitalCommRing \u03b1\n[PROOFSTEP]\nlet zero := e.zero\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\n\u22a2 NonUnitalCommRing \u03b1\n[PROOFSTEP]\nlet add := e.add\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\n\u22a2 NonUnitalCommRing \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 NonUnitalCommRing \u03b1\n[PROOFSTEP]\nlet neg := e.Neg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\n\u22a2 NonUnitalCommRing \u03b1\n[PROOFSTEP]\nlet sub := e.sub\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\n\u22a2 NonUnitalCommRing \u03b1\n[PROOFSTEP]\nlet nsmul := e.smul \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\n\u22a2 NonUnitalCommRing \u03b1\n[PROOFSTEP]\nlet zsmul := e.smul \u2124\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 NonUnitalCommRing \u03b1\n[PROOFSTEP]\napply e.injective.nonUnitalCommRing _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x + y) = \u2191e x + \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x : \u03b1), \u2191e (-x) = -\u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x - y) = \u2191e x - \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase zsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2200 (x : \u03b1) (n : \u2124), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d + y\u271d) = \u2191e x\u271d + \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d : \u03b1\n\u22a2 \u2191e (-x\u271d) = -\u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d - y\u271d) = \u2191e x\u271d - \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase zsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : NonUnitalCommRing \u03b2\nzero : Zero \u03b1 := Equiv.zero e\nadd : Add \u03b1 := Equiv.add e\nmul : Mul \u03b1 := Equiv.mul e\nneg : Neg \u03b1 := Equiv.Neg e\nsub : Sub \u03b1 := Equiv.sub e\nnsmul : SMul \u2115 \u03b1 := Equiv.smul e \u2115\nzsmul : SMul \u2124 \u03b1 := Equiv.smul e \u2124\nx\u271d : \u03b1\nn\u271d : \u2124\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\n\u22a2 CommRing \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 CommRing \u03b1\n[PROOFSTEP]\nlet add_group_with_one := e.addGroupWithOne\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\n\u22a2 CommRing \u03b1\n[PROOFSTEP]\nlet npow := e.pow \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 CommRing \u03b1\n[PROOFSTEP]\napply e.injective.commRing _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x + y) = \u2191e x + \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x : \u03b1), \u2191e (-x) = -\u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x - y) = \u2191e x - \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase zsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x : \u03b1) (n : \u2124), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (x ^ n) = \u2191e x ^ n\n[PROOFSTEP]\nintros\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (n : \u2115), \u2191e \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase int_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2200 (n : \u2124), \u2191e \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d + y\u271d) = \u2191e x\u271d + \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d : \u03b1\n\u22a2 \u2191e (-x\u271d) = -\u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d - y\u271d) = \u2191e x\u271d - \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase zsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d : \u03b1\nn\u271d : \u2124\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (x\u271d ^ n\u271d) = \u2191e x\u271d ^ n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nn\u271d : \u2115\n\u22a2 \u2191e \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase int_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : CommRing \u03b2\nmul : Mul \u03b1 := Equiv.mul e\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nn\u271d : \u2124\n\u22a2 \u2191e \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\n\u22a2 DivisionRing \u03b1\n[PROOFSTEP]\nlet add_group_with_one := e.addGroupWithOne\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\n\u22a2 DivisionRing \u03b1\n[PROOFSTEP]\nlet inv := e.Inv\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\n\u22a2 DivisionRing \u03b1\n[PROOFSTEP]\nlet div := e.div\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\n\u22a2 DivisionRing \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 DivisionRing \u03b1\n[PROOFSTEP]\nlet npow := e.pow \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 DivisionRing \u03b1\n[PROOFSTEP]\nlet zpow := e.pow \u2124\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 DivisionRing \u03b1\n[PROOFSTEP]\nlet rat_cast := e.RatCast\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\n\u22a2 DivisionRing \u03b1\n[PROOFSTEP]\nlet qsmul := e.smul \u211a\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 DivisionRing \u03b1\n[PROOFSTEP]\napply e.injective.divisionRing _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x + y) = \u2191e x + \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x : \u03b1), \u2191e (-x) = -\u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x - y) = \u2191e x - \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase inv\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x : \u03b1), \u2191e x\u207b\u00b9 = (\u2191e x)\u207b\u00b9\n[PROOFSTEP]\nintros\n[GOAL]\ncase div\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x / y) = \u2191e x / \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase zsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x : \u03b1) (n : \u2124), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase qsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x : \u03b1) (n : \u211a), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (x ^ n) = \u2191e x ^ n\n[PROOFSTEP]\nintros\n[GOAL]\ncase zpow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x : \u03b1) (n : \u2124), \u2191e (x ^ n) = \u2191e x ^ n\n[PROOFSTEP]\nintros\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (n : \u2115), \u2191e \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase int_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (n : \u2124), \u2191e \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase rat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (n : \u211a), \u2191e \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d + y\u271d) = \u2191e x\u271d + \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d : \u03b1\n\u22a2 \u2191e (-x\u271d) = -\u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d - y\u271d) = \u2191e x\u271d - \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase inv\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d : \u03b1\n\u22a2 \u2191e x\u271d\u207b\u00b9 = (\u2191e x\u271d)\u207b\u00b9\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase div\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d / y\u271d) = \u2191e x\u271d / \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase zsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d : \u03b1\nn\u271d : \u2124\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase qsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d : \u03b1\nn\u271d : \u211a\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (x\u271d ^ n\u271d) = \u2191e x\u271d ^ n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase zpow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d : \u03b1\nn\u271d : \u2124\n\u22a2 \u2191e (x\u271d ^ n\u271d) = \u2191e x\u271d ^ n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nn\u271d : \u2115\n\u22a2 \u2191e \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase int_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nn\u271d : \u2124\n\u22a2 \u2191e \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase rat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : DivisionRing \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nn\u271d : \u211a\n\u22a2 \u2191e \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\n\u22a2 Field \u03b1\n[PROOFSTEP]\nlet add_group_with_one := e.addGroupWithOne\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\n\u22a2 Field \u03b1\n[PROOFSTEP]\nlet neg := e.Neg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\n\u22a2 Field \u03b1\n[PROOFSTEP]\nlet inv := e.Inv\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\n\u22a2 Field \u03b1\n[PROOFSTEP]\nlet div := e.div\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\n\u22a2 Field \u03b1\n[PROOFSTEP]\nlet mul := e.mul\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\n\u22a2 Field \u03b1\n[PROOFSTEP]\nlet npow := e.pow \u2115\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\n\u22a2 Field \u03b1\n[PROOFSTEP]\nlet zpow := e.pow \u2124\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\n\u22a2 Field \u03b1\n[PROOFSTEP]\nlet rat_cast := e.RatCast\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\n\u22a2 Field \u03b1\n[PROOFSTEP]\nlet qsmul := e.smul \u211a\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 Field \u03b1\n[PROOFSTEP]\napply e.injective.field _\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x + y) = \u2191e x + \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x : \u03b1), \u2191e (-x) = -\u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x - y) = \u2191e x - \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase inv\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x : \u03b1), \u2191e x\u207b\u00b9 = (\u2191e x)\u207b\u00b9\n[PROOFSTEP]\nintros\n[GOAL]\ncase div\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x y : \u03b1), \u2191e (x / y) = \u2191e x / \u2191e y\n[PROOFSTEP]\nintros\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase zsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x : \u03b1) (n : \u2124), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase qsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x : \u03b1) (n : \u211a), \u2191e (n \u2022 x) = n \u2022 \u2191e x\n[PROOFSTEP]\nintros\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x : \u03b1) (n : \u2115), \u2191e (x ^ n) = \u2191e x ^ n\n[PROOFSTEP]\nintros\n[GOAL]\ncase zpow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (x : \u03b1) (n : \u2124), \u2191e (x ^ n) = \u2191e x ^ n\n[PROOFSTEP]\nintros\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (n : \u2115), \u2191e \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase int_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (n : \u2124), \u2191e \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase rat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2200 (n : \u211a), \u2191e \u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2191e 0 = 0\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase one\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\n\u22a2 \u2191e 1 = 1\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase add\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d + y\u271d) = \u2191e x\u271d + \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase mul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d * y\u271d) = \u2191e x\u271d * \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d : \u03b1\n\u22a2 \u2191e (-x\u271d) = -\u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase sub\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d - y\u271d) = \u2191e x\u271d - \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase inv\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d : \u03b1\n\u22a2 \u2191e x\u271d\u207b\u00b9 = (\u2191e x\u271d)\u207b\u00b9\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase div\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d y\u271d : \u03b1\n\u22a2 \u2191e (x\u271d / y\u271d) = \u2191e x\u271d / \u2191e y\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase zsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d : \u03b1\nn\u271d : \u2124\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase qsmul\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d : \u03b1\nn\u271d : \u211a\n\u22a2 \u2191e (n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191e x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase npow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d : \u03b1\nn\u271d : \u2115\n\u22a2 \u2191e (x\u271d ^ n\u271d) = \u2191e x\u271d ^ n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase zpow\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nx\u271d : \u03b1\nn\u271d : \u2124\n\u22a2 \u2191e (x\u271d ^ n\u271d) = \u2191e x\u271d ^ n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase nat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nn\u271d : \u2115\n\u22a2 \u2191e \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase int_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nn\u271d : \u2124\n\u22a2 \u2191e \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\ncase rat_cast\n\u03b1 : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Field \u03b2\nadd_group_with_one : AddGroupWithOne \u03b1 := Equiv.addGroupWithOne e\nneg : Neg \u03b1 := Equiv.Neg e\ninv : Inv \u03b1 := Equiv.Inv e\ndiv : Div \u03b1 := Equiv.div e\nmul : Mul \u03b1 := Equiv.mul e\nnpow : Pow \u03b1 \u2115 := Equiv.pow e \u2115\nzpow : Pow \u03b1 \u2124 := Equiv.pow e \u2124\nrat_cast : RatCast \u03b1 := Equiv.RatCast e\nqsmul : SMul \u211a \u03b1 := Equiv.smul e \u211a\nn\u271d : \u211a\n\u22a2 \u2191e \u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulAction R \u03b2\nsrc\u271d : SMul R \u03b1 := Equiv.smul e R\n\u22a2 \u2200 (b : \u03b1), 1 \u2022 b = b\n[PROOFSTEP]\nsimp [smul_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\ne : \u03b1 \u2243 \u03b2\ninst\u271d : MulAction R \u03b2\nsrc\u271d : SMul R \u03b1 := Equiv.smul e R\n\u22a2 \u2200 (x y : R) (b : \u03b1), (x * y) \u2022 b = x \u2022 y \u2022 b\n[PROOFSTEP]\nsimp [smul_def, mul_smul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\ne : \u03b1 \u2243 \u03b2\ninst\u271d : AddCommMonoid \u03b2\n\u22a2 [inst : DistribMulAction R \u03b2] \u2192 DistribMulAction R \u03b1\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : AddCommMonoid \u03b2\ninst\u271d : DistribMulAction R \u03b2\n\u22a2 DistribMulAction R \u03b1\n[PROOFSTEP]\nletI := Equiv.addCommMonoid e\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : AddCommMonoid \u03b2\ninst\u271d : DistribMulAction R \u03b2\nthis : AddCommMonoid \u03b1 := Equiv.addCommMonoid e\n\u22a2 DistribMulAction R \u03b1\n[PROOFSTEP]\nexact\n ({ Equiv.mulAction R e with\n smul_zero := by simp [zero_def, smul_def]\n smul_add := by simp [add_def, smul_def, smul_add] } :\n DistribMulAction R \u03b1)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : AddCommMonoid \u03b2\ninst\u271d : DistribMulAction R \u03b2\nthis : AddCommMonoid \u03b1 := Equiv.addCommMonoid e\nsrc\u271d : MulAction R \u03b1 := Equiv.mulAction R e\n\u22a2 \u2200 (a : R), a \u2022 0 = 0\n[PROOFSTEP]\nsimp [zero_def, smul_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : AddCommMonoid \u03b2\ninst\u271d : DistribMulAction R \u03b2\nthis : AddCommMonoid \u03b1 := Equiv.addCommMonoid e\nsrc\u271d : MulAction R \u03b1 := Equiv.mulAction R e\n\u22a2 \u2200 (a : R) (x y : \u03b1), a \u2022 (x + y) = a \u2022 x + a \u2022 y\n[PROOFSTEP]\nsimp [add_def, smul_def, smul_add]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d : AddCommMonoid \u03b2\n\u22a2 let addCommMonoid := Equiv.addCommMonoid e;\n [inst : Module R \u03b2] \u2192 Module R \u03b1\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : AddCommMonoid \u03b2\naddCommMonoid\u271d : AddCommMonoid \u03b1 := Equiv.addCommMonoid e\ninst\u271d : Module R \u03b2\n\u22a2 Module R \u03b1\n[PROOFSTEP]\nexact\n ({ Equiv.distribMulAction R e with\n zero_smul := by simp [smul_def, zero_smul, zero_def]\n add_smul := by simp [add_def, smul_def, add_smul] } :\n Module R \u03b1)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : AddCommMonoid \u03b2\naddCommMonoid\u271d : AddCommMonoid \u03b1 := Equiv.addCommMonoid e\ninst\u271d : Module R \u03b2\nsrc\u271d : DistribMulAction R \u03b1 := Equiv.distribMulAction R e\n\u22a2 \u2200 (r s : R) (x : \u03b1), (r + s) \u2022 x = r \u2022 x + s \u2022 x\n[PROOFSTEP]\nsimp [add_def, smul_def, add_smul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : AddCommMonoid \u03b2\naddCommMonoid\u271d : AddCommMonoid \u03b1 := Equiv.addCommMonoid e\ninst\u271d : Module R \u03b2\nsrc\u271d : DistribMulAction R \u03b1 := Equiv.distribMulAction R e\n\u22a2 \u2200 (x : \u03b1), 0 \u2022 x = 0\n[PROOFSTEP]\nsimp [smul_def, zero_smul, zero_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : AddCommMonoid \u03b2\ninst\u271d : Module R \u03b2\n\u22a2 Sort ?u.508797\n[PROOFSTEP]\nlet addCommMonoid := Equiv.addCommMonoid e\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : AddCommMonoid \u03b2\ninst\u271d : Module R \u03b2\naddCommMonoid : AddCommMonoid \u03b1 := Equiv.addCommMonoid e\n\u22a2 Sort ?u.508797\n[PROOFSTEP]\nlet module := Equiv.module R e\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : AddCommMonoid \u03b2\ninst\u271d : Module R \u03b2\naddCommMonoid : AddCommMonoid \u03b1 := Equiv.addCommMonoid e\nmodule : Module R \u03b1 := Equiv.module R e\n\u22a2 Sort ?u.508797\n[PROOFSTEP]\nexact \u03b1 \u2243\u2097[R] \u03b2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : AddCommMonoid \u03b2\ninst\u271d : Module R \u03b2\n\u22a2 let addCommMonoid := Equiv.addCommMonoid e;\n let module := Equiv.module R e;\n \u03b1 \u2243\u2097[R] \u03b2\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : AddCommMonoid \u03b2\ninst\u271d : Module R \u03b2\naddCommMonoid\u271d : AddCommMonoid \u03b1 := Equiv.addCommMonoid e\nmodule\u271d : Module R \u03b1 := Equiv.module R e\n\u22a2 \u03b1 \u2243\u2097[R] \u03b2\n[PROOFSTEP]\nexact\n { Equiv.addEquiv e with\n map_smul' := fun r x => by\n apply e.symm.injective\n simp\n exact Iff.mpr (apply_eq_iff_eq_symm_apply _) rfl }\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : AddCommMonoid \u03b2\ninst\u271d : Module R \u03b2\naddCommMonoid\u271d : AddCommMonoid \u03b1 := Equiv.addCommMonoid e\nmodule\u271d : Module R \u03b1 := Equiv.module R e\nsrc\u271d : let mul := Equiv.add e;\n\u03b1 \u2243+ \u03b2 :=\n addEquiv e\nr : R\nx : \u03b1\n\u22a2 AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' := (_ : \u2200 (x y : \u03b1), toFun src\u271d.toEquiv (x + y) = toFun src\u271d.toEquiv x + toFun src\u271d.toEquiv y) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' := (_ : \u2200 (x y : \u03b1), toFun src\u271d.toEquiv (x + y) = toFun src\u271d.toEquiv x + toFun src\u271d.toEquiv y) }\n x\n[PROOFSTEP]\napply e.symm.injective\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : AddCommMonoid \u03b2\ninst\u271d : Module R \u03b2\naddCommMonoid\u271d : AddCommMonoid \u03b1 := Equiv.addCommMonoid e\nmodule\u271d : Module R \u03b1 := Equiv.module R e\nsrc\u271d : let mul := Equiv.add e;\n\u03b1 \u2243+ \u03b2 :=\n addEquiv e\nr : R\nx : \u03b1\n\u22a2 \u2191e.symm\n (AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' := (_ : \u2200 (x y : \u03b1), toFun src\u271d.toEquiv (x + y) = toFun src\u271d.toEquiv x + toFun src\u271d.toEquiv y) }\n (r \u2022 x)) =\n \u2191e.symm\n (\u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' := (_ : \u2200 (x y : \u03b1), toFun src\u271d.toEquiv (x + y) = toFun src\u271d.toEquiv x + toFun src\u271d.toEquiv y) }\n x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : AddCommMonoid \u03b2\ninst\u271d : Module R \u03b2\naddCommMonoid\u271d : AddCommMonoid \u03b1 := Equiv.addCommMonoid e\nmodule\u271d : Module R \u03b1 := Equiv.module R e\nsrc\u271d : let mul := Equiv.add e;\n\u03b1 \u2243+ \u03b2 :=\n addEquiv e\nr : R\nx : \u03b1\n\u22a2 \u2191(addEquiv e).toEquiv (r \u2022 x) = r \u2022 \u2191(addEquiv e).toEquiv x\n[PROOFSTEP]\nexact Iff.mpr (apply_eq_iff_eq_symm_apply _) rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d : Semiring \u03b2\n\u22a2 let semiring := Equiv.semiring e;\n [inst : Algebra R \u03b2] \u2192 Algebra R \u03b1\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\nsemiring\u271d : Semiring \u03b1 := Equiv.semiring e\ninst\u271d : Algebra R \u03b2\n\u22a2 Algebra R \u03b1\n[PROOFSTEP]\nfapply RingHom.toAlgebra'\n[GOAL]\ncase i\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\nsemiring\u271d : Semiring \u03b1 := Equiv.semiring e\ninst\u271d : Algebra R \u03b2\n\u22a2 R \u2192+* \u03b1\n[PROOFSTEP]\nexact ((ringEquiv e).symm : \u03b2 \u2192+* \u03b1).comp (algebraMap R \u03b2)\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\nsemiring\u271d : Semiring \u03b1 := Equiv.semiring e\ninst\u271d : Algebra R \u03b2\n\u22a2 \u2200 (c : R) (x : (fun x => \u03b1) c),\n \u2191(RingHom.comp (\u2191(RingEquiv.symm (ringEquiv e))) (algebraMap R \u03b2)) c * x =\n x * \u2191(RingHom.comp (\u2191(RingEquiv.symm (ringEquiv e))) (algebraMap R \u03b2)) c\n[PROOFSTEP]\nintro r x\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\nsemiring\u271d : Semiring \u03b1 := Equiv.semiring e\ninst\u271d : Algebra R \u03b2\nr : R\nx : \u03b1\n\u22a2 \u2191(RingHom.comp (\u2191(RingEquiv.symm (ringEquiv e))) (algebraMap R \u03b2)) r * x =\n x * \u2191(RingHom.comp (\u2191(RingEquiv.symm (ringEquiv e))) (algebraMap R \u03b2)) r\n[PROOFSTEP]\nsimp only [Function.comp_apply, RingHom.coe_comp]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\nsemiring\u271d : Semiring \u03b1 := Equiv.semiring e\ninst\u271d : Algebra R \u03b2\nr : R\nx : \u03b1\n\u22a2 \u2191\u2191(RingEquiv.symm (ringEquiv e)) (\u2191(algebraMap R \u03b2) r) * x =\n x * \u2191\u2191(RingEquiv.symm (ringEquiv e)) (\u2191(algebraMap R \u03b2) r)\n[PROOFSTEP]\nhave p := ringEquiv_symm_apply e\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\nsemiring\u271d : Semiring \u03b1 := Equiv.semiring e\ninst\u271d : Algebra R \u03b2\nr : R\nx : \u03b1\np : \u2200 (b : \u03b2), \u2191(RingEquiv.symm (ringEquiv e)) b = \u2191e.symm b\n\u22a2 \u2191\u2191(RingEquiv.symm (ringEquiv e)) (\u2191(algebraMap R \u03b2) r) * x =\n x * \u2191\u2191(RingEquiv.symm (ringEquiv e)) (\u2191(algebraMap R \u03b2) r)\n[PROOFSTEP]\ndsimp at p \n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\nsemiring\u271d : Semiring \u03b1 := Equiv.semiring e\ninst\u271d : Algebra R \u03b2\nr : R\nx : \u03b1\np : \u2200 (b : \u03b2), \u2191(RingEquiv.symm (ringEquiv e)) b = \u2191e.symm b\n\u22a2 \u2191\u2191(RingEquiv.symm (ringEquiv e)) (\u2191(algebraMap R \u03b2) r) * x =\n x * \u2191\u2191(RingEquiv.symm (ringEquiv e)) (\u2191(algebraMap R \u03b2) r)\n[PROOFSTEP]\nerw [p]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\nsemiring\u271d : Semiring \u03b1 := Equiv.semiring e\ninst\u271d : Algebra R \u03b2\nr : R\nx : \u03b1\np : \u2200 (b : \u03b2), \u2191(RingEquiv.symm (ringEquiv e)) b = \u2191e.symm b\n\u22a2 \u2191e.symm (\u2191(algebraMap R \u03b2) r) * x = x * \u2191e.symm (\u2191(algebraMap R \u03b2) r)\n[PROOFSTEP]\nclear p\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\nsemiring\u271d : Semiring \u03b1 := Equiv.semiring e\ninst\u271d : Algebra R \u03b2\nr : R\nx : \u03b1\n\u22a2 \u2191e.symm (\u2191(algebraMap R \u03b2) r) * x = x * \u2191e.symm (\u2191(algebraMap R \u03b2) r)\n[PROOFSTEP]\napply (ringEquiv e).injective\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\nsemiring\u271d : Semiring \u03b1 := Equiv.semiring e\ninst\u271d : Algebra R \u03b2\nr : R\nx : \u03b1\n\u22a2 \u2191(ringEquiv e) (\u2191e.symm (\u2191(algebraMap R \u03b2) r) * x) = \u2191(ringEquiv e) (x * \u2191e.symm (\u2191(algebraMap R \u03b2) r))\n[PROOFSTEP]\nsimp only [(ringEquiv e).map_mul]\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\nsemiring\u271d : Semiring \u03b1 := Equiv.semiring e\ninst\u271d : Algebra R \u03b2\nr : R\nx : \u03b1\n\u22a2 \u2191(ringEquiv e) (\u2191e.symm (\u2191(algebraMap R \u03b2) r)) * \u2191(ringEquiv e) x =\n \u2191(ringEquiv e) x * \u2191(ringEquiv e) (\u2191e.symm (\u2191(algebraMap R \u03b2) r))\n[PROOFSTEP]\nsimp [Algebra.commutes]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\ninst\u271d : Algebra R \u03b2\n\u22a2 Sort ?u.532380\n[PROOFSTEP]\nlet semiring := Equiv.semiring e\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\ninst\u271d : Algebra R \u03b2\nsemiring : Semiring \u03b1 := Equiv.semiring e\n\u22a2 Sort ?u.532380\n[PROOFSTEP]\nlet algebra := Equiv.algebra R e\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\ninst\u271d : Algebra R \u03b2\nsemiring : Semiring \u03b1 := Equiv.semiring e\nalgebra : Algebra R \u03b1 := Equiv.algebra R e\n\u22a2 Sort ?u.532380\n[PROOFSTEP]\nexact \u03b1 \u2243\u2090[R] \u03b2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\ninst\u271d : Algebra R \u03b2\n\u22a2 let semiring := Equiv.semiring e;\n let algebra := Equiv.algebra R e;\n \u03b1 \u2243\u2090[R] \u03b2\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\ninst\u271d : Algebra R \u03b2\nsemiring\u271d : Semiring \u03b1 := Equiv.semiring e\nalgebra\u271d : Algebra R \u03b1 := Equiv.algebra R e\n\u22a2 \u03b1 \u2243\u2090[R] \u03b2\n[PROOFSTEP]\nexact\n { Equiv.ringEquiv e with\n commutes' := fun r => by\n apply e.symm.injective\n simp\n rfl }\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\ninst\u271d : Algebra R \u03b2\nsemiring\u271d : Semiring \u03b1 := Equiv.semiring e\nalgebra\u271d : Algebra R \u03b1 := Equiv.algebra R e\nsrc\u271d : let add := Equiv.add e;\nlet mul := Equiv.mul e;\n\u03b1 \u2243+* \u03b2 :=\n ringEquiv e\nr : R\n\u22a2 toFun src\u271d.toEquiv (\u2191(algebraMap R \u03b1) r) = \u2191(algebraMap R \u03b2) r\n[PROOFSTEP]\napply e.symm.injective\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\ninst\u271d : Algebra R \u03b2\nsemiring\u271d : Semiring \u03b1 := Equiv.semiring e\nalgebra\u271d : Algebra R \u03b1 := Equiv.algebra R e\nsrc\u271d : let add := Equiv.add e;\nlet mul := Equiv.mul e;\n\u03b1 \u2243+* \u03b2 :=\n ringEquiv e\nr : R\n\u22a2 \u2191e.symm (toFun src\u271d.toEquiv (\u2191(algebraMap R \u03b1) r)) = \u2191e.symm (\u2191(algebraMap R \u03b2) r)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ne\u271d : \u03b1 \u2243 \u03b2\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\ne : \u03b1 \u2243 \u03b2\ninst\u271d\u00b9 : Semiring \u03b2\ninst\u271d : Algebra R \u03b2\nsemiring\u271d : Semiring \u03b1 := Equiv.semiring e\nalgebra\u271d : Algebra R \u03b1 := Equiv.algebra R e\nsrc\u271d : let add := Equiv.add e;\nlet mul := Equiv.mul e;\n\u03b1 \u2243+* \u03b2 :=\n ringEquiv e\nr : R\n\u22a2 \u2191(algebraMap R \u03b1) r = \u2191e.symm (\u2191(algebraMap R \u03b2) r)\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Logic.Equiv.TransferInstance", "llama_tokens": 65693, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754371026367, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.44236386167027303}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace R\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\ne : R\nh : HasSum (fun n => \u2191(expSeries \ud835\udd5c R n) fun x_1 => fst x) e\n\u22a2 HasSum (fun n => fst (\u2191(expSeries \ud835\udd5c (tsze R M) n) fun x_1 => x)) e\n[PROOFSTEP]\nsimpa [expSeries_apply_eq] using h\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => \u2191(expSeries \ud835\udd5c R n) fun x_1 => fst x) e\n\u22a2 HasSum (fun n => snd (\u2191(expSeries \ud835\udd5c (tsze R M) n) fun x_1 => x)) (e \u2022 snd x)\n[PROOFSTEP]\nsimp_rw [expSeries_apply_eq] at *\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => (\u2191(Nat.factorial n))\u207b\u00b9 \u2022 fst x ^ n) e\n\u22a2 HasSum (fun n => snd ((\u2191(Nat.factorial n))\u207b\u00b9 \u2022 x ^ n)) (e \u2022 snd x)\n[PROOFSTEP]\nconv =>\n congr\n ext n\n rw [snd_smul, snd_pow_of_smul_comm _ _ hx, nsmul_eq_smul_cast \ud835\udd5c n, smul_smul, inv_mul_eq_div, \u2190 inv_div, \u2190 smul_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => (\u2191(Nat.factorial n))\u207b\u00b9 \u2022 fst x ^ n) e\n| HasSum (fun n => snd ((\u2191(Nat.factorial n))\u207b\u00b9 \u2022 x ^ n)) (e \u2022 snd x)\n[PROOFSTEP]\n congr\n ext n\n rw [snd_smul, snd_pow_of_smul_comm _ _ hx, nsmul_eq_smul_cast \ud835\udd5c n, smul_smul, inv_mul_eq_div, \u2190 inv_div, \u2190 smul_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => (\u2191(Nat.factorial n))\u207b\u00b9 \u2022 fst x ^ n) e\n| HasSum (fun n => snd ((\u2191(Nat.factorial n))\u207b\u00b9 \u2022 x ^ n)) (e \u2022 snd x)\n[PROOFSTEP]\n congr\n ext n\n rw [snd_smul, snd_pow_of_smul_comm _ _ hx, nsmul_eq_smul_cast \ud835\udd5c n, smul_smul, inv_mul_eq_div, \u2190 inv_div, \u2190 smul_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => (\u2191(Nat.factorial n))\u207b\u00b9 \u2022 fst x ^ n) e\n| HasSum (fun n => snd ((\u2191(Nat.factorial n))\u207b\u00b9 \u2022 x ^ n)) (e \u2022 snd x)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase f\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => (\u2191(Nat.factorial n))\u207b\u00b9 \u2022 fst x ^ n) e\n| fun n => snd ((\u2191(Nat.factorial n))\u207b\u00b9 \u2022 x ^ n)\ncase a\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => (\u2191(Nat.factorial n))\u207b\u00b9 \u2022 fst x ^ n) e\n| e \u2022 snd x\n[PROOFSTEP]\next n\n[GOAL]\ncase f.h\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => (\u2191(Nat.factorial n))\u207b\u00b9 \u2022 fst x ^ n) e\nn : \u2115\n| snd ((\u2191(Nat.factorial n))\u207b\u00b9 \u2022 x ^ n)\ncase a\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => (\u2191(Nat.factorial n))\u207b\u00b9 \u2022 fst x ^ n) e\n| e \u2022 snd x\n[PROOFSTEP]\nrw [snd_smul, snd_pow_of_smul_comm _ _ hx, nsmul_eq_smul_cast \ud835\udd5c n, smul_smul, inv_mul_eq_div, \u2190 inv_div, \u2190 smul_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => (\u2191(Nat.factorial n))\u207b\u00b9 \u2022 fst x ^ n) e\n\u22a2 HasSum (fun n => ((\u2191(Nat.factorial n) / \u2191n)\u207b\u00b9 \u2022 fst x ^ Nat.pred n) \u2022 snd x) (e \u2022 snd x)\n[PROOFSTEP]\napply HasSum.smul_const\n[GOAL]\ncase hf\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => (\u2191(Nat.factorial n))\u207b\u00b9 \u2022 fst x ^ n) e\n\u22a2 HasSum (fun z => (\u2191(Nat.factorial z) / \u2191z)\u207b\u00b9 \u2022 fst x ^ Nat.pred z) e\n[PROOFSTEP]\nrw [\u2190 hasSum_nat_add_iff' 1]\n[GOAL]\ncase hf\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => (\u2191(Nat.factorial n))\u207b\u00b9 \u2022 fst x ^ n) e\n\u22a2 HasSum (fun n => (\u2191(Nat.factorial (n + 1)) / \u2191(n + 1))\u207b\u00b9 \u2022 fst x ^ Nat.pred (n + 1))\n (e - Finset.sum (Finset.range 1) fun i => (\u2191(Nat.factorial i) / \u2191i)\u207b\u00b9 \u2022 fst x ^ Nat.pred i)\n[PROOFSTEP]\nrw [Finset.range_one, Finset.sum_singleton, Nat.cast_zero, div_zero, inv_zero, zero_smul, sub_zero]\n[GOAL]\ncase hf\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => (\u2191(Nat.factorial n))\u207b\u00b9 \u2022 fst x ^ n) e\n\u22a2 HasSum (fun n => (\u2191(Nat.factorial (n + 1)) / \u2191(n + 1))\u207b\u00b9 \u2022 fst x ^ Nat.pred (n + 1)) e\n[PROOFSTEP]\nsimp_rw [\u2190 Nat.succ_eq_add_one, Nat.pred_succ, Nat.factorial_succ, Nat.cast_mul, \u2190 Nat.succ_eq_add_one,\n mul_div_cancel_left _ ((@Nat.cast_ne_zero \ud835\udd5c _ _ _).mpr <| Nat.succ_ne_zero _)]\n[GOAL]\ncase hf\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => (\u2191(Nat.factorial n))\u207b\u00b9 \u2022 fst x ^ n) e\n\u22a2 HasSum (fun n => (\u2191(Nat.factorial n))\u207b\u00b9 \u2022 fst x ^ n) e\n[PROOFSTEP]\nexact h\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => \u2191(expSeries \ud835\udd5c R n) fun x_1 => fst x) e\n\u22a2 HasSum (fun n => \u2191(expSeries \ud835\udd5c (tsze R M) n) fun x_1 => x) (inl e + inr (e \u2022 snd x))\n[PROOFSTEP]\nsimpa only [inl_fst_add_inr_snd_eq] using\n (hasSum_inl _ <| hasSum_fst_expSeries \ud835\udd5c x h).add (hasSum_inr _ <| hasSum_snd_expSeries_of_smul_comm \ud835\udd5c x hx h)\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2076 : NormedRing R\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : NormedAlgebra \ud835\udd5c R\ninst\u271d\u00b9\u00b3 : Module R M\ninst\u271d\u00b9\u00b2 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u00b9 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c M\ninst\u271d\u2079 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2078 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : TopologicalRing R\ninst\u271d\u2075 : TopologicalAddGroup M\ninst\u271d\u2074 : ContinuousSMul R M\ninst\u271d\u00b3 : ContinuousSMul R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b2 : CompleteSpace R\ninst\u271d\u00b9 : T2Space R\ninst\u271d : T2Space M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\n\u22a2 exp \ud835\udd5c x = inl (exp \ud835\udd5c (fst x)) + inr (exp \ud835\udd5c (fst x) \u2022 snd x)\n[PROOFSTEP]\nsimp_rw [exp, FormalMultilinearSeries.sum]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2076 : NormedRing R\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : NormedAlgebra \ud835\udd5c R\ninst\u271d\u00b9\u00b3 : Module R M\ninst\u271d\u00b9\u00b2 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u00b9 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c M\ninst\u271d\u2079 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2078 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : TopologicalRing R\ninst\u271d\u2075 : TopologicalAddGroup M\ninst\u271d\u2074 : ContinuousSMul R M\ninst\u271d\u00b3 : ContinuousSMul R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b2 : CompleteSpace R\ninst\u271d\u00b9 : T2Space R\ninst\u271d : T2Space M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\n\u22a2 (\u2211' (n : \u2115), \u2191(expSeries \ud835\udd5c (tsze R M) n) fun x_1 => x) =\n inl (\u2211' (n : \u2115), \u2191(expSeries \ud835\udd5c R n) fun x_1 => fst x) +\n inr ((\u2211' (n : \u2115), \u2191(expSeries \ud835\udd5c R n) fun x_1 => fst x) \u2022 snd x)\n[PROOFSTEP]\nrefine' (hasSum_expSeries_of_smul_comm \ud835\udd5c x hx _).tsum_eq\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2076 : NormedRing R\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : NormedAlgebra \ud835\udd5c R\ninst\u271d\u00b9\u00b3 : Module R M\ninst\u271d\u00b9\u00b2 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u00b9 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c M\ninst\u271d\u2079 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2078 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : TopologicalRing R\ninst\u271d\u2075 : TopologicalAddGroup M\ninst\u271d\u2074 : ContinuousSMul R M\ninst\u271d\u00b3 : ContinuousSMul R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b2 : CompleteSpace R\ninst\u271d\u00b9 : T2Space R\ninst\u271d : T2Space M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\n\u22a2 HasSum (fun n => \u2191(expSeries \ud835\udd5c R n) fun x_1 => fst x) (\u2211' (n : \u2115), \u2191(expSeries \ud835\udd5c R n) fun x_1 => fst x)\n[PROOFSTEP]\nexact expSeries_hasSum_exp _\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2076 : NormedRing R\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : NormedAlgebra \ud835\udd5c R\ninst\u271d\u00b9\u00b3 : Module R M\ninst\u271d\u00b9\u00b2 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u00b9 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c M\ninst\u271d\u2079 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2078 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : TopologicalRing R\ninst\u271d\u2075 : TopologicalAddGroup M\ninst\u271d\u2074 : ContinuousSMul R M\ninst\u271d\u00b3 : ContinuousSMul R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b2 : CompleteSpace R\ninst\u271d\u00b9 : T2Space R\ninst\u271d : T2Space M\nx : R\n\u22a2 exp \ud835\udd5c (inl x) = inl (exp \ud835\udd5c x)\n[PROOFSTEP]\nrw [exp_def_of_smul_comm, snd_inl, fst_inl, smul_zero, inr_zero, add_zero]\n[GOAL]\ncase hx\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2076 : NormedRing R\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : NormedAlgebra \ud835\udd5c R\ninst\u271d\u00b9\u00b3 : Module R M\ninst\u271d\u00b9\u00b2 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u00b9 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c M\ninst\u271d\u2079 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2078 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : TopologicalRing R\ninst\u271d\u2075 : TopologicalAddGroup M\ninst\u271d\u2074 : ContinuousSMul R M\ninst\u271d\u00b3 : ContinuousSMul R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b2 : CompleteSpace R\ninst\u271d\u00b9 : T2Space R\ninst\u271d : T2Space M\nx : R\n\u22a2 MulOpposite.op (fst (inl x)) \u2022 snd (inl x) = fst (inl x) \u2022 snd (inl x)\n[PROOFSTEP]\nrw [snd_inl, fst_inl, smul_zero, smul_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2076 : NormedRing R\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : NormedAlgebra \ud835\udd5c R\ninst\u271d\u00b9\u00b3 : Module R M\ninst\u271d\u00b9\u00b2 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u00b9 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c M\ninst\u271d\u2079 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2078 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : TopologicalRing R\ninst\u271d\u2075 : TopologicalAddGroup M\ninst\u271d\u2074 : ContinuousSMul R M\ninst\u271d\u00b3 : ContinuousSMul R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b2 : CompleteSpace R\ninst\u271d\u00b9 : T2Space R\ninst\u271d : T2Space M\nm : M\n\u22a2 exp \ud835\udd5c (inr m) = 1 + inr m\n[PROOFSTEP]\nrw [exp_def_of_smul_comm, snd_inr, fst_inr, exp_zero, one_smul, inl_one]\n[GOAL]\ncase hx\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2076 : NormedRing R\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : NormedAlgebra \ud835\udd5c R\ninst\u271d\u00b9\u00b3 : Module R M\ninst\u271d\u00b9\u00b2 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u00b9 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c M\ninst\u271d\u2079 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2078 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : TopologicalRing R\ninst\u271d\u2075 : TopologicalAddGroup M\ninst\u271d\u2074 : ContinuousSMul R M\ninst\u271d\u00b3 : ContinuousSMul R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b2 : CompleteSpace R\ninst\u271d\u00b9 : T2Space R\ninst\u271d : T2Space M\nm : M\n\u22a2 MulOpposite.op (fst (inr m)) \u2022 snd (inr m) = fst (inr m) \u2022 snd (inr m)\n[PROOFSTEP]\nrw [snd_inr, fst_inr, MulOpposite.op_zero, zero_smul, zero_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedCommRing R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : NormedAlgebra \ud835\udd5c R\ninst\u271d\u00b9\u00b9 : Module R M\ninst\u271d\u00b9\u2070 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2079 : IsCentralScalar R M\ninst\u271d\u2078 : Module \ud835\udd5c M\ninst\u271d\u2077 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : TopologicalRing R\ninst\u271d\u2074 : TopologicalAddGroup M\ninst\u271d\u00b3 : ContinuousSMul R M\ninst\u271d\u00b2 : CompleteSpace R\ninst\u271d\u00b9 : T2Space R\ninst\u271d : T2Space M\nx : tsze R M\n\u22a2 fst (exp \ud835\udd5c x) = exp \ud835\udd5c (fst x)\n[PROOFSTEP]\nrw [exp_def, fst_add, fst_inl, fst_inr, add_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedCommRing R\ninst\u271d\u00b9\u00b3 : AddCommGroup M\ninst\u271d\u00b9\u00b2 : NormedAlgebra \ud835\udd5c R\ninst\u271d\u00b9\u00b9 : Module R M\ninst\u271d\u00b9\u2070 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2079 : IsCentralScalar R M\ninst\u271d\u2078 : Module \ud835\udd5c M\ninst\u271d\u2077 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : TopologicalRing R\ninst\u271d\u2074 : TopologicalAddGroup M\ninst\u271d\u00b3 : ContinuousSMul R M\ninst\u271d\u00b2 : CompleteSpace R\ninst\u271d\u00b9 : T2Space R\ninst\u271d : T2Space M\nx : tsze R M\n\u22a2 snd (exp \ud835\udd5c x) = exp \ud835\udd5c (fst x) \u2022 snd x\n[PROOFSTEP]\nrw [exp_def, snd_add, snd_inl, snd_inr, zero_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedCommRing R\ninst\u271d\u00b9\u2074 : AddCommGroup M\ninst\u271d\u00b9\u00b3 : NormedAlgebra \ud835\udd5c R\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2070 : IsCentralScalar R M\ninst\u271d\u2079 : Module \ud835\udd5c M\ninst\u271d\u2078 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : TopologicalRing R\ninst\u271d\u2075 : TopologicalAddGroup M\ninst\u271d\u2074 : ContinuousSMul R M\ninst\u271d\u00b3 : CompleteSpace R\ninst\u271d\u00b2 : T2Space R\ninst\u271d\u00b9 : T2Space M\nx : tsze R M\ninst\u271d : Invertible (fst x)\n\u22a2 x = fst x \u2022 exp \ud835\udd5c (\u215f(fst x) \u2022 inr (snd x))\n[PROOFSTEP]\nrw [\u2190 inr_smul, exp_inr, smul_add, \u2190 inl_one, \u2190 inl_smul, \u2190 inr_smul, smul_eq_mul, mul_one, smul_smul, mul_invOf_self,\n one_smul, inl_fst_add_inr_snd_eq]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.TrivSqZeroExt", "llama_tokens": 10352, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737963569014, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.4422334080790111}} {"text": "[GOAL]\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\n\u22a2 \u2200 (a\u2081 a\u2082 b\u2081 b\u2082 : K),\n Setoid.r a\u2081 b\u2081 \u2192 Setoid.r a\u2082 b\u2082 \u2192 (fun a b => \u2203 c, c \u2022 b = a) a\u2081 a\u2082 = (fun a b => \u2203 c, c \u2022 b = a) b\u2081 b\u2082\n[PROOFSTEP]\nrintro _ _ a b \u27e8c, rfl\u27e9 \u27e8d, rfl\u27e9\n[GOAL]\ncase intro.intro\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\n\u22a2 (fun a b => \u2203 c, c \u2022 b = a) ((fun m => m \u2022 a) c) ((fun m => m \u2022 b) d) = (fun a b => \u2203 c, c \u2022 b = a) a b\n[PROOFSTEP]\next\n[GOAL]\ncase intro.intro.a\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\n\u22a2 (fun a b => \u2203 c, c \u2022 b = a) ((fun m => m \u2022 a) c) ((fun m => m \u2022 b) d) \u2194 (fun a b => \u2203 c, c \u2022 b = a) a b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.a.mp\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\n\u22a2 (fun a b => \u2203 c, c \u2022 b = a) ((fun m => m \u2022 a) c) ((fun m => m \u2022 b) d) \u2192 (fun a b => \u2203 c, c \u2022 b = a) a b\n[PROOFSTEP]\nrintro \u27e8e, he\u27e9\n[GOAL]\ncase intro.intro.a.mp.intro\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\ne : A\nhe : e \u2022 (fun m => m \u2022 b) d = (fun m => m \u2022 a) c\n\u22a2 \u2203 c, c \u2022 b = a\n[PROOFSTEP]\nuse(c\u207b\u00b9 : A\u02e3) * e * d\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\ne : A\nhe : e \u2022 (fun m => m \u2022 b) d = (fun m => m \u2022 a) c\n\u22a2 (\u2191c\u207b\u00b9 * e * \u2191d) \u2022 b = a\n[PROOFSTEP]\napply_fun fun t => c\u207b\u00b9 \u2022 t at he \n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\ne : A\nhe : c\u207b\u00b9 \u2022 e \u2022 (fun m => m \u2022 b) d = c\u207b\u00b9 \u2022 (fun m => m \u2022 a) c\n\u22a2 (\u2191c\u207b\u00b9 * e * \u2191d) \u2022 b = a\n[PROOFSTEP]\nsimpa [mul_smul] using he\n[GOAL]\ncase intro.intro.a.mpr\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\n\u22a2 (fun a b => \u2203 c, c \u2022 b = a) a b \u2192 (fun a b => \u2203 c, c \u2022 b = a) ((fun m => m \u2022 a) c) ((fun m => m \u2022 b) d)\n[PROOFSTEP]\nrintro \u27e8e, he\u27e9\n[GOAL]\ncase intro.intro.a.mpr.intro\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\ne : A\nhe : e \u2022 b = a\n\u22a2 \u2203 c_1, c_1 \u2022 (fun m => m \u2022 b) d = (fun m => m \u2022 a) c\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro.a.mpr.intro\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\ne : A\nhe : e \u2022 b = a\n\u22a2 \u2203 c_1, c_1 \u2022 d \u2022 b = c \u2022 a\n[PROOFSTEP]\nuse(d\u207b\u00b9 : A\u02e3) * c * e\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\ne : A\nhe : e \u2022 b = a\n\u22a2 (\u2191d\u207b\u00b9 * \u2191c * e) \u2022 d \u2022 b = c \u2022 a\n[PROOFSTEP]\nerw [\u2190 he, \u2190 mul_smul, \u2190 mul_smul]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\ne : A\nhe : e \u2022 b = a\n\u22a2 (\u2191d\u207b\u00b9 * \u2191c * e * \u2191d) \u2022 b = (\u2191c * e) \u2022 b\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e_a\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\ne : A\nhe : e \u2022 b = a\n\u22a2 \u2191d\u207b\u00b9 * \u2191c * e * \u2191d = \u2191c * e\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase h.e_a\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\ne : A\nhe : e \u2022 b = a\n\u22a2 \u2191d * (\u2191d\u207b\u00b9 * \u2191c * e) = \u2191c * e\n[PROOFSTEP]\nsimp only [\u2190 mul_assoc, \u2190 Units.val_mul, mul_inv_self, one_mul]\n[GOAL]\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\n\u22a2 \u2200 (a\u2081 a\u2082 b\u2081 b\u2082 : K),\n Setoid.r a\u2081 b\u2081 \u2192\n Setoid.r a\u2082 b\u2082 \u2192 (fun a b => Quotient.mk'' (a * b)) a\u2081 a\u2082 = (fun a b => Quotient.mk'' (a * b)) b\u2081 b\u2082\n[PROOFSTEP]\nrintro _ _ a b \u27e8c, rfl\u27e9 \u27e8d, rfl\u27e9\n[GOAL]\ncase intro.intro\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\n\u22a2 (fun a b => Quotient.mk'' (a * b)) ((fun m => m \u2022 a) c) ((fun m => m \u2022 b) d) = (fun a b => Quotient.mk'' (a * b)) a b\n[PROOFSTEP]\napply Quotient.sound'\n[GOAL]\ncase intro.intro.a\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\n\u22a2 Setoid.r ((fun m => m \u2022 a) c * (fun m => m \u2022 b) d) (a * b)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro.a\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\n\u22a2 Setoid.r (c \u2022 a * d \u2022 b) (a * b)\n[PROOFSTEP]\nuse c * d\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\n\u22a2 (fun m => m \u2022 (a * b)) (c * d) = c \u2022 a * d \u2022 b\n[PROOFSTEP]\nsimp only [mul_smul, Algebra.smul_def, Units.smul_def, RingHom.map_mul, Units.val_mul]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : A\u02e3\n\u22a2 \u2191(algebraMap A K) \u2191c * (\u2191(algebraMap A K) \u2191d * (a * b)) = \u2191(algebraMap A K) \u2191c * a * (\u2191(algebraMap A K) \u2191d * b)\n[PROOFSTEP]\nring\n[GOAL]\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx : ValueGroup A K\n\u22a2 \u2200 (a b : K), Setoid.r a b \u2192 (fun a => Quotient.mk'' a\u207b\u00b9) a = (fun a => Quotient.mk'' a\u207b\u00b9) b\n[PROOFSTEP]\nrintro _ a \u27e8b, rfl\u27e9\n[GOAL]\ncase intro\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx : ValueGroup A K\na : K\nb : A\u02e3\n\u22a2 (fun a => Quotient.mk'' a\u207b\u00b9) ((fun m => m \u2022 a) b) = (fun a => Quotient.mk'' a\u207b\u00b9) a\n[PROOFSTEP]\napply Quotient.sound'\n[GOAL]\ncase intro.a\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx : ValueGroup A K\na : K\nb : A\u02e3\n\u22a2 Setoid.r ((fun m => m \u2022 a) b)\u207b\u00b9 a\u207b\u00b9\n[PROOFSTEP]\nuse b\u207b\u00b9\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx : ValueGroup A K\na : K\nb : A\u02e3\n\u22a2 (fun m => m \u2022 a\u207b\u00b9) b\u207b\u00b9 = ((fun m => m \u2022 a) b)\u207b\u00b9\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b2 : CommRing A\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A K\nx : ValueGroup A K\na : K\nb : A\u02e3\n\u22a2 b\u207b\u00b9 \u2022 a\u207b\u00b9 = (b \u2022 a)\u207b\u00b9\n[PROOFSTEP]\nrw [Units.smul_def, Units.smul_def, Algebra.smul_def, Algebra.smul_def, mul_inv, map_units_inv]\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\n\u22a2 a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\nrcases a with \u27e8a\u27e9\n[GOAL]\ncase mk\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b : ValueGroup A K\na : K\n\u22a2 Quot.mk Setoid.r a \u2264 b \u2228 b \u2264 Quot.mk Setoid.r a\n[PROOFSTEP]\nrcases b with \u27e8b\u27e9\n[GOAL]\ncase mk.mk\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\na b : K\n\u22a2 Quot.mk Setoid.r a \u2264 Quot.mk Setoid.r b \u2228 Quot.mk Setoid.r b \u2264 Quot.mk Setoid.r a\n[PROOFSTEP]\nobtain \u27e8xa, ya, hya, rfl\u27e9 : \u2203 a b : A, _ := IsFractionRing.div_surjective a\n[GOAL]\ncase mk.mk.intro.intro.intro\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b\u271d : ValueGroup A K\nb : K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\n\u22a2 Quot.mk Setoid.r (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya) \u2264 Quot.mk Setoid.r b \u2228\n Quot.mk Setoid.r b \u2264 Quot.mk Setoid.r (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya)\n[PROOFSTEP]\nobtain \u27e8xb, yb, hyb, rfl\u27e9 : \u2203 a b : A, _ := IsFractionRing.div_surjective b\n[GOAL]\ncase mk.mk.intro.intro.intro.intro.intro.intro\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\n\u22a2 Quot.mk Setoid.r (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya) \u2264\n Quot.mk Setoid.r (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2228\n Quot.mk Setoid.r (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2264\n Quot.mk Setoid.r (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya)\n[PROOFSTEP]\nhave : (algebraMap A K) ya \u2260 0 := IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors hya\n[GOAL]\ncase mk.mk.intro.intro.intro.intro.intro.intro\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis : \u2191(algebraMap A K) ya \u2260 0\n\u22a2 Quot.mk Setoid.r (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya) \u2264\n Quot.mk Setoid.r (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2228\n Quot.mk Setoid.r (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2264\n Quot.mk Setoid.r (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya)\n[PROOFSTEP]\nhave : (algebraMap A K) yb \u2260 0 := IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors hyb\n[GOAL]\ncase mk.mk.intro.intro.intro.intro.intro.intro\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\n\u22a2 Quot.mk Setoid.r (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya) \u2264\n Quot.mk Setoid.r (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2228\n Quot.mk Setoid.r (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2264\n Quot.mk Setoid.r (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya)\n[PROOFSTEP]\nobtain \u27e8c, h | h\u27e9 := ValuationRing.cond (xa * yb) (xb * ya)\n[GOAL]\ncase mk.mk.intro.intro.intro.intro.intro.intro.intro.inl\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xa * yb * c = xb * ya\n\u22a2 Quot.mk Setoid.r (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya) \u2264\n Quot.mk Setoid.r (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2228\n Quot.mk Setoid.r (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2264\n Quot.mk Setoid.r (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya)\n[PROOFSTEP]\nright\n[GOAL]\ncase mk.mk.intro.intro.intro.intro.intro.intro.intro.inl.h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xa * yb * c = xb * ya\n\u22a2 Quot.mk Setoid.r (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2264\n Quot.mk Setoid.r (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya)\n[PROOFSTEP]\nuse c\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xa * yb * c = xb * ya\n\u22a2 c \u2022 (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya) = \u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb\n[PROOFSTEP]\nrw [Algebra.smul_def]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xa * yb * c = xb * ya\n\u22a2 \u2191(algebraMap A K) c * (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya) = \u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xa * yb * c = xb * ya\n\u22a2 \u2191(algebraMap A K) c * \u2191(algebraMap A K) xa * \u2191(algebraMap A K) yb = \u2191(algebraMap A K) xb * \u2191(algebraMap A K) ya\n[PROOFSTEP]\nsimp only [\u2190 RingHom.map_mul, \u2190 h]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xa * yb * c = xb * ya\n\u22a2 \u2191(algebraMap A K) (c * xa * yb) = \u2191(algebraMap A K) (xa * yb * c)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.h.e_6.h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xa * yb * c = xb * ya\n\u22a2 c * xa * yb = xa * yb * c\n[PROOFSTEP]\nring\n[GOAL]\ncase mk.mk.intro.intro.intro.intro.intro.intro.intro.inr\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xb * ya * c = xa * yb\n\u22a2 Quot.mk Setoid.r (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya) \u2264\n Quot.mk Setoid.r (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2228\n Quot.mk Setoid.r (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2264\n Quot.mk Setoid.r (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya)\n[PROOFSTEP]\nleft\n[GOAL]\ncase mk.mk.intro.intro.intro.intro.intro.intro.intro.inr.h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xb * ya * c = xa * yb\n\u22a2 Quot.mk Setoid.r (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya) \u2264\n Quot.mk Setoid.r (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb)\n[PROOFSTEP]\nuse c\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xb * ya * c = xa * yb\n\u22a2 c \u2022 (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) = \u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya\n[PROOFSTEP]\nrw [Algebra.smul_def]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xb * ya * c = xa * yb\n\u22a2 \u2191(algebraMap A K) c * (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) = \u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xb * ya * c = xa * yb\n\u22a2 \u2191(algebraMap A K) c * \u2191(algebraMap A K) xb * \u2191(algebraMap A K) ya = \u2191(algebraMap A K) xa * \u2191(algebraMap A K) yb\n[PROOFSTEP]\nsimp only [\u2190 RingHom.map_mul, \u2190 h]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xb * ya * c = xa * yb\n\u22a2 \u2191(algebraMap A K) (c * xb * ya) = \u2191(algebraMap A K) (xb * ya * c)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.h.e_6.h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : ValueGroup A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xb * ya * c = xa * yb\n\u22a2 c * xb * ya = xb * ya * c\n[PROOFSTEP]\nring\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 \u2200 (a : ValueGroup A K), a \u2264 a\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\ncase mk\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d\u00b9 : ValueGroup A K\na\u271d : K\n\u22a2 Quot.mk Setoid.r a\u271d \u2264 Quot.mk Setoid.r a\u271d\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d\u00b9 : ValueGroup A K\na\u271d : K\n\u22a2 1 \u2022 a\u271d = a\u271d\n[PROOFSTEP]\nrw [one_smul]\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 \u2200 (a b c : ValueGroup A K), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n[PROOFSTEP]\nrintro \u27e8a\u27e9 \u27e8b\u27e9 \u27e8c\u27e9 \u27e8e, rfl\u27e9 \u27e8f, rfl\u27e9\n[GOAL]\ncase mk.mk.mk.intro.intro\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d c\u271d : ValueGroup A K\nc : K\ne f : A\n\u22a2 Quot.mk Setoid.r (e \u2022 f \u2022 c) \u2264 Quot.mk Setoid.r c\n[PROOFSTEP]\nuse e * f\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d c\u271d : ValueGroup A K\nc : K\ne f : A\n\u22a2 (e * f) \u2022 c = e \u2022 f \u2022 c\n[PROOFSTEP]\nrw [mul_smul]\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 \u2200 (a b : ValueGroup A K), a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\nrintro \u27e8a\u27e9 \u27e8b\u27e9 \u27e8e, rfl\u27e9 \u27e8f, hf\u27e9\n[GOAL]\ncase mk.mk.intro.intro\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\nb : K\ne f : A\nhf : f \u2022 e \u2022 b = b\n\u22a2 Quot.mk Setoid.r (e \u2022 b) = Quot.mk Setoid.r b\n[PROOFSTEP]\nby_cases hb : b = 0\n[GOAL]\ncase pos\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\nb : K\ne f : A\nhf : f \u2022 e \u2022 b = b\nhb : b = 0\n\u22a2 Quot.mk Setoid.r (e \u2022 b) = Quot.mk Setoid.r b\n[PROOFSTEP]\nsimp [hb]\n[GOAL]\ncase neg\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\nb : K\ne f : A\nhf : f \u2022 e \u2022 b = b\nhb : \u00acb = 0\n\u22a2 Quot.mk Setoid.r (e \u2022 b) = Quot.mk Setoid.r b\n[PROOFSTEP]\nhave : IsUnit e := by\n apply isUnit_of_dvd_one\n use f\n rw [mul_comm]\n rw [\u2190 mul_smul, Algebra.smul_def] at hf \n nth_rw 2 [\u2190 one_mul b] at hf \n rw [\u2190 (algebraMap A K).map_one] at hf \n exact IsFractionRing.injective _ _ (mul_right_cancel\u2080 hb hf).symm\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\nb : K\ne f : A\nhf : f \u2022 e \u2022 b = b\nhb : \u00acb = 0\n\u22a2 IsUnit e\n[PROOFSTEP]\napply isUnit_of_dvd_one\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\nb : K\ne f : A\nhf : f \u2022 e \u2022 b = b\nhb : \u00acb = 0\n\u22a2 e \u2223 1\n[PROOFSTEP]\nuse f\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\nb : K\ne f : A\nhf : f \u2022 e \u2022 b = b\nhb : \u00acb = 0\n\u22a2 1 = e * f\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\nb : K\ne f : A\nhf : f \u2022 e \u2022 b = b\nhb : \u00acb = 0\n\u22a2 1 = f * e\n[PROOFSTEP]\nrw [\u2190 mul_smul, Algebra.smul_def] at hf \n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\nb : K\ne f : A\nhf : \u2191(algebraMap A K) (f * e) * b = b\nhb : \u00acb = 0\n\u22a2 1 = f * e\n[PROOFSTEP]\nnth_rw 2 [\u2190 one_mul b] at hf \n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\nb : K\ne f : A\nhf : \u2191(algebraMap A K) (f * e) * b = 1 * b\nhb : \u00acb = 0\n\u22a2 1 = f * e\n[PROOFSTEP]\nrw [\u2190 (algebraMap A K).map_one] at hf \n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\nb : K\ne f : A\nhf : \u2191(algebraMap A K) (f * e) * b = \u2191(algebraMap A K) 1 * b\nhb : \u00acb = 0\n\u22a2 1 = f * e\n[PROOFSTEP]\nexact IsFractionRing.injective _ _ (mul_right_cancel\u2080 hb hf).symm\n[GOAL]\ncase neg\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\nb : K\ne f : A\nhf : f \u2022 e \u2022 b = b\nhb : \u00acb = 0\nthis : IsUnit e\n\u22a2 Quot.mk Setoid.r (e \u2022 b) = Quot.mk Setoid.r b\n[PROOFSTEP]\napply Quotient.sound'\n[GOAL]\ncase neg.a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\nb : K\ne f : A\nhf : f \u2022 e \u2022 b = b\nhb : \u00acb = 0\nthis : IsUnit e\n\u22a2 Setoid.r (e \u2022 b) b\n[PROOFSTEP]\nexact \u27e8this.unit, rfl\u27e9\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 DecidableRel fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nclassical infer_instance\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 DecidableRel fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 \u2200 (a b c : ValueGroup A K), a * b * c = a * (b * c)\n[PROOFSTEP]\nrintro \u27e8a\u27e9 \u27e8b\u27e9 \u27e8c\u27e9\n[GOAL]\ncase mk.mk.mk\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\nb\u271d : ValueGroup A K\nb : K\nc\u271d : ValueGroup A K\nc : K\n\u22a2 Quot.mk Setoid.r a * Quot.mk Setoid.r b * Quot.mk Setoid.r c =\n Quot.mk Setoid.r a * (Quot.mk Setoid.r b * Quot.mk Setoid.r c)\n[PROOFSTEP]\napply Quotient.sound'\n[GOAL]\ncase mk.mk.mk.a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\nb\u271d : ValueGroup A K\nb : K\nc\u271d : ValueGroup A K\nc : K\n\u22a2 Setoid.r (a * b * c) (a * (b * c))\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\ncase mk.mk.mk.a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\nb\u271d : ValueGroup A K\nb : K\nc\u271d : ValueGroup A K\nc : K\n\u22a2 Setoid.r (a * (b * c)) (a * (b * c))\n[PROOFSTEP]\napply Setoid.refl'\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 \u2200 (a : ValueGroup A K), 1 * a = a\n[PROOFSTEP]\nrintro \u27e8a\u27e9\n[GOAL]\ncase mk\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\n\u22a2 1 * Quot.mk Setoid.r a = Quot.mk Setoid.r a\n[PROOFSTEP]\napply Quotient.sound'\n[GOAL]\ncase mk.a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\n\u22a2 Setoid.r (1 * a) a\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\ncase mk.a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\n\u22a2 Setoid.r a a\n[PROOFSTEP]\napply Setoid.refl'\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 \u2200 (a : ValueGroup A K), a * 1 = a\n[PROOFSTEP]\nrintro \u27e8a\u27e9\n[GOAL]\ncase mk\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\n\u22a2 Quot.mk Setoid.r a * 1 = Quot.mk Setoid.r a\n[PROOFSTEP]\napply Quotient.sound'\n[GOAL]\ncase mk.a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\n\u22a2 Setoid.r (a * 1) a\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\ncase mk.a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\n\u22a2 Setoid.r a a\n[PROOFSTEP]\napply Setoid.refl'\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 \u2200 (a b : ValueGroup A K), a * b = b * a\n[PROOFSTEP]\nrintro \u27e8a\u27e9 \u27e8b\u27e9\n[GOAL]\ncase mk.mk\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\nb\u271d : ValueGroup A K\nb : K\n\u22a2 Quot.mk Setoid.r a * Quot.mk Setoid.r b = Quot.mk Setoid.r b * Quot.mk Setoid.r a\n[PROOFSTEP]\napply Quotient.sound'\n[GOAL]\ncase mk.mk.a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\nb\u271d : ValueGroup A K\nb : K\n\u22a2 Setoid.r (a * b) (b * a)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase mk.mk.a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\nb\u271d : ValueGroup A K\nb : K\n\u22a2 Setoid.r (b * a) (b * a)\n[PROOFSTEP]\napply Setoid.refl'\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 \u2200 (a b : ValueGroup A K), a \u2264 b \u2192 \u2200 (c : ValueGroup A K), c * a \u2264 c * b\n[PROOFSTEP]\nrintro \u27e8a\u27e9 \u27e8b\u27e9 \u27e8c, rfl\u27e9 \u27e8d\u27e9\n[GOAL]\ncase mk.mk.intro.mk\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\nb : K\nc : A\nc\u271d : ValueGroup A K\nd : K\n\u22a2 Quot.mk Setoid.r d * Quot.mk Setoid.r (c \u2022 b) \u2264 Quot.mk Setoid.r d * Quot.mk Setoid.r b\n[PROOFSTEP]\nuse c\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\nb : K\nc : A\nc\u271d : ValueGroup A K\nd : K\n\u22a2 c \u2022 (d * b) = d * c \u2022 b\n[PROOFSTEP]\nsimp only [Algebra.smul_def]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d b\u271d : ValueGroup A K\nb : K\nc : A\nc\u271d : ValueGroup A K\nd : K\n\u22a2 \u2191(algebraMap A K) c * (d * b) = d * (\u2191(algebraMap A K) c * b)\n[PROOFSTEP]\nring\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 \u2200 (a : ValueGroup A K), 0 * a = 0\n[PROOFSTEP]\nrintro \u27e8a\u27e9\n[GOAL]\ncase mk\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\n\u22a2 0 * Quot.mk Setoid.r a = 0\n[PROOFSTEP]\napply Quotient.sound'\n[GOAL]\ncase mk.a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\n\u22a2 Setoid.r (0 * a) 0\n[PROOFSTEP]\nrw [zero_mul]\n[GOAL]\ncase mk.a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\n\u22a2 Setoid.r 0 0\n[PROOFSTEP]\napply Setoid.refl'\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 \u2200 (a : ValueGroup A K), a * 0 = 0\n[PROOFSTEP]\nrintro \u27e8a\u27e9\n[GOAL]\ncase mk\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\n\u22a2 Quot.mk Setoid.r a * 0 = 0\n[PROOFSTEP]\napply Quotient.sound'\n[GOAL]\ncase mk.a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\n\u22a2 Setoid.r (a * 0) 0\n[PROOFSTEP]\nrw [mul_zero]\n[GOAL]\ncase mk.a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\n\u22a2 Setoid.r 0 0\n[PROOFSTEP]\napply Setoid.refl'\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 0 \u2022 1 = 0\n[PROOFSTEP]\nrw [zero_smul]\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 \u2203 x y, x \u2260 y\n[PROOFSTEP]\nuse 0, 1\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 0 \u2260 1\n[PROOFSTEP]\nintro c\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nc : 0 = 1\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8d, hd\u27e9 := Quotient.exact' c\n[GOAL]\ncase h.intro\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nc : 0 = 1\nd : A\u02e3\nhd : (fun m => m \u2022 1) d = 0\n\u22a2 False\n[PROOFSTEP]\napply_fun fun t => d\u207b\u00b9 \u2022 t at hd \n[GOAL]\ncase h.intro\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nc : 0 = 1\nd : A\u02e3\nhd : d\u207b\u00b9 \u2022 (fun m => m \u2022 1) d = d\u207b\u00b9 \u2022 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [inv_smul_smul, smul_zero, one_ne_zero] at hd \n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\napply Quotient.sound'\n[GOAL]\ncase a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 Setoid.r 0\u207b\u00b9 0\n[PROOFSTEP]\nrw [inv_zero]\n[GOAL]\ncase a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 Setoid.r 0 0\n[PROOFSTEP]\napply Setoid.refl'\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 \u2200 (a : ValueGroup A K), a \u2260 0 \u2192 a * a\u207b\u00b9 = 1\n[PROOFSTEP]\nrintro \u27e8a\u27e9 ha\n[GOAL]\ncase mk\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\nha : Quot.mk Setoid.r a \u2260 0\n\u22a2 Quot.mk Setoid.r a * (Quot.mk Setoid.r a)\u207b\u00b9 = 1\n[PROOFSTEP]\napply Quotient.sound'\n[GOAL]\ncase mk.a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\nha : Quot.mk Setoid.r a \u2260 0\n\u22a2 Setoid.r (a * a\u207b\u00b9) 1\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\nha : Quot.mk Setoid.r a \u2260 0\n\u22a2 (fun m => m \u2022 1) 1 = a * a\u207b\u00b9\n[PROOFSTEP]\nsimp only [one_smul, ne_eq]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\nha : Quot.mk Setoid.r a \u2260 0\n\u22a2 1 = a * a\u207b\u00b9\n[PROOFSTEP]\napply (mul_inv_cancel _).symm\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\nha : Quot.mk Setoid.r a \u2260 0\n\u22a2 a \u2260 0\n[PROOFSTEP]\ncontrapose ha\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\nha : \u00aca \u2260 0\n\u22a2 \u00acQuot.mk Setoid.r a \u2260 0\n[PROOFSTEP]\nsimp only [Classical.not_not] at ha \u22a2\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\nha : a = 0\n\u22a2 Quot.mk Setoid.r a = 0\n[PROOFSTEP]\nrw [ha]\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na\u271d : ValueGroup A K\na : K\nha : a = 0\n\u22a2 Quot.mk Setoid.r 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 \u2200 (x y : K),\n ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (x + y) \u2264\n max\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n x)\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n y)\n[PROOFSTEP]\nintro a b\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na b : K\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (a + b) \u2264\n max\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n a)\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n b)\n[PROOFSTEP]\nobtain \u27e8xa, ya, hya, rfl\u27e9 : \u2203 a b : A, _ := IsFractionRing.div_surjective a\n[GOAL]\ncase intro.intro.intro\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nb : K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya + b) \u2264\n max\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya))\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n b)\n[PROOFSTEP]\nobtain \u27e8xb, yb, hyb, rfl\u27e9 : \u2203 a b : A, _ := IsFractionRing.div_surjective b\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya + \u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2264\n max\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya))\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb))\n[PROOFSTEP]\nhave : (algebraMap A K) ya \u2260 0 := IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors hya\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis : \u2191(algebraMap A K) ya \u2260 0\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya + \u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2264\n max\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya))\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb))\n[PROOFSTEP]\nhave : (algebraMap A K) yb \u2260 0 := IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors hyb\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya + \u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2264\n max\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya))\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb))\n[PROOFSTEP]\nobtain \u27e8c, h | h\u27e9 := ValuationRing.cond (xa * yb) (xb * ya)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.inl\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xa * yb * c = xb * ya\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya + \u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2264\n max\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya))\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb))\ncase intro.intro.intro.intro.intro.intro.intro.inr\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xb * ya * c = xa * yb\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya + \u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2264\n max\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya))\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.inl\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xa * yb * c = xb * ya\n\u22a2 Quotient.mk'' (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya + \u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2264\n max (Quotient.mk'' (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya))\n (Quotient.mk'' (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb))\n[PROOFSTEP]\napply le_trans _ (le_max_left _ _)\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xa * yb * c = xb * ya\n\u22a2 Quotient.mk'' (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya + \u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2264\n Quotient.mk'' (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya)\n[PROOFSTEP]\nuse c + 1\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xa * yb * c = xb * ya\n\u22a2 (c + 1) \u2022 (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya) =\n \u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya + \u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb\n[PROOFSTEP]\nrw [Algebra.smul_def]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xa * yb * c = xb * ya\n\u22a2 \u2191(algebraMap A K) (c + 1) * (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya) =\n \u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya + \u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xa * yb * c = xb * ya\n\u22a2 (\u2191(algebraMap A K) c + 1) * \u2191(algebraMap A K) xa * (\u2191(algebraMap A K) ya * \u2191(algebraMap A K) yb) =\n (\u2191(algebraMap A K) xa * \u2191(algebraMap A K) yb + \u2191(algebraMap A K) xb * \u2191(algebraMap A K) ya) * \u2191(algebraMap A K) ya\n[PROOFSTEP]\nsimp only [\u2190 RingHom.map_mul, \u2190 RingHom.map_add, \u2190 (algebraMap A K).map_one, \u2190 h]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xa * yb * c = xb * ya\n\u22a2 \u2191(algebraMap A K) ((c + 1) * xa * (ya * yb)) = \u2191(algebraMap A K) ((xa * yb + xa * yb * c) * ya)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.h.e_6.h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xa * yb * c = xb * ya\n\u22a2 (c + 1) * xa * (ya * yb) = (xa * yb + xa * yb * c) * ya\n[PROOFSTEP]\nring\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.inr\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xb * ya * c = xa * yb\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya + \u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2264\n max\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya))\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb))\n[PROOFSTEP]\napply le_trans _ (le_max_right _ _)\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xb * ya * c = xa * yb\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya + \u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) \u2264\n ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) },\n map_one' :=\n (_ :\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1 =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) } 1),\n map_mul' :=\n (_ :\n \u2200 (x x_1 : K),\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1) =\n ZeroHom.toFun { toFun := Quotient.mk'', map_zero' := (_ : Quotient.mk'' 0 = Quotient.mk'' 0) }\n (x * x_1)) })\n (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb)\n[PROOFSTEP]\nuse c + 1\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xb * ya * c = xa * yb\n\u22a2 (c + 1) \u2022 (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) =\n \u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya + \u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb\n[PROOFSTEP]\nrw [Algebra.smul_def]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xb * ya * c = xa * yb\n\u22a2 \u2191(algebraMap A K) (c + 1) * (\u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb) =\n \u2191(algebraMap A K) xa / \u2191(algebraMap A K) ya + \u2191(algebraMap A K) xb / \u2191(algebraMap A K) yb\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xb * ya * c = xa * yb\n\u22a2 (\u2191(algebraMap A K) c + 1) * \u2191(algebraMap A K) xb * (\u2191(algebraMap A K) ya * \u2191(algebraMap A K) yb) =\n (\u2191(algebraMap A K) xa * \u2191(algebraMap A K) yb + \u2191(algebraMap A K) xb * \u2191(algebraMap A K) ya) * \u2191(algebraMap A K) yb\n[PROOFSTEP]\nsimp only [\u2190 RingHom.map_mul, \u2190 RingHom.map_add, \u2190 (algebraMap A K).map_one, \u2190 h]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xb * ya * c = xa * yb\n\u22a2 \u2191(algebraMap A K) ((c + 1) * xb * (ya * yb)) = \u2191(algebraMap A K) ((xb * ya * c + xb * ya) * yb)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.h.e_6.h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nxa ya : A\nhya : ya \u2208 nonZeroDivisors A\nxb yb : A\nhyb : yb \u2208 nonZeroDivisors A\nthis\u271d : \u2191(algebraMap A K) ya \u2260 0\nthis : \u2191(algebraMap A K) yb \u2260 0\nc : A\nh : xb * ya * c = xa * yb\n\u22a2 (c + 1) * xb * (ya * yb) = (xb * ya * c + xb * ya) * yb\n[PROOFSTEP]\nring\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nx : K\n\u22a2 x \u2208 Valuation.integer (valuation A K) \u2194 \u2203 a, \u2191(algebraMap A K) a = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nx : K\n\u22a2 x \u2208 Valuation.integer (valuation A K) \u2192 \u2203 a, \u2191(algebraMap A K) a = x\n[PROOFSTEP]\nrintro \u27e8c, rfl\u27e9\n[GOAL]\ncase mp.intro\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nc : A\n\u22a2 \u2203 a, \u2191(algebraMap A K) a = c \u2022 1\n[PROOFSTEP]\nuse c\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nc : A\n\u22a2 \u2191(algebraMap A K) c = c \u2022 1\n[PROOFSTEP]\nrw [Algebra.smul_def, mul_one]\n[GOAL]\ncase mpr\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nx : K\n\u22a2 (\u2203 a, \u2191(algebraMap A K) a = x) \u2192 x \u2208 Valuation.integer (valuation A K)\n[PROOFSTEP]\nrintro \u27e8c, rfl\u27e9\n[GOAL]\ncase mpr.intro\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nc : A\n\u22a2 \u2191(algebraMap A K) c \u2208 Valuation.integer (valuation A K)\n[PROOFSTEP]\nuse c\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nc : A\n\u22a2 c \u2022 1 = \u2191(algebraMap A K) c\n[PROOFSTEP]\nrw [Algebra.smul_def, mul_one]\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nx\u271d\u00b9 x\u271d : A\n\u22a2 (fun a => { val := \u2191(algebraMap A K) a, property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x\u271d\u00b9 * x\u271d) =\n (fun a => { val := \u2191(algebraMap A K) a, property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x\u271d\u00b9 *\n (fun a =>\n { val := \u2191(algebraMap A K) a, property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x\u271d\n[PROOFSTEP]\next1\n[GOAL]\ncase a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nx\u271d\u00b9 x\u271d : A\n\u22a2 \u2191((fun a => { val := \u2191(algebraMap A K) a, property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x\u271d\u00b9 * x\u271d)) =\n \u2191((fun a =>\n { val := \u2191(algebraMap A K) a, property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x\u271d\u00b9 *\n (fun a =>\n { val := \u2191(algebraMap A K) a, property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x\u271d)\n[PROOFSTEP]\nexact (algebraMap A K).map_mul _ _\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a, property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n 0 =\n 0\n[PROOFSTEP]\next1\n[GOAL]\ncase a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 \u2191(MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a, property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n 0) =\n \u21910\n[PROOFSTEP]\nexact (algebraMap A K).map_zero\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nx\u271d\u00b9 x\u271d : A\n\u22a2 MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a, property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n (x\u271d\u00b9 + x\u271d) =\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a, property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n x\u271d\u00b9 +\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a, property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n x\u271d\n[PROOFSTEP]\next1\n[GOAL]\ncase a\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nx\u271d\u00b9 x\u271d : A\n\u22a2 \u2191(MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a, property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n (x\u271d\u00b9 + x\u271d)) =\n \u2191(MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a, property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n x\u271d\u00b9 +\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a, property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n x\u271d)\n[PROOFSTEP]\nexact (algebraMap A K).map_add _ _\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 Function.Bijective\n \u2191(let_fun this :=\n {\n toMulHom :=\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) },\n map_zero' :=\n (_ :\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x x_1 : A),\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n (x + x_1) =\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x +\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x_1) };\n this)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 Function.Injective\n \u2191(let_fun this :=\n {\n toMulHom :=\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) },\n map_zero' :=\n (_ :\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x x_1 : A),\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n (x + x_1) =\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x +\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x_1) };\n this)\n[PROOFSTEP]\nintro x y h\n[GOAL]\ncase left\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nx y : A\nh :\n \u2191(let_fun this :=\n {\n toMulHom :=\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) },\n map_zero' :=\n (_ :\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x x_1 : A),\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n (x + x_1) =\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x +\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x_1) };\n this)\n x =\n \u2191(let_fun this :=\n {\n toMulHom :=\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) },\n map_zero' :=\n (_ :\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x x_1 : A),\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n (x + x_1) =\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x +\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x_1) };\n this)\n y\n\u22a2 x = y\n[PROOFSTEP]\napply_fun (algebraMap (valuation A K).integer K) at h \n[GOAL]\ncase left\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nx y : A\nh :\n \u2191(algebraMap { x // x \u2208 Valuation.integer (valuation A K) } K)\n (\u2191(let_fun this :=\n {\n toMulHom :=\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) },\n map_zero' :=\n (_ :\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x x_1 : A),\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n (x + x_1) =\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x +\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x_1) };\n this)\n x) =\n \u2191(algebraMap { x // x \u2208 Valuation.integer (valuation A K) } K)\n (\u2191(let_fun this :=\n {\n toMulHom :=\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) },\n map_zero' :=\n (_ :\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x x_1 : A),\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n (x + x_1) =\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x +\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x_1) };\n this)\n y)\n\u22a2 x = y\n[PROOFSTEP]\nexact IsFractionRing.injective _ _ h\n[GOAL]\ncase right\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 Function.Surjective\n \u2191(let_fun this :=\n {\n toMulHom :=\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) },\n map_zero' :=\n (_ :\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x x_1 : A),\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n (x + x_1) =\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x +\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x_1) };\n this)\n[PROOFSTEP]\nrintro \u27e8-, ha\u27e9\n[GOAL]\ncase right.mk\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nval\u271d : K\nha : val\u271d \u2208 Valuation.integer (valuation A K)\n\u22a2 \u2203 a,\n \u2191(let_fun this :=\n {\n toMulHom :=\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) },\n map_zero' :=\n (_ :\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x x_1 : A),\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n (x + x_1) =\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x +\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x_1) };\n this)\n a =\n { val := val\u271d, property := ha }\n[PROOFSTEP]\nrw [mem_integer_iff] at ha \n[GOAL]\ncase right.mk\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nval\u271d : K\nha\u271d : val\u271d \u2208 Valuation.integer (valuation A K)\nha : \u2203 a, \u2191(algebraMap A K) a = val\u271d\n\u22a2 \u2203 a,\n \u2191(let_fun this :=\n {\n toMulHom :=\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) },\n map_zero' :=\n (_ :\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x x_1 : A),\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n (x + x_1) =\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x +\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x_1) };\n this)\n a =\n { val := val\u271d, property := ha\u271d }\n[PROOFSTEP]\nobtain \u27e8a, rfl\u27e9 := ha\n[GOAL]\ncase right.mk.intro\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\na : A\nha : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)\n\u22a2 \u2203 a_1,\n \u2191(let_fun this :=\n {\n toMulHom :=\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) },\n map_zero' :=\n (_ :\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_1 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_1) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_1) }\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x x_1 : A),\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n (x + x_1) =\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x +\n MulHom.toFun\n {\n toFun := fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) },\n map_mul' :=\n (_ :\n \u2200 (x x_2 : A),\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n (x * x_2) =\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x *\n (fun a =>\n { val := \u2191(algebraMap A K) a,\n property := (_ : \u2191(algebraMap A K) a \u2208 Valuation.integer (valuation A K)) })\n x_2) }\n x_1) };\n this)\n a_1 =\n { val := \u2191(algebraMap A K) a, property := ha }\n[PROOFSTEP]\nexact \u27e8a, rfl\u27e9\n[GOAL]\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\n\u22a2 Valuation.integer (valuation A K) = RingHom.range (algebraMap A K)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2075 : CommRing A\nK : Type v\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : IsFractionRing A K\nx\u271d : K\n\u22a2 x\u271d \u2208 Valuation.integer (valuation A K) \u2194 x\u271d \u2208 RingHom.range (algebraMap A K)\n[PROOFSTEP]\nexact mem_integer_iff _ _ _\n[GOAL]\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : ValuationRing A\n\u22a2 \u2200 (a : A), IsUnit a \u2228 IsUnit (1 - a)\n[PROOFSTEP]\nintro a\n[GOAL]\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : ValuationRing A\na : A\n\u22a2 IsUnit a \u2228 IsUnit (1 - a)\n[PROOFSTEP]\nobtain \u27e8c, h | h\u27e9 := ValuationRing.cond a (1 - a)\n[GOAL]\ncase intro.inl\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : ValuationRing A\na c : A\nh : a * c = 1 - a\n\u22a2 IsUnit a \u2228 IsUnit (1 - a)\n[PROOFSTEP]\nleft\n[GOAL]\ncase intro.inl.h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : ValuationRing A\na c : A\nh : a * c = 1 - a\n\u22a2 IsUnit a\n[PROOFSTEP]\napply isUnit_of_mul_eq_one _ (c + 1)\n[GOAL]\ncase intro.inl.h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : ValuationRing A\na c : A\nh : a * c = 1 - a\n\u22a2 a * (c + 1) = 1\n[PROOFSTEP]\nsimp [mul_add, h]\n[GOAL]\ncase intro.inr\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : ValuationRing A\na c : A\nh : (1 - a) * c = a\n\u22a2 IsUnit a \u2228 IsUnit (1 - a)\n[PROOFSTEP]\nright\n[GOAL]\ncase intro.inr.h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : ValuationRing A\na c : A\nh : (1 - a) * c = a\n\u22a2 IsUnit (1 - a)\n[PROOFSTEP]\napply isUnit_of_mul_eq_one _ (c + 1)\n[GOAL]\ncase intro.inr.h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : ValuationRing A\na c : A\nh : (1 - a) * c = a\n\u22a2 (1 - a) * (c + 1) = 1\n[PROOFSTEP]\nsimp [mul_add, h]\n[GOAL]\nA : Type u\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : CompleteLattice (Ideal A) := inferInstance\n\u22a2 \u2200 (a b : Ideal A), a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\nintro \u03b1 \u03b2\n[GOAL]\nA : Type u\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : CompleteLattice (Ideal A) := inferInstance\n\u03b1 \u03b2 : Ideal A\n\u22a2 \u03b1 \u2264 \u03b2 \u2228 \u03b2 \u2264 \u03b1\n[PROOFSTEP]\nby_cases h : \u03b1 \u2264 \u03b2\n[GOAL]\ncase pos\nA : Type u\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : CompleteLattice (Ideal A) := inferInstance\n\u03b1 \u03b2 : Ideal A\nh : \u03b1 \u2264 \u03b2\n\u22a2 \u03b1 \u2264 \u03b2 \u2228 \u03b2 \u2264 \u03b1\n[PROOFSTEP]\nexact Or.inl h\n[GOAL]\ncase neg\nA : Type u\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : CompleteLattice (Ideal A) := inferInstance\n\u03b1 \u03b2 : Ideal A\nh : \u00ac\u03b1 \u2264 \u03b2\n\u22a2 \u03b1 \u2264 \u03b2 \u2228 \u03b2 \u2264 \u03b1\n[PROOFSTEP]\nerw [not_forall] at h \n[GOAL]\ncase neg\nA : Type u\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : CompleteLattice (Ideal A) := inferInstance\n\u03b1 \u03b2 : Ideal A\nh : \u2203 x, \u00ac(x \u2208 \u03b1 \u2192 x \u2208 \u03b2)\n\u22a2 \u03b1 \u2264 \u03b2 \u2228 \u03b2 \u2264 \u03b1\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\nA : Type u\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : CompleteLattice (Ideal A) := inferInstance\n\u03b1 \u03b2 : Ideal A\nh : \u2203 x, x \u2208 \u03b1 \u2227 \u00acx \u2208 \u03b2\n\u22a2 \u03b1 \u2264 \u03b2 \u2228 \u03b2 \u2264 \u03b1\n[PROOFSTEP]\nobtain \u27e8a, h\u2081, h\u2082\u27e9 := h\n[GOAL]\ncase neg.intro.intro\nA : Type u\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : CompleteLattice (Ideal A) := inferInstance\n\u03b1 \u03b2 : Ideal A\na : A\nh\u2081 : a \u2208 \u03b1\nh\u2082 : \u00aca \u2208 \u03b2\n\u22a2 \u03b1 \u2264 \u03b2 \u2228 \u03b2 \u2264 \u03b1\n[PROOFSTEP]\nright\n[GOAL]\ncase neg.intro.intro.h\nA : Type u\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : CompleteLattice (Ideal A) := inferInstance\n\u03b1 \u03b2 : Ideal A\na : A\nh\u2081 : a \u2208 \u03b1\nh\u2082 : \u00aca \u2208 \u03b2\n\u22a2 \u03b2 \u2264 \u03b1\n[PROOFSTEP]\nintro b hb\n[GOAL]\ncase neg.intro.intro.h\nA : Type u\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : CompleteLattice (Ideal A) := inferInstance\n\u03b1 \u03b2 : Ideal A\na : A\nh\u2081 : a \u2208 \u03b1\nh\u2082 : \u00aca \u2208 \u03b2\nb : A\nhb : b \u2208 \u03b2\n\u22a2 b \u2208 \u03b1\n[PROOFSTEP]\nobtain \u27e8c, h | h\u27e9 := ValuationRing.cond a b\n[GOAL]\ncase neg.intro.intro.h.intro.inl\nA : Type u\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : CompleteLattice (Ideal A) := inferInstance\n\u03b1 \u03b2 : Ideal A\na : A\nh\u2081 : a \u2208 \u03b1\nh\u2082 : \u00aca \u2208 \u03b2\nb : A\nhb : b \u2208 \u03b2\nc : A\nh : a * c = b\n\u22a2 b \u2208 \u03b1\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase neg.intro.intro.h.intro.inl\nA : Type u\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : CompleteLattice (Ideal A) := inferInstance\n\u03b1 \u03b2 : Ideal A\na : A\nh\u2081 : a \u2208 \u03b1\nh\u2082 : \u00aca \u2208 \u03b2\nb : A\nhb : b \u2208 \u03b2\nc : A\nh : a * c = b\n\u22a2 a * c \u2208 \u03b1\n[PROOFSTEP]\nexact Ideal.mul_mem_right _ _ h\u2081\n[GOAL]\ncase neg.intro.intro.h.intro.inr\nA : Type u\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : CompleteLattice (Ideal A) := inferInstance\n\u03b1 \u03b2 : Ideal A\na : A\nh\u2081 : a \u2208 \u03b1\nh\u2082 : \u00aca \u2208 \u03b2\nb : A\nhb : b \u2208 \u03b2\nc : A\nh : b * c = a\n\u22a2 b \u2208 \u03b1\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.intro.intro.h.intro.inr.h\nA : Type u\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : CompleteLattice (Ideal A) := inferInstance\n\u03b1 \u03b2 : Ideal A\na : A\nh\u2081 : a \u2208 \u03b1\nh\u2082 : \u00aca \u2208 \u03b2\nb : A\nhb : b \u2208 \u03b2\nc : A\nh : b * c = a\n\u22a2 False\n[PROOFSTEP]\napply h\u2082\n[GOAL]\ncase neg.intro.intro.h.intro.inr.h\nA : Type u\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : CompleteLattice (Ideal A) := inferInstance\n\u03b1 \u03b2 : Ideal A\na : A\nh\u2081 : a \u2208 \u03b1\nh\u2082 : \u00aca \u2208 \u03b2\nb : A\nhb : b \u2208 \u03b2\nc : A\nh : b * c = a\n\u22a2 a \u2208 \u03b2\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase neg.intro.intro.h.intro.inr.h\nA : Type u\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : ValuationRing A\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : CompleteLattice (Ideal A) := inferInstance\n\u03b1 \u03b2 : Ideal A\na : A\nh\u2081 : a \u2208 \u03b1\nh\u2082 : \u00aca \u2208 \u03b2\nb : A\nhb : b \u2208 \u03b2\nc : A\nh : b * c = a\n\u22a2 b * c \u2208 \u03b2\n[PROOFSTEP]\napply Ideal.mul_mem_right _ _ hb\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\n\u22a2 ValuationRing R \u2194 IsTotal R fun x x_1 => x \u2223 x_1\n[PROOFSTEP]\nclassical\nrefine \u27e8fun H => \u27e8fun a b => ?_\u27e9, fun H => \u27e8fun a b => ?_\u27e9\u27e9\n\u00b7 obtain \u27e8c, rfl | rfl\u27e9 := ValuationRing.cond a b <;> simp\n\u00b7 obtain \u27e8c, rfl\u27e9 | \u27e8c, rfl\u27e9 := @IsTotal.total _ _ H a b <;> use c <;> simp\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\n\u22a2 ValuationRing R \u2194 IsTotal R fun x x_1 => x \u2223 x_1\n[PROOFSTEP]\nrefine \u27e8fun H => \u27e8fun a b => ?_\u27e9, fun H => \u27e8fun a b => ?_\u27e9\u27e9\n[GOAL]\ncase refine_1\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : ValuationRing R\na b : R\n\u22a2 a \u2223 b \u2228 b \u2223 a\n[PROOFSTEP]\nobtain \u27e8c, rfl | rfl\u27e9 := ValuationRing.cond a b\n[GOAL]\ncase refine_1.intro.inl\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : ValuationRing R\na c : R\n\u22a2 a \u2223 a * c \u2228 a * c \u2223 a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_1.intro.inr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : ValuationRing R\nb c : R\n\u22a2 b * c \u2223 b \u2228 b \u2223 b * c\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_2\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : IsTotal R fun x x_1 => x \u2223 x_1\na b : R\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nobtain \u27e8c, rfl\u27e9 | \u27e8c, rfl\u27e9 := @IsTotal.total _ _ H a b\n[GOAL]\ncase refine_2.inl.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : IsTotal R fun x x_1 => x \u2223 x_1\na c : R\n\u22a2 \u2203 c_1, a * c_1 = a * c \u2228 a * c * c_1 = a\n[PROOFSTEP]\nuse c\n[GOAL]\ncase refine_2.inr.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : IsTotal R fun x x_1 => x \u2223 x_1\nb c : R\n\u22a2 \u2203 c_1, b * c * c_1 = b \u2228 b * c_1 = b * c\n[PROOFSTEP]\nuse c\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : IsTotal R fun x x_1 => x \u2223 x_1\na c : R\n\u22a2 a * c = a * c \u2228 a * c * c = a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : IsTotal R fun x x_1 => x \u2223 x_1\nb c : R\n\u22a2 b * c * c = b \u2228 b * c = b * c\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\n\u22a2 ValuationRing R \u2194 IsTotal (Ideal R) fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nclassical\nrefine' \u27e8fun _ => \u27e8le_total\u27e9, fun H => iff_dvd_total.mpr \u27e8fun a b => _\u27e9\u27e9\nhave := @IsTotal.total _ _ H (Ideal.span { a }) (Ideal.span { b })\nsimp_rw [Ideal.span_singleton_le_span_singleton] at this \nexact this.symm\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\n\u22a2 ValuationRing R \u2194 IsTotal (Ideal R) fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nrefine' \u27e8fun _ => \u27e8le_total\u27e9, fun H => iff_dvd_total.mpr \u27e8fun a b => _\u27e9\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : IsTotal (Ideal R) fun x x_1 => x \u2264 x_1\na b : R\n\u22a2 a \u2223 b \u2228 b \u2223 a\n[PROOFSTEP]\nhave := @IsTotal.total _ _ H (Ideal.span { a }) (Ideal.span { b })\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : IsTotal (Ideal R) fun x x_1 => x \u2264 x_1\na b : R\nthis : Ideal.span {a} \u2264 Ideal.span {b} \u2228 Ideal.span {b} \u2264 Ideal.span {a}\n\u22a2 a \u2223 b \u2228 b \u2223 a\n[PROOFSTEP]\nsimp_rw [Ideal.span_singleton_le_span_singleton] at this \n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : IsTotal (Ideal R) fun x x_1 => x \u2264 x_1\na b : R\nthis : b \u2223 a \u2228 a \u2223 b\n\u22a2 a \u2223 b \u2228 b \u2223 a\n[PROOFSTEP]\nexact this.symm\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : ValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\n\u22a2 Associated p q\n[PROOFSTEP]\nhave := dvd_total p q\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : ValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nthis : p \u2223 q \u2228 q \u2223 p\n\u22a2 Associated p q\n[PROOFSTEP]\nrw [Irreducible.dvd_comm hp hq, or_self_iff] at this \n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : ValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nthis : q \u2223 p\n\u22a2 Associated p q\n[PROOFSTEP]\nexact associated_of_dvd_dvd (Irreducible.dvd_symm hq hp this) this\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\n\u22a2 ValuationRing R \u2194 \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\n\u22a2 ValuationRing R \u2192 \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\n[PROOFSTEP]\nintro H x\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : ValuationRing R\nx : K\n\u22a2 IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\n[PROOFSTEP]\nobtain \u27e8x : R, y, hy, rfl\u27e9 := IsFractionRing.div_surjective (A := R) x\n[GOAL]\ncase mp.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : ValuationRing R\nx y : R\nhy : y \u2208 nonZeroDivisors R\n\u22a2 IsLocalization.IsInteger R (\u2191(algebraMap R K) x / \u2191(algebraMap R K) y) \u2228\n IsLocalization.IsInteger R (\u2191(algebraMap R K) x / \u2191(algebraMap R K) y)\u207b\u00b9\n[PROOFSTEP]\nhave := (map_ne_zero_iff _ (IsFractionRing.injective R K)).mpr (nonZeroDivisors.ne_zero hy)\n[GOAL]\ncase mp.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : ValuationRing R\nx y : R\nhy : y \u2208 nonZeroDivisors R\nthis : \u2191(algebraMap R K) y \u2260 0\n\u22a2 IsLocalization.IsInteger R (\u2191(algebraMap R K) x / \u2191(algebraMap R K) y) \u2228\n IsLocalization.IsInteger R (\u2191(algebraMap R K) x / \u2191(algebraMap R K) y)\u207b\u00b9\n[PROOFSTEP]\nobtain \u27e8s, rfl | rfl\u27e9 := ValuationRing.cond x y\n[GOAL]\ncase mp.intro.intro.intro.intro.inl\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : ValuationRing R\nx s : R\nhy : x * s \u2208 nonZeroDivisors R\nthis : \u2191(algebraMap R K) (x * s) \u2260 0\n\u22a2 IsLocalization.IsInteger R (\u2191(algebraMap R K) x / \u2191(algebraMap R K) (x * s)) \u2228\n IsLocalization.IsInteger R (\u2191(algebraMap R K) x / \u2191(algebraMap R K) (x * s))\u207b\u00b9\n[PROOFSTEP]\nexact Or.inr \u27e8s, eq_inv_of_mul_eq_one_left <| by rwa [mul_div, div_eq_one_iff_eq, map_mul, mul_comm]\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : ValuationRing R\nx s : R\nhy : x * s \u2208 nonZeroDivisors R\nthis : \u2191(algebraMap R K) (x * s) \u2260 0\n\u22a2 \u2191(algebraMap R K) s * (\u2191(algebraMap R K) x / \u2191(algebraMap R K) (x * s)) = 1\n[PROOFSTEP]\nrwa [mul_div, div_eq_one_iff_eq, map_mul, mul_comm]\n[GOAL]\ncase mp.intro.intro.intro.intro.inr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : ValuationRing R\ny : R\nhy : y \u2208 nonZeroDivisors R\nthis : \u2191(algebraMap R K) y \u2260 0\ns : R\n\u22a2 IsLocalization.IsInteger R (\u2191(algebraMap R K) (y * s) / \u2191(algebraMap R K) y) \u2228\n IsLocalization.IsInteger R (\u2191(algebraMap R K) (y * s) / \u2191(algebraMap R K) y)\u207b\u00b9\n[PROOFSTEP]\nexact Or.inl \u27e8s, by rwa [eq_div_iff, map_mul, mul_comm]\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : ValuationRing R\ny : R\nhy : y \u2208 nonZeroDivisors R\nthis : \u2191(algebraMap R K) y \u2260 0\ns : R\n\u22a2 \u2191(algebraMap R K) s = \u2191(algebraMap R K) (y * s) / \u2191(algebraMap R K) y\n[PROOFSTEP]\nrwa [eq_div_iff, map_mul, mul_comm]\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\n\u22a2 (\u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9) \u2192 ValuationRing R\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\n\u22a2 ValuationRing R\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.cond'\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\n\u22a2 \u2200 (a b : R), \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nintro a b\n[GOAL]\ncase mpr.cond'\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\na b : R\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\na b : R\nha : a = 0\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nsubst ha\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\nb : R\n\u22a2 \u2203 c, 0 * c = b \u2228 b * c = 0\n[PROOFSTEP]\nexact \u27e80, Or.inr <| mul_zero b\u27e9\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\na b : R\nha : \u00aca = 0\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nby_cases hb : b = 0\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\na b : R\nha : \u00aca = 0\nhb : b = 0\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nsubst hb\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\na : R\nha : \u00aca = 0\n\u22a2 \u2203 c, a * c = 0 \u2228 0 * c = a\n[PROOFSTEP]\nexact \u27e80, Or.inl <| mul_zero a\u27e9\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\na b : R\nha : \u00aca = 0\nhb : \u00acb = 0\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nreplace ha := (map_ne_zero_iff _ (IsFractionRing.injective R K)).mpr ha\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\na b : R\nhb : \u00acb = 0\nha : \u2191(algebraMap R K) a \u2260 0\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nreplace hb := (map_ne_zero_iff _ (IsFractionRing.injective R K)).mpr hb\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\na b : R\nha : \u2191(algebraMap R K) a \u2260 0\nhb : \u2191(algebraMap R K) b \u2260 0\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nobtain \u27e8c, e\u27e9 | \u27e8c, e\u27e9 := H (algebraMap R K a / algebraMap R K b)\n[GOAL]\ncase neg.inl.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\na b : R\nha : \u2191(algebraMap R K) a \u2260 0\nhb : \u2191(algebraMap R K) b \u2260 0\nc : R\ne : \u2191(algebraMap R K) c = \u2191(algebraMap R K) a / \u2191(algebraMap R K) b\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nrw [eq_div_iff hb, \u2190 map_mul, (IsFractionRing.injective R K).eq_iff, mul_comm] at e \n[GOAL]\ncase neg.inl.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\na b : R\nha : \u2191(algebraMap R K) a \u2260 0\nhb : \u2191(algebraMap R K) b \u2260 0\nc : R\ne : b * c = a\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nexact \u27e8c, Or.inr e\u27e9\n[GOAL]\ncase neg.inr.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\na b : R\nha : \u2191(algebraMap R K) a \u2260 0\nhb : \u2191(algebraMap R K) b \u2260 0\nc : R\ne : \u2191(algebraMap R K) c = (\u2191(algebraMap R K) a / \u2191(algebraMap R K) b)\u207b\u00b9\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nrw [inv_div, eq_div_iff ha, \u2190 map_mul, (IsFractionRing.injective R K).eq_iff, mul_comm c] at e \n[GOAL]\ncase neg.inr.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nH : \u2200 (x : K), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\na b : R\nha : \u2191(algebraMap R K) a \u2260 0\nhb : \u2191(algebraMap R K) b \u2260 0\nc : R\ne : a * c = b\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nexact \u27e8c, Or.inl e\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : ValuationRing R\n\u22a2 IsBezout R\n[PROOFSTEP]\nclassical\nrw [IsBezout.iff_span_pair_isPrincipal]\nintro x y\nrw [Ideal.span_insert]\ncases' le_total (Ideal.span { x } : Ideal R) (Ideal.span { y }) with h h\n\u00b7 erw [sup_eq_right.mpr h]; exact \u27e8\u27e8_, rfl\u27e9\u27e9\n\u00b7 erw [sup_eq_left.mpr h]; exact \u27e8\u27e8_, rfl\u27e9\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : ValuationRing R\n\u22a2 IsBezout R\n[PROOFSTEP]\nrw [IsBezout.iff_span_pair_isPrincipal]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : ValuationRing R\n\u22a2 \u2200 (x y : R), Submodule.IsPrincipal (Ideal.span {x, y})\n[PROOFSTEP]\nintro x y\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : ValuationRing R\nx y : R\n\u22a2 Submodule.IsPrincipal (Ideal.span {x, y})\n[PROOFSTEP]\nrw [Ideal.span_insert]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : ValuationRing R\nx y : R\n\u22a2 Submodule.IsPrincipal (Ideal.span {x} \u2294 Ideal.span {y})\n[PROOFSTEP]\ncases' le_total (Ideal.span { x } : Ideal R) (Ideal.span { y }) with h h\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : ValuationRing R\nx y : R\nh : Ideal.span {x} \u2264 Ideal.span {y}\n\u22a2 Submodule.IsPrincipal (Ideal.span {x} \u2294 Ideal.span {y})\n[PROOFSTEP]\nerw [sup_eq_right.mpr h]\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : ValuationRing R\nx y : R\nh : Ideal.span {x} \u2264 Ideal.span {y}\n\u22a2 Submodule.IsPrincipal (Ideal.span {y})\n[PROOFSTEP]\nexact \u27e8\u27e8_, rfl\u27e9\u27e9\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : ValuationRing R\nx y : R\nh : Ideal.span {y} \u2264 Ideal.span {x}\n\u22a2 Submodule.IsPrincipal (Ideal.span {x} \u2294 Ideal.span {y})\n[PROOFSTEP]\nerw [sup_eq_left.mpr h]\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\nK : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : ValuationRing R\nx y : R\nh : Ideal.span {y} \u2264 Ideal.span {x}\n\u22a2 Submodule.IsPrincipal (Ideal.span {x})\n[PROOFSTEP]\nexact \u27e8\u27e8_, rfl\u27e9\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\n\u22a2 ValuationRing R \u2194 LocalRing R \u2227 IsBezout R\n[PROOFSTEP]\nclassical\nrefine \u27e8fun H => \u27e8inferInstance, inferInstance\u27e9, ?_\u27e9\nrintro \u27e8h\u2081, h\u2082\u27e9\nrefine iff_dvd_total.mpr \u27e8fun a b => ?_\u27e9\nobtain \u27e8g, e : _ = Ideal.span _\u27e9 := IsBezout.span_pair_isPrincipal a b\nobtain \u27e8a, rfl\u27e9 :=\n Ideal.mem_span_singleton'.mp (show a \u2208 Ideal.span { g } by rw [\u2190 e]; exact Ideal.subset_span (by simp))\nobtain \u27e8b, rfl\u27e9 :=\n Ideal.mem_span_singleton'.mp (show b \u2208 Ideal.span { g } by rw [\u2190 e]; exact Ideal.subset_span (by simp))\nobtain \u27e8x, y, e'\u27e9 :=\n Ideal.mem_span_pair.mp (show g \u2208 Ideal.span {a * g, b * g} by rw [e]; exact Ideal.subset_span (by simp))\ncases' eq_or_ne g 0 with h h\n\u00b7 simp [h]\nhave : x * a + y * b = 1 := by apply mul_left_injective\u2080 h; convert e' using 1 <;> ring\ncases' LocalRing.isUnit_or_isUnit_of_add_one this with h' h'\nleft\nswap\nright\nall_goals exact mul_dvd_mul_right (isUnit_iff_forall_dvd.mp (isUnit_of_mul_isUnit_right h') _) _\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\n\u22a2 ValuationRing R \u2194 LocalRing R \u2227 IsBezout R\n[PROOFSTEP]\nrefine \u27e8fun H => \u27e8inferInstance, inferInstance\u27e9, ?_\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\n\u22a2 LocalRing R \u2227 IsBezout R \u2192 ValuationRing R\n[PROOFSTEP]\nrintro \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\n\u22a2 ValuationRing R\n[PROOFSTEP]\nrefine iff_dvd_total.mpr \u27e8fun a b => ?_\u27e9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\na b : R\n\u22a2 a \u2223 b \u2228 b \u2223 a\n[PROOFSTEP]\nobtain \u27e8g, e : _ = Ideal.span _\u27e9 := IsBezout.span_pair_isPrincipal a b\n[GOAL]\ncase intro.mk.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\na b g : R\ne : Ideal.span {a, b} = Ideal.span {g}\n\u22a2 a \u2223 b \u2228 b \u2223 a\n[PROOFSTEP]\nobtain \u27e8a, rfl\u27e9 :=\n Ideal.mem_span_singleton'.mp (show a \u2208 Ideal.span { g } by rw [\u2190 e]; exact Ideal.subset_span (by simp))\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\na b g : R\ne : Ideal.span {a, b} = Ideal.span {g}\n\u22a2 a \u2208 Ideal.span {g}\n[PROOFSTEP]\nrw [\u2190 e]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\na b g : R\ne : Ideal.span {a, b} = Ideal.span {g}\n\u22a2 a \u2208 Ideal.span {a, b}\n[PROOFSTEP]\nexact Ideal.subset_span (by simp)\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\na b g : R\ne : Ideal.span {a, b} = Ideal.span {g}\n\u22a2 a \u2208 {a, b}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.mk.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\nb g a : R\ne : Ideal.span {a * g, b} = Ideal.span {g}\n\u22a2 a * g \u2223 b \u2228 b \u2223 a * g\n[PROOFSTEP]\nobtain \u27e8b, rfl\u27e9 :=\n Ideal.mem_span_singleton'.mp (show b \u2208 Ideal.span { g } by rw [\u2190 e]; exact Ideal.subset_span (by simp))\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\nb g a : R\ne : Ideal.span {a * g, b} = Ideal.span {g}\n\u22a2 b \u2208 Ideal.span {g}\n[PROOFSTEP]\nrw [\u2190 e]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\nb g a : R\ne : Ideal.span {a * g, b} = Ideal.span {g}\n\u22a2 b \u2208 Ideal.span {a * g, b}\n[PROOFSTEP]\nexact Ideal.subset_span (by simp)\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\nb g a : R\ne : Ideal.span {a * g, b} = Ideal.span {g}\n\u22a2 b \u2208 {a * g, b}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.mk.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\n\u22a2 a * g \u2223 b * g \u2228 b * g \u2223 a * g\n[PROOFSTEP]\nobtain \u27e8x, y, e'\u27e9 :=\n Ideal.mem_span_pair.mp (show g \u2208 Ideal.span {a * g, b * g} by rw [e]; exact Ideal.subset_span (by simp))\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\n\u22a2 g \u2208 Ideal.span {a * g, b * g}\n[PROOFSTEP]\nrw [e]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\n\u22a2 g \u2208 Ideal.span {g}\n[PROOFSTEP]\nexact Ideal.subset_span (by simp)\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\n\u22a2 g \u2208 {g}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\n\u22a2 a * g \u2223 b * g \u2228 b * g \u2223 a * g\n[PROOFSTEP]\ncases' eq_or_ne g 0 with h h\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.inl\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g = 0\n\u22a2 a * g \u2223 b * g \u2228 b * g \u2223 a * g\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.inr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\n\u22a2 a * g \u2223 b * g \u2228 b * g \u2223 a * g\n[PROOFSTEP]\nhave : x * a + y * b = 1 := by apply mul_left_injective\u2080 h; convert e' using 1 <;> ring\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\n\u22a2 x * a + y * b = 1\n[PROOFSTEP]\napply mul_left_injective\u2080 h\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\n\u22a2 (fun a => a * g) (x * a + y * b) = (fun a => a * g) 1\n[PROOFSTEP]\nconvert e' using 1\n[GOAL]\ncase h.e'_2\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\n\u22a2 (fun a => a * g) (x * a + y * b) = x * (a * g) + y * (b * g)\n[PROOFSTEP]\nring\n[GOAL]\ncase h.e'_3\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\n\u22a2 (fun a => a * g) 1 = g\n[PROOFSTEP]\nring\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.inr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\nthis : x * a + y * b = 1\n\u22a2 a * g \u2223 b * g \u2228 b * g \u2223 a * g\n[PROOFSTEP]\ncases' LocalRing.isUnit_or_isUnit_of_add_one this with h' h'\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.inr.inl\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\nthis : x * a + y * b = 1\nh' : IsUnit (x * a)\n\u22a2 a * g \u2223 b * g \u2228 b * g \u2223 a * g\ncase intro.mk.intro.intro.intro.intro.intro.inr.inr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\nthis : x * a + y * b = 1\nh' : IsUnit (y * b)\n\u22a2 a * g \u2223 b * g \u2228 b * g \u2223 a * g\n[PROOFSTEP]\nleft\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.inr.inl.h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\nthis : x * a + y * b = 1\nh' : IsUnit (x * a)\n\u22a2 a * g \u2223 b * g\ncase intro.mk.intro.intro.intro.intro.intro.inr.inr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\nthis : x * a + y * b = 1\nh' : IsUnit (y * b)\n\u22a2 a * g \u2223 b * g \u2228 b * g \u2223 a * g\n[PROOFSTEP]\nswap\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.inr.inr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\nthis : x * a + y * b = 1\nh' : IsUnit (y * b)\n\u22a2 a * g \u2223 b * g \u2228 b * g \u2223 a * g\ncase intro.mk.intro.intro.intro.intro.intro.inr.inl.h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\nthis : x * a + y * b = 1\nh' : IsUnit (x * a)\n\u22a2 a * g \u2223 b * g\n[PROOFSTEP]\nright\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.inr.inr.h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\nthis : x * a + y * b = 1\nh' : IsUnit (y * b)\n\u22a2 b * g \u2223 a * g\ncase intro.mk.intro.intro.intro.intro.intro.inr.inl.h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\nthis : x * a + y * b = 1\nh' : IsUnit (x * a)\n\u22a2 a * g \u2223 b * g\n[PROOFSTEP]\nall_goals exact mul_dvd_mul_right (isUnit_iff_forall_dvd.mp (isUnit_of_mul_isUnit_right h') _) _\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.inr.inr.h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\nthis : x * a + y * b = 1\nh' : IsUnit (y * b)\n\u22a2 b * g \u2223 a * g\n[PROOFSTEP]\nexact mul_dvd_mul_right (isUnit_iff_forall_dvd.mp (isUnit_of_mul_isUnit_right h') _) _\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.inr.inl.h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nh\u2081 : LocalRing R\nh\u2082 : IsBezout R\ng a b : R\ne : Ideal.span {a * g, b * g} = Ideal.span {g}\nx y : R\ne' : x * (a * g) + y * (b * g) = g\nh : g \u2260 0\nthis : x * a + y * b = 1\nh' : IsUnit (x * a)\n\u22a2 a * g \u2223 b * g\n[PROOFSTEP]\nexact mul_dvd_mul_right (isUnit_iff_forall_dvd.mp (isUnit_of_mul_isUnit_right h') _) _\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : IsDomain R\u271d\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R\u271d K\ninst\u271d\u00b2 : IsFractionRing R\u271d K\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 List.TFAE\n [ValuationRing R, \u2200 (x : FractionRing R), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9,\n IsTotal R fun x x_1 => x \u2223 x_1, IsTotal (Ideal R) fun x x_1 => x \u2264 x_1, LocalRing R \u2227 IsBezout R]\n[PROOFSTEP]\ntfae_have 1 \u2194 2\n[GOAL]\ncase tfae_1_iff_2\nR\u271d : Type u_1\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : IsDomain R\u271d\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R\u271d K\ninst\u271d\u00b2 : IsFractionRing R\u271d K\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 ValuationRing R \u2194 \u2200 (x : FractionRing R), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\n[PROOFSTEP]\nexact iff_isInteger_or_isInteger R _\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : IsDomain R\u271d\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R\u271d K\ninst\u271d\u00b2 : IsFractionRing R\u271d K\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ntfae_1_iff_2 : ValuationRing R \u2194 \u2200 (x : FractionRing R), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\n\u22a2 List.TFAE\n [ValuationRing R, \u2200 (x : FractionRing R), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9,\n IsTotal R fun x x_1 => x \u2223 x_1, IsTotal (Ideal R) fun x x_1 => x \u2264 x_1, LocalRing R \u2227 IsBezout R]\n[PROOFSTEP]\ntfae_have 1 \u2194 3\n[GOAL]\ncase tfae_1_iff_3\nR\u271d : Type u_1\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : IsDomain R\u271d\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R\u271d K\ninst\u271d\u00b2 : IsFractionRing R\u271d K\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ntfae_1_iff_2 : ValuationRing R \u2194 \u2200 (x : FractionRing R), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\n\u22a2 ValuationRing R \u2194 IsTotal R fun x x_1 => x \u2223 x_1\n[PROOFSTEP]\nexact iff_dvd_total\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : IsDomain R\u271d\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R\u271d K\ninst\u271d\u00b2 : IsFractionRing R\u271d K\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ntfae_1_iff_2 : ValuationRing R \u2194 \u2200 (x : FractionRing R), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\ntfae_1_iff_3 : ValuationRing R \u2194 IsTotal R fun x x_1 => x \u2223 x_1\n\u22a2 List.TFAE\n [ValuationRing R, \u2200 (x : FractionRing R), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9,\n IsTotal R fun x x_1 => x \u2223 x_1, IsTotal (Ideal R) fun x x_1 => x \u2264 x_1, LocalRing R \u2227 IsBezout R]\n[PROOFSTEP]\ntfae_have 1 \u2194 4\n[GOAL]\ncase tfae_1_iff_4\nR\u271d : Type u_1\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : IsDomain R\u271d\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R\u271d K\ninst\u271d\u00b2 : IsFractionRing R\u271d K\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ntfae_1_iff_2 : ValuationRing R \u2194 \u2200 (x : FractionRing R), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\ntfae_1_iff_3 : ValuationRing R \u2194 IsTotal R fun x x_1 => x \u2223 x_1\n\u22a2 ValuationRing R \u2194 IsTotal (Ideal R) fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nexact iff_ideal_total\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : IsDomain R\u271d\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R\u271d K\ninst\u271d\u00b2 : IsFractionRing R\u271d K\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ntfae_1_iff_2 : ValuationRing R \u2194 \u2200 (x : FractionRing R), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\ntfae_1_iff_3 : ValuationRing R \u2194 IsTotal R fun x x_1 => x \u2223 x_1\ntfae_1_iff_4 : ValuationRing R \u2194 IsTotal (Ideal R) fun x x_1 => x \u2264 x_1\n\u22a2 List.TFAE\n [ValuationRing R, \u2200 (x : FractionRing R), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9,\n IsTotal R fun x x_1 => x \u2223 x_1, IsTotal (Ideal R) fun x x_1 => x \u2264 x_1, LocalRing R \u2227 IsBezout R]\n[PROOFSTEP]\ntfae_have 1 \u2194 5\n[GOAL]\ncase tfae_1_iff_5\nR\u271d : Type u_1\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : IsDomain R\u271d\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R\u271d K\ninst\u271d\u00b2 : IsFractionRing R\u271d K\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ntfae_1_iff_2 : ValuationRing R \u2194 \u2200 (x : FractionRing R), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\ntfae_1_iff_3 : ValuationRing R \u2194 IsTotal R fun x x_1 => x \u2223 x_1\ntfae_1_iff_4 : ValuationRing R \u2194 IsTotal (Ideal R) fun x x_1 => x \u2264 x_1\n\u22a2 ValuationRing R \u2194 LocalRing R \u2227 IsBezout R\n[PROOFSTEP]\nexact iff_local_bezout_domain\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : IsDomain R\u271d\nK : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra R\u271d K\ninst\u271d\u00b2 : IsFractionRing R\u271d K\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ntfae_1_iff_2 : ValuationRing R \u2194 \u2200 (x : FractionRing R), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9\ntfae_1_iff_3 : ValuationRing R \u2194 IsTotal R fun x x_1 => x \u2223 x_1\ntfae_1_iff_4 : ValuationRing R \u2194 IsTotal (Ideal R) fun x x_1 => x \u2264 x_1\ntfae_1_iff_5 : ValuationRing R \u2194 LocalRing R \u2227 IsBezout R\n\u22a2 List.TFAE\n [ValuationRing R, \u2200 (x : FractionRing R), IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9,\n IsTotal R fun x x_1 => x \u2223 x_1, IsTotal (Ideal R) fun x x_1 => x \u2264 x_1, LocalRing R \u2227 IsBezout R]\n[PROOFSTEP]\ntfae_finish\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : ValuationRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : IsDomain S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\na b : S\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nobtain \u27e8\u27e8a, rfl\u27e9, \u27e8b, rfl\u27e9\u27e9 := hf a, hf b\n[GOAL]\ncase intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : ValuationRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : IsDomain S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\na b : R\n\u22a2 \u2203 c, \u2191f a * c = \u2191f b \u2228 \u2191f b * c = \u2191f a\n[PROOFSTEP]\nobtain \u27e8c, rfl | rfl\u27e9 := ValuationRing.cond a b\n[GOAL]\ncase intro.intro.intro.inl\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : ValuationRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : IsDomain S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\na c : R\n\u22a2 \u2203 c_1, \u2191f a * c_1 = \u2191f (a * c) \u2228 \u2191f (a * c) * c_1 = \u2191f a\ncase intro.intro.intro.inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : ValuationRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : IsDomain S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nb c : R\n\u22a2 \u2203 c_1, \u2191f (b * c) * c_1 = \u2191f b \u2228 \u2191f b * c_1 = \u2191f (b * c)\n[PROOFSTEP]\nexacts [\u27e8f c, Or.inl <| (map_mul _ _ _).symm\u27e9, \u27e8f c, Or.inr <| (map_mul _ _ _).symm\u27e9]\n[GOAL]\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\n\u22a2 ValuationRing \ud835\udcaa\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cond'\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\n\u22a2 \u2200 (a b : \ud835\udcaa), \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nintro a b\n[GOAL]\ncase cond'\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\ncases' le_total (v (algebraMap \ud835\udcaa K a)) (v (algebraMap \ud835\udcaa K b)) with h h\n[GOAL]\ncase cond'.inl\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) a) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) b)\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 := Valuation.Integers.dvd_of_le hh h\n[GOAL]\ncase cond'.inl.intro\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) a) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) b)\nc : \ud835\udcaa\nhc : a = b * c\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nuse c\n[GOAL]\ncase h\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) a) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) b)\nc : \ud835\udcaa\nhc : a = b * c\n\u22a2 a * c = b \u2228 b * c = a\n[PROOFSTEP]\nexact Or.inr hc.symm\n[GOAL]\ncase cond'.inr\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) b) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) a)\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 := Valuation.Integers.dvd_of_le hh h\n[GOAL]\ncase cond'.inr.intro\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) b) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) a)\nc : \ud835\udcaa\nhc : b = a * c\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nuse c\n[GOAL]\ncase h\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) b) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) a)\nc : \ud835\udcaa\nhc : b = a * c\n\u22a2 a * c = b \u2228 b * c = a\n[PROOFSTEP]\nexact Or.inl hc.symm\n[GOAL]\nK : Type u\ninst\u271d : Field K\n\u22a2 ValuationRing K\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cond'\nK : Type u\ninst\u271d : Field K\n\u22a2 \u2200 (a b : K), \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nintro a b\n[GOAL]\ncase cond'\nK : Type u\ninst\u271d : Field K\na b : K\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nby_cases b = 0\n[GOAL]\ncase cond'\nK : Type u\ninst\u271d : Field K\na b : K\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nby_cases b = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\na b : K\nh : b = 0\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\na b : K\nh : b = 0\n\u22a2 a * 0 = b \u2228 b * 0 = a\n[PROOFSTEP]\nleft\n[GOAL]\ncase h.h\nK : Type u\ninst\u271d : Field K\na b : K\nh : b = 0\n\u22a2 a * 0 = b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\na b : K\nh : \u00acb = 0\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nuse a * b\u207b\u00b9\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\na b : K\nh : \u00acb = 0\n\u22a2 a * (a * b\u207b\u00b9) = b \u2228 b * (a * b\u207b\u00b9) = a\n[PROOFSTEP]\nright\n[GOAL]\ncase h.h\nK : Type u\ninst\u271d : Field K\na b : K\nh : \u00acb = 0\n\u22a2 b * (a * b\u207b\u00b9) = a\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase h.h\nK : Type u\ninst\u271d : Field K\na b : K\nh : \u00acb = 0\n\u22a2 b * a = a * b\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\n\u22a2 ValuationRing A\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cond'\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\n\u22a2 \u2200 (a b : A), \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nintro a b\n[GOAL]\ncase cond'\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\na b : A\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\na b : A\nha : a = 0\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\na b : A\nha : a = 0\n\u22a2 a * 0 = b \u2228 b * 0 = a\n[PROOFSTEP]\nright\n[GOAL]\ncase h.h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\na b : A\nha : a = 0\n\u22a2 b * 0 = a\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\ncase neg\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\na b : A\nha : \u00aca = 0\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nby_cases hb : b = 0\n[GOAL]\ncase pos\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\na b : A\nha : \u00aca = 0\nhb : b = 0\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\na b : A\nha : \u00aca = 0\nhb : b = 0\n\u22a2 a * 0 = b \u2228 b * 0 = a\n[PROOFSTEP]\nleft\n[GOAL]\ncase h.h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\na b : A\nha : \u00aca = 0\nhb : b = 0\n\u22a2 a * 0 = b\n[PROOFSTEP]\nsimp [hb]\n[GOAL]\ncase neg\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\na b : A\nha : \u00aca = 0\nhb : \u00acb = 0\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nobtain \u27e8\u03d6, h\u03d6\u27e9 := DiscreteValuationRing.exists_irreducible A\n[GOAL]\ncase neg.intro\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\na b : A\nha : \u00aca = 0\nhb : \u00acb = 0\n\u03d6 : A\nh\u03d6 : Irreducible \u03d6\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nobtain \u27e8m, u, rfl\u27e9 := DiscreteValuationRing.eq_unit_mul_pow_irreducible ha h\u03d6\n[GOAL]\ncase neg.intro.intro.intro\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\nb : A\nhb : \u00acb = 0\n\u03d6 : A\nh\u03d6 : Irreducible \u03d6\nm : \u2115\nu : A\u02e3\nha : \u00ac\u2191u * \u03d6 ^ m = 0\n\u22a2 \u2203 c, \u2191u * \u03d6 ^ m * c = b \u2228 b * c = \u2191u * \u03d6 ^ m\n[PROOFSTEP]\nobtain \u27e8n, v, rfl\u27e9 := DiscreteValuationRing.eq_unit_mul_pow_irreducible hb h\u03d6\n[GOAL]\ncase neg.intro.intro.intro.intro.intro\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\n\u03d6 : A\nh\u03d6 : Irreducible \u03d6\nm : \u2115\nu : A\u02e3\nha : \u00ac\u2191u * \u03d6 ^ m = 0\nn : \u2115\nv : A\u02e3\nhb : \u00ac\u2191v * \u03d6 ^ n = 0\n\u22a2 \u2203 c, \u2191u * \u03d6 ^ m * c = \u2191v * \u03d6 ^ n \u2228 \u2191v * \u03d6 ^ n * c = \u2191u * \u03d6 ^ m\n[PROOFSTEP]\ncases' le_total m n with h h\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.inl\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\n\u03d6 : A\nh\u03d6 : Irreducible \u03d6\nm : \u2115\nu : A\u02e3\nha : \u00ac\u2191u * \u03d6 ^ m = 0\nn : \u2115\nv : A\u02e3\nhb : \u00ac\u2191v * \u03d6 ^ n = 0\nh : m \u2264 n\n\u22a2 \u2203 c, \u2191u * \u03d6 ^ m * c = \u2191v * \u03d6 ^ n \u2228 \u2191v * \u03d6 ^ n * c = \u2191u * \u03d6 ^ m\n[PROOFSTEP]\nuse(u\u207b\u00b9 * v : A\u02e3) * \u03d6 ^ (n - m)\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\n\u03d6 : A\nh\u03d6 : Irreducible \u03d6\nm : \u2115\nu : A\u02e3\nha : \u00ac\u2191u * \u03d6 ^ m = 0\nn : \u2115\nv : A\u02e3\nhb : \u00ac\u2191v * \u03d6 ^ n = 0\nh : m \u2264 n\n\u22a2 \u2191u * \u03d6 ^ m * (\u2191(u\u207b\u00b9 * v) * \u03d6 ^ (n - m)) = \u2191v * \u03d6 ^ n \u2228 \u2191v * \u03d6 ^ n * (\u2191(u\u207b\u00b9 * v) * \u03d6 ^ (n - m)) = \u2191u * \u03d6 ^ m\n[PROOFSTEP]\nleft\n[GOAL]\ncase h.h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\n\u03d6 : A\nh\u03d6 : Irreducible \u03d6\nm : \u2115\nu : A\u02e3\nha : \u00ac\u2191u * \u03d6 ^ m = 0\nn : \u2115\nv : A\u02e3\nhb : \u00ac\u2191v * \u03d6 ^ n = 0\nh : m \u2264 n\n\u22a2 \u2191u * \u03d6 ^ m * (\u2191(u\u207b\u00b9 * v) * \u03d6 ^ (n - m)) = \u2191v * \u03d6 ^ n\n[PROOFSTEP]\nsimp_rw [mul_comm (u : A), Units.val_mul, \u2190 mul_assoc, mul_assoc _ (u : A)]\n[GOAL]\ncase h.h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\n\u03d6 : A\nh\u03d6 : Irreducible \u03d6\nm : \u2115\nu : A\u02e3\nha : \u00ac\u2191u * \u03d6 ^ m = 0\nn : \u2115\nv : A\u02e3\nhb : \u00ac\u2191v * \u03d6 ^ n = 0\nh : m \u2264 n\n\u22a2 \u03d6 ^ m * (\u2191u * \u2191u\u207b\u00b9) * \u2191v * \u03d6 ^ (n - m) = \u2191v * \u03d6 ^ n\n[PROOFSTEP]\nsimp only [Units.mul_inv, mul_one, mul_comm _ (v : A), mul_assoc, \u2190 pow_add]\n[GOAL]\ncase h.h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\n\u03d6 : A\nh\u03d6 : Irreducible \u03d6\nm : \u2115\nu : A\u02e3\nha : \u00ac\u2191u * \u03d6 ^ m = 0\nn : \u2115\nv : A\u02e3\nhb : \u00ac\u2191v * \u03d6 ^ n = 0\nh : m \u2264 n\n\u22a2 \u2191v * \u03d6 ^ (m + (n - m)) = \u2191v * \u03d6 ^ n\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase h.h.e_a.e_a\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\n\u03d6 : A\nh\u03d6 : Irreducible \u03d6\nm : \u2115\nu : A\u02e3\nha : \u00ac\u2191u * \u03d6 ^ m = 0\nn : \u2115\nv : A\u02e3\nhb : \u00ac\u2191v * \u03d6 ^ n = 0\nh : m \u2264 n\n\u22a2 m + (n - m) = n\n[PROOFSTEP]\nexact Nat.add_sub_of_le h\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.inr\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\n\u03d6 : A\nh\u03d6 : Irreducible \u03d6\nm : \u2115\nu : A\u02e3\nha : \u00ac\u2191u * \u03d6 ^ m = 0\nn : \u2115\nv : A\u02e3\nhb : \u00ac\u2191v * \u03d6 ^ n = 0\nh : n \u2264 m\n\u22a2 \u2203 c, \u2191u * \u03d6 ^ m * c = \u2191v * \u03d6 ^ n \u2228 \u2191v * \u03d6 ^ n * c = \u2191u * \u03d6 ^ m\n[PROOFSTEP]\nuse(v\u207b\u00b9 * u : A\u02e3) * \u03d6 ^ (m - n)\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\n\u03d6 : A\nh\u03d6 : Irreducible \u03d6\nm : \u2115\nu : A\u02e3\nha : \u00ac\u2191u * \u03d6 ^ m = 0\nn : \u2115\nv : A\u02e3\nhb : \u00ac\u2191v * \u03d6 ^ n = 0\nh : n \u2264 m\n\u22a2 \u2191u * \u03d6 ^ m * (\u2191(v\u207b\u00b9 * u) * \u03d6 ^ (m - n)) = \u2191v * \u03d6 ^ n \u2228 \u2191v * \u03d6 ^ n * (\u2191(v\u207b\u00b9 * u) * \u03d6 ^ (m - n)) = \u2191u * \u03d6 ^ m\n[PROOFSTEP]\nright\n[GOAL]\ncase h.h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\n\u03d6 : A\nh\u03d6 : Irreducible \u03d6\nm : \u2115\nu : A\u02e3\nha : \u00ac\u2191u * \u03d6 ^ m = 0\nn : \u2115\nv : A\u02e3\nhb : \u00ac\u2191v * \u03d6 ^ n = 0\nh : n \u2264 m\n\u22a2 \u2191v * \u03d6 ^ n * (\u2191(v\u207b\u00b9 * u) * \u03d6 ^ (m - n)) = \u2191u * \u03d6 ^ m\n[PROOFSTEP]\nsimp_rw [mul_comm (v : A), Units.val_mul, \u2190 mul_assoc, mul_assoc _ (v : A)]\n[GOAL]\ncase h.h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\n\u03d6 : A\nh\u03d6 : Irreducible \u03d6\nm : \u2115\nu : A\u02e3\nha : \u00ac\u2191u * \u03d6 ^ m = 0\nn : \u2115\nv : A\u02e3\nhb : \u00ac\u2191v * \u03d6 ^ n = 0\nh : n \u2264 m\n\u22a2 \u03d6 ^ n * (\u2191v * \u2191v\u207b\u00b9) * \u2191u * \u03d6 ^ (m - n) = \u2191u * \u03d6 ^ m\n[PROOFSTEP]\nsimp only [Units.mul_inv, mul_one, mul_comm _ (u : A), mul_assoc, \u2190 pow_add]\n[GOAL]\ncase h.h\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\n\u03d6 : A\nh\u03d6 : Irreducible \u03d6\nm : \u2115\nu : A\u02e3\nha : \u00ac\u2191u * \u03d6 ^ m = 0\nn : \u2115\nv : A\u02e3\nhb : \u00ac\u2191v * \u03d6 ^ n = 0\nh : n \u2264 m\n\u22a2 \u2191u * \u03d6 ^ (n + (m - n)) = \u2191u * \u03d6 ^ m\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase h.h.e_a.e_a\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DiscreteValuationRing A\n\u03d6 : A\nh\u03d6 : Irreducible \u03d6\nm : \u2115\nu : A\u02e3\nha : \u00ac\u2191u * \u03d6 ^ m = 0\nn : \u2115\nv : A\u02e3\nhb : \u00ac\u2191v * \u03d6 ^ n = 0\nh : n \u2264 m\n\u22a2 n + (m - n) = m\n[PROOFSTEP]\nexact Nat.add_sub_of_le h\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Valuation.ValuationRing", "llama_tokens": 80998, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389817407016, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.44221165340145335}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\nl : Filter \u03b1\n\u22a2 Set.univ \u2208 {s | Rel.core r s \u2208 l}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\nl : Filter \u03b1\nx\u271d y\u271d : Set \u03b2\nhs : x\u271d \u2208 {s | Rel.core r s \u2208 l}\nht : y\u271d \u2208 {s | Rel.core r s \u2208 l}\n\u22a2 x\u271d \u2229 y\u271d \u2208 {s | Rel.core r s \u2208 l}\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\nl : Filter \u03b1\nx\u271d y\u271d : Set \u03b2\nhs : x\u271d \u2208 {s | Rel.core r s \u2208 l}\nht : y\u271d \u2208 {s | Rel.core r s \u2208 l}\n\u22a2 Rel.core r (x\u271d \u2229 y\u271d) \u2208 l\n[PROOFSTEP]\nconvert inter_mem hs ht\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\nl : Filter \u03b1\nx\u271d y\u271d : Set \u03b2\nhs : x\u271d \u2208 {s | Rel.core r s \u2208 l}\nht : y\u271d \u2208 {s | Rel.core r s \u2208 l}\n\u22a2 Rel.core r (x\u271d \u2229 y\u271d) = Rel.core r x\u271d \u2229 Rel.core r y\u271d\n[PROOFSTEP]\nrw [\u2190 Rel.core_inter]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nl : Filter \u03b1\n\u22a2 (rmap s (rmap r l)).sets = (rmap (Rel.comp r s) l).sets\n[PROOFSTEP]\nsimp [rmap_sets, Set.preimage, Rel.core_comp]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nl : Filter \u03b3\n\u22a2 (rcomap r (rcomap s l)).sets = (rcomap (Rel.comp r s) l).sets\n[PROOFSTEP]\next t\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nl : Filter \u03b3\nt : Set \u03b1\n\u22a2 t \u2208 (rcomap r (rcomap s l)).sets \u2194 t \u2208 (rcomap (Rel.comp r s) l).sets\n[PROOFSTEP]\nsimp [rcomap_sets, Rel.image, Rel.core_comp]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nl : Filter \u03b3\nt : Set \u03b1\n\u22a2 (\u2203 x, (\u2203 x_1, x_1 \u2208 l \u2227 Rel.core s x_1 \u2286 x) \u2227 Rel.core r x \u2286 t) \u2194 \u2203 x, x \u2208 l \u2227 Rel.core r (Rel.core s x) \u2286 t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nl : Filter \u03b3\nt : Set \u03b1\n\u22a2 (\u2203 x, (\u2203 x_1, x_1 \u2208 l \u2227 Rel.core s x_1 \u2286 x) \u2227 Rel.core r x \u2286 t) \u2192 \u2203 x, x \u2208 l \u2227 Rel.core r (Rel.core s x) \u2286 t\n[PROOFSTEP]\nrintro \u27e8u, \u27e8v, vsets, hv\u27e9, h\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nl : Filter \u03b3\nt : Set \u03b1\nu : Set \u03b2\nh : Rel.core r u \u2286 t\nv : Set \u03b3\nvsets : v \u2208 l\nhv : Rel.core s v \u2286 u\n\u22a2 \u2203 x, x \u2208 l \u2227 Rel.core r (Rel.core s x) \u2286 t\n[PROOFSTEP]\nexact \u27e8v, vsets, Set.Subset.trans (Rel.core_mono _ hv) h\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nl : Filter \u03b3\nt : Set \u03b1\n\u22a2 (\u2203 x, x \u2208 l \u2227 Rel.core r (Rel.core s x) \u2286 t) \u2192 \u2203 x, (\u2203 x_1, x_1 \u2208 l \u2227 Rel.core s x_1 \u2286 x) \u2227 Rel.core r x \u2286 t\n[PROOFSTEP]\nrintro \u27e8t, tsets, ht\u27e9\n[GOAL]\ncase h.mpr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nl : Filter \u03b3\nt\u271d : Set \u03b1\nt : Set \u03b3\ntsets : t \u2208 l\nht : Rel.core r (Rel.core s t) \u2286 t\u271d\n\u22a2 \u2203 x, (\u2203 x_1, x_1 \u2208 l \u2227 Rel.core s x_1 \u2286 x) \u2227 Rel.core r x \u2286 t\u271d\n[PROOFSTEP]\nexact \u27e8Rel.core s t, \u27e8t, tsets, Set.Subset.rfl\u27e9, ht\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\n\u22a2 RTendsto r l\u2081 l\u2082 \u2194 l\u2081 \u2264 rcomap r l\u2082\n[PROOFSTEP]\nrw [rtendsto_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\n\u22a2 (\u2200 (s : Set \u03b2), s \u2208 l\u2082 \u2192 Rel.core r s \u2208 l\u2081) \u2194 l\u2081 \u2264 rcomap r l\u2082\n[PROOFSTEP]\nsimp_rw [\u2190 l\u2082.mem_sets]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\n\u22a2 (\u2200 (s : Set \u03b2), s \u2208 l\u2082.sets \u2192 Rel.core r s \u2208 l\u2081) \u2194 l\u2081 \u2264 rcomap r l\u2082\n[PROOFSTEP]\nsimp [Filter.le_def, rcomap, Rel.mem_image]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\n\u22a2 (\u2200 (s : Set \u03b2), s \u2208 l\u2082 \u2192 Rel.core r s \u2208 l\u2081) \u2194 \u2200 (x : Set \u03b1) (x_1 : Set \u03b2), x_1 \u2208 l\u2082 \u2192 Rel.core r x_1 \u2286 x \u2192 x \u2208 l\u2081\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\n\u22a2 (\u2200 (s : Set \u03b2), s \u2208 l\u2082 \u2192 Rel.core r s \u2208 l\u2081) \u2192 \u2200 (x : Set \u03b1) (x_1 : Set \u03b2), x_1 \u2208 l\u2082 \u2192 Rel.core r x_1 \u2286 x \u2192 x \u2208 l\u2081\n[PROOFSTEP]\nexact fun h s t tl\u2082 => mem_of_superset (h t tl\u2082)\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\n\u22a2 (\u2200 (x : Set \u03b1) (x_1 : Set \u03b2), x_1 \u2208 l\u2082 \u2192 Rel.core r x_1 \u2286 x \u2192 x \u2208 l\u2081) \u2192 \u2200 (s : Set \u03b2), s \u2208 l\u2082 \u2192 Rel.core r s \u2208 l\u2081\n[PROOFSTEP]\nexact fun h t tl\u2082 => h _ t tl\u2082 Set.Subset.rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nl : Filter \u03b3\nt : Set \u03b1\n\u22a2 t \u2208 rcomap' r (rcomap' s l) \u2194 t \u2208 rcomap' (Rel.comp r s) l\n[PROOFSTEP]\nsimp only [mem_rcomap', Rel.preimage_comp]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nl : Filter \u03b3\nt : Set \u03b1\n\u22a2 (\u2203 t_1, (\u2203 t, t \u2208 l \u2227 Rel.preimage s t \u2286 t_1) \u2227 Rel.preimage r t_1 \u2286 t) \u2194\n \u2203 t_1, t_1 \u2208 l \u2227 Rel.preimage r (Rel.preimage s t_1) \u2286 t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nl : Filter \u03b3\nt : Set \u03b1\n\u22a2 (\u2203 t_1, (\u2203 t, t \u2208 l \u2227 Rel.preimage s t \u2286 t_1) \u2227 Rel.preimage r t_1 \u2286 t) \u2192\n \u2203 t_1, t_1 \u2208 l \u2227 Rel.preimage r (Rel.preimage s t_1) \u2286 t\n[PROOFSTEP]\nrintro \u27e8u, \u27e8v, vsets, hv\u27e9, h\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nl : Filter \u03b3\nt : Set \u03b1\nu : Set \u03b2\nh : Rel.preimage r u \u2286 t\nv : Set \u03b3\nvsets : v \u2208 l\nhv : Rel.preimage s v \u2286 u\n\u22a2 \u2203 t_1, t_1 \u2208 l \u2227 Rel.preimage r (Rel.preimage s t_1) \u2286 t\n[PROOFSTEP]\nexact \u27e8v, vsets, (Rel.preimage_mono _ hv).trans h\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nl : Filter \u03b3\nt : Set \u03b1\n\u22a2 (\u2203 t_1, t_1 \u2208 l \u2227 Rel.preimage r (Rel.preimage s t_1) \u2286 t) \u2192\n \u2203 t_1, (\u2203 t, t \u2208 l \u2227 Rel.preimage s t \u2286 t_1) \u2227 Rel.preimage r t_1 \u2286 t\n[PROOFSTEP]\nrintro \u27e8t, tsets, ht\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nl : Filter \u03b3\nt\u271d : Set \u03b1\nt : Set \u03b3\ntsets : t \u2208 l\nht : Rel.preimage r (Rel.preimage s t) \u2286 t\u271d\n\u22a2 \u2203 t, (\u2203 t_1, t_1 \u2208 l \u2227 Rel.preimage s t_1 \u2286 t) \u2227 Rel.preimage r t \u2286 t\u271d\n[PROOFSTEP]\nexact \u27e8s.preimage t, \u27e8t, tsets, Set.Subset.rfl\u27e9, ht\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\n\u22a2 RTendsto' r l\u2081 l\u2082 \u2194 \u2200 (s : Set \u03b2), s \u2208 l\u2082 \u2192 Rel.preimage r s \u2208 l\u2081\n[PROOFSTEP]\nunfold RTendsto' rcomap'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\n\u22a2 l\u2081 \u2264\n { sets := Rel.image (fun s t => Rel.preimage r s \u2286 t) l\u2082.sets,\n univ_sets := (_ : \u2203 x, x \u2208 l\u2082.sets \u2227 (fun s t => Rel.preimage r s \u2286 t) x Set.univ),\n sets_of_superset :=\n (_ :\n \u2200 {x y : Set \u03b1},\n x \u2208 Rel.image (fun s t => Rel.preimage r s \u2286 t) l\u2082.sets \u2192\n x \u2286 y \u2192 y \u2208 Rel.image (fun s t => Rel.preimage r s \u2286 t) l\u2082.sets),\n inter_sets :=\n (_ :\n \u2200 {x y : Set \u03b1},\n x \u2208 Rel.image (fun s t => Rel.preimage r s \u2286 t) l\u2082.sets \u2192\n y \u2208 Rel.image (fun s t => Rel.preimage r s \u2286 t) l\u2082.sets \u2192\n x \u2229 y \u2208 Rel.image (fun s t => Rel.preimage r s \u2286 t) l\u2082.sets) } \u2194\n \u2200 (s : Set \u03b2), s \u2208 l\u2082 \u2192 Rel.preimage r s \u2208 l\u2081\n[PROOFSTEP]\nsimp [le_def, Rel.mem_image]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\n\u22a2 (\u2200 (x : Set \u03b1) (x_1 : Set \u03b2), x_1 \u2208 l\u2082 \u2192 Rel.preimage r x_1 \u2286 x \u2192 x \u2208 l\u2081) \u2194\n \u2200 (s : Set \u03b2), s \u2208 l\u2082 \u2192 Rel.preimage r s \u2208 l\u2081\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\n\u22a2 (\u2200 (x : Set \u03b1) (x_1 : Set \u03b2), x_1 \u2208 l\u2082 \u2192 Rel.preimage r x_1 \u2286 x \u2192 x \u2208 l\u2081) \u2192\n \u2200 (s : Set \u03b2), s \u2208 l\u2082 \u2192 Rel.preimage r s \u2208 l\u2081\n[PROOFSTEP]\nexact fun h s hs => h _ _ hs Set.Subset.rfl\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nr : Rel \u03b1 \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\n\u22a2 (\u2200 (s : Set \u03b2), s \u2208 l\u2082 \u2192 Rel.preimage r s \u2208 l\u2081) \u2192\n \u2200 (x : Set \u03b1) (x_1 : Set \u03b2), x_1 \u2208 l\u2082 \u2192 Rel.preimage r x_1 \u2286 x \u2192 x \u2208 l\u2081\n[PROOFSTEP]\nexact fun h s t ht => mem_of_superset (h t ht)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 Tendsto f l\u2081 l\u2082 \u2194 RTendsto (Function.graph f) l\u2081 l\u2082\n[PROOFSTEP]\nsimp [tendsto_def, Function.graph, rtendsto_def, Rel.core, Set.preimage]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 Tendsto f l\u2081 l\u2082 \u2194 RTendsto' (Function.graph f) l\u2081 l\u2082\n[PROOFSTEP]\nsimp [tendsto_def, Function.graph, rtendsto'_def, Rel.preimage_def, Set.preimage]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : Filter \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 pmap (PFun.res f s) l = map f (l \u2293 \ud835\udcdf s)\n[PROOFSTEP]\next t\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : Filter \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nt : Set \u03b2\n\u22a2 t \u2208 pmap (PFun.res f s) l \u2194 t \u2208 map f (l \u2293 \ud835\udcdf s)\n[PROOFSTEP]\nsimp only [PFun.core_res, mem_pmap, mem_map, mem_inf_principal, imp_iff_not_or]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : Filter \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nt : Set \u03b2\n\u22a2 s\u1d9c \u222a f \u207b\u00b9' t \u2208 l \u2194 {x | \u00acx \u2208 s \u2228 x \u2208 f \u207b\u00b9' t} \u2208 l\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 Tendsto f (l\u2081 \u2293 \ud835\udcdf s) l\u2082 \u2194 PTendsto (PFun.res f s) l\u2081 l\u2082\n[PROOFSTEP]\nsimp only [Tendsto, PTendsto, pmap_res]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 Tendsto f l\u2081 l\u2082 \u2194 PTendsto (PFun.res f Set.univ) l\u2081 l\u2082\n[PROOFSTEP]\nrw [\u2190 tendsto_iff_ptendsto]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 Tendsto f l\u2081 l\u2082 \u2194 Tendsto f (l\u2081 \u2293 \ud835\udcdf Set.univ) l\u2082\n[PROOFSTEP]\nsimp [principal_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192. \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\n\u22a2 PTendsto' f l\u2081 l\u2082 \u2192 PTendsto f l\u2081 l\u2082\n[PROOFSTEP]\nrw [ptendsto_def, ptendsto'_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192. \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\n\u22a2 (\u2200 (s : Set \u03b2), s \u2208 l\u2082 \u2192 PFun.preimage f s \u2208 l\u2081) \u2192 \u2200 (s : Set \u03b2), s \u2208 l\u2082 \u2192 PFun.core f s \u2208 l\u2081\n[PROOFSTEP]\nexact fun h s sl\u2082 => mem_of_superset (h s sl\u2082) (PFun.preimage_subset_core _ _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192. \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\nh : PFun.Dom f \u2208 l\u2081\n\u22a2 PTendsto f l\u2081 l\u2082 \u2192 PTendsto' f l\u2081 l\u2082\n[PROOFSTEP]\nrw [ptendsto_def, ptendsto'_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192. \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\nh : PFun.Dom f \u2208 l\u2081\n\u22a2 (\u2200 (s : Set \u03b2), s \u2208 l\u2082 \u2192 PFun.core f s \u2208 l\u2081) \u2192 \u2200 (s : Set \u03b2), s \u2208 l\u2082 \u2192 PFun.preimage f s \u2208 l\u2081\n[PROOFSTEP]\nintro h' s sl\u2082\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192. \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\nh : PFun.Dom f \u2208 l\u2081\nh' : \u2200 (s : Set \u03b2), s \u2208 l\u2082 \u2192 PFun.core f s \u2208 l\u2081\ns : Set \u03b2\nsl\u2082 : s \u2208 l\u2082\n\u22a2 PFun.preimage f s \u2208 l\u2081\n[PROOFSTEP]\nrw [PFun.preimage_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192. \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\nh : PFun.Dom f \u2208 l\u2081\nh' : \u2200 (s : Set \u03b2), s \u2208 l\u2082 \u2192 PFun.core f s \u2208 l\u2081\ns : Set \u03b2\nsl\u2082 : s \u2208 l\u2082\n\u22a2 PFun.core f s \u2229 PFun.Dom f \u2208 l\u2081\n[PROOFSTEP]\nexact inter_mem (h' s sl\u2082) h\n", "meta": {"mathlib_filename": "Mathlib.Order.Filter.Partial", "llama_tokens": 5445, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506526772883, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4421685753019218}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\n\u22a2 \u00acBddAbove s \u2194 \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 \u00acy \u2264 x\n[PROOFSTEP]\nsimp [BddAbove, upperBounds, Set.Nonempty]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\u271d\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\n\u22a2 \u00acBddAbove s \u2194 \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x < y\n[PROOFSTEP]\nsimp only [not_bddAbove_iff', not_le]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\nh : IsLUB s a\n\u22a2 a \u2264 b \u2194 b \u2208 upperBounds s\n[PROOFSTEP]\nrw [h.upperBounds_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\nh : IsLUB s a\n\u22a2 a \u2264 b \u2194 b \u2208 Ici a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\nh : IsGLB s a\n\u22a2 b \u2264 a \u2194 b \u2208 lowerBounds s\n[PROOFSTEP]\nrw [h.lowerBounds_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\nh : IsGLB s a\n\u22a2 b \u2264 a \u2194 b \u2208 Iic a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns\u271d t\u271d : Set \u03b1\na\u271d b a : \u03b1\ns t : Set \u03b1\n\u22a2 IsLeast (s \u222a t) a \u2194 IsLeast s a \u2227 a \u2208 lowerBounds t \u2228 a \u2208 lowerBounds s \u2227 IsLeast t a\n[PROOFSTEP]\nsimp [IsLeast, lowerBounds_union, or_and_right, and_comm (a := a \u2208 t), and_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ninst\u271d : SemilatticeSup \u03b3\ns t : Set \u03b3\n\u22a2 BddAbove s \u2192 BddAbove t \u2192 BddAbove (s \u222a t)\n[PROOFSTEP]\nrintro \u27e8bs, hs\u27e9 \u27e8bt, ht\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ninst\u271d : SemilatticeSup \u03b3\ns t : Set \u03b3\nbs : \u03b3\nhs : bs \u2208 upperBounds s\nbt : \u03b3\nht : bt \u2208 upperBounds t\n\u22a2 BddAbove (s \u222a t)\n[PROOFSTEP]\nuse bs \u2294 bt\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ninst\u271d : SemilatticeSup \u03b3\ns t : Set \u03b3\nbs : \u03b3\nhs : bs \u2208 upperBounds s\nbt : \u03b3\nht : bt \u2208 upperBounds t\n\u22a2 bs \u2294 bt \u2208 upperBounds (s \u222a t)\n[PROOFSTEP]\nrw [upperBounds_union]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ninst\u271d : SemilatticeSup \u03b3\ns t : Set \u03b3\nbs : \u03b3\nhs : bs \u2208 upperBounds s\nbt : \u03b3\nht : bt \u2208 upperBounds t\n\u22a2 bs \u2294 bt \u2208 upperBounds s \u2229 upperBounds t\n[PROOFSTEP]\nexact \u27e8upperBounds_mono_mem le_sup_left hs, upperBounds_mono_mem le_sup_right ht\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t\u271d : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : LinearOrder \u03b3\na b : \u03b3\ns t : Set \u03b3\nha : IsLeast s a\nhb : IsLeast t b\n\u22a2 min a b \u2208 s \u222a t\n[PROOFSTEP]\ncases' le_total a b with h h\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t\u271d : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : LinearOrder \u03b3\na b : \u03b3\ns t : Set \u03b3\nha : IsLeast s a\nhb : IsLeast t b\nh : a \u2264 b\n\u22a2 min a b \u2208 s \u222a t\n[PROOFSTEP]\nsimp [h, ha.1, hb.1]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t\u271d : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : LinearOrder \u03b3\na b : \u03b3\ns t : Set \u03b3\nha : IsLeast s a\nhb : IsLeast t b\nh : b \u2264 a\n\u22a2 min a b \u2208 s \u222a t\n[PROOFSTEP]\nsimp [h, ha.1, hb.1]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t\u271d : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : LinearOrder \u03b3\na b : \u03b3\ns t : Set \u03b3\nha : IsGreatest s a\nhb : IsGreatest t b\n\u22a2 max a b \u2208 s \u222a t\n[PROOFSTEP]\ncases' le_total a b with h h\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t\u271d : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : LinearOrder \u03b3\na b : \u03b3\ns t : Set \u03b3\nha : IsGreatest s a\nhb : IsGreatest t b\nh : a \u2264 b\n\u22a2 max a b \u2208 s \u222a t\n[PROOFSTEP]\nsimp [h, ha.1, hb.1]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t\u271d : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : LinearOrder \u03b3\na b : \u03b3\ns t : Set \u03b3\nha : IsGreatest s a\nhb : IsGreatest t b\nh : b \u2264 a\n\u22a2 max a b \u2208 s \u222a t\n[PROOFSTEP]\nsimp [h, ha.1, hb.1]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\na b : \u03b1\ninst\u271d : SemilatticeSup \u03b3\ns : Set \u03b3\nx\u2080 : \u03b3\n\u22a2 BddAbove s \u2194 \u2203 x, x\u2080 \u2264 x \u2227 \u2200 (y : \u03b3), y \u2208 s \u2192 y \u2264 x\n[PROOFSTEP]\nrw [bddAbove_def, exists_ge_and_iff_exists]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\na b : \u03b1\ninst\u271d : SemilatticeSup \u03b3\ns : Set \u03b3\nx\u2080 : \u03b3\n\u22a2 Monotone fun x => \u2200 (y : \u03b3), y \u2208 s \u2192 y \u2264 x\n[PROOFSTEP]\nexact Monotone.ball fun x _ => monotone_le\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : PartialOrder \u03b3\nj i : \u03b3\nhj : IsLUB (Iio i) j\n\u22a2 j = i \u2228 Iio i = Iic j\n[PROOFSTEP]\ncases' eq_or_lt_of_le (lub_Iio_le i hj) with hj_eq_i hj_lt_i\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : PartialOrder \u03b3\nj i : \u03b3\nhj : IsLUB (Iio i) j\nhj_eq_i : j = i\n\u22a2 j = i \u2228 Iio i = Iic j\n[PROOFSTEP]\nexact Or.inl hj_eq_i\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : PartialOrder \u03b3\nj i : \u03b3\nhj : IsLUB (Iio i) j\nhj_lt_i : j < i\n\u22a2 j = i \u2228 Iio i = Iic j\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : PartialOrder \u03b3\nj i : \u03b3\nhj : IsLUB (Iio i) j\nhj_lt_i : j < i\n\u22a2 Iio i = Iic j\n[PROOFSTEP]\nexact Set.ext fun k => \u27e8fun hk_lt => hj.1 hk_lt, fun hk_le_j => lt_of_le_of_lt hk_le_j hj_lt_i\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : LinearOrder \u03b3\ni : \u03b3\n\u22a2 \u2203 j, IsLUB (Iio i) j\n[PROOFSTEP]\nby_cases h_exists_lt : \u2203 j, j \u2208 upperBounds (Iio i) \u2227 j < i\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : LinearOrder \u03b3\ni : \u03b3\nh_exists_lt : \u2203 j, j \u2208 upperBounds (Iio i) \u2227 j < i\n\u22a2 \u2203 j, IsLUB (Iio i) j\n[PROOFSTEP]\nobtain \u27e8j, hj_ub, hj_lt_i\u27e9 := h_exists_lt\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : LinearOrder \u03b3\ni j : \u03b3\nhj_ub : j \u2208 upperBounds (Iio i)\nhj_lt_i : j < i\n\u22a2 \u2203 j, IsLUB (Iio i) j\n[PROOFSTEP]\nexact \u27e8j, hj_ub, fun k hk_ub => hk_ub hj_lt_i\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : LinearOrder \u03b3\ni : \u03b3\nh_exists_lt : \u00ac\u2203 j, j \u2208 upperBounds (Iio i) \u2227 j < i\n\u22a2 \u2203 j, IsLUB (Iio i) j\n[PROOFSTEP]\nrefine' \u27e8i, fun j hj => le_of_lt hj, _\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : LinearOrder \u03b3\ni : \u03b3\nh_exists_lt : \u00ac\u2203 j, j \u2208 upperBounds (Iio i) \u2227 j < i\n\u22a2 i \u2208 lowerBounds (upperBounds (Iio i))\n[PROOFSTEP]\nrw [mem_lowerBounds]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : LinearOrder \u03b3\ni : \u03b3\nh_exists_lt : \u00ac\u2203 j, j \u2208 upperBounds (Iio i) \u2227 j < i\n\u22a2 \u2200 (x : \u03b3), x \u2208 upperBounds (Iio i) \u2192 i \u2264 x\n[PROOFSTEP]\nby_contra h\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : LinearOrder \u03b3\ni : \u03b3\nh_exists_lt : \u00ac\u2203 j, j \u2208 upperBounds (Iio i) \u2227 j < i\nh : \u00ac\u2200 (x : \u03b3), x \u2208 upperBounds (Iio i) \u2192 i \u2264 x\n\u22a2 False\n[PROOFSTEP]\nrefine' h_exists_lt _\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : LinearOrder \u03b3\ni : \u03b3\nh_exists_lt : \u00ac\u2203 j, j \u2208 upperBounds (Iio i) \u2227 j < i\nh : \u00ac\u2200 (x : \u03b3), x \u2208 upperBounds (Iio i) \u2192 i \u2264 x\n\u22a2 \u2203 j, j \u2208 upperBounds (Iio i) \u2227 j < i\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : LinearOrder \u03b3\ni : \u03b3\nh_exists_lt : \u00ac\u2203 j, j \u2208 upperBounds (Iio i) \u2227 j < i\nh : \u2203 x, x \u2208 upperBounds (Iio i) \u2227 x < i\n\u22a2 \u2203 j, j \u2208 upperBounds (Iio i) \u2227 j < i\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ns t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b3\ninst\u271d : DenselyOrdered \u03b3\na b : \u03b3\nh : a < b\nx : \u03b3\nhx : x \u2208 lowerBounds (Ioo a b)\n\u22a2 x \u2264 a\n[PROOFSTEP]\ncases' eq_or_lt_of_le (le_sup_right : a \u2264 x \u2294 a) with h\u2081 h\u2082\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ns t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b3\ninst\u271d : DenselyOrdered \u03b3\na b : \u03b3\nh : a < b\nx : \u03b3\nhx : x \u2208 lowerBounds (Ioo a b)\nh\u2081 : a = x \u2294 a\n\u22a2 x \u2264 a\n[PROOFSTEP]\nexact h\u2081.symm \u25b8 le_sup_left\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ns t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b3\ninst\u271d : DenselyOrdered \u03b3\na b : \u03b3\nh : a < b\nx : \u03b3\nhx : x \u2208 lowerBounds (Ioo a b)\nh\u2082 : a < x \u2294 a\n\u22a2 x \u2264 a\n[PROOFSTEP]\nobtain \u27e8y, lty, ylt\u27e9 := exists_between h\u2082\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ns t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b3\ninst\u271d : DenselyOrdered \u03b3\na b : \u03b3\nh : a < b\nx : \u03b3\nhx : x \u2208 lowerBounds (Ioo a b)\nh\u2082 : a < x \u2294 a\ny : \u03b3\nlty : a < y\nylt : y < x \u2294 a\n\u22a2 x \u2264 a\n[PROOFSTEP]\napply (not_lt_of_le (sup_le (hx \u27e8lty, ylt.trans_le (sup_le _ h.le)\u27e9) lty.le) ylt).elim\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ns t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b3\ninst\u271d : DenselyOrdered \u03b3\na b : \u03b3\nh : a < b\nx : \u03b3\nhx : x \u2208 lowerBounds (Ioo a b)\nh\u2082 : a < x \u2294 a\ny : \u03b3\nlty : a < y\nylt : y < x \u2294 a\n\u22a2 x \u2264 b\n[PROOFSTEP]\nobtain \u27e8u, au, ub\u27e9 := exists_between h\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ns t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b3\ninst\u271d : DenselyOrdered \u03b3\na b : \u03b3\nh : a < b\nx : \u03b3\nhx : x \u2208 lowerBounds (Ioo a b)\nh\u2082 : a < x \u2294 a\ny : \u03b3\nlty : a < y\nylt : y < x \u2294 a\nu : \u03b3\nau : a < u\nub : u < b\n\u22a2 x \u2264 b\n[PROOFSTEP]\napply (hx \u27e8au, ub\u27e9).trans ub.le\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ns t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d\u00b9 : SemilatticeInf \u03b3\ninst\u271d : DenselyOrdered \u03b3\na b : \u03b3\nhab : a < b\n\u22a2 IsLUB (Ioo a b) b\n[PROOFSTEP]\nsimpa only [dual_Ioo] using isGLB_Ioo hab.dual\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ns t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d\u00b9 : SemilatticeInf \u03b3\ninst\u271d : DenselyOrdered \u03b3\na b : \u03b3\nhab : a < b\n\u22a2 IsLUB (Ico a b) b\n[PROOFSTEP]\nsimpa only [dual_Ioc] using isGLB_Ioc hab.dual\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\n\u22a2 BddBelow s \u2227 BddAbove s \u2194 \u2203 a b, s \u2286 Icc a b\n[PROOFSTEP]\nsimp [Ici_inter_Iic.symm, subset_inter_iff, bddBelow_iff_subset_Ici, bddAbove_iff_subset_Iic, exists_and_left,\n exists_and_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\n\u22a2 IsGreatest univ a \u2194 IsTop a\n[PROOFSTEP]\nsimp [IsGreatest, mem_upperBounds, IsTop]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b3\ninst\u271d : OrderTop \u03b3\n\u22a2 upperBounds univ = {\u22a4}\n[PROOFSTEP]\nrw [isGreatest_univ.upperBounds_eq, Ici_top]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : NoMaxOrder \u03b1\n\u22a2 \u00acBddAbove univ\n[PROOFSTEP]\nsimp [BddAbove]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\n\u22a2 upperBounds \u2205 = univ\n[PROOFSTEP]\nsimp only [upperBounds, eq_univ_iff_forall, mem_setOf_eq, ball_empty_iff, forall_true_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : Nonempty \u03b1\n\u22a2 BddAbove \u2205\n[PROOFSTEP]\nsimp only [BddAbove, upperBounds_empty, univ_nonempty]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : Nonempty \u03b1\n\u22a2 BddBelow \u2205\n[PROOFSTEP]\nsimp only [BddBelow, lowerBounds_empty, univ_nonempty]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\n\u22a2 IsGLB \u2205 a \u2194 IsTop a\n[PROOFSTEP]\nsimp [IsGLB]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : NoMinOrder \u03b1\nhs : IsLUB s a\na' : \u03b1\nha' : a' < a\nh : s = \u2205\n\u22a2 a' \u2208 upperBounds s\n[PROOFSTEP]\nrw [h, upperBounds_empty]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : NoMinOrder \u03b1\nhs : IsLUB s a\na' : \u03b1\nha' : a' < a\nh : s = \u2205\n\u22a2 a' \u2208 univ\n[PROOFSTEP]\nexact mem_univ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\na\u271d b : \u03b1\ninst\u271d : SemilatticeSup \u03b3\na : \u03b3\ns : Set \u03b3\n\u22a2 BddAbove (insert a s) \u2194 BddAbove s\n[PROOFSTEP]\nsimp only [insert_eq, bddAbove_union, bddAbove_singleton, true_and_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\na\u271d b : \u03b1\ninst\u271d : SemilatticeInf \u03b3\na : \u03b3\ns : Set \u03b3\n\u22a2 BddBelow (insert a s) \u2194 BddBelow s\n[PROOFSTEP]\nsimp only [insert_eq, bddBelow_union, bddBelow_singleton, true_and_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : SemilatticeSup \u03b3\na b : \u03b3\ns : Set \u03b3\nhs : IsLUB s b\n\u22a2 IsLUB (insert a s) (a \u2294 b)\n[PROOFSTEP]\nrw [insert_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : SemilatticeSup \u03b3\na b : \u03b3\ns : Set \u03b3\nhs : IsLUB s b\n\u22a2 IsLUB ({a} \u222a s) (a \u2294 b)\n[PROOFSTEP]\nexact isLUB_singleton.union hs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : SemilatticeInf \u03b3\na b : \u03b3\ns : Set \u03b3\nhs : IsGLB s b\n\u22a2 IsGLB (insert a s) (a \u2293 b)\n[PROOFSTEP]\nrw [insert_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : SemilatticeInf \u03b3\na b : \u03b3\ns : Set \u03b3\nhs : IsGLB s b\n\u22a2 IsGLB ({a} \u222a s) (a \u2293 b)\n[PROOFSTEP]\nexact isGLB_singleton.union hs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : LinearOrder \u03b3\na b : \u03b3\ns : Set \u03b3\nhs : IsGreatest s b\n\u22a2 IsGreatest (insert a s) (max a b)\n[PROOFSTEP]\nrw [insert_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : LinearOrder \u03b3\na b : \u03b3\ns : Set \u03b3\nhs : IsGreatest s b\n\u22a2 IsGreatest ({a} \u222a s) (max a b)\n[PROOFSTEP]\nexact isGreatest_singleton.union hs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : LinearOrder \u03b3\na b : \u03b3\ns : Set \u03b3\nhs : IsLeast s b\n\u22a2 IsLeast (insert a s) (min a b)\n[PROOFSTEP]\nrw [insert_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : LinearOrder \u03b3\na b : \u03b3\ns : Set \u03b3\nhs : IsLeast s b\n\u22a2 IsLeast ({a} \u222a s) (min a b)\n[PROOFSTEP]\nexact isLeast_singleton.union hs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns\u271d t : Set \u03b1\na\u271d b a : \u03b1\ns : Set \u03b1\n\u22a2 upperBounds (insert a s) = Ici a \u2229 upperBounds s\n[PROOFSTEP]\nrw [insert_eq, upperBounds_union, upperBounds_singleton]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns\u271d t : Set \u03b1\na\u271d b a : \u03b1\ns : Set \u03b1\n\u22a2 lowerBounds (insert a s) = Iic a \u2229 lowerBounds s\n[PROOFSTEP]\nrw [insert_eq, lowerBounds_union, lowerBounds_singleton]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na b : \u03b1\nh : IsLUB s a\n\u22a2 b < a \u2194 \u2203 c, c \u2208 s \u2227 b < c\n[PROOFSTEP]\nsimp_rw [\u2190 not_le, isLUB_le_iff h, mem_upperBounds, not_forall, not_le, exists_prop]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nHf : MonotoneOn f t\na : \u03b1\nHst\u271d Hst : s \u2286 t\n\u22a2 f '' (upperBounds s \u2229 t) \u2286 upperBounds (f '' s)\n[PROOFSTEP]\nrintro _ \u27e8a, ha, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nHf : MonotoneOn f t\na\u271d : \u03b1\nHst\u271d Hst : s \u2286 t\na : \u03b1\nha : a \u2208 upperBounds s \u2229 t\n\u22a2 f a \u2208 upperBounds (f '' s)\n[PROOFSTEP]\nexact Hf.mem_upperBounds_image Hst ha.1 ha.2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nHf : Monotone f\na : \u03b1\ns : Set \u03b1\n\u22a2 f '' upperBounds s \u2286 upperBounds (f '' s)\n[PROOFSTEP]\nrintro _ \u27e8a, ha, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nHf : Monotone f\na\u271d : \u03b1\ns : Set \u03b1\na : \u03b1\nha : a \u2208 upperBounds s\n\u22a2 f a \u2208 upperBounds (f '' s)\n[PROOFSTEP]\nexact Hf.mem_upperBounds_image ha\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\n\u22a2 image2 f (upperBounds s) (upperBounds t) \u2286 upperBounds (image2 f s t)\n[PROOFSTEP]\nrintro _ \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\na : \u03b1\nb : \u03b2\nha : a \u2208 upperBounds s\nhb : b \u2208 upperBounds t\n\u22a2 f a b \u2208 upperBounds (image2 f s t)\n[PROOFSTEP]\nexact mem_upperBounds_image2 h\u2080 h\u2081 ha hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\n\u22a2 image2 f (lowerBounds s) (lowerBounds t) \u2286 lowerBounds (image2 f s t)\n[PROOFSTEP]\nrintro _ \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\na : \u03b1\nb : \u03b2\nha : a \u2208 lowerBounds s\nhb : b \u2208 lowerBounds t\n\u22a2 f a b \u2208 lowerBounds (image2 f s t)\n[PROOFSTEP]\nexact mem_lowerBounds_image2 h\u2080 h\u2081 ha hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\n\u22a2 BddAbove s \u2192 BddAbove t \u2192 BddAbove (Set.image2 f s t)\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\na : \u03b1\nha : a \u2208 upperBounds s\nb : \u03b2\nhb : b \u2208 upperBounds t\n\u22a2 BddAbove (Set.image2 f s t)\n[PROOFSTEP]\nexact \u27e8f a b, mem_upperBounds_image2 h\u2080 h\u2081 ha hb\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\n\u22a2 BddBelow s \u2192 BddBelow t \u2192 BddBelow (Set.image2 f s t)\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\na : \u03b1\nha : a \u2208 lowerBounds s\nb : \u03b2\nhb : b \u2208 lowerBounds t\n\u22a2 BddBelow (Set.image2 f s t)\n[PROOFSTEP]\nexact \u27e8f a b, mem_lowerBounds_image2 h\u2080 h\u2081 ha hb\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\n\u22a2 image2 f (upperBounds s) (lowerBounds t) \u2286 upperBounds (image2 f s t)\n[PROOFSTEP]\nrintro _ \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\na : \u03b1\nb : \u03b2\nha : a \u2208 upperBounds s\nhb : b \u2208 lowerBounds t\n\u22a2 f a b \u2208 upperBounds (image2 f s t)\n[PROOFSTEP]\nexact mem_upperBounds_image2_of_mem_upperBounds_of_mem_lowerBounds h\u2080 h\u2081 ha hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\n\u22a2 image2 f (lowerBounds s) (upperBounds t) \u2286 lowerBounds (image2 f s t)\n[PROOFSTEP]\nrintro _ \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\na : \u03b1\nb : \u03b2\nha : a \u2208 lowerBounds s\nhb : b \u2208 upperBounds t\n\u22a2 f a b \u2208 lowerBounds (image2 f s t)\n[PROOFSTEP]\nexact mem_lowerBounds_image2_of_mem_lowerBounds_of_mem_upperBounds h\u2080 h\u2081 ha hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\n\u22a2 BddAbove s \u2192 BddBelow t \u2192 BddAbove (Set.image2 f s t)\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\na : \u03b1\nha : a \u2208 upperBounds s\nb : \u03b2\nhb : b \u2208 lowerBounds t\n\u22a2 BddAbove (Set.image2 f s t)\n[PROOFSTEP]\nexact \u27e8f a b, mem_upperBounds_image2_of_mem_upperBounds_of_mem_lowerBounds h\u2080 h\u2081 ha hb\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\n\u22a2 BddBelow s \u2192 BddAbove t \u2192 BddBelow (Set.image2 f s t)\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\na : \u03b1\nha : a \u2208 lowerBounds s\nb : \u03b2\nhb : b \u2208 upperBounds t\n\u22a2 BddBelow (Set.image2 f s t)\n[PROOFSTEP]\nexact \u27e8f a b, mem_lowerBounds_image2_of_mem_lowerBounds_of_mem_upperBounds h\u2080 h\u2081 ha hb\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\n\u22a2 image2 f (lowerBounds s) (lowerBounds t) \u2286 upperBounds (image2 f s t)\n[PROOFSTEP]\nrintro _ \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\na : \u03b1\nb : \u03b2\nha : a \u2208 lowerBounds s\nhb : b \u2208 lowerBounds t\n\u22a2 f a b \u2208 upperBounds (image2 f s t)\n[PROOFSTEP]\nexact mem_upperBounds_image2_of_mem_lowerBounds h\u2080 h\u2081 ha hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\n\u22a2 image2 f (upperBounds s) (upperBounds t) \u2286 lowerBounds (image2 f s t)\n[PROOFSTEP]\nrintro _ \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\na : \u03b1\nb : \u03b2\nha : a \u2208 upperBounds s\nhb : b \u2208 upperBounds t\n\u22a2 f a b \u2208 lowerBounds (image2 f s t)\n[PROOFSTEP]\nexact mem_lowerBounds_image2_of_mem_upperBounds h\u2080 h\u2081 ha hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\n\u22a2 BddBelow s \u2192 BddBelow t \u2192 BddAbove (Set.image2 f s t)\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\na : \u03b1\nha : a \u2208 lowerBounds s\nb : \u03b2\nhb : b \u2208 lowerBounds t\n\u22a2 BddAbove (Set.image2 f s t)\n[PROOFSTEP]\nexact \u27e8f a b, mem_upperBounds_image2_of_mem_lowerBounds h\u2080 h\u2081 ha hb\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\n\u22a2 BddAbove s \u2192 BddAbove t \u2192 BddBelow (Set.image2 f s t)\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Antitone (f a)\na : \u03b1\nha : a \u2208 upperBounds s\nb : \u03b2\nhb : b \u2208 upperBounds t\n\u22a2 BddBelow (Set.image2 f s t)\n[PROOFSTEP]\nexact \u27e8f a b, mem_lowerBounds_image2_of_mem_upperBounds h\u2080 h\u2081 ha hb\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\n\u22a2 image2 f (lowerBounds s) (upperBounds t) \u2286 upperBounds (image2 f s t)\n[PROOFSTEP]\nrintro _ \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\na : \u03b1\nb : \u03b2\nha : a \u2208 lowerBounds s\nhb : b \u2208 upperBounds t\n\u22a2 f a b \u2208 upperBounds (image2 f s t)\n[PROOFSTEP]\nexact mem_upperBounds_image2_of_mem_upperBounds_of_mem_upperBounds h\u2080 h\u2081 ha hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\n\u22a2 image2 f (upperBounds s) (lowerBounds t) \u2286 lowerBounds (image2 f s t)\n[PROOFSTEP]\nrintro _ \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\na : \u03b1\nb : \u03b2\nha : a \u2208 upperBounds s\nhb : b \u2208 lowerBounds t\n\u22a2 f a b \u2208 lowerBounds (image2 f s t)\n[PROOFSTEP]\nexact mem_lowerBounds_image2_of_mem_lowerBounds_of_mem_lowerBounds h\u2080 h\u2081 ha hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\n\u22a2 BddBelow s \u2192 BddAbove t \u2192 BddAbove (Set.image2 f s t)\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\na : \u03b1\nha : a \u2208 lowerBounds s\nb : \u03b2\nhb : b \u2208 upperBounds t\n\u22a2 BddAbove (Set.image2 f s t)\n[PROOFSTEP]\nexact \u27e8f a b, mem_upperBounds_image2_of_mem_upperBounds_of_mem_upperBounds h\u2080 h\u2081 ha hb\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\n\u22a2 BddAbove s \u2192 BddBelow t \u2192 BddBelow (Set.image2 f s t)\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Antitone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\na : \u03b1\nha : a \u2208 upperBounds s\nb : \u03b2\nhb : b \u2208 lowerBounds t\n\u22a2 BddBelow (Set.image2 f s t)\n[PROOFSTEP]\nexact \u27e8f a b, mem_lowerBounds_image2_of_mem_lowerBounds_of_mem_lowerBounds h\u2080 h\u2081 ha hb\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03c0 : \u03b1 \u2192 Type u_1\ninst\u271d : (a : \u03b1) \u2192 Preorder (\u03c0 a)\nF : \u03b9 \u2192 (a : \u03b1) \u2192 \u03c0 a\n\u22a2 BddAbove (range F) \u2194 \u2200 (a : \u03b1), BddAbove (range fun i => F i a)\n[PROOFSTEP]\nsimp only [bddAbove_pi, \u2190 range_comp]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03c0 : \u03b1 \u2192 Type u_1\ninst\u271d : (a : \u03b1) \u2192 Preorder (\u03c0 a)\nF : \u03b9 \u2192 (a : \u03b1) \u2192 \u03c0 a\n\u22a2 (\u2200 (a : \u03b1), BddAbove (range (eval a \u2218 F))) \u2194 \u2200 (a : \u03b1), BddAbove (range fun i => F i a)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03c0 : \u03b1 \u2192 Type u_1\ninst\u271d : (a : \u03b1) \u2192 Preorder (\u03c0 a)\ns : Set ((a : \u03b1) \u2192 \u03c0 a)\nf : (a : \u03b1) \u2192 \u03c0 a\n\u22a2 IsLUB s f \u2194 \u2200 (a : \u03b1), IsLUB (eval a '' s) (f a)\n[PROOFSTEP]\nclassical\nrefine' \u27e8fun H a => \u27e8(Function.monotone_eval a).mem_upperBounds_image H.1, fun b hb => _\u27e9, fun H => \u27e8_, _\u27e9\u27e9\n\u00b7 suffices h : Function.update f a b \u2208 upperBounds s from Function.update_same a b f \u25b8 H.2 h a\n refine' fun g hg => le_update_iff.2 \u27e8hb <| mem_image_of_mem _ hg, fun i _ => H.1 hg i\u27e9\n\u00b7 exact fun g hg a => (H a).1 (mem_image_of_mem _ hg)\n\u00b7 exact fun g hg a => (H a).2 ((Function.monotone_eval a).mem_upperBounds_image hg)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03c0 : \u03b1 \u2192 Type u_1\ninst\u271d : (a : \u03b1) \u2192 Preorder (\u03c0 a)\ns : Set ((a : \u03b1) \u2192 \u03c0 a)\nf : (a : \u03b1) \u2192 \u03c0 a\n\u22a2 IsLUB s f \u2194 \u2200 (a : \u03b1), IsLUB (eval a '' s) (f a)\n[PROOFSTEP]\nrefine' \u27e8fun H a => \u27e8(Function.monotone_eval a).mem_upperBounds_image H.1, fun b hb => _\u27e9, fun H => \u27e8_, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03c0 : \u03b1 \u2192 Type u_1\ninst\u271d : (a : \u03b1) \u2192 Preorder (\u03c0 a)\ns : Set ((a : \u03b1) \u2192 \u03c0 a)\nf : (a : \u03b1) \u2192 \u03c0 a\nH : IsLUB s f\na : \u03b1\nb : \u03c0 a\nhb : b \u2208 upperBounds (eval a '' s)\n\u22a2 f a \u2264 b\n[PROOFSTEP]\nsuffices h : Function.update f a b \u2208 upperBounds s from Function.update_same a b f \u25b8 H.2 h a\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03c0 : \u03b1 \u2192 Type u_1\ninst\u271d : (a : \u03b1) \u2192 Preorder (\u03c0 a)\ns : Set ((a : \u03b1) \u2192 \u03c0 a)\nf : (a : \u03b1) \u2192 \u03c0 a\nH : IsLUB s f\na : \u03b1\nb : \u03c0 a\nhb : b \u2208 upperBounds (eval a '' s)\n\u22a2 update f a b \u2208 upperBounds s\n[PROOFSTEP]\nrefine' fun g hg => le_update_iff.2 \u27e8hb <| mem_image_of_mem _ hg, fun i _ => H.1 hg i\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03c0 : \u03b1 \u2192 Type u_1\ninst\u271d : (a : \u03b1) \u2192 Preorder (\u03c0 a)\ns : Set ((a : \u03b1) \u2192 \u03c0 a)\nf : (a : \u03b1) \u2192 \u03c0 a\nH : \u2200 (a : \u03b1), IsLUB (eval a '' s) (f a)\n\u22a2 f \u2208 upperBounds s\n[PROOFSTEP]\nexact fun g hg a => (H a).1 (mem_image_of_mem _ hg)\n[GOAL]\ncase refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03c0 : \u03b1 \u2192 Type u_1\ninst\u271d : (a : \u03b1) \u2192 Preorder (\u03c0 a)\ns : Set ((a : \u03b1) \u2192 \u03c0 a)\nf : (a : \u03b1) \u2192 \u03c0 a\nH : \u2200 (a : \u03b1), IsLUB (eval a '' s) (f a)\n\u22a2 f \u2208 lowerBounds (upperBounds s)\n[PROOFSTEP]\nexact fun g hg a => (H a).2 ((Function.monotone_eval a).mem_upperBounds_image hg)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set (\u03b1 \u00d7 \u03b2)\np : \u03b1 \u00d7 \u03b2\n\u22a2 IsLUB s p \u2194 IsLUB (Prod.fst '' s) p.fst \u2227 IsLUB (Prod.snd '' s) p.snd\n[PROOFSTEP]\nrefine'\n \u27e8fun H =>\n \u27e8\u27e8monotone_fst.mem_upperBounds_image H.1, fun a ha => _\u27e9, \u27e8monotone_snd.mem_upperBounds_image H.1, fun a ha => _\u27e9\u27e9,\n fun H => \u27e8_, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set (\u03b1 \u00d7 \u03b2)\np : \u03b1 \u00d7 \u03b2\nH : IsLUB s p\na : \u03b1\nha : a \u2208 upperBounds (Prod.fst '' s)\n\u22a2 p.fst \u2264 a\n[PROOFSTEP]\nsuffices h : (a, p.2) \u2208 upperBounds s from (H.2 h).1\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set (\u03b1 \u00d7 \u03b2)\np : \u03b1 \u00d7 \u03b2\nH : IsLUB s p\na : \u03b1\nha : a \u2208 upperBounds (Prod.fst '' s)\n\u22a2 (a, p.snd) \u2208 upperBounds s\n[PROOFSTEP]\nexact fun q hq => \u27e8ha <| mem_image_of_mem _ hq, (H.1 hq).2\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set (\u03b1 \u00d7 \u03b2)\np : \u03b1 \u00d7 \u03b2\nH : IsLUB s p\na : \u03b2\nha : a \u2208 upperBounds (Prod.snd '' s)\n\u22a2 p.snd \u2264 a\n[PROOFSTEP]\nsuffices h : (p.1, a) \u2208 upperBounds s from (H.2 h).2\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set (\u03b1 \u00d7 \u03b2)\np : \u03b1 \u00d7 \u03b2\nH : IsLUB s p\na : \u03b2\nha : a \u2208 upperBounds (Prod.snd '' s)\n\u22a2 (p.fst, a) \u2208 upperBounds s\n[PROOFSTEP]\nexact fun q hq => \u27e8(H.1 hq).1, ha <| mem_image_of_mem _ hq\u27e9\n[GOAL]\ncase refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set (\u03b1 \u00d7 \u03b2)\np : \u03b1 \u00d7 \u03b2\nH : IsLUB (Prod.fst '' s) p.fst \u2227 IsLUB (Prod.snd '' s) p.snd\n\u22a2 p \u2208 upperBounds s\n[PROOFSTEP]\nexact fun q hq => \u27e8H.1.1 <| mem_image_of_mem _ hq, H.2.1 <| mem_image_of_mem _ hq\u27e9\n[GOAL]\ncase refine'_4\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set (\u03b1 \u00d7 \u03b2)\np : \u03b1 \u00d7 \u03b2\nH : IsLUB (Prod.fst '' s) p.fst \u2227 IsLUB (Prod.snd '' s) p.snd\n\u22a2 p \u2208 lowerBounds (upperBounds s)\n[PROOFSTEP]\nexact fun q hq => \u27e8H.1.2 <| monotone_fst.mem_upperBounds_image hq, H.2.2 <| monotone_snd.mem_upperBounds_image hq\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Order.Bounds.Basic", "llama_tokens": 17864, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185944046238981, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.44199869132595804}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX Y Z : Bipointed\nf : Bipointed.Hom X Y\ng : Bipointed.Hom Y Z\n\u22a2 (g.toFun \u2218 f.toFun) X.toProd.fst = Z.toProd.fst\n[PROOFSTEP]\nrw [Function.comp_apply, f.map_fst, g.map_fst]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX Y Z : Bipointed\nf : Bipointed.Hom X Y\ng : Bipointed.Hom Y Z\n\u22a2 (g.toFun \u2218 f.toFun) X.toProd.snd = Z.toProd.snd\n[PROOFSTEP]\nrw [Function.comp_apply, f.map_snd, g.map_snd]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX : Pointed\nY : Bipointed\nf : pointedToBipointedFst.obj X \u27f6 Y\n\u22a2 (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point := (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n f) =\n f\n[PROOFSTEP]\napply Bipointed.Hom.ext\n[GOAL]\ncase toFun\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX : Pointed\nY : Bipointed\nf : pointedToBipointedFst.obj X \u27f6 Y\n\u22a2 ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point := (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n f)).toFun =\n f.toFun\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase toFun.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX : Pointed\nY : Bipointed\nf : pointedToBipointedFst.obj X \u27f6 Y\nx : (pointedToBipointedFst.obj X).X\n\u22a2 Bipointed.Hom.toFun\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point := (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n f))\n x =\n Bipointed.Hom.toFun f x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase toFun.h.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX : Pointed\nY : Bipointed\nf : pointedToBipointedFst.obj X \u27f6 Y\n\u22a2 Bipointed.Hom.toFun\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point := (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n f))\n none =\n Bipointed.Hom.toFun f none\n[PROOFSTEP]\nexact f.map_snd.symm\n[GOAL]\ncase toFun.h.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX : Pointed\nY : Bipointed\nf : pointedToBipointedFst.obj X \u27f6 Y\nval\u271d : X.X\n\u22a2 Bipointed.Hom.toFun\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point := (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n f))\n (some val\u271d) =\n Bipointed.Hom.toFun f (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX'\u271d X\u271d : Pointed\nY\u271d : Bipointed\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 bipointedToPointedFst.obj Y\u271d\n\u22a2 \u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point := (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedFst.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedFst.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n f) =\n f) })\n X'\u271d Y\u271d).symm\n (f \u226b g) =\n pointedToBipointedFst.map f \u226b\n \u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedFst.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedFst.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n f) =\n f) })\n X\u271d Y\u271d).symm\n g\n[PROOFSTEP]\napply Bipointed.Hom.ext\n[GOAL]\ncase toFun\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX'\u271d X\u271d : Pointed\nY\u271d : Bipointed\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 bipointedToPointedFst.obj Y\u271d\n\u22a2 (\u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedFst.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedFst.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n f) =\n f) })\n X'\u271d Y\u271d).symm\n (f \u226b g)).toFun =\n (pointedToBipointedFst.map f \u226b\n \u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedFst.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst =\n Y.toProd.fst) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedFst.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n f) =\n f) })\n X\u271d Y\u271d).symm\n g).toFun\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase toFun.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX'\u271d X\u271d : Pointed\nY\u271d : Bipointed\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 bipointedToPointedFst.obj Y\u271d\nx : (pointedToBipointedFst.obj X'\u271d).X\n\u22a2 Bipointed.Hom.toFun\n (\u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedFst.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedFst.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n f) =\n f) })\n X'\u271d Y\u271d).symm\n (f \u226b g))\n x =\n Bipointed.Hom.toFun\n (pointedToBipointedFst.map f \u226b\n \u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedFst.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst =\n Y.toProd.fst) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedFst.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n f) =\n f) })\n X\u271d Y\u271d).symm\n g)\n x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase toFun.h.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX'\u271d X\u271d : Pointed\nY\u271d : Bipointed\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 bipointedToPointedFst.obj Y\u271d\n\u22a2 Bipointed.Hom.toFun\n (\u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedFst.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedFst.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n f) =\n f) })\n X'\u271d Y\u271d).symm\n (f \u226b g))\n none =\n Bipointed.Hom.toFun\n (pointedToBipointedFst.map f \u226b\n \u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedFst.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst =\n Y.toProd.fst) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedFst.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n f) =\n f) })\n X\u271d Y\u271d).symm\n g)\n none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase toFun.h.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX'\u271d X\u271d : Pointed\nY\u271d : Bipointed\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 bipointedToPointedFst.obj Y\u271d\nval\u271d : X'\u271d.X\n\u22a2 Bipointed.Hom.toFun\n (\u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedFst.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedFst.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n f) =\n f) })\n X'\u271d Y\u271d).symm\n (f \u226b g))\n (some val\u271d) =\n Bipointed.Hom.toFun\n (pointedToBipointedFst.map f \u226b\n \u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun) (pointedToBipointedFst.obj X).toProd.snd) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedFst.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst =\n Y.toProd.fst) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedFst.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedFst.obj X).toProd.fst = Y.toProd.fst) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.snd f.toFun,\n map_fst := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedFst.obj Y).point),\n map_snd :=\n (_ :\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd =\n (fun o => Option.elim o Y.toProd.snd f.toFun)\n (pointedToBipointedFst.obj X).toProd.snd) })\n f) =\n f) })\n X\u271d Y\u271d).symm\n g)\n (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX : Pointed\nY : Bipointed\nf : pointedToBipointedSnd.obj X \u27f6 Y\n\u22a2 (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point := (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n f) =\n f\n[PROOFSTEP]\napply Bipointed.Hom.ext\n[GOAL]\ncase toFun\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX : Pointed\nY : Bipointed\nf : pointedToBipointedSnd.obj X \u27f6 Y\n\u22a2 ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point := (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n f)).toFun =\n f.toFun\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase toFun.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX : Pointed\nY : Bipointed\nf : pointedToBipointedSnd.obj X \u27f6 Y\nx : (pointedToBipointedSnd.obj X).X\n\u22a2 Bipointed.Hom.toFun\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point := (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n f))\n x =\n Bipointed.Hom.toFun f x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase toFun.h.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX : Pointed\nY : Bipointed\nf : pointedToBipointedSnd.obj X \u27f6 Y\n\u22a2 Bipointed.Hom.toFun\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point := (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n f))\n none =\n Bipointed.Hom.toFun f none\n[PROOFSTEP]\nexact f.map_fst.symm\n[GOAL]\ncase toFun.h.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX : Pointed\nY : Bipointed\nf : pointedToBipointedSnd.obj X \u27f6 Y\nval\u271d : X.X\n\u22a2 Bipointed.Hom.toFun\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point := (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n f))\n (some val\u271d) =\n Bipointed.Hom.toFun f (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX'\u271d X\u271d : Pointed\nY\u271d : Bipointed\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 bipointedToPointedSnd.obj Y\u271d\n\u22a2 \u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point := (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedSnd.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedSnd.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n f) =\n f) })\n X'\u271d Y\u271d).symm\n (f \u226b g) =\n pointedToBipointedSnd.map f \u226b\n \u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedSnd.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedSnd.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n f) =\n f) })\n X\u271d Y\u271d).symm\n g\n[PROOFSTEP]\napply Bipointed.Hom.ext\n[GOAL]\ncase toFun\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX'\u271d X\u271d : Pointed\nY\u271d : Bipointed\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 bipointedToPointedSnd.obj Y\u271d\n\u22a2 (\u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedSnd.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedSnd.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n f) =\n f) })\n X'\u271d Y\u271d).symm\n (f \u226b g)).toFun =\n (pointedToBipointedSnd.map f \u226b\n \u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedSnd.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd =\n Y.toProd.snd) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedSnd.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd :=\n (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n f) =\n f) })\n X\u271d Y\u271d).symm\n g).toFun\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase toFun.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX'\u271d X\u271d : Pointed\nY\u271d : Bipointed\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 bipointedToPointedSnd.obj Y\u271d\nx : (pointedToBipointedSnd.obj X'\u271d).X\n\u22a2 Bipointed.Hom.toFun\n (\u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedSnd.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedSnd.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n f) =\n f) })\n X'\u271d Y\u271d).symm\n (f \u226b g))\n x =\n Bipointed.Hom.toFun\n (pointedToBipointedSnd.map f \u226b\n \u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedSnd.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd =\n Y.toProd.snd) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedSnd.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd :=\n (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n f) =\n f) })\n X\u271d Y\u271d).symm\n g)\n x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase toFun.h.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX'\u271d X\u271d : Pointed\nY\u271d : Bipointed\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 bipointedToPointedSnd.obj Y\u271d\n\u22a2 Bipointed.Hom.toFun\n (\u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedSnd.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedSnd.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n f) =\n f) })\n X'\u271d Y\u271d).symm\n (f \u226b g))\n none =\n Bipointed.Hom.toFun\n (pointedToBipointedSnd.map f \u226b\n \u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedSnd.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd =\n Y.toProd.snd) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedSnd.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd :=\n (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n f) =\n f) })\n X\u271d Y\u271d).symm\n g)\n none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase toFun.h.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nX'\u271d X\u271d : Pointed\nY\u271d : Bipointed\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 bipointedToPointedSnd.obj Y\u271d\nval\u271d : X'\u271d.X\n\u22a2 Bipointed.Hom.toFun\n (\u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedSnd.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedSnd.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n f) =\n f) })\n X'\u271d Y\u271d).symm\n (f \u226b g))\n (some val\u271d) =\n Bipointed.Hom.toFun\n (pointedToBipointedSnd.map f \u226b\n \u2191((fun X Y =>\n {\n toFun := fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ : Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) },\n invFun := fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun) (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) },\n left_inv :=\n (_ :\n \u2200 (f : pointedToBipointedSnd.obj X \u27f6 Y),\n (fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd := (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n ((fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd =\n Y.toProd.snd) })\n f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 bipointedToPointedSnd.obj Y),\n (fun f =>\n { toFun := f.toFun \u2218 some,\n map_point :=\n (_ :\n Bipointed.Hom.toFun f (pointedToBipointedSnd.obj X).toProd.snd = Y.toProd.snd) })\n ((fun f =>\n { toFun := fun o => Option.elim o Y.toProd.fst f.toFun,\n map_fst :=\n (_ :\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst =\n (fun o => Option.elim o Y.toProd.fst f.toFun)\n (pointedToBipointedSnd.obj X).toProd.fst),\n map_snd :=\n (_ : Pointed.Hom.toFun f X.point = (bipointedToPointedSnd.obj Y).point) })\n f) =\n f) })\n X\u271d Y\u271d).symm\n g)\n (some val\u271d)\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Category.Bipointed", "llama_tokens": 23526, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943925708561, "lm_q2_score": 0.615087862571909, "lm_q1q2_score": 0.4419986889825671}} {"text": "[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\n\u22a2 IsJacobson R \u2194 \u2200 (P : Ideal R), IsPrime P \u2192 jacobson P = P\n[PROOFSTEP]\nrefine' isJacobson_iff.trans \u27e8fun h I hI => h I hI.isRadical, _\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\n\u22a2 (\u2200 (P : Ideal R), IsPrime P \u2192 jacobson P = P) \u2192 \u2200 (I : Ideal R), IsRadical I \u2192 jacobson I = I\n[PROOFSTEP]\nrefine fun h I hI \u21a6 le_antisymm (fun x hx \u21a6 ?_) (fun x hx \u21a6 mem_sInf.mpr fun _ hJ \u21a6 hJ.left hx)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d : Ideal R\nh : \u2200 (P : Ideal R), IsPrime P \u2192 jacobson P = P\nI : Ideal R\nhI : IsRadical I\nx : R\nhx : x \u2208 jacobson I\n\u22a2 x \u2208 I\n[PROOFSTEP]\nrw [\u2190 hI.radical, radical_eq_sInf I, mem_sInf]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d : Ideal R\nh : \u2200 (P : Ideal R), IsPrime P \u2192 jacobson P = P\nI : Ideal R\nhI : IsRadical I\nx : R\nhx : x \u2208 jacobson I\n\u22a2 \u2200 \u2983I_1 : Ideal R\u2984, I_1 \u2208 {J | I \u2264 J \u2227 IsPrime J} \u2192 x \u2208 I_1\n[PROOFSTEP]\nintro P hP\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d : Ideal R\nh : \u2200 (P : Ideal R), IsPrime P \u2192 jacobson P = P\nI : Ideal R\nhI : IsRadical I\nx : R\nhx : x \u2208 jacobson I\nP : Ideal R\nhP : P \u2208 {J | I \u2264 J \u2227 IsPrime J}\n\u22a2 x \u2208 P\n[PROOFSTEP]\nrw [Set.mem_setOf_eq] at hP \n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d : Ideal R\nh : \u2200 (P : Ideal R), IsPrime P \u2192 jacobson P = P\nI : Ideal R\nhI : IsRadical I\nx : R\nhx : x \u2208 jacobson I\nP : Ideal R\nhP : I \u2264 P \u2227 IsPrime P\n\u22a2 x \u2208 P\n[PROOFSTEP]\nerw [mem_sInf] at hx \n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d : Ideal R\nh : \u2200 (P : Ideal R), IsPrime P \u2192 jacobson P = P\nI : Ideal R\nhI : IsRadical I\nx : R\nhx : \u2200 \u2983I_1 : Ideal R\u2984, I_1 \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 x \u2208 I_1\nP : Ideal R\nhP : I \u2264 P \u2227 IsPrime P\n\u22a2 x \u2208 P\n[PROOFSTEP]\nerw [\u2190 h P hP.right, mem_sInf]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d : Ideal R\nh : \u2200 (P : Ideal R), IsPrime P \u2192 jacobson P = P\nI : Ideal R\nhI : IsRadical I\nx : R\nhx : \u2200 \u2983I_1 : Ideal R\u2984, I_1 \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 x \u2208 I_1\nP : Ideal R\nhP : I \u2264 P \u2227 IsPrime P\n\u22a2 \u2200 \u2983I : Ideal R\u2984, I \u2208 {J | P \u2264 J \u2227 IsMaximal J} \u2192 x \u2208 I\n[PROOFSTEP]\nexact fun J hJ => hx \u27e8le_trans hP.left hJ.left, hJ.right\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI\u271d : Ideal R\nK : Type u_3\ninst\u271d : Field K\nI : Ideal K\nx\u271d : IsRadical I\nh : I = \u22a4\n\u22a2 jacobson I = I\n[PROOFSTEP]\nrw [h, jacobson_eq_top_iff]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nH : IsJacobson R\n\u22a2 (\u2203 f, Function.Surjective \u2191f) \u2192 IsJacobson S\n[PROOFSTEP]\nrintro \u27e8f, hf\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nH : IsJacobson R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\n\u22a2 IsJacobson S\n[PROOFSTEP]\nrw [isJacobson_iff_sInf_maximal]\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nH : IsJacobson R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\n\u22a2 \u2200 {I : Ideal S}, IsPrime I \u2192 \u2203 M, (\u2200 (J : Ideal S), J \u2208 M \u2192 IsMaximal J \u2228 J = \u22a4) \u2227 I = sInf M\n[PROOFSTEP]\nintro p hp\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nH : IsJacobson R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\np : Ideal S\nhp : IsPrime p\n\u22a2 \u2203 M, (\u2200 (J : Ideal S), J \u2208 M \u2192 IsMaximal J \u2228 J = \u22a4) \u2227 p = sInf M\n[PROOFSTEP]\nuse map f '' {J : Ideal R | comap f p \u2264 J \u2227 J.IsMaximal}\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nH : IsJacobson R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\np : Ideal S\nhp : IsPrime p\n\u22a2 (\u2200 (J : Ideal S), J \u2208 map f '' {J | comap f p \u2264 J \u2227 IsMaximal J} \u2192 IsMaximal J \u2228 J = \u22a4) \u2227\n p = sInf (map f '' {J | comap f p \u2264 J \u2227 IsMaximal J})\n[PROOFSTEP]\nuse fun j \u27e8J, hJ, hmap\u27e9 => hmap \u25b8 (map_eq_top_or_isMaximal_of_surjective f hf hJ.right).symm\n[GOAL]\ncase right\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nH : IsJacobson R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\np : Ideal S\nhp : IsPrime p\n\u22a2 p = sInf (map f '' {J | comap f p \u2264 J \u2227 IsMaximal J})\n[PROOFSTEP]\nhave : p = map f (comap f p).jacobson :=\n (IsJacobson.out' _ <| hp.isRadical.comap f).symm \u25b8 (map_comap_of_surjective f hf p).symm\n[GOAL]\ncase right\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nH : IsJacobson R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\np : Ideal S\nhp : IsPrime p\nthis : p = map f (jacobson (comap f p))\n\u22a2 p = sInf (map f '' {J | comap f p \u2264 J \u2227 IsMaximal J})\n[PROOFSTEP]\nexact this.trans (map_sInf hf fun J \u27e8hJ, _\u27e9 => le_trans (Ideal.ker_le_comap f) hJ)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : IsJacobson R\n\u22a2 Function.Surjective \u2191(Quotient.mk I)\n[PROOFSTEP]\nrintro \u27e8x\u27e9\n[GOAL]\ncase mk\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : IsJacobson R\nb\u271d : R \u29f8 I\nx : R\n\u22a2 \u2203 a, \u2191(Quotient.mk I) a = Quot.mk Setoid.r x\n[PROOFSTEP]\nuse x\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : IsJacobson R\nb\u271d : R \u29f8 I\nx : R\n\u22a2 \u2191(Quotient.mk I) x = Quot.mk Setoid.r x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\n\u22a2 IsJacobson S\n[PROOFSTEP]\nrw [isJacobson_iff_prime_eq]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\n\u22a2 \u2200 (P : Ideal S), IsPrime P \u2192 jacobson P = P\n[PROOFSTEP]\nintro P hP\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\n\u22a2 jacobson P = P\n[PROOFSTEP]\nby_cases hP_top : comap (algebraMap R S) P = \u22a4\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : comap (algebraMap R S) P = \u22a4\n\u22a2 jacobson P = P\n[PROOFSTEP]\nsimp [comap_eq_top_iff.1 hP_top]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\n\u22a2 jacobson P = P\n[PROOFSTEP]\nhaveI : Nontrivial (R \u29f8 comap (algebraMap R S) P) := Quotient.nontrivial hP_top\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\n\u22a2 jacobson P = P\n[PROOFSTEP]\nrw [jacobson_eq_iff_jacobson_quotient_eq_bot]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nrefine' eq_bot_of_comap_eq_bot (isIntegral_quotient_of_isIntegral hRS) _\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\n\u22a2 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) (jacobson \u22a5) = \u22a5\n[PROOFSTEP]\nrw [eq_bot_iff, \u2190\n jacobson_eq_iff_jacobson_quotient_eq_bot.1\n ((isJacobson_iff_prime_eq.1 hR) (comap (algebraMap R S) P) (comap_isPrime _ _)),\n comap_jacobson]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\n\u22a2 sInf (comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) '' {J | \u22a5 \u2264 J \u2227 IsMaximal J}) \u2264 jacobson \u22a5\n[PROOFSTEP]\nrefine' sInf_le_sInf fun J hJ => _\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\nJ : Ideal (R \u29f8 comap (algebraMap R S) P)\nhJ : J \u2208 {J | \u22a5 \u2264 J \u2227 IsMaximal J}\n\u22a2 J \u2208 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) '' {J | \u22a5 \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\nsimp only [true_and_iff, Set.mem_image, bot_le, Set.mem_setOf_eq]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\nJ : Ideal (R \u29f8 comap (algebraMap R S) P)\nhJ : J \u2208 {J | \u22a5 \u2264 J \u2227 IsMaximal J}\n\u22a2 \u2203 x, IsMaximal x \u2227 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) x = J\n[PROOFSTEP]\nhave : J.IsMaximal := by simpa using hJ\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\nJ : Ideal (R \u29f8 comap (algebraMap R S) P)\nhJ : J \u2208 {J | \u22a5 \u2264 J \u2227 IsMaximal J}\n\u22a2 IsMaximal J\n[PROOFSTEP]\nsimpa using hJ\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis\u271d : Nontrivial (R \u29f8 comap (algebraMap R S) P)\nJ : Ideal (R \u29f8 comap (algebraMap R S) P)\nhJ : J \u2208 {J | \u22a5 \u2264 J \u2227 IsMaximal J}\nthis : IsMaximal J\n\u22a2 \u2203 x, IsMaximal x \u2227 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) x = J\n[PROOFSTEP]\nexact\n exists_ideal_over_maximal_of_isIntegral (isIntegral_quotient_of_isIntegral hRS) J\n (comap_bot_le_of_injective _ algebraMap_quotient_injective)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nhI : IsRadical I\n\u22a2 Disjoint \u2191(powers y) \u2191I \u2194 \u00acy \u2208 I.toAddSubmonoid\n[PROOFSTEP]\nrefine' \u27e8fun h => Set.disjoint_left.1 h (mem_powers _), fun h => disjoint_iff.mpr (eq_bot_iff.mpr _)\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nhI : IsRadical I\nh : \u00acy \u2208 I.toAddSubmonoid\n\u22a2 \u2191(powers y) \u2293 \u2191I \u2264 \u22a5\n[PROOFSTEP]\nrintro x \u27e8\u27e8n, rfl\u27e9, hx'\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nhI : IsRadical I\nh : \u00acy \u2208 I.toAddSubmonoid\nn : \u2115\nhx' : (fun x x_1 => x ^ x_1) y n \u2208 \u2191I\n\u22a2 (fun x x_1 => x ^ x_1) y n \u2208 \u22a5\n[PROOFSTEP]\nexact h (hI <| mem_radical_of_pow_mem <| le_radical hx')\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\n\u22a2 IsMaximal J \u2194 IsMaximal (comap (algebraMap R S) J) \u2227 \u00acy \u2208 comap (algebraMap R S) J\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\n\u22a2 IsMaximal J \u2192 IsMaximal (comap (algebraMap R S) J) \u2227 \u00acy \u2208 comap (algebraMap R S) J\n[PROOFSTEP]\nrefine' fun h => \u27e8_, fun hy => h.ne_top (Ideal.eq_top_of_isUnit_mem _ hy (map_units _ \u27e8y, Submonoid.mem_powers _\u27e9))\u27e9\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal J\n\u22a2 IsMaximal (comap (algebraMap R S) J)\n[PROOFSTEP]\nhave hJ : J.IsPrime := IsMaximal.isPrime h\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal J\nhJ : IsPrime J\n\u22a2 IsMaximal (comap (algebraMap R S) J)\n[PROOFSTEP]\nrw [isPrime_iff_isPrime_disjoint (Submonoid.powers y)] at hJ \n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal J\nhJ : IsPrime (comap (algebraMap R S) J) \u2227 Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) J)\n\u22a2 IsMaximal (comap (algebraMap R S) J)\n[PROOFSTEP]\nhave : y \u2209 (comap (algebraMap R S) J).1 := Set.disjoint_left.1 hJ.right (Submonoid.mem_powers _)\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal J\nhJ : IsPrime (comap (algebraMap R S) J) \u2227 Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) J)\nthis : \u00acy \u2208 (comap (algebraMap R S) J).toAddSubmonoid\n\u22a2 IsMaximal (comap (algebraMap R S) J)\n[PROOFSTEP]\nerw [\u2190 H.out hJ.left.isRadical, mem_sInf] at this \n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal J\nhJ : IsPrime (comap (algebraMap R S) J) \u2227 Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) J)\nthis : \u00ac\u2200 \u2983I : Ideal R\u2984, I \u2208 {J_1 | comap (algebraMap R S) J \u2264 J_1 \u2227 IsMaximal J_1} \u2192 y \u2208 I\n\u22a2 IsMaximal (comap (algebraMap R S) J)\n[PROOFSTEP]\npush_neg at this \n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal J\nhJ : IsPrime (comap (algebraMap R S) J) \u2227 Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) J)\nthis : Exists fun \u2983I\u2984 => I \u2208 {J_1 | comap (algebraMap R S) J \u2264 J_1 \u2227 IsMaximal J_1} \u2227 \u00acy \u2208 I\n\u22a2 IsMaximal (comap (algebraMap R S) J)\n[PROOFSTEP]\nrcases this with \u27e8I, hI, hI'\u27e9\n[GOAL]\ncase mp.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal J\nhJ : IsPrime (comap (algebraMap R S) J) \u2227 Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) J)\nI : Ideal R\nhI : I \u2208 {J_1 | comap (algebraMap R S) J \u2264 J_1 \u2227 IsMaximal J_1}\nhI' : \u00acy \u2208 I\n\u22a2 IsMaximal (comap (algebraMap R S) J)\n[PROOFSTEP]\nconvert hI.right\n[GOAL]\ncase h.e'_3.h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal J\nhJ : IsPrime (comap (algebraMap R S) J) \u2227 Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) J)\nI : Ideal R\nhI : I \u2208 {J_1 | comap (algebraMap R S) J \u2264 J_1 \u2227 IsMaximal J_1}\nhI' : \u00acy \u2208 I\ne_2\u271d : CommSemiring.toSemiring = Ring.toSemiring\n\u22a2 comap (algebraMap R S) J = I\n[PROOFSTEP]\nby_cases hJ : J = map (algebraMap R S) I\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal J\nhJ\u271d : IsPrime (comap (algebraMap R S) J) \u2227 Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) J)\nI : Ideal R\nhI : I \u2208 {J_1 | comap (algebraMap R S) J \u2264 J_1 \u2227 IsMaximal J_1}\nhI' : \u00acy \u2208 I\ne_2\u271d : CommSemiring.toSemiring = Ring.toSemiring\nhJ : J = map (algebraMap R S) I\n\u22a2 comap (algebraMap R S) J = I\n[PROOFSTEP]\nrw [hJ, comap_map_of_isPrime_disjoint (powers y) S I (IsMaximal.isPrime hI.right)]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal J\nhJ\u271d : IsPrime (comap (algebraMap R S) J) \u2227 Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) J)\nI : Ideal R\nhI : I \u2208 {J_1 | comap (algebraMap R S) J \u2264 J_1 \u2227 IsMaximal J_1}\nhI' : \u00acy \u2208 I\ne_2\u271d : CommSemiring.toSemiring = Ring.toSemiring\nhJ : J = map (algebraMap R S) I\n\u22a2 Disjoint \u2191(powers y) \u2191I\n[PROOFSTEP]\nrwa [disjoint_powers_iff_not_mem y hI.right.isPrime.isRadical]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal J\nhJ\u271d : IsPrime (comap (algebraMap R S) J) \u2227 Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) J)\nI : Ideal R\nhI : I \u2208 {J_1 | comap (algebraMap R S) J \u2264 J_1 \u2227 IsMaximal J_1}\nhI' : \u00acy \u2208 I\ne_2\u271d : CommSemiring.toSemiring = Ring.toSemiring\nhJ : \u00acJ = map (algebraMap R S) I\n\u22a2 comap (algebraMap R S) J = I\n[PROOFSTEP]\nhave hI_p : (map (algebraMap R S) I).IsPrime :=\n by\n refine' isPrime_of_isPrime_disjoint (powers y) _ I hI.right.isPrime _\n rwa [disjoint_powers_iff_not_mem y hI.right.isPrime.isRadical]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal J\nhJ\u271d : IsPrime (comap (algebraMap R S) J) \u2227 Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) J)\nI : Ideal R\nhI : I \u2208 {J_1 | comap (algebraMap R S) J \u2264 J_1 \u2227 IsMaximal J_1}\nhI' : \u00acy \u2208 I\ne_2\u271d : CommSemiring.toSemiring = Ring.toSemiring\nhJ : \u00acJ = map (algebraMap R S) I\n\u22a2 IsPrime (map (algebraMap R S) I)\n[PROOFSTEP]\nrefine' isPrime_of_isPrime_disjoint (powers y) _ I hI.right.isPrime _\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal J\nhJ\u271d : IsPrime (comap (algebraMap R S) J) \u2227 Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) J)\nI : Ideal R\nhI : I \u2208 {J_1 | comap (algebraMap R S) J \u2264 J_1 \u2227 IsMaximal J_1}\nhI' : \u00acy \u2208 I\ne_2\u271d : CommSemiring.toSemiring = Ring.toSemiring\nhJ : \u00acJ = map (algebraMap R S) I\n\u22a2 Disjoint \u2191(powers y) \u2191I\n[PROOFSTEP]\nrwa [disjoint_powers_iff_not_mem y hI.right.isPrime.isRadical]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal J\nhJ\u271d : IsPrime (comap (algebraMap R S) J) \u2227 Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) J)\nI : Ideal R\nhI : I \u2208 {J_1 | comap (algebraMap R S) J \u2264 J_1 \u2227 IsMaximal J_1}\nhI' : \u00acy \u2208 I\ne_2\u271d : CommSemiring.toSemiring = Ring.toSemiring\nhJ : \u00acJ = map (algebraMap R S) I\nhI_p : IsPrime (map (algebraMap R S) I)\n\u22a2 comap (algebraMap R S) J = I\n[PROOFSTEP]\nhave : J \u2264 map (algebraMap R S) I := map_comap (Submonoid.powers y) S J \u25b8 map_mono hI.left\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal J\nhJ\u271d : IsPrime (comap (algebraMap R S) J) \u2227 Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) J)\nI : Ideal R\nhI : I \u2208 {J_1 | comap (algebraMap R S) J \u2264 J_1 \u2227 IsMaximal J_1}\nhI' : \u00acy \u2208 I\ne_2\u271d : CommSemiring.toSemiring = Ring.toSemiring\nhJ : \u00acJ = map (algebraMap R S) I\nhI_p : IsPrime (map (algebraMap R S) I)\nthis : J \u2264 map (algebraMap R S) I\n\u22a2 comap (algebraMap R S) J = I\n[PROOFSTEP]\nexact absurd (h.1.2 _ (lt_of_le_of_ne this hJ)) hI_p.1\n[GOAL]\ncase mpr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\n\u22a2 IsMaximal (comap (algebraMap R S) J) \u2227 \u00acy \u2208 comap (algebraMap R S) J \u2192 IsMaximal J\n[PROOFSTEP]\nrefine' fun h => \u27e8\u27e8fun hJ => h.1.ne_top (eq_top_iff.2 _), fun I hI => _\u27e9\u27e9\n[GOAL]\ncase mpr.refine'_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal (comap (algebraMap R S) J) \u2227 \u00acy \u2208 comap (algebraMap R S) J\nhJ : J = \u22a4\n\u22a2 \u22a4 \u2264 comap (algebraMap R S) J\n[PROOFSTEP]\nrwa [eq_top_iff, \u2190 (IsLocalization.orderEmbedding (powers y) S).le_iff_le] at hJ \n[GOAL]\ncase mpr.refine'_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal (comap (algebraMap R S) J) \u2227 \u00acy \u2208 comap (algebraMap R S) J\nI : Ideal S\nhI : J < I\n\u22a2 I = \u22a4\n[PROOFSTEP]\nhave := congr_arg (map (algebraMap R S)) (h.1.1.2 _ \u27e8comap_mono (le_of_lt hI), ?_\u27e9)\n[GOAL]\ncase mpr.refine'_2.refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal (comap (algebraMap R S) J) \u2227 \u00acy \u2208 comap (algebraMap R S) J\nI : Ideal S\nhI : J < I\nthis : map (algebraMap R S) (comap (algebraMap R S) I) = map (algebraMap R S) \u22a4\n\u22a2 I = \u22a4\ncase mpr.refine'_2.refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal (comap (algebraMap R S) J) \u2227 \u00acy \u2208 comap (algebraMap R S) J\nI : Ideal S\nhI : J < I\n\u22a2 \u00ac\u2191(comap (algebraMap R S) I) \u2286 \u2191(comap (algebraMap R S) J)\n[PROOFSTEP]\nrwa [map_comap (powers y) S I, map_top] at this \n[GOAL]\ncase mpr.refine'_2.refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal (comap (algebraMap R S) J) \u2227 \u00acy \u2208 comap (algebraMap R S) J\nI : Ideal S\nhI : J < I\n\u22a2 \u00ac\u2191(comap (algebraMap R S) I) \u2286 \u2191(comap (algebraMap R S) J)\n[PROOFSTEP]\nrefine' fun hI' => hI.right _\n[GOAL]\ncase mpr.refine'_2.refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal (comap (algebraMap R S) J) \u2227 \u00acy \u2208 comap (algebraMap R S) J\nI : Ideal S\nhI : J < I\nhI' : \u2191(comap (algebraMap R S) I) \u2286 \u2191(comap (algebraMap R S) J)\n\u22a2 \u2191I \u2286 \u2191J\n[PROOFSTEP]\nrw [\u2190 map_comap (powers y) S I, \u2190 map_comap (powers y) S J]\n[GOAL]\ncase mpr.refine'_2.refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nJ : Ideal S\nh : IsMaximal (comap (algebraMap R S) J) \u2227 \u00acy \u2208 comap (algebraMap R S) J\nI : Ideal S\nhI : J < I\nhI' : \u2191(comap (algebraMap R S) I) \u2286 \u2191(comap (algebraMap R S) J)\n\u22a2 \u2191(map (algebraMap R S) (comap (algebraMap R S) I)) \u2286 \u2191(map (algebraMap R S) (comap (algebraMap R S) J))\n[PROOFSTEP]\nexact map_mono hI'\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : Away y S\ninst\u271d : IsJacobson R\nI : Ideal R\nhI : IsMaximal I\nhy : \u00acy \u2208 I\n\u22a2 IsMaximal (map (algebraMap R S) I)\n[PROOFSTEP]\nrw [isMaximal_iff_isMaximal_disjoint S y,\n comap_map_of_isPrime_disjoint (powers y) S I (IsMaximal.isPrime hI)\n ((disjoint_powers_iff_not_mem y hI.isPrime.isRadical).2 hy)]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : Away y S\ninst\u271d : IsJacobson R\nI : Ideal R\nhI : IsMaximal I\nhy : \u00acy \u2208 I\n\u22a2 IsMaximal I \u2227 \u00acy \u2208 I\n[PROOFSTEP]\nexact \u27e8hI, hy\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\n\u22a2 IsJacobson S\n[PROOFSTEP]\nrw [isJacobson_iff_prime_eq]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\n\u22a2 \u2200 (P : Ideal S), IsPrime P \u2192 jacobson P = P\n[PROOFSTEP]\nrefine' fun P' hP' => le_antisymm _ le_jacobson\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP' : IsPrime P'\n\u22a2 jacobson P' \u2264 P'\n[PROOFSTEP]\nobtain \u27e8hP', hPM\u27e9 := (IsLocalization.isPrime_iff_isPrime_disjoint (powers y) S P').mp hP'\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\n\u22a2 jacobson P' \u2264 P'\n[PROOFSTEP]\nhave hP := H.out hP'.isRadical\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\n\u22a2 jacobson P' \u2264 P'\n[PROOFSTEP]\nrefine'\n (IsLocalization.map_comap (powers y) S P'.jacobson).ge.trans\n ((map_mono _).trans (IsLocalization.map_comap (powers y) S P').le)\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\n\u22a2 comap (algebraMap R S) (jacobson P') \u2264 comap (algebraMap R S) P'\n[PROOFSTEP]\nhave : sInf {I : Ideal R | comap (algebraMap R S) P' \u2264 I \u2227 I.IsMaximal \u2227 y \u2209 I} \u2264 comap (algebraMap R S) P' :=\n by\n intro x hx\n have hxy : x * y \u2208 (comap (algebraMap R S) P').jacobson :=\n by\n rw [Ideal.jacobson, mem_sInf]\n intro J hJ\n by_cases y \u2208 J\n \u00b7 exact J.mul_mem_left x h\n \u00b7 exact J.mul_mem_right y ((mem_sInf.1 hx) \u27e8hJ.left, \u27e8hJ.right, h\u27e9\u27e9)\n rw [hP] at hxy \n cases' hP'.mem_or_mem hxy with hxy hxy\n \u00b7 exact hxy\n \u00b7 exact (hPM.le_bot \u27e8Submonoid.mem_powers _, hxy\u27e9).elim\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\n\u22a2 sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I} \u2264 comap (algebraMap R S) P'\n[PROOFSTEP]\nintro x hx\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nx : R\nhx : x \u2208 sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I}\n\u22a2 x \u2208 comap (algebraMap R S) P'\n[PROOFSTEP]\nhave hxy : x * y \u2208 (comap (algebraMap R S) P').jacobson :=\n by\n rw [Ideal.jacobson, mem_sInf]\n intro J hJ\n by_cases y \u2208 J\n \u00b7 exact J.mul_mem_left x h\n \u00b7 exact J.mul_mem_right y ((mem_sInf.1 hx) \u27e8hJ.left, \u27e8hJ.right, h\u27e9\u27e9)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nx : R\nhx : x \u2208 sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I}\n\u22a2 x * y \u2208 jacobson (comap (algebraMap R S) P')\n[PROOFSTEP]\nrw [Ideal.jacobson, mem_sInf]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nx : R\nhx : x \u2208 sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I}\n\u22a2 \u2200 \u2983I : Ideal R\u2984, I \u2208 {J | comap (algebraMap R S) P' \u2264 J \u2227 IsMaximal J} \u2192 x * y \u2208 I\n[PROOFSTEP]\nintro J hJ\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nx : R\nhx : x \u2208 sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I}\nJ : Ideal R\nhJ : J \u2208 {J | comap (algebraMap R S) P' \u2264 J \u2227 IsMaximal J}\n\u22a2 x * y \u2208 J\n[PROOFSTEP]\nby_cases y \u2208 J\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nx : R\nhx : x \u2208 sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I}\nJ : Ideal R\nhJ : J \u2208 {J | comap (algebraMap R S) P' \u2264 J \u2227 IsMaximal J}\n\u22a2 x * y \u2208 J\n[PROOFSTEP]\nby_cases y \u2208 J\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nx : R\nhx : x \u2208 sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I}\nJ : Ideal R\nhJ : J \u2208 {J | comap (algebraMap R S) P' \u2264 J \u2227 IsMaximal J}\nh : y \u2208 J\n\u22a2 x * y \u2208 J\n[PROOFSTEP]\nexact J.mul_mem_left x h\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nx : R\nhx : x \u2208 sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I}\nJ : Ideal R\nhJ : J \u2208 {J | comap (algebraMap R S) P' \u2264 J \u2227 IsMaximal J}\nh : \u00acy \u2208 J\n\u22a2 x * y \u2208 J\n[PROOFSTEP]\nexact J.mul_mem_right y ((mem_sInf.1 hx) \u27e8hJ.left, \u27e8hJ.right, h\u27e9\u27e9)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nx : R\nhx : x \u2208 sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I}\nhxy : x * y \u2208 jacobson (comap (algebraMap R S) P')\n\u22a2 x \u2208 comap (algebraMap R S) P'\n[PROOFSTEP]\nrw [hP] at hxy \n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nx : R\nhx : x \u2208 sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I}\nhxy : x * y \u2208 comap (algebraMap R S) P'\n\u22a2 x \u2208 comap (algebraMap R S) P'\n[PROOFSTEP]\ncases' hP'.mem_or_mem hxy with hxy hxy\n[GOAL]\ncase inl\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nx : R\nhx : x \u2208 sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I}\nhxy\u271d : x * y \u2208 comap (algebraMap R S) P'\nhxy : x \u2208 comap (algebraMap R S) P'\n\u22a2 x \u2208 comap (algebraMap R S) P'\n[PROOFSTEP]\nexact hxy\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nx : R\nhx : x \u2208 sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I}\nhxy\u271d : x * y \u2208 comap (algebraMap R S) P'\nhxy : y \u2208 comap (algebraMap R S) P'\n\u22a2 x \u2208 comap (algebraMap R S) P'\n[PROOFSTEP]\nexact (hPM.le_bot \u27e8Submonoid.mem_powers _, hxy\u27e9).elim\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nthis : sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I} \u2264 comap (algebraMap R S) P'\n\u22a2 comap (algebraMap R S) (jacobson P') \u2264 comap (algebraMap R S) P'\n[PROOFSTEP]\nrefine' le_trans _ this\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nthis : sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I} \u2264 comap (algebraMap R S) P'\n\u22a2 comap (algebraMap R S) (jacobson P') \u2264 sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I}\n[PROOFSTEP]\nrw [Ideal.jacobson, comap_sInf', sInf_eq_iInf]\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nthis : sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I} \u2264 comap (algebraMap R S) P'\n\u22a2 \u2a05 (I : Ideal R) (_ : I \u2208 comap (algebraMap R S) '' {J | P' \u2264 J \u2227 IsMaximal J}), I \u2264\n \u2a05 (a : Ideal R) (_ : a \u2208 {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I}), a\n[PROOFSTEP]\nrefine' iInf_le_iInf_of_subset fun I hI => \u27e8map (algebraMap R S) I, \u27e8_, _\u27e9\u27e9\n[GOAL]\ncase intro.refine'_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nthis : sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I} \u2264 comap (algebraMap R S) P'\nI : Ideal R\nhI : I \u2208 {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I}\n\u22a2 map (algebraMap R S) I \u2208 {J | P' \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\nexact\n \u27e8le_trans (le_of_eq (IsLocalization.map_comap (powers y) S P').symm) (map_mono hI.1),\n isMaximal_of_isMaximal_disjoint y _ hI.2.1 hI.2.2\u27e9\n[GOAL]\ncase intro.refine'_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nI\u271d : Ideal R\ny : R\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Away y S\nH : IsJacobson R\nP' : Ideal S\nhP'\u271d : IsPrime P'\nhP' : IsPrime (comap (algebraMap R S) P')\nhPM : Disjoint \u2191(powers y) \u2191(comap (algebraMap R S) P')\nhP : jacobson (comap (algebraMap R S) P') = comap (algebraMap R S) P'\nthis : sInf {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I} \u2264 comap (algebraMap R S) P'\nI : Ideal R\nhI : I \u2208 {I | comap (algebraMap R S) P' \u2264 I \u2227 IsMaximal I \u2227 \u00acy \u2208 I}\n\u22a2 comap (algebraMap R S) (map (algebraMap R S) I) = I\n[PROOFSTEP]\nexact\n IsLocalization.comap_map_of_isPrime_disjoint _ S I (IsMaximal.isPrime hI.2.1)\n ((disjoint_powers_iff_not_mem y hI.2.1.isPrime.isRadical).2 hI.2.2)\n[GOAL]\nS : Type u_1\nT : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\ng : S \u2192+* T\nu : Set S\nx : S\nhx : x \u2208 Subring.closure u\n\u22a2 \u2191g x \u2208 Subring.closure (\u2191g '' u)\n[PROOFSTEP]\nrw [Subring.mem_closure] at hx \u22a2\n[GOAL]\nS : Type u_1\nT : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\ng : S \u2192+* T\nu : Set S\nx : S\nhx : \u2200 (S_1 : Subring S), u \u2286 \u2191S_1 \u2192 x \u2208 S_1\n\u22a2 \u2200 (S_1 : Subring ((fun x => T) x)), \u2191g '' u \u2286 \u2191S_1 \u2192 \u2191g x \u2208 S_1\n[PROOFSTEP]\nintro T\u2081 h\u2081\n[GOAL]\nS : Type u_1\nT : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\ng : S \u2192+* T\nu : Set S\nx : S\nhx : \u2200 (S_1 : Subring S), u \u2286 \u2191S_1 \u2192 x \u2208 S_1\nT\u2081 : Subring ((fun x => T) x)\nh\u2081 : \u2191g '' u \u2286 \u2191T\u2081\n\u22a2 \u2191g x \u2208 T\u2081\n[PROOFSTEP]\nrw [\u2190 Subring.mem_comap]\n[GOAL]\nS : Type u_1\nT : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\ng : S \u2192+* T\nu : Set S\nx : S\nhx : \u2200 (S_1 : Subring S), u \u2286 \u2191S_1 \u2192 x \u2208 S_1\nT\u2081 : Subring ((fun x => T) x)\nh\u2081 : \u2191g '' u \u2286 \u2191T\u2081\n\u22a2 x \u2208 Subring.comap g T\u2081\n[PROOFSTEP]\napply hx\n[GOAL]\ncase a\nS : Type u_1\nT : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\ng : S \u2192+* T\nu : Set S\nx : S\nhx : \u2200 (S_1 : Subring S), u \u2286 \u2191S_1 \u2192 x \u2208 S_1\nT\u2081 : Subring ((fun x => T) x)\nh\u2081 : \u2191g '' u \u2286 \u2191T\u2081\n\u22a2 u \u2286 \u2191(Subring.comap g T\u2081)\n[PROOFSTEP]\nsimp only [Subring.coe_comap, \u2190 Set.image_subset_iff, SetLike.mem_coe]\n[GOAL]\ncase a\nS : Type u_1\nT : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\ng : S \u2192+* T\nu : Set S\nx : S\nhx : \u2200 (S_1 : Subring S), u \u2286 \u2191S_1 \u2192 x \u2208 S_1\nT\u2081 : Subring ((fun x => T) x)\nh\u2081 : \u2191g '' u \u2286 \u2191T\u2081\n\u22a2 \u2191g '' u \u2286 \u2191T\u2081\n[PROOFSTEP]\nexact h\u2081\n[GOAL]\nR : Type u_1\ninst\u271d : Ring R\np : R[X]\n\u22a2 p \u2208 Subring.closure (insert X {f | degree f \u2264 0})\n[PROOFSTEP]\nrefine' Polynomial.induction_on p _ _ _\n[GOAL]\ncase refine'_1\nR : Type u_1\ninst\u271d : Ring R\np : R[X]\n\u22a2 \u2200 (a : R), \u2191C a \u2208 Subring.closure (insert X {f | degree f \u2264 0})\n[PROOFSTEP]\nintro r\n[GOAL]\ncase refine'_1\nR : Type u_1\ninst\u271d : Ring R\np : R[X]\nr : R\n\u22a2 \u2191C r \u2208 Subring.closure (insert X {f | degree f \u2264 0})\n[PROOFSTEP]\napply Subring.subset_closure\n[GOAL]\ncase refine'_1.a\nR : Type u_1\ninst\u271d : Ring R\np : R[X]\nr : R\n\u22a2 \u2191C r \u2208 insert X {f | degree f \u2264 0}\n[PROOFSTEP]\napply Set.mem_insert_of_mem\n[GOAL]\ncase refine'_1.a.a\nR : Type u_1\ninst\u271d : Ring R\np : R[X]\nr : R\n\u22a2 \u2191C r \u2208 {f | degree f \u2264 0}\n[PROOFSTEP]\nexact degree_C_le\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d : Ring R\np : R[X]\n\u22a2 \u2200 (p q : R[X]),\n p \u2208 Subring.closure (insert X {f | degree f \u2264 0}) \u2192\n q \u2208 Subring.closure (insert X {f | degree f \u2264 0}) \u2192 p + q \u2208 Subring.closure (insert X {f | degree f \u2264 0})\n[PROOFSTEP]\nintros p1 p2 h1 h2\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d : Ring R\np p1 p2 : R[X]\nh1 : p1 \u2208 Subring.closure (insert X {f | degree f \u2264 0})\nh2 : p2 \u2208 Subring.closure (insert X {f | degree f \u2264 0})\n\u22a2 p1 + p2 \u2208 Subring.closure (insert X {f | degree f \u2264 0})\n[PROOFSTEP]\nexact Subring.add_mem _ h1 h2\n[GOAL]\ncase refine'_3\nR : Type u_1\ninst\u271d : Ring R\np : R[X]\n\u22a2 \u2200 (n : \u2115) (a : R),\n \u2191C a * X ^ n \u2208 Subring.closure (insert X {f | degree f \u2264 0}) \u2192\n \u2191C a * X ^ (n + 1) \u2208 Subring.closure (insert X {f | degree f \u2264 0})\n[PROOFSTEP]\nintros n r hr\n[GOAL]\ncase refine'_3\nR : Type u_1\ninst\u271d : Ring R\np : R[X]\nn : \u2115\nr : R\nhr : \u2191C r * X ^ n \u2208 Subring.closure (insert X {f | degree f \u2264 0})\n\u22a2 \u2191C r * X ^ (n + 1) \u2208 Subring.closure (insert X {f | degree f \u2264 0})\n[PROOFSTEP]\nrw [pow_succ', \u2190 mul_assoc]\n[GOAL]\ncase refine'_3\nR : Type u_1\ninst\u271d : Ring R\np : R[X]\nn : \u2115\nr : R\nhr : \u2191C r * X ^ n \u2208 Subring.closure (insert X {f | degree f \u2264 0})\n\u22a2 \u2191C r * X ^ n * X \u2208 Subring.closure (insert X {f | degree f \u2264 0})\n[PROOFSTEP]\napply Subring.mul_mem _ hr\n[GOAL]\ncase refine'_3\nR : Type u_1\ninst\u271d : Ring R\np : R[X]\nn : \u2115\nr : R\nhr : \u2191C r * X ^ n \u2208 Subring.closure (insert X {f | degree f \u2264 0})\n\u22a2 X \u2208 Subring.closure (insert X {f | degree f \u2264 0})\n[PROOFSTEP]\napply Subring.subset_closure\n[GOAL]\ncase refine'_3.a\nR : Type u_1\ninst\u271d : Ring R\np : R[X]\nn : \u2115\nr : R\nhr : \u2191C r * X ^ n \u2208 Subring.closure (insert X {f | degree f \u2264 0})\n\u22a2 X \u2208 insert X {f | degree f \u2264 0}\n[PROOFSTEP]\napply Set.mem_insert\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\n\u22a2 RingHom.IsIntegral\n (IsLocalization.map S\u2098 (quotientMap P C (_ : comap C P \u2264 comap C P))\n (_ :\n Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) \u2264\n Submonoid.comap (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))))\n[PROOFSTEP]\nlet P' : Ideal R := P.comap C\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\n\u22a2 RingHom.IsIntegral\n (IsLocalization.map S\u2098 (quotientMap P C (_ : comap C P \u2264 comap C P))\n (_ :\n Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) \u2264\n Submonoid.comap (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))))\n[PROOFSTEP]\nlet M : Submonoid (R \u29f8 P') := Submonoid.powers (pX.map (Quotient.mk (P.comap (C : R \u2192+* R[X])))).leadingCoeff\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\n\u22a2 RingHom.IsIntegral\n (IsLocalization.map S\u2098 (quotientMap P C (_ : comap C P \u2264 comap C P))\n (_ :\n Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) \u2264\n Submonoid.comap (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))))\n[PROOFSTEP]\nlet M' : Submonoid (R[X] \u29f8 P) :=\n (Submonoid.powers (pX.map (Quotient.mk (P.comap (C : R \u2192+* R[X])))).leadingCoeff).map (quotientMap P C le_rfl)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u22a2 RingHom.IsIntegral\n (IsLocalization.map S\u2098 (quotientMap P C (_ : comap C P \u2264 comap C P))\n (_ :\n Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) \u2264\n Submonoid.comap (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))))\n[PROOFSTEP]\nlet \u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C le_rfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u22a2 RingHom.IsIntegral\n (IsLocalization.map S\u2098 (quotientMap P C (_ : comap C P \u2264 comap C P))\n (_ :\n Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) \u2264\n Submonoid.comap (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))))\n[PROOFSTEP]\nlet \u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 M.le_comap_map\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\n\u22a2 RingHom.IsIntegral\n (IsLocalization.map S\u2098 (quotientMap P C (_ : comap C P \u2264 comap C P))\n (_ :\n Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) \u2264\n Submonoid.comap (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))))\n[PROOFSTEP]\nhave h\u03c6' : \u03c6.comp (Quotient.mk P') = (Quotient.mk P).comp C := rfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\n\u22a2 RingHom.IsIntegral\n (IsLocalization.map S\u2098 (quotientMap P C (_ : comap C P \u2264 comap C P))\n (_ :\n Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) \u2264\n Submonoid.comap (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))))\n[PROOFSTEP]\nintro p\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np : S\u2098\n\u22a2 RingHom.IsIntegralElem\n (IsLocalization.map S\u2098 (quotientMap P C (_ : comap C P \u2264 comap C P))\n (_ :\n Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) \u2264\n Submonoid.comap (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))))\n p\n[PROOFSTEP]\nobtain \u27e8\u27e8p', \u27e8q, hq\u27e9\u27e9, hp\u27e9 := IsLocalization.surj M' p\n[GOAL]\ncase intro.mk.mk\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp :\n p * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) \u2191(p', { val := q, property := hq }).snd =\n \u2191(algebraMap (R[X] \u29f8 P) S\u2098) (p', { val := q, property := hq }).fst\n\u22a2 RingHom.IsIntegralElem\n (IsLocalization.map S\u2098 (quotientMap P C (_ : comap C P \u2264 comap C P))\n (_ :\n Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) \u2264\n Submonoid.comap (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))))\n p\n[PROOFSTEP]\nsuffices \u03c6'.IsIntegralElem (algebraMap (R[X] \u29f8 P) S\u2098 p')\n by\n obtain \u27e8q', hq', rfl\u27e9 := hq\n obtain \u27e8q'', hq''\u27e9 := isUnit_iff_exists_inv'.1 (IsLocalization.map_units R\u2098 (\u27e8q', hq'\u27e9 : M))\n refine' \u03c6'.isIntegral_of_isIntegral_mul_unit p (algebraMap (R[X] \u29f8 P) S\u2098 (\u03c6 q')) q'' _ (hp.symm \u25b8 this)\n rw [\u2190 \u03c6'.map_one, \u2190 congr_arg \u03c6' hq'', \u03c6'.map_mul, \u2190 \u03c6'.comp_apply]\n simp only [IsLocalization.map_comp _]\n rw [RingHom.comp_apply]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp :\n p * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) \u2191(p', { val := q, property := hq }).snd =\n \u2191(algebraMap (R[X] \u29f8 P) S\u2098) (p', { val := q, property := hq }).fst\nthis : RingHom.IsIntegralElem \u03c6' (\u2191(algebraMap (R[X] \u29f8 P) S\u2098) p')\n\u22a2 RingHom.IsIntegralElem\n (IsLocalization.map S\u2098 (quotientMap P C (_ : comap C P \u2264 comap C P))\n (_ :\n Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) \u2264\n Submonoid.comap (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))))\n p\n[PROOFSTEP]\nobtain \u27e8q', hq', rfl\u27e9 := hq\n[GOAL]\ncase intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np : S\u2098\np' : R[X] \u29f8 P\nthis : RingHom.IsIntegralElem \u03c6' (\u2191(algebraMap (R[X] \u29f8 P) S\u2098) p')\nq' : R \u29f8 comap C P\nhq' : q' \u2208 \u2191(Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\nhp :\n p *\n \u2191(algebraMap (R[X] \u29f8 P) S\u2098)\n \u2191(p',\n { val := \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q',\n property :=\n (_ :\n \u2203 a,\n a \u2208 \u2191(Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))) \u2227\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) a =\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q') }).snd =\n \u2191(algebraMap (R[X] \u29f8 P) S\u2098)\n (p',\n { val := \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q',\n property :=\n (_ :\n \u2203 a,\n a \u2208 \u2191(Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))) \u2227\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) a =\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q') }).fst\n\u22a2 RingHom.IsIntegralElem\n (IsLocalization.map S\u2098 (quotientMap P C (_ : comap C P \u2264 comap C P))\n (_ :\n Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) \u2264\n Submonoid.comap (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))))\n p\n[PROOFSTEP]\nobtain \u27e8q'', hq''\u27e9 := isUnit_iff_exists_inv'.1 (IsLocalization.map_units R\u2098 (\u27e8q', hq'\u27e9 : M))\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np : S\u2098\np' : R[X] \u29f8 P\nthis : RingHom.IsIntegralElem \u03c6' (\u2191(algebraMap (R[X] \u29f8 P) S\u2098) p')\nq' : R \u29f8 comap C P\nhq' : q' \u2208 \u2191(Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\nhp :\n p *\n \u2191(algebraMap (R[X] \u29f8 P) S\u2098)\n \u2191(p',\n { val := \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q',\n property :=\n (_ :\n \u2203 a,\n a \u2208 \u2191(Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))) \u2227\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) a =\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q') }).snd =\n \u2191(algebraMap (R[X] \u29f8 P) S\u2098)\n (p',\n { val := \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q',\n property :=\n (_ :\n \u2203 a,\n a \u2208 \u2191(Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))) \u2227\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) a =\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q') }).fst\nq'' : R\u2098\nhq'' : q'' * \u2191(algebraMap (R \u29f8 P') R\u2098) \u2191{ val := q', property := hq' } = 1\n\u22a2 RingHom.IsIntegralElem\n (IsLocalization.map S\u2098 (quotientMap P C (_ : comap C P \u2264 comap C P))\n (_ :\n Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) \u2264\n Submonoid.comap (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))))\n p\n[PROOFSTEP]\nrefine' \u03c6'.isIntegral_of_isIntegral_mul_unit p (algebraMap (R[X] \u29f8 P) S\u2098 (\u03c6 q')) q'' _ (hp.symm \u25b8 this)\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np : S\u2098\np' : R[X] \u29f8 P\nthis : RingHom.IsIntegralElem \u03c6' (\u2191(algebraMap (R[X] \u29f8 P) S\u2098) p')\nq' : R \u29f8 comap C P\nhq' : q' \u2208 \u2191(Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\nhp :\n p *\n \u2191(algebraMap (R[X] \u29f8 P) S\u2098)\n \u2191(p',\n { val := \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q',\n property :=\n (_ :\n \u2203 a,\n a \u2208 \u2191(Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))) \u2227\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) a =\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q') }).snd =\n \u2191(algebraMap (R[X] \u29f8 P) S\u2098)\n (p',\n { val := \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q',\n property :=\n (_ :\n \u2203 a,\n a \u2208 \u2191(Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))) \u2227\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) a =\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q') }).fst\nq'' : R\u2098\nhq'' : q'' * \u2191(algebraMap (R \u29f8 P') R\u2098) \u2191{ val := q', property := hq' } = 1\n\u22a2 \u2191\u03c6' q'' * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) (\u2191\u03c6 q') = 1\n[PROOFSTEP]\nrw [\u2190 \u03c6'.map_one, \u2190 congr_arg \u03c6' hq'', \u03c6'.map_mul, \u2190 \u03c6'.comp_apply]\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np : S\u2098\np' : R[X] \u29f8 P\nthis : RingHom.IsIntegralElem \u03c6' (\u2191(algebraMap (R[X] \u29f8 P) S\u2098) p')\nq' : R \u29f8 comap C P\nhq' : q' \u2208 \u2191(Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\nhp :\n p *\n \u2191(algebraMap (R[X] \u29f8 P) S\u2098)\n \u2191(p',\n { val := \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q',\n property :=\n (_ :\n \u2203 a,\n a \u2208 \u2191(Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))) \u2227\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) a =\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q') }).snd =\n \u2191(algebraMap (R[X] \u29f8 P) S\u2098)\n (p',\n { val := \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q',\n property :=\n (_ :\n \u2203 a,\n a \u2208 \u2191(Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))) \u2227\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) a =\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q') }).fst\nq'' : R\u2098\nhq'' : q'' * \u2191(algebraMap (R \u29f8 P') R\u2098) \u2191{ val := q', property := hq' } = 1\n\u22a2 \u2191\u03c6' q'' * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) (\u2191\u03c6 q') =\n \u2191\u03c6' q'' * \u2191(RingHom.comp \u03c6' (algebraMap (R \u29f8 P') R\u2098)) \u2191{ val := q', property := hq' }\n[PROOFSTEP]\nsimp only [IsLocalization.map_comp _]\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np : S\u2098\np' : R[X] \u29f8 P\nthis : RingHom.IsIntegralElem \u03c6' (\u2191(algebraMap (R[X] \u29f8 P) S\u2098) p')\nq' : R \u29f8 comap C P\nhq' : q' \u2208 \u2191(Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\nhp :\n p *\n \u2191(algebraMap (R[X] \u29f8 P) S\u2098)\n \u2191(p',\n { val := \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q',\n property :=\n (_ :\n \u2203 a,\n a \u2208 \u2191(Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))) \u2227\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) a =\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q') }).snd =\n \u2191(algebraMap (R[X] \u29f8 P) S\u2098)\n (p',\n { val := \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q',\n property :=\n (_ :\n \u2203 a,\n a \u2208 \u2191(Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))) \u2227\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) a =\n \u2191(quotientMap P C (_ : comap C P \u2264 comap C P)) q') }).fst\nq'' : R\u2098\nhq'' : q'' * \u2191(algebraMap (R \u29f8 P') R\u2098) \u2191{ val := q', property := hq' } = 1\n\u22a2 \u2191(IsLocalization.map S\u2098 (quotientMap P C (_ : P' \u2264 P')) (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))) q'' *\n \u2191(algebraMap (R[X] \u29f8 P) S\u2098) (\u2191(quotientMap P C (_ : P' \u2264 P')) q') =\n \u2191(IsLocalization.map S\u2098 (quotientMap P C (_ : P' \u2264 P')) (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))) q'' *\n \u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (quotientMap P C (_ : P' \u2264 P'))) q'\n[PROOFSTEP]\nrw [RingHom.comp_apply]\n[GOAL]\ncase intro.mk.mk\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp :\n p * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) \u2191(p', { val := q, property := hq }).snd =\n \u2191(algebraMap (R[X] \u29f8 P) S\u2098) (p', { val := q, property := hq }).fst\n\u22a2 RingHom.IsIntegralElem \u03c6' (\u2191(algebraMap (R[X] \u29f8 P) S\u2098) p')\n[PROOFSTEP]\ndsimp at hp \n[GOAL]\ncase intro.mk.mk\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\n\u22a2 RingHom.IsIntegralElem \u03c6' (\u2191(algebraMap (R[X] \u29f8 P) S\u2098) p')\n[PROOFSTEP]\nrefine'\n @isIntegral_of_mem_closure'' R\u2098 _ S\u2098 _ \u03c6'\n ((algebraMap (R[X] \u29f8 P) S\u2098).comp (Quotient.mk P) '' insert X {p | p.degree \u2264 0}) _ ((algebraMap (R[X] \u29f8 P) S\u2098) p') _\n[GOAL]\ncase intro.mk.mk.refine'_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\n\u22a2 \u2200 (x : S\u2098),\n x \u2208 \u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) '' insert X {p | degree p \u2264 0} \u2192\n RingHom.IsIntegralElem \u03c6' x\n[PROOFSTEP]\nrintro x \u27e8p, hp, rfl\u27e9\n[GOAL]\ncase intro.mk.mk.refine'_1.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp\u271d : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhp : p \u2208 insert X {p | degree p \u2264 0}\n\u22a2 RingHom.IsIntegralElem \u03c6' (\u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) p)\n[PROOFSTEP]\nsimp only [Set.mem_insert_iff] at hp \n[GOAL]\ncase intro.mk.mk.refine'_1.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp\u271d : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhp : p = X \u2228 p \u2208 {p | degree p \u2264 0}\n\u22a2 RingHom.IsIntegralElem \u03c6' (\u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) p)\n[PROOFSTEP]\ncases' hp with hy hy\n[GOAL]\ncase intro.mk.mk.refine'_1.intro.intro.inl\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhy : p = X\n\u22a2 RingHom.IsIntegralElem \u03c6' (\u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) p)\n[PROOFSTEP]\nrw [hy]\n[GOAL]\ncase intro.mk.mk.refine'_1.intro.intro.inl\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhy : p = X\n\u22a2 RingHom.IsIntegralElem \u03c6' (\u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) X)\n[PROOFSTEP]\nrefine' \u03c6.isIntegralElem_localization_at_leadingCoeff ((Quotient.mk P) X) (pX.map (Quotient.mk P')) _ M _\n[GOAL]\ncase intro.mk.mk.refine'_1.intro.intro.inl.refine'_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhy : p = X\n\u22a2 eval\u2082 \u03c6 (\u2191(Quotient.mk P) X) (Polynomial.map (Quotient.mk P') pX) = 0\n[PROOFSTEP]\nrwa [eval\u2082_map, h\u03c6', \u2190 hom_eval\u2082, Quotient.eq_zero_iff_mem, eval\u2082_C_X]\n[GOAL]\ncase intro.mk.mk.refine'_1.intro.intro.inl.refine'_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhy : p = X\n\u22a2 Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX) \u2208 M\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhy : p = X\n\u22a2 (fun x x_1 => x ^ x_1) (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) 1 =\n Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\n[PROOFSTEP]\nsimp only [pow_one]\n[GOAL]\ncase intro.mk.mk.refine'_1.intro.intro.inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhy : p \u2208 {p | degree p \u2264 0}\n\u22a2 RingHom.IsIntegralElem \u03c6' (\u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) p)\n[PROOFSTEP]\nrw [Set.mem_setOf_eq, degree_le_zero_iff] at hy \n[GOAL]\ncase intro.mk.mk.refine'_1.intro.intro.inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhy : p = \u2191C (coeff p 0)\n\u22a2 RingHom.IsIntegralElem \u03c6' (\u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) p)\n[PROOFSTEP]\nrw [hy]\n[GOAL]\ncase intro.mk.mk.refine'_1.intro.intro.inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhy : p = \u2191C (coeff p 0)\n\u22a2 RingHom.IsIntegralElem \u03c6' (\u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) (\u2191C (coeff p 0)))\n[PROOFSTEP]\nuse X - C (algebraMap (R \u29f8 P') R\u2098 ((Quotient.mk P') (p.coeff 0)))\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhy : p = \u2191C (coeff p 0)\n\u22a2 Monic (X - \u2191C (\u2191(algebraMap (R \u29f8 P') R\u2098) (\u2191(Quotient.mk P') (coeff p 0)))) \u2227\n eval\u2082 \u03c6' (\u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) (\u2191C (coeff p 0)))\n (X - \u2191C (\u2191(algebraMap (R \u29f8 P') R\u2098) (\u2191(Quotient.mk P') (coeff p 0)))) =\n 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhy : p = \u2191C (coeff p 0)\n\u22a2 Monic (X - \u2191C (\u2191(algebraMap (R \u29f8 P') R\u2098) (\u2191(Quotient.mk P') (coeff p 0))))\n[PROOFSTEP]\napply monic_X_sub_C\n[GOAL]\ncase h.right\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhy : p = \u2191C (coeff p 0)\n\u22a2 eval\u2082 \u03c6' (\u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) (\u2191C (coeff p 0)))\n (X - \u2191C (\u2191(algebraMap (R \u29f8 P') R\u2098) (\u2191(Quotient.mk P') (coeff p 0)))) =\n 0\n[PROOFSTEP]\nsimp only [eval\u2082_sub, eval\u2082_X, eval\u2082_C]\n[GOAL]\ncase h.right\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhy : p = \u2191C (coeff p 0)\n\u22a2 \u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) (\u2191C (coeff p 0)) -\n \u2191(IsLocalization.map S\u2098 (quotientMap P C (_ : P' \u2264 P')) (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M)))\n (\u2191(algebraMap (R \u29f8 comap C P) R\u2098) (\u2191(Quotient.mk (comap C P)) (coeff p 0))) =\n 0\n[PROOFSTEP]\nrw [sub_eq_zero, \u2190 \u03c6'.comp_apply]\n[GOAL]\ncase h.right\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhy : p = \u2191C (coeff p 0)\n\u22a2 \u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) (\u2191C (coeff p 0)) =\n \u2191(RingHom.comp \u03c6' (algebraMap (R \u29f8 comap C P) R\u2098)) (\u2191(Quotient.mk (comap C P)) (coeff p 0))\n[PROOFSTEP]\nsimp only [IsLocalization.map_comp _]\n[GOAL]\ncase h.right\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\np : R[X]\nhy : p = \u2191C (coeff p 0)\n\u22a2 \u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) (\u2191C (coeff p 0)) =\n \u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (quotientMap P C (_ : P' \u2264 P'))) (\u2191(Quotient.mk (comap C P)) (coeff p 0))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.mk.mk.refine'_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np : S\u2098\np' q : R[X] \u29f8 P\nhq : q \u2208 M'\nhp : p * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p'\n\u22a2 \u2191(algebraMap (R[X] \u29f8 P) S\u2098) p' \u2208\n Subring.closure (\u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) '' insert X {p | degree p \u2264 0})\n[PROOFSTEP]\nobtain \u27e8p, rfl\u27e9 := Quotient.mk_surjective p'\n[GOAL]\ncase intro.mk.mk.refine'_2.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\nq : R[X] \u29f8 P\nhq : q \u2208 M'\np : R[X]\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) (\u2191(Quotient.mk P) p)\n\u22a2 \u2191(algebraMap (R[X] \u29f8 P) S\u2098) (\u2191(Quotient.mk P) p) \u2208\n Subring.closure (\u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) '' insert X {p | degree p \u2264 0})\n[PROOFSTEP]\nrw [\u2190 RingHom.comp_apply]\n[GOAL]\ncase intro.mk.mk.refine'_2.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\nq : R[X] \u29f8 P\nhq : q \u2208 M'\np : R[X]\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) (\u2191(Quotient.mk P) p)\n\u22a2 \u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) p \u2208\n Subring.closure (\u2191(RingHom.comp (algebraMap (R[X] \u29f8 P) S\u2098) (Quotient.mk P)) '' insert X {p | degree p \u2264 0})\n[PROOFSTEP]\napply Subring.mem_closure_image_of\n[GOAL]\ncase intro.mk.mk.refine'_2.intro.hx\nR : Type u_1\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\nP : Ideal R[X]\npX : R[X]\nhpX : pX \u2208 P\ninst\u271d\u00b3 : Algebra (R \u29f8 comap C P) R\u2098\ninst\u271d\u00b2 : IsLocalization.Away (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)) R\u2098\ninst\u271d\u00b9 : Algebra (R[X] \u29f8 P) S\u2098\ninst\u271d :\n IsLocalization\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))))\n S\u2098\nP' : Ideal R := comap C P\nM : Submonoid (R \u29f8 P') := Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX))\nM' : Submonoid (R[X] \u29f8 P) :=\n Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.powers (Polynomial.leadingCoeff (Polynomial.map (Quotient.mk (comap C P)) pX)))\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map S\u2098 \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M))\nh\u03c6' : RingHom.comp \u03c6 (Quotient.mk P') = RingHom.comp (Quotient.mk P) C\np\u271d : S\u2098\nq : R[X] \u29f8 P\nhq : q \u2208 M'\np : R[X]\nhp : p\u271d * \u2191(algebraMap (R[X] \u29f8 P) S\u2098) q = \u2191(algebraMap (R[X] \u29f8 P) S\u2098) (\u2191(Quotient.mk P) p)\n\u22a2 p \u2208 Subring.closure (insert X {p | degree p \u2264 0})\n[PROOFSTEP]\napply Polynomial.mem_closure_X_union_C\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nhave hM : ((Submonoid.powers x).map \u03c6 : Submonoid S) \u2264 nonZeroDivisors S :=\n map_le_nonZeroDivisors_of_injective \u03c6 h\u03c6 (powers_le_nonZeroDivisors_of_noZeroDivisors hx)\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nletI : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors _ hM\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nlet \u03c6' : R\u2098 \u2192+* S\u2098 := IsLocalization.map _ \u03c6 (Submonoid.powers x).le_comap_map\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nsuffices \u2200 I : Ideal S\u2098, I.IsMaximal \u2192 (I.comap (algebraMap S S\u2098)).IsMaximal\n by\n have h\u03d5' : comap (algebraMap S S\u2098) (\u22a5 : Ideal S\u2098) = (\u22a5 : Ideal S) :=\n by\n rw [\u2190 RingHom.ker_eq_comap_bot, \u2190 RingHom.injective_iff_ker_eq_bot]\n exact IsLocalization.injective S\u2098 hM\n have hS\u2098 : IsJacobson S\u2098 := isJacobson_of_isIntegral' \u03c6' h\u03c6' (isJacobson_localization x)\n refine' eq_bot_iff.mpr (le_trans _ (le_of_eq h\u03d5'))\n rw [\u2190 hS\u2098.out isRadical_bot_of_noZeroDivisors, comap_jacobson]\n exact\n sInf_le_sInf fun j hj =>\n \u27e8bot_le,\n let \u27e8J, hJ\u27e9 := hj\n hJ.2 \u25b8 this J hJ.1.2\u27e9\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nthis : \u2200 (I : Ideal S\u2098), IsMaximal I \u2192 IsMaximal (comap (algebraMap S S\u2098) I)\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nhave h\u03d5' : comap (algebraMap S S\u2098) (\u22a5 : Ideal S\u2098) = (\u22a5 : Ideal S) :=\n by\n rw [\u2190 RingHom.ker_eq_comap_bot, \u2190 RingHom.injective_iff_ker_eq_bot]\n exact IsLocalization.injective S\u2098 hM\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nthis : \u2200 (I : Ideal S\u2098), IsMaximal I \u2192 IsMaximal (comap (algebraMap S S\u2098) I)\n\u22a2 comap (algebraMap S S\u2098) \u22a5 = \u22a5\n[PROOFSTEP]\nrw [\u2190 RingHom.ker_eq_comap_bot, \u2190 RingHom.injective_iff_ker_eq_bot]\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nthis : \u2200 (I : Ideal S\u2098), IsMaximal I \u2192 IsMaximal (comap (algebraMap S S\u2098) I)\n\u22a2 Function.Injective \u2191(algebraMap S S\u2098)\n[PROOFSTEP]\nexact IsLocalization.injective S\u2098 hM\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nthis : \u2200 (I : Ideal S\u2098), IsMaximal I \u2192 IsMaximal (comap (algebraMap S S\u2098) I)\nh\u03d5' : comap (algebraMap S S\u2098) \u22a5 = \u22a5\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nhave hS\u2098 : IsJacobson S\u2098 := isJacobson_of_isIntegral' \u03c6' h\u03c6' (isJacobson_localization x)\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nthis : \u2200 (I : Ideal S\u2098), IsMaximal I \u2192 IsMaximal (comap (algebraMap S S\u2098) I)\nh\u03d5' : comap (algebraMap S S\u2098) \u22a5 = \u22a5\nhS\u2098 : IsJacobson S\u2098\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nrefine' eq_bot_iff.mpr (le_trans _ (le_of_eq h\u03d5'))\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nthis : \u2200 (I : Ideal S\u2098), IsMaximal I \u2192 IsMaximal (comap (algebraMap S S\u2098) I)\nh\u03d5' : comap (algebraMap S S\u2098) \u22a5 = \u22a5\nhS\u2098 : IsJacobson S\u2098\n\u22a2 jacobson \u22a5 \u2264 comap (algebraMap S S\u2098) \u22a5\n[PROOFSTEP]\nrw [\u2190 hS\u2098.out isRadical_bot_of_noZeroDivisors, comap_jacobson]\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nthis : \u2200 (I : Ideal S\u2098), IsMaximal I \u2192 IsMaximal (comap (algebraMap S S\u2098) I)\nh\u03d5' : comap (algebraMap S S\u2098) \u22a5 = \u22a5\nhS\u2098 : IsJacobson S\u2098\n\u22a2 jacobson \u22a5 \u2264 sInf (comap (algebraMap S S\u2098) '' {J | \u22a5 \u2264 J \u2227 IsMaximal J})\n[PROOFSTEP]\nexact\n sInf_le_sInf fun j hj =>\n \u27e8bot_le,\n let \u27e8J, hJ\u27e9 := hj\n hJ.2 \u25b8 this J hJ.1.2\u27e9\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\n\u22a2 \u2200 (I : Ideal S\u2098), IsMaximal I \u2192 IsMaximal (comap (algebraMap S S\u2098) I)\n[PROOFSTEP]\nintro I hI\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nI : Ideal S\u2098\nhI : IsMaximal I\n\u22a2 IsMaximal (comap (algebraMap S S\u2098) I)\n[PROOFSTEP]\nhaveI : (I.comap (algebraMap S S\u2098)).IsPrime := comap_isPrime _ I\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nI : Ideal S\u2098\nhI : IsMaximal I\nthis : IsPrime (comap (algebraMap S S\u2098) I)\n\u22a2 IsMaximal (comap (algebraMap S S\u2098) I)\n[PROOFSTEP]\nhaveI : (I.comap \u03c6').IsPrime := comap_isPrime \u03c6' I\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d\u00b9 : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nI : Ideal S\u2098\nhI : IsMaximal I\nthis\u271d : IsPrime (comap (algebraMap S S\u2098) I)\nthis : IsPrime (comap \u03c6' I)\n\u22a2 IsMaximal (comap (algebraMap S S\u2098) I)\n[PROOFSTEP]\nhaveI : (\u22a5 : Ideal (S \u29f8 I.comap (algebraMap S S\u2098))).IsPrime := bot_prime\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d\u00b2 : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nI : Ideal S\u2098\nhI : IsMaximal I\nthis\u271d\u00b9 : IsPrime (comap (algebraMap S S\u2098) I)\nthis\u271d : IsPrime (comap \u03c6' I)\nthis : IsPrime \u22a5\n\u22a2 IsMaximal (comap (algebraMap S S\u2098) I)\n[PROOFSTEP]\nhave hcomm : \u03c6'.comp (algebraMap R R\u2098) = (algebraMap S S\u2098).comp \u03c6 := IsLocalization.map_comp _\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d\u00b2 : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nI : Ideal S\u2098\nhI : IsMaximal I\nthis\u271d\u00b9 : IsPrime (comap (algebraMap S S\u2098) I)\nthis\u271d : IsPrime (comap \u03c6' I)\nthis : IsPrime \u22a5\nhcomm : RingHom.comp \u03c6' (algebraMap R R\u2098) = RingHom.comp (algebraMap S S\u2098) \u03c6\n\u22a2 IsMaximal (comap (algebraMap S S\u2098) I)\n[PROOFSTEP]\nlet f := quotientMap (I.comap (algebraMap S S\u2098)) \u03c6 le_rfl\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d\u00b2 : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nI : Ideal S\u2098\nhI : IsMaximal I\nthis\u271d\u00b9 : IsPrime (comap (algebraMap S S\u2098) I)\nthis\u271d : IsPrime (comap \u03c6' I)\nthis : IsPrime \u22a5\nhcomm : RingHom.comp \u03c6' (algebraMap R R\u2098) = RingHom.comp (algebraMap S S\u2098) \u03c6\nf : R \u29f8 comap \u03c6 (comap (algebraMap S S\u2098) I) \u2192+* S \u29f8 comap (algebraMap S S\u2098) I :=\n quotientMap (comap (algebraMap S S\u2098) I) \u03c6\n (_ : comap \u03c6 (comap (algebraMap S S\u2098) I) \u2264 comap \u03c6 (comap (algebraMap S S\u2098) I))\n\u22a2 IsMaximal (comap (algebraMap S S\u2098) I)\n[PROOFSTEP]\nlet g := quotientMap I (algebraMap S S\u2098) le_rfl\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d\u00b2 : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nI : Ideal S\u2098\nhI : IsMaximal I\nthis\u271d\u00b9 : IsPrime (comap (algebraMap S S\u2098) I)\nthis\u271d : IsPrime (comap \u03c6' I)\nthis : IsPrime \u22a5\nhcomm : RingHom.comp \u03c6' (algebraMap R R\u2098) = RingHom.comp (algebraMap S S\u2098) \u03c6\nf : R \u29f8 comap \u03c6 (comap (algebraMap S S\u2098) I) \u2192+* S \u29f8 comap (algebraMap S S\u2098) I :=\n quotientMap (comap (algebraMap S S\u2098) I) \u03c6\n (_ : comap \u03c6 (comap (algebraMap S S\u2098) I) \u2264 comap \u03c6 (comap (algebraMap S S\u2098) I))\ng : S \u29f8 comap (algebraMap S S\u2098) I \u2192+* S\u2098 \u29f8 I :=\n quotientMap I (algebraMap S S\u2098) (_ : comap (algebraMap S S\u2098) I \u2264 comap (algebraMap S S\u2098) I)\n\u22a2 IsMaximal (comap (algebraMap S S\u2098) I)\n[PROOFSTEP]\nhave := isMaximal_comap_of_isIntegral_of_isMaximal' \u03c6' h\u03c6' I hI\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d\u00b3 : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nI : Ideal S\u2098\nhI : IsMaximal I\nthis\u271d\u00b2 : IsPrime (comap (algebraMap S S\u2098) I)\nthis\u271d\u00b9 : IsPrime (comap \u03c6' I)\nthis\u271d : IsPrime \u22a5\nhcomm : RingHom.comp \u03c6' (algebraMap R R\u2098) = RingHom.comp (algebraMap S S\u2098) \u03c6\nf : R \u29f8 comap \u03c6 (comap (algebraMap S S\u2098) I) \u2192+* S \u29f8 comap (algebraMap S S\u2098) I :=\n quotientMap (comap (algebraMap S S\u2098) I) \u03c6\n (_ : comap \u03c6 (comap (algebraMap S S\u2098) I) \u2264 comap \u03c6 (comap (algebraMap S S\u2098) I))\ng : S \u29f8 comap (algebraMap S S\u2098) I \u2192+* S\u2098 \u29f8 I :=\n quotientMap I (algebraMap S S\u2098) (_ : comap (algebraMap S S\u2098) I \u2264 comap (algebraMap S S\u2098) I)\nthis : IsMaximal (comap \u03c6' I)\n\u22a2 IsMaximal (comap (algebraMap S S\u2098) I)\n[PROOFSTEP]\nhave := ((isMaximal_iff_isMaximal_disjoint R\u2098 x _).1 this).left\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d\u2074 : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nI : Ideal S\u2098\nhI : IsMaximal I\nthis\u271d\u00b3 : IsPrime (comap (algebraMap S S\u2098) I)\nthis\u271d\u00b2 : IsPrime (comap \u03c6' I)\nthis\u271d\u00b9 : IsPrime \u22a5\nhcomm : RingHom.comp \u03c6' (algebraMap R R\u2098) = RingHom.comp (algebraMap S S\u2098) \u03c6\nf : R \u29f8 comap \u03c6 (comap (algebraMap S S\u2098) I) \u2192+* S \u29f8 comap (algebraMap S S\u2098) I :=\n quotientMap (comap (algebraMap S S\u2098) I) \u03c6\n (_ : comap \u03c6 (comap (algebraMap S S\u2098) I) \u2264 comap \u03c6 (comap (algebraMap S S\u2098) I))\ng : S \u29f8 comap (algebraMap S S\u2098) I \u2192+* S\u2098 \u29f8 I :=\n quotientMap I (algebraMap S S\u2098) (_ : comap (algebraMap S S\u2098) I \u2264 comap (algebraMap S S\u2098) I)\nthis\u271d : IsMaximal (comap \u03c6' I)\nthis : IsMaximal (comap (algebraMap R R\u2098) (comap \u03c6' I))\n\u22a2 IsMaximal (comap (algebraMap S S\u2098) I)\n[PROOFSTEP]\nhave : ((I.comap (algebraMap S S\u2098)).comap \u03c6).IsMaximal := by rwa [comap_comap, hcomm, \u2190 comap_comap] at this \n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d\u2074 : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nI : Ideal S\u2098\nhI : IsMaximal I\nthis\u271d\u00b3 : IsPrime (comap (algebraMap S S\u2098) I)\nthis\u271d\u00b2 : IsPrime (comap \u03c6' I)\nthis\u271d\u00b9 : IsPrime \u22a5\nhcomm : RingHom.comp \u03c6' (algebraMap R R\u2098) = RingHom.comp (algebraMap S S\u2098) \u03c6\nf : R \u29f8 comap \u03c6 (comap (algebraMap S S\u2098) I) \u2192+* S \u29f8 comap (algebraMap S S\u2098) I :=\n quotientMap (comap (algebraMap S S\u2098) I) \u03c6\n (_ : comap \u03c6 (comap (algebraMap S S\u2098) I) \u2264 comap \u03c6 (comap (algebraMap S S\u2098) I))\ng : S \u29f8 comap (algebraMap S S\u2098) I \u2192+* S\u2098 \u29f8 I :=\n quotientMap I (algebraMap S S\u2098) (_ : comap (algebraMap S S\u2098) I \u2264 comap (algebraMap S S\u2098) I)\nthis\u271d : IsMaximal (comap \u03c6' I)\nthis : IsMaximal (comap (algebraMap R R\u2098) (comap \u03c6' I))\n\u22a2 IsMaximal (comap \u03c6 (comap (algebraMap S S\u2098) I))\n[PROOFSTEP]\nrwa [comap_comap, hcomm, \u2190 comap_comap] at this \n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d\u2075 : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nI : Ideal S\u2098\nhI : IsMaximal I\nthis\u271d\u2074 : IsPrime (comap (algebraMap S S\u2098) I)\nthis\u271d\u00b3 : IsPrime (comap \u03c6' I)\nthis\u271d\u00b2 : IsPrime \u22a5\nhcomm : RingHom.comp \u03c6' (algebraMap R R\u2098) = RingHom.comp (algebraMap S S\u2098) \u03c6\nf : R \u29f8 comap \u03c6 (comap (algebraMap S S\u2098) I) \u2192+* S \u29f8 comap (algebraMap S S\u2098) I :=\n quotientMap (comap (algebraMap S S\u2098) I) \u03c6\n (_ : comap \u03c6 (comap (algebraMap S S\u2098) I) \u2264 comap \u03c6 (comap (algebraMap S S\u2098) I))\ng : S \u29f8 comap (algebraMap S S\u2098) I \u2192+* S\u2098 \u29f8 I :=\n quotientMap I (algebraMap S S\u2098) (_ : comap (algebraMap S S\u2098) I \u2264 comap (algebraMap S S\u2098) I)\nthis\u271d\u00b9 : IsMaximal (comap \u03c6' I)\nthis\u271d : IsMaximal (comap (algebraMap R R\u2098) (comap \u03c6' I))\nthis : IsMaximal (comap \u03c6 (comap (algebraMap S S\u2098) I))\n\u22a2 IsMaximal (comap (algebraMap S S\u2098) I)\n[PROOFSTEP]\nrw [\u2190 bot_quotient_isMaximal_iff] at this \u22a2\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d\u2075 : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nI : Ideal S\u2098\nhI : IsMaximal I\nthis\u271d\u2074 : IsPrime (comap (algebraMap S S\u2098) I)\nthis\u271d\u00b3 : IsPrime (comap \u03c6' I)\nthis\u271d\u00b2 : IsPrime \u22a5\nhcomm : RingHom.comp \u03c6' (algebraMap R R\u2098) = RingHom.comp (algebraMap S S\u2098) \u03c6\nf : R \u29f8 comap \u03c6 (comap (algebraMap S S\u2098) I) \u2192+* S \u29f8 comap (algebraMap S S\u2098) I :=\n quotientMap (comap (algebraMap S S\u2098) I) \u03c6\n (_ : comap \u03c6 (comap (algebraMap S S\u2098) I) \u2264 comap \u03c6 (comap (algebraMap S S\u2098) I))\ng : S \u29f8 comap (algebraMap S S\u2098) I \u2192+* S\u2098 \u29f8 I :=\n quotientMap I (algebraMap S S\u2098) (_ : comap (algebraMap S S\u2098) I \u2264 comap (algebraMap S S\u2098) I)\nthis\u271d\u00b9 : IsMaximal (comap \u03c6' I)\nthis\u271d : IsMaximal (comap (algebraMap R R\u2098) (comap \u03c6' I))\nthis : IsMaximal \u22a5\n\u22a2 IsMaximal \u22a5\n[PROOFSTEP]\nrefine'\n isMaximal_of_isIntegral_of_isMaximal_comap' f _ \u22a5\n ((eq_bot_iff.2 (comap_bot_le_of_injective f quotientMap_injective)).symm \u25b8 this)\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d\u2075 : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nI : Ideal S\u2098\nhI : IsMaximal I\nthis\u271d\u2074 : IsPrime (comap (algebraMap S S\u2098) I)\nthis\u271d\u00b3 : IsPrime (comap \u03c6' I)\nthis\u271d\u00b2 : IsPrime \u22a5\nhcomm : RingHom.comp \u03c6' (algebraMap R R\u2098) = RingHom.comp (algebraMap S S\u2098) \u03c6\nf : R \u29f8 comap \u03c6 (comap (algebraMap S S\u2098) I) \u2192+* S \u29f8 comap (algebraMap S S\u2098) I :=\n quotientMap (comap (algebraMap S S\u2098) I) \u03c6\n (_ : comap \u03c6 (comap (algebraMap S S\u2098) I) \u2264 comap \u03c6 (comap (algebraMap S S\u2098) I))\ng : S \u29f8 comap (algebraMap S S\u2098) I \u2192+* S\u2098 \u29f8 I :=\n quotientMap I (algebraMap S S\u2098) (_ : comap (algebraMap S S\u2098) I \u2264 comap (algebraMap S S\u2098) I)\nthis\u271d\u00b9 : IsMaximal (comap \u03c6' I)\nthis\u271d : IsMaximal (comap (algebraMap R R\u2098) (comap \u03c6' I))\nthis : IsMaximal \u22a5\n\u22a2 RingHom.IsIntegral f\n[PROOFSTEP]\nexact\n f.isIntegral_tower_bot_of_isIntegral g quotientMap_injective\n ((comp_quotientMap_eq_of_comp_eq hcomm I).symm \u25b8\n RingHom.isIntegral_trans _ _\n (RingHom.isIntegral_of_surjective _\n (IsLocalization.surjective_quotientMap_of_maximal_of_localization (Submonoid.powers x) R\u2098\n (by rwa [comap_comap, hcomm, \u2190 bot_quotient_isMaximal_iff])))\n (RingHom.isIntegral_quotient_of_isIntegral _ h\u03c6'))\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\u2098\u271d\ninst\u271d\u2079 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsJacobson R\nR\u2098 : Type u_6\nS\u2098 : Type u_7\ninst\u271d\u2075 : CommRing R\u2098\ninst\u271d\u2074 : CommRing S\u2098\n\u03c6 : R \u2192+* S\nh\u03c6 : Function.Injective \u2191\u03c6\nx : R\nhx : x \u2260 0\ninst\u271d\u00b3 : Algebra R R\u2098\ninst\u271d\u00b2 : IsLocalization.Away x R\u2098\ninst\u271d\u00b9 : Algebra S S\u2098\ninst\u271d : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\nh\u03c6' :\n RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\nhM : Submonoid.map \u03c6 (Submonoid.powers x) \u2264 nonZeroDivisors S\nthis\u271d\u2075 : IsDomain S\u2098 := IsLocalization.isDomain_of_le_nonZeroDivisors S hM\n\u03c6' : R\u2098 \u2192+* S\u2098 :=\n IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x)))\nI : Ideal S\u2098\nhI : IsMaximal I\nthis\u271d\u2074 : IsPrime (comap (algebraMap S S\u2098) I)\nthis\u271d\u00b3 : IsPrime (comap \u03c6' I)\nthis\u271d\u00b2 : IsPrime \u22a5\nhcomm : RingHom.comp \u03c6' (algebraMap R R\u2098) = RingHom.comp (algebraMap S S\u2098) \u03c6\nf : R \u29f8 comap \u03c6 (comap (algebraMap S S\u2098) I) \u2192+* S \u29f8 comap (algebraMap S S\u2098) I :=\n quotientMap (comap (algebraMap S S\u2098) I) \u03c6\n (_ : comap \u03c6 (comap (algebraMap S S\u2098) I) \u2264 comap \u03c6 (comap (algebraMap S S\u2098) I))\ng : S \u29f8 comap (algebraMap S S\u2098) I \u2192+* S\u2098 \u29f8 I :=\n quotientMap I (algebraMap S S\u2098) (_ : comap (algebraMap S S\u2098) I \u2264 comap (algebraMap S S\u2098) I)\nthis\u271d\u00b9 : IsMaximal (comap \u03c6' I)\nthis\u271d : IsMaximal (comap (algebraMap R R\u2098) (comap \u03c6' I))\nthis : IsMaximal \u22a5\n\u22a2 IsMaximal (comap (algebraMap R R\u2098) (comap \u03c6' I))\n[PROOFSTEP]\nrwa [comap_comap, hcomm, \u2190 bot_quotient_isMaximal_iff]\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\ninst\u271d\u00b3 : CommRing S\u2098\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\n\u22a2 jacobson P = P\n[PROOFSTEP]\nby_cases Pb : P = \u22a5\n[GOAL]\ncase pos\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\ninst\u271d\u00b3 : CommRing S\u2098\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : P = \u22a5\n\u22a2 jacobson P = P\n[PROOFSTEP]\nexact Pb.symm \u25b8 jacobson_bot_polynomial_of_jacobson_bot (hR.out isRadical_bot_of_noZeroDivisors)\n[GOAL]\ncase neg\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\ninst\u271d\u00b3 : CommRing S\u2098\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\n\u22a2 jacobson P = P\n[PROOFSTEP]\nrw [jacobson_eq_iff_jacobson_quotient_eq_bot]\n[GOAL]\ncase neg\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\ninst\u271d\u00b3 : CommRing S\u2098\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nlet P' := P.comap (C : R \u2192+* R[X])\n[GOAL]\ncase neg\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\ninst\u271d\u00b3 : CommRing S\u2098\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\nP' : Ideal R := comap C P\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nhaveI : P'.IsPrime := comap_isPrime C P\n[GOAL]\ncase neg\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\ninst\u271d\u00b3 : CommRing S\u2098\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\nP' : Ideal R := comap C P\nthis : IsPrime P'\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nhaveI hR' : IsJacobson (R \u29f8 P') := by infer_instance\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\ninst\u271d\u00b3 : CommRing S\u2098\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\nP' : Ideal R := comap C P\nthis : IsPrime P'\n\u22a2 IsJacobson (R \u29f8 P')\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase neg\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\ninst\u271d\u00b3 : CommRing S\u2098\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\nP' : Ideal R := comap C P\nthis : IsPrime P'\nhR' : IsJacobson (R \u29f8 P')\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nobtain \u27e8p, pP, p0\u27e9 := exists_nonzero_mem_of_ne_bot Pb hP\n[GOAL]\ncase neg.intro.intro\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\ninst\u271d\u00b3 : CommRing S\u2098\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\nP' : Ideal R := comap C P\nthis : IsPrime P'\nhR' : IsJacobson (R \u29f8 P')\np : R[X]\npP : p \u2208 P\np0 : Polynomial.map (Quotient.mk (comap C P)) p \u2260 0\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nlet x := (Polynomial.map (Quotient.mk P') p).leadingCoeff\n[GOAL]\ncase neg.intro.intro\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\ninst\u271d\u00b3 : CommRing S\u2098\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\nP' : Ideal R := comap C P\nthis : IsPrime P'\nhR' : IsJacobson (R \u29f8 P')\np : R[X]\npP : p \u2208 P\np0 : Polynomial.map (Quotient.mk (comap C P)) p \u2260 0\nx : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') p)\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nhave hx : x \u2260 0 := by rwa [Ne.def, leadingCoeff_eq_zero]\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\ninst\u271d\u00b3 : CommRing S\u2098\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\nP' : Ideal R := comap C P\nthis : IsPrime P'\nhR' : IsJacobson (R \u29f8 P')\np : R[X]\npP : p \u2208 P\np0 : Polynomial.map (Quotient.mk (comap C P)) p \u2260 0\nx : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') p)\n\u22a2 x \u2260 0\n[PROOFSTEP]\nrwa [Ne.def, leadingCoeff_eq_zero]\n[GOAL]\ncase neg.intro.intro\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\ninst\u271d\u00b3 : CommRing S\u2098\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\nP' : Ideal R := comap C P\nthis : IsPrime P'\nhR' : IsJacobson (R \u29f8 P')\np : R[X]\npP : p \u2208 P\np0 : Polynomial.map (Quotient.mk (comap C P)) p \u2260 0\nx : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') p)\nhx : x \u2260 0\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nlet \u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := Ideal.quotientMap P (C : R \u2192+* R[X]) le_rfl\n[GOAL]\ncase neg.intro.intro\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\ninst\u271d\u00b3 : CommRing S\u2098\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\nP' : Ideal R := comap C P\nthis : IsPrime P'\nhR' : IsJacobson (R \u29f8 P')\np : R[X]\npP : p \u2208 P\np0 : Polynomial.map (Quotient.mk (comap C P)) p \u2260 0\nx : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') p)\nhx : x \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nlet h\u03c6 : Function.Injective \u2191\u03c6 := quotientMap_injective\n[GOAL]\ncase neg.intro.intro\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\ninst\u271d\u00b3 : CommRing S\u2098\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\nP' : Ideal R := comap C P\nthis : IsPrime P'\nhR' : IsJacobson (R \u29f8 P')\np : R[X]\npP : p \u2208 P\np0 : Polynomial.map (Quotient.mk (comap C P)) p \u2260 0\nx : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') p)\nhx : x \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nh\u03c6 : Function.Injective \u2191\u03c6 := quotientMap_injective\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nlet R\u2098 := Localization.Away x\n[GOAL]\ncase neg.intro.intro\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\u271d\ninst\u271d\u00b3 : CommRing S\u2098\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\nP' : Ideal R := comap C P\nthis : IsPrime P'\nhR' : IsJacobson (R \u29f8 P')\np : R[X]\npP : p \u2208 P\np0 : Polynomial.map (Quotient.mk (comap C P)) p \u2260 0\nx : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') p)\nhx : x \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nh\u03c6 : Function.Injective \u2191\u03c6 := quotientMap_injective\nR\u2098 : Type u_5 := Localization.Away x\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nlet S\u2098 := (Localization ((Submonoid.powers x).map \u03c6 : Submonoid (R[X] \u29f8 P)))\n[GOAL]\ncase neg.intro.intro\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\u271d\ninst\u271d\u00b3 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\nP' : Ideal R := comap C P\nthis : IsPrime P'\nhR' : IsJacobson (R \u29f8 P')\np : R[X]\npP : p \u2208 P\np0 : Polynomial.map (Quotient.mk (comap C P)) p \u2260 0\nx : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') p)\nhx : x \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nh\u03c6 : Function.Injective \u2191\u03c6 := quotientMap_injective\nR\u2098 : Type u_5 := Localization.Away x\nS\u2098 : Type u_5 := Localization (Submonoid.map \u03c6 (Submonoid.powers x))\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nrefine' jacobson_bot_of_integral_localization (S := R[X] \u29f8 P) (R := R \u29f8 P') R\u2098 S\u2098 _ h\u03c6 _ hx _\n[GOAL]\ncase neg.intro.intro\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\u271d\ninst\u271d\u00b3 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\nP' : Ideal R := comap C P\nthis : IsPrime P'\nhR' : IsJacobson (R \u29f8 P')\np : R[X]\npP : p \u2208 P\np0 : Polynomial.map (Quotient.mk (comap C P)) p \u2260 0\nx : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') p)\nhx : x \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nh\u03c6 : Function.Injective \u2191\u03c6 := quotientMap_injective\nR\u2098 : Type u_5 := Localization.Away x\nS\u2098 : Type u_5 := Localization (Submonoid.map \u03c6 (Submonoid.powers x))\n\u22a2 RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\n[PROOFSTEP]\nhaveI islocS\u2098 : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098 := by infer_instance\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\u271d\ninst\u271d\u00b3 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\nP' : Ideal R := comap C P\nthis : IsPrime P'\nhR' : IsJacobson (R \u29f8 P')\np : R[X]\npP : p \u2208 P\np0 : Polynomial.map (Quotient.mk (comap C P)) p \u2260 0\nx : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') p)\nhx : x \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nh\u03c6 : Function.Injective \u2191\u03c6 := quotientMap_injective\nR\u2098 : Type u_5 := Localization.Away x\nS\u2098 : Type u_5 := Localization (Submonoid.map \u03c6 (Submonoid.powers x))\n\u22a2 IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase neg.intro.intro\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : IsDomain S\nR\u2098\u271d : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u2074 : CommRing R\u2098\u271d\ninst\u271d\u00b3 : CommRing S\u2098\u271d\nR : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhR : IsJacobson R\nP : Ideal R[X]\ninst\u271d : IsPrime P\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nPb : \u00acP = \u22a5\nP' : Ideal R := comap C P\nthis : IsPrime P'\nhR' : IsJacobson (R \u29f8 P')\np : R[X]\npP : p \u2208 P\np0 : Polynomial.map (Quotient.mk (comap C P)) p \u2260 0\nx : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') p)\nhx : x \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nh\u03c6 : Function.Injective \u2191\u03c6 := quotientMap_injective\nR\u2098 : Type u_5 := Localization.Away x\nS\u2098 : Type u_5 := Localization (Submonoid.map \u03c6 (Submonoid.powers x))\nislocS\u2098 : IsLocalization (Submonoid.map \u03c6 (Submonoid.powers x)) S\u2098\n\u22a2 RingHom.IsIntegral\n (IsLocalization.map S\u2098 \u03c6 (_ : Submonoid.powers x \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 (Submonoid.powers x))))\n[PROOFSTEP]\nexact @isIntegral_isLocalization_polynomial_quotient R _ R\u2098 S\u2098 _ _ P p pP _ _ _ islocS\u2098\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\n\u22a2 IsJacobson R[X]\n[PROOFSTEP]\nrw [isJacobson_iff_prime_eq]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\n\u22a2 \u2200 (P : Ideal R[X]), IsPrime P \u2192 jacobson P = P\n[PROOFSTEP]\nintro I hI\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\n\u22a2 jacobson I = I\n[PROOFSTEP]\nlet R' : Subring (R[X] \u29f8 I) := ((Quotient.mk I).comp C).range\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\n\u22a2 jacobson I = I\n[PROOFSTEP]\nlet i : R \u2192+* R' := ((Quotient.mk I).comp C).rangeRestrict\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\n\u22a2 jacobson I = I\n[PROOFSTEP]\nhave hi : Function.Surjective \u2191i := ((Quotient.mk I).comp C).rangeRestrict_surjective\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\n\u22a2 jacobson I = I\n[PROOFSTEP]\nhave hi' : RingHom.ker (mapRingHom i) \u2264 I\n[GOAL]\ncase hi'\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\n\u22a2 RingHom.ker (mapRingHom i) \u2264 I\n[PROOFSTEP]\nintro f hf\n[GOAL]\ncase hi'\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nf : R[X]\nhf : f \u2208 RingHom.ker (mapRingHom i)\n\u22a2 f \u2208 I\n[PROOFSTEP]\napply polynomial_mem_ideal_of_coeff_mem_ideal I f\n[GOAL]\ncase hi'\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nf : R[X]\nhf : f \u2208 RingHom.ker (mapRingHom i)\n\u22a2 \u2200 (n : \u2115), coeff f n \u2208 comap C I\n[PROOFSTEP]\nintro n\n[GOAL]\ncase hi'\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nf : R[X]\nhf : f \u2208 RingHom.ker (mapRingHom i)\nn : \u2115\n\u22a2 coeff f n \u2208 comap C I\n[PROOFSTEP]\nreplace hf := congrArg (fun g : Polynomial ((Quotient.mk I).comp C).range => g.coeff n) hf\n[GOAL]\ncase hi'\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nf : R[X]\nn : \u2115\nhf : (fun g => coeff g n) (\u2191(mapRingHom i) f) = (fun g => coeff g n) 0\n\u22a2 coeff f n \u2208 comap C I\n[PROOFSTEP]\nchange (Polynomial.map ((Quotient.mk I).comp C).rangeRestrict f).coeff n = 0 at hf \n[GOAL]\ncase hi'\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nf : R[X]\nn : \u2115\nhf : coeff (Polynomial.map (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)) f) n = 0\n\u22a2 coeff f n \u2208 comap C I\n[PROOFSTEP]\nrw [coeff_map, Subtype.ext_iff] at hf \n[GOAL]\ncase hi'\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nf : R[X]\nn : \u2115\nhf : \u2191(\u2191(RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)) (coeff f n)) = \u21910\n\u22a2 coeff f n \u2208 comap C I\n[PROOFSTEP]\nrwa [mem_comap, \u2190 Quotient.eq_zero_iff_mem, \u2190 RingHom.comp_apply]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nhi' : RingHom.ker (mapRingHom i) \u2264 I\n\u22a2 jacobson I = I\n[PROOFSTEP]\nhave R'_jacob : IsJacobson R' := isJacobson_of_surjective \u27e8i, hi\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nhi' : RingHom.ker (mapRingHom i) \u2264 I\nR'_jacob : IsJacobson { x // x \u2208 R' }\n\u22a2 jacobson I = I\n[PROOFSTEP]\nlet J := map (mapRingHom i) I\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nhi' : RingHom.ker (mapRingHom i) \u2264 I\nR'_jacob : IsJacobson { x // x \u2208 R' }\nJ : Ideal { x // x \u2208 R' }[X] := map (mapRingHom i) I\n\u22a2 jacobson I = I\n[PROOFSTEP]\nhave h_surj : Function.Surjective (mapRingHom i) := Polynomial.map_surjective i hi\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nhi' : RingHom.ker (mapRingHom i) \u2264 I\nR'_jacob : IsJacobson { x // x \u2208 R' }\nJ : Ideal { x // x \u2208 R' }[X] := map (mapRingHom i) I\nh_surj : Function.Surjective \u2191(mapRingHom i)\n\u22a2 jacobson I = I\n[PROOFSTEP]\nhave : IsPrime J := map_isPrime_of_surjective h_surj hi'\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nhi' : RingHom.ker (mapRingHom i) \u2264 I\nR'_jacob : IsJacobson { x // x \u2208 R' }\nJ : Ideal { x // x \u2208 R' }[X] := map (mapRingHom i) I\nh_surj : Function.Surjective \u2191(mapRingHom i)\nthis : IsPrime J\n\u22a2 jacobson I = I\n[PROOFSTEP]\nsuffices h : J.jacobson = J\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nhi' : RingHom.ker (mapRingHom i) \u2264 I\nR'_jacob : IsJacobson { x // x \u2208 R' }\nJ : Ideal { x // x \u2208 R' }[X] := map (mapRingHom i) I\nh_surj : Function.Surjective \u2191(mapRingHom i)\nthis : IsPrime J\nh : jacobson J = J\n\u22a2 jacobson I = I\n[PROOFSTEP]\nreplace h := congrArg (comap (Polynomial.mapRingHom i)) h\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nhi' : RingHom.ker (mapRingHom i) \u2264 I\nR'_jacob : IsJacobson { x // x \u2208 R' }\nJ : Ideal { x // x \u2208 R' }[X] := map (mapRingHom i) I\nh_surj : Function.Surjective \u2191(mapRingHom i)\nthis : IsPrime J\nh : comap (mapRingHom i) (jacobson J) = comap (mapRingHom i) J\n\u22a2 jacobson I = I\n[PROOFSTEP]\nrw [\u2190 map_jacobson_of_surjective h_surj hi', comap_map_of_surjective _ h_surj, comap_map_of_surjective _ h_surj] at h \n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nhi' : RingHom.ker (mapRingHom i) \u2264 I\nR'_jacob : IsJacobson { x // x \u2208 R' }\nJ : Ideal { x // x \u2208 R' }[X] := map (mapRingHom i) I\nh_surj : Function.Surjective \u2191(mapRingHom i)\nthis : IsPrime J\nh : jacobson I \u2294 comap (mapRingHom i) \u22a5 = I \u2294 comap (mapRingHom i) \u22a5\n\u22a2 jacobson I = I\n[PROOFSTEP]\nrefine le_antisymm ?_ le_jacobson\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nhi' : RingHom.ker (mapRingHom i) \u2264 I\nR'_jacob : IsJacobson { x // x \u2208 R' }\nJ : Ideal { x // x \u2208 R' }[X] := map (mapRingHom i) I\nh_surj : Function.Surjective \u2191(mapRingHom i)\nthis : IsPrime J\nh : jacobson I \u2294 comap (mapRingHom i) \u22a5 = I \u2294 comap (mapRingHom i) \u22a5\n\u22a2 jacobson I \u2264 I\n[PROOFSTEP]\nexact le_trans (le_sup_of_le_left le_rfl) (le_trans (le_of_eq h) (sup_le le_rfl hi'))\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nhi' : RingHom.ker (mapRingHom i) \u2264 I\nR'_jacob : IsJacobson { x // x \u2208 R' }\nJ : Ideal { x // x \u2208 R' }[X] := map (mapRingHom i) I\nh_surj : Function.Surjective \u2191(mapRingHom i)\nthis : IsPrime J\n\u22a2 jacobson J = J\n[PROOFSTEP]\napply isJacobson_polynomial_of_domain R' J\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nhR : IsJacobson R\nI : Ideal R[X]\nhI : IsPrime I\nR' : Subring (R[X] \u29f8 I) := RingHom.range (RingHom.comp (Quotient.mk I) C)\ni : R \u2192+* { x // x \u2208 R' } := RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi : Function.Surjective \u2191i\nhi' : RingHom.ker (mapRingHom i) \u2264 I\nR'_jacob : IsJacobson { x // x \u2208 R' }\nJ : Ideal { x // x \u2208 R' }[X] := map (mapRingHom i) I\nh_surj : Function.Surjective \u2191(mapRingHom i)\nthis : IsPrime J\n\u22a2 \u2200 (x : { x // x \u2208 R' }), \u2191C x \u2208 J \u2192 x = 0\n[PROOFSTEP]\nexact (eq_zero_of_polynomial_mem_map_range I)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\n\u22a2 IsJacobson R[X] \u2194 IsJacobson R\n[PROOFSTEP]\nrefine' \u27e8_, isJacobson_polynomial_of_isJacobson\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\n\u22a2 IsJacobson R[X] \u2192 IsJacobson R\n[PROOFSTEP]\nintro H\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nH : IsJacobson R[X]\n\u22a2 IsJacobson R\n[PROOFSTEP]\nexact\n isJacobson_of_surjective\n \u27e8eval\u2082RingHom (RingHom.id _) 1, fun x => \u27e8C x, by simp only [coe_eval\u2082RingHom, RingHom.id_apply, eval\u2082_C]\u27e9\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9 : CommRing R\u2098\ninst\u271d : CommRing S\u2098\nH : IsJacobson R[X]\nx : R\n\u22a2 \u2191(eval\u2082RingHom (RingHom.id R) 1) (\u2191C x) = x\n[PROOFSTEP]\nsimp only [coe_eval\u2082RingHom, RingHom.id_apply, eval\u2082_C]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\n\u22a2 IsMaximal (comap C P)\n[PROOFSTEP]\nlet P' := comap (C : R \u2192+* R[X]) P\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\n\u22a2 IsMaximal (comap C P)\n[PROOFSTEP]\nhaveI hP'_prime : P'.IsPrime := comap_isPrime C P\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\n\u22a2 IsMaximal (comap C P)\n[PROOFSTEP]\nobtain \u27e8\u27e8m, hmem_P\u27e9, hm\u27e9 := Submodule.nonzero_mem_of_bot_lt (bot_lt_of_maximal P polynomial_not_isField)\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\n\u22a2 IsMaximal (comap C P)\n[PROOFSTEP]\nhave hm' : m \u2260 0 := by simpa [Submodule.coe_eq_zero] using hm\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\n\u22a2 m \u2260 0\n[PROOFSTEP]\nsimpa [Submodule.coe_eq_zero] using hm\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u22a2 IsMaximal (comap C P)\n[PROOFSTEP]\nlet \u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P (C : R \u2192+* R[X]) le_rfl\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u22a2 IsMaximal (comap C P)\n[PROOFSTEP]\nlet a : R \u29f8 P' := (m.map (Quotient.mk P')).leadingCoeff\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\n\u22a2 IsMaximal (comap C P)\n[PROOFSTEP]\nlet M : Submonoid (R \u29f8 P') := Submonoid.powers a\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u22a2 IsMaximal (comap C P)\n[PROOFSTEP]\nrw [\u2190 bot_quotient_isMaximal_iff]\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u22a2 IsMaximal \u22a5\n[PROOFSTEP]\nhave hp0 : a \u2260 0 := fun hp0' =>\n hm' <|\n map_injective (Quotient.mk (P.comap (C : R \u2192+* R[X]) : Ideal R))\n ((injective_iff_map_eq_zero (Quotient.mk (P.comap (C : R \u2192+* R[X]) : Ideal R))).2 fun x hx => by\n rwa [Quotient.eq_zero_iff_mem, (by rwa [eq_bot_iff] : (P.comap C : Ideal R) = \u22a5)] at hx )\n (by simpa only [leadingCoeff_eq_zero, Polynomial.map_zero] using hp0')\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0' : a = 0\nx : R\nhx : \u2191(Quotient.mk (comap C P)) x = 0\n\u22a2 x = 0\n[PROOFSTEP]\nrwa [Quotient.eq_zero_iff_mem, (by rwa [eq_bot_iff] : (P.comap C : Ideal R) = \u22a5)] at hx \n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0' : a = 0\nx : R\nhx : x \u2208 comap C P\n\u22a2 comap C P = \u22a5\n[PROOFSTEP]\nrwa [eq_bot_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0' : a = 0\n\u22a2 Polynomial.map (Quotient.mk (comap C P)) m = Polynomial.map (Quotient.mk (comap C P)) 0\n[PROOFSTEP]\nsimpa only [leadingCoeff_eq_zero, Polynomial.map_zero] using hp0'\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\n\u22a2 IsMaximal \u22a5\n[PROOFSTEP]\nhave hM : (0 : R \u29f8 P') \u2209 M := fun \u27e8n, hn\u27e9 => hp0 (pow_eq_zero hn)\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\n\u22a2 IsMaximal \u22a5\n[PROOFSTEP]\nsuffices (\u22a5 : Ideal (Localization M)).IsMaximal\n by\n rw [\u2190\n IsLocalization.comap_map_of_isPrime_disjoint M (Localization M) \u22a5 bot_prime\n (disjoint_iff_inf_le.mpr fun x hx => hM (hx.2 \u25b8 hx.1))]\n refine' ((isMaximal_iff_isMaximal_disjoint (Localization M) _ _).mp (by rwa [map_bot])).1\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nthis : IsMaximal \u22a5\n\u22a2 IsMaximal \u22a5\n[PROOFSTEP]\nrw [\u2190\n IsLocalization.comap_map_of_isPrime_disjoint M (Localization M) \u22a5 bot_prime\n (disjoint_iff_inf_le.mpr fun x hx => hM (hx.2 \u25b8 hx.1))]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nthis : IsMaximal \u22a5\n\u22a2 IsMaximal (comap (algebraMap (R \u29f8 P') (Localization M)) (map (algebraMap (R \u29f8 P') (Localization M)) \u22a5))\n[PROOFSTEP]\nrefine' ((isMaximal_iff_isMaximal_disjoint (Localization M) _ _).mp (by rwa [map_bot])).1\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nthis : IsMaximal \u22a5\n\u22a2 IsMaximal (map (algebraMap (R \u29f8 P') (Localization M)) \u22a5)\n[PROOFSTEP]\nrwa [map_bot]\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\n\u22a2 IsMaximal \u22a5\n[PROOFSTEP]\nlet M' : Submonoid (R[X] \u29f8 P) := M.map \u03c6\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\n\u22a2 IsMaximal \u22a5\n[PROOFSTEP]\nhave hM' : (0 : R[X] \u29f8 P) \u2209 M' := fun \u27e8z, hz\u27e9 => hM (quotientMap_injective (_root_.trans hz.2 \u03c6.map_zero.symm) \u25b8 hz.1)\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nhM' : \u00ac0 \u2208 M'\n\u22a2 IsMaximal \u22a5\n[PROOFSTEP]\nhaveI : IsDomain (Localization M') := IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hM')\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nhM' : \u00ac0 \u2208 M'\nthis : IsDomain (Localization M')\n\u22a2 IsMaximal \u22a5\n[PROOFSTEP]\nsuffices (\u22a5 : Ideal (Localization M')).IsMaximal\n by\n rw [le_antisymm bot_le\n (comap_bot_le_of_injective _\n (IsLocalization.map_injective_of_injective M (Localization M) (Localization M') quotientMap_injective))]\n refine' isMaximal_comap_of_isIntegral_of_isMaximal' _ _ \u22a5 this\n have isloc : IsLocalization (Submonoid.map \u03c6 M) (Localization M') := by infer_instance\n exact @isIntegral_isLocalization_polynomial_quotient R _ (Localization M) (Localization M') _ _ P m hmem_P _ _ _ isloc\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nhM' : \u00ac0 \u2208 M'\nthis\u271d : IsDomain (Localization M')\nthis : IsMaximal \u22a5\n\u22a2 IsMaximal \u22a5\n[PROOFSTEP]\nrw [le_antisymm bot_le\n (comap_bot_le_of_injective _\n (IsLocalization.map_injective_of_injective M (Localization M) (Localization M') quotientMap_injective))]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nhM' : \u00ac0 \u2208 M'\nthis\u271d : IsDomain (Localization M')\nthis : IsMaximal \u22a5\n\u22a2 IsMaximal\n (comap\n (IsLocalization.map (Localization M') (quotientMap P C (_ : comap C P \u2264 comap C P))\n (_ :\n M \u2264\n Submonoid.comap (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P)) M)))\n \u22a5)\n[PROOFSTEP]\nrefine' isMaximal_comap_of_isIntegral_of_isMaximal' _ _ \u22a5 this\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nhM' : \u00ac0 \u2208 M'\nthis\u271d : IsDomain (Localization M')\nthis : IsMaximal \u22a5\n\u22a2 RingHom.IsIntegral\n (IsLocalization.map (Localization M') (quotientMap P C (_ : comap C P \u2264 comap C P))\n (_ :\n M \u2264\n Submonoid.comap (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P)) M)))\n[PROOFSTEP]\nhave isloc : IsLocalization (Submonoid.map \u03c6 M) (Localization M') := by infer_instance\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nhM' : \u00ac0 \u2208 M'\nthis\u271d : IsDomain (Localization M')\nthis : IsMaximal \u22a5\n\u22a2 IsLocalization (Submonoid.map \u03c6 M) (Localization M')\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nhM' : \u00ac0 \u2208 M'\nthis\u271d : IsDomain (Localization M')\nthis : IsMaximal \u22a5\nisloc : IsLocalization (Submonoid.map \u03c6 M) (Localization M')\n\u22a2 RingHom.IsIntegral\n (IsLocalization.map (Localization M') (quotientMap P C (_ : comap C P \u2264 comap C P))\n (_ :\n M \u2264\n Submonoid.comap (quotientMap P C (_ : comap C P \u2264 comap C P))\n (Submonoid.map (quotientMap P C (_ : comap C P \u2264 comap C P)) M)))\n[PROOFSTEP]\nexact @isIntegral_isLocalization_polynomial_quotient R _ (Localization M) (Localization M') _ _ P m hmem_P _ _ _ isloc\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nhM' : \u00ac0 \u2208 M'\nthis : IsDomain (Localization M')\n\u22a2 IsMaximal \u22a5\n[PROOFSTEP]\nrw [(map_bot.symm : (\u22a5 : Ideal (Localization M')) = map (algebraMap (R[X] \u29f8 P) (Localization M')) \u22a5)]\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nhM' : \u00ac0 \u2208 M'\nthis : IsDomain (Localization M')\n\u22a2 IsMaximal (map (algebraMap (R[X] \u29f8 P) (Localization M')) \u22a5)\n[PROOFSTEP]\nlet bot_maximal := (bot_quotient_isMaximal_iff _).mpr hP\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nhM' : \u00ac0 \u2208 M'\nthis : IsDomain (Localization M')\nbot_maximal : IsMaximal \u22a5 := Iff.mpr (bot_quotient_isMaximal_iff P) hP\n\u22a2 IsMaximal (map (algebraMap (R[X] \u29f8 P) (Localization M')) \u22a5)\n[PROOFSTEP]\nrefine' map.isMaximal (algebraMap (R[X] \u29f8 P) (Localization M')) _ bot_maximal\n[GOAL]\ncase intro.mk\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nhM' : \u00ac0 \u2208 M'\nthis : IsDomain (Localization M')\nbot_maximal : IsMaximal \u22a5 := Iff.mpr (bot_quotient_isMaximal_iff P) hP\n\u22a2 Function.Bijective \u2191(algebraMap (R[X] \u29f8 P) (Localization M'))\n[PROOFSTEP]\napply IsField.localization_map_bijective hM'\n[GOAL]\ncase intro.mk.hR\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\nhP'_prime : IsPrime P'\nm : R[X]\nhmem_P : m \u2208 P\nhm : { val := m, property := hmem_P } \u2260 0\nhm' : m \u2260 0\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') m)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\nhp0 : a \u2260 0\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nhM' : \u00ac0 \u2208 M'\nthis : IsDomain (Localization M')\nbot_maximal : IsMaximal \u22a5 := Iff.mpr (bot_quotient_isMaximal_iff P) hP\n\u22a2 IsField (R[X] \u29f8 P)\n[PROOFSTEP]\nrwa [\u2190 Quotient.maximal_ideal_iff_isField_quotient, \u2190 bot_quotient_isMaximal_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\n\u22a2 RingHom.IsIntegral (RingHom.comp (Quotient.mk P) C)\n[PROOFSTEP]\nrefine' (isIntegral_quotientMap_iff _).mp _\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\n\u22a2 RingHom.IsIntegral (quotientMap P C (_ : comap C P \u2264 comap C P))\n[PROOFSTEP]\nlet P' : Ideal R := P.comap C\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\n\u22a2 RingHom.IsIntegral (quotientMap P C (_ : comap C P \u2264 comap C P))\n[PROOFSTEP]\nobtain \u27e8pX, hpX, hp0\u27e9 := exists_nonzero_mem_of_ne_bot (ne_of_lt (bot_lt_of_maximal P polynomial_not_isField)).symm hP'\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\n\u22a2 RingHom.IsIntegral (quotientMap P C (_ : comap C P \u2264 comap C P))\n[PROOFSTEP]\nlet a : R \u29f8 P' := (pX.map (Quotient.mk P')).leadingCoeff\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\n\u22a2 RingHom.IsIntegral (quotientMap P C (_ : comap C P \u2264 comap C P))\n[PROOFSTEP]\nlet M : Submonoid (R \u29f8 P') := Submonoid.powers a\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u22a2 RingHom.IsIntegral (quotientMap P C (_ : comap C P \u2264 comap C P))\n[PROOFSTEP]\nlet \u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C le_rfl\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\n\u22a2 RingHom.IsIntegral (quotientMap P C (_ : comap C P \u2264 comap C P))\n[PROOFSTEP]\nhaveI hP'_prime : P'.IsPrime := comap_isPrime C P\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nhP'_prime : IsPrime P'\n\u22a2 RingHom.IsIntegral (quotientMap P C (_ : comap C P \u2264 comap C P))\n[PROOFSTEP]\nhave hM : (0 : R \u29f8 P') \u2209 M := fun \u27e8n, hn\u27e9 => hp0 <| leadingCoeff_eq_zero.mp (pow_eq_zero hn)\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nhP'_prime : IsPrime P'\nhM : \u00ac0 \u2208 M\n\u22a2 RingHom.IsIntegral (quotientMap P C (_ : comap C P \u2264 comap C P))\n[PROOFSTEP]\nlet M' : Submonoid (R[X] \u29f8 P) := M.map \u03c6\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nhP'_prime : IsPrime P'\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\n\u22a2 RingHom.IsIntegral (quotientMap P C (_ : comap C P \u2264 comap C P))\n[PROOFSTEP]\nrefine' \u03c6.isIntegral_tower_bot_of_isIntegral (algebraMap _ (Localization M')) _ _\n[GOAL]\ncase intro.intro.refine'_1\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nhP'_prime : IsPrime P'\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\n\u22a2 Function.Injective \u2191(algebraMap (R[X] \u29f8 P) (Localization M'))\n[PROOFSTEP]\nrefine'\n IsLocalization.injective (Localization M') (show M' \u2264 _ from le_nonZeroDivisors_of_noZeroDivisors fun hM' => hM _)\n[GOAL]\ncase intro.intro.refine'_1\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nhP'_prime : IsPrime P'\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nhM' : 0 \u2208 M'\n\u22a2 0 \u2208 M\n[PROOFSTEP]\nexact\n let \u27e8z, zM, z0\u27e9 := hM'\n quotientMap_injective (_root_.trans z0 \u03c6.map_zero.symm) \u25b8 zM\n[GOAL]\ncase intro.intro.refine'_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nhP'_prime : IsPrime P'\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\n\u22a2 RingHom.IsIntegral (RingHom.comp (algebraMap (R[X] \u29f8 P) (Localization M')) \u03c6)\n[PROOFSTEP]\nsuffices :\n RingHom.comp (algebraMap (R[X] \u29f8 P) (Localization M')) \u03c6 =\n (IsLocalization.map (Localization M') \u03c6 M.le_comap_map).comp (algebraMap (R \u29f8 P') (Localization M))\n[GOAL]\ncase intro.intro.refine'_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nhP'_prime : IsPrime P'\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nthis :\n RingHom.comp (algebraMap (R[X] \u29f8 P) (Localization M')) \u03c6 =\n RingHom.comp (IsLocalization.map (Localization M') \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M)))\n (algebraMap (R \u29f8 P') (Localization M))\n\u22a2 RingHom.IsIntegral (RingHom.comp (algebraMap (R[X] \u29f8 P) (Localization M')) \u03c6)\ncase this\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nhP'_prime : IsPrime P'\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\n\u22a2 RingHom.comp (algebraMap (R[X] \u29f8 P) (Localization M')) \u03c6 =\n RingHom.comp (IsLocalization.map (Localization M') \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M)))\n (algebraMap (R \u29f8 P') (Localization M))\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase intro.intro.refine'_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nhP'_prime : IsPrime P'\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nthis :\n RingHom.comp (algebraMap (R[X] \u29f8 P) (Localization M')) \u03c6 =\n RingHom.comp (IsLocalization.map (Localization M') \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M)))\n (algebraMap (R \u29f8 P') (Localization M))\n\u22a2 RingHom.IsIntegral\n (RingHom.comp (IsLocalization.map (Localization M') \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M)))\n (algebraMap (R \u29f8 P') (Localization M)))\ncase this\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nhP'_prime : IsPrime P'\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\n\u22a2 RingHom.comp (algebraMap (R[X] \u29f8 P) (Localization M')) \u03c6 =\n RingHom.comp (IsLocalization.map (Localization M') \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M)))\n (algebraMap (R \u29f8 P') (Localization M))\n[PROOFSTEP]\nrefine'\n RingHom.isIntegral_trans (algebraMap (R \u29f8 P') (Localization M))\n (IsLocalization.map (Localization M') \u03c6 M.le_comap_map) _ _\n[GOAL]\ncase intro.intro.refine'_2.refine'_1\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nhP'_prime : IsPrime P'\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nthis :\n RingHom.comp (algebraMap (R[X] \u29f8 P) (Localization M')) \u03c6 =\n RingHom.comp (IsLocalization.map (Localization M') \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M)))\n (algebraMap (R \u29f8 P') (Localization M))\n\u22a2 RingHom.IsIntegral (algebraMap (R \u29f8 P') (Localization M))\n[PROOFSTEP]\nexact\n (algebraMap (R \u29f8 P') (Localization M)).isIntegral_of_surjective\n (IsField.localization_map_bijective hM\n ((Quotient.maximal_ideal_iff_isField_quotient _).mp (isMaximal_comap_C_of_isMaximal P hP'))).2\n[GOAL]\ncase intro.intro.refine'_2.refine'_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nhP'_prime : IsPrime P'\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nthis :\n RingHom.comp (algebraMap (R[X] \u29f8 P) (Localization M')) \u03c6 =\n RingHom.comp (IsLocalization.map (Localization M') \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M)))\n (algebraMap (R \u29f8 P') (Localization M))\n\u22a2 RingHom.IsIntegral (IsLocalization.map (Localization M') \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M)))\n[PROOFSTEP]\nhave isloc : IsLocalization M' (Localization M') := by infer_instance\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nhP'_prime : IsPrime P'\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nthis :\n RingHom.comp (algebraMap (R[X] \u29f8 P) (Localization M')) \u03c6 =\n RingHom.comp (IsLocalization.map (Localization M') \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M)))\n (algebraMap (R \u29f8 P') (Localization M))\n\u22a2 IsLocalization M' (Localization M')\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase intro.intro.refine'_2.refine'_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nhP'_prime : IsPrime P'\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\nthis :\n RingHom.comp (algebraMap (R[X] \u29f8 P) (Localization M')) \u03c6 =\n RingHom.comp (IsLocalization.map (Localization M') \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M)))\n (algebraMap (R \u29f8 P') (Localization M))\nisloc : IsLocalization M' (Localization M')\n\u22a2 RingHom.IsIntegral (IsLocalization.map (Localization M') \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M)))\n[PROOFSTEP]\nexact @isIntegral_isLocalization_polynomial_quotient R _ (Localization M) (Localization M') _ _ P pX hpX _ _ _ isloc\n[GOAL]\ncase this\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\ninst\u271d : Nontrivial R\nhR : IsJacobson R\nhP' : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX \u2208 P\nhp0 : Polynomial.map (Quotient.mk (comap C P)) pX \u2260 0\na : R \u29f8 P' := Polynomial.leadingCoeff (Polynomial.map (Quotient.mk P') pX)\nM : Submonoid (R \u29f8 P') := Submonoid.powers a\n\u03c6 : R \u29f8 P' \u2192+* R[X] \u29f8 P := quotientMap P C (_ : P' \u2264 P')\nhP'_prime : IsPrime P'\nhM : \u00ac0 \u2208 M\nM' : Submonoid (R[X] \u29f8 P) := Submonoid.map \u03c6 M\n\u22a2 RingHom.comp (algebraMap (R[X] \u29f8 P) (Localization M')) \u03c6 =\n RingHom.comp (IsLocalization.map (Localization M') \u03c6 (_ : M \u2264 Submonoid.comap \u03c6 (Submonoid.map \u03c6 M)))\n (algebraMap (R \u29f8 P') (Localization M))\n[PROOFSTEP]\nrw [IsLocalization.map_comp M.le_comap_map]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\n\u22a2 RingHom.IsIntegral (RingHom.comp (Quotient.mk P) C)\n[PROOFSTEP]\nlet P' : Ideal R := P.comap C\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\n\u22a2 RingHom.IsIntegral (RingHom.comp (Quotient.mk P) C)\n[PROOFSTEP]\nhaveI : P'.IsPrime := comap_isPrime C P\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis : IsPrime P'\n\u22a2 RingHom.IsIntegral (RingHom.comp (Quotient.mk P) C)\n[PROOFSTEP]\nlet f : R[X] \u2192+* Polynomial (R \u29f8 P') := Polynomial.mapRingHom (Quotient.mk P')\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\n\u22a2 RingHom.IsIntegral (RingHom.comp (Quotient.mk P) C)\n[PROOFSTEP]\nhave hf : Function.Surjective \u2191f := map_surjective (Quotient.mk P') Quotient.mk_surjective\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\n\u22a2 RingHom.IsIntegral (RingHom.comp (Quotient.mk P) C)\n[PROOFSTEP]\nhave hPJ : P = (P.map f).comap f := by\n rw [comap_map_of_surjective _ hf]\n refine' le_antisymm (le_sup_of_le_left le_rfl) (sup_le le_rfl _)\n refine' fun p hp => polynomial_mem_ideal_of_coeff_mem_ideal P p fun n => Quotient.eq_zero_iff_mem.mp _\n simpa only [coeff_map, coe_mapRingHom] using (Polynomial.ext_iff.mp hp) n\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\n\u22a2 P = comap f (map f P)\n[PROOFSTEP]\nrw [comap_map_of_surjective _ hf]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\n\u22a2 P = P \u2294 comap f \u22a5\n[PROOFSTEP]\nrefine' le_antisymm (le_sup_of_le_left le_rfl) (sup_le le_rfl _)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\n\u22a2 comap f \u22a5 \u2264 P\n[PROOFSTEP]\nrefine' fun p hp => polynomial_mem_ideal_of_coeff_mem_ideal P p fun n => Quotient.eq_zero_iff_mem.mp _\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\np : R[X]\nhp : p \u2208 comap f \u22a5\nn : \u2115\n\u22a2 \u2191(Quotient.mk (comap C P)) (coeff p n) = 0\n[PROOFSTEP]\nsimpa only [coeff_map, coe_mapRingHom] using (Polynomial.ext_iff.mp hp) n\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\nhPJ : P = comap f (map f P)\n\u22a2 RingHom.IsIntegral (RingHom.comp (Quotient.mk P) C)\n[PROOFSTEP]\nrefine' RingHom.isIntegral_tower_bot_of_isIntegral _ _ (injective_quotient_le_comap_map P) _\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\nhPJ : P = comap f (map f P)\n\u22a2 RingHom.IsIntegral\n (RingHom.comp\n (quotientMap (map (mapRingHom (Quotient.mk (comap C P))) P) (mapRingHom (Quotient.mk (comap C P)))\n (_ : P \u2264 comap (mapRingHom (Quotient.mk (comap C P))) (map (mapRingHom (Quotient.mk (comap C P))) P)))\n (RingHom.comp (Quotient.mk P) C))\n[PROOFSTEP]\nrw [\u2190 quotient_mk_maps_eq]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\nhPJ : P = comap f (map f P)\n\u22a2 RingHom.IsIntegral\n (RingHom.comp (RingHom.comp (Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)) C)\n (Quotient.mk (comap C P)))\n[PROOFSTEP]\nrefine RingHom.isIntegral_trans _ _ ((Quotient.mk P').isIntegral_of_surjective Quotient.mk_surjective) ?_\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\nhPJ : P = comap f (map f P)\n\u22a2 RingHom.IsIntegral (RingHom.comp (Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)) C)\n[PROOFSTEP]\nhave : IsMaximal (map (mapRingHom (Quotient.mk (comap C P))) P)\n[GOAL]\ncase this\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\nhPJ : P = comap f (map f P)\n\u22a2 IsMaximal (map (mapRingHom (Quotient.mk (comap C P))) P)\n[PROOFSTEP]\nexact\n Or.recOn (map_eq_top_or_isMaximal_of_surjective f hf hP)\n (fun h => absurd (_root_.trans (h \u25b8 hPJ : P = comap f \u22a4) comap_top : P = \u22a4) hP.ne_top) id\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis\u271d : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\nhPJ : P = comap f (map f P)\nthis : IsMaximal (map (mapRingHom (Quotient.mk (comap C P))) P)\n\u22a2 RingHom.IsIntegral (RingHom.comp (Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)) C)\n[PROOFSTEP]\napply quotient_mk_comp_C_isIntegral_of_jacobson' _ ?_ (fun x hx => ?_)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis\u271d : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\nhPJ : P = comap f (map f P)\nthis : IsMaximal (map (mapRingHom (Quotient.mk (comap C P))) P)\n\u22a2 IsJacobson (R \u29f8 comap C P)\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis\u271d : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\nhPJ : P = comap f (map f P)\nthis : IsMaximal (map (mapRingHom (Quotient.mk (comap C P))) P)\nx : R \u29f8 comap C P\nhx : \u2191C x \u2208 map (mapRingHom (Quotient.mk (comap C P))) P\n\u22a2 x = 0\n[PROOFSTEP]\nany_goals exact Ideal.isJacobson_quotient\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis\u271d : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\nhPJ : P = comap f (map f P)\nthis : IsMaximal (map (mapRingHom (Quotient.mk (comap C P))) P)\n\u22a2 IsJacobson (R \u29f8 comap C P)\n[PROOFSTEP]\nexact Ideal.isJacobson_quotient\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis\u271d : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\nhPJ : P = comap f (map f P)\nthis : IsMaximal (map (mapRingHom (Quotient.mk (comap C P))) P)\nx : R \u29f8 comap C P\nhx : \u2191C x \u2208 map (mapRingHom (Quotient.mk (comap C P))) P\n\u22a2 x = 0\n[PROOFSTEP]\nexact Ideal.isJacobson_quotient\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis\u271d : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\nhPJ : P = comap f (map f P)\nthis : IsMaximal (map (mapRingHom (Quotient.mk (comap C P))) P)\nx : R \u29f8 comap C P\nhx : \u2191C x \u2208 map (mapRingHom (Quotient.mk (comap C P))) P\n\u22a2 x = 0\n[PROOFSTEP]\nobtain \u27e8z, rfl\u27e9 := Quotient.mk_surjective x\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nP' : Ideal R := comap C P\nthis\u271d : IsPrime P'\nf : R[X] \u2192+* (R \u29f8 P')[X] := mapRingHom (Quotient.mk P')\nhf : Function.Surjective \u2191f\nhPJ : P = comap f (map f P)\nthis : IsMaximal (map (mapRingHom (Quotient.mk (comap C P))) P)\nz : R\nhx : \u2191C (\u2191(Quotient.mk (comap C P)) z) \u2208 map (mapRingHom (Quotient.mk (comap C P))) P\n\u22a2 \u2191(Quotient.mk (comap C P)) z = 0\n[PROOFSTEP]\nrwa [Quotient.eq_zero_iff_mem, mem_comap, hPJ, mem_comap, coe_mapRingHom, map_C]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\n\u22a2 IsMaximal (comap C P)\n[PROOFSTEP]\nrw [\u2190 @mk_ker _ _ P, RingHom.ker_eq_comap_bot, comap_comap]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\n\u22a2 IsMaximal (comap (RingHom.comp (Quotient.mk P) C) \u22a5)\n[PROOFSTEP]\nexact\n isMaximal_comap_of_isIntegral_of_isMaximal' _ (quotient_mk_comp_C_isIntegral_of_jacobson P) \u22a5\n ((bot_quotient_isMaximal_iff _).mpr hP)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP\u271d : Ideal R[X]\nhP\u271d : IsMaximal P\u271d\nP : Ideal R[X]\nhP : IsMaximal P\n\u22a2 IsMaximal (comap C P)\n[PROOFSTEP]\nhaveI := hP\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP\u271d : Ideal R[X]\nhP\u271d : IsMaximal P\u271d\nP : Ideal R[X]\nhP this : IsMaximal P\n\u22a2 IsMaximal (comap C P)\n[PROOFSTEP]\nexact isMaximal_comap_C_of_isJacobson P\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nS : Type u_2\ninst\u271d : Field S\nf : R[X] \u2192+* S\nhf : Function.Surjective \u2191f\n\u22a2 RingHom.IsIntegral (RingHom.comp f C)\n[PROOFSTEP]\nhaveI : f.ker.IsMaximal := RingHom.ker_isMaximal_of_surjective f hf\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nS : Type u_2\ninst\u271d : Field S\nf : R[X] \u2192+* S\nhf : Function.Surjective \u2191f\nthis : IsMaximal (RingHom.ker f)\n\u22a2 RingHom.IsIntegral (RingHom.comp f C)\n[PROOFSTEP]\nlet g : R[X] \u29f8 (RingHom.ker f) \u2192+* S := Ideal.Quotient.lift (RingHom.ker f) f fun _ h => h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nS : Type u_2\ninst\u271d : Field S\nf : R[X] \u2192+* S\nhf : Function.Surjective \u2191f\nthis : IsMaximal (RingHom.ker f)\ng : R[X] \u29f8 RingHom.ker f \u2192+* S :=\n Quotient.lift (RingHom.ker f) f (_ : \u2200 (x : R[X]), x \u2208 RingHom.ker f \u2192 x \u2208 RingHom.ker f)\n\u22a2 RingHom.IsIntegral (RingHom.comp f C)\n[PROOFSTEP]\nhave hfg : g.comp (Quotient.mk (RingHom.ker f)) = f := ringHom_ext' rfl rfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nS : Type u_2\ninst\u271d : Field S\nf : R[X] \u2192+* S\nhf : Function.Surjective \u2191f\nthis : IsMaximal (RingHom.ker f)\ng : R[X] \u29f8 RingHom.ker f \u2192+* S :=\n Quotient.lift (RingHom.ker f) f (_ : \u2200 (x : R[X]), x \u2208 RingHom.ker f \u2192 x \u2208 RingHom.ker f)\nhfg : RingHom.comp g (Quotient.mk (RingHom.ker f)) = f\n\u22a2 RingHom.IsIntegral (RingHom.comp f C)\n[PROOFSTEP]\nrw [\u2190 hfg, RingHom.comp_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nS : Type u_2\ninst\u271d : Field S\nf : R[X] \u2192+* S\nhf : Function.Surjective \u2191f\nthis : IsMaximal (RingHom.ker f)\ng : R[X] \u29f8 RingHom.ker f \u2192+* S :=\n Quotient.lift (RingHom.ker f) f (_ : \u2200 (x : R[X]), x \u2208 RingHom.ker f \u2192 x \u2208 RingHom.ker f)\nhfg : RingHom.comp g (Quotient.mk (RingHom.ker f)) = f\n\u22a2 RingHom.IsIntegral (RingHom.comp g (RingHom.comp (Quotient.mk (RingHom.ker f)) C))\n[PROOFSTEP]\nrefine\n RingHom.isIntegral_trans _ g (quotient_mk_comp_C_isIntegral_of_jacobson (RingHom.ker f))\n (g.isIntegral_of_surjective ?_)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nS : Type u_2\ninst\u271d : Field S\nf : R[X] \u2192+* S\nhf : Function.Surjective \u2191f\nthis : IsMaximal (RingHom.ker f)\ng : R[X] \u29f8 RingHom.ker f \u2192+* S :=\n Quotient.lift (RingHom.ker f) f (_ : \u2200 (x : R[X]), x \u2208 RingHom.ker f \u2192 x \u2208 RingHom.ker f)\nhfg : RingHom.comp g (Quotient.mk (RingHom.ker f)) = f\n\u22a2 Function.Surjective \u2191g\n[PROOFSTEP]\nrw [\u2190 hfg] at hf \n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nS : Type u_2\ninst\u271d : Field S\nf : R[X] \u2192+* S\nthis : IsMaximal (RingHom.ker f)\ng : R[X] \u29f8 RingHom.ker f \u2192+* S :=\n Quotient.lift (RingHom.ker f) f (_ : \u2200 (x : R[X]), x \u2208 RingHom.ker f \u2192 x \u2208 RingHom.ker f)\nhf : Function.Surjective \u2191(RingHom.comp g (Quotient.mk (RingHom.ker f)))\nhfg : RingHom.comp g (Quotient.mk (RingHom.ker f)) = f\n\u22a2 Function.Surjective \u2191g\n[PROOFSTEP]\nnorm_num at hf \n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsJacobson R\nP : Ideal R[X]\nhP : IsMaximal P\nS : Type u_2\ninst\u271d : Field S\nf : R[X] \u2192+* S\nthis : IsMaximal (RingHom.ker f)\ng : R[X] \u29f8 RingHom.ker f \u2192+* S :=\n Quotient.lift (RingHom.ker f) f (_ : \u2200 (x : R[X]), x \u2208 RingHom.ker f \u2192 x \u2208 RingHom.ker f)\nhf :\n Function.Surjective\n (\u2191(Quotient.lift (RingHom.ker f) f (_ : \u2200 (x : R[X]), x \u2208 RingHom.ker f \u2192 x \u2208 RingHom.ker f)) \u2218\n \u2191(Quotient.mk (RingHom.ker f)))\nhfg : RingHom.comp g (Quotient.mk (RingHom.ker f)) = f\n\u22a2 Function.Surjective \u2191g\n[PROOFSTEP]\nexact Function.Surjective.of_comp hf\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\n\u03b9 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03b9\ninst\u271d : IsJacobson R\n\u22a2 IsJacobson (MvPolynomial \u03b9 R)\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\n\u03b9 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03b9\ninst\u271d : IsJacobson R\nval\u271d : Fintype \u03b9\n\u22a2 IsJacobson (MvPolynomial \u03b9 R)\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\n\u03b9 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03b9\ninst\u271d : IsJacobson R\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9\n\u22a2 IsJacobson (MvPolynomial \u03b9 R)\n[PROOFSTEP]\nlet e := Fintype.equivFin \u03b9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\n\u03b9 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03b9\ninst\u271d : IsJacobson R\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9\ne : \u03b9 \u2243 Fin (Fintype.card \u03b9) := Fintype.equivFin \u03b9\n\u22a2 IsJacobson (MvPolynomial \u03b9 R)\n[PROOFSTEP]\nrw [isJacobson_iso (renameEquiv R e).toRingEquiv]\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\n\u03b9 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03b9\ninst\u271d : IsJacobson R\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9\ne : \u03b9 \u2243 Fin (Fintype.card \u03b9) := Fintype.equivFin \u03b9\n\u22a2 IsJacobson (MvPolynomial (Fin (Fintype.card \u03b9)) R)\n[PROOFSTEP]\nexact isJacobson_MvPolynomial_fin _\n[GOAL]\nn : \u2115\nR : Type u\nS : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nsrc\u271d : S \u2192+* S[X] := Polynomial.C\nr : R\n\u22a2 OneHom.toFun\n (\u2191\u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n map_add' :=\n (_ : \u2200 (x y : S), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) })\n (\u2191(algebraMap R S) r) =\n \u2191(algebraMap R S[X]) r\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\n\u22a2 RingHom.IsIntegral (algebraMap R (T \u29f8 P))\n[PROOFSTEP]\nlet Q := P.comap v.toAlgHom.toRingHom\n[GOAL]\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\n\u22a2 RingHom.IsIntegral (algebraMap R (T \u29f8 P))\n[PROOFSTEP]\nhave hw : Ideal.map v Q = P := map_comap_of_surjective v v.surjective P\n[GOAL]\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\n\u22a2 RingHom.IsIntegral (algebraMap R (T \u29f8 P))\n[PROOFSTEP]\nhaveI hQ : IsMaximal Q := comap_isMaximal_of_surjective _ v.surjective\n[GOAL]\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\n\u22a2 RingHom.IsIntegral (algebraMap R (T \u29f8 P))\n[PROOFSTEP]\nlet w : (S[X] \u29f8 Q) \u2243\u2090[R] (T \u29f8 P) := Ideal.quotientEquivAlg Q P v hw.symm\n[GOAL]\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\n\u22a2 RingHom.IsIntegral (algebraMap R (T \u29f8 P))\n[PROOFSTEP]\nlet Q' := Q.comap (Polynomial.C)\n[GOAL]\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\n\u22a2 RingHom.IsIntegral (algebraMap R (T \u29f8 P))\n[PROOFSTEP]\nlet w' : (S \u29f8 Q') \u2192\u2090[R] (S[X] \u29f8 Q) := Ideal.quotientMap\u2090 Q (C\u2090 R S) le_rfl\n[GOAL]\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\n\u22a2 RingHom.IsIntegral (algebraMap R (T \u29f8 P))\n[PROOFSTEP]\nhave h_eq : algebraMap R (T \u29f8 P) = w.toRingEquiv.toRingHom.comp (w'.toRingHom.comp (algebraMap R (S \u29f8 Q'))) :=\n by\n ext r\n simp only [AlgEquiv.toAlgHom_eq_coe, AlgHom.toRingHom_eq_coe, AlgEquiv.toRingEquiv_eq_coe, RingEquiv.toRingHom_eq_coe,\n AlgHom.comp_algebraMap_of_tower, coe_comp, coe_coe, AlgEquiv.coe_ringEquiv, Function.comp_apply, AlgEquiv.commutes]\n[GOAL]\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\n\u22a2 algebraMap R (T \u29f8 P) =\n RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\n[PROOFSTEP]\next r\n[GOAL]\ncase a\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\nr : R\n\u22a2 \u2191(algebraMap R (T \u29f8 P)) r =\n \u2191(RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))) r\n[PROOFSTEP]\nsimp only [AlgEquiv.toAlgHom_eq_coe, AlgHom.toRingHom_eq_coe, AlgEquiv.toRingEquiv_eq_coe, RingEquiv.toRingHom_eq_coe,\n AlgHom.comp_algebraMap_of_tower, coe_comp, coe_coe, AlgEquiv.coe_ringEquiv, Function.comp_apply, AlgEquiv.commutes]\n[GOAL]\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\nh_eq :\n algebraMap R (T \u29f8 P) =\n RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\n\u22a2 RingHom.IsIntegral (algebraMap R (T \u29f8 P))\n[PROOFSTEP]\nrw [h_eq]\n[GOAL]\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\nh_eq :\n algebraMap R (T \u29f8 P) =\n RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\n\u22a2 RingHom.IsIntegral\n (RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q'))))\n[PROOFSTEP]\napply RingHom.isIntegral_trans\n[GOAL]\ncase hf\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\nh_eq :\n algebraMap R (T \u29f8 P) =\n RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\n\u22a2 RingHom.IsIntegral (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\ncase hg\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\nh_eq :\n algebraMap R (T \u29f8 P) =\n RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\n\u22a2 RingHom.IsIntegral (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w))\n[PROOFSTEP]\napply RingHom.isIntegral_trans\n[GOAL]\ncase hf.hf\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\nh_eq :\n algebraMap R (T \u29f8 P) =\n RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\n\u22a2 RingHom.IsIntegral (algebraMap R (S \u29f8 Q'))\n[PROOFSTEP]\napply IH\n[GOAL]\ncase hf.hf.a\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\nh_eq :\n algebraMap R (T \u29f8 P) =\n RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\n\u22a2 IsMaximal Q'\n[PROOFSTEP]\napply Polynomial.isMaximal_comap_C_of_isJacobson'\n[GOAL]\ncase hf.hf.a.hP\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\nh_eq :\n algebraMap R (T \u29f8 P) =\n RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\n\u22a2 IsMaximal Q\n[PROOFSTEP]\nexact hQ\n[GOAL]\ncase hf.hg\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\nh_eq :\n algebraMap R (T \u29f8 P) =\n RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\n\u22a2 RingHom.IsIntegral \u2191w'\n[PROOFSTEP]\nsuffices : w'.toRingHom = Ideal.quotientMap Q (Polynomial.C) le_rfl\n[GOAL]\ncase hf.hg\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\nh_eq :\n algebraMap R (T \u29f8 P) =\n RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\nthis : \u2191w' = quotientMap Q Polynomial.C (_ : comap Polynomial.C Q \u2264 comap Polynomial.C Q)\n\u22a2 RingHom.IsIntegral \u2191w'\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase hf.hg\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\nh_eq :\n algebraMap R (T \u29f8 P) =\n RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\nthis : \u2191w' = quotientMap Q Polynomial.C (_ : comap Polynomial.C Q \u2264 comap Polynomial.C Q)\n\u22a2 RingHom.IsIntegral (quotientMap Q Polynomial.C (_ : comap Polynomial.C Q \u2264 comap Polynomial.C Q))\n[PROOFSTEP]\nrw [isIntegral_quotientMap_iff _]\n[GOAL]\ncase hf.hg\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\nh_eq :\n algebraMap R (T \u29f8 P) =\n RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\nthis : \u2191w' = quotientMap Q Polynomial.C (_ : comap Polynomial.C Q \u2264 comap Polynomial.C Q)\n\u22a2 RingHom.IsIntegral (RingHom.comp (Quotient.mk Q) Polynomial.C)\n[PROOFSTEP]\napply Polynomial.quotient_mk_comp_C_isIntegral_of_jacobson\n[GOAL]\ncase this\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\nh_eq :\n algebraMap R (T \u29f8 P) =\n RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\n\u22a2 \u2191w' = quotientMap Q Polynomial.C (_ : comap Polynomial.C Q \u2264 comap Polynomial.C Q)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase hg\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\nh_eq :\n algebraMap R (T \u29f8 P) =\n RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\n\u22a2 RingHom.IsIntegral (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w))\n[PROOFSTEP]\napply RingHom.isIntegral_of_surjective\n[GOAL]\ncase hg.hf\nn : \u2115\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : IsJacobson S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nIH : \u2200 (Q : Ideal S), IsMaximal Q \u2192 RingHom.IsIntegral (algebraMap R (S \u29f8 Q))\nv : S[X] \u2243\u2090[R] T\nP : Ideal T\nhP : IsMaximal P\nQ : Ideal S[X] := comap (\u2191\u2191v) P\nhw : map v Q = P\nhQ : IsMaximal Q\nw : (S[X] \u29f8 Q) \u2243\u2090[R] T \u29f8 P := quotientEquivAlg Q P v (_ : P = map v Q)\nQ' : Ideal S := comap Polynomial.C Q\nw' : S \u29f8 Q' \u2192\u2090[R] S[X] \u29f8 Q := quotientMap\u2090 Q (Ideal.MvPolynomial.C\u2090 R S) (_ : Q' \u2264 Q')\nh_eq :\n algebraMap R (T \u29f8 P) =\n RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv w)) (RingHom.comp (\u2191w') (algebraMap R (S \u29f8 Q')))\n\u22a2 Function.Surjective \u2191(RingEquiv.toRingHom (AlgEquiv.toRingEquiv w))\n[PROOFSTEP]\nexact w.surjective\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal (MvPolynomial (Fin n) R)\nhP : IsMaximal P\n\u22a2 RingHom.IsIntegral (algebraMap R (MvPolynomial (Fin n) R \u29f8 P))\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP\u271d : Ideal (MvPolynomial (Fin n) R)\nhP\u271d : IsMaximal P\u271d\nP : Ideal (MvPolynomial (Fin Nat.zero) R)\nhP : IsMaximal P\n\u22a2 RingHom.IsIntegral (algebraMap R (MvPolynomial (Fin Nat.zero) R \u29f8 P))\n[PROOFSTEP]\napply RingHom.isIntegral_of_surjective\n[GOAL]\ncase zero.hf\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP\u271d : Ideal (MvPolynomial (Fin n) R)\nhP\u271d : IsMaximal P\u271d\nP : Ideal (MvPolynomial (Fin Nat.zero) R)\nhP : IsMaximal P\n\u22a2 Function.Surjective \u2191(algebraMap R (MvPolynomial (Fin Nat.zero) R \u29f8 P))\n[PROOFSTEP]\napply Function.Surjective.comp Quotient.mk_surjective\n[GOAL]\ncase zero.hf\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP\u271d : Ideal (MvPolynomial (Fin n) R)\nhP\u271d : IsMaximal P\u271d\nP : Ideal (MvPolynomial (Fin Nat.zero) R)\nhP : IsMaximal P\n\u22a2 Function.Surjective fun x => \u2191(algebraMap R (MvPolynomial (Fin Nat.zero) R)) x\n[PROOFSTEP]\nexact C_surjective (Fin 0)\n[GOAL]\ncase succ\nn\u271d : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP\u271d : Ideal (MvPolynomial (Fin n\u271d) R)\nhP\u271d : IsMaximal P\u271d\nn : \u2115\nIH :\n \u2200 (P : Ideal (MvPolynomial (Fin n) R)), IsMaximal P \u2192 RingHom.IsIntegral (algebraMap R (MvPolynomial (Fin n) R \u29f8 P))\nP : Ideal (MvPolynomial (Fin (Nat.succ n)) R)\nhP : IsMaximal P\n\u22a2 RingHom.IsIntegral (algebraMap R (MvPolynomial (Fin (Nat.succ n)) R \u29f8 P))\n[PROOFSTEP]\napply aux_IH IH (finSuccEquiv R n).symm P hP\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal (MvPolynomial (Fin n) R)\nhP : IsMaximal P\n\u22a2 RingHom.IsIntegral (RingHom.comp (Quotient.mk P) MvPolynomial.C)\n[PROOFSTEP]\nchange RingHom.IsIntegral (algebraMap R (MvPolynomial (Fin n) R \u29f8 P))\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal (MvPolynomial (Fin n) R)\nhP : IsMaximal P\n\u22a2 RingHom.IsIntegral (algebraMap R (MvPolynomial (Fin n) R \u29f8 P))\n[PROOFSTEP]\napply quotient_mk_comp_C_isIntegral_of_jacobson'\n[GOAL]\ncase hP\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsJacobson R\nP : Ideal (MvPolynomial (Fin n) R)\nhP : IsMaximal P\n\u22a2 IsMaximal P\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\n\u22a2 RingHom.IsIntegral (RingHom.comp f MvPolynomial.C)\n[PROOFSTEP]\ncases nonempty_fintype \u03c3\n[GOAL]\ncase intro\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\n\u22a2 RingHom.IsIntegral (RingHom.comp f MvPolynomial.C)\n[PROOFSTEP]\nhave e := (Fintype.equivFin \u03c3).symm\n[GOAL]\ncase intro\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\n\u22a2 RingHom.IsIntegral (RingHom.comp f MvPolynomial.C)\n[PROOFSTEP]\nlet f' : MvPolynomial (Fin _) R \u2192+* S := f.comp (renameEquiv R e).toRingEquiv.toRingHom\n[GOAL]\ncase intro\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\nf' : MvPolynomial (Fin (Fintype.card \u03c3)) R \u2192+* S :=\n RingHom.comp f (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e)))\n\u22a2 RingHom.IsIntegral (RingHom.comp f MvPolynomial.C)\n[PROOFSTEP]\nhave hf' := Function.Surjective.comp hf (renameEquiv R e).surjective\n[GOAL]\ncase intro\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\nf' : MvPolynomial (Fin (Fintype.card \u03c3)) R \u2192+* S :=\n RingHom.comp f (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e)))\nhf' : Function.Surjective (\u2191f \u2218 \u2191(renameEquiv R e))\n\u22a2 RingHom.IsIntegral (RingHom.comp f MvPolynomial.C)\n[PROOFSTEP]\nchange Function.Surjective \u2191f' at hf' \n[GOAL]\ncase intro\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\nf' : MvPolynomial (Fin (Fintype.card \u03c3)) R \u2192+* S :=\n RingHom.comp f (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e)))\nhf' : Function.Surjective \u2191f'\n\u22a2 RingHom.IsIntegral (RingHom.comp f MvPolynomial.C)\n[PROOFSTEP]\nhave : (f'.comp C).IsIntegral :=\n by\n haveI : f'.ker.IsMaximal := ker_isMaximal_of_surjective f' hf'\n let g : MvPolynomial _ R \u29f8 (RingHom.ker f') \u2192+* S := Ideal.Quotient.lift (RingHom.ker f') f' fun _ h => h\n have hfg : g.comp (Quotient.mk (RingHom.ker f')) = f' := ringHom_ext (fun r => rfl) fun i => rfl\n rw [\u2190 hfg, RingHom.comp_assoc]\n refine'\n RingHom.isIntegral_trans _ g (quotient_mk_comp_C_isIntegral_of_jacobson (RingHom.ker f'))\n (g.isIntegral_of_surjective _)\n rw [\u2190 hfg] at hf' \n norm_num at hf' \n exact Function.Surjective.of_comp hf'\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\nf' : MvPolynomial (Fin (Fintype.card \u03c3)) R \u2192+* S :=\n RingHom.comp f (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e)))\nhf' : Function.Surjective \u2191f'\n\u22a2 RingHom.IsIntegral (RingHom.comp f' MvPolynomial.C)\n[PROOFSTEP]\nhaveI : f'.ker.IsMaximal := ker_isMaximal_of_surjective f' hf'\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\nf' : MvPolynomial (Fin (Fintype.card \u03c3)) R \u2192+* S :=\n RingHom.comp f (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e)))\nhf' : Function.Surjective \u2191f'\nthis : IsMaximal (ker f')\n\u22a2 RingHom.IsIntegral (RingHom.comp f' MvPolynomial.C)\n[PROOFSTEP]\nlet g : MvPolynomial _ R \u29f8 (RingHom.ker f') \u2192+* S := Ideal.Quotient.lift (RingHom.ker f') f' fun _ h => h\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\nf' : MvPolynomial (Fin (Fintype.card \u03c3)) R \u2192+* S :=\n RingHom.comp f (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e)))\nhf' : Function.Surjective \u2191f'\nthis : IsMaximal (ker f')\ng : MvPolynomial (Fin (Fintype.card \u03c3)) R \u29f8 ker f' \u2192+* S :=\n Quotient.lift (ker f') f' (_ : \u2200 (x : MvPolynomial (Fin (Fintype.card \u03c3)) R), x \u2208 ker f' \u2192 x \u2208 ker f')\n\u22a2 RingHom.IsIntegral (RingHom.comp f' MvPolynomial.C)\n[PROOFSTEP]\nhave hfg : g.comp (Quotient.mk (RingHom.ker f')) = f' := ringHom_ext (fun r => rfl) fun i => rfl\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\nf' : MvPolynomial (Fin (Fintype.card \u03c3)) R \u2192+* S :=\n RingHom.comp f (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e)))\nhf' : Function.Surjective \u2191f'\nthis : IsMaximal (ker f')\ng : MvPolynomial (Fin (Fintype.card \u03c3)) R \u29f8 ker f' \u2192+* S :=\n Quotient.lift (ker f') f' (_ : \u2200 (x : MvPolynomial (Fin (Fintype.card \u03c3)) R), x \u2208 ker f' \u2192 x \u2208 ker f')\nhfg : RingHom.comp g (Quotient.mk (ker f')) = f'\n\u22a2 RingHom.IsIntegral (RingHom.comp f' MvPolynomial.C)\n[PROOFSTEP]\nrw [\u2190 hfg, RingHom.comp_assoc]\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\nf' : MvPolynomial (Fin (Fintype.card \u03c3)) R \u2192+* S :=\n RingHom.comp f (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e)))\nhf' : Function.Surjective \u2191f'\nthis : IsMaximal (ker f')\ng : MvPolynomial (Fin (Fintype.card \u03c3)) R \u29f8 ker f' \u2192+* S :=\n Quotient.lift (ker f') f' (_ : \u2200 (x : MvPolynomial (Fin (Fintype.card \u03c3)) R), x \u2208 ker f' \u2192 x \u2208 ker f')\nhfg : RingHom.comp g (Quotient.mk (ker f')) = f'\n\u22a2 RingHom.IsIntegral (RingHom.comp g (RingHom.comp (Quotient.mk (ker f')) MvPolynomial.C))\n[PROOFSTEP]\nrefine'\n RingHom.isIntegral_trans _ g (quotient_mk_comp_C_isIntegral_of_jacobson (RingHom.ker f'))\n (g.isIntegral_of_surjective _)\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\nf' : MvPolynomial (Fin (Fintype.card \u03c3)) R \u2192+* S :=\n RingHom.comp f (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e)))\nhf' : Function.Surjective \u2191f'\nthis : IsMaximal (ker f')\ng : MvPolynomial (Fin (Fintype.card \u03c3)) R \u29f8 ker f' \u2192+* S :=\n Quotient.lift (ker f') f' (_ : \u2200 (x : MvPolynomial (Fin (Fintype.card \u03c3)) R), x \u2208 ker f' \u2192 x \u2208 ker f')\nhfg : RingHom.comp g (Quotient.mk (ker f')) = f'\n\u22a2 Function.Surjective \u2191g\n[PROOFSTEP]\nrw [\u2190 hfg] at hf' \n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\nf' : MvPolynomial (Fin (Fintype.card \u03c3)) R \u2192+* S :=\n RingHom.comp f (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e)))\nthis : IsMaximal (ker f')\ng : MvPolynomial (Fin (Fintype.card \u03c3)) R \u29f8 ker f' \u2192+* S :=\n Quotient.lift (ker f') f' (_ : \u2200 (x : MvPolynomial (Fin (Fintype.card \u03c3)) R), x \u2208 ker f' \u2192 x \u2208 ker f')\nhf' : Function.Surjective \u2191(RingHom.comp g (Quotient.mk (ker f')))\nhfg : RingHom.comp g (Quotient.mk (ker f')) = f'\n\u22a2 Function.Surjective \u2191g\n[PROOFSTEP]\nnorm_num at hf' \n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\nf' : MvPolynomial (Fin (Fintype.card \u03c3)) R \u2192+* S :=\n RingHom.comp f (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e)))\nthis : IsMaximal (ker f')\ng : MvPolynomial (Fin (Fintype.card \u03c3)) R \u29f8 ker f' \u2192+* S :=\n Quotient.lift (ker f') f' (_ : \u2200 (x : MvPolynomial (Fin (Fintype.card \u03c3)) R), x \u2208 ker f' \u2192 x \u2208 ker f')\nhf' :\n Function.Surjective\n (\u2191(Quotient.lift (ker f') (RingHom.comp f \u2191\u2191(renameEquiv R e))\n (_ :\n \u2200 (a : MvPolynomial (Fin (Fintype.card \u03c3)) R), a \u2208 ker f' \u2192 \u2191(RingHom.comp f \u2191\u2191(renameEquiv R e)) a = 0)) \u2218\n \u2191(Quotient.mk (ker (RingHom.comp f \u2191\u2191(renameEquiv R e)))))\nhfg : RingHom.comp g (Quotient.mk (ker f')) = f'\n\u22a2 Function.Surjective \u2191g\n[PROOFSTEP]\nexact Function.Surjective.of_comp hf'\n[GOAL]\ncase intro\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\nf' : MvPolynomial (Fin (Fintype.card \u03c3)) R \u2192+* S :=\n RingHom.comp f (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e)))\nhf' : Function.Surjective \u2191f'\nthis : RingHom.IsIntegral (RingHom.comp f' MvPolynomial.C)\n\u22a2 RingHom.IsIntegral (RingHom.comp f MvPolynomial.C)\n[PROOFSTEP]\nrw [RingHom.comp_assoc] at this \n[GOAL]\ncase intro\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\nf' : MvPolynomial (Fin (Fintype.card \u03c3)) R \u2192+* S :=\n RingHom.comp f (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e)))\nhf' : Function.Surjective \u2191f'\nthis :\n RingHom.IsIntegral\n (RingHom.comp f (RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e))) MvPolynomial.C))\n\u22a2 RingHom.IsIntegral (RingHom.comp f MvPolynomial.C)\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_5.h.e'_8\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\nf' : MvPolynomial (Fin (Fintype.card \u03c3)) R \u2192+* S :=\n RingHom.comp f (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e)))\nhf' : Function.Surjective \u2191f'\nthis :\n RingHom.IsIntegral\n (RingHom.comp f (RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e))) MvPolynomial.C))\n\u22a2 MvPolynomial.C = RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e))) MvPolynomial.C\n[PROOFSTEP]\nrefine' RingHom.ext fun x => _\n[GOAL]\ncase h.e'_5.h.e'_8\nn : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsJacobson R\n\u03c3 : Type u_2\ninst\u271d\u00b9 : _root_.Finite \u03c3\nS : Type u_3\ninst\u271d : Field S\nf : MvPolynomial \u03c3 R \u2192+* S\nhf : Function.Surjective \u2191f\nval\u271d : Fintype \u03c3\ne : Fin (Fintype.card \u03c3) \u2243 \u03c3\nf' : MvPolynomial (Fin (Fintype.card \u03c3)) R \u2192+* S :=\n RingHom.comp f (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e)))\nhf' : Function.Surjective \u2191f'\nthis :\n RingHom.IsIntegral\n (RingHom.comp f (RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e))) MvPolynomial.C))\nx : R\n\u22a2 \u2191MvPolynomial.C x = \u2191(RingHom.comp (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (renameEquiv R e))) MvPolynomial.C) x\n[PROOFSTEP]\nexact ((renameEquiv R e).commutes' x).symm\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Jacobson", "llama_tokens": 126868, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943925708561, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.4419986839122783}} {"text": "[GOAL]\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nsrc\u271d : L \u2192\u2097[R] UniversalEnvelopingAlgebra R L := LinearMap.comp (AlgHom.toLinearMap (mkAlgHom R L)) \u03b9\u209c\nx y : L\n\u22a2 AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : L),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n \u2045x, y\u2046 =\n \u2045AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : L),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n x,\n AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : L),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n y\u2046\n[PROOFSTEP]\nsuffices mkAlgHom R L (\u03b9\u209c \u2045x, y\u2046 + \u03b9\u209c y * \u03b9\u209c x) = mkAlgHom R L (\u03b9\u209c x * \u03b9\u209c y) by rw [AlgHom.map_mul] at this ;\n simp [LieRing.of_associative_ring_bracket, \u2190 this]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nsrc\u271d : L \u2192\u2097[R] UniversalEnvelopingAlgebra R L := LinearMap.comp (AlgHom.toLinearMap (mkAlgHom R L)) \u03b9\u209c\nx y : L\nthis : \u2191(mkAlgHom R L) (\u2191\u03b9\u209c \u2045x, y\u2046 + \u2191\u03b9\u209c y * \u2191\u03b9\u209c x) = \u2191(mkAlgHom R L) (\u2191\u03b9\u209c x * \u2191\u03b9\u209c y)\n\u22a2 AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : L),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n \u2045x, y\u2046 =\n \u2045AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : L),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n x,\n AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : L),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n y\u2046\n[PROOFSTEP]\nrw [AlgHom.map_mul] at this \n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nsrc\u271d : L \u2192\u2097[R] UniversalEnvelopingAlgebra R L := LinearMap.comp (AlgHom.toLinearMap (mkAlgHom R L)) \u03b9\u209c\nx y : L\nthis : \u2191(mkAlgHom R L) (\u2191\u03b9\u209c \u2045x, y\u2046 + \u2191\u03b9\u209c y * \u2191\u03b9\u209c x) = \u2191(mkAlgHom R L) (\u2191\u03b9\u209c x) * \u2191(mkAlgHom R L) (\u2191\u03b9\u209c y)\n\u22a2 AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : L),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n \u2045x, y\u2046 =\n \u2045AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : L),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n x,\n AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : L),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n y\u2046\n[PROOFSTEP]\nsimp [LieRing.of_associative_ring_bracket, \u2190 this]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nsrc\u271d : L \u2192\u2097[R] UniversalEnvelopingAlgebra R L := LinearMap.comp (AlgHom.toLinearMap (mkAlgHom R L)) \u03b9\u209c\nx y : L\n\u22a2 \u2191(mkAlgHom R L) (\u2191\u03b9\u209c \u2045x, y\u2046 + \u2191\u03b9\u209c y * \u2191\u03b9\u209c x) = \u2191(mkAlgHom R L) (\u2191\u03b9\u209c x * \u2191\u03b9\u209c y)\n[PROOFSTEP]\nexact RingQuot.mkAlgHom_rel _ (Rel.lie_compat x y)\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf\u271d f : L \u2192\u2097\u2045R\u2046 A\n\u22a2 \u2200 \u2983x y : TensorAlgebra R L\u2984, Rel R L x y \u2192 \u2191(\u2191(TensorAlgebra.lift R) \u2191f) x = \u2191(\u2191(TensorAlgebra.lift R) \u2191f) y\n[PROOFSTEP]\nintro a b h\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf\u271d f : L \u2192\u2097\u2045R\u2046 A\na b : TensorAlgebra R L\nh : Rel R L a b\n\u22a2 \u2191(\u2191(TensorAlgebra.lift R) \u2191f) a = \u2191(\u2191(TensorAlgebra.lift R) \u2191f) b\n[PROOFSTEP]\ninduction' h with x y\n[GOAL]\ncase lie_compat\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf\u271d f : L \u2192\u2097\u2045R\u2046 A\na b : TensorAlgebra R L\nx y : L\n\u22a2 \u2191(\u2191(TensorAlgebra.lift R) \u2191f) (\u2191\u03b9\u209c \u2045x, y\u2046 + \u2191\u03b9\u209c y * \u2191\u03b9\u209c x) = \u2191(\u2191(TensorAlgebra.lift R) \u2191f) (\u2191\u03b9\u209c x * \u2191\u03b9\u209c y)\n[PROOFSTEP]\nsimp only [LieRing.of_associative_ring_bracket, map_add, TensorAlgebra.lift_\u03b9_apply, LieHom.coe_toLinearMap,\n LieHom.map_lie, map_mul, sub_add_cancel]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf\u271d f : L \u2192\u2097\u2045R\u2046 A\n\u22a2 (fun F => LieHom.comp (AlgHom.toLieHom F) (\u03b9 R))\n ((fun f =>\n \u2191(RingQuot.liftAlgHom R)\n { val := \u2191(TensorAlgebra.lift R) \u2191f,\n property :=\n (_ :\n \u2200 \u2983a b : TensorAlgebra R L\u2984,\n Rel R L a b \u2192 \u2191(\u2191(TensorAlgebra.lift R) \u2191f) a = \u2191(\u2191(TensorAlgebra.lift R) \u2191f) b) })\n f) =\n f\n[PROOFSTEP]\next\n -- Porting note: was\n -- simp only [\u03b9, mkAlgHom, TensorAlgebra.lift_\u03b9_apply, LieHom.coe_toLinearMap,\n -- LinearMap.toFun_eq_coe, LinearMap.coe_comp, LieHom.coe_comp, AlgHom.coe_toLieHom,\n -- LieHom.coe_mk, Function.comp_apply, AlgHom.toLinearMap_apply,\n -- RingQuot.liftAlgHom_mkAlgHom_apply]\n[GOAL]\ncase h\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf\u271d f : L \u2192\u2097\u2045R\u2046 A\nx\u271d : L\n\u22a2 \u2191((fun F => LieHom.comp (AlgHom.toLieHom F) (\u03b9 R))\n ((fun f =>\n \u2191(RingQuot.liftAlgHom R)\n { val := \u2191(TensorAlgebra.lift R) \u2191f,\n property :=\n (_ :\n \u2200 \u2983a b : TensorAlgebra R L\u2984,\n Rel R L a b \u2192 \u2191(\u2191(TensorAlgebra.lift R) \u2191f) a = \u2191(\u2191(TensorAlgebra.lift R) \u2191f) b) })\n f))\n x\u271d =\n \u2191f x\u271d\n[PROOFSTEP]\nsimp only [LieHom.coe_comp, Function.comp_apply, AlgHom.coe_toLieHom, UniversalEnvelopingAlgebra.\u03b9_apply, mkAlgHom]\n[GOAL]\ncase h\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf\u271d f : L \u2192\u2097\u2045R\u2046 A\nx\u271d : L\n\u22a2 \u2191(\u2191(RingQuot.liftAlgHom R)\n { val := \u2191(TensorAlgebra.lift R) \u2191f,\n property :=\n (_ :\n \u2200 \u2983a b : TensorAlgebra R L\u2984,\n Rel R L a b \u2192 \u2191(\u2191(TensorAlgebra.lift R) \u2191f) a = \u2191(\u2191(TensorAlgebra.lift R) \u2191f) b) })\n (\u2191(RingQuot.mkAlgHom R (Rel R L)) (\u2191\u03b9\u209c x\u271d)) =\n \u2191f x\u271d\n[PROOFSTEP]\nrw [RingQuot.liftAlgHom_mkAlgHom_apply]\n[GOAL]\ncase h\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf\u271d f : L \u2192\u2097\u2045R\u2046 A\nx\u271d : L\n\u22a2 \u2191(\u2191(TensorAlgebra.lift R) \u2191f) (\u2191\u03b9\u209c x\u271d) = \u2191f x\u271d\n[PROOFSTEP]\nsimp only [TensorAlgebra.lift_\u03b9_apply, LieHom.coe_toLinearMap]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\nF : UniversalEnvelopingAlgebra R L \u2192\u2090[R] A\n\u22a2 (fun f =>\n \u2191(RingQuot.liftAlgHom R)\n { val := \u2191(TensorAlgebra.lift R) \u2191f,\n property :=\n (_ :\n \u2200 \u2983a b : TensorAlgebra R L\u2984,\n Rel R L a b \u2192 \u2191(\u2191(TensorAlgebra.lift R) \u2191f) a = \u2191(\u2191(TensorAlgebra.lift R) \u2191f) b) })\n ((fun F => LieHom.comp (AlgHom.toLieHom F) (\u03b9 R)) F) =\n F\n[PROOFSTEP]\napply RingQuot.ringQuot_ext'\n[GOAL]\ncase w\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\nF : UniversalEnvelopingAlgebra R L \u2192\u2090[R] A\n\u22a2 AlgHom.comp\n ((fun f =>\n \u2191(RingQuot.liftAlgHom R)\n { val := \u2191(TensorAlgebra.lift R) \u2191f,\n property :=\n (_ :\n \u2200 \u2983a b : TensorAlgebra R L\u2984,\n Rel R L a b \u2192 \u2191(\u2191(TensorAlgebra.lift R) \u2191f) a = \u2191(\u2191(TensorAlgebra.lift R) \u2191f) b) })\n ((fun F => LieHom.comp (AlgHom.toLieHom F) (\u03b9 R)) F))\n (RingQuot.mkAlgHom R (Rel R L)) =\n AlgHom.comp F (RingQuot.mkAlgHom R (Rel R L))\n[PROOFSTEP]\next\n -- Porting note: was\n -- simp only [\u03b9, mkAlgHom, TensorAlgebra.lift_\u03b9_apply, LieHom.coe_toLinearMap,\n -- LinearMap.toFun_eq_coe, LinearMap.coe_comp, LieHom.coe_linearMap_comp,\n -- AlgHom.comp_toLinearMap, Function.comp_apply, AlgHom.toLinearMap_apply,\n -- RingQuot.liftAlgHom_mkAlgHom_apply, AlgHom.coe_toLieHom, LieHom.coe_mk]\n[GOAL]\ncase w.w.h\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\nF : UniversalEnvelopingAlgebra R L \u2192\u2090[R] A\nx\u271d : L\n\u22a2 \u2191(LinearMap.comp\n (AlgHom.toLinearMap\n (AlgHom.comp\n ((fun f =>\n \u2191(RingQuot.liftAlgHom R)\n { val := \u2191(TensorAlgebra.lift R) \u2191f,\n property :=\n (_ :\n \u2200 \u2983a b : TensorAlgebra R L\u2984,\n Rel R L a b \u2192 \u2191(\u2191(TensorAlgebra.lift R) \u2191f) a = \u2191(\u2191(TensorAlgebra.lift R) \u2191f) b) })\n ((fun F => LieHom.comp (AlgHom.toLieHom F) (\u03b9 R)) F))\n (RingQuot.mkAlgHom R (Rel R L))))\n \u03b9\u209c)\n x\u271d =\n \u2191(LinearMap.comp (AlgHom.toLinearMap (AlgHom.comp F (RingQuot.mkAlgHom R (Rel R L)))) \u03b9\u209c) x\u271d\n[PROOFSTEP]\nsimp [mkAlgHom]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\nx : L\n\u22a2 \u2191(\u2191(lift R) f) (\u2191(\u03b9 R) x) = \u2191f x\n[PROOFSTEP]\nrw [\u2190 Function.comp_apply (f := lift R f) (g := \u03b9 R) (x := x), \u03b9_comp_lift]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\nx : L\n\u22a2 \u2191(\u2191(lift R) f) (\u2191(mkAlgHom R L) (\u2191\u03b9\u209c x)) = \u2191f x\n[PROOFSTEP]\nsimpa using lift_\u03b9_apply R f x\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\ng : UniversalEnvelopingAlgebra R L \u2192\u2090[R] A\n\u22a2 \u2191g \u2218 \u2191(\u03b9 R) = \u2191f \u2194 g = \u2191(lift R) f\n[PROOFSTEP]\nrefine' Iff.trans _ (lift R).symm_apply_eq\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\ng : UniversalEnvelopingAlgebra R L \u2192\u2090[R] A\n\u22a2 \u2191g \u2218 \u2191(\u03b9 R) = \u2191f \u2194 \u2191(lift R).symm g = f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\ng : UniversalEnvelopingAlgebra R L \u2192\u2090[R] A\n\u22a2 \u2191g \u2218 \u2191(\u03b9 R) = \u2191f \u2192 \u2191(lift R).symm g = f\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\ng : UniversalEnvelopingAlgebra R L \u2192\u2090[R] A\nh : \u2191g \u2218 \u2191(\u03b9 R) = \u2191f\n\u22a2 \u2191(lift R).symm g = f\n[PROOFSTEP]\next\n[GOAL]\ncase mp.h\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\ng : UniversalEnvelopingAlgebra R L \u2192\u2090[R] A\nh : \u2191g \u2218 \u2191(\u03b9 R) = \u2191f\nx\u271d : L\n\u22a2 \u2191(\u2191(lift R).symm g) x\u271d = \u2191f x\u271d\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\ncase mpr\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\ng : UniversalEnvelopingAlgebra R L \u2192\u2090[R] A\n\u22a2 \u2191(lift R).symm g = f \u2192 \u2191g \u2218 \u2191(\u03b9 R) = \u2191f\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\ng : UniversalEnvelopingAlgebra R L \u2192\u2090[R] A\nh : \u2191(lift R).symm g = f\n\u22a2 \u2191g \u2218 \u2191(\u03b9 R) = \u2191f\n[PROOFSTEP]\next\n[GOAL]\ncase mpr.h\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\ng : UniversalEnvelopingAlgebra R L \u2192\u2090[R] A\nh : \u2191(lift R).symm g = f\nx\u271d : L\n\u22a2 (\u2191g \u2218 \u2191(\u03b9 R)) x\u271d = \u2191f x\u271d\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\ng\u2081 g\u2082 : UniversalEnvelopingAlgebra R L \u2192\u2090[R] A\nh : LieHom.comp (AlgHom.toLieHom g\u2081) (\u03b9 R) = LieHom.comp (AlgHom.toLieHom g\u2082) (\u03b9 R)\n\u22a2 \u2191(lift R).symm g\u2081 = \u2191(lift R).symm g\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\ng\u2081 g\u2082 : UniversalEnvelopingAlgebra R L \u2192\u2090[R] A\nh : LieHom.comp (AlgHom.toLieHom g\u2081) (\u03b9 R) = LieHom.comp (AlgHom.toLieHom g\u2082) (\u03b9 R)\nx\u271d : L\n\u22a2 \u2191(\u2191(lift R).symm g\u2081) x\u271d = \u2191(\u2191(lift R).symm g\u2082) x\u271d\n[PROOFSTEP]\nsimp [h]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Lie.UniversalEnveloping", "llama_tokens": 6692, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8080672227971211, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.4418011800714372}} {"text": "[GOAL]\nS : Type u\ninst\u271d : Semiring S\n\u22a2 pochhammer S 1 = X\n[PROOFSTEP]\nsimp [pochhammer]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\n\u22a2 pochhammer S (n + 1) = X * comp (pochhammer S n) (X + 1)\n[PROOFSTEP]\nrw [pochhammer]\n[GOAL]\nS : Type u\ninst\u271d\u00b9 : Semiring S\nT : Type v\ninst\u271d : Semiring T\nf : S \u2192+* T\nn : \u2115\n\u22a2 map f (pochhammer S n) = pochhammer T n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nS : Type u\ninst\u271d\u00b9 : Semiring S\nT : Type v\ninst\u271d : Semiring T\nf : S \u2192+* T\n\u22a2 map f (pochhammer S Nat.zero) = pochhammer T Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nS : Type u\ninst\u271d\u00b9 : Semiring S\nT : Type v\ninst\u271d : Semiring T\nf : S \u2192+* T\nn : \u2115\nih : map f (pochhammer S n) = pochhammer T n\n\u22a2 map f (pochhammer S (Nat.succ n)) = pochhammer T (Nat.succ n)\n[PROOFSTEP]\nsimp [ih, pochhammer_succ_left, map_comp]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn k : \u2115\n\u22a2 \u2191(eval k (pochhammer \u2115 n)) = eval (\u2191k) (pochhammer S n)\n[PROOFSTEP]\nrw [\u2190 pochhammer_map (algebraMap \u2115 S), eval_map, \u2190 eq_natCast (algebraMap \u2115 S), eval\u2082_at_nat_cast, Nat.cast_id,\n eq_natCast]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\n\u22a2 eval 0 (pochhammer S n) = if n = 0 then 1 else 0\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nS : Type u\ninst\u271d : Semiring S\n\u22a2 eval 0 (pochhammer S Nat.zero) = if Nat.zero = 0 then 1 else 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nS : Type u\ninst\u271d : Semiring S\nn\u271d : \u2115\n\u22a2 eval 0 (pochhammer S (Nat.succ n\u271d)) = if Nat.succ n\u271d = 0 then 1 else 0\n[PROOFSTEP]\nsimp [X_mul, Nat.succ_ne_zero, pochhammer_succ_left]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\n\u22a2 eval 0 (pochhammer S 0) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\nh : n \u2260 0\n\u22a2 eval 0 (pochhammer S n) = 0\n[PROOFSTEP]\nsimp [pochhammer_eval_zero, h]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\n\u22a2 pochhammer S (n + 1) = pochhammer S n * (X + \u2191n)\n[PROOFSTEP]\nsuffices h : pochhammer \u2115 (n + 1) = pochhammer \u2115 n * (X + (n : \u2115[X]))\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\nh : pochhammer \u2115 (n + 1) = pochhammer \u2115 n * (X + \u2191n)\n\u22a2 pochhammer S (n + 1) = pochhammer S n * (X + \u2191n)\n[PROOFSTEP]\napply_fun Polynomial.map (algebraMap \u2115 S) at h \n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\nh : map (algebraMap \u2115 S) (pochhammer \u2115 (n + 1)) = map (algebraMap \u2115 S) (pochhammer \u2115 n * (X + \u2191n))\n\u22a2 pochhammer S (n + 1) = pochhammer S n * (X + \u2191n)\n[PROOFSTEP]\nsimpa only [pochhammer_map, Polynomial.map_mul, Polynomial.map_add, map_X, Polynomial.map_nat_cast] using h\n[GOAL]\ncase h\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\n\u22a2 pochhammer \u2115 (n + 1) = pochhammer \u2115 n * (X + \u2191n)\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase h.zero\nS : Type u\ninst\u271d : Semiring S\n\u22a2 pochhammer \u2115 (Nat.zero + 1) = pochhammer \u2115 Nat.zero * (X + \u2191Nat.zero)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.succ\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\nih : pochhammer \u2115 (n + 1) = pochhammer \u2115 n * (X + \u2191n)\n\u22a2 pochhammer \u2115 (Nat.succ n + 1) = pochhammer \u2115 (Nat.succ n) * (X + \u2191(Nat.succ n))\n[PROOFSTEP]\nconv_lhs =>\n rw [pochhammer_succ_left, ih, mul_comp, \u2190 mul_assoc, \u2190 pochhammer_succ_left, add_comp, X_comp, nat_cast_comp,\n add_assoc, add_comm (1 : \u2115[X]), \u2190 Nat.cast_succ]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\nih : pochhammer \u2115 (n + 1) = pochhammer \u2115 n * (X + \u2191n)\n| pochhammer \u2115 (Nat.succ n + 1)\n[PROOFSTEP]\nrw [pochhammer_succ_left, ih, mul_comp, \u2190 mul_assoc, \u2190 pochhammer_succ_left, add_comp, X_comp, nat_cast_comp, add_assoc,\n add_comm (1 : \u2115[X]), \u2190 Nat.cast_succ]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\nih : pochhammer \u2115 (n + 1) = pochhammer \u2115 n * (X + \u2191n)\n| pochhammer \u2115 (Nat.succ n + 1)\n[PROOFSTEP]\nrw [pochhammer_succ_left, ih, mul_comp, \u2190 mul_assoc, \u2190 pochhammer_succ_left, add_comp, X_comp, nat_cast_comp, add_assoc,\n add_comm (1 : \u2115[X]), \u2190 Nat.cast_succ]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\nih : pochhammer \u2115 (n + 1) = pochhammer \u2115 n * (X + \u2191n)\n| pochhammer \u2115 (Nat.succ n + 1)\n[PROOFSTEP]\nrw [pochhammer_succ_left, ih, mul_comp, \u2190 mul_assoc, \u2190 pochhammer_succ_left, add_comp, X_comp, nat_cast_comp, add_assoc,\n add_comm (1 : \u2115[X]), \u2190 Nat.cast_succ]\n[GOAL]\nS\u271d : Type u\ninst\u271d\u00b9 : Semiring S\u271d\nS : Type u_1\ninst\u271d : Semiring S\nn : \u2115\nk : S\n\u22a2 eval k (pochhammer S (n + 1)) = eval k (pochhammer S n) * (k + \u2191n)\n[PROOFSTEP]\nrw [pochhammer_succ_right, mul_add, eval_add, eval_mul_X, \u2190 Nat.cast_comm, \u2190 C_eq_nat_cast, eval_C_mul, Nat.cast_comm, \u2190\n mul_add]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\n\u22a2 comp (pochhammer S (n + 1)) (X + 1) = pochhammer S (n + 1) + (n + 1) \u2022 comp (pochhammer S n) (X + 1)\n[PROOFSTEP]\nsuffices (pochhammer \u2115 (n + 1)).comp (X + 1) = pochhammer \u2115 (n + 1) + (n + 1) * (pochhammer \u2115 n).comp (X + 1) by\n simpa [map_comp] using congr_arg (Polynomial.map (Nat.castRingHom S)) this\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\nthis : comp (pochhammer \u2115 (n + 1)) (X + 1) = pochhammer \u2115 (n + 1) + (\u2191n + 1) * comp (pochhammer \u2115 n) (X + 1)\n\u22a2 comp (pochhammer S (n + 1)) (X + 1) = pochhammer S (n + 1) + (n + 1) \u2022 comp (pochhammer S n) (X + 1)\n[PROOFSTEP]\nsimpa [map_comp] using congr_arg (Polynomial.map (Nat.castRingHom S)) this\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\n\u22a2 comp (pochhammer \u2115 (n + 1)) (X + 1) = pochhammer \u2115 (n + 1) + (\u2191n + 1) * comp (pochhammer \u2115 n) (X + 1)\n[PROOFSTEP]\nnth_rw 2 [pochhammer_succ_left]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\n\u22a2 comp (pochhammer \u2115 (n + 1)) (X + 1) = X * comp (pochhammer \u2115 n) (X + 1) + (\u2191n + 1) * comp (pochhammer \u2115 n) (X + 1)\n[PROOFSTEP]\nrw [\u2190 add_mul, pochhammer_succ_right \u2115 n, mul_comp, mul_comm, add_comp, X_comp, nat_cast_comp, add_comm, \u2190 add_assoc]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\n\u22a2 (\u2191n + X + 1) * comp (pochhammer \u2115 n) (X + 1) = (X + (\u2191n + 1)) * comp (pochhammer \u2115 n) (X + 1)\n[PROOFSTEP]\nring\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\np q : S[X]\nn : \u2115\n\u22a2 comp (p * (X + \u2191n)) q = comp p q * (q + \u2191n)\n[PROOFSTEP]\nrw [mul_add, add_comp, mul_X_comp, \u2190 Nat.cast_comm, nat_cast_mul_comp, Nat.cast_comm, mul_add]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn m : \u2115\n\u22a2 pochhammer S n * comp (pochhammer S m) (X + \u2191n) = pochhammer S (n + m)\n[PROOFSTEP]\ninduction' m with m ih\n[GOAL]\ncase zero\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\n\u22a2 pochhammer S n * comp (pochhammer S Nat.zero) (X + \u2191n) = pochhammer S (n + Nat.zero)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nS : Type u\ninst\u271d : Semiring S\nn m : \u2115\nih : pochhammer S n * comp (pochhammer S m) (X + \u2191n) = pochhammer S (n + m)\n\u22a2 pochhammer S n * comp (pochhammer S (Nat.succ m)) (X + \u2191n) = pochhammer S (n + Nat.succ m)\n[PROOFSTEP]\nrw [pochhammer_succ_right, Polynomial.mul_X_add_nat_cast_comp, \u2190 mul_assoc, ih, Nat.succ_eq_add_one, \u2190 add_assoc,\n pochhammer_succ_right, Nat.cast_add, add_assoc]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn : \u2115\n\u22a2 eval (n + 1) (pochhammer \u2115 0) = Nat.ascFactorial n 0\n[PROOFSTEP]\nrw [pochhammer_zero, eval_one, Nat.ascFactorial_zero]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn t : \u2115\n\u22a2 eval (n + 1) (pochhammer \u2115 (t + 1)) = Nat.ascFactorial n (t + 1)\n[PROOFSTEP]\nrw [pochhammer_succ_right, eval_mul, pochhammer_nat_eq_ascFactorial n t]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn t : \u2115\n\u22a2 Nat.ascFactorial n t * eval (n + 1) (X + \u2191t) = Nat.ascFactorial n (t + 1)\n[PROOFSTEP]\nsimp only [eval_add, eval_X, eval_nat_cast, Nat.cast_id]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\nn t : \u2115\n\u22a2 Nat.ascFactorial n t * (n + 1 + t) = Nat.ascFactorial n (t + 1)\n[PROOFSTEP]\nrw [Nat.ascFactorial_succ, add_right_comm, mul_comm]\n[GOAL]\nS : Type u\ninst\u271d : Semiring S\na b : \u2115\n\u22a2 eval a (pochhammer \u2115 b) = Nat.descFactorial (a + b - 1) b\n[PROOFSTEP]\ncases' b with b\n[GOAL]\ncase zero\nS : Type u\ninst\u271d : Semiring S\na : \u2115\n\u22a2 eval a (pochhammer \u2115 Nat.zero) = Nat.descFactorial (a + Nat.zero - 1) Nat.zero\n[PROOFSTEP]\nrw [Nat.descFactorial_zero, pochhammer_zero, Polynomial.eval_one]\n[GOAL]\ncase succ\nS : Type u\ninst\u271d : Semiring S\na b : \u2115\n\u22a2 eval a (pochhammer \u2115 (Nat.succ b)) = Nat.descFactorial (a + Nat.succ b - 1) (Nat.succ b)\n[PROOFSTEP]\nrw [Nat.add_succ, Nat.succ_sub_succ, tsub_zero]\n[GOAL]\ncase succ\nS : Type u\ninst\u271d : Semiring S\na b : \u2115\n\u22a2 eval a (pochhammer \u2115 (Nat.succ b)) = Nat.descFactorial (a + b) (Nat.succ b)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase succ.zero\nS : Type u\ninst\u271d : Semiring S\nb : \u2115\n\u22a2 eval Nat.zero (pochhammer \u2115 (Nat.succ b)) = Nat.descFactorial (Nat.zero + b) (Nat.succ b)\n[PROOFSTEP]\nsimp only [Nat.zero_eq, ne_eq, Nat.succ_ne_zero, not_false_iff, pochhammer_ne_zero_eval_zero, zero_add,\n Nat.descFactorial_succ, le_refl, tsub_eq_zero_of_le, zero_mul]\n[GOAL]\ncase succ.succ\nS : Type u\ninst\u271d : Semiring S\nb n\u271d : \u2115\n\u22a2 eval (Nat.succ n\u271d) (pochhammer \u2115 (Nat.succ b)) = Nat.descFactorial (Nat.succ n\u271d + b) (Nat.succ b)\n[PROOFSTEP]\nrw [Nat.succ_add, \u2190 Nat.add_succ, Nat.add_descFactorial_eq_ascFactorial, pochhammer_nat_eq_ascFactorial]\n[GOAL]\nS : Type u_1\ninst\u271d : StrictOrderedSemiring S\nn : \u2115\ns : S\nh : 0 < s\n\u22a2 0 < eval s (pochhammer S n)\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nS : Type u_1\ninst\u271d : StrictOrderedSemiring S\ns : S\nh : 0 < s\n\u22a2 0 < eval s (pochhammer S Nat.zero)\n[PROOFSTEP]\nsimp only [Nat.zero_eq, pochhammer_zero, eval_one]\n[GOAL]\ncase zero\nS : Type u_1\ninst\u271d : StrictOrderedSemiring S\ns : S\nh : 0 < s\n\u22a2 0 < 1\n[PROOFSTEP]\nexact zero_lt_one\n[GOAL]\ncase succ\nS : Type u_1\ninst\u271d : StrictOrderedSemiring S\ns : S\nh : 0 < s\nn : \u2115\nih : 0 < eval s (pochhammer S n)\n\u22a2 0 < eval s (pochhammer S (Nat.succ n))\n[PROOFSTEP]\nrw [pochhammer_succ_right, mul_add, eval_add, \u2190 Nat.cast_comm, eval_nat_cast_mul, eval_mul_X, Nat.cast_comm, \u2190 mul_add]\n[GOAL]\ncase succ\nS : Type u_1\ninst\u271d : StrictOrderedSemiring S\ns : S\nh : 0 < s\nn : \u2115\nih : 0 < eval s (pochhammer S n)\n\u22a2 0 < eval s (pochhammer S n) * (s + \u2191n)\n[PROOFSTEP]\nexact mul_pos ih (lt_of_lt_of_le h ((le_add_iff_nonneg_right _).mpr (Nat.cast_nonneg n)))\n[GOAL]\nS\u271d : Type u_1\ninst\u271d\u00b9 : Semiring S\u271d\nr n\u271d : \u2115\nS : Type u_2\ninst\u271d : Semiring S\nn : \u2115\n\u22a2 eval 1 (pochhammer S n) = \u2191n !\n[PROOFSTEP]\nrw_mod_cast [pochhammer_nat_eq_ascFactorial, Nat.zero_ascFactorial]\n[GOAL]\nS\u271d : Type u_1\ninst\u271d\u00b9 : Semiring S\u271d\nr\u271d n\u271d : \u2115\nS : Type u_2\ninst\u271d : Semiring S\nr n : \u2115\n\u22a2 \u2191r ! * eval (\u2191r + 1) (pochhammer S n) = \u2191(r + n)!\n[PROOFSTEP]\nrw_mod_cast [pochhammer_nat_eq_ascFactorial, Nat.factorial_mul_ascFactorial]\n[GOAL]\nS : Type u_1\ninst\u271d : Semiring S\nr\u271d n r : \u2115\n\u22a2 0 * eval (0 + 1) (pochhammer \u2115 r) = (0 + r) * eval 0 (pochhammer \u2115 r)\n[PROOFSTEP]\nby_cases h : r = 0\n[GOAL]\ncase pos\nS : Type u_1\ninst\u271d : Semiring S\nr\u271d n r : \u2115\nh : r = 0\n\u22a2 0 * eval (0 + 1) (pochhammer \u2115 r) = (0 + r) * eval 0 (pochhammer \u2115 r)\n[PROOFSTEP]\nsimp only [h, zero_mul, zero_add]\n[GOAL]\ncase neg\nS : Type u_1\ninst\u271d : Semiring S\nr\u271d n r : \u2115\nh : \u00acr = 0\n\u22a2 0 * eval (0 + 1) (pochhammer \u2115 r) = (0 + r) * eval 0 (pochhammer \u2115 r)\n[PROOFSTEP]\nsimp only [pochhammer_eval_zero, zero_mul, if_neg h, mul_zero]\n[GOAL]\nS : Type u_1\ninst\u271d : Semiring S\nr\u271d n r k : \u2115\n\u22a2 (k + 1) * eval (k + 1 + 1) (pochhammer \u2115 r) = (k + 1 + r) * eval (k + 1) (pochhammer \u2115 r)\n[PROOFSTEP]\nsimp only [pochhammer_nat_eq_ascFactorial, Nat.succ_ascFactorial, add_right_comm]\n[GOAL]\nS : Type u_1\ninst\u271d : Semiring S\nr\u271d n\u271d r n : \u2115\n\u22a2 \u2191n * eval (\u2191n + 1) (pochhammer S r) = (\u2191n + \u2191r) * eval (\u2191n) (pochhammer S r)\n[PROOFSTEP]\nexact_mod_cast congr_arg Nat.cast (pochhammer_nat_eval_succ r n)\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Polynomial.Pochhammer", "llama_tokens": 5654, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056295505783, "lm_q2_score": 0.6406358411176238, "lm_q1q2_score": 0.44159389177424796}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\n\u22a2 (fun x => eval x P) ~[atTop] fun x => leadingCoeff P * x ^ natDegree P\n[PROOFSTEP]\nby_cases h : P = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nh : P = 0\n\u22a2 (fun x => eval x P) ~[atTop] fun x => leadingCoeff P * x ^ natDegree P\n[PROOFSTEP]\nsimp [h, IsEquivalent.refl]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nh : \u00acP = 0\n\u22a2 (fun x => eval x P) ~[atTop] fun x => leadingCoeff P * x ^ natDegree P\n[PROOFSTEP]\nsimp only [Polynomial.eval_eq_sum_range, sum_range_succ]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nh : \u00acP = 0\n\u22a2 (fun x =>\n (Finset.sum (range (natDegree P)) fun i => coeff P i * x ^ i) + coeff P (natDegree P) * x ^ natDegree P) ~[atTop]\n fun x => leadingCoeff P * x ^ natDegree P\n[PROOFSTEP]\nexact\n IsLittleO.add_isEquivalent\n (IsLittleO.sum fun i hi =>\n IsLittleO.const_mul_left\n ((IsLittleO.const_mul_right fun hz => h <| leadingCoeff_eq_zero.mp hz) <|\n isLittleO_pow_pow_atTop_of_lt (mem_range.mp hi))\n _)\n IsEquivalent.refl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\n\u22a2 Tendsto (fun x => eval x P) atTop atTop \u2194 0 < degree P \u2227 0 \u2264 leadingCoeff P\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => tendsto_atTop_of_leadingCoeff_nonneg P h.1 h.2\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nh : Tendsto (fun x => eval x P) atTop atTop\n\u22a2 0 < degree P \u2227 0 \u2264 leadingCoeff P\n[PROOFSTEP]\nhave : Tendsto (fun x => P.leadingCoeff * x ^ P.natDegree) atTop atTop := (isEquivalent_atTop_lead P).tendsto_atTop h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nh : Tendsto (fun x => eval x P) atTop atTop\nthis : Tendsto (fun x => leadingCoeff P * x ^ natDegree P) atTop atTop\n\u22a2 0 < degree P \u2227 0 \u2264 leadingCoeff P\n[PROOFSTEP]\nrw [tendsto_const_mul_pow_atTop_iff, \u2190 pos_iff_ne_zero, natDegree_pos_iff_degree_pos] at this \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nh : Tendsto (fun x => eval x P) atTop atTop\nthis : 0 < degree P \u2227 0 < leadingCoeff P\n\u22a2 0 < degree P \u2227 0 \u2264 leadingCoeff P\n[PROOFSTEP]\nexact \u27e8this.1, this.2.le\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\n\u22a2 Tendsto (fun x => eval x P) atTop atBot \u2194 0 < degree P \u2227 leadingCoeff P \u2264 0\n[PROOFSTEP]\nsimp only [\u2190 tendsto_neg_atTop_iff, \u2190 eval_neg, tendsto_atTop_iff_leadingCoeff_nonneg, degree_neg, leadingCoeff_neg,\n neg_nonneg]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : 0 < degree P\n\u22a2 Tendsto (fun x => |eval x P|) atTop atTop\n[PROOFSTEP]\ncases' le_total 0 P.leadingCoeff with hP hP\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : 0 < degree P\nhP : 0 \u2264 leadingCoeff P\n\u22a2 Tendsto (fun x => |eval x P|) atTop atTop\n[PROOFSTEP]\nexact tendsto_abs_atTop_atTop.comp (P.tendsto_atTop_of_leadingCoeff_nonneg hdeg hP)\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : 0 < degree P\nhP : leadingCoeff P \u2264 0\n\u22a2 Tendsto (fun x => |eval x P|) atTop atTop\n[PROOFSTEP]\nexact tendsto_abs_atBot_atTop.comp (P.tendsto_atBot_of_leadingCoeff_nonpos hdeg hP)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\n\u22a2 (IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => |eval x P|) \u2194 degree P \u2264 0\n[PROOFSTEP]\nrefine'\n \u27e8fun h => _, fun h =>\n \u27e8|P.coeff 0|,\n eventually_map.mpr\n (eventually_of_forall\n (forall_imp (fun _ => le_of_eq) fun x =>\n congr_arg abs <| _root_.trans (congr_arg (eval x) (eq_C_of_degree_le_zero h)) eval_C))\u27e9\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nh : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => |eval x P|\n\u22a2 degree P \u2264 0\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nh : 0 < degree P\n\u22a2 \u00acIsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => |eval x P|\n[PROOFSTEP]\nexact not_isBoundedUnder_of_tendsto_atTop (abs_tendsto_atTop P h)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nc : \ud835\udd5c\n\u22a2 Tendsto (fun x => eval x P) atTop (\ud835\udcdd c) \u2194 leadingCoeff P = c \u2227 degree P \u2264 0\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nc : \ud835\udd5c\nh : Tendsto (fun x => eval x P) atTop (\ud835\udcdd c)\n\u22a2 leadingCoeff P = c \u2227 degree P \u2264 0\n[PROOFSTEP]\nhave := P.isEquivalent_atTop_lead.tendsto_nhds h\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nc : \ud835\udd5c\nh : Tendsto (fun x => eval x P) atTop (\ud835\udcdd c)\nthis : Tendsto (fun x => leadingCoeff P * x ^ natDegree P) atTop (\ud835\udcdd c)\n\u22a2 leadingCoeff P = c \u2227 degree P \u2264 0\n[PROOFSTEP]\nby_cases hP : P.leadingCoeff = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nc : \ud835\udd5c\nh : Tendsto (fun x => eval x P) atTop (\ud835\udcdd c)\nthis : Tendsto (fun x => leadingCoeff P * x ^ natDegree P) atTop (\ud835\udcdd c)\nhP : leadingCoeff P = 0\n\u22a2 leadingCoeff P = c \u2227 degree P \u2264 0\n[PROOFSTEP]\nsimp only [hP, zero_mul, tendsto_const_nhds_iff] at this \n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nc : \ud835\udd5c\nh : Tendsto (fun x => eval x P) atTop (\ud835\udcdd c)\nhP : leadingCoeff P = 0\nthis : 0 = c\n\u22a2 leadingCoeff P = c \u2227 degree P \u2264 0\n[PROOFSTEP]\nrefine' \u27e8_root_.trans hP this, by simp [leadingCoeff_eq_zero.1 hP]\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nc : \ud835\udd5c\nh : Tendsto (fun x => eval x P) atTop (\ud835\udcdd c)\nhP : leadingCoeff P = 0\nthis : 0 = c\n\u22a2 degree P \u2264 0\n[PROOFSTEP]\nsimp [leadingCoeff_eq_zero.1 hP]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nc : \ud835\udd5c\nh : Tendsto (fun x => eval x P) atTop (\ud835\udcdd c)\nthis : Tendsto (fun x => leadingCoeff P * x ^ natDegree P) atTop (\ud835\udcdd c)\nhP : \u00acleadingCoeff P = 0\n\u22a2 leadingCoeff P = c \u2227 degree P \u2264 0\n[PROOFSTEP]\nrw [tendsto_const_mul_pow_nhds_iff hP, natDegree_eq_zero_iff_degree_le_zero] at this \n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nc : \ud835\udd5c\nh : Tendsto (fun x => eval x P) atTop (\ud835\udcdd c)\nthis : degree P \u2264 0 \u2227 leadingCoeff P = c\nhP : \u00acleadingCoeff P = 0\n\u22a2 leadingCoeff P = c \u2227 degree P \u2264 0\n[PROOFSTEP]\nexact this.symm\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nc : \ud835\udd5c\nh : leadingCoeff P = c \u2227 degree P \u2264 0\n\u22a2 Tendsto (fun x => eval x P) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nrefine' P.isEquivalent_atTop_lead.symm.tendsto_nhds _\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nc : \ud835\udd5c\nh : leadingCoeff P = c \u2227 degree P \u2264 0\n\u22a2 Tendsto (fun x => leadingCoeff P * x ^ natDegree P) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nhave : P.natDegree = 0 := natDegree_eq_zero_iff_degree_le_zero.2 h.2\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nc : \ud835\udd5c\nh : leadingCoeff P = c \u2227 degree P \u2264 0\nthis : natDegree P = 0\n\u22a2 Tendsto (fun x => leadingCoeff P * x ^ natDegree P) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nsimp only [h.1, this, pow_zero, mul_one]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nc : \ud835\udd5c\nh : leadingCoeff P = c \u2227 degree P \u2264 0\nthis : natDegree P = 0\n\u22a2 Tendsto (fun x => c) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nexact tendsto_const_nhds\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\n\u22a2 (fun x => eval x P / eval x Q) ~[atTop] fun x =>\n leadingCoeff P / leadingCoeff Q * x ^ (\u2191(natDegree P) - \u2191(natDegree Q))\n[PROOFSTEP]\nby_cases hP : P = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhP : P = 0\n\u22a2 (fun x => eval x P / eval x Q) ~[atTop] fun x =>\n leadingCoeff P / leadingCoeff Q * x ^ (\u2191(natDegree P) - \u2191(natDegree Q))\n[PROOFSTEP]\nsimp [hP, IsEquivalent.refl]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhP : \u00acP = 0\n\u22a2 (fun x => eval x P / eval x Q) ~[atTop] fun x =>\n leadingCoeff P / leadingCoeff Q * x ^ (\u2191(natDegree P) - \u2191(natDegree Q))\n[PROOFSTEP]\nby_cases hQ : Q = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhP : \u00acP = 0\nhQ : Q = 0\n\u22a2 (fun x => eval x P / eval x Q) ~[atTop] fun x =>\n leadingCoeff P / leadingCoeff Q * x ^ (\u2191(natDegree P) - \u2191(natDegree Q))\n[PROOFSTEP]\nsimp [hQ, IsEquivalent.refl]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhP : \u00acP = 0\nhQ : \u00acQ = 0\n\u22a2 (fun x => eval x P / eval x Q) ~[atTop] fun x =>\n leadingCoeff P / leadingCoeff Q * x ^ (\u2191(natDegree P) - \u2191(natDegree Q))\n[PROOFSTEP]\nrefine'\n (P.isEquivalent_atTop_lead.symm.div Q.isEquivalent_atTop_lead.symm).symm.trans\n (EventuallyEq.isEquivalent ((eventually_gt_atTop 0).mono fun x hx => _))\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhP : \u00acP = 0\nhQ : \u00acQ = 0\nx : \ud835\udd5c\nhx : 0 < x\n\u22a2 (fun x => leadingCoeff P * x ^ natDegree P / (leadingCoeff Q * x ^ natDegree Q)) x =\n (fun x => leadingCoeff P / leadingCoeff Q * x ^ (\u2191(natDegree P) - \u2191(natDegree Q))) x\n[PROOFSTEP]\nsimp [\u2190 div_mul_div_comm, hP, hQ, zpow_sub\u2080 hx.ne.symm]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree P < degree Q\n\u22a2 Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nby_cases hP : P = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree P < degree Q\nhP : P = 0\n\u22a2 Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp [hP, tendsto_const_nhds]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree P < degree Q\nhP : \u00acP = 0\n\u22a2 Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 natDegree_lt_natDegree_iff hP] at hdeg \n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : natDegree P < natDegree Q\nhP : \u00acP = 0\n\u22a2 Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' (isEquivalent_atTop_div P Q).symm.tendsto_nhds _\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : natDegree P < natDegree Q\nhP : \u00acP = 0\n\u22a2 Tendsto (fun x => leadingCoeff P / leadingCoeff Q * x ^ (\u2191(natDegree P) - \u2191(natDegree Q))) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 mul_zero]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : natDegree P < natDegree Q\nhP : \u00acP = 0\n\u22a2 Tendsto (fun x => leadingCoeff P / leadingCoeff Q * x ^ (\u2191(natDegree P) - \u2191(natDegree Q))) atTop (\ud835\udcdd (?neg\u271d * 0))\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : natDegree P < natDegree Q\nhP : \u00acP = 0\n\u22a2 \ud835\udd5c\n[PROOFSTEP]\nrefine' (tendsto_zpow_atTop_zero _).const_mul _\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : natDegree P < natDegree Q\nhP : \u00acP = 0\n\u22a2 \u2191(natDegree P) - \u2191(natDegree Q) < 0\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhQ : Q \u2260 0\n\u22a2 Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0) \u2194 degree P < degree Q\n[PROOFSTEP]\nrefine' \u27e8fun h => _, div_tendsto_zero_of_degree_lt P Q\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhQ : Q \u2260 0\nh : Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\n\u22a2 degree P < degree Q\n[PROOFSTEP]\nby_cases hPQ : P.leadingCoeff / Q.leadingCoeff = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhQ : Q \u2260 0\nh : Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\nhPQ : leadingCoeff P / leadingCoeff Q = 0\n\u22a2 degree P < degree Q\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, inv_eq_zero, mul_eq_zero] at hPQ \n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhQ : Q \u2260 0\nh : Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\nhPQ : leadingCoeff P = 0 \u2228 leadingCoeff Q = 0\n\u22a2 degree P < degree Q\n[PROOFSTEP]\ncases' hPQ with hP0 hQ0\n[GOAL]\ncase pos.inl\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhQ : Q \u2260 0\nh : Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\nhP0 : leadingCoeff P = 0\n\u22a2 degree P < degree Q\n[PROOFSTEP]\nrw [leadingCoeff_eq_zero.1 hP0, degree_zero]\n[GOAL]\ncase pos.inl\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhQ : Q \u2260 0\nh : Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\nhP0 : leadingCoeff P = 0\n\u22a2 \u22a5 < degree Q\n[PROOFSTEP]\nexact bot_lt_iff_ne_bot.2 fun hQ' => hQ (degree_eq_bot.1 hQ')\n[GOAL]\ncase pos.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhQ : Q \u2260 0\nh : Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\nhQ0 : leadingCoeff Q = 0\n\u22a2 degree P < degree Q\n[PROOFSTEP]\nexact absurd (leadingCoeff_eq_zero.1 hQ0) hQ\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhQ : Q \u2260 0\nh : Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\nhPQ : \u00acleadingCoeff P / leadingCoeff Q = 0\n\u22a2 degree P < degree Q\n[PROOFSTEP]\nhave := (isEquivalent_atTop_div P Q).tendsto_nhds h\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhQ : Q \u2260 0\nh : Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\nhPQ : \u00acleadingCoeff P / leadingCoeff Q = 0\nthis : Tendsto (fun x => leadingCoeff P / leadingCoeff Q * x ^ (\u2191(natDegree P) - \u2191(natDegree Q))) atTop (\ud835\udcdd 0)\n\u22a2 degree P < degree Q\n[PROOFSTEP]\nrw [tendsto_const_mul_zpow_atTop_nhds_iff hPQ] at this \n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhQ : Q \u2260 0\nh : Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\nhPQ : \u00acleadingCoeff P / leadingCoeff Q = 0\nthis :\n \u2191(natDegree P) - \u2191(natDegree Q) = 0 \u2227 leadingCoeff P / leadingCoeff Q = 0 \u2228\n \u2191(natDegree P) - \u2191(natDegree Q) < 0 \u2227 0 = 0\n\u22a2 degree P < degree Q\n[PROOFSTEP]\ncases' this with h h\n[GOAL]\ncase neg.inl\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhQ : Q \u2260 0\nh\u271d : Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\nhPQ : \u00acleadingCoeff P / leadingCoeff Q = 0\nh : \u2191(natDegree P) - \u2191(natDegree Q) = 0 \u2227 leadingCoeff P / leadingCoeff Q = 0\n\u22a2 degree P < degree Q\n[PROOFSTEP]\nexact absurd h.2 hPQ\n[GOAL]\ncase neg.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhQ : Q \u2260 0\nh\u271d : Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\nhPQ : \u00acleadingCoeff P / leadingCoeff Q = 0\nh : \u2191(natDegree P) - \u2191(natDegree Q) < 0 \u2227 0 = 0\n\u22a2 degree P < degree Q\n[PROOFSTEP]\nrw [sub_lt_iff_lt_add, zero_add, Int.ofNat_lt] at h \n[GOAL]\ncase neg.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhQ : Q \u2260 0\nh\u271d : Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd 0)\nhPQ : \u00acleadingCoeff P / leadingCoeff Q = 0\nh : natDegree P < natDegree Q \u2227 0 = 0\n\u22a2 degree P < degree Q\n[PROOFSTEP]\nexact degree_lt_degree h.1\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree P = degree Q\n\u22a2 Tendsto (fun x => eval x P / eval x Q) atTop (\ud835\udcdd (leadingCoeff P / leadingCoeff Q))\n[PROOFSTEP]\nrefine' (isEquivalent_atTop_div P Q).symm.tendsto_nhds _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree P = degree Q\n\u22a2 Tendsto (fun x => leadingCoeff P / leadingCoeff Q * x ^ (\u2191(natDegree P) - \u2191(natDegree Q))) atTop\n (\ud835\udcdd (leadingCoeff P / leadingCoeff Q))\n[PROOFSTEP]\nrw [show (P.natDegree : \u2124) = Q.natDegree by simp [hdeg, natDegree]]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree P = degree Q\n\u22a2 \u2191(natDegree P) = \u2191(natDegree Q)\n[PROOFSTEP]\nsimp [hdeg, natDegree]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree P = degree Q\n\u22a2 Tendsto (fun x => leadingCoeff P / leadingCoeff Q * x ^ (\u2191(natDegree Q) - \u2191(natDegree Q))) atTop\n (\ud835\udcdd (leadingCoeff P / leadingCoeff Q))\n[PROOFSTEP]\nsimp [tendsto_const_nhds]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree Q < degree P\nhpos : 0 < leadingCoeff P / leadingCoeff Q\n\u22a2 Tendsto (fun x => eval x P / eval x Q) atTop atTop\n[PROOFSTEP]\nhave hQ : Q \u2260 0 := fun h => by\n simp only [h, div_zero, leadingCoeff_zero] at hpos \n exact hpos.false\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree Q < degree P\nhpos : 0 < leadingCoeff P / leadingCoeff Q\nh : Q = 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [h, div_zero, leadingCoeff_zero] at hpos \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree Q < degree P\nh : Q = 0\nhpos : 0 < 0\n\u22a2 False\n[PROOFSTEP]\nexact hpos.false\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree Q < degree P\nhpos : 0 < leadingCoeff P / leadingCoeff Q\nhQ : Q \u2260 0\n\u22a2 Tendsto (fun x => eval x P / eval x Q) atTop atTop\n[PROOFSTEP]\nrw [\u2190 natDegree_lt_natDegree_iff hQ] at hdeg \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : natDegree Q < natDegree P\nhpos : 0 < leadingCoeff P / leadingCoeff Q\nhQ : Q \u2260 0\n\u22a2 Tendsto (fun x => eval x P / eval x Q) atTop atTop\n[PROOFSTEP]\nrefine' (isEquivalent_atTop_div P Q).symm.tendsto_atTop _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : natDegree Q < natDegree P\nhpos : 0 < leadingCoeff P / leadingCoeff Q\nhQ : Q \u2260 0\n\u22a2 Tendsto (fun x => leadingCoeff P / leadingCoeff Q * x ^ (\u2191(natDegree P) - \u2191(natDegree Q))) atTop atTop\n[PROOFSTEP]\napply Tendsto.const_mul_atTop hpos\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : natDegree Q < natDegree P\nhpos : 0 < leadingCoeff P / leadingCoeff Q\nhQ : Q \u2260 0\n\u22a2 Tendsto (fun x => x ^ (\u2191(natDegree P) - \u2191(natDegree Q))) atTop atTop\n[PROOFSTEP]\napply tendsto_zpow_atTop_atTop\n[GOAL]\ncase hn\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : natDegree Q < natDegree P\nhpos : 0 < leadingCoeff P / leadingCoeff Q\nhQ : Q \u2260 0\n\u22a2 0 < \u2191(natDegree P) - \u2191(natDegree Q)\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree Q < degree P\nhneg : leadingCoeff P / leadingCoeff Q < 0\n\u22a2 Tendsto (fun x => eval x P / eval x Q) atTop atBot\n[PROOFSTEP]\nhave hQ : Q \u2260 0 := fun h => by\n simp only [h, div_zero, leadingCoeff_zero] at hneg \n exact hneg.false\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree Q < degree P\nhneg : leadingCoeff P / leadingCoeff Q < 0\nh : Q = 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [h, div_zero, leadingCoeff_zero] at hneg \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree Q < degree P\nh : Q = 0\nhneg : 0 < 0\n\u22a2 False\n[PROOFSTEP]\nexact hneg.false\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree Q < degree P\nhneg : leadingCoeff P / leadingCoeff Q < 0\nhQ : Q \u2260 0\n\u22a2 Tendsto (fun x => eval x P / eval x Q) atTop atBot\n[PROOFSTEP]\nrw [\u2190 natDegree_lt_natDegree_iff hQ] at hdeg \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : natDegree Q < natDegree P\nhneg : leadingCoeff P / leadingCoeff Q < 0\nhQ : Q \u2260 0\n\u22a2 Tendsto (fun x => eval x P / eval x Q) atTop atBot\n[PROOFSTEP]\nrefine' (isEquivalent_atTop_div P Q).symm.tendsto_atBot _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : natDegree Q < natDegree P\nhneg : leadingCoeff P / leadingCoeff Q < 0\nhQ : Q \u2260 0\n\u22a2 Tendsto (fun x => leadingCoeff P / leadingCoeff Q * x ^ (\u2191(natDegree P) - \u2191(natDegree Q))) atTop atBot\n[PROOFSTEP]\napply Tendsto.neg_const_mul_atTop hneg\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : natDegree Q < natDegree P\nhneg : leadingCoeff P / leadingCoeff Q < 0\nhQ : Q \u2260 0\n\u22a2 Tendsto (fun x => x ^ (\u2191(natDegree P) - \u2191(natDegree Q))) atTop atTop\n[PROOFSTEP]\napply tendsto_zpow_atTop_atTop\n[GOAL]\ncase hn\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : natDegree Q < natDegree P\nhneg : leadingCoeff P / leadingCoeff Q < 0\nhQ : Q \u2260 0\n\u22a2 0 < \u2191(natDegree P) - \u2191(natDegree Q)\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree Q < degree P\nhQ : Q \u2260 0\n\u22a2 Tendsto (fun x => |eval x P / eval x Q|) atTop atTop\n[PROOFSTEP]\nby_cases h : 0 \u2264 P.leadingCoeff / Q.leadingCoeff\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree Q < degree P\nhQ : Q \u2260 0\nh : 0 \u2264 leadingCoeff P / leadingCoeff Q\n\u22a2 Tendsto (fun x => |eval x P / eval x Q|) atTop atTop\n[PROOFSTEP]\nexact tendsto_abs_atTop_atTop.comp (P.div_tendsto_atTop_of_degree_gt Q hdeg hQ h)\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree Q < degree P\nhQ : Q \u2260 0\nh : \u00ac0 \u2264 leadingCoeff P / leadingCoeff Q\n\u22a2 Tendsto (fun x => |eval x P / eval x Q|) atTop atTop\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nhdeg : degree Q < degree P\nhQ : Q \u2260 0\nh : leadingCoeff P / leadingCoeff Q < 0\n\u22a2 Tendsto (fun x => |eval x P / eval x Q|) atTop atTop\n[PROOFSTEP]\nexact tendsto_abs_atBot_atTop.comp (P.div_tendsto_atBot_of_degree_gt Q hdeg hQ h.le)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nh : degree P \u2264 degree Q\n\u22a2 (fun x => eval x P) =O[atTop] fun x => eval x Q\n[PROOFSTEP]\nby_cases hp : P = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nh : degree P \u2264 degree Q\nhp : P = 0\n\u22a2 (fun x => eval x P) =O[atTop] fun x => eval x Q\n[PROOFSTEP]\nsimpa [hp] using isBigO_zero (fun x => eval x Q) atTop\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nh : degree P \u2264 degree Q\nhp : \u00acP = 0\n\u22a2 (fun x => eval x P) =O[atTop] fun x => eval x Q\n[PROOFSTEP]\nhave hq : Q \u2260 0 := ne_zero_of_degree_ge_degree h hp\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nh : degree P \u2264 degree Q\nhp : \u00acP = 0\nhq : Q \u2260 0\n\u22a2 (fun x => eval x P) =O[atTop] fun x => eval x Q\n[PROOFSTEP]\nhave hPQ : \u2200\u1da0 x : \ud835\udd5c in atTop, eval x Q = 0 \u2192 eval x P = 0 :=\n Filter.mem_of_superset (Polynomial.eventually_no_roots Q hq) fun x h h' => absurd h' h\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nh : degree P \u2264 degree Q\nhp : \u00acP = 0\nhq : Q \u2260 0\nhPQ : \u2200\u1da0 (x : \ud835\udd5c) in atTop, eval x Q = 0 \u2192 eval x P = 0\n\u22a2 (fun x => eval x P) =O[atTop] fun x => eval x Q\n[PROOFSTEP]\ncases' le_iff_lt_or_eq.mp h with h h\n[GOAL]\ncase neg.inl\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nh\u271d : degree P \u2264 degree Q\nhp : \u00acP = 0\nhq : Q \u2260 0\nhPQ : \u2200\u1da0 (x : \ud835\udd5c) in atTop, eval x Q = 0 \u2192 eval x P = 0\nh : degree P < degree Q\n\u22a2 (fun x => eval x P) =O[atTop] fun x => eval x Q\n[PROOFSTEP]\nexact isBigO_of_div_tendsto_nhds hPQ 0 (div_tendsto_zero_of_degree_lt P Q h)\n[GOAL]\ncase neg.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NormedLinearOrderedField \ud835\udd5c\nP Q : \ud835\udd5c[X]\ninst\u271d : OrderTopology \ud835\udd5c\nh\u271d : degree P \u2264 degree Q\nhp : \u00acP = 0\nhq : Q \u2260 0\nhPQ : \u2200\u1da0 (x : \ud835\udd5c) in atTop, eval x Q = 0 \u2192 eval x P = 0\nh : degree P = degree Q\n\u22a2 (fun x => eval x P) =O[atTop] fun x => eval x Q\n[PROOFSTEP]\nexact isBigO_of_div_tendsto_nhds hPQ _ (div_tendsto_leadingCoeff_div_of_degree_eq P Q h)\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.Polynomials", "llama_tokens": 11879, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056040203135, "lm_q2_score": 0.640635861701035, "lm_q1q2_score": 0.441593889606906}} {"text": "[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\n\u22a2 Multiset.map (fun x => Int.natAbs (valMinAbs (a * \u2191x))) (Ico 1 (succ (p / 2))).val =\n Multiset.map (fun a => a) (Ico 1 (succ (p / 2))).val\n[PROOFSTEP]\nhave he : \u2200 {x}, x \u2208 Ico 1 (p / 2).succ \u2192 x \u2260 0 \u2227 x \u2264 p / 2 := by\n simp (config := { contextual := true }) [Nat.lt_succ_iff, Nat.succ_le_iff, pos_iff_ne_zero]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\n\u22a2 \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\n[PROOFSTEP]\nsimp (config := { contextual := true }) [Nat.lt_succ_iff, Nat.succ_le_iff, pos_iff_ne_zero]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\n\u22a2 Multiset.map (fun x => Int.natAbs (valMinAbs (a * \u2191x))) (Ico 1 (succ (p / 2))).val =\n Multiset.map (fun a => a) (Ico 1 (succ (p / 2))).val\n[PROOFSTEP]\nhave hep : \u2200 {x}, x \u2208 Ico 1 (p / 2).succ \u2192 x < p := fun hx =>\n lt_of_le_of_lt (he hx).2 (Nat.div_lt_self hp.1.pos (by decide))\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nx\u271d : \u2115\nhx : x\u271d \u2208 Ico 1 (succ (p / 2))\n\u22a2 1 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\n\u22a2 Multiset.map (fun x => Int.natAbs (valMinAbs (a * \u2191x))) (Ico 1 (succ (p / 2))).val =\n Multiset.map (fun a => a) (Ico 1 (succ (p / 2))).val\n[PROOFSTEP]\nhave hpe : \u2200 {x}, x \u2208 Ico 1 (p / 2).succ \u2192 \u00acp \u2223 x := fun hx hpx =>\n not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero (he hx).1) hpx) (hep hx)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\n\u22a2 Multiset.map (fun x => Int.natAbs (valMinAbs (a * \u2191x))) (Ico 1 (succ (p / 2))).val =\n Multiset.map (fun a => a) (Ico 1 (succ (p / 2))).val\n[PROOFSTEP]\nhave hmem : \u2200 (x : \u2115) (hx : x \u2208 Ico 1 (p / 2).succ), (a * x : ZMod p).valMinAbs.natAbs \u2208 Ico 1 (p / 2).succ :=\n by\n intro x hx\n simp [hap, CharP.cast_eq_zero_iff (ZMod p) p, hpe hx, lt_succ_iff, succ_le_iff, pos_iff_ne_zero,\n natAbs_valMinAbs_le _]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\n\u22a2 \u2200 (x : \u2115), x \u2208 Ico 1 (succ (p / 2)) \u2192 Int.natAbs (valMinAbs (a * \u2191x)) \u2208 Ico 1 (succ (p / 2))\n[PROOFSTEP]\nintro x hx\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\nx : \u2115\nhx : x \u2208 Ico 1 (succ (p / 2))\n\u22a2 Int.natAbs (valMinAbs (a * \u2191x)) \u2208 Ico 1 (succ (p / 2))\n[PROOFSTEP]\nsimp [hap, CharP.cast_eq_zero_iff (ZMod p) p, hpe hx, lt_succ_iff, succ_le_iff, pos_iff_ne_zero, natAbs_valMinAbs_le _]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\nhmem : \u2200 (x : \u2115), x \u2208 Ico 1 (succ (p / 2)) \u2192 Int.natAbs (valMinAbs (a * \u2191x)) \u2208 Ico 1 (succ (p / 2))\n\u22a2 Multiset.map (fun x => Int.natAbs (valMinAbs (a * \u2191x))) (Ico 1 (succ (p / 2))).val =\n Multiset.map (fun a => a) (Ico 1 (succ (p / 2))).val\n[PROOFSTEP]\nhave hsurj : \u2200 (b : \u2115) (hb : b \u2208 Ico 1 (p / 2).succ), \u2203 x \u2208 Ico 1 (p / 2).succ, b = (a * x : ZMod p).valMinAbs.natAbs :=\n by\n intro b hb\n refine' \u27e8(b / a : ZMod p).valMinAbs.natAbs, mem_Ico.mpr \u27e8_, _\u27e9, _\u27e9\n \u00b7 apply Nat.pos_of_ne_zero\n simp only [div_eq_mul_inv, hap, CharP.cast_eq_zero_iff (ZMod p) p, hpe hb, not_false_iff, valMinAbs_eq_zero,\n inv_eq_zero, Int.natAbs_eq_zero, Ne.def, _root_.mul_eq_zero, or_self_iff]\n \u00b7 apply lt_succ_of_le; apply natAbs_valMinAbs_le\n \u00b7 rw [nat_cast_natAbs_valMinAbs]\n split_ifs\n \u00b7\n erw [mul_div_cancel' _ hap, valMinAbs_def_pos, val_cast_of_lt (hep hb), if_pos (le_of_lt_succ (mem_Ico.1 hb).2),\n Int.natAbs_ofNat]\n \u00b7\n erw [mul_neg, mul_div_cancel' _ hap, natAbs_valMinAbs_neg, valMinAbs_def_pos, val_cast_of_lt (hep hb),\n if_pos (le_of_lt_succ (mem_Ico.1 hb).2), Int.natAbs_ofNat]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\nhmem : \u2200 (x : \u2115), x \u2208 Ico 1 (succ (p / 2)) \u2192 Int.natAbs (valMinAbs (a * \u2191x)) \u2208 Ico 1 (succ (p / 2))\n\u22a2 \u2200 (b : \u2115), b \u2208 Ico 1 (succ (p / 2)) \u2192 \u2203 x, x \u2208 Ico 1 (succ (p / 2)) \u2227 b = Int.natAbs (valMinAbs (a * \u2191x))\n[PROOFSTEP]\nintro b hb\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\nhmem : \u2200 (x : \u2115), x \u2208 Ico 1 (succ (p / 2)) \u2192 Int.natAbs (valMinAbs (a * \u2191x)) \u2208 Ico 1 (succ (p / 2))\nb : \u2115\nhb : b \u2208 Ico 1 (succ (p / 2))\n\u22a2 \u2203 x, x \u2208 Ico 1 (succ (p / 2)) \u2227 b = Int.natAbs (valMinAbs (a * \u2191x))\n[PROOFSTEP]\nrefine' \u27e8(b / a : ZMod p).valMinAbs.natAbs, mem_Ico.mpr \u27e8_, _\u27e9, _\u27e9\n[GOAL]\ncase refine'_1\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\nhmem : \u2200 (x : \u2115), x \u2208 Ico 1 (succ (p / 2)) \u2192 Int.natAbs (valMinAbs (a * \u2191x)) \u2208 Ico 1 (succ (p / 2))\nb : \u2115\nhb : b \u2208 Ico 1 (succ (p / 2))\n\u22a2 1 \u2264 Int.natAbs (valMinAbs (\u2191b / a))\n[PROOFSTEP]\napply Nat.pos_of_ne_zero\n[GOAL]\ncase refine'_1.a\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\nhmem : \u2200 (x : \u2115), x \u2208 Ico 1 (succ (p / 2)) \u2192 Int.natAbs (valMinAbs (a * \u2191x)) \u2208 Ico 1 (succ (p / 2))\nb : \u2115\nhb : b \u2208 Ico 1 (succ (p / 2))\n\u22a2 Int.natAbs (valMinAbs (\u2191b / a)) \u2260 0\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, hap, CharP.cast_eq_zero_iff (ZMod p) p, hpe hb, not_false_iff, valMinAbs_eq_zero,\n inv_eq_zero, Int.natAbs_eq_zero, Ne.def, _root_.mul_eq_zero, or_self_iff]\n[GOAL]\ncase refine'_2\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\nhmem : \u2200 (x : \u2115), x \u2208 Ico 1 (succ (p / 2)) \u2192 Int.natAbs (valMinAbs (a * \u2191x)) \u2208 Ico 1 (succ (p / 2))\nb : \u2115\nhb : b \u2208 Ico 1 (succ (p / 2))\n\u22a2 Int.natAbs (valMinAbs (\u2191b / a)) < succ (p / 2)\n[PROOFSTEP]\napply lt_succ_of_le\n[GOAL]\ncase refine'_2.a\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\nhmem : \u2200 (x : \u2115), x \u2208 Ico 1 (succ (p / 2)) \u2192 Int.natAbs (valMinAbs (a * \u2191x)) \u2208 Ico 1 (succ (p / 2))\nb : \u2115\nhb : b \u2208 Ico 1 (succ (p / 2))\n\u22a2 Int.natAbs (valMinAbs (\u2191b / a)) \u2264 p / 2\n[PROOFSTEP]\napply natAbs_valMinAbs_le\n[GOAL]\ncase refine'_3\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\nhmem : \u2200 (x : \u2115), x \u2208 Ico 1 (succ (p / 2)) \u2192 Int.natAbs (valMinAbs (a * \u2191x)) \u2208 Ico 1 (succ (p / 2))\nb : \u2115\nhb : b \u2208 Ico 1 (succ (p / 2))\n\u22a2 b = Int.natAbs (valMinAbs (a * \u2191(Int.natAbs (valMinAbs (\u2191b / a)))))\n[PROOFSTEP]\nrw [nat_cast_natAbs_valMinAbs]\n[GOAL]\ncase refine'_3\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\nhmem : \u2200 (x : \u2115), x \u2208 Ico 1 (succ (p / 2)) \u2192 Int.natAbs (valMinAbs (a * \u2191x)) \u2208 Ico 1 (succ (p / 2))\nb : \u2115\nhb : b \u2208 Ico 1 (succ (p / 2))\n\u22a2 b = Int.natAbs (valMinAbs (a * if val (\u2191b / a) \u2264 p / 2 then \u2191b / a else -(\u2191b / a)))\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\nhmem : \u2200 (x : \u2115), x \u2208 Ico 1 (succ (p / 2)) \u2192 Int.natAbs (valMinAbs (a * \u2191x)) \u2208 Ico 1 (succ (p / 2))\nb : \u2115\nhb : b \u2208 Ico 1 (succ (p / 2))\nh\u271d : val (\u2191b / a) \u2264 p / 2\n\u22a2 b = Int.natAbs (valMinAbs (a * (\u2191b / a)))\n[PROOFSTEP]\nerw [mul_div_cancel' _ hap, valMinAbs_def_pos, val_cast_of_lt (hep hb), if_pos (le_of_lt_succ (mem_Ico.1 hb).2),\n Int.natAbs_ofNat]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\nhmem : \u2200 (x : \u2115), x \u2208 Ico 1 (succ (p / 2)) \u2192 Int.natAbs (valMinAbs (a * \u2191x)) \u2208 Ico 1 (succ (p / 2))\nb : \u2115\nhb : b \u2208 Ico 1 (succ (p / 2))\nh\u271d : \u00acval (\u2191b / a) \u2264 p / 2\n\u22a2 b = Int.natAbs (valMinAbs (a * -(\u2191b / a)))\n[PROOFSTEP]\nerw [mul_neg, mul_div_cancel' _ hap, natAbs_valMinAbs_neg, valMinAbs_def_pos, val_cast_of_lt (hep hb),\n if_pos (le_of_lt_succ (mem_Ico.1 hb).2), Int.natAbs_ofNat]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\nhmem : \u2200 (x : \u2115), x \u2208 Ico 1 (succ (p / 2)) \u2192 Int.natAbs (valMinAbs (a * \u2191x)) \u2208 Ico 1 (succ (p / 2))\nhsurj : \u2200 (b : \u2115), b \u2208 Ico 1 (succ (p / 2)) \u2192 \u2203 x, x \u2208 Ico 1 (succ (p / 2)) \u2227 b = Int.natAbs (valMinAbs (a * \u2191x))\n\u22a2 Multiset.map (fun x => Int.natAbs (valMinAbs (a * \u2191x))) (Ico 1 (succ (p / 2))).val =\n Multiset.map (fun a => a) (Ico 1 (succ (p / 2))).val\n[PROOFSTEP]\nsimp only [\u2190 exists_prop] at hsurj \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : ZMod p\nhap : a \u2260 0\nhe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x \u2260 0 \u2227 x \u2264 p / 2\nhep : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 x < p\nhpe : \u2200 {x : \u2115}, x \u2208 Ico 1 (succ (p / 2)) \u2192 \u00acp \u2223 x\nhmem : \u2200 (x : \u2115), x \u2208 Ico 1 (succ (p / 2)) \u2192 Int.natAbs (valMinAbs (a * \u2191x)) \u2208 Ico 1 (succ (p / 2))\nhsurj : \u2200 (b : \u2115), b \u2208 Ico 1 (succ (p / 2)) \u2192 \u2203 x _h, b = Int.natAbs (valMinAbs (a * \u2191x))\n\u22a2 Multiset.map (fun x => Int.natAbs (valMinAbs (a * \u2191x))) (Ico 1 (succ (p / 2))).val =\n Multiset.map (fun a => a) (Ico 1 (succ (p / 2))).val\n[PROOFSTEP]\nexact\n Multiset.map_eq_map_of_bij_of_nodup _ _ (Finset.nodup _) (Finset.nodup _)\n (fun x _ => (a * x : ZMod p).valMinAbs.natAbs) hmem (fun _ _ => rfl)\n (inj_on_of_surj_on_of_card_le _ hmem hsurj le_rfl) hsurj\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\n\u22a2 \u2191(a ^ (p / 2)) * \u2191(p / 2)! = \u2191(\u220f x in Ico 1 (succ (p / 2)), a * \u2191x)\n[PROOFSTEP]\nrw [prod_mul_distrib, \u2190 prod_natCast, prod_Ico_id_eq_factorial, prod_const, card_Ico, succ_sub_one]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\n\u22a2 \u2191(a ^ (p / 2)) * \u2191(p / 2)! = \u2191(a ^ (p / 2) * \u2191(p / 2)!)\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\n\u22a2 \u2191(\u220f x in Ico 1 (succ (p / 2)), a * \u2191x) = \u220f x in Ico 1 (succ (p / 2)), \u2191(val (\u2191a * \u2191x))\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\nx\u271d\u00b9 : \u2115\nx\u271d : x\u271d\u00b9 \u2208 Ico 1 (succ (p / 2))\n\u22a2 \u2191(val (\u2191a * \u2191x\u271d\u00b9)) = (if val (\u2191a * \u2191x\u271d\u00b9) \u2264 p / 2 then 1 else -1) * \u2191(Int.natAbs (valMinAbs (\u2191a * \u2191x\u271d\u00b9)))\n[PROOFSTEP]\nsimp only [nat_cast_natAbs_valMinAbs]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\nx\u271d\u00b9 : \u2115\nx\u271d : x\u271d\u00b9 \u2208 Ico 1 (succ (p / 2))\n\u22a2 \u2191(val (\u2191a * \u2191x\u271d\u00b9)) =\n (if val (\u2191a * \u2191x\u271d\u00b9) \u2264 p / 2 then 1 else -1) * if val (\u2191a * \u2191x\u271d\u00b9) \u2264 p / 2 then \u2191a * \u2191x\u271d\u00b9 else -(\u2191a * \u2191x\u271d\u00b9)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\nx\u271d\u00b9 : \u2115\nx\u271d : x\u271d\u00b9 \u2208 Ico 1 (succ (p / 2))\nh\u271d : val (\u2191a * \u2191x\u271d\u00b9) \u2264 p / 2\n\u22a2 \u2191(val (\u2191a * \u2191x\u271d\u00b9)) = 1 * (\u2191a * \u2191x\u271d\u00b9)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\nx\u271d\u00b9 : \u2115\nx\u271d : x\u271d\u00b9 \u2208 Ico 1 (succ (p / 2))\nh\u271d : \u00acval (\u2191a * \u2191x\u271d\u00b9) \u2264 p / 2\n\u22a2 \u2191(val (\u2191a * \u2191x\u271d\u00b9)) = -1 * -(\u2191a * \u2191x\u271d\u00b9)\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\n\u22a2 \u220f x in Ico 1 (succ (p / 2)), (if val (\u2191a * \u2191x) \u2264 p / 2 then 1 else -1) * \u2191(Int.natAbs (valMinAbs (\u2191a * \u2191x))) =\n (-1) ^ Finset.card (filter (fun x => \u00acval (\u2191a * \u2191x) \u2264 p / 2) (Ico 1 (succ (p / 2)))) *\n \u220f x in Ico 1 (succ (p / 2)), \u2191(Int.natAbs (valMinAbs (\u2191a * \u2191x)))\n[PROOFSTEP]\nhave :\n (\u220f x in Ico 1 (p / 2).succ, if (a * x : ZMod p).val \u2264 p / 2 then (1 : ZMod p) else -1) =\n \u220f x in (Ico 1 (p / 2).succ).filter fun x : \u2115 => \u00ac(a * x : ZMod p).val \u2264 p / 2, -1 :=\n prod_bij_ne_one (fun x _ _ => x) (fun x => by split_ifs <;> simp_all (config := { contextual := true }))\n (fun _ _ _ _ _ _ => id) (fun b h _ => \u27e8b, by simp_all [-not_le]\u27e9) (by intros; split_ifs at * <;> simp_all)\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\nx : \u2115\n\u22a2 \u2200 (h\u2081 : x \u2208 Ico 1 (succ (p / 2))) (h\u2082 : (if val (\u2191a * \u2191x) \u2264 p / 2 then 1 else -1) \u2260 1),\n (fun x x_1 x_2 => x) x h\u2081 h\u2082 \u2208 filter (fun x => \u00acval (\u2191a * \u2191x) \u2264 p / 2) (Ico 1 (succ (p / 2)))\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\nx : \u2115\nh\u271d : val (\u2191a * \u2191x) \u2264 p / 2\n\u22a2 \u2200 (h\u2081 : x \u2208 Ico 1 (succ (p / 2))) (h\u2082 : (if val (\u2191a * \u2191x) \u2264 p / 2 then 1 else -1) \u2260 1),\n (fun x x_1 x_2 => x) x h\u2081 h\u2082 \u2208 filter (fun x => \u00acval (\u2191a * \u2191x) \u2264 p / 2) (Ico 1 (succ (p / 2)))\n[PROOFSTEP]\nsimp_all (config := { contextual := true })\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\nx : \u2115\nh\u271d : \u00acval (\u2191a * \u2191x) \u2264 p / 2\n\u22a2 \u2200 (h\u2081 : x \u2208 Ico 1 (succ (p / 2))) (h\u2082 : (if val (\u2191a * \u2191x) \u2264 p / 2 then 1 else -1) \u2260 1),\n (fun x x_1 x_2 => x) x h\u2081 h\u2082 \u2208 filter (fun x => \u00acval (\u2191a * \u2191x) \u2264 p / 2) (Ico 1 (succ (p / 2)))\n[PROOFSTEP]\nsimp_all (config := { contextual := true })\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\nb : \u2115\nh : b \u2208 filter (fun x => \u00acval (\u2191a * \u2191x) \u2264 p / 2) (Ico 1 (succ (p / 2)))\nx\u271d : -1 \u2260 1\n\u22a2 \u2203 h\u2081 h\u2082, b = (fun x x_1 x_2 => x) b h\u2081 h\u2082\n[PROOFSTEP]\nsimp_all [-not_le]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\n\u22a2 \u2200 (a_1 : \u2115),\n a_1 \u2208 Ico 1 (succ (p / 2)) \u2192\n (if val (\u2191a * \u2191a_1) \u2264 p / 2 then 1 else -1) \u2260 1 \u2192 (if val (\u2191a * \u2191a_1) \u2264 p / 2 then 1 else -1) = -1\n[PROOFSTEP]\nintros\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\na\u271d : \u2115\nh\u2081\u271d : a\u271d \u2208 Ico 1 (succ (p / 2))\nh\u2082\u271d : (if val (\u2191a * \u2191a\u271d) \u2264 p / 2 then 1 else -1) \u2260 1\n\u22a2 (if val (\u2191a * \u2191a\u271d) \u2264 p / 2 then 1 else -1) = -1\n[PROOFSTEP]\nsplit_ifs at *\n[GOAL]\ncase pos\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\na\u271d : \u2115\nh\u2081\u271d : a\u271d \u2208 Ico 1 (succ (p / 2))\nh\u271d : val (\u2191a * \u2191a\u271d) \u2264 p / 2\nh\u2082\u271d : 1 \u2260 1\n\u22a2 1 = -1\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\na\u271d : \u2115\nh\u2081\u271d : a\u271d \u2208 Ico 1 (succ (p / 2))\nh\u271d : \u00acval (\u2191a * \u2191a\u271d) \u2264 p / 2\nh\u2082\u271d : -1 \u2260 1\n\u22a2 -1 = -1\n[PROOFSTEP]\nsimp_all\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\nthis :\n (\u220f x in Ico 1 (succ (p / 2)), if val (\u2191a * \u2191x) \u2264 p / 2 then 1 else -1) =\n \u220f x in filter (fun x => \u00acval (\u2191a * \u2191x) \u2264 p / 2) (Ico 1 (succ (p / 2))), -1\n\u22a2 \u220f x in Ico 1 (succ (p / 2)), (if val (\u2191a * \u2191x) \u2264 p / 2 then 1 else -1) * \u2191(Int.natAbs (valMinAbs (\u2191a * \u2191x))) =\n (-1) ^ Finset.card (filter (fun x => \u00acval (\u2191a * \u2191x) \u2264 p / 2) (Ico 1 (succ (p / 2)))) *\n \u220f x in Ico 1 (succ (p / 2)), \u2191(Int.natAbs (valMinAbs (\u2191a * \u2191x)))\n[PROOFSTEP]\nrw [prod_mul_distrib, this]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\nthis :\n (\u220f x in Ico 1 (succ (p / 2)), if val (\u2191a * \u2191x) \u2264 p / 2 then 1 else -1) =\n \u220f x in filter (fun x => \u00acval (\u2191a * \u2191x) \u2264 p / 2) (Ico 1 (succ (p / 2))), -1\n\u22a2 (\u220f x in filter (fun x => \u00acval (\u2191a * \u2191x) \u2264 p / 2) (Ico 1 (succ (p / 2))), -1) *\n \u220f x in Ico 1 (succ (p / 2)), \u2191(Int.natAbs (valMinAbs (\u2191a * \u2191x))) =\n (-1) ^ Finset.card (filter (fun x => \u00acval (\u2191a * \u2191x) \u2264 p / 2) (Ico 1 (succ (p / 2)))) *\n \u220f x in Ico 1 (succ (p / 2)), \u2191(Int.natAbs (valMinAbs (\u2191a * \u2191x)))\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\n\u22a2 (-1) ^ Finset.card (filter (fun x => \u00acval (\u2191a * \u2191x) \u2264 p / 2) (Ico 1 (succ (p / 2)))) *\n \u220f x in Ico 1 (succ (p / 2)), \u2191(Int.natAbs (valMinAbs (\u2191a * \u2191x))) =\n (-1) ^ Finset.card (filter (fun x => \u00acval (\u2191a * \u2191x) \u2264 p / 2) (Ico 1 (succ (p / 2)))) * \u2191(p / 2)!\n[PROOFSTEP]\nrw [\u2190 prod_natCast, Finset.prod_eq_multiset_prod, Ico_map_valMinAbs_natAbs_eq_Ico_map_id p a hap, \u2190\n Finset.prod_eq_multiset_prod, prod_Ico_id_eq_factorial]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\n\u22a2 \u2191(p / 2)! \u2260 0\n[PROOFSTEP]\nrw [Ne.def, CharP.cast_eq_zero_iff (ZMod p) p, hp.1.dvd_factorial, not_le]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\n\u22a2 p / 2 < p\n[PROOFSTEP]\nexact Nat.div_lt_self hp.1.pos (by decide)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\n\u22a2 1 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : \u2124\nhap : \u2191a \u2260 0\n\u22a2 \u2191a ^ (p / 2) * \u2191(p / 2)! =\n \u2191((-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2))))) * \u2191(p / 2)!\n[PROOFSTEP]\nsimpa using gauss_lemma_aux\u2081 p hap\n[GOAL]\np : \u2115\nh : Fact (Nat.Prime p)\na : \u2124\nhp : p \u2260 2\nha0 : \u2191a \u2260 0\n\u22a2 legendreSym p a = (-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2))))\n[PROOFSTEP]\nreplace hp : Odd p := h.out.odd_of_ne_two hp\n[GOAL]\np : \u2115\nh : Fact (Nat.Prime p)\na : \u2124\nha0 : \u2191a \u2260 0\nhp : Odd p\n\u22a2 legendreSym p a = (-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2))))\n[PROOFSTEP]\nhave :\n (legendreSym p a : ZMod p) =\n (((-1) ^ ((Ico 1 (p / 2).succ).filter fun x : \u2115 => p / 2 < (a * x : ZMod p).val).card : \u2124) : ZMod p) :=\n by rw [legendreSym.eq_pow, gauss_lemma_aux p ha0]\n[GOAL]\np : \u2115\nh : Fact (Nat.Prime p)\na : \u2124\nha0 : \u2191a \u2260 0\nhp : Odd p\n\u22a2 \u2191(legendreSym p a) = \u2191((-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2)))))\n[PROOFSTEP]\nrw [legendreSym.eq_pow, gauss_lemma_aux p ha0]\n[GOAL]\np : \u2115\nh : Fact (Nat.Prime p)\na : \u2124\nha0 : \u2191a \u2260 0\nhp : Odd p\nthis : \u2191(legendreSym p a) = \u2191((-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2)))))\n\u22a2 legendreSym p a = (-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2))))\n[PROOFSTEP]\ncases legendreSym.eq_one_or_neg_one p ha0\n[GOAL]\ncase inl\np : \u2115\nh : Fact (Nat.Prime p)\na : \u2124\nha0 : \u2191a \u2260 0\nhp : Odd p\nthis : \u2191(legendreSym p a) = \u2191((-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2)))))\nh\u271d : legendreSym p a = 1\n\u22a2 legendreSym p a = (-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2))))\n[PROOFSTEP]\ncases neg_one_pow_eq_or \u2124 ((Ico 1 (p / 2).succ).filter fun x : \u2115 => p / 2 < (a * x : ZMod p).val).card\n[GOAL]\ncase inr\np : \u2115\nh : Fact (Nat.Prime p)\na : \u2124\nha0 : \u2191a \u2260 0\nhp : Odd p\nthis : \u2191(legendreSym p a) = \u2191((-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2)))))\nh\u271d : legendreSym p a = -1\n\u22a2 legendreSym p a = (-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2))))\n[PROOFSTEP]\ncases neg_one_pow_eq_or \u2124 ((Ico 1 (p / 2).succ).filter fun x : \u2115 => p / 2 < (a * x : ZMod p).val).card\n[GOAL]\ncase inl.inl\np : \u2115\nh : Fact (Nat.Prime p)\na : \u2124\nha0 : \u2191a \u2260 0\nhp : Odd p\nthis : \u2191(legendreSym p a) = \u2191((-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2)))))\nh\u271d\u00b9 : legendreSym p a = 1\nh\u271d : (-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2)))) = 1\n\u22a2 legendreSym p a = (-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2))))\n[PROOFSTEP]\nsimp_all [ne_neg_self hp one_ne_zero, (ne_neg_self hp one_ne_zero).symm]\n[GOAL]\ncase inl.inr\np : \u2115\nh : Fact (Nat.Prime p)\na : \u2124\nha0 : \u2191a \u2260 0\nhp : Odd p\nthis : \u2191(legendreSym p a) = \u2191((-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2)))))\nh\u271d\u00b9 : legendreSym p a = 1\nh\u271d : (-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2)))) = -1\n\u22a2 legendreSym p a = (-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2))))\n[PROOFSTEP]\nsimp_all [ne_neg_self hp one_ne_zero, (ne_neg_self hp one_ne_zero).symm]\n[GOAL]\ncase inr.inl\np : \u2115\nh : Fact (Nat.Prime p)\na : \u2124\nha0 : \u2191a \u2260 0\nhp : Odd p\nthis : \u2191(legendreSym p a) = \u2191((-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2)))))\nh\u271d\u00b9 : legendreSym p a = -1\nh\u271d : (-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2)))) = 1\n\u22a2 legendreSym p a = (-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2))))\n[PROOFSTEP]\nsimp_all [ne_neg_self hp one_ne_zero, (ne_neg_self hp one_ne_zero).symm]\n[GOAL]\ncase inr.inr\np : \u2115\nh : Fact (Nat.Prime p)\na : \u2124\nha0 : \u2191a \u2260 0\nhp : Odd p\nthis : \u2191(legendreSym p a) = \u2191((-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2)))))\nh\u271d\u00b9 : legendreSym p a = -1\nh\u271d : (-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2)))) = -1\n\u22a2 legendreSym p a = (-1) ^ Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2))))\n[PROOFSTEP]\nsimp_all [ne_neg_self hp one_ne_zero, (ne_neg_self hp one_ne_zero).symm]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp2\u271d : Fact (p % 2 = 1)\na : \u2115\nhap : \u2191a \u2260 0\nhp2 : \u2191p = \u21911\n\u22a2 \u2191(\u2211 x in Ico 1 (succ (p / 2)), a * x) = \u2191(\u2211 x in Ico 1 (succ (p / 2)), (a * x % p + p * (a * x / p)))\n[PROOFSTEP]\nsimp only [mod_add_div]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp2\u271d : Fact (p % 2 = 1)\na : \u2115\nhap : \u2191a \u2260 0\nhp2 : \u2191p = \u21911\n\u22a2 \u2191(\u2211 x in Ico 1 (succ (p / 2)), (a * x % p + p * (a * x / p))) =\n \u2191(\u2211 x in Ico 1 (succ (p / 2)), val \u2191(a * x)) + \u2191(\u2211 x in Ico 1 (succ (p / 2)), a * x / p)\n[PROOFSTEP]\nsimp only [val_nat_cast]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp2\u271d : Fact (p % 2 = 1)\na : \u2115\nhap : \u2191a \u2260 0\nhp2 : \u2191p = \u21911\n\u22a2 \u2191(\u2211 x in Ico 1 (succ (p / 2)), (a * x % p + p * (a * x / p))) =\n \u2191(\u2211 x in Ico 1 (succ (p / 2)), a * x % p) + \u2191(\u2211 x in Ico 1 (succ (p / 2)), a * x / p)\n[PROOFSTEP]\nsimp [sum_add_distrib, mul_sum.symm, Nat.cast_add, Nat.cast_mul, Nat.cast_sum, hp2]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp2\u271d : Fact (p % 2 = 1)\na : \u2115\nhap : \u2191a \u2260 0\nhp2 : \u2191p = \u21911\n\u22a2 \u2191(\u2211 x in Ico 1 (succ (p / 2)), val \u2191(a * x)) =\n \u2211 x in Ico 1 (succ (p / 2)), \u2191(valMinAbs (\u2191a * \u2191x) + \u2191(if val (\u2191a * \u2191x) \u2264 p / 2 then 0 else p))\n[PROOFSTEP]\nsimp only [(val_eq_ite_valMinAbs _).symm]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp2\u271d : Fact (p % 2 = 1)\na : \u2115\nhap : \u2191a \u2260 0\nhp2 : \u2191p = \u21911\n\u22a2 \u2191(\u2211 x in Ico 1 (succ (p / 2)), val \u2191(a * x)) = \u2211 x in Ico 1 (succ (p / 2)), \u2191\u2191(val (\u2191a * \u2191x))\n[PROOFSTEP]\nsimp [Nat.cast_sum]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp2\u271d : Fact (p % 2 = 1)\na : \u2115\nhap : \u2191a \u2260 0\nhp2 : \u2191p = \u21911\n\u22a2 \u2211 x in Ico 1 (succ (p / 2)), \u2191(valMinAbs (\u2191a * \u2191x) + \u2191(if val (\u2191a * \u2191x) \u2264 p / 2 then 0 else p)) =\n \u2191(Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2))))) +\n \u2191(\u2211 x in Ico 1 (succ (p / 2)), Int.natAbs (valMinAbs (\u2191a * \u2191x)))\n[PROOFSTEP]\nsimp [add_comm, sum_add_distrib, Finset.sum_ite, hp2, Nat.cast_sum]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp2\u271d : Fact (p % 2 = 1)\na : \u2115\nhap : \u2191a \u2260 0\nhp2 : \u2191p = \u21911\n\u22a2 \u2191(Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2))))) +\n \u2191(\u2211 x in Ico 1 (succ (p / 2)), Int.natAbs (valMinAbs (\u2191a * \u2191x))) =\n \u2191(Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2))))) + \u2191(\u2211 x in Ico 1 (succ (p / 2)), x)\n[PROOFSTEP]\nrw [Finset.sum_eq_multiset_sum, Ico_map_valMinAbs_natAbs_eq_Ico_map_id p a hap, \u2190 Finset.sum_eq_multiset_sum]\n[GOAL]\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (p % 2 = 1)\na : \u2115\nha2\u271d : a % 2 = 1\nhap : \u2191a \u2260 0\nha2 : \u2191a = \u21911\n\u22a2 \u2191(Finset.card (filter (fun x => p / 2 < val (\u2191a * \u2191x)) (Ico 1 (succ (p / 2))))) -\n \u2191(\u2211 x in Ico 1 (succ (p / 2)), x * a / p) =\n 0\n[PROOFSTEP]\nsimpa [add_left_comm, sub_eq_add_neg, Finset.mul_sum.symm, mul_comm, ha2, Nat.cast_sum, add_neg_eq_iff_eq_add.symm,\n neg_eq_self_mod_two, add_assoc] using Eq.symm (eisenstein_lemma_aux\u2081 p hap)\n[GOAL]\na b c : \u2115\nhb0 : 0 < b\nhc : a / b \u2264 c\n\u22a2 a / b = Finset.card (Ico 1 (succ (a / b)))\n[PROOFSTEP]\nsimp\n[GOAL]\na b c : \u2115\nhb0 : 0 < b\nhc : a / b \u2264 c\nx : \u2115\n\u22a2 x \u2208 Ico 1 (succ (a / b)) \u2194 x \u2208 filter (fun x => x * b \u2264 a) (Ico 1 (succ c))\n[PROOFSTEP]\nhave : x * b \u2264 a \u2192 x \u2264 c := fun h => le_trans (by rwa [le_div_iff_mul_le hb0]) hc\n[GOAL]\na b c : \u2115\nhb0 : 0 < b\nhc : a / b \u2264 c\nx : \u2115\nh : x * b \u2264 a\n\u22a2 x \u2264 a / b\n[PROOFSTEP]\nrwa [le_div_iff_mul_le hb0]\n[GOAL]\na b c : \u2115\nhb0 : 0 < b\nhc : a / b \u2264 c\nx : \u2115\nthis : x * b \u2264 a \u2192 x \u2264 c\n\u22a2 x \u2208 Ico 1 (succ (a / b)) \u2194 x \u2208 filter (fun x => x * b \u2264 a) (Ico 1 (succ c))\n[PROOFSTEP]\nsimp [lt_succ_iff, le_div_iff_mul_le hb0]\n[GOAL]\na b c : \u2115\nhb0 : 0 < b\nhc : a / b \u2264 c\nx : \u2115\nthis : x * b \u2264 a \u2192 x \u2264 c\n\u22a2 x * b \u2264 a \u2192 1 \u2264 x \u2192 x \u2264 c\n[PROOFSTEP]\ntauto\n[GOAL]\np q : \u2115\nhp0 : p = 0\n\u22a2 \u2211 a in Ico 1 (succ (p / 2)), a * q / p =\n Finset.card (filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\n[PROOFSTEP]\nsimp [hp0, Finset.ext_iff]\n[GOAL]\np q : \u2115\nhp0 : \u00acp = 0\n\u22a2 \u2211 a in Ico 1 (succ (p / 2)), Finset.card (filter (fun x => x * p \u2264 a * q) (Ico 1 (succ (q / 2)))) =\n Finset.card (filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\n[PROOFSTEP]\nrw [\u2190 card_sigma]\n[GOAL]\np q : \u2115\nhp0 : \u00acp = 0\n\u22a2 Finset.card (Finset.sigma (Ico 1 (succ (p / 2))) fun a => filter (fun x => x * p \u2264 a * q) (Ico 1 (succ (q / 2)))) =\n Finset.card (filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\n[PROOFSTEP]\nexact\n card_congr (fun a _ => \u27e8a.1, a.2\u27e9)\n (by\n simp (config := { contextual := true }) only [mem_filter, mem_sigma, and_self_iff, forall_true_iff, mem_product])\n (fun \u27e8_, _\u27e9 \u27e8_, _\u27e9 => by\n simp (config := { contextual := true }) only [Prod.mk.inj_iff, eq_self_iff_true, and_self_iff, heq_iff_eq,\n forall_true_iff])\n fun \u27e8b\u2081, b\u2082\u27e9 h =>\n \u27e8\u27e8b\u2081, b\u2082\u27e9, by\n revert h\n simp (config := { contextual := true }) only [mem_filter, eq_self_iff_true, exists_prop_of_true, mem_sigma,\n and_self_iff, forall_true_iff, mem_product]\u27e9\n[GOAL]\np q : \u2115\nhp0 : \u00acp = 0\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115)\n (ha : a \u2208 Finset.sigma (Ico 1 (succ (p / 2))) fun a => filter (fun x => x * p \u2264 a * q) (Ico 1 (succ (q / 2)))),\n (fun a x => (a.fst, a.snd)) a ha \u2208\n filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2)))\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [mem_filter, mem_sigma, and_self_iff, forall_true_iff, mem_product]\n[GOAL]\np q : \u2115\nhp0 : \u00acp = 0\nx\u271d\u00b9 x\u271d : (_ : \u2115) \u00d7 \u2115\nfst\u271d\u00b9 snd\u271d\u00b9 fst\u271d snd\u271d : \u2115\n\u22a2 \u2200\n (ha :\n { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 } \u2208\n Finset.sigma (Ico 1 (succ (p / 2))) fun a => filter (fun x => x * p \u2264 a * q) (Ico 1 (succ (q / 2))))\n (hb :\n { fst := fst\u271d, snd := snd\u271d } \u2208\n Finset.sigma (Ico 1 (succ (p / 2))) fun a => filter (fun x => x * p \u2264 a * q) (Ico 1 (succ (q / 2)))),\n (fun a x => (a.fst, a.snd)) { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 } ha =\n (fun a x => (a.fst, a.snd)) { fst := fst\u271d, snd := snd\u271d } hb \u2192\n { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 } = { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [Prod.mk.inj_iff, eq_self_iff_true, and_self_iff, heq_iff_eq,\n forall_true_iff]\n[GOAL]\np q : \u2115\nhp0 : \u00acp = 0\nx\u271d : \u2115 \u00d7 \u2115\nb\u2081 b\u2082 : \u2115\nh : (b\u2081, b\u2082) \u2208 filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2)))\n\u22a2 \u2203 ha, (fun a x => (a.fst, a.snd)) { fst := b\u2081, snd := b\u2082 } ha = (b\u2081, b\u2082)\n[PROOFSTEP]\nrevert h\n[GOAL]\np q : \u2115\nhp0 : \u00acp = 0\nx\u271d : \u2115 \u00d7 \u2115\nb\u2081 b\u2082 : \u2115\n\u22a2 (b\u2081, b\u2082) \u2208 filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))) \u2192\n \u2203 ha, (fun a x => (a.fst, a.snd)) { fst := b\u2081, snd := b\u2082 } ha = (b\u2081, b\u2082)\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [mem_filter, eq_self_iff_true, exists_prop_of_true, mem_sigma,\n and_self_iff, forall_true_iff, mem_product]\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\n\u22a2 \u2211 a in Ico 1 (succ (p / 2)), a * q / p + \u2211 a in Ico 1 (succ (q / 2)), a * p / q = p / 2 * (q / 2)\n[PROOFSTEP]\nhave hswap :\n ((Ico 1 (q / 2).succ \u00d7\u02e2 Ico 1 (p / 2).succ).filter fun x : \u2115 \u00d7 \u2115 => x.2 * q \u2264 x.1 * p).card =\n ((Ico 1 (p / 2).succ \u00d7\u02e2 Ico 1 (q / 2).succ).filter fun x : \u2115 \u00d7 \u2115 => x.1 * q \u2264 x.2 * p).card :=\n card_congr (fun x _ => Prod.swap x)\n (fun \u27e8_, _\u27e9 => by\n simp (config := { contextual := true }) only [mem_filter, and_self_iff, Prod.swap_prod_mk, forall_true_iff,\n mem_product])\n (fun \u27e8_, _\u27e9 \u27e8_, _\u27e9 => by\n simp (config := { contextual := true }) only [Prod.mk.inj_iff, eq_self_iff_true, and_self_iff, Prod.swap_prod_mk,\n forall_true_iff])\n fun \u27e8x\u2081, x\u2082\u27e9 h =>\n \u27e8\u27e8x\u2082, x\u2081\u27e9, by\n revert h\n simp (config := { contextual := true }) only [mem_filter, eq_self_iff_true, and_self_iff, exists_prop_of_true,\n Prod.swap_prod_mk, forall_true_iff, mem_product]\u27e9\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nx\u271d : \u2115 \u00d7 \u2115\nfst\u271d snd\u271d : \u2115\n\u22a2 \u2200 (ha : (fst\u271d, snd\u271d) \u2208 filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))),\n (fun x x_1 => Prod.swap x) (fst\u271d, snd\u271d) ha \u2208\n filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2)))\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [mem_filter, and_self_iff, Prod.swap_prod_mk, forall_true_iff, mem_product]\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nx\u271d\u00b9 x\u271d : \u2115 \u00d7 \u2115\nfst\u271d\u00b9 snd\u271d\u00b9 fst\u271d snd\u271d : \u2115\n\u22a2 \u2200 (ha : (fst\u271d\u00b9, snd\u271d\u00b9) \u2208 filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2))))\n (hb : (fst\u271d, snd\u271d) \u2208 filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))),\n (fun x x_1 => Prod.swap x) (fst\u271d\u00b9, snd\u271d\u00b9) ha = (fun x x_1 => Prod.swap x) (fst\u271d, snd\u271d) hb \u2192\n (fst\u271d\u00b9, snd\u271d\u00b9) = (fst\u271d, snd\u271d)\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [Prod.mk.inj_iff, eq_self_iff_true, and_self_iff, Prod.swap_prod_mk,\n forall_true_iff]\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nx\u271d : \u2115 \u00d7 \u2115\nx\u2081 x\u2082 : \u2115\nh : (x\u2081, x\u2082) \u2208 filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2)))\n\u22a2 \u2203 ha, (fun x x_1 => Prod.swap x) (x\u2082, x\u2081) ha = (x\u2081, x\u2082)\n[PROOFSTEP]\nrevert h\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nx\u271d : \u2115 \u00d7 \u2115\nx\u2081 x\u2082 : \u2115\n\u22a2 (x\u2081, x\u2082) \u2208 filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))) \u2192\n \u2203 ha, (fun x x_1 => Prod.swap x) (x\u2082, x\u2081) ha = (x\u2081, x\u2082)\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [mem_filter, eq_self_iff_true, and_self_iff, exists_prop_of_true,\n Prod.swap_prod_mk, forall_true_iff, mem_product]\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\n\u22a2 \u2211 a in Ico 1 (succ (p / 2)), a * q / p + \u2211 a in Ico 1 (succ (q / 2)), a * p / q = p / 2 * (q / 2)\n[PROOFSTEP]\nhave hdisj :\n Disjoint ((Ico 1 (p / 2).succ \u00d7\u02e2 Ico 1 (q / 2).succ).filter fun x : \u2115 \u00d7 \u2115 => x.2 * p \u2264 x.1 * q)\n ((Ico 1 (p / 2).succ \u00d7\u02e2 Ico 1 (q / 2).succ).filter fun x : \u2115 \u00d7 \u2115 => x.1 * q \u2264 x.2 * p) :=\n by\n apply disjoint_filter.2 fun x hx hpq hqp => ?_\n have hxp : x.1 < p :=\n lt_of_le_of_lt (show x.1 \u2264 p / 2 by simp_all only [lt_succ_iff, mem_Ico, mem_product])\n (Nat.div_lt_self hp.1.pos (by decide))\n have : (x.1 : ZMod p) = 0 := by simpa [hq0] using congr_arg ((\u2191) : \u2115 \u2192 ZMod p) (le_antisymm hpq hqp)\n apply_fun ZMod.val at this \n rw [val_cast_of_lt hxp, val_zero] at this \n simp only [this, nonpos_iff_eq_zero, mem_Ico, one_ne_zero, false_and_iff, mem_product] at hx \n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\n\u22a2 Disjoint (filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\n (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\n[PROOFSTEP]\napply disjoint_filter.2 fun x hx hpq hqp => ?_\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nx : \u2115 \u00d7 \u2115\nhx : x \u2208 Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))\nhpq : x.snd * p \u2264 x.fst * q\nhqp : x.fst * q \u2264 x.snd * p\n\u22a2 False\n[PROOFSTEP]\nhave hxp : x.1 < p :=\n lt_of_le_of_lt (show x.1 \u2264 p / 2 by simp_all only [lt_succ_iff, mem_Ico, mem_product])\n (Nat.div_lt_self hp.1.pos (by decide))\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nx : \u2115 \u00d7 \u2115\nhx : x \u2208 Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))\nhpq : x.snd * p \u2264 x.fst * q\nhqp : x.fst * q \u2264 x.snd * p\n\u22a2 x.fst \u2264 p / 2\n[PROOFSTEP]\nsimp_all only [lt_succ_iff, mem_Ico, mem_product]\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nx : \u2115 \u00d7 \u2115\nhx : x \u2208 Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))\nhpq : x.snd * p \u2264 x.fst * q\nhqp : x.fst * q \u2264 x.snd * p\n\u22a2 1 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nx : \u2115 \u00d7 \u2115\nhx : x \u2208 Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))\nhpq : x.snd * p \u2264 x.fst * q\nhqp : x.fst * q \u2264 x.snd * p\nhxp : x.fst < p\n\u22a2 False\n[PROOFSTEP]\nhave : (x.1 : ZMod p) = 0 := by simpa [hq0] using congr_arg ((\u2191) : \u2115 \u2192 ZMod p) (le_antisymm hpq hqp)\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nx : \u2115 \u00d7 \u2115\nhx : x \u2208 Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))\nhpq : x.snd * p \u2264 x.fst * q\nhqp : x.fst * q \u2264 x.snd * p\nhxp : x.fst < p\n\u22a2 \u2191x.fst = 0\n[PROOFSTEP]\nsimpa [hq0] using congr_arg ((\u2191) : \u2115 \u2192 ZMod p) (le_antisymm hpq hqp)\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nx : \u2115 \u00d7 \u2115\nhx : x \u2208 Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))\nhpq : x.snd * p \u2264 x.fst * q\nhqp : x.fst * q \u2264 x.snd * p\nhxp : x.fst < p\nthis : \u2191x.fst = 0\n\u22a2 False\n[PROOFSTEP]\napply_fun ZMod.val at this \n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nx : \u2115 \u00d7 \u2115\nhx : x \u2208 Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))\nhpq : x.snd * p \u2264 x.fst * q\nhqp : x.fst * q \u2264 x.snd * p\nhxp : x.fst < p\nthis : val \u2191x.fst = val 0\n\u22a2 False\n[PROOFSTEP]\nrw [val_cast_of_lt hxp, val_zero] at this \n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nx : \u2115 \u00d7 \u2115\nhx : x \u2208 Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))\nhpq : x.snd * p \u2264 x.fst * q\nhqp : x.fst * q \u2264 x.snd * p\nhxp : x.fst < p\nthis : x.fst = 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [this, nonpos_iff_eq_zero, mem_Ico, one_ne_zero, false_and_iff, mem_product] at hx \n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nhdisj :\n Disjoint (filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\n (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\n\u22a2 \u2211 a in Ico 1 (succ (p / 2)), a * q / p + \u2211 a in Ico 1 (succ (q / 2)), a * p / q = p / 2 * (q / 2)\n[PROOFSTEP]\nhave hunion :\n (((Ico 1 (p / 2).succ \u00d7\u02e2 Ico 1 (q / 2).succ).filter fun x : \u2115 \u00d7 \u2115 => x.2 * p \u2264 x.1 * q) \u222a\n (Ico 1 (p / 2).succ \u00d7\u02e2 Ico 1 (q / 2).succ).filter fun x : \u2115 \u00d7 \u2115 => x.1 * q \u2264 x.2 * p) =\n Ico 1 (p / 2).succ \u00d7\u02e2 Ico 1 (q / 2).succ :=\n Finset.ext fun x => by\n have := le_total (x.2 * p) (x.1 * q)\n simp only [mem_union, mem_filter, mem_Ico, mem_product]\n tauto\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nhdisj :\n Disjoint (filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\n (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nx : \u2115 \u00d7 \u2115\n\u22a2 x \u2208\n filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))) \u222a\n filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))) \u2194\n x \u2208 Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))\n[PROOFSTEP]\nhave := le_total (x.2 * p) (x.1 * q)\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nhdisj :\n Disjoint (filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\n (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nx : \u2115 \u00d7 \u2115\nthis : x.snd * p \u2264 x.fst * q \u2228 x.fst * q \u2264 x.snd * p\n\u22a2 x \u2208\n filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))) \u222a\n filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))) \u2194\n x \u2208 Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))\n[PROOFSTEP]\nsimp only [mem_union, mem_filter, mem_Ico, mem_product]\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nhdisj :\n Disjoint (filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\n (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nx : \u2115 \u00d7 \u2115\nthis : x.snd * p \u2264 x.fst * q \u2228 x.fst * q \u2264 x.snd * p\n\u22a2 ((1 \u2264 x.fst \u2227 x.fst < succ (p / 2)) \u2227 1 \u2264 x.snd \u2227 x.snd < succ (q / 2)) \u2227 x.snd * p \u2264 x.fst * q \u2228\n ((1 \u2264 x.fst \u2227 x.fst < succ (p / 2)) \u2227 1 \u2264 x.snd \u2227 x.snd < succ (q / 2)) \u2227 x.fst * q \u2264 x.snd * p \u2194\n (1 \u2264 x.fst \u2227 x.fst < succ (p / 2)) \u2227 1 \u2264 x.snd \u2227 x.snd < succ (q / 2)\n[PROOFSTEP]\ntauto\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nhdisj :\n Disjoint (filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\n (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nhunion :\n filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))) \u222a\n filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))) =\n Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))\n\u22a2 \u2211 a in Ico 1 (succ (p / 2)), a * q / p + \u2211 a in Ico 1 (succ (q / 2)), a * p / q = p / 2 * (q / 2)\n[PROOFSTEP]\nrw [sum_Ico_eq_card_lt, sum_Ico_eq_card_lt, hswap, \u2190 card_disjoint_union hdisj, hunion, card_product]\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq0 : \u2191q \u2260 0\nhswap :\n Finset.card (filter (fun x => x.snd * q \u2264 x.fst * p) (Ico 1 (succ (q / 2)) \u00d7\u02e2 Ico 1 (succ (p / 2)))) =\n Finset.card (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nhdisj :\n Disjoint (filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\n (filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))))\nhunion :\n filter (fun x => x.snd * p \u2264 x.fst * q) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))) \u222a\n filter (fun x => x.fst * q \u2264 x.snd * p) (Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))) =\n Ico 1 (succ (p / 2)) \u00d7\u02e2 Ico 1 (succ (q / 2))\n\u22a2 Finset.card (Ico 1 (succ (p / 2))) * Finset.card (Ico 1 (succ (q / 2))) = p / 2 * (q / 2)\n[PROOFSTEP]\nsimp only [card_Ico, tsub_zero, succ_sub_succ_eq_sub]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\na : \u2115\nha1 : a % 2 = 1\nha0 : \u2191a \u2260 0\n\u22a2 legendreSym p \u2191a = (-1) ^ \u2211 x in Ico 1 (succ (p / 2)), x * a / p\n[PROOFSTEP]\nhaveI hp' : Fact (p % 2 = 1) := \u27e8Nat.Prime.mod_two_eq_one_iff_ne_two.mpr hp\u27e9\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\na : \u2115\nha1 : a % 2 = 1\nha0 : \u2191a \u2260 0\nhp' : Fact (p % 2 = 1)\n\u22a2 legendreSym p \u2191a = (-1) ^ \u2211 x in Ico 1 (succ (p / 2)), x * a / p\n[PROOFSTEP]\nhave ha0' : ((a : \u2124) : ZMod p) \u2260 0 := by norm_cast\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\na : \u2115\nha1 : a % 2 = 1\nha0 : \u2191a \u2260 0\nhp' : Fact (p % 2 = 1)\n\u22a2 \u2191\u2191a \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\na : \u2115\nha1 : a % 2 = 1\nha0 : \u2191a \u2260 0\nhp' : Fact (p % 2 = 1)\nha0' : \u2191\u2191a \u2260 0\n\u22a2 legendreSym p \u2191a = (-1) ^ \u2211 x in Ico 1 (succ (p / 2)), x * a / p\n[PROOFSTEP]\nrw [neg_one_pow_eq_pow_mod_two, gauss_lemma hp ha0', neg_one_pow_eq_pow_mod_two,\n (by norm_cast : ((a : \u2124) : ZMod p) = (a : ZMod p)), show _ = _ from eisenstein_lemma_aux p ha1 ha0]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\na : \u2115\nha1 : a % 2 = 1\nha0 : \u2191a \u2260 0\nhp' : Fact (p % 2 = 1)\nha0' : \u2191\u2191a \u2260 0\n\u22a2 \u2191\u2191a = \u2191a\n[PROOFSTEP]\nnorm_cast\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas", "llama_tokens": 24729, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929002541068, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.4414904388651884}} {"text": "[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\n\u22a2 Con (M \u00d7 { x // x \u2208 S })\n[PROOFSTEP]\nrefine\n { r := fun a b : M \u00d7 S \u21a6 \u2203 c : S, \u2191c * (\u2191b.2 * a.1) = c * (a.2 * b.1)\n iseqv := \u27e8fun a \u21a6 \u27e81, rfl\u27e9, fun \u27e8c, hc\u27e9 \u21a6 \u27e8c, hc.symm\u27e9, ?_\u27e9\n mul' := ?_ }\n[GOAL]\ncase refine_1\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\n\u22a2 \u2200 {x y z : M \u00d7 { x // x \u2208 S }},\n (\u2203 c, \u2191c * (\u2191y.snd * x.fst) = \u2191c * (\u2191x.snd * y.fst)) \u2192\n (\u2203 c, \u2191c * (\u2191z.snd * y.fst) = \u2191c * (\u2191y.snd * z.fst)) \u2192 \u2203 c, \u2191c * (\u2191z.snd * x.fst) = \u2191c * (\u2191x.snd * z.fst)\n[PROOFSTEP]\nrintro a b c \u27e8t\u2081, ht\u2081\u27e9 \u27e8t\u2082, ht\u2082\u27e9\n[GOAL]\ncase refine_1.intro.intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na b c : M \u00d7 { x // x \u2208 S }\nt\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191t\u2081 * (\u2191b.snd * a.fst) = \u2191t\u2081 * (\u2191a.snd * b.fst)\nt\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191t\u2082 * (\u2191c.snd * b.fst) = \u2191t\u2082 * (\u2191b.snd * c.fst)\n\u22a2 \u2203 c_1, \u2191c_1 * (\u2191c.snd * a.fst) = \u2191c_1 * (\u2191a.snd * c.fst)\n[PROOFSTEP]\nuse t\u2082 * t\u2081 * b.2\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na b c : M \u00d7 { x // x \u2208 S }\nt\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191t\u2081 * (\u2191b.snd * a.fst) = \u2191t\u2081 * (\u2191a.snd * b.fst)\nt\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191t\u2082 * (\u2191c.snd * b.fst) = \u2191t\u2082 * (\u2191b.snd * c.fst)\n\u22a2 \u2191(t\u2082 * t\u2081 * b.snd) * (\u2191c.snd * a.fst) = \u2191(t\u2082 * t\u2081 * b.snd) * (\u2191a.snd * c.fst)\n[PROOFSTEP]\nsimp only [Submonoid.coe_mul]\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na b c : M \u00d7 { x // x \u2208 S }\nt\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191t\u2081 * (\u2191b.snd * a.fst) = \u2191t\u2081 * (\u2191a.snd * b.fst)\nt\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191t\u2082 * (\u2191c.snd * b.fst) = \u2191t\u2082 * (\u2191b.snd * c.fst)\n\u22a2 \u2191t\u2082 * \u2191t\u2081 * \u2191b.snd * (\u2191c.snd * a.fst) = \u2191t\u2082 * \u2191t\u2081 * \u2191b.snd * (\u2191a.snd * c.fst)\n[PROOFSTEP]\ncalc\n (t\u2082 * t\u2081 * b.2 : M) * (c.2 * a.1) = t\u2082 * c.2 * (t\u2081 * (b.2 * a.1)) := by ac_rfl\n _ = t\u2081 * a.2 * (t\u2082 * (c.2 * b.1)) := by rw [ht\u2081]; ac_rfl\n _ = t\u2082 * t\u2081 * b.2 * (a.2 * c.1) := by rw [ht\u2082]; ac_rfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na b c : M \u00d7 { x // x \u2208 S }\nt\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191t\u2081 * (\u2191b.snd * a.fst) = \u2191t\u2081 * (\u2191a.snd * b.fst)\nt\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191t\u2082 * (\u2191c.snd * b.fst) = \u2191t\u2082 * (\u2191b.snd * c.fst)\n\u22a2 \u2191t\u2082 * \u2191t\u2081 * \u2191b.snd * (\u2191c.snd * a.fst) = \u2191t\u2082 * \u2191c.snd * (\u2191t\u2081 * (\u2191b.snd * a.fst))\n[PROOFSTEP]\nac_rfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na b c : M \u00d7 { x // x \u2208 S }\nt\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191t\u2081 * (\u2191b.snd * a.fst) = \u2191t\u2081 * (\u2191a.snd * b.fst)\nt\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191t\u2082 * (\u2191c.snd * b.fst) = \u2191t\u2082 * (\u2191b.snd * c.fst)\n\u22a2 \u2191t\u2082 * \u2191c.snd * (\u2191t\u2081 * (\u2191b.snd * a.fst)) = \u2191t\u2081 * \u2191a.snd * (\u2191t\u2082 * (\u2191c.snd * b.fst))\n[PROOFSTEP]\nrw [ht\u2081]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na b c : M \u00d7 { x // x \u2208 S }\nt\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191t\u2081 * (\u2191b.snd * a.fst) = \u2191t\u2081 * (\u2191a.snd * b.fst)\nt\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191t\u2082 * (\u2191c.snd * b.fst) = \u2191t\u2082 * (\u2191b.snd * c.fst)\n\u22a2 \u2191t\u2082 * \u2191c.snd * (\u2191t\u2081 * (\u2191a.snd * b.fst)) = \u2191t\u2081 * \u2191a.snd * (\u2191t\u2082 * (\u2191c.snd * b.fst))\n[PROOFSTEP]\nac_rfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na b c : M \u00d7 { x // x \u2208 S }\nt\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191t\u2081 * (\u2191b.snd * a.fst) = \u2191t\u2081 * (\u2191a.snd * b.fst)\nt\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191t\u2082 * (\u2191c.snd * b.fst) = \u2191t\u2082 * (\u2191b.snd * c.fst)\n\u22a2 \u2191t\u2081 * \u2191a.snd * (\u2191t\u2082 * (\u2191c.snd * b.fst)) = \u2191t\u2082 * \u2191t\u2081 * \u2191b.snd * (\u2191a.snd * c.fst)\n[PROOFSTEP]\nrw [ht\u2082]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na b c : M \u00d7 { x // x \u2208 S }\nt\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191t\u2081 * (\u2191b.snd * a.fst) = \u2191t\u2081 * (\u2191a.snd * b.fst)\nt\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191t\u2082 * (\u2191c.snd * b.fst) = \u2191t\u2082 * (\u2191b.snd * c.fst)\n\u22a2 \u2191t\u2081 * \u2191a.snd * (\u2191t\u2082 * (\u2191b.snd * c.fst)) = \u2191t\u2082 * \u2191t\u2081 * \u2191b.snd * (\u2191a.snd * c.fst)\n[PROOFSTEP]\nac_rfl\n[GOAL]\ncase refine_2\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\n\u22a2 \u2200 {w x y z : M \u00d7 { x // x \u2208 S }}, Setoid.r w x \u2192 Setoid.r y z \u2192 Setoid.r (w * y) (x * z)\n[PROOFSTEP]\nrintro a b c d \u27e8t\u2081, ht\u2081\u27e9 \u27e8t\u2082, ht\u2082\u27e9\n[GOAL]\ncase refine_2.intro.intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na b c d : M \u00d7 { x // x \u2208 S }\nt\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191t\u2081 * (\u2191b.snd * a.fst) = \u2191t\u2081 * (\u2191a.snd * b.fst)\nt\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191t\u2082 * (\u2191d.snd * c.fst) = \u2191t\u2082 * (\u2191c.snd * d.fst)\n\u22a2 Setoid.r (a * c) (b * d)\n[PROOFSTEP]\nuse t\u2082 * t\u2081\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na b c d : M \u00d7 { x // x \u2208 S }\nt\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191t\u2081 * (\u2191b.snd * a.fst) = \u2191t\u2081 * (\u2191a.snd * b.fst)\nt\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191t\u2082 * (\u2191d.snd * c.fst) = \u2191t\u2082 * (\u2191c.snd * d.fst)\n\u22a2 \u2191(t\u2082 * t\u2081) * (\u2191(b * d).snd * (a * c).fst) = \u2191(t\u2082 * t\u2081) * (\u2191(a * c).snd * (b * d).fst)\n[PROOFSTEP]\ncalc\n (t\u2082 * t\u2081 : M) * (b.2 * d.2 * (a.1 * c.1)) = t\u2082 * (d.2 * c.1) * (t\u2081 * (b.2 * a.1)) := by ac_rfl\n _ = (t\u2082 * t\u2081 : M) * (a.2 * c.2 * (b.1 * d.1)) := by rw [ht\u2081, ht\u2082]; ac_rfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na b c d : M \u00d7 { x // x \u2208 S }\nt\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191t\u2081 * (\u2191b.snd * a.fst) = \u2191t\u2081 * (\u2191a.snd * b.fst)\nt\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191t\u2082 * (\u2191d.snd * c.fst) = \u2191t\u2082 * (\u2191c.snd * d.fst)\n\u22a2 \u2191t\u2082 * \u2191t\u2081 * (\u2191b.snd * \u2191d.snd * (a.fst * c.fst)) = \u2191t\u2082 * (\u2191d.snd * c.fst) * (\u2191t\u2081 * (\u2191b.snd * a.fst))\n[PROOFSTEP]\nac_rfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na b c d : M \u00d7 { x // x \u2208 S }\nt\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191t\u2081 * (\u2191b.snd * a.fst) = \u2191t\u2081 * (\u2191a.snd * b.fst)\nt\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191t\u2082 * (\u2191d.snd * c.fst) = \u2191t\u2082 * (\u2191c.snd * d.fst)\n\u22a2 \u2191t\u2082 * (\u2191d.snd * c.fst) * (\u2191t\u2081 * (\u2191b.snd * a.fst)) = \u2191t\u2082 * \u2191t\u2081 * (\u2191a.snd * \u2191c.snd * (b.fst * d.fst))\n[PROOFSTEP]\nrw [ht\u2081, ht\u2082]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na b c d : M \u00d7 { x // x \u2208 S }\nt\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191t\u2081 * (\u2191b.snd * a.fst) = \u2191t\u2081 * (\u2191a.snd * b.fst)\nt\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191t\u2082 * (\u2191d.snd * c.fst) = \u2191t\u2082 * (\u2191c.snd * d.fst)\n\u22a2 \u2191t\u2082 * (\u2191c.snd * d.fst) * (\u2191t\u2081 * (\u2191a.snd * b.fst)) = \u2191t\u2082 * \u2191t\u2081 * (\u2191a.snd * \u2191c.snd * (b.fst * d.fst))\n[PROOFSTEP]\nac_rfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx\u271d : { x // x \u2208 S }\n\u22a2 \u21911 * (\u2191(\u2191x\u271d, x\u271d).snd * 1.fst) = \u21911 * (\u21911.snd * (\u2191x\u271d, x\u271d).fst)\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nb : Con (M \u00d7 { x // x \u2208 S })\nH : b \u2208 {c | \u2200 (y : { x // x \u2208 S }), \u2191c 1 (\u2191y, y)}\nx\u271d\u00b2 x\u271d\u00b9 : M \u00d7 { x // x \u2208 S }\np : M\nq : { x // x \u2208 S }\nx : M\ny : { x // x \u2208 S }\nx\u271d : \u2191(r' S) (p, q) (x, y)\nt : { x // x \u2208 S }\nht : \u2191t * (\u2191(x, y).snd * (p, q).fst) = \u2191t * (\u2191(p, q).snd * (x, y).fst)\n\u22a2 \u2191b (p, q) (x, y)\n[PROOFSTEP]\nrw [\u2190 one_mul (p, q), \u2190 one_mul (x, y)]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nb : Con (M \u00d7 { x // x \u2208 S })\nH : b \u2208 {c | \u2200 (y : { x // x \u2208 S }), \u2191c 1 (\u2191y, y)}\nx\u271d\u00b2 x\u271d\u00b9 : M \u00d7 { x // x \u2208 S }\np : M\nq : { x // x \u2208 S }\nx : M\ny : { x // x \u2208 S }\nx\u271d : \u2191(r' S) (p, q) (x, y)\nt : { x // x \u2208 S }\nht : \u2191t * (\u2191(x, y).snd * (p, q).fst) = \u2191t * (\u2191(p, q).snd * (x, y).fst)\n\u22a2 \u2191b (1 * (p, q)) (1 * (x, y))\n[PROOFSTEP]\nrefine b.trans (b.mul (H (t * y)) (b.refl _)) ?_\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nb : Con (M \u00d7 { x // x \u2208 S })\nH : b \u2208 {c | \u2200 (y : { x // x \u2208 S }), \u2191c 1 (\u2191y, y)}\nx\u271d\u00b2 x\u271d\u00b9 : M \u00d7 { x // x \u2208 S }\np : M\nq : { x // x \u2208 S }\nx : M\ny : { x // x \u2208 S }\nx\u271d : \u2191(r' S) (p, q) (x, y)\nt : { x // x \u2208 S }\nht : \u2191t * (\u2191(x, y).snd * (p, q).fst) = \u2191t * (\u2191(p, q).snd * (x, y).fst)\n\u22a2 \u2191b ((\u2191(t * y), t * y) * (p, q)) (1 * (x, y))\n[PROOFSTEP]\nconvert b.symm (b.mul (H (t * q)) (b.refl (x, y))) using 1\n[GOAL]\ncase h.e'_3\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nb : Con (M \u00d7 { x // x \u2208 S })\nH : b \u2208 {c | \u2200 (y : { x // x \u2208 S }), \u2191c 1 (\u2191y, y)}\nx\u271d\u00b2 x\u271d\u00b9 : M \u00d7 { x // x \u2208 S }\np : M\nq : { x // x \u2208 S }\nx : M\ny : { x // x \u2208 S }\nx\u271d : \u2191(r' S) (p, q) (x, y)\nt : { x // x \u2208 S }\nht : \u2191t * (\u2191(x, y).snd * (p, q).fst) = \u2191t * (\u2191(p, q).snd * (x, y).fst)\n\u22a2 (\u2191(t * y), t * y) * (p, q) = (\u2191(t * q), t * q) * (x, y)\n[PROOFSTEP]\ndsimp only [Prod.mk_mul_mk, Submonoid.coe_mul] at ht \u22a2\n[GOAL]\ncase h.e'_3\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nb : Con (M \u00d7 { x // x \u2208 S })\nH : b \u2208 {c | \u2200 (y : { x // x \u2208 S }), \u2191c 1 (\u2191y, y)}\nx\u271d\u00b2 x\u271d\u00b9 : M \u00d7 { x // x \u2208 S }\np : M\nq : { x // x \u2208 S }\nx : M\ny : { x // x \u2208 S }\nx\u271d : \u2191(r' S) (p, q) (x, y)\nt : { x // x \u2208 S }\nht : \u2191t * (\u2191y * p) = \u2191t * (\u2191q * x)\n\u22a2 (\u2191t * \u2191y * p, t * y * q) = (\u2191t * \u2191q * x, t * q * y)\n[PROOFSTEP]\nsimp_rw [mul_assoc, ht, mul_comm y q]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx y : M \u00d7 { x // x \u2208 S }\n\u22a2 \u2191(r S) x y \u2194 \u2203 c, \u2191c * (\u2191y.snd * x.fst) = \u2191c * (\u2191x.snd * y.fst)\n[PROOFSTEP]\nrw [r_eq_r' S]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx y : M \u00d7 { x // x \u2208 S }\n\u22a2 \u2191(r' S) x y \u2194 \u2203 c, \u2191c * (\u2191y.snd * x.fst) = \u2191c * (\u2191x.snd * y.fst)\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx y z : Localization S\n\u22a2 Localization.mul S (Localization.mul S x y) z = Localization.mul S x (Localization.mul S y z)\n[PROOFSTEP]\nrw [Localization.mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx y z : Localization S\n\u22a2 Mul.mul (Mul.mul x y) z = Mul.mul x (Mul.mul y z)\n[PROOFSTEP]\napply (r S).commMonoid.mul_assoc\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx : Localization S\n\u22a2 Localization.mul S (Localization.one S) x = x\n[PROOFSTEP]\nrw [Localization.mul, Localization.one]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx : Localization S\n\u22a2 Mul.mul One.one x = x\n[PROOFSTEP]\napply (r S).commMonoid.one_mul\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx : Localization S\n\u22a2 Localization.mul S x (Localization.one S) = x\n[PROOFSTEP]\nrw [Localization.mul, Localization.one]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx : Localization S\n\u22a2 Mul.mul x One.one = x\n[PROOFSTEP]\napply (r S).commMonoid.mul_one\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx : Localization S\n\u22a2 Localization.npow S 0 x = Localization.one S\n[PROOFSTEP]\nrw [Localization.npow, Localization.one]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx : Localization S\n\u22a2 Monoid.npow 0 x = One.one\n[PROOFSTEP]\napply (r S).commMonoid.npow_zero\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nn : \u2115\nx : Localization S\n\u22a2 Localization.npow S (Nat.succ n) x = Localization.mul S x (Localization.npow S n x)\n[PROOFSTEP]\nrw [Localization.npow, Localization.mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nn : \u2115\nx : Localization S\n\u22a2 Monoid.npow (Nat.succ n) x = Mul.mul x (Monoid.npow n x)\n[PROOFSTEP]\napply (r S).commMonoid.npow_succ\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx y : Localization S\n\u22a2 Localization.mul S x y = Localization.mul S y x\n[PROOFSTEP]\nrw [Localization.mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx y : Localization S\n\u22a2 Mul.mul x y = Mul.mul y x\n[PROOFSTEP]\napply (r S).commMonoid.mul_comm\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\np : Localization S \u2192 Sort u\nf : (a : M) \u2192 (b : { x // x \u2208 S }) \u2192 p (mk a b)\nH : \u2200 {a c : M} {b d : { x // x \u2208 S }} (h : \u2191(r S) (a, b) (c, d)), (_ : mk a b = mk c d) \u25b8 f a b = f c d\nx : Localization S\ny : M \u00d7 { x // x \u2208 S }\n\u22a2 mk y.fst y.snd = Quot.mk Setoid.r y\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\np : Localization S \u2192 Sort u\nf : (a : M) \u2192 (b : { x // x \u2208 S }) \u2192 p (mk a b)\nH : \u2200 {a c : M} {b d : { x // x \u2208 S }} (h : \u2191(r S) (a, b) (c, d)), (_ : mk a b = mk c d) \u25b8 f a b = f c d\nx : Localization S\ny z : M \u00d7 { x // x \u2208 S }\nh : Setoid.r y z\n\u22a2 (_ : Quot.mk Setoid.r y = Quot.mk Setoid.r z) \u25b8 (fun y => (_ : mk y.fst y.snd = mk y.fst y.snd) \u25b8 f y.fst y.snd) y =\n (fun y => (_ : mk y.fst y.snd = mk y.fst y.snd) \u25b8 f y.fst y.snd) z\n[PROOFSTEP]\ncases y\n[GOAL]\ncase mk\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\np : Localization S \u2192 Sort u\nf : (a : M) \u2192 (b : { x // x \u2208 S }) \u2192 p (mk a b)\nH : \u2200 {a c : M} {b d : { x // x \u2208 S }} (h : \u2191(r S) (a, b) (c, d)), (_ : mk a b = mk c d) \u25b8 f a b = f c d\nx : Localization S\nz : M \u00d7 { x // x \u2208 S }\nfst\u271d : M\nsnd\u271d : { x // x \u2208 S }\nh : Setoid.r (fst\u271d, snd\u271d) z\n\u22a2 (_ : Quot.mk Setoid.r (fst\u271d, snd\u271d) = Quot.mk Setoid.r z) \u25b8\n (fun y => (_ : mk y.fst y.snd = mk y.fst y.snd) \u25b8 f y.fst y.snd) (fst\u271d, snd\u271d) =\n (fun y => (_ : mk y.fst y.snd = mk y.fst y.snd) \u25b8 f y.fst y.snd) z\n[PROOFSTEP]\ncases z\n[GOAL]\ncase mk.mk\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\np : Localization S \u2192 Sort u\nf : (a : M) \u2192 (b : { x // x \u2208 S }) \u2192 p (mk a b)\nH : \u2200 {a c : M} {b d : { x // x \u2208 S }} (h : \u2191(r S) (a, b) (c, d)), (_ : mk a b = mk c d) \u25b8 f a b = f c d\nx : Localization S\nfst\u271d\u00b9 : M\nsnd\u271d\u00b9 : { x // x \u2208 S }\nfst\u271d : M\nsnd\u271d : { x // x \u2208 S }\nh : Setoid.r (fst\u271d\u00b9, snd\u271d\u00b9) (fst\u271d, snd\u271d)\n\u22a2 (_ : Quot.mk Setoid.r (fst\u271d\u00b9, snd\u271d\u00b9) = Quot.mk Setoid.r (fst\u271d, snd\u271d)) \u25b8\n (fun y => (_ : mk y.fst y.snd = mk y.fst y.snd) \u25b8 f y.fst y.snd) (fst\u271d\u00b9, snd\u271d\u00b9) =\n (fun y => (_ : mk y.fst y.snd = mk y.fst y.snd) \u25b8 f y.fst y.snd) (fst\u271d, snd\u271d)\n[PROOFSTEP]\nexact H h\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na c : M\nb d : { x // x \u2208 S }\n\u22a2 Localization.mul S (mk a b) (mk c d) = mk (a * c) (b * d)\n[PROOFSTEP]\nrw [Localization.mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na c : M\nb d : { x // x \u2208 S }\n\u22a2 Mul.mul (mk a b) (mk c d) = mk (a * c) (b * d)\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\n\u22a2 mk 1 1 = Localization.one S\n[PROOFSTEP]\nrw [Localization.one]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\n\u22a2 mk 1 1 = One.one\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nn : \u2115\na : M\nb : { x // x \u2208 S }\n\u22a2 Localization.npow S n (mk a b) = mk (a ^ n) (b ^ n)\n[PROOFSTEP]\nrw [Localization.npow]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nn : \u2115\na : M\nb : { x // x \u2208 S }\n\u22a2 Monoid.npow n (mk a b) = mk (a ^ n) (b ^ n)\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\np : Sort u\nx : Localization S\nf : M \u2192 { x // x \u2208 S } \u2192 p\nH : \u2200 {a c : M} {b d : { x // x \u2208 S }}, \u2191(r S) (a, b) (c, d) \u2192 f a b = f c d\na\u271d c\u271d : M\nb\u271d d\u271d : { x // x \u2208 S }\nh : \u2191(r S) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ : mk a\u271d b\u271d = mk c\u271d d\u271d) \u25b8 f a\u271d b\u271d = f c\u271d d\u271d\n[PROOFSTEP]\nsimpa only [eq_rec_constant] using H h\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx y : M \u00d7 { x // x \u2208 S }\nh : \u2191y.snd * x.fst = \u2191x.snd * y.fst\n\u22a2 \u21911 * (\u2191y.snd * x.fst) = \u21911 * (\u2191x.snd * y.fst)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na : { x // x \u2208 S }\n\u22a2 mk (\u2191a) a = 1\n[PROOFSTEP]\nsymm\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na : { x // x \u2208 S }\n\u22a2 1 = mk (\u2191a) a\n[PROOFSTEP]\nrw [\u2190 mk_one, mk_eq_mk_iff]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\na : { x // x \u2208 S }\n\u22a2 \u2191(r S) (1, 1) (\u2191a, a)\n[PROOFSTEP]\nexact one_rel a\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b3 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b2 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b9 : SMul R M\ninst\u271d : IsScalarTower R M M\nc : R\nz : Localization S\na a' : M\nb b' : { x // x \u2208 S }\nh : \u2191(r S) (a, b) (a', b')\n\u22a2 \u2191(r S) (c \u2022 a, b) (c \u2022 a', b')\n[PROOFSTEP]\ncases' b with b hb\n[GOAL]\ncase mk\nM : Type u_1\ninst\u271d\u2074 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b3 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b2 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b9 : SMul R M\ninst\u271d : IsScalarTower R M M\nc : R\nz : Localization S\na a' : M\nb' : { x // x \u2208 S }\nb : M\nhb : b \u2208 S\nh : \u2191(r S) (a, { val := b, property := hb }) (a', b')\n\u22a2 \u2191(r S) (c \u2022 a, { val := b, property := hb }) (c \u2022 a', b')\n[PROOFSTEP]\ncases' b' with b' hb'\n[GOAL]\ncase mk.mk\nM : Type u_1\ninst\u271d\u2074 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b3 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b2 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b9 : SMul R M\ninst\u271d : IsScalarTower R M M\nc : R\nz : Localization S\na a' b : M\nhb : b \u2208 S\nb' : M\nhb' : b' \u2208 S\nh : \u2191(r S) (a, { val := b, property := hb }) (a', { val := b', property := hb' })\n\u22a2 \u2191(r S) (c \u2022 a, { val := b, property := hb }) (c \u2022 a', { val := b', property := hb' })\n[PROOFSTEP]\nrw [r_eq_r'] at h \u22a2\n[GOAL]\ncase mk.mk\nM : Type u_1\ninst\u271d\u2074 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b3 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b2 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b9 : SMul R M\ninst\u271d : IsScalarTower R M M\nc : R\nz : Localization S\na a' b : M\nhb : b \u2208 S\nb' : M\nhb' : b' \u2208 S\nh : \u2191(r' S) (a, { val := b, property := hb }) (a', { val := b', property := hb' })\n\u22a2 \u2191(r' S) (c \u2022 a, { val := b, property := hb }) (c \u2022 a', { val := b', property := hb' })\n[PROOFSTEP]\ncases' h with t ht\n[GOAL]\ncase mk.mk.intro\nM : Type u_1\ninst\u271d\u2074 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b3 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b2 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b9 : SMul R M\ninst\u271d : IsScalarTower R M M\nc : R\nz : Localization S\na a' b : M\nhb : b \u2208 S\nb' : M\nhb' : b' \u2208 S\nt : { x // x \u2208 S }\nht :\n \u2191t * (\u2191(a', { val := b', property := hb' }).snd * (a, { val := b, property := hb }).fst) =\n \u2191t * (\u2191(a, { val := b, property := hb }).snd * (a', { val := b', property := hb' }).fst)\n\u22a2 \u2191(r' S) (c \u2022 a, { val := b, property := hb }) (c \u2022 a', { val := b', property := hb' })\n[PROOFSTEP]\nuse t\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2074 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b3 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b2 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b9 : SMul R M\ninst\u271d : IsScalarTower R M M\nc : R\nz : Localization S\na a' b : M\nhb : b \u2208 S\nb' : M\nhb' : b' \u2208 S\nt : { x // x \u2208 S }\nht :\n \u2191t * (\u2191(a', { val := b', property := hb' }).snd * (a, { val := b, property := hb }).fst) =\n \u2191t * (\u2191(a, { val := b, property := hb }).snd * (a', { val := b', property := hb' }).fst)\n\u22a2 \u2191t * (\u2191(c \u2022 a', { val := b', property := hb' }).snd * (c \u2022 a, { val := b, property := hb }).fst) =\n \u2191t * (\u2191(c \u2022 a, { val := b, property := hb }).snd * (c \u2022 a', { val := b', property := hb' }).fst)\n[PROOFSTEP]\ndsimp only [Subtype.coe_mk] at ht \u22a2\n -- TODO: this definition should take `SMulCommClass R M M` instead of `IsScalarTower R M M` if\n -- we ever want to generalize to the non-commutative case.\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2074 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b3 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b2 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b9 : SMul R M\ninst\u271d : IsScalarTower R M M\nc : R\nz : Localization S\na a' b : M\nhb : b \u2208 S\nb' : M\nhb' : b' \u2208 S\nt : { x // x \u2208 S }\nht : \u2191t * (b' * a) = \u2191t * (b * a')\n\u22a2 \u2191t * (b' * c \u2022 a) = \u2191t * (b * c \u2022 a')\n[PROOFSTEP]\nhaveI : SMulCommClass R M M := \u27e8fun r m\u2081 m\u2082 \u21a6 by simp_rw [smul_eq_mul, mul_comm m\u2081, smul_mul_assoc]\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b3 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b2 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b9 : SMul R M\ninst\u271d : IsScalarTower R M M\nc : R\nz : Localization S\na a' b : M\nhb : b \u2208 S\nb' : M\nhb' : b' \u2208 S\nt : { x // x \u2208 S }\nht : \u2191t * (b' * a) = \u2191t * (b * a')\nr : R\nm\u2081 m\u2082 : M\n\u22a2 r \u2022 m\u2081 \u2022 m\u2082 = m\u2081 \u2022 r \u2022 m\u2082\n[PROOFSTEP]\nsimp_rw [smul_eq_mul, mul_comm m\u2081, smul_mul_assoc]\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2074 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b3 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b2 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b9 : SMul R M\ninst\u271d : IsScalarTower R M M\nc : R\nz : Localization S\na a' b : M\nhb : b \u2208 S\nb' : M\nhb' : b' \u2208 S\nt : { x // x \u2208 S }\nht : \u2191t * (b' * a) = \u2191t * (b * a')\nthis : SMulCommClass R M M\n\u22a2 \u2191t * (b' * c \u2022 a) = \u2191t * (b * c \u2022 a')\n[PROOFSTEP]\nsimp only [mul_smul_comm, ht]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b3 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b2 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b9 : SMul R M\ninst\u271d : IsScalarTower R M M\nc : R\na : M\nb : { x // x \u2208 S }\n\u22a2 c \u2022 mk a b = mk (c \u2022 a) b\n[PROOFSTEP]\nsimp only [HSMul.hSMul, instHSMul, SMul.smul, instSMulLocalization, Localization.smul]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b3 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b2 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b9 : SMul R M\ninst\u271d : IsScalarTower R M M\nc : R\na : M\nb : { x // x \u2208 S }\n\u22a2 liftOn (mk a b) (fun a b => mk (SMul.smul c a) b)\n (_ :\n \u2200 {a c_1 : M} {b d : { x // x \u2208 S }},\n \u2191(r S) (a, b) (c_1, d) \u2192 (fun a b => mk (SMul.smul c a) b) a b = (fun a b => mk (SMul.smul c a) b) c_1 d) =\n mk (SMul.smul c a) b\n[PROOFSTEP]\nshow liftOn (mk a b) (fun a b => mk (c \u2022 a) b) _ = _\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b3 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b2 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b9 : SMul R M\ninst\u271d : IsScalarTower R M M\nc : R\na : M\nb : { x // x \u2208 S }\n\u22a2 liftOn (mk a b) (fun a b => mk (c \u2022 a) b)\n (_ :\n \u2200 {a c_1 : M} {b d : { x // x \u2208 S }},\n \u2191(r S) (a, b) (c_1, d) \u2192 (fun a b => mk (SMul.smul c a) b) a b = (fun a b => mk (SMul.smul c a) b) c_1 d) =\n mk (SMul.smul c a) b\n[PROOFSTEP]\nexact liftOn_mk (fun a b => mk (c \u2022 a) b) _ a b\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2076 : CommMonoid N\nP : Type u_3\ninst\u271d\u2075 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u2074 : SMul R\u2081 M\ninst\u271d\u00b3 : SMul R\u2082 M\ninst\u271d\u00b2 : IsScalarTower R\u2081 M M\ninst\u271d\u00b9 : IsScalarTower R\u2082 M M\ninst\u271d : SMulCommClass R\u2081 R\u2082 M\ns : R\u2081\nt : R\u2082\nr : M\nx : { x // x \u2208 S }\n\u22a2 s \u2022 t \u2022 mk (r, x).fst (r, x).snd = t \u2022 s \u2022 mk (r, x).fst (r, x).snd\n[PROOFSTEP]\nsimp only [smul_mk, smul_comm s t r]\n[GOAL]\nM : Type u_1\ninst\u271d\u2078 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2077 : CommMonoid N\nP : Type u_3\ninst\u271d\u2076 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u2075 : SMul R\u2081 M\ninst\u271d\u2074 : SMul R\u2082 M\ninst\u271d\u00b3 : IsScalarTower R\u2081 M M\ninst\u271d\u00b2 : IsScalarTower R\u2082 M M\ninst\u271d\u00b9 : SMul R\u2081 R\u2082\ninst\u271d : IsScalarTower R\u2081 R\u2082 M\ns : R\u2081\nt : R\u2082\nr : M\nx : { x // x \u2208 S }\n\u22a2 (s \u2022 t) \u2022 mk (r, x).fst (r, x).snd = s \u2022 t \u2022 mk (r, x).fst (r, x).snd\n[PROOFSTEP]\nsimp only [smul_mk, smul_assoc s t r]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b3 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b2 : CommMonoid P\nR\u271d : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : Type u_7\ninst\u271d\u00b9 : SMul R M\ninst\u271d : IsScalarTower R M M\ns : R\nr\u2081 : M\nx\u2081 : { x // x \u2208 S }\nr\u2082 : M\nx\u2082 : { x // x \u2208 S }\n\u22a2 s \u2022 mk (r\u2081, x\u2081).fst (r\u2081, x\u2081).snd \u2022 mk (r\u2082, x\u2082).fst (r\u2082, x\u2082).snd =\n mk (r\u2081, x\u2081).fst (r\u2081, x\u2081).snd \u2022 s \u2022 mk (r\u2082, x\u2082).fst (r\u2082, x\u2082).snd\n[PROOFSTEP]\nsimp only [smul_mk, smul_eq_mul, mk_mul, mul_comm r\u2081, smul_mul_assoc]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b3 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b2 : CommMonoid P\nR\u271d : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : Type u_7\ninst\u271d\u00b9 : SMul R M\ninst\u271d : IsScalarTower R M M\ns : R\nr\u2081 : M\nx\u2081 : { x // x \u2208 S }\nr\u2082 : M\nx\u2082 : { x // x \u2208 S }\n\u22a2 (s \u2022 mk (r\u2081, x\u2081).fst (r\u2081, x\u2081).snd) \u2022 mk (r\u2082, x\u2082).fst (r\u2082, x\u2082).snd =\n s \u2022 mk (r\u2081, x\u2081).fst (r\u2081, x\u2081).snd \u2022 mk (r\u2082, x\u2082).fst (r\u2082, x\u2082).snd\n[PROOFSTEP]\nsimp only [smul_mk, smul_eq_mul, mk_mul, smul_mul_assoc]\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2076 : CommMonoid N\nP : Type u_3\ninst\u271d\u2075 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u2074 : SMul R M\ninst\u271d\u00b3 : SMul R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b2 : IsScalarTower R M M\ninst\u271d\u00b9 : IsScalarTower R\u1d50\u1d52\u1d56 M M\ninst\u271d : IsCentralScalar R M\ns : R\nr : M\nx : { x // x \u2208 S }\n\u22a2 MulOpposite.op s \u2022 mk (r, x).fst (r, x).snd = s \u2022 mk (r, x).fst (r, x).snd\n[PROOFSTEP]\nsimp only [smul_mk, op_smul_eq_smul]\n[GOAL]\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : MulAction R M\ninst\u271d : IsScalarTower R M M\n\u22a2 \u2200 (fst : M) (snd : { x // x \u2208 S }), 1 \u2022 mk (fst, snd).fst (fst, snd).snd = mk (fst, snd).fst (fst, snd).snd\n[PROOFSTEP]\nintros\n[GOAL]\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : MulAction R M\ninst\u271d : IsScalarTower R M M\nfst\u271d : M\nsnd\u271d : { x // x \u2208 S }\n\u22a2 1 \u2022 mk (fst\u271d, snd\u271d).fst (fst\u271d, snd\u271d).snd = mk (fst\u271d, snd\u271d).fst (fst\u271d, snd\u271d).snd\n[PROOFSTEP]\nsimp only [Localization.smul_mk, one_smul]\n[GOAL]\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : MulAction R M\ninst\u271d : IsScalarTower R M M\ns\u2081 s\u2082 : R\n\u22a2 \u2200 (fst : M) (snd : { x // x \u2208 S }),\n (s\u2081 * s\u2082) \u2022 mk (fst, snd).fst (fst, snd).snd = s\u2081 \u2022 s\u2082 \u2022 mk (fst, snd).fst (fst, snd).snd\n[PROOFSTEP]\nintros\n[GOAL]\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : MulAction R M\ninst\u271d : IsScalarTower R M M\ns\u2081 s\u2082 : R\nfst\u271d : M\nsnd\u271d : { x // x \u2208 S }\n\u22a2 (s\u2081 * s\u2082) \u2022 mk (fst\u271d, snd\u271d).fst (fst\u271d, snd\u271d).snd = s\u2081 \u2022 s\u2082 \u2022 mk (fst\u271d, snd\u271d).fst (fst\u271d, snd\u271d).snd\n[PROOFSTEP]\nsimp only [Localization.smul_mk, mul_smul]\n[GOAL]\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : MulDistribMulAction R M\ninst\u271d : IsScalarTower R M M\ns : R\nx y : Localization S\nr\u2081 : M\nx\u2081 : { x // x \u2208 S }\nr\u2082 : M\nx\u2082 : { x // x \u2208 S }\n\u22a2 s \u2022 (mk (r\u2081, x\u2081).fst (r\u2081, x\u2081).snd * mk (r\u2082, x\u2082).fst (r\u2082, x\u2082).snd) =\n s \u2022 mk (r\u2081, x\u2081).fst (r\u2081, x\u2081).snd * s \u2022 mk (r\u2082, x\u2082).fst (r\u2082, x\u2082).snd\n[PROOFSTEP]\nsimp only [Localization.smul_mk, Localization.mk_mul, smul_mul']\n[GOAL]\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nR : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : MulDistribMulAction R M\ninst\u271d : IsScalarTower R M M\ns : R\n\u22a2 s \u2022 1 = 1\n[PROOFSTEP]\nsimp only [\u2190 Localization.mk_one, Localization.smul_mk, smul_one]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf g : LocalizationMap S N\nh : \u2200 (x : M), \u2191(toMap f) x = \u2191(toMap g) x\n\u22a2 f = g\n[PROOFSTEP]\nrcases f with \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.mk\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\ng : LocalizationMap S N\ntoOneHom\u271d : OneHom M N\nmap_mul'\u271d : \u2200 (x y : M), OneHom.toFun toOneHom\u271d (x * y) = OneHom.toFun toOneHom\u271d x * OneHom.toFun toOneHom\u271d y\nmap_units'\u271d : \u2200 (y : { x // x \u2208 S }), IsUnit (OneHom.toFun \u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d } \u2191y)\nsurj'\u271d :\n \u2200 (z : N),\n \u2203 x,\n z * OneHom.toFun \u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d } \u2191x.snd =\n OneHom.toFun (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }) x.fst\neq_iff_exists'\u271d :\n \u2200 (x y : M),\n OneHom.toFun (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }) x =\n OneHom.toFun (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }) y \u2194\n \u2203 c, \u2191c * x = \u2191c * y\nh :\n \u2200 (x : M),\n \u2191(toMap\n { toMonoidHom := { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }, map_units' := map_units'\u271d,\n surj' := surj'\u271d, eq_iff_exists' := eq_iff_exists'\u271d })\n x =\n \u2191(toMap g) x\n\u22a2 { toMonoidHom := { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }, map_units' := map_units'\u271d, surj' := surj'\u271d,\n eq_iff_exists' := eq_iff_exists'\u271d } =\n g\n[PROOFSTEP]\nrcases g with \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.mk.mk.mk\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\ntoOneHom\u271d\u00b9 : OneHom M N\nmap_mul'\u271d\u00b9 : \u2200 (x y : M), OneHom.toFun toOneHom\u271d\u00b9 (x * y) = OneHom.toFun toOneHom\u271d\u00b9 x * OneHom.toFun toOneHom\u271d\u00b9 y\nmap_units'\u271d\u00b9 : \u2200 (y : { x // x \u2208 S }), IsUnit (OneHom.toFun \u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } \u2191y)\nsurj'\u271d\u00b9 :\n \u2200 (z : N),\n \u2203 x,\n z * OneHom.toFun \u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } \u2191x.snd =\n OneHom.toFun (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }) x.fst\neq_iff_exists'\u271d\u00b9 :\n \u2200 (x y : M),\n OneHom.toFun (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }) x =\n OneHom.toFun (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }) y \u2194\n \u2203 c, \u2191c * x = \u2191c * y\ntoOneHom\u271d : OneHom M N\nmap_mul'\u271d : \u2200 (x y : M), OneHom.toFun toOneHom\u271d (x * y) = OneHom.toFun toOneHom\u271d x * OneHom.toFun toOneHom\u271d y\nmap_units'\u271d : \u2200 (y : { x // x \u2208 S }), IsUnit (OneHom.toFun \u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d } \u2191y)\nsurj'\u271d :\n \u2200 (z : N),\n \u2203 x,\n z * OneHom.toFun \u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d } \u2191x.snd =\n OneHom.toFun (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }) x.fst\neq_iff_exists'\u271d :\n \u2200 (x y : M),\n OneHom.toFun (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }) x =\n OneHom.toFun (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }) y \u2194\n \u2203 c, \u2191c * x = \u2191c * y\nh :\n \u2200 (x : M),\n \u2191(toMap\n { toMonoidHom := { toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }, map_units' := map_units'\u271d\u00b9,\n surj' := surj'\u271d\u00b9, eq_iff_exists' := eq_iff_exists'\u271d\u00b9 })\n x =\n \u2191(toMap\n { toMonoidHom := { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }, map_units' := map_units'\u271d,\n surj' := surj'\u271d, eq_iff_exists' := eq_iff_exists'\u271d })\n x\n\u22a2 { toMonoidHom := { toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }, map_units' := map_units'\u271d\u00b9, surj' := surj'\u271d\u00b9,\n eq_iff_exists' := eq_iff_exists'\u271d\u00b9 } =\n { toMonoidHom := { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }, map_units' := map_units'\u271d, surj' := surj'\u271d,\n eq_iff_exists' := eq_iff_exists'\u271d }\n[PROOFSTEP]\nsimp only [mk.injEq, MonoidHom.mk.injEq]\n[GOAL]\ncase mk.mk.mk.mk\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\ntoOneHom\u271d\u00b9 : OneHom M N\nmap_mul'\u271d\u00b9 : \u2200 (x y : M), OneHom.toFun toOneHom\u271d\u00b9 (x * y) = OneHom.toFun toOneHom\u271d\u00b9 x * OneHom.toFun toOneHom\u271d\u00b9 y\nmap_units'\u271d\u00b9 : \u2200 (y : { x // x \u2208 S }), IsUnit (OneHom.toFun \u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } \u2191y)\nsurj'\u271d\u00b9 :\n \u2200 (z : N),\n \u2203 x,\n z * OneHom.toFun \u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } \u2191x.snd =\n OneHom.toFun (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }) x.fst\neq_iff_exists'\u271d\u00b9 :\n \u2200 (x y : M),\n OneHom.toFun (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }) x =\n OneHom.toFun (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }) y \u2194\n \u2203 c, \u2191c * x = \u2191c * y\ntoOneHom\u271d : OneHom M N\nmap_mul'\u271d : \u2200 (x y : M), OneHom.toFun toOneHom\u271d (x * y) = OneHom.toFun toOneHom\u271d x * OneHom.toFun toOneHom\u271d y\nmap_units'\u271d : \u2200 (y : { x // x \u2208 S }), IsUnit (OneHom.toFun \u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d } \u2191y)\nsurj'\u271d :\n \u2200 (z : N),\n \u2203 x,\n z * OneHom.toFun \u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d } \u2191x.snd =\n OneHom.toFun (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }) x.fst\neq_iff_exists'\u271d :\n \u2200 (x y : M),\n OneHom.toFun (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }) x =\n OneHom.toFun (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }) y \u2194\n \u2203 c, \u2191c * x = \u2191c * y\nh :\n \u2200 (x : M),\n \u2191(toMap\n { toMonoidHom := { toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }, map_units' := map_units'\u271d\u00b9,\n surj' := surj'\u271d\u00b9, eq_iff_exists' := eq_iff_exists'\u271d\u00b9 })\n x =\n \u2191(toMap\n { toMonoidHom := { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }, map_units' := map_units'\u271d,\n surj' := surj'\u271d, eq_iff_exists' := eq_iff_exists'\u271d })\n x\n\u22a2 toOneHom\u271d\u00b9 = toOneHom\u271d\n[PROOFSTEP]\nexact OneHom.ext h\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nz : N\n\u22a2 \u2191(toMap f) (sec f z).fst = \u2191(toMap f) \u2191(sec f z).snd * z\n[PROOFSTEP]\nrw [mul_comm, sec_spec]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : M \u2192* N\nh : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191f \u2191y)\ny : { x // x \u2208 S }\nw z : N\n\u22a2 w * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict f S) h) y)\u207b\u00b9 = z \u2194 w = \u2191f \u2191y * z\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : M \u2192* N\nh : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191f \u2191y)\ny : { x // x \u2208 S }\nw z : N\n\u22a2 \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict f S) h) y)\u207b\u00b9 * w = z \u2194 w = \u2191f \u2191y * z\n[PROOFSTEP]\nexact Units.inv_mul_eq_iff_eq_mul (IsUnit.liftRight (f.restrict S) h y)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : M \u2192* N\nh : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191f \u2191y)\ny : { x // x \u2208 S }\nw z : N\n\u22a2 z = w * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict f S) h) y)\u207b\u00b9 \u2194 z * \u2191f \u2191y = w\n[PROOFSTEP]\nrw [eq_comm, mul_inv_left h, mul_comm, eq_comm]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : M \u2192* N\nh : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191f \u2191y)\nx\u2081 x\u2082 : M\ny\u2081 y\u2082 : { x // x \u2208 S }\n\u22a2 \u2191f x\u2081 * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict f S) h) y\u2081)\u207b\u00b9 =\n \u2191f x\u2082 * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict f S) h) y\u2082)\u207b\u00b9 \u2194\n \u2191f (x\u2081 * \u2191y\u2082) = \u2191f (x\u2082 * \u2191y\u2081)\n[PROOFSTEP]\nrw [mul_inv_right h, mul_assoc, mul_comm _ (f y\u2082), \u2190 mul_assoc, mul_inv_left h, mul_comm x\u2082, f.map_mul, f.map_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : M \u2192* N\nhf : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191f \u2191y)\ny z : { x // x \u2208 S }\nh : (\u2191(IsUnit.liftRight (MonoidHom.restrict f S) hf) y)\u207b\u00b9 = (\u2191(IsUnit.liftRight (MonoidHom.restrict f S) hf) z)\u207b\u00b9\n\u22a2 \u2191f \u2191y = \u2191f \u2191z\n[PROOFSTEP]\nrw [\u2190 mul_one (f y), eq_comm, \u2190 mul_inv_left hf y (f z) 1, h]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : M \u2192* N\nhf : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191f \u2191y)\ny z : { x // x \u2208 S }\nh : (\u2191(IsUnit.liftRight (MonoidHom.restrict f S) hf) y)\u207b\u00b9 = (\u2191(IsUnit.liftRight (MonoidHom.restrict f S) hf) z)\u207b\u00b9\n\u22a2 \u2191f \u2191z * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict f S) hf) z)\u207b\u00b9 = 1\n[PROOFSTEP]\nexact Units.inv_mul (IsUnit.liftRight (f.restrict S) hf z)\u207b\u00b9\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : M \u2192* N\nh : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191f \u2191y)\ny : { x // x \u2208 S }\nz : N\nH : \u2191f \u2191y * z = 1\n\u22a2 \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict f S) h) y)\u207b\u00b9 = z\n[PROOFSTEP]\nrw [\u2190 one_mul _\u207b\u00b9, Units.val_mul, mul_inv_left]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : M \u2192* N\nh : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191f \u2191y)\ny : { x // x \u2208 S }\nz : N\nH : \u2191f \u2191y * z = 1\n\u22a2 \u21911 = \u2191f \u2191y * z\n[PROOFSTEP]\nexact H.symm\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx y : M\nc : { x // x \u2208 S }\nh : \u2191(toMap f) (\u2191c * x) = \u2191(toMap f) (\u2191c * y)\n\u22a2 \u2191(toMap f) x = \u2191(toMap f) y\n[PROOFSTEP]\nrw [f.toMap.map_mul, f.toMap.map_mul] at h \n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx y : M\nc : { x // x \u2208 S }\nh : \u2191(toMap f) \u2191c * \u2191(toMap f) x = \u2191(toMap f) \u2191c * \u2191(toMap f) y\n\u22a2 \u2191(toMap f) x = \u2191(toMap f) y\n[PROOFSTEP]\ncases' f.map_units c with u hu\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx y : M\nc : { x // x \u2208 S }\nh : \u2191(toMap f) \u2191c * \u2191(toMap f) x = \u2191(toMap f) \u2191c * \u2191(toMap f) y\nu : ((fun x => N) \u2191c)\u02e3\nhu : \u2191u = \u2191(toMap f) \u2191c\n\u22a2 \u2191(toMap f) x = \u2191(toMap f) y\n[PROOFSTEP]\nrw [\u2190 hu] at h \n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx y : M\nc : { x // x \u2208 S }\nu : ((fun x => N) \u2191c)\u02e3\nh : \u2191u * \u2191(toMap f) x = \u2191u * \u2191(toMap f) y\nhu : \u2191u = \u2191(toMap f) \u2191c\n\u22a2 \u2191(toMap f) x = \u2191(toMap f) y\n[PROOFSTEP]\nexact (Units.mul_right_inj u).1 h\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx y : M\nc : { x // x \u2208 S }\nh : \u2191(toMap f) (x * \u2191c) = \u2191(toMap f) (y * \u2191c)\n\u22a2 \u2191(toMap f) (\u2191(?m.701298 h) * x) = \u2191(toMap f) (\u2191(?m.701298 h) * y)\n[PROOFSTEP]\nrw [mul_comm _ x, mul_comm _ y, h]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx\u2081 x\u2082 : M\ny\u2081 y\u2082 : { x // x \u2208 S }\n\u22a2 \u2191(toMap f) (x\u2081 * x\u2082) =\n \u2191(toMap f) \u2191(y\u2081 * y\u2082) *\n (\u2191(toMap f) x\u2081 *\n \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict (toMap f) S) (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap f) \u2191y)))\n y\u2081)\u207b\u00b9 *\n (\u2191(toMap f) x\u2082 *\n \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict (toMap f) S) (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap f) \u2191y)))\n y\u2082)\u207b\u00b9))\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 mul_assoc, mul_inv_right f.map_units, mul_assoc, mul_assoc, mul_comm _ (f.toMap x\u2082), \u2190 mul_assoc, \u2190\n mul_assoc, mul_inv_right f.map_units, Submonoid.coe_mul, f.toMap.map_mul, f.toMap.map_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx\u2081 x\u2082 : M\ny\u2081 y\u2082 : { x // x \u2208 S }\n\u22a2 \u2191(toMap f) x\u2081 * \u2191(toMap f) x\u2082 * \u2191(toMap f) \u2191y\u2082 * \u2191(toMap f) \u2191y\u2081 =\n \u2191(toMap f) \u2191y\u2081 * \u2191(toMap f) \u2191y\u2082 * \u2191(toMap f) x\u2081 * \u2191(toMap f) x\u2082\n[PROOFSTEP]\nac_rfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx : M\n\u22a2 mk' f x 1 = \u2191(toMap f) x\n[PROOFSTEP]\nrw [mk', MonoidHom.map_one]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx : M\n\u22a2 \u2191(toMap f) x * \u21911\u207b\u00b9 = \u2191(toMap f) x\n[PROOFSTEP]\nexact mul_one _\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nz : N\n\u22a2 \u2191(toMap f) (sec f z).fst *\n \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict (toMap f) S) (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap f) \u2191y)))\n (sec f z).snd)\u207b\u00b9 =\n z\n[PROOFSTEP]\nrw [\u2190 sec_spec, mul_inv_left, mul_comm]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(toMap f) x *\n \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict (toMap f) S) (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap f) \u2191y)))\n y)\u207b\u00b9 *\n \u2191(toMap f) \u2191y =\n \u2191(toMap f) x\n[PROOFSTEP]\nrw [mul_assoc, mul_comm _ (f.toMap y), \u2190 mul_assoc, mul_inv_left, mul_comm]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(toMap f) \u2191y * mk' f x y = \u2191(toMap f) x\n[PROOFSTEP]\nrw [mul_comm, mk'_spec]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx : M\ny : { x // x \u2208 S }\nz : N\nH : z = mk' f x y\n\u22a2 z * \u2191(toMap f) \u2191y = \u2191(toMap f) x\n[PROOFSTEP]\nrw [H, mk'_spec]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx : M\ny : { x // x \u2208 S }\nz : N\nH : z * \u2191(toMap f) \u2191y = \u2191(toMap f) x\n\u22a2 z = mk' f x y\n[PROOFSTEP]\nerw [mul_inv_right, H]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx : M\ny : { x // x \u2208 S }\nz : N\n\u22a2 mk' f x y = z \u2194 \u2191(toMap f) x = z * \u2191(toMap f) \u2191y\n[PROOFSTEP]\nrw [eq_comm, eq_mk'_iff_mul_eq, eq_comm]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx\u2081 x\u2082 : M\ny\u2081 y\u2082 : { x // x \u2208 S }\nH : mk' f x\u2081 y\u2081 = mk' f x\u2082 y\u2082\n\u22a2 \u2191(toMap f) (\u2191y\u2082 * x\u2081) = \u2191(toMap f) (\u2191y\u2081 * x\u2082)\n[PROOFSTEP]\nrw [f.toMap.map_mul, f.toMap.map_mul, f.mk'_eq_iff_eq_mul.1 H, \u2190 mul_assoc, mk'_spec', mul_comm ((toMap f) x\u2082) _]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx\u2081 x\u2082 : M\ny\u2081 y\u2082 : { x // x \u2208 S }\nH : \u2191(toMap f) (\u2191y\u2082 * x\u2081) = \u2191(toMap f) (\u2191y\u2081 * x\u2082)\n\u22a2 mk' f x\u2081 y\u2081 = mk' f x\u2082 y\u2082\n[PROOFSTEP]\nrw [mk'_eq_iff_eq_mul, mk', mul_assoc, mul_comm _ (f.toMap y\u2081), \u2190 mul_assoc, \u2190 f.toMap.map_mul, mul_comm x\u2082, \u2190 H, \u2190\n mul_comm x\u2081, f.toMap.map_mul, mul_inv_right f.map_units]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx\u2081 x\u2082 : M\ny\u2081 y\u2082 : { x // x \u2208 S }\n\u22a2 mk' f x\u2081 y\u2081 = mk' f x\u2082 y\u2082 \u2194 \u2191(toMap f) (x\u2081 * \u2191y\u2082) = \u2191(toMap f) (x\u2082 * \u2191y\u2081)\n[PROOFSTEP]\nsimp only [f.mk'_eq_iff_eq, mul_comm]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\na\u2081 b\u2081 : M\na\u2082 b\u2082 : { x // x \u2208 S }\n\u22a2 mk' f a\u2081 a\u2082 = mk' f b\u2081 b\u2082 \u2194 \u2191(Localization.r S) (a\u2081, a\u2082) (b\u2081, b\u2082)\n[PROOFSTEP]\nrw [f.eq, Localization.r_iff_exists]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\na\u2081 b\u2081 : M\na\u2082 b\u2082 : { x // x \u2208 S }\nH : b\u2081 * \u2191a\u2082 = a\u2081 * \u2191b\u2082\n\u22a2 \u2191a\u2082 * b\u2081 = \u2191b\u2082 * a\u2081\n[PROOFSTEP]\nsimpa only [mul_comm] using H\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ny : { x // x \u2208 S }\n\u22a2 \u2191(toMap f) \u2191y *\n \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict (toMap f) S) (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap f) \u2191y))) y)\u207b\u00b9 =\n 1\n[PROOFSTEP]\nrw [mul_inv_left, mul_one]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx\u2081 x\u2082 : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(toMap f) x\u2081 * mk' f x\u2082 y = mk' f (x\u2081 * x\u2082) y\n[PROOFSTEP]\nrw [\u2190 mk'_one, \u2190 mk'_mul, one_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx\u2081 x\u2082 : M\ny : { x // x \u2208 S }\n\u22a2 mk' f x\u2082 y * \u2191(toMap f) x\u2081 = mk' f (x\u2081 * x\u2082) y\n[PROOFSTEP]\nrw [mul_comm, mul_mk'_eq_mk'_of_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(toMap f) x * mk' f 1 y = mk' f x y\n[PROOFSTEP]\nrw [mul_mk'_eq_mk'_of_mul, mul_one]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx : M\ny : { x // x \u2208 S }\n\u22a2 mk' f (x * \u2191y) y = \u2191(toMap f) x\n[PROOFSTEP]\nrw [\u2190 mul_mk'_one_eq_mk', f.toMap.map_mul, mul_assoc, mul_mk'_one_eq_mk', mk'_self', mul_one]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\nx : M\ny : { x // x \u2208 S }\n\u22a2 mk' f (\u2191y * x) y = \u2191(toMap f) x\n[PROOFSTEP]\nrw [mul_comm, mk'_mul_cancel_right]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : M\nh : \u2191(toMap f) x = \u2191(toMap f) y\n\u22a2 \u2191g x = \u2191g y\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 := f.eq_iff_exists.1 h\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : M\nh : \u2191(toMap f) x = \u2191(toMap f) y\nc : { x // x \u2208 S }\nhc : \u2191c * x = \u2191c * y\n\u22a2 \u2191g x = \u2191g y\n[PROOFSTEP]\nrw [\u2190 one_mul (g x), \u2190 IsUnit.liftRight_inv_mul (g.restrict S) hg c]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : M\nh : \u2191(toMap f) x = \u2191(toMap f) y\nc : { x // x \u2208 S }\nhc : \u2191c * x = \u2191c * y\n\u22a2 \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) c)\u207b\u00b9 * \u2191(MonoidHom.restrict g S) c * \u2191g x = \u2191g y\n[PROOFSTEP]\nshow _ * g c * _ = _\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : M\nh : \u2191(toMap f) x = \u2191(toMap f) y\nc : { x // x \u2208 S }\nhc : \u2191c * x = \u2191c * y\n\u22a2 \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) c)\u207b\u00b9 * \u2191g \u2191c * \u2191g x = \u2191g y\n[PROOFSTEP]\nrw [mul_assoc, \u2190 g.map_mul, hc, mul_comm, mul_inv_left hg, g.map_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\n\u22a2 (fun z => \u2191g (sec f z).fst * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) (sec f z).snd)\u207b\u00b9) 1 = 1\n[PROOFSTEP]\nrw [mul_inv_left, mul_one]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\n\u22a2 \u2191g (sec f 1).fst = \u2191g \u2191(sec f 1).snd\n[PROOFSTEP]\nexact f.eq_of_eq hg (by rw [\u2190 sec_spec, one_mul])\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\n\u22a2 \u2191(toMap f) (sec f 1).fst = \u2191(toMap f) \u2191(sec f 1).snd\n[PROOFSTEP]\nrw [\u2190 sec_spec, one_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : N\n\u22a2 OneHom.toFun\n { toFun := fun z => \u2191g (sec f z).fst * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) (sec f z).snd)\u207b\u00b9,\n map_one' :=\n (_ : (fun z => \u2191g (sec f z).fst * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) (sec f z).snd)\u207b\u00b9) 1 = 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun z => \u2191g (sec f z).fst * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) (sec f z).snd)\u207b\u00b9,\n map_one' :=\n (_ :\n (fun z => \u2191g (sec f z).fst * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) (sec f z).snd)\u207b\u00b9) 1 = 1) }\n x *\n OneHom.toFun\n { toFun := fun z => \u2191g (sec f z).fst * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) (sec f z).snd)\u207b\u00b9,\n map_one' :=\n (_ :\n (fun z => \u2191g (sec f z).fst * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) (sec f z).snd)\u207b\u00b9) 1 = 1) }\n y\n[PROOFSTEP]\ndsimp only\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : N\n\u22a2 \u2191g (sec f (x * y)).fst * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) (sec f (x * y)).snd)\u207b\u00b9 =\n \u2191g (sec f x).fst * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) (sec f x).snd)\u207b\u00b9 *\n (\u2191g (sec f y).fst * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) (sec f y).snd)\u207b\u00b9)\n[PROOFSTEP]\nrw [mul_inv_left hg, \u2190 mul_assoc, \u2190 mul_assoc, mul_inv_right hg, mul_comm _ (g (f.sec y).1), \u2190 mul_assoc, \u2190 mul_assoc,\n mul_inv_right hg]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : N\n\u22a2 \u2191g (sec f (x * y)).fst * \u2191g \u2191(sec f y).snd * \u2191g \u2191(sec f x).snd =\n \u2191g (sec f y).fst * \u2191g \u2191(sec f (x * y)).snd * \u2191g (sec f x).fst\n[PROOFSTEP]\nrepeat' rw [\u2190 g.map_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : N\n\u22a2 \u2191g (sec f (x * y)).fst * \u2191g \u2191(sec f y).snd * \u2191g \u2191(sec f x).snd =\n \u2191g (sec f y).fst * \u2191g \u2191(sec f (x * y)).snd * \u2191g (sec f x).fst\n[PROOFSTEP]\nrw [\u2190 g.map_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : N\n\u22a2 \u2191g ((sec f (x * y)).fst * \u2191(sec f y).snd) * \u2191g \u2191(sec f x).snd =\n \u2191g (sec f y).fst * \u2191g \u2191(sec f (x * y)).snd * \u2191g (sec f x).fst\n[PROOFSTEP]\nrw [\u2190 g.map_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : N\n\u22a2 \u2191g ((sec f (x * y)).fst * \u2191(sec f y).snd * \u2191(sec f x).snd) =\n \u2191g (sec f y).fst * \u2191g \u2191(sec f (x * y)).snd * \u2191g (sec f x).fst\n[PROOFSTEP]\nrw [\u2190 g.map_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : N\n\u22a2 \u2191g ((sec f (x * y)).fst * \u2191(sec f y).snd * \u2191(sec f x).snd) =\n \u2191g ((sec f y).fst * \u2191(sec f (x * y)).snd) * \u2191g (sec f x).fst\n[PROOFSTEP]\nrw [\u2190 g.map_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : N\n\u22a2 \u2191g ((sec f (x * y)).fst * \u2191(sec f y).snd * \u2191(sec f x).snd) = \u2191g ((sec f y).fst * \u2191(sec f (x * y)).snd * (sec f x).fst)\n[PROOFSTEP]\nrw [\u2190 g.map_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : N\n\u22a2 \u2191g ((sec f (x * y)).fst * \u2191(sec f y).snd * \u2191(sec f x).snd) = \u2191g ((sec f y).fst * \u2191(sec f (x * y)).snd * (sec f x).fst)\n[PROOFSTEP]\nexact f.eq_of_eq hg (by simp_rw [f.toMap.map_mul, sec_spec']; ac_rfl)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : N\n\u22a2 \u2191(toMap f) ((sec f (x * y)).fst * \u2191(sec f y).snd * \u2191(sec f x).snd) =\n \u2191(toMap f) ((sec f y).fst * \u2191(sec f (x * y)).snd * (sec f x).fst)\n[PROOFSTEP]\nsimp_rw [f.toMap.map_mul, sec_spec']\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : N\n\u22a2 \u2191(toMap f) \u2191(sec f (x * y)).snd * (x * y) * \u2191(toMap f) \u2191(sec f y).snd * \u2191(toMap f) \u2191(sec f x).snd =\n \u2191(toMap f) \u2191(sec f y).snd * y * \u2191(toMap f) \u2191(sec f (x * y)).snd * (\u2191(toMap f) \u2191(sec f x).snd * x)\n[PROOFSTEP]\nac_rfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(toMap f) ((sec f (mk' f x y)).fst * \u2191y) = \u2191(toMap f) (x * \u2191(sec f (mk' f x y)).snd)\n[PROOFSTEP]\nsimp_rw [f.toMap.map_mul, sec_spec', mul_assoc, f.mk'_spec, mul_comm]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nz : N\nw v : (fun x => P) z\n\u22a2 \u2191(lift f hg) z * w = v \u2194 \u2191g (sec f z).fst * w = \u2191g \u2191(sec f z).snd * v\n[PROOFSTEP]\nerw [mul_comm, \u2190 mul_assoc, mul_inv_left hg, mul_comm]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx : M\nv : P\ny : { x // x \u2208 S }\n\u22a2 \u2191(lift f hg) (mk' f x y) = v \u2194 \u2191g x = \u2191g \u2191y * v\n[PROOFSTEP]\nrw [f.lift_mk' hg]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx : M\nv : P\ny : { x // x \u2208 S }\n\u22a2 \u2191g x * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) y)\u207b\u00b9 = v \u2194 \u2191g x = \u2191g \u2191y * v\n[PROOFSTEP]\nexact mul_inv_left hg _ _ _\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nz : N\n\u22a2 \u2191(lift f hg) z * \u2191g \u2191(sec f z).snd = \u2191g (sec f z).fst\n[PROOFSTEP]\nerw [mul_assoc, IsUnit.liftRight_inv_mul, mul_one]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nz : N\n\u22a2 \u2191g \u2191(sec f z).snd * \u2191(lift f hg) z = \u2191g (sec f z).fst\n[PROOFSTEP]\nrw [mul_comm, lift_mul_right]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx : M\n\u22a2 \u2191(lift f hg) (\u2191(toMap f) x) = \u2191g x\n[PROOFSTEP]\nrw [lift_spec, \u2190 g.map_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx : M\n\u22a2 \u2191g (sec f (\u2191(toMap f) x)).fst = \u2191g (\u2191(sec f (\u2191(toMap f) x)).snd * x)\n[PROOFSTEP]\nexact f.eq_of_eq hg (by rw [sec_spec', f.toMap.map_mul])\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx : M\n\u22a2 \u2191(toMap f) (sec f (\u2191(toMap f) x)).fst = \u2191(toMap f) (\u2191(sec f (\u2191(toMap f) x)).snd * x)\n[PROOFSTEP]\nrw [sec_spec', f.toMap.map_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx y : M \u00d7 { x // x \u2208 S }\n\u22a2 \u2191(lift f hg) (mk' f x.fst x.snd) = \u2191(lift f hg) (mk' f y.fst y.snd) \u2194 \u2191g (x.fst * \u2191y.snd) = \u2191g (y.fst * \u2191x.snd)\n[PROOFSTEP]\nrw [lift_mk', lift_mk', mul_inv hg]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\n\u22a2 MonoidHom.comp (lift f hg) (toMap f) = g\n[PROOFSTEP]\next\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx\u271d : M\n\u22a2 \u2191(MonoidHom.comp (lift f hg) (toMap f)) x\u271d = \u2191g x\u271d\n[PROOFSTEP]\nexact f.lift_eq hg _\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nj : N \u2192* P\n\u22a2 lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(MonoidHom.comp j (toMap f)) \u2191y)) = j\n[PROOFSTEP]\next\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nj : N \u2192* P\nx\u271d : N\n\u22a2 \u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(MonoidHom.comp j (toMap f)) \u2191y))) x\u271d = \u2191j x\u271d\n[PROOFSTEP]\nrw [lift_spec]\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nj : N \u2192* P\nx\u271d : N\n\u22a2 \u2191(MonoidHom.comp j (toMap f)) (sec f x\u271d).fst = \u2191(MonoidHom.comp j (toMap f)) \u2191(sec f x\u271d).snd * \u2191j x\u271d\n[PROOFSTEP]\nshow j _ = j _ * _\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nj : N \u2192* P\nx\u271d : N\n\u22a2 \u2191j (\u2191(toMap f) (sec f x\u271d).fst) = \u2191j (\u2191(toMap f) \u2191(sec f x\u271d).snd) * \u2191j x\u271d\n[PROOFSTEP]\nerw [\u2190 j.map_mul, sec_spec']\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nj k : N \u2192* P\nh : \u2200 (a : M), \u2191(MonoidHom.comp j (toMap f)) a = \u2191(MonoidHom.comp k (toMap f)) a\n\u22a2 j = k\n[PROOFSTEP]\nrw [\u2190 f.lift_of_comp j, \u2190 f.lift_of_comp k]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nj k : N \u2192* P\nh : \u2200 (a : M), \u2191(MonoidHom.comp j (toMap f)) a = \u2191(MonoidHom.comp k (toMap f)) a\n\u22a2 lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(MonoidHom.comp j (toMap f)) \u2191y)) =\n lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(MonoidHom.comp k (toMap f)) \u2191y))\n[PROOFSTEP]\ncongr 1 with x\n[GOAL]\ncase e_g.h\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nj k : N \u2192* P\nh : \u2200 (a : M), \u2191(MonoidHom.comp j (toMap f)) a = \u2191(MonoidHom.comp k (toMap f)) a\nx : M\n\u22a2 \u2191(MonoidHom.comp j (toMap f)) x = \u2191(MonoidHom.comp k (toMap f)) x\n[PROOFSTEP]\nexact h x\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nj : N \u2192* P\nhj : \u2200 (x : M), \u2191j (\u2191(toMap f) x) = \u2191g x\n\u22a2 lift f hg = j\n[PROOFSTEP]\next\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nj : N \u2192* P\nhj : \u2200 (x : M), \u2191j (\u2191(toMap f) x) = \u2191g x\nx\u271d : N\n\u22a2 \u2191(lift f hg) x\u271d = \u2191j x\u271d\n[PROOFSTEP]\nrw [lift_spec, \u2190 hj, \u2190 hj, \u2190 j.map_mul]\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nj : N \u2192* P\nhj : \u2200 (x : M), \u2191j (\u2191(toMap f) x) = \u2191g x\nx\u271d : N\n\u22a2 \u2191j (\u2191(toMap f) (sec f x\u271d).fst) = \u2191j (\u2191(toMap f) \u2191(sec f x\u271d).snd * x\u271d)\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h.h\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nj : N \u2192* P\nhj : \u2200 (x : M), \u2191j (\u2191(toMap f) x) = \u2191g x\nx\u271d : N\n\u22a2 \u2191(toMap f) (sec f x\u271d).fst = \u2191(toMap f) \u2191(sec f x\u271d).snd * x\u271d\n[PROOFSTEP]\nrw [\u2190 sec_spec']\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\n\u22a2 \u2191(lift k (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap f) \u2191y)))\n (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z) =\n z\n[PROOFSTEP]\nrw [lift_spec]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\n\u22a2 \u2191(toMap f) (sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).fst =\n \u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * z\n[PROOFSTEP]\ncases' f.surj z with x hx\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap f) (sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).fst =\n \u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * z\n[PROOFSTEP]\nconv_rhs =>\n congr\n next => skip\n rw [f.eq_mk'_iff_mul_eq.2 hx]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n| \u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * z\n[PROOFSTEP]\n congr\n next => skip\n rw [f.eq_mk'_iff_mul_eq.2 hx]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n| \u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * z\n[PROOFSTEP]\n congr\n next => skip\n rw [f.eq_mk'_iff_mul_eq.2 hx]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n| \u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * z\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n| \u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd\ncase a\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n| z\n[PROOFSTEP]\nnext => skip\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n| \u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd\n[PROOFSTEP]\nskip\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n| \u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd\n[PROOFSTEP]\nskip\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n| \u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n| z\n[PROOFSTEP]\nrw [f.eq_mk'_iff_mul_eq.2 hx]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap f) (sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).fst =\n \u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * mk' f x.fst x.snd\n[PROOFSTEP]\nrw [mk', \u2190 mul_assoc, mul_inv_right f.map_units, \u2190 f.toMap.map_mul, \u2190 f.toMap.map_mul]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap f) ((sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).fst * \u2191x.snd) =\n \u2191(toMap f) (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * x.fst)\n[PROOFSTEP]\napply k.eq_of_eq f.map_units\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap k) ((sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).fst * \u2191x.snd) =\n \u2191(toMap k) (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * x.fst)\n[PROOFSTEP]\nrw [k.toMap.map_mul, k.toMap.map_mul, \u2190 sec_spec, mul_assoc, lift_spec_mul]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap k) (sec f z).fst *\n (\u2191(toMap k) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191(toMap k) \u2191x.snd) =\n \u2191(toMap k) \u2191(sec f z).snd *\n (\u2191(toMap k) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191(toMap k) x.fst)\n[PROOFSTEP]\nrepeat' rw [\u2190 k.toMap.map_mul]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap k) (sec f z).fst *\n (\u2191(toMap k) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191(toMap k) \u2191x.snd) =\n \u2191(toMap k) \u2191(sec f z).snd *\n (\u2191(toMap k) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191(toMap k) x.fst)\n[PROOFSTEP]\nrw [\u2190 k.toMap.map_mul]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap k) (sec f z).fst *\n \u2191(toMap k) (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191x.snd) =\n \u2191(toMap k) \u2191(sec f z).snd *\n (\u2191(toMap k) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191(toMap k) x.fst)\n[PROOFSTEP]\nrw [\u2190 k.toMap.map_mul]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap k)\n ((sec f z).fst * (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191x.snd)) =\n \u2191(toMap k) \u2191(sec f z).snd *\n (\u2191(toMap k) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191(toMap k) x.fst)\n[PROOFSTEP]\nrw [\u2190 k.toMap.map_mul]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap k)\n ((sec f z).fst * (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191x.snd)) =\n \u2191(toMap k) \u2191(sec f z).snd *\n \u2191(toMap k) (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * x.fst)\n[PROOFSTEP]\nrw [\u2190 k.toMap.map_mul]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap k)\n ((sec f z).fst * (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191x.snd)) =\n \u2191(toMap k)\n (\u2191(sec f z).snd * (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * x.fst))\n[PROOFSTEP]\nrw [\u2190 k.toMap.map_mul]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap k)\n ((sec f z).fst * (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191x.snd)) =\n \u2191(toMap k)\n (\u2191(sec f z).snd * (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * x.fst))\n[PROOFSTEP]\napply f.eq_of_eq k.map_units\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap f)\n ((sec f z).fst * (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191x.snd)) =\n \u2191(toMap f)\n (\u2191(sec f z).snd * (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * x.fst))\n[PROOFSTEP]\nrepeat' rw [f.toMap.map_mul]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap f)\n ((sec f z).fst * (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191x.snd)) =\n \u2191(toMap f)\n (\u2191(sec f z).snd * (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * x.fst))\n[PROOFSTEP]\nrw [f.toMap.map_mul]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap f) (sec f z).fst *\n \u2191(toMap f) (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191x.snd) =\n \u2191(toMap f)\n (\u2191(sec f z).snd * (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * x.fst))\n[PROOFSTEP]\nrw [f.toMap.map_mul]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap f) (sec f z).fst *\n (\u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191(toMap f) \u2191x.snd) =\n \u2191(toMap f)\n (\u2191(sec f z).snd * (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * x.fst))\n[PROOFSTEP]\nrw [f.toMap.map_mul]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap f) (sec f z).fst *\n (\u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191(toMap f) \u2191x.snd) =\n \u2191(toMap f) \u2191(sec f z).snd *\n \u2191(toMap f) (\u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * x.fst)\n[PROOFSTEP]\nrw [f.toMap.map_mul]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap f) (sec f z).fst *\n (\u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191(toMap f) \u2191x.snd) =\n \u2191(toMap f) \u2191(sec f z).snd *\n (\u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191(toMap f) x.fst)\n[PROOFSTEP]\nrw [f.toMap.map_mul]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap f) (sec f z).fst *\n (\u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191(toMap f) \u2191x.snd) =\n \u2191(toMap f) \u2191(sec f z).snd *\n (\u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191(toMap f) x.fst)\n[PROOFSTEP]\nrw [sec_spec', \u2190 hx]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nk : LocalizationMap S P\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191(toMap f) \u2191(sec f z).snd * z *\n (\u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd * \u2191(toMap f) \u2191x.snd) =\n \u2191(toMap f) \u2191(sec f z).snd *\n (\u2191(toMap f) \u2191(sec k (\u2191(lift f (_ : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191(toMap k) \u2191y))) z)).snd *\n (z * \u2191(toMap f) \u2191x.snd))\n[PROOFSTEP]\nac_rfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\n\u22a2 Surjective \u2191(lift f hg) \u2194 \u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\n\u22a2 Surjective \u2191(lift f hg) \u2192 \u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst\n[PROOFSTEP]\nintro H v\n[GOAL]\ncase mp\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Surjective \u2191(lift f hg)\nv : P\n\u22a2 \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst\n[PROOFSTEP]\nobtain \u27e8z, hz\u27e9 := H v\n[GOAL]\ncase mp.intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Surjective \u2191(lift f hg)\nv : P\nz : N\nhz : \u2191(lift f hg) z = v\n\u22a2 \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := f.surj z\n[GOAL]\ncase mp.intro.intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Surjective \u2191(lift f hg)\nv : P\nz : N\nhz : \u2191(lift f hg) z = v\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst\n[PROOFSTEP]\nuse x\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Surjective \u2191(lift f hg)\nv : P\nz : N\nhz : \u2191(lift f hg) z = v\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 v * \u2191g \u2191x.snd = \u2191g x.fst\n[PROOFSTEP]\nrw [\u2190 hz, f.eq_mk'_iff_mul_eq.2 hx, lift_mk', mul_assoc, mul_comm _ (g \u2191x.2)]\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Surjective \u2191(lift f hg)\nv : P\nz : N\nhz : \u2191(lift f hg) z = v\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191g x.fst * (\u2191g \u2191x.snd * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) x.snd)\u207b\u00b9) = \u2191g x.fst\n[PROOFSTEP]\nerw [IsUnit.mul_liftRight_inv (g.restrict S) hg, mul_one]\n[GOAL]\ncase mpr\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\n\u22a2 (\u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst) \u2192 Surjective \u2191(lift f hg)\n[PROOFSTEP]\nintro H v\n[GOAL]\ncase mpr\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : \u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst\nv : P\n\u22a2 \u2203 a, \u2191(lift f hg) a = v\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := H v\n[GOAL]\ncase mpr.intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : \u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst\nv : P\nx : M \u00d7 { x // x \u2208 S }\nhx : v * \u2191g \u2191x.snd = \u2191g x.fst\n\u22a2 \u2203 a, \u2191(lift f hg) a = v\n[PROOFSTEP]\nuse f.mk' x.1 x.2\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : \u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst\nv : P\nx : M \u00d7 { x // x \u2208 S }\nhx : v * \u2191g \u2191x.snd = \u2191g x.fst\n\u22a2 \u2191(lift f hg) (mk' f x.fst x.snd) = v\n[PROOFSTEP]\nrw [lift_mk', mul_inv_left hg, mul_comm, \u2190 hx]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\n\u22a2 Injective \u2191(lift f hg) \u2194 \u2200 (x y : M), \u2191(toMap f) x = \u2191(toMap f) y \u2194 \u2191g x = \u2191g y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\n\u22a2 Injective \u2191(lift f hg) \u2192 \u2200 (x y : M), \u2191(toMap f) x = \u2191(toMap f) y \u2194 \u2191g x = \u2191g y\n[PROOFSTEP]\nintro H x y\n[GOAL]\ncase mp\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Injective \u2191(lift f hg)\nx y : M\n\u22a2 \u2191(toMap f) x = \u2191(toMap f) y \u2194 \u2191g x = \u2191g y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.mp\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Injective \u2191(lift f hg)\nx y : M\n\u22a2 \u2191(toMap f) x = \u2191(toMap f) y \u2192 \u2191g x = \u2191g y\n[PROOFSTEP]\nexact f.eq_of_eq hg\n[GOAL]\ncase mp.mpr\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Injective \u2191(lift f hg)\nx y : M\n\u22a2 \u2191g x = \u2191g y \u2192 \u2191(toMap f) x = \u2191(toMap f) y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp.mpr\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Injective \u2191(lift f hg)\nx y : M\nh : \u2191g x = \u2191g y\n\u22a2 \u2191(toMap f) x = \u2191(toMap f) y\n[PROOFSTEP]\nrw [\u2190 f.lift_eq hg, \u2190 f.lift_eq hg] at h \n[GOAL]\ncase mp.mpr\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Injective \u2191(lift f hg)\nx y : M\nh : \u2191(lift f hg) (\u2191(toMap f) x) = \u2191(lift f hg) (\u2191(toMap f) y)\n\u22a2 \u2191(toMap f) x = \u2191(toMap f) y\n[PROOFSTEP]\nexact H h\n[GOAL]\ncase mpr\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\n\u22a2 (\u2200 (x y : M), \u2191(toMap f) x = \u2191(toMap f) y \u2194 \u2191g x = \u2191g y) \u2192 Injective \u2191(lift f hg)\n[PROOFSTEP]\nintro H z w h\n[GOAL]\ncase mpr\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : \u2200 (x y : M), \u2191(toMap f) x = \u2191(toMap f) y \u2194 \u2191g x = \u2191g y\nz w : N\nh : \u2191(lift f hg) z = \u2191(lift f hg) w\n\u22a2 z = w\n[PROOFSTEP]\nobtain \u27e8_, _\u27e9 := f.surj z\n[GOAL]\ncase mpr.intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : \u2200 (x y : M), \u2191(toMap f) x = \u2191(toMap f) y \u2194 \u2191g x = \u2191g y\nz w : N\nh : \u2191(lift f hg) z = \u2191(lift f hg) w\nw\u271d : M \u00d7 { x // x \u2208 S }\nh\u271d : z * \u2191(toMap f) \u2191w\u271d.snd = \u2191(toMap f) w\u271d.fst\n\u22a2 z = w\n[PROOFSTEP]\nobtain \u27e8_, _\u27e9 := f.surj w\n[GOAL]\ncase mpr.intro.intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : \u2200 (x y : M), \u2191(toMap f) x = \u2191(toMap f) y \u2194 \u2191g x = \u2191g y\nz w : N\nh : \u2191(lift f hg) z = \u2191(lift f hg) w\nw\u271d\u00b9 : M \u00d7 { x // x \u2208 S }\nh\u271d\u00b9 : z * \u2191(toMap f) \u2191w\u271d\u00b9.snd = \u2191(toMap f) w\u271d\u00b9.fst\nw\u271d : M \u00d7 { x // x \u2208 S }\nh\u271d : w * \u2191(toMap f) \u2191w\u271d.snd = \u2191(toMap f) w\u271d.fst\n\u22a2 z = w\n[PROOFSTEP]\nrw [\u2190 f.mk'_sec z, \u2190 f.mk'_sec w]\n[GOAL]\ncase mpr.intro.intro\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : \u2200 (x y : M), \u2191(toMap f) x = \u2191(toMap f) y \u2194 \u2191g x = \u2191g y\nz w : N\nh : \u2191(lift f hg) z = \u2191(lift f hg) w\nw\u271d\u00b9 : M \u00d7 { x // x \u2208 S }\nh\u271d\u00b9 : z * \u2191(toMap f) \u2191w\u271d\u00b9.snd = \u2191(toMap f) w\u271d\u00b9.fst\nw\u271d : M \u00d7 { x // x \u2208 S }\nh\u271d : w * \u2191(toMap f) \u2191w\u271d.snd = \u2191(toMap f) w\u271d.fst\n\u22a2 mk' f (sec f z).fst (sec f z).snd = mk' f (sec f w).fst (sec f w).snd\n[PROOFSTEP]\nexact (mul_inv f.map_units).2 ((H _ _).2 <| (mul_inv hg).1 h)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(map f hy k) (mk' f x y) = mk' k (\u2191g x) { val := \u2191g \u2191y, property := (_ : \u2191g \u2191y \u2208 T) }\n[PROOFSTEP]\nrw [map, lift_mk', mul_inv_left]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(MonoidHom.comp (toMap k) g) x =\n \u2191(MonoidHom.comp (toMap k) g) \u2191y * mk' k (\u2191g x) { val := \u2191g \u2191y, property := (_ : \u2191g \u2191y \u2208 T) }\n[PROOFSTEP]\nshow k.toMap (g x) = k.toMap (g y) * _\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(toMap k) (\u2191g x) = \u2191(toMap k) (\u2191g \u2191y) * mk' k (\u2191g x) { val := \u2191g \u2191y, property := (_ : \u2191g \u2191y \u2208 T) }\n[PROOFSTEP]\nrw [mul_mk'_eq_mk'_of_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(toMap k) (\u2191g x) = mk' k (\u2191g \u2191y * \u2191g x) { val := \u2191g \u2191y, property := (_ : \u2191g \u2191y \u2208 T) }\n[PROOFSTEP]\nexact (k.mk'_mul_cancel_left (g x) \u27e8g y, hy y\u27e9).symm\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nz : N\n\u22a2 \u2191(toMap k) (\u2191g \u2191(sec f z).snd) * \u2191(map f hy k) z = \u2191(toMap k) (\u2191g (sec f z).fst)\n[PROOFSTEP]\nrw [mul_comm, f.map_mul_right]\n[GOAL]\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d\u00b2 : CommMonoid Q\nk : LocalizationMap T Q\nA : Type u_5\ninst\u271d\u00b9 : CommMonoid A\nU : Submonoid A\nR : Type u_6\ninst\u271d : CommMonoid R\nj : LocalizationMap U R\nl : P \u2192* A\nhl : \u2200 (w : { x // x \u2208 T }), \u2191l \u2191w \u2208 U\n\u22a2 MonoidHom.comp (map k hl j) (map f hy k) = map f (_ : \u2200 (x : { x // x \u2208 S }), \u2191(MonoidHom.comp l g) \u2191x \u2208 U) j\n[PROOFSTEP]\next z\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d\u00b2 : CommMonoid Q\nk : LocalizationMap T Q\nA : Type u_5\ninst\u271d\u00b9 : CommMonoid A\nU : Submonoid A\nR : Type u_6\ninst\u271d : CommMonoid R\nj : LocalizationMap U R\nl : P \u2192* A\nhl : \u2200 (w : { x // x \u2208 T }), \u2191l \u2191w \u2208 U\nz : N\n\u22a2 \u2191(MonoidHom.comp (map k hl j) (map f hy k)) z =\n \u2191(map f (_ : \u2200 (x : { x // x \u2208 S }), \u2191(MonoidHom.comp l g) \u2191x \u2208 U) j) z\n[PROOFSTEP]\nshow j.toMap _ * _ = j.toMap (l _) * _\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d\u00b2 : CommMonoid Q\nk : LocalizationMap T Q\nA : Type u_5\ninst\u271d\u00b9 : CommMonoid A\nU : Submonoid A\nR : Type u_6\ninst\u271d : CommMonoid R\nj : LocalizationMap U R\nl : P \u2192* A\nhl : \u2200 (w : { x // x \u2208 T }), \u2191l \u2191w \u2208 U\nz : N\n\u22a2 \u2191(toMap j) (\u2191l (sec k (\u2191(map f hy k) z)).fst) *\n \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict (MonoidHom.comp (toMap j) l) T)\n (_ : \u2200 (y : { x // x \u2208 T }), IsUnit (\u2191(toMap j) \u2191{ val := \u2191l \u2191y, property := (_ : \u2191l \u2191y \u2208 U) })))\n (sec k (\u2191(map f hy k) z)).snd)\u207b\u00b9 =\n \u2191(toMap j) (\u2191l (\u2191g (sec f z).fst)) *\n \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict (MonoidHom.comp (toMap j) (MonoidHom.comp l g)) S)\n (_ :\n \u2200 (y : { x // x \u2208 S }),\n IsUnit\n (\u2191(toMap j)\n \u2191{ val := \u2191(MonoidHom.comp l g) \u2191y, property := (_ : \u2191(MonoidHom.comp l g) \u2191y \u2208 U) })))\n (sec f z).snd)\u207b\u00b9\n[PROOFSTEP]\nrw [mul_inv_left, \u2190 mul_assoc, mul_inv_right]\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d\u00b2 : CommMonoid Q\nk : LocalizationMap T Q\nA : Type u_5\ninst\u271d\u00b9 : CommMonoid A\nU : Submonoid A\nR : Type u_6\ninst\u271d : CommMonoid R\nj : LocalizationMap U R\nl : P \u2192* A\nhl : \u2200 (w : { x // x \u2208 T }), \u2191l \u2191w \u2208 U\nz : N\n\u22a2 \u2191(toMap j) (\u2191l (sec k (\u2191(map f hy k) z)).fst) * \u2191(MonoidHom.comp (toMap j) (MonoidHom.comp l g)) \u2191(sec f z).snd =\n \u2191(MonoidHom.comp (toMap j) l) \u2191(sec k (\u2191(map f hy k) z)).snd * \u2191(toMap j) (\u2191l (\u2191g (sec f z).fst))\n[PROOFSTEP]\nshow j.toMap _ * j.toMap (l (g _)) = j.toMap (l _) * _\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d\u00b2 : CommMonoid Q\nk : LocalizationMap T Q\nA : Type u_5\ninst\u271d\u00b9 : CommMonoid A\nU : Submonoid A\nR : Type u_6\ninst\u271d : CommMonoid R\nj : LocalizationMap U R\nl : P \u2192* A\nhl : \u2200 (w : { x // x \u2208 T }), \u2191l \u2191w \u2208 U\nz : N\n\u22a2 \u2191(toMap j) (\u2191l (sec k (\u2191(map f hy k) z)).fst) * \u2191(toMap j) (\u2191l (\u2191g \u2191(sec f z).snd)) =\n \u2191(toMap j) (\u2191l \u2191(sec k (\u2191(map f hy k) z)).snd) * \u2191(toMap j) (\u2191l (\u2191g (sec f z).fst))\n[PROOFSTEP]\nrw [\u2190 j.toMap.map_mul, \u2190 j.toMap.map_mul, \u2190 l.map_mul, \u2190 l.map_mul]\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d\u00b2 : CommMonoid Q\nk : LocalizationMap T Q\nA : Type u_5\ninst\u271d\u00b9 : CommMonoid A\nU : Submonoid A\nR : Type u_6\ninst\u271d : CommMonoid R\nj : LocalizationMap U R\nl : P \u2192* A\nhl : \u2200 (w : { x // x \u2208 T }), \u2191l \u2191w \u2208 U\nz : N\n\u22a2 \u2191(toMap j) (\u2191l ((sec k (\u2191(map f hy k) z)).fst * \u2191g \u2191(sec f z).snd)) =\n \u2191(toMap j) (\u2191l (\u2191(sec k (\u2191(map f hy k) z)).snd * \u2191g (sec f z).fst))\n[PROOFSTEP]\nexact k.comp_eq_of_eq hl j (by rw [k.toMap.map_mul, k.toMap.map_mul, sec_spec', mul_assoc, map_mul_right])\n[GOAL]\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d\u00b2 : CommMonoid Q\nk : LocalizationMap T Q\nA : Type u_5\ninst\u271d\u00b9 : CommMonoid A\nU : Submonoid A\nR : Type u_6\ninst\u271d : CommMonoid R\nj : LocalizationMap U R\nl : P \u2192* A\nhl : \u2200 (w : { x // x \u2208 T }), \u2191l \u2191w \u2208 U\nz : N\n\u22a2 \u2191(toMap k) ((sec k (\u2191(map f hy k) z)).fst * \u2191g \u2191(sec f z).snd) =\n \u2191(toMap k) (\u2191(sec k (\u2191(map f hy k) z)).snd * \u2191g (sec f z).fst)\n[PROOFSTEP]\nrw [k.toMap.map_mul, k.toMap.map_mul, sec_spec', mul_assoc, map_mul_right]\n[GOAL]\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d\u00b2 : CommMonoid Q\nk : LocalizationMap T Q\nA : Type u_5\ninst\u271d\u00b9 : CommMonoid A\nU : Submonoid A\nR : Type u_6\ninst\u271d : CommMonoid R\nj : LocalizationMap U R\nl : P \u2192* A\nhl : \u2200 (w : { x // x \u2208 T }), \u2191l \u2191w \u2208 U\nx : N\n\u22a2 \u2191(map k hl j) (\u2191(map f hy k) x) = \u2191(map f (_ : \u2200 (x : { x // x \u2208 S }), \u2191(MonoidHom.comp l g) \u2191x \u2208 U) j) x\n[PROOFSTEP]\nrw [\u2190 @map_comp_map M _ S N _ P _ f g T hy Q _ k A _ U R _ j l hl]\n[GOAL]\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d\u00b2 : CommMonoid Q\nk : LocalizationMap T Q\nA : Type u_5\ninst\u271d\u00b9 : CommMonoid A\nU : Submonoid A\nR : Type u_6\ninst\u271d : CommMonoid R\nj : LocalizationMap U R\nl : P \u2192* A\nhl : \u2200 (w : { x // x \u2208 T }), \u2191l \u2191w \u2208 U\nx : N\n\u22a2 \u2191(map k hl j) (\u2191(map f hy k) x) = \u2191(MonoidHom.comp (map k hl j) (map f hy k)) x\n[PROOFSTEP]\nsimp only [MonoidHom.coe_comp, comp_apply]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg\u271d : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nx : M\nF : AwayMap x N\nhg : IsUnit (\u2191g x)\ny : { x_1 // x_1 \u2208 powers x }\n\u22a2 IsUnit (\u2191g \u2191y)\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := y.2\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg\u271d : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nx : M\nF : AwayMap x N\nhg : IsUnit (\u2191g x)\ny : { x_1 // x_1 \u2208 powers x }\nn : \u2115\nhn : (fun x x_1 => x ^ x_1) x n = \u2191y\n\u22a2 IsUnit (\u2191g \u2191y)\n[PROOFSTEP]\nrw [\u2190 hn, g.map_pow]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg\u271d : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nx : M\nF : AwayMap x N\nhg : IsUnit (\u2191g x)\ny : { x_1 // x_1 \u2208 powers x }\nn : \u2115\nhn : (fun x x_1 => x ^ x_1) x n = \u2191y\n\u22a2 IsUnit (\u2191g x ^ n)\n[PROOFSTEP]\nexact IsUnit.pow n hg\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nx : M\nF : AwayMap x N\ny : M\nG : AwayMap (x * y) P\n\u22a2 \u2191(toMap G) x * mk' G y { val := x * y, property := (_ : x * y \u2208 powers (x * y)) } = 1\n[PROOFSTEP]\nrw [mul_mk'_eq_mk'_of_mul, mk'_self]\n[GOAL]\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nA : Type u_4\ninst\u271d\u00b2 : AddCommMonoid A\nx : A\nB : Type u_5\ninst\u271d\u00b9 : AddCommMonoid B\nF : AwayMap x B\nC : Type u_6\ninst\u271d : AddCommMonoid C\ng : A \u2192+ C\nhg : IsAddUnit (\u2191g x)\ny : { x_1 // x_1 \u2208 multiples x }\n\u22a2 IsAddUnit (\u2191g \u2191y)\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := y.2\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nA : Type u_4\ninst\u271d\u00b2 : AddCommMonoid A\nx : A\nB : Type u_5\ninst\u271d\u00b9 : AddCommMonoid B\nF : AwayMap x B\nC : Type u_6\ninst\u271d : AddCommMonoid C\ng : A \u2192+ C\nhg : IsAddUnit (\u2191g x)\ny : { x_1 // x_1 \u2208 multiples x }\nn : \u2115\nhn : (fun i => i \u2022 x) n = \u2191y\n\u22a2 IsAddUnit (\u2191g \u2191y)\n[PROOFSTEP]\nrw [\u2190 hn]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nA : Type u_4\ninst\u271d\u00b2 : AddCommMonoid A\nx : A\nB : Type u_5\ninst\u271d\u00b9 : AddCommMonoid B\nF : AwayMap x B\nC : Type u_6\ninst\u271d : AddCommMonoid C\ng : A \u2192+ C\nhg : IsAddUnit (\u2191g x)\ny : { x_1 // x_1 \u2208 multiples x }\nn : \u2115\nhn : (fun i => i \u2022 x) n = \u2191y\n\u22a2 IsAddUnit (\u2191g ((fun i => i \u2022 x) n))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nA : Type u_4\ninst\u271d\u00b2 : AddCommMonoid A\nx : A\nB : Type u_5\ninst\u271d\u00b9 : AddCommMonoid B\nF : AwayMap x B\nC : Type u_6\ninst\u271d : AddCommMonoid C\ng : A \u2192+ C\nhg : IsAddUnit (\u2191g x)\ny : { x_1 // x_1 \u2208 multiples x }\nn : \u2115\nhn : (fun i => i \u2022 x) n = \u2191y\n\u22a2 IsAddUnit (\u2191g (n \u2022 x))\n[PROOFSTEP]\nrw [g.map_nsmul]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nA : Type u_4\ninst\u271d\u00b2 : AddCommMonoid A\nx : A\nB : Type u_5\ninst\u271d\u00b9 : AddCommMonoid B\nF : AwayMap x B\nC : Type u_6\ninst\u271d : AddCommMonoid C\ng : A \u2192+ C\nhg : IsAddUnit (\u2191g x)\ny : { x_1 // x_1 \u2208 multiples x }\nn : \u2115\nhn : (fun i => i \u2022 x) n = \u2191y\n\u22a2 IsAddUnit (n \u2022 \u2191g x)\n[PROOFSTEP]\nexact IsAddUnit.map (nsmulAddMonoidHom n : C \u2192+ C) hg\n[GOAL]\nM : Type u_1\ninst\u271d\u2075 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u2074 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b3 : CommMonoid P\nA : Type u_4\ninst\u271d\u00b2 : AddCommMonoid A\nx : A\nB : Type u_5\ninst\u271d\u00b9 : AddCommMonoid B\nF : AwayMap x B\nC : Type u_6\ninst\u271d : AddCommMonoid C\ng : A \u2192+ C\ny : A\nG : AwayMap (x + y) C\n\u22a2 \u2191(toMap G) x + mk' G y { val := x + y, property := (_ : x + y \u2208 multiples (x + y)) } = 0\n[PROOFSTEP]\nrw [add_mk'_eq_mk'_of_add, mk'_self]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : N \u2243* P\nv : P\nz : N\nhz : \u2191k.toEquiv z = v\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 v * \u2191k (\u2191(toMap f) \u2191x.snd) = \u2191k (\u2191(toMap f) x.fst)\n[PROOFSTEP]\nrw [\u2190 hx, k.map_mul, \u2190 hz]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : N \u2243* P\nv : P\nz : N\nhz : \u2191k.toEquiv z = v\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191k.toEquiv z * \u2191k (\u2191(toMap f) \u2191x.snd) = \u2191k z * \u2191k (\u2191(toMap f) \u2191x.snd)\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : LocalizationMap S P\nx : M\n\u22a2 \u2191(toMap (ofMulEquivOfLocalizations f (mulEquivOfLocalizations f k))) x = \u2191(toMap k) x\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : N \u2243* P\nx : N\n\u22a2 \u2191(mulEquivOfLocalizations f (ofMulEquivOfLocalizations f k)) x = \u2191k x\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\n\u22a2 ofMulEquivOfLocalizations f (MulEquiv.refl N) = f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nx\u271d : M\n\u22a2 \u2191(toMap (ofMulEquivOfLocalizations f (MulEquiv.refl N))) x\u271d = \u2191(toMap f) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : N \u2243* P\nj : P \u2243* Q\n\u22a2 toMap (ofMulEquivOfLocalizations f (MulEquiv.trans k j)) =\n MonoidHom.comp (MulEquiv.toMonoidHom j) (toMap (ofMulEquivOfLocalizations f k))\n[PROOFSTEP]\next\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : N \u2243* P\nj : P \u2243* Q\nx\u271d : M\n\u22a2 \u2191(toMap (ofMulEquivOfLocalizations f (MulEquiv.trans k j))) x\u271d =\n \u2191(MonoidHom.comp (MulEquiv.toMonoidHom j) (toMap (ofMulEquivOfLocalizations f k))) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : P \u2243* M\nH : Submonoid.map (MulEquiv.toMonoidHom k) T = S\nH' : comap (MulEquiv.toMonoidHom k) S = T :=\n H \u25b8 SetLike.coe_injective (Set.preimage_image_eq T.carrier (Equiv.injective k.toEquiv))\nz : N\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\nv : P\nhv : \u2191k.toEquiv v = x.fst\nw : P\nhw : \u2191k.toEquiv w = \u2191x.snd\n\u22a2 z * \u2191(toMap f) (\u2191k.toEquiv w) = \u2191(toMap f) (\u2191k.toEquiv v)\n[PROOFSTEP]\nerw [hv, hw, hx]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : P \u2243* M\nH : Submonoid.map (MulEquiv.toMonoidHom k) T = S\nH' : comap (MulEquiv.toMonoidHom k) S = T :=\n H \u25b8 SetLike.coe_injective (Set.preimage_image_eq T.carrier (Equiv.injective k.toEquiv))\nx y : P\n\u22a2 \u2191(toMap f) (\u2191(MulEquiv.toMonoidHom k) x) = \u2191(toMap f) (\u2191(MulEquiv.toMonoidHom k) y) \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nerw [f.eq_iff_exists]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : P \u2243* M\nH : Submonoid.map (MulEquiv.toMonoidHom k) T = S\nH' : comap (MulEquiv.toMonoidHom k) S = T :=\n H \u25b8 SetLike.coe_injective (Set.preimage_image_eq T.carrier (Equiv.injective k.toEquiv))\nx y : P\n\u22a2 (\u2203 c, \u2191c * \u2191(MulEquiv.toMonoidHom k) x = \u2191c * \u2191(MulEquiv.toMonoidHom k) y) \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nexact\n \u27e8fun \u27e8c, hc\u27e9 \u21a6\n let \u27e8d, hd\u27e9 := k.toEquiv.surjective c\n \u27e8\u27e8d, H' \u25b8 show k d \u2208 S from hd.symm \u25b8 c.2\u27e9, by erw [\u2190 hd, \u2190 k.map_mul, \u2190 k.map_mul] at hc ;\n exact k.toEquiv.injective hc\u27e9,\n fun \u27e8c, hc\u27e9 \u21a6 \u27e8\u27e8k c, H \u25b8 Set.mem_image_of_mem k c.2\u27e9, by erw [\u2190 k.map_mul]; rw [hc, k.map_mul]; rfl\u27e9\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : P \u2243* M\nH : Submonoid.map (MulEquiv.toMonoidHom k) T = S\nH' : comap (MulEquiv.toMonoidHom k) S = T :=\n H \u25b8 SetLike.coe_injective (Set.preimage_image_eq T.carrier (Equiv.injective k.toEquiv))\nx y : P\nx\u271d : \u2203 c, \u2191c * \u2191(MulEquiv.toMonoidHom k) x = \u2191c * \u2191(MulEquiv.toMonoidHom k) y\nc : { x // x \u2208 S }\nhc : \u2191c * \u2191(MulEquiv.toMonoidHom k) x = \u2191c * \u2191(MulEquiv.toMonoidHom k) y\nd : P\nhd : \u2191k.toEquiv d = \u2191c\n\u22a2 \u2191{ val := d, property := (_ : d \u2208 T) } * x = \u2191{ val := d, property := (_ : d \u2208 T) } * y\n[PROOFSTEP]\nerw [\u2190 hd, \u2190 k.map_mul, \u2190 k.map_mul] at hc \n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : P \u2243* M\nH : Submonoid.map (MulEquiv.toMonoidHom k) T = S\nH' : comap (MulEquiv.toMonoidHom k) S = T :=\n H \u25b8 SetLike.coe_injective (Set.preimage_image_eq T.carrier (Equiv.injective k.toEquiv))\nx y : P\nx\u271d : \u2203 c, \u2191c * \u2191(MulEquiv.toMonoidHom k) x = \u2191c * \u2191(MulEquiv.toMonoidHom k) y\nc : { x // x \u2208 S }\nd : P\nhc : \u2191k (d * x) = \u2191k (d * y)\nhd : \u2191k.toEquiv d = \u2191c\n\u22a2 \u2191{ val := d, property := (_ : d \u2208 T) } * x = \u2191{ val := d, property := (_ : d \u2208 T) } * y\n[PROOFSTEP]\nexact k.toEquiv.injective hc\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : P \u2243* M\nH : Submonoid.map (MulEquiv.toMonoidHom k) T = S\nH' : comap (MulEquiv.toMonoidHom k) S = T :=\n H \u25b8 SetLike.coe_injective (Set.preimage_image_eq T.carrier (Equiv.injective k.toEquiv))\nx y : P\nx\u271d : \u2203 c, \u2191c * x = \u2191c * y\nc : { x // x \u2208 T }\nhc : \u2191c * x = \u2191c * y\n\u22a2 \u2191{ val := \u2191k \u2191c, property := (_ : \u2191k \u2191c \u2208 S) } * \u2191(MulEquiv.toMonoidHom k) x =\n \u2191{ val := \u2191k \u2191c, property := (_ : \u2191k \u2191c \u2208 S) } * \u2191(MulEquiv.toMonoidHom k) y\n[PROOFSTEP]\nerw [\u2190 k.map_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : P \u2243* M\nH : Submonoid.map (MulEquiv.toMonoidHom k) T = S\nH' : comap (MulEquiv.toMonoidHom k) S = T :=\n H \u25b8 SetLike.coe_injective (Set.preimage_image_eq T.carrier (Equiv.injective k.toEquiv))\nx y : P\nx\u271d : \u2203 c, \u2191c * x = \u2191c * y\nc : { x // x \u2208 T }\nhc : \u2191c * x = \u2191c * y\n\u22a2 \u2191k (\u2191c * x) = \u2191{ val := \u2191k \u2191c, property := (_ : \u2191k \u2191c \u2208 S) } * \u2191(MulEquiv.toMonoidHom k) y\n[PROOFSTEP]\nrw [hc, k.map_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nk : P \u2243* M\nH : Submonoid.map (MulEquiv.toMonoidHom k) T = S\nH' : comap (MulEquiv.toMonoidHom k) S = T :=\n H \u25b8 SetLike.coe_injective (Set.preimage_image_eq T.carrier (Equiv.injective k.toEquiv))\nx y : P\nx\u271d : \u2203 c, \u2191c * x = \u2191c * y\nc : { x // x \u2208 T }\nhc : \u2191c * x = \u2191c * y\n\u22a2 \u2191k \u2191c * \u2191k y = \u2191{ val := \u2191k \u2191c, property := (_ : \u2191k \u2191c \u2208 S) } * \u2191(MulEquiv.toMonoidHom k) y\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\n\u22a2 ofMulEquivOfDom f (_ : Submonoid.map (MulEquiv.toMonoidHom (MulEquiv.refl M)) S = S) = f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_3\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d : CommMonoid Q\nx\u271d : M\n\u22a2 \u2191(toMap (ofMulEquivOfDom f (_ : Submonoid.map (MulEquiv.toMonoidHom (MulEquiv.refl M)) S = S))) x\u271d = \u2191(toMap f) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nsrc\u271d : M \u2192* Con.Quotient (r S) := MonoidHom.comp (Con.mk' (r S)) (MonoidHom.inl M { x // x \u2208 S })\nx y : M\n\u22a2 OneHom.toFun { toFun := fun x => mk x 1, map_one' := (_ : mk 1 1 = 1) } (x * y) =\n OneHom.toFun { toFun := fun x => mk x 1, map_one' := (_ : mk 1 1 = 1) } x *\n OneHom.toFun { toFun := fun x => mk x 1, map_one' := (_ : mk 1 1 = 1) } y\n[PROOFSTEP]\ndsimp only\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nsrc\u271d : M \u2192* Con.Quotient (r S) := MonoidHom.comp (Con.mk' (r S)) (MonoidHom.inl M { x // x \u2208 S })\nx y : M\n\u22a2 mk (x * y) 1 = mk x 1 * mk y 1\n[PROOFSTEP]\nrw [mk_mul, mul_one]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nsrc\u271d : M \u2192* Con.Quotient (r S) := MonoidHom.comp (Con.mk' (r S)) (MonoidHom.inl M { x // x \u2208 S })\ny : { x // x \u2208 S }\n\u22a2 OneHom.toFun\n \u2191{ toOneHom := { toFun := fun x => mk x 1, map_one' := (_ : mk 1 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : M),\n OneHom.toFun { toFun := fun x => mk x 1, map_one' := (_ : mk 1 1 = 1) } (x * y) =\n OneHom.toFun { toFun := fun x => mk x 1, map_one' := (_ : mk 1 1 = 1) } x *\n OneHom.toFun { toFun := fun x => mk x 1, map_one' := (_ : mk 1 1 = 1) } y) }\n \u2191y *\n mk 1 y =\n 1\n[PROOFSTEP]\ndsimp only\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nsrc\u271d : M \u2192* Con.Quotient (r S) := MonoidHom.comp (Con.mk' (r S)) (MonoidHom.inl M { x // x \u2208 S })\ny : { x // x \u2208 S }\n\u22a2 mk (\u2191y) 1 * mk 1 y = 1\n[PROOFSTEP]\nrw [mk_mul, mul_one, one_mul, mk_self]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nsrc\u271d : M \u2192* Con.Quotient (r S) := MonoidHom.comp (Con.mk' (r S)) (MonoidHom.inl M { x // x \u2208 S })\nz : Localization S\nx : M \u00d7 { x // x \u2208 S }\n\u22a2 mk x.fst x.snd *\n OneHom.toFun\n \u2191{ toOneHom := { toFun := fun x => mk x 1, map_one' := (_ : mk 1 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : M),\n OneHom.toFun { toFun := fun x => mk x 1, map_one' := (_ : mk 1 1 = 1) } (x * y) =\n OneHom.toFun { toFun := fun x => mk x 1, map_one' := (_ : mk 1 1 = 1) } x *\n OneHom.toFun { toFun := fun x => mk x 1, map_one' := (_ : mk 1 1 = 1) } y) }\n \u2191x.snd =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := fun x => mk x 1, map_one' := (_ : mk 1 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : M),\n OneHom.toFun { toFun := fun x => mk x 1, map_one' := (_ : mk 1 1 = 1) } (x * y) =\n OneHom.toFun { toFun := fun x => mk x 1, map_one' := (_ : mk 1 1 = 1) } x *\n OneHom.toFun { toFun := fun x => mk x 1, map_one' := (_ : mk 1 1 = 1) } y) })\n x.fst\n[PROOFSTEP]\ndsimp only\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nsrc\u271d : M \u2192* Con.Quotient (r S) := MonoidHom.comp (Con.mk' (r S)) (MonoidHom.inl M { x // x \u2208 S })\nz : Localization S\nx : M \u00d7 { x // x \u2208 S }\n\u22a2 mk x.fst x.snd * mk (\u2191x.snd) 1 = mk x.fst 1\n[PROOFSTEP]\nrw [mk_mul, mul_comm x.fst, \u2190 mk_mul, mk_self, one_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nsrc\u271d : M \u2192* Con.Quotient (r S) := MonoidHom.comp (Con.mk' (r S)) (MonoidHom.inl M { x // x \u2208 S })\nx y : M\n\u22a2 (\u2203 c, \u2191c * (1 * x) = \u2191c * (1 * y)) \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nrw [one_mul, one_mul]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx : M\ny : { x // x \u2208 S }\n\u22a2 mk x y * \u2191(Submonoid.LocalizationMap.toMap (monoidOf S)) \u2191y = \u2191(Submonoid.LocalizationMap.toMap (monoidOf S)) x\n[PROOFSTEP]\nrw [\u2190 mk_one_eq_monoidOf_mk, \u2190 mk_one_eq_monoidOf_mk, mk_mul x y y 1, mul_comm y 1]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx : M\ny : { x // x \u2208 S }\n\u22a2 mk (x * \u2191y) (1 * y) = mk x 1\n[PROOFSTEP]\nconv => rhs; rw [\u2190 mul_one 1]; rw [\u2190 mul_one x]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx : M\ny : { x // x \u2208 S }\n| mk (x * \u2191y) (1 * y) = mk x 1\n[PROOFSTEP]\nrhs; rw [\u2190 mul_one 1]; rw [\u2190 mul_one x]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx : M\ny : { x // x \u2208 S }\n| mk (x * \u2191y) (1 * y) = mk x 1\n[PROOFSTEP]\nrhs; rw [\u2190 mul_one 1]; rw [\u2190 mul_one x]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx : M\ny : { x // x \u2208 S }\n| mk (x * \u2191y) (1 * y) = mk x 1\n[PROOFSTEP]\nrhs\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx : M\ny : { x // x \u2208 S }\n| mk x 1\n[PROOFSTEP]\nrw [\u2190 mul_one 1]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx : M\ny : { x // x \u2208 S }\n| mk x (1 * 1)\n[PROOFSTEP]\nrw [\u2190 mul_one x]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nx : M\ny : { x // x \u2208 S }\n\u22a2 mk (x * \u2191y) (1 * y) = mk (x * 1) (1 * 1)\n[PROOFSTEP]\nexact mk_eq_mk_iff.2 (Con.symm _ <| (Localization.r S).mul (Con.refl _ (x, 1)) <| one_rel _)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\np : Sort u\nf : M \u2192 { x // x \u2208 S } \u2192 p\nH : \u2200 {a c : M} {b d : { x // x \u2208 S }}, \u2191(r S) (a, b) (c, d) \u2192 f a b = f c d\na : M\nb : { x // x \u2208 S }\n\u22a2 liftOn (Submonoid.LocalizationMap.mk' (monoidOf S) a b) f H = f a b\n[PROOFSTEP]\nrw [\u2190 mk_eq_monoidOf_mk', liftOn_mk]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\np : Sort u_4\nf : M \u2192 { x // x \u2208 S } \u2192 M \u2192 { x // x \u2208 S } \u2192 p\nH :\n \u2200 {a a' : M} {b b' : { x // x \u2208 S }} {c c' : M} {d d' : { x // x \u2208 S }},\n \u2191(r S) (a, b) (a', b') \u2192 \u2191(r S) (c, d) (c', d') \u2192 f a b c d = f a' b' c' d'\na c : M\nb d : { x // x \u2208 S }\n\u22a2 liftOn\u2082 (Submonoid.LocalizationMap.mk' (monoidOf S) a b) (Submonoid.LocalizationMap.mk' (monoidOf S) c d) f H =\n f a b c d\n[PROOFSTEP]\nrw [\u2190 mk_eq_monoidOf_mk', liftOn\u2082_mk]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : Submonoid.LocalizationMap S N\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(mulEquivOfQuotient f) (mk x y) = Submonoid.LocalizationMap.mk' f x y\n[PROOFSTEP]\nrw [mk_eq_monoidOf_mk'_apply]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : Submonoid.LocalizationMap S N\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(mulEquivOfQuotient f) (Submonoid.LocalizationMap.mk' (monoidOf S) x y) = Submonoid.LocalizationMap.mk' f x y\n[PROOFSTEP]\nexact mulEquivOfQuotient_mk' _ _\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : Submonoid.LocalizationMap S N\nx : M\n\u22a2 \u2191(mulEquivOfQuotient f) (\u2191(Submonoid.LocalizationMap.toMap (monoidOf S)) x) = \u2191(Submonoid.LocalizationMap.toMap f) x\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : Submonoid.LocalizationMap S N\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(MulEquiv.symm (mulEquivOfQuotient f)) (Submonoid.LocalizationMap.mk' f x y) = mk x y\n[PROOFSTEP]\nrw [mk_eq_monoidOf_mk'_apply]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : Submonoid.LocalizationMap S N\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(MulEquiv.symm (mulEquivOfQuotient f)) (Submonoid.LocalizationMap.mk' f x y) =\n Submonoid.LocalizationMap.mk' (monoidOf S) x y\n[PROOFSTEP]\nexact mulEquivOfQuotient_symm_mk' _ _\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : Submonoid.LocalizationMap S N\nx : M\n\u22a2 mk = Submonoid.LocalizationMap.mk' (monoidOf x)\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoidWithZero N\nP : Type u_3\ninst\u271d : CommMonoidWithZero P\nx : { x // x \u2208 S }\n\u22a2 \u2191(0, 1).snd * (0, x).fst = \u2191(0, x).snd * (0, 1).fst\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoidWithZero N\nP : Type u_3\ninst\u271d : CommMonoidWithZero P\nx : Localization S\ny : M \u00d7 { x // x \u2208 S }\n\u22a2 0 * mk y.fst y.snd = 0\n[PROOFSTEP]\nsimp only [\u2190 Localization.mk_zero y.2, mk_mul, mk_eq_mk_iff, mul_zero, zero_mul, r_of_eq]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoidWithZero N\nP : Type u_3\ninst\u271d : CommMonoidWithZero P\nx : Localization S\ny : M \u00d7 { x // x \u2208 S }\n\u22a2 mk y.fst y.snd * 0 = 0\n[PROOFSTEP]\nsimp only [\u2190 Localization.mk_zero y.2, mk_mul, mk_eq_mk_iff, mul_zero, zero_mul, r_of_eq]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoidWithZero N\nP : Type u_3\ninst\u271d : CommMonoidWithZero P\np : Type u_4\nf : M \u2192 { x // x \u2208 S } \u2192 p\nH : \u2200 {a c : M} {b d : { x // x \u2208 S }}, \u2191(r S) (a, b) (c, d) \u2192 f a b = f c d\n\u22a2 liftOn 0 f H = f 0 1\n[PROOFSTEP]\nrw [\u2190 mk_zero 1, liftOn_mk]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoidWithZero N\nP : Type u_3\ninst\u271d : CommMonoidWithZero P\nf : LocalizationMap S N\n\u22a2 \u2191(toMap f) (sec f 0).fst = 0\n[PROOFSTEP]\nrw [LocalizationMap.sec_spec', mul_zero]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoidWithZero N\nP : Type u_3\ninst\u271d : CommMonoidWithZero P\nf : LocalizationWithZeroMap S N\ng : M \u2192*\u2080 P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nsrc\u271d : N \u2192* P := LocalizationMap.lift f.toLocalizationMap hg\n\u22a2 OneHom.toFun (\u2191src\u271d) 0 = 0\n[PROOFSTEP]\nerw [LocalizationMap.lift_spec f.toLocalizationMap hg 0 0]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoidWithZero N\nP : Type u_3\ninst\u271d : CommMonoidWithZero P\nf : LocalizationWithZeroMap S N\ng : M \u2192*\u2080 P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nsrc\u271d : N \u2192* P := LocalizationMap.lift f.toLocalizationMap hg\n\u22a2 \u2191\u2191g (LocalizationMap.sec f.toLocalizationMap 0).fst = \u2191\u2191g \u2191(LocalizationMap.sec f.toLocalizationMap 0).snd * 0\n[PROOFSTEP]\nrw [mul_zero, \u2190 map_zero g, \u2190 g.toMonoidHom_coe]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoidWithZero N\nP : Type u_3\ninst\u271d : CommMonoidWithZero P\nf : LocalizationWithZeroMap S N\ng : M \u2192*\u2080 P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nsrc\u271d : N \u2192* P := LocalizationMap.lift f.toLocalizationMap hg\n\u22a2 \u2191\u2191g (LocalizationMap.sec f.toLocalizationMap 0).fst = OneHom.toFun (\u2191\u2191g) 0\n[PROOFSTEP]\nrefine f.toLocalizationMap.eq_of_eq hg ?_\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoidWithZero N\nP : Type u_3\ninst\u271d : CommMonoidWithZero P\nf : LocalizationWithZeroMap S N\ng : M \u2192*\u2080 P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nsrc\u271d : N \u2192* P := LocalizationMap.lift f.toLocalizationMap hg\n\u22a2 \u2191(LocalizationMap.toMap f.toLocalizationMap) (LocalizationMap.sec f.toLocalizationMap 0).fst =\n \u2191(LocalizationMap.toMap f.toLocalizationMap) 0\n[PROOFSTEP]\nrw [LocalizationMap.sec_zero_fst]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoidWithZero N\nP : Type u_3\ninst\u271d : CommMonoidWithZero P\nf : LocalizationWithZeroMap S N\ng : M \u2192*\u2080 P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nsrc\u271d : N \u2192* P := LocalizationMap.lift f.toLocalizationMap hg\n\u22a2 0 = \u2191(LocalizationMap.toMap f.toLocalizationMap) 0\n[PROOFSTEP]\nexact f.toMonoidWithZeroHom.map_zero.symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 b : { x // x \u2208 s }\nc d : \u03b1\nh : (fun a => mk a b) c = (fun a => mk a b) d\n\u22a2 c = d\n[PROOFSTEP]\nhave : Nonempty s := One.nonempty\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 b : { x // x \u2208 s }\nc d : \u03b1\nh : (fun a => mk a b) c = (fun a => mk a b) d\nthis : Nonempty { x // x \u2208 s }\n\u22a2 c = d\n[PROOFSTEP]\nsimpa [-mk_eq_monoidOf_mk', mk_eq_mk_iff, r_iff_exists] using h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\n\u22a2 mk a\u2081 a\u2082 = mk b\u2081 b\u2082 \u2194 \u2191b\u2082 * a\u2081 = \u2191a\u2082 * b\u2081\n[PROOFSTEP]\nhave : Nonempty s := One.nonempty\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\nthis : Nonempty { x // x \u2208 s }\n\u22a2 mk a\u2081 a\u2082 = mk b\u2081 b\u2082 \u2194 \u2191b\u2082 * a\u2081 = \u2191a\u2082 * b\u2081\n[PROOFSTEP]\nsimp_rw [mk_eq_mk_iff, r_iff_exists, mul_left_cancel_iff, exists_const]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081\u271d : \u03b1\na\u2082\u271d b\u2082\u271d : { x // x \u2208 s }\na b : Localization s\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\nc\u2081 d\u2081 : \u03b1\nc\u2082 d\u2082 : { x // x \u2208 s }\nhab : \u2191(r s) (a\u2081, a\u2082) (b\u2081, b\u2082)\nhcd : \u2191(r s) (c\u2081, c\u2082) (d\u2081, d\u2082)\n\u22a2 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 \u2264 \u2191a\u2082 * b\u2081) a\u2081 a\u2082 c\u2081 c\u2082 \u2194 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 \u2264 \u2191a\u2082 * b\u2081) b\u2081 b\u2082 d\u2081 d\u2082\n[PROOFSTEP]\nobtain \u27e8e, he\u27e9 := r_iff_exists.1 hab\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081\u271d : \u03b1\na\u2082\u271d b\u2082\u271d : { x // x \u2208 s }\na b : Localization s\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\nc\u2081 d\u2081 : \u03b1\nc\u2082 d\u2082 : { x // x \u2208 s }\nhab : \u2191(r s) (a\u2081, a\u2082) (b\u2081, b\u2082)\nhcd : \u2191(r s) (c\u2081, c\u2082) (d\u2081, d\u2082)\ne : { x // x \u2208 s }\nhe : \u2191e * (\u2191(b\u2081, b\u2082).snd * (a\u2081, a\u2082).fst) = \u2191e * (\u2191(a\u2081, a\u2082).snd * (b\u2081, b\u2082).fst)\n\u22a2 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 \u2264 \u2191a\u2082 * b\u2081) a\u2081 a\u2082 c\u2081 c\u2082 \u2194 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 \u2264 \u2191a\u2082 * b\u2081) b\u2081 b\u2082 d\u2081 d\u2082\n[PROOFSTEP]\nobtain \u27e8f, hf\u27e9 := r_iff_exists.1 hcd\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081\u271d : \u03b1\na\u2082\u271d b\u2082\u271d : { x // x \u2208 s }\na b : Localization s\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\nc\u2081 d\u2081 : \u03b1\nc\u2082 d\u2082 : { x // x \u2208 s }\nhab : \u2191(r s) (a\u2081, a\u2082) (b\u2081, b\u2082)\nhcd : \u2191(r s) (c\u2081, c\u2082) (d\u2081, d\u2082)\ne : { x // x \u2208 s }\nhe : \u2191e * (\u2191(b\u2081, b\u2082).snd * (a\u2081, a\u2082).fst) = \u2191e * (\u2191(a\u2081, a\u2082).snd * (b\u2081, b\u2082).fst)\nf : { x // x \u2208 s }\nhf : \u2191f * (\u2191(d\u2081, d\u2082).snd * (c\u2081, c\u2082).fst) = \u2191f * (\u2191(c\u2081, c\u2082).snd * (d\u2081, d\u2082).fst)\n\u22a2 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 \u2264 \u2191a\u2082 * b\u2081) a\u2081 a\u2082 c\u2081 c\u2082 \u2194 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 \u2264 \u2191a\u2082 * b\u2081) b\u2081 b\u2082 d\u2081 d\u2082\n[PROOFSTEP]\nsimp only [mul_right_inj] at he hf \n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081\u271d : \u03b1\na\u2082\u271d b\u2082\u271d : { x // x \u2208 s }\na b : Localization s\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\nc\u2081 d\u2081 : \u03b1\nc\u2082 d\u2082 : { x // x \u2208 s }\nhab : \u2191(r s) (a\u2081, a\u2082) (b\u2081, b\u2082)\nhcd : \u2191(r s) (c\u2081, c\u2082) (d\u2081, d\u2082)\ne f : { x // x \u2208 s }\nhe : \u2191b\u2082 * a\u2081 = \u2191a\u2082 * b\u2081\nhf : \u2191d\u2082 * c\u2081 = \u2191c\u2082 * d\u2081\n\u22a2 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 \u2264 \u2191a\u2082 * b\u2081) a\u2081 a\u2082 c\u2081 c\u2082 \u2194 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 \u2264 \u2191a\u2082 * b\u2081) b\u2081 b\u2082 d\u2081 d\u2082\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081\u271d : \u03b1\na\u2082\u271d b\u2082\u271d : { x // x \u2208 s }\na b : Localization s\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\nc\u2081 d\u2081 : \u03b1\nc\u2082 d\u2082 : { x // x \u2208 s }\nhab : \u2191(r s) (a\u2081, a\u2082) (b\u2081, b\u2082)\nhcd : \u2191(r s) (c\u2081, c\u2082) (d\u2081, d\u2082)\ne f : { x // x \u2208 s }\nhe : \u2191b\u2082 * a\u2081 = \u2191a\u2082 * b\u2081\nhf : \u2191d\u2082 * c\u2081 = \u2191c\u2082 * d\u2081\n\u22a2 \u2191c\u2082 * a\u2081 \u2264 \u2191a\u2082 * c\u2081 \u2194 \u2191d\u2082 * b\u2081 \u2264 \u2191b\u2082 * d\u2081\n[PROOFSTEP]\nrw [\u2190 mul_le_mul_iff_right, mul_right_comm, \u2190 hf, mul_right_comm, mul_right_comm (a\u2082 : \u03b1), mul_le_mul_iff_right, \u2190\n mul_le_mul_iff_left, mul_left_comm, he, mul_left_comm, mul_left_comm (b\u2082 : \u03b1), mul_le_mul_iff_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081\u271d : \u03b1\na\u2082\u271d b\u2082\u271d : { x // x \u2208 s }\na b : Localization s\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\nc\u2081 d\u2081 : \u03b1\nc\u2082 d\u2082 : { x // x \u2208 s }\nhab : \u2191(r s) (a\u2081, a\u2082) (b\u2081, b\u2082)\nhcd : \u2191(r s) (c\u2081, c\u2082) (d\u2081, d\u2082)\n\u22a2 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 < \u2191a\u2082 * b\u2081) a\u2081 a\u2082 c\u2081 c\u2082 \u2194 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 < \u2191a\u2082 * b\u2081) b\u2081 b\u2082 d\u2081 d\u2082\n[PROOFSTEP]\nobtain \u27e8e, he\u27e9 := r_iff_exists.1 hab\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081\u271d : \u03b1\na\u2082\u271d b\u2082\u271d : { x // x \u2208 s }\na b : Localization s\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\nc\u2081 d\u2081 : \u03b1\nc\u2082 d\u2082 : { x // x \u2208 s }\nhab : \u2191(r s) (a\u2081, a\u2082) (b\u2081, b\u2082)\nhcd : \u2191(r s) (c\u2081, c\u2082) (d\u2081, d\u2082)\ne : { x // x \u2208 s }\nhe : \u2191e * (\u2191(b\u2081, b\u2082).snd * (a\u2081, a\u2082).fst) = \u2191e * (\u2191(a\u2081, a\u2082).snd * (b\u2081, b\u2082).fst)\n\u22a2 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 < \u2191a\u2082 * b\u2081) a\u2081 a\u2082 c\u2081 c\u2082 \u2194 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 < \u2191a\u2082 * b\u2081) b\u2081 b\u2082 d\u2081 d\u2082\n[PROOFSTEP]\nobtain \u27e8f, hf\u27e9 := r_iff_exists.1 hcd\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081\u271d : \u03b1\na\u2082\u271d b\u2082\u271d : { x // x \u2208 s }\na b : Localization s\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\nc\u2081 d\u2081 : \u03b1\nc\u2082 d\u2082 : { x // x \u2208 s }\nhab : \u2191(r s) (a\u2081, a\u2082) (b\u2081, b\u2082)\nhcd : \u2191(r s) (c\u2081, c\u2082) (d\u2081, d\u2082)\ne : { x // x \u2208 s }\nhe : \u2191e * (\u2191(b\u2081, b\u2082).snd * (a\u2081, a\u2082).fst) = \u2191e * (\u2191(a\u2081, a\u2082).snd * (b\u2081, b\u2082).fst)\nf : { x // x \u2208 s }\nhf : \u2191f * (\u2191(d\u2081, d\u2082).snd * (c\u2081, c\u2082).fst) = \u2191f * (\u2191(c\u2081, c\u2082).snd * (d\u2081, d\u2082).fst)\n\u22a2 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 < \u2191a\u2082 * b\u2081) a\u2081 a\u2082 c\u2081 c\u2082 \u2194 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 < \u2191a\u2082 * b\u2081) b\u2081 b\u2082 d\u2081 d\u2082\n[PROOFSTEP]\nsimp only [mul_right_inj] at he hf \n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081\u271d : \u03b1\na\u2082\u271d b\u2082\u271d : { x // x \u2208 s }\na b : Localization s\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\nc\u2081 d\u2081 : \u03b1\nc\u2082 d\u2082 : { x // x \u2208 s }\nhab : \u2191(r s) (a\u2081, a\u2082) (b\u2081, b\u2082)\nhcd : \u2191(r s) (c\u2081, c\u2082) (d\u2081, d\u2082)\ne f : { x // x \u2208 s }\nhe : \u2191b\u2082 * a\u2081 = \u2191a\u2082 * b\u2081\nhf : \u2191d\u2082 * c\u2081 = \u2191c\u2082 * d\u2081\n\u22a2 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 < \u2191a\u2082 * b\u2081) a\u2081 a\u2082 c\u2081 c\u2082 \u2194 (fun a\u2081 a\u2082 b\u2081 b\u2082 => \u2191b\u2082 * a\u2081 < \u2191a\u2082 * b\u2081) b\u2081 b\u2082 d\u2081 d\u2082\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081\u271d : \u03b1\na\u2082\u271d b\u2082\u271d : { x // x \u2208 s }\na b : Localization s\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\nc\u2081 d\u2081 : \u03b1\nc\u2082 d\u2082 : { x // x \u2208 s }\nhab : \u2191(r s) (a\u2081, a\u2082) (b\u2081, b\u2082)\nhcd : \u2191(r s) (c\u2081, c\u2082) (d\u2081, d\u2082)\ne f : { x // x \u2208 s }\nhe : \u2191b\u2082 * a\u2081 = \u2191a\u2082 * b\u2081\nhf : \u2191d\u2082 * c\u2081 = \u2191c\u2082 * d\u2081\n\u22a2 \u2191c\u2082 * a\u2081 < \u2191a\u2082 * c\u2081 \u2194 \u2191d\u2082 * b\u2081 < \u2191b\u2082 * d\u2081\n[PROOFSTEP]\nrw [\u2190 mul_lt_mul_iff_right, mul_right_comm, \u2190 hf, mul_right_comm, mul_right_comm (a\u2082 : \u03b1), mul_lt_mul_iff_right, \u2190\n mul_lt_mul_iff_left, mul_left_comm, he, mul_left_comm, mul_left_comm (b\u2082 : \u03b1), mul_lt_mul_iff_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\na\u271d b\u271d c\u271d : Localization s\na b c : \u03b1 \u00d7 { x // x \u2208 s }\nhab : mk a.fst a.snd \u2264 mk b.fst b.snd\nhbc : mk b.fst b.snd \u2264 mk c.fst c.snd\n\u22a2 mk a.fst a.snd \u2264 mk c.fst c.snd\n[PROOFSTEP]\nsimp only [mk_le_mk] at hab hbc \u22a2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\na\u271d b\u271d c\u271d : Localization s\na b c : \u03b1 \u00d7 { x // x \u2208 s }\nhab : \u2191b.snd * a.fst \u2264 \u2191a.snd * b.fst\nhbc : \u2191c.snd * b.fst \u2264 \u2191b.snd * c.fst\n\u22a2 \u2191c.snd * a.fst \u2264 \u2191a.snd * c.fst\n[PROOFSTEP]\nrefine' le_of_mul_le_mul_left' _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\na\u271d b\u271d c\u271d : Localization s\na b c : \u03b1 \u00d7 { x // x \u2208 s }\nhab : \u2191b.snd * a.fst \u2264 \u2191a.snd * b.fst\nhbc : \u2191c.snd * b.fst \u2264 \u2191b.snd * c.fst\n\u22a2 \u03b1\n[PROOFSTEP]\nexact \u2191b.2\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\na\u271d b\u271d c\u271d : Localization s\na b c : \u03b1 \u00d7 { x // x \u2208 s }\nhab : \u2191b.snd * a.fst \u2264 \u2191a.snd * b.fst\nhbc : \u2191c.snd * b.fst \u2264 \u2191b.snd * c.fst\n\u22a2 \u2191b.snd * (\u2191c.snd * a.fst) \u2264 \u2191b.snd * (\u2191a.snd * c.fst)\n[PROOFSTEP]\nrw [mul_left_comm]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\na\u271d b\u271d c\u271d : Localization s\na b c : \u03b1 \u00d7 { x // x \u2208 s }\nhab : \u2191b.snd * a.fst \u2264 \u2191a.snd * b.fst\nhbc : \u2191c.snd * b.fst \u2264 \u2191b.snd * c.fst\n\u22a2 \u2191c.snd * (\u2191b.snd * a.fst) \u2264 \u2191b.snd * (\u2191a.snd * c.fst)\n[PROOFSTEP]\nrefine' (mul_le_mul_left' hab _).trans _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\na\u271d b\u271d c\u271d : Localization s\na b c : \u03b1 \u00d7 { x // x \u2208 s }\nhab : \u2191b.snd * a.fst \u2264 \u2191a.snd * b.fst\nhbc : \u2191c.snd * b.fst \u2264 \u2191b.snd * c.fst\n\u22a2 \u2191c.snd * (\u2191a.snd * b.fst) \u2264 \u2191b.snd * (\u2191a.snd * c.fst)\n[PROOFSTEP]\nrwa [mul_left_comm, mul_left_comm (b.2 : \u03b1), mul_le_mul_iff_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\na b : Localization s\n\u22a2 a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\ninduction' a using Localization.rec with a\u2081 a\u2082\n[GOAL]\ncase f\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081 : \u03b1\na\u2082\u271d b\u2082 : { x // x \u2208 s }\nb : Localization s\na\u2081 : \u03b1\na\u2082 : { x // x \u2208 s }\n\u22a2 mk a\u2081 a\u2082 \u2264 b \u2192 b \u2264 mk a\u2081 a\u2082 \u2192 mk a\u2081 a\u2082 = b\ncase H\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\na b : Localization s\na\u271d c\u271d : \u03b1\nb\u271d d\u271d : { x // x \u2208 s }\nh\u271d : \u2191(r s) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ : mk c\u271d d\u271d \u2264 b \u2192 b \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d = b) = (_ : mk c\u271d d\u271d \u2264 b \u2192 b \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d = b)\n[PROOFSTEP]\ninduction' b using Localization.rec with b\u2081 b\u2082\n[GOAL]\ncase f.f\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081\u271d : \u03b1\na\u2082\u271d b\u2082\u271d : { x // x \u2208 s }\na\u2081 : \u03b1\na\u2082 : { x // x \u2208 s }\nb\u2081 : \u03b1\nb\u2082 : { x // x \u2208 s }\n\u22a2 mk a\u2081 a\u2082 \u2264 mk b\u2081 b\u2082 \u2192 mk b\u2081 b\u2082 \u2264 mk a\u2081 a\u2082 \u2192 mk a\u2081 a\u2082 = mk b\u2081 b\u2082\ncase f.H\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081 : \u03b1\na\u2082\u271d b\u2082 : { x // x \u2208 s }\nb : Localization s\na\u2081 : \u03b1\na\u2082 : { x // x \u2208 s }\na\u271d c\u271d : \u03b1\nb\u271d d\u271d : { x // x \u2208 s }\nh\u271d : \u2191(r s) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ : mk a\u2081 a\u2082 \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d \u2264 mk a\u2081 a\u2082 \u2192 mk a\u2081 a\u2082 = mk c\u271d d\u271d) =\n (_ : mk a\u2081 a\u2082 \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d \u2264 mk a\u2081 a\u2082 \u2192 mk a\u2081 a\u2082 = mk c\u271d d\u271d)\ncase H\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\na b : Localization s\na\u271d c\u271d : \u03b1\nb\u271d d\u271d : { x // x \u2208 s }\nh\u271d : \u2191(r s) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ : mk c\u271d d\u271d \u2264 b \u2192 b \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d = b) = (_ : mk c\u271d d\u271d \u2264 b \u2192 b \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d = b)\n[PROOFSTEP]\nsimp_rw [mk_le_mk, mk_eq_mk_iff, r_iff_exists]\n[GOAL]\ncase f.f\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081\u271d : \u03b1\na\u2082\u271d b\u2082\u271d : { x // x \u2208 s }\na\u2081 : \u03b1\na\u2082 : { x // x \u2208 s }\nb\u2081 : \u03b1\nb\u2082 : { x // x \u2208 s }\n\u22a2 \u2191b\u2082 * a\u2081 \u2264 \u2191a\u2082 * b\u2081 \u2192 \u2191a\u2082 * b\u2081 \u2264 \u2191b\u2082 * a\u2081 \u2192 \u2203 c, \u2191c * (\u2191b\u2082 * a\u2081) = \u2191c * (\u2191a\u2082 * b\u2081)\ncase f.H\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081 : \u03b1\na\u2082\u271d b\u2082 : { x // x \u2208 s }\nb : Localization s\na\u2081 : \u03b1\na\u2082 : { x // x \u2208 s }\na\u271d c\u271d : \u03b1\nb\u271d d\u271d : { x // x \u2208 s }\nh\u271d : \u2191(r s) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ : mk a\u2081 a\u2082 \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d \u2264 mk a\u2081 a\u2082 \u2192 mk a\u2081 a\u2082 = mk c\u271d d\u271d) =\n (_ : mk a\u2081 a\u2082 \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d \u2264 mk a\u2081 a\u2082 \u2192 mk a\u2081 a\u2082 = mk c\u271d d\u271d)\ncase H\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\na b : Localization s\na\u271d c\u271d : \u03b1\nb\u271d d\u271d : { x // x \u2208 s }\nh\u271d : \u2191(r s) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ : mk c\u271d d\u271d \u2264 b \u2192 b \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d = b) = (_ : mk c\u271d d\u271d \u2264 b \u2192 b \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d = b)\n[PROOFSTEP]\nexact fun hab hba => \u27e81, by rw [hab.antisymm hba]\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081\u271d : \u03b1\na\u2082\u271d b\u2082\u271d : { x // x \u2208 s }\na\u2081 : \u03b1\na\u2082 : { x // x \u2208 s }\nb\u2081 : \u03b1\nb\u2082 : { x // x \u2208 s }\nhab : \u2191b\u2082 * a\u2081 \u2264 \u2191a\u2082 * b\u2081\nhba : \u2191a\u2082 * b\u2081 \u2264 \u2191b\u2082 * a\u2081\n\u22a2 \u21911 * (\u2191b\u2082 * a\u2081) = \u21911 * (\u2191a\u2082 * b\u2081)\n[PROOFSTEP]\nrw [hab.antisymm hba]\n[GOAL]\ncase f.H\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081 : \u03b1\na\u2082\u271d b\u2082 : { x // x \u2208 s }\nb : Localization s\na\u2081 : \u03b1\na\u2082 : { x // x \u2208 s }\na\u271d c\u271d : \u03b1\nb\u271d d\u271d : { x // x \u2208 s }\nh\u271d : \u2191(r s) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ : mk a\u2081 a\u2082 \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d \u2264 mk a\u2081 a\u2082 \u2192 mk a\u2081 a\u2082 = mk c\u271d d\u271d) =\n (_ : mk a\u2081 a\u2082 \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d \u2264 mk a\u2081 a\u2082 \u2192 mk a\u2081 a\u2082 = mk c\u271d d\u271d)\ncase H\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\na b : Localization s\na\u271d c\u271d : \u03b1\nb\u271d d\u271d : { x // x \u2208 s }\nh\u271d : \u2191(r s) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ : mk c\u271d d\u271d \u2264 b \u2192 b \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d = b) = (_ : mk c\u271d d\u271d \u2264 b \u2192 b \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d = b)\n[PROOFSTEP]\nall_goals intros; rfl\n[GOAL]\ncase f.H\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081 : \u03b1\na\u2082\u271d b\u2082 : { x // x \u2208 s }\nb : Localization s\na\u2081 : \u03b1\na\u2082 : { x // x \u2208 s }\na\u271d c\u271d : \u03b1\nb\u271d d\u271d : { x // x \u2208 s }\nh\u271d : \u2191(r s) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ : mk a\u2081 a\u2082 \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d \u2264 mk a\u2081 a\u2082 \u2192 mk a\u2081 a\u2082 = mk c\u271d d\u271d) =\n (_ : mk a\u2081 a\u2082 \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d \u2264 mk a\u2081 a\u2082 \u2192 mk a\u2081 a\u2082 = mk c\u271d d\u271d)\n[PROOFSTEP]\nintros\n[GOAL]\ncase f.H\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081\u271d b\u2081 : \u03b1\na\u2082\u271d b\u2082 : { x // x \u2208 s }\nb : Localization s\na\u2081 : \u03b1\na\u2082 : { x // x \u2208 s }\na\u271d c\u271d : \u03b1\nb\u271d d\u271d : { x // x \u2208 s }\nh\u271d : \u2191(r s) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ : mk a\u2081 a\u2082 \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d \u2264 mk a\u2081 a\u2082 \u2192 mk a\u2081 a\u2082 = mk c\u271d d\u271d) =\n (_ : mk a\u2081 a\u2082 \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d \u2264 mk a\u2081 a\u2082 \u2192 mk a\u2081 a\u2082 = mk c\u271d d\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase H\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\na b : Localization s\na\u271d c\u271d : \u03b1\nb\u271d d\u271d : { x // x \u2208 s }\nh\u271d : \u2191(r s) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ : mk c\u271d d\u271d \u2264 b \u2192 b \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d = b) = (_ : mk c\u271d d\u271d \u2264 b \u2192 b \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d = b)\n[PROOFSTEP]\nintros\n[GOAL]\ncase H\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\na b : Localization s\na\u271d c\u271d : \u03b1\nb\u271d d\u271d : { x // x \u2208 s }\nh\u271d : \u2191(r s) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ : mk c\u271d d\u271d \u2264 b \u2192 b \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d = b) = (_ : mk c\u271d d\u271d \u2264 b \u2192 b \u2264 mk c\u271d d\u271d \u2192 mk c\u271d d\u271d = b)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\nsrc\u271d\u00b9 : CommMonoid (Localization s) := commMonoid s\nsrc\u271d : PartialOrder (Localization s) := partialOrder\na\u271d b\u271d : Localization s\na b : \u03b1 \u00d7 { x // x \u2208 s }\nhab : mk a.fst a.snd \u2264 mk b.fst b.snd\nc\u271d : Localization s\nc : \u03b1 \u00d7 { x // x \u2208 s }\n\u22a2 mk c.fst c.snd * mk a.fst a.snd \u2264 mk c.fst c.snd * mk b.fst b.snd\n[PROOFSTEP]\nsimp only [mk_mul, mk_le_mk, Submonoid.coe_mul, mul_mul_mul_comm _ _ c.1] at hab \u22a2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\nsrc\u271d\u00b9 : CommMonoid (Localization s) := commMonoid s\nsrc\u271d : PartialOrder (Localization s) := partialOrder\na\u271d b\u271d : Localization s\na b : \u03b1 \u00d7 { x // x \u2208 s }\nc\u271d : Localization s\nc : \u03b1 \u00d7 { x // x \u2208 s }\nhab : \u2191b.snd * a.fst \u2264 \u2191a.snd * b.fst\n\u22a2 \u2191c.snd * c.fst * (\u2191b.snd * a.fst) \u2264 \u2191c.snd * c.fst * (\u2191a.snd * b.fst)\n[PROOFSTEP]\nexact mul_le_mul_left' hab _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\nsrc\u271d\u00b9 : CommMonoid (Localization s) := commMonoid s\nsrc\u271d : PartialOrder (Localization s) := partialOrder\na\u271d b\u271d c\u271d : Localization s\na b c : \u03b1 \u00d7 { x // x \u2208 s }\nhab : mk a.fst a.snd * mk b.fst b.snd \u2264 mk a.fst a.snd * mk c.fst c.snd\n\u22a2 mk b.fst b.snd \u2264 mk c.fst c.snd\n[PROOFSTEP]\nsimp only [mk_mul, mk_le_mk, Submonoid.coe_mul, mul_mul_mul_comm _ _ a.1] at hab \u22a2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 : { x // x \u2208 s }\nsrc\u271d\u00b9 : CommMonoid (Localization s) := commMonoid s\nsrc\u271d : PartialOrder (Localization s) := partialOrder\na\u271d b\u271d c\u271d : Localization s\na b c : \u03b1 \u00d7 { x // x \u2208 s }\nhab : \u2191a.snd * a.fst * (\u2191c.snd * b.fst) \u2264 \u2191a.snd * a.fst * (\u2191b.snd * c.fst)\n\u22a2 \u2191c.snd * b.fst \u2264 \u2191b.snd * c.fst\n[PROOFSTEP]\nexact le_of_mul_le_mul_left' hab\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\na\u2081 b\u2081 : \u03b1\na\u2082 b\u2082 b\u271d : { x // x \u2208 s }\na b : \u03b1\n\u22a2 \u2191{ toFun := fun a => mk a b\u271d, inj' := (_ : Injective fun a => mk a b\u271d) } a \u2264\n \u2191{ toFun := fun a => mk a b\u271d, inj' := (_ : Injective fun a => mk a b\u271d) } b \u2194\n a \u2264 b\n[PROOFSTEP]\nsimp [-mk_eq_monoidOf_mk', mk_le_mk]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\nsrc\u271d : OrderedCancelCommMonoid (Localization s) := orderedCancelCommMonoid\na b : Localization s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 { x // x \u2208 s }\n\u22a2 mk x\u271d\u00b9.fst x\u271d\u00b9.snd \u2264 mk x\u271d.fst x\u271d.snd \u2228 mk x\u271d.fst x\u271d.snd \u2264 mk x\u271d\u00b9.fst x\u271d\u00b9.snd\n[PROOFSTEP]\nsimp_rw [mk_le_mk]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedCancelCommMonoid \u03b1\ns : Submonoid \u03b1\nsrc\u271d : OrderedCancelCommMonoid (Localization s) := orderedCancelCommMonoid\na b : Localization s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 { x // x \u2208 s }\n\u22a2 \u2191x\u271d.snd * x\u271d\u00b9.fst \u2264 \u2191x\u271d\u00b9.snd * x\u271d.fst \u2228 \u2191x\u271d\u00b9.snd * x\u271d.fst \u2264 \u2191x\u271d.snd * x\u271d\u00b9.fst\n[PROOFSTEP]\nexact le_total _ _\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.MonoidLocalization", "llama_tokens": 77436, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833945721304, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.44139840113830675}} {"text": "[GOAL]\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj : ConnectedComponents J\n\u22a2 Nonempty (Component j)\n[PROOFSTEP]\ninduction j using Quotient.inductionOn'\n[GOAL]\ncase h\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\na\u271d : J\n\u22a2 Nonempty (Component (Quotient.mk'' a\u271d))\n[PROOFSTEP]\nexact \u27e8\u27e8_, rfl\u27e9\u27e9\n[GOAL]\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj : ConnectedComponents J\n\u22a2 IsConnected (Component j)\n[PROOFSTEP]\napply isConnected_of_zigzag\n[GOAL]\ncase h\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj : ConnectedComponents J\n\u22a2 \u2200 (j\u2081 j\u2082 : Component j), \u2203 l, List.Chain Zag j\u2081 l \u2227 List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 []) = j\u2082\n[PROOFSTEP]\nrintro \u27e8j\u2081, hj\u2081\u27e9\n \u27e8j\u2082, rfl\u27e9\n -- We know that the underlying objects j\u2081 j\u2082 have some zigzag between them in `J`\n[GOAL]\ncase h.mk.mk\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj\u2081 j\u2082 : J\nhj\u2081 : Quotient.mk'' j\u2081 = Quotient.mk'' j\u2082\n\u22a2 \u2203 l,\n List.Chain Zag { obj := j\u2081, property := hj\u2081 } l \u2227\n List.getLast ({ obj := j\u2081, property := hj\u2081 } :: l) (_ : { obj := j\u2081, property := hj\u2081 } :: l \u2260 []) =\n { obj := j\u2082, property := (_ : Quotient.mk'' j\u2082 = Quotient.mk'' j\u2082) }\n[PROOFSTEP]\nhave h\u2081\u2082 : Zigzag j\u2081 j\u2082 := Quotient.exact' hj\u2081\n[GOAL]\ncase h.mk.mk\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj\u2081 j\u2082 : J\nhj\u2081 : Quotient.mk'' j\u2081 = Quotient.mk'' j\u2082\nh\u2081\u2082 : Zigzag j\u2081 j\u2082\n\u22a2 \u2203 l,\n List.Chain Zag { obj := j\u2081, property := hj\u2081 } l \u2227\n List.getLast ({ obj := j\u2081, property := hj\u2081 } :: l) (_ : { obj := j\u2081, property := hj\u2081 } :: l \u2260 []) =\n { obj := j\u2082, property := (_ : Quotient.mk'' j\u2082 = Quotient.mk'' j\u2082) }\n[PROOFSTEP]\nrcases List.exists_chain_of_relationReflTransGen h\u2081\u2082 with\n \u27e8l, hl\u2081, hl\u2082\u27e9\n -- Everything which has a zigzag to j\u2082 can be lifted to the same component as `j\u2082`.\n[GOAL]\ncase h.mk.mk.intro.intro\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj\u2081 j\u2082 : J\nhj\u2081 : Quotient.mk'' j\u2081 = Quotient.mk'' j\u2082\nh\u2081\u2082 : Zigzag j\u2081 j\u2082\nl : List J\nhl\u2081 : List.Chain Zag j\u2081 l\nhl\u2082 : List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 []) = j\u2082\n\u22a2 \u2203 l,\n List.Chain Zag { obj := j\u2081, property := hj\u2081 } l \u2227\n List.getLast ({ obj := j\u2081, property := hj\u2081 } :: l) (_ : { obj := j\u2081, property := hj\u2081 } :: l \u2260 []) =\n { obj := j\u2082, property := (_ : Quotient.mk'' j\u2082 = Quotient.mk'' j\u2082) }\n[PROOFSTEP]\nlet f : \u2200 x, Zigzag x j\u2082 \u2192 Component (Quotient.mk'' j\u2082) := fun x h =>\n \u27e8x, Quotient.sound' h\u27e9\n -- Everything in our chosen zigzag from `j\u2081` to `j\u2082` has a zigzag to `j\u2082`.\n[GOAL]\ncase h.mk.mk.intro.intro\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj\u2081 j\u2082 : J\nhj\u2081 : Quotient.mk'' j\u2081 = Quotient.mk'' j\u2082\nh\u2081\u2082 : Zigzag j\u2081 j\u2082\nl : List J\nhl\u2081 : List.Chain Zag j\u2081 l\nhl\u2082 : List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 []) = j\u2082\nf : (x : J) \u2192 Zigzag x j\u2082 \u2192 Component (Quotient.mk'' j\u2082) :=\n fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j\u2082) }\n\u22a2 \u2203 l,\n List.Chain Zag { obj := j\u2081, property := hj\u2081 } l \u2227\n List.getLast ({ obj := j\u2081, property := hj\u2081 } :: l) (_ : { obj := j\u2081, property := hj\u2081 } :: l \u2260 []) =\n { obj := j\u2082, property := (_ : Quotient.mk'' j\u2082 = Quotient.mk'' j\u2082) }\n[PROOFSTEP]\nhave hf : \u2200 a : J, a \u2208 l \u2192 Zigzag a j\u2082 := by\n intro i hi\n apply List.Chain.induction (fun t => Zigzag t j\u2082) _ hl\u2081 hl\u2082 _ _ _ (List.mem_of_mem_tail hi)\n \u00b7 intro j k\n apply Relation.ReflTransGen.head\n \u00b7 apply Relation.ReflTransGen.refl\n[GOAL]\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj\u2081 j\u2082 : J\nhj\u2081 : Quotient.mk'' j\u2081 = Quotient.mk'' j\u2082\nh\u2081\u2082 : Zigzag j\u2081 j\u2082\nl : List J\nhl\u2081 : List.Chain Zag j\u2081 l\nhl\u2082 : List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 []) = j\u2082\nf : (x : J) \u2192 Zigzag x j\u2082 \u2192 Component (Quotient.mk'' j\u2082) :=\n fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j\u2082) }\n\u22a2 \u2200 (a : J), a \u2208 l \u2192 Zigzag a j\u2082\n[PROOFSTEP]\nintro i hi\n[GOAL]\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj\u2081 j\u2082 : J\nhj\u2081 : Quotient.mk'' j\u2081 = Quotient.mk'' j\u2082\nh\u2081\u2082 : Zigzag j\u2081 j\u2082\nl : List J\nhl\u2081 : List.Chain Zag j\u2081 l\nhl\u2082 : List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 []) = j\u2082\nf : (x : J) \u2192 Zigzag x j\u2082 \u2192 Component (Quotient.mk'' j\u2082) :=\n fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j\u2082) }\ni : J\nhi : i \u2208 l\n\u22a2 Zigzag i j\u2082\n[PROOFSTEP]\napply List.Chain.induction (fun t => Zigzag t j\u2082) _ hl\u2081 hl\u2082 _ _ _ (List.mem_of_mem_tail hi)\n[GOAL]\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj\u2081 j\u2082 : J\nhj\u2081 : Quotient.mk'' j\u2081 = Quotient.mk'' j\u2082\nh\u2081\u2082 : Zigzag j\u2081 j\u2082\nl : List J\nhl\u2081 : List.Chain Zag j\u2081 l\nhl\u2082 : List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 []) = j\u2082\nf : (x : J) \u2192 Zigzag x j\u2082 \u2192 Component (Quotient.mk'' j\u2082) :=\n fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j\u2082) }\ni : J\nhi : i \u2208 l\n\u22a2 \u2200 \u2983x y : J\u2984, Zag x y \u2192 (fun t => Zigzag t j\u2082) y \u2192 (fun t => Zigzag t j\u2082) x\n[PROOFSTEP]\nintro j k\n[GOAL]\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj\u2081 j\u2082 : J\nhj\u2081 : Quotient.mk'' j\u2081 = Quotient.mk'' j\u2082\nh\u2081\u2082 : Zigzag j\u2081 j\u2082\nl : List J\nhl\u2081 : List.Chain Zag j\u2081 l\nhl\u2082 : List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 []) = j\u2082\nf : (x : J) \u2192 Zigzag x j\u2082 \u2192 Component (Quotient.mk'' j\u2082) :=\n fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j\u2082) }\ni : J\nhi : i \u2208 l\nj k : J\n\u22a2 Zag j k \u2192 (fun t => Zigzag t j\u2082) k \u2192 (fun t => Zigzag t j\u2082) j\n[PROOFSTEP]\napply Relation.ReflTransGen.head\n[GOAL]\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj\u2081 j\u2082 : J\nhj\u2081 : Quotient.mk'' j\u2081 = Quotient.mk'' j\u2082\nh\u2081\u2082 : Zigzag j\u2081 j\u2082\nl : List J\nhl\u2081 : List.Chain Zag j\u2081 l\nhl\u2082 : List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 []) = j\u2082\nf : (x : J) \u2192 Zigzag x j\u2082 \u2192 Component (Quotient.mk'' j\u2082) :=\n fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j\u2082) }\ni : J\nhi : i \u2208 l\n\u22a2 Zigzag j\u2082 j\u2082\n[PROOFSTEP]\napply Relation.ReflTransGen.refl\n[GOAL]\ncase h.mk.mk.intro.intro\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj\u2081 j\u2082 : J\nhj\u2081 : Quotient.mk'' j\u2081 = Quotient.mk'' j\u2082\nh\u2081\u2082 : Zigzag j\u2081 j\u2082\nl : List J\nhl\u2081 : List.Chain Zag j\u2081 l\nhl\u2082 : List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 []) = j\u2082\nf : (x : J) \u2192 Zigzag x j\u2082 \u2192 Component (Quotient.mk'' j\u2082) :=\n fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j\u2082) }\nhf : \u2200 (a : J), a \u2208 l \u2192 Zigzag a j\u2082\n\u22a2 \u2203 l,\n List.Chain Zag { obj := j\u2081, property := hj\u2081 } l \u2227\n List.getLast ({ obj := j\u2081, property := hj\u2081 } :: l) (_ : { obj := j\u2081, property := hj\u2081 } :: l \u2260 []) =\n { obj := j\u2082, property := (_ : Quotient.mk'' j\u2082 = Quotient.mk'' j\u2082) }\n[PROOFSTEP]\nrefine' \u27e8l.pmap f hf, _, _\u27e9\n[GOAL]\ncase h.mk.mk.intro.intro.refine'_1\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj\u2081 j\u2082 : J\nhj\u2081 : Quotient.mk'' j\u2081 = Quotient.mk'' j\u2082\nh\u2081\u2082 : Zigzag j\u2081 j\u2082\nl : List J\nhl\u2081 : List.Chain Zag j\u2081 l\nhl\u2082 : List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 []) = j\u2082\nf : (x : J) \u2192 Zigzag x j\u2082 \u2192 Component (Quotient.mk'' j\u2082) :=\n fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j\u2082) }\nhf : \u2200 (a : J), a \u2208 l \u2192 Zigzag a j\u2082\n\u22a2 List.Chain Zag { obj := j\u2081, property := hj\u2081 } (List.pmap f l hf)\n[PROOFSTEP]\nrefine' @List.chain_pmap_of_chain _ _ _ _ _ f (fun x y _ _ h => _) _ _ hl\u2081 h\u2081\u2082 _\n[GOAL]\ncase h.mk.mk.intro.intro.refine'_1\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj\u2081 j\u2082 : J\nhj\u2081 : Quotient.mk'' j\u2081 = Quotient.mk'' j\u2082\nh\u2081\u2082 : Zigzag j\u2081 j\u2082\nl : List J\nhl\u2081 : List.Chain Zag j\u2081 l\nhl\u2082 : List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 []) = j\u2082\nf : (x : J) \u2192 Zigzag x j\u2082 \u2192 Component (Quotient.mk'' j\u2082) :=\n fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j\u2082) }\nhf : \u2200 (a : J), a \u2208 l \u2192 Zigzag a j\u2082\nx y : J\nx\u271d\u00b9 : Zigzag x j\u2082\nx\u271d : Zigzag y j\u2082\nh : Zag x y\n\u22a2 Zag (f x x\u271d\u00b9) (f y x\u271d)\n[PROOFSTEP]\nexact zag_of_zag_obj (Component.\u03b9 _) h\n[GOAL]\ncase h.mk.mk.intro.intro.refine'_2\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj\u2081 j\u2082 : J\nhj\u2081 : Quotient.mk'' j\u2081 = Quotient.mk'' j\u2082\nh\u2081\u2082 : Zigzag j\u2081 j\u2082\nl : List J\nhl\u2081 : List.Chain Zag j\u2081 l\nhl\u2082 : List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 []) = j\u2082\nf : (x : J) \u2192 Zigzag x j\u2082 \u2192 Component (Quotient.mk'' j\u2082) :=\n fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j\u2082) }\nhf : \u2200 (a : J), a \u2208 l \u2192 Zigzag a j\u2082\n\u22a2 List.getLast ({ obj := j\u2081, property := hj\u2081 } :: List.pmap f l hf)\n (_ : { obj := j\u2081, property := hj\u2081 } :: List.pmap f l hf \u2260 []) =\n { obj := j\u2082, property := (_ : Quotient.mk'' j\u2082 = Quotient.mk'' j\u2082) }\n[PROOFSTEP]\nerw [List.getLast_pmap _ f (j\u2081 :: l) (by simpa [h\u2081\u2082] using hf) (List.cons_ne_nil _ _)]\n[GOAL]\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj\u2081 j\u2082 : J\nhj\u2081 : Quotient.mk'' j\u2081 = Quotient.mk'' j\u2082\nh\u2081\u2082 : Zigzag j\u2081 j\u2082\nl : List J\nhl\u2081 : List.Chain Zag j\u2081 l\nhl\u2082 : List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 []) = j\u2082\nf : (x : J) \u2192 Zigzag x j\u2082 \u2192 Component (Quotient.mk'' j\u2082) :=\n fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j\u2082) }\nhf : \u2200 (a : J), a \u2208 l \u2192 Zigzag a j\u2082\n\u22a2 \u2200 (a : J), a \u2208 j\u2081 :: l \u2192 Zigzag a j\u2082\n[PROOFSTEP]\nsimpa [h\u2081\u2082] using hf\n[GOAL]\ncase h.mk.mk.intro.intro.refine'_2\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj\u2081 j\u2082 : J\nhj\u2081 : Quotient.mk'' j\u2081 = Quotient.mk'' j\u2082\nh\u2081\u2082 : Zigzag j\u2081 j\u2082\nl : List J\nhl\u2081 : List.Chain Zag j\u2081 l\nhl\u2082 : List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 []) = j\u2082\nf : (x : J) \u2192 Zigzag x j\u2082 \u2192 Component (Quotient.mk'' j\u2082) :=\n fun x h => { obj := x, property := (_ : Quotient.mk'' x = Quotient.mk'' j\u2082) }\nhf : \u2200 (a : J), a \u2208 l \u2192 Zigzag a j\u2082\n\u22a2 f (List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 [])) (_ : Zigzag (List.getLast (j\u2081 :: l) (_ : j\u2081 :: l \u2260 [])) j\u2082) =\n { obj := j\u2082, property := (_ : Quotient.mk'' j\u2082 = Quotient.mk'' j\u2082) }\n[PROOFSTEP]\nexact FullSubcategory.ext _ _ hl\u2082\n[GOAL]\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\n\u22a2 {X Y : Decomposed J} \u2192 ((decomposedTo J).obj X \u27f6 (decomposedTo J).obj Y) \u2192 (X \u27f6 Y)\n[PROOFSTEP]\nrintro \u27e8j', X, hX\u27e9 \u27e8k', Y, hY\u27e9 f\n[GOAL]\ncase mk.mk.mk.mk\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj' : ConnectedComponents J\nX : J\nhX : Quotient.mk'' X = j'\nk' : ConnectedComponents J\nY : J\nhY : Quotient.mk'' Y = k'\nf :\n (decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } \u27f6\n (decomposedTo J).obj { fst := k', snd := { obj := Y, property := hY } }\n\u22a2 { fst := j', snd := { obj := X, property := hX } } \u27f6 { fst := k', snd := { obj := Y, property := hY } }\n[PROOFSTEP]\ndsimp at f \n[GOAL]\ncase mk.mk.mk.mk\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj' : ConnectedComponents J\nX : J\nhX : Quotient.mk'' X = j'\nk' : ConnectedComponents J\nY : J\nhY : Quotient.mk'' Y = k'\nf : X \u27f6 Y\n\u22a2 { fst := j', snd := { obj := X, property := hX } } \u27f6 { fst := k', snd := { obj := Y, property := hY } }\n[PROOFSTEP]\nhave : j' = k'\n[GOAL]\ncase this\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj' : ConnectedComponents J\nX : J\nhX : Quotient.mk'' X = j'\nk' : ConnectedComponents J\nY : J\nhY : Quotient.mk'' Y = k'\nf : X \u27f6 Y\n\u22a2 j' = k'\ncase mk.mk.mk.mk\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj' : ConnectedComponents J\nX : J\nhX : Quotient.mk'' X = j'\nk' : ConnectedComponents J\nY : J\nhY : Quotient.mk'' Y = k'\nf : X \u27f6 Y\nthis : j' = k'\n\u22a2 { fst := j', snd := { obj := X, property := hX } } \u27f6 { fst := k', snd := { obj := Y, property := hY } }\n[PROOFSTEP]\nrw [\u2190 hX, \u2190 hY, Quotient.eq'']\n[GOAL]\ncase this\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj' : ConnectedComponents J\nX : J\nhX : Quotient.mk'' X = j'\nk' : ConnectedComponents J\nY : J\nhY : Quotient.mk'' Y = k'\nf : X \u27f6 Y\n\u22a2 Setoid.r X Y\ncase mk.mk.mk.mk\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj' : ConnectedComponents J\nX : J\nhX : Quotient.mk'' X = j'\nk' : ConnectedComponents J\nY : J\nhY : Quotient.mk'' Y = k'\nf : X \u27f6 Y\nthis : j' = k'\n\u22a2 { fst := j', snd := { obj := X, property := hX } } \u27f6 { fst := k', snd := { obj := Y, property := hY } }\n[PROOFSTEP]\nexact Relation.ReflTransGen.single (Or.inl \u27e8f\u27e9)\n[GOAL]\ncase mk.mk.mk.mk\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj' : ConnectedComponents J\nX : J\nhX : Quotient.mk'' X = j'\nk' : ConnectedComponents J\nY : J\nhY : Quotient.mk'' Y = k'\nf : X \u27f6 Y\nthis : j' = k'\n\u22a2 { fst := j', snd := { obj := X, property := hX } } \u27f6 { fst := k', snd := { obj := Y, property := hY } }\n[PROOFSTEP]\nsubst this\n[GOAL]\ncase mk.mk.mk.mk\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj' : ConnectedComponents J\nX : J\nhX : Quotient.mk'' X = j'\nY : J\nf : X \u27f6 Y\nhY : Quotient.mk'' Y = j'\n\u22a2 { fst := j', snd := { obj := X, property := hX } } \u27f6 { fst := j', snd := { obj := Y, property := hY } }\n[PROOFSTEP]\nexact Sigma.SigmaHom.mk f\n[GOAL]\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\n\u22a2 \u2200 {X Y : Decomposed J} (f : (decomposedTo J).obj X \u27f6 (decomposedTo J).obj Y),\n (decomposedTo J).map\n (Sigma.casesOn (motive := fun x =>\n {Y : Decomposed J} \u2192 ((decomposedTo J).obj x \u27f6 (decomposedTo J).obj Y) \u2192 (x \u27f6 Y)) X\n (fun j' snd =>\n @FullSubcategory.casesOn J (fun k => Quotient.mk'' k = j')\n (fun x =>\n {Y : Decomposed J} \u2192\n ((decomposedTo J).obj { fst := j', snd := x } \u27f6 (decomposedTo J).obj Y) \u2192\n ({ fst := j', snd := x } \u27f6 Y))\n snd fun X hX {Y} =>\n Sigma.casesOn (motive := fun x =>\n ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } \u27f6 (decomposedTo J).obj x) \u2192\n ({ fst := j', snd := { obj := X, property := hX } } \u27f6 x))\n Y fun k' snd =>\n FullSubcategory.casesOn (motive := fun x =>\n ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } \u27f6\n (decomposedTo J).obj { fst := k', snd := x }) \u2192\n ({ fst := j', snd := { obj := X, property := hX } } \u27f6 { fst := k', snd := x }))\n snd fun Y hY f =>\n Eq.ndrec (motive := fun k' =>\n (hY : Quotient.mk'' Y = k') \u2192\n { fst := j', snd := { obj := X, property := hX } } \u27f6\n { fst := k', snd := { obj := Y, property := hY } })\n (fun hY => Sigma.SigmaHom.mk f) (_ : j' = k') hY)\n f) =\n f\n[PROOFSTEP]\nrintro \u27e8j', X, hX\u27e9 \u27e8_, Y, rfl\u27e9 f\n[GOAL]\ncase mk.mk.mk.mk\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj' : ConnectedComponents J\nX : J\nhX : Quotient.mk'' X = j'\nY : J\nf :\n (decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } \u27f6\n (decomposedTo J).obj\n { fst := Quotient.mk'' Y, snd := { obj := Y, property := (_ : Quotient.mk'' Y = Quotient.mk'' Y) } }\n\u22a2 (decomposedTo J).map\n (Sigma.casesOn (motive := fun x =>\n {Y : Decomposed J} \u2192 ((decomposedTo J).obj x \u27f6 (decomposedTo J).obj Y) \u2192 (x \u27f6 Y))\n { fst := j', snd := { obj := X, property := hX } }\n (fun j' snd =>\n @FullSubcategory.casesOn J (fun k => Quotient.mk'' k = j')\n (fun x =>\n {Y : Decomposed J} \u2192\n ((decomposedTo J).obj { fst := j', snd := x } \u27f6 (decomposedTo J).obj Y) \u2192 ({ fst := j', snd := x } \u27f6 Y))\n snd fun X hX {Y} =>\n Sigma.casesOn (motive := fun x =>\n ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } \u27f6 (decomposedTo J).obj x) \u2192\n ({ fst := j', snd := { obj := X, property := hX } } \u27f6 x))\n Y fun k' snd =>\n FullSubcategory.casesOn (motive := fun x =>\n ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } \u27f6\n (decomposedTo J).obj { fst := k', snd := x }) \u2192\n ({ fst := j', snd := { obj := X, property := hX } } \u27f6 { fst := k', snd := x }))\n snd fun Y hY f =>\n Eq.ndrec (motive := fun k' =>\n (hY : Quotient.mk'' Y = k') \u2192\n { fst := j', snd := { obj := X, property := hX } } \u27f6\n { fst := k', snd := { obj := Y, property := hY } })\n (fun hY => Sigma.SigmaHom.mk f) (_ : j' = k') hY)\n f) =\n f\n[PROOFSTEP]\nhave : Quotient.mk'' Y = j' := by\n rw [\u2190 hX, Quotient.eq'']\n exact Relation.ReflTransGen.single (Or.inr \u27e8f\u27e9)\n[GOAL]\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj' : ConnectedComponents J\nX : J\nhX : Quotient.mk'' X = j'\nY : J\nf :\n (decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } \u27f6\n (decomposedTo J).obj\n { fst := Quotient.mk'' Y, snd := { obj := Y, property := (_ : Quotient.mk'' Y = Quotient.mk'' Y) } }\n\u22a2 Quotient.mk'' Y = j'\n[PROOFSTEP]\nrw [\u2190 hX, Quotient.eq'']\n[GOAL]\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj' : ConnectedComponents J\nX : J\nhX : Quotient.mk'' X = j'\nY : J\nf :\n (decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } \u27f6\n (decomposedTo J).obj\n { fst := Quotient.mk'' Y, snd := { obj := Y, property := (_ : Quotient.mk'' Y = Quotient.mk'' Y) } }\n\u22a2 Setoid.r Y X\n[PROOFSTEP]\nexact Relation.ReflTransGen.single (Or.inr \u27e8f\u27e9)\n[GOAL]\ncase mk.mk.mk.mk\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj' : ConnectedComponents J\nX : J\nhX : Quotient.mk'' X = j'\nY : J\nf :\n (decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } \u27f6\n (decomposedTo J).obj\n { fst := Quotient.mk'' Y, snd := { obj := Y, property := (_ : Quotient.mk'' Y = Quotient.mk'' Y) } }\nthis : Quotient.mk'' Y = j'\n\u22a2 (decomposedTo J).map\n (Sigma.casesOn (motive := fun x =>\n {Y : Decomposed J} \u2192 ((decomposedTo J).obj x \u27f6 (decomposedTo J).obj Y) \u2192 (x \u27f6 Y))\n { fst := j', snd := { obj := X, property := hX } }\n (fun j' snd =>\n @FullSubcategory.casesOn J (fun k => Quotient.mk'' k = j')\n (fun x =>\n {Y : Decomposed J} \u2192\n ((decomposedTo J).obj { fst := j', snd := x } \u27f6 (decomposedTo J).obj Y) \u2192 ({ fst := j', snd := x } \u27f6 Y))\n snd fun X hX {Y} =>\n Sigma.casesOn (motive := fun x =>\n ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } \u27f6 (decomposedTo J).obj x) \u2192\n ({ fst := j', snd := { obj := X, property := hX } } \u27f6 x))\n Y fun k' snd =>\n FullSubcategory.casesOn (motive := fun x =>\n ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } \u27f6\n (decomposedTo J).obj { fst := k', snd := x }) \u2192\n ({ fst := j', snd := { obj := X, property := hX } } \u27f6 { fst := k', snd := x }))\n snd fun Y hY f =>\n Eq.ndrec (motive := fun k' =>\n (hY : Quotient.mk'' Y = k') \u2192\n { fst := j', snd := { obj := X, property := hX } } \u27f6\n { fst := k', snd := { obj := Y, property := hY } })\n (fun hY => Sigma.SigmaHom.mk f) (_ : j' = k') hY)\n f) =\n f\n[PROOFSTEP]\nsubst this\n[GOAL]\ncase mk.mk.mk.mk\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nX Y : J\nhX : Quotient.mk'' X = Quotient.mk'' Y\nf :\n (decomposedTo J).obj { fst := Quotient.mk'' Y, snd := { obj := X, property := hX } } \u27f6\n (decomposedTo J).obj\n { fst := Quotient.mk'' Y, snd := { obj := Y, property := (_ : Quotient.mk'' Y = Quotient.mk'' Y) } }\n\u22a2 (decomposedTo J).map\n (Sigma.casesOn (motive := fun x =>\n {Y : Decomposed J} \u2192 ((decomposedTo J).obj x \u27f6 (decomposedTo J).obj Y) \u2192 (x \u27f6 Y))\n { fst := Quotient.mk'' Y, snd := { obj := X, property := hX } }\n (fun j' snd =>\n @FullSubcategory.casesOn J (fun k => Quotient.mk'' k = j')\n (fun x =>\n {Y : Decomposed J} \u2192\n ((decomposedTo J).obj { fst := j', snd := x } \u27f6 (decomposedTo J).obj Y) \u2192 ({ fst := j', snd := x } \u27f6 Y))\n snd fun X hX {Y} =>\n Sigma.casesOn (motive := fun x =>\n ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } \u27f6 (decomposedTo J).obj x) \u2192\n ({ fst := j', snd := { obj := X, property := hX } } \u27f6 x))\n Y fun k' snd =>\n FullSubcategory.casesOn (motive := fun x =>\n ((decomposedTo J).obj { fst := j', snd := { obj := X, property := hX } } \u27f6\n (decomposedTo J).obj { fst := k', snd := x }) \u2192\n ({ fst := j', snd := { obj := X, property := hX } } \u27f6 { fst := k', snd := x }))\n snd fun Y hY f =>\n Eq.ndrec (motive := fun k' =>\n (hY : Quotient.mk'' Y = k') \u2192\n { fst := j', snd := { obj := X, property := hX } } \u27f6\n { fst := k', snd := { obj := Y, property := hY } })\n (fun hY => Sigma.SigmaHom.mk f) (_ : j' = k') hY)\n f) =\n f\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\n\u22a2 \u2200 {X Y : Decomposed J}, Function.Injective (decomposedTo J).map\n[PROOFSTEP]\nrintro \u27e8_, j, rfl\u27e9 \u27e8_, k, hY\u27e9 \u27e8f\u27e9 \u27e8_\u27e9 rfl\n[GOAL]\ncase mk.mk.mk.mk.mk.mk\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nC : Type u\u2082\ninst\u271d : Category.{u\u2081, u\u2082} C\nj k : J\nhY : Quotient.mk'' k = Quotient.mk'' j\nf : { obj := j, property := (_ : Quotient.mk'' j = Quotient.mk'' j) } \u27f6 { obj := k, property := hY }\n\u22a2 Sigma.SigmaHom.mk f = Sigma.SigmaHom.mk ((decomposedTo J).map (Sigma.SigmaHom.mk f))\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.ConnectedComponents", "llama_tokens": 9918, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432182679956, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.44126079299635834}} {"text": "[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131\u271d \u2131 : Filtration \u03b9 m0\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nx : E\ni j : \u03b9\nx\u271d : i \u2264 j\n\u22a2 \u03bc[(fun x_1 x_2 => x) j|\u2191\u2131 i] =\u1d50[\u03bc] (fun x_1 x_2 => x) i\n[PROOFSTEP]\nrw [condexp_const (\u2131.le _)]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u03b9 m0\ninst\u271d\u00b9 : OrderBot \u03b9\n\u2131 : Filtration \u03b9 m0\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192 E\nhf : StronglyMeasurable f\nhfint : Integrable f\n\u22a2 Martingale (fun x => f) \u2131 \u03bc\n[PROOFSTEP]\nrefine' \u27e8fun i => hf.mono <| \u2131.mono bot_le, fun i j _ => _\u27e9\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u03b9 m0\ninst\u271d\u00b9 : OrderBot \u03b9\n\u2131 : Filtration \u03b9 m0\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192 E\nhf : StronglyMeasurable f\nhfint : Integrable f\ni j : \u03b9\nx\u271d : i \u2264 j\n\u22a2 \u03bc[(fun x => f) j|\u2191\u2131 i] =\u1d50[\u03bc] (fun x => f) i\n[PROOFSTEP]\nrw [condexp_of_stronglyMeasurable (\u2131.le _) (hf.mono <| \u2131.mono bot_le) hfint]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131\u271d \u2131 : Filtration \u03b9 m0\n\u03bc : Measure \u03a9\ni j : \u03b9\nx\u271d : i \u2264 j\n\u22a2 \u03bc[OfNat.ofNat 0 j|\u2191\u2131 i] =\u1d50[\u03bc] OfNat.ofNat 0 i\n[PROOFSTEP]\nrw [Pi.zero_apply, condexp_zero]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131\u271d \u2131 : Filtration \u03b9 m0\n\u03bc : Measure \u03a9\ni j : \u03b9\nx\u271d : i \u2264 j\n\u22a2 0 =\u1d50[\u03bc] OfNat.ofNat 0 i\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131\u271d \u2131 : Filtration \u03b9 m0\n\u03bc : Measure \u03a9\ni j : \u03b9\nx\u271d : i \u2264 j\n\u22a2 0 =\u1d50[\u03bc] 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nhf : Martingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc = \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 @set_integral_condexp _ _ _ _ _ (\u2131 i) m0 _ _ _ (\u2131.le i) _ (hf.integrable j) hs]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nhf : Martingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc = \u222b (x : \u03a9) in s, (\u03bc[f j|\u2191\u2131 i]) x \u2202\u03bc\n[PROOFSTEP]\nrefine' set_integral_congr_ae (\u2131.le i s hs) _\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nhf : Martingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, x \u2208 s \u2192 f i x = (\u03bc[f j|\u2191\u2131 i]) x\n[PROOFSTEP]\nfilter_upwards [hf.2 i j hij] with _ heq _ using heq.symm\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nhf : Martingale f \u2131 \u03bc\nhg : Martingale g \u2131 \u03bc\n\u22a2 Martingale (f + g) \u2131 \u03bc\n[PROOFSTEP]\nrefine' \u27e8hf.adapted.add hg.adapted, fun i j hij => _\u27e9\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nhf : Martingale f \u2131 \u03bc\nhg : Martingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 \u03bc[(f + g) j|\u2191\u2131 i] =\u1d50[\u03bc] (f + g) i\n[PROOFSTEP]\nexact (condexp_add (hf.integrable j) (hg.integrable j)).trans ((hf.2 i j hij).add (hg.2 i j hij))\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nhf : Martingale f \u2131 \u03bc\nhg : Martingale g \u2131 \u03bc\n\u22a2 Martingale (f - g) \u2131 \u03bc\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nhf : Martingale f \u2131 \u03bc\nhg : Martingale g \u2131 \u03bc\n\u22a2 Martingale (f + -g) \u2131 \u03bc\n[PROOFSTEP]\nexact hf.add hg.neg\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nc : \u211d\nhf : Martingale f \u2131 \u03bc\n\u22a2 Martingale (c \u2022 f) \u2131 \u03bc\n[PROOFSTEP]\nrefine' \u27e8hf.adapted.smul c, fun i j hij => _\u27e9\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nc : \u211d\nhf : Martingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 \u03bc[(c \u2022 f) j|\u2191\u2131 i] =\u1d50[\u03bc] (c \u2022 f) i\n[PROOFSTEP]\nrefine' (condexp_smul c (f j)).trans ((hf.2 i j hij).mono fun x hx => _)\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nc : \u211d\nhf : Martingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\nx : \u03a9\nhx : (\u03bc[f j|\u2191\u2131 i]) x = f i x\n\u22a2 (c \u2022 \u03bc[f j|\u2191\u2131 i]) x = (c \u2022 f) i x\n[PROOFSTEP]\nrw [Pi.smul_apply, hx, Pi.smul_apply, Pi.smul_apply]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 set_integral_condexp (\u2131.le i) (hf.integrable j) hs]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (x : \u03a9) in s, (\u03bc[f j|\u2191\u2131 i]) x \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc\n[PROOFSTEP]\nrefine' set_integral_mono_ae integrable_condexp.integrableOn (hf.integrable i).integrableOn _\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 (fun x => (\u03bc[f j|\u2191\u2131 i]) x) \u2264\u1d50[\u03bc] fun \u03c9 => f i \u03c9\n[PROOFSTEP]\nfilter_upwards [hf.2.1 i j hij] with _ heq using heq\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Supermartingale f \u2131 \u03bc\nhg : Supermartingale g \u2131 \u03bc\n\u22a2 Supermartingale (f + g) \u2131 \u03bc\n[PROOFSTEP]\nrefine' \u27e8hf.1.add hg.1, fun i j hij => _, fun i => (hf.2.2 i).add (hg.2.2 i)\u27e9\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Supermartingale f \u2131 \u03bc\nhg : Supermartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 \u03bc[(f + g) j|\u2191\u2131 i] \u2264\u1d50[\u03bc] (f + g) i\n[PROOFSTEP]\nrefine' (condexp_add (hf.integrable j) (hg.integrable j)).le.trans _\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Supermartingale f \u2131 \u03bc\nhg : Supermartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 \u03bc[f j|\u2191\u2131 i] + \u03bc[g j|\u2191\u2131 i] \u2264\u1d50[\u03bc] (f + g) i\n[PROOFSTEP]\nfilter_upwards [hf.2.1 i j hij, hg.2.1 i j hij]\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Supermartingale f \u2131 \u03bc\nhg : Supermartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 \u2200 (a : \u03a9), (\u03bc[f j|\u2191\u2131 i]) a \u2264 f i a \u2192 (\u03bc[g j|\u2191\u2131 i]) a \u2264 g i a \u2192 (\u03bc[f j|\u2191\u2131 i] + \u03bc[g j|\u2191\u2131 i]) a \u2264 (f + g) i a\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Supermartingale f \u2131 \u03bc\nhg : Supermartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\na\u271d\u00b2 : \u03a9\na\u271d\u00b9 : (\u03bc[f j|\u2191\u2131 i]) a\u271d\u00b2 \u2264 f i a\u271d\u00b2\na\u271d : (\u03bc[g j|\u2191\u2131 i]) a\u271d\u00b2 \u2264 g i a\u271d\u00b2\n\u22a2 (\u03bc[f j|\u2191\u2131 i] + \u03bc[g j|\u2191\u2131 i]) a\u271d\u00b2 \u2264 (f + g) i a\u271d\u00b2\n[PROOFSTEP]\nrefine' add_le_add _ _\n[GOAL]\ncase h.refine'_1\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Supermartingale f \u2131 \u03bc\nhg : Supermartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\na\u271d\u00b2 : \u03a9\na\u271d\u00b9 : (\u03bc[f j|\u2191\u2131 i]) a\u271d\u00b2 \u2264 f i a\u271d\u00b2\na\u271d : (\u03bc[g j|\u2191\u2131 i]) a\u271d\u00b2 \u2264 g i a\u271d\u00b2\n\u22a2 (\u03bc[f j|\u2191\u2131 i]) a\u271d\u00b2 \u2264 f i a\u271d\u00b2\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h.refine'_2\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Supermartingale f \u2131 \u03bc\nhg : Supermartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\na\u271d\u00b2 : \u03a9\na\u271d\u00b9 : (\u03bc[f j|\u2191\u2131 i]) a\u271d\u00b2 \u2264 f i a\u271d\u00b2\na\u271d : (\u03bc[g j|\u2191\u2131 i]) a\u271d\u00b2 \u2264 g i a\u271d\u00b2\n\u22a2 (\u03bc[g j|\u2191\u2131 i]) a\u271d\u00b2 \u2264 g i a\u271d\u00b2\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Supermartingale f \u2131 \u03bc\n\u22a2 Submartingale (-f) \u2131 \u03bc\n[PROOFSTEP]\nrefine' \u27e8hf.1.neg, fun i j hij => _, fun i => (hf.2.2 i).neg\u27e9\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 (-f) i \u2264\u1d50[\u03bc] \u03bc[(-f) j|\u2191\u2131 i]\n[PROOFSTEP]\nrefine' EventuallyLE.trans _ (condexp_neg (f j)).symm.le\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 (-f) i \u2264\u1d50[\u03bc] -\u03bc[f j|\u2191\u2131 i]\n[PROOFSTEP]\nfilter_upwards [hf.2.1 i j hij] with _ _\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\na\u271d\u00b9 : \u03a9\na\u271d : (\u03bc[f j|\u2191\u2131 i]) a\u271d\u00b9 \u2264 f i a\u271d\u00b9\n\u22a2 (-f) i a\u271d\u00b9 \u2264 (-\u03bc[f j|\u2191\u2131 i]) a\u271d\u00b9\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\n\u22a2 Submartingale (f + g) \u2131 \u03bc\n[PROOFSTEP]\nrefine' \u27e8hf.1.add hg.1, fun i j hij => _, fun i => (hf.2.2 i).add (hg.2.2 i)\u27e9\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 (f + g) i \u2264\u1d50[\u03bc] \u03bc[(f + g) j|\u2191\u2131 i]\n[PROOFSTEP]\nrefine' EventuallyLE.trans _ (condexp_add (hf.integrable j) (hg.integrable j)).symm.le\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 (f + g) i \u2264\u1d50[\u03bc] \u03bc[f j|\u2191\u2131 i] + \u03bc[g j|\u2191\u2131 i]\n[PROOFSTEP]\nfilter_upwards [hf.2.1 i j hij, hg.2.1 i j hij]\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 \u2200 (a : \u03a9), f i a \u2264 (\u03bc[f j|\u2191\u2131 i]) a \u2192 g i a \u2264 (\u03bc[g j|\u2191\u2131 i]) a \u2192 (f + g) i a \u2264 (\u03bc[f j|\u2191\u2131 i] + \u03bc[g j|\u2191\u2131 i]) a\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\na\u271d\u00b2 : \u03a9\na\u271d\u00b9 : f i a\u271d\u00b2 \u2264 (\u03bc[f j|\u2191\u2131 i]) a\u271d\u00b2\na\u271d : g i a\u271d\u00b2 \u2264 (\u03bc[g j|\u2191\u2131 i]) a\u271d\u00b2\n\u22a2 (f + g) i a\u271d\u00b2 \u2264 (\u03bc[f j|\u2191\u2131 i] + \u03bc[g j|\u2191\u2131 i]) a\u271d\u00b2\n[PROOFSTEP]\nrefine' add_le_add _ _\n[GOAL]\ncase h.refine'_1\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\na\u271d\u00b2 : \u03a9\na\u271d\u00b9 : f i a\u271d\u00b2 \u2264 (\u03bc[f j|\u2191\u2131 i]) a\u271d\u00b2\na\u271d : g i a\u271d\u00b2 \u2264 (\u03bc[g j|\u2191\u2131 i]) a\u271d\u00b2\n\u22a2 f i a\u271d\u00b2 \u2264 (\u03bc[f j|\u2191\u2131 i]) a\u271d\u00b2\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h.refine'_2\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\na\u271d\u00b2 : \u03a9\na\u271d\u00b9 : f i a\u271d\u00b2 \u2264 (\u03bc[f j|\u2191\u2131 i]) a\u271d\u00b2\na\u271d : g i a\u271d\u00b2 \u2264 (\u03bc[g j|\u2191\u2131 i]) a\u271d\u00b2\n\u22a2 g i a\u271d\u00b2 \u2264 (\u03bc[g j|\u2191\u2131 i]) a\u271d\u00b2\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Submartingale f \u2131 \u03bc\n\u22a2 Supermartingale (-f) \u2131 \u03bc\n[PROOFSTEP]\nrefine' \u27e8hf.1.neg, fun i j hij => (condexp_neg (f j)).le.trans _, fun i => (hf.2.2 i).neg\u27e9\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Submartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 -\u03bc[f j|\u2191\u2131 i] \u2264\u1d50[\u03bc] (-f) i\n[PROOFSTEP]\nfilter_upwards [hf.2.1 i j hij] with _ _\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Submartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\na\u271d\u00b9 : \u03a9\na\u271d : f i a\u271d\u00b9 \u2264 (\u03bc[f j|\u2191\u2131 i]) a\u271d\u00b9\n\u22a2 (-\u03bc[f j|\u2191\u2131 i]) a\u271d\u00b9 \u2264 (-f) i a\u271d\u00b9\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 neg_le_neg_iff, \u2190 integral_neg, \u2190 integral_neg]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (a : \u03a9) in s, -f j a \u2202\u03bc \u2264 \u222b (a : \u03a9) in s, -f i a \u2202\u03bc\n[PROOFSTEP]\nexact Supermartingale.set_integral_le hf.neg hij hs\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Submartingale f \u2131 \u03bc\nhg : Supermartingale g \u2131 \u03bc\n\u22a2 Submartingale (f - g) \u2131 \u03bc\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Submartingale f \u2131 \u03bc\nhg : Supermartingale g \u2131 \u03bc\n\u22a2 Submartingale (f + -g) \u2131 \u03bc\n[PROOFSTEP]\nexact hf.add hg.neg\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf g : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\n\u22a2 Submartingale (f \u2294 g) \u2131 \u03bc\n[PROOFSTEP]\nrefine'\n \u27e8fun i => @StronglyMeasurable.sup _ _ _ _ (\u2131 i) _ _ _ (hf.adapted i) (hg.adapted i), fun i j hij => _, fun i =>\n Integrable.sup (hf.integrable _) (hg.integrable _)\u27e9\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf g : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 (f \u2294 g) i \u2264\u1d50[\u03bc] \u03bc[(f \u2294 g) j|\u2191\u2131 i]\n[PROOFSTEP]\nrefine' EventuallyLE.sup_le _ _\n[GOAL]\ncase refine'_1\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf g : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 (fun i_1 => f i i_1) \u2264\u1d50[\u03bc] \u03bc[(f \u2294 g) j|\u2191\u2131 i]\n[PROOFSTEP]\nexact\n EventuallyLE.trans (hf.2.1 i j hij)\n (condexp_mono (hf.integrable _) (Integrable.sup (hf.integrable j) (hg.integrable j))\n (eventually_of_forall fun x => le_max_left _ _))\n[GOAL]\ncase refine'_2\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf g : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 (fun i_1 => g i i_1) \u2264\u1d50[\u03bc] \u03bc[(f \u2294 g) j|\u2191\u2131 i]\n[PROOFSTEP]\nexact\n EventuallyLE.trans (hg.2.1 i j hij)\n (condexp_mono (hg.integrable _) (Integrable.sup (hf.integrable j) (hg.integrable j))\n (eventually_of_forall fun x => le_max_right _ _))\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \u2131 f\nhint : \u2200 (i : \u03b9), Integrable (f i)\nhf : \u2200 (i j : \u03b9), i \u2264 j \u2192 \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc\n\u22a2 Submartingale f \u2131 \u03bc\n[PROOFSTEP]\nrefine' \u27e8hadp, fun i j hij => _, hint\u27e9\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \u2131 f\nhint : \u2200 (i : \u03b9), Integrable (f i)\nhf : \u2200 (i j : \u03b9), i \u2264 j \u2192 \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 f i \u2264\u1d50[\u03bc] \u03bc[f j|\u2191\u2131 i]\n[PROOFSTEP]\nsuffices f i \u2264\u1d50[\u03bc.trim (\u2131.le i)] \u03bc[f j|\u2131 i] by exact ae_le_of_ae_le_trim this\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \u2131 f\nhint : \u2200 (i : \u03b9), Integrable (f i)\nhf : \u2200 (i j : \u03b9), i \u2264 j \u2192 \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc\ni j : \u03b9\nhij : i \u2264 j\nthis : f i \u2264\u1d50[Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0)] \u03bc[f j|\u2191\u2131 i]\n\u22a2 f i \u2264\u1d50[\u03bc] \u03bc[f j|\u2191\u2131 i]\n[PROOFSTEP]\nexact ae_le_of_ae_le_trim this\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \u2131 f\nhint : \u2200 (i : \u03b9), Integrable (f i)\nhf : \u2200 (i j : \u03b9), i \u2264 j \u2192 \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 f i \u2264\u1d50[Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0)] \u03bc[f j|\u2191\u2131 i]\n[PROOFSTEP]\nsuffices 0 \u2264\u1d50[\u03bc.trim (\u2131.le i)] \u03bc[f j|\u2131 i] - f i\n by\n filter_upwards [this] with x hx\n rwa [\u2190 sub_nonneg]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \u2131 f\nhint : \u2200 (i : \u03b9), Integrable (f i)\nhf : \u2200 (i j : \u03b9), i \u2264 j \u2192 \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc\ni j : \u03b9\nhij : i \u2264 j\nthis : 0 \u2264\u1d50[Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0)] \u03bc[f j|\u2191\u2131 i] - f i\n\u22a2 f i \u2264\u1d50[Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0)] \u03bc[f j|\u2191\u2131 i]\n[PROOFSTEP]\nfilter_upwards [this] with x hx\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \u2131 f\nhint : \u2200 (i : \u03b9), Integrable (f i)\nhf : \u2200 (i j : \u03b9), i \u2264 j \u2192 \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc\ni j : \u03b9\nhij : i \u2264 j\nthis : 0 \u2264\u1d50[Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0)] \u03bc[f j|\u2191\u2131 i] - f i\nx : \u03a9\nhx : OfNat.ofNat 0 x \u2264 (\u03bc[f j|\u2191\u2131 i] - f i) x\n\u22a2 f i x \u2264 (\u03bc[f j|\u2191\u2131 i]) x\n[PROOFSTEP]\nrwa [\u2190 sub_nonneg]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \u2131 f\nhint : \u2200 (i : \u03b9), Integrable (f i)\nhf : \u2200 (i j : \u03b9), i \u2264 j \u2192 \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 0 \u2264\u1d50[Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0)] \u03bc[f j|\u2191\u2131 i] - f i\n[PROOFSTEP]\nrefine'\n ae_nonneg_of_forall_set_integral_nonneg\n ((integrable_condexp.sub (hint i)).trim _ (stronglyMeasurable_condexp.sub <| hadp i)) fun s hs _ => _\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \u2131 f\nhint : \u2200 (i : \u03b9), Integrable (f i)\nhf : \u2200 (i j : \u03b9), i \u2264 j \u2192 \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc\ni j : \u03b9\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\nx\u271d : \u2191\u2191(Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0)) s < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03a9) in s, (\u03bc[f j|\u2191\u2131 i] - f i) x \u2202Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0)\n[PROOFSTEP]\nspecialize hf i j hij s hs\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \u2131 f\nhint : \u2200 (i : \u03b9), Integrable (f i)\ni j : \u03b9\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\nx\u271d : \u2191\u2191(Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0)) s < \u22a4\nhf : \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc\n\u22a2 0 \u2264 \u222b (x : \u03a9) in s, (\u03bc[f j|\u2191\u2131 i] - f i) x \u2202Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0)\n[PROOFSTEP]\nrwa [\u2190 set_integral_trim _ (stronglyMeasurable_condexp.sub <| hadp i) hs,\n integral_sub' integrable_condexp.integrableOn (hint i).integrableOn, sub_nonneg,\n set_integral_condexp (\u2131.le i) (hint j) hs]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \u2131 f\nhint : \u2200 (i : \u03b9), Integrable (f i)\nhf : \u2200 (i j : \u03b9), i \u2264 j \u2192 0 \u2264\u1d50[\u03bc] \u03bc[f j - f i|\u2191\u2131 i]\n\u22a2 Submartingale f \u2131 \u03bc\n[PROOFSTEP]\nrefine' \u27e8hadp, fun i j hij => _, hint\u27e9\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \u2131 f\nhint : \u2200 (i : \u03b9), Integrable (f i)\nhf : \u2200 (i j : \u03b9), i \u2264 j \u2192 0 \u2264\u1d50[\u03bc] \u03bc[f j - f i|\u2191\u2131 i]\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 f i \u2264\u1d50[\u03bc] \u03bc[f j|\u2191\u2131 i]\n[PROOFSTEP]\nrw [\u2190 condexp_of_stronglyMeasurable (\u2131.le _) (hadp _) (hint _), \u2190 eventually_sub_nonneg]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \u2131 f\nhint : \u2200 (i : \u03b9), Integrable (f i)\nhf : \u2200 (i j : \u03b9), i \u2264 j \u2192 0 \u2264\u1d50[\u03bc] \u03bc[f j - f i|\u2191\u2131 i]\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[f j|\u2191\u2131 i] - \u03bc[f i|\u2191\u2131 i]\n[PROOFSTEP]\nexact EventuallyLE.trans (hf i j hij) (condexp_sub (hint _) (hint _)).le\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[f j - f i|\u2191\u2131 i]\n[PROOFSTEP]\nby_cases h : SigmaFinite (\u03bc.trim (\u2131.le i))\n[GOAL]\ncase pos\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\nh : SigmaFinite (Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0))\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[f j - f i|\u2191\u2131 i]\ncase neg\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\nh : \u00acSigmaFinite (Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0))\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[f j - f i|\u2191\u2131 i]\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\nh : \u00acSigmaFinite (Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0))\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[f j - f i|\u2191\u2131 i]\n[PROOFSTEP]\nrw [condexp_of_not_sigmaFinite (\u2131.le i) h]\n[GOAL]\ncase pos\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\nh : SigmaFinite (Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0))\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[f j - f i|\u2191\u2131 i]\n[PROOFSTEP]\nrefine' EventuallyLE.trans _ (condexp_sub (hf.integrable _) (hf.integrable _)).symm.le\n[GOAL]\ncase pos\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\nh : SigmaFinite (Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0))\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[f j|\u2191\u2131 i] - \u03bc[f i|\u2191\u2131 i]\n[PROOFSTEP]\nrw [eventually_sub_nonneg, condexp_of_stronglyMeasurable (\u2131.le _) (hf.adapted _) (hf.integrable _)]\n[GOAL]\ncase pos\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nf : \u03b9 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\nh : SigmaFinite (Measure.trim \u03bc (_ : \u2191\u2131 i \u2264 m0))\n\u22a2 f i \u2264\u1d50[\u03bc] \u03bc[f j|\u2191\u2131 i]\n[PROOFSTEP]\nexact hf.2.1 i j hij\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Supermartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\n\u22a2 Supermartingale (f - g) \u2131 \u03bc\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nhf : Supermartingale f \u2131 \u03bc\nhg : Submartingale g \u2131 \u03bc\n\u22a2 Supermartingale (f + -g) \u2131 \u03bc\n[PROOFSTEP]\nexact hf.add hg.neg\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Supermartingale f \u2131 \u03bc\n\u22a2 Supermartingale (c \u2022 f) \u2131 \u03bc\n[PROOFSTEP]\nrefine' \u27e8hf.1.smul c, fun i j hij => _, fun i => (hf.2.2 i).smul c\u27e9\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 \u03bc[(c \u2022 f) j|\u2191\u2131 i] \u2264\u1d50[\u03bc] (c \u2022 f) i\n[PROOFSTEP]\nrefine' (condexp_smul c (f j)).le.trans _\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\n\u22a2 c \u2022 \u03bc[f j|\u2191\u2131 i] \u2264\u1d50[\u03bc] (c \u2022 f) i\n[PROOFSTEP]\nfilter_upwards [hf.2.1 i j hij] with _ hle\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\na\u271d : \u03a9\nhle : (\u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 f i a\u271d\n\u22a2 (c \u2022 \u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 (c \u2022 f) i a\u271d\n[PROOFSTEP]\nsimp_rw [Pi.smul_apply]\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Supermartingale f \u2131 \u03bc\ni j : \u03b9\nhij : i \u2264 j\na\u271d : \u03a9\nhle : (\u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 f i a\u271d\n\u22a2 c \u2022 (\u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 c \u2022 f i a\u271d\n[PROOFSTEP]\nexact smul_le_smul_of_nonneg hle hc\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : c \u2264 0\nhf : Supermartingale f \u2131 \u03bc\n\u22a2 Submartingale (c \u2022 f) \u2131 \u03bc\n[PROOFSTEP]\nrw [\u2190 neg_neg c, (by ext (i x); simp : - -c \u2022 f = -(-c \u2022 f))]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : c \u2264 0\nhf : Supermartingale f \u2131 \u03bc\n\u22a2 - -c \u2022 f = -(-c \u2022 f)\n[PROOFSTEP]\next (i x)\n[GOAL]\ncase h.h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : c \u2264 0\nhf : Supermartingale f \u2131 \u03bc\ni : \u03b9\nx : \u03a9\n\u22a2 (- -c \u2022 f) i x = (-(-c \u2022 f)) i x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : c \u2264 0\nhf : Supermartingale f \u2131 \u03bc\n\u22a2 Submartingale (-(-c \u2022 f)) \u2131 \u03bc\n[PROOFSTEP]\nexact (hf.smul_nonneg <| neg_nonneg.2 hc).neg\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Submartingale f \u2131 \u03bc\n\u22a2 Submartingale (c \u2022 f) \u2131 \u03bc\n[PROOFSTEP]\nrw [\u2190 neg_neg c, (by ext (i x); simp : - -c \u2022 f = -(c \u2022 -f))]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Submartingale f \u2131 \u03bc\n\u22a2 - -c \u2022 f = -(c \u2022 -f)\n[PROOFSTEP]\next (i x)\n[GOAL]\ncase h.h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Submartingale f \u2131 \u03bc\ni : \u03b9\nx : \u03a9\n\u22a2 (- -c \u2022 f) i x = (-(c \u2022 -f)) i x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : 0 \u2264 c\nhf : Submartingale f \u2131 \u03bc\n\u22a2 Submartingale (-(c \u2022 -f)) \u2131 \u03bc\n[PROOFSTEP]\nexact Supermartingale.neg (hf.neg.smul_nonneg hc)\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : c \u2264 0\nhf : Submartingale f \u2131 \u03bc\n\u22a2 Supermartingale (c \u2022 f) \u2131 \u03bc\n[PROOFSTEP]\nrw [\u2190 neg_neg c, (by ext (i x); simp : - -c \u2022 f = -(-c \u2022 f))]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : c \u2264 0\nhf : Submartingale f \u2131 \u03bc\n\u22a2 - -c \u2022 f = -(-c \u2022 f)\n[PROOFSTEP]\next (i x)\n[GOAL]\ncase h.h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : c \u2264 0\nhf : Submartingale f \u2131 \u03bc\ni : \u03b9\nx : \u03a9\n\u22a2 (- -c \u2022 f) i x = (-(-c \u2022 f)) i x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2077 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\nF : Type u_4\ninst\u271d\u00b3 : NormedLatticeAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : OrderedSMul \u211d F\nf : \u03b9 \u2192 \u03a9 \u2192 F\nc : \u211d\nhc : c \u2264 0\nhf : Submartingale f \u2131 \u03bc\n\u22a2 Supermartingale (-(-c \u2022 f)) \u2131 \u03bc\n[PROOFSTEP]\nexact (hf.smul_nonneg <| neg_nonneg.2 hc).neg\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115) (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f (i + 1) \u03c9 \u2202\u03bc\n\u22a2 Submartingale f \ud835\udca2 \u03bc\n[PROOFSTEP]\nrefine' submartingale_of_set_integral_le hadp hint fun i j hij s hs => _\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115) (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f (i + 1) \u03c9 \u2202\u03bc\ni j : \u2115\nhij : i \u2264 j\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc\n[PROOFSTEP]\ninduction' hij with k hk\u2081 hk\u2082\n[GOAL]\ncase refl\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115) (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f (i + 1) \u03c9 \u2202\u03bc\ni j : \u2115\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase step\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115) (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f (i + 1) \u03c9 \u2202\u03bc\ni j : \u2115\ns : Set \u03a9\nhs : MeasurableSet s\nk : \u2115\nhk\u2081 : Nat.le i k\nhk\u2082 : \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f k \u03c9 \u2202\u03bc\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f (Nat.succ k) \u03c9 \u2202\u03bc\n[PROOFSTEP]\nexact le_trans hk\u2082 (hf k s (\ud835\udca2.mono hk\u2081 _ hs))\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115) (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, f (i + 1) \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc\n\u22a2 Supermartingale f \ud835\udca2 \u03bc\n[PROOFSTEP]\nrw [\u2190 neg_neg f]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115) (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, f (i + 1) \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc\n\u22a2 Supermartingale (- -f) \ud835\udca2 \u03bc\n[PROOFSTEP]\nrefine' (submartingale_of_set_integral_le_succ hadp.neg (fun i => (hint i).neg) _).neg\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115) (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, f (i + 1) \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc\n\u22a2 \u2200 (i : \u2115) (s : Set \u03a9), MeasurableSet s \u2192 \u222b (\u03c9 : \u03a9) in s, (-f) i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, (-f) (i + 1) \u03c9 \u2202\u03bc\n[PROOFSTEP]\nsimpa only [integral_neg, Pi.neg_apply, neg_le_neg_iff]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), f i \u2264\u1d50[\u03bc] \u03bc[f (i + 1)|\u2191\ud835\udca2 i]\n\u22a2 Submartingale f \ud835\udca2 \u03bc\n[PROOFSTEP]\nrefine' submartingale_of_set_integral_le_succ hadp hint fun i s hs => _\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), f i \u2264\u1d50[\u03bc] \u03bc[f (i + 1)|\u2191\ud835\udca2 i]\ni : \u2115\ns : Set \u03a9\nhs : MeasurableSet s\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f (i + 1) \u03c9 \u2202\u03bc\n[PROOFSTEP]\nhave : \u222b \u03c9 in s, f (i + 1) \u03c9 \u2202\u03bc = \u222b \u03c9 in s, (\u03bc[f (i + 1)|\ud835\udca2 i]) \u03c9 \u2202\u03bc := (set_integral_condexp (\ud835\udca2.le i) (hint _) hs).symm\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), f i \u2264\u1d50[\u03bc] \u03bc[f (i + 1)|\u2191\ud835\udca2 i]\ni : \u2115\ns : Set \u03a9\nhs : MeasurableSet s\nthis : \u222b (\u03c9 : \u03a9) in s, f (i + 1) \u03c9 \u2202\u03bc = \u222b (\u03c9 : \u03a9) in s, (\u03bc[f (i + 1)|\u2191\ud835\udca2 i]) \u03c9 \u2202\u03bc\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f (i + 1) \u03c9 \u2202\u03bc\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), f i \u2264\u1d50[\u03bc] \u03bc[f (i + 1)|\u2191\ud835\udca2 i]\ni : \u2115\ns : Set \u03a9\nhs : MeasurableSet s\nthis : \u222b (\u03c9 : \u03a9) in s, f (i + 1) \u03c9 \u2202\u03bc = \u222b (\u03c9 : \u03a9) in s, (\u03bc[f (i + 1)|\u2191\ud835\udca2 i]) \u03c9 \u2202\u03bc\n\u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, (\u03bc[f (i + 1)|\u2191\ud835\udca2 i]) \u03c9 \u2202\u03bc\n[PROOFSTEP]\nexact set_integral_mono_ae (hint i).integrableOn integrable_condexp.integrableOn (hf i)\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1)|\u2191\ud835\udca2 i] \u2264\u1d50[\u03bc] f i\n\u22a2 Supermartingale f \ud835\udca2 \u03bc\n[PROOFSTEP]\nrw [\u2190 neg_neg f]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1)|\u2191\ud835\udca2 i] \u2264\u1d50[\u03bc] f i\n\u22a2 Supermartingale (- -f) \ud835\udca2 \u03bc\n[PROOFSTEP]\nrefine' (submartingale_nat hadp.neg (fun i => (hint i).neg) fun i => EventuallyLE.trans _ (condexp_neg _).symm.le).neg\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1)|\u2191\ud835\udca2 i] \u2264\u1d50[\u03bc] f i\ni : \u2115\n\u22a2 (-f) i \u2264\u1d50[\u03bc] -\u03bc[fun i_1 => f (i + 1) i_1|\u2191\ud835\udca2 i]\n[PROOFSTEP]\nfilter_upwards [hf i] with x hx using neg_le_neg hx\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), 0 \u2264\u1d50[\u03bc] \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]\n\u22a2 Submartingale f \ud835\udca2 \u03bc\n[PROOFSTEP]\nrefine' submartingale_nat hadp hint fun i => _\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), 0 \u2264\u1d50[\u03bc] \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]\ni : \u2115\n\u22a2 f i \u2264\u1d50[\u03bc] \u03bc[f (i + 1)|\u2191\ud835\udca2 i]\n[PROOFSTEP]\nrw [\u2190 condexp_of_stronglyMeasurable (\ud835\udca2.le _) (hadp _) (hint _), \u2190 eventually_sub_nonneg]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), 0 \u2264\u1d50[\u03bc] \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[f (i + 1)|\u2191\ud835\udca2 i] - \u03bc[f i|\u2191\ud835\udca2 i]\n[PROOFSTEP]\nexact EventuallyLE.trans (hf i) (condexp_sub (hint _) (hint _)).le\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), 0 \u2264\u1d50[\u03bc] \u03bc[f i - f (i + 1)|\u2191\ud835\udca2 i]\n\u22a2 Supermartingale f \ud835\udca2 \u03bc\n[PROOFSTEP]\nrw [\u2190 neg_neg f]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), 0 \u2264\u1d50[\u03bc] \u03bc[f i - f (i + 1)|\u2191\ud835\udca2 i]\n\u22a2 Supermartingale (- -f) \ud835\udca2 \u03bc\n[PROOFSTEP]\nrefine' (submartingale_of_condexp_sub_nonneg_nat hadp.neg (fun i => (hint i).neg) _).neg\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), 0 \u2264\u1d50[\u03bc] \u03bc[f i - f (i + 1)|\u2191\ud835\udca2 i]\n\u22a2 \u2200 (i : \u2115), 0 \u2264\u1d50[\u03bc] \u03bc[(-f) (i + 1) - (-f) i|\u2191\ud835\udca2 i]\n[PROOFSTEP]\nsimpa only [Pi.zero_apply, Pi.neg_apply, neg_sub_neg]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\n\u22a2 Martingale f \ud835\udca2 \u03bc\n[PROOFSTEP]\nrefine'\n martingale_iff.2\n \u27e8supermartingale_of_condexp_sub_nonneg_nat hadp hint fun i => _,\n submartingale_of_condexp_sub_nonneg_nat hadp hint fun i => (hf i).symm.le\u27e9\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[f i - f (i + 1)|\u2191\ud835\udca2 i]\n[PROOFSTEP]\nrw [\u2190 neg_sub]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[-(f (i + 1) - f i)|\u2191\ud835\udca2 i]\n[PROOFSTEP]\nrefine' (EventuallyEq.trans _ (condexp_neg _).symm).le\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\n\u22a2 0 =\u1d50[\u03bc] -\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]\n[PROOFSTEP]\nfilter_upwards [hf i] with x hx\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\nx : \u03a9\nhx : (\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x = OfNat.ofNat 0 x\n\u22a2 OfNat.ofNat 0 x = (-\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x\n[PROOFSTEP]\nsimpa only [Pi.zero_apply, Pi.neg_apply, zero_eq_neg]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : SigmaFiniteFiltration \u03bc \ud835\udca2\nf : \u2115 \u2192 \u03a9 \u2192 E\nhfmgle : Submartingale f \ud835\udca2 \u03bc\nhfadp : Adapted \ud835\udca2 fun n => f (n + 1)\nn : \u2115\n\u22a2 f 0 \u2264\u1d50[\u03bc] f n\n[PROOFSTEP]\ninduction' n with k ih\n[GOAL]\ncase zero\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : SigmaFiniteFiltration \u03bc \ud835\udca2\nf : \u2115 \u2192 \u03a9 \u2192 E\nhfmgle : Submartingale f \ud835\udca2 \u03bc\nhfadp : Adapted \ud835\udca2 fun n => f (n + 1)\n\u22a2 f 0 \u2264\u1d50[\u03bc] f Nat.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : SigmaFiniteFiltration \u03bc \ud835\udca2\nf : \u2115 \u2192 \u03a9 \u2192 E\nhfmgle : Submartingale f \ud835\udca2 \u03bc\nhfadp : Adapted \ud835\udca2 fun n => f (n + 1)\nk : \u2115\nih : f 0 \u2264\u1d50[\u03bc] f k\n\u22a2 f 0 \u2264\u1d50[\u03bc] f (Nat.succ k)\n[PROOFSTEP]\nexact\n ih.trans\n ((hfmgle.2.1 k (k + 1) k.le_succ).trans_eq <|\n Germ.coe_eq.mp <| congr_arg Germ.ofFun <| condexp_of_stronglyMeasurable (\ud835\udca2.le _) (hfadp _) <| hfmgle.integrable _)\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : SigmaFiniteFiltration \u03bc \ud835\udca2\nf : \u2115 \u2192 \u03a9 \u2192 E\nhfmgle : Supermartingale f \ud835\udca2 \u03bc\nhfadp : Adapted \ud835\udca2 fun n => f (n + 1)\nn : \u2115\n\u22a2 f n \u2264\u1d50[\u03bc] f 0\n[PROOFSTEP]\ninduction' n with k ih\n[GOAL]\ncase zero\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : SigmaFiniteFiltration \u03bc \ud835\udca2\nf : \u2115 \u2192 \u03a9 \u2192 E\nhfmgle : Supermartingale f \ud835\udca2 \u03bc\nhfadp : Adapted \ud835\udca2 fun n => f (n + 1)\n\u22a2 f Nat.zero \u2264\u1d50[\u03bc] f 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d\u00b9 : Preorder E\ninst\u271d : SigmaFiniteFiltration \u03bc \ud835\udca2\nf : \u2115 \u2192 \u03a9 \u2192 E\nhfmgle : Supermartingale f \ud835\udca2 \u03bc\nhfadp : Adapted \ud835\udca2 fun n => f (n + 1)\nk : \u2115\nih : f k \u2264\u1d50[\u03bc] f 0\n\u22a2 f (Nat.succ k) \u2264\u1d50[\u03bc] f 0\n[PROOFSTEP]\nexact\n ((Germ.coe_eq.mp <|\n congr_arg Germ.ofFun <|\n condexp_of_stronglyMeasurable (\ud835\udca2.le _) (hfadp _) <| hfmgle.integrable _).symm.trans_le\n (hfmgle.2.1 k (k + 1) k.le_succ)).trans\n ih\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : SigmaFiniteFiltration \u03bc \ud835\udca2\nf : \u2115 \u2192 \u03a9 \u2192 E\nhfmgle : Martingale f \ud835\udca2 \u03bc\nhfadp : Adapted \ud835\udca2 fun n => f (n + 1)\nn : \u2115\n\u22a2 f n =\u1d50[\u03bc] f 0\n[PROOFSTEP]\ninduction' n with k ih\n[GOAL]\ncase zero\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : SigmaFiniteFiltration \u03bc \ud835\udca2\nf : \u2115 \u2192 \u03a9 \u2192 E\nhfmgle : Martingale f \ud835\udca2 \u03bc\nhfadp : Adapted \ud835\udca2 fun n => f (n + 1)\n\u22a2 f Nat.zero =\u1d50[\u03bc] f 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : SigmaFiniteFiltration \u03bc \ud835\udca2\nf : \u2115 \u2192 \u03a9 \u2192 E\nhfmgle : Martingale f \ud835\udca2 \u03bc\nhfadp : Adapted \ud835\udca2 fun n => f (n + 1)\nk : \u2115\nih : f k =\u1d50[\u03bc] f 0\n\u22a2 f (Nat.succ k) =\u1d50[\u03bc] f 0\n[PROOFSTEP]\nexact\n ((Germ.coe_eq.mp\n (congr_arg Germ.ofFun <| condexp_of_stronglyMeasurable (\ud835\udca2.le _) (hfadp _) (hfmgle.integrable _))).symm.trans\n (hfmgle.2 k (k + 1) k.le_succ)).trans\n ih\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\n\u22a2 Submartingale (fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) \ud835\udca2 \u03bc\n[PROOFSTEP]\nhave h\u03bebdd : \u2200 i, \u2203 C, \u2200 \u03c9, |\u03be i \u03c9| \u2264 C := fun i => \u27e8R, fun \u03c9 => (abs_of_nonneg (hnonneg i \u03c9)).trans_le (hbdd i \u03c9)\u27e9\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\n\u22a2 Submartingale (fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) \ud835\udca2 \u03bc\n[PROOFSTEP]\nhave hint : \u2200 m, Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k)) \u03bc := fun m =>\n integrable_finset_sum' _ fun i _ =>\n Integrable.bdd_mul ((hf.integrable _).sub (hf.integrable _)) h\u03be.stronglyMeasurable.aestronglyMeasurable (h\u03bebdd _)\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\n\u22a2 Submartingale (fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) \ud835\udca2 \u03bc\n[PROOFSTEP]\nhave hadp : Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k) :=\n by\n intro m\n refine' Finset.stronglyMeasurable_sum' _ fun i hi => _\n rw [Finset.mem_range] at hi \n exact\n (h\u03be.stronglyMeasurable_le hi.le).mul\n ((hf.adapted.stronglyMeasurable_le (Nat.succ_le_of_lt hi)).sub (hf.adapted.stronglyMeasurable_le hi.le))\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\n\u22a2 Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)\n[PROOFSTEP]\nintro m\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nm : \u2115\n\u22a2 StronglyMeasurable ((fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) m)\n[PROOFSTEP]\nrefine' Finset.stronglyMeasurable_sum' _ fun i hi => _\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nm i : \u2115\nhi : i \u2208 Finset.range m\n\u22a2 StronglyMeasurable (\u03be i * (f (i + 1) - f i))\n[PROOFSTEP]\nrw [Finset.mem_range] at hi \n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nm i : \u2115\nhi : i < m\n\u22a2 StronglyMeasurable (\u03be i * (f (i + 1) - f i))\n[PROOFSTEP]\nexact\n (h\u03be.stronglyMeasurable_le hi.le).mul\n ((hf.adapted.stronglyMeasurable_le (Nat.succ_le_of_lt hi)).sub (hf.adapted.stronglyMeasurable_le hi.le))\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nhadp : Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)\n\u22a2 Submartingale (fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)) \ud835\udca2 \u03bc\n[PROOFSTEP]\nrefine' submartingale_of_condexp_sub_nonneg_nat hadp hint fun i => _\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nhadp : Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[\u2211 k in Finset.range (i + 1), \u03be k * (f (k + 1) - f k) - \u2211 k in Finset.range i, \u03be k * (f (k + 1) - f k)|\u2191\ud835\udca2 i]\n[PROOFSTEP]\nsimp only [\u2190 Finset.sum_Ico_eq_sub _ (Nat.le_succ _), Finset.sum_apply, Pi.mul_apply, Pi.sub_apply,\n Nat.Ico_succ_singleton, Finset.sum_singleton]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nR : \u211d\n\u03be f : \u2115 \u2192 \u03a9 \u2192 \u211d\nhf : Submartingale f \ud835\udca2 \u03bc\nh\u03be : Adapted \ud835\udca2 \u03be\nhbdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u03be n \u03c9 \u2264 R\nhnonneg : \u2200 (n : \u2115) (\u03c9 : \u03a9), 0 \u2264 \u03be n \u03c9\nh\u03bebdd : \u2200 (i : \u2115), \u2203 C, \u2200 (\u03c9 : \u03a9), |\u03be i \u03c9| \u2264 C\nhint : \u2200 (m : \u2115), Integrable (\u2211 k in Finset.range m, \u03be k * (f (k + 1) - f k))\nhadp : Adapted \ud835\udca2 fun n => \u2211 k in Finset.range n, \u03be k * (f (k + 1) - f k)\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[\u03be i * (f (i + 1) - f i)|\u2191\ud835\udca2 i]\n[PROOFSTEP]\nexact\n EventuallyLE.trans\n (EventuallyLE.mul_nonneg (eventually_of_forall (hnonneg _)) (hf.condexp_sub_nonneg (Nat.le_succ _)))\n (condexp_stronglyMeasurable_mul (h\u03be _)\n (((hf.integrable _).sub (hf.integrable _)).bdd_mul h\u03be.stronglyMeasurable.aestronglyMeasurable (h\u03bebdd _))\n ((hf.integrable _).sub (hf.integrable _))).symm.le\n", "meta": {"mathlib_filename": "Mathlib.Probability.Martingale.Basic", "llama_tokens": 37958, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.441260785677873}} {"text": "[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\ns\u2081 : Sorted r l\u2081\ns\u2082 : Sorted r l\u2082\n\u22a2 l\u2081 = l\u2082\n[PROOFSTEP]\ninduction' s\u2081 with a l\u2081 h\u2081 s\u2081 IH generalizing l\u2082\n[GOAL]\ncase nil\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082\u271d : List \u03b1\np\u271d : l\u2081 ~ l\u2082\u271d\ns\u2082\u271d : Sorted r l\u2082\u271d\nl\u2082 : List \u03b1\np : [] ~ l\u2082\ns\u2082 : Sorted r l\u2082\n\u22a2 [] = l\u2082\n[PROOFSTEP]\nexact p.nil_eq\n[GOAL]\ncase cons\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081\u271d l\u2082\u271d : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\u271d\ns\u2082\u271d : Sorted r l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a'\ns\u2081 : Pairwise r l\u2081\nIH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082\nl\u2082 : List \u03b1\np : a :: l\u2081 ~ l\u2082\ns\u2082 : Sorted r l\u2082\n\u22a2 a :: l\u2081 = l\u2082\n[PROOFSTEP]\nhave : a \u2208 l\u2082 := p.subset (mem_cons_self _ _)\n[GOAL]\ncase cons\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081\u271d l\u2082\u271d : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\u271d\ns\u2082\u271d : Sorted r l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a'\ns\u2081 : Pairwise r l\u2081\nIH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082\nl\u2082 : List \u03b1\np : a :: l\u2081 ~ l\u2082\ns\u2082 : Sorted r l\u2082\nthis : a \u2208 l\u2082\n\u22a2 a :: l\u2081 = l\u2082\n[PROOFSTEP]\nrcases mem_split this with \u27e8u\u2082, v\u2082, rfl\u27e9\n[GOAL]\ncase cons.intro.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081\u271d l\u2082 : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nl\u2081 : List \u03b1\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a'\ns\u2081 : Pairwise r l\u2081\nIH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082\nu\u2082 v\u2082 : List \u03b1\np : a :: l\u2081 ~ u\u2082 ++ a :: v\u2082\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\n\u22a2 a :: l\u2081 = u\u2082 ++ a :: v\u2082\n[PROOFSTEP]\nhave p' := (perm_cons a).1 (p.trans perm_middle)\n[GOAL]\ncase cons.intro.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081\u271d l\u2082 : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nl\u2081 : List \u03b1\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a'\ns\u2081 : Pairwise r l\u2081\nIH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082\nu\u2082 v\u2082 : List \u03b1\np : a :: l\u2081 ~ u\u2082 ++ a :: v\u2082\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\np' : l\u2081 ~ u\u2082 ++ v\u2082\n\u22a2 a :: l\u2081 = u\u2082 ++ a :: v\u2082\n[PROOFSTEP]\nobtain rfl := IH p' (s\u2082.sublist <| by simp)\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081\u271d l\u2082 : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nl\u2081 : List \u03b1\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a'\ns\u2081 : Pairwise r l\u2081\nIH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082\nu\u2082 v\u2082 : List \u03b1\np : a :: l\u2081 ~ u\u2082 ++ a :: v\u2082\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\np' : l\u2081 ~ u\u2082 ++ v\u2082\n\u22a2 u\u2082 ++ v\u2082 <+ u\u2082 ++ a :: v\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.intro.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\n\u22a2 a :: (u\u2082 ++ v\u2082) = u\u2082 ++ a :: v\u2082\n[PROOFSTEP]\nchange a :: u\u2082 ++ v\u2082 = u\u2082 ++ ([a] ++ v\u2082)\n[GOAL]\ncase cons.intro.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\n\u22a2 a :: u\u2082 ++ v\u2082 = u\u2082 ++ ([a] ++ v\u2082)\n[PROOFSTEP]\nrw [\u2190 append_assoc]\n[GOAL]\ncase cons.intro.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\n\u22a2 a :: u\u2082 ++ v\u2082 = u\u2082 ++ [a] ++ v\u2082\n[PROOFSTEP]\ncongr\n[GOAL]\ncase cons.intro.intro.e_a\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\n\u22a2 a :: u\u2082 = u\u2082 ++ [a]\n[PROOFSTEP]\nhave : \u2200 (x : \u03b1) (_ : x \u2208 u\u2082), x = a := fun x m =>\n antisymm ((pairwise_append.1 s\u2082).2.2 _ m a (mem_cons_self _ _)) (h\u2081 _ (by simp [m]))\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\nx : \u03b1\nm : x \u2208 u\u2082\n\u22a2 x \u2208 u\u2082 ++ v\u2082\n[PROOFSTEP]\nsimp [m]\n[GOAL]\ncase cons.intro.intro.e_a\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis\u271d : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\nthis : \u2200 (x : \u03b1), x \u2208 u\u2082 \u2192 x = a\n\u22a2 a :: u\u2082 = u\u2082 ++ [a]\n[PROOFSTEP]\nrw [(@eq_replicate _ a (length u\u2082 + 1) (a :: u\u2082)).2, (@eq_replicate _ a (length u\u2082 + 1) (u\u2082 ++ [a])).2]\n[GOAL]\ncase cons.intro.intro.e_a\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis\u271d : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\nthis : \u2200 (x : \u03b1), x \u2208 u\u2082 \u2192 x = a\n\u22a2 length (u\u2082 ++ [a]) = length u\u2082 + 1 \u2227 \u2200 (b : \u03b1), b \u2208 u\u2082 ++ [a] \u2192 b = a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons.intro.intro.e_a\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis\u271d : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\nthis : \u2200 (x : \u03b1), x \u2208 u\u2082 \u2192 x = a\n\u22a2 length (a :: u\u2082) = length u\u2082 + 1 \u2227 \u2200 (b : \u03b1), b \u2208 a :: u\u2082 \u2192 b = a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons.intro.intro.e_a.left\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis\u271d : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\nthis : \u2200 (x : \u03b1), x \u2208 u\u2082 \u2192 x = a\n\u22a2 length (u\u2082 ++ [a]) = length u\u2082 + 1\n[PROOFSTEP]\nsimp [iff_true_intro this, or_comm]\n[GOAL]\ncase cons.intro.intro.e_a.right\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis\u271d : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\nthis : \u2200 (x : \u03b1), x \u2208 u\u2082 \u2192 x = a\n\u22a2 \u2200 (b : \u03b1), b \u2208 u\u2082 ++ [a] \u2192 b = a\n[PROOFSTEP]\nsimp [iff_true_intro this, or_comm]\n[GOAL]\ncase cons.intro.intro.e_a.left\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis\u271d : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\nthis : \u2200 (x : \u03b1), x \u2208 u\u2082 \u2192 x = a\n\u22a2 length (a :: u\u2082) = length u\u2082 + 1\n[PROOFSTEP]\nsimp [iff_true_intro this, or_comm]\n[GOAL]\ncase cons.intro.intro.e_a.right\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis\u271d : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\nthis : \u2200 (x : \u03b1), x \u2208 u\u2082 \u2192 x = a\n\u22a2 \u2200 (b : \u03b1), b \u2208 a :: u\u2082 \u2192 b = a\n[PROOFSTEP]\nsimp [iff_true_intro this, or_comm]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 <+~ l\u2082\ns\u2081 : Sorted r l\u2081\ns\u2082 : Sorted r l\u2082\n\u22a2 l\u2081 <+ l\u2082\n[PROOFSTEP]\nlet \u27e8_, h, h'\u27e9 := p\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 <+~ l\u2082\ns\u2081 : Sorted r l\u2081\ns\u2082 : Sorted r l\u2082\nw\u271d : List \u03b1\nh : w\u271d ~ l\u2081\nh' : w\u271d <+ l\u2082\n\u22a2 l\u2081 <+ l\u2082\n[PROOFSTEP]\nrwa [\u2190 eq_of_perm_of_sorted h (s\u2082.sublist h') s\u2081]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : IsRefl \u03b1 r\nl : List \u03b1\nh : Sorted r l\na b : Fin (length l)\nhab : a \u2264 b\n\u22a2 r (get l a) (get l b)\n[PROOFSTEP]\nrcases hab.eq_or_lt with (rfl | hlt)\n[GOAL]\ncase inl\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : IsRefl \u03b1 r\nl : List \u03b1\nh : Sorted r l\na : Fin (length l)\nhab : a \u2264 a\n\u22a2 r (get l a) (get l a)\ncase inr\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : IsRefl \u03b1 r\nl : List \u03b1\nh : Sorted r l\na b : Fin (length l)\nhab : a \u2264 b\nhlt : a < b\n\u22a2 r (get l a) (get l b)\n[PROOFSTEP]\nexacts [refl _, h.rel_get_of_lt hlt]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nh : Sorted r l\nk : \u2115\nx y : \u03b1\nhx : x \u2208 take k l\nhy : y \u2208 drop k l\n\u22a2 r x y\n[PROOFSTEP]\nobtain \u27e8\u27e8iy, hiy\u27e9, rfl\u27e9 := get_of_mem hy\n[GOAL]\ncase intro.mk\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nh : Sorted r l\nk : \u2115\nx : \u03b1\nhx : x \u2208 take k l\niy : \u2115\nhiy : iy < length (drop k l)\nhy : get (drop k l) { val := iy, isLt := hiy } \u2208 drop k l\n\u22a2 r x (get (drop k l) { val := iy, isLt := hiy })\n[PROOFSTEP]\nobtain \u27e8\u27e8ix, hix\u27e9, rfl\u27e9 := get_of_mem hx\n[GOAL]\ncase intro.mk.intro.mk\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nh : Sorted r l\nk iy : \u2115\nhiy : iy < length (drop k l)\nhy : get (drop k l) { val := iy, isLt := hiy } \u2208 drop k l\nix : \u2115\nhix : ix < length (take k l)\nhx : get (take k l) { val := ix, isLt := hix } \u2208 take k l\n\u22a2 r (get (take k l) { val := ix, isLt := hix }) (get (drop k l) { val := iy, isLt := hiy })\n[PROOFSTEP]\nrw [get_take', get_drop']\n[GOAL]\ncase intro.mk.intro.mk\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nh : Sorted r l\nk iy : \u2115\nhiy : iy < length (drop k l)\nhy : get (drop k l) { val := iy, isLt := hiy } \u2208 drop k l\nix : \u2115\nhix : ix < length (take k l)\nhx : get (take k l) { val := ix, isLt := hix } \u2208 take k l\n\u22a2 r (get l { val := \u2191{ val := ix, isLt := hix }, isLt := (_ : \u2191{ val := ix, isLt := hix } < length l) })\n (get l { val := k + \u2191{ val := iy, isLt := hiy }, isLt := (_ : k + \u2191{ val := iy, isLt := hiy } < length l) })\n[PROOFSTEP]\nrw [length_take] at hix \n[GOAL]\ncase intro.mk.intro.mk\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nh : Sorted r l\nk iy : \u2115\nhiy : iy < length (drop k l)\nhy : get (drop k l) { val := iy, isLt := hiy } \u2208 drop k l\nix : \u2115\nhix\u271d : ix < length (take k l)\nhix : ix < min k (length l)\nhx : get (take k l) { val := ix, isLt := hix\u271d } \u2208 take k l\n\u22a2 r (get l { val := \u2191{ val := ix, isLt := hix\u271d }, isLt := (_ : \u2191{ val := ix, isLt := hix\u271d } < length l) })\n (get l { val := k + \u2191{ val := iy, isLt := hiy }, isLt := (_ : k + \u2191{ val := iy, isLt := hiy } < length l) })\n[PROOFSTEP]\nexact h.rel_nthLe_of_lt _ _ (ix.lt_add_right _ _ (lt_min_iff.mp hix).left)\n[GOAL]\nn : \u2115\n\u03b1 : Type uu\ninst\u271d : Preorder \u03b1\nf : Fin n \u2192 \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 Sorted r (ofFn f) \u2194 ((fun x x_1 => x < x_1) \u21d2 r) f f\n[PROOFSTEP]\nsimp_rw [Sorted, pairwise_iff_get, get_ofFn, Relator.LiftFun]\n[GOAL]\nn : \u2115\n\u03b1 : Type uu\ninst\u271d : Preorder \u03b1\nf : Fin n \u2192 \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 (\u2200 (i j : Fin (length (ofFn f))),\n i < j \u2192 r (f (\u2191(Fin.castIso (_ : length (ofFn f) = n)) i)) (f (\u2191(Fin.castIso (_ : length (ofFn f) = n)) j))) \u2194\n \u2200 \u2983a b : Fin n\u2984, a < b \u2192 r (f a) (f b)\n[PROOFSTEP]\nexact Iff.symm (Fin.castIso _).surjective.forall\u2082\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nhd : \u03b1\ntl : List \u03b1\na : \u03b1\n\u22a2 length (orderedInsert r a (hd :: tl)) = length (hd :: tl) + 1\n[PROOFSTEP]\ndsimp [orderedInsert]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nhd : \u03b1\ntl : List \u03b1\na : \u03b1\n\u22a2 length (if r a hd then a :: hd :: tl else hd :: orderedInsert r a tl) = Nat.succ (length tl) + 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nhd : \u03b1\ntl : List \u03b1\na : \u03b1\nh\u271d : r a hd\n\u22a2 length (a :: hd :: tl) = Nat.succ (length tl) + 1\n[PROOFSTEP]\nsimp [orderedInsert_length tl]\n[GOAL]\ncase neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nhd : \u03b1\ntl : List \u03b1\na : \u03b1\nh\u271d : \u00acr a hd\n\u22a2 length (hd :: orderedInsert r a tl) = Nat.succ (length tl) + 1\n[PROOFSTEP]\nsimp [orderedInsert_length tl]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\n\u22a2 orderedInsert r a (b :: l) =\n takeWhile (fun b => decide \u00acr a b) (b :: l) ++ a :: dropWhile (fun b => decide \u00acr a b) (b :: l)\n[PROOFSTEP]\ndsimp only [orderedInsert]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\n\u22a2 (if r a b then a :: b :: l else b :: orderedInsert r a l) =\n takeWhile (fun b => decide \u00acr a b) (b :: l) ++ a :: dropWhile (fun b => decide \u00acr a b) (b :: l)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\nh : r a b\n\u22a2 a :: b :: l = takeWhile (fun b => decide \u00acr a b) (b :: l) ++ a :: dropWhile (fun b => decide \u00acr a b) (b :: l)\n[PROOFSTEP]\nsimp [takeWhile, dropWhile, *, orderedInsert_eq_take_drop a l]\n[GOAL]\ncase neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\nh : \u00acr a b\n\u22a2 b :: orderedInsert r a l =\n takeWhile (fun b => decide \u00acr a b) (b :: l) ++ a :: dropWhile (fun b => decide \u00acr a b) (b :: l)\n[PROOFSTEP]\nsimp [takeWhile, dropWhile, *, orderedInsert_eq_take_drop a l]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\n\u22a2 orderedInsert r a (b :: l) ~ a :: b :: l\n[PROOFSTEP]\nby_cases h : a \u227c b\n[GOAL]\ncase pos\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\nh : r a b\n\u22a2 orderedInsert r a (b :: l) ~ a :: b :: l\n[PROOFSTEP]\nsimp [orderedInsert, h]\n[GOAL]\ncase neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\nh : \u00acr a b\n\u22a2 orderedInsert r a (b :: l) ~ a :: b :: l\n[PROOFSTEP]\nsimpa [orderedInsert, h] using ((perm_orderedInsert a l).cons _).trans (Perm.swap _ _ _)\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableRel r\ninst\u271d : DecidableEq \u03b1\nL : List \u03b1\na b : \u03b1\n\u22a2 count a (orderedInsert r b L) = count a L + if a = b then 1 else 0\n[PROOFSTEP]\nrw [(L.perm_orderedInsert r b).count_eq, count_cons]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableRel r\ninst\u271d : DecidableEq \u03b1\nL : List \u03b1\na b : \u03b1\n\u22a2 (if a = b then Nat.succ (count a L) else count a L) = count a L + if a = b then 1 else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableRel r\ninst\u271d : DecidableEq \u03b1\nL : List \u03b1\na b : \u03b1\nh\u271d : a = b\n\u22a2 Nat.succ (count a L) = count a L + 1\n[PROOFSTEP]\nsimp only [Nat.succ_eq_add_one, add_zero]\n[GOAL]\ncase neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableRel r\ninst\u271d : DecidableEq \u03b1\nL : List \u03b1\na b : \u03b1\nh\u271d : \u00aca = b\n\u22a2 count a L = count a L + 0\n[PROOFSTEP]\nsimp only [Nat.succ_eq_add_one, add_zero]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nb : \u03b1\nl : List \u03b1\n\u22a2 insertionSort r (b :: l) ~ b :: l\n[PROOFSTEP]\nsimpa [insertionSort] using (perm_orderedInsert _ _ _).trans ((perm_insertionSort l).cons b)\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\nh : Sorted r (a :: b :: l)\n\u22a2 insertionSort r (a :: b :: l) = a :: b :: l\n[PROOFSTEP]\nrw [insertionSort, Sorted.insertionSort_eq, orderedInsert, if_pos]\n[GOAL]\ncase hc\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\nh : Sorted r (a :: b :: l)\n\u22a2 r a b\n\u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r a b : \u03b1 l : List \u03b1 h : Sorted r (a :: b :: l) \u22a2 Sorted r (b :: l)\n[PROOFSTEP]\nexacts [rel_of_sorted_cons h _ (mem_cons_self _ _), h.tail]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na b : \u03b1\nl : List \u03b1\nh : Sorted r (b :: l)\n\u22a2 Sorted r (List.orderedInsert r a (b :: l))\n[PROOFSTEP]\nby_cases h' : a \u227c b\n[GOAL]\ncase pos\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na b : \u03b1\nl : List \u03b1\nh : Sorted r (b :: l)\nh' : r a b\n\u22a2 Sorted r (List.orderedInsert r a (b :: l))\n[PROOFSTEP]\nrw [List.orderedInsert, if_pos h', sorted_cons]\n[GOAL]\ncase pos\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na b : \u03b1\nl : List \u03b1\nh : Sorted r (b :: l)\nh' : r a b\n\u22a2 (\u2200 (b_1 : \u03b1), b_1 \u2208 b :: l \u2192 r a b_1) \u2227 Sorted r (b :: l)\n[PROOFSTEP]\nexact \u27e8forall_mem_cons.2 \u27e8h', fun c hc => _root_.trans h' (rel_of_sorted_cons h _ hc)\u27e9, h\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na b : \u03b1\nl : List \u03b1\nh : Sorted r (b :: l)\nh' : \u00acr a b\n\u22a2 Sorted r (List.orderedInsert r a (b :: l))\n[PROOFSTEP]\nsuffices \u2200 b' : \u03b1, b' \u2208 List.orderedInsert r a l \u2192 r b b' by simpa [orderedInsert, h', h.of_cons.orderedInsert a l]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na b : \u03b1\nl : List \u03b1\nh : Sorted r (b :: l)\nh' : \u00acr a b\nthis : \u2200 (b' : \u03b1), b' \u2208 List.orderedInsert r a l \u2192 r b b'\n\u22a2 Sorted r (List.orderedInsert r a (b :: l))\n[PROOFSTEP]\nsimpa [orderedInsert, h', h.of_cons.orderedInsert a l]\n[GOAL]\ncase neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na b : \u03b1\nl : List \u03b1\nh : Sorted r (b :: l)\nh' : \u00acr a b\n\u22a2 \u2200 (b' : \u03b1), b' \u2208 List.orderedInsert r a l \u2192 r b b'\n[PROOFSTEP]\nintro b' bm\n[GOAL]\ncase neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na b : \u03b1\nl : List \u03b1\nh : Sorted r (b :: l)\nh' : \u00acr a b\nb' : \u03b1\nbm : b' \u2208 List.orderedInsert r a l\n\u22a2 r b b'\n[PROOFSTEP]\ncases' show b' = a \u2228 b' \u2208 l by simpa using (perm_orderedInsert _ _ _).subset bm with be bm\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na b : \u03b1\nl : List \u03b1\nh : Sorted r (b :: l)\nh' : \u00acr a b\nb' : \u03b1\nbm : b' \u2208 List.orderedInsert r a l\n\u22a2 b' = a \u2228 b' \u2208 l\n[PROOFSTEP]\nsimpa using (perm_orderedInsert _ _ _).subset bm\n[GOAL]\ncase neg.inl\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na b : \u03b1\nl : List \u03b1\nh : Sorted r (b :: l)\nh' : \u00acr a b\nb' : \u03b1\nbm : b' \u2208 List.orderedInsert r a l\nbe : b' = a\n\u22a2 r b b'\n[PROOFSTEP]\nsubst b'\n[GOAL]\ncase neg.inl\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na b : \u03b1\nl : List \u03b1\nh : Sorted r (b :: l)\nh' : \u00acr a b\nbm : a \u2208 List.orderedInsert r a l\n\u22a2 r b a\n[PROOFSTEP]\nexact (total_of r _ _).resolve_left h'\n[GOAL]\ncase neg.inr\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na b : \u03b1\nl : List \u03b1\nh : Sorted r (b :: l)\nh' : \u00acr a b\nb' : \u03b1\nbm\u271d : b' \u2208 List.orderedInsert r a l\nbm : b' \u2208 l\n\u22a2 r b b'\n[PROOFSTEP]\nexact rel_of_sorted_cons h _ bm\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl l\u2081 l\u2082 : List \u03b1\nh : split l = (l\u2081, l\u2082)\n\u22a2 split (a :: l) = (a :: l\u2082, l\u2081)\n[PROOFSTEP]\nrw [split, h]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl l\u2081' l\u2082' : List \u03b1\nh : split (a :: l) = (l\u2081', l\u2082')\n\u22a2 length l\u2081' \u2264 length (a :: l) \u2227 length l\u2082' \u2264 length (a :: l)\n[PROOFSTEP]\ncases' e : split l with l\u2081 l\u2082\n[GOAL]\ncase mk\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl l\u2081' l\u2082' : List \u03b1\nh : split (a :: l) = (l\u2081', l\u2082')\nl\u2081 l\u2082 : List \u03b1\ne : split l = (l\u2081, l\u2082)\n\u22a2 length l\u2081' \u2264 length (a :: l) \u2227 length l\u2082' \u2264 length (a :: l)\n[PROOFSTEP]\ninjection (split_cons_of_eq _ e).symm.trans h\n[GOAL]\ncase mk\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl l\u2081' l\u2082' : List \u03b1\nh : split (a :: l) = (l\u2081', l\u2082')\nl\u2081 l\u2082 : List \u03b1\ne : split l = (l\u2081, l\u2082)\nfst_eq\u271d : a :: l\u2082 = l\u2081'\nsnd_eq\u271d : l\u2081 = l\u2082'\n\u22a2 length l\u2081' \u2264 length (a :: l) \u2227 length l\u2082' \u2264 length (a :: l)\n[PROOFSTEP]\nsubsts l\u2081' l\u2082'\n[GOAL]\ncase mk\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl l\u2081 l\u2082 : List \u03b1\ne : split l = (l\u2081, l\u2082)\nh : split (a :: l) = (a :: l\u2082, l\u2081)\n\u22a2 length (a :: l\u2082) \u2264 length (a :: l) \u2227 length l\u2081 \u2264 length (a :: l)\n[PROOFSTEP]\ncases' length_split_le e with h\u2081 h\u2082\n[GOAL]\ncase mk.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl l\u2081 l\u2082 : List \u03b1\ne : split l = (l\u2081, l\u2082)\nh : split (a :: l) = (a :: l\u2082, l\u2081)\nh\u2081 : length l\u2081 \u2264 length l\nh\u2082 : length l\u2082 \u2264 length l\n\u22a2 length (a :: l\u2082) \u2264 length (a :: l) \u2227 length l\u2081 \u2264 length (a :: l)\n[PROOFSTEP]\nexact \u27e8Nat.succ_le_succ h\u2082, Nat.le_succ_of_le h\u2081\u27e9\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl l\u2081 l\u2082 : List \u03b1\nh : split (a :: b :: l) = (l\u2081, l\u2082)\n\u22a2 length l\u2081 < length (a :: b :: l) \u2227 length l\u2082 < length (a :: b :: l)\n[PROOFSTEP]\ncases' e : split l with l\u2081' l\u2082'\n[GOAL]\ncase mk\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl l\u2081 l\u2082 : List \u03b1\nh : split (a :: b :: l) = (l\u2081, l\u2082)\nl\u2081' l\u2082' : List \u03b1\ne : split l = (l\u2081', l\u2082')\n\u22a2 length l\u2081 < length (a :: b :: l) \u2227 length l\u2082 < length (a :: b :: l)\n[PROOFSTEP]\ninjection (split_cons_of_eq _ (split_cons_of_eq _ e)).symm.trans h\n[GOAL]\ncase mk\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl l\u2081 l\u2082 : List \u03b1\nh : split (a :: b :: l) = (l\u2081, l\u2082)\nl\u2081' l\u2082' : List \u03b1\ne : split l = (l\u2081', l\u2082')\nfst_eq\u271d : a :: l\u2081' = l\u2081\nsnd_eq\u271d : b :: l\u2082' = l\u2082\n\u22a2 length l\u2081 < length (a :: b :: l) \u2227 length l\u2082 < length (a :: b :: l)\n[PROOFSTEP]\nsubsts l\u2081 l\u2082\n[GOAL]\ncase mk\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl l\u2081' l\u2082' : List \u03b1\ne : split l = (l\u2081', l\u2082')\nh : split (a :: b :: l) = (a :: l\u2081', b :: l\u2082')\n\u22a2 length (a :: l\u2081') < length (a :: b :: l) \u2227 length (b :: l\u2082') < length (a :: b :: l)\n[PROOFSTEP]\ncases' length_split_le e with h\u2081 h\u2082\n[GOAL]\ncase mk.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl l\u2081' l\u2082' : List \u03b1\ne : split l = (l\u2081', l\u2082')\nh : split (a :: b :: l) = (a :: l\u2081', b :: l\u2082')\nh\u2081 : length l\u2081' \u2264 length l\nh\u2082 : length l\u2082' \u2264 length l\n\u22a2 length (a :: l\u2081') < length (a :: b :: l) \u2227 length (b :: l\u2082') < length (a :: b :: l)\n[PROOFSTEP]\nexact \u27e8Nat.succ_le_succ (Nat.succ_le_succ h\u2081), Nat.succ_le_succ (Nat.succ_le_succ h\u2082)\u27e9\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl l\u2081' l\u2082' : List \u03b1\nh : split (a :: l) = (l\u2081', l\u2082')\n\u22a2 a :: l ~ l\u2081' ++ l\u2082'\n[PROOFSTEP]\ncases' e : split l with l\u2081 l\u2082\n[GOAL]\ncase mk\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl l\u2081' l\u2082' : List \u03b1\nh : split (a :: l) = (l\u2081', l\u2082')\nl\u2081 l\u2082 : List \u03b1\ne : split l = (l\u2081, l\u2082)\n\u22a2 a :: l ~ l\u2081' ++ l\u2082'\n[PROOFSTEP]\ninjection (split_cons_of_eq _ e).symm.trans h\n[GOAL]\ncase mk\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl l\u2081' l\u2082' : List \u03b1\nh : split (a :: l) = (l\u2081', l\u2082')\nl\u2081 l\u2082 : List \u03b1\ne : split l = (l\u2081, l\u2082)\nfst_eq\u271d : a :: l\u2082 = l\u2081'\nsnd_eq\u271d : l\u2081 = l\u2082'\n\u22a2 a :: l ~ l\u2081' ++ l\u2082'\n[PROOFSTEP]\nsubsts l\u2081' l\u2082'\n[GOAL]\ncase mk\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl l\u2081 l\u2082 : List \u03b1\ne : split l = (l\u2081, l\u2082)\nh : split (a :: l) = (a :: l\u2082, l\u2081)\n\u22a2 a :: l ~ a :: l\u2082 ++ l\u2081\n[PROOFSTEP]\nexact ((perm_split e).trans perm_append_comm).cons a\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\n\u22a2 List \u03b1\n[PROOFSTEP]\nlet ls := (split (a :: b :: l))\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\nls : List \u03b1 \u00d7 List \u03b1 := split (a :: b :: l)\n\u22a2 List \u03b1\n[PROOFSTEP]\nhave e : split (a :: b :: l) = \u27e8ls.1, ls.2\u27e9 := rfl\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\nls : List \u03b1 \u00d7 List \u03b1 := split (a :: b :: l)\ne : split (a :: b :: l) = (ls.fst, ls.snd)\n\u22a2 List \u03b1\n[PROOFSTEP]\nhave h := length_split_lt e\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\nls : List \u03b1 \u00d7 List \u03b1 := split (a :: b :: l)\ne : split (a :: b :: l) = (ls.fst, ls.snd)\nh : length ls.fst < length (a :: b :: l) \u2227 length ls.snd < length (a :: b :: l)\n\u22a2 List \u03b1\n[PROOFSTEP]\nhave := h.1\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\nls : List \u03b1 \u00d7 List \u03b1 := split (a :: b :: l)\ne : split (a :: b :: l) = (ls.fst, ls.snd)\nh : length ls.fst < length (a :: b :: l) \u2227 length ls.snd < length (a :: b :: l)\nthis : length ls.fst < length (a :: b :: l)\n\u22a2 List \u03b1\n[PROOFSTEP]\nhave := h.2\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\nls : List \u03b1 \u00d7 List \u03b1 := split (a :: b :: l)\ne : split (a :: b :: l) = (ls.fst, ls.snd)\nh : length ls.fst < length (a :: b :: l) \u2227 length ls.snd < length (a :: b :: l)\nthis\u271d : length ls.fst < length (a :: b :: l)\nthis : length ls.snd < length (a :: b :: l)\n\u22a2 List \u03b1\n[PROOFSTEP]\nexact merge r (mergeSort ls.1) (mergeSort ls.2)\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl l\u2081 l\u2082 : List \u03b1\nh : split (a :: b :: l) = (l\u2081, l\u2082)\n\u22a2 mergeSort r (a :: b :: l) = merge r (mergeSort r l\u2081) (mergeSort r l\u2082)\n[PROOFSTEP]\nsimp only [mergeSort, h]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\n\u22a2 merge r [] [] ~ [] ++ []\n[PROOFSTEP]\nsimp [merge]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nb : \u03b1\nl' : List \u03b1\n\u22a2 merge r [] (b :: l') ~ [] ++ b :: l'\n[PROOFSTEP]\nsimp [merge]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl : List \u03b1\n\u22a2 merge r (a :: l) [] ~ a :: l ++ []\n[PROOFSTEP]\nsimp [merge]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\n\u22a2 merge r (a :: l) (b :: l') ~ a :: l ++ b :: l'\n[PROOFSTEP]\nby_cases h : a \u227c b\n[GOAL]\ncase pos\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh : r a b\n\u22a2 merge r (a :: l) (b :: l') ~ a :: l ++ b :: l'\n[PROOFSTEP]\nsimpa [merge, h] using perm_merge _ _\n[GOAL]\ncase neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh : \u00acr a b\n\u22a2 merge r (a :: l) (b :: l') ~ a :: l ++ b :: l'\n[PROOFSTEP]\nsuffices b :: merge r (a :: l) l' ~ a :: (l ++ b :: l') by simpa [merge, h]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh : \u00acr a b\nthis : b :: merge r (a :: l) l' ~ a :: (l ++ b :: l')\n\u22a2 merge r (a :: l) (b :: l') ~ a :: l ++ b :: l'\n[PROOFSTEP]\nsimpa [merge, h]\n[GOAL]\ncase neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh : \u00acr a b\n\u22a2 b :: merge r (a :: l) l' ~ a :: (l ++ b :: l')\n[PROOFSTEP]\nexact ((perm_merge _ _).cons _).trans ((swap _ _ _).trans (perm_middle.symm.cons _))\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\n\u22a2 mergeSort r [] ~ []\n[PROOFSTEP]\nsimp [mergeSort]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\n\u22a2 mergeSort r [a] ~ [a]\n[PROOFSTEP]\nsimp [mergeSort]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\n\u22a2 mergeSort r (a :: b :: l) ~ a :: b :: l\n[PROOFSTEP]\ncases' e : split (a :: b :: l) with l\u2081 l\u2082\n[GOAL]\ncase mk\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl l\u2081 l\u2082 : List \u03b1\ne : split (a :: b :: l) = (l\u2081, l\u2082)\n\u22a2 mergeSort r (a :: b :: l) ~ a :: b :: l\n[PROOFSTEP]\ncases' length_split_lt e with h\u2081 h\u2082\n[GOAL]\ncase mk.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl l\u2081 l\u2082 : List \u03b1\ne : split (a :: b :: l) = (l\u2081, l\u2082)\nh\u2081 : length l\u2081 < length (a :: b :: l)\nh\u2082 : length l\u2082 < length (a :: b :: l)\n\u22a2 mergeSort r (a :: b :: l) ~ a :: b :: l\n[PROOFSTEP]\nrw [mergeSort_cons_cons r e]\n[GOAL]\ncase mk.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl l\u2081 l\u2082 : List \u03b1\ne : split (a :: b :: l) = (l\u2081, l\u2082)\nh\u2081 : length l\u2081 < length (a :: b :: l)\nh\u2082 : length l\u2082 < length (a :: b :: l)\n\u22a2 merge r (mergeSort r l\u2081) (mergeSort r l\u2082) ~ a :: b :: l\n[PROOFSTEP]\napply (perm_merge r _ _).trans\n[GOAL]\ncase mk.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl l\u2081 l\u2082 : List \u03b1\ne : split (a :: b :: l) = (l\u2081, l\u2082)\nh\u2081 : length l\u2081 < length (a :: b :: l)\nh\u2082 : length l\u2082 < length (a :: b :: l)\n\u22a2 mergeSort r l\u2081 ++ mergeSort r l\u2082 ~ a :: b :: l\n[PROOFSTEP]\nexact ((perm_mergeSort l\u2081).append (perm_mergeSort l\u2082)).trans (perm_split e).symm\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\nx\u271d\u00b9 x\u271d : Sorted r []\n\u22a2 Sorted r (List.merge r [] [])\n[PROOFSTEP]\nsimp [List.merge]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\nb : \u03b1\nl' : List \u03b1\nx\u271d : Sorted r []\nh\u2082 : Sorted r (b :: l')\n\u22a2 Sorted r (List.merge r [] (b :: l'))\n[PROOFSTEP]\nsimpa [List.merge] using h\u2082\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nh\u2081 : Sorted r (a :: l)\nx\u271d : Sorted r []\n\u22a2 Sorted r (List.merge r (a :: l) [])\n[PROOFSTEP]\nsimpa [List.merge] using h\u2081\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\n\u22a2 Sorted r (List.merge r (a :: l) (b :: l'))\n[PROOFSTEP]\nby_cases h : a \u227c b\n[GOAL]\ncase pos\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : r a b\n\u22a2 Sorted r (List.merge r (a :: l) (b :: l'))\n[PROOFSTEP]\nsuffices \u2200 (b' : \u03b1) (_ : b' \u2208 List.merge r l (b :: l')), r a b' by simpa [List.merge, h, h\u2081.of_cons.merge h\u2082]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : r a b\nthis : \u2200 (b' : \u03b1), b' \u2208 List.merge r l (b :: l') \u2192 r a b'\n\u22a2 Sorted r (List.merge r (a :: l) (b :: l'))\n[PROOFSTEP]\nsimpa [List.merge, h, h\u2081.of_cons.merge h\u2082]\n[GOAL]\ncase pos\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : r a b\n\u22a2 \u2200 (b' : \u03b1), b' \u2208 List.merge r l (b :: l') \u2192 r a b'\n[PROOFSTEP]\nintro b' bm\n[GOAL]\ncase pos\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : r a b\nb' : \u03b1\nbm : b' \u2208 List.merge r l (b :: l')\n\u22a2 r a b'\n[PROOFSTEP]\nrcases show b' = b \u2228 b' \u2208 l \u2228 b' \u2208 l' by simpa [or_left_comm] using (perm_merge _ _ _).subset bm with (be | bl | bl')\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : r a b\nb' : \u03b1\nbm : b' \u2208 List.merge r l (b :: l')\n\u22a2 b' = b \u2228 b' \u2208 l \u2228 b' \u2208 l'\n[PROOFSTEP]\nsimpa [or_left_comm] using (perm_merge _ _ _).subset bm\n[GOAL]\ncase pos.inl\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : r a b\nb' : \u03b1\nbm : b' \u2208 List.merge r l (b :: l')\nbe : b' = b\n\u22a2 r a b'\n[PROOFSTEP]\nsubst b'\n[GOAL]\ncase pos.inl\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : r a b\nbm : b \u2208 List.merge r l (b :: l')\n\u22a2 r a b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.inr.inl\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : r a b\nb' : \u03b1\nbm : b' \u2208 List.merge r l (b :: l')\nbl : b' \u2208 l\n\u22a2 r a b'\n[PROOFSTEP]\nexact rel_of_sorted_cons h\u2081 _ bl\n[GOAL]\ncase pos.inr.inr\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : r a b\nb' : \u03b1\nbm : b' \u2208 List.merge r l (b :: l')\nbl' : b' \u2208 l'\n\u22a2 r a b'\n[PROOFSTEP]\nexact _root_.trans h (rel_of_sorted_cons h\u2082 _ bl')\n[GOAL]\ncase neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : \u00acr a b\n\u22a2 Sorted r (List.merge r (a :: l) (b :: l'))\n[PROOFSTEP]\nsuffices \u2200 (b' : \u03b1) (_ : b' \u2208 List.merge r (a :: l) l'), r b b' by simpa [List.merge, h, h\u2081.merge h\u2082.of_cons]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : \u00acr a b\nthis : \u2200 (b' : \u03b1), b' \u2208 List.merge r (a :: l) l' \u2192 r b b'\n\u22a2 Sorted r (List.merge r (a :: l) (b :: l'))\n[PROOFSTEP]\nsimpa [List.merge, h, h\u2081.merge h\u2082.of_cons]\n[GOAL]\ncase neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : \u00acr a b\n\u22a2 \u2200 (b' : \u03b1), b' \u2208 List.merge r (a :: l) l' \u2192 r b b'\n[PROOFSTEP]\nintro b' bm\n[GOAL]\ncase neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : \u00acr a b\nb' : \u03b1\nbm : b' \u2208 List.merge r (a :: l) l'\n\u22a2 r b b'\n[PROOFSTEP]\nhave ba : b \u227c a := (total_of r _ _).resolve_left h\n[GOAL]\ncase neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : \u00acr a b\nb' : \u03b1\nbm : b' \u2208 List.merge r (a :: l) l'\nba : r b a\n\u22a2 r b b'\n[PROOFSTEP]\nhave : b' = a \u2228 b' \u2208 l \u2228 b' \u2208 l' := by simpa using (perm_merge _ _ _).subset bm\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : \u00acr a b\nb' : \u03b1\nbm : b' \u2208 List.merge r (a :: l) l'\nba : r b a\n\u22a2 b' = a \u2228 b' \u2208 l \u2228 b' \u2208 l'\n[PROOFSTEP]\nsimpa using (perm_merge _ _ _).subset bm\n[GOAL]\ncase neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : \u00acr a b\nb' : \u03b1\nbm : b' \u2208 List.merge r (a :: l) l'\nba : r b a\nthis : b' = a \u2228 b' \u2208 l \u2228 b' \u2208 l'\n\u22a2 r b b'\n[PROOFSTEP]\nrcases this with (be | bl | bl')\n[GOAL]\ncase neg.inl\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : \u00acr a b\nb' : \u03b1\nbm : b' \u2208 List.merge r (a :: l) l'\nba : r b a\nbe : b' = a\n\u22a2 r b b'\n[PROOFSTEP]\nsubst b'\n[GOAL]\ncase neg.inl\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : \u00acr a b\nba : r b a\nbm : a \u2208 List.merge r (a :: l) l'\n\u22a2 r b a\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg.inr.inl\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : \u00acr a b\nb' : \u03b1\nbm : b' \u2208 List.merge r (a :: l) l'\nba : r b a\nbl : b' \u2208 l\n\u22a2 r b b'\n[PROOFSTEP]\nexact _root_.trans ba (rel_of_sorted_cons h\u2081 _ bl)\n[GOAL]\ncase neg.inr.inr\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\nb : \u03b1\nl' : List \u03b1\nh\u2081 : Sorted r (a :: l)\nh\u2082 : Sorted r (b :: l')\nh : \u00acr a b\nb' : \u03b1\nbm : b' \u2208 List.merge r (a :: l) l'\nba : r b a\nbl' : b' \u2208 l'\n\u22a2 r b b'\n[PROOFSTEP]\nexact rel_of_sorted_cons h\u2082 _ bl'\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\n\u22a2 Sorted r (mergeSort r [])\n[PROOFSTEP]\nsimp [mergeSort]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\n\u22a2 Sorted r (mergeSort r [a])\n[PROOFSTEP]\nsimp [mergeSort]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na b : \u03b1\nl : List \u03b1\n\u22a2 Sorted r (mergeSort r (a :: b :: l))\n[PROOFSTEP]\ncases' e : split (a :: b :: l) with l\u2081 l\u2082\n[GOAL]\ncase mk\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na b : \u03b1\nl l\u2081 l\u2082 : List \u03b1\ne : split (a :: b :: l) = (l\u2081, l\u2082)\n\u22a2 Sorted r (mergeSort r (a :: b :: l))\n[PROOFSTEP]\ncases' length_split_lt e with h\u2081 h\u2082\n[GOAL]\ncase mk.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na b : \u03b1\nl l\u2081 l\u2082 : List \u03b1\ne : split (a :: b :: l) = (l\u2081, l\u2082)\nh\u2081 : length l\u2081 < length (a :: b :: l)\nh\u2082 : length l\u2082 < length (a :: b :: l)\n\u22a2 Sorted r (mergeSort r (a :: b :: l))\n[PROOFSTEP]\nrw [mergeSort_cons_cons r e]\n[GOAL]\ncase mk.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidableRel r\ninst\u271d\u00b9 : IsTotal \u03b1 r\ninst\u271d : IsTrans \u03b1 r\na b : \u03b1\nl l\u2081 l\u2082 : List \u03b1\ne : split (a :: b :: l) = (l\u2081, l\u2082)\nh\u2081 : length l\u2081 < length (a :: b :: l)\nh\u2082 : length l\u2082 < length (a :: b :: l)\n\u22a2 Sorted r (merge r (mergeSort r l\u2081) (mergeSort r l\u2082))\n[PROOFSTEP]\nexact (sorted_mergeSort l\u2081).merge (sorted_mergeSort l\u2082)\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\n\u22a2 mergeSort r [] = []\n[PROOFSTEP]\nrw [List.mergeSort]\n[GOAL]\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\n\u22a2 mergeSort r [a] = [a]\n[PROOFSTEP]\nrw [List.mergeSort]\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Sort", "llama_tokens": 19387, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419958239132, "lm_q2_score": 0.6370308013713524, "lm_q1q2_score": 0.4412342856631603}} {"text": "[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX X' Y Y' : C\nf : X \u27f6 Y\ng : X' \u27f6 Y'\n\u22a2 tensorHom \u212c f g \u226b (BinaryFan.braiding (\u212c Y Y').isLimit (\u212c Y' Y).isLimit).hom =\n (BinaryFan.braiding (\u212c X X').isLimit (\u212c X' X).isLimit).hom \u226b tensorHom \u212c g f\n[PROOFSTEP]\ndsimp [tensorHom, Limits.BinaryFan.braiding]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX X' Y Y' : C\nf : X \u27f6 Y\ng : X' \u27f6 Y'\n\u22a2 IsLimit.lift (\u212c Y Y').isLimit (BinaryFan.mk (BinaryFan.fst (\u212c X X').cone \u226b f) (BinaryFan.snd (\u212c X X').cone \u226b g)) \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c Y Y').isLimit (IsLimit.swapBinaryFan (\u212c Y' Y).isLimit)).hom =\n (IsLimit.conePointUniqueUpToIso (\u212c X X').isLimit (IsLimit.swapBinaryFan (\u212c X' X).isLimit)).hom \u226b\n IsLimit.lift (\u212c Y' Y).isLimit (BinaryFan.mk (BinaryFan.fst (\u212c X' X).cone \u226b g) (BinaryFan.snd (\u212c X' X).cone \u226b f))\n[PROOFSTEP]\napply (\u212c _ _).isLimit.hom_ext\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX X' Y Y' : C\nf : X \u27f6 Y\ng : X' \u27f6 Y'\n\u22a2 \u2200 (j : Discrete WalkingPair),\n (IsLimit.lift (\u212c Y Y').isLimit (BinaryFan.mk (BinaryFan.fst (\u212c X X').cone \u226b f) (BinaryFan.snd (\u212c X X').cone \u226b g)) \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c Y Y').isLimit (IsLimit.swapBinaryFan (\u212c Y' Y).isLimit)).hom) \u226b\n NatTrans.app (\u212c Y' Y).cone.\u03c0 j =\n ((IsLimit.conePointUniqueUpToIso (\u212c X X').isLimit (IsLimit.swapBinaryFan (\u212c X' X).isLimit)).hom \u226b\n IsLimit.lift (\u212c Y' Y).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c X' X).cone \u226b g) (BinaryFan.snd (\u212c X' X).cone \u226b f))) \u226b\n NatTrans.app (\u212c Y' Y).cone.\u03c0 j\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX X' Y Y' : C\nf : X \u27f6 Y\ng : X' \u27f6 Y'\n\u22a2 (IsLimit.lift (\u212c Y Y').isLimit (BinaryFan.mk (BinaryFan.fst (\u212c X X').cone \u226b f) (BinaryFan.snd (\u212c X X').cone \u226b g)) \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c Y Y').isLimit (IsLimit.swapBinaryFan (\u212c Y' Y).isLimit)).hom) \u226b\n NatTrans.app (\u212c Y' Y).cone.\u03c0 { as := WalkingPair.left } =\n ((IsLimit.conePointUniqueUpToIso (\u212c X X').isLimit (IsLimit.swapBinaryFan (\u212c X' X).isLimit)).hom \u226b\n IsLimit.lift (\u212c Y' Y).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c X' X).cone \u226b g) (BinaryFan.snd (\u212c X' X).cone \u226b f))) \u226b\n NatTrans.app (\u212c Y' Y).cone.\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\ndsimp [Limits.IsLimit.conePointUniqueUpToIso]\n[GOAL]\ncase mk.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX X' Y Y' : C\nf : X \u27f6 Y\ng : X' \u27f6 Y'\n\u22a2 (IsLimit.lift (\u212c Y Y').isLimit (BinaryFan.mk (BinaryFan.fst (\u212c X X').cone \u226b f) (BinaryFan.snd (\u212c X X').cone \u226b g)) \u226b\n IsLimit.lift (\u212c Y' Y).isLimit (BinaryFan.swap (\u212c Y Y').cone)) \u226b\n BinaryFan.fst (\u212c Y' Y).cone =\n (IsLimit.lift (\u212c X' X).isLimit (BinaryFan.swap (\u212c X X').cone) \u226b\n IsLimit.lift (\u212c Y' Y).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c X' X).cone \u226b g) (BinaryFan.snd (\u212c X' X).cone \u226b f))) \u226b\n BinaryFan.fst (\u212c Y' Y).cone\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX X' Y Y' : C\nf : X \u27f6 Y\ng : X' \u27f6 Y'\n\u22a2 (IsLimit.lift (\u212c Y Y').isLimit (BinaryFan.mk (BinaryFan.fst (\u212c X X').cone \u226b f) (BinaryFan.snd (\u212c X X').cone \u226b g)) \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c Y Y').isLimit (IsLimit.swapBinaryFan (\u212c Y' Y).isLimit)).hom) \u226b\n NatTrans.app (\u212c Y' Y).cone.\u03c0 { as := WalkingPair.right } =\n ((IsLimit.conePointUniqueUpToIso (\u212c X X').isLimit (IsLimit.swapBinaryFan (\u212c X' X).isLimit)).hom \u226b\n IsLimit.lift (\u212c Y' Y).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c X' X).cone \u226b g) (BinaryFan.snd (\u212c X' X).cone \u226b f))) \u226b\n NatTrans.app (\u212c Y' Y).cone.\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\ndsimp [Limits.IsLimit.conePointUniqueUpToIso]\n[GOAL]\ncase mk.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX X' Y Y' : C\nf : X \u27f6 Y\ng : X' \u27f6 Y'\n\u22a2 (IsLimit.lift (\u212c Y Y').isLimit (BinaryFan.mk (BinaryFan.fst (\u212c X X').cone \u226b f) (BinaryFan.snd (\u212c X X').cone \u226b g)) \u226b\n IsLimit.lift (\u212c Y' Y).isLimit (BinaryFan.swap (\u212c Y Y').cone)) \u226b\n BinaryFan.snd (\u212c Y' Y).cone =\n (IsLimit.lift (\u212c X' X).isLimit (BinaryFan.swap (\u212c X X').cone) \u226b\n IsLimit.lift (\u212c Y' Y).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c X' X).cone \u226b g) (BinaryFan.snd (\u212c X' X).cone \u226b f))) \u226b\n BinaryFan.snd (\u212c Y' Y).cone\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 (BinaryFan.associatorOfLimitCone \u212c X Y Z).hom \u226b\n (BinaryFan.braiding (\u212c X (tensorObj \u212c Y Z)).isLimit (\u212c (tensorObj \u212c Y Z) X).isLimit).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y Z X).hom =\n tensorHom \u212c (BinaryFan.braiding (\u212c X Y).isLimit (\u212c Y X).isLimit).hom (\ud835\udfd9 Z) \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y X Z).hom \u226b\n tensorHom \u212c (\ud835\udfd9 Y) (BinaryFan.braiding (\u212c X Z).isLimit (\u212c Z X).isLimit).hom\n[PROOFSTEP]\ndsimp [tensorHom, Limits.BinaryFan.braiding]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 (BinaryFan.associatorOfLimitCone \u212c X Y Z).hom \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X (tensorObj \u212c Y Z)).isLimit\n (IsLimit.swapBinaryFan (\u212c (tensorObj \u212c Y Z) X).isLimit)).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y Z X).hom =\n IsLimit.lift (\u212c (\u212c Y X).cone.pt Z).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Y).cone.pt Z).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Y).isLimit (IsLimit.swapBinaryFan (\u212c Y X).isLimit)).hom)\n (BinaryFan.snd (\u212c (\u212c X Y).cone.pt Z).cone \u226b \ud835\udfd9 Z)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y X Z).hom \u226b\n IsLimit.lift (\u212c Y (\u212c Z X).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c Y (\u212c X Z).cone.pt).cone \u226b \ud835\udfd9 Y)\n (BinaryFan.snd (\u212c Y (\u212c X Z).cone.pt).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Z).isLimit (IsLimit.swapBinaryFan (\u212c Z X).isLimit)).hom))\n[PROOFSTEP]\napply (\u212c _ _).isLimit.hom_ext\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 \u2200 (j : Discrete WalkingPair),\n ((BinaryFan.associatorOfLimitCone \u212c X Y Z).hom \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X (tensorObj \u212c Y Z)).isLimit\n (IsLimit.swapBinaryFan (\u212c (tensorObj \u212c Y Z) X).isLimit)).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y Z X).hom) \u226b\n NatTrans.app (\u212c Y (\u212c Z X).cone.pt).cone.\u03c0 j =\n (IsLimit.lift (\u212c (\u212c Y X).cone.pt Z).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Y).cone.pt Z).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Y).isLimit (IsLimit.swapBinaryFan (\u212c Y X).isLimit)).hom)\n (BinaryFan.snd (\u212c (\u212c X Y).cone.pt Z).cone \u226b \ud835\udfd9 Z)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y X Z).hom \u226b\n IsLimit.lift (\u212c Y (\u212c Z X).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c Y (\u212c X Z).cone.pt).cone \u226b \ud835\udfd9 Y)\n (BinaryFan.snd (\u212c Y (\u212c X Z).cone.pt).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Z).isLimit (IsLimit.swapBinaryFan (\u212c Z X).isLimit)).hom))) \u226b\n NatTrans.app (\u212c Y (\u212c Z X).cone.pt).cone.\u03c0 j\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 ((BinaryFan.associatorOfLimitCone \u212c X Y Z).hom \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X (tensorObj \u212c Y Z)).isLimit\n (IsLimit.swapBinaryFan (\u212c (tensorObj \u212c Y Z) X).isLimit)).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y Z X).hom) \u226b\n NatTrans.app (\u212c Y (\u212c Z X).cone.pt).cone.\u03c0 { as := WalkingPair.left } =\n (IsLimit.lift (\u212c (\u212c Y X).cone.pt Z).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Y).cone.pt Z).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Y).isLimit (IsLimit.swapBinaryFan (\u212c Y X).isLimit)).hom)\n (BinaryFan.snd (\u212c (\u212c X Y).cone.pt Z).cone \u226b \ud835\udfd9 Z)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y X Z).hom \u226b\n IsLimit.lift (\u212c Y (\u212c Z X).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c Y (\u212c X Z).cone.pt).cone \u226b \ud835\udfd9 Y)\n (BinaryFan.snd (\u212c Y (\u212c X Z).cone.pt).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Z).isLimit (IsLimit.swapBinaryFan (\u212c Z X).isLimit)).hom))) \u226b\n NatTrans.app (\u212c Y (\u212c Z X).cone.pt).cone.\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\ndsimp [Limits.IsLimit.conePointUniqueUpToIso]\n[GOAL]\ncase mk.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 ((BinaryFan.associatorOfLimitCone \u212c X Y Z).hom \u226b\n IsLimit.lift (\u212c (tensorObj \u212c Y Z) X).isLimit (BinaryFan.swap (\u212c X (tensorObj \u212c Y Z)).cone) \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y Z X).hom) \u226b\n BinaryFan.fst (\u212c Y (\u212c Z X).cone.pt).cone =\n (IsLimit.lift (\u212c (\u212c Y X).cone.pt Z).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Y).cone.pt Z).cone \u226b IsLimit.lift (\u212c Y X).isLimit (BinaryFan.swap (\u212c X Y).cone))\n (BinaryFan.snd (\u212c (\u212c X Y).cone.pt Z).cone \u226b \ud835\udfd9 Z)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y X Z).hom \u226b\n IsLimit.lift (\u212c Y (\u212c Z X).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c Y (\u212c X Z).cone.pt).cone \u226b \ud835\udfd9 Y)\n (BinaryFan.snd (\u212c Y (\u212c X Z).cone.pt).cone \u226b\n IsLimit.lift (\u212c Z X).isLimit (BinaryFan.swap (\u212c X Z).cone)))) \u226b\n BinaryFan.fst (\u212c Y (\u212c Z X).cone.pt).cone\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 ((BinaryFan.associatorOfLimitCone \u212c X Y Z).hom \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X (tensorObj \u212c Y Z)).isLimit\n (IsLimit.swapBinaryFan (\u212c (tensorObj \u212c Y Z) X).isLimit)).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y Z X).hom) \u226b\n NatTrans.app (\u212c Y (\u212c Z X).cone.pt).cone.\u03c0 { as := WalkingPair.right } =\n (IsLimit.lift (\u212c (\u212c Y X).cone.pt Z).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Y).cone.pt Z).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Y).isLimit (IsLimit.swapBinaryFan (\u212c Y X).isLimit)).hom)\n (BinaryFan.snd (\u212c (\u212c X Y).cone.pt Z).cone \u226b \ud835\udfd9 Z)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y X Z).hom \u226b\n IsLimit.lift (\u212c Y (\u212c Z X).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c Y (\u212c X Z).cone.pt).cone \u226b \ud835\udfd9 Y)\n (BinaryFan.snd (\u212c Y (\u212c X Z).cone.pt).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Z).isLimit (IsLimit.swapBinaryFan (\u212c Z X).isLimit)).hom))) \u226b\n NatTrans.app (\u212c Y (\u212c Z X).cone.pt).cone.\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\napply (\u212c _ _).isLimit.hom_ext\n[GOAL]\ncase mk.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 \u2200 (j : Discrete WalkingPair),\n (((BinaryFan.associatorOfLimitCone \u212c X Y Z).hom \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X (tensorObj \u212c Y Z)).isLimit\n (IsLimit.swapBinaryFan (\u212c (tensorObj \u212c Y Z) X).isLimit)).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y Z X).hom) \u226b\n NatTrans.app (\u212c Y (\u212c Z X).cone.pt).cone.\u03c0 { as := WalkingPair.right }) \u226b\n NatTrans.app (\u212c Z X).cone.\u03c0 j =\n ((IsLimit.lift (\u212c (\u212c Y X).cone.pt Z).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Y).cone.pt Z).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Y).isLimit (IsLimit.swapBinaryFan (\u212c Y X).isLimit)).hom)\n (BinaryFan.snd (\u212c (\u212c X Y).cone.pt Z).cone \u226b \ud835\udfd9 Z)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y X Z).hom \u226b\n IsLimit.lift (\u212c Y (\u212c Z X).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c Y (\u212c X Z).cone.pt).cone \u226b \ud835\udfd9 Y)\n (BinaryFan.snd (\u212c Y (\u212c X Z).cone.pt).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Z).isLimit (IsLimit.swapBinaryFan (\u212c Z X).isLimit)).hom))) \u226b\n NatTrans.app (\u212c Y (\u212c Z X).cone.pt).cone.\u03c0 { as := WalkingPair.right }) \u226b\n NatTrans.app (\u212c Z X).cone.\u03c0 j\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.right.mk.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 (((BinaryFan.associatorOfLimitCone \u212c X Y Z).hom \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X (tensorObj \u212c Y Z)).isLimit\n (IsLimit.swapBinaryFan (\u212c (tensorObj \u212c Y Z) X).isLimit)).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y Z X).hom) \u226b\n NatTrans.app (\u212c Y (\u212c Z X).cone.pt).cone.\u03c0 { as := WalkingPair.right }) \u226b\n NatTrans.app (\u212c Z X).cone.\u03c0 { as := WalkingPair.left } =\n ((IsLimit.lift (\u212c (\u212c Y X).cone.pt Z).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Y).cone.pt Z).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Y).isLimit (IsLimit.swapBinaryFan (\u212c Y X).isLimit)).hom)\n (BinaryFan.snd (\u212c (\u212c X Y).cone.pt Z).cone \u226b \ud835\udfd9 Z)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y X Z).hom \u226b\n IsLimit.lift (\u212c Y (\u212c Z X).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c Y (\u212c X Z).cone.pt).cone \u226b \ud835\udfd9 Y)\n (BinaryFan.snd (\u212c Y (\u212c X Z).cone.pt).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Z).isLimit (IsLimit.swapBinaryFan (\u212c Z X).isLimit)).hom))) \u226b\n NatTrans.app (\u212c Y (\u212c Z X).cone.pt).cone.\u03c0 { as := WalkingPair.right }) \u226b\n NatTrans.app (\u212c Z X).cone.\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\ndsimp [Limits.IsLimit.conePointUniqueUpToIso]\n[GOAL]\ncase mk.right.mk.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 (((BinaryFan.associatorOfLimitCone \u212c X Y Z).hom \u226b\n IsLimit.lift (\u212c (tensorObj \u212c Y Z) X).isLimit (BinaryFan.swap (\u212c X (tensorObj \u212c Y Z)).cone) \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y Z X).hom) \u226b\n BinaryFan.snd (\u212c Y (\u212c Z X).cone.pt).cone) \u226b\n BinaryFan.fst (\u212c Z X).cone =\n ((IsLimit.lift (\u212c (\u212c Y X).cone.pt Z).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Y).cone.pt Z).cone \u226b IsLimit.lift (\u212c Y X).isLimit (BinaryFan.swap (\u212c X Y).cone))\n (BinaryFan.snd (\u212c (\u212c X Y).cone.pt Z).cone \u226b \ud835\udfd9 Z)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y X Z).hom \u226b\n IsLimit.lift (\u212c Y (\u212c Z X).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c Y (\u212c X Z).cone.pt).cone \u226b \ud835\udfd9 Y)\n (BinaryFan.snd (\u212c Y (\u212c X Z).cone.pt).cone \u226b\n IsLimit.lift (\u212c Z X).isLimit (BinaryFan.swap (\u212c X Z).cone)))) \u226b\n BinaryFan.snd (\u212c Y (\u212c Z X).cone.pt).cone) \u226b\n BinaryFan.fst (\u212c Z X).cone\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.right.mk.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 (((BinaryFan.associatorOfLimitCone \u212c X Y Z).hom \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X (tensorObj \u212c Y Z)).isLimit\n (IsLimit.swapBinaryFan (\u212c (tensorObj \u212c Y Z) X).isLimit)).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y Z X).hom) \u226b\n NatTrans.app (\u212c Y (\u212c Z X).cone.pt).cone.\u03c0 { as := WalkingPair.right }) \u226b\n NatTrans.app (\u212c Z X).cone.\u03c0 { as := WalkingPair.right } =\n ((IsLimit.lift (\u212c (\u212c Y X).cone.pt Z).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Y).cone.pt Z).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Y).isLimit (IsLimit.swapBinaryFan (\u212c Y X).isLimit)).hom)\n (BinaryFan.snd (\u212c (\u212c X Y).cone.pt Z).cone \u226b \ud835\udfd9 Z)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y X Z).hom \u226b\n IsLimit.lift (\u212c Y (\u212c Z X).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c Y (\u212c X Z).cone.pt).cone \u226b \ud835\udfd9 Y)\n (BinaryFan.snd (\u212c Y (\u212c X Z).cone.pt).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Z).isLimit (IsLimit.swapBinaryFan (\u212c Z X).isLimit)).hom))) \u226b\n NatTrans.app (\u212c Y (\u212c Z X).cone.pt).cone.\u03c0 { as := WalkingPair.right }) \u226b\n NatTrans.app (\u212c Z X).cone.\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\ndsimp [Limits.IsLimit.conePointUniqueUpToIso]\n[GOAL]\ncase mk.right.mk.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 (((BinaryFan.associatorOfLimitCone \u212c X Y Z).hom \u226b\n IsLimit.lift (\u212c (tensorObj \u212c Y Z) X).isLimit (BinaryFan.swap (\u212c X (tensorObj \u212c Y Z)).cone) \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y Z X).hom) \u226b\n BinaryFan.snd (\u212c Y (\u212c Z X).cone.pt).cone) \u226b\n BinaryFan.snd (\u212c Z X).cone =\n ((IsLimit.lift (\u212c (\u212c Y X).cone.pt Z).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Y).cone.pt Z).cone \u226b IsLimit.lift (\u212c Y X).isLimit (BinaryFan.swap (\u212c X Y).cone))\n (BinaryFan.snd (\u212c (\u212c X Y).cone.pt Z).cone \u226b \ud835\udfd9 Z)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c Y X Z).hom \u226b\n IsLimit.lift (\u212c Y (\u212c Z X).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c Y (\u212c X Z).cone.pt).cone \u226b \ud835\udfd9 Y)\n (BinaryFan.snd (\u212c Y (\u212c X Z).cone.pt).cone \u226b\n IsLimit.lift (\u212c Z X).isLimit (BinaryFan.swap (\u212c X Z).cone)))) \u226b\n BinaryFan.snd (\u212c Y (\u212c Z X).cone.pt).cone) \u226b\n BinaryFan.snd (\u212c Z X).cone\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 (BinaryFan.associatorOfLimitCone \u212c X Y Z).inv \u226b\n (BinaryFan.braiding (\u212c (tensorObj \u212c X Y) Z).isLimit (\u212c Z (tensorObj \u212c X Y)).isLimit).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Z X Y).inv =\n tensorHom \u212c (\ud835\udfd9 X) (BinaryFan.braiding (\u212c Y Z).isLimit (\u212c Z Y).isLimit).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c X Z Y).inv \u226b\n tensorHom \u212c (BinaryFan.braiding (\u212c X Z).isLimit (\u212c Z X).isLimit).hom (\ud835\udfd9 Y)\n[PROOFSTEP]\ndsimp [tensorHom, Limits.BinaryFan.braiding]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 (BinaryFan.associatorOfLimitCone \u212c X Y Z).inv \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c (tensorObj \u212c X Y) Z).isLimit\n (IsLimit.swapBinaryFan (\u212c Z (tensorObj \u212c X Y)).isLimit)).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Z X Y).inv =\n IsLimit.lift (\u212c X (\u212c Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c Y Z).cone.pt).cone \u226b \ud835\udfd9 X)\n (BinaryFan.snd (\u212c X (\u212c Y Z).cone.pt).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c Y Z).isLimit (IsLimit.swapBinaryFan (\u212c Z Y).isLimit)).hom)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c X Z Y).inv \u226b\n IsLimit.lift (\u212c (\u212c Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Z).cone.pt Y).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Z).isLimit (IsLimit.swapBinaryFan (\u212c Z X).isLimit)).hom)\n (BinaryFan.snd (\u212c (\u212c X Z).cone.pt Y).cone \u226b \ud835\udfd9 Y))\n[PROOFSTEP]\napply (\u212c _ _).isLimit.hom_ext\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 \u2200 (j : Discrete WalkingPair),\n ((BinaryFan.associatorOfLimitCone \u212c X Y Z).inv \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c (tensorObj \u212c X Y) Z).isLimit\n (IsLimit.swapBinaryFan (\u212c Z (tensorObj \u212c X Y)).isLimit)).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Z X Y).inv) \u226b\n NatTrans.app (\u212c (\u212c Z X).cone.pt Y).cone.\u03c0 j =\n (IsLimit.lift (\u212c X (\u212c Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c Y Z).cone.pt).cone \u226b \ud835\udfd9 X)\n (BinaryFan.snd (\u212c X (\u212c Y Z).cone.pt).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c Y Z).isLimit (IsLimit.swapBinaryFan (\u212c Z Y).isLimit)).hom)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c X Z Y).inv \u226b\n IsLimit.lift (\u212c (\u212c Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Z).cone.pt Y).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Z).isLimit (IsLimit.swapBinaryFan (\u212c Z X).isLimit)).hom)\n (BinaryFan.snd (\u212c (\u212c X Z).cone.pt Y).cone \u226b \ud835\udfd9 Y))) \u226b\n NatTrans.app (\u212c (\u212c Z X).cone.pt Y).cone.\u03c0 j\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 ((BinaryFan.associatorOfLimitCone \u212c X Y Z).inv \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c (tensorObj \u212c X Y) Z).isLimit\n (IsLimit.swapBinaryFan (\u212c Z (tensorObj \u212c X Y)).isLimit)).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Z X Y).inv) \u226b\n NatTrans.app (\u212c (\u212c Z X).cone.pt Y).cone.\u03c0 { as := WalkingPair.left } =\n (IsLimit.lift (\u212c X (\u212c Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c Y Z).cone.pt).cone \u226b \ud835\udfd9 X)\n (BinaryFan.snd (\u212c X (\u212c Y Z).cone.pt).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c Y Z).isLimit (IsLimit.swapBinaryFan (\u212c Z Y).isLimit)).hom)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c X Z Y).inv \u226b\n IsLimit.lift (\u212c (\u212c Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Z).cone.pt Y).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Z).isLimit (IsLimit.swapBinaryFan (\u212c Z X).isLimit)).hom)\n (BinaryFan.snd (\u212c (\u212c X Z).cone.pt Y).cone \u226b \ud835\udfd9 Y))) \u226b\n NatTrans.app (\u212c (\u212c Z X).cone.pt Y).cone.\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\napply (\u212c _ _).isLimit.hom_ext\n[GOAL]\ncase mk.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 \u2200 (j : Discrete WalkingPair),\n (((BinaryFan.associatorOfLimitCone \u212c X Y Z).inv \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c (tensorObj \u212c X Y) Z).isLimit\n (IsLimit.swapBinaryFan (\u212c Z (tensorObj \u212c X Y)).isLimit)).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Z X Y).inv) \u226b\n NatTrans.app (\u212c (\u212c Z X).cone.pt Y).cone.\u03c0 { as := WalkingPair.left }) \u226b\n NatTrans.app (\u212c Z X).cone.\u03c0 j =\n ((IsLimit.lift (\u212c X (\u212c Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c Y Z).cone.pt).cone \u226b \ud835\udfd9 X)\n (BinaryFan.snd (\u212c X (\u212c Y Z).cone.pt).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c Y Z).isLimit (IsLimit.swapBinaryFan (\u212c Z Y).isLimit)).hom)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c X Z Y).inv \u226b\n IsLimit.lift (\u212c (\u212c Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Z).cone.pt Y).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Z).isLimit (IsLimit.swapBinaryFan (\u212c Z X).isLimit)).hom)\n (BinaryFan.snd (\u212c (\u212c X Z).cone.pt Y).cone \u226b \ud835\udfd9 Y))) \u226b\n NatTrans.app (\u212c (\u212c Z X).cone.pt Y).cone.\u03c0 { as := WalkingPair.left }) \u226b\n NatTrans.app (\u212c Z X).cone.\u03c0 j\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.left.mk.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 (((BinaryFan.associatorOfLimitCone \u212c X Y Z).inv \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c (tensorObj \u212c X Y) Z).isLimit\n (IsLimit.swapBinaryFan (\u212c Z (tensorObj \u212c X Y)).isLimit)).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Z X Y).inv) \u226b\n NatTrans.app (\u212c (\u212c Z X).cone.pt Y).cone.\u03c0 { as := WalkingPair.left }) \u226b\n NatTrans.app (\u212c Z X).cone.\u03c0 { as := WalkingPair.left } =\n ((IsLimit.lift (\u212c X (\u212c Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c Y Z).cone.pt).cone \u226b \ud835\udfd9 X)\n (BinaryFan.snd (\u212c X (\u212c Y Z).cone.pt).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c Y Z).isLimit (IsLimit.swapBinaryFan (\u212c Z Y).isLimit)).hom)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c X Z Y).inv \u226b\n IsLimit.lift (\u212c (\u212c Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Z).cone.pt Y).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Z).isLimit (IsLimit.swapBinaryFan (\u212c Z X).isLimit)).hom)\n (BinaryFan.snd (\u212c (\u212c X Z).cone.pt Y).cone \u226b \ud835\udfd9 Y))) \u226b\n NatTrans.app (\u212c (\u212c Z X).cone.pt Y).cone.\u03c0 { as := WalkingPair.left }) \u226b\n NatTrans.app (\u212c Z X).cone.\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\ndsimp [BinaryFan.associatorOfLimitCone, BinaryFan.associator, Limits.IsLimit.conePointUniqueUpToIso]\n[GOAL]\ncase mk.left.mk.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 ((IsLimit.lift (\u212c (\u212c X Y).cone.pt Z).isLimit (BinaryFan.assocInv (\u212c X Y).isLimit (\u212c X (\u212c Y Z).cone.pt).cone) \u226b\n IsLimit.lift (\u212c Z (tensorObj \u212c X Y)).isLimit (BinaryFan.swap (\u212c (tensorObj \u212c X Y) Z).cone) \u226b\n IsLimit.lift (\u212c (\u212c Z X).cone.pt Y).isLimit\n (BinaryFan.assocInv (\u212c Z X).isLimit (\u212c Z (\u212c X Y).cone.pt).cone)) \u226b\n BinaryFan.fst (\u212c (\u212c Z X).cone.pt Y).cone) \u226b\n BinaryFan.fst (\u212c Z X).cone =\n ((IsLimit.lift (\u212c X (\u212c Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c Y Z).cone.pt).cone \u226b \ud835\udfd9 X)\n (BinaryFan.snd (\u212c X (\u212c Y Z).cone.pt).cone \u226b IsLimit.lift (\u212c Z Y).isLimit (BinaryFan.swap (\u212c Y Z).cone))) \u226b\n IsLimit.lift (\u212c (\u212c X Z).cone.pt Y).isLimit (BinaryFan.assocInv (\u212c X Z).isLimit (\u212c X (\u212c Z Y).cone.pt).cone) \u226b\n IsLimit.lift (\u212c (\u212c Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Z).cone.pt Y).cone \u226b IsLimit.lift (\u212c Z X).isLimit (BinaryFan.swap (\u212c X Z).cone))\n (BinaryFan.snd (\u212c (\u212c X Z).cone.pt Y).cone \u226b \ud835\udfd9 Y))) \u226b\n BinaryFan.fst (\u212c (\u212c Z X).cone.pt Y).cone) \u226b\n BinaryFan.fst (\u212c Z X).cone\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.left.mk.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 (((BinaryFan.associatorOfLimitCone \u212c X Y Z).inv \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c (tensorObj \u212c X Y) Z).isLimit\n (IsLimit.swapBinaryFan (\u212c Z (tensorObj \u212c X Y)).isLimit)).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Z X Y).inv) \u226b\n NatTrans.app (\u212c (\u212c Z X).cone.pt Y).cone.\u03c0 { as := WalkingPair.left }) \u226b\n NatTrans.app (\u212c Z X).cone.\u03c0 { as := WalkingPair.right } =\n ((IsLimit.lift (\u212c X (\u212c Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c Y Z).cone.pt).cone \u226b \ud835\udfd9 X)\n (BinaryFan.snd (\u212c X (\u212c Y Z).cone.pt).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c Y Z).isLimit (IsLimit.swapBinaryFan (\u212c Z Y).isLimit)).hom)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c X Z Y).inv \u226b\n IsLimit.lift (\u212c (\u212c Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Z).cone.pt Y).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Z).isLimit (IsLimit.swapBinaryFan (\u212c Z X).isLimit)).hom)\n (BinaryFan.snd (\u212c (\u212c X Z).cone.pt Y).cone \u226b \ud835\udfd9 Y))) \u226b\n NatTrans.app (\u212c (\u212c Z X).cone.pt Y).cone.\u03c0 { as := WalkingPair.left }) \u226b\n NatTrans.app (\u212c Z X).cone.\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\ndsimp [BinaryFan.associatorOfLimitCone, BinaryFan.associator, Limits.IsLimit.conePointUniqueUpToIso]\n[GOAL]\ncase mk.left.mk.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 ((IsLimit.lift (\u212c (\u212c X Y).cone.pt Z).isLimit (BinaryFan.assocInv (\u212c X Y).isLimit (\u212c X (\u212c Y Z).cone.pt).cone) \u226b\n IsLimit.lift (\u212c Z (tensorObj \u212c X Y)).isLimit (BinaryFan.swap (\u212c (tensorObj \u212c X Y) Z).cone) \u226b\n IsLimit.lift (\u212c (\u212c Z X).cone.pt Y).isLimit\n (BinaryFan.assocInv (\u212c Z X).isLimit (\u212c Z (\u212c X Y).cone.pt).cone)) \u226b\n BinaryFan.fst (\u212c (\u212c Z X).cone.pt Y).cone) \u226b\n BinaryFan.snd (\u212c Z X).cone =\n ((IsLimit.lift (\u212c X (\u212c Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c Y Z).cone.pt).cone \u226b \ud835\udfd9 X)\n (BinaryFan.snd (\u212c X (\u212c Y Z).cone.pt).cone \u226b IsLimit.lift (\u212c Z Y).isLimit (BinaryFan.swap (\u212c Y Z).cone))) \u226b\n IsLimit.lift (\u212c (\u212c X Z).cone.pt Y).isLimit (BinaryFan.assocInv (\u212c X Z).isLimit (\u212c X (\u212c Z Y).cone.pt).cone) \u226b\n IsLimit.lift (\u212c (\u212c Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Z).cone.pt Y).cone \u226b IsLimit.lift (\u212c Z X).isLimit (BinaryFan.swap (\u212c X Z).cone))\n (BinaryFan.snd (\u212c (\u212c X Z).cone.pt Y).cone \u226b \ud835\udfd9 Y))) \u226b\n BinaryFan.fst (\u212c (\u212c Z X).cone.pt Y).cone) \u226b\n BinaryFan.snd (\u212c Z X).cone\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 ((BinaryFan.associatorOfLimitCone \u212c X Y Z).inv \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c (tensorObj \u212c X Y) Z).isLimit\n (IsLimit.swapBinaryFan (\u212c Z (tensorObj \u212c X Y)).isLimit)).hom \u226b\n (BinaryFan.associatorOfLimitCone \u212c Z X Y).inv) \u226b\n NatTrans.app (\u212c (\u212c Z X).cone.pt Y).cone.\u03c0 { as := WalkingPair.right } =\n (IsLimit.lift (\u212c X (\u212c Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c Y Z).cone.pt).cone \u226b \ud835\udfd9 X)\n (BinaryFan.snd (\u212c X (\u212c Y Z).cone.pt).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c Y Z).isLimit (IsLimit.swapBinaryFan (\u212c Z Y).isLimit)).hom)) \u226b\n (BinaryFan.associatorOfLimitCone \u212c X Z Y).inv \u226b\n IsLimit.lift (\u212c (\u212c Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Z).cone.pt Y).cone \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c X Z).isLimit (IsLimit.swapBinaryFan (\u212c Z X).isLimit)).hom)\n (BinaryFan.snd (\u212c (\u212c X Z).cone.pt Y).cone \u226b \ud835\udfd9 Y))) \u226b\n NatTrans.app (\u212c (\u212c Z X).cone.pt Y).cone.\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\ndsimp [BinaryFan.associatorOfLimitCone, BinaryFan.associator, Limits.IsLimit.conePointUniqueUpToIso]\n[GOAL]\ncase mk.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y Z : C\n\u22a2 (IsLimit.lift (\u212c (\u212c X Y).cone.pt Z).isLimit (BinaryFan.assocInv (\u212c X Y).isLimit (\u212c X (\u212c Y Z).cone.pt).cone) \u226b\n IsLimit.lift (\u212c Z (tensorObj \u212c X Y)).isLimit (BinaryFan.swap (\u212c (tensorObj \u212c X Y) Z).cone) \u226b\n IsLimit.lift (\u212c (\u212c Z X).cone.pt Y).isLimit (BinaryFan.assocInv (\u212c Z X).isLimit (\u212c Z (\u212c X Y).cone.pt).cone)) \u226b\n BinaryFan.snd (\u212c (\u212c Z X).cone.pt Y).cone =\n (IsLimit.lift (\u212c X (\u212c Z Y).cone.pt).isLimit\n (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c Y Z).cone.pt).cone \u226b \ud835\udfd9 X)\n (BinaryFan.snd (\u212c X (\u212c Y Z).cone.pt).cone \u226b IsLimit.lift (\u212c Z Y).isLimit (BinaryFan.swap (\u212c Y Z).cone))) \u226b\n IsLimit.lift (\u212c (\u212c X Z).cone.pt Y).isLimit (BinaryFan.assocInv (\u212c X Z).isLimit (\u212c X (\u212c Z Y).cone.pt).cone) \u226b\n IsLimit.lift (\u212c (\u212c Z X).cone.pt Y).isLimit\n (BinaryFan.mk\n (BinaryFan.fst (\u212c (\u212c X Z).cone.pt Y).cone \u226b IsLimit.lift (\u212c Z X).isLimit (BinaryFan.swap (\u212c X Z).cone))\n (BinaryFan.snd (\u212c (\u212c X Z).cone.pt Y).cone \u226b \ud835\udfd9 Y))) \u226b\n BinaryFan.snd (\u212c (\u212c Z X).cone.pt Y).cone\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y : C\n\u22a2 (BinaryFan.braiding (\u212c X Y).isLimit (\u212c Y X).isLimit).hom \u226b (BinaryFan.braiding (\u212c Y X).isLimit (\u212c X Y).isLimit).hom =\n \ud835\udfd9 (tensorObj \u212c X Y)\n[PROOFSTEP]\ndsimp [tensorHom, Limits.BinaryFan.braiding]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y : C\n\u22a2 (IsLimit.conePointUniqueUpToIso (\u212c X Y).isLimit (IsLimit.swapBinaryFan (\u212c Y X).isLimit)).hom \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c Y X).isLimit (IsLimit.swapBinaryFan (\u212c X Y).isLimit)).hom =\n \ud835\udfd9 (tensorObj \u212c X Y)\n[PROOFSTEP]\napply (\u212c _ _).isLimit.hom_ext\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y : C\n\u22a2 \u2200 (j : Discrete WalkingPair),\n ((IsLimit.conePointUniqueUpToIso (\u212c X Y).isLimit (IsLimit.swapBinaryFan (\u212c Y X).isLimit)).hom \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c Y X).isLimit (IsLimit.swapBinaryFan (\u212c X Y).isLimit)).hom) \u226b\n NatTrans.app (\u212c X Y).cone.\u03c0 j =\n \ud835\udfd9 (tensorObj \u212c X Y) \u226b NatTrans.app (\u212c X Y).cone.\u03c0 j\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y : C\n\u22a2 ((IsLimit.conePointUniqueUpToIso (\u212c X Y).isLimit (IsLimit.swapBinaryFan (\u212c Y X).isLimit)).hom \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c Y X).isLimit (IsLimit.swapBinaryFan (\u212c X Y).isLimit)).hom) \u226b\n NatTrans.app (\u212c X Y).cone.\u03c0 { as := WalkingPair.left } =\n \ud835\udfd9 (tensorObj \u212c X Y) \u226b NatTrans.app (\u212c X Y).cone.\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\ndsimp [Limits.IsLimit.conePointUniqueUpToIso]\n[GOAL]\ncase mk.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y : C\n\u22a2 (IsLimit.lift (\u212c Y X).isLimit (BinaryFan.swap (\u212c X Y).cone) \u226b\n IsLimit.lift (\u212c X Y).isLimit (BinaryFan.swap (\u212c Y X).cone)) \u226b\n BinaryFan.fst (\u212c X Y).cone =\n \ud835\udfd9 (tensorObj \u212c X Y) \u226b BinaryFan.fst (\u212c X Y).cone\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y : C\n\u22a2 ((IsLimit.conePointUniqueUpToIso (\u212c X Y).isLimit (IsLimit.swapBinaryFan (\u212c Y X).isLimit)).hom \u226b\n (IsLimit.conePointUniqueUpToIso (\u212c Y X).isLimit (IsLimit.swapBinaryFan (\u212c X Y).isLimit)).hom) \u226b\n NatTrans.app (\u212c X Y).cone.\u03c0 { as := WalkingPair.right } =\n \ud835\udfd9 (tensorObj \u212c X Y) \u226b NatTrans.app (\u212c X Y).cone.\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\ndsimp [Limits.IsLimit.conePointUniqueUpToIso]\n[GOAL]\ncase mk.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y : C\n\u22a2 (IsLimit.lift (\u212c Y X).isLimit (BinaryFan.swap (\u212c X Y).cone) \u226b\n IsLimit.lift (\u212c X Y).isLimit (BinaryFan.swap (\u212c Y X).cone)) \u226b\n BinaryFan.snd (\u212c X Y).cone =\n \ud835\udfd9 (tensorObj \u212c X Y) \u226b BinaryFan.snd (\u212c X Y).cone\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Symmetric", "llama_tokens": 17225, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802264851919, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4412047195332403}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nm : M\nf : \u03b1 \u2192\u2080 M\n\u22a2 (a, m) \u2208 graph f \u2194 \u2191f a = m \u2227 m \u2260 0\n[PROOFSTEP]\nsimp_rw [graph, mem_map, mem_support_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nm : M\nf : \u03b1 \u2192\u2080 M\n\u22a2 (\u2203 a_1,\n \u2191f a_1 \u2260 0 \u2227\n \u2191{ toFun := fun a => (a, \u2191f a),\n inj' := (_ : \u2200 (x x_1 : \u03b1), (fun a => (a, \u2191f a)) x = (fun a => (a, \u2191f a)) x_1 \u2192 x = x_1) }\n a_1 =\n (a, m)) \u2194\n \u2191f a = m \u2227 m \u2260 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nm : M\nf : \u03b1 \u2192\u2080 M\n\u22a2 (\u2203 a_1,\n \u2191f a_1 \u2260 0 \u2227\n \u2191{ toFun := fun a => (a, \u2191f a),\n inj' := (_ : \u2200 (x x_1 : \u03b1), (fun a => (a, \u2191f a)) x = (fun a => (a, \u2191f a)) x_1 \u2192 x = x_1) }\n a_1 =\n (a, m)) \u2192\n \u2191f a = m \u2227 m \u2260 0\n[PROOFSTEP]\nrintro \u27e8b, ha, rfl, -\u27e9\n[GOAL]\ncase mp.intro.intro.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nf : \u03b1 \u2192\u2080 M\nha : \u2191f a \u2260 0\n\u22a2 \u2191f a = \u2191f a \u2227 \u2191f a \u2260 0\n[PROOFSTEP]\nexact \u27e8rfl, ha\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nm : M\nf : \u03b1 \u2192\u2080 M\n\u22a2 \u2191f a = m \u2227 m \u2260 0 \u2192\n \u2203 a_2,\n \u2191f a_2 \u2260 0 \u2227\n \u2191{ toFun := fun a => (a, \u2191f a),\n inj' := (_ : \u2200 (x x_1 : \u03b1), (fun a => (a, \u2191f a)) x = (fun a => (a, \u2191f a)) x_1 \u2192 x = x_1) }\n a_2 =\n (a, m)\n[PROOFSTEP]\nrintro \u27e8rfl, ha\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nf : \u03b1 \u2192\u2080 M\nha : \u2191f a \u2260 0\n\u22a2 \u2203 a_1,\n \u2191f a_1 \u2260 0 \u2227\n \u2191{ toFun := fun a => (a, \u2191f a),\n inj' := (_ : \u2200 (x x_1 : \u03b1), (fun a => (a, \u2191f a)) x = (fun a => (a, \u2191f a)) x_1 \u2192 x = x_1) }\n a_1 =\n (a, \u2191f a)\n[PROOFSTEP]\nexact \u27e8a, ha, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nc : \u03b1 \u00d7 M\nf : \u03b1 \u2192\u2080 M\n\u22a2 c \u2208 graph f \u2194 \u2191f c.fst = c.snd \u2227 c.snd \u2260 0\n[PROOFSTEP]\ncases c\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\nfst\u271d : \u03b1\nsnd\u271d : M\n\u22a2 (fst\u271d, snd\u271d) \u2208 graph f \u2194 \u2191f (fst\u271d, snd\u271d).fst = (fst\u271d, snd\u271d).snd \u2227 (fst\u271d, snd\u271d).snd \u2260 0\n[PROOFSTEP]\nexact mk_mem_graph_iff\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192\u2080 M\n\u22a2 image Prod.fst (graph f) = f.support\n[PROOFSTEP]\nclassical simp only [graph, map_eq_image, image_image, Embedding.coeFn_mk, (\u00b7 \u2218 \u00b7), image_id']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192\u2080 M\n\u22a2 image Prod.fst (graph f) = f.support\n[PROOFSTEP]\nsimp only [graph, map_eq_image, image_image, Embedding.coeFn_mk, (\u00b7 \u2218 \u00b7), image_id']\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\u271d\n\u03b1 : Type u_13\nM : Type u_14\ninst\u271d : Zero M\n\u22a2 Injective graph\n[PROOFSTEP]\nintro f g h\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\u271d\n\u03b1 : Type u_13\nM : Type u_14\ninst\u271d : Zero M\nf g : \u03b1 \u2192\u2080 M\nh : graph f = graph g\n\u22a2 f = g\n[PROOFSTEP]\nclassical\nhave hsup : f.support = g.support := by rw [\u2190 image_fst_graph, h, image_fst_graph]\nrefine' ext_iff'.2 \u27e8hsup, fun x hx => apply_eq_of_mem_graph <| h.symm \u25b8 _\u27e9\nexact mk_mem_graph _ (hsup \u25b8 hx)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\u271d\n\u03b1 : Type u_13\nM : Type u_14\ninst\u271d : Zero M\nf g : \u03b1 \u2192\u2080 M\nh : graph f = graph g\n\u22a2 f = g\n[PROOFSTEP]\nhave hsup : f.support = g.support := by rw [\u2190 image_fst_graph, h, image_fst_graph]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\u271d\n\u03b1 : Type u_13\nM : Type u_14\ninst\u271d : Zero M\nf g : \u03b1 \u2192\u2080 M\nh : graph f = graph g\n\u22a2 f.support = g.support\n[PROOFSTEP]\nrw [\u2190 image_fst_graph, h, image_fst_graph]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\u271d\n\u03b1 : Type u_13\nM : Type u_14\ninst\u271d : Zero M\nf g : \u03b1 \u2192\u2080 M\nh : graph f = graph g\nhsup : f.support = g.support\n\u22a2 f = g\n[PROOFSTEP]\nrefine' ext_iff'.2 \u27e8hsup, fun x hx => apply_eq_of_mem_graph <| h.symm \u25b8 _\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\u271d\n\u03b1 : Type u_13\nM : Type u_14\ninst\u271d : Zero M\nf g : \u03b1 \u2192\u2080 M\nh : graph f = graph g\nhsup : f.support = g.support\nx : \u03b1\nhx : x \u2208 f.support\n\u22a2 (x, \u2191g x) \u2208 graph g\n[PROOFSTEP]\nexact mk_mem_graph _ (hsup \u25b8 hx)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\n\u22a2 graph 0 = \u2205\n[PROOFSTEP]\nsimp [graph]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2243 N\nhf : \u2191f 0 = 0\nhf' : \u2191f.symm 0 = 0\nx : \u03b1 \u2192\u2080 M\n\u22a2 mapRange (\u2191f.symm) hf' (mapRange (\u2191f) hf x) = x\n[PROOFSTEP]\nrw [\u2190 mapRange_comp _ _ _ _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2243 N\nhf : \u2191f 0 = 0\nhf' : \u2191f.symm 0 = 0\nx : \u03b1 \u2192\u2080 M\n\u22a2 mapRange (\u2191f.symm \u2218 \u2191f) ?h x = x\n[PROOFSTEP]\nsimp_rw [Equiv.symm_comp_self]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2243 N\nhf : \u2191f 0 = 0\nhf' : \u2191f.symm 0 = 0\nx : \u03b1 \u2192\u2080 M\n\u22a2 (\u2191f.symm \u2218 \u2191f) 0 = 0\n[PROOFSTEP]\nsimp_rw [Equiv.symm_comp_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2243 N\nhf : \u2191f 0 = 0\nhf' : \u2191f.symm 0 = 0\nx : \u03b1 \u2192\u2080 M\n\u22a2 mapRange id (_ : id 0 = 0) x = x\n[PROOFSTEP]\nexact mapRange_id _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2243 N\nhf : \u2191f 0 = 0\nhf' : \u2191f.symm 0 = 0\nx : \u03b1 \u2192\u2080 M\n\u22a2 id 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2243 N\nhf : \u2191f 0 = 0\nhf' : \u2191f.symm 0 = 0\nx : \u03b1 \u2192\u2080 N\n\u22a2 mapRange (\u2191f) hf (mapRange (\u2191f.symm) hf' x) = x\n[PROOFSTEP]\nrw [\u2190 mapRange_comp _ _ _ _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2243 N\nhf : \u2191f 0 = 0\nhf' : \u2191f.symm 0 = 0\nx : \u03b1 \u2192\u2080 N\n\u22a2 mapRange (\u2191f \u2218 \u2191f.symm) ?h x = x\n[PROOFSTEP]\nsimp_rw [Equiv.self_comp_symm]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2243 N\nhf : \u2191f 0 = 0\nhf' : \u2191f.symm 0 = 0\nx : \u03b1 \u2192\u2080 N\n\u22a2 (\u2191f \u2218 \u2191f.symm) 0 = 0\n[PROOFSTEP]\nsimp_rw [Equiv.self_comp_symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2243 N\nhf : \u2191f 0 = 0\nhf' : \u2191f.symm 0 = 0\nx : \u03b1 \u2192\u2080 N\n\u22a2 mapRange id (_ : id 0 = 0) x = x\n[PROOFSTEP]\nexact mapRange_id _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2243 N\nhf : \u2191f 0 = 0\nhf' : \u2191f.symm 0 = 0\nx : \u03b1 \u2192\u2080 N\n\u22a2 id 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2243 N\nhf : \u2191f 0 = 0\nhf' : \u2191f.symm 0 = 0\nf\u2082 : N \u2243 P\nhf\u2082 : \u2191f\u2082 0 = 0\nhf\u2082' : \u2191f\u2082.symm 0 = 0\n\u22a2 \u2191(f.trans f\u2082) 0 = 0\n[PROOFSTEP]\nrw [Equiv.trans_apply, hf, hf\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2243 N\nhf : \u2191f 0 = 0\nhf' : \u2191f.symm 0 = 0\nf\u2082 : N \u2243 P\nhf\u2082 : \u2191f\u2082 0 = 0\nhf\u2082' : \u2191f\u2082.symm 0 = 0\n\u22a2 \u2191(f.trans f\u2082).symm 0 = 0\n[PROOFSTEP]\nrw [Equiv.symm_trans_apply, hf\u2082', hf']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : ZeroHom N P\nf\u2082 : ZeroHom M N\n\u22a2 (\u2191f \u2218 \u2191f\u2082) 0 = 0\n[PROOFSTEP]\nsimp only [comp_apply, map_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : AddCommMonoid P\nf : M \u2192+ N\na b : \u03b1 \u2192\u2080 M\n\u22a2 ZeroHom.toFun { toFun := mapRange \u2191f (_ : \u2191f 0 = 0), map_zero' := (_ : mapRange \u2191f (_ : \u2191f 0 = 0) 0 = 0) } (a + b) =\n ZeroHom.toFun { toFun := mapRange \u2191f (_ : \u2191f 0 = 0), map_zero' := (_ : mapRange \u2191f (_ : \u2191f 0 = 0) 0 = 0) } a +\n ZeroHom.toFun { toFun := mapRange \u2191f (_ : \u2191f 0 = 0), map_zero' := (_ : mapRange \u2191f (_ : \u2191f 0 = 0) 0 = 0) } b\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : AddCommMonoid P\nf : M \u2192+ N\na b : \u03b1 \u2192\u2080 M\n\u22a2 mapRange \u2191f (_ : \u2191f 0 = 0) (a + b) = mapRange \u2191f (_ : \u2191f 0 = 0) a + mapRange \u2191f (_ : \u2191f 0 = 0) b\n[PROOFSTEP]\nexact mapRange_add f.map_add _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : AddCommMonoid P\nf : N \u2192+ P\nf\u2082 : M \u2192+ N\n\u22a2 (\u2191f \u2218 \u2191f\u2082) 0 = 0\n[PROOFSTEP]\nsimp only [comp_apply, map_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : AddCommMonoid P\nf : M \u2243+ N\nsrc\u271d : (\u03b1 \u2192\u2080 M) \u2192+ \u03b1 \u2192\u2080 N := addMonoidHom (AddEquiv.toAddMonoidHom f)\nx : \u03b1 \u2192\u2080 M\n\u22a2 mapRange \u2191(AddEquiv.symm f) (_ : \u2191(AddEquiv.symm f) 0 = 0) (mapRange \u2191f (_ : \u2191f 0 = 0) x) = x\n[PROOFSTEP]\nrw [\u2190 mapRange_comp _ _ _ _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : AddCommMonoid P\nf : M \u2243+ N\nsrc\u271d : (\u03b1 \u2192\u2080 M) \u2192+ \u03b1 \u2192\u2080 N := addMonoidHom (AddEquiv.toAddMonoidHom f)\nx : \u03b1 \u2192\u2080 M\n\u22a2 mapRange (\u2191(AddEquiv.symm f) \u2218 \u2191f) ?h x = x\n[PROOFSTEP]\nsimp_rw [AddEquiv.symm_comp_self]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : AddCommMonoid P\nf : M \u2243+ N\nsrc\u271d : (\u03b1 \u2192\u2080 M) \u2192+ \u03b1 \u2192\u2080 N := addMonoidHom (AddEquiv.toAddMonoidHom f)\nx : \u03b1 \u2192\u2080 M\n\u22a2 (\u2191(AddEquiv.symm f) \u2218 \u2191f) 0 = 0\n[PROOFSTEP]\nsimp_rw [AddEquiv.symm_comp_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : AddCommMonoid P\nf : M \u2243+ N\nsrc\u271d : (\u03b1 \u2192\u2080 M) \u2192+ \u03b1 \u2192\u2080 N := addMonoidHom (AddEquiv.toAddMonoidHom f)\nx : \u03b1 \u2192\u2080 M\n\u22a2 mapRange id (_ : id 0 = 0) x = x\n[PROOFSTEP]\nexact mapRange_id _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : AddCommMonoid P\nf : M \u2243+ N\nsrc\u271d : (\u03b1 \u2192\u2080 M) \u2192+ \u03b1 \u2192\u2080 N := addMonoidHom (AddEquiv.toAddMonoidHom f)\nx : \u03b1 \u2192\u2080 M\n\u22a2 id 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : AddCommMonoid P\nf : M \u2243+ N\nsrc\u271d : (\u03b1 \u2192\u2080 M) \u2192+ \u03b1 \u2192\u2080 N := addMonoidHom (AddEquiv.toAddMonoidHom f)\nx : \u03b1 \u2192\u2080 N\n\u22a2 mapRange \u2191f (_ : \u2191f 0 = 0) (mapRange \u2191(AddEquiv.symm f) (_ : \u2191(AddEquiv.symm f) 0 = 0) x) = x\n[PROOFSTEP]\nrw [\u2190 mapRange_comp _ _ _ _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : AddCommMonoid P\nf : M \u2243+ N\nsrc\u271d : (\u03b1 \u2192\u2080 M) \u2192+ \u03b1 \u2192\u2080 N := addMonoidHom (AddEquiv.toAddMonoidHom f)\nx : \u03b1 \u2192\u2080 N\n\u22a2 mapRange (\u2191f \u2218 \u2191(AddEquiv.symm f)) ?h x = x\n[PROOFSTEP]\nsimp_rw [AddEquiv.self_comp_symm]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : AddCommMonoid P\nf : M \u2243+ N\nsrc\u271d : (\u03b1 \u2192\u2080 M) \u2192+ \u03b1 \u2192\u2080 N := addMonoidHom (AddEquiv.toAddMonoidHom f)\nx : \u03b1 \u2192\u2080 N\n\u22a2 (\u2191f \u2218 \u2191(AddEquiv.symm f)) 0 = 0\n[PROOFSTEP]\nsimp_rw [AddEquiv.self_comp_symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : AddCommMonoid P\nf : M \u2243+ N\nsrc\u271d : (\u03b1 \u2192\u2080 M) \u2192+ \u03b1 \u2192\u2080 N := addMonoidHom (AddEquiv.toAddMonoidHom f)\nx : \u03b1 \u2192\u2080 N\n\u22a2 mapRange id (_ : id 0 = 0) x = x\n[PROOFSTEP]\nexact mapRange_id _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : AddCommMonoid P\nf : M \u2243+ N\nsrc\u271d : (\u03b1 \u2192\u2080 M) \u2192+ \u03b1 \u2192\u2080 N := addMonoidHom (AddEquiv.toAddMonoidHom f)\nx : \u03b1 \u2192\u2080 N\n\u22a2 id 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : AddCommMonoid P\nf : M \u2243+ N\nf\u2082 : N \u2243+ P\n\u22a2 (\u2191f\u2082 \u2218 \u2191f) 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2243 \u03b2\nl : \u03b1 \u2192\u2080 M\na : \u03b2\n\u22a2 a \u2208 map (Equiv.toEmbedding f) l.support \u2194 (fun a => \u2191l (\u2191f.symm a)) a \u2260 0\n[PROOFSTEP]\nsimp only [Finset.mem_map_equiv, mem_support_toFun]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2243 \u03b2\nl : \u03b1 \u2192\u2080 M\na : \u03b2\n\u22a2 toFun l (\u2191f.symm a) \u2260 0 \u2194 \u2191l (\u2191f.symm a) \u2260 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nl : \u03b1 \u2192\u2080 M\n\u22a2 equivMapDomain (Equiv.refl \u03b1) l = l\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nl : \u03b1 \u2192\u2080 M\nx : \u03b1\n\u22a2 \u2191(equivMapDomain (Equiv.refl \u03b1) l) x = \u2191l x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\n\u22a2 equivMapDomain (Equiv.refl \u03b1) = id\n[PROOFSTEP]\next x\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nx : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\n\u22a2 \u2191(equivMapDomain (Equiv.refl \u03b1) x) a\u271d = \u2191(id x) a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2243 \u03b2\ng : \u03b2 \u2243 \u03b3\nl : \u03b1 \u2192\u2080 M\n\u22a2 equivMapDomain (f.trans g) l = equivMapDomain g (equivMapDomain f l)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2243 \u03b2\ng : \u03b2 \u2243 \u03b3\nl : \u03b1 \u2192\u2080 M\nx : \u03b3\n\u22a2 \u2191(equivMapDomain (f.trans g) l) x = \u2191(equivMapDomain g (equivMapDomain f l)) x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2243 \u03b2\ng : \u03b2 \u2243 \u03b3\n\u22a2 equivMapDomain (f.trans g) = equivMapDomain g \u2218 equivMapDomain f\n[PROOFSTEP]\next x\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2243 \u03b2\ng : \u03b2 \u2243 \u03b3\nx : \u03b1 \u2192\u2080 M\na\u271d : \u03b3\n\u22a2 \u2191(equivMapDomain (f.trans g) x) a\u271d = \u2191((equivMapDomain g \u2218 equivMapDomain f) x) a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2243 \u03b2\na : \u03b1\nb : M\n\u22a2 equivMapDomain f (single a b) = single (\u2191f a) b\n[PROOFSTEP]\nclassical\next x\nsimp only [single_apply, Equiv.apply_eq_iff_eq_symm_apply, equivMapDomain_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2243 \u03b2\na : \u03b1\nb : M\n\u22a2 equivMapDomain f (single a b) = single (\u2191f a) b\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2243 \u03b2\na : \u03b1\nb : M\nx : \u03b2\n\u22a2 \u2191(equivMapDomain f (single a b)) x = \u2191(single (\u2191f a) b) x\n[PROOFSTEP]\nsimp only [single_apply, Equiv.apply_eq_iff_eq_symm_apply, equivMapDomain_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2243 \u03b2\n\u22a2 equivMapDomain f 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2243 \u03b2\na\u271d : \u03b2\n\u22a2 \u2191(equivMapDomain f 0) a\u271d = \u21910 a\u271d\n[PROOFSTEP]\nsimp only [equivMapDomain_apply, coe_zero, Pi.zero_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2243 \u03b2\n\u22a2 (\u03b1 \u2192\u2080 M) \u2243 (\u03b2 \u2192\u2080 M)\n[PROOFSTEP]\nrefine' \u27e8equivMapDomain f, equivMapDomain f.symm, fun f => _, fun f => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2243 \u03b2\nf : \u03b1 \u2192\u2080 M\n\u22a2 equivMapDomain f\u271d.symm (equivMapDomain f\u271d f) = f\n[PROOFSTEP]\next x\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2243 \u03b2\nf : \u03b2 \u2192\u2080 M\n\u22a2 equivMapDomain f\u271d (equivMapDomain f\u271d.symm f) = f\n[PROOFSTEP]\next x\n[GOAL]\ncase refine'_1.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2243 \u03b2\nf : \u03b1 \u2192\u2080 M\nx : \u03b1\n\u22a2 \u2191(equivMapDomain f\u271d.symm (equivMapDomain f\u271d f)) x = \u2191f x\n[PROOFSTEP]\nsimp only [equivMapDomain_apply, Equiv.symm_symm, Equiv.symm_apply_apply, Equiv.apply_symm_apply]\n[GOAL]\ncase refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2243 \u03b2\nf : \u03b2 \u2192\u2080 M\nx : \u03b2\n\u22a2 \u2191(equivMapDomain f\u271d (equivMapDomain f\u271d.symm f)) x = \u2191f x\n[PROOFSTEP]\nsimp only [equivMapDomain_apply, Equiv.symm_symm, Equiv.symm_apply_apply, Equiv.apply_symm_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nx : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 \u2191(mapDomain f x) (f a) = \u2191x a\n[PROOFSTEP]\nrw [mapDomain, sum_apply, sum, Finset.sum_eq_single a, single_eq_same]\n[GOAL]\ncase h\u2080\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nx : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 \u2200 (b : \u03b1), b \u2208 x.support \u2192 b \u2260 a \u2192 \u2191(single (f b) (\u2191x b)) (f a) = 0\n[PROOFSTEP]\nintro b _ hba\n[GOAL]\ncase h\u2080\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nx : \u03b1 \u2192\u2080 M\na b : \u03b1\na\u271d : b \u2208 x.support\nhba : b \u2260 a\n\u22a2 \u2191(single (f b) (\u2191x b)) (f a) = 0\n[PROOFSTEP]\nexact single_eq_of_ne (hf.ne hba)\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nx : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 \u00aca \u2208 x.support \u2192 \u2191(single (f a) (\u2191x a)) (f a) = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nx : \u03b1 \u2192\u2080 M\na : \u03b1\nh : \u00aca \u2208 x.support\n\u22a2 \u2191(single (f a) (\u2191x a)) (f a) = 0\n[PROOFSTEP]\nrw [not_mem_support_iff.1 h, single_zero, zero_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\nx : \u03b1 \u2192\u2080 M\na : \u03b2\nh : \u00aca \u2208 Set.range f\n\u22a2 \u2191(mapDomain f x) a = 0\n[PROOFSTEP]\nrw [mapDomain, sum_apply, sum]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\nx : \u03b1 \u2192\u2080 M\na : \u03b2\nh : \u00aca \u2208 Set.range f\n\u22a2 \u2211 a_1 in x.support, \u2191(single (f a_1) (\u2191x a_1)) a = 0\n[PROOFSTEP]\nexact Finset.sum_eq_zero fun a' _ => single_eq_of_ne fun eq => h <| eq \u25b8 Set.mem_range_self _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\n\u22a2 mapDomain (g \u2218 f) v = mapDomain g (mapDomain f v)\n[PROOFSTEP]\nrefine' ((sum_sum_index _ _).trans _).symm\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\n\u22a2 \u2200 (a : \u03b2), single (g a) 0 = 0\n[PROOFSTEP]\nintro\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\na\u271d : \u03b2\n\u22a2 single (g a\u271d) 0 = 0\n[PROOFSTEP]\nexact single_zero _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\n\u22a2 \u2200 (a : \u03b2) (b\u2081 b\u2082 : M), single (g a) (b\u2081 + b\u2082) = single (g a) b\u2081 + single (g a) b\u2082\n[PROOFSTEP]\nintro\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\na\u271d : \u03b2\n\u22a2 \u2200 (b\u2081 b\u2082 : M), single (g a\u271d) (b\u2081 + b\u2082) = single (g a\u271d) b\u2081 + single (g a\u271d) b\u2082\n[PROOFSTEP]\nexact single_add _\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\n\u22a2 (sum v fun a b => sum (single (f a) b) fun a => single (g a)) = mapDomain (g \u2218 f) v\n[PROOFSTEP]\nrefine' sum_congr fun _ _ => sum_single_index _\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nx\u271d\u00b9 : \u03b1\nx\u271d : x\u271d\u00b9 \u2208 v.support\n\u22a2 single (g (f x\u271d\u00b9)) 0 = 0\n[PROOFSTEP]\nexact single_zero _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf g : \u03b1 \u2192 \u03b2\nh : \u2200 (x : \u03b1), x \u2208 v.support \u2192 f x = g x\nx\u271d : \u03b1\nH : x\u271d \u2208 v.support\n\u22a2 (fun a => single (f a)) x\u271d (\u2191v x\u271d) = (fun a => single (g a)) x\u271d (\u2191v x\u271d)\n[PROOFSTEP]\nsimp only [h _ H]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2243 \u03b2\nx : \u03b1 \u2192\u2080 M\na : \u03b2\n\u22a2 \u2191(mapDomain (\u2191f) x) a = \u2191x (\u2191f.symm a)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 f.apply_symm_apply a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2243 \u03b2\nx : \u03b1 \u2192\u2080 M\na : \u03b2\n| \u2191(mapDomain (\u2191f) x) a\n[PROOFSTEP]\nrw [\u2190 f.apply_symm_apply a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2243 \u03b2\nx : \u03b1 \u2192\u2080 M\na : \u03b2\n| \u2191(mapDomain (\u2191f) x) a\n[PROOFSTEP]\nrw [\u2190 f.apply_symm_apply a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2243 \u03b2\nx : \u03b1 \u2192\u2080 M\na : \u03b2\n| \u2191(mapDomain (\u2191f) x) a\n[PROOFSTEP]\nrw [\u2190 f.apply_symm_apply a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2243 \u03b2\nx : \u03b1 \u2192\u2080 M\na : \u03b2\n\u22a2 \u2191(mapDomain (\u2191f) x) (\u2191f (\u2191f.symm a)) = \u2191x (\u2191f.symm a)\n[PROOFSTEP]\nexact mapDomain_apply f.injective _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\n\u22a2 (Finset.biUnion s.support fun a => {f a}) \u2286 image f s.support\n[PROOFSTEP]\nrw [Finset.biUnion_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1 \u2192\u2080 M\nhS : \u2191x.support \u2286 S\nhf : Set.InjOn f S\na : \u03b1\nha : a \u2208 S\n\u22a2 \u2191(mapDomain f x) (f a) = \u2191x a\n[PROOFSTEP]\nclassical\nrw [mapDomain, sum_apply, sum]\nsimp_rw [single_apply]\nby_cases hax : a \u2208 x.support\n\u00b7 rw [\u2190 Finset.add_sum_erase _ _ hax, if_pos rfl]\n convert add_zero (x a)\n refine' Finset.sum_eq_zero fun i hi => if_neg _\n exact (hf.mono hS).ne (Finset.mem_of_mem_erase hi) hax (Finset.ne_of_mem_erase hi)\n\u00b7 rw [not_mem_support_iff.1 hax]\n refine' Finset.sum_eq_zero fun i hi => if_neg _\n exact hf.ne (hS hi) ha (ne_of_mem_of_not_mem hi hax)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1 \u2192\u2080 M\nhS : \u2191x.support \u2286 S\nhf : Set.InjOn f S\na : \u03b1\nha : a \u2208 S\n\u22a2 \u2191(mapDomain f x) (f a) = \u2191x a\n[PROOFSTEP]\nrw [mapDomain, sum_apply, sum]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1 \u2192\u2080 M\nhS : \u2191x.support \u2286 S\nhf : Set.InjOn f S\na : \u03b1\nha : a \u2208 S\n\u22a2 \u2211 a_1 in x.support, \u2191(single (f a_1) (\u2191x a_1)) (f a) = \u2191x a\n[PROOFSTEP]\nsimp_rw [single_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1 \u2192\u2080 M\nhS : \u2191x.support \u2286 S\nhf : Set.InjOn f S\na : \u03b1\nha : a \u2208 S\n\u22a2 (\u2211 x_1 in x.support, if f x_1 = f a then \u2191x x_1 else 0) = \u2191x a\n[PROOFSTEP]\nby_cases hax : a \u2208 x.support\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1 \u2192\u2080 M\nhS : \u2191x.support \u2286 S\nhf : Set.InjOn f S\na : \u03b1\nha : a \u2208 S\nhax : a \u2208 x.support\n\u22a2 (\u2211 x_1 in x.support, if f x_1 = f a then \u2191x x_1 else 0) = \u2191x a\n[PROOFSTEP]\nrw [\u2190 Finset.add_sum_erase _ _ hax, if_pos rfl]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1 \u2192\u2080 M\nhS : \u2191x.support \u2286 S\nhf : Set.InjOn f S\na : \u03b1\nha : a \u2208 S\nhax : a \u2208 x.support\n\u22a2 (\u2191x a + \u2211 x_1 in Finset.erase x.support a, if f x_1 = f a then \u2191x x_1 else 0) = \u2191x a\n[PROOFSTEP]\nconvert add_zero (x a)\n[GOAL]\ncase h.e'_2.h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1 \u2192\u2080 M\nhS : \u2191x.support \u2286 S\nhf : Set.InjOn f S\na : \u03b1\nha : a \u2208 S\nhax : a \u2208 x.support\n\u22a2 (\u2211 x_1 in Finset.erase x.support a, if f x_1 = f a then \u2191x x_1 else 0) = 0\n[PROOFSTEP]\nrefine' Finset.sum_eq_zero fun i hi => if_neg _\n[GOAL]\ncase h.e'_2.h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1 \u2192\u2080 M\nhS : \u2191x.support \u2286 S\nhf : Set.InjOn f S\na : \u03b1\nha : a \u2208 S\nhax : a \u2208 x.support\ni : \u03b1\nhi : i \u2208 Finset.erase x.support a\n\u22a2 \u00acf i = f a\n[PROOFSTEP]\nexact (hf.mono hS).ne (Finset.mem_of_mem_erase hi) hax (Finset.ne_of_mem_erase hi)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1 \u2192\u2080 M\nhS : \u2191x.support \u2286 S\nhf : Set.InjOn f S\na : \u03b1\nha : a \u2208 S\nhax : \u00aca \u2208 x.support\n\u22a2 (\u2211 x_1 in x.support, if f x_1 = f a then \u2191x x_1 else 0) = \u2191x a\n[PROOFSTEP]\nrw [not_mem_support_iff.1 hax]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1 \u2192\u2080 M\nhS : \u2191x.support \u2286 S\nhf : Set.InjOn f S\na : \u03b1\nha : a \u2208 S\nhax : \u00aca \u2208 x.support\n\u22a2 (\u2211 x_1 in x.support, if f x_1 = f a then \u2191x x_1 else 0) = 0\n[PROOFSTEP]\nrefine' Finset.sum_eq_zero fun i hi => if_neg _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1 \u2192\u2080 M\nhS : \u2191x.support \u2286 S\nhf : Set.InjOn f S\na : \u03b1\nha : a \u2208 S\nhax : \u00aca \u2208 x.support\ni : \u03b1\nhi : i \u2208 x.support\n\u22a2 \u00acf i = f a\n[PROOFSTEP]\nexact hf.ne (hS hi) ha (ne_of_mem_of_not_mem hi hax)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\n\u22a2 image f s.support \u2286 (mapDomain f s).support\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nx : \u03b2\nhx : x \u2208 image f s.support\n\u22a2 x \u2208 (mapDomain f s).support\n[PROOFSTEP]\nsimp only [mem_image, exists_prop, mem_support_iff, Ne.def] at hx \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nx : \u03b2\nhx : \u2203 a, \u00ac\u2191s a = 0 \u2227 f a = x\n\u22a2 x \u2208 (mapDomain f s).support\n[PROOFSTEP]\nrcases hx with \u27e8hx_w, hx_h_left, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 f hx_w \u2208 (mapDomain f s).support\n[PROOFSTEP]\nsimp only [mem_support_iff, Ne.def]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 \u00ac\u2191(mapDomain f s) (f hx_w) = 0\n[PROOFSTEP]\nrw [mapDomain_apply' (\u2191s.support : Set _) _ _ hf]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 \u00ac\u2191s hx_w = 0\n[PROOFSTEP]\nexact hx_h_left\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 hx_w \u2208 \u2191s.support\n[PROOFSTEP]\nsimp only [mem_coe, mem_support_iff, Ne.def]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 \u00ac\u2191s hx_w = 0\n[PROOFSTEP]\nexact hx_h_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 \u2191s.support \u2286 \u2191s.support\n[PROOFSTEP]\nexact Subset.refl _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv\u271d v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\n\u22a2 embDomain f v = mapDomain (\u2191f) v\n[PROOFSTEP]\next a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv\u271d v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b2\n\u22a2 \u2191(embDomain f v) a = \u2191(mapDomain (\u2191f) v) a\n[PROOFSTEP]\nby_cases h : a \u2208 Set.range f\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv\u271d v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b2\nh : a \u2208 Set.range \u2191f\n\u22a2 \u2191(embDomain f v) a = \u2191(mapDomain (\u2191f) v) a\n[PROOFSTEP]\nrcases h with \u27e8a, rfl\u27e9\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv\u271d v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 \u2191(embDomain f v) (\u2191f a) = \u2191(mapDomain (\u2191f) v) (\u2191f a)\n[PROOFSTEP]\nrw [mapDomain_apply f.injective, embDomain_apply]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv\u271d v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b2\nh : \u00aca \u2208 Set.range \u2191f\n\u22a2 \u2191(embDomain f v) a = \u2191(mapDomain (\u2191f) v) a\n[PROOFSTEP]\nrw [mapDomain_notin_range, embDomain_notin_range]\n[GOAL]\ncase neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv\u271d v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b2\nh : \u00aca \u2208 Set.range \u2191f\n\u22a2 \u00aca \u2208 Set.range \u2191f\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv\u271d v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b2\nh : \u00aca \u2208 Set.range \u2191f\n\u22a2 \u00aca \u2208 Set.range \u2191f\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : CommMonoid N\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nh : \u03b2 \u2192 M \u2192 N\nhf : Injective f\n\u22a2 prod (mapDomain f s) h = prod s fun a b => h (f a) b\n[PROOFSTEP]\nrw [\u2190 Function.Embedding.coeFn_mk f hf, \u2190 embDomain_eq_mapDomain, prod_embDomain]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\n\u22a2 Injective (mapDomain f)\n[PROOFSTEP]\nintro v\u2081 v\u2082 eq\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081\u271d v\u2082\u271d : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nv\u2081 v\u2082 : \u03b1 \u2192\u2080 M\neq : mapDomain f v\u2081 = mapDomain f v\u2082\n\u22a2 v\u2081 = v\u2082\n[PROOFSTEP]\next a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081\u271d v\u2082\u271d : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nv\u2081 v\u2082 : \u03b1 \u2192\u2080 M\neq : mapDomain f v\u2081 = mapDomain f v\u2082\na : \u03b1\n\u22a2 \u2191v\u2081 a = \u2191v\u2082 a\n[PROOFSTEP]\nhave : mapDomain f v\u2081 (f a) = mapDomain f v\u2082 (f a) := by rw [eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081\u271d v\u2082\u271d : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nv\u2081 v\u2082 : \u03b1 \u2192\u2080 M\neq : mapDomain f v\u2081 = mapDomain f v\u2082\na : \u03b1\n\u22a2 \u2191(mapDomain f v\u2081) (f a) = \u2191(mapDomain f v\u2082) (f a)\n[PROOFSTEP]\nrw [eq]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081\u271d v\u2082\u271d : \u03b1 \u2192\u2080 M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nv\u2081 v\u2082 : \u03b1 \u2192\u2080 M\neq : mapDomain f v\u2081 = mapDomain f v\u2082\na : \u03b1\nthis : \u2191(mapDomain f v\u2081) (f a) = \u2191(mapDomain f v\u2082) (f a)\n\u22a2 \u2191v\u2081 a = \u2191v\u2082 a\n[PROOFSTEP]\nrwa [mapDomain_apply hf, mapDomain_apply hf] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u2192 \u03b2\ng : M \u2192+ N\n\u22a2 AddMonoidHom.comp (addMonoidHom f) (mapRange.addMonoidHom g) =\n AddMonoidHom.comp (mapRange.addMonoidHom g) (addMonoidHom f)\n[PROOFSTEP]\next\n[GOAL]\ncase H.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u2192 \u03b2\ng : M \u2192+ N\nx\u271d\u00b9 : \u03b1\nx\u271d : M\na\u271d : \u03b2\n\u22a2 \u2191(\u2191(AddMonoidHom.comp (AddMonoidHom.comp (addMonoidHom f) (mapRange.addMonoidHom g)) (singleAddHom x\u271d\u00b9)) x\u271d) a\u271d =\n \u2191(\u2191(AddMonoidHom.comp (AddMonoidHom.comp (mapRange.addMonoidHom g) (addMonoidHom f)) (singleAddHom x\u271d\u00b9)) x\u271d) a\u271d\n[PROOFSTEP]\nsimp only [AddMonoidHom.coe_comp, Finsupp.mapRange_single, Finsupp.mapDomain.addMonoidHom_apply,\n Finsupp.singleAddHom_apply, eq_self_iff_true, Function.comp_apply, Finsupp.mapDomain_single,\n Finsupp.mapRange.addMonoidHom_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : AddCommMonoid \u03b2\nf : \u03b9 \u2192\u2080 \u03b1\ni : \u03b9\na : \u03b1\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (i : \u03b9), g i 0 = 0\nhgg : \u2200 (j : \u03b9) (a\u2081 a\u2082 : \u03b1), g j (a\u2081 + a\u2082) = g j a\u2081 + g j a\u2082\n\u22a2 sum (update f i a) g + g i (\u2191f i) = sum f g + g i a\n[PROOFSTEP]\nrw [update_eq_erase_add_single, sum_add_index' hg hgg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : AddCommMonoid \u03b2\nf : \u03b9 \u2192\u2080 \u03b1\ni : \u03b9\na : \u03b1\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (i : \u03b9), g i 0 = 0\nhgg : \u2200 (j : \u03b9) (a\u2081 a\u2082 : \u03b1), g j (a\u2081 + a\u2082) = g j a\u2081 + g j a\u2082\n\u22a2 ((sum (erase i f) fun a => g a) + sum (single i a) fun a => g a) + g i (\u2191f i) = sum f g + g i a\n[PROOFSTEP]\nconv_rhs => rw [\u2190 Finsupp.update_self f i]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : AddCommMonoid \u03b2\nf : \u03b9 \u2192\u2080 \u03b1\ni : \u03b9\na : \u03b1\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (i : \u03b9), g i 0 = 0\nhgg : \u2200 (j : \u03b9) (a\u2081 a\u2082 : \u03b1), g j (a\u2081 + a\u2082) = g j a\u2081 + g j a\u2082\n| sum f g + g i a\n[PROOFSTEP]\nrw [\u2190 Finsupp.update_self f i]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : AddCommMonoid \u03b2\nf : \u03b9 \u2192\u2080 \u03b1\ni : \u03b9\na : \u03b1\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (i : \u03b9), g i 0 = 0\nhgg : \u2200 (j : \u03b9) (a\u2081 a\u2082 : \u03b1), g j (a\u2081 + a\u2082) = g j a\u2081 + g j a\u2082\n| sum f g + g i a\n[PROOFSTEP]\nrw [\u2190 Finsupp.update_self f i]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : AddCommMonoid \u03b2\nf : \u03b9 \u2192\u2080 \u03b1\ni : \u03b9\na : \u03b1\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (i : \u03b9), g i 0 = 0\nhgg : \u2200 (j : \u03b9) (a\u2081 a\u2082 : \u03b1), g j (a\u2081 + a\u2082) = g j a\u2081 + g j a\u2082\n| sum f g + g i a\n[PROOFSTEP]\nrw [\u2190 Finsupp.update_self f i]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : AddCommMonoid \u03b2\nf : \u03b9 \u2192\u2080 \u03b1\ni : \u03b9\na : \u03b1\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (i : \u03b9), g i 0 = 0\nhgg : \u2200 (j : \u03b9) (a\u2081 a\u2082 : \u03b1), g j (a\u2081 + a\u2082) = g j a\u2081 + g j a\u2082\n\u22a2 ((sum (erase i f) fun a => g a) + sum (single i a) fun a => g a) + g i (\u2191f i) = sum (update f i (\u2191f i)) g + g i a\n[PROOFSTEP]\nrw [update_eq_erase_add_single, sum_add_index' hg hgg, add_assoc, add_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : AddCommMonoid \u03b2\nf : \u03b9 \u2192\u2080 \u03b1\ni : \u03b9\na : \u03b1\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (i : \u03b9), g i 0 = 0\nhgg : \u2200 (j : \u03b9) (a\u2081 a\u2082 : \u03b1), g j (a\u2081 + a\u2082) = g j a\u2081 + g j a\u2082\n\u22a2 (sum (erase i f) fun a => g a) + ((sum (single i a) fun a => g a) + g i (\u2191f i)) =\n (sum (erase i f) fun a => g a) + ((sum (single i (\u2191f i)) fun a => g a) + g i a)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : AddCommMonoid \u03b2\nf : \u03b9 \u2192\u2080 \u03b1\ni : \u03b9\na : \u03b1\ng : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhg : \u2200 (i : \u03b9), g i 0 = 0\nhgg : \u2200 (j : \u03b9) (a\u2081 a\u2082 : \u03b1), g j (a\u2081 + a\u2082) = g j a\u2081 + g j a\u2082\n\u22a2 (sum (single i a) fun a => g a) + g i (\u2191f i) = (sum (single i (\u2191f i)) fun a => g a) + g i a\n[PROOFSTEP]\nrw [add_comm, sum_single_index (hg _), sum_single_index (hg _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Set.InjOn f S\n\u22a2 Set.InjOn (mapDomain f) {w | \u2191w.support \u2286 S}\n[PROOFSTEP]\nintro v\u2081 hv\u2081 v\u2082 hv\u2082 eq\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081\u271d v\u2082\u271d : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Set.InjOn f S\nv\u2081 : \u03b1 \u2192\u2080 M\nhv\u2081 : v\u2081 \u2208 {w | \u2191w.support \u2286 S}\nv\u2082 : \u03b1 \u2192\u2080 M\nhv\u2082 : v\u2082 \u2208 {w | \u2191w.support \u2286 S}\neq : mapDomain f v\u2081 = mapDomain f v\u2082\n\u22a2 v\u2081 = v\u2082\n[PROOFSTEP]\next a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081\u271d v\u2082\u271d : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Set.InjOn f S\nv\u2081 : \u03b1 \u2192\u2080 M\nhv\u2081 : v\u2081 \u2208 {w | \u2191w.support \u2286 S}\nv\u2082 : \u03b1 \u2192\u2080 M\nhv\u2082 : v\u2082 \u2208 {w | \u2191w.support \u2286 S}\neq : mapDomain f v\u2081 = mapDomain f v\u2082\na : \u03b1\n\u22a2 \u2191v\u2081 a = \u2191v\u2082 a\n[PROOFSTEP]\nclassical\nby_cases h : a \u2208 v\u2081.support \u222a v\u2082.support\n\u00b7\n rw [\u2190 mapDomain_apply' S _ hv\u2081 hf _, \u2190 mapDomain_apply' S _ hv\u2082 hf _, eq] <;>\n \u00b7 apply Set.union_subset hv\u2081 hv\u2082\n exact_mod_cast h\n\u00b7 simp only [not_or, mem_union, not_not, mem_support_iff] at h \n simp [h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081\u271d v\u2082\u271d : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Set.InjOn f S\nv\u2081 : \u03b1 \u2192\u2080 M\nhv\u2081 : v\u2081 \u2208 {w | \u2191w.support \u2286 S}\nv\u2082 : \u03b1 \u2192\u2080 M\nhv\u2082 : v\u2082 \u2208 {w | \u2191w.support \u2286 S}\neq : mapDomain f v\u2081 = mapDomain f v\u2082\na : \u03b1\n\u22a2 \u2191v\u2081 a = \u2191v\u2082 a\n[PROOFSTEP]\nby_cases h : a \u2208 v\u2081.support \u222a v\u2082.support\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081\u271d v\u2082\u271d : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Set.InjOn f S\nv\u2081 : \u03b1 \u2192\u2080 M\nhv\u2081 : v\u2081 \u2208 {w | \u2191w.support \u2286 S}\nv\u2082 : \u03b1 \u2192\u2080 M\nhv\u2082 : v\u2082 \u2208 {w | \u2191w.support \u2286 S}\neq : mapDomain f v\u2081 = mapDomain f v\u2082\na : \u03b1\nh : a \u2208 v\u2081.support \u222a v\u2082.support\n\u22a2 \u2191v\u2081 a = \u2191v\u2082 a\n[PROOFSTEP]\nrw [\u2190 mapDomain_apply' S _ hv\u2081 hf _, \u2190 mapDomain_apply' S _ hv\u2082 hf _, eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081\u271d v\u2082\u271d : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Set.InjOn f S\nv\u2081 : \u03b1 \u2192\u2080 M\nhv\u2081 : v\u2081 \u2208 {w | \u2191w.support \u2286 S}\nv\u2082 : \u03b1 \u2192\u2080 M\nhv\u2082 : v\u2082 \u2208 {w | \u2191w.support \u2286 S}\neq : mapDomain f v\u2081 = mapDomain f v\u2082\na : \u03b1\nh : a \u2208 v\u2081.support \u222a v\u2082.support\n\u22a2 a \u2208 S\n[PROOFSTEP]\napply Set.union_subset hv\u2081 hv\u2082\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081\u271d v\u2082\u271d : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Set.InjOn f S\nv\u2081 : \u03b1 \u2192\u2080 M\nhv\u2081 : v\u2081 \u2208 {w | \u2191w.support \u2286 S}\nv\u2082 : \u03b1 \u2192\u2080 M\nhv\u2082 : v\u2082 \u2208 {w | \u2191w.support \u2286 S}\neq : mapDomain f v\u2081 = mapDomain f v\u2082\na : \u03b1\nh : a \u2208 v\u2081.support \u222a v\u2082.support\n\u22a2 a \u2208 \u2191v\u2081.support \u222a \u2191v\u2082.support\n[PROOFSTEP]\nexact_mod_cast h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081\u271d v\u2082\u271d : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Set.InjOn f S\nv\u2081 : \u03b1 \u2192\u2080 M\nhv\u2081 : v\u2081 \u2208 {w | \u2191w.support \u2286 S}\nv\u2082 : \u03b1 \u2192\u2080 M\nhv\u2082 : v\u2082 \u2208 {w | \u2191w.support \u2286 S}\neq : mapDomain f v\u2081 = mapDomain f v\u2082\na : \u03b1\nh : a \u2208 v\u2081.support \u222a v\u2082.support\n\u22a2 a \u2208 S\n[PROOFSTEP]\napply Set.union_subset hv\u2081 hv\u2082\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081\u271d v\u2082\u271d : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Set.InjOn f S\nv\u2081 : \u03b1 \u2192\u2080 M\nhv\u2081 : v\u2081 \u2208 {w | \u2191w.support \u2286 S}\nv\u2082 : \u03b1 \u2192\u2080 M\nhv\u2082 : v\u2082 \u2208 {w | \u2191w.support \u2286 S}\neq : mapDomain f v\u2081 = mapDomain f v\u2082\na : \u03b1\nh : a \u2208 v\u2081.support \u222a v\u2082.support\n\u22a2 a \u2208 \u2191v\u2081.support \u222a \u2191v\u2082.support\n[PROOFSTEP]\nexact_mod_cast h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081\u271d v\u2082\u271d : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Set.InjOn f S\nv\u2081 : \u03b1 \u2192\u2080 M\nhv\u2081 : v\u2081 \u2208 {w | \u2191w.support \u2286 S}\nv\u2082 : \u03b1 \u2192\u2080 M\nhv\u2082 : v\u2082 \u2208 {w | \u2191w.support \u2286 S}\neq : mapDomain f v\u2081 = mapDomain f v\u2082\na : \u03b1\nh : \u00aca \u2208 v\u2081.support \u222a v\u2082.support\n\u22a2 \u2191v\u2081 a = \u2191v\u2082 a\n[PROOFSTEP]\nsimp only [not_or, mem_union, not_not, mem_support_iff] at h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS\u271d : Type u_12\ninst\u271d : AddCommMonoid M\nv v\u2081\u271d v\u2082\u271d : \u03b1 \u2192\u2080 M\nS : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Set.InjOn f S\nv\u2081 : \u03b1 \u2192\u2080 M\nhv\u2081 : v\u2081 \u2208 {w | \u2191w.support \u2286 S}\nv\u2082 : \u03b1 \u2192\u2080 M\nhv\u2082 : v\u2082 \u2208 {w | \u2191w.support \u2286 S}\neq : mapDomain f v\u2081 = mapDomain f v\u2082\na : \u03b1\nh : \u2191v\u2081 a = 0 \u2227 \u2191v\u2082 a = 0\n\u22a2 \u2191v\u2081 a = \u2191v\u2082 a\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\u271d\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\u271d\nM : Type u_13\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2243 \u03b2\nl : \u03b1 \u2192\u2080 M\n\u22a2 equivMapDomain f l = mapDomain (\u2191f) l\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\u271d\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\u271d\nM : Type u_13\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2243 \u03b2\nl : \u03b1 \u2192\u2080 M\nx : \u03b2\n\u22a2 \u2191(equivMapDomain f l) x = \u2191(mapDomain (\u2191f) l) x\n[PROOFSTEP]\nsimp [mapDomain_equiv_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192 \u03b2\nl : \u03b2 \u2192\u2080 M\nhf : Set.InjOn f (f \u207b\u00b9' \u2191l.support)\n\u22a2 \u2200 (a : \u03b1), a \u2208 preimage l.support f hf \u2194 (fun a => \u2191l (f a)) a \u2260 0\n[PROOFSTEP]\nintro a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192 \u03b2\nl : \u03b2 \u2192\u2080 M\nhf : Set.InjOn f (f \u207b\u00b9' \u2191l.support)\na : \u03b1\n\u22a2 a \u2208 preimage l.support f hf \u2194 (fun a => \u2191l (f a)) a \u2260 0\n[PROOFSTEP]\nsimp only [Finset.mem_def.symm, Finset.mem_preimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192 \u03b2\nl : \u03b2 \u2192\u2080 M\nhf : Set.InjOn f (f \u207b\u00b9' \u2191l.support)\na : \u03b1\n\u22a2 f a \u2208 l.support \u2194 \u2191l (f a) \u2260 0\n[PROOFSTEP]\nexact l.mem_support_toFun (f a)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u2192 \u03b2\nl : \u03b2 \u2192\u2080 M\ng : \u03b2 \u2192 M \u2192 N\nhf : Set.BijOn f (f \u207b\u00b9' \u2191l.support) \u2191l.support\n\u22a2 sum (comapDomain f l (_ : Set.InjOn f (f \u207b\u00b9' \u2191l.support))) (g \u2218 f) = sum l g\n[PROOFSTEP]\nsimp only [sum, comapDomain_apply, (\u00b7 \u2218 \u00b7), comapDomain]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u2192 \u03b2\nl : \u03b2 \u2192\u2080 M\ng : \u03b2 \u2192 M \u2192 N\nhf : Set.BijOn f (f \u207b\u00b9' \u2191l.support) \u2191l.support\n\u22a2 \u2211 x in preimage l.support f (_ : Set.InjOn f (f \u207b\u00b9' \u2191l.support)),\n g (f x)\n (\u2191{ support := preimage l.support f (_ : Set.InjOn f (f \u207b\u00b9' \u2191l.support)), toFun := fun a => \u2191l (f a),\n mem_support_toFun :=\n (_ :\n \u2200 (a : \u03b1),\n a \u2208 preimage l.support f (_ : Set.InjOn f (f \u207b\u00b9' \u2191l.support)) \u2194 (fun a => \u2191l (f a)) a \u2260 0) }\n x) =\n \u2211 x in l.support, g x (\u2191l x)\n[PROOFSTEP]\nexact Finset.sum_preimage_of_bij f _ hf fun x => g x (l x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192 \u03b2\nl : \u03b2 \u2192\u2080 M\nhf : Set.BijOn f (f \u207b\u00b9' \u2191l.support) \u2191l.support\n\u22a2 comapDomain f l (_ : Set.InjOn f (f \u207b\u00b9' \u2191l.support)) = 0 \u2192 l = 0\n[PROOFSTEP]\nrw [\u2190 support_eq_empty, \u2190 support_eq_empty, comapDomain]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192 \u03b2\nl : \u03b2 \u2192\u2080 M\nhf : Set.BijOn f (f \u207b\u00b9' \u2191l.support) \u2191l.support\n\u22a2 { support := preimage l.support f (_ : Set.InjOn f (f \u207b\u00b9' \u2191l.support)), toFun := fun a => \u2191l (f a),\n mem_support_toFun :=\n (_ :\n \u2200 (a : \u03b1),\n a \u2208 preimage l.support f (_ : Set.InjOn f (f \u207b\u00b9' \u2191l.support)) \u2194 (fun a => \u2191l (f a)) a \u2260 0) }.support =\n \u2205 \u2192\n l.support = \u2205\n[PROOFSTEP]\nsimp only [Finset.ext_iff, Finset.not_mem_empty, iff_false_iff, mem_preimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192 \u03b2\nl : \u03b2 \u2192\u2080 M\nhf : Set.BijOn f (f \u207b\u00b9' \u2191l.support) \u2191l.support\n\u22a2 (\u2200 (a : \u03b1), \u00acf a \u2208 l.support) \u2192 \u2200 (a : \u03b2), \u00aca \u2208 l.support\n[PROOFSTEP]\nintro h a ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192 \u03b2\nl : \u03b2 \u2192\u2080 M\nhf : Set.BijOn f (f \u207b\u00b9' \u2191l.support) \u2191l.support\nh : \u2200 (a : \u03b1), \u00acf a \u2208 l.support\na : \u03b2\nha : a \u2208 l.support\n\u22a2 False\n[PROOFSTEP]\ncases' hf.2.2 ha with b hb\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192 \u03b2\nl : \u03b2 \u2192\u2080 M\nhf : Set.BijOn f (f \u207b\u00b9' \u2191l.support) \u2191l.support\nh : \u2200 (a : \u03b1), \u00acf a \u2208 l.support\na : \u03b2\nha : a \u2208 l.support\nb : \u03b1\nhb : b \u2208 f \u207b\u00b9' \u2191l.support \u2227 f b = a\n\u22a2 False\n[PROOFSTEP]\nexact h b (hb.2.symm \u25b8 ha)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192 \u03b2\nhif : optParam (Set.InjOn f (f \u207b\u00b9' \u21910.support)) (_ : Set.InjOn f (f \u207b\u00b9' \u2191\u2205))\n\u22a2 comapDomain f 0 hif = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192 \u03b2\nhif : optParam (Set.InjOn f (f \u207b\u00b9' \u21910.support)) (_ : Set.InjOn f (f \u207b\u00b9' \u2191\u2205))\na\u271d : \u03b1\n\u22a2 \u2191(comapDomain f 0 hif) a\u271d = \u21910 a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nm : M\nhif : Set.InjOn f (f \u207b\u00b9' \u2191(single (f a) m).support)\n\u22a2 comapDomain f (single (f a) m) hif = single a m\n[PROOFSTEP]\nrcases eq_or_ne m 0 with (rfl | hm)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhif : Set.InjOn f (f \u207b\u00b9' \u2191(single (f a) 0).support)\n\u22a2 comapDomain f (single (f a) 0) hif = single a 0\n[PROOFSTEP]\nsimp only [single_zero, comapDomain_zero]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nm : M\nhif : Set.InjOn f (f \u207b\u00b9' \u2191(single (f a) m).support)\nhm : m \u2260 0\n\u22a2 comapDomain f (single (f a) m) hif = single a m\n[PROOFSTEP]\nrw [eq_single_iff, comapDomain_apply, comapDomain_support, \u2190 Finset.coe_subset, coe_preimage,\n support_single_ne_zero _ hm, coe_singleton, coe_singleton, single_eq_same]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nm : M\nhif : Set.InjOn f (f \u207b\u00b9' \u2191(single (f a) m).support)\nhm : m \u2260 0\n\u22a2 f \u207b\u00b9' {f a} \u2286 {a} \u2227 m = m\n[PROOFSTEP]\nrw [support_single_ne_zero _ hm, coe_singleton] at hif \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nm : M\nhif : Set.InjOn f (f \u207b\u00b9' {f a})\nhm : m \u2260 0\n\u22a2 f \u207b\u00b9' {f a} \u2286 {a} \u2227 m = m\n[PROOFSTEP]\nexact \u27e8fun x hx => hif hx rfl hx, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u2192 \u03b2\nv\u2081 v\u2082 : \u03b2 \u2192\u2080 M\nhv\u2081 : Set.InjOn f (f \u207b\u00b9' \u2191v\u2081.support)\nhv\u2082 : Set.InjOn f (f \u207b\u00b9' \u2191v\u2082.support)\nhv\u2081\u2082 : Set.InjOn f (f \u207b\u00b9' \u2191(v\u2081 + v\u2082).support)\n\u22a2 comapDomain f (v\u2081 + v\u2082) hv\u2081\u2082 = comapDomain f v\u2081 hv\u2081 + comapDomain f v\u2082 hv\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u2192 \u03b2\nv\u2081 v\u2082 : \u03b2 \u2192\u2080 M\nhv\u2081 : Set.InjOn f (f \u207b\u00b9' \u2191v\u2081.support)\nhv\u2082 : Set.InjOn f (f \u207b\u00b9' \u2191v\u2082.support)\nhv\u2081\u2082 : Set.InjOn f (f \u207b\u00b9' \u2191(v\u2081 + v\u2082).support)\na\u271d : \u03b1\n\u22a2 \u2191(comapDomain f (v\u2081 + v\u2082) hv\u2081\u2082) a\u271d = \u2191(comapDomain f v\u2081 hv\u2081 + comapDomain f v\u2082 hv\u2082) a\u271d\n[PROOFSTEP]\nsimp only [comapDomain_apply, coe_add, Pi.add_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nl : \u03b2 \u2192\u2080 M\nhl : \u2191l.support \u2286 Set.range f\n\u22a2 mapDomain f (comapDomain f l (_ : Set.InjOn f (f \u207b\u00b9' \u2191l.support))) = l\n[PROOFSTEP]\next a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nl : \u03b2 \u2192\u2080 M\nhl : \u2191l.support \u2286 Set.range f\na : \u03b2\n\u22a2 \u2191(mapDomain f (comapDomain f l (_ : Set.InjOn f (f \u207b\u00b9' \u2191l.support)))) a = \u2191l a\n[PROOFSTEP]\nby_cases h_cases : a \u2208 Set.range f\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nl : \u03b2 \u2192\u2080 M\nhl : \u2191l.support \u2286 Set.range f\na : \u03b2\nh_cases : a \u2208 Set.range f\n\u22a2 \u2191(mapDomain f (comapDomain f l (_ : Set.InjOn f (f \u207b\u00b9' \u2191l.support)))) a = \u2191l a\n[PROOFSTEP]\nrcases Set.mem_range.1 h_cases with \u27e8b, hb\u27e9\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nl : \u03b2 \u2192\u2080 M\nhl : \u2191l.support \u2286 Set.range f\na : \u03b2\nh_cases : a \u2208 Set.range f\nb : \u03b1\nhb : f b = a\n\u22a2 \u2191(mapDomain f (comapDomain f l (_ : Set.InjOn f (f \u207b\u00b9' \u2191l.support)))) a = \u2191l a\n[PROOFSTEP]\nrw [hb.symm, mapDomain_apply hf, comapDomain_apply]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nl : \u03b2 \u2192\u2080 M\nhl : \u2191l.support \u2286 Set.range f\na : \u03b2\nh_cases : \u00aca \u2208 Set.range f\n\u22a2 \u2191(mapDomain f (comapDomain f l (_ : Set.InjOn f (f \u207b\u00b9' \u2191l.support)))) a = \u2191l a\n[PROOFSTEP]\nrw [mapDomain_notin_range _ _ h_cases]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nl : \u03b2 \u2192\u2080 M\nhl : \u2191l.support \u2286 Set.range f\na : \u03b2\nh_cases : \u00aca \u2208 Set.range f\n\u22a2 0 = \u2191l a\n[PROOFSTEP]\nby_contra h_contr\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nl : \u03b2 \u2192\u2080 M\nhl : \u2191l.support \u2286 Set.range f\na : \u03b2\nh_cases : \u00aca \u2208 Set.range f\nh_contr : \u00ac0 = \u2191l a\n\u22a2 False\n[PROOFSTEP]\napply h_cases (hl <| Finset.mem_coe.2 <| mem_support_iff.2 fun h => h_contr h.symm)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : Option \u03b1 \u2192\u2080 M\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 Option.some \u207b\u00b9' \u2191f.support \u2192\n \u2200 \u2983x\u2082 : \u03b1\u2984, x\u2082 \u2208 Option.some \u207b\u00b9' \u2191f.support \u2192 Option.some x\u271d = Option.some x\u2082 \u2192 x\u271d = x\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\n\u22a2 some 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d : \u03b1\n\u22a2 \u2191(some 0) a\u271d = \u21910 a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nf g : Option \u03b1 \u2192\u2080 M\n\u22a2 some (f + g) = some f + some g\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nf g : Option \u03b1 \u2192\u2080 M\na\u271d : \u03b1\n\u22a2 \u2191(some (f + g)) a\u271d = \u2191(some f + some g) a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nm : M\n\u22a2 some (single none m) = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nm : M\na\u271d : \u03b1\n\u22a2 \u2191(some (single none m)) a\u271d = \u21910 a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nm : M\n\u22a2 some (single (Option.some a) m) = single a m\n[PROOFSTEP]\nclassical\next b\nsimp [single_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nm : M\n\u22a2 some (single (Option.some a) m) = single a m\n[PROOFSTEP]\next b\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nm : M\nb : \u03b1\n\u22a2 \u2191(some (single (Option.some a) m)) b = \u2191(single a m) b\n[PROOFSTEP]\nsimp [single_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\n\u22a2 prod f b = b none (\u2191f none) * prod (some f) fun a => b (Option.some a)\n[PROOFSTEP]\nclassical\napply induction_linear f\n\u00b7 simp [some_zero, h_zero]\n\u00b7 intro f\u2081 f\u2082 h\u2081 h\u2082\n rw [Finsupp.prod_add_index, h\u2081, h\u2082, some_add, Finsupp.prod_add_index]\n simp only [h_add, Pi.add_apply, Finsupp.coe_add]\n rw [mul_mul_mul_comm]\n all_goals simp [h_zero, h_add]\n\u00b7 rintro (_ | a) m <;> simp [h_zero, h_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\n\u22a2 prod f b = b none (\u2191f none) * prod (some f) fun a => b (Option.some a)\n[PROOFSTEP]\napply induction_linear f\n[GOAL]\ncase h0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\n\u22a2 prod 0 b = b none (\u21910 none) * prod (some 0) fun a => b (Option.some a)\n[PROOFSTEP]\nsimp [some_zero, h_zero]\n[GOAL]\ncase hadd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\n\u22a2 \u2200 (f g : Option \u03b1 \u2192\u2080 M),\n (prod f b = b none (\u2191f none) * prod (some f) fun a => b (Option.some a)) \u2192\n (prod g b = b none (\u2191g none) * prod (some g) fun a => b (Option.some a)) \u2192\n prod (f + g) b = b none (\u2191(f + g) none) * prod (some (f + g)) fun a => b (Option.some a)\n[PROOFSTEP]\nintro f\u2081 f\u2082 h\u2081 h\u2082\n[GOAL]\ncase hadd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 prod (f\u2081 + f\u2082) b = b none (\u2191(f\u2081 + f\u2082) none) * prod (some (f\u2081 + f\u2082)) fun a => b (Option.some a)\n[PROOFSTEP]\nrw [Finsupp.prod_add_index, h\u2081, h\u2082, some_add, Finsupp.prod_add_index]\n[GOAL]\ncase hadd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 (b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)) *\n (b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)) =\n b none (\u2191(f\u2081 + f\u2082) none) * ((prod (some f\u2081) fun a => b (Option.some a)) * prod (some f\u2082) fun a => b (Option.some a))\ncase hadd.h_zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : \u03b1), a \u2208 (some f\u2081).support \u222a (some f\u2082).support \u2192 b (Option.some a) 0 = 1\ncase hadd.h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : \u03b1),\n a \u2208 (some f\u2081).support \u222a (some f\u2082).support \u2192\n \u2200 (b\u2081 b\u2082 : M), b (Option.some a) (b\u2081 + b\u2082) = b (Option.some a) b\u2081 * b (Option.some a) b\u2082\ncase hadd.h_zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : Option \u03b1), a \u2208 f\u2081.support \u222a f\u2082.support \u2192 b a 0 = 1\ncase hadd.h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : Option \u03b1), a \u2208 f\u2081.support \u222a f\u2082.support \u2192 \u2200 (b\u2081 b\u2082 : M), b a (b\u2081 + b\u2082) = b a b\u2081 * b a b\u2082\n[PROOFSTEP]\nsimp only [h_add, Pi.add_apply, Finsupp.coe_add]\n[GOAL]\ncase hadd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 (b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)) *\n (b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)) =\n b none (\u2191f\u2081 none) * b none (\u2191f\u2082 none) *\n ((prod (some f\u2081) fun a => b (Option.some a)) * prod (some f\u2082) fun a => b (Option.some a))\ncase hadd.h_zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : \u03b1), a \u2208 (some f\u2081).support \u222a (some f\u2082).support \u2192 b (Option.some a) 0 = 1\ncase hadd.h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : \u03b1),\n a \u2208 (some f\u2081).support \u222a (some f\u2082).support \u2192\n \u2200 (b\u2081 b\u2082 : M), b (Option.some a) (b\u2081 + b\u2082) = b (Option.some a) b\u2081 * b (Option.some a) b\u2082\ncase hadd.h_zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : Option \u03b1), a \u2208 f\u2081.support \u222a f\u2082.support \u2192 b a 0 = 1\ncase hadd.h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : Option \u03b1), a \u2208 f\u2081.support \u222a f\u2082.support \u2192 \u2200 (b\u2081 b\u2082 : M), b a (b\u2081 + b\u2082) = b a b\u2081 * b a b\u2082\n[PROOFSTEP]\nrw [mul_mul_mul_comm]\n[GOAL]\ncase hadd.h_zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : \u03b1), a \u2208 (some f\u2081).support \u222a (some f\u2082).support \u2192 b (Option.some a) 0 = 1\ncase hadd.h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : \u03b1),\n a \u2208 (some f\u2081).support \u222a (some f\u2082).support \u2192\n \u2200 (b\u2081 b\u2082 : M), b (Option.some a) (b\u2081 + b\u2082) = b (Option.some a) b\u2081 * b (Option.some a) b\u2082\ncase hadd.h_zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : Option \u03b1), a \u2208 f\u2081.support \u222a f\u2082.support \u2192 b a 0 = 1\ncase hadd.h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : Option \u03b1), a \u2208 f\u2081.support \u222a f\u2082.support \u2192 \u2200 (b\u2081 b\u2082 : M), b a (b\u2081 + b\u2082) = b a b\u2081 * b a b\u2082\n[PROOFSTEP]\nall_goals simp [h_zero, h_add]\n[GOAL]\ncase hadd.h_zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : \u03b1), a \u2208 (some f\u2081).support \u222a (some f\u2082).support \u2192 b (Option.some a) 0 = 1\n[PROOFSTEP]\nsimp [h_zero, h_add]\n[GOAL]\ncase hadd.h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : \u03b1),\n a \u2208 (some f\u2081).support \u222a (some f\u2082).support \u2192\n \u2200 (b\u2081 b\u2082 : M), b (Option.some a) (b\u2081 + b\u2082) = b (Option.some a) b\u2081 * b (Option.some a) b\u2082\n[PROOFSTEP]\nsimp [h_zero, h_add]\n[GOAL]\ncase hadd.h_zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : Option \u03b1), a \u2208 f\u2081.support \u222a f\u2082.support \u2192 b a 0 = 1\n[PROOFSTEP]\nsimp [h_zero, h_add]\n[GOAL]\ncase hadd.h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nf\u2081 f\u2082 : Option \u03b1 \u2192\u2080 M\nh\u2081 : prod f\u2081 b = b none (\u2191f\u2081 none) * prod (some f\u2081) fun a => b (Option.some a)\nh\u2082 : prod f\u2082 b = b none (\u2191f\u2082 none) * prod (some f\u2082) fun a => b (Option.some a)\n\u22a2 \u2200 (a : Option \u03b1), a \u2208 f\u2081.support \u222a f\u2082.support \u2192 \u2200 (b\u2081 b\u2082 : M), b a (b\u2081 + b\u2082) = b a b\u2081 * b a b\u2082\n[PROOFSTEP]\nsimp [h_zero, h_add]\n[GOAL]\ncase hsingle\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\n\u22a2 \u2200 (a : Option \u03b1) (b_1 : M),\n prod (single a b_1) b = b none (\u2191(single a b_1) none) * prod (some (single a b_1)) fun a => b (Option.some a)\n[PROOFSTEP]\nrintro (_ | a) m\n[GOAL]\ncase hsingle.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\nm : M\n\u22a2 prod (single none m) b = b none (\u2191(single none m) none) * prod (some (single none m)) fun a => b (Option.some a)\n[PROOFSTEP]\nsimp [h_zero, h_add]\n[GOAL]\ncase hsingle.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : CommMonoid N\nf : Option \u03b1 \u2192\u2080 M\nb : Option \u03b1 \u2192 M \u2192 N\nh_zero : \u2200 (o : Option \u03b1), b o 0 = 1\nh_add : \u2200 (o : Option \u03b1) (m\u2081 m\u2082 : M), b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082\na : \u03b1\nm : M\n\u22a2 prod (single (Option.some a) m) b =\n b none (\u2191(single (Option.some a) m) none) * prod (some (single (Option.some a) m)) fun a => b (Option.some a)\n[PROOFSTEP]\nsimp [h_zero, h_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np\u271d : \u03b1 \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 a \u2208 Finset.filter (fun a => p a) f.support \u2194 (fun a => if p a then \u2191f a else 0) a \u2260 0\n[PROOFSTEP]\nsimp only\n -- porting note: necessary to beta reduce to activate `split_ifs`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np\u271d : \u03b1 \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 a \u2208 Finset.filter (fun a => p a) f.support \u2194 (if p a then \u2191f a else 0) \u2260 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np\u271d : \u03b1 \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : p a\n\u22a2 a \u2208 Finset.filter (fun a => p a) f.support \u2194 \u2191f a \u2260 0\n[PROOFSTEP]\nsimp only [h, @mem_filter _ _ (Classical.decPred p), mem_support_iff]\n -- porting note: I needed to provide the instance explicitly\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np\u271d : \u03b1 \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : p a\n\u22a2 \u2191f a \u2260 0 \u2227 True \u2194 \u2191f a \u2260 0\n[PROOFSTEP]\ntauto\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np\u271d : \u03b1 \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : \u00acp a\n\u22a2 a \u2208 Finset.filter (fun a => p a) f.support \u2194 0 \u2260 0\n[PROOFSTEP]\nsimp only [h, @mem_filter _ _ (Classical.decPred p), mem_support_iff]\n -- porting note: I needed to provide the instance explicitly\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np\u271d : \u03b1 \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : \u00acp a\n\u22a2 \u2191f a \u2260 0 \u2227 False \u2194 0 \u2260 0\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nD : Decidable (p a)\n\u22a2 \u2191(filter p f) a = if p a then \u2191f a else 0\n[PROOFSTEP]\nrw [Subsingleton.elim D]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nD : Decidable (p a)\n\u22a2 \u2191(filter p f) a = if p a then \u2191f a else 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\n\u22a2 filter p f = 0 \u2194 \u2200 (x : \u03b1), p x \u2192 \u2191f x = 0\n[PROOFSTEP]\nsimp only [FunLike.ext_iff, filter_eq_indicator, zero_apply, Set.indicator_apply_eq_zero, Set.mem_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\n\u22a2 filter p f = f \u2194 \u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 p x\n[PROOFSTEP]\nsimp only [FunLike.ext_iff, filter_eq_indicator, Set.indicator_apply_eq_self, Set.mem_setOf_eq, not_imp_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\nD : DecidablePred p\n\u22a2 (filter p f).support = Finset.filter p f.support\n[PROOFSTEP]\nrw [Subsingleton.elim D]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\nD : DecidablePred p\n\u22a2 (filter p f).support = Finset.filter p f.support\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\n\u22a2 filter p 0 = 0\n[PROOFSTEP]\nclassical rw [\u2190 support_eq_empty, support_filter, support_zero, Finset.filter_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\n\u22a2 filter p 0 = 0\n[PROOFSTEP]\nrw [\u2190 support_eq_empty, support_filter, support_zero, Finset.filter_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\ninst\u271d : CommMonoid N\ng : \u03b1 \u2192 M \u2192 N\n\u22a2 prod (filter p f) g = \u220f x in (filter p f).support, g x (\u2191f x)\n[PROOFSTEP]\nclassical\nrefine' Finset.prod_congr rfl fun x hx => _\nrw [support_filter, Finset.mem_filter] at hx \nrw [filter_apply_pos _ _ hx.2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\ninst\u271d : CommMonoid N\ng : \u03b1 \u2192 M \u2192 N\n\u22a2 prod (filter p f) g = \u220f x in (filter p f).support, g x (\u2191f x)\n[PROOFSTEP]\nrefine' Finset.prod_congr rfl fun x hx => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\ninst\u271d : CommMonoid N\ng : \u03b1 \u2192 M \u2192 N\nx : \u03b1\nhx : x \u2208 (filter p f).support\n\u22a2 g x (\u2191(filter p f) x) = g x (\u2191f x)\n[PROOFSTEP]\nrw [support_filter, Finset.mem_filter] at hx \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\ninst\u271d : CommMonoid N\ng : \u03b1 \u2192 M \u2192 N\nx : \u03b1\nhx : x \u2208 f.support \u2227 p x\n\u22a2 g x (\u2191(filter p f) x) = g x (\u2191f x)\n[PROOFSTEP]\nrw [filter_apply_pos _ _ hx.2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\ninst\u271d : CommMonoid N\ng : \u03b1 \u2192 M \u2192 N\n\u22a2 prod (filter p f) g * prod (filter (fun a => \u00acp a) f) g = prod f g\n[PROOFSTEP]\nclassical simp_rw [prod_filter_index, support_filter, Finset.prod_filter_mul_prod_filter_not, Finsupp.prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\ninst\u271d : CommMonoid N\ng : \u03b1 \u2192 M \u2192 N\n\u22a2 prod (filter p f) g * prod (filter (fun a => \u00acp a) f) g = prod f g\n[PROOFSTEP]\nsimp_rw [prod_filter_index, support_filter, Finset.prod_filter_mul_prod_filter_not, Finsupp.prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\ny : M\n\u22a2 y \u2208 frange f \u2194 y \u2260 0 \u2227 \u2203 x, \u2191f x = y\n[PROOFSTEP]\nrw [frange, @Finset.mem_image _ _ (Classical.decEq _) _ f.support]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\ny : M\n\u22a2 (\u2203 a, a \u2208 f.support \u2227 \u2191f a = y) \u2194 y \u2260 0 \u2227 \u2203 x, \u2191f x = y\n[PROOFSTEP]\nexact\n \u27e8fun \u27e8x, hx1, hx2\u27e9 => \u27e8hx2 \u25b8 mem_support_iff.1 hx1, x, hx2\u27e9, fun \u27e8hy, x, hx\u27e9 =>\n \u27e8x, mem_support_iff.2 (hx.symm \u25b8 hy), hx\u27e9\u27e9\n -- porting note: maybe there is a better way to fix this, but (1) it wasn't seeing past `frange`\n -- the definition, and (2) it needed the `Classical.decEq` instance again.\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nx : \u03b1\ny r : M\nhr : r \u2208 frange (single x y)\nt : r \u2260 0\nht1 : \u03b1\nht2 : \u2191(single x y) ht1 = r\n\u22a2 \u2191(single x y) ht1 \u2208 {y}\n[PROOFSTEP]\nclassical\nrw [single_apply] at ht2 \u22a2\nsplit_ifs at ht2 \u22a2\n\u00b7 exact Finset.mem_singleton_self _\n\u00b7 exact (t ht2.symm).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nx : \u03b1\ny r : M\nhr : r \u2208 frange (single x y)\nt : r \u2260 0\nht1 : \u03b1\nht2 : \u2191(single x y) ht1 = r\n\u22a2 \u2191(single x y) ht1 \u2208 {y}\n[PROOFSTEP]\nrw [single_apply] at ht2 \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nx : \u03b1\ny r : M\nhr : r \u2208 frange (single x y)\nt : r \u2260 0\nht1 : \u03b1\nht2\u271d : \u2191(single x y) ht1 = r\nht2 : (if x = ht1 then y else 0) = r\n\u22a2 (if x = ht1 then y else 0) \u2208 {y}\n[PROOFSTEP]\nsplit_ifs at ht2 \u22a2\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nx : \u03b1\ny r : M\nhr : r \u2208 frange (single x y)\nt : r \u2260 0\nht1 : \u03b1\nht2\u271d : \u2191(single x y) ht1 = r\nh\u271d : x = ht1\nht2 : y = r\n\u22a2 y \u2208 {y}\n[PROOFSTEP]\nexact Finset.mem_singleton_self _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nx : \u03b1\ny r : M\nhr : r \u2208 frange (single x y)\nt : r \u2260 0\nht1 : \u03b1\nht2\u271d : \u2191(single x y) ht1 = r\nh\u271d : \u00acx = ht1\nht2 : 0 = r\n\u22a2 0 \u2208 {y}\n[PROOFSTEP]\nexact (t ht2.symm).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np\u271d p : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\na : Subtype p\n\u22a2 a \u2208 Finset.subtype p f.support \u2194 (\u2191f \u2218 Subtype.val) a \u2260 0\n[PROOFSTEP]\nsimp only [@mem_subtype _ _ (Classical.decPred p), mem_support_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np\u271d p : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\na : Subtype p\n\u22a2 \u2191f \u2191a \u2260 0 \u2194 (\u2191f \u2218 Subtype.val) a \u2260 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np : \u03b1 \u2192 Prop\nD : DecidablePred p\nf : \u03b1 \u2192\u2080 M\n\u22a2 (subtypeDomain p f).support = Finset.subtype p f.support\n[PROOFSTEP]\nrw [Subsingleton.elim D]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np : \u03b1 \u2192 Prop\nD : DecidablePred p\nf : \u03b1 \u2192\u2080 M\n\u22a2 (subtypeDomain p f).support = Finset.subtype p f.support\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\n\u22a2 subtypeDomain p f = 0 \u2194 \u2200 (x : \u03b1), p x \u2192 \u2191f x = 0\n[PROOFSTEP]\nclassical simp_rw [\u2190 support_eq_empty, support_subtypeDomain, subtype_eq_empty, not_mem_support_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\n\u22a2 subtypeDomain p f = 0 \u2194 \u2200 (x : \u03b1), p x \u2192 \u2191f x = 0\n[PROOFSTEP]\nsimp_rw [\u2190 support_eq_empty, support_subtypeDomain, subtype_eq_empty, not_mem_support_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\nhf : \u2200 (x : \u03b1), x \u2208 f.support \u2192 p x\nH : \u2200 (x : \u03b1), p x \u2192 \u2191f x = 0\nx : \u03b1\n\u22a2 \u2191f x = \u21910 x\n[PROOFSTEP]\nclassical exact if hx : p x then H x hx else not_mem_support_iff.1 <| mt (hf x) hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\nhf : \u2200 (x : \u03b1), x \u2208 f.support \u2192 p x\nH : \u2200 (x : \u03b1), p x \u2192 \u2191f x = 0\nx : \u03b1\n\u22a2 \u2191f x = \u21910 x\n[PROOFSTEP]\nexact if hx : p x then H x hx else not_mem_support_iff.1 <| mt (hf x) hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192\u2080 M\nhf : \u2200 (x : \u03b1), x \u2208 f.support \u2192 p x\nH : f = 0\nx : \u03b1\nx\u271d : p x\n\u22a2 \u2191f x = 0\n[PROOFSTEP]\nsimp [H]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\np : \u03b1 \u2192 Prop\ninst\u271d : CommMonoid N\nv : \u03b1 \u2192\u2080 M\nh : \u03b1 \u2192 M \u2192 N\nhp : \u2200 (x : \u03b1), x \u2208 v.support \u2192 p x\nx\u271d : Subtype p\n\u22a2 \u2200 (ha : x\u271d \u2208 (subtypeDomain p v).support), (fun p_1 x => \u2191p_1) x\u271d ha \u2208 v.support\n[PROOFSTEP]\nclassical exact mem_subtype.1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\np : \u03b1 \u2192 Prop\ninst\u271d : CommMonoid N\nv : \u03b1 \u2192\u2080 M\nh : \u03b1 \u2192 M \u2192 N\nhp : \u2200 (x : \u03b1), x \u2208 v.support \u2192 p x\nx\u271d : Subtype p\n\u22a2 \u2200 (ha : x\u271d \u2208 (subtypeDomain p v).support), (fun p_1 x => \u2191p_1) x\u271d ha \u2208 v.support\n[PROOFSTEP]\nexact mem_subtype.1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\np : \u03b1 \u2192 Prop\ninst\u271d : CommMonoid N\nv : \u03b1 \u2192\u2080 M\nh : \u03b1 \u2192 M \u2192 N\nhp : \u2200 (x : \u03b1), x \u2208 v.support \u2192 p x\nb : \u03b1\nhb : b \u2208 v.support\n\u22a2 { val := b, property := (_ : p b) } \u2208 (subtypeDomain p v).support\n[PROOFSTEP]\nclassical exact mem_subtype.2 hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\np : \u03b1 \u2192 Prop\ninst\u271d : CommMonoid N\nv : \u03b1 \u2192\u2080 M\nh : \u03b1 \u2192 M \u2192 N\nhp : \u2200 (x : \u03b1), x \u2208 v.support \u2192 p x\nb : \u03b1\nhb : b \u2208 v.support\n\u22a2 { val := b, property := (_ : p b) } \u2208 (subtypeDomain p v).support\n[PROOFSTEP]\nexact mem_subtype.2 hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\np\u271d p : \u03b1 \u2192 Prop\nD : DecidablePred p\nf : \u03b1 \u2192\u2080 M\n\u22a2 (filter p f).support = Finset.filter p f.support\n[PROOFSTEP]\nrw [Subsingleton.elim D]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\np\u271d p : \u03b1 \u2192 Prop\nD : DecidablePred p\nf : \u03b1 \u2192\u2080 M\n\u22a2 (filter p f).support = Finset.filter p f.support\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\np\u271d p : \u03b1 \u2192 Prop\nD : DecidablePred p\nf : \u03b1 \u2192\u2080 M\nx : \u03b1\nhx : x \u2208 Finset.filter p f.support\n\u22a2 single x (\u2191(filter p f) x) = single x (\u2191f x)\n[PROOFSTEP]\nrw [filter_apply_pos _ _ (mem_filter.1 hx).2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\ns : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nh : a \u2208 (Multiset.sum 0).support\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\ns : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\n\u22a2 \u2200 \u2983a_1 : \u03b1 \u2192\u2080 M\u2984 {s : Multiset (\u03b1 \u2192\u2080 M)},\n (a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support) \u2192\n a \u2208 (Multiset.sum (a_1 ::\u2098 s)).support \u2192 \u2203 f, f \u2208 a_1 ::\u2098 s \u2227 a \u2208 f.support\n[PROOFSTEP]\nintro f s ih ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\ns\u271d : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nf : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support\nha : a \u2208 (Multiset.sum (f ::\u2098 s)).support\n\u22a2 \u2203 f_1, f_1 \u2208 f ::\u2098 s \u2227 a \u2208 f_1.support\n[PROOFSTEP]\nby_cases h : a \u2208 f.support\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\ns\u271d : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nf : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support\nha : a \u2208 (Multiset.sum (f ::\u2098 s)).support\nh : a \u2208 f.support\n\u22a2 \u2203 f_1, f_1 \u2208 f ::\u2098 s \u2227 a \u2208 f_1.support\n[PROOFSTEP]\nexact \u27e8f, Multiset.mem_cons_self _ _, h\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\ns\u271d : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nf : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support\nha : a \u2208 (Multiset.sum (f ::\u2098 s)).support\nh : \u00aca \u2208 f.support\n\u22a2 \u2203 f_1, f_1 \u2208 f ::\u2098 s \u2227 a \u2208 f_1.support\n[PROOFSTEP]\nsimp only [Multiset.sum_cons, mem_support_iff, add_apply, not_mem_support_iff.1 h, zero_add] at ha \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\ns\u271d : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nf : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support\nh : \u00aca \u2208 f.support\nha : \u2191(Multiset.sum s) a \u2260 0\n\u22a2 \u2203 f_1, f_1 \u2208 f ::\u2098 s \u2227 a \u2208 f_1.support\n[PROOFSTEP]\nrcases ih (mem_support_iff.2 ha) with \u27e8f', h\u2080, h\u2081\u27e9\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\ns\u271d : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nf : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support\nh : \u00aca \u2208 f.support\nha : \u2191(Multiset.sum s) a \u2260 0\nf' : \u03b1 \u2192\u2080 M\nh\u2080 : f' \u2208 s\nh\u2081 : a \u2208 f'.support\n\u22a2 \u2203 f_1, f_1 \u2208 f ::\u2098 s \u2227 a \u2208 f_1.support\n[PROOFSTEP]\nexact \u27e8f', Multiset.mem_cons_of_mem h\u2080, h\u2081\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\nx : \u03b1\ny : \u03b2\n\u22a2 \u2191(\u2191(Finsupp.curry f) x) y = \u2191f (x, y)\n[PROOFSTEP]\nclassical\nhave : \u2200 b : \u03b1 \u00d7 \u03b2, single b.fst (single b.snd (f b)) x y = if b = (x, y) then f b else 0 :=\n by\n rintro \u27e8b\u2081, b\u2082\u27e9\n simp [single_apply, ite_apply, Prod.ext_iff, ite_and]\n split_ifs <;> simp [single_apply, *]\nrw [Finsupp.curry, sum_apply, sum_apply, Finsupp.sum, Finset.sum_eq_single, this, if_pos rfl]\n\u00b7 intro b _ b_ne\n rw [this b, if_neg b_ne]\n\u00b7 intro hxy\n rw [this (x, y), if_pos rfl, not_mem_support_iff.mp hxy]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\nx : \u03b1\ny : \u03b2\n\u22a2 \u2191(\u2191(Finsupp.curry f) x) y = \u2191f (x, y)\n[PROOFSTEP]\nhave : \u2200 b : \u03b1 \u00d7 \u03b2, single b.fst (single b.snd (f b)) x y = if b = (x, y) then f b else 0 :=\n by\n rintro \u27e8b\u2081, b\u2082\u27e9\n simp [single_apply, ite_apply, Prod.ext_iff, ite_and]\n split_ifs <;> simp [single_apply, *]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\nx : \u03b1\ny : \u03b2\n\u22a2 \u2200 (b : \u03b1 \u00d7 \u03b2), \u2191(\u2191(single b.fst (single b.snd (\u2191f b))) x) y = if b = (x, y) then \u2191f b else 0\n[PROOFSTEP]\nrintro \u27e8b\u2081, b\u2082\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\nx : \u03b1\ny : \u03b2\nb\u2081 : \u03b1\nb\u2082 : \u03b2\n\u22a2 \u2191(\u2191(single (b\u2081, b\u2082).fst (single (b\u2081, b\u2082).snd (\u2191f (b\u2081, b\u2082)))) x) y = if (b\u2081, b\u2082) = (x, y) then \u2191f (b\u2081, b\u2082) else 0\n[PROOFSTEP]\nsimp [single_apply, ite_apply, Prod.ext_iff, ite_and]\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\nx : \u03b1\ny : \u03b2\nb\u2081 : \u03b1\nb\u2082 : \u03b2\n\u22a2 \u2191(if b\u2081 = x then single b\u2082 (\u2191f (b\u2081, b\u2082)) else 0) y = if b\u2081 = x then if b\u2082 = y then \u2191f (b\u2081, b\u2082) else 0 else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\nx : \u03b1\ny : \u03b2\nb\u2081 : \u03b1\nb\u2082 : \u03b2\nh\u271d\u00b9 : b\u2081 = x\nh\u271d : b\u2082 = y\n\u22a2 \u2191(single b\u2082 (\u2191f (b\u2081, b\u2082))) y = \u2191f (b\u2081, b\u2082)\n[PROOFSTEP]\nsimp [single_apply, *]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\nx : \u03b1\ny : \u03b2\nb\u2081 : \u03b1\nb\u2082 : \u03b2\nh\u271d\u00b9 : b\u2081 = x\nh\u271d : \u00acb\u2082 = y\n\u22a2 \u2191(single b\u2082 (\u2191f (b\u2081, b\u2082))) y = 0\n[PROOFSTEP]\nsimp [single_apply, *]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\nx : \u03b1\ny : \u03b2\nb\u2081 : \u03b1\nb\u2082 : \u03b2\nh\u271d : \u00acb\u2081 = x\n\u22a2 \u21910 y = 0\n[PROOFSTEP]\nsimp [single_apply, *]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\nx : \u03b1\ny : \u03b2\nthis : \u2200 (b : \u03b1 \u00d7 \u03b2), \u2191(\u2191(single b.fst (single b.snd (\u2191f b))) x) y = if b = (x, y) then \u2191f b else 0\n\u22a2 \u2191(\u2191(Finsupp.curry f) x) y = \u2191f (x, y)\n[PROOFSTEP]\nrw [Finsupp.curry, sum_apply, sum_apply, Finsupp.sum, Finset.sum_eq_single, this, if_pos rfl]\n[GOAL]\ncase h\u2080\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\nx : \u03b1\ny : \u03b2\nthis : \u2200 (b : \u03b1 \u00d7 \u03b2), \u2191(\u2191(single b.fst (single b.snd (\u2191f b))) x) y = if b = (x, y) then \u2191f b else 0\n\u22a2 \u2200 (b : \u03b1 \u00d7 \u03b2), b \u2208 f.support \u2192 b \u2260 (x, y) \u2192 \u2191(\u2191(single b.fst (single b.snd (\u2191f b))) x) y = 0\n[PROOFSTEP]\nintro b _ b_ne\n[GOAL]\ncase h\u2080\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\nx : \u03b1\ny : \u03b2\nthis : \u2200 (b : \u03b1 \u00d7 \u03b2), \u2191(\u2191(single b.fst (single b.snd (\u2191f b))) x) y = if b = (x, y) then \u2191f b else 0\nb : \u03b1 \u00d7 \u03b2\na\u271d : b \u2208 f.support\nb_ne : b \u2260 (x, y)\n\u22a2 \u2191(\u2191(single b.fst (single b.snd (\u2191f b))) x) y = 0\n[PROOFSTEP]\nrw [this b, if_neg b_ne]\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\nx : \u03b1\ny : \u03b2\nthis : \u2200 (b : \u03b1 \u00d7 \u03b2), \u2191(\u2191(single b.fst (single b.snd (\u2191f b))) x) y = if b = (x, y) then \u2191f b else 0\n\u22a2 \u00ac(x, y) \u2208 f.support \u2192 \u2191(\u2191(single (x, y).fst (single (x, y).snd (\u2191f (x, y)))) x) y = 0\n[PROOFSTEP]\nintro hxy\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\nx : \u03b1\ny : \u03b2\nthis : \u2200 (b : \u03b1 \u00d7 \u03b2), \u2191(\u2191(single b.fst (single b.snd (\u2191f b))) x) y = if b = (x, y) then \u2191f b else 0\nhxy : \u00ac(x, y) \u2208 f.support\n\u22a2 \u2191(\u2191(single (x, y).fst (single (x, y).snd (\u2191f (x, y)))) x) y = 0\n[PROOFSTEP]\nrw [this (x, y), if_pos rfl, not_mem_support_iff.mp hxy]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\ng : \u03b1 \u2192 \u03b2 \u2192 M \u2192 N\nhg\u2080 : \u2200 (a : \u03b1) (b : \u03b2), g a b 0 = 0\nhg\u2081 : \u2200 (a : \u03b1) (b : \u03b2) (c\u2080 c\u2081 : M), g a b (c\u2080 + c\u2081) = g a b c\u2080 + g a b c\u2081\n\u22a2 (sum (Finsupp.curry f) fun a f => sum f (g a)) = sum f fun p c => g p.fst p.snd c\n[PROOFSTEP]\nrw [Finsupp.curry]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\ng : \u03b1 \u2192 \u03b2 \u2192 M \u2192 N\nhg\u2080 : \u2200 (a : \u03b1) (b : \u03b2), g a b 0 = 0\nhg\u2081 : \u2200 (a : \u03b1) (b : \u03b2) (c\u2080 c\u2081 : M), g a b (c\u2080 + c\u2081) = g a b c\u2080 + g a b c\u2081\n\u22a2 (sum (sum f fun p c => single p.fst (single p.snd c)) fun a f => sum f (g a)) = sum f fun p c => g p.fst p.snd c\n[PROOFSTEP]\ntrans\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\ng : \u03b1 \u2192 \u03b2 \u2192 M \u2192 N\nhg\u2080 : \u2200 (a : \u03b1) (b : \u03b2), g a b 0 = 0\nhg\u2081 : \u2200 (a : \u03b1) (b : \u03b2) (c\u2080 c\u2081 : M), g a b (c\u2080 + c\u2081) = g a b c\u2080 + g a b c\u2081\n\u22a2 (sum (sum f fun p c => single p.fst (single p.snd c)) fun a f => sum f (g a)) = ?m.537688\n[PROOFSTEP]\nexact\n sum_sum_index (fun a => sum_zero_index) fun a b\u2080 b\u2081 => sum_add_index' (fun a => hg\u2080 _ _) fun c d\u2080 d\u2081 => hg\u2081 _ _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\ng : \u03b1 \u2192 \u03b2 \u2192 M \u2192 N\nhg\u2080 : \u2200 (a : \u03b1) (b : \u03b2), g a b 0 = 0\nhg\u2081 : \u2200 (a : \u03b1) (b : \u03b2) (c\u2080 c\u2081 : M), g a b (c\u2080 + c\u2081) = g a b c\u2080 + g a b c\u2081\n\u22a2 (sum f fun a b => sum (single a.fst (single a.snd b)) fun a f => sum f (g a)) = sum f fun p c => g p.fst p.snd c\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_g\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\ng : \u03b1 \u2192 \u03b2 \u2192 M \u2192 N\nhg\u2080 : \u2200 (a : \u03b1) (b : \u03b2), g a b 0 = 0\nhg\u2081 : \u2200 (a : \u03b1) (b : \u03b2) (c\u2080 c\u2081 : M), g a b (c\u2080 + c\u2081) = g a b c\u2080 + g a b c\u2081\n\u22a2 (fun a b => sum (single a.fst (single a.snd b)) fun a f => sum f (g a)) = fun p c => g p.fst p.snd c\n[PROOFSTEP]\nfunext p c\n[GOAL]\ncase e_g.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\ng : \u03b1 \u2192 \u03b2 \u2192 M \u2192 N\nhg\u2080 : \u2200 (a : \u03b1) (b : \u03b2), g a b 0 = 0\nhg\u2081 : \u2200 (a : \u03b1) (b : \u03b2) (c\u2080 c\u2081 : M), g a b (c\u2080 + c\u2081) = g a b c\u2080 + g a b c\u2081\np : \u03b1 \u00d7 \u03b2\nc : M\n\u22a2 (sum (single p.fst (single p.snd c)) fun a f => sum f (g a)) = g p.fst p.snd c\n[PROOFSTEP]\ntrans\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\ng : \u03b1 \u2192 \u03b2 \u2192 M \u2192 N\nhg\u2080 : \u2200 (a : \u03b1) (b : \u03b2), g a b 0 = 0\nhg\u2081 : \u2200 (a : \u03b1) (b : \u03b2) (c\u2080 c\u2081 : M), g a b (c\u2080 + c\u2081) = g a b c\u2080 + g a b c\u2081\np : \u03b1 \u00d7 \u03b2\nc : M\n\u22a2 (sum (single p.fst (single p.snd c)) fun a f => sum f (g a)) = ?m.538719\n[PROOFSTEP]\nexact sum_single_index sum_zero_index\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\ng : \u03b1 \u2192 \u03b2 \u2192 M \u2192 N\nhg\u2080 : \u2200 (a : \u03b1) (b : \u03b2), g a b 0 = 0\nhg\u2081 : \u2200 (a : \u03b1) (b : \u03b2) (c\u2080 c\u2081 : M), g a b (c\u2080 + c\u2081) = g a b c\u2080 + g a b c\u2081\np : \u03b1 \u00d7 \u03b2\nc : M\n\u22a2 sum (single p.snd c) (g p.fst) = g p.fst p.snd c\n[PROOFSTEP]\nexact sum_single_index (hg\u2080 _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\n\u22a2 Finsupp.uncurry (Finsupp.curry f) = f\n[PROOFSTEP]\nrw [Finsupp.uncurry, sum_curry_index]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\n\u22a2 (sum f fun p c => single (p.fst, p.snd) c) = f\n[PROOFSTEP]\nsimp_rw [Prod.mk.eta, sum_single]\n[GOAL]\ncase hg\u2080\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\n\u22a2 \u2200 (a : \u03b1) (b : \u03b2), single (a, b) 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase hg\u2080\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : \u03b2\n\u22a2 single (a\u271d, b\u271d) 0 = 0\n[PROOFSTEP]\napply single_zero\n[GOAL]\ncase hg\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\n\u22a2 \u2200 (a : \u03b1) (b : \u03b2) (c\u2080 c\u2081 : M), single (a, b) (c\u2080 + c\u2081) = single (a, b) c\u2080 + single (a, b) c\u2081\n[PROOFSTEP]\nintros\n[GOAL]\ncase hg\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : \u03b2\nc\u2080\u271d c\u2081\u271d : M\n\u22a2 single (a\u271d, b\u271d) (c\u2080\u271d + c\u2081\u271d) = single (a\u271d, b\u271d) c\u2080\u271d + single (a\u271d, b\u271d) c\u2081\u271d\n[PROOFSTEP]\napply single_add\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u2192\u2080 \u03b2 \u2192\u2080 M\n\u22a2 Finsupp.curry (Finsupp.uncurry f) = f\n[PROOFSTEP]\nsimp only [Finsupp.curry, Finsupp.uncurry, sum_sum_index, sum_zero_index, sum_add_index, sum_single_index, single_zero,\n single_add, eq_self_iff_true, forall_true_iff, forall\u2083_true_iff, Prod.mk.eta, (single_sum _ _ _).symm, sum_single]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\np : \u03b1 \u2192 Prop\n\u22a2 Finsupp.curry (filter (fun a => p a.fst) f) = filter p (Finsupp.curry f)\n[PROOFSTEP]\nclassical\nrw [Finsupp.curry, Finsupp.curry, Finsupp.sum, Finsupp.sum, filter_sum, support_filter, sum_filter]\nrefine' Finset.sum_congr rfl _\nrintro \u27e8a\u2081, a\u2082\u27e9 _\ndsimp only\nsplit_ifs with h\n\u00b7 rw [filter_apply_pos, filter_single_of_pos] <;> exact h\n\u00b7 rwa [filter_single_of_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\np : \u03b1 \u2192 Prop\n\u22a2 Finsupp.curry (filter (fun a => p a.fst) f) = filter p (Finsupp.curry f)\n[PROOFSTEP]\nrw [Finsupp.curry, Finsupp.curry, Finsupp.sum, Finsupp.sum, filter_sum, support_filter, sum_filter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\np : \u03b1 \u2192 Prop\n\u22a2 (\u2211 a in f.support, if p a.fst then single a.fst (single a.snd (\u2191(filter (fun a => p a.fst) f) a)) else 0) =\n \u2211 a in f.support, filter p (single a.fst (single a.snd (\u2191f a)))\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\np : \u03b1 \u2192 Prop\n\u22a2 \u2200 (x : \u03b1 \u00d7 \u03b2),\n x \u2208 f.support \u2192\n (if p x.fst then single x.fst (single x.snd (\u2191(filter (fun a => p a.fst) f) x)) else 0) =\n filter p (single x.fst (single x.snd (\u2191f x)))\n[PROOFSTEP]\nrintro \u27e8a\u2081, a\u2082\u27e9 _\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\np : \u03b1 \u2192 Prop\na\u2081 : \u03b1\na\u2082 : \u03b2\na\u271d : (a\u2081, a\u2082) \u2208 f.support\n\u22a2 (if p (a\u2081, a\u2082).fst then single (a\u2081, a\u2082).fst (single (a\u2081, a\u2082).snd (\u2191(filter (fun a => p a.fst) f) (a\u2081, a\u2082))) else 0) =\n filter p (single (a\u2081, a\u2082).fst (single (a\u2081, a\u2082).snd (\u2191f (a\u2081, a\u2082))))\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\np : \u03b1 \u2192 Prop\na\u2081 : \u03b1\na\u2082 : \u03b2\na\u271d : (a\u2081, a\u2082) \u2208 f.support\n\u22a2 (if p a\u2081 then single a\u2081 (single a\u2082 (\u2191(filter (fun a => p a.fst) f) (a\u2081, a\u2082))) else 0) =\n filter p (single a\u2081 (single a\u2082 (\u2191f (a\u2081, a\u2082))))\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\np : \u03b1 \u2192 Prop\na\u2081 : \u03b1\na\u2082 : \u03b2\na\u271d : (a\u2081, a\u2082) \u2208 f.support\nh : p a\u2081\n\u22a2 single a\u2081 (single a\u2082 (\u2191(filter (fun a => p a.fst) f) (a\u2081, a\u2082))) = filter p (single a\u2081 (single a\u2082 (\u2191f (a\u2081, a\u2082))))\n[PROOFSTEP]\nrw [filter_apply_pos, filter_single_of_pos]\n[GOAL]\ncase pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\np : \u03b1 \u2192 Prop\na\u2081 : \u03b1\na\u2082 : \u03b2\na\u271d : (a\u2081, a\u2082) \u2208 f.support\nh : p a\u2081\n\u22a2 p a\u2081\n[PROOFSTEP]\nexact h\n[GOAL]\ncase pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\np : \u03b1 \u2192 Prop\na\u2081 : \u03b1\na\u2082 : \u03b2\na\u271d : (a\u2081, a\u2082) \u2208 f.support\nh : p a\u2081\n\u22a2 p (a\u2081, a\u2082).fst\n[PROOFSTEP]\nexact h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\np : \u03b1 \u2192 Prop\na\u2081 : \u03b1\na\u2082 : \u03b2\na\u271d : (a\u2081, a\u2082) \u2208 f.support\nh : \u00acp a\u2081\n\u22a2 0 = filter p (single a\u2081 (single a\u2082 (\u2191f (a\u2081, a\u2082))))\n[PROOFSTEP]\nrwa [filter_single_of_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\n\u22a2 (Finsupp.curry f).support \u2286 image Prod.fst f.support\n[PROOFSTEP]\nrw [\u2190 Finset.biUnion_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\n\u22a2 (Finsupp.curry f).support \u2286 Finset.biUnion f.support fun a => {a.fst}\n[PROOFSTEP]\nrefine' Finset.Subset.trans support_sum _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u00d7 \u03b2 \u2192\u2080 M\n\u22a2 (Finset.biUnion f.support fun a => (single a.fst (single a.snd (\u2191f a))).support) \u2286\n Finset.biUnion f.support fun a => {a.fst}\n[PROOFSTEP]\nrefine' Finset.biUnion_mono fun a _ => support_single_subset\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2192\u2080 \u03b3\ng : \u03b2 \u2192\u2080 \u03b3\n\u22a2 Finset (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b1\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2192\u2080 \u03b3\ng : \u03b2 \u2192\u2080 \u03b3\nthis : DecidableEq \u03b1\n\u22a2 Finset (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b2\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2192\u2080 \u03b3\ng : \u03b2 \u2192\u2080 \u03b3\nthis\u271d : DecidableEq \u03b1\nthis : DecidableEq \u03b2\n\u22a2 Finset (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\nexact f.support.map \u27e8_, Sum.inl_injective\u27e9 \u222a g.support.map \u27e8_, Sum.inr_injective\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2192\u2080 \u03b3\ng : \u03b2 \u2192\u2080 \u03b3\nab : \u03b1 \u2295 \u03b2\nh : Sum.elim (\u2191f) (\u2191g) ab \u2260 0\n\u22a2 ab \u2208\n map { toFun := Sum.inl, inj' := (_ : Injective Sum.inl) } f.support \u222a\n map { toFun := Sum.inr, inj' := (_ : Injective Sum.inr) } g.support\n[PROOFSTEP]\ncases' ab with a b\n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2192\u2080 \u03b3\ng : \u03b2 \u2192\u2080 \u03b3\na : \u03b1\nh : Sum.elim (\u2191f) (\u2191g) (Sum.inl a) \u2260 0\n\u22a2 Sum.inl a \u2208\n map { toFun := Sum.inl, inj' := (_ : Injective Sum.inl) } f.support \u222a\n map { toFun := Sum.inr, inj' := (_ : Injective Sum.inr) } g.support\n[PROOFSTEP]\nletI := Classical.decEq \u03b1\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2192\u2080 \u03b3\ng : \u03b2 \u2192\u2080 \u03b3\nb : \u03b2\nh : Sum.elim (\u2191f) (\u2191g) (Sum.inr b) \u2260 0\n\u22a2 Sum.inr b \u2208\n map { toFun := Sum.inl, inj' := (_ : Injective Sum.inl) } f.support \u222a\n map { toFun := Sum.inr, inj' := (_ : Injective Sum.inr) } g.support\n[PROOFSTEP]\nletI := Classical.decEq \u03b1\n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2192\u2080 \u03b3\ng : \u03b2 \u2192\u2080 \u03b3\na : \u03b1\nh : Sum.elim (\u2191f) (\u2191g) (Sum.inl a) \u2260 0\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\n\u22a2 Sum.inl a \u2208\n map { toFun := Sum.inl, inj' := (_ : Injective Sum.inl) } f.support \u222a\n map { toFun := Sum.inr, inj' := (_ : Injective Sum.inr) } g.support\n[PROOFSTEP]\nletI := Classical.decEq \u03b2\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2192\u2080 \u03b3\ng : \u03b2 \u2192\u2080 \u03b3\nb : \u03b2\nh : Sum.elim (\u2191f) (\u2191g) (Sum.inr b) \u2260 0\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\n\u22a2 Sum.inr b \u2208\n map { toFun := Sum.inl, inj' := (_ : Injective Sum.inl) } f.support \u222a\n map { toFun := Sum.inr, inj' := (_ : Injective Sum.inr) } g.support\n[PROOFSTEP]\nletI := Classical.decEq \u03b2\n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2192\u2080 \u03b3\ng : \u03b2 \u2192\u2080 \u03b3\na : \u03b1\nh : Sum.elim (\u2191f) (\u2191g) (Sum.inl a) \u2260 0\nthis\u271d : DecidableEq \u03b1 := Classical.decEq \u03b1\nthis : DecidableEq \u03b2 := Classical.decEq \u03b2\n\u22a2 Sum.inl a \u2208\n map { toFun := Sum.inl, inj' := (_ : Injective Sum.inl) } f.support \u222a\n map { toFun := Sum.inr, inj' := (_ : Injective Sum.inr) } g.support\n[PROOFSTEP]\nsimp only [Sum.elim_inl, Sum.elim_inr] at h \n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2192\u2080 \u03b3\ng : \u03b2 \u2192\u2080 \u03b3\nb : \u03b2\nh : Sum.elim (\u2191f) (\u2191g) (Sum.inr b) \u2260 0\nthis\u271d : DecidableEq \u03b1 := Classical.decEq \u03b1\nthis : DecidableEq \u03b2 := Classical.decEq \u03b2\n\u22a2 Sum.inr b \u2208\n map { toFun := Sum.inl, inj' := (_ : Injective Sum.inl) } f.support \u222a\n map { toFun := Sum.inr, inj' := (_ : Injective Sum.inr) } g.support\n[PROOFSTEP]\nsimp only [Sum.elim_inl, Sum.elim_inr] at h \n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2192\u2080 \u03b3\ng : \u03b2 \u2192\u2080 \u03b3\na : \u03b1\nh : \u2191f a \u2260 0\nthis\u271d : DecidableEq \u03b1 := Classical.decEq \u03b1\nthis : DecidableEq \u03b2 := Classical.decEq \u03b2\n\u22a2 Sum.inl a \u2208\n map { toFun := Sum.inl, inj' := (_ : Injective Sum.inl) } f.support \u222a\n map { toFun := Sum.inr, inj' := (_ : Injective Sum.inr) } g.support\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2192\u2080 \u03b3\ng : \u03b2 \u2192\u2080 \u03b3\nb : \u03b2\nh : \u2191g b \u2260 0\nthis\u271d : DecidableEq \u03b1 := Classical.decEq \u03b1\nthis : DecidableEq \u03b2 := Classical.decEq \u03b2\n\u22a2 Sum.inr b \u2208\n map { toFun := Sum.inl, inj' := (_ : Injective Sum.inl) } f.support \u222a\n map { toFun := Sum.inr, inj' := (_ : Injective Sum.inr) } g.support\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2295 \u03b2 \u2192\u2080 \u03b3\n\u22a2 (fun fg => sumElim fg.fst fg.snd)\n ((fun f =>\n (comapDomain Sum.inl f (_ : Set.InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191f.support)),\n comapDomain Sum.inr f (_ : Set.InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191f.support))))\n f) =\n f\n[PROOFSTEP]\next ab\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2295 \u03b2 \u2192\u2080 \u03b3\nab : \u03b1 \u2295 \u03b2\n\u22a2 \u2191((fun fg => sumElim fg.fst fg.snd)\n ((fun f =>\n (comapDomain Sum.inl f (_ : Set.InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191f.support)),\n comapDomain Sum.inr f (_ : Set.InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191f.support))))\n f))\n ab =\n \u2191f ab\n[PROOFSTEP]\ncases' ab with a b\n[GOAL]\ncase h.inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2295 \u03b2 \u2192\u2080 \u03b3\na : \u03b1\n\u22a2 \u2191((fun fg => sumElim fg.fst fg.snd)\n ((fun f =>\n (comapDomain Sum.inl f (_ : Set.InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191f.support)),\n comapDomain Sum.inr f (_ : Set.InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191f.support))))\n f))\n (Sum.inl a) =\n \u2191f (Sum.inl a)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nf : \u03b1 \u2295 \u03b2 \u2192\u2080 \u03b3\nb : \u03b2\n\u22a2 \u2191((fun fg => sumElim fg.fst fg.snd)\n ((fun f =>\n (comapDomain Sum.inl f (_ : Set.InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191f.support)),\n comapDomain Sum.inr f (_ : Set.InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191f.support))))\n f))\n (Sum.inr b) =\n \u2191f (Sum.inr b)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nfg : (\u03b1 \u2192\u2080 \u03b3) \u00d7 (\u03b2 \u2192\u2080 \u03b3)\n\u22a2 (fun f =>\n (comapDomain Sum.inl f (_ : Set.InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191f.support)),\n comapDomain Sum.inr f (_ : Set.InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191f.support))))\n ((fun fg => sumElim fg.fst fg.snd) fg) =\n fg\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081.h\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nfg : (\u03b1 \u2192\u2080 \u03b3) \u00d7 (\u03b2 \u2192\u2080 \u03b3)\na\u271d : \u03b1\n\u22a2 \u2191((fun f =>\n (comapDomain Sum.inl f (_ : Set.InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191f.support)),\n comapDomain Sum.inr f (_ : Set.InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191f.support))))\n ((fun fg => sumElim fg.fst fg.snd) fg)).fst\n a\u271d =\n \u2191fg.fst a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082.h\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\n\u03b3 : Type u_15\ninst\u271d : Zero \u03b3\nfg : (\u03b1 \u2192\u2080 \u03b3) \u00d7 (\u03b2 \u2192\u2080 \u03b3)\na\u271d : \u03b2\n\u22a2 \u2191((fun f =>\n (comapDomain Sum.inl f (_ : Set.InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191f.support)),\n comapDomain Sum.inr f (_ : Set.InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191f.support))))\n ((fun fg => sumElim fg.fst fg.snd) fg)).snd\n a\u271d =\n \u2191fg.snd a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddMonoid M\n\u03b1 : Type u_13\n\u03b2 : Type u_14\nsrc\u271d : (\u03b1 \u2295 \u03b2 \u2192\u2080 M) \u2243 (\u03b1 \u2192\u2080 M) \u00d7 (\u03b2 \u2192\u2080 M) := sumFinsuppEquivProdFinsupp\n\u22a2 \u2200 (x y : \u03b1 \u2295 \u03b2 \u2192\u2080 M),\n Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (x + y) =\n Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n x +\n Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n y\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddMonoid M\n\u03b1 : Type u_13\n\u03b2 : Type u_14\nsrc\u271d : (\u03b1 \u2295 \u03b2 \u2192\u2080 M) \u2243 (\u03b1 \u2192\u2080 M) \u00d7 (\u03b2 \u2192\u2080 M) := sumFinsuppEquivProdFinsupp\nx\u271d y\u271d : \u03b1 \u2295 \u03b2 \u2192\u2080 M\n\u22a2 Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (x\u271d + y\u271d) =\n Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n x\u271d +\n Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n y\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081.h\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddMonoid M\n\u03b1 : Type u_13\n\u03b2 : Type u_14\nsrc\u271d : (\u03b1 \u2295 \u03b2 \u2192\u2080 M) \u2243 (\u03b1 \u2192\u2080 M) \u00d7 (\u03b2 \u2192\u2080 M) := sumFinsuppEquivProdFinsupp\nx\u271d y\u271d : \u03b1 \u2295 \u03b2 \u2192\u2080 M\na\u271d : \u03b1\n\u22a2 \u2191(Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (x\u271d + y\u271d)).fst\n a\u271d =\n \u2191(Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n x\u271d +\n Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n y\u271d).fst\n a\u271d\n[PROOFSTEP]\nsimp only [Equiv.toFun_as_coe, Prod.fst_add, Prod.snd_add, add_apply, snd_sumFinsuppEquivProdFinsupp,\n fst_sumFinsuppEquivProdFinsupp]\n[GOAL]\ncase h\u2082.h\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddMonoid M\n\u03b1 : Type u_13\n\u03b2 : Type u_14\nsrc\u271d : (\u03b1 \u2295 \u03b2 \u2192\u2080 M) \u2243 (\u03b1 \u2192\u2080 M) \u00d7 (\u03b2 \u2192\u2080 M) := sumFinsuppEquivProdFinsupp\nx\u271d y\u271d : \u03b1 \u2295 \u03b2 \u2192\u2080 M\na\u271d : \u03b2\n\u22a2 \u2191(Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (x\u271d + y\u271d)).snd\n a\u271d =\n \u2191(Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n x\u271d +\n Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n y\u271d).snd\n a\u271d\n[PROOFSTEP]\nsimp only [Equiv.toFun_as_coe, Prod.fst_add, Prod.snd_add, add_apply, snd_sumFinsuppEquivProdFinsupp,\n fst_sumFinsuppEquivProdFinsupp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : MonoidWithZero R\ninst\u271d : MulActionWithZero R M\na b : \u03b1\nf : \u03b1 \u2192 M\nr : R\n\u22a2 \u2191(single a r) b \u2022 f a = \u2191(single a (r \u2022 f b)) b\n[PROOFSTEP]\nby_cases h : a = b\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : MonoidWithZero R\ninst\u271d : MulActionWithZero R M\na b : \u03b1\nf : \u03b1 \u2192 M\nr : R\nh : a = b\n\u22a2 \u2191(single a r) b \u2022 f a = \u2191(single a (r \u2022 f b)) b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : MonoidWithZero R\ninst\u271d : MulActionWithZero R M\na b : \u03b1\nf : \u03b1 \u2192 M\nr : R\nh : \u00aca = b\n\u22a2 \u2191(single a r) b \u2022 f a = \u2191(single a (r \u2022 f b)) b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192\u2080 M\n\u22a2 1 \u2022 f = f\n[PROOFSTEP]\nrw [comapSMul_def, one_smul_eq_id, mapDomain_id]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : AddCommMonoid M\ng g' : G\nf : \u03b1 \u2192\u2080 M\n\u22a2 (g * g') \u2022 f = g \u2022 g' \u2022 f\n[PROOFSTEP]\nrw [comapSMul_def, comapSMul_def, comapSMul_def, \u2190 comp_smul_left, mapDomain_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : AddCommMonoid M\ng : G\n\u22a2 g \u2022 0 = 0\n[PROOFSTEP]\next a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : AddCommMonoid M\ng : G\na : \u03b1\n\u22a2 \u2191(g \u2022 0) a = \u21910 a\n[PROOFSTEP]\nsimp only [comapSMul_def]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : AddCommMonoid M\ng : G\na : \u03b1\n\u22a2 \u2191(mapDomain (fun x => g \u2022 x) 0) a = \u21910 a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : AddCommMonoid M\ng : G\nf f' : \u03b1 \u2192\u2080 M\n\u22a2 g \u2022 (f + f') = g \u2022 f + g \u2022 f'\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : AddCommMonoid M\ng : G\nf f' : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\n\u22a2 \u2191(g \u2022 (f + f')) a\u271d = \u2191(g \u2022 f + g \u2022 f') a\u271d\n[PROOFSTEP]\nsimp only [comapSMul_def]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : AddCommMonoid M\ng : G\nf f' : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\n\u22a2 \u2191(mapDomain (fun x => g \u2022 x) (f + f')) a\u271d = \u2191(mapDomain (fun x => g \u2022 x) f + mapDomain (fun x => g \u2022 x) f') a\u271d\n[PROOFSTEP]\nsimp [mapDomain_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : AddCommMonoid M\ng : G\nf : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 \u2191(g \u2022 f) a = \u2191f (g\u207b\u00b9 \u2022 a)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 smul_inv_smul g a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : AddCommMonoid M\ng : G\nf : \u03b1 \u2192\u2080 M\na : \u03b1\n| \u2191(g \u2022 f) a\n[PROOFSTEP]\nrw [\u2190 smul_inv_smul g a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : AddCommMonoid M\ng : G\nf : \u03b1 \u2192\u2080 M\na : \u03b1\n| \u2191(g \u2022 f) a\n[PROOFSTEP]\nrw [\u2190 smul_inv_smul g a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : AddCommMonoid M\ng : G\nf : \u03b1 \u2192\u2080 M\na : \u03b1\n| \u2191(g \u2022 f) a\n[PROOFSTEP]\nrw [\u2190 smul_inv_smul g a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : AddCommMonoid M\ng : G\nf : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 \u2191(g \u2022 f) (g \u2022 g\u207b\u00b9 \u2022 a) = \u2191f (g\u207b\u00b9 \u2022 a)\n[PROOFSTEP]\nexact mapDomain_apply (MulAction.injective g) _ (g\u207b\u00b9 \u2022 a)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : SMulZeroClass R M\na : R\n\u22a2 a \u2022 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : SMulZeroClass R M\na : R\na\u271d : \u03b1\n\u22a2 \u2191(a \u2022 0) a\u271d = \u21910 a\u271d\n[PROOFSTEP]\napply smul_zero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b3 : Nonempty \u03b1\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : SMulZeroClass R M\ninst\u271d : FaithfulSMul R M\nm\u2081\u271d m\u2082\u271d : R\nh : \u2200 (a : \u03b1 \u2192\u2080 M), m\u2081\u271d \u2022 a = m\u2082\u271d \u2022 a\na : \u03b1\nm : M\n\u22a2 m\u2081\u271d \u2022 m = m\u2082\u271d \u2022 m\n[PROOFSTEP]\nsimpa using FunLike.congr_fun (h (single a m)) a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddMonoid M\ninst\u271d : SMulZeroClass R M\nb : R\ng : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 a \u2208 (b \u2022 g).support \u2192 a \u2208 g.support\n[PROOFSTEP]\nsimp only [smul_apply, mem_support_iff, Ne.def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddMonoid M\ninst\u271d : SMulZeroClass R M\nb : R\ng : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 \u00acb \u2022 \u2191g a = 0 \u2192 \u00ac\u2191g a = 0\n[PROOFSTEP]\nexact mt fun h => h.symm \u25b8 smul_zero _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nb : R\nhb : b \u2260 0\ng : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 a \u2208 (b \u2022 g).support \u2194 a \u2208 g.support\n[PROOFSTEP]\nsimp [Finsupp.smul_apply, hb]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\nx\u271d : Monoid R\ninst\u271d\u00b3 : AddMonoid M\ninst\u271d\u00b2 : DistribMulAction R M\ninst\u271d\u00b9 : AddMonoid N\ninst\u271d : DistribMulAction R N\nf : M \u2192 N\nhf : f 0 = 0\nc : R\nv : \u03b1 \u2192\u2080 M\nhsmul : \u2200 (x : M), f (c \u2022 x) = c \u2022 f x\n\u22a2 mapRange f hf (c \u2022 v) = c \u2022 mapRange f hf v\n[PROOFSTEP]\nerw [\u2190 mapRange_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\nx\u271d : Monoid R\ninst\u271d\u00b3 : AddMonoid M\ninst\u271d\u00b2 : DistribMulAction R M\ninst\u271d\u00b9 : AddMonoid N\ninst\u271d : DistribMulAction R N\nf : M \u2192 N\nhf : f 0 = 0\nc : R\nv : \u03b1 \u2192\u2080 M\nhsmul : \u2200 (x : M), f (c \u2022 x) = c \u2022 f x\n\u22a2 mapRange (f \u2218 (fun x x_1 => x \u2022 x_1) c) ?h v = c \u2022 mapRange f hf v\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\nx\u271d : Monoid R\ninst\u271d\u00b3 : AddMonoid M\ninst\u271d\u00b2 : DistribMulAction R M\ninst\u271d\u00b9 : AddMonoid N\ninst\u271d : DistribMulAction R N\nf : M \u2192 N\nhf : f 0 = 0\nc : R\nv : \u03b1 \u2192\u2080 M\nhsmul : \u2200 (x : M), f (c \u2022 x) = c \u2022 f x\n\u22a2 (f \u2218 (fun x x_1 => x \u2022 x_1) c) 0 = 0\n[PROOFSTEP]\nhave : f \u2218 (\u00b7 \u2022 \u00b7) c = (\u00b7 \u2022 \u00b7) c \u2218 f := funext hsmul\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\nx\u271d : Monoid R\ninst\u271d\u00b3 : AddMonoid M\ninst\u271d\u00b2 : DistribMulAction R M\ninst\u271d\u00b9 : AddMonoid N\ninst\u271d : DistribMulAction R N\nf : M \u2192 N\nhf : f 0 = 0\nc : R\nv : \u03b1 \u2192\u2080 M\nhsmul : \u2200 (x : M), f (c \u2022 x) = c \u2022 f x\nthis : f \u2218 (fun x x_1 => x \u2022 x_1) c = (fun x x_1 => x \u2022 x_1) c \u2218 f\n\u22a2 mapRange (f \u2218 (fun x x_1 => x \u2022 x_1) c) ?h v = c \u2022 mapRange f hf v\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\nx\u271d : Monoid R\ninst\u271d\u00b3 : AddMonoid M\ninst\u271d\u00b2 : DistribMulAction R M\ninst\u271d\u00b9 : AddMonoid N\ninst\u271d : DistribMulAction R N\nf : M \u2192 N\nhf : f 0 = 0\nc : R\nv : \u03b1 \u2192\u2080 M\nhsmul : \u2200 (x : M), f (c \u2022 x) = c \u2022 f x\n\u22a2 (f \u2218 (fun x x_1 => x \u2022 x_1) c) 0 = 0\n[PROOFSTEP]\nsimp_rw [this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\nx\u271d : Monoid R\ninst\u271d\u00b3 : AddMonoid M\ninst\u271d\u00b2 : DistribMulAction R M\ninst\u271d\u00b9 : AddMonoid N\ninst\u271d : DistribMulAction R N\nf : M \u2192 N\nhf : f 0 = 0\nc : R\nv : \u03b1 \u2192\u2080 M\nhsmul : \u2200 (x : M), f (c \u2022 x) = c \u2022 f x\nthis : f \u2218 (fun x x_1 => x \u2022 x_1) c = (fun x x_1 => x \u2022 x_1) c \u2218 f\n\u22a2 mapRange ((fun x => c \u2022 x) \u2218 f) (_ : ((fun x => c \u2022 x) \u2218 f) 0 = 0) v = c \u2022 mapRange f hf v\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\nx\u271d : Monoid R\ninst\u271d\u00b3 : AddMonoid M\ninst\u271d\u00b2 : DistribMulAction R M\ninst\u271d\u00b9 : AddMonoid N\ninst\u271d : DistribMulAction R N\nf : M \u2192 N\nhf : f 0 = 0\nc : R\nv : \u03b1 \u2192\u2080 M\nhsmul : \u2200 (x : M), f (c \u2022 x) = c \u2022 f x\n\u22a2 (f \u2218 (fun x x_1 => x \u2022 x_1) c) 0 = 0\n[PROOFSTEP]\napply mapRange_comp\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\nx\u271d : Monoid R\ninst\u271d\u00b3 : AddMonoid M\ninst\u271d\u00b2 : DistribMulAction R M\ninst\u271d\u00b9 : AddMonoid N\ninst\u271d : DistribMulAction R N\nf : M \u2192 N\nhf : f 0 = 0\nc : R\nv : \u03b1 \u2192\u2080 M\nhsmul : \u2200 (x : M), f (c \u2022 x) = c \u2022 f x\n\u22a2 (f \u2218 (fun x x_1 => x \u2022 x_1) c) 0 = 0\n[PROOFSTEP]\nsimp only [Function.comp_apply, smul_zero, hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Semiring R\na : \u03b1\nb : R\n\u22a2 b \u2022 single a 1 = single a b\n[PROOFSTEP]\nrw [smul_single, smul_eq_mul, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddMonoid M\ninst\u271d\u00b9 : Monoid R\ninst\u271d : DistribMulAction R M\nf : \u03b1 \u2192 \u03b2\nr : R\nv : \u03b2 \u2192\u2080 M\nhfv : Set.InjOn f (f \u207b\u00b9' \u2191v.support)\nhfrv : optParam (Set.InjOn f (f \u207b\u00b9' \u2191(r \u2022 v).support)) (_ : Set.InjOn f (f \u207b\u00b9' \u2191(r \u2022 v).support))\n\u22a2 comapDomain f (r \u2022 v) hfrv = r \u2022 comapDomain f v hfv\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : AddMonoid M\ninst\u271d\u00b9 : Monoid R\ninst\u271d : DistribMulAction R M\nf : \u03b1 \u2192 \u03b2\nr : R\nv : \u03b2 \u2192\u2080 M\nhfv : Set.InjOn f (f \u207b\u00b9' \u2191v.support)\nhfrv : optParam (Set.InjOn f (f \u207b\u00b9' \u2191(r \u2022 v).support)) (_ : Set.InjOn f (f \u207b\u00b9' \u2191(r \u2022 v).support))\na\u271d : \u03b1\n\u22a2 \u2191(comapDomain f (r \u2022 v) hfrv) a\u271d = \u2191(r \u2022 comapDomain f v hfv) a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : DistribMulAction R M\ninst\u271d : DistribMulAction R N\na : \u03b1\nsrc\u271d : M \u2192+ \u03b1 \u2192\u2080 M := singleAddHom a\nk : R\nm : M\n\u22a2 ZeroHom.toFun (\u2191src\u271d) (k \u2022 m) = k \u2022 ZeroHom.toFun (\u2191src\u271d) m\n[PROOFSTEP]\nsimp only\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : DistribMulAction R M\ninst\u271d : DistribMulAction R N\na : \u03b1\nsrc\u271d : M \u2192+ \u03b1 \u2192\u2080 M := singleAddHom a\nk : R\nm : M\n\u22a2 ZeroHom.toFun (\u2191(singleAddHom a)) (k \u2022 m) = k \u2022 ZeroHom.toFun (\u2191(singleAddHom a)) m\n[PROOFSTEP]\nshow singleAddHom a (k \u2022 m) = k \u2022 singleAddHom a m\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : DistribMulAction R M\ninst\u271d : DistribMulAction R N\na : \u03b1\nsrc\u271d : M \u2192+ \u03b1 \u2192\u2080 M := singleAddHom a\nk : R\nm : M\n\u22a2 \u2191(singleAddHom a) (k \u2022 m) = k \u2022 \u2191(singleAddHom a) m\n[PROOFSTEP]\nchange\n Finsupp.single a (k \u2022 m) =\n k \u2022\n (Finsupp.single a m)\n -- porting note: because `singleAddHom_apply` is missing\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : DistribMulAction R M\ninst\u271d : DistribMulAction R N\na : \u03b1\nsrc\u271d : M \u2192+ \u03b1 \u2192\u2080 M := singleAddHom a\nk : R\nm : M\n\u22a2 Finsupp.single a (k \u2022 m) = k \u2022 Finsupp.single a m\n[PROOFSTEP]\nsimp only [smul_single]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ns : Set \u03b1\nM : Type u_13\ninst\u271d : AddCommMonoid M\nf : \u2191s \u2192\u2080 M\n\u22a2 \u2191(embDomain (Embedding.subtype fun x => x \u2208 s) f).support \u2286 s\n[PROOFSTEP]\nrw [support_embDomain, Finset.coe_map, Set.image_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ns : Set \u03b1\nM : Type u_13\ninst\u271d : AddCommMonoid M\nf : \u2191s \u2192\u2080 M\n\u22a2 \u2191f.support \u2286 \u2191(Embedding.subtype fun x => x \u2208 s) \u207b\u00b9' s\n[PROOFSTEP]\nexact fun x _ => x.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ns : Set \u03b1\nM : Type u_13\ninst\u271d : AddCommMonoid M\n\u22a2 LeftInverse\n (fun f =>\n { val := embDomain (Embedding.subtype fun x => x \u2208 s) f,\n property := (_ : \u2191(embDomain (Embedding.subtype fun x => x \u2208 s) f).support \u2286 s) })\n fun f => subtypeDomain (fun x => x \u2208 s) \u2191f\n[PROOFSTEP]\nrintro \u27e8f, hf\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ns : Set \u03b1\nM : Type u_13\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192\u2080 M\nhf : \u2191f.support \u2286 s\n\u22a2 (fun f =>\n { val := embDomain (Embedding.subtype fun x => x \u2208 s) f,\n property := (_ : \u2191(embDomain (Embedding.subtype fun x => x \u2208 s) f).support \u2286 s) })\n ((fun f => subtypeDomain (fun x => x \u2208 s) \u2191f) { val := f, property := hf }) =\n { val := f, property := hf }\n[PROOFSTEP]\next a\n[GOAL]\ncase mk.a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ns : Set \u03b1\nM : Type u_13\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192\u2080 M\nhf : \u2191f.support \u2286 s\na : \u03b1\n\u22a2 \u2191\u2191((fun f =>\n { val := embDomain (Embedding.subtype fun x => x \u2208 s) f,\n property := (_ : \u2191(embDomain (Embedding.subtype fun x => x \u2208 s) f).support \u2286 s) })\n ((fun f => subtypeDomain (fun x => x \u2208 s) \u2191f) { val := f, property := hf }))\n a =\n \u2191\u2191{ val := f, property := hf } a\n[PROOFSTEP]\nby_cases h : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ns : Set \u03b1\nM : Type u_13\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192\u2080 M\nhf : \u2191f.support \u2286 s\na : \u03b1\nh : a \u2208 s\n\u22a2 \u2191\u2191((fun f =>\n { val := embDomain (Embedding.subtype fun x => x \u2208 s) f,\n property := (_ : \u2191(embDomain (Embedding.subtype fun x => x \u2208 s) f).support \u2286 s) })\n ((fun f => subtypeDomain (fun x => x \u2208 s) \u2191f) { val := f, property := hf }))\n a =\n \u2191\u2191{ val := f, property := hf } a\n[PROOFSTEP]\nlift a to s using h\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ns : Set \u03b1\nM : Type u_13\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192\u2080 M\nhf : \u2191f.support \u2286 s\na : { x // x \u2208 s }\n\u22a2 \u2191\u2191((fun f =>\n { val := embDomain (Embedding.subtype fun x => x \u2208 s) f,\n property := (_ : \u2191(embDomain (Embedding.subtype fun x => x \u2208 s) f).support \u2286 s) })\n ((fun f => subtypeDomain (fun x => x \u2208 s) \u2191f) { val := f, property := hf }))\n \u2191a =\n \u2191\u2191{ val := f, property := hf } \u2191a\n[PROOFSTEP]\nexact embDomain_apply _ _ _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ns : Set \u03b1\nM : Type u_13\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192\u2080 M\nhf : \u2191f.support \u2286 s\na : \u03b1\nh : \u00aca \u2208 s\n\u22a2 \u2191\u2191((fun f =>\n { val := embDomain (Embedding.subtype fun x => x \u2208 s) f,\n property := (_ : \u2191(embDomain (Embedding.subtype fun x => x \u2208 s) f).support \u2286 s) })\n ((fun f => subtypeDomain (fun x => x \u2208 s) \u2191f) { val := f, property := hf }))\n a =\n \u2191\u2191{ val := f, property := hf } a\n[PROOFSTEP]\nrw [embDomain_notin_range, eq_comm, \u2190 Finsupp.not_mem_support_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ns : Set \u03b1\nM : Type u_13\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192\u2080 M\nhf : \u2191f.support \u2286 s\na : \u03b1\nh : \u00aca \u2208 s\n\u22a2 \u00aca \u2208 (\u2191{ val := f, property := hf }).support\n[PROOFSTEP]\nexact fun hs => h <| hf hs\n[GOAL]\ncase neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM\u271d : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ns : Set \u03b1\nM : Type u_13\ninst\u271d : AddCommMonoid M\nf : \u03b1 \u2192\u2080 M\nhf : \u2191f.support \u2286 s\na : \u03b1\nh : \u00aca \u2208 s\n\u22a2 \u00aca \u2208 Set.range \u2191(Embedding.subtype fun x => x \u2208 s)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\ne : \u03b1 \u2243 \u03b2\nv : \u03b1 \u2192\u2080 M\n\u22a2 equivMapDomain e.symm (equivMapDomain e v) = v\n[PROOFSTEP]\nsimp only [\u2190 equivMapDomain_trans, Equiv.self_trans_symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\ne : \u03b1 \u2243 \u03b2\nv : \u03b1 \u2192\u2080 M\n\u22a2 equivMapDomain (Equiv.refl \u03b1) v = v\n[PROOFSTEP]\nexact equivMapDomain_refl _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\ne : \u03b1 \u2243 \u03b2\n\u22a2 Function.RightInverse (equivMapDomain e.symm) (equivMapDomain e)\n[PROOFSTEP]\nintro v\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\ne : \u03b1 \u2243 \u03b2\nv : \u03b2 \u2192\u2080 M\n\u22a2 equivMapDomain e (equivMapDomain e.symm v) = v\n[PROOFSTEP]\nsimp only [\u2190 equivMapDomain_trans, Equiv.symm_trans_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\ne : \u03b1 \u2243 \u03b2\nv : \u03b2 \u2192\u2080 M\n\u22a2 equivMapDomain (Equiv.refl \u03b2) v = v\n[PROOFSTEP]\nexact equivMapDomain_refl _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\ne : \u03b1 \u2243 \u03b2\na b : \u03b1 \u2192\u2080 M\n\u22a2 Equiv.toFun\n { toFun := equivMapDomain e, invFun := equivMapDomain e.symm,\n left_inv := (_ : \u2200 (v : \u03b1 \u2192\u2080 M), equivMapDomain e.symm (equivMapDomain e v) = v),\n right_inv := (_ : \u2200 (v : \u03b2 \u2192\u2080 M), equivMapDomain e (equivMapDomain e.symm v) = v) }\n (a + b) =\n Equiv.toFun\n { toFun := equivMapDomain e, invFun := equivMapDomain e.symm,\n left_inv := (_ : \u2200 (v : \u03b1 \u2192\u2080 M), equivMapDomain e.symm (equivMapDomain e v) = v),\n right_inv := (_ : \u2200 (v : \u03b2 \u2192\u2080 M), equivMapDomain e (equivMapDomain e.symm v) = v) }\n a +\n Equiv.toFun\n { toFun := equivMapDomain e, invFun := equivMapDomain e.symm,\n left_inv := (_ : \u2200 (v : \u03b1 \u2192\u2080 M), equivMapDomain e.symm (equivMapDomain e v) = v),\n right_inv := (_ : \u2200 (v : \u03b2 \u2192\u2080 M), equivMapDomain e (equivMapDomain e.symm v) = v) }\n b\n[PROOFSTEP]\nsimp only [equivMapDomain_eq_mapDomain]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\ne : \u03b1 \u2243 \u03b2\na b : \u03b1 \u2192\u2080 M\n\u22a2 mapDomain (\u2191e) (a + b) = mapDomain (\u2191e) a + mapDomain (\u2191e) b\n[PROOFSTEP]\nexact mapDomain_add\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\ni : \u03b9\nx : \u03b1s i\n\u22a2 \u2191(split l i) x = \u2191l { fst := i, snd := x }\n[PROOFSTEP]\ndsimp only [split]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\ni : \u03b9\nx : \u03b1s i\n\u22a2 \u2191(comapDomain (Sigma.mk i) l\n (_ :\n \u2200 (x : \u03b1s i),\n x \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192\n \u2200 (x_2 : \u03b1s i),\n x_2 \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192 { fst := i, snd := x } = { fst := i, snd := x_2 } \u2192 x = x_2))\n x =\n \u2191l { fst := i, snd := x }\n[PROOFSTEP]\nrw [comapDomain_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\ni : \u03b9\n\u22a2 i \u2208 splitSupport l \u2194 split l i \u2260 0\n[PROOFSTEP]\nrw [splitSupport, @mem_image _ _ (Classical.decEq _), Ne.def, \u2190 support_eq_empty, \u2190 Ne.def, \u2190\n Finset.nonempty_iff_ne_empty, split, comapDomain, Finset.Nonempty]\n -- porting note: had to add the `Classical.decEq` instance manually\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\ni : \u03b9\n\u22a2 (\u2203 a, a \u2208 l.support \u2227 a.fst = i) \u2194\n \u2203 x,\n x \u2208\n {\n support :=\n preimage l.support (Sigma.mk i)\n (_ :\n \u2200 (x : \u03b1s i),\n x \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192\n \u2200 (x_2 : \u03b1s i),\n x_2 \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192 { fst := i, snd := x } = { fst := i, snd := x_2 } \u2192 x = x_2),\n toFun := fun a => \u2191l { fst := i, snd := a },\n mem_support_toFun :=\n (_ :\n \u2200 (a : \u03b1s i),\n a \u2208\n preimage l.support (Sigma.mk i)\n (_ :\n \u2200 (x : \u03b1s i),\n x \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192\n \u2200 (x_2 : \u03b1s i),\n x_2 \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192\n { fst := i, snd := x } = { fst := i, snd := x_2 } \u2192 x = x_2) \u2194\n (fun a => \u2191l { fst := i, snd := a }) a \u2260 0) }.support\n[PROOFSTEP]\nsimp only [exists_prop, Finset.mem_preimage, exists_and_right, exists_eq_right, mem_support_iff, Sigma.exists, Ne.def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d\u00b9 : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\ninst\u271d : Zero N\ng : (i : \u03b9) \u2192 (\u03b1s i \u2192\u2080 M) \u2192 N\nhg : \u2200 (i : \u03b9) (x : \u03b1s i \u2192\u2080 M), x = 0 \u2194 g i x = 0\n\u22a2 \u2200 (a : \u03b9), a \u2208 splitSupport l \u2194 (fun i => g i (split l i)) a \u2260 0\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d\u00b9 : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\ninst\u271d : Zero N\ng : (i : \u03b9) \u2192 (\u03b1s i \u2192\u2080 M) \u2192 N\nhg : \u2200 (i : \u03b9) (x : \u03b1s i \u2192\u2080 M), x = 0 \u2194 g i x = 0\ni : \u03b9\n\u22a2 i \u2208 splitSupport l \u2194 (fun i => g i (split l i)) i \u2260 0\n[PROOFSTEP]\nrw [mem_splitSupport_iff_nonzero, not_iff_not, hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\n\u22a2 l.support = Finset.sigma (splitSupport l) fun i => (split l i).support\n[PROOFSTEP]\nsimp only [Finset.ext_iff, splitSupport, split, comapDomain, @mem_image _ _ (Classical.decEq _), mem_preimage,\n Sigma.forall, mem_sigma]\n -- porting note: had to add the `Classical.decEq` instance manually\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\n\u22a2 \u2200 (a : \u03b9) (b : \u03b1s a),\n { fst := a, snd := b } \u2208 l.support \u2194 (\u2203 a_1, a_1 \u2208 l.support \u2227 a_1.fst = a) \u2227 { fst := a, snd := b } \u2208 l.support\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d\u00b9 : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\ninst\u271d : AddCommMonoid N\nf : (i : \u03b9) \u00d7 \u03b1s i \u2192 M \u2192 N\n\u22a2 sum l f = \u2211 i in splitSupport l, sum (split l i) fun a b => f { fst := i, snd := a } b\n[PROOFSTEP]\nsimp only [sum, sigma_support, sum_sigma, split_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d\u00b2 : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\n\u03b7 : Type u_14\ninst\u271d\u00b9 : Fintype \u03b7\n\u03b9s : \u03b7 \u2192 Type u_15\ninst\u271d : Zero \u03b1\nf : (j : \u03b7) \u00d7 \u03b9s j \u2192\u2080 \u03b1\n\u22a2 (fun f =>\n onFinset (Finset.sigma univ fun j => (f j).support) (fun ji => \u2191(f ji.fst) ji.snd)\n (_ :\n \u2200 (g : (j : \u03b7) \u00d7 \u03b9s j),\n (fun ji => \u2191(f ji.fst) ji.snd) g \u2260 0 \u2192 g \u2208 Finset.sigma univ fun j => (f j).support))\n (split f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d\u00b2 : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\n\u03b7 : Type u_14\ninst\u271d\u00b9 : Fintype \u03b7\n\u03b9s : \u03b7 \u2192 Type u_15\ninst\u271d : Zero \u03b1\nf : (j : \u03b7) \u00d7 \u03b9s j \u2192\u2080 \u03b1\na\u271d : (j : \u03b7) \u00d7 \u03b9s j\n\u22a2 \u2191((fun f =>\n onFinset (Finset.sigma univ fun j => (f j).support) (fun ji => \u2191(f ji.fst) ji.snd)\n (_ :\n \u2200 (g : (j : \u03b7) \u00d7 \u03b9s j),\n (fun ji => \u2191(f ji.fst) ji.snd) g \u2260 0 \u2192 g \u2208 Finset.sigma univ fun j => (f j).support))\n (split f))\n a\u271d =\n \u2191f a\u271d\n[PROOFSTEP]\nsimp [split]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d\u00b2 : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\n\u03b7 : Type u_14\ninst\u271d\u00b9 : Fintype \u03b7\n\u03b9s : \u03b7 \u2192 Type u_15\ninst\u271d : Zero \u03b1\nf : (j : \u03b7) \u2192 \u03b9s j \u2192\u2080 \u03b1\n\u22a2 split\n ((fun f =>\n onFinset (Finset.sigma univ fun j => (f j).support) (fun ji => \u2191(f ji.fst) ji.snd)\n (_ :\n \u2200 (g : (j : \u03b7) \u00d7 \u03b9s j),\n (fun ji => \u2191(f ji.fst) ji.snd) g \u2260 0 \u2192 g \u2208 Finset.sigma univ fun j => (f j).support))\n f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d\u00b2 : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\n\u03b7 : Type u_14\ninst\u271d\u00b9 : Fintype \u03b7\n\u03b9s : \u03b7 \u2192 Type u_15\ninst\u271d : Zero \u03b1\nf : (j : \u03b7) \u2192 \u03b9s j \u2192\u2080 \u03b1\nx\u271d : \u03b7\na\u271d : \u03b9s x\u271d\n\u22a2 \u2191(split\n ((fun f =>\n onFinset (Finset.sigma univ fun j => (f j).support) (fun ji => \u2191(f ji.fst) ji.snd)\n (_ :\n \u2200 (g : (j : \u03b7) \u00d7 \u03b9s j),\n (fun ji => \u2191(f ji.fst) ji.snd) g \u2260 0 \u2192 g \u2208 Finset.sigma univ fun j => (f j).support))\n f)\n x\u271d)\n a\u271d =\n \u2191(f x\u271d) a\u271d\n[PROOFSTEP]\nsimp [split]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d\u00b3 : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\n\u03b7 : Type u_14\ninst\u271d\u00b2 : Fintype \u03b7\n\u03b9s\u271d : \u03b7 \u2192 Type u_15\ninst\u271d\u00b9 : Zero \u03b1\u271d\n\u03b1 : Type u_16\n\u03b9s : \u03b7 \u2192 Type u_17\ninst\u271d : AddMonoid \u03b1\nsrc\u271d : ((j : \u03b7) \u00d7 \u03b9s j \u2192\u2080 \u03b1) \u2243 ((j : \u03b7) \u2192 \u03b9s j \u2192\u2080 \u03b1) := sigmaFinsuppEquivPiFinsupp\nf g : (j : \u03b7) \u00d7 \u03b9s j \u2192\u2080 \u03b1\n\u22a2 Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (f + g) =\n Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n f +\n Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n g\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\n\u03b1s : \u03b9 \u2192 Type u_13\ninst\u271d\u00b3 : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\n\u03b7 : Type u_14\ninst\u271d\u00b2 : Fintype \u03b7\n\u03b9s\u271d : \u03b7 \u2192 Type u_15\ninst\u271d\u00b9 : Zero \u03b1\u271d\n\u03b1 : Type u_16\n\u03b9s : \u03b7 \u2192 Type u_17\ninst\u271d : AddMonoid \u03b1\nsrc\u271d : ((j : \u03b7) \u00d7 \u03b9s j \u2192\u2080 \u03b1) \u2243 ((j : \u03b7) \u2192 \u03b9s j \u2192\u2080 \u03b1) := sigmaFinsuppEquivPiFinsupp\nf g : (j : \u03b7) \u00d7 \u03b9s j \u2192\u2080 \u03b1\nx\u271d : \u03b7\na\u271d : \u03b9s x\u271d\n\u22a2 \u2191(Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (f + g) x\u271d)\n a\u271d =\n \u2191((Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n f +\n Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n g)\n x\u271d)\n a\u271d\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Finsupp.Basic", "llama_tokens": 89754, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.44082714616822716}} {"text": "[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\nv : V\n\u22a2 filter (fun d => d.fst = v) univ = image (dartOfNeighborSet G v) univ\n[PROOFSTEP]\next d\n[GOAL]\ncase a\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\nv : V\nd : Dart G\n\u22a2 d \u2208 filter (fun d => d.fst = v) univ \u2194 d \u2208 image (dartOfNeighborSet G v) univ\n[PROOFSTEP]\nsimp only [mem_image, true_and_iff, mem_filter, SetCoe.exists, mem_univ, exists_prop_of_true]\n[GOAL]\ncase a\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\nv : V\nd : Dart G\n\u22a2 d.fst = v \u2194 \u2203 x h, dartOfNeighborSet G v { val := x, property := h } = d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mp\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\nv : V\nd : Dart G\n\u22a2 d.fst = v \u2192 \u2203 x h, dartOfNeighborSet G v { val := x, property := h } = d\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase a.mp\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\nd : Dart G\n\u22a2 \u2203 x h, dartOfNeighborSet G d.fst { val := x, property := h } = d\n[PROOFSTEP]\nexact \u27e8_, d.is_adj, by ext <;> rfl\u27e9\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\nd : Dart G\n\u22a2 dartOfNeighborSet G d.fst { val := d.snd, property := (_ : Adj G d.fst d.snd) } = d\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\u2081\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\nd : Dart G\n\u22a2 (dartOfNeighborSet G d.fst { val := d.snd, property := (_ : Adj G d.fst d.snd) }).toProd.fst = d.fst\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.h\u2082\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\nd : Dart G\n\u22a2 (dartOfNeighborSet G d.fst { val := d.snd, property := (_ : Adj G d.fst d.snd) }).toProd.snd = d.snd\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.mpr\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\nv : V\nd : Dart G\n\u22a2 (\u2203 x h, dartOfNeighborSet G v { val := x, property := h } = d) \u2192 d.fst = v\n[PROOFSTEP]\nrintro \u27e8e, he, rfl\u27e9\n[GOAL]\ncase a.mpr.intro.intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\nv e : V\nhe : e \u2208 neighborSet G v\n\u22a2 (dartOfNeighborSet G v { val := e, property := he }).toProd.fst = v\n[PROOFSTEP]\nrfl\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\nv : V\n\u22a2 card (filter (fun d => d.fst = v) univ) = degree G v\n[PROOFSTEP]\nsimpa only [dart_fst_fiber, Finset.card_univ, card_neighborSet_eq_degree] using\n card_image_of_injective univ (G.dartOfNeighborSet_injective v)\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableRel G.Adj\ninst\u271d : Fintype (Sym2 V)\n\u22a2 Fintype.card (Dart G) = \u2211 v : V, degree G v\n[PROOFSTEP]\nhaveI := Classical.decEq V\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableRel G.Adj\ninst\u271d : Fintype (Sym2 V)\nthis : DecidableEq V\n\u22a2 Fintype.card (Dart G) = \u2211 v : V, degree G v\n[PROOFSTEP]\nsimp only [\u2190 card_univ, \u2190 dart_fst_fiber_card_eq_degree]\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableRel G.Adj\ninst\u271d : Fintype (Sym2 V)\nthis : DecidableEq V\n\u22a2 card univ = \u2211 x : V, card (filter (fun d => d.fst = x) univ)\n[PROOFSTEP]\nexact card_eq_sum_card_fiberwise (by simp)\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableRel G.Adj\ninst\u271d : Fintype (Sym2 V)\nthis : DecidableEq V\n\u22a2 \u2200 (x : Dart G), x \u2208 univ \u2192 x.fst \u2208 univ\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\nd d' : Dart G\n\u22a2 d' \u2208 filter (fun d' => edge d' = edge d) univ \u2194 d' \u2208 {d, symm d}\n[PROOFSTEP]\nsimpa using dart_edge_eq_iff d' d\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\ne : Sym2 V\nh : e \u2208 edgeSet G\n\u22a2 card (filter (fun d => Dart.edge d = e) univ) = 2\n[PROOFSTEP]\nrefine' Sym2.ind (fun v w h => _) e h\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\ne : Sym2 V\nh\u271d : e \u2208 edgeSet G\nv w : V\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) \u2208 edgeSet G\n\u22a2 card (filter (fun d => Dart.edge d = Quotient.mk (Sym2.Rel.setoid V) (v, w)) univ) = 2\n[PROOFSTEP]\nlet d : G.Dart := \u27e8(v, w), h\u27e9\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\ne : Sym2 V\nh\u271d : e \u2208 edgeSet G\nv w : V\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) \u2208 edgeSet G\nd : Dart G := { toProd := (v, w), is_adj := h }\n\u22a2 card (filter (fun d => Dart.edge d = Quotient.mk (Sym2.Rel.setoid V) (v, w)) univ) = 2\n[PROOFSTEP]\nconvert congr_arg card d.edge_fiber\n[GOAL]\ncase h.e'_3\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\ne : Sym2 V\nh\u271d : e \u2208 edgeSet G\nv w : V\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) \u2208 edgeSet G\nd : Dart G := { toProd := (v, w), is_adj := h }\n\u22a2 2 = card {d, Dart.symm d}\n[PROOFSTEP]\nrw [card_insert_of_not_mem, card_singleton]\n[GOAL]\ncase h.e'_3\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\ne : Sym2 V\nh\u271d : e \u2208 edgeSet G\nv w : V\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) \u2208 edgeSet G\nd : Dart G := { toProd := (v, w), is_adj := h }\n\u22a2 \u00acd \u2208 {Dart.symm d}\n[PROOFSTEP]\nrw [mem_singleton]\n[GOAL]\ncase h.e'_3\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\ne : Sym2 V\nh\u271d : e \u2208 edgeSet G\nv w : V\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) \u2208 edgeSet G\nd : Dart G := { toProd := (v, w), is_adj := h }\n\u22a2 \u00acd = Dart.symm d\n[PROOFSTEP]\nexact d.symm_ne.symm\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\n\u22a2 Fintype.card (Dart G) = 2 * card (edgeFinset G)\n[PROOFSTEP]\nrw [\u2190 card_univ]\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\n\u22a2 card univ = 2 * card (edgeFinset G)\n[PROOFSTEP]\nrw [@card_eq_sum_card_fiberwise _ _ _ Dart.edge _ G.edgeFinset fun d _h => by rw [mem_edgeFinset]; apply Dart.edge_mem]\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\nd : Dart G\n_h : d \u2208 univ\n\u22a2 Dart.edge d \u2208 edgeFinset G\n[PROOFSTEP]\nrw [mem_edgeFinset]\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\nd : Dart G\n_h : d \u2208 univ\n\u22a2 Dart.edge d \u2208 edgeSet G\n[PROOFSTEP]\napply Dart.edge_mem\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\n\u22a2 \u2211 a in edgeFinset G, card (filter (fun x => Dart.edge x = a) univ) = 2 * card (edgeFinset G)\n[PROOFSTEP]\nrw [\u2190 mul_comm, sum_const_nat]\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\n\u22a2 \u2200 (x : Sym2 V), x \u2208 edgeFinset G \u2192 card (filter (fun x_1 => Dart.edge x_1 = x) univ) = 2\n[PROOFSTEP]\nintro e h\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\ne : Sym2 V\nh : e \u2208 edgeFinset G\n\u22a2 card (filter (fun x => Dart.edge x = e) univ) = 2\n[PROOFSTEP]\napply G.dart_edge_fiber_card e\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableRel G.Adj\ninst\u271d\u00b9 : Fintype (Sym2 V)\ninst\u271d : DecidableEq V\ne : Sym2 V\nh : e \u2208 edgeFinset G\n\u22a2 e \u2208 edgeSet G\n[PROOFSTEP]\nrwa [\u2190 mem_edgeFinset]\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\n\u22a2 Even (card (filter (fun v => Odd (degree G v)) univ))\n[PROOFSTEP]\nclassical\nhave h := congr_arg (fun n => \u2191n : \u2115 \u2192 ZMod 2) G.sum_degrees_eq_twice_card_edges\nsimp only [ZMod.nat_cast_self, zero_mul, Nat.cast_mul] at h \nrw [Nat.cast_sum, \u2190 sum_filter_ne_zero] at h \nrw [@sum_congr _ _ _ _ (fun v => (G.degree v : ZMod 2)) (fun _v => (1 : ZMod 2)) _ rfl] at h \n\u00b7 simp only [filter_congr, mul_one, nsmul_eq_mul, sum_const, Ne.def] at h \n rw [\u2190 ZMod.eq_zero_iff_even]\n convert h\n exact ZMod.ne_zero_iff_odd.symm\n\u00b7 intro v\n simp only [true_and_iff, mem_filter, mem_univ, Ne.def]\n rw [ZMod.eq_zero_iff_even, ZMod.eq_one_iff_odd, Nat.odd_iff_not_even, imp_self]\n trivial\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\n\u22a2 Even (card (filter (fun v => Odd (degree G v)) univ))\n[PROOFSTEP]\nhave h := congr_arg (fun n => \u2191n : \u2115 \u2192 ZMod 2) G.sum_degrees_eq_twice_card_edges\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nh : (fun n => \u2191n) (\u2211 v : V, degree G v) = (fun n => \u2191n) (2 * card (edgeFinset G))\n\u22a2 Even (card (filter (fun v => Odd (degree G v)) univ))\n[PROOFSTEP]\nsimp only [ZMod.nat_cast_self, zero_mul, Nat.cast_mul] at h \n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nh : \u2191(\u2211 v : V, degree G v) = 0\n\u22a2 Even (card (filter (fun v => Odd (degree G v)) univ))\n[PROOFSTEP]\nrw [Nat.cast_sum, \u2190 sum_filter_ne_zero] at h \n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nh\u271d : \u2211 x : V, \u2191(degree G x) = 0\nh : \u2211 x in filter (fun x => \u2191(degree G x) \u2260 0) univ, \u2191(degree G x) = 0\n\u22a2 Even (card (filter (fun v => Odd (degree G v)) univ))\n[PROOFSTEP]\nrw [@sum_congr _ _ _ _ (fun v => (G.degree v : ZMod 2)) (fun _v => (1 : ZMod 2)) _ rfl] at h \n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nh\u271d : \u2211 x : V, \u2191(degree G x) = 0\nh : \u2211 _v in filter (fun x => \u2191(degree G x) \u2260 0) univ, 1 = 0\n\u22a2 Even (card (filter (fun v => Odd (degree G v)) univ))\n[PROOFSTEP]\nsimp only [filter_congr, mul_one, nsmul_eq_mul, sum_const, Ne.def] at h \n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nh\u271d : \u2211 x : V, \u2191(degree G x) = 0\nh : \u2191(card (filter (fun x => \u00ac\u2191(degree G x) = 0) univ)) = 0\n\u22a2 Even (card (filter (fun v => Odd (degree G v)) univ))\n[PROOFSTEP]\nrw [\u2190 ZMod.eq_zero_iff_even]\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nh\u271d : \u2211 x : V, \u2191(degree G x) = 0\nh : \u2191(card (filter (fun x => \u00ac\u2191(degree G x) = 0) univ)) = 0\n\u22a2 \u2191(card (filter (fun v => Odd (degree G v)) univ)) = 0\n[PROOFSTEP]\nconvert h\n[GOAL]\ncase h.e'_2.h.e'_3.h.e'_2.h.e'_2.h.a\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nh\u271d : \u2211 x : V, \u2191(degree G x) = 0\nh : \u2191(card (filter (fun x => \u00ac\u2191(degree G x) = 0) univ)) = 0\nx\u271d : V\n\u22a2 Odd (degree G x\u271d) \u2194 \u00ac\u2191(degree G x\u271d) = 0\n[PROOFSTEP]\nexact ZMod.ne_zero_iff_odd.symm\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nh\u271d : \u2211 x : V, \u2191(degree G x) = 0\nh : \u2211 x in filter (fun x => \u2191(degree G x) \u2260 0) univ, \u2191(degree G x) = 0\n\u22a2 \u2200 (x : V), x \u2208 filter (fun x => \u2191(degree G x) \u2260 0) univ \u2192 \u2191(degree G x) = 1\n[PROOFSTEP]\nintro v\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nh\u271d : \u2211 x : V, \u2191(degree G x) = 0\nh : \u2211 x in filter (fun x => \u2191(degree G x) \u2260 0) univ, \u2191(degree G x) = 0\nv : V\n\u22a2 v \u2208 filter (fun x => \u2191(degree G x) \u2260 0) univ \u2192 \u2191(degree G v) = 1\n[PROOFSTEP]\nsimp only [true_and_iff, mem_filter, mem_univ, Ne.def]\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nh\u271d : \u2211 x : V, \u2191(degree G x) = 0\nh : \u2211 x in filter (fun x => \u2191(degree G x) \u2260 0) univ, \u2191(degree G x) = 0\nv : V\n\u22a2 \u00ac\u2191(degree G v) = 0 \u2192 \u2191(degree G v) = 1\n[PROOFSTEP]\nrw [ZMod.eq_zero_iff_even, ZMod.eq_one_iff_odd, Nat.odd_iff_not_even, imp_self]\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nh\u271d : \u2211 x : V, \u2191(degree G x) = 0\nh : \u2211 x in filter (fun x => \u2191(degree G x) \u2260 0) univ, \u2191(degree G x) = 0\nv : V\n\u22a2 True\n[PROOFSTEP]\ntrivial\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\n\u22a2 Odd (card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ))\n[PROOFSTEP]\nrcases G.even_card_odd_degree_vertices with \u27e8k, hg\u27e9\n[GOAL]\ncase intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\n\u22a2 Odd (card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ))\n[PROOFSTEP]\nhave hk : 0 < k :=\n by\n have hh : (filter (fun v : V => Odd (G.degree v)) univ).Nonempty :=\n by\n use v\n simp only [true_and_iff, mem_filter, mem_univ]\n exact h\n rwa [\u2190 card_pos, hg, \u2190 two_mul, zero_lt_mul_left] at hh \n exact zero_lt_two\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\n\u22a2 0 < k\n[PROOFSTEP]\nhave hh : (filter (fun v : V => Odd (G.degree v)) univ).Nonempty :=\n by\n use v\n simp only [true_and_iff, mem_filter, mem_univ]\n exact h\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\n\u22a2 Finset.Nonempty (filter (fun v => Odd (degree G v)) univ)\n[PROOFSTEP]\nuse v\n[GOAL]\ncase h\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\n\u22a2 v \u2208 filter (fun v => Odd (degree G v)) univ\n[PROOFSTEP]\nsimp only [true_and_iff, mem_filter, mem_univ]\n[GOAL]\ncase h\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\n\u22a2 Odd (degree G v)\n[PROOFSTEP]\nexact h\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\nhh : Finset.Nonempty (filter (fun v => Odd (degree G v)) univ)\n\u22a2 0 < k\n[PROOFSTEP]\nrwa [\u2190 card_pos, hg, \u2190 two_mul, zero_lt_mul_left] at hh \n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\nhh : 0 < 2 * k\n\u22a2 0 < 2\n[PROOFSTEP]\nexact zero_lt_two\n[GOAL]\ncase intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\nhk : 0 < k\n\u22a2 Odd (card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ))\n[PROOFSTEP]\nhave hc : (fun w : V => w \u2260 v \u2227 Odd (G.degree w)) = fun w : V => Odd (G.degree w) \u2227 w \u2260 v :=\n by\n ext w\n rw [and_comm]\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\nhk : 0 < k\n\u22a2 (fun w => w \u2260 v \u2227 Odd (degree G w)) = fun w => Odd (degree G w) \u2227 w \u2260 v\n[PROOFSTEP]\next w\n[GOAL]\ncase h.a\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\nhk : 0 < k\nw : V\n\u22a2 w \u2260 v \u2227 Odd (degree G w) \u2194 Odd (degree G w) \u2227 w \u2260 v\n[PROOFSTEP]\nrw [and_comm]\n[GOAL]\ncase intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\nhk : 0 < k\nhc : (fun w => w \u2260 v \u2227 Odd (degree G w)) = fun w => Odd (degree G w) \u2227 w \u2260 v\n\u22a2 Odd (card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ))\n[PROOFSTEP]\nsimp only [hc, filter_congr]\n[GOAL]\ncase intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\nhk : 0 < k\nhc : (fun w => w \u2260 v \u2227 Odd (degree G w)) = fun w => Odd (degree G w) \u2227 w \u2260 v\n\u22a2 Odd (card (filter (fun w => Odd (degree G w) \u2227 w \u2260 v) univ))\n[PROOFSTEP]\nrw [\u2190 filter_filter, filter_ne', card_erase_of_mem]\n[GOAL]\ncase intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\nhk : 0 < k\nhc : (fun w => w \u2260 v \u2227 Odd (degree G w)) = fun w => Odd (degree G w) \u2227 w \u2260 v\n\u22a2 Odd (card (filter (fun w => Odd (degree G w)) univ) - 1)\n[PROOFSTEP]\nrefine' \u27e8k - 1, tsub_eq_of_eq_add <| hg.trans _\u27e9\n[GOAL]\ncase intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\nhk : 0 < k\nhc : (fun w => w \u2260 v \u2227 Odd (degree G w)) = fun w => Odd (degree G w) \u2227 w \u2260 v\n\u22a2 k + k = 2 * (k - 1) + 1 + 1\n[PROOFSTEP]\nrw [add_assoc, one_add_one_eq_two, \u2190 Nat.mul_succ, \u2190 two_mul]\n[GOAL]\ncase intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\nhk : 0 < k\nhc : (fun w => w \u2260 v \u2227 Odd (degree G w)) = fun w => Odd (degree G w) \u2227 w \u2260 v\n\u22a2 2 * k = 2 * Nat.succ (k - 1)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase intro.e_a\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\nhk : 0 < k\nhc : (fun w => w \u2260 v \u2227 Odd (degree G w)) = fun w => Odd (degree G w) \u2227 w \u2260 v\n\u22a2 k = Nat.succ (k - 1)\n[PROOFSTEP]\nexact (tsub_add_cancel_of_le <| Nat.succ_le_iff.2 hk).symm\n[GOAL]\ncase intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nk : \u2115\nhg : card (filter (fun v => Odd (degree G v)) univ) = k + k\nhk : 0 < k\nhc : (fun w => w \u2260 v \u2227 Odd (degree G w)) = fun w => Odd (degree G w) \u2227 w \u2260 v\n\u22a2 v \u2208 filter (fun w => Odd (degree G w)) univ\n[PROOFSTEP]\nsimpa only [true_and_iff, mem_filter, mem_univ]\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\n\u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w)\n[PROOFSTEP]\nhaveI := Classical.decEq V\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\n\u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w)\n[PROOFSTEP]\nrcases G.odd_card_odd_degree_vertices_ne v h with \u27e8k, hg\u27e9\n[GOAL]\ncase intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\nk : \u2115\nhg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1\n\u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w)\n[PROOFSTEP]\nhave hg' : (filter (fun w : V => w \u2260 v \u2227 Odd (G.degree w)) univ).card > 0 :=\n by\n rw [hg]\n apply Nat.succ_pos\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\nk : \u2115\nhg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1\n\u22a2 card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) > 0\n[PROOFSTEP]\nrw [hg]\n[GOAL]\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\nk : \u2115\nhg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1\n\u22a2 2 * k + 1 > 0\n[PROOFSTEP]\napply Nat.succ_pos\n[GOAL]\ncase intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\nk : \u2115\nhg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1\nhg' : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) > 0\n\u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w)\n[PROOFSTEP]\nrcases card_pos.mp hg' with \u27e8w, hw\u27e9\n[GOAL]\ncase intro.intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\nk : \u2115\nhg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1\nhg' : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) > 0\nw : V\nhw : w \u2208 filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ\n\u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w)\n[PROOFSTEP]\nsimp only [true_and_iff, mem_filter, mem_univ, Ne.def] at hw \n[GOAL]\ncase intro.intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\nk : \u2115\nhg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1\nhg' : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) > 0\nw : V\nhw : \u00acw = v \u2227 Odd (degree G w)\n\u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w)\n[PROOFSTEP]\nexact \u27e8w, hw\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SimpleGraph.DegreeSum", "llama_tokens": 10714, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583124210895, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.440827138179427}} {"text": "[GOAL]\n\u03b1 : Type u\n\u22a2 \u2200 (t\u2081 : Set (Ultrafilter \u03b1)),\n t\u2081 \u2208 ultrafilterBasis \u03b1 \u2192\n \u2200 (t\u2082 : Set (Ultrafilter \u03b1)),\n t\u2082 \u2208 ultrafilterBasis \u03b1 \u2192\n \u2200 (x : Ultrafilter \u03b1), x \u2208 t\u2081 \u2229 t\u2082 \u2192 \u2203 t\u2083, t\u2083 \u2208 ultrafilterBasis \u03b1 \u2227 x \u2208 t\u2083 \u2227 t\u2083 \u2286 t\u2081 \u2229 t\u2082\n[PROOFSTEP]\nrintro _ \u27e8a, rfl\u27e9 _ \u27e8b, rfl\u27e9 u \u27e8ua, ub\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\na b : Set \u03b1\nu : Ultrafilter \u03b1\nua : u \u2208 (fun s => {u | s \u2208 u}) a\nub : u \u2208 (fun s => {u | s \u2208 u}) b\n\u22a2 \u2203 t\u2083, t\u2083 \u2208 ultrafilterBasis \u03b1 \u2227 u \u2208 t\u2083 \u2227 t\u2083 \u2286 (fun s => {u | s \u2208 u}) a \u2229 (fun s => {u | s \u2208 u}) b\n[PROOFSTEP]\nrefine' \u27e8_, \u27e8a \u2229 b, rfl\u27e9, inter_mem ua ub, fun v hv => \u27e8_, _\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u\na b : Set \u03b1\nu : Ultrafilter \u03b1\nua : u \u2208 (fun s => {u | s \u2208 u}) a\nub : u \u2208 (fun s => {u | s \u2208 u}) b\nv : Ultrafilter \u03b1\nhv : v \u2208 (fun s => {u | s \u2208 u}) (a \u2229 b)\n\u22a2 v \u2208 (fun s => {u | s \u2208 u}) a\n[PROOFSTEP]\napply mem_of_superset hv\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u\na b : Set \u03b1\nu : Ultrafilter \u03b1\nua : u \u2208 (fun s => {u | s \u2208 u}) a\nub : u \u2208 (fun s => {u | s \u2208 u}) b\nv : Ultrafilter \u03b1\nhv : v \u2208 (fun s => {u | s \u2208 u}) (a \u2229 b)\n\u22a2 v \u2208 (fun s => {u | s \u2208 u}) b\n[PROOFSTEP]\napply mem_of_superset hv\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u\na b : Set \u03b1\nu : Ultrafilter \u03b1\nua : u \u2208 (fun s => {u | s \u2208 u}) a\nub : u \u2208 (fun s => {u | s \u2208 u}) b\nv : Ultrafilter \u03b1\nhv : v \u2208 (fun s => {u | s \u2208 u}) (a \u2229 b)\n\u22a2 a \u2229 b \u2286 a\n[PROOFSTEP]\nsimp [inter_subset_right a b]\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u\na b : Set \u03b1\nu : Ultrafilter \u03b1\nua : u \u2208 (fun s => {u | s \u2208 u}) a\nub : u \u2208 (fun s => {u | s \u2208 u}) b\nv : Ultrafilter \u03b1\nhv : v \u2208 (fun s => {u | s \u2208 u}) (a \u2229 b)\n\u22a2 a \u2229 b \u2286 b\n[PROOFSTEP]\nsimp [inter_subset_right a b]\n[GOAL]\n\u03b1 : Type u\ns : Set \u03b1\n\u22a2 IsClosed {u | s \u2208 u}\n[PROOFSTEP]\nrw [\u2190 isOpen_compl_iff]\n[GOAL]\n\u03b1 : Type u\ns : Set \u03b1\n\u22a2 IsOpen {u | s \u2208 u}\u1d9c\n[PROOFSTEP]\nconvert ultrafilter_isOpen_basic s\u1d9c using 1\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u\ns : Set \u03b1\n\u22a2 {u | s \u2208 u}\u1d9c = {u | s\u1d9c \u2208 u}\n[PROOFSTEP]\next u\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u\ns : Set \u03b1\nu : Ultrafilter \u03b1\n\u22a2 u \u2208 {u | s \u2208 u}\u1d9c \u2194 u \u2208 {u | s\u1d9c \u2208 u}\n[PROOFSTEP]\nexact Ultrafilter.compl_mem_iff_not_mem.symm\n[GOAL]\n\u03b1 : Type u\nu : Ultrafilter (Ultrafilter \u03b1)\nx : Ultrafilter \u03b1\n\u22a2 \u2191u \u2264 \ud835\udcdd x \u2194 x = joinM u\n[PROOFSTEP]\nrw [eq_comm, \u2190 Ultrafilter.coe_le_coe]\n[GOAL]\n\u03b1 : Type u\nu : Ultrafilter (Ultrafilter \u03b1)\nx : Ultrafilter \u03b1\n\u22a2 \u2191u \u2264 \ud835\udcdd x \u2194 \u2191(joinM u) \u2264 \u2191x\n[PROOFSTEP]\nchange \u2191u \u2264 \ud835\udcdd x \u2194 \u2200 s \u2208 x, {v : Ultrafilter \u03b1 | s \u2208 v} \u2208 u\n[GOAL]\n\u03b1 : Type u\nu : Ultrafilter (Ultrafilter \u03b1)\nx : Ultrafilter \u03b1\n\u22a2 \u2191u \u2264 \ud835\udcdd x \u2194 \u2200 (s : Set \u03b1), s \u2208 x \u2192 {v | s \u2208 v} \u2208 u\n[PROOFSTEP]\nsimp only [TopologicalSpace.nhds_generateFrom, le_iInf_iff, ultrafilterBasis, le_principal_iff, mem_setOf_eq]\n[GOAL]\n\u03b1 : Type u\nu : Ultrafilter (Ultrafilter \u03b1)\nx : Ultrafilter \u03b1\n\u22a2 (\u2200 (i : Set (Ultrafilter \u03b1)), (x \u2208 i \u2227 i \u2208 range fun s => {u | s \u2208 u}) \u2192 i \u2208 \u2191u) \u2194\n \u2200 (s : Set \u03b1), s \u2208 x \u2192 {v | s \u2208 v} \u2208 u\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\nu : Ultrafilter (Ultrafilter \u03b1)\nx : Ultrafilter \u03b1\n\u22a2 (\u2200 (i : Set (Ultrafilter \u03b1)), (x \u2208 i \u2227 i \u2208 range fun s => {u | s \u2208 u}) \u2192 i \u2208 \u2191u) \u2192\n \u2200 (s : Set \u03b1), s \u2208 x \u2192 {v | s \u2208 v} \u2208 u\n[PROOFSTEP]\nintro h a ha\n[GOAL]\ncase mp\n\u03b1 : Type u\nu : Ultrafilter (Ultrafilter \u03b1)\nx : Ultrafilter \u03b1\nh : \u2200 (i : Set (Ultrafilter \u03b1)), (x \u2208 i \u2227 i \u2208 range fun s => {u | s \u2208 u}) \u2192 i \u2208 \u2191u\na : Set \u03b1\nha : a \u2208 x\n\u22a2 {v | a \u2208 v} \u2208 u\n[PROOFSTEP]\nexact h _ \u27e8ha, a, rfl\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\nu : Ultrafilter (Ultrafilter \u03b1)\nx : Ultrafilter \u03b1\n\u22a2 (\u2200 (s : Set \u03b1), s \u2208 x \u2192 {v | s \u2208 v} \u2208 u) \u2192\n \u2200 (i : Set (Ultrafilter \u03b1)), (x \u2208 i \u2227 i \u2208 range fun s => {u | s \u2208 u}) \u2192 i \u2208 \u2191u\n[PROOFSTEP]\nrintro h a \u27e8xi, a, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u\nu : Ultrafilter (Ultrafilter \u03b1)\nx : Ultrafilter \u03b1\nh : \u2200 (s : Set \u03b1), s \u2208 x \u2192 {v | s \u2208 v} \u2208 u\na : Set \u03b1\nxi : x \u2208 (fun s => {u | s \u2208 u}) a\n\u22a2 (fun s => {u | s \u2208 u}) a \u2208 \u2191u\n[PROOFSTEP]\nexact h _ xi\n[GOAL]\n\u03b1 : Type u\n\u22a2 TotallyDisconnectedSpace (Ultrafilter \u03b1)\n[PROOFSTEP]\nrw [totallyDisconnectedSpace_iff_connectedComponent_singleton]\n[GOAL]\n\u03b1 : Type u\n\u22a2 \u2200 (x : Ultrafilter \u03b1), connectedComponent x = {x}\n[PROOFSTEP]\nintro A\n[GOAL]\n\u03b1 : Type u\nA : Ultrafilter \u03b1\n\u22a2 connectedComponent A = {A}\n[PROOFSTEP]\nsimp only [Set.eq_singleton_iff_unique_mem, mem_connectedComponent, true_and_iff]\n[GOAL]\n\u03b1 : Type u\nA : Ultrafilter \u03b1\n\u22a2 \u2200 (x : Ultrafilter \u03b1), x \u2208 connectedComponent A \u2192 x = A\n[PROOFSTEP]\nintro B hB\n[GOAL]\n\u03b1 : Type u\nA B : Ultrafilter \u03b1\nhB : B \u2208 connectedComponent A\n\u22a2 B = A\n[PROOFSTEP]\nrw [\u2190 Ultrafilter.coe_le_coe]\n[GOAL]\n\u03b1 : Type u\nA B : Ultrafilter \u03b1\nhB : B \u2208 connectedComponent A\n\u22a2 \u2191B \u2264 \u2191A\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\u03b1 : Type u\nA B : Ultrafilter \u03b1\nhB : B \u2208 connectedComponent A\ns : Set \u03b1\nhs : s \u2208 \u2191A\n\u22a2 s \u2208 \u2191B\n[PROOFSTEP]\nrw [connectedComponent_eq_iInter_clopen, Set.mem_iInter] at hB \n[GOAL]\n\u03b1 : Type u\nA B : Ultrafilter \u03b1\nhB : \u2200 (i : { Z // IsClopen Z \u2227 A \u2208 Z }), B \u2208 \u2191i\ns : Set \u03b1\nhs : s \u2208 \u2191A\n\u22a2 s \u2208 \u2191B\n[PROOFSTEP]\nlet Z := {F : Ultrafilter \u03b1 | s \u2208 F}\n[GOAL]\n\u03b1 : Type u\nA B : Ultrafilter \u03b1\nhB : \u2200 (i : { Z // IsClopen Z \u2227 A \u2208 Z }), B \u2208 \u2191i\ns : Set \u03b1\nhs : s \u2208 \u2191A\nZ : Set (Ultrafilter \u03b1) := {F | s \u2208 F}\n\u22a2 s \u2208 \u2191B\n[PROOFSTEP]\nhave hZ : IsClopen Z := \u27e8ultrafilter_isOpen_basic s, ultrafilter_isClosed_basic s\u27e9\n[GOAL]\n\u03b1 : Type u\nA B : Ultrafilter \u03b1\nhB : \u2200 (i : { Z // IsClopen Z \u2227 A \u2208 Z }), B \u2208 \u2191i\ns : Set \u03b1\nhs : s \u2208 \u2191A\nZ : Set (Ultrafilter \u03b1) := {F | s \u2208 F}\nhZ : IsClopen Z\n\u22a2 s \u2208 \u2191B\n[PROOFSTEP]\nexact hB \u27e8Z, hZ, hs\u27e9\n[GOAL]\n\u03b1 : Type u\nb : Ultrafilter \u03b1\n\u22a2 Tendsto pure (\u2191b) (\ud835\udcdd b)\n[PROOFSTEP]\nrw [Tendsto, \u2190 coe_map, ultrafilter_converges_iff]\n[GOAL]\n\u03b1 : Type u\nb : Ultrafilter \u03b1\n\u22a2 b = joinM (map pure b)\n[PROOFSTEP]\next s\n[GOAL]\ncase h\n\u03b1 : Type u\nb : Ultrafilter \u03b1\ns : Set \u03b1\n\u22a2 s \u2208 b \u2194 s \u2208 joinM (map pure b)\n[PROOFSTEP]\nchange s \u2208 b \u2194 {t | s \u2208 t} \u2208 map pure b\n[GOAL]\ncase h\n\u03b1 : Type u\nb : Ultrafilter \u03b1\ns : Set \u03b1\n\u22a2 s \u2208 b \u2194 {t | s \u2208 t} \u2208 map pure b\n[PROOFSTEP]\nsimp_rw [mem_map, preimage_setOf_eq, mem_pure, setOf_mem_eq]\n[GOAL]\n\u03b1 : Type u\nb : Ultrafilter \u03b1\n\u22a2 comap pure (\ud835\udcdd b) \u2264 \u2191b\n[PROOFSTEP]\nrw [TopologicalSpace.nhds_generateFrom]\n[GOAL]\n\u03b1 : Type u\nb : Ultrafilter \u03b1\n\u22a2 comap pure (\u2a05 (s : Set (Ultrafilter \u03b1)) (_ : s \u2208 {s | b \u2208 s \u2227 s \u2208 ultrafilterBasis \u03b1}), \ud835\udcdf s) \u2264 \u2191b\n[PROOFSTEP]\nsimp only [comap_iInf, comap_principal]\n[GOAL]\n\u03b1 : Type u\nb : Ultrafilter \u03b1\n\u22a2 \u2a05 (i : Set (Ultrafilter \u03b1)) (_ : i \u2208 {s | b \u2208 s \u2227 s \u2208 ultrafilterBasis \u03b1}), \ud835\udcdf (pure \u207b\u00b9' i) \u2264 \u2191b\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\u03b1 : Type u\nb : Ultrafilter \u03b1\ns : Set \u03b1\nhs : s \u2208 \u2191b\n\u22a2 s \u2208 \u2a05 (i : Set (Ultrafilter \u03b1)) (_ : i \u2208 {s | b \u2208 s \u2227 s \u2208 ultrafilterBasis \u03b1}), \ud835\udcdf (pure \u207b\u00b9' i)\n[PROOFSTEP]\nrw [\u2190 le_principal_iff]\n[GOAL]\n\u03b1 : Type u\nb : Ultrafilter \u03b1\ns : Set \u03b1\nhs : s \u2208 \u2191b\n\u22a2 \u2a05 (i : Set (Ultrafilter \u03b1)) (_ : i \u2208 {s | b \u2208 s \u2227 s \u2208 ultrafilterBasis \u03b1}), \ud835\udcdf (pure \u207b\u00b9' i) \u2264 \ud835\udcdf s\n[PROOFSTEP]\nrefine' iInf_le_of_le {u | s \u2208 u} _\n[GOAL]\n\u03b1 : Type u\nb : Ultrafilter \u03b1\ns : Set \u03b1\nhs : s \u2208 \u2191b\n\u22a2 \u2a05 (_ : {u | s \u2208 u} \u2208 {s | b \u2208 s \u2227 s \u2208 ultrafilterBasis \u03b1}), \ud835\udcdf (pure \u207b\u00b9' {u | s \u2208 u}) \u2264 \ud835\udcdf s\n[PROOFSTEP]\nrefine' iInf_le_of_le \u27e8hs, \u27e8s, rfl\u27e9\u27e9 _\n[GOAL]\n\u03b1 : Type u\nb : Ultrafilter \u03b1\ns : Set \u03b1\nhs : s \u2208 \u2191b\n\u22a2 \ud835\udcdf (pure \u207b\u00b9' {u | s \u2208 u}) \u2264 \ud835\udcdf s\n[PROOFSTEP]\nexact principal_mono.2 fun a => id\n[GOAL]\n\u03b1 : Type u\n\u22a2 Function.Injective pure\n[PROOFSTEP]\nintro x y h\n[GOAL]\n\u03b1 : Type u\nx y : \u03b1\nh : pure x = pure y\n\u22a2 x = y\n[PROOFSTEP]\nhave : { x } \u2208 (pure x : Ultrafilter \u03b1) := singleton_mem_pure\n[GOAL]\n\u03b1 : Type u\nx y : \u03b1\nh : pure x = pure y\nthis : {x} \u2208 pure x\n\u22a2 x = y\n[PROOFSTEP]\nrw [h] at this \n[GOAL]\n\u03b1 : Type u\nx y : \u03b1\nh : pure x = pure y\nthis : {x} \u2208 pure y\n\u22a2 x = y\n[PROOFSTEP]\nexact (mem_singleton_iff.mp (mem_pure.mp this)).symm\n[GOAL]\n\u03b1 : Type u\n\u22a2 induced pure Ultrafilter.topologicalSpace = \u22a5\n[PROOFSTEP]\napply eq_bot_of_singletons_open\n[GOAL]\ncase h\n\u03b1 : Type u\n\u22a2 \u2200 (x : \u03b1), IsOpen {x}\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h\n\u03b1 : Type u\nx : \u03b1\n\u22a2 IsOpen {x}\n[PROOFSTEP]\nuse{u : Ultrafilter \u03b1 | { x } \u2208 u}, ultrafilter_isOpen_basic _\n[GOAL]\ncase right\n\u03b1 : Type u\nx : \u03b1\n\u22a2 pure \u207b\u00b9' {u | {x} \u2208 u} = {x}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : T2Space \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 Ultrafilter.extend f \u2218 pure = f\n[PROOFSTEP]\nletI : TopologicalSpace \u03b1 := \u22a5\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : T2Space \u03b3\nf : \u03b1 \u2192 \u03b3\nthis : TopologicalSpace \u03b1 := \u22a5\n\u22a2 Ultrafilter.extend f \u2218 pure = f\n[PROOFSTEP]\nhaveI : DiscreteTopology \u03b1 := \u27e8rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : T2Space \u03b3\nf : \u03b1 \u2192 \u03b3\nthis\u271d : TopologicalSpace \u03b1 := \u22a5\nthis : DiscreteTopology \u03b1\n\u22a2 Ultrafilter.extend f \u2218 pure = f\n[PROOFSTEP]\nexact funext (denseInducing_pure.extend_eq continuous_of_discreteTopology)\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 Continuous (Ultrafilter.extend f)\n[PROOFSTEP]\nhave h : \u2200 b : Ultrafilter \u03b1, \u2203 c, Tendsto f (comap pure (\ud835\udcdd b)) (\ud835\udcdd c) :=\n fun b =>\n -- b.map f is an ultrafilter on \u03b3, which is compact, so it converges to some c in \u03b3.\n let \u27e8c, _, h'\u27e9 := isCompact_univ.ultrafilter_le_nhds (b.map f) (by rw [le_principal_iff]; exact univ_mem)\n \u27e8c, le_trans (map_mono (ultrafilter_comap_pure_nhds _)) h'\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nb : Ultrafilter \u03b1\n\u22a2 \u2191(Ultrafilter.map f b) \u2264 \ud835\udcdf univ\n[PROOFSTEP]\nrw [le_principal_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nb : Ultrafilter \u03b1\n\u22a2 univ \u2208 \u2191(Ultrafilter.map f b)\n[PROOFSTEP]\nexact univ_mem\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nh : \u2200 (b : Ultrafilter \u03b1), \u2203 c, Tendsto f (comap pure (\ud835\udcdd b)) (\ud835\udcdd c)\n\u22a2 Continuous (Ultrafilter.extend f)\n[PROOFSTEP]\nletI : TopologicalSpace \u03b1 := \u22a5\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nh : \u2200 (b : Ultrafilter \u03b1), \u2203 c, Tendsto f (comap pure (\ud835\udcdd b)) (\ud835\udcdd c)\nthis : TopologicalSpace \u03b1 := \u22a5\n\u22a2 Continuous (Ultrafilter.extend f)\n[PROOFSTEP]\nhaveI : NormalSpace \u03b3 := normalOfCompactT2\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nh : \u2200 (b : Ultrafilter \u03b1), \u2203 c, Tendsto f (comap pure (\ud835\udcdd b)) (\ud835\udcdd c)\nthis\u271d : TopologicalSpace \u03b1 := \u22a5\nthis : NormalSpace \u03b3\n\u22a2 Continuous (Ultrafilter.extend f)\n[PROOFSTEP]\nexact denseInducing_pure.continuous_extend h\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nb : Ultrafilter \u03b1\nc : \u03b3\nh : Ultrafilter.extend f b = c\n\u22a2 \u2191(Ultrafilter.map f b) \u2264 \ud835\udcdd c\n[PROOFSTEP]\nlet b' : Ultrafilter (Ultrafilter \u03b1) := b.map pure\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nb : Ultrafilter \u03b1\nc : \u03b3\nh : Ultrafilter.extend f b = c\nb' : Ultrafilter (Ultrafilter \u03b1) := Ultrafilter.map pure b\n\u22a2 \u2191(Ultrafilter.map f b) \u2264 \ud835\udcdd c\n[PROOFSTEP]\nhave t : \u2191b' \u2264 \ud835\udcdd b := ultrafilter_converges_iff.mpr (bind_pure _).symm\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nb : Ultrafilter \u03b1\nc : \u03b3\nh : Ultrafilter.extend f b = c\nb' : Ultrafilter (Ultrafilter \u03b1) := Ultrafilter.map pure b\nt : \u2191b' \u2264 \ud835\udcdd b\n\u22a2 \u2191(Ultrafilter.map f b) \u2264 \ud835\udcdd c\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nb : Ultrafilter \u03b1\nc : \u03b3\nh : Ultrafilter.extend f b = c\nb' : Ultrafilter (Ultrafilter \u03b1) := Ultrafilter.map pure b\nt : \u2191b' \u2264 \ud835\udcdd b\n\u22a2 \u2191(Ultrafilter.map f b) \u2264 \ud835\udcdd (Ultrafilter.extend f b)\n[PROOFSTEP]\nhave := (continuous_ultrafilter_extend f).tendsto b\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nb : Ultrafilter \u03b1\nc : \u03b3\nh : Ultrafilter.extend f b = c\nb' : Ultrafilter (Ultrafilter \u03b1) := Ultrafilter.map pure b\nt : \u2191b' \u2264 \ud835\udcdd b\nthis : Tendsto (Ultrafilter.extend f) (\ud835\udcdd b) (\ud835\udcdd (Ultrafilter.extend f b))\n\u22a2 \u2191(Ultrafilter.map f b) \u2264 \ud835\udcdd (Ultrafilter.extend f b)\n[PROOFSTEP]\nrefine' le_trans _ (le_trans (map_mono t) this)\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nb : Ultrafilter \u03b1\nc : \u03b3\nh : Ultrafilter.extend f b = c\nb' : Ultrafilter (Ultrafilter \u03b1) := Ultrafilter.map pure b\nt : \u2191b' \u2264 \ud835\udcdd b\nthis : Tendsto (Ultrafilter.extend f) (\ud835\udcdd b) (\ud835\udcdd (Ultrafilter.extend f b))\n\u22a2 \u2191(Ultrafilter.map f b) \u2264 map (Ultrafilter.extend f) \u2191b'\n[PROOFSTEP]\nchange _ \u2264 map (Ultrafilter.extend f \u2218 pure) \u2191b\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nb : Ultrafilter \u03b1\nc : \u03b3\nh : Ultrafilter.extend f b = c\nb' : Ultrafilter (Ultrafilter \u03b1) := Ultrafilter.map pure b\nt : \u2191b' \u2264 \ud835\udcdd b\nthis : Tendsto (Ultrafilter.extend f) (\ud835\udcdd b) (\ud835\udcdd (Ultrafilter.extend f b))\n\u22a2 \u2191(Ultrafilter.map f b) \u2264 map (Ultrafilter.extend f \u2218 pure) \u2191b\n[PROOFSTEP]\nrw [ultrafilter_extend_extends]\n[GOAL]\n\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nb : Ultrafilter \u03b1\nc : \u03b3\nh : Ultrafilter.extend f b = c\nb' : Ultrafilter (Ultrafilter \u03b1) := Ultrafilter.map pure b\nt : \u2191b' \u2264 \ud835\udcdd b\nthis : Tendsto (Ultrafilter.extend f) (\ud835\udcdd b) (\ud835\udcdd (Ultrafilter.extend f b))\n\u22a2 \u2191(Ultrafilter.map f b) \u2264 map f \u2191b\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 TopologicalSpace (StoneCech \u03b1)\n[PROOFSTEP]\nunfold StoneCech\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 TopologicalSpace (Quotient (stoneCechSetoid \u03b1))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Inhabited \u03b1\n\u22a2 Inhabited (StoneCech \u03b1)\n[PROOFSTEP]\nunfold StoneCech\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Inhabited \u03b1\n\u22a2 Inhabited (Quotient (stoneCechSetoid \u03b1))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b3 : Type u\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : T2Space \u03b3\ninst\u271d\u00b2 : CompactSpace \u03b3\n\u03b3' : Type u\ninst\u271d\u00b9 : TopologicalSpace \u03b3'\ninst\u271d : T2Space \u03b3'\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\ng\u2081 g\u2082 : StoneCech \u03b1 \u2192 \u03b3'\nh\u2081 : Continuous g\u2081\nh\u2082 : Continuous g\u2082\nh : g\u2081 \u2218 stoneCechUnit = g\u2082 \u2218 stoneCechUnit\n\u22a2 g\u2081 = g\u2082\n[PROOFSTEP]\napply Continuous.ext_on denseRange_stoneCechUnit h\u2081 h\u2082\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b3 : Type u\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : T2Space \u03b3\ninst\u271d\u00b2 : CompactSpace \u03b3\n\u03b3' : Type u\ninst\u271d\u00b9 : TopologicalSpace \u03b3'\ninst\u271d : T2Space \u03b3'\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\ng\u2081 g\u2082 : StoneCech \u03b1 \u2192 \u03b3'\nh\u2081 : Continuous g\u2081\nh\u2082 : Continuous g\u2082\nh : g\u2081 \u2218 stoneCechUnit = g\u2082 \u2218 stoneCechUnit\n\u22a2 EqOn g\u2081 g\u2082 (range stoneCechUnit)\n[PROOFSTEP]\nrintro x \u27e8x, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b3 : Type u\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : T2Space \u03b3\ninst\u271d\u00b2 : CompactSpace \u03b3\n\u03b3' : Type u\ninst\u271d\u00b9 : TopologicalSpace \u03b3'\ninst\u271d : T2Space \u03b3'\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\ng\u2081 g\u2082 : StoneCech \u03b1 \u2192 \u03b3'\nh\u2081 : Continuous g\u2081\nh\u2082 : Continuous g\u2082\nh : g\u2081 \u2218 stoneCechUnit = g\u2082 \u2218 stoneCechUnit\nx : \u03b1\n\u22a2 g\u2081 (stoneCechUnit x) = g\u2082 (stoneCechUnit x)\n[PROOFSTEP]\napply congr_fun h x\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nu : Ultrafilter \u03b1\nx : \u03b1\nux : \u2191u \u2264 \ud835\udcdd x\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\n\u22a2 Ultrafilter.extend f u = Ultrafilter.extend f (pure x)\n[PROOFSTEP]\nskip\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nu : Ultrafilter \u03b1\nx : \u03b1\nux : \u2191u \u2264 \ud835\udcdd x\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\n\u22a2 Ultrafilter.extend f u = Ultrafilter.extend f (pure x)\n[PROOFSTEP]\ntrans f x\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nu : Ultrafilter \u03b1\nx : \u03b1\nux : \u2191u \u2264 \ud835\udcdd x\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\n\u22a2 Ultrafilter.extend f u = f x\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nu : Ultrafilter \u03b1\nx : \u03b1\nux : \u2191u \u2264 \ud835\udcdd x\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\n\u22a2 f x = Ultrafilter.extend f (pure x)\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nu : Ultrafilter \u03b1\nx : \u03b1\nux : \u2191u \u2264 \ud835\udcdd x\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\n\u22a2 f x = Ultrafilter.extend f (pure x)\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nu : Ultrafilter \u03b1\nx : \u03b1\nux : \u2191u \u2264 \ud835\udcdd x\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\n\u22a2 Ultrafilter.extend f u = f x\n[PROOFSTEP]\nsymm\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nu : Ultrafilter \u03b1\nx : \u03b1\nux : \u2191u \u2264 \ud835\udcdd x\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\n\u22a2 Ultrafilter.extend f (pure x) = f x\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nu : Ultrafilter \u03b1\nx : \u03b1\nux : \u2191u \u2264 \ud835\udcdd x\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\n\u22a2 Ultrafilter.extend f u = f x\n[PROOFSTEP]\nall_goals refine' ultrafilter_extend_eq_iff.mpr (le_trans (map_mono _) (hf.tendsto _))\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nu : Ultrafilter \u03b1\nx : \u03b1\nux : \u2191u \u2264 \ud835\udcdd x\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\n\u22a2 Ultrafilter.extend f (pure x) = f x\n[PROOFSTEP]\nrefine' ultrafilter_extend_eq_iff.mpr (le_trans (map_mono _) (hf.tendsto _))\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nu : Ultrafilter \u03b1\nx : \u03b1\nux : \u2191u \u2264 \ud835\udcdd x\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\n\u22a2 Ultrafilter.extend f u = f x\n[PROOFSTEP]\nrefine' ultrafilter_extend_eq_iff.mpr (le_trans (map_mono _) (hf.tendsto _))\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nu : Ultrafilter \u03b1\nx : \u03b1\nux : \u2191u \u2264 \ud835\udcdd x\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\n\u22a2 \u2191(pure x) \u2264 \ud835\udcdd x\n[PROOFSTEP]\napply pure_le_nhds\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nu : Ultrafilter \u03b1\nx : \u03b1\nux : \u2191u \u2264 \ud835\udcdd x\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\n\u22a2 \u2191u \u2264 \ud835\udcdd x\n[PROOFSTEP]\nexact ux\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nx : \u03b1\ng : Ultrafilter \u03b1\ngx : \u2191g \u2264 \ud835\udcdd x\n\u22a2 Tendsto stoneCechUnit (\u2191g) (\ud835\udcdd (stoneCechUnit x))\n[PROOFSTEP]\nhave : (g.map pure).toFilter \u2264 \ud835\udcdd g := by\n rw [ultrafilter_converges_iff]\n exact (bind_pure _).symm\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nx : \u03b1\ng : Ultrafilter \u03b1\ngx : \u2191g \u2264 \ud835\udcdd x\n\u22a2 \u2191(Ultrafilter.map pure g) \u2264 \ud835\udcdd g\n[PROOFSTEP]\nrw [ultrafilter_converges_iff]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nx : \u03b1\ng : Ultrafilter \u03b1\ngx : \u2191g \u2264 \ud835\udcdd x\n\u22a2 g = joinM (Ultrafilter.map pure g)\n[PROOFSTEP]\nexact (bind_pure _).symm\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nx : \u03b1\ng : Ultrafilter \u03b1\ngx : \u2191g \u2264 \ud835\udcdd x\nthis : \u2191(Ultrafilter.map pure g) \u2264 \ud835\udcdd g\n\u22a2 Tendsto stoneCechUnit (\u2191g) (\ud835\udcdd (stoneCechUnit x))\n[PROOFSTEP]\nhave : (g.map stoneCechUnit : Filter (StoneCech \u03b1)) \u2264 \ud835\udcdd \u27e6g\u27e7 :=\n continuousAt_iff_ultrafilter.mp (continuous_quotient_mk'.tendsto g) _ this\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nx : \u03b1\ng : Ultrafilter \u03b1\ngx : \u2191g \u2264 \ud835\udcdd x\nthis\u271d : \u2191(Ultrafilter.map pure g) \u2264 \ud835\udcdd g\nthis : \u2191(Ultrafilter.map stoneCechUnit g) \u2264 \ud835\udcdd (Quotient.mk (stoneCechSetoid \u03b1) g)\n\u22a2 Tendsto stoneCechUnit (\u2191g) (\ud835\udcdd (stoneCechUnit x))\n[PROOFSTEP]\nrwa [show \u27e6g\u27e7 = \u27e6pure x\u27e7 from Quotient.sound <| convergent_eqv_pure gx] at this \n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 T2Space (StoneCech \u03b1)\n[PROOFSTEP]\nrw [t2_iff_ultrafilter]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 \u2200 {x y : StoneCech \u03b1} (f : Ultrafilter (StoneCech \u03b1)), \u2191f \u2264 \ud835\udcdd x \u2192 \u2191f \u2264 \ud835\udcdd y \u2192 x = y\n[PROOFSTEP]\nrintro \u27e8x\u27e9 \u27e8y\u27e9 g gx gy\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nx\u271d : StoneCech \u03b1\nx : Ultrafilter \u03b1\ny\u271d : StoneCech \u03b1\ny : Ultrafilter \u03b1\ng : Ultrafilter (StoneCech \u03b1)\ngx : \u2191g \u2264 \ud835\udcdd (Quot.mk Setoid.r x)\ngy : \u2191g \u2264 \ud835\udcdd (Quot.mk Setoid.r y)\n\u22a2 Quot.mk Setoid.r x = Quot.mk Setoid.r y\n[PROOFSTEP]\napply Quotient.sound\n[GOAL]\ncase mk.mk.a\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nx\u271d : StoneCech \u03b1\nx : Ultrafilter \u03b1\ny\u271d : StoneCech \u03b1\ny : Ultrafilter \u03b1\ng : Ultrafilter (StoneCech \u03b1)\ngx : \u2191g \u2264 \ud835\udcdd (Quot.mk Setoid.r x)\ngy : \u2191g \u2264 \ud835\udcdd (Quot.mk Setoid.r y)\n\u22a2 x \u2248 y\n[PROOFSTEP]\nintro \u03b3 t\u03b3 h\u2081 h\u2082 f hf\n[GOAL]\ncase mk.mk.a\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nx\u271d : StoneCech \u03b1\nx : Ultrafilter \u03b1\ny\u271d : StoneCech \u03b1\ny : Ultrafilter \u03b1\ng : Ultrafilter (StoneCech \u03b1)\ngx : \u2191g \u2264 \ud835\udcdd (Quot.mk Setoid.r x)\ngy : \u2191g \u2264 \ud835\udcdd (Quot.mk Setoid.r y)\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\n\u22a2 Ultrafilter.extend f x = Ultrafilter.extend f y\n[PROOFSTEP]\nskip\n[GOAL]\ncase mk.mk.a\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nx\u271d : StoneCech \u03b1\nx : Ultrafilter \u03b1\ny\u271d : StoneCech \u03b1\ny : Ultrafilter \u03b1\ng : Ultrafilter (StoneCech \u03b1)\ngx : \u2191g \u2264 \ud835\udcdd (Quot.mk Setoid.r x)\ngy : \u2191g \u2264 \ud835\udcdd (Quot.mk Setoid.r y)\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\n\u22a2 Ultrafilter.extend f x = Ultrafilter.extend f y\n[PROOFSTEP]\nlet ff := stoneCechExtend hf\n[GOAL]\ncase mk.mk.a\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nx\u271d : StoneCech \u03b1\nx : Ultrafilter \u03b1\ny\u271d : StoneCech \u03b1\ny : Ultrafilter \u03b1\ng : Ultrafilter (StoneCech \u03b1)\ngx : \u2191g \u2264 \ud835\udcdd (Quot.mk Setoid.r x)\ngy : \u2191g \u2264 \ud835\udcdd (Quot.mk Setoid.r y)\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\nff : StoneCech \u03b1 \u2192 \u03b3 := stoneCechExtend hf\n\u22a2 Ultrafilter.extend f x = Ultrafilter.extend f y\n[PROOFSTEP]\nchange ff \u27e6x\u27e7 = ff \u27e6y\u27e7\n[GOAL]\ncase mk.mk.a\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nx\u271d : StoneCech \u03b1\nx : Ultrafilter \u03b1\ny\u271d : StoneCech \u03b1\ny : Ultrafilter \u03b1\ng : Ultrafilter (StoneCech \u03b1)\ngx : \u2191g \u2264 \ud835\udcdd (Quot.mk Setoid.r x)\ngy : \u2191g \u2264 \ud835\udcdd (Quot.mk Setoid.r y)\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\nff : StoneCech \u03b1 \u2192 \u03b3 := stoneCechExtend hf\n\u22a2 ff (Quotient.mk (stoneCechSetoid \u03b1) x) = ff (Quotient.mk (stoneCechSetoid \u03b1) y)\n[PROOFSTEP]\nhave lim := fun (z : Ultrafilter \u03b1) (gz : (g : Filter (StoneCech \u03b1)) \u2264 \ud835\udcdd \u27e6z\u27e7) =>\n ((continuous_stoneCechExtend hf).tendsto _).mono_left gz\n[GOAL]\ncase mk.mk.a\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nx\u271d : StoneCech \u03b1\nx : Ultrafilter \u03b1\ny\u271d : StoneCech \u03b1\ny : Ultrafilter \u03b1\ng : Ultrafilter (StoneCech \u03b1)\ngx : \u2191g \u2264 \ud835\udcdd (Quot.mk Setoid.r x)\ngy : \u2191g \u2264 \ud835\udcdd (Quot.mk Setoid.r y)\n\u03b3 : Type u\nt\u03b3 : TopologicalSpace \u03b3\nh\u2081 : T2Space \u03b3\nh\u2082 : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : Continuous f\nff : StoneCech \u03b1 \u2192 \u03b3 := stoneCechExtend hf\nlim :\n \u2200 (z : Ultrafilter \u03b1),\n \u2191g \u2264 \ud835\udcdd (Quotient.mk (stoneCechSetoid \u03b1) z) \u2192\n Tendsto (stoneCechExtend hf) (\u2191g) (\ud835\udcdd (stoneCechExtend hf (Quotient.mk (stoneCechSetoid \u03b1) z)))\n\u22a2 ff (Quotient.mk (stoneCechSetoid \u03b1) x) = ff (Quotient.mk (stoneCechSetoid \u03b1) y)\n[PROOFSTEP]\nexact tendsto_nhds_unique (lim x gx) (lim y gy)\n", "meta": {"mathlib_filename": "Mathlib.Topology.StoneCech", "llama_tokens": 11109, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.712232184238947, "lm_q2_score": 0.6187804407739559, "lm_q1q2_score": 0.44071534489677294}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192\u209b \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u22a2 \u222b\u207b (a : \u03b1), \u2191f a \u2202\u03bc = lintegral f \u03bc\n[PROOFSTEP]\nrw [MeasureTheory.lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192\u209b \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u22a2 \u2a06 (g : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g \u2264 fun a => \u2191f a), lintegral g \u03bc = lintegral f \u03bc\n[PROOFSTEP]\nexact le_antisymm (iSup\u2082_le fun g hg => lintegral_mono hg <| le_rfl) (le_iSup\u2082_of_le f le_rfl le_rfl)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nh\u03bc\u03bd : \u03bc \u2264 \u03bd\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264 g\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), g a \u2202\u03bd\n[PROOFSTEP]\nrw [lintegral, lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nh\u03bc\u03bd : \u03bc \u2264 \u03bd\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264 g\n\u22a2 \u2a06 (g : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g \u2264 fun a => f a), SimpleFunc.lintegral g \u03bc \u2264\n \u2a06 (g_1 : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g_1 \u2264 fun a => g a), SimpleFunc.lintegral g_1 \u03bd\n[PROOFSTEP]\nexact iSup_mono fun \u03c6 => iSup_mono' fun h\u03c6 => \u27e8le_trans h\u03c6 hfg, lintegral_mono (le_refl \u03c6) h\u03bc\u03bd\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 (g : \u03b1 \u2192 \u211d\u22650\u221e) (_ : Measurable g) (_ : g \u2264 f), \u222b\u207b (a : \u03b1), g a \u2202\u03bc = \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 (g : \u03b1 \u2192 \u211d\u22650\u221e) (_ : Measurable g) (_ : g \u2264 f), \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nexact iSup_le fun i => iSup_le fun _ => iSup_le fun h'i => lintegral_mono h'i\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2264 \u2a06 (g : \u03b1 \u2192 \u211d\u22650\u221e) (_ : Measurable g) (_ : g \u2264 f), \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 (g : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g \u2264 fun a => f a), SimpleFunc.lintegral g \u03bc \u2264\n \u2a06 (g : \u03b1 \u2192 \u211d\u22650\u221e) (_ : Measurable g) (_ : g \u2264 f), \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nrefine' iSup\u2082_le fun i hi => le_iSup\u2082_of_le i i.measurable <| le_iSup_of_le hi _\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ni : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhi : \u2191i \u2264 fun a => f a\n\u22a2 SimpleFunc.lintegral i \u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191i a \u2202\u03bc\n[PROOFSTEP]\nexact le_of_eq (i.lintegral_eq_lintegral _).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : \u211d\u22650\u221e\n\u22a2 \u222b\u207b (x : \u03b1), c \u2202\u03bc = c * \u2191\u2191\u03bc univ\n[PROOFSTEP]\nrw [\u2190 SimpleFunc.const_lintegral, \u2190 SimpleFunc.lintegral_eq_lintegral, SimpleFunc.coe_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : \u211d\u22650\u221e\n\u22a2 \u222b\u207b (x : \u03b1), c \u2202\u03bc = \u222b\u207b (a : \u03b1), Function.const \u03b1 c a \u2202\u03bc\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u22a2 \u222b\u207b (x : \u03b1), 0 \u2202\u03bc = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u22a2 \u222b\u207b (x : \u03b1), 1 \u2202\u03bc = \u2191\u2191\u03bc univ\n[PROOFSTEP]\nrw [lintegral_const, one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nc : \u211d\u22650\u221e\n\u22a2 \u222b\u207b (x : \u03b1) in s, c \u2202\u03bc = c * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [lintegral_const, Measure.restrict_apply_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\n\u22a2 \u222b\u207b (x : \u03b1) in s, 1 \u2202\u03bc = \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [set_lintegral_const, one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\ns : Set \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, c \u2202\u03bc < \u22a4\n[PROOFSTEP]\nrw [lintegral_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\ns : Set \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 c * \u2191\u2191(Measure.restrict \u03bc s) univ < \u22a4\n[PROOFSTEP]\nexact ENNReal.mul_lt_top hc (measure_ne_top (\u03bc.restrict s) univ)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), c \u2202\u03bc < \u22a4\n[PROOFSTEP]\nsimpa only [Measure.restrict_univ] using set_lintegral_const_lt_top (univ : Set \u03b1) hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2203 g, Measurable g \u2227 g \u2264 f \u2227 \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\ncases' eq_or_ne (\u222b\u207b a, f a \u2202\u03bc) 0 with h\u2080 h\u2080\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u2080 : \u222b\u207b (a : \u03b1), f a \u2202\u03bc = 0\n\u22a2 \u2203 g, Measurable g \u2227 g \u2264 f \u2227 \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nexact \u27e80, measurable_zero, zero_le f, h\u2080.trans lintegral_zero.symm\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u2080 : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 0\n\u22a2 \u2203 g, Measurable g \u2227 g \u2264 f \u2227 \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nrcases exists_seq_strictMono_tendsto' h\u2080.bot_lt with \u27e8L, _, hLf, hL_tendsto\u27e9\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u2080 : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 0\nL : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictMono L\nhLf : \u2200 (n : \u2115), L n \u2208 Ioo \u22a5 (\u222b\u207b (a : \u03b1), f a \u2202\u03bc)\nhL_tendsto : Tendsto L atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\n\u22a2 \u2203 g, Measurable g \u2227 g \u2264 f \u2227 \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nhave : \u2200 n, \u2203 g : \u03b1 \u2192 \u211d\u22650\u221e, Measurable g \u2227 g \u2264 f \u2227 L n < \u222b\u207b a, g a \u2202\u03bc :=\n by\n intro n\n simpa only [\u2190 iSup_lintegral_measurable_le_eq_lintegral f, lt_iSup_iff, exists_prop] using (hLf n).2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u2080 : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 0\nL : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictMono L\nhLf : \u2200 (n : \u2115), L n \u2208 Ioo \u22a5 (\u222b\u207b (a : \u03b1), f a \u2202\u03bc)\nhL_tendsto : Tendsto L atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\n\u22a2 \u2200 (n : \u2115), \u2203 g, Measurable g \u2227 g \u2264 f \u2227 L n < \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u2080 : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 0\nL : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictMono L\nhLf : \u2200 (n : \u2115), L n \u2208 Ioo \u22a5 (\u222b\u207b (a : \u03b1), f a \u2202\u03bc)\nhL_tendsto : Tendsto L atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\nn : \u2115\n\u22a2 \u2203 g, Measurable g \u2227 g \u2264 f \u2227 L n < \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nsimpa only [\u2190 iSup_lintegral_measurable_le_eq_lintegral f, lt_iSup_iff, exists_prop] using (hLf n).2\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u2080 : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 0\nL : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictMono L\nhLf : \u2200 (n : \u2115), L n \u2208 Ioo \u22a5 (\u222b\u207b (a : \u03b1), f a \u2202\u03bc)\nhL_tendsto : Tendsto L atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\nthis : \u2200 (n : \u2115), \u2203 g, Measurable g \u2227 g \u2264 f \u2227 L n < \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2203 g, Measurable g \u2227 g \u2264 f \u2227 \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nchoose g hgm hgf hLg using this\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u2080 : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 0\nL : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictMono L\nhLf : \u2200 (n : \u2115), L n \u2208 Ioo \u22a5 (\u222b\u207b (a : \u03b1), f a \u2202\u03bc)\nhL_tendsto : Tendsto L atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhgm : \u2200 (n : \u2115), Measurable (g n)\nhgf : \u2200 (n : \u2115), g n \u2264 f\nhLg : \u2200 (n : \u2115), L n < \u222b\u207b (a : \u03b1), g n a \u2202\u03bc\n\u22a2 \u2203 g, Measurable g \u2227 g \u2264 f \u2227 \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nrefine' \u27e8fun x => \u2a06 n, g n x, measurable_iSup hgm, fun x => iSup_le fun n => hgf n x, le_antisymm _ _\u27e9\n[GOAL]\ncase inr.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u2080 : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 0\nL : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictMono L\nhLf : \u2200 (n : \u2115), L n \u2208 Ioo \u22a5 (\u222b\u207b (a : \u03b1), f a \u2202\u03bc)\nhL_tendsto : Tendsto L atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhgm : \u2200 (n : \u2115), Measurable (g n)\nhgf : \u2200 (n : \u2115), g n \u2264 f\nhLg : \u2200 (n : \u2115), L n < \u222b\u207b (a : \u03b1), g n a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), (fun x => \u2a06 (n : \u2115), g n x) a \u2202\u03bc\n[PROOFSTEP]\nrefine' le_of_tendsto' hL_tendsto fun n => (hLg n).le.trans <| lintegral_mono fun x => _\n[GOAL]\ncase inr.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u2080 : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 0\nL : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictMono L\nhLf : \u2200 (n : \u2115), L n \u2208 Ioo \u22a5 (\u222b\u207b (a : \u03b1), f a \u2202\u03bc)\nhL_tendsto : Tendsto L atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhgm : \u2200 (n : \u2115), Measurable (g n)\nhgf : \u2200 (n : \u2115), g n \u2264 f\nhLg : \u2200 (n : \u2115), L n < \u222b\u207b (a : \u03b1), g n a \u2202\u03bc\nn : \u2115\nx : \u03b1\n\u22a2 g n x \u2264 (fun x => \u2a06 (n : \u2115), g n x) x\n[PROOFSTEP]\nexact le_iSup (fun n => g n x) n\n[GOAL]\ncase inr.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u2080 : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 0\nL : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictMono L\nhLf : \u2200 (n : \u2115), L n \u2208 Ioo \u22a5 (\u222b\u207b (a : \u03b1), f a \u2202\u03bc)\nhL_tendsto : Tendsto L atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhgm : \u2200 (n : \u2115), Measurable (g n)\nhgf : \u2200 (n : \u2115), g n \u2264 f\nhLg : \u2200 (n : \u2115), L n < \u222b\u207b (a : \u03b1), g n a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), (fun x => \u2a06 (n : \u2115), g n x) a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nexact lintegral_mono fun x => iSup_le fun n => hgf n x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc =\n \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc\n[PROOFSTEP]\nrw [lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u22a2 \u2a06 (g : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g \u2264 fun a => f a), SimpleFunc.lintegral g \u03bc =\n \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc\n[PROOFSTEP]\nrefine' le_antisymm (iSup\u2082_le fun \u03c6 h\u03c6 => _) (iSup_mono' fun \u03c6 => \u27e8\u03c6.map ((\u2191) : \u211d\u22650 \u2192 \u211d\u22650\u221e), le_rfl\u27e9)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc\n[PROOFSTEP]\nby_cases h : \u2200\u1d50 a \u2202\u03bc, \u03c6 a \u2260 \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc\n[PROOFSTEP]\nlet \u03c8 := \u03c6.map ENNReal.toNNReal\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map ENNReal.toNNReal \u03c6\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc\n[PROOFSTEP]\nreplace h : \u03c8.map ((\u2191) : \u211d\u22650 \u2192 \u211d\u22650\u221e) =\u1d50[\u03bc] \u03c6 := h.mono fun a => ENNReal.coe_toNNReal\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map ENNReal.toNNReal \u03c6\nh : \u2191(SimpleFunc.map ENNReal.some \u03c8) =\u1d50[\u03bc] \u2191\u03c6\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc\n[PROOFSTEP]\nhave : \u2200 x, \u2191(\u03c8 x) \u2264 f x := fun x => le_trans ENNReal.coe_toNNReal_le_self (h\u03c6 x)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map ENNReal.toNNReal \u03c6\nh : \u2191(SimpleFunc.map ENNReal.some \u03c8) =\u1d50[\u03bc] \u2191\u03c6\nthis : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc\n[PROOFSTEP]\nexact le_iSup_of_le (\u03c6.map ENNReal.toNNReal) (le_iSup_of_le this (ge_of_eq <| lintegral_congr h))\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc\n[PROOFSTEP]\nhave h_meas : \u03bc (\u03c6 \u207b\u00b9' {\u221e}) \u2260 0 := mt measure_zero_iff_ae_nmem.1 h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\n\u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264\n \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc\n[PROOFSTEP]\nrefine' le_trans le_top (ge_of_eq <| (iSup_eq_top _).2 fun b hb => _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\n\u22a2 \u2203 i, b < \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : \u2203 n : \u2115, b < n * \u03bc (\u03c6 \u207b\u00b9' {\u221e})\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\n\u22a2 \u2203 n, b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\n\u22a2 \u2203 i, b < \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc\n[PROOFSTEP]\nexact exists_nat_mul_gt h_meas (ne_of_lt hb)\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\n\u22a2 \u2203 i, b < \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc\n[PROOFSTEP]\nuse(const \u03b1 (n : \u211d\u22650)).restrict (\u03c6 \u207b\u00b9' {\u221e})\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\n\u22a2 b <\n \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191(restrict (const \u03b1 \u2191n) (\u2191\u03c6 \u207b\u00b9' {\u22a4})) x) \u2264 f x),\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (restrict (const \u03b1 \u2191n) (\u2191\u03c6 \u207b\u00b9' {\u22a4}))) \u03bc\n[PROOFSTEP]\nsimp only [lt_iSup_iff, exists_prop, coe_restrict, \u03c6.measurableSet_preimage, coe_const, ENNReal.coe_indicator,\n map_coe_ennreal_restrict, SimpleFunc.map_const, ENNReal.coe_nat, restrict_const_lintegral]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\n\u22a2 (\u2200 (x : \u03b1), indicator (\u2191\u03c6 \u207b\u00b9' {\u22a4}) (fun x => \u2191(Function.const \u03b1 (\u2191n) x)) x \u2264 f x) \u2227 b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\n[PROOFSTEP]\nrefine' \u27e8indicator_le fun x hx => le_trans _ (h\u03c6 _), hn\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\nx : \u03b1\nhx : x \u2208 \u2191\u03c6 \u207b\u00b9' {\u22a4}\n\u22a2 \u2191(Function.const \u03b1 (\u2191n) x) \u2264 \u2191\u03c6 x\n[PROOFSTEP]\nsimp only [mem_preimage, mem_singleton_iff] at hx \n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nh : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4\nh_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0\nb : \u211d\u22650\u221e\nhb : b < \u22a4\nn : \u2115\nhn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})\nx : \u03b1\nhx : \u2191\u03c6 x = \u22a4\n\u22a2 \u2191(Function.const \u03b1 (\u2191n) x) \u2264 \u2191\u03c6 x\n[PROOFSTEP]\nsimp only [hx, le_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 \u03c6,\n (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\n[PROOFSTEP]\nrw [lintegral_eq_nnreal] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 \u03c6,\n (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\n[PROOFSTEP]\nhave := ENNReal.lt_add_right h h\u03b5\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis :\n \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc <\n (\u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc) + \u03b5\n\u22a2 \u2203 \u03c6,\n (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\n[PROOFSTEP]\nerw [ENNReal.biSup_add] at this <;> [skip; exact \u27e80, fun x => zero_le _\u27e9]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis :\n \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc <\n (\u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc) + \u03b5\n\u22a2 \u2203 \u03c6,\n (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\n[PROOFSTEP]\nerw [ENNReal.biSup_add] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis :\n \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc <\n \u2a06 (i : \u03b1 \u2192\u209b \u211d\u22650) (_ : i \u2208 fun \u03c6 => \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x),\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc + \u03b5\n\u22a2 \u2203 \u03c6,\n (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\n[PROOFSTEP]\nskip\n[GOAL]\ncase hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis :\n \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc <\n (\u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc) + \u03b5\n\u22a2 Set.Nonempty fun \u03c6 => \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\n[PROOFSTEP]\nexact \u27e80, fun x => zero_le _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis :\n \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc <\n \u2a06 (i : \u03b1 \u2192\u209b \u211d\u22650) (_ : i \u2208 fun \u03c6 => \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x),\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc + \u03b5\n\u22a2 \u2203 \u03c6,\n (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\n[PROOFSTEP]\nsimp_rw [lt_iSup_iff, iSup_lt_iff, iSup_le_iff] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis :\n \u2203 i h b,\n b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc + \u03b5 \u2227\n \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u22a2 \u2203 \u03c6,\n (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\n[PROOFSTEP]\nrcases this with \u27e8\u03c6, hle : \u2200 x, \u2191(\u03c6 x) \u2264 f x, b, hb\u03c6, hb\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u22a2 \u2203 \u03c6,\n (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227\n \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\n[PROOFSTEP]\nrefine' \u27e8\u03c6, hle, fun \u03c8 h\u03c8 => _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\n[PROOFSTEP]\nhave : (map (\u2191) \u03c6).lintegral \u03bc \u2260 \u221e := ne_top_of_le_ne_top h (by exact le_iSup\u2082 (\u03b1 := \u211d\u22650\u221e) \u03c6 hle)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2264\n \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc\n[PROOFSTEP]\nexact le_iSup\u2082 (\u03b1 := \u211d\u22650\u221e) \u03c6 hle\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\n[PROOFSTEP]\nrw [\u2190 ENNReal.add_lt_add_iff_left this, \u2190 add_lintegral, \u2190 SimpleFunc.map_add @ENNReal.coe_add]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c6 + (\u03c8 - \u03c6))) \u03bc <\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\n[PROOFSTEP]\nrefine' (hb _ fun x => le_trans _ (max_le (hle x) (h\u03c8 x))).trans_lt hb\u03c6\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\nx : \u03b1\n\u22a2 \u2191(\u2191(\u03c6 + (\u03c8 - \u03c6)) x) \u2264 max \u2191(\u2191\u03c6 x) \u2191(\u2191\u03c8 x)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\nx : \u03b1\n\u22a2 \u2191(\u03c6 + (\u03c8 - \u03c6)) x \u2264 max (\u2191\u03c6 x) (\u2191\u03c8 x)\n[PROOFSTEP]\nsimp only [add_apply, sub_apply, add_tsub_eq_max]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nhle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nb : \u211d\u22650\u221e\nhb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5\nhb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4\nx : \u03b1\n\u22a2 max (\u2191\u03c6 x) (\u2191\u03c8 x) \u2264 max (\u2191\u03c6 x) (\u2191\u03c8 x)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Sort u_5\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 (i : \u03b9), \u222b\u207b (a : \u03b1), f i a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2a06 (i : \u03b9), f i a \u2202\u03bc\n[PROOFSTEP]\nsimp only [\u2190 iSup_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Sort u_5\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 (i : \u03b9), \u222b\u207b (a : \u03b1), f i a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc\n[PROOFSTEP]\nexact (monotone_lintegral \u03bc).le_map_iSup\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Sort u_5\n\u03b9' : \u03b9 \u2192 Sort u_6\nf : (i : \u03b9) \u2192 \u03b9' i \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 (i : \u03b9) (j : \u03b9' i), \u222b\u207b (a : \u03b1), f i j a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2a06 (i : \u03b9) (j : \u03b9' i), f i j a \u2202\u03bc\n[PROOFSTEP]\nconvert (monotone_lintegral \u03bc).le_map_iSup\u2082 f with a\n[GOAL]\ncase h.e'_4.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Sort u_5\n\u03b9' : \u03b9 \u2192 Sort u_6\nf : (i : \u03b9) \u2192 \u03b9' i \u2192 \u03b1 \u2192 \u211d\u22650\u221e\na : \u03b1\n\u22a2 \u2a06 (i : \u03b9) (j : \u03b9' i), f i j a = iSup (fun i => \u2a06 (j : \u03b9' i), f i j) a\n[PROOFSTEP]\nsimp only [iSup_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Sort u_5\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), \u2a05 (i : \u03b9), f i a \u2202\u03bc \u2264 \u2a05 (i : \u03b9), \u222b\u207b (a : \u03b1), f i a \u2202\u03bc\n[PROOFSTEP]\nsimp only [\u2190 iInf_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Sort u_5\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), iInf (fun i => f i) a \u2202\u03bc \u2264 \u2a05 (i : \u03b9), \u222b\u207b (a : \u03b1), f i a \u2202\u03bc\n[PROOFSTEP]\nexact (monotone_lintegral \u03bc).map_iInf_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Sort u_5\n\u03b9' : \u03b9 \u2192 Sort u_6\nf : (i : \u03b9) \u2192 \u03b9' i \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), \u2a05 (i : \u03b9) (h : \u03b9' i), f i h a \u2202\u03bc \u2264 \u2a05 (i : \u03b9) (h : \u03b9' i), \u222b\u207b (a : \u03b1), f i h a \u2202\u03bc\n[PROOFSTEP]\nconvert (monotone_lintegral \u03bc).map_iInf\u2082_le f with a\n[GOAL]\ncase h.e'_3.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Sort u_5\n\u03b9' : \u03b9 \u2192 Sort u_6\nf : (i : \u03b9) \u2192 \u03b9' i \u2192 \u03b1 \u2192 \u211d\u22650\u221e\na : \u03b1\n\u22a2 \u2a05 (i : \u03b9) (h : \u03b9' i), f i h a = iInf (fun i => \u2a05 (j : \u03b9' i), f i j) a\n[PROOFSTEP]\nsimp only [iInf_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 g a\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nrcases exists_measurable_superset_of_null h with \u27e8t, hts, ht, ht0\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 g a\nt : Set \u03b1\nhts : {x | (fun a => f a \u2264 g a) x}\u1d9c \u2286 t\nht : MeasurableSet t\nht0 : \u2191\u2191\u03bc t = 0\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nhave : \u2200\u1d50 x \u2202\u03bc, x \u2209 t := measure_zero_iff_ae_nmem.1 ht0\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 g a\nt : Set \u03b1\nhts : {x | (fun a => f a \u2264 g a) x}\u1d9c \u2286 t\nht : MeasurableSet t\nht0 : \u2191\u2191\u03bc t = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral, lintegral]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 g a\nt : Set \u03b1\nhts : {x | (fun a => f a \u2264 g a) x}\u1d9c \u2286 t\nht : MeasurableSet t\nht0 : \u2191\u2191\u03bc t = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\n\u22a2 \u2a06 (g : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g \u2264 fun a => f a), SimpleFunc.lintegral g \u03bc \u2264\n \u2a06 (g_1 : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g_1 \u2264 fun a => g a), SimpleFunc.lintegral g_1 \u03bc\n[PROOFSTEP]\nrefine' iSup_le fun s => iSup_le fun hfs => le_iSup_of_le (s.restrict t\u1d9c) <| le_iSup_of_le _ _\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 g a\nt : Set \u03b1\nhts : {x | (fun a => f a \u2264 g a) x}\u1d9c \u2286 t\nht : MeasurableSet t\nht0 : \u2191\u2191\u03bc t = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhfs : \u2191s \u2264 fun a => f a\n\u22a2 \u2191(restrict s t\u1d9c) \u2264 fun a => g a\n[PROOFSTEP]\nintro a\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 g a\nt : Set \u03b1\nhts : {x | (fun a => f a \u2264 g a) x}\u1d9c \u2286 t\nht : MeasurableSet t\nht0 : \u2191\u2191\u03bc t = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhfs : \u2191s \u2264 fun a => f a\na : \u03b1\n\u22a2 \u2191(restrict s t\u1d9c) a \u2264 (fun a => g a) a\n[PROOFSTEP]\nby_cases h : a \u2208 t\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 g a\nt : Set \u03b1\nhts : {x | (fun a => f a \u2264 g a) x}\u1d9c \u2286 t\nht : MeasurableSet t\nht0 : \u2191\u2191\u03bc t = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhfs : \u2191s \u2264 fun a => f a\na : \u03b1\nh : a \u2208 t\n\u22a2 \u2191(restrict s t\u1d9c) a \u2264 (fun a => g a) a\n[PROOFSTEP]\nsimp [h, restrict_apply s ht.compl, ht.compl]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 g a\nt : Set \u03b1\nhts : {x | (fun a => f a \u2264 g a) x}\u1d9c \u2286 t\nht : MeasurableSet t\nht0 : \u2191\u2191\u03bc t = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhfs : \u2191s \u2264 fun a => f a\na : \u03b1\nh : \u00aca \u2208 t\n\u22a2 \u2191(restrict s t\u1d9c) a \u2264 (fun a => g a) a\n[PROOFSTEP]\nsimp [h, restrict_apply s ht.compl, ht.compl]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 g a\nt : Set \u03b1\nhts : {x | (fun a => f a \u2264 g a) x}\u1d9c \u2286 t\nht : MeasurableSet t\nht0 : \u2191\u2191\u03bc t = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhfs : \u2191s \u2264 fun a => f a\na : \u03b1\nh : \u00aca \u2208 t\n\u22a2 \u2191s a \u2264 g a\n[PROOFSTEP]\nexact le_trans (hfs a) (_root_.by_contradiction fun hnfg => h (hts hnfg))\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 g a\nt : Set \u03b1\nhts : {x | (fun a => f a \u2264 g a) x}\u1d9c \u2286 t\nht : MeasurableSet t\nht0 : \u2191\u2191\u03bc t = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhfs : \u2191s \u2264 fun a => f a\n\u22a2 SimpleFunc.lintegral s \u03bc \u2264 SimpleFunc.lintegral (restrict s t\u1d9c) \u03bc\n[PROOFSTEP]\nrefine' le_of_eq (SimpleFunc.lintegral_congr <| this.mono fun a hnt => _)\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 g a\nt : Set \u03b1\nhts : {x | (fun a => f a \u2264 g a) x}\u1d9c \u2286 t\nht : MeasurableSet t\nht0 : \u2191\u2191\u03bc t = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhfs : \u2191s \u2264 fun a => f a\na : \u03b1\nhnt : \u00aca \u2208 t\n\u22a2 \u2191s a = \u2191(restrict s t\u1d9c) a\n[PROOFSTEP]\nby_cases hat : a \u2208 t\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 g a\nt : Set \u03b1\nhts : {x | (fun a => f a \u2264 g a) x}\u1d9c \u2286 t\nht : MeasurableSet t\nht0 : \u2191\u2191\u03bc t = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhfs : \u2191s \u2264 fun a => f a\na : \u03b1\nhnt : \u00aca \u2208 t\nhat : a \u2208 t\n\u22a2 \u2191s a = \u2191(restrict s t\u1d9c) a\n[PROOFSTEP]\nsimp [hat, restrict_apply s ht.compl]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 g a\nt : Set \u03b1\nhts : {x | (fun a => f a \u2264 g a) x}\u1d9c \u2286 t\nht : MeasurableSet t\nht0 : \u2191\u2191\u03bc t = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhfs : \u2191s \u2264 fun a => f a\na : \u03b1\nhnt hat : \u00aca \u2208 t\n\u22a2 \u2191s a = \u2191(restrict s t\u1d9c) a\n[PROOFSTEP]\nsimp [hat, restrict_apply s ht.compl]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 g a\nt : Set \u03b1\nhts : {x | (fun a => f a \u2264 g a) x}\u1d9c \u2286 t\nht : MeasurableSet t\nht0 : \u2191\u2191\u03bc t = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhfs : \u2191s \u2264 fun a => f a\na : \u03b1\nhnt : \u00aca \u2208 t\nhat : a \u2208 t\n\u22a2 \u2191s a = 0\n[PROOFSTEP]\nexact (hnt hat).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (a : \u03b1), f a = g a\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns t : Set \u03b1\nh : s =\u1d50[\u03bc] t\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in t, f x \u2202\u03bc\n[PROOFSTEP]\nrw [Measure.restrict_congr_set h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = g x\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_congr_ae]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = g x\n\u22a2 (fun x => f x) =\u1d50[Measure.restrict \u03bc s] fun a => g a\n[PROOFSTEP]\nrw [EventuallyEq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = g x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, f x = g x\n[PROOFSTEP]\nrwa [ae_restrict_iff' hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\n\u22a2 \u222b\u207b (x : \u03b1), ENNReal.ofReal (f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [\u2190 ofReal_norm_eq_coe_nnnorm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\n\u22a2 \u222b\u207b (x : \u03b1), ENNReal.ofReal (f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), ENNReal.ofReal \u2016f x\u2016 \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_mono fun x => ENNReal.ofReal_le_ofReal _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nx : \u03b1\n\u22a2 f x \u2264 \u2016f x\u2016\n[PROOFSTEP]\nrw [Real.norm_eq_abs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nx : \u03b1\n\u22a2 f x \u2264 |f x|\n[PROOFSTEP]\nexact le_abs_self (f x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nh_nonneg : 0 \u2264\u1d50[\u03bc] f\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202\u03bc = \u222b\u207b (x : \u03b1), ENNReal.ofReal (f x) \u2202\u03bc\n[PROOFSTEP]\napply lintegral_congr_ae\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nh_nonneg : 0 \u2264\u1d50[\u03bc] f\n\u22a2 (fun a => \u2191\u2016f a\u2016\u208a) =\u1d50[\u03bc] fun a => ENNReal.ofReal (f a)\n[PROOFSTEP]\nfilter_upwards [h_nonneg] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nh_nonneg : 0 \u2264\u1d50[\u03bc] f\nx : \u03b1\nhx : OfNat.ofNat 0 x \u2264 f x\n\u22a2 \u2191\u2016f x\u2016\u208a = ENNReal.ofReal (f x)\n[PROOFSTEP]\nrw [Real.nnnorm_of_nonneg hx, ENNReal.ofReal_eq_coe_nnreal hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (n : \u2115), f n a \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nset c : \u211d\u22650 \u2192 \u211d\u22650\u221e := (\u2191)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (n : \u2115), f n a \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nset F := fun a : \u03b1 => \u2a06 n, f n a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\n\u22a2 lintegral \u03bc F = \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nhave _ : Measurable F := measurable_iSup hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\n\u22a2 lintegral \u03bc F = \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nrefine' le_antisymm _ (iSup_lintegral_le _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\n\u22a2 lintegral \u03bc F \u2264 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_eq_nnreal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\n\u22a2 \u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2a06 (n : \u2115), f n x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2264\n \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nrefine' iSup_le fun s => iSup_le fun hsf => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nrefine' ENNReal.le_of_forall_lt_one_mul_le fun a ha => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\na : \u211d\u22650\u221e\nha : a < 1\n\u22a2 a * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nrcases ENNReal.lt_iff_exists_coe.1 ha with \u27e8r, rfl, _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha : \u2191r < 1\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nhave ha : r < 1 := ENNReal.coe_lt_coe.1 ha\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nlet rs := s.map fun a => r * a\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nhave eq_rs : (const \u03b1 r : \u03b1 \u2192\u209b \u211d\u22650\u221e) * map c s = rs.map c :=\n by\n ext1 a\n exact ENNReal.coe_mul.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\n\u22a2 const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\n[PROOFSTEP]\next1 a\n[GOAL]\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\na : \u03b1\n\u22a2 \u2191(const \u03b1 \u2191r * SimpleFunc.map c s) a = \u2191(SimpleFunc.map c rs) a\n[PROOFSTEP]\nexact ENNReal.coe_mul.symm\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nhave eq : \u2200 p, rs.map c \u207b\u00b9' { p } = \u22c3 n, rs.map c \u207b\u00b9' { p } \u2229 {a | p \u2264 f n a} :=\n by\n intro p\n rw [\u2190 inter_iUnion]; nth_rw 1 [\u2190 inter_univ (map c rs \u207b\u00b9' { p })]\n refine' Set.ext fun x => and_congr_right fun hx => true_iff_iff.2 _\n by_cases p_eq : p = 0\n \u00b7 simp [p_eq]\n simp [-ENNReal.coe_mul] at hx \n subst hx\n have : r * s x \u2260 0 := by rwa [Ne, \u2190 ENNReal.coe_eq_zero]\n have : s x \u2260 0 := by\n refine' mt _ this\n intro h\n rw [h, mul_zero]\n have : (rs.map c) x < \u2a06 n : \u2115, f n x :=\n by\n refine' lt_of_lt_of_le (ENNReal.coe_lt_coe.2 _) (hsf x)\n suffices : r * s x < 1 * s x\n simpa\n exact mul_lt_mul_of_pos_right ha (pos_iff_ne_zero.2 this)\n rcases lt_iSup_iff.1 this with \u27e8i, hi\u27e9\n exact mem_iUnion.2 \u27e8i, le_of_lt hi\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\n\u22a2 \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\n[PROOFSTEP]\nintro p\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\n\u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\n[PROOFSTEP]\nrw [\u2190 inter_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\n\u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 \u22c3 (i : \u2115), {a | p \u2264 f i a}\n[PROOFSTEP]\nnth_rw 1 [\u2190 inter_univ (map c rs \u207b\u00b9' { p })]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\n\u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 univ = \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 \u22c3 (i : \u2115), {a | p \u2264 f i a}\n[PROOFSTEP]\nrefine' Set.ext fun x => and_congr_right fun hx => true_iff_iff.2 _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\nhx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p}\n\u22a2 x \u2208 \u22c3 (i : \u2115), {a | p \u2264 f i a}\n[PROOFSTEP]\nby_cases p_eq : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\nhx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p}\np_eq : p = 0\n\u22a2 x \u2208 \u22c3 (i : \u2115), {a | p \u2264 f i a}\n[PROOFSTEP]\nsimp [p_eq]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\nhx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p}\np_eq : \u00acp = 0\n\u22a2 x \u2208 \u22c3 (i : \u2115), {a | p \u2264 f i a}\n[PROOFSTEP]\nsimp [-ENNReal.coe_mul] at hx \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\np_eq : \u00acp = 0\nhx : \u2191(r * \u2191s x) = p\n\u22a2 x \u2208 \u22c3 (i : \u2115), {a | p \u2264 f i a}\n[PROOFSTEP]\nsubst hx\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\n\u22a2 x \u2208 \u22c3 (i : \u2115), {a | \u2191(r * \u2191s x) \u2264 f i a}\n[PROOFSTEP]\nhave : r * s x \u2260 0 := by rwa [Ne, \u2190 ENNReal.coe_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\n\u22a2 r * \u2191s x \u2260 0\n[PROOFSTEP]\nrwa [Ne, \u2190 ENNReal.coe_eq_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\n\u22a2 x \u2208 \u22c3 (i : \u2115), {a | \u2191(r * \u2191s x) \u2264 f i a}\n[PROOFSTEP]\nhave : s x \u2260 0 := by\n refine' mt _ this\n intro h\n rw [h, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\n\u22a2 \u2191s x \u2260 0\n[PROOFSTEP]\nrefine' mt _ this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\n\u22a2 \u2191s x = 0 \u2192 r * \u2191s x = 0\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\nh : \u2191s x = 0\n\u22a2 r * \u2191s x = 0\n[PROOFSTEP]\nrw [h, mul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 x \u2208 \u22c3 (i : \u2115), {a | \u2191(r * \u2191s x) \u2264 f i a}\n[PROOFSTEP]\nhave : (rs.map c) x < \u2a06 n : \u2115, f n x :=\n by\n refine' lt_of_lt_of_le (ENNReal.coe_lt_coe.2 _) (hsf x)\n suffices : r * s x < 1 * s x\n simpa\n exact mul_lt_mul_of_pos_right ha (pos_iff_ne_zero.2 this)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 \u2191(SimpleFunc.map c rs) x < \u2a06 (n : \u2115), f n x\n[PROOFSTEP]\nrefine' lt_of_lt_of_le (ENNReal.coe_lt_coe.2 _) (hsf x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 \u2191rs x < \u2191s x\n[PROOFSTEP]\nsuffices : r * s x < 1 * s x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d\u00b9 : r * \u2191s x \u2260 0\nthis\u271d : \u2191s x \u2260 0\nthis : r * \u2191s x < 1 * \u2191s x\n\u22a2 \u2191rs x < \u2191s x\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 r * \u2191s x < 1 * \u2191s x\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 r * \u2191s x < 1 * \u2191s x\n[PROOFSTEP]\nexact mul_lt_mul_of_pos_right ha (pos_iff_ne_zero.2 this)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d\u00b9 : r * \u2191s x \u2260 0\nthis\u271d : \u2191s x \u2260 0\nthis : \u2191(SimpleFunc.map c rs) x < \u2a06 (n : \u2115), f n x\n\u22a2 x \u2208 \u22c3 (i : \u2115), {a | \u2191(r * \u2191s x) \u2264 f i a}\n[PROOFSTEP]\nrcases lt_iSup_iff.1 this with \u27e8i, hi\u27e9\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d\u00b9 : r * \u2191s x \u2260 0\nthis\u271d : \u2191s x \u2260 0\nthis : \u2191(SimpleFunc.map c rs) x < \u2a06 (n : \u2115), f n x\ni : \u2115\nhi : \u2191(SimpleFunc.map c rs) x < f i x\n\u22a2 x \u2208 \u22c3 (i : \u2115), {a | \u2191(r * \u2191s x) \u2264 f i a}\n[PROOFSTEP]\nexact mem_iUnion.2 \u27e8i, le_of_lt hi\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nhave mono : \u2200 r : \u211d\u22650\u221e, Monotone fun n => rs.map c \u207b\u00b9' { r } \u2229 {a | r \u2264 f n a} :=\n by\n intro r i j h\n refine' inter_subset_inter (Subset.refl _) _\n intro x (hx : r \u2264 f i x)\n exact le_trans hx (h_mono h x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\n\u22a2 \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\n[PROOFSTEP]\nintro r i j h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nr : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\n\u22a2 (fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) i \u2264\n (fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) j\n[PROOFSTEP]\nrefine' inter_subset_inter (Subset.refl _) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nr : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\n\u22a2 {a | r \u2264 f i a} \u2286 {a | r \u2264 f j a}\n[PROOFSTEP]\nintro x (hx : r \u2264 f i x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nr : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\nx : \u03b1\nhx : r \u2264 f i x\n\u22a2 x \u2208 {a | r \u2264 f j a}\n[PROOFSTEP]\nexact le_trans hx (h_mono h x)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nhave h_meas : \u2200 n, MeasurableSet {a : \u03b1 | (\u21d1(map c rs)) a \u2264 f n a} := fun n =>\n measurableSet_le (SimpleFunc.measurable _) (hf n)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\ncalc\n (r : \u211d\u22650\u221e) * (s.map c).lintegral \u03bc = \u2211 r in (rs.map c).range, r * \u03bc (rs.map c \u207b\u00b9' { r }) := by\n rw [\u2190 const_mul_lintegral, eq_rs, SimpleFunc.lintegral]\n _ = \u2211 r in (rs.map c).range, r * \u03bc (\u22c3 n, rs.map c \u207b\u00b9' { r } \u2229 {a | r \u2264 f n a}) := by simp only [(eq _).symm]\n _ = \u2211 r in (rs.map c).range, \u2a06 n, r * \u03bc (rs.map c \u207b\u00b9' { r } \u2229 {a | r \u2264 f n a}) :=\n (Finset.sum_congr rfl fun x _ => by rw [measure_iUnion_eq_iSup (directed_of_sup <| mono x), ENNReal.mul_iSup])\n _ = \u2a06 n, \u2211 r in (rs.map c).range, r * \u03bc (rs.map c \u207b\u00b9' { r } \u2229 {a | r \u2264 f n a}) :=\n by\n rw [ENNReal.finset_sum_iSup_nat]\n intro p i j h\n exact mul_le_mul_left' (measure_mono <| mono p h) _\n _ \u2264 \u2a06 n : \u2115, ((rs.map c).restrict {a | (rs.map c) a \u2264 f n a}).lintegral \u03bc :=\n by\n refine' iSup_mono fun n => _\n rw [restrict_lintegral _ (h_meas n)]\n \u00b7 refine' le_of_eq (Finset.sum_congr rfl fun r _ => _)\n congr 2 with a\n refine' and_congr_right _\n simp (config := { contextual := true })\n _ \u2264 \u2a06 n, \u222b\u207b a, f n a \u2202\u03bc := by\n refine' iSup_mono fun n => _\n rw [\u2190 SimpleFunc.lintegral_eq_lintegral]\n refine' lintegral_mono fun a => _\n simp only [map_apply] at h_meas \n simp only [coe_map, restrict_apply _ (h_meas _), (\u00b7 \u2218 \u00b7)]\n exact indicator_apply_le id\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map c s) \u03bc =\n \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r})\n[PROOFSTEP]\nrw [\u2190 const_mul_lintegral, eq_rs, SimpleFunc.lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r}) =\n \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a})\n[PROOFSTEP]\nsimp only [(eq _).symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nx : \u211d\u22650\u221e\nx\u271d : x \u2208 SimpleFunc.range (SimpleFunc.map c rs)\n\u22a2 x * \u2191\u2191\u03bc (\u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {x} \u2229 {a | x \u2264 f n a}) =\n \u2a06 (n : \u2115), x * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {x} \u2229 {a | x \u2264 f n a})\n[PROOFSTEP]\nrw [measure_iUnion_eq_iSup (directed_of_sup <| mono x), ENNReal.mul_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), \u2a06 (n : \u2115), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) =\n \u2a06 (n : \u2115), \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a})\n[PROOFSTEP]\nrw [ENNReal.finset_sum_iSup_nat]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2200 (a : \u211d\u22650\u221e), Monotone fun n => a * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {a} \u2229 {a_1 | a \u2264 f n a_1})\n[PROOFSTEP]\nintro p i j h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\np : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\n\u22a2 (fun n => p * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a})) i \u2264\n (fun n => p * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a})) j\n[PROOFSTEP]\nexact mul_le_mul_left' (measure_mono <| mono p h) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2a06 (n : \u2115), \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264\n \u2a06 (n : \u2115), SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc\n[PROOFSTEP]\nrefine' iSup_mono fun n => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264\n SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc\n[PROOFSTEP]\nrw [restrict_lintegral _ (h_meas n)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264\n \u2211 r in SimpleFunc.range (SimpleFunc.map c rs),\n r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a})\n[PROOFSTEP]\nrefine' le_of_eq (Finset.sum_congr rfl fun r _ => _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\nr : \u211d\u22650\u221e\nx\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs)\n\u22a2 r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) =\n r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a})\n[PROOFSTEP]\ncongr 2 with a\n[GOAL]\ncase e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\nr : \u211d\u22650\u221e\nx\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs)\na : \u03b1\n\u22a2 a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} \u2194\n a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n[PROOFSTEP]\nrefine' and_congr_right _\n[GOAL]\ncase e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\nr : \u211d\u22650\u221e\nx\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs)\na : \u03b1\n\u22a2 a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2192 (a \u2208 {a | r \u2264 f n a} \u2194 a \u2208 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a})\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2a06 (n : \u2115), SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc \u2264\n \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nrefine' iSup_mono fun n => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc \u2264 \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 SimpleFunc.lintegral_eq_lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_mono fun a => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\na : \u03b1\n\u22a2 \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2264 f n a\n[PROOFSTEP]\nsimp only [map_apply] at h_meas \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(r * \u2191s a) \u2264 f n a}\nn : \u2115\na : \u03b1\n\u22a2 \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2264 f n a\n[PROOFSTEP]\nsimp only [coe_map, restrict_apply _ (h_meas _), (\u00b7 \u2218 \u00b7)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 (n : \u2115), f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 (n : \u2115), f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 (n : \u2115), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(r * \u2191s a) \u2264 f n a}\nn : \u2115\na : \u03b1\n\u22a2 indicator {a | \u2191(r * \u2191s a) \u2264 f n a} (fun x => \u2191(r * \u2191s x)) a \u2264 f n a\n[PROOFSTEP]\nexact indicator_apply_le id\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (n : \u2115), f n a \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [\u2190 iSup_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nlet p : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun _ f' => Monotone f'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\np : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f'\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nhave hp : \u2200\u1d50 x \u2202\u03bc, p x fun i => f i x := h_mono\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\np : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nhave h_ae_seq_mono : Monotone (aeSeq hf p) := by\n intro n m hnm x\n by_cases hx : x \u2208 aeSeqSet hf p\n \u00b7 exact aeSeq.prop_of_mem_aeSeqSet hf hx hnm\n \u00b7 simp only [aeSeq, hx, if_false]\n exact le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\np : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\n\u22a2 Monotone (aeSeq hf p)\n[PROOFSTEP]\nintro n m hnm x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\np : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nn m : \u2115\nhnm : n \u2264 m\nx : \u03b1\n\u22a2 aeSeq hf p n x \u2264 aeSeq hf p m x\n[PROOFSTEP]\nby_cases hx : x \u2208 aeSeqSet hf p\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\np : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nn m : \u2115\nhnm : n \u2264 m\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 aeSeq hf p n x \u2264 aeSeq hf p m x\n[PROOFSTEP]\nexact aeSeq.prop_of_mem_aeSeqSet hf hx hnm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\np : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nn m : \u2115\nhnm : n \u2264 m\nx : \u03b1\nhx : \u00acx \u2208 aeSeqSet hf p\n\u22a2 aeSeq hf p n x \u2264 aeSeq hf p m x\n[PROOFSTEP]\nsimp only [aeSeq, hx, if_false]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\np : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nn m : \u2115\nhnm : n \u2264 m\nx : \u03b1\nhx : \u00acx \u2208 aeSeqSet hf p\n\u22a2 Nonempty.some (_ : Nonempty \u211d\u22650\u221e) \u2264 Nonempty.some (_ : Nonempty \u211d\u22650\u221e)\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\np : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_mono : Monotone (aeSeq hf p)\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_congr_ae (aeSeq.iSup hf hp).symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\np : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_mono : Monotone (aeSeq hf p)\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun n => aeSeq hf (fun x => p x) n) a \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [iSup_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\np : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_mono : Monotone (aeSeq hf p)\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (i : \u2115), aeSeq hf (fun x f' => Monotone f') i a \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nrw [@lintegral_iSup _ _ \u03bc _ (aeSeq.measurable hf p) h_ae_seq_mono]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\np : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_mono : Monotone (aeSeq hf p)\n\u22a2 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), aeSeq hf p n a \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\np : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_mono : Monotone (aeSeq hf p)\n\u22a2 (fun n => \u222b\u207b (a : \u03b1), aeSeq hf p n a \u2202\u03bc) = fun n => \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nexact funext fun n => lintegral_congr_ae (aeSeq.aeSeq_n_eq_fun_n_ae hf hp n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nF : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (F x))\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b1), f n x \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), F x \u2202\u03bc))\n[PROOFSTEP]\nhave : Monotone fun n => \u222b\u207b x, f n x \u2202\u03bc := fun i j hij => lintegral_mono_ae (h_mono.mono fun x hx => hx hij)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nF : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (F x))\nthis : Monotone fun n => \u222b\u207b (x : \u03b1), f n x \u2202\u03bc\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b1), f n x \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), F x \u2202\u03bc))\n[PROOFSTEP]\nsuffices key : \u222b\u207b x, F x \u2202\u03bc = \u2a06 n, \u222b\u207b x, f n x \u2202\u03bc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nF : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (F x))\nthis : Monotone fun n => \u222b\u207b (x : \u03b1), f n x \u2202\u03bc\nkey : \u222b\u207b (x : \u03b1), F x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), f n x \u2202\u03bc\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b1), f n x \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), F x \u2202\u03bc))\n[PROOFSTEP]\nrw [key]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nF : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (F x))\nthis : Monotone fun n => \u222b\u207b (x : \u03b1), f n x \u2202\u03bc\nkey : \u222b\u207b (x : \u03b1), F x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), f n x \u2202\u03bc\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b1), f n x \u2202\u03bc) atTop (\ud835\udcdd (\u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), f n x \u2202\u03bc))\n[PROOFSTEP]\nexact tendsto_atTop_iSup this\n[GOAL]\ncase key\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nF : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (F x))\nthis : Monotone fun n => \u222b\u207b (x : \u03b1), f n x \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), F x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), f n x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lintegral_iSup' hf h_mono]\n[GOAL]\ncase key\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nF : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (F x))\nthis : Monotone fun n => \u222b\u207b (x : \u03b1), f n x \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), F x \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2a06 (n : \u2115), f n a \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_congr_ae _\n[GOAL]\ncase key\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nF : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), AEMeasurable (f n)\nh_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (F x))\nthis : Monotone fun n => \u222b\u207b (x : \u03b1), f n x \u2202\u03bc\n\u22a2 (fun x => F x) =\u1d50[\u03bc] fun a => \u2a06 (n : \u2115), f n a\n[PROOFSTEP]\nfilter_upwards [h_mono, h_tendsto] with _ hx_mono hx_tendsto using\n tendsto_nhds_unique hx_tendsto (tendsto_atTop_iSup hx_mono)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2a06 (n : \u2115), \u2191(eapprox f n) a \u2202\u03bc\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 (fun a => f a) = fun a => \u2a06 (n : \u2115), \u2191(eapprox f n) a\n[PROOFSTEP]\next a\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\n\u22a2 f a = \u2a06 (n : \u2115), \u2191(eapprox f n) a\n[PROOFSTEP]\nrw [iSup_eapprox_apply f hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (n : \u2115), \u2191(eapprox f n) a \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), \u2191(eapprox f n) a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_iSup]\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u2200 (n : \u2115), Measurable fun a => \u2191(eapprox f n) a\n[PROOFSTEP]\nmeasurability\n[GOAL]\ncase h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 Monotone fun n a => \u2191(eapprox f n) a\n[PROOFSTEP]\nintro i j h\n[GOAL]\ncase h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ni j : \u2115\nh : i \u2264 j\n\u22a2 (fun n a => \u2191(eapprox f n) a) i \u2264 (fun n a => \u2191(eapprox f n) a) j\n[PROOFSTEP]\nexact monotone_eapprox f h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), \u2191(eapprox f n) a \u2202\u03bc = \u2a06 (n : \u2115), SimpleFunc.lintegral (eapprox f n) \u03bc\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 (fun n => \u222b\u207b (a : \u03b1), \u2191(eapprox f n) a \u2202\u03bc) = fun n => SimpleFunc.lintegral (eapprox f n) \u03bc\n[PROOFSTEP]\next n\n[GOAL]\ncase e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nn : \u2115\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(eapprox f n) a \u2202\u03bc = SimpleFunc.lintegral (eapprox f n) \u03bc\n[PROOFSTEP]\nrw [(eapprox f n).lintegral_eq_lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5\n[PROOFSTEP]\nrcases exists_between h\u03b5.bot_lt with \u27e8\u03b5\u2082, h\u03b5\u20820 : 0 < \u03b5\u2082, h\u03b5\u2082\u03b5\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5\n[PROOFSTEP]\nrcases exists_between h\u03b5\u20820 with \u27e8\u03b5\u2081, h\u03b5\u20810, h\u03b5\u2081\u2082\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5\n[PROOFSTEP]\nrcases exists_simpleFunc_forall_lintegral_sub_lt_of_pos h h\u03b5\u20810.ne' with \u27e8\u03c6, _, h\u03c6\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5\n[PROOFSTEP]\nrcases \u03c6.exists_forall_le with \u27e8C, hC\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5\n[PROOFSTEP]\nuse(\u03b5\u2082 - \u03b5\u2081) / C, ENNReal.div_pos_iff.2 \u27e8(tsub_pos_iff_lt.2 h\u03b5\u2081\u2082).ne', ENNReal.coe_ne_top\u27e9\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\n\u22a2 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5\n[PROOFSTEP]\nrefine' fun s hs => lt_of_le_of_lt _ h\u03b5\u2082\u03b5\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u03b5\u2082\n[PROOFSTEP]\nsimp only [lintegral_eq_nnreal, iSup_le_iff]\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u22a2 \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650),\n (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) (Measure.restrict \u03bc s) \u2264 \u03b5\u2082\n[PROOFSTEP]\nintro \u03c8 h\u03c8\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c8) (Measure.restrict \u03bc s) \u2264 \u03b5\u2082\n[PROOFSTEP]\ncalc\n (map (\u2191) \u03c8).lintegral (\u03bc.restrict s) \u2264\n (map (\u2191) \u03c6).lintegral (\u03bc.restrict s) + (map (\u2191) (\u03c8 - \u03c6)).lintegral (\u03bc.restrict s) :=\n by\n rw [\u2190 SimpleFunc.add_lintegral, \u2190 SimpleFunc.map_add @ENNReal.coe_add]\n refine' SimpleFunc.lintegral_mono (fun x => _) le_rfl\n simp only [add_tsub_eq_max, le_max_right, coe_map, Function.comp_apply, SimpleFunc.coe_add, SimpleFunc.coe_sub,\n Pi.add_apply, Pi.sub_apply, WithTop.coe_max (\u03c6 x) (\u03c8 x), ENNReal.some]\n _ \u2264 (map (\u2191) \u03c6).lintegral (\u03bc.restrict s) + \u03b5\u2081 :=\n by\n refine' add_le_add le_rfl (le_trans _ (h\u03c6 _ h\u03c8).le)\n exact SimpleFunc.lintegral_mono le_rfl Measure.restrict_le_self\n _ \u2264 (SimpleFunc.const \u03b1 (C : \u211d\u22650\u221e)).lintegral (\u03bc.restrict s) + \u03b5\u2081 :=\n (add_le_add (SimpleFunc.lintegral_mono (fun x => by exact coe_le_coe.2 (hC x)) le_rfl) le_rfl)\n _ = C * \u03bc s + \u03b5\u2081 := by\n simp only [\u2190 SimpleFunc.lintegral_eq_lintegral, coe_const, lintegral_const, Measure.restrict_apply,\n MeasurableSet.univ, univ_inter, Function.const]\n _ \u2264 C * ((\u03b5\u2082 - \u03b5\u2081) / C) + \u03b5\u2081 := by gcongr\n _ \u2264 \u03b5\u2082 - \u03b5\u2081 + \u03b5\u2081 := by gcongr; apply mul_div_le\n _ = \u03b5\u2082 := tsub_add_cancel_of_le h\u03b5\u2081\u2082.le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c8) (Measure.restrict \u03bc s) \u2264\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) (Measure.restrict \u03bc s) +\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) (Measure.restrict \u03bc s)\n[PROOFSTEP]\nrw [\u2190 SimpleFunc.add_lintegral, \u2190 SimpleFunc.map_add @ENNReal.coe_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c8) (Measure.restrict \u03bc s) \u2264\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c6 + (\u03c8 - \u03c6))) (Measure.restrict \u03bc s)\n[PROOFSTEP]\nrefine' SimpleFunc.lintegral_mono (fun x => _) le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nx : \u03b1\n\u22a2 \u2191(SimpleFunc.map ENNReal.some \u03c8) x \u2264 \u2191(SimpleFunc.map ENNReal.some (\u03c6 + (\u03c8 - \u03c6))) x\n[PROOFSTEP]\nsimp only [add_tsub_eq_max, le_max_right, coe_map, Function.comp_apply, SimpleFunc.coe_add, SimpleFunc.coe_sub,\n Pi.add_apply, Pi.sub_apply, WithTop.coe_max (\u03c6 x) (\u03c8 x), ENNReal.some]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) (Measure.restrict \u03bc s) +\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) (Measure.restrict \u03bc s) \u2264\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) (Measure.restrict \u03bc s) + \u03b5\u2081\n[PROOFSTEP]\nrefine' add_le_add le_rfl (le_trans _ (h\u03c6 _ h\u03c8).le)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) (Measure.restrict \u03bc s) \u2264\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc\n[PROOFSTEP]\nexact SimpleFunc.lintegral_mono le_rfl Measure.restrict_le_self\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\nx : \u03b1\n\u22a2 \u2191(SimpleFunc.map ENNReal.some \u03c6) x \u2264 \u2191(const \u03b1 \u2191C) x\n[PROOFSTEP]\nexact coe_le_coe.2 (hC x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 SimpleFunc.lintegral (const \u03b1 \u2191C) (Measure.restrict \u03bc s) + \u03b5\u2081 = \u2191C * \u2191\u2191\u03bc s + \u03b5\u2081\n[PROOFSTEP]\nsimp only [\u2190 SimpleFunc.lintegral_eq_lintegral, coe_const, lintegral_const, Measure.restrict_apply, MeasurableSet.univ,\n univ_inter, Function.const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 \u2191C * \u2191\u2191\u03bc s + \u03b5\u2081 \u2264 \u2191C * ((\u03b5\u2082 - \u03b5\u2081) / \u2191C) + \u03b5\u2081\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 \u2191C * ((\u03b5\u2082 - \u03b5\u2081) / \u2191C) + \u03b5\u2081 \u2264 \u03b5\u2082 - \u03b5\u2081 + \u03b5\u2081\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b5\u2082 : \u211d\u22650\u221e\nh\u03b5\u20820 : 0 < \u03b5\u2082\nh\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5\n\u03b5\u2081 : \u211d\u22650\u221e\nh\u03b5\u20810 : 0 < \u03b5\u2081\nh\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\nleft\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x\nh\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081\nC : \u211d\u22650\nhC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C\n\u03c8 : \u03b1 \u2192\u209b \u211d\u22650\nh\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x\n\u22a2 \u2191C * ((\u03b5\u2082 - \u03b5\u2081) / \u2191C) \u2264 \u03b5\u2082 - \u03b5\u2081\n[PROOFSTEP]\napply mul_div_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhl : Tendsto (\u2191\u2191\u03bc \u2218 s) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1) in s i, f x \u2202\u03bc) l (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp only [ENNReal.nhds_zero, tendsto_iInf, tendsto_principal, mem_Iio, \u2190 pos_iff_ne_zero] at hl \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhl : \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, (\u2191\u2191\u03bc \u2218 s) a < i\n\u22a2 \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, \u222b\u207b (x : \u03b1) in s a, f x \u2202\u03bc < i\n[PROOFSTEP]\nintro \u03b5 \u03b50\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhl : \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, (\u2191\u2191\u03bc \u2218 s) a < i\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (a : \u03b9) in l, \u222b\u207b (x : \u03b1) in s a, f x \u2202\u03bc < \u03b5\n[PROOFSTEP]\nrcases exists_pos_set_lintegral_lt_of_measure_lt h \u03b50.ne' with \u27e8\u03b4, \u03b40, h\u03b4\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nl : Filter \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhl : \u2200 (i : \u211d\u22650\u221e), 0 < i \u2192 \u2200\u1da0 (a : \u03b9) in l, (\u2191\u2191\u03bc \u2218 s) a < i\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\u22650\u221e\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5\n\u22a2 \u2200\u1da0 (a : \u03b9) in l, \u222b\u207b (x : \u03b1) in s a, f x \u2202\u03bc < \u03b5\n[PROOFSTEP]\nexact (hl \u03b4 \u03b40).mono fun i => h\u03b4 _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc\n[PROOFSTEP]\nsimp only [lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 (\u2a06 (g : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g \u2264 fun a => f a), SimpleFunc.lintegral g \u03bc) +\n \u2a06 (g_1 : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g_1 \u2264 fun a => g a), SimpleFunc.lintegral g_1 \u03bc \u2264\n \u2a06 (g_1 : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g_1 \u2264 fun a => f a + g a), SimpleFunc.lintegral g_1 \u03bc\n[PROOFSTEP]\nrefine'\n ENNReal.biSup_add_biSup_le' (p := fun h : \u03b1 \u2192\u209b \u211d\u22650\u221e => h \u2264 f) (q := fun h : \u03b1 \u2192\u209b \u211d\u22650\u221e => h \u2264 g) \u27e80, zero_le f\u27e9\n \u27e80, zero_le g\u27e9 fun f' hf' g' hg' => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhf' : (fun h => \u2191h \u2264 f) f'\ng' : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhg' : (fun h => \u2191h \u2264 g) g'\n\u22a2 SimpleFunc.lintegral f' \u03bc + SimpleFunc.lintegral g' \u03bc \u2264\n \u2a06 (g_1 : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g_1 \u2264 fun a => f a + g a), SimpleFunc.lintegral g_1 \u03bc\n[PROOFSTEP]\nexact le_iSup\u2082_of_le (f' + g') (add_le_add hf' hg') (add_lintegral _ _).ge\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\n\u22a2 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2a06 (n : \u2115), \u2191(eapprox f n) a) + \u2a06 (n : \u2115), \u2191(eapprox g n) a \u2202\u03bc\n[PROOFSTEP]\nsimp only [iSup_eapprox_apply, hf, hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\n\u22a2 \u222b\u207b (a : \u03b1), (\u2a06 (n : \u2115), \u2191(eapprox f n) a) + \u2a06 (n : \u2115), \u2191(eapprox g n) a \u2202\u03bc =\n \u222b\u207b (a : \u03b1), \u2a06 (n : \u2115), (\u2191(eapprox f n) + \u2191(eapprox g n)) a \u2202\u03bc\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\n\u22a2 (fun a => (\u2a06 (n : \u2115), \u2191(eapprox f n) a) + \u2a06 (n : \u2115), \u2191(eapprox g n) a) = fun a =>\n \u2a06 (n : \u2115), (\u2191(eapprox f n) + \u2191(eapprox g n)) a\n[PROOFSTEP]\nfunext a\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\na : \u03b1\n\u22a2 (\u2a06 (n : \u2115), \u2191(eapprox f n) a) + \u2a06 (n : \u2115), \u2191(eapprox g n) a = \u2a06 (n : \u2115), (\u2191(eapprox f n) + \u2191(eapprox g n)) a\n[PROOFSTEP]\nrw [ENNReal.iSup_add_iSup_of_monotone]\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\na : \u03b1\n\u22a2 \u2a06 (a_1 : \u2115), \u2191(eapprox f a_1) a + \u2191(eapprox g a_1) a = \u2a06 (n : \u2115), (\u2191(eapprox f n) + \u2191(eapprox g n)) a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase e_f.h.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\na : \u03b1\n\u22a2 Monotone fun n => \u2191(eapprox f n) a\n[PROOFSTEP]\nintro i j h\n[GOAL]\ncase e_f.h.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\na : \u03b1\ni j : \u2115\nh : i \u2264 j\n\u22a2 (fun n => \u2191(eapprox f n) a) i \u2264 (fun n => \u2191(eapprox f n) a) j\n[PROOFSTEP]\nexact monotone_eapprox _ h a\n[GOAL]\ncase e_f.h.hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\na : \u03b1\n\u22a2 Monotone fun n => \u2191(eapprox g n) a\n[PROOFSTEP]\nintro i j h\n[GOAL]\ncase e_f.h.hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\na : \u03b1\ni j : \u2115\nh : i \u2264 j\n\u22a2 (fun n => \u2191(eapprox g n) a) i \u2264 (fun n => \u2191(eapprox g n) a) j\n[PROOFSTEP]\nexact monotone_eapprox _ h a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (n : \u2115), (\u2191(eapprox f n) + \u2191(eapprox g n)) a \u2202\u03bc =\n \u2a06 (n : \u2115), SimpleFunc.lintegral (eapprox f n) \u03bc + SimpleFunc.lintegral (eapprox g n) \u03bc\n[PROOFSTEP]\nrw [lintegral_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\n\u22a2 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2191(eapprox f n) + \u2191(eapprox g n)) a \u2202\u03bc =\n \u2a06 (n : \u2115), SimpleFunc.lintegral (eapprox f n) \u03bc + SimpleFunc.lintegral (eapprox g n) \u03bc\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\n\u22a2 (fun n => \u222b\u207b (a : \u03b1), (\u2191(eapprox f n) + \u2191(eapprox g n)) a \u2202\u03bc) = fun n =>\n SimpleFunc.lintegral (eapprox f n) \u03bc + SimpleFunc.lintegral (eapprox g n) \u03bc\n[PROOFSTEP]\nfunext n\n[GOAL]\ncase e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nn : \u2115\n\u22a2 \u222b\u207b (a : \u03b1), (\u2191(eapprox f n) + \u2191(eapprox g n)) a \u2202\u03bc =\n SimpleFunc.lintegral (eapprox f n) \u03bc + SimpleFunc.lintegral (eapprox g n) \u03bc\n[PROOFSTEP]\nrw [\u2190 SimpleFunc.add_lintegral, \u2190 SimpleFunc.lintegral_eq_lintegral]\n[GOAL]\ncase e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nn : \u2115\n\u22a2 \u222b\u207b (a : \u03b1), (\u2191(eapprox f n) + \u2191(eapprox g n)) a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191(eapprox f n + eapprox g n) a \u2202\u03bc\n[PROOFSTEP]\nrfl\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\n\u22a2 \u2200 (n : \u2115), Measurable fun a => (\u2191(eapprox f n) + \u2191(eapprox g n)) a\n[PROOFSTEP]\nmeasurability\n[GOAL]\ncase h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\n\u22a2 Monotone fun n a => (\u2191(eapprox f n) + \u2191(eapprox g n)) a\n[PROOFSTEP]\nintro i j h a\n[GOAL]\ncase h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\ni j : \u2115\nh : i \u2264 j\na : \u03b1\n\u22a2 (fun n a => (\u2191(eapprox f n) + \u2191(eapprox g n)) a) i a \u2264 (fun n a => (\u2191(eapprox f n) + \u2191(eapprox g n)) a) j a\n[PROOFSTEP]\nexact add_le_add (monotone_eapprox _ h _) (monotone_eapprox _ h _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\n\u22a2 \u2a06 (n : \u2115), SimpleFunc.lintegral (eapprox f n) \u03bc + SimpleFunc.lintegral (eapprox g n) \u03bc =\n (\u2a06 (n : \u2115), SimpleFunc.lintegral (eapprox f n) \u03bc) + \u2a06 (n : \u2115), SimpleFunc.lintegral (eapprox g n) \u03bc\n[PROOFSTEP]\nrefine' (ENNReal.iSup_add_iSup_of_monotone _ _).symm\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\n\u22a2 Monotone fun n => SimpleFunc.lintegral (eapprox f n) \u03bc\n[PROOFSTEP]\nintro i j h\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\ni j : \u2115\nh : i \u2264 j\n\u22a2 (fun n => SimpleFunc.lintegral (eapprox f n) \u03bc) i \u2264 (fun n => SimpleFunc.lintegral (eapprox f n) \u03bc) j\n[PROOFSTEP]\nexact SimpleFunc.lintegral_mono (monotone_eapprox _ h) (le_refl \u03bc)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\n\u22a2 Monotone fun n => SimpleFunc.lintegral (eapprox g n) \u03bc\n[PROOFSTEP]\nintro i j h\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\ni j : \u2115\nh : i \u2264 j\n\u22a2 (fun n => SimpleFunc.lintegral (eapprox g n) \u03bc) i \u2264 (fun n => SimpleFunc.lintegral (eapprox g n) \u03bc) j\n[PROOFSTEP]\nexact SimpleFunc.lintegral_mono (monotone_eapprox _ h) (le_refl \u03bc)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\n\u22a2 (\u2a06 (n : \u2115), SimpleFunc.lintegral (eapprox f n) \u03bc) + \u2a06 (n : \u2115), SimpleFunc.lintegral (eapprox g n) \u03bc =\n \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_eq_iSup_eapprox_lintegral hf, lintegral_eq_iSup_eapprox_lintegral hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc = \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nrefine' le_antisymm _ (le_lintegral_add _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nrcases exists_measurable_le_lintegral_eq \u03bc fun a => f a + g a with \u27e8\u03c6, h\u03c6m, h\u03c6_le, h\u03c6_eq\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng \u03c6 : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03c6m : Measurable \u03c6\nh\u03c6_le : \u03c6 \u2264 fun a => f a + g a\nh\u03c6_eq : \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u03c6 a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\ncalc\n \u222b\u207b a, f a + g a \u2202\u03bc = \u222b\u207b a, \u03c6 a \u2202\u03bc := h\u03c6_eq\n _ \u2264 \u222b\u207b a, f a + (\u03c6 a - f a) \u2202\u03bc := (lintegral_mono fun a => le_add_tsub)\n _ = \u222b\u207b a, f a \u2202\u03bc + \u222b\u207b a, \u03c6 a - f a \u2202\u03bc := (lintegral_add_aux hf (h\u03c6m.sub hf))\n _ \u2264 \u222b\u207b a, f a \u2202\u03bc + \u222b\u207b a, g a \u2202\u03bc := add_le_add_left (lintegral_mono fun a => tsub_le_iff_left.2 <| h\u03c6_le a) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc = \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_congr_ae hf.ae_eq_mk, \u2190 lintegral_add_left hf.measurable_mk,\n lintegral_congr_ae (hf.ae_eq_mk.add (ae_eq_refl g))]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhg : AEMeasurable g\n\u22a2 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc = \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nsimpa only [add_comm] using lintegral_add_left' hg f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202c \u2022 \u03bc = c * \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nsimp only [lintegral, iSup_subtype', SimpleFunc.lintegral_smul, ENNReal.mul_iSup, smul_eq_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202Measure.sum \u03bc = \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), f a \u2202\u03bc i\n[PROOFSTEP]\nsimp only [lintegral, iSup_subtype', SimpleFunc.lintegral_sum, ENNReal.tsum_eq_iSup_sum]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u2a06 (x : { i // \u2191i \u2264 fun a => f a }) (s : Finset \u03b9), \u2211 i in s, SimpleFunc.lintegral (\u2191x) (\u03bc i) =\n \u2a06 (s : Finset \u03b9), \u2211 a in s, \u2a06 (x : { i // \u2191i \u2264 fun a => f a }), SimpleFunc.lintegral (\u2191x) (\u03bc a)\n[PROOFSTEP]\nrw [iSup_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u2a06 (j : Finset \u03b9) (i : { i // \u2191i \u2264 fun a => f a }), \u2211 i_1 in j, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) =\n \u2a06 (s : Finset \u03b9), \u2211 a in s, \u2a06 (x : { i // \u2191i \u2264 fun a => f a }), SimpleFunc.lintegral (\u2191x) (\u03bc a)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 (fun j => \u2a06 (i : { i // \u2191i \u2264 fun a => f a }), \u2211 i_1 in j, SimpleFunc.lintegral (\u2191i) (\u03bc i_1)) = fun s =>\n \u2211 a in s, \u2a06 (x : { i // \u2191i \u2264 fun a => f a }), SimpleFunc.lintegral (\u2191x) (\u03bc a)\n[PROOFSTEP]\nfunext s\n[GOAL]\ncase e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : Finset \u03b9\n\u22a2 \u2a06 (i : { i // \u2191i \u2264 fun a => f a }), \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) =\n \u2211 a in s, \u2a06 (x : { i // \u2191i \u2264 fun a => f a }), SimpleFunc.lintegral (\u2191x) (\u03bc a)\n[PROOFSTEP]\ninduction' s using Finset.induction_on with i s hi hs\n[GOAL]\ncase e_s.h.empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u2a06 (i : { i // \u2191i \u2264 fun a => f a }), \u2211 i_1 in \u2205, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) =\n \u2211 a in \u2205, \u2a06 (x : { i // \u2191i \u2264 fun a => f a }), SimpleFunc.lintegral (\u2191x) (\u03bc a)\n[PROOFSTEP]\napply bot_unique\n[GOAL]\ncase e_s.h.empty.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u2a06 (i : { i // \u2191i \u2264 fun a => f a }), \u2211 i_1 in \u2205, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) \u2264 \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e_s.h.insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nhs :\n \u2a06 (i : { i // \u2191i \u2264 fun a => f a }), \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) =\n \u2211 a in s, \u2a06 (x : { i // \u2191i \u2264 fun a => f a }), SimpleFunc.lintegral (\u2191x) (\u03bc a)\n\u22a2 \u2a06 (i_1 : { i // \u2191i \u2264 fun a => f a }), \u2211 i in insert i s, SimpleFunc.lintegral (\u2191i_1) (\u03bc i) =\n \u2211 a in insert i s, \u2a06 (x : { i // \u2191i \u2264 fun a => f a }), SimpleFunc.lintegral (\u2191x) (\u03bc a)\n[PROOFSTEP]\nsimp only [Finset.sum_insert hi, \u2190 hs]\n[GOAL]\ncase e_s.h.insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nhs :\n \u2a06 (i : { i // \u2191i \u2264 fun a => f a }), \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) =\n \u2211 a in s, \u2a06 (x : { i // \u2191i \u2264 fun a => f a }), SimpleFunc.lintegral (\u2191x) (\u03bc a)\n\u22a2 \u2a06 (i_1 : { i // \u2191i \u2264 fun a => f a }),\n SimpleFunc.lintegral (\u2191i_1) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191i_1) (\u03bc i) =\n (\u2a06 (x : { i // \u2191i \u2264 fun a => f a }), SimpleFunc.lintegral (\u2191x) (\u03bc i)) +\n \u2a06 (i : { i // \u2191i \u2264 fun a => f a }), \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1)\n[PROOFSTEP]\nrefine' (ENNReal.iSup_add_iSup _).symm\n[GOAL]\ncase e_s.h.insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nhs :\n \u2a06 (i : { i // \u2191i \u2264 fun a => f a }), \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) =\n \u2211 a in s, \u2a06 (x : { i // \u2191i \u2264 fun a => f a }), SimpleFunc.lintegral (\u2191x) (\u03bc a)\n\u22a2 \u2200 (i_1 j : { i // \u2191i \u2264 fun a => f a }),\n \u2203 k,\n SimpleFunc.lintegral (\u2191i_1) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191j) (\u03bc i) \u2264\n SimpleFunc.lintegral (\u2191k) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191k) (\u03bc i)\n[PROOFSTEP]\nintro \u03c6 \u03c8\n[GOAL]\ncase e_s.h.insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03b9 : Type u_5\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nhs :\n \u2a06 (i : { i // \u2191i \u2264 fun a => f a }), \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) =\n \u2211 a in s, \u2a06 (x : { i // \u2191i \u2264 fun a => f a }), SimpleFunc.lintegral (\u2191x) (\u03bc a)\n\u03c6 \u03c8 : { i // \u2191i \u2264 fun a => f a }\n\u22a2 \u2203 k,\n SimpleFunc.lintegral (\u2191\u03c6) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191\u03c8) (\u03bc i) \u2264\n SimpleFunc.lintegral (\u2191k) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191k) (\u03bc i)\n[PROOFSTEP]\nexact\n \u27e8\u27e8\u03c6 \u2294 \u03c8, fun x => sup_le (\u03c6.2 x) (\u03c8.2 x)\u27e9,\n add_le_add (SimpleFunc.lintegral_mono le_sup_left le_rfl)\n (Finset.sum_le_sum fun j _ => SimpleFunc.lintegral_mono le_sup_right le_rfl)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc \u03bd : Measure \u03b1\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202(\u03bc + \u03bd) = \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), f a \u2202\u03bd\n[PROOFSTEP]\nsimpa [tsum_fintype] using lintegral_sum_measure f fun b => cond b \u03bc \u03bd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nm : MeasurableSpace \u03b1\ns : Finset \u03b9\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i = \u2211 i in s, \u222b\u207b (a : \u03b1), f a \u2202\u03bc i\n[PROOFSTEP]\nrw [\u2190 Measure.sum_coe_finset, lintegral_sum_measure, \u2190 Finset.tsum_subtype']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nm : MeasurableSpace \u03b1\ns : Finset \u03b9\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 \u2211' (i : { x // x \u2208 s }), \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2191i = \u2211' (x : \u2191\u2191s), \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2191x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a \u22020 \u2264 \u22a5\n[PROOFSTEP]\nsimp [lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (x : \u03b1) in \u2205, f x \u2202\u03bc = 0\n[PROOFSTEP]\nrw [Measure.restrict_empty, lintegral_zero_measure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [Measure.restrict_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhs' : \u2191\u2191\u03bc s = 0\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = 0\n[PROOFSTEP]\nconvert lintegral_zero_measure _\n[GOAL]\ncase h.e'_2.h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhs' : \u2191\u2191\u03bc s = 0\n\u22a2 Measure.restrict \u03bc s = 0\n[PROOFSTEP]\nexact Measure.restrict_eq_zero.2 hs'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Finset \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), b \u2208 s \u2192 AEMeasurable (f b)\n\u22a2 \u222b\u207b (a : \u03b1), \u2211 b in s, f b a \u2202\u03bc = \u2211 b in s, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\ninduction' s using Finset.induction_on with a s has ih\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Finset \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : \u2200 (b : \u03b2), b \u2208 s \u2192 AEMeasurable (f b)\nhf : \u2200 (b : \u03b2), b \u2208 \u2205 \u2192 AEMeasurable (f b)\n\u22a2 \u222b\u207b (a : \u03b1), \u2211 b in \u2205, f b a \u2202\u03bc = \u2211 b in \u2205, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns\u271d : Finset \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : \u2200 (b : \u03b2), b \u2208 s\u271d \u2192 AEMeasurable (f b)\na : \u03b2\ns : Finset \u03b2\nhas : \u00aca \u2208 s\nih : (\u2200 (b : \u03b2), b \u2208 s \u2192 AEMeasurable (f b)) \u2192 \u222b\u207b (a : \u03b1), \u2211 b in s, f b a \u2202\u03bc = \u2211 b in s, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\nhf : \u2200 (b : \u03b2), b \u2208 insert a s \u2192 AEMeasurable (f b)\n\u22a2 \u222b\u207b (a_1 : \u03b1), \u2211 b in insert a s, f b a_1 \u2202\u03bc = \u2211 b in insert a s, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nsimp only [Finset.sum_insert has]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns\u271d : Finset \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : \u2200 (b : \u03b2), b \u2208 s\u271d \u2192 AEMeasurable (f b)\na : \u03b2\ns : Finset \u03b2\nhas : \u00aca \u2208 s\nih : (\u2200 (b : \u03b2), b \u2208 s \u2192 AEMeasurable (f b)) \u2192 \u222b\u207b (a : \u03b1), \u2211 b in s, f b a \u2202\u03bc = \u2211 b in s, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\nhf : \u2200 (b : \u03b2), b \u2208 insert a s \u2192 AEMeasurable (f b)\n\u22a2 \u222b\u207b (a_1 : \u03b1), f a a_1 + \u2211 b in s, f b a_1 \u2202\u03bc = \u222b\u207b (a_1 : \u03b1), f a a_1 \u2202\u03bc + \u2211 b in s, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nrw [Finset.forall_mem_insert] at hf \n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns\u271d : Finset \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : \u2200 (b : \u03b2), b \u2208 s\u271d \u2192 AEMeasurable (f b)\na : \u03b2\ns : Finset \u03b2\nhas : \u00aca \u2208 s\nih : (\u2200 (b : \u03b2), b \u2208 s \u2192 AEMeasurable (f b)) \u2192 \u222b\u207b (a : \u03b1), \u2211 b in s, f b a \u2202\u03bc = \u2211 b in s, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\nhf : AEMeasurable (f a) \u2227 \u2200 (x : \u03b2), x \u2208 s \u2192 AEMeasurable (f x)\n\u22a2 \u222b\u207b (a_1 : \u03b1), f a a_1 + \u2211 b in s, f b a_1 \u2202\u03bc = \u222b\u207b (a_1 : \u03b1), f a a_1 \u2202\u03bc + \u2211 b in s, \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_add_left' hf.1, ih hf.2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2a06 (n : \u2115), \u2191(const \u03b1 r * eapprox f n) a \u2202\u03bc\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 (fun a => r * f a) = fun a => \u2a06 (n : \u2115), \u2191(const \u03b1 r * eapprox f n) a\n[PROOFSTEP]\nfunext a\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\n\u22a2 r * f a = \u2a06 (n : \u2115), \u2191(const \u03b1 r * eapprox f n) a\n[PROOFSTEP]\nrw [\u2190 iSup_eapprox_apply f hf, ENNReal.mul_iSup]\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\n\u22a2 \u2a06 (i : \u2115), r * \u2191(eapprox f i) a = \u2a06 (n : \u2115), \u2191(const \u03b1 r * eapprox f n) a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (n : \u2115), \u2191(const \u03b1 r * eapprox f n) a \u2202\u03bc = \u2a06 (n : \u2115), r * SimpleFunc.lintegral (eapprox f n) \u03bc\n[PROOFSTEP]\nrw [lintegral_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), \u2191(const \u03b1 r * eapprox f n) a \u2202\u03bc = \u2a06 (n : \u2115), r * SimpleFunc.lintegral (eapprox f n) \u03bc\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 (fun n => \u222b\u207b (a : \u03b1), \u2191(const \u03b1 r * eapprox f n) a \u2202\u03bc) = fun n => r * SimpleFunc.lintegral (eapprox f n) \u03bc\n[PROOFSTEP]\nfunext n\n[GOAL]\ncase e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nn : \u2115\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(const \u03b1 r * eapprox f n) a \u2202\u03bc = r * SimpleFunc.lintegral (eapprox f n) \u03bc\n[PROOFSTEP]\nrw [\u2190 SimpleFunc.const_mul_lintegral, \u2190 SimpleFunc.lintegral_eq_lintegral]\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u2200 (n : \u2115), Measurable fun a => \u2191(const \u03b1 r * eapprox f n) a\n[PROOFSTEP]\nintro n\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nn : \u2115\n\u22a2 Measurable fun a => \u2191(const \u03b1 r * eapprox f n) a\n[PROOFSTEP]\nexact SimpleFunc.measurable _\n[GOAL]\ncase h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 Monotone fun n a => \u2191(const \u03b1 r * eapprox f n) a\n[PROOFSTEP]\nintro i j h a\n[GOAL]\ncase h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ni j : \u2115\nh : i \u2264 j\na : \u03b1\n\u22a2 (fun n a => \u2191(const \u03b1 r * eapprox f n) a) i a \u2264 (fun n a => \u2191(const \u03b1 r * eapprox f n) a) j a\n[PROOFSTEP]\nexact mul_le_mul_left' (monotone_eapprox _ h _) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u2a06 (n : \u2115), r * SimpleFunc.lintegral (eapprox f n) \u03bc = r * \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 ENNReal.mul_iSup, lintegral_eq_iSup_eapprox_lintegral hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u22a2 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc = r * \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nhave A : \u222b\u207b a, f a \u2202\u03bc = \u222b\u207b a, hf.mk f a \u2202\u03bc := lintegral_congr_ae hf.ae_eq_mk\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nA : \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk f hf a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc = r * \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nhave B : \u222b\u207b a, r * f a \u2202\u03bc = \u222b\u207b a, r * hf.mk f a \u2202\u03bc := lintegral_congr_ae (EventuallyEq.fun_comp hf.ae_eq_mk _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nA : \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk f hf a \u2202\u03bc\nB : \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc = \u222b\u207b (a : \u03b1), r * AEMeasurable.mk f hf a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc = r * \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nrw [A, B, lintegral_const_mul _ hf.measurable_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 r * \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral, ENNReal.mul_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 (i : \u03b1 \u2192\u209b \u211d\u22650\u221e), r * \u2a06 (_ : \u2191i \u2264 fun a => f a), SimpleFunc.lintegral i \u03bc \u2264 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc\n[PROOFSTEP]\nrefine' iSup_le fun s => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\n\u22a2 r * \u2a06 (_ : \u2191s \u2264 fun a => f a), SimpleFunc.lintegral s \u03bc \u2264 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc\n[PROOFSTEP]\nrw [ENNReal.mul_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\n\u22a2 \u2a06 (_ : \u2191s \u2264 fun a => f a), r * SimpleFunc.lintegral s \u03bc \u2264 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc\n[PROOFSTEP]\nsimp only [iSup_le_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\n\u22a2 (\u2191s \u2264 fun a => f a) \u2192 r * SimpleFunc.lintegral s \u03bc \u2264 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc\n[PROOFSTEP]\nintro hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhs : \u2191s \u2264 fun a => f a\n\u22a2 r * SimpleFunc.lintegral s \u03bc \u2264 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 SimpleFunc.const_mul_lintegral, lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhs : \u2191s \u2264 fun a => f a\n\u22a2 SimpleFunc.lintegral (const \u03b1 r * s) \u03bc \u2264 \u2a06 (g : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g \u2264 fun a => r * f a), SimpleFunc.lintegral g \u03bc\n[PROOFSTEP]\nrefine' le_iSup_of_le (const \u03b1 r * s) (le_iSup_of_le (fun x => _) le_rfl)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhs : \u2191s \u2264 fun a => f a\nx : \u03b1\n\u22a2 \u2191(const \u03b1 r * s) x \u2264 (fun a => r * f a) x\n[PROOFSTEP]\nexact mul_le_mul_left' (hs x) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhr : r \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc = r * \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nby_cases h : r = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhr : r \u2260 \u22a4\nh : r = 0\n\u22a2 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc = r * \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhr : r \u2260 \u22a4\nh : \u00acr = 0\n\u22a2 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc = r * \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\napply le_antisymm _ (lintegral_const_mul_le r f)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhr : r \u2260 \u22a4\nh : \u00acr = 0\n\u22a2 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc \u2264 r * \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nhave rinv : r * r\u207b\u00b9 = 1 := ENNReal.mul_inv_cancel h hr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhr : r \u2260 \u22a4\nh : \u00acr = 0\nrinv : r * r\u207b\u00b9 = 1\n\u22a2 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc \u2264 r * \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nhave rinv' : r\u207b\u00b9 * r = 1 := by\n rw [mul_comm]\n exact rinv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhr : r \u2260 \u22a4\nh : \u00acr = 0\nrinv : r * r\u207b\u00b9 = 1\n\u22a2 r\u207b\u00b9 * r = 1\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhr : r \u2260 \u22a4\nh : \u00acr = 0\nrinv : r * r\u207b\u00b9 = 1\n\u22a2 r * r\u207b\u00b9 = 1\n[PROOFSTEP]\nexact rinv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhr : r \u2260 \u22a4\nh : \u00acr = 0\nrinv : r * r\u207b\u00b9 = 1\nrinv' : r\u207b\u00b9 * r = 1\n\u22a2 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc \u2264 r * \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nhave := lintegral_const_mul_le (\u03bc := \u03bc) r\u207b\u00b9 fun x => r * f x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhr : r \u2260 \u22a4\nh : \u00acr = 0\nrinv : r * r\u207b\u00b9 = 1\nrinv' : r\u207b\u00b9 * r = 1\nthis : r\u207b\u00b9 * \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), r\u207b\u00b9 * (r * f a) \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc \u2264 r * \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nsimp [(mul_assoc _ _ _).symm, rinv'] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhr : r \u2260 \u22a4\nh : \u00acr = 0\nrinv : r * r\u207b\u00b9 = 1\nrinv' : r\u207b\u00b9 * r = 1\nthis : r\u207b\u00b9 * \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), r * f a \u2202\u03bc \u2264 r * \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nsimpa [(mul_assoc _ _ _).symm, rinv] using mul_le_mul_left' this r\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1), f a * r \u2202\u03bc = (\u222b\u207b (a : \u03b1), f a \u2202\u03bc) * r\n[PROOFSTEP]\nsimp_rw [mul_comm, lintegral_const_mul r hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u22a2 \u222b\u207b (a : \u03b1), f a * r \u2202\u03bc = (\u222b\u207b (a : \u03b1), f a \u2202\u03bc) * r\n[PROOFSTEP]\nsimp_rw [mul_comm, lintegral_const_mul'' r hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 (\u222b\u207b (a : \u03b1), f a \u2202\u03bc) * r \u2264 \u222b\u207b (a : \u03b1), f a * r \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [mul_comm, lintegral_const_mul_le r f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhr : r \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), f a * r \u2202\u03bc = (\u222b\u207b (a : \u03b1), f a \u2202\u03bc) * r\n[PROOFSTEP]\nsimp_rw [mul_comm, lintegral_const_mul' r f hr]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\n\u03b2 : Type u_5\ninst\u271d : MeasurableSpace \u03b2\n\u03bd : Measure \u03b2\nf : \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\n\u22a2 \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f x * g y \u2202\u03bd \u2202\u03bc = (\u222b\u207b (x : \u03b1), f x \u2202\u03bc) * \u222b\u207b (y : \u03b2), g y \u2202\u03bd\n[PROOFSTEP]\nsimp [lintegral_const_mul'' _ hg, lintegral_mul_const'' _ hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf f' : \u03b1 \u2192 \u03b2\nh\u271d : f =\u1d50[\u03bc] f'\ng : \u03b2 \u2192 \u211d\u22650\u221e\na : \u03b1\nh : f a = f' a\n\u22a2 (fun a => g (f a)) a = (fun a => g (f' a)) a\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf f' : \u03b1 \u2192 \u03b2\nh\u271d : f =\u1d50[\u03bc] f'\ng : \u03b2 \u2192 \u211d\u22650\u221e\na : \u03b1\nh : f a = f' a\n\u22a2 g (f a) = g (f' a)\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf\u2081 f\u2081' : \u03b1 \u2192 \u03b2\nf\u2082 f\u2082' : \u03b1 \u2192 \u03b3\nh\u2081\u271d : f\u2081 =\u1d50[\u03bc] f\u2081'\nh\u2082\u271d : f\u2082 =\u1d50[\u03bc] f\u2082'\ng : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e\nx\u271d : \u03b1\nh\u2082 : f\u2082 x\u271d = f\u2082' x\u271d\nh\u2081 : f\u2081 x\u271d = f\u2081' x\u271d\n\u22a2 (fun a => g (f\u2081 a) (f\u2082 a)) x\u271d = (fun a => g (f\u2081' a) (f\u2082' a)) x\u271d\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf\u2081 f\u2081' : \u03b1 \u2192 \u03b2\nf\u2082 f\u2082' : \u03b1 \u2192 \u03b3\nh\u2081\u271d : f\u2081 =\u1d50[\u03bc] f\u2081'\nh\u2082\u271d : f\u2082 =\u1d50[\u03bc] f\u2082'\ng : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e\nx\u271d : \u03b1\nh\u2082 : f\u2082 x\u271d = f\u2082' x\u271d\nh\u2081 : f\u2081 x\u271d = f\u2081' x\u271d\n\u22a2 g (f\u2081 x\u271d) (f\u2082 x\u271d) = g (f\u2081' x\u271d) (f\u2082' x\u271d)\n[PROOFSTEP]\nrw [h\u2081, h\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), indicator s f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc\n[PROOFSTEP]\nsimp only [lintegral, \u2190 restrict_lintegral_eq_lintegral_restrict _ hs, iSup_subtype']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2a06 (x : { i // \u2191i \u2264 fun a => indicator s f a }), SimpleFunc.lintegral (\u2191x) \u03bc =\n \u2a06 (x : { i // \u2191i \u2264 fun a => f a }), SimpleFunc.lintegral (restrict (\u2191x) s) \u03bc\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2a06 (x : { i // \u2191i \u2264 fun a => indicator s f a }), SimpleFunc.lintegral (\u2191x) \u03bc \u2264\n \u2a06 (x : { i // \u2191i \u2264 fun a => f a }), SimpleFunc.lintegral (restrict (\u2191x) s) \u03bc\n[PROOFSTEP]\nrefine' iSup_mono' (Subtype.forall.2 fun \u03c6 h\u03c6 => _)\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2a06 (x : { i // \u2191i \u2264 fun a => f a }), SimpleFunc.lintegral (restrict (\u2191x) s) \u03bc \u2264\n \u2a06 (x : { i // \u2191i \u2264 fun a => indicator s f a }), SimpleFunc.lintegral (\u2191x) \u03bc\n[PROOFSTEP]\nrefine' iSup_mono' (Subtype.forall.2 fun \u03c6 h\u03c6 => _)\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => indicator s f a\n\u22a2 \u2203 i', SimpleFunc.lintegral (\u2191{ val := \u03c6, property := h\u03c6 }) \u03bc \u2264 SimpleFunc.lintegral (restrict (\u2191i') s) \u03bc\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u03c6, le_trans h\u03c6 (indicator_le_self _ _)\u27e9, _\u27e9\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => indicator s f a\n\u22a2 SimpleFunc.lintegral (\u2191{ val := \u03c6, property := h\u03c6 }) \u03bc \u2264\n SimpleFunc.lintegral (restrict (\u2191{ val := \u03c6, property := (_ : \u2191\u03c6 \u2264 fun a => f a) }) s) \u03bc\n[PROOFSTEP]\nrefine' SimpleFunc.lintegral_mono (fun x => _) le_rfl\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => indicator s f a\nx : \u03b1\n\u22a2 \u2191\u2191{ val := \u03c6, property := h\u03c6 } x \u2264 \u2191(restrict (\u2191{ val := \u03c6, property := (_ : \u2191\u03c6 \u2264 fun a => f a) }) s) x\n[PROOFSTEP]\nby_cases hx : x \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => indicator s f a\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2191\u2191{ val := \u03c6, property := h\u03c6 } x \u2264 \u2191(restrict (\u2191{ val := \u03c6, property := (_ : \u2191\u03c6 \u2264 fun a => f a) }) s) x\n[PROOFSTEP]\nsimp [hx, hs, le_refl]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => indicator s f a\nx : \u03b1\nhx : \u00acx \u2208 s\n\u22a2 \u2191\u2191{ val := \u03c6, property := h\u03c6 } x \u2264 \u2191(restrict (\u2191{ val := \u03c6, property := (_ : \u2191\u03c6 \u2264 fun a => f a) }) s) x\n[PROOFSTEP]\napply le_trans (h\u03c6 x)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => indicator s f a\nx : \u03b1\nhx : \u00acx \u2208 s\n\u22a2 (fun a => indicator s f a) x \u2264 \u2191(restrict (\u2191{ val := \u03c6, property := (_ : \u2191\u03c6 \u2264 fun a => f a) }) s) x\n[PROOFSTEP]\nsimp [hx, hs, le_refl]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\n\u22a2 \u2203 i', SimpleFunc.lintegral (restrict (\u2191{ val := \u03c6, property := h\u03c6 }) s) \u03bc \u2264 SimpleFunc.lintegral (\u2191i') \u03bc\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u03c6.restrict s, fun x => _\u27e9, le_rfl\u27e9\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nh\u03c6 : \u2191\u03c6 \u2264 fun a => f a\nx : \u03b1\n\u22a2 \u2191(restrict \u03c6 s) x \u2264 (fun a => indicator s f a) x\n[PROOFSTEP]\nsimp [h\u03c6 x, hs, indicator_le_indicator]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), indicator s f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lintegral_congr_ae (indicator_ae_eq_of_ae_eq_set hs.toMeasurable_ae_eq),\n lintegral_indicator _ (measurableSet_toMeasurable _ _), Measure.restrict_congr_set hs.toMeasurable_ae_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : NullMeasurableSet s\nc : \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), indicator s (fun x => c) a \u2202\u03bc = c * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [lintegral_indicator\u2080 _ hs, set_lintegral_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nr : \u211d\u22650\u221e\n\u22a2 \u222b\u207b (x : \u03b1) in {x | f x = r}, f x \u2202\u03bc = r * \u2191\u2191\u03bc {x | f x = r}\n[PROOFSTEP]\nhave : \u2200\u1d50 x \u2202\u03bc, x \u2208 {x | f x = r} \u2192 f x = r := ae_of_all \u03bc fun _ hx => hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nr : \u211d\u22650\u221e\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 {x | f x = r} \u2192 f x = r\n\u22a2 \u222b\u207b (x : \u03b1) in {x | f x = r}, f x \u2202\u03bc = r * \u2191\u2191\u03bc {x | f x = r}\n[PROOFSTEP]\nrw [set_lintegral_congr_fun _ this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nr : \u211d\u22650\u221e\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 {x | f x = r} \u2192 f x = r\n\u22a2 \u222b\u207b (x : \u03b1) in {x | f x = r}, r \u2202\u03bc = r * \u2191\u2191\u03bc {x | f x = r}\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nr : \u211d\u22650\u221e\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 {x | f x = r} \u2192 f x = r\n\u22a2 MeasurableSet {x | f x = r}\n[PROOFSTEP]\nrw [lintegral_const, Measure.restrict_apply MeasurableSet.univ, Set.univ_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nr : \u211d\u22650\u221e\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 {x | f x = r} \u2192 f x = r\n\u22a2 MeasurableSet {x | f x = r}\n[PROOFSTEP]\nexact hf (measurableSet_singleton r)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhle : f \u2264\u1d50[\u03bc] g\nhg : AEMeasurable g\n\u03b5 : \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u03b5 * \u2191\u2191\u03bc {x | f x + \u03b5 \u2264 g x} \u2264 \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nrcases exists_measurable_le_lintegral_eq \u03bc f with \u27e8\u03c6, h\u03c6m, h\u03c6_le, h\u03c6_eq\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhle : f \u2264\u1d50[\u03bc] g\nhg : AEMeasurable g\n\u03b5 : \u211d\u22650\u221e\n\u03c6 : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03c6m : Measurable \u03c6\nh\u03c6_le : \u03c6 \u2264 f\nh\u03c6_eq : \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u03c6 a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u03b5 * \u2191\u2191\u03bc {x | f x + \u03b5 \u2264 g x} \u2264 \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\ncalc\n \u222b\u207b x, f x \u2202\u03bc + \u03b5 * \u03bc {x | f x + \u03b5 \u2264 g x} = \u222b\u207b x, \u03c6 x \u2202\u03bc + \u03b5 * \u03bc {x | f x + \u03b5 \u2264 g x} := by rw [h\u03c6_eq]\n _ \u2264 \u222b\u207b x, \u03c6 x \u2202\u03bc + \u03b5 * \u03bc {x | \u03c6 x + \u03b5 \u2264 g x} := by\n gcongr\n exact measure_mono fun x => (add_le_add_right (h\u03c6_le _) _).trans\n _ = \u222b\u207b x, \u03c6 x + indicator {x | \u03c6 x + \u03b5 \u2264 g x} (fun _ => \u03b5) x \u2202\u03bc :=\n by\n rw [lintegral_add_left h\u03c6m, lintegral_indicator\u2080, set_lintegral_const]\n exact measurableSet_le (h\u03c6m.nullMeasurable.measurable'.add_const _) hg.nullMeasurable\n _ \u2264 \u222b\u207b x, g x \u2202\u03bc := lintegral_mono_ae (hle.mono fun x hx\u2081 => ?_)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhle : f \u2264\u1d50[\u03bc] g\nhg : AEMeasurable g\n\u03b5 : \u211d\u22650\u221e\n\u03c6 : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03c6m : Measurable \u03c6\nh\u03c6_le : \u03c6 \u2264 f\nh\u03c6_eq : \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u03c6 a \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5 * \u2191\u2191\u03bc {x | f x + \u03b5 \u2264 g x} = \u222b\u207b (x : \u03b1), \u03c6 x \u2202\u03bc + \u03b5 * \u2191\u2191\u03bc {x | f x + \u03b5 \u2264 g x}\n[PROOFSTEP]\nrw [h\u03c6_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhle : f \u2264\u1d50[\u03bc] g\nhg : AEMeasurable g\n\u03b5 : \u211d\u22650\u221e\n\u03c6 : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03c6m : Measurable \u03c6\nh\u03c6_le : \u03c6 \u2264 f\nh\u03c6_eq : \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u03c6 a \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u03c6 x \u2202\u03bc + \u03b5 * \u2191\u2191\u03bc {x | f x + \u03b5 \u2264 g x} \u2264 \u222b\u207b (x : \u03b1), \u03c6 x \u2202\u03bc + \u03b5 * \u2191\u2191\u03bc {x | \u03c6 x + \u03b5 \u2264 g x}\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase bc.bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhle : f \u2264\u1d50[\u03bc] g\nhg : AEMeasurable g\n\u03b5 : \u211d\u22650\u221e\n\u03c6 : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03c6m : Measurable \u03c6\nh\u03c6_le : \u03c6 \u2264 f\nh\u03c6_eq : \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u03c6 a \u2202\u03bc\n\u22a2 \u2191\u2191\u03bc {x | f x + \u03b5 \u2264 g x} \u2264 \u2191\u2191\u03bc {x | \u03c6 x + \u03b5 \u2264 g x}\n[PROOFSTEP]\nexact measure_mono fun x => (add_le_add_right (h\u03c6_le _) _).trans\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhle : f \u2264\u1d50[\u03bc] g\nhg : AEMeasurable g\n\u03b5 : \u211d\u22650\u221e\n\u03c6 : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03c6m : Measurable \u03c6\nh\u03c6_le : \u03c6 \u2264 f\nh\u03c6_eq : \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u03c6 a \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u03c6 x \u2202\u03bc + \u03b5 * \u2191\u2191\u03bc {x | \u03c6 x + \u03b5 \u2264 g x} = \u222b\u207b (x : \u03b1), \u03c6 x + indicator {x | \u03c6 x + \u03b5 \u2264 g x} (fun x => \u03b5) x \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_add_left h\u03c6m, lintegral_indicator\u2080, set_lintegral_const]\n[GOAL]\ncase hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhle : f \u2264\u1d50[\u03bc] g\nhg : AEMeasurable g\n\u03b5 : \u211d\u22650\u221e\n\u03c6 : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03c6m : Measurable \u03c6\nh\u03c6_le : \u03c6 \u2264 f\nh\u03c6_eq : \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u03c6 a \u2202\u03bc\n\u22a2 NullMeasurableSet {x | \u03c6 x + \u03b5 \u2264 g x}\n[PROOFSTEP]\nexact measurableSet_le (h\u03c6m.nullMeasurable.measurable'.add_const _) hg.nullMeasurable\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhle : f \u2264\u1d50[\u03bc] g\nhg : AEMeasurable g\n\u03b5 : \u211d\u22650\u221e\n\u03c6 : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03c6m : Measurable \u03c6\nh\u03c6_le : \u03c6 \u2264 f\nh\u03c6_eq : \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u03c6 a \u2202\u03bc\nx : \u03b1\nhx\u2081 : f x \u2264 g x\n\u22a2 \u03c6 x + indicator {x | \u03c6 x + \u03b5 \u2264 g x} (fun x => \u03b5) x \u2264 g x\n[PROOFSTEP]\nsimp only [indicator_apply]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhle : f \u2264\u1d50[\u03bc] g\nhg : AEMeasurable g\n\u03b5 : \u211d\u22650\u221e\n\u03c6 : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03c6m : Measurable \u03c6\nh\u03c6_le : \u03c6 \u2264 f\nh\u03c6_eq : \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u03c6 a \u2202\u03bc\nx : \u03b1\nhx\u2081 : f x \u2264 g x\n\u22a2 (\u03c6 x + if x \u2208 {x | \u03c6 x + \u03b5 \u2264 g x} then \u03b5 else 0) \u2264 g x\n[PROOFSTEP]\nsplit_ifs with hx\u2082\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhle : f \u2264\u1d50[\u03bc] g\nhg : AEMeasurable g\n\u03b5 : \u211d\u22650\u221e\n\u03c6 : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03c6m : Measurable \u03c6\nh\u03c6_le : \u03c6 \u2264 f\nh\u03c6_eq : \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u03c6 a \u2202\u03bc\nx : \u03b1\nhx\u2081 : f x \u2264 g x\nhx\u2082 : x \u2208 {x | \u03c6 x + \u03b5 \u2264 g x}\n\u22a2 \u03c6 x + \u03b5 \u2264 g x\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhle : f \u2264\u1d50[\u03bc] g\nhg : AEMeasurable g\n\u03b5 : \u211d\u22650\u221e\n\u03c6 : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03c6m : Measurable \u03c6\nh\u03c6_le : \u03c6 \u2264 f\nh\u03c6_eq : \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u03c6 a \u2202\u03bc\nx : \u03b1\nhx\u2081 : f x \u2264 g x\nhx\u2082 : \u00acx \u2208 {x | \u03c6 x + \u03b5 \u2264 g x}\n\u22a2 \u03c6 x + 0 \u2264 g x\n[PROOFSTEP]\nexacts [hx\u2082, (add_zero _).trans_le <| (h\u03c6_le x).trans hx\u2081]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u22a2 \u03b5 * \u2191\u2191\u03bc {x | \u03b5 \u2264 f x} \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nsimpa only [lintegral_zero, zero_add] using\n lintegral_add_mul_meas_add_le_le_lintegral (ae_of_all _ fun x => zero_le (f x)) hf \u03b5\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nh\u03bcf : \u2191\u2191\u03bc {x | f x = \u22a4} \u2260 0\n\u22a2 \u22a4 = \u22a4 * \u2191\u2191\u03bc {x | \u22a4 \u2264 f x}\n[PROOFSTEP]\nsimp [mul_eq_top, h\u03bcf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f\nh\u03bcf : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x = \u22a4} \u2260 0\n\u22a2 \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x = \u22a4} \u2264 \u2191\u2191(Measure.restrict \u03bc s) {x | f x = \u22a4}\n[PROOFSTEP]\nrw [\u2190 setOf_inter_eq_sep]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f\nh\u03bcf : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x = \u22a4} \u2260 0\n\u22a2 \u2191\u2191\u03bc ({a | f a = \u22a4} \u2229 s) \u2264 \u2191\u2191(Measure.restrict \u03bc s) {x | f x = \u22a4}\n[PROOFSTEP]\nexact Measure.le_restrict_apply _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5' : \u03b5 \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 f x} * \u03b5 \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5' : \u03b5 \u2260 \u22a4\n\u22a2 \u03b5 * \u2191\u2191\u03bc {x | \u03b5 \u2264 f x} \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nexact mul_meas_ge_le_lintegral\u2080 hf \u03b5\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\u22a2 f =\u1d50[\u03bc] g\n[PROOFSTEP]\nhave : \u2200 n : \u2115, \u2200\u1d50 x \u2202\u03bc, g x < f x + (n : \u211d\u22650\u221e)\u207b\u00b9 := by\n intro n\n simp only [ae_iff, not_lt]\n have : \u222b\u207b x, f x \u2202\u03bc + (\u2191n)\u207b\u00b9 * \u03bc {x : \u03b1 | f x + (n : \u211d\u22650\u221e)\u207b\u00b9 \u2264 g x} \u2264 \u222b\u207b x, f x \u2202\u03bc :=\n (lintegral_add_mul_meas_add_le_le_lintegral hfg hg n\u207b\u00b9).trans hgf\n rw [(ENNReal.cancel_of_ne hf).add_le_iff_nonpos_right, nonpos_iff_eq_zero, mul_eq_zero] at this \n exact this.resolve_left (ENNReal.inv_ne_zero.2 (ENNReal.nat_ne_top _))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\u22a2 \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nn : \u2115\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nsimp only [ae_iff, not_lt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc {a | f a + (\u2191n)\u207b\u00b9 \u2264 g a} = 0\n[PROOFSTEP]\nhave : \u222b\u207b x, f x \u2202\u03bc + (\u2191n)\u207b\u00b9 * \u03bc {x : \u03b1 | f x + (n : \u211d\u22650\u221e)\u207b\u00b9 \u2264 g x} \u2264 \u222b\u207b x, f x \u2202\u03bc :=\n (lintegral_add_mul_meas_add_le_le_lintegral hfg hg n\u207b\u00b9).trans hgf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nn : \u2115\nthis : \u222b\u207b (x : \u03b1), f x \u2202\u03bc + (\u2191n)\u207b\u00b9 * \u2191\u2191\u03bc {x | f x + (\u2191n)\u207b\u00b9 \u2264 g x} \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\u22a2 \u2191\u2191\u03bc {a | f a + (\u2191n)\u207b\u00b9 \u2264 g a} = 0\n[PROOFSTEP]\nrw [(ENNReal.cancel_of_ne hf).add_le_iff_nonpos_right, nonpos_iff_eq_zero, mul_eq_zero] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nn : \u2115\nthis : (\u2191n)\u207b\u00b9 = 0 \u2228 \u2191\u2191\u03bc {x | f x + (\u2191n)\u207b\u00b9 \u2264 g x} = 0\n\u22a2 \u2191\u2191\u03bc {a | f a + (\u2191n)\u207b\u00b9 \u2264 g a} = 0\n[PROOFSTEP]\nexact this.resolve_left (ENNReal.inv_ne_zero.2 (ENNReal.nat_ne_top _))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nthis : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9\n\u22a2 f =\u1d50[\u03bc] g\n[PROOFSTEP]\nrefine' hfg.mp ((ae_all_iff.2 this).mono fun x hlt hle => hle.antisymm _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nthis : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9\nx : \u03b1\nhlt : \u2200 (i : \u2115), g x < f x + (\u2191i)\u207b\u00b9\nhle : f x \u2264 g x\n\u22a2 g x \u2264 f x\n[PROOFSTEP]\nsuffices : Tendsto (fun n : \u2115 => f x + (n : \u211d\u22650\u221e)\u207b\u00b9) atTop (\ud835\udcdd (f x))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nthis\u271d : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9\nx : \u03b1\nhlt : \u2200 (i : \u2115), g x < f x + (\u2191i)\u207b\u00b9\nhle : f x \u2264 g x\nthis : Tendsto (fun n => f x + (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd (f x))\n\u22a2 g x \u2264 f x\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nthis : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9\nx : \u03b1\nhlt : \u2200 (i : \u2115), g x < f x + (\u2191i)\u207b\u00b9\nhle : f x \u2264 g x\n\u22a2 Tendsto (fun n => f x + (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd (f x))\n[PROOFSTEP]\nexact ge_of_tendsto' this fun i => (hlt i).le\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhfg : f \u2264\u1d50[\u03bc] g\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhg : AEMeasurable g\nhgf : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nthis : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < f x + (\u2191n)\u207b\u00b9\nx : \u03b1\nhlt : \u2200 (i : \u2115), g x < f x + (\u2191i)\u207b\u00b9\nhle : f x \u2264 g x\n\u22a2 Tendsto (fun n => f x + (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd (f x))\n[PROOFSTEP]\nsimpa only [inv_top, add_zero] using tendsto_const_nhds.add (ENNReal.tendsto_inv_iff.2 ENNReal.tendsto_nat_nhds_top)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u22a2 \u222b\u207b (x : \u03b1), 0 \u2202\u03bc \u2260 \u22a4\n[PROOFSTEP]\nsimp [lintegral_zero, zero_ne_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 0 < \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2194 0 < \u2191\u2191\u03bc (support f)\n[PROOFSTEP]\nsimp [pos_iff_ne_zero, hf, Filter.EventuallyEq, ae_iff, Function.support]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f n a \u2264 f (Nat.succ n) a\ns : Set \u03b1\nhs : {a | \u00ac\u2200 (i : \u2115), f i a \u2264 f (Nat.succ i) a} \u2286 s \u2227 MeasurableSet s \u2227 \u2191\u2191\u03bc s = 0\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e := fun n a => if a \u2208 s then 0 else f n a\ng_eq_f : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), g n a = f n a\na : \u03b1\nha : \u2200 (n : \u2115), g n a = f n a\n\u22a2 (fun a => \u2a06 (n : \u2115), f n a) a = (fun a => \u2a06 (n : \u2115), g n a) a\n[PROOFSTEP]\nsimp only [ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f n a \u2264 f (Nat.succ n) a\ns : Set \u03b1\nhs : {a | \u00ac\u2200 (i : \u2115), f i a \u2264 f (Nat.succ i) a} \u2286 s \u2227 MeasurableSet s \u2227 \u2191\u2191\u03bc s = 0\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e := fun n a => if a \u2208 s then 0 else f n a\ng_eq_f : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), g n a = f n a\nn : \u2115\na : \u03b1\nh : a \u2208 s\n\u22a2 g n a \u2264 g (n + 1) a\n[PROOFSTEP]\nsimp [if_pos h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f n a \u2264 f (Nat.succ n) a\ns : Set \u03b1\nhs : {a | \u00ac\u2200 (i : \u2115), f i a \u2264 f (Nat.succ i) a} \u2286 s \u2227 MeasurableSet s \u2227 \u2191\u2191\u03bc s = 0\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e := fun n a => if a \u2208 s then 0 else f n a\ng_eq_f : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), g n a = f n a\nn : \u2115\na : \u03b1\nh : \u00aca \u2208 s\n\u22a2 g n a \u2264 g (n + 1) a\n[PROOFSTEP]\nsimp only [if_neg h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f n a \u2264 f (Nat.succ n) a\ns : Set \u03b1\nhs : {a | \u00ac\u2200 (i : \u2115), f i a \u2264 f (Nat.succ i) a} \u2286 s \u2227 MeasurableSet s \u2227 \u2191\u2191\u03bc s = 0\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e := fun n a => if a \u2208 s then 0 else f n a\ng_eq_f : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), g n a = f n a\nn : \u2115\na : \u03b1\nh : \u00aca \u2208 s\n\u22a2 f n a \u2264 f (n + 1) a\n[PROOFSTEP]\nhave := hs.1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f n a \u2264 f (Nat.succ n) a\ns : Set \u03b1\nhs : {a | \u00ac\u2200 (i : \u2115), f i a \u2264 f (Nat.succ i) a} \u2286 s \u2227 MeasurableSet s \u2227 \u2191\u2191\u03bc s = 0\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e := fun n a => if a \u2208 s then 0 else f n a\ng_eq_f : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), g n a = f n a\nn : \u2115\na : \u03b1\nh : \u00aca \u2208 s\nthis : {a | \u00ac\u2200 (i : \u2115), f i a \u2264 f (Nat.succ i) a} \u2286 s\n\u22a2 f n a \u2264 f (n + 1) a\n[PROOFSTEP]\nrw [subset_def] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f n a \u2264 f (Nat.succ n) a\ns : Set \u03b1\nhs : {a | \u00ac\u2200 (i : \u2115), f i a \u2264 f (Nat.succ i) a} \u2286 s \u2227 MeasurableSet s \u2227 \u2191\u2191\u03bc s = 0\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e := fun n a => if a \u2208 s then 0 else f n a\ng_eq_f : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), g n a = f n a\nn : \u2115\na : \u03b1\nh : \u00aca \u2208 s\nthis : \u2200 (x : \u03b1), x \u2208 {a | \u00ac\u2200 (i : \u2115), f i a \u2264 f (Nat.succ i) a} \u2192 x \u2208 s\n\u22a2 f n a \u2264 f (n + 1) a\n[PROOFSTEP]\nhave := mt (this a) h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f n a \u2264 f (Nat.succ n) a\ns : Set \u03b1\nhs : {a | \u00ac\u2200 (i : \u2115), f i a \u2264 f (Nat.succ i) a} \u2286 s \u2227 MeasurableSet s \u2227 \u2191\u2191\u03bc s = 0\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e := fun n a => if a \u2208 s then 0 else f n a\ng_eq_f : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), g n a = f n a\nn : \u2115\na : \u03b1\nh : \u00aca \u2208 s\nthis\u271d : \u2200 (x : \u03b1), x \u2208 {a | \u00ac\u2200 (i : \u2115), f i a \u2264 f (Nat.succ i) a} \u2192 x \u2208 s\nthis : \u00aca \u2208 {a | \u00ac\u2200 (i : \u2115), f i a \u2264 f (Nat.succ i) a}\n\u22a2 f n a \u2264 f (n + 1) a\n[PROOFSTEP]\nsimp only [Classical.not_not, mem_setOf_eq] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f n a \u2264 f (Nat.succ n) a\ns : Set \u03b1\nhs : {a | \u00ac\u2200 (i : \u2115), f i a \u2264 f (Nat.succ i) a} \u2286 s \u2227 MeasurableSet s \u2227 \u2191\u2191\u03bc s = 0\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e := fun n a => if a \u2208 s then 0 else f n a\ng_eq_f : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), g n a = f n a\nn : \u2115\na : \u03b1\nh : \u00aca \u2208 s\nthis\u271d : \u2200 (x : \u03b1), x \u2208 {a | \u00ac\u2200 (i : \u2115), f i a \u2264 f (Nat.succ i) a} \u2192 x \u2208 s\nthis : \u2200 (i : \u2115), f i a \u2264 f (Nat.succ i) a\n\u22a2 f n a \u2264 f (n + 1) a\n[PROOFSTEP]\nexact this n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, f n a \u2264 f (Nat.succ n) a\ns : Set \u03b1\nhs : {a | \u00ac\u2200 (i : \u2115), f i a \u2264 f (Nat.succ i) a} \u2286 s \u2227 MeasurableSet s \u2227 \u2191\u2191\u03bc s = 0\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e := fun n a => if a \u2208 s then 0 else f n a\ng_eq_f : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), g n a = f n a\n\u22a2 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), g n a \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nsimp only [lintegral_congr_ae (g_eq_f.mono fun _a ha => ha _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhg : AEMeasurable g\nhg_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\nh_le : g \u2264\u1d50[\u03bc] f\n\u22a2 \u222b\u207b (a : \u03b1), f a - g a \u2202\u03bc = \u222b\u207b (a : \u03b1), f a \u2202\u03bc - \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\nrefine' ENNReal.eq_sub_of_add_eq hg_fin _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhg : AEMeasurable g\nhg_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\nh_le : g \u2264\u1d50[\u03bc] f\n\u22a2 \u222b\u207b (a : \u03b1), f a - g a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc = \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lintegral_add_right' _ hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhg : AEMeasurable g\nhg_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\nh_le : g \u2264\u1d50[\u03bc] f\n\u22a2 \u222b\u207b (a : \u03b1), f a - g a + g a \u2202\u03bc = \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nexact lintegral_congr_ae (h_le.mono fun x hx => tsub_add_cancel_of_le hx)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc - \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), g x - f x \u2202\u03bc\n[PROOFSTEP]\nrw [tsub_le_iff_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), g x - f x \u2202\u03bc + \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nby_cases hfi : \u222b\u207b x, f x \u2202\u03bc = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), g x - f x \u2202\u03bc + \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [hfi, add_top]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhfi : \u00ac\u222b\u207b (x : \u03b1), f x \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), g x - f x \u2202\u03bc + \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lintegral_add_right' _ hf]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhfi : \u00ac\u222b\u207b (x : \u03b1), f x \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), g a - f a + f a \u2202\u03bc\n[PROOFSTEP]\nexact lintegral_mono fun x => le_tsub_add\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh_le : f \u2264\u1d50[\u03bc] g\nh : \u2203\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 g x\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc < \u222b\u207b (x : \u03b1), g x \u2202\u03bc\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh_le : f \u2264\u1d50[\u03bc] g\nh : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\u22a2 \u00ac\u2203\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 g x\n[PROOFSTEP]\nsimp only [not_frequently, Ne.def, Classical.not_not]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh_le : f \u2264\u1d50[\u03bc] g\nh : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x = g x\n[PROOFSTEP]\nexact ae_eq_of_ae_le_of_lintegral_le h_le hfi hg h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u03bc \u2260 0\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < g x\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc < \u222b\u207b (x : \u03b1), g x \u2202\u03bc\n[PROOFSTEP]\nrw [Ne.def, \u2190 Measure.measure_univ_eq_zero] at h\u03bc \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < g x\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc < \u222b\u207b (x : \u03b1), g x \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_strict_mono_of_ae_le_of_ae_lt_on hg hfi (ae_le_of_ae_lt h) h\u03bc _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u00ac\u2191\u2191\u03bc univ = 0\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < g x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 univ \u2192 f x < g x\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 0\nhg : Measurable g\nhfi : \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2260 \u22a4\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x < g x\n\u22a2 Measure.restrict \u03bc s \u2260 0\n[PROOFSTEP]\nsimp [hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_meas : \u2200 (n : \u2115), Measurable (f n)\nh_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n\nh_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4\nfn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 (n : \u2115), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nfn_le_f0' : \u2a05 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\nn : \u2115\na : \u03b1\nh : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\n\u22a2 f n a \u2264 f 0 a\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_meas : \u2200 (n : \u2115), Measurable (f n)\nh_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n\nh_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4\nfn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 (n : \u2115), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nfn_le_f0' : \u2a05 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\na : \u03b1\nh : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\n\u22a2 f Nat.zero a \u2264 f 0 a\n[PROOFSTEP]\n{exact le_rfl\n}\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_meas : \u2200 (n : \u2115), Measurable (f n)\nh_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n\nh_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4\nfn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 (n : \u2115), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nfn_le_f0' : \u2a05 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\na : \u03b1\nh : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\n\u22a2 f Nat.zero a \u2264 f 0 a\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_meas : \u2200 (n : \u2115), Measurable (f n)\nh_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n\nh_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4\nfn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 (n : \u2115), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nfn_le_f0' : \u2a05 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\na : \u03b1\nh : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\nn : \u2115\nih : f n a \u2264 f 0 a\n\u22a2 f (Nat.succ n) a \u2264 f 0 a\n[PROOFSTEP]\n{exact le_trans (h n) ih\n}\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_meas : \u2200 (n : \u2115), Measurable (f n)\nh_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n\nh_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4\nfn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 (n : \u2115), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nfn_le_f0' : \u2a05 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\na : \u03b1\nh : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\nn : \u2115\nih : f n a \u2264 f 0 a\n\u22a2 f (Nat.succ n) a \u2264 f 0 a\n[PROOFSTEP]\nexact le_trans (h n) ih\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_meas : \u2200 (n : \u2115), AEMeasurable (f n)\n\u22a2 \u222b\u207b (a : \u03b1), liminf (fun n => f n a) atTop \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n), f i a \u2202\u03bc\n[PROOFSTEP]\nsimp only [liminf_eq_iSup_iInf_of_nat]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2a05 (n : \u2115), \u222b\u207b (a : \u03b1), \u2a06 (i : \u2115) (_ : i \u2265 n), f i a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2a05 (n : \u2115), \u2a06 (i : \u2115) (_ : i \u2265 n), f i a \u2202\u03bc\n[PROOFSTEP]\nrefine' (lintegral_iInf _ _ _).symm\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2200 (n : \u2115), Measurable fun a => \u2a06 (i : \u2115) (_ : i \u2265 n), f i a\n[PROOFSTEP]\nintro n\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\nn : \u2115\n\u22a2 Measurable fun a => \u2a06 (i : \u2115) (_ : i \u2265 n), f i a\n[PROOFSTEP]\nexact measurable_biSup _ (to_countable _) hf_meas\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 Antitone fun n a => \u2a06 (i : \u2115) (_ : i \u2265 n), f i a\n[PROOFSTEP]\nintro n m hnm a\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\nn m : \u2115\nhnm : n \u2264 m\na : \u03b1\n\u22a2 (fun n a => \u2a06 (i : \u2115) (_ : i \u2265 n), f i a) m a \u2264 (fun n a => \u2a06 (i : \u2115) (_ : i \u2265 n), f i a) n a\n[PROOFSTEP]\nexact iSup_le_iSup_of_subset fun i hi => le_trans hnm hi\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (i : \u2115) (_ : i \u2265 0), f i a \u2202\u03bc \u2260 \u22a4\n[PROOFSTEP]\nrefine' ne_top_of_le_ne_top h_fin (lintegral_mono_ae _)\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2a06 (i : \u2115) (_ : i \u2265 0), f i a \u2264 g a\n[PROOFSTEP]\nrefine' (ae_all_iff.2 h_bound).mono fun n hn => _\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\nn : \u03b1\nhn : \u2200 (i : \u2115), f i n \u2264 g n\n\u22a2 \u2a06 (i : \u2115) (_ : i \u2265 0), f i n \u2264 g n\n[PROOFSTEP]\nexact iSup_le fun i => iSup_le fun _ => hn i\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : \u2200 (n : \u2115), Measurable (f n)\nh_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g\nh_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2a05 (n : \u2115), \u2a06 (i : \u2115) (_ : i \u2265 n), f i a \u2202\u03bc = \u222b\u207b (a : \u03b1), limsup (fun n => f n a) atTop \u2202\u03bc\n[PROOFSTEP]\nsimp only [limsup_eq_iInf_iSup_of_nat]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nF : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200 (n : \u2115), AEMeasurable (F n)\nh_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), F n a \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\n[PROOFSTEP]\nhave : \u2200 n, \u222b\u207b a, F n a \u2202\u03bc = \u222b\u207b a, (hF_meas n).mk (F n) a \u2202\u03bc := fun n => lintegral_congr_ae (hF_meas n).ae_eq_mk\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nF : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200 (n : \u2115), AEMeasurable (F n)\nh_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nthis : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), F n a \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\n[PROOFSTEP]\nsimp_rw [this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nF : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200 (n : \u2115), AEMeasurable (F n)\nh_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nthis : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\n[PROOFSTEP]\napply tendsto_lintegral_of_dominated_convergence bound (fun n => (hF_meas n).measurable_mk) _ h_fin\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nF : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200 (n : \u2115), AEMeasurable (F n)\nh_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nthis : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (\ud835\udcdd (f a))\n[PROOFSTEP]\nhave : \u2200 n, \u2200\u1d50 a \u2202\u03bc, (hF_meas n).mk (F n) a = F n a := fun n => (hF_meas n).ae_eq_mk.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nF : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200 (n : \u2115), AEMeasurable (F n)\nh_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nthis\u271d : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc\nthis : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (\ud835\udcdd (f a))\n[PROOFSTEP]\nhave : \u2200\u1d50 a \u2202\u03bc, \u2200 n, (hF_meas n).mk (F n) a = F n a := ae_all_iff.mpr this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nF : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200 (n : \u2115), AEMeasurable (F n)\nh_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nthis\u271d\u00b9 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc\nthis\u271d : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\nthis : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (\ud835\udcdd (f a))\n[PROOFSTEP]\nfilter_upwards [this, h_lim] with a H H'\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nF : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200 (n : \u2115), AEMeasurable (F n)\nh_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nthis\u271d\u00b9 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc\nthis\u271d : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\nthis : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\na : \u03b1\nH : \u2200 (n : \u2115), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\nH' : Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\n\u22a2 Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (\ud835\udcdd (f a))\n[PROOFSTEP]\nsimp_rw [H]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nF : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200 (n : \u2115), AEMeasurable (F n)\nh_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nthis\u271d\u00b9 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc\nthis\u271d : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\nthis : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\na : \u03b1\nH : \u2200 (n : \u2115), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a\nH' : Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\n\u22a2 Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\n[PROOFSTEP]\nexact H'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nF : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200 (n : \u2115), AEMeasurable (F n)\nh_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nthis : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc\n\u22a2 \u2200 (n : \u2115), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) \u2264\u1d50[\u03bc] bound\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nF : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200 (n : \u2115), AEMeasurable (F n)\nh_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nthis : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc\nn : \u2115\n\u22a2 AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) \u2264\u1d50[\u03bc] bound\n[PROOFSTEP]\nfilter_upwards [h_bound n, (hF_meas n).ae_eq_mk] with a H H'\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nF : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200 (n : \u2115), AEMeasurable (F n)\nh_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a))\nthis : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc\nn : \u2115\na : \u03b1\nH : F n a \u2264 bound a\nH' : F n a = AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a\n\u22a2 AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2264 bound a\n[PROOFSTEP]\nrwa [H'] at H \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\n\u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), F n a \u2202\u03bc) l (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\n[PROOFSTEP]\nrw [tendsto_iff_seq_tendsto]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\n\u22a2 \u2200 (x : \u2115 \u2192 \u03b9), Tendsto x atTop l \u2192 Tendsto ((fun n => \u222b\u207b (a : \u03b1), F n a \u2202\u03bc) \u2218 x) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\n[PROOFSTEP]\nintro x xl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\n\u22a2 Tendsto ((fun n => \u222b\u207b (a : \u03b1), F n a \u2202\u03bc) \u2218 x) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\n[PROOFSTEP]\nhave hxl := by\n rw [tendsto_atTop'] at xl \n exact xl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\n\u22a2 ?m.1156041\n[PROOFSTEP]\nrw [tendsto_atTop'] at xl \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl\u271d : Tendsto x atTop l\nxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\n\u22a2 ?m.1156041\n[PROOFSTEP]\nexact xl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\n\u22a2 Tendsto ((fun n => \u222b\u207b (a : \u03b1), F n a \u2202\u03bc) \u2218 x) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\n[PROOFSTEP]\nhave h := inter_mem hF_meas h_bound\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nh : {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x} \u2208 l\n\u22a2 Tendsto ((fun n => \u222b\u207b (a : \u03b1), F n a \u2202\u03bc) \u2218 x) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\n[PROOFSTEP]\nreplace h := hxl _ h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nh :\n \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\n\u22a2 Tendsto ((fun n => \u222b\u207b (a : \u03b1), F n a \u2202\u03bc) \u2218 x) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\n[PROOFSTEP]\nrcases h with \u27e8k, h\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nk : \u2115\nh : \u2200 (b : \u2115), b \u2265 k \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\n\u22a2 Tendsto ((fun n => \u222b\u207b (a : \u03b1), F n a \u2202\u03bc) \u2218 x) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\n[PROOFSTEP]\nrw [\u2190 tendsto_add_atTop_iff_nat k]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nk : \u2115\nh : \u2200 (b : \u2115), b \u2265 k \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\n\u22a2 Tendsto (fun n => ((fun n => \u222b\u207b (a : \u03b1), F n a \u2202\u03bc) \u2218 x) (n + k)) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc))\n[PROOFSTEP]\nrefine' tendsto_lintegral_of_dominated_convergence _ _ _ _ _\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nk : \u2115\nh : \u2200 (b : \u2115), b \u2265 k \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\n\u22a2 \u03b1 \u2192 \u211d\u22650\u221e\n[PROOFSTEP]\nexact bound\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nk : \u2115\nh : \u2200 (b : \u2115), b \u2265 k \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\n\u22a2 \u2200 (n : \u2115), Measurable fun a => F (x (n + k)) a\n[PROOFSTEP]\nintro\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nk : \u2115\nh : \u2200 (b : \u2115), b \u2265 k \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\nn\u271d : \u2115\n\u22a2 Measurable fun a => F (x (n\u271d + k)) a\n[PROOFSTEP]\nrefine' (h _ _).1\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nk : \u2115\nh : \u2200 (b : \u2115), b \u2265 k \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\nn\u271d : \u2115\n\u22a2 n\u271d + k \u2265 k\n[PROOFSTEP]\nexact Nat.le_add_left _ _\n[GOAL]\ncase intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nk : \u2115\nh : \u2200 (b : \u2115), b \u2265 k \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\n\u22a2 \u2200 (n : \u2115), (fun a => F (x (n + k)) a) \u2264\u1d50[\u03bc] bound\n[PROOFSTEP]\nintro\n[GOAL]\ncase intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nk : \u2115\nh : \u2200 (b : \u2115), b \u2265 k \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\nn\u271d : \u2115\n\u22a2 (fun a => F (x (n\u271d + k)) a) \u2264\u1d50[\u03bc] bound\n[PROOFSTEP]\nrefine' (h _ _).2\n[GOAL]\ncase intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nk : \u2115\nh : \u2200 (b : \u2115), b \u2265 k \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\nn\u271d : \u2115\n\u22a2 n\u271d + k \u2265 k\n[PROOFSTEP]\nexact Nat.le_add_left _ _\n[GOAL]\ncase intro.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nk : \u2115\nh : \u2200 (b : \u2115), b \u2265 k \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\n\u22a2 \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.refine'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nk : \u2115\nh : \u2200 (b : \u2115), b \u2265 k \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F (x (n + k)) a) atTop (\ud835\udcdd (f a))\n[PROOFSTEP]\nrefine' h_lim.mono fun a h_lim => _\n[GOAL]\ncase intro.refine'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nk : \u2115\nh : \u2200 (b : \u2115), b \u2265 k \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\na : \u03b1\nh_lim : Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\n\u22a2 Tendsto (fun n => F (x (n + k)) a) atTop (\ud835\udcdd (f a))\n[PROOFSTEP]\napply @Tendsto.comp _ _ _ (fun n => x (n + k)) fun n => F n a\n[GOAL]\ncase intro.refine'_5.hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nk : \u2115\nh : \u2200 (b : \u2115), b \u2265 k \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\na : \u03b1\nh_lim : Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\n\u22a2 Tendsto (fun n => F n a) ?intro.refine'_5.y (\ud835\udcdd (f a))\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.refine'_5.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nk : \u2115\nh : \u2200 (b : \u2115), b \u2265 k \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\na : \u03b1\nh_lim : Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\n\u22a2 Tendsto (fun n => x (n + k)) atTop l\n[PROOFSTEP]\nrw [tendsto_add_atTop_iff_nat]\n[GOAL]\ncase intro.refine'_5.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nF : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nf bound : \u03b1 \u2192 \u211d\u22650\u221e\nhF_meas : \u2200\u1da0 (n : \u03b9) in l, Measurable (F n)\nh_bound : \u2200\u1da0 (n : \u03b9) in l, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a\nh_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4\nh_lim\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\nx : \u2115 \u2192 \u03b9\nxl : Tendsto x atTop l\nhxl : \u2200 (s : Set \u03b9), s \u2208 l \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 x b \u2208 s\nk : \u2115\nh : \u2200 (b : \u2115), b \u2265 k \u2192 x b \u2208 {x | (fun n => Measurable (F n)) x} \u2229 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, F n a \u2264 bound a) x}\na : \u03b1\nh_lim : Tendsto (fun n => F n a) l (\ud835\udcdd (f a))\n\u22a2 Tendsto x atTop l\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), Measurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (b : \u03b2), f b a \u2202\u03bc = \u2a06 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\ncases nonempty_encodable \u03b2\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), Measurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\nval\u271d : Encodable \u03b2\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (b : \u03b2), f b a \u2202\u03bc = \u2a06 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b2\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), Measurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (b : \u03b2), f b a \u2202\u03bc = \u2a06 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nsimp [iSup_of_empty]\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), Measurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (b : \u03b2), f b a \u2202\u03bc = \u2a06 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\ninhabit \u03b2\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), Measurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (b : \u03b2), f b a \u2202\u03bc = \u2a06 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nhave : \u2200 a, \u2a06 b, f b a = \u2a06 n, f (h_directed.sequence f n) a :=\n by\n intro a\n refine' le_antisymm (iSup_le fun b => _) (iSup_le fun n => le_iSup (fun n => f n a) _)\n exact le_iSup_of_le (encode b + 1) (h_directed.le_sequence b a)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), Measurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\n\u22a2 \u2200 (a : \u03b1), \u2a06 (b : \u03b2), f b a = \u2a06 (n : \u2115), f (Directed.sequence f h_directed n) a\n[PROOFSTEP]\nintro a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), Measurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\na : \u03b1\n\u22a2 \u2a06 (b : \u03b2), f b a = \u2a06 (n : \u2115), f (Directed.sequence f h_directed n) a\n[PROOFSTEP]\nrefine' le_antisymm (iSup_le fun b => _) (iSup_le fun n => le_iSup (fun n => f n a) _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), Measurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 f b a \u2264 \u2a06 (n : \u2115), f (Directed.sequence f h_directed n) a\n[PROOFSTEP]\nexact le_iSup_of_le (encode b + 1) (h_directed.le_sequence b a)\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), Measurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\nthis : \u2200 (a : \u03b1), \u2a06 (b : \u03b2), f b a = \u2a06 (n : \u2115), f (Directed.sequence f h_directed n) a\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (b : \u03b2), f b a \u2202\u03bc = \u2a06 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\ncalc\n \u222b\u207b a, \u2a06 b, f b a \u2202\u03bc = \u222b\u207b a, \u2a06 n, f (h_directed.sequence f n) a \u2202\u03bc := by simp only [this]\n _ = \u2a06 n, \u222b\u207b a, f (h_directed.sequence f n) a \u2202\u03bc := (lintegral_iSup (fun n => hf _) h_directed.sequence_mono)\n _ = \u2a06 b, \u222b\u207b a, f b a \u2202\u03bc := by\n refine' le_antisymm (iSup_le fun n => _) (iSup_le fun b => _)\n \u00b7 exact le_iSup (fun b => \u222b\u207b a, f b a \u2202\u03bc) _\n \u00b7 exact le_iSup_of_le (encode b + 1) (lintegral_mono <| h_directed.le_sequence b)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), Measurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\nthis : \u2200 (a : \u03b1), \u2a06 (b : \u03b2), f b a = \u2a06 (n : \u2115), f (Directed.sequence f h_directed n) a\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (b : \u03b2), f b a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2a06 (n : \u2115), f (Directed.sequence f h_directed n) a \u2202\u03bc\n[PROOFSTEP]\nsimp only [this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), Measurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\nthis : \u2200 (a : \u03b1), \u2a06 (b : \u03b2), f b a = \u2a06 (n : \u2115), f (Directed.sequence f h_directed n) a\n\u22a2 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f (Directed.sequence f h_directed n) a \u2202\u03bc = \u2a06 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nrefine' le_antisymm (iSup_le fun n => _) (iSup_le fun b => _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), Measurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\nthis : \u2200 (a : \u03b1), \u2a06 (b : \u03b2), f b a = \u2a06 (n : \u2115), f (Directed.sequence f h_directed n) a\nn : \u2115\n\u22a2 \u222b\u207b (a : \u03b1), f (Directed.sequence f h_directed n) a \u2202\u03bc \u2264 \u2a06 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nexact le_iSup (fun b => \u222b\u207b a, f b a \u2202\u03bc) _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), Measurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\nthis : \u2200 (a : \u03b1), \u2a06 (b : \u03b2), f b a = \u2a06 (n : \u2115), f (Directed.sequence f h_directed n) a\nb : \u03b2\n\u22a2 \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2264 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f (Directed.sequence f h_directed n) a \u2202\u03bc\n[PROOFSTEP]\nexact le_iSup_of_le (encode b + 1) (lintegral_mono <| h_directed.le_sequence b)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (b : \u03b2), f b a \u2202\u03bc = \u2a06 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [\u2190 iSup_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nlet p : \u03b1 \u2192 (\u03b2 \u2192 ENNReal) \u2192 Prop := fun x f' => Directed LE.le f'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nhave hp : \u2200\u1d50 x \u2202\u03bc, p x fun i => f i x := by\n filter_upwards [] with x i j\n obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := h_directed i j\n exact \u27e8z, hz\u2081 x, hz\u2082 x\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\n[PROOFSTEP]\nfilter_upwards [] with x i j\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nx : \u03b1\ni j : \u03b2\n\u22a2 \u2203 z, (fun i => f i x) i \u2264 (fun i => f i x) z \u2227 (fun i => f i x) j \u2264 (fun i => f i x) z\n[PROOFSTEP]\nobtain \u27e8z, hz\u2081, hz\u2082\u27e9 := h_directed i j\n[GOAL]\ncase h.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nx : \u03b1\ni j z : \u03b2\nhz\u2081 : f i \u2264 f z\nhz\u2082 : f j \u2264 f z\n\u22a2 \u2203 z, (fun i => f i x) i \u2264 (fun i => f i x) z \u2227 (fun i => f i x) j \u2264 (fun i => f i x) z\n[PROOFSTEP]\nexact \u27e8z, hz\u2081 x, hz\u2082 x\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nhave h_ae_seq_directed : Directed LE.le (aeSeq hf p) :=\n by\n intro b\u2081 b\u2082\n obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := h_directed b\u2081 b\u2082\n refine' \u27e8z, _, _\u27e9 <;>\n \u00b7 intro x\n by_cases hx : x \u2208 aeSeqSet hf p\n \u00b7 repeat' rw [aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx]\n apply_rules [hz\u2081, hz\u2082]\n \u00b7 simp only [aeSeq, hx, if_false]\n exact le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\n\u22a2 Directed LE.le (aeSeq hf p)\n[PROOFSTEP]\nintro b\u2081 b\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 : \u03b2\n\u22a2 \u2203 z, aeSeq hf p b\u2081 \u2264 aeSeq hf p z \u2227 aeSeq hf p b\u2082 \u2264 aeSeq hf p z\n[PROOFSTEP]\nobtain \u27e8z, hz\u2081, hz\u2082\u27e9 := h_directed b\u2081 b\u2082\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\n\u22a2 \u2203 z, aeSeq hf p b\u2081 \u2264 aeSeq hf p z \u2227 aeSeq hf p b\u2082 \u2264 aeSeq hf p z\n[PROOFSTEP]\nrefine' \u27e8z, _, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\n\u22a2 aeSeq hf p b\u2081 \u2264 aeSeq hf p z\n[PROOFSTEP]\nintro x\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\n\u22a2 aeSeq hf p b\u2081 x \u2264 aeSeq hf p z x\n[PROOFSTEP]\nby_cases hx : x \u2208 aeSeqSet hf p\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 aeSeq hf p b\u2081 x \u2264 aeSeq hf p z x\n[PROOFSTEP]\nrepeat' rw [aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 aeSeq hf p b\u2081 x \u2264 aeSeq hf p z x\n[PROOFSTEP]\nrw [aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 f b\u2081 x \u2264 aeSeq hf p z x\n[PROOFSTEP]\nrw [aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 f b\u2081 x \u2264 f z x\n[PROOFSTEP]\nrw [aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 f b\u2081 x \u2264 f z x\n[PROOFSTEP]\napply_rules [hz\u2081, hz\u2082]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : \u00acx \u2208 aeSeqSet hf p\n\u22a2 aeSeq hf p b\u2081 x \u2264 aeSeq hf p z x\n[PROOFSTEP]\nsimp only [aeSeq, hx, if_false]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : \u00acx \u2208 aeSeqSet hf p\n\u22a2 Nonempty.some (_ : Nonempty \u211d\u22650\u221e) \u2264 Nonempty.some (_ : Nonempty \u211d\u22650\u221e)\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\n\u22a2 aeSeq hf p b\u2082 \u2264 aeSeq hf p z\n[PROOFSTEP]\nintro x\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\n\u22a2 aeSeq hf p b\u2082 x \u2264 aeSeq hf p z x\n[PROOFSTEP]\nby_cases hx : x \u2208 aeSeqSet hf p\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 aeSeq hf p b\u2082 x \u2264 aeSeq hf p z x\n[PROOFSTEP]\nrepeat' rw [aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 aeSeq hf p b\u2082 x \u2264 aeSeq hf p z x\n[PROOFSTEP]\nrw [aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 f b\u2082 x \u2264 aeSeq hf p z x\n[PROOFSTEP]\nrw [aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 f b\u2082 x \u2264 f z x\n[PROOFSTEP]\nrw [aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\n\u22a2 f b\u2082 x \u2264 f z x\n[PROOFSTEP]\napply_rules [hz\u2081, hz\u2082]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : \u00acx \u2208 aeSeqSet hf p\n\u22a2 aeSeq hf p b\u2082 x \u2264 aeSeq hf p z x\n[PROOFSTEP]\nsimp only [aeSeq, hx, if_false]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nb\u2081 b\u2082 z : \u03b2\nhz\u2081 : f b\u2081 \u2264 f z\nhz\u2082 : f b\u2082 \u2264 f z\nx : \u03b1\nhx : \u00acx \u2208 aeSeqSet hf p\n\u22a2 Nonempty.some (_ : Nonempty \u211d\u22650\u221e) \u2264 Nonempty.some (_ : Nonempty \u211d\u22650\u221e)\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nconvert lintegral_iSup_directed_of_measurable (aeSeq.measurable hf p) h_ae_seq_directed using 1\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2a06 (b : \u03b2), aeSeq hf p b a \u2202?m.1165752\n[PROOFSTEP]\nsimp_rw [\u2190 iSup_apply]\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\n\u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc =\n \u222b\u207b (a : \u03b1), iSup (fun i => aeSeq hf (fun x f' => Directed LE.le f') i) a \u2202?m.1165752\n[PROOFSTEP]\nrw [lintegral_congr_ae (aeSeq.iSup hf hp).symm]\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\n\u22a2 \u2a06 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc = \u2a06 (b : \u03b2), \u222b\u207b (a : \u03b1), aeSeq hf p b a \u2202\u03bc\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e'_3.e_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\n\u22a2 (fun b => \u222b\u207b (a : \u03b1), f b a \u2202\u03bc) = fun b => \u222b\u207b (a : \u03b1), aeSeq hf p b a \u2202\u03bc\n[PROOFSTEP]\next1 b\n[GOAL]\ncase h.e'_3.e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\nb : \u03b2\n\u22a2 \u222b\u207b (a : \u03b1), f b a \u2202\u03bc = \u222b\u207b (a : \u03b1), aeSeq hf p b a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_congr_ae]\n[GOAL]\ncase h.e'_3.e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\nb : \u03b2\n\u22a2 (fun a => f b a) =\u1d50[\u03bc] fun a => aeSeq hf p b a\n[PROOFSTEP]\napply EventuallyEq.symm\n[GOAL]\ncase h.e'_3.e_s.h.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (b : \u03b2), AEMeasurable (f b)\nh_directed : Directed (fun x x_1 => x \u2264 x_1) f\np : \u03b1 \u2192 (\u03b2 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Directed LE.le f'\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x\nh_ae_seq_directed : Directed LE.le (aeSeq hf p)\nb : \u03b2\n\u22a2 (fun a => aeSeq hf p b a) =\u1d50[\u03bc] fun a => f b a\n[PROOFSTEP]\nrefine' aeSeq.aeSeq_n_eq_fun_n_ae hf hp _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\n\u22a2 \u222b\u207b (a : \u03b1), \u2211' (i : \u03b2), f i a \u2202\u03bc = \u2211' (i : \u03b2), \u222b\u207b (a : \u03b1), f i a \u2202\u03bc\n[PROOFSTEP]\nsimp only [ENNReal.tsum_eq_iSup_sum]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 (s : Finset \u03b2), \u2211 i in s, f i a \u2202\u03bc = \u2a06 (s : Finset \u03b2), \u2211 i in s, \u222b\u207b (a : \u03b1), f i a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_iSup_directed]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\n\u22a2 \u2a06 (b : Finset \u03b2), \u222b\u207b (a : \u03b1), \u2211 i in b, f i a \u2202\u03bc = \u2a06 (s : Finset \u03b2), \u2211 i in s, \u222b\u207b (a : \u03b1), f i a \u2202\u03bc\n[PROOFSTEP]\nsimp [lintegral_finset_sum' _ fun i _ => hf i]\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\n\u22a2 \u2200 (b : Finset \u03b2), AEMeasurable fun a => \u2211 i in b, f i a\n[PROOFSTEP]\nintro b\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\nb : Finset \u03b2\n\u22a2 AEMeasurable fun a => \u2211 i in b, f i a\n[PROOFSTEP]\nexact Finset.aemeasurable_sum _ fun i _ => hf i\n[GOAL]\ncase h_directed\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\n\u22a2 Directed (fun x x_1 => x \u2264 x_1) fun s a => \u2211 i in s, f i a\n[PROOFSTEP]\nintro s t\n[GOAL]\ncase h_directed\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\ns t : Finset \u03b2\n\u22a2 \u2203 z,\n (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) s) ((fun s a => \u2211 i in s, f i a) z) \u2227\n (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) t) ((fun s a => \u2211 i in s, f i a) z)\n[PROOFSTEP]\nuse s \u222a t\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\ns t : Finset \u03b2\n\u22a2 (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) s) ((fun s a => \u2211 i in s, f i a) (s \u222a t)) \u2227\n (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) t) ((fun s a => \u2211 i in s, f i a) (s \u222a t))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\ns t : Finset \u03b2\n\u22a2 (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) s) ((fun s a => \u2211 i in s, f i a) (s \u222a t))\n[PROOFSTEP]\nexact fun a => Finset.sum_le_sum_of_subset (Finset.subset_union_left _ _)\n[GOAL]\ncase h.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b2), AEMeasurable (f i)\ns t : Finset \u03b2\n\u22a2 (fun x x_1 => x \u2264 x_1) ((fun s a => \u2211 i in s, f i a) t) ((fun s a => \u2211 i in s, f i a) (s \u222a t))\n[PROOFSTEP]\nexact fun a => Finset.sum_le_sum_of_subset (Finset.subset_union_right _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\ns : \u03b2 \u2192 Set \u03b1\nhm : \u2200 (i : \u03b2), NullMeasurableSet (s i)\nhd : Pairwise (AEDisjoint \u03bc on s)\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1) in \u22c3 (i : \u03b2), s i, f a \u2202\u03bc = \u2211' (i : \u03b2), \u222b\u207b (a : \u03b1) in s i, f a \u2202\u03bc\n[PROOFSTEP]\nsimp only [Measure.restrict_iUnion_ae hd hm, lintegral_sum_measure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nt : Set \u03b2\ns : \u03b2 \u2192 Set \u03b1\nht : Set.Countable t\nhm : \u2200 (i : \u03b2), i \u2208 t \u2192 NullMeasurableSet (s i)\nhd : Set.Pairwise t (AEDisjoint \u03bc on s)\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1) in \u22c3 (i : \u03b2) (_ : i \u2208 t), s i, f a \u2202\u03bc = \u2211' (i : \u2191t), \u222b\u207b (a : \u03b1) in s \u2191i, f a \u2202\u03bc\n[PROOFSTEP]\nhaveI := ht.toEncodable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nt : Set \u03b2\ns : \u03b2 \u2192 Set \u03b1\nht : Set.Countable t\nhm : \u2200 (i : \u03b2), i \u2208 t \u2192 NullMeasurableSet (s i)\nhd : Set.Pairwise t (AEDisjoint \u03bc on s)\nf : \u03b1 \u2192 \u211d\u22650\u221e\nthis : Encodable \u2191t\n\u22a2 \u222b\u207b (a : \u03b1) in \u22c3 (i : \u03b2) (_ : i \u2208 t), s i, f a \u2202\u03bc = \u2211' (i : \u2191t), \u222b\u207b (a : \u03b1) in s \u2191i, f a \u2202\u03bc\n[PROOFSTEP]\nrw [biUnion_eq_iUnion, lintegral_iUnion\u2080 (SetCoe.forall'.1 hm) (hd.subtype _ _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Finset \u03b2\nt : \u03b2 \u2192 Set \u03b1\nhd : Set.Pairwise (\u2191s) (AEDisjoint \u03bc on t)\nhm : \u2200 (b : \u03b2), b \u2208 s \u2192 NullMeasurableSet (t b)\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1) in \u22c3 (b : \u03b2) (_ : b \u2208 s), t b, f a \u2202\u03bc = \u2211 b in s, \u222b\u207b (a : \u03b1) in t b, f a \u2202\u03bc\n[PROOFSTEP]\nsimp only [\u2190 Finset.mem_coe, lintegral_biUnion\u2080 s.countable_toSet hm hd, \u2190 Finset.tsum_subtype']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\ns : \u03b2 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1) in \u22c3 (i : \u03b2), s i, f a \u2202\u03bc \u2264 \u2211' (i : \u03b2), \u222b\u207b (a : \u03b1) in s i, f a \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lintegral_sum_measure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\ns : \u03b2 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1) in \u22c3 (i : \u03b2), s i, f a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a \u2202sum fun i => Measure.restrict \u03bc (s i)\n[PROOFSTEP]\nexact lintegral_mono' restrict_iUnion_le le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nA B : Set \u03b1\nhB : MeasurableSet B\nhAB : Disjoint A B\n\u22a2 \u222b\u207b (a : \u03b1) in A \u222a B, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in A, f a \u2202\u03bc + \u222b\u207b (a : \u03b1) in B, f a \u2202\u03bc\n[PROOFSTEP]\nrw [restrict_union hAB hB, lintegral_add_measure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns t : Set \u03b1\n\u22a2 \u222b\u207b (a : \u03b1) in s \u222a t, f a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc + \u222b\u207b (a : \u03b1) in t, f a \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lintegral_add_measure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns t : Set \u03b1\n\u22a2 \u222b\u207b (a : \u03b1) in s \u222a t, f a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a \u2202(Measure.restrict \u03bc s + Measure.restrict \u03bc t)\n[PROOFSTEP]\nexact lintegral_mono' (restrict_union_le _ _) le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nB : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nA : Set \u03b1\nhB : MeasurableSet B\n\u22a2 \u222b\u207b (x : \u03b1) in A \u2229 B, f x \u2202\u03bc + \u222b\u207b (x : \u03b1) in A \\ B, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in A, f x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lintegral_add_measure, restrict_inter_add_diff _ hB]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nA : Set \u03b1\nhA : MeasurableSet A\n\u22a2 \u222b\u207b (x : \u03b1) in A, f x \u2202\u03bc + \u222b\u207b (x : \u03b1) in A\u1d9c, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lintegral_add_measure, Measure.restrict_add_restrict_compl hA]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\n\u22a2 \u222b\u207b (x : \u03b1), max (f x) (g x) \u2202\u03bc = \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in {x | g x < f x}, f x \u2202\u03bc\n[PROOFSTEP]\nhave hm : MeasurableSet {x | f x \u2264 g x} := measurableSet_le hf hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nhm : MeasurableSet {x | f x \u2264 g x}\n\u22a2 \u222b\u207b (x : \u03b1), max (f x) (g x) \u2202\u03bc = \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in {x | g x < f x}, f x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lintegral_add_compl (fun x => max (f x) (g x)) hm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nhm : MeasurableSet {x | f x \u2264 g x}\n\u22a2 \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}, max (f x) (g x) \u2202\u03bc + \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}\u1d9c, max (f x) (g x) \u2202\u03bc =\n \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in {x | g x < f x}, f x \u2202\u03bc\n[PROOFSTEP]\nsimp only [\u2190 compl_setOf, \u2190 not_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nhm : MeasurableSet {x | f x \u2264 g x}\n\u22a2 \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}, max (f x) (g x) \u2202\u03bc + \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}\u1d9c, max (f x) (g x) \u2202\u03bc =\n \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in {a | f a \u2264 g a}\u1d9c, f x \u2202\u03bc\n[PROOFSTEP]\nrefine' congr_arg\u2082 (\u00b7 + \u00b7) (set_lintegral_congr_fun hm _) (set_lintegral_congr_fun hm.compl _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nhm : MeasurableSet {x | f x \u2264 g x}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 {x | f x \u2264 g x} \u2192 max (f x) (g x) = g x\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nhm : MeasurableSet {x | f x \u2264 g x}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 {x | f x \u2264 g x}\u1d9c \u2192 max (f x) (g x) = f x\n[PROOFSTEP]\nexacts [ae_of_all _ fun x => max_eq_right (a := f x) (b := g x),\n ae_of_all _ fun x (hx : \u00acf x \u2264 g x) => max_eq_left (not_le.1 hx).le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\ns : Set \u03b1\n\u22a2 \u222b\u207b (x : \u03b1) in s, max (f x) (g x) \u2202\u03bc =\n \u222b\u207b (x : \u03b1) in s \u2229 {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in s \u2229 {x | g x < f x}, f x \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_max hf hg, restrict_restrict, restrict_restrict, inter_comm s, inter_comm s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\ns : Set \u03b1\n\u22a2 MeasurableSet {x | g x < f x}\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\ns : Set \u03b1\n\u22a2 MeasurableSet {x | f x \u2264 g x}\n[PROOFSTEP]\nexacts [measurableSet_lt hg hf, measurableSet_le hf hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\n\u22a2 \u222b\u207b (a : \u03b2), f a \u2202Measure.map g \u03bc = \u222b\u207b (a : \u03b1), f (g a) \u2202\u03bc\n[PROOFSTEP]\nerw [lintegral_eq_iSup_eapprox_lintegral hf, lintegral_eq_iSup_eapprox_lintegral (hf.comp hg)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\n\u22a2 \u2a06 (n : \u2115), SimpleFunc.lintegral (eapprox f n) (Measure.map g \u03bc) =\n \u2a06 (n : \u2115), SimpleFunc.lintegral (eapprox (f \u2218 g) n) \u03bc\n[PROOFSTEP]\ncongr with n : 1\n[GOAL]\ncase e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\nn : \u2115\n\u22a2 SimpleFunc.lintegral (eapprox f n) (Measure.map g \u03bc) = SimpleFunc.lintegral (eapprox (f \u2218 g) n) \u03bc\n[PROOFSTEP]\nconvert SimpleFunc.lintegral_map _ hg\n[GOAL]\ncase h.e'_3.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\nn : \u2115\n\u22a2 eapprox (f \u2218 g) n = comp (eapprox f n) g hg\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h.e'_3.h.e'_3.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\nn : \u2115\nx : \u03b1\n\u22a2 \u2191(eapprox (f \u2218 g) n) x = \u2191(comp (eapprox f n) g hg) x\n[PROOFSTEP]\nsimp only [eapprox_comp hf hg, coe_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\nhg : AEMeasurable g\n\u22a2 \u222b\u207b (a : \u03b2), AEMeasurable.mk f hf a \u2202Measure.map g \u03bc =\n \u222b\u207b (a : \u03b2), AEMeasurable.mk f hf a \u2202Measure.map (AEMeasurable.mk g hg) \u03bc\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_\u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\nhg : AEMeasurable g\n\u22a2 Measure.map g \u03bc = Measure.map (AEMeasurable.mk g hg) \u03bc\n[PROOFSTEP]\nexact Measure.map_congr hg.ae_eq_mk\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhg : Measurable g\n\u22a2 \u222b\u207b (a : \u03b2), f a \u2202Measure.map g \u03bc \u2264 \u222b\u207b (a : \u03b1), f (g a) \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 iSup_lintegral_measurable_le_eq_lintegral, \u2190 iSup_lintegral_measurable_le_eq_lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhg : Measurable g\n\u22a2 \u2a06 (g_1 : \u03b2 \u2192 \u211d\u22650\u221e) (_ : Measurable g_1) (_ : g_1 \u2264 fun a => f a), \u222b\u207b (a : \u03b2), g_1 a \u2202Measure.map g \u03bc \u2264\n \u2a06 (g_1 : \u03b1 \u2192 \u211d\u22650\u221e) (_ : Measurable g_1) (_ : g_1 \u2264 fun a => f (g a)), \u222b\u207b (a : \u03b1), g_1 a \u2202\u03bc\n[PROOFSTEP]\nrefine' iSup\u2082_le fun i hi => iSup_le fun h'i => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhg : Measurable g\ni : \u03b2 \u2192 \u211d\u22650\u221e\nhi : Measurable i\nh'i : i \u2264 fun a => f a\n\u22a2 \u222b\u207b (a : \u03b2), i a \u2202Measure.map g \u03bc \u2264\n \u2a06 (g_1 : \u03b1 \u2192 \u211d\u22650\u221e) (_ : Measurable g_1) (_ : g_1 \u2264 fun a => f (g a)), \u222b\u207b (a : \u03b1), g_1 a \u2202\u03bc\n[PROOFSTEP]\nrefine' le_iSup\u2082_of_le (i \u2218 g) (hi.comp hg) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\nhg : Measurable g\ni : \u03b2 \u2192 \u211d\u22650\u221e\nhi : Measurable i\nh'i : i \u2264 fun a => f a\n\u22a2 \u222b\u207b (a : \u03b2), i a \u2202Measure.map g \u03bc \u2264 \u2a06 (_ : i \u2218 g \u2264 fun a => f (g a)), \u222b\u207b (a : \u03b1), (i \u2218 g) a \u2202\u03bc\n[PROOFSTEP]\nexact le_iSup_of_le (fun x => h'i (g x)) (le_of_eq (lintegral_map hi hg))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhs : MeasurableSet s\nhf : Measurable f\nhg : Measurable g\n\u22a2 \u222b\u207b (y : \u03b2) in s, f y \u2202Measure.map g \u03bc = \u222b\u207b (x : \u03b1) in g \u207b\u00b9' s, f (g x) \u2202\u03bc\n[PROOFSTEP]\nrw [restrict_map hg hs, lintegral_map hf hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhf : Measurable f\nhs : MeasurableSet s\nc : \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), indicator s (fun x => c) (f a) \u2202\u03bc = c * \u2191\u2191\u03bc (f \u207b\u00b9' s)\n[PROOFSTEP]\nerw [lintegral_comp (measurable_const.indicator hs) hf, lintegral_indicator_const hs, Measure.map_apply hf hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurableEmbedding g\nf : \u03b2 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b2), f a \u2202Measure.map g \u03bc = \u222b\u207b (a : \u03b1), f (g a) \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral, lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurableEmbedding g\nf : \u03b2 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 (g_1 : \u03b2 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g_1 \u2264 fun a => f a), SimpleFunc.lintegral g_1 (Measure.map g \u03bc) =\n \u2a06 (g_1 : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g_1 \u2264 fun a => f (g a)), SimpleFunc.lintegral g_1 \u03bc\n[PROOFSTEP]\nrefine' le_antisymm (iSup\u2082_le fun f\u2080 hf\u2080 => _) (iSup\u2082_le fun f\u2080 hf\u2080 => _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurableEmbedding g\nf : \u03b2 \u2192 \u211d\u22650\u221e\nf\u2080 : \u03b2 \u2192\u209b \u211d\u22650\u221e\nhf\u2080 : \u2191f\u2080 \u2264 fun a => f a\n\u22a2 SimpleFunc.lintegral f\u2080 (Measure.map g \u03bc) \u2264\n \u2a06 (g_1 : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g_1 \u2264 fun a => f (g a)), SimpleFunc.lintegral g_1 \u03bc\n[PROOFSTEP]\nrw [SimpleFunc.lintegral_map _ hg.measurable]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurableEmbedding g\nf : \u03b2 \u2192 \u211d\u22650\u221e\nf\u2080 : \u03b2 \u2192\u209b \u211d\u22650\u221e\nhf\u2080 : \u2191f\u2080 \u2264 fun a => f a\n\u22a2 SimpleFunc.lintegral (comp f\u2080 g (_ : Measurable g)) \u03bc \u2264\n \u2a06 (g_1 : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g_1 \u2264 fun a => f (g a)), SimpleFunc.lintegral g_1 \u03bc\n[PROOFSTEP]\nhave : (f\u2080.comp g hg.measurable : \u03b1 \u2192 \u211d\u22650\u221e) \u2264 f \u2218 g := fun x => hf\u2080 (g x)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurableEmbedding g\nf : \u03b2 \u2192 \u211d\u22650\u221e\nf\u2080 : \u03b2 \u2192\u209b \u211d\u22650\u221e\nhf\u2080 : \u2191f\u2080 \u2264 fun a => f a\nthis : \u2191(comp f\u2080 g (_ : Measurable g)) \u2264 f \u2218 g\n\u22a2 SimpleFunc.lintegral (comp f\u2080 g (_ : Measurable g)) \u03bc \u2264\n \u2a06 (g_1 : \u03b1 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g_1 \u2264 fun a => f (g a)), SimpleFunc.lintegral g_1 \u03bc\n[PROOFSTEP]\nexact le_iSup_of_le (comp f\u2080 g hg.measurable) (by exact le_iSup (\u03b1 := \u211d\u22650\u221e) _ this)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurableEmbedding g\nf : \u03b2 \u2192 \u211d\u22650\u221e\nf\u2080 : \u03b2 \u2192\u209b \u211d\u22650\u221e\nhf\u2080 : \u2191f\u2080 \u2264 fun a => f a\nthis : \u2191(comp f\u2080 g (_ : Measurable g)) \u2264 f \u2218 g\n\u22a2 SimpleFunc.lintegral (comp f\u2080 g (_ : Measurable g)) \u03bc \u2264\n \u2a06 (_ : \u2191(comp f\u2080 g (_ : Measurable g)) \u2264 fun a => f (g a)), SimpleFunc.lintegral (comp f\u2080 g (_ : Measurable g)) \u03bc\n[PROOFSTEP]\nexact le_iSup (\u03b1 := \u211d\u22650\u221e) _ this\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurableEmbedding g\nf : \u03b2 \u2192 \u211d\u22650\u221e\nf\u2080 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhf\u2080 : \u2191f\u2080 \u2264 fun a => f (g a)\n\u22a2 SimpleFunc.lintegral f\u2080 \u03bc \u2264 \u2a06 (g_1 : \u03b2 \u2192\u209b \u211d\u22650\u221e) (_ : \u2191g_1 \u2264 fun a => f a), SimpleFunc.lintegral g_1 (Measure.map g \u03bc)\n[PROOFSTEP]\nrw [\u2190 f\u2080.extend_comp_eq hg (const _ 0), \u2190 SimpleFunc.lintegral_map, \u2190 SimpleFunc.lintegral_eq_lintegral, \u2190 lintegral]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurableEmbedding g\nf : \u03b2 \u2192 \u211d\u22650\u221e\nf\u2080 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhf\u2080 : \u2191f\u2080 \u2264 fun a => f (g a)\n\u22a2 \u222b\u207b (a : \u03b2), \u2191(SimpleFunc.extend f\u2080 g hg (const \u03b2 0)) a \u2202Measure.map g \u03bc \u2264 \u222b\u207b (a : \u03b2), f a \u2202Measure.map g \u03bc\n[PROOFSTEP]\nrefine' lintegral_mono_ae (hg.ae_map_iff.2 <| eventually_of_forall fun x => _)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurableEmbedding g\nf : \u03b2 \u2192 \u211d\u22650\u221e\nf\u2080 : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhf\u2080 : \u2191f\u2080 \u2264 fun a => f (g a)\nx : \u03b1\n\u22a2 \u2191(SimpleFunc.extend f\u2080 g hg (const \u03b2 0)) (g x) \u2264 f (g x)\n[PROOFSTEP]\nexact (extend_apply _ _ _ _).trans_le (hf\u2080 _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nmb : MeasurableSpace \u03b2\n\u03bd : Measure \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nf : \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1), f (g a) \u2202\u03bc = \u222b\u207b (b : \u03b2), f b \u2202\u03bd\n[PROOFSTEP]\nrw [\u2190 hg.map_eq, lintegral_map hf hg.measurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nmb : MeasurableSpace \u03b2\n\u03bd : Measure \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nhge : MeasurableEmbedding g\nf : \u03b2 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f (g a) \u2202\u03bc = \u222b\u207b (b : \u03b2), f b \u2202\u03bd\n[PROOFSTEP]\nrw [\u2190 hg.map_eq, hge.lintegral_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nmb : MeasurableSpace \u03b2\n\u03bd : Measure \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\ns : Set \u03b2\nhs : MeasurableSet s\nf : \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1) in g \u207b\u00b9' s, f (g a) \u2202\u03bc = \u222b\u207b (b : \u03b2) in s, f b \u2202\u03bd\n[PROOFSTEP]\nrw [\u2190 hg.map_eq, set_lintegral_map hs hf hg.measurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nmb : MeasurableSpace \u03b2\n\u03bd : Measure \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nhge : MeasurableEmbedding g\nf : \u03b2 \u2192 \u211d\u22650\u221e\ns : Set \u03b2\n\u22a2 \u222b\u207b (a : \u03b1) in g \u207b\u00b9' s, f (g a) \u2202\u03bc = \u222b\u207b (b : \u03b2) in s, f b \u2202\u03bd\n[PROOFSTEP]\nrw [\u2190 hg.map_eq, hge.restrict_map, hge.lintegral_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nmb : MeasurableSpace \u03b2\n\u03bd : Measure \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nhge : MeasurableEmbedding g\nf : \u03b2 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\n\u22a2 \u222b\u207b (a : \u03b1) in s, f (g a) \u2202\u03bc = \u222b\u207b (b : \u03b2) in g '' s, f b \u2202\u03bd\n[PROOFSTEP]\nrw [\u2190 hg.set_lintegral_comp_preimage_emb hge, preimage_image_eq _ hge.injective]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202dirac a = f a\n[PROOFSTEP]\nsimp [lintegral_congr_ae (ae_eq_dirac' hf)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202dirac a = f a\n[PROOFSTEP]\nsimp [lintegral_congr_ae (ae_eq_dirac f)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : Decidable (a \u2208 s)\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202dirac a = if a \u2208 s then f a else 0\n[PROOFSTEP]\nrw [restrict_dirac' hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : Decidable (a \u2208 s)\n\u22a2 (\u222b\u207b (x : \u03b1), f x \u2202if a \u2208 s then dirac a else 0) = if a \u2208 s then f a else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : Decidable (a \u2208 s)\nh\u271d : a \u2208 s\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202dirac a = f a\n[PROOFSTEP]\nexact lintegral_dirac' _ hf\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : Decidable (a \u2208 s)\nh\u271d : \u00aca \u2208 s\n\u22a2 \u222b\u207b (x : \u03b1), f x \u22020 = 0\n[PROOFSTEP]\nexact lintegral_zero_measure _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : Decidable (a \u2208 s)\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202dirac a = if a \u2208 s then f a else 0\n[PROOFSTEP]\nrw [restrict_dirac]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : Decidable (a \u2208 s)\n\u22a2 (\u222b\u207b (x : \u03b1), f x \u2202if a \u2208 s then dirac a else 0) = if a \u2208 s then f a else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : Decidable (a \u2208 s)\nh\u271d : a \u2208 s\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202dirac a = f a\n[PROOFSTEP]\nexact lintegral_dirac _ _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : Decidable (a \u2208 s)\nh\u271d : \u00aca \u2208 s\n\u22a2 \u222b\u207b (x : \u03b1), f x \u22020 = 0\n[PROOFSTEP]\nexact lintegral_zero_measure _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202count = \u2211' (a : \u03b1), f a\n[PROOFSTEP]\nrw [count, lintegral_sum_measure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u2211' (i : \u03b1), \u222b\u207b (a : \u03b1), f a \u2202dirac i = \u2211' (a : \u03b1), f a\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 (fun i => \u222b\u207b (a : \u03b1), f a \u2202dirac i) = fun a => f a\n[PROOFSTEP]\nexact funext fun a => lintegral_dirac' a hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202count = \u2211' (a : \u03b1), f a\n[PROOFSTEP]\nrw [count, lintegral_sum_measure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2211' (i : \u03b1), \u222b\u207b (a : \u03b1), f a \u2202dirac i = \u2211' (a : \u03b1), f a\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 (fun i => \u222b\u207b (a : \u03b1), f a \u2202dirac i) = fun a => f a\n[PROOFSTEP]\nexact funext fun a => lintegral_dirac a f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nc : \u211d\u22650\u221e\n\u22a2 \u2211' (x : \u03b1), c = c * \u2191\u2191count univ\n[PROOFSTEP]\nrw [\u2190 lintegral_count, lintegral_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\u221e\na_mble : Measurable a\nc : \u211d\u22650\u221e\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\u221e\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u03b5_ne_top : \u03b5 \u2260 \u22a4\n\u22a2 \u2191\u2191count {i | \u03b5 \u2264 a i} \u2264 c / \u03b5\n[PROOFSTEP]\nrw [\u2190 lintegral_count] at tsum_le_c \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\u221e\na_mble : Measurable a\nc : \u211d\u22650\u221e\ntsum_le_c\u271d : \u2211' (i : \u03b1), a i \u2264 c\ntsum_le_c : \u222b\u207b (a_1 : \u03b1), a a_1 \u2202count \u2264 c\n\u03b5 : \u211d\u22650\u221e\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u03b5_ne_top : \u03b5 \u2260 \u22a4\n\u22a2 \u2191\u2191count {i | \u03b5 \u2264 a i} \u2264 c / \u03b5\n[PROOFSTEP]\napply (MeasureTheory.meas_ge_le_lintegral_div a_mble.aemeasurable \u03b5_ne_zero \u03b5_ne_top).trans\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\u221e\na_mble : Measurable a\nc : \u211d\u22650\u221e\ntsum_le_c\u271d : \u2211' (i : \u03b1), a i \u2264 c\ntsum_le_c : \u222b\u207b (a_1 : \u03b1), a a_1 \u2202count \u2264 c\n\u03b5 : \u211d\u22650\u221e\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u03b5_ne_top : \u03b5 \u2260 \u22a4\n\u22a2 (\u222b\u207b (a_1 : \u03b1), a a_1 \u2202count) / \u03b5 \u2264 c / \u03b5\n[PROOFSTEP]\nexact ENNReal.div_le_div tsum_le_c rfl.le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\na_mble : Measurable a\na_summable : Summable a\nc : \u211d\u22650\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u22a2 \u2191\u2191count {i | \u03b5 \u2264 a i} \u2264 \u2191c / \u2191\u03b5\n[PROOFSTEP]\nrw [show (fun i => \u03b5 \u2264 a i) = fun i => (\u03b5 : \u211d\u22650\u221e) \u2264 ((\u2191) \u2218 a) i\n by\n funext i\n simp only [ENNReal.coe_le_coe, Function.comp]]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\na_mble : Measurable a\na_summable : Summable a\nc : \u211d\u22650\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u22a2 (fun i => \u03b5 \u2264 a i) = fun i => \u2191\u03b5 \u2264 (ENNReal.some \u2218 a) i\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\na_mble : Measurable a\na_summable : Summable a\nc : \u211d\u22650\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\n\u03b5_ne_zero : \u03b5 \u2260 0\ni : \u03b1\n\u22a2 (\u03b5 \u2264 a i) = (\u2191\u03b5 \u2264 (ENNReal.some \u2218 a) i)\n[PROOFSTEP]\nsimp only [ENNReal.coe_le_coe, Function.comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\na_mble : Measurable a\na_summable : Summable a\nc : \u211d\u22650\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u22a2 \u2191\u2191count {i | \u2191\u03b5 \u2264 (ENNReal.some \u2218 a) i} \u2264 \u2191c / \u2191\u03b5\n[PROOFSTEP]\napply\n ENNReal.count_const_le_le_of_tsum_le (measurable_coe_nnreal_ennreal.comp a_mble) _ (by exact_mod_cast \u03b5_ne_zero)\n (@ENNReal.coe_ne_top \u03b5)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\na_mble : Measurable a\na_summable : Summable a\nc : \u211d\u22650\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u22a2 \u2191\u03b5 \u2260 0\n[PROOFSTEP]\nexact_mod_cast \u03b5_ne_zero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\na_mble : Measurable a\na_summable : Summable a\nc : \u211d\u22650\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u22a2 \u2211' (i : \u03b1), (ENNReal.some \u2218 a) i \u2264 \u2191c\n[PROOFSTEP]\nconvert ENNReal.coe_le_coe.mpr tsum_le_c\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1 \u2192 \u211d\u22650\na_mble : Measurable a\na_summable : Summable a\nc : \u211d\u22650\ntsum_le_c : \u2211' (i : \u03b1), a i \u2264 c\n\u03b5 : \u211d\u22650\n\u03b5_ne_zero : \u03b5 \u2260 0\n\u22a2 \u2211' (i : \u03b1), (ENNReal.some \u2218 a) i = \u2191(\u2211' (i : \u03b1), a i)\n[PROOFSTEP]\nerw [ENNReal.tsum_coe_eq a_summable.hasSum]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u2211' (a : \u03b1), f a * \u2191\u2191\u03bc {a}\n[PROOFSTEP]\nconv_lhs => rw [\u2190 sum_smul_dirac \u03bc, lintegral_sum_measure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n| \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 sum_smul_dirac \u03bc, lintegral_sum_measure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n| \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 sum_smul_dirac \u03bc, lintegral_sum_measure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n| \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 sum_smul_dirac \u03bc, lintegral_sum_measure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2211' (i : \u03b1), \u222b\u207b (a : \u03b1), f a \u2202\u2191\u2191\u03bc {i} \u2022 dirac i = \u2211' (a : \u03b1), f a * \u2191\u2191\u03bc {a}\n[PROOFSTEP]\ncongr 1 with a : 1\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\na : \u03b1\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u2191\u2191\u03bc {a} \u2022 dirac a = f a * \u2191\u2191\u03bc {a}\n[PROOFSTEP]\nrw [lintegral_smul_measure, lintegral_dirac, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u03b1\n\u22a2 \u222b\u207b (x : \u03b1) in {a}, f x \u2202\u03bc = f a * \u2191\u2191\u03bc {a}\n[PROOFSTEP]\nsimp only [restrict_singleton, lintegral_smul_measure, lintegral_dirac' _ hf, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\na : \u03b1\n\u22a2 \u222b\u207b (x : \u03b1) in {a}, f x \u2202\u03bc = f a * \u2191\u2191\u03bc {a}\n[PROOFSTEP]\nsimp only [restrict_singleton, lintegral_smul_measure, lintegral_dirac, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : Set.Countable s\n\u22a2 \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in \u22c3 (x : \u03b1) (_ : x \u2208 s), {x}, f a \u2202\u03bc\n[PROOFSTEP]\nrw [biUnion_of_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : Set.Countable s\n\u22a2 \u2211' (a : \u2191s), \u222b\u207b (x : \u03b1) in {\u2191a}, f x \u2202\u03bc = \u2211' (a : \u2191s), f \u2191a * \u2191\u2191\u03bc {\u2191a}\n[PROOFSTEP]\nsimp only [lintegral_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1\ns : Set \u03b1\nh : \u00aca \u2208 s\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (x : \u03b1) in insert a s, f x \u2202\u03bc = f a * \u2191\u2191\u03bc {a} + \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 union_singleton, lintegral_union (measurableSet_singleton a), lintegral_singleton, add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1\ns : Set \u03b1\nh : \u00aca \u2208 s\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 Disjoint s {a}\n[PROOFSTEP]\nrwa [disjoint_singleton_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (x : \u03b1) in \u2191s, f x \u2202\u03bc = \u2211 x in s, f x * \u2191\u2191\u03bc {x}\n[PROOFSTEP]\nsimp only [lintegral_countable _ s.countable_toSet, \u2190 Finset.tsum_subtype']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b1\ninst\u271d : Fintype \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc = \u2211 x : \u03b1, f x * \u2191\u2191\u03bc {x}\n[PROOFSTEP]\nrw [\u2190 lintegral_finset, Finset.coe_univ, Measure.restrict_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\n[PROOFSTEP]\nsimp_rw [ae_iff, ENNReal.not_lt_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc {a | f a = \u22a4} = 0\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh : \u00ac\u2191\u2191\u03bc {a | f a = \u22a4} = 0\n\u22a2 False\n[PROOFSTEP]\napply h2f.lt_top.not_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh : \u00ac\u2191\u2191\u03bc {a | f a = \u22a4} = 0\n\u22a2 \u22a4 \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nhave : (f \u207b\u00b9' {\u221e}).indicator \u22a4 \u2264 f := by\n intro x\n by_cases hx : x \u2208 f \u207b\u00b9' {\u221e} <;> [simpa [indicator_of_mem hx]; simp [indicator_of_not_mem hx]]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh : \u00ac\u2191\u2191\u03bc {a | f a = \u22a4} = 0\n\u22a2 indicator (f \u207b\u00b9' {\u22a4}) \u22a4 \u2264 f\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh : \u00ac\u2191\u2191\u03bc {a | f a = \u22a4} = 0\nx : \u03b1\n\u22a2 indicator (f \u207b\u00b9' {\u22a4}) \u22a4 x \u2264 f x\n[PROOFSTEP]\nby_cases hx : x \u2208 f \u207b\u00b9' {\u221e} <;> [simpa [indicator_of_mem hx]; simp [indicator_of_not_mem hx]]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh : \u00ac\u2191\u2191\u03bc {a | f a = \u22a4} = 0\nx : \u03b1\n\u22a2 indicator (f \u207b\u00b9' {\u22a4}) \u22a4 x \u2264 f x\n[PROOFSTEP]\nby_cases hx : x \u2208 f \u207b\u00b9' {\u221e}\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh : \u00ac\u2191\u2191\u03bc {a | f a = \u22a4} = 0\nx : \u03b1\nhx : x \u2208 f \u207b\u00b9' {\u22a4}\n\u22a2 indicator (f \u207b\u00b9' {\u22a4}) \u22a4 x \u2264 f x\n[PROOFSTEP]\nsimpa [indicator_of_mem hx]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh : \u00ac\u2191\u2191\u03bc {a | f a = \u22a4} = 0\nx : \u03b1\nhx : \u00acx \u2208 f \u207b\u00b9' {\u22a4}\n\u22a2 indicator (f \u207b\u00b9' {\u22a4}) \u22a4 x \u2264 f x\n[PROOFSTEP]\nsimp [indicator_of_not_mem hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh : \u00ac\u2191\u2191\u03bc {a | f a = \u22a4} = 0\nthis : indicator (f \u207b\u00b9' {\u22a4}) \u22a4 \u2264 f\n\u22a2 \u22a4 \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nconvert lintegral_mono this\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh : \u00ac\u2191\u2191\u03bc {a | f a = \u22a4} = 0\nthis : indicator (f \u207b\u00b9' {\u22a4}) \u22a4 \u2264 f\n\u22a2 \u22a4 = \u222b\u207b (a : \u03b1), indicator (f \u207b\u00b9' {\u22a4}) \u22a4 a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_indicator _ (hf (measurableSet_singleton \u221e))]\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh : \u00ac\u2191\u2191\u03bc {a | f a = \u22a4} = 0\nthis : indicator (f \u207b\u00b9' {\u22a4}) \u22a4 \u2264 f\n\u22a2 \u22a4 = \u222b\u207b (a : \u03b1) in f \u207b\u00b9' {\u22a4}, \u22a4 a \u2202\u03bc\n[PROOFSTEP]\nsimp [ENNReal.top_mul', preimage, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nh2f : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), AEMeasurable.mk f hf x \u2202\u03bc \u2260 \u22a4\n[PROOFSTEP]\nrwa [\u2190 lintegral_congr_ae hf.ae_eq_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nh2f : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh2f_meas : \u222b\u207b (x : \u03b1), AEMeasurable.mk f hf x \u2202\u03bc \u2260 \u22a4\nx : \u03b1\nhx : f x = AEMeasurable.mk f hf x\nh : AEMeasurable.mk f hf x < \u22a4\n\u22a2 f x < \u22a4\n[PROOFSTEP]\nrwa [hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\nhbdd : BddAbove (f '' s)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191(f x) \u2202\u03bc < \u22a4\n[PROOFSTEP]\nobtain \u27e8M, hM\u27e9 := hbdd\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\nM : \u211d\u22650\nhM : M \u2208 upperBounds (f '' s)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191(f x) \u2202\u03bc < \u22a4\n[PROOFSTEP]\nrw [mem_upperBounds] at hM \n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\nM : \u211d\u22650\nhM : \u2200 (x : \u211d\u22650), x \u2208 f '' s \u2192 x \u2264 M\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191(f x) \u2202\u03bc < \u22a4\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (set_lintegral_mono hf.coe_nnreal_ennreal (@measurable_const _ _ _ _ \u2191M) _) _\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\nM : \u211d\u22650\nhM : \u2200 (x : \u211d\u22650), x \u2208 f '' s \u2192 x \u2264 M\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2191(f x) \u2264 \u2191M\n[PROOFSTEP]\nsimpa using hM\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\nM : \u211d\u22650\nhM : \u2200 (x : \u211d\u22650), x \u2208 f '' s \u2192 x \u2264 M\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191M \u2202\u03bc < \u22a4\n[PROOFSTEP]\nrw [lintegral_const]\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\nM : \u211d\u22650\nhM : \u2200 (x : \u211d\u22650), x \u2208 f '' s \u2192 x \u2264 M\n\u22a2 \u2191M * \u2191\u2191(Measure.restrict \u03bc s) univ < \u22a4\n[PROOFSTEP]\nrefine' ENNReal.mul_lt_top ENNReal.coe_lt_top.ne _\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\nM : \u211d\u22650\nhM : \u2200 (x : \u211d\u22650), x \u2208 f '' s \u2192 x \u2264 M\n\u22a2 \u2191\u2191(Measure.restrict \u03bc s) univ \u2260 \u22a4\n[PROOFSTEP]\nsimp [hs]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\u271d\n\u03bc\u271d \u03bd : Measure \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03bc_fin : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_bdd : \u2203 c, \u2200 (x : \u03b1), f x \u2264 \u2191c\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc < \u22a4\n[PROOFSTEP]\ncases' f_bdd with c hc\n[GOAL]\ncase intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\u271d\n\u03bc\u271d \u03bd : Measure \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03bc_fin : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\nhc : \u2200 (x : \u03b1), f x \u2264 \u2191c\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc < \u22a4\n[PROOFSTEP]\napply lt_of_le_of_lt (@lintegral_mono _ _ \u03bc _ _ hc)\n[GOAL]\ncase intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\u271d\n\u03bc\u271d \u03bd : Measure \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03bc_fin : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\nhc : \u2200 (x : \u03b1), f x \u2264 \u2191c\n\u22a2 \u222b\u207b (a : \u03b1), \u2191c \u2202\u03bc < \u22a4\n[PROOFSTEP]\nrw [lintegral_const]\n[GOAL]\ncase intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\u271d\n\u03bc\u271d \u03bd : Measure \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03bc_fin : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\nhc : \u2200 (x : \u03b1), f x \u2264 \u2191c\n\u22a2 \u2191c * \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nexact ENNReal.mul_lt_top ENNReal.coe_lt_top.ne \u03bc_fin.measure_univ_lt_top.ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 (fun s x => \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc) \u2205 (_ : MeasurableSet \u2205) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : f =\u1da0[ae \u03bc] g\n\u22a2 withDensity \u03bc f = withDensity \u03bc g\n[PROOFSTEP]\nrefine Measure.ext fun s hs => ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : f =\u1da0[ae \u03bc] g\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity \u03bc f) s = \u2191\u2191(withDensity \u03bc g) s\n[PROOFSTEP]\nrw [withDensity_apply _ hs, withDensity_apply _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : f =\u1da0[ae \u03bc] g\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\n[PROOFSTEP]\nexact lintegral_congr_ae (ae_restrict_of_ae h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 withDensity \u03bc (f + g) = withDensity \u03bc f + withDensity \u03bc g\n[PROOFSTEP]\nrefine' Measure.ext fun s hs => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity \u03bc (f + g)) s = \u2191\u2191(withDensity \u03bc f + withDensity \u03bc g) s\n[PROOFSTEP]\nrw [withDensity_apply _ hs, Measure.add_apply, withDensity_apply _ hs, withDensity_apply _ hs, \u2190 lintegral_add_left hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, (f + g) a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, f a + g a \u2202\u03bc\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhg : Measurable g\n\u22a2 withDensity \u03bc (f + g) = withDensity \u03bc f + withDensity \u03bc g\n[PROOFSTEP]\nsimpa only [add_comm] using withDensity_add_left hg f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 withDensity (\u03bc + \u03bd) f = withDensity \u03bc f + withDensity \u03bd f\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity (\u03bc + \u03bd) f) s = \u2191\u2191(withDensity \u03bc f + withDensity \u03bd f) s\n[PROOFSTEP]\nsimp only [withDensity_apply f hs, restrict_add, lintegral_add_measure, Measure.add_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nm : MeasurableSpace \u03b1\n\u03bc : \u03b9 \u2192 Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 withDensity (sum \u03bc) f = sum fun n => withDensity (\u03bc n) f\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\n\u03b9 : Type u_5\nm : MeasurableSpace \u03b1\n\u03bc : \u03b9 \u2192 Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity (sum \u03bc) f) s = \u2191\u2191(sum fun n => withDensity (\u03bc n) f) s\n[PROOFSTEP]\nsimp_rw [sum_apply _ hs, withDensity_apply f hs, restrict_sum \u03bc hs, lintegral_sum_measure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 withDensity \u03bc (r \u2022 f) = r \u2022 withDensity \u03bc f\n[PROOFSTEP]\nrefine' Measure.ext fun s hs => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity \u03bc (r \u2022 f)) s = \u2191\u2191(r \u2022 withDensity \u03bc f) s\n[PROOFSTEP]\nrw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs, smul_eq_mul, \u2190\n lintegral_const_mul r hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, (r \u2022 f) a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, r * f a \u2202\u03bc\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhr : r \u2260 \u22a4\n\u22a2 withDensity \u03bc (r \u2022 f) = r \u2022 withDensity \u03bc f\n[PROOFSTEP]\nrefine' Measure.ext fun s hs => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhr : r \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity \u03bc (r \u2022 f)) s = \u2191\u2191(r \u2022 withDensity \u03bc f) s\n[PROOFSTEP]\nrw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs, smul_eq_mul, \u2190\n lintegral_const_mul' r f hr]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhr : r \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, (r \u2022 f) a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, r * f a \u2202\u03bc\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2191\u2191(withDensity \u03bc f) univ < \u22a4\n[PROOFSTEP]\nrwa [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ, lt_top_iff_ne_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 withDensity \u03bc f \u226a \u03bc\n[PROOFSTEP]\nrefine' AbsolutelyContinuous.mk fun s hs\u2081 hs\u2082 => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs\u2081 : MeasurableSet s\nhs\u2082 : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191(withDensity \u03bc f) s = 0\n[PROOFSTEP]\nrw [withDensity_apply _ hs\u2081]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs\u2081 : MeasurableSet s\nhs\u2082 : \u2191\u2191\u03bc s = 0\n\u22a2 \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = 0\n[PROOFSTEP]\nexact set_lintegral_measure_zero _ _ hs\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u22a2 withDensity \u03bc 0 = 0\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity \u03bc 0) s = \u2191\u21910 s\n[PROOFSTEP]\nsimp [withDensity_apply _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u22a2 withDensity \u03bc 1 = \u03bc\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity \u03bc 1) s = \u2191\u2191\u03bc s\n[PROOFSTEP]\nsimp [withDensity_apply _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (i : \u2115), Measurable (f i)\n\u22a2 withDensity \u03bc (\u2211' (n : \u2115), f n) = sum fun n => withDensity \u03bc (f n)\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (i : \u2115), Measurable (f i)\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity \u03bc (\u2211' (n : \u2115), f n)) s = \u2191\u2191(sum fun n => withDensity \u03bc (f n)) s\n[PROOFSTEP]\nsimp_rw [sum_apply _ hs, withDensity_apply _ hs]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (i : \u2115), Measurable (f i)\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, tsum (fun n => f n) a \u2202\u03bc = \u2211' (i : \u2115), \u222b\u207b (a : \u03b1) in s, f i a \u2202\u03bc\n[PROOFSTEP]\nchange \u222b\u207b x in s, (\u2211' n, f n) x \u2202\u03bc = \u2211' i : \u2115, \u222b\u207b x, f i x \u2202\u03bc.restrict s\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (i : \u2115), Measurable (f i)\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (x : \u03b1) in s, tsum (fun n => f n) x \u2202\u03bc = \u2211' (i : \u2115), \u222b\u207b (x : \u03b1) in s, f i x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lintegral_tsum fun i => (h i).aemeasurable]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (i : \u2115), Measurable (f i)\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (x : \u03b1) in s, tsum (fun n => f n) x \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, \u2211' (i : \u2115), f i a \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_congr fun x => tsum_apply (Pi.summable.2 fun _ => ENNReal.summable)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 withDensity \u03bc (indicator s f) = withDensity (Measure.restrict \u03bc s) f\n[PROOFSTEP]\next1 t ht\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(withDensity \u03bc (indicator s f)) t = \u2191\u2191(withDensity (Measure.restrict \u03bc s) f) t\n[PROOFSTEP]\nrw [withDensity_apply _ ht, lintegral_indicator _ hs, restrict_comm hs, \u2190 withDensity_apply _ ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 withDensity \u03bc (indicator s 1) = Measure.restrict \u03bc s\n[PROOFSTEP]\nrw [withDensity_indicator hs, withDensity_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\n\u22a2 (withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098 withDensity \u03bc fun x => ENNReal.ofReal (-f x)\n[PROOFSTEP]\nset S : Set \u03b1 := {x | f x < 0}\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nS : Set \u03b1 := {x | f x < 0}\n\u22a2 (withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098 withDensity \u03bc fun x => ENNReal.ofReal (-f x)\n[PROOFSTEP]\nhave hS : MeasurableSet S := measurableSet_lt hf measurable_const\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nS : Set \u03b1 := {x | f x < 0}\nhS : MeasurableSet S\n\u22a2 (withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098 withDensity \u03bc fun x => ENNReal.ofReal (-f x)\n[PROOFSTEP]\nrefine' \u27e8S, hS, _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nS : Set \u03b1 := {x | f x < 0}\nhS : MeasurableSet S\n\u22a2 \u2191\u2191(withDensity \u03bc fun x => ENNReal.ofReal (f x)) S = 0\n[PROOFSTEP]\nrw [withDensity_apply _ hS, lintegral_eq_zero_iff hf.ennreal_ofReal, EventuallyEq]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nS : Set \u03b1 := {x | f x < 0}\nhS : MeasurableSet S\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc S, ENNReal.ofReal (f x) = OfNat.ofNat 0 x\n[PROOFSTEP]\nexact (ae_restrict_mem hS).mono fun x hx => ENNReal.ofReal_eq_zero.2 (le_of_lt hx)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nS : Set \u03b1 := {x | f x < 0}\nhS : MeasurableSet S\n\u22a2 \u2191\u2191(withDensity \u03bc fun x => ENNReal.ofReal (-f x)) S\u1d9c = 0\n[PROOFSTEP]\nrw [withDensity_apply _ hS.compl, lintegral_eq_zero_iff hf.neg.ennreal_ofReal, EventuallyEq]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable f\nS : Set \u03b1 := {x | f x < 0}\nhS : MeasurableSet S\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc S\u1d9c, ENNReal.ofReal (-f x) = OfNat.ofNat 0 x\n[PROOFSTEP]\nexact (ae_restrict_mem hS.compl).mono fun x hx => ENNReal.ofReal_eq_zero.2 (not_lt.1 <| mt neg_pos.1 hx)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 Measure.restrict (withDensity \u03bc f) s = withDensity (Measure.restrict \u03bc s) f\n[PROOFSTEP]\next1 t ht\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.restrict (withDensity \u03bc f) s) t = \u2191\u2191(withDensity (Measure.restrict \u03bc s) f) t\n[PROOFSTEP]\nrw [restrict_apply ht, withDensity_apply _ ht, withDensity_apply _ (ht.inter hs), restrict_restrict ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nh : withDensity \u03bc f = 0\n\u22a2 f =\u1da0[ae \u03bc] 0\n[PROOFSTEP]\nrw [\u2190 lintegral_eq_zero_iff' hf, \u2190 set_lintegral_univ, \u2190 withDensity_apply _ MeasurableSet.univ, h, Measure.coe_zero,\n Pi.zero_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\n\u22a2 \u2191\u2191(withDensity \u03bc f) s = 0 \u2194 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\n\u22a2 \u2191\u2191(withDensity \u03bc f) s = 0 \u2192 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\n[PROOFSTEP]\nlet t := toMeasurable (\u03bc.withDensity f) s\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\n[PROOFSTEP]\napply measure_mono_null (inter_subset_inter_right _ (subset_toMeasurable (\u03bc.withDensity f) s))\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0\n[PROOFSTEP]\nhave A : \u03bc.withDensity f t = 0 := by rw [measure_toMeasurable, hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\n\u22a2 \u2191\u2191(withDensity \u03bc f) t = 0\n[PROOFSTEP]\nrw [measure_toMeasurable, hs]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191(withDensity \u03bc f) t = 0\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0\n[PROOFSTEP]\nrw [withDensity_apply f (measurableSet_toMeasurable _ s), lintegral_eq_zero_iff hf, EventuallyEq, ae_restrict_iff,\n ae_iff] at A \n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = OfNat.ofNat 0 a)} = 0\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (toMeasurable (withDensity \u03bc f) s), f x = OfNat.ofNat 0 x\n\u22a2 MeasurableSet {x | f x = OfNat.ofNat 0 x}\n[PROOFSTEP]\nswap\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (toMeasurable (withDensity \u03bc f) s), f x = OfNat.ofNat 0 x\n\u22a2 MeasurableSet {x | f x = OfNat.ofNat 0 x}\n[PROOFSTEP]\nexact hf (measurableSet_singleton 0)\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = OfNat.ofNat 0 a)} = 0\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0\n[PROOFSTEP]\nsimp only [Pi.zero_apply, mem_setOf_eq, Filter.mem_mk] at A \n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)} = 0\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0\n[PROOFSTEP]\nconvert A using 2\n[GOAL]\ncase h.e'_2.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)} = 0\n\u22a2 {x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s = {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)}\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_2.h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191(withDensity \u03bc f) s = 0\nt : Set \u03b1 := toMeasurable (withDensity \u03bc f) s\nA : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)} = 0\nx : \u03b1\n\u22a2 x \u2208 {x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s \u2194 x \u2208 {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)}\n[PROOFSTEP]\nsimp only [and_comm, exists_prop, mem_inter_iff, iff_self_iff, mem_setOf_eq, mem_compl_iff, not_forall]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\n\u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 \u2192 \u2191\u2191(withDensity \u03bc f) s = 0\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\n\u22a2 \u2191\u2191(withDensity \u03bc f) s = 0\n[PROOFSTEP]\nlet t := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\n\u22a2 \u2191\u2191(withDensity \u03bc f) s = 0\n[PROOFSTEP]\nhave A : s \u2286 t \u222a {x | f x = 0} := by\n intro x hx\n rcases eq_or_ne (f x) 0 with (fx | fx)\n \u00b7 simp only [fx, mem_union, mem_setOf_eq, eq_self_iff_true, or_true_iff]\n \u00b7 left\n apply subset_toMeasurable _ _\n exact \u27e8fx, hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\n\u22a2 s \u2286 t \u222a {x | f x = 0}\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 t \u222a {x | f x = 0}\n[PROOFSTEP]\nrcases eq_or_ne (f x) 0 with (fx | fx)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nx : \u03b1\nhx : x \u2208 s\nfx : f x = 0\n\u22a2 x \u2208 t \u222a {x | f x = 0}\n[PROOFSTEP]\nsimp only [fx, mem_union, mem_setOf_eq, eq_self_iff_true, or_true_iff]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nx : \u03b1\nhx : x \u2208 s\nfx : f x \u2260 0\n\u22a2 x \u2208 t \u222a {x | f x = 0}\n[PROOFSTEP]\nleft\n[GOAL]\ncase inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nx : \u03b1\nhx : x \u2208 s\nfx : f x \u2260 0\n\u22a2 x \u2208 t\n[PROOFSTEP]\napply subset_toMeasurable _ _\n[GOAL]\ncase inr.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nx : \u03b1\nhx : x \u2208 s\nfx : f x \u2260 0\n\u22a2 x \u2208 {x | f x \u2260 0} \u2229 s\n[PROOFSTEP]\nexact \u27e8fx, hx\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\n\u22a2 \u2191\u2191(withDensity \u03bc f) s = 0\n[PROOFSTEP]\napply measure_mono_null A (measure_union_null _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\n\u22a2 \u2191\u2191(withDensity \u03bc f) t = 0\n[PROOFSTEP]\napply withDensity_absolutelyContinuous\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\n\u22a2 \u2191\u2191\u03bc t = 0\n[PROOFSTEP]\nrwa [measure_toMeasurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\n\u22a2 \u2191\u2191(withDensity \u03bc f) {x | f x = 0} = 0\n[PROOFSTEP]\nhave M : MeasurableSet {x : \u03b1 | f x = 0} := hf (measurableSet_singleton _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\nM : MeasurableSet {x | f x = 0}\n\u22a2 \u2191\u2191(withDensity \u03bc f) {x | f x = 0} = 0\n[PROOFSTEP]\nrw [withDensity_apply _ M, lintegral_eq_zero_iff hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\nM : MeasurableSet {x | f x = 0}\n\u22a2 f =\u1da0[ae (Measure.restrict \u03bc {x | f x = 0})] 0\n[PROOFSTEP]\nfilter_upwards [ae_restrict_mem M]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : Measurable f\nhs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0\nt : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s)\nA : s \u2286 t \u222a {x | f x = 0}\nM : MeasurableSet {x | f x = 0}\n\u22a2 \u2200 (a : \u03b1), f a = 0 \u2192 f a = OfNat.ofNat 0 a\n[PROOFSTEP]\nsimp only [imp_self, Pi.zero_apply, imp_true_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202withDensity \u03bc f, p x) \u2194 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 0 \u2192 p x\n[PROOFSTEP]\nrw [ae_iff, ae_iff, withDensity_apply_eq_zero hf, iff_iff_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 (\u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 {a | \u00acp a}) = 0) = (\u2191\u2191\u03bc {a | \u00ac(f a \u2260 0 \u2192 p a)} = 0)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 {x | f x \u2260 0} \u2229 {a | \u00acp a} = {a | \u00ac(f a \u2260 0 \u2192 p a)}\n[PROOFSTEP]\next x\n[GOAL]\ncase e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nx : \u03b1\n\u22a2 x \u2208 {x | f x \u2260 0} \u2229 {a | \u00acp a} \u2194 x \u2208 {a | \u00ac(f a \u2260 0 \u2192 p a)}\n[PROOFSTEP]\nsimp only [exists_prop, mem_inter_iff, iff_self_iff, mem_setOf_eq, not_forall]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202withDensity \u03bc f, p x) \u2194 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f x \u2260 0}, p x\n[PROOFSTEP]\nrw [ae_withDensity_iff hf, ae_restrict_iff']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 0 \u2192 p x) \u2194 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 {x | f x \u2260 0} \u2192 p x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 MeasurableSet {x | f x \u2260 0}\n[PROOFSTEP]\nexact hf (measurableSet_singleton 0).compl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 AEMeasurable g \u2194 AEMeasurable fun x => \u2191(f x) * g x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 AEMeasurable g \u2192 AEMeasurable fun x => \u2191(f x) * g x\n[PROOFSTEP]\nrintro \u27e8g', g'meas, hg'\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\n\u22a2 AEMeasurable fun x => \u2191(f x) * g x\n[PROOFSTEP]\nhave A : MeasurableSet {x : \u03b1 | f x \u2260 0} := (hf (measurableSet_singleton 0)).compl\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 AEMeasurable fun x => \u2191(f x) * g x\n[PROOFSTEP]\nrefine' \u27e8fun x => f x * g' x, hf.coe_nnreal_ennreal.smul g'meas, _\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] fun x => \u2191(f x) * g' x\n[PROOFSTEP]\napply ae_of_ae_restrict_of_ae_restrict_compl {x | f x \u2260 0}\n[GOAL]\ncase mp.intro.intro.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f x \u2260 0}, (fun x => \u2191(f x) * g x) x = (fun x => \u2191(f x) * g' x) x\n[PROOFSTEP]\nrw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal] at hg' \n[GOAL]\ncase mp.intro.intro.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f x \u2260 0}, (fun x => \u2191(f x) * g x) x = (fun x => \u2191(f x) * g' x) x\n[PROOFSTEP]\nrw [ae_restrict_iff' A]\n[GOAL]\ncase mp.intro.intro.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 {x | f x \u2260 0} \u2192 (fun x => \u2191(f x) * g x) x = (fun x => \u2191(f x) * g' x) x\n[PROOFSTEP]\nfilter_upwards [hg']\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200 (a : \u03b1), (\u2191(f a) \u2260 0 \u2192 g a = g' a) \u2192 f a \u2260 0 \u2192 \u2191(f a) * g a = \u2191(f a) * g' a\n[PROOFSTEP]\nintro a ha h'a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\n\u22a2 \u2191(f a) * g a = \u2191(f a) * g' a\n[PROOFSTEP]\nhave : (f a : \u211d\u22650\u221e) \u2260 0 := by simpa only [Ne.def, coe_eq_zero] using h'a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\n\u22a2 \u2191(f a) \u2260 0\n[PROOFSTEP]\nsimpa only [Ne.def, coe_eq_zero] using h'a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\nthis : \u2191(f a) \u2260 0\n\u22a2 \u2191(f a) * g a = \u2191(f a) * g' a\n[PROOFSTEP]\nrw [ha this]\n[GOAL]\ncase mp.intro.intro.htc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f x \u2260 0}\u1d9c, (fun x => \u2191(f x) * g x) x = (fun x => \u2191(f x) * g' x) x\n[PROOFSTEP]\nfilter_upwards [ae_restrict_mem A.compl]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200 (a : \u03b1), a \u2208 {x | f x \u2260 0}\u1d9c \u2192 \u2191(f a) * g a = \u2191(f a) * g' a\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\nx : \u03b1\nhx : x \u2208 {x | f x \u2260 0}\u1d9c\n\u22a2 \u2191(f x) * g x = \u2191(f x) * g' x\n[PROOFSTEP]\nsimp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx \n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\nx : \u03b1\nhx : f x = 0\n\u22a2 \u2191(f x) * g x = \u2191(f x) * g' x\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 (AEMeasurable fun x => \u2191(f x) * g x) \u2192 AEMeasurable g\n[PROOFSTEP]\nrintro \u27e8g', g'meas, hg'\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g'\n\u22a2 AEMeasurable g\n[PROOFSTEP]\nrefine' \u27e8fun x => ((f x)\u207b\u00b9 : \u211d\u22650\u221e) * g' x, hf.coe_nnreal_ennreal.inv.smul g'meas, _\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g'\n\u22a2 g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] fun x => (\u2191(f x))\u207b\u00b9 * g' x\n[PROOFSTEP]\nrw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal]\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g'\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = (\u2191(f x))\u207b\u00b9 * g' x\n[PROOFSTEP]\nfilter_upwards [hg']\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g'\n\u22a2 \u2200 (a : \u03b1), \u2191(f a) * g a = g' a \u2192 \u2191(f a) \u2260 0 \u2192 g a = (\u2191(f a))\u207b\u00b9 * g' a\n[PROOFSTEP]\nintro x hx h'x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g'\nx : \u03b1\nhx : \u2191(f x) * g x = g' x\nh'x : \u2191(f x) \u2260 0\n\u22a2 g x = (\u2191(f x))\u207b\u00b9 * g' x\n[PROOFSTEP]\nrw [\u2190 hx, \u2190 mul_assoc, ENNReal.inv_mul_cancel h'x ENNReal.coe_ne_top, one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh_mf : Measurable f\n\u22a2 \u2200 {g : \u03b1 \u2192 \u211d\u22650\u221e}, Measurable g \u2192 \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc\n[PROOFSTEP]\napply Measurable.ennreal_induction\n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh_mf : Measurable f\n\u22a2 \u2200 (c : \u211d\u22650\u221e) \u2983s : Set \u03b1\u2984,\n MeasurableSet s \u2192\n \u222b\u207b (a : \u03b1), indicator s (fun x => c) a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * indicator s fun x => c) a \u2202\u03bc\n[PROOFSTEP]\nintro c s h_ms\n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh_mf : Measurable f\nc : \u211d\u22650\u221e\ns : Set \u03b1\nh_ms : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), indicator s (fun x => c) a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * indicator s fun x => c) a \u2202\u03bc\n[PROOFSTEP]\nsimp [*, mul_comm _ c, \u2190 indicator_mul_right]\n[GOAL]\ncase h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh_mf : Measurable f\n\u22a2 \u2200 \u2983f_1 g : \u03b1 \u2192 \u211d\u22650\u221e\u2984,\n Disjoint (support f_1) (support g) \u2192\n Measurable f_1 \u2192\n Measurable g \u2192\n \u222b\u207b (a : \u03b1), f_1 a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * f_1) a \u2202\u03bc \u2192\n \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2192\n \u222b\u207b (a : \u03b1), (f_1 + g) a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * (f_1 + g)) a \u2202\u03bc\n[PROOFSTEP]\nintro g h _ h_mea_g _ h_ind_g h_ind_h\n[GOAL]\ncase h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh_mf : Measurable f\ng h : \u03b1 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (support g) (support h)\nh_mea_g : Measurable g\na\u271d : Measurable h\nh_ind_g : \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc\nh_ind_h : \u222b\u207b (a : \u03b1), h a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * h) a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), (g + h) a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * (g + h)) a \u2202\u03bc\n[PROOFSTEP]\nsimp [mul_add, *, Measurable.mul]\n[GOAL]\ncase h_iSup\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh_mf : Measurable f\n\u22a2 \u2200 \u2983f_1 : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\u2984,\n (\u2200 (n : \u2115), Measurable (f_1 n)) \u2192\n Monotone f_1 \u2192\n (\u2200 (n : \u2115), \u222b\u207b (a : \u03b1), f_1 n a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * f_1 n) a \u2202\u03bc) \u2192\n \u222b\u207b (a : \u03b1), (fun x => \u2a06 (n : \u2115), f_1 n x) a \u2202Measure.withDensity \u03bc f =\n \u222b\u207b (a : \u03b1), (f * fun x => \u2a06 (n : \u2115), f_1 n x) a \u2202\u03bc\n[PROOFSTEP]\nintro g h_mea_g h_mono_g h_ind\n[GOAL]\ncase h_iSup\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh_mf : Measurable f\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_mea_g : \u2200 (n : \u2115), Measurable (g n)\nh_mono_g : Monotone g\nh_ind : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), g n a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g n) a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), (fun x => \u2a06 (n : \u2115), g n x) a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * fun x => \u2a06 (n : \u2115), g n x) a \u2202\u03bc\n[PROOFSTEP]\nhave : Monotone fun n a => f a * g n a := fun m n hmn x => mul_le_mul_left' (h_mono_g hmn x) _\n[GOAL]\ncase h_iSup\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh_mf : Measurable f\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_mea_g : \u2200 (n : \u2115), Measurable (g n)\nh_mono_g : Monotone g\nh_ind : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), g n a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g n) a \u2202\u03bc\nthis : Monotone fun n a => f a * g n a\n\u22a2 \u222b\u207b (a : \u03b1), (fun x => \u2a06 (n : \u2115), g n x) a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * fun x => \u2a06 (n : \u2115), g n x) a \u2202\u03bc\n[PROOFSTEP]\nsimp [lintegral_iSup, ENNReal.mul_iSup, h_mf.mul (h_mea_g _), *]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202Measure.withDensity \u03bc f = \u222b\u207b (x : \u03b1) in s, (f * g) x \u2202\u03bc\n[PROOFSTEP]\nrw [restrict_withDensity hs, lintegral_withDensity_eq_lintegral_mul _ hf hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhg : AEMeasurable g\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc\n[PROOFSTEP]\nlet f' := hf.mk f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhg : AEMeasurable g\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc\n[PROOFSTEP]\nhave : \u03bc.withDensity f = \u03bc.withDensity f' := withDensity_congr_ae hf.ae_eq_mk\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhg : AEMeasurable g\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc\n[PROOFSTEP]\nrw [this] at hg \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f' = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc\n[PROOFSTEP]\nlet g' := hg.mk g\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f' = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc\n[PROOFSTEP]\ncalc\n \u222b\u207b a, g a \u2202\u03bc.withDensity f' = \u222b\u207b a, g' a \u2202\u03bc.withDensity f' := lintegral_congr_ae hg.ae_eq_mk\n _ = \u222b\u207b a, (f' * g') a \u2202\u03bc := (lintegral_withDensity_eq_lintegral_mul _ hf.measurable_mk hg.measurable_mk)\n _ = \u222b\u207b a, (f' * g) a \u2202\u03bc := by\n apply lintegral_congr_ae\n apply ae_of_ae_restrict_of_ae_restrict_compl {x | f' x \u2260 0}\n \u00b7 have Z := hg.ae_eq_mk\n rw [EventuallyEq, ae_withDensity_iff_ae_restrict hf.measurable_mk] at Z \n filter_upwards [Z]\n intro x hx\n simp only [hx, Pi.mul_apply]\n \u00b7 have M : MeasurableSet {x : \u03b1 | f' x \u2260 0}\u1d9c := (hf.measurable_mk (measurableSet_singleton 0).compl).compl\n filter_upwards [ae_restrict_mem M]\n intro x hx\n simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx \n simp only [hx, zero_mul, Pi.mul_apply]\n _ = \u222b\u207b a : \u03b1, (f * g) a \u2202\u03bc := by\n apply lintegral_congr_ae\n filter_upwards [hf.ae_eq_mk]\n intro x hx\n simp only [hx, Pi.mul_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 \u222b\u207b (a : \u03b1), (f' * g') a \u2202\u03bc = \u222b\u207b (a : \u03b1), (f' * g) a \u2202\u03bc\n[PROOFSTEP]\napply lintegral_congr_ae\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 (fun a => (f' * g') a) =\u1d50[\u03bc] fun a => (f' * g) a\n[PROOFSTEP]\napply ae_of_ae_restrict_of_ae_restrict_compl {x | f' x \u2260 0}\n[GOAL]\ncase h.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x\n[PROOFSTEP]\nhave Z := hg.ae_eq_mk\n[GOAL]\ncase h.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nZ : g =\u1d50[Measure.withDensity \u03bc f'] AEMeasurable.mk g hg\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x\n[PROOFSTEP]\nrw [EventuallyEq, ae_withDensity_iff_ae_restrict hf.measurable_mk] at Z \n[GOAL]\ncase h.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nZ : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | AEMeasurable.mk f hf x \u2260 0}, g x = AEMeasurable.mk g hg x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x\n[PROOFSTEP]\nfilter_upwards [Z]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nZ : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | AEMeasurable.mk f hf x \u2260 0}, g x = AEMeasurable.mk g hg x\n\u22a2 \u2200 (a : \u03b1),\n g a = AEMeasurable.mk g hg a \u2192 (AEMeasurable.mk f hf * AEMeasurable.mk g hg) a = (AEMeasurable.mk f hf * g) a\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nZ : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | AEMeasurable.mk f hf x \u2260 0}, g x = AEMeasurable.mk g hg x\nx : \u03b1\nhx : g x = AEMeasurable.mk g hg x\n\u22a2 (AEMeasurable.mk f hf * AEMeasurable.mk g hg) x = (AEMeasurable.mk f hf * g) x\n[PROOFSTEP]\nsimp only [hx, Pi.mul_apply]\n[GOAL]\ncase h.htc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}\u1d9c, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x\n[PROOFSTEP]\nhave M : MeasurableSet {x : \u03b1 | f' x \u2260 0}\u1d9c := (hf.measurable_mk (measurableSet_singleton 0).compl).compl\n[GOAL]\ncase h.htc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nM : MeasurableSet {x | f' x \u2260 0}\u1d9c\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}\u1d9c, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x\n[PROOFSTEP]\nfilter_upwards [ae_restrict_mem M]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nM : MeasurableSet {x | f' x \u2260 0}\u1d9c\n\u22a2 \u2200 (a : \u03b1),\n a \u2208 {x | AEMeasurable.mk f hf x \u2260 0}\u1d9c \u2192\n (AEMeasurable.mk f hf * AEMeasurable.mk g hg) a = (AEMeasurable.mk f hf * g) a\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nM : MeasurableSet {x | f' x \u2260 0}\u1d9c\nx : \u03b1\nhx : x \u2208 {x | AEMeasurable.mk f hf x \u2260 0}\u1d9c\n\u22a2 (AEMeasurable.mk f hf * AEMeasurable.mk g hg) x = (AEMeasurable.mk f hf * g) x\n[PROOFSTEP]\nsimp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx \n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nM : MeasurableSet {x | f' x \u2260 0}\u1d9c\nx : \u03b1\nhx : AEMeasurable.mk f hf x = 0\n\u22a2 (AEMeasurable.mk f hf * AEMeasurable.mk g hg) x = (AEMeasurable.mk f hf * g) x\n[PROOFSTEP]\nsimp only [hx, zero_mul, Pi.mul_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 \u222b\u207b (a : \u03b1), (f' * g) a \u2202\u03bc = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc\n[PROOFSTEP]\napply lintegral_congr_ae\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 (fun a => (f' * g) a) =\u1d50[\u03bc] fun a => (f * g) a\n[PROOFSTEP]\nfilter_upwards [hf.ae_eq_mk]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\n\u22a2 \u2200 (a : \u03b1), f a = AEMeasurable.mk f hf a \u2192 (AEMeasurable.mk f hf * g) a = (f * g) a\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity \u03bc f = Measure.withDensity \u03bc f'\ng' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg\nx : \u03b1\nhx : f x = AEMeasurable.mk f hf x\n\u22a2 (AEMeasurable.mk f hf * g) x = (f * g) x\n[PROOFSTEP]\nsimp only [hx, Pi.mul_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f \u2264 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 iSup_lintegral_measurable_le_eq_lintegral, \u2190 iSup_lintegral_measurable_le_eq_lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 (g_1 : \u03b1 \u2192 \u211d\u22650\u221e) (_ : Measurable g_1) (_ : g_1 \u2264 fun a => g a), \u222b\u207b (a : \u03b1), g_1 a \u2202Measure.withDensity \u03bc f \u2264\n \u2a06 (g_1 : \u03b1 \u2192 \u211d\u22650\u221e) (_ : Measurable g_1) (_ : g_1 \u2264 fun a => (f * g) a), \u222b\u207b (a : \u03b1), g_1 a \u2202\u03bc\n[PROOFSTEP]\nrefine' iSup\u2082_le fun i i_meas => iSup_le fun hi => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => g a\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202Measure.withDensity \u03bc f \u2264\n \u2a06 (g_1 : \u03b1 \u2192 \u211d\u22650\u221e) (_ : Measurable g_1) (_ : g_1 \u2264 fun a => (f * g) a), \u222b\u207b (a : \u03b1), g_1 a \u2202\u03bc\n[PROOFSTEP]\nhave A : f * i \u2264 f * g := fun x => mul_le_mul_left' (hi x) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => g a\nA : f * i \u2264 f * g\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202Measure.withDensity \u03bc f \u2264\n \u2a06 (g_1 : \u03b1 \u2192 \u211d\u22650\u221e) (_ : Measurable g_1) (_ : g_1 \u2264 fun a => (f * g) a), \u222b\u207b (a : \u03b1), g_1 a \u2202\u03bc\n[PROOFSTEP]\nrefine' le_iSup\u2082_of_le (f * i) (f_meas.mul i_meas) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => g a\nA : f * i \u2264 f * g\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202Measure.withDensity \u03bc f \u2264 \u2a06 (_ : f * i \u2264 fun a => (f * g) a), \u222b\u207b (a : \u03b1), (f * i) a \u2202\u03bc\n[PROOFSTEP]\nexact le_iSup_of_le A (le_of_eq (lintegral_withDensity_eq_lintegral_mul _ f_meas i_meas))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc\n[PROOFSTEP]\nrefine' le_antisymm (lintegral_withDensity_le_lintegral_mul \u03bc f_meas g) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f\n[PROOFSTEP]\nrw [\u2190 iSup_lintegral_measurable_le_eq_lintegral, \u2190 iSup_lintegral_measurable_le_eq_lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 (g_1 : \u03b1 \u2192 \u211d\u22650\u221e) (_ : Measurable g_1) (_ : g_1 \u2264 fun a => (f * g) a), \u222b\u207b (a : \u03b1), g_1 a \u2202\u03bc \u2264\n \u2a06 (g_1 : \u03b1 \u2192 \u211d\u22650\u221e) (_ : Measurable g_1) (_ : g_1 \u2264 fun a => g a), \u222b\u207b (a : \u03b1), g_1 a \u2202Measure.withDensity \u03bc f\n[PROOFSTEP]\nrefine' iSup\u2082_le fun i i_meas => iSup_le fun hi => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264\n \u2a06 (g_1 : \u03b1 \u2192 \u211d\u22650\u221e) (_ : Measurable g_1) (_ : g_1 \u2264 fun a => g a), \u222b\u207b (a : \u03b1), g_1 a \u2202Measure.withDensity \u03bc f\n[PROOFSTEP]\nhave A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g := by\n intro x\n dsimp\n rw [mul_comm, \u2190 div_eq_mul_inv]\n exact div_le_of_le_mul' (hi x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\n\u22a2 (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nx : \u03b1\n\u22a2 (fun x => (f x)\u207b\u00b9 * i x) x \u2264 g x\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nx : \u03b1\n\u22a2 (f x)\u207b\u00b9 * i x \u2264 g x\n[PROOFSTEP]\nrw [mul_comm, \u2190 div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nx : \u03b1\n\u22a2 i x / f x \u2264 g x\n[PROOFSTEP]\nexact div_le_of_le_mul' (hi x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264\n \u2a06 (g_1 : \u03b1 \u2192 \u211d\u22650\u221e) (_ : Measurable g_1) (_ : g_1 \u2264 fun a => g a), \u222b\u207b (a : \u03b1), g_1 a \u2202Measure.withDensity \u03bc f\n[PROOFSTEP]\nrefine' le_iSup_of_le (fun x => (f x)\u207b\u00b9 * i x) (le_iSup_of_le (f_meas.inv.mul i_meas) _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264\n \u2a06 (_ : (fun x => (f x)\u207b\u00b9 * i x) \u2264 fun a => g a), \u222b\u207b (a : \u03b1), (fun x => (f x)\u207b\u00b9 * i x) a \u2202Measure.withDensity \u03bc f\n[PROOFSTEP]\nrefine' le_iSup_of_le A _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), (fun x => (f x)\u207b\u00b9 * i x) a \u2202Measure.withDensity \u03bc f\n[PROOFSTEP]\nrw [lintegral_withDensity_eq_lintegral_mul _ f_meas (f_meas.inv.mul i_meas)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), (f * fun a => (f a)\u207b\u00b9 * i a) a \u2202\u03bc\n[PROOFSTEP]\napply lintegral_mono_ae\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, i a \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) a\n[PROOFSTEP]\nfilter_upwards [hf]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\n\u22a2 \u2200 (a : \u03b1), f a < \u22a4 \u2192 i a \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) a\n[PROOFSTEP]\nintro x h'x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\n\u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x\n[PROOFSTEP]\nrcases eq_or_ne (f x) 0 with (hx | hx)\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x = 0\n\u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x\n[PROOFSTEP]\nhave := hi x\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x = 0\nthis : i x \u2264 (fun a => (f * g) a) x\n\u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x\n[PROOFSTEP]\nsimp only [hx, zero_mul, Pi.mul_apply, nonpos_iff_eq_zero] at this \n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x = 0\nthis : i x = 0\n\u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x \u2260 0\n\u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x\n[PROOFSTEP]\napply le_of_eq _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x \u2260 0\n\u22a2 i x = (f * fun a => (f a)\u207b\u00b9 * i a) x\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng i : \u03b1 \u2192 \u211d\u22650\u221e\ni_meas : Measurable i\nhi : i \u2264 fun a => (f * g) a\nA : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g\nx : \u03b1\nh'x : f x < \u22a4\nhx : f x \u2260 0\n\u22a2 i x = f x * ((f x)\u207b\u00b9 * i x)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, ENNReal.mul_inv_cancel hx h'x.ne, one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, f x < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in s, g a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1) in s, (f * g) a \u2202\u03bc\n[PROOFSTEP]\nrw [restrict_withDensity hs, lintegral_withDensity_eq_lintegral_mul_non_measurable _ f_meas hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nh'f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc\n[PROOFSTEP]\nlet f' := hf.mk f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nh'f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc\n[PROOFSTEP]\ncalc\n \u222b\u207b a, g a \u2202\u03bc.withDensity f = \u222b\u207b a, g a \u2202\u03bc.withDensity f' := by rw [withDensity_congr_ae hf.ae_eq_mk]\n _ = \u222b\u207b a, (f' * g) a \u2202\u03bc :=\n by\n apply lintegral_withDensity_eq_lintegral_mul_non_measurable _ hf.measurable_mk\n filter_upwards [h'f, hf.ae_eq_mk]\n intro x hx h'x\n rwa [\u2190 h'x]\n _ = \u222b\u207b a, (f * g) a \u2202\u03bc := by\n apply lintegral_congr_ae\n filter_upwards [hf.ae_eq_mk]\n intro x hx\n simp only [hx, Pi.mul_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nh'f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f'\n[PROOFSTEP]\nrw [withDensity_congr_ae hf.ae_eq_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nh'f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\n\u22a2 \u222b\u207b (a : \u03b1), g a \u2202Measure.withDensity \u03bc f' = \u222b\u207b (a : \u03b1), (f' * g) a \u2202\u03bc\n[PROOFSTEP]\napply lintegral_withDensity_eq_lintegral_mul_non_measurable _ hf.measurable_mk\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nh'f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, AEMeasurable.mk f hf x < \u22a4\n[PROOFSTEP]\nfilter_upwards [h'f, hf.ae_eq_mk]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nh'f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\n\u22a2 \u2200 (a : \u03b1), f a < \u22a4 \u2192 f a = AEMeasurable.mk f hf a \u2192 AEMeasurable.mk f hf a < \u22a4\n[PROOFSTEP]\nintro x hx h'x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nh'f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nx : \u03b1\nhx : f x < \u22a4\nh'x : f x = AEMeasurable.mk f hf x\n\u22a2 AEMeasurable.mk f hf x < \u22a4\n[PROOFSTEP]\nrwa [\u2190 h'x]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nh'f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\n\u22a2 \u222b\u207b (a : \u03b1), (f' * g) a \u2202\u03bc = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc\n[PROOFSTEP]\napply lintegral_congr_ae\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nh'f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\n\u22a2 (fun a => (f' * g) a) =\u1d50[\u03bc] fun a => (f * g) a\n[PROOFSTEP]\nfilter_upwards [hf.ae_eq_mk]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nh'f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\n\u22a2 \u2200 (a : \u03b1), f a = AEMeasurable.mk f hf a \u2192 (AEMeasurable.mk f hf * g) a = (f * g) a\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nh'f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf\nx : \u03b1\nhx : f x = AEMeasurable.mk f hf x\n\u22a2 (AEMeasurable.mk f hf * g) x = (f * g) x\n[PROOFSTEP]\nsimp only [hx, Pi.mul_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhf : AEMeasurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhs : MeasurableSet s\nh'f : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, f x < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in s, g a \u2202Measure.withDensity \u03bc f = \u222b\u207b (a : \u03b1) in s, (f * g) a \u2202\u03bc\n[PROOFSTEP]\nrw [restrict_withDensity hs, lintegral_withDensity_eq_lintegral_mul_non_measurable\u2080 _ hf h'f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\n\u22a2 Measure.withDensity \u03bc (f * g) = Measure.withDensity (Measure.withDensity \u03bc f) g\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.withDensity \u03bc (f * g)) s = \u2191\u2191(Measure.withDensity (Measure.withDensity \u03bc f) g) s\n[PROOFSTEP]\nsimp [withDensity_apply _ hs, restrict_withDensity hs, lintegral_withDensity_eq_lintegral_mul _ hf hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5\n[PROOFSTEP]\nset s : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5\n[PROOFSTEP]\nhave : \u2200 n, \u03bc (s n) < \u221e := fun n => (measure_mono <| disjointed_subset _ _).trans_lt (measure_spanningSets_lt_top \u03bc n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5\n[PROOFSTEP]\nobtain \u27e8\u03b4, \u03b4pos, \u03b4sum\u27e9 : \u2203 \u03b4 : \u2115 \u2192 \u211d\u22650, (\u2200 i, 0 < \u03b4 i) \u2227 (\u2211' i, \u03bc (s i) * \u03b4 i) < \u03b5\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u22a2 \u2203 \u03b4, (\u2200 (i : \u2115), 0 < \u03b4 i) \u2227 \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5\n[PROOFSTEP]\nexact ENNReal.exists_pos_tsum_mul_lt_of_countable \u03b50 _ fun n => (this n).ne\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5\n[PROOFSTEP]\nset N : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\nN : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5\n[PROOFSTEP]\nhave hN_meas : Measurable N := measurable_spanningSetsIndex \u03bc\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\nN : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\nhN_meas : Measurable N\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5\n[PROOFSTEP]\nhave hNs : \u2200 n, N \u207b\u00b9' { n } = s n := preimage_spanningSetsIndex_singleton \u03bc\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\nN : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\nhN_meas : Measurable N\nhNs : \u2200 (n : \u2115), N \u207b\u00b9' {n} = s n\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u03b5\n[PROOFSTEP]\nrefine' \u27e8\u03b4 \u2218 N, fun x => \u03b4pos _, measurable_from_nat.comp hN_meas, _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\nN : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\nhN_meas : Measurable N\nhNs : \u2200 (n : \u2115), N \u207b\u00b9' {n} = s n\n\u22a2 \u222b\u207b (x : \u03b1), \u2191((\u03b4 \u2218 N) x) \u2202\u03bc < \u03b5\n[PROOFSTEP]\nerw [lintegral_comp measurable_from_nat.coe_nnreal_ennreal hN_meas]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ns : \u2115 \u2192 Set \u03b1 := disjointed (spanningSets \u03bc)\nthis : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\n\u03b4sum : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) * \u2191(\u03b4 i) < \u03b5\nN : \u03b1 \u2192 \u2115 := spanningSetsIndex \u03bc\nhN_meas : Measurable N\nhNs : \u2200 (n : \u2115), N \u207b\u00b9' {n} = s n\n\u22a2 \u222b\u207b (a : \u2115), \u2191(\u03b4 a) \u2202Measure.map N \u03bc < \u03b5\n[PROOFSTEP]\nsimpa [hNs, lintegral_countable', measurable_spanningSetsIndex, mul_comm] using \u03b4sum\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nrefine' @Measurable.ennreal_induction \u03b1 m (fun f => \u222b\u207b a, f a \u2202\u03bc.trim hm = \u222b\u207b a, f a \u2202\u03bc) _ _ _ f hf\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u2200 (c : \u211d\u22650\u221e) \u2983s : Set \u03b1\u2984,\n MeasurableSet s \u2192 (fun f => \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202\u03bc) (indicator s fun x => c)\n[PROOFSTEP]\nintro c s hs\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nc : \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), indicator s (fun x => c) a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), indicator s (fun x => c) a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_indicator _ hs, lintegral_indicator _ (hm s hs), set_lintegral_const, set_lintegral_const]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nc : \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 c * \u2191\u2191(Measure.trim \u03bc hm) s = c * \u2191\u2191\u03bc s\n[PROOFSTEP]\nsuffices h_trim_s : \u03bc.trim hm s = \u03bc s\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nc : \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nh_trim_s : \u2191\u2191(Measure.trim \u03bc hm) s = \u2191\u2191\u03bc s\n\u22a2 c * \u2191\u2191(Measure.trim \u03bc hm) s = c * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [h_trim_s]\n[GOAL]\ncase h_trim_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nc : \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.trim \u03bc hm) s = \u2191\u2191\u03bc s\n[PROOFSTEP]\nexact trim_measurableSet_eq hm hs\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 \u211d\u22650\u221e\u2984,\n Disjoint (support f) (support g) \u2192\n Measurable f \u2192\n Measurable g \u2192\n (fun f => \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202\u03bc) f \u2192\n (fun f => \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202\u03bc) g \u2192\n (fun f => \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202\u03bc) (f + g)\n[PROOFSTEP]\nintro f g _ hf _ hf_prop hg_prop\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (support f) (support g)\nhf : Measurable f\na\u271d : Measurable g\nhf_prop : \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202\u03bc\nhg_prop : \u222b\u207b (a : \u03b1), g a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), (f + g) a \u2202\u03bc\n[PROOFSTEP]\nhave h_m := lintegral_add_left (\u03bc := Measure.trim \u03bc hm) hf g\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (support f) (support g)\nhf : Measurable f\na\u271d : Measurable g\nhf_prop : \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202\u03bc\nhg_prop : \u222b\u207b (a : \u03b1), g a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\nh_m : \u222b\u207b (a : \u03b1), f a + g a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm + \u222b\u207b (a : \u03b1), g a \u2202Measure.trim \u03bc hm\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), (f + g) a \u2202\u03bc\n[PROOFSTEP]\nhave h_m0 := lintegral_add_left (\u03bc := \u03bc) (Measurable.mono hf hm le_rfl) g\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\nf g : \u03b1 \u2192 \u211d\u22650\u221e\na\u271d\u00b9 : Disjoint (support f) (support g)\nhf : Measurable f\na\u271d : Measurable g\nhf_prop : \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202\u03bc\nhg_prop : \u222b\u207b (a : \u03b1), g a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\nh_m : \u222b\u207b (a : \u03b1), f a + g a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm + \u222b\u207b (a : \u03b1), g a \u2202Measure.trim \u03bc hm\nh_m0 : \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc = \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), (f + g) a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), (f + g) a \u2202\u03bc\n[PROOFSTEP]\nrwa [hf_prop, hg_prop, \u2190 h_m0] at h_m \n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u2200 \u2983f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\u2984,\n (\u2200 (n : \u2115), Measurable (f n)) \u2192\n Monotone f \u2192\n (\u2200 (n : \u2115), (fun f => \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202\u03bc) (f n)) \u2192\n (fun f => \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202\u03bc) fun x => \u2a06 (n : \u2115), f n x\n[PROOFSTEP]\nintro f hf hf_mono hf_prop\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nhf_mono : Monotone f\nhf_prop : \u2200 (n : \u2115), (fun f => \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202\u03bc) (f n)\n\u22a2 \u222b\u207b (a : \u03b1), (fun x => \u2a06 (n : \u2115), f n x) a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), (fun x => \u2a06 (n : \u2115), f n x) a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_iSup hf hf_mono]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nhf_mono : Monotone f\nhf_prop : \u2200 (n : \u2115), (fun f => \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202\u03bc) (f n)\n\u22a2 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), (fun x => \u2a06 (n : \u2115), f n x) a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_iSup (fun n => Measurable.mono (hf n) hm le_rfl) hf_mono]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nhf_mono : Monotone f\nhf_prop : \u2200 (n : \u2115), (fun f => \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202\u03bc) (f n)\n\u22a2 \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202Measure.trim \u03bc hm = \u2a06 (n : \u2115), \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\ncongr\n[GOAL]\ncase refine'_3.e_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nhf\u271d : Measurable f\u271d\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nhf_mono : Monotone f\nhf_prop : \u2200 (n : \u2115), (fun f => \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202\u03bc) (f n)\n\u22a2 (fun n => \u222b\u207b (a : \u03b1), f n a \u2202Measure.trim \u03bc hm) = fun n => \u222b\u207b (a : \u03b1), f n a \u2202\u03bc\n[PROOFSTEP]\nexact funext fun n => hf_prop n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202Measure.trim \u03bc hm = \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_congr_ae (ae_eq_of_ae_eq_trim hf.ae_eq_mk), lintegral_congr_ae hf.ae_eq_mk,\n lintegral_trim hm hf.measurable_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 (n : \u2115), S n) \u2264 \u2a06 (n : \u2115), f (S n)\n\u22a2 f univ \u2264 C\n[PROOFSTEP]\nlet S := @spanningSets _ m (\u03bc.trim hm) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 (n : \u2115), S n) \u2264 \u2a06 (n : \u2115), f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\n\u22a2 f univ \u2264 C\n[PROOFSTEP]\nhave hS_mono : Monotone S := @monotone_spanningSets _ m (\u03bc.trim hm) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 (n : \u2115), S n) \u2264 \u2a06 (n : \u2115), f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_mono : Monotone S\n\u22a2 f univ \u2264 C\n[PROOFSTEP]\nhave hS_meas : \u2200 n, MeasurableSet[m] (S n) := @measurable_spanningSets _ m (\u03bc.trim hm) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 (n : \u2115), S n) \u2264 \u2a06 (n : \u2115), f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_mono : Monotone S\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\n\u22a2 f univ \u2264 C\n[PROOFSTEP]\nrw [\u2190 @iUnion_spanningSets _ m (\u03bc.trim hm)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 (n : \u2115), S n) \u2264 \u2a06 (n : \u2115), f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_mono : Monotone S\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\n\u22a2 f (\u22c3 (i : \u2115), spanningSets (Measure.trim \u03bc hm) i) \u2264 C\n[PROOFSTEP]\nrefine' (h_F_lim S hS_meas hS_mono).trans _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 (n : \u2115), S n) \u2264 \u2a06 (n : \u2115), f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_mono : Monotone S\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\n\u22a2 \u2a06 (n : \u2115), f (S n) \u2264 C\n[PROOFSTEP]\nrefine' iSup_le fun n => hf (S n) (hS_meas n) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 (n : \u2115), S n) \u2264 \u2a06 (n : \u2115), f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_mono : Monotone S\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (S n) \u2260 \u22a4\n[PROOFSTEP]\nexact ((le_trim hm).trans_lt (@measure_spanningSets_lt_top _ m (\u03bc.trim hm) _ n)).ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C\n[PROOFSTEP]\nhave : \u222b\u207b x in univ, f x \u2202\u03bc = \u222b\u207b x, f x \u2202\u03bc := by simp only [Measure.restrict_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\n\u22a2 \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nsimp only [Measure.restrict_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc \u2264 C\n[PROOFSTEP]\nrefine' univ_le_of_forall_fin_meas_le hm C hf fun S hS_meas hS_mono => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 \u222b\u207b (x : \u03b1) in \u22c3 (n : \u2115), S n, f x \u2202\u03bc \u2264 \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lintegral_indicator]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 (n : \u2115), S n) (fun x => f x) a \u2202\u03bc \u2264 \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc\ncase hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 MeasurableSet (\u22c3 (n : \u2115), S n)\n[PROOFSTEP]\nswap\n[GOAL]\ncase hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 MeasurableSet (\u22c3 (n : \u2115), S n)\n[PROOFSTEP]\nexact hm (\u22c3 n, S n) (@MeasurableSet.iUnion _ _ m _ _ hS_meas)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 (n : \u2115), S n) (fun x => f x) a \u2202\u03bc \u2264 \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc\n[PROOFSTEP]\nhave h_integral_indicator : \u2a06 n, \u222b\u207b x in S n, f x \u2202\u03bc = \u2a06 n, \u222b\u207b x, (S n).indicator f x \u2202\u03bc :=\n by\n congr\n ext1 n\n rw [lintegral_indicator _ (hm _ (hS_meas n))]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\n\u22a2 (fun n => \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc) = fun n => \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\n[PROOFSTEP]\next1 n\n[GOAL]\ncase e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nn : \u2115\n\u22a2 \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_indicator _ (hm _ (hS_meas n))]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 (n : \u2115), S n) (fun x => f x) a \u2202\u03bc \u2264 \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc\n[PROOFSTEP]\nrw [h_integral_indicator, \u2190 lintegral_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), indicator (\u22c3 (n : \u2115), S n) (fun x => f x) a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2a06 (n : \u2115), indicator (S n) f a \u2202\u03bc\n[PROOFSTEP]\nrefine' le_of_eq (lintegral_congr fun x => _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\n\u22a2 indicator (\u22c3 (n : \u2115), S n) (fun x => f x) x = \u2a06 (n : \u2115), indicator (S n) f x\n[PROOFSTEP]\nsimp_rw [indicator_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\n\u22a2 (if x \u2208 \u22c3 (n : \u2115), S n then f x else 0) = \u2a06 (n : \u2115), if x \u2208 S n then f x else 0\n[PROOFSTEP]\nby_cases hx_mem : x \u2208 iUnion S\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\n\u22a2 (if x \u2208 \u22c3 (n : \u2115), S n then f x else 0) = \u2a06 (n : \u2115), if x \u2208 S n then f x else 0\n[PROOFSTEP]\nsimp only [hx_mem, if_true]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\n\u22a2 f x = \u2a06 (n : \u2115), if x \u2208 S n then f x else 0\n[PROOFSTEP]\nobtain \u27e8n, hxn\u27e9 := mem_iUnion.mp hx_mem\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\nn : \u2115\nhxn : x \u2208 S n\n\u22a2 f x = \u2a06 (n : \u2115), if x \u2208 S n then f x else 0\n[PROOFSTEP]\nrefine' le_antisymm (_root_.trans _ (le_iSup _ n)) (iSup_le fun i => _)\n[GOAL]\ncase pos.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\nn : \u2115\nhxn : x \u2208 S n\n\u22a2 f x \u2264 if x \u2208 S n then f x else 0\n[PROOFSTEP]\nsimp only [hxn, le_refl, if_true]\n[GOAL]\ncase pos.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\nn : \u2115\nhxn : x \u2208 S n\ni : \u2115\n\u22a2 (if x \u2208 S i then f x else 0) \u2264 f x\n[PROOFSTEP]\nby_cases hxi : x \u2208 S i\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\nn : \u2115\nhxn : x \u2208 S n\ni : \u2115\nhxi : x \u2208 S i\n\u22a2 (if x \u2208 S i then f x else 0) \u2264 f x\n[PROOFSTEP]\nsimp [hxi]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : x \u2208 iUnion S\nn : \u2115\nhxn : x \u2208 S n\ni : \u2115\nhxi : \u00acx \u2208 S i\n\u22a2 (if x \u2208 S i then f x else 0) \u2264 f x\n[PROOFSTEP]\nsimp [hxi]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : \u00acx \u2208 iUnion S\n\u22a2 (if x \u2208 \u22c3 (n : \u2115), S n then f x else 0) = \u2a06 (n : \u2115), if x \u2208 S n then f x else 0\n[PROOFSTEP]\nsimp only [hx_mem, if_false]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : \u00acx \u2208 iUnion S\n\u22a2 0 = \u2a06 (n : \u2115), if x \u2208 S n then f x else 0\n[PROOFSTEP]\nrw [mem_iUnion] at hx_mem \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : \u00ac\u2203 i, x \u2208 S i\n\u22a2 0 = \u2a06 (n : \u2115), if x \u2208 S n then f x else 0\n[PROOFSTEP]\npush_neg at hx_mem \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : \u2200 (i : \u2115), \u00acx \u2208 S i\n\u22a2 0 = \u2a06 (n : \u2115), if x \u2208 S n then f x else 0\n[PROOFSTEP]\nrefine' le_antisymm (zero_le _) (iSup_le fun n => _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nx : \u03b1\nhx_mem : \u2200 (i : \u2115), \u00acx \u2208 S i\nn : \u2115\n\u22a2 (if x \u2208 S n then f x else 0) \u2264 0\n[PROOFSTEP]\nsimp only [hx_mem n, if_false, nonpos_iff_eq_zero]\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\n\u22a2 \u2200 (n : \u2115), Measurable fun x => indicator (S n) f x\n[PROOFSTEP]\nexact fun n => hf_meas.indicator (hm _ (hS_meas n))\n[GOAL]\ncase h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\n\u22a2 Monotone fun n x => indicator (S n) f x\n[PROOFSTEP]\nintro n\u2081 n\u2082 hn\u2081\u2082 a\n[GOAL]\ncase h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\n\u22a2 (fun n x => indicator (S n) f x) n\u2081 a \u2264 (fun n x => indicator (S n) f x) n\u2082 a\n[PROOFSTEP]\nsimp_rw [indicator_apply]\n[GOAL]\ncase h_mono\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\n\u22a2 (if a \u2208 S n\u2081 then f a else 0) \u2264 if a \u2208 S n\u2082 then f a else 0\n[PROOFSTEP]\nsplit_ifs with h h_1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\nh : a \u2208 S n\u2081\nh_1 : a \u2208 S n\u2082\n\u22a2 f a \u2264 f a\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\nh : a \u2208 S n\u2081\nh_1 : \u00aca \u2208 S n\u2082\n\u22a2 f a \u2264 0\n[PROOFSTEP]\nexact absurd (mem_of_mem_of_subset h (hS_mono hn\u2081\u2082)) h_1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\nh : \u00aca \u2208 S n\u2081\nh\u271d : a \u2208 S n\u2082\n\u22a2 0 \u2264 f a\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : Measurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nthis : \u222b\u207b (x : \u03b1) in univ, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\nS : \u2115 \u2192 Set \u03b1\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nhS_mono : Monotone S\nh_integral_indicator : \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1) in S n, f x \u2202\u03bc = \u2a06 (n : \u2115), \u222b\u207b (x : \u03b1), indicator (S n) f x \u2202\u03bc\nn\u2081 n\u2082 : \u2115\nhn\u2081\u2082 : n\u2081 \u2264 n\u2082\na : \u03b1\nh : \u00aca \u2208 S n\u2081\nh\u271d : \u00aca \u2208 S n\u2082\n\u22a2 0 \u2264 0\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C\n[PROOFSTEP]\nlet f' := hf_meas.mk f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C\n[PROOFSTEP]\nhave hf' : \u2200 s, MeasurableSet[m] s \u2192 \u03bc s \u2260 \u221e \u2192 \u222b\u207b x in s, f' x \u2202\u03bc \u2264 C :=\n by\n refine' fun s hs h\u03bcs => (le_of_eq _).trans (hf s hs h\u03bcs)\n refine' lintegral_congr_ae (ae_restrict_of_ae (hf_meas.ae_eq_mk.mono fun x hx => _))\n dsimp only\n rw [hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f' x \u2202\u03bc \u2264 C\n[PROOFSTEP]\nrefine' fun s hs h\u03bcs => (le_of_eq _).trans (hf s hs h\u03bcs)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, f' x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_congr_ae (ae_restrict_of_ae (hf_meas.ae_eq_mk.mono fun x hx => _))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : \u03b1\nhx : f x = AEMeasurable.mk f hf_meas x\n\u22a2 (fun x => f' x) x = (fun x => f x) x\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : \u03b1\nhx : f x = AEMeasurable.mk f hf_meas x\n\u22a2 AEMeasurable.mk f hf_meas x = f x\n[PROOFSTEP]\nrw [hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\nhf' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f' x \u2202\u03bc \u2264 C\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 C\n[PROOFSTEP]\nrw [lintegral_congr_ae hf_meas.ae_eq_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\nf' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf_meas\nhf' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f' x \u2202\u03bc \u2264 C\n\u22a2 \u222b\u207b (a : \u03b1), AEMeasurable.mk f hf_meas a \u2202\u03bc \u2264 C\n[PROOFSTEP]\nexact lintegral_le_of_forall_fin_meas_le_of_measurable hm C hf_meas.measurable_mk hf'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : OpensMeasurableSpace E\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nC : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf_meas : AEMeasurable f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C\n\u22a2 SigmaFinite (Measure.trim \u03bc (_ : inst\u271d\u00b9 \u2264 inst\u271d\u00b9))\n[PROOFSTEP]\nrwa [trim_eq_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\ninduction' f using MeasureTheory.SimpleFunc.induction with c s hs f\u2081 f\u2082 _ h\u2081 h\u2082 generalizing L\n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nsimp only [hs, const_zero, coe_piecewise, coe_const, SimpleFunc.coe_zero, univ_inter, piecewise_eq_indicator,\n lintegral_indicator, lintegral_const, Measure.restrict_apply', ENNReal.coe_indicator, Function.const_apply] at hL \n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nhave c_ne_zero : c \u2260 0 := by\n intro hc\n simp only [hc, ENNReal.coe_zero, zero_mul, not_lt_zero] at hL \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\n\u22a2 c \u2260 0\n[PROOFSTEP]\nintro hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nhc : c = 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [hc, ENNReal.coe_zero, zero_mul, not_lt_zero] at hL \n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nhave : L / c < \u03bc s := by\n rwa [ENNReal.div_lt_iff, mul_comm]\n \u00b7 simp only [c_ne_zero, Ne.def, coe_eq_zero, not_false_iff, true_or_iff]\n \u00b7 simp only [Ne.def, coe_ne_top, not_false_iff, true_or_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\n\u22a2 L / \u2191c < \u2191\u2191\u03bc s\n[PROOFSTEP]\nrwa [ENNReal.div_lt_iff, mul_comm]\n[GOAL]\ncase h0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\n\u22a2 \u2191c \u2260 0 \u2228 L \u2260 0\n[PROOFSTEP]\nsimp only [c_ne_zero, Ne.def, coe_eq_zero, not_false_iff, true_or_iff]\n[GOAL]\ncase ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\n\u22a2 \u2191c \u2260 \u22a4 \u2228 L \u2260 \u22a4\n[PROOFSTEP]\nsimp only [Ne.def, coe_ne_top, not_false_iff, true_or_iff]\n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nobtain \u27e8t, ht, ts, mlt, t_top\u27e9 : \u2203 t : Set \u03b1, MeasurableSet t \u2227 t \u2286 s \u2227 L / \u2191c < \u03bc t \u2227 \u03bc t < \u221e :=\n Measure.exists_subset_measure_lt_top hs this\n[GOAL]\ncase h_ind.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nrefine' \u27e8piecewise t ht (const \u03b1 c) (const \u03b1 0), fun x => _, _, _\u27e9\n[GOAL]\ncase h_ind.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\nx : \u03b1\n\u22a2 \u2191(piecewise t ht (const \u03b1 c) (const \u03b1 0)) x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x\n[PROOFSTEP]\nrefine indicator_le_indicator_of_subset ts (fun x => ?_) x\n[GOAL]\ncase h_ind.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\nx\u271d x : \u03b1\n\u22a2 0 \u2264 \u2191(const \u03b1 c) x\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase h_ind.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise t ht (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc < \u22a4\n[PROOFSTEP]\nsimp only [ht, const_zero, coe_piecewise, coe_const, SimpleFunc.coe_zero, univ_inter, piecewise_eq_indicator,\n ENNReal.coe_indicator, Function.const_apply, lintegral_indicator, lintegral_const, Measure.restrict_apply',\n ENNReal.mul_lt_top ENNReal.coe_ne_top t_top.ne]\n[GOAL]\ncase h_ind.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 L < \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise t ht (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc\n[PROOFSTEP]\nsimp only [ht, const_zero, coe_piecewise, coe_const, SimpleFunc.coe_zero, piecewise_eq_indicator, ENNReal.coe_indicator,\n Function.const_apply, lintegral_indicator, lintegral_const, Measure.restrict_apply', univ_inter]\n[GOAL]\ncase h_ind.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 L < \u2191c * \u2191\u2191\u03bc t\n[PROOFSTEP]\nrwa [mul_comm, \u2190 ENNReal.div_lt_iff]\n[GOAL]\ncase h_ind.intro.intro.intro.intro.refine'_3.h0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u2191c \u2260 0 \u2228 L \u2260 0\n[PROOFSTEP]\nsimp only [c_ne_zero, Ne.def, coe_eq_zero, not_false_iff, true_or_iff]\n[GOAL]\ncase h_ind.intro.intro.intro.intro.refine'_3.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nL : \u211d\u22650\u221e\nhL : L < \u2191c * \u2191\u2191\u03bc s\nc_ne_zero : c \u2260 0\nthis : L / \u2191c < \u2191\u2191\u03bc s\nt : Set \u03b1\nht : MeasurableSet t\nts : t \u2286 s\nmlt : L / \u2191c < \u2191\u2191\u03bc t\nt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u2191c \u2260 \u22a4 \u2228 L \u2260 \u22a4\n[PROOFSTEP]\nsimp only [Ne.def, coe_ne_top, not_false_iff, true_or_iff]\n[GOAL]\ncase h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nreplace hL : L < \u222b\u207b x, f\u2081 x \u2202\u03bc + \u222b\u207b x, f\u2082 x \u2202\u03bc\n[GOAL]\ncase hL\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc\n\u22a2 L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\n[PROOFSTEP]\nrwa [\u2190 lintegral_add_left f\u2081.measurable.coe_nnreal_ennreal]\n[GOAL]\ncase h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nby_cases hf\u2081 : \u222b\u207b x, f\u2081 x \u2202\u03bc = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nsimp only [hf\u2081, zero_add] at hL \n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nrcases h\u2082 hL with \u27e8g, g_le, g_top, gL\u27e9\n[GOAL]\ncase pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\ng : \u03b1 \u2192\u209b \u211d\u22650\ng_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x\ng_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4\ngL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nrefine' \u27e8g, fun x => (g_le x).trans _, g_top, gL\u27e9\n[GOAL]\ncase pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\ng : \u03b1 \u2192\u209b \u211d\u22650\ng_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x\ng_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4\ngL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nx : \u03b1\n\u22a2 \u2191f\u2082 x \u2264 \u2191(f\u2081 + f\u2082) x\n[PROOFSTEP]\nsimp only [SimpleFunc.coe_add, Pi.add_apply, le_add_iff_nonneg_left, zero_le']\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nby_cases hf\u2082 : \u222b\u207b x, f\u2082 x \u2202\u03bc = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nsimp only [hf\u2082, add_zero] at hL \n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nrcases h\u2081 hL with \u27e8g, g_le, g_top, gL\u27e9\n[GOAL]\ncase pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\ng : \u03b1 \u2192\u209b \u211d\u22650\ng_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x\ng_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4\ngL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nrefine' \u27e8g, fun x => (g_le x).trans _, g_top, gL\u27e9\n[GOAL]\ncase pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\ng : \u03b1 \u2192\u209b \u211d\u22650\ng_le : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x\ng_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4\ngL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nx : \u03b1\n\u22a2 \u2191f\u2081 x \u2264 \u2191(f\u2081 + f\u2082) x\n[PROOFSTEP]\nsimp only [SimpleFunc.coe_add, Pi.add_apply, le_add_iff_nonneg_right, zero_le']\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nobtain \u27e8L\u2081, L\u2082, hL\u2081, hL\u2082, hL\u27e9 : \u2203 L\u2081 L\u2082 : \u211d\u22650\u221e, (L\u2081 < \u222b\u207b x, f\u2081 x \u2202\u03bc) \u2227 (L\u2082 < \u222b\u207b x, f\u2082 x \u2202\u03bc) \u2227 L < L\u2081 + L\u2082 :=\n ENNReal.exists_lt_add_of_lt_add hL hf\u2081 hf\u2082\n[GOAL]\ncase neg.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nrcases h\u2081 hL\u2081 with \u27e8g\u2081, g\u2081_le, g\u2081_top, hg\u2081\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nrcases h\u2082 hL\u2082 with \u27e8g\u2082, g\u2082_le, g\u2082_top, hg\u2082\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nrefine' \u27e8g\u2081 + g\u2082, fun x => add_le_add (g\u2081_le x) (g\u2082_le x), _, _\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc < \u22a4\n[PROOFSTEP]\napply lt_of_le_of_lt _ (add_lt_top.2 \u27e8g\u2081_top, g\u2082_top\u27e9)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lintegral_add_left g\u2081.measurable.coe_nnreal_ennreal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191(\u2191g\u2081 a) + \u2191(\u2191g\u2082 a) \u2202\u03bc\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 L < \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc\n[PROOFSTEP]\napply hL.trans ((ENNReal.add_lt_add hg\u2081 hg\u2082).trans_le _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lintegral_add_left g\u2081.measurable.coe_nnreal_ennreal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nL\u271d : \u211d\u22650\u221e\nhL\u271d\u00b9 : L\u271d < \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (support \u2191f\u2081) (support \u2191f\u2082)\nh\u2081 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2081 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nh\u2082 :\n \u2200 {L : \u211d\u22650\u221e},\n L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2192\n \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 \u2191f\u2082 x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\nL : \u211d\u22650\u221e\nhL\u271d : L < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhf\u2081 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc = 0\nhf\u2082 : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc = 0\nL\u2081 L\u2082 : \u211d\u22650\u221e\nhL\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc\nhL\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc\nhL : L < L\u2081 + L\u2082\ng\u2081 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2081_le : \u2200 (x : \u03b1), \u2191g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc < \u22a4\nhg\u2081 : L\u2081 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2081 x) \u2202\u03bc\ng\u2082 : \u03b1 \u2192\u209b \u211d\u22650\ng\u2082_le : \u2200 (x : \u03b1), \u2191g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082_top : \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc < \u22a4\nhg\u2082 : L\u2082 < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2082 x) \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(\u2191g\u2081 a) + \u2191(\u2191g\u2082 a) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(g\u2081 + g\u2082) x) \u2202\u03bc\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\nhL : L < \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [lintegral_eq_nnreal, lt_iSup_iff] at hL \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\nhL : \u2203 i i_1, L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nrcases hL with \u27e8g\u2080, hg\u2080, g\u2080L\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nhave h'L : L < \u222b\u207b x, g\u2080 x \u2202\u03bc := by\n convert g\u2080L\n rw [\u2190 SimpleFunc.lintegral_eq_lintegral]\n rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\n\u22a2 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc\n[PROOFSTEP]\nconvert g\u2080L\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc = SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\n[PROOFSTEP]\nrw [\u2190 SimpleFunc.lintegral_eq_lintegral]\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191(SimpleFunc.map ENNReal.some g\u2080) a \u2202\u03bc\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\nh'L : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nrcases SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral h'L with \u27e8g, hg, gL, gtop\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nL : \u211d\u22650\u221e\ng\u2080 : \u03b1 \u2192\u209b \u211d\u22650\nhg\u2080 : \u2200 (x : \u03b1), \u2191(\u2191g\u2080 x) \u2264 \u2191(f x)\ng\u2080L : L < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some g\u2080) \u03bc\nh'L : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g\u2080 x) \u2202\u03bc\ng : \u03b1 \u2192\u209b \u211d\u22650\nhg : \u2200 (x : \u03b1), \u2191g x \u2264 \u2191g\u2080 x\ngL : \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4\ngtop : L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191g x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc < \u22a4 \u2227 L < \u222b\u207b (x : \u03b1), \u2191(\u2191g x) \u2202\u03bc\n[PROOFSTEP]\nexact \u27e8g, fun x => (hg x).trans (coe_le_coe.1 (hg\u2080 x)), gL, gtop\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 \u2203 \u03bd, IsFiniteMeasure \u03bd \u2227 \u03bc \u226a \u03bd\n[PROOFSTEP]\nobtain \u27e8g, gpos, gmeas, hg\u27e9 : \u2203 g : \u03b1 \u2192 \u211d\u22650, (\u2200 x : \u03b1, 0 < g x) \u2227 Measurable g \u2227 \u222b\u207b x : \u03b1, \u2191(g x) \u2202\u03bc < 1 :=\n exists_pos_lintegral_lt_of_sigmaFinite \u03bc one_ne_zero\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\n\u22a2 \u2203 \u03bd, IsFiniteMeasure \u03bd \u2227 \u03bc \u226a \u03bd\n[PROOFSTEP]\nrefine' \u27e8\u03bc.withDensity fun x => g x, isFiniteMeasure_withDensity hg.ne_top, _\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\n\u22a2 \u03bc \u226a Measure.withDensity \u03bc fun x => \u2191(g x)\n[PROOFSTEP]\nhave : \u03bc = (\u03bc.withDensity fun x => g x).withDensity fun x => (g x)\u207b\u00b9 :=\n by\n have A : ((fun x : \u03b1 => (g x : \u211d\u22650\u221e)) * fun x : \u03b1 => (g x : \u211d\u22650\u221e)\u207b\u00b9) = 1 :=\n by\n ext1 x\n exact ENNReal.mul_inv_cancel (ENNReal.coe_ne_zero.2 (gpos x).ne') ENNReal.coe_ne_top\n rw [\u2190 withDensity_mul _ gmeas.coe_nnreal_ennreal gmeas.coe_nnreal_ennreal.inv, A, withDensity_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\n\u22a2 \u03bc = Measure.withDensity (Measure.withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9\n[PROOFSTEP]\nhave A : ((fun x : \u03b1 => (g x : \u211d\u22650\u221e)) * fun x : \u03b1 => (g x : \u211d\u22650\u221e)\u207b\u00b9) = 1 :=\n by\n ext1 x\n exact ENNReal.mul_inv_cancel (ENNReal.coe_ne_zero.2 (gpos x).ne') ENNReal.coe_ne_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\n\u22a2 ((fun x => \u2191(g x)) * fun x => (\u2191(g x))\u207b\u00b9) = 1\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\nx : \u03b1\n\u22a2 ((fun x => \u2191(g x)) * fun x => (\u2191(g x))\u207b\u00b9) x = OfNat.ofNat 1 x\n[PROOFSTEP]\nexact ENNReal.mul_inv_cancel (ENNReal.coe_ne_zero.2 (gpos x).ne') ENNReal.coe_ne_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\nA : ((fun x => \u2191(g x)) * fun x => (\u2191(g x))\u207b\u00b9) = 1\n\u22a2 \u03bc = Measure.withDensity (Measure.withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 withDensity_mul _ gmeas.coe_nnreal_ennreal gmeas.coe_nnreal_ennreal.inv, A, withDensity_one]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\nthis : \u03bc = Measure.withDensity (Measure.withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9\n\u22a2 \u03bc \u226a Measure.withDensity \u03bc fun x => \u2191(g x)\n[PROOFSTEP]\nnth_rw 1 [this]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ng : \u03b1 \u2192 \u211d\u22650\ngpos : \u2200 (x : \u03b1), 0 < g x\ngmeas : Measurable g\nhg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < 1\nthis : \u03bc = Measure.withDensity (Measure.withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9\n\u22a2 (Measure.withDensity (Measure.withDensity \u03bc fun x => \u2191(g x)) fun x => (\u2191(g x))\u207b\u00b9) \u226a\n Measure.withDensity \u03bc fun x => \u2191(g x)\n[PROOFSTEP]\nexact withDensity_absolutelyContinuous _ _\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Integral.Lebesgue", "llama_tokens": 235780, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195269001831, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.4406099943510988}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n (fderivWithin \ud835\udd5c (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I))) (range \u2191I))\n (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I) \u226b LocalHomeomorph.extend (\u2191j) I).source\n[PROOFSTEP]\nhave h : ((i.1.extend I).symm \u226b j.1.extend I).source \u2286 range I := by rw [i.1.extend_coord_change_source];\n apply image_subset_range\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\n\u22a2 (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I) \u226b LocalHomeomorph.extend (\u2191j) I).source \u2286 range \u2191I\n[PROOFSTEP]\nrw [i.1.extend_coord_change_source]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\n\u22a2 \u2191I '' (LocalHomeomorph.symm \u2191i \u226b\u2095 \u2191j).toLocalEquiv.source \u2286 range \u2191I\n[PROOFSTEP]\napply image_subset_range\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\nh : (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I) \u226b LocalHomeomorph.extend (\u2191j) I).source \u2286 range \u2191I\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n (fderivWithin \ud835\udd5c (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I))) (range \u2191I))\n (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I) \u226b LocalHomeomorph.extend (\u2191j) I).source\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\nh : (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I) \u226b LocalHomeomorph.extend (\u2191j) I).source \u2286 range \u2191I\nx : E\nhx : x \u2208 (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I) \u226b LocalHomeomorph.extend (\u2191j) I).source\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4\n (fderivWithin \ud835\udd5c (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I))) (range \u2191I))\n (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I) \u226b LocalHomeomorph.extend (\u2191j) I).source x\n[PROOFSTEP]\nrefine' (ContDiffWithinAt.fderivWithin_right _ I.unique_diff le_top <| h hx).mono h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\nh : (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I) \u226b LocalHomeomorph.extend (\u2191j) I).source \u2286 range \u2191I\nx : E\nhx : x \u2208 (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I) \u226b LocalHomeomorph.extend (\u2191j) I).source\n\u22a2 ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (range \u2191I) x\n[PROOFSTEP]\nrefine'\n (LocalHomeomorph.contDiffOn_extend_coord_change I (subset_maximalAtlas I j.2) (subset_maximalAtlas I i.2) x\n hx).mono_of_mem\n _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\nh : (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I) \u226b LocalHomeomorph.extend (\u2191j) I).source \u2286 range \u2191I\nx : E\nhx : x \u2208 (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I) \u226b LocalHomeomorph.extend (\u2191j) I).source\n\u22a2 (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I) \u226b LocalHomeomorph.extend (\u2191j) I).source \u2208 \ud835\udcdd[range \u2191I] x\n[PROOFSTEP]\nexact i.1.extend_coord_change_source_mem_nhdsWithin j.1 I hx\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni : \u2191(atlas H M)\nx : M\nhx : x \u2208 (fun i => (\u2191i).source) i\nv : E\n\u22a2 \u2191((fun i j x =>\n fderivWithin \ud835\udd5c (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (range \u2191I) (\u2191(LocalHomeomorph.extend (\u2191i) I) x))\n i i x)\n v =\n v\n[PROOFSTEP]\nsimp only\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni : \u2191(atlas H M)\nx : M\nhx : x \u2208 (fun i => (\u2191i).source) i\nv : E\n\u22a2 \u2191(fderivWithin \ud835\udd5c (\u2191(LocalHomeomorph.extend (\u2191i) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I))) (range \u2191I)\n (\u2191(LocalHomeomorph.extend (\u2191i) I) x))\n v =\n v\n[PROOFSTEP]\nrw [Filter.EventuallyEq.fderivWithin_eq, fderivWithin_id', ContinuousLinearMap.id_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni : \u2191(atlas H M)\nx : M\nhx : x \u2208 (fun i => (\u2191i).source) i\nv : E\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (range \u2191I) (\u2191(LocalHomeomorph.extend (\u2191i) I) x)\n[PROOFSTEP]\nexact I.unique_diff_at_image\n[GOAL]\ncase hs\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni : \u2191(atlas H M)\nx : M\nhx : x \u2208 (fun i => (\u2191i).source) i\nv : E\n\u22a2 \u2191(LocalHomeomorph.extend (\u2191i) I) \u2218\n \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)) =\u1da0[\ud835\udcdd[range \u2191I] \u2191(LocalHomeomorph.extend (\u2191i) I) x]\n fun x => x\n[PROOFSTEP]\nfilter_upwards [i.1.extend_target_mem_nhdsWithin I hx] with y hy\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni : \u2191(atlas H M)\nx : M\nhx : x \u2208 (fun i => (\u2191i).source) i\nv y : E\nhy : y \u2208 (LocalHomeomorph.extend (\u2191i) I).target\n\u22a2 (\u2191(LocalHomeomorph.extend (\u2191i) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I))) y = y\n[PROOFSTEP]\nexact (i.1.extend I).right_inv hy\n[GOAL]\ncase hx\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni : \u2191(atlas H M)\nx : M\nhx : x \u2208 (fun i => (\u2191i).source) i\nv : E\n\u22a2 (\u2191(LocalHomeomorph.extend (\u2191i) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (\u2191(LocalHomeomorph.extend (\u2191i) I) x) =\n \u2191(LocalHomeomorph.extend (\u2191i) I) x\n[PROOFSTEP]\nsimp_rw [Function.comp_apply, i.1.extend_left_inv I hx]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\n\u22a2 ContinuousOn\n ((fun i j x =>\n fderivWithin \ud835\udd5c (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (range \u2191I) (\u2191(LocalHomeomorph.extend (\u2191i) I) x))\n i j)\n ((fun i => (\u2191i).source) i \u2229 (fun i => (\u2191i).source) j)\n[PROOFSTEP]\nrefine' (contDiffOn_fderiv_coord_change I i j).continuousOn.comp ((i.1.continuousOn_extend I).mono _) _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\n\u22a2 (fun i => (\u2191i).source) i \u2229 (fun i => (\u2191i).source) j \u2286 (LocalHomeomorph.extend (\u2191i) I).source\n[PROOFSTEP]\nrw [i.1.extend_source]\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\n\u22a2 (fun i => (\u2191i).source) i \u2229 (fun i => (\u2191i).source) j \u2286 (\u2191i).source\n[PROOFSTEP]\nexact inter_subset_left _ _\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\n\u22a2 MapsTo (fun x => \u2191(LocalHomeomorph.extend (\u2191i) I) x) ((fun i => (\u2191i).source) i \u2229 (fun i => (\u2191i).source) j)\n (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I) \u226b LocalHomeomorph.extend (\u2191j) I).source\n[PROOFSTEP]\nsimp_rw [\u2190 i.1.extend_image_source_inter, mapsTo_image]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 \u2200 (i j k : \u2191(atlas H M)) (x : M),\n x \u2208 (fun i => (\u2191i).source) i \u2229 (fun i => (\u2191i).source) j \u2229 (fun i => (\u2191i).source) k \u2192\n \u2200 (v : E),\n \u2191((fun i j x =>\n fderivWithin \ud835\udd5c (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (range \u2191I) (\u2191(LocalHomeomorph.extend (\u2191i) I) x))\n j k x)\n (\u2191((fun i j x =>\n fderivWithin \ud835\udd5c\n (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I))) (range \u2191I)\n (\u2191(LocalHomeomorph.extend (\u2191i) I) x))\n i j x)\n v) =\n \u2191((fun i j x =>\n fderivWithin \ud835\udd5c (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (range \u2191I) (\u2191(LocalHomeomorph.extend (\u2191i) I) x))\n i k x)\n v\n[PROOFSTEP]\nrintro i j k x \u27e8\u27e8hxi, hxj\u27e9, hxk\u27e9 v\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j k : \u2191(atlas H M)\nx : M\nhxk : x \u2208 (fun i => (\u2191i).source) k\nhxi : x \u2208 (fun i => (\u2191i).source) i\nhxj : x \u2208 (fun i => (\u2191i).source) j\nv : E\n\u22a2 \u2191((fun i j x =>\n fderivWithin \ud835\udd5c (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (range \u2191I) (\u2191(LocalHomeomorph.extend (\u2191i) I) x))\n j k x)\n (\u2191((fun i j x =>\n fderivWithin \ud835\udd5c (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (range \u2191I) (\u2191(LocalHomeomorph.extend (\u2191i) I) x))\n i j x)\n v) =\n \u2191((fun i j x =>\n fderivWithin \ud835\udd5c (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (range \u2191I) (\u2191(LocalHomeomorph.extend (\u2191i) I) x))\n i k x)\n v\n[PROOFSTEP]\nrw [fderivWithin_fderivWithin, Filter.EventuallyEq.fderivWithin_eq]\n[GOAL]\ncase intro.intro.hs\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j k : \u2191(atlas H M)\nx : M\nhxk : x \u2208 (fun i => (\u2191i).source) k\nhxi : x \u2208 (fun i => (\u2191i).source) i\nhxj : x \u2208 (fun i => (\u2191i).source) j\nv : E\n\u22a2 (\u2191(LocalHomeomorph.extend (\u2191k) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191j) I))) \u2218\n \u2191(LocalHomeomorph.extend (\u2191j) I) \u2218\n \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)) =\u1da0[\ud835\udcdd[range \u2191I] \u2191(LocalHomeomorph.extend (\u2191i) I) x]\n \u2191(LocalHomeomorph.extend (\u2191k) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I))\n[PROOFSTEP]\nhave := i.1.extend_preimage_mem_nhds I hxi (j.1.extend_source_mem_nhds I hxj)\n[GOAL]\ncase intro.intro.hs\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j k : \u2191(atlas H M)\nx : M\nhxk : x \u2208 (fun i => (\u2191i).source) k\nhxi : x \u2208 (fun i => (\u2191i).source) i\nhxj : x \u2208 (fun i => (\u2191i).source) j\nv : E\nthis :\n \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)) \u207b\u00b9' (LocalHomeomorph.extend (\u2191j) I).source \u2208\n \ud835\udcdd (\u2191(LocalHomeomorph.extend (\u2191i) I) x)\n\u22a2 (\u2191(LocalHomeomorph.extend (\u2191k) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191j) I))) \u2218\n \u2191(LocalHomeomorph.extend (\u2191j) I) \u2218\n \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)) =\u1da0[\ud835\udcdd[range \u2191I] \u2191(LocalHomeomorph.extend (\u2191i) I) x]\n \u2191(LocalHomeomorph.extend (\u2191k) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I))\n[PROOFSTEP]\nfilter_upwards [nhdsWithin_le_nhds this] with y hy\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j k : \u2191(atlas H M)\nx : M\nhxk : x \u2208 (fun i => (\u2191i).source) k\nhxi : x \u2208 (fun i => (\u2191i).source) i\nhxj : x \u2208 (fun i => (\u2191i).source) j\nv : E\nthis :\n \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)) \u207b\u00b9' (LocalHomeomorph.extend (\u2191j) I).source \u2208\n \ud835\udcdd (\u2191(LocalHomeomorph.extend (\u2191i) I) x)\ny : E\nhy : y \u2208 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)) \u207b\u00b9' (LocalHomeomorph.extend (\u2191j) I).source\n\u22a2 ((\u2191(LocalHomeomorph.extend (\u2191k) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191j) I))) \u2218\n \u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n y =\n (\u2191(LocalHomeomorph.extend (\u2191k) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I))) y\n[PROOFSTEP]\nsimp_rw [Function.comp_apply, (j.1.extend I).left_inv hy]\n[GOAL]\ncase intro.intro.hx\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j k : \u2191(atlas H M)\nx : M\nhxk : x \u2208 (fun i => (\u2191i).source) k\nhxi : x \u2208 (fun i => (\u2191i).source) i\nhxj : x \u2208 (fun i => (\u2191i).source) j\nv : E\n\u22a2 ((\u2191(LocalHomeomorph.extend (\u2191k) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191j) I))) \u2218\n \u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (\u2191(LocalHomeomorph.extend (\u2191i) I) x) =\n (\u2191(LocalHomeomorph.extend (\u2191k) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (\u2191(LocalHomeomorph.extend (\u2191i) I) x)\n[PROOFSTEP]\nsimp_rw [Function.comp_apply, i.1.extend_left_inv I hxi, j.1.extend_left_inv I hxj]\n[GOAL]\ncase intro.intro.hg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j k : \u2191(atlas H M)\nx : M\nhxk : x \u2208 (fun i => (\u2191i).source) k\nhxi : x \u2208 (fun i => (\u2191i).source) i\nhxj : x \u2208 (fun i => (\u2191i).source) j\nv : E\n\u22a2 DifferentiableWithinAt \ud835\udd5c (\u2191(LocalHomeomorph.extend (\u2191k) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191j) I)))\n (range \u2191I) (\u2191(LocalHomeomorph.extend (\u2191j) I) x)\n[PROOFSTEP]\nexact\n (contDiffWithinAt_extend_coord_change' I (subset_maximalAtlas I k.2) (subset_maximalAtlas I j.2) hxk\n hxj).differentiableWithinAt\n le_top\n[GOAL]\ncase intro.intro.hf\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j k : \u2191(atlas H M)\nx : M\nhxk : x \u2208 (fun i => (\u2191i).source) k\nhxi : x \u2208 (fun i => (\u2191i).source) i\nhxj : x \u2208 (fun i => (\u2191i).source) j\nv : E\n\u22a2 DifferentiableWithinAt \ud835\udd5c (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (range \u2191I) (\u2191(LocalHomeomorph.extend (\u2191i) I) x)\n[PROOFSTEP]\nexact\n (contDiffWithinAt_extend_coord_change' I (subset_maximalAtlas I j.2) (subset_maximalAtlas I i.2) hxj\n hxi).differentiableWithinAt\n le_top\n[GOAL]\ncase intro.intro.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j k : \u2191(atlas H M)\nx : M\nhxk : x \u2208 (fun i => (\u2191i).source) k\nhxi : x \u2208 (fun i => (\u2191i).source) i\nhxj : x \u2208 (fun i => (\u2191i).source) j\nv : E\n\u22a2 MapsTo (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I))) (range \u2191I) (range \u2191I)\n[PROOFSTEP]\nintro x _\n[GOAL]\ncase intro.intro.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j k : \u2191(atlas H M)\nx\u271d : M\nhxk : x\u271d \u2208 (fun i => (\u2191i).source) k\nhxi : x\u271d \u2208 (fun i => (\u2191i).source) i\nhxj : x\u271d \u2208 (fun i => (\u2191i).source) j\nv x : E\na\u271d : x \u2208 range \u2191I\n\u22a2 (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I))) x \u2208 range \u2191I\n[PROOFSTEP]\nexact mem_range_self _\n[GOAL]\ncase intro.intro.hxs\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j k : \u2191(atlas H M)\nx : M\nhxk : x \u2208 (fun i => (\u2191i).source) k\nhxi : x \u2208 (fun i => (\u2191i).source) i\nhxj : x \u2208 (fun i => (\u2191i).source) j\nv : E\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (range \u2191I) (\u2191(LocalHomeomorph.extend (\u2191i) I) x)\n[PROOFSTEP]\nexact I.unique_diff_at_image\n[GOAL]\ncase intro.intro.hy\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j k : \u2191(atlas H M)\nx : M\nhxk : x \u2208 (fun i => (\u2191i).source) k\nhxi : x \u2208 (fun i => (\u2191i).source) i\nhxj : x \u2208 (fun i => (\u2191i).source) j\nv : E\n\u22a2 (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (\u2191(LocalHomeomorph.extend (\u2191i) I) x) =\n \u2191(LocalHomeomorph.extend (\u2191j) I) x\n[PROOFSTEP]\nrw [Function.comp_apply, i.1.extend_left_inv I hxi]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np q : TM\n\u22a2 p \u2208 (chartAt (ModelProd H E) q).toLocalEquiv.source \u2194 p.proj \u2208 (chartAt H q.proj).toLocalEquiv.source\n[PROOFSTEP]\nsimp only [FiberBundle.chartedSpace_chartAt, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : H \u00d7 E\nq : TM\n\u22a2 p \u2208 (chartAt (ModelProd H E) q).toLocalEquiv.target \u2194 p.fst \u2208 (chartAt H q.proj).toLocalEquiv.target\n[PROOFSTEP]\nsimp only [FiberBundle.chartedSpace_chartAt, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : H \u00d7 E\nq : TM\n\u22a2 p.fst \u2208 (chartAt H q.proj).toLocalEquiv.target \u2227\n (\u2191(LocalEquiv.symm (LocalEquiv.prod (chartAt H q.proj).toLocalEquiv (LocalEquiv.refl E))) p).fst \u2208\n (chartAt H q.proj).toLocalEquiv.source \u2194\n p.fst \u2208 (chartAt H q.proj).toLocalEquiv.target\n[PROOFSTEP]\nrw [LocalEquiv.prod_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : H \u00d7 E\nq : TM\n\u22a2 p.fst \u2208 (chartAt H q.proj).toLocalEquiv.target \u2227\n (\u2191(LocalEquiv.prod (LocalEquiv.symm (chartAt H q.proj).toLocalEquiv) (LocalEquiv.symm (LocalEquiv.refl E)))\n p).fst \u2208\n (chartAt H q.proj).toLocalEquiv.source \u2194\n p.fst \u2208 (chartAt H q.proj).toLocalEquiv.target\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [and_iff_left_iff_imp, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nb b' x : F\n\u22a2 VectorBundleCore.coordChange (tangentBundleCore \ud835\udcd8(\ud835\udd5c, F) F) (achart F b) (achart F b') x = 1\n[PROOFSTEP]\nsimpa only [tangentBundleCore_coordChange, mfld_simps] using fderivWithin_id uniqueDiffWithinAt_univ\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nb b' : F\n\u22a2 Trivialization.symmL \ud835\udd5c (trivializationAt F (TangentSpace \ud835\udcd8(\ud835\udd5c, F)) b) b' = 1\n[PROOFSTEP]\nrw [TangentBundle.trivializationAt_symmL, coordChange_model_space]\n[GOAL]\ncase hb\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nb b' : F\n\u22a2 b' \u2208 (trivializationAt F (TangentSpace \ud835\udcd8(\ud835\udd5c, F)) b).baseSet\n[PROOFSTEP]\napply mem_univ\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nb b' : F\n\u22a2 Trivialization.continuousLinearMapAt \ud835\udd5c (trivializationAt F (TangentSpace \ud835\udcd8(\ud835\udd5c, F)) b) b' = 1\n[PROOFSTEP]\nrw [TangentBundle.trivializationAt_continuousLinearMapAt, coordChange_model_space]\n[GOAL]\ncase hb\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nb b' : F\n\u22a2 b' \u2208 (trivializationAt F (TangentSpace \ud835\udcd8(\ud835\udd5c, F)) b).baseSet\n[PROOFSTEP]\napply mem_univ\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 VectorBundleCore.IsSmooth (tangentBundleCore I M) I\n[PROOFSTEP]\nrefine' \u27e8fun i j => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\n\u22a2 SmoothOn I \ud835\udcd8(\ud835\udd5c, E \u2192L[\ud835\udd5c] E) (VectorBundleCore.coordChange (tangentBundleCore I M) i j)\n (VectorBundleCore.baseSet (tangentBundleCore I M) i \u2229 VectorBundleCore.baseSet (tangentBundleCore I M) j)\n[PROOFSTEP]\nrw [SmoothOn, contMDiffOn_iff_source_of_mem_maximalAtlas (subset_maximalAtlas I i.2), contMDiffOn_iff_contDiffOn]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n (VectorBundleCore.coordChange (tangentBundleCore I M) i j \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (\u2191(LocalHomeomorph.extend (\u2191i) I) ''\n (VectorBundleCore.baseSet (tangentBundleCore I M) i \u2229 VectorBundleCore.baseSet (tangentBundleCore I M) j))\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\n\u22a2 VectorBundleCore.baseSet (tangentBundleCore I M) i \u2229 VectorBundleCore.baseSet (tangentBundleCore I M) j \u2286 (\u2191i).source\n[PROOFSTEP]\nrefine' ((contDiffOn_fderiv_coord_change I i j).congr fun x hx => _).mono _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\nx : E\nhx : x \u2208 (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I) \u226b LocalHomeomorph.extend (\u2191j) I).source\n\u22a2 (VectorBundleCore.coordChange (tangentBundleCore I M) i j \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I))) x =\n fderivWithin \ud835\udd5c (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I))) (range \u2191I) x\n[PROOFSTEP]\nrw [LocalEquiv.trans_source'] at hx \n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\nx : E\nhx :\n x \u2208\n (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)).source \u2229\n \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)) \u207b\u00b9'\n ((LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)).target \u2229 (LocalHomeomorph.extend (\u2191j) I).source)\n\u22a2 (VectorBundleCore.coordChange (tangentBundleCore I M) i j \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I))) x =\n fderivWithin \ud835\udd5c (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I))) (range \u2191I) x\n[PROOFSTEP]\nsimp_rw [Function.comp_apply, tangentBundleCore_coordChange, (i.1.extend I).right_inv hx.1]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\n\u22a2 \u2191(LocalHomeomorph.extend (\u2191i) I) ''\n (VectorBundleCore.baseSet (tangentBundleCore I M) i \u2229 VectorBundleCore.baseSet (tangentBundleCore I M) j) \u2286\n (LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I) \u226b LocalHomeomorph.extend (\u2191j) I).source\n[PROOFSTEP]\nexact (i.1.extend_image_source_inter j.1 I).subset\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\ni j : \u2191(atlas H M)\n\u22a2 VectorBundleCore.baseSet (tangentBundleCore I M) i \u2229 VectorBundleCore.baseSet (tangentBundleCore I M) j \u2286 (\u2191i).source\n[PROOFSTEP]\napply inter_subset_left\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : TangentBundle I H\n\u22a2 (chartAt (ModelProd H E) p).toLocalEquiv = Equiv.toLocalEquiv (TotalSpace.toProd H E)\n[PROOFSTEP]\next x : 1\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np x : TangentBundle I H\n\u22a2 \u2191(chartAt (ModelProd H E) p).toLocalEquiv x = \u2191(Equiv.toLocalEquiv (TotalSpace.toProd H E)) x\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\u2081\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np x : TangentBundle I H\n\u22a2 (\u2191(chartAt (ModelProd H E) p).toLocalEquiv x).fst = (\u2191(Equiv.toLocalEquiv (TotalSpace.toProd H E)) x).fst\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.h\u2082\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np x : TangentBundle I H\n\u22a2 (\u2191(chartAt (ModelProd H E) p).toLocalEquiv x).snd = (\u2191(Equiv.toLocalEquiv (TotalSpace.toProd H E)) x).snd\n[PROOFSTEP]\nexact (tangentBundleCore I H).coordChange_self (achart _ x.1) x.1 (mem_achart_source H x.1) x.2\n[GOAL]\ncase hsymm\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : TangentBundle I H\nx : ModelProd H E\n\u22a2 \u2191(LocalEquiv.symm (chartAt (ModelProd H E) p).toLocalEquiv) x =\n \u2191(LocalEquiv.symm (Equiv.toLocalEquiv (TotalSpace.toProd H E))) x\n[PROOFSTEP]\nrefine congr_arg (TotalSpace.mk _) ?_\n[GOAL]\ncase hsymm\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : TangentBundle I H\nx : ModelProd H E\n\u22a2 FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p.proj)\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (\u2191(LocalEquiv.symm\n (LocalHomeomorph.restrOpen (chartAt (ModelProd H E) (\u2191(chartAt (H \u00d7 E) p) p))\n (chartAt (H \u00d7 E) p).toLocalEquiv.target\n (_ : IsOpen (chartAt (H \u00d7 E) p).toLocalEquiv.target)).toLocalEquiv)\n x).fst)\n (\u2191(LocalEquiv.symm\n (LocalHomeomorph.restrOpen (chartAt (ModelProd H E) (\u2191(chartAt (H \u00d7 E) p) p))\n (chartAt (H \u00d7 E) p).toLocalEquiv.target\n (_ : IsOpen (chartAt (H \u00d7 E) p).toLocalEquiv.target)).toLocalEquiv)\n x).fst\n (\u2191(LocalEquiv.symm\n (LocalHomeomorph.restrOpen (chartAt (ModelProd H E) (\u2191(chartAt (H \u00d7 E) p) p))\n (chartAt (H \u00d7 E) p).toLocalEquiv.target\n (_ : IsOpen (chartAt (H \u00d7 E) p).toLocalEquiv.target)).toLocalEquiv)\n x).snd =\n x.snd\n[PROOFSTEP]\nexact (tangentBundleCore I H).coordChange_self (achart _ x.1) x.1 (mem_achart_source H x.1) x.2\n[GOAL]\ncase hs\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : TangentBundle I H\n\u22a2 (chartAt (ModelProd H E) p).toLocalEquiv.source = (Equiv.toLocalEquiv (TotalSpace.toProd H E)).source\n[PROOFSTEP]\nsimp_rw [TangentBundle.chartAt, FiberBundleCore.localTriv, FiberBundleCore.localTrivAsLocalEquiv,\n VectorBundleCore.toFiberBundleCore_baseSet, tangentBundleCore_baseSet]\n[GOAL]\ncase hs\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : TangentBundle I H\n\u22a2 ({\n toLocalEquiv :=\n {\n toFun := fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd),\n invFun := fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (achart H p.proj)\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd },\n source :=\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source,\n target := (\u2191(achart H p.proj)).source \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 \u2983x : FiberBundleCore.TotalSpace (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\u2984,\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source \u2192\n (fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n x \u2208\n (\u2191(achart H p.proj)).source \u00d7\u02e2 univ),\n map_target' :=\n (_ :\n \u2200 \u2983x : H \u00d7 E\u2984,\n x \u2208 (\u2191(achart H p.proj)).source \u00d7\u02e2 univ \u2192\n (fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source),\n left_inv' :=\n (_ :\n \u2200 \u2983x : FiberBundleCore.TotalSpace (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\u2984,\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source \u2192\n (fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n ((fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n x) =\n x),\n right_inv' :=\n (_ :\n \u2200 \u2983x : H \u00d7 E\u2984,\n x \u2208 (\u2191(achart H p.proj)).source \u00d7\u02e2 univ \u2192\n (fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n ((fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n x) =\n x) },\n open_source :=\n (_ :\n IsOpen\n {\n toFun := fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd),\n invFun := fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (achart H p.proj)\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n p_1.fst)\n p_1.fst p_1.snd },\n source :=\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source,\n target := (\u2191(achart H p.proj)).source \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200\n \u2983x :\n FiberBundleCore.TotalSpace (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\u2984,\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source \u2192\n (fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n x \u2208\n (\u2191(achart H p.proj)).source \u00d7\u02e2 univ),\n map_target' :=\n (_ :\n \u2200 \u2983x : H \u00d7 E\u2984,\n x \u2208 (\u2191(achart H p.proj)).source \u00d7\u02e2 univ \u2192\n (fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source),\n left_inv' :=\n (_ :\n \u2200\n \u2983x :\n FiberBundleCore.TotalSpace (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\u2984,\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source \u2192\n (fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n ((fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n x) =\n x),\n right_inv' :=\n (_ :\n \u2200 \u2983x : H \u00d7 E\u2984,\n x \u2208 (\u2191(achart H p.proj)).source \u00d7\u02e2 univ \u2192\n (fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n ((fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n x) =\n x) }.source),\n open_target :=\n (_ :\n IsOpen\n {\n toFun := fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd),\n invFun := fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (achart H p.proj)\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n p_1.fst)\n p_1.fst p_1.snd },\n source :=\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source,\n target := (\u2191(achart H p.proj)).source \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200\n \u2983x :\n FiberBundleCore.TotalSpace (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\u2984,\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source \u2192\n (fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n x \u2208\n (\u2191(achart H p.proj)).source \u00d7\u02e2 univ),\n map_target' :=\n (_ :\n \u2200 \u2983x : H \u00d7 E\u2984,\n x \u2208 (\u2191(achart H p.proj)).source \u00d7\u02e2 univ \u2192\n (fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source),\n left_inv' :=\n (_ :\n \u2200\n \u2983x :\n FiberBundleCore.TotalSpace (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\u2984,\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source \u2192\n (fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n ((fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n x) =\n x),\n right_inv' :=\n (_ :\n \u2200 \u2983x : H \u00d7 E\u2984,\n x \u2208 (\u2191(achart H p.proj)).source \u00d7\u02e2 univ \u2192\n (fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n ((fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n x) =\n x) }.target),\n continuous_toFun :=\n (_ :\n ContinuousOn\n \u2191{\n toFun := fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd),\n invFun := fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (achart H p.proj)\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n p_1.fst)\n p_1.fst p_1.snd },\n source :=\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source,\n target := (\u2191(achart H p.proj)).source \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200\n \u2983x :\n FiberBundleCore.TotalSpace (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\u2984,\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source \u2192\n (fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n x \u2208\n (\u2191(achart H p.proj)).source \u00d7\u02e2 univ),\n map_target' :=\n (_ :\n \u2200 \u2983x : H \u00d7 E\u2984,\n x \u2208 (\u2191(achart H p.proj)).source \u00d7\u02e2 univ \u2192\n (fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source),\n left_inv' :=\n (_ :\n \u2200\n \u2983x :\n FiberBundleCore.TotalSpace (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\u2984,\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source \u2192\n (fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n ((fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n x) =\n x),\n right_inv' :=\n (_ :\n \u2200 \u2983x : H \u00d7 E\u2984,\n x \u2208 (\u2191(achart H p.proj)).source \u00d7\u02e2 univ \u2192\n (fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n ((fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n x) =\n x) }\n {\n toFun := fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd),\n invFun := fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (achart H p.proj)\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n p_1.fst)\n p_1.fst p_1.snd },\n source :=\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source,\n target := (\u2191(achart H p.proj)).source \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200\n \u2983x :\n FiberBundleCore.TotalSpace (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\u2984,\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source \u2192\n (fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n x \u2208\n (\u2191(achart H p.proj)).source \u00d7\u02e2 univ),\n map_target' :=\n (_ :\n \u2200 \u2983x : H \u00d7 E\u2984,\n x \u2208 (\u2191(achart H p.proj)).source \u00d7\u02e2 univ \u2192\n (fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source),\n left_inv' :=\n (_ :\n \u2200\n \u2983x :\n FiberBundleCore.TotalSpace (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\u2984,\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source \u2192\n (fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n ((fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n x) =\n x),\n right_inv' :=\n (_ :\n \u2200 \u2983x : H \u00d7 E\u2984,\n x \u2208 (\u2191(achart H p.proj)).source \u00d7\u02e2 univ \u2192\n (fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n ((fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n x) =\n x) }.source),\n continuous_invFun :=\n (_ :\n ContinuousOn\n {\n toFun := fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd),\n invFun := fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (achart H p.proj)\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n p_1.fst)\n p_1.fst p_1.snd },\n source :=\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source,\n target := (\u2191(achart H p.proj)).source \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200\n \u2983x :\n FiberBundleCore.TotalSpace (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\u2984,\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source \u2192\n (fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n x \u2208\n (\u2191(achart H p.proj)).source \u00d7\u02e2 univ),\n map_target' :=\n (_ :\n \u2200 \u2983x : H \u00d7 E\u2984,\n x \u2208 (\u2191(achart H p.proj)).source \u00d7\u02e2 univ \u2192\n (fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source),\n left_inv' :=\n (_ :\n \u2200\n \u2983x :\n FiberBundleCore.TotalSpace (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\u2984,\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source \u2192\n (fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n ((fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n x) =\n x),\n right_inv' :=\n (_ :\n \u2200 \u2983x : H \u00d7 E\u2984,\n x \u2208 (\u2191(achart H p.proj)).source \u00d7\u02e2 univ \u2192\n (fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n ((fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n x) =\n x) }.invFun\n {\n toFun := fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd),\n invFun := fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (achart H p.proj)\n (FiberBundleCore.indexAt (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n p_1.fst)\n p_1.fst p_1.snd },\n source :=\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source,\n target := (\u2191(achart H p.proj)).source \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200\n \u2983x :\n FiberBundleCore.TotalSpace (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\u2984,\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source \u2192\n (fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n x \u2208\n (\u2191(achart H p.proj)).source \u00d7\u02e2 univ),\n map_target' :=\n (_ :\n \u2200 \u2983x : H \u00d7 E\u2984,\n x \u2208 (\u2191(achart H p.proj)).source \u00d7\u02e2 univ \u2192\n (fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source),\n left_inv' :=\n (_ :\n \u2200\n \u2983x :\n FiberBundleCore.TotalSpace (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\u2984,\n x \u2208\n FiberBundleCore.proj (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) \u207b\u00b9'\n (\u2191(achart H p.proj)).source \u2192\n (fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n ((fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n x) =\n x),\n right_inv' :=\n (_ :\n \u2200 \u2983x : H \u00d7 E\u2984,\n x \u2208 (\u2191(achart H p.proj)).source \u00d7\u02e2 univ \u2192\n (fun p_1 =>\n (p_1.proj,\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.proj)\n (achart H p.proj) p_1.proj p_1.snd))\n ((fun p_1 =>\n { proj := p_1.fst,\n snd :=\n FiberBundleCore.coordChange\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H))\n (achart H p.proj)\n (FiberBundleCore.indexAt\n (VectorBundleCore.toFiberBundleCore (tangentBundleCore I H)) p_1.fst)\n p_1.fst p_1.snd })\n x) =\n x) }.target) } \u226b\u2095\n LocalHomeomorph.prod (chartAt H p.proj) (LocalHomeomorph.refl E)).toLocalEquiv.source =\n (Equiv.toLocalEquiv (TotalSpace.toProd H E)).source\n[PROOFSTEP]\nsimp only [mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : TangentBundle I H\n\u22a2 \u2191(chartAt (ModelProd H E) p) = \u2191(TotalSpace.toProd H E)\n[PROOFSTEP]\nrw [\u2190 LocalHomeomorph.coe_coe, tangentBundle_model_space_chartAt]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : TangentBundle I H\n\u22a2 \u2191(Equiv.toLocalEquiv (TotalSpace.toProd H E)) = \u2191(TotalSpace.toProd H E)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : TangentBundle I H\n\u22a2 \u2191(LocalHomeomorph.symm (chartAt (ModelProd H E) p)) = \u2191(TotalSpace.toProd H E).symm\n[PROOFSTEP]\nrw [\u2190 LocalHomeomorph.coe_coe, LocalHomeomorph.symm_toLocalEquiv, tangentBundle_model_space_chartAt]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : TangentBundle I H\n\u22a2 \u2191(LocalEquiv.symm (Equiv.toLocalEquiv (TotalSpace.toProd H E))) = \u2191(TotalSpace.toProd H E).symm\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nx x' z : H\n\u22a2 VectorBundleCore.coordChange (tangentBundleCore I H) (achart H x) (achart H x') z = ContinuousLinearMap.id \ud835\udd5c E\n[PROOFSTEP]\next v\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nx x' z : H\nv : E\n\u22a2 \u2191(VectorBundleCore.coordChange (tangentBundleCore I H) (achart H x) (achart H x') z) v =\n \u2191(ContinuousLinearMap.id \ud835\udd5c E) v\n[PROOFSTEP]\nexact (tangentBundleCore I H).coordChange_self (achart _ z) z (mem_univ _) v\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nsrc\u271d : (TotalSpace E fun x => E) \u2243 H \u00d7 E := TotalSpace.toProd H E\n\u22a2 Continuous\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }.toFun\n[PROOFSTEP]\nlet p : TangentBundle I H := \u27e8I.symm (0 : E), (0 : E)\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nsrc\u271d : (TotalSpace E fun x => E) \u2243 H \u00d7 E := TotalSpace.toProd H E\np : TangentBundle I H := { proj := \u2191(ModelWithCorners.symm I) 0, snd := 0 }\n\u22a2 Continuous\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }.toFun\n[PROOFSTEP]\nhave : Continuous (chartAt (ModelProd H E) p) :=\n by\n rw [continuous_iff_continuousOn_univ]\n convert (chartAt (ModelProd H E) p).continuousOn\n simp only [TangentSpace.fiberBundle, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nsrc\u271d : (TotalSpace E fun x => E) \u2243 H \u00d7 E := TotalSpace.toProd H E\np : TangentBundle I H := { proj := \u2191(ModelWithCorners.symm I) 0, snd := 0 }\n\u22a2 Continuous \u2191(chartAt (ModelProd H E) p)\n[PROOFSTEP]\nrw [continuous_iff_continuousOn_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nsrc\u271d : (TotalSpace E fun x => E) \u2243 H \u00d7 E := TotalSpace.toProd H E\np : TangentBundle I H := { proj := \u2191(ModelWithCorners.symm I) 0, snd := 0 }\n\u22a2 ContinuousOn (\u2191(chartAt (ModelProd H E) p)) univ\n[PROOFSTEP]\nconvert (chartAt (ModelProd H E) p).continuousOn\n[GOAL]\ncase h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nsrc\u271d : (TotalSpace E fun x => E) \u2243 H \u00d7 E := TotalSpace.toProd H E\np : TangentBundle I H := { proj := \u2191(ModelWithCorners.symm I) 0, snd := 0 }\n\u22a2 univ = (chartAt (ModelProd H E) p).toLocalEquiv.source\n[PROOFSTEP]\nsimp only [TangentSpace.fiberBundle, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nsrc\u271d : (TotalSpace E fun x => E) \u2243 H \u00d7 E := TotalSpace.toProd H E\np : TangentBundle I H := { proj := \u2191(ModelWithCorners.symm I) 0, snd := 0 }\nthis : Continuous \u2191(chartAt (ModelProd H E) p)\n\u22a2 Continuous\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }.toFun\n[PROOFSTEP]\nsimpa only [mfld_simps] using this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nsrc\u271d : (TotalSpace E fun x => E) \u2243 H \u00d7 E := TotalSpace.toProd H E\n\u22a2 Continuous\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }.invFun\n[PROOFSTEP]\nlet p : TangentBundle I H := \u27e8I.symm (0 : E), (0 : E)\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nsrc\u271d : (TotalSpace E fun x => E) \u2243 H \u00d7 E := TotalSpace.toProd H E\np : TangentBundle I H := { proj := \u2191(ModelWithCorners.symm I) 0, snd := 0 }\n\u22a2 Continuous\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }.invFun\n[PROOFSTEP]\nhave : Continuous (chartAt (ModelProd H E) p).symm :=\n by\n rw [continuous_iff_continuousOn_univ]\n convert (chartAt (ModelProd H E) p).symm.continuousOn\n simp only [mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nsrc\u271d : (TotalSpace E fun x => E) \u2243 H \u00d7 E := TotalSpace.toProd H E\np : TangentBundle I H := { proj := \u2191(ModelWithCorners.symm I) 0, snd := 0 }\n\u22a2 Continuous \u2191(LocalHomeomorph.symm (chartAt (ModelProd H E) p))\n[PROOFSTEP]\nrw [continuous_iff_continuousOn_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nsrc\u271d : (TotalSpace E fun x => E) \u2243 H \u00d7 E := TotalSpace.toProd H E\np : TangentBundle I H := { proj := \u2191(ModelWithCorners.symm I) 0, snd := 0 }\n\u22a2 ContinuousOn (\u2191(LocalHomeomorph.symm (chartAt (ModelProd H E) p))) univ\n[PROOFSTEP]\nconvert (chartAt (ModelProd H E) p).symm.continuousOn\n[GOAL]\ncase h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nsrc\u271d : (TotalSpace E fun x => E) \u2243 H \u00d7 E := TotalSpace.toProd H E\np : TangentBundle I H := { proj := \u2191(ModelWithCorners.symm I) 0, snd := 0 }\n\u22a2 univ = (LocalHomeomorph.symm (chartAt (ModelProd H E) p)).toLocalEquiv.source\n[PROOFSTEP]\nsimp only [mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nsrc\u271d : (TotalSpace E fun x => E) \u2243 H \u00d7 E := TotalSpace.toProd H E\np : TangentBundle I H := { proj := \u2191(ModelWithCorners.symm I) 0, snd := 0 }\nthis : Continuous \u2191(LocalHomeomorph.symm (chartAt (ModelProd H E) p))\n\u22a2 Continuous\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }.invFun\n[PROOFSTEP]\nsimpa only [mfld_simps] using this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nN : Type u_9\nx\u2080 x : H\ny\u2080 y : H'\n\u03d5 : E \u2192L[\ud835\udd5c] E'\n\u22a2 inCoordinates E (TangentSpace I) E' (TangentSpace I') x\u2080 x y\u2080 y \u03d5 = \u03d5\n[PROOFSTEP]\nerw [VectorBundleCore.inCoordinates_eq]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nN : Type u_9\nx\u2080 x : H\ny\u2080 y : H'\n\u03d5 : E \u2192L[\ud835\udd5c] E'\n\u22a2 comp\n (VectorBundleCore.coordChange (tangentBundleCore I' H') (VectorBundleCore.indexAt (tangentBundleCore I' H') y)\n (VectorBundleCore.indexAt (tangentBundleCore I' H') y\u2080) y)\n (comp \u03d5\n (VectorBundleCore.coordChange (tangentBundleCore I H) (VectorBundleCore.indexAt (tangentBundleCore I H) x\u2080)\n (VectorBundleCore.indexAt (tangentBundleCore I H) x) x)) =\n \u03d5\n[PROOFSTEP]\ntry trivial\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nN : Type u_9\nx\u2080 x : H\ny\u2080 y : H'\n\u03d5 : E \u2192L[\ud835\udd5c] E'\n\u22a2 comp\n (VectorBundleCore.coordChange (tangentBundleCore I' H') (VectorBundleCore.indexAt (tangentBundleCore I' H') y)\n (VectorBundleCore.indexAt (tangentBundleCore I' H') y\u2080) y)\n (comp \u03d5\n (VectorBundleCore.coordChange (tangentBundleCore I H) (VectorBundleCore.indexAt (tangentBundleCore I H) x\u2080)\n (VectorBundleCore.indexAt (tangentBundleCore I H) x) x)) =\n \u03d5\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase hx\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nN : Type u_9\nx\u2080 x : H\ny\u2080 y : H'\n\u03d5 : E \u2192L[\ud835\udd5c] E'\n\u22a2 x \u2208 VectorBundleCore.baseSet (tangentBundleCore I H) (VectorBundleCore.indexAt (tangentBundleCore I H) x\u2080)\n[PROOFSTEP]\ntry trivial\n[GOAL]\ncase hx\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nN : Type u_9\nx\u2080 x : H\ny\u2080 y : H'\n\u03d5 : E \u2192L[\ud835\udd5c] E'\n\u22a2 x \u2208 VectorBundleCore.baseSet (tangentBundleCore I H) (VectorBundleCore.indexAt (tangentBundleCore I H) x\u2080)\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase hy\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nN : Type u_9\nx\u2080 x : H\ny\u2080 y : H'\n\u03d5 : E \u2192L[\ud835\udd5c] E'\n\u22a2 y \u2208 VectorBundleCore.baseSet (tangentBundleCore I' H') (VectorBundleCore.indexAt (tangentBundleCore I' H') y\u2080)\n[PROOFSTEP]\ntry trivial\n[GOAL]\ncase hy\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nN : Type u_9\nx\u2080 x : H\ny\u2080 y : H'\n\u03d5 : E \u2192L[\ud835\udd5c] E'\n\u22a2 y \u2208 VectorBundleCore.baseSet (tangentBundleCore I' H') (VectorBundleCore.indexAt (tangentBundleCore I' H') y\u2080)\n[PROOFSTEP]\ntrivial\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nN : Type u_9\nx\u2080 x : H\ny\u2080 y : H'\n\u03d5 : E \u2192L[\ud835\udd5c] E'\n\u22a2 comp\n (VectorBundleCore.coordChange (tangentBundleCore I' H') (VectorBundleCore.indexAt (tangentBundleCore I' H') y)\n (VectorBundleCore.indexAt (tangentBundleCore I' H') y\u2080) y)\n (comp \u03d5\n (VectorBundleCore.coordChange (tangentBundleCore I H) (VectorBundleCore.indexAt (tangentBundleCore I H) x\u2080)\n (VectorBundleCore.indexAt (tangentBundleCore I H) x) x)) =\n \u03d5\n[PROOFSTEP]\nsimp_rw [tangentBundleCore_indexAt, tangentBundleCore_coordChange_model_space, ContinuousLinearMap.id_comp,\n ContinuousLinearMap.comp_id]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nE' : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH : Type u_4\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nH' : Type u_5\ninst\u271d\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_6\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace H M\ninst\u271d\u2075 : SmoothManifoldWithCorners I M\nM' : Type u_7\ninst\u271d\u2074 : TopologicalSpace M'\ninst\u271d\u00b3 : ChartedSpace H' M'\ninst\u271d\u00b2 : SmoothManifoldWithCorners I' M'\nF : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nN : Type u_9\nf : N \u2192 H\ng : N \u2192 H'\n\u03d5 : N \u2192 E \u2192L[\ud835\udd5c] E'\nx\u2080 : N\n\u22a2 inTangentCoordinates I I' f g \u03d5 x\u2080 = \u03d5\n[PROOFSTEP]\nsimp_rw [inTangentCoordinates, inCoordinates_tangent_bundle_core_model_space]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\nH : Type u_2\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_3\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nx : M\n\u22a2 PathConnectedSpace (TangentSpace I x)\n[PROOFSTEP]\nunfold TangentSpace\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\nH : Type u_2\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_3\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nx : M\n\u22a2 PathConnectedSpace E\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Manifold.VectorBundle.Tangent", "llama_tokens": 47222, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4405082100948128}} {"text": "[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : IsScalarTower F K L\nf : F[X]\ninst\u271d : IsSplittingField F L f\n\u22a2 Splits (algebraMap K L) (Polynomial.map (algebraMap F K) f)\n[PROOFSTEP]\nrw [splits_map_iff, \u2190 IsScalarTower.algebraMap_eq]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : IsScalarTower F K L\nf : F[X]\ninst\u271d : IsSplittingField F L f\n\u22a2 Splits (algebraMap F L) f\n[PROOFSTEP]\nexact splits L f\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : IsScalarTower F K L\nf : F[X]\ninst\u271d : IsSplittingField F L f\n\u22a2 Subalgebra.restrictScalars F (Algebra.adjoin K (rootSet (Polynomial.map (algebraMap F K) f) L)) =\n Subalgebra.restrictScalars F \u22a4\n[PROOFSTEP]\nrw [rootSet, map_map, \u2190 IsScalarTower.algebraMap_eq, Subalgebra.restrictScalars_top, eq_top_iff, \u2190 adjoin_rootSet L f,\n Algebra.adjoin_le_iff]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : IsScalarTower F K L\nf : F[X]\ninst\u271d : IsSplittingField F L f\n\u22a2 rootSet f L \u2286\n \u2191(Subalgebra.restrictScalars F (Algebra.adjoin K \u2191(Multiset.toFinset (roots (Polynomial.map (algebraMap F L) f)))))\n[PROOFSTEP]\nexact fun x hx => @Algebra.subset_adjoin K _ _ _ _ _ _ hx\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : IsScalarTower F K L\nf : K[X]\ninst\u271d : IsSplittingField K L f\nh : Splits (RingHom.id K) f\n\u22a2 \u22a4 = \u22a5\n[PROOFSTEP]\nrw [eq_bot_iff, \u2190 adjoin_rootSet L f, rootSet, roots_map (algebraMap K L) h, Algebra.adjoin_le_iff]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : IsScalarTower F K L\nf : K[X]\ninst\u271d : IsSplittingField K L f\nh : Splits (RingHom.id K) f\n\u22a2 \u2191(Multiset.toFinset (Multiset.map (\u2191(algebraMap K L)) (roots f))) \u2286 \u2191\u22a5\n[PROOFSTEP]\nintro y hy\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : IsScalarTower F K L\nf : K[X]\ninst\u271d : IsSplittingField K L f\nh : Splits (RingHom.id K) f\ny : L\nhy : y \u2208 \u2191(Multiset.toFinset (Multiset.map (\u2191(algebraMap K L)) (roots f)))\n\u22a2 y \u2208 \u2191\u22a5\n[PROOFSTEP]\nrw [Multiset.toFinset_map, Finset.mem_coe, Finset.mem_image] at hy \n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : IsScalarTower F K L\nf : K[X]\ninst\u271d : IsSplittingField K L f\nh : Splits (RingHom.id K) f\ny : L\nhy\u271d : y \u2208 \u2191(Multiset.toFinset (Multiset.map (\u2191(algebraMap K L)) (roots f)))\nhy : \u2203 a, a \u2208 Multiset.toFinset (roots f) \u2227 \u2191(algebraMap K L) a = y\n\u22a2 y \u2208 \u2191\u22a5\n[PROOFSTEP]\nobtain \u27e8x : K, -, hxy : algebraMap K L x = y\u27e9 := hy\n[GOAL]\ncase intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : IsScalarTower F K L\nf : K[X]\ninst\u271d : IsSplittingField K L f\nh : Splits (RingHom.id K) f\ny : L\nhy : y \u2208 \u2191(Multiset.toFinset (Multiset.map (\u2191(algebraMap K L)) (roots f)))\nx : K\nhxy : \u2191(algebraMap K L) x = y\n\u22a2 y \u2208 \u2191\u22a5\n[PROOFSTEP]\nrw [\u2190 hxy]\n[GOAL]\ncase intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : IsScalarTower F K L\nf : K[X]\ninst\u271d : IsSplittingField K L f\nh : Splits (RingHom.id K) f\ny : L\nhy : y \u2208 \u2191(Multiset.toFinset (Multiset.map (\u2191(algebraMap K L)) (roots f)))\nx : K\nhxy : \u2191(algebraMap K L) x = y\n\u22a2 \u2191(algebraMap K L) x \u2208 \u2191\u22a5\n[PROOFSTEP]\nexact SetLike.mem_coe.2 <| Subalgebra.algebraMap_mem _ _\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : IsScalarTower F K L\nf : K[X]\ninst\u271d : IsSplittingField K L f\nh : \u22a4 = \u22a5\n\u22a2 Splits\n (RingEquiv.toRingHom\n (RingEquiv.trans (RingEquiv.ofBijective (algebraMap K L) (_ : Function.Bijective \u2191(algebraMap K L)))\n (RingEquiv.symm (RingEquiv.ofBijective (algebraMap K L) (_ : Function.Bijective \u2191(algebraMap K L))))))\n f\n[PROOFSTEP]\nrw [RingEquiv.toRingHom_trans]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : IsScalarTower F K L\nf : K[X]\ninst\u271d : IsSplittingField K L f\nh : \u22a4 = \u22a5\n\u22a2 Splits\n (RingHom.comp\n (RingEquiv.toRingHom\n (RingEquiv.symm (RingEquiv.ofBijective (algebraMap K L) (_ : Function.Bijective \u2191(algebraMap K L)))))\n (RingEquiv.toRingHom (RingEquiv.ofBijective (algebraMap K L) (_ : Function.Bijective \u2191(algebraMap K L)))))\n f\n[PROOFSTEP]\nexact splits_comp_of_splits _ _ (splits L f)\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra F L\ninst\u271d\u00b2 : IsScalarTower F K L\nf g : F[X]\nhf : f \u2260 0\nhg : g \u2260 0\ninst\u271d\u00b9 : IsSplittingField F K f\ninst\u271d : IsSplittingField K L (Polynomial.map (algebraMap F K) g)\n\u22a2 Algebra.adjoin F (rootSet (f * g) L) = \u22a4\n[PROOFSTEP]\nrw [rootSet, Polynomial.map_mul, roots_mul (mul_ne_zero (map_ne_zero hf : f.map (algebraMap F L) \u2260 0) (map_ne_zero hg)),\n Multiset.toFinset_add, Finset.coe_union, Algebra.adjoin_union_eq_adjoin_adjoin, IsScalarTower.algebraMap_eq F K L, \u2190\n map_map, roots_map (algebraMap K L) ((splits_id_iff_splits <| algebraMap F K).2 <| splits K f), Multiset.toFinset_map,\n Finset.coe_image, Algebra.adjoin_algebraMap, \u2190 rootSet, adjoin_rootSet, Algebra.map_top,\n IsScalarTower.adjoin_range_toAlgHom, \u2190 map_map, \u2190 rootSet, adjoin_rootSet, Subalgebra.restrictScalars_top]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\nf : K[X]\ninst\u271d : IsSplittingField K L f\nhf : Splits (algebraMap K F) f\nhf0 : f = 0\n\u22a2 L \u2192\u2090[K] { x // x \u2208 \u22a5 }\n[PROOFSTEP]\nrw [\u2190 (splits_iff L f).1 (show f.Splits (RingHom.id K) from hf0.symm \u25b8 splits_zero _)]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\nf : K[X]\ninst\u271d : IsSplittingField K L f\nhf : Splits (algebraMap K F) f\nhf0 : f = 0\n\u22a2 L \u2192\u2090[K] { x // x \u2208 \u22a4 }\n[PROOFSTEP]\nexact Algebra.toTop\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\nf : K[X]\ninst\u271d : IsSplittingField K L f\nhf : Splits (algebraMap K F) f\nhf0 : \u00acf = 0\n\u22a2 { x // x \u2208 \u22a4 } \u2192\u2090[K] F\n[PROOFSTEP]\nrw [\u2190 adjoin_rootSet L f]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\nf : K[X]\ninst\u271d : IsSplittingField K L f\nhf : Splits (algebraMap K F) f\nhf0 : \u00acf = 0\n\u22a2 { x // x \u2208 Algebra.adjoin K (rootSet f L) } \u2192\u2090[K] F\n[PROOFSTEP]\nexact\n Classical.choice\n (lift_of_splits _ fun y hy =>\n have : aeval y f = 0 :=\n (eval\u2082_eq_eval_map _).trans <| (mem_roots <| map_ne_zero hf0).1 (Multiset.mem_toFinset.mp hy)\n \u27e8isAlgebraic_iff_isIntegral.1 \u27e8f, hf0, this\u27e9, splits_of_splits_of_dvd _ hf0 hf <| minpoly.dvd _ _ this\u27e9)\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\nf : K[X]\ninst\u271d : IsSplittingField K L f\ny : L\nhy : y \u2208 rootSet f L\nhf : f = 0\n\u22a2 IsIntegral K y\n[PROOFSTEP]\nrw [hf, rootSet_zero] at hy \n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\nf : K[X]\ninst\u271d : IsSplittingField K L f\ny : L\nhy : y \u2208 \u2205\nhf : f = 0\n\u22a2 IsIntegral K y\n[PROOFSTEP]\ncases hy\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\np : K[X]\nf : F \u2243\u2090[K] L\ninst\u271d : IsSplittingField K F p\n\u22a2 IsSplittingField K L p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase splits'\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\np : K[X]\nf : F \u2243\u2090[K] L\ninst\u271d : IsSplittingField K F p\n\u22a2 Splits (algebraMap K L) p\n[PROOFSTEP]\nrw [\u2190 f.toAlgHom.comp_algebraMap]\n[GOAL]\ncase splits'\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\np : K[X]\nf : F \u2243\u2090[K] L\ninst\u271d : IsSplittingField K F p\n\u22a2 Splits (RingHom.comp (\u2191\u2191f) (algebraMap K F)) p\n[PROOFSTEP]\nexact splits_comp_of_splits _ _ (splits F p)\n[GOAL]\ncase adjoin_rootSet'\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\np : K[X]\nf : F \u2243\u2090[K] L\ninst\u271d : IsSplittingField K F p\n\u22a2 Algebra.adjoin K (rootSet p L) = \u22a4\n[PROOFSTEP]\nrw [\u2190 (Algebra.range_top_iff_surjective f.toAlgHom).mpr f.surjective, adjoin_rootSet_eq_range (splits F p),\n adjoin_rootSet F p]\n[GOAL]\nF\u271d : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\np : K[X]\nF : IntermediateField K L\nh : Splits (algebraMap K L) p\nhF : \u2200 (x : L), x \u2208 rootSet p L \u2192 x \u2208 F\n\u22a2 Splits (algebraMap K { x // x \u2208 F }) p\n[PROOFSTEP]\nsimp_rw [rootSet_def, Finset.mem_coe, Multiset.mem_toFinset] at hF \n[GOAL]\nF\u271d : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\np : K[X]\nF : IntermediateField K L\nh : Splits (algebraMap K L) p\nhF : \u2200 (x : L), x \u2208 roots (Polynomial.map (algebraMap K L) p) \u2192 x \u2208 F\n\u22a2 Splits (algebraMap K { x // x \u2208 F }) p\n[PROOFSTEP]\nrw [splits_iff_exists_multiset]\n[GOAL]\nF\u271d : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\np : K[X]\nF : IntermediateField K L\nh : Splits (algebraMap K L) p\nhF : \u2200 (x : L), x \u2208 roots (Polynomial.map (algebraMap K L) p) \u2192 x \u2208 F\n\u22a2 \u2203 s,\n Polynomial.map (algebraMap K { x // x \u2208 F }) p =\n \u2191C (\u2191(algebraMap K { x // x \u2208 F }) (leadingCoeff p)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\n[PROOFSTEP]\nrefine' \u27e8Multiset.pmap Subtype.mk _ hF, map_injective _ (algebraMap F L).injective _\u27e9\n[GOAL]\nF\u271d : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\np : K[X]\nF : IntermediateField K L\nh : Splits (algebraMap K L) p\nhF : \u2200 (x : L), x \u2208 roots (Polynomial.map (algebraMap K L) p) \u2192 x \u2208 F\n\u22a2 Polynomial.map (algebraMap { x // x \u2208 F } L) (Polynomial.map (algebraMap K { x // x \u2208 F }) p) =\n Polynomial.map (algebraMap { x // x \u2208 F } L)\n (\u2191C (\u2191(algebraMap K { x // x \u2208 F }) (leadingCoeff p)) *\n Multiset.prod\n (Multiset.map (fun a => X - \u2191C a) (Multiset.pmap Subtype.mk (roots (Polynomial.map (algebraMap K L) p)) hF)))\n[PROOFSTEP]\nconv_lhs =>\n rw [Polynomial.map_map, \u2190 IsScalarTower.algebraMap_eq, eq_prod_roots_of_splits h, \u2190 Multiset.pmap_eq_map _ _ _ hF]\n[GOAL]\nF\u271d : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\np : K[X]\nF : IntermediateField K L\nh : Splits (algebraMap K L) p\nhF : \u2200 (x : L), x \u2208 roots (Polynomial.map (algebraMap K L) p) \u2192 x \u2208 F\n| Polynomial.map (algebraMap { x // x \u2208 F } L) (Polynomial.map (algebraMap K { x // x \u2208 F }) p)\n[PROOFSTEP]\nrw [Polynomial.map_map, \u2190 IsScalarTower.algebraMap_eq, eq_prod_roots_of_splits h, \u2190 Multiset.pmap_eq_map _ _ _ hF]\n[GOAL]\nF\u271d : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\np : K[X]\nF : IntermediateField K L\nh : Splits (algebraMap K L) p\nhF : \u2200 (x : L), x \u2208 roots (Polynomial.map (algebraMap K L) p) \u2192 x \u2208 F\n| Polynomial.map (algebraMap { x // x \u2208 F } L) (Polynomial.map (algebraMap K { x // x \u2208 F }) p)\n[PROOFSTEP]\nrw [Polynomial.map_map, \u2190 IsScalarTower.algebraMap_eq, eq_prod_roots_of_splits h, \u2190 Multiset.pmap_eq_map _ _ _ hF]\n[GOAL]\nF\u271d : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\np : K[X]\nF : IntermediateField K L\nh : Splits (algebraMap K L) p\nhF : \u2200 (x : L), x \u2208 roots (Polynomial.map (algebraMap K L) p) \u2192 x \u2208 F\n| Polynomial.map (algebraMap { x // x \u2208 F } L) (Polynomial.map (algebraMap K { x // x \u2208 F }) p)\n[PROOFSTEP]\nrw [Polynomial.map_map, \u2190 IsScalarTower.algebraMap_eq, eq_prod_roots_of_splits h, \u2190 Multiset.pmap_eq_map _ _ _ hF]\n[GOAL]\nF\u271d : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\np : K[X]\nF : IntermediateField K L\nh : Splits (algebraMap K L) p\nhF : \u2200 (x : L), x \u2208 roots (Polynomial.map (algebraMap K L) p) \u2192 x \u2208 F\n\u22a2 \u2191C (\u2191(algebraMap K L) (leadingCoeff p)) *\n Multiset.prod (Multiset.pmap (fun a x => X - \u2191C a) (roots (Polynomial.map (algebraMap K L) p)) hF) =\n Polynomial.map (algebraMap { x // x \u2208 F } L)\n (\u2191C (\u2191(algebraMap K { x // x \u2208 F }) (leadingCoeff p)) *\n Multiset.prod\n (Multiset.map (fun a => X - \u2191C a) (Multiset.pmap Subtype.mk (roots (Polynomial.map (algebraMap K L) p)) hF)))\n[PROOFSTEP]\nsimp_rw [Polynomial.map_mul, Polynomial.map_multiset_prod, Multiset.map_pmap, Polynomial.map_sub, map_C, map_X]\n[GOAL]\nF\u271d : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\np : K[X]\nF : IntermediateField K L\nh : Splits (algebraMap K L) p\nhF : \u2200 (x : L), x \u2208 roots (Polynomial.map (algebraMap K L) p) \u2192 x \u2208 F\n\u22a2 \u2191C (\u2191(algebraMap K L) (leadingCoeff p)) *\n Multiset.prod (Multiset.pmap (fun a x => X - \u2191C a) (roots (Polynomial.map (algebraMap K L) p)) hF) =\n \u2191C (\u2191(algebraMap { x // x \u2208 F } L) (\u2191(algebraMap K { x // x \u2208 F }) (leadingCoeff p))) *\n Multiset.prod\n (Multiset.pmap (fun a h => X - \u2191C (\u2191(algebraMap { x // x \u2208 F } L) { val := a, property := h }))\n (roots (Polynomial.map (algebraMap K L) p)) hF)\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.FieldTheory.SplittingField.IsSplittingField", "llama_tokens": 6723, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943603346811, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4405082037353805}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t : Set E\nx : E\n\u22a2 x \u2208 balancedCore \ud835\udd5c s \u2194 \u2203 t, Balanced \ud835\udd5c t \u2227 t \u2286 s \u2227 x \u2208 t\n[PROOFSTEP]\nsimp_rw [balancedCore, mem_sUnion, mem_setOf_eq, and_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns\u271d t : Set E\nx : E\ns : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\n\u22a2 a \u2022 balancedCore \ud835\udd5c s \u2286 balancedCore \ud835\udd5c s\n[PROOFSTEP]\nrintro x \u27e8y, hy, rfl\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns\u271d t : Set E\nx : E\ns : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nhy : y \u2208 balancedCore \ud835\udd5c s\n\u22a2 (fun x => a \u2022 x) y \u2208 balancedCore \ud835\udd5c s\n[PROOFSTEP]\nrw [mem_balancedCore_iff] at hy \n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns\u271d t : Set E\nx : E\ns : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nhy : \u2203 t, Balanced \ud835\udd5c t \u2227 t \u2286 s \u2227 y \u2208 t\n\u22a2 (fun x => a \u2022 x) y \u2208 balancedCore \ud835\udd5c s\n[PROOFSTEP]\nrcases hy with \u27e8t, ht1, ht2, hy\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns\u271d t\u271d : Set E\nx : E\ns : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nt : Set E\nht1 : Balanced \ud835\udd5c t\nht2 : t \u2286 s\nhy : y \u2208 t\n\u22a2 (fun x => a \u2022 x) y \u2208 balancedCore \ud835\udd5c s\n[PROOFSTEP]\nexact \u27e8t, \u27e8ht1, ht2\u27e9, ht1 a ha (smul_mem_smul_set hy)\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t : Set E\nx : E\nht : Balanced \ud835\udd5c t\nh : s \u2286 t\n\u22a2 balancedHull \ud835\udd5c s \u2286 t\n[PROOFSTEP]\nintros x hx\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t : Set E\nx\u271d : E\nht : Balanced \ud835\udd5c t\nh : s \u2286 t\nx : E\nhx : x \u2208 balancedHull \ud835\udd5c s\n\u22a2 x \u2208 t\n[PROOFSTEP]\nobtain \u27e8r, hr, y, hy, rfl\u27e9 := mem_balancedHull_iff.1 hx\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t : Set E\nx : E\nht : Balanced \ud835\udd5c t\nh : s \u2286 t\nr : \ud835\udd5c\nhr : \u2016r\u2016 \u2264 1\ny : E\nhy : y \u2208 s\nhx : (fun x => r \u2022 x) y \u2208 balancedHull \ud835\udd5c s\n\u22a2 (fun x => r \u2022 x) y \u2208 t\n[PROOFSTEP]\nexact ht.smul_mem hr (h hy)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d s : Set E\n\u22a2 Balanced \ud835\udd5c (balancedHull \ud835\udd5c s)\n[PROOFSTEP]\nintro a ha\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d s : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\n\u22a2 a \u2022 balancedHull \ud835\udd5c s \u2286 balancedHull \ud835\udd5c s\n[PROOFSTEP]\nsimp_rw [balancedHull, smul_set_iUnion\u2082, subset_def, mem_iUnion\u2082]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d s : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\n\u22a2 \u2200 (x : E), (\u2203 i j, x \u2208 a \u2022 i \u2022 s) \u2192 \u2203 i j, x \u2208 i \u2022 s\n[PROOFSTEP]\nrintro x \u27e8r, hr, hx\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d s : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\nx : E\nr : \ud835\udd5c\nhr : \u2016r\u2016 \u2264 1\nhx : x \u2208 a \u2022 r \u2022 s\n\u22a2 \u2203 i j, x \u2208 i \u2022 s\n[PROOFSTEP]\nrw [\u2190 smul_assoc] at hx \n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d s : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\nx : E\nr : \ud835\udd5c\nhr : \u2016r\u2016 \u2264 1\nhx\u271d : x \u2208 a \u2022 r \u2022 s\nhx : x \u2208 (a \u2022 r) \u2022 s\n\u22a2 \u2203 i j, x \u2208 i \u2022 s\n[PROOFSTEP]\nexact \u27e8a \u2022 r, (SeminormedRing.norm_mul _ _).trans (mul_le_one ha (norm_nonneg r) hr), hx\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\n\u22a2 balancedCoreAux \ud835\udd5c \u2205 = \u2205\n[PROOFSTEP]\nsimp_rw [balancedCoreAux, iInter\u2082_eq_empty_iff, smul_set_empty]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\n\u22a2 \u2200 (a : E), \u2203 i h, \u00aca \u2208 \u2205\n[PROOFSTEP]\nexact fun _ => \u27e81, norm_one.ge, not_mem_empty _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t s : Set E\nx : E\nhx : x \u2208 balancedCoreAux \ud835\udd5c s\n\u22a2 x \u2208 s\n[PROOFSTEP]\nsimpa only [one_smul] using mem_balancedCoreAux_iff.1 hx 1 norm_one.ge\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nh0 : 0 \u2208 balancedCoreAux \ud835\udd5c s\n\u22a2 Balanced \ud835\udd5c (balancedCoreAux \ud835\udd5c s)\n[PROOFSTEP]\nrintro a ha x \u27e8y, hy, rfl\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nh0 : 0 \u2208 balancedCoreAux \ud835\udd5c s\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nhy : y \u2208 balancedCoreAux \ud835\udd5c s\n\u22a2 (fun x => a \u2022 x) y \u2208 balancedCoreAux \ud835\udd5c s\n[PROOFSTEP]\nobtain rfl | h := eq_or_ne a 0\n[GOAL]\ncase intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nh0 : 0 \u2208 balancedCoreAux \ud835\udd5c s\ny : E\nhy : y \u2208 balancedCoreAux \ud835\udd5c s\nha : \u20160\u2016 \u2264 1\n\u22a2 (fun x => 0 \u2022 x) y \u2208 balancedCoreAux \ud835\udd5c s\n[PROOFSTEP]\nsimp_rw [zero_smul, h0]\n[GOAL]\ncase intro.intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nh0 : 0 \u2208 balancedCoreAux \ud835\udd5c s\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nhy : y \u2208 balancedCoreAux \ud835\udd5c s\nh : a \u2260 0\n\u22a2 (fun x => a \u2022 x) y \u2208 balancedCoreAux \ud835\udd5c s\n[PROOFSTEP]\nrw [mem_balancedCoreAux_iff] at hy \u22a2\n[GOAL]\ncase intro.intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nh0 : 0 \u2208 balancedCoreAux \ud835\udd5c s\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nhy : \u2200 (r : \ud835\udd5c), 1 \u2264 \u2016r\u2016 \u2192 y \u2208 r \u2022 s\nh : a \u2260 0\n\u22a2 \u2200 (r : \ud835\udd5c), 1 \u2264 \u2016r\u2016 \u2192 (fun x => a \u2022 x) y \u2208 r \u2022 s\n[PROOFSTEP]\nintro r hr\n[GOAL]\ncase intro.intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nh0 : 0 \u2208 balancedCoreAux \ud835\udd5c s\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nhy : \u2200 (r : \ud835\udd5c), 1 \u2264 \u2016r\u2016 \u2192 y \u2208 r \u2022 s\nh : a \u2260 0\nr : \ud835\udd5c\nhr : 1 \u2264 \u2016r\u2016\n\u22a2 (fun x => a \u2022 x) y \u2208 r \u2022 s\n[PROOFSTEP]\nhave h'' : 1 \u2264 \u2016a\u207b\u00b9 \u2022 r\u2016 := by\n rw [norm_smul, norm_inv]\n exact one_le_mul_of_one_le_of_one_le (one_le_inv (norm_pos_iff.mpr h) ha) hr\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nh0 : 0 \u2208 balancedCoreAux \ud835\udd5c s\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nhy : \u2200 (r : \ud835\udd5c), 1 \u2264 \u2016r\u2016 \u2192 y \u2208 r \u2022 s\nh : a \u2260 0\nr : \ud835\udd5c\nhr : 1 \u2264 \u2016r\u2016\n\u22a2 1 \u2264 \u2016a\u207b\u00b9 \u2022 r\u2016\n[PROOFSTEP]\nrw [norm_smul, norm_inv]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nh0 : 0 \u2208 balancedCoreAux \ud835\udd5c s\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nhy : \u2200 (r : \ud835\udd5c), 1 \u2264 \u2016r\u2016 \u2192 y \u2208 r \u2022 s\nh : a \u2260 0\nr : \ud835\udd5c\nhr : 1 \u2264 \u2016r\u2016\n\u22a2 1 \u2264 \u2016a\u2016\u207b\u00b9 * \u2016r\u2016\n[PROOFSTEP]\nexact one_le_mul_of_one_le_of_one_le (one_le_inv (norm_pos_iff.mpr h) ha) hr\n[GOAL]\ncase intro.intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nh0 : 0 \u2208 balancedCoreAux \ud835\udd5c s\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nhy : \u2200 (r : \ud835\udd5c), 1 \u2264 \u2016r\u2016 \u2192 y \u2208 r \u2022 s\nh : a \u2260 0\nr : \ud835\udd5c\nhr : 1 \u2264 \u2016r\u2016\nh'' : 1 \u2264 \u2016a\u207b\u00b9 \u2022 r\u2016\n\u22a2 (fun x => a \u2022 x) y \u2208 r \u2022 s\n[PROOFSTEP]\nhave h' := hy (a\u207b\u00b9 \u2022 r) h''\n[GOAL]\ncase intro.intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nh0 : 0 \u2208 balancedCoreAux \ud835\udd5c s\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\ny : E\nhy : \u2200 (r : \ud835\udd5c), 1 \u2264 \u2016r\u2016 \u2192 y \u2208 r \u2022 s\nh : a \u2260 0\nr : \ud835\udd5c\nhr : 1 \u2264 \u2016r\u2016\nh'' : 1 \u2264 \u2016a\u207b\u00b9 \u2022 r\u2016\nh' : y \u2208 (a\u207b\u00b9 \u2022 r) \u2022 s\n\u22a2 (fun x => a \u2022 x) y \u2208 r \u2022 s\n[PROOFSTEP]\nrwa [smul_assoc, mem_inv_smul_set_iff\u2080 h] at h' \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nh : t \u2286 s\nht : Balanced \ud835\udd5c t\n\u22a2 t \u2286 balancedCoreAux \ud835\udd5c s\n[PROOFSTEP]\nrefine' fun x hx => mem_balancedCoreAux_iff.2 fun r hr => _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nh : t \u2286 s\nht : Balanced \ud835\udd5c t\nx : E\nhx : x \u2208 t\nr : \ud835\udd5c\nhr : 1 \u2264 \u2016r\u2016\n\u22a2 x \u2208 r \u2022 s\n[PROOFSTEP]\nrw [mem_smul_set_iff_inv_smul_mem\u2080 (norm_pos_iff.mp <| zero_lt_one.trans_le hr)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nh : t \u2286 s\nht : Balanced \ud835\udd5c t\nx : E\nhx : x \u2208 t\nr : \ud835\udd5c\nhr : 1 \u2264 \u2016r\u2016\n\u22a2 r\u207b\u00b9 \u2022 x \u2208 s\n[PROOFSTEP]\nrefine' h (ht.smul_mem _ hx)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nh : t \u2286 s\nht : Balanced \ud835\udd5c t\nx : E\nhx : x \u2208 t\nr : \ud835\udd5c\nhr : 1 \u2264 \u2016r\u2016\n\u22a2 \u2016r\u207b\u00b9\u2016 \u2264 1\n[PROOFSTEP]\nrw [norm_inv]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nh : t \u2286 s\nht : Balanced \ud835\udd5c t\nx : E\nhx : x \u2208 t\nr : \ud835\udd5c\nhr : 1 \u2264 \u2016r\u2016\n\u22a2 \u2016r\u2016\u207b\u00b9 \u2264 1\n[PROOFSTEP]\nexact inv_le_one hr\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nhs : 0 \u2208 s\n\u22a2 balancedCore \ud835\udd5c s = \u22c2 (r : \ud835\udd5c) (_ : 1 \u2264 \u2016r\u2016), r \u2022 s\n[PROOFSTEP]\nrefine' balancedCore_subset_balancedCoreAux.antisymm _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nhs : 0 \u2208 s\n\u22a2 balancedCoreAux \ud835\udd5c s \u2286 balancedCore \ud835\udd5c s\n[PROOFSTEP]\nrefine' (balancedCoreAux_balanced _).subset_balancedCore_of_subset (balancedCoreAux_subset s)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nhs : 0 \u2208 s\n\u22a2 0 \u2208 balancedCoreAux \ud835\udd5c s\n[PROOFSTEP]\nexact balancedCore_subset_balancedCoreAux (balancedCore_zero_mem hs)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nht : 0 \u2208 t\nhst : \u2200 (a : \ud835\udd5c), \u2016a\u2016 \u2264 1 \u2192 a \u2022 s \u2286 t\n\u22a2 s \u2286 balancedCore \ud835\udd5c t\n[PROOFSTEP]\nrw [balancedCore_eq_iInter ht]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nht : 0 \u2208 t\nhst : \u2200 (a : \ud835\udd5c), \u2016a\u2016 \u2264 1 \u2192 a \u2022 s \u2286 t\n\u22a2 s \u2286 \u22c2 (r : \ud835\udd5c) (_ : 1 \u2264 \u2016r\u2016), r \u2022 t\n[PROOFSTEP]\nrefine' subset_iInter\u2082 fun a ha => _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nht : 0 \u2208 t\nhst : \u2200 (a : \ud835\udd5c), \u2016a\u2016 \u2264 1 \u2192 a \u2022 s \u2286 t\na : \ud835\udd5c\nha : 1 \u2264 \u2016a\u2016\n\u22a2 s \u2286 a \u2022 t\n[PROOFSTEP]\nrw [\u2190 smul_inv_smul\u2080 (norm_pos_iff.mp <| zero_lt_one.trans_le ha) s]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nht : 0 \u2208 t\nhst : \u2200 (a : \ud835\udd5c), \u2016a\u2016 \u2264 1 \u2192 a \u2022 s \u2286 t\na : \ud835\udd5c\nha : 1 \u2264 \u2016a\u2016\n\u22a2 a \u2022 a\u207b\u00b9 \u2022 s \u2286 a \u2022 t\n[PROOFSTEP]\nrefine' smul_set_mono (hst _ _)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nht : 0 \u2208 t\nhst : \u2200 (a : \ud835\udd5c), \u2016a\u2016 \u2264 1 \u2192 a \u2022 s \u2286 t\na : \ud835\udd5c\nha : 1 \u2264 \u2016a\u2016\n\u22a2 \u2016a\u207b\u00b9\u2016 \u2264 1\n[PROOFSTEP]\nrw [norm_inv]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t : Set E\nht : 0 \u2208 t\nhst : \u2200 (a : \ud835\udd5c), \u2016a\u2016 \u2264 1 \u2192 a \u2022 s \u2286 t\na : \ud835\udd5c\nha : 1 \u2264 \u2016a\u2016\n\u22a2 \u2016a\u2016\u207b\u00b9 \u2264 1\n[PROOFSTEP]\nexact inv_le_one ha\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : IsClosed U\n\u22a2 IsClosed (balancedCore \ud835\udd5c U)\n[PROOFSTEP]\nby_cases h : (0 : E) \u2208 U\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : IsClosed U\nh : 0 \u2208 U\n\u22a2 IsClosed (balancedCore \ud835\udd5c U)\n[PROOFSTEP]\nrw [balancedCore_eq_iInter h]\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : IsClosed U\nh : 0 \u2208 U\n\u22a2 IsClosed (\u22c2 (r : \ud835\udd5c) (_ : 1 \u2264 \u2016r\u2016), r \u2022 U)\n[PROOFSTEP]\nrefine' isClosed_iInter fun a => _\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : IsClosed U\nh : 0 \u2208 U\na : \ud835\udd5c\n\u22a2 IsClosed (\u22c2 (_ : 1 \u2264 \u2016a\u2016), a \u2022 U)\n[PROOFSTEP]\nrefine' isClosed_iInter fun ha => _\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : IsClosed U\nh : 0 \u2208 U\na : \ud835\udd5c\nha : 1 \u2264 \u2016a\u2016\n\u22a2 IsClosed (a \u2022 U)\n[PROOFSTEP]\nhave ha' := lt_of_lt_of_le zero_lt_one ha\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : IsClosed U\nh : 0 \u2208 U\na : \ud835\udd5c\nha : 1 \u2264 \u2016a\u2016\nha' : 0 < \u2016a\u2016\n\u22a2 IsClosed (a \u2022 U)\n[PROOFSTEP]\nrw [norm_pos_iff] at ha' \n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : IsClosed U\nh : 0 \u2208 U\na : \ud835\udd5c\nha : 1 \u2264 \u2016a\u2016\nha' : a \u2260 0\n\u22a2 IsClosed (a \u2022 U)\n[PROOFSTEP]\nexact isClosedMap_smul_of_ne_zero ha' U hU\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : IsClosed U\nh : \u00ac0 \u2208 U\n\u22a2 IsClosed (balancedCore \ud835\udd5c U)\n[PROOFSTEP]\nhave : balancedCore \ud835\udd5c U = \u2205 := by\n contrapose! h\n exact balancedCore_nonempty_iff.mp (Set.nonempty_iff_ne_empty.2 h)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : IsClosed U\nh : \u00ac0 \u2208 U\n\u22a2 balancedCore \ud835\udd5c U = \u2205\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : IsClosed U\nh : balancedCore \ud835\udd5c U \u2260 \u2205\n\u22a2 0 \u2208 U\n[PROOFSTEP]\nexact balancedCore_nonempty_iff.mp (Set.nonempty_iff_ne_empty.2 h)\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : IsClosed U\nh : \u00ac0 \u2208 U\nthis : balancedCore \ud835\udd5c U = \u2205\n\u22a2 IsClosed (balancedCore \ud835\udd5c U)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : IsClosed U\nh : \u00ac0 \u2208 U\nthis : balancedCore \ud835\udd5c U = \u2205\n\u22a2 IsClosed \u2205\n[PROOFSTEP]\nexact isClosed_empty\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\n\u22a2 balancedCore \ud835\udd5c U \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nobtain \u27e8r, V, hr, hV, hrVU\u27e9 :\n \u2203 (r : \u211d) (V : Set E), 0 < r \u2227 V \u2208 \ud835\udcdd (0 : E) \u2227 \u2200 (c : \ud835\udd5c) (y : E), \u2016c\u2016 < r \u2192 y \u2208 V \u2192 c \u2022 y \u2208 U :=\n by\n have h : Filter.Tendsto (fun x : \ud835\udd5c \u00d7 E => x.fst \u2022 x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0) :=\n continuous_smul.tendsto' (0, 0) _ (smul_zero _)\n simpa only [\u2190 Prod.exists', \u2190 Prod.forall', \u2190 and_imp, \u2190 and_assoc, exists_prop] using\n h.basis_left (NormedAddCommGroup.nhds_zero_basis_norm_lt.prod_nhds (\ud835\udcdd _).basis_sets) U hU\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\n\u22a2 \u2203 r V, 0 < r \u2227 V \u2208 \ud835\udcdd 0 \u2227 \u2200 (c : \ud835\udd5c) (y : E), \u2016c\u2016 < r \u2192 y \u2208 V \u2192 c \u2022 y \u2208 U\n[PROOFSTEP]\nhave h : Filter.Tendsto (fun x : \ud835\udd5c \u00d7 E => x.fst \u2022 x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0) :=\n continuous_smul.tendsto' (0, 0) _ (smul_zero _)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst \u2022 x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\n\u22a2 \u2203 r V, 0 < r \u2227 V \u2208 \ud835\udcdd 0 \u2227 \u2200 (c : \ud835\udd5c) (y : E), \u2016c\u2016 < r \u2192 y \u2208 V \u2192 c \u2022 y \u2208 U\n[PROOFSTEP]\nsimpa only [\u2190 Prod.exists', \u2190 Prod.forall', \u2190 and_imp, \u2190 and_assoc, exists_prop] using\n h.basis_left (NormedAddCommGroup.nhds_zero_basis_norm_lt.prod_nhds (\ud835\udcdd _).basis_sets) U hU\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nr : \u211d\nV : Set E\nhr : 0 < r\nhV : V \u2208 \ud835\udcdd 0\nhrVU : \u2200 (c : \ud835\udd5c) (y : E), \u2016c\u2016 < r \u2192 y \u2208 V \u2192 c \u2022 y \u2208 U\n\u22a2 balancedCore \ud835\udd5c U \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nrcases NormedField.exists_norm_lt \ud835\udd5c hr with \u27e8y, hy\u2080, hyr\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nr : \u211d\nV : Set E\nhr : 0 < r\nhV : V \u2208 \ud835\udcdd 0\nhrVU : \u2200 (c : \ud835\udd5c) (y : E), \u2016c\u2016 < r \u2192 y \u2208 V \u2192 c \u2022 y \u2208 U\ny : \ud835\udd5c\nhy\u2080 : 0 < \u2016y\u2016\nhyr : \u2016y\u2016 < r\n\u22a2 balancedCore \ud835\udd5c U \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nrw [norm_pos_iff] at hy\u2080 \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nr : \u211d\nV : Set E\nhr : 0 < r\nhV : V \u2208 \ud835\udcdd 0\nhrVU : \u2200 (c : \ud835\udd5c) (y : E), \u2016c\u2016 < r \u2192 y \u2208 V \u2192 c \u2022 y \u2208 U\ny : \ud835\udd5c\nhy\u2080 : y \u2260 0\nhyr : \u2016y\u2016 < r\n\u22a2 balancedCore \ud835\udd5c U \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nhave : y \u2022 V \u2208 \ud835\udcdd (0 : E) := (set_smul_mem_nhds_zero_iff hy\u2080).mpr hV\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nr : \u211d\nV : Set E\nhr : 0 < r\nhV : V \u2208 \ud835\udcdd 0\nhrVU : \u2200 (c : \ud835\udd5c) (y : E), \u2016c\u2016 < r \u2192 y \u2208 V \u2192 c \u2022 y \u2208 U\ny : \ud835\udd5c\nhy\u2080 : y \u2260 0\nhyr : \u2016y\u2016 < r\nthis : y \u2022 V \u2208 \ud835\udcdd 0\n\u22a2 balancedCore \ud835\udd5c U \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nrefine' Filter.mem_of_superset this (subset_balancedCore (mem_of_mem_nhds hU) fun a ha => _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nr : \u211d\nV : Set E\nhr : 0 < r\nhV : V \u2208 \ud835\udcdd 0\nhrVU : \u2200 (c : \ud835\udd5c) (y : E), \u2016c\u2016 < r \u2192 y \u2208 V \u2192 c \u2022 y \u2208 U\ny : \ud835\udd5c\nhy\u2080 : y \u2260 0\nhyr : \u2016y\u2016 < r\nthis : y \u2022 V \u2208 \ud835\udcdd 0\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\n\u22a2 a \u2022 y \u2022 V \u2286 U\n[PROOFSTEP]\nrw [smul_smul]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nr : \u211d\nV : Set E\nhr : 0 < r\nhV : V \u2208 \ud835\udcdd 0\nhrVU : \u2200 (c : \ud835\udd5c) (y : E), \u2016c\u2016 < r \u2192 y \u2208 V \u2192 c \u2022 y \u2208 U\ny : \ud835\udd5c\nhy\u2080 : y \u2260 0\nhyr : \u2016y\u2016 < r\nthis : y \u2022 V \u2208 \ud835\udcdd 0\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\n\u22a2 (a * y) \u2022 V \u2286 U\n[PROOFSTEP]\nrintro _ \u27e8z, hz, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nr : \u211d\nV : Set E\nhr : 0 < r\nhV : V \u2208 \ud835\udcdd 0\nhrVU : \u2200 (c : \ud835\udd5c) (y : E), \u2016c\u2016 < r \u2192 y \u2208 V \u2192 c \u2022 y \u2208 U\ny : \ud835\udd5c\nhy\u2080 : y \u2260 0\nhyr : \u2016y\u2016 < r\nthis : y \u2022 V \u2208 \ud835\udcdd 0\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\nz : E\nhz : z \u2208 V\n\u22a2 (fun x => (a * y) \u2022 x) z \u2208 U\n[PROOFSTEP]\nrefine' hrVU _ _ _ hz\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nr : \u211d\nV : Set E\nhr : 0 < r\nhV : V \u2208 \ud835\udcdd 0\nhrVU : \u2200 (c : \ud835\udd5c) (y : E), \u2016c\u2016 < r \u2192 y \u2208 V \u2192 c \u2022 y \u2208 U\ny : \ud835\udd5c\nhy\u2080 : y \u2260 0\nhyr : \u2016y\u2016 < r\nthis : y \u2022 V \u2208 \ud835\udcdd 0\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\nz : E\nhz : z \u2208 V\n\u22a2 \u2016a * y\u2016 < r\n[PROOFSTEP]\nrw [norm_mul, \u2190 one_mul r]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5c E\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nr : \u211d\nV : Set E\nhr : 0 < r\nhV : V \u2208 \ud835\udcdd 0\nhrVU : \u2200 (c : \ud835\udd5c) (y : E), \u2016c\u2016 < r \u2192 y \u2208 V \u2192 c \u2022 y \u2208 U\ny : \ud835\udd5c\nhy\u2080 : y \u2260 0\nhyr : \u2016y\u2016 < r\nthis : y \u2022 V \u2208 \ud835\udcdd 0\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\nz : E\nhz : z \u2208 V\n\u22a2 \u2016a\u2016 * \u2016y\u2016 < 1 * r\n[PROOFSTEP]\nexact mul_lt_mul' ha hyr (norm_nonneg y) one_pos\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c E\nU : Set E\ninst\u271d : RegularSpace E\n\u22a2 HasBasis (\ud835\udcdd 0) (fun s => s \u2208 \ud835\udcdd 0 \u2227 IsClosed s \u2227 Balanced \ud835\udd5c s) id\n[PROOFSTEP]\nrefine' (closed_nhds_basis 0).to_hasBasis (fun s hs => _) fun s hs => \u27e8s, \u27e8hs.1, hs.2.1\u27e9, rfl.subset\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c E\nU : Set E\ninst\u271d : RegularSpace E\ns : Set E\nhs : s \u2208 \ud835\udcdd 0 \u2227 IsClosed s\n\u22a2 \u2203 i', (i' \u2208 \ud835\udcdd 0 \u2227 IsClosed i' \u2227 Balanced \ud835\udd5c i') \u2227 id i' \u2286 id s\n[PROOFSTEP]\nrefine' \u27e8balancedCore \ud835\udd5c s, \u27e8balancedCore_mem_nhds_zero hs.1, _\u27e9, balancedCore_subset s\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c E\nU : Set E\ninst\u271d : RegularSpace E\ns : Set E\nhs : s \u2208 \ud835\udcdd 0 \u2227 IsClosed s\n\u22a2 IsClosed (balancedCore \ud835\udd5c s) \u2227 Balanced \ud835\udd5c (balancedCore \ud835\udd5c s)\n[PROOFSTEP]\nexact \u27e8hs.2.balancedCore, balancedCore_balanced s\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "llama_tokens": 11952, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702761768248, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.4404769697392529}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\nx : FreeAlgebra R X\n\u22a2 star (star x) = x\n[PROOFSTEP]\nunfold Star.star\n[GOAL]\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\nx : FreeAlgebra R X\n\u22a2 { star := MulOpposite.unop \u2218 \u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) }.1\n ({ star := MulOpposite.unop \u2218 \u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) }.1 x) =\n x\n[PROOFSTEP]\nsimp only [Function.comp_apply]\n[GOAL]\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\nx : FreeAlgebra R X\n\u22a2 MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) x))) = x\n[PROOFSTEP]\nrefine' FreeAlgebra.induction R X _ _ _ _ x\n[GOAL]\ncase refine'_1\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\nx : FreeAlgebra R X\n\u22a2 \u2200 (r : R),\n MulOpposite.unop\n (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R))\n (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (\u2191(algebraMap R (FreeAlgebra R X)) r)))) =\n \u2191(algebraMap R (FreeAlgebra R X)) r\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\nx : FreeAlgebra R X\nr\u271d : R\n\u22a2 MulOpposite.unop\n (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R))\n (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (\u2191(algebraMap R (FreeAlgebra R X)) r\u271d)))) =\n \u2191(algebraMap R (FreeAlgebra R X)) r\u271d\n[PROOFSTEP]\nsimp only [AlgHom.commutes, MulOpposite.algebraMap_apply, MulOpposite.unop_op]\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\nx : FreeAlgebra R X\n\u22a2 \u2200 (x : X),\n MulOpposite.unop\n (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (\u03b9 R x)))) =\n \u03b9 R x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\nx : FreeAlgebra R X\nx\u271d : X\n\u22a2 MulOpposite.unop\n (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (\u03b9 R x\u271d)))) =\n \u03b9 R x\u271d\n[PROOFSTEP]\nsimp only [lift_\u03b9_apply, Function.comp_apply, MulOpposite.unop_op]\n[GOAL]\ncase refine'_3\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\nx : FreeAlgebra R X\n\u22a2 \u2200 (a b : FreeAlgebra R X),\n MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) a))) =\n a \u2192\n MulOpposite.unop\n (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) b))) =\n b \u2192\n MulOpposite.unop\n (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (a * b)))) =\n a * b\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\nx a\u271d\u00b2 b\u271d : FreeAlgebra R X\na\u271d\u00b9 :\n MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) a\u271d\u00b2))) =\n a\u271d\u00b2\na\u271d :\n MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) b\u271d))) =\n b\u271d\n\u22a2 MulOpposite.unop\n (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (a\u271d\u00b2 * b\u271d)))) =\n a\u271d\u00b2 * b\u271d\n[PROOFSTEP]\nsimp only [*, map_mul, MulOpposite.unop_mul]\n[GOAL]\ncase refine'_4\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\nx : FreeAlgebra R X\n\u22a2 \u2200 (a b : FreeAlgebra R X),\n MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) a))) =\n a \u2192\n MulOpposite.unop\n (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) b))) =\n b \u2192\n MulOpposite.unop\n (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (a + b)))) =\n a + b\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_4\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\nx a\u271d\u00b2 b\u271d : FreeAlgebra R X\na\u271d\u00b9 :\n MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) a\u271d\u00b2))) =\n a\u271d\u00b2\na\u271d :\n MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) b\u271d))) =\n b\u271d\n\u22a2 MulOpposite.unop\n (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (MulOpposite.unop (\u2191(\u2191(lift R) (MulOpposite.op \u2218 \u03b9 R)) (a\u271d\u00b2 + b\u271d)))) =\n a\u271d\u00b2 + b\u271d\n[PROOFSTEP]\nsimp only [*, map_add, MulOpposite.unop_add]\n[GOAL]\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\na b : FreeAlgebra R X\n\u22a2 star (a * b) = star b * star a\n[PROOFSTEP]\nsimp only [Function.comp_apply, map_mul, MulOpposite.unop_mul]\n[GOAL]\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\na b : FreeAlgebra R X\n\u22a2 star (a + b) = star a + star b\n[PROOFSTEP]\nsimp only [Function.comp_apply, map_add, MulOpposite.unop_add]\n[GOAL]\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\nx : X\n\u22a2 star (\u03b9 R x) = \u03b9 R x\n[PROOFSTEP]\nsimp [star, Star.star]\n[GOAL]\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\nr : R\n\u22a2 star (\u2191(algebraMap R (FreeAlgebra R X)) r) = \u2191(algebraMap R (FreeAlgebra R X)) r\n[PROOFSTEP]\nsimp [star, Star.star]\n[GOAL]\nR : Type u_1\ninst\u271d : CommSemiring R\nX : Type u_2\nsrc\u271d : FreeAlgebra R X \u2243+* (FreeAlgebra R X)\u1d50\u1d52\u1d56 := starRingEquiv\nr : R\n\u22a2 Equiv.toFun src\u271d.toEquiv (\u2191(algebraMap R (FreeAlgebra R X)) r) = \u2191(algebraMap R (FreeAlgebra R X)\u1d50\u1d52\u1d56) r\n[PROOFSTEP]\nsimp [star_algebraMap]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Star.Free", "llama_tokens": 2764, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124811, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.44045085685899604}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 Icc a b = 0 \u2194 \u00aca \u2264 b\n[PROOFSTEP]\nrw [Icc, Finset.val_eq_zero, Finset.Icc_eq_empty_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 Ico a b = 0 \u2194 \u00aca < b\n[PROOFSTEP]\nrw [Ico, Finset.val_eq_zero, Finset.Ico_eq_empty_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 Ioc a b = 0 \u2194 \u00aca < b\n[PROOFSTEP]\nrw [Ioc, Finset.val_eq_zero, Finset.Ioc_eq_empty_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DenselyOrdered \u03b1\n\u22a2 Ioo a b = 0 \u2194 \u00aca < b\n[PROOFSTEP]\nrw [Ioo, Finset.val_eq_zero, Finset.Ioo_eq_empty_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 Ico a a = 0\n[PROOFSTEP]\nrw [Ico, Finset.Ico_self, Finset.empty_val]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 Ioc a a = 0\n[PROOFSTEP]\nrw [Ioc, Finset.Ioc_self, Finset.empty_val]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 Ioo a a = 0\n[PROOFSTEP]\nrw [Ioo, Finset.Ioo_self, Finset.empty_val]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidablePred fun x => x < c\nhca : c \u2264 a\n\u22a2 filter (fun x => x < c) (Ico a b) = \u2205\n[PROOFSTEP]\nrw [Ico, \u2190 Finset.filter_val, Finset.Ico_filter_lt_of_le_left hca]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidablePred fun x => x < c\nhca : c \u2264 a\n\u22a2 \u2205.val = \u2205\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidablePred fun x => x < c\nhbc : b \u2264 c\n\u22a2 filter (fun x => x < c) (Ico a b) = Ico a b\n[PROOFSTEP]\nrw [Ico, \u2190 Finset.filter_val, Finset.Ico_filter_lt_of_right_le hbc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidablePred fun x => x < c\nhcb : c \u2264 b\n\u22a2 filter (fun x => x < c) (Ico a b) = Ico a c\n[PROOFSTEP]\nrw [Ico, \u2190 Finset.filter_val, Finset.Ico_filter_lt_of_le_right hcb]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidablePred fun x => x < c\nhcb : c \u2264 b\n\u22a2 (Finset.Ico a c).val = Ico a c\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidablePred ((fun x x_1 => x \u2264 x_1) c)\nhca : c \u2264 a\n\u22a2 filter (fun x => c \u2264 x) (Ico a b) = Ico a b\n[PROOFSTEP]\nrw [Ico, \u2190 Finset.filter_val, Finset.Ico_filter_le_of_le_left hca]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidablePred ((fun x x_1 => x \u2264 x_1) b)\n\u22a2 filter (fun x => b \u2264 x) (Ico a b) = \u2205\n[PROOFSTEP]\nrw [Ico, \u2190 Finset.filter_val, Finset.Ico_filter_le_of_right_le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidablePred ((fun x x_1 => x \u2264 x_1) b)\n\u22a2 \u2205.val = \u2205\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidablePred ((fun x x_1 => x \u2264 x_1) c)\nhac : a \u2264 c\n\u22a2 filter (fun x => c \u2264 x) (Ico a b) = Ico c b\n[PROOFSTEP]\nrw [Ico, \u2190 Finset.filter_val, Finset.Ico_filter_le_of_left_le hac]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidablePred ((fun x x_1 => x \u2264 x_1) c)\nhac : a \u2264 c\n\u22a2 (Finset.Ico c b).val = Ico c b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b a : \u03b1\n\u22a2 Icc a a = {a}\n[PROOFSTEP]\nrw [Icc, Finset.Icc_self, Finset.singleton_val]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 b ::\u2098 Ico a b = Icc a b\n[PROOFSTEP]\nclassical\nrw [Ico, \u2190 Finset.insert_val_of_not_mem right_not_mem_Ico, Finset.Ico_insert_right h]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 b ::\u2098 Ico a b = Icc a b\n[PROOFSTEP]\nrw [Ico, \u2190 Finset.insert_val_of_not_mem right_not_mem_Ico, Finset.Ico_insert_right h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 (Finset.Icc a b).val = Icc a b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\nh : a < b\n\u22a2 a ::\u2098 Ioo a b = Ico a b\n[PROOFSTEP]\nclassical\nrw [Ioo, \u2190 Finset.insert_val_of_not_mem left_not_mem_Ioo, Finset.Ioo_insert_left h]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\nh : a < b\n\u22a2 a ::\u2098 Ioo a b = Ico a b\n[PROOFSTEP]\nrw [Ioo, \u2190 Finset.insert_val_of_not_mem left_not_mem_Ioo, Finset.Ioo_insert_left h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\nh : a < b\n\u22a2 (Finset.Ico a b).val = Ico a b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c d : \u03b1\nh : b \u2264 c\nx : \u03b1\nhab : x \u2208 Ico a b\nhbc : x \u2208 Ico c d\n\u22a2 False\n[PROOFSTEP]\nrw [mem_Ico] at hab hbc \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c d : \u03b1\nh : b \u2264 c\nx : \u03b1\nhab : a \u2264 x \u2227 x < b\nhbc : c \u2264 x \u2227 x < d\n\u22a2 False\n[PROOFSTEP]\nexact hab.2.not_le (h.trans hbc.1)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b : \u03b1\ninst\u271d : DecidablePred fun x => x \u2264 a\nhab : a < b\n\u22a2 filter (fun x => x \u2264 a) (Ico a b) = {a}\n[PROOFSTEP]\nrw [Ico, \u2190 Finset.filter_val, Finset.Ico_filter_le_left hab]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b : \u03b1\ninst\u271d : DecidablePred fun x => x \u2264 a\nhab : a < b\n\u22a2 {a}.val = {a}\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\n\u22a2 Ico a b + Ico b c = Ico a c\n[PROOFSTEP]\nrw [add_eq_union_iff_disjoint.2 (Ico_disjoint_Ico le_rfl), Ico, Ico, Ico, \u2190 Finset.union_val,\n Finset.Ico_union_Ico_eq_Ico hab hbc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c d : \u03b1\n\u22a2 Ico a b \u2229 Ico c d = Ico (max a c) (min b d)\n[PROOFSTEP]\nrw [Ico, Ico, Ico, \u2190 Finset.inter_val, Finset.Ico_inter_Ico]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 filter (fun x => x < c) (Ico a b) = Ico a (min b c)\n[PROOFSTEP]\nrw [Ico, Ico, \u2190 Finset.filter_val, Finset.Ico_filter_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 filter (fun x => c \u2264 x) (Ico a b) = Ico (max a c) b\n[PROOFSTEP]\nrw [Ico, Ico, \u2190 Finset.filter_val, Finset.Ico_filter_le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 Ico a b - Ico a c = Ico (max a c) b\n[PROOFSTEP]\nrw [Ico, Ico, Ico, \u2190 Finset.sdiff_val, Finset.Ico_diff_Ico_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 Ico a b - Ico c b = Ico a (min b c)\n[PROOFSTEP]\nrw [Ico, Ico, Ico, \u2190 Finset.sdiff_val, Finset.Ico_diff_Ico_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map ((fun x x_1 => x + x_1) c) (Icc a b) = Icc (c + a) (c + b)\n[PROOFSTEP]\nclassical rw [Icc, Icc, \u2190 Finset.image_add_left_Icc, Finset.image_val,\n ((Finset.nodup _).map <| add_right_injective c).dedup]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map ((fun x x_1 => x + x_1) c) (Icc a b) = Icc (c + a) (c + b)\n[PROOFSTEP]\nrw [Icc, Icc, \u2190 Finset.image_add_left_Icc, Finset.image_val, ((Finset.nodup _).map <| add_right_injective c).dedup]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map ((fun x x_1 => x + x_1) c) (Ico a b) = Ico (c + a) (c + b)\n[PROOFSTEP]\nclassical rw [Ico, Ico, \u2190 Finset.image_add_left_Ico, Finset.image_val,\n ((Finset.nodup _).map <| add_right_injective c).dedup]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map ((fun x x_1 => x + x_1) c) (Ico a b) = Ico (c + a) (c + b)\n[PROOFSTEP]\nrw [Ico, Ico, \u2190 Finset.image_add_left_Ico, Finset.image_val, ((Finset.nodup _).map <| add_right_injective c).dedup]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map ((fun x x_1 => x + x_1) c) (Ioc a b) = Ioc (c + a) (c + b)\n[PROOFSTEP]\nclassical rw [Ioc, Ioc, \u2190 Finset.image_add_left_Ioc, Finset.image_val,\n ((Finset.nodup _).map <| add_right_injective c).dedup]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map ((fun x x_1 => x + x_1) c) (Ioc a b) = Ioc (c + a) (c + b)\n[PROOFSTEP]\nrw [Ioc, Ioc, \u2190 Finset.image_add_left_Ioc, Finset.image_val, ((Finset.nodup _).map <| add_right_injective c).dedup]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map ((fun x x_1 => x + x_1) c) (Ioo a b) = Ioo (c + a) (c + b)\n[PROOFSTEP]\nclassical rw [Ioo, Ioo, \u2190 Finset.image_add_left_Ioo, Finset.image_val,\n ((Finset.nodup _).map <| add_right_injective c).dedup]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map ((fun x x_1 => x + x_1) c) (Ioo a b) = Ioo (c + a) (c + b)\n[PROOFSTEP]\nrw [Ioo, Ioo, \u2190 Finset.image_add_left_Ioo, Finset.image_val, ((Finset.nodup _).map <| add_right_injective c).dedup]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (fun x => x + c) (Icc a b) = Icc (a + c) (b + c)\n[PROOFSTEP]\nsimp_rw [add_comm _ c]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (fun x => c + x) (Icc a b) = Icc (c + a) (c + b)\n[PROOFSTEP]\nexact map_add_left_Icc _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (fun x => x + c) (Ico a b) = Ico (a + c) (b + c)\n[PROOFSTEP]\nsimp_rw [add_comm _ c]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (fun x => c + x) (Ico a b) = Ico (c + a) (c + b)\n[PROOFSTEP]\nexact map_add_left_Ico _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (fun x => x + c) (Ioc a b) = Ioc (a + c) (b + c)\n[PROOFSTEP]\nsimp_rw [add_comm _ c]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (fun x => c + x) (Ioc a b) = Ioc (c + a) (c + b)\n[PROOFSTEP]\nexact map_add_left_Ioc _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (fun x => x + c) (Ioo a b) = Ioo (a + c) (b + c)\n[PROOFSTEP]\nsimp_rw [add_comm _ c]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (fun x => c + x) (Ioo a b) = Ioo (c + a) (c + b)\n[PROOFSTEP]\nexact map_add_left_Ioo _ _ _\n", "meta": {"mathlib_filename": "Mathlib.Data.Multiset.LocallyFinite", "llama_tokens": 5730, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105454764747, "lm_q2_score": 0.6619228691808011, "lm_q1q2_score": 0.4404504574449501}} {"text": "[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nf g : Valuation R \u0393\u2080\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8_, _\u27e9, _\u27e9, _\u27e9 := f\n[GOAL]\ncase mk.mk.mk\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\ng : Valuation R \u0393\u2080\ntoFun\u271d : R \u2192 \u0393\u2080\nmap_zero'\u271d : toFun\u271d 0 = 0\nmap_one'\u271d : ZeroHom.toFun { toFun := toFun\u271d, map_zero' := map_zero'\u271d } 1 = 1\nmap_mul'\u271d :\n \u2200 (x y : R),\n ZeroHom.toFun { toFun := toFun\u271d, map_zero' := map_zero'\u271d } (x * y) =\n ZeroHom.toFun { toFun := toFun\u271d, map_zero' := map_zero'\u271d } x *\n ZeroHom.toFun { toFun := toFun\u271d, map_zero' := map_zero'\u271d } y\nmap_add_le_max'\u271d :\n \u2200 (x y : R),\n ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := toFun\u271d, map_zero' := map_zero'\u271d }, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d })\n (x + y) \u2264\n max\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := toFun\u271d, map_zero' := map_zero'\u271d }, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d })\n x)\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := toFun\u271d, map_zero' := map_zero'\u271d }, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d })\n y)\nh :\n (fun f => f.toFun)\n {\n toMonoidWithZeroHom :=\n { toZeroHom := { toFun := toFun\u271d, map_zero' := map_zero'\u271d }, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d },\n map_add_le_max' := map_add_le_max'\u271d } =\n (fun f => f.toFun) g\n\u22a2 {\n toMonoidWithZeroHom :=\n { toZeroHom := { toFun := toFun\u271d, map_zero' := map_zero'\u271d }, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d },\n map_add_le_max' := map_add_le_max'\u271d } =\n g\n[PROOFSTEP]\ncongr\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\n\u22a2 v.toFun = \u2191v\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\n\u22a2 \u2191v.toMonoidWithZeroHom = \u2191v\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\nx y z : R\n\u22a2 \u2200 (x y : R), \u2191v (x + y) \u2264 \u2191v x \u2228 \u2191v (x + y) \u2264 \u2191v y\n[PROOFSTEP]\nintro x y\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\nx\u271d y\u271d z x y : R\n\u22a2 \u2191v (x + y) \u2264 \u2191v x \u2228 \u2191v (x + y) \u2264 \u2191v y\n[PROOFSTEP]\nrw [\u2190 le_max_iff, \u2190 ge_iff_le]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\nx\u271d y\u271d z x y : R\n\u22a2 max (\u2191v x) (\u2191v y) \u2265 \u2191v (x + y)\n[PROOFSTEP]\napply map_add\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\nx y z : R\n\u03b9 : Type u_7\ns : Finset \u03b9\nf : \u03b9 \u2192 R\ng : \u0393\u2080\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191v (f i) \u2264 g\n\u22a2 \u2191v (\u2211 i in s, f i) \u2264 g\n[PROOFSTEP]\nrefine' Finset.induction_on s (fun _ => v.map_zero \u25b8 zero_le') (fun a s has ih hf => _) hf\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\nx y z : R\n\u03b9 : Type u_7\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\ng : \u0393\u2080\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2191v (f i) \u2264 g\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2191v (f i) \u2264 g) \u2192 \u2191v (\u2211 i in s, f i) \u2264 g\nhf : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2191v (f i) \u2264 g\n\u22a2 \u2191v (\u2211 i in insert a s, f i) \u2264 g\n[PROOFSTEP]\nrw [Finset.forall_mem_insert] at hf \n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\nx y z : R\n\u03b9 : Type u_7\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\ng : \u0393\u2080\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2191v (f i) \u2264 g\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2191v (f i) \u2264 g) \u2192 \u2191v (\u2211 i in s, f i) \u2264 g\nhf : \u2191v (f a) \u2264 g \u2227 \u2200 (x : \u03b9), x \u2208 s \u2192 \u2191v (f x) \u2264 g\n\u22a2 \u2191v (\u2211 i in insert a s, f i) \u2264 g\n[PROOFSTEP]\nrw [Finset.sum_insert has]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\nx y z : R\n\u03b9 : Type u_7\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\ng : \u0393\u2080\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2191v (f i) \u2264 g\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2191v (f i) \u2264 g) \u2192 \u2191v (\u2211 i in s, f i) \u2264 g\nhf : \u2191v (f a) \u2264 g \u2227 \u2200 (x : \u03b9), x \u2208 s \u2192 \u2191v (f x) \u2264 g\n\u22a2 \u2191v (f a + \u2211 x in s, f x) \u2264 g\n[PROOFSTEP]\nexact v.map_add_le hf.1 (ih hf.2)\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\nx y z : R\n\u03b9 : Type u_7\ns : Finset \u03b9\nf : \u03b9 \u2192 R\ng : \u0393\u2080\nhg : g \u2260 0\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191v (f i) < g\n\u22a2 \u2191v (\u2211 i in s, f i) < g\n[PROOFSTEP]\nrefine' Finset.induction_on s (fun _ => v.map_zero \u25b8 (zero_lt_iff.2 hg)) (fun a s has ih hf => _) hf\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\nx y z : R\n\u03b9 : Type u_7\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\ng : \u0393\u2080\nhg : g \u2260 0\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2191v (f i) < g\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2191v (f i) < g) \u2192 \u2191v (\u2211 i in s, f i) < g\nhf : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2191v (f i) < g\n\u22a2 \u2191v (\u2211 i in insert a s, f i) < g\n[PROOFSTEP]\nrw [Finset.forall_mem_insert] at hf \n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\nx y z : R\n\u03b9 : Type u_7\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\ng : \u0393\u2080\nhg : g \u2260 0\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2191v (f i) < g\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2191v (f i) < g) \u2192 \u2191v (\u2211 i in s, f i) < g\nhf : \u2191v (f a) < g \u2227 \u2200 (x : \u03b9), x \u2208 s \u2192 \u2191v (f x) < g\n\u22a2 \u2191v (\u2211 i in insert a s, f i) < g\n[PROOFSTEP]\nrw [Finset.sum_insert has]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\nx y z : R\n\u03b9 : Type u_7\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\ng : \u0393\u2080\nhg : g \u2260 0\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2191v (f i) < g\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2191v (f i) < g) \u2192 \u2191v (\u2211 i in s, f i) < g\nhf : \u2191v (f a) < g \u2227 \u2200 (x : \u03b9), x \u2208 s \u2192 \u2191v (f x) < g\n\u22a2 \u2191v (f a + \u2211 x in s, f x) < g\n[PROOFSTEP]\nexact v.map_add_lt hf.1 (ih hf.2)\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2075 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u2074 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv\u271d : Valuation R \u0393\u2080\nx\u271d y\u271d z : R\nS : Type u_7\ninst\u271d : Ring S\nf : S \u2192+* R\nv : Valuation R \u0393\u2080\nsrc\u271d : S \u2192*\u2080 \u0393\u2080 := MonoidWithZeroHom.comp v.toMonoidWithZeroHom (RingHom.toMonoidWithZeroHom f)\nx y : S\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := \u2191v \u2218 \u2191f, map_zero' := (_ : ZeroHom.toFun (\u2191src\u271d) 0 = 0) },\n map_one' := (_ : ZeroHom.toFun (\u2191src\u271d) 1 = 1),\n map_mul' :=\n (_ : \u2200 (x y : S), ZeroHom.toFun (\u2191src\u271d) (x * y) = ZeroHom.toFun (\u2191src\u271d) x * ZeroHom.toFun (\u2191src\u271d) y) })\n (x + y) \u2264\n max\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := \u2191v \u2218 \u2191f, map_zero' := (_ : ZeroHom.toFun (\u2191src\u271d) 0 = 0) },\n map_one' := (_ : ZeroHom.toFun (\u2191src\u271d) 1 = 1),\n map_mul' :=\n (_ : \u2200 (x y : S), ZeroHom.toFun (\u2191src\u271d) (x * y) = ZeroHom.toFun (\u2191src\u271d) x * ZeroHom.toFun (\u2191src\u271d) y) })\n x)\n (ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := \u2191v \u2218 \u2191f, map_zero' := (_ : ZeroHom.toFun (\u2191src\u271d) 0 = 0) },\n map_one' := (_ : ZeroHom.toFun (\u2191src\u271d) 1 = 1),\n map_mul' :=\n (_ : \u2200 (x y : S), ZeroHom.toFun (\u2191src\u271d) (x * y) = ZeroHom.toFun (\u2191src\u271d) x * ZeroHom.toFun (\u2191src\u271d) y) })\n y)\n[PROOFSTEP]\nsimp only [comp_apply, map_add, f.map_add]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx\u271d y\u271d z x y : R\n\u22a2 \u2191v (x - y) = \u2191v (x + -y)\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx\u271d y\u271d z x y : R\n\u22a2 max (\u2191v x) (\u2191v (-y)) = max (\u2191v x) (\u2191v y)\n[PROOFSTEP]\nrw [map_neg]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx\u271d y\u271d z x y : R\ng : (fun x => \u0393\u2080) x\nhx : \u2191v x \u2264 g\nhy : \u2191v y \u2264 g\n\u22a2 \u2191v (x - y) \u2264 g\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx\u271d y\u271d z x y : R\ng : (fun x => \u0393\u2080) x\nhx : \u2191v x \u2264 g\nhy : \u2191v y \u2264 g\n\u22a2 \u2191v (x + -y) \u2264 g\n[PROOFSTEP]\nexact v.map_add_le hx (le_trans (le_of_eq (v.map_neg y)) hy)\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v x \u2260 \u2191v y\n\u22a2 \u2191v (x + y) = max (\u2191v x) (\u2191v y)\n[PROOFSTEP]\nsuffices : \u00acv (x + y) < max (v x) (v y)\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v x \u2260 \u2191v y\nthis : \u00ac\u2191v (x + y) < max (\u2191v x) (\u2191v y)\n\u22a2 \u2191v (x + y) = max (\u2191v x) (\u2191v y)\ncase this\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v x \u2260 \u2191v y\n\u22a2 \u00ac\u2191v (x + y) < max (\u2191v x) (\u2191v y)\n[PROOFSTEP]\nexact or_iff_not_imp_right.1 (le_iff_eq_or_lt.1 (v.map_add x y)) this\n[GOAL]\ncase this\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v x \u2260 \u2191v y\n\u22a2 \u00ac\u2191v (x + y) < max (\u2191v x) (\u2191v y)\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase this\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v x \u2260 \u2191v y\nh' : \u2191v (x + y) < max (\u2191v x) (\u2191v y)\n\u22a2 False\n[PROOFSTEP]\nwlog vyx : v y < v x generalizing x y\n[GOAL]\ncase this.inr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v x \u2260 \u2191v y\nh' : \u2191v (x + y) < max (\u2191v x) (\u2191v y)\nthis : \u2200 {x y : R}, \u2191v x \u2260 \u2191v y \u2192 \u2191v (x + y) < max (\u2191v x) (\u2191v y) \u2192 \u2191v y < \u2191v x \u2192 False\nvyx : \u00ac\u2191v y < \u2191v x\n\u22a2 False\n[PROOFSTEP]\nrefine' this h.symm _ (h.lt_or_lt.resolve_right vyx)\n[GOAL]\ncase this.inr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v x \u2260 \u2191v y\nh' : \u2191v (x + y) < max (\u2191v x) (\u2191v y)\nthis : \u2200 {x y : R}, \u2191v x \u2260 \u2191v y \u2192 \u2191v (x + y) < max (\u2191v x) (\u2191v y) \u2192 \u2191v y < \u2191v x \u2192 False\nvyx : \u00ac\u2191v y < \u2191v x\n\u22a2 \u2191v (y + x) < max (\u2191v y) (\u2191v x)\n[PROOFSTEP]\nrwa [add_comm, max_comm]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx\u271d y\u271d z x y : R\nh : \u2191v x \u2260 \u2191v y\nh' : \u2191v (x + y) < max (\u2191v x) (\u2191v y)\nvyx : \u2191v y < \u2191v x\n\u22a2 False\n[PROOFSTEP]\nrw [max_eq_left_of_lt vyx] at h' \n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx\u271d y\u271d z x y : R\nh : \u2191v x \u2260 \u2191v y\nh' : \u2191v (x + y) < \u2191v x\nvyx : \u2191v y < \u2191v x\n\u22a2 False\n[PROOFSTEP]\napply lt_irrefl (v x)\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx\u271d y\u271d z x y : R\nh : \u2191v x \u2260 \u2191v y\nh' : \u2191v (x + y) < \u2191v x\nvyx : \u2191v y < \u2191v x\n\u22a2 \u2191v x < \u2191v x\n[PROOFSTEP]\ncalc\n v x = v (x + y - y) := by simp\n _ \u2264 max (v <| x + y) (v y) := (map_sub _ _ _)\n _ < v x := max_lt h' vyx\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx\u271d y\u271d z x y : R\nh : \u2191v x \u2260 \u2191v y\nh' : \u2191v (x + y) < \u2191v x\nvyx : \u2191v y < \u2191v x\n\u22a2 \u2191v x = \u2191v (x + y - y)\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v y < \u2191v x\n\u22a2 \u2191v (x + y) = \u2191v x\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v y < \u2191v x\n\u22a2 \u2191v (y + x) = \u2191v x\n[PROOFSTEP]\nexact map_add_eq_of_lt_right _ h\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v (y - x) < \u2191v x\n\u22a2 \u2191v y = \u2191v x\n[PROOFSTEP]\nhave := Valuation.map_add_of_distinct_val v (ne_of_gt h).symm\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v (y - x) < \u2191v x\nthis : \u2191v (y - x + x) = max (\u2191v (y - x)) (\u2191v x)\n\u22a2 \u2191v y = \u2191v x\n[PROOFSTEP]\nrw [max_eq_right (le_of_lt h)] at this \n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v (y - x) < \u2191v x\nthis : \u2191v (y - x + x) = \u2191v x\n\u22a2 \u2191v y = \u2191v x\n[PROOFSTEP]\nsimpa using this\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v x < 1\n\u22a2 \u2191v (1 + x) = 1\n[PROOFSTEP]\nrw [\u2190 v.map_one] at h \n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v x < \u2191v 1\n\u22a2 \u2191v (1 + x) = 1\n[PROOFSTEP]\nsimpa only [v.map_one] using v.map_add_eq_of_lt_left h\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v x < 1\n\u22a2 \u2191v (1 - x) = 1\n[PROOFSTEP]\nrw [\u2190 v.map_one, \u2190 v.map_neg] at h \n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v (-x) < \u2191v 1\n\u22a2 \u2191v (1 - x) = 1\n[PROOFSTEP]\nrw [sub_eq_add_neg 1 x]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y z : R\nh : \u2191v (-x) < \u2191v 1\n\u22a2 \u2191v (1 + -x) = 1\n[PROOFSTEP]\nsimpa only [v.map_one, v.map_neg] using v.map_add_eq_of_lt_left h\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv\u271d : Valuation R \u0393\u2080\nx\u271d y z : R\nv : Valuation K \u0393\u2080\nx : K\nh : x \u2260 0\n\u22a2 1 < \u2191v x \u2194 \u2191v x\u207b\u00b9 < 1\n[PROOFSTEP]\nsimpa using (inv_lt_inv\u2080 (v.ne_zero_iff.2 h) one_ne_zero).symm\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv\u271d : Valuation R \u0393\u2080\nx y z : R\nv : Valuation R \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\n\u22a2 0 \u2208\n { carrier := {x | \u2191v x < \u2191\u03b3},\n add_mem' := (_ : \u2200 {x y : R}, x \u2208 {x | \u2191v x < \u2191\u03b3} \u2192 y \u2208 {x | \u2191v x < \u2191\u03b3} \u2192 \u2191v (x + y) < \u2191\u03b3) }.carrier\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv\u271d : Valuation R \u0393\u2080\nx y z : R\nv : Valuation R \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx\u271d : R\nx_in :\n x\u271d \u2208\n {\n toAddSubsemigroup :=\n { carrier := {x | \u2191v x < \u2191\u03b3},\n add_mem' := (_ : \u2200 {x y : R}, x \u2208 {x | \u2191v x < \u2191\u03b3} \u2192 y \u2208 {x | \u2191v x < \u2191\u03b3} \u2192 \u2191v (x + y) < \u2191\u03b3) },\n zero_mem' := (_ : \u2191v 0 < \u2191\u03b3) }.toAddSubsemigroup.carrier\n\u22a2 -x\u271d \u2208\n {\n toAddSubsemigroup :=\n { carrier := {x | \u2191v x < \u2191\u03b3},\n add_mem' := (_ : \u2200 {x y : R}, x \u2208 {x | \u2191v x < \u2191\u03b3} \u2192 y \u2208 {x | \u2191v x < \u2191\u03b3} \u2192 \u2191v (x + y) < \u2191\u03b3) },\n zero_mem' := (_ : \u2191v 0 < \u2191\u03b3) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrwa [Set.mem_setOf, map_neg]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv v\u2081 : Valuation R \u0393\u2080\nv\u2082 : Valuation R \u0393'\u2080\nv\u2083 : Valuation R \u0393''\u2080\nv' : Valuation R \u0393\u2080\nh : v = v'\n\u22a2 IsEquiv v v'\n[PROOFSTEP]\nsubst h\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv v\u2081 : Valuation R \u0393\u2080\nv\u2082 : Valuation R \u0393'\u2080\nv\u2083 : Valuation R \u0393''\u2080\n\u22a2 IsEquiv v v\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv v\u2081 : Valuation R \u0393\u2080\nv\u2082 : Valuation R \u0393'\u2080\nv\u2083 : Valuation R \u0393''\u2080\nv' : Valuation R \u0393\u2080\nf : \u0393\u2080 \u2192*\u2080 \u0393'\u2080\nhf : Monotone \u2191f\ninf : Injective \u2191f\nh : IsEquiv v v'\nH : StrictMono \u2191f := Monotone.strictMono_of_injective hf inf\nr s : R\n\u22a2 \u2191f (\u2191v r) \u2264 \u2191f (\u2191v s) \u2194 \u2191v r \u2264 \u2191v s\n[PROOFSTEP]\nrw [H.le_iff_le]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv v\u2081 : Valuation R \u0393\u2080\nv\u2082 : Valuation R \u0393'\u2080\nv\u2083 : Valuation R \u0393''\u2080\nv' : Valuation R \u0393\u2080\nf : \u0393\u2080 \u2192*\u2080 \u0393'\u2080\nhf : Monotone \u2191f\ninf : Injective \u2191f\nh : IsEquiv v v'\nH : StrictMono \u2191f := Monotone.strictMono_of_injective hf inf\nr s : R\n\u22a2 \u2191v' r \u2264 \u2191v' s \u2194 \u2191f (\u2191v' r) \u2264 \u2191f (\u2191v' s)\n[PROOFSTEP]\nrw [H.le_iff_le]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv v\u2081 : Valuation R \u0393\u2080\nv\u2082 : Valuation R \u0393'\u2080\nv\u2083 : Valuation R \u0393''\u2080\nh : IsEquiv v\u2081 v\u2082\nr s : R\n\u22a2 \u2191v\u2081 r = \u2191v\u2081 s \u2194 \u2191v\u2082 r = \u2191v\u2082 s\n[PROOFSTEP]\nsimpa only [le_antisymm_iff] using and_congr (h r s) (h s r)\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv v\u2081 : Valuation R \u0393\u2080\nv\u2082 : Valuation R \u0393'\u2080\nv\u2083 : Valuation R \u0393''\u2080\nh : IsEquiv v\u2081 v\u2082\nr : R\n\u22a2 \u2191v\u2081 r \u2260 0 \u2194 \u2191v\u2082 r \u2260 0\n[PROOFSTEP]\nhave : v\u2081 r \u2260 v\u2081 0 \u2194 v\u2082 r \u2260 v\u2082 0 := not_congr h.val_eq\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv v\u2081 : Valuation R \u0393\u2080\nv\u2082 : Valuation R \u0393'\u2080\nv\u2083 : Valuation R \u0393''\u2080\nh : IsEquiv v\u2081 v\u2082\nr : R\nthis : \u2191v\u2081 r \u2260 \u2191v\u2081 0 \u2194 \u2191v\u2082 r \u2260 \u2191v\u2082 0\n\u22a2 \u2191v\u2081 r \u2260 0 \u2194 \u2191v\u2082 r \u2260 0\n[PROOFSTEP]\nrwa [v\u2081.map_zero, v\u2082.map_zero] at this \n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\n\u22a2 IsEquiv v v'\n[PROOFSTEP]\nintro x y\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\n\u22a2 \u2191v x \u2264 \u2191v y \u2194 \u2191v' x \u2264 \u2191v' y\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : y = 0\n\u22a2 \u2191v x \u2264 \u2191v y \u2194 \u2191v' x \u2264 \u2191v' y\n[PROOFSTEP]\nsimp [hy, zero_iff]\n[GOAL]\ncase neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u00acy = 0\n\u22a2 \u2191v x \u2264 \u2191v y \u2194 \u2191v' x \u2264 \u2191v' y\n[PROOFSTEP]\nrw [show y = 1 * y by rw [one_mul]]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u00acy = 0\n\u22a2 y = 1 * y\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\ncase neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u00acy = 0\n\u22a2 \u2191v x \u2264 \u2191v (1 * y) \u2194 \u2191v' x \u2264 \u2191v' (1 * y)\n[PROOFSTEP]\nrw [\u2190 inv_mul_cancel_right\u2080 hy x]\n[GOAL]\ncase neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u00acy = 0\n\u22a2 \u2191v (x * y\u207b\u00b9 * y) \u2264 \u2191v (1 * y) \u2194 \u2191v' (x * y\u207b\u00b9 * y) \u2264 \u2191v' (1 * y)\n[PROOFSTEP]\niterate 2 rw [v.map_mul _ y, v'.map_mul _ y]\n[GOAL]\ncase neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u00acy = 0\n\u22a2 \u2191v (x * y\u207b\u00b9 * y) \u2264 \u2191v (1 * y) \u2194 \u2191v' (x * y\u207b\u00b9 * y) \u2264 \u2191v' (1 * y)\n[PROOFSTEP]\nrw [v.map_mul _ y, v'.map_mul _ y]\n[GOAL]\ncase neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u00acy = 0\n\u22a2 \u2191v (x * y\u207b\u00b9) * \u2191v y \u2264 \u2191v (1 * y) \u2194 \u2191v' (x * y\u207b\u00b9) * \u2191v' y \u2264 \u2191v' (1 * y)\n[PROOFSTEP]\nrw [v.map_mul _ y, v'.map_mul _ y]\n[GOAL]\ncase neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u00acy = 0\n\u22a2 \u2191v (x * y\u207b\u00b9) * \u2191v y \u2264 \u2191v 1 * \u2191v y \u2194 \u2191v' (x * y\u207b\u00b9) * \u2191v' y \u2264 \u2191v' 1 * \u2191v' y\n[PROOFSTEP]\nrw [v.map_one, v'.map_one]\n[GOAL]\ncase neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u00acy = 0\n\u22a2 \u2191v (x * y\u207b\u00b9) * \u2191v y \u2264 1 * \u2191v y \u2194 \u2191v' (x * y\u207b\u00b9) * \u2191v' y \u2264 1 * \u2191v' y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase neg.mp\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u00acy = 0\n\u22a2 \u2191v (x * y\u207b\u00b9) * \u2191v y \u2264 1 * \u2191v y \u2192 \u2191v' (x * y\u207b\u00b9) * \u2191v' y \u2264 1 * \u2191v' y\n[PROOFSTEP]\nintro H\n[GOAL]\ncase neg.mpr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u00acy = 0\n\u22a2 \u2191v' (x * y\u207b\u00b9) * \u2191v' y \u2264 1 * \u2191v' y \u2192 \u2191v (x * y\u207b\u00b9) * \u2191v y \u2264 1 * \u2191v y\n[PROOFSTEP]\nintro H\n[GOAL]\ncase neg.mp\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u00acy = 0\nH : \u2191v (x * y\u207b\u00b9) * \u2191v y \u2264 1 * \u2191v y\n\u22a2 \u2191v' (x * y\u207b\u00b9) * \u2191v' y \u2264 1 * \u2191v' y\n[PROOFSTEP]\napply mul_le_mul_right'\n[GOAL]\ncase neg.mp.bc\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u00acy = 0\nH : \u2191v (x * y\u207b\u00b9) * \u2191v y \u2264 1 * \u2191v y\n\u22a2 \u2191v' (x * y\u207b\u00b9) \u2264 1\n[PROOFSTEP]\nreplace hy := v.ne_zero_iff.mpr hy\n[GOAL]\ncase neg.mp.bc\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nH : \u2191v (x * y\u207b\u00b9) * \u2191v y \u2264 1 * \u2191v y\nhy : \u2191v y \u2260 0\n\u22a2 \u2191v' (x * y\u207b\u00b9) \u2264 1\n[PROOFSTEP]\nreplace H := le_of_le_mul_right hy H\n[GOAL]\ncase neg.mp.bc\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u2191v y \u2260 0\nH : \u2191v (x * y\u207b\u00b9) \u2264 1\n\u22a2 \u2191v' (x * y\u207b\u00b9) \u2264 1\n[PROOFSTEP]\nrwa [h] at H \n[GOAL]\ncase neg.mpr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u00acy = 0\nH : \u2191v' (x * y\u207b\u00b9) * \u2191v' y \u2264 1 * \u2191v' y\n\u22a2 \u2191v (x * y\u207b\u00b9) * \u2191v y \u2264 1 * \u2191v y\n[PROOFSTEP]\napply mul_le_mul_right'\n[GOAL]\ncase neg.mpr.bc\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u00acy = 0\nH : \u2191v' (x * y\u207b\u00b9) * \u2191v' y \u2264 1 * \u2191v' y\n\u22a2 \u2191v (x * y\u207b\u00b9) \u2264 1\n[PROOFSTEP]\nreplace hy := v'.ne_zero_iff.mpr hy\n[GOAL]\ncase neg.mpr.bc\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nH : \u2191v' (x * y\u207b\u00b9) * \u2191v' y \u2264 1 * \u2191v' y\nhy : \u2191v' y \u2260 0\n\u22a2 \u2191v (x * y\u207b\u00b9) \u2264 1\n[PROOFSTEP]\nreplace H := le_of_le_mul_right hy H\n[GOAL]\ncase neg.mpr.bc\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\nx y : K\nhy : \u2191v' y \u2260 0\nH : \u2191v' (x * y\u207b\u00b9) \u2264 1\n\u22a2 \u2191v (x * y\u207b\u00b9) \u2264 1\n[PROOFSTEP]\nrwa [h]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : IsEquiv v v'\nx : K\n\u22a2 \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\n[PROOFSTEP]\nsimpa using h x 1\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 IsEquiv v v' \u2192 \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\n[PROOFSTEP]\nintro h x\n[GOAL]\ncase mp\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : IsEquiv v v'\nx : K\n\u22a2 \u2191v x = 1 \u2194 \u2191v' x = 1\n[PROOFSTEP]\nsimpa using @IsEquiv.val_eq _ _ _ _ _ _ v v' h x 1\n[GOAL]\ncase mpr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 (\u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1) \u2192 IsEquiv v v'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\n\u22a2 IsEquiv v v'\n[PROOFSTEP]\napply isEquiv_of_val_le_one\n[GOAL]\ncase mpr.h\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\n\u22a2 \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\n[PROOFSTEP]\nintro x\n[GOAL]\ncase mpr.h\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\n\u22a2 \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.h.mp\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\n\u22a2 \u2191v x \u2264 1 \u2192 \u2191v' x \u2264 1\n[PROOFSTEP]\nintro hx\n[GOAL]\ncase mpr.h.mp\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v x \u2264 1\n\u22a2 \u2191v' x \u2264 1\n[PROOFSTEP]\ncases' lt_or_eq_of_le hx with hx' hx'\n[GOAL]\ncase mpr.h.mp.inl\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v x \u2264 1\nhx' : \u2191v x < 1\n\u22a2 \u2191v' x \u2264 1\n[PROOFSTEP]\nhave : v (1 + x) = 1 := by\n rw [\u2190 v.map_one]\n apply map_add_eq_of_lt_left\n simpa\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v x \u2264 1\nhx' : \u2191v x < 1\n\u22a2 \u2191v (1 + x) = 1\n[PROOFSTEP]\nrw [\u2190 v.map_one]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v x \u2264 1\nhx' : \u2191v x < 1\n\u22a2 \u2191v (1 + x) = \u2191v 1\n[PROOFSTEP]\napply map_add_eq_of_lt_left\n[GOAL]\ncase h\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v x \u2264 1\nhx' : \u2191v x < 1\n\u22a2 \u2191v x < \u2191v 1\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase mpr.h.mp.inl\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v x \u2264 1\nhx' : \u2191v x < 1\nthis : \u2191v (1 + x) = 1\n\u22a2 \u2191v' x \u2264 1\n[PROOFSTEP]\nrw [h] at this \n[GOAL]\ncase mpr.h.mp.inl\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v x \u2264 1\nhx' : \u2191v x < 1\nthis : \u2191v' (1 + x) = 1\n\u22a2 \u2191v' x \u2264 1\n[PROOFSTEP]\nrw [show x = -1 + (1 + x) by simp]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v x \u2264 1\nhx' : \u2191v x < 1\nthis : \u2191v' (1 + x) = 1\n\u22a2 x = -1 + (1 + x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr.h.mp.inl\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v x \u2264 1\nhx' : \u2191v x < 1\nthis : \u2191v' (1 + x) = 1\n\u22a2 \u2191v' (-1 + (1 + x)) \u2264 1\n[PROOFSTEP]\nrefine' le_trans (v'.map_add _ _) _\n[GOAL]\ncase mpr.h.mp.inl\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v x \u2264 1\nhx' : \u2191v x < 1\nthis : \u2191v' (1 + x) = 1\n\u22a2 max (\u2191v' (-1)) (\u2191v' (1 + x)) \u2264 1\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase mpr.h.mp.inr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v x \u2264 1\nhx' : \u2191v x = 1\n\u22a2 \u2191v' x \u2264 1\n[PROOFSTEP]\nrw [h] at hx' \n[GOAL]\ncase mpr.h.mp.inr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v x \u2264 1\nhx' : \u2191v' x = 1\n\u22a2 \u2191v' x \u2264 1\n[PROOFSTEP]\nexact le_of_eq hx'\n[GOAL]\ncase mpr.h.mpr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\n\u22a2 \u2191v' x \u2264 1 \u2192 \u2191v x \u2264 1\n[PROOFSTEP]\nintro hx\n[GOAL]\ncase mpr.h.mpr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v' x \u2264 1\n\u22a2 \u2191v x \u2264 1\n[PROOFSTEP]\ncases' lt_or_eq_of_le hx with hx' hx'\n[GOAL]\ncase mpr.h.mpr.inl\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v' x \u2264 1\nhx' : \u2191v' x < 1\n\u22a2 \u2191v x \u2264 1\n[PROOFSTEP]\nhave : v' (1 + x) = 1 := by\n rw [\u2190 v'.map_one]\n apply map_add_eq_of_lt_left\n simpa\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v' x \u2264 1\nhx' : \u2191v' x < 1\n\u22a2 \u2191v' (1 + x) = 1\n[PROOFSTEP]\nrw [\u2190 v'.map_one]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v' x \u2264 1\nhx' : \u2191v' x < 1\n\u22a2 \u2191v' (1 + x) = \u2191v' 1\n[PROOFSTEP]\napply map_add_eq_of_lt_left\n[GOAL]\ncase h\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v' x \u2264 1\nhx' : \u2191v' x < 1\n\u22a2 \u2191v' x < \u2191v' 1\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase mpr.h.mpr.inl\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v' x \u2264 1\nhx' : \u2191v' x < 1\nthis : \u2191v' (1 + x) = 1\n\u22a2 \u2191v x \u2264 1\n[PROOFSTEP]\nrw [\u2190 h] at this \n[GOAL]\ncase mpr.h.mpr.inl\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v' x \u2264 1\nhx' : \u2191v' x < 1\nthis : \u2191v (1 + x) = 1\n\u22a2 \u2191v x \u2264 1\n[PROOFSTEP]\nrw [show x = -1 + (1 + x) by simp]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v' x \u2264 1\nhx' : \u2191v' x < 1\nthis : \u2191v (1 + x) = 1\n\u22a2 x = -1 + (1 + x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr.h.mpr.inl\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v' x \u2264 1\nhx' : \u2191v' x < 1\nthis : \u2191v (1 + x) = 1\n\u22a2 \u2191v (-1 + (1 + x)) \u2264 1\n[PROOFSTEP]\nrefine' le_trans (v.map_add _ _) _\n[GOAL]\ncase mpr.h.mpr.inl\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v' x \u2264 1\nhx' : \u2191v' x < 1\nthis : \u2191v (1 + x) = 1\n\u22a2 max (\u2191v (-1)) (\u2191v (1 + x)) \u2264 1\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase mpr.h.mpr.inr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v' x \u2264 1\nhx' : \u2191v' x = 1\n\u22a2 \u2191v x \u2264 1\n[PROOFSTEP]\nrw [\u2190 h] at hx' \n[GOAL]\ncase mpr.h.mpr.inr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\nx : K\nhx : \u2191v' x \u2264 1\nhx' : \u2191v x = 1\n\u22a2 \u2191v x \u2264 1\n[PROOFSTEP]\nexact le_of_eq hx'\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 IsEquiv v v' \u2192 \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\n[PROOFSTEP]\nintro h x\n[GOAL]\ncase mp\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : IsEquiv v v'\nx : K\n\u22a2 \u2191v x < 1 \u2194 \u2191v' x < 1\n[PROOFSTEP]\nsimp only [lt_iff_le_and_ne, and_congr ((isEquiv_iff_val_le_one _ _).1 h) ((isEquiv_iff_val_eq_one _ _).1 h).not]\n[GOAL]\ncase mpr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 (\u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1) \u2192 IsEquiv v v'\n[PROOFSTEP]\nrw [isEquiv_iff_val_eq_one]\n[GOAL]\ncase mpr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 (\u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1) \u2192 \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\n[PROOFSTEP]\nintro h x\n[GOAL]\ncase mpr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\n\u22a2 \u2191v x = 1 \u2194 \u2191v' x = 1\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : x = 0\n\u22a2 \u2191v x = 1 \u2194 \u2191v' x = 1\n[PROOFSTEP]\nrw [(zero_iff _).2 hx, (zero_iff _).2 hx]\n[GOAL]\ncase pos\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : x = 0\n\u22a2 0 = 1 \u2194 0 = 1\n[PROOFSTEP]\nsimp only [zero_ne_one]\n[GOAL]\ncase neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\n\u22a2 \u2191v x = 1 \u2194 \u2191v' x = 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase neg.mp\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\n\u22a2 \u2191v x = 1 \u2192 \u2191v' x = 1\n[PROOFSTEP]\nintro hh\n[GOAL]\ncase neg.mp\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x = 1\n\u22a2 \u2191v' x = 1\n[PROOFSTEP]\nby_contra h_1\n[GOAL]\ncase neg.mp\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x = 1\nh_1 : \u00ac\u2191v' x = 1\n\u22a2 False\n[PROOFSTEP]\ncases ne_iff_lt_or_gt.1 h_1 with\n| inl h_2 => simpa [hh, lt_self_iff_false] using h.2 h_2\n| inr h_2 =>\n rw [\u2190 inv_one, \u2190 inv_eq_iff_eq_inv, \u2190 map_inv\u2080] at hh \n exact hh.not_lt (h.2 ((one_lt_val_iff v' hx).1 h_2))\n[GOAL]\ncase neg.mp\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x = 1\nh_1 : \u00ac\u2191v' x = 1\nx\u271d : \u2191v' x < 1 \u2228 \u2191v' x > 1\n\u22a2 False\n[PROOFSTEP]\ncases ne_iff_lt_or_gt.1 h_1 with\n| inl h_2 => simpa [hh, lt_self_iff_false] using h.2 h_2\n| inr h_2 =>\n rw [\u2190 inv_one, \u2190 inv_eq_iff_eq_inv, \u2190 map_inv\u2080] at hh \n exact hh.not_lt (h.2 ((one_lt_val_iff v' hx).1 h_2))\n[GOAL]\ncase neg.mp.inl\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x = 1\nh_1 : \u00ac\u2191v' x = 1\nh_2 : \u2191v' x < 1\n\u22a2 False\n[PROOFSTEP]\n\n| inl h_2 => simpa [hh, lt_self_iff_false] using h.2 h_2\n[GOAL]\ncase neg.mp.inl\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x = 1\nh_1 : \u00ac\u2191v' x = 1\nh_2 : \u2191v' x < 1\n\u22a2 False\n[PROOFSTEP]\nsimpa [hh, lt_self_iff_false] using h.2 h_2\n[GOAL]\ncase neg.mp.inr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x = 1\nh_1 : \u00ac\u2191v' x = 1\nh_2 : \u2191v' x > 1\n\u22a2 False\n[PROOFSTEP]\n\n| inr h_2 =>\n rw [\u2190 inv_one, \u2190 inv_eq_iff_eq_inv, \u2190 map_inv\u2080] at hh \n exact hh.not_lt (h.2 ((one_lt_val_iff v' hx).1 h_2))\n[GOAL]\ncase neg.mp.inr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x = 1\nh_1 : \u00ac\u2191v' x = 1\nh_2 : \u2191v' x > 1\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 inv_one, \u2190 inv_eq_iff_eq_inv, \u2190 map_inv\u2080] at hh \n[GOAL]\ncase neg.mp.inr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x\u207b\u00b9 = 1\nh_1 : \u00ac\u2191v' x = 1\nh_2 : \u2191v' x > 1\n\u22a2 False\n[PROOFSTEP]\nexact hh.not_lt (h.2 ((one_lt_val_iff v' hx).1 h_2))\n[GOAL]\ncase neg.mpr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\n\u22a2 \u2191v' x = 1 \u2192 \u2191v x = 1\n[PROOFSTEP]\nintro hh\n[GOAL]\ncase neg.mpr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x = 1\n\u22a2 \u2191v x = 1\n[PROOFSTEP]\nby_contra h_1\n[GOAL]\ncase neg.mpr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x = 1\nh_1 : \u00ac\u2191v x = 1\n\u22a2 False\n[PROOFSTEP]\ncases ne_iff_lt_or_gt.1 h_1 with\n| inl h_2 => simpa [hh, lt_self_iff_false] using h.1 h_2\n| inr h_2 =>\n rw [\u2190 inv_one, \u2190 inv_eq_iff_eq_inv, \u2190 map_inv\u2080] at hh \n exact hh.not_lt (h.1 ((one_lt_val_iff v hx).1 h_2))\n[GOAL]\ncase neg.mpr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x = 1\nh_1 : \u00ac\u2191v x = 1\nx\u271d : \u2191v x < 1 \u2228 \u2191v x > 1\n\u22a2 False\n[PROOFSTEP]\ncases ne_iff_lt_or_gt.1 h_1 with\n| inl h_2 => simpa [hh, lt_self_iff_false] using h.1 h_2\n| inr h_2 =>\n rw [\u2190 inv_one, \u2190 inv_eq_iff_eq_inv, \u2190 map_inv\u2080] at hh \n exact hh.not_lt (h.1 ((one_lt_val_iff v hx).1 h_2))\n[GOAL]\ncase neg.mpr.inl\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x = 1\nh_1 : \u00ac\u2191v x = 1\nh_2 : \u2191v x < 1\n\u22a2 False\n[PROOFSTEP]\n\n| inl h_2 => simpa [hh, lt_self_iff_false] using h.1 h_2\n[GOAL]\ncase neg.mpr.inl\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x = 1\nh_1 : \u00ac\u2191v x = 1\nh_2 : \u2191v x < 1\n\u22a2 False\n[PROOFSTEP]\nsimpa [hh, lt_self_iff_false] using h.1 h_2\n[GOAL]\ncase neg.mpr.inr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x = 1\nh_1 : \u00ac\u2191v x = 1\nh_2 : \u2191v x > 1\n\u22a2 False\n[PROOFSTEP]\n\n| inr h_2 =>\n rw [\u2190 inv_one, \u2190 inv_eq_iff_eq_inv, \u2190 map_inv\u2080] at hh \n exact hh.not_lt (h.1 ((one_lt_val_iff v hx).1 h_2))\n[GOAL]\ncase neg.mpr.inr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x = 1\nh_1 : \u00ac\u2191v x = 1\nh_2 : \u2191v x > 1\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 inv_one, \u2190 inv_eq_iff_eq_inv, \u2190 map_inv\u2080] at hh \n[GOAL]\ncase neg.mpr.inr\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x\u207b\u00b9 = 1\nh_1 : \u00ac\u2191v x = 1\nh_2 : \u2191v x > 1\n\u22a2 False\n[PROOFSTEP]\nexact hh.not_lt (h.1 ((one_lt_val_iff v hx).1 h_2))\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v (x - 1) < 1 \u2194 \u2191v' (x - 1) < 1\n[PROOFSTEP]\nrw [isEquiv_iff_val_lt_one]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 (\u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1) \u2194 \u2200 {x : K}, \u2191v (x - 1) < 1 \u2194 \u2191v' (x - 1) < 1\n[PROOFSTEP]\nexact (Equiv.subRight 1).surjective.forall\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 List.TFAE\n [IsEquiv v v', \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1, \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1, \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1,\n \u2200 {x : K}, \u2191v (x - 1) < 1 \u2194 \u2191v' (x - 1) < 1]\n[PROOFSTEP]\ntfae_have 1 \u2194 2\n[GOAL]\ncase tfae_1_iff_2\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\n[PROOFSTEP]\napply isEquiv_iff_val_le_one\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\ntfae_1_iff_2 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\n\u22a2 List.TFAE\n [IsEquiv v v', \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1, \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1, \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1,\n \u2200 {x : K}, \u2191v (x - 1) < 1 \u2194 \u2191v' (x - 1) < 1]\n[PROOFSTEP]\ntfae_have 1 \u2194 3\n[GOAL]\ncase tfae_1_iff_3\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\ntfae_1_iff_2 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\n\u22a2 IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\n[PROOFSTEP]\napply isEquiv_iff_val_eq_one\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\ntfae_1_iff_2 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\ntfae_1_iff_3 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\n\u22a2 List.TFAE\n [IsEquiv v v', \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1, \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1, \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1,\n \u2200 {x : K}, \u2191v (x - 1) < 1 \u2194 \u2191v' (x - 1) < 1]\n[PROOFSTEP]\ntfae_have 1 \u2194 4\n[GOAL]\ncase tfae_1_iff_4\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\ntfae_1_iff_2 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\ntfae_1_iff_3 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\n\u22a2 IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\n[PROOFSTEP]\napply isEquiv_iff_val_lt_one\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\ntfae_1_iff_2 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\ntfae_1_iff_3 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\ntfae_1_iff_4 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\n\u22a2 List.TFAE\n [IsEquiv v v', \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1, \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1, \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1,\n \u2200 {x : K}, \u2191v (x - 1) < 1 \u2194 \u2191v' (x - 1) < 1]\n[PROOFSTEP]\ntfae_have 1 \u2194 5\n[GOAL]\ncase tfae_1_iff_5\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\ntfae_1_iff_2 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\ntfae_1_iff_3 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\ntfae_1_iff_4 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\n\u22a2 IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v (x - 1) < 1 \u2194 \u2191v' (x - 1) < 1\n[PROOFSTEP]\napply isEquiv_iff_val_sub_one_lt_one\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\ntfae_1_iff_2 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1\ntfae_1_iff_3 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1\ntfae_1_iff_4 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\ntfae_1_iff_5 : IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v (x - 1) < 1 \u2194 \u2191v' (x - 1) < 1\n\u22a2 List.TFAE\n [IsEquiv v v', \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191v' x \u2264 1, \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1, \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1,\n \u2200 {x : K}, \u2191v (x - 1) < 1 \u2194 \u2191v' (x - 1) < 1]\n[PROOFSTEP]\ntfae_finish\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2076 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u2075 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\ninst\u271d\u00b9 : Nontrivial \u0393\u2080\ninst\u271d : NoZeroDivisors \u0393\u2080\nh : supp v = \u22a4\n\u22a2 \u2191v 1 = 0\n[PROOFSTEP]\nrw [\u2190 mem_supp_iff, h]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2076 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u2075 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\ninst\u271d\u00b9 : Nontrivial \u0393\u2080\ninst\u271d : NoZeroDivisors \u0393\u2080\nh : supp v = \u22a4\n\u22a2 1 \u2208 \u22a4\n[PROOFSTEP]\nexact Submodule.mem_top\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2076 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u2075 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\ninst\u271d\u00b9 : Nontrivial \u0393\u2080\ninst\u271d : NoZeroDivisors \u0393\u2080\nx y : R\nhxy : x * y \u2208 supp v\n\u22a2 x \u2208 supp v \u2228 y \u2208 supp v\n[PROOFSTEP]\nsimp only [mem_supp_iff] at hxy \u22a2\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2076 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u2075 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\ninst\u271d\u00b9 : Nontrivial \u0393\u2080\ninst\u271d : NoZeroDivisors \u0393\u2080\nx y : R\nhxy : \u2191v (x * y) = 0\n\u22a2 \u2191v x = 0 \u2228 \u2191v y = 0\n[PROOFSTEP]\nrw [v.map_mul x y] at hxy \n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2076 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u2075 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\ninst\u271d\u00b9 : Nontrivial \u0393\u2080\ninst\u271d : NoZeroDivisors \u0393\u2080\nx y : R\nhxy : \u2191v x * \u2191v y = 0\n\u22a2 \u2191v x = 0 \u2228 \u2191v y = 0\n[PROOFSTEP]\nexact eq_zero_or_eq_zero_of_mul_eq_zero hxy\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\na s : R\nh : s \u2208 supp v\n\u22a2 \u2191v (a + s) = \u2191v a\n[PROOFSTEP]\nhave aux : \u2200 a s, v s = 0 \u2192 v (a + s) \u2264 v a := by\n intro a' s' h'\n refine' le_trans (v.map_add a' s') (max_le le_rfl _)\n simp [h']\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\na s : R\nh : s \u2208 supp v\n\u22a2 \u2200 (a s : R), \u2191v s = 0 \u2192 \u2191v (a + s) \u2264 \u2191v a\n[PROOFSTEP]\nintro a' s' h'\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\na s : R\nh : s \u2208 supp v\na' s' : R\nh' : \u2191v s' = 0\n\u22a2 \u2191v (a' + s') \u2264 \u2191v a'\n[PROOFSTEP]\nrefine' le_trans (v.map_add a' s') (max_le le_rfl _)\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\na s : R\nh : s \u2208 supp v\na' s' : R\nh' : \u2191v s' = 0\n\u22a2 \u2191v s' \u2264 \u2191v a'\n[PROOFSTEP]\nsimp [h']\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\na s : R\nh : s \u2208 supp v\naux : \u2200 (a s : R), \u2191v s = 0 \u2192 \u2191v (a + s) \u2264 \u2191v a\n\u22a2 \u2191v (a + s) = \u2191v a\n[PROOFSTEP]\napply le_antisymm (aux a s h)\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\na s : R\nh : s \u2208 supp v\naux : \u2200 (a s : R), \u2191v s = 0 \u2192 \u2191v (a + s) \u2264 \u2191v a\n\u22a2 \u2191v a \u2264 \u2191v (a + s)\n[PROOFSTEP]\ncalc\n v a = v (a + s + -s) := by simp\n _ \u2264 v (a + s) := aux (a + s) (-s) (by rwa [\u2190 Ideal.neg_mem_iff] at h )\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\na s : R\nh : s \u2208 supp v\naux : \u2200 (a s : R), \u2191v s = 0 \u2192 \u2191v (a + s) \u2264 \u2191v a\n\u22a2 \u2191v a = \u2191v (a + s + -s)\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2074 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b3 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\na s : R\nh : s \u2208 supp v\naux : \u2200 (a s : R), \u2191v s = 0 \u2192 \u2191v (a + s) \u2264 \u2191v a\n\u22a2 \u2191v (-s) = 0\n[PROOFSTEP]\nrwa [\u2190 Ideal.neg_mem_iff] at h \n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u2075 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u2074 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393\u2080\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero \u0393'\u2080\nv : Valuation R \u0393\u2080\nS : Type u_7\ninst\u271d : CommRing S\nf : S \u2192+* R\nx : S\n\u22a2 x \u2208 supp (comap f v) \u2194 x \u2208 Ideal.comap f (supp v)\n[PROOFSTEP]\nrw [mem_supp_iff, Ideal.mem_comap, mem_supp_iff, comap_apply]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR\u271d : Type u_3\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080\u271d : Type u_4\n\u0393'\u2080 : Type u_5\nR : Type ?u.4327439\n\u0393\u2080 : Type ?u.4327442\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedAddCommMonoidWithTop \u0393\u2080\nf g : AddValuation R \u0393\u2080\n\u22a2 (fun v => v.toFun) f = (fun v => v.toFun) g \u2192 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nK : Type u_1\nF : Type u_2\nR\u271d : Type u_3\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080\u271d : Type u_4\n\u0393'\u2080 : Type u_5\nR : Type ?u.4327439\n\u0393\u2080 : Type ?u.4327442\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedAddCommMonoidWithTop \u0393\u2080\ng : AddValuation R \u0393\u2080\ntoMonoidWithZeroHom\u271d : R \u2192*\u2080 Multiplicative \u0393\u2080\u1d52\u1d48\nmap_add_le_max'\u271d :\n \u2200 (x y : R),\n ZeroHom.toFun (\u2191toMonoidWithZeroHom\u271d) (x + y) \u2264\n max (ZeroHom.toFun (\u2191toMonoidWithZeroHom\u271d) x) (ZeroHom.toFun (\u2191toMonoidWithZeroHom\u271d) y)\n\u22a2 (fun v => v.toFun) { toMonoidWithZeroHom := toMonoidWithZeroHom\u271d, map_add_le_max' := map_add_le_max'\u271d } =\n (fun v => v.toFun) g \u2192\n { toMonoidWithZeroHom := toMonoidWithZeroHom\u271d, map_add_le_max' := map_add_le_max'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nK : Type u_1\nF : Type u_2\nR\u271d : Type u_3\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080\u271d : Type u_4\n\u0393'\u2080 : Type u_5\nR : Type ?u.4327439\n\u0393\u2080 : Type ?u.4327442\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedAddCommMonoidWithTop \u0393\u2080\ntoMonoidWithZeroHom\u271d\u00b9 : R \u2192*\u2080 Multiplicative \u0393\u2080\u1d52\u1d48\nmap_add_le_max'\u271d\u00b9 :\n \u2200 (x y : R),\n ZeroHom.toFun (\u2191toMonoidWithZeroHom\u271d\u00b9) (x + y) \u2264\n max (ZeroHom.toFun (\u2191toMonoidWithZeroHom\u271d\u00b9) x) (ZeroHom.toFun (\u2191toMonoidWithZeroHom\u271d\u00b9) y)\ntoMonoidWithZeroHom\u271d : R \u2192*\u2080 Multiplicative \u0393\u2080\u1d52\u1d48\nmap_add_le_max'\u271d :\n \u2200 (x y : R),\n ZeroHom.toFun (\u2191toMonoidWithZeroHom\u271d) (x + y) \u2264\n max (ZeroHom.toFun (\u2191toMonoidWithZeroHom\u271d) x) (ZeroHom.toFun (\u2191toMonoidWithZeroHom\u271d) y)\n\u22a2 (fun v => v.toFun) { toMonoidWithZeroHom := toMonoidWithZeroHom\u271d\u00b9, map_add_le_max' := map_add_le_max'\u271d\u00b9 } =\n (fun v => v.toFun) { toMonoidWithZeroHom := toMonoidWithZeroHom\u271d, map_add_le_max' := map_add_le_max'\u271d } \u2192\n { toMonoidWithZeroHom := toMonoidWithZeroHom\u271d\u00b9, map_add_le_max' := map_add_le_max'\u271d\u00b9 } =\n { toMonoidWithZeroHom := toMonoidWithZeroHom\u271d, map_add_le_max' := map_add_le_max'\u271d }\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedAddCommMonoidWithTop \u0393\u2080\ninst\u271d : LinearOrderedAddCommMonoidWithTop \u0393'\u2080\nv : AddValuation R \u0393\u2080\nx y z : R\n\u22a2 \u2200 (x y : R), \u2191v x \u2264 \u2191v (x + y) \u2228 \u2191v y \u2264 \u2191v (x + y)\n[PROOFSTEP]\nintro x y\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedAddCommMonoidWithTop \u0393\u2080\ninst\u271d : LinearOrderedAddCommMonoidWithTop \u0393'\u2080\nv : AddValuation R \u0393\u2080\nx\u271d y\u271d z x y : R\n\u22a2 \u2191v x \u2264 \u2191v (x + y) \u2228 \u2191v y \u2264 \u2191v (x + y)\n[PROOFSTEP]\nrw [\u2190 @min_le_iff _ _ (v x) (v y) (v (x + y)), \u2190 ge_iff_le]\n[GOAL]\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedAddCommMonoidWithTop \u0393\u2080\ninst\u271d : LinearOrderedAddCommMonoidWithTop \u0393'\u2080\nv : AddValuation R \u0393\u2080\nx\u271d y\u271d z x y : R\n\u22a2 \u2191v (x + y) \u2265 min (\u2191v x) (\u2191v y)\n[PROOFSTEP]\napply map_add\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Valuation.Basic", "llama_tokens": 40810, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8289388167733099, "lm_q2_score": 0.5312093733737562, "lm_q1q2_score": 0.44034006942333287}} {"text": "[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9 : HasImage f\ng : B \u27f6 C\ninst\u271d : HasKernel g\nw : f \u226b g = 0\n\u22a2 kernel.lift g f w \u226b kernel.\u03b9 g = f\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9 : HasImage f\ng : B \u27f6 C\ninst\u271d : HasKernel g\nw : f \u226b g = 0\n\u22a2 Mono (imageToKernel f g w)\n[PROOFSTEP]\ndsimp only [imageToKernel]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9 : HasImage f\ng : B \u27f6 C\ninst\u271d : HasKernel g\nw : f \u226b g = 0\n\u22a2 Mono (Subobject.ofLE (imageSubobject f) (kernelSubobject g) (_ : imageSubobject f \u2264 kernelSubobject g))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9 : HasImage f\ng : B \u27f6 C\ninst\u271d : HasKernel g\nw : f \u226b g = 0\n\u22a2 imageToKernel f g w \u226b Subobject.arrow (kernelSubobject g) = Subobject.arrow (imageSubobject f)\n[PROOFSTEP]\nsimp [imageToKernel]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b2 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9 : HasKernel g\ninst\u271d : ConcreteCategory V\nw : f \u226b g = 0\nx : (forget V).obj (Subobject.underlying.obj (imageSubobject f))\n\u22a2 \u2191(Subobject.arrow (kernelSubobject g)) (\u2191(imageToKernel f g w) x) = \u2191(Subobject.arrow (imageSubobject f)) x\n[PROOFSTEP]\nrw [\u2190 comp_apply, imageToKernel_arrow]\n -- This is less useful as a `simp` lemma than it initially appears,\n -- as it \"loses\" the information the morphism factors through the image.\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9 : HasImage f\ng : B \u27f6 C\ninst\u271d : HasKernel g\nw : f \u226b g = 0\n\u22a2 factorThruImageSubobject f \u226b imageToKernel f g w = factorThruKernelSubobject g f w\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9 : HasImage f\ng : B \u27f6 C\ninst\u271d : HasKernel g\nw : f \u226b g = 0\n\u22a2 (factorThruImageSubobject f \u226b imageToKernel f g w) \u226b Subobject.arrow (kernelSubobject g) =\n factorThruKernelSubobject g f w \u226b Subobject.arrow (kernelSubobject g)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasZeroObject V\nw : 0 \u226b g = 0\n\u22a2 imageToKernel 0 g w = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasZeroObject V\nw : 0 \u226b g = 0\n\u22a2 imageToKernel 0 g w \u226b Subobject.arrow (kernelSubobject g) = 0 \u226b Subobject.arrow (kernelSubobject g)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b2 : Category.{v, u} V\ninst\u271d\u00b9 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d : HasImages V\nw : f \u226b 0 = 0\n\u22a2 imageToKernel f 0 w = Subobject.arrow (imageSubobject f) \u226b inv (Subobject.arrow (kernelSubobject 0))\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b2 : Category.{v, u} V\ninst\u271d\u00b9 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d : HasImages V\nw : f \u226b 0 = 0\n\u22a2 imageToKernel f 0 w \u226b Subobject.arrow (kernelSubobject 0) =\n (Subobject.arrow (imageSubobject f) \u226b inv (Subobject.arrow (kernelSubobject 0))) \u226b\n Subobject.arrow (kernelSubobject 0)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasImages V\nD : V\nh : C \u27f6 D\nw : f \u226b g = 0\n\u22a2 f \u226b g \u226b h = 0\n[PROOFSTEP]\nsimp [reassoc_of% w]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasImages V\nD : V\nh : C \u27f6 D\nw : f \u226b g = 0\n\u22a2 imageToKernel f (g \u226b h) (_ : f \u226b g \u226b h = 0) =\n imageToKernel f g w \u226b\n Subobject.ofLE (kernelSubobject g) (kernelSubobject (g \u226b h)) (_ : kernelSubobject g \u2264 kernelSubobject (g \u226b h))\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasImages V\nD : V\nh : C \u27f6 D\nw : f \u226b g = 0\n\u22a2 imageToKernel f (g \u226b h) (_ : f \u226b g \u226b h = 0) \u226b Subobject.arrow (kernelSubobject (g \u226b h)) =\n (imageToKernel f g w \u226b\n Subobject.ofLE (kernelSubobject g) (kernelSubobject (g \u226b h))\n (_ : kernelSubobject g \u2264 kernelSubobject (g \u226b h))) \u226b\n Subobject.arrow (kernelSubobject (g \u226b h))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasImages V\nZ : V\nh : Z \u27f6 A\nw : f \u226b g = 0\n\u22a2 (h \u226b f) \u226b g = 0\n[PROOFSTEP]\nsimp [w]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasImages V\nZ : V\nh : Z \u27f6 A\nw : f \u226b g = 0\n\u22a2 imageToKernel (h \u226b f) g (_ : (h \u226b f) \u226b g = 0) =\n Subobject.ofLE (imageSubobject (h \u226b f)) (imageSubobject f) (_ : imageSubobject (h \u226b f) \u2264 imageSubobject f) \u226b\n imageToKernel f g w\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasImages V\nZ : V\nh : Z \u27f6 A\nw : f \u226b g = 0\n\u22a2 imageToKernel (h \u226b f) g (_ : (h \u226b f) \u226b g = 0) \u226b Subobject.arrow (kernelSubobject g) =\n (Subobject.ofLE (imageSubobject (h \u226b f)) (imageSubobject f) (_ : imageSubobject (h \u226b f) \u2264 imageSubobject f) \u226b\n imageToKernel f g w) \u226b\n Subobject.arrow (kernelSubobject g)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b2 : HasKernels V\ninst\u271d\u00b9 : HasImages V\nD : V\nh : C \u27f6 D\ninst\u271d : Mono h\nw : f \u226b g \u226b h = 0\n\u22a2 (f \u226b g) \u226b h = 0 \u226b h\n[PROOFSTEP]\nsimpa using w\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b2 : HasKernels V\ninst\u271d\u00b9 : HasImages V\nD : V\nh : C \u27f6 D\ninst\u271d : Mono h\nw : f \u226b g \u226b h = 0\n\u22a2 imageToKernel f (g \u226b h) w =\n imageToKernel f g (_ : f \u226b g = 0) \u226b\n (Subobject.isoOfEq (kernelSubobject (g \u226b h)) (kernelSubobject g)\n (_ : kernelSubobject (g \u226b h) = kernelSubobject g)).inv\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b2 : HasKernels V\ninst\u271d\u00b9 : HasImages V\nD : V\nh : C \u27f6 D\ninst\u271d : Mono h\nw : f \u226b g \u226b h = 0\n\u22a2 imageToKernel f (g \u226b h) w \u226b Subobject.arrow (kernelSubobject (g \u226b h)) =\n (imageToKernel f g (_ : f \u226b g = 0) \u226b\n (Subobject.isoOfEq (kernelSubobject (g \u226b h)) (kernelSubobject g)\n (_ : kernelSubobject (g \u226b h) = kernelSubobject g)).inv) \u226b\n Subobject.arrow (kernelSubobject (g \u226b h))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b2 : HasKernels V\ninst\u271d\u00b9 : HasImages V\nZ : V\nh : Z \u27f6 A\ninst\u271d : Epi h\nw : (h \u226b f) \u226b g = 0\n\u22a2 h \u226b f \u226b g = h \u226b 0\n[PROOFSTEP]\nsimpa using w\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b2 : HasKernels V\ninst\u271d\u00b9 : HasImages V\nZ : V\nh : Z \u27f6 A\ninst\u271d : Epi h\nw : (h \u226b f) \u226b g = 0\n\u22a2 imageToKernel (h \u226b f) g w =\n Subobject.ofLE (imageSubobject (h \u226b f)) (imageSubobject f) (_ : imageSubobject (h \u226b f) \u2264 imageSubobject f) \u226b\n imageToKernel f g (_ : f \u226b g = 0)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b2 : HasKernels V\ninst\u271d\u00b9 : HasImages V\nZ : V\nh : Z \u27f6 A\ninst\u271d : Epi h\nw : (h \u226b f) \u226b g = 0\n\u22a2 imageToKernel (h \u226b f) g w \u226b Subobject.arrow (kernelSubobject g) =\n (Subobject.ofLE (imageSubobject (h \u226b f)) (imageSubobject f) (_ : imageSubobject (h \u226b f) \u2264 imageSubobject f) \u226b\n imageToKernel f g (_ : f \u226b g = 0)) \u226b\n Subobject.arrow (kernelSubobject g)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasEqualizers V\ninst\u271d : HasImages V\nZ : V\ni : B \u2245 Z\nw : (f \u226b i.hom) \u226b i.inv \u226b g = 0\n\u22a2 f \u226b g = 0\n[PROOFSTEP]\nsimpa using w\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasEqualizers V\ninst\u271d : HasImages V\nZ : V\ni : B \u2245 Z\nw : (f \u226b i.hom) \u226b i.inv \u226b g = 0\n\u22a2 imageToKernel (f \u226b i.hom) (i.inv \u226b g) w =\n (imageSubobjectCompIso f i.hom).hom \u226b imageToKernel f g (_ : f \u226b g = 0) \u226b (kernelSubobjectIsoComp i.inv g).inv\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasEqualizers V\ninst\u271d : HasImages V\nZ : V\ni : B \u2245 Z\nw : (f \u226b i.hom) \u226b i.inv \u226b g = 0\n\u22a2 imageToKernel (f \u226b i.hom) (i.inv \u226b g) w \u226b Subobject.arrow (kernelSubobject (i.inv \u226b g)) =\n ((imageSubobjectCompIso f i.hom).hom \u226b imageToKernel f g (_ : f \u226b g = 0) \u226b (kernelSubobjectIsoComp i.inv g).inv) \u226b\n Subobject.arrow (kernelSubobject (i.inv \u226b g))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b2 : HasKernels V\ninst\u271d\u00b9 : HasZeroObject V\ninst\u271d : Mono g\n\u22a2 0 \u226b g = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasImages V\ninst\u271d : Epi f\n\u22a2 f \u226b 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasImages V\ninst\u271d : Epi f\n\u22a2 Epi (imageToKernel f 0 (_ : f \u226b 0 = 0))\n[PROOFSTEP]\nsimp only [imageToKernel_zero_right]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasImages V\ninst\u271d : Epi f\n\u22a2 Epi (Subobject.arrow (imageSubobject f) \u226b inv (Subobject.arrow (kernelSubobject 0)))\n[PROOFSTEP]\nhaveI := epi_image_of_epi f\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasImages V\ninst\u271d : Epi f\nthis : Epi (image.\u03b9 f)\n\u22a2 Epi (Subobject.arrow (imageSubobject f) \u226b inv (Subobject.arrow (kernelSubobject 0)))\n[PROOFSTEP]\nrw [\u2190 imageSubobject_arrow]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\ninst\u271d\u00b9 : HasImages V\ninst\u271d : Epi f\nthis : Epi (image.\u03b9 f)\n\u22a2 Epi (((imageSubobjectIso f).hom \u226b image.\u03b9 f) \u226b inv (Subobject.arrow (kernelSubobject 0)))\n[PROOFSTEP]\nrefine' @epi_comp _ _ _ _ _ _ (epi_comp _ _) _ _\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b2 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9 : HasKernel g\nw : f \u226b g = 0\ninst\u271d : HasCokernel (imageToKernel f g w)\nD : V\nk : Subobject.underlying.obj (kernelSubobject g) \u27f6 D\np : imageToKernel f g w \u226b k = 0\n\u22a2 \u03c0 f g w \u226b desc f g w k p = k\n[PROOFSTEP]\nsimp [homology.\u03c0, homology.desc]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2078 : Category.{v, u} V\ninst\u271d\u2077 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u2076 : HasImage f\ng : B \u27f6 C\ninst\u271d\u2075 : HasKernel g\nw : f \u226b g = 0\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f 0 (_ : f \u226b 0 = 0))\ninst\u271d\u00b2 : HasCokernel f\ninst\u271d\u00b9 : HasCokernel (image.\u03b9 f)\ninst\u271d : Epi (factorThruImage f)\n\u22a2 imageToKernel f 0 (_ : f \u226b 0 = 0) \u226b (kernelSubobjectIso 0 \u226a\u226b kernelZeroIsoSource).hom =\n (imageSubobjectIso f).hom \u226b image.\u03b9 f\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2077 : Category.{v, u} V\ninst\u271d\u2076 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u2075 : HasImage f\ng : B \u27f6 C\ninst\u271d\u2074 : HasKernel g\nw : f \u226b g = 0\ninst\u271d\u00b3 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b2 : HasZeroObject V\ninst\u271d\u00b9 : HasImage 0\ninst\u271d : HasCokernel (imageToKernel 0 0 (_ : 0 \u226b 0 = 0))\n\u22a2 imageToKernel 0 0 (_ : 0 \u226b 0 = 0) \u226b Subobject.arrow (kernelSubobject 0) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2074 : Category.{v, u} V\ninst\u271d\u00b9\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u00b2 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u00b9 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2070 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u2079 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u2078 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u2077 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u2076 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u2075 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2074 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u00b3 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u00b2 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u00b9 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\np : \u03b1.right = \u03b2.left\n\u22a2 imageToKernel f g w \u226b kernelSubobjectMap \u03b2 = imageSubobjectMap \u03b1 \u226b imageToKernel f' g' w'\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2074 : Category.{v, u} V\ninst\u271d\u00b9\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u00b2 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u00b9 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2070 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u2079 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u2078 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u2077 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u2076 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u2075 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2074 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u00b3 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u00b2 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u00b9 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\np : \u03b1.right = \u03b2.left\n\u22a2 (imageToKernel f g w \u226b kernelSubobjectMap \u03b2) \u226b Subobject.arrow (kernelSubobject g') =\n (imageSubobjectMap \u03b1 \u226b imageToKernel f' g' w') \u226b Subobject.arrow (kernelSubobject g')\n[PROOFSTEP]\nsimp [p]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\np : \u03b1.right = \u03b2.left\n\u22a2 imageToKernel f g w \u226b kernelSubobjectMap \u03b2 \u226b cokernel.\u03c0 (imageToKernel f' g' w') = 0\n[PROOFSTEP]\nrw [imageSubobjectMap_comp_imageToKernel_assoc w w' \u03b1 \u03b2 p]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\np : \u03b1.right = \u03b2.left\n\u22a2 imageSubobjectMap \u03b1 \u226b imageToKernel f' g' w' \u226b cokernel.\u03c0 (imageToKernel f' g' w') = 0\n[PROOFSTEP]\nsimp only [cokernel.condition, comp_zero]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\np : \u03b1.right = \u03b2.left\n\u22a2 \u03c0 f g w \u226b map w w' \u03b1 \u03b2 p = kernelSubobjectMap \u03b2 \u226b \u03c0 f' g' w'\n[PROOFSTEP]\nsimp only [homology.\u03c0, homology.map, cokernel.\u03c0_desc]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b2\u2070 : Category.{v, u} V\ninst\u271d\u00b9\u2079 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2078 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2077 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2076 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2075 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u2074 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b3 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b2 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u00b9 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u00b9\u2070 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2079 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2078 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2077 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2076 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2075 : HasCokernel (imageToKernel f g w)\ninst\u271d\u2074 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b3 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\ninst\u271d : ConcreteCategory V\np : \u03b1.right = \u03b2.left\nx : (forget V).obj (Subobject.underlying.obj (kernelSubobject g))\n\u22a2 \u2191(map w w' \u03b1 \u03b2 p) (\u2191(\u03c0 f g w) x) = \u2191(\u03c0 f' g' w') (\u2191(kernelSubobjectMap \u03b2) x)\n[PROOFSTEP]\nsimp only [\u2190 comp_apply, homology.\u03c0_map w w' \u03b1 \u03b2 p]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\np : \u03b1.right = \u03b2.left\nD : V\nk : Subobject.underlying.obj (kernelSubobject g') \u27f6 D\nz : imageToKernel f' g' w' \u226b k = 0\n\u22a2 imageToKernel f g w \u226b kernelSubobjectMap \u03b2 \u226b k = 0\n[PROOFSTEP]\nsimp only [imageSubobjectMap_comp_imageToKernel_assoc w w' \u03b1 \u03b2 p, z, comp_zero]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\np : \u03b1.right = \u03b2.left\nD : V\nk : Subobject.underlying.obj (kernelSubobject g') \u27f6 D\nz : imageToKernel f' g' w' \u226b k = 0\n\u22a2 map w w' \u03b1 \u03b2 p \u226b desc f' g' w' k z =\n desc f g w (kernelSubobjectMap \u03b2 \u226b k) (_ : imageToKernel f g w \u226b kernelSubobjectMap \u03b2 \u226b k = 0)\n[PROOFSTEP]\next\n[GOAL]\ncase p\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\np : \u03b1.right = \u03b2.left\nD : V\nk : Subobject.underlying.obj (kernelSubobject g') \u27f6 D\nz : imageToKernel f' g' w' \u226b k = 0\n\u22a2 \u03c0 f g w \u226b map w w' \u03b1 \u03b2 p \u226b desc f' g' w' k z =\n \u03c0 f g w \u226b desc f g w (kernelSubobjectMap \u03b2 \u226b k) (_ : imageToKernel f g w \u226b kernelSubobjectMap \u03b2 \u226b k = 0)\n[PROOFSTEP]\nsimp only [homology.\u03c0_desc, homology.\u03c0_map_assoc]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\n\u22a2 map w w (\ud835\udfd9 (Arrow.mk f)) (\ud835\udfd9 (Arrow.mk g)) (_ : (\ud835\udfd9 (Arrow.mk f)).right = (\ud835\udfd9 (Arrow.mk f)).right) = \ud835\udfd9 (homology f g w)\n[PROOFSTEP]\next\n[GOAL]\ncase p\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\n\u22a2 \u03c0 f g w \u226b map w w (\ud835\udfd9 (Arrow.mk f)) (\ud835\udfd9 (Arrow.mk g)) (_ : (\ud835\udfd9 (Arrow.mk f)).right = (\ud835\udfd9 (Arrow.mk f)).right) =\n \u03c0 f g w \u226b \ud835\udfd9 (homology f g w)\n[PROOFSTEP]\nsimp only [homology.\u03c0_map, kernelSubobjectMap_id, Category.id_comp, Category.comp_id]\n[GOAL]\n\u03b9 : Type u_1\nV\u271d : Type u\ninst\u271d\u00b2\u2070 : Category.{v, u} V\u271d\ninst\u271d\u00b9\u2079 : HasZeroMorphisms V\u271d\nA B C : V\u271d\nf : A \u27f6 B\ninst\u271d\u00b9\u2078 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2077 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\u271d\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2076 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2075 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u2074 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081\u271d B\u2081\u271d C\u2081\u271d : V\u271d\nf\u2081\u271d : A\u2081\u271d \u27f6 B\u2081\u271d\ninst\u271d\u00b9\u00b3 : HasImage f\u2081\u271d\ng\u2081\u271d : B\u2081\u271d \u27f6 C\u2081\u271d\ninst\u271d\u00b9\u00b2 : HasKernel g\u2081\u271d\nw\u2081 : f\u2081\u271d \u226b g\u2081\u271d = 0\nA\u2082\u271d B\u2082\u271d C\u2082\u271d : V\u271d\nf\u2082\u271d : A\u2082\u271d \u27f6 B\u2082\u271d\ninst\u271d\u00b9\u00b9 : HasImage f\u2082\u271d\ng\u2082\u271d : B\u2082\u271d \u27f6 C\u2082\u271d\ninst\u271d\u00b9\u2070 : HasKernel g\u2082\u271d\nw\u2082 : f\u2082\u271d \u226b g\u2082\u271d = 0\nA\u2083\u271d B\u2083\u271d C\u2083\u271d : V\u271d\nf\u2083\u271d : A\u2083\u271d \u27f6 B\u2083\u271d\ninst\u271d\u2079 : HasImage f\u2083\u271d\ng\u2083\u271d : B\u2083\u271d \u27f6 C\u2083\u271d\ninst\u271d\u2078 : HasKernel g\u2083\u271d\nw\u2083 : f\u2083\u271d \u226b g\u2083\u271d = 0\n\u03b1\u2081\u271d : Arrow.mk f\u2081\u271d \u27f6 Arrow.mk f\u2082\u271d\ninst\u271d\u2077 : HasImageMap \u03b1\u2081\u271d\n\u03b2\u2081\u271d : Arrow.mk g\u2081\u271d \u27f6 Arrow.mk g\u2082\u271d\n\u03b1\u2082\u271d : Arrow.mk f\u2082\u271d \u27f6 Arrow.mk f\u2083\u271d\ninst\u271d\u2076 : HasImageMap \u03b1\u2082\u271d\n\u03b2\u2082\u271d : Arrow.mk g\u2082\u271d \u27f6 Arrow.mk g\u2083\u271d\ninst\u271d\u2075 : HasCokernel (imageToKernel f g w)\ninst\u271d\u2074 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b3 : HasCokernel (imageToKernel f\u2081\u271d g\u2081\u271d w\u2081)\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2082\u271d g\u2082\u271d w\u2082)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2083\u271d g\u2083\u271d w\u2083)\nV : Type u_2\ninst\u271d : Category.{u_3, u_2} V\nA\u2081 B\u2081 C\u2081 A\u2082 B\u2082 C\u2082 A\u2083 B\u2083 C\u2083 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\nf\u2082 : A\u2082 \u27f6 B\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\nf\u2083 : A\u2083 \u27f6 B\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\np\u2081 : \u03b1\u2081.right = \u03b2\u2081.left\np\u2082 : \u03b1\u2082.right = \u03b2\u2082.left\n\u22a2 (\u03b1\u2081 \u226b \u03b1\u2082).right = (\u03b2\u2081 \u226b \u03b2\u2082).left\n[PROOFSTEP]\nsimp only [Arrow.comp_left, Arrow.comp_right, p\u2081, p\u2082]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\np\u2081 : \u03b1\u2081.right = \u03b2\u2081.left\np\u2082 : \u03b1\u2082.right = \u03b2\u2082.left\n\u22a2 map w\u2081 w\u2082 \u03b1\u2081 \u03b2\u2081 p\u2081 \u226b map w\u2082 w\u2083 \u03b1\u2082 \u03b2\u2082 p\u2082 = map w\u2081 w\u2083 (\u03b1\u2081 \u226b \u03b1\u2082) (\u03b2\u2081 \u226b \u03b2\u2082) (_ : (\u03b1\u2081 \u226b \u03b1\u2082).right = (\u03b2\u2081 \u226b \u03b2\u2082).left)\n[PROOFSTEP]\next\n[GOAL]\ncase p\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\np\u2081 : \u03b1\u2081.right = \u03b2\u2081.left\np\u2082 : \u03b1\u2082.right = \u03b2\u2082.left\n\u22a2 \u03c0 f\u2081 g\u2081 w\u2081 \u226b map w\u2081 w\u2082 \u03b1\u2081 \u03b2\u2081 p\u2081 \u226b map w\u2082 w\u2083 \u03b1\u2082 \u03b2\u2082 p\u2082 =\n \u03c0 f\u2081 g\u2081 w\u2081 \u226b map w\u2081 w\u2083 (\u03b1\u2081 \u226b \u03b1\u2082) (\u03b2\u2081 \u226b \u03b2\u2082) (_ : (\u03b1\u2081 \u226b \u03b1\u2082).right = (\u03b2\u2081 \u226b \u03b2\u2082).left)\n[PROOFSTEP]\nsimp only [kernelSubobjectMap_comp, homology.\u03c0_map_assoc, homology.\u03c0_map, Category.assoc]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1\u271d : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\u271d\n\u03b2\u271d : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\n\u03b1 : Arrow.mk f\u2081 \u2245 Arrow.mk f\u2082\n\u03b2 : Arrow.mk g\u2081 \u2245 Arrow.mk g\u2082\np : \u03b1.hom.right = \u03b2.hom.left\n\u22a2 \u03b1.inv.right = \u03b2.inv.left\n[PROOFSTEP]\nrw [\u2190 cancel_mono \u03b1.hom.right, \u2190 Comma.comp_right, \u03b1.inv_hom_id, Comma.id_right, p, \u2190 Comma.comp_left, \u03b2.inv_hom_id,\n Comma.id_left]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1\u271d : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\u271d\n\u03b2\u271d : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\n\u03b1 : Arrow.mk f\u2081 \u2245 Arrow.mk f\u2082\n\u03b2 : Arrow.mk g\u2081 \u2245 Arrow.mk g\u2082\np : \u03b1.hom.right = \u03b2.hom.left\n\u22a2 \ud835\udfd9 (Arrow.mk f\u2082).right = \ud835\udfd9 (Arrow.mk g\u2082).left\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1\u271d : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\u271d\n\u03b2\u271d : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\n\u03b1 : Arrow.mk f\u2081 \u2245 Arrow.mk f\u2082\n\u03b2 : Arrow.mk g\u2081 \u2245 Arrow.mk g\u2082\np : \u03b1.hom.right = \u03b2.hom.left\n\u22a2 map w\u2081 w\u2082 \u03b1.hom \u03b2.hom p \u226b map w\u2082 w\u2081 \u03b1.inv \u03b2.inv (_ : \u03b1.inv.right = \u03b2.inv.left) = \ud835\udfd9 (homology f\u2081 g\u2081 w\u2081)\n[PROOFSTEP]\nrw [homology.map_comp, \u2190 homology.map_id]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1\u271d : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\u271d\n\u03b2\u271d : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\n\u03b1 : Arrow.mk f\u2081 \u2245 Arrow.mk f\u2082\n\u03b2 : Arrow.mk g\u2081 \u2245 Arrow.mk g\u2082\np : \u03b1.hom.right = \u03b2.hom.left\n\u22a2 map w\u2081 w\u2081 (\u03b1.hom \u226b \u03b1.inv) (\u03b2.hom \u226b \u03b2.inv) (_ : (\u03b1.hom \u226b \u03b1.inv).right = (\u03b2.hom \u226b \u03b2.inv).left) =\n map w\u2081 w\u2081 (\ud835\udfd9 (Arrow.mk f\u2081)) (\ud835\udfd9 (Arrow.mk g\u2081)) (_ : (\ud835\udfd9 (Arrow.mk f\u2081)).right = (\ud835\udfd9 (Arrow.mk f\u2081)).right)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_\u03b1\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1\u271d : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\u271d\n\u03b2\u271d : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\n\u03b1 : Arrow.mk f\u2081 \u2245 Arrow.mk f\u2082\n\u03b2 : Arrow.mk g\u2081 \u2245 Arrow.mk g\u2082\np : \u03b1.hom.right = \u03b2.hom.left\n\u22a2 \u03b1.hom \u226b \u03b1.inv = \ud835\udfd9 (Arrow.mk f\u2081)\n[PROOFSTEP]\nsimp only [Iso.hom_inv_id]\n[GOAL]\ncase e_\u03b2\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1\u271d : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\u271d\n\u03b2\u271d : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\n\u03b1 : Arrow.mk f\u2081 \u2245 Arrow.mk f\u2082\n\u03b2 : Arrow.mk g\u2081 \u2245 Arrow.mk g\u2082\np : \u03b1.hom.right = \u03b2.hom.left\n\u22a2 \u03b2.hom \u226b \u03b2.inv = \ud835\udfd9 (Arrow.mk g\u2081)\n[PROOFSTEP]\nsimp only [Iso.hom_inv_id]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1\u271d : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\u271d\n\u03b2\u271d : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\n\u03b1 : Arrow.mk f\u2081 \u2245 Arrow.mk f\u2082\n\u03b2 : Arrow.mk g\u2081 \u2245 Arrow.mk g\u2082\np : \u03b1.hom.right = \u03b2.hom.left\n\u22a2 map w\u2082 w\u2081 \u03b1.inv \u03b2.inv (_ : \u03b1.inv.right = \u03b2.inv.left) \u226b map w\u2081 w\u2082 \u03b1.hom \u03b2.hom p = \ud835\udfd9 (homology f\u2082 g\u2082 w\u2082)\n[PROOFSTEP]\nrw [homology.map_comp, \u2190 homology.map_id]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1\u271d : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\u271d\n\u03b2\u271d : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\n\u03b1 : Arrow.mk f\u2081 \u2245 Arrow.mk f\u2082\n\u03b2 : Arrow.mk g\u2081 \u2245 Arrow.mk g\u2082\np : \u03b1.hom.right = \u03b2.hom.left\n\u22a2 map w\u2082 w\u2082 (\u03b1.inv \u226b \u03b1.hom) (\u03b2.inv \u226b \u03b2.hom) (_ : (\u03b1.inv \u226b \u03b1.hom).right = (\u03b2.inv \u226b \u03b2.hom).left) =\n map w\u2082 w\u2082 (\ud835\udfd9 (Arrow.mk f\u2082)) (\ud835\udfd9 (Arrow.mk g\u2082)) (_ : (\ud835\udfd9 (Arrow.mk f\u2082)).right = (\ud835\udfd9 (Arrow.mk f\u2082)).right)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_\u03b1\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1\u271d : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\u271d\n\u03b2\u271d : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\n\u03b1 : Arrow.mk f\u2081 \u2245 Arrow.mk f\u2082\n\u03b2 : Arrow.mk g\u2081 \u2245 Arrow.mk g\u2082\np : \u03b1.hom.right = \u03b2.hom.left\n\u22a2 \u03b1.inv \u226b \u03b1.hom = \ud835\udfd9 (Arrow.mk f\u2082)\n[PROOFSTEP]\nsimp only [Iso.inv_hom_id]\n[GOAL]\ncase e_\u03b2\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9\u2079 : Category.{v, u} V\ninst\u271d\u00b9\u2078 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ninst\u271d\u00b9\u2077 : HasImage f\ng : B \u27f6 C\ninst\u271d\u00b9\u2076 : HasKernel g\nw : f \u226b g = 0\nA' B' C' : V\nf' : A' \u27f6 B'\ninst\u271d\u00b9\u2075 : HasImage f'\ng' : B' \u27f6 C'\ninst\u271d\u00b9\u2074 : HasKernel g'\nw' : f' \u226b g' = 0\n\u03b1\u271d : Arrow.mk f \u27f6 Arrow.mk f'\ninst\u271d\u00b9\u00b3 : HasImageMap \u03b1\u271d\n\u03b2\u271d : Arrow.mk g \u27f6 Arrow.mk g'\nA\u2081 B\u2081 C\u2081 : V\nf\u2081 : A\u2081 \u27f6 B\u2081\ninst\u271d\u00b9\u00b2 : HasImage f\u2081\ng\u2081 : B\u2081 \u27f6 C\u2081\ninst\u271d\u00b9\u00b9 : HasKernel g\u2081\nw\u2081 : f\u2081 \u226b g\u2081 = 0\nA\u2082 B\u2082 C\u2082 : V\nf\u2082 : A\u2082 \u27f6 B\u2082\ninst\u271d\u00b9\u2070 : HasImage f\u2082\ng\u2082 : B\u2082 \u27f6 C\u2082\ninst\u271d\u2079 : HasKernel g\u2082\nw\u2082 : f\u2082 \u226b g\u2082 = 0\nA\u2083 B\u2083 C\u2083 : V\nf\u2083 : A\u2083 \u27f6 B\u2083\ninst\u271d\u2078 : HasImage f\u2083\ng\u2083 : B\u2083 \u27f6 C\u2083\ninst\u271d\u2077 : HasKernel g\u2083\nw\u2083 : f\u2083 \u226b g\u2083 = 0\n\u03b1\u2081 : Arrow.mk f\u2081 \u27f6 Arrow.mk f\u2082\ninst\u271d\u2076 : HasImageMap \u03b1\u2081\n\u03b2\u2081 : Arrow.mk g\u2081 \u27f6 Arrow.mk g\u2082\n\u03b1\u2082 : Arrow.mk f\u2082 \u27f6 Arrow.mk f\u2083\ninst\u271d\u2075 : HasImageMap \u03b1\u2082\n\u03b2\u2082 : Arrow.mk g\u2082 \u27f6 Arrow.mk g\u2083\ninst\u271d\u2074 : HasCokernel (imageToKernel f g w)\ninst\u271d\u00b3 : HasCokernel (imageToKernel f' g' w')\ninst\u271d\u00b2 : HasCokernel (imageToKernel f\u2081 g\u2081 w\u2081)\ninst\u271d\u00b9 : HasCokernel (imageToKernel f\u2082 g\u2082 w\u2082)\ninst\u271d : HasCokernel (imageToKernel f\u2083 g\u2083 w\u2083)\n\u03b1 : Arrow.mk f\u2081 \u2245 Arrow.mk f\u2082\n\u03b2 : Arrow.mk g\u2081 \u2245 Arrow.mk g\u2082\np : \u03b1.hom.right = \u03b2.hom.left\n\u22a2 \u03b2.inv \u226b \u03b2.hom = \ud835\udfd9 (Arrow.mk g\u2082)\n[PROOFSTEP]\nsimp only [Iso.inv_hom_id]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\nf' : A \u27f6 B\ng' : B \u27f6 C\nw' : f' \u226b g' = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\npf : f = f'\npg : g = g'\n\u22a2 (\ud835\udfed V).map (\ud835\udfd9 (Arrow.mk f).left) \u226b (Arrow.mk f').hom = (Arrow.mk f).hom \u226b (\ud835\udfed V).map (\ud835\udfd9 (Arrow.mk f).right)\n[PROOFSTEP]\naesop_cat\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\nf' : A \u27f6 B\ng' : B \u27f6 C\nw' : f' \u226b g' = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\npf : f = f'\npg : g = g'\n\u22a2 (\ud835\udfed V).map (\ud835\udfd9 (Arrow.mk g).left) \u226b (Arrow.mk g').hom = (Arrow.mk g).hom \u226b (\ud835\udfed V).map (\ud835\udfd9 (Arrow.mk g).right)\n[PROOFSTEP]\naesop_cat\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\nf' : A \u27f6 B\ng' : B \u27f6 C\nw' : f' \u226b g' = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\npf : f = f'\npg : g = g'\n\u22a2 (\ud835\udfed V).map (\ud835\udfd9 (Arrow.mk f').left) \u226b (Arrow.mk f).hom = (Arrow.mk f').hom \u226b (\ud835\udfed V).map (\ud835\udfd9 (Arrow.mk f').right)\n[PROOFSTEP]\naesop_cat\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\nf' : A \u27f6 B\ng' : B \u27f6 C\nw' : f' \u226b g' = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\npf : f = f'\npg : g = g'\n\u22a2 (\ud835\udfed V).map (\ud835\udfd9 (Arrow.mk g').left) \u226b (Arrow.mk g).hom = (Arrow.mk g').hom \u226b (\ud835\udfed V).map (\ud835\udfd9 (Arrow.mk g').right)\n[PROOFSTEP]\naesop_cat\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\nf' : A \u27f6 B\ng' : B \u27f6 C\nw' : f' \u226b g' = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\npf : f = f'\npg : g = g'\n\u22a2 map w w' (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right))\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right))\n (_ :\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right =\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right) \u226b\n map w' w (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f').left) (\ud835\udfd9 (Arrow.mk f').right))\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g').left) (\ud835\udfd9 (Arrow.mk g').right))\n (_ :\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f').left) (\ud835\udfd9 (Arrow.mk f').right)).right =\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f').left) (\ud835\udfd9 (Arrow.mk f').right)).right) =\n \ud835\udfd9 (homology f g w)\n[PROOFSTEP]\nobtain rfl := pf\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\ng' : B \u27f6 C\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\npg : g = g'\nw' : f \u226b g' = 0\n\u22a2 map w w' (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right))\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right))\n (_ :\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right =\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right) \u226b\n map w' w (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right))\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g').left) (\ud835\udfd9 (Arrow.mk g').right))\n (_ :\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right =\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right) =\n \ud835\udfd9 (homology f g w)\n[PROOFSTEP]\nobtain rfl := pg\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\nw' : f \u226b g = 0\n\u22a2 map w w' (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right))\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right))\n (_ :\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right =\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right) \u226b\n map w' w (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right))\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right))\n (_ :\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right =\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right) =\n \ud835\udfd9 (homology f g w)\n[PROOFSTEP]\nrw [homology.map_comp, \u2190 homology.map_id]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\nw' : f \u226b g = 0\n\u22a2 map w w\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right) \u226b\n CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right))\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right) \u226b\n CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right))\n (_ :\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right) \u226b\n CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right =\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right) \u226b\n CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right)).left) =\n map w w (\ud835\udfd9 (Arrow.mk f)) (\ud835\udfd9 (Arrow.mk g)) (_ : (\ud835\udfd9 (Arrow.mk f)).right = (\ud835\udfd9 (Arrow.mk f)).right)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_\u03b1\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\nw' : f \u226b g = 0\n\u22a2 CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right) \u226b\n CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right) =\n \ud835\udfd9 (Arrow.mk f)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase e_\u03b2\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\nw' : f \u226b g = 0\n\u22a2 CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right) \u226b\n CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right) =\n \ud835\udfd9 (Arrow.mk g)\n[PROOFSTEP]\naesop_cat\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\nf' : A \u27f6 B\ng' : B \u27f6 C\nw' : f' \u226b g' = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\npf : f = f'\npg : g = g'\n\u22a2 map w' w (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f').left) (\ud835\udfd9 (Arrow.mk f').right))\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g').left) (\ud835\udfd9 (Arrow.mk g').right))\n (_ :\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f').left) (\ud835\udfd9 (Arrow.mk f').right)).right =\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f').left) (\ud835\udfd9 (Arrow.mk f').right)).right) \u226b\n map w w' (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right))\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right))\n (_ :\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right =\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right) =\n \ud835\udfd9 (homology f' g' w')\n[PROOFSTEP]\nobtain rfl := pf\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\ng' : B \u27f6 C\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\npg : g = g'\nw' : f \u226b g' = 0\n\u22a2 map w' w (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right))\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g').left) (\ud835\udfd9 (Arrow.mk g').right))\n (_ :\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right =\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right) \u226b\n map w w' (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right))\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right))\n (_ :\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right =\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right) =\n \ud835\udfd9 (homology f g' w')\n[PROOFSTEP]\nobtain rfl := pg\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\nw' : f \u226b g = 0\n\u22a2 map w' w (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right))\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right))\n (_ :\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right =\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right) \u226b\n map w w' (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right))\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right))\n (_ :\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right =\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right) =\n \ud835\udfd9 (homology f g w')\n[PROOFSTEP]\nrw [homology.map_comp, \u2190 homology.map_id]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\nw' : f \u226b g = 0\n\u22a2 map w' w'\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right) \u226b\n CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right))\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right) \u226b\n CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right))\n (_ :\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right) \u226b\n CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right)).right =\n (CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right) \u226b\n CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right)).left) =\n map w' w' (\ud835\udfd9 (Arrow.mk f)) (\ud835\udfd9 (Arrow.mk g)) (_ : (\ud835\udfd9 (Arrow.mk f)).right = (\ud835\udfd9 (Arrow.mk f)).right)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_\u03b1\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\nw' : f \u226b g = 0\n\u22a2 CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right) \u226b\n CommaMorphism.mk (\ud835\udfd9 (Arrow.mk f).left) (\ud835\udfd9 (Arrow.mk f).right) =\n \ud835\udfd9 (Arrow.mk f)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase e_\u03b2\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw : f \u226b g = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasCokernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasImageMaps V\nw' : f \u226b g = 0\n\u22a2 CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right) \u226b\n CommaMorphism.mk (\ud835\udfd9 (Arrow.mk g).left) (\ud835\udfd9 (Arrow.mk g).right) =\n \ud835\udfd9 (Arrow.mk g)\n[PROOFSTEP]\naesop_cat\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasImages V\nw : f \u226b g = 0\n\u22a2 image.\u03b9 f \u226b g = 0\n[PROOFSTEP]\next\n[GOAL]\ncase w\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasImages V\nw : f \u226b g = 0\n\u22a2 factorThruImage f \u226b image.\u03b9 f \u226b g = factorThruImage f \u226b 0\n[PROOFSTEP]\nsimpa using w\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasImages V\nw : f \u226b g = 0\n\u22a2 (imageSubobjectIso f).hom \u226b imageToKernel' f g w = imageToKernel f g w \u226b (kernelSubobjectIso g).hom\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasImages V\nw : f \u226b g = 0\n\u22a2 ((imageSubobjectIso f).hom \u226b imageToKernel' f g w) \u226b equalizer.\u03b9 g 0 =\n (imageToKernel f g w \u226b (kernelSubobjectIso g).hom) \u226b equalizer.\u03b9 g 0\n[PROOFSTEP]\nsimp [imageToKernel']\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasImages V\nw : f \u226b g = 0\n\u22a2 imageToKernel' f g w \u226b (kernelSubobjectIso g).inv = (imageSubobjectIso f).inv \u226b imageToKernel f g w\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasImages V\nw : f \u226b g = 0\n\u22a2 (imageToKernel' f g w \u226b (kernelSubobjectIso g).inv) \u226b Subobject.arrow (kernelSubobject g) =\n ((imageSubobjectIso f).inv \u226b imageToKernel f g w) \u226b Subobject.arrow (kernelSubobject g)\n[PROOFSTEP]\nsimp [imageToKernel']\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b2 : HasKernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasCokernels V\nw : f \u226b g = 0\n\u22a2 imageToKernel f g w \u226b (kernelSubobjectIso g).hom = (imageSubobjectIso f).hom \u226b imageToKernel' f g w\n[PROOFSTEP]\nsimp only [imageSubobjectIso_imageToKernel']\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b2 : HasKernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasCokernels V\nw : f \u226b g = 0\n\u22a2 imageToKernel' f g w \u226b (kernelSubobjectIso g).inv = (imageSubobjectIso f).inv \u226b imageToKernel f g w\n[PROOFSTEP]\nsimp only [imageToKernel'_kernelSubobjectIso]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b2 : HasKernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasCokernels V\nw : f \u226b g = 0\n\u22a2 cokernel.map (imageToKernel f g w) (imageToKernel' f g w) (imageSubobjectIso f).hom (kernelSubobjectIso g).hom\n (_ : imageToKernel f g w \u226b (kernelSubobjectIso g).hom = (imageSubobjectIso f).hom \u226b imageToKernel' f g w) \u226b\n cokernel.map (imageToKernel' f g w) (imageToKernel f g w) (imageSubobjectIso f).inv (kernelSubobjectIso g).inv\n (_ : imageToKernel' f g w \u226b (kernelSubobjectIso g).inv = (imageSubobjectIso f).inv \u226b imageToKernel f g w) =\n \ud835\udfd9 (homology f g w)\n[PROOFSTEP]\napply coequalizer.hom_ext\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b2 : HasKernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasCokernels V\nw : f \u226b g = 0\n\u22a2 coequalizer.\u03c0 (imageToKernel f g w) 0 \u226b\n cokernel.map (imageToKernel f g w) (imageToKernel' f g w) (imageSubobjectIso f).hom (kernelSubobjectIso g).hom\n (_ : imageToKernel f g w \u226b (kernelSubobjectIso g).hom = (imageSubobjectIso f).hom \u226b imageToKernel' f g w) \u226b\n cokernel.map (imageToKernel' f g w) (imageToKernel f g w) (imageSubobjectIso f).inv (kernelSubobjectIso g).inv\n (_ : imageToKernel' f g w \u226b (kernelSubobjectIso g).inv = (imageSubobjectIso f).inv \u226b imageToKernel f g w) =\n coequalizer.\u03c0 (imageToKernel f g w) 0 \u226b \ud835\udfd9 (homology f g w)\n[PROOFSTEP]\nsimp only [Iso.hom_inv_id_assoc, cokernel.\u03c0_desc, cokernel.\u03c0_desc_assoc, Category.assoc, coequalizer_as_cokernel]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b2 : HasKernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasCokernels V\nw : f \u226b g = 0\n\u22a2 cokernel.\u03c0 (imageToKernel f g w) = cokernel.\u03c0 (imageToKernel f g w) \u226b \ud835\udfd9 (homology f g w)\n[PROOFSTEP]\nexact (Category.comp_id _).symm\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b2 : HasKernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasCokernels V\nw : f \u226b g = 0\n\u22a2 cokernel.map (imageToKernel' f g w) (imageToKernel f g w) (imageSubobjectIso f).inv (kernelSubobjectIso g).inv\n (_ : imageToKernel' f g w \u226b (kernelSubobjectIso g).inv = (imageSubobjectIso f).inv \u226b imageToKernel f g w) \u226b\n cokernel.map (imageToKernel f g w) (imageToKernel' f g w) (imageSubobjectIso f).hom (kernelSubobjectIso g).hom\n (_ : imageToKernel f g w \u226b (kernelSubobjectIso g).hom = (imageSubobjectIso f).hom \u226b imageToKernel' f g w) =\n \ud835\udfd9 (cokernel (imageToKernel' f g w))\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b2 : HasKernels V\ninst\u271d\u00b9 : HasImages V\ninst\u271d : HasCokernels V\nw : f \u226b g = 0\n\u22a2 coequalizer.\u03c0 (imageToKernel' f g w) 0 \u226b\n cokernel.map (imageToKernel' f g w) (imageToKernel f g w) (imageSubobjectIso f).inv (kernelSubobjectIso g).inv\n (_ : imageToKernel' f g w \u226b (kernelSubobjectIso g).inv = (imageSubobjectIso f).inv \u226b imageToKernel f g w) \u226b\n cokernel.map (imageToKernel f g w) (imageToKernel' f g w) (imageSubobjectIso f).hom (kernelSubobjectIso g).hom\n (_ : imageToKernel f g w \u226b (kernelSubobjectIso g).hom = (imageSubobjectIso f).hom \u226b imageToKernel' f g w) =\n coequalizer.\u03c0 (imageToKernel' f g w) 0 \u226b \ud835\udfd9 (cokernel (imageToKernel' f g w))\n[PROOFSTEP]\nsimp only [Iso.inv_hom_id_assoc, cokernel.\u03c0_desc, Category.comp_id, cokernel.\u03c0_desc_assoc, Category.assoc]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasImages V\ninst\u271d\u00b9 : HasCokernels V\ninst\u271d : HasEqualizers V\nw : f \u226b g = 0\n\u22a2 homology f g w \u2245 cokernel (kernel.lift g f w)\n[PROOFSTEP]\nrefine' homologyIsoCokernelImageToKernel' f g w \u226a\u226b _\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasImages V\ninst\u271d\u00b9 : HasCokernels V\ninst\u271d : HasEqualizers V\nw : f \u226b g = 0\n\u22a2 cokernel (imageToKernel' f g w) \u2245 cokernel (kernel.lift g f w)\n[PROOFSTEP]\nhave p : factorThruImage f \u226b imageToKernel' f g w = kernel.lift g f w :=\n by\n ext\n simp [imageToKernel']\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasImages V\ninst\u271d\u00b9 : HasCokernels V\ninst\u271d : HasEqualizers V\nw : f \u226b g = 0\n\u22a2 factorThruImage f \u226b imageToKernel' f g w = kernel.lift g f w\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasImages V\ninst\u271d\u00b9 : HasCokernels V\ninst\u271d : HasEqualizers V\nw : f \u226b g = 0\n\u22a2 (factorThruImage f \u226b imageToKernel' f g w) \u226b equalizer.\u03b9 g 0 = kernel.lift g f w \u226b equalizer.\u03b9 g 0\n[PROOFSTEP]\nsimp [imageToKernel']\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b3 : HasKernels V\ninst\u271d\u00b2 : HasImages V\ninst\u271d\u00b9 : HasCokernels V\ninst\u271d : HasEqualizers V\nw : f \u226b g = 0\np : factorThruImage f \u226b imageToKernel' f g w = kernel.lift g f w\n\u22a2 cokernel (imageToKernel' f g w) \u2245 cokernel (kernel.lift g f w)\n[PROOFSTEP]\nexact (cokernelEpiComp _ _).symm \u226a\u226b cokernelIsoOfEq p\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Homology.ImageToKernel", "llama_tokens": 33702, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.4398710541647257}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nfinsetIcc : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nmem_Icc : \u2200 (a b x : \u03b1), x \u2208 finsetIcc a b \u2194 a \u2264 x \u2227 x \u2264 b\na b x : \u03b1\n\u22a2 x \u2208 (fun a b => filter (fun x => \u00acb \u2264 x) (finsetIcc a b)) a b \u2194 a \u2264 x \u2227 x < b\n[PROOFSTEP]\nrw [Finset.mem_filter, mem_Icc, and_assoc, lt_iff_le_not_le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nfinsetIcc : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nmem_Icc : \u2200 (a b x : \u03b1), x \u2208 finsetIcc a b \u2194 a \u2264 x \u2227 x \u2264 b\na b x : \u03b1\n\u22a2 x \u2208 (fun a b => filter (fun x => \u00acx \u2264 a) (finsetIcc a b)) a b \u2194 a < x \u2227 x \u2264 b\n[PROOFSTEP]\nrw [Finset.mem_filter, mem_Icc, and_right_comm, lt_iff_le_not_le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nfinsetIcc : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nmem_Icc : \u2200 (a b x : \u03b1), x \u2208 finsetIcc a b \u2194 a \u2264 x \u2227 x \u2264 b\na b x : \u03b1\n\u22a2 x \u2208 (fun a b => filter (fun x => \u00acx \u2264 a \u2227 \u00acb \u2264 x) (finsetIcc a b)) a b \u2194 a < x \u2227 x < b\n[PROOFSTEP]\nrw [Finset.mem_filter, mem_Icc, and_and_and_comm, lt_iff_le_not_le, lt_iff_le_not_le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : DecidableEq \u03b1\nfinsetIcc : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nmem_Icc : \u2200 (a b x : \u03b1), x \u2208 finsetIcc a b \u2194 a \u2264 x \u2227 x \u2264 b\na b x : \u03b1\n\u22a2 x \u2208 (fun a b => filter (fun x => x \u2260 b) (finsetIcc a b)) a b \u2194 a \u2264 x \u2227 x < b\n[PROOFSTEP]\nrw [Finset.mem_filter, mem_Icc, and_assoc, lt_iff_le_and_ne]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : DecidableEq \u03b1\nfinsetIcc : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nmem_Icc : \u2200 (a b x : \u03b1), x \u2208 finsetIcc a b \u2194 a \u2264 x \u2227 x \u2264 b\na b x : \u03b1\n\u22a2 x \u2208 (fun a b => filter (fun x => a \u2260 x) (finsetIcc a b)) a b \u2194 a < x \u2227 x \u2264 b\n[PROOFSTEP]\nrw [Finset.mem_filter, mem_Icc, and_right_comm, lt_iff_le_and_ne]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : DecidableEq \u03b1\nfinsetIcc : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nmem_Icc : \u2200 (a b x : \u03b1), x \u2208 finsetIcc a b \u2194 a \u2264 x \u2227 x \u2264 b\na b x : \u03b1\n\u22a2 x \u2208 (fun a b => filter (fun x => a \u2260 x \u2227 x \u2260 b) (finsetIcc a b)) a b \u2194 a < x \u2227 x < b\n[PROOFSTEP]\nrw [Finset.mem_filter, mem_Icc, and_and_and_comm, lt_iff_le_and_ne, lt_iff_le_and_ne]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nfinsetIci : \u03b1 \u2192 Finset \u03b1\nmem_Ici : \u2200 (a x : \u03b1), x \u2208 finsetIci a \u2194 a \u2264 x\na x : \u03b1\n\u22a2 x \u2208 (fun a => filter (fun x => \u00acx \u2264 a) (finsetIci a)) a \u2194 a < x\n[PROOFSTEP]\nrw [mem_filter, mem_Ici, lt_iff_le_not_le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : DecidableEq \u03b1\nfinsetIci : \u03b1 \u2192 Finset \u03b1\nmem_Ici : \u2200 (a x : \u03b1), x \u2208 finsetIci a \u2194 a \u2264 x\na x : \u03b1\n\u22a2 x \u2208 (fun a => filter (fun x => a \u2260 x) (finsetIci a)) a \u2194 a < x\n[PROOFSTEP]\nrw [mem_filter, mem_Ici, lt_iff_le_and_ne]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nfinsetIic : \u03b1 \u2192 Finset \u03b1\nmem_Iic : \u2200 (a x : \u03b1), x \u2208 finsetIic a \u2194 x \u2264 a\na x : \u03b1\n\u22a2 x \u2208 (fun a => filter (fun x => \u00aca \u2264 x) (finsetIic a)) a \u2194 x < a\n[PROOFSTEP]\nrw [mem_filter, mem_Iic, lt_iff_le_not_le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : DecidableEq \u03b1\nfinsetIic : \u03b1 \u2192 Finset \u03b1\nmem_Iic : \u2200 (a x : \u03b1), x \u2208 finsetIic a \u2194 x \u2264 a\na x : \u03b1\n\u22a2 x \u2208 (fun a => filter (fun x => x \u2260 a) (finsetIic a)) a \u2194 x < a\n[PROOFSTEP]\nrw [mem_filter, mem_Iic, lt_iff_le_and_ne]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : OrderTop \u03b1\na\u271d x\u271d a x : \u03b1\n\u22a2 x \u2208 (fun b => Icc b \u22a4) a \u2194 a \u2264 x\n[PROOFSTEP]\nrw [mem_Icc, and_iff_left le_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : OrderTop \u03b1\na\u271d x\u271d a x : \u03b1\n\u22a2 x \u2208 (fun b => Ioc b \u22a4) a \u2194 a < x\n[PROOFSTEP]\nrw [mem_Ioc, and_iff_left le_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nb x\u271d a x : \u03b1\n\u22a2 x \u2208 Icc \u22a5 a \u2194 x \u2264 a\n[PROOFSTEP]\nrw [mem_Icc, and_iff_right bot_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nb x\u271d a x : \u03b1\n\u22a2 x \u2208 Ico \u22a5 a \u2194 x < a\n[PROOFSTEP]\nrw [mem_Ico, and_iff_right bot_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b x : \u03b1\n\u22a2 x \u2208 Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\n[PROOFSTEP]\nrw [Icc, \u2190 Finset.mem_def, Finset.mem_Icc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b x : \u03b1\n\u22a2 x \u2208 Ico a b \u2194 a \u2264 x \u2227 x < b\n[PROOFSTEP]\nrw [Ico, \u2190 Finset.mem_def, Finset.mem_Ico]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b x : \u03b1\n\u22a2 x \u2208 Ioc a b \u2194 a < x \u2227 x \u2264 b\n[PROOFSTEP]\nrw [Ioc, \u2190 Finset.mem_def, Finset.mem_Ioc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b x : \u03b1\n\u22a2 x \u2208 Ioo a b \u2194 a < x \u2227 x < b\n[PROOFSTEP]\nrw [Ioo, \u2190 Finset.mem_def, Finset.mem_Ioo]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\na x : \u03b1\n\u22a2 x \u2208 Ici a \u2194 a \u2264 x\n[PROOFSTEP]\nrw [Ici, \u2190 Finset.mem_def, Finset.mem_Ici]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\na x : \u03b1\n\u22a2 x \u2208 Ioi a \u2194 a < x\n[PROOFSTEP]\nrw [Ioi, \u2190 Finset.mem_def, Finset.mem_Ioi]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\nb x : \u03b1\n\u22a2 x \u2208 Iic b \u2194 x \u2264 b\n[PROOFSTEP]\nrw [Iic, \u2190 Finset.mem_def, Finset.mem_Iic]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\nb x : \u03b1\n\u22a2 x \u2208 Iio b \u2194 x < b\n[PROOFSTEP]\nrw [Iio, \u2190 Finset.mem_def, Finset.mem_Iio]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh : \u2200 (a b : \u03b1), Set.Finite (Set.Icc a b)\na b x : \u03b1\n\u22a2 x \u2208 (fun a b => Set.Finite.toFinset (_ : Set.Finite (Set.Icc a b))) a b \u2194 a \u2264 x \u2227 x \u2264 b\n[PROOFSTEP]\nrw [Set.Finite.mem_toFinset, Set.mem_Icc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\na b x : \u03b1\n\u22a2 x \u2208 (fun a b => Set.toFinset (Set.Icc a b)) a b \u2194 a \u2264 x \u2227 x \u2264 b\n[PROOFSTEP]\nsimp only [Set.mem_toFinset, Set.mem_Icc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\na b x : \u03b1\n\u22a2 x \u2208 (fun a b => Set.toFinset (Set.Ico a b)) a b \u2194 a \u2264 x \u2227 x < b\n[PROOFSTEP]\nsimp only [Set.mem_toFinset, Set.mem_Ico]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\na b x : \u03b1\n\u22a2 x \u2208 (fun a b => Set.toFinset (Set.Ioc a b)) a b \u2194 a < x \u2227 x \u2264 b\n[PROOFSTEP]\nsimp only [Set.mem_toFinset, Set.mem_Ioc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\na b x : \u03b1\n\u22a2 x \u2208 (fun a b => Set.toFinset (Set.Ioo a b)) a b \u2194 a < x \u2227 x < b\n[PROOFSTEP]\nsimp only [Set.mem_toFinset, Set.mem_Ioo]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080 h\u2081 : LocallyFiniteOrder \u03b1\n\u22a2 h\u2080 = h\u2081\n[PROOFSTEP]\ncases' h\u2080 with h\u2080_finset_Icc h\u2080_finset_Ico h\u2080_finset_Ioc h\u2080_finset_Ioo h\u2080_finset_mem_Icc h\u2080_finset_mem_Ico\n h\u2080_finset_mem_Ioc h\u2080_finset_mem_Ioo\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2081 : LocallyFiniteOrder \u03b1\nh\u2080_finset_Icc h\u2080_finset_Ico h\u2080_finset_Ioc h\u2080_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2080_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2080_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2080_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioo a b \u2194 a < x \u2227 x < b\n\u22a2 { finsetIcc := h\u2080_finset_Icc, finsetIco := h\u2080_finset_Ico, finsetIoc := h\u2080_finset_Ioc, finsetIoo := h\u2080_finset_Ioo,\n finset_mem_Icc := h\u2080_finset_mem_Icc, finset_mem_Ico := h\u2080_finset_mem_Ico, finset_mem_Ioc := h\u2080_finset_mem_Ioc,\n finset_mem_Ioo := h\u2080_finset_mem_Ioo } =\n h\u2081\n[PROOFSTEP]\ncases' h\u2081 with h\u2081_finset_Icc h\u2081_finset_Ico h\u2081_finset_Ioc h\u2081_finset_Ioo h\u2081_finset_mem_Icc h\u2081_finset_mem_Ico\n h\u2081_finset_mem_Ioc h\u2081_finset_mem_Ioo\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Icc h\u2080_finset_Ico h\u2080_finset_Ioc h\u2080_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2080_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2080_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2080_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioo a b \u2194 a < x \u2227 x < b\nh\u2081_finset_Icc h\u2081_finset_Ico h\u2081_finset_Ioc h\u2081_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2081_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2081_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2081_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioo a b \u2194 a < x \u2227 x < b\n\u22a2 { finsetIcc := h\u2080_finset_Icc, finsetIco := h\u2080_finset_Ico, finsetIoc := h\u2080_finset_Ioc, finsetIoo := h\u2080_finset_Ioo,\n finset_mem_Icc := h\u2080_finset_mem_Icc, finset_mem_Ico := h\u2080_finset_mem_Ico, finset_mem_Ioc := h\u2080_finset_mem_Ioc,\n finset_mem_Ioo := h\u2080_finset_mem_Ioo } =\n { finsetIcc := h\u2081_finset_Icc, finsetIco := h\u2081_finset_Ico, finsetIoc := h\u2081_finset_Ioc, finsetIoo := h\u2081_finset_Ioo,\n finset_mem_Icc := h\u2081_finset_mem_Icc, finset_mem_Ico := h\u2081_finset_mem_Ico, finset_mem_Ioc := h\u2081_finset_mem_Ioc,\n finset_mem_Ioo := h\u2081_finset_mem_Ioo }\n[PROOFSTEP]\nhave hIcc : h\u2080_finset_Icc = h\u2081_finset_Icc := by\n ext a b x\n rw [h\u2080_finset_mem_Icc, h\u2081_finset_mem_Icc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Icc h\u2080_finset_Ico h\u2080_finset_Ioc h\u2080_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2080_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2080_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2080_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioo a b \u2194 a < x \u2227 x < b\nh\u2081_finset_Icc h\u2081_finset_Ico h\u2081_finset_Ioc h\u2081_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2081_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2081_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2081_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioo a b \u2194 a < x \u2227 x < b\n\u22a2 h\u2080_finset_Icc = h\u2081_finset_Icc\n[PROOFSTEP]\next a b x\n[GOAL]\ncase h.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Icc h\u2080_finset_Ico h\u2080_finset_Ioc h\u2080_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2080_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2080_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2080_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioo a b \u2194 a < x \u2227 x < b\nh\u2081_finset_Icc h\u2081_finset_Ico h\u2081_finset_Ioc h\u2081_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2081_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2081_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2081_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioo a b \u2194 a < x \u2227 x < b\na b x : \u03b1\n\u22a2 x \u2208 h\u2080_finset_Icc a b \u2194 x \u2208 h\u2081_finset_Icc a b\n[PROOFSTEP]\nrw [h\u2080_finset_mem_Icc, h\u2081_finset_mem_Icc]\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Icc h\u2080_finset_Ico h\u2080_finset_Ioc h\u2080_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2080_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2080_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2080_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioo a b \u2194 a < x \u2227 x < b\nh\u2081_finset_Icc h\u2081_finset_Ico h\u2081_finset_Ioc h\u2081_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2081_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2081_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2081_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioo a b \u2194 a < x \u2227 x < b\nhIcc : h\u2080_finset_Icc = h\u2081_finset_Icc\n\u22a2 { finsetIcc := h\u2080_finset_Icc, finsetIco := h\u2080_finset_Ico, finsetIoc := h\u2080_finset_Ioc, finsetIoo := h\u2080_finset_Ioo,\n finset_mem_Icc := h\u2080_finset_mem_Icc, finset_mem_Ico := h\u2080_finset_mem_Ico, finset_mem_Ioc := h\u2080_finset_mem_Ioc,\n finset_mem_Ioo := h\u2080_finset_mem_Ioo } =\n { finsetIcc := h\u2081_finset_Icc, finsetIco := h\u2081_finset_Ico, finsetIoc := h\u2081_finset_Ioc, finsetIoo := h\u2081_finset_Ioo,\n finset_mem_Icc := h\u2081_finset_mem_Icc, finset_mem_Ico := h\u2081_finset_mem_Ico, finset_mem_Ioc := h\u2081_finset_mem_Ioc,\n finset_mem_Ioo := h\u2081_finset_mem_Ioo }\n[PROOFSTEP]\nhave hIco : h\u2080_finset_Ico = h\u2081_finset_Ico := by\n ext a b x\n rw [h\u2080_finset_mem_Ico, h\u2081_finset_mem_Ico]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Icc h\u2080_finset_Ico h\u2080_finset_Ioc h\u2080_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2080_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2080_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2080_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioo a b \u2194 a < x \u2227 x < b\nh\u2081_finset_Icc h\u2081_finset_Ico h\u2081_finset_Ioc h\u2081_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2081_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2081_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2081_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioo a b \u2194 a < x \u2227 x < b\nhIcc : h\u2080_finset_Icc = h\u2081_finset_Icc\n\u22a2 h\u2080_finset_Ico = h\u2081_finset_Ico\n[PROOFSTEP]\next a b x\n[GOAL]\ncase h.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Icc h\u2080_finset_Ico h\u2080_finset_Ioc h\u2080_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2080_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2080_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2080_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioo a b \u2194 a < x \u2227 x < b\nh\u2081_finset_Icc h\u2081_finset_Ico h\u2081_finset_Ioc h\u2081_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2081_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2081_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2081_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioo a b \u2194 a < x \u2227 x < b\nhIcc : h\u2080_finset_Icc = h\u2081_finset_Icc\na b x : \u03b1\n\u22a2 x \u2208 h\u2080_finset_Ico a b \u2194 x \u2208 h\u2081_finset_Ico a b\n[PROOFSTEP]\nrw [h\u2080_finset_mem_Ico, h\u2081_finset_mem_Ico]\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Icc h\u2080_finset_Ico h\u2080_finset_Ioc h\u2080_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2080_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2080_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2080_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioo a b \u2194 a < x \u2227 x < b\nh\u2081_finset_Icc h\u2081_finset_Ico h\u2081_finset_Ioc h\u2081_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2081_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2081_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2081_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioo a b \u2194 a < x \u2227 x < b\nhIcc : h\u2080_finset_Icc = h\u2081_finset_Icc\nhIco : h\u2080_finset_Ico = h\u2081_finset_Ico\n\u22a2 { finsetIcc := h\u2080_finset_Icc, finsetIco := h\u2080_finset_Ico, finsetIoc := h\u2080_finset_Ioc, finsetIoo := h\u2080_finset_Ioo,\n finset_mem_Icc := h\u2080_finset_mem_Icc, finset_mem_Ico := h\u2080_finset_mem_Ico, finset_mem_Ioc := h\u2080_finset_mem_Ioc,\n finset_mem_Ioo := h\u2080_finset_mem_Ioo } =\n { finsetIcc := h\u2081_finset_Icc, finsetIco := h\u2081_finset_Ico, finsetIoc := h\u2081_finset_Ioc, finsetIoo := h\u2081_finset_Ioo,\n finset_mem_Icc := h\u2081_finset_mem_Icc, finset_mem_Ico := h\u2081_finset_mem_Ico, finset_mem_Ioc := h\u2081_finset_mem_Ioc,\n finset_mem_Ioo := h\u2081_finset_mem_Ioo }\n[PROOFSTEP]\nhave hIoc : h\u2080_finset_Ioc = h\u2081_finset_Ioc := by\n ext a b x\n rw [h\u2080_finset_mem_Ioc, h\u2081_finset_mem_Ioc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Icc h\u2080_finset_Ico h\u2080_finset_Ioc h\u2080_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2080_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2080_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2080_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioo a b \u2194 a < x \u2227 x < b\nh\u2081_finset_Icc h\u2081_finset_Ico h\u2081_finset_Ioc h\u2081_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2081_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2081_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2081_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioo a b \u2194 a < x \u2227 x < b\nhIcc : h\u2080_finset_Icc = h\u2081_finset_Icc\nhIco : h\u2080_finset_Ico = h\u2081_finset_Ico\n\u22a2 h\u2080_finset_Ioc = h\u2081_finset_Ioc\n[PROOFSTEP]\next a b x\n[GOAL]\ncase h.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Icc h\u2080_finset_Ico h\u2080_finset_Ioc h\u2080_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2080_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2080_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2080_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioo a b \u2194 a < x \u2227 x < b\nh\u2081_finset_Icc h\u2081_finset_Ico h\u2081_finset_Ioc h\u2081_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2081_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2081_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2081_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioo a b \u2194 a < x \u2227 x < b\nhIcc : h\u2080_finset_Icc = h\u2081_finset_Icc\nhIco : h\u2080_finset_Ico = h\u2081_finset_Ico\na b x : \u03b1\n\u22a2 x \u2208 h\u2080_finset_Ioc a b \u2194 x \u2208 h\u2081_finset_Ioc a b\n[PROOFSTEP]\nrw [h\u2080_finset_mem_Ioc, h\u2081_finset_mem_Ioc]\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Icc h\u2080_finset_Ico h\u2080_finset_Ioc h\u2080_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2080_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2080_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2080_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioo a b \u2194 a < x \u2227 x < b\nh\u2081_finset_Icc h\u2081_finset_Ico h\u2081_finset_Ioc h\u2081_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2081_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2081_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2081_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioo a b \u2194 a < x \u2227 x < b\nhIcc : h\u2080_finset_Icc = h\u2081_finset_Icc\nhIco : h\u2080_finset_Ico = h\u2081_finset_Ico\nhIoc : h\u2080_finset_Ioc = h\u2081_finset_Ioc\n\u22a2 { finsetIcc := h\u2080_finset_Icc, finsetIco := h\u2080_finset_Ico, finsetIoc := h\u2080_finset_Ioc, finsetIoo := h\u2080_finset_Ioo,\n finset_mem_Icc := h\u2080_finset_mem_Icc, finset_mem_Ico := h\u2080_finset_mem_Ico, finset_mem_Ioc := h\u2080_finset_mem_Ioc,\n finset_mem_Ioo := h\u2080_finset_mem_Ioo } =\n { finsetIcc := h\u2081_finset_Icc, finsetIco := h\u2081_finset_Ico, finsetIoc := h\u2081_finset_Ioc, finsetIoo := h\u2081_finset_Ioo,\n finset_mem_Icc := h\u2081_finset_mem_Icc, finset_mem_Ico := h\u2081_finset_mem_Ico, finset_mem_Ioc := h\u2081_finset_mem_Ioc,\n finset_mem_Ioo := h\u2081_finset_mem_Ioo }\n[PROOFSTEP]\nhave hIoo : h\u2080_finset_Ioo = h\u2081_finset_Ioo := by\n ext a b x\n rw [h\u2080_finset_mem_Ioo, h\u2081_finset_mem_Ioo]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Icc h\u2080_finset_Ico h\u2080_finset_Ioc h\u2080_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2080_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2080_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2080_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioo a b \u2194 a < x \u2227 x < b\nh\u2081_finset_Icc h\u2081_finset_Ico h\u2081_finset_Ioc h\u2081_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2081_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2081_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2081_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioo a b \u2194 a < x \u2227 x < b\nhIcc : h\u2080_finset_Icc = h\u2081_finset_Icc\nhIco : h\u2080_finset_Ico = h\u2081_finset_Ico\nhIoc : h\u2080_finset_Ioc = h\u2081_finset_Ioc\n\u22a2 h\u2080_finset_Ioo = h\u2081_finset_Ioo\n[PROOFSTEP]\next a b x\n[GOAL]\ncase h.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Icc h\u2080_finset_Ico h\u2080_finset_Ioc h\u2080_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2080_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2080_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2080_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioo a b \u2194 a < x \u2227 x < b\nh\u2081_finset_Icc h\u2081_finset_Ico h\u2081_finset_Ioc h\u2081_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2081_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2081_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2081_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioo a b \u2194 a < x \u2227 x < b\nhIcc : h\u2080_finset_Icc = h\u2081_finset_Icc\nhIco : h\u2080_finset_Ico = h\u2081_finset_Ico\nhIoc : h\u2080_finset_Ioc = h\u2081_finset_Ioc\na b x : \u03b1\n\u22a2 x \u2208 h\u2080_finset_Ioo a b \u2194 x \u2208 h\u2081_finset_Ioo a b\n[PROOFSTEP]\nrw [h\u2080_finset_mem_Ioo, h\u2081_finset_mem_Ioo]\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Icc h\u2080_finset_Ico h\u2080_finset_Ioc h\u2080_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2080_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2080_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2080_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2080_finset_Ioo a b \u2194 a < x \u2227 x < b\nh\u2081_finset_Icc h\u2081_finset_Ico h\u2081_finset_Ioc h\u2081_finset_Ioo : \u03b1 \u2192 \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Icc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Icc a b \u2194 a \u2264 x \u2227 x \u2264 b\nh\u2081_finset_mem_Ico : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ico a b \u2194 a \u2264 x \u2227 x < b\nh\u2081_finset_mem_Ioc : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioc a b \u2194 a < x \u2227 x \u2264 b\nh\u2081_finset_mem_Ioo : \u2200 (a b x : \u03b1), x \u2208 h\u2081_finset_Ioo a b \u2194 a < x \u2227 x < b\nhIcc : h\u2080_finset_Icc = h\u2081_finset_Icc\nhIco : h\u2080_finset_Ico = h\u2081_finset_Ico\nhIoc : h\u2080_finset_Ioc = h\u2081_finset_Ioc\nhIoo : h\u2080_finset_Ioo = h\u2081_finset_Ioo\n\u22a2 { finsetIcc := h\u2080_finset_Icc, finsetIco := h\u2080_finset_Ico, finsetIoc := h\u2080_finset_Ioc, finsetIoo := h\u2080_finset_Ioo,\n finset_mem_Icc := h\u2080_finset_mem_Icc, finset_mem_Ico := h\u2080_finset_mem_Ico, finset_mem_Ioc := h\u2080_finset_mem_Ioc,\n finset_mem_Ioo := h\u2080_finset_mem_Ioo } =\n { finsetIcc := h\u2081_finset_Icc, finsetIco := h\u2081_finset_Ico, finsetIoc := h\u2081_finset_Ioc, finsetIoo := h\u2081_finset_Ioo,\n finset_mem_Icc := h\u2081_finset_mem_Icc, finset_mem_Ico := h\u2081_finset_mem_Ico, finset_mem_Ioc := h\u2081_finset_mem_Ioc,\n finset_mem_Ioo := h\u2081_finset_mem_Ioo }\n[PROOFSTEP]\nsimp_rw [hIcc, hIco, hIoc, hIoo]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080 h\u2081 : LocallyFiniteOrderTop \u03b1\n\u22a2 h\u2080 = h\u2081\n[PROOFSTEP]\ncases' h\u2080 with h\u2080_finset_Ioi h\u2080_finset_Ici h\u2080_finset_mem_Ici h\u2080_finset_mem_Ioi\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2081 : LocallyFiniteOrderTop \u03b1\nh\u2080_finset_Ioi h\u2080_finset_Ici : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Ici : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ici a \u2194 a \u2264 x\nh\u2080_finset_mem_Ioi : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ioi a \u2194 a < x\n\u22a2 { finsetIoi := h\u2080_finset_Ioi, finsetIci := h\u2080_finset_Ici, finset_mem_Ici := h\u2080_finset_mem_Ici,\n finset_mem_Ioi := h\u2080_finset_mem_Ioi } =\n h\u2081\n[PROOFSTEP]\ncases' h\u2081 with h\u2081_finset_Ioi h\u2081_finset_Ici h\u2081_finset_mem_Ici h\u2081_finset_mem_Ioi\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Ioi h\u2080_finset_Ici : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Ici : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ici a \u2194 a \u2264 x\nh\u2080_finset_mem_Ioi : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ioi a \u2194 a < x\nh\u2081_finset_Ioi h\u2081_finset_Ici : \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Ici : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Ici a \u2194 a \u2264 x\nh\u2081_finset_mem_Ioi : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Ioi a \u2194 a < x\n\u22a2 { finsetIoi := h\u2080_finset_Ioi, finsetIci := h\u2080_finset_Ici, finset_mem_Ici := h\u2080_finset_mem_Ici,\n finset_mem_Ioi := h\u2080_finset_mem_Ioi } =\n { finsetIoi := h\u2081_finset_Ioi, finsetIci := h\u2081_finset_Ici, finset_mem_Ici := h\u2081_finset_mem_Ici,\n finset_mem_Ioi := h\u2081_finset_mem_Ioi }\n[PROOFSTEP]\nhave hIci : h\u2080_finset_Ici = h\u2081_finset_Ici := by\n ext a b\n rw [h\u2080_finset_mem_Ici, h\u2081_finset_mem_Ici]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Ioi h\u2080_finset_Ici : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Ici : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ici a \u2194 a \u2264 x\nh\u2080_finset_mem_Ioi : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ioi a \u2194 a < x\nh\u2081_finset_Ioi h\u2081_finset_Ici : \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Ici : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Ici a \u2194 a \u2264 x\nh\u2081_finset_mem_Ioi : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Ioi a \u2194 a < x\n\u22a2 h\u2080_finset_Ici = h\u2081_finset_Ici\n[PROOFSTEP]\next a b\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Ioi h\u2080_finset_Ici : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Ici : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ici a \u2194 a \u2264 x\nh\u2080_finset_mem_Ioi : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ioi a \u2194 a < x\nh\u2081_finset_Ioi h\u2081_finset_Ici : \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Ici : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Ici a \u2194 a \u2264 x\nh\u2081_finset_mem_Ioi : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Ioi a \u2194 a < x\na b : \u03b1\n\u22a2 b \u2208 h\u2080_finset_Ici a \u2194 b \u2208 h\u2081_finset_Ici a\n[PROOFSTEP]\nrw [h\u2080_finset_mem_Ici, h\u2081_finset_mem_Ici]\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Ioi h\u2080_finset_Ici : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Ici : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ici a \u2194 a \u2264 x\nh\u2080_finset_mem_Ioi : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ioi a \u2194 a < x\nh\u2081_finset_Ioi h\u2081_finset_Ici : \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Ici : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Ici a \u2194 a \u2264 x\nh\u2081_finset_mem_Ioi : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Ioi a \u2194 a < x\nhIci : h\u2080_finset_Ici = h\u2081_finset_Ici\n\u22a2 { finsetIoi := h\u2080_finset_Ioi, finsetIci := h\u2080_finset_Ici, finset_mem_Ici := h\u2080_finset_mem_Ici,\n finset_mem_Ioi := h\u2080_finset_mem_Ioi } =\n { finsetIoi := h\u2081_finset_Ioi, finsetIci := h\u2081_finset_Ici, finset_mem_Ici := h\u2081_finset_mem_Ici,\n finset_mem_Ioi := h\u2081_finset_mem_Ioi }\n[PROOFSTEP]\nhave hIoi : h\u2080_finset_Ioi = h\u2081_finset_Ioi := by\n ext a b\n rw [h\u2080_finset_mem_Ioi, h\u2081_finset_mem_Ioi]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Ioi h\u2080_finset_Ici : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Ici : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ici a \u2194 a \u2264 x\nh\u2080_finset_mem_Ioi : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ioi a \u2194 a < x\nh\u2081_finset_Ioi h\u2081_finset_Ici : \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Ici : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Ici a \u2194 a \u2264 x\nh\u2081_finset_mem_Ioi : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Ioi a \u2194 a < x\nhIci : h\u2080_finset_Ici = h\u2081_finset_Ici\n\u22a2 h\u2080_finset_Ioi = h\u2081_finset_Ioi\n[PROOFSTEP]\next a b\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Ioi h\u2080_finset_Ici : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Ici : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ici a \u2194 a \u2264 x\nh\u2080_finset_mem_Ioi : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ioi a \u2194 a < x\nh\u2081_finset_Ioi h\u2081_finset_Ici : \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Ici : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Ici a \u2194 a \u2264 x\nh\u2081_finset_mem_Ioi : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Ioi a \u2194 a < x\nhIci : h\u2080_finset_Ici = h\u2081_finset_Ici\na b : \u03b1\n\u22a2 b \u2208 h\u2080_finset_Ioi a \u2194 b \u2208 h\u2081_finset_Ioi a\n[PROOFSTEP]\nrw [h\u2080_finset_mem_Ioi, h\u2081_finset_mem_Ioi]\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Ioi h\u2080_finset_Ici : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Ici : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ici a \u2194 a \u2264 x\nh\u2080_finset_mem_Ioi : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Ioi a \u2194 a < x\nh\u2081_finset_Ioi h\u2081_finset_Ici : \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Ici : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Ici a \u2194 a \u2264 x\nh\u2081_finset_mem_Ioi : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Ioi a \u2194 a < x\nhIci : h\u2080_finset_Ici = h\u2081_finset_Ici\nhIoi : h\u2080_finset_Ioi = h\u2081_finset_Ioi\n\u22a2 { finsetIoi := h\u2080_finset_Ioi, finsetIci := h\u2080_finset_Ici, finset_mem_Ici := h\u2080_finset_mem_Ici,\n finset_mem_Ioi := h\u2080_finset_mem_Ioi } =\n { finsetIoi := h\u2081_finset_Ioi, finsetIci := h\u2081_finset_Ici, finset_mem_Ici := h\u2081_finset_mem_Ici,\n finset_mem_Ioi := h\u2081_finset_mem_Ioi }\n[PROOFSTEP]\nsimp_rw [hIci, hIoi]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080 h\u2081 : LocallyFiniteOrderBot \u03b1\n\u22a2 h\u2080 = h\u2081\n[PROOFSTEP]\ncases' h\u2080 with h\u2080_finset_Iio h\u2080_finset_Iic h\u2080_finset_mem_Iic h\u2080_finset_mem_Iio\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2081 : LocallyFiniteOrderBot \u03b1\nh\u2080_finset_Iio h\u2080_finset_Iic : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Iic : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iic a \u2194 x \u2264 a\nh\u2080_finset_mem_Iio : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iio a \u2194 x < a\n\u22a2 { finsetIio := h\u2080_finset_Iio, finsetIic := h\u2080_finset_Iic, finset_mem_Iic := h\u2080_finset_mem_Iic,\n finset_mem_Iio := h\u2080_finset_mem_Iio } =\n h\u2081\n[PROOFSTEP]\ncases' h\u2081 with h\u2081_finset_Iio h\u2081_finset_Iic h\u2081_finset_mem_Iic h\u2081_finset_mem_Iio\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Iio h\u2080_finset_Iic : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Iic : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iic a \u2194 x \u2264 a\nh\u2080_finset_mem_Iio : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iio a \u2194 x < a\nh\u2081_finset_Iio h\u2081_finset_Iic : \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Iic : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Iic a \u2194 x \u2264 a\nh\u2081_finset_mem_Iio : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Iio a \u2194 x < a\n\u22a2 { finsetIio := h\u2080_finset_Iio, finsetIic := h\u2080_finset_Iic, finset_mem_Iic := h\u2080_finset_mem_Iic,\n finset_mem_Iio := h\u2080_finset_mem_Iio } =\n { finsetIio := h\u2081_finset_Iio, finsetIic := h\u2081_finset_Iic, finset_mem_Iic := h\u2081_finset_mem_Iic,\n finset_mem_Iio := h\u2081_finset_mem_Iio }\n[PROOFSTEP]\nhave hIic : h\u2080_finset_Iic = h\u2081_finset_Iic := by\n ext a b\n rw [h\u2080_finset_mem_Iic, h\u2081_finset_mem_Iic]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Iio h\u2080_finset_Iic : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Iic : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iic a \u2194 x \u2264 a\nh\u2080_finset_mem_Iio : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iio a \u2194 x < a\nh\u2081_finset_Iio h\u2081_finset_Iic : \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Iic : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Iic a \u2194 x \u2264 a\nh\u2081_finset_mem_Iio : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Iio a \u2194 x < a\n\u22a2 h\u2080_finset_Iic = h\u2081_finset_Iic\n[PROOFSTEP]\next a b\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Iio h\u2080_finset_Iic : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Iic : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iic a \u2194 x \u2264 a\nh\u2080_finset_mem_Iio : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iio a \u2194 x < a\nh\u2081_finset_Iio h\u2081_finset_Iic : \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Iic : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Iic a \u2194 x \u2264 a\nh\u2081_finset_mem_Iio : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Iio a \u2194 x < a\na b : \u03b1\n\u22a2 b \u2208 h\u2080_finset_Iic a \u2194 b \u2208 h\u2081_finset_Iic a\n[PROOFSTEP]\nrw [h\u2080_finset_mem_Iic, h\u2081_finset_mem_Iic]\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Iio h\u2080_finset_Iic : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Iic : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iic a \u2194 x \u2264 a\nh\u2080_finset_mem_Iio : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iio a \u2194 x < a\nh\u2081_finset_Iio h\u2081_finset_Iic : \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Iic : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Iic a \u2194 x \u2264 a\nh\u2081_finset_mem_Iio : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Iio a \u2194 x < a\nhIic : h\u2080_finset_Iic = h\u2081_finset_Iic\n\u22a2 { finsetIio := h\u2080_finset_Iio, finsetIic := h\u2080_finset_Iic, finset_mem_Iic := h\u2080_finset_mem_Iic,\n finset_mem_Iio := h\u2080_finset_mem_Iio } =\n { finsetIio := h\u2081_finset_Iio, finsetIic := h\u2081_finset_Iic, finset_mem_Iic := h\u2081_finset_mem_Iic,\n finset_mem_Iio := h\u2081_finset_mem_Iio }\n[PROOFSTEP]\nhave hIio : h\u2080_finset_Iio = h\u2081_finset_Iio := by\n ext a b\n rw [h\u2080_finset_mem_Iio, h\u2081_finset_mem_Iio]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Iio h\u2080_finset_Iic : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Iic : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iic a \u2194 x \u2264 a\nh\u2080_finset_mem_Iio : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iio a \u2194 x < a\nh\u2081_finset_Iio h\u2081_finset_Iic : \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Iic : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Iic a \u2194 x \u2264 a\nh\u2081_finset_mem_Iio : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Iio a \u2194 x < a\nhIic : h\u2080_finset_Iic = h\u2081_finset_Iic\n\u22a2 h\u2080_finset_Iio = h\u2081_finset_Iio\n[PROOFSTEP]\next a b\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Iio h\u2080_finset_Iic : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Iic : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iic a \u2194 x \u2264 a\nh\u2080_finset_mem_Iio : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iio a \u2194 x < a\nh\u2081_finset_Iio h\u2081_finset_Iic : \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Iic : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Iic a \u2194 x \u2264 a\nh\u2081_finset_mem_Iio : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Iio a \u2194 x < a\nhIic : h\u2080_finset_Iic = h\u2081_finset_Iic\na b : \u03b1\n\u22a2 b \u2208 h\u2080_finset_Iio a \u2194 b \u2208 h\u2081_finset_Iio a\n[PROOFSTEP]\nrw [h\u2080_finset_mem_Iio, h\u2081_finset_mem_Iio]\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nh\u2080_finset_Iio h\u2080_finset_Iic : \u03b1 \u2192 Finset \u03b1\nh\u2080_finset_mem_Iic : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iic a \u2194 x \u2264 a\nh\u2080_finset_mem_Iio : \u2200 (a x : \u03b1), x \u2208 h\u2080_finset_Iio a \u2194 x < a\nh\u2081_finset_Iio h\u2081_finset_Iic : \u03b1 \u2192 Finset \u03b1\nh\u2081_finset_mem_Iic : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Iic a \u2194 x \u2264 a\nh\u2081_finset_mem_Iio : \u2200 (a x : \u03b1), x \u2208 h\u2081_finset_Iio a \u2194 x < a\nhIic : h\u2080_finset_Iic = h\u2081_finset_Iic\nhIio : h\u2080_finset_Iio = h\u2081_finset_Iio\n\u22a2 { finsetIio := h\u2080_finset_Iio, finsetIic := h\u2080_finset_Iic, finset_mem_Iic := h\u2080_finset_mem_Iic,\n finset_mem_Iio := h\u2080_finset_mem_Iio } =\n { finsetIio := h\u2081_finset_Iio, finsetIic := h\u2081_finset_Iic, finset_mem_Iic := h\u2081_finset_mem_Iic,\n finset_mem_Iio := h\u2081_finset_mem_Iio }\n[PROOFSTEP]\nsimp_rw [hIic, hIio]\n -- Should this be called `LocallyFiniteOrder.lift`?\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b2\nf : \u03b1 \u21aao \u03b2\na b x : \u03b1\n\u22a2 x \u2208 (fun a b => preimage (Icc (\u2191f a) (\u2191f b)) \u2191f (_ : Set.InjOn (\u2191f.toEmbedding) (\u2191f \u207b\u00b9' \u2191(Icc (\u2191f a) (\u2191f b))))) a b \u2194\n a \u2264 x \u2227 x \u2264 b\n[PROOFSTEP]\nrw [mem_preimage, mem_Icc, f.le_iff_le, f.le_iff_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b2\nf : \u03b1 \u21aao \u03b2\na b x : \u03b1\n\u22a2 x \u2208 (fun a b => preimage (Ico (\u2191f a) (\u2191f b)) \u2191f (_ : Set.InjOn (\u2191f.toEmbedding) (\u2191f \u207b\u00b9' \u2191(Ico (\u2191f a) (\u2191f b))))) a b \u2194\n a \u2264 x \u2227 x < b\n[PROOFSTEP]\nrw [mem_preimage, mem_Ico, f.le_iff_le, f.lt_iff_lt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b2\nf : \u03b1 \u21aao \u03b2\na b x : \u03b1\n\u22a2 x \u2208 (fun a b => preimage (Ioc (\u2191f a) (\u2191f b)) \u2191f (_ : Set.InjOn (\u2191f.toEmbedding) (\u2191f \u207b\u00b9' \u2191(Ioc (\u2191f a) (\u2191f b))))) a b \u2194\n a < x \u2227 x \u2264 b\n[PROOFSTEP]\nrw [mem_preimage, mem_Ioc, f.lt_iff_lt, f.le_iff_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b2\nf : \u03b1 \u21aao \u03b2\na b x : \u03b1\n\u22a2 x \u2208 (fun a b => preimage (Ioo (\u2191f a) (\u2191f b)) \u2191f (_ : Set.InjOn (\u2191f.toEmbedding) (\u2191f \u207b\u00b9' \u2191(Ioo (\u2191f a) (\u2191f b))))) a b \u2194\n a < x \u2227 x < b\n[PROOFSTEP]\nrw [mem_preimage, mem_Ioo, f.lt_iff_lt, f.lt_iff_lt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\n\u22a2 Icc (\u2191toDual a) (\u2191toDual b) = map (Equiv.toEmbedding toDual) (Icc b a)\n[PROOFSTEP]\nrefine' Eq.trans _ map_refl.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\n\u22a2 Icc (\u2191toDual a) (\u2191toDual b) = Icc b a\n[PROOFSTEP]\next c\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\nc : (fun x => \u03b1\u1d52\u1d48) a\n\u22a2 c \u2208 Icc (\u2191toDual a) (\u2191toDual b) \u2194 c \u2208 Icc b a\n[PROOFSTEP]\nrw [mem_Icc, mem_Icc (\u03b1 := \u03b1)]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\nc : (fun x => \u03b1\u1d52\u1d48) a\n\u22a2 \u2191toDual a \u2264 c \u2227 c \u2264 \u2191toDual b \u2194 b \u2264 c \u2227 c \u2264 a\n[PROOFSTEP]\nexact and_comm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\n\u22a2 Ico (\u2191toDual a) (\u2191toDual b) = map (Equiv.toEmbedding toDual) (Ioc b a)\n[PROOFSTEP]\nrefine' Eq.trans _ map_refl.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\n\u22a2 Ico (\u2191toDual a) (\u2191toDual b) = Ioc b a\n[PROOFSTEP]\next c\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\nc : (fun x => \u03b1\u1d52\u1d48) a\n\u22a2 c \u2208 Ico (\u2191toDual a) (\u2191toDual b) \u2194 c \u2208 Ioc b a\n[PROOFSTEP]\nrw [mem_Ico, mem_Ioc (\u03b1 := \u03b1)]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\nc : (fun x => \u03b1\u1d52\u1d48) a\n\u22a2 \u2191toDual a \u2264 c \u2227 c < \u2191toDual b \u2194 b < c \u2227 c \u2264 a\n[PROOFSTEP]\nexact and_comm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\n\u22a2 Ioc (\u2191toDual a) (\u2191toDual b) = map (Equiv.toEmbedding toDual) (Ico b a)\n[PROOFSTEP]\nrefine' Eq.trans _ map_refl.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\n\u22a2 Ioc (\u2191toDual a) (\u2191toDual b) = Ico b a\n[PROOFSTEP]\next c\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\nc : (fun x => \u03b1\u1d52\u1d48) a\n\u22a2 c \u2208 Ioc (\u2191toDual a) (\u2191toDual b) \u2194 c \u2208 Ico b a\n[PROOFSTEP]\nrw [mem_Ioc, mem_Ico (\u03b1 := \u03b1)]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\nc : (fun x => \u03b1\u1d52\u1d48) a\n\u22a2 \u2191toDual a < c \u2227 c \u2264 \u2191toDual b \u2194 b \u2264 c \u2227 c < a\n[PROOFSTEP]\nexact and_comm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\n\u22a2 Ioo (\u2191toDual a) (\u2191toDual b) = map (Equiv.toEmbedding toDual) (Ioo b a)\n[PROOFSTEP]\nrefine' Eq.trans _ map_refl.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\n\u22a2 Ioo (\u2191toDual a) (\u2191toDual b) = Ioo b a\n[PROOFSTEP]\next c\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\nc : (fun x => \u03b1\u1d52\u1d48) a\n\u22a2 c \u2208 Ioo (\u2191toDual a) (\u2191toDual b) \u2194 c \u2208 Ioo b a\n[PROOFSTEP]\nrw [mem_Ioo, mem_Ioo (\u03b1 := \u03b1)]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na b : \u03b1\nc : (fun x => \u03b1\u1d52\u1d48) a\n\u22a2 \u2191toDual a < c \u2227 c < \u2191toDual b \u2194 b < c \u2227 c < a\n[PROOFSTEP]\nexact and_comm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\n\u22a2 Icc (\u2191ofDual a) (\u2191ofDual b) = map (Equiv.toEmbedding ofDual) (Icc b a)\n[PROOFSTEP]\nrefine' Eq.trans _ map_refl.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\n\u22a2 Icc (\u2191ofDual a) (\u2191ofDual b) = Icc b a\n[PROOFSTEP]\next c\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\nc : (fun x => \u03b1) a\n\u22a2 c \u2208 Icc (\u2191ofDual a) (\u2191ofDual b) \u2194 c \u2208 Icc b a\n[PROOFSTEP]\nrw [mem_Icc, mem_Icc (\u03b1 := \u03b1\u1d52\u1d48)]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\nc : (fun x => \u03b1) a\n\u22a2 \u2191ofDual a \u2264 c \u2227 c \u2264 \u2191ofDual b \u2194 b \u2264 c \u2227 c \u2264 a\n[PROOFSTEP]\nexact and_comm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\n\u22a2 Ico (\u2191ofDual a) (\u2191ofDual b) = map (Equiv.toEmbedding ofDual) (Ioc b a)\n[PROOFSTEP]\nrefine' Eq.trans _ map_refl.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\n\u22a2 Ico (\u2191ofDual a) (\u2191ofDual b) = Ioc b a\n[PROOFSTEP]\next c\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\nc : (fun x => \u03b1) a\n\u22a2 c \u2208 Ico (\u2191ofDual a) (\u2191ofDual b) \u2194 c \u2208 Ioc b a\n[PROOFSTEP]\nrw [mem_Ico, mem_Ioc (\u03b1 := \u03b1\u1d52\u1d48)]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\nc : (fun x => \u03b1) a\n\u22a2 \u2191ofDual a \u2264 c \u2227 c < \u2191ofDual b \u2194 b < c \u2227 c \u2264 a\n[PROOFSTEP]\nexact and_comm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\n\u22a2 Ioc (\u2191ofDual a) (\u2191ofDual b) = map (Equiv.toEmbedding ofDual) (Ico b a)\n[PROOFSTEP]\nrefine' Eq.trans _ map_refl.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\n\u22a2 Ioc (\u2191ofDual a) (\u2191ofDual b) = Ico b a\n[PROOFSTEP]\next c\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\nc : (fun x => \u03b1) a\n\u22a2 c \u2208 Ioc (\u2191ofDual a) (\u2191ofDual b) \u2194 c \u2208 Ico b a\n[PROOFSTEP]\nrw [mem_Ioc, mem_Ico (\u03b1 := \u03b1\u1d52\u1d48)]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\nc : (fun x => \u03b1) a\n\u22a2 \u2191ofDual a < c \u2227 c \u2264 \u2191ofDual b \u2194 b \u2264 c \u2227 c < a\n[PROOFSTEP]\nexact and_comm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\n\u22a2 Ioo (\u2191ofDual a) (\u2191ofDual b) = map (Equiv.toEmbedding ofDual) (Ioo b a)\n[PROOFSTEP]\nrefine' Eq.trans _ map_refl.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\n\u22a2 Ioo (\u2191ofDual a) (\u2191ofDual b) = Ioo b a\n[PROOFSTEP]\next c\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\nc : (fun x => \u03b1) a\n\u22a2 c \u2208 Ioo (\u2191ofDual a) (\u2191ofDual b) \u2194 c \u2208 Ioo b a\n[PROOFSTEP]\nrw [mem_Ioo, mem_Ioo (\u03b1 := \u03b1\u1d52\u1d48)]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\nc : (fun x => \u03b1) a\n\u22a2 \u2191ofDual a < c \u2227 c < \u2191ofDual b \u2194 b < c \u2227 c < a\n[PROOFSTEP]\nexact and_comm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : LocallyFiniteOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b2\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\na b x : \u03b1 \u00d7 \u03b2\n\u22a2 x \u2208 (fun a b => Icc a.fst b.fst \u00d7\u02e2 Icc a.snd b.snd) a b \u2194 a \u2264 x \u2227 x \u2264 b\n[PROOFSTEP]\nrw [mem_product, mem_Icc, mem_Icc, and_and_and_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : LocallyFiniteOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b2\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\na b x : \u03b1 \u00d7 \u03b2\n\u22a2 (a.fst \u2264 x.fst \u2227 a.snd \u2264 x.snd) \u2227 x.fst \u2264 b.fst \u2227 x.snd \u2264 b.snd \u2194 a \u2264 x \u2227 x \u2264 b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderTop \u03b2\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\na x : \u03b1 \u00d7 \u03b2\n\u22a2 x \u2208 (fun a => Ici a.fst \u00d7\u02e2 Ici a.snd) a \u2194 a \u2264 x\n[PROOFSTEP]\nrw [mem_product, mem_Ici, mem_Ici]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderTop \u03b2\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\na x : \u03b1 \u00d7 \u03b2\n\u22a2 a.fst \u2264 x.fst \u2227 a.snd \u2264 x.snd \u2194 a \u2264 x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : LocallyFiniteOrderBot \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b2\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\na x : \u03b1 \u00d7 \u03b2\n\u22a2 x \u2208 (fun a => Iic a.fst \u00d7\u02e2 Iic a.snd) a \u2194 x \u2264 a\n[PROOFSTEP]\nrw [mem_product, mem_Iic, mem_Iic]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : LocallyFiniteOrderBot \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b2\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\na x : \u03b1 \u00d7 \u03b2\n\u22a2 x.fst \u2264 a.fst \u2227 x.snd \u2264 a.snd \u2194 x \u2264 a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nx : \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 (\u2203 a, p a \u2227 Option.some a = Option.some x) \u2194 p x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nx : \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 (\u2203 a, p a \u2227 Option.some a = Option.some x) \u2192 p x\n[PROOFSTEP]\nrintro \u27e8x', hx, hx'\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nx : \u03b1\np : \u03b1 \u2192 Prop\nx' : \u03b1\nhx : p x'\nhx' : Option.some x' = Option.some x\n\u22a2 p x\n[PROOFSTEP]\nobtain rfl := Option.some_inj.mp hx'\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\np : \u03b1 \u2192 Prop\nx' : \u03b1\nhx : p x'\nhx' : Option.some x' = Option.some x'\n\u22a2 p x'\n[PROOFSTEP]\nexact hx\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nx : \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 p x \u2192 \u2203 a, p a \u2227 Option.some a = Option.some x\n[PROOFSTEP]\nexact fun h => \u27e8x, h, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b x : WithTop \u03b1\na : \u03b1\n\u22a2 \u22a4 \u2208\n (fun a b =>\n match a, b with\n | none, none => {\u22a4}\n | none, Option.some b => \u2205\n | Option.some a, none => \u2191insertNone (Ici a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Icc a b))\n \u2191a \u22a4 \u2194\n \u2191a \u2264 \u22a4 \u2227 \u22a4 \u2264 \u22a4\n[PROOFSTEP]\nsimp [WithTop.some, WithTop.top, insertNone]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b x\u271d : WithTop \u03b1\na x : \u03b1\n\u22a2 \u2191x \u2208\n (fun a b =>\n match a, b with\n | none, none => {\u22a4}\n | none, Option.some b => \u2205\n | Option.some a, none => \u2191insertNone (Ici a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Icc a b))\n \u2191a \u22a4 \u2194\n \u2191a \u2264 \u2191x \u2227 \u2191x \u2264 \u22a4\n[PROOFSTEP]\nsimp only [some, le_eq_subset, some_le_some, le_top, and_true]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b x\u271d : WithTop \u03b1\na x : \u03b1\n\u22a2 Option.some x \u2208 \u2191insertNone (Ici a) \u2194 a \u2264 x\n[PROOFSTEP]\nrw [some_mem_insertNone]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b x\u271d : WithTop \u03b1\na x : \u03b1\n\u22a2 x \u2208 Ici a \u2194 a \u2264 x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d x : WithTop \u03b1\na b : \u03b1\n\u22a2 \u22a4 \u2208\n (fun a b =>\n match a, b with\n | none, none => {\u22a4}\n | none, Option.some b => \u2205\n | Option.some a, none => \u2191insertNone (Ici a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Icc a b))\n \u2191a \u2191b \u2194\n \u2191a \u2264 \u22a4 \u2227 \u22a4 \u2264 \u2191b\n[PROOFSTEP]\nsimp only [some, le_eq_subset, mem_map, mem_Icc, le_top, top_le_iff, and_false, iff_false, not_exists, not_and, and_imp,\n Embedding.some, forall_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d x\u271d : WithTop \u03b1\na b x : \u03b1\n\u22a2 \u2191x \u2208\n (fun a b =>\n match a, b with\n | none, none => {\u22a4}\n | none, Option.some b => \u2205\n | Option.some a, none => \u2191insertNone (Ici a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Icc a b))\n \u2191a \u2191b \u2194\n \u2191a \u2264 \u2191x \u2227 \u2191x \u2264 \u2191b\n[PROOFSTEP]\nsimp only [some, le_eq_subset, Embedding.some, mem_map, mem_Icc, Embedding.coeFn_mk, some_le_some, aux]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b x : WithTop \u03b1\na : \u03b1\n\u22a2 \u22a4 \u2208\n (fun a b =>\n match a, b with\n | none, x => \u2205\n | Option.some a, none => Finset.map Embedding.some (Ici a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Ico a b))\n \u2191a \u22a4 \u2194\n \u2191a \u2264 \u22a4 \u2227 \u22a4 < \u22a4\n[PROOFSTEP]\nsimp [some, Embedding.some]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b x\u271d : WithTop \u03b1\na x : \u03b1\n\u22a2 \u2191x \u2208\n (fun a b =>\n match a, b with\n | none, x => \u2205\n | Option.some a, none => Finset.map Embedding.some (Ici a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Ico a b))\n \u2191a \u22a4 \u2194\n \u2191a \u2264 \u2191x \u2227 \u2191x < \u22a4\n[PROOFSTEP]\nsimp only [some, Embedding.some, mem_map, mem_Ici, Embedding.coeFn_mk, some_le_some, aux, top, some_lt_none, and_true]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d x : WithTop \u03b1\na b : \u03b1\n\u22a2 \u22a4 \u2208\n (fun a b =>\n match a, b with\n | none, x => \u2205\n | Option.some a, none => Finset.map Embedding.some (Ici a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Ico a b))\n \u2191a \u2191b \u2194\n \u2191a \u2264 \u22a4 \u2227 \u22a4 < \u2191b\n[PROOFSTEP]\nsimp [some, Embedding.some]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d x\u271d : WithTop \u03b1\na b x : \u03b1\n\u22a2 \u2191x \u2208\n (fun a b =>\n match a, b with\n | none, x => \u2205\n | Option.some a, none => Finset.map Embedding.some (Ici a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Ico a b))\n \u2191a \u2191b \u2194\n \u2191a \u2264 \u2191x \u2227 \u2191x < \u2191b\n[PROOFSTEP]\nsimp [some, Embedding.some, aux]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b x : WithTop \u03b1\na : \u03b1\n\u22a2 \u22a4 \u2208\n (fun a b =>\n match a, b with\n | none, x => \u2205\n | Option.some a, none => \u2191insertNone (Ioi a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Ioc a b))\n \u2191a \u22a4 \u2194\n \u2191a < \u22a4 \u2227 \u22a4 \u2264 \u22a4\n[PROOFSTEP]\nsimp [some, insertNone, top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b x\u271d : WithTop \u03b1\na x : \u03b1\n\u22a2 \u2191x \u2208\n (fun a b =>\n match a, b with\n | none, x => \u2205\n | Option.some a, none => \u2191insertNone (Ioi a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Ioc a b))\n \u2191a \u22a4 \u2194\n \u2191a < \u2191x \u2227 \u2191x \u2264 \u22a4\n[PROOFSTEP]\nsimp [some, Embedding.some, insertNone, aux]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d x : WithTop \u03b1\na b : \u03b1\n\u22a2 \u22a4 \u2208\n (fun a b =>\n match a, b with\n | none, x => \u2205\n | Option.some a, none => \u2191insertNone (Ioi a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Ioc a b))\n \u2191a \u2191b \u2194\n \u2191a < \u22a4 \u2227 \u22a4 \u2264 \u2191b\n[PROOFSTEP]\nsimp [some, Embedding.some, insertNone]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d x\u271d : WithTop \u03b1\na b x : \u03b1\n\u22a2 \u2191x \u2208\n (fun a b =>\n match a, b with\n | none, x => \u2205\n | Option.some a, none => \u2191insertNone (Ioi a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Ioc a b))\n \u2191a \u2191b \u2194\n \u2191a < \u2191x \u2227 \u2191x \u2264 \u2191b\n[PROOFSTEP]\nsimp [some, Embedding.some, insertNone, aux]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b x : WithTop \u03b1\na : \u03b1\n\u22a2 \u22a4 \u2208\n (fun a b =>\n match a, b with\n | none, x => \u2205\n | Option.some a, none => Finset.map Embedding.some (Ioi a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Ioo a b))\n \u2191a \u22a4 \u2194\n \u2191a < \u22a4 \u2227 \u22a4 < \u22a4\n[PROOFSTEP]\nsimp [some, Embedding.some, insertNone]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b x\u271d : WithTop \u03b1\na x : \u03b1\n\u22a2 \u2191x \u2208\n (fun a b =>\n match a, b with\n | none, x => \u2205\n | Option.some a, none => Finset.map Embedding.some (Ioi a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Ioo a b))\n \u2191a \u22a4 \u2194\n \u2191a < \u2191x \u2227 \u2191x < \u22a4\n[PROOFSTEP]\nsimp [some, Embedding.some, insertNone, aux, top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d x : WithTop \u03b1\na b : \u03b1\n\u22a2 \u22a4 \u2208\n (fun a b =>\n match a, b with\n | none, x => \u2205\n | Option.some a, none => Finset.map Embedding.some (Ioi a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Ioo a b))\n \u2191a \u2191b \u2194\n \u2191a < \u22a4 \u2227 \u22a4 < \u2191b\n[PROOFSTEP]\nsimp [some, Embedding.some, insertNone]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d x\u271d : WithTop \u03b1\na b x : \u03b1\n\u22a2 \u2191x \u2208\n (fun a b =>\n match a, b with\n | none, x => \u2205\n | Option.some a, none => Finset.map Embedding.some (Ioi a)\n | Option.some a, Option.some b => Finset.map Embedding.some (Ioo a b))\n \u2191a \u2191b \u2194\n \u2191a < \u2191x \u2227 \u2191x < \u2191b\n[PROOFSTEP]\nsimp [some, Embedding.some, insertNone, aux]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b2\nf : \u03b1 \u2243o \u03b2\n\u22a2 \u2200 (a b x : \u03b1), x \u2208 (fun a b => map (Equiv.toEmbedding (symm f).toEquiv) (Icc (\u2191f a) (\u2191f b))) a b \u2194 a \u2264 x \u2227 x \u2264 b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b2\nf : \u03b1 \u2243o \u03b2\n\u22a2 \u2200 (a b x : \u03b1), x \u2208 (fun a b => map (Equiv.toEmbedding (symm f).toEquiv) (Ico (\u2191f a) (\u2191f b))) a b \u2194 a \u2264 x \u2227 x < b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b2\nf : \u03b1 \u2243o \u03b2\n\u22a2 \u2200 (a b x : \u03b1), x \u2208 (fun a b => map (Equiv.toEmbedding (symm f).toEquiv) (Ioc (\u2191f a) (\u2191f b))) a b \u2194 a < x \u2227 x \u2264 b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrder \u03b2\nf : \u03b1 \u2243o \u03b2\n\u22a2 \u2200 (a b x : \u03b1), x \u2208 (fun a b => map (Equiv.toEmbedding (symm f).toEquiv) (Ioo (\u2191f a) (\u2191f b))) a b \u2194 a < x \u2227 x < b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrderTop \u03b2\nf : \u03b1 \u2243o \u03b2\n\u22a2 \u2200 (a x : \u03b1), x \u2208 (fun a => map (Equiv.toEmbedding (symm f).toEquiv) (Ici (\u2191f a))) a \u2194 a \u2264 x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrderTop \u03b2\nf : \u03b1 \u2243o \u03b2\n\u22a2 \u2200 (a x : \u03b1), x \u2208 (fun a => map (Equiv.toEmbedding (symm f).toEquiv) (Ioi (\u2191f a))) a \u2194 a < x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrderBot \u03b2\nf : \u03b1 \u2243o \u03b2\n\u22a2 \u2200 (a x : \u03b1), x \u2208 (fun a => map (Equiv.toEmbedding (symm f).toEquiv) (Iic (\u2191f a))) a \u2194 x \u2264 a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : LocallyFiniteOrderBot \u03b2\nf : \u03b1 \u2243o \u03b2\n\u22a2 \u2200 (a x : \u03b1), x \u2208 (fun a => map (Equiv.toEmbedding (symm f).toEquiv) (Iio (\u2191f a))) a \u2194 x < a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b x : Subtype p\n\u22a2 x \u2208 (fun a b => Finset.subtype p (Icc \u2191a \u2191b)) a b \u2194 a \u2264 x \u2227 x \u2264 b\n[PROOFSTEP]\nsimp_rw [Finset.mem_subtype, mem_Icc, Subtype.coe_le_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b x : Subtype p\n\u22a2 x \u2208 (fun a b => Finset.subtype p (Ico \u2191a \u2191b)) a b \u2194 a \u2264 x \u2227 x < b\n[PROOFSTEP]\nsimp_rw [Finset.mem_subtype, mem_Ico, Subtype.coe_le_coe, Subtype.coe_lt_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b x : Subtype p\n\u22a2 x \u2208 (fun a b => Finset.subtype p (Ioc \u2191a \u2191b)) a b \u2194 a < x \u2227 x \u2264 b\n[PROOFSTEP]\nsimp_rw [Finset.mem_subtype, mem_Ioc, Subtype.coe_le_coe, Subtype.coe_lt_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b x : Subtype p\n\u22a2 x \u2208 (fun a b => Finset.subtype p (Ioo \u2191a \u2191b)) a b \u2194 a < x \u2227 x < b\n[PROOFSTEP]\nsimp_rw [Finset.mem_subtype, mem_Ioo, Subtype.coe_lt_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderTop \u03b1\na x : Subtype p\n\u22a2 x \u2208 (fun a => Finset.subtype p (Ici \u2191a)) a \u2194 a \u2264 x\n[PROOFSTEP]\nsimp_rw [Finset.mem_subtype, mem_Ici, Subtype.coe_le_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderTop \u03b1\na x : Subtype p\n\u22a2 x \u2208 (fun a => Finset.subtype p (Ioi \u2191a)) a \u2194 a < x\n[PROOFSTEP]\nsimp_rw [Finset.mem_subtype, mem_Ioi, Subtype.coe_lt_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderBot \u03b1\na x : Subtype p\n\u22a2 x \u2208 (fun a => Finset.subtype p (Iic \u2191a)) a \u2194 x \u2264 a\n[PROOFSTEP]\nsimp_rw [Finset.mem_subtype, mem_Iic, Subtype.coe_le_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderBot \u03b1\na x : Subtype p\n\u22a2 x \u2208 (fun a => Finset.subtype p (Iio \u2191a)) a \u2194 x < a\n[PROOFSTEP]\nsimp_rw [Finset.mem_subtype, mem_Iio, Subtype.coe_lt_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\n\u22a2 map (Embedding.subtype p) (Icc a b) = Icc \u2191a \u2191b\n[PROOFSTEP]\nrw [subtype_Icc_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\n\u22a2 map (Embedding.subtype p) (Finset.subtype p (Icc \u2191a \u2191b)) = Icc \u2191a \u2191b\n[PROOFSTEP]\nrefine' Finset.subtype_map_of_mem fun x hx => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\nx : \u03b1\nhx : x \u2208 Icc \u2191a \u2191b\n\u22a2 p x\n[PROOFSTEP]\nrw [mem_Icc] at hx \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\nx : \u03b1\nhx : \u2191a \u2264 x \u2227 x \u2264 \u2191b\n\u22a2 p x\n[PROOFSTEP]\nexact hp hx.1 hx.2 a.prop b.prop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\n\u22a2 map (Embedding.subtype p) (Ico a b) = Ico \u2191a \u2191b\n[PROOFSTEP]\nrw [subtype_Ico_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\n\u22a2 map (Embedding.subtype p) (Finset.subtype p (Ico \u2191a \u2191b)) = Ico \u2191a \u2191b\n[PROOFSTEP]\nrefine' Finset.subtype_map_of_mem fun x hx => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\nx : \u03b1\nhx : x \u2208 Ico \u2191a \u2191b\n\u22a2 p x\n[PROOFSTEP]\nrw [mem_Ico] at hx \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\nx : \u03b1\nhx : \u2191a \u2264 x \u2227 x < \u2191b\n\u22a2 p x\n[PROOFSTEP]\nexact hp hx.1 hx.2.le a.prop b.prop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\n\u22a2 map (Embedding.subtype p) (Ioc a b) = Ioc \u2191a \u2191b\n[PROOFSTEP]\nrw [subtype_Ioc_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\n\u22a2 map (Embedding.subtype p) (Finset.subtype p (Ioc \u2191a \u2191b)) = Ioc \u2191a \u2191b\n[PROOFSTEP]\nrefine' Finset.subtype_map_of_mem fun x hx => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\nx : \u03b1\nhx : x \u2208 Ioc \u2191a \u2191b\n\u22a2 p x\n[PROOFSTEP]\nrw [mem_Ioc] at hx \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\nx : \u03b1\nhx : \u2191a < x \u2227 x \u2264 \u2191b\n\u22a2 p x\n[PROOFSTEP]\nexact hp hx.1.le hx.2 a.prop b.prop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\n\u22a2 map (Embedding.subtype p) (Ioo a b) = Ioo \u2191a \u2191b\n[PROOFSTEP]\nrw [subtype_Ioo_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\n\u22a2 map (Embedding.subtype p) (Finset.subtype p (Ioo \u2191a \u2191b)) = Ioo \u2191a \u2191b\n[PROOFSTEP]\nrefine' Finset.subtype_map_of_mem fun x hx => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\nx : \u03b1\nhx : x \u2208 Ioo \u2191a \u2191b\n\u22a2 p x\n[PROOFSTEP]\nrw [mem_Ioo] at hx \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrder \u03b1\na b : Subtype p\nhp : \u2200 \u2983a b x : \u03b1\u2984, a \u2264 x \u2192 x \u2264 b \u2192 p a \u2192 p b \u2192 p x\nx : \u03b1\nhx : \u2191a < x \u2227 x < \u2191b\n\u22a2 p x\n[PROOFSTEP]\nexact hp hx.1.le hx.2.le a.prop b.prop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderTop \u03b1\na : Subtype p\nhp : \u2200 \u2983a x : \u03b1\u2984, a \u2264 x \u2192 p a \u2192 p x\n\u22a2 map (Embedding.subtype p) (Ici a) = Ici \u2191a\n[PROOFSTEP]\nrw [subtype_Ici_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderTop \u03b1\na : Subtype p\nhp : \u2200 \u2983a x : \u03b1\u2984, a \u2264 x \u2192 p a \u2192 p x\n\u22a2 map (Embedding.subtype p) (Finset.subtype p (Ici \u2191a)) = Ici \u2191a\n[PROOFSTEP]\nexact Finset.subtype_map_of_mem fun x hx => hp (mem_Ici.1 hx) a.prop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderTop \u03b1\na : Subtype p\nhp : \u2200 \u2983a x : \u03b1\u2984, a \u2264 x \u2192 p a \u2192 p x\n\u22a2 map (Embedding.subtype p) (Ioi a) = Ioi \u2191a\n[PROOFSTEP]\nrw [subtype_Ioi_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderTop \u03b1\na : Subtype p\nhp : \u2200 \u2983a x : \u03b1\u2984, a \u2264 x \u2192 p a \u2192 p x\n\u22a2 map (Embedding.subtype p) (Finset.subtype p (Ioi \u2191a)) = Ioi \u2191a\n[PROOFSTEP]\nexact Finset.subtype_map_of_mem fun x hx => hp (mem_Ioi.1 hx).le a.prop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderBot \u03b1\na : Subtype p\nhp : \u2200 \u2983a x : \u03b1\u2984, x \u2264 a \u2192 p a \u2192 p x\n\u22a2 map (Embedding.subtype p) (Iic a) = Iic \u2191a\n[PROOFSTEP]\nrw [subtype_Iic_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderBot \u03b1\na : Subtype p\nhp : \u2200 \u2983a x : \u03b1\u2984, x \u2264 a \u2192 p a \u2192 p x\n\u22a2 map (Embedding.subtype p) (Finset.subtype p (Iic \u2191a)) = Iic \u2191a\n[PROOFSTEP]\nexact Finset.subtype_map_of_mem fun x hx => hp (mem_Iic.1 hx) a.prop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderBot \u03b1\na : Subtype p\nhp : \u2200 \u2983a x : \u03b1\u2984, x \u2264 a \u2192 p a \u2192 p x\n\u22a2 map (Embedding.subtype p) (Iio a) = Iio \u2191a\n[PROOFSTEP]\nrw [subtype_Iio_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderBot \u03b1\na : Subtype p\nhp : \u2200 \u2983a x : \u03b1\u2984, x \u2264 a \u2192 p a \u2192 p x\n\u22a2 map (Embedding.subtype p) (Finset.subtype p (Iio \u2191a)) = Iio \u2191a\n[PROOFSTEP]\nexact Finset.subtype_map_of_mem fun x hx => hp (mem_Iio.1 hx).le a.prop\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\u271d\np : \u03b1\u271d \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\n\u03b1 : Type u_3\ns : Set \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : OrderBot \u03b1\nh : Set.Finite s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 Finite.toFinset h\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\u271d\np : \u03b1\u271d \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\n\u03b1 : Type u_3\ns : Set \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\n\u22a2 Set.Finite s \u2194 BddBelow s \u2227 BddAbove s\n[PROOFSTEP]\nobtain (rfl | hs) := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\u271d\np : \u03b1\u271d \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\n\u03b1 : Type u_3\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\n\u22a2 Set.Finite \u2205 \u2194 BddBelow \u2205 \u2227 BddAbove \u2205\n[PROOFSTEP]\nsimp only [Set.finite_empty, bddBelow_empty, bddAbove_empty, and_self]\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\u271d\np : \u03b1\u271d \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\n\u03b1 : Type u_3\ns : Set \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nhs : Set.Nonempty s\n\u22a2 Set.Finite s \u2194 BddBelow s \u2227 BddAbove s\n[PROOFSTEP]\nexact\n \u27e8fun h \u21a6\n \u27e8\u27e8h.toFinset.inf' (by simpa) id, fun x hx \u21a6 Finset.inf'_le id (by simpa)\u27e9,\n \u27e8h.toFinset.sup' (by simpa) id, fun x hx \u21a6 Finset.le_sup' id (by simpa)\u27e9\u27e9,\n fun \u27e8\u27e8a, ha\u27e9, \u27e8b, hb\u27e9\u27e9 \u21a6 (Set.finite_Icc a b).subset (fun x hx \u21a6 \u27e8ha hx, hb hx\u27e9)\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\u271d\np : \u03b1\u271d \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\n\u03b1 : Type u_3\ns : Set \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nhs : Set.Nonempty s\nh : Set.Finite s\n\u22a2 Finset.Nonempty (Finite.toFinset h)\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\u271d\np : \u03b1\u271d \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\n\u03b1 : Type u_3\ns : Set \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nhs : Set.Nonempty s\nh : Set.Finite s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 Finite.toFinset h\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\u271d\np : \u03b1\u271d \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\n\u03b1 : Type u_3\ns : Set \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nhs : Set.Nonempty s\nh : Set.Finite s\n\u22a2 Finset.Nonempty (Finite.toFinset h)\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\u271d\np : \u03b1\u271d \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\n\u03b1 : Type u_3\ns : Set \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nhs : Set.Nonempty s\nh : Set.Finite s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 Finite.toFinset h\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na : { x // x \u2264 y }\n\u22a2 y \u2264 y\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na : { x // x \u2264 y }\n\u22a2 y \u2264 y\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // x \u2264 y }\n\u22a2 b \u2208 (fun a => Icc a { val := y, property := (_ : y \u2264 y) }) a \u2194 a \u2264 b\n[PROOFSTEP]\nsimp only [Finset.mem_Icc, and_iff_left_iff_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // x \u2264 y }\n\u22a2 a \u2264 b \u2192 b \u2264 { val := y, property := (_ : y \u2264 y) }\n[PROOFSTEP]\nexact fun _ => b.property\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // x \u2264 y }\n\u22a2 b \u2208 (fun a => Ioc a { val := y, property := (_ : y \u2264 y) }) a \u2194 a < b\n[PROOFSTEP]\nsimp only [Finset.mem_Ioc, and_iff_left_iff_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // x \u2264 y }\n\u22a2 a < b \u2192 b \u2264 { val := y, property := (_ : y \u2264 y) }\n[PROOFSTEP]\nexact fun _ => b.property\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // x < y }\n\u22a2 b \u2208 (fun a => Finset.subtype (fun x => x < y) (Ico (\u2191a) y)) a \u2194 a \u2264 b\n[PROOFSTEP]\nsimp only [Finset.mem_subtype, Finset.mem_Ico, Subtype.coe_le_coe, and_iff_left_iff_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // x < y }\n\u22a2 a \u2264 b \u2192 \u2191b < y\n[PROOFSTEP]\nexact fun _ => b.property\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // x < y }\n\u22a2 b \u2208 (fun a => Finset.subtype (fun x => x < y) (Ioo (\u2191a) y)) a \u2194 a < b\n[PROOFSTEP]\nsimp only [Finset.mem_subtype, Finset.mem_Ioo, Subtype.coe_lt_coe, and_iff_left_iff_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // x < y }\n\u22a2 a < b \u2192 \u2191b < y\n[PROOFSTEP]\nexact fun _ => b.property\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na : { x // y \u2264 x }\n\u22a2 y \u2264 y\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na : { x // y \u2264 x }\n\u22a2 y \u2264 y\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // y \u2264 x }\n\u22a2 b \u2208 (fun a => Icc { val := y, property := (_ : y \u2264 y) } a) a \u2194 b \u2264 a\n[PROOFSTEP]\nsimp only [Finset.mem_Icc, and_iff_right_iff_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // y \u2264 x }\n\u22a2 b \u2264 a \u2192 { val := y, property := (_ : y \u2264 y) } \u2264 b\n[PROOFSTEP]\nexact fun _ => b.property\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // y \u2264 x }\n\u22a2 b \u2208 (fun a => Ico { val := y, property := (_ : y \u2264 y) } a) a \u2194 b < a\n[PROOFSTEP]\nsimp only [Finset.mem_Ico, and_iff_right_iff_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // y \u2264 x }\n\u22a2 b < a \u2192 { val := y, property := (_ : y \u2264 y) } \u2264 b\n[PROOFSTEP]\nexact fun _ => b.property\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // y < x }\n\u22a2 b \u2208 (fun a => Finset.subtype (fun x => y < x) (Ioc y \u2191a)) a \u2194 b \u2264 a\n[PROOFSTEP]\nsimp only [Finset.mem_subtype, Finset.mem_Ioc, Subtype.coe_le_coe, and_iff_right_iff_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // y < x }\n\u22a2 b \u2264 a \u2192 y < \u2191b\n[PROOFSTEP]\nexact fun _ => b.property\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // y < x }\n\u22a2 b \u2208 (fun a => Finset.subtype (fun x => y < x) (Ioo y \u2191a)) a \u2194 b < a\n[PROOFSTEP]\nsimp only [Finset.mem_subtype, Finset.mem_Ioo, Subtype.coe_lt_coe, and_iff_right_iff_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : LocallyFiniteOrder \u03b1\na b : { x // y < x }\n\u22a2 b < a \u2192 y < \u2191b\n[PROOFSTEP]\nexact fun _ => b.property\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\n\u22a2 Finite { x // x \u2264 y }\n[PROOFSTEP]\napply Set.Finite.to_subtype\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\n\u22a2 Set.Finite fun x => Preorder.toLE.1 x y\n[PROOFSTEP]\nconvert (Finset.Iic y).finite_toSet using 1\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\n\u22a2 (fun x => Preorder.toLE.1 x y) = \u2191(Iic y)\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\nx\u271d : \u03b1\n\u22a2 (x\u271d \u2208 fun x => Preorder.toLE.1 x y) \u2194 x\u271d \u2208 \u2191(Iic y)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\nx\u271d : \u03b1\n\u22a2 (x\u271d \u2208 fun x => Preorder.toLE.1 x y) \u2194 x\u271d \u2264 y\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\n\u22a2 Finite { x // x < y }\n[PROOFSTEP]\napply Set.Finite.to_subtype\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\n\u22a2 Set.Finite fun x => Preorder.toLT.1 x y\n[PROOFSTEP]\nconvert (Finset.Iio y).finite_toSet using 1\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\n\u22a2 (fun x => Preorder.toLT.1 x y) = \u2191(Iio y)\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\nx\u271d : \u03b1\n\u22a2 (x\u271d \u2208 fun x => Preorder.toLT.1 x y) \u2194 x\u271d \u2208 \u2191(Iio y)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\nx\u271d : \u03b1\n\u22a2 (x\u271d \u2208 fun x => Preorder.toLT.1 x y) \u2194 x\u271d < y\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\n\u22a2 Finite { x // y \u2264 x }\n[PROOFSTEP]\napply Set.Finite.to_subtype\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\n\u22a2 Set.Finite fun x => Preorder.toLE.1 y x\n[PROOFSTEP]\nconvert (Finset.Ici y).finite_toSet using 1\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\n\u22a2 (fun x => Preorder.toLE.1 y x) = \u2191(Ici y)\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\nx\u271d : \u03b1\n\u22a2 (x\u271d \u2208 fun x => Preorder.toLE.1 y x) \u2194 x\u271d \u2208 \u2191(Ici y)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\nx\u271d : \u03b1\n\u22a2 (x\u271d \u2208 fun x => Preorder.toLE.1 y x) \u2194 y \u2264 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\n\u22a2 Finite { x // y < x }\n[PROOFSTEP]\napply Set.Finite.to_subtype\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\n\u22a2 Set.Finite fun x => Preorder.toLT.1 y x\n[PROOFSTEP]\nconvert (Finset.Ioi y).finite_toSet using 1\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\n\u22a2 (fun x => Preorder.toLT.1 y x) = \u2191(Ioi y)\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\nx\u271d : \u03b1\n\u22a2 (x\u271d \u2208 fun x => Preorder.toLT.1 y x) \u2194 x\u271d \u2208 \u2191(Ioi y)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ny : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\nx\u271d : \u03b1\n\u22a2 (x\u271d \u2208 fun x => Preorder.toLT.1 y x) \u2194 y < x\u271d\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Order.LocallyFinite", "llama_tokens": 41333, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300698514778, "lm_q2_score": 0.6261241632752914, "lm_q1q2_score": 0.4398710521614886}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : SMul \ud835\udd5c \u03b1\ninst\u271d : SMul \ud835\udd5c \u03b2\ns\u271d : Set E\nf : E \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\ns : Set \u03b2\nhs : Convex \ud835\udd5c s\n\u22a2 \u2200 \u2983x : \u03b2\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : \u03b2\u2984,\n y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 _root_.id (a \u2022 x + b \u2022 y) \u2264 a \u2022 _root_.id x + b \u2022 _root_.id y\n[PROOFSTEP]\nintros\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : SMul \ud835\udd5c \u03b1\ninst\u271d : SMul \ud835\udd5c \u03b2\ns\u271d : Set E\nf : E \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\ns : Set \u03b2\nhs : Convex \ud835\udd5c s\nx\u271d : \u03b2\na\u271d\u2075 : x\u271d \u2208 s\ny\u271d : \u03b2\na\u271d\u2074 : y\u271d \u2208 s\na\u271d\u00b3 b\u271d : \ud835\udd5c\na\u271d\u00b2 : 0 \u2264 a\u271d\u00b3\na\u271d\u00b9 : 0 \u2264 b\u271d\na\u271d : a\u271d\u00b3 + b\u271d = 1\n\u22a2 _root_.id (a\u271d\u00b3 \u2022 x\u271d + b\u271d \u2022 y\u271d) \u2264 a\u271d\u00b3 \u2022 _root_.id x\u271d + b\u271d \u2022 _root_.id y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : SMul \ud835\udd5c \u03b1\ninst\u271d : SMul \ud835\udd5c \u03b2\ns\u271d : Set E\nf : E \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\ns : Set \u03b2\nhs : Convex \ud835\udd5c s\n\u22a2 \u2200 \u2983x : \u03b2\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : \u03b2\u2984,\n y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 a \u2022 _root_.id x + b \u2022 _root_.id y \u2264 _root_.id (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nintros\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : SMul \ud835\udd5c \u03b1\ninst\u271d : SMul \ud835\udd5c \u03b2\ns\u271d : Set E\nf : E \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\ns : Set \u03b2\nhs : Convex \ud835\udd5c s\nx\u271d : \u03b2\na\u271d\u2075 : x\u271d \u2208 s\ny\u271d : \u03b2\na\u271d\u2074 : y\u271d \u2208 s\na\u271d\u00b3 b\u271d : \ud835\udd5c\na\u271d\u00b2 : 0 \u2264 a\u271d\u00b3\na\u271d\u00b9 : 0 \u2264 b\u271d\na\u271d : a\u271d\u00b3 + b\u271d = 1\n\u22a2 a\u271d\u00b3 \u2022 _root_.id x\u271d + b\u271d \u2022 _root_.id y\u271d \u2264 _root_.id (a\u271d\u00b3 \u2022 x\u271d + b\u271d \u2022 y\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : DistribMulAction \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nhg : ConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 f x + b \u2022 f y + (a \u2022 g x + b \u2022 g y) = a \u2022 (f x + g x) + b \u2022 (f y + g y)\n[PROOFSTEP]\nrw [smul_add, smul_add, add_add_add_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nr : \u03b2\nx : E\nhx : x \u2208 {x | x \u2208 s \u2227 f x \u2264 r}\ny : E\nhy : y \u2208 {x | x \u2208 s \u2227 f x \u2264 r}\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 f x + b \u2022 f y \u2264 a \u2022 r + b \u2022 r\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nr : \u03b2\nx : E\nhx : x \u2208 {x | x \u2208 s \u2227 f x \u2264 r}\ny : E\nhy : y \u2208 {x | x \u2208 s \u2227 f x \u2264 r}\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 f x \u2264 r\ncase h\u2082.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nr : \u03b2\nx : E\nhx : x \u2208 {x | x \u2208 s \u2227 f x \u2264 r}\ny : E\nhy : y \u2208 {x | x \u2208 s \u2227 f x \u2264 r}\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 f y \u2264 r\n[PROOFSTEP]\nexact hx.2\n[GOAL]\ncase h\u2082.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nr : \u03b2\nx : E\nhx : x \u2208 {x | x \u2208 s \u2227 f x \u2264 r}\ny : E\nhy : y \u2208 {x | x \u2208 s \u2227 f x \u2264 r}\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 f y \u2264 r\n[PROOFSTEP]\nexact hy.2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\n\u22a2 Convex \ud835\udd5c {p | p.fst \u2208 s \u2227 f p.fst \u2264 p.snd}\n[PROOFSTEP]\nrintro \u27e8x, r\u27e9 \u27e8hx, hr\u27e9 \u27e8y, t\u27e9 \u27e8hy, ht\u27e9 a b ha hb hab\n[GOAL]\ncase mk.intro.mk.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx : E\nr : \u03b2\nhx : (x, r).fst \u2208 s\nhr : f (x, r).fst \u2264 (x, r).snd\ny : E\nt : \u03b2\nhy : (y, t).fst \u2208 s\nht : f (y, t).fst \u2264 (y, t).snd\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 (x, r) + b \u2022 (y, t) \u2208 {p | p.fst \u2208 s \u2227 f p.fst \u2264 p.snd}\n[PROOFSTEP]\nrefine' \u27e8hf.1 hx hy ha hb hab, _\u27e9\n[GOAL]\ncase mk.intro.mk.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx : E\nr : \u03b2\nhx : (x, r).fst \u2208 s\nhr : f (x, r).fst \u2264 (x, r).snd\ny : E\nt : \u03b2\nhy : (y, t).fst \u2208 s\nht : f (y, t).fst \u2264 (y, t).snd\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 f (a \u2022 (x, r) + b \u2022 (y, t)).fst \u2264 (a \u2022 (x, r) + b \u2022 (y, t)).snd\n[PROOFSTEP]\ncalc\n f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y := hf.2 hx hy ha hb hab\n _ \u2264 a \u2022 r + b \u2022 t := by gcongr\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx : E\nr : \u03b2\nhx : (x, r).fst \u2208 s\nhr : f (x, r).fst \u2264 (x, r).snd\ny : E\nt : \u03b2\nhy : (y, t).fst \u2208 s\nht : f (y, t).fst \u2264 (y, t).snd\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 f x + b \u2022 f y \u2264 a \u2022 r + b \u2022 t\n[PROOFSTEP]\ngcongr\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nc x : E\nhx : x \u2208 (fun z => c + z) \u207b\u00b9' s\ny : E\nhy : y \u2208 (fun z => c + z) \u207b\u00b9' s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 f (c + (a \u2022 x + b \u2022 y)) = f (a \u2022 (c + x) + b \u2022 (c + y))\n[PROOFSTEP]\nrw [smul_add, smul_add, add_add_add_comm, Convex.combo_self hab]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nc : E\n\u22a2 ConvexOn \ud835\udd5c ((fun z => c + z) \u207b\u00b9' s) (f \u2218 fun z => z + c)\n[PROOFSTEP]\nsimpa only [add_comm c] using hf.translate_right c\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\n\u22a2 ConvexOn \ud835\udd5c s f \u2194\n Convex \ud835\udd5c s \u2227\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nrefine' and_congr_right' \u27e8fun h x hx y hy a b ha hb hab => h hx hy ha.le hb.le hab, fun h x hx y hy a b ha hb hab => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nobtain rfl | ha' := ha.eq_or_lt\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nb : \ud835\udd5c\nhb : 0 \u2264 b\nha : 0 \u2264 0\nhab : 0 + b = 1\n\u22a2 f (0 \u2022 x + b \u2022 y) \u2264 0 \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nrw [zero_add] at hab \n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nb : \ud835\udd5c\nhb : 0 \u2264 b\nha : 0 \u2264 0\nhab : b = 1\n\u22a2 f (0 \u2022 x + b \u2022 y) \u2264 0 \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nha : 0 \u2264 0\nhb : 0 \u2264 1\n\u22a2 f (0 \u2022 x + 1 \u2022 y) \u2264 0 \u2022 f x + 1 \u2022 f y\n[PROOFSTEP]\nsimp_rw [zero_smul, zero_add, one_smul, le_rfl]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nha' : 0 < a\n\u22a2 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nobtain rfl | hb' := hb.eq_or_lt\n[GOAL]\ncase inr.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na : \ud835\udd5c\nha : 0 \u2264 a\nha' : 0 < a\nhb : 0 \u2264 0\nhab : a + 0 = 1\n\u22a2 f (a \u2022 x + 0 \u2022 y) \u2264 a \u2022 f x + 0 \u2022 f y\n[PROOFSTEP]\nrw [add_zero] at hab \n[GOAL]\ncase inr.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na : \ud835\udd5c\nha : 0 \u2264 a\nha' : 0 < a\nhb : 0 \u2264 0\nhab : a = 1\n\u22a2 f (a \u2022 x + 0 \u2022 y) \u2264 a \u2022 f x + 0 \u2022 f y\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inr.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhb : 0 \u2264 0\nha : 0 \u2264 1\nha' : 0 < 1\n\u22a2 f (1 \u2022 x + 0 \u2022 y) \u2264 1 \u2022 f x + 0 \u2022 f y\n[PROOFSTEP]\nsimp_rw [zero_smul, add_zero, one_smul, le_rfl]\n[GOAL]\ncase inr.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nha' : 0 < a\nhb' : 0 < b\n\u22a2 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nexact h hx hy ha' hb' hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\n\u22a2 ConvexOn \ud835\udd5c s f \u2194\n Convex \ud835\udd5c s \u2227\n Set.Pairwise s fun x y => \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nrw [convexOn_iff_forall_pos]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\n\u22a2 (Convex \ud835\udd5c s \u2227\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y) \u2194\n Convex \ud835\udd5c s \u2227\n Set.Pairwise s fun x y => \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nrefine' and_congr_right' \u27e8fun h x hx y hy _ a b ha hb hab => h hx hy ha hb hab, fun h x hx y hy a b ha hb hab => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh : Set.Pairwise s fun x y => \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nobtain rfl | hxy := eq_or_ne x y\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh : Set.Pairwise s fun x y => \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhy : x \u2208 s\n\u22a2 f (a \u2022 x + b \u2022 x) \u2264 a \u2022 f x + b \u2022 f x\n[PROOFSTEP]\nrw [Convex.combo_self hab, Convex.combo_self hab]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh : Set.Pairwise s fun x y => \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhxy : x \u2260 y\n\u22a2 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nexact h hx hy hxy ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\nf : E \u2192\u2097[\ud835\udd5c] \u03b2\ns : Set E\nhs : Convex \ud835\udd5c s\nx\u271d\u2078 : E\nx\u271d\u2077 : x\u271d\u2078 \u2208 s\nx\u271d\u2076 : E\nx\u271d\u2075 : x\u271d\u2076 \u2208 s\nx\u271d\u2074 x\u271d\u00b3 : \ud835\udd5c\nx\u271d\u00b2 : 0 \u2264 x\u271d\u2074\nx\u271d\u00b9 : 0 \u2264 x\u271d\u00b3\nx\u271d : x\u271d\u2074 + x\u271d\u00b3 = 1\n\u22a2 \u2191f (x\u271d\u2074 \u2022 x\u271d\u2078 + x\u271d\u00b3 \u2022 x\u271d\u2076) \u2264 x\u271d\u2074 \u2022 \u2191f x\u271d\u2078 + x\u271d\u00b3 \u2022 \u2191f x\u271d\u2076\n[PROOFSTEP]\nrw [f.map_add, f.map_smul, f.map_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\nf : E \u2192\u2097[\ud835\udd5c] \u03b2\ns : Set E\nhs : Convex \ud835\udd5c s\nx\u271d\u2078 : E\nx\u271d\u2077 : x\u271d\u2078 \u2208 s\nx\u271d\u2076 : E\nx\u271d\u2075 : x\u271d\u2076 \u2208 s\nx\u271d\u2074 x\u271d\u00b3 : \ud835\udd5c\nx\u271d\u00b2 : 0 \u2264 x\u271d\u2074\nx\u271d\u00b9 : 0 \u2264 x\u271d\u00b3\nx\u271d : x\u271d\u2074 + x\u271d\u00b3 = 1\n\u22a2 x\u271d\u2074 \u2022 \u2191f x\u271d\u2078 + x\u271d\u00b3 \u2022 \u2191f x\u271d\u2076 \u2264 \u2191f (x\u271d\u2074 \u2022 x\u271d\u2078 + x\u271d\u00b3 \u2022 x\u271d\u2076)\n[PROOFSTEP]\nrw [f.map_add, f.map_smul, f.map_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nr : \u03b2\nx : E\nhx : x \u2208 {x | x \u2208 s \u2227 f x < r}\ny : E\nhy : y \u2208 {x | x \u2208 s \u2227 f x < r}\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 f x + b \u2022 f y \u2264 a \u2022 r + b \u2022 r\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nr : \u03b2\nx : E\nhx : x \u2208 {x | x \u2208 s \u2227 f x < r}\ny : E\nhy : y \u2208 {x | x \u2208 s \u2227 f x < r}\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f x \u2264 r\ncase h\u2082.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nr : \u03b2\nx : E\nhx : x \u2208 {x | x \u2208 s \u2227 f x < r}\ny : E\nhy : y \u2208 {x | x \u2208 s \u2227 f x < r}\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f y \u2264 r\n[PROOFSTEP]\nexact hx.2.le\n[GOAL]\ncase h\u2082.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nr : \u03b2\nx : E\nhx : x \u2208 {x | x \u2208 s \u2227 f x < r}\ny : E\nhy : y \u2208 {x | x \u2208 s \u2227 f x < r}\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f y \u2264 r\n[PROOFSTEP]\nexact hy.2.le\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : LinearOrder E\ns : Set E\nf : E \u2192 \u03b2\nhs : Convex \ud835\udd5c s\nhf :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\n\u22a2 ConvexOn \ud835\udd5c s f\n[PROOFSTEP]\nrefine'\n convexOn_iff_pairwise_pos.2\n \u27e8hs, fun x hx y hy hxy a b ha hb hab => _\u27e9\n -- Porting note: without clearing the stray variables, `wlog` gives a bad term.\n -- See https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/wlog.20.2316495\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : LinearOrder E\ns : Set E\nf : E \u2192 \u03b2\nhs : Convex \ud835\udd5c s\nhf :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nclear! \u03b1 F \u03b9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b2 : Type u_5\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : LinearOrder E\ns : Set E\nf : E \u2192 \u03b2\nhs : Convex \ud835\udd5c s\nhf :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nwlog h : x < y\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b2 : Type u_5\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : LinearOrder E\ns : Set E\nf : E \u2192 \u03b2\nhs : Convex \ud835\udd5c s\nhf :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nthis :\n \u2200 {\ud835\udd5c : Type u_1} {E : Type u_2} {\u03b2 : Type u_5} [inst : OrderedSemiring \ud835\udd5c] [inst_1 : AddCommMonoid E]\n [inst_2 : OrderedAddCommMonoid \u03b2] [inst_3 : Module \ud835\udd5c E] [inst_4 : Module \ud835\udd5c \u03b2] [inst_5 : LinearOrder E] {s : Set E}\n {f : E \u2192 \u03b2},\n Convex \ud835\udd5c s \u2192\n (\u2200 \u2983x : E\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y) \u2192\n \u2200 (x : E),\n x \u2208 s \u2192\n \u2200 (y : E),\n y \u2208 s \u2192 x \u2260 y \u2192 \u2200 (a b : \ud835\udd5c), 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 x < y \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nh : \u00acx < y\n\u22a2 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nrw [add_comm (a \u2022 x), add_comm (a \u2022 f x)]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b2 : Type u_5\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : LinearOrder E\ns : Set E\nf : E \u2192 \u03b2\nhs : Convex \ud835\udd5c s\nhf :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nthis :\n \u2200 {\ud835\udd5c : Type u_1} {E : Type u_2} {\u03b2 : Type u_5} [inst : OrderedSemiring \ud835\udd5c] [inst_1 : AddCommMonoid E]\n [inst_2 : OrderedAddCommMonoid \u03b2] [inst_3 : Module \ud835\udd5c E] [inst_4 : Module \ud835\udd5c \u03b2] [inst_5 : LinearOrder E] {s : Set E}\n {f : E \u2192 \u03b2},\n Convex \ud835\udd5c s \u2192\n (\u2200 \u2983x : E\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y) \u2192\n \u2200 (x : E),\n x \u2208 s \u2192\n \u2200 (y : E),\n y \u2208 s \u2192 x \u2260 y \u2192 \u2200 (a b : \ud835\udd5c), 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 x < y \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nh : \u00acx < y\n\u22a2 f (b \u2022 y + a \u2022 x) \u2264 b \u2022 f y + a \u2022 f x\n[PROOFSTEP]\nrw [add_comm] at hab \n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b2 : Type u_5\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : LinearOrder E\ns : Set E\nf : E \u2192 \u03b2\nhs : Convex \ud835\udd5c s\nhf :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : b + a = 1\nthis :\n \u2200 {\ud835\udd5c : Type u_1} {E : Type u_2} {\u03b2 : Type u_5} [inst : OrderedSemiring \ud835\udd5c] [inst_1 : AddCommMonoid E]\n [inst_2 : OrderedAddCommMonoid \u03b2] [inst_3 : Module \ud835\udd5c E] [inst_4 : Module \ud835\udd5c \u03b2] [inst_5 : LinearOrder E] {s : Set E}\n {f : E \u2192 \u03b2},\n Convex \ud835\udd5c s \u2192\n (\u2200 \u2983x : E\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y) \u2192\n \u2200 (x : E),\n x \u2208 s \u2192\n \u2200 (y : E),\n y \u2208 s \u2192 x \u2260 y \u2192 \u2200 (a b : \ud835\udd5c), 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 x < y \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nh : \u00acx < y\n\u22a2 f (b \u2022 y + a \u2022 x) \u2264 b \u2022 f y + a \u2022 f x\n[PROOFSTEP]\nrefine' this hs hf y hy x hx hxy.symm b a hb ha hab (hxy.lt_or_lt.resolve_left h)\n[GOAL]\n\ud835\udd5c\u271d : Type u_1\nE\u271d : Type u_2\n\u03b2\u271d : Type u_5\ninst\u271d\u00b9\u00b9 : OrderedSemiring \ud835\udd5c\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid E\u271d\ninst\u271d\u2079 : OrderedAddCommMonoid \u03b2\u271d\ninst\u271d\u2078 : Module \ud835\udd5c\u271d E\u271d\ninst\u271d\u2077 : Module \ud835\udd5c\u271d \u03b2\u271d\ninst\u271d\u2076 : LinearOrder E\u271d\ns\u271d : Set E\u271d\nf\u271d : E\u271d \u2192 \u03b2\u271d\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b2 : Type u_5\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : LinearOrder E\ns : Set E\nf : E \u2192 \u03b2\nhs : Convex \ud835\udd5c s\nhf :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nh : x < y\n\u22a2 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nexact hf hx hy h ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : LinearOrder E\ns : Set E\nf : E \u2192 \u03b2\nhs : Convex \ud835\udd5c s\nhf :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\n\u22a2 StrictConvexOn \ud835\udd5c s f\n[PROOFSTEP]\nrefine'\n \u27e8hs, fun x hx y hy hxy a b ha hb hab => _\u27e9\n -- Porting note: without clearing the stray variables, `wlog` gives a bad term.\n -- See https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/wlog.20.2316495\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : LinearOrder E\ns : Set E\nf : E \u2192 \u03b2\nhs : Convex \ud835\udd5c s\nhf :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nclear! \u03b1 F \u03b9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b2 : Type u_5\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : LinearOrder E\ns : Set E\nf : E \u2192 \u03b2\nhs : Convex \ud835\udd5c s\nhf :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nwlog h : x < y\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b2 : Type u_5\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : LinearOrder E\ns : Set E\nf : E \u2192 \u03b2\nhs : Convex \ud835\udd5c s\nhf :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nthis :\n \u2200 {\ud835\udd5c : Type u_1} {E : Type u_2} {\u03b2 : Type u_5} [inst : OrderedSemiring \ud835\udd5c] [inst_1 : AddCommMonoid E]\n [inst_2 : OrderedAddCommMonoid \u03b2] [inst_3 : Module \ud835\udd5c E] [inst_4 : Module \ud835\udd5c \u03b2] [inst_5 : LinearOrder E] {s : Set E}\n {f : E \u2192 \u03b2},\n Convex \ud835\udd5c s \u2192\n (\u2200 \u2983x : E\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y) \u2192\n \u2200 (x : E),\n x \u2208 s \u2192\n \u2200 (y : E),\n y \u2208 s \u2192 x \u2260 y \u2192 \u2200 (a b : \ud835\udd5c), 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 x < y \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\nh : \u00acx < y\n\u22a2 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nrw [add_comm (a \u2022 x), add_comm (a \u2022 f x)]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b2 : Type u_5\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : LinearOrder E\ns : Set E\nf : E \u2192 \u03b2\nhs : Convex \ud835\udd5c s\nhf :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nthis :\n \u2200 {\ud835\udd5c : Type u_1} {E : Type u_2} {\u03b2 : Type u_5} [inst : OrderedSemiring \ud835\udd5c] [inst_1 : AddCommMonoid E]\n [inst_2 : OrderedAddCommMonoid \u03b2] [inst_3 : Module \ud835\udd5c E] [inst_4 : Module \ud835\udd5c \u03b2] [inst_5 : LinearOrder E] {s : Set E}\n {f : E \u2192 \u03b2},\n Convex \ud835\udd5c s \u2192\n (\u2200 \u2983x : E\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y) \u2192\n \u2200 (x : E),\n x \u2208 s \u2192\n \u2200 (y : E),\n y \u2208 s \u2192 x \u2260 y \u2192 \u2200 (a b : \ud835\udd5c), 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 x < y \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\nh : \u00acx < y\n\u22a2 f (b \u2022 y + a \u2022 x) < b \u2022 f y + a \u2022 f x\n[PROOFSTEP]\nrw [add_comm] at hab \n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b2 : Type u_5\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : LinearOrder E\ns : Set E\nf : E \u2192 \u03b2\nhs : Convex \ud835\udd5c s\nhf :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : b + a = 1\nthis :\n \u2200 {\ud835\udd5c : Type u_1} {E : Type u_2} {\u03b2 : Type u_5} [inst : OrderedSemiring \ud835\udd5c] [inst_1 : AddCommMonoid E]\n [inst_2 : OrderedAddCommMonoid \u03b2] [inst_3 : Module \ud835\udd5c E] [inst_4 : Module \ud835\udd5c \u03b2] [inst_5 : LinearOrder E] {s : Set E}\n {f : E \u2192 \u03b2},\n Convex \ud835\udd5c s \u2192\n (\u2200 \u2983x : E\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y) \u2192\n \u2200 (x : E),\n x \u2208 s \u2192\n \u2200 (y : E),\n y \u2208 s \u2192 x \u2260 y \u2192 \u2200 (a b : \ud835\udd5c), 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 x < y \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\nh : \u00acx < y\n\u22a2 f (b \u2022 y + a \u2022 x) < b \u2022 f y + a \u2022 f x\n[PROOFSTEP]\nrefine' this hs hf y hy x hx hxy.symm b a hb ha hab (hxy.lt_or_lt.resolve_left h)\n[GOAL]\n\ud835\udd5c\u271d : Type u_1\nE\u271d : Type u_2\n\u03b2\u271d : Type u_5\ninst\u271d\u00b9\u00b9 : OrderedSemiring \ud835\udd5c\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid E\u271d\ninst\u271d\u2079 : OrderedAddCommMonoid \u03b2\u271d\ninst\u271d\u2078 : Module \ud835\udd5c\u271d E\u271d\ninst\u271d\u2077 : Module \ud835\udd5c\u271d \u03b2\u271d\ninst\u271d\u2076 : LinearOrder E\u271d\ns\u271d : Set E\u271d\nf\u271d : E\u271d \u2192 \u03b2\u271d\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b2 : Type u_5\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : LinearOrder E\ns : Set E\nf : E \u2192 \u03b2\nhs : Convex \ud835\udd5c s\nhf :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nh : x < y\n\u22a2 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nexact hf hx hy h ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2077 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2076 : AddCommMonoid E\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : F \u2192 \u03b2\ns : Set F\nhf : ConvexOn \ud835\udd5c s f\ng : E \u2192\u2097[\ud835\udd5c] F\nx : E\nhx : x \u2208 \u2191g \u207b\u00b9' s\ny : E\nhy : y \u2208 \u2191g \u207b\u00b9' s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 f (\u2191g (a \u2022 x + b \u2022 y)) = f (a \u2022 \u2191g x + b \u2022 \u2191g y)\n[PROOFSTEP]\nrw [g.map_add, g.map_smul, g.map_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : DistribMulAction \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nhg : ConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 f x + b \u2022 f y + (a \u2022 g x + b \u2022 g y) = a \u2022 (f x + g x) + b \u2022 (f y + g y)\n[PROOFSTEP]\nrw [smul_add, smul_add, add_add_add_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : DistribMulAction \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nhg : StrictConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 f x + b \u2022 f y + (a \u2022 g x + b \u2022 g y) = a \u2022 (f x + g x) + b \u2022 (f y + g y)\n[PROOFSTEP]\nrw [smul_add, smul_add, add_add_add_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\np q : E \u00d7 \u03b2\nhp : p.fst \u2208 s \u2227 f p.fst < p.snd\nhq : q.fst \u2208 s \u2227 f q.fst \u2264 q.snd\n\u22a2 openSegment \ud835\udd5c p q \u2286 {p | p.fst \u2208 s \u2227 f p.fst < p.snd}\n[PROOFSTEP]\nrintro _ \u27e8a, b, ha, hb, hab, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\np q : E \u00d7 \u03b2\nhp : p.fst \u2208 s \u2227 f p.fst < p.snd\nhq : q.fst \u2208 s \u2227 f q.fst \u2264 q.snd\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 p + b \u2022 q \u2208 {p | p.fst \u2208 s \u2227 f p.fst < p.snd}\n[PROOFSTEP]\nrefine' \u27e8hf.1 hp.1 hq.1 ha.le hb.le hab, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\np q : E \u00d7 \u03b2\nhp : p.fst \u2208 s \u2227 f p.fst < p.snd\nhq : q.fst \u2208 s \u2227 f q.fst \u2264 q.snd\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f (a \u2022 p + b \u2022 q).fst < (a \u2022 p + b \u2022 q).snd\n[PROOFSTEP]\ncalc\n f (a \u2022 p.1 + b \u2022 q.1) \u2264 a \u2022 f p.1 + b \u2022 f q.1 := hf.2 hp.1 hq.1 ha.le hb.le hab\n _ < a \u2022 p.2 + b \u2022 q.2 := add_lt_add_of_lt_of_le (smul_lt_smul_of_pos hp.2 ha) (smul_le_smul_of_nonneg hq.2 hb.le)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nhg : ConvexOn \ud835\udd5c s g\n\u22a2 ConvexOn \ud835\udd5c s (f \u2294 g)\n[PROOFSTEP]\nrefine' \u27e8hf.left, fun x hx y hy a b ha hb hab => sup_le _ _\u27e9\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nhg : ConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 (f \u2294 g) x + b \u2022 (f \u2294 g) y\n[PROOFSTEP]\ncalc\n f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y := hf.right hx hy ha hb hab\n _ \u2264 a \u2022 (f x \u2294 g x) + b \u2022 (f y \u2294 g y) := by gcongr <;> apply le_sup_left\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nhg : ConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 f x + b \u2022 f y \u2264 a \u2022 (f x \u2294 g x) + b \u2022 (f y \u2294 g y)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nhg : ConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 f x \u2264 f x \u2294 g x\n[PROOFSTEP]\napply le_sup_left\n[GOAL]\ncase h\u2082.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nhg : ConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 f y \u2264 f y \u2294 g y\n[PROOFSTEP]\napply le_sup_left\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nhg : ConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 g (a \u2022 x + b \u2022 y) \u2264 a \u2022 (f \u2294 g) x + b \u2022 (f \u2294 g) y\n[PROOFSTEP]\ncalc\n g (a \u2022 x + b \u2022 y) \u2264 a \u2022 g x + b \u2022 g y := hg.right hx hy ha hb hab\n _ \u2264 a \u2022 (f x \u2294 g x) + b \u2022 (f y \u2294 g y) := by gcongr <;> apply le_sup_right\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nhg : ConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 g x + b \u2022 g y \u2264 a \u2022 (f x \u2294 g x) + b \u2022 (f y \u2294 g y)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nhg : ConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 g x \u2264 f x \u2294 g x\n[PROOFSTEP]\napply le_sup_right\n[GOAL]\ncase h\u2082.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nhg : ConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 g y \u2264 f y \u2294 g y\n[PROOFSTEP]\napply le_sup_right\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nhg : StrictConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 f x + b \u2022 f y \u2264 a \u2022 (f x \u2294 g x) + b \u2022 (f y \u2294 g y)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nhg : StrictConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f x \u2264 f x \u2294 g x\n[PROOFSTEP]\napply le_sup_left\n[GOAL]\ncase h\u2082.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nhg : StrictConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f y \u2264 f y \u2294 g y\n[PROOFSTEP]\napply le_sup_left\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nhg : StrictConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 g x + b \u2022 g y \u2264 a \u2022 (f x \u2294 g x) + b \u2022 (f y \u2294 g y)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nhg : StrictConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 g x \u2264 f x \u2294 g x\n[PROOFSTEP]\napply le_sup_right\n[GOAL]\ncase h\u2082.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nhg : StrictConvexOn \ud835\udd5c s g\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 g y \u2264 f y \u2294 g y\n[PROOFSTEP]\napply le_sup_right\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx y : E\nhx : x \u2208 s\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 f x + b \u2022 f y \u2264 a \u2022 max (f x) (f y) + b \u2022 max (f x) (f y)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx y : E\nhx : x \u2208 s\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 f x \u2264 max (f x) (f y)\n[PROOFSTEP]\napply le_max_left\n[GOAL]\ncase h\u2082.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx y : E\nhx : x \u2208 s\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 f y \u2264 max (f x) (f y)\n[PROOFSTEP]\napply le_max_right\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nx y : E\nhx : x \u2208 s\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 f x + b \u2022 f y \u2264 a \u2022 max (f x) (f y) + b \u2022 max (f x) (f y)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nx y : E\nhx : x \u2208 s\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f x \u2264 max (f x) (f y)\n[PROOFSTEP]\napply le_max_left\n[GOAL]\ncase h\u2082.h\u2081\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nx y : E\nhx : x \u2208 s\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f y \u2264 max (f x) (f y)\n[PROOFSTEP]\napply le_max_right\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx y : E\na b : \ud835\udd5c\nhx : x \u2208 s\nhy : y \u2208 s\nha : 0 \u2264 a\nhb : 0 < b\nhab : a + b = 1\nhfx : f x \u2264 f (a \u2022 x + b \u2022 y)\n\u22a2 f (a \u2022 x + b \u2022 y) \u2264 f y\n[PROOFSTEP]\nrw [add_comm] at hab hfx \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx y : E\na b : \ud835\udd5c\nhx : x \u2208 s\nhy : y \u2208 s\nha : 0 \u2264 a\nhb : 0 < b\nhab : b + a = 1\nhfx : f x \u2264 f (b \u2022 y + a \u2022 x)\n\u22a2 f (b \u2022 y + a \u2022 x) \u2264 f y\n[PROOFSTEP]\nexact hf.le_left_of_right_le' hy hx hb ha hab hfx\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx y z : E\nhx : x \u2208 s\nhy : y \u2208 s\nhz : z \u2208 openSegment \ud835\udd5c x y\nhyz : f y \u2264 f z\n\u22a2 f z \u2264 f x\n[PROOFSTEP]\nobtain \u27e8a, b, ha, hb, hab, rfl\u27e9 := hz\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx y : E\nhx : x \u2208 s\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhyz : f y \u2264 f (a \u2022 x + b \u2022 y)\n\u22a2 f (a \u2022 x + b \u2022 y) \u2264 f x\n[PROOFSTEP]\nexact hf.le_left_of_right_le' hx hy ha hb.le hab hyz\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx y z : E\nhx : x \u2208 s\nhy : y \u2208 s\nhz : z \u2208 openSegment \ud835\udd5c x y\nhxz : f x \u2264 f z\n\u22a2 f z \u2264 f y\n[PROOFSTEP]\nobtain \u27e8a, b, ha, hb, hab, rfl\u27e9 := hz\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx y : E\nhx : x \u2208 s\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhxz : f x \u2264 f (a \u2022 x + b \u2022 y)\n\u22a2 f (a \u2022 x + b \u2022 y) \u2264 f y\n[PROOFSTEP]\nexact hf.le_right_of_left_le' hx hy ha.le hb hab hxz\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx y : E\na b : \ud835\udd5c\nhx : x \u2208 s\nhy : y \u2208 s\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhfx : f x < f (a \u2022 x + b \u2022 y)\n\u22a2 f (a \u2022 x + b \u2022 y) < f y\n[PROOFSTEP]\nrw [add_comm] at hab hfx \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx y : E\na b : \ud835\udd5c\nhx : x \u2208 s\nhy : y \u2208 s\nha : 0 < a\nhb : 0 < b\nhab : b + a = 1\nhfx : f x < f (b \u2022 y + a \u2022 x)\n\u22a2 f (b \u2022 y + a \u2022 x) < f y\n[PROOFSTEP]\nexact hf.lt_left_of_right_lt' hy hx hb ha hab hfx\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx y z : E\nhx : x \u2208 s\nhy : y \u2208 s\nhz : z \u2208 openSegment \ud835\udd5c x y\nhyz : f y < f z\n\u22a2 f z < f x\n[PROOFSTEP]\nobtain \u27e8a, b, ha, hb, hab, rfl\u27e9 := hz\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx y : E\nhx : x \u2208 s\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhyz : f y < f (a \u2022 x + b \u2022 y)\n\u22a2 f (a \u2022 x + b \u2022 y) < f x\n[PROOFSTEP]\nexact hf.lt_left_of_right_lt' hx hy ha hb hab hyz\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx y z : E\nhx : x \u2208 s\nhy : y \u2208 s\nhz : z \u2208 openSegment \ud835\udd5c x y\nhxz : f x < f z\n\u22a2 f z < f y\n[PROOFSTEP]\nobtain \u27e8a, b, ha, hb, hab, rfl\u27e9 := hz\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhf : ConvexOn \ud835\udd5c s f\nx y : E\nhx : x \u2208 s\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhxz : f x < f (a \u2022 x + b \u2022 y)\n\u22a2 f (a \u2022 x + b \u2022 y) < f y\n[PROOFSTEP]\nexact hf.lt_right_of_left_lt' hx hy ha hb hab hxz\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\n\u22a2 ConvexOn \ud835\udd5c s (-f) \u2194 ConcaveOn \ud835\udd5c s f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\n\u22a2 ConvexOn \ud835\udd5c s (-f) \u2192 ConcaveOn \ud835\udd5c s f\n[PROOFSTEP]\nrintro \u27e8hconv, h\u27e9\n[GOAL]\ncase mp.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhconv : Convex \ud835\udd5c s\nh :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 (-f) (a \u2022 x + b \u2022 y) \u2264 a \u2022 (-f) x + b \u2022 (-f) y\n\u22a2 ConcaveOn \ud835\udd5c s f\n[PROOFSTEP]\nrefine' \u27e8hconv, fun x hx y hy a b ha hb hab => _\u27e9\n[GOAL]\ncase mp.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhconv : Convex \ud835\udd5c s\nh :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 (-f) (a \u2022 x + b \u2022 y) \u2264 a \u2022 (-f) x + b \u2022 (-f) y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 f x + b \u2022 f y \u2264 f (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nsimp [neg_apply, neg_le, add_comm] at h \n[GOAL]\ncase mp.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhconv : Convex \ud835\udd5c s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 a \u2022 f x + b \u2022 f y \u2264 f (a \u2022 x + b \u2022 y)\n\u22a2 a \u2022 f x + b \u2022 f y \u2264 f (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nexact h hx hy ha hb hab\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\n\u22a2 ConcaveOn \ud835\udd5c s f \u2192 ConvexOn \ud835\udd5c s (-f)\n[PROOFSTEP]\nrintro \u27e8hconv, h\u27e9\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhconv : Convex \ud835\udd5c s\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 a \u2022 f x + b \u2022 f y \u2264 f (a \u2022 x + b \u2022 y)\n\u22a2 ConvexOn \ud835\udd5c s (-f)\n[PROOFSTEP]\nrefine' \u27e8hconv, fun x hx y hy a b ha hb hab => _\u27e9\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhconv : Convex \ud835\udd5c s\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 a \u2022 f x + b \u2022 f y \u2264 f (a \u2022 x + b \u2022 y)\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 (-f) (a \u2022 x + b \u2022 y) \u2264 a \u2022 (-f) x + b \u2022 (-f) y\n[PROOFSTEP]\nrw [\u2190 neg_le_neg_iff]\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhconv : Convex \ud835\udd5c s\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 a \u2022 f x + b \u2022 f y \u2264 f (a \u2022 x + b \u2022 y)\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 -(a \u2022 (-f) x + b \u2022 (-f) y) \u2264 -(-f) (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nsimp_rw [neg_add, Pi.neg_apply, smul_neg, neg_neg]\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhconv : Convex \ud835\udd5c s\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 a \u2022 f x + b \u2022 f y \u2264 f (a \u2022 x + b \u2022 y)\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 f x + b \u2022 f y \u2264 f (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nexact h hx hy ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\n\u22a2 ConcaveOn \ud835\udd5c s (-f) \u2194 ConvexOn \ud835\udd5c s f\n[PROOFSTEP]\nrw [\u2190 neg_convexOn_iff, neg_neg f]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\n\u22a2 StrictConvexOn \ud835\udd5c s (-f) \u2194 StrictConcaveOn \ud835\udd5c s f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\n\u22a2 StrictConvexOn \ud835\udd5c s (-f) \u2192 StrictConcaveOn \ud835\udd5c s f\n[PROOFSTEP]\nrintro \u27e8hconv, h\u27e9\n[GOAL]\ncase mp.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhconv : Convex \ud835\udd5c s\nh :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x \u2260 y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 (-f) (a \u2022 x + b \u2022 y) < a \u2022 (-f) x + b \u2022 (-f) y\n\u22a2 StrictConcaveOn \ud835\udd5c s f\n[PROOFSTEP]\nrefine' \u27e8hconv, fun x hx y hy hxy a b ha hb hab => _\u27e9\n[GOAL]\ncase mp.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhconv : Convex \ud835\udd5c s\nh :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x \u2260 y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 (-f) (a \u2022 x + b \u2022 y) < a \u2022 (-f) x + b \u2022 (-f) y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 f x + b \u2022 f y < f (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nsimp [neg_apply, neg_lt, add_comm] at h \n[GOAL]\ncase mp.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhconv : Convex \ud835\udd5c s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nh :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u00acx = y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 f x + b \u2022 f y < f (a \u2022 x + b \u2022 y)\n\u22a2 a \u2022 f x + b \u2022 f y < f (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nexact h hx hy hxy ha hb hab\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\n\u22a2 StrictConcaveOn \ud835\udd5c s f \u2192 StrictConvexOn \ud835\udd5c s (-f)\n[PROOFSTEP]\nrintro \u27e8hconv, h\u27e9\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhconv : Convex \ud835\udd5c s\nh :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x \u2260 y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 f x + b \u2022 f y < f (a \u2022 x + b \u2022 y)\n\u22a2 StrictConvexOn \ud835\udd5c s (-f)\n[PROOFSTEP]\nrefine' \u27e8hconv, fun x hx y hy hxy a b ha hb hab => _\u27e9\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhconv : Convex \ud835\udd5c s\nh :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x \u2260 y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 f x + b \u2022 f y < f (a \u2022 x + b \u2022 y)\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 (-f) (a \u2022 x + b \u2022 y) < a \u2022 (-f) x + b \u2022 (-f) y\n[PROOFSTEP]\nrw [\u2190 neg_lt_neg_iff]\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhconv : Convex \ud835\udd5c s\nh :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x \u2260 y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 f x + b \u2022 f y < f (a \u2022 x + b \u2022 y)\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 -(a \u2022 (-f) x + b \u2022 (-f) y) < -(-f) (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nsimp_rw [neg_add, Pi.neg_apply, smul_neg, neg_neg]\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\nhconv : Convex \ud835\udd5c s\nh :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x \u2260 y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 f x + b \u2022 f y < f (a \u2022 x + b \u2022 y)\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 f x + b \u2022 f y < f (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nexact h hx hy hxy ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : OrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c \u03b2\ns : Set E\nf g : E \u2192 \u03b2\n\u22a2 StrictConcaveOn \ud835\udd5c s (-f) \u2194 StrictConvexOn \ud835\udd5c s f\n[PROOFSTEP]\nrw [\u2190 neg_strictConvexOn_iff, neg_neg f]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCancelCommMonoid E\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nc x : E\nhx : x \u2208 (fun z => c + z) \u207b\u00b9' s\ny : E\nhy : y \u2208 (fun z => c + z) \u207b\u00b9' s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f (c + (a \u2022 x + b \u2022 y)) = f (a \u2022 (c + x) + b \u2022 (c + y))\n[PROOFSTEP]\nrw [smul_add, smul_add, add_add_add_comm, Convex.combo_self hab]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCancelCommMonoid E\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nc : E\n\u22a2 StrictConvexOn \ud835\udd5c ((fun z => c + z) \u207b\u00b9' s) (f \u2218 fun z => z + c)\n[PROOFSTEP]\nsimpa only [add_comm] using hf.translate_right c\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCancelCommMonoid E\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nhf : StrictConcaveOn \ud835\udd5c s f\nc : E\n\u22a2 StrictConcaveOn \ud835\udd5c ((fun z => c + z) \u207b\u00b9' s) (f \u2218 fun z => z + c)\n[PROOFSTEP]\nsimpa only [add_comm] using hf.translate_right c\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : OrderedCommSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c \u03b2\ninst\u271d : OrderedSMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nc : \ud835\udd5c\nhc : 0 \u2264 c\nhf : ConvexOn \ud835\udd5c s f\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 c \u2022 (a \u2022 f x + b \u2022 f y) = a \u2022 c \u2022 f x + b \u2022 c \u2022 f y\n[PROOFSTEP]\nrw [smul_add, smul_comm c, smul_comm c]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2076 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Module \ud835\udd5c F\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : F \u2192 \u03b2\ng : E \u2192\u1d43[\ud835\udd5c] F\ns : Set F\nhf : ConvexOn \ud835\udd5c s f\nx : E\nhx : x \u2208 \u2191g \u207b\u00b9' s\ny : E\nhy : y \u2208 \u2191g \u207b\u00b9' s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 f (\u2191g (a \u2022 x + b \u2022 y)) = f (a \u2022 \u2191g x + b \u2022 \u2191g y)\n[PROOFSTEP]\nrw [Convex.combo_affine_apply hab]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\n\u22a2 (\u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y) \u2192\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : E\u2984,\n y \u2208 s \u2192\n \u2200 \u2983a b : \ud835\udd5c\u2984,\n 0 \u2264 a \u2192\n 0 \u2264 b \u2192\n 0 < a + b \u2192 f ((a / (a + b)) \u2022 x + (b / (a + b)) \u2022 y) \u2264 (a / (a + b)) \u2022 f x + (b / (a + b)) \u2022 f y\n[PROOFSTEP]\nintro h x hx y hy a b ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : 0 < a + b\n\u22a2 f ((a / (a + b)) \u2022 x + (b / (a + b)) \u2022 y) \u2264 (a / (a + b)) \u2022 f x + (b / (a + b)) \u2022 f y\n[PROOFSTEP]\napply h hx hy (div_nonneg ha hab.le) (div_nonneg hb hab.le)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh : \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : 0 < a + b\n\u22a2 a / (a + b) + b / (a + b) = 1\n[PROOFSTEP]\nrw [\u2190 add_div, div_self hab.ne']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\n\u22a2 (\u2200 \u2983x : E\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : E\u2984,\n y \u2208 s \u2192\n \u2200 \u2983a b : \ud835\udd5c\u2984,\n 0 \u2264 a \u2192\n 0 \u2264 b \u2192\n 0 < a + b \u2192 f ((a / (a + b)) \u2022 x + (b / (a + b)) \u2022 y) \u2264 (a / (a + b)) \u2022 f x + (b / (a + b)) \u2022 f y) \u2192\n \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nintro h x hx y hy a b ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : E\u2984,\n y \u2208 s \u2192\n \u2200 \u2983a b : \ud835\udd5c\u2984,\n 0 \u2264 a \u2192\n 0 \u2264 b \u2192 0 < a + b \u2192 f ((a / (a + b)) \u2022 x + (b / (a + b)) \u2022 y) \u2264 (a / (a + b)) \u2022 f x + (b / (a + b)) \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 f (a \u2022 x + b \u2022 y) \u2264 a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nsimpa [hab, zero_lt_one] using h hx hy ha hb\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\n\u22a2 (\u2200 \u2983x : E\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x \u2260 y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y) \u2192\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : E\u2984,\n y \u2208 s \u2192\n x \u2260 y \u2192\n \u2200 \u2983a b : \ud835\udd5c\u2984,\n 0 < a \u2192 0 < b \u2192 f ((a / (a + b)) \u2022 x + (b / (a + b)) \u2022 y) < (a / (a + b)) \u2022 f x + (b / (a + b)) \u2022 f y\n[PROOFSTEP]\nintro h x hx y hy hxy a b ha hb\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x \u2260 y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\n\u22a2 f ((a / (a + b)) \u2022 x + (b / (a + b)) \u2022 y) < (a / (a + b)) \u2022 f x + (b / (a + b)) \u2022 f y\n[PROOFSTEP]\nhave hab := add_pos ha hb\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x \u2260 y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : 0 < a + b\n\u22a2 f ((a / (a + b)) \u2022 x + (b / (a + b)) \u2022 y) < (a / (a + b)) \u2022 f x + (b / (a + b)) \u2022 f y\n[PROOFSTEP]\napply h hx hy hxy (div_pos ha hab) (div_pos hb hab)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x \u2260 y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : 0 < a + b\n\u22a2 a / (a + b) + b / (a + b) = 1\n[PROOFSTEP]\nrw [\u2190 add_div, div_self hab.ne']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\n\u22a2 (\u2200 \u2983x : E\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : E\u2984,\n y \u2208 s \u2192\n x \u2260 y \u2192\n \u2200 \u2983a b : \ud835\udd5c\u2984,\n 0 < a \u2192 0 < b \u2192 f ((a / (a + b)) \u2022 x + (b / (a + b)) \u2022 y) < (a / (a + b)) \u2022 f x + (b / (a + b)) \u2022 f y) \u2192\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 x \u2260 y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nintro h x hx y hy hxy a b ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c \u03b2\ns : Set E\nf : E \u2192 \u03b2\nh :\n \u2200 \u2983x : E\u2984,\n x \u2208 s \u2192\n \u2200 \u2983y : E\u2984,\n y \u2208 s \u2192\n x \u2260 y \u2192\n \u2200 \u2983a b : \ud835\udd5c\u2984,\n 0 < a \u2192 0 < b \u2192 f ((a / (a + b)) \u2022 x + (b / (a + b)) \u2022 y) < (a / (a + b)) \u2022 f x + (b / (a + b)) \u2022 f y\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y\n[PROOFSTEP]\nsimpa [hab, zero_lt_one] using h hx hy hxy ha hb\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConcaveOn \ud835\udd5c univ \u2191f\n\u22a2 StrictConvexOn \ud835\udd5c univ \u2191(symm f)\n[PROOFSTEP]\nrefine \u27e8convex_univ, fun x _ y _ hxy a b ha hb hab => ?_\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 \u2191(symm f) (a \u2022 x + b \u2022 y) < a \u2022 \u2191(symm f) x + b \u2022 \u2191(symm f) y\n[PROOFSTEP]\nobtain \u27e8x', hx''\u27e9 := f.surjective.exists.mp \u27e8x, rfl\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\n\u22a2 \u2191(symm f) (a \u2022 x + b \u2022 y) < a \u2022 \u2191(symm f) x + b \u2022 \u2191(symm f) y\n[PROOFSTEP]\nobtain \u27e8y', hy''\u27e9 := f.surjective.exists.mp \u27e8y, rfl\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 \u2191(symm f) (a \u2022 x + b \u2022 y) < a \u2022 \u2191(symm f) x + b \u2022 \u2191(symm f) y\n[PROOFSTEP]\nhave hxy' : x' \u2260 y' := by rw [\u2190 f.injective.ne_iff, \u2190 hx'', \u2190 hy'']; exact hxy\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 x' \u2260 y'\n[PROOFSTEP]\nrw [\u2190 f.injective.ne_iff, \u2190 hx'', \u2190 hy'']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 x \u2260 y\n[PROOFSTEP]\nexact hxy\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\nhxy' : x' \u2260 y'\n\u22a2 \u2191(symm f) (a \u2022 x + b \u2022 y) < a \u2022 \u2191(symm f) x + b \u2022 \u2191(symm f) y\n[PROOFSTEP]\nsimp only [hx'', hy'', OrderIso.symm_apply_apply, gt_iff_lt]\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\nhxy' : x' \u2260 y'\n\u22a2 \u2191(symm f) (a \u2022 \u2191f x' + b \u2022 \u2191f y') < a \u2022 x' + b \u2022 y'\n[PROOFSTEP]\nrw [\u2190 f.lt_iff_lt, OrderIso.apply_symm_apply]\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\nhxy' : x' \u2260 y'\n\u22a2 a \u2022 \u2191f x' + b \u2022 \u2191f y' < \u2191f (a \u2022 x' + b \u2022 y')\n[PROOFSTEP]\nexact hf.2 (by simp : x' \u2208 univ) (by simp : y' \u2208 univ) hxy' ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\nhxy' : x' \u2260 y'\n\u22a2 x' \u2208 univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\nhxy' : x' \u2260 y'\n\u22a2 y' \u2208 univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConcaveOn \ud835\udd5c univ \u2191f\n\u22a2 ConvexOn \ud835\udd5c univ \u2191(symm f)\n[PROOFSTEP]\nrefine \u27e8convex_univ, fun x _ y _ a b ha hb hab => ?_\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 \u2191(symm f) (a \u2022 x + b \u2022 y) \u2264 a \u2022 \u2191(symm f) x + b \u2022 \u2191(symm f) y\n[PROOFSTEP]\nobtain \u27e8x', hx''\u27e9 := f.surjective.exists.mp \u27e8x, rfl\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\n\u22a2 \u2191(symm f) (a \u2022 x + b \u2022 y) \u2264 a \u2022 \u2191(symm f) x + b \u2022 \u2191(symm f) y\n[PROOFSTEP]\nobtain \u27e8y', hy''\u27e9 := f.surjective.exists.mp \u27e8y, rfl\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 \u2191(symm f) (a \u2022 x + b \u2022 y) \u2264 a \u2022 \u2191(symm f) x + b \u2022 \u2191(symm f) y\n[PROOFSTEP]\nsimp only [hx'', hy'', OrderIso.symm_apply_apply, gt_iff_lt]\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 \u2191(symm f) (a \u2022 \u2191f x' + b \u2022 \u2191f y') \u2264 a \u2022 x' + b \u2022 y'\n[PROOFSTEP]\nrw [\u2190 f.le_iff_le, OrderIso.apply_symm_apply]\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 a \u2022 \u2191f x' + b \u2022 \u2191f y' \u2264 \u2191f (a \u2022 x' + b \u2022 y')\n[PROOFSTEP]\nexact hf.2 (by simp : x' \u2208 univ) (by simp : y' \u2208 univ) ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 x' \u2208 univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConcaveOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 y' \u2208 univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConvexOn \ud835\udd5c univ \u2191f\n\u22a2 StrictConcaveOn \ud835\udd5c univ \u2191(symm f)\n[PROOFSTEP]\nrefine \u27e8convex_univ, fun x _ y _ hxy a b ha hb hab => ?_\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 \u2191(symm f) x + b \u2022 \u2191(symm f) y < \u2191(symm f) (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nobtain \u27e8x', hx''\u27e9 := f.surjective.exists.mp \u27e8x, rfl\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\n\u22a2 a \u2022 \u2191(symm f) x + b \u2022 \u2191(symm f) y < \u2191(symm f) (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nobtain \u27e8y', hy''\u27e9 := f.surjective.exists.mp \u27e8y, rfl\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 a \u2022 \u2191(symm f) x + b \u2022 \u2191(symm f) y < \u2191(symm f) (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nhave hxy' : x' \u2260 y' := by rw [\u2190 f.injective.ne_iff, \u2190 hx'', \u2190 hy'']; exact hxy\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 x' \u2260 y'\n[PROOFSTEP]\nrw [\u2190 f.injective.ne_iff, \u2190 hx'', \u2190 hy'']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 x \u2260 y\n[PROOFSTEP]\nexact hxy\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\nhxy' : x' \u2260 y'\n\u22a2 a \u2022 \u2191(symm f) x + b \u2022 \u2191(symm f) y < \u2191(symm f) (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nsimp only [hx'', hy'', OrderIso.symm_apply_apply, gt_iff_lt]\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\nhxy' : x' \u2260 y'\n\u22a2 a \u2022 x' + b \u2022 y' < \u2191(symm f) (a \u2022 \u2191f x' + b \u2022 \u2191f y')\n[PROOFSTEP]\nrw [\u2190 f.lt_iff_lt, OrderIso.apply_symm_apply]\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\nhxy' : x' \u2260 y'\n\u22a2 \u2191f (a \u2022 x' + b \u2022 y') < a \u2022 \u2191f x' + b \u2022 \u2191f y'\n[PROOFSTEP]\nexact hf.2 (by simp : x' \u2208 univ) (by simp : y' \u2208 univ) hxy' ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\nhxy' : x' \u2260 y'\n\u22a2 x' \u2208 univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : StrictConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\nhxy' : x' \u2260 y'\n\u22a2 y' \u2208 univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConvexOn \ud835\udd5c univ \u2191f\n\u22a2 ConcaveOn \ud835\udd5c univ \u2191(symm f)\n[PROOFSTEP]\nrefine \u27e8convex_univ, fun x _ y _ a b ha hb hab => ?_\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 \u2191(symm f) x + b \u2022 \u2191(symm f) y \u2264 \u2191(symm f) (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nobtain \u27e8x', hx''\u27e9 := f.surjective.exists.mp \u27e8x, rfl\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\n\u22a2 a \u2022 \u2191(symm f) x + b \u2022 \u2191(symm f) y \u2264 \u2191(symm f) (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nobtain \u27e8y', hy''\u27e9 := f.surjective.exists.mp \u27e8y, rfl\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 a \u2022 \u2191(symm f) x + b \u2022 \u2191(symm f) y \u2264 \u2191(symm f) (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nsimp only [hx'', hy'', OrderIso.symm_apply_apply, gt_iff_lt]\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 a \u2022 x' + b \u2022 y' \u2264 \u2191(symm f) (a \u2022 \u2191f x' + b \u2022 \u2191f y')\n[PROOFSTEP]\nrw [\u2190 f.le_iff_le, OrderIso.apply_symm_apply]\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 \u2191f (a \u2022 x' + b \u2022 y') \u2264 a \u2022 \u2191f x' + b \u2022 \u2191f y'\n[PROOFSTEP]\nexact hf.2 (by simp : x' \u2208 univ) (by simp : y' \u2208 univ) ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 x' \u2208 univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : SMul \ud835\udd5c \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nf : \u03b1 \u2243o \u03b2\nhf : ConvexOn \ud835\udd5c univ \u2191f\nx : \u03b2\nx\u271d\u00b9 : x \u2208 univ\ny : \u03b2\nx\u271d : y \u2208 univ\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nx' : \u03b1\nhx'' : x = \u2191f x'\ny' : \u03b1\nhy'' : y = \u2191f y'\n\u22a2 y' \u2208 univ\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.Function", "llama_tokens": 51921, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390747, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.43983706372037396}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\n\ud835\udd5c' : Type u_2\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nE : Type u_3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c' E\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c \ud835\udd5c' E\nF : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c'] F\ns : Set E\nx : E\ng' : E \u2192L[\ud835\udd5c] F\nh : HasFDerivWithinAt f g' s x\nH : restrictScalars \ud835\udd5c f' = g'\n\u22a2 HasFDerivWithinAt f f' s x\n[PROOFSTEP]\nrw [\u2190 H] at h \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\n\ud835\udd5c' : Type u_2\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nE : Type u_3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c' E\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c \ud835\udd5c' E\nF : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c'] F\ns : Set E\nx : E\ng' : E \u2192L[\ud835\udd5c] F\nh : HasFDerivWithinAt f (restrictScalars \ud835\udd5c f') s x\nH : restrictScalars \ud835\udd5c f' = g'\n\u22a2 HasFDerivWithinAt f f' s x\n[PROOFSTEP]\nexact h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\n\ud835\udd5c' : Type u_2\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nE : Type u_3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c' E\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c \ud835\udd5c' E\nF : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c'] F\ns : Set E\nx : E\ng' : E \u2192L[\ud835\udd5c] F\nh : HasFDerivAt f g' x\nH : restrictScalars \ud835\udd5c f' = g'\n\u22a2 HasFDerivAt f f' x\n[PROOFSTEP]\nrw [\u2190 H] at h \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\n\ud835\udd5c' : Type u_2\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nE : Type u_3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c' E\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c \ud835\udd5c' E\nF : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c'] F\ns : Set E\nx : E\ng' : E \u2192L[\ud835\udd5c] F\nh : HasFDerivAt f (restrictScalars \ud835\udd5c f') x\nH : restrictScalars \ud835\udd5c f' = g'\n\u22a2 HasFDerivAt f f' x\n[PROOFSTEP]\nexact h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\n\ud835\udd5c' : Type u_2\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nE : Type u_3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c' E\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c \ud835\udd5c' E\nF : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c'] F\ns : Set E\nx : E\nhf : DifferentiableWithinAt \ud835\udd5c f s x\nhs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 DifferentiableWithinAt \ud835\udd5c' f s x \u2194 \u2203 g', restrictScalars \ud835\udd5c g' = fderivWithin \ud835\udd5c f s x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\n\ud835\udd5c' : Type u_2\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nE : Type u_3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c' E\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c \ud835\udd5c' E\nF : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c'] F\ns : Set E\nx : E\nhf : DifferentiableWithinAt \ud835\udd5c f s x\nhs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 DifferentiableWithinAt \ud835\udd5c' f s x \u2192 \u2203 g', restrictScalars \ud835\udd5c g' = fderivWithin \ud835\udd5c f s x\n[PROOFSTEP]\nrintro \u27e8g', hg'\u27e9\n[GOAL]\ncase mp.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\n\ud835\udd5c' : Type u_2\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nE : Type u_3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c' E\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c \ud835\udd5c' E\nF : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c'] F\ns : Set E\nx : E\nhf : DifferentiableWithinAt \ud835\udd5c f s x\nhs : UniqueDiffWithinAt \ud835\udd5c s x\ng' : E \u2192L[\ud835\udd5c'] F\nhg' : HasFDerivWithinAt f g' s x\n\u22a2 \u2203 g', restrictScalars \ud835\udd5c g' = fderivWithin \ud835\udd5c f s x\n[PROOFSTEP]\nexact \u27e8g', hs.eq (hg'.restrictScalars \ud835\udd5c) hf.hasFDerivWithinAt\u27e9\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\n\ud835\udd5c' : Type u_2\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nE : Type u_3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c' E\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c \ud835\udd5c' E\nF : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c'] F\ns : Set E\nx : E\nhf : DifferentiableWithinAt \ud835\udd5c f s x\nhs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 (\u2203 g', restrictScalars \ud835\udd5c g' = fderivWithin \ud835\udd5c f s x) \u2192 DifferentiableWithinAt \ud835\udd5c' f s x\n[PROOFSTEP]\nrintro \u27e8f', hf'\u27e9\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\n\ud835\udd5c' : Type u_2\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nE : Type u_3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c' E\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c \ud835\udd5c' E\nF : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nf : E \u2192 F\nf'\u271d : E \u2192L[\ud835\udd5c'] F\ns : Set E\nx : E\nhf : DifferentiableWithinAt \ud835\udd5c f s x\nhs : UniqueDiffWithinAt \ud835\udd5c s x\nf' : E \u2192L[\ud835\udd5c'] F\nhf' : restrictScalars \ud835\udd5c f' = fderivWithin \ud835\udd5c f s x\n\u22a2 DifferentiableWithinAt \ud835\udd5c' f s x\n[PROOFSTEP]\nexact \u27e8f', hasFDerivWithinAt_of_restrictScalars \ud835\udd5c hf.hasFDerivWithinAt hf'\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\n\ud835\udd5c' : Type u_2\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nE : Type u_3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c' E\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c \ud835\udd5c' E\nF : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c'] F\ns : Set E\nx : E\nhf : DifferentiableAt \ud835\udd5c f x\n\u22a2 DifferentiableAt \ud835\udd5c' f x \u2194 \u2203 g', restrictScalars \ud835\udd5c g' = fderiv \ud835\udd5c f x\n[PROOFSTEP]\nrw [\u2190 differentiableWithinAt_univ, \u2190 fderivWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\n\ud835\udd5c' : Type u_2\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nE : Type u_3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c' E\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c \ud835\udd5c' E\nF : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c'] F\ns : Set E\nx : E\nhf : DifferentiableAt \ud835\udd5c f x\n\u22a2 DifferentiableWithinAt \ud835\udd5c' f univ x \u2194 \u2203 g', restrictScalars \ud835\udd5c g' = fderivWithin \ud835\udd5c f univ x\n[PROOFSTEP]\nexact differentiableWithinAt_iff_restrictScalars \ud835\udd5c hf.differentiableWithinAt uniqueDiffWithinAt_univ\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "llama_tokens": 3450, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7931059609645723, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.4397538309389129}} {"text": "[GOAL]\nG : Type u\ninst\u271d : Group G\n\u22a2 commutator G = Subgroup.closure (commutatorSet G)\n[PROOFSTEP]\nsimp [commutator, Subgroup.commutator_def, commutatorSet]\n[GOAL]\nG : Type u\ninst\u271d : Group G\n\u22a2 commutator G = Subgroup.normalClosure (commutatorSet G)\n[PROOFSTEP]\nsimp [commutator, Subgroup.commutator_def', commutatorSet]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite \u2191(commutatorSet G)\n\u22a2 Group.FG { x // x \u2208 commutator G }\n[PROOFSTEP]\nrw [commutator_eq_closure]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite \u2191(commutatorSet G)\n\u22a2 Group.FG { x // x \u2208 Subgroup.closure (commutatorSet G) }\n[PROOFSTEP]\napply Group.closure_finite_fg\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite \u2191(commutatorSet G)\n\u22a2 Group.rank { x // x \u2208 commutator G } \u2264 Nat.card \u2191(commutatorSet G)\n[PROOFSTEP]\nrw [Subgroup.rank_congr (commutator_eq_closure G)]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite \u2191(commutatorSet G)\n\u22a2 Group.rank { x // x \u2208 Subgroup.closure (commutatorSet G) } \u2264 Nat.card \u2191(commutatorSet G)\n[PROOFSTEP]\napply Subgroup.rank_closure_finite_le_nat_card\n[GOAL]\nG : Type u\ninst\u271d : Group G\n\u22a2 \u2045centralizer \u2191(commutator G), centralizer \u2191(commutator G)\u2046 \u2264 Subgroup.center G\n[PROOFSTEP]\nrw [\u2190 Subgroup.centralizer_univ, \u2190 Subgroup.coe_top, \u2190 Subgroup.commutator_eq_bot_iff_le_centralizer]\n[GOAL]\nG : Type u\ninst\u271d : Group G\n\u22a2 \u2045\u2045centralizer \u2191(commutator G), centralizer \u2191(commutator G)\u2046, \u22a4\u2046 = \u22a5\n[PROOFSTEP]\nsuffices \u2045\u2045\u22a4, centralizer (commutator G : Set G)\u2046, centralizer (commutator G : Set G)\u2046 = \u22a5\n by\n refine' Subgroup.commutator_commutator_eq_bot_of_rotate _ this\n rwa [Subgroup.commutator_comm (centralizer (commutator G : Set G))]\n[GOAL]\nG : Type u\ninst\u271d : Group G\nthis : \u2045\u2045\u22a4, centralizer \u2191(commutator G)\u2046, centralizer \u2191(commutator G)\u2046 = \u22a5\n\u22a2 \u2045\u2045centralizer \u2191(commutator G), centralizer \u2191(commutator G)\u2046, \u22a4\u2046 = \u22a5\n[PROOFSTEP]\nrefine' Subgroup.commutator_commutator_eq_bot_of_rotate _ this\n[GOAL]\nG : Type u\ninst\u271d : Group G\nthis : \u2045\u2045\u22a4, centralizer \u2191(commutator G)\u2046, centralizer \u2191(commutator G)\u2046 = \u22a5\n\u22a2 \u2045\u2045centralizer \u2191(commutator G), \u22a4\u2046, centralizer \u2191(commutator G)\u2046 = \u22a5\n[PROOFSTEP]\nrwa [Subgroup.commutator_comm (centralizer (commutator G : Set G))]\n[GOAL]\nG : Type u\ninst\u271d : Group G\n\u22a2 \u2045\u2045\u22a4, centralizer \u2191(commutator G)\u2046, centralizer \u2191(commutator G)\u2046 = \u22a5\n[PROOFSTEP]\nrw [Subgroup.commutator_comm, Subgroup.commutator_eq_bot_iff_le_centralizer]\n[GOAL]\nG : Type u\ninst\u271d : Group G\n\u22a2 centralizer \u2191(commutator G) \u2264 centralizer \u2191\u2045\u22a4, centralizer \u2191(commutator G)\u2046\n[PROOFSTEP]\nexact Set.centralizer_subset (Subgroup.commutator_mono le_top le_top)\n[GOAL]\nG : Type u\ninst\u271d : Group G\nsrc\u271d : Group (G \u29f8 commutator G) := QuotientGroup.Quotient.group (commutator G)\nx y : Abelianization G\na b : G\n\u22a2 \u2045b\u207b\u00b9, a\u207b\u00b9\u2046 = ((fun x x_1 => x * x_1) a b)\u207b\u00b9 * (fun x x_1 => x * x_1) b a\n[PROOFSTEP]\ngroup\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nA : Type v\ninst\u271d : CommGroup A\nf : G \u2192* A\n\u22a2 commutator G \u2264 MonoidHom.ker f\n[PROOFSTEP]\nrw [commutator_eq_closure, Subgroup.closure_le]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nA : Type v\ninst\u271d : CommGroup A\nf : G \u2192* A\n\u22a2 commutatorSet G \u2286 \u2191(MonoidHom.ker f)\n[PROOFSTEP]\nrintro x \u27e8p, q, rfl\u27e9\n[GOAL]\ncase intro.intro\nG : Type u\ninst\u271d\u00b9 : Group G\nA : Type v\ninst\u271d : CommGroup A\nf : G \u2192* A\np q : G\n\u22a2 \u2045p, q\u2046 \u2208 \u2191(MonoidHom.ker f)\n[PROOFSTEP]\nsimp [MonoidHom.mem_ker, mul_right_comm (f p) (f q), commutatorElement_def]\n[GOAL]\nG : Type u\ninst\u271d\u00b2 inst\u271d\u00b9 : Group G\nH : Type v\ninst\u271d : Group H\ne : G \u2243* H\n\u22a2 Function.LeftInverse \u2191(Abelianization.map (toMonoidHom (symm e))) \u2191(Abelianization.map (toMonoidHom e))\n[PROOFSTEP]\nrintro \u27e8a\u27e9\n[GOAL]\ncase mk\nG : Type u\ninst\u271d\u00b2 inst\u271d\u00b9 : Group G\nH : Type v\ninst\u271d : Group H\ne : G \u2243* H\nx\u271d : Abelianization G\na : G\n\u22a2 \u2191(Abelianization.map (toMonoidHom (symm e))) (\u2191(Abelianization.map (toMonoidHom e)) (Quot.mk Setoid.r a)) =\n Quot.mk Setoid.r a\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u\ninst\u271d\u00b2 inst\u271d\u00b9 : Group G\nH : Type v\ninst\u271d : Group H\ne : G \u2243* H\n\u22a2 Function.RightInverse \u2191(Abelianization.map (toMonoidHom (symm e))) \u2191(Abelianization.map (toMonoidHom e))\n[PROOFSTEP]\nrintro \u27e8a\u27e9\n[GOAL]\ncase mk\nG : Type u\ninst\u271d\u00b2 inst\u271d\u00b9 : Group G\nH : Type v\ninst\u271d : Group H\ne : G \u2243* H\nx\u271d : Abelianization H\na : H\n\u22a2 \u2191(Abelianization.map (toMonoidHom e)) (\u2191(Abelianization.map (toMonoidHom (symm e))) (Quot.mk Setoid.r a)) =\n Quot.mk Setoid.r a\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nH : Type u_1\ninst\u271d : CommGroup H\nsrc\u271d : H \u2192* Abelianization H := of\n\u22a2 Function.RightInverse \u2191(\u2191lift (MonoidHom.id H)) \u2191of\n[PROOFSTEP]\nrintro \u27e8a\u27e9\n[GOAL]\ncase mk\nG : Type u\ninst\u271d\u00b9 : Group G\nH : Type u_1\ninst\u271d : CommGroup H\nsrc\u271d : H \u2192* Abelianization H := of\nx\u271d : Abelianization H\na : H\n\u22a2 \u2191of (\u2191(\u2191lift (MonoidHom.id H)) (Quot.mk Setoid.r a)) = Quot.mk Setoid.r a\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite \u2191(commutatorSet G)\n\u22a2 Group.rank { x // x \u2208 closureCommutatorRepresentatives G } \u2264 2 * Nat.card \u2191(commutatorSet G)\n[PROOFSTEP]\nrw [two_mul]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite \u2191(commutatorSet G)\n\u22a2 Group.rank { x // x \u2208 closureCommutatorRepresentatives G } \u2264 Nat.card \u2191(commutatorSet G) + Nat.card \u2191(commutatorSet G)\n[PROOFSTEP]\nexact\n (Subgroup.rank_closure_finite_le_nat_card _).trans\n ((Set.card_union_le _ _).trans\n (add_le_add ((Finite.card_image_le _).trans (Finite.card_range_le _))\n ((Finite.card_image_le _).trans (Finite.card_range_le _))))\n[GOAL]\nG : Type u\ninst\u271d : Group G\n\u22a2 \u2191(Subgroup.subtype (closureCommutatorRepresentatives G)) ''\n commutatorSet { x // x \u2208 closureCommutatorRepresentatives G } =\n commutatorSet G\n[PROOFSTEP]\napply Set.Subset.antisymm\n[GOAL]\ncase h\u2081\nG : Type u\ninst\u271d : Group G\n\u22a2 \u2191(Subgroup.subtype (closureCommutatorRepresentatives G)) ''\n commutatorSet { x // x \u2208 closureCommutatorRepresentatives G } \u2286\n commutatorSet G\n[PROOFSTEP]\nrintro - \u27e8-, \u27e8g\u2081, g\u2082, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h\u2081.intro.intro.intro.intro\nG : Type u\ninst\u271d : Group G\ng\u2081 g\u2082 : { x // x \u2208 closureCommutatorRepresentatives G }\n\u22a2 \u2191(Subgroup.subtype (closureCommutatorRepresentatives G)) \u2045g\u2081, g\u2082\u2046 \u2208 commutatorSet G\n[PROOFSTEP]\nexact \u27e8g\u2081, g\u2082, rfl\u27e9\n[GOAL]\ncase h\u2082\nG : Type u\ninst\u271d : Group G\n\u22a2 commutatorSet G \u2286\n \u2191(Subgroup.subtype (closureCommutatorRepresentatives G)) ''\n commutatorSet { x // x \u2208 closureCommutatorRepresentatives G }\n[PROOFSTEP]\nexact fun g hg =>\n \u27e8_,\n \u27e8\u27e8_, subset_closure (Or.inl \u27e8_, \u27e8\u27e8g, hg\u27e9, rfl\u27e9, rfl\u27e9)\u27e9, \u27e8_, subset_closure (Or.inr \u27e8_, \u27e8\u27e8g, hg\u27e9, rfl\u27e9, rfl\u27e9)\u27e9, rfl\u27e9,\n hg.choose_spec.choose_spec\u27e9\n[GOAL]\nG : Type u\ninst\u271d : Group G\n\u22a2 Nat.card \u2191(commutatorSet { x // x \u2208 closureCommutatorRepresentatives G }) = Nat.card \u2191(commutatorSet G)\n[PROOFSTEP]\nrw [\u2190 image_commutatorSet_closureCommutatorRepresentatives G]\n[GOAL]\nG : Type u\ninst\u271d : Group G\n\u22a2 Nat.card \u2191(commutatorSet { x // x \u2208 closureCommutatorRepresentatives G }) =\n Nat.card\n \u2191(\u2191(Subgroup.subtype (closureCommutatorRepresentatives G)) ''\n commutatorSet { x // x \u2208 closureCommutatorRepresentatives G })\n[PROOFSTEP]\nexact Nat.card_congr (Equiv.Set.image _ _ (subtype_injective _))\n[GOAL]\nG : Type u\ninst\u271d : Group G\n\u22a2 Nat.card { x // x \u2208 _root_.commutator { x // x \u2208 closureCommutatorRepresentatives G } } =\n Nat.card { x // x \u2208 _root_.commutator G }\n[PROOFSTEP]\nrw [commutator_eq_closure G, \u2190 image_commutatorSet_closureCommutatorRepresentatives, \u2190 MonoidHom.map_closure, \u2190\n commutator_eq_closure]\n[GOAL]\nG : Type u\ninst\u271d : Group G\n\u22a2 Nat.card { x // x \u2208 _root_.commutator { x // x \u2208 closureCommutatorRepresentatives G } } =\n Nat.card\n { x //\n x \u2208\n map (Subgroup.subtype (closureCommutatorRepresentatives G))\n (_root_.commutator { x // x \u2208 closureCommutatorRepresentatives G }) }\n[PROOFSTEP]\nexact Nat.card_congr (Equiv.Set.image _ _ (subtype_injective _))\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite \u2191(commutatorSet G)\n\u22a2 Finite \u2191(commutatorSet { x // x \u2208 closureCommutatorRepresentatives G })\n[PROOFSTEP]\napply Nat.finite_of_card_ne_zero\n[GOAL]\ncase h\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite \u2191(commutatorSet G)\n\u22a2 Nat.card \u2191(commutatorSet { x // x \u2208 closureCommutatorRepresentatives G }) \u2260 0\n[PROOFSTEP]\nrw [card_commutatorSet_closureCommutatorRepresentatives]\n[GOAL]\ncase h\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite \u2191(commutatorSet G)\n\u22a2 Nat.card \u2191(commutatorSet G) \u2260 0\n[PROOFSTEP]\nexact Finite.card_pos.ne'\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Abelianization", "llama_tokens": 3715, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.43973113673657854}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b3\u00b2 : Semiring R\nS : Type u_2\ninst\u271d\u00b3\u00b9 : Semiring S\nR\u2082 : Type u_3\ninst\u271d\u00b3\u2070 : Semiring R\u2082\nS\u2082 : Type u_4\ninst\u271d\u00b2\u2079 : Semiring S\u2082\nM : Type u_5\nN : Type u_6\nP : Type u_7\nM\u2082 : Type u_8\nN\u2082 : Type u_9\nP\u2082 : Type u_10\nN\u2097 : Type u_11\nP\u2097 : Type u_12\nM' : Type u_13\nN' : Type u_14\nP' : Type u_15\ninst\u271d\u00b2\u2078 : AddCommMonoid M\ninst\u271d\u00b2\u2077 : AddCommMonoid N\ninst\u271d\u00b2\u2076 : AddCommMonoid P\ninst\u271d\u00b2\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b2\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b2\u00b3 : AddCommMonoid P\u2082\ninst\u271d\u00b2\u00b2 : AddCommMonoid N\u2097\ninst\u271d\u00b2\u00b9 : AddCommMonoid P\u2097\ninst\u271d\u00b2\u2070 : AddCommGroup M'\ninst\u271d\u00b9\u2079 : AddCommGroup N'\ninst\u271d\u00b9\u2078 : AddCommGroup P'\ninst\u271d\u00b9\u2077 : Module R M\ninst\u271d\u00b9\u2076 : Module S N\ninst\u271d\u00b9\u2075 : Module R\u2082 P\ninst\u271d\u00b9\u2074 : Module S\u2082 P\ninst\u271d\u00b9\u00b3 : Module R M\u2082\ninst\u271d\u00b9\u00b2 : Module S N\u2082\ninst\u271d\u00b9\u00b9 : Module R P\u2082\ninst\u271d\u00b9\u2070 : Module S\u2082 P\u2082\ninst\u271d\u2079 : Module R P\u2097\ninst\u271d\u2078 : Module S P\u2097\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : Module S N'\ninst\u271d\u2075 : Module R\u2082 P'\ninst\u271d\u2074 : Module S\u2082 P'\ninst\u271d\u00b3 : SMulCommClass S\u2082 R\u2082 P\ninst\u271d\u00b2 : SMulCommClass S R P\u2097\ninst\u271d\u00b9 : SMulCommClass S\u2082 R\u2082 P'\ninst\u271d : SMulCommClass S\u2082 R P\u2082\n\u03c1\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2081\u2082 : S \u2192+* S\u2082\nf : M \u2192\u209b\u2097[\u03c1\u2081\u2082] N \u2192\u209b\u2097[\u03c3\u2081\u2082] P\nn : N\nm\u2081 m\u2082 : M\n\u22a2 (fun n m => \u2191(\u2191f m) n) n (m\u2081 + m\u2082) = (fun n m => \u2191(\u2191f m) n) n m\u2081 + (fun n m => \u2191(\u2191f m) n) n m\u2082\n[PROOFSTEP]\nsimp only [map_add, add_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3\u00b2 : Semiring R\nS : Type u_2\ninst\u271d\u00b3\u00b9 : Semiring S\nR\u2082 : Type u_3\ninst\u271d\u00b3\u2070 : Semiring R\u2082\nS\u2082 : Type u_4\ninst\u271d\u00b2\u2079 : Semiring S\u2082\nM : Type u_5\nN : Type u_6\nP : Type u_7\nM\u2082 : Type u_8\nN\u2082 : Type u_9\nP\u2082 : Type u_10\nN\u2097 : Type u_11\nP\u2097 : Type u_12\nM' : Type u_13\nN' : Type u_14\nP' : Type u_15\ninst\u271d\u00b2\u2078 : AddCommMonoid M\ninst\u271d\u00b2\u2077 : AddCommMonoid N\ninst\u271d\u00b2\u2076 : AddCommMonoid P\ninst\u271d\u00b2\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b2\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b2\u00b3 : AddCommMonoid P\u2082\ninst\u271d\u00b2\u00b2 : AddCommMonoid N\u2097\ninst\u271d\u00b2\u00b9 : AddCommMonoid P\u2097\ninst\u271d\u00b2\u2070 : AddCommGroup M'\ninst\u271d\u00b9\u2079 : AddCommGroup N'\ninst\u271d\u00b9\u2078 : AddCommGroup P'\ninst\u271d\u00b9\u2077 : Module R M\ninst\u271d\u00b9\u2076 : Module S N\ninst\u271d\u00b9\u2075 : Module R\u2082 P\ninst\u271d\u00b9\u2074 : Module S\u2082 P\ninst\u271d\u00b9\u00b3 : Module R M\u2082\ninst\u271d\u00b9\u00b2 : Module S N\u2082\ninst\u271d\u00b9\u00b9 : Module R P\u2082\ninst\u271d\u00b9\u2070 : Module S\u2082 P\u2082\ninst\u271d\u2079 : Module R P\u2097\ninst\u271d\u2078 : Module S P\u2097\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : Module S N'\ninst\u271d\u2075 : Module R\u2082 P'\ninst\u271d\u2074 : Module S\u2082 P'\ninst\u271d\u00b3 : SMulCommClass S\u2082 R\u2082 P\ninst\u271d\u00b2 : SMulCommClass S R P\u2097\ninst\u271d\u00b9 : SMulCommClass S\u2082 R\u2082 P'\ninst\u271d : SMulCommClass S\u2082 R P\u2082\n\u03c1\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2081\u2082 : S \u2192+* S\u2082\nf : M \u2192\u209b\u2097[\u03c1\u2081\u2082] N \u2192\u209b\u2097[\u03c3\u2081\u2082] P\nc : R\nn : N\nm : M\n\u22a2 (fun n m => \u2191(\u2191f m) n) n (c \u2022 m) = \u2191\u03c1\u2081\u2082 c \u2022 (fun n m => \u2191(\u2191f m) n) n m\n[PROOFSTEP]\nsimp only [map_smul\u209b\u2097, smul_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3\u00b2 : Semiring R\nS : Type u_2\ninst\u271d\u00b3\u00b9 : Semiring S\nR\u2082 : Type u_3\ninst\u271d\u00b3\u2070 : Semiring R\u2082\nS\u2082 : Type u_4\ninst\u271d\u00b2\u2079 : Semiring S\u2082\nM : Type u_5\nN : Type u_6\nP : Type u_7\nM\u2082 : Type u_8\nN\u2082 : Type u_9\nP\u2082 : Type u_10\nN\u2097 : Type u_11\nP\u2097 : Type u_12\nM' : Type u_13\nN' : Type u_14\nP' : Type u_15\ninst\u271d\u00b2\u2078 : AddCommMonoid M\ninst\u271d\u00b2\u2077 : AddCommMonoid N\ninst\u271d\u00b2\u2076 : AddCommMonoid P\ninst\u271d\u00b2\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b2\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b2\u00b3 : AddCommMonoid P\u2082\ninst\u271d\u00b2\u00b2 : AddCommMonoid N\u2097\ninst\u271d\u00b2\u00b9 : AddCommMonoid P\u2097\ninst\u271d\u00b2\u2070 : AddCommGroup M'\ninst\u271d\u00b9\u2079 : AddCommGroup N'\ninst\u271d\u00b9\u2078 : AddCommGroup P'\ninst\u271d\u00b9\u2077 : Module R M\ninst\u271d\u00b9\u2076 : Module S N\ninst\u271d\u00b9\u2075 : Module R\u2082 P\ninst\u271d\u00b9\u2074 : Module S\u2082 P\ninst\u271d\u00b9\u00b3 : Module R M\u2082\ninst\u271d\u00b9\u00b2 : Module S N\u2082\ninst\u271d\u00b9\u00b9 : Module R P\u2082\ninst\u271d\u00b9\u2070 : Module S\u2082 P\u2082\ninst\u271d\u2079 : Module R P\u2097\ninst\u271d\u2078 : Module S P\u2097\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : Module S N'\ninst\u271d\u2075 : Module R\u2082 P'\ninst\u271d\u2074 : Module S\u2082 P'\ninst\u271d\u00b3 : SMulCommClass S\u2082 R\u2082 P\ninst\u271d\u00b2 : SMulCommClass S R P\u2097\ninst\u271d\u00b9 : SMulCommClass S\u2082 R\u2082 P'\ninst\u271d : SMulCommClass S\u2082 R P\u2082\n\u03c1\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2081\u2082 : S \u2192+* S\u2082\nf g : M \u2192\u209b\u2097[\u03c1\u2081\u2082] N \u2192\u209b\u2097[\u03c3\u2081\u2082] P\nH : flip f = flip g\nm : M\nn : N\n\u22a2 \u2191(\u2191(flip f) n) m = \u2191(\u2191(flip g) n) m\n[PROOFSTEP]\nrw [H]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3\u00b2 : Semiring R\nS : Type u_2\ninst\u271d\u00b3\u00b9 : Semiring S\nR\u2082 : Type u_3\ninst\u271d\u00b3\u2070 : Semiring R\u2082\nS\u2082 : Type u_4\ninst\u271d\u00b2\u2079 : Semiring S\u2082\nM : Type u_5\nN : Type u_6\nP : Type u_7\nM\u2082 : Type u_8\nN\u2082 : Type u_9\nP\u2082 : Type u_10\nN\u2097 : Type u_11\nP\u2097 : Type u_12\nM' : Type u_13\nN' : Type u_14\nP' : Type u_15\ninst\u271d\u00b2\u2078 : AddCommMonoid M\ninst\u271d\u00b2\u2077 : AddCommMonoid N\ninst\u271d\u00b2\u2076 : AddCommMonoid P\ninst\u271d\u00b2\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b2\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b2\u00b3 : AddCommMonoid P\u2082\ninst\u271d\u00b2\u00b2 : AddCommMonoid N\u2097\ninst\u271d\u00b2\u00b9 : AddCommMonoid P\u2097\ninst\u271d\u00b2\u2070 : AddCommGroup M'\ninst\u271d\u00b9\u2079 : AddCommGroup N'\ninst\u271d\u00b9\u2078 : AddCommGroup P'\ninst\u271d\u00b9\u2077 : Module R M\ninst\u271d\u00b9\u2076 : Module S N\ninst\u271d\u00b9\u2075 : Module R\u2082 P\ninst\u271d\u00b9\u2074 : Module S\u2082 P\ninst\u271d\u00b9\u00b3 : Module R M\u2082\ninst\u271d\u00b9\u00b2 : Module S N\u2082\ninst\u271d\u00b9\u00b9 : Module R P\u2082\ninst\u271d\u00b9\u2070 : Module S\u2082 P\u2082\ninst\u271d\u2079 : Module R P\u2097\ninst\u271d\u2078 : Module S P\u2097\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : Module S N'\ninst\u271d\u2075 : Module R\u2082 P'\ninst\u271d\u2074 : Module S\u2082 P'\ninst\u271d\u00b3 : SMulCommClass S\u2082 R\u2082 P\ninst\u271d\u00b2 : SMulCommClass S R P\u2097\ninst\u271d\u00b9 : SMulCommClass S\u2082 R\u2082 P'\ninst\u271d : SMulCommClass S\u2082 R P\u2082\n\u03c1\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2081\u2082 : S \u2192+* S\u2082\nf : M \u2192\u209b\u2097[\u03c1\u2081\u2082] N \u2192\u209b\u2097[\u03c3\u2081\u2082] P\nq : Submodule S N\nm\u2081 m\u2082 : M\nx\u271d : { x // x \u2208 q }\n\u22a2 \u2191((fun m => domRestrict (\u2191f m) q) (m\u2081 + m\u2082)) x\u271d =\n \u2191((fun m => domRestrict (\u2191f m) q) m\u2081 + (fun m => domRestrict (\u2191f m) q) m\u2082) x\u271d\n[PROOFSTEP]\nsimp only [map_add, domRestrict_apply, add_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3\u00b2 : Semiring R\nS : Type u_2\ninst\u271d\u00b3\u00b9 : Semiring S\nR\u2082 : Type u_3\ninst\u271d\u00b3\u2070 : Semiring R\u2082\nS\u2082 : Type u_4\ninst\u271d\u00b2\u2079 : Semiring S\u2082\nM : Type u_5\nN : Type u_6\nP : Type u_7\nM\u2082 : Type u_8\nN\u2082 : Type u_9\nP\u2082 : Type u_10\nN\u2097 : Type u_11\nP\u2097 : Type u_12\nM' : Type u_13\nN' : Type u_14\nP' : Type u_15\ninst\u271d\u00b2\u2078 : AddCommMonoid M\ninst\u271d\u00b2\u2077 : AddCommMonoid N\ninst\u271d\u00b2\u2076 : AddCommMonoid P\ninst\u271d\u00b2\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b2\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b2\u00b3 : AddCommMonoid P\u2082\ninst\u271d\u00b2\u00b2 : AddCommMonoid N\u2097\ninst\u271d\u00b2\u00b9 : AddCommMonoid P\u2097\ninst\u271d\u00b2\u2070 : AddCommGroup M'\ninst\u271d\u00b9\u2079 : AddCommGroup N'\ninst\u271d\u00b9\u2078 : AddCommGroup P'\ninst\u271d\u00b9\u2077 : Module R M\ninst\u271d\u00b9\u2076 : Module S N\ninst\u271d\u00b9\u2075 : Module R\u2082 P\ninst\u271d\u00b9\u2074 : Module S\u2082 P\ninst\u271d\u00b9\u00b3 : Module R M\u2082\ninst\u271d\u00b9\u00b2 : Module S N\u2082\ninst\u271d\u00b9\u00b9 : Module R P\u2082\ninst\u271d\u00b9\u2070 : Module S\u2082 P\u2082\ninst\u271d\u2079 : Module R P\u2097\ninst\u271d\u2078 : Module S P\u2097\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : Module S N'\ninst\u271d\u2075 : Module R\u2082 P'\ninst\u271d\u2074 : Module S\u2082 P'\ninst\u271d\u00b3 : SMulCommClass S\u2082 R\u2082 P\ninst\u271d\u00b2 : SMulCommClass S R P\u2097\ninst\u271d\u00b9 : SMulCommClass S\u2082 R\u2082 P'\ninst\u271d : SMulCommClass S\u2082 R P\u2082\n\u03c1\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2081\u2082 : S \u2192+* S\u2082\nf : M \u2192\u209b\u2097[\u03c1\u2081\u2082] N \u2192\u209b\u2097[\u03c3\u2081\u2082] P\nq : Submodule S N\nc : R\nm : M\nx\u271d : { x // x \u2208 q }\n\u22a2 \u2191(AddHom.toFun\n { toFun := fun m => domRestrict (\u2191f m) q,\n map_add' :=\n (_ :\n \u2200 (m\u2081 m\u2082 : M),\n (fun m => domRestrict (\u2191f m) q) (m\u2081 + m\u2082) =\n (fun m => domRestrict (\u2191f m) q) m\u2081 + (fun m => domRestrict (\u2191f m) q) m\u2082) }\n (c \u2022 m))\n x\u271d =\n \u2191(\u2191\u03c1\u2081\u2082 c \u2022\n AddHom.toFun\n { toFun := fun m => domRestrict (\u2191f m) q,\n map_add' :=\n (_ :\n \u2200 (m\u2081 m\u2082 : M),\n (fun m => domRestrict (\u2191f m) q) (m\u2081 + m\u2082) =\n (fun m => domRestrict (\u2191f m) q) m\u2081 + (fun m => domRestrict (\u2191f m) q) m\u2082) }\n m)\n x\u271d\n[PROOFSTEP]\nsimp only [f.map_smul\u209b\u2097, domRestrict_apply, smul_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\n\u22a2 compl\u2082 f id = f\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nx\u271d\u00b9 : M\nx\u271d : N\n\u22a2 \u2191(\u2191(compl\u2082 f id) x\u271d\u00b9) x\u271d = \u2191(\u2191f x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nrw [compl\u2082_apply, id_coe, id.def]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\n\u22a2 compl\u2081\u2082 f id id = f\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\nx\u271d\u00b9 : M\u2097\nx\u271d : N\u2097\n\u22a2 \u2191(\u2191(compl\u2081\u2082 f id id) x\u271d\u00b9) x\u271d = \u2191(\u2191f x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nsimp_rw [compl\u2081\u2082_apply, id_coe, id.def]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u2191g\nh\u1d63 : Function.Surjective \u2191g'\n\u22a2 compl\u2081\u2082 f\u2081 g g' = compl\u2081\u2082 f\u2082 g g' \u2194 f\u2081 = f\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u2191g\nh\u1d63 : Function.Surjective \u2191g'\n\u22a2 compl\u2081\u2082 f\u2081 g g' = compl\u2081\u2082 f\u2082 g g' \u2192 f\u2081 = f\u2082\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u2191g\nh\u1d63 : Function.Surjective \u2191g'\n\u22a2 f\u2081 = f\u2082 \u2192 compl\u2081\u2082 f\u2081 g g' = compl\u2081\u2082 f\u2082 g g'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u2191g\nh\u1d63 : Function.Surjective \u2191g'\nh : compl\u2081\u2082 f\u2081 g g' = compl\u2081\u2082 f\u2082 g g'\n\u22a2 f\u2081 = f\u2082\n[PROOFSTEP]\next x y\n[GOAL]\ncase mp.h.h\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u2191g\nh\u1d63 : Function.Surjective \u2191g'\nh : compl\u2081\u2082 f\u2081 g g' = compl\u2081\u2082 f\u2082 g g'\nx : M\u2097\ny : N\u2097\n\u22a2 \u2191(\u2191f\u2081 x) y = \u2191(\u2191f\u2082 x) y\n[PROOFSTEP]\ncases' h\u2097 x with x' hx\n[GOAL]\ncase mp.h.h.intro\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u2191g\nh\u1d63 : Function.Surjective \u2191g'\nh : compl\u2081\u2082 f\u2081 g g' = compl\u2081\u2082 f\u2082 g g'\nx : M\u2097\ny : N\u2097\nx' : Q\u2097\nhx : \u2191g x' = x\n\u22a2 \u2191(\u2191f\u2081 x) y = \u2191(\u2191f\u2082 x) y\n[PROOFSTEP]\nsubst hx\n[GOAL]\ncase mp.h.h.intro\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u2191g\nh\u1d63 : Function.Surjective \u2191g'\nh : compl\u2081\u2082 f\u2081 g g' = compl\u2081\u2082 f\u2082 g g'\ny : N\u2097\nx' : Q\u2097\n\u22a2 \u2191(\u2191f\u2081 (\u2191g x')) y = \u2191(\u2191f\u2082 (\u2191g x')) y\n[PROOFSTEP]\ncases' h\u1d63 y with y' hy\n[GOAL]\ncase mp.h.h.intro.intro\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u2191g\nh\u1d63 : Function.Surjective \u2191g'\nh : compl\u2081\u2082 f\u2081 g g' = compl\u2081\u2082 f\u2082 g g'\ny : N\u2097\nx' : Q\u2097\ny' : Q\u2097'\nhy : \u2191g' y' = y\n\u22a2 \u2191(\u2191f\u2081 (\u2191g x')) y = \u2191(\u2191f\u2082 (\u2191g x')) y\n[PROOFSTEP]\nsubst hy\n[GOAL]\ncase mp.h.h.intro.intro\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u2191g\nh\u1d63 : Function.Surjective \u2191g'\nh : compl\u2081\u2082 f\u2081 g g' = compl\u2081\u2082 f\u2082 g g'\nx' : Q\u2097\ny' : Q\u2097'\n\u22a2 \u2191(\u2191f\u2081 (\u2191g x')) (\u2191g' y') = \u2191(\u2191f\u2082 (\u2191g x')) (\u2191g' y')\n[PROOFSTEP]\nconvert LinearMap.congr_fun\u2082 h x' y' using 0\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 f\u2082 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u2191g\nh\u1d63 : Function.Surjective \u2191g'\nh : f\u2081 = f\u2082\n\u22a2 compl\u2081\u2082 f\u2081 g g' = compl\u2081\u2082 f\u2082 g g'\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nf\u2081 : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097\ng : Q\u2097 \u2192\u2097[R] M\u2097\ng' : Q\u2097' \u2192\u2097[R] N\u2097\nh\u2097 : Function.Surjective \u2191g\nh\u1d63 : Function.Surjective \u2191g'\n\u22a2 compl\u2081\u2082 f\u2081 g g' = compl\u2081\u2082 f\u2081 g g'\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2\u00b3 : CommSemiring R\nR\u2082 : Type u_2\ninst\u271d\u00b2\u00b2 : CommSemiring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b2\u00b9 : CommSemiring R\u2083\nR\u2084 : Type u_4\ninst\u271d\u00b2\u2070 : CommSemiring R\u2084\nM : Type u_5\nN : Type u_6\nP : Type u_7\nQ : Type u_8\nM\u2097 : Type u_9\nN\u2097 : Type u_10\nP\u2097 : Type u_11\nQ\u2097 : Type u_12\nQ\u2097' : Type u_13\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : AddCommMonoid N\ninst\u271d\u00b9\u2077 : AddCommMonoid P\ninst\u271d\u00b9\u2076 : AddCommMonoid Q\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2097\ninst\u271d\u00b9\u2074 : AddCommMonoid N\u2097\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\u2097\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\u2097\ninst\u271d\u00b9\u00b9 : AddCommMonoid Q\u2097'\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R\u2082 N\ninst\u271d\u2078 : Module R\u2083 P\ninst\u271d\u2077 : Module R\u2084 Q\ninst\u271d\u2076 : Module R M\u2097\ninst\u271d\u2075 : Module R N\u2097\ninst\u271d\u2074 : Module R P\u2097\ninst\u271d\u00b3 : Module R Q\u2097\ninst\u271d\u00b2 : Module R Q\u2097'\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2084\u2082 : R\u2084 \u2192+* R\u2082\n\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2084\u2082 \u03c3\u2082\u2083 \u03c3\u2084\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2083] N \u2192\u209b\u2097[\u03c3\u2082\u2083] P\nr s : R\nm : M\n\u22a2 (fun x x_1 => x \u2022 x_1) s (r \u2022 m) = r \u2022 (fun x x_1 => x \u2022 x_1) s m\n[PROOFSTEP]\nsimp only [smul_smul, smul_eq_mul, mul_comm]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.BilinearMap", "llama_tokens": 13867, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7981867777396211, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.4394875188962752}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b2 : One M\ninst\u271d\u00b9 : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b1\ninst\u271d : Decidable (a \u2208 s)\n\u22a2 mulIndicator s f a = if a \u2208 s then f a else 1\n[PROOFSTEP]\nunfold mulIndicator\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b2 : One M\ninst\u271d\u00b9 : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b1\ninst\u271d : Decidable (a \u2208 s)\n\u22a2 (if a \u2208 s then f a else 1) = if a \u2208 s then f a else 1\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b1\n\u22a2 mulIndicator s f a = 1 \u2228 mulIndicator s f a = f a\n[PROOFSTEP]\nby_cases h : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b1\nh : a \u2208 s\n\u22a2 mulIndicator s f a = 1 \u2228 mulIndicator s f a = f a\n[PROOFSTEP]\nexact Or.inr (mulIndicator_of_mem h f)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b1\nh : \u00aca \u2208 s\n\u22a2 mulIndicator s f a = 1 \u2228 mulIndicator s f a = f a\n[PROOFSTEP]\nexact Or.inl (mulIndicator_of_not_mem h f)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\nthis : Decidable (a \u2208 s) := Classical.dec (a \u2208 s)\n\u22a2 \u00aca \u2208 s \u2192 1 = f a \u2194 \u00aca \u2208 s \u2192 f a = 1\n[PROOFSTEP]\nrw [@eq_comm _ (f a)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\n\u22a2 mulIndicator s f = f \u2194 mulSupport f \u2286 s\n[PROOFSTEP]\nsimp only [funext_iff, subset_def, mem_mulSupport, mulIndicator_apply_eq_self, not_imp_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\nh1 : mulIndicator s f = f\nh2 : s \u2286 t\n\u22a2 mulIndicator t f = f\n[PROOFSTEP]\nrw [mulIndicator_eq_self] at h1 \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\nh1 : mulSupport f \u2286 s\nh2 : s \u2286 t\n\u22a2 mulSupport f \u2286 t\n[PROOFSTEP]\nexact Subset.trans h1 h2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\n\u22a2 (mulIndicator s f = fun x => 1) \u2194 Disjoint (mulSupport f) s\n[PROOFSTEP]\nsimp only [funext_iff, mulIndicator_apply_eq_one, Set.disjoint_left, mem_mulSupport, not_imp_not]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na\u271d a : \u03b1\n\u22a2 mulIndicator s f a \u2260 1 \u2194 a \u2208 s \u2229 mulSupport f\n[PROOFSTEP]\nsimp only [Ne.def, mulIndicator_apply_eq_one, not_imp, mem_inter_iff, mem_mulSupport]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na x : \u03b1\n\u22a2 x \u2208 mulSupport (mulIndicator s f) \u2194 x \u2208 s \u2229 mulSupport f\n[PROOFSTEP]\nsimp [Function.mem_mulSupport, mulIndicator_apply_eq_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\nh : EqOn f g s\nx : \u03b1\n\u22a2 mulIndicator s f x = mulIndicator s g x\n[PROOFSTEP]\nsimp only [mulIndicator]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\nh : EqOn f g s\nx : \u03b1\n\u22a2 (if x \u2208 s then f x else 1) = if x \u2208 s then g x else 1\n[PROOFSTEP]\nsplit_ifs with h_1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\nh : EqOn f g s\nx : \u03b1\nh_1 : x \u2208 s\n\u22a2 f x = g x\n[PROOFSTEP]\nexact h h_1\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\nh : EqOn f g s\nx : \u03b1\nh_1 : \u00acx \u2208 s\n\u22a2 1 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\n\u22a2 Disjoint (mulSupport fun x => 1) s\n[PROOFSTEP]\nsimp only [mulSupport_one, empty_disjoint]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\n\u22a2 mulIndicator s (mulIndicator t f) x = mulIndicator (s \u2229 t) f x\n[PROOFSTEP]\nsimp only [mulIndicator]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\n\u22a2 (if x \u2208 s then if x \u2208 t then f x else 1 else 1) = if x \u2208 s \u2229 t then f x else 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nh\u271d\u00b2 : x \u2208 s\nh\u271d\u00b9 : x \u2208 t\nh\u271d : x \u2208 s \u2229 t\n\u22a2 f x = f x\n[PROOFSTEP]\nsimp_all (config := { contextual := true })\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nh\u271d\u00b2 : x \u2208 s\nh\u271d\u00b9 : x \u2208 t\nh\u271d : \u00acx \u2208 s \u2229 t\n\u22a2 f x = 1\n[PROOFSTEP]\nsimp_all (config := { contextual := true })\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nh\u271d\u00b2 : x \u2208 s\nh\u271d\u00b9 : \u00acx \u2208 t\nh\u271d : x \u2208 s \u2229 t\n\u22a2 1 = f x\n[PROOFSTEP]\nsimp_all (config := { contextual := true })\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nh\u271d\u00b2 : x \u2208 s\nh\u271d\u00b9 : \u00acx \u2208 t\nh\u271d : \u00acx \u2208 s \u2229 t\n\u22a2 1 = 1\n[PROOFSTEP]\nsimp_all (config := { contextual := true })\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nh\u271d\u00b9 : \u00acx \u2208 s\nh\u271d : x \u2208 s \u2229 t\n\u22a2 1 = f x\n[PROOFSTEP]\nsimp_all (config := { contextual := true })\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nh\u271d\u00b9 : \u00acx \u2208 s\nh\u271d : \u00acx \u2208 s \u2229 t\n\u22a2 1 = 1\n[PROOFSTEP]\nsimp_all (config := { contextual := true })\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 M\n\u22a2 mulIndicator (s \u2229 mulSupport f) f = mulIndicator s f\n[PROOFSTEP]\nrw [\u2190 mulIndicator_mulIndicator, mulIndicator_mulSupport]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b2 : One M\ninst\u271d\u00b9 : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nh : M \u2192 \u03b2\nf : \u03b1 \u2192 M\ns : Set \u03b1\nx : \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\n\u22a2 h (mulIndicator s f x) = piecewise s (h \u2218 f) (const \u03b1 (h 1)) x\n[PROOFSTEP]\nletI := Classical.decPred (\u00b7 \u2208 s)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b2 : One M\ninst\u271d\u00b9 : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nh : M \u2192 \u03b2\nf : \u03b1 \u2192 M\ns : Set \u03b1\nx : \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nthis : DecidablePred fun x => x \u2208 s := Classical.decPred fun x => x \u2208 s\n\u22a2 h (mulIndicator s f x) = piecewise s (h \u2218 f) (const \u03b1 (h 1)) x\n[PROOFSTEP]\nconvert s.apply_piecewise f (const \u03b1 1) (fun _ => h) (x := x) using 2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 M\nx : \u03b2\n\u22a2 mulIndicator (f \u207b\u00b9' s) (g \u2218 f) x = mulIndicator s g (f x)\n[PROOFSTEP]\nsimp only [mulIndicator, Function.comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 M\nx : \u03b2\n\u22a2 (if x \u2208 f \u207b\u00b9' s then g (f x) else 1) = if f x \u2208 s then g (f x) else 1\n[PROOFSTEP]\nsplit_ifs with h h' h''\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 M\nx : \u03b2\nh : x \u2208 f \u207b\u00b9' s\nh' : f x \u2208 s\n\u22a2 g (f x) = g (f x)\n[PROOFSTEP]\nfirst\n| rfl\n| contradiction\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 M\nx : \u03b2\nh : x \u2208 f \u207b\u00b9' s\nh' : f x \u2208 s\n\u22a2 g (f x) = g (f x)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 M\nx : \u03b2\nh : x \u2208 f \u207b\u00b9' s\nh' : \u00acf x \u2208 s\n\u22a2 g (f x) = 1\n[PROOFSTEP]\nfirst\n| rfl\n| contradiction\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 M\nx : \u03b2\nh : x \u2208 f \u207b\u00b9' s\nh' : \u00acf x \u2208 s\n\u22a2 g (f x) = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 M\nx : \u03b2\nh : x \u2208 f \u207b\u00b9' s\nh' : \u00acf x \u2208 s\n\u22a2 g (f x) = 1\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 M\nx : \u03b2\nh : \u00acx \u2208 f \u207b\u00b9' s\nh'' : f x \u2208 s\n\u22a2 1 = g (f x)\n[PROOFSTEP]\nfirst\n| rfl\n| contradiction\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 M\nx : \u03b2\nh : \u00acx \u2208 f \u207b\u00b9' s\nh'' : f x \u2208 s\n\u22a2 1 = g (f x)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 M\nx : \u03b2\nh : \u00acx \u2208 f \u207b\u00b9' s\nh'' : f x \u2208 s\n\u22a2 1 = g (f x)\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 M\nx : \u03b2\nh : \u00acx \u2208 f \u207b\u00b9' s\nh'' : \u00acf x \u2208 s\n\u22a2 1 = 1\n[PROOFSTEP]\nfirst\n| rfl\n| contradiction\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 M\nx : \u03b2\nh : \u00acx \u2208 f \u207b\u00b9' s\nh'' : \u00acf x \u2208 s\n\u22a2 1 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf : \u03b2 \u2192 M\ng : \u03b1 \u2192 \u03b2\nhg : Injective g\nx : \u03b1\n\u22a2 mulIndicator (g '' s) f (g x) = mulIndicator s (f \u2218 g) x\n[PROOFSTEP]\nrw [\u2190 mulIndicator_comp_right, preimage_image_eq _ hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g\u271d : \u03b1 \u2192 M\na : \u03b1\ng : M \u2192 N\nhg : g 1 = 1\n\u22a2 mulIndicator s (g \u2218 f) = g \u2218 mulIndicator s f\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g\u271d : \u03b1 \u2192 M\na : \u03b1\ng : M \u2192 N\nhg : g 1 = 1\nx\u271d : \u03b1\n\u22a2 mulIndicator s (g \u2218 f) x\u271d = (g \u2218 mulIndicator s f) x\u271d\n[PROOFSTEP]\nsimp only [mulIndicator]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g\u271d : \u03b1 \u2192 M\na : \u03b1\ng : M \u2192 N\nhg : g 1 = 1\nx\u271d : \u03b1\n\u22a2 (if x\u271d \u2208 s then (g \u2218 f) x\u271d else 1) = (g \u2218 mulIndicator s f) x\u271d\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g\u271d : \u03b1 \u2192 M\na : \u03b1\ng : M \u2192 N\nhg : g 1 = 1\nx\u271d : \u03b1\nh\u271d : x\u271d \u2208 s\n\u22a2 (g \u2218 f) x\u271d = (g \u2218 mulIndicator s f) x\u271d\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g\u271d : \u03b1 \u2192 M\na : \u03b1\ng : M \u2192 N\nhg : g 1 = 1\nx\u271d : \u03b1\nh\u271d : \u00acx\u271d \u2208 s\n\u22a2 1 = (g \u2218 mulIndicator s f) x\u271d\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\ns : Set M\n\u22a2 mulIndicator t 1 \u207b\u00b9' s \u2208 {univ, \u2205}\n[PROOFSTEP]\nclassical\nrw [mulIndicator_one', preimage_one]\nsplit_ifs <;> simp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\ns : Set M\n\u22a2 mulIndicator t 1 \u207b\u00b9' s \u2208 {univ, \u2205}\n[PROOFSTEP]\nrw [mulIndicator_one', preimage_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\ns : Set M\n\u22a2 (if 1 \u2208 s then univ else \u2205) \u2208 {univ, \u2205}\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\ns : Set M\nh\u271d : 1 \u2208 s\n\u22a2 univ \u2208 {univ, \u2205}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\ns : Set M\nh\u271d : \u00ac1 \u2208 s\n\u22a2 \u2205 \u2208 {univ, \u2205}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : One M\ninst\u271d\u00b2 : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na\u271d : \u03b1\nU : Set \u03b1\ns : Set M\na : M\ninst\u271d\u00b9 : Decidable (a \u2208 s)\ninst\u271d : Decidable (1 \u2208 s)\n\u22a2 (mulIndicator U fun x => a) \u207b\u00b9' s = (if a \u2208 s then U else \u2205) \u222a if 1 \u2208 s then U\u1d9c else \u2205\n[PROOFSTEP]\nrw [mulIndicator_preimage, preimage_one, preimage_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : One M\ninst\u271d\u00b2 : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na\u271d : \u03b1\nU : Set \u03b1\ns : Set M\na : M\ninst\u271d\u00b9 : Decidable (a \u2208 s)\ninst\u271d : Decidable (1 \u2208 s)\n\u22a2 Set.ite U (if a \u2208 s then univ else \u2205) (if 1 \u2208 s then univ else \u2205) = (if a \u2208 s then U else \u2205) \u222a if 1 \u2208 s then U\u1d9c else \u2205\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : One M\ninst\u271d\u00b2 : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na\u271d : \u03b1\nU : Set \u03b1\ns : Set M\na : M\ninst\u271d\u00b9 : Decidable (a \u2208 s)\ninst\u271d : Decidable (1 \u2208 s)\nh\u271d\u00b9 : a \u2208 s\nh\u271d : 1 \u2208 s\n\u22a2 Set.ite U univ univ = U \u222a U\u1d9c\n[PROOFSTEP]\nsimp [\u2190 compl_eq_univ_diff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : One M\ninst\u271d\u00b2 : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na\u271d : \u03b1\nU : Set \u03b1\ns : Set M\na : M\ninst\u271d\u00b9 : Decidable (a \u2208 s)\ninst\u271d : Decidable (1 \u2208 s)\nh\u271d\u00b9 : a \u2208 s\nh\u271d : \u00ac1 \u2208 s\n\u22a2 Set.ite U univ \u2205 = U \u222a \u2205\n[PROOFSTEP]\nsimp [\u2190 compl_eq_univ_diff]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : One M\ninst\u271d\u00b2 : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na\u271d : \u03b1\nU : Set \u03b1\ns : Set M\na : M\ninst\u271d\u00b9 : Decidable (a \u2208 s)\ninst\u271d : Decidable (1 \u2208 s)\nh\u271d\u00b9 : \u00aca \u2208 s\nh\u271d : 1 \u2208 s\n\u22a2 Set.ite U \u2205 univ = \u2205 \u222a U\u1d9c\n[PROOFSTEP]\nsimp [\u2190 compl_eq_univ_diff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : One M\ninst\u271d\u00b2 : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na\u271d : \u03b1\nU : Set \u03b1\ns : Set M\na : M\ninst\u271d\u00b9 : Decidable (a \u2208 s)\ninst\u271d : Decidable (1 \u2208 s)\nh\u271d\u00b9 : \u00aca \u2208 s\nh\u271d : \u00ac1 \u2208 s\n\u22a2 Set.ite U \u2205 \u2205 = \u2205 \u222a \u2205\n[PROOFSTEP]\nsimp [\u2190 compl_eq_univ_diff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na\u271d : \u03b1\nU : Set \u03b1\ns : Set M\na : M\n\u22a2 (mulIndicator U fun x => a) \u207b\u00b9' s \u2208 {univ, U, U\u1d9c, \u2205}\n[PROOFSTEP]\nclassical\nrw [mulIndicator_const_preimage_eq_union]\nsplit_ifs <;> simp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na\u271d : \u03b1\nU : Set \u03b1\ns : Set M\na : M\n\u22a2 (mulIndicator U fun x => a) \u207b\u00b9' s \u2208 {univ, U, U\u1d9c, \u2205}\n[PROOFSTEP]\nrw [mulIndicator_const_preimage_eq_union]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na\u271d : \u03b1\nU : Set \u03b1\ns : Set M\na : M\n\u22a2 ((if a \u2208 s then U else \u2205) \u222a if 1 \u2208 s then U\u1d9c else \u2205) \u2208 {univ, U, U\u1d9c, \u2205}\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na\u271d : \u03b1\nU : Set \u03b1\ns : Set M\na : M\nh\u271d\u00b9 : a \u2208 s\nh\u271d : 1 \u2208 s\n\u22a2 U \u222a U\u1d9c \u2208 {univ, U, U\u1d9c, \u2205}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na\u271d : \u03b1\nU : Set \u03b1\ns : Set M\na : M\nh\u271d\u00b9 : a \u2208 s\nh\u271d : \u00ac1 \u2208 s\n\u22a2 U \u222a \u2205 \u2208 {univ, U, U\u1d9c, \u2205}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na\u271d : \u03b1\nU : Set \u03b1\ns : Set M\na : M\nh\u271d\u00b9 : \u00aca \u2208 s\nh\u271d : 1 \u2208 s\n\u22a2 \u2205 \u222a U\u1d9c \u2208 {univ, U, U\u1d9c, \u2205}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf g : \u03b1 \u2192 M\na\u271d : \u03b1\nU : Set \u03b1\ns : Set M\na : M\nh\u271d\u00b9 : \u00aca \u2208 s\nh\u271d : \u00ac1 \u2208 s\n\u22a2 \u2205 \u222a \u2205 \u2208 {univ, U, U\u1d9c, \u2205}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 M\nt : Set M\nht : \u00ac1 \u2208 t\n\u22a2 mulIndicator s f \u207b\u00b9' t = f \u207b\u00b9' t \u2229 s\n[PROOFSTEP]\nsimp [mulIndicator_preimage, Pi.one_def, Set.preimage_const_of_not_mem ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nr : M\ns : Set \u03b1\nf : \u03b1 \u2192 M\n\u22a2 r \u2208 range (mulIndicator s f) \u2194 r = 1 \u2227 s \u2260 univ \u2228 r \u2208 f '' s\n[PROOFSTEP]\nsimp only [mem_range, mulIndicator, ne_eq, mem_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nr : M\ns : Set \u03b1\nf : \u03b1 \u2192 M\n\u22a2 (\u2203 y, (if y \u2208 s then f y else 1) = r) \u2194 r = 1 \u2227 \u00acs = univ \u2228 \u2203 x, x \u2208 s \u2227 f x = r\n[PROOFSTEP]\nrw [eq_univ_iff_forall, not_forall]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nr : M\ns : Set \u03b1\nf : \u03b1 \u2192 M\n\u22a2 (\u2203 y, (if y \u2208 s then f y else 1) = r) \u2194 (r = 1 \u2227 \u2203 x, \u00acx \u2208 s) \u2228 \u2203 x, x \u2208 s \u2227 f x = r\n[PROOFSTEP]\nrefine \u27e8?_, ?_\u27e9\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nr : M\ns : Set \u03b1\nf : \u03b1 \u2192 M\n\u22a2 (\u2203 y, (if y \u2208 s then f y else 1) = r) \u2192 (r = 1 \u2227 \u2203 x, \u00acx \u2208 s) \u2228 \u2203 x, x \u2208 s \u2227 f x = r\n[PROOFSTEP]\nrintro \u27e8y, hy\u27e9\n[GOAL]\ncase refine_1.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nr : M\ns : Set \u03b1\nf : \u03b1 \u2192 M\ny : \u03b1\nhy : (if y \u2208 s then f y else 1) = r\n\u22a2 (r = 1 \u2227 \u2203 x, \u00acx \u2208 s) \u2228 \u2203 x, x \u2208 s \u2227 f x = r\n[PROOFSTEP]\nsplit_ifs at hy with hys\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nr : M\ns : Set \u03b1\nf : \u03b1 \u2192 M\ny : \u03b1\nhys : y \u2208 s\nhy : f y = r\n\u22a2 (r = 1 \u2227 \u2203 x, \u00acx \u2208 s) \u2228 \u2203 x, x \u2208 s \u2227 f x = r\n[PROOFSTEP]\ntauto\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nr : M\ns : Set \u03b1\nf : \u03b1 \u2192 M\ny : \u03b1\nhys : \u00acy \u2208 s\nhy : 1 = r\n\u22a2 (r = 1 \u2227 \u2203 x, \u00acx \u2208 s) \u2228 \u2203 x, x \u2208 s \u2227 f x = r\n[PROOFSTEP]\nleft\n[GOAL]\ncase neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nr : M\ns : Set \u03b1\nf : \u03b1 \u2192 M\ny : \u03b1\nhys : \u00acy \u2208 s\nhy : 1 = r\n\u22a2 r = 1 \u2227 \u2203 x, \u00acx \u2208 s\n[PROOFSTEP]\ntauto\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nr : M\ns : Set \u03b1\nf : \u03b1 \u2192 M\n\u22a2 ((r = 1 \u2227 \u2203 x, \u00acx \u2208 s) \u2228 \u2203 x, x \u2208 s \u2227 f x = r) \u2192 \u2203 y, (if y \u2208 s then f y else 1) = r\n[PROOFSTEP]\nrintro (\u27e8hr, \u27e8x, hx\u27e9\u27e9 | \u27e8x, \u27e8hx, hxs\u27e9\u27e9)\n[GOAL]\ncase refine_2.inl.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nr : M\ns : Set \u03b1\nf : \u03b1 \u2192 M\nhr : r = 1\nx : \u03b1\nhx : \u00acx \u2208 s\n\u22a2 \u2203 y, (if y \u2208 s then f y else 1) = r\n[PROOFSTEP]\nuse x\n[GOAL]\ncase refine_2.inr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nr : M\ns : Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : x \u2208 s\nhxs : f x = r\n\u22a2 \u2203 y, (if y \u2208 s then f y else 1) = r\n[PROOFSTEP]\nuse x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nr : M\ns : Set \u03b1\nf : \u03b1 \u2192 M\nhr : r = 1\nx : \u03b1\nhx : \u00acx \u2208 s\n\u22a2 (if x \u2208 s then f x else 1) = r\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nr : M\ns : Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : x \u2208 s\nhxs : f x = r\n\u22a2 (if x \u2208 s then f x else 1) = r\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nr : M\ns : Set \u03b1\nf : \u03b1 \u2192 M\nhr : r = 1\nx : \u03b1\nhx : \u00acx \u2208 s\n\u22a2 1 = r\n[PROOFSTEP]\ntauto\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nr : M\ns : Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : x \u2208 s\nhxs : f x = r\n\u22a2 f x = r\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\nr : M \u2192 M \u2192 Prop\nh1 : r 1 1\nha : a \u2208 s \u2192 r (f a) (g a)\n\u22a2 r (mulIndicator s f a) (mulIndicator s g a)\n[PROOFSTEP]\nsimp only [mulIndicator]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\nr : M \u2192 M \u2192 Prop\nh1 : r 1 1\nha : a \u2208 s \u2192 r (f a) (g a)\n\u22a2 r (if a \u2208 s then f a else 1) (if a \u2208 s then g a else 1)\n[PROOFSTEP]\nsplit_ifs with has\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\nr : M \u2192 M \u2192 Prop\nh1 : r 1 1\nha : a \u2208 s \u2192 r (f a) (g a)\nhas : a \u2208 s\n\u22a2 r (f a) (g a)\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\nr : M \u2192 M \u2192 Prop\nh1 : r 1 1\nha : a \u2208 s \u2192 r (f a) (g a)\nhas : \u00aca \u2208 s\n\u22a2 r 1 1\n[PROOFSTEP]\nexacts [ha has, h1]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\nf : \u03b1 \u2192 M\ns t : Set \u03b1\na : \u03b1\n\u22a2 mulIndicator (s \u222a t) f a * mulIndicator (s \u2229 t) f a = mulIndicator s f a * mulIndicator t f a\n[PROOFSTEP]\nby_cases hs : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\nf : \u03b1 \u2192 M\ns t : Set \u03b1\na : \u03b1\nhs : a \u2208 s\n\u22a2 mulIndicator (s \u222a t) f a * mulIndicator (s \u2229 t) f a = mulIndicator s f a * mulIndicator t f a\n[PROOFSTEP]\nby_cases ht : a \u2208 t\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\nf : \u03b1 \u2192 M\ns t : Set \u03b1\na : \u03b1\nhs : \u00aca \u2208 s\n\u22a2 mulIndicator (s \u222a t) f a * mulIndicator (s \u2229 t) f a = mulIndicator s f a * mulIndicator t f a\n[PROOFSTEP]\nby_cases ht : a \u2208 t\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\nf : \u03b1 \u2192 M\ns t : Set \u03b1\na : \u03b1\nhs : a \u2208 s\nht : a \u2208 t\n\u22a2 mulIndicator (s \u222a t) f a * mulIndicator (s \u2229 t) f a = mulIndicator s f a * mulIndicator t f a\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\nf : \u03b1 \u2192 M\ns t : Set \u03b1\na : \u03b1\nhs : a \u2208 s\nht : \u00aca \u2208 t\n\u22a2 mulIndicator (s \u222a t) f a * mulIndicator (s \u2229 t) f a = mulIndicator s f a * mulIndicator t f a\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\nf : \u03b1 \u2192 M\ns t : Set \u03b1\na : \u03b1\nhs : \u00aca \u2208 s\nht : a \u2208 t\n\u22a2 mulIndicator (s \u222a t) f a * mulIndicator (s \u2229 t) f a = mulIndicator s f a * mulIndicator t f a\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t\u271d : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\nf : \u03b1 \u2192 M\ns t : Set \u03b1\na : \u03b1\nhs : \u00aca \u2208 s\nht : \u00aca \u2208 t\n\u22a2 mulIndicator (s \u222a t) f a * mulIndicator (s \u2229 t) f a = mulIndicator s f a * mulIndicator t f a\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na : \u03b1\nh : \u00aca \u2208 s \u2229 t\nf : \u03b1 \u2192 M\n\u22a2 mulIndicator (s \u222a t) f a = mulIndicator s f a * mulIndicator t f a\n[PROOFSTEP]\nrw [\u2190 mulIndicator_union_mul_inter_apply f s t, mulIndicator_of_not_mem h, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\n\u22a2 (mulIndicator s fun a => f a * g a) = fun a => mulIndicator s f a * mulIndicator s g a\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\nx\u271d : \u03b1\n\u22a2 mulIndicator s (fun a => f a * g a) x\u271d = mulIndicator s f x\u271d * mulIndicator s g x\u271d\n[PROOFSTEP]\nsimp only [mulIndicator]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\nx\u271d : \u03b1\n\u22a2 (if x\u271d \u2208 s then f x\u271d * g x\u271d else 1) = (if x\u271d \u2208 s then f x\u271d else 1) * if x\u271d \u2208 s then g x\u271d else 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\nx\u271d : \u03b1\nh\u271d : x\u271d \u2208 s\n\u22a2 f x\u271d * g x\u271d = f x\u271d * g x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\nx\u271d : \u03b1\nh\u271d : \u00acx\u271d \u2208 s\n\u22a2 1 = 1 * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b1\nha : a \u2208 s\n\u22a2 mulIndicator s\u1d9c f a * mulIndicator s f a = f a\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 mulIndicator s\u1d9c f a * mulIndicator s f a = f a\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b1\nha : a \u2208 s\n\u22a2 mulIndicator s f a * mulIndicator s\u1d9c f a = f a\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\na\u271d : \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 mulIndicator s f a * mulIndicator s\u1d9c f a = f a\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\nf g : \u03b1 \u2192 M\nh : Disjoint (mulSupport f) (mulSupport g)\n\u22a2 mulIndicator (mulSupport f) (f * g) = f\n[PROOFSTEP]\nrefine' (mulIndicator_congr fun x hx => _).trans mulIndicator_mulSupport\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\nf g : \u03b1 \u2192 M\nh : Disjoint (mulSupport f) (mulSupport g)\nx : \u03b1\nhx : x \u2208 mulSupport f\n\u22a2 (f * g) x = f x\n[PROOFSTEP]\nhave : g x = 1 := nmem_mulSupport.1 (disjoint_left.1 h hx)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\nf g : \u03b1 \u2192 M\nh : Disjoint (mulSupport f) (mulSupport g)\nx : \u03b1\nhx : x \u2208 mulSupport f\nthis : g x = 1\n\u22a2 (f * g) x = f x\n[PROOFSTEP]\nrw [Pi.mul_apply, this, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\nf g : \u03b1 \u2192 M\nh : Disjoint (mulSupport f) (mulSupport g)\n\u22a2 mulIndicator (mulSupport g) (f * g) = g\n[PROOFSTEP]\nrefine' (mulIndicator_congr fun x hx => _).trans mulIndicator_mulSupport\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\nf g : \u03b1 \u2192 M\nh : Disjoint (mulSupport f) (mulSupport g)\nx : \u03b1\nhx : x \u2208 mulSupport g\n\u22a2 (f * g) x = g x\n[PROOFSTEP]\nhave : f x = 1 := nmem_mulSupport.1 (disjoint_right.1 h hx)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulOneClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\nf g : \u03b1 \u2192 M\nh : Disjoint (mulSupport f) (mulSupport g)\nx : \u03b1\nhx : x \u2208 mulSupport g\nthis : f x = 1\n\u22a2 (f * g) x = g x\n[PROOFSTEP]\nrw [Pi.mul_apply, this, one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : MulOneClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nf g : \u03b1 \u2192 M\n\u22a2 mulIndicator s f * mulIndicator s\u1d9c g = piecewise s f g\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : MulOneClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nf g : \u03b1 \u2192 M\nx : \u03b1\n\u22a2 (mulIndicator s f * mulIndicator s\u1d9c g) x = piecewise s f g x\n[PROOFSTEP]\nby_cases h : x \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : MulOneClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nf g : \u03b1 \u2192 M\nx : \u03b1\nh : x \u2208 s\n\u22a2 (mulIndicator s f * mulIndicator s\u1d9c g) x = piecewise s f g x\n[PROOFSTEP]\nrw [piecewise_eq_of_mem _ _ _ h, Pi.mul_apply, Set.mulIndicator_of_mem h,\n Set.mulIndicator_of_not_mem (Set.not_mem_compl_iff.2 h), mul_one]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : MulOneClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nf g : \u03b1 \u2192 M\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 (mulIndicator s f * mulIndicator s\u1d9c g) x = piecewise s f g x\n[PROOFSTEP]\nrw [piecewise_eq_of_not_mem _ _ _ h, Pi.mul_apply, Set.mulIndicator_of_not_mem h,\n Set.mulIndicator_of_mem (Set.mem_compl h), one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\nA : Type u_6\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : Monoid M\ninst\u271d : DistribMulAction M A\ns : Set \u03b1\nr : \u03b1 \u2192 M\nf : \u03b1 \u2192 A\nx : \u03b1\n\u22a2 indicator s (fun x => r x \u2022 f x) x = r x \u2022 indicator s f x\n[PROOFSTEP]\ndsimp only [indicator]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\nA : Type u_6\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : Monoid M\ninst\u271d : DistribMulAction M A\ns : Set \u03b1\nr : \u03b1 \u2192 M\nf : \u03b1 \u2192 A\nx : \u03b1\n\u22a2 (if x \u2208 s then r x \u2022 f x else 0) = r x \u2022 if x \u2208 s then f x else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\nA : Type u_6\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : Monoid M\ninst\u271d : DistribMulAction M A\ns : Set \u03b1\nr : \u03b1 \u2192 M\nf : \u03b1 \u2192 A\nx : \u03b1\nh\u271d : x \u2208 s\n\u22a2 r x \u2022 f x = r x \u2022 f x\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\nA : Type u_6\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : Monoid M\ninst\u271d : DistribMulAction M A\ns : Set \u03b1\nr : \u03b1 \u2192 M\nf : \u03b1 \u2192 A\nx : \u03b1\nh\u271d : \u00acx \u2208 s\n\u22a2 0 = r x \u2022 0\n[PROOFSTEP]\nexacts [rfl, (smul_zero (r x)).symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\nA : Type u_6\ninst\u271d\u00b2 : Zero A\ninst\u271d\u00b9 : Zero M\ninst\u271d : SMulWithZero M A\ns : Set \u03b1\nr : \u03b1 \u2192 M\nf : \u03b1 \u2192 A\nx : \u03b1\n\u22a2 indicator s (fun x => r x \u2022 f x) x = indicator s r x \u2022 f x\n[PROOFSTEP]\ndsimp only [indicator]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\nA : Type u_6\ninst\u271d\u00b2 : Zero A\ninst\u271d\u00b9 : Zero M\ninst\u271d : SMulWithZero M A\ns : Set \u03b1\nr : \u03b1 \u2192 M\nf : \u03b1 \u2192 A\nx : \u03b1\n\u22a2 (if x \u2208 s then r x \u2022 f x else 0) = (if x \u2208 s then r x else 0) \u2022 f x\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\nA : Type u_6\ninst\u271d\u00b2 : Zero A\ninst\u271d\u00b9 : Zero M\ninst\u271d : SMulWithZero M A\ns : Set \u03b1\nr : \u03b1 \u2192 M\nf : \u03b1 \u2192 A\nx : \u03b1\nh\u271d : x \u2208 s\n\u22a2 r x \u2022 f x = r x \u2022 f x\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\nA : Type u_6\ninst\u271d\u00b2 : Zero A\ninst\u271d\u00b9 : Zero M\ninst\u271d : SMulWithZero M A\ns : Set \u03b1\nr : \u03b1 \u2192 M\nf : \u03b1 \u2192 A\nx : \u03b1\nh\u271d : \u00acx \u2208 s\n\u22a2 0 = 0 \u2022 f x\n[PROOFSTEP]\nexacts [rfl, (zero_smul _ (f x)).symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\ninst\u271d : Group G\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 G\na : \u03b1\ns : Set \u03b1\nf : \u03b1 \u2192 G\n\u22a2 mulIndicator s\u1d9c f = f / mulIndicator s f\n[PROOFSTEP]\nrw [div_eq_mul_inv, mulIndicator_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\ninst\u271d : Group G\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 G\na : \u03b1\nh : s \u2286 t\nf : \u03b1 \u2192 G\n\u22a2 mulIndicator (t \\ s) f * mulIndicator s f = mulIndicator t f\n[PROOFSTEP]\nrw [Pi.mul_def, \u2190 mulIndicator_union_of_disjoint, diff_union_self, union_eq_self_of_subset_right h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\ninst\u271d : Group G\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 G\na : \u03b1\nh : s \u2286 t\nf : \u03b1 \u2192 G\n\u22a2 Disjoint (t \\ s) s\n[PROOFSTEP]\nexact disjoint_sdiff_self_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\ninst\u271d : Group G\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 G\na : \u03b1\nh : s \u2286 t\nf : \u03b1 \u2192 G\n\u22a2 mulIndicator (t \\ s) f = mulIndicator t f / mulIndicator s f\n[PROOFSTEP]\nrw [mulIndicator_diff h, div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : One N\nf : \u03b1 \u2192 N\ng : \u03b1 \u2192 N \u2192 M\ns t : Finset \u03b1\nh : s \u2286 t\nhg : \u2200 (a : \u03b1), g a 1 = 1\n\u22a2 \u220f i in s, g i (f i) = \u220f i in t, g i (mulIndicator (\u2191s) f i)\n[PROOFSTEP]\nrw [\u2190 Finset.prod_subset h _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : One N\nf : \u03b1 \u2192 N\ng : \u03b1 \u2192 N \u2192 M\ns t : Finset \u03b1\nh : s \u2286 t\nhg : \u2200 (a : \u03b1), g a 1 = 1\n\u22a2 \u220f i in s, g i (f i) = \u220f x in s, g x (mulIndicator (\u2191s) f x)\n[PROOFSTEP]\napply Finset.prod_congr rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : One N\nf : \u03b1 \u2192 N\ng : \u03b1 \u2192 N \u2192 M\ns t : Finset \u03b1\nh : s \u2286 t\nhg : \u2200 (a : \u03b1), g a 1 = 1\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 g x (f x) = g x (mulIndicator (\u2191s) f x)\n[PROOFSTEP]\nintro i hi\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : One N\nf : \u03b1 \u2192 N\ng : \u03b1 \u2192 N \u2192 M\ns t : Finset \u03b1\nh : s \u2286 t\nhg : \u2200 (a : \u03b1), g a 1 = 1\ni : \u03b1\nhi : i \u2208 s\n\u22a2 g i (f i) = g i (mulIndicator (\u2191s) f i)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : One N\nf : \u03b1 \u2192 N\ng : \u03b1 \u2192 N \u2192 M\ns t : Finset \u03b1\nh : s \u2286 t\nhg : \u2200 (a : \u03b1), g a 1 = 1\ni : \u03b1\nhi : i \u2208 s\n\u22a2 f i = mulIndicator (\u2191s) f i\n[PROOFSTEP]\nsymm\n -- Porting note: This did not use to need the implicit argument\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : One N\nf : \u03b1 \u2192 N\ng : \u03b1 \u2192 N \u2192 M\ns t : Finset \u03b1\nh : s \u2286 t\nhg : \u2200 (a : \u03b1), g a 1 = 1\ni : \u03b1\nhi : i \u2208 s\n\u22a2 mulIndicator (\u2191s) f i = f i\n[PROOFSTEP]\nexact mulIndicator_of_mem (\u03b1 := \u03b1) hi f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : One N\nf : \u03b1 \u2192 N\ng : \u03b1 \u2192 N \u2192 M\ns t : Finset \u03b1\nh : s \u2286 t\nhg : \u2200 (a : \u03b1), g a 1 = 1\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 \u00acx \u2208 s \u2192 g x (mulIndicator (\u2191s) f x) = 1\n[PROOFSTEP]\nrefine' fun i _ hn => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : One N\nf : \u03b1 \u2192 N\ng : \u03b1 \u2192 N \u2192 M\ns t : Finset \u03b1\nh : s \u2286 t\nhg : \u2200 (a : \u03b1), g a 1 = 1\ni : \u03b1\nx\u271d : i \u2208 t\nhn : \u00aci \u2208 s\n\u22a2 g i (mulIndicator (\u2191s) f i) = 1\n[PROOFSTEP]\nconvert hg i\n[GOAL]\ncase h.e'_2.h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : One N\nf : \u03b1 \u2192 N\ng : \u03b1 \u2192 N \u2192 M\ns t : Finset \u03b1\nh : s \u2286 t\nhg : \u2200 (a : \u03b1), g a 1 = 1\ni : \u03b1\nx\u271d : i \u2208 t\nhn : \u00aci \u2208 s\n\u22a2 mulIndicator (\u2191s) f i = 1\n[PROOFSTEP]\nexact mulIndicator_of_not_mem (\u03b1 := \u03b1) hn f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : CommMonoid M\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 M\nt : \u03b9 \u2192 Set \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d : DecidablePred fun i => g i \u2208 t i\n\u22a2 \u220f i in s, mulIndicator (t i) (f i) (g i) = \u220f i in Finset.filter (fun i => g i \u2208 t i) s, f i (g i)\n[PROOFSTEP]\nrefine' (Finset.prod_filter_mul_prod_filter_not s (fun i => g i \u2208 t i) _).symm.trans _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : CommMonoid M\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 M\nt : \u03b9 \u2192 Set \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d : DecidablePred fun i => g i \u2208 t i\n\u22a2 (\u220f x in Finset.filter (fun i => g i \u2208 t i) s, mulIndicator (t x) (f x) (g x)) *\n \u220f x in Finset.filter (fun x => \u00acg x \u2208 t x) s, mulIndicator (t x) (f x) (g x) =\n \u220f i in Finset.filter (fun i => g i \u2208 t i) s, f i (g i)\n[PROOFSTEP]\nrefine' Eq.trans _ (mul_one _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : CommMonoid M\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 M\nt : \u03b9 \u2192 Set \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d : DecidablePred fun i => g i \u2208 t i\n\u22a2 (\u220f x in Finset.filter (fun i => g i \u2208 t i) s, mulIndicator (t x) (f x) (g x)) *\n \u220f x in Finset.filter (fun x => \u00acg x \u2208 t x) s, mulIndicator (t x) (f x) (g x) =\n (\u220f i in Finset.filter (fun i => g i \u2208 t i) s, f i (g i)) * 1\n[PROOFSTEP]\nexact\n congr_arg\u2082 (\u00b7 * \u00b7) (Finset.prod_congr rfl fun x hx => mulIndicator_of_mem (Finset.mem_filter.1 hx).2 _)\n (Finset.prod_eq_one fun x hx => mulIndicator_of_not_mem (Finset.mem_filter.1 hx).2 _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : CommMonoid M\nI : Finset \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\n\u22a2 (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)) \u2192\n mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) f = fun a => \u220f i in I, mulIndicator (s i) f a\n[PROOFSTEP]\nclassical\nrefine' Finset.induction_on I _ _\n\u00b7 intro\n funext\n simp\nintro a I haI ih hI\nfunext\nrw [Finset.prod_insert haI, Finset.set_biUnion_insert, mulIndicator_union_of_not_mem_inter, ih _]\n\u00b7 intro i hi j hj hij\n exact hI i (Finset.mem_insert_of_mem hi) j (Finset.mem_insert_of_mem hj) hij\nsimp only [not_exists, exists_prop, mem_iUnion, mem_inter_iff, not_and]\nintro hx a' ha'\nrefine' disjoint_left.1 (hI a (Finset.mem_insert_self _ _) a' (Finset.mem_insert_of_mem ha') _) hx\nexact (ne_of_mem_of_not_mem ha' haI).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : CommMonoid M\nI : Finset \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\n\u22a2 (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)) \u2192\n mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) f = fun a => \u220f i in I, mulIndicator (s i) f a\n[PROOFSTEP]\nrefine' Finset.induction_on I _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : CommMonoid M\nI : Finset \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\n\u22a2 (\u2200 (i : \u03b9), i \u2208 \u2205 \u2192 \u2200 (j : \u03b9), j \u2208 \u2205 \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)) \u2192\n mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 \u2205), s i) f = fun a => \u220f i in \u2205, mulIndicator (s i) f a\n[PROOFSTEP]\nintro\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : CommMonoid M\nI : Finset \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\na\u271d : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 \u2200 (j : \u03b9), j \u2208 \u2205 \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)\n\u22a2 mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 \u2205), s i) f = fun a => \u220f i in \u2205, mulIndicator (s i) f a\n[PROOFSTEP]\nfunext\n[GOAL]\ncase refine'_1.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : CommMonoid M\nI : Finset \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\na\u271d : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 \u2200 (j : \u03b9), j \u2208 \u2205 \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)\nx\u271d : \u03b1\n\u22a2 mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 \u2205), s i) f x\u271d = \u220f i in \u2205, mulIndicator (s i) f x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : CommMonoid M\nI : Finset \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\n\u22a2 \u2200 \u2983a : \u03b9\u2984 {s_1 : Finset \u03b9},\n \u00aca \u2208 s_1 \u2192\n ((\u2200 (i : \u03b9), i \u2208 s_1 \u2192 \u2200 (j : \u03b9), j \u2208 s_1 \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)) \u2192\n mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 s_1), s i) f = fun a => \u220f i in s_1, mulIndicator (s i) f a) \u2192\n (\u2200 (i : \u03b9), i \u2208 insert a s_1 \u2192 \u2200 (j : \u03b9), j \u2208 insert a s_1 \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)) \u2192\n mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 insert a s_1), s i) f = fun a_4 =>\n \u220f i in insert a s_1, mulIndicator (s i) f a_4\n[PROOFSTEP]\nintro a I haI ih hI\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : CommMonoid M\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b9\nI : Finset \u03b9\nhaI : \u00aca \u2208 I\nih :\n (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)) \u2192\n mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) f = fun a => \u220f i in I, mulIndicator (s i) f a\nhI : \u2200 (i : \u03b9), i \u2208 insert a I \u2192 \u2200 (j : \u03b9), j \u2208 insert a I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)\n\u22a2 mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 insert a I), s i) f = fun a_1 => \u220f i in insert a I, mulIndicator (s i) f a_1\n[PROOFSTEP]\nfunext\n[GOAL]\ncase refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : CommMonoid M\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b9\nI : Finset \u03b9\nhaI : \u00aca \u2208 I\nih :\n (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)) \u2192\n mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) f = fun a => \u220f i in I, mulIndicator (s i) f a\nhI : \u2200 (i : \u03b9), i \u2208 insert a I \u2192 \u2200 (j : \u03b9), j \u2208 insert a I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)\nx\u271d : \u03b1\n\u22a2 mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 insert a I), s i) f x\u271d = \u220f i in insert a I, mulIndicator (s i) f x\u271d\n[PROOFSTEP]\nrw [Finset.prod_insert haI, Finset.set_biUnion_insert, mulIndicator_union_of_not_mem_inter, ih _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : CommMonoid M\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b9\nI : Finset \u03b9\nhaI : \u00aca \u2208 I\nih :\n (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)) \u2192\n mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) f = fun a => \u220f i in I, mulIndicator (s i) f a\nhI : \u2200 (i : \u03b9), i \u2208 insert a I \u2192 \u2200 (j : \u03b9), j \u2208 insert a I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)\nx\u271d : \u03b1\n\u22a2 \u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)\n[PROOFSTEP]\nintro i hi j hj hij\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : CommMonoid M\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b9\nI : Finset \u03b9\nhaI : \u00aca \u2208 I\nih :\n (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)) \u2192\n mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) f = fun a => \u220f i in I, mulIndicator (s i) f a\nhI : \u2200 (i : \u03b9), i \u2208 insert a I \u2192 \u2200 (j : \u03b9), j \u2208 insert a I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)\nx\u271d : \u03b1\ni : \u03b9\nhi : i \u2208 I\nj : \u03b9\nhj : j \u2208 I\nhij : i \u2260 j\n\u22a2 Disjoint (s i) (s j)\n[PROOFSTEP]\nexact hI i (Finset.mem_insert_of_mem hi) j (Finset.mem_insert_of_mem hj) hij\n[GOAL]\ncase refine'_2.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : CommMonoid M\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b9\nI : Finset \u03b9\nhaI : \u00aca \u2208 I\nih :\n (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)) \u2192\n mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) f = fun a => \u220f i in I, mulIndicator (s i) f a\nhI : \u2200 (i : \u03b9), i \u2208 insert a I \u2192 \u2200 (j : \u03b9), j \u2208 insert a I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)\nx\u271d : \u03b1\n\u22a2 \u00acx\u271d \u2208 s a \u2229 \u22c3 (x : \u03b9) (_ : x \u2208 I), s x\n[PROOFSTEP]\nsimp only [not_exists, exists_prop, mem_iUnion, mem_inter_iff, not_and]\n[GOAL]\ncase refine'_2.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : CommMonoid M\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b9\nI : Finset \u03b9\nhaI : \u00aca \u2208 I\nih :\n (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)) \u2192\n mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) f = fun a => \u220f i in I, mulIndicator (s i) f a\nhI : \u2200 (i : \u03b9), i \u2208 insert a I \u2192 \u2200 (j : \u03b9), j \u2208 insert a I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 s a \u2192 \u2200 (x : \u03b9), x \u2208 I \u2192 \u00acx\u271d \u2208 s x\n[PROOFSTEP]\nintro hx a' ha'\n[GOAL]\ncase refine'_2.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : CommMonoid M\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b9\nI : Finset \u03b9\nhaI : \u00aca \u2208 I\nih :\n (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)) \u2192\n mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) f = fun a => \u220f i in I, mulIndicator (s i) f a\nhI : \u2200 (i : \u03b9), i \u2208 insert a I \u2192 \u2200 (j : \u03b9), j \u2208 insert a I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)\nx\u271d : \u03b1\nhx : x\u271d \u2208 s a\na' : \u03b9\nha' : a' \u2208 I\n\u22a2 \u00acx\u271d \u2208 s a'\n[PROOFSTEP]\nrefine' disjoint_left.1 (hI a (Finset.mem_insert_self _ _) a' (Finset.mem_insert_of_mem ha') _) hx\n[GOAL]\ncase refine'_2.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : CommMonoid M\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\na : \u03b9\nI : Finset \u03b9\nhaI : \u00aca \u2208 I\nih :\n (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)) \u2192\n mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) f = fun a => \u220f i in I, mulIndicator (s i) f a\nhI : \u2200 (i : \u03b9), i \u2208 insert a I \u2192 \u2200 (j : \u03b9), j \u2208 insert a I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)\nx\u271d : \u03b1\nhx : x\u271d \u2208 s a\na' : \u03b9\nha' : a' \u2208 I\n\u22a2 a \u2260 a'\n[PROOFSTEP]\nexact (ne_of_mem_of_not_mem ha' haI).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : CommMonoid M\nI : Finset \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nh : \u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 Disjoint (s i) (s j)\nx : \u03b1\n\u22a2 mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) f x = \u220f i in I, mulIndicator (s i) f x\n[PROOFSTEP]\nrw [Set.mulIndicator_finset_biUnion I s h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\n\u22a2 (indicator s fun a => f a * g a) = fun a => indicator s f a * indicator s g a\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\nx\u271d : \u03b1\n\u22a2 indicator s (fun a => f a * g a) x\u271d = indicator s f x\u271d * indicator s g x\u271d\n[PROOFSTEP]\nsimp only [indicator]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\nx\u271d : \u03b1\n\u22a2 (if x\u271d \u2208 s then f x\u271d * g x\u271d else 0) = (if x\u271d \u2208 s then f x\u271d else 0) * if x\u271d \u2208 s then g x\u271d else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\nx\u271d : \u03b1\nh\u271d : x\u271d \u2208 s\n\u22a2 f x\u271d * g x\u271d = f x\u271d * g x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\nx\u271d : \u03b1\nh\u271d : \u00acx\u271d \u2208 s\n\u22a2 0 = 0 * 0\n[PROOFSTEP]\nrw [mul_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\n\u22a2 indicator s (fun a => f a * g a) a = indicator s f a * g a\n[PROOFSTEP]\nsimp only [indicator]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\n\u22a2 (if a \u2208 s then f a * g a else 0) = (if a \u2208 s then f a else 0) * g a\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\nh\u271d : a \u2208 s\n\u22a2 f a * g a = f a * g a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\nh\u271d : \u00aca \u2208 s\n\u22a2 0 = 0 * g a\n[PROOFSTEP]\nrw [zero_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\n\u22a2 indicator s (fun a => f a * g a) a = f a * indicator s g a\n[PROOFSTEP]\nsimp only [indicator]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\n\u22a2 (if a \u2208 s then f a * g a else 0) = f a * if a \u2208 s then g a else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\nh\u271d : a \u2208 s\n\u22a2 f a * g a = f a * g a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns\u271d t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\ns : Set \u03b1\nf g : \u03b1 \u2192 M\nh\u271d : \u00aca \u2208 s\n\u22a2 0 = f a * 0\n[PROOFSTEP]\nrw [mul_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\nt1 t2 : Set \u03b1\nf g : \u03b1 \u2192 M\nx : \u03b1\n\u22a2 indicator (t1 \u2229 t2) (fun x => f x * g x) x = indicator t1 f x * indicator t2 g x\n[PROOFSTEP]\nrw [\u2190 Set.indicator_indicator]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\nt1 t2 : Set \u03b1\nf g : \u03b1 \u2192 M\nx : \u03b1\n\u22a2 indicator t1 (indicator t2 fun x => f x * g x) x = indicator t1 f x * indicator t2 g x\n[PROOFSTEP]\nsimp_rw [indicator]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\nt1 t2 : Set \u03b1\nf g : \u03b1 \u2192 M\nx : \u03b1\n\u22a2 (if x \u2208 t1 then if x \u2208 t2 then f x * g x else 0 else 0) = (if x \u2208 t1 then f x else 0) * if x \u2208 t2 then g x else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\nt1 t2 : Set \u03b1\nf g : \u03b1 \u2192 M\nx : \u03b1\nh\u271d\u00b9 : x \u2208 t1\nh\u271d : x \u2208 t2\n\u22a2 f x * g x = f x * g x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\nt1 t2 : Set \u03b1\nf g : \u03b1 \u2192 M\nx : \u03b1\nh\u271d\u00b9 : x \u2208 t1\nh\u271d : \u00acx \u2208 t2\n\u22a2 0 = f x * 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\nt1 t2 : Set \u03b1\nf g : \u03b1 \u2192 M\nx : \u03b1\nh\u271d\u00b9 : \u00acx \u2208 t1\nh\u271d : x \u2208 t2\n\u22a2 0 = 0 * g x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroClass M\ns t : Set \u03b1\nf\u271d g\u271d : \u03b1 \u2192 M\na : \u03b1\nt1 t2 : Set \u03b1\nf g : \u03b1 \u2192 M\nx : \u03b1\nh\u271d\u00b9 : \u00acx \u2208 t1\nh\u271d : \u00acx \u2208 t2\n\u22a2 0 = 0 * 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns t : Set \u03b1\nx\u271d : \u03b1\n\u22a2 indicator (s \u2229 t) 1 x\u271d = (indicator s 1 * indicator t 1) x\u271d\n[PROOFSTEP]\nsimp only [\u2190 inter_indicator_mul, Pi.mul_apply, Pi.one_apply, one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns t : Set \u03b1\nx\u271d : \u03b1\n\u22a2 indicator (s \u2229 t) 1 x\u271d = indicator (s \u2229 t) (fun x => 1) x\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 indicator (s \u00d7\u02e2 t) 1 (x, y) = indicator s 1 x * indicator t 1 y\n[PROOFSTEP]\nsimp_rw [indicator, mem_prod_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 (if x \u2208 s \u2227 y \u2208 t then OfNat.ofNat 1 (x, y) else 0) =\n (if x \u2208 s then OfNat.ofNat 1 x else 0) * if y \u2208 t then OfNat.ofNat 1 y else 0\n[PROOFSTEP]\nsplit_ifs with h\u2080\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\ny : \u03b2\nh\u2080 : x \u2208 s \u2227 y \u2208 t\nh\u271d\u00b9 : x \u2208 s\nh\u271d : y \u2208 t\n\u22a2 OfNat.ofNat 1 (x, y) = OfNat.ofNat 1 x * OfNat.ofNat 1 y\n[PROOFSTEP]\nsimp only [Pi.one_apply, mul_one, mul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\ny : \u03b2\nh\u2080 : x \u2208 s \u2227 y \u2208 t\nh\u271d\u00b9 : x \u2208 s\nh\u271d : \u00acy \u2208 t\n\u22a2 OfNat.ofNat 1 (x, y) = OfNat.ofNat 1 x * 0\n[PROOFSTEP]\nsimp only [Pi.one_apply, mul_one, mul_zero]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\ny : \u03b2\nh\u2080 : x \u2208 s \u2227 y \u2208 t\nh\u271d\u00b9 : \u00acx \u2208 s\nh\u271d : y \u2208 t\n\u22a2 OfNat.ofNat 1 (x, y) = 0 * OfNat.ofNat 1 y\n[PROOFSTEP]\nsimp only [Pi.one_apply, mul_one, mul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\ny : \u03b2\nh\u2080 : x \u2208 s \u2227 y \u2208 t\nh\u271d\u00b9 : \u00acx \u2208 s\nh\u271d : \u00acy \u2208 t\n\u22a2 OfNat.ofNat 1 (x, y) = 0 * 0\n[PROOFSTEP]\nsimp only [Pi.one_apply, mul_one, mul_zero]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\ny : \u03b2\nh\u2080 : \u00ac(x \u2208 s \u2227 y \u2208 t)\nh\u271d\u00b9 : x \u2208 s\nh\u271d : y \u2208 t\n\u22a2 0 = OfNat.ofNat 1 x * OfNat.ofNat 1 y\n[PROOFSTEP]\nsimp only [Pi.one_apply, mul_one, mul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\ny : \u03b2\nh\u2080 : \u00ac(x \u2208 s \u2227 y \u2208 t)\nh\u271d\u00b9 : x \u2208 s\nh\u271d : \u00acy \u2208 t\n\u22a2 0 = OfNat.ofNat 1 x * 0\n[PROOFSTEP]\nsimp only [Pi.one_apply, mul_one, mul_zero]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\ny : \u03b2\nh\u2080 : \u00ac(x \u2208 s \u2227 y \u2208 t)\nh\u271d\u00b9 : \u00acx \u2208 s\nh\u271d : y \u2208 t\n\u22a2 0 = 0 * OfNat.ofNat 1 y\n[PROOFSTEP]\nsimp only [Pi.one_apply, mul_one, mul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\ny : \u03b2\nh\u2080 : \u00ac(x \u2208 s \u2227 y \u2208 t)\nh\u271d\u00b9 : \u00acx \u2208 s\nh\u271d : \u00acy \u2208 t\n\u22a2 0 = 0 * 0\n[PROOFSTEP]\nsimp only [Pi.one_apply, mul_one, mul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\ny : \u03b2\nh\u2080 : x \u2208 s \u2227 y \u2208 t\nh\u271d\u00b9 : x \u2208 s\nh\u271d : \u00acy \u2208 t\n\u22a2 1 = 0\n[PROOFSTEP]\ntauto\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\ny : \u03b2\nh\u2080 : x \u2208 s \u2227 y \u2208 t\nh\u271d\u00b9 : \u00acx \u2208 s\nh\u271d : y \u2208 t\n\u22a2 1 = 0\n[PROOFSTEP]\ntauto\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\ny : \u03b2\nh\u2080 : x \u2208 s \u2227 y \u2208 t\nh\u271d\u00b9 : \u00acx \u2208 s\nh\u271d : \u00acy \u2208 t\n\u22a2 1 = 0\n[PROOFSTEP]\ntauto\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : MulZeroOneClass M\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\ny : \u03b2\nh\u2080 : \u00ac(x \u2208 s \u2227 y \u2208 t)\nh\u271d\u00b9 : x \u2208 s\nh\u271d : y \u2208 t\n\u22a2 0 = 1\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : MulZeroOneClass M\ninst\u271d : Nontrivial M\nU : Set \u03b1\nx : \u03b1\n\u22a2 indicator U 1 x = 0 \u2194 \u00acx \u2208 U\n[PROOFSTEP]\nclassical simp [indicator_apply, imp_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : MulZeroOneClass M\ninst\u271d : Nontrivial M\nU : Set \u03b1\nx : \u03b1\n\u22a2 indicator U 1 x = 0 \u2194 \u00acx \u2208 U\n[PROOFSTEP]\nsimp [indicator_apply, imp_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : MulZeroOneClass M\ninst\u271d : Nontrivial M\nU : Set \u03b1\nx : \u03b1\n\u22a2 indicator U 1 x = 1 \u2194 x \u2208 U\n[PROOFSTEP]\nclassical simp [indicator_apply, imp_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : MulZeroOneClass M\ninst\u271d : Nontrivial M\nU : Set \u03b1\nx : \u03b1\n\u22a2 indicator U 1 x = 1 \u2194 x \u2208 U\n[PROOFSTEP]\nsimp [indicator_apply, imp_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : MulZeroOneClass M\ninst\u271d : Nontrivial M\nU V : Set \u03b1\nh : indicator U 1 = indicator V 1\n\u22a2 U = V\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : MulZeroOneClass M\ninst\u271d : Nontrivial M\nU V : Set \u03b1\nh : indicator U 1 = indicator V 1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 U \u2194 x\u271d \u2208 V\n[PROOFSTEP]\nsimp_rw [\u2190 indicator_eq_one_iff_mem M, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\ny : M\ninst\u271d : LE M\nhfg : a \u2208 s \u2192 f a \u2264 y\nhg : \u00aca \u2208 s \u2192 1 \u2264 y\n\u22a2 mulIndicator s f a \u2264 y\n[PROOFSTEP]\nby_cases ha : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\ny : M\ninst\u271d : LE M\nhfg : a \u2208 s \u2192 f a \u2264 y\nhg : \u00aca \u2208 s \u2192 1 \u2264 y\nha : a \u2208 s\n\u22a2 mulIndicator s f a \u2264 y\n[PROOFSTEP]\nsimpa [ha] using hfg ha\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\ny : M\ninst\u271d : LE M\nhfg : a \u2208 s \u2192 f a \u2264 y\nhg : \u00aca \u2208 s \u2192 1 \u2264 y\nha : \u00aca \u2208 s\n\u22a2 mulIndicator s f a \u2264 y\n[PROOFSTEP]\nsimpa [ha] using hg ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b2 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\n\u22a2 mulIndicator (\u22c3 (i : \u03b9), s i) f x = \u2a06 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nby_cases hx : x \u2208 \u22c3 i, s i\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b2 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : x \u2208 \u22c3 (i : \u03b9), s i\n\u22a2 mulIndicator (\u22c3 (i : \u03b9), s i) f x = \u2a06 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nrw [mulIndicator_of_mem hx]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b2 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : x \u2208 \u22c3 (i : \u03b9), s i\n\u22a2 f x = \u2a06 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nrw [mem_iUnion] at hx \n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b2 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : \u2203 i, x \u2208 s i\n\u22a2 f x = \u2a06 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nrefine' le_antisymm _ (iSup_le fun i => mulIndicator_le_self' (fun x _ => h1 \u25b8 bot_le) x)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b2 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : \u2203 i, x \u2208 s i\n\u22a2 f x \u2264 \u2a06 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nrcases hx with \u27e8i, hi\u27e9\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b2 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\ni : \u03b9\nhi : x \u2208 s i\n\u22a2 f x \u2264 \u2a06 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nexact le_iSup_of_le i (ge_of_eq <| mulIndicator_of_mem hi _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b2 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : \u00acx \u2208 \u22c3 (i : \u03b9), s i\n\u22a2 mulIndicator (\u22c3 (i : \u03b9), s i) f x = \u2a06 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nrw [mulIndicator_of_not_mem hx]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b2 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : \u00acx \u2208 \u22c3 (i : \u03b9), s i\n\u22a2 1 = \u2a06 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nsimp only [mem_iUnion, not_exists] at hx \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b2 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : \u2200 (x_1 : \u03b9), \u00acx \u2208 s x_1\n\u22a2 1 = \u2a06 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nsimp [hx, \u2190 h1]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b3 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\n\u22a2 mulIndicator (\u22c2 (i : \u03b9), s i) f x = \u2a05 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nby_cases hx : x \u2208 \u22c2 i, s i\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b3 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : x \u2208 \u22c2 (i : \u03b9), s i\n\u22a2 mulIndicator (\u22c2 (i : \u03b9), s i) f x = \u2a05 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nrw [mulIndicator_of_mem hx]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b3 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : x \u2208 \u22c2 (i : \u03b9), s i\n\u22a2 f x = \u2a05 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nrw [mem_iInter] at hx \n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b3 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : \u2200 (i : \u03b9), x \u2208 s i\n\u22a2 f x = \u2a05 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nrefine le_antisymm ?_ (by simp only [mulIndicator_of_mem (hx _), ciInf_const, le_refl])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b3 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : \u2200 (i : \u03b9), x \u2208 s i\n\u22a2 \u2a05 (i : \u03b9), mulIndicator (s i) f x \u2264 f x\n[PROOFSTEP]\nsimp only [mulIndicator_of_mem (hx _), ciInf_const, le_refl]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b3 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : \u2200 (i : \u03b9), x \u2208 s i\n\u22a2 f x \u2264 \u2a05 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nexact le_iInf (fun j \u21a6 by simp only [mulIndicator_of_mem (hx j), le_refl])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b3 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : \u2200 (i : \u03b9), x \u2208 s i\nj : \u03b9\n\u22a2 f x \u2264 mulIndicator (s j) f x\n[PROOFSTEP]\nsimp only [mulIndicator_of_mem (hx j), le_refl]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b3 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : \u00acx \u2208 \u22c2 (i : \u03b9), s i\n\u22a2 mulIndicator (\u22c2 (i : \u03b9), s i) f x = \u2a05 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nrw [mulIndicator_of_not_mem hx]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b3 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : \u00acx \u2208 \u22c2 (i : \u03b9), s i\n\u22a2 1 = \u2a05 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nsimp only [mem_iInter, not_exists, not_forall] at hx \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b3 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nhx : \u2203 x_1, \u00acx \u2208 s x_1\n\u22a2 1 = \u2a05 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nrcases hx with \u27e8j, hj\u27e9\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b3 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nj : \u03b9\nhj : \u00acx \u2208 s j\n\u22a2 1 = \u2a05 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nrefine le_antisymm (by simp only [\u2190 h1, le_iInf_iff, bot_le, forall_const]) ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b3 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nj : \u03b9\nhj : \u00acx \u2208 s j\n\u22a2 1 \u2264 \u2a05 (i : \u03b9), mulIndicator (s i) f x\n[PROOFSTEP]\nsimp only [\u2190 h1, le_iInf_iff, bot_le, forall_const]\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : One M\u271d\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 M\u271d\na : \u03b1\ny : M\u271d\ninst\u271d\u00b3 : Preorder M\u271d\n\u03b9 : Sort u_6\nM : Type u_7\ninst\u271d\u00b2 : Nonempty \u03b9\ninst\u271d\u00b9 : CompleteLattice M\ninst\u271d : One M\nh1 : \u22a5 = 1\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 M\nx : \u03b1\nj : \u03b9\nhj : \u00acx \u2208 s j\n\u22a2 \u2a05 (i : \u03b9), mulIndicator (s i) f x \u2264 1\n[PROOFSTEP]\nsimpa [mulIndicator_of_not_mem hj] using (iInf_le (fun i \u21a6 (s i).mulIndicator f) j) x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 indicator s f \u2264 indicator {x | 0 \u2264 f x} f\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 indicator s f x \u2264 indicator {x | 0 \u2264 f x} f x\n[PROOFSTEP]\nclassical\nsimp_rw [indicator_apply]\nsplit_ifs with h_1 h_2 h_3\n\u00b7 exact le_rfl\n\u00b7 exact (not_le.mp h_2).le\n\u00b7 exact h_3\n\u00b7 exact le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 indicator s f x \u2264 indicator {x | 0 \u2264 f x} f x\n[PROOFSTEP]\nsimp_rw [indicator_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 (if x \u2208 s then f x else 0) \u2264 if x \u2208 {x | 0 \u2264 f x} then f x else 0\n[PROOFSTEP]\nsplit_ifs with h_1 h_2 h_3\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\nh_1 : x \u2208 s\nh_2 : x \u2208 {x | 0 \u2264 f x}\n\u22a2 f x \u2264 f x\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\nh_1 : x \u2208 s\nh_2 : \u00acx \u2208 {x | 0 \u2264 f x}\n\u22a2 f x \u2264 0\n[PROOFSTEP]\nexact (not_le.mp h_2).le\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\nh_1 : \u00acx \u2208 s\nh_3 : x \u2208 {x | 0 \u2264 f x}\n\u22a2 0 \u2264 f x\n[PROOFSTEP]\nexact h_3\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : Zero \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\nh_1 : \u00acx \u2208 s\nh_3 : \u00acx \u2208 {x | 0 \u2264 f x}\n\u22a2 0 \u2264 0\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM\u271d : Type u_4\nN\u271d : Type u_5\nM : Type u_6\nN : Type u_7\ninst\u271d\u00b9 : MulOneClass M\ninst\u271d : MulOneClass N\nf : M \u2192* N\ns : Set \u03b1\ng : \u03b1 \u2192 M\nx : \u03b1\n\u22a2 \u2191f (Set.mulIndicator s g x) = Set.mulIndicator s (\u2191f \u2218 g) x\n[PROOFSTEP]\nsimp [Set.mulIndicator_comp_of_one]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.IndicatorFunction", "llama_tokens": 37742, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.6406358617010351, "lm_q1q2_score": 0.4394438201925288}} {"text": "[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : MulZeroClass G\u2080\na : G\u2080\n\u22a2 SemiconjBy a 0 0\n[PROOFSTEP]\nsimp only [SemiconjBy, mul_zero, zero_mul]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : MulZeroClass G\u2080\nx y : G\u2080\n\u22a2 SemiconjBy 0 x y\n[PROOFSTEP]\nsimp only [SemiconjBy, mul_zero, zero_mul]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na x y x' y' : G\u2080\nha : a = 0\n\u22a2 SemiconjBy a\u207b\u00b9 x y \u2194 SemiconjBy a y x\n[PROOFSTEP]\nsimp only [ha, inv_zero, SemiconjBy.zero_left]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na x y x' y' : G\u2080\nh : SemiconjBy a x y\n\u22a2 SemiconjBy a x\u207b\u00b9 y\u207b\u00b9\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na x y x' y' : G\u2080\nh : SemiconjBy a x y\nha : a = 0\n\u22a2 SemiconjBy a x\u207b\u00b9 y\u207b\u00b9\n[PROOFSTEP]\nsimp only [ha, zero_left]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na x y x' y' : G\u2080\nh : SemiconjBy a x y\nha : \u00aca = 0\n\u22a2 SemiconjBy a x\u207b\u00b9 y\u207b\u00b9\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na x y x' y' : G\u2080\nh : SemiconjBy a x y\nha : \u00aca = 0\nhx : x = 0\n\u22a2 SemiconjBy a x\u207b\u00b9 y\u207b\u00b9\n[PROOFSTEP]\nsubst x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na y x' y' : G\u2080\nha : \u00aca = 0\nh : SemiconjBy a 0 y\n\u22a2 SemiconjBy a 0\u207b\u00b9 y\u207b\u00b9\n[PROOFSTEP]\nsimp only [SemiconjBy, mul_zero, @eq_comm _ _ (y * a), mul_eq_zero] at h \n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na y x' y' : G\u2080\nha : \u00aca = 0\nh : y = 0 \u2228 a = 0\n\u22a2 SemiconjBy a 0\u207b\u00b9 y\u207b\u00b9\n[PROOFSTEP]\nsimp [h.resolve_right ha]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na x y x' y' : G\u2080\nh : SemiconjBy a x y\nha : \u00aca = 0\nhx : \u00acx = 0\n\u22a2 SemiconjBy a x\u207b\u00b9 y\u207b\u00b9\n[PROOFSTEP]\nhave := mul_ne_zero ha hx\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na x y x' y' : G\u2080\nh : SemiconjBy a x y\nha : \u00aca = 0\nhx : \u00acx = 0\nthis : a * x \u2260 0\n\u22a2 SemiconjBy a x\u207b\u00b9 y\u207b\u00b9\n[PROOFSTEP]\nrw [h.eq, mul_ne_zero_iff] at this \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na x y x' y' : G\u2080\nh : SemiconjBy a x y\nha : \u00aca = 0\nhx : \u00acx = 0\nthis : y \u2260 0 \u2227 a \u2260 0\n\u22a2 SemiconjBy a x\u207b\u00b9 y\u207b\u00b9\n[PROOFSTEP]\nexact @units_inv_right _ _ _ (Units.mk0 x hx) (Units.mk0 y this.1) h\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na x y x' y' : G\u2080\nh : SemiconjBy a x y\nh' : SemiconjBy a x' y'\n\u22a2 SemiconjBy a (x / x') (y / y')\n[PROOFSTEP]\nrw [div_eq_mul_inv, div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na x y x' y' : G\u2080\nh : SemiconjBy a x y\nh' : SemiconjBy a x' y'\n\u22a2 SemiconjBy a (x * x'\u207b\u00b9) (y * y'\u207b\u00b9)\n[PROOFSTEP]\nexact h.mul_right h'.inv_right\u2080\n", "meta": {"mathlib_filename": "Mathlib.Algebra.GroupWithZero.Semiconj", "llama_tokens": 1978, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.6406358411176238, "lm_q1q2_score": 0.43944380607334904}} {"text": "[GOAL]\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\n\u22a2 x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y)\n[PROOFSTEP]\nsimp only [rpow_def, Complex.cpow_def]\n[GOAL]\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\n\u22a2 (if \u2191x = 0 then if \u2191y = 0 then 1 else 0 else Complex.exp (Complex.log \u2191x * \u2191y)).re =\n if x = 0 then if y = 0 then 1 else 0 else exp (log x * y)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b3 : \u2191x = 0\nh\u271d\u00b2 : \u2191y = 0\nh\u271d\u00b9 : x = 0\nh\u271d : y = 0\n\u22a2 1.re = 1\n[PROOFSTEP]\nsimp_all [(Complex.ofReal_log hx).symm, -Complex.ofReal_mul, -IsROrC.ofReal_mul, (Complex.ofReal_mul _ _).symm,\n Complex.exp_ofReal_re, Complex.ofReal_eq_zero]\n[GOAL]\ncase neg\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b3 : \u2191x = 0\nh\u271d\u00b2 : \u2191y = 0\nh\u271d\u00b9 : x = 0\nh\u271d : \u00acy = 0\n\u22a2 1.re = 0\n[PROOFSTEP]\nsimp_all [(Complex.ofReal_log hx).symm, -Complex.ofReal_mul, -IsROrC.ofReal_mul, (Complex.ofReal_mul _ _).symm,\n Complex.exp_ofReal_re, Complex.ofReal_eq_zero]\n[GOAL]\ncase neg\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b2 : \u2191x = 0\nh\u271d\u00b9 : \u2191y = 0\nh\u271d : \u00acx = 0\n\u22a2 1.re = exp (log x * y)\n[PROOFSTEP]\nsimp_all [(Complex.ofReal_log hx).symm, -Complex.ofReal_mul, -IsROrC.ofReal_mul, (Complex.ofReal_mul _ _).symm,\n Complex.exp_ofReal_re, Complex.ofReal_eq_zero]\n[GOAL]\ncase pos\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b3 : \u2191x = 0\nh\u271d\u00b2 : \u00ac\u2191y = 0\nh\u271d\u00b9 : x = 0\nh\u271d : y = 0\n\u22a2 0.re = 1\n[PROOFSTEP]\nsimp_all [(Complex.ofReal_log hx).symm, -Complex.ofReal_mul, -IsROrC.ofReal_mul, (Complex.ofReal_mul _ _).symm,\n Complex.exp_ofReal_re, Complex.ofReal_eq_zero]\n[GOAL]\ncase neg\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b3 : \u2191x = 0\nh\u271d\u00b2 : \u00ac\u2191y = 0\nh\u271d\u00b9 : x = 0\nh\u271d : \u00acy = 0\n\u22a2 0.re = 0\n[PROOFSTEP]\nsimp_all [(Complex.ofReal_log hx).symm, -Complex.ofReal_mul, -IsROrC.ofReal_mul, (Complex.ofReal_mul _ _).symm,\n Complex.exp_ofReal_re, Complex.ofReal_eq_zero]\n[GOAL]\ncase neg\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b2 : \u2191x = 0\nh\u271d\u00b9 : \u00ac\u2191y = 0\nh\u271d : \u00acx = 0\n\u22a2 0.re = exp (log x * y)\n[PROOFSTEP]\nsimp_all [(Complex.ofReal_log hx).symm, -Complex.ofReal_mul, -IsROrC.ofReal_mul, (Complex.ofReal_mul _ _).symm,\n Complex.exp_ofReal_re, Complex.ofReal_eq_zero]\n[GOAL]\ncase pos\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b2 : \u00ac\u2191x = 0\nh\u271d\u00b9 : x = 0\nh\u271d : y = 0\n\u22a2 (Complex.exp (Complex.log \u2191x * \u2191y)).re = 1\n[PROOFSTEP]\nsimp_all [(Complex.ofReal_log hx).symm, -Complex.ofReal_mul, -IsROrC.ofReal_mul, (Complex.ofReal_mul _ _).symm,\n Complex.exp_ofReal_re, Complex.ofReal_eq_zero]\n[GOAL]\ncase neg\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b2 : \u00ac\u2191x = 0\nh\u271d\u00b9 : x = 0\nh\u271d : \u00acy = 0\n\u22a2 (Complex.exp (Complex.log \u2191x * \u2191y)).re = 0\n[PROOFSTEP]\nsimp_all [(Complex.ofReal_log hx).symm, -Complex.ofReal_mul, -IsROrC.ofReal_mul, (Complex.ofReal_mul _ _).symm,\n Complex.exp_ofReal_re, Complex.ofReal_eq_zero]\n[GOAL]\ncase neg\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b9 : \u00ac\u2191x = 0\nh\u271d : \u00acx = 0\n\u22a2 (Complex.exp (Complex.log \u2191x * \u2191y)).re = exp (log x * y)\n[PROOFSTEP]\nsimp_all [(Complex.ofReal_log hx).symm, -Complex.ofReal_mul, -IsROrC.ofReal_mul, (Complex.ofReal_mul _ _).symm,\n Complex.exp_ofReal_re, Complex.ofReal_eq_zero]\n[GOAL]\nx : \u211d\nhx : 0 < x\ny : \u211d\n\u22a2 x ^ y = exp (log x * y)\n[PROOFSTEP]\nrw [rpow_def_of_nonneg (le_of_lt hx), if_neg (ne_of_gt hx)]\n[GOAL]\nx y : \u211d\n\u22a2 exp (x * y) = exp x ^ y\n[PROOFSTEP]\nrw [rpow_def_of_pos (exp_pos _), log_exp]\n[GOAL]\nx : \u211d\n\u22a2 exp 1 ^ x = exp x\n[PROOFSTEP]\nrw [\u2190 exp_mul, one_mul]\n[GOAL]\nx y : \u211d\nhx : 0 \u2264 x\n\u22a2 x ^ y = 0 \u2194 x = 0 \u2227 y \u2260 0\n[PROOFSTEP]\nsimp only [rpow_def_of_nonneg hx]\n[GOAL]\nx y : \u211d\nhx : 0 \u2264 x\n\u22a2 (if x = 0 then if y = 0 then 1 else 0 else exp (log x * y)) = 0 \u2194 x = 0 \u2227 y \u2260 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nx y : \u211d\nhx : 0 \u2264 x\nh\u271d\u00b9 : x = 0\nh\u271d : y = 0\n\u22a2 1 = 0 \u2194 x = 0 \u2227 y \u2260 0\n[PROOFSTEP]\nsimp [*, exp_ne_zero]\n[GOAL]\ncase neg\nx y : \u211d\nhx : 0 \u2264 x\nh\u271d\u00b9 : x = 0\nh\u271d : \u00acy = 0\n\u22a2 0 = 0 \u2194 x = 0 \u2227 y \u2260 0\n[PROOFSTEP]\nsimp [*, exp_ne_zero]\n[GOAL]\ncase neg\nx y : \u211d\nhx : 0 \u2264 x\nh\u271d : \u00acx = 0\n\u22a2 exp (log x * y) = 0 \u2194 x = 0 \u2227 y \u2260 0\n[PROOFSTEP]\nsimp [*, exp_ne_zero]\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\n\u22a2 x ^ y = exp (log x * y) * cos (y * \u03c0)\n[PROOFSTEP]\nrw [rpow_def, Complex.cpow_def, if_neg]\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\n\u22a2 (Complex.exp (Complex.log \u2191x * \u2191y)).re = exp (log x * y) * cos (y * \u03c0)\ncase hnc x : \u211d hx : x < 0 y : \u211d \u22a2 \u00ac\u2191x = 0\n[PROOFSTEP]\nhave : Complex.log x * y = \u2191(log (-x) * y) + \u2191(y * \u03c0) * Complex.I :=\n by\n simp only [Complex.log, abs_of_neg hx, Complex.arg_ofReal_of_neg hx, Complex.abs_ofReal, Complex.ofReal_mul]\n ring\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\n\u22a2 Complex.log \u2191x * \u2191y = \u2191(log (-x) * y) + \u2191(y * \u03c0) * Complex.I\n[PROOFSTEP]\nsimp only [Complex.log, abs_of_neg hx, Complex.arg_ofReal_of_neg hx, Complex.abs_ofReal, Complex.ofReal_mul]\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\n\u22a2 (\u2191(log (-x)) + \u2191\u03c0 * Complex.I) * \u2191y = \u2191(log (-x)) * \u2191y + \u2191y * \u2191\u03c0 * Complex.I\n[PROOFSTEP]\nring\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\nthis : Complex.log \u2191x * \u2191y = \u2191(log (-x) * y) + \u2191(y * \u03c0) * Complex.I\n\u22a2 (Complex.exp (Complex.log \u2191x * \u2191y)).re = exp (log x * y) * cos (y * \u03c0)\n[PROOFSTEP]\nrw [this, Complex.exp_add_mul_I, \u2190 Complex.ofReal_exp, \u2190 Complex.ofReal_cos, \u2190 Complex.ofReal_sin, mul_add, \u2190\n Complex.ofReal_mul, \u2190 mul_assoc, \u2190 Complex.ofReal_mul, Complex.add_re, Complex.ofReal_re, Complex.mul_re,\n Complex.I_re, Complex.ofReal_im, Real.log_neg_eq_log]\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\nthis : Complex.log \u2191x * \u2191y = \u2191(log (-x) * y) + \u2191(y * \u03c0) * Complex.I\n\u22a2 exp (log x * y) * cos (y * \u03c0) + ((\u2191(exp (log x * y) * sin (y * \u03c0))).re * 0 - 0 * Complex.I.im) =\n exp (log x * y) * cos (y * \u03c0)\n[PROOFSTEP]\nring\n[GOAL]\ncase hnc\nx : \u211d\nhx : x < 0\ny : \u211d\n\u22a2 \u00ac\u2191x = 0\n[PROOFSTEP]\nrw [Complex.ofReal_eq_zero]\n[GOAL]\ncase hnc\nx : \u211d\nhx : x < 0\ny : \u211d\n\u22a2 \u00acx = 0\n[PROOFSTEP]\nexact ne_of_lt hx\n[GOAL]\nx : \u211d\nhx : x \u2264 0\ny : \u211d\n\u22a2 x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) * cos (y * \u03c0)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nx : \u211d\nhx : x \u2264 0\ny : \u211d\nh : x = 0\nh\u271d : y = 0\n\u22a2 x ^ y = 1\n[PROOFSTEP]\nsimp [rpow_def, *]\n[GOAL]\ncase neg\nx : \u211d\nhx : x \u2264 0\ny : \u211d\nh : x = 0\nh\u271d : \u00acy = 0\n\u22a2 x ^ y = 0\n[PROOFSTEP]\nsimp [rpow_def, *]\n[GOAL]\ncase neg\nx : \u211d\nhx : x \u2264 0\ny : \u211d\nh : \u00acx = 0\n\u22a2 x ^ y = exp (log x * y) * cos (y * \u03c0)\n[PROOFSTEP]\nsimp [rpow_def, *]\n[GOAL]\ncase neg\nx : \u211d\nhx : x \u2264 0\ny : \u211d\nh : \u00acx = 0\n\u22a2 (\u2191x ^ \u2191y).re = exp (log x * y) * cos (y * \u03c0)\n[PROOFSTEP]\nexact rpow_def_of_neg (lt_of_le_of_ne hx h) _\n[GOAL]\nx : \u211d\nhx : 0 < x\ny : \u211d\n\u22a2 0 < x ^ y\n[PROOFSTEP]\nrw [rpow_def_of_pos hx]\n[GOAL]\nx : \u211d\nhx : 0 < x\ny : \u211d\n\u22a2 0 < exp (log x * y)\n[PROOFSTEP]\napply exp_pos\n[GOAL]\nx : \u211d\n\u22a2 x ^ 0 = 1\n[PROOFSTEP]\nsimp [rpow_def]\n[GOAL]\nx : \u211d\n\u22a2 0 < x ^ 0\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\nh : x \u2260 0\n\u22a2 0 ^ x = 0\n[PROOFSTEP]\nsimp [rpow_def, *]\n[GOAL]\nx a : \u211d\n\u22a2 0 ^ x = a \u2194 x \u2260 0 \u2227 a = 0 \u2228 x = 0 \u2227 a = 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nx a : \u211d\n\u22a2 0 ^ x = a \u2192 x \u2260 0 \u2227 a = 0 \u2228 x = 0 \u2227 a = 1\n[PROOFSTEP]\nintro hyp\n[GOAL]\ncase mp\nx a : \u211d\nhyp : 0 ^ x = a\n\u22a2 x \u2260 0 \u2227 a = 0 \u2228 x = 0 \u2227 a = 1\n[PROOFSTEP]\nsimp only [rpow_def, Complex.ofReal_zero] at hyp \n[GOAL]\ncase mp\nx a : \u211d\nhyp : (0 ^ \u2191x).re = a\n\u22a2 x \u2260 0 \u2227 a = 0 \u2228 x = 0 \u2227 a = 1\n[PROOFSTEP]\nby_cases x = 0\n[GOAL]\ncase mp\nx a : \u211d\nhyp : (0 ^ \u2191x).re = a\n\u22a2 x \u2260 0 \u2227 a = 0 \u2228 x = 0 \u2227 a = 1\n[PROOFSTEP]\nby_cases x = 0\n[GOAL]\ncase pos\nx a : \u211d\nhyp : (0 ^ \u2191x).re = a\nh : x = 0\n\u22a2 x \u2260 0 \u2227 a = 0 \u2228 x = 0 \u2227 a = 1\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\na : \u211d\nhyp : (0 ^ \u21910).re = a\n\u22a2 0 \u2260 0 \u2227 a = 0 \u2228 0 = 0 \u2227 a = 1\n[PROOFSTEP]\nsimp only [Complex.one_re, Complex.ofReal_zero, Complex.cpow_zero] at hyp \n[GOAL]\ncase pos\na : \u211d\nhyp : 1 = a\n\u22a2 0 \u2260 0 \u2227 a = 0 \u2228 0 = 0 \u2227 a = 1\n[PROOFSTEP]\nexact Or.inr \u27e8rfl, hyp.symm\u27e9\n[GOAL]\ncase neg\nx a : \u211d\nhyp : (0 ^ \u2191x).re = a\nh : \u00acx = 0\n\u22a2 x \u2260 0 \u2227 a = 0 \u2228 x = 0 \u2227 a = 1\n[PROOFSTEP]\nrw [Complex.zero_cpow (Complex.ofReal_ne_zero.mpr h)] at hyp \n[GOAL]\ncase neg\nx a : \u211d\nhyp : 0.re = a\nh : \u00acx = 0\n\u22a2 x \u2260 0 \u2227 a = 0 \u2228 x = 0 \u2227 a = 1\n[PROOFSTEP]\nexact Or.inl \u27e8h, hyp.symm\u27e9\n[GOAL]\ncase mpr\nx a : \u211d\n\u22a2 x \u2260 0 \u2227 a = 0 \u2228 x = 0 \u2227 a = 1 \u2192 0 ^ x = a\n[PROOFSTEP]\nrintro (\u27e8h, rfl\u27e9 | \u27e8rfl, rfl\u27e9)\n[GOAL]\ncase mpr.inl.intro\nx : \u211d\nh : x \u2260 0\n\u22a2 0 ^ x = 0\n[PROOFSTEP]\nexact zero_rpow h\n[GOAL]\ncase mpr.inr.intro\n\u22a2 0 ^ 0 = 1\n[PROOFSTEP]\nexact rpow_zero _\n[GOAL]\nx a : \u211d\n\u22a2 a = 0 ^ x \u2194 x \u2260 0 \u2227 a = 0 \u2228 x = 0 \u2227 a = 1\n[PROOFSTEP]\nrw [\u2190 zero_rpow_eq_iff, eq_comm]\n[GOAL]\nx : \u211d\n\u22a2 x ^ 1 = x\n[PROOFSTEP]\nsimp [rpow_def]\n[GOAL]\nx : \u211d\n\u22a2 1 ^ x = 1\n[PROOFSTEP]\nsimp [rpow_def]\n[GOAL]\nx : \u211d\n\u22a2 0 ^ x \u2264 1\n[PROOFSTEP]\nby_cases h : x = 0\n[GOAL]\ncase pos\nx : \u211d\nh : x = 0\n\u22a2 0 ^ x \u2264 1\n[PROOFSTEP]\nsimp [h, zero_le_one]\n[GOAL]\ncase neg\nx : \u211d\nh : \u00acx = 0\n\u22a2 0 ^ x \u2264 1\n[PROOFSTEP]\nsimp [h, zero_le_one]\n[GOAL]\nx : \u211d\n\u22a2 0 \u2264 0 ^ x\n[PROOFSTEP]\nby_cases h : x = 0\n[GOAL]\ncase pos\nx : \u211d\nh : x = 0\n\u22a2 0 \u2264 0 ^ x\n[PROOFSTEP]\nsimp [h, zero_le_one]\n[GOAL]\ncase neg\nx : \u211d\nh : \u00acx = 0\n\u22a2 0 \u2264 0 ^ x\n[PROOFSTEP]\nsimp [h, zero_le_one]\n[GOAL]\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\n\u22a2 0 \u2264 x ^ y\n[PROOFSTEP]\nrw [rpow_def_of_nonneg hx]\n[GOAL]\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\n\u22a2 0 \u2264 if x = 0 then if y = 0 then 1 else 0 else exp (log x * y)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b9 : x = 0\nh\u271d : y = 0\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nsimp only [zero_le_one, le_refl, le_of_lt (exp_pos _)]\n[GOAL]\ncase neg\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b9 : x = 0\nh\u271d : \u00acy = 0\n\u22a2 0 \u2264 0\n[PROOFSTEP]\nsimp only [zero_le_one, le_refl, le_of_lt (exp_pos _)]\n[GOAL]\ncase neg\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d : \u00acx = 0\n\u22a2 0 \u2264 exp (log x * y)\n[PROOFSTEP]\nsimp only [zero_le_one, le_refl, le_of_lt (exp_pos _)]\n[GOAL]\nx y : \u211d\nhx_nonneg : 0 \u2264 x\n\u22a2 |x ^ y| = |x| ^ y\n[PROOFSTEP]\nhave h_rpow_nonneg : 0 \u2264 x ^ y := Real.rpow_nonneg_of_nonneg hx_nonneg _\n[GOAL]\nx y : \u211d\nhx_nonneg : 0 \u2264 x\nh_rpow_nonneg : 0 \u2264 x ^ y\n\u22a2 |x ^ y| = |x| ^ y\n[PROOFSTEP]\nrw [abs_eq_self.mpr hx_nonneg, abs_eq_self.mpr h_rpow_nonneg]\n[GOAL]\nx y : \u211d\n\u22a2 |x ^ y| \u2264 |x| ^ y\n[PROOFSTEP]\ncases' le_or_lt 0 x with hx hx\n[GOAL]\ncase inl\nx y : \u211d\nhx : 0 \u2264 x\n\u22a2 |x ^ y| \u2264 |x| ^ y\n[PROOFSTEP]\nrw [abs_rpow_of_nonneg hx]\n[GOAL]\ncase inr\nx y : \u211d\nhx : x < 0\n\u22a2 |x ^ y| \u2264 |x| ^ y\n[PROOFSTEP]\nrw [abs_of_neg hx, rpow_def_of_neg hx, rpow_def_of_pos (neg_pos.2 hx), log_neg_eq_log, abs_mul, abs_of_pos (exp_pos _)]\n[GOAL]\ncase inr\nx y : \u211d\nhx : x < 0\n\u22a2 exp (log x * y) * |cos (y * \u03c0)| \u2264 exp (log x * y)\n[PROOFSTEP]\nexact mul_le_of_le_one_right (exp_pos _).le (abs_cos_le_one _)\n[GOAL]\nx y : \u211d\n\u22a2 |x ^ y| \u2264 exp (log x * y)\n[PROOFSTEP]\nrefine' (abs_rpow_le_abs_rpow x y).trans _\n[GOAL]\nx y : \u211d\n\u22a2 |x| ^ y \u2264 exp (log x * y)\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nx y : \u211d\nhx : x = 0\n\u22a2 |x| ^ y \u2264 exp (log x * y)\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\nx y : \u211d\nhx : x = 0\nhy : y = 0\n\u22a2 |x| ^ y \u2264 exp (log x * y)\n[PROOFSTEP]\nsimp [hx, hy, zero_le_one]\n[GOAL]\ncase neg\nx y : \u211d\nhx : x = 0\nhy : \u00acy = 0\n\u22a2 |x| ^ y \u2264 exp (log x * y)\n[PROOFSTEP]\nsimp [hx, hy, zero_le_one]\n[GOAL]\ncase neg\nx y : \u211d\nhx : \u00acx = 0\n\u22a2 |x| ^ y \u2264 exp (log x * y)\n[PROOFSTEP]\nrw [rpow_def_of_pos (abs_pos.2 hx), log_abs]\n[GOAL]\nx y : \u211d\nhx_nonneg : 0 \u2264 x\n\u22a2 \u2016x ^ y\u2016 = \u2016x\u2016 ^ y\n[PROOFSTEP]\nsimp_rw [Real.norm_eq_abs]\n[GOAL]\nx y : \u211d\nhx_nonneg : 0 \u2264 x\n\u22a2 |x ^ y| = |x| ^ y\n[PROOFSTEP]\nexact abs_rpow_of_nonneg hx_nonneg\n[GOAL]\nx y\u271d z\u271d : \u211d\nhx : 0 < x\ny z : \u211d\n\u22a2 x ^ (y + z) = x ^ y * x ^ z\n[PROOFSTEP]\nsimp only [rpow_def_of_pos hx, mul_add, exp_add]\n[GOAL]\nx y z : \u211d\nhx : 0 \u2264 x\nh : y + z \u2260 0\n\u22a2 x ^ (y + z) = x ^ y * x ^ z\n[PROOFSTEP]\nrcases hx.eq_or_lt with (rfl | pos)\n[GOAL]\ncase inl\ny z : \u211d\nh : y + z \u2260 0\nhx : 0 \u2264 0\n\u22a2 0 ^ (y + z) = 0 ^ y * 0 ^ z\n[PROOFSTEP]\nrw [zero_rpow h, zero_eq_mul]\n[GOAL]\ncase inl\ny z : \u211d\nh : y + z \u2260 0\nhx : 0 \u2264 0\n\u22a2 0 ^ y = 0 \u2228 0 ^ z = 0\n[PROOFSTEP]\nhave : y \u2260 0 \u2228 z \u2260 0 := not_and_or.1 fun \u27e8hy, hz\u27e9 => h <| hy.symm \u25b8 hz.symm \u25b8 zero_add 0\n[GOAL]\ncase inl\ny z : \u211d\nh : y + z \u2260 0\nhx : 0 \u2264 0\nthis : y \u2260 0 \u2228 z \u2260 0\n\u22a2 0 ^ y = 0 \u2228 0 ^ z = 0\n[PROOFSTEP]\nexact this.imp zero_rpow zero_rpow\n[GOAL]\ncase inr\nx y z : \u211d\nhx : 0 \u2264 x\nh : y + z \u2260 0\npos : 0 < x\n\u22a2 x ^ (y + z) = x ^ y * x ^ z\n[PROOFSTEP]\nexact rpow_add pos _ _\n[GOAL]\nx y z : \u211d\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nhz : 0 \u2264 z\n\u22a2 x ^ (y + z) = x ^ y * x ^ z\n[PROOFSTEP]\nrcases hy.eq_or_lt with (rfl | hy)\n[GOAL]\ncase inl\nx z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhy : 0 \u2264 0\n\u22a2 x ^ (0 + z) = x ^ 0 * x ^ z\n[PROOFSTEP]\nrw [zero_add, rpow_zero, one_mul]\n[GOAL]\ncase inr\nx y z : \u211d\nhx : 0 \u2264 x\nhy\u271d : 0 \u2264 y\nhz : 0 \u2264 z\nhy : 0 < y\n\u22a2 x ^ (y + z) = x ^ y * x ^ z\n[PROOFSTEP]\nexact rpow_add' hx (ne_of_gt <| add_pos_of_pos_of_nonneg hy hz)\n[GOAL]\nx\u271d y\u271d z\u271d x : \u211d\nhx : 0 \u2264 x\ny z : \u211d\n\u22a2 x ^ y * x ^ z \u2264 x ^ (y + z)\n[PROOFSTEP]\nrcases le_iff_eq_or_lt.1 hx with (H | pos)\n[GOAL]\ncase inl\nx\u271d y\u271d z\u271d x : \u211d\nhx : 0 \u2264 x\ny z : \u211d\nH : 0 = x\n\u22a2 x ^ y * x ^ z \u2264 x ^ (y + z)\n[PROOFSTEP]\nby_cases h : y + z = 0\n[GOAL]\ncase pos\nx\u271d y\u271d z\u271d x : \u211d\nhx : 0 \u2264 x\ny z : \u211d\nH : 0 = x\nh : y + z = 0\n\u22a2 x ^ y * x ^ z \u2264 x ^ (y + z)\n[PROOFSTEP]\nsimp only [H.symm, h, rpow_zero]\n[GOAL]\ncase pos\nx\u271d y\u271d z\u271d x : \u211d\nhx : 0 \u2264 x\ny z : \u211d\nH : 0 = x\nh : y + z = 0\n\u22a2 0 ^ y * 0 ^ z \u2264 1\n[PROOFSTEP]\ncalc\n (0 : \u211d) ^ y * 0 ^ z \u2264 1 * 1 := mul_le_mul (zero_rpow_le_one y) (zero_rpow_le_one z) (zero_rpow_nonneg z) zero_le_one\n _ = 1 := by simp\n[GOAL]\nx\u271d y\u271d z\u271d x : \u211d\nhx : 0 \u2264 x\ny z : \u211d\nH : 0 = x\nh : y + z = 0\n\u22a2 1 * 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nx\u271d y\u271d z\u271d x : \u211d\nhx : 0 \u2264 x\ny z : \u211d\nH : 0 = x\nh : \u00acy + z = 0\n\u22a2 x ^ y * x ^ z \u2264 x ^ (y + z)\n[PROOFSTEP]\nsimp [rpow_add', \u2190 H, h]\n[GOAL]\ncase inr\nx\u271d y\u271d z\u271d x : \u211d\nhx : 0 \u2264 x\ny z : \u211d\npos : 0 < x\n\u22a2 x ^ y * x ^ z \u2264 x ^ (y + z)\n[PROOFSTEP]\nsimp [rpow_add pos]\n[GOAL]\nx y z : \u211d\n\u03b9 : Type u_1\na : \u211d\nha : 0 \u2264 a\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\nh : \u2200 (x : \u03b9), x \u2208 s \u2192 0 \u2264 f x\n\u22a2 a ^ \u2211 x in s, f x = \u220f x in s, a ^ f x\n[PROOFSTEP]\ninduction' s using Finset.cons_induction with i s hi ihs\n[GOAL]\ncase empty\nx y z : \u211d\n\u03b9 : Type u_1\na : \u211d\nha : 0 \u2264 a\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\nh\u271d : \u2200 (x : \u03b9), x \u2208 s \u2192 0 \u2264 f x\nh : \u2200 (x : \u03b9), x \u2208 \u2205 \u2192 0 \u2264 f x\n\u22a2 a ^ \u2211 x in \u2205, f x = \u220f x in \u2205, a ^ f x\n[PROOFSTEP]\nrw [sum_empty, Finset.prod_empty, rpow_zero]\n[GOAL]\ncase cons\nx y z : \u211d\n\u03b9 : Type u_1\na : \u211d\nha : 0 \u2264 a\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u211d\nh\u271d : \u2200 (x : \u03b9), x \u2208 s\u271d \u2192 0 \u2264 f x\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nihs : (\u2200 (x : \u03b9), x \u2208 s \u2192 0 \u2264 f x) \u2192 a ^ \u2211 x in s, f x = \u220f x in s, a ^ f x\nh : \u2200 (x : \u03b9), x \u2208 cons i s hi \u2192 0 \u2264 f x\n\u22a2 a ^ \u2211 x in cons i s hi, f x = \u220f x in cons i s hi, a ^ f x\n[PROOFSTEP]\nrw [forall_mem_cons] at h \n[GOAL]\ncase cons\nx y z : \u211d\n\u03b9 : Type u_1\na : \u211d\nha : 0 \u2264 a\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u211d\nh\u271d : \u2200 (x : \u03b9), x \u2208 s\u271d \u2192 0 \u2264 f x\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nihs : (\u2200 (x : \u03b9), x \u2208 s \u2192 0 \u2264 f x) \u2192 a ^ \u2211 x in s, f x = \u220f x in s, a ^ f x\nh : 0 \u2264 f i \u2227 \u2200 (x : \u03b9), x \u2208 s \u2192 0 \u2264 f x\n\u22a2 a ^ \u2211 x in cons i s hi, f x = \u220f x in cons i s hi, a ^ f x\n[PROOFSTEP]\nrw [sum_cons, prod_cons, \u2190 ihs h.2, rpow_add_of_nonneg ha h.1 (sum_nonneg h.2)]\n[GOAL]\nx\u271d y\u271d z x : \u211d\nhx : 0 \u2264 x\ny : \u211d\n\u22a2 x ^ (-y) = (x ^ y)\u207b\u00b9\n[PROOFSTEP]\nsimp only [rpow_def_of_nonneg hx]\n[GOAL]\nx\u271d y\u271d z x : \u211d\nhx : 0 \u2264 x\ny : \u211d\n\u22a2 (if x = 0 then if -y = 0 then 1 else 0 else exp (log x * -y)) =\n (if x = 0 then if y = 0 then 1 else 0 else exp (log x * y))\u207b\u00b9\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nx\u271d y\u271d z x : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b2 : x = 0\nh\u271d\u00b9 : -y = 0\nh\u271d : y = 0\n\u22a2 1 = 1\u207b\u00b9\n[PROOFSTEP]\nsimp_all [exp_neg]\n[GOAL]\ncase neg\nx\u271d y\u271d z x : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b2 : x = 0\nh\u271d\u00b9 : -y = 0\nh\u271d : \u00acy = 0\n\u22a2 1 = 0\u207b\u00b9\n[PROOFSTEP]\nsimp_all [exp_neg]\n[GOAL]\ncase pos\nx\u271d y\u271d z x : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b2 : x = 0\nh\u271d\u00b9 : \u00ac-y = 0\nh\u271d : y = 0\n\u22a2 0 = 1\u207b\u00b9\n[PROOFSTEP]\nsimp_all [exp_neg]\n[GOAL]\ncase neg\nx\u271d y\u271d z x : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b2 : x = 0\nh\u271d\u00b9 : \u00ac-y = 0\nh\u271d : \u00acy = 0\n\u22a2 0 = 0\u207b\u00b9\n[PROOFSTEP]\nsimp_all [exp_neg]\n[GOAL]\ncase neg\nx\u271d y\u271d z x : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d : \u00acx = 0\n\u22a2 exp (log x * -y) = (exp (log x * y))\u207b\u00b9\n[PROOFSTEP]\nsimp_all [exp_neg]\n[GOAL]\nx\u271d y\u271d z\u271d x : \u211d\nhx : 0 < x\ny z : \u211d\n\u22a2 x ^ (y - z) = x ^ y / x ^ z\n[PROOFSTEP]\nsimp only [sub_eq_add_neg, rpow_add hx, rpow_neg (le_of_lt hx), div_eq_mul_inv]\n[GOAL]\nx\u271d y\u271d z\u271d x : \u211d\nhx : 0 \u2264 x\ny z : \u211d\nh : y - z \u2260 0\n\u22a2 x ^ (y - z) = x ^ y / x ^ z\n[PROOFSTEP]\nsimp only [sub_eq_add_neg] at h \u22a2\n[GOAL]\nx\u271d y\u271d z\u271d x : \u211d\nhx : 0 \u2264 x\ny z : \u211d\nh : y + -z \u2260 0\n\u22a2 x ^ (y + -z) = x ^ y / x ^ z\n[PROOFSTEP]\nsimp only [rpow_add' hx h, rpow_neg hx, div_eq_mul_inv]\n[GOAL]\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\n\u22a2 \u2191(x ^ y) = \u2191x ^ \u2191y\n[PROOFSTEP]\nsimp only [Real.rpow_def_of_nonneg hx, Complex.cpow_def, ofReal_eq_zero]\n[GOAL]\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\n\u22a2 \u2191(if x = 0 then if y = 0 then 1 else 0 else Real.exp (Real.log x * y)) =\n if x = 0 then if y = 0 then 1 else 0 else exp (log \u2191x * \u2191y)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b9 : x = 0\nh\u271d : y = 0\n\u22a2 \u21911 = 1\n[PROOFSTEP]\nsimp [Complex.ofReal_log hx]\n[GOAL]\ncase neg\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d\u00b9 : x = 0\nh\u271d : \u00acy = 0\n\u22a2 \u21910 = 0\n[PROOFSTEP]\nsimp [Complex.ofReal_log hx]\n[GOAL]\ncase neg\nx : \u211d\nhx : 0 \u2264 x\ny : \u211d\nh\u271d : \u00acx = 0\n\u22a2 \u2191(Real.exp (Real.log x * y)) = exp (log \u2191x * \u2191y)\n[PROOFSTEP]\nsimp [Complex.ofReal_log hx]\n[GOAL]\nx : \u211d\nhx : x \u2264 0\ny : \u2102\n\u22a2 \u2191x ^ y = (-\u2191x) ^ y * exp (\u2191\u03c0 * I * y)\n[PROOFSTEP]\nrcases hx.eq_or_lt with (rfl | hlt)\n[GOAL]\ncase inl\ny : \u2102\nhx : 0 \u2264 0\n\u22a2 \u21910 ^ y = (-\u21910) ^ y * exp (\u2191\u03c0 * I * y)\n[PROOFSTEP]\nrcases eq_or_ne y 0 with (rfl | hy)\n[GOAL]\ncase inl.inl\nhx : 0 \u2264 0\n\u22a2 \u21910 ^ 0 = (-\u21910) ^ 0 * exp (\u2191\u03c0 * I * 0)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inl.inr\ny : \u2102\nhx : 0 \u2264 0\nhy : y \u2260 0\n\u22a2 \u21910 ^ y = (-\u21910) ^ y * exp (\u2191\u03c0 * I * y)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inr\nx : \u211d\nhx : x \u2264 0\ny : \u2102\nhlt : x < 0\n\u22a2 \u2191x ^ y = (-\u2191x) ^ y * exp (\u2191\u03c0 * I * y)\n[PROOFSTEP]\nhave hne : (x : \u2102) \u2260 0 := ofReal_ne_zero.mpr hlt.ne\n[GOAL]\ncase inr\nx : \u211d\nhx : x \u2264 0\ny : \u2102\nhlt : x < 0\nhne : \u2191x \u2260 0\n\u22a2 \u2191x ^ y = (-\u2191x) ^ y * exp (\u2191\u03c0 * I * y)\n[PROOFSTEP]\nrw [cpow_def_of_ne_zero hne, cpow_def_of_ne_zero (neg_ne_zero.2 hne), \u2190 exp_add, \u2190 add_mul, log, log, abs.map_neg,\n arg_ofReal_of_neg hlt, \u2190 ofReal_neg, arg_ofReal_of_nonneg (neg_nonneg.2 hx), ofReal_zero, zero_mul, add_zero]\n[GOAL]\nz : \u2102\nhz : z \u2260 0\nw : \u2102\n\u22a2 \u2191abs (z ^ w) = \u2191abs z ^ w.re / Real.exp (arg z * w.im)\n[PROOFSTEP]\nrw [cpow_def_of_ne_zero hz, abs_exp, mul_re, log_re, log_im, Real.exp_sub, Real.rpow_def_of_pos (abs.pos hz)]\n[GOAL]\nz w : \u2102\nh : z = 0 \u2192 w.re = 0 \u2192 w = 0\n\u22a2 \u2191abs (z ^ w) = \u2191abs z ^ w.re / Real.exp (arg z * w.im)\n[PROOFSTEP]\nrcases ne_or_eq z 0 with (hz | rfl) <;> [exact abs_cpow_of_ne_zero hz w; rw [map_zero]]\n[GOAL]\nz w : \u2102\nh : z = 0 \u2192 w.re = 0 \u2192 w = 0\n\u22a2 \u2191abs (z ^ w) = \u2191abs z ^ w.re / Real.exp (arg z * w.im)\n[PROOFSTEP]\nrcases ne_or_eq z 0 with (hz | rfl)\n[GOAL]\ncase inl\nz w : \u2102\nh : z = 0 \u2192 w.re = 0 \u2192 w = 0\nhz : z \u2260 0\n\u22a2 \u2191abs (z ^ w) = \u2191abs z ^ w.re / Real.exp (arg z * w.im)\n[PROOFSTEP]\nexact abs_cpow_of_ne_zero hz w\n[GOAL]\ncase inr\nw : \u2102\nh : 0 = 0 \u2192 w.re = 0 \u2192 w = 0\n\u22a2 \u2191abs (0 ^ w) = \u2191abs 0 ^ w.re / Real.exp (arg 0 * w.im)\n[PROOFSTEP]\nrw [map_zero]\n[GOAL]\ncase inr\nw : \u2102\nh : 0 = 0 \u2192 w.re = 0 \u2192 w = 0\n\u22a2 \u2191abs (0 ^ w) = 0 ^ w.re / Real.exp (arg 0 * w.im)\n[PROOFSTEP]\ncases' eq_or_ne w.re 0 with hw hw\n[GOAL]\ncase inr.inl\nw : \u2102\nh : 0 = 0 \u2192 w.re = 0 \u2192 w = 0\nhw : w.re = 0\n\u22a2 \u2191abs (0 ^ w) = 0 ^ w.re / Real.exp (arg 0 * w.im)\n[PROOFSTEP]\nsimp [hw, h rfl hw]\n[GOAL]\ncase inr.inr\nw : \u2102\nh : 0 = 0 \u2192 w.re = 0 \u2192 w = 0\nhw : w.re \u2260 0\n\u22a2 \u2191abs (0 ^ w) = 0 ^ w.re / Real.exp (arg 0 * w.im)\n[PROOFSTEP]\nrw [Real.zero_rpow hw, zero_div, zero_cpow, map_zero]\n[GOAL]\ncase inr.inr\nw : \u2102\nh : 0 = 0 \u2192 w.re = 0 \u2192 w = 0\nhw : w.re \u2260 0\n\u22a2 w \u2260 0\n[PROOFSTEP]\nexact ne_of_apply_ne re hw\n[GOAL]\nz w : \u2102\n\u22a2 \u2191abs (z ^ w) \u2264 \u2191abs z ^ w.re / Real.exp (arg z * w.im)\n[PROOFSTEP]\nrcases ne_or_eq z 0 with (hz | rfl) <;> [exact (abs_cpow_of_ne_zero hz w).le; rw [map_zero]]\n[GOAL]\nz w : \u2102\n\u22a2 \u2191abs (z ^ w) \u2264 \u2191abs z ^ w.re / Real.exp (arg z * w.im)\n[PROOFSTEP]\nrcases ne_or_eq z 0 with (hz | rfl)\n[GOAL]\ncase inl\nz w : \u2102\nhz : z \u2260 0\n\u22a2 \u2191abs (z ^ w) \u2264 \u2191abs z ^ w.re / Real.exp (arg z * w.im)\n[PROOFSTEP]\nexact (abs_cpow_of_ne_zero hz w).le\n[GOAL]\ncase inr\nw : \u2102\n\u22a2 \u2191abs (0 ^ w) \u2264 \u2191abs 0 ^ w.re / Real.exp (arg 0 * w.im)\n[PROOFSTEP]\nrw [map_zero]\n[GOAL]\ncase inr\nw : \u2102\n\u22a2 \u2191abs (0 ^ w) \u2264 0 ^ w.re / Real.exp (arg 0 * w.im)\n[PROOFSTEP]\nrcases eq_or_ne w 0 with (rfl | hw)\n[GOAL]\ncase inr.inl\n\u22a2 \u2191abs (0 ^ 0) \u2264 0 ^ 0.re / Real.exp (arg 0 * 0.im)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\nw : \u2102\nhw : w \u2260 0\n\u22a2 \u2191abs (0 ^ w) \u2264 0 ^ w.re / Real.exp (arg 0 * w.im)\n[PROOFSTEP]\nrw [zero_cpow hw, map_zero]\n[GOAL]\ncase inr.inr\nw : \u2102\nhw : w \u2260 0\n\u22a2 0 \u2264 0 ^ w.re / Real.exp (arg 0 * w.im)\n[PROOFSTEP]\nexact div_nonneg (Real.rpow_nonneg_of_nonneg le_rfl _) (Real.exp_pos _).le\n[GOAL]\nx : \u2102\ny : \u211d\n\u22a2 \u2191abs (x ^ \u2191y) = \u2191abs x ^ y\n[PROOFSTEP]\nrcases eq_or_ne x 0 with (rfl | hx) <;> [rcases eq_or_ne y 0 with (rfl | hy); skip]\n[GOAL]\nx : \u2102\ny : \u211d\n\u22a2 \u2191abs (x ^ \u2191y) = \u2191abs x ^ y\n[PROOFSTEP]\nrcases eq_or_ne x 0 with (rfl | hx)\n[GOAL]\ncase inl\ny : \u211d\n\u22a2 \u2191abs (0 ^ \u2191y) = \u2191abs 0 ^ y\n[PROOFSTEP]\nrcases eq_or_ne y 0 with (rfl | hy)\n[GOAL]\ncase inr\nx : \u2102\ny : \u211d\nhx : x \u2260 0\n\u22a2 \u2191abs (x ^ \u2191y) = \u2191abs x ^ y\n[PROOFSTEP]\nskip\n[GOAL]\ncase inl.inl\n\u22a2 \u2191abs (0 ^ \u21910) = \u2191abs 0 ^ 0\n[PROOFSTEP]\nsimp [*, abs_cpow_of_ne_zero]\n[GOAL]\ncase inl.inr\ny : \u211d\nhy : y \u2260 0\n\u22a2 \u2191abs (0 ^ \u2191y) = \u2191abs 0 ^ y\n[PROOFSTEP]\nsimp [*, abs_cpow_of_ne_zero]\n[GOAL]\ncase inr\nx : \u2102\ny : \u211d\nhx : x \u2260 0\n\u22a2 \u2191abs (x ^ \u2191y) = \u2191abs x ^ y\n[PROOFSTEP]\nsimp [*, abs_cpow_of_ne_zero]\n[GOAL]\nx : \u2102\nn : \u2115\n\u22a2 \u2191abs (x ^ (\u2191n)\u207b\u00b9) = \u2191abs x ^ (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 abs_cpow_real]\n[GOAL]\nx : \u2102\nn : \u2115\n\u22a2 \u2191abs (x ^ (\u2191n)\u207b\u00b9) = \u2191abs (x ^ \u2191(\u2191n)\u207b\u00b9)\n[PROOFSTEP]\nsimp [-abs_cpow_real]\n[GOAL]\nx : \u211d\nhx : 0 < x\ny : \u2102\n\u22a2 \u2191abs (\u2191x ^ y) = x ^ y.re\n[PROOFSTEP]\nrw [abs_cpow_of_ne_zero (ofReal_ne_zero.mpr hx.ne'), arg_ofReal_of_nonneg hx.le, zero_mul, Real.exp_zero, div_one,\n abs_of_nonneg hx.le]\n[GOAL]\nx : \u211d\nhx : 0 \u2264 x\ny : \u2102\nhy : y.re \u2260 0\n\u22a2 \u2191abs (\u2191x ^ y) = x ^ y.re\n[PROOFSTEP]\nrcases hx.eq_or_lt with (rfl | hlt)\n[GOAL]\ncase inl\ny : \u2102\nhy : y.re \u2260 0\nhx : 0 \u2264 0\n\u22a2 \u2191abs (\u21910 ^ y) = 0 ^ y.re\n[PROOFSTEP]\nrw [ofReal_zero, zero_cpow, map_zero, Real.zero_rpow hy]\n[GOAL]\ncase inl\ny : \u2102\nhy : y.re \u2260 0\nhx : 0 \u2264 0\n\u22a2 y \u2260 0\n[PROOFSTEP]\nexact ne_of_apply_ne re hy\n[GOAL]\ncase inr\nx : \u211d\nhx : 0 \u2264 x\ny : \u2102\nhy : y.re \u2260 0\nhlt : 0 < x\n\u22a2 \u2191abs (\u2191x ^ y) = x ^ y.re\n[PROOFSTEP]\nexact abs_cpow_eq_rpow_re_of_pos hlt y\n[GOAL]\nx\u271d y\u271d z\u271d x : \u211d\nhx : 0 \u2264 x\ny z : \u211d\n\u22a2 x ^ (y * z) = (x ^ y) ^ z\n[PROOFSTEP]\nrw [\u2190 Complex.ofReal_inj, Complex.ofReal_cpow (rpow_nonneg_of_nonneg hx _), Complex.ofReal_cpow hx, Complex.ofReal_mul,\n Complex.cpow_mul, Complex.ofReal_cpow hx]\n[GOAL]\ncase h\u2081\nx\u271d y\u271d z\u271d x : \u211d\nhx : 0 \u2264 x\ny z : \u211d\n\u22a2 -\u03c0 < (Complex.log \u2191x * \u2191y).im\n[PROOFSTEP]\nsimp only [(Complex.ofReal_mul _ _).symm, (Complex.ofReal_log hx).symm, Complex.ofReal_im, neg_lt_zero, pi_pos,\n le_of_lt pi_pos]\n[GOAL]\ncase h\u2082\nx\u271d y\u271d z\u271d x : \u211d\nhx : 0 \u2264 x\ny z : \u211d\n\u22a2 (Complex.log \u2191x * \u2191y).im \u2264 \u03c0\n[PROOFSTEP]\nsimp only [(Complex.ofReal_mul _ _).symm, (Complex.ofReal_log hx).symm, Complex.ofReal_im, neg_lt_zero, pi_pos,\n le_of_lt pi_pos]\n[GOAL]\nx\u271d y\u271d z x : \u211d\nhx : x \u2260 0\ny : \u211d\nn : \u2124\n\u22a2 x ^ (y + \u2191n) = x ^ y * x ^ n\n[PROOFSTEP]\nrw [rpow_def, rpow_def, Complex.ofReal_add, Complex.cpow_add _ _ (Complex.ofReal_ne_zero.mpr hx),\n Complex.ofReal_int_cast, Complex.cpow_int_cast, \u2190 Complex.ofReal_zpow, mul_comm, Complex.ofReal_mul_re, mul_comm]\n[GOAL]\nx\u271d y\u271d z x : \u211d\nhx : x \u2260 0\ny : \u211d\nn : \u2115\n\u22a2 x ^ (y + \u2191n) = x ^ y * x ^ n\n[PROOFSTEP]\nsimpa using rpow_add_int hx y n\n[GOAL]\nx\u271d y\u271d z x : \u211d\nhx : x \u2260 0\ny : \u211d\nn : \u2115\n\u22a2 x ^ (y - \u2191n) = x ^ y / x ^ n\n[PROOFSTEP]\nsimpa using rpow_add_int hx y (-n)\n[GOAL]\nx\u271d y\u271d z x : \u211d\nhx : x \u2260 0\ny : \u211d\nn : \u2115\n\u22a2 x ^ (y - \u2191n) = x ^ y / x ^ n\n[PROOFSTEP]\nsimpa using rpow_sub_int hx y n\n[GOAL]\nx\u271d y\u271d z x : \u211d\nhx : x \u2260 0\ny : \u211d\n\u22a2 x ^ (y + 1) = x ^ y * x\n[PROOFSTEP]\nsimpa using rpow_add_nat hx y 1\n[GOAL]\nx\u271d y\u271d z x : \u211d\nhx : x \u2260 0\ny : \u211d\n\u22a2 x ^ (y - 1) = x ^ y / x\n[PROOFSTEP]\nsimpa using rpow_sub_nat hx y 1\n[GOAL]\nx\u271d y z x : \u211d\nn : \u2124\n\u22a2 x ^ \u2191n = x ^ n\n[PROOFSTEP]\nsimp only [rpow_def, \u2190 Complex.ofReal_zpow, Complex.cpow_int_cast, Complex.ofReal_int_cast, Complex.ofReal_re]\n[GOAL]\nx\u271d y z x : \u211d\nn : \u2115\n\u22a2 x ^ \u2191n = x ^ n\n[PROOFSTEP]\nsimpa using rpow_int_cast x n\n[GOAL]\nx\u271d y z x : \u211d\n\u22a2 x ^ 2 = x ^ 2\n[PROOFSTEP]\nrw [\u2190 rpow_nat_cast]\n[GOAL]\nx\u271d y z x : \u211d\n\u22a2 x ^ 2 = x ^ \u21912\n[PROOFSTEP]\nsimp only [Nat.cast_ofNat]\n[GOAL]\nx\u271d y z x : \u211d\n\u22a2 x ^ (-1) = x\u207b\u00b9\n[PROOFSTEP]\nsuffices H : x ^ ((-1 : \u2124) : \u211d) = x\u207b\u00b9\n[GOAL]\nx\u271d y z x : \u211d\nH : x ^ \u2191(-1) = x\u207b\u00b9\n\u22a2 x ^ (-1) = x\u207b\u00b9\n[PROOFSTEP]\nrwa [Int.cast_neg, Int.cast_one] at H \n[GOAL]\ncase H\nx\u271d y z x : \u211d\n\u22a2 x ^ \u2191(-1) = x\u207b\u00b9\n[PROOFSTEP]\nsimp only [rpow_int_cast, zpow_one, zpow_neg]\n[GOAL]\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\n\u22a2 (x * y) ^ z = x ^ z * y ^ z\n[PROOFSTEP]\niterate 2 rw [Real.rpow_def_of_nonneg]; split_ifs with h_ifs <;> simp_all\n[GOAL]\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\n\u22a2 (x * y) ^ z = x ^ z * y ^ z\n[PROOFSTEP]\nrw [Real.rpow_def_of_nonneg]\n[GOAL]\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\n\u22a2 (if x * y = 0 then if z = 0 then 1 else 0 else exp (log (x * y) * z)) = x ^ z * y ^ z\n[PROOFSTEP]\nsplit_ifs with h_ifs\n[GOAL]\ncase pos\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : x * y = 0\nh\u271d : z = 0\n\u22a2 1 = x ^ z * y ^ z\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase neg\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : x * y = 0\nh\u271d : \u00acz = 0\n\u22a2 0 = x ^ z * y ^ z\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase neg\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00acx * y = 0\n\u22a2 exp (log (x * y) * z) = x ^ z * y ^ z\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase neg\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : x = 0 \u2228 y = 0\nh\u271d : \u00acz = 0\n\u22a2 x ^ z = 0 \u2228 y ^ z = 0\ncase neg\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00ac(x = 0 \u2228 y = 0)\n\u22a2 exp (log (x * y) * z) = x ^ z * y ^ z\ncase hx x\u271d y\u271d z\u271d x y z : \u211d h : 0 \u2264 x h\u2081 : 0 \u2264 y \u22a2 0 \u2264 x * y\n[PROOFSTEP]\nrw [Real.rpow_def_of_nonneg]\n[GOAL]\ncase neg\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : x = 0 \u2228 y = 0\nh\u271d : \u00acz = 0\n\u22a2 (if x = 0 then if z = 0 then 1 else 0 else exp (log x * z)) = 0 \u2228 y ^ z = 0\n[PROOFSTEP]\nsplit_ifs with h_ifs\n[GOAL]\ncase pos\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs\u271d : x = 0 \u2228 y = 0\nh\u271d : \u00acz = 0\nh_ifs : x = 0\n\u22a2 0 = 0 \u2228 y ^ z = 0\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase neg\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs\u271d : x = 0 \u2228 y = 0\nh\u271d : \u00acz = 0\nh_ifs : \u00acx = 0\n\u22a2 exp (log x * z) = 0 \u2228 y ^ z = 0\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase neg.hx\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : x = 0 \u2228 y = 0\nh\u271d : \u00acz = 0\n\u22a2 0 \u2264 x\n[PROOFSTEP]\nexact h\n[GOAL]\ncase neg\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00ac(x = 0 \u2228 y = 0)\n\u22a2 exp (log (x * y) * z) = x ^ z * y ^ z\n[PROOFSTEP]\nrw [not_or] at h_ifs \n[GOAL]\ncase neg\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00acx = 0 \u2227 \u00acy = 0\n\u22a2 exp (log (x * y) * z) = x ^ z * y ^ z\n[PROOFSTEP]\nhave hx : 0 < x := by\n cases' lt_or_eq_of_le h with h\u2082 h\u2082\n \u00b7 exact h\u2082\n exfalso\n apply h_ifs.1\n exact Eq.symm h\u2082\n[GOAL]\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00acx = 0 \u2227 \u00acy = 0\n\u22a2 0 < x\n[PROOFSTEP]\ncases' lt_or_eq_of_le h with h\u2082 h\u2082\n[GOAL]\ncase inl\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00acx = 0 \u2227 \u00acy = 0\nh\u2082 : 0 < x\n\u22a2 0 < x\n[PROOFSTEP]\nexact h\u2082\n[GOAL]\ncase inr\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00acx = 0 \u2227 \u00acy = 0\nh\u2082 : 0 = x\n\u22a2 0 < x\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase inr.h\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00acx = 0 \u2227 \u00acy = 0\nh\u2082 : 0 = x\n\u22a2 False\n[PROOFSTEP]\napply h_ifs.1\n[GOAL]\ncase inr.h\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00acx = 0 \u2227 \u00acy = 0\nh\u2082 : 0 = x\n\u22a2 x = 0\n[PROOFSTEP]\nexact Eq.symm h\u2082\n[GOAL]\ncase neg\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00acx = 0 \u2227 \u00acy = 0\nhx : 0 < x\n\u22a2 exp (log (x * y) * z) = x ^ z * y ^ z\n[PROOFSTEP]\nhave hy : 0 < y := by\n cases' lt_or_eq_of_le h\u2081 with h\u2082 h\u2082\n \u00b7 exact h\u2082\n exfalso\n apply h_ifs.2\n exact Eq.symm h\u2082\n[GOAL]\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00acx = 0 \u2227 \u00acy = 0\nhx : 0 < x\n\u22a2 0 < y\n[PROOFSTEP]\ncases' lt_or_eq_of_le h\u2081 with h\u2082 h\u2082\n[GOAL]\ncase inl\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00acx = 0 \u2227 \u00acy = 0\nhx : 0 < x\nh\u2082 : 0 < y\n\u22a2 0 < y\n[PROOFSTEP]\nexact h\u2082\n[GOAL]\ncase inr\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00acx = 0 \u2227 \u00acy = 0\nhx : 0 < x\nh\u2082 : 0 = y\n\u22a2 0 < y\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase inr.h\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00acx = 0 \u2227 \u00acy = 0\nhx : 0 < x\nh\u2082 : 0 = y\n\u22a2 False\n[PROOFSTEP]\napply h_ifs.2\n[GOAL]\ncase inr.h\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00acx = 0 \u2227 \u00acy = 0\nhx : 0 < x\nh\u2082 : 0 = y\n\u22a2 y = 0\n[PROOFSTEP]\nexact Eq.symm h\u2082\n[GOAL]\ncase neg\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\nh_ifs : \u00acx = 0 \u2227 \u00acy = 0\nhx : 0 < x\nhy : 0 < y\n\u22a2 exp (log (x * y) * z) = x ^ z * y ^ z\n[PROOFSTEP]\nrw [log_mul (ne_of_gt hx) (ne_of_gt hy), add_mul, exp_add, rpow_def_of_pos hx, rpow_def_of_pos hy]\n[GOAL]\ncase hx\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : 0 \u2264 y\n\u22a2 0 \u2264 x * y\n[PROOFSTEP]\nexact mul_nonneg h h\u2081\n[GOAL]\nx y\u271d z : \u211d\nhx : 0 \u2264 x\ny : \u211d\n\u22a2 x\u207b\u00b9 ^ y = (x ^ y)\u207b\u00b9\n[PROOFSTEP]\nsimp only [\u2190 rpow_neg_one, \u2190 rpow_mul hx, mul_comm]\n[GOAL]\nx y z\u271d : \u211d\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nz : \u211d\n\u22a2 (x / y) ^ z = x ^ z / y ^ z\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, mul_rpow hx (inv_nonneg.2 hy), inv_rpow hy]\n[GOAL]\nx\u271d y\u271d z x : \u211d\nhx : 0 < x\ny : \u211d\n\u22a2 log (x ^ y) = y * log x\n[PROOFSTEP]\napply exp_injective\n[GOAL]\ncase a\nx\u271d y\u271d z x : \u211d\nhx : 0 < x\ny : \u211d\n\u22a2 exp (log (x ^ y)) = exp (y * log x)\n[PROOFSTEP]\nrw [exp_log (rpow_pos_of_pos hx y), \u2190 exp_log hx, mul_comm, rpow_def_of_pos (exp_pos (log x)) y]\n[GOAL]\nx y z : \u211d\nhx : 0 \u2264 x\nhxy : x < y\nhz : 0 < z\n\u22a2 x ^ z < y ^ z\n[PROOFSTEP]\nrw [le_iff_eq_or_lt] at hx \n[GOAL]\nx y z : \u211d\nhx : 0 = x \u2228 0 < x\nhxy : x < y\nhz : 0 < z\n\u22a2 x ^ z < y ^ z\n[PROOFSTEP]\ncases' hx with hx hx\n[GOAL]\ncase inl\nx y z : \u211d\nhxy : x < y\nhz : 0 < z\nhx : 0 = x\n\u22a2 x ^ z < y ^ z\n[PROOFSTEP]\nrw [\u2190 hx, zero_rpow (ne_of_gt hz)]\n[GOAL]\ncase inl\nx y z : \u211d\nhxy : x < y\nhz : 0 < z\nhx : 0 = x\n\u22a2 0 < y ^ z\n[PROOFSTEP]\nexact rpow_pos_of_pos (by rwa [\u2190 hx] at hxy ) _\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhz : 0 < z\nhx : 0 = x\n\u22a2 0 < y\n[PROOFSTEP]\nrwa [\u2190 hx] at hxy \n[GOAL]\ncase inr\nx y z : \u211d\nhxy : x < y\nhz : 0 < z\nhx : 0 < x\n\u22a2 x ^ z < y ^ z\n[PROOFSTEP]\nrw [rpow_def_of_pos hx, rpow_def_of_pos (lt_trans hx hxy), exp_lt_exp]\n[GOAL]\ncase inr\nx y z : \u211d\nhxy : x < y\nhz : 0 < z\nhx : 0 < x\n\u22a2 log x * z < log y * z\n[PROOFSTEP]\nexact mul_lt_mul_of_pos_right (log_lt_log hx hxy) hz\n[GOAL]\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : x \u2264 y\nh\u2082 : 0 \u2264 z\n\u22a2 x ^ z \u2264 y ^ z\n[PROOFSTEP]\nrcases eq_or_lt_of_le h\u2081 with (rfl | h\u2081')\n[GOAL]\ncase inl\nx\u271d y z\u271d x z : \u211d\nh : 0 \u2264 x\nh\u2082 : 0 \u2264 z\nh\u2081 : x \u2264 x\n\u22a2 x ^ z \u2264 x ^ z\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : x \u2264 y\nh\u2082 : 0 \u2264 z\nh\u2081' : x < y\n\u22a2 x ^ z \u2264 y ^ z\n[PROOFSTEP]\nrcases eq_or_lt_of_le h\u2082 with (rfl | h\u2082')\n[GOAL]\ncase inr.inl\nx\u271d y\u271d z x y : \u211d\nh : 0 \u2264 x\nh\u2081 : x \u2264 y\nh\u2081' : x < y\nh\u2082 : 0 \u2264 0\n\u22a2 x ^ 0 \u2264 y ^ 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\nx\u271d y\u271d z\u271d x y z : \u211d\nh : 0 \u2264 x\nh\u2081 : x \u2264 y\nh\u2082 : 0 \u2264 z\nh\u2081' : x < y\nh\u2082' : 0 < z\n\u22a2 x ^ z \u2264 y ^ z\n[PROOFSTEP]\nexact le_of_lt (rpow_lt_rpow h h\u2081' h\u2082')\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\n\u22a2 x \u2264 y ^ z\u207b\u00b9 \u2194 y \u2264 x ^ z\n[PROOFSTEP]\nhave hz' : 0 < -z := by rwa [lt_neg, neg_zero]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\n\u22a2 0 < -z\n[PROOFSTEP]\nrwa [lt_neg, neg_zero]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\n\u22a2 x \u2264 y ^ z\u207b\u00b9 \u2194 y \u2264 x ^ z\n[PROOFSTEP]\nhave hxz : 0 < x ^ (-z) := Real.rpow_pos_of_pos hx _\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\n\u22a2 x \u2264 y ^ z\u207b\u00b9 \u2194 y \u2264 x ^ z\n[PROOFSTEP]\nhave hyz : 0 < y ^ z\u207b\u00b9 := Real.rpow_pos_of_pos hy _\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z\u207b\u00b9\n\u22a2 x \u2264 y ^ z\u207b\u00b9 \u2194 y \u2264 x ^ z\n[PROOFSTEP]\nrw [\u2190 Real.rpow_le_rpow_iff hx.le hyz.le hz', \u2190 Real.rpow_mul hy.le]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z\u207b\u00b9\n\u22a2 x ^ (-z) \u2264 y ^ (z\u207b\u00b9 * -z) \u2194 y \u2264 x ^ z\n[PROOFSTEP]\nsimp only [ne_of_lt hz, Real.rpow_neg_one, mul_neg, inv_mul_cancel, Ne.def, not_false_iff]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z\u207b\u00b9\n\u22a2 x ^ (-z) \u2264 y\u207b\u00b9 \u2194 y \u2264 x ^ z\n[PROOFSTEP]\nrw [le_inv hxz hy, \u2190 Real.rpow_neg_one, \u2190 Real.rpow_mul hx.le]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z\u207b\u00b9\n\u22a2 y \u2264 x ^ (-z * -1) \u2194 y \u2264 x ^ z\n[PROOFSTEP]\nsimp\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\n\u22a2 x < y ^ z\u207b\u00b9 \u2194 y < x ^ z\n[PROOFSTEP]\nhave hz' : 0 < -z := by rwa [lt_neg, neg_zero]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\n\u22a2 0 < -z\n[PROOFSTEP]\nrwa [lt_neg, neg_zero]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\n\u22a2 x < y ^ z\u207b\u00b9 \u2194 y < x ^ z\n[PROOFSTEP]\nhave hxz : 0 < x ^ (-z) := Real.rpow_pos_of_pos hx _\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\n\u22a2 x < y ^ z\u207b\u00b9 \u2194 y < x ^ z\n[PROOFSTEP]\nhave hyz : 0 < y ^ z\u207b\u00b9 := Real.rpow_pos_of_pos hy _\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z\u207b\u00b9\n\u22a2 x < y ^ z\u207b\u00b9 \u2194 y < x ^ z\n[PROOFSTEP]\nrw [\u2190 Real.rpow_lt_rpow_iff hx.le hyz.le hz', \u2190 Real.rpow_mul hy.le]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z\u207b\u00b9\n\u22a2 x ^ (-z) < y ^ (z\u207b\u00b9 * -z) \u2194 y < x ^ z\n[PROOFSTEP]\nsimp only [ne_of_lt hz, Real.rpow_neg_one, mul_neg, inv_mul_cancel, Ne.def, not_false_iff]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z\u207b\u00b9\n\u22a2 x ^ (-z) < y\u207b\u00b9 \u2194 y < x ^ z\n[PROOFSTEP]\nrw [lt_inv hxz hy, \u2190 Real.rpow_neg_one, \u2190 Real.rpow_mul hx.le]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\nhz' : 0 < -z\nhxz : 0 < x ^ (-z)\nhyz : 0 < y ^ z\u207b\u00b9\n\u22a2 y < x ^ (-z * -1) \u2194 y < x ^ z\n[PROOFSTEP]\nsimp\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\n\u22a2 x ^ z\u207b\u00b9 < y \u2194 y ^ z < x\n[PROOFSTEP]\nconvert lt_rpow_inv_iff_of_neg (Real.rpow_pos_of_pos hx z\u207b\u00b9) (Real.rpow_pos_of_pos hy z) hz\n[GOAL]\ncase h.e'_1.h.e'_4\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\n\u22a2 y = (y ^ z) ^ z\u207b\u00b9\n[PROOFSTEP]\nsimp [\u2190 Real.rpow_mul hx.le, \u2190 Real.rpow_mul hy.le, ne_of_lt hz]\n[GOAL]\ncase h.e'_2.h.e'_4\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\n\u22a2 x = (x ^ z\u207b\u00b9) ^ z\n[PROOFSTEP]\nsimp [\u2190 Real.rpow_mul hx.le, \u2190 Real.rpow_mul hy.le, ne_of_lt hz]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\n\u22a2 x ^ z\u207b\u00b9 \u2264 y \u2194 y ^ z \u2264 x\n[PROOFSTEP]\nconvert le_rpow_inv_iff_of_neg (Real.rpow_pos_of_pos hx z\u207b\u00b9) (Real.rpow_pos_of_pos hy z) hz\n[GOAL]\ncase h.e'_1.h.e'_4\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\n\u22a2 y = (y ^ z) ^ z\u207b\u00b9\n[PROOFSTEP]\nsimp [\u2190 Real.rpow_mul hx.le, \u2190 Real.rpow_mul hy.le, ne_of_lt hz]\n[GOAL]\ncase h.e'_2.h.e'_4\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\n\u22a2 x = (x ^ z\u207b\u00b9) ^ z\n[PROOFSTEP]\nsimp [\u2190 Real.rpow_mul hx.le, \u2190 Real.rpow_mul hy.le, ne_of_lt hz]\n[GOAL]\nx y z : \u211d\nhx : 1 < x\nhyz : y < z\n\u22a2 x ^ y < x ^ z\n[PROOFSTEP]\nrepeat' rw [rpow_def_of_pos (lt_trans zero_lt_one hx)]\n[GOAL]\nx y z : \u211d\nhx : 1 < x\nhyz : y < z\n\u22a2 x ^ y < x ^ z\n[PROOFSTEP]\nrw [rpow_def_of_pos (lt_trans zero_lt_one hx)]\n[GOAL]\nx y z : \u211d\nhx : 1 < x\nhyz : y < z\n\u22a2 exp (log x * y) < x ^ z\n[PROOFSTEP]\nrw [rpow_def_of_pos (lt_trans zero_lt_one hx)]\n[GOAL]\nx y z : \u211d\nhx : 1 < x\nhyz : y < z\n\u22a2 exp (log x * y) < exp (log x * z)\n[PROOFSTEP]\nrw [rpow_def_of_pos (lt_trans zero_lt_one hx)]\n[GOAL]\nx y z : \u211d\nhx : 1 < x\nhyz : y < z\n\u22a2 exp (log x * y) < exp (log x * z)\n[PROOFSTEP]\nrw [exp_lt_exp]\n[GOAL]\nx y z : \u211d\nhx : 1 < x\nhyz : y < z\n\u22a2 log x * y < log x * z\n[PROOFSTEP]\nexact mul_lt_mul_of_pos_left hyz (log_pos hx)\n[GOAL]\nx y z : \u211d\nhx : 1 \u2264 x\nhyz : y \u2264 z\n\u22a2 x ^ y \u2264 x ^ z\n[PROOFSTEP]\nrepeat' rw [rpow_def_of_pos (lt_of_lt_of_le zero_lt_one hx)]\n[GOAL]\nx y z : \u211d\nhx : 1 \u2264 x\nhyz : y \u2264 z\n\u22a2 x ^ y \u2264 x ^ z\n[PROOFSTEP]\nrw [rpow_def_of_pos (lt_of_lt_of_le zero_lt_one hx)]\n[GOAL]\nx y z : \u211d\nhx : 1 \u2264 x\nhyz : y \u2264 z\n\u22a2 exp (log x * y) \u2264 x ^ z\n[PROOFSTEP]\nrw [rpow_def_of_pos (lt_of_lt_of_le zero_lt_one hx)]\n[GOAL]\nx y z : \u211d\nhx : 1 \u2264 x\nhyz : y \u2264 z\n\u22a2 exp (log x * y) \u2264 exp (log x * z)\n[PROOFSTEP]\nrw [rpow_def_of_pos (lt_of_lt_of_le zero_lt_one hx)]\n[GOAL]\nx y z : \u211d\nhx : 1 \u2264 x\nhyz : y \u2264 z\n\u22a2 exp (log x * y) \u2264 exp (log x * z)\n[PROOFSTEP]\nrw [exp_le_exp]\n[GOAL]\nx y z : \u211d\nhx : 1 \u2264 x\nhyz : y \u2264 z\n\u22a2 log x * y \u2264 log x * z\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_left hyz (log_nonneg hx)\n[GOAL]\nx\u271d y\u271d z\u271d x y z : \u211d\nhy : 0 < y\nhxy : y < x\nhz : z < 0\n\u22a2 x ^ z < y ^ z\n[PROOFSTEP]\nhave hx : 0 < x := hy.trans hxy\n[GOAL]\nx\u271d y\u271d z\u271d x y z : \u211d\nhy : 0 < y\nhxy : y < x\nhz : z < 0\nhx : 0 < x\n\u22a2 x ^ z < y ^ z\n[PROOFSTEP]\nrw [\u2190 neg_neg z, Real.rpow_neg (le_of_lt hx) (-z), Real.rpow_neg (le_of_lt hy) (-z),\n inv_lt_inv (rpow_pos_of_pos hx _) (rpow_pos_of_pos hy _)]\n[GOAL]\nx\u271d y\u271d z\u271d x y z : \u211d\nhy : 0 < y\nhxy : y < x\nhz : z < 0\nhx : 0 < x\n\u22a2 y ^ (-z) < x ^ (-z)\n[PROOFSTEP]\nexact Real.rpow_lt_rpow (by positivity) hxy <| neg_pos_of_neg hz\n[GOAL]\nx\u271d y\u271d z\u271d x y z : \u211d\nhy : 0 < y\nhxy : y < x\nhz : z < 0\nhx : 0 < x\n\u22a2 0 \u2264 y\n[PROOFSTEP]\npositivity\n[GOAL]\nx\u271d y\u271d z x y : \u211d\nhy : 0 < y\nhxy : y \u2264 x\nhz : z \u2264 0\n\u22a2 x ^ z \u2264 y ^ z\n[PROOFSTEP]\nrcases ne_or_eq z 0 with hz_zero | rfl\n[GOAL]\ncase inl\nx\u271d y\u271d z x y : \u211d\nhy : 0 < y\nhxy : y \u2264 x\nhz : z \u2264 0\nhz_zero : z \u2260 0\n\u22a2 x ^ z \u2264 y ^ z\ncase inr x\u271d y\u271d x y : \u211d hy : 0 < y hxy : y \u2264 x hz : 0 \u2264 0 \u22a2 x ^ 0 \u2264 y ^ 0\n[PROOFSTEP]\ncase inl =>\n rcases ne_or_eq x y with hxy' | rfl\n case inl =>\n exact le_of_lt <| rpow_lt_rpow_of_exponent_neg hy (Ne.lt_of_le (id (Ne.symm hxy')) hxy) (Ne.lt_of_le hz_zero hz)\n case inr => simp\n[GOAL]\nx\u271d y\u271d z x y : \u211d\nhy : 0 < y\nhxy : y \u2264 x\nhz : z \u2264 0\nhz_zero : z \u2260 0\n\u22a2 x ^ z \u2264 y ^ z\n[PROOFSTEP]\ncase inl =>\n rcases ne_or_eq x y with hxy' | rfl\n case inl =>\n exact le_of_lt <| rpow_lt_rpow_of_exponent_neg hy (Ne.lt_of_le (id (Ne.symm hxy')) hxy) (Ne.lt_of_le hz_zero hz)\n case inr => simp\n[GOAL]\nx\u271d y\u271d z x y : \u211d\nhy : 0 < y\nhxy : y \u2264 x\nhz : z \u2264 0\nhz_zero : z \u2260 0\n\u22a2 x ^ z \u2264 y ^ z\n[PROOFSTEP]\nrcases ne_or_eq x y with hxy' | rfl\n[GOAL]\ncase inl\nx\u271d y\u271d z x y : \u211d\nhy : 0 < y\nhxy : y \u2264 x\nhz : z \u2264 0\nhz_zero : z \u2260 0\nhxy' : x \u2260 y\n\u22a2 x ^ z \u2264 y ^ z\ncase inr x\u271d y z x : \u211d hz : z \u2264 0 hz_zero : z \u2260 0 hy : 0 < x hxy : x \u2264 x \u22a2 x ^ z \u2264 x ^ z\n[PROOFSTEP]\ncase inl =>\n exact le_of_lt <| rpow_lt_rpow_of_exponent_neg hy (Ne.lt_of_le (id (Ne.symm hxy')) hxy) (Ne.lt_of_le hz_zero hz)\n[GOAL]\nx\u271d y\u271d z x y : \u211d\nhy : 0 < y\nhxy : y \u2264 x\nhz : z \u2264 0\nhz_zero : z \u2260 0\nhxy' : x \u2260 y\n\u22a2 x ^ z \u2264 y ^ z\n[PROOFSTEP]\ncase inl =>\n exact le_of_lt <| rpow_lt_rpow_of_exponent_neg hy (Ne.lt_of_le (id (Ne.symm hxy')) hxy) (Ne.lt_of_le hz_zero hz)\n[GOAL]\nx\u271d y\u271d z x y : \u211d\nhy : 0 < y\nhxy : y \u2264 x\nhz : z \u2264 0\nhz_zero : z \u2260 0\nhxy' : x \u2260 y\n\u22a2 x ^ z \u2264 y ^ z\n[PROOFSTEP]\nexact le_of_lt <| rpow_lt_rpow_of_exponent_neg hy (Ne.lt_of_le (id (Ne.symm hxy')) hxy) (Ne.lt_of_le hz_zero hz)\n[GOAL]\ncase inr\nx\u271d y z x : \u211d\nhz : z \u2264 0\nhz_zero : z \u2260 0\nhy : 0 < x\nhxy : x \u2264 x\n\u22a2 x ^ z \u2264 x ^ z\n[PROOFSTEP]\ncase inr => simp\n[GOAL]\nx\u271d y z x : \u211d\nhz : z \u2264 0\nhz_zero : z \u2260 0\nhy : 0 < x\nhxy : x \u2264 x\n\u22a2 x ^ z \u2264 x ^ z\n[PROOFSTEP]\ncase inr => simp\n[GOAL]\nx\u271d y z x : \u211d\nhz : z \u2264 0\nhz_zero : z \u2260 0\nhy : 0 < x\nhxy : x \u2264 x\n\u22a2 x ^ z \u2264 x ^ z\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nx\u271d y\u271d x y : \u211d\nhy : 0 < y\nhxy : y \u2264 x\nhz : 0 \u2264 0\n\u22a2 x ^ 0 \u2264 y ^ 0\n[PROOFSTEP]\ncase inr => simp\n[GOAL]\nx\u271d y\u271d x y : \u211d\nhy : 0 < y\nhxy : y \u2264 x\nhz : 0 \u2264 0\n\u22a2 x ^ 0 \u2264 y ^ 0\n[PROOFSTEP]\ncase inr => simp\n[GOAL]\nx\u271d y\u271d x y : \u211d\nhy : 0 < y\nhxy : y \u2264 x\nhz : 0 \u2264 0\n\u22a2 x ^ 0 \u2264 y ^ 0\n[PROOFSTEP]\nsimp\n[GOAL]\nx y z : \u211d\nhx : 1 < x\n\u22a2 x ^ y \u2264 x ^ z \u2194 y \u2264 z\n[PROOFSTEP]\nhave x_pos : 0 < x := lt_trans zero_lt_one hx\n[GOAL]\nx y z : \u211d\nhx : 1 < x\nx_pos : 0 < x\n\u22a2 x ^ y \u2264 x ^ z \u2194 y \u2264 z\n[PROOFSTEP]\nrw [\u2190 log_le_log (rpow_pos_of_pos x_pos y) (rpow_pos_of_pos x_pos z), log_rpow x_pos, log_rpow x_pos,\n mul_le_mul_right (log_pos hx)]\n[GOAL]\nx y z : \u211d\nhx : 1 < x\n\u22a2 x ^ y < x ^ z \u2194 y < z\n[PROOFSTEP]\nrw [lt_iff_not_le, rpow_le_rpow_left_iff hx, lt_iff_not_le]\n[GOAL]\nx y z : \u211d\nhx0 : 0 < x\nhx1 : x < 1\nhyz : z < y\n\u22a2 x ^ y < x ^ z\n[PROOFSTEP]\nrepeat' rw [rpow_def_of_pos hx0]\n[GOAL]\nx y z : \u211d\nhx0 : 0 < x\nhx1 : x < 1\nhyz : z < y\n\u22a2 x ^ y < x ^ z\n[PROOFSTEP]\nrw [rpow_def_of_pos hx0]\n[GOAL]\nx y z : \u211d\nhx0 : 0 < x\nhx1 : x < 1\nhyz : z < y\n\u22a2 exp (log x * y) < x ^ z\n[PROOFSTEP]\nrw [rpow_def_of_pos hx0]\n[GOAL]\nx y z : \u211d\nhx0 : 0 < x\nhx1 : x < 1\nhyz : z < y\n\u22a2 exp (log x * y) < exp (log x * z)\n[PROOFSTEP]\nrw [rpow_def_of_pos hx0]\n[GOAL]\nx y z : \u211d\nhx0 : 0 < x\nhx1 : x < 1\nhyz : z < y\n\u22a2 exp (log x * y) < exp (log x * z)\n[PROOFSTEP]\nrw [exp_lt_exp]\n[GOAL]\nx y z : \u211d\nhx0 : 0 < x\nhx1 : x < 1\nhyz : z < y\n\u22a2 log x * y < log x * z\n[PROOFSTEP]\nexact mul_lt_mul_of_neg_left hyz (log_neg hx0 hx1)\n[GOAL]\nx y z : \u211d\nhx0 : 0 < x\nhx1 : x \u2264 1\nhyz : z \u2264 y\n\u22a2 x ^ y \u2264 x ^ z\n[PROOFSTEP]\nrepeat' rw [rpow_def_of_pos hx0]\n[GOAL]\nx y z : \u211d\nhx0 : 0 < x\nhx1 : x \u2264 1\nhyz : z \u2264 y\n\u22a2 x ^ y \u2264 x ^ z\n[PROOFSTEP]\nrw [rpow_def_of_pos hx0]\n[GOAL]\nx y z : \u211d\nhx0 : 0 < x\nhx1 : x \u2264 1\nhyz : z \u2264 y\n\u22a2 exp (log x * y) \u2264 x ^ z\n[PROOFSTEP]\nrw [rpow_def_of_pos hx0]\n[GOAL]\nx y z : \u211d\nhx0 : 0 < x\nhx1 : x \u2264 1\nhyz : z \u2264 y\n\u22a2 exp (log x * y) \u2264 exp (log x * z)\n[PROOFSTEP]\nrw [rpow_def_of_pos hx0]\n[GOAL]\nx y z : \u211d\nhx0 : 0 < x\nhx1 : x \u2264 1\nhyz : z \u2264 y\n\u22a2 exp (log x * y) \u2264 exp (log x * z)\n[PROOFSTEP]\nrw [exp_le_exp]\n[GOAL]\nx y z : \u211d\nhx0 : 0 < x\nhx1 : x \u2264 1\nhyz : z \u2264 y\n\u22a2 log x * y \u2264 log x * z\n[PROOFSTEP]\nexact mul_le_mul_of_nonpos_left hyz (log_nonpos (le_of_lt hx0) hx1)\n[GOAL]\nx y z : \u211d\nhx0 : 0 < x\nhx1 : x < 1\n\u22a2 x ^ y \u2264 x ^ z \u2194 z \u2264 y\n[PROOFSTEP]\nrw [\u2190 log_le_log (rpow_pos_of_pos hx0 y) (rpow_pos_of_pos hx0 z), log_rpow hx0, log_rpow hx0,\n mul_le_mul_right_of_neg (log_neg hx0 hx1)]\n[GOAL]\nx y z : \u211d\nhx0 : 0 < x\nhx1 : x < 1\n\u22a2 x ^ y < x ^ z \u2194 z < y\n[PROOFSTEP]\nrw [lt_iff_not_le, rpow_le_rpow_left_iff_of_base_lt_one hx0 hx1, lt_iff_not_le]\n[GOAL]\nx\u271d y z\u271d x z : \u211d\nhx1 : 0 \u2264 x\nhx2 : x < 1\nhz : 0 < z\n\u22a2 x ^ z < 1\n[PROOFSTEP]\nrw [\u2190 one_rpow z]\n[GOAL]\nx\u271d y z\u271d x z : \u211d\nhx1 : 0 \u2264 x\nhx2 : x < 1\nhz : 0 < z\n\u22a2 x ^ z < 1 ^ z\n[PROOFSTEP]\nexact rpow_lt_rpow hx1 hx2 hz\n[GOAL]\nx\u271d y z\u271d x z : \u211d\nhx1 : 0 \u2264 x\nhx2 : x \u2264 1\nhz : 0 \u2264 z\n\u22a2 x ^ z \u2264 1\n[PROOFSTEP]\nrw [\u2190 one_rpow z]\n[GOAL]\nx\u271d y z\u271d x z : \u211d\nhx1 : 0 \u2264 x\nhx2 : x \u2264 1\nhz : 0 \u2264 z\n\u22a2 x ^ z \u2264 1 ^ z\n[PROOFSTEP]\nexact rpow_le_rpow hx1 hx2 hz\n[GOAL]\nx\u271d y z\u271d x z : \u211d\nhx : 1 < x\nhz : z < 0\n\u22a2 x ^ z < 1\n[PROOFSTEP]\nconvert rpow_lt_rpow_of_exponent_lt hx hz\n[GOAL]\ncase h.e'_4\nx\u271d y z\u271d x z : \u211d\nhx : 1 < x\nhz : z < 0\n\u22a2 1 = x ^ 0\n[PROOFSTEP]\nexact (rpow_zero x).symm\n[GOAL]\nx\u271d y z\u271d x z : \u211d\nhx : 1 \u2264 x\nhz : z \u2264 0\n\u22a2 x ^ z \u2264 1\n[PROOFSTEP]\nconvert rpow_le_rpow_of_exponent_le hx hz\n[GOAL]\ncase h.e'_4\nx\u271d y z\u271d x z : \u211d\nhx : 1 \u2264 x\nhz : z \u2264 0\n\u22a2 1 = x ^ 0\n[PROOFSTEP]\nexact (rpow_zero x).symm\n[GOAL]\nx\u271d y z\u271d x z : \u211d\nhx : 1 < x\nhz : 0 < z\n\u22a2 1 < x ^ z\n[PROOFSTEP]\nrw [\u2190 one_rpow z]\n[GOAL]\nx\u271d y z\u271d x z : \u211d\nhx : 1 < x\nhz : 0 < z\n\u22a2 1 ^ z < x ^ z\n[PROOFSTEP]\nexact rpow_lt_rpow zero_le_one hx hz\n[GOAL]\nx\u271d y z\u271d x z : \u211d\nhx : 1 \u2264 x\nhz : 0 \u2264 z\n\u22a2 1 \u2264 x ^ z\n[PROOFSTEP]\nrw [\u2190 one_rpow z]\n[GOAL]\nx\u271d y z\u271d x z : \u211d\nhx : 1 \u2264 x\nhz : 0 \u2264 z\n\u22a2 1 ^ z \u2264 x ^ z\n[PROOFSTEP]\nexact rpow_le_rpow zero_le_one hx hz\n[GOAL]\nx y z : \u211d\nhx1 : 0 < x\nhx2 : x < 1\nhz : z < 0\n\u22a2 1 < x ^ z\n[PROOFSTEP]\nconvert rpow_lt_rpow_of_exponent_gt hx1 hx2 hz\n[GOAL]\ncase h.e'_3\nx y z : \u211d\nhx1 : 0 < x\nhx2 : x < 1\nhz : z < 0\n\u22a2 1 = x ^ 0\n[PROOFSTEP]\nexact (rpow_zero x).symm\n[GOAL]\nx y z : \u211d\nhx1 : 0 < x\nhx2 : x \u2264 1\nhz : z \u2264 0\n\u22a2 1 \u2264 x ^ z\n[PROOFSTEP]\nconvert rpow_le_rpow_of_exponent_ge hx1 hx2 hz\n[GOAL]\ncase h.e'_3\nx y z : \u211d\nhx1 : 0 < x\nhx2 : x \u2264 1\nhz : z \u2264 0\n\u22a2 1 = x ^ 0\n[PROOFSTEP]\nexact (rpow_zero x).symm\n[GOAL]\nx y z : \u211d\nhx : 0 < x\n\u22a2 x ^ y < 1 \u2194 1 < x \u2227 y < 0 \u2228 x < 1 \u2227 0 < y\n[PROOFSTEP]\nrw [rpow_def_of_pos hx, exp_lt_one_iff, mul_neg_iff, log_pos_iff hx, log_neg_iff hx]\n[GOAL]\nx y z : \u211d\nhx : 0 \u2264 x\n\u22a2 x ^ y < 1 \u2194 x = 0 \u2227 y \u2260 0 \u2228 1 < x \u2227 y < 0 \u2228 x < 1 \u2227 0 < y\n[PROOFSTEP]\nrcases hx.eq_or_lt with (rfl | hx)\n[GOAL]\ncase inl\ny z : \u211d\nhx : 0 \u2264 0\n\u22a2 0 ^ y < 1 \u2194 0 = 0 \u2227 y \u2260 0 \u2228 1 < 0 \u2227 y < 0 \u2228 0 < 1 \u2227 0 < y\n[PROOFSTEP]\nrcases _root_.em (y = 0) with (rfl | hy)\n[GOAL]\ncase inl.inl\nz : \u211d\nhx : 0 \u2264 0\n\u22a2 0 ^ 0 < 1 \u2194 0 = 0 \u2227 0 \u2260 0 \u2228 1 < 0 \u2227 0 < 0 \u2228 0 < 1 \u2227 0 < 0\n[PROOFSTEP]\nsimp [*, lt_irrefl, zero_lt_one]\n[GOAL]\ncase inl.inr\ny z : \u211d\nhx : 0 \u2264 0\nhy : \u00acy = 0\n\u22a2 0 ^ y < 1 \u2194 0 = 0 \u2227 y \u2260 0 \u2228 1 < 0 \u2227 y < 0 \u2228 0 < 1 \u2227 0 < y\n[PROOFSTEP]\nsimp [*, lt_irrefl, zero_lt_one]\n[GOAL]\ncase inr\nx y z : \u211d\nhx\u271d : 0 \u2264 x\nhx : 0 < x\n\u22a2 x ^ y < 1 \u2194 x = 0 \u2227 y \u2260 0 \u2228 1 < x \u2227 y < 0 \u2228 x < 1 \u2227 0 < y\n[PROOFSTEP]\nsimp [rpow_lt_one_iff_of_pos hx, hx.ne.symm]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\n\u22a2 1 < x ^ y \u2194 1 < x \u2227 0 < y \u2228 x < 1 \u2227 y < 0\n[PROOFSTEP]\nrw [rpow_def_of_pos hx, one_lt_exp_iff, mul_pos_iff, log_pos_iff hx, log_neg_iff hx]\n[GOAL]\nx y z : \u211d\nhx : 0 \u2264 x\n\u22a2 1 < x ^ y \u2194 1 < x \u2227 0 < y \u2228 0 < x \u2227 x < 1 \u2227 y < 0\n[PROOFSTEP]\nrcases hx.eq_or_lt with (rfl | hx)\n[GOAL]\ncase inl\ny z : \u211d\nhx : 0 \u2264 0\n\u22a2 1 < 0 ^ y \u2194 1 < 0 \u2227 0 < y \u2228 0 < 0 \u2227 0 < 1 \u2227 y < 0\n[PROOFSTEP]\nrcases _root_.em (y = 0) with (rfl | hy)\n[GOAL]\ncase inl.inl\nz : \u211d\nhx : 0 \u2264 0\n\u22a2 1 < 0 ^ 0 \u2194 1 < 0 \u2227 0 < 0 \u2228 0 < 0 \u2227 0 < 1 \u2227 0 < 0\n[PROOFSTEP]\nsimp [*, lt_irrefl, (zero_lt_one' \u211d).not_lt]\n[GOAL]\ncase inl.inr\ny z : \u211d\nhx : 0 \u2264 0\nhy : \u00acy = 0\n\u22a2 1 < 0 ^ y \u2194 1 < 0 \u2227 0 < y \u2228 0 < 0 \u2227 0 < 1 \u2227 y < 0\n[PROOFSTEP]\nsimp [*, lt_irrefl, (zero_lt_one' \u211d).not_lt]\n[GOAL]\ncase inr\nx y z : \u211d\nhx\u271d : 0 \u2264 x\nhx : 0 < x\n\u22a2 1 < x ^ y \u2194 1 < x \u2227 0 < y \u2228 0 < x \u2227 x < 1 \u2227 y < 0\n[PROOFSTEP]\nsimp [one_lt_rpow_iff_of_pos hx, hx]\n[GOAL]\nx y z : \u211d\nhx0 : 0 \u2264 x\nhx1 : x \u2264 1\nhz : 0 \u2264 z\nhyz : z \u2264 y\n\u22a2 x ^ y \u2264 x ^ z\n[PROOFSTEP]\nrcases eq_or_lt_of_le hx0 with (rfl | hx0')\n[GOAL]\ncase inl\ny z : \u211d\nhz : 0 \u2264 z\nhyz : z \u2264 y\nhx0 : 0 \u2264 0\nhx1 : 0 \u2264 1\n\u22a2 0 ^ y \u2264 0 ^ z\n[PROOFSTEP]\nrcases eq_or_lt_of_le hz with (rfl | hz')\n[GOAL]\ncase inl.inl\ny : \u211d\nhx0 : 0 \u2264 0\nhx1 : 0 \u2264 1\nhz : 0 \u2264 0\nhyz : 0 \u2264 y\n\u22a2 0 ^ y \u2264 0 ^ 0\n[PROOFSTEP]\nexact (rpow_zero 0).symm \u25b8 rpow_le_one hx0 hx1 hyz\n[GOAL]\ncase inl.inr\ny z : \u211d\nhz : 0 \u2264 z\nhyz : z \u2264 y\nhx0 : 0 \u2264 0\nhx1 : 0 \u2264 1\nhz' : 0 < z\n\u22a2 0 ^ y \u2264 0 ^ z\n[PROOFSTEP]\nrw [zero_rpow, zero_rpow]\n[GOAL]\ncase inl.inr\ny z : \u211d\nhz : 0 \u2264 z\nhyz : z \u2264 y\nhx0 : 0 \u2264 0\nhx1 : 0 \u2264 1\nhz' : 0 < z\n\u22a2 z \u2260 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase inl.inr\ny z : \u211d\nhz : 0 \u2264 z\nhyz : z \u2264 y\nhx0 : 0 \u2264 0\nhx1 : 0 \u2264 1\nhz' : 0 < z\n\u22a2 y \u2260 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase inr\nx y z : \u211d\nhx0 : 0 \u2264 x\nhx1 : x \u2264 1\nhz : 0 \u2264 z\nhyz : z \u2264 y\nhx0' : 0 < x\n\u22a2 x ^ y \u2264 x ^ z\n[PROOFSTEP]\nexact rpow_le_rpow_of_exponent_ge hx0' hx1 hyz\n[GOAL]\nx\u271d y z x : \u211d\nhx : x \u2260 0\n\u22a2 InjOn (fun y => y ^ x) {y | 0 \u2264 y}\n[PROOFSTEP]\nrintro y hy z hz (hyz : y ^ x = z ^ x)\n[GOAL]\nx\u271d y\u271d z\u271d x : \u211d\nhx : x \u2260 0\ny : \u211d\nhy : y \u2208 {y | 0 \u2264 y}\nz : \u211d\nhz : z \u2208 {y | 0 \u2264 y}\nhyz : y ^ x = z ^ x\n\u22a2 y = z\n[PROOFSTEP]\nrw [\u2190 rpow_one y, \u2190 rpow_one z, \u2190 _root_.mul_inv_cancel hx, rpow_mul hy, rpow_mul hz, hyz]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\n\u22a2 x \u2264 y ^ z \u2194 log x \u2264 z * log y\n[PROOFSTEP]\nrw [\u2190 Real.log_le_log hx (Real.rpow_pos_of_pos hy z), Real.log_rpow hy]\n[GOAL]\nx y z : \u211d\nhx : 0 \u2264 x\nhy : 0 < y\nh : log x \u2264 z * log y\n\u22a2 x \u2264 y ^ z\n[PROOFSTEP]\nobtain hx | rfl := hx.lt_or_eq\n[GOAL]\ncase inl\nx y z : \u211d\nhx\u271d : 0 \u2264 x\nhy : 0 < y\nh : log x \u2264 z * log y\nhx : 0 < x\n\u22a2 x \u2264 y ^ z\n[PROOFSTEP]\nexact (le_rpow_iff_log_le hx hy).2 h\n[GOAL]\ncase inr\ny z : \u211d\nhy : 0 < y\nhx : 0 \u2264 0\nh : log 0 \u2264 z * log y\n\u22a2 0 \u2264 y ^ z\n[PROOFSTEP]\nexact (Real.rpow_pos_of_pos hy z).le\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhy : 0 < y\n\u22a2 x < y ^ z \u2194 log x < z * log y\n[PROOFSTEP]\nrw [\u2190 Real.log_lt_log_iff hx (Real.rpow_pos_of_pos hy z), Real.log_rpow hy]\n[GOAL]\nx y z : \u211d\nhx : 0 \u2264 x\nhy : 0 < y\nh : log x < z * log y\n\u22a2 x < y ^ z\n[PROOFSTEP]\nobtain hx | rfl := hx.lt_or_eq\n[GOAL]\ncase inl\nx y z : \u211d\nhx\u271d : 0 \u2264 x\nhy : 0 < y\nh : log x < z * log y\nhx : 0 < x\n\u22a2 x < y ^ z\n[PROOFSTEP]\nexact (lt_rpow_iff_log_lt hx hy).2 h\n[GOAL]\ncase inr\ny z : \u211d\nhy : 0 < y\nhx : 0 \u2264 0\nh : log 0 < z * log y\n\u22a2 0 < y ^ z\n[PROOFSTEP]\nexact Real.rpow_pos_of_pos hy z\n[GOAL]\nx y z : \u211d\nhx : 0 < x\n\u22a2 x ^ y \u2264 1 \u2194 1 \u2264 x \u2227 y \u2264 0 \u2228 x \u2264 1 \u2227 0 \u2264 y\n[PROOFSTEP]\nrw [rpow_def_of_pos hx, exp_le_one_iff, mul_nonpos_iff, log_nonneg_iff hx, log_nonpos_iff hx]\n[GOAL]\nx\u271d y z x t : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\nht : 0 < t\n\u22a2 |log x * x ^ t| < 1 / t\n[PROOFSTEP]\nrw [lt_div_iff ht]\n[GOAL]\nx\u271d y z x t : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\nht : 0 < t\n\u22a2 |log x * x ^ t| * t < 1\n[PROOFSTEP]\nhave := abs_log_mul_self_lt (x ^ t) (rpow_pos_of_pos h1 t) (rpow_le_one h1.le h2 ht.le)\n[GOAL]\nx\u271d y z x t : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\nht : 0 < t\nthis : |log (x ^ t) * x ^ t| < 1\n\u22a2 |log x * x ^ t| * t < 1\n[PROOFSTEP]\nrwa [log_rpow h1, mul_assoc, abs_mul, abs_of_pos ht, mul_comm] at this \n[GOAL]\nx\u271d y z x : \u211d\nhx : 0 \u2264 x\nn : \u2115\nhn : n \u2260 0\n\u22a2 (x ^ n) ^ (\u2191n)\u207b\u00b9 = x\n[PROOFSTEP]\nhave hn0 : (n : \u211d) \u2260 0 := Nat.cast_ne_zero.2 hn\n[GOAL]\nx\u271d y z x : \u211d\nhx : 0 \u2264 x\nn : \u2115\nhn : n \u2260 0\nhn0 : \u2191n \u2260 0\n\u22a2 (x ^ n) ^ (\u2191n)\u207b\u00b9 = x\n[PROOFSTEP]\nrw [\u2190 rpow_nat_cast, \u2190 rpow_mul hx, mul_inv_cancel hn0, rpow_one]\n[GOAL]\nx\u271d y z x : \u211d\nhx : 0 \u2264 x\nn : \u2115\nhn : n \u2260 0\n\u22a2 (x ^ (\u2191n)\u207b\u00b9) ^ n = x\n[PROOFSTEP]\nhave hn0 : (n : \u211d) \u2260 0 := Nat.cast_ne_zero.2 hn\n[GOAL]\nx\u271d y z x : \u211d\nhx : 0 \u2264 x\nn : \u2115\nhn : n \u2260 0\nhn0 : \u2191n \u2260 0\n\u22a2 (x ^ (\u2191n)\u207b\u00b9) ^ n = x\n[PROOFSTEP]\nrw [\u2190 rpow_nat_cast, \u2190 rpow_mul hx, inv_mul_cancel hn0, rpow_one]\n[GOAL]\nx y z b : \u211d\nhb : 1 < b\n\u22a2 StrictMono (rpow b)\n[PROOFSTEP]\nshow StrictMono (fun (x : \u211d) => b ^ x)\n[GOAL]\nx y z b : \u211d\nhb : 1 < b\n\u22a2 StrictMono fun x => b ^ x\n[PROOFSTEP]\nsimp_rw [Real.rpow_def_of_pos (zero_lt_one.trans hb)]\n[GOAL]\nx y z b : \u211d\nhb : 1 < b\n\u22a2 StrictMono fun x => exp (log b * x)\n[PROOFSTEP]\nexact exp_strictMono.comp <| StrictMono.const_mul strictMono_id <| Real.log_pos hb\n[GOAL]\nx y z b : \u211d\nhb : 1 \u2264 b\n\u22a2 Monotone (rpow b)\n[PROOFSTEP]\nrcases lt_or_eq_of_le hb with hb | rfl\n[GOAL]\ncase inl\nx y z b : \u211d\nhb\u271d : 1 \u2264 b\nhb : 1 < b\n\u22a2 Monotone (rpow b)\ncase inr x y z : \u211d hb : 1 \u2264 1 \u22a2 Monotone (rpow 1)\n[PROOFSTEP]\ncase inl => exact (strictMono_rpow_of_base_gt_one hb).monotone\n[GOAL]\nx y z b : \u211d\nhb\u271d : 1 \u2264 b\nhb : 1 < b\n\u22a2 Monotone (rpow b)\n[PROOFSTEP]\ncase inl => exact (strictMono_rpow_of_base_gt_one hb).monotone\n[GOAL]\nx y z b : \u211d\nhb\u271d : 1 \u2264 b\nhb : 1 < b\n\u22a2 Monotone (rpow b)\n[PROOFSTEP]\nexact (strictMono_rpow_of_base_gt_one hb).monotone\n[GOAL]\ncase inr\nx y z : \u211d\nhb : 1 \u2264 1\n\u22a2 Monotone (rpow 1)\n[PROOFSTEP]\ncase inr => intro _ _ _; simp\n[GOAL]\nx y z : \u211d\nhb : 1 \u2264 1\n\u22a2 Monotone (rpow 1)\n[PROOFSTEP]\ncase inr => intro _ _ _; simp\n[GOAL]\nx y z : \u211d\nhb : 1 \u2264 1\n\u22a2 Monotone (rpow 1)\n[PROOFSTEP]\nintro _ _ _\n[GOAL]\nx y z : \u211d\nhb : 1 \u2264 1\na\u271d\u00b9 b\u271d : \u211d\na\u271d : a\u271d\u00b9 \u2264 b\u271d\n\u22a2 rpow 1 a\u271d\u00b9 \u2264 rpow 1 b\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nx y z b : \u211d\nhb\u2080 : 0 < b\nhb\u2081 : b < 1\n\u22a2 StrictAnti (rpow b)\n[PROOFSTEP]\nshow StrictAnti (fun (x : \u211d) => b ^ x)\n[GOAL]\nx y z b : \u211d\nhb\u2080 : 0 < b\nhb\u2081 : b < 1\n\u22a2 StrictAnti fun x => b ^ x\n[PROOFSTEP]\nsimp_rw [Real.rpow_def_of_pos hb\u2080]\n[GOAL]\nx y z b : \u211d\nhb\u2080 : 0 < b\nhb\u2081 : b < 1\n\u22a2 StrictAnti fun x => exp (log b * x)\n[PROOFSTEP]\nexact exp_strictMono.comp_strictAnti <| StrictMono.const_mul_of_neg strictMono_id <| Real.log_neg hb\u2080 hb\u2081\n[GOAL]\nx y z b : \u211d\nhb\u2080 : 0 < b\nhb\u2081 : b \u2264 1\n\u22a2 Antitone (rpow b)\n[PROOFSTEP]\nrcases lt_or_eq_of_le hb\u2081 with hb\u2081 | rfl\n[GOAL]\ncase inl\nx y z b : \u211d\nhb\u2080 : 0 < b\nhb\u2081\u271d : b \u2264 1\nhb\u2081 : b < 1\n\u22a2 Antitone (rpow b)\ncase inr x y z : \u211d hb\u2080 : 0 < 1 hb\u2081 : 1 \u2264 1 \u22a2 Antitone (rpow 1)\n[PROOFSTEP]\ncase inl => exact (strictAnti_rpow_of_base_lt_one hb\u2080 hb\u2081).antitone\n[GOAL]\nx y z b : \u211d\nhb\u2080 : 0 < b\nhb\u2081\u271d : b \u2264 1\nhb\u2081 : b < 1\n\u22a2 Antitone (rpow b)\n[PROOFSTEP]\ncase inl => exact (strictAnti_rpow_of_base_lt_one hb\u2080 hb\u2081).antitone\n[GOAL]\nx y z b : \u211d\nhb\u2080 : 0 < b\nhb\u2081\u271d : b \u2264 1\nhb\u2081 : b < 1\n\u22a2 Antitone (rpow b)\n[PROOFSTEP]\nexact (strictAnti_rpow_of_base_lt_one hb\u2080 hb\u2081).antitone\n[GOAL]\ncase inr\nx y z : \u211d\nhb\u2080 : 0 < 1\nhb\u2081 : 1 \u2264 1\n\u22a2 Antitone (rpow 1)\n[PROOFSTEP]\ncase inr => intro _ _ _; simp\n[GOAL]\nx y z : \u211d\nhb\u2080 : 0 < 1\nhb\u2081 : 1 \u2264 1\n\u22a2 Antitone (rpow 1)\n[PROOFSTEP]\ncase inr => intro _ _ _; simp\n[GOAL]\nx y z : \u211d\nhb\u2080 : 0 < 1\nhb\u2081 : 1 \u2264 1\n\u22a2 Antitone (rpow 1)\n[PROOFSTEP]\nintro _ _ _\n[GOAL]\nx y z : \u211d\nhb\u2080 : 0 < 1\nhb\u2081 : 1 \u2264 1\na\u271d\u00b9 b\u271d : \u211d\na\u271d : a\u271d\u00b9 \u2264 b\u271d\n\u22a2 rpow 1 b\u271d \u2264 rpow 1 a\u271d\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\nz x\u271d y x : \u211d\n\u22a2 sqrt x = x ^ (1 / 2)\n[PROOFSTEP]\nobtain h | h := le_or_lt 0 x\n[GOAL]\ncase inl\nz x\u271d y x : \u211d\nh : 0 \u2264 x\n\u22a2 sqrt x = x ^ (1 / 2)\n[PROOFSTEP]\nrw [\u2190 mul_self_inj_of_nonneg (sqrt_nonneg _) (rpow_nonneg_of_nonneg h _), mul_self_sqrt h, \u2190 sq, \u2190 rpow_nat_cast, \u2190\n rpow_mul h]\n[GOAL]\ncase inl\nz x\u271d y x : \u211d\nh : 0 \u2264 x\n\u22a2 x = x ^ (1 / 2 * \u21912)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr\nz x\u271d y x : \u211d\nh : x < 0\n\u22a2 sqrt x = x ^ (1 / 2)\n[PROOFSTEP]\nhave : 1 / (2 : \u211d) * \u03c0 = \u03c0 / (2 : \u211d)\n[GOAL]\ncase this\nz x\u271d y x : \u211d\nh : x < 0\n\u22a2 1 / 2 * \u03c0 = \u03c0 / 2\ncase inr z x\u271d y x : \u211d h : x < 0 this : 1 / 2 * \u03c0 = \u03c0 / 2 \u22a2 sqrt x = x ^ (1 / 2)\n[PROOFSTEP]\nring\n[GOAL]\ncase inr\nz x\u271d y x : \u211d\nh : x < 0\nthis : 1 / 2 * \u03c0 = \u03c0 / 2\n\u22a2 sqrt x = x ^ (1 / 2)\n[PROOFSTEP]\nrw [sqrt_eq_zero_of_nonpos h.le, rpow_def_of_neg h, this, cos_pi_div_two, mul_zero]\n[GOAL]\nz x\u271d y x r : \u211d\nhx : 0 \u2264 x\n\u22a2 x ^ (r / 2) = sqrt x ^ r\n[PROOFSTEP]\nrw [sqrt_eq_rpow, \u2190 rpow_mul hx]\n[GOAL]\nz x\u271d y x r : \u211d\nhx : 0 \u2264 x\n\u22a2 x ^ (r / 2) = x ^ (1 / 2 * r)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nz x\u271d y x r : \u211d\nhx : 0 \u2264 x\n\u22a2 r / 2 = 1 / 2 * r\n[PROOFSTEP]\nring\n[GOAL]\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y\n[PROOFSTEP]\nhave hn' : 0 < (n : \u211d) := by exact_mod_cast hn.bot_lt\n[GOAL]\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\n\u22a2 0 < \u2191n\n[PROOFSTEP]\nexact_mod_cast hn.bot_lt\n[GOAL]\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y\n[PROOFSTEP]\nobtain \u27e8q, hxq, hqy\u27e9 := exists_rat_btwn (rpow_lt_rpow (le_max_left 0 x) (max_lt hy h) <| inv_pos.mpr hn')\n[GOAL]\ncase intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nhxq : max 0 x ^ (\u2191n)\u207b\u00b9 < \u2191q\nhqy : \u2191q < y ^ (\u2191n)\u207b\u00b9\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y\n[PROOFSTEP]\nhave := rpow_nonneg_of_nonneg (le_max_left 0 x) n\u207b\u00b9\n[GOAL]\ncase intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nhxq : max 0 x ^ (\u2191n)\u207b\u00b9 < \u2191q\nhqy : \u2191q < y ^ (\u2191n)\u207b\u00b9\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y\n[PROOFSTEP]\nhave hq := this.trans_lt hxq\n[GOAL]\ncase intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nhxq : max 0 x ^ (\u2191n)\u207b\u00b9 < \u2191q\nhqy : \u2191q < y ^ (\u2191n)\u207b\u00b9\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y\n[PROOFSTEP]\nreplace hxq := rpow_lt_rpow this hxq hn'\n[GOAL]\ncase intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nhqy : \u2191q < y ^ (\u2191n)\u207b\u00b9\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : (max 0 x ^ (\u2191n)\u207b\u00b9) ^ \u2191n < \u2191q ^ \u2191n\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y\n[PROOFSTEP]\nreplace hqy := rpow_lt_rpow hq.le hqy hn'\n[GOAL]\ncase intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : (max 0 x ^ (\u2191n)\u207b\u00b9) ^ \u2191n < \u2191q ^ \u2191n\nhqy : \u2191q ^ \u2191n < (y ^ (\u2191n)\u207b\u00b9) ^ \u2191n\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y\n[PROOFSTEP]\nrw [rpow_nat_cast, rpow_nat_cast, rpow_nat_inv_pow_nat _ hn] at hxq hqy \n[GOAL]\ncase intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : max 0 x < \u2191q ^ n\nhqy : \u2191q ^ n < y\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y\n[PROOFSTEP]\nexact \u27e8q, by exact_mod_cast hq, (le_max_right _ _).trans_lt hxq, hqy\u27e9\n[GOAL]\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : max 0 x < \u2191q ^ n\nhqy : \u2191q ^ n < y\n\u22a2 0 < q\n[PROOFSTEP]\nexact_mod_cast hq\n[GOAL]\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : max 0 x < \u2191q ^ n\nhqy : \u2191q ^ n < (y ^ (\u2191n)\u207b\u00b9) ^ n\n\u22a2 0 \u2264 y\n[PROOFSTEP]\nexact hy.le\n[GOAL]\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211d\nh : x < y\nhy : 0 < y\nhn' : 0 < \u2191n\nq : \u211a\nthis : 0 \u2264 max 0 x ^ (\u2191n)\u207b\u00b9\nhq : 0 < \u2191q\nhxq : (max 0 x ^ (\u2191n)\u207b\u00b9) ^ n < \u2191q ^ n\nhqy : \u2191q ^ n < (y ^ (\u2191n)\u207b\u00b9) ^ n\n\u22a2 0 \u2264 max 0 x\n[PROOFSTEP]\nexact le_max_left _ _\n[GOAL]\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211a\nh : x < y\nhy : 0 < y\n\u22a2 \u2203 q, 0 < q \u2227 x < q ^ n \u2227 q ^ n < y\n[PROOFSTEP]\napply_mod_cast exists_rat_pow_btwn_rat_aux hn x y\n[GOAL]\ncase a\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211a\nh : x < y\nhy : 0 < y\n\u22a2 x < y\n[PROOFSTEP]\nassumption\n[GOAL]\ncase a\nz x\u271d y\u271d : \u211d\nn : \u2115\nhn : n \u2260 0\nx y : \u211a\nh : x < y\nhy : 0 < y\n\u22a2 0 < y\n[PROOFSTEP]\nassumption\n[GOAL]\nz x\u271d y\u271d : \u211d\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nhn : n \u2260 0\nx y : \u03b1\nh : x < y\nhy : 0 < y\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y\n[PROOFSTEP]\nobtain \u27e8q\u2082, hx\u2082, hy\u2082\u27e9 := exists_rat_btwn (max_lt h hy)\n[GOAL]\ncase intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nhn : n \u2260 0\nx y : \u03b1\nh : x < y\nhy : 0 < y\nq\u2082 : \u211a\nhx\u2082 : max x 0 < \u2191q\u2082\nhy\u2082 : \u2191q\u2082 < y\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y\n[PROOFSTEP]\nobtain \u27e8q\u2081, hx\u2081, hq\u2081\u2082\u27e9 := exists_rat_btwn hx\u2082\n[GOAL]\ncase intro.intro.intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nhn : n \u2260 0\nx y : \u03b1\nh : x < y\nhy : 0 < y\nq\u2082 : \u211a\nhx\u2082 : max x 0 < \u2191q\u2082\nhy\u2082 : \u2191q\u2082 < y\nq\u2081 : \u211a\nhx\u2081 : max x 0 < \u2191q\u2081\nhq\u2081\u2082 : \u2191q\u2081 < \u2191q\u2082\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y\n[PROOFSTEP]\nhave : (0 : \u03b1) < q\u2082 := (le_max_right _ _).trans_lt hx\u2082\n[GOAL]\ncase intro.intro.intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nhn : n \u2260 0\nx y : \u03b1\nh : x < y\nhy : 0 < y\nq\u2082 : \u211a\nhx\u2082 : max x 0 < \u2191q\u2082\nhy\u2082 : \u2191q\u2082 < y\nq\u2081 : \u211a\nhx\u2081 : max x 0 < \u2191q\u2081\nhq\u2081\u2082 : \u2191q\u2081 < \u2191q\u2082\nthis : 0 < \u2191q\u2082\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y\n[PROOFSTEP]\nnorm_cast at hq\u2081\u2082 this \n[GOAL]\ncase intro.intro.intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nhn : n \u2260 0\nx y : \u03b1\nh : x < y\nhy : 0 < y\nq\u2082 : \u211a\nhx\u2082 : max x 0 < \u2191q\u2082\nhy\u2082 : \u2191q\u2082 < y\nq\u2081 : \u211a\nhx\u2081 : max x 0 < \u2191q\u2081\nhq\u2081\u2082 : q\u2081 < q\u2082\nthis : 0 < q\u2082\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y\n[PROOFSTEP]\nobtain \u27e8q, hq, hq\u2081, hq\u2082\u27e9 := exists_rat_pow_btwn_rat hn hq\u2081\u2082 this\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nz x\u271d y\u271d : \u211d\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nhn : n \u2260 0\nx y : \u03b1\nh : x < y\nhy : 0 < y\nq\u2082 : \u211a\nhx\u2082 : max x 0 < \u2191q\u2082\nhy\u2082 : \u2191q\u2082 < y\nq\u2081 : \u211a\nhx\u2081 : max x 0 < \u2191q\u2081\nhq\u2081\u2082 : q\u2081 < q\u2082\nthis : 0 < q\u2082\nq : \u211a\nhq : 0 < q\nhq\u2081 : q\u2081 < q ^ n\nhq\u2082 : q ^ n < q\u2082\n\u22a2 \u2203 q, 0 < q \u2227 x < \u2191q ^ n \u2227 \u2191q ^ n < y\n[PROOFSTEP]\nrefine' \u27e8q, hq, (le_max_left _ _).trans_lt <| hx\u2081.trans _, hy\u2082.trans' _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_1\nz x\u271d y\u271d : \u211d\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nhn : n \u2260 0\nx y : \u03b1\nh : x < y\nhy : 0 < y\nq\u2082 : \u211a\nhx\u2082 : max x 0 < \u2191q\u2082\nhy\u2082 : \u2191q\u2082 < y\nq\u2081 : \u211a\nhx\u2081 : max x 0 < \u2191q\u2081\nhq\u2081\u2082 : q\u2081 < q\u2082\nthis : 0 < q\u2082\nq : \u211a\nhq : 0 < q\nhq\u2081 : q\u2081 < q ^ n\nhq\u2082 : q ^ n < q\u2082\n\u22a2 \u2191q\u2081 < \u2191q ^ n\n[PROOFSTEP]\nassumption_mod_cast\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_2\nz x\u271d y\u271d : \u211d\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nhn : n \u2260 0\nx y : \u03b1\nh : x < y\nhy : 0 < y\nq\u2082 : \u211a\nhx\u2082 : max x 0 < \u2191q\u2082\nhy\u2082 : \u2191q\u2082 < y\nq\u2081 : \u211a\nhx\u2081 : max x 0 < \u2191q\u2081\nhq\u2081\u2082 : q\u2081 < q\u2082\nthis : 0 < q\u2082\nq : \u211a\nhq : 0 < q\nhq\u2081 : q\u2081 < q ^ n\nhq\u2082 : q ^ n < q\u2082\n\u22a2 \u2191q ^ n < \u2191q\u2082\n[PROOFSTEP]\nassumption_mod_cast\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.Pow.Real", "llama_tokens": 35866, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389817407016, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.439393136389277}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nhf : Injective2 f\n\u22a2 f a b \u2208 image2 f s t \u2192 a \u2208 s \u2227 b \u2208 t\n[PROOFSTEP]\nrintro \u27e8a', b', ha', hb', h\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a'\u271d : \u03b1\nb b'\u271d : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nhf : Injective2 f\na' : \u03b1\nb' : \u03b2\nha' : a' \u2208 s\nhb' : b' \u2208 t\nh : f a' b' = f a b\n\u22a2 a \u2208 s \u2227 b \u2208 t\n[PROOFSTEP]\nrcases hf h with \u27e8rfl, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na'\u271d : \u03b1\nb'\u271d : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nhf : Injective2 f\na' : \u03b1\nb' : \u03b2\nha' : a' \u2208 s\nhb' : b' \u2208 t\nh : f a' b' = f a' b'\n\u22a2 a' \u2208 s \u2227 b' \u2208 t\n[PROOFSTEP]\nexact \u27e8ha', hb'\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nhs : s \u2286 s'\nht : t \u2286 t'\n\u22a2 image2 f s t \u2286 image2 f s' t'\n[PROOFSTEP]\nrintro _ \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nhs : s \u2286 s'\nht : t \u2286 t'\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\n\u22a2 f a b \u2208 image2 f s' t'\n[PROOFSTEP]\nexact mem_image2_of_mem (hs ha) (ht hb)\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f s t \u2286 u \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 (fun b => f a b) '' t \u2286 u\n[PROOFSTEP]\nsimp_rw [image2_subset_iff, image_subset_iff, subset_def, mem_preimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f s t \u2286 u \u2194 \u2200 (b : \u03b2), b \u2208 t \u2192 (fun a => f a b) '' s \u2286 u\n[PROOFSTEP]\nsimp_rw [image2_subset_iff, image_subset_iff, subset_def, mem_preimage, @forall\u2082_swap \u03b1]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\na : \u03b3\n\u22a2 a \u2208 (fun x => f x.fst x.snd) '' s \u00d7\u02e2 t \u2192 a \u2208 image2 f s t\n[PROOFSTEP]\nrintro \u27e8_, _, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nw\u271d : \u03b1 \u00d7 \u03b2\nleft\u271d : w\u271d \u2208 s \u00d7\u02e2 t\n\u22a2 (fun x => f x.fst x.snd) w\u271d \u2208 image2 f s t\n[PROOFSTEP]\nexact \u27e8_, _, (mem_prod.1 \u2039_\u203a).1, (mem_prod.1 \u2039_\u203a).2, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\na : \u03b3\n\u22a2 a \u2208 image2 f s t \u2192 a \u2208 (fun x => f x.fst x.snd) '' s \u00d7\u02e2 t\n[PROOFSTEP]\nrintro \u27e8_, _, _, _, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nw\u271d\u00b9 : \u03b1\nw\u271d : \u03b2\nleft\u271d\u00b9 : w\u271d\u00b9 \u2208 s\nleft\u271d : w\u271d \u2208 t\n\u22a2 f w\u271d\u00b9 w\u271d \u2208 (fun x => f x.fst x.snd) '' s \u00d7\u02e2 t\n[PROOFSTEP]\nexact \u27e8(_, _), \u27e8\u2039_\u203a, \u2039_\u203a\u27e9, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 \u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 image2 Prod.mk s t \u2194 x \u2208 s \u00d7\u02e2 t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Set \u03b1\nt\u271d t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 image2 (fun a b => f (a, b)) s t = f '' s \u00d7\u02e2 t\n[PROOFSTEP]\nsimp [\u2190 image_uncurry_prod, uncurry]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Set \u03b1\nt\u271d t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 image2 f s t = image2 (fun a b => f b a) t s\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Set \u03b1\nt\u271d t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\ns : Set \u03b1\nt : Set \u03b2\nx\u271d : \u03b3\n\u22a2 x\u271d \u2208 image2 f s t \u2194 x\u271d \u2208 image2 (fun a b => f b a) t s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Set \u03b1\nt\u271d t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\ns : Set \u03b1\nt : Set \u03b2\nx\u271d : \u03b3\n\u22a2 x\u271d \u2208 image2 f s t \u2192 x\u271d \u2208 image2 (fun a b => f b a) t s\n[PROOFSTEP]\nrintro \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Set \u03b1\nt\u271d t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\ns : Set \u03b1\nt : Set \u03b2\nx\u271d : \u03b3\n\u22a2 x\u271d \u2208 image2 (fun a b => f b a) t s \u2192 x\u271d \u2208 image2 f s t\n[PROOFSTEP]\nrintro \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Set \u03b1\nt\u271d t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\n\u22a2 f a b \u2208 image2 (fun a b => f b a) t s\n[PROOFSTEP]\nexact \u27e8b, a, hb, ha, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Set \u03b1\nt\u271d t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\ns : Set \u03b1\nt : Set \u03b2\na : \u03b2\nb : \u03b1\nha : a \u2208 t\nhb : b \u2208 s\n\u22a2 (fun a b => f b a) a b \u2208 image2 f s t\n[PROOFSTEP]\nexact \u27e8b, a, hb, ha, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f (s \u222a s') t = image2 f s t \u222a image2 f s' t\n[PROOFSTEP]\next c\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nc : \u03b3\n\u22a2 c \u2208 image2 f (s \u222a s') t \u2194 c \u2208 image2 f s t \u222a image2 f s' t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nc : \u03b3\n\u22a2 c \u2208 image2 f (s \u222a s') t \u2192 c \u2208 image2 f s t \u222a image2 f s' t\n[PROOFSTEP]\nrintro \u27e8a, b, ha | ha, hb, rfl\u27e9 <;> [left; right]\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nc : \u03b3\n\u22a2 c \u2208 image2 f (s \u222a s') t \u2192 c \u2208 image2 f s t \u222a image2 f s' t\n[PROOFSTEP]\nrintro \u27e8a, b, ha | ha, hb, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.inl.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\n\u22a2 f a b \u2208 image2 f s t \u222a image2 f s' t\n[PROOFSTEP]\nleft\n[GOAL]\ncase h.mp.intro.intro.intro.inr.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\na : \u03b1\nb : \u03b2\nha : a \u2208 s'\nhb : b \u2208 t\n\u22a2 f a b \u2208 image2 f s t \u222a image2 f s' t\n[PROOFSTEP]\nright\n[GOAL]\ncase h.mp.intro.intro.intro.inl.intro.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\n\u22a2 f a b \u2208 image2 f s t\n[PROOFSTEP]\nexact \u27e8_, _, \u2039_\u203a, \u2039_\u203a, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.inr.intro.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\na : \u03b1\nb : \u03b2\nha : a \u2208 s'\nhb : b \u2208 t\n\u22a2 f a b \u2208 image2 f s' t\n[PROOFSTEP]\nexact \u27e8_, _, \u2039_\u203a, \u2039_\u203a, rfl\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nc : \u03b3\n\u22a2 c \u2208 image2 f s t \u222a image2 f s' t \u2192 c \u2208 image2 f (s \u222a s') t\n[PROOFSTEP]\nrintro (\u27e8_, _, _, _, rfl\u27e9 | \u27e8_, _, _, _, rfl\u27e9)\n[GOAL]\ncase h.mpr.inl.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nw\u271d\u00b9 : \u03b1\nw\u271d : \u03b2\nleft\u271d\u00b9 : w\u271d\u00b9 \u2208 s\nleft\u271d : w\u271d \u2208 t\n\u22a2 f w\u271d\u00b9 w\u271d \u2208 image2 f (s \u222a s') t\n[PROOFSTEP]\nrefine' \u27e8_, _, _, \u2039_\u203a, rfl\u27e9\n[GOAL]\ncase h.mpr.inr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nw\u271d\u00b9 : \u03b1\nw\u271d : \u03b2\nleft\u271d\u00b9 : w\u271d\u00b9 \u2208 s'\nleft\u271d : w\u271d \u2208 t\n\u22a2 f w\u271d\u00b9 w\u271d \u2208 image2 f (s \u222a s') t\n[PROOFSTEP]\nrefine' \u27e8_, _, _, \u2039_\u203a, rfl\u27e9\n[GOAL]\ncase h.mpr.inl.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nw\u271d\u00b9 : \u03b1\nw\u271d : \u03b2\nleft\u271d\u00b9 : w\u271d\u00b9 \u2208 s\nleft\u271d : w\u271d \u2208 t\n\u22a2 w\u271d\u00b9 \u2208 s \u222a s'\n[PROOFSTEP]\nsimp [mem_union, *]\n[GOAL]\ncase h.mpr.inr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nw\u271d\u00b9 : \u03b1\nw\u271d : \u03b2\nleft\u271d\u00b9 : w\u271d\u00b9 \u2208 s'\nleft\u271d : w\u271d \u2208 t\n\u22a2 w\u271d\u00b9 \u2208 s \u222a s'\n[PROOFSTEP]\nsimp [mem_union, *]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f s (t \u222a t') = image2 f s t \u222a image2 f s t'\n[PROOFSTEP]\nrw [\u2190 image2_swap, image2_union_left, image2_swap f, image2_swap f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nhf : Injective2 f\n\u22a2 image2 f (s \u2229 s') t = image2 f s t \u2229 image2 f s' t\n[PROOFSTEP]\nsimp_rw [\u2190 image_uncurry_prod, inter_prod, image_inter hf.uncurry]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nhf : Injective2 f\n\u22a2 image2 f s (t \u2229 t') = image2 f s t \u2229 image2 f s t'\n[PROOFSTEP]\nsimp_rw [\u2190 image_uncurry_prod, prod_inter, image_inter hf.uncurry]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 \u2200 (x : \u03b3), x \u2208 image2 f \u2205 t \u2194 x \u2208 \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 \u2200 (x : \u03b3), x \u2208 image2 f s \u2205 \u2194 x \u2208 \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f s t = \u2205 \u2194 s = \u2205 \u2228 t = \u2205\n[PROOFSTEP]\nrw [\u2190 not_nonempty_iff_eq_empty, image2_nonempty_iff, not_and_or]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 \u00acSet.Nonempty s \u2228 \u00acSet.Nonempty t \u2194 s = \u2205 \u2228 t = \u2205\n[PROOFSTEP]\nsimp [not_nonempty_iff_eq_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nhs : Set.Subsingleton s\nht : Set.Subsingleton t\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u22a2 Set.Subsingleton (Set.image2 f s t)\n[PROOFSTEP]\nrw [\u2190 image_prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nhs : Set.Subsingleton s\nht : Set.Subsingleton t\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u22a2 Set.Subsingleton ((fun x => f x.fst x.snd) '' s \u00d7\u02e2 t)\n[PROOFSTEP]\napply (hs.prod ht).image\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f (s \u2229 s') t \u2286 image2 f s t \u2229 image2 f s' t\n[PROOFSTEP]\nrintro _ \u27e8a, b, \u27e8h1a, h2a\u27e9, hb, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\na : \u03b1\nb : \u03b2\nh1a : a \u2208 s\nh2a : a \u2208 s'\nhb : b \u2208 t\n\u22a2 f a b \u2208 image2 f s t \u2229 image2 f s' t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.intro.intro.intro.left\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\na : \u03b1\nb : \u03b2\nh1a : a \u2208 s\nh2a : a \u2208 s'\nhb : b \u2208 t\n\u22a2 f a b \u2208 image2 f s t\n[PROOFSTEP]\nexact \u27e8_, _, \u2039_\u203a, \u2039_\u203a, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.right\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\na : \u03b1\nb : \u03b2\nh1a : a \u2208 s\nh2a : a \u2208 s'\nhb : b \u2208 t\n\u22a2 f a b \u2208 image2 f s' t\n[PROOFSTEP]\nexact \u27e8_, _, \u2039_\u203a, \u2039_\u203a, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f s (t \u2229 t') \u2286 image2 f s t \u2229 image2 f s t'\n[PROOFSTEP]\nrintro _ \u27e8a, b, ha, \u27e8h1b, h2b\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nh1b : b \u2208 t\nh2b : b \u2208 t'\n\u22a2 f a b \u2208 image2 f s t \u2229 image2 f s t'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.intro.intro.intro.left\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nh1b : b \u2208 t\nh2b : b \u2208 t'\n\u22a2 f a b \u2208 image2 f s t\n[PROOFSTEP]\nexact \u27e8_, _, \u2039_\u203a, \u2039_\u203a, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.right\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nh1b : b \u2208 t\nh2b : b \u2208 t'\n\u22a2 f a b \u2208 image2 f s t'\n[PROOFSTEP]\nexact \u27e8_, _, \u2039_\u203a, \u2039_\u203a, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nx : \u03b3\n\u22a2 x \u2208 image2 f {a} t \u2194 x \u2208 f a '' t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nx : \u03b3\n\u22a2 x \u2208 image2 f s {b} \u2194 x \u2208 (fun a => f a b) '' s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f {a} {b} = {f a b}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f (insert a s) t = (fun b => f a b) '' t \u222a image2 f s t\n[PROOFSTEP]\nrw [insert_eq, image2_union_left, image2_singleton_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f s (insert b t) = (fun a => f a b) '' s \u222a image2 f s t\n[PROOFSTEP]\nrw [insert_eq, image2_union_right, image2_singleton_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 f a b = f' a b\n\u22a2 image2 f s t = image2 f' s t\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 f a b = f' a b\nx\u271d : \u03b3\n\u22a2 x\u271d \u2208 image2 f s t \u2194 x\u271d \u2208 image2 f' s t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 f a b = f' a b\nx\u271d : \u03b3\n\u22a2 x\u271d \u2208 image2 f s t \u2192 x\u271d \u2208 image2 f' s t\n[PROOFSTEP]\nrintro \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 f a b = f' a b\nx\u271d : \u03b3\n\u22a2 x\u271d \u2208 image2 f' s t \u2192 x\u271d \u2208 image2 f s t\n[PROOFSTEP]\nrintro \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 f a b = f' a b\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\n\u22a2 f a b \u2208 image2 f' s t\n[PROOFSTEP]\nrefine' \u27e8a, b, ha, hb, by rw [h a ha b hb]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 f a b = f' a b\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\n\u22a2 f' a b = f a b\n[PROOFSTEP]\nrw [h a ha b hb]\n[GOAL]\ncase h.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 f a b = f' a b\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\n\u22a2 f' a b \u2208 image2 f s t\n[PROOFSTEP]\nrefine' \u27e8a, b, ha, hb, by rw [h a ha b hb]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 f a b = f' a b\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\n\u22a2 f a b = f' a b\n[PROOFSTEP]\nrw [h a ha b hb]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 \u2200 (c : \u03b3), c \u2208 u \u2192 g a b c = g' a b c\n\u22a2 image3 g s t u = image3 g' s t u\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 \u2200 (c : \u03b3), c \u2208 u \u2192 g a b c = g' a b c\nx : \u03b4\n\u22a2 x \u2208 image3 g s t u \u2194 x \u2208 image3 g' s t u\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 \u2200 (c : \u03b3), c \u2208 u \u2192 g a b c = g' a b c\nx : \u03b4\n\u22a2 x \u2208 image3 g s t u \u2192 x \u2208 image3 g' s t u\n[PROOFSTEP]\nrintro \u27e8a, b, c, ha, hb, hc, rfl\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 \u2200 (c : \u03b3), c \u2208 u \u2192 g a b c = g' a b c\nx : \u03b4\n\u22a2 x \u2208 image3 g' s t u \u2192 x \u2208 image3 g s t u\n[PROOFSTEP]\nrintro \u27e8a, b, c, ha, hb, hc, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 \u2200 (c : \u03b3), c \u2208 u \u2192 g a b c = g' a b c\na : \u03b1\nb : \u03b2\nc : \u03b3\nha : a \u2208 s\nhb : b \u2208 t\nhc : c \u2208 u\n\u22a2 g a b c \u2208 image3 g' s t u\n[PROOFSTEP]\nexact \u27e8a, b, c, ha, hb, hc, by rw [h a ha b hb c hc]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 \u2200 (c : \u03b3), c \u2208 u \u2192 g a b c = g' a b c\na : \u03b1\nb : \u03b2\nc : \u03b3\nha : a \u2208 s\nhb : b \u2208 t\nhc : c \u2208 u\n\u22a2 g' a b c = g a b c\n[PROOFSTEP]\nrw [h a ha b hb c hc]\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 \u2200 (c : \u03b3), c \u2208 u \u2192 g a b c = g' a b c\na : \u03b1\nb : \u03b2\nc : \u03b3\nha : a \u2208 s\nhb : b \u2208 t\nhc : c \u2208 u\n\u22a2 g' a b c \u2208 image3 g s t u\n[PROOFSTEP]\nexact \u27e8a, b, c, ha, hb, hc, by rw [h a ha b hb c hc]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 \u2200 (c : \u03b3), c \u2208 u \u2192 g a b c = g' a b c\na : \u03b1\nb : \u03b2\nc : \u03b3\nha : a \u2208 s\nhb : b \u2208 t\nhc : c \u2208 u\n\u22a2 g a b c = g' a b c\n[PROOFSTEP]\nrw [h a ha b hb c hc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b4 \u2192 \u03b3 \u2192 \u03b5\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b4\n\u22a2 image2 f (image2 g s t) u = image3 (fun a b c => f (g a b) c) s t u\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b4 \u2192 \u03b3 \u2192 \u03b5\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nx\u271d : \u03b5\n\u22a2 x\u271d \u2208 image2 f (image2 g s t) u \u2194 x\u271d \u2208 image3 (fun a b c => f (g a b) c) s t u\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b4 \u2192 \u03b3 \u2192 \u03b5\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nx\u271d : \u03b5\n\u22a2 x\u271d \u2208 image2 f (image2 g s t) u \u2192 x\u271d \u2208 image3 (fun a b c => f (g a b) c) s t u\n[PROOFSTEP]\nrintro \u27e8_, c, \u27e8a, b, ha, hb, rfl\u27e9, hc, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nf : \u03b4 \u2192 \u03b3 \u2192 \u03b5\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nc : \u03b3\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\nhc : c \u2208 u\n\u22a2 f (g a b) c \u2208 image3 (fun a b c => f (g a b) c) s t u\n[PROOFSTEP]\nrefine' \u27e8a, b, c, ha, hb, hc, rfl\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b4 \u2192 \u03b3 \u2192 \u03b5\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nx\u271d : \u03b5\n\u22a2 x\u271d \u2208 image3 (fun a b c => f (g a b) c) s t u \u2192 x\u271d \u2208 image2 f (image2 g s t) u\n[PROOFSTEP]\nrintro \u27e8a, b, c, ha, hb, hc, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nf : \u03b4 \u2192 \u03b3 \u2192 \u03b5\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nha : a \u2208 s\nhb : b \u2208 t\nhc : c \u2208 u\n\u22a2 (fun a b c => f (g a b) c) a b c \u2208 image2 f (image2 g s t) u\n[PROOFSTEP]\nrefine' \u27e8_, c, \u27e8a, b, ha, hb, rfl\u27e9, hc, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b4 \u2192 \u03b5\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\n\u22a2 image2 f s (image2 g t u) = image3 (fun a b c => f a (g b c)) s t u\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b4 \u2192 \u03b5\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nx\u271d : \u03b5\n\u22a2 x\u271d \u2208 image2 f s (image2 g t u) \u2194 x\u271d \u2208 image3 (fun a b c => f a (g b c)) s t u\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b4 \u2192 \u03b5\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nx\u271d : \u03b5\n\u22a2 x\u271d \u2208 image2 f s (image2 g t u) \u2192 x\u271d \u2208 image3 (fun a b c => f a (g b c)) s t u\n[PROOFSTEP]\nrintro \u27e8a, _, ha, \u27e8b, c, hb, hc, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b4 \u2192 \u03b5\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\na : \u03b1\nha : a \u2208 s\nb : \u03b2\nc : \u03b3\nhb : b \u2208 t\nhc : c \u2208 u\n\u22a2 f a (g b c) \u2208 image3 (fun a b c => f a (g b c)) s t u\n[PROOFSTEP]\nrefine' \u27e8a, b, c, ha, hb, hc, rfl\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b4 \u2192 \u03b5\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nx\u271d : \u03b5\n\u22a2 x\u271d \u2208 image3 (fun a b c => f a (g b c)) s t u \u2192 x\u271d \u2208 image2 f s (image2 g t u)\n[PROOFSTEP]\nrintro \u27e8a, b, c, ha, hb, hc, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b4 \u2192 \u03b5\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nha : a \u2208 s\nhb : b \u2208 t\nhc : c \u2208 u\n\u22a2 (fun a b c => f a (g b c)) a b c \u2208 image2 f s (image2 g t u)\n[PROOFSTEP]\nrefine' \u27e8a, _, ha, \u27e8b, c, hb, hc, rfl\u27e9, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b4\n\u22a2 g '' image2 f s t = image2 (fun a b => g (f a b)) s t\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b4\nx\u271d : \u03b4\n\u22a2 x\u271d \u2208 g '' image2 f s t \u2194 x\u271d \u2208 image2 (fun a b => g (f a b)) s t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b4\nx\u271d : \u03b4\n\u22a2 x\u271d \u2208 g '' image2 f s t \u2192 x\u271d \u2208 image2 (fun a b => g (f a b)) s t\n[PROOFSTEP]\nrintro \u27e8_, \u27e8a, b, ha, hb, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b4\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\n\u22a2 g (f a b) \u2208 image2 (fun a b => g (f a b)) s t\n[PROOFSTEP]\nrefine' \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b4\nx\u271d : \u03b4\n\u22a2 x\u271d \u2208 image2 (fun a b => g (f a b)) s t \u2192 x\u271d \u2208 g '' image2 f s t\n[PROOFSTEP]\nrintro \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b4\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\n\u22a2 (fun a b => g (f a b)) a b \u2208 g '' image2 f s t\n[PROOFSTEP]\nrefine' \u27e8_, \u27e8a, b, ha, hb, rfl\u27e9, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b4\ng : \u03b1 \u2192 \u03b3\n\u22a2 image2 f (g '' s) t = image2 (fun a b => f (g a) b) s t\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b4\ng : \u03b1 \u2192 \u03b3\nx\u271d : \u03b4\n\u22a2 x\u271d \u2208 image2 f (g '' s) t \u2194 x\u271d \u2208 image2 (fun a b => f (g a) b) s t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b4\ng : \u03b1 \u2192 \u03b3\nx\u271d : \u03b4\n\u22a2 x\u271d \u2208 image2 f (g '' s) t \u2192 x\u271d \u2208 image2 (fun a b => f (g a) b) s t\n[PROOFSTEP]\nrintro \u27e8_, b, \u27e8a, ha, rfl\u27e9, hb, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b4\ng : \u03b1 \u2192 \u03b3\nb : \u03b2\na : \u03b1\nha : a \u2208 s\nhb : b \u2208 t\n\u22a2 f (g a) b \u2208 image2 (fun a b => f (g a) b) s t\n[PROOFSTEP]\nrefine' \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b4\ng : \u03b1 \u2192 \u03b3\nx\u271d : \u03b4\n\u22a2 x\u271d \u2208 image2 (fun a b => f (g a) b) s t \u2192 x\u271d \u2208 image2 f (g '' s) t\n[PROOFSTEP]\nrintro \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b4\ng : \u03b1 \u2192 \u03b3\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\n\u22a2 (fun a b => f (g a) b) a b \u2208 image2 f (g '' s) t\n[PROOFSTEP]\nrefine' \u27e8_, b, \u27e8a, ha, rfl\u27e9, hb, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b3 \u2192 \u03b4\ng : \u03b2 \u2192 \u03b3\n\u22a2 image2 f s (g '' t) = image2 (fun a b => f a (g b)) s t\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b3 \u2192 \u03b4\ng : \u03b2 \u2192 \u03b3\nx\u271d : \u03b4\n\u22a2 x\u271d \u2208 image2 f s (g '' t) \u2194 x\u271d \u2208 image2 (fun a b => f a (g b)) s t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b3 \u2192 \u03b4\ng : \u03b2 \u2192 \u03b3\nx\u271d : \u03b4\n\u22a2 x\u271d \u2208 image2 f s (g '' t) \u2192 x\u271d \u2208 image2 (fun a b => f a (g b)) s t\n[PROOFSTEP]\nrintro \u27e8a, _, ha, \u27e8b, hb, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b3 \u2192 \u03b4\ng : \u03b2 \u2192 \u03b3\na : \u03b1\nha : a \u2208 s\nb : \u03b2\nhb : b \u2208 t\n\u22a2 f a (g b) \u2208 image2 (fun a b => f a (g b)) s t\n[PROOFSTEP]\nrefine' \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b3 \u2192 \u03b4\ng : \u03b2 \u2192 \u03b3\nx\u271d : \u03b4\n\u22a2 x\u271d \u2208 image2 (fun a b => f a (g b)) s t \u2192 x\u271d \u2208 image2 f s (g '' t)\n[PROOFSTEP]\nrintro \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b3 \u2192 \u03b4\ng : \u03b2 \u2192 \u03b3\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\n\u22a2 (fun a b => f a (g b)) a b \u2208 image2 f s (g '' t)\n[PROOFSTEP]\nrefine' \u27e8a, _, ha, \u27e8b, hb, rfl\u27e9, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : Set.Nonempty t\n\u22a2 image2 (fun x x_1 => x) s t = s\n[PROOFSTEP]\nsimp [nonempty_def.mp h, ext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : Set.Nonempty s\n\u22a2 image2 (fun x y => y) s t = t\n[PROOFSTEP]\nsimp [nonempty_def.mp h, ext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b4 \u2192 \u03b3 \u2192 \u03b5\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nf' : \u03b1 \u2192 \u03b5' \u2192 \u03b5\ng' : \u03b2 \u2192 \u03b3 \u2192 \u03b5'\nh_assoc : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), f (g a b) c = f' a (g' b c)\n\u22a2 image2 f (image2 g s t) u = image2 f' s (image2 g' t u)\n[PROOFSTEP]\nsimp only [image2_image2_left, image2_image2_right, h_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\ng : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh_comm : \u2200 (a : \u03b1) (b : \u03b2), f a b = g b a\n\u22a2 image2 (fun a b => f b a) t s = image2 g t s\n[PROOFSTEP]\nsimp_rw [h_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b4 \u2192 \u03b5\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nf' : \u03b1 \u2192 \u03b3 \u2192 \u03b4'\ng' : \u03b2 \u2192 \u03b4' \u2192 \u03b5\nh_left_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), f a (g b c) = g' b (f' a c)\n\u22a2 image2 f s (image2 g t u) = image2 g' t (image2 f' s u)\n[PROOFSTEP]\nrw [image2_swap f', image2_swap f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b4 \u2192 \u03b5\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nf' : \u03b1 \u2192 \u03b3 \u2192 \u03b4'\ng' : \u03b2 \u2192 \u03b4' \u2192 \u03b5\nh_left_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), f a (g b c) = g' b (f' a c)\n\u22a2 image2 (fun a b => f b a) (image2 g t u) s = image2 g' t (image2 (fun a b => f' b a) u s)\n[PROOFSTEP]\nexact image2_assoc fun _ _ _ => h_left_comm _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b4 \u2192 \u03b3 \u2192 \u03b5\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nf' : \u03b1 \u2192 \u03b3 \u2192 \u03b4'\ng' : \u03b4' \u2192 \u03b2 \u2192 \u03b5\nh_right_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), f (g a b) c = g' (f' a c) b\n\u22a2 image2 f (image2 g s t) u = image2 g' (image2 f' s u) t\n[PROOFSTEP]\nrw [image2_swap g, image2_swap g']\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b4 \u2192 \u03b3 \u2192 \u03b5\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nf' : \u03b1 \u2192 \u03b3 \u2192 \u03b4'\ng' : \u03b4' \u2192 \u03b2 \u2192 \u03b5\nh_right_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), f (g a b) c = g' (f' a c) b\n\u22a2 image2 f (image2 (fun a b => g b a) t s) u = image2 (fun a b => g' b a) t (image2 f' s u)\n[PROOFSTEP]\nexact image2_assoc fun _ _ _ => h_right_comm _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\n\u22a2 image2 f (image2 g s t) (image2 h u v) = image2 f' (image2 g' s u) (image2 h' t v)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\nx\u271d : \u03bd\n\u22a2 x\u271d \u2208 image2 f (image2 g s t) (image2 h u v) \u2194 x\u271d \u2208 image2 f' (image2 g' s u) (image2 h' t v)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\nx\u271d : \u03bd\n\u22a2 x\u271d \u2208 image2 f (image2 g s t) (image2 h u v) \u2192 x\u271d \u2208 image2 f' (image2 g' s u) (image2 h' t v)\n[PROOFSTEP]\nrintro \u27e8_, _, \u27e8a, b, ha, hb, rfl\u27e9, \u27e8c, d, hc, hd, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd\u271d d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\nc : \u03b3\nd : \u03b4\nhc : c \u2208 u\nhd : d \u2208 v\n\u22a2 f (g a b) (h c d) \u2208 image2 f' (image2 g' s u) (image2 h' t v)\n[PROOFSTEP]\nexact \u27e8_, _, \u27e8a, c, ha, hc, rfl\u27e9, \u27e8b, d, hb, hd, rfl\u27e9, (h_comm _ _ _ _).symm\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\nx\u271d : \u03bd\n\u22a2 x\u271d \u2208 image2 f' (image2 g' s u) (image2 h' t v) \u2192 x\u271d \u2208 image2 f (image2 g s t) (image2 h u v)\n[PROOFSTEP]\nrintro \u27e8_, _, \u27e8a, c, ha, hc, rfl\u27e9, \u27e8b, d, hb, hd, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd\u271d d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\na : \u03b1\nc : \u03b3\nha : a \u2208 s\nhc : c \u2208 u\nb : \u03b2\nd : \u03b4\nhb : b \u2208 t\nhd : d \u2208 v\n\u22a2 f' (g' a c) (h' b d) \u2208 image2 f (image2 g s t) (image2 h u v)\n[PROOFSTEP]\nexact \u27e8_, _, \u27e8a, b, ha, hb, rfl\u27e9, \u27e8c, d, hc, hd, rfl\u27e9, h_comm _ _ _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\ng : \u03b3 \u2192 \u03b4\nf' : \u03b1' \u2192 \u03b2' \u2192 \u03b4\ng\u2081 : \u03b1 \u2192 \u03b1'\ng\u2082 : \u03b2 \u2192 \u03b2'\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2), g (f a b) = f' (g\u2081 a) (g\u2082 b)\n\u22a2 g '' image2 f s t = image2 f' (g\u2081 '' s) (g\u2082 '' t)\n[PROOFSTEP]\nsimp_rw [image_image2, image2_image_left, image2_image_right, h_distrib]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\ng : \u03b3 \u2192 \u03b4\nf' : \u03b1' \u2192 \u03b2 \u2192 \u03b4\ng' : \u03b1 \u2192 \u03b1'\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2), g (f a b) = f' (g' a) b\n\u22a2 image2 f' ((fun a => g' a) '' s) ((fun b => b) '' t) = image2 f' (g' '' s) t\n[PROOFSTEP]\nrw [image_id']\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\ng : \u03b3 \u2192 \u03b4\nf' : \u03b1 \u2192 \u03b2' \u2192 \u03b4\ng' : \u03b2 \u2192 \u03b2'\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2), g (f a b) = f' a (g' b)\n\u22a2 image2 f' ((fun a => a) '' s) ((fun b => g' b) '' t) = image2 f' s (g' '' t)\n[PROOFSTEP]\nrw [image_id']\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b4 \u2192 \u03b5\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03b2'\nf\u2082 : \u03b1 \u2192 \u03b3 \u2192 \u03b3'\ng' : \u03b2' \u2192 \u03b3' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), f a (g b c) = g' (f\u2081 a b) (f\u2082 a c)\n\u22a2 image2 f s (image2 g t u) \u2286 image2 g' (image2 f\u2081 s t) (image2 f\u2082 s u)\n[PROOFSTEP]\nrintro _ \u27e8a, _, ha, \u27e8b, c, hb, hc, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b4 \u2192 \u03b5\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03b2'\nf\u2082 : \u03b1 \u2192 \u03b3 \u2192 \u03b3'\ng' : \u03b2' \u2192 \u03b3' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), f a (g b c) = g' (f\u2081 a b) (f\u2082 a c)\na : \u03b1\nha : a \u2208 s\nb : \u03b2\nc : \u03b3\nhb : b \u2208 t\nhc : c \u2208 u\n\u22a2 f a (g b c) \u2208 image2 g' (image2 f\u2081 s t) (image2 f\u2082 s u)\n[PROOFSTEP]\nrw [h_distrib]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b4 \u2192 \u03b5\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nf\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03b2'\nf\u2082 : \u03b1 \u2192 \u03b3 \u2192 \u03b3'\ng' : \u03b2' \u2192 \u03b3' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), f a (g b c) = g' (f\u2081 a b) (f\u2082 a c)\na : \u03b1\nha : a \u2208 s\nb : \u03b2\nc : \u03b3\nhb : b \u2208 t\nhc : c \u2208 u\n\u22a2 g' (f\u2081 a b) (f\u2082 a c) \u2208 image2 g' (image2 f\u2081 s t) (image2 f\u2082 s u)\n[PROOFSTEP]\nexact mem_image2_of_mem (mem_image2_of_mem ha hb) (mem_image2_of_mem ha hc)\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b4 \u2192 \u03b3 \u2192 \u03b5\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nf\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nf\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\ng' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), f (g a b) c = g' (f\u2081 a c) (f\u2082 b c)\n\u22a2 image2 f (image2 g s t) u \u2286 image2 g' (image2 f\u2081 s u) (image2 f\u2082 t u)\n[PROOFSTEP]\nrintro _ \u27e8_, c, \u27e8a, b, ha, hb, rfl\u27e9, hc, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nf : \u03b4 \u2192 \u03b3 \u2192 \u03b5\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nf\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nf\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\ng' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), f (g a b) c = g' (f\u2081 a c) (f\u2082 b c)\nc : \u03b3\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\nhc : c \u2208 u\n\u22a2 f (g a b) c \u2208 image2 g' (image2 f\u2081 s u) (image2 f\u2082 t u)\n[PROOFSTEP]\nrw [h_distrib]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd d' : \u03b4\nf : \u03b4 \u2192 \u03b3 \u2192 \u03b5\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nf\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nf\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\ng' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), f (g a b) c = g' (f\u2081 a c) (f\u2082 b c)\nc : \u03b3\na : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\nhc : c \u2208 u\n\u22a2 g' (f\u2081 a c) (f\u2082 b c) \u2208 image2 g' (image2 f\u2081 s u) (image2 f\u2082 t u)\n[PROOFSTEP]\nexact mem_image2_of_mem (mem_image2_of_mem ha hc) (mem_image2_of_mem hb hc)\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\ng : \u03b3 \u2192 \u03b4\nf' : \u03b2' \u2192 \u03b1' \u2192 \u03b4\ng\u2081 : \u03b2 \u2192 \u03b2'\ng\u2082 : \u03b1 \u2192 \u03b1'\nh_antidistrib : \u2200 (a : \u03b1) (b : \u03b2), g (f a b) = f' (g\u2081 b) (g\u2082 a)\n\u22a2 g '' image2 f s t = image2 f' (g\u2081 '' t) (g\u2082 '' s)\n[PROOFSTEP]\nrw [image2_swap f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\ng : \u03b3 \u2192 \u03b4\nf' : \u03b2' \u2192 \u03b1' \u2192 \u03b4\ng\u2081 : \u03b2 \u2192 \u03b2'\ng\u2082 : \u03b1 \u2192 \u03b1'\nh_antidistrib : \u2200 (a : \u03b1) (b : \u03b2), g (f a b) = f' (g\u2081 b) (g\u2082 a)\n\u22a2 g '' image2 (fun a b => f b a) t s = image2 f' (g\u2081 '' t) (g\u2082 '' s)\n[PROOFSTEP]\nexact image_image2_distrib fun _ _ => h_antidistrib _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\ng : \u03b3 \u2192 \u03b4\nf' : \u03b2' \u2192 \u03b1 \u2192 \u03b4\ng' : \u03b2 \u2192 \u03b2'\nh_antidistrib : \u2200 (a : \u03b1) (b : \u03b2), g (f a b) = f' (g' b) a\n\u22a2 image2 f' ((fun b => g' b) '' t) ((fun a => a) '' s) = image2 f' (g' '' t) s\n[PROOFSTEP]\nrw [image_id']\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\ng : \u03b3 \u2192 \u03b4\nf' : \u03b2 \u2192 \u03b1' \u2192 \u03b4\ng' : \u03b1 \u2192 \u03b1'\nh_antidistrib : \u2200 (a : \u03b1) (b : \u03b2), g (f a b) = f' b (g' a)\n\u22a2 image2 f' ((fun b => b) '' t) ((fun a => g' a) '' s) = image2 f' t (g' '' s)\n[PROOFSTEP]\nrw [image_id']\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt\u271d t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\na : \u03b1\nh : \u2200 (b : \u03b2), f a b = b\nt : Set \u03b2\n\u22a2 image2 f {a} t = t\n[PROOFSTEP]\nrw [image2_singleton_left, show f a = id from funext h, image_id]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\nb : \u03b2\nh : \u2200 (a : \u03b1), f a b = a\ns : Set \u03b1\n\u22a2 image2 f s {b} = s\n[PROOFSTEP]\nrw [image2_singleton_right, funext h, image_id']\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f (s \u2229 s') (t \u222a t') \u2286 image2 f s t \u222a image2 f s' t'\n[PROOFSTEP]\nrw [image2_union_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f (s \u2229 s') t \u222a image2 f (s \u2229 s') t' \u2286 image2 f s t \u222a image2 f s' t'\n[PROOFSTEP]\nexact union_subset_union (image2_subset_right <| inter_subset_left _ _) (image2_subset_right <| inter_subset_right _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f (s \u222a s') (t \u2229 t') \u2286 image2 f s t \u222a image2 f s' t'\n[PROOFSTEP]\nrw [image2_union_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f s (t \u2229 t') \u222a image2 f s' (t \u2229 t') \u2286 image2 f s t \u222a image2 f s' t'\n[PROOFSTEP]\nexact union_subset_union (image2_subset_left <| inter_subset_left _ _) (image2_subset_left <| inter_subset_right _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Set \u03b1\nt\u271d t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nhf : \u2200 (a b : \u03b1), f a b = f b a\n\u22a2 image2 f (s \u2229 t) (s \u222a t) \u2286 image2 f s t\n[PROOFSTEP]\nrw [inter_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Set \u03b1\nt\u271d t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nhf : \u2200 (a b : \u03b1), f a b = f b a\n\u22a2 image2 f (t \u2229 s) (s \u222a t) \u2286 image2 f s t\n[PROOFSTEP]\nexact image2_inter_union_subset_union.trans (union_subset (image2_comm hf).subset Subset.rfl)\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Set \u03b1\nt\u271d t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nhf : \u2200 (a b : \u03b1), f a b = f b a\n\u22a2 image2 f (s \u222a t) (s \u2229 t) \u2286 image2 f s t\n[PROOFSTEP]\nrw [image2_comm hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns\u271d s' : Set \u03b1\nt\u271d t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nhf : \u2200 (a b : \u03b1), f a b = f b a\n\u22a2 image2 (fun b a => f b a) (s \u2229 t) (s \u222a t) \u2286 image2 f s t\n[PROOFSTEP]\nexact image2_inter_union_subset hf\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.NAry", "llama_tokens": 40413, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850402140659, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.4393224841039215}} {"text": "[GOAL]\nM : Type u \u2192 Type v\ninst\u271d\u00b2 : Monad M\n\u03c9 \u03b1 \u03b2 : Type u\ninst\u271d\u00b9 : Monoid \u03c9\ninst\u271d : LawfulMonad M\n\u22a2 \u2200 {\u03b1 : Type u} (x : WriterT \u03c9 M \u03b1), id <$> x = x\n[PROOFSTEP]\nintros\n[GOAL]\nM : Type u \u2192 Type v\ninst\u271d\u00b2 : Monad M\n\u03c9 \u03b1 \u03b2 : Type u\ninst\u271d\u00b9 : Monoid \u03c9\ninst\u271d : LawfulMonad M\n\u03b1\u271d : Type u\nx\u271d : WriterT \u03c9 M \u03b1\u271d\n\u22a2 id <$> x\u271d = x\u271d\n[PROOFSTEP]\nsimp [Functor.map, WriterT.mk]\n[GOAL]\nM : Type u \u2192 Type v\ninst\u271d\u00b2 : Monad M\n\u03c9 \u03b1 \u03b2 : Type u\ninst\u271d\u00b9 : Monoid \u03c9\ninst\u271d : LawfulMonad M\n\u22a2 \u2200 {\u03b1 \u03b2 : Type u} (x : \u03b1) (f : \u03b1 \u2192 WriterT \u03c9 M \u03b2), pure x >>= f = f x\n[PROOFSTEP]\nintros\n[GOAL]\nM : Type u \u2192 Type v\ninst\u271d\u00b2 : Monad M\n\u03c9 \u03b1 \u03b2 : Type u\ninst\u271d\u00b9 : Monoid \u03c9\ninst\u271d : LawfulMonad M\n\u03b1\u271d \u03b2\u271d : Type u\nx\u271d : \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 WriterT \u03c9 M \u03b2\u271d\n\u22a2 pure x\u271d >>= f\u271d = f\u271d x\u271d\n[PROOFSTEP]\nsimp [Bind.bind, Pure.pure, WriterT.mk]\n[GOAL]\nM : Type u \u2192 Type v\ninst\u271d\u00b2 : Monad M\n\u03c9 \u03b1 \u03b2 : Type u\ninst\u271d\u00b9 : Monoid \u03c9\ninst\u271d : LawfulMonad M\n\u22a2 \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : WriterT \u03c9 M \u03b1) (f : \u03b1 \u2192 WriterT \u03c9 M \u03b2) (g : \u03b2 \u2192 WriterT \u03c9 M \u03b3),\n x >>= f >>= g = x >>= fun x => f x >>= g\n[PROOFSTEP]\nintros\n[GOAL]\nM : Type u \u2192 Type v\ninst\u271d\u00b2 : Monad M\n\u03c9 \u03b1 \u03b2 : Type u\ninst\u271d\u00b9 : Monoid \u03c9\ninst\u271d : LawfulMonad M\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : WriterT \u03c9 M \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 WriterT \u03c9 M \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 WriterT \u03c9 M \u03b3\u271d\n\u22a2 x\u271d >>= f\u271d >>= g\u271d = x\u271d >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\nsimp [Bind.bind, mul_assoc, WriterT.mk, \u2190 bind_pure_comp]\n[GOAL]\nM : Type u \u2192 Type v\ninst\u271d\u00b2 : Monad M\n\u03c9 \u03b1 \u03b2 : Type u\ninst\u271d\u00b9 : Monoid \u03c9\ninst\u271d : LawfulMonad M\n\u22a2 \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : WriterT \u03c9 M \u03b1),\n (do\n let y \u2190 x\n pure (f y)) =\n f <$> x\n[PROOFSTEP]\nintros\n[GOAL]\nM : Type u \u2192 Type v\ninst\u271d\u00b2 : Monad M\n\u03c9 \u03b1 \u03b2 : Type u\ninst\u271d\u00b9 : Monoid \u03c9\ninst\u271d : LawfulMonad M\n\u03b1\u271d \u03b2\u271d : Type u\nf\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : WriterT \u03c9 M \u03b1\u271d\n\u22a2 (do\n let y \u2190 x\u271d\n pure (f\u271d y)) =\n f\u271d <$> x\u271d\n[PROOFSTEP]\nsimp [Bind.bind, Functor.map, Pure.pure, WriterT.mk, bind_pure_comp]\n", "meta": {"mathlib_filename": "Mathlib.Control.Monad.Writer", "llama_tokens": 1088, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.43932247639970834}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nsrc\u271d : V \u2192\u2097[R] W := f.linear\n\u22a2 Continuous\n {\n toAddHom :=\n { toFun := \u2191f.linear,\n map_add' :=\n (_ :\n \u2200 (x y : V),\n AddHom.toFun src\u271d.toAddHom (x + y) = AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : V),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom.toFun\n[PROOFSTEP]\nrw [AffineMap.continuous_linear_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nsrc\u271d : V \u2192\u2097[R] W := f.linear\n\u22a2 Continuous \u2191f.toAffineMap\n[PROOFSTEP]\nexact f.cont\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\n\u22a2 \u2191(contLinear f) = f.linear\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nx\u271d : V\n\u22a2 \u2191\u2191(contLinear f) x\u271d = \u2191f.linear x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\n\u22a2 contLinear f = 0 \u2194 \u2203 q, f = const R P q\n[PROOFSTEP]\nhave h\u2081 : f.contLinear = 0 \u2194 (f : P \u2192\u1d43[R] Q).linear = 0 :=\n by\n refine' \u27e8fun h => _, fun h => _\u27e9 <;> ext\n \u00b7 rw [\u2190 coe_contLinear_eq_linear, h]; rfl\n \u00b7 rw [\u2190 coe_linear_eq_coe_contLinear, h]; rfl\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\n\u22a2 contLinear f = 0 \u2194 f.linear = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh : contLinear f = 0\n\u22a2 f.linear = 0\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh : f.linear = 0\n\u22a2 contLinear f = 0\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_1.h\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh : contLinear f = 0\nx\u271d : V\n\u22a2 \u2191f.linear x\u271d = \u21910 x\u271d\n[PROOFSTEP]\nrw [\u2190 coe_contLinear_eq_linear, h]\n[GOAL]\ncase refine'_1.h\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh : contLinear f = 0\nx\u271d : V\n\u22a2 \u2191\u21910 x\u271d = \u21910 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.h\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh : f.linear = 0\nx\u271d : V\n\u22a2 \u2191(contLinear f) x\u271d = \u21910 x\u271d\n[PROOFSTEP]\nrw [\u2190 coe_linear_eq_coe_contLinear, h]\n[GOAL]\ncase refine'_2.h\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh : f.linear = 0\nx\u271d : V\n\u22a2 \u21910 x\u271d = \u21910 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh\u2081 : contLinear f = 0 \u2194 f.linear = 0\n\u22a2 contLinear f = 0 \u2194 \u2203 q, f = const R P q\n[PROOFSTEP]\nhave h\u2082 : \u2200 q : Q, f = const R P q \u2194 (f : P \u2192\u1d43[R] Q) = AffineMap.const R P q :=\n by\n intro q\n refine' \u27e8fun h => _, fun h => _\u27e9 <;> ext\n \u00b7 rw [h]; rfl\n \u00b7 rw [\u2190 coe_to_affineMap, h]; rfl\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh\u2081 : contLinear f = 0 \u2194 f.linear = 0\n\u22a2 \u2200 (q : Q), f = const R P q \u2194 f.toAffineMap = AffineMap.const R P q\n[PROOFSTEP]\nintro q\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh\u2081 : contLinear f = 0 \u2194 f.linear = 0\nq : Q\n\u22a2 f = const R P q \u2194 f.toAffineMap = AffineMap.const R P q\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh\u2081 : contLinear f = 0 \u2194 f.linear = 0\nq : Q\nh : f = const R P q\n\u22a2 f.toAffineMap = AffineMap.const R P q\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh\u2081 : contLinear f = 0 \u2194 f.linear = 0\nq : Q\nh : f.toAffineMap = AffineMap.const R P q\n\u22a2 f = const R P q\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_1.h\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh\u2081 : contLinear f = 0 \u2194 f.linear = 0\nq : Q\nh : f = const R P q\np\u271d : P\n\u22a2 \u2191f.toAffineMap p\u271d = \u2191(AffineMap.const R P q) p\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase refine'_1.h\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh\u2081 : contLinear f = 0 \u2194 f.linear = 0\nq : Q\nh : f = const R P q\np\u271d : P\n\u22a2 \u2191(const R P q).toAffineMap p\u271d = \u2191(AffineMap.const R P q) p\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.h\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh\u2081 : contLinear f = 0 \u2194 f.linear = 0\nq : Q\nh : f.toAffineMap = AffineMap.const R P q\nx\u271d : P\n\u22a2 \u2191f x\u271d = \u2191(const R P q) x\u271d\n[PROOFSTEP]\nrw [\u2190 coe_to_affineMap, h]\n[GOAL]\ncase refine'_2.h\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh\u2081 : contLinear f = 0 \u2194 f.linear = 0\nq : Q\nh : f.toAffineMap = AffineMap.const R P q\nx\u271d : P\n\u22a2 \u2191(AffineMap.const R P q) x\u271d = \u2191(const R P q) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh\u2081 : contLinear f = 0 \u2194 f.linear = 0\nh\u2082 : \u2200 (q : Q), f = const R P q \u2194 f.toAffineMap = AffineMap.const R P q\n\u22a2 contLinear f = 0 \u2194 \u2203 q, f = const R P q\n[PROOFSTEP]\nsimp_rw [h\u2081, h\u2082]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nh\u2081 : contLinear f = 0 \u2194 f.linear = 0\nh\u2082 : \u2200 (q : Q), f = const R P q \u2194 f.toAffineMap = AffineMap.const R P q\n\u22a2 f.linear = 0 \u2194 \u2203 q, f.toAffineMap = AffineMap.const R P q\n[PROOFSTEP]\nexact (f : P \u2192\u1d43[R] Q).linear_eq_zero_iff_exists_const\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192L[R] W\n\u22a2 contLinear (ContinuousLinearMap.toContinuousAffineMap f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192L[R] W\nx\u271d : V\n\u22a2 \u2191(contLinear (ContinuousLinearMap.toContinuousAffineMap f)) x\u271d = \u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[R] W\n\u22a2 \u2191f = \u2191(contLinear f) + Function.const V (\u2191f 0)\n[PROOFSTEP]\nrcases f with \u27e8f, h\u27e9\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192\u1d43[R] W\nh : Continuous f.toFun\n\u22a2 \u2191{ toAffineMap := f, cont := h } =\n \u2191(contLinear { toAffineMap := f, cont := h }) + Function.const V (\u2191{ toAffineMap := f, cont := h } 0)\n[PROOFSTEP]\nrw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add, \u2190 Function.const_def]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\nh : \u2191f 0 = 0\n\u22a2 \u2016f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016\n[PROOFSTEP]\nrw [norm_def]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\nh : \u2191f 0 = 0\n\u22a2 max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016 = max 0 \u2016contLinear f\u2016\n[PROOFSTEP]\nrw [h, norm_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\n\u22a2 (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) 0 = 0\n[PROOFSTEP]\nsimp [(ContinuousAffineMap.zero_apply)]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d f g : V \u2192A[\ud835\udd5c] W\n\u22a2 (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g\n[PROOFSTEP]\nsimp only [coe_add, max_le_iff]\n -- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`\n -- suffices, but now they don't fire.\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d f g : V \u2192A[\ud835\udd5c] W\n\u22a2 \u2016(\u2191f + \u2191g) 0\u2016 \u2264 max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016 + max \u2016\u2191g 0\u2016 \u2016contLinear g\u2016 \u2227\n \u2016contLinear (f + g)\u2016 \u2264 max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016 + max \u2016\u2191g 0\u2016 \u2016contLinear g\u2016\n[PROOFSTEP]\nrw [add_contLinear]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d f g : V \u2192A[\ud835\udd5c] W\n\u22a2 \u2016(\u2191f + \u2191g) 0\u2016 \u2264 max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016 + max \u2016\u2191g 0\u2016 \u2016contLinear g\u2016 \u2227\n \u2016contLinear f + contLinear g\u2016 \u2264 max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016 + max \u2016\u2191g 0\u2016 \u2016contLinear g\u2016\n[PROOFSTEP]\nexact\n \u27e8(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),\n (norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d f : V \u2192A[\ud835\udd5c] W\n\u22a2 (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (-f) = (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f\n[PROOFSTEP]\nsimp [(ContinuousAffineMap.neg_apply)]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d f : V \u2192A[\ud835\udd5c] W\nh\u2080 :\n AddGroupSeminorm.toFun\n { toFun := fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016, map_zero' := (_ : max \u2016\u21910 0\u2016 \u20160\u2016 = 0),\n add_le' :=\n (_ :\n \u2200 (f g : V \u2192A[\ud835\udd5c] W),\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g),\n neg' := (_ : \u2200 (f : V \u2192A[\ud835\udd5c] W), max \u2016\u2191(-f) 0\u2016 \u2016-contLinear f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) }\n f =\n 0\n\u22a2 f = 0\n[PROOFSTEP]\nrcases max_eq_iff.mp h\u2080 with (\u27e8h\u2081, h\u2082\u27e9 | \u27e8h\u2081, h\u2082\u27e9)\n[GOAL]\ncase inl.intro\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d f : V \u2192A[\ud835\udd5c] W\nh\u2080 :\n AddGroupSeminorm.toFun\n { toFun := fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016, map_zero' := (_ : max \u2016\u21910 0\u2016 \u20160\u2016 = 0),\n add_le' :=\n (_ :\n \u2200 (f g : V \u2192A[\ud835\udd5c] W),\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g),\n neg' := (_ : \u2200 (f : V \u2192A[\ud835\udd5c] W), max \u2016\u2191(-f) 0\u2016 \u2016-contLinear f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) }\n f =\n 0\nh\u2081 : \u2016\u2191f 0\u2016 = 0\nh\u2082 : \u2016contLinear f\u2016 \u2264 \u2016\u2191f 0\u2016\n\u22a2 f = 0\n[PROOFSTEP]\nrw [h\u2081] at h\u2082 \n[GOAL]\ncase inr.intro\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d f : V \u2192A[\ud835\udd5c] W\nh\u2080 :\n AddGroupSeminorm.toFun\n { toFun := fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016, map_zero' := (_ : max \u2016\u21910 0\u2016 \u20160\u2016 = 0),\n add_le' :=\n (_ :\n \u2200 (f g : V \u2192A[\ud835\udd5c] W),\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g),\n neg' := (_ : \u2200 (f : V \u2192A[\ud835\udd5c] W), max \u2016\u2191(-f) 0\u2016 \u2016-contLinear f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) }\n f =\n 0\nh\u2081 : \u2016contLinear f\u2016 = 0\nh\u2082 : \u2016\u2191f 0\u2016 \u2264 \u2016contLinear f\u2016\n\u22a2 f = 0\n[PROOFSTEP]\nrw [h\u2081] at h\u2082 \n[GOAL]\ncase inl.intro\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d f : V \u2192A[\ud835\udd5c] W\nh\u2080 :\n AddGroupSeminorm.toFun\n { toFun := fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016, map_zero' := (_ : max \u2016\u21910 0\u2016 \u20160\u2016 = 0),\n add_le' :=\n (_ :\n \u2200 (f g : V \u2192A[\ud835\udd5c] W),\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g),\n neg' := (_ : \u2200 (f : V \u2192A[\ud835\udd5c] W), max \u2016\u2191(-f) 0\u2016 \u2016-contLinear f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) }\n f =\n 0\nh\u2081 : \u2016\u2191f 0\u2016 = 0\nh\u2082 : \u2016contLinear f\u2016 \u2264 0\n\u22a2 f = 0\n[PROOFSTEP]\nrw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h\u2082 \n[GOAL]\ncase inl.intro\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d f : V \u2192A[\ud835\udd5c] W\nh\u2080 :\n AddGroupSeminorm.toFun\n { toFun := fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016, map_zero' := (_ : max \u2016\u21910 0\u2016 \u20160\u2016 = 0),\n add_le' :=\n (_ :\n \u2200 (f g : V \u2192A[\ud835\udd5c] W),\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g),\n neg' := (_ : \u2200 (f : V \u2192A[\ud835\udd5c] W), max \u2016\u2191(-f) 0\u2016 \u2016-contLinear f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) }\n f =\n 0\nh\u2081 : \u2016\u2191f 0\u2016 = 0\nh\u2082 : \u2203 q, f = const \ud835\udd5c V q\n\u22a2 f = 0\n[PROOFSTEP]\nobtain \u27e8q, rfl\u27e9 := h\u2082\n[GOAL]\ncase inl.intro.intro\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\nq : W\nh\u2080 :\n AddGroupSeminorm.toFun\n { toFun := fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016, map_zero' := (_ : max \u2016\u21910 0\u2016 \u20160\u2016 = 0),\n add_le' :=\n (_ :\n \u2200 (f g : V \u2192A[\ud835\udd5c] W),\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g),\n neg' := (_ : \u2200 (f : V \u2192A[\ud835\udd5c] W), max \u2016\u2191(-f) 0\u2016 \u2016-contLinear f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) }\n (const \ud835\udd5c V q) =\n 0\nh\u2081 : \u2016\u2191(const \ud835\udd5c V q) 0\u2016 = 0\n\u22a2 const \ud835\udd5c V q = 0\n[PROOFSTEP]\nsimp only [norm_eq_zero] at h\u2081 \n[GOAL]\ncase inl.intro.intro\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\nq : W\nh\u2080 :\n AddGroupSeminorm.toFun\n { toFun := fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016, map_zero' := (_ : max \u2016\u21910 0\u2016 \u20160\u2016 = 0),\n add_le' :=\n (_ :\n \u2200 (f g : V \u2192A[\ud835\udd5c] W),\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g),\n neg' := (_ : \u2200 (f : V \u2192A[\ud835\udd5c] W), max \u2016\u2191(-f) 0\u2016 \u2016-contLinear f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) }\n (const \ud835\udd5c V q) =\n 0\nh\u2081 : \u2191(const \ud835\udd5c V q) 0 = 0\n\u22a2 const \ud835\udd5c V q = 0\n[PROOFSTEP]\nrw [coe_const, Function.const_apply] at h\u2081 \n[GOAL]\ncase inl.intro.intro\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\nq : W\nh\u2080 :\n AddGroupSeminorm.toFun\n { toFun := fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016, map_zero' := (_ : max \u2016\u21910 0\u2016 \u20160\u2016 = 0),\n add_le' :=\n (_ :\n \u2200 (f g : V \u2192A[\ud835\udd5c] W),\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g),\n neg' := (_ : \u2200 (f : V \u2192A[\ud835\udd5c] W), max \u2016\u2191(-f) 0\u2016 \u2016-contLinear f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) }\n (const \ud835\udd5c V q) =\n 0\nh\u2081 : q = 0\n\u22a2 const \ud835\udd5c V q = 0\n[PROOFSTEP]\nrw [h\u2081]\n[GOAL]\ncase inl.intro.intro\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\nq : W\nh\u2080 :\n AddGroupSeminorm.toFun\n { toFun := fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016, map_zero' := (_ : max \u2016\u21910 0\u2016 \u20160\u2016 = 0),\n add_le' :=\n (_ :\n \u2200 (f g : V \u2192A[\ud835\udd5c] W),\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g),\n neg' := (_ : \u2200 (f : V \u2192A[\ud835\udd5c] W), max \u2016\u2191(-f) 0\u2016 \u2016-contLinear f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) }\n (const \ud835\udd5c V q) =\n 0\nh\u2081 : q = 0\n\u22a2 const \ud835\udd5c V 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr.intro\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d f : V \u2192A[\ud835\udd5c] W\nh\u2080 :\n AddGroupSeminorm.toFun\n { toFun := fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016, map_zero' := (_ : max \u2016\u21910 0\u2016 \u20160\u2016 = 0),\n add_le' :=\n (_ :\n \u2200 (f g : V \u2192A[\ud835\udd5c] W),\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g),\n neg' := (_ : \u2200 (f : V \u2192A[\ud835\udd5c] W), max \u2016\u2191(-f) 0\u2016 \u2016-contLinear f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) }\n f =\n 0\nh\u2081 : \u2016contLinear f\u2016 = 0\nh\u2082 : \u2016\u2191f 0\u2016 \u2264 0\n\u22a2 f = 0\n[PROOFSTEP]\nrw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h\u2081 \n[GOAL]\ncase inr.intro\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d f : V \u2192A[\ud835\udd5c] W\nh\u2080 :\n AddGroupSeminorm.toFun\n { toFun := fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016, map_zero' := (_ : max \u2016\u21910 0\u2016 \u20160\u2016 = 0),\n add_le' :=\n (_ :\n \u2200 (f g : V \u2192A[\ud835\udd5c] W),\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g),\n neg' := (_ : \u2200 (f : V \u2192A[\ud835\udd5c] W), max \u2016\u2191(-f) 0\u2016 \u2016-contLinear f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) }\n f =\n 0\nh\u2081 : \u2203 q, f = const \ud835\udd5c V q\nh\u2082 : \u2016\u2191f 0\u2016 \u2264 0\n\u22a2 f = 0\n[PROOFSTEP]\nobtain \u27e8q, rfl\u27e9 := h\u2081\n[GOAL]\ncase inr.intro.intro\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\nq : W\nh\u2080 :\n AddGroupSeminorm.toFun\n { toFun := fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016, map_zero' := (_ : max \u2016\u21910 0\u2016 \u20160\u2016 = 0),\n add_le' :=\n (_ :\n \u2200 (f g : V \u2192A[\ud835\udd5c] W),\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g),\n neg' := (_ : \u2200 (f : V \u2192A[\ud835\udd5c] W), max \u2016\u2191(-f) 0\u2016 \u2016-contLinear f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) }\n (const \ud835\udd5c V q) =\n 0\nh\u2082 : \u2016\u2191(const \ud835\udd5c V q) 0\u2016 \u2264 0\n\u22a2 const \ud835\udd5c V q = 0\n[PROOFSTEP]\nsimp only [norm_le_zero_iff] at h\u2082 \n[GOAL]\ncase inr.intro.intro\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\nq : W\nh\u2080 :\n AddGroupSeminorm.toFun\n { toFun := fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016, map_zero' := (_ : max \u2016\u21910 0\u2016 \u20160\u2016 = 0),\n add_le' :=\n (_ :\n \u2200 (f g : V \u2192A[\ud835\udd5c] W),\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g),\n neg' := (_ : \u2200 (f : V \u2192A[\ud835\udd5c] W), max \u2016\u2191(-f) 0\u2016 \u2016-contLinear f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) }\n (const \ud835\udd5c V q) =\n 0\nh\u2082 : \u2191(const \ud835\udd5c V q) 0 = 0\n\u22a2 const \ud835\udd5c V q = 0\n[PROOFSTEP]\nrw [coe_const, Function.const_apply] at h\u2082 \n[GOAL]\ncase inr.intro.intro\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\nq : W\nh\u2080 :\n AddGroupSeminorm.toFun\n { toFun := fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016, map_zero' := (_ : max \u2016\u21910 0\u2016 \u20160\u2016 = 0),\n add_le' :=\n (_ :\n \u2200 (f g : V \u2192A[\ud835\udd5c] W),\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g),\n neg' := (_ : \u2200 (f : V \u2192A[\ud835\udd5c] W), max \u2016\u2191(-f) 0\u2016 \u2016-contLinear f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) }\n (const \ud835\udd5c V q) =\n 0\nh\u2082 : q = 0\n\u22a2 const \ud835\udd5c V q = 0\n[PROOFSTEP]\nrw [h\u2082]\n[GOAL]\ncase inr.intro.intro\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\nq : W\nh\u2080 :\n AddGroupSeminorm.toFun\n { toFun := fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016, map_zero' := (_ : max \u2016\u21910 0\u2016 \u20160\u2016 = 0),\n add_le' :=\n (_ :\n \u2200 (f g : V \u2192A[\ud835\udd5c] W),\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) (f + g) \u2264\n (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) f + (fun f => max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) g),\n neg' := (_ : \u2200 (f : V \u2192A[\ud835\udd5c] W), max \u2016\u2191(-f) 0\u2016 \u2016-contLinear f\u2016 = max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016) }\n (const \ud835\udd5c V q) =\n 0\nh\u2082 : q = 0\n\u22a2 const \ud835\udd5c V 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d : V \u2192A[\ud835\udd5c] W\nt : \ud835\udd5c\nf : V \u2192A[\ud835\udd5c] W\n\u22a2 \u2016t \u2022 f\u2016 \u2264 \u2016t\u2016 * \u2016f\u2016\n[PROOFSTEP]\nsimp only [norm_def, (smul_contLinear), norm_smul]\n -- Porting note: previously all these rewrites were in the `simp only`,\n -- but now they don't fire.\n -- (in fact, `norm_smul` fires, but only once rather than twice!)\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d : V \u2192A[\ud835\udd5c] W\nt : \ud835\udd5c\nf : V \u2192A[\ud835\udd5c] W\n\u22a2 max \u2016\u2191(t \u2022 f) 0\u2016 (\u2016t\u2016 * \u2016contLinear f\u2016) \u2264 \u2016t\u2016 * max \u2016\u2191f 0\u2016 \u2016contLinear f\u2016\n[PROOFSTEP]\nrw [coe_smul, Pi.smul_apply, norm_smul, \u2190 mul_max_of_nonneg _ _ (norm_nonneg t)]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\ng : W\u2082 \u2192A[\ud835\udd5c] V\n\u22a2 \u2016comp f g\u2016 \u2264 \u2016f\u2016 * \u2016g\u2016 + \u2016\u2191f 0\u2016\n[PROOFSTEP]\nrw [norm_def, max_le_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\ng : W\u2082 \u2192A[\ud835\udd5c] V\n\u22a2 \u2016\u2191(comp f g) 0\u2016 \u2264 \u2016f\u2016 * \u2016g\u2016 + \u2016\u2191f 0\u2016 \u2227 \u2016contLinear (comp f g)\u2016 \u2264 \u2016f\u2016 * \u2016g\u2016 + \u2016\u2191f 0\u2016\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\ng : W\u2082 \u2192A[\ud835\udd5c] V\n\u22a2 \u2016\u2191(comp f g) 0\u2016 \u2264 \u2016f\u2016 * \u2016g\u2016 + \u2016\u2191f 0\u2016\n[PROOFSTEP]\ncalc\n \u2016f.comp g 0\u2016 = \u2016f (g 0)\u2016 := by simp\n _ = \u2016f.contLinear (g 0) + f 0\u2016 := by rw [f.decomp]; simp\n _ \u2264 \u2016f.contLinear\u2016 * \u2016g 0\u2016 + \u2016f 0\u2016 := ((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))\n _ \u2264 \u2016f\u2016 * \u2016g\u2016 + \u2016f 0\u2016 :=\n add_le_add_right (mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\ng : W\u2082 \u2192A[\ud835\udd5c] V\n\u22a2 \u2016\u2191(comp f g) 0\u2016 = \u2016\u2191f (\u2191g 0)\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\ng : W\u2082 \u2192A[\ud835\udd5c] V\n\u22a2 \u2016\u2191f (\u2191g 0)\u2016 = \u2016\u2191(contLinear f) (\u2191g 0) + \u2191f 0\u2016\n[PROOFSTEP]\nrw [f.decomp]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\ng : W\u2082 \u2192A[\ud835\udd5c] V\n\u22a2 \u2016(\u2191(contLinear f) + Function.const V (\u2191f 0)) (\u2191g 0)\u2016 =\n \u2016\u2191(contLinear f) (\u2191g 0) + (\u2191(contLinear f) + Function.const V (\u2191f 0)) 0\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\ng : W\u2082 \u2192A[\ud835\udd5c] V\n\u22a2 \u2016contLinear (comp f g)\u2016 \u2264 \u2016f\u2016 * \u2016g\u2016 + \u2016\u2191f 0\u2016\n[PROOFSTEP]\ncalc\n \u2016(f.comp g).contLinear\u2016 \u2264 \u2016f.contLinear\u2016 * \u2016g.contLinear\u2016 :=\n (g.comp_contLinear f).symm \u25b8 f.contLinear.op_norm_comp_le _\n _ \u2264 \u2016f\u2016 * \u2016g\u2016 := (mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))\n _ \u2264 \u2016f\u2016 * \u2016g\u2016 + \u2016f 0\u2016 := by rw [le_add_iff_nonneg_right]; apply norm_nonneg\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\ng : W\u2082 \u2192A[\ud835\udd5c] V\n\u22a2 \u2016f\u2016 * \u2016g\u2016 \u2264 \u2016f\u2016 * \u2016g\u2016 + \u2016\u2191f 0\u2016\n[PROOFSTEP]\nrw [le_add_iff_nonneg_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\ng : W\u2082 \u2192A[\ud835\udd5c] V\n\u22a2 0 \u2264 \u2016\u2191f 0\u2016\n[PROOFSTEP]\napply norm_nonneg\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d f : V \u2192A[\ud835\udd5c] W\n\u22a2 (fun p => ContinuousLinearMap.toContinuousAffineMap p.snd + const \ud835\udd5c V p.fst)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_1 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_1) = (fun f => (\u2191f 0, contLinear f)) (x + x_1)) },\n map_smul' :=\n (_ :\n \u2200 (x : \ud835\udd5c) (x_1 : V \u2192A[\ud835\udd5c] W),\n AddHom.toFun\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_2 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_2) = (fun f => (\u2191f 0, contLinear f)) (x + x_2)) }\n (x \u2022 x_1) =\n AddHom.toFun\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_2 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_2) = (fun f => (\u2191f 0, contLinear f)) (x + x_2)) }\n (x \u2022 x_1)) }.toAddHom\n f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d f : V \u2192A[\ud835\udd5c] W\nx\u271d : V\n\u22a2 \u2191((fun p => ContinuousLinearMap.toContinuousAffineMap p.snd + const \ud835\udd5c V p.fst)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_1 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_1) = (fun f => (\u2191f 0, contLinear f)) (x + x_1)) },\n map_smul' :=\n (_ :\n \u2200 (x : \ud835\udd5c) (x_1 : V \u2192A[\ud835\udd5c] W),\n AddHom.toFun\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_2 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_2) =\n (fun f => (\u2191f 0, contLinear f)) (x + x_2)) }\n (x \u2022 x_1) =\n AddHom.toFun\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_2 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_2) =\n (fun f => (\u2191f 0, contLinear f)) (x + x_2)) }\n (x \u2022 x_1)) }.toAddHom\n f))\n x\u271d =\n \u2191f x\u271d\n[PROOFSTEP]\nrw [f.decomp]\n -- Porting note: previously `simp` closed the goal, but now we need to rewrite:\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d f : V \u2192A[\ud835\udd5c] W\nx\u271d : V\n\u22a2 \u2191((fun p => ContinuousLinearMap.toContinuousAffineMap p.snd + const \ud835\udd5c V p.fst)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_1 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_1) = (fun f => (\u2191f 0, contLinear f)) (x + x_1)) },\n map_smul' :=\n (_ :\n \u2200 (x : \ud835\udd5c) (x_1 : V \u2192A[\ud835\udd5c] W),\n AddHom.toFun\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_2 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_2) =\n (fun f => (\u2191f 0, contLinear f)) (x + x_2)) }\n (x \u2022 x_1) =\n AddHom.toFun\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_2 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_2) =\n (fun f => (\u2191f 0, contLinear f)) (x + x_2)) }\n (x \u2022 x_1)) }.toAddHom\n f))\n x\u271d =\n (\u2191(contLinear f) + Function.const V (\u2191f 0)) x\u271d\n[PROOFSTEP]\nsimp only [coe_add, ContinuousLinearMap.coe_toContinuousAffineMap, Pi.add_apply]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d f : V \u2192A[\ud835\udd5c] W\nx\u271d : V\n\u22a2 \u2191(contLinear f) x\u271d + \u2191(const \ud835\udd5c V (\u2191f 0)) x\u271d = \u2191(contLinear f) x\u271d + Function.const V (\u2191f 0) x\u271d\n[PROOFSTEP]\nrw [ContinuousAffineMap.coe_const, Function.const_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : V \u2192A[\ud835\udd5c] W\n\u22a2 Function.RightInverse (fun p => ContinuousLinearMap.toContinuousAffineMap p.snd + const \ud835\udd5c V p.fst)\n {\n toAddHom :=\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_1 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_1) = (fun f => (\u2191f 0, contLinear f)) (x + x_1)) },\n map_smul' :=\n (_ :\n \u2200 (x : \ud835\udd5c) (x_1 : V \u2192A[\ud835\udd5c] W),\n AddHom.toFun\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_2 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_2) = (fun f => (\u2191f 0, contLinear f)) (x + x_2)) }\n (x \u2022 x_1) =\n AddHom.toFun\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_2 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_2) = (fun f => (\u2191f 0, contLinear f)) (x + x_2)) }\n (x \u2022 x_1)) }.toAddHom.toFun\n[PROOFSTEP]\nrintro \u27e8v, f\u27e9\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d : V \u2192A[\ud835\udd5c] W\nv : W\nf : V \u2192L[\ud835\udd5c] W\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_1 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_1) = (fun f => (\u2191f 0, contLinear f)) (x + x_1)) },\n map_smul' :=\n (_ :\n \u2200 (x : \ud835\udd5c) (x_1 : V \u2192A[\ud835\udd5c] W),\n AddHom.toFun\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_2 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_2) = (fun f => (\u2191f 0, contLinear f)) (x + x_2)) }\n (x \u2022 x_1) =\n AddHom.toFun\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_2 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_2) = (fun f => (\u2191f 0, contLinear f)) (x + x_2)) }\n (x \u2022 x_1)) }.toAddHom\n ((fun p => ContinuousLinearMap.toContinuousAffineMap p.snd + const \ud835\udd5c V p.fst) (v, f)) =\n (v, f)\n[PROOFSTEP]\next\n[GOAL]\ncase mk.h\u2081\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d : V \u2192A[\ud835\udd5c] W\nv : W\nf : V \u2192L[\ud835\udd5c] W\n\u22a2 (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_1 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_1) = (fun f => (\u2191f 0, contLinear f)) (x + x_1)) },\n map_smul' :=\n (_ :\n \u2200 (x : \ud835\udd5c) (x_1 : V \u2192A[\ud835\udd5c] W),\n AddHom.toFun\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_2 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_2) = (fun f => (\u2191f 0, contLinear f)) (x + x_2)) }\n (x \u2022 x_1) =\n AddHom.toFun\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_2 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_2) = (fun f => (\u2191f 0, contLinear f)) (x + x_2)) }\n (x \u2022 x_1)) }.toAddHom\n ((fun p => ContinuousLinearMap.toContinuousAffineMap p.snd + const \ud835\udd5c V p.fst) (v, f))).fst =\n (v, f).fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.h\u2082.h\n\ud835\udd5c : Type u_1\nR : Type u_2\nV : Type u_3\nW : Type u_4\nW\u2082 : Type u_5\nP : Type u_6\nQ : Type u_7\nQ\u2082 : Type u_8\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf\u271d : V \u2192A[\ud835\udd5c] W\nv : W\nf : V \u2192L[\ud835\udd5c] W\nx\u271d : V\n\u22a2 \u2191(AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_1 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_1) = (fun f => (\u2191f 0, contLinear f)) (x + x_1)) },\n map_smul' :=\n (_ :\n \u2200 (x : \ud835\udd5c) (x_1 : V \u2192A[\ud835\udd5c] W),\n AddHom.toFun\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_2 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_2) =\n (fun f => (\u2191f 0, contLinear f)) (x + x_2)) }\n (x \u2022 x_1) =\n AddHom.toFun\n { toFun := fun f => (\u2191f 0, contLinear f),\n map_add' :=\n (_ :\n \u2200 (x x_2 : V \u2192A[\ud835\udd5c] W),\n (fun f => (\u2191f 0, contLinear f)) (x + x_2) =\n (fun f => (\u2191f 0, contLinear f)) (x + x_2)) }\n (x \u2022 x_1)) }.toAddHom\n ((fun p => ContinuousLinearMap.toContinuousAffineMap p.snd + const \ud835\udd5c V p.fst) (v, f))).snd\n x\u271d =\n \u2191(v, f).snd x\u271d\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.ContinuousAffineMap", "llama_tokens": 32196, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506526772884, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.4392722264466967}} {"text": "[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\n\u22a2 Disjoint f g \u2192 Disjoint g f\n[PROOFSTEP]\nsimp only [Disjoint, or_comm, imp_self]\n[GOAL]\n\u03b1 : Type u_1\nf g h\u271d : Perm \u03b1\nh : Disjoint f g\nx : \u03b1\nhf : \u2191f x = x\nhg : \u2191f (\u2191g x) = \u2191g x\n\u22a2 \u2191(f * g) x = \u2191(g * f) x\n[PROOFSTEP]\nsimp [mul_apply, hf, hg]\n[GOAL]\n\u03b1 : Type u_1\nf g h\u271d : Perm \u03b1\nh : Disjoint f g\nx : \u03b1\nhf : \u2191f x = x\nhg : \u2191g (\u2191g x) = \u2191g x\n\u22a2 \u2191(f * g) x = \u2191(g * f) x\n[PROOFSTEP]\nsimp [mul_apply, hf, g.injective hg]\n[GOAL]\n\u03b1 : Type u_1\nf g h\u271d : Perm \u03b1\nh : Disjoint f g\nx : \u03b1\nhg : \u2191g x = x\nhf : \u2191f (\u2191f x) = \u2191f x\n\u22a2 \u2191(f * g) x = \u2191(g * f) x\n[PROOFSTEP]\nsimp [mul_apply, f.injective hf, hg]\n[GOAL]\n\u03b1 : Type u_1\nf g h\u271d : Perm \u03b1\nh : Disjoint f g\nx : \u03b1\nhg : \u2191g x = x\nhf : \u2191g (\u2191f x) = \u2191f x\n\u22a2 \u2191(f * g) x = \u2191(g * f) x\n[PROOFSTEP]\nsimp [mul_apply, hf, hg]\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\n\u22a2 Disjoint f f \u2194 f = 1\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => h.symm \u25b8 disjoint_one_left 1\u27e9\n[GOAL]\n\u03b1 : Type u_1\nf g h\u271d : Perm \u03b1\nh : Disjoint f f\n\u22a2 f = 1\n[PROOFSTEP]\next x\n[GOAL]\ncase H\n\u03b1 : Type u_1\nf g h\u271d : Perm \u03b1\nh : Disjoint f f\nx : \u03b1\n\u22a2 \u2191f x = \u21911 x\n[PROOFSTEP]\ncases' h x with hx hx\n[GOAL]\ncase H.inl\n\u03b1 : Type u_1\nf g h\u271d : Perm \u03b1\nh : Disjoint f f\nx : \u03b1\nhx : \u2191f x = x\n\u22a2 \u2191f x = \u21911 x\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase H.inr\n\u03b1 : Type u_1\nf g h\u271d : Perm \u03b1\nh : Disjoint f f\nx : \u03b1\nhx : \u2191f x = x\n\u22a2 \u2191f x = \u21911 x\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\n\u03b1 : Type u_1\nf g h\u271d : Perm \u03b1\nh : Disjoint f g\n\u22a2 Disjoint f\u207b\u00b9 g\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\nf g h\u271d : Perm \u03b1\nh : Disjoint f g\nx : \u03b1\n\u22a2 \u2191f\u207b\u00b9 x = x \u2228 \u2191g x = x\n[PROOFSTEP]\nrw [inv_eq_iff_eq, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\nf g h\u271d : Perm \u03b1\nh : Disjoint f g\nx : \u03b1\n\u22a2 \u2191f x = x \u2228 \u2191g x = x\n[PROOFSTEP]\nexact h x\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\n\u22a2 Disjoint f\u207b\u00b9 g \u2194 Disjoint f g\n[PROOFSTEP]\nrefine' \u27e8fun h => _, Disjoint.inv_left\u27e9\n[GOAL]\n\u03b1 : Type u_1\nf g h\u271d : Perm \u03b1\nh : Disjoint f\u207b\u00b9 g\n\u22a2 Disjoint f g\n[PROOFSTEP]\nconvert h.inv_left\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\n\u22a2 Disjoint f g\u207b\u00b9 \u2194 Disjoint f g\n[PROOFSTEP]\nrw [disjoint_comm, disjoint_inv_left_iff, disjoint_comm]\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nH1 : Disjoint f h\nH2 : Disjoint g h\nx : \u03b1\n\u22a2 \u2191(f * g) x = x \u2228 \u2191h x = x\n[PROOFSTEP]\ncases H1 x\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nH1 : Disjoint f h\nH2 : Disjoint g h\nx : \u03b1\nh\u271d : \u2191f x = x\n\u22a2 \u2191(f * g) x = x \u2228 \u2191h x = x\n[PROOFSTEP]\ncases H2 x\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nH1 : Disjoint f h\nH2 : Disjoint g h\nx : \u03b1\nh\u271d : \u2191h x = x\n\u22a2 \u2191(f * g) x = x \u2228 \u2191h x = x\n[PROOFSTEP]\ncases H2 x\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nH1 : Disjoint f h\nH2 : Disjoint g h\nx : \u03b1\nh\u271d\u00b9 : \u2191f x = x\nh\u271d : \u2191g x = x\n\u22a2 \u2191(f * g) x = x \u2228 \u2191h x = x\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nH1 : Disjoint f h\nH2 : Disjoint g h\nx : \u03b1\nh\u271d\u00b9 : \u2191f x = x\nh\u271d : \u2191h x = x\n\u22a2 \u2191(f * g) x = x \u2228 \u2191h x = x\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nH1 : Disjoint f h\nH2 : Disjoint g h\nx : \u03b1\nh\u271d\u00b9 : \u2191h x = x\nh\u271d : \u2191g x = x\n\u22a2 \u2191(f * g) x = x \u2228 \u2191h x = x\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nH1 : Disjoint f h\nH2 : Disjoint g h\nx : \u03b1\nh\u271d\u00b9 h\u271d : \u2191h x = x\n\u22a2 \u2191(f * g) x = x \u2228 \u2191h x = x\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nH1 : Disjoint f g\nH2 : Disjoint f h\n\u22a2 Disjoint f (g * h)\n[PROOFSTEP]\nrw [disjoint_comm]\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nH1 : Disjoint f g\nH2 : Disjoint f h\n\u22a2 Disjoint (g * h) f\n[PROOFSTEP]\nexact H1.symm.mul_left H2.symm\n[GOAL]\n\u03b1 : Type u_1\nf g h\u271d h : Perm \u03b1\nx\u271d : \u03b1\n\u22a2 \u2191(h * f * h\u207b\u00b9) x\u271d = x\u271d \u2228 \u2191(h * g * h\u207b\u00b9) x\u271d = x\u271d \u2194 \u2191f (\u2191h\u207b\u00b9 x\u271d) = \u2191h\u207b\u00b9 x\u271d \u2228 \u2191g (\u2191h\u207b\u00b9 x\u271d) = \u2191h\u207b\u00b9 x\u271d\n[PROOFSTEP]\nsimp only [mul_apply, eq_inv_iff_eq]\n[GOAL]\n\u03b1 : Type u_1\nf g h\u271d : Perm \u03b1\nl : List (Perm \u03b1)\nh : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 Disjoint f g\n\u22a2 Disjoint f (List.prod l)\n[PROOFSTEP]\ninduction' l with g l ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\nf g h\u271d\u00b9 : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 Disjoint f g\nh : \u2200 (g : Perm \u03b1), g \u2208 [] \u2192 Disjoint f g\n\u22a2 Disjoint f (List.prod [])\n[PROOFSTEP]\nexact disjoint_one_right _\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nf g\u271d h\u271d\u00b9 : Perm \u03b1\nl\u271d : List (Perm \u03b1)\nh\u271d : \u2200 (g : Perm \u03b1), g \u2208 l\u271d \u2192 Disjoint f g\ng : Perm \u03b1\nl : List (Perm \u03b1)\nih : (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 Disjoint f g) \u2192 Disjoint f (List.prod l)\nh : \u2200 (g_1 : Perm \u03b1), g_1 \u2208 g :: l \u2192 Disjoint f g_1\n\u22a2 Disjoint f (List.prod (g :: l))\n[PROOFSTEP]\nrw [List.prod_cons]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nf g\u271d h\u271d\u00b9 : Perm \u03b1\nl\u271d : List (Perm \u03b1)\nh\u271d : \u2200 (g : Perm \u03b1), g \u2208 l\u271d \u2192 Disjoint f g\ng : Perm \u03b1\nl : List (Perm \u03b1)\nih : (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 Disjoint f g) \u2192 Disjoint f (List.prod l)\nh : \u2200 (g_1 : Perm \u03b1), g_1 \u2208 g :: l \u2192 Disjoint f g_1\n\u22a2 Disjoint f (g * List.prod l)\n[PROOFSTEP]\nexact (h _ (List.mem_cons_self _ _)).mul_right (ih fun g hg => h g (List.mem_cons_of_mem _ hg))\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nl : List (Perm \u03b1)\nh1 : \u00ac1 \u2208 l\nh2 : List.Pairwise Disjoint l\n\u22a2 List.Nodup l\n[PROOFSTEP]\nrefine' List.Pairwise.imp_of_mem _ h2\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nl : List (Perm \u03b1)\nh1 : \u00ac1 \u2208 l\nh2 : List.Pairwise Disjoint l\n\u22a2 \u2200 {a b : Perm \u03b1}, a \u2208 l \u2192 b \u2208 l \u2192 Disjoint a b \u2192 a \u2260 b\n[PROOFSTEP]\nintro \u03c4 \u03c3 h_mem _ h_disjoint _\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nl : List (Perm \u03b1)\nh1 : \u00ac1 \u2208 l\nh2 : List.Pairwise Disjoint l\n\u03c4 \u03c3 : Perm \u03b1\nh_mem : \u03c4 \u2208 l\na\u271d\u00b9 : \u03c3 \u2208 l\nh_disjoint : Disjoint \u03c4 \u03c3\na\u271d : \u03c4 = \u03c3\n\u22a2 False\n[PROOFSTEP]\nsubst \u03c4\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nl : List (Perm \u03b1)\nh1 : \u00ac1 \u2208 l\nh2 : List.Pairwise Disjoint l\n\u03c3 : Perm \u03b1\na\u271d h_mem : \u03c3 \u2208 l\nh_disjoint : Disjoint \u03c3 \u03c3\n\u22a2 False\n[PROOFSTEP]\nsuffices (\u03c3 : Perm \u03b1) = 1 by\n rw [this] at h_mem \n exact h1 h_mem\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nl : List (Perm \u03b1)\nh1 : \u00ac1 \u2208 l\nh2 : List.Pairwise Disjoint l\n\u03c3 : Perm \u03b1\na\u271d h_mem : \u03c3 \u2208 l\nh_disjoint : Disjoint \u03c3 \u03c3\nthis : \u03c3 = 1\n\u22a2 False\n[PROOFSTEP]\nrw [this] at h_mem \n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nl : List (Perm \u03b1)\nh1 : \u00ac1 \u2208 l\nh2 : List.Pairwise Disjoint l\n\u03c3 : Perm \u03b1\na\u271d : \u03c3 \u2208 l\nh_mem : 1 \u2208 l\nh_disjoint : Disjoint \u03c3 \u03c3\nthis : \u03c3 = 1\n\u22a2 False\n[PROOFSTEP]\nexact h1 h_mem\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nl : List (Perm \u03b1)\nh1 : \u00ac1 \u2208 l\nh2 : List.Pairwise Disjoint l\n\u03c3 : Perm \u03b1\na\u271d h_mem : \u03c3 \u2208 l\nh_disjoint : Disjoint \u03c3 \u03c3\n\u22a2 \u03c3 = 1\n[PROOFSTEP]\nexact ext fun a => (or_self_iff _).mp (h_disjoint a)\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nx : \u03b1\nhfx : \u2191f x = x\nn : \u2115\n\u22a2 \u2191(f ^ (n + 1)) x = x\n[PROOFSTEP]\nrw [pow_succ', mul_apply, hfx, pow_apply_eq_self_of_apply_eq_self hfx n]\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nx : \u03b1\nhfx : \u2191f x = x\nn : \u2115\n\u22a2 \u2191(f ^ Int.negSucc n) x = x\n[PROOFSTEP]\nrw [zpow_negSucc, inv_eq_iff_eq, pow_apply_eq_self_of_apply_eq_self hfx]\n[GOAL]\n\u03b1 : Type u_1\nf g h\u271d : Perm \u03b1\nx : \u03b1\nhffx : \u2191f (\u2191f x) = x\nn : \u2115\nh : \u2191(f ^ n) x = x\n\u22a2 \u2191(f ^ (n + 1)) x = \u2191f x\n[PROOFSTEP]\nrw [pow_succ, mul_apply, h]\n[GOAL]\n\u03b1 : Type u_1\nf g h\u271d : Perm \u03b1\nx : \u03b1\nhffx : \u2191f (\u2191f x) = x\nn : \u2115\nh : \u2191(f ^ n) x = \u2191f x\n\u22a2 \u2191(f ^ (n + 1)) x = x\n[PROOFSTEP]\nrw [pow_succ, mul_apply, h, hffx]\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nx : \u03b1\nhffx : \u2191f (\u2191f x) = x\nn : \u2115\n\u22a2 \u2191(f ^ Int.negSucc n) x = x \u2228 \u2191(f ^ Int.negSucc n) x = \u2191f x\n[PROOFSTEP]\nrw [zpow_negSucc, inv_eq_iff_eq, \u2190 f.injective.eq_iff, \u2190 mul_apply, \u2190 pow_succ, eq_comm, inv_eq_iff_eq, \u2190 mul_apply, \u2190\n pow_succ', @eq_comm _ x, or_comm]\n[GOAL]\n\u03b1 : Type u_1\nf g h : Perm \u03b1\nx : \u03b1\nhffx : \u2191f (\u2191f x) = x\nn : \u2115\n\u22a2 \u2191(f ^ (n + 1 + 1)) x = x \u2228 \u2191(f ^ (n + 1 + 1)) x = \u2191f x\n[PROOFSTEP]\nexact pow_apply_eq_of_apply_apply_eq_self hffx _\n[GOAL]\n\u03b1 : Type u_1\nf g h \u03c3 \u03c4 : Perm \u03b1\nh\u03c3\u03c4 : Disjoint \u03c3 \u03c4\na : \u03b1\n\u22a2 \u2191(\u03c3 * \u03c4) a = a \u2194 \u2191\u03c3 a = a \u2227 \u2191\u03c4 a = a\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => by rw [mul_apply, h.2, h.1]\u27e9\n[GOAL]\n\u03b1 : Type u_1\nf g h\u271d \u03c3 \u03c4 : Perm \u03b1\nh\u03c3\u03c4 : Disjoint \u03c3 \u03c4\na : \u03b1\nh : \u2191\u03c3 a = a \u2227 \u2191\u03c4 a = a\n\u22a2 \u2191(\u03c3 * \u03c4) a = a\n[PROOFSTEP]\nrw [mul_apply, h.2, h.1]\n[GOAL]\n\u03b1 : Type u_1\nf g h\u271d \u03c3 \u03c4 : Perm \u03b1\nh\u03c3\u03c4 : Disjoint \u03c3 \u03c4\na : \u03b1\nh : \u2191(\u03c3 * \u03c4) a = a\n\u22a2 \u2191\u03c3 a = a \u2227 \u2191\u03c4 a = a\n[PROOFSTEP]\ncases' h\u03c3\u03c4 a with h\u03c3 h\u03c4\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nf g h\u271d \u03c3 \u03c4 : Perm \u03b1\nh\u03c3\u03c4 : Disjoint \u03c3 \u03c4\na : \u03b1\nh : \u2191(\u03c3 * \u03c4) a = a\nh\u03c3 : \u2191\u03c3 a = a\n\u22a2 \u2191\u03c3 a = a \u2227 \u2191\u03c4 a = a\n[PROOFSTEP]\nexact \u27e8h\u03c3, \u03c3.injective (h.trans h\u03c3.symm)\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nf g h\u271d \u03c3 \u03c4 : Perm \u03b1\nh\u03c3\u03c4 : Disjoint \u03c3 \u03c4\na : \u03b1\nh : \u2191(\u03c3 * \u03c4) a = a\nh\u03c4 : \u2191\u03c4 a = a\n\u22a2 \u2191\u03c3 a = a \u2227 \u2191\u03c4 a = a\n[PROOFSTEP]\nexact \u27e8(congr_arg \u03c3 h\u03c4).symm.trans h, h\u03c4\u27e9\n[GOAL]\n\u03b1 : Type u_1\nf g h \u03c3 \u03c4 : Perm \u03b1\nh\u03c3\u03c4 : Disjoint \u03c3 \u03c4\n\u22a2 \u03c3 * \u03c4 = 1 \u2194 \u03c3 = 1 \u2227 \u03c4 = 1\n[PROOFSTEP]\nsimp_rw [ext_iff, one_apply, h\u03c3\u03c4.mul_apply_eq_iff, forall_and]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\n\u22a2 \u2191(\u2191ofSubtype (swap x y)) z = \u2191(swap \u2191x \u2191y) z\n[PROOFSTEP]\nby_cases hz : p z\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : p z\n\u22a2 \u2191(\u2191ofSubtype (swap x y)) z = \u2191(swap \u2191x \u2191y) z\n[PROOFSTEP]\nrw [swap_apply_def, ofSubtype_apply_of_mem _ hz]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : p z\n\u22a2 \u2191(\u2191(swap x y) { val := z, property := hz }) = if z = \u2191x then \u2191y else if z = \u2191y then \u2191x else z\n[PROOFSTEP]\nsplit_ifs with hzx hzy\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : p z\nhzx : z = \u2191x\n\u22a2 \u2191(\u2191(swap x y) { val := z, property := hz }) = \u2191y\n[PROOFSTEP]\nsimp_rw [hzx, Subtype.coe_eta, swap_apply_left]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : p z\nhzx : \u00acz = \u2191x\nhzy : z = \u2191y\n\u22a2 \u2191(\u2191(swap x y) { val := z, property := hz }) = \u2191x\n[PROOFSTEP]\nsimp_rw [hzy, Subtype.coe_eta, swap_apply_right]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : p z\nhzx : \u00acz = \u2191x\nhzy : \u00acz = \u2191y\n\u22a2 \u2191(\u2191(swap x y) { val := z, property := hz }) = z\n[PROOFSTEP]\nrw [swap_apply_of_ne_of_ne]\n[GOAL]\ncase neg.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : p z\nhzx : \u00acz = \u2191x\nhzy : \u00acz = \u2191y\n\u22a2 { val := z, property := hz } \u2260 x\n[PROOFSTEP]\nsimp [Subtype.ext_iff, *]\n[GOAL]\ncase neg.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : p z\nhzx : \u00acz = \u2191x\nhzy : \u00acz = \u2191y\n\u22a2 { val := z, property := hz } \u2260 y\n[PROOFSTEP]\nsimp [Subtype.ext_iff, *]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : \u00acp z\n\u22a2 \u2191(\u2191ofSubtype (swap x y)) z = \u2191(swap \u2191x \u2191y) z\n[PROOFSTEP]\nrw [ofSubtype_apply_of_not_mem _ hz, swap_apply_of_ne_of_ne]\n[GOAL]\ncase neg.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : \u00acp z\n\u22a2 z \u2260 \u2191x\ncase neg.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : \u00acp z\n\u22a2 z \u2260 \u2191y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase neg.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : \u00acp z\nh : z = \u2191x\n\u22a2 False\ncase neg.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : \u00acp z\n\u22a2 z \u2260 \u2191y\n[PROOFSTEP]\napply hz\n[GOAL]\ncase neg.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : \u00acp z\nh : z = \u2191x\n\u22a2 p z\ncase neg.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : \u00acp z\n\u22a2 z \u2260 \u2191y\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase neg.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : \u00acp z\nh : z = \u2191x\n\u22a2 p \u2191x\ncase neg.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : \u00acp z\n\u22a2 z \u2260 \u2191y\n[PROOFSTEP]\nexact Subtype.prop x\n[GOAL]\ncase neg.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : \u00acp z\n\u22a2 z \u2260 \u2191y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase neg.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : \u00acp z\nh : z = \u2191y\n\u22a2 False\n[PROOFSTEP]\napply hz\n[GOAL]\ncase neg.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : \u00acp z\nh : z = \u2191y\n\u22a2 p z\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase neg.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : Subtype p\nz : \u03b1\nhz : \u00acp z\nh : z = \u2191y\n\u22a2 p \u2191y\n[PROOFSTEP]\nexact Subtype.prop y\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nf : Perm (Subtype p)\nh : IsSwap f\nx : \u03b1\nhx : p x\ny : \u03b1\nhy : p y\nhxy :\n { val := x, property := hx } \u2260 { val := y, property := hy } \u2227\n f = swap { val := x, property := hx } { val := y, property := hy }\n\u22a2 x \u2260 y\n[PROOFSTEP]\nsimp only [Ne.def, Subtype.ext_iff] at hxy \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nf : Perm (Subtype p)\nh : IsSwap f\nx : \u03b1\nhx : p x\ny : \u03b1\nhy : p y\nhxy : \u00acx = y \u2227 f = swap { val := x, property := hx } { val := y, property := hy }\n\u22a2 x \u2260 y\n[PROOFSTEP]\nexact hxy.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nf : Perm (Subtype p)\nh : IsSwap f\nx : \u03b1\nhx : p x\ny : \u03b1\nhy : p y\nhxy :\n { val := x, property := hx } \u2260 { val := y, property := hy } \u2227\n f = swap { val := x, property := hx } { val := y, property := hy }\n\u22a2 \u2191ofSubtype f = swap x y\n[PROOFSTEP]\nrw [hxy.2, ofSubtype_swap_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nhy : \u2191(swap x (\u2191f x) * f) y \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nsimp only [swap_apply_def, mul_apply, f.injective.eq_iff] at *\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nhy : (if \u2191f y = x then \u2191f x else if y = x then x else \u2191f y) \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nby_cases h : f y = x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nhy : (if \u2191f y = x then \u2191f x else if y = x then x else \u2191f y) \u2260 y\nh : \u2191f y = x\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase pos.left\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nhy : (if \u2191f y = x then \u2191f x else if y = x then x else \u2191f y) \u2260 y\nh : \u2191f y = x\n\u22a2 \u2191f y \u2260 y\n[PROOFSTEP]\nintro\n[GOAL]\ncase pos.right\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nhy : (if \u2191f y = x then \u2191f x else if y = x then x else \u2191f y) \u2260 y\nh : \u2191f y = x\n\u22a2 y \u2260 x\n[PROOFSTEP]\nintro\n[GOAL]\ncase pos.left\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nhy : (if \u2191f y = x then \u2191f x else if y = x then x else \u2191f y) \u2260 y\nh : \u2191f y = x\na\u271d : \u2191f y = y\n\u22a2 False\n[PROOFSTEP]\nsimp_all only [if_true, eq_self_iff_true, not_true, Ne.def]\n[GOAL]\ncase pos.right\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nhy : (if \u2191f y = x then \u2191f x else if y = x then x else \u2191f y) \u2260 y\nh : \u2191f y = x\na\u271d : y = x\n\u22a2 False\n[PROOFSTEP]\nsimp_all only [if_true, eq_self_iff_true, not_true, Ne.def]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nhy : (if \u2191f y = x then \u2191f x else if y = x then x else \u2191f y) \u2260 y\nh : \u00ac\u2191f y = x\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nsplit_ifs at hy with h h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d : \u00ac\u2191f y = x\nh : \u2191f y = x\nhy : \u2191f x \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\ntry {subst x\n}\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d : \u00ac\u2191f y = x\nh : \u2191f y = x\nhy : \u2191f x \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\n{subst x\n}\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d : \u00ac\u2191f y = x\nh : \u2191f y = x\nhy : \u2191f x \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nsubst x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d\u00b9 : \u00ac\u2191f y = x\nh\u271d : \u00ac\u2191f y = x\nh : y = x\nhy : x \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\ntry {subst x\n}\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d\u00b9 : \u00ac\u2191f y = x\nh\u271d : \u00ac\u2191f y = x\nh : y = x\nhy : x \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\n{subst x\n}\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d\u00b9 : \u00ac\u2191f y = x\nh\u271d : \u00ac\u2191f y = x\nh : y = x\nhy : x \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nsubst x\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d\u00b9 : \u00ac\u2191f y = x\nh\u271d : \u00ac\u2191f y = x\nh : \u00acy = x\nhy : \u2191f y \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\ntry {subst x\n}\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d\u00b9 : \u00ac\u2191f y = x\nh\u271d : \u00ac\u2191f y = x\nh : \u00acy = x\nhy : \u2191f y \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\n{subst x\n}\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d\u00b9 : \u00ac\u2191f y = x\nh\u271d : \u00ac\u2191f y = x\nh : \u00acy = x\nhy : \u2191f y \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nsubst x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d : \u00ac\u2191f y = x\nh : \u2191f y = x\nhy : \u2191f x \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\ntry {simp [*] at *\n}\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d : \u00ac\u2191f y = x\nh : \u2191f y = x\nhy : \u2191f x \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\n{simp [*] at *\n}\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d : \u00ac\u2191f y = x\nh : \u2191f y = x\nhy : \u2191f x \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nsimp [*] at *\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d\u00b9 : \u00ac\u2191f y = x\nh\u271d : \u00ac\u2191f y = x\nh : y = x\nhy : x \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\ntry {simp [*] at *\n}\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d\u00b9 : \u00ac\u2191f y = x\nh\u271d : \u00ac\u2191f y = x\nh : y = x\nhy : x \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\n{simp [*] at *\n}\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d\u00b9 : \u00ac\u2191f y = x\nh\u271d : \u00ac\u2191f y = x\nh : y = x\nhy : x \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nsimp [*] at *\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d\u00b9 : \u00ac\u2191f y = x\nh\u271d : \u00ac\u2191f y = x\nh : \u00acy = x\nhy : \u2191f y \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\ntry {simp [*] at *\n}\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d\u00b9 : \u00ac\u2191f y = x\nh\u271d : \u00ac\u2191f y = x\nh : \u00acy = x\nhy : \u2191f y \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\n{simp [*] at *\n}\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\nh\u271d\u00b9 : \u00ac\u2191f y = x\nh\u271d : \u00ac\u2191f y = x\nh : \u00acy = x\nhy : \u2191f y \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nsimp [*] at *\n[GOAL]\n\u03b1 : Type u_1\np q : Perm \u03b1\n\u22a2 {x | \u2191p\u207b\u00b9 x \u2260 x} = {x | \u2191p x \u2260 x}\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\np q : Perm \u03b1\nx : \u03b1\n\u22a2 x \u2208 {x | \u2191p\u207b\u00b9 x \u2260 x} \u2194 x \u2208 {x | \u2191p x \u2260 x}\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq, Ne.def]\n[GOAL]\ncase h\n\u03b1 : Type u_1\np q : Perm \u03b1\nx : \u03b1\n\u22a2 \u00ac\u2191p\u207b\u00b9 x = x \u2194 \u00ac\u2191p x = x\n[PROOFSTEP]\nrw [inv_def, symm_apply_eq, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\np\u271d q p : Perm \u03b1\na : \u03b1\n\u22a2 \u2191p a \u2208 {x | \u2191p x \u2260 x} \u2194 a \u2208 {x | \u2191p x \u2260 x}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\np q : Perm \u03b1\nn : \u2124\n\u22a2 {x | \u2191(p ^ n) x \u2260 x} \u2286 {x | \u2191p x \u2260 x}\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\np q : Perm \u03b1\nn : \u2124\nx : \u03b1\n\u22a2 x \u2208 {x | \u2191(p ^ n) x \u2260 x} \u2192 x \u2208 {x | \u2191p x \u2260 x}\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq, Ne.def]\n[GOAL]\n\u03b1 : Type u_1\np q : Perm \u03b1\nn : \u2124\nx : \u03b1\n\u22a2 \u00ac\u2191(p ^ n) x = x \u2192 \u00ac\u2191p x = x\n[PROOFSTEP]\nintro hx H\n[GOAL]\n\u03b1 : Type u_1\np q : Perm \u03b1\nn : \u2124\nx : \u03b1\nhx : \u00ac\u2191(p ^ n) x = x\nH : \u2191p x = x\n\u22a2 False\n[PROOFSTEP]\nsimp [zpow_apply_eq_self_of_apply_eq_self H] at hx \n[GOAL]\n\u03b1 : Type u_1\np q : Perm \u03b1\n\u22a2 {x | \u2191(p * q) x \u2260 x} \u2286 {x | \u2191p x \u2260 x} \u222a {x | \u2191q x \u2260 x}\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\np q : Perm \u03b1\nx : \u03b1\n\u22a2 x \u2208 {x | \u2191(p * q) x \u2260 x} \u2192 x \u2208 {x | \u2191p x \u2260 x} \u222a {x | \u2191q x \u2260 x}\n[PROOFSTEP]\nsimp only [Perm.coe_mul, Function.comp_apply, Ne.def, Set.mem_union, Set.mem_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\np q : Perm \u03b1\nx : \u03b1\n\u22a2 \u00ac\u2191p (\u2191q x) = x \u2192 \u00ac\u2191p x = x \u2228 \u00ac\u2191q x = x\n[PROOFSTEP]\nby_cases hq : q x = x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\np q : Perm \u03b1\nx : \u03b1\nhq : \u2191q x = x\n\u22a2 \u00ac\u2191p (\u2191q x) = x \u2192 \u00ac\u2191p x = x \u2228 \u00ac\u2191q x = x\n[PROOFSTEP]\nsimp [hq]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\np q : Perm \u03b1\nx : \u03b1\nhq : \u00ac\u2191q x = x\n\u22a2 \u00ac\u2191p (\u2191q x) = x \u2192 \u00ac\u2191p x = x \u2228 \u00ac\u2191q x = x\n[PROOFSTEP]\nsimp [hq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx : \u03b1\n\u22a2 x \u2208 support f \u2194 \u2191f x \u2260 x\n[PROOFSTEP]\nrw [support, mem_filter, and_iff_right (mem_univ x)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx : \u03b1\n\u22a2 \u00acx \u2208 support f \u2194 \u2191f x = x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\n\u22a2 \u2191(support f) = {x | \u2191f x \u2260 x}\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(support f) \u2194 x\u271d \u2208 {x | \u2191f x \u2260 x}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g \u03c3 : Perm \u03b1\n\u22a2 support \u03c3 = \u2205 \u2194 \u03c3 = 1\n[PROOFSTEP]\nsimp_rw [Finset.ext_iff, mem_support, Finset.not_mem_empty, iff_false_iff, not_not, Equiv.Perm.ext_iff, one_apply]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\n\u22a2 support 1 = \u2205\n[PROOFSTEP]\nrw [support_eq_empty_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : support f \u2286 support g\nh' : \u2200 (x : \u03b1), x \u2208 support g \u2192 \u2191f x = \u2191g x\n\u22a2 f = g\n[PROOFSTEP]\next x\n[GOAL]\ncase H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : support f \u2286 support g\nh' : \u2200 (x : \u03b1), x \u2208 support g \u2192 \u2191f x = \u2191g x\nx : \u03b1\n\u22a2 \u2191f x = \u2191g x\n[PROOFSTEP]\nby_cases hx : x \u2208 g.support\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : support f \u2286 support g\nh' : \u2200 (x : \u03b1), x \u2208 support g \u2192 \u2191f x = \u2191g x\nx : \u03b1\nhx : x \u2208 support g\n\u22a2 \u2191f x = \u2191g x\n[PROOFSTEP]\nexact h' x hx\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : support f \u2286 support g\nh' : \u2200 (x : \u03b1), x \u2208 support g \u2192 \u2191f x = \u2191g x\nx : \u03b1\nhx : \u00acx \u2208 support g\n\u22a2 \u2191f x = \u2191g x\n[PROOFSTEP]\nrw [not_mem_support.mp hx, \u2190 not_mem_support]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : support f \u2286 support g\nh' : \u2200 (x : \u03b1), x \u2208 support g \u2192 \u2191f x = \u2191g x\nx : \u03b1\nhx : \u00acx \u2208 support g\n\u22a2 \u00acx \u2208 support f\n[PROOFSTEP]\nexact fun H => hx (h H)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g\u271d f g : Perm \u03b1\nx : \u03b1\n\u22a2 x \u2208 support (f * g) \u2192 x \u2208 support f \u2294 support g\n[PROOFSTEP]\nsimp only [sup_eq_union]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g\u271d f g : Perm \u03b1\nx : \u03b1\n\u22a2 x \u2208 support (f * g) \u2192 x \u2208 support f \u222a support g\n[PROOFSTEP]\nrw [mem_union, mem_support, mem_support, mem_support, mul_apply, \u2190 not_and_or, not_imp_not]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g\u271d f g : Perm \u03b1\nx : \u03b1\n\u22a2 \u2191f x = x \u2227 \u2191g x = x \u2192 \u2191f (\u2191g x) = x\n[PROOFSTEP]\nrintro \u27e8hf, hg\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g\u271d f g : Perm \u03b1\nx : \u03b1\nhf : \u2191f x = x\nhg : \u2191g x = x\n\u22a2 \u2191f (\u2191g x) = x\n[PROOFSTEP]\nrw [hg, hf]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nx : \u03b1\nhx : x \u2208 support (List.prod l)\n\u22a2 \u2203 f, f \u2208 l \u2227 x \u2208 support f\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nx : \u03b1\nhx : \u2200 (f : Perm \u03b1), f \u2208 l \u2192 \u00acx \u2208 support f\n\u22a2 \u00acx \u2208 support (List.prod l)\n[PROOFSTEP]\nsimp_rw [mem_support, not_not] at hx \u22a2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nx : \u03b1\nhx : \u2200 (f : Perm \u03b1), f \u2208 l \u2192 \u2191f x = x\n\u22a2 \u2191(List.prod l) x = x\n[PROOFSTEP]\ninduction' l with f l ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nx : \u03b1\nhx\u271d : \u2200 (f : Perm \u03b1), f \u2208 l \u2192 \u2191f x = x\nhx : \u2200 (f : Perm \u03b1), f \u2208 [] \u2192 \u2191f x = x\n\u22a2 \u2191(List.prod []) x = x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g : Perm \u03b1\nl\u271d : List (Perm \u03b1)\nx : \u03b1\nhx\u271d : \u2200 (f : Perm \u03b1), f \u2208 l\u271d \u2192 \u2191f x = x\nf : Perm \u03b1\nl : List (Perm \u03b1)\nih : (\u2200 (f : Perm \u03b1), f \u2208 l \u2192 \u2191f x = x) \u2192 \u2191(List.prod l) x = x\nhx : \u2200 (f_1 : Perm \u03b1), f_1 \u2208 f :: l \u2192 \u2191f_1 x = x\n\u22a2 \u2191(List.prod (f :: l)) x = x\n[PROOFSTEP]\nrw [List.prod_cons, mul_apply, ih, hx]\n[GOAL]\ncase cons._\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g : Perm \u03b1\nl\u271d : List (Perm \u03b1)\nx : \u03b1\nhx\u271d : \u2200 (f : Perm \u03b1), f \u2208 l\u271d \u2192 \u2191f x = x\nf : Perm \u03b1\nl : List (Perm \u03b1)\nih : (\u2200 (f : Perm \u03b1), f \u2208 l \u2192 \u2191f x = x) \u2192 \u2191(List.prod l) x = x\nhx : \u2200 (f_1 : Perm \u03b1), f_1 \u2208 f :: l \u2192 \u2191f_1 x = x\n\u22a2 f \u2208 f :: l\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g : Perm \u03b1\nl\u271d : List (Perm \u03b1)\nx : \u03b1\nhx\u271d : \u2200 (f : Perm \u03b1), f \u2208 l\u271d \u2192 \u2191f x = x\nf : Perm \u03b1\nl : List (Perm \u03b1)\nih : (\u2200 (f : Perm \u03b1), f \u2208 l \u2192 \u2191f x = x) \u2192 \u2191(List.prod l) x = x\nhx : \u2200 (f_1 : Perm \u03b1), f_1 \u2208 f :: l \u2192 \u2191f_1 x = x\n\u22a2 \u2200 (f : Perm \u03b1), f \u2208 l \u2192 \u2191f x = x\n[PROOFSTEP]\nsimp only [List.find?, List.mem_cons, true_or]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g : Perm \u03b1\nl\u271d : List (Perm \u03b1)\nx : \u03b1\nhx\u271d : \u2200 (f : Perm \u03b1), f \u2208 l\u271d \u2192 \u2191f x = x\nf : Perm \u03b1\nl : List (Perm \u03b1)\nih : (\u2200 (f : Perm \u03b1), f \u2208 l \u2192 \u2191f x = x) \u2192 \u2191(List.prod l) x = x\nhx : \u2200 (f_1 : Perm \u03b1), f_1 \u2208 f :: l \u2192 \u2191f_1 x = x\n\u22a2 \u2200 (f : Perm \u03b1), f \u2208 l \u2192 \u2191f x = x\n[PROOFSTEP]\nintros f' hf'\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g : Perm \u03b1\nl\u271d : List (Perm \u03b1)\nx : \u03b1\nhx\u271d : \u2200 (f : Perm \u03b1), f \u2208 l\u271d \u2192 \u2191f x = x\nf : Perm \u03b1\nl : List (Perm \u03b1)\nih : (\u2200 (f : Perm \u03b1), f \u2208 l \u2192 \u2191f x = x) \u2192 \u2191(List.prod l) x = x\nhx : \u2200 (f_1 : Perm \u03b1), f_1 \u2208 f :: l \u2192 \u2191f_1 x = x\nf' : Perm \u03b1\nhf' : f' \u2208 l\n\u22a2 \u2191f' x = x\n[PROOFSTEP]\nrefine' hx f' _\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g : Perm \u03b1\nl\u271d : List (Perm \u03b1)\nx : \u03b1\nhx\u271d : \u2200 (f : Perm \u03b1), f \u2208 l\u271d \u2192 \u2191f x = x\nf : Perm \u03b1\nl : List (Perm \u03b1)\nih : (\u2200 (f : Perm \u03b1), f \u2208 l \u2192 \u2191f x = x) \u2192 \u2191(List.prod l) x = x\nhx : \u2200 (f_1 : Perm \u03b1), f_1 \u2208 f :: l \u2192 \u2191f_1 x = x\nf' : Perm \u03b1\nhf' : f' \u2208 l\n\u22a2 f' \u2208 f :: l\n[PROOFSTEP]\nsimp only [List.find?, List.mem_cons]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g : Perm \u03b1\nl\u271d : List (Perm \u03b1)\nx : \u03b1\nhx\u271d : \u2200 (f : Perm \u03b1), f \u2208 l\u271d \u2192 \u2191f x = x\nf : Perm \u03b1\nl : List (Perm \u03b1)\nih : (\u2200 (f : Perm \u03b1), f \u2208 l \u2192 \u2191f x = x) \u2192 \u2191(List.prod l) x = x\nhx : \u2200 (f_1 : Perm \u03b1), f_1 \u2208 f :: l \u2192 \u2191f_1 x = x\nf' : Perm \u03b1\nhf' : f' \u2208 l\n\u22a2 f' = f \u2228 f' \u2208 l\n[PROOFSTEP]\nexact Or.inr hf'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g \u03c3 : Perm \u03b1\n\u22a2 support \u03c3\u207b\u00b9 = support \u03c3\n[PROOFSTEP]\nsimp_rw [Finset.ext_iff, mem_support, not_iff_not, inv_eq_iff_eq.trans eq_comm, imp_true_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx : \u03b1\n\u22a2 \u2191f x \u2208 support f \u2194 x \u2208 support f\n[PROOFSTEP]\nrw [mem_support, mem_support, Ne.def, Ne.def, apply_eq_iff_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nn : \u2115\nx : \u03b1\n\u22a2 \u2191f (\u2191(f ^ n) x) = \u2191(f ^ n) x \u2194 \u2191f x = x\n[PROOFSTEP]\nrw [\u2190 mul_apply, Commute.self_pow f, mul_apply, apply_eq_iff_eq]\n -- @[simp] -- Porting note: simp can prove this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nn : \u2115\nx : \u03b1\n\u22a2 \u2191(f ^ n) x \u2208 support f \u2194 x \u2208 support f\n[PROOFSTEP]\nsimp only [mem_support, ne_eq, apply_pow_apply_eq_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nn : \u2124\nx : \u03b1\n\u22a2 \u2191f (\u2191(f ^ n) x) = \u2191(f ^ n) x \u2194 \u2191f x = x\n[PROOFSTEP]\nrw [\u2190 mul_apply, Commute.self_zpow f, mul_apply, apply_eq_iff_eq]\n -- @[simp] -- Porting note: simp can prove this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nn : \u2124\nx : \u03b1\n\u22a2 \u2191(f ^ n) x \u2208 support f \u2194 x \u2208 support f\n[PROOFSTEP]\nsimp only [mem_support, ne_eq, apply_zpow_apply_eq_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : \u2200 (x : \u03b1), x \u2208 support f \u2229 support g \u2192 \u2191f x = \u2191g x\nk : \u2115\n\u22a2 \u2200 (x : \u03b1), x \u2208 support f \u2229 support g \u2192 \u2191(f ^ k) x = \u2191(g ^ k) x\n[PROOFSTEP]\ninduction' k with k hk\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : \u2200 (x : \u03b1), x \u2208 support f \u2229 support g \u2192 \u2191f x = \u2191g x\n\u22a2 \u2200 (x : \u03b1), x \u2208 support f \u2229 support g \u2192 \u2191(f ^ Nat.zero) x = \u2191(g ^ Nat.zero) x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : \u2200 (x : \u03b1), x \u2208 support f \u2229 support g \u2192 \u2191f x = \u2191g x\nk : \u2115\nhk : \u2200 (x : \u03b1), x \u2208 support f \u2229 support g \u2192 \u2191(f ^ k) x = \u2191(g ^ k) x\n\u22a2 \u2200 (x : \u03b1), x \u2208 support f \u2229 support g \u2192 \u2191(f ^ Nat.succ k) x = \u2191(g ^ Nat.succ k) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : \u2200 (x : \u03b1), x \u2208 support f \u2229 support g \u2192 \u2191f x = \u2191g x\nk : \u2115\nhk : \u2200 (x : \u03b1), x \u2208 support f \u2229 support g \u2192 \u2191(f ^ k) x = \u2191(g ^ k) x\nx : \u03b1\nhx : x \u2208 support f \u2229 support g\n\u22a2 \u2191(f ^ Nat.succ k) x = \u2191(g ^ Nat.succ k) x\n[PROOFSTEP]\nrw [pow_succ', mul_apply, pow_succ', mul_apply, h _ hx, hk]\n[GOAL]\ncase succ.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : \u2200 (x : \u03b1), x \u2208 support f \u2229 support g \u2192 \u2191f x = \u2191g x\nk : \u2115\nhk : \u2200 (x : \u03b1), x \u2208 support f \u2229 support g \u2192 \u2191(f ^ k) x = \u2191(g ^ k) x\nx : \u03b1\nhx : x \u2208 support f \u2229 support g\n\u22a2 \u2191g x \u2208 support f \u2229 support g\n[PROOFSTEP]\nrwa [mem_inter, apply_mem_support, \u2190 h _ hx, apply_mem_support, \u2190 mem_inter]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\n\u22a2 Disjoint f g \u2194 _root_.Disjoint (support f) (support g)\n[PROOFSTEP]\nsimp [disjoint_iff_eq_or_eq, disjoint_iff, disjoint_iff, Finset.ext_iff, not_and_or, imp_iff_not_or]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : Disjoint f g\n\u22a2 support (f * g) = support f \u222a support g\n[PROOFSTEP]\nrefine' le_antisymm (support_mul_le _ _) fun a => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : Disjoint f g\na : \u03b1\n\u22a2 a \u2208 support f \u222a support g \u2192 a \u2208 support (f * g)\n[PROOFSTEP]\nrw [mem_union, mem_support, mem_support, mem_support, mul_apply, \u2190 not_and_or, not_imp_not]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : Disjoint f g\na : \u03b1\n\u22a2 \u2191f (\u2191g a) = a \u2192 \u2191f a = a \u2227 \u2191g a = a\n[PROOFSTEP]\nexact\n (h a).elim (fun hf h => \u27e8hf, f.apply_eq_iff_eq.mp (h.trans hf.symm)\u27e9) fun hg h => \u27e8(congr_arg f hg).symm.trans h, hg\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh : List.Pairwise Disjoint l\n\u22a2 support (List.prod l) = List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support l)\n[PROOFSTEP]\ninduction' l with hd tl hl\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : List.Pairwise Disjoint l\nh : List.Pairwise Disjoint []\n\u22a2 support (List.prod []) = List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : List.Pairwise Disjoint l\nhd : Perm \u03b1\ntl : List (Perm \u03b1)\nhl : List.Pairwise Disjoint tl \u2192 support (List.prod tl) = List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support tl)\nh : List.Pairwise Disjoint (hd :: tl)\n\u22a2 support (List.prod (hd :: tl)) = List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support (hd :: tl))\n[PROOFSTEP]\nrw [List.pairwise_cons] at h \n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : List.Pairwise Disjoint l\nhd : Perm \u03b1\ntl : List (Perm \u03b1)\nhl : List.Pairwise Disjoint tl \u2192 support (List.prod tl) = List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support tl)\nh : (\u2200 (a' : Perm \u03b1), a' \u2208 tl \u2192 Disjoint hd a') \u2227 List.Pairwise Disjoint tl\n\u22a2 support (List.prod (hd :: tl)) = List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support (hd :: tl))\n[PROOFSTEP]\nhave : Disjoint hd tl.prod := disjoint_prod_right _ h.left\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : List.Pairwise Disjoint l\nhd : Perm \u03b1\ntl : List (Perm \u03b1)\nhl : List.Pairwise Disjoint tl \u2192 support (List.prod tl) = List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support tl)\nh : (\u2200 (a' : Perm \u03b1), a' \u2208 tl \u2192 Disjoint hd a') \u2227 List.Pairwise Disjoint tl\nthis : Disjoint hd (List.prod tl)\n\u22a2 support (List.prod (hd :: tl)) = List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support (hd :: tl))\n[PROOFSTEP]\nsimp [this.support_mul, hl h.right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\n\u22a2 support (List.prod l) \u2264 List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support l)\n[PROOFSTEP]\ninduction' l with hd tl hl\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\n\u22a2 support (List.prod []) \u2264 List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g hd : Perm \u03b1\ntl : List (Perm \u03b1)\nhl : support (List.prod tl) \u2264 List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support tl)\n\u22a2 support (List.prod (hd :: tl)) \u2264 List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support (hd :: tl))\n[PROOFSTEP]\nrw [List.prod_cons, List.map_cons, List.foldr_cons]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g hd : Perm \u03b1\ntl : List (Perm \u03b1)\nhl : support (List.prod tl) \u2264 List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support tl)\n\u22a2 support (hd * List.prod tl) \u2264 support hd \u2294 List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support tl)\n[PROOFSTEP]\nrefine' (support_mul_le hd tl.prod).trans _\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g hd : Perm \u03b1\ntl : List (Perm \u03b1)\nhl : support (List.prod tl) \u2264 List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support tl)\n\u22a2 support hd \u2294 support (List.prod tl) \u2264 support hd \u2294 List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (List.map support tl)\n[PROOFSTEP]\nexact sup_le_sup le_rfl hl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\n\u22a2 support (swap x y) = {x, y}\n[PROOFSTEP]\next z\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}\n[PROOFSTEP]\nby_cases hx : z = x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\nhx : z = x\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\nhx : \u00acz = x\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}\n[PROOFSTEP]\nany_goals simpa [hx] using h.symm\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\nhx : z = x\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}\n[PROOFSTEP]\nsimpa [hx] using h.symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\nhx : \u00acz = x\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}\n[PROOFSTEP]\nsimpa [hx] using h.symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\nhx : \u00acz = x\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}\n[PROOFSTEP]\nby_cases hy : z = y\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\nhx : \u00acz = x\nhy : z = y\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}\n[PROOFSTEP]\nsimp [swap_apply_of_ne_of_ne, hx, hy]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\nhx : \u00acz = x\nhy : z = y\n\u22a2 \u00acx = y\n[PROOFSTEP]\nexact h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\nhx : \u00acz = x\nhy : \u00acz = y\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}\n[PROOFSTEP]\nsimp [swap_apply_of_ne_of_ne, hx, hy]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\n\u22a2 support (swap x y) = {x, y} \u2194 x \u2260 y\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => support_swap h\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : support (swap x y) = {x, y}\n\u22a2 x \u2260 y\n[PROOFSTEP]\nby_contra'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : support (swap x y) = {x, y}\nthis : x = y\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 this] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : support (swap x x) = {x, x}\nthis : x = y\n\u22a2 False\n[PROOFSTEP]\nsimp only [swap_self, support_refl, pair_eq_singleton] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nthis : x = y\nh : \u2205 = {x}\n\u22a2 False\n[PROOFSTEP]\nhave : x \u2208 \u2205 := by\n rw [h]\n exact mem_singleton.mpr rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nthis : x = y\nh : \u2205 = {x}\n\u22a2 x \u2208 \u2205\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nthis : x = y\nh : \u2205 = {x}\n\u22a2 x \u2208 {x}\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nthis : x = y\nh : \u2205 = {x}\n\u22a2 Membership \u03b1 (Finset \u03b1)\n[PROOFSTEP]\nexact mem_singleton.mpr rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nthis\u271d : x = y\nh : \u2205 = {x}\nthis : x \u2208 \u2205\n\u22a2 False\n[PROOFSTEP]\nhave := (Finset.ne_empty_of_mem this)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nthis\u271d\u00b9 : x = y\nh : \u2205 = {x}\nthis\u271d : x \u2208 \u2205\nthis : \u2205 \u2260 \u2205\n\u22a2 False\n[PROOFSTEP]\nexact this rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y z : \u03b1\nh : List.Nodup [x, y, z]\n\u22a2 support (swap x y * swap y z) = {x, y, z}\n[PROOFSTEP]\nsimp only [List.not_mem_nil, and_true_iff, List.mem_cons, not_false_iff, List.nodup_cons, List.mem_singleton,\n and_self_iff, List.nodup_nil] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y z : \u03b1\nh : \u00ac(x = y \u2228 x = z \u2228 False) \u2227 \u00ac(y = z \u2228 False)\n\u22a2 support (swap x y * swap y z) = {x, y, z}\n[PROOFSTEP]\npush_neg at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y z : \u03b1\nh : (x \u2260 y \u2227 x \u2260 z \u2227 \u00acFalse) \u2227 y \u2260 z \u2227 \u00acFalse\n\u22a2 support (swap x y * swap y z) = {x, y, z}\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y z : \u03b1\nh : (x \u2260 y \u2227 x \u2260 z \u2227 \u00acFalse) \u2227 y \u2260 z \u2227 \u00acFalse\n\u22a2 support (swap x y * swap y z) \u2264 {x, y, z}\n[PROOFSTEP]\nconvert support_mul_le (swap x y) (swap y z) using 1\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y z : \u03b1\nh : (x \u2260 y \u2227 x \u2260 z \u2227 \u00acFalse) \u2227 y \u2260 z \u2227 \u00acFalse\n\u22a2 {x, y, z} = support (swap x y) \u2294 support (swap y z)\n[PROOFSTEP]\nrw [support_swap h.left.left, support_swap h.right.left]\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y z : \u03b1\nh : (x \u2260 y \u2227 x \u2260 z \u2227 \u00acFalse) \u2227 y \u2260 z \u2227 \u00acFalse\n\u22a2 {x, y, z} = {x, y} \u2294 {y, z}\n[PROOFSTEP]\nsimp only [sup_eq_union]\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y z : \u03b1\nh : (x \u2260 y \u2227 x \u2260 z \u2227 \u00acFalse) \u2227 y \u2260 z \u2227 \u00acFalse\n\u22a2 {x, y, z} = {x, y} \u222a {y, z}\n[PROOFSTEP]\nsimp only [mem_singleton, mem_insert, union_insert, insert_union, mem_union, true_or, or_true, insert_eq_of_mem]\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y z : \u03b1\nh : (x \u2260 y \u2227 x \u2260 z \u2227 \u00acFalse) \u2227 y \u2260 z \u2227 \u00acFalse\n\u22a2 {x, y, z} = insert x ({y} \u222a {z})\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y z : \u03b1\nh : (x \u2260 y \u2227 x \u2260 z \u2227 \u00acFalse) \u2227 y \u2260 z \u2227 \u00acFalse\n\u22a2 {x, y, z} \u2264 support (swap x y * swap y z)\n[PROOFSTEP]\nintro\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y z : \u03b1\nh : (x \u2260 y \u2227 x \u2260 z \u2227 \u00acFalse) \u2227 y \u2260 z \u2227 \u00acFalse\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 {x, y, z} \u2192 a\u271d \u2208 support (swap x y * swap y z)\n[PROOFSTEP]\nsimp only [mem_insert, mem_singleton]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y z : \u03b1\nh : (x \u2260 y \u2227 x \u2260 z \u2227 \u00acFalse) \u2227 y \u2260 z \u2227 \u00acFalse\na\u271d : \u03b1\n\u22a2 a\u271d = x \u2228 a\u271d = y \u2228 a\u271d = z \u2192 a\u271d \u2208 support (swap x y * swap y z)\n[PROOFSTEP]\nrintro (rfl | rfl | rfl | _)\n[GOAL]\ncase a.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\ny z a\u271d : \u03b1\nh : (a\u271d \u2260 y \u2227 a\u271d \u2260 z \u2227 \u00acFalse) \u2227 y \u2260 z \u2227 \u00acFalse\n\u22a2 a\u271d \u2208 support (swap a\u271d y * swap y z)\n[PROOFSTEP]\nsimp [swap_apply_of_ne_of_ne, h.left.left, h.left.left.symm, h.left.right.symm, h.left.right.left.symm,\n h.right.left.symm]\n[GOAL]\ncase a.inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx z a\u271d : \u03b1\nh : (x \u2260 a\u271d \u2227 x \u2260 z \u2227 \u00acFalse) \u2227 a\u271d \u2260 z \u2227 \u00acFalse\n\u22a2 a\u271d \u2208 support (swap x a\u271d * swap a\u271d z)\n[PROOFSTEP]\nsimp [swap_apply_of_ne_of_ne, h.left.left, h.left.left.symm, h.left.right.symm, h.left.right.left.symm,\n h.right.left.symm]\n[GOAL]\ncase a.inr.inr.refl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y z : \u03b1\nh : (x \u2260 y \u2227 x \u2260 z \u2227 \u00acFalse) \u2227 y \u2260 z \u2227 \u00acFalse\n\u22a2 z \u2208 support (swap x y * swap y z)\n[PROOFSTEP]\nsimp [swap_apply_of_ne_of_ne, h.left.left, h.left.left.symm, h.left.right.symm, h.left.right.left.symm,\n h.right.left.symm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx y : \u03b1\n\u22a2 support f \\ {x, y} \u2264 support (swap x y * f)\n[PROOFSTEP]\nintro\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx y a\u271d : \u03b1\n\u22a2 a\u271d \u2208 support f \\ {x, y} \u2192 a\u271d \u2208 support (swap x y * f)\n[PROOFSTEP]\nsimp only [and_imp, Perm.coe_mul, Function.comp_apply, Ne.def, mem_support, mem_insert, mem_sdiff, mem_singleton]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx y a\u271d : \u03b1\n\u22a2 \u00ac\u2191f a\u271d = a\u271d \u2192 \u00ac(a\u271d = x \u2228 a\u271d = y) \u2192 \u00ac\u2191(swap x y) (\u2191f a\u271d) = a\u271d\n[PROOFSTEP]\npush_neg\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx y a\u271d : \u03b1\n\u22a2 \u2191f a\u271d \u2260 a\u271d \u2192 a\u271d \u2260 x \u2227 a\u271d \u2260 y \u2192 \u2191(swap x y) (\u2191f a\u271d) \u2260 a\u271d\n[PROOFSTEP]\nrintro ha \u27e8hx, hy\u27e9 H\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx y a\u271d : \u03b1\nha : \u2191f a\u271d \u2260 a\u271d\nhx : a\u271d \u2260 x\nhy : a\u271d \u2260 y\nH : \u2191(swap x y) (\u2191f a\u271d) = a\u271d\n\u22a2 False\n[PROOFSTEP]\nrw [swap_apply_eq_iff, swap_apply_of_ne_of_ne hx hy] at H \n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx y a\u271d : \u03b1\nha : \u2191f a\u271d \u2260 a\u271d\nhx : a\u271d \u2260 x\nhy : a\u271d \u2260 y\nH : \u2191f a\u271d = a\u271d\n\u22a2 False\n[PROOFSTEP]\nexact ha H\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx : \u03b1\nh : \u2191f (\u2191f x) \u2260 x\n\u22a2 support (swap x (\u2191f x) * f) = support f \\ {x}\n[PROOFSTEP]\nby_cases hx : f x = x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx : \u03b1\nh : \u2191f (\u2191f x) \u2260 x\nhx : \u2191f x = x\n\u22a2 support (swap x (\u2191f x) * f) = support f \\ {x}\n[PROOFSTEP]\nsimp [hx, sdiff_singleton_eq_erase, not_mem_support.mpr hx, erase_eq_of_not_mem]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx : \u03b1\nh : \u2191f (\u2191f x) \u2260 x\nhx : \u00ac\u2191f x = x\n\u22a2 support (swap x (\u2191f x) * f) = support f \\ {x}\n[PROOFSTEP]\next z\n[GOAL]\ncase neg.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx : \u03b1\nh : \u2191f (\u2191f x) \u2260 x\nhx : \u00ac\u2191f x = x\nz : \u03b1\n\u22a2 z \u2208 support (swap x (\u2191f x) * f) \u2194 z \u2208 support f \\ {x}\n[PROOFSTEP]\nby_cases hzx : z = x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx : \u03b1\nh : \u2191f (\u2191f x) \u2260 x\nhx : \u00ac\u2191f x = x\nz : \u03b1\nhzx : z = x\n\u22a2 z \u2208 support (swap x (\u2191f x) * f) \u2194 z \u2208 support f \\ {x}\n[PROOFSTEP]\nsimp [hzx]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx : \u03b1\nh : \u2191f (\u2191f x) \u2260 x\nhx : \u00ac\u2191f x = x\nz : \u03b1\nhzx : \u00acz = x\n\u22a2 z \u2208 support (swap x (\u2191f x) * f) \u2194 z \u2208 support f \\ {x}\n[PROOFSTEP]\nby_cases hzf : z = f x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx : \u03b1\nh : \u2191f (\u2191f x) \u2260 x\nhx : \u00ac\u2191f x = x\nz : \u03b1\nhzx : \u00acz = x\nhzf : z = \u2191f x\n\u22a2 z \u2208 support (swap x (\u2191f x) * f) \u2194 z \u2208 support f \\ {x}\n[PROOFSTEP]\nsimp [hzf, hx, h, swap_apply_of_ne_of_ne]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx : \u03b1\nh : \u2191f (\u2191f x) \u2260 x\nhx : \u00ac\u2191f x = x\nz : \u03b1\nhzx : \u00acz = x\nhzf : \u00acz = \u2191f x\n\u22a2 z \u2208 support (swap x (\u2191f x) * f) \u2194 z \u2208 support f \\ {x}\n[PROOFSTEP]\nby_cases hzfx : f z = x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx : \u03b1\nh : \u2191f (\u2191f x) \u2260 x\nhx : \u00ac\u2191f x = x\nz : \u03b1\nhzx : \u00acz = x\nhzf : \u00acz = \u2191f x\nhzfx : \u2191f z = x\n\u22a2 z \u2208 support (swap x (\u2191f x) * f) \u2194 z \u2208 support f \\ {x}\n[PROOFSTEP]\nsimp [Ne.symm hzx, hzx, Ne.symm hzf, hzfx]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx : \u03b1\nh : \u2191f (\u2191f x) \u2260 x\nhx : \u00ac\u2191f x = x\nz : \u03b1\nhzx : \u00acz = x\nhzf : \u00acz = \u2191f x\nhzfx : \u00ac\u2191f z = x\n\u22a2 z \u2208 support (swap x (\u2191f x) * f) \u2194 z \u2208 support f \\ {x}\n[PROOFSTEP]\nsimp [Ne.symm hzx, hzx, Ne.symm hzf, hzfx, f.injective.ne hzx, swap_apply_of_ne_of_ne]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhy : y \u2208 support (swap x (\u2191f x) * f)\n\u22a2 y \u2208 support f \u2227 y \u2260 x\n[PROOFSTEP]\nsimp only [mem_support, swap_apply_def, mul_apply, f.injective.eq_iff] at *\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhy : (if \u2191f y = x then \u2191f x else if y = x then x else \u2191f y) \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nby_cases h : f y = x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhy : (if \u2191f y = x then \u2191f x else if y = x then x else \u2191f y) \u2260 y\nh : \u2191f y = x\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase pos.left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhy : (if \u2191f y = x then \u2191f x else if y = x then x else \u2191f y) \u2260 y\nh : \u2191f y = x\n\u22a2 \u2191f y \u2260 y\n[PROOFSTEP]\nintro\n[GOAL]\ncase pos.right\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhy : (if \u2191f y = x then \u2191f x else if y = x then x else \u2191f y) \u2260 y\nh : \u2191f y = x\n\u22a2 y \u2260 x\n[PROOFSTEP]\nintro\n[GOAL]\ncase pos.left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhy : (if \u2191f y = x then \u2191f x else if y = x then x else \u2191f y) \u2260 y\nh : \u2191f y = x\na\u271d : \u2191f y = y\n\u22a2 False\n[PROOFSTEP]\nsimp_all only [if_true, eq_self_iff_true, not_true, Ne.def]\n[GOAL]\ncase pos.right\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhy : (if \u2191f y = x then \u2191f x else if y = x then x else \u2191f y) \u2260 y\nh : \u2191f y = x\na\u271d : y = x\n\u22a2 False\n[PROOFSTEP]\nsimp_all only [if_true, eq_self_iff_true, not_true, Ne.def]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhy : (if \u2191f y = x then \u2191f x else if y = x then x else \u2191f y) \u2260 y\nh : \u00ac\u2191f y = x\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nsplit_ifs at hy with hf heq\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : \u00ac\u2191f y = x\nhf : \u2191f y = x\nhy : \u2191f x \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nsimp_all only [not_true]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : \u00ac\u2191f y = x\nhf : \u00ac\u2191f y = x\nheq : y = x\nhy : x \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nsimp_all only [not_true]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : \u00ac\u2191f y = x\nhf : \u00ac\u2191f y = x\nheq : \u00acy = x\nhy : \u2191f y \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nsimp_all only [not_true]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : \u00ac\u2191f x = x\nheq : y = x\nhy : x \u2260 x\n\u22a2 \u2191f x \u2260 x \u2227 x \u2260 x\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhf : \u00ac\u2191f y = x\nheq : \u00acy = x\nhy : \u2191f y \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nexact \u27e8h, hy\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhf : \u00ac\u2191f y = x\nheq : \u00acy = x\nhy : \u2191f y \u2260 y\n\u22a2 \u2191f y \u2260 y \u2227 y \u2260 x\n[PROOFSTEP]\nrefine' \u27e8hy, heq\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh : f \u2208 l\nhl : List.Pairwise Disjoint l\n\u22a2 \u2200 (x : \u03b1), x \u2208 support f \u2192 \u2191f x = \u2191(List.prod l) x\n[PROOFSTEP]\ninduction' l with hd tl IH\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : f \u2208 l\nhl\u271d : List.Pairwise Disjoint l\nh : f \u2208 []\nhl : List.Pairwise Disjoint []\n\u22a2 \u2200 (x : \u03b1), x \u2208 support f \u2192 \u2191f x = \u2191(List.prod []) x\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : f \u2208 l\nhl\u271d : List.Pairwise Disjoint l\nhd : Perm \u03b1\ntl : List (Perm \u03b1)\nIH : f \u2208 tl \u2192 List.Pairwise Disjoint tl \u2192 \u2200 (x : \u03b1), x \u2208 support f \u2192 \u2191f x = \u2191(List.prod tl) x\nh : f \u2208 hd :: tl\nhl : List.Pairwise Disjoint (hd :: tl)\n\u22a2 \u2200 (x : \u03b1), x \u2208 support f \u2192 \u2191f x = \u2191(List.prod (hd :: tl)) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : f \u2208 l\nhl\u271d : List.Pairwise Disjoint l\nhd : Perm \u03b1\ntl : List (Perm \u03b1)\nIH : f \u2208 tl \u2192 List.Pairwise Disjoint tl \u2192 \u2200 (x : \u03b1), x \u2208 support f \u2192 \u2191f x = \u2191(List.prod tl) x\nh : f \u2208 hd :: tl\nhl : List.Pairwise Disjoint (hd :: tl)\nx : \u03b1\nhx : x \u2208 support f\n\u22a2 \u2191f x = \u2191(List.prod (hd :: tl)) x\n[PROOFSTEP]\nrw [List.pairwise_cons] at hl \n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : f \u2208 l\nhl\u271d : List.Pairwise Disjoint l\nhd : Perm \u03b1\ntl : List (Perm \u03b1)\nIH : f \u2208 tl \u2192 List.Pairwise Disjoint tl \u2192 \u2200 (x : \u03b1), x \u2208 support f \u2192 \u2191f x = \u2191(List.prod tl) x\nh : f \u2208 hd :: tl\nhl : (\u2200 (a' : Perm \u03b1), a' \u2208 tl \u2192 Disjoint hd a') \u2227 List.Pairwise Disjoint tl\nx : \u03b1\nhx : x \u2208 support f\n\u22a2 \u2191f x = \u2191(List.prod (hd :: tl)) x\n[PROOFSTEP]\nrw [List.mem_cons] at h \n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : f \u2208 l\nhl\u271d : List.Pairwise Disjoint l\nhd : Perm \u03b1\ntl : List (Perm \u03b1)\nIH : f \u2208 tl \u2192 List.Pairwise Disjoint tl \u2192 \u2200 (x : \u03b1), x \u2208 support f \u2192 \u2191f x = \u2191(List.prod tl) x\nh : f = hd \u2228 f \u2208 tl\nhl : (\u2200 (a' : Perm \u03b1), a' \u2208 tl \u2192 Disjoint hd a') \u2227 List.Pairwise Disjoint tl\nx : \u03b1\nhx : x \u2208 support f\n\u22a2 \u2191f x = \u2191(List.prod (hd :: tl)) x\n[PROOFSTEP]\nrcases h with (rfl | h)\n[GOAL]\ncase cons.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh : f \u2208 l\nhl\u271d : List.Pairwise Disjoint l\ntl : List (Perm \u03b1)\nIH : f \u2208 tl \u2192 List.Pairwise Disjoint tl \u2192 \u2200 (x : \u03b1), x \u2208 support f \u2192 \u2191f x = \u2191(List.prod tl) x\nx : \u03b1\nhx : x \u2208 support f\nhl : (\u2200 (a' : Perm \u03b1), a' \u2208 tl \u2192 Disjoint f a') \u2227 List.Pairwise Disjoint tl\n\u22a2 \u2191f x = \u2191(List.prod (f :: tl)) x\n[PROOFSTEP]\nrw [List.prod_cons, mul_apply, not_mem_support.mp ((disjoint_prod_right tl hl.left).mem_imp hx)]\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : f \u2208 l\nhl\u271d : List.Pairwise Disjoint l\nhd : Perm \u03b1\ntl : List (Perm \u03b1)\nIH : f \u2208 tl \u2192 List.Pairwise Disjoint tl \u2192 \u2200 (x : \u03b1), x \u2208 support f \u2192 \u2191f x = \u2191(List.prod tl) x\nhl : (\u2200 (a' : Perm \u03b1), a' \u2208 tl \u2192 Disjoint hd a') \u2227 List.Pairwise Disjoint tl\nx : \u03b1\nhx : x \u2208 support f\nh : f \u2208 tl\n\u22a2 \u2191f x = \u2191(List.prod (hd :: tl)) x\n[PROOFSTEP]\nrw [List.prod_cons, mul_apply, \u2190 IH h hl.right _ hx, eq_comm, \u2190 not_mem_support]\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : f \u2208 l\nhl\u271d : List.Pairwise Disjoint l\nhd : Perm \u03b1\ntl : List (Perm \u03b1)\nIH : f \u2208 tl \u2192 List.Pairwise Disjoint tl \u2192 \u2200 (x : \u03b1), x \u2208 support f \u2192 \u2191f x = \u2191(List.prod tl) x\nhl : (\u2200 (a' : Perm \u03b1), a' \u2208 tl \u2192 Disjoint hd a') \u2227 List.Pairwise Disjoint tl\nx : \u03b1\nhx : x \u2208 support f\nh : f \u2208 tl\n\u22a2 \u00ac\u2191f x \u2208 support hd\n[PROOFSTEP]\nrefine' (hl.left _ h).symm.mem_imp _\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : f \u2208 l\nhl\u271d : List.Pairwise Disjoint l\nhd : Perm \u03b1\ntl : List (Perm \u03b1)\nIH : f \u2208 tl \u2192 List.Pairwise Disjoint tl \u2192 \u2200 (x : \u03b1), x \u2208 support f \u2192 \u2191f x = \u2191(List.prod tl) x\nhl : (\u2200 (a' : Perm \u03b1), a' \u2208 tl \u2192 Disjoint hd a') \u2227 List.Pairwise Disjoint tl\nx : \u03b1\nhx : x \u2208 support f\nh : f \u2208 tl\n\u22a2 \u2191f x \u2208 support f\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g x y : Perm \u03b1\nh : Disjoint f g\nhf : support x \u2264 support f\nhg : support y \u2264 support g\n\u22a2 Disjoint x y\n[PROOFSTEP]\nrw [disjoint_iff_disjoint_support] at h \u22a2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g x y : Perm \u03b1\nh\u271d : Disjoint f g\nh : _root_.Disjoint (support f) (support g)\nhf : support x \u2264 support f\nhg : support y \u2264 support g\n\u22a2 _root_.Disjoint (support x) (support y)\n[PROOFSTEP]\nexact h.mono hf hg\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh : f \u2208 l\nhl : List.Pairwise Disjoint l\n\u22a2 support f \u2264 support (List.prod l)\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh : f \u2208 l\nhl : List.Pairwise Disjoint l\nx : \u03b1\nhx : x \u2208 support f\n\u22a2 x \u2208 support (List.prod l)\n[PROOFSTEP]\nrwa [mem_support, \u2190 eq_on_support_mem_disjoint h hl _ hx, \u2190 mem_support]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\n\u22a2 support (extendDomain g f) = map (asEmbedding f) (support g)\n[PROOFSTEP]\next b\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\n\u22a2 b \u2208 support (extendDomain g f) \u2194 b \u2208 map (asEmbedding f) (support g)\n[PROOFSTEP]\nsimp only [exists_prop, Function.Embedding.coeFn_mk, toEmbedding_apply, mem_map, Ne.def, Function.Embedding.trans_apply,\n mem_support]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\n\u22a2 \u00ac\u2191(extendDomain g f) b = b \u2194 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b\n[PROOFSTEP]\nby_cases pb : p b\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\n\u22a2 \u00ac\u2191(extendDomain g f) b = b \u2194 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b\n[PROOFSTEP]\nrw [extendDomain_apply_subtype _ _ pb]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\n\u22a2 \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b \u2194 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase pos.mp\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\n\u22a2 \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b \u2192 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b\n[PROOFSTEP]\nrintro h\n[GOAL]\ncase pos.mp\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nh : \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b\n[PROOFSTEP]\nrefine' \u27e8f.symm \u27e8b, pb\u27e9, _, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nh : \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 \u2191(asEmbedding f) (\u2191f.symm { val := b, property := pb }) = b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos.mp\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nh : \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 \u00ac\u2191g (\u2191f.symm { val := b, property := pb }) = \u2191f.symm { val := b, property := pb }\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase pos.mp\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nh : \u2191g (\u2191f.symm { val := b, property := pb }) = \u2191f.symm { val := b, property := pb }\n\u22a2 \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase pos.mpr\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\n\u22a2 (\u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b) \u2192 \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n[PROOFSTEP]\nrintro \u27e8a, ha, hb\u27e9\n[GOAL]\ncase pos.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\na : \u03b1\nha : \u00ac\u2191g a = a\nhb : \u2191(asEmbedding f) a = b\n\u22a2 \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n[PROOFSTEP]\ncontrapose! ha\n[GOAL]\ncase pos.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\na : \u03b1\nhb : \u2191(asEmbedding f) a = b\nha : \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 \u2191g a = a\n[PROOFSTEP]\nobtain rfl : a = f.symm \u27e8b, pb\u27e9 := by\n rw [eq_symm_apply]\n exact Subtype.coe_injective hb\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\na : \u03b1\nhb : \u2191(asEmbedding f) a = b\nha : \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 a = \u2191f.symm { val := b, property := pb }\n[PROOFSTEP]\nrw [eq_symm_apply]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\na : \u03b1\nhb : \u2191(asEmbedding f) a = b\nha : \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 \u2191f a = { val := b, property := pb }\n[PROOFSTEP]\nexact Subtype.coe_injective hb\n[GOAL]\ncase pos.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nha : \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\nhb : \u2191(asEmbedding f) (\u2191f.symm { val := b, property := pb }) = b\n\u22a2 \u2191g (\u2191f.symm { val := b, property := pb }) = \u2191f.symm { val := b, property := pb }\n[PROOFSTEP]\nrw [eq_symm_apply]\n[GOAL]\ncase pos.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nha : \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\nhb : \u2191(asEmbedding f) (\u2191f.symm { val := b, property := pb }) = b\n\u22a2 \u2191f (\u2191g (\u2191f.symm { val := b, property := pb })) = { val := b, property := pb }\n[PROOFSTEP]\nexact Subtype.coe_injective ha\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : \u00acp b\n\u22a2 \u00ac\u2191(extendDomain g f) b = b \u2194 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b\n[PROOFSTEP]\nrw [extendDomain_apply_not_subtype _ _ pb]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : \u00acp b\n\u22a2 \u00acb = b \u2194 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b\n[PROOFSTEP]\nsimp only [not_exists, false_iff_iff, not_and, eq_self_iff_true, not_true]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : \u00acp b\n\u22a2 \u2200 (x : \u03b1), \u00ac\u2191g x = x \u2192 \u00ac\u2191(asEmbedding f) x = b\n[PROOFSTEP]\nrintro a _ rfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\na : \u03b1\na\u271d : \u00ac\u2191g a = a\npb : \u00acp (\u2191(asEmbedding f) a)\n\u22a2 False\n[PROOFSTEP]\nexact pb (Subtype.prop _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\n\u22a2 card (support (extendDomain g f)) = card (support g)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\n\u22a2 card (support f) = 0 \u2194 f = 1\n[PROOFSTEP]\nrw [Finset.card_eq_zero, support_eq_empty_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : f \u2260 1\n\u22a2 1 < card (support f)\n[PROOFSTEP]\nsimp_rw [one_lt_card_iff, mem_support, \u2190 not_or]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : f \u2260 1\n\u22a2 \u2203 a b, \u00ac(\u2191f a = a \u2228 \u2191f b = b \u2228 a = b)\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : \u2200 (a b : \u03b1), \u2191f a = a \u2228 \u2191f b = b \u2228 a = b\n\u22a2 f = 1\n[PROOFSTEP]\next a\n[GOAL]\ncase H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : \u2200 (a b : \u03b1), \u2191f a = a \u2228 \u2191f b = b \u2228 a = b\na : \u03b1\n\u22a2 \u2191f a = \u21911 a\n[PROOFSTEP]\nspecialize h (f a) a\n[GOAL]\ncase H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\na : \u03b1\nh : \u2191f (\u2191f a) = \u2191f a \u2228 \u2191f a = a \u2228 \u2191f a = a\n\u22a2 \u2191f a = \u21911 a\n[PROOFSTEP]\nrwa [apply_eq_iff_eq, or_self_iff, or_self_iff] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\n\u22a2 card (support f) \u2260 1\n[PROOFSTEP]\nby_cases h : f = 1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : f = 1\n\u22a2 card (support f) \u2260 1\n[PROOFSTEP]\nexact ne_of_eq_of_ne (card_support_eq_zero.mpr h) zero_ne_one\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : \u00acf = 1\n\u22a2 card (support f) \u2260 1\n[PROOFSTEP]\nexact ne_of_gt (one_lt_card_support_of_ne_one h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\n\u22a2 card (support f) \u2264 1 \u2194 f = 1\n[PROOFSTEP]\nrw [le_iff_lt_or_eq, Nat.lt_succ_iff, le_zero_iff, card_support_eq_zero, or_iff_not_imp_right,\n imp_iff_right f.card_support_ne_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nx : \u03b1\nhx : \u2191f x \u2260 x\nh : support f \u2286 support (swap x (\u2191f x) * f)\n\u22a2 \u00ac\u2191(swap x (\u2191f x) * f) x \u2260 x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhxy : x \u2260 y\n\u22a2 Multiset.Nodup (x ::\u2098 y ::\u2098 0)\n[PROOFSTEP]\nsimp [hxy]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhxy : x \u2260 y\n\u22a2 support (swap x y) = { val := x ::\u2098 y ::\u2098 0, nodup := (_ : Multiset.Nodup (x ::\u2098 {y})) }\n[PROOFSTEP]\nsimp [support_swap hxy, *, Finset.ext_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\n\u22a2 card (support f) = 2 \u2194 IsSwap f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\n\u22a2 card (support f) = 2 \u2192 IsSwap f\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\n\u22a2 IsSwap f \u2192 card (support f) = 2\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\n\u22a2 IsSwap f\n[PROOFSTEP]\nobtain \u27e8x, t, hmem, hins, ht\u27e9 := card_eq_succ.1 h\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx : \u03b1\nt : Finset \u03b1\nhmem : \u00acx \u2208 t\nhins : insert x t = support f\nht : card t = 1\n\u22a2 IsSwap f\n[PROOFSTEP]\nobtain \u27e8y, rfl\u27e9 := card_eq_one.1 ht\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx \u2208 {y}\nhins : {x, y} = support f\nht : card {y} = 1\n\u22a2 IsSwap f\n[PROOFSTEP]\nrw [mem_singleton] at hmem \n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\n\u22a2 IsSwap f\n[PROOFSTEP]\nrefine' \u27e8x, y, hmem, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\n\u22a2 f = swap x y\n[PROOFSTEP]\next a\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\n\u22a2 \u2191f a = \u2191(swap x y) a\n[PROOFSTEP]\nhave key : \u2200 b, f b \u2260 b \u2194 _ := fun b => by rw [\u2190 mem_support, \u2190 hins, mem_insert, mem_singleton]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na b : \u03b1\n\u22a2 \u2191f b \u2260 b \u2194 ?m.195733 b\n[PROOFSTEP]\nrw [\u2190 mem_support, \u2190 hins, mem_insert, mem_singleton]\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = y\n\u22a2 \u2191f a = \u2191(swap x y) a\n[PROOFSTEP]\nby_cases ha : f a = a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = y\nha : \u2191f a = a\n\u22a2 \u2191f a = \u2191(swap x y) a\n[PROOFSTEP]\nhave ha' := not_or.mp (mt (key a).mpr (not_not.mpr ha))\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = y\nha : \u2191f a = a\nha' : \u00aca = x \u2227 \u00aca = y\n\u22a2 \u2191f a = \u2191(swap x y) a\n[PROOFSTEP]\nrw [ha, swap_apply_of_ne_of_ne ha'.1 ha'.2]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = y\nha : \u00ac\u2191f a = a\n\u22a2 \u2191f a = \u2191(swap x y) a\n[PROOFSTEP]\nhave ha' := (key (f a)).mp (mt f.apply_eq_iff_eq.mp ha)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = y\nha : \u00ac\u2191f a = a\nha' : \u2191f a = x \u2228 \u2191f a = y\n\u22a2 \u2191f a = \u2191(swap x y) a\n[PROOFSTEP]\nobtain rfl | rfl := (key a).mp ha\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\ny : \u03b1\nht : card {y} = 1\na : \u03b1\nha : \u00ac\u2191f a = a\nhmem : \u00aca = y\nhins : {a, y} = support f\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = a \u2228 b = y\nha' : \u2191f a = a \u2228 \u2191f a = y\n\u22a2 \u2191f a = \u2191(swap a y) a\n[PROOFSTEP]\nrw [Or.resolve_left ha' ha, swap_apply_left]\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx a : \u03b1\nha : \u00ac\u2191f a = a\nhmem : \u00acx = a\nhins : {x, a} = support f\nht : card {a} = 1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = a\nha' : \u2191f a = x \u2228 \u2191f a = a\n\u22a2 \u2191f a = \u2191(swap x a) a\n[PROOFSTEP]\nrw [Or.resolve_right ha' ha, swap_apply_right]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : IsSwap f\n\u22a2 card (support f) = 2\n[PROOFSTEP]\nobtain \u27e8x, y, hxy, rfl\u27e9 := h\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhxy : x \u2260 y\n\u22a2 card (support (swap x y)) = 2\n[PROOFSTEP]\nexact card_support_swap hxy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : Disjoint f g\n\u22a2 card (support (f * g)) = card (support f) + card (support g)\n[PROOFSTEP]\nrw [\u2190 Finset.card_disjoint_union]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : Disjoint f g\n\u22a2 card (support (f * g)) = card (support f \u222a support g)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : Disjoint f g\n\u22a2 support (f * g) = support f \u222a support g\n[PROOFSTEP]\next\n[GOAL]\ncase e_s.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : Disjoint f g\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 support (f * g) \u2194 a\u271d \u2208 support f \u222a support g\n[PROOFSTEP]\nsimp [h.support_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nh : Disjoint f g\n\u22a2 _root_.Disjoint (support f) (support g)\n[PROOFSTEP]\nsimpa using h.disjoint_support\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh : List.Pairwise Disjoint l\n\u22a2 card (support (List.prod l)) = List.sum (List.map (card \u2218 support) l)\n[PROOFSTEP]\ninduction' l with a t ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : List.Pairwise Disjoint l\nh : List.Pairwise Disjoint []\n\u22a2 card (support (List.prod [])) = List.sum (List.map (card \u2218 support) [])\n[PROOFSTEP]\nexact card_support_eq_zero.mpr rfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : List.Pairwise Disjoint l\na : Perm \u03b1\nt : List (Perm \u03b1)\nih : List.Pairwise Disjoint t \u2192 card (support (List.prod t)) = List.sum (List.map (card \u2218 support) t)\nh : List.Pairwise Disjoint (a :: t)\n\u22a2 card (support (List.prod (a :: t))) = List.sum (List.map (card \u2218 support) (a :: t))\n[PROOFSTEP]\nobtain \u27e8ha, ht\u27e9 := List.pairwise_cons.1 h\n[GOAL]\ncase cons.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : List.Pairwise Disjoint l\na : Perm \u03b1\nt : List (Perm \u03b1)\nih : List.Pairwise Disjoint t \u2192 card (support (List.prod t)) = List.sum (List.map (card \u2218 support) t)\nh : List.Pairwise Disjoint (a :: t)\nha : \u2200 (a' : Perm \u03b1), a' \u2208 t \u2192 Disjoint a a'\nht : List.Pairwise Disjoint t\n\u22a2 card (support (List.prod (a :: t))) = List.sum (List.map (card \u2218 support) (a :: t))\n[PROOFSTEP]\nrw [List.prod_cons, List.map_cons, List.sum_cons, \u2190 ih ht]\n[GOAL]\ncase cons.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nl : List (Perm \u03b1)\nh\u271d : List.Pairwise Disjoint l\na : Perm \u03b1\nt : List (Perm \u03b1)\nih : List.Pairwise Disjoint t \u2192 card (support (List.prod t)) = List.sum (List.map (card \u2218 support) t)\nh : List.Pairwise Disjoint (a :: t)\nha : \u2200 (a' : Perm \u03b1), a' \u2208 t \u2192 Disjoint a a'\nht : List.Pairwise Disjoint t\n\u22a2 card (support (a * List.prod t)) = (card \u2218 support) a + card (support (List.prod t))\n[PROOFSTEP]\nexact (disjoint_prod_right _ ha).card_support_mul\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : Perm \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2194 \u2191f x \u2208 s\n\u22a2 support (subtypePerm f h) = filter (fun x => decide (\u2191f \u2191x \u2260 \u2191x) = true) (attach s)\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : Perm \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2194 \u2191f x \u2208 s\na\u271d : { x // x \u2208 s }\n\u22a2 a\u271d \u2208 support (subtypePerm f h) \u2194 a\u271d \u2208 filter (fun x => decide (\u2191f \u2191x \u2260 \u2191x) = true) (attach s)\n[PROOFSTEP]\nsimp [Subtype.ext_iff]\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Perm.Support", "llama_tokens": 37836, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056295505783, "lm_q2_score": 0.6370307875894138, "lm_q1q2_score": 0.4391089080824216}} {"text": "[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\n\u22a2 (m.fst - 1, m.snd) \u2208 Finset.erase b m\n[PROOFSTEP]\nrw [mem_right] at h \n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 b \u2227 (m.fst - 1, m.snd) \u2208 b\n\u22a2 (m.fst - 1, m.snd) \u2208 Finset.erase b m\n[PROOFSTEP]\napply Finset.mem_erase_of_ne_of_mem _ h.2\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 b \u2227 (m.fst - 1, m.snd) \u2208 b\n\u22a2 (m.fst - 1, m.snd) \u2260 m\n[PROOFSTEP]\nexact ne_of_apply_ne Prod.fst (pred_ne_self m.1)\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 left b\n\u22a2 (m.fst, m.snd - 1) \u2208 Finset.erase b m\n[PROOFSTEP]\nrw [mem_left] at h \n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 b \u2227 (m.fst, m.snd - 1) \u2208 b\n\u22a2 (m.fst, m.snd - 1) \u2208 Finset.erase b m\n[PROOFSTEP]\napply Finset.mem_erase_of_ne_of_mem _ h.2\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 b \u2227 (m.fst, m.snd - 1) \u2208 b\n\u22a2 (m.fst, m.snd - 1) \u2260 m\n[PROOFSTEP]\nexact ne_of_apply_ne Prod.snd (pred_ne_self m.2)\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 left b\n\u22a2 2 \u2264 Finset.card b\n[PROOFSTEP]\nhave w\u2081 : m \u2208 b := (Finset.mem_inter.1 h).1\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 left b\nw\u2081 : m \u2208 b\n\u22a2 2 \u2264 Finset.card b\n[PROOFSTEP]\nhave w\u2082 : (m.1, m.2 - 1) \u2208 b.erase m := snd_pred_mem_erase_of_mem_left h\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 left b\nw\u2081 : m \u2208 b\nw\u2082 : (m.fst, m.snd - 1) \u2208 Finset.erase b m\n\u22a2 2 \u2264 Finset.card b\n[PROOFSTEP]\nhave i\u2081 := Finset.card_erase_lt_of_mem w\u2081\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 left b\nw\u2081 : m \u2208 b\nw\u2082 : (m.fst, m.snd - 1) \u2208 Finset.erase b m\ni\u2081 : Finset.card (Finset.erase b m) < Finset.card b\n\u22a2 2 \u2264 Finset.card b\n[PROOFSTEP]\nhave i\u2082 := Nat.lt_of_le_of_lt (Nat.zero_le _) (Finset.card_erase_lt_of_mem w\u2082)\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 left b\nw\u2081 : m \u2208 b\nw\u2082 : (m.fst, m.snd - 1) \u2208 Finset.erase b m\ni\u2081 : Finset.card (Finset.erase b m) < Finset.card b\ni\u2082 : 0 < Finset.card (Finset.erase b m)\n\u22a2 2 \u2264 Finset.card b\n[PROOFSTEP]\nexact Nat.lt_of_le_of_lt i\u2082 i\u2081\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\n\u22a2 2 \u2264 Finset.card b\n[PROOFSTEP]\nhave w\u2081 : m \u2208 b := (Finset.mem_inter.1 h).1\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\nw\u2081 : m \u2208 b\n\u22a2 2 \u2264 Finset.card b\n[PROOFSTEP]\nhave w\u2082 := fst_pred_mem_erase_of_mem_right h\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\nw\u2081 : m \u2208 b\nw\u2082 : (m.fst - 1, m.snd) \u2208 Finset.erase b m\n\u22a2 2 \u2264 Finset.card b\n[PROOFSTEP]\nhave i\u2081 := Finset.card_erase_lt_of_mem w\u2081\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\nw\u2081 : m \u2208 b\nw\u2082 : (m.fst - 1, m.snd) \u2208 Finset.erase b m\ni\u2081 : Finset.card (Finset.erase b m) < Finset.card b\n\u22a2 2 \u2264 Finset.card b\n[PROOFSTEP]\nhave i\u2082 := Nat.lt_of_le_of_lt (Nat.zero_le _) (Finset.card_erase_lt_of_mem w\u2082)\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\nw\u2081 : m \u2208 b\nw\u2082 : (m.fst - 1, m.snd) \u2208 Finset.erase b m\ni\u2081 : Finset.card (Finset.erase b m) < Finset.card b\ni\u2082 : 0 < Finset.card (Finset.erase b m)\n\u22a2 2 \u2264 Finset.card b\n[PROOFSTEP]\nexact Nat.lt_of_le_of_lt i\u2082 i\u2081\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 left b\n\u22a2 Finset.card (moveLeft b m) + 2 = Finset.card b\n[PROOFSTEP]\ndsimp [moveLeft]\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 left b\n\u22a2 Finset.card (Finset.erase (Finset.erase b m) (m.fst, m.snd - 1)) + 2 = Finset.card b\n[PROOFSTEP]\nrw [Finset.card_erase_of_mem (snd_pred_mem_erase_of_mem_left h)]\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 left b\n\u22a2 Finset.card (Finset.erase b m) - 1 + 2 = Finset.card b\n[PROOFSTEP]\nrw [Finset.card_erase_of_mem (Finset.mem_of_mem_inter_left h)]\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 left b\n\u22a2 Finset.card b - 1 - 1 + 2 = Finset.card b\n[PROOFSTEP]\nexact tsub_add_cancel_of_le (card_of_mem_left h)\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\n\u22a2 Finset.card (moveRight b m) + 2 = Finset.card b\n[PROOFSTEP]\ndsimp [moveRight]\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\n\u22a2 Finset.card (Finset.erase (Finset.erase b m) (m.fst - 1, m.snd)) + 2 = Finset.card b\n[PROOFSTEP]\nrw [Finset.card_erase_of_mem (fst_pred_mem_erase_of_mem_right h)]\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\n\u22a2 Finset.card (Finset.erase b m) - 1 + 2 = Finset.card b\n[PROOFSTEP]\nrw [Finset.card_erase_of_mem (Finset.mem_of_mem_inter_left h)]\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\n\u22a2 Finset.card b - 1 - 1 + 2 = Finset.card b\n[PROOFSTEP]\nexact tsub_add_cancel_of_le (card_of_mem_right h)\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 left b\n\u22a2 Finset.card (moveLeft b m) / 2 < Finset.card b / 2\n[PROOFSTEP]\nsimp [\u2190 moveLeft_card h, lt_add_one]\n[GOAL]\nb : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\n\u22a2 Finset.card (moveRight b m) / 2 < Finset.card b / 2\n[PROOFSTEP]\nsimp [\u2190 moveRight_card h, lt_add_one]\n[GOAL]\ns\u271d t\u271d : Board\nm : t\u271d \u2208 (fun s => Finset.image (moveLeft s) (left s)) s\u271d\n\u22a2 (fun s => Finset.card s / 2) t\u271d < (fun s => Finset.card s / 2) s\u271d\n[PROOFSTEP]\nsimp only [Finset.mem_image, Prod.exists] at m \n[GOAL]\ns\u271d t\u271d : Board\nm : \u2203 a b, (a, b) \u2208 left s\u271d \u2227 moveLeft s\u271d (a, b) = t\u271d\n\u22a2 (fun s => Finset.card s / 2) t\u271d < (fun s => Finset.card s / 2) s\u271d\n[PROOFSTEP]\nrcases m with \u27e8_, _, \u27e8h, rfl\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro\ns\u271d : Board\nw\u271d\u00b9 w\u271d : \u2124\nh : (w\u271d\u00b9, w\u271d) \u2208 left s\u271d\n\u22a2 (fun s => Finset.card s / 2) (moveLeft s\u271d (w\u271d\u00b9, w\u271d)) < (fun s => Finset.card s / 2) s\u271d\n[PROOFSTEP]\nexact moveLeft_smaller h\n[GOAL]\ns\u271d t\u271d : Board\nm : t\u271d \u2208 (fun s => Finset.image (moveRight s) (right s)) s\u271d\n\u22a2 (fun s => Finset.card s / 2) t\u271d < (fun s => Finset.card s / 2) s\u271d\n[PROOFSTEP]\nsimp only [Finset.mem_image, Prod.exists] at m \n[GOAL]\ns\u271d t\u271d : Board\nm : \u2203 a b, (a, b) \u2208 right s\u271d \u2227 moveRight s\u271d (a, b) = t\u271d\n\u22a2 (fun s => Finset.card s / 2) t\u271d < (fun s => Finset.card s / 2) s\u271d\n[PROOFSTEP]\nrcases m with \u27e8_, _, \u27e8h, rfl\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro\ns\u271d : Board\nw\u271d\u00b9 w\u271d : \u2124\nh : (w\u271d\u00b9, w\u271d) \u2208 right s\u271d\n\u22a2 (fun s => Finset.card s / 2) (moveRight s\u271d (w\u271d\u00b9, w\u271d)) < (fun s => Finset.card s / 2) s\u271d\n[PROOFSTEP]\nexact moveRight_smaller h\n[GOAL]\nb : Domineering.Board\n\u22a2 Short (domineering b)\n[PROOFSTEP]\ndsimp [domineering]\n[GOAL]\nb : Domineering.Board\n\u22a2 Short (ofState b)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 Short domineering.one\n[PROOFSTEP]\ndsimp [domineering.one]\n[GOAL]\n\u22a2 Short (domineering (List.toFinset [(0, 0), (0, 1)]))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 Short domineering.L\n[PROOFSTEP]\ndsimp [domineering.L]\n[GOAL]\n\u22a2 Short (domineering (List.toFinset [(0, 2), (0, 1), (0, 0), (1, 0)]))\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.SetTheory.Game.Domineering", "llama_tokens": 3298, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149868676283, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4388344150398635}} {"text": "[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhcs : \u2191\u2191\u03bc s \u2260 0\n\u22a2 \u2191\u2191(\u03bc[|s]) Set.univ = 1\n[PROOFSTEP]\nrw [cond, Measure.smul_apply, Measure.restrict_apply MeasurableSet.univ, Set.univ_inter]\n[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhcs : \u2191\u2191\u03bc s \u2260 0\n\u22a2 (\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 \u2191\u2191\u03bc s = 1\n[PROOFSTEP]\nexact ENNReal.inv_mul_cancel hcs (measure_ne_top _ s)\n[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\n\u22a2 \u03bc[|\u2205] = 0\n[PROOFSTEP]\nsimp [cond]\n[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\n\u22a2 \u03bc[|Set.univ] = \u03bc\n[PROOFSTEP]\nsimp [cond, measure_univ, Measure.restrict_univ]\n[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t\u271d : Set \u03a9\nhms : MeasurableSet s\nt : Set \u03a9\n\u22a2 \u2191\u2191(\u03bc[|s]) t = (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191\u03bc (s \u2229 t)\n[PROOFSTEP]\nrw [cond, Measure.smul_apply, Measure.restrict_apply' hms, Set.inter_comm, smul_eq_mul]\n[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t\u271d : Set \u03a9\nhms : MeasurableSet s\nt : Set \u03a9\n\u22a2 \u2191\u2191(\u03bc[|s]) (s \u2229 t) = \u2191\u2191(\u03bc[|s]) t\n[PROOFSTEP]\nrw [cond_apply _ hms, \u2190 Set.inter_assoc, Set.inter_self, \u2190 cond_apply _ hms]\n[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\nhms : MeasurableSet s\nhcst : \u2191\u2191(\u03bc[|s]) t \u2260 0\n\u22a2 0 < \u2191\u2191\u03bc (s \u2229 t)\n[PROOFSTEP]\nrefine' pos_iff_ne_zero.mpr (right_ne_zero_of_mul _)\n[GOAL]\ncase refine'_1\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\nhms : MeasurableSet s\nhcst : \u2191\u2191(\u03bc[|s]) t \u2260 0\n\u22a2 \u211d\u22650\u221e\n[PROOFSTEP]\nexact (\u03bc s)\u207b\u00b9\n[GOAL]\ncase refine'_2\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\nhms : MeasurableSet s\nhcst : \u2191\u2191(\u03bc[|s]) t \u2260 0\n\u22a2 (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191\u03bc (s \u2229 t) \u2260 0\n[PROOFSTEP]\nconvert hcst\n[GOAL]\ncase h.e'_2\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\nhms : MeasurableSet s\nhcst : \u2191\u2191(\u03bc[|s]) t \u2260 0\n\u22a2 (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191(\u03bc[|s]) t\n[PROOFSTEP]\nsimp [hms, Set.inter_comm, cond]\n[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhms : MeasurableSet s\nhci : \u2191\u2191\u03bc (s \u2229 t) \u2260 0\n\u22a2 0 < \u2191\u2191(\u03bc[|s]) t\n[PROOFSTEP]\nrw [cond_apply _ hms]\n[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhms : MeasurableSet s\nhci : \u2191\u2191\u03bc (s \u2229 t) \u2260 0\n\u22a2 0 < (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191\u03bc (s \u2229 t)\n[PROOFSTEP]\nrefine' ENNReal.mul_pos _ hci\n[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhms : MeasurableSet s\nhci : \u2191\u2191\u03bc (s \u2229 t) \u2260 0\n\u22a2 (\u2191\u2191\u03bc s)\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nexact ENNReal.inv_ne_zero.mpr (measure_ne_top _ _)\n[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\nhms : MeasurableSet s\nhmt : MeasurableSet t\nhcs : \u2191\u2191\u03bc s \u2260 \u22a4\nhci : \u2191\u2191\u03bc (s \u2229 t) \u2260 0\n\u22a2 \u03bc[|s][|t] = \u03bc[|s \u2229 t]\n[PROOFSTEP]\nhave hcs' : \u03bc s \u2260 0 := (\u03bc.toOuterMeasure.pos_of_subset_ne_zero (Set.inter_subset_left _ _) hci).ne'\n[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\nhms : MeasurableSet s\nhmt : MeasurableSet t\nhcs : \u2191\u2191\u03bc s \u2260 \u22a4\nhci : \u2191\u2191\u03bc (s \u2229 t) \u2260 0\nhcs' : \u2191\u2191\u03bc s \u2260 0\n\u22a2 \u03bc[|s][|t] = \u03bc[|s \u2229 t]\n[PROOFSTEP]\next u\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\nhms : MeasurableSet s\nhmt : MeasurableSet t\nhcs : \u2191\u2191\u03bc s \u2260 \u22a4\nhci : \u2191\u2191\u03bc (s \u2229 t) \u2260 0\nhcs' : \u2191\u2191\u03bc s \u2260 0\nu : Set \u03a9\na\u271d : MeasurableSet u\n\u22a2 \u2191\u2191(\u03bc[|s][|t]) u = \u2191\u2191(\u03bc[|s \u2229 t]) u\n[PROOFSTEP]\nsimp [*, hms.inter hmt, cond_apply, \u2190 mul_assoc, \u2190 Set.inter_assoc, ENNReal.mul_inv, mul_comm, \u2190 mul_assoc,\n ENNReal.mul_inv_cancel]\n[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t\u271d : Set \u03a9\nhms : MeasurableSet s\nhcs : \u2191\u2191\u03bc s \u2260 0\nhcs' : \u2191\u2191\u03bc s \u2260 \u22a4\nt : Set \u03a9\n\u22a2 \u2191\u2191(\u03bc[|s]) t * \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 t)\n[PROOFSTEP]\nrw [cond_apply \u03bc hms t, mul_comm, \u2190 mul_assoc, ENNReal.mul_inv_cancel hcs hcs', one_mul]\n[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhms : MeasurableSet s\nhcs : \u2191\u2191\u03bc s \u2260 0\nhcs' : \u2191\u2191\u03bc s\u1d9c \u2260 0\n\u22a2 \u2191\u2191(\u03bc[|s]) t * \u2191\u2191\u03bc s + \u2191\u2191(\u03bc[|s\u1d9c]) t * \u2191\u2191\u03bc s\u1d9c = \u2191\u2191\u03bc t\n[PROOFSTEP]\nrw [cond_mul_eq_inter \u03bc hms hcs, cond_mul_eq_inter \u03bc hms.compl hcs', Set.inter_comm _ t, Set.inter_comm _ t]\n[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhms : MeasurableSet s\nhcs : \u2191\u2191\u03bc s \u2260 0\nhcs' : \u2191\u2191\u03bc s\u1d9c \u2260 0\n\u22a2 \u2191\u2191\u03bc (t \u2229 s) + \u2191\u2191\u03bc (t \u2229 s\u1d9c) = \u2191\u2191\u03bc t\n[PROOFSTEP]\nexact measure_inter_add_diff t hms\n[GOAL]\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhms : MeasurableSet s\nhmt : MeasurableSet t\n\u22a2 \u2191\u2191(\u03bc[|s]) t = (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191(\u03bc[|t]) s * \u2191\u2191\u03bc t\n[PROOFSTEP]\nby_cases ht : \u03bc t = 0\n[GOAL]\ncase pos\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhms : MeasurableSet s\nhmt : MeasurableSet t\nht : \u2191\u2191\u03bc t = 0\n\u22a2 \u2191\u2191(\u03bc[|s]) t = (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191(\u03bc[|t]) s * \u2191\u2191\u03bc t\n[PROOFSTEP]\nsimp [cond, ht, Measure.restrict_apply hmt, Or.inr (measure_inter_null_of_null_left s ht)]\n[GOAL]\ncase neg\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhms : MeasurableSet s\nhmt : MeasurableSet t\nht : \u00ac\u2191\u2191\u03bc t = 0\n\u22a2 \u2191\u2191(\u03bc[|s]) t = (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191(\u03bc[|t]) s * \u2191\u2191\u03bc t\n[PROOFSTEP]\nrw [mul_assoc, cond_mul_eq_inter \u03bc hmt ht s, Set.inter_comm, cond_apply _ hms]\n", "meta": {"mathlib_filename": "Mathlib.Probability.ConditionalProbability", "llama_tokens": 2840, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743505760728, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.43867661572913275}} {"text": "[GOAL]\np\u271d p : \u2115 \u2192 Prop\nn : \u2115\n\u22a2 \u2115\n[PROOFSTEP]\nclassical exact\n if h : Set.Finite (setOf p) then (h.toFinset.sort (\u00b7 \u2264 \u00b7)).getD n 0\n else @Nat.Subtype.orderIsoOfNat (setOf p) (Set.Infinite.to_subtype h) n\n[GOAL]\np\u271d p : \u2115 \u2192 Prop\nn : \u2115\n\u22a2 \u2115\n[PROOFSTEP]\nexact\n if h : Set.Finite (setOf p) then (h.toFinset.sort (\u00b7 \u2264 \u00b7)).getD n 0\n else @Nat.Subtype.orderIsoOfNat (setOf p) (Set.Infinite.to_subtype h) n\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Finite (setOf p)\nn : \u2115\nhn : card (Set.Finite.toFinset hf) \u2264 n\n\u22a2 nth p n = 0\n[PROOFSTEP]\nrw [nth, dif_pos hf, List.getD_eq_default]\n[GOAL]\ncase hn\np : \u2115 \u2192 Prop\nhf : Set.Finite (setOf p)\nn : \u2115\nhn : card (Set.Finite.toFinset hf) \u2264 n\n\u22a2 List.length (sort (fun x x_1 => x \u2264 x_1) (Set.Finite.toFinset hf)) \u2264 n\n[PROOFSTEP]\nrwa [Finset.length_sort]\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Finite (setOf p)\nn : \u2115\nhn : n < card (Set.Finite.toFinset hf)\n\u22a2 nth p n =\n \u2191(orderEmbOfFin (Set.Finite.toFinset hf) (_ : card (Set.Finite.toFinset hf) = card (Set.Finite.toFinset hf)))\n { val := n, isLt := hn }\n[PROOFSTEP]\nrw [nth_eq_getD_sort hf, Finset.orderEmbOfFin_apply, List.getD_eq_get]\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Finite (setOf p)\nn : \u2115\nhn : n < card (Set.Finite.toFinset hf)\n\u22a2 List.get (sort (fun x x_1 => x \u2264 x_1) (Set.Finite.toFinset hf)) { val := n, isLt := ?hn } =\n List.nthLe (sort (fun x x_1 => x \u2264 x_1) (Set.Finite.toFinset hf)) \u2191{ val := n, isLt := hn }\n (_ : \u2191{ val := n, isLt := hn } < List.length (sort (fun x x_1 => x \u2264 x_1) (Set.Finite.toFinset hf)))\ncase hn\np : \u2115 \u2192 Prop\nhf : Set.Finite (setOf p)\nn : \u2115\nhn : n < card (Set.Finite.toFinset hf)\n\u22a2 n < List.length (sort (fun x x_1 => x \u2264 x_1) (Set.Finite.toFinset hf))\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Finite (setOf p)\n\u22a2 StrictMonoOn (nth p) (Set.Iio (card (Set.Finite.toFinset hf)))\n[PROOFSTEP]\nrintro m (hm : m < _) n (hn : n < _) h\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Finite (setOf p)\nm : \u2115\nhm : m < card (Set.Finite.toFinset hf)\nn : \u2115\nhn : n < card (Set.Finite.toFinset hf)\nh : m < n\n\u22a2 nth p m < nth p n\n[PROOFSTEP]\nsimp only [nth_eq_orderEmbOfFin, *]\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Finite (setOf p)\nm : \u2115\nhm : m < card (Set.Finite.toFinset hf)\nn : \u2115\nhn : n < card (Set.Finite.toFinset hf)\nh : m < n\n\u22a2 \u2191(orderEmbOfFin (Set.Finite.toFinset (_ : Set.Finite (setOf p)))\n (_ :\n card (Set.Finite.toFinset (_ : Set.Finite (setOf p))) =\n card (Set.Finite.toFinset (_ : Set.Finite (setOf p)))))\n { val := m, isLt := (_ : m < card (Set.Finite.toFinset hf)) } <\n \u2191(orderEmbOfFin (Set.Finite.toFinset (_ : Set.Finite (setOf p)))\n (_ :\n card (Set.Finite.toFinset (_ : Set.Finite (setOf p))) =\n card (Set.Finite.toFinset (_ : Set.Finite (setOf p)))))\n { val := n, isLt := (_ : n < card (Set.Finite.toFinset hf)) }\n[PROOFSTEP]\nexact OrderEmbedding.strictMono _ h\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Finite (setOf p)\n\u22a2 Set.range (nth p) = insert 0 (setOf p)\n[PROOFSTEP]\nsimpa only [\u2190 nth_eq_getD_sort hf, mem_sort, Set.Finite.mem_toFinset] using\n Set.range_list_getD (hf.toFinset.sort (\u00b7 \u2264 \u00b7)) 0\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Finite (setOf p)\n\u22a2 nth p '' Set.Iio (card (Set.Finite.toFinset hf)) =\n Set.range\n \u2191(orderEmbOfFin (Set.Finite.toFinset hf) (_ : card (Set.Finite.toFinset hf) = card (Set.Finite.toFinset hf)))\n[PROOFSTEP]\next x\n[GOAL]\ncase h\np : \u2115 \u2192 Prop\nhf : Set.Finite (setOf p)\nx : \u2115\n\u22a2 x \u2208 nth p '' Set.Iio (card (Set.Finite.toFinset hf)) \u2194\n x \u2208\n Set.range\n \u2191(orderEmbOfFin (Set.Finite.toFinset hf) (_ : card (Set.Finite.toFinset hf) = card (Set.Finite.toFinset hf)))\n[PROOFSTEP]\nsimp only [Set.mem_image, Set.mem_range, Fin.exists_iff, \u2190 nth_eq_orderEmbOfFin hf, Set.mem_Iio, exists_prop]\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Finite (setOf p)\n\u22a2 Set.range\n \u2191(orderEmbOfFin (Set.Finite.toFinset hf) (_ : card (Set.Finite.toFinset hf) = card (Set.Finite.toFinset hf))) =\n setOf p\n[PROOFSTEP]\nrw [range_orderEmbOfFin, Set.Finite.coe_toFinset]\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Finite (setOf p)\nx : \u2115\nh : p x\n\u22a2 \u2203 n, n < card (Set.Finite.toFinset hf) \u2227 nth p n = x\n[PROOFSTEP]\nrwa [\u2190 @Set.mem_setOf_eq _ _ p, \u2190 image_nth_Iio_card hf] at h \n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Infinite (setOf p)\nn : \u2115\n\u22a2 nth p n = \u2191(\u2191(Subtype.orderIsoOfNat (setOf p)) n)\n[PROOFSTEP]\nrw [nth, dif_neg hf]\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Infinite (setOf p)\n\u22a2 StrictMono (nth p)\n[PROOFSTEP]\nrw [nth_eq_orderIsoOfNat hf]\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Infinite (setOf p)\n\u22a2 StrictMono (Subtype.val \u2218 \u2191(Subtype.orderIsoOfNat (setOf p)))\n[PROOFSTEP]\nexact (Subtype.strictMono_coe _).comp (OrderIso.strictMono _)\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Infinite (setOf p)\n\u22a2 Set.range (nth p) = setOf p\n[PROOFSTEP]\nrw [nth_eq_orderIsoOfNat hf]\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Infinite (setOf p)\n\u22a2 Set.range (Subtype.val \u2218 \u2191(Subtype.orderIsoOfNat (setOf p))) = setOf p\n[PROOFSTEP]\nhaveI := hf.to_subtype\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Infinite (setOf p)\nthis : Infinite \u2191(setOf p)\n\u22a2 Set.range (Subtype.val \u2218 \u2191(Subtype.orderIsoOfNat (setOf p))) = setOf p\n[PROOFSTEP]\nclassical exact Nat.Subtype.coe_comp_ofNat_range\n[GOAL]\np : \u2115 \u2192 Prop\nhf : Set.Infinite (setOf p)\nthis : Infinite \u2191(setOf p)\n\u22a2 Set.range (Subtype.val \u2218 \u2191(Subtype.orderIsoOfNat (setOf p))) = setOf p\n[PROOFSTEP]\nexact Nat.Subtype.coe_comp_ofNat_range\n[GOAL]\np : \u2115 \u2192 Prop\nx : \u2115\nh : p x\n\u22a2 \u2203 n, (\u2200 (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)) \u2227 nth p n = x\n[PROOFSTEP]\nrefine' (setOf p).finite_or_infinite.elim (fun hf => _) fun hf => _\n[GOAL]\ncase refine'_1\np : \u2115 \u2192 Prop\nx : \u2115\nh : p x\nhf : Set.Finite (setOf p)\n\u22a2 \u2203 n, (\u2200 (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)) \u2227 nth p n = x\n[PROOFSTEP]\nrcases exists_lt_card_finite_nth_eq hf h with \u27e8n, hn, hx\u27e9\n[GOAL]\ncase refine'_1.intro.intro\np : \u2115 \u2192 Prop\nx : \u2115\nh : p x\nhf : Set.Finite (setOf p)\nn : \u2115\nhn : n < card (Set.Finite.toFinset hf)\nhx : nth p n = x\n\u22a2 \u2203 n, (\u2200 (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)) \u2227 nth p n = x\n[PROOFSTEP]\nexact \u27e8n, fun _ => hn, hx\u27e9\n[GOAL]\ncase refine'_2\np : \u2115 \u2192 Prop\nx : \u2115\nh : p x\nhf : Set.Infinite (setOf p)\n\u22a2 \u2203 n, (\u2200 (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)) \u2227 nth p n = x\n[PROOFSTEP]\nrw [\u2190 @Set.mem_setOf_eq _ _ p, \u2190 range_nth_of_infinite hf] at h \n[GOAL]\ncase refine'_2\np : \u2115 \u2192 Prop\nx : \u2115\nh : x \u2208 Set.range (nth p)\nhf : Set.Infinite (setOf p)\n\u22a2 \u2203 n, (\u2200 (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)) \u2227 nth p n = x\n[PROOFSTEP]\nrcases h with \u27e8n, hx\u27e9\n[GOAL]\ncase refine'_2.intro\np : \u2115 \u2192 Prop\nx : \u2115\nhf : Set.Infinite (setOf p)\nn : \u2115\nhx : nth p n = x\n\u22a2 \u2203 n, (\u2200 (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)) \u2227 nth p n = x\n[PROOFSTEP]\nexact \u27e8n, fun hf' => absurd hf' hf, hx\u27e9\n[GOAL]\np\u271d p : \u2115 \u2192 Prop\nn : \u2115\n\u22a2 nth p n = sInf {x | p x \u2227 \u2200 (k : \u2115), k < n \u2192 nth p k < x}\n[PROOFSTEP]\nby_cases hn : \u2200 hf : (setOf p).Finite, n < hf.toFinset.card\n[GOAL]\ncase pos\np\u271d p : \u2115 \u2192 Prop\nn : \u2115\nhn : \u2200 (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)\n\u22a2 nth p n = sInf {x | p x \u2227 \u2200 (k : \u2115), k < n \u2192 nth p k < x}\n[PROOFSTEP]\nexact (isLeast_nth hn).csInf_eq.symm\n[GOAL]\ncase neg\np\u271d p : \u2115 \u2192 Prop\nn : \u2115\nhn : \u00ac\u2200 (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)\n\u22a2 nth p n = sInf {x | p x \u2227 \u2200 (k : \u2115), k < n \u2192 nth p k < x}\n[PROOFSTEP]\npush_neg at hn \n[GOAL]\ncase neg\np\u271d p : \u2115 \u2192 Prop\nn : \u2115\nhn : \u2203 h, card (Set.Finite.toFinset (_ : Set.Finite (setOf p))) \u2264 n\n\u22a2 nth p n = sInf {x | p x \u2227 \u2200 (k : \u2115), k < n \u2192 nth p k < x}\n[PROOFSTEP]\nrcases hn with \u27e8hf, hn\u27e9\n[GOAL]\ncase neg.intro\np\u271d p : \u2115 \u2192 Prop\nn : \u2115\nhf : Set.Finite (setOf p)\nhn : card (Set.Finite.toFinset (_ : Set.Finite (setOf p))) \u2264 n\n\u22a2 nth p n = sInf {x | p x \u2227 \u2200 (k : \u2115), k < n \u2192 nth p k < x}\n[PROOFSTEP]\nrw [nth_of_card_le _ hn]\n[GOAL]\ncase neg.intro\np\u271d p : \u2115 \u2192 Prop\nn : \u2115\nhf : Set.Finite (setOf p)\nhn : card (Set.Finite.toFinset (_ : Set.Finite (setOf p))) \u2264 n\n\u22a2 0 = sInf {x | p x \u2227 \u2200 (k : \u2115), k < n \u2192 nth p k < x}\n[PROOFSTEP]\nrefine' ((congr_arg sInf <| Set.eq_empty_of_forall_not_mem fun k hk => _).trans sInf_empty).symm\n[GOAL]\ncase neg.intro\np\u271d p : \u2115 \u2192 Prop\nn : \u2115\nhf : Set.Finite (setOf p)\nhn : card (Set.Finite.toFinset (_ : Set.Finite (setOf p))) \u2264 n\nk : \u2115\nhk : k \u2208 {x | p x \u2227 \u2200 (k : \u2115), k < n \u2192 nth p k < x}\n\u22a2 False\n[PROOFSTEP]\nrcases exists_lt_card_nth_eq hk.1 with \u27e8k, hlt, rfl\u27e9\n[GOAL]\ncase neg.intro.intro.intro\np\u271d p : \u2115 \u2192 Prop\nn : \u2115\nhf : Set.Finite (setOf p)\nhn : card (Set.Finite.toFinset (_ : Set.Finite (setOf p))) \u2264 n\nk : \u2115\nhlt : \u2200 (hf : Set.Finite (setOf p)), k < card (Set.Finite.toFinset hf)\nhk : nth p k \u2208 {x | p x \u2227 \u2200 (k : \u2115), k < n \u2192 nth p k < x}\n\u22a2 False\n[PROOFSTEP]\nexact (hk.2 _ ((hlt hf).trans_le hn)).false\n[GOAL]\np : \u2115 \u2192 Prop\n\u22a2 nth p 0 = sInf (setOf p)\n[PROOFSTEP]\nrw [nth_eq_sInf]\n[GOAL]\np : \u2115 \u2192 Prop\n\u22a2 sInf {x | p x \u2227 \u2200 (k : \u2115), k < 0 \u2192 nth p k < x} = sInf (setOf p)\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115 \u2192 Prop\nh : p 0\n\u22a2 nth p 0 = 0\n[PROOFSTEP]\nsimp [nth_zero, h]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nh : \u2203 n, p n\n\u22a2 nth p 0 = Nat.find h\n[PROOFSTEP]\nrw [nth_zero]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nh : \u2203 n, p n\n\u22a2 sInf (setOf p) = Nat.find h\n[PROOFSTEP]\nconvert Nat.sInf_def h\n[GOAL]\np : \u2115 \u2192 Prop\nn : \u2115\n\u22a2 nth p n = 0 \u2194 p 0 \u2227 n = 0 \u2228 \u2203 hf, card (Set.Finite.toFinset hf) \u2264 n\n[PROOFSTEP]\nrefine' \u27e8fun h => _, _\u27e9\n[GOAL]\ncase refine'_1\np : \u2115 \u2192 Prop\nn : \u2115\nh : nth p n = 0\n\u22a2 p 0 \u2227 n = 0 \u2228 \u2203 hf, card (Set.Finite.toFinset hf) \u2264 n\n[PROOFSTEP]\nsimp only [or_iff_not_imp_right, not_exists, not_le]\n[GOAL]\ncase refine'_1\np : \u2115 \u2192 Prop\nn : \u2115\nh : nth p n = 0\n\u22a2 (\u2200 (x : Set.Finite (setOf p)), n < card (Set.Finite.toFinset x)) \u2192 p 0 \u2227 n = 0\n[PROOFSTEP]\nexact fun hn => \u27e8h \u25b8 nth_mem _ hn, nonpos_iff_eq_zero.1 <| h \u25b8 le_nth hn\u27e9\n[GOAL]\ncase refine'_2\np : \u2115 \u2192 Prop\nn : \u2115\n\u22a2 (p 0 \u2227 n = 0 \u2228 \u2203 hf, card (Set.Finite.toFinset hf) \u2264 n) \u2192 nth p n = 0\n[PROOFSTEP]\nrintro (\u27e8h\u2080, rfl\u27e9 | \u27e8hf, hle\u27e9)\n[GOAL]\ncase refine'_2.inl.intro\np : \u2115 \u2192 Prop\nh\u2080 : p 0\n\u22a2 nth p 0 = 0\ncase refine'_2.inr.intro\np : \u2115 \u2192 Prop\nn : \u2115\nhf : Set.Finite (setOf p)\nhle : card (Set.Finite.toFinset hf) \u2264 n\n\u22a2 nth p n = 0\n[PROOFSTEP]\nexacts [nth_zero_of_zero h\u2080, nth_of_card_le hf hle]\n[GOAL]\np : \u2115 \u2192 Prop\nh\u2080 : \u00acp 0\na b : \u2115\nhab : a \u2264 b\nha : nth p a = 0\n\u22a2 nth p b = 0\n[PROOFSTEP]\nsimp only [nth_eq_zero, h\u2080, false_and_iff, false_or_iff] at ha \u22a2\n[GOAL]\np : \u2115 \u2192 Prop\nh\u2080 : \u00acp 0\na b : \u2115\nhab : a \u2264 b\nha : \u2203 hf, card (Set.Finite.toFinset hf) \u2264 a\n\u22a2 \u2203 hf, card (Set.Finite.toFinset hf) \u2264 b\n[PROOFSTEP]\nexact ha.imp fun hf hle => hle.trans hab\n[GOAL]\np : \u2115 \u2192 Prop\nk a : \u2115\nh : a < nth p (k + 1)\nha : p a\n\u22a2 a \u2264 nth p k\n[PROOFSTEP]\ncases' (setOf p).finite_or_infinite with hf hf\n[GOAL]\ncase inl\np : \u2115 \u2192 Prop\nk a : \u2115\nh : a < nth p (k + 1)\nha : p a\nhf : Set.Finite (setOf p)\n\u22a2 a \u2264 nth p k\n[PROOFSTEP]\nrcases exists_lt_card_finite_nth_eq hf ha with \u27e8n, hn, rfl\u27e9\n[GOAL]\ncase inl.intro.intro\np : \u2115 \u2192 Prop\nk : \u2115\nhf : Set.Finite (setOf p)\nn : \u2115\nhn : n < card (Set.Finite.toFinset hf)\nh : nth p n < nth p (k + 1)\nha : p (nth p n)\n\u22a2 nth p n \u2264 nth p k\n[PROOFSTEP]\ncases' lt_or_le (k + 1) hf.toFinset.card with hk hk\n[GOAL]\ncase inl.intro.intro.inl\np : \u2115 \u2192 Prop\nk : \u2115\nhf : Set.Finite (setOf p)\nn : \u2115\nhn : n < card (Set.Finite.toFinset hf)\nh : nth p n < nth p (k + 1)\nha : p (nth p n)\nhk : k + 1 < card (Set.Finite.toFinset hf)\n\u22a2 nth p n \u2264 nth p k\n[PROOFSTEP]\nrwa [(nth_strictMonoOn hf).lt_iff_lt hn hk, lt_succ_iff, \u2190\n (nth_strictMonoOn hf).le_iff_le hn (k.lt_succ_self.trans hk)] at h \n[GOAL]\ncase inl.intro.intro.inr\np : \u2115 \u2192 Prop\nk : \u2115\nhf : Set.Finite (setOf p)\nn : \u2115\nhn : n < card (Set.Finite.toFinset hf)\nh : nth p n < nth p (k + 1)\nha : p (nth p n)\nhk : card (Set.Finite.toFinset hf) \u2264 k + 1\n\u22a2 nth p n \u2264 nth p k\n[PROOFSTEP]\nrw [nth_of_card_le _ hk] at h \n[GOAL]\ncase inl.intro.intro.inr\np : \u2115 \u2192 Prop\nk : \u2115\nhf : Set.Finite (setOf p)\nn : \u2115\nhn : n < card (Set.Finite.toFinset hf)\nh : nth p n < 0\nha : p (nth p n)\nhk : card (Set.Finite.toFinset hf) \u2264 k + 1\n\u22a2 nth p n \u2264 nth p k\n[PROOFSTEP]\nexact absurd h (zero_le _).not_lt\n[GOAL]\ncase inr\np : \u2115 \u2192 Prop\nk a : \u2115\nh : a < nth p (k + 1)\nha : p a\nhf : Set.Infinite (setOf p)\n\u22a2 a \u2264 nth p k\n[PROOFSTEP]\nrcases subset_range_nth ha with \u27e8n, rfl\u27e9\n[GOAL]\ncase inr.intro\np : \u2115 \u2192 Prop\nk : \u2115\nhf : Set.Infinite (setOf p)\nn : \u2115\nh : nth p n < nth p (k + 1)\nha : p (nth p n)\n\u22a2 nth p n \u2264 nth p k\n[PROOFSTEP]\nrwa [nth_lt_nth hf, lt_succ_iff, \u2190 nth_le_nth hf] at h \n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 count p (nth p 0) = 0\n[PROOFSTEP]\nrw [count_eq_card_filter_range, card_eq_zero, filter_eq_empty_iff, nth_zero]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 \u2200 (x : \u2115), x \u2208 range (sInf (setOf p)) \u2192 \u00acp x\n[PROOFSTEP]\nexact fun n h\u2081 h\u2082 => (mem_range.1 h\u2081).not_le (Nat.sInf_le h\u2082)\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nk : \u2115\n\u22a2 filter p (range (nth p (k + 1))) \u2286 insert (nth p k) (filter p (range (nth p k)))\n[PROOFSTEP]\nintro a ha\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nk a : \u2115\nha : a \u2208 filter p (range (nth p (k + 1)))\n\u22a2 a \u2208 insert (nth p k) (filter p (range (nth p k)))\n[PROOFSTEP]\nsimp only [mem_insert, mem_filter, mem_range] at ha \u22a2\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nk a : \u2115\nha : a < nth p (k + 1) \u2227 p a\n\u22a2 a = nth p k \u2228 a < nth p k \u2227 p a\n[PROOFSTEP]\nexact (le_nth_of_lt_nth_succ ha.1 ha.2).eq_or_lt.imp_right fun h => \u27e8h, ha.2\u27e9\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nk : \u2115\nhlt : \u2200 (hf : Set.Finite (setOf p)), k + 1 < card (Set.Finite.toFinset hf)\n\u22a2 filter p (range (nth p (k + 1))) = insert (nth p k) (filter p (range (nth p k)))\n[PROOFSTEP]\nrefine' (filter_range_nth_subset_insert p k).antisymm fun a ha => _\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nk : \u2115\nhlt : \u2200 (hf : Set.Finite (setOf p)), k + 1 < card (Set.Finite.toFinset hf)\na : \u2115\nha : a \u2208 insert (nth p k) (filter p (range (nth p k)))\n\u22a2 a \u2208 filter p (range (nth p (k + 1)))\n[PROOFSTEP]\nsimp only [mem_insert, mem_filter, mem_range] at ha \u22a2\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nk : \u2115\nhlt : \u2200 (hf : Set.Finite (setOf p)), k + 1 < card (Set.Finite.toFinset hf)\na : \u2115\nha : a = nth p k \u2228 a < nth p k \u2227 p a\n\u22a2 a < nth p (k + 1) \u2227 p a\n[PROOFSTEP]\nhave : nth p k < nth p (k + 1) := nth_lt_nth' k.lt_succ_self hlt\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nk : \u2115\nhlt : \u2200 (hf : Set.Finite (setOf p)), k + 1 < card (Set.Finite.toFinset hf)\na : \u2115\nha : a = nth p k \u2228 a < nth p k \u2227 p a\nthis : nth p k < nth p (k + 1)\n\u22a2 a < nth p (k + 1) \u2227 p a\n[PROOFSTEP]\nrcases ha with (rfl | \u27e8hlt, hpa\u27e9)\n[GOAL]\ncase inl\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nk : \u2115\nhlt : \u2200 (hf : Set.Finite (setOf p)), k + 1 < card (Set.Finite.toFinset hf)\nthis : nth p k < nth p (k + 1)\n\u22a2 nth p k < nth p (k + 1) \u2227 p (nth p k)\n[PROOFSTEP]\nexact \u27e8this, nth_mem _ fun hf => k.lt_succ_self.trans (hlt hf)\u27e9\n[GOAL]\ncase inr.intro\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nk : \u2115\nhlt\u271d : \u2200 (hf : Set.Finite (setOf p)), k + 1 < card (Set.Finite.toFinset hf)\na : \u2115\nthis : nth p k < nth p (k + 1)\nhlt : a < nth p k\nhpa : p a\n\u22a2 a < nth p (k + 1) \u2227 p a\n[PROOFSTEP]\nexact \u27e8hlt.trans this, hpa\u27e9\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\nhn : \u2200 (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)\n\u22a2 count p (nth p n) = n\n[PROOFSTEP]\ninduction' n with k ihk\n[GOAL]\ncase zero\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\nhn\u271d : \u2200 (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)\nhn : \u2200 (hf : Set.Finite (setOf p)), zero < card (Set.Finite.toFinset hf)\n\u22a2 count p (nth p zero) = zero\n[PROOFSTEP]\nexact count_nth_zero _\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\nhn\u271d : \u2200 (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)\nk : \u2115\nihk : (\u2200 (hf : Set.Finite (setOf p)), k < card (Set.Finite.toFinset hf)) \u2192 count p (nth p k) = k\nhn : \u2200 (hf : Set.Finite (setOf p)), succ k < card (Set.Finite.toFinset hf)\n\u22a2 count p (nth p (succ k)) = succ k\n[PROOFSTEP]\nrw [count_eq_card_filter_range, filter_range_nth_eq_insert hn, card_insert_of_not_mem, \u2190 count_eq_card_filter_range,\n ihk fun hf => lt_of_succ_lt (hn hf)]\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\nhn\u271d : \u2200 (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)\nk : \u2115\nihk : (\u2200 (hf : Set.Finite (setOf p)), k < card (Set.Finite.toFinset hf)) \u2192 count p (nth p k) = k\nhn : \u2200 (hf : Set.Finite (setOf p)), succ k < card (Set.Finite.toFinset hf)\n\u22a2 \u00acnth p k \u2208 filter p (range (nth p k))\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\nhn : \u2200 (hf : Set.Finite (setOf p)), n < card (Set.Finite.toFinset hf)\n\u22a2 count p (nth p n + 1) = n + 1\n[PROOFSTEP]\nrw [count_succ, count_nth hn, if_pos (nth_mem _ hn)]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn k : \u2115\nh : k < count p n\n\u22a2 nth p k < n\n[PROOFSTEP]\nrefine' (count_monotone p).reflect_lt _\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn k : \u2115\nh : k < count p n\n\u22a2 count p (nth p k) < count p n\n[PROOFSTEP]\nrwa [count_nth]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn k : \u2115\nh : k < count p n\n\u22a2 \u2200 (hf : Set.Finite (setOf p)), k < card (Set.Finite.toFinset hf)\n[PROOFSTEP]\nexact fun hf => h.trans_le (count_le_card hf n)\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 nth p (count p n) = sInf {i | p i \u2227 n \u2264 i}\n[PROOFSTEP]\nrefine' (nth_eq_sInf _ _).trans (congr_arg sInf _)\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 {x | p x \u2227 \u2200 (k : \u2115), k < count p n \u2192 nth p k < x} = {i | p i \u2227 n \u2264 i}\n[PROOFSTEP]\nrefine' Set.ext fun a => and_congr_right fun hpa => _\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn a : \u2115\nhpa : p a\n\u22a2 (\u2200 (k : \u2115), k < count p n \u2192 nth p k < a) \u2194 n \u2264 a\n[PROOFSTEP]\nrefine' \u27e8fun h => not_lt.1 fun ha => _, fun hn k hk => lt_of_lt_of_le (nth_lt_of_lt_count hk) hn\u27e9\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn a : \u2115\nhpa : p a\nh : \u2200 (k : \u2115), k < count p n \u2192 nth p k < a\nha : a < n\n\u22a2 False\n[PROOFSTEP]\nhave hn : nth p (count p a) < a := h _ (count_strict_mono hpa ha)\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn a : \u2115\nhpa : p a\nh : \u2200 (k : \u2115), k < count p n \u2192 nth p k < a\nha : a < n\nhn : nth p (count p a) < a\n\u22a2 False\n[PROOFSTEP]\nrwa [nth_count hpa, lt_self_iff_false] at hn \n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Nth", "llama_tokens": 9081, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833841649232, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.438428552492269}} {"text": "[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.41, u_3} E\nF G : Karoubi C \u2964 D\n\u03c6 : F \u27f6 G\nP : Karoubi C\n\u22a2 NatTrans.app \u03c6 P = F.map (decompId_i P) \u226b NatTrans.app \u03c6 (Karoubi.mk P.X (\ud835\udfd9 P.X)) \u226b G.map (decompId_p P)\n[PROOFSTEP]\nrw [\u2190 \u03c6.naturality, \u2190 assoc, \u2190 F.map_comp]\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.41, u_3} E\nF G : Karoubi C \u2964 D\n\u03c6 : F \u27f6 G\nP : Karoubi C\n\u22a2 NatTrans.app \u03c6 P = F.map (decompId_i P \u226b decompId_p P) \u226b NatTrans.app \u03c6 P\n[PROOFSTEP]\nconv =>\n lhs\n rw [\u2190 id_comp (\u03c6.app P), \u2190 F.map_id]\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.41, u_3} E\nF G : Karoubi C \u2964 D\n\u03c6 : F \u27f6 G\nP : Karoubi C\n| NatTrans.app \u03c6 P = F.map (decompId_i P \u226b decompId_p P) \u226b NatTrans.app \u03c6 P\n[PROOFSTEP]\n lhs\n rw [\u2190 id_comp (\u03c6.app P), \u2190 F.map_id]\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.41, u_3} E\nF G : Karoubi C \u2964 D\n\u03c6 : F \u27f6 G\nP : Karoubi C\n| NatTrans.app \u03c6 P = F.map (decompId_i P \u226b decompId_p P) \u226b NatTrans.app \u03c6 P\n[PROOFSTEP]\n lhs\n rw [\u2190 id_comp (\u03c6.app P), \u2190 F.map_id]\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.41, u_3} E\nF G : Karoubi C \u2964 D\n\u03c6 : F \u27f6 G\nP : Karoubi C\n| NatTrans.app \u03c6 P = F.map (decompId_i P \u226b decompId_p P) \u226b NatTrans.app \u03c6 P\n[PROOFSTEP]\nlhs\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.41, u_3} E\nF G : Karoubi C \u2964 D\n\u03c6 : F \u27f6 G\nP : Karoubi C\n| NatTrans.app \u03c6 P\n[PROOFSTEP]\nrw [\u2190 id_comp (\u03c6.app P), \u2190 F.map_id]\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.41, u_3} E\nF G : Karoubi C \u2964 D\n\u03c6 : F \u27f6 G\nP : Karoubi C\n\u22a2 F.map (\ud835\udfd9 P) \u226b NatTrans.app \u03c6 P = F.map (decompId_i P \u226b decompId_p P) \u226b NatTrans.app \u03c6 P\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.41, u_3} E\nF G : Karoubi C \u2964 D\n\u03c6 : F \u27f6 G\nP : Karoubi C\n\u22a2 \ud835\udfd9 P = decompId_i P \u226b decompId_p P\n[PROOFSTEP]\napply decompId\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.1588, u_1} C\ninst\u271d\u00b9 : Category.{?u.1592, u_2} D\ninst\u271d : Category.{?u.1596, u_3} E\nF : C \u2964 Karoubi D\nP : Karoubi C\n\u22a2 (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n[PROOFSTEP]\nsimpa only [F.map_comp, hom_ext_iff] using F.congr_map P.idem\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.1588, u_1} C\ninst\u271d\u00b9 : Category.{?u.1592, u_2} D\ninst\u271d : Category.{?u.1596, u_3} E\nF : C \u2964 Karoubi D\nX\u271d Y\u271d : Karoubi C\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (F.map f.f).f =\n ((fun P => Karoubi.mk (F.obj P.X).X (F.map P.p).f) X\u271d).p \u226b\n (F.map f.f).f \u226b ((fun P => Karoubi.mk (F.obj P.X).X (F.map P.p).f) Y\u271d).p\n[PROOFSTEP]\nsimpa only [F.map_comp, hom_ext_iff] using F.congr_map f.comm\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nP : Karoubi C\n\u22a2 (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f =\n ((obj F).obj P).p \u226b ((F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f) \u226b ((obj G).obj P).p\n[PROOFSTEP]\nhave h := \u03c6.naturality P.p\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nP : Karoubi C\nh : F.map P.p \u226b NatTrans.app \u03c6 P.X = NatTrans.app \u03c6 P.X \u226b G.map P.p\n\u22a2 (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f =\n ((obj F).obj P).p \u226b ((F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f) \u226b ((obj G).obj P).p\n[PROOFSTEP]\nhave h' := F.congr_map P.idem\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nP : Karoubi C\nh : F.map P.p \u226b NatTrans.app \u03c6 P.X = NatTrans.app \u03c6 P.X \u226b G.map P.p\nh' : F.map (P.p \u226b P.p) = F.map P.p\n\u22a2 (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f =\n ((obj F).obj P).p \u226b ((F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f) \u226b ((obj G).obj P).p\n[PROOFSTEP]\nsimp only [hom_ext_iff, Karoubi.comp_f, F.map_comp] at h h' \n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n\u22a2 (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f =\n ((obj F).obj P).p \u226b ((F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f) \u226b ((obj G).obj P).p\n[PROOFSTEP]\nsimp only [obj_obj_p, assoc, \u2190 h]\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n\u22a2 (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (F.map P.p).f \u226b (F.map P.p).f \u226b (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f\n[PROOFSTEP]\nslice_rhs 1 3 => rw [h', h']\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (F.map P.p).f \u226b (F.map P.p).f \u226b (F.map P.p).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (NatTrans.app \u03c6 P.X).f\n[PROOFSTEP]\nrw [h', h']\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (F.map P.p).f \u226b (F.map P.p).f \u226b (F.map P.p).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (NatTrans.app \u03c6 P.X).f\n[PROOFSTEP]\nrw [h', h']\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (F.map P.p).f \u226b (F.map P.p).f \u226b (F.map P.p).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (NatTrans.app \u03c6 P.X).f\n[PROOFSTEP]\nrw [h', h']\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 (obj F).map f \u226b (fun P => Hom.mk ((F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f)) x\u271d =\n (fun P => Hom.mk ((F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f)) x\u271d\u00b9 \u226b (obj G).map f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 ((obj F).map f \u226b (fun P => Hom.mk ((F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f)) x\u271d).f =\n ((fun P => Hom.mk ((F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f)) x\u271d\u00b9 \u226b (obj G).map f).f\n[PROOFSTEP]\ndsimp [obj]\n[GOAL]\ncase h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 (F.map f.f).f \u226b (F.map x\u271d.p).f \u226b (NatTrans.app \u03c6 x\u271d.X).f =\n ((F.map x\u271d\u00b9.p).f \u226b (NatTrans.app \u03c6 x\u271d\u00b9.X).f) \u226b (G.map f.f).f\n[PROOFSTEP]\nhave h := \u03c6.naturality f.f\n[GOAL]\ncase h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : F.map f.f \u226b NatTrans.app \u03c6 x\u271d.X = NatTrans.app \u03c6 x\u271d\u00b9.X \u226b G.map f.f\n\u22a2 (F.map f.f).f \u226b (F.map x\u271d.p).f \u226b (NatTrans.app \u03c6 x\u271d.X).f =\n ((F.map x\u271d\u00b9.p).f \u226b (NatTrans.app \u03c6 x\u271d\u00b9.X).f) \u226b (G.map f.f).f\n[PROOFSTEP]\nhave h' := F.congr_map (comp_p f)\n[GOAL]\ncase h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : F.map f.f \u226b NatTrans.app \u03c6 x\u271d.X = NatTrans.app \u03c6 x\u271d\u00b9.X \u226b G.map f.f\nh' : F.map (f.f \u226b x\u271d.p) = F.map f.f\n\u22a2 (F.map f.f).f \u226b (F.map x\u271d.p).f \u226b (NatTrans.app \u03c6 x\u271d.X).f =\n ((F.map x\u271d\u00b9.p).f \u226b (NatTrans.app \u03c6 x\u271d\u00b9.X).f) \u226b (G.map f.f).f\n[PROOFSTEP]\nhave h'' := F.congr_map (p_comp f)\n[GOAL]\ncase h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : F.map f.f \u226b NatTrans.app \u03c6 x\u271d.X = NatTrans.app \u03c6 x\u271d\u00b9.X \u226b G.map f.f\nh' : F.map (f.f \u226b x\u271d.p) = F.map f.f\nh'' : F.map (x\u271d\u00b9.p \u226b f.f) = F.map f.f\n\u22a2 (F.map f.f).f \u226b (F.map x\u271d.p).f \u226b (NatTrans.app \u03c6 x\u271d.X).f =\n ((F.map x\u271d\u00b9.p).f \u226b (NatTrans.app \u03c6 x\u271d\u00b9.X).f) \u226b (G.map f.f).f\n[PROOFSTEP]\nsimp only [hom_ext_iff, Functor.map_comp, comp_f] at h h' h'' \u22a2\n[GOAL]\ncase h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n\u22a2 (F.map f.f).f \u226b (F.map x\u271d.p).f \u226b (NatTrans.app \u03c6 x\u271d.X).f =\n ((F.map x\u271d\u00b9.p).f \u226b (NatTrans.app \u03c6 x\u271d\u00b9.X).f) \u226b (G.map f.f).f\n[PROOFSTEP]\nslice_rhs 2 3 => rw [\u2190 h]\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (F.map x\u271d\u00b9.p).f\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (F.map x\u271d\u00b9.p).f\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (F.map x\u271d\u00b9.p).f\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n\u22a2 (F.map f.f).f \u226b (F.map x\u271d.p).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f\n[PROOFSTEP]\nslice_lhs 1 2 => rw [h']\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (F.map f.f).f \u226b (F.map x\u271d.p).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (NatTrans.app \u03c6 x\u271d.X).f\n[PROOFSTEP]\nrw [h']\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (F.map f.f).f \u226b (F.map x\u271d.p).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (NatTrans.app \u03c6 x\u271d.X).f\n[PROOFSTEP]\nrw [h']\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (F.map f.f).f \u226b (F.map x\u271d.p).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (NatTrans.app \u03c6 x\u271d.X).f\n[PROOFSTEP]\nrw [h']\n[GOAL]\ncase h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n\u22a2 (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f\n[PROOFSTEP]\nslice_rhs 1 2 => rw [h'']\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (NatTrans.app \u03c6 x\u271d.X).f\n[PROOFSTEP]\nrw [h'']\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (NatTrans.app \u03c6 x\u271d.X).f\n[PROOFSTEP]\nrw [h'']\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.7089, u_1} C\ninst\u271d\u00b9 : Category.{?u.7093, u_2} D\ninst\u271d : Category.{?u.7097, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\nx\u271d\u00b9 x\u271d : Karoubi C\nf : x\u271d\u00b9 \u27f6 x\u271d\nh : (F.map f.f).f \u226b (NatTrans.app \u03c6 x\u271d.X).f = (NatTrans.app \u03c6 x\u271d\u00b9.X).f \u226b (G.map f.f).f\nh' : (F.map f.f).f \u226b (F.map x\u271d.p).f = (F.map f.f).f\nh'' : (F.map x\u271d\u00b9.p).f \u226b (F.map f.f).f = (F.map f.f).f\n| (NatTrans.app \u03c6 x\u271d.X).f\n[PROOFSTEP]\nrw [h'']\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF : C \u2964 Karoubi D\n\u22a2 { obj := FunctorExtension\u2081.obj, map := fun {X Y} => FunctorExtension\u2081.map }.map (\ud835\udfd9 F) =\n \ud835\udfd9 ({ obj := FunctorExtension\u2081.obj, map := fun {X Y} => FunctorExtension\u2081.map }.obj F)\n[PROOFSTEP]\next P\n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF : C \u2964 Karoubi D\nP : Karoubi C\n\u22a2 (NatTrans.app ({ obj := FunctorExtension\u2081.obj, map := fun {X Y} => FunctorExtension\u2081.map }.map (\ud835\udfd9 F)) P).f =\n (NatTrans.app (\ud835\udfd9 ({ obj := FunctorExtension\u2081.obj, map := fun {X Y} => FunctorExtension\u2081.map }.obj F)) P).f\n[PROOFSTEP]\nexact comp_p (F.map P.p)\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\n\u22a2 { obj := FunctorExtension\u2081.obj, map := fun {X Y} => FunctorExtension\u2081.map }.map (\u03c6 \u226b \u03c6') =\n { obj := FunctorExtension\u2081.obj, map := fun {X Y} => FunctorExtension\u2081.map }.map \u03c6 \u226b\n { obj := FunctorExtension\u2081.obj, map := fun {X Y} => FunctorExtension\u2081.map }.map \u03c6'\n[PROOFSTEP]\next P\n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\n\u22a2 (NatTrans.app ({ obj := FunctorExtension\u2081.obj, map := fun {X Y} => FunctorExtension\u2081.map }.map (\u03c6 \u226b \u03c6')) P).f =\n (NatTrans.app\n ({ obj := FunctorExtension\u2081.obj, map := fun {X Y} => FunctorExtension\u2081.map }.map \u03c6 \u226b\n { obj := FunctorExtension\u2081.obj, map := fun {X Y} => FunctorExtension\u2081.map }.map \u03c6')\n P).f\n[PROOFSTEP]\nsimp only [comp_f, FunctorExtension\u2081.map_app_f, NatTrans.comp_app, assoc]\n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\n\u22a2 (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f \u226b (NatTrans.app \u03c6' P.X).f =\n (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f \u226b (NatTrans.app \u03c6' P.X).f\n[PROOFSTEP]\nhave h := \u03c6.naturality P.p\n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : F.map P.p \u226b NatTrans.app \u03c6 P.X = NatTrans.app \u03c6 P.X \u226b G.map P.p\n\u22a2 (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f \u226b (NatTrans.app \u03c6' P.X).f =\n (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f \u226b (NatTrans.app \u03c6' P.X).f\n[PROOFSTEP]\nhave h' := F.congr_map P.idem\n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : F.map P.p \u226b NatTrans.app \u03c6 P.X = NatTrans.app \u03c6 P.X \u226b G.map P.p\nh' : F.map (P.p \u226b P.p) = F.map P.p\n\u22a2 (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f \u226b (NatTrans.app \u03c6' P.X).f =\n (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f \u226b (NatTrans.app \u03c6' P.X).f\n[PROOFSTEP]\nsimp only [hom_ext_iff, comp_f, F.map_comp] at h h' \n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n\u22a2 (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f \u226b (NatTrans.app \u03c6' P.X).f =\n (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f \u226b (NatTrans.app \u03c6' P.X).f\n[PROOFSTEP]\nslice_rhs 2 3 => rw [\u2190 h]\n[GOAL]\ncase a.a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\ncase a.a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (NatTrans.app \u03c6' P.X).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (F.map P.p).f\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase a.a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\ncase a.a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (NatTrans.app \u03c6' P.X).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (F.map P.p).f\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase a.a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\ncase a.a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (NatTrans.app \u03c6' P.X).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (F.map P.p).f\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n\u22a2 (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f \u226b (NatTrans.app \u03c6' P.X).f =\n (F.map P.p).f \u226b ((F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f) \u226b (NatTrans.app \u03c6' P.X).f\n[PROOFSTEP]\nslice_rhs 1 2 => rw [h']\n[GOAL]\ncase a.a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (F.map P.p).f \u226b (F.map P.p).f\ncase a.a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (NatTrans.app \u03c6 P.X).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (NatTrans.app \u03c6' P.X).f\n[PROOFSTEP]\nrw [h']\n[GOAL]\ncase a.a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (F.map P.p).f \u226b (F.map P.p).f\ncase a.a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (NatTrans.app \u03c6 P.X).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (NatTrans.app \u03c6' P.X).f\n[PROOFSTEP]\nrw [h']\n[GOAL]\ncase a.a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (F.map P.p).f \u226b (F.map P.p).f\ncase a.a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (NatTrans.app \u03c6 P.X).f\ncase a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n| (NatTrans.app \u03c6' P.X).f\n[PROOFSTEP]\nrw [h']\n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.20838, u_1} C\ninst\u271d\u00b9 : Category.{?u.20842, u_2} D\ninst\u271d : Category.{?u.20846, u_3} E\nF G H : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u03c6' : G \u27f6 H\nP : Karoubi C\nh : (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f = (NatTrans.app \u03c6 P.X).f \u226b (G.map P.p).f\nh' : (F.map P.p).f \u226b (F.map P.p).f = (F.map P.p).f\n\u22a2 (F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f \u226b (NatTrans.app \u03c6' P.X).f =\n ((F.map P.p).f \u226b (NatTrans.app \u03c6 P.X).f) \u226b (NatTrans.app \u03c6' P.X).f\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.29797, u_3} E\n\u22a2 functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) = \ud835\udfed (C \u2964 Karoubi D)\n[PROOFSTEP]\nrefine' Functor.ext _ _\n[GOAL]\ncase refine'_1\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.29797, u_3} E\n\u22a2 \u2200 (X : C \u2964 Karoubi D),\n (functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj X =\n (\ud835\udfed (C \u2964 Karoubi D)).obj X\n[PROOFSTEP]\nintro F\n[GOAL]\ncase refine'_1\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.29797, u_3} E\nF : C \u2964 Karoubi D\n\u22a2 (functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F =\n (\ud835\udfed (C \u2964 Karoubi D)).obj F\n[PROOFSTEP]\nrefine' Functor.ext _ _\n[GOAL]\ncase refine'_1.refine'_1\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.29797, u_3} E\nF : C \u2964 Karoubi D\n\u22a2 \u2200 (X : C),\n ((functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X =\n ((\ud835\udfed (C \u2964 Karoubi D)).obj F).obj X\n[PROOFSTEP]\nintro X\n[GOAL]\ncase refine'_1.refine'_1\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.29797, u_3} E\nF : C \u2964 Karoubi D\nX : C\n\u22a2 ((functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X =\n ((\ud835\udfed (C \u2964 Karoubi D)).obj F).obj X\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_1.refine'_1.h_X\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.29797, u_3} E\nF : C \u2964 Karoubi D\nX : C\n\u22a2 (((functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).X =\n (((\ud835\udfed (C \u2964 Karoubi D)).obj F).obj X).X\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_1.refine'_1.h_p\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.29797, u_3} E\nF : C \u2964 Karoubi D\nX : C\n\u22a2 (((functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X).p \u226b\n eqToHom (_ : (F.obj X).X = (F.obj X).X) =\n eqToHom (_ : (F.obj X).X = (F.obj X).X) \u226b (((\ud835\udfed (C \u2964 Karoubi D)).obj F).obj X).p\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_1.refine'_2\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.29797, u_3} E\nF : C \u2964 Karoubi D\n\u22a2 \u2200 (X Y : C) (f : X \u27f6 Y),\n ((functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).map f =\n eqToHom\n (_ :\n ((functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj X =\n ((\ud835\udfed (C \u2964 Karoubi D)).obj F).obj X) \u226b\n ((\ud835\udfed (C \u2964 Karoubi D)).obj F).map f \u226b\n eqToHom\n (_ :\n ((\ud835\udfed (C \u2964 Karoubi D)).obj F).obj Y =\n ((functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F).obj Y)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase refine'_2\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.29797, u_3} E\n\u22a2 \u2200 (X Y : C \u2964 Karoubi D) (f : X \u27f6 Y),\n (functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).map f =\n eqToHom\n (_ :\n (functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj X =\n (\ud835\udfed (C \u2964 Karoubi D)).obj X) \u226b\n (\ud835\udfed (C \u2964 Karoubi D)).map f \u226b\n eqToHom\n (_ :\n (\ud835\udfed (C \u2964 Karoubi D)).obj Y =\n (functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj Y)\n[PROOFSTEP]\nintro F G \u03c6\n[GOAL]\ncase refine'_2\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.29797, u_3} E\nF G : C \u2964 Karoubi D\n\u03c6 : F \u27f6 G\n\u22a2 (functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).map \u03c6 =\n eqToHom\n (_ :\n (functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj F =\n (\ud835\udfed (C \u2964 Karoubi D)).obj F) \u226b\n (\ud835\udfed (C \u2964 Karoubi D)).map \u03c6 \u226b\n eqToHom\n (_ :\n (\ud835\udfed (C \u2964 Karoubi D)).obj G =\n (functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)).obj G)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.49904, u_1} C\ninst\u271d\u00b9 : Category.{?u.49908, u_2} D\ninst\u271d : Category.{?u.49912, u_3} E\nG : Karoubi C \u2964 Karoubi D\nP : Karoubi C\n\u22a2 (G.map (decompId_p P)).f =\n ((((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) \u22d9 functorExtension\u2081 C D).obj G).obj P).p \u226b\n (G.map (decompId_p P)).f \u226b (((\ud835\udfed (Karoubi C \u2964 Karoubi D)).obj G).obj P).p\n[PROOFSTEP]\nsimpa only [hom_ext_iff, G.map_comp, G.map_id] using\n G.congr_map (show P.decompId_p = (toKaroubi C).map P.p \u226b P.decompId_p \u226b \ud835\udfd9 _ by simp)\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.49904, u_1} C\ninst\u271d\u00b9 : Category.{?u.49908, u_2} D\ninst\u271d : Category.{?u.49912, u_3} E\nG : Karoubi C \u2964 Karoubi D\nP : Karoubi C\n\u22a2 decompId_p P = (toKaroubi C).map P.p \u226b decompId_p P \u226b \ud835\udfd9 P\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.49904, u_1} C\ninst\u271d\u00b9 : Category.{?u.49908, u_2} D\ninst\u271d : Category.{?u.49912, u_3} E\nG : Karoubi C \u2964 Karoubi D\nP Q : Karoubi C\nf : P \u27f6 Q\n\u22a2 (((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) \u22d9 functorExtension\u2081 C D).obj G).map f \u226b\n (fun P => Hom.mk (G.map (decompId_p P)).f) Q =\n (fun P => Hom.mk (G.map (decompId_p P)).f) P \u226b ((\ud835\udfed (Karoubi C \u2964 Karoubi D)).obj G).map f\n[PROOFSTEP]\nsimpa only [hom_ext_iff, G.map_comp] using (G.congr_map (decompId_p_naturality f)).symm\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.49904, u_1} C\ninst\u271d\u00b9 : Category.{?u.49908, u_2} D\ninst\u271d : Category.{?u.49912, u_3} E\nG : Karoubi C \u2964 Karoubi D\nP : Karoubi C\n\u22a2 (G.map (decompId_i P)).f =\n (((\ud835\udfed (Karoubi C \u2964 Karoubi D)).obj G).obj P).p \u226b\n (G.map (decompId_i P)).f \u226b\n ((((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) \u22d9 functorExtension\u2081 C D).obj G).obj P).p\n[PROOFSTEP]\nsimpa only [hom_ext_iff, G.map_comp, G.map_id] using\n G.congr_map (show P.decompId_i = \ud835\udfd9 _ \u226b P.decompId_i \u226b (toKaroubi C).map P.p by simp)\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.49904, u_1} C\ninst\u271d\u00b9 : Category.{?u.49908, u_2} D\ninst\u271d : Category.{?u.49912, u_3} E\nG : Karoubi C \u2964 Karoubi D\nP : Karoubi C\n\u22a2 decompId_i P = \ud835\udfd9 P \u226b decompId_i P \u226b (toKaroubi C).map P.p\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.49904, u_1} C\ninst\u271d\u00b9 : Category.{?u.49908, u_2} D\ninst\u271d : Category.{?u.49912, u_3} E\nG : Karoubi C \u2964 Karoubi D\nP Q : Karoubi C\nf : P \u27f6 Q\n\u22a2 ((\ud835\udfed (Karoubi C \u2964 Karoubi D)).obj G).map f \u226b (fun P => Hom.mk (G.map (decompId_i P)).f) Q =\n (fun P => Hom.mk (G.map (decompId_i P)).f) P \u226b\n (((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) \u22d9 functorExtension\u2081 C D).obj G).map f\n[PROOFSTEP]\nsimpa only [hom_ext_iff, G.map_comp] using G.congr_map (decompId_i_naturality f)\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.49904, u_1} C\ninst\u271d\u00b9 : Category.{?u.49908, u_2} D\ninst\u271d : Category.{?u.49912, u_3} E\nG : Karoubi C \u2964 Karoubi D\n\u22a2 ((NatTrans.mk fun P => Hom.mk (G.map (decompId_p P)).f) \u226b NatTrans.mk fun P => Hom.mk (G.map (decompId_i P)).f) =\n \ud835\udfd9 (((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) \u22d9 functorExtension\u2081 C D).obj G)\n[PROOFSTEP]\next P\n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.49904, u_1} C\ninst\u271d\u00b9 : Category.{?u.49908, u_2} D\ninst\u271d : Category.{?u.49912, u_3} E\nG : Karoubi C \u2964 Karoubi D\nP : Karoubi C\n\u22a2 (NatTrans.app\n ((NatTrans.mk fun P => Hom.mk (G.map (decompId_p P)).f) \u226b NatTrans.mk fun P => Hom.mk (G.map (decompId_i P)).f)\n P).f =\n (NatTrans.app (\ud835\udfd9 (((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) \u22d9 functorExtension\u2081 C D).obj G))\n P).f\n[PROOFSTEP]\nsimpa only [hom_ext_iff, G.map_comp, G.map_id] using G.congr_map P.decomp_p.symm\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.49904, u_1} C\ninst\u271d\u00b9 : Category.{?u.49908, u_2} D\ninst\u271d : Category.{?u.49912, u_3} E\nG : Karoubi C \u2964 Karoubi D\n\u22a2 ((NatTrans.mk fun P => Hom.mk (G.map (decompId_i P)).f) \u226b NatTrans.mk fun P => Hom.mk (G.map (decompId_p P)).f) =\n \ud835\udfd9 ((\ud835\udfed (Karoubi C \u2964 Karoubi D)).obj G)\n[PROOFSTEP]\next P\n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.49904, u_1} C\ninst\u271d\u00b9 : Category.{?u.49908, u_2} D\ninst\u271d : Category.{?u.49912, u_3} E\nG : Karoubi C \u2964 Karoubi D\nP : Karoubi C\n\u22a2 (NatTrans.app\n ((NatTrans.mk fun P => Hom.mk (G.map (decompId_i P)).f) \u226b NatTrans.mk fun P => Hom.mk (G.map (decompId_p P)).f)\n P).f =\n (NatTrans.app (\ud835\udfd9 ((\ud835\udfed (Karoubi C \u2964 Karoubi D)).obj G)) P).f\n[PROOFSTEP]\nsimpa only [hom_ext_iff, G.map_comp, G.map_id] using G.congr_map P.decompId.symm\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.49904, u_1} C\ninst\u271d\u00b9 : Category.{?u.49908, u_2} D\ninst\u271d : Category.{?u.49912, u_3} E\nX Y : Karoubi C \u2964 Karoubi D\n\u03c6 : X \u27f6 Y\n\u22a2 ((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) \u22d9 functorExtension\u2081 C D).map \u03c6 \u226b\n ((fun G =>\n Iso.mk (NatTrans.mk fun P => Hom.mk (G.map (decompId_p P)).f)\n (NatTrans.mk fun P => Hom.mk (G.map (decompId_i P)).f))\n Y).hom =\n ((fun G =>\n Iso.mk (NatTrans.mk fun P => Hom.mk (G.map (decompId_p P)).f)\n (NatTrans.mk fun P => Hom.mk (G.map (decompId_i P)).f))\n X).hom \u226b\n (\ud835\udfed (Karoubi C \u2964 Karoubi D)).map \u03c6\n[PROOFSTEP]\next P\n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.49904, u_1} C\ninst\u271d\u00b9 : Category.{?u.49908, u_2} D\ninst\u271d : Category.{?u.49912, u_3} E\nX Y : Karoubi C \u2964 Karoubi D\n\u03c6 : X \u27f6 Y\nP : Karoubi C\n\u22a2 (NatTrans.app\n (((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) \u22d9 functorExtension\u2081 C D).map \u03c6 \u226b\n ((fun G =>\n Iso.mk (NatTrans.mk fun P => Hom.mk (G.map (decompId_p P)).f)\n (NatTrans.mk fun P => Hom.mk (G.map (decompId_i P)).f))\n Y).hom)\n P).f =\n (NatTrans.app\n (((fun G =>\n Iso.mk (NatTrans.mk fun P => Hom.mk (G.map (decompId_p P)).f)\n (NatTrans.mk fun P => Hom.mk (G.map (decompId_i P)).f))\n X).hom \u226b\n (\ud835\udfed (Karoubi C \u2964 Karoubi D)).map \u03c6)\n P).f\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.49904, u_1} C\ninst\u271d\u00b9 : Category.{?u.49908, u_2} D\ninst\u271d : Category.{?u.49912, u_3} E\nX Y : Karoubi C \u2964 Karoubi D\n\u03c6 : X \u27f6 Y\nP : Karoubi C\n\u22a2 ((X.map ((toKaroubi C).map P.p)).f \u226b (NatTrans.app \u03c6 ((toKaroubi C).obj P.X)).f) \u226b (Y.map (decompId_p P)).f =\n (X.map (decompId_p P)).f \u226b (NatTrans.app \u03c6 P).f\n[PROOFSTEP]\nrw [natTrans_eq \u03c6 P, P.decomp_p]\n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.49904, u_1} C\ninst\u271d\u00b9 : Category.{?u.49908, u_2} D\ninst\u271d : Category.{?u.49912, u_3} E\nX Y : Karoubi C \u2964 Karoubi D\n\u03c6 : X \u27f6 Y\nP : Karoubi C\n\u22a2 ((X.map (decompId_p P \u226b decompId_i P)).f \u226b (NatTrans.app \u03c6 ((toKaroubi C).obj P.X)).f) \u226b (Y.map (decompId_p P)).f =\n (X.map (decompId_p P)).f \u226b (X.map (decompId_i P) \u226b NatTrans.app \u03c6 (Karoubi.mk P.X (\ud835\udfd9 P.X)) \u226b Y.map (decompId_p P)).f\n[PROOFSTEP]\nsimp only [Functor.map_comp, comp_f, assoc]\n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.49904, u_1} C\ninst\u271d\u00b9 : Category.{?u.49908, u_2} D\ninst\u271d : Category.{?u.49912, u_3} E\nX Y : Karoubi C \u2964 Karoubi D\n\u03c6 : X \u27f6 Y\nP : Karoubi C\n\u22a2 (X.map (decompId_p P)).f \u226b\n (X.map (decompId_i P)).f \u226b (NatTrans.app \u03c6 ((toKaroubi C).obj P.X)).f \u226b (Y.map (decompId_p P)).f =\n (X.map (decompId_p P)).f \u226b\n (X.map (decompId_i P)).f \u226b (NatTrans.app \u03c6 (Karoubi.mk P.X (\ud835\udfd9 P.X))).f \u226b (Y.map (decompId_p P)).f\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.61683, u_1} C\ninst\u271d\u00b9 : Category.{?u.61687, u_2} D\ninst\u271d : Category.{?u.61691, u_3} E\nF : C \u2964 Karoubi D\n\u22a2 (functorExtension\u2081 C D).map (NatTrans.app (functorExtension\u2081_comp_whiskeringLeft_toKaroubi_iso C D).symm.hom F) \u226b\n NatTrans.app (KaroubiUniversal\u2081.counitIso C D).hom ((functorExtension\u2081 C D).obj F) =\n \ud835\udfd9 ((functorExtension\u2081 C D).obj F)\n[PROOFSTEP]\next P\n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.61683, u_1} C\ninst\u271d\u00b9 : Category.{?u.61687, u_2} D\ninst\u271d : Category.{?u.61691, u_3} E\nF : C \u2964 Karoubi D\nP : Karoubi C\n\u22a2 (NatTrans.app\n ((functorExtension\u2081 C D).map\n (NatTrans.app (functorExtension\u2081_comp_whiskeringLeft_toKaroubi_iso C D).symm.hom F) \u226b\n NatTrans.app (KaroubiUniversal\u2081.counitIso C D).hom ((functorExtension\u2081 C D).obj F))\n P).f =\n (NatTrans.app (\ud835\udfd9 ((functorExtension\u2081 C D).obj F)) P).f\n[PROOFSTEP]\ndsimp [FunctorExtension\u2081.map, KaroubiUniversal\u2081.counitIso]\n[GOAL]\ncase w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.61683, u_1} C\ninst\u271d\u00b9 : Category.{?u.61687, u_2} D\ninst\u271d : Category.{?u.61691, u_3} E\nF : C \u2964 Karoubi D\nP : Karoubi C\n\u22a2 ((F.map P.p).f \u226b\n (NatTrans.app\n (NatTrans.app\n (eqToHom\n (_ :\n \ud835\udfed (C \u2964 Karoubi D) =\n functorExtension\u2081 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C)))\n F)\n P.X).f) \u226b\n (F.map P.p).f =\n (F.map P.p).f\n[PROOFSTEP]\nsimp only [eqToHom_app, Functor.id_obj, Functor.comp_obj, functorExtension\u2081_obj, whiskeringLeft_obj_obj, eqToHom_f,\n FunctorExtension\u2081.obj_obj_X, toKaroubi_obj_X, eqToHom_refl, comp_id, comp_p, \u2190 comp_f, \u2190 F.map_comp, P.idem]\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{u_6, u_3} E\nF : C \u2964 Karoubi D\nG : D \u2964 Karoubi E\n\u22a2 (functorExtension\u2081 C E).obj (F \u22d9 (functorExtension\u2081 D E).obj G) =\n (functorExtension\u2081 C D).obj F \u22d9 (functorExtension\u2081 D E).obj G\n[PROOFSTEP]\nrefine' Functor.ext _ _\n[GOAL]\ncase refine'_1\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{u_6, u_3} E\nF : C \u2964 Karoubi D\nG : D \u2964 Karoubi E\n\u22a2 \u2200 (X : Karoubi C),\n ((functorExtension\u2081 C E).obj (F \u22d9 (functorExtension\u2081 D E).obj G)).obj X =\n ((functorExtension\u2081 C D).obj F \u22d9 (functorExtension\u2081 D E).obj G).obj X\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase refine'_2\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{u_6, u_3} E\nF : C \u2964 Karoubi D\nG : D \u2964 Karoubi E\n\u22a2 \u2200 (X Y : Karoubi C) (f : X \u27f6 Y),\n ((functorExtension\u2081 C E).obj (F \u22d9 (functorExtension\u2081 D E).obj G)).map f =\n eqToHom\n (_ :\n ((functorExtension\u2081 C E).obj (F \u22d9 (functorExtension\u2081 D E).obj G)).obj X =\n ((functorExtension\u2081 C D).obj F \u22d9 (functorExtension\u2081 D E).obj G).obj X) \u226b\n ((functorExtension\u2081 C D).obj F \u22d9 (functorExtension\u2081 D E).obj G).map f \u226b\n eqToHom\n (_ :\n ((functorExtension\u2081 C D).obj F \u22d9 (functorExtension\u2081 D E).obj G).obj Y =\n ((functorExtension\u2081 C E).obj (F \u22d9 (functorExtension\u2081 D E).obj G)).obj Y)\n[PROOFSTEP]\nintro X Y f\n[GOAL]\ncase refine'_2\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{u_6, u_3} E\nF : C \u2964 Karoubi D\nG : D \u2964 Karoubi E\nX Y : Karoubi C\nf : X \u27f6 Y\n\u22a2 ((functorExtension\u2081 C E).obj (F \u22d9 (functorExtension\u2081 D E).obj G)).map f =\n eqToHom\n (_ :\n ((functorExtension\u2081 C E).obj (F \u22d9 (functorExtension\u2081 D E).obj G)).obj X =\n ((functorExtension\u2081 C D).obj F \u22d9 (functorExtension\u2081 D E).obj G).obj X) \u226b\n ((functorExtension\u2081 C D).obj F \u22d9 (functorExtension\u2081 D E).obj G).map f \u226b\n eqToHom\n (_ :\n ((functorExtension\u2081 C D).obj F \u22d9 (functorExtension\u2081 D E).obj G).obj Y =\n ((functorExtension\u2081 C E).obj (F \u22d9 (functorExtension\u2081 D E).obj G)).obj Y)\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_2.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{u_6, u_3} E\nF : C \u2964 Karoubi D\nG : D \u2964 Karoubi E\nX Y : Karoubi C\nf : X \u27f6 Y\n\u22a2 (((functorExtension\u2081 C E).obj (F \u22d9 (functorExtension\u2081 D E).obj G)).map f).f =\n (eqToHom\n (_ :\n ((functorExtension\u2081 C E).obj (F \u22d9 (functorExtension\u2081 D E).obj G)).obj X =\n ((functorExtension\u2081 C D).obj F \u22d9 (functorExtension\u2081 D E).obj G).obj X) \u226b\n ((functorExtension\u2081 C D).obj F \u22d9 (functorExtension\u2081 D E).obj G).map f \u226b\n eqToHom\n (_ :\n ((functorExtension\u2081 C D).obj F \u22d9 (functorExtension\u2081 D E).obj G).obj Y =\n ((functorExtension\u2081 C E).obj (F \u22d9 (functorExtension\u2081 D E).obj G)).obj Y)).f\n[PROOFSTEP]\nsimp only [eqToHom_refl, id_comp, comp_id]\n[GOAL]\ncase refine'_2.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{u_6, u_3} E\nF : C \u2964 Karoubi D\nG : D \u2964 Karoubi E\nX Y : Karoubi C\nf : X \u27f6 Y\n\u22a2 (((functorExtension\u2081 C E).obj (F \u22d9 (functorExtension\u2081 D E).obj G)).map f).f =\n (((functorExtension\u2081 C D).obj F \u22d9 (functorExtension\u2081 D E).obj G).map f).f\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{u_4, u_1} C\ninst\u271d\u00b9 : Category.{u_5, u_2} D\ninst\u271d : Category.{?u.72126, u_3} E\n\u22a2 functorExtension\u2082 C D \u22d9 (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) =\n (whiskeringRight C D (Karoubi D)).obj (toKaroubi D)\n[PROOFSTEP]\nsimp only [functorExtension\u2082, Functor.assoc, functorExtension\u2081_comp_whiskeringLeft_toKaroubi, Functor.comp_id]\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b3 : Category.{?u.76565, u_1} C\ninst\u271d\u00b2 : Category.{?u.76569, u_2} D\ninst\u271d\u00b9 : Category.{?u.76573, u_3} E\ninst\u271d : IsIdempotentComplete D\n\u22a2 IsEquivalence (functorExtension\u2082 C D)\n[PROOFSTEP]\nrw [\u2190 karoubiUniversal\u2082_functor_eq]\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b3 : Category.{?u.76565, u_1} C\ninst\u271d\u00b2 : Category.{?u.76569, u_2} D\ninst\u271d\u00b9 : Category.{?u.76573, u_3} E\ninst\u271d : IsIdempotentComplete D\n\u22a2 IsEquivalence (karoubiUniversal\u2082 C D).functor\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b3 : Category.{?u.81595, u_1} C\ninst\u271d\u00b2 : Category.{?u.81599, u_2} D\ninst\u271d\u00b9 : Category.{?u.81603, u_3} E\ninst\u271d : IsIdempotentComplete D\n\u22a2 IsEquivalence (functorExtension C D)\n[PROOFSTEP]\nrw [\u2190 karoubiUniversal_functor_eq]\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b3 : Category.{?u.81595, u_1} C\ninst\u271d\u00b2 : Category.{?u.81599, u_2} D\ninst\u271d\u00b9 : Category.{?u.81603, u_3} E\ninst\u271d : IsIdempotentComplete D\n\u22a2 IsEquivalence (karoubiUniversal C D).functor\n[PROOFSTEP]\ninfer_instance\n -- porting note: added to avoid a timeout in the following definition\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b3 : Category.{u_4, u_1} C\ninst\u271d\u00b2 : Category.{u_5, u_2} D\ninst\u271d\u00b9 : Category.{?u.82137, u_3} E\ninst\u271d : IsIdempotentComplete D\n\u22a2 (whiskeringLeft C (Karoubi C) D).obj (toKaroubi C) \u22d9\n (whiskeringRight C D (Karoubi D)).obj (toKaroubi D) \u22d9\n (whiskeringRight C (Karoubi D) D).obj (Functor.inv (toKaroubi D)) =\n (karoubiUniversal C D).inverse\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b3 : Category.{?u.84023, u_1} C\ninst\u271d\u00b2 : Category.{?u.84027, u_2} D\ninst\u271d\u00b9 : Category.{?u.84031, u_3} E\ninst\u271d : IsIdempotentComplete D\n\u22a2 IsEquivalence\n ((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C) \u22d9\n (whiskeringRight C D (Karoubi D)).obj (toKaroubi D) \u22d9\n (whiskeringRight C (Karoubi D) D).obj (Functor.inv (toKaroubi D)))\n[PROOFSTEP]\nrw [isEquivalence_whiskeringLeft_obj_toKaroubi_aux]\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b3 : Category.{?u.84023, u_1} C\ninst\u271d\u00b2 : Category.{?u.84027, u_2} D\ninst\u271d\u00b9 : Category.{?u.84031, u_3} E\ninst\u271d : IsIdempotentComplete D\n\u22a2 IsEquivalence (karoubiUniversal C D).inverse\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b3 : Category.{u_4, u_1} C\ninst\u271d\u00b2 : Category.{u_5, u_2} D\ninst\u271d\u00b9 : Category.{?u.87671, u_3} E\ninst\u271d : IsIdempotentComplete D\nF G : Karoubi C \u2964 D\n\u03c4 : toKaroubi C \u22d9 F \u27f6 toKaroubi C \u22d9 G\nP : Karoubi C\n\u22a2 NatTrans.app (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage \u03c4) P =\n F.map (decompId_i P) \u226b NatTrans.app \u03c4 P.X \u226b G.map (decompId_p P)\n[PROOFSTEP]\nrw [natTrans_eq]\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b3 : Category.{u_4, u_1} C\ninst\u271d\u00b2 : Category.{u_5, u_2} D\ninst\u271d\u00b9 : Category.{?u.87671, u_3} E\ninst\u271d : IsIdempotentComplete D\nF G : Karoubi C \u2964 D\n\u03c4 : toKaroubi C \u22d9 F \u27f6 toKaroubi C \u22d9 G\nP : Karoubi C\n\u22a2 F.map (decompId_i P) \u226b\n NatTrans.app (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage \u03c4) (Karoubi.mk P.X (\ud835\udfd9 P.X)) \u226b\n G.map (decompId_p P) =\n F.map (decompId_i P) \u226b NatTrans.app \u03c4 P.X \u226b G.map (decompId_p P)\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b3 : Category.{u_4, u_1} C\ninst\u271d\u00b2 : Category.{u_5, u_2} D\ninst\u271d\u00b9 : Category.{?u.87671, u_3} E\ninst\u271d : IsIdempotentComplete D\nF G : Karoubi C \u2964 D\n\u03c4 : toKaroubi C \u22d9 F \u27f6 toKaroubi C \u22d9 G\nP : Karoubi C\n\u22a2 NatTrans.app (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage \u03c4) (Karoubi.mk P.X (\ud835\udfd9 P.X)) =\n NatTrans.app \u03c4 P.X\n[PROOFSTEP]\nrw [\u2190 congr_app (((whiskeringLeft _ _ _).obj (toKaroubi _)).image_preimage \u03c4) P.X]\n[GOAL]\ncase e_a.e_a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b3 : Category.{u_4, u_1} C\ninst\u271d\u00b2 : Category.{u_5, u_2} D\ninst\u271d\u00b9 : Category.{?u.87671, u_3} E\ninst\u271d : IsIdempotentComplete D\nF G : Karoubi C \u2964 D\n\u03c4 : toKaroubi C \u22d9 F \u27f6 toKaroubi C \u22d9 G\nP : Karoubi C\n\u22a2 NatTrans.app (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage \u03c4) (Karoubi.mk P.X (\ud835\udfd9 P.X)) =\n NatTrans.app\n (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).map\n (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage \u03c4))\n P.X\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase e_a.e_a\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b3 : Category.{u_4, u_1} C\ninst\u271d\u00b2 : Category.{u_5, u_2} D\ninst\u271d\u00b9 : Category.{?u.87671, u_3} E\ninst\u271d : IsIdempotentComplete D\nF G : Karoubi C \u2964 D\n\u03c4 : toKaroubi C \u22d9 F \u27f6 toKaroubi C \u22d9 G\nP : Karoubi C\n\u22a2 NatTrans.app (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage \u03c4) (Karoubi.mk P.X (\ud835\udfd9 P.X)) =\n NatTrans.app (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage \u03c4) ((toKaroubi C).obj P.X)\n[PROOFSTEP]\ncongr\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Idempotents.FunctorExtension", "llama_tokens": 31692, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707281, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4382624122449829}} {"text": "[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\n\u22a2 IsLimit (limitCone F)\n[PROOFSTEP]\nrefine'\n IsLimit.ofFaithful (forget (AlgebraCat R))\n (Types.limitConeIsLimit.{v, w} _)\n -- Porting note: in mathlib3 the function term\n -- `fun v => \u27e8fun j => ((forget (AlgebraCat R)).mapCone s).\u03c0.app j v`\n -- was provided by unification, and the last argument `(fun s => _)` was `(fun s => rfl)`.\n (fun s => \u27e8\u27e8\u27e8\u27e8fun v => \u27e8fun j => ((forget (AlgebraCat R)).mapCone s).\u03c0.app j v, _\u27e9, _\u27e9, _\u27e9, _, _\u27e9, _\u27e9) (fun s => _)\n[GOAL]\ncase refine'_1\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\nv : \u2191s.pt\n\u22a2 (fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v) \u2208 Functor.sections (F \u22d9 forget (AlgebraCatMax R))\n[PROOFSTEP]\nintro j j' f\n[GOAL]\ncase refine'_1\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\nv : \u2191s.pt\nj j' : J\nf : j \u27f6 j'\n\u22a2 (F \u22d9 forget (AlgebraCatMax R)).map f ((fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v) j) =\n (fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v) j'\n[PROOFSTEP]\nexact FunLike.congr_fun (Cone.w s f) v\n[GOAL]\ncase refine'_2\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\n\u22a2 (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property := (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1\n[PROOFSTEP]\napply Subtype.ext\n[GOAL]\ncase refine'_2.a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\n\u22a2 \u2191((fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property := (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1) =\n \u21911\n[PROOFSTEP]\next j\n[GOAL]\ncase refine'_2.a.h\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\nj : J\n\u22a2 \u2191((fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1)\n j =\n \u21911 j\n[PROOFSTEP]\nsimp [forget_map_eq_coe, AlgHom.map_one, Functor.mapCone_\u03c0_app]\n[GOAL]\ncase refine'_2.a.h\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\nj : J\n\u22a2 1 = \u21911 j\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_3\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\n\u22a2 \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase refine'_3\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\nx y : \u2191s.pt\n\u22a2 OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property := (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y\n[PROOFSTEP]\napply Subtype.ext\n[GOAL]\ncase refine'_3.a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\nx y : \u2191s.pt\n\u22a2 \u2191(OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y)) =\n \u2191(OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y)\n[PROOFSTEP]\next j\n[GOAL]\ncase refine'_3.a.h\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\nx y : \u2191s.pt\nj : J\n\u22a2 \u2191(OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y))\n j =\n \u2191(OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y)\n j\n[PROOFSTEP]\nsimp [forget_map_eq_coe, AlgHom.map_mul, Functor.mapCone_\u03c0_app]\n[GOAL]\ncase refine'_3.a.h\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\nx y : \u2191s.pt\nj : J\n\u22a2 \u2191(NatTrans.app s.\u03c0 j) x * \u2191(NatTrans.app s.\u03c0 j) y =\n \u2191({ val := fun j => \u2191(NatTrans.app s.\u03c0 j) x,\n property :=\n (_ : (fun x => x \u2208 Functor.sections (F \u22d9 forget (AlgebraCatMax R))) fun j => \u2191(NatTrans.app s.\u03c0 j) x) } *\n { val := fun j => \u2191(NatTrans.app s.\u03c0 j) y,\n property :=\n (_ : (fun x => x \u2208 Functor.sections (F \u22d9 forget (AlgebraCatMax R))) fun j => \u2191(NatTrans.app s.\u03c0 j) y) })\n j\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_4\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\n\u22a2 OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n 0 =\n 0\n[PROOFSTEP]\nsimp [forget_map_eq_coe, AlgHom.map_zero, Functor.mapCone_\u03c0_app]\n[GOAL]\ncase refine'_4\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\n\u22a2 { val := fun j => 0, property := (_ : (fun x => x \u2208 Functor.sections (F \u22d9 forget (AlgebraCatMax R))) fun j => 0) } = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_5\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\n\u22a2 \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n (x + y) =\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n x +\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n y\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase refine'_5\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\nx y : \u2191s.pt\n\u22a2 OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n (x + y) =\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n x +\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n y\n[PROOFSTEP]\nsimp [forget_map_eq_coe, AlgHom.map_add, Functor.mapCone_\u03c0_app]\n[GOAL]\ncase refine'_5\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\nx y : \u2191s.pt\n\u22a2 { val := fun j => \u2191(NatTrans.app s.\u03c0 j) x + \u2191(NatTrans.app s.\u03c0 j) y,\n property :=\n (_ :\n (fun x => x \u2208 Functor.sections (F \u22d9 forget (AlgebraCatMax R))) fun j =>\n \u2191(NatTrans.app s.\u03c0 j) x + \u2191(NatTrans.app s.\u03c0 j) y) } =\n { val := fun j => \u2191(NatTrans.app s.\u03c0 j) x,\n property :=\n (_ : (fun x => x \u2208 Functor.sections (F \u22d9 forget (AlgebraCatMax R))) fun j => \u2191(NatTrans.app s.\u03c0 j) x) } +\n { val := fun j => \u2191(NatTrans.app s.\u03c0 j) y,\n property :=\n (_ : (fun x => x \u2208 Functor.sections (F \u22d9 forget (AlgebraCatMax R))) fun j => \u2191(NatTrans.app s.\u03c0 j) y) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_6\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\n\u22a2 \u2200 (r : R),\n OneHom.toFun\n (\u2191\u2191{\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n (x + y) =\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n x +\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n y) })\n (\u2191(algebraMap R \u2191s.pt) r) =\n \u2191(algebraMap R \u2191(limitCone F).pt) r\n[PROOFSTEP]\nintro r\n[GOAL]\ncase refine'_6\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\nr : R\n\u22a2 OneHom.toFun\n (\u2191\u2191{\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n (x + y) =\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n x +\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n y) })\n (\u2191(algebraMap R \u2191s.pt) r) =\n \u2191(algebraMap R \u2191(limitCone F).pt) r\n[PROOFSTEP]\napply Subtype.ext\n[GOAL]\ncase refine'_6.a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\nr : R\n\u22a2 \u2191(OneHom.toFun\n (\u2191\u2191{\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n (x + y) =\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n x +\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n y) })\n (\u2191(algebraMap R \u2191s.pt) r)) =\n \u2191(\u2191(algebraMap R \u2191(limitCone F).pt) r)\n[PROOFSTEP]\next j\n[GOAL]\ncase refine'_6.a.h\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\nr : R\nj : J\n\u22a2 \u2191(OneHom.toFun\n (\u2191\u2191{\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n (x + y) =\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n x +\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n y) })\n (\u2191(algebraMap R \u2191s.pt) r))\n j =\n \u2191(\u2191(algebraMap R \u2191(limitCone F).pt) r) j\n[PROOFSTEP]\nexact (s.\u03c0.app j).commutes r\n[GOAL]\ncase refine'_7\nR : Type u\ninst\u271d\u00b9 : CommRing R\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 AlgebraCatMax R\ns : Cone F\n\u22a2 (forget (AlgebraCat R)).map\n ((fun s =>\n {\n toRingHom :=\n {\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n (x + y) =\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n x +\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n y) },\n commutes' :=\n (_ :\n \u2200 (r : R),\n OneHom.toFun\n (\u2191\u2191{\n toMonoidHom :=\n {\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n { val := fun j => NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j\n v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j\n v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j\n v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v = \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j\n v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0\n j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0\n j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n (x + y) =\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0\n j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app\n ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app\n ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n x +\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0\n j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app\n ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun v =>\n {\n val := fun j =>\n NatTrans.app ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) },\n map_one' :=\n (_ :\n (fun v =>\n {\n val := fun j =>\n NatTrans.app\n ((forget (AlgebraCat R)).mapCone s).\u03c0 j v,\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n \u2191(NatTrans.app s.\u03c0 j \u226b F.map f) v =\n \u2191(NatTrans.app s.\u03c0 j') v) })\n 1 =\n 1) }\n y) })\n y) })\n (\u2191(algebraMap R \u2191s.pt) r) =\n \u2191(algebraMap R \u2191(limitCone F).pt) r) })\n s) =\n IsLimit.lift (Types.limitConeIsLimit (F \u22d9 forget (AlgebraCat R))) ((forget (AlgebraCat R)).mapCone s)\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.AlgebraCat.Limits", "llama_tokens": 50418, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321720225278, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.43808535931888676}} {"text": "[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\n\u22a2 IsometryEquiv (weightedSumSquares \u2102 w') (weightedSumSquares \u2102 fun i => if w' i = 0 then 0 else 1)\n[PROOFSTEP]\nlet w i := if h : w' i = 0 then (1 : Units \u2102) else Units.mk0 (w' i) h\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\n\u22a2 IsometryEquiv (weightedSumSquares \u2102 w') (weightedSumSquares \u2102 fun i => if w' i = 0 then 0 else 1)\n[PROOFSTEP]\nhave hw' : \u2200 i : \u03b9, (w i : \u2102) ^ (-(1 / 2 : \u2102)) \u2260 0 := by\n intro i hi\n exact (w i).ne_zero ((Complex.cpow_eq_zero_iff _ _).1 hi).1\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\n\u22a2 \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\n[PROOFSTEP]\nintro i hi\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\ni : \u03b9\nhi : \u2191(w i) ^ (-(1 / 2)) = 0\n\u22a2 False\n[PROOFSTEP]\nexact (w i).ne_zero ((Complex.cpow_eq_zero_iff _ _).1 hi).1\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\n\u22a2 IsometryEquiv (weightedSumSquares \u2102 w') (weightedSumSquares \u2102 fun i => if w' i = 0 then 0 else 1)\n[PROOFSTEP]\nconvert\n (weightedSumSquares \u2102 w').isometryEquivBasisRepr\n ((Pi.basisFun \u2102 \u03b9).unitsSMul fun i => (isUnit_iff_ne_zero.2 <| hw' i).unit)\n[GOAL]\ncase h.e'_10\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\n\u22a2 (weightedSumSquares \u2102 fun i => if w' i = 0 then 0 else 1) =\n basisRepr (weightedSumSquares \u2102 w')\n (Basis.unitsSMul (Pi.basisFun \u2102 \u03b9) fun i => IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2)))))\n[PROOFSTEP]\next1 v\n[GOAL]\ncase h.e'_10.H\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\n\u22a2 \u2191(weightedSumSquares \u2102 fun i => if w' i = 0 then 0 else 1) v =\n \u2191(basisRepr (weightedSumSquares \u2102 w')\n (Basis.unitsSMul (Pi.basisFun \u2102 \u03b9) fun i => IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))))\n v\n[PROOFSTEP]\nerw [basisRepr_apply, weightedSumSquares_apply, weightedSumSquares_apply]\n[GOAL]\ncase h.e'_10.H\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\n\u22a2 \u2211 i : \u03b9, (if w' i = 0 then 0 else 1) \u2022 (v i * v i) =\n \u2211 i : \u03b9,\n w' i \u2022\n (Finset.sum univ\n (fun i =>\n v i \u2022 \u2191(Basis.unitsSMul (Pi.basisFun \u2102 \u03b9) fun i => IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) i)\n i *\n Finset.sum univ\n (fun i =>\n v i \u2022 \u2191(Basis.unitsSMul (Pi.basisFun \u2102 \u03b9) fun i => IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) i)\n i)\n[PROOFSTEP]\nrefine' sum_congr rfl fun j hj => _\n[GOAL]\ncase h.e'_10.H\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\n\u22a2 (if w' j = 0 then 0 else 1) \u2022 (v j * v j) =\n w' j \u2022\n (Finset.sum univ\n (fun i =>\n v i \u2022 \u2191(Basis.unitsSMul (Pi.basisFun \u2102 \u03b9) fun i => IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) i)\n j *\n Finset.sum univ\n (fun i =>\n v i \u2022 \u2191(Basis.unitsSMul (Pi.basisFun \u2102 \u03b9) fun i => IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) i)\n j)\n[PROOFSTEP]\nhave hsum :\n (\u2211 i : \u03b9, v i \u2022 ((isUnit_iff_ne_zero.2 <| hw' i).unit : \u2102) \u2022 (Pi.basisFun \u2102 \u03b9) i) j = v j \u2022 w j ^ (-(1 / 2 : \u2102)) :=\n by\n rw [Finset.sum_apply, sum_eq_single j, Pi.basisFun_apply, IsUnit.unit_spec, LinearMap.stdBasis_apply, Pi.smul_apply,\n Pi.smul_apply, Function.update_same, smul_eq_mul, smul_eq_mul, smul_eq_mul, mul_one]\n intro i _ hij\n rw [Pi.basisFun_apply, LinearMap.stdBasis_apply, Pi.smul_apply, Pi.smul_apply, Function.update_noteq hij.symm,\n Pi.zero_apply, smul_eq_mul, smul_eq_mul, mul_zero, mul_zero]\n intro hj'; exact False.elim (hj' hj)\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\n\u22a2 Finset.sum univ (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) i) j =\n v j \u2022 \u2191(w j) ^ (-(1 / 2))\n[PROOFSTEP]\nrw [Finset.sum_apply, sum_eq_single j, Pi.basisFun_apply, IsUnit.unit_spec, LinearMap.stdBasis_apply, Pi.smul_apply,\n Pi.smul_apply, Function.update_same, smul_eq_mul, smul_eq_mul, smul_eq_mul, mul_one]\n[GOAL]\ncase h\u2080\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\n\u22a2 \u2200 (b : \u03b9), b \u2208 univ \u2192 b \u2260 j \u2192 (v b \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w b) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) b) j = 0\ncase h\u2081\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\n\u22a2 \u00acj \u2208 univ \u2192 (v j \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w j) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) j) j = 0\n[PROOFSTEP]\nintro i _ hij\n[GOAL]\ncase h\u2080\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\ni : \u03b9\na\u271d : i \u2208 univ\nhij : i \u2260 j\n\u22a2 (v i \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) i) j = 0\ncase h\u2081\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\n\u22a2 \u00acj \u2208 univ \u2192 (v j \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w j) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) j) j = 0\n[PROOFSTEP]\nrw [Pi.basisFun_apply, LinearMap.stdBasis_apply, Pi.smul_apply, Pi.smul_apply, Function.update_noteq hij.symm,\n Pi.zero_apply, smul_eq_mul, smul_eq_mul, mul_zero, mul_zero]\n[GOAL]\ncase h\u2081\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\n\u22a2 \u00acj \u2208 univ \u2192 (v j \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w j) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) j) j = 0\n[PROOFSTEP]\nintro hj'\n[GOAL]\ncase h\u2081\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\nhj' : \u00acj \u2208 univ\n\u22a2 (v j \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w j) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) j) j = 0\n[PROOFSTEP]\nexact False.elim (hj' hj)\n[GOAL]\ncase h.e'_10.H\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n Finset.sum univ (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) i) j =\n v j \u2022 \u2191(w j) ^ (-(1 / 2))\n\u22a2 (if w' j = 0 then 0 else 1) \u2022 (v j * v j) =\n w' j \u2022\n (Finset.sum univ\n (fun i =>\n v i \u2022 \u2191(Basis.unitsSMul (Pi.basisFun \u2102 \u03b9) fun i => IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) i)\n j *\n Finset.sum univ\n (fun i =>\n v i \u2022 \u2191(Basis.unitsSMul (Pi.basisFun \u2102 \u03b9) fun i => IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) i)\n j)\n[PROOFSTEP]\nsimp_rw [Basis.unitsSMul_apply]\n[GOAL]\ncase h.e'_10.H\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n Finset.sum univ (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) i) j =\n v j \u2022 \u2191(w j) ^ (-(1 / 2))\n\u22a2 (if w' j = 0 then 0 else 1) \u2022 (v j * v j) =\n w' j \u2022\n (Finset.sum univ (fun x => v x \u2022 IsUnit.unit (_ : IsUnit (\u2191(w x) ^ (-(1 / 2)))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) x) j *\n Finset.sum univ (fun x => v x \u2022 IsUnit.unit (_ : IsUnit (\u2191(w x) ^ (-(1 / 2)))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) x) j)\n[PROOFSTEP]\nerw [hsum, smul_eq_mul]\n[GOAL]\ncase h.e'_10.H\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n Finset.sum univ (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) i) j =\n v j \u2022 \u2191(w j) ^ (-(1 / 2))\n\u22a2 (if w' j = 0 then 0 else 1) * (v j * v j) = w' j \u2022 (v j \u2022 \u2191(w j) ^ (-(1 / 2)) * v j \u2022 \u2191(w j) ^ (-(1 / 2)))\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n Finset.sum univ (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) i) j =\n v j \u2022 \u2191(w j) ^ (-(1 / 2))\nh : w' j = 0\n\u22a2 0 * (v j * v j) = w' j \u2022 (v j \u2022 \u21911 ^ (-(1 / 2)) * v j \u2022 \u21911 ^ (-(1 / 2)))\n[PROOFSTEP]\nsimp only [h, zero_smul, zero_mul]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n Finset.sum univ (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) i) j =\n v j \u2022 \u2191(w j) ^ (-(1 / 2))\nh : \u00acw' j = 0\n\u22a2 1 * (v j * v j) = w' j \u2022 (v j \u2022 \u2191(Units.mk0 (w' j) h) ^ (-(1 / 2)) * v j \u2022 \u2191(Units.mk0 (w' j) h) ^ (-(1 / 2)))\n[PROOFSTEP]\nhave hww' : w' j = w j := by simp only [dif_neg h, Units.val_mk0]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n Finset.sum univ (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) i) j =\n v j \u2022 \u2191(w j) ^ (-(1 / 2))\nh : \u00acw' j = 0\n\u22a2 w' j = \u2191(w j)\n[PROOFSTEP]\nsimp only [dif_neg h, Units.val_mk0]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n Finset.sum univ (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) i) j =\n v j \u2022 \u2191(w j) ^ (-(1 / 2))\nh : \u00acw' j = 0\nhww' : w' j = \u2191(w j)\n\u22a2 1 * (v j * v j) = w' j \u2022 (v j \u2022 \u2191(Units.mk0 (w' j) h) ^ (-(1 / 2)) * v j \u2022 \u2191(Units.mk0 (w' j) h) ^ (-(1 / 2)))\n[PROOFSTEP]\nsimp only [one_mul, Units.val_mk0, smul_eq_mul]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n Finset.sum univ (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) i) j =\n v j \u2022 \u2191(w j) ^ (-(1 / 2))\nh : \u00acw' j = 0\nhww' : w' j = \u2191(w j)\n\u22a2 v j * v j = w' j * (v j * w' j ^ (-(1 / 2)) * (v j * w' j ^ (-(1 / 2))))\n[PROOFSTEP]\nrw [hww']\n[GOAL]\ncase neg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n Finset.sum univ (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) i) j =\n v j \u2022 \u2191(w j) ^ (-(1 / 2))\nh : \u00acw' j = 0\nhww' : w' j = \u2191(w j)\n\u22a2 v j * v j = \u2191(w j) * (v j * \u2191(w j) ^ (-(1 / 2)) * (v j * \u2191(w j) ^ (-(1 / 2))))\n[PROOFSTEP]\nsuffices v j * v j = w j ^ (-(1 / 2 : \u2102)) * w j ^ (-(1 / 2 : \u2102)) * w j * v j * v j by rw [this]; ring\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n Finset.sum univ (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) i) j =\n v j \u2022 \u2191(w j) ^ (-(1 / 2))\nh : \u00acw' j = 0\nhww' : w' j = \u2191(w j)\nthis : v j * v j = \u2191(w j) ^ (-(1 / 2)) * \u2191(w j) ^ (-(1 / 2)) * \u2191(w j) * v j * v j\n\u22a2 v j * v j = \u2191(w j) * (v j * \u2191(w j) ^ (-(1 / 2)) * (v j * \u2191(w j) ^ (-(1 / 2))))\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n Finset.sum univ (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) i) j =\n v j \u2022 \u2191(w j) ^ (-(1 / 2))\nh : \u00acw' j = 0\nhww' : w' j = \u2191(w j)\nthis : v j * v j = \u2191(w j) ^ (-(1 / 2)) * \u2191(w j) ^ (-(1 / 2)) * \u2191(w j) * v j * v j\n\u22a2 \u2191(w j) ^ (-(1 / 2)) * \u2191(w j) ^ (-(1 / 2)) * \u2191(w j) * v j * v j =\n \u2191(w j) * (v j * \u2191(w j) ^ (-(1 / 2)) * (v j * \u2191(w j) ^ (-(1 / 2))))\n[PROOFSTEP]\nring\n[GOAL]\ncase neg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n Finset.sum univ (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) i) j =\n v j \u2022 \u2191(w j) ^ (-(1 / 2))\nh : \u00acw' j = 0\nhww' : w' j = \u2191(w j)\n\u22a2 v j * v j = \u2191(w j) ^ (-(1 / 2)) * \u2191(w j) ^ (-(1 / 2)) * \u2191(w j) * v j * v j\n[PROOFSTEP]\nrw [\u2190 Complex.cpow_add _ _ (w j).ne_zero, show -(1 / 2 : \u2102) + -(1 / 2) = -1 by simp [\u2190 two_mul], Complex.cpow_neg_one,\n inv_mul_cancel (w j).ne_zero, one_mul]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw' : \u03b9 \u2192 \u2102\nw : \u03b9 \u2192 \u2102\u02e3 := fun i => if h : w' i = 0 then 1 else Units.mk0 (w' i) h\nhw' : \u2200 (i : \u03b9), \u2191(w i) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u2102\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n Finset.sum univ (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191(w i) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u2102 \u03b9) i) j =\n v j \u2022 \u2191(w j) ^ (-(1 / 2))\nh : \u00acw' j = 0\nhww' : w' j = \u2191(w j)\n\u22a2 -(1 / 2) + -(1 / 2) = -1\n[PROOFSTEP]\nsimp [\u2190 two_mul]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u2102\u02e3\n\u22a2 IsometryEquiv (weightedSumSquares \u2102 w) (weightedSumSquares \u2102 1)\n[PROOFSTEP]\nsimpa using isometryEquivSumSquares ((\u2191) \u2218 w)\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.QuadraticForm.Complex", "llama_tokens": 8420, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195385342971, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.4378532261023457}} {"text": "[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nS : Submonoid R\nL : Type u\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\n\u22a2 #L \u2264 #R\n[PROOFSTEP]\nclassical\ncases fintypeOrInfinite R\n\u00b7 exact Cardinal.mk_le_of_surjective (IsArtinianRing.localization_surjective S _)\nerw [\u2190 Cardinal.mul_eq_self <| Cardinal.aleph0_le_mk R]\nset f : R \u00d7 R \u2192 L := fun aa => IsLocalization.mk' _ aa.1 (if h : aa.2 \u2208 S then \u27e8aa.2, h\u27e9 else 1)\nrefine' @Cardinal.mk_le_of_surjective _ _ f fun a => _\nobtain \u27e8x, y, h\u27e9 := IsLocalization.mk'_surjective S a\nuse(x, y)\ndsimp\nrwa [dif_pos <| show \u2191y \u2208 S from y.2, SetLike.eta]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nS : Submonoid R\nL : Type u\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\n\u22a2 #L \u2264 #R\n[PROOFSTEP]\ncases fintypeOrInfinite R\n[GOAL]\ncase inl\nR : Type u\ninst\u271d\u00b3 : CommRing R\nS : Submonoid R\nL : Type u\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nval\u271d : Fintype R\n\u22a2 #L \u2264 #R\n[PROOFSTEP]\nexact Cardinal.mk_le_of_surjective (IsArtinianRing.localization_surjective S _)\n[GOAL]\ncase inr\nR : Type u\ninst\u271d\u00b3 : CommRing R\nS : Submonoid R\nL : Type u\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nval\u271d : Infinite R\n\u22a2 #L \u2264 #R\n[PROOFSTEP]\nerw [\u2190 Cardinal.mul_eq_self <| Cardinal.aleph0_le_mk R]\n[GOAL]\ncase inr\nR : Type u\ninst\u271d\u00b3 : CommRing R\nS : Submonoid R\nL : Type u\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nval\u271d : Infinite R\n\u22a2 #L \u2264 #R * #R\n[PROOFSTEP]\nset f : R \u00d7 R \u2192 L := fun aa => IsLocalization.mk' _ aa.1 (if h : aa.2 \u2208 S then \u27e8aa.2, h\u27e9 else 1)\n[GOAL]\ncase inr\nR : Type u\ninst\u271d\u00b3 : CommRing R\nS : Submonoid R\nL : Type u\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nval\u271d : Infinite R\nf : R \u00d7 R \u2192 L := fun aa => mk' L aa.fst (if h : aa.snd \u2208 S then { val := aa.snd, property := h } else 1)\n\u22a2 #L \u2264 #R * #R\n[PROOFSTEP]\nrefine' @Cardinal.mk_le_of_surjective _ _ f fun a => _\n[GOAL]\ncase inr\nR : Type u\ninst\u271d\u00b3 : CommRing R\nS : Submonoid R\nL : Type u\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nval\u271d : Infinite R\nf : R \u00d7 R \u2192 L := fun aa => mk' L aa.fst (if h : aa.snd \u2208 S then { val := aa.snd, property := h } else 1)\na : L\n\u22a2 \u2203 a_1, f a_1 = a\n[PROOFSTEP]\nobtain \u27e8x, y, h\u27e9 := IsLocalization.mk'_surjective S a\n[GOAL]\ncase inr.intro.intro\nR : Type u\ninst\u271d\u00b3 : CommRing R\nS : Submonoid R\nL : Type u\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nval\u271d : Infinite R\nf : R \u00d7 R \u2192 L := fun aa => mk' L aa.fst (if h : aa.snd \u2208 S then { val := aa.snd, property := h } else 1)\na : L\nx : R\ny : { x // x \u2208 S }\nh : mk' L x y = a\n\u22a2 \u2203 a_1, f a_1 = a\n[PROOFSTEP]\nuse(x, y)\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b3 : CommRing R\nS : Submonoid R\nL : Type u\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nval\u271d : Infinite R\nf : R \u00d7 R \u2192 L := fun aa => mk' L aa.fst (if h : aa.snd \u2208 S then { val := aa.snd, property := h } else 1)\na : L\nx : R\ny : { x // x \u2208 S }\nh : mk' L x y = a\n\u22a2 f (x, \u2191y) = a\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b3 : CommRing R\nS : Submonoid R\nL : Type u\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nval\u271d : Infinite R\nf : R \u00d7 R \u2192 L := fun aa => mk' L aa.fst (if h : aa.snd \u2208 S then { val := aa.snd, property := h } else 1)\na : L\nx : R\ny : { x // x \u2208 S }\nh : mk' L x y = a\n\u22a2 mk' L x (if h : \u2191y \u2208 S then { val := \u2191y, property := h } else 1) = a\n[PROOFSTEP]\nrwa [dif_pos <| show \u2191y \u2208 S from y.2, SetLike.eta]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Localization.Cardinality", "llama_tokens": 1690, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195152660688, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.43785321222440315}} {"text": "[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Category.{?u.28, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 g.op \u226b f.op = 0\n[PROOFSTEP]\nrw [\u2190 op_comp, w, op_zero]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Category.{?u.28, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 f \u226b factorThruImage g = 0\n[PROOFSTEP]\nrw [\u2190 cancel_mono (image.\u03b9 g), Category.assoc, image.fac, w, zero_comp]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 imageToKernel g.op f.op (_ : g.op \u226b f.op = 0) =\n (imageSubobjectIso g.op \u226a\u226b (imageOpOp g).symm).hom \u226b\n (cokernel.desc f (factorThruImage g) (_ : f \u226b factorThruImage g = 0)).op \u226b\n (kernelSubobjectIso f.op \u226a\u226b kernelOpOp f).inv\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 imageToKernel g.op f.op (_ : g.op \u226b f.op = 0) \u226b Subobject.arrow (kernelSubobject f.op) =\n ((imageSubobjectIso g.op \u226a\u226b (imageOpOp g).symm).hom \u226b\n (cokernel.desc f (factorThruImage g) (_ : f \u226b factorThruImage g = 0)).op \u226b\n (kernelSubobjectIso f.op \u226a\u226b kernelOpOp f).inv) \u226b\n Subobject.arrow (kernelSubobject f.op)\n[PROOFSTEP]\nsimp only [Iso.trans_hom, Iso.symm_hom, Iso.trans_inv, kernelOpOp_inv, Category.assoc, imageToKernel_arrow,\n kernelSubobject_arrow', kernel.lift_\u03b9, \u2190 op_comp, cokernel.\u03c0_desc, \u2190 imageSubobject_arrow, \u2190\n imageUnopOp_inv_comp_op_factorThruImage g.op]\n[GOAL]\ncase h\nV : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 (imageSubobjectIso g.op).hom \u226b (imageUnopOp g.op).inv \u226b (factorThruImage g.op.unop).op =\n (imageSubobjectIso g.op).hom \u226b (imageOpOp g).inv \u226b (factorThruImage g).op\n[PROOFSTEP]\nrfl\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Category.{?u.7815, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\u1d52\u1d56\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 g.unop \u226b f.unop = 0\n[PROOFSTEP]\nrw [\u2190 unop_comp, w, unop_zero]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Category.{?u.7815, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\u1d52\u1d56\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 f \u226b factorThruImage g = 0\n[PROOFSTEP]\nrw [\u2190 cancel_mono (image.\u03b9 g), Category.assoc, image.fac, w, zero_comp]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\u1d52\u1d56\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 imageToKernel g.unop f.unop (_ : g.unop \u226b f.unop = 0) =\n (imageSubobjectIso g.unop \u226a\u226b (imageUnopUnop g).symm).hom \u226b\n (cokernel.desc f (factorThruImage g) (_ : f \u226b factorThruImage g = 0)).unop \u226b\n (kernelSubobjectIso f.unop \u226a\u226b kernelUnopUnop f).inv\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\u1d52\u1d56\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 imageToKernel g.unop f.unop (_ : g.unop \u226b f.unop = 0) \u226b Subobject.arrow (kernelSubobject f.unop) =\n ((imageSubobjectIso g.unop \u226a\u226b (imageUnopUnop g).symm).hom \u226b\n (cokernel.desc f (factorThruImage g) (_ : f \u226b factorThruImage g = 0)).unop \u226b\n (kernelSubobjectIso f.unop \u226a\u226b kernelUnopUnop f).inv) \u226b\n Subobject.arrow (kernelSubobject f.unop)\n[PROOFSTEP]\ndsimp only [imageUnopUnop]\n[GOAL]\ncase h\nV : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\u1d52\u1d56\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 imageToKernel g.unop f.unop (_ : g.unop \u226b f.unop = 0) \u226b Subobject.arrow (kernelSubobject f.unop) =\n ((imageSubobjectIso g.unop \u226a\u226b (Iso.unop (imageUnopOp g)).symm).hom \u226b\n (cokernel.desc f (factorThruImage g) (_ : f \u226b factorThruImage g = 0)).unop \u226b\n (kernelSubobjectIso f.unop \u226a\u226b kernelUnopUnop f).inv) \u226b\n Subobject.arrow (kernelSubobject f.unop)\n[PROOFSTEP]\nsimp only [Iso.trans_hom, Iso.symm_hom, Iso.trans_inv, kernelUnopUnop_inv, Category.assoc, imageToKernel_arrow,\n kernelSubobject_arrow', kernel.lift_\u03b9, cokernel.\u03c0_desc, Iso.unop_inv, \u2190 unop_comp,\n factorThruImage_comp_imageUnopOp_inv, Quiver.Hom.unop_op, imageSubobject_arrow]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Category.{?u.16638, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 g.op \u226b f.op = 0\n[PROOFSTEP]\nrw [\u2190 op_comp, w, op_zero]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Category.{?u.16638, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 cokernel.desc f g w = cokernel.desc f (factorThruImage g) (_ : f \u226b factorThruImage g = 0) \u226b image.\u03b9 g\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type u_1\ninst\u271d\u00b9 : Category.{?u.16638, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 coequalizer.\u03c0 f 0 \u226b cokernel.desc f g w =\n coequalizer.\u03c0 f 0 \u226b cokernel.desc f (factorThruImage g) (_ : f \u226b factorThruImage g = 0) \u226b image.\u03b9 g\n[PROOFSTEP]\nsimp only [image.fac, cokernel.\u03c0_desc, cokernel.\u03c0_desc_assoc]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Category.{?u.28807, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\u1d52\u1d56\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 g.unop \u226b f.unop = 0\n[PROOFSTEP]\nrw [\u2190 unop_comp, w, unop_zero]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Category.{?u.28807, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\u1d52\u1d56\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 cokernel.desc f g w = cokernel.desc f (factorThruImage g) (_ : f \u226b factorThruImage g = 0) \u226b image.\u03b9 g\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type u_1\ninst\u271d\u00b9 : Category.{?u.28807, u_1} V\ninst\u271d : Abelian V\nX Y Z : V\u1d52\u1d56\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 coequalizer.\u03c0 f 0 \u226b cokernel.desc f g w =\n coequalizer.\u03c0 f 0 \u226b cokernel.desc f (factorThruImage g) (_ : f \u226b factorThruImage g = 0) \u226b image.\u03b9 g\n[PROOFSTEP]\nsimp only [image.fac, cokernel.\u03c0_desc, cokernel.\u03c0_desc_assoc]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.40782, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : HomologicalComplex V c\ni j : \u03b9\nhij : \u00acComplexShape.Rel (ComplexShape.symm c) i j\n\u22a2 (fun i j => (d X j i).op) i j = 0\n[PROOFSTEP]\nsimp only\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.40782, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : HomologicalComplex V c\ni j : \u03b9\nhij : \u00acComplexShape.Rel (ComplexShape.symm c) i j\n\u22a2 (d X j i).op = 0\n[PROOFSTEP]\nrw [X.shape j i hij, op_zero]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.40782, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : HomologicalComplex V c\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : \u03b9\nx\u271d\u00b9 : ComplexShape.Rel (ComplexShape.symm c) x\u271d\u2074 x\u271d\u00b3\nx\u271d : ComplexShape.Rel (ComplexShape.symm c) x\u271d\u00b3 x\u271d\u00b2\n\u22a2 (fun i j => (d X j i).op) x\u271d\u2074 x\u271d\u00b3 \u226b (fun i j => (d X j i).op) x\u271d\u00b3 x\u271d\u00b2 = 0\n[PROOFSTEP]\nrw [\u2190 op_comp, X.d_comp_d, op_zero]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.42509, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : HomologicalComplex V (ComplexShape.symm c)\ni j : \u03b9\nhij : \u00acComplexShape.Rel c i j\n\u22a2 (fun i j => (d X j i).op) i j = 0\n[PROOFSTEP]\nsimp only\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.42509, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : HomologicalComplex V (ComplexShape.symm c)\ni j : \u03b9\nhij : \u00acComplexShape.Rel c i j\n\u22a2 (d X j i).op = 0\n[PROOFSTEP]\nrw [X.shape j i hij, op_zero]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.42509, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : HomologicalComplex V (ComplexShape.symm c)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : \u03b9\nx\u271d\u00b9 : ComplexShape.Rel c x\u271d\u2074 x\u271d\u00b3\nx\u271d : ComplexShape.Rel c x\u271d\u00b3 x\u271d\u00b2\n\u22a2 (fun i j => (d X j i).op) x\u271d\u2074 x\u271d\u00b3 \u226b (fun i j => (d X j i).op) x\u271d\u00b3 x\u271d\u00b2 = 0\n[PROOFSTEP]\nrw [\u2190 op_comp, X.d_comp_d, op_zero]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.44206, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : HomologicalComplex V\u1d52\u1d56 c\ni j : \u03b9\nhij : \u00acComplexShape.Rel (ComplexShape.symm c) i j\n\u22a2 (fun i j => (d X j i).unop) i j = 0\n[PROOFSTEP]\nsimp only\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.44206, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : HomologicalComplex V\u1d52\u1d56 c\ni j : \u03b9\nhij : \u00acComplexShape.Rel (ComplexShape.symm c) i j\n\u22a2 (d X j i).unop = 0\n[PROOFSTEP]\nrw [X.shape j i hij, unop_zero]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.44206, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : HomologicalComplex V\u1d52\u1d56 c\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : \u03b9\nx\u271d\u00b9 : ComplexShape.Rel (ComplexShape.symm c) x\u271d\u2074 x\u271d\u00b3\nx\u271d : ComplexShape.Rel (ComplexShape.symm c) x\u271d\u00b3 x\u271d\u00b2\n\u22a2 (fun i j => (d X j i).unop) x\u271d\u2074 x\u271d\u00b3 \u226b (fun i j => (d X j i).unop) x\u271d\u00b3 x\u271d\u00b2 = 0\n[PROOFSTEP]\nrw [\u2190 unop_comp, X.d_comp_d, unop_zero]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.45915, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : HomologicalComplex V\u1d52\u1d56 (ComplexShape.symm c)\ni j : \u03b9\nhij : \u00acComplexShape.Rel c i j\n\u22a2 (fun i j => (d X j i).unop) i j = 0\n[PROOFSTEP]\nsimp only\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.45915, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : HomologicalComplex V\u1d52\u1d56 (ComplexShape.symm c)\ni j : \u03b9\nhij : \u00acComplexShape.Rel c i j\n\u22a2 (d X j i).unop = 0\n[PROOFSTEP]\nrw [X.shape j i hij, unop_zero]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.45915, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : HomologicalComplex V\u1d52\u1d56 (ComplexShape.symm c)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : \u03b9\nx\u271d\u00b9 : ComplexShape.Rel c x\u271d\u2074 x\u271d\u00b3\nx\u271d : ComplexShape.Rel c x\u271d\u00b3 x\u271d\u00b2\n\u22a2 (fun i j => (d X j i).unop) x\u271d\u2074 x\u271d\u00b3 \u226b (fun i j => (d X j i).unop) x\u271d\u00b3 x\u271d\u00b2 = 0\n[PROOFSTEP]\nrw [\u2190 unop_comp, X.d_comp_d, unop_zero]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.47656, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX\u271d Y\u271d : (HomologicalComplex V c)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\ni j : \u03b9\nx\u271d : ComplexShape.Rel (ComplexShape.symm c) i j\n\u22a2 (fun i => (Hom.f f.unop i).op) i \u226b d ((fun X => HomologicalComplex.op X.unop) Y\u271d) i j =\n d ((fun X => HomologicalComplex.op X.unop) X\u271d) i j \u226b (fun i => (Hom.f f.unop i).op) j\n[PROOFSTEP]\nsimp only [op_d, \u2190 op_comp, f.unop.comm]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.55812, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX\u271d Y\u271d : HomologicalComplex V\u1d52\u1d56 (ComplexShape.symm c)\nf : X\u271d \u27f6 Y\u271d\ni j : \u03b9\nx\u271d : ComplexShape.Rel c i j\n\u22a2 (fun i => (Hom.f f i).unop) i \u226b d (HomologicalComplex.unopSymm X\u271d) i j =\n d (HomologicalComplex.unopSymm Y\u271d) i j \u226b (fun i => (Hom.f f i).unop) j\n[PROOFSTEP]\nsimp only [unopSymm_d, \u2190 unop_comp, f.comm]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.63659, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : (HomologicalComplex V c)\u1d52\u1d56\ni j : \u03b9\nx\u271d : ComplexShape.Rel c i j\n\u22a2 ((fun i => Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)) i).hom \u226b\n d X.unop i j =\n d (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i j \u226b\n ((fun i => Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)) j).hom\n[PROOFSTEP]\nsimp only [Iso.refl_hom, Category.id_comp, unopSymm_d, op_d, Quiver.Hom.unop_op, Category.comp_id]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.63659, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\n\u22a2 \u2200 {X Y : (HomologicalComplex V c)\u1d52\u1d56} (f : X \u27f6 Y),\n (\ud835\udfed (HomologicalComplex V c)\u1d52\u1d56).map f \u226b\n ((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n Y).hom =\n ((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n X).hom \u226b\n (opFunctor V c \u22d9 opInverse V c).map f\n[PROOFSTEP]\nintro X Y f\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.63659, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX Y : (HomologicalComplex V c)\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 (\ud835\udfed (HomologicalComplex V c)\u1d52\u1d56).map f \u226b\n ((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n Y).hom =\n ((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n X).hom \u226b\n (opFunctor V c \u22d9 opInverse V c).map f\n[PROOFSTEP]\nrefine' Quiver.Hom.unop_inj _\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.63659, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX Y : (HomologicalComplex V c)\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 ((\ud835\udfed (HomologicalComplex V c)\u1d52\u1d56).map f \u226b\n ((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n Y).hom).unop =\n (((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n X).hom \u226b\n (opFunctor V c \u22d9 opInverse V c).map f).unop\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.63659, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX Y : (HomologicalComplex V c)\u1d52\u1d56\nf : X \u27f6 Y\nx : \u03b9\n\u22a2 Hom.f\n ((\ud835\udfed (HomologicalComplex V c)\u1d52\u1d56).map f \u226b\n ((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n Y).hom).unop\n x =\n Hom.f\n (((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n X).hom \u226b\n (opFunctor V c \u22d9 opInverse V c).map f).unop\n x\n[PROOFSTEP]\nsimp only [Quiver.Hom.unop_op, Functor.id_map, Iso.op_hom, Functor.comp_map, unop_comp, comp_f,\n Hom.isoOfComponents_hom_f]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.63659, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX Y : (HomologicalComplex V c)\u1d52\u1d56\nf : X \u27f6 Y\nx : \u03b9\n\u22a2 (Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op Y.unop)) x)).hom \u226b\n Hom.f f.unop x =\n Hom.f ((opInverse V c).map ((opFunctor V c).map f)).unop x \u226b\n (Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) x)).hom\n[PROOFSTEP]\nerw [Category.id_comp, Category.comp_id (f.unop.f x)]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.82165, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : (HomologicalComplex V c)\u1d52\u1d56\n\u22a2 (opFunctor V c).map (NatTrans.app (opUnitIso V c).hom X) \u226b\n NatTrans.app (opCounitIso V c).hom ((opFunctor V c).obj X) =\n \ud835\udfd9 ((opFunctor V c).obj X)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.82165, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : (HomologicalComplex V c)\u1d52\u1d56\ni\u271d : \u03b9\n\u22a2 Hom.f\n ((opFunctor V c).map (NatTrans.app (opUnitIso V c).hom X) \u226b\n NatTrans.app (opCounitIso V c).hom ((opFunctor V c).obj X))\n i\u271d =\n Hom.f (\ud835\udfd9 ((opFunctor V c).obj X)) i\u271d\n[PROOFSTEP]\nsimp only [opUnitIso, opCounitIso, NatIso.ofComponents_hom_app, Iso.op_hom, comp_f, opFunctor_map_f, Quiver.Hom.unop_op,\n Hom.isoOfComponents_hom_f]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.82165, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : (HomologicalComplex V c)\u1d52\u1d56\ni\u271d : \u03b9\n\u22a2 (Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i\u271d)).hom.op \u226b\n (Iso.refl (HomologicalComplex.X ((opInverse V c \u22d9 opFunctor V c).obj ((opFunctor V c).obj X)) i\u271d)).hom =\n Hom.f (\ud835\udfd9 ((opFunctor V c).obj X)) i\u271d\n[PROOFSTEP]\nexact Category.comp_id _\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.83999, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX\u271d Y\u271d : (HomologicalComplex V\u1d52\u1d56 c)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\ni j : \u03b9\nx\u271d : ComplexShape.Rel (ComplexShape.symm c) i j\n\u22a2 (fun i => (Hom.f f.unop i).unop) i \u226b d ((fun X => HomologicalComplex.unop X.unop) Y\u271d) i j =\n d ((fun X => HomologicalComplex.unop X.unop) X\u271d) i j \u226b (fun i => (Hom.f f.unop i).unop) j\n[PROOFSTEP]\nsimp only [unop_d, \u2190 unop_comp, f.unop.comm]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.92175, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX\u271d Y\u271d : HomologicalComplex V (ComplexShape.symm c)\nf : X\u271d \u27f6 Y\u271d\ni j : \u03b9\nx\u271d : ComplexShape.Rel c i j\n\u22a2 (fun i => (Hom.f f i).op) i \u226b d (HomologicalComplex.opSymm X\u271d) i j =\n d (HomologicalComplex.opSymm Y\u271d) i j \u226b (fun i => (Hom.f f i).op) j\n[PROOFSTEP]\nsimp only [opSymm_d, \u2190 op_comp, f.comm]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.100003, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : (HomologicalComplex V\u1d52\u1d56 c)\u1d52\u1d56\ni j : \u03b9\nx\u271d : ComplexShape.Rel c i j\n\u22a2 ((fun i => Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)) i).hom \u226b\n d X.unop i j =\n d (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i j \u226b\n ((fun i => Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)) j).hom\n[PROOFSTEP]\nsimp only [Iso.refl_hom, Category.id_comp, unopSymm_d, op_d, Quiver.Hom.unop_op, Category.comp_id]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.100003, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\n\u22a2 \u2200 {X Y : (HomologicalComplex V\u1d52\u1d56 c)\u1d52\u1d56} (f : X \u27f6 Y),\n (\ud835\udfed (HomologicalComplex V\u1d52\u1d56 c)\u1d52\u1d56).map f \u226b\n ((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n Y).hom =\n ((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n X).hom \u226b\n (unopFunctor V c \u22d9 unopInverse V c).map f\n[PROOFSTEP]\nintro X Y f\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.100003, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX Y : (HomologicalComplex V\u1d52\u1d56 c)\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 (\ud835\udfed (HomologicalComplex V\u1d52\u1d56 c)\u1d52\u1d56).map f \u226b\n ((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n Y).hom =\n ((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n X).hom \u226b\n (unopFunctor V c \u22d9 unopInverse V c).map f\n[PROOFSTEP]\nrefine' Quiver.Hom.unop_inj _\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.100003, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX Y : (HomologicalComplex V\u1d52\u1d56 c)\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 ((\ud835\udfed (HomologicalComplex V\u1d52\u1d56 c)\u1d52\u1d56).map f \u226b\n ((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n Y).hom).unop =\n (((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n X).hom \u226b\n (unopFunctor V c \u22d9 unopInverse V c).map f).unop\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.100003, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX Y : (HomologicalComplex V\u1d52\u1d56 c)\u1d52\u1d56\nf : X \u27f6 Y\nx : \u03b9\n\u22a2 Hom.f\n ((\ud835\udfed (HomologicalComplex V\u1d52\u1d56 c)\u1d52\u1d56).map f \u226b\n ((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n Y).hom).unop\n x =\n Hom.f\n (((fun X =>\n Iso.op\n (Hom.isoOfComponents fun i =>\n Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)))\n X).hom \u226b\n (unopFunctor V c \u22d9 unopInverse V c).map f).unop\n x\n[PROOFSTEP]\nsimp only [Quiver.Hom.unop_op, Functor.id_map, Iso.op_hom, Functor.comp_map, unop_comp, comp_f,\n Hom.isoOfComponents_hom_f]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.100003, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX Y : (HomologicalComplex V\u1d52\u1d56 c)\u1d52\u1d56\nf : X \u27f6 Y\nx : \u03b9\n\u22a2 (Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op Y.unop)) x)).hom \u226b\n Hom.f f.unop x =\n Hom.f ((unopInverse V c).map ((unopFunctor V c).map f)).unop x \u226b\n (Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) x)).hom\n[PROOFSTEP]\nerw [Category.id_comp, Category.comp_id (f.unop.f x)]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.118561, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : (HomologicalComplex V\u1d52\u1d56 c)\u1d52\u1d56\n\u22a2 (unopFunctor V c).map (NatTrans.app (unopUnitIso V c).hom X) \u226b\n NatTrans.app (unopCounitIso V c).hom ((unopFunctor V c).obj X) =\n \ud835\udfd9 ((unopFunctor V c).obj X)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.118561, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : (HomologicalComplex V\u1d52\u1d56 c)\u1d52\u1d56\ni\u271d : \u03b9\n\u22a2 Hom.f\n ((unopFunctor V c).map (NatTrans.app (unopUnitIso V c).hom X) \u226b\n NatTrans.app (unopCounitIso V c).hom ((unopFunctor V c).obj X))\n i\u271d =\n Hom.f (\ud835\udfd9 ((unopFunctor V c).obj X)) i\u271d\n[PROOFSTEP]\nsimp only [opUnitIso, opCounitIso, NatIso.ofComponents_hom_app, Iso.op_hom, comp_f, opFunctor_map_f, Quiver.Hom.unop_op,\n Hom.isoOfComponents_hom_f]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.118561, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Preadditive V\nX : (HomologicalComplex V\u1d52\u1d56 c)\u1d52\u1d56\ni\u271d : \u03b9\n\u22a2 Hom.f ((unopFunctor V c).map (NatTrans.app (unopUnitIso V c).hom X)) i\u271d \u226b\n Hom.f (NatTrans.app (unopCounitIso V c).hom ((unopFunctor V c).obj X)) i\u271d =\n Hom.f (\ud835\udfd9 ((unopFunctor V c).obj X)) i\u271d\n[PROOFSTEP]\nexact Category.comp_id _\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.125717, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Abelian V\nC : HomologicalComplex V c\ni : \u03b9\n\u22a2 (dFrom C i).op \u226b (dTo C i).op = 0\n[PROOFSTEP]\nrw [\u2190 op_comp, C.dTo_comp_dFrom i, op_zero]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u_2\ninst\u271d\u00b9 : Category.{?u.129295, u_2} V\nc : ComplexShape \u03b9\ninst\u271d : Abelian V\nC\u271d : HomologicalComplex V c\ni : \u03b9\nC : HomologicalComplex V\u1d52\u1d56 c\n\u22a2 (dFrom C i).unop \u226b (dTo C i).unop = 0\n[PROOFSTEP]\nrw [\u2190 unop_comp, C.dTo_comp_dFrom i, unop_zero]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Homology.Opposite", "llama_tokens": 10879, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.6261241772283035, "lm_q1q2_score": 0.4378200990586688}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx y\u271d y : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (y \u2208 pi univ fun i => Ici (x i)) \u2194 y \u2208 Ici x\n[PROOFSTEP]\nsimp [Pi.le_def]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx y\u271d y : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (y \u2208 pi univ fun i => Iic (x i)) \u2194 y \u2208 Iic x\n[PROOFSTEP]\nsimp [Pi.le_def]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx y\u271d y : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (y \u2208 pi univ fun i => Icc (x i) (y\u271d i)) \u2194 y \u2208 Icc x y\u271d\n[PROOFSTEP]\nsimp [Pi.le_def, forall_and]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nhm : x i\u2080 \u2264 m\n\u22a2 (pi univ fun i => Ioc (update x i\u2080 m i) (y i)) = {z | m < z i\u2080} \u2229 pi univ fun i => Ioc (x i) (y i)\n[PROOFSTEP]\nhave : Ioc m (y i\u2080) = Ioi m \u2229 Ioc (x i\u2080) (y i\u2080) := by\n rw [\u2190 Ioi_inter_Iic, \u2190 Ioi_inter_Iic, \u2190 inter_assoc, inter_eq_self_of_subset_left (Ioi_subset_Ioi hm)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nhm : x i\u2080 \u2264 m\n\u22a2 Ioc m (y i\u2080) = Ioi m \u2229 Ioc (x i\u2080) (y i\u2080)\n[PROOFSTEP]\nrw [\u2190 Ioi_inter_Iic, \u2190 Ioi_inter_Iic, \u2190 inter_assoc, inter_eq_self_of_subset_left (Ioi_subset_Ioi hm)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nhm : x i\u2080 \u2264 m\nthis : Ioc m (y i\u2080) = Ioi m \u2229 Ioc (x i\u2080) (y i\u2080)\n\u22a2 (pi univ fun i => Ioc (update x i\u2080 m i) (y i)) = {z | m < z i\u2080} \u2229 pi univ fun i => Ioc (x i) (y i)\n[PROOFSTEP]\nsimp_rw [univ_pi_update i\u2080 _ _ fun i z \u21a6 Ioc z (y i), \u2190 pi_inter_compl ({ i\u2080 } : Set \u03b9), singleton_pi', \u2190 inter_assoc,\n this]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nhm : x i\u2080 \u2264 m\nthis : Ioc m (y i\u2080) = Ioi m \u2229 Ioc (x i\u2080) (y i\u2080)\n\u22a2 ({x_1 | x_1 i\u2080 \u2208 Ioi m \u2229 Ioc (x i\u2080) (y i\u2080)} \u2229 pi {i\u2080}\u1d9c fun j => Ioc (x j) (y j)) =\n {z | m < z i\u2080} \u2229 {x_1 | x_1 i\u2080 \u2208 Ioc (x i\u2080) (y i\u2080)} \u2229 pi {i\u2080}\u1d9c fun j => Ioc (x j) (y j)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nhm : m \u2264 y i\u2080\n\u22a2 (pi univ fun i => Ioc (x i) (update y i\u2080 m i)) = {z | z i\u2080 \u2264 m} \u2229 pi univ fun i => Ioc (x i) (y i)\n[PROOFSTEP]\nhave : Ioc (x i\u2080) m = Iic m \u2229 Ioc (x i\u2080) (y i\u2080) := by\n rw [\u2190 Ioi_inter_Iic, \u2190 Ioi_inter_Iic, inter_left_comm, inter_eq_self_of_subset_left (Iic_subset_Iic.2 hm)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nhm : m \u2264 y i\u2080\n\u22a2 Ioc (x i\u2080) m = Iic m \u2229 Ioc (x i\u2080) (y i\u2080)\n[PROOFSTEP]\nrw [\u2190 Ioi_inter_Iic, \u2190 Ioi_inter_Iic, inter_left_comm, inter_eq_self_of_subset_left (Iic_subset_Iic.2 hm)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nhm : m \u2264 y i\u2080\nthis : Ioc (x i\u2080) m = Iic m \u2229 Ioc (x i\u2080) (y i\u2080)\n\u22a2 (pi univ fun i => Ioc (x i) (update y i\u2080 m i)) = {z | z i\u2080 \u2264 m} \u2229 pi univ fun i => Ioc (x i) (y i)\n[PROOFSTEP]\nsimp_rw [univ_pi_update i\u2080 y m fun i z \u21a6 Ioc (x i) z, \u2190 pi_inter_compl ({ i\u2080 } : Set \u03b9), singleton_pi', \u2190 inter_assoc,\n this]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nhm : m \u2264 y i\u2080\nthis : Ioc (x i\u2080) m = Iic m \u2229 Ioc (x i\u2080) (y i\u2080)\n\u22a2 ({x_1 | x_1 i\u2080 \u2208 Iic m \u2229 Ioc (x i\u2080) (y i\u2080)} \u2229 pi {i\u2080}\u1d9c fun j => Ioc (x j) (y j)) =\n {z | z i\u2080 \u2264 m} \u2229 {x_1 | x_1 i\u2080 \u2208 Ioc (x i\u2080) (y i\u2080)} \u2229 pi {i\u2080}\u1d9c fun j => Ioc (x j) (y j)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\n\u22a2 Disjoint (pi univ fun i => Ioc (x i) (update y i\u2080 m i)) (pi univ fun i => Ioc (update x i\u2080 m i) (y i))\n[PROOFSTEP]\nrw [disjoint_left]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\n\u22a2 \u2200 \u2983a : (i : \u03b9) \u2192 \u03b1 i\u2984,\n (a \u2208 pi univ fun i => Ioc (x i) (update y i\u2080 m i)) \u2192 \u00aca \u2208 pi univ fun i => Ioc (update x i\u2080 m i) (y i)\n[PROOFSTEP]\nrintro z h\u2081 h\u2082\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nz : (i : \u03b9) \u2192 \u03b1 i\nh\u2081 : z \u2208 pi univ fun i => Ioc (x i) (update y i\u2080 m i)\nh\u2082 : z \u2208 pi univ fun i => Ioc (update x i\u2080 m i) (y i)\n\u22a2 False\n[PROOFSTEP]\nrefine' (h\u2081 i\u2080 (mem_univ _)).2.not_lt _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nz : (i : \u03b9) \u2192 \u03b1 i\nh\u2081 : z \u2208 pi univ fun i => Ioc (x i) (update y i\u2080 m i)\nh\u2082 : z \u2208 pi univ fun i => Ioc (update x i\u2080 m i) (y i)\n\u22a2 update y i\u2080 m i\u2080 < z i\u2080\n[PROOFSTEP]\nsimpa only [Function.update_same] using (h\u2082 i\u2080 (mem_univ _)).1\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na b : \u03b1 i\n\u22a2 update f i '' Icc a b = Icc (update f i a) (update f i b)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na b : \u03b1 i\nx : (a : \u03b9) \u2192 \u03b1 a\n\u22a2 x \u2208 update f i '' Icc a b \u2194 x \u2208 Icc (update f i a) (update f i b)\n[PROOFSTEP]\nrw [\u2190 Set.pi_univ_Icc]\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na b : \u03b1 i\nx : (a : \u03b9) \u2192 \u03b1 a\n\u22a2 x \u2208 update f i '' Icc a b \u2194 x \u2208 pi univ fun i_1 => Icc (update f i a i_1) (update f i b i_1)\n[PROOFSTEP]\nrefine' \u27e8_, fun h => \u27e8x i, _, _\u27e9\u27e9\n[GOAL]\ncase h.refine'_1\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na b : \u03b1 i\nx : (a : \u03b9) \u2192 \u03b1 a\n\u22a2 x \u2208 update f i '' Icc a b \u2192 x \u2208 pi univ fun i_1 => Icc (update f i a i_1) (update f i b i_1)\n[PROOFSTEP]\nrintro \u27e8c, hc, rfl\u27e9\n[GOAL]\ncase h.refine'_1.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na b c : \u03b1 i\nhc : c \u2208 Icc a b\n\u22a2 update f i c \u2208 pi univ fun i_1 => Icc (update f i a i_1) (update f i b i_1)\n[PROOFSTEP]\nsimpa [update_le_update_iff]\n[GOAL]\ncase h.refine'_2\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na b : \u03b1 i\nx : (a : \u03b9) \u2192 \u03b1 a\nh : x \u2208 pi univ fun i_1 => Icc (update f i a i_1) (update f i b i_1)\n\u22a2 x i \u2208 Icc a b\n[PROOFSTEP]\nsimpa only [Function.update_same] using h i (mem_univ i)\n[GOAL]\ncase h.refine'_3\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na b : \u03b1 i\nx : (a : \u03b9) \u2192 \u03b1 a\nh : x \u2208 pi univ fun i_1 => Icc (update f i a i_1) (update f i b i_1)\n\u22a2 update f i (x i) = x\n[PROOFSTEP]\next j\n[GOAL]\ncase h.refine'_3.h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na b : \u03b1 i\nx : (a : \u03b9) \u2192 \u03b1 a\nh : x \u2208 pi univ fun i_1 => Icc (update f i a i_1) (update f i b i_1)\nj : \u03b9\n\u22a2 update f i (x i) j = x j\n[PROOFSTEP]\nobtain rfl | hij := eq_or_ne i j\n[GOAL]\ncase h.refine'_3.h.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na b : \u03b1 i\nx : (a : \u03b9) \u2192 \u03b1 a\nh : x \u2208 pi univ fun i_1 => Icc (update f i a i_1) (update f i b i_1)\n\u22a2 update f i (x i) i = x i\n[PROOFSTEP]\nexact Function.update_same _ _ _\n[GOAL]\ncase h.refine'_3.h.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na b : \u03b1 i\nx : (a : \u03b9) \u2192 \u03b1 a\nh : x \u2208 pi univ fun i_1 => Icc (update f i a i_1) (update f i b i_1)\nj : \u03b9\nhij : i \u2260 j\n\u22a2 update f i (x i) j = x j\n[PROOFSTEP]\nsimpa only [Function.update_noteq hij.symm, le_antisymm_iff] using h j (mem_univ j)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na b : \u03b1 i\n\u22a2 update f i '' Ico a b = Ico (update f i a) (update f i b)\n[PROOFSTEP]\nrw [\u2190 Icc_diff_right, \u2190 Icc_diff_right, image_diff (update_injective _ _), image_singleton, image_update_Icc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na b : \u03b1 i\n\u22a2 update f i '' Ioc a b = Ioc (update f i a) (update f i b)\n[PROOFSTEP]\nrw [\u2190 Icc_diff_left, \u2190 Icc_diff_left, image_diff (update_injective _ _), image_singleton, image_update_Icc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na b : \u03b1 i\n\u22a2 update f i '' Ioo a b = Ioo (update f i a) (update f i b)\n[PROOFSTEP]\nrw [\u2190 Ico_diff_left, \u2190 Ico_diff_left, image_diff (update_injective _ _), image_singleton, image_update_Ico]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na : \u03b1 i\n\u22a2 update f i '' Icc a (f i) = Icc (update f i a) f\n[PROOFSTEP]\nsimpa using image_update_Icc f i a (f i)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na : \u03b1 i\n\u22a2 update f i '' Ico a (f i) = Ico (update f i a) f\n[PROOFSTEP]\nsimpa using image_update_Ico f i a (f i)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na : \u03b1 i\n\u22a2 update f i '' Ioc a (f i) = Ioc (update f i a) f\n[PROOFSTEP]\nsimpa using image_update_Ioc f i a (f i)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na : \u03b1 i\n\u22a2 update f i '' Ioo a (f i) = Ioo (update f i a) f\n[PROOFSTEP]\nsimpa using image_update_Ioo f i a (f i)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\nb : \u03b1 i\n\u22a2 update f i '' Icc (f i) b = Icc f (update f i b)\n[PROOFSTEP]\nsimpa using image_update_Icc f i (f i) b\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\nb : \u03b1 i\n\u22a2 update f i '' Ico (f i) b = Ico f (update f i b)\n[PROOFSTEP]\nsimpa using image_update_Ico f i (f i) b\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\nb : \u03b1 i\n\u22a2 update f i '' Ioc (f i) b = Ioc f (update f i b)\n[PROOFSTEP]\nsimpa using image_update_Ioc f i (f i) b\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\nb : \u03b1 i\n\u22a2 update f i '' Ioo (f i) b = Ioo f (update f i b)\n[PROOFSTEP]\nsimpa using image_update_Ioo f i (f i) b\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Lattice (\u03b1 i)\ninst\u271d : DecidableEq \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na b : \u03b1 i\n\u22a2 Icc (update f i (a \u2293 b)) (update f i (a \u2294 b)) = uIcc (update f i a) (update f i b)\n[PROOFSTEP]\nsimp_rw [uIcc, update_sup, update_inf]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Lattice (\u03b1 i)\ninst\u271d : DecidableEq \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\na : \u03b1 i\n\u22a2 update f i '' uIcc a (f i) = uIcc (update f i a) f\n[PROOFSTEP]\nsimpa using image_update_uIcc f i a (f i)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Lattice (\u03b1 i)\ninst\u271d : DecidableEq \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\nb : \u03b1 i\n\u22a2 update f i '' uIcc (f i) b = uIcc f (update f i b)\n[PROOFSTEP]\nsimpa using image_update_uIcc f i (f i) b\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nhm : m \u2208 Icc (x i\u2080) (y i\u2080)\n\u22a2 ((pi univ fun i => Ioc (x i) (update y i\u2080 m i)) \u222a pi univ fun i => Ioc (update x i\u2080 m i) (y i)) =\n pi univ fun i => Ioc (x i) (y i)\n[PROOFSTEP]\nsimp_rw [pi_univ_Ioc_update_left hm.1, pi_univ_Ioc_update_right hm.2, \u2190 union_inter_distrib_right, \u2190 setOf_or, le_or_lt,\n setOf_true, univ_inter]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nx y x' y' : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (Icc x y \\ pi univ fun i => Ioo (x' i) (y' i)) \u2286\n (\u22c3 (i : \u03b9), Icc x (update y i (x' i))) \u222a \u22c3 (i : \u03b9), Icc (update x i (y' i)) y\n[PROOFSTEP]\nrintro a \u27e8\u27e8hxa, hay\u27e9, ha'\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nx y x' y' a : (i : \u03b9) \u2192 \u03b1 i\nha' : \u00aca \u2208 pi univ fun i => Ioo (x' i) (y' i)\nhxa : x \u2264 a\nhay : a \u2264 y\n\u22a2 a \u2208 (\u22c3 (i : \u03b9), Icc x (update y i (x' i))) \u222a \u22c3 (i : \u03b9), Icc (update x i (y' i)) y\n[PROOFSTEP]\nsimp at ha' \n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nx y x' y' a : (i : \u03b9) \u2192 \u03b1 i\nhxa : x \u2264 a\nhay : a \u2264 y\nha' : \u2203 x, x' x < a x \u2192 y' x \u2264 a x\n\u22a2 a \u2208 (\u22c3 (i : \u03b9), Icc x (update y i (x' i))) \u222a \u22c3 (i : \u03b9), Icc (update x i (y' i)) y\n[PROOFSTEP]\nsimp [le_update_iff, update_le_iff, hxa, hay, hxa _, hay _, \u2190 exists_or]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nx y x' y' a : (i : \u03b9) \u2192 \u03b1 i\nhxa : x \u2264 a\nhay : a \u2264 y\nha' : \u2203 x, x' x < a x \u2192 y' x \u2264 a x\n\u22a2 \u2203 x, a x \u2264 x' x \u2228 y' x \u2264 a x\n[PROOFSTEP]\nrcases ha' with \u27e8w, hw\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nx y x' y' a : (i : \u03b9) \u2192 \u03b1 i\nhxa : x \u2264 a\nhay : a \u2264 y\nw : \u03b9\nhw : x' w < a w \u2192 y' w \u2264 a w\n\u22a2 \u2203 x, a x \u2264 x' x \u2228 y' x \u2264 a x\n[PROOFSTEP]\napply Exists.intro w\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nx y x' y' a : (i : \u03b9) \u2192 \u03b1 i\nhxa : x \u2264 a\nhay : a \u2264 y\nw : \u03b9\nhw : x' w < a w \u2192 y' w \u2264 a w\n\u22a2 a w \u2264 x' w \u2228 y' w \u2264 a w\n[PROOFSTEP]\ncases lt_or_le (x' w) (a w)\n[GOAL]\ncase intro.intro.intro.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nx y x' y' a : (i : \u03b9) \u2192 \u03b1 i\nhxa : x \u2264 a\nhay : a \u2264 y\nw : \u03b9\nhw : x' w < a w \u2192 y' w \u2264 a w\nh\u271d : x' w < a w\n\u22a2 a w \u2264 x' w \u2228 y' w \u2264 a w\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase intro.intro.intro.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nx y x' y' a : (i : \u03b9) \u2192 \u03b1 i\nhxa : x \u2264 a\nhay : a \u2264 y\nw : \u03b9\nhw : x' w < a w \u2192 y' w \u2264 a w\nh\u271d : a w \u2264 x' w\n\u22a2 a w \u2264 x' w \u2228 y' w \u2264 a w\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nx y z : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (Icc x z \\ pi univ fun i => Ioc (y i) (z i)) \u2286 \u22c3 (i : \u03b9), Icc x (update z i (y i))\n[PROOFSTEP]\nrintro a \u27e8\u27e8hax, haz\u27e9, hay\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nx y z a : (i : \u03b9) \u2192 \u03b1 i\nhay : \u00aca \u2208 pi univ fun i => Ioc (y i) (z i)\nhax : x \u2264 a\nhaz : a \u2264 z\n\u22a2 a \u2208 \u22c3 (i : \u03b9), Icc x (update z i (y i))\n[PROOFSTEP]\nsimpa [not_and_or, hax, le_update_iff, haz _] using hay\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Intervals.Pi", "llama_tokens": 8349, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124812, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.43764356260626475}} {"text": "[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf g : { x // x \u2208 Lp E p }\n\u22a2 \u2191\u2191f \u2264\u1d50[\u03bc] \u2191\u2191g \u2194 f \u2264 g\n[PROOFSTEP]\nrw [\u2190 Subtype.coe_le_coe, \u2190 AEEqFun.coeFn_le]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf : { x // x \u2208 Lp E p }\n\u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191f \u2194 0 \u2264 f\n[PROOFSTEP]\nrw [\u2190 coeFn_le]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf : { x // x \u2208 Lp E p }\n\u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191f \u2194 \u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191f\n[PROOFSTEP]\nhave h0 := Lp.coeFn_zero E p \u03bc\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf : { x // x \u2208 Lp E p }\nh0 : \u2191\u21910 =\u1d50[\u03bc] 0\n\u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191f \u2194 \u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf : { x // x \u2208 Lp E p }\nh0 : \u2191\u21910 =\u1d50[\u03bc] 0\n\u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191f \u2192 \u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191f\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf : { x // x \u2208 Lp E p }\nh0 : \u2191\u21910 =\u1d50[\u03bc] 0\n\u22a2 \u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191f \u2192 0 \u2264\u1d50[\u03bc] \u2191\u2191f\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf : { x // x \u2208 Lp E p }\nh0 : \u2191\u21910 =\u1d50[\u03bc] 0\nh : 0 \u2264\u1d50[\u03bc] \u2191\u2191f\n\u22a2 \u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191f\n[PROOFSTEP]\nfilter_upwards [h, h0] with _ _ h2\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf : { x // x \u2208 Lp E p }\nh0 : \u2191\u21910 =\u1d50[\u03bc] 0\nh : \u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191f\n\u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191f\n[PROOFSTEP]\nfilter_upwards [h, h0] with _ _ h2\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf : { x // x \u2208 Lp E p }\nh0 : \u2191\u21910 =\u1d50[\u03bc] 0\nh : 0 \u2264\u1d50[\u03bc] \u2191\u2191f\na\u271d\u00b9 : \u03b1\na\u271d : OfNat.ofNat 0 a\u271d\u00b9 \u2264 \u2191\u2191f a\u271d\u00b9\nh2 : \u2191\u21910 a\u271d\u00b9 = OfNat.ofNat 0 a\u271d\u00b9\n\u22a2 \u2191\u21910 a\u271d\u00b9 \u2264 \u2191\u2191f a\u271d\u00b9\n[PROOFSTEP]\nrwa [h2]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf : { x // x \u2208 Lp E p }\nh0 : \u2191\u21910 =\u1d50[\u03bc] 0\nh : \u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191f\na\u271d\u00b9 : \u03b1\na\u271d : \u2191\u21910 a\u271d\u00b9 \u2264 \u2191\u2191f a\u271d\u00b9\nh2 : \u2191\u21910 a\u271d\u00b9 = OfNat.ofNat 0 a\u271d\u00b9\n\u22a2 OfNat.ofNat 0 a\u271d\u00b9 \u2264 \u2191\u2191f a\u271d\u00b9\n[PROOFSTEP]\nrwa [\u2190 h2]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\n\u22a2 CovariantClass { x // x \u2208 Lp E p } { x // x \u2208 Lp E p } (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nrefine' \u27e8fun f g\u2081 g\u2082 hg\u2081\u2082 => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf g\u2081 g\u2082 : { x // x \u2208 Lp E p }\nhg\u2081\u2082 : g\u2081 \u2264 g\u2082\n\u22a2 f + g\u2081 \u2264 f + g\u2082\n[PROOFSTEP]\nrw [\u2190 coeFn_le] at hg\u2081\u2082 \u22a2\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf g\u2081 g\u2082 : { x // x \u2208 Lp E p }\nhg\u2081\u2082 : \u2191\u2191g\u2081 \u2264\u1d50[\u03bc] \u2191\u2191g\u2082\n\u22a2 \u2191\u2191(f + g\u2081) \u2264\u1d50[\u03bc] \u2191\u2191(f + g\u2082)\n[PROOFSTEP]\nfilter_upwards [coeFn_add f g\u2081, coeFn_add f g\u2082, hg\u2081\u2082] with _ h1 h2 h3\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf g\u2081 g\u2082 : { x // x \u2208 Lp E p }\nhg\u2081\u2082 : \u2191\u2191g\u2081 \u2264\u1d50[\u03bc] \u2191\u2191g\u2082\na\u271d : \u03b1\nh1 : \u2191\u2191(f + g\u2081) a\u271d = (\u2191\u2191f + \u2191\u2191g\u2081) a\u271d\nh2 : \u2191\u2191(f + g\u2082) a\u271d = (\u2191\u2191f + \u2191\u2191g\u2082) a\u271d\nh3 : \u2191\u2191g\u2081 a\u271d \u2264 \u2191\u2191g\u2082 a\u271d\n\u22a2 \u2191\u2191(f + g\u2081) a\u271d \u2264 \u2191\u2191(f + g\u2082) a\u271d\n[PROOFSTEP]\nrw [h1, h2, Pi.add_apply, Pi.add_apply]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf g\u2081 g\u2082 : { x // x \u2208 Lp E p }\nhg\u2081\u2082 : \u2191\u2191g\u2081 \u2264\u1d50[\u03bc] \u2191\u2191g\u2082\na\u271d : \u03b1\nh1 : \u2191\u2191(f + g\u2081) a\u271d = (\u2191\u2191f + \u2191\u2191g\u2081) a\u271d\nh2 : \u2191\u2191(f + g\u2082) a\u271d = (\u2191\u2191f + \u2191\u2191g\u2082) a\u271d\nh3 : \u2191\u2191g\u2081 a\u271d \u2264 \u2191\u2191g\u2082 a\u271d\n\u22a2 \u2191\u2191f a\u271d + \u2191\u2191g\u2081 a\u271d \u2264 \u2191\u2191f a\u271d + \u2191\u2191g\u2082 a\u271d\n[PROOFSTEP]\nexact add_le_add le_rfl h3\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf g : \u03b1 \u2192\u2098[\u03bc] E\nhf : f \u2208 Lp E p\nhg : g \u2208 Lp E p\n\u22a2 f \u2294 g \u2208 Lp E p\n[PROOFSTEP]\nrw [mem_Lp_iff_mem\u2112p] at *\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf g : \u03b1 \u2192\u2098[\u03bc] E\nhf : Mem\u2112p (\u2191f) p\nhg : Mem\u2112p (\u2191g) p\n\u22a2 Mem\u2112p (\u2191(f \u2294 g)) p\n[PROOFSTEP]\nexact (mem\u2112p_congr_ae (AEEqFun.coeFn_sup _ _)).mpr (hf.sup hg)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf g : \u03b1 \u2192\u2098[\u03bc] E\nhf : f \u2208 Lp E p\nhg : g \u2208 Lp E p\n\u22a2 f \u2293 g \u2208 Lp E p\n[PROOFSTEP]\nrw [mem_Lp_iff_mem\u2112p] at *\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : NormedLatticeAddCommGroup E\nf g : \u03b1 \u2192\u2098[\u03bc] E\nhf : Mem\u2112p (\u2191f) p\nhg : Mem\u2112p (\u2191g) p\n\u22a2 Mem\u2112p (\u2191(f \u2293 g)) p\n[PROOFSTEP]\nexact (mem\u2112p_congr_ae (AEEqFun.coeFn_inf _ _)).mpr (hf.inf hg)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedLatticeAddCommGroup E\ninst\u271d : Fact (1 \u2264 p)\nsrc\u271d\u00b9 : Lattice { x // x \u2208 Lp E p } := instLattice\nsrc\u271d : NormedAddCommGroup { x // x \u2208 Lp E p } := instNormedAddCommGroup\nf g : { x // x \u2208 Lp E p }\nhfg : |f| \u2264 |g|\n\u22a2 \u2016f\u2016 \u2264 \u2016g\u2016\n[PROOFSTEP]\nrw [\u2190 coeFn_le] at hfg \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedLatticeAddCommGroup E\ninst\u271d : Fact (1 \u2264 p)\nsrc\u271d\u00b9 : Lattice { x // x \u2208 Lp E p } := instLattice\nsrc\u271d : NormedAddCommGroup { x // x \u2208 Lp E p } := instNormedAddCommGroup\nf g : { x // x \u2208 Lp E p }\nhfg : \u2191\u2191|f| \u2264\u1d50[\u03bc] \u2191\u2191|g|\n\u22a2 \u2016f\u2016 \u2264 \u2016g\u2016\n[PROOFSTEP]\nsimp_rw [Lp.norm_def, ENNReal.toReal_le_toReal (Lp.snorm_ne_top f) (Lp.snorm_ne_top g)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedLatticeAddCommGroup E\ninst\u271d : Fact (1 \u2264 p)\nsrc\u271d\u00b9 : Lattice { x // x \u2208 Lp E p } := instLattice\nsrc\u271d : NormedAddCommGroup { x // x \u2208 Lp E p } := instNormedAddCommGroup\nf g : { x // x \u2208 Lp E p }\nhfg : \u2191\u2191|f| \u2264\u1d50[\u03bc] \u2191\u2191|g|\n\u22a2 snorm (\u2191\u2191f) p \u03bc \u2264 snorm (\u2191\u2191g) p \u03bc\n[PROOFSTEP]\nrefine' snorm_mono_ae _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedLatticeAddCommGroup E\ninst\u271d : Fact (1 \u2264 p)\nsrc\u271d\u00b9 : Lattice { x // x \u2208 Lp E p } := instLattice\nsrc\u271d : NormedAddCommGroup { x // x \u2208 Lp E p } := instNormedAddCommGroup\nf g : { x // x \u2208 Lp E p }\nhfg : \u2191\u2191|f| \u2264\u1d50[\u03bc] \u2191\u2191|g|\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191\u2191f x\u2016 \u2264 \u2016\u2191\u2191g x\u2016\n[PROOFSTEP]\nfilter_upwards [hfg, Lp.coeFn_abs f, Lp.coeFn_abs g] with x hx hxf hxg\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedLatticeAddCommGroup E\ninst\u271d : Fact (1 \u2264 p)\nsrc\u271d\u00b9 : Lattice { x // x \u2208 Lp E p } := instLattice\nsrc\u271d : NormedAddCommGroup { x // x \u2208 Lp E p } := instNormedAddCommGroup\nf g : { x // x \u2208 Lp E p }\nhfg : \u2191\u2191|f| \u2264\u1d50[\u03bc] \u2191\u2191|g|\nx : \u03b1\nhx : \u2191\u2191|f| x \u2264 \u2191\u2191|g| x\nhxf : \u2191\u2191|f| x = |\u2191\u2191f x|\nhxg : \u2191\u2191|g| x = |\u2191\u2191g x|\n\u22a2 \u2016\u2191\u2191f x\u2016 \u2264 \u2016\u2191\u2191g x\u2016\n[PROOFSTEP]\nrw [hxf, hxg] at hx \n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedLatticeAddCommGroup E\ninst\u271d : Fact (1 \u2264 p)\nsrc\u271d\u00b9 : Lattice { x // x \u2208 Lp E p } := instLattice\nsrc\u271d : NormedAddCommGroup { x // x \u2208 Lp E p } := instNormedAddCommGroup\nf g : { x // x \u2208 Lp E p }\nhfg : \u2191\u2191|f| \u2264\u1d50[\u03bc] \u2191\u2191|g|\nx : \u03b1\nhx : |\u2191\u2191f x| \u2264 |\u2191\u2191g x|\nhxf : \u2191\u2191|f| x = |\u2191\u2191f x|\nhxg : \u2191\u2191|g| x = |\u2191\u2191g x|\n\u22a2 \u2016\u2191\u2191f x\u2016 \u2264 \u2016\u2191\u2191g x\u2016\n[PROOFSTEP]\nexact HasSolidNorm.solid hx\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.LpOrder", "llama_tokens": 4368, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.743167997235783, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4376435559066508}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\n\u22a2 t' D i i j = (pullbackSymmetry (f D i i) (f D i j)).hom\n[PROOFSTEP]\nhave eq\u2081 := D.t_fac i i j\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\neq\u2081 : t' D i i j \u226b pullback.snd = pullback.fst \u226b t D i i\n\u22a2 t' D i i j = (pullbackSymmetry (f D i i) (f D i j)).hom\n[PROOFSTEP]\nhave eq\u2082 := (IsIso.eq_comp_inv (D.f i i)).mpr (@pullback.condition _ _ _ _ _ _ (D.f i j) _)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\neq\u2081 : t' D i i j \u226b pullback.snd = pullback.fst \u226b t D i i\neq\u2082 : pullback.fst = (pullback.snd \u226b f D i j) \u226b inv (f D i i)\n\u22a2 t' D i i j = (pullbackSymmetry (f D i i) (f D i j)).hom\n[PROOFSTEP]\nrw [D.t_id, Category.comp_id, eq\u2082] at eq\u2081 \n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\neq\u2081 : t' D i i j \u226b pullback.snd = (pullback.snd \u226b f D i j) \u226b inv (f D i i)\neq\u2082 : pullback.fst = (pullback.snd \u226b f D i j) \u226b inv (f D i i)\n\u22a2 t' D i i j = (pullbackSymmetry (f D i i) (f D i j)).hom\n[PROOFSTEP]\nhave eq\u2083 := (IsIso.eq_comp_inv (D.f i i)).mp eq\u2081\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\neq\u2081 : t' D i i j \u226b pullback.snd = (pullback.snd \u226b f D i j) \u226b inv (f D i i)\neq\u2082 : pullback.fst = (pullback.snd \u226b f D i j) \u226b inv (f D i i)\neq\u2083 : (t' D i i j \u226b pullback.snd) \u226b f D i i = pullback.snd \u226b f D i j\n\u22a2 t' D i i j = (pullbackSymmetry (f D i i) (f D i j)).hom\n[PROOFSTEP]\nrw [Category.assoc, \u2190 pullback.condition, \u2190 Category.assoc] at eq\u2083 \n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\neq\u2081 : t' D i i j \u226b pullback.snd = (pullback.snd \u226b f D i j) \u226b inv (f D i i)\neq\u2082 : pullback.fst = (pullback.snd \u226b f D i j) \u226b inv (f D i i)\neq\u2083 : (t' D i i j \u226b pullback.fst) \u226b f D i j = pullback.snd \u226b f D i j\n\u22a2 t' D i i j = (pullbackSymmetry (f D i i) (f D i j)).hom\n[PROOFSTEP]\nexact Mono.right_cancellation _ _ ((Mono.right_cancellation _ _ eq\u2083).trans (pullbackSymmetry_hom_comp_fst _ _).symm)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\n\u22a2 t' D j i i = pullback.fst \u226b t D j i \u226b inv pullback.snd\n[PROOFSTEP]\nrw [\u2190 Category.assoc, \u2190 D.t_fac]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\n\u22a2 t' D j i i = (t' D j i i \u226b pullback.snd) \u226b inv pullback.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\n\u22a2 t' D i j i = pullback.fst \u226b t D i j \u226b inv pullback.snd\n[PROOFSTEP]\nrw [\u2190 Category.assoc, \u2190 D.t_fac]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\n\u22a2 t' D i j i = (t' D i j i \u226b pullback.snd) \u226b inv pullback.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\n\u22a2 t D i j \u226b t D j i = \ud835\udfd9 (V D (i, j))\n[PROOFSTEP]\nhave eq : (pullbackSymmetry (D.f i i) (D.f i j)).hom = pullback.snd \u226b inv pullback.fst := by simp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\n\u22a2 (pullbackSymmetry (f D i i) (f D i j)).hom = pullback.snd \u226b inv pullback.fst\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\neq : (pullbackSymmetry (f D i i) (f D i j)).hom = pullback.snd \u226b inv pullback.fst\n\u22a2 t D i j \u226b t D j i = \ud835\udfd9 (V D (i, j))\n[PROOFSTEP]\nhave := D.cocycle i j i\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\neq : (pullbackSymmetry (f D i i) (f D i j)).hom = pullback.snd \u226b inv pullback.fst\nthis : t' D i j i \u226b t' D j i i \u226b t' D i i j = \ud835\udfd9 (pullback (f D i j) (f D i i))\n\u22a2 t D i j \u226b t D j i = \ud835\udfd9 (V D (i, j))\n[PROOFSTEP]\nrw [D.t'_iij, D.t'_jii, D.t'_iji, fst_eq_snd_of_mono_eq, eq] at this \n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\neq : (pullbackSymmetry (f D i i) (f D i j)).hom = pullback.snd \u226b inv pullback.fst\nthis :\n (pullback.fst \u226b t D i j \u226b inv pullback.snd) \u226b\n (pullback.snd \u226b t D j i \u226b inv pullback.snd) \u226b pullback.snd \u226b inv pullback.fst =\n \ud835\udfd9 (pullback (f D i j) (f D i i))\n\u22a2 t D i j \u226b t D j i = \ud835\udfd9 (V D (i, j))\n[PROOFSTEP]\nsimp only [Category.assoc, IsIso.inv_hom_id_assoc] at this \n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\neq : (pullbackSymmetry (f D i i) (f D i j)).hom = pullback.snd \u226b inv pullback.fst\nthis : pullback.fst \u226b t D i j \u226b t D j i \u226b inv pullback.fst = \ud835\udfd9 (pullback (f D i j) (f D i i))\n\u22a2 t D i j \u226b t D j i = \ud835\udfd9 (V D (i, j))\n[PROOFSTEP]\nrw [\u2190 IsIso.eq_inv_comp, \u2190 Category.assoc, IsIso.comp_inv_eq] at this \n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j : D.J\neq : (pullbackSymmetry (f D i i) (f D i j)).hom = pullback.snd \u226b inv pullback.fst\nthis : t D i j \u226b t D j i = (inv pullback.fst \u226b \ud835\udfd9 (pullback (f D i j) (f D i i))) \u226b pullback.fst\n\u22a2 t D i j \u226b t D j i = \ud835\udfd9 (V D (i, j))\n[PROOFSTEP]\nsimpa using this\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j k : D.J\n\u22a2 t' D i j k \u226b (pullbackSymmetry (f D j k) (f D j i)).hom \u226b t' D j i k \u226b (pullbackSymmetry (f D i k) (f D i j)).hom =\n \ud835\udfd9 (pullback (f D i j) (f D i k))\n[PROOFSTEP]\nrw [\u2190 cancel_mono (pullback.fst : pullback (D.f i j) (D.f i k) \u27f6 _)]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j k : D.J\n\u22a2 (t' D i j k \u226b (pullbackSymmetry (f D j k) (f D j i)).hom \u226b t' D j i k \u226b (pullbackSymmetry (f D i k) (f D i j)).hom) \u226b\n pullback.fst =\n \ud835\udfd9 (pullback (f D i j) (f D i k)) \u226b pullback.fst\n[PROOFSTEP]\nsimp [t_fac, t_fac_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j k : D.J\n\u22a2 (t' D j k i \u226b t' D k i j) \u226b t' D i j k = \ud835\udfd9 (pullback (f D j k) (f D j i))\n[PROOFSTEP]\nsimpa using D.cocycle _ _ _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j k : D.J\n\u22a2 t' D j k i \u226b t' D k i j =\n (pullbackSymmetry (f D j k) (f D j i)).hom \u226b t' D j i k \u226b (pullbackSymmetry (f D i k) (f D i j)).hom\n[PROOFSTEP]\ntrans inv (D.t' i j k)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j k : D.J\n\u22a2 t' D j k i \u226b t' D k i j = inv (t' D i j k)\n[PROOFSTEP]\nexact IsIso.eq_inv_of_hom_inv_id (D.cocycle _ _ _)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j k : D.J\n\u22a2 inv (t' D i j k) =\n (pullbackSymmetry (f D j k) (f D j i)).hom \u226b t' D j i k \u226b (pullbackSymmetry (f D i k) (f D i j)).hom\n[PROOFSTEP]\nrw [\u2190 cancel_mono (pullback.fst : pullback (D.f i j) (D.f i k) \u27f6 _)]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\ni j k : D.J\n\u22a2 inv (t' D i j k) \u226b pullback.fst =\n ((pullbackSymmetry (f D j k) (f D j i)).hom \u226b t' D j i k \u226b (pullbackSymmetry (f D i k) (f D i j)).hom) \u226b\n pullback.fst\n[PROOFSTEP]\nsimp [t_fac, t_fac_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b9 : Category.{v, u\u2082} C'\nD : GlueData C\ninst\u271d : HasMulticoequalizer (diagram D)\ni j : D.J\n\u22a2 f D i j \u226b \u03b9 D i = (t D i j \u226b f D j i) \u226b \u03b9 D j\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} C'\nD : GlueData C\ninst\u271d\u00b9 : HasMulticoequalizer (diagram D)\ninst\u271d : HasColimits C\n\u22a2 Epi (\u03c0 D)\n[PROOFSTEP]\nunfold \u03c0\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} C'\nD : GlueData C\ninst\u271d\u00b9 : HasMulticoequalizer (diagram D)\ninst\u271d : HasColimits C\n\u22a2 Epi (Multicoequalizer.sigma\u03c0 (diagram D))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD\u271d : GlueData C\nD : GlueData (Type u_1)\nx : glued D\n\u22a2 \u2203 i y, \u03b9 D i y = x\n[PROOFSTEP]\ndelta CategoryTheory.GlueData.\u03b9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD\u271d : GlueData C\nD : GlueData (Type u_1)\nx : glued D\n\u22a2 \u2203 i y, Multicoequalizer.\u03c0 (diagram D) i y = x\n[PROOFSTEP]\nsimp_rw [\u2190 Multicoequalizer.\u03b9_sigma\u03c0 D.diagram]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD\u271d : GlueData C\nD : GlueData (Type u_1)\nx : glued D\n\u22a2 \u2203 i y, (Sigma.\u03b9 (diagram D).right i \u226b Multicoequalizer.sigma\u03c0 (diagram D)) y = x\n[PROOFSTEP]\nrcases D.types_\u03c0_surjective x with\n \u27e8x', rfl\u27e9\n --have := colimit.isoColimitCocone (Types.coproductColimitCocone _)\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD\u271d : GlueData C\nD : GlueData (Type u_1)\nx' : sigmaOpens D\n\u22a2 \u2203 i y, (Sigma.\u03b9 (diagram D).right i \u226b Multicoequalizer.sigma\u03c0 (diagram D)) y = \u03c0 D x'\n[PROOFSTEP]\nrw [\u2190\n show (colimit.isoColimitCocone (Types.coproductColimitCocone _)).inv _ = x' from\n ConcreteCategory.congr_hom (colimit.isoColimitCocone (Types.coproductColimitCocone _)).hom_inv_id x']\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD\u271d : GlueData C\nD : GlueData (Type u_1)\nx' : sigmaOpens D\n\u22a2 \u2203 i y,\n (Sigma.\u03b9 (diagram D).right i \u226b Multicoequalizer.sigma\u03c0 (diagram D)) y =\n \u03c0 D\n ((colimit.isoColimitCocone (Types.coproductColimitCocone D.U)).inv\n ((colimit.isoColimitCocone (Types.coproductColimitCocone D.U)).hom x'))\n[PROOFSTEP]\nrcases(colimit.isoColimitCocone (Types.coproductColimitCocone _)).hom x' with \u27e8i, y\u27e9\n[GOAL]\ncase intro.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD\u271d : GlueData C\nD : GlueData (Type u_1)\nx' : sigmaOpens D\ni : D.J\ny : U D i\n\u22a2 \u2203 i_1 y_1,\n (Sigma.\u03b9 (diagram D).right i_1 \u226b Multicoequalizer.sigma\u03c0 (diagram D)) y_1 =\n \u03c0 D ((colimit.isoColimitCocone (Types.coproductColimitCocone D.U)).inv { fst := i, snd := y })\n[PROOFSTEP]\nexact\n \u27e8i, y, by\n simp [\u2190 Multicoequalizer.\u03b9_sigma\u03c0, -Multicoequalizer.\u03b9_sigma\u03c0]\n rfl\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD\u271d : GlueData C\nD : GlueData (Type u_1)\nx' : sigmaOpens D\ni : D.J\ny : U D i\n\u22a2 (Sigma.\u03b9 (diagram D).right i \u226b Multicoequalizer.sigma\u03c0 (diagram D)) y =\n \u03c0 D ((colimit.isoColimitCocone (Types.coproductColimitCocone D.U)).inv { fst := i, snd := y })\n[PROOFSTEP]\nsimp [\u2190 Multicoequalizer.\u03b9_sigma\u03c0, -Multicoequalizer.\u03b9_sigma\u03c0]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD\u271d : GlueData C\nD : GlueData (Type u_1)\nx' : sigmaOpens D\ni : D.J\ny : U D i\n\u22a2 Multicoequalizer.sigma\u03c0 (diagram D) (Sigma.\u03b9 D.U i y) =\n \u03c0 D ((colimit.isoColimitCocone (Types.coproductColimitCocone D.U)).inv { fst := i, snd := y })\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ni : D.J\n\u22a2 (fun i j => F.map (t D i j)) i i = \ud835\udfd9 ((fun i => F.obj (V D i)) (i, i))\n[PROOFSTEP]\nsimp [D.t_id i]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ni j k : D.J\n\u22a2 (fun i j k =>\n (PreservesPullback.iso F (f D i j) (f D i k)).inv \u226b\n F.map (t' D i j k) \u226b (PreservesPullback.iso F (f D j k) (f D j i)).hom)\n i j k \u226b\n pullback.snd =\n pullback.fst \u226b (fun i j => F.map (t D i j)) i j\n[PROOFSTEP]\nsimpa [Iso.inv_comp_eq] using congr_arg (fun f => F.map f) (D.t_fac i j k)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ni j k : D.J\n\u22a2 (fun i j k =>\n (PreservesPullback.iso F (f D i j) (f D i k)).inv \u226b\n F.map (t' D i j k) \u226b (PreservesPullback.iso F (f D j k) (f D j i)).hom)\n i j k \u226b\n (fun i j k =>\n (PreservesPullback.iso F (f D i j) (f D i k)).inv \u226b\n F.map (t' D i j k) \u226b (PreservesPullback.iso F (f D j k) (f D j i)).hom)\n j k i \u226b\n (fun i j k =>\n (PreservesPullback.iso F (f D i j) (f D i k)).inv \u226b\n F.map (t' D i j k) \u226b (PreservesPullback.iso F (f D j k) (f D j i)).hom)\n k i j =\n \ud835\udfd9 (pullback ((fun i j => F.map (f D i j)) i j) ((fun i j => F.map (f D i j)) i k))\n[PROOFSTEP]\nsimp only [Category.assoc, Iso.hom_inv_id_assoc, \u2190 Functor.map_comp_assoc, D.cocycle, Iso.inv_hom_id,\n CategoryTheory.Functor.map_id, Category.id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\n\u22a2 \u2200 {X Y : WalkingMultispan (diagram D).fstFrom (diagram D).sndFrom} (f : X \u27f6 Y),\n (MultispanIndex.multispan (diagram D) \u22d9 F).map f \u226b\n ((fun x =>\n match x with\n | WalkingMultispan.left a =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.left a))\n | WalkingMultispan.right b =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.right b)))\n Y).hom =\n ((fun x =>\n match x with\n | WalkingMultispan.left a =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.left a))\n | WalkingMultispan.right b =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.right b)))\n X).hom \u226b\n (MultispanIndex.multispan (diagram (mapGlueData D F))).map f\n[PROOFSTEP]\nrintro (\u27e8_, _\u27e9 | _) _ (_ | _ | _)\n[GOAL]\ncase left.mk.id\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\nfst\u271d snd\u271d : D.J\n\u22a2 (MultispanIndex.multispan (diagram D) \u22d9 F).map (WalkingMultispan.Hom.id (WalkingMultispan.left (fst\u271d, snd\u271d))) \u226b\n ((fun x =>\n match x with\n | WalkingMultispan.left a =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.left a))\n | WalkingMultispan.right b =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.right b)))\n (WalkingMultispan.left (fst\u271d, snd\u271d))).hom =\n ((fun x =>\n match x with\n | WalkingMultispan.left a =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.left a))\n | WalkingMultispan.right b =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.right b)))\n (WalkingMultispan.left (fst\u271d, snd\u271d))).hom \u226b\n (MultispanIndex.multispan (diagram (mapGlueData D F))).map\n (WalkingMultispan.Hom.id (WalkingMultispan.left (fst\u271d, snd\u271d)))\n[PROOFSTEP]\nerw [Category.comp_id, Category.id_comp, Functor.map_id]\n[GOAL]\ncase left.mk.id\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\nfst\u271d snd\u271d : D.J\n\u22a2 \ud835\udfd9 ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.left (fst\u271d, snd\u271d))) =\n (MultispanIndex.multispan (diagram (mapGlueData D F))).map\n (WalkingMultispan.Hom.id (WalkingMultispan.left (fst\u271d, snd\u271d)))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase left.mk.fst\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\nfst\u271d snd\u271d : D.J\n\u22a2 (MultispanIndex.multispan (diagram D) \u22d9 F).map (WalkingMultispan.Hom.fst (fst\u271d, snd\u271d)) \u226b\n ((fun x =>\n match x with\n | WalkingMultispan.left a =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.left a))\n | WalkingMultispan.right b =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.right b)))\n (WalkingMultispan.right (MultispanIndex.fstFrom (diagram D) (fst\u271d, snd\u271d)))).hom =\n ((fun x =>\n match x with\n | WalkingMultispan.left a =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.left a))\n | WalkingMultispan.right b =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.right b)))\n (WalkingMultispan.left (fst\u271d, snd\u271d))).hom \u226b\n (MultispanIndex.multispan (diagram (mapGlueData D F))).map (WalkingMultispan.Hom.fst (fst\u271d, snd\u271d))\n[PROOFSTEP]\nerw [Category.comp_id, Category.id_comp]\n[GOAL]\ncase left.mk.fst\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\nfst\u271d snd\u271d : D.J\n\u22a2 (MultispanIndex.multispan (diagram D) \u22d9 F).map (WalkingMultispan.Hom.fst (fst\u271d, snd\u271d)) =\n (MultispanIndex.multispan (diagram (mapGlueData D F))).map (WalkingMultispan.Hom.fst (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase left.mk.snd\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\nfst\u271d snd\u271d : D.J\n\u22a2 (MultispanIndex.multispan (diagram D) \u22d9 F).map (WalkingMultispan.Hom.snd (fst\u271d, snd\u271d)) \u226b\n ((fun x =>\n match x with\n | WalkingMultispan.left a =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.left a))\n | WalkingMultispan.right b =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.right b)))\n (WalkingMultispan.right (MultispanIndex.sndFrom (diagram D) (fst\u271d, snd\u271d)))).hom =\n ((fun x =>\n match x with\n | WalkingMultispan.left a =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.left a))\n | WalkingMultispan.right b =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.right b)))\n (WalkingMultispan.left (fst\u271d, snd\u271d))).hom \u226b\n (MultispanIndex.multispan (diagram (mapGlueData D F))).map (WalkingMultispan.Hom.snd (fst\u271d, snd\u271d))\n[PROOFSTEP]\nerw [Category.comp_id, Category.id_comp, Functor.map_comp]\n[GOAL]\ncase left.mk.snd\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\nfst\u271d snd\u271d : D.J\n\u22a2 F.map (t D fst\u271d snd\u271d) \u226b F.map (f D snd\u271d fst\u271d) =\n (MultispanIndex.multispan (diagram (mapGlueData D F))).map (WalkingMultispan.Hom.snd (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right.id\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\na\u271d : (diagram D).R\n\u22a2 (MultispanIndex.multispan (diagram D) \u22d9 F).map (WalkingMultispan.Hom.id (WalkingMultispan.right a\u271d)) \u226b\n ((fun x =>\n match x with\n | WalkingMultispan.left a =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.left a))\n | WalkingMultispan.right b =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.right b)))\n (WalkingMultispan.right a\u271d)).hom =\n ((fun x =>\n match x with\n | WalkingMultispan.left a =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.left a))\n | WalkingMultispan.right b =>\n Iso.refl ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.right b)))\n (WalkingMultispan.right a\u271d)).hom \u226b\n (MultispanIndex.multispan (diagram (mapGlueData D F))).map (WalkingMultispan.Hom.id (WalkingMultispan.right a\u271d))\n[PROOFSTEP]\nerw [Category.comp_id, Category.id_comp, Functor.map_id]\n[GOAL]\ncase right.id\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\na\u271d : (diagram D).R\n\u22a2 \ud835\udfd9 ((MultispanIndex.multispan (diagram D) \u22d9 F).obj (WalkingMultispan.right a\u271d)) =\n (MultispanIndex.multispan (diagram (mapGlueData D F))).map (WalkingMultispan.Hom.id (WalkingMultispan.right a\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b9 : HasMulticoequalizer (diagram D)\ninst\u271d : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni : D.J\n\u22a2 F.map (\u03b9 D i) \u226b (gluedIso D F).hom = \u03b9 (mapGlueData D F) i\n[PROOFSTEP]\nhaveI : HasColimit (MultispanIndex.multispan (diagram (mapGlueData D F))) := inferInstance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b9 : HasMulticoequalizer (diagram D)\ninst\u271d : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni : D.J\nthis : HasColimit (MultispanIndex.multispan (diagram (mapGlueData D F)))\n\u22a2 F.map (\u03b9 D i) \u226b (gluedIso D F).hom = \u03b9 (mapGlueData D F) i\n[PROOFSTEP]\nerw [\u03b9_preservesColimitsIso_hom_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b9 : HasMulticoequalizer (diagram D)\ninst\u271d : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni : D.J\nthis : HasColimit (MultispanIndex.multispan (diagram (mapGlueData D F)))\n\u22a2 colimit.\u03b9 (MultispanIndex.multispan (diagram D) \u22d9 F) (WalkingMultispan.right i) \u226b\n (HasColimit.isoOfNatIso (diagramIso D F)).hom =\n \u03b9 (mapGlueData D F) i\n[PROOFSTEP]\nrw [HasColimit.isoOfNatIso_\u03b9_hom]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b9 : HasMulticoequalizer (diagram D)\ninst\u271d : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni : D.J\nthis : HasColimit (MultispanIndex.multispan (diagram (mapGlueData D F)))\n\u22a2 NatTrans.app (diagramIso D F).hom (WalkingMultispan.right i) \u226b\n colimit.\u03b9 (MultispanIndex.multispan (diagram (mapGlueData D F))) (WalkingMultispan.right i) =\n \u03b9 (mapGlueData D F) i\n[PROOFSTEP]\nerw [Category.id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b9 : HasMulticoequalizer (diagram D)\ninst\u271d : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni : D.J\nthis : HasColimit (MultispanIndex.multispan (diagram (mapGlueData D F)))\n\u22a2 colimit.\u03b9 (MultispanIndex.multispan (diagram (mapGlueData D F))) (WalkingMultispan.right i) = \u03b9 (mapGlueData D F) i\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b9 : HasMulticoequalizer (diagram D)\ninst\u271d : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni : D.J\n\u22a2 \u03b9 (mapGlueData D F) i \u226b (gluedIso D F).inv = F.map (\u03b9 D i)\n[PROOFSTEP]\nrw [Iso.comp_inv_eq, \u03b9_gluedIso_hom]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\n\u22a2 IsLimit (vPullbackCone D i j)\n[PROOFSTEP]\napply isLimitOfReflects F\n[GOAL]\ncase t\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\n\u22a2 IsLimit (F.mapCone (vPullbackCone D i j))\n[PROOFSTEP]\napply (isLimitMapConePullbackConeEquiv _ _).symm _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\n\u22a2 IsLimit\n (PullbackCone.mk (F.map (f D i j)) (F.map (t D i j \u226b f D j i))\n (_ : F.map (f D i j) \u226b F.map (\u03b9 D i) = F.map (t D i j \u226b f D j i) \u226b F.map (\u03b9 D j)))\n[PROOFSTEP]\nlet e : cospan (F.map (D.\u03b9 i)) (F.map (D.\u03b9 j)) \u2245 cospan ((D.mapGlueData F).\u03b9 i) ((D.mapGlueData F).\u03b9 j)\n[GOAL]\ncase e\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\n\u22a2 cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) := ?e\n\u22a2 IsLimit\n (PullbackCone.mk (F.map (f D i j)) (F.map (t D i j \u226b f D j i))\n (_ : F.map (f D i j) \u226b F.map (\u03b9 D i) = F.map (t D i j \u226b f D j i) \u226b F.map (\u03b9 D j)))\n[PROOFSTEP]\nexact\n NatIso.ofComponents\n (fun x => by\n cases x\n exacts [D.gluedIso F, Iso.refl _])\n (by rintro (_ | _) (_ | _) (_ | _ | _) <;> simp)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\nx : WalkingCospan\n\u22a2 (cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245 (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase none\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\n\u22a2 (cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj none \u2245 (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj none\ncase some\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\nval\u271d : WalkingPair\n\u22a2 (cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val\u271d) \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj (some val\u271d)\n[PROOFSTEP]\nexacts [D.gluedIso F, Iso.refl _]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\n\u22a2 \u2200 {X Y : WalkingCospan} (f : X \u27f6 Y),\n (cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).map f \u226b\n ((fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val)))\n (_ : x = x))\n Y).hom =\n ((fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val)))\n (_ : x = x))\n X).hom \u226b\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).map f\n[PROOFSTEP]\nrintro (_ | _) (_ | _) (_ | _ | _)\n[GOAL]\ncase none.none.id\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\n\u22a2 (cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).map (WidePullbackShape.Hom.id none) \u226b\n ((fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val)))\n (_ : x = x))\n none).hom =\n ((fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val)))\n (_ : x = x))\n none).hom \u226b\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).map (WidePullbackShape.Hom.id none)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.none.term.left\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\n\u22a2 (cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).map (WidePullbackShape.Hom.term WalkingPair.left) \u226b\n ((fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val)))\n (_ : x = x))\n none).hom =\n ((fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val)))\n (_ : x = x))\n (some WalkingPair.left)).hom \u226b\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).map (WidePullbackShape.Hom.term WalkingPair.left)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.none.term.right\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\n\u22a2 (cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).map (WidePullbackShape.Hom.term WalkingPair.right) \u226b\n ((fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val)))\n (_ : x = x))\n none).hom =\n ((fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val)))\n (_ : x = x))\n (some WalkingPair.right)).hom \u226b\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).map (WidePullbackShape.Hom.term WalkingPair.right)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.some.id\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\nval\u271d : WalkingPair\n\u22a2 (cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).map (WidePullbackShape.Hom.id (some val\u271d)) \u226b\n ((fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val)))\n (_ : x = x))\n (some val\u271d)).hom =\n ((fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val)))\n (_ : x = x))\n (some val\u271d)).hom \u226b\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).map (WidePullbackShape.Hom.id (some val\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 IsLimit\n (PullbackCone.mk (F.map (f D i j)) (F.map (t D i j \u226b f D j i))\n (_ : F.map (f D i j) \u226b F.map (\u03b9 D i) = F.map (t D i j \u226b f D j i) \u226b F.map (\u03b9 D j)))\n[PROOFSTEP]\napply IsLimit.postcomposeHomEquiv e _ _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 IsLimit\n ((Cones.postcompose e.hom).obj\n (PullbackCone.mk (F.map (f D i j)) (F.map (t D i j \u226b f D j i))\n (_ : F.map (f D i j) \u226b F.map (\u03b9 D i) = F.map (t D i j \u226b f D j i) \u226b F.map (\u03b9 D j))))\n[PROOFSTEP]\napply hc.ofIsoLimit\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 vPullbackCone (mapGlueData D F) i j \u2245\n (Cones.postcompose e.hom).obj\n (PullbackCone.mk (F.map (f D i j)) (F.map (t D i j \u226b f D j i))\n (_ : F.map (f D i j) \u226b F.map (\u03b9 D i) = F.map (t D i j \u226b f D j i) \u226b F.map (\u03b9 D j)))\n[PROOFSTEP]\nrefine' Cones.ext (Iso.refl _) _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 \u2200 (j_1 : WalkingCospan),\n NatTrans.app (vPullbackCone (mapGlueData D F) i j).\u03c0 j_1 =\n (Iso.refl (vPullbackCone (mapGlueData D F) i j).pt).hom \u226b\n NatTrans.app\n ((Cones.postcompose e.hom).obj\n (PullbackCone.mk (F.map (f D i j)) (F.map (t D i j \u226b f D j i))\n (_ : F.map (f D i j) \u226b F.map (\u03b9 D i) = F.map (t D i j \u226b f D j i) \u226b F.map (\u03b9 D j)))).\u03c0\n j_1\n[PROOFSTEP]\nrintro (_ | _ | _)\n[GOAL]\ncase none\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 NatTrans.app (vPullbackCone (mapGlueData D F) i j).\u03c0 none =\n (Iso.refl (vPullbackCone (mapGlueData D F) i j).pt).hom \u226b\n NatTrans.app\n ((Cones.postcompose e.hom).obj\n (PullbackCone.mk (F.map (f D i j)) (F.map (t D i j \u226b f D j i))\n (_ : F.map (f D i j) \u226b F.map (\u03b9 D i) = F.map (t D i j \u226b f D j i) \u226b F.map (\u03b9 D j)))).\u03c0\n none\ncase some.left\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 NatTrans.app (vPullbackCone (mapGlueData D F) i j).\u03c0 (some WalkingPair.left) =\n (Iso.refl (vPullbackCone (mapGlueData D F) i j).pt).hom \u226b\n NatTrans.app\n ((Cones.postcompose e.hom).obj\n (PullbackCone.mk (F.map (f D i j)) (F.map (t D i j \u226b f D j i))\n (_ : F.map (f D i j) \u226b F.map (\u03b9 D i) = F.map (t D i j \u226b f D j i) \u226b F.map (\u03b9 D j)))).\u03c0\n (some WalkingPair.left)\ncase some.right\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 NatTrans.app (vPullbackCone (mapGlueData D F) i j).\u03c0 (some WalkingPair.right) =\n (Iso.refl (vPullbackCone (mapGlueData D F) i j).pt).hom \u226b\n NatTrans.app\n ((Cones.postcompose e.hom).obj\n (PullbackCone.mk (F.map (f D i j)) (F.map (t D i j \u226b f D j i))\n (_ : F.map (f D i j) \u226b F.map (\u03b9 D i) = F.map (t D i j \u226b f D j i) \u226b F.map (\u03b9 D j)))).\u03c0\n (some WalkingPair.right)\n[PROOFSTEP]\nchange _ = _ \u226b (_ \u226b _) \u226b _\n[GOAL]\ncase none\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 NatTrans.app (vPullbackCone (mapGlueData D F) i j).\u03c0 none =\n (Iso.refl (vPullbackCone (mapGlueData D F) i j).pt).hom \u226b\n (F.map (f D i j) \u226b F.map (\u03b9 D i)) \u226b NatTrans.app e.hom none\ncase some.left\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 NatTrans.app (vPullbackCone (mapGlueData D F) i j).\u03c0 (some WalkingPair.left) =\n (Iso.refl (vPullbackCone (mapGlueData D F) i j).pt).hom \u226b\n NatTrans.app\n ((Cones.postcompose e.hom).obj\n (PullbackCone.mk (F.map (f D i j)) (F.map (t D i j \u226b f D j i))\n (_ : F.map (f D i j) \u226b F.map (\u03b9 D i) = F.map (t D i j \u226b f D j i) \u226b F.map (\u03b9 D j)))).\u03c0\n (some WalkingPair.left)\ncase some.right\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 NatTrans.app (vPullbackCone (mapGlueData D F) i j).\u03c0 (some WalkingPair.right) =\n (Iso.refl (vPullbackCone (mapGlueData D F) i j).pt).hom \u226b\n NatTrans.app\n ((Cones.postcompose e.hom).obj\n (PullbackCone.mk (F.map (f D i j)) (F.map (t D i j \u226b f D j i))\n (_ : F.map (f D i j) \u226b F.map (\u03b9 D i) = F.map (t D i j \u226b f D j i) \u226b F.map (\u03b9 D j)))).\u03c0\n (some WalkingPair.right)\n[PROOFSTEP]\nall_goals change _ = \ud835\udfd9 _ \u226b _ \u226b _; aesop_cat\n[GOAL]\ncase none\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 NatTrans.app (vPullbackCone (mapGlueData D F) i j).\u03c0 none =\n (Iso.refl (vPullbackCone (mapGlueData D F) i j).pt).hom \u226b\n (F.map (f D i j) \u226b F.map (\u03b9 D i)) \u226b NatTrans.app e.hom none\n[PROOFSTEP]\nchange _ = \ud835\udfd9 _ \u226b _ \u226b _\n[GOAL]\ncase none\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 NatTrans.app (vPullbackCone (mapGlueData D F) i j).\u03c0 none =\n \ud835\udfd9 (((Functor.const WalkingCospan).obj (vPullbackCone (mapGlueData D F) i j).pt).obj none) \u226b\n (F.map (f D i j) \u226b F.map (\u03b9 D i)) \u226b NatTrans.app e.hom none\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase some.left\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 NatTrans.app (vPullbackCone (mapGlueData D F) i j).\u03c0 (some WalkingPair.left) =\n (Iso.refl (vPullbackCone (mapGlueData D F) i j).pt).hom \u226b\n NatTrans.app\n ((Cones.postcompose e.hom).obj\n (PullbackCone.mk (F.map (f D i j)) (F.map (t D i j \u226b f D j i))\n (_ : F.map (f D i j) \u226b F.map (\u03b9 D i) = F.map (t D i j \u226b f D j i) \u226b F.map (\u03b9 D j)))).\u03c0\n (some WalkingPair.left)\n[PROOFSTEP]\nchange _ = \ud835\udfd9 _ \u226b _ \u226b _\n[GOAL]\ncase some.left\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 NatTrans.app (vPullbackCone (mapGlueData D F) i j).\u03c0 (some WalkingPair.left) =\n \ud835\udfd9 (((Functor.const WalkingCospan).obj (vPullbackCone (mapGlueData D F) i j).pt).obj (some WalkingPair.left)) \u226b\n NatTrans.app\n (PullbackCone.mk (F.map (f D i j)) (F.map (t D i j \u226b f D j i))\n (_ : F.map (f D i j) \u226b F.map (\u03b9 D i) = F.map (t D i j \u226b f D j i) \u226b F.map (\u03b9 D j))).\u03c0\n (some WalkingPair.left) \u226b\n NatTrans.app e.hom (some WalkingPair.left)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase some.right\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 NatTrans.app (vPullbackCone (mapGlueData D F) i j).\u03c0 (some WalkingPair.right) =\n (Iso.refl (vPullbackCone (mapGlueData D F) i j).pt).hom \u226b\n NatTrans.app\n ((Cones.postcompose e.hom).obj\n (PullbackCone.mk (F.map (f D i j)) (F.map (t D i j \u226b f D j i))\n (_ : F.map (f D i j) \u226b F.map (\u03b9 D i) = F.map (t D i j \u226b f D j i) \u226b F.map (\u03b9 D j)))).\u03c0\n (some WalkingPair.right)\n[PROOFSTEP]\nchange _ = \ud835\udfd9 _ \u226b _ \u226b _\n[GOAL]\ncase some.right\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} C'\nD : GlueData C\nF : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\ninst\u271d\u00b2 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni j : D.J\ninst\u271d : ReflectsLimit (cospan (\u03b9 D i) (\u03b9 D j)) F\nhc : IsLimit (vPullbackCone (mapGlueData D F) i j)\ne : cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j)) \u2245 cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j) :=\n NatIso.ofComponents fun x =>\n Option.casesOn (motive := fun t =>\n x = t \u2192\n ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj x \u2245\n (cospan (\u03b9 (mapGlueData D F) i) (\u03b9 (mapGlueData D F) j)).obj x))\n x (fun h => (_ : none = x) \u25b8 gluedIso D F)\n (fun val h => (_ : some val = x) \u25b8 Iso.refl ((cospan (F.map (\u03b9 D i)) (F.map (\u03b9 D j))).obj (some val))) (_ : x = x)\n\u22a2 NatTrans.app (vPullbackCone (mapGlueData D F) i j).\u03c0 (some WalkingPair.right) =\n \ud835\udfd9 (((Functor.const WalkingCospan).obj (vPullbackCone (mapGlueData D F) i j).pt).obj (some WalkingPair.right)) \u226b\n NatTrans.app\n (PullbackCone.mk (F.map (f D i j)) (F.map (t D i j \u226b f D j i))\n (_ : F.map (f D i j) \u226b F.map (\u03b9 D i) = F.map (t D i j \u226b f D j i) \u226b F.map (\u03b9 D j))).\u03c0\n (some WalkingPair.right) \u226b\n NatTrans.app e.hom (some WalkingPair.right)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u2074 : Category.{v, u\u2082} C'\nD : GlueData C\nF\u271d : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\u271d\ninst\u271d\u00b3 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b2 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\u271d\nF : C \u2964 Type v\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ninst\u271d : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\nx : F.obj (glued D)\n\u22a2 \u2203 i y, F.map (\u03b9 D i) y = x\n[PROOFSTEP]\nlet e := D.gluedIso F\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u2074 : Category.{v, u\u2082} C'\nD : GlueData C\nF\u271d : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\u271d\ninst\u271d\u00b3 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b2 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\u271d\nF : C \u2964 Type v\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ninst\u271d : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\nx : F.obj (glued D)\ne : F.obj (glued D) \u2245 glued (mapGlueData D F) := gluedIso D F\n\u22a2 \u2203 i y, F.map (\u03b9 D i) y = x\n[PROOFSTEP]\nobtain \u27e8i, y, eq\u27e9 := (D.mapGlueData F).types_\u03b9_jointly_surjective (e.hom x)\n[GOAL]\ncase intro.intro\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u2074 : Category.{v, u\u2082} C'\nD : GlueData C\nF\u271d : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\u271d\ninst\u271d\u00b3 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b2 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\u271d\nF : C \u2964 Type v\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ninst\u271d : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\nx : F.obj (glued D)\ne : F.obj (glued D) \u2245 glued (mapGlueData D F) := gluedIso D F\ni : (mapGlueData D F).J\ny : U (mapGlueData D F) i\neq : \u03b9 (mapGlueData D F) i y = e.hom x\n\u22a2 \u2203 i y, F.map (\u03b9 D i) y = x\n[PROOFSTEP]\nreplace eq := congr_arg e.inv eq\n[GOAL]\ncase intro.intro\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u2074 : Category.{v, u\u2082} C'\nD : GlueData C\nF\u271d : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\u271d\ninst\u271d\u00b3 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b2 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\u271d\nF : C \u2964 Type v\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ninst\u271d : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\nx : F.obj (glued D)\ne : F.obj (glued D) \u2245 glued (mapGlueData D F) := gluedIso D F\ni : (mapGlueData D F).J\ny : U (mapGlueData D F) i\neq : e.inv (\u03b9 (mapGlueData D F) i y) = e.inv (e.hom x)\n\u22a2 \u2203 i y, F.map (\u03b9 D i) y = x\n[PROOFSTEP]\nchange ((D.mapGlueData F).\u03b9 i \u226b e.inv) y = (e.hom \u226b e.inv) x at eq \n[GOAL]\ncase intro.intro\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u2074 : Category.{v, u\u2082} C'\nD : GlueData C\nF\u271d : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\u271d\ninst\u271d\u00b3 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b2 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\u271d\nF : C \u2964 Type v\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ninst\u271d : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\nx : F.obj (glued D)\ne : F.obj (glued D) \u2245 glued (mapGlueData D F) := gluedIso D F\ni : (mapGlueData D F).J\ny : U (mapGlueData D F) i\neq : (\u03b9 (mapGlueData D F) i \u226b e.inv) y = (e.hom \u226b e.inv) x\n\u22a2 \u2203 i y, F.map (\u03b9 D i) y = x\n[PROOFSTEP]\nrw [e.hom_inv_id, D.\u03b9_gluedIso_inv] at eq \n[GOAL]\ncase intro.intro\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v, u\u2081} C\nC' : Type u\u2082\ninst\u271d\u2074 : Category.{v, u\u2082} C'\nD : GlueData C\nF\u271d : C \u2964 C'\nH : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\u271d\ninst\u271d\u00b3 : HasMulticoequalizer (diagram D)\ninst\u271d\u00b2 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\u271d\nF : C \u2964 Type v\ninst\u271d\u00b9 : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ninst\u271d : (i j k : D.J) \u2192 PreservesLimit (cospan (f D i j) (f D i k)) F\nx : F.obj (glued D)\ne : F.obj (glued D) \u2245 glued (mapGlueData D F) := gluedIso D F\ni : (mapGlueData D F).J\ny : U (mapGlueData D F) i\neq : F.map (\u03b9 D i) y = \ud835\udfd9 (F.obj (glued D)) x\n\u22a2 \u2203 i y, F.map (\u03b9 D i) y = x\n[PROOFSTEP]\nexact \u27e8i, y, eq\u27e9\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.GlueData", "llama_tokens": 28782, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.43739733428408634}} {"text": "[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI : Ideal R\n\u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I\n[PROOFSTEP]\nrefine'\n IsNoetherian.induction (P := fun I => \u2203 Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) \u2264 I)\n (fun (M : Ideal R) hgt => _) I\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\n\u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M\n[PROOFSTEP]\nby_cases h_prM : M.IsPrime\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : Ideal.IsPrime M\n\u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M\n[PROOFSTEP]\nuse{\u27e8M, h_prM\u27e9}\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : Ideal.IsPrime M\n\u22a2 Multiset.prod (Multiset.map asIdeal {{ asIdeal := M, IsPrime := h_prM }}) \u2264 M\n[PROOFSTEP]\nrw [Multiset.map_singleton, Multiset.prod_singleton]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\n\u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M\n[PROOFSTEP]\nby_cases htop : M = \u22a4\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : M = \u22a4\n\u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M\n[PROOFSTEP]\nrw [htop]\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : M = \u22a4\n\u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) \u22a4\n[PROOFSTEP]\nexact \u27e80, le_top\u27e9\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\n\u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M\n[PROOFSTEP]\nhave lt_add : \u2200 (z) (_ : z \u2209 M), M < M + span R { z } :=\n by\n intro z hz\n refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _\n rw [m_eq]\n exact Ideal.mem_sup_right (mem_span_singleton_self z)\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\n\u22a2 \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z}\n[PROOFSTEP]\nintro z hz\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\nz : R\nhz : \u00acz \u2208 M\n\u22a2 M < M + span R {z}\n[PROOFSTEP]\nrefine' lt_of_le_of_ne le_sup_left fun m_eq => hz _\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\nz : R\nhz : \u00acz \u2208 M\nm_eq : M = M + span R {z}\n\u22a2 z \u2208 M\n[PROOFSTEP]\nrw [m_eq]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\nz : R\nhz : \u00acz \u2208 M\nm_eq : M = M + span R {z}\n\u22a2 z \u2208 M + span R {z}\n[PROOFSTEP]\nexact Ideal.mem_sup_right (mem_span_singleton_self z)\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\nlt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z}\n\u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M\n[PROOFSTEP]\nobtain \u27e8x, hx, y, hy, hxy\u27e9 := (Ideal.not_isPrime_iff.mp h_prM).resolve_left htop\n[GOAL]\ncase neg.intro.intro.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\nlt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z}\nx : R\nhx : \u00acx \u2208 M\ny : R\nhy : \u00acy \u2208 M\nhxy : x * y \u2208 M\n\u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M\n[PROOFSTEP]\nobtain \u27e8Wx, h_Wx\u27e9 := hgt (M + span R { x }) (lt_add _ hx)\n[GOAL]\ncase neg.intro.intro.intro.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\nlt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z}\nx : R\nhx : \u00acx \u2208 M\ny : R\nhy : \u00acy \u2208 M\nhxy : x * y \u2208 M\nWx : Multiset (PrimeSpectrum R)\nh_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x}\n\u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M\n[PROOFSTEP]\nobtain \u27e8Wy, h_Wy\u27e9 := hgt (M + span R { y }) (lt_add _ hy)\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\nlt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z}\nx : R\nhx : \u00acx \u2208 M\ny : R\nhy : \u00acy \u2208 M\nhxy : x * y \u2208 M\nWx : Multiset (PrimeSpectrum R)\nh_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x}\nWy : Multiset (PrimeSpectrum R)\nh_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y}\n\u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M\n[PROOFSTEP]\nuse Wx + Wy\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\nlt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z}\nx : R\nhx : \u00acx \u2208 M\ny : R\nhy : \u00acy \u2208 M\nhxy : x * y \u2208 M\nWx : Multiset (PrimeSpectrum R)\nh_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x}\nWy : Multiset (PrimeSpectrum R)\nh_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y}\n\u22a2 Multiset.prod (Multiset.map asIdeal (Wx + Wy)) \u2264 M\n[PROOFSTEP]\nrw [Multiset.map_add, Multiset.prod_add]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\nlt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z}\nx : R\nhx : \u00acx \u2208 M\ny : R\nhy : \u00acy \u2208 M\nhxy : x * y \u2208 M\nWx : Multiset (PrimeSpectrum R)\nh_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x}\nWy : Multiset (PrimeSpectrum R)\nh_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y}\n\u22a2 Multiset.prod (Multiset.map asIdeal Wx) * Multiset.prod (Multiset.map asIdeal Wy) \u2264 M\n[PROOFSTEP]\napply le_trans (Submodule.mul_le_mul h_Wx h_Wy)\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\nlt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z}\nx : R\nhx : \u00acx \u2208 M\ny : R\nhy : \u00acy \u2208 M\nhxy : x * y \u2208 M\nWx : Multiset (PrimeSpectrum R)\nh_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x}\nWy : Multiset (PrimeSpectrum R)\nh_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y}\n\u22a2 (M + span R {x}) * (M + span R {y}) \u2264 M\n[PROOFSTEP]\nrw [add_mul]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\nlt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z}\nx : R\nhx : \u00acx \u2208 M\ny : R\nhy : \u00acy \u2208 M\nhxy : x * y \u2208 M\nWx : Multiset (PrimeSpectrum R)\nh_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x}\nWy : Multiset (PrimeSpectrum R)\nh_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y}\n\u22a2 M * (M + span R {y}) + span R {x} * (M + span R {y}) \u2264 M\n[PROOFSTEP]\napply sup_le (show M * (M + span R { y }) \u2264 M from Ideal.mul_le_right)\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\nlt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z}\nx : R\nhx : \u00acx \u2208 M\ny : R\nhy : \u00acy \u2208 M\nhxy : x * y \u2208 M\nWx : Multiset (PrimeSpectrum R)\nh_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x}\nWy : Multiset (PrimeSpectrum R)\nh_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y}\n\u22a2 span R {x} * (M + span R {y}) \u2264 M\n[PROOFSTEP]\nrw [mul_add]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\nlt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z}\nx : R\nhx : \u00acx \u2208 M\ny : R\nhy : \u00acy \u2208 M\nhxy : x * y \u2208 M\nWx : Multiset (PrimeSpectrum R)\nh_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x}\nWy : Multiset (PrimeSpectrum R)\nh_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y}\n\u22a2 span R {x} * M + span R {x} * span R {y} \u2264 M\n[PROOFSTEP]\napply sup_le (show span R { x } * M \u2264 M from Ideal.mul_le_left)\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nI M : Ideal R\nhgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J\nh_prM : \u00acIdeal.IsPrime M\nhtop : \u00acM = \u22a4\nlt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z}\nx : R\nhx : \u00acx \u2208 M\ny : R\nhy : \u00acy \u2208 M\nhxy : x * y \u2208 M\nWx : Multiset (PrimeSpectrum R)\nh_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x}\nWy : Multiset (PrimeSpectrum R)\nh_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y}\n\u22a2 span R {x} * span R {y} \u2264 M\n[PROOFSTEP]\nrwa [span_mul_span, Set.singleton_mul_singleton, span_singleton_le_iff_mem]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI : Ideal A\nh_nzI : I \u2260 \u22a5\n\u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5\n[PROOFSTEP]\nrevert h_nzI\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI : Ideal A\n\u22a2 I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5\n[PROOFSTEP]\nrefine'\n IsNoetherian.induction (P := fun I =>\n I \u2260 \u22a5 \u2192 \u2203 Z : Multiset (PrimeSpectrum A), Multiset.prod (Z.map asIdeal) \u2264 I \u2227 Multiset.prod (Z.map asIdeal) \u2260 \u22a5)\n (fun (M : Ideal A) hgt => _) I\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\n\u22a2 (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) M\n[PROOFSTEP]\nintro h_nzM\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\n\u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 M \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5\n[PROOFSTEP]\nhave hA_nont : Nontrivial A\n[GOAL]\ncase hA_nont\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\n\u22a2 Nontrivial A\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\n\u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 M \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5\n[PROOFSTEP]\napply IsDomain.toNontrivial\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\n\u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 M \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5\n[PROOFSTEP]\nby_cases h_topM : M = \u22a4\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : M = \u22a4\n\u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 M \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5\n[PROOFSTEP]\nrcases h_topM with rfl\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI : Ideal A\nhA_nont : Nontrivial A\nhgt :\n \u2200 (J : Submodule A A),\n J > \u22a4 \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : \u22a4 \u2260 \u22a5\n\u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 \u22a4 \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5\n[PROOFSTEP]\nobtain \u27e8p_id, h_nzp, h_pp\u27e9 : \u2203 p : Ideal A, p \u2260 \u22a5 \u2227 p.IsPrime := by apply Ring.not_isField_iff_exists_prime.mp h_fA\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI : Ideal A\nhA_nont : Nontrivial A\nhgt :\n \u2200 (J : Submodule A A),\n J > \u22a4 \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : \u22a4 \u2260 \u22a5\n\u22a2 \u2203 p, p \u2260 \u22a5 \u2227 Ideal.IsPrime p\n[PROOFSTEP]\napply Ring.not_isField_iff_exists_prime.mp h_fA\n[GOAL]\ncase pos.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI : Ideal A\nhA_nont : Nontrivial A\nhgt :\n \u2200 (J : Submodule A A),\n J > \u22a4 \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : \u22a4 \u2260 \u22a5\np_id : Ideal A\nh_nzp : p_id \u2260 \u22a5\nh_pp : Ideal.IsPrime p_id\n\u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 \u22a4 \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5\n[PROOFSTEP]\nuse({\u27e8p_id, h_pp\u27e9} : Multiset (PrimeSpectrum A)), le_top\n[GOAL]\ncase right\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI : Ideal A\nhA_nont : Nontrivial A\nhgt :\n \u2200 (J : Submodule A A),\n J > \u22a4 \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : \u22a4 \u2260 \u22a5\np_id : Ideal A\nh_nzp : p_id \u2260 \u22a5\nh_pp : Ideal.IsPrime p_id\n\u22a2 Multiset.prod (Multiset.map asIdeal {{ asIdeal := p_id, IsPrime := h_pp }}) \u2260 \u22a5\n[PROOFSTEP]\nrwa [Multiset.map_singleton, Multiset.prod_singleton]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\n\u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 M \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5\n[PROOFSTEP]\nby_cases h_prM : M.IsPrime\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : Ideal.IsPrime M\n\u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 M \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5\n[PROOFSTEP]\nuse({\u27e8M, h_prM\u27e9} : Multiset (PrimeSpectrum A))\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : Ideal.IsPrime M\n\u22a2 Multiset.prod (Multiset.map asIdeal {{ asIdeal := M, IsPrime := h_prM }}) \u2264 M \u2227\n Multiset.prod (Multiset.map asIdeal {{ asIdeal := M, IsPrime := h_prM }}) \u2260 \u22a5\n[PROOFSTEP]\nrw [Multiset.map_singleton, Multiset.prod_singleton]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : Ideal.IsPrime M\n\u22a2 { asIdeal := M, IsPrime := h_prM }.asIdeal \u2264 M \u2227 { asIdeal := M, IsPrime := h_prM }.asIdeal \u2260 \u22a5\n[PROOFSTEP]\nexact \u27e8le_rfl, h_nzM\u27e9\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\n\u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 M \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5\n[PROOFSTEP]\nobtain \u27e8x, hx, y, hy, h_xy\u27e9 := (Ideal.not_isPrime_iff.mp h_prM).resolve_left h_topM\n[GOAL]\ncase neg.intro.intro.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\n\u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 M \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5\n[PROOFSTEP]\nhave lt_add : \u2200 (z) (_ : z \u2209 M), M < M + span A { z } :=\n by\n intro z hz\n refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _\n rw [m_eq]\n exact mem_sup_right (mem_span_singleton_self z)\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\n\u22a2 \u2200 (z : A), \u00acz \u2208 M \u2192 M < M + span A {z}\n[PROOFSTEP]\nintro z hz\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nz : A\nhz : \u00acz \u2208 M\n\u22a2 M < M + span A {z}\n[PROOFSTEP]\nrefine' lt_of_le_of_ne le_sup_left fun m_eq => hz _\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nz : A\nhz : \u00acz \u2208 M\nm_eq : M = M + span A {z}\n\u22a2 z \u2208 M\n[PROOFSTEP]\nrw [m_eq]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nz : A\nhz : \u00acz \u2208 M\nm_eq : M = M + span A {z}\n\u22a2 z \u2208 M + span A {z}\n[PROOFSTEP]\nexact mem_sup_right (mem_span_singleton_self z)\n[GOAL]\ncase neg.intro.intro.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nlt_add : \u2200 (z : A), \u00acz \u2208 M \u2192 M < M + span A {z}\n\u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 M \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5\n[PROOFSTEP]\nobtain \u27e8Wx, h_Wx_le, h_Wx_ne\u27e9 := hgt (M + span A { x }) (lt_add _ hx) (ne_bot_of_gt (lt_add _ hx))\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nlt_add : \u2200 (z : A), \u00acz \u2208 M \u2192 M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span A {x}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wx) \u2260 \u22a5\n\u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 M \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5\n[PROOFSTEP]\nobtain \u27e8Wy, h_Wy_le, h_Wx_ne\u27e9 := hgt (M + span A { y }) (lt_add _ hy) (ne_bot_of_gt (lt_add _ hy))\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nlt_add : \u2200 (z : A), \u00acz \u2208 M \u2192 M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span A {x}\nh_Wx_ne\u271d : Multiset.prod (Multiset.map asIdeal Wx) \u2260 \u22a5\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) \u2260 \u22a5\n\u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 M \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5\n[PROOFSTEP]\nuse Wx + Wy\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nlt_add : \u2200 (z : A), \u00acz \u2208 M \u2192 M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span A {x}\nh_Wx_ne\u271d : Multiset.prod (Multiset.map asIdeal Wx) \u2260 \u22a5\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) \u2260 \u22a5\n\u22a2 Multiset.prod (Multiset.map asIdeal (Wx + Wy)) \u2264 M \u2227 Multiset.prod (Multiset.map asIdeal (Wx + Wy)) \u2260 \u22a5\n[PROOFSTEP]\nrw [Multiset.map_add, Multiset.prod_add]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nlt_add : \u2200 (z : A), \u00acz \u2208 M \u2192 M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span A {x}\nh_Wx_ne\u271d : Multiset.prod (Multiset.map asIdeal Wx) \u2260 \u22a5\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) \u2260 \u22a5\n\u22a2 Multiset.prod (Multiset.map asIdeal Wx) * Multiset.prod (Multiset.map asIdeal Wy) \u2264 M \u2227\n Multiset.prod (Multiset.map asIdeal Wx) * Multiset.prod (Multiset.map asIdeal Wy) \u2260 \u22a5\n[PROOFSTEP]\nrefine' \u27e8le_trans (Submodule.mul_le_mul h_Wx_le h_Wy_le) _, mt Ideal.mul_eq_bot.mp _\u27e9\n[GOAL]\ncase h.refine'_1\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nlt_add : \u2200 (z : A), \u00acz \u2208 M \u2192 M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span A {x}\nh_Wx_ne\u271d : Multiset.prod (Multiset.map asIdeal Wx) \u2260 \u22a5\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) \u2260 \u22a5\n\u22a2 (M + span A {x}) * (M + span A {y}) \u2264 M\n[PROOFSTEP]\nrw [add_mul]\n[GOAL]\ncase h.refine'_1\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nlt_add : \u2200 (z : A), \u00acz \u2208 M \u2192 M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span A {x}\nh_Wx_ne\u271d : Multiset.prod (Multiset.map asIdeal Wx) \u2260 \u22a5\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) \u2260 \u22a5\n\u22a2 M * (M + span A {y}) + span A {x} * (M + span A {y}) \u2264 M\n[PROOFSTEP]\napply sup_le (show M * (M + span A { y }) \u2264 M from Ideal.mul_le_right)\n[GOAL]\ncase h.refine'_1\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nlt_add : \u2200 (z : A), \u00acz \u2208 M \u2192 M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span A {x}\nh_Wx_ne\u271d : Multiset.prod (Multiset.map asIdeal Wx) \u2260 \u22a5\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) \u2260 \u22a5\n\u22a2 span A {x} * (M + span A {y}) \u2264 M\n[PROOFSTEP]\nrw [mul_add]\n[GOAL]\ncase h.refine'_1\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nlt_add : \u2200 (z : A), \u00acz \u2208 M \u2192 M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span A {x}\nh_Wx_ne\u271d : Multiset.prod (Multiset.map asIdeal Wx) \u2260 \u22a5\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) \u2260 \u22a5\n\u22a2 span A {x} * M + span A {x} * span A {y} \u2264 M\n[PROOFSTEP]\napply sup_le (show span A { x } * M \u2264 M from Ideal.mul_le_left)\n[GOAL]\ncase h.refine'_1\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nlt_add : \u2200 (z : A), \u00acz \u2208 M \u2192 M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span A {x}\nh_Wx_ne\u271d : Multiset.prod (Multiset.map asIdeal Wx) \u2260 \u22a5\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) \u2260 \u22a5\n\u22a2 span A {x} * span A {y} \u2264 M\n[PROOFSTEP]\nrwa [span_mul_span, Set.singleton_mul_singleton, span_singleton_le_iff_mem]\n[GOAL]\ncase h.refine'_2\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nlt_add : \u2200 (z : A), \u00acz \u2208 M \u2192 M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span A {x}\nh_Wx_ne\u271d : Multiset.prod (Multiset.map asIdeal Wx) \u2260 \u22a5\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) \u2260 \u22a5\n\u22a2 \u00ac(Multiset.prod (Multiset.map asIdeal Wx) = \u22a5 \u2228 Multiset.prod (Multiset.map asIdeal Wy) = \u22a5)\n[PROOFSTEP]\nrintro (hx | hy)\n[GOAL]\ncase h.refine'_2.inl\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx\u271d : \u00acx \u2208 M\ny : A\nhy : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nlt_add : \u2200 (z : A), \u00acz \u2208 M \u2192 M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span A {x}\nh_Wx_ne\u271d : Multiset.prod (Multiset.map asIdeal Wx) \u2260 \u22a5\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) \u2260 \u22a5\nhx : Multiset.prod (Multiset.map asIdeal Wx) = \u22a5\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase h.refine'_2.inr\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nh_fA : \u00acIsField A\nI M : Ideal A\nhgt :\n \u2200 (J : Submodule A A),\n J > M \u2192\n (fun I => I \u2260 \u22a5 \u2192 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I \u2227 Multiset.prod (Multiset.map asIdeal Z) \u2260 \u22a5) J\nh_nzM : M \u2260 \u22a5\nhA_nont : Nontrivial A\nh_topM : \u00acM = \u22a4\nh_prM : \u00acIdeal.IsPrime M\nx : A\nhx : \u00acx \u2208 M\ny : A\nhy\u271d : \u00acy \u2208 M\nh_xy : x * y \u2208 M\nlt_add : \u2200 (z : A), \u00acz \u2208 M \u2192 M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span A {x}\nh_Wx_ne\u271d : Multiset.prod (Multiset.map asIdeal Wx) \u2260 \u22a5\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) \u2260 \u22a5\nhy : Multiset.prod (Multiset.map asIdeal Wy) = \u22a5\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\n\u22a2 NoetherianSpace (PrimeSpectrum R)\n[PROOFSTEP]\napply ((noetherianSpace_TFAE <| PrimeSpectrum R).out 0 1).mpr\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\n\u22a2 WellFounded fun s t => s < t\n[PROOFSTEP]\nhave H := \u2039IsNoetherianRing R\u203a\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nH : IsNoetherianRing R\n\u22a2 WellFounded fun s t => s < t\n[PROOFSTEP]\nrw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at H \n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsNoetherianRing R\nA : Type u\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : IsNoetherianRing A\nH : WellFounded fun x x_1 => x > x_1\n\u22a2 WellFounded fun s t => s < t\n[PROOFSTEP]\nexact (closedsEmbedding R).dual.wellFounded H\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian", "llama_tokens": 19884, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7826624789529375, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.43698158945286686}} {"text": "[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 min a b = if a \u2264 b then a else b\n[PROOFSTEP]\nrw [LinearOrder.min_def a]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 max a b = if a \u2264 b then b else a\n[PROOFSTEP]\nrw [LinearOrder.max_def a]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 min a b \u2264 a\n[PROOFSTEP]\nif h : a \u2264 b then simp [min_def, if_pos h, le_refl] else simp [min_def, if_neg h]; exact le_of_not_le h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 min a b \u2264 a\n[PROOFSTEP]\nsimp [min_def, if_pos h, le_refl]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : \u00aca \u2264 b\n\u22a2 min a b \u2264 a\n[PROOFSTEP]\nsimp [min_def, if_neg h]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : \u00aca \u2264 b\n\u22a2 b \u2264 a\n[PROOFSTEP]\nexact le_of_not_le h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 min a b \u2264 b\n[PROOFSTEP]\nif h : a \u2264 b then simp [min_def, if_pos h]; exact h else simp [min_def, if_neg h, le_refl]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 min a b \u2264 b\n[PROOFSTEP]\nsimp [min_def, if_pos h]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 a \u2264 b\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : \u00aca \u2264 b\n\u22a2 min a b \u2264 b\n[PROOFSTEP]\nsimp [min_def, if_neg h, le_refl]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : c \u2264 a\nh\u2082 : c \u2264 b\n\u22a2 c \u2264 min a b\n[PROOFSTEP]\nif h : a \u2264 b then simp [min_def, if_pos h]; exact h\u2081 else simp [min_def, if_neg h]; exact h\u2082\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : c \u2264 a\nh\u2082 : c \u2264 b\nh : a \u2264 b\n\u22a2 c \u2264 min a b\n[PROOFSTEP]\nsimp [min_def, if_pos h]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : c \u2264 a\nh\u2082 : c \u2264 b\nh : a \u2264 b\n\u22a2 c \u2264 a\n[PROOFSTEP]\nexact h\u2081\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : c \u2264 a\nh\u2082 : c \u2264 b\nh : \u00aca \u2264 b\n\u22a2 c \u2264 min a b\n[PROOFSTEP]\nsimp [min_def, if_neg h]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : c \u2264 a\nh\u2082 : c \u2264 b\nh : \u00aca \u2264 b\n\u22a2 c \u2264 b\n[PROOFSTEP]\nexact h\u2082\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 a \u2264 max a b\n[PROOFSTEP]\nif h : a \u2264 b then simp [max_def, if_pos h]; exact h else simp [max_def, if_neg h, le_refl]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 a \u2264 max a b\n[PROOFSTEP]\nsimp [max_def, if_pos h]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 a \u2264 b\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : \u00aca \u2264 b\n\u22a2 a \u2264 max a b\n[PROOFSTEP]\nsimp [max_def, if_neg h, le_refl]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 b \u2264 max a b\n[PROOFSTEP]\nif h : a \u2264 b then simp [max_def, if_pos h, le_refl] else simp [max_def, if_neg h]; exact le_of_not_le h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 b \u2264 max a b\n[PROOFSTEP]\nsimp [max_def, if_pos h, le_refl]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : \u00aca \u2264 b\n\u22a2 b \u2264 max a b\n[PROOFSTEP]\nsimp [max_def, if_neg h]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : \u00aca \u2264 b\n\u22a2 b \u2264 a\n[PROOFSTEP]\nexact le_of_not_le h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : a \u2264 c\nh\u2082 : b \u2264 c\n\u22a2 max a b \u2264 c\n[PROOFSTEP]\nif h : a \u2264 b then simp [max_def, if_pos h]; exact h\u2082 else simp [max_def, if_neg h]; exact h\u2081\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : a \u2264 c\nh\u2082 : b \u2264 c\nh : a \u2264 b\n\u22a2 max a b \u2264 c\n[PROOFSTEP]\nsimp [max_def, if_pos h]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : a \u2264 c\nh\u2082 : b \u2264 c\nh : a \u2264 b\n\u22a2 b \u2264 c\n[PROOFSTEP]\nexact h\u2082\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : a \u2264 c\nh\u2082 : b \u2264 c\nh : \u00aca \u2264 b\n\u22a2 max a b \u2264 c\n[PROOFSTEP]\nsimp [max_def, if_neg h]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : a \u2264 c\nh\u2082 : b \u2264 c\nh : \u00aca \u2264 b\n\u22a2 a \u2264 c\n[PROOFSTEP]\nexact h\u2081\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 min (min a b) c = min a (min b c)\n[PROOFSTEP]\napply eq_min\n[GOAL]\ncase h\u2081\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 min (min a b) c \u2264 a\n[PROOFSTEP]\napply le_trans\n[GOAL]\ncase h\u2081.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 min (min a b) c \u2264 ?h\u2081.b\ncase h\u2081.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 ?h\u2081.b \u2264 a\ncase h\u2081.b \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c : \u03b1 \u22a2 \u03b1\n[PROOFSTEP]\napply min_le_left\n[GOAL]\ncase h\u2081.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 min a b \u2264 a\n[PROOFSTEP]\napply min_le_left\n[GOAL]\ncase h\u2082\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 min (min a b) c \u2264 min b c\n[PROOFSTEP]\napply le_min\n[GOAL]\ncase h\u2082.h\u2081\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 min (min a b) c \u2264 b\ncase h\u2082.h\u2082 \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c : \u03b1 \u22a2 min (min a b) c \u2264 c\n[PROOFSTEP]\napply le_trans\n[GOAL]\ncase h\u2082.h\u2081.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 min (min a b) c \u2264 ?h\u2082.h\u2081.b\ncase h\u2082.h\u2081.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 ?h\u2082.h\u2081.b \u2264 b\ncase h\u2082.h\u2081.b\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 \u03b1\ncase h\u2082.h\u2082 \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c : \u03b1 \u22a2 min (min a b) c \u2264 c\n[PROOFSTEP]\napply min_le_left\n[GOAL]\ncase h\u2082.h\u2081.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 min a b \u2264 b\ncase h\u2082.h\u2082 \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c : \u03b1 \u22a2 min (min a b) c \u2264 c\n[PROOFSTEP]\napply min_le_right\n[GOAL]\ncase h\u2082.h\u2082\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 min (min a b) c \u2264 c\n[PROOFSTEP]\napply min_le_right\n[GOAL]\ncase h\u2083\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 \u2200 {d : \u03b1}, d \u2264 a \u2192 d \u2264 min b c \u2192 d \u2264 min (min a b) c\n[PROOFSTEP]\nintro d h\u2081 h\u2082\n[GOAL]\ncase h\u2083\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\nh\u2081 : d \u2264 a\nh\u2082 : d \u2264 min b c\n\u22a2 d \u2264 min (min a b) c\n[PROOFSTEP]\napply le_min\n[GOAL]\ncase h\u2083.h\u2081\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\nh\u2081 : d \u2264 a\nh\u2082 : d \u2264 min b c\n\u22a2 d \u2264 min a b\ncase h\u2083.h\u2082 \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c d : \u03b1 h\u2081 : d \u2264 a h\u2082 : d \u2264 min b c \u22a2 d \u2264 c\n[PROOFSTEP]\napply le_min h\u2081\n[GOAL]\ncase h\u2083.h\u2081\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\nh\u2081 : d \u2264 a\nh\u2082 : d \u2264 min b c\n\u22a2 d \u2264 b\ncase h\u2083.h\u2082 \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c d : \u03b1 h\u2081 : d \u2264 a h\u2082 : d \u2264 min b c \u22a2 d \u2264 c\n[PROOFSTEP]\napply le_trans h\u2082\n[GOAL]\ncase h\u2083.h\u2081\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\nh\u2081 : d \u2264 a\nh\u2082 : d \u2264 min b c\n\u22a2 min b c \u2264 b\ncase h\u2083.h\u2082 \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c d : \u03b1 h\u2081 : d \u2264 a h\u2082 : d \u2264 min b c \u22a2 d \u2264 c\n[PROOFSTEP]\napply min_le_left\n[GOAL]\ncase h\u2083.h\u2082\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\nh\u2081 : d \u2264 a\nh\u2082 : d \u2264 min b c\n\u22a2 d \u2264 c\n[PROOFSTEP]\napply le_trans h\u2082\n[GOAL]\ncase h\u2083.h\u2082\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\nh\u2081 : d \u2264 a\nh\u2082 : d \u2264 min b c\n\u22a2 min b c \u2264 c\n[PROOFSTEP]\napply min_le_right\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 min a a = a\n[PROOFSTEP]\nsimp [min_def]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 min a b = a\n[PROOFSTEP]\napply Eq.symm\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 a = min a b\n[PROOFSTEP]\napply eq_min (le_refl _) h\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 \u2200 {d : \u03b1}, d \u2264 a \u2192 d \u2264 b \u2192 d \u2264 a\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nd\u271d : \u03b1\na\u271d\u00b9 : d\u271d \u2264 a\na\u271d : d\u271d \u2264 b\n\u22a2 d\u271d \u2264 a\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 max (max a b) c = max a (max b c)\n[PROOFSTEP]\napply eq_max\n[GOAL]\ncase h\u2081\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 a \u2264 max (max a b) c\n[PROOFSTEP]\napply le_trans\n[GOAL]\ncase h\u2081.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 a \u2264 ?h\u2081.b\ncase h\u2081.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 ?h\u2081.b \u2264 max (max a b) c\ncase h\u2081.b \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c : \u03b1 \u22a2 \u03b1\n[PROOFSTEP]\napply le_max_left a b\n[GOAL]\ncase h\u2081.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 max a b \u2264 max (max a b) c\n[PROOFSTEP]\napply le_max_left\n[GOAL]\ncase h\u2082\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 max b c \u2264 max (max a b) c\n[PROOFSTEP]\napply max_le\n[GOAL]\ncase h\u2082.h\u2081\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 b \u2264 max (max a b) c\ncase h\u2082.h\u2082 \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c : \u03b1 \u22a2 c \u2264 max (max a b) c\n[PROOFSTEP]\napply le_trans\n[GOAL]\ncase h\u2082.h\u2081.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 b \u2264 ?h\u2082.h\u2081.b\ncase h\u2082.h\u2081.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 ?h\u2082.h\u2081.b \u2264 max (max a b) c\ncase h\u2082.h\u2081.b\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 \u03b1\ncase h\u2082.h\u2082 \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c : \u03b1 \u22a2 c \u2264 max (max a b) c\n[PROOFSTEP]\napply le_max_right a b\n[GOAL]\ncase h\u2082.h\u2081.a\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 max a b \u2264 max (max a b) c\ncase h\u2082.h\u2082 \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c : \u03b1 \u22a2 c \u2264 max (max a b) c\n[PROOFSTEP]\napply le_max_left\n[GOAL]\ncase h\u2082.h\u2082\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 c \u2264 max (max a b) c\n[PROOFSTEP]\napply le_max_right\n[GOAL]\ncase h\u2083\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\n\u22a2 \u2200 {d : \u03b1}, a \u2264 d \u2192 max b c \u2264 d \u2192 max (max a b) c \u2264 d\n[PROOFSTEP]\nintro d h\u2081 h\u2082\n[GOAL]\ncase h\u2083\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\nh\u2081 : a \u2264 d\nh\u2082 : max b c \u2264 d\n\u22a2 max (max a b) c \u2264 d\n[PROOFSTEP]\napply max_le\n[GOAL]\ncase h\u2083.h\u2081\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\nh\u2081 : a \u2264 d\nh\u2082 : max b c \u2264 d\n\u22a2 max a b \u2264 d\ncase h\u2083.h\u2082 \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c d : \u03b1 h\u2081 : a \u2264 d h\u2082 : max b c \u2264 d \u22a2 c \u2264 d\n[PROOFSTEP]\napply max_le h\u2081\n[GOAL]\ncase h\u2083.h\u2081\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\nh\u2081 : a \u2264 d\nh\u2082 : max b c \u2264 d\n\u22a2 b \u2264 d\ncase h\u2083.h\u2082 \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b c d : \u03b1 h\u2081 : a \u2264 d h\u2082 : max b c \u2264 d \u22a2 c \u2264 d\n[PROOFSTEP]\napply le_trans (le_max_left _ _) h\u2082\n[GOAL]\ncase h\u2083.h\u2082\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\nh\u2081 : a \u2264 d\nh\u2082 : max b c \u2264 d\n\u22a2 c \u2264 d\n[PROOFSTEP]\napply le_trans (le_max_right _ _) h\u2082\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 max a a = a\n[PROOFSTEP]\nsimp [max_def]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : b \u2264 a\n\u22a2 max a b = a\n[PROOFSTEP]\napply Eq.symm\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : b \u2264 a\n\u22a2 a = max a b\n[PROOFSTEP]\napply eq_max (le_refl _) h\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : b \u2264 a\n\u22a2 \u2200 {d : \u03b1}, a \u2264 d \u2192 b \u2264 d \u2192 a \u2264 d\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : b \u2264 a\nd\u271d : \u03b1\na\u271d\u00b9 : a \u2264 d\u271d\na\u271d : b \u2264 d\u271d\n\u22a2 a \u2264 d\u271d\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : a < b\nh\u2082 : a < c\nh : b \u2264 c\n\u22a2 a < min b c\n[PROOFSTEP]\nrwa [min_eq_left h]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : a < b\nh\u2082 : a < c\nh : b > c\n\u22a2 a < min b c\n[PROOFSTEP]\nrwa [min_eq_right_of_lt h]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : a < c\nh\u2082 : b < c\nh : a \u2264 b\n\u22a2 max a b < c\n[PROOFSTEP]\nrwa [max_eq_right h]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : a < c\nh\u2082 : b < c\nh : a > b\n\u22a2 max a b < c\n[PROOFSTEP]\nrwa [max_eq_left_of_lt h]\n", "meta": {"mathlib_filename": "Mathlib.Init.Algebra.Functions", "llama_tokens": 5459, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494678483918, "lm_q2_score": 0.6370307875894138, "lm_q1q2_score": 0.4369709297500003}} {"text": "[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 IsProbabilityMeasure (\u2191(condKernelReal \u03c1) a)\n[PROOFSTEP]\nrw [condKernelReal]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 IsProbabilityMeasure\n (\u2191{ val := fun a => StieltjesFunction.measure (condCdf \u03c1 a),\n property := (_ : Measurable fun a => StieltjesFunction.measure (condCdf \u03c1 a)) }\n a)\n[PROOFSTEP]\nexact instIsProbabilityMeasure \u03c1 a\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (Iic x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n[PROOFSTEP]\nsimp_rw [condKernelReal_Iic]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n[PROOFSTEP]\nexact set_lintegral_condCdf \u03c1 x hs\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nsimp only [measure_univ, lintegral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter, one_mul,\n Measure.fst_apply hs, \u2190 prod_univ]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 univ\n[PROOFSTEP]\nrw [\u2190 set_lintegral_univ, set_lintegral_condKernelReal_univ \u03c1 MeasurableSet.univ, univ_prod_univ]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n[PROOFSTEP]\napply MeasurableSpace.induction_on_inter (borel_eq_generateFrom_Iic \u211d) isPiSystem_Iic _ _ _ _ ht\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) \u2205 \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205)\n[PROOFSTEP]\nsimp only [measure_empty, lintegral_const, zero_mul, prod_empty]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u211d), t \u2208 range Iic \u2192 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n[PROOFSTEP]\nrintro t \u27e8q, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nq : \u211d\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (Iic q) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic q)\n[PROOFSTEP]\nexact set_lintegral_condKernelReal_Iic \u03c1 q hs\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u211d),\n MeasurableSet t \u2192\n \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u2192\n \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u1d9c \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t\u1d9c)\n[PROOFSTEP]\nintro t ht ht_lintegral\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u1d9c \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t\u1d9c)\n[PROOFSTEP]\ncalc\n \u222b\u207b a in s, condKernelReal \u03c1 a t\u1d9c \u2202\u03c1.fst = \u222b\u207b a in s, condKernelReal \u03c1 a univ - condKernelReal \u03c1 a t \u2202\u03c1.fst := by\n congr with a; rw [measure_compl ht (measure_ne_top (condKernelReal \u03c1 a) _)]\n _ = \u222b\u207b a in s, condKernelReal \u03c1 a univ \u2202\u03c1.fst - \u222b\u207b a in s, condKernelReal \u03c1 a t \u2202\u03c1.fst :=\n by\n rw [lintegral_sub (kernel.measurable_coe (condKernelReal \u03c1) ht)]\n \u00b7 rw [ht_lintegral]\n exact measure_ne_top \u03c1 _\n \u00b7 exact eventually_of_forall fun a => measure_mono (subset_univ _)\n _ = \u03c1 (s \u00d7\u02e2 univ) - \u03c1 (s \u00d7\u02e2 t) := by rw [set_lintegral_condKernelReal_univ \u03c1 hs, ht_lintegral]\n _ = \u03c1 (s \u00d7\u02e2 t\u1d9c) := by\n rw [\u2190 measure_diff _ (hs.prod ht) (measure_ne_top \u03c1 _)]\n \u00b7 rw [prod_diff_prod, compl_eq_univ_diff]\n simp only [diff_self, empty_prod, union_empty]\n \u00b7 rw [prod_subset_prod_iff]\n exact Or.inl \u27e8subset_rfl, subset_univ t\u27e9\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u1d9c \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1\n[PROOFSTEP]\ncongr with a\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u1d9c = \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\n[PROOFSTEP]\nrw [measure_compl ht (measure_ne_top (condKernelReal \u03c1 a) _)]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -\n \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1\n[PROOFSTEP]\nrw [lintegral_sub (kernel.measurable_coe (condKernelReal \u03c1) ht)]\n[GOAL]\ncase hg_fin\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 \u2260 \u22a4\n[PROOFSTEP]\nrw [ht_lintegral]\n[GOAL]\ncase hg_fin\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 t) \u2260 \u22a4\n[PROOFSTEP]\nexact measure_ne_top \u03c1 _\n[GOAL]\ncase h_le\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) t) \u2264\u1d50[Measure.restrict (Measure.fst \u03c1) s] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n[PROOFSTEP]\nexact eventually_of_forall fun a => measure_mono (subset_univ _)\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -\n \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 =\n \u2191\u2191\u03c1 (s \u00d7\u02e2 univ) - \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n[PROOFSTEP]\nrw [set_lintegral_condKernelReal_univ \u03c1 hs, ht_lintegral]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 univ) - \u2191\u2191\u03c1 (s \u00d7\u02e2 t) = \u2191\u2191\u03c1 (s \u00d7\u02e2 t\u1d9c)\n[PROOFSTEP]\nrw [\u2190 measure_diff _ (hs.prod ht) (measure_ne_top \u03c1 _)]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 univ \\ s \u00d7\u02e2 t) = \u2191\u2191\u03c1 (s \u00d7\u02e2 t\u1d9c)\n[PROOFSTEP]\nrw [prod_diff_prod, compl_eq_univ_diff]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 (univ \\ t) \u222a (s \\ s) \u00d7\u02e2 univ) = \u2191\u2191\u03c1 (s \u00d7\u02e2 (univ \\ t))\n[PROOFSTEP]\nsimp only [diff_self, empty_prod, union_empty]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 s \u00d7\u02e2 t \u2286 s \u00d7\u02e2 univ\n[PROOFSTEP]\nrw [prod_subset_prod_iff]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt\u271d : Set \u211d\nht\u271d : MeasurableSet t\u271d\nt : Set \u211d\nht : MeasurableSet t\nht_lintegral : \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n\u22a2 s \u2286 s \u2227 t \u2286 univ \u2228 s = \u2205 \u2228 t = \u2205\n[PROOFSTEP]\nexact Or.inl \u27e8subset_rfl, subset_univ t\u27e9\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u211d),\n Pairwise (Disjoint on f) \u2192\n (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n (\u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)) \u2192\n \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 (i : \u2115), f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nintro f hf_disj hf_meas hf_eq\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 (i : \u2115), f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nsimp_rw [measure_iUnion hf_disj hf_meas]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2211' (i : \u2115), \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 \u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nrw [lintegral_tsum fun i => (kernel.measurable_coe _ (hf_meas i)).aemeasurable.restrict, prod_iUnion, measure_iUnion]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\u22a2 \u2211' (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2211' (i : \u2115), \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n[PROOFSTEP]\nsimp_rw [hf_eq]\n[GOAL]\ncase hn\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\u22a2 Pairwise (Disjoint on fun i => s \u00d7\u02e2 f i)\n[PROOFSTEP]\nintro i j hij\n[GOAL]\ncase hn\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\ni j : \u2115\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => s \u00d7\u02e2 f i) i j\n[PROOFSTEP]\nrw [Function.onFun, disjoint_prod]\n[GOAL]\ncase hn\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\ni j : \u2115\nhij : i \u2260 j\n\u22a2 Disjoint s s \u2228 Disjoint (f i) (f j)\n[PROOFSTEP]\nexact Or.inr (hf_disj hij)\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (f i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 f i)\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u00d7\u02e2 f i)\n[PROOFSTEP]\nexact fun i => MeasurableSet.prod hs (hf_meas i)\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 s} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 s\n[PROOFSTEP]\napply MeasurableSpace.induction_on_inter generateFrom_prod.symm isPiSystem_prod _ _ _ _ hs\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u2205} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 \u2205\n[PROOFSTEP]\nsimp only [mem_empty_iff_false, setOf_false, measure_empty, lintegral_const, zero_mul]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set (\u03b1 \u00d7 \u211d)),\n t \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) {s | MeasurableSet s} {t | MeasurableSet t} \u2192\n \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\n[PROOFSTEP]\nintro t ht\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : t \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) {s | MeasurableSet s} {t | MeasurableSet t}\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\n[PROOFSTEP]\nrw [mem_image2] at ht \n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : \u2203 a b, a \u2208 {s | MeasurableSet s} \u2227 b \u2208 {t | MeasurableSet t} \u2227 a \u00d7\u02e2 b = t\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\n[PROOFSTEP]\nobtain \u27e8t\u2081, t\u2082, ht\u2081, ht\u2082, rfl\u27e9 := ht\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)\n[PROOFSTEP]\nhave h_prod_eq_snd : \u2200 a \u2208 t\u2081, {x : \u211d | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 :=\n by\n intro a ha\n simp only [ha, prod_mk_mem_set_prod_eq, true_and_iff, setOf_mem_eq]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\n\u22a2 \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\n[PROOFSTEP]\nintro a ha\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\na : \u03b1\nha : a \u2208 t\u2081\n\u22a2 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\n[PROOFSTEP]\nsimp only [ha, prod_mk_mem_set_prod_eq, true_and_iff, setOf_mem_eq]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)\n[PROOFSTEP]\ncases' eq_empty_or_nonempty t\u2082 with h h\n[GOAL]\ncase intro.intro.intro.intro.inl\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : t\u2082 = \u2205\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)\n[PROOFSTEP]\nsimp only [h, prod_empty, mem_empty_iff_false, setOf_false, measure_empty, lintegral_const, zero_mul]\n[GOAL]\ncase intro.intro.intro.intro.inr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)\n[PROOFSTEP]\nrw [\u2190 lintegral_add_compl _ ht\u2081]\n[GOAL]\ncase intro.intro.intro.intro.inr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\n\u22a2 \u222b\u207b (x : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 +\n \u222b\u207b (x : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)\n[PROOFSTEP]\nhave h_eq1 :\n \u222b\u207b a in t\u2081, condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202\u03c1.fst = \u222b\u207b a in t\u2081, condKernelReal \u03c1 a t\u2082 \u2202\u03c1.fst :=\n by\n refine' set_lintegral_congr_fun ht\u2081 (eventually_of_forall fun a ha => _)\n rw [h_prod_eq_snd a ha]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\n\u22a2 \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\n[PROOFSTEP]\nrefine' set_lintegral_congr_fun ht\u2081 (eventually_of_forall fun a ha => _)\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\na : \u03b1\nha : a \u2208 t\u2081\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082\n[PROOFSTEP]\nrw [h_prod_eq_snd a ha]\n[GOAL]\ncase intro.intro.intro.intro.inr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (x : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 +\n \u222b\u207b (x : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)\n[PROOFSTEP]\nhave h_eq2 : \u222b\u207b a in t\u2081\u1d9c, condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202\u03c1.fst = 0 :=\n by\n suffices h_eq_zero : \u2200 a \u2208 t\u2081\u1d9c, condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0\n \u00b7 rw [set_lintegral_congr_fun ht\u2081.compl (eventually_of_forall h_eq_zero)]\n simp only [lintegral_const, zero_mul]\n intro a hat\u2081\n rw [mem_compl_iff] at hat\u2081 \n simp only [hat\u2081, prod_mk_mem_set_prod_eq, false_and_iff, setOf_false, measure_empty]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (a : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = 0\n[PROOFSTEP]\nsuffices h_eq_zero : \u2200 a \u2208 t\u2081\u1d9c, condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\nh_eq_zero : \u2200 (a : \u03b1), a \u2208 t\u2081\u1d9c \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0\n\u22a2 \u222b\u207b (a : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = 0\n[PROOFSTEP]\nrw [set_lintegral_congr_fun ht\u2081.compl (eventually_of_forall h_eq_zero)]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\nh_eq_zero : \u2200 (a : \u03b1), a \u2208 t\u2081\u1d9c \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0\n\u22a2 \u222b\u207b (x : \u03b1) in t\u2081\u1d9c, 0 \u2202Measure.fst \u03c1 = 0\n[PROOFSTEP]\nsimp only [lintegral_const, zero_mul]\n[GOAL]\ncase h_eq_zero\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\n\u22a2 \u2200 (a : \u03b1), a \u2208 t\u2081\u1d9c \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0\n[PROOFSTEP]\nintro a hat\u2081\n[GOAL]\ncase h_eq_zero\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\na : \u03b1\nhat\u2081 : a \u2208 t\u2081\u1d9c\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0\n[PROOFSTEP]\nrw [mem_compl_iff] at hat\u2081 \n[GOAL]\ncase h_eq_zero\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\na : \u03b1\nhat\u2081 : \u00aca \u2208 t\u2081\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0\n[PROOFSTEP]\nsimp only [hat\u2081, prod_mk_mem_set_prod_eq, false_and_iff, setOf_false, measure_empty]\n[GOAL]\ncase intro.intro.intro.intro.inr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\nh_eq2 : \u222b\u207b (a : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = 0\n\u22a2 \u222b\u207b (x : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 +\n \u222b\u207b (x : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)\n[PROOFSTEP]\nrw [h_eq1, h_eq2, add_zero]\n[GOAL]\ncase intro.intro.intro.intro.inr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt\u2081 : Set \u03b1\nt\u2082 : Set \u211d\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\nh_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082\nh : Set.Nonempty t\u2082\nh_eq1 :\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1\nh_eq2 : \u222b\u207b (a : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = 0\n\u22a2 \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082)\n[PROOFSTEP]\nexact set_lintegral_condKernelReal_prod \u03c1 ht\u2081 ht\u2082\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set (\u03b1 \u00d7 \u211d)),\n MeasurableSet t \u2192\n \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t \u2192\n \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u1d9c} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\u1d9c\n[PROOFSTEP]\nintro t ht ht_eq\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u1d9c} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\u1d9c\n[PROOFSTEP]\ncalc\n \u222b\u207b a, condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t\u1d9c} \u2202\u03c1.fst = \u222b\u207b a, condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t}\u1d9c \u2202\u03c1.fst := rfl\n _ = \u222b\u207b a, condKernelReal \u03c1 a univ - condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t} \u2202\u03c1.fst :=\n by\n congr with a : 1\n exact measure_compl (measurable_prod_mk_left ht) (measure_ne_top (condKernelReal \u03c1 a) _)\n _ = \u222b\u207b a, condKernelReal \u03c1 a univ \u2202\u03c1.fst - \u222b\u207b a, condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t} \u2202\u03c1.fst :=\n by\n have h_le : (fun a => condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t}) \u2264\u1d50[\u03c1.fst] fun a => condKernelReal \u03c1 a univ :=\n eventually_of_forall fun a => measure_mono (subset_univ _)\n rw [lintegral_sub _ _ h_le]\n \u00b7 exact kernel.measurable_kernel_prod_mk_left ht\n refine' ((lintegral_mono_ae h_le).trans_lt _).ne\n rw [lintegral_condKernelReal_univ]\n exact measure_lt_top \u03c1 univ\n _ = \u03c1 univ - \u03c1 t := by rw [ht_eq, lintegral_condKernelReal_univ]\n _ = \u03c1 t\u1d9c := (measure_compl ht (measure_ne_top _ _)).symm\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}\u1d9c \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1\n[PROOFSTEP]\ncongr with a : 1\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}\u1d9c =\n \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}\n[PROOFSTEP]\nexact measure_compl (measurable_prod_mk_left ht) (measure_ne_top (condKernelReal \u03c1 a) _)\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -\n \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1\n[PROOFSTEP]\nhave h_le : (fun a => condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t}) \u2264\u1d50[\u03c1.fst] fun a => condKernelReal \u03c1 a univ :=\n eventually_of_forall fun a => measure_mono (subset_univ _)\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\nh_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -\n \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1\n[PROOFSTEP]\nrw [lintegral_sub _ _ h_le]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\nh_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n\u22a2 Measurable fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}\n[PROOFSTEP]\nexact kernel.measurable_kernel_prod_mk_left ht\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\nh_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 \u2260 \u22a4\n[PROOFSTEP]\nrefine' ((lintegral_mono_ae h_le).trans_lt _).ne\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\nh_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n\u22a2 \u222b\u207b (a : \u03b1), (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ) a \u2202Measure.fst \u03c1 < \u22a4\n[PROOFSTEP]\nrw [lintegral_condKernelReal_univ]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\nh_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ\n\u22a2 \u2191\u2191\u03c1 univ < \u22a4\n[PROOFSTEP]\nexact measure_lt_top \u03c1 univ\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u211d)\nht : MeasurableSet t\nht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -\n \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 =\n \u2191\u2191\u03c1 univ - \u2191\u2191\u03c1 t\n[PROOFSTEP]\nrw [ht_eq, lintegral_condKernelReal_univ]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2200 (f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)),\n Pairwise (Disjoint on f) \u2192\n (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n (\u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)) \u2192\n \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nintro f hf_disj hf_meas hf_eq\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nhave h_eq : \u2200 a, {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} :=\n by\n intro a\n ext1 x\n simp only [mem_iUnion, mem_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\n\u22a2 \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\n[PROOFSTEP]\nintro a\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\na : \u03b1\n\u22a2 {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\na : \u03b1\nx : \u211d\n\u22a2 x \u2208 {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} \u2194 x \u2208 \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\n[PROOFSTEP]\nsimp only [mem_iUnion, mem_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nsimp_rw [h_eq]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 (i : \u2115), {x | (a, x) \u2208 f i}) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nhave h_disj : \u2200 a, Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i}) :=\n by\n intro a i j hij\n have h_disj := hf_disj hij\n rw [Function.onFun, disjoint_iff_inter_eq_empty] at h_disj \u22a2\n ext1 x\n simp only [mem_inter_iff, mem_setOf_eq, mem_empty_iff_false, iff_false_iff]\n intro h_mem_both\n suffices (a, x) \u2208 \u2205 by rwa [mem_empty_iff_false] at this \n rwa [\u2190 h_disj, mem_inter_iff]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\n\u22a2 \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i})\n[PROOFSTEP]\nintro a i j hij\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => {x | (a, x) \u2208 f i}) i j\n[PROOFSTEP]\nhave h_disj := hf_disj hij\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : (Disjoint on f) i j\n\u22a2 (Disjoint on fun i => {x | (a, x) \u2208 f i}) i j\n[PROOFSTEP]\nrw [Function.onFun, disjoint_iff_inter_eq_empty] at h_disj \u22a2\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\n\u22a2 {x | (a, x) \u2208 f i} \u2229 {x | (a, x) \u2208 f j} = \u2205\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\nx : \u211d\n\u22a2 x \u2208 {x | (a, x) \u2208 f i} \u2229 {x | (a, x) \u2208 f j} \u2194 x \u2208 \u2205\n[PROOFSTEP]\nsimp only [mem_inter_iff, mem_setOf_eq, mem_empty_iff_false, iff_false_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\nx : \u211d\n\u22a2 \u00ac((a, x) \u2208 f i \u2227 (a, x) \u2208 f j)\n[PROOFSTEP]\nintro h_mem_both\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\nx : \u211d\nh_mem_both : (a, x) \u2208 f i \u2227 (a, x) \u2208 f j\n\u22a2 False\n[PROOFSTEP]\nsuffices (a, x) \u2208 \u2205 by rwa [mem_empty_iff_false] at this \n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\nx : \u211d\nh_mem_both : (a, x) \u2208 f i \u2227 (a, x) \u2208 f j\nthis : (a, x) \u2208 \u2205\n\u22a2 False\n[PROOFSTEP]\nrwa [mem_empty_iff_false] at this \n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\na : \u03b1\ni j : \u2115\nhij : i \u2260 j\nh_disj : f i \u2229 f j = \u2205\nx : \u211d\nh_mem_both : (a, x) \u2208 f i \u2227 (a, x) \u2208 f j\n\u22a2 (a, x) \u2208 \u2205\n[PROOFSTEP]\nrwa [\u2190 h_disj, mem_inter_iff]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\nh_disj : \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i})\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 (i : \u2115), {x | (a, x) \u2208 f i}) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\ncalc\n \u222b\u207b a, condKernelReal \u03c1 a (\u22c3 i, {x | (a, x) \u2208 f i}) \u2202\u03c1.fst =\n \u222b\u207b a, \u2211' i, condKernelReal \u03c1 a {x | (a, x) \u2208 f i} \u2202\u03c1.fst :=\n by\n congr with a : 1\n rw [measure_iUnion (h_disj a) fun i => measurable_prod_mk_left (hf_meas i)]\n _ = \u2211' i, \u222b\u207b a, condKernelReal \u03c1 a {x | (a, x) \u2208 f i} \u2202\u03c1.fst :=\n (lintegral_tsum fun i => (kernel.measurable_kernel_prod_mk_left (hf_meas i)).aemeasurable)\n _ = \u2211' i, \u03c1 (f i) := by simp_rw [hf_eq]\n _ = \u03c1 (iUnion f) := (measure_iUnion hf_disj hf_meas).symm\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\nh_disj : \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i})\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 (i : \u2115), {x | (a, x) \u2208 f i}) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1), \u2211' (i : \u2115), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1\n[PROOFSTEP]\ncongr with a : 1\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\nh_disj : \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i})\na : \u03b1\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 (i : \u2115), {x | (a, x) \u2208 f i}) = \u2211' (i : \u2115), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i}\n[PROOFSTEP]\nrw [measure_iUnion (h_disj a) fun i => measurable_prod_mk_left (hf_meas i)]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)\nh_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 (i : \u2115), f i} = \u22c3 (i : \u2115), {x | (a, x) \u2208 f i}\nh_disj : \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i})\n\u22a2 \u2211' (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2211' (i : \u2115), \u2191\u2191\u03c1 (f i)\n[PROOFSTEP]\nsimp_rw [hf_eq]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d\n\u03b3 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 const \u03b3 \u03c1 = const \u03b3 (Measure.fst \u03c1) \u2297\u2096 prodMkLeft \u03b3 (condKernelReal \u03c1)\n[PROOFSTEP]\next a s hs : 2\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d\n\u03b3 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b3\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(const \u03b3 \u03c1) a) s = \u2191\u2191(\u2191(const \u03b3 (Measure.fst \u03c1) \u2297\u2096 prodMkLeft \u03b3 (condKernelReal \u03c1)) a) s\n[PROOFSTEP]\nrw [kernel.compProd_apply _ _ _ hs, kernel.const_apply, kernel.const_apply]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d\n\u03b3 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b3\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 s = \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(prodMkLeft \u03b3 (condKernelReal \u03c1)) (a, b)) {c | (b, c) \u2208 s} \u2202Measure.fst \u03c1\n[PROOFSTEP]\nsimp_rw [kernel.prodMkLeft_apply]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d\n\u03b3 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b3\ns : Set (\u03b1 \u00d7 \u211d)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 s = \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) b) {c | (b, c) \u2208 s} \u2202Measure.fst \u03c1\n[PROOFSTEP]\nrw [lintegral_condKernelReal_mem \u03c1 hs]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (condKernelReal \u03c1)) ()\n[PROOFSTEP]\nrw [\u2190 kernel.const_eq_compProd_real Unit \u03c1, kernel.const_apply]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u211d \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (y : \u211d), f (a, y) \u2202\u2191(condKernelReal \u03c1) a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1 \u00d7 \u211d), f x \u2202\u03c1\n[PROOFSTEP]\nnth_rw 3 [measure_eq_compProd_real \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u211d \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (y : \u211d), f (a, y) \u2202\u2191(condKernelReal \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b\u207b (x : \u03b1 \u00d7 \u211d), f x \u2202\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (condKernelReal \u03c1)) ()\n[PROOFSTEP]\nrw [kernel.lintegral_compProd _ _ _ hf, kernel.const_apply]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u211d \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (y : \u211d), f (a, y) \u2202\u2191(condKernelReal \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b\u207b (b : \u03b1), \u222b\u207b (c : \u211d), f (b, c) \u2202\u2191(kernel.prodMkLeft Unit (condKernelReal \u03c1)) ((), b) \u2202Measure.fst \u03c1\n[PROOFSTEP]\nsimp_rw [kernel.prodMkLeft_apply]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2191\u2191(\u2191(condKernelReal \u03c1) a) s = 1\n[PROOFSTEP]\nhave h :\n \u03c1 {x | x.snd \u2208 s\u1d9c} = (kernel.const Unit \u03c1.fst \u2297\u2096 kernel.prodMkLeft Unit (condKernelReal \u03c1)) () {x | x.snd \u2208 s\u1d9c} := by\n rw [\u2190 measure_eq_compProd_real]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\n\u22a2 \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} =\n \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (condKernelReal \u03c1)) ()) {x | x.snd \u2208 s\u1d9c}\n[PROOFSTEP]\nrw [\u2190 measure_eq_compProd_real]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\nh :\n \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} =\n \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (condKernelReal \u03c1)) ()) {x | x.snd \u2208 s\u1d9c}\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2191\u2191(\u2191(condKernelReal \u03c1) a) s = 1\n[PROOFSTEP]\nrw [h\u03c1, kernel.compProd_apply] at h \n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\nh :\n 0 =\n \u222b\u207b (b : \u03b1),\n \u2191\u2191(\u2191(kernel.prodMkLeft Unit (condKernelReal \u03c1)) ((), b))\n {c | (b, c) \u2208 {x | x.snd \u2208 s\u1d9c}} \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) ()\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2191\u2191(\u2191(condKernelReal \u03c1) a) s = 1\ncase hs\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\nh : 0 = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (condKernelReal \u03c1)) ()) {x | x.snd \u2208 s\u1d9c}\n\u22a2 MeasurableSet {x | x.snd \u2208 s\u1d9c}\n[PROOFSTEP]\nswap\n[GOAL]\ncase hs\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\nh : 0 = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (condKernelReal \u03c1)) ()) {x | x.snd \u2208 s\u1d9c}\n\u22a2 MeasurableSet {x | x.snd \u2208 s\u1d9c}\n[PROOFSTEP]\nexact measurable_snd hs.compl\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\nh :\n 0 =\n \u222b\u207b (b : \u03b1),\n \u2191\u2191(\u2191(kernel.prodMkLeft Unit (condKernelReal \u03c1)) ((), b))\n {c | (b, c) \u2208 {x | x.snd \u2208 s\u1d9c}} \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) ()\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2191\u2191(\u2191(condKernelReal \u03c1) a) s = 1\n[PROOFSTEP]\nrw [eq_comm, lintegral_eq_zero_iff] at h \n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\nh :\n (fun b =>\n \u2191\u2191(\u2191(kernel.prodMkLeft Unit (condKernelReal \u03c1)) ((), b))\n {c | (b, c) \u2208 {x | x.snd \u2208 s\u1d9c}}) =\u1d50[\u2191(kernel.const Unit (Measure.fst \u03c1)) ()]\n 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2191\u2191(\u2191(condKernelReal \u03c1) a) s = 1\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\nh :\n \u222b\u207b (b : \u03b1),\n \u2191\u2191(\u2191(kernel.prodMkLeft Unit (condKernelReal \u03c1)) ((), b))\n {c | (b, c) \u2208 {x | x.snd \u2208 s\u1d9c}} \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) () =\n 0\n\u22a2 Measurable fun b => \u2191\u2191(\u2191(kernel.prodMkLeft Unit (condKernelReal \u03c1)) ((), b)) {c | (b, c) \u2208 {x | x.snd \u2208 s\u1d9c}}\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\nh :\n \u222b\u207b (b : \u03b1),\n \u2191\u2191(\u2191(kernel.prodMkLeft Unit (condKernelReal \u03c1)) ((), b))\n {c | (b, c) \u2208 {x | x.snd \u2208 s\u1d9c}} \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) () =\n 0\n\u22a2 Measurable fun b => \u2191\u2191(\u2191(kernel.prodMkLeft Unit (condKernelReal \u03c1)) ((), b)) {c | (b, c) \u2208 {x | x.snd \u2208 s\u1d9c}}\n[PROOFSTEP]\nsimp_rw [kernel.prodMkLeft_apply']\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\nh :\n \u222b\u207b (b : \u03b1),\n \u2191\u2191(\u2191(kernel.prodMkLeft Unit (condKernelReal \u03c1)) ((), b))\n {c | (b, c) \u2208 {x | x.snd \u2208 s\u1d9c}} \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) () =\n 0\n\u22a2 Measurable fun b => \u2191\u2191(\u2191(condKernelReal \u03c1) b) {c | (b, c) \u2208 {x | x.snd \u2208 s\u1d9c}}\n[PROOFSTEP]\nsimp only [mem_compl_iff, mem_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\nh :\n \u222b\u207b (b : \u03b1),\n \u2191\u2191(\u2191(kernel.prodMkLeft Unit (condKernelReal \u03c1)) ((), b))\n {c | (b, c) \u2208 {x | x.snd \u2208 s\u1d9c}} \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) () =\n 0\n\u22a2 Measurable fun b => \u2191\u2191(\u2191(condKernelReal \u03c1) b) {c | \u00acc \u2208 s}\n[PROOFSTEP]\nexact kernel.measurable_coe _ hs.compl\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\nh :\n (fun b =>\n \u2191\u2191(\u2191(kernel.prodMkLeft Unit (condKernelReal \u03c1)) ((), b))\n {c | (b, c) \u2208 {x | x.snd \u2208 s\u1d9c}}) =\u1d50[\u2191(kernel.const Unit (Measure.fst \u03c1)) ()]\n 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2191\u2191(\u2191(condKernelReal \u03c1) a) s = 1\n[PROOFSTEP]\nrw [kernel.const_apply] at h \n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\nh :\n (fun b => \u2191\u2191(\u2191(kernel.prodMkLeft Unit (condKernelReal \u03c1)) ((), b)) {c | (b, c) \u2208 {x | x.snd \u2208 s\u1d9c}}) =\u1d50[Measure.fst \u03c1]\n 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2191\u2191(\u2191(condKernelReal \u03c1) a) s = 1\n[PROOFSTEP]\nsimp only [mem_compl_iff, mem_setOf_eq, kernel.prodMkLeft_apply'] at h \n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\nh : (fun b => \u2191\u2191(\u2191(condKernelReal \u03c1) b) {c | \u00acc \u2208 s}) =\u1d50[Measure.fst \u03c1] 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2191\u2191(\u2191(condKernelReal \u03c1) a) s = 1\n[PROOFSTEP]\nfilter_upwards [h] with a ha\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\nh : (fun b => \u2191\u2191(\u2191(condKernelReal \u03c1) b) {c | \u00acc \u2208 s}) =\u1d50[Measure.fst \u03c1] 0\na : \u03b1\nha : \u2191\u2191(\u2191(condKernelReal \u03c1) a) {c | \u00acc \u2208 s} = OfNat.ofNat 0 a\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) s = 1\n[PROOFSTEP]\nchange condKernelReal \u03c1 a s\u1d9c = 0 at ha \n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u211d\nhs : MeasurableSet s\nh\u03c1 : \u2191\u2191\u03c1 {x | x.snd \u2208 s\u1d9c} = 0\nh : (fun b => \u2191\u2191(\u2191(condKernelReal \u03c1) b) {c | \u00acc \u2208 s}) =\u1d50[Measure.fst \u03c1] 0\na : \u03b1\nha : \u2191\u2191(\u2191(condKernelReal \u03c1) a) s\u1d9c = 0\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) s = 1\n[PROOFSTEP]\nrwa [prob_compl_eq_zero_iff hs] at ha \n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7\n[PROOFSTEP]\nobtain \u27e8f, hf\u27e9 := exists_measurableEmbedding_real \u03a9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7\n[PROOFSTEP]\nlet \u03c1' : Measure (\u03b1 \u00d7 \u211d) :=\n \u03c1.map\n (Prod.map id f)\n -- The general idea is to define `\u03b7 = kernel.comapRight (condKernelReal \u03c1') hf`. There is\n -- however an issue: that `\u03b7` may not be a Markov kernel since its value is only a\n -- probability distribution almost everywhere with respect to `\u03c1.fst`, not everywhere.\n -- We modify it to obtain a Markov kernel which is almost everywhere equal.\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7\n[PROOFSTEP]\nlet \u03c1_set := (toMeasurable \u03c1.fst {a | condKernelReal \u03c1' a (range f) = 1}\u1d9c)\u1d9c\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7\n[PROOFSTEP]\nhave hm : MeasurableSet \u03c1_set := (measurableSet_toMeasurable _ _).compl\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7\n[PROOFSTEP]\nhave h_eq_one_of_mem : \u2200 a \u2208 \u03c1_set, condKernelReal \u03c1' a (range f) = 1 :=\n by\n intro a ha\n rw [mem_compl_iff] at ha \n have h_ss := subset_toMeasurable \u03c1.fst {a : \u03b1 | condKernelReal \u03c1' a (range f) = 1}\u1d9c\n suffices ha' : a \u2209 {a : \u03b1 | condKernelReal \u03c1' a (range f) = 1}\u1d9c\n \u00b7 rwa [not_mem_compl_iff] at ha' \n exact not_mem_subset h_ss ha\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\n\u22a2 \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\n[PROOFSTEP]\nintro a ha\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\na : \u03b1\nha : a \u2208 \u03c1_set\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\n[PROOFSTEP]\nrw [mem_compl_iff] at ha \n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\na : \u03b1\nha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\n[PROOFSTEP]\nhave h_ss := subset_toMeasurable \u03c1.fst {a : \u03b1 | condKernelReal \u03c1' a (range f) = 1}\u1d9c\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\na : \u03b1\nha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\nh_ss :\n {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u2286\n toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\n[PROOFSTEP]\nsuffices ha' : a \u2209 {a : \u03b1 | condKernelReal \u03c1' a (range f) = 1}\u1d9c\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\na : \u03b1\nha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\nh_ss :\n {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u2286\n toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\nha' : \u00aca \u2208 {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\n[PROOFSTEP]\nrwa [not_mem_compl_iff] at ha' \n[GOAL]\ncase ha'\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\na : \u03b1\nha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\nh_ss :\n {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u2286\n toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\n\u22a2 \u00aca \u2208 {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c\n[PROOFSTEP]\nexact not_mem_subset h_ss ha\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7\n[PROOFSTEP]\nhave h_prod_embed : MeasurableEmbedding (Prod.map (id : \u03b1 \u2192 \u03b1) f) := MeasurableEmbedding.id.prod_mk hf\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7\n[PROOFSTEP]\nhave h_fst : \u03c1'.fst = \u03c1.fst := by\n ext1 u hu\n rw [Measure.fst_apply hu, Measure.fst_apply hu, Measure.map_apply h_prod_embed.measurable (measurable_fst hu)]\n rfl\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\n\u22a2 Measure.fst \u03c1' = Measure.fst \u03c1\n[PROOFSTEP]\next1 u hu\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(Measure.fst \u03c1') u = \u2191\u2191(Measure.fst \u03c1) u\n[PROOFSTEP]\nrw [Measure.fst_apply hu, Measure.fst_apply hu, Measure.map_apply h_prod_embed.measurable (measurable_fst hu)]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' (Prod.fst \u207b\u00b9' u)) = \u2191\u2191\u03c1 (Prod.fst \u207b\u00b9' u)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7\n[PROOFSTEP]\nhave h_ae : \u2200\u1d50 a \u2202\u03c1.fst, a \u2208 \u03c1_set := by\n rw [ae_iff]\n simp only [not_mem_compl_iff, setOf_mem_eq, measure_toMeasurable]\n change \u03c1.fst {a : \u03b1 | a \u2209 {a' : \u03b1 | condKernelReal \u03c1' a' (range f) = 1}} = 0\n rw [\u2190 ae_iff, \u2190 h_fst]\n refine' ae_condKernelReal_eq_one \u03c1' hf.measurableSet_range _\n rw [Measure.map_apply h_prod_embed.measurable]\n swap; \u00b7 exact measurable_snd hf.measurableSet_range.compl\n convert measure_empty (\u03b1 := \u03b1 \u00d7 \u03a9)\n ext1 x\n simp only [mem_compl_iff, mem_range, preimage_setOf_eq, Prod_map, mem_setOf_eq, mem_empty_iff_false, iff_false_iff,\n Classical.not_not, exists_apply_eq_apply]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\n[PROOFSTEP]\nrw [ae_iff]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191(Measure.fst \u03c1) {a | \u00aca \u2208 \u03c1_set} = 0\n[PROOFSTEP]\nsimp only [not_mem_compl_iff, setOf_mem_eq, measure_toMeasurable]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191(Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c = 0\n[PROOFSTEP]\nchange \u03c1.fst {a : \u03b1 | a \u2209 {a' : \u03b1 | condKernelReal \u03c1' a' (range f) = 1}} = 0\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191(Measure.fst \u03c1) {a | \u00aca \u2208 {a' | \u2191\u2191(\u2191(condKernelReal \u03c1') a') (range f) = 1}} = 0\n[PROOFSTEP]\nrw [\u2190 ae_iff, \u2190 h_fst]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1', a \u2208 {a' | \u2191\u2191(\u2191(condKernelReal \u03c1') a') (range f) = 1}\n[PROOFSTEP]\nrefine' ae_condKernelReal_eq_one \u03c1' hf.measurableSet_range _\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191\u03c1' {x | x.snd \u2208 (range f)\u1d9c} = 0\n[PROOFSTEP]\nrw [Measure.map_apply h_prod_embed.measurable]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' {x | x.snd \u2208 (range f)\u1d9c}) = 0\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 MeasurableSet {x | x.snd \u2208 (range f)\u1d9c}\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 MeasurableSet {x | x.snd \u2208 (range f)\u1d9c}\n[PROOFSTEP]\nexact measurable_snd hf.measurableSet_range.compl\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' {x | x.snd \u2208 (range f)\u1d9c}) = 0\n[PROOFSTEP]\nconvert measure_empty (\u03b1 := \u03b1 \u00d7 \u03a9)\n[GOAL]\ncase h.e'_2.h.e'_3\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\n\u22a2 Prod.map id f \u207b\u00b9' {x | x.snd \u2208 (range f)\u1d9c} = \u2205\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h.e'_2.h.e'_3.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nx : \u03b1 \u00d7 \u03a9\n\u22a2 x \u2208 Prod.map id f \u207b\u00b9' {x | x.snd \u2208 (range f)\u1d9c} \u2194 x \u2208 \u2205\n[PROOFSTEP]\nsimp only [mem_compl_iff, mem_range, preimage_setOf_eq, Prod_map, mem_setOf_eq, mem_empty_iff_false, iff_false_iff,\n Classical.not_not, exists_apply_eq_apply]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7\n[PROOFSTEP]\nclassical\nobtain \u27e8x\u2080, hx\u2080\u27e9 : \u2203 x, x \u2208 range f := range_nonempty _\nlet \u03b7' :=\n kernel.piecewise hm (condKernelReal \u03c1')\n (kernel.deterministic (fun _ => x\u2080) measurable_const)\n -- We show that `kernel.comapRight \u03b7' hf` is a suitable Markov kernel.\nrefine' \u27e8kernel.comapRight \u03b7' hf, _, _\u27e9\n\u00b7 refine' kernel.IsMarkovKernel.comapRight _ _ fun a => _\n rw [kernel.piecewise_apply']\n split_ifs with h_mem\n \u00b7 exact h_eq_one_of_mem _ h_mem\n \u00b7 rw [kernel.deterministic_apply' _ _ hf.measurableSet_range, Set.indicator_apply, if_pos hx\u2080]\nhave : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed :=\n by\n ext c t ht : 2\n rw [kernel.const_apply, kernel.comapRight_apply' _ _ _ ht, kernel.const_apply,\n Measure.map_apply h_prod_embed.measurable (h_prod_embed.measurableSet_image.mpr ht)]\n congr with x : 1\n rw [\u2190 @Prod.mk.eta _ _ x]\n simp only [id.def, mem_preimage, Prod.map_mk, mem_image, Prod.mk.inj_iff, Prod.exists]\n refine' \u27e8fun h => \u27e8x.1, x.2, h, rfl, rfl\u27e9, _\u27e9\n rintro \u27e8a, b, h_mem, rfl, hf_eq\u27e9\n rwa [hf.injective hf_eq] at h_mem \nrw [this, kernel.const_eq_compProd_real _ \u03c1']\next c t ht : 2\nrw [kernel.comapRight_apply' _ _ _ ht, kernel.compProd_apply _ _ _ (h_prod_embed.measurableSet_image.mpr ht),\n kernel.const_apply, h_fst, kernel.compProd_apply _ _ _ ht, kernel.const_apply]\nrefine' lintegral_congr_ae _\nfilter_upwards [h_ae] with a ha\nrw [kernel.prodMkLeft_apply', kernel.prodMkLeft_apply', kernel.comapRight_apply']\nswap\n\u00b7 exact measurable_prod_mk_left ht\nhave h1 : {c : \u211d | (a, c) \u2208 Prod.map id f '' t} = f '' {c : \u03a9 | (a, c) \u2208 t} :=\n by\n ext1 x\n simp only [Prod_map, id.def, mem_image, Prod.mk.inj_iff, Prod.exists, mem_setOf_eq]\n constructor\n \u00b7 rintro \u27e8a', b, h_mem, rfl, hf_eq\u27e9\n exact \u27e8b, h_mem, hf_eq\u27e9\n \u00b7 rintro \u27e8b, h_mem, hf_eq\u27e9\n exact \u27e8a, b, h_mem, rfl, hf_eq\u27e9\nhave h2 : condKernelReal \u03c1' (c, a).snd = \u03b7' (c, a).snd := by rw [kernel.piecewise_apply, if_pos ha]\nrw [h1, h2]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7\n[PROOFSTEP]\nobtain \u27e8x\u2080, hx\u2080\u27e9 : \u2203 x, x \u2208 range f := range_nonempty _\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7\n[PROOFSTEP]\nlet \u03b7' :=\n kernel.piecewise hm (condKernelReal \u03c1')\n (kernel.deterministic (fun _ => x\u2080) measurable_const)\n -- We show that `kernel.comapRight \u03b7' hf` is a suitable Markov kernel.\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\n\u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7\n[PROOFSTEP]\nrefine' \u27e8kernel.comapRight \u03b7' hf, _, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\n\u22a2 IsMarkovKernel (kernel.comapRight \u03b7' hf)\n[PROOFSTEP]\nrefine' kernel.IsMarkovKernel.comapRight _ _ fun a => _\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\na : \u03b1\n\u22a2 \u2191\u2191(\u2191\u03b7' a) (range f) = 1\n[PROOFSTEP]\nrw [kernel.piecewise_apply']\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\na : \u03b1\n\u22a2 (if a \u2208 \u03c1_set then \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f)\n else \u2191\u2191(\u2191(kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) a) (range f)) =\n 1\n[PROOFSTEP]\nsplit_ifs with h_mem\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\na : \u03b1\nh_mem : a \u2208 \u03c1_set\n\u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\n[PROOFSTEP]\nexact h_eq_one_of_mem _ h_mem\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\na : \u03b1\nh_mem : \u00aca \u2208 \u03c1_set\n\u22a2 \u2191\u2191(\u2191(kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) a) (range f) = 1\n[PROOFSTEP]\nrw [kernel.deterministic_apply' _ _ hf.measurableSet_range, Set.indicator_apply, if_pos hx\u2080]\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\n\u22a2 kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)\n[PROOFSTEP]\nhave : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed :=\n by\n ext c t ht : 2\n rw [kernel.const_apply, kernel.comapRight_apply' _ _ _ ht, kernel.const_apply,\n Measure.map_apply h_prod_embed.measurable (h_prod_embed.measurableSet_image.mpr ht)]\n congr with x : 1\n rw [\u2190 @Prod.mk.eta _ _ x]\n simp only [id.def, mem_preimage, Prod.map_mk, mem_image, Prod.mk.inj_iff, Prod.exists]\n refine' \u27e8fun h => \u27e8x.1, x.2, h, rfl, rfl\u27e9, _\u27e9\n rintro \u27e8a, b, h_mem, rfl, hf_eq\u27e9\n rwa [hf.injective hf_eq] at h_mem \n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\n\u22a2 kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\n[PROOFSTEP]\next c t ht : 2\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\n\u22a2 \u2191\u2191(\u2191(kernel.const \u03b3 \u03c1) c) t = \u2191\u2191(\u2191(kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed) c) t\n[PROOFSTEP]\nrw [kernel.const_apply, kernel.comapRight_apply' _ _ _ ht, kernel.const_apply,\n Measure.map_apply h_prod_embed.measurable (h_prod_embed.measurableSet_image.mpr ht)]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03c1 t = \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' (Prod.map id f '' t))\n[PROOFSTEP]\ncongr with x : 1\n[GOAL]\ncase h.h.e_a.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u03b1 \u00d7 \u03a9\n\u22a2 x \u2208 t \u2194 x \u2208 Prod.map id f \u207b\u00b9' (Prod.map id f '' t)\n[PROOFSTEP]\nrw [\u2190 @Prod.mk.eta _ _ x]\n[GOAL]\ncase h.h.e_a.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u03b1 \u00d7 \u03a9\n\u22a2 (x.fst, x.snd) \u2208 t \u2194 (x.fst, x.snd) \u2208 Prod.map id f \u207b\u00b9' (Prod.map id f '' t)\n[PROOFSTEP]\nsimp only [id.def, mem_preimage, Prod.map_mk, mem_image, Prod.mk.inj_iff, Prod.exists]\n[GOAL]\ncase h.h.e_a.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u03b1 \u00d7 \u03a9\n\u22a2 (x.fst, x.snd) \u2208 t \u2194 \u2203 a b, (a, b) \u2208 t \u2227 a = x.fst \u2227 f b = f x.snd\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8x.1, x.2, h, rfl, rfl\u27e9, _\u27e9\n[GOAL]\ncase h.h.e_a.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u03b1 \u00d7 \u03a9\n\u22a2 (\u2203 a b, (a, b) \u2208 t \u2227 a = x.fst \u2227 f b = f x.snd) \u2192 (x.fst, x.snd) \u2208 t\n[PROOFSTEP]\nrintro \u27e8a, b, h_mem, rfl, hf_eq\u27e9\n[GOAL]\ncase h.h.e_a.h.intro.intro.intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u03b1 \u00d7 \u03a9\nb : \u03a9\nhf_eq : f b = f x.snd\nh_mem : (x.fst, b) \u2208 t\n\u22a2 (x.fst, x.snd) \u2208 t\n[PROOFSTEP]\nrwa [hf.injective hf_eq] at h_mem \n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\n\u22a2 kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)\n[PROOFSTEP]\nrw [this, kernel.const_eq_compProd_real _ \u03c1']\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\n\u22a2 kernel.comapRight (kernel.const \u03b3 (Measure.fst \u03c1') \u2297\u2096 kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) h_prod_embed =\n kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)\n[PROOFSTEP]\next c t ht : 2\n[GOAL]\ncase intro.intro.refine'_2.h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\n\u22a2 \u2191\u2191(\u2191(kernel.comapRight (kernel.const \u03b3 (Measure.fst \u03c1') \u2297\u2096 kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) h_prod_embed) c)\n t =\n \u2191\u2191(\u2191(kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)) c) t\n[PROOFSTEP]\nrw [kernel.comapRight_apply' _ _ _ ht, kernel.compProd_apply _ _ _ (h_prod_embed.measurableSet_image.mpr ht),\n kernel.const_apply, h_fst, kernel.compProd_apply _ _ _ ht, kernel.const_apply]\n[GOAL]\ncase intro.intro.refine'_2.h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\n\u22a2 \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) (c, b)) {c | (b, c) \u2208 Prod.map id f '' t} \u2202Measure.fst \u03c1 =\n \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)) (c, b)) {c | (b, c) \u2208 t} \u2202Measure.fst \u03c1\n[PROOFSTEP]\nrefine' lintegral_congr_ae _\n[GOAL]\ncase intro.intro.refine'_2.h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\n\u22a2 (fun b => \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) (c, b)) {c | (b, c) \u2208 Prod.map id f '' t}) =\u1d50[Measure.fst \u03c1]\n fun b => \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)) (c, b)) {c | (b, c) \u2208 t}\n[PROOFSTEP]\nfilter_upwards [h_ae] with a ha\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c)\u1d9c\n\u22a2 \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (condKernelReal (Measure.map (Prod.map id f) \u03c1))) (c, a)) {c | (a, c) \u2208 Prod.map id f '' t} =\n \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3\n (kernel.comapRight\n (kernel.piecewise hm (condKernelReal (Measure.map (Prod.map id f) \u03c1))\n (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)))\n hf))\n (c, a))\n {c | (a, c) \u2208 t}\n[PROOFSTEP]\nrw [kernel.prodMkLeft_apply', kernel.prodMkLeft_apply', kernel.comapRight_apply']\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c)\u1d9c\n\u22a2 \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) (c, a).snd) {c | (a, c) \u2208 Prod.map id f '' t} =\n \u2191\u2191(\u2191(kernel.piecewise hm (condKernelReal (Measure.map (Prod.map id f) \u03c1))\n (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)))\n (c, a).snd)\n (f '' {c | (a, c) \u2208 t})\ncase h.ht\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c)\u1d9c\n\u22a2 MeasurableSet {c | (a, c) \u2208 t}\n[PROOFSTEP]\nswap\n[GOAL]\ncase h.ht\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c)\u1d9c\n\u22a2 MeasurableSet {c | (a, c) \u2208 t}\n[PROOFSTEP]\nexact measurable_prod_mk_left ht\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c)\u1d9c\n\u22a2 \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) (c, a).snd) {c | (a, c) \u2208 Prod.map id f '' t} =\n \u2191\u2191(\u2191(kernel.piecewise hm (condKernelReal (Measure.map (Prod.map id f) \u03c1))\n (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)))\n (c, a).snd)\n (f '' {c | (a, c) \u2208 t})\n[PROOFSTEP]\nhave h1 : {c : \u211d | (a, c) \u2208 Prod.map id f '' t} = f '' {c : \u03a9 | (a, c) \u2208 t} :=\n by\n ext1 x\n simp only [Prod_map, id.def, mem_image, Prod.mk.inj_iff, Prod.exists, mem_setOf_eq]\n constructor\n \u00b7 rintro \u27e8a', b, h_mem, rfl, hf_eq\u27e9\n exact \u27e8b, h_mem, hf_eq\u27e9\n \u00b7 rintro \u27e8b, h_mem, hf_eq\u27e9\n exact \u27e8a, b, h_mem, rfl, hf_eq\u27e9\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c)\u1d9c\n\u22a2 {c | (a, c) \u2208 Prod.map id f '' t} = f '' {c | (a, c) \u2208 t}\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c)\u1d9c\nx : \u211d\n\u22a2 x \u2208 {c | (a, c) \u2208 Prod.map id f '' t} \u2194 x \u2208 f '' {c | (a, c) \u2208 t}\n[PROOFSTEP]\nsimp only [Prod_map, id.def, mem_image, Prod.mk.inj_iff, Prod.exists, mem_setOf_eq]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c)\u1d9c\nx : \u211d\n\u22a2 (\u2203 a_1 b, (a_1, b) \u2208 t \u2227 a_1 = a \u2227 f b = x) \u2194 \u2203 x_1, (a, x_1) \u2208 t \u2227 f x_1 = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c)\u1d9c\nx : \u211d\n\u22a2 (\u2203 a_1 b, (a_1, b) \u2208 t \u2227 a_1 = a \u2227 f b = x) \u2192 \u2203 x_1, (a, x_1) \u2208 t \u2227 f x_1 = x\n[PROOFSTEP]\nrintro \u27e8a', b, h_mem, rfl, hf_eq\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\nx : \u211d\na' : \u03b1\nb : \u03a9\nh_mem : (a', b) \u2208 t\nhf_eq : f b = x\nha : a' \u2208 (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c)\u1d9c\n\u22a2 \u2203 x_1, (a', x_1) \u2208 t \u2227 f x_1 = x\n[PROOFSTEP]\nexact \u27e8b, h_mem, hf_eq\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c)\u1d9c\nx : \u211d\n\u22a2 (\u2203 x_1, (a, x_1) \u2208 t \u2227 f x_1 = x) \u2192 \u2203 a_2 b, (a_2, b) \u2208 t \u2227 a_2 = a \u2227 f b = x\n[PROOFSTEP]\nrintro \u27e8b, h_mem, hf_eq\u27e9\n[GOAL]\ncase h.mpr.intro.intro\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c)\u1d9c\nx : \u211d\nb : \u03a9\nh_mem : (a, b) \u2208 t\nhf_eq : f b = x\n\u22a2 \u2203 a_1 b, (a_1, b) \u2208 t \u2227 a_1 = a \u2227 f b = x\n[PROOFSTEP]\nexact \u27e8a, b, h_mem, rfl, hf_eq\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c)\u1d9c\nh1 : {c | (a, c) \u2208 Prod.map id f '' t} = f '' {c | (a, c) \u2208 t}\n\u22a2 \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) (c, a).snd) {c | (a, c) \u2208 Prod.map id f '' t} =\n \u2191\u2191(\u2191(kernel.piecewise hm (condKernelReal (Measure.map (Prod.map id f) \u03c1))\n (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)))\n (c, a).snd)\n (f '' {c | (a, c) \u2208 t})\n[PROOFSTEP]\nhave h2 : condKernelReal \u03c1' (c, a).snd = \u03b7' (c, a).snd := by rw [kernel.piecewise_apply, if_pos ha]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c)\u1d9c\nh1 : {c | (a, c) \u2208 Prod.map id f '' t} = f '' {c | (a, c) \u2208 t}\n\u22a2 \u2191(condKernelReal \u03c1') (c, a).snd = \u2191\u03b7' (c, a).snd\n[PROOFSTEP]\nrw [kernel.piecewise_apply, if_pos ha]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2076 : TopologicalSpace \u03a9\ninst\u271d\u2075 : PolishSpace \u03a9\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : BorelSpace \u03a9\ninst\u271d\u00b2 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b9 : IsFiniteMeasure \u03c1\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b3\nf : \u03a9 \u2192 \u211d\nhf : MeasurableEmbedding f\n\u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1\n\u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c\nhm : MeasurableSet \u03c1_set\nh_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1\nh_prod_embed : MeasurableEmbedding (Prod.map id f)\nh_fst : Measure.fst \u03c1' = Measure.fst \u03c1\nh_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set\nx\u2080 : \u211d\nhx\u2080 : x\u2080 \u2208 range f\n\u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=\n kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080))\nthis : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed\nc : \u03b3\nt : Set (\u03b1 \u00d7 \u03a9)\nht : MeasurableSet t\na : \u03b1\nha : a \u2208 (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c)\u1d9c\nh1 : {c | (a, c) \u2208 Prod.map id f '' t} = f '' {c | (a, c) \u2208 t}\nh2 : \u2191(condKernelReal \u03c1') (c, a).snd = \u2191\u03b7' (c, a).snd\n\u22a2 \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) (c, a).snd) {c | (a, c) \u2208 Prod.map id f '' t} =\n \u2191\u2191(\u2191(kernel.piecewise hm (condKernelReal (Measure.map (Prod.map id f) \u03c1))\n (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)))\n (c, a).snd)\n (f '' {c | (a, c) \u2208 t})\n[PROOFSTEP]\nrw [h1, h2]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 Measure.condKernel \u03c1 =\n Exists.choose (_ : \u2203 \u03b7 _h, kernel.const Unit \u03c1 = kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03b7)\n[PROOFSTEP]\nrw [MeasureTheory.Measure.condKernel]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 IsMarkovKernel (Measure.condKernel \u03c1)\n[PROOFSTEP]\nrw [condKernel_def]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 IsMarkovKernel\n (Exists.choose (_ : \u2203 \u03b7 _h, kernel.const Unit \u03c1 = kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03b7))\n[PROOFSTEP]\nexact (exists_cond_kernel \u03c1 Unit).choose_spec.choose\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 const Unit \u03c1 = const Unit (Measure.fst \u03c1) \u2297\u2096 prodMkLeft Unit (Measure.condKernel \u03c1)\n[PROOFSTEP]\nsimp_rw [condKernel_def]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 const Unit \u03c1 =\n const Unit (Measure.fst \u03c1) \u2297\u2096\n prodMkLeft Unit (Exists.choose (_ : \u2203 \u03b7 _h, const Unit \u03c1 = const Unit (Measure.fst \u03c1) \u2297\u2096 prodMkLeft Unit \u03b7))\n[PROOFSTEP]\nexact (exists_cond_kernel \u03c1 Unit).choose_spec.choose_spec\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()\n[PROOFSTEP]\nrw [\u2190 kernel.const_unit_eq_compProd, kernel.const_apply]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d\n\u03b3 : Type u_3\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 const \u03b3 \u03c1 = const \u03b3 (Measure.fst \u03c1) \u2297\u2096 prodMkLeft \u03b3 (Measure.condKernel \u03c1)\n[PROOFSTEP]\next a s hs : 2\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2077 : TopologicalSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : MeasurableSpace \u03a9\ninst\u271d\u2074 : BorelSpace \u03a9\ninst\u271d\u00b3 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d\n\u03b3 : Type u_3\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b3\ns : Set (\u03b1 \u00d7 \u03a9)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(const \u03b3 \u03c1) a) s = \u2191\u2191(\u2191(const \u03b3 (Measure.fst \u03c1) \u2297\u2096 prodMkLeft \u03b3 (Measure.condKernel \u03c1)) a) s\n[PROOFSTEP]\nsimpa only [kernel.const_apply, kernel.compProd_apply _ _ _ hs, kernel.prodMkLeft_apply'] using\n kernel.ext_iff'.mp (kernel.const_unit_eq_compProd \u03c1) () s hs\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u03a9)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(Measure.condKernel \u03c1) a) {x | (a, x) \u2208 s} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 s\n[PROOFSTEP]\nconv_rhs => rw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u03a9)\nhs : MeasurableSet s\n| \u2191\u2191\u03c1 s\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u03a9)\nhs : MeasurableSet s\n| \u2191\u2191\u03c1 s\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u03a9)\nhs : MeasurableSet s\n| \u2191\u2191\u03c1 s\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set (\u03b1 \u00d7 \u03a9)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(Measure.condKernel \u03c1) a) {x | (a, x) \u2208 s} \u2202Measure.fst \u03c1 =\n \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()) s\n[PROOFSTEP]\nsimp_rw [kernel.compProd_apply _ _ _ hs, kernel.const_apply, kernel.prodMkLeft_apply]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(Measure.condKernel \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n[PROOFSTEP]\nhave : \u03c1 (s \u00d7\u02e2 t) = ((kernel.const Unit \u03c1.fst \u2297\u2096 kernel.prodMkLeft Unit \u03c1.condKernel) ()) (s \u00d7\u02e2 t) := by congr;\n exact measure_eq_compProd \u03c1\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 t) = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()) (s \u00d7\u02e2 t)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_self.e_self\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\n\u22a2 \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()\n[PROOFSTEP]\nexact measure_eq_compProd \u03c1\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nthis :\n \u2191\u2191\u03c1 (s \u00d7\u02e2 t) = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()) (s \u00d7\u02e2 t)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(Measure.condKernel \u03c1) a) t \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 t)\n[PROOFSTEP]\nrw [this, kernel.compProd_apply _ _ _ (hs.prod ht)]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nthis :\n \u2191\u2191\u03c1 (s \u00d7\u02e2 t) = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()) (s \u00d7\u02e2 t)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(Measure.condKernel \u03c1) a) t \u2202Measure.fst \u03c1 =\n \u222b\u207b (b : \u03b1),\n \u2191\u2191(\u2191(kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ((), b))\n {c | (b, c) \u2208 s \u00d7\u02e2 t} \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) ()\n[PROOFSTEP]\nsimp only [prod_mk_mem_set_prod_eq, kernel.lintegral_const, kernel.prodMkLeft_apply]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nthis :\n \u2191\u2191\u03c1 (s \u00d7\u02e2 t) = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()) (s \u00d7\u02e2 t)\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(Measure.condKernel \u03c1) a) t \u2202Measure.fst \u03c1 =\n \u222b\u207b (x : \u03b1), \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | x \u2208 s \u2227 c \u2208 t} \u2202Measure.fst \u03c1\n[PROOFSTEP]\nrw [\u2190 lintegral_indicator _ hs]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nthis :\n \u2191\u2191\u03c1 (s \u00d7\u02e2 t) = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()) (s \u00d7\u02e2 t)\n\u22a2 \u222b\u207b (a : \u03b1), indicator s (fun a => \u2191\u2191(\u2191(Measure.condKernel \u03c1) a) t) a \u2202Measure.fst \u03c1 =\n \u222b\u207b (x : \u03b1), \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | x \u2208 s \u2227 c \u2208 t} \u2202Measure.fst \u03c1\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nthis :\n \u2191\u2191\u03c1 (s \u00d7\u02e2 t) = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()) (s \u00d7\u02e2 t)\n\u22a2 (fun a => indicator s (fun a => \u2191\u2191(\u2191(Measure.condKernel \u03c1) a) t) a) = fun x =>\n \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | x \u2208 s \u2227 c \u2208 t}\n[PROOFSTEP]\next1 x\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nthis :\n \u2191\u2191\u03c1 (s \u00d7\u02e2 t) = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()) (s \u00d7\u02e2 t)\nx : \u03b1\n\u22a2 indicator s (fun a => \u2191\u2191(\u2191(Measure.condKernel \u03c1) a) t) x = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | x \u2208 s \u2227 c \u2208 t}\n[PROOFSTEP]\nclassical\nrw [indicator_apply]\nsplit_ifs with hx\n\u00b7 simp only [hx, if_true, true_and_iff, setOf_mem_eq]\n\u00b7 simp only [hx, if_false, false_and_iff, setOf_false, measure_empty]\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nthis :\n \u2191\u2191\u03c1 (s \u00d7\u02e2 t) = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()) (s \u00d7\u02e2 t)\nx : \u03b1\n\u22a2 indicator s (fun a => \u2191\u2191(\u2191(Measure.condKernel \u03c1) a) t) x = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | x \u2208 s \u2227 c \u2208 t}\n[PROOFSTEP]\nrw [indicator_apply]\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nthis :\n \u2191\u2191\u03c1 (s \u00d7\u02e2 t) = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()) (s \u00d7\u02e2 t)\nx : \u03b1\n\u22a2 (if x \u2208 s then \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) t else 0) = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | x \u2208 s \u2227 c \u2208 t}\n[PROOFSTEP]\nsplit_ifs with hx\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nthis :\n \u2191\u2191\u03c1 (s \u00d7\u02e2 t) = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()) (s \u00d7\u02e2 t)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) t = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | x \u2208 s \u2227 c \u2208 t}\n[PROOFSTEP]\nsimp only [hx, if_true, true_and_iff, setOf_mem_eq]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nthis :\n \u2191\u2191\u03c1 (s \u00d7\u02e2 t) = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()) (s \u00d7\u02e2 t)\nx : \u03b1\nhx : \u00acx \u2208 s\n\u22a2 0 = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | x \u2208 s \u2227 c \u2208 t}\n[PROOFSTEP]\nsimp only [hx, if_false, false_and_iff, setOf_false, measure_empty]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (\u03c9 : \u03a9), f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1 \u00d7 \u03a9), f x \u2202\u03c1\n[PROOFSTEP]\nconv_rhs => rw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n| \u222b\u207b (x : \u03b1 \u00d7 \u03a9), f x \u2202\u03c1\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n| \u222b\u207b (x : \u03b1 \u00d7 \u03a9), f x \u2202\u03c1\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n| \u222b\u207b (x : \u03b1 \u00d7 \u03a9), f x \u2202\u03c1\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (\u03c9 : \u03a9), f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b\u207b (x : \u03b1 \u00d7 \u03a9), f x \u2202\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()\n[PROOFSTEP]\nrw [kernel.lintegral_compProd _ _ _ hf, kernel.const_apply]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (\u03c9 : \u03a9), f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b\u207b (b : \u03b1), \u222b\u207b (c : \u03a9), f (b, c) \u2202\u2191(kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ((), b) \u2202Measure.fst \u03c1\n[PROOFSTEP]\nsimp_rw [kernel.prodMkLeft_apply]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u222b\u207b (\u03c9 : \u03a9) in t, f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b\u207b (x : \u03b1 \u00d7 \u03a9) in s \u00d7\u02e2 t, f x \u2202\u03c1\n[PROOFSTEP]\nconv_rhs => rw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\n| \u222b\u207b (x : \u03b1 \u00d7 \u03a9) in s \u00d7\u02e2 t, f x \u2202\u03c1\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\n| \u222b\u207b (x : \u03b1 \u00d7 \u03a9) in s \u00d7\u02e2 t, f x \u2202\u03c1\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\n| \u222b\u207b (x : \u03b1 \u00d7 \u03a9) in s \u00d7\u02e2 t, f x \u2202\u03c1\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u222b\u207b (\u03c9 : \u03a9) in t, f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b\u207b (x : \u03b1 \u00d7 \u03a9) in s \u00d7\u02e2 t,\n f x \u2202\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()\n[PROOFSTEP]\nrw [\u2190 kernel.restrict_apply _ (hs.prod ht), \u2190 kernel.compProd_restrict hs ht, kernel.lintegral_compProd _ _ _ hf,\n kernel.restrict_apply]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u222b\u207b (\u03c9 : \u03a9) in t, f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b\u207b (b : \u03b1) in s,\n \u222b\u207b (c : \u03a9),\n f\n (b,\n c) \u2202\u2191(kernel.restrict (kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ht)\n ((), b) \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) ()\n[PROOFSTEP]\nconv_rhs => enter [2, b, 1]; rw [kernel.restrict_apply _ ht]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\n| \u222b\u207b (b : \u03b1) in s,\n \u222b\u207b (c : \u03a9),\n f\n (b,\n c) \u2202\u2191(kernel.restrict (kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ht)\n ((), b) \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) ()\n[PROOFSTEP]\nenter [2, b, 1]; rw [kernel.restrict_apply _ ht]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\n| \u222b\u207b (b : \u03b1) in s,\n \u222b\u207b (c : \u03a9),\n f\n (b,\n c) \u2202\u2191(kernel.restrict (kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ht)\n ((), b) \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) ()\n[PROOFSTEP]\nenter [2, b, 1]; rw [kernel.restrict_apply _ ht]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\n| \u222b\u207b (b : \u03b1) in s,\n \u222b\u207b (c : \u03a9),\n f\n (b,\n c) \u2202\u2191(kernel.restrict (kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ht)\n ((), b) \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) ()\n[PROOFSTEP]\nenter [2, b, 1]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nb : \u03b1\n| \u2191(kernel.restrict (kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ht) ((), b)\n[PROOFSTEP]\nrw [kernel.restrict_apply _ ht]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u222b\u207b (\u03c9 : \u03a9), f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1 \u00d7 \u03a9) in s \u00d7\u02e2 univ, f x \u2202\u03c1\n[PROOFSTEP]\nrw [\u2190 set_lintegral_condKernel \u03c1 hf hs MeasurableSet.univ]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u222b\u207b (\u03c9 : \u03a9), f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, \u222b\u207b (\u03c9 : \u03a9) in univ, f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1\n[PROOFSTEP]\nsimp_rw [Measure.restrict_univ]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nt : Set \u03a9\nht : MeasurableSet t\n\u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (\u03c9 : \u03a9) in t, f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1 \u00d7 \u03a9) in univ \u00d7\u02e2 t, f x \u2202\u03c1\n[PROOFSTEP]\nrw [\u2190 set_lintegral_condKernel \u03c1 hf MeasurableSet.univ ht]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : PolishSpace \u03a9\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : BorelSpace \u03a9\ninst\u271d\u00b9 : Nonempty \u03a9\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nt : Set \u03a9\nht : MeasurableSet t\n\u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (\u03c9 : \u03a9) in t, f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in univ, \u222b\u207b (\u03c9 : \u03a9) in t, f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1\n[PROOFSTEP]\nsimp_rw [Measure.restrict_univ]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf : AEStronglyMeasurable f \u03c1\n\u22a2 AEStronglyMeasurable (fun x => \u222b (y : \u03a9), f (x, y) \u2202\u2191(Measure.condKernel \u03c1) x) (Measure.fst \u03c1)\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1] at hf \n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf : AEStronglyMeasurable f (\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ())\n\u22a2 AEStronglyMeasurable (fun x => \u222b (y : \u03a9), f (x, y) \u2202\u2191(Measure.condKernel \u03c1) x) (Measure.fst \u03c1)\n[PROOFSTEP]\nexact AEStronglyMeasurable.integral_kernel_compProd hf\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf : Integrable f\n\u22a2 \u222b (a : \u03b1), \u222b (x : \u03a9), f (a, x) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 = \u222b (\u03c9 : \u03b1 \u00d7 \u03a9), f \u03c9 \u2202\u03c1\n[PROOFSTEP]\nconv_rhs => rw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf : Integrable f\n| \u222b (\u03c9 : \u03b1 \u00d7 \u03a9), f \u03c9 \u2202\u03c1\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf : Integrable f\n| \u222b (\u03c9 : \u03b1 \u00d7 \u03a9), f \u03c9 \u2202\u03c1\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf : Integrable f\n| \u222b (\u03c9 : \u03b1 \u00d7 \u03a9), f \u03c9 \u2202\u03c1\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf : Integrable f\n\u22a2 \u222b (a : \u03b1), \u222b (x : \u03a9), f (a, x) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b (\u03c9 : \u03b1 \u00d7 \u03a9), f \u03c9 \u2202\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()\n[PROOFSTEP]\nhave hf' : Integrable f ((kernel.const Unit \u03c1.fst \u2297\u2096 kernel.prodMkLeft Unit \u03c1.condKernel) ()) := by\n rwa [measure_eq_compProd \u03c1] at hf \n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf : Integrable f\n\u22a2 Integrable f\n[PROOFSTEP]\nrwa [measure_eq_compProd \u03c1] at hf \n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf : Integrable f\nhf' : Integrable f\n\u22a2 \u222b (a : \u03b1), \u222b (x : \u03a9), f (a, x) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b (\u03c9 : \u03b1 \u00d7 \u03a9), f \u03c9 \u2202\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()\n[PROOFSTEP]\nrw [integral_compProd hf', kernel.const_apply]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf : Integrable f\nhf' : Integrable f\n\u22a2 \u222b (a : \u03b1), \u222b (x : \u03a9), f (a, x) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b (x : \u03b1), \u222b (y : \u03a9), f (x, y) \u2202\u2191(kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ((), x) \u2202Measure.fst \u03c1\n[PROOFSTEP]\nsimp_rw [kernel.prodMkLeft_apply]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nhf : IntegrableOn f (s \u00d7\u02e2 t)\n\u22a2 \u222b (a : \u03b1) in s, \u222b (\u03c9 : \u03a9) in t, f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 = \u222b (x : \u03b1 \u00d7 \u03a9) in s \u00d7\u02e2 t, f x \u2202\u03c1\n[PROOFSTEP]\nconv_rhs => rw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nhf : IntegrableOn f (s \u00d7\u02e2 t)\n| \u222b (x : \u03b1 \u00d7 \u03a9) in s \u00d7\u02e2 t, f x \u2202\u03c1\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nhf : IntegrableOn f (s \u00d7\u02e2 t)\n| \u222b (x : \u03b1 \u00d7 \u03a9) in s \u00d7\u02e2 t, f x \u2202\u03c1\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nhf : IntegrableOn f (s \u00d7\u02e2 t)\n| \u222b (x : \u03b1 \u00d7 \u03a9) in s \u00d7\u02e2 t, f x \u2202\u03c1\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nhf : IntegrableOn f (s \u00d7\u02e2 t)\n\u22a2 \u222b (a : \u03b1) in s, \u222b (\u03c9 : \u03a9) in t, f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b (x : \u03b1 \u00d7 \u03a9) in s \u00d7\u02e2 t,\n f x \u2202\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ()\n[PROOFSTEP]\nrw [set_integral_compProd hs ht]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nhf : IntegrableOn f (s \u00d7\u02e2 t)\n\u22a2 \u222b (a : \u03b1) in s, \u222b (\u03c9 : \u03a9) in t, f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b (x : \u03b1) in s,\n \u222b (y : \u03a9) in t,\n f (x, y) \u2202\u2191(kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ((), x) \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) ()\n[PROOFSTEP]\nsimp_rw [kernel.prodMkLeft_apply, kernel.const_apply]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03a9\nht : MeasurableSet t\nhf : IntegrableOn f (s \u00d7\u02e2 t)\n\u22a2 IntegrableOn (fun x => f x) (s \u00d7\u02e2 t)\n[PROOFSTEP]\nrwa [measure_eq_compProd \u03c1] at hf \n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : IntegrableOn f (s \u00d7\u02e2 univ)\n\u22a2 \u222b (a : \u03b1) in s, \u222b (\u03c9 : \u03a9), f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 = \u222b (x : \u03b1 \u00d7 \u03a9) in s \u00d7\u02e2 univ, f x \u2202\u03c1\n[PROOFSTEP]\nrw [\u2190 set_integral_condKernel hs MeasurableSet.univ hf]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : IntegrableOn f (s \u00d7\u02e2 univ)\n\u22a2 \u222b (a : \u03b1) in s, \u222b (\u03c9 : \u03a9), f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b (a : \u03b1) in s, \u222b (\u03c9 : \u03a9) in univ, f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1\n[PROOFSTEP]\nsimp_rw [Measure.restrict_univ]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nt : Set \u03a9\nht : MeasurableSet t\nhf : IntegrableOn f (univ \u00d7\u02e2 t)\n\u22a2 \u222b (a : \u03b1), \u222b (\u03c9 : \u03a9) in t, f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 = \u222b (x : \u03b1 \u00d7 \u03a9) in univ \u00d7\u02e2 t, f x \u2202\u03c1\n[PROOFSTEP]\nrw [\u2190 set_integral_condKernel MeasurableSet.univ ht hf]\n[GOAL]\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03a9 : Type u_2\ninst\u271d\u2079 : TopologicalSpace \u03a9\ninst\u271d\u2078 : PolishSpace \u03a9\ninst\u271d\u2077 : MeasurableSpace \u03a9\ninst\u271d\u2076 : BorelSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\n\u03c1\u271d : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d\u2074 : IsFiniteMeasure \u03c1\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nt : Set \u03a9\nht : MeasurableSet t\nhf : IntegrableOn f (univ \u00d7\u02e2 t)\n\u22a2 \u222b (a : \u03b1), \u222b (\u03c9 : \u03a9) in t, f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1 =\n \u222b (a : \u03b1) in univ, \u222b (\u03c9 : \u03a9) in t, f (a, \u03c9) \u2202\u2191(Measure.condKernel \u03c1) a \u2202Measure.fst \u03c1\n[PROOFSTEP]\nsimp_rw [Measure.restrict_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf : AEStronglyMeasurable f \u03c1\n\u22a2 ((\u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Integrable fun \u03c9 => f (a, \u03c9)) \u2227\n Integrable fun a => \u222b (\u03c9 : \u03a9), \u2016f (a, \u03c9)\u2016 \u2202\u2191(Measure.condKernel \u03c1) a) \u2194\n Integrable f\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1] at hf \n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf : AEStronglyMeasurable f (\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ())\n\u22a2 ((\u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Integrable fun \u03c9 => f (a, \u03c9)) \u2227\n Integrable fun a => \u222b (\u03c9 : \u03a9), \u2016f (a, \u03c9)\u2016 \u2202\u2191(Measure.condKernel \u03c1) a) \u2194\n Integrable f\n[PROOFSTEP]\nconv_rhs => rw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf : AEStronglyMeasurable f (\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ())\n| Integrable f\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf : AEStronglyMeasurable f (\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ())\n| Integrable f\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf : AEStronglyMeasurable f (\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ())\n| Integrable f\n[PROOFSTEP]\nrw [measure_eq_compProd \u03c1]\n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf : AEStronglyMeasurable f (\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ())\n\u22a2 ((\u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Integrable fun \u03c9 => f (a, \u03c9)) \u2227\n Integrable fun a => \u222b (\u03c9 : \u03a9), \u2016f (a, \u03c9)\u2016 \u2202\u2191(Measure.condKernel \u03c1) a) \u2194\n Integrable f\n[PROOFSTEP]\nrw [integrable_compProd_iff hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf : AEStronglyMeasurable f (\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ())\n\u22a2 ((\u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Integrable fun \u03c9 => f (a, \u03c9)) \u2227\n Integrable fun a => \u222b (\u03c9 : \u03a9), \u2016f (a, \u03c9)\u2016 \u2202\u2191(Measure.condKernel \u03c1) a) \u2194\n (\u2200\u1d50 (x : \u03b1) \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) (), Integrable fun y => f (x, y)) \u2227\n Integrable fun x => \u222b (y : \u03a9), \u2016f (x, y)\u2016 \u2202\u2191(kernel.prodMkLeft Unit (Measure.condKernel \u03c1)) ((), x)\n[PROOFSTEP]\nsimp_rw [kernel.prodMkLeft_apply, kernel.const_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf_int : Integrable f\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Integrable fun \u03c9 => f (a, \u03c9)\n[PROOFSTEP]\nhave hf_ae : AEStronglyMeasurable f \u03c1 := hf_int.1\n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf_int : Integrable f\nhf_ae : AEStronglyMeasurable f \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Integrable fun \u03c9 => f (a, \u03c9)\n[PROOFSTEP]\nrw [\u2190 hf_ae.ae_integrable_condKernel_iff] at hf_int \n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf_int :\n (\u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Integrable fun \u03c9 => f (a, \u03c9)) \u2227\n Integrable fun a => \u222b (\u03c9 : \u03a9), \u2016f (a, \u03c9)\u2016 \u2202\u2191(Measure.condKernel \u03c1) a\nhf_ae : AEStronglyMeasurable f \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Integrable fun \u03c9 => f (a, \u03c9)\n[PROOFSTEP]\nexact hf_int.1\n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf_int : Integrable f\n\u22a2 Integrable fun x => \u222b (y : \u03a9), \u2016f (x, y)\u2016 \u2202\u2191(Measure.condKernel \u03c1) x\n[PROOFSTEP]\nhave hf_ae : AEStronglyMeasurable f \u03c1 := hf_int.1\n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf_int : Integrable f\nhf_ae : AEStronglyMeasurable f \u03c1\n\u22a2 Integrable fun x => \u222b (y : \u03a9), \u2016f (x, y)\u2016 \u2202\u2191(Measure.condKernel \u03c1) x\n[PROOFSTEP]\nrw [\u2190 hf_ae.ae_integrable_condKernel_iff] at hf_int \n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 F\nhf_int :\n (\u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Integrable fun \u03c9 => f (a, \u03c9)) \u2227\n Integrable fun a => \u222b (\u03c9 : \u03a9), \u2016f (a, \u03c9)\u2016 \u2202\u2191(Measure.condKernel \u03c1) a\nhf_ae : AEStronglyMeasurable f \u03c1\n\u22a2 Integrable fun x => \u222b (y : \u03a9), \u2016f (x, y)\u2016 \u2202\u2191(Measure.condKernel \u03c1) x\n[PROOFSTEP]\nexact hf_int.2\n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf_int : Integrable f\n\u22a2 Integrable fun x => \u2016\u222b (y : \u03a9), f (x, y) \u2202\u2191(Measure.condKernel \u03c1) x\u2016\n[PROOFSTEP]\nrefine' hf_int.integral_norm_condKernel.mono hf_int.1.integral_condKernel.norm _\n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf_int : Integrable f\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1,\n \u2016\u2016\u222b (y : \u03a9), f (a, y) \u2202\u2191(Measure.condKernel \u03c1) a\u2016\u2016 \u2264 \u2016\u222b (y : \u03a9), \u2016f (a, y)\u2016 \u2202\u2191(Measure.condKernel \u03c1) a\u2016\n[PROOFSTEP]\nrefine' eventually_of_forall fun x => _\n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf_int : Integrable f\nx : \u03b1\n\u22a2 \u2016\u2016\u222b (y : \u03a9), f (x, y) \u2202\u2191(Measure.condKernel \u03c1) x\u2016\u2016 \u2264 \u2016\u222b (y : \u03a9), \u2016f (x, y)\u2016 \u2202\u2191(Measure.condKernel \u03c1) x\u2016\n[PROOFSTEP]\nrw [norm_norm]\n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf_int : Integrable f\nx : \u03b1\n\u22a2 \u2016\u222b (y : \u03a9), f (x, y) \u2202\u2191(Measure.condKernel \u03c1) x\u2016 \u2264 \u2016\u222b (y : \u03a9), \u2016f (x, y)\u2016 \u2202\u2191(Measure.condKernel \u03c1) x\u2016\n[PROOFSTEP]\nrefine' (norm_integral_le_integral_norm _).trans_eq (Real.norm_of_nonneg _).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03a9 : Type u_2\nE : Type u_3\nF : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03a9\ninst\u271d\u2078 : TopologicalSpace \u03a9\ninst\u271d\u2077 : BorelSpace \u03a9\ninst\u271d\u2076 : PolishSpace \u03a9\ninst\u271d\u2075 : Nonempty \u03a9\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03c1 : Measure (\u03b1 \u00d7 \u03a9)\ninst\u271d : IsFiniteMeasure \u03c1\nf : \u03b1 \u00d7 \u03a9 \u2192 E\nhf_int : Integrable f\nx : \u03b1\n\u22a2 0 \u2264 \u222b (a : \u03a9), \u2016f (x, a)\u2016 \u2202\u2191(Measure.condKernel \u03c1) x\n[PROOFSTEP]\nexact integral_nonneg_of_ae (eventually_of_forall fun y => norm_nonneg _)\n", "meta": {"mathlib_filename": "Mathlib.Probability.Kernel.Disintegration", "llama_tokens": 88426, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494421679929, "lm_q2_score": 0.6370307875894139, "lm_q1q2_score": 0.43697091339079563}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\n\u22a2 card (A / C) * card B \u2264 card (A / B) * card (B / C)\n[PROOFSTEP]\nrw [\u2190 card_product (A / B), \u2190 mul_one ((A / B) \u00d7\u02e2 (B / C)).card]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\n\u22a2 card (A / C) * card B \u2264 card ((A / B) \u00d7\u02e2 (B / C)) * 1\n[PROOFSTEP]\nrefine'\n card_mul_le_card_mul (fun b ac \u21a6 ac.1 * ac.2 = b) (fun x hx \u21a6 _) fun x _ \u21a6\n card_le_one_iff.2 fun hu hv \u21a6 ((mem_bipartiteBelow _).1 hu).2.symm.trans ((mem_bipartiteBelow _).1 hv).2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\nx : \u03b1\nhx : x \u2208 A / C\n\u22a2 card B \u2264 card (bipartiteAbove (fun b ac => ac.fst * ac.snd = b) ((A / B) \u00d7\u02e2 (B / C)) x)\n[PROOFSTEP]\nobtain \u27e8a, c, ha, hc, rfl\u27e9 := mem_div.1 hx\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\na c : \u03b1\nha : a \u2208 A\nhc : c \u2208 C\nhx : a / c \u2208 A / C\n\u22a2 card B \u2264 card (bipartiteAbove (fun b ac => ac.fst * ac.snd = b) ((A / B) \u00d7\u02e2 (B / C)) (a / c))\n[PROOFSTEP]\nrefine' card_le_card_of_inj_on (fun b \u21a6 (a / b, b / c)) (fun b hb \u21a6 _) fun b\u2081 _ b\u2082 _ h \u21a6 _\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\na c : \u03b1\nha : a \u2208 A\nhc : c \u2208 C\nhx : a / c \u2208 A / C\nb : \u03b1\nhb : b \u2208 B\n\u22a2 (fun b => (a / b, b / c)) b \u2208 bipartiteAbove (fun b ac => ac.fst * ac.snd = b) ((A / B) \u00d7\u02e2 (B / C)) (a / c)\n[PROOFSTEP]\nrw [mem_bipartiteAbove]\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\na c : \u03b1\nha : a \u2208 A\nhc : c \u2208 C\nhx : a / c \u2208 A / C\nb : \u03b1\nhb : b \u2208 B\n\u22a2 (fun b => (a / b, b / c)) b \u2208 (A / B) \u00d7\u02e2 (B / C) \u2227\n ((fun b => (a / b, b / c)) b).fst * ((fun b => (a / b, b / c)) b).snd = a / c\n[PROOFSTEP]\nexact \u27e8mk_mem_product (div_mem_div ha hb) (div_mem_div hb hc), div_mul_div_cancel' _ _ _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\na c : \u03b1\nha : a \u2208 A\nhc : c \u2208 C\nhx : a / c \u2208 A / C\nb\u2081 : \u03b1\nx\u271d\u00b9 : b\u2081 \u2208 B\nb\u2082 : \u03b1\nx\u271d : b\u2082 \u2208 B\nh : (fun b => (a / b, b / c)) b\u2081 = (fun b => (a / b, b / c)) b\u2082\n\u22a2 b\u2081 = b\u2082\n[PROOFSTEP]\nexact div_right_injective (Prod.ext_iff.1 h).1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\n\u22a2 card (A / C) * card B \u2264 card (A * B) * card (B * C)\n[PROOFSTEP]\nrw [\u2190 div_inv_eq_mul, \u2190 card_inv B, \u2190 card_inv (B * C), mul_inv, \u2190 div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\n\u22a2 card (A / C) * card B\u207b\u00b9 \u2264 card (A / B\u207b\u00b9) * card (B\u207b\u00b9 / C)\n[PROOFSTEP]\nexact card_div_mul_le_card_div_mul_card_div _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\n\u22a2 card (A * C) * card B \u2264 card (A / B) * card (B * C)\n[PROOFSTEP]\nrw [\u2190 div_inv_eq_mul, \u2190 div_inv_eq_mul B]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\n\u22a2 card (A / C\u207b\u00b9) * card B \u2264 card (A / B) * card (B / C\u207b\u00b9)\n[PROOFSTEP]\nexact card_div_mul_le_card_div_mul_card_div _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\n\u22a2 card (A * C) * card B \u2264 card (A * B) * card (B / C)\n[PROOFSTEP]\nrw [\u2190 div_inv_eq_mul, div_eq_mul_inv B]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\n\u22a2 card (A / C\u207b\u00b9) * card B \u2264 card (A * B) * card (B * C\u207b\u00b9)\n[PROOFSTEP]\nexact card_div_mul_le_card_mul_mul_card_mul _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d C : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\n\u22a2 card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\n[PROOFSTEP]\ninduction' C using Finset.induction_on with x C _ ih\n[GOAL]\ncase empty\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\n\u22a2 card (A * B * \u2205) * card A \u2264 card (A * B) * card (A * \u2205)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\n\u22a2 card (A * B * insert x C) * card A \u2264 card (A * B) * card (A * insert x C)\n[PROOFSTEP]\nset A' := A \u2229 (A * C / { x }) with hA'\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\n\u22a2 card (A * B * insert x C) * card A \u2264 card (A * B) * card (A * insert x C)\n[PROOFSTEP]\nset C' := insert x C with hC'\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\n\u22a2 card (A * B * C') * card A \u2264 card (A * B) * card (A * C')\n[PROOFSTEP]\nhave h\u2080 : A' * { x } = A * { x } \u2229 (A * C) := by rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\n\u22a2 A' * {x} = A * {x} \u2229 (A * C)\n[PROOFSTEP]\nrw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\nh\u2080 : A' * {x} = A * {x} \u2229 (A * C)\n\u22a2 card (A * B * C') * card A \u2264 card (A * B) * card (A * C')\n[PROOFSTEP]\nhave h\u2081 : A * B * C' = A * B * C \u222a (A * B * { x }) \\ (A' * B * { x }) :=\n by\n rw [hC', insert_eq, union_comm, mul_union]\n refine' (sup_sdiff_eq_sup _).symm\n rw [mul_right_comm, mul_right_comm A, h\u2080]\n exact mul_subset_mul_right (inter_subset_right _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\nh\u2080 : A' * {x} = A * {x} \u2229 (A * C)\n\u22a2 A * B * C' = A * B * C \u222a (A * B * {x}) \\ (A' * B * {x})\n[PROOFSTEP]\nrw [hC', insert_eq, union_comm, mul_union]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\nh\u2080 : A' * {x} = A * {x} \u2229 (A * C)\n\u22a2 A * B * C \u222a A * B * {x} = A * B * C \u222a (A * B * {x}) \\ (A' * B * {x})\n[PROOFSTEP]\nrefine' (sup_sdiff_eq_sup _).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\nh\u2080 : A' * {x} = A * {x} \u2229 (A * C)\n\u22a2 A' * B * {x} \u2264 A * B * C\n[PROOFSTEP]\nrw [mul_right_comm, mul_right_comm A, h\u2080]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\nh\u2080 : A' * {x} = A * {x} \u2229 (A * C)\n\u22a2 A * {x} \u2229 (A * C) * B \u2264 A * C * B\n[PROOFSTEP]\nexact mul_subset_mul_right (inter_subset_right _ _)\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\nh\u2080 : A' * {x} = A * {x} \u2229 (A * C)\nh\u2081 : A * B * C' = A * B * C \u222a (A * B * {x}) \\ (A' * B * {x})\n\u22a2 card (A * B * C') * card A \u2264 card (A * B) * card (A * C')\n[PROOFSTEP]\nhave h\u2082 : A' * B * { x } \u2286 A * B * { x } := mul_subset_mul_right (mul_subset_mul_right <| inter_subset_left _ _)\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\nh\u2080 : A' * {x} = A * {x} \u2229 (A * C)\nh\u2081 : A * B * C' = A * B * C \u222a (A * B * {x}) \\ (A' * B * {x})\nh\u2082 : A' * B * {x} \u2286 A * B * {x}\n\u22a2 card (A * B * C') * card A \u2264 card (A * B) * card (A * C')\n[PROOFSTEP]\nhave h\u2083 : (A * B * C').card \u2264 (A * B * C).card + (A * B).card - (A' * B).card :=\n by\n rw [h\u2081]\n refine' (card_union_le _ _).trans_eq _\n rw [card_sdiff h\u2082, \u2190 add_tsub_assoc_of_le (card_le_of_subset h\u2082), card_mul_singleton, card_mul_singleton]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\nh\u2080 : A' * {x} = A * {x} \u2229 (A * C)\nh\u2081 : A * B * C' = A * B * C \u222a (A * B * {x}) \\ (A' * B * {x})\nh\u2082 : A' * B * {x} \u2286 A * B * {x}\n\u22a2 card (A * B * C') \u2264 card (A * B * C) + card (A * B) - card (A' * B)\n[PROOFSTEP]\nrw [h\u2081]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\nh\u2080 : A' * {x} = A * {x} \u2229 (A * C)\nh\u2081 : A * B * C' = A * B * C \u222a (A * B * {x}) \\ (A' * B * {x})\nh\u2082 : A' * B * {x} \u2286 A * B * {x}\n\u22a2 card (A * B * C \u222a (A * B * {x}) \\ (A' * B * {x})) \u2264 card (A * B * C) + card (A * B) - card (A' * B)\n[PROOFSTEP]\nrefine' (card_union_le _ _).trans_eq _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\nh\u2080 : A' * {x} = A * {x} \u2229 (A * C)\nh\u2081 : A * B * C' = A * B * C \u222a (A * B * {x}) \\ (A' * B * {x})\nh\u2082 : A' * B * {x} \u2286 A * B * {x}\n\u22a2 card (A * B * C) + card ((A * B * {x}) \\ (A' * B * {x})) = card (A * B * C) + card (A * B) - card (A' * B)\n[PROOFSTEP]\nrw [card_sdiff h\u2082, \u2190 add_tsub_assoc_of_le (card_le_of_subset h\u2082), card_mul_singleton, card_mul_singleton]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\nh\u2080 : A' * {x} = A * {x} \u2229 (A * C)\nh\u2081 : A * B * C' = A * B * C \u222a (A * B * {x}) \\ (A' * B * {x})\nh\u2082 : A' * B * {x} \u2286 A * B * {x}\nh\u2083 : card (A * B * C') \u2264 card (A * B * C) + card (A * B) - card (A' * B)\n\u22a2 card (A * B * C') * card A \u2264 card (A * B) * card (A * C')\n[PROOFSTEP]\nrefine' (mul_le_mul_right' h\u2083 _).trans _\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\nh\u2080 : A' * {x} = A * {x} \u2229 (A * C)\nh\u2081 : A * B * C' = A * B * C \u222a (A * B * {x}) \\ (A' * B * {x})\nh\u2082 : A' * B * {x} \u2286 A * B * {x}\nh\u2083 : card (A * B * C') \u2264 card (A * B * C) + card (A * B) - card (A' * B)\n\u22a2 (card (A * B * C) + card (A * B) - card (A' * B)) * card A \u2264 card (A * B) * card (A * C')\n[PROOFSTEP]\nrw [tsub_mul, add_mul]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\nh\u2080 : A' * {x} = A * {x} \u2229 (A * C)\nh\u2081 : A * B * C' = A * B * C \u222a (A * B * {x}) \\ (A' * B * {x})\nh\u2082 : A' * B * {x} \u2286 A * B * {x}\nh\u2083 : card (A * B * C') \u2264 card (A * B * C) + card (A * B) - card (A' * B)\n\u22a2 card (A * B * C) * card A + card (A * B) * card A - card (A' * B) * card A \u2264 card (A * B) * card (A * C')\n[PROOFSTEP]\nrefine' (tsub_le_tsub (add_le_add_right ih _) <| hA _ <| inter_subset_left _ _).trans_eq _\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C\u271d : Finset \u03b1\nhA : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nx : \u03b1\nC : Finset \u03b1\na\u271d : \u00acx \u2208 C\nih : card (A * B * C) * card A \u2264 card (A * B) * card (A * C)\nA' : Finset \u03b1 := A \u2229 (A * C / {x})\nhA' : A' = A \u2229 (A * C / {x})\nC' : Finset \u03b1 := insert x C\nhC' : C' = insert x C\nh\u2080 : A' * {x} = A * {x} \u2229 (A * C)\nh\u2081 : A * B * C' = A * B * C \u222a (A * B * {x}) \\ (A' * B * {x})\nh\u2082 : A' * B * {x} \u2286 A * B * {x}\nh\u2083 : card (A * B * C') \u2264 card (A * B * C) + card (A * B) - card (A' * B)\n\u22a2 card (A * B) * card (A * C) + card (A * B) * card A - card (A * B) * card (A \u2229 (A * C / {x})) =\n card (A * B) * card (A * C')\n[PROOFSTEP]\nrw [\u2190 mul_add, \u2190 mul_tsub, \u2190 hA', hC', insert_eq, mul_union, \u2190 card_mul_singleton A x, \u2190 card_mul_singleton A' x,\n add_comm (card _), h\u2080, eq_tsub_of_add_eq (card_union_add_card_inter _ _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhA : Finset.Nonempty A\nhAB : A \u2286 B\nh : \u2200 (A' : Finset \u03b1), A' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (A * C)) / \u2191(card A) \u2264 \u2191(card (A' * C)) / \u2191(card A')\n\u22a2 \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * C) * card A' \u2264 card (A' * C) * card A\n[PROOFSTEP]\nrintro A' hAA'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhA : Finset.Nonempty A\nhAB : A \u2286 B\nh : \u2200 (A' : Finset \u03b1), A' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (A * C)) / \u2191(card A) \u2264 \u2191(card (A' * C)) / \u2191(card A')\nA' : Finset \u03b1\nhAA' : A' \u2286 A\n\u22a2 card (A * C) * card A' \u2264 card (A' * C) * card A\n[PROOFSTEP]\nobtain rfl | hA' := A'.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhA : Finset.Nonempty A\nhAB : A \u2286 B\nh : \u2200 (A' : Finset \u03b1), A' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (A * C)) / \u2191(card A) \u2264 \u2191(card (A' * C)) / \u2191(card A')\nhAA' : \u2205 \u2286 A\n\u22a2 card (A * C) * card \u2205 \u2264 card (\u2205 * C) * card A\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhA : Finset.Nonempty A\nhAB : A \u2286 B\nh : \u2200 (A' : Finset \u03b1), A' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (A * C)) / \u2191(card A) \u2264 \u2191(card (A' * C)) / \u2191(card A')\nA' : Finset \u03b1\nhAA' : A' \u2286 A\nhA' : Finset.Nonempty A'\n\u22a2 card (A * C) * card A' \u2264 card (A' * C) * card A\n[PROOFSTEP]\nhave hA\u2080 : (0 : \u211a\u22650) < A.card := cast_pos.2 hA.card_pos\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhA : Finset.Nonempty A\nhAB : A \u2286 B\nh : \u2200 (A' : Finset \u03b1), A' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (A * C)) / \u2191(card A) \u2264 \u2191(card (A' * C)) / \u2191(card A')\nA' : Finset \u03b1\nhAA' : A' \u2286 A\nhA' : Finset.Nonempty A'\nhA\u2080 : 0 < \u2191(card A)\n\u22a2 card (A * C) * card A' \u2264 card (A' * C) * card A\n[PROOFSTEP]\nhave hA\u2080' : (0 : \u211a\u22650) < A'.card := cast_pos.2 hA'.card_pos\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhA : Finset.Nonempty A\nhAB : A \u2286 B\nh : \u2200 (A' : Finset \u03b1), A' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (A * C)) / \u2191(card A) \u2264 \u2191(card (A' * C)) / \u2191(card A')\nA' : Finset \u03b1\nhAA' : A' \u2286 A\nhA' : Finset.Nonempty A'\nhA\u2080 : 0 < \u2191(card A)\nhA\u2080' : 0 < \u2191(card A')\n\u22a2 card (A * C) * card A' \u2264 card (A' * C) * card A\n[PROOFSTEP]\nexact_mod_cast\n (div_le_div_iff hA\u2080 hA\u2080').1 (h _ <| mem_erase_of_ne_of_mem hA'.ne_empty <| mem_powerset.2 <| hAA'.trans hAB)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\n\u22a2 card (A * C) * card B \u2264 card (A * B) * card (B * C)\n[PROOFSTEP]\nobtain rfl | hB := B.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B C\u271d A C : Finset \u03b1\n\u22a2 card (A * C) * card \u2205 \u2264 card (A * \u2205) * card (\u2205 * C)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\nhB : Finset.Nonempty B\n\u22a2 card (A * C) * card B \u2264 card (A * B) * card (B * C)\n[PROOFSTEP]\nhave hB' : B \u2208 B.powerset.erase \u2205 := mem_erase_of_ne_of_mem hB.ne_empty (mem_powerset_self _)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\nhB : Finset.Nonempty B\nhB' : B \u2208 erase (powerset B) \u2205\n\u22a2 card (A * C) * card B \u2264 card (A * B) * card (B * C)\n[PROOFSTEP]\nobtain \u27e8U, hU, hUA\u27e9 := exists_min_image (B.powerset.erase \u2205) (fun U \u21a6 (U * A).card / U.card : _ \u2192 \u211a\u22650) \u27e8B, hB'\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\nhB : Finset.Nonempty B\nhB' : B \u2208 erase (powerset B) \u2205\nU : Finset \u03b1\nhU : U \u2208 erase (powerset B) \u2205\nhUA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (U * A)) / \u2191(card U) \u2264 \u2191(card (x' * A)) / \u2191(card x')\n\u22a2 card (A * C) * card B \u2264 card (A * B) * card (B * C)\n[PROOFSTEP]\nrw [mem_erase, mem_powerset, \u2190 nonempty_iff_ne_empty] at hU \n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\nhB : Finset.Nonempty B\nhB' : B \u2208 erase (powerset B) \u2205\nU : Finset \u03b1\nhU : Finset.Nonempty U \u2227 U \u2286 B\nhUA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (U * A)) / \u2191(card U) \u2264 \u2191(card (x' * A)) / \u2191(card x')\n\u22a2 card (A * C) * card B \u2264 card (A * B) * card (B * C)\n[PROOFSTEP]\nrefine' cast_le.1 (_ : (_ : \u211a\u22650) \u2264 _)\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\nhB : Finset.Nonempty B\nhB' : B \u2208 erase (powerset B) \u2205\nU : Finset \u03b1\nhU : Finset.Nonempty U \u2227 U \u2286 B\nhUA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (U * A)) / \u2191(card U) \u2264 \u2191(card (x' * A)) / \u2191(card x')\n\u22a2 \u2191(card (A * C) * card B) \u2264 \u2191(card (A * B) * card (B * C))\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\nhB : Finset.Nonempty B\nhB' : B \u2208 erase (powerset B) \u2205\nU : Finset \u03b1\nhU : Finset.Nonempty U \u2227 U \u2286 B\nhUA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (U * A)) / \u2191(card U) \u2264 \u2191(card (x' * A)) / \u2191(card x')\n\u22a2 \u2191(card (A * C)) * \u2191(card B) \u2264 \u2191(card (A * B)) * \u2191(card (B * C))\n[PROOFSTEP]\nrefine' (le_div_iff <| cast_pos.2 hB.card_pos).1 _\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\nhB : Finset.Nonempty B\nhB' : B \u2208 erase (powerset B) \u2205\nU : Finset \u03b1\nhU : Finset.Nonempty U \u2227 U \u2286 B\nhUA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (U * A)) / \u2191(card U) \u2264 \u2191(card (x' * A)) / \u2191(card x')\n\u22a2 \u2191(card (A * C)) \u2264 \u2191(card (A * B)) * \u2191(card (B * C)) / \u2191(card B)\n[PROOFSTEP]\nrw [mul_div_right_comm, mul_comm _ B]\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\nhB : Finset.Nonempty B\nhB' : B \u2208 erase (powerset B) \u2205\nU : Finset \u03b1\nhU : Finset.Nonempty U \u2227 U \u2286 B\nhUA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (U * A)) / \u2191(card U) \u2264 \u2191(card (x' * A)) / \u2191(card x')\n\u22a2 \u2191(card (A * C)) \u2264 \u2191(card (B * A)) / \u2191(card B) * \u2191(card (B * C))\n[PROOFSTEP]\nrefine' (cast_le.2 <| card_le_card_mul_left _ hU.1).trans _\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\nhB : Finset.Nonempty B\nhB' : B \u2208 erase (powerset B) \u2205\nU : Finset \u03b1\nhU : Finset.Nonempty U \u2227 U \u2286 B\nhUA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (U * A)) / \u2191(card U) \u2264 \u2191(card (x' * A)) / \u2191(card x')\n\u22a2 \u2191(card (U * (A * C))) \u2264 \u2191(card (B * A)) / \u2191(card B) * \u2191(card (B * C))\n[PROOFSTEP]\nrefine'\n le_trans _\n (mul_le_mul (hUA _ hB') (cast_le.2 <| card_le_of_subset <| mul_subset_mul_right hU.2) (zero_le _) (zero_le _))\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\nhB : Finset.Nonempty B\nhB' : B \u2208 erase (powerset B) \u2205\nU : Finset \u03b1\nhU : Finset.Nonempty U \u2227 U \u2286 B\nhUA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (U * A)) / \u2191(card U) \u2264 \u2191(card (x' * A)) / \u2191(card x')\n\u22a2 \u2191(card (U * (A * C))) \u2264 \u2191(card (U * A)) / \u2191(card U) * \u2191(card (U * C))\n[PROOFSTEP]\nrw [\u2190 mul_div_right_comm, \u2190 mul_assoc]\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\nhB : Finset.Nonempty B\nhB' : B \u2208 erase (powerset B) \u2205\nU : Finset \u03b1\nhU : Finset.Nonempty U \u2227 U \u2286 B\nhUA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (U * A)) / \u2191(card U) \u2264 \u2191(card (x' * A)) / \u2191(card x')\n\u22a2 \u2191(card (U * A * C)) \u2264 \u2191(card (U * A)) * \u2191(card (U * C)) / \u2191(card U)\n[PROOFSTEP]\nrefine' (le_div_iff <| cast_pos.2 hU.1.card_pos).2 _\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\nhB : Finset.Nonempty B\nhB' : B \u2208 erase (powerset B) \u2205\nU : Finset \u03b1\nhU : Finset.Nonempty U \u2227 U \u2286 B\nhUA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset B) \u2205 \u2192 \u2191(card (U * A)) / \u2191(card U) \u2264 \u2191(card (x' * A)) / \u2191(card x')\n\u22a2 \u2191(card (U * A * C)) * \u2191(card U) \u2264 \u2191(card (U * A)) * \u2191(card (U * C))\n[PROOFSTEP]\nexact_mod_cast mul_pluennecke_petridis C (mul_aux hU.1 hU.2 hUA)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\n\u22a2 card (A * C) * card B \u2264 card (A / B) * card (B / C)\n[PROOFSTEP]\nrw [div_eq_mul_inv, \u2190 card_inv B, \u2190 card_inv (B / C), inv_div', div_inv_eq_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\n\u22a2 card (A * C) * card B\u207b\u00b9 \u2264 card (A * B\u207b\u00b9) * card (B\u207b\u00b9 * C)\n[PROOFSTEP]\nexact card_mul_mul_card_le_card_mul_mul_card_mul _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\n\u22a2 card (A / C) * card B \u2264 card (A * B) * card (B / C)\n[PROOFSTEP]\nrw [div_eq_mul_inv, div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\n\u22a2 card (A * C\u207b\u00b9) * card B \u2264 card (A * B) * card (B * C\u207b\u00b9)\n[PROOFSTEP]\nexact card_mul_mul_card_le_card_mul_mul_card_mul _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\n\u22a2 card (A / C) * card B \u2264 card (A / B) * card (B * C)\n[PROOFSTEP]\nrw [\u2190 div_inv_eq_mul, div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA\u271d B\u271d C\u271d A B C : Finset \u03b1\n\u22a2 card (A * C\u207b\u00b9) * card B \u2264 card (A / B) * card (B / C\u207b\u00b9)\n[PROOFSTEP]\nexact card_mul_mul_le_card_div_mul_card_div _ _ _\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : CommGroup \u03b1\u271d\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\nA\u271d B\u271d C : Finset \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : AddCommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A + B) * card A' \u2264 card (A' + B) * card A\nn : \u2115\n\u22a2 \u2191(card (A + n \u2022 B)) \u2264 (\u2191(card (A + B)) / \u2191(card A)) ^ n * \u2191(card A)\n[PROOFSTEP]\nobtain rfl | hA := A.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : CommGroup \u03b1\u271d\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\nA B\u271d C : Finset \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : AddCommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nB : Finset \u03b1\nn : \u2115\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 \u2205 \u2192 card (\u2205 + B) * card A' \u2264 card (A' + B) * card \u2205\n\u22a2 \u2191(card (\u2205 + n \u2022 B)) \u2264 (\u2191(card (\u2205 + B)) / \u2191(card \u2205)) ^ n * \u2191(card \u2205)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : CommGroup \u03b1\u271d\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\nA\u271d B\u271d C : Finset \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : AddCommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A + B) * card A' \u2264 card (A' + B) * card A\nn : \u2115\nhA : Finset.Nonempty A\n\u22a2 \u2191(card (A + n \u2022 B)) \u2264 (\u2191(card (A + B)) / \u2191(card A)) ^ n * \u2191(card A)\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase inr.zero\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : CommGroup \u03b1\u271d\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\nA\u271d B\u271d C : Finset \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : AddCommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A + B) * card A' \u2264 card (A' + B) * card A\nhA : Finset.Nonempty A\n\u22a2 \u2191(card (A + zero \u2022 B)) \u2264 (\u2191(card (A + B)) / \u2191(card A)) ^ zero * \u2191(card A)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.succ\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : CommGroup \u03b1\u271d\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\nA\u271d B\u271d C : Finset \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : AddCommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A + B) * card A' \u2264 card (A' + B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A + n \u2022 B)) \u2264 (\u2191(card (A + B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 \u2191(card (A + succ n \u2022 B)) \u2264 (\u2191(card (A + B)) / \u2191(card A)) ^ succ n * \u2191(card A)\n[PROOFSTEP]\nrw [succ_nsmul, \u2190 add_assoc, _root_.pow_succ, mul_assoc, \u2190 mul_div_right_comm, le_div_iff, \u2190 cast_mul]\n[GOAL]\ncase inr.succ\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : CommGroup \u03b1\u271d\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\nA\u271d B\u271d C : Finset \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : AddCommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A + B) * card A' \u2264 card (A' + B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A + n \u2022 B)) \u2264 (\u2191(card (A + B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 \u2191(card (A + B + n \u2022 B) * card A) \u2264 \u2191(card (A + B)) * ((\u2191(card (A + B)) / \u2191(card A)) ^ n * \u2191(card A))\ncase inr.succ\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : CommGroup \u03b1\u271d\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\nA\u271d B\u271d C : Finset \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : AddCommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A + B) * card A' \u2264 card (A' + B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A + n \u2022 B)) \u2264 (\u2191(card (A + B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 0 < \u2191(card A)\n[PROOFSTEP]\nswap\n[GOAL]\ncase inr.succ\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : CommGroup \u03b1\u271d\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\nA\u271d B\u271d C : Finset \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : AddCommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A + B) * card A' \u2264 card (A' + B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A + n \u2022 B)) \u2264 (\u2191(card (A + B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 0 < \u2191(card A)\ncase inr.succ\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : CommGroup \u03b1\u271d\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\nA\u271d B\u271d C : Finset \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : AddCommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A + B) * card A' \u2264 card (A' + B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A + n \u2022 B)) \u2264 (\u2191(card (A + B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 \u2191(card (A + B + n \u2022 B) * card A) \u2264 \u2191(card (A + B)) * ((\u2191(card (A + B)) / \u2191(card A)) ^ n * \u2191(card A))\n[PROOFSTEP]\nexact cast_pos.2 hA.card_pos\n[GOAL]\ncase inr.succ\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : CommGroup \u03b1\u271d\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\nA\u271d B\u271d C : Finset \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : AddCommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A + B) * card A' \u2264 card (A' + B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A + n \u2022 B)) \u2264 (\u2191(card (A + B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 \u2191(card (A + B + n \u2022 B) * card A) \u2264 \u2191(card (A + B)) * ((\u2191(card (A + B)) / \u2191(card A)) ^ n * \u2191(card A))\n[PROOFSTEP]\nrefine' (cast_le.2 <| add_pluennecke_petridis _ hAB).trans _\n[GOAL]\ncase inr.succ\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : CommGroup \u03b1\u271d\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\nA\u271d B\u271d C : Finset \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : AddCommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A + B) * card A' \u2264 card (A' + B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A + n \u2022 B)) \u2264 (\u2191(card (A + B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 \u2191(card (A + B) * card (A + n \u2022 B)) \u2264 \u2191(card (A + B)) * ((\u2191(card (A + B)) / \u2191(card A)) ^ n * \u2191(card A))\n[PROOFSTEP]\nrw [cast_mul]\n[GOAL]\ncase inr.succ\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : CommGroup \u03b1\u271d\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\nA\u271d B\u271d C : Finset \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : AddCommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A + B) * card A' \u2264 card (A' + B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A + n \u2022 B)) \u2264 (\u2191(card (A + B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 \u2191(card (A + B)) * \u2191(card (A + n \u2022 B)) \u2264 \u2191(card (A + B)) * ((\u2191(card (A + B)) / \u2191(card A)) ^ n * \u2191(card A))\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nn : \u2115\n\u22a2 \u2191(card (A * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A)\n[PROOFSTEP]\nobtain rfl | hA := A.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nB C : Finset \u03b1\nn : \u2115\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 \u2205 \u2192 card (\u2205 * B) * card A' \u2264 card (A' * B) * card \u2205\n\u22a2 \u2191(card (\u2205 * B ^ n)) \u2264 (\u2191(card (\u2205 * B)) / \u2191(card \u2205)) ^ n * \u2191(card \u2205)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nn : \u2115\nhA : Finset.Nonempty A\n\u22a2 \u2191(card (A * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A)\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase inr.zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nhA : Finset.Nonempty A\n\u22a2 \u2191(card (A * B ^ zero)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ zero * \u2191(card A)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 \u2191(card (A * B ^ succ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ succ n * \u2191(card A)\n[PROOFSTEP]\nrw [_root_.pow_succ, \u2190 mul_assoc, _root_.pow_succ, @mul_assoc \u211a\u22650, \u2190 mul_div_right_comm, le_div_iff, \u2190 cast_mul]\n[GOAL]\ncase inr.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 \u2191(card (A * B * B ^ n) * card A) \u2264 \u2191(card (A * B)) * ((\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A))\ncase inr.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 0 < \u2191(card A)\n[PROOFSTEP]\nswap\n[GOAL]\ncase inr.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 0 < \u2191(card A)\ncase inr.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 \u2191(card (A * B * B ^ n) * card A) \u2264 \u2191(card (A * B)) * ((\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A))\n[PROOFSTEP]\nexact cast_pos.2 hA.card_pos\n[GOAL]\ncase inr.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 \u2191(card (A * B * B ^ n) * card A) \u2264 \u2191(card (A * B)) * ((\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A))\n[PROOFSTEP]\nrefine' (cast_le.2 <| mul_pluennecke_petridis _ hAB).trans _\n[GOAL]\ncase inr.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 \u2191(card (A * B) * card (A * B ^ n)) \u2264 \u2191(card (A * B)) * ((\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A))\n[PROOFSTEP]\nrw [cast_mul]\n[GOAL]\ncase inr.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B C : Finset \u03b1\nhAB : \u2200 (A' : Finset \u03b1), A' \u2286 A \u2192 card (A * B) * card A' \u2264 card (A' * B) * card A\nhA : Finset.Nonempty A\nn : \u2115\nih : \u2191(card (A * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A)\n\u22a2 \u2191(card (A * B)) * \u2191(card (A * B ^ n)) \u2264 \u2191(card (A * B)) * ((\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A))\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\n\u22a2 \u2191(card (B ^ m / B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A)\n[PROOFSTEP]\nhave hA' : A \u2208 A.powerset.erase \u2205 := mem_erase_of_ne_of_mem hA.ne_empty (mem_powerset_self _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\n\u22a2 \u2191(card (B ^ m / B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A)\n[PROOFSTEP]\nobtain \u27e8C, hC, hCA\u27e9 := exists_min_image (A.powerset.erase \u2205) (fun C \u21a6 (C * B).card / C.card : _ \u2192 \u211a\u22650) \u27e8A, hA'\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : C \u2208 erase (powerset A) \u2205\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m / B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A)\n[PROOFSTEP]\nrw [mem_erase, mem_powerset, \u2190 nonempty_iff_ne_empty] at hC \n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m / B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A)\n[PROOFSTEP]\nrefine' (mul_le_mul_right <| cast_pos.2 hC.1.card_pos).1 _\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m / B ^ n)) * \u2191(card C) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m / B ^ n) * card C) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)\n[PROOFSTEP]\nrefine' (cast_le.2 <| card_div_mul_le_card_mul_mul_card_mul _ _ _).trans _\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m * C) * card (C * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m * C)) * \u2191(card (C * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)\n[PROOFSTEP]\nrw [mul_comm _ C]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (C * B ^ m)) * \u2191(card (C * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)\n[PROOFSTEP]\nrefine'\n (mul_le_mul (card_mul_pow_le (mul_aux hC.1 hC.2 hCA) _) (card_mul_pow_le (mul_aux hC.1 hC.2 hCA) _) (zero_le _)\n (zero_le _)).trans\n _\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 (\u2191(card (C * B)) / \u2191(card C)) ^ m * \u2191(card C) * ((\u2191(card (C * B)) / \u2191(card C)) ^ n * \u2191(card C)) \u2264\n (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)\n[PROOFSTEP]\nrw [mul_mul_mul_comm, \u2190 pow_add, \u2190 mul_assoc]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 (\u2191(card (C * B)) / \u2191(card C)) ^ (m + n) * \u2191(card C) * \u2191(card C) \u2264\n (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)\n[PROOFSTEP]\ngcongr((?_ ^ _) * Nat.cast ?_) * _\n[GOAL]\ncase intro.intro.bc.h\u2081.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (A * B)) / \u2191(card A)\n[PROOFSTEP]\nexact hCA _ hA'\n[GOAL]\ncase intro.intro.bc.h\u2082.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 card C \u2264 card A\n[PROOFSTEP]\nexact card_le_of_subset hC.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\n\u22a2 \u2191(card (B ^ m / B ^ n)) \u2264 (\u2191(card (A / B)) / \u2191(card A)) ^ (m + n) * \u2191(card A)\n[PROOFSTEP]\nrw [\u2190 card_inv, inv_div', \u2190 inv_pow, \u2190 inv_pow, div_eq_mul_inv A]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\n\u22a2 \u2191(card (B\u207b\u00b9 ^ m / B\u207b\u00b9 ^ n)) \u2264 (\u2191(card (A * B\u207b\u00b9)) / \u2191(card A)) ^ (m + n) * \u2191(card A)\n[PROOFSTEP]\nexact card_pow_div_pow_le hA _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nn : \u2115\n\u22a2 \u2191(card (B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ n * \u2191(card A)\n[PROOFSTEP]\nsimpa only [_root_.pow_zero, div_one] using card_pow_div_pow_le hA _ _ 0\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nn : \u2115\n\u22a2 \u2191(card (B ^ n)) \u2264 (\u2191(card (A / B)) / \u2191(card A)) ^ n * \u2191(card A)\n[PROOFSTEP]\nsimpa only [_root_.pow_zero, div_one] using card_pow_div_pow_le' hA _ _ 0\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.Additive.PluenneckeRuzsa", "llama_tokens": 22061, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754371026367, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.43692238317636206}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\n\u22a2 g = 0\n[PROOFSTEP]\nlet u := kernel.\u03b9 g \u226b i\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\n\u22a2 g = 0\n[PROOFSTEP]\nhaveI : Mono u := mono_comp _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\n\u22a2 g = 0\n[PROOFSTEP]\nhaveI hu := normalMonoOfMono u\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\nhu : NormalMono u\n\u22a2 g = 0\n[PROOFSTEP]\nlet h := hu.g\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\nhu : NormalMono u\nh : Q \u27f6 NormalMono.Z u := NormalMono.g\n\u22a2 g = 0\n[PROOFSTEP]\nobtain \u27e8t, ht\u27e9 := kernel.lift' g p hpg\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\nhu : NormalMono u\nh : Q \u27f6 NormalMono.Z u := NormalMono.g\nt : P \u27f6 kernel g\nht : t \u226b kernel.\u03b9 g = p\n\u22a2 g = 0\n[PROOFSTEP]\nhave fh : f \u226b h = 0\n[GOAL]\ncase fh\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\nhu : NormalMono u\nh : Q \u27f6 NormalMono.Z u := NormalMono.g\nt : P \u27f6 kernel g\nht : t \u226b kernel.\u03b9 g = p\n\u22a2 f \u226b h = 0\ncase mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\nhu : NormalMono u\nh : Q \u27f6 NormalMono.Z u := NormalMono.g\nt : P \u27f6 kernel g\nht : t \u226b kernel.\u03b9 g = p\nfh : f \u226b h = 0\n\u22a2 g = 0\n[PROOFSTEP]\ncalc\n f \u226b h = (p \u226b i) \u226b h := (Abelian.image.fac f).symm \u25b8 rfl\n _ = ((t \u226b kernel.\u03b9 g) \u226b i) \u226b h := (ht \u25b8 rfl)\n _ = t \u226b u \u226b h := by simp only [Category.assoc]\n _ = t \u226b 0 := (hu.w \u25b8 rfl)\n _ = 0 :=\n HasZeroMorphisms.comp_zero _\n _\n -- h factors through the cokernel of f via some l.\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\nhu : NormalMono u\nh : Q \u27f6 NormalMono.Z u := NormalMono.g\nt : P \u27f6 kernel g\nht : t \u226b kernel.\u03b9 g = p\n\u22a2 ((t \u226b kernel.\u03b9 g) \u226b i) \u226b h = t \u226b u \u226b h\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\nhu : NormalMono u\nh : Q \u27f6 NormalMono.Z u := NormalMono.g\nt : P \u27f6 kernel g\nht : t \u226b kernel.\u03b9 g = p\nfh : f \u226b h = 0\n\u22a2 g = 0\n[PROOFSTEP]\nobtain \u27e8l, hl\u27e9 := cokernel.desc' f h fh\n[GOAL]\ncase mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\nhu : NormalMono u\nh : Q \u27f6 NormalMono.Z u := NormalMono.g\nt : P \u27f6 kernel g\nht : t \u226b kernel.\u03b9 g = p\nfh : f \u226b h = 0\nl : cokernel f \u27f6 NormalMono.Z u\nhl : cokernel.\u03c0 f \u226b l = h\n\u22a2 g = 0\n[PROOFSTEP]\nhave hih : i \u226b h = 0\n[GOAL]\ncase hih\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\nhu : NormalMono u\nh : Q \u27f6 NormalMono.Z u := NormalMono.g\nt : P \u27f6 kernel g\nht : t \u226b kernel.\u03b9 g = p\nfh : f \u226b h = 0\nl : cokernel f \u27f6 NormalMono.Z u\nhl : cokernel.\u03c0 f \u226b l = h\n\u22a2 i \u226b h = 0\ncase mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\nhu : NormalMono u\nh : Q \u27f6 NormalMono.Z u := NormalMono.g\nt : P \u27f6 kernel g\nht : t \u226b kernel.\u03b9 g = p\nfh : f \u226b h = 0\nl : cokernel f \u27f6 NormalMono.Z u\nhl : cokernel.\u03c0 f \u226b l = h\nhih : i \u226b h = 0\n\u22a2 g = 0\n[PROOFSTEP]\ncalc\n i \u226b h = i \u226b cokernel.\u03c0 f \u226b l := hl \u25b8 rfl\n _ = 0 \u226b l := by rw [\u2190 Category.assoc, kernel.condition]\n _ = 0 := zero_comp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\nhu : NormalMono u\nh : Q \u27f6 NormalMono.Z u := NormalMono.g\nt : P \u27f6 kernel g\nht : t \u226b kernel.\u03b9 g = p\nfh : f \u226b h = 0\nl : cokernel f \u27f6 NormalMono.Z u\nhl : cokernel.\u03c0 f \u226b l = h\n\u22a2 i \u226b cokernel.\u03c0 f \u226b l = 0 \u226b l\n[PROOFSTEP]\nrw [\u2190 Category.assoc, kernel.condition]\n[GOAL]\ncase mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\nhu : NormalMono u\nh : Q \u27f6 NormalMono.Z u := NormalMono.g\nt : P \u27f6 kernel g\nht : t \u226b kernel.\u03b9 g = p\nfh : f \u226b h = 0\nl : cokernel f \u27f6 NormalMono.Z u\nhl : cokernel.\u03c0 f \u226b l = h\nhih : i \u226b h = 0\n\u22a2 g = 0\n[PROOFSTEP]\nobtain \u27e8s, hs\u27e9 := NormalMono.lift' u i hih\n[GOAL]\ncase mk.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\nhu : NormalMono u\nh : Q \u27f6 NormalMono.Z u := NormalMono.g\nt : P \u27f6 kernel g\nht : t \u226b kernel.\u03b9 g = p\nfh : f \u226b h = 0\nl : cokernel f \u27f6 NormalMono.Z u\nhl : cokernel.\u03c0 f \u226b l = h\nhih : i \u226b h = 0\ns : kernel (cokernel.\u03c0 f) \u27f6 kernel g\nhs : s \u226b u = i\n\u22a2 g = 0\n[PROOFSTEP]\nhave hs' : (s \u226b kernel.\u03b9 g) \u226b i = \ud835\udfd9 I \u226b i := by rw [Category.assoc, hs, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\nhu : NormalMono u\nh : Q \u27f6 NormalMono.Z u := NormalMono.g\nt : P \u27f6 kernel g\nht : t \u226b kernel.\u03b9 g = p\nfh : f \u226b h = 0\nl : cokernel f \u27f6 NormalMono.Z u\nhl : cokernel.\u03c0 f \u226b l = h\nhih : i \u226b h = 0\ns : kernel (cokernel.\u03c0 f) \u27f6 kernel g\nhs : s \u226b u = i\n\u22a2 (s \u226b kernel.\u03b9 g) \u226b i = \ud835\udfd9 I \u226b i\n[PROOFSTEP]\nrw [Category.assoc, hs, Category.id_comp]\n[GOAL]\ncase mk.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis : Mono u\nhu : NormalMono u\nh : Q \u27f6 NormalMono.Z u := NormalMono.g\nt : P \u27f6 kernel g\nht : t \u226b kernel.\u03b9 g = p\nfh : f \u226b h = 0\nl : cokernel f \u27f6 NormalMono.Z u\nhl : cokernel.\u03c0 f \u226b l = h\nhih : i \u226b h = 0\ns : kernel (cokernel.\u03c0 f) \u27f6 kernel g\nhs : s \u226b u = i\nhs' : (s \u226b kernel.\u03b9 g) \u226b i = \ud835\udfd9 I \u226b i\n\u22a2 g = 0\n[PROOFSTEP]\nhaveI : Epi (kernel.\u03b9 g) :=\n epi_of_epi_fac\n ((cancel_mono _).1 hs')\n -- ker g is an epimorphism, but ker g \u226b g = 0 = ker g \u226b 0, so g = 0 as required.\n[GOAL]\ncase mk.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.image f\np : P \u27f6 Abelian.image f := Abelian.factorThruImage f\ni : kernel (cokernel.\u03c0 f) \u27f6 Q := kernel.\u03b9 (cokernel.\u03c0 f)\nR : C\ng : I \u27f6 R\nhpg : p \u226b g = 0\nu : kernel g \u27f6 Q := kernel.\u03b9 g \u226b i\nthis\u271d : Mono u\nhu : NormalMono u\nh : Q \u27f6 NormalMono.Z u := NormalMono.g\nt : P \u27f6 kernel g\nht : t \u226b kernel.\u03b9 g = p\nfh : f \u226b h = 0\nl : cokernel f \u27f6 NormalMono.Z u\nhl : cokernel.\u03c0 f \u226b l = h\nhih : i \u226b h = 0\ns : kernel (cokernel.\u03c0 f) \u27f6 kernel g\nhs : s \u226b u = i\nhs' : (s \u226b kernel.\u03b9 g) \u226b i = \ud835\udfd9 I \u226b i\nthis : Epi (kernel.\u03b9 g)\n\u22a2 g = 0\n[PROOFSTEP]\nexact zero_of_epi_comp _ (kernel.condition g)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\n\u22a2 g = 0\n[PROOFSTEP]\nlet u := p \u226b cokernel.\u03c0 g\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\n\u22a2 g = 0\n[PROOFSTEP]\nhaveI : Epi u := epi_comp _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\n\u22a2 g = 0\n[PROOFSTEP]\nhaveI hu := normalEpiOfEpi u\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\n\u22a2 g = 0\n[PROOFSTEP]\nlet h := hu.g\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\n\u22a2 g = 0\n[PROOFSTEP]\nobtain \u27e8t, ht\u27e9 := cokernel.desc' g i hgi\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\nt : cokernel g \u27f6 Q\nht : cokernel.\u03c0 g \u226b t = i\n\u22a2 g = 0\n[PROOFSTEP]\nhave hf : h \u226b f = 0\n[GOAL]\ncase hf\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\nt : cokernel g \u27f6 Q\nht : cokernel.\u03c0 g \u226b t = i\n\u22a2 h \u226b f = 0\ncase mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\nt : cokernel g \u27f6 Q\nht : cokernel.\u03c0 g \u226b t = i\nhf : h \u226b f = 0\n\u22a2 g = 0\n[PROOFSTEP]\ncalc\n h \u226b f = h \u226b p \u226b i := (Abelian.coimage.fac f).symm \u25b8 rfl\n _ = h \u226b p \u226b cokernel.\u03c0 g \u226b t := (ht \u25b8 rfl)\n _ = h \u226b u \u226b t := by simp only [Category.assoc]\n _ = 0 \u226b t := by rw [\u2190 Category.assoc, hu.w]\n _ = 0 := zero_comp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\nt : cokernel g \u27f6 Q\nht : cokernel.\u03c0 g \u226b t = i\n\u22a2 h \u226b p \u226b cokernel.\u03c0 g \u226b t = h \u226b u \u226b t\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\nt : cokernel g \u27f6 Q\nht : cokernel.\u03c0 g \u226b t = i\n\u22a2 h \u226b u \u226b t = 0 \u226b t\n[PROOFSTEP]\nrw [\u2190 Category.assoc, hu.w]\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\nt : cokernel g \u27f6 Q\nht : cokernel.\u03c0 g \u226b t = i\nhf : h \u226b f = 0\n\u22a2 g = 0\n[PROOFSTEP]\nobtain \u27e8l, hl\u27e9 := kernel.lift' f h hf\n[GOAL]\ncase mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\nt : cokernel g \u27f6 Q\nht : cokernel.\u03c0 g \u226b t = i\nhf : h \u226b f = 0\nl : NormalEpi.W u \u27f6 kernel f\nhl : l \u226b kernel.\u03b9 f = h\n\u22a2 g = 0\n[PROOFSTEP]\nhave hhp : h \u226b p = 0\n[GOAL]\ncase hhp\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\nt : cokernel g \u27f6 Q\nht : cokernel.\u03c0 g \u226b t = i\nhf : h \u226b f = 0\nl : NormalEpi.W u \u27f6 kernel f\nhl : l \u226b kernel.\u03b9 f = h\n\u22a2 h \u226b p = 0\ncase mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\nt : cokernel g \u27f6 Q\nht : cokernel.\u03c0 g \u226b t = i\nhf : h \u226b f = 0\nl : NormalEpi.W u \u27f6 kernel f\nhl : l \u226b kernel.\u03b9 f = h\nhhp : h \u226b p = 0\n\u22a2 g = 0\n[PROOFSTEP]\ncalc\n h \u226b p = (l \u226b kernel.\u03b9 f) \u226b p := hl \u25b8 rfl\n _ = l \u226b 0 := by rw [Category.assoc, cokernel.condition]\n _ = 0 := comp_zero\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\nt : cokernel g \u27f6 Q\nht : cokernel.\u03c0 g \u226b t = i\nhf : h \u226b f = 0\nl : NormalEpi.W u \u27f6 kernel f\nhl : l \u226b kernel.\u03b9 f = h\n\u22a2 (l \u226b kernel.\u03b9 f) \u226b p = l \u226b 0\n[PROOFSTEP]\nrw [Category.assoc, cokernel.condition]\n[GOAL]\ncase mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\nt : cokernel g \u27f6 Q\nht : cokernel.\u03c0 g \u226b t = i\nhf : h \u226b f = 0\nl : NormalEpi.W u \u27f6 kernel f\nhl : l \u226b kernel.\u03b9 f = h\nhhp : h \u226b p = 0\n\u22a2 g = 0\n[PROOFSTEP]\nobtain \u27e8s, hs\u27e9 := NormalEpi.desc' u p hhp\n[GOAL]\ncase mk.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\nt : cokernel g \u27f6 Q\nht : cokernel.\u03c0 g \u226b t = i\nhf : h \u226b f = 0\nl : NormalEpi.W u \u27f6 kernel f\nhl : l \u226b kernel.\u03b9 f = h\nhhp : h \u226b p = 0\ns : cokernel g \u27f6 cokernel (kernel.\u03b9 f)\nhs : u \u226b s = p\n\u22a2 g = 0\n[PROOFSTEP]\nhave hs' : p \u226b cokernel.\u03c0 g \u226b s = p \u226b \ud835\udfd9 I := by rw [\u2190 Category.assoc, hs, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\nt : cokernel g \u27f6 Q\nht : cokernel.\u03c0 g \u226b t = i\nhf : h \u226b f = 0\nl : NormalEpi.W u \u27f6 kernel f\nhl : l \u226b kernel.\u03b9 f = h\nhhp : h \u226b p = 0\ns : cokernel g \u27f6 cokernel (kernel.\u03b9 f)\nhs : u \u226b s = p\n\u22a2 p \u226b cokernel.\u03c0 g \u226b s = p \u226b \ud835\udfd9 I\n[PROOFSTEP]\nrw [\u2190 Category.assoc, hs, Category.comp_id]\n[GOAL]\ncase mk.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\nt : cokernel g \u27f6 Q\nht : cokernel.\u03c0 g \u226b t = i\nhf : h \u226b f = 0\nl : NormalEpi.W u \u27f6 kernel f\nhl : l \u226b kernel.\u03b9 f = h\nhhp : h \u226b p = 0\ns : cokernel g \u27f6 cokernel (kernel.\u03b9 f)\nhs : u \u226b s = p\nhs' : p \u226b cokernel.\u03c0 g \u226b s = p \u226b \ud835\udfd9 I\n\u22a2 g = 0\n[PROOFSTEP]\nhaveI : Mono (cokernel.\u03c0 g) :=\n mono_of_mono_fac\n ((cancel_epi _).1 hs')\n -- coker g is a monomorphism, but g \u226b coker g = 0 = 0 \u226b coker g, so g = 0 as required.\n[GOAL]\ncase mk.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nP Q : C\nf : P \u27f6 Q\nI : C := Abelian.coimage f\ni : Abelian.coimage f \u27f6 Q := Abelian.factorThruCoimage f\np : P \u27f6 cokernel (kernel.\u03b9 f) := cokernel.\u03c0 (kernel.\u03b9 f)\nR : C\ng : R \u27f6 I\nhgi : g \u226b i = 0\nu : P \u27f6 cokernel g := p \u226b cokernel.\u03c0 g\nthis\u271d : Epi u\nhu : NormalEpi u\nh : NormalEpi.W u \u27f6 P := NormalEpi.g\nt : cokernel g \u27f6 Q\nht : cokernel.\u03c0 g \u226b t = i\nhf : h \u226b f = 0\nl : NormalEpi.W u \u27f6 kernel f\nhl : l \u226b kernel.\u03b9 f = h\nhhp : h \u226b p = 0\ns : cokernel g \u27f6 cokernel (kernel.\u03b9 f)\nhs : u \u226b s = p\nhs' : p \u226b cokernel.\u03c0 g \u226b s = p \u226b \ud835\udfd9 I\nthis : Mono (cokernel.\u03c0 g)\n\u22a2 g = 0\n[PROOFSTEP]\nexact zero_of_comp_mono _ (cokernel.condition g)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\n\u22a2 Mono (r A)\n[PROOFSTEP]\nlet hl : IsLimit (KernelFork.of\u03b9 (diag A) (cokernel.condition (diag A))) :=\n monoIsKernelOfCokernel _ (colimit.isColimit _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhl : IsLimit (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)) :=\n monoIsKernelOfCokernel (colimit.cocone (parallelPair (diag A) 0)) (colimit.isColimit (parallelPair (diag A) 0))\n\u22a2 Mono (r A)\n[PROOFSTEP]\napply NormalEpiCategory.mono_of_cancel_zero\n[GOAL]\ncase hf\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhl : IsLimit (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)) :=\n monoIsKernelOfCokernel (colimit.cocone (parallelPair (diag A) 0)) (colimit.isColimit (parallelPair (diag A) 0))\n\u22a2 \u2200 (Z : C) (g : Z \u27f6 A), g \u226b r A = 0 \u2192 g = 0\n[PROOFSTEP]\nintro Z x hx\n[GOAL]\ncase hf\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhl : IsLimit (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)) :=\n monoIsKernelOfCokernel (colimit.cocone (parallelPair (diag A) 0)) (colimit.isColimit (parallelPair (diag A) 0))\nZ : C\nx : Z \u27f6 A\nhx : x \u226b r A = 0\n\u22a2 x = 0\n[PROOFSTEP]\nhave hxx : (x \u226b prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A)) \u226b cokernel.\u03c0 (diag A) = 0 := by rw [Category.assoc, hx]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhl : IsLimit (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)) :=\n monoIsKernelOfCokernel (colimit.cocone (parallelPair (diag A) 0)) (colimit.isColimit (parallelPair (diag A) 0))\nZ : C\nx : Z \u27f6 A\nhx : x \u226b r A = 0\n\u22a2 (x \u226b prod.lift (\ud835\udfd9 A) 0) \u226b cokernel.\u03c0 (diag A) = 0\n[PROOFSTEP]\nrw [Category.assoc, hx]\n[GOAL]\ncase hf\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhl : IsLimit (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)) :=\n monoIsKernelOfCokernel (colimit.cocone (parallelPair (diag A) 0)) (colimit.isColimit (parallelPair (diag A) 0))\nZ : C\nx : Z \u27f6 A\nhx : x \u226b r A = 0\nhxx : (x \u226b prod.lift (\ud835\udfd9 A) 0) \u226b cokernel.\u03c0 (diag A) = 0\n\u22a2 x = 0\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := KernelFork.IsLimit.lift' hl _ hxx\n[GOAL]\ncase hf.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhl : IsLimit (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)) :=\n monoIsKernelOfCokernel (colimit.cocone (parallelPair (diag A) 0)) (colimit.isColimit (parallelPair (diag A) 0))\nZ : C\nx : Z \u27f6 A\nhx : x \u226b r A = 0\nhxx : (x \u226b prod.lift (\ud835\udfd9 A) 0) \u226b cokernel.\u03c0 (diag A) = 0\ny : Z \u27f6 (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)).pt\nhy : y \u226b Fork.\u03b9 (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)) = x \u226b prod.lift (\ud835\udfd9 A) 0\n\u22a2 x = 0\n[PROOFSTEP]\nrw [KernelFork.\u03b9_of\u03b9] at hy \n[GOAL]\ncase hf.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhl : IsLimit (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)) :=\n monoIsKernelOfCokernel (colimit.cocone (parallelPair (diag A) 0)) (colimit.isColimit (parallelPair (diag A) 0))\nZ : C\nx : Z \u27f6 A\nhx : x \u226b r A = 0\nhxx : (x \u226b prod.lift (\ud835\udfd9 A) 0) \u226b cokernel.\u03c0 (diag A) = 0\ny : Z \u27f6 (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)).pt\nhy : y \u226b diag A = x \u226b prod.lift (\ud835\udfd9 A) 0\n\u22a2 x = 0\n[PROOFSTEP]\nhave hyy : y = 0 := by\n erw [\u2190 Category.comp_id y, \u2190 Limits.prod.lift_snd (\ud835\udfd9 A) (\ud835\udfd9 A), \u2190 Category.assoc, hy, Category.assoc, prod.lift_snd,\n HasZeroMorphisms.comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhl : IsLimit (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)) :=\n monoIsKernelOfCokernel (colimit.cocone (parallelPair (diag A) 0)) (colimit.isColimit (parallelPair (diag A) 0))\nZ : C\nx : Z \u27f6 A\nhx : x \u226b r A = 0\nhxx : (x \u226b prod.lift (\ud835\udfd9 A) 0) \u226b cokernel.\u03c0 (diag A) = 0\ny : Z \u27f6 (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)).pt\nhy : y \u226b diag A = x \u226b prod.lift (\ud835\udfd9 A) 0\n\u22a2 y = 0\n[PROOFSTEP]\nerw [\u2190 Category.comp_id y, \u2190 Limits.prod.lift_snd (\ud835\udfd9 A) (\ud835\udfd9 A), \u2190 Category.assoc, hy, Category.assoc, prod.lift_snd,\n HasZeroMorphisms.comp_zero]\n[GOAL]\ncase hf.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhl : IsLimit (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)) :=\n monoIsKernelOfCokernel (colimit.cocone (parallelPair (diag A) 0)) (colimit.isColimit (parallelPair (diag A) 0))\nZ : C\nx : Z \u27f6 A\nhx : x \u226b r A = 0\nhxx : (x \u226b prod.lift (\ud835\udfd9 A) 0) \u226b cokernel.\u03c0 (diag A) = 0\ny : Z \u27f6 (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)).pt\nhy : y \u226b diag A = x \u226b prod.lift (\ud835\udfd9 A) 0\nhyy : y = 0\n\u22a2 x = 0\n[PROOFSTEP]\nhaveI : Mono (prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A)) := mono_of_mono_fac (prod.lift_fst _ _)\n[GOAL]\ncase hf.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhl : IsLimit (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)) :=\n monoIsKernelOfCokernel (colimit.cocone (parallelPair (diag A) 0)) (colimit.isColimit (parallelPair (diag A) 0))\nZ : C\nx : Z \u27f6 A\nhx : x \u226b r A = 0\nhxx : (x \u226b prod.lift (\ud835\udfd9 A) 0) \u226b cokernel.\u03c0 (diag A) = 0\ny : Z \u27f6 (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)).pt\nhy : y \u226b diag A = x \u226b prod.lift (\ud835\udfd9 A) 0\nhyy : y = 0\nthis : Mono (prod.lift (\ud835\udfd9 A) 0)\n\u22a2 x = 0\n[PROOFSTEP]\napply (cancel_mono (prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A))).1\n[GOAL]\ncase hf.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhl : IsLimit (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)) :=\n monoIsKernelOfCokernel (colimit.cocone (parallelPair (diag A) 0)) (colimit.isColimit (parallelPair (diag A) 0))\nZ : C\nx : Z \u27f6 A\nhx : x \u226b r A = 0\nhxx : (x \u226b prod.lift (\ud835\udfd9 A) 0) \u226b cokernel.\u03c0 (diag A) = 0\ny : Z \u27f6 (KernelFork.of\u03b9 (diag A) (_ : diag A \u226b cokernel.\u03c0 (diag A) = 0)).pt\nhy : y \u226b diag A = x \u226b prod.lift (\ud835\udfd9 A) 0\nhyy : y = 0\nthis : Mono (prod.lift (\ud835\udfd9 A) 0)\n\u22a2 x \u226b prod.lift (\ud835\udfd9 A) 0 = 0 \u226b prod.lift (\ud835\udfd9 A) 0\n[PROOFSTEP]\nrw [\u2190 hy, hyy, zero_comp, zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\n\u22a2 Epi (r A)\n[PROOFSTEP]\nhave hlp : prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A) \u226b Limits.prod.snd = 0 := prod.lift_snd _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\n\u22a2 Epi (r A)\n[PROOFSTEP]\nlet hp1 : IsLimit (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A)) hlp) :=\n by\n refine' Fork.IsLimit.mk _ (fun s => Fork.\u03b9 s \u226b Limits.prod.fst) _ _\n \u00b7 intro s\n apply prod.hom_ext <;> simp\n \u00b7 intro s m h\n haveI : Mono (prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A)) := mono_of_mono_fac (prod.lift_fst _ _)\n apply (cancel_mono (prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A))).1\n convert h\n apply prod.hom_ext <;> simp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\n\u22a2 IsLimit (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp)\n[PROOFSTEP]\nrefine' Fork.IsLimit.mk _ (fun s => Fork.\u03b9 s \u226b Limits.prod.fst) _ _\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\n\u22a2 \u2200 (s : Fork prod.snd 0), (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s\n[PROOFSTEP]\nintro s\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\ns : Fork prod.snd 0\n\u22a2 (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s\n[PROOFSTEP]\napply prod.hom_ext\n[GOAL]\ncase refine'_1.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\ns : Fork prod.snd 0\n\u22a2 ((fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp)) \u226b prod.fst = Fork.\u03b9 s \u226b prod.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_1.h\u2082\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\ns : Fork prod.snd 0\n\u22a2 ((fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp)) \u226b prod.snd = Fork.\u03b9 s \u226b prod.snd\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\n\u22a2 \u2200 (s : Fork prod.snd 0) (m : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt),\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s \u2192 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s\n[PROOFSTEP]\nintro s m h\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\ns : Fork prod.snd 0\nm : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt\nh : m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s\n\u22a2 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s\n[PROOFSTEP]\nhaveI : Mono (prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A)) := mono_of_mono_fac (prod.lift_fst _ _)\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\ns : Fork prod.snd 0\nm : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt\nh : m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s\nthis : Mono (prod.lift (\ud835\udfd9 A) 0)\n\u22a2 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s\n[PROOFSTEP]\napply (cancel_mono (prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A))).1\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\ns : Fork prod.snd 0\nm : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt\nh : m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s\nthis : Mono (prod.lift (\ud835\udfd9 A) 0)\n\u22a2 m \u226b prod.lift (\ud835\udfd9 A) 0 = (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b prod.lift (\ud835\udfd9 A) 0\n[PROOFSTEP]\nconvert h\n[GOAL]\ncase h.e'_3.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\ns : Fork prod.snd 0\nm : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt\nh : m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s\nthis : Mono (prod.lift (\ud835\udfd9 A) 0)\ne_1\u271d : (s.pt \u27f6 A \u2a2f A) = (s.pt \u27f6 (parallelPair prod.snd 0).obj WalkingParallelPair.zero)\n\u22a2 (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b prod.lift (\ud835\udfd9 A) 0 = Fork.\u03b9 s\n[PROOFSTEP]\napply prod.hom_ext\n[GOAL]\ncase h.e'_3.h.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\ns : Fork prod.snd 0\nm : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt\nh : m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s\nthis : Mono (prod.lift (\ud835\udfd9 A) 0)\ne_1\u271d : (s.pt \u27f6 A \u2a2f A) = (s.pt \u27f6 (parallelPair prod.snd 0).obj WalkingParallelPair.zero)\n\u22a2 ((fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b prod.lift (\ud835\udfd9 A) 0) \u226b prod.fst = Fork.\u03b9 s \u226b prod.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3.h.h\u2082\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\ns : Fork prod.snd 0\nm : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt\nh : m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s\nthis : Mono (prod.lift (\ud835\udfd9 A) 0)\ne_1\u271d : (s.pt \u27f6 A \u2a2f A) = (s.pt \u27f6 (parallelPair prod.snd 0).obj WalkingParallelPair.zero)\n\u22a2 ((fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b prod.lift (\ud835\udfd9 A) 0) \u226b prod.snd = Fork.\u03b9 s \u226b prod.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\nhp1 : IsLimit (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) :=\n Fork.IsLimit.mk (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) (fun s => Fork.\u03b9 s \u226b prod.fst)\n (_ :\n \u2200 (s : Fork prod.snd 0),\n (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s)\n (_ :\n \u2200 (s : Fork prod.snd 0) (m : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt),\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s \u2192 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s)\n\u22a2 Epi (r A)\n[PROOFSTEP]\nlet hp2 : IsColimit (CokernelCofork.of\u03c0 (Limits.prod.snd : A \u2a2f A \u27f6 A) hlp) := epiIsCokernelOfKernel _ hp1\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\nhp1 : IsLimit (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) :=\n Fork.IsLimit.mk (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) (fun s => Fork.\u03b9 s \u226b prod.fst)\n (_ :\n \u2200 (s : Fork prod.snd 0),\n (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s)\n (_ :\n \u2200 (s : Fork prod.snd 0) (m : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt),\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s \u2192 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s)\nhp2 : IsColimit (CokernelCofork.of\u03c0 prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) hp1\n\u22a2 Epi (r A)\n[PROOFSTEP]\napply NormalMonoCategory.epi_of_zero_cancel\n[GOAL]\ncase hf\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\nhp1 : IsLimit (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) :=\n Fork.IsLimit.mk (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) (fun s => Fork.\u03b9 s \u226b prod.fst)\n (_ :\n \u2200 (s : Fork prod.snd 0),\n (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s)\n (_ :\n \u2200 (s : Fork prod.snd 0) (m : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt),\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s \u2192 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s)\nhp2 : IsColimit (CokernelCofork.of\u03c0 prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) hp1\n\u22a2 \u2200 (Z : C) (g : cokernel (diag A) \u27f6 Z), r A \u226b g = 0 \u2192 g = 0\n[PROOFSTEP]\nintro Z z hz\n[GOAL]\ncase hf\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\nhp1 : IsLimit (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) :=\n Fork.IsLimit.mk (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) (fun s => Fork.\u03b9 s \u226b prod.fst)\n (_ :\n \u2200 (s : Fork prod.snd 0),\n (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s)\n (_ :\n \u2200 (s : Fork prod.snd 0) (m : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt),\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s \u2192 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s)\nhp2 : IsColimit (CokernelCofork.of\u03c0 prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) hp1\nZ : C\nz : cokernel (diag A) \u27f6 Z\nhz : r A \u226b z = 0\n\u22a2 z = 0\n[PROOFSTEP]\nhave h : prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A) \u226b cokernel.\u03c0 (diag A) \u226b z = 0 := by rw [\u2190 Category.assoc, hz]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\nhp1 : IsLimit (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) :=\n Fork.IsLimit.mk (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) (fun s => Fork.\u03b9 s \u226b prod.fst)\n (_ :\n \u2200 (s : Fork prod.snd 0),\n (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s)\n (_ :\n \u2200 (s : Fork prod.snd 0) (m : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt),\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s \u2192 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s)\nhp2 : IsColimit (CokernelCofork.of\u03c0 prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) hp1\nZ : C\nz : cokernel (diag A) \u27f6 Z\nhz : r A \u226b z = 0\n\u22a2 prod.lift (\ud835\udfd9 A) 0 \u226b cokernel.\u03c0 (diag A) \u226b z = 0\n[PROOFSTEP]\nrw [\u2190 Category.assoc, hz]\n[GOAL]\ncase hf\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\nhp1 : IsLimit (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) :=\n Fork.IsLimit.mk (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) (fun s => Fork.\u03b9 s \u226b prod.fst)\n (_ :\n \u2200 (s : Fork prod.snd 0),\n (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s)\n (_ :\n \u2200 (s : Fork prod.snd 0) (m : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt),\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s \u2192 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s)\nhp2 : IsColimit (CokernelCofork.of\u03c0 prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) hp1\nZ : C\nz : cokernel (diag A) \u27f6 Z\nhz : r A \u226b z = 0\nh : prod.lift (\ud835\udfd9 A) 0 \u226b cokernel.\u03c0 (diag A) \u226b z = 0\n\u22a2 z = 0\n[PROOFSTEP]\nobtain \u27e8t, ht\u27e9 := CokernelCofork.IsColimit.desc' hp2 _ h\n[GOAL]\ncase hf.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\nhp1 : IsLimit (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) :=\n Fork.IsLimit.mk (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) (fun s => Fork.\u03b9 s \u226b prod.fst)\n (_ :\n \u2200 (s : Fork prod.snd 0),\n (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s)\n (_ :\n \u2200 (s : Fork prod.snd 0) (m : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt),\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s \u2192 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s)\nhp2 : IsColimit (CokernelCofork.of\u03c0 prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) hp1\nZ : C\nz : cokernel (diag A) \u27f6 Z\nhz : r A \u226b z = 0\nh : prod.lift (\ud835\udfd9 A) 0 \u226b cokernel.\u03c0 (diag A) \u226b z = 0\nt : (CokernelCofork.of\u03c0 prod.snd hlp).pt \u27f6 Z\nht : Cofork.\u03c0 (CokernelCofork.of\u03c0 prod.snd hlp) \u226b t = cokernel.\u03c0 (diag A) \u226b z\n\u22a2 z = 0\n[PROOFSTEP]\nrw [CokernelCofork.\u03c0_of\u03c0] at ht \n[GOAL]\ncase hf.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\nhp1 : IsLimit (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) :=\n Fork.IsLimit.mk (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) (fun s => Fork.\u03b9 s \u226b prod.fst)\n (_ :\n \u2200 (s : Fork prod.snd 0),\n (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s)\n (_ :\n \u2200 (s : Fork prod.snd 0) (m : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt),\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s \u2192 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s)\nhp2 : IsColimit (CokernelCofork.of\u03c0 prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) hp1\nZ : C\nz : cokernel (diag A) \u27f6 Z\nhz : r A \u226b z = 0\nh : prod.lift (\ud835\udfd9 A) 0 \u226b cokernel.\u03c0 (diag A) \u226b z = 0\nt : (CokernelCofork.of\u03c0 prod.snd hlp).pt \u27f6 Z\nht : prod.snd \u226b t = cokernel.\u03c0 (diag A) \u226b z\n\u22a2 z = 0\n[PROOFSTEP]\nhave htt : t = 0 := by\n rw [\u2190 Category.id_comp t]\n change \ud835\udfd9 A \u226b t = 0\n rw [\u2190 Limits.prod.lift_snd (\ud835\udfd9 A) (\ud835\udfd9 A), Category.assoc, ht, \u2190 Category.assoc, cokernel.condition, zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\nhp1 : IsLimit (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) :=\n Fork.IsLimit.mk (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) (fun s => Fork.\u03b9 s \u226b prod.fst)\n (_ :\n \u2200 (s : Fork prod.snd 0),\n (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s)\n (_ :\n \u2200 (s : Fork prod.snd 0) (m : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt),\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s \u2192 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s)\nhp2 : IsColimit (CokernelCofork.of\u03c0 prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) hp1\nZ : C\nz : cokernel (diag A) \u27f6 Z\nhz : r A \u226b z = 0\nh : prod.lift (\ud835\udfd9 A) 0 \u226b cokernel.\u03c0 (diag A) \u226b z = 0\nt : (CokernelCofork.of\u03c0 prod.snd hlp).pt \u27f6 Z\nht : prod.snd \u226b t = cokernel.\u03c0 (diag A) \u226b z\n\u22a2 t = 0\n[PROOFSTEP]\nrw [\u2190 Category.id_comp t]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\nhp1 : IsLimit (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) :=\n Fork.IsLimit.mk (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) (fun s => Fork.\u03b9 s \u226b prod.fst)\n (_ :\n \u2200 (s : Fork prod.snd 0),\n (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s)\n (_ :\n \u2200 (s : Fork prod.snd 0) (m : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt),\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s \u2192 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s)\nhp2 : IsColimit (CokernelCofork.of\u03c0 prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) hp1\nZ : C\nz : cokernel (diag A) \u27f6 Z\nhz : r A \u226b z = 0\nh : prod.lift (\ud835\udfd9 A) 0 \u226b cokernel.\u03c0 (diag A) \u226b z = 0\nt : (CokernelCofork.of\u03c0 prod.snd hlp).pt \u27f6 Z\nht : prod.snd \u226b t = cokernel.\u03c0 (diag A) \u226b z\n\u22a2 \ud835\udfd9 (CokernelCofork.of\u03c0 prod.snd hlp).pt \u226b t = 0\n[PROOFSTEP]\nchange \ud835\udfd9 A \u226b t = 0\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\nhp1 : IsLimit (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) :=\n Fork.IsLimit.mk (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) (fun s => Fork.\u03b9 s \u226b prod.fst)\n (_ :\n \u2200 (s : Fork prod.snd 0),\n (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s)\n (_ :\n \u2200 (s : Fork prod.snd 0) (m : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt),\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s \u2192 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s)\nhp2 : IsColimit (CokernelCofork.of\u03c0 prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) hp1\nZ : C\nz : cokernel (diag A) \u27f6 Z\nhz : r A \u226b z = 0\nh : prod.lift (\ud835\udfd9 A) 0 \u226b cokernel.\u03c0 (diag A) \u226b z = 0\nt : (CokernelCofork.of\u03c0 prod.snd hlp).pt \u27f6 Z\nht : prod.snd \u226b t = cokernel.\u03c0 (diag A) \u226b z\n\u22a2 \ud835\udfd9 A \u226b t = 0\n[PROOFSTEP]\nrw [\u2190 Limits.prod.lift_snd (\ud835\udfd9 A) (\ud835\udfd9 A), Category.assoc, ht, \u2190 Category.assoc, cokernel.condition, zero_comp]\n[GOAL]\ncase hf.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\nhp1 : IsLimit (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) :=\n Fork.IsLimit.mk (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) (fun s => Fork.\u03b9 s \u226b prod.fst)\n (_ :\n \u2200 (s : Fork prod.snd 0),\n (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s)\n (_ :\n \u2200 (s : Fork prod.snd 0) (m : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt),\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s \u2192 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s)\nhp2 : IsColimit (CokernelCofork.of\u03c0 prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) hp1\nZ : C\nz : cokernel (diag A) \u27f6 Z\nhz : r A \u226b z = 0\nh : prod.lift (\ud835\udfd9 A) 0 \u226b cokernel.\u03c0 (diag A) \u226b z = 0\nt : (CokernelCofork.of\u03c0 prod.snd hlp).pt \u27f6 Z\nht : prod.snd \u226b t = cokernel.\u03c0 (diag A) \u226b z\nhtt : t = 0\n\u22a2 z = 0\n[PROOFSTEP]\napply (cancel_epi (cokernel.\u03c0 (diag A))).1\n[GOAL]\ncase hf.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nA : C\nhlp : prod.lift (\ud835\udfd9 A) 0 \u226b prod.snd = 0\nhp1 : IsLimit (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) :=\n Fork.IsLimit.mk (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) (fun s => Fork.\u03b9 s \u226b prod.fst)\n (_ :\n \u2200 (s : Fork prod.snd 0),\n (fun s => Fork.\u03b9 s \u226b prod.fst) s \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s)\n (_ :\n \u2200 (s : Fork prod.snd 0) (m : s.pt \u27f6 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp).pt),\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) = Fork.\u03b9 s \u2192 m = (fun s => Fork.\u03b9 s \u226b prod.fst) s)\nhp2 : IsColimit (CokernelCofork.of\u03c0 prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.of\u03b9 (prod.lift (\ud835\udfd9 A) 0) hlp) hp1\nZ : C\nz : cokernel (diag A) \u27f6 Z\nhz : r A \u226b z = 0\nh : prod.lift (\ud835\udfd9 A) 0 \u226b cokernel.\u03c0 (diag A) \u226b z = 0\nt : (CokernelCofork.of\u03c0 prod.snd hlp).pt \u27f6 Z\nht : prod.snd \u226b t = cokernel.\u03c0 (diag A) \u226b z\nhtt : t = 0\n\u22a2 cokernel.\u03c0 (diag A) \u226b z = cokernel.\u03c0 (diag A) \u226b 0\n[PROOFSTEP]\nrw [\u2190 ht, htt, comp_zero, comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX : C\n\u22a2 diag X \u226b \u03c3 = 0\n[PROOFSTEP]\nrw [cokernel.condition_assoc, zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX : C\n\u22a2 prod.lift (\ud835\udfd9 X) 0 \u226b \u03c3 = \ud835\udfd9 X\n[PROOFSTEP]\nrw [\u2190 Category.assoc, IsIso.hom_inv_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\nf : X \u27f6 Y\n\u22a2 prod.lift (\ud835\udfd9 X) 0 \u226b prod.map f f = f \u226b prod.lift (\ud835\udfd9 Y) 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX : C\n\u22a2 cokernel.\u03c0 (diag X) \u226b (asIso (r X)).symm.hom = \u03c3\n[PROOFSTEP]\nrw [Iso.symm_hom, asIso_inv]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\nf : X \u27f6 Y\n\u22a2 \u03c3 \u226b f = prod.map f f \u226b \u03c3\n[PROOFSTEP]\nobtain \u27e8g, hg\u27e9 :=\n CokernelCofork.IsColimit.desc' isColimit\u03c3 (Limits.prod.map f f \u226b \u03c3) (by rw [prod.diag_map_assoc, diag_\u03c3, comp_zero])\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\nf : X \u27f6 Y\n\u22a2 diag X \u226b prod.map f f \u226b \u03c3 = 0\n[PROOFSTEP]\nrw [prod.diag_map_assoc, diag_\u03c3, comp_zero]\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\nf : X \u27f6 Y\ng : (CokernelCofork.of\u03c0 \u03c3 (_ : diag X \u226b \u03c3 = 0)).pt \u27f6 Y\nhg : Cofork.\u03c0 (CokernelCofork.of\u03c0 \u03c3 (_ : diag X \u226b \u03c3 = 0)) \u226b g = prod.map f f \u226b \u03c3\n\u22a2 \u03c3 \u226b f = prod.map f f \u226b \u03c3\n[PROOFSTEP]\nsuffices hfg : f = g\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\nf : X \u27f6 Y\ng : (CokernelCofork.of\u03c0 \u03c3 (_ : diag X \u226b \u03c3 = 0)).pt \u27f6 Y\nhg : Cofork.\u03c0 (CokernelCofork.of\u03c0 \u03c3 (_ : diag X \u226b \u03c3 = 0)) \u226b g = prod.map f f \u226b \u03c3\nhfg : f = g\n\u22a2 \u03c3 \u226b f = prod.map f f \u226b \u03c3\n[PROOFSTEP]\nrw [\u2190 hg, Cofork.\u03c0_of\u03c0, hfg]\n[GOAL]\ncase hfg\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\nf : X \u27f6 Y\ng : (CokernelCofork.of\u03c0 \u03c3 (_ : diag X \u226b \u03c3 = 0)).pt \u27f6 Y\nhg : Cofork.\u03c0 (CokernelCofork.of\u03c0 \u03c3 (_ : diag X \u226b \u03c3 = 0)) \u226b g = prod.map f f \u226b \u03c3\n\u22a2 f = g\n[PROOFSTEP]\ncalc\n f = f \u226b prod.lift (\ud835\udfd9 Y) 0 \u226b \u03c3 := by rw [lift_\u03c3, Category.comp_id]\n _ = prod.lift (\ud835\udfd9 X) 0 \u226b Limits.prod.map f f \u226b \u03c3 := by rw [lift_map_assoc]\n _ = prod.lift (\ud835\udfd9 X) 0 \u226b \u03c3 \u226b g := by rw [\u2190 hg, CokernelCofork.\u03c0_of\u03c0]\n _ = g := by rw [\u2190 Category.assoc, lift_\u03c3, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\nf : X \u27f6 Y\ng : (CokernelCofork.of\u03c0 \u03c3 (_ : diag X \u226b \u03c3 = 0)).pt \u27f6 Y\nhg : Cofork.\u03c0 (CokernelCofork.of\u03c0 \u03c3 (_ : diag X \u226b \u03c3 = 0)) \u226b g = prod.map f f \u226b \u03c3\n\u22a2 f = f \u226b prod.lift (\ud835\udfd9 Y) 0 \u226b \u03c3\n[PROOFSTEP]\nrw [lift_\u03c3, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\nf : X \u27f6 Y\ng : (CokernelCofork.of\u03c0 \u03c3 (_ : diag X \u226b \u03c3 = 0)).pt \u27f6 Y\nhg : Cofork.\u03c0 (CokernelCofork.of\u03c0 \u03c3 (_ : diag X \u226b \u03c3 = 0)) \u226b g = prod.map f f \u226b \u03c3\n\u22a2 f \u226b prod.lift (\ud835\udfd9 Y) 0 \u226b \u03c3 = prod.lift (\ud835\udfd9 X) 0 \u226b prod.map f f \u226b \u03c3\n[PROOFSTEP]\nrw [lift_map_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\nf : X \u27f6 Y\ng : (CokernelCofork.of\u03c0 \u03c3 (_ : diag X \u226b \u03c3 = 0)).pt \u27f6 Y\nhg : Cofork.\u03c0 (CokernelCofork.of\u03c0 \u03c3 (_ : diag X \u226b \u03c3 = 0)) \u226b g = prod.map f f \u226b \u03c3\n\u22a2 prod.lift (\ud835\udfd9 X) 0 \u226b prod.map f f \u226b \u03c3 = prod.lift (\ud835\udfd9 X) 0 \u226b \u03c3 \u226b g\n[PROOFSTEP]\nrw [\u2190 hg, CokernelCofork.\u03c0_of\u03c0]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\nf : X \u27f6 Y\ng : (CokernelCofork.of\u03c0 \u03c3 (_ : diag X \u226b \u03c3 = 0)).pt \u27f6 Y\nhg : Cofork.\u03c0 (CokernelCofork.of\u03c0 \u03c3 (_ : diag X \u226b \u03c3 = 0)) \u226b g = prod.map f f \u226b \u03c3\n\u22a2 prod.lift (\ud835\udfd9 X) 0 \u226b \u03c3 \u226b g = g\n[PROOFSTEP]\nrw [\u2190 Category.assoc, lift_\u03c3, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n\u22a2 a - 0 = a\n[PROOFSTEP]\nrw [sub_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n\u22a2 prod.lift a 0 \u226b \u03c3 = a\n[PROOFSTEP]\nconv_lhs =>\n congr; congr; rw [\u2190 Category.comp_id a]\n case a.g => rw [show 0 = a \u226b (0 : Y \u27f6 Y) by simp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n| prod.lift a 0 \u226b \u03c3\n[PROOFSTEP]\n congr; congr; rw [\u2190 Category.comp_id a]\n case a.g => rw [show 0 = a \u226b (0 : Y \u27f6 Y) by simp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n| prod.lift a 0 \u226b \u03c3\n[PROOFSTEP]\n congr; congr; rw [\u2190 Category.comp_id a]\n case a.g => rw [show 0 = a \u226b (0 : Y \u27f6 Y) by simp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n| prod.lift a 0 \u226b \u03c3\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n| prod.lift a 0\ncase a C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : NonPreadditiveAbelian C X Y : C a : X \u27f6 Y | \u03c3\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.f\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n| a\ncase a.g\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n| 0\ncase a C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : NonPreadditiveAbelian C X Y : C a : X \u27f6 Y | \u03c3\n[PROOFSTEP]\nrw [\u2190 Category.comp_id a]\n[GOAL]\ncase a.f\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n| a \u226b \ud835\udfd9 Y\ncase a.g\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n| 0\ncase a C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : NonPreadditiveAbelian C X Y : C a : X \u27f6 Y | \u03c3\n[PROOFSTEP]\ncase a.g => rw [show 0 = a \u226b (0 : Y \u27f6 Y) by simp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n| 0\n[PROOFSTEP]\nrw [show 0 = a \u226b (0 : Y \u27f6 Y) by simp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n| 0\n[PROOFSTEP]\nrw [show 0 = a \u226b (0 : Y \u27f6 Y) by simp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n| 0\n[PROOFSTEP]\nrw [show 0 = a \u226b (0 : Y \u27f6 Y) by simp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n\u22a2 0 = a \u226b 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n\u22a2 prod.lift (a \u226b \ud835\udfd9 Y) (a \u226b 0) \u226b \u03c3 = a\n[PROOFSTEP]\nrw [\u2190 prod.comp_lift, Category.assoc, lift_\u03c3, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n\u22a2 a - a = 0\n[PROOFSTEP]\nrw [sub_def, \u2190 Category.comp_id a, \u2190 prod.comp_lift, Category.assoc, diag_\u03c3, comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b c d : X \u27f6 Y\n\u22a2 prod.lift a b - prod.lift c d = prod.lift (a - c) (b - d)\n[PROOFSTEP]\nsimp only [sub_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b c d : X \u27f6 Y\n\u22a2 prod.lift (prod.lift a b) (prod.lift c d) \u226b \u03c3 = prod.lift (prod.lift a c \u226b \u03c3) (prod.lift b d \u226b \u03c3)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b c d : X \u27f6 Y\n\u22a2 (prod.lift (prod.lift a b) (prod.lift c d) \u226b \u03c3) \u226b prod.fst =\n prod.lift (prod.lift a c \u226b \u03c3) (prod.lift b d \u226b \u03c3) \u226b prod.fst\n[PROOFSTEP]\nrw [Category.assoc, \u03c3_comp, prod.lift_map_assoc, prod.lift_fst, prod.lift_fst, prod.lift_fst]\n[GOAL]\ncase h\u2082\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b c d : X \u27f6 Y\n\u22a2 (prod.lift (prod.lift a b) (prod.lift c d) \u226b \u03c3) \u226b prod.snd =\n prod.lift (prod.lift a c \u226b \u03c3) (prod.lift b d \u226b \u03c3) \u226b prod.snd\n[PROOFSTEP]\nrw [Category.assoc, \u03c3_comp, prod.lift_map_assoc, prod.lift_snd, prod.lift_snd, prod.lift_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b c d : X \u27f6 Y\n\u22a2 a - c - (b - d) = a - b - (c - d)\n[PROOFSTEP]\nrw [sub_def, \u2190 lift_sub_lift, sub_def, Category.assoc, \u03c3_comp, prod.lift_map_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b c d : X \u27f6 Y\n\u22a2 prod.lift (prod.lift a b \u226b \u03c3) (prod.lift c d \u226b \u03c3) \u226b \u03c3 = a - b - (c - d)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n\u22a2 -a - b = -b - a\n[PROOFSTEP]\nconv_lhs => rw [neg_def, \u2190 sub_zero b, sub_sub_sub, sub_zero, \u2190 neg_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| -a - b\n[PROOFSTEP]\nrw [neg_def, \u2190 sub_zero b, sub_sub_sub, sub_zero, \u2190 neg_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| -a - b\n[PROOFSTEP]\nrw [neg_def, \u2190 sub_zero b, sub_sub_sub, sub_zero, \u2190 neg_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| -a - b\n[PROOFSTEP]\nrw [neg_def, \u2190 sub_zero b, sub_sub_sub, sub_zero, \u2190 neg_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n\u22a2 - -a = a\n[PROOFSTEP]\nrw [neg_def, neg_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n\u22a2 0 - (0 - a) = a\n[PROOFSTEP]\nconv_lhs => congr; rw [\u2190 sub_self a]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n| 0 - (0 - a)\n[PROOFSTEP]\ncongr; rw [\u2190 sub_self a]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n| 0 - (0 - a)\n[PROOFSTEP]\ncongr; rw [\u2190 sub_self a]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n| 0 - (0 - a)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n| 0\ncase a C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : NonPreadditiveAbelian C X Y : C a : X \u27f6 Y | 0 - a\n[PROOFSTEP]\nrw [\u2190 sub_self a]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n\u22a2 a - a - (0 - a) = a\n[PROOFSTEP]\nrw [sub_sub_sub, sub_zero, sub_self, sub_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n\u22a2 a + b = b + a\n[PROOFSTEP]\nrw [add_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n\u22a2 a - -b = b + a\n[PROOFSTEP]\nconv_lhs => rw [\u2190 neg_neg a]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| a - -b\n[PROOFSTEP]\nrw [\u2190 neg_neg a]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| a - -b\n[PROOFSTEP]\nrw [\u2190 neg_neg a]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| a - -b\n[PROOFSTEP]\nrw [\u2190 neg_neg a]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n\u22a2 - -a - -b = b + a\n[PROOFSTEP]\nrw [neg_def, neg_def, neg_def, sub_sub_sub]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n\u22a2 0 - 0 - (0 - a - b) = b + a\n[PROOFSTEP]\nconv_lhs =>\n congr\n next => skip\n rw [\u2190 neg_def, neg_sub]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| 0 - 0 - (0 - a - b)\n[PROOFSTEP]\n congr\n next => skip\n rw [\u2190 neg_def, neg_sub]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| 0 - 0 - (0 - a - b)\n[PROOFSTEP]\n congr\n next => skip\n rw [\u2190 neg_def, neg_sub]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| 0 - 0 - (0 - a - b)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| 0 - 0\ncase a C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : NonPreadditiveAbelian C X Y : C a b : X \u27f6 Y | 0 - a - b\n[PROOFSTEP]\nnext => skip\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| 0 - 0\n[PROOFSTEP]\nskip\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| 0 - 0\n[PROOFSTEP]\nskip\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| 0 - 0\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| 0 - a - b\n[PROOFSTEP]\nrw [\u2190 neg_def, neg_sub]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n\u22a2 0 - 0 - (-b - a) = b + a\n[PROOFSTEP]\nrw [sub_sub_sub, add_def, \u2190 neg_def, neg_neg b, neg_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n\u22a2 a + -b = a - b\n[PROOFSTEP]\nrw [add_def, neg_neg]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n\u22a2 a + -a = 0\n[PROOFSTEP]\nrw [add_neg, sub_self]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n\u22a2 -a + a = 0\n[PROOFSTEP]\nrw [add_comm, add_neg_self]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n\u22a2 -(a - b) = -a + b\n[PROOFSTEP]\nrw [neg_def, neg_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n\u22a2 0 - (a - b) = 0 - a + b\n[PROOFSTEP]\nconv_lhs => rw [\u2190 sub_self (0 : X \u27f6 Y)]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| 0 - (a - b)\n[PROOFSTEP]\nrw [\u2190 sub_self (0 : X \u27f6 Y)]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| 0 - (a - b)\n[PROOFSTEP]\nrw [\u2190 sub_self (0 : X \u27f6 Y)]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n| 0 - (a - b)\n[PROOFSTEP]\nrw [\u2190 sub_self (0 : X \u27f6 Y)]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n\u22a2 0 - 0 - (a - b) = 0 - a + b\n[PROOFSTEP]\nrw [sub_sub_sub, add_def, neg_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b : X \u27f6 Y\n\u22a2 -(a + b) = -a - b\n[PROOFSTEP]\nrw [add_def, neg_sub', add_neg]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b c : X \u27f6 Y\n\u22a2 a - b + c = a - (b - c)\n[PROOFSTEP]\nrw [add_def, neg_def, sub_sub_sub, sub_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b c : X \u27f6 Y\n\u22a2 a + b + c = a + (b + c)\n[PROOFSTEP]\nconv_lhs => congr; rw [add_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b c : X \u27f6 Y\n| a + b + c\n[PROOFSTEP]\ncongr; rw [add_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b c : X \u27f6 Y\n| a + b + c\n[PROOFSTEP]\ncongr; rw [add_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b c : X \u27f6 Y\n| a + b + c\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b c : X \u27f6 Y\n| a + b\ncase a C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : NonPreadditiveAbelian C X Y : C a b c : X \u27f6 Y | c\n[PROOFSTEP]\nrw [add_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na b c : X \u27f6 Y\n\u22a2 a - -b + c = a + (b + c)\n[PROOFSTEP]\nrw [sub_add, \u2190 add_neg, neg_sub', neg_neg]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n\u22a2 a + 0 = a\n[PROOFSTEP]\nrw [add_def, neg_def, sub_self, sub_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y Z : C\nf : X \u27f6 Y\ng h : Y \u27f6 Z\n\u22a2 f \u226b (g - h) = f \u226b g - f \u226b h\n[PROOFSTEP]\nrw [sub_def, \u2190 Category.assoc, prod.comp_lift, sub_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 (f - g) \u226b h = f \u226b h - g \u226b h\n[PROOFSTEP]\nrw [sub_def, Category.assoc, \u03c3_comp, \u2190 Category.assoc, prod.lift_map, sub_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y Z : C\nf : X \u27f6 Y\ng h : Y \u27f6 Z\n\u22a2 f \u226b (g + h) = f \u226b g + f \u226b h\n[PROOFSTEP]\nrw [add_def, comp_sub, neg_def, comp_sub, comp_zero, add_def, neg_def]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 (f + g) \u226b h = f \u226b h + g \u226b h\n[PROOFSTEP]\nrw [add_def, sub_comp, neg_def, sub_comp, zero_comp, add_def, neg_def]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Abelian.NonPreadditive", "llama_tokens": 32104, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7879311956428946, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.43688457640494116}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\n\u22a2 EckmannHilton.IsUnital (fun x x_1 => leftAdd X Y x x_1) 0\n[PROOFSTEP]\nhave hr : \u2200 f : X \u27f6 Y, biprod.lift (0 : X \u27f6 Y) f = f \u226b biprod.inr :=\n by\n intro f\n ext\n \u00b7 aesop_cat\n \u00b7 simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\n\u22a2 \u2200 (f : X \u27f6 Y), biprod.lift 0 f = f \u226b biprod.inr\n[PROOFSTEP]\nintro f\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf : X \u27f6 Y\n\u22a2 biprod.lift 0 f = f \u226b biprod.inr\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf : X \u27f6 Y\n\u22a2 biprod.lift 0 f \u226b biprod.fst = (f \u226b biprod.inr) \u226b biprod.fst\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf : X \u27f6 Y\n\u22a2 biprod.lift 0 f \u226b biprod.snd = (f \u226b biprod.inr) \u226b biprod.snd\n[PROOFSTEP]\nsimp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nhr : \u2200 (f : X \u27f6 Y), biprod.lift 0 f = f \u226b biprod.inr\n\u22a2 EckmannHilton.IsUnital (fun x x_1 => leftAdd X Y x x_1) 0\n[PROOFSTEP]\nhave hl : \u2200 f : X \u27f6 Y, biprod.lift f (0 : X \u27f6 Y) = f \u226b biprod.inl :=\n by\n intro f\n ext\n \u00b7 aesop_cat\n \u00b7 simp [biprod.lift_snd, Category.assoc, biprod.inl_snd, comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nhr : \u2200 (f : X \u27f6 Y), biprod.lift 0 f = f \u226b biprod.inr\n\u22a2 \u2200 (f : X \u27f6 Y), biprod.lift f 0 = f \u226b biprod.inl\n[PROOFSTEP]\nintro f\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nhr : \u2200 (f : X \u27f6 Y), biprod.lift 0 f = f \u226b biprod.inr\nf : X \u27f6 Y\n\u22a2 biprod.lift f 0 = f \u226b biprod.inl\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nhr : \u2200 (f : X \u27f6 Y), biprod.lift 0 f = f \u226b biprod.inr\nf : X \u27f6 Y\n\u22a2 biprod.lift f 0 \u226b biprod.fst = (f \u226b biprod.inl) \u226b biprod.fst\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nhr : \u2200 (f : X \u27f6 Y), biprod.lift 0 f = f \u226b biprod.inr\nf : X \u27f6 Y\n\u22a2 biprod.lift f 0 \u226b biprod.snd = (f \u226b biprod.inl) \u226b biprod.snd\n[PROOFSTEP]\nsimp [biprod.lift_snd, Category.assoc, biprod.inl_snd, comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nhr : \u2200 (f : X \u27f6 Y), biprod.lift 0 f = f \u226b biprod.inr\nhl : \u2200 (f : X \u27f6 Y), biprod.lift f 0 = f \u226b biprod.inl\n\u22a2 EckmannHilton.IsUnital (fun x x_1 => leftAdd X Y x x_1) 0\n[PROOFSTEP]\nexact\n \u27e8\u27e8fun f => by simp [hr f, leftAdd, Category.assoc, Category.comp_id, biprod.inr_desc]\u27e9,\n \u27e8fun f => by simp [hl f, leftAdd, Category.assoc, Category.comp_id, biprod.inl_desc]\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nhr : \u2200 (f : X \u27f6 Y), biprod.lift 0 f = f \u226b biprod.inr\nhl : \u2200 (f : X \u27f6 Y), biprod.lift f 0 = f \u226b biprod.inl\nf : X \u27f6 Y\n\u22a2 leftAdd X Y 0 f = f\n[PROOFSTEP]\nsimp [hr f, leftAdd, Category.assoc, Category.comp_id, biprod.inr_desc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nhr : \u2200 (f : X \u27f6 Y), biprod.lift 0 f = f \u226b biprod.inr\nhl : \u2200 (f : X \u27f6 Y), biprod.lift f 0 = f \u226b biprod.inl\nf : X \u27f6 Y\n\u22a2 leftAdd X Y f 0 = f\n[PROOFSTEP]\nsimp [hl f, leftAdd, Category.assoc, Category.comp_id, biprod.inl_desc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\n\u22a2 EckmannHilton.IsUnital (fun x x_1 => rightAdd X Y x x_1) 0\n[PROOFSTEP]\nhave h\u2082 : \u2200 f : X \u27f6 Y, biprod.desc (0 : X \u27f6 Y) f = biprod.snd \u226b f :=\n by\n intro f\n ext\n \u00b7 aesop_cat\n \u00b7 simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\n\u22a2 \u2200 (f : X \u27f6 Y), biprod.desc 0 f = biprod.snd \u226b f\n[PROOFSTEP]\nintro f\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf : X \u27f6 Y\n\u22a2 biprod.desc 0 f = biprod.snd \u226b f\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf : X \u27f6 Y\n\u22a2 biprod.inl \u226b biprod.desc 0 f = biprod.inl \u226b biprod.snd \u226b f\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf : X \u27f6 Y\n\u22a2 biprod.inr \u226b biprod.desc 0 f = biprod.inr \u226b biprod.snd \u226b f\n[PROOFSTEP]\nsimp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh\u2082 : \u2200 (f : X \u27f6 Y), biprod.desc 0 f = biprod.snd \u226b f\n\u22a2 EckmannHilton.IsUnital (fun x x_1 => rightAdd X Y x x_1) 0\n[PROOFSTEP]\nhave h\u2081 : \u2200 f : X \u27f6 Y, biprod.desc f (0 : X \u27f6 Y) = biprod.fst \u226b f :=\n by\n intro f\n ext\n \u00b7 aesop_cat\n \u00b7 simp only [biprod.inr_desc, BinaryBicone.inr_fst_assoc, zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh\u2082 : \u2200 (f : X \u27f6 Y), biprod.desc 0 f = biprod.snd \u226b f\n\u22a2 \u2200 (f : X \u27f6 Y), biprod.desc f 0 = biprod.fst \u226b f\n[PROOFSTEP]\nintro f\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh\u2082 : \u2200 (f : X \u27f6 Y), biprod.desc 0 f = biprod.snd \u226b f\nf : X \u27f6 Y\n\u22a2 biprod.desc f 0 = biprod.fst \u226b f\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh\u2082 : \u2200 (f : X \u27f6 Y), biprod.desc 0 f = biprod.snd \u226b f\nf : X \u27f6 Y\n\u22a2 biprod.inl \u226b biprod.desc f 0 = biprod.inl \u226b biprod.fst \u226b f\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh\u2082 : \u2200 (f : X \u27f6 Y), biprod.desc 0 f = biprod.snd \u226b f\nf : X \u27f6 Y\n\u22a2 biprod.inr \u226b biprod.desc f 0 = biprod.inr \u226b biprod.fst \u226b f\n[PROOFSTEP]\nsimp only [biprod.inr_desc, BinaryBicone.inr_fst_assoc, zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh\u2082 : \u2200 (f : X \u27f6 Y), biprod.desc 0 f = biprod.snd \u226b f\nh\u2081 : \u2200 (f : X \u27f6 Y), biprod.desc f 0 = biprod.fst \u226b f\n\u22a2 EckmannHilton.IsUnital (fun x x_1 => rightAdd X Y x x_1) 0\n[PROOFSTEP]\nexact\n \u27e8\u27e8fun f => by simp [h\u2082 f, rightAdd, biprod.lift_snd_assoc, Category.id_comp]\u27e9,\n \u27e8fun f => by simp [h\u2081 f, rightAdd, biprod.lift_fst_assoc, Category.id_comp]\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh\u2082 : \u2200 (f : X \u27f6 Y), biprod.desc 0 f = biprod.snd \u226b f\nh\u2081 : \u2200 (f : X \u27f6 Y), biprod.desc f 0 = biprod.fst \u226b f\nf : X \u27f6 Y\n\u22a2 rightAdd X Y 0 f = f\n[PROOFSTEP]\nsimp [h\u2082 f, rightAdd, biprod.lift_snd_assoc, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh\u2082 : \u2200 (f : X \u27f6 Y), biprod.desc 0 f = biprod.snd \u226b f\nh\u2081 : \u2200 (f : X \u27f6 Y), biprod.desc f 0 = biprod.fst \u226b f\nf : X \u27f6 Y\n\u22a2 rightAdd X Y f 0 = f\n[PROOFSTEP]\nsimp [h\u2081 f, rightAdd, biprod.lift_fst_assoc, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\n\u22a2 leftAdd X Y (rightAdd X Y f g) (rightAdd X Y h k) = rightAdd X Y (leftAdd X Y f h) (leftAdd X Y g k)\n[PROOFSTEP]\nlet diag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\n\u22a2 leftAdd X Y (rightAdd X Y f g) (rightAdd X Y h k) = rightAdd X Y (leftAdd X Y f h) (leftAdd X Y g k)\n[PROOFSTEP]\nhave hd\u2081 : biprod.inl \u226b diag = biprod.lift f h := by ext <;> simp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\n\u22a2 biprod.inl \u226b diag = biprod.lift f h\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\n\u22a2 (biprod.inl \u226b diag) \u226b biprod.fst = biprod.lift f h \u226b biprod.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\n\u22a2 (biprod.inl \u226b diag) \u226b biprod.snd = biprod.lift f h \u226b biprod.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\nhd\u2081 : biprod.inl \u226b diag = biprod.lift f h\n\u22a2 leftAdd X Y (rightAdd X Y f g) (rightAdd X Y h k) = rightAdd X Y (leftAdd X Y f h) (leftAdd X Y g k)\n[PROOFSTEP]\nhave hd\u2082 : biprod.inr \u226b diag = biprod.lift g k := by ext <;> simp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\nhd\u2081 : biprod.inl \u226b diag = biprod.lift f h\n\u22a2 biprod.inr \u226b diag = biprod.lift g k\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\nhd\u2081 : biprod.inl \u226b diag = biprod.lift f h\n\u22a2 (biprod.inr \u226b diag) \u226b biprod.fst = biprod.lift g k \u226b biprod.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\nhd\u2081 : biprod.inl \u226b diag = biprod.lift f h\n\u22a2 (biprod.inr \u226b diag) \u226b biprod.snd = biprod.lift g k \u226b biprod.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\nhd\u2081 : biprod.inl \u226b diag = biprod.lift f h\nhd\u2082 : biprod.inr \u226b diag = biprod.lift g k\n\u22a2 leftAdd X Y (rightAdd X Y f g) (rightAdd X Y h k) = rightAdd X Y (leftAdd X Y f h) (leftAdd X Y g k)\n[PROOFSTEP]\nhave h\u2081 : biprod.lift (f +\u1d63 g) (h +\u1d63 k) = biprod.lift (\ud835\udfd9 X) (\ud835\udfd9 X) \u226b diag := by ext <;> aesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\nhd\u2081 : biprod.inl \u226b diag = biprod.lift f h\nhd\u2082 : biprod.inr \u226b diag = biprod.lift g k\n\u22a2 biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) = biprod.lift (\ud835\udfd9 X) (\ud835\udfd9 X) \u226b diag\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\nhd\u2081 : biprod.inl \u226b diag = biprod.lift f h\nhd\u2082 : biprod.inr \u226b diag = biprod.lift g k\n\u22a2 biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) \u226b biprod.fst = (biprod.lift (\ud835\udfd9 X) (\ud835\udfd9 X) \u226b diag) \u226b biprod.fst\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\nhd\u2081 : biprod.inl \u226b diag = biprod.lift f h\nhd\u2082 : biprod.inr \u226b diag = biprod.lift g k\n\u22a2 biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) \u226b biprod.snd = (biprod.lift (\ud835\udfd9 X) (\ud835\udfd9 X) \u226b diag) \u226b biprod.snd\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\nhd\u2081 : biprod.inl \u226b diag = biprod.lift f h\nhd\u2082 : biprod.inr \u226b diag = biprod.lift g k\nh\u2081 : biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) = biprod.lift (\ud835\udfd9 X) (\ud835\udfd9 X) \u226b diag\n\u22a2 leftAdd X Y (rightAdd X Y f g) (rightAdd X Y h k) = rightAdd X Y (leftAdd X Y f h) (leftAdd X Y g k)\n[PROOFSTEP]\nhave h\u2082 : diag \u226b biprod.desc (\ud835\udfd9 Y) (\ud835\udfd9 Y) = biprod.desc (f +\u2097 h) (g +\u2097 k) := by\n ext <;> simp [reassoc_of% hd\u2081, reassoc_of% hd\u2082]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\nhd\u2081 : biprod.inl \u226b diag = biprod.lift f h\nhd\u2082 : biprod.inr \u226b diag = biprod.lift g k\nh\u2081 : biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) = biprod.lift (\ud835\udfd9 X) (\ud835\udfd9 X) \u226b diag\n\u22a2 diag \u226b biprod.desc (\ud835\udfd9 Y) (\ud835\udfd9 Y) = biprod.desc (leftAdd X Y f h) (leftAdd X Y g k)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\nhd\u2081 : biprod.inl \u226b diag = biprod.lift f h\nhd\u2082 : biprod.inr \u226b diag = biprod.lift g k\nh\u2081 : biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) = biprod.lift (\ud835\udfd9 X) (\ud835\udfd9 X) \u226b diag\n\u22a2 biprod.inl \u226b diag \u226b biprod.desc (\ud835\udfd9 Y) (\ud835\udfd9 Y) = biprod.inl \u226b biprod.desc (leftAdd X Y f h) (leftAdd X Y g k)\n[PROOFSTEP]\nsimp [reassoc_of% hd\u2081, reassoc_of% hd\u2082]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\nhd\u2081 : biprod.inl \u226b diag = biprod.lift f h\nhd\u2082 : biprod.inr \u226b diag = biprod.lift g k\nh\u2081 : biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) = biprod.lift (\ud835\udfd9 X) (\ud835\udfd9 X) \u226b diag\n\u22a2 biprod.inr \u226b diag \u226b biprod.desc (\ud835\udfd9 Y) (\ud835\udfd9 Y) = biprod.inr \u226b biprod.desc (leftAdd X Y f h) (leftAdd X Y g k)\n[PROOFSTEP]\nsimp [reassoc_of% hd\u2081, reassoc_of% hd\u2082]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nf g h k : X \u27f6 Y\ndiag : X \u229e X \u27f6 Y \u229e Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\nhd\u2081 : biprod.inl \u226b diag = biprod.lift f h\nhd\u2082 : biprod.inr \u226b diag = biprod.lift g k\nh\u2081 : biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) = biprod.lift (\ud835\udfd9 X) (\ud835\udfd9 X) \u226b diag\nh\u2082 : diag \u226b biprod.desc (\ud835\udfd9 Y) (\ud835\udfd9 Y) = biprod.desc (leftAdd X Y f h) (leftAdd X Y g k)\n\u22a2 leftAdd X Y (rightAdd X Y f g) (rightAdd X Y h k) = rightAdd X Y (leftAdd X Y f h) (leftAdd X Y g k)\n[PROOFSTEP]\nrw [leftAdd, h\u2081, Category.assoc, h\u2082, rightAdd]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 (f + g) \u226b h = f \u226b h + g \u226b h\n[PROOFSTEP]\nsimp only [add_eq_right_addition, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 biprod.lift (\ud835\udfd9 X) (\ud835\udfd9 X) \u226b biprod.desc f g \u226b h = biprod.lift (\ud835\udfd9 X) (\ud835\udfd9 X) \u226b biprod.desc (f \u226b h) (g \u226b h)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 biprod.desc f g \u226b h = biprod.desc (f \u226b h) (g \u226b h)\n[PROOFSTEP]\next\n[GOAL]\ncase e_a.h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 biprod.inl \u226b biprod.desc f g \u226b h = biprod.inl \u226b biprod.desc (f \u226b h) (g \u226b h)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e_a.h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 biprod.inr \u226b biprod.desc f g \u226b h = biprod.inr \u226b biprod.desc (f \u226b h) (g \u226b h)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y Z : C\nf : X \u27f6 Y\ng h : Y \u27f6 Z\n\u22a2 f \u226b (g + h) = f \u226b g + f \u226b h\n[PROOFSTEP]\nsimp only [add_eq_left_addition, \u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y Z : C\nf : X \u27f6 Y\ng h : Y \u27f6 Z\n\u22a2 (f \u226b biprod.lift g h) \u226b biprod.desc (\ud835\udfd9 Z) (\ud835\udfd9 Z) = biprod.lift (f \u226b g) (f \u226b h) \u226b biprod.desc (\ud835\udfd9 Z) (\ud835\udfd9 Z)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y Z : C\nf : X \u27f6 Y\ng h : Y \u27f6 Z\n\u22a2 f \u226b biprod.lift g h = biprod.lift (f \u226b g) (f \u226b h)\n[PROOFSTEP]\next\n[GOAL]\ncase e_a.h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y Z : C\nf : X \u27f6 Y\ng h : Y \u27f6 Z\n\u22a2 (f \u226b biprod.lift g h) \u226b biprod.fst = biprod.lift (f \u226b g) (f \u226b h) \u226b biprod.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e_a.h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproducts C\nX Y Z : C\nf : X \u27f6 Y\ng h : Y \u27f6 Z\n\u22a2 (f \u226b biprod.lift g h) \u226b biprod.snd = biprod.lift (f \u226b g) (f \u226b h) \u226b biprod.snd\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Preadditive.OfBiproducts", "llama_tokens": 9361, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850402140659, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.4367259733135803}} {"text": "[GOAL]\n\u03b1 : Type u_1\nX Y : Discrete \u03b1\n\u22a2 Fintype (X \u27f6 Y)\n[PROOFSTEP]\napply ULift.fintype\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.FinCategory", "llama_tokens": 46, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7461389930307512, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.43656678601832916}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d : Countable \u03b2\nm : OuterMeasure \u03b1\ns : \u03b2 \u2192 Set \u03b1\nh : \u2200 (i : \u03b2), \u2191m (s i) = 0\n\u22a2 \u2191m (\u22c3 (i : \u03b2), s i) = 0\n[PROOFSTEP]\nsimpa [h] using m.iUnion s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\n\u03b9 : Prop\ns : \u03b9 \u2192 Set \u03b1\n\u22a2 (\u2200 (i : \u03b9), \u2191m (s i) = 0) \u2192 \u2191m (\u22c3 (i : \u03b9), s i) = 0\n[PROOFSTEP]\nby_cases i : \u03b9\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\n\u03b9 : Prop\ns : \u03b9 \u2192 Set \u03b1\ni : \u03b9\n\u22a2 (\u2200 (i : \u03b9), \u2191m (s i) = 0) \u2192 \u2191m (\u22c3 (i : \u03b9), s i) = 0\n[PROOFSTEP]\nsimp [i]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\n\u03b9 : Prop\ns : \u03b9 \u2192 Set \u03b1\ni : \u00ac\u03b9\n\u22a2 (\u2200 (i : \u03b9), \u2191m (s i) = 0) \u2192 \u2191m (\u22c3 (i : \u03b9), s i) = 0\n[PROOFSTEP]\nsimp [i]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\n\u03b9 : Prop\ns : \u03b9 \u2192 Set \u03b1\ni : \u03b9\n\u22a2 (\u2200 (i : \u03b9), \u2191m (s i) = 0) \u2192 \u2191m (s (_ : \u03b9)) = 0\n[PROOFSTEP]\nexact (fun h => h (Iff.mpr (Iff.of_eq (eq_true i)) trivial))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b2\nhs : Set.Countable s\nt : \u03b2 \u2192 Set \u03b1\n\u22a2 \u2191m (\u22c3 (i : \u03b2) (_ : i \u2208 s), t i) = 0 \u2194 \u2200 (i : \u03b2), i \u2208 s \u2192 \u2191m (t i) = 0\n[PROOFSTEP]\nhaveI := hs.toEncodable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b2\nhs : Set.Countable s\nt : \u03b2 \u2192 Set \u03b1\nthis : Encodable \u2191s\n\u22a2 \u2191m (\u22c3 (i : \u03b2) (_ : i \u2208 s), t i) = 0 \u2194 \u2200 (i : \u03b2), i \u2208 s \u2192 \u2191m (t i) = 0\n[PROOFSTEP]\nrw [biUnion_eq_iUnion, iUnion_null_iff, SetCoe.forall']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\nS : Set (Set \u03b1)\nhS : Set.Countable S\n\u22a2 \u2191m (\u22c3\u2080 S) = 0 \u2194 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191m s = 0\n[PROOFSTEP]\nrw [sUnion_eq_biUnion, m.biUnion_null_iff hS]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nm : OuterMeasure \u03b1\ns : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 \u2191m u = 0\n\u22a2 \u2191m s = 0\n[PROOFSTEP]\nchoose! u hxu hu\u2080 using hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nm : OuterMeasure \u03b1\ns : Set \u03b1\nu : \u03b1 \u2192 Set \u03b1\nhxu : \u2200 (x : \u03b1), x \u2208 s \u2192 u x \u2208 \ud835\udcdd[s] x\nhu\u2080 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2191m (u x) = 0\n\u22a2 \u2191m s = 0\n[PROOFSTEP]\nchoose t ht using TopologicalSpace.countable_cover_nhdsWithin hxu\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nm : OuterMeasure \u03b1\ns : Set \u03b1\nu : \u03b1 \u2192 Set \u03b1\nhxu : \u2200 (x : \u03b1), x \u2208 s \u2192 u x \u2208 \ud835\udcdd[s] x\nhu\u2080 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2191m (u x) = 0\nt : Set \u03b1\nht : t \u2286 s \u2227 Set.Countable t \u2227 s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), u x\n\u22a2 \u2191m s = 0\n[PROOFSTEP]\nrcases ht with \u27e8ts, t_count, ht\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nm : OuterMeasure \u03b1\ns : Set \u03b1\nu : \u03b1 \u2192 Set \u03b1\nhxu : \u2200 (x : \u03b1), x \u2208 s \u2192 u x \u2208 \ud835\udcdd[s] x\nhu\u2080 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2191m (u x) = 0\nt : Set \u03b1\nts : t \u2286 s\nt_count : Set.Countable t\nht : s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), u x\n\u22a2 \u2191m s = 0\n[PROOFSTEP]\napply m.mono_null ht\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nm : OuterMeasure \u03b1\ns : Set \u03b1\nu : \u03b1 \u2192 Set \u03b1\nhxu : \u2200 (x : \u03b1), x \u2208 s \u2192 u x \u2208 \ud835\udcdd[s] x\nhu\u2080 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2191m (u x) = 0\nt : Set \u03b1\nts : t \u2286 s\nt_count : Set.Countable t\nht : s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), u x\n\u22a2 \u2191m (\u22c3 (x : \u03b1) (_ : x \u2208 t), u x) = 0\n[PROOFSTEP]\nexact (m.biUnion_null_iff t_count).2 fun x hx => hu\u2080 x (ts hx)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nm : OuterMeasure \u03b1\ns : Set \u03b1\nhs : \u2191m s \u2260 0\n\u22a2 \u2203 x, x \u2208 s \u2227 \u2200 (t : Set \u03b1), t \u2208 \ud835\udcdd[s] x \u2192 0 < \u2191m t\n[PROOFSTEP]\ncontrapose! hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nm : OuterMeasure \u03b1\ns : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 \u2191m t \u2264 0\n\u22a2 \u2191m s = 0\n[PROOFSTEP]\nsimp only [nonpos_iff_eq_zero] at hs \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nm : OuterMeasure \u03b1\ns : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 \u2191m t = 0\n\u22a2 \u2191m s = 0\n[PROOFSTEP]\nexact m.null_of_locally_null s hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b9 : Type u_5\nm : OuterMeasure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nh0 : Tendsto (fun k => \u2191m ((\u22c3 (n : \u03b9), s n) \\ s k)) l (\ud835\udcdd 0)\n\u22a2 \u2191m (\u22c3 (n : \u03b9), s n) = \u2a06 (n : \u03b9), \u2191m (s n)\n[PROOFSTEP]\nset S := \u22c3 n, s n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b9 : Type u_5\nm : OuterMeasure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nS : Set \u03b1 := \u22c3 (n : \u03b9), s n\nh0 : Tendsto (fun k => \u2191m (S \\ s k)) l (\ud835\udcdd 0)\n\u22a2 \u2191m S = \u2a06 (n : \u03b9), \u2191m (s n)\n[PROOFSTEP]\nset M := \u2a06 n, m (s n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b9 : Type u_5\nm : OuterMeasure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nS : Set \u03b1 := \u22c3 (n : \u03b9), s n\nh0 : Tendsto (fun k => \u2191m (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 (n : \u03b9), \u2191m (s n)\n\u22a2 \u2191m S = M\n[PROOFSTEP]\nhave hsS : \u2200 {k}, s k \u2286 S := fun {k} => subset_iUnion _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b9 : Type u_5\nm : OuterMeasure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nS : Set \u03b1 := \u22c3 (n : \u03b9), s n\nh0 : Tendsto (fun k => \u2191m (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 (n : \u03b9), \u2191m (s n)\nhsS : \u2200 {k : \u03b9}, s k \u2286 S\n\u22a2 \u2191m S = M\n[PROOFSTEP]\nrefine' le_antisymm _ (iSup_le fun n => m.mono hsS)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b9 : Type u_5\nm : OuterMeasure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nS : Set \u03b1 := \u22c3 (n : \u03b9), s n\nh0 : Tendsto (fun k => \u2191m (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 (n : \u03b9), \u2191m (s n)\nhsS : \u2200 {k : \u03b9}, s k \u2286 S\n\u22a2 \u2191m S \u2264 M\n[PROOFSTEP]\nhave A : \u2200 k, m S \u2264 M + m (S \\ s k) := fun k =>\n calc\n m S = m (s k \u222a S \\ s k) := by rw [union_diff_self, union_eq_self_of_subset_left hsS]\n _ \u2264 m (s k) + m (S \\ s k) := (m.union _ _)\n _ \u2264 M + m (S \\ s k) := add_le_add_right (le_iSup (m.measureOf \u2218 s) k) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b9 : Type u_5\nm : OuterMeasure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nS : Set \u03b1 := \u22c3 (n : \u03b9), s n\nh0 : Tendsto (fun k => \u2191m (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 (n : \u03b9), \u2191m (s n)\nhsS : \u2200 {k : \u03b9}, s k \u2286 S\nk : \u03b9\n\u22a2 \u2191m S = \u2191m (s k \u222a S \\ s k)\n[PROOFSTEP]\nrw [union_diff_self, union_eq_self_of_subset_left hsS]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b9 : Type u_5\nm : OuterMeasure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nS : Set \u03b1 := \u22c3 (n : \u03b9), s n\nh0 : Tendsto (fun k => \u2191m (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 (n : \u03b9), \u2191m (s n)\nhsS : \u2200 {k : \u03b9}, s k \u2286 S\nA : \u2200 (k : \u03b9), \u2191m S \u2264 M + \u2191m (S \\ s k)\n\u22a2 \u2191m S \u2264 M\n[PROOFSTEP]\nhave B : Tendsto (fun k => M + m (S \\ s k)) l (\ud835\udcdd (M + 0)) := tendsto_const_nhds.add h0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b9 : Type u_5\nm : OuterMeasure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nS : Set \u03b1 := \u22c3 (n : \u03b9), s n\nh0 : Tendsto (fun k => \u2191m (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 (n : \u03b9), \u2191m (s n)\nhsS : \u2200 {k : \u03b9}, s k \u2286 S\nA : \u2200 (k : \u03b9), \u2191m S \u2264 M + \u2191m (S \\ s k)\nB : Tendsto (fun k => M + \u2191m (S \\ s k)) l (\ud835\udcdd (M + 0))\n\u22a2 \u2191m S \u2264 M\n[PROOFSTEP]\nrw [add_zero] at B \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b9 : Type u_5\nm : OuterMeasure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nS : Set \u03b1 := \u22c3 (n : \u03b9), s n\nh0 : Tendsto (fun k => \u2191m (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 (n : \u03b9), \u2191m (s n)\nhsS : \u2200 {k : \u03b9}, s k \u2286 S\nA : \u2200 (k : \u03b9), \u2191m S \u2264 M + \u2191m (S \\ s k)\nB : Tendsto (fun k => M + \u2191m (S \\ s k)) l (\ud835\udcdd M)\n\u22a2 \u2191m S \u2264 M\n[PROOFSTEP]\nexact ge_of_tendsto' B A\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\n\u22a2 \u2191m (\u22c3 (n : \u2115), s n) = \u2a06 (n : \u2115), \u2191m (s n)\n[PROOFSTEP]\nrefine' m.iUnion_of_tendsto_zero atTop _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\n\u22a2 Tendsto (fun k => \u2191m ((\u22c3 (n : \u2115), s n) \\ s k)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\n\u22a2 \u2191m ((\u22c3 (n : \u2115), s n) \\ s n) \u2264 \u2211' (k : \u2115), \u2191m (s (k + n + 1) \\ s (k + n))\n[PROOFSTEP]\nrefine'\n (m.mono _).trans\n (m.iUnion _)\n -- Current goal: `(\u22c3 k, s k) \\ s n \u2286 \u22c3 k, s (k + n + 1) \\ s (k + n)`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\n\u22a2 (\u22c3 (n : \u2115), s n) \\ s n \u2286 \u22c3 (i : \u2115), s (i + n + 1) \\ s (i + n)\n[PROOFSTEP]\nhave h' : Monotone s := @monotone_nat_of_le_succ (Set \u03b1) _ _ h_mono\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\n\u22a2 (\u22c3 (n : \u2115), s n) \\ s n \u2286 \u22c3 (i : \u2115), s (i + n + 1) \\ s (i + n)\n[PROOFSTEP]\nsimp only [diff_subset_iff, iUnion_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\n\u22a2 \u2200 (i : \u2115), s i \u2286 s n \u222a \u22c3 (i : \u2115), s (i + n + 1) \\ s (i + n)\n[PROOFSTEP]\nintro i x hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\n\u22a2 x \u2208 s n \u222a \u22c3 (i : \u2115), s (i + n + 1) \\ s (i + n)\n[PROOFSTEP]\nhave : \u2203 i, x \u2208 s i := by exists i\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\n\u22a2 \u2203 i, x \u2208 s i\n[PROOFSTEP]\nexists i\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\nthis : \u2203 i, x \u2208 s i\n\u22a2 x \u2208 s n \u222a \u22c3 (i : \u2115), s (i + n + 1) \\ s (i + n)\n[PROOFSTEP]\nrcases Nat.findX this with \u27e8j, hj, hlt\u27e9\n[GOAL]\ncase mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\ni : \u2115\nx : \u03b1\nhx : x \u2208 s i\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\n\u22a2 x \u2208 s n \u222a \u22c3 (i : \u2115), s (i + n + 1) \\ s (i + n)\n[PROOFSTEP]\nclear hx i\n[GOAL]\ncase mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\n\u22a2 x \u2208 s n \u222a \u22c3 (i : \u2115), s (i + n + 1) \\ s (i + n)\n[PROOFSTEP]\ncases' le_or_lt j n with hjn hnj\n[GOAL]\ncase mk.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhjn : j \u2264 n\n\u22a2 x \u2208 s n \u222a \u22c3 (i : \u2115), s (i + n + 1) \\ s (i + n)\n[PROOFSTEP]\nexact Or.inl (h' hjn hj)\n[GOAL]\ncase mk.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhnj : n < j\n\u22a2 x \u2208 s n \u222a \u22c3 (i : \u2115), s (i + n + 1) \\ s (i + n)\n[PROOFSTEP]\nhave : j - (n + 1) + n + 1 = j := by rw [add_assoc, tsub_add_cancel_of_le hnj.nat_succ_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhnj : n < j\n\u22a2 j - (n + 1) + n + 1 = j\n[PROOFSTEP]\nrw [add_assoc, tsub_add_cancel_of_le hnj.nat_succ_le]\n[GOAL]\ncase mk.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis\u271d : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n\u22a2 x \u2208 s n \u222a \u22c3 (i : \u2115), s (i + n + 1) \\ s (i + n)\n[PROOFSTEP]\nrefine' Or.inr (mem_iUnion.2 \u27e8j - (n + 1), _, hlt _ _\u27e9)\n[GOAL]\ncase mk.intro.inr.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis\u271d : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n\u22a2 x \u2208 s (j - (n + 1) + n + 1)\n[PROOFSTEP]\nrwa [this]\n[GOAL]\ncase mk.intro.inr.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\nh_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1)\nh0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4\ninst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i\nn : \u2115\nh' : Monotone s\nx : \u03b1\nthis\u271d : \u2203 i, x \u2208 s i\nj : \u2115\nhj : x \u2208 s j\nhlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n\u22a2 j - (n + 1) + n < j\n[PROOFSTEP]\nrw [\u2190 Nat.succ_le_iff, Nat.succ_eq_add_one, this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\nt s : Set \u03b1\n\u22a2 \u2191m t \u2264 \u2191m (t \u2229 s) + \u2191m (t \\ s)\n[PROOFSTEP]\nconvert m.union _ _\n[GOAL]\ncase h.e'_3.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\nt s : Set \u03b1\n\u22a2 t = t \u2229 s \u222a t \\ s\n[PROOFSTEP]\nrw [inter_union_diff t s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns t : Set \u03b1\nht : \u2191m t = 0\n\u22a2 \u2191m (s \\ t) = \u2191m s\n[PROOFSTEP]\nrefine' le_antisymm (m.mono <| diff_subset _ _) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns t : Set \u03b1\nht : \u2191m t = 0\n\u22a2 \u2191m s \u2264 \u2191m (s \\ t)\n[PROOFSTEP]\ncalc\n m s \u2264 m (s \u2229 t) + m (s \\ t) := le_inter_add_diff _\n _ \u2264 m t + m (s \\ t) := (add_le_add_right (m.mono <| inter_subset_right _ _) _)\n _ = m (s \\ t) := by rw [ht, zero_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns t : Set \u03b1\nht : \u2191m t = 0\n\u22a2 \u2191m t + \u2191m (s \\ t) = \u2191m (s \\ t)\n[PROOFSTEP]\nrw [ht, zero_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : \u2191m s\u2081 = 0\nh\u2082 : \u2191m s\u2082 = 0\n\u22a2 \u2191m (s\u2081 \u222a s\u2082) = 0\n[PROOFSTEP]\nsimpa [h\u2081, h\u2082] using m.union s\u2081 s\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm \u03bc\u2081 \u03bc\u2082 : OuterMeasure \u03b1\nh : (fun \u03bc s => \u2191\u03bc s) \u03bc\u2081 = (fun \u03bc s => \u2191\u03bc s) \u03bc\u2082\n\u22a2 \u03bc\u2081 = \u03bc\u2082\n[PROOFSTEP]\ncases \u03bc\u2081\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm \u03bc\u2082 : OuterMeasure \u03b1\nmeasureOf\u271d : Set \u03b1 \u2192 \u211d\u22650\u221e\nempty\u271d : measureOf\u271d \u2205 = 0\nmono\u271d : \u2200 {s\u2081 s\u2082 : Set \u03b1}, s\u2081 \u2286 s\u2082 \u2192 measureOf\u271d s\u2081 \u2264 measureOf\u271d s\u2082\niUnion_nat\u271d : \u2200 (s : \u2115 \u2192 Set \u03b1), measureOf\u271d (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), measureOf\u271d (s i)\nh :\n (fun \u03bc s => \u2191\u03bc s) { measureOf := measureOf\u271d, empty := empty\u271d, mono := mono\u271d, iUnion_nat := iUnion_nat\u271d } =\n (fun \u03bc s => \u2191\u03bc s) \u03bc\u2082\n\u22a2 { measureOf := measureOf\u271d, empty := empty\u271d, mono := mono\u271d, iUnion_nat := iUnion_nat\u271d } = \u03bc\u2082\n[PROOFSTEP]\ncases \u03bc\u2082\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\nmeasureOf\u271d\u00b9 : Set \u03b1 \u2192 \u211d\u22650\u221e\nempty\u271d\u00b9 : measureOf\u271d\u00b9 \u2205 = 0\nmono\u271d\u00b9 : \u2200 {s\u2081 s\u2082 : Set \u03b1}, s\u2081 \u2286 s\u2082 \u2192 measureOf\u271d\u00b9 s\u2081 \u2264 measureOf\u271d\u00b9 s\u2082\niUnion_nat\u271d\u00b9 : \u2200 (s : \u2115 \u2192 Set \u03b1), measureOf\u271d\u00b9 (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), measureOf\u271d\u00b9 (s i)\nmeasureOf\u271d : Set \u03b1 \u2192 \u211d\u22650\u221e\nempty\u271d : measureOf\u271d \u2205 = 0\nmono\u271d : \u2200 {s\u2081 s\u2082 : Set \u03b1}, s\u2081 \u2286 s\u2082 \u2192 measureOf\u271d s\u2081 \u2264 measureOf\u271d s\u2082\niUnion_nat\u271d : \u2200 (s : \u2115 \u2192 Set \u03b1), measureOf\u271d (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), measureOf\u271d (s i)\nh :\n (fun \u03bc s => \u2191\u03bc s) { measureOf := measureOf\u271d\u00b9, empty := empty\u271d\u00b9, mono := mono\u271d\u00b9, iUnion_nat := iUnion_nat\u271d\u00b9 } =\n (fun \u03bc s => \u2191\u03bc s) { measureOf := measureOf\u271d, empty := empty\u271d, mono := mono\u271d, iUnion_nat := iUnion_nat\u271d }\n\u22a2 { measureOf := measureOf\u271d\u00b9, empty := empty\u271d\u00b9, mono := mono\u271d\u00b9, iUnion_nat := iUnion_nat\u271d\u00b9 } =\n { measureOf := measureOf\u271d, empty := empty\u271d, mono := mono\u271d, iUnion_nat := iUnion_nat\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm \u03bc\u2081 \u03bc\u2082 : OuterMeasure \u03b1\nh : \u2200 (s : Set \u03b1), Set.Nonempty s \u2192 \u2191\u03bc\u2081 s = \u2191\u03bc\u2082 s\ns : Set \u03b1\nhe : s = \u2205\n\u22a2 \u2191\u03bc\u2081 s = \u2191\u03bc\u2082 s\n[PROOFSTEP]\nrw [he, empty', empty']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\n\u22a2 \u2200 {s\u2081 s\u2082 : Set \u03b1}, s\u2081 \u2286 s\u2082 \u2192 (fun x => 0) s\u2081 \u2264 (fun x => 0) s\u2082\n[PROOFSTEP]\nintro _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\ns\u2081\u271d s\u2082\u271d : Set \u03b1\na\u271d : s\u2081\u271d \u2286 s\u2082\u271d\n\u22a2 (fun x => 0) s\u2081\u271d \u2264 (fun x => 0) s\u2082\u271d\n[PROOFSTEP]\nexact le_refl 0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm m\u2081 m\u2082 : OuterMeasure \u03b1\n\u22a2 \u2191m\u2081 \u2205 + \u2191m\u2082 \u2205 = 0\n[PROOFSTEP]\nsimp [OuterMeasure.empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : SMul R' \u211d\u22650\u221e\ninst\u271d : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\nc : R\nm : OuterMeasure \u03b1\n\u22a2 (fun s => c \u2022 \u2191m s) \u2205 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : SMul R' \u211d\u22650\u221e\ninst\u271d : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\nc : R\nm : OuterMeasure \u03b1\n\u22a2 c \u2022 0 = 0\n[PROOFSTEP]\nrw [\u2190 smul_one_mul c]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : SMul R' \u211d\u22650\u221e\ninst\u271d : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\nc : R\nm : OuterMeasure \u03b1\n\u22a2 c \u2022 1 * 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : SMul R' \u211d\u22650\u221e\ninst\u271d : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\nc : R\nm : OuterMeasure \u03b1\ns t : Set \u03b1\nh : s \u2286 t\n\u22a2 (fun s => c \u2022 \u2191m s) s \u2264 (fun s => c \u2022 \u2191m s) t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : SMul R' \u211d\u22650\u221e\ninst\u271d : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\nc : R\nm : OuterMeasure \u03b1\ns t : Set \u03b1\nh : s \u2286 t\n\u22a2 c \u2022 \u2191m s \u2264 c \u2022 \u2191m t\n[PROOFSTEP]\nrw [\u2190 smul_one_mul c, \u2190 smul_one_mul c (m t)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : SMul R' \u211d\u22650\u221e\ninst\u271d : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\nc : R\nm : OuterMeasure \u03b1\ns t : Set \u03b1\nh : s \u2286 t\n\u22a2 c \u2022 1 * \u2191m s \u2264 c \u2022 1 * \u2191m t\n[PROOFSTEP]\nexact ENNReal.mul_left_mono (m.mono h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : SMul R' \u211d\u22650\u221e\ninst\u271d : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\nc : R\nm : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\n\u22a2 (fun s => c \u2022 \u2191m s) (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), (fun s => c \u2022 \u2191m s) (s i)\n[PROOFSTEP]\nsimp_rw [\u2190 smul_one_mul c (m _), ENNReal.tsum_mul_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : SMul R' \u211d\u22650\u221e\ninst\u271d : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\nc : R\nm : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\n\u22a2 c \u2022 1 * \u2191m (\u22c3 (i : \u2115), s i) \u2264 c \u2022 1 * \u2211' (i : \u2115), \u2191m (s i)\n[PROOFSTEP]\nexact ENNReal.mul_left_mono (m.iUnion_nat _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm a : OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191\u22a5 s \u2264 \u2191a s\n[PROOFSTEP]\nsimp only [coe_zero, Pi.zero_apply, coe_bot, zero_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms\u271d : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\nms : Set (OuterMeasure \u03b1)\nf : \u2115 \u2192 Set \u03b1\nm : OuterMeasure \u03b1\nhm : m \u2208 ms\ni : \u2115\n\u22a2 \u2191m (f i) \u2264 \u2a06 (m : OuterMeasure \u03b1) (_ : m \u2208 ms), \u2191m (f i)\n[PROOFSTEP]\napply le_iSup\u2082 m hm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms\u271d : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\nsrc\u271d : OrderBot (OuterMeasure \u03b1) := OuterMeasure.orderBot\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\nhm : m \u2208 ms\ns : Set \u03b1\n\u22a2 \u2191m s \u2264 \u2191(sSup ms) s\n[PROOFSTEP]\napply le_iSup\u2082 m hm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_5\nf : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191(\u2a06 (i : \u03b9), f i) s = \u2a06 (i : \u03b9), \u2191(f i) s\n[PROOFSTEP]\nrw [iSup, sSup_apply, iSup_range, iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_5\nf : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191(\u2a06 (i : \u03b9), f i) s = iSup (fun i => \u2191(f i)) s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm m\u2081 m\u2082 : OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191(m\u2081 \u2294 m\u2082) s = \u2191m\u2081 s \u2294 \u2191m\u2082 s\n[PROOFSTEP]\nhave := iSup_apply (fun b => cond b m\u2081 m\u2082) s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm m\u2081 m\u2082 : OuterMeasure \u03b1\ns : Set \u03b1\nthis : \u2191(\u2a06 (i : Bool), bif i then m\u2081 else m\u2082) s = \u2a06 (i : Bool), \u2191(bif i then m\u2081 else m\u2082) s\n\u22a2 \u2191(m\u2081 \u2294 m\u2082) s = \u2191m\u2081 s \u2294 \u2191m\u2082 s\n[PROOFSTEP]\nrwa [iSup_bool_eq, iSup_bool_eq] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\ninst\u271d\u00b9 : SMul R \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\n\u03b9 : Sort u_5\nf : \u03b9 \u2192 OuterMeasure \u03b1\nc : R\ns : Set \u03b1\n\u22a2 \u2191(c \u2022 \u2a06 (i : \u03b9), f i) s = \u2191(\u2a06 (i : \u03b9), c \u2022 f i) s\n[PROOFSTEP]\nsimp only [smul_apply, iSup_apply, ENNReal.smul_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b2 : Type ?u.69709\nf : \u03b1 \u2192 \u03b2\nm : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b2\n\u22a2 (fun s => \u2191m (f \u207b\u00b9' s)) (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), (fun s => \u2191m (f \u207b\u00b9' s)) (s i)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b2 : Type ?u.69709\nf : \u03b1 \u2192 \u03b2\nm : OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b2\n\u22a2 \u2191m (\u22c3 (i : \u2115), f \u207b\u00b9' s i) \u2264 \u2211' (i : \u2115), \u2191m (f \u207b\u00b9' s i)\n[PROOFSTEP]\napply m.iUnion_nat fun i => f \u207b\u00b9' s i\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b2 : Type u_5\nf : \u03b1 \u2192 \u03b2\nm m' : OuterMeasure \u03b1\ns : Set \u03b2\n\u22a2 \u2191(\u2191(map f) (m \u2294 m')) s = \u2191(\u2191(map f) m \u2294 \u2191(map f) m') s\n[PROOFSTEP]\nsimp only [map_apply, sup_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b2 : Type u_5\n\u03b9 : Sort u_6\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b2\n\u22a2 \u2191(\u2191(map f) (\u2a06 (i : \u03b9), m i)) s = \u2191(\u2a06 (i : \u03b9), \u2191(map f) (m i)) s\n[PROOFSTEP]\nsimp only [map_apply, iSup_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\n\u22a2 LawfulFunctor OuterMeasure\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase map_const\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.80543}, Functor.mapConst = Functor.map \u2218 const \u03b2\n[PROOFSTEP]\nintros\n[GOAL]\ncase id_map\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\n\u22a2 \u2200 {\u03b1 : Type ?u.80543} (x : OuterMeasure \u03b1), id <$> x = x\n[PROOFSTEP]\nintros\n[GOAL]\ncase comp_map\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\n\u22a2 \u2200 {\u03b1 \u03b2 \u03b3 : Type ?u.80543} (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 \u03b3) (x : OuterMeasure \u03b1), (h \u2218 g) <$> x = h <$> g <$> x\n[PROOFSTEP]\nintros\n[GOAL]\ncase map_const\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.80543\n\u22a2 Functor.mapConst = Functor.map \u2218 const \u03b2\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase id_map\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\n\u03b1\u271d : Type ?u.80543\nx\u271d : OuterMeasure \u03b1\u271d\n\u22a2 id <$> x\u271d = x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase comp_map\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.80543\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nh\u271d : \u03b2\u271d \u2192 \u03b3\u271d\nx\u271d : OuterMeasure \u03b1\u271d\n\u22a2 (h\u271d \u2218 g\u271d) <$> x\u271d = h\u271d <$> g\u271d <$> x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\na : \u03b1\n\u22a2 (fun s => indicator s (fun x => 1) a) \u2205 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\na : \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u00aca \u2208 \u22c3 (n : \u2115), s n\n\u22a2 (fun s => indicator s (fun x => 1) a) (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), (fun s => indicator s (fun x => 1) a) (s i)\n[PROOFSTEP]\nsimp only [indicator_of_not_mem hs, zero_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\n\u03b9 : Type ?u.84633\nf : \u03b9 \u2192 OuterMeasure \u03b1\n\u22a2 (fun s => \u2211' (i : \u03b9), \u2191(f i) s) \u2205 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\n\u03b9 : Type ?u.84633\nf : \u03b9 \u2192 OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\n\u22a2 (fun s => \u2211' (i : \u03b9), \u2191(f i) s) (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), (fun s => \u2211' (i : \u03b9), \u2191(f i) s) (s i)\n[PROOFSTEP]\nrw [ENNReal.tsum_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\n\u03b9 : Type ?u.84633\nf : \u03b9 \u2192 OuterMeasure \u03b1\ns : \u2115 \u2192 Set \u03b1\n\u22a2 (fun s => \u2211' (i : \u03b9), \u2191(f i) s) (\u22c3 (i : \u2115), s i) \u2264 \u2211' (b : \u03b9) (a : \u2115), \u2191(f b) (s a)\n[PROOFSTEP]\nexact ENNReal.tsum_le_tsum fun i => (f i).iUnion_nat _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\na : \u211d\u22650\u221e\nb : \u03b1\ns : Set \u03b1\n\u22a2 \u2191(a \u2022 dirac b) s = indicator s (fun x => a) b\n[PROOFSTEP]\nsimp only [smul_apply, smul_eq_mul, dirac_apply, \u2190 indicator_mul_right _ fun _ => a, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b2 : Type ?u.88170\nf : \u03b1 \u2192 \u03b2\nm : OuterMeasure \u03b2\n\u22a2 (fun s => \u2191m (f '' s)) \u2205 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b2 : Type ?u.88170\nf : \u03b1 \u2192 \u03b2\nm : OuterMeasure \u03b2\ns : \u2115 \u2192 Set \u03b1\n\u22a2 (fun s => \u2191m (f '' s)) (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), (fun s => \u2191m (f '' s)) (s i)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b2 : Type ?u.88170\nf : \u03b1 \u2192 \u03b2\nm : OuterMeasure \u03b2\ns : \u2115 \u2192 Set \u03b1\n\u22a2 \u2191m (f '' \u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), \u2191m (f '' s i)\n[PROOFSTEP]\nrw [image_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b2 : Type ?u.88170\nf : \u03b1 \u2192 \u03b2\nm : OuterMeasure \u03b2\ns : \u2115 \u2192 Set \u03b1\n\u22a2 \u2191m (\u22c3 (i : \u2115), f '' s i) \u2264 \u2211' (i : \u2115), \u2191m (f '' s i)\n[PROOFSTEP]\napply m.iUnion_nat\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b2 : Type u_5\n\u03b9 : Sort u_6\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b1\n\u22a2 \u2191(\u2191(comap f) (\u2a06 (i : \u03b9), m i)) s = \u2191(\u2a06 (i : \u03b9), \u2191(comap f) (m i)) s\n[PROOFSTEP]\nsimp only [comap_apply, iSup_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ns t : Set \u03b1\nm : OuterMeasure \u03b1\n\u22a2 \u2191(\u2191(restrict s) m) t = \u2191m (t \u2229 s)\n[PROOFSTEP]\nsimp [restrict]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ns t : Set \u03b1\nh : s \u2286 t\nm m' : OuterMeasure \u03b1\nhm : m \u2264 m'\nu : Set \u03b1\n\u22a2 \u2191(\u2191(restrict s) m) u \u2264 \u2191(\u2191(restrict t) m') u\n[PROOFSTEP]\nsimp only [restrict_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\ns t : Set \u03b1\nh : s \u2286 t\nm m' : OuterMeasure \u03b1\nhm : m \u2264 m'\nu : Set \u03b1\n\u22a2 \u2191m (u \u2229 s) \u2264 \u2191m' (u \u2229 t)\n[PROOFSTEP]\nexact (hm _).trans (m'.mono <| inter_subset_inter_right _ h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191(\u2191(restrict univ) m) s = \u2191m s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191(\u2191(restrict \u2205) m) s = \u21910 s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b9 : Sort u_5\ns : Set \u03b1\nm : \u03b9 \u2192 OuterMeasure \u03b1\n\u22a2 \u2191(restrict s) (\u2a06 (i : \u03b9), m i) = \u2a06 (i : \u03b9), \u2191(restrict s) (m i)\n[PROOFSTEP]\nsimp [restrict]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b2 : Type u_5\nf : \u03b1 \u2192 \u03b2\nm : OuterMeasure \u03b2\ns : Set \u03b2\n\u22a2 f '' (f \u207b\u00b9' s) = Subtype.val '' (Subtype.val \u207b\u00b9' s)\n[PROOFSTEP]\nsimp only [image_preimage_eq_inter_range, Subtype.range_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\n\u03b2 : Type u_5\nma : OuterMeasure \u03b1\nmb : OuterMeasure \u03b2\nf : \u03b1 \u2192 \u03b2\nh : \u2191(map f) ma \u2264 mb\ns : Set \u03b2\n\u22a2 \u2191(\u2191(map f) ma) s \u2264 \u2191(\u2191(restrict (range f)) mb) s\n[PROOFSTEP]\nsimpa using h (s \u2229 range f)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b2 : Type u_5\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nm : OuterMeasure \u03b2\ns : Set \u03b2\n\u22a2 \u2191(\u2191(map f) (\u2191(comap f) m)) s = \u2191m s\n[PROOFSTEP]\nrw [map_apply, comap_apply, hf.image_preimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm\u271d : OuterMeasure \u03b1\n\u03b2 : Type u_5\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nm : OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191(\u2191(comap f) (\u2191(map f) m)) s = \u2191m s\n[PROOFSTEP]\nrw [comap_apply, map_apply, hf.preimage_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\ns : Set \u03b1\nh : Set.Nonempty s\na : \u03b1\nas : a \u2208 s\n\u22a2 \u22a4 \u2264 \u2191(\u22a4 \u2022 dirac a) s\n[PROOFSTEP]\nsimp [smul_dirac_apply, as]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\ns : Set \u03b1\nh : s = \u2205\n\u22a2 \u2191\u22a4 s = \u2a05 (_ : s = \u2205), 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\ns : Set \u03b1\nh : Set.Nonempty s\n\u22a2 \u2191\u22a4 s = \u2a05 (_ : s = \u2205), 0\n[PROOFSTEP]\nsimp [h, h.ne_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2191(\u2191(comap f) \u22a4) s = \u2191\u22a4 s\n[PROOFSTEP]\nrw [comap_apply, top_apply hs, top_apply (hs.image _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u22a2 \u2191(\u2191(map f) \u22a4) s = \u2191(\u2191(restrict (range f)) \u22a4) s\n[PROOFSTEP]\nrw [map_apply, restrict_apply, \u2190 image_preimage_eq_inter_range, top_apply', top_apply', Set.image_eq_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nms : Set (OuterMeasure \u03b1)\nm : OuterMeasure \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\n\u22a2 \u2191(map f) \u22a4 = \u22a4\n[PROOFSTEP]\nrw [map_top, hf.range_eq, restrict_univ]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\n\u22a2 \u2211' (i : \u2115), m ((fun x => \u2205) i) \u2264 0\n[PROOFSTEP]\nsimp [m_empty]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u22a2 \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2211' (i : \u2115), \u03bc (s i) < \u22a4 \u2192 \u03bc (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), \u03bc (s i) + \u2191\u03b5\n[PROOFSTEP]\nintro \u03b5 h\u03b5 (hb : (\u2211' i, \u03bc (s i)) < \u221e)\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u22a2 \u03bc (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), \u03bc (s i) + \u2191\u03b5\n[PROOFSTEP]\nrcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 h\u03b5).ne' \u2115 with \u27e8\u03b5', h\u03b5', hl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\n\u22a2 \u03bc (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), \u03bc (s i) + \u2191\u03b5\n[PROOFSTEP]\nrefine' le_trans _ (add_le_add_left (le_of_lt hl) _)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\n\u22a2 \u03bc (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), \u03bc (s i) + \u2211' (i : \u2115), \u2191(\u03b5' i)\n[PROOFSTEP]\nrw [\u2190 ENNReal.tsum_add]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\n\u22a2 \u03bc (\u22c3 (i : \u2115), s i) \u2264 \u2211' (a : \u2115), (\u03bc (s a) + \u2191(\u03b5' a))\n[PROOFSTEP]\nchoose f hf using\n show \u2200 i, \u2203 f : \u2115 \u2192 Set \u03b1, (s i \u2286 \u22c3 i, f i) \u2227 (\u2211' i, m (f i)) < \u03bc (s i) + \u03b5' i\n by\n intro i\n have : \u03bc (s i) < \u03bc (s i) + \u03b5' i :=\n ENNReal.lt_add_right (ne_top_of_le_ne_top hb.ne <| ENNReal.le_tsum _) (by simpa using (h\u03b5' i).ne')\n rcases iInf_lt_iff.mp this with \u27e8t, ht\u27e9\n exists t\n contrapose! ht\n exact le_iInf ht\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\n\u22a2 \u2200 (i : \u2115), \u2203 f, s i \u2286 \u22c3 (i : \u2115), f i \u2227 \u2211' (i : \u2115), m (f i) < \u03bc (s i) + \u2191(\u03b5' i)\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\ni : \u2115\n\u22a2 \u2203 f, s i \u2286 \u22c3 (i : \u2115), f i \u2227 \u2211' (i : \u2115), m (f i) < \u03bc (s i) + \u2191(\u03b5' i)\n[PROOFSTEP]\nhave : \u03bc (s i) < \u03bc (s i) + \u03b5' i :=\n ENNReal.lt_add_right (ne_top_of_le_ne_top hb.ne <| ENNReal.le_tsum _) (by simpa using (h\u03b5' i).ne')\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\ni : \u2115\n\u22a2 \u2191(\u03b5' i) \u2260 0\n[PROOFSTEP]\nsimpa using (h\u03b5' i).ne'\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\ni : \u2115\nthis : \u03bc (s i) < \u03bc (s i) + \u2191(\u03b5' i)\n\u22a2 \u2203 f, s i \u2286 \u22c3 (i : \u2115), f i \u2227 \u2211' (i : \u2115), m (f i) < \u03bc (s i) + \u2191(\u03b5' i)\n[PROOFSTEP]\nrcases iInf_lt_iff.mp this with \u27e8t, ht\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\ni : \u2115\nthis : \u03bc (s i) < \u03bc (s i) + \u2191(\u03b5' i)\nt : \u2115 \u2192 Set \u03b1\nht : \u2a05 (_ : s i \u2286 \u22c3 (i : \u2115), t i), \u2211' (i : \u2115), m (t i) < \u03bc (s i) + \u2191(\u03b5' i)\n\u22a2 \u2203 f, s i \u2286 \u22c3 (i : \u2115), f i \u2227 \u2211' (i : \u2115), m (f i) < \u03bc (s i) + \u2191(\u03b5' i)\n[PROOFSTEP]\nexists t\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\ni : \u2115\nthis : \u03bc (s i) < \u03bc (s i) + \u2191(\u03b5' i)\nt : \u2115 \u2192 Set \u03b1\nht : \u2a05 (_ : s i \u2286 \u22c3 (i : \u2115), t i), \u2211' (i : \u2115), m (t i) < \u03bc (s i) + \u2191(\u03b5' i)\n\u22a2 s i \u2286 \u22c3 (i : \u2115), t i \u2227 \u2211' (i : \u2115), m (t i) < \u03bc (s i) + \u2191(\u03b5' i)\n[PROOFSTEP]\ncontrapose! ht\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\ni : \u2115\nthis : \u03bc (s i) < \u03bc (s i) + \u2191(\u03b5' i)\nt : \u2115 \u2192 Set \u03b1\nht :\n s i \u2286 \u22c3 (i : \u2115), t i \u2192\n (fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)) (s i) + \u2191(\u03b5' i) \u2264 \u2211' (i : \u2115), m (t i)\n\u22a2 (fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)) (s i) + \u2191(\u03b5' i) \u2264\n \u2a05 (_ : s i \u2286 \u22c3 (i : \u2115), t i), \u2211' (i : \u2115), m (t i)\n[PROOFSTEP]\nexact le_iInf ht\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1\nhf : \u2200 (i : \u2115), s i \u2286 \u22c3 (i_1 : \u2115), f i i_1 \u2227 \u2211' (i_1 : \u2115), m (f i i_1) < \u03bc (s i) + \u2191(\u03b5' i)\n\u22a2 \u03bc (\u22c3 (i : \u2115), s i) \u2264 \u2211' (a : \u2115), (\u03bc (s a) + \u2191(\u03b5' a))\n[PROOFSTEP]\nrefine' le_trans _ (ENNReal.tsum_le_tsum fun i => le_of_lt (hf i).2)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1\nhf : \u2200 (i : \u2115), s i \u2286 \u22c3 (i_1 : \u2115), f i i_1 \u2227 \u2211' (i_1 : \u2115), m (f i i_1) < \u03bc (s i) + \u2191(\u03b5' i)\n\u22a2 \u03bc (\u22c3 (i : \u2115), s i) \u2264 \u2211' (a : \u2115) (i : \u2115), m (f a i)\n[PROOFSTEP]\nrw [\u2190 ENNReal.tsum_prod, \u2190 Nat.pairEquiv.symm.tsum_eq]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1\nhf : \u2200 (i : \u2115), s i \u2286 \u22c3 (i_1 : \u2115), f i i_1 \u2227 \u2211' (i_1 : \u2115), m (f i i_1) < \u03bc (s i) + \u2191(\u03b5' i)\n\u22a2 \u03bc (\u22c3 (i : \u2115), s i) \u2264 \u2211' (c : \u2115), m (f (\u2191Nat.pairEquiv.symm c).fst (\u2191Nat.pairEquiv.symm c).snd)\n[PROOFSTEP]\nrefine' iInf_le_of_le _ (iInf_le _ _)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1\nhf : \u2200 (i : \u2115), s i \u2286 \u22c3 (i_1 : \u2115), f i i_1 \u2227 \u2211' (i_1 : \u2115), m (f i i_1) < \u03bc (s i) + \u2191(\u03b5' i)\n\u22a2 \u22c3 (i : \u2115), s i \u2286 \u22c3 (i : \u2115), f (\u2191Nat.pairEquiv.symm i).fst (\u2191Nat.pairEquiv.symm i).snd\n[PROOFSTEP]\napply iUnion_subset\n[GOAL]\ncase intro.intro.h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1\nhf : \u2200 (i : \u2115), s i \u2286 \u22c3 (i_1 : \u2115), f i i_1 \u2227 \u2211' (i_1 : \u2115), m (f i i_1) < \u03bc (s i) + \u2191(\u03b5' i)\n\u22a2 \u2200 (i : \u2115), s i \u2286 \u22c3 (i : \u2115), f (\u2191Nat.pairEquiv.symm i).fst (\u2191Nat.pairEquiv.symm i).snd\n[PROOFSTEP]\nintro i\n[GOAL]\ncase intro.intro.h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1\nhf : \u2200 (i : \u2115), s i \u2286 \u22c3 (i_1 : \u2115), f i i_1 \u2227 \u2211' (i_1 : \u2115), m (f i i_1) < \u03bc (s i) + \u2191(\u03b5' i)\ni : \u2115\n\u22a2 s i \u2286 \u22c3 (i : \u2115), f (\u2191Nat.pairEquiv.symm i).fst (\u2191Nat.pairEquiv.symm i).snd\n[PROOFSTEP]\napply Subset.trans (hf i).1\n[GOAL]\ncase intro.intro.h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1\nhf : \u2200 (i : \u2115), s i \u2286 \u22c3 (i_1 : \u2115), f i i_1 \u2227 \u2211' (i_1 : \u2115), m (f i i_1) < \u03bc (s i) + \u2191(\u03b5' i)\ni : \u2115\n\u22a2 \u22c3 (i_1 : \u2115), f i i_1 \u2286 \u22c3 (i : \u2115), f (\u2191Nat.pairEquiv.symm i).fst (\u2191Nat.pairEquiv.symm i).snd\n[PROOFSTEP]\napply iUnion_subset\n[GOAL]\ncase intro.intro.h.h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1\nhf : \u2200 (i : \u2115), s i \u2286 \u22c3 (i_1 : \u2115), f i i_1 \u2227 \u2211' (i_1 : \u2115), m (f i i_1) < \u03bc (s i) + \u2191(\u03b5' i)\ni : \u2115\n\u22a2 \u2200 (i_1 : \u2115), f i i_1 \u2286 \u22c3 (i : \u2115), f (\u2191Nat.pairEquiv.symm i).fst (\u2191Nat.pairEquiv.symm i).snd\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.h.h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1\nhf : \u2200 (i : \u2115), s i \u2286 \u22c3 (i_1 : \u2115), f i i_1 \u2227 \u2211' (i_1 : \u2115), m (f i i_1) < \u03bc (s i) + \u2191(\u03b5' i)\ni : \u2115\n\u22a2 \u2200 (i_1 : \u2115), f i i_1 \u2286 \u22c3 (i : \u2115), f (Nat.unpair i).fst (Nat.unpair i).snd\n[PROOFSTEP]\nrw [iUnion_unpair]\n[GOAL]\ncase intro.intro.h.h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1\nhf : \u2200 (i : \u2115), s i \u2286 \u22c3 (i_1 : \u2115), f i i_1 \u2227 \u2211' (i_1 : \u2115), m (f i i_1) < \u03bc (s i) + \u2191(\u03b5' i)\ni : \u2115\n\u22a2 \u2200 (i_1 : \u2115), f i i_1 \u2286 \u22c3 (i : \u2115) (j : \u2115), f i j\n[PROOFSTEP]\nintro j\n[GOAL]\ncase intro.intro.h.h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03bc : Set \u03b1 \u2192 \u211d\u22650\u221e := fun s => \u2a05 (f : \u2115 \u2192 Set \u03b1) (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), m (f i)\ns : \u2115 \u2192 Set \u03b1\n\u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u03b5\nhb : \u2211' (i : \u2115), \u03bc (s i) < \u22a4\n\u03b5' : \u2115 \u2192 \u211d\u22650\nh\u03b5' : \u2200 (i : \u2115), 0 < \u03b5' i\nhl : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b5\nf : \u2115 \u2192 \u2115 \u2192 Set \u03b1\nhf : \u2200 (i : \u2115), s i \u2286 \u22c3 (i_1 : \u2115), f i i_1 \u2227 \u2211' (i_1 : \u2115), m (f i i_1) < \u03bc (s i) + \u2191(\u03b5' i)\ni j : \u2115\n\u22a2 f i j \u2286 \u22c3 (i : \u2115) (j : \u2115), f i j\n[PROOFSTEP]\napply subset_iUnion\u2082 i\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nf : \u2115 \u2192 Set \u03b1 := fun i => Nat.casesOn i s fun x => \u2205\n\u22a2 \u2200 (b' : \u2115), b' \u2260 0 \u2192 m (f b') = 0\n[PROOFSTEP]\nrintro (_ | i)\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nf : \u2115 \u2192 Set \u03b1 := fun i => Nat.casesOn i s fun x => \u2205\n\u22a2 Nat.zero \u2260 0 \u2192 m (f Nat.zero) = 0\ncase succ\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nf : \u2115 \u2192 Set \u03b1 := fun i => Nat.casesOn i s fun x => \u2205\ni : \u2115\n\u22a2 Nat.succ i \u2260 0 \u2192 m (f (Nat.succ i)) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nf : \u2115 \u2192 Set \u03b1 := fun i => Nat.casesOn i s fun x => \u2205\ni : \u2115\n\u22a2 Nat.succ i \u2260 0 \u2192 m (f (Nat.succ i)) = 0\n[PROOFSTEP]\nsimp [m_empty]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\n\u22a2 \u2191(OuterMeasure.ofFunction m m_empty) (s \u222a t) =\n \u2191(OuterMeasure.ofFunction m m_empty) s + \u2191(OuterMeasure.ofFunction m m_empty) t\n[PROOFSTEP]\nrefine' le_antisymm (OuterMeasure.union _ _ _) (le_iInf fun f => le_iInf fun hf => _)\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 (i : \u2115), f i\n\u22a2 \u2191(OuterMeasure.ofFunction m m_empty) s + \u2191(OuterMeasure.ofFunction m m_empty) t \u2264 \u2211' (i : \u2115), m (f i)\n[PROOFSTEP]\nset \u03bc := OuterMeasure.ofFunction m m_empty\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 (i : \u2115), f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)\n[PROOFSTEP]\nrcases Classical.em (\u2203 i, (s \u2229 f i).Nonempty \u2227 (t \u2229 f i).Nonempty) with (\u27e8i, hs, ht\u27e9 | he)\n[GOAL]\ncase inl.intro.intro\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 (i : \u2115), f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\ni : \u2115\nhs : Set.Nonempty (s \u2229 f i)\nht : Set.Nonempty (t \u2229 f i)\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)\n[PROOFSTEP]\ncalc\n \u03bc s + \u03bc t \u2264 \u221e := le_top\n _ = m (f i) := (h (f i) hs ht).symm\n _ \u2264 \u2211' i, m (f i) := ENNReal.le_tsum i\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 (i : \u2115), f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\nhe : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i)\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)\n[PROOFSTEP]\nset I := fun s => {i : \u2115 | (s \u2229 f i).Nonempty}\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 (i : \u2115), f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\nhe : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i)\nI : Set \u03b1 \u2192 Set \u2115 := fun s => {i | Set.Nonempty (s \u2229 f i)}\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)\n[PROOFSTEP]\nhave hd : Disjoint (I s) (I t) := disjoint_iff_inf_le.mpr fun i hi => he \u27e8i, hi\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 (i : \u2115), f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\nhe : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i)\nI : Set \u03b1 \u2192 Set \u2115 := fun s => {i | Set.Nonempty (s \u2229 f i)}\nhd : Disjoint (I s) (I t)\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)\n[PROOFSTEP]\nhave hI : \u2200 (u) (_ : u \u2286 s \u222a t), \u03bc u \u2264 \u2211' i : I u, \u03bc (f i) := fun u hu =>\n calc\n \u03bc u \u2264 \u03bc (\u22c3 i : I u, f i) :=\n \u03bc.mono fun x hx =>\n let \u27e8i, hi\u27e9 := mem_iUnion.1 (hf (hu hx))\n mem_iUnion.2 \u27e8\u27e8i, \u27e8x, hx, hi\u27e9\u27e9, hi\u27e9\n _ \u2264 \u2211' i : I u, \u03bc (f i) := \u03bc.iUnion _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : \u2200 (u : Set \u03b1), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 m u = \u22a4\nf : \u2115 \u2192 Set \u03b1\nhf : s \u222a t \u2286 \u22c3 (i : \u2115), f i\n\u03bc : OuterMeasure \u03b1 := OuterMeasure.ofFunction m m_empty\nhe : \u00ac\u2203 i, Set.Nonempty (s \u2229 f i) \u2227 Set.Nonempty (t \u2229 f i)\nI : Set \u03b1 \u2192 Set \u2115 := fun s => {i | Set.Nonempty (s \u2229 f i)}\nhd : Disjoint (I s) (I t)\nhI : \u2200 (u : Set \u03b1), u \u2286 s \u222a t \u2192 \u2191\u03bc u \u2264 \u2211' (i : \u2191(I u)), \u2191\u03bc (f \u2191i)\n\u22a2 \u2191\u03bc s + \u2191\u03bc t \u2264 \u2211' (i : \u2115), m (f i)\n[PROOFSTEP]\ncalc\n \u03bc s + \u03bc t \u2264 (\u2211' i : I s, \u03bc (f i)) + \u2211' i : I t, \u03bc (f i) :=\n add_le_add (hI _ <| subset_union_left _ _) (hI _ <| subset_union_right _ _)\n _ = \u2211' i : \u2191(I s \u222a I t), \u03bc (f i) :=\n (tsum_union_disjoint (f := fun i => \u03bc (f i)) hd ENNReal.summable ENNReal.summable).symm\n _ \u2264 \u2211' i, \u03bc (f i) :=\n (tsum_le_tsum_of_inj (\u2191) Subtype.coe_injective (fun _ _ => zero_le _) (fun _ => le_rfl) ENNReal.summable\n ENNReal.summable)\n _ \u2264 \u2211' i, m (f i) := ENNReal.tsum_le_tsum fun i => ofFunction_le _\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type ?u.151391\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\n\u22a2 (fun s => m (f '' s)) \u2205 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type ?u.151391\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\n\u22a2 m \u2205 = 0\n[PROOFSTEP]\nsimp [m_empty]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\n\u22a2 \u2191(comap f) (OuterMeasure.ofFunction m m_empty) =\n OuterMeasure.ofFunction (fun s => m (f '' s)) (_ : (fun s => m (f '' s)) \u2205 = 0)\n[PROOFSTEP]\nrefine' le_antisymm (le_ofFunction.2 fun s => _) fun s => _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\n\u22a2 \u2191(\u2191(comap f) (OuterMeasure.ofFunction m m_empty)) s \u2264 m (f '' s)\n[PROOFSTEP]\nrw [comap_apply]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\n\u22a2 \u2191(OuterMeasure.ofFunction m m_empty) (f '' s) \u2264 m (f '' s)\n[PROOFSTEP]\napply ofFunction_le\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\n\u22a2 \u2191(OuterMeasure.ofFunction (fun s => m (f '' s)) (_ : (fun s => m (f '' s)) \u2205 = 0)) s \u2264\n \u2191(\u2191(comap f) (OuterMeasure.ofFunction m m_empty)) s\n[PROOFSTEP]\nrw [comap_apply, ofFunction_apply, ofFunction_apply]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\n\u22a2 \u2a05 (t : \u2115 \u2192 Set \u03b2) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), m (f '' t n) \u2264\n \u2a05 (t : \u2115 \u2192 Set \u03b1) (_ : f '' s \u2286 iUnion t), \u2211' (n : \u2115), m (t n)\n[PROOFSTEP]\nrefine' iInf_mono' fun t => \u27e8fun k => f \u207b\u00b9' t k, _\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\n\u22a2 \u2a05 (_ : s \u2286 \u22c3 (k : \u2115), f \u207b\u00b9' t k), \u2211' (n : \u2115), m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264\n \u2a05 (_ : f '' s \u2286 iUnion t), \u2211' (n : \u2115), m (t n)\n[PROOFSTEP]\nrefine' iInf_mono' fun ht => _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f '' s \u2286 iUnion t\n\u22a2 \u2203 i, \u2211' (n : \u2115), m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264 \u2211' (n : \u2115), m (t n)\n[PROOFSTEP]\nrw [Set.image_subset_iff, preimage_iUnion] at ht \n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : s \u2286 \u22c3 (i : \u2115), f \u207b\u00b9' t i\n\u22a2 \u2203 i, \u2211' (n : \u2115), m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264 \u2211' (n : \u2115), m (t n)\n[PROOFSTEP]\nrefine' \u27e8ht, ENNReal.tsum_le_tsum fun n => _\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : Monotone m \u2228 Surjective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : s \u2286 \u22c3 (i : \u2115), f \u207b\u00b9' t i\nn : \u2115\n\u22a2 m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264 m (t n)\n[PROOFSTEP]\ncases' h with hl hr\n[GOAL]\ncase refine'_2.inl\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : s \u2286 \u22c3 (i : \u2115), f \u207b\u00b9' t i\nn : \u2115\nhl : Monotone m\n\u22a2 m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264 m (t n)\ncase refine'_2.inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : s \u2286 \u22c3 (i : \u2115), f \u207b\u00b9' t i\nn : \u2115\nhr : Surjective f\n\u22a2 m (f '' (fun k => f \u207b\u00b9' t k) n) \u2264 m (t n)\n[PROOFSTEP]\nexacts [hl (image_preimage_subset _ _), (congr_arg m (hr.image_preimage (t n))).le]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u22a2 \u2191(\u2191(map f) (OuterMeasure.ofFunction m m_empty)) s \u2264 m (f \u207b\u00b9' s)\n[PROOFSTEP]\nrw [map_apply]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u22a2 \u2191(OuterMeasure.ofFunction m m_empty) (f \u207b\u00b9' s) \u2264 m (f \u207b\u00b9' s)\n[PROOFSTEP]\napply ofFunction_le\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\n\u22a2 \u2191(map f) (OuterMeasure.ofFunction m m_empty) = OuterMeasure.ofFunction (fun s => m (f \u207b\u00b9' s)) m_empty\n[PROOFSTEP]\nrefine' (map_ofFunction_le _).antisymm fun s => _\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\n\u22a2 \u2191(OuterMeasure.ofFunction (fun s => m (f \u207b\u00b9' s)) m_empty) s \u2264 \u2191(\u2191(map f) (OuterMeasure.ofFunction m m_empty)) s\n[PROOFSTEP]\nsimp only [ofFunction_apply, map_apply, le_iInf_iff]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\n\u22a2 \u2200 (i : \u2115 \u2192 Set \u03b1),\n f \u207b\u00b9' s \u2286 iUnion i \u2192 \u2a05 (t : \u2115 \u2192 Set \u03b2) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), m (f \u207b\u00b9' t n) \u2264 \u2211' (n : \u2115), m (i n)\n[PROOFSTEP]\nintro t ht\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 \u2a05 (t : \u2115 \u2192 Set \u03b2) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), m (f \u207b\u00b9' t n) \u2264 \u2211' (n : \u2115), m (t n)\n[PROOFSTEP]\nrefine' iInf_le_of_le (fun n => (range f)\u1d9c \u222a f '' t n) (iInf_le_of_le _ _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 s \u2286 \u22c3 (n : \u2115), (range f)\u1d9c \u222a f '' t n\n[PROOFSTEP]\nrw [\u2190 union_iUnion, \u2190 inter_subset, \u2190 image_preimage_eq_inter_range, \u2190 image_iUnion]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 f '' (f \u207b\u00b9' s) \u2286 f '' \u22c3 (i : \u2115), t i\n[PROOFSTEP]\nexact image_subset _ ht\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 \u2211' (n : \u2115), m (f \u207b\u00b9' (fun n => (range f)\u1d9c \u222a f '' t n) n) \u2264 \u2211' (n : \u2115), m (t n)\n[PROOFSTEP]\nrefine' ENNReal.tsum_le_tsum fun n => le_of_eq _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\nn : \u2115\n\u22a2 m (f \u207b\u00b9' (fun n => (range f)\u1d9c \u222a f '' t n) n) = m (t n)\n[PROOFSTEP]\nsimp [hf.preimage_image]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 (fun t => m (t \u2229 s)) \u2205 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 m \u2205 = 0\n[PROOFSTEP]\nsimp [m_empty]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 \u2191(restrict s) (OuterMeasure.ofFunction m m_empty) =\n OuterMeasure.ofFunction (fun t => m (t \u2229 s)) (_ : (fun t => m (t \u2229 s)) \u2205 = 0)\n[PROOFSTEP]\nrw [restrict]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 \u2191(LinearMap.comp (map Subtype.val) (comap Subtype.val)) (OuterMeasure.ofFunction m m_empty) =\n OuterMeasure.ofFunction (fun t => m (t \u2229 s)) (_ : (fun t => m (t \u2229 s)) \u2205 = 0)\n[PROOFSTEP]\nsimp only [LinearMap.comp_apply]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 \u2191(map Subtype.val) (\u2191(comap Subtype.val) (OuterMeasure.ofFunction m m_empty)) =\n OuterMeasure.ofFunction (fun t => m (t \u2229 s)) (_ : (fun t => m (t \u2229 s)) \u2205 = 0)\n[PROOFSTEP]\nrw [comap_ofFunction _ (Or.inl hm)]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nhm : Monotone m\n\u22a2 \u2191(map Subtype.val)\n (OuterMeasure.ofFunction (fun s_1 => m (Subtype.val '' s_1)) (_ : (fun s_1 => m (Subtype.val '' s_1)) \u2205 = 0)) =\n OuterMeasure.ofFunction (fun t => m (t \u2229 s)) (_ : (fun t => m (t \u2229 s)) \u2205 = 0)\n[PROOFSTEP]\nsimp only [map_ofFunction Subtype.coe_injective, Subtype.image_preimage_coe]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 (c \u2022 m) \u2205 = 0\n[PROOFSTEP]\nsimp [m_empty]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 c \u2022 OuterMeasure.ofFunction m m_empty = OuterMeasure.ofFunction (c \u2022 m) (_ : c \u2022 m \u2205 = 0)\n[PROOFSTEP]\next1 s\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ns : Set \u03b1\n\u22a2 \u2191(c \u2022 OuterMeasure.ofFunction m m_empty) s = \u2191(OuterMeasure.ofFunction (c \u2022 m) (_ : c \u2022 m \u2205 = 0)) s\n[PROOFSTEP]\nhaveI : Nonempty { t : \u2115 \u2192 Set \u03b1 // s \u2286 \u22c3 i, t i } := \u27e8\u27e8fun _ => s, subset_iUnion (fun _ => s) 0\u27e9\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ns : Set \u03b1\nthis : Nonempty { t // s \u2286 \u22c3 (i : \u2115), t i }\n\u22a2 \u2191(c \u2022 OuterMeasure.ofFunction m m_empty) s = \u2191(OuterMeasure.ofFunction (c \u2022 m) (_ : c \u2022 m \u2205 = 0)) s\n[PROOFSTEP]\nsimp only [smul_apply, ofFunction_apply, ENNReal.tsum_mul_left, Pi.smul_apply, smul_eq_mul, iInf_subtype']\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ns : Set \u03b1\nthis : Nonempty { t // s \u2286 \u22c3 (i : \u2115), t i }\n\u22a2 c * \u2a05 (x : { i // s \u2286 iUnion i }), \u2211' (n : \u2115), m (\u2191x n) = \u2a05 (x : { i // s \u2286 iUnion i }), c * \u2211' (i : \u2115), m (\u2191x i)\n[PROOFSTEP]\nrw [ENNReal.iInf_mul_left fun h => (hc h).elim]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 (fun s => \u2a06 (_ : Set.Nonempty s), m s) \u2205 = 0\n[PROOFSTEP]\nsimp [Set.not_nonempty_empty]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\n\u22a2 \u2191(boundedBy m) s = \u2191(OuterMeasure.ofFunction m m_empty) s\n[PROOFSTEP]\nhave : (fun s : Set \u03b1 => \u2a06 _ : s.Nonempty, m s) = m := by\n ext1 t\n cases' t.eq_empty_or_nonempty with h h <;> simp [h, Set.not_nonempty_empty, m_empty]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\n\u22a2 (fun s => \u2a06 (_ : Set.Nonempty s), m s) = m\n[PROOFSTEP]\next1 t\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\n\u22a2 \u2a06 (_ : Set.Nonempty t), m t = m t\n[PROOFSTEP]\ncases' t.eq_empty_or_nonempty with h h\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : t = \u2205\n\u22a2 \u2a06 (_ : Set.Nonempty t), m t = m t\n[PROOFSTEP]\nsimp [h, Set.not_nonempty_empty, m_empty]\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns t : Set \u03b1\nh : Set.Nonempty t\n\u22a2 \u2a06 (_ : Set.Nonempty t), m t = m t\n[PROOFSTEP]\nsimp [h, Set.not_nonempty_empty, m_empty]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nm_empty : m \u2205 = 0\ns : Set \u03b1\nthis : (fun s => \u2a06 (_ : Set.Nonempty s), m s) = m\n\u22a2 \u2191(boundedBy m) s = \u2191(OuterMeasure.ofFunction m m_empty) s\n[PROOFSTEP]\nsimp [boundedBy, this]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\n\u22a2 \u2191(boundedBy m) s = \u2a05 (t : \u2115 \u2192 Set \u03b1) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (t n)\n[PROOFSTEP]\nsimp [boundedBy, ofFunction_apply]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nm_empty : m \u2205 = 0\nm_mono : \u2200 \u2983t : Set \u03b1\u2984, s \u2286 t \u2192 m s \u2264 m t\nm_subadd : \u2200 (s : \u2115 \u2192 Set \u03b1), m (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), m (s i)\n\u22a2 \u2191(boundedBy m) s = m s\n[PROOFSTEP]\nrw [boundedBy_eq_ofFunction m_empty, ofFunction_eq s m_mono m_subadd]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : OuterMeasure \u03b1\n\u22a2 \u03bc \u2264 boundedBy m \u2194 \u2200 (s : Set \u03b1), \u2191\u03bc s \u2264 m s\n[PROOFSTEP]\nrw [boundedBy, le_ofFunction, forall_congr']\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : OuterMeasure \u03b1\n\u22a2 \u2200 (a : Set \u03b1), \u2191\u03bc a \u2264 \u2a06 (_ : Set.Nonempty a), m a \u2194 \u2191\u03bc a \u2264 m a\n[PROOFSTEP]\nintro s\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191\u03bc s \u2264 \u2a06 (_ : Set.Nonempty s), m s \u2194 \u2191\u03bc s \u2264 m s\n[PROOFSTEP]\ncases' s.eq_empty_or_nonempty with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : OuterMeasure \u03b1\ns : Set \u03b1\nh : s = \u2205\n\u22a2 \u2191\u03bc s \u2264 \u2a06 (_ : Set.Nonempty s), m s \u2194 \u2191\u03bc s \u2264 m s\n[PROOFSTEP]\nsimp [h, Set.not_nonempty_empty]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : OuterMeasure \u03b1\ns : Set \u03b1\nh : Set.Nonempty s\n\u22a2 \u2191\u03bc s \u2264 \u2a06 (_ : Set.Nonempty s), m s \u2194 \u2191\u03bc s \u2264 m s\n[PROOFSTEP]\nsimp [h, Set.not_nonempty_empty]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : OuterMeasure \u03b1\n\u22a2 \u03bc \u2264 boundedBy m \u2194 \u2200 (s : Set \u03b1), Set.Nonempty s \u2192 \u2191\u03bc s \u2264 m s\n[PROOFSTEP]\nrw [le_boundedBy, forall_congr']\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : OuterMeasure \u03b1\n\u22a2 \u2200 (a : Set \u03b1), \u2191\u03bc a \u2264 m a \u2194 Set.Nonempty a \u2192 \u2191\u03bc a \u2264 m a\n[PROOFSTEP]\nintro s\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191\u03bc s \u2264 m s \u2194 Set.Nonempty s \u2192 \u2191\u03bc s \u2264 m s\n[PROOFSTEP]\ncases' s.eq_empty_or_nonempty with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : OuterMeasure \u03b1\ns : Set \u03b1\nh : s = \u2205\n\u22a2 \u2191\u03bc s \u2264 m s \u2194 Set.Nonempty s \u2192 \u2191\u03bc s \u2264 m s\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : OuterMeasure \u03b1\ns : Set \u03b1\nh : Set.Nonempty s\n\u22a2 \u2191\u03bc s \u2264 m s \u2194 Set.Nonempty s \u2192 \u2191\u03bc s \u2264 m s\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 boundedBy \u22a4 = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff, le_boundedBy']\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2200 (s : Set \u03b1), Set.Nonempty s \u2192 \u2191\u22a4 s \u2264 \u22a4 s\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2191\u22a4 s \u2264 \u22a4 s\n[PROOFSTEP]\nrw [top_apply hs]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u22a4 \u2264 \u22a4 s\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 boundedBy 0 = 0\n[PROOFSTEP]\nrw [\u2190 coe_bot, eq_bot_iff]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 boundedBy 0 \u2264 \u22a5\n[PROOFSTEP]\napply boundedBy_le\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 c \u2022 boundedBy m = boundedBy (c \u2022 m)\n[PROOFSTEP]\nsimp only [boundedBy, smul_ofFunction hc]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 OuterMeasure.ofFunction (c \u2022 fun s => \u2a06 (_ : Set.Nonempty s), m s) (_ : c \u2022 \u2a06 (_ : Set.Nonempty \u2205), m \u2205 = 0) =\n OuterMeasure.ofFunction (fun s => \u2a06 (_ : Set.Nonempty s), (c \u2022 m) s) (_ : \u2a06 (_ : Set.Nonempty \u2205), (c \u2022 m) \u2205 = 0)\n[PROOFSTEP]\ncongr 1 with s : 1\n[GOAL]\ncase e_m.h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ns : Set \u03b1\n\u22a2 (c \u2022 fun s => \u2a06 (_ : Set.Nonempty s), m s) s = \u2a06 (_ : Set.Nonempty s), (c \u2022 m) s\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | hs)\n[GOAL]\ncase e_m.h.inl\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 (c \u2022 fun s => \u2a06 (_ : Set.Nonempty s), m s) \u2205 = \u2a06 (_ : Set.Nonempty \u2205), (c \u2022 m) \u2205\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase e_m.h.inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 (c \u2022 fun s => \u2a06 (_ : Set.Nonempty s), m s) s = \u2a06 (_ : Set.Nonempty s), (c \u2022 m) s\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : (Monotone fun s => m \u2191s) \u2228 Surjective f\n\u22a2 \u2191(comap f) (boundedBy m) = boundedBy fun s => m (f '' s)\n[PROOFSTEP]\nrefine' (comap_ofFunction _ _).trans _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : (Monotone fun s => m \u2191s) \u2228 Surjective f\n\u22a2 (Monotone fun s => \u2a06 (_ : Set.Nonempty s), m s) \u2228 Surjective f\n[PROOFSTEP]\nrefine' h.imp (fun H s t hst => iSup_le fun hs => _) id\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : (Monotone fun s => m \u2191s) \u2228 Surjective f\nH : Monotone fun s => m \u2191s\ns t : Set \u03b1\nhst : s \u2264 t\nhs : Set.Nonempty s\n\u22a2 m s \u2264 (fun s => \u2a06 (_ : Set.Nonempty s), m s) t\n[PROOFSTEP]\nhave ht : t.Nonempty := hs.mono hst\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : (Monotone fun s => m \u2191s) \u2228 Surjective f\nH : Monotone fun s => m \u2191s\ns t : Set \u03b1\nhst : s \u2264 t\nhs : Set.Nonempty s\nht : Set.Nonempty t\n\u22a2 m s \u2264 (fun s => \u2a06 (_ : Set.Nonempty s), m s) t\n[PROOFSTEP]\nexact (@H \u27e8s, hs\u27e9 \u27e8t, ht\u27e9 hst).trans (le_iSup (fun _ : t.Nonempty => m t) ht)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : (Monotone fun s => m \u2191s) \u2228 Surjective f\n\u22a2 OuterMeasure.ofFunction (fun s => \u2a06 (_ : Set.Nonempty (f '' s)), m (f '' s))\n (_ : (fun s => \u2a06 (_ : Set.Nonempty (f '' s)), m (f '' s)) \u2205 = 0) =\n boundedBy fun s => m (f '' s)\n[PROOFSTEP]\ndsimp only [boundedBy]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : (Monotone fun s => m \u2191s) \u2228 Surjective f\n\u22a2 OuterMeasure.ofFunction (fun s => \u2a06 (_ : Set.Nonempty (f '' s)), m (f '' s))\n (_ : \u2a06 (_ : Set.Nonempty (f '' \u2205)), m (f '' \u2205) = 0) =\n OuterMeasure.ofFunction (fun s => \u2a06 (_ : Set.Nonempty s), m (f '' s)) (_ : \u2a06 (_ : Set.Nonempty \u2205), m (f '' \u2205) = 0)\n[PROOFSTEP]\ncongr with s : 1\n[GOAL]\ncase refine'_2.e_m.h\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1\nh : (Monotone fun s => m \u2191s) \u2228 Surjective f\ns : Set \u03b2\n\u22a2 \u2a06 (_ : Set.Nonempty (f '' s)), m (f '' s) = \u2a06 (_ : Set.Nonempty s), m (f '' s)\n[PROOFSTEP]\nrw [nonempty_image_iff]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns s\u2081 s\u2082 : Set \u03b1\n\u22a2 IsCaratheodory m \u2205\n[PROOFSTEP]\nsimp [IsCaratheodory, m.empty, diff_empty]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns s\u2081 s\u2082 : Set \u03b1\n\u22a2 IsCaratheodory m s\u2081 \u2192 IsCaratheodory m s\u2081\u1d9c\n[PROOFSTEP]\nsimp [IsCaratheodory, diff_eq, add_comm]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns s\u2081 s\u2082 : Set \u03b1\nh : IsCaratheodory m s\u1d9c\n\u22a2 IsCaratheodory m s\n[PROOFSTEP]\nsimpa using isCaratheodory_compl m h\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns s\u2081 s\u2082 : Set \u03b1\nh\u2081 : IsCaratheodory m s\u2081\nh\u2082 : IsCaratheodory m s\u2082\nt : Set \u03b1\n\u22a2 \u2191m t = \u2191m (t \u2229 (s\u2081 \u222a s\u2082)) + \u2191m (t \\ (s\u2081 \u222a s\u2082))\n[PROOFSTEP]\nrw [h\u2081 t, h\u2082 (t \u2229 s\u2081), h\u2082 (t \\ s\u2081), h\u2081 (t \u2229 (s\u2081 \u222a s\u2082)), inter_diff_assoc _ _ s\u2081, Set.inter_assoc _ _ s\u2081,\n inter_eq_self_of_subset_right (Set.subset_union_left _ _), union_diff_left, h\u2082 (t \u2229 s\u2081)]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns s\u2081 s\u2082 : Set \u03b1\nh\u2081 : IsCaratheodory m s\u2081\nh\u2082 : IsCaratheodory m s\u2082\nt : Set \u03b1\n\u22a2 \u2191m (t \u2229 s\u2081 \u2229 s\u2082) + \u2191m ((t \u2229 s\u2081) \\ s\u2082) + (\u2191m (t \\ s\u2081 \u2229 s\u2082) + \u2191m ((t \\ s\u2081) \\ s\u2082)) =\n \u2191m (t \u2229 s\u2081 \u2229 s\u2082) + \u2191m ((t \u2229 s\u2081) \\ s\u2082) + \u2191m (t \u2229 (s\u2082 \\ s\u2081)) + \u2191m (t \\ (s\u2081 \u222a s\u2082))\n[PROOFSTEP]\nsimp [diff_eq, add_assoc]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns s\u2081 s\u2082 : Set \u03b1\nh : s\u2081 \u2229 s\u2082 \u2286 \u2205\nh\u2081 : IsCaratheodory m s\u2081\nt : Set \u03b1\n\u22a2 \u2191m (t \u2229 (s\u2081 \u222a s\u2082)) = \u2191m (t \u2229 s\u2081) + \u2191m (t \u2229 s\u2082)\n[PROOFSTEP]\nrw [h\u2081, Set.inter_assoc, Set.union_inter_cancel_left, inter_diff_assoc, union_diff_cancel_left h]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nx\u271d : \u2200 (i : \u2115), i < 0 \u2192 IsCaratheodory m (s i)\n\u22a2 IsCaratheodory m (\u22c3 (i : \u2115) (_ : i < 0), s i)\n[PROOFSTEP]\nsimp [Nat.not_lt_zero]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nn : \u2115\nh : \u2200 (i : \u2115), i < n + 1 \u2192 IsCaratheodory m (s i)\n\u22a2 IsCaratheodory m (\u22c3 (i : \u2115) (_ : i < n + 1), s i)\n[PROOFSTEP]\nrw [biUnion_lt_succ]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nn : \u2115\nh : \u2200 (i : \u2115), i < n + 1 \u2192 IsCaratheodory m (s i)\n\u22a2 IsCaratheodory m ((\u22c3 (k : \u2115) (_ : k < n), s k) \u222a s n)\n[PROOFSTEP]\nexact\n isCaratheodory_union m (isCaratheodory_iUnion_lt fun i hi => h i <| lt_of_lt_of_le hi <| Nat.le_succ _)\n (h n (le_refl (n + 1)))\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns s\u2081 s\u2082 : Set \u03b1\nh\u2081 : IsCaratheodory m s\u2081\nh\u2082 : IsCaratheodory m s\u2082\n\u22a2 IsCaratheodory m (s\u2081 \u2229 s\u2082)\n[PROOFSTEP]\nrw [\u2190 isCaratheodory_compl_iff, Set.compl_inter]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns s\u2081 s\u2082 : Set \u03b1\nh\u2081 : IsCaratheodory m s\u2081\nh\u2082 : IsCaratheodory m s\u2082\n\u22a2 IsCaratheodory m (s\u2081\u1d9c \u222a s\u2082\u1d9c)\n[PROOFSTEP]\nexact isCaratheodory_union _ (isCaratheodory_compl _ h\u2081) (isCaratheodory_compl _ h\u2082)\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 \u2211 i in Finset.range 0, \u2191m (t \u2229 s i) = \u2191m (t \u2229 \u22c3 (i : \u2115) (_ : i < 0), s i)\n[PROOFSTEP]\nsimp [Nat.not_lt_zero, m.empty]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nn : \u2115\n\u22a2 \u2211 i in Finset.range (Nat.succ n), \u2191m (t \u2229 s i) = \u2191m (t \u2229 \u22c3 (i : \u2115) (_ : i < Nat.succ n), s i)\n[PROOFSTEP]\nrw [biUnion_lt_succ, Finset.sum_range_succ, Set.union_comm, isCaratheodory_sum h hd, m.measure_inter_union _ (h n),\n add_comm]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nn : \u2115\n\u22a2 s n \u2229 \u22c3 (k : \u2115) (_ : k < n), s k \u2286 \u2205\n[PROOFSTEP]\nintro a\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nn : \u2115\na : \u03b1\n\u22a2 a \u2208 s n \u2229 \u22c3 (k : \u2115) (_ : k < n), s k \u2192 a \u2208 \u2205\n[PROOFSTEP]\nsimpa using fun (h\u2081 : a \u2208 s n) i (hi : i < n) h\u2082 => (hd (ne_of_gt hi)).le_bot \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 IsCaratheodory m (\u22c3 (i : \u2115), s i)\n[PROOFSTEP]\napply (isCaratheodory_iff_le' m).mpr\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 \u2200 (t : Set \u03b1), \u2191m (t \u2229 \u22c3 (i : \u2115), s i) + \u2191m (t \\ \u22c3 (i : \u2115), s i) \u2264 \u2191m t\n[PROOFSTEP]\nintro t\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 \u2191m (t \u2229 \u22c3 (i : \u2115), s i) + \u2191m (t \\ \u22c3 (i : \u2115), s i) \u2264 \u2191m t\n[PROOFSTEP]\nhave hp : m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, m (t \u2229 \u22c3 i < n, s i) :=\n by\n convert m.iUnion fun i => t \u2229 s i using 1\n \u00b7 simp [inter_iUnion]\n \u00b7 simp [ENNReal.tsum_eq_iSup_nat, isCaratheodory_sum m h hd]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 \u2191m (t \u2229 \u22c3 (i : \u2115), s i) \u2264 \u2a06 (n : \u2115), \u2191m (t \u2229 \u22c3 (i : \u2115) (_ : i < n), s i)\n[PROOFSTEP]\nconvert m.iUnion fun i => t \u2229 s i using 1\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 \u2191m (t \u2229 \u22c3 (i : \u2115), s i) = \u2191m (\u22c3 (i : \u2115), t \u2229 s i)\n[PROOFSTEP]\nsimp [inter_iUnion]\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 \u2a06 (n : \u2115), \u2191m (t \u2229 \u22c3 (i : \u2115) (_ : i < n), s i) = \u2211' (i : \u2115), \u2191m (t \u2229 s i)\n[PROOFSTEP]\nsimp [ENNReal.tsum_eq_iSup_nat, isCaratheodory_sum m h hd]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : \u2191m (t \u2229 \u22c3 (i : \u2115), s i) \u2264 \u2a06 (n : \u2115), \u2191m (t \u2229 \u22c3 (i : \u2115) (_ : i < n), s i)\n\u22a2 \u2191m (t \u2229 \u22c3 (i : \u2115), s i) + \u2191m (t \\ \u22c3 (i : \u2115), s i) \u2264 \u2191m t\n[PROOFSTEP]\nrefine' le_trans (add_le_add_right hp _) _\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : \u2191m (t \u2229 \u22c3 (i : \u2115), s i) \u2264 \u2a06 (n : \u2115), \u2191m (t \u2229 \u22c3 (i : \u2115) (_ : i < n), s i)\n\u22a2 (\u2a06 (n : \u2115), \u2191m (t \u2229 \u22c3 (i : \u2115) (_ : i < n), s i)) + \u2191m (t \\ \u22c3 (i : \u2115), s i) \u2264 \u2191m t\n[PROOFSTEP]\nrw [ENNReal.iSup_add]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : \u2191m (t \u2229 \u22c3 (i : \u2115), s i) \u2264 \u2a06 (n : \u2115), \u2191m (t \u2229 \u22c3 (i : \u2115) (_ : i < n), s i)\n\u22a2 \u2a06 (b : \u2115), \u2191m (t \u2229 \u22c3 (i : \u2115) (_ : i < b), s i) + \u2191m (t \\ \u22c3 (i : \u2115), s i) \u2264 \u2191m t\n[PROOFSTEP]\nrefine' iSup_le fun n => le_trans (add_le_add_left _ _) (ge_of_eq (isCaratheodory_iUnion_lt m (fun i _ => h i) _))\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : \u2191m (t \u2229 \u22c3 (i : \u2115), s i) \u2264 \u2a06 (n : \u2115), \u2191m (t \u2229 \u22c3 (i : \u2115) (_ : i < n), s i)\nn : \u2115\n\u22a2 \u2191m (t \\ \u22c3 (i : \u2115), s i) \u2264 \u2191m (t \\ \u22c3 (i : \u2115) (_ : i < n), s i)\n[PROOFSTEP]\nrefine' m.mono (diff_subset_diff_right _)\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : \u2191m (t \u2229 \u22c3 (i : \u2115), s i) \u2264 \u2a06 (n : \u2115), \u2191m (t \u2229 \u22c3 (i : \u2115) (_ : i < n), s i)\nn : \u2115\n\u22a2 \u22c3 (i : \u2115) (_ : i < n), s i \u2286 \u22c3 (i : \u2115), s i\n[PROOFSTEP]\nexact iUnion\u2082_subset fun i _ => subset_iUnion _ i\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 \u2191m (\u22c3 (i : \u2115), s i) = \u2211' (i : \u2115), \u2191m (s i)\n[PROOFSTEP]\nrefine' le_antisymm (m.iUnion_nat s) _\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 \u2211' (i : \u2115), \u2191m (s i) \u2264 \u2191m (\u22c3 (i : \u2115), s i)\n[PROOFSTEP]\nrw [ENNReal.tsum_eq_iSup_nat]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 \u2a06 (i : \u2115), \u2211 a in Finset.range i, \u2191m (s a) \u2264 \u2191m (\u22c3 (i : \u2115), s i)\n[PROOFSTEP]\nrefine' iSup_le fun n => _\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nn : \u2115\n\u22a2 \u2211 a in Finset.range n, \u2191m (s a) \u2264 \u2191m (\u22c3 (i : \u2115), s i)\n[PROOFSTEP]\nhave := @isCaratheodory_sum _ m _ h hd univ n\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nn : \u2115\nthis : \u2211 i in Finset.range n, \u2191m (univ \u2229 s i) = \u2191m (univ \u2229 \u22c3 (i : \u2115) (_ : i < n), s i)\n\u22a2 \u2211 a in Finset.range n, \u2191m (s a) \u2264 \u2191m (\u22c3 (i : \u2115), s i)\n[PROOFSTEP]\nsimp at this \n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nn : \u2115\nthis : \u2211 x in Finset.range n, \u2191m (s x) = \u2191m (\u22c3 (i : \u2115) (_ : i < n), s i)\n\u22a2 \u2211 a in Finset.range n, \u2191m (s a) \u2264 \u2191m (\u22c3 (i : \u2115), s i)\n[PROOFSTEP]\nsimp [this]\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), IsCaratheodory m (s i)\nhd : Pairwise (Disjoint on s)\nn : \u2115\nthis : \u2211 x in Finset.range n, \u2191m (s x) = \u2191m (\u22c3 (i : \u2115) (_ : i < n), s i)\n\u22a2 \u2191m (\u22c3 (i : \u2115) (_ : i < n), s i) \u2264 \u2191m (\u22c3 (i : \u2115), s i)\n[PROOFSTEP]\nexact m.mono (iUnion\u2082_subset fun i _ => subset_iUnion _ i)\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns s\u2081 s\u2082 : Set \u03b1\nf\u271d : \u2115 \u2192 Set \u03b1\nf : Pairwise (Disjoint on f\u271d)\nhf : \u2200 (i : \u2115), IsCaratheodory m (f\u271d i)\nhn : Set \u03b1\n\u22a2 \u2191m hn = \u2191m (hn \u2229 \u22c3 (i : \u2115), f\u271d i) + \u2191m (hn \\ \u22c3 (i : \u2115), f\u271d i)\n[PROOFSTEP]\napply isCaratheodory_iUnion_nat m hf f\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns s\u2081 s\u2082 : Set \u03b1\n\u22a2 MeasurableSpace \u03b1\n[PROOFSTEP]\napply MeasurableSpace.DynkinSystem.toMeasurableSpace (caratheodoryDynkin m)\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns s\u2081 s\u2082 : Set \u03b1\n\u22a2 \u2200 (s\u2081 s\u2082 : Set \u03b1),\n MeasurableSpace.DynkinSystem.Has (caratheodoryDynkin m) s\u2081 \u2192\n MeasurableSpace.DynkinSystem.Has (caratheodoryDynkin m) s\u2082 \u2192\n MeasurableSpace.DynkinSystem.Has (caratheodoryDynkin m) (s\u2081 \u2229 s\u2082)\n[PROOFSTEP]\nintro s\u2081 s\u2082\n[GOAL]\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns s\u2081\u271d s\u2082\u271d s\u2081 s\u2082 : Set \u03b1\n\u22a2 MeasurableSpace.DynkinSystem.Has (caratheodoryDynkin m) s\u2081 \u2192\n MeasurableSpace.DynkinSystem.Has (caratheodoryDynkin m) s\u2082 \u2192\n MeasurableSpace.DynkinSystem.Has (caratheodoryDynkin m) (s\u2081 \u2229 s\u2082)\n[PROOFSTEP]\napply isCaratheodory_inter\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\n\u22a2 MeasurableSet s\n[PROOFSTEP]\napply (isCaratheodory_iff_le _).mpr\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\n\u22a2 \u2200 (t : Set \u03b1),\n \u2191(OuterMeasure.ofFunction m h\u2080) (t \u2229 s) + \u2191(OuterMeasure.ofFunction m h\u2080) (t \\ s) \u2264\n \u2191(OuterMeasure.ofFunction m h\u2080) t\n[PROOFSTEP]\nrefine' fun t => le_iInf fun f => le_iInf fun hf => _\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 (i : \u2115), f i\n\u22a2 \u2191(OuterMeasure.ofFunction m h\u2080) (t \u2229 s) + \u2191(OuterMeasure.ofFunction m h\u2080) (t \\ s) \u2264 \u2211' (i : \u2115), m (f i)\n[PROOFSTEP]\nrefine'\n le_trans\n (add_le_add ((iInf_le_of_le fun i => f i \u2229 s) <| iInf_le _ _) ((iInf_le_of_le fun i => f i \\ s) <| iInf_le _ _)) _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 (i : \u2115), f i\n\u22a2 t \u2229 s \u2286 \u22c3 (i : \u2115), (fun i => f i \u2229 s) i\n[PROOFSTEP]\nrw [\u2190 iUnion_inter]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 (i : \u2115), f i\n\u22a2 t \u2229 s \u2286 (\u22c3 (i : \u2115), f i) \u2229 s\n[PROOFSTEP]\nexact inter_subset_inter_left _ hf\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 (i : \u2115), f i\n\u22a2 t \\ s \u2286 \u22c3 (i : \u2115), (fun i => f i \\ s) i\n[PROOFSTEP]\nrw [\u2190 iUnion_diff]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 (i : \u2115), f i\n\u22a2 t \\ s \u2286 (\u22c3 (i : \u2115), f i) \\ s\n[PROOFSTEP]\nexact diff_subset_diff_left hf\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 (i : \u2115), f i\n\u22a2 \u2211' (i : \u2115), m ((fun i => f i \u2229 s) i) + \u2211' (i : \u2115), m ((fun i => f i \\ s) i) \u2264 \u2211' (i : \u2115), m (f i)\n[PROOFSTEP]\nrw [\u2190 ENNReal.tsum_add]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 (i : \u2115), f i\n\u22a2 \u2211' (a : \u2115), (m ((fun i => f i \u2229 s) a) + m ((fun i => f i \\ s) a)) \u2264 \u2211' (i : \u2115), m (f i)\n[PROOFSTEP]\nexact ENNReal.tsum_le_tsum fun i => hs _\n[GOAL]\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\n\u22a2 MeasurableSet s\n[PROOFSTEP]\napply ofFunction_caratheodory\n[GOAL]\ncase hs\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\n\u22a2 \u2200 (t : Set \u03b1),\n (\u2a06 (_ : Set.Nonempty (t \u2229 s)), m (t \u2229 s)) + \u2a06 (_ : Set.Nonempty (t \\ s)), m (t \\ s) \u2264 \u2a06 (_ : Set.Nonempty t), m t\n[PROOFSTEP]\nintro t\n[GOAL]\ncase hs\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\n\u22a2 (\u2a06 (_ : Set.Nonempty (t \u2229 s)), m (t \u2229 s)) + \u2a06 (_ : Set.Nonempty (t \\ s)), m (t \\ s) \u2264 \u2a06 (_ : Set.Nonempty t), m t\n[PROOFSTEP]\ncases' t.eq_empty_or_nonempty with h h\n[GOAL]\ncase hs.inl\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nh : t = \u2205\n\u22a2 (\u2a06 (_ : Set.Nonempty (t \u2229 s)), m (t \u2229 s)) + \u2a06 (_ : Set.Nonempty (t \\ s)), m (t \\ s) \u2264 \u2a06 (_ : Set.Nonempty t), m t\n[PROOFSTEP]\nsimp [h, Set.not_nonempty_empty]\n[GOAL]\ncase hs.inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nh : Set.Nonempty t\n\u22a2 (\u2a06 (_ : Set.Nonempty (t \u2229 s)), m (t \u2229 s)) + \u2a06 (_ : Set.Nonempty (t \\ s)), m (t \\ s) \u2264 \u2a06 (_ : Set.Nonempty t), m t\n[PROOFSTEP]\nconvert le_trans _ (hs t)\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nh : Set.Nonempty t\n\u22a2 \u2a06 (_ : Set.Nonempty t), m t = m t\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase hs.inr.convert_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nh : Set.Nonempty t\n\u22a2 (\u2a06 (_ : Set.Nonempty (t \u2229 s)), m (t \u2229 s)) + \u2a06 (_ : Set.Nonempty (t \\ s)), m (t \\ s) \u2264 m (t \u2229 s) + m (t \\ s)\n[PROOFSTEP]\nexact add_le_add iSup_const_le iSup_const_le\n[GOAL]\n\u03b1 : Type u_1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : t = \u2205\n\u22a2 \u2191\u22a4 (t \u2229 s) + \u2191\u22a4 (t \\ s) \u2264 \u2191\u22a4 t\n[PROOFSTEP]\nsimp [ht]\n[GOAL]\n\u03b1 : Type u_1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : Set.Nonempty t\n\u22a2 \u2191\u22a4 (t \u2229 s) + \u2191\u22a4 (t \\ s) \u2264 \u2191\u22a4 t\n[PROOFSTEP]\nsimp only [ht, top_apply, le_top]\n[GOAL]\n\u03b1 : Type u_1\nm\u2081 m\u2082 : OuterMeasure \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nhs\u2081 : s \u2208 (fun m => {t | MeasurableSet t}) (OuterMeasure.caratheodory m\u2081)\nhs\u2082 : s \u2208 (fun m => {t | MeasurableSet t}) (OuterMeasure.caratheodory m\u2082)\n\u22a2 \u2191(m\u2081 + m\u2082) t = \u2191(m\u2081 + m\u2082) (t \u2229 s) + \u2191(m\u2081 + m\u2082) (t \\ s)\n[PROOFSTEP]\nsimp [hs\u2081 t, hs\u2082 t, add_left_comm, add_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nm : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nh : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191(sum m) t = \u2191(sum m) (t \u2229 s) + \u2191(sum m) (t \\ s)\n[PROOFSTEP]\nsimp [fun i => MeasurableSpace.measurableSet_iInf.1 h i t, ENNReal.tsum_add]\n[GOAL]\n\u03b1 : Type u_1\na : \u211d\u22650\u221e\nm : OuterMeasure \u03b1\ns : Set \u03b1\nh : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191(a \u2022 m) t = \u2191(a \u2022 m) (t \u2229 s) + \u2191(a \u2022 m) (t \\ s)\n[PROOFSTEP]\nsimp [smul_apply]\n[GOAL]\n\u03b1 : Type u_1\na : \u211d\u22650\u221e\nm : OuterMeasure \u03b1\ns : Set \u03b1\nh : MeasurableSet s\nt : Set \u03b1\n\u22a2 a * \u2191m t = a * \u2191m (t \u2229 s) + a * \u2191m (t \\ s)\n[PROOFSTEP]\nrw [(isCaratheodory_iff m).mp h t]\n[GOAL]\n\u03b1 : Type u_1\na : \u211d\u22650\u221e\nm : OuterMeasure \u03b1\ns : Set \u03b1\nh : MeasurableSet s\nt : Set \u03b1\n\u22a2 a * (\u2191m (t \u2229 s) + \u2191m (t \\ s)) = a * \u2191m (t \u2229 s) + a * \u2191m (t \\ s)\n[PROOFSTEP]\nsimp [mul_add]\n[GOAL]\n\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191(dirac a) t = \u2191(dirac a) (t \u2229 s) + \u2191(dirac a) (t \\ s)\n[PROOFSTEP]\nby_cases ht : a \u2208 t\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : a \u2208 t\n\u22a2 \u2191(dirac a) t = \u2191(dirac a) (t \u2229 s) + \u2191(dirac a) (t \\ s)\ncase neg\n\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : \u00aca \u2208 t\n\u22a2 \u2191(dirac a) t = \u2191(dirac a) (t \u2229 s) + \u2191(dirac a) (t \\ s)\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : \u00aca \u2208 t\n\u22a2 \u2191(dirac a) t = \u2191(dirac a) (t \u2229 s) + \u2191(dirac a) (t \\ s)\n[PROOFSTEP]\nsimp [ht]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : a \u2208 t\n\u22a2 \u2191(dirac a) t = \u2191(dirac a) (t \u2229 s) + \u2191(dirac a) (t \\ s)\n[PROOFSTEP]\nby_cases hs : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : a \u2208 t\nhs : a \u2208 s\n\u22a2 \u2191(dirac a) t = \u2191(dirac a) (t \u2229 s) + \u2191(dirac a) (t \\ s)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : a \u2208 t\nhs : \u00aca \u2208 s\n\u22a2 \u2191(dirac a) t = \u2191(dirac a) (t \u2229 s) + \u2191(dirac a) (t \\ s)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u22a2 sInf m = boundedBy (sInfGen m)\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u22a2 sInf m \u2264 boundedBy (sInfGen m)\n[PROOFSTEP]\nrefine' le_boundedBy.2 fun s => le_iInf\u2082 fun \u03bc h\u03bc => _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\ns : Set \u03b1\n\u03bc : OuterMeasure \u03b1\nh\u03bc : \u03bc \u2208 m\n\u22a2 \u2191(sInf m) s \u2264 \u2191\u03bc s\n[PROOFSTEP]\napply sInf_le h\u03bc\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u22a2 boundedBy (sInfGen m) \u2264 sInf m\n[PROOFSTEP]\nrefine' le_sInf _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u22a2 \u2200 (b : OuterMeasure \u03b1), b \u2208 m \u2192 boundedBy (sInfGen m) \u2264 b\n[PROOFSTEP]\nintro \u03bc h\u03bc t\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u03bc : OuterMeasure \u03b1\nh\u03bc : \u03bc \u2208 m\nt : Set \u03b1\n\u22a2 \u2191(boundedBy (sInfGen m)) t \u2264 \u2191\u03bc t\n[PROOFSTEP]\nrefine' le_trans (boundedBy_le t) (iInf\u2082_le \u03bc h\u03bc)\n[GOAL]\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\nh : Set.Nonempty m\nt : Set \u03b1\n\u22a2 \u2a06 (_ : Set.Nonempty t), sInfGen m t = \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 m), \u2191\u03bc t\n[PROOFSTEP]\nrcases t.eq_empty_or_nonempty with (rfl | ht)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\nh : Set.Nonempty m\n\u22a2 \u2a06 (_ : Set.Nonempty \u2205), sInfGen m \u2205 = \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 m), \u2191\u03bc \u2205\n[PROOFSTEP]\nrcases h with \u27e8\u03bc, h\u03bc\u27e9\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u03bc : OuterMeasure \u03b1\nh\u03bc : \u03bc \u2208 m\n\u22a2 \u2a06 (_ : Set.Nonempty \u2205), sInfGen m \u2205 = \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 m), \u2191\u03bc \u2205\n[PROOFSTEP]\nrw [eq_false Set.not_nonempty_empty, iSup_false, eq_comm]\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u03bc : OuterMeasure \u03b1\nh\u03bc : \u03bc \u2208 m\n\u22a2 \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 m), \u2191\u03bc \u2205 = \u22a5\n[PROOFSTEP]\nsimp_rw [empty']\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u03bc : OuterMeasure \u03b1\nh\u03bc : \u03bc \u2208 m\n\u22a2 \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 m), 0 = \u22a5\n[PROOFSTEP]\napply bot_unique\n[GOAL]\ncase inl.intro.h\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\n\u03bc : OuterMeasure \u03b1\nh\u03bc : \u03bc \u2208 m\n\u22a2 \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 m), 0 \u2264 \u22a5\n[PROOFSTEP]\nrefine' iInf_le_of_le \u03bc (iInf_le _ h\u03bc)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\nh : Set.Nonempty m\nt : Set \u03b1\nht : Set.Nonempty t\n\u22a2 \u2a06 (_ : Set.Nonempty t), sInfGen m t = \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 m), \u2191\u03bc t\n[PROOFSTEP]\nsimp [ht, sInfGen_def]\n[GOAL]\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\ns : Set \u03b1\nh : Set.Nonempty m\n\u22a2 \u2191(sInf m) s = \u2a05 (t : \u2115 \u2192 Set \u03b1) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 m), \u2191\u03bc (t n)\n[PROOFSTEP]\nsimp_rw [sInf_eq_boundedBy_sInfGen, boundedBy_apply, iSup_sInfGen_nonempty h]\n[GOAL]\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\ns : Set \u03b1\nh : Set.Nonempty s\nhm : m = \u2205\n\u22a2 \u2191(sInf m) s = \u2a05 (t : \u2115 \u2192 Set \u03b1) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 m), \u2191\u03bc (t n)\n[PROOFSTEP]\nsimp [hm, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\ninst\u271d : Nonempty \u03b9\nm : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191(\u2a05 (i : \u03b9), m i) s = \u2a05 (t : \u2115 \u2192 Set \u03b1) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 (i : \u03b9), \u2191(m i) (t n)\n[PROOFSTEP]\nrw [iInf, sInf_apply (range_nonempty m)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\ninst\u271d : Nonempty \u03b9\nm : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2a05 (t : \u2115 \u2192 Set \u03b1) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 range m), \u2191\u03bc (t n) =\n \u2a05 (t : \u2115 \u2192 Set \u03b1) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 (i : \u03b9), \u2191(m i) (t n)\n[PROOFSTEP]\nsimp only [iInf_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\nm : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2191(\u2a05 (i : \u03b9), m i) s = \u2a05 (t : \u2115 \u2192 Set \u03b1) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 (i : \u03b9), \u2191(m i) (t n)\n[PROOFSTEP]\nrw [iInf, sInf_apply' hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\nm : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2a05 (t : \u2115 \u2192 Set \u03b1) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 range fun i => m i), \u2191\u03bc (t n) =\n \u2a05 (t : \u2115 \u2192 Set \u03b1) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 (i : \u03b9), \u2191(m i) (t n)\n[PROOFSTEP]\nsimp only [iInf_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nI : Set \u03b9\nhI : Set.Nonempty I\nm : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 I), m i) s =\n \u2a05 (t : \u2115 \u2192 Set \u03b1) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 (i : \u03b9) (_ : i \u2208 I), \u2191(m i) (t n)\n[PROOFSTEP]\nhaveI := hI.to_subtype\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nI : Set \u03b9\nhI : Set.Nonempty I\nm : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nthis : Nonempty \u2191I\n\u22a2 \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 I), m i) s =\n \u2a05 (t : \u2115 \u2192 Set \u03b1) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 (i : \u03b9) (_ : i \u2208 I), \u2191(m i) (t n)\n[PROOFSTEP]\nsimp only [\u2190 iInf_subtype'', iInf_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nI : Set \u03b9\nm : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 I), m i) s =\n \u2a05 (t : \u2115 \u2192 Set \u03b1) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 (i : \u03b9) (_ : i \u2208 I), \u2191(m i) (t n)\n[PROOFSTEP]\nsimp only [\u2190 iInf_subtype'', iInf_apply' _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\n\u22a2 \u2191(comap f) (\u2a05 (i : \u03b9), m i) = \u2a05 (i : \u03b9), \u2191(comap f) (m i)\n[PROOFSTEP]\nrefine' ext_nonempty fun s hs => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2191(\u2191(comap f) (\u2a05 (i : \u03b9), m i)) s = \u2191(\u2a05 (i : \u03b9), \u2191(comap f) (m i)) s\n[PROOFSTEP]\nrefine' ((comap_mono f).map_iInf_le s).antisymm _\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2191(\u2a05 (i : \u03b9), \u2191(comap f) (m i)) s \u2264 \u2191(\u2191(comap f) (\u2a05 (i : \u03b9), m i)) s\n[PROOFSTEP]\nsimp only [comap_apply, iInf_apply' _ hs, iInf_apply' _ (hs.image _), le_iInf_iff, Set.image_subset_iff,\n preimage_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2200 (i : \u2115 \u2192 Set \u03b2),\n s \u2286 \u22c3 (i_1 : \u2115), f \u207b\u00b9' i i_1 \u2192\n \u2a05 (t : \u2115 \u2192 Set \u03b1) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 (i : \u03b9), \u2191(m i) (f '' t n) \u2264\n \u2211' (n : \u2115), \u2a05 (i_2 : \u03b9), \u2191(m i_2) (i n)\n[PROOFSTEP]\nrefine' fun t ht => iInf_le_of_le _ (iInf_le_of_le ht <| ENNReal.tsum_le_tsum fun k => _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b1\nhs : Set.Nonempty s\nt : \u2115 \u2192 Set \u03b2\nht : s \u2286 \u22c3 (i : \u2115), f \u207b\u00b9' t i\nk : \u2115\n\u22a2 \u2a05 (i : \u03b9), \u2191(m i) (f '' (f \u207b\u00b9' t k)) \u2264 \u2a05 (i : \u03b9), \u2191(m i) (t k)\n[PROOFSTEP]\nexact iInf_mono fun i => (m i).mono (image_preimage_subset _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nm : \u03b9 \u2192 OuterMeasure \u03b1\n\u22a2 \u2191(map f) (\u2a05 (i : \u03b9), m i) = \u2191(restrict (range f)) (\u2a05 (i : \u03b9), \u2191(map f) (m i))\n[PROOFSTEP]\nrefine' Eq.trans _ (map_comap _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nm : \u03b9 \u2192 OuterMeasure \u03b1\n\u22a2 \u2191(map f) (\u2a05 (i : \u03b9), m i) = \u2191(map f) (\u2191(comap f) (\u2a05 (i : \u03b9), \u2191(map f) (m i)))\n[PROOFSTEP]\nsimp only [comap_iInf, comap_map hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\n\u22a2 \u2191(map f) (\u2a05 (i : \u03b9), \u2191(comap f) (m i)) = \u2a05 (i : \u03b9), \u2191(map f) (\u2191(comap f) (m i))\n[PROOFSTEP]\nrefine' (map_iInf_le _ _).antisymm fun s => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\n\u22a2 \u2191(\u2a05 (i : \u03b9), \u2191(map f) (\u2191(comap f) (m i))) s \u2264 \u2191(\u2191(map f) (\u2a05 (i : \u03b9), \u2191(comap f) (m i))) s\n[PROOFSTEP]\nsimp only [map_apply, comap_apply, iInf_apply, le_iInf_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\n\u22a2 \u2200 (i : \u2115 \u2192 Set \u03b1),\n f \u207b\u00b9' s \u2286 iUnion i \u2192\n \u2a05 (t : \u2115 \u2192 Set \u03b2) (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 (i : \u03b9), \u2191(m i) (f '' (f \u207b\u00b9' t n)) \u2264\n \u2211' (n : \u2115), \u2a05 (i_2 : \u03b9), \u2191(m i_2) (f '' i n)\n[PROOFSTEP]\nrefine' fun t ht => iInf_le_of_le (fun n => f '' t n \u222a (range f)\u1d9c) (iInf_le_of_le _ _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 s \u2286 \u22c3 (n : \u2115), f '' t n \u222a (range f)\u1d9c\n[PROOFSTEP]\nrw [\u2190 iUnion_union, Set.union_comm, \u2190 inter_subset, \u2190 image_iUnion, \u2190 image_preimage_eq_inter_range]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 f '' (f \u207b\u00b9' s) \u2286 f '' \u22c3 (i : \u2115), t i\n[PROOFSTEP]\nexact image_subset _ ht\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\n\u22a2 \u2211' (n : \u2115), \u2a05 (i : \u03b9), \u2191(m i) (f '' (f \u207b\u00b9' (fun n => f '' t n \u222a (range f)\u1d9c) n)) \u2264\n \u2211' (n : \u2115), \u2a05 (i : \u03b9), \u2191(m i) (f '' t n)\n[PROOFSTEP]\nrefine' ENNReal.tsum_le_tsum fun n => iInf_mono fun i => (m i).mono _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\nn : \u2115\ni : \u03b9\n\u22a2 f '' (f \u207b\u00b9' (fun n => f '' t n \u222a (range f)\u1d9c) n) \u2286 f '' t n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Type u_3\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\ns : Set \u03b2\nt : \u2115 \u2192 Set \u03b1\nht : f \u207b\u00b9' s \u2286 iUnion t\nn : \u2115\ni : \u03b9\n\u22a2 f \u207b\u00b9' (f '' t n) \u2286 f \u207b\u00b9' (f '' t n)\n[PROOFSTEP]\nexact subset_refl _\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\nI : Set \u03b9\nhI : Set.Nonempty I\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\n\u22a2 \u2191(map f) (\u2a05 (i : \u03b9) (_ : i \u2208 I), \u2191(comap f) (m i)) = \u2a05 (i : \u03b9) (_ : i \u2208 I), \u2191(map f) (\u2191(comap f) (m i))\n[PROOFSTEP]\nhaveI := hI.to_subtype\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\nI : Set \u03b9\nhI : Set.Nonempty I\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\nthis : Nonempty \u2191I\n\u22a2 \u2191(map f) (\u2a05 (i : \u03b9) (_ : i \u2208 I), \u2191(comap f) (m i)) = \u2a05 (i : \u03b9) (_ : i \u2208 I), \u2191(map f) (\u2191(comap f) (m i))\n[PROOFSTEP]\nrw [\u2190 iInf_subtype'', \u2190 iInf_subtype'']\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\nI : Set \u03b9\nhI : Set.Nonempty I\nf : \u03b1 \u2192 \u03b2\nm : \u03b9 \u2192 OuterMeasure \u03b2\nthis : Nonempty \u2191I\n\u22a2 \u2191(map f) (\u2a05 (i : \u2191I), \u2191(comap f) (m \u2191i)) = \u2a05 (i : \u2191I), \u2191(map f) (\u2191(comap f) (m \u2191i))\n[PROOFSTEP]\nexact map_iInf_comap _\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\ns : Set \u03b1\nm : \u03b9 \u2192 OuterMeasure \u03b1\n\u22a2 \u2191(restrict s) (\u2a05 (i : \u03b9), \u2191(restrict s) (m i)) = \u2191(restrict (range Subtype.val)) (\u2a05 (i : \u03b9), \u2191(restrict s) (m i))\n[PROOFSTEP]\nrw [Subtype.range_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nI : Set \u03b9\nhI : Set.Nonempty I\ns : Set \u03b1\nm : \u03b9 \u2192 OuterMeasure \u03b1\n\u22a2 \u2191(restrict s) (\u2a05 (i : \u03b9) (_ : i \u2208 I), m i) = \u2a05 (i : \u03b9) (_ : i \u2208 I), \u2191(restrict s) (m i)\n[PROOFSTEP]\nhaveI := hI.to_subtype\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nI : Set \u03b9\nhI : Set.Nonempty I\ns : Set \u03b1\nm : \u03b9 \u2192 OuterMeasure \u03b1\nthis : Nonempty \u2191I\n\u22a2 \u2191(restrict s) (\u2a05 (i : \u03b9) (_ : i \u2208 I), m i) = \u2a05 (i : \u03b9) (_ : i \u2208 I), \u2191(restrict s) (m i)\n[PROOFSTEP]\nrw [\u2190 iInf_subtype'', \u2190 iInf_subtype'']\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nI : Set \u03b9\nhI : Set.Nonempty I\ns : Set \u03b1\nm : \u03b9 \u2192 OuterMeasure \u03b1\nthis : Nonempty \u2191I\n\u22a2 \u2191(restrict s) (\u2a05 (i : \u2191I), m \u2191i) = \u2a05 (i : \u2191I), \u2191(restrict s) (m \u2191i)\n[PROOFSTEP]\nexact restrict_iInf _ _\n[GOAL]\n\u03b1 : Type u_1\nm : Set (OuterMeasure \u03b1)\ns : Set \u03b1\nhm : Set.Nonempty m\n\u22a2 \u2191(restrict s) (sInf m) = sInf (\u2191(restrict s) '' m)\n[PROOFSTEP]\nsimp only [sInf_eq_iInf, restrict_biInf, hm, iInf_image]\n[GOAL]\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\ns : \u03b1\nh : P s\n\u22a2 extend m s = m s h\n[PROOFSTEP]\nsimp [extend, h]\n[GOAL]\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\ns : \u03b1\nh : \u00acP s\n\u22a2 extend m s = \u22a4\n[PROOFSTEP]\nsimp [extend, h]\n[GOAL]\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nR : Type u_2\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e\nc : R\nhc : c \u2260 0\n\u22a2 c \u2022 extend m = extend fun s h => c \u2022 m s h\n[PROOFSTEP]\next1 s\n[GOAL]\ncase h\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nR : Type u_2\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e\nc : R\nhc : c \u2260 0\ns : \u03b1\n\u22a2 (c \u2022 extend m) s = extend (fun s h => c \u2022 m s h) s\n[PROOFSTEP]\ndsimp [extend]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nR : Type u_2\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e\nc : R\nhc : c \u2260 0\ns : \u03b1\n\u22a2 c \u2022 \u2a05 (h : P s), m s h = \u2a05 (h : P s), c \u2022 m s h\n[PROOFSTEP]\nby_cases h : P s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nR : Type u_2\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e\nc : R\nhc : c \u2260 0\ns : \u03b1\nh : P s\n\u22a2 c \u2022 \u2a05 (h : P s), m s h = \u2a05 (h : P s), c \u2022 m s h\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nR : Type u_2\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : NoZeroSMulDivisors R \u211d\u22650\u221e\nc : R\nhc : c \u2260 0\ns : \u03b1\nh : \u00acP s\n\u22a2 c \u2022 \u2a05 (h : P s), m s h = \u2a05 (h : P s), c \u2022 m s h\n[PROOFSTEP]\nsimp [h, ENNReal.smul_top, hc]\n[GOAL]\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\ns : \u03b1\nh : P s\n\u22a2 m s h \u2264 extend m s\n[PROOFSTEP]\nsimp only [extend, le_iInf_iff]\n[GOAL]\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\ns : \u03b1\nh : P s\n\u22a2 \u2200 (i : P s), m s h \u2264 m s i\n[PROOFSTEP]\nintro\n[GOAL]\n\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\ns : \u03b1\nh i\u271d : P s\n\u22a2 m s h \u2264 m s i\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU\u271d :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\nf : \u2115 \u2192 Set \u03b1\nhm : \u2200 (i : \u2115), P (f i)\nmU : m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\n\u22a2 \u2211' (i : \u2115), m (f i) (_ : P (f i)) = \u2211' (i : \u2115), extend m (f i)\n[PROOFSTEP]\ncongr with i\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU\u271d :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\nf : \u2115 \u2192 Set \u03b1\nhm : \u2200 (i : \u2115), P (f i)\nmU : m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\ni : \u2115\n\u22a2 m (f i) (_ : P (f i)) = extend m (f i)\n[PROOFSTEP]\nrw [extend_eq]\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : \u2115 \u2192 Set \u03b1\n\u22a2 extend m (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), extend m (s i)\n[PROOFSTEP]\nby_cases h : \u2200 i, P (s i)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), P (s i)\n\u22a2 extend m (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), extend m (s i)\n[PROOFSTEP]\nrw [extend_eq _ (PU h), congr_arg tsum _]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), P (s i)\n\u22a2 m (\u22c3 (i : \u2115), s i) (_ : P (\u22c3 (i : \u2115), s i)) \u2264 tsum ?m.289896\n[PROOFSTEP]\napply msU h\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), P (s i)\n\u22a2 (fun i => extend m (s i)) = fun i => m (s i) (_ : P (s i))\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), P (s i)\ni : \u2115\n\u22a2 extend m (s i) = m (s i) (_ : P (s i))\n[PROOFSTEP]\napply extend_eq _ (h i)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : \u2115 \u2192 Set \u03b1\nh : \u00ac\u2200 (i : \u2115), P (s i)\n\u22a2 extend m (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), extend m (s i)\n[PROOFSTEP]\ncases' not_forall.1 h with i hi\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : \u2115 \u2192 Set \u03b1\nh : \u00ac\u2200 (i : \u2115), P (s i)\ni : \u2115\nhi : \u00acP (s i)\n\u22a2 extend m (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), extend m (s i)\n[PROOFSTEP]\nexact le_trans (le_iInf fun h => hi.elim h) (ENNReal.le_tsum i)\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : P s\u2081\nhs : s\u2081 \u2286 s\u2082\n\u22a2 extend m s\u2081 \u2264 extend m s\u2082\n[PROOFSTEP]\nrefine' le_iInf _\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : P s\u2081\nhs : s\u2081 \u2286 s\u2082\n\u22a2 \u2200 (i : (fun s => P s) s\u2082), extend m s\u2081 \u2264 m s\u2082 i\n[PROOFSTEP]\nintro h\u2082\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : P s\u2081\nhs : s\u2081 \u2286 s\u2082\nh\u2082 : (fun s => P s) s\u2082\n\u22a2 extend m s\u2081 \u2264 m s\u2082 h\u2082\n[PROOFSTEP]\nrw [extend_eq m h\u2081]\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : P s\u2081\nhs : s\u2081 \u2286 s\u2082\nh\u2082 : (fun s => P s) s\u2082\n\u22a2 m s\u2081 h\u2081 \u2264 m s\u2082 h\u2082\n[PROOFSTEP]\nexact m_mono h\u2081 h\u2082 hs\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\n\u03b2 : Type u_2\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nhm : \u2200 (i : \u03b2), P (f i)\n\u22a2 extend m (\u22c3 (i : \u03b2), f i) = \u2211' (i : \u03b2), extend m (f i)\n[PROOFSTEP]\ncases nonempty_encodable \u03b2\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\n\u03b2 : Type u_2\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nhm : \u2200 (i : \u03b2), P (f i)\nval\u271d : Encodable \u03b2\n\u22a2 extend m (\u22c3 (i : \u03b2), f i) = \u2211' (i : \u03b2), extend m (f i)\n[PROOFSTEP]\nrw [\u2190 Encodable.iUnion_decode\u2082, \u2190 tsum_iUnion_decode\u2082]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\n\u03b2 : Type u_2\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nhm : \u2200 (i : \u03b2), P (f i)\nval\u271d : Encodable \u03b2\n\u22a2 extend m (\u22c3 (i : \u2115) (b : \u03b2) (_ : b \u2208 Encodable.decode\u2082 \u03b2 i), f b) =\n \u2211' (i : \u2115), extend m (\u22c3 (b : \u03b2) (_ : b \u2208 Encodable.decode\u2082 \u03b2 i), f b)\n[PROOFSTEP]\nexact\n extend_iUnion_nat PU (fun n => Encodable.iUnion_decode\u2082_cases P0 hm) (mU _ (Encodable.iUnion_decode\u2082_disjoint_on hd))\n[GOAL]\ncase intro.m0\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\n\u03b2 : Type u_2\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nhm : \u2200 (i : \u03b2), P (f i)\nval\u271d : Encodable \u03b2\n\u22a2 extend m \u2205 = 0\n[PROOFSTEP]\nexact extend_empty P0 m0\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns\u2081 s\u2082 : Set \u03b1\nhd : Disjoint s\u2081 s\u2082\nh\u2081 : P s\u2081\nh\u2082 : P s\u2082\n\u22a2 extend m (s\u2081 \u222a s\u2082) = extend m s\u2081 + extend m s\u2082\n[PROOFSTEP]\nrw [union_eq_iUnion, extend_iUnion P0 m0 PU mU (pairwise_disjoint_on_bool.2 hd) (Bool.forall_bool.2 \u27e8h\u2082, h\u2081\u27e9),\n tsum_fintype]\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns\u2081 s\u2082 : Set \u03b1\nhd : Disjoint s\u2081 s\u2082\nh\u2081 : P s\u2081\nh\u2082 : P s\u2082\n\u22a2 \u2211 b : Bool, extend m (bif b then s\u2081 else s\u2082) = extend m s\u2081 + extend m s\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\n\u22a2 \u2191(inducedOuterMeasure m P0 m0) s = \u2a05 (t : Set \u03b1) (ht : P t) (_ : s \u2286 t), m t ht\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\n\u22a2 \u2191(inducedOuterMeasure m P0 m0) s \u2264 \u2a05 (t : Set \u03b1) (ht : P t) (_ : s \u2286 t), m t ht\n[PROOFSTEP]\nsimp only [le_iInf_iff]\n[GOAL]\ncase a\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\n\u22a2 \u2200 (i : Set \u03b1) (i_1 : P i), s \u2286 i \u2192 \u2191(inducedOuterMeasure m P0 m0) s \u2264 m i i_1\n[PROOFSTEP]\nintro t ht hs\n[GOAL]\ncase a\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns t : Set \u03b1\nht : P t\nhs : s \u2286 t\n\u22a2 \u2191(inducedOuterMeasure m P0 m0) s \u2264 m t ht\n[PROOFSTEP]\nrefine' le_trans (mono' _ hs) _\n[GOAL]\ncase a\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns t : Set \u03b1\nht : P t\nhs : s \u2286 t\n\u22a2 \u2191(inducedOuterMeasure m P0 m0) t \u2264 m t ht\n[PROOFSTEP]\nexact le_of_eq (inducedOuterMeasure_eq' _ msU m_mono _)\n[GOAL]\ncase a\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\n\u22a2 \u2a05 (t : Set \u03b1) (ht : P t) (_ : s \u2286 t), m t ht \u2264 \u2191(inducedOuterMeasure m P0 m0) s\n[PROOFSTEP]\nrefine' le_iInf _\n[GOAL]\ncase a\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\n\u22a2 \u2200 (i : \u2115 \u2192 Set \u03b1),\n \u2a05 (t : Set \u03b1) (ht : P t) (_ : s \u2286 t), m t ht \u2264 \u2a05 (_ : s \u2286 \u22c3 (i_1 : \u2115), i i_1), \u2211' (i_1 : \u2115), extend m (i i_1)\n[PROOFSTEP]\nintro f\n[GOAL]\ncase a\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\n\u22a2 \u2a05 (t : Set \u03b1) (ht : P t) (_ : s \u2286 t), m t ht \u2264 \u2a05 (_ : s \u2286 \u22c3 (i : \u2115), f i), \u2211' (i : \u2115), extend m (f i)\n[PROOFSTEP]\nrefine' le_iInf _\n[GOAL]\ncase a\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\n\u22a2 s \u2286 \u22c3 (i : \u2115), f i \u2192 \u2a05 (t : Set \u03b1) (ht : P t) (_ : s \u2286 t), m t ht \u2264 \u2211' (i : \u2115), extend m (f i)\n[PROOFSTEP]\nintro hf\n[GOAL]\ncase a\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : s \u2286 \u22c3 (i : \u2115), f i\n\u22a2 \u2a05 (t : Set \u03b1) (ht : P t) (_ : s \u2286 t), m t ht \u2264 \u2211' (i : \u2115), extend m (f i)\n[PROOFSTEP]\nrefine' le_trans _ (extend_iUnion_le_tsum_nat' _ msU _)\n[GOAL]\ncase a\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : s \u2286 \u22c3 (i : \u2115), f i\n\u22a2 \u2a05 (t : Set \u03b1) (ht : P t) (_ : s \u2286 t), m t ht \u2264 extend m (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nrefine' le_iInf _\n[GOAL]\ncase a\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : s \u2286 \u22c3 (i : \u2115), f i\n\u22a2 \u2200 (i : (fun s => P s) (\u22c3 (i : \u2115), f i)), \u2a05 (t : Set \u03b1) (ht : P t) (_ : s \u2286 t), m t ht \u2264 m (\u22c3 (i : \u2115), f i) i\n[PROOFSTEP]\nintro h2f\n[GOAL]\ncase a\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : s \u2286 \u22c3 (i : \u2115), f i\nh2f : (fun s => P s) (\u22c3 (i : \u2115), f i)\n\u22a2 \u2a05 (t : Set \u03b1) (ht : P t) (_ : s \u2286 t), m t ht \u2264 m (\u22c3 (i : \u2115), f i) h2f\n[PROOFSTEP]\nrefine' iInf_le_of_le _ (iInf_le_of_le h2f <| iInf_le _ hf)\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\nf : \u03b1 \u2243 \u03b1\nPm : \u2200 (s : Set \u03b1), P (\u2191f \u207b\u00b9' s) \u2194 P s\nmm : \u2200 (s : Set \u03b1) (hs : P s), m (\u2191f \u207b\u00b9' s) (_ : P (\u2191f \u207b\u00b9' s)) = m s hs\nA : Set \u03b1\n\u22a2 \u2191(inducedOuterMeasure m P0 m0) (\u2191f \u207b\u00b9' A) = \u2191(inducedOuterMeasure m P0 m0) A\n[PROOFSTEP]\nrw [inducedOuterMeasure_eq_iInf _ msU m_mono, inducedOuterMeasure_eq_iInf _ msU m_mono]\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\nf : \u03b1 \u2243 \u03b1\nPm : \u2200 (s : Set \u03b1), P (\u2191f \u207b\u00b9' s) \u2194 P s\nmm : \u2200 (s : Set \u03b1) (hs : P s), m (\u2191f \u207b\u00b9' s) (_ : P (\u2191f \u207b\u00b9' s)) = m s hs\nA : Set \u03b1\n\u22a2 \u2a05 (t : Set \u03b1) (ht : P t) (_ : \u2191f \u207b\u00b9' A \u2286 t), m t ht = \u2a05 (t : Set \u03b1) (ht : P t) (_ : A \u2286 t), m t ht\n[PROOFSTEP]\nsymm\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\nf : \u03b1 \u2243 \u03b1\nPm : \u2200 (s : Set \u03b1), P (\u2191f \u207b\u00b9' s) \u2194 P s\nmm : \u2200 (s : Set \u03b1) (hs : P s), m (\u2191f \u207b\u00b9' s) (_ : P (\u2191f \u207b\u00b9' s)) = m s hs\nA : Set \u03b1\n\u22a2 \u2a05 (t : Set \u03b1) (ht : P t) (_ : A \u2286 t), m t ht = \u2a05 (t : Set \u03b1) (ht : P t) (_ : \u2191f \u207b\u00b9' A \u2286 t), m t ht\n[PROOFSTEP]\nrefine' f.injective.preimage_surjective.iInf_congr (preimage f) fun s => _\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\nf : \u03b1 \u2243 \u03b1\nPm : \u2200 (s : Set \u03b1), P (\u2191f \u207b\u00b9' s) \u2194 P s\nmm : \u2200 (s : Set \u03b1) (hs : P s), m (\u2191f \u207b\u00b9' s) (_ : P (\u2191f \u207b\u00b9' s)) = m s hs\nA s : Set \u03b1\n\u22a2 \u2a05 (ht : P (\u2191f \u207b\u00b9' s)) (_ : \u2191f \u207b\u00b9' A \u2286 \u2191f \u207b\u00b9' s), m (\u2191f \u207b\u00b9' s) ht = \u2a05 (ht : P s) (_ : A \u2286 s), m s ht\n[PROOFSTEP]\nrefine' iInf_congr_Prop (Pm s) _\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\nf : \u03b1 \u2243 \u03b1\nPm : \u2200 (s : Set \u03b1), P (\u2191f \u207b\u00b9' s) \u2194 P s\nmm : \u2200 (s : Set \u03b1) (hs : P s), m (\u2191f \u207b\u00b9' s) (_ : P (\u2191f \u207b\u00b9' s)) = m s hs\nA s : Set \u03b1\n\u22a2 \u2200 (x : P s), \u2a05 (_ : \u2191f \u207b\u00b9' A \u2286 \u2191f \u207b\u00b9' s), m (\u2191f \u207b\u00b9' s) (_ : P (\u2191f \u207b\u00b9' s)) = \u2a05 (_ : A \u2286 s), m s x\n[PROOFSTEP]\nintro hs\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\nf : \u03b1 \u2243 \u03b1\nPm : \u2200 (s : Set \u03b1), P (\u2191f \u207b\u00b9' s) \u2194 P s\nmm : \u2200 (s : Set \u03b1) (hs : P s), m (\u2191f \u207b\u00b9' s) (_ : P (\u2191f \u207b\u00b9' s)) = m s hs\nA s : Set \u03b1\nhs : P s\n\u22a2 \u2a05 (_ : \u2191f \u207b\u00b9' A \u2286 \u2191f \u207b\u00b9' s), m (\u2191f \u207b\u00b9' s) (_ : P (\u2191f \u207b\u00b9' s)) = \u2a05 (_ : A \u2286 s), m s hs\n[PROOFSTEP]\nrefine' iInf_congr_Prop f.surjective.preimage_subset_preimage_iff _\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\nf : \u03b1 \u2243 \u03b1\nPm : \u2200 (s : Set \u03b1), P (\u2191f \u207b\u00b9' s) \u2194 P s\nmm : \u2200 (s : Set \u03b1) (hs : P s), m (\u2191f \u207b\u00b9' s) (_ : P (\u2191f \u207b\u00b9' s)) = m s hs\nA s : Set \u03b1\nhs : P s\n\u22a2 A \u2286 s \u2192 m (\u2191f \u207b\u00b9' s) (_ : P (\u2191f \u207b\u00b9' s)) = m s hs\n[PROOFSTEP]\nintro _\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\nf : \u03b1 \u2243 \u03b1\nPm : \u2200 (s : Set \u03b1), P (\u2191f \u207b\u00b9' s) \u2194 P s\nmm : \u2200 (s : Set \u03b1) (hs : P s), m (\u2191f \u207b\u00b9' s) (_ : P (\u2191f \u207b\u00b9' s)) = m s hs\nA s : Set \u03b1\nhs : P s\nx\u271d : A \u2286 s\n\u22a2 m (\u2191f \u207b\u00b9' s) (_ : P (\u2191f \u207b\u00b9' s)) = m s hs\n[PROOFSTEP]\nexact mm s hs\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nhs : \u2191(inducedOuterMeasure m P0 m0) s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 t _ht, s \u2286 t \u2227 \u2191(inducedOuterMeasure m P0 m0) t \u2264 \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n[PROOFSTEP]\nhave h := ENNReal.lt_add_right hs h\u03b5\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nhs : \u2191(inducedOuterMeasure m P0 m0) s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u2191(inducedOuterMeasure m P0 m0) s < \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n\u22a2 \u2203 t _ht, s \u2286 t \u2227 \u2191(inducedOuterMeasure m P0 m0) t \u2264 \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n[PROOFSTEP]\nconv at h =>\n lhs\n rw [inducedOuterMeasure_eq_iInf _ msU m_mono]\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nhs : \u2191(inducedOuterMeasure m P0 m0) s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u2191(inducedOuterMeasure m P0 m0) s < \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n| \u2191(inducedOuterMeasure m P0 m0) s < \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n[PROOFSTEP]\n lhs\n rw [inducedOuterMeasure_eq_iInf _ msU m_mono]\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nhs : \u2191(inducedOuterMeasure m P0 m0) s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u2191(inducedOuterMeasure m P0 m0) s < \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n| \u2191(inducedOuterMeasure m P0 m0) s < \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n[PROOFSTEP]\n lhs\n rw [inducedOuterMeasure_eq_iInf _ msU m_mono]\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nhs : \u2191(inducedOuterMeasure m P0 m0) s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u2191(inducedOuterMeasure m P0 m0) s < \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n| \u2191(inducedOuterMeasure m P0 m0) s < \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n[PROOFSTEP]\nlhs\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nhs : \u2191(inducedOuterMeasure m P0 m0) s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u2191(inducedOuterMeasure m P0 m0) s < \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n| \u2191(inducedOuterMeasure m P0 m0) s\n[PROOFSTEP]\nrw [inducedOuterMeasure_eq_iInf _ msU m_mono]\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nhs : \u2191(inducedOuterMeasure m P0 m0) s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u2a05 (t : Set \u03b1) (ht : P t) (_ : s \u2286 t), m t ht < \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n\u22a2 \u2203 t _ht, s \u2286 t \u2227 \u2191(inducedOuterMeasure m P0 m0) t \u2264 \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n[PROOFSTEP]\nsimp only [iInf_lt_iff] at h \n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nhs : \u2191(inducedOuterMeasure m P0 m0) s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u2203 i h i_1, m i (_ : P i) < \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n\u22a2 \u2203 t _ht, s \u2286 t \u2227 \u2191(inducedOuterMeasure m P0 m0) t \u2264 \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n[PROOFSTEP]\nrcases h with \u27e8t, h1t, h2t, h3t\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nhs : \u2191(inducedOuterMeasure m P0 m0) s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nt : Set \u03b1\nh1t : P t\nh2t : s \u2286 t\nh3t : m t (_ : P t) < \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n\u22a2 \u2203 t _ht, s \u2286 t \u2227 \u2191(inducedOuterMeasure m P0 m0) t \u2264 \u2191(inducedOuterMeasure m P0 m0) s + \u03b5\n[PROOFSTEP]\nexact \u27e8t, h1t, h2t, le_trans (le_of_eq <| inducedOuterMeasure_eq' _ msU m_mono h1t) (le_of_lt h3t)\u27e9\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\n\u22a2 MeasurableSet s \u2194\n \u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264\n \u2191(inducedOuterMeasure m P0 m0) t\n[PROOFSTEP]\nrw [isCaratheodory_iff_le]\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\n\u22a2 (\u2200 (t : Set \u03b1),\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264\n \u2191(inducedOuterMeasure m P0 m0) t) \u2194\n \u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264\n \u2191(inducedOuterMeasure m P0 m0) t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\n\u22a2 (\u2200 (t : Set \u03b1),\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264\n \u2191(inducedOuterMeasure m P0 m0) t) \u2192\n \u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264\n \u2191(inducedOuterMeasure m P0 m0) t\n[PROOFSTEP]\nintro h t _ht\n[GOAL]\ncase mp\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nh :\n \u2200 (t : Set \u03b1),\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) t\nt : Set \u03b1\n_ht : P t\n\u22a2 \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) t\n[PROOFSTEP]\nexact h t\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\n\u22a2 (\u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264\n \u2191(inducedOuterMeasure m P0 m0) t) \u2192\n \u2200 (t : Set \u03b1),\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) t\n[PROOFSTEP]\nintro h u\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nh :\n \u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) t\nu : Set \u03b1\n\u22a2 \u2191(inducedOuterMeasure m P0 m0) (u \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (u \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) u\n[PROOFSTEP]\nconv_rhs => rw [inducedOuterMeasure_eq_iInf _ msU m_mono]\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nh :\n \u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) t\nu : Set \u03b1\n| \u2191(inducedOuterMeasure m P0 m0) u\n[PROOFSTEP]\nrw [inducedOuterMeasure_eq_iInf _ msU m_mono]\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nh :\n \u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) t\nu : Set \u03b1\n| \u2191(inducedOuterMeasure m P0 m0) u\n[PROOFSTEP]\nrw [inducedOuterMeasure_eq_iInf _ msU m_mono]\n[GOAL]\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nh :\n \u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) t\nu : Set \u03b1\n| \u2191(inducedOuterMeasure m P0 m0) u\n[PROOFSTEP]\nrw [inducedOuterMeasure_eq_iInf _ msU m_mono]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nh :\n \u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) t\nu : Set \u03b1\n\u22a2 \u2191(inducedOuterMeasure m P0 m0) (u \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (u \\ s) \u2264\n \u2a05 (t : Set \u03b1) (ht : P t) (_ : u \u2286 t), m t ht\n[PROOFSTEP]\nrefine' le_iInf _\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nh :\n \u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) t\nu : Set \u03b1\n\u22a2 \u2200 (i : Set \u03b1),\n \u2191(inducedOuterMeasure m P0 m0) (u \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (u \\ s) \u2264 \u2a05 (ht : P i) (_ : u \u2286 i), m i ht\n[PROOFSTEP]\nintro t\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nh :\n \u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) t\nu t : Set \u03b1\n\u22a2 \u2191(inducedOuterMeasure m P0 m0) (u \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (u \\ s) \u2264 \u2a05 (ht : P t) (_ : u \u2286 t), m t ht\n[PROOFSTEP]\nrefine' le_iInf _\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nh :\n \u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) t\nu t : Set \u03b1\n\u22a2 \u2200 (i : P t), \u2191(inducedOuterMeasure m P0 m0) (u \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (u \\ s) \u2264 \u2a05 (_ : u \u2286 t), m t i\n[PROOFSTEP]\nintro ht\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nh :\n \u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) t\nu t : Set \u03b1\nht : P t\n\u22a2 \u2191(inducedOuterMeasure m P0 m0) (u \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (u \\ s) \u2264 \u2a05 (_ : u \u2286 t), m t ht\n[PROOFSTEP]\nrefine' le_iInf _\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nh :\n \u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) t\nu t : Set \u03b1\nht : P t\n\u22a2 u \u2286 t \u2192 \u2191(inducedOuterMeasure m P0 m0) (u \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (u \\ s) \u2264 m t ht\n[PROOFSTEP]\nintro h2t\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nh :\n \u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) t\nu t : Set \u03b1\nht : P t\nh2t : u \u2286 t\n\u22a2 \u2191(inducedOuterMeasure m P0 m0) (u \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (u \\ s) \u2264 m t ht\n[PROOFSTEP]\nrefine' le_trans _ (le_trans (h t ht) <| le_of_eq <| inducedOuterMeasure_eq' _ msU m_mono ht)\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nP : Set \u03b1 \u2192 Prop\nm : (s : Set \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\nP0 : P \u2205\nm0 : m \u2205 P0 = 0\nPU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984, (\u2200 (i : \u2115), P (f i)) \u2192 P (\u22c3 (i : \u2115), f i)\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n Pairwise (Disjoint on f) \u2192 m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) = \u2211' (i : \u2115), m (f i) (_ : P (f i))\nmsU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), P (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : P (\u22c3 (i : \u2115), f i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : P (f i))\nm_mono : \u2200 \u2983s\u2081 s\u2082 : Set \u03b1\u2984 (hs\u2081 : P s\u2081) (hs\u2082 : P s\u2082), s\u2081 \u2286 s\u2082 \u2192 m s\u2081 hs\u2081 \u2264 m s\u2082 hs\u2082\ns : Set \u03b1\nh :\n \u2200 (t : Set \u03b1),\n P t \u2192\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s) \u2264 \u2191(inducedOuterMeasure m P0 m0) t\nu t : Set \u03b1\nht : P t\nh2t : u \u2286 t\n\u22a2 \u2191(inducedOuterMeasure m P0 m0) (u \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (u \\ s) \u2264\n \u2191(inducedOuterMeasure m P0 m0) (t \u2229 s) + \u2191(inducedOuterMeasure m P0 m0) (t \\ s)\n[PROOFSTEP]\nrefine' add_le_add (mono' _ <| Set.inter_subset_inter_left _ h2t) (mono' _ <| diff_subset_diff_left h2t)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n Pairwise (Disjoint on f) \u2192\n m (\u22c3 (i : \u2115), f i) (_ : MeasurableSet (\u22c3 (b : \u2115), f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\n\u22a2 extend m s\u2081 \u2264 extend m s\u2082\n[PROOFSTEP]\nrefine' le_iInf _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n Pairwise (Disjoint on f) \u2192\n m (\u22c3 (i : \u2115), f i) (_ : MeasurableSet (\u22c3 (b : \u2115), f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\n\u22a2 \u2200 (i : (fun s => MeasurableSet s) s\u2082), extend m s\u2081 \u2264 m s\u2082 i\n[PROOFSTEP]\nintro h\u2082\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n Pairwise (Disjoint on f) \u2192\n m (\u22c3 (i : \u2115), f i) (_ : MeasurableSet (\u22c3 (b : \u2115), f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\nh\u2082 : (fun s => MeasurableSet s) s\u2082\n\u22a2 extend m s\u2081 \u2264 m s\u2082 h\u2082\n[PROOFSTEP]\nhave := extend_union MeasurableSet.empty m0 MeasurableSet.iUnion mU disjoint_sdiff_self_right h\u2081 (h\u2082.diff h\u2081)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n Pairwise (Disjoint on f) \u2192\n m (\u22c3 (i : \u2115), f i) (_ : MeasurableSet (\u22c3 (b : \u2115), f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\nh\u2082 : (fun s => MeasurableSet s) s\u2082\nthis : extend m (s\u2081 \u222a s\u2082 \\ s\u2081) = extend m s\u2081 + extend m (s\u2082 \\ s\u2081)\n\u22a2 extend m s\u2081 \u2264 m s\u2082 h\u2082\n[PROOFSTEP]\nrw [union_diff_cancel hs] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n Pairwise (Disjoint on f) \u2192\n m (\u22c3 (i : \u2115), f i) (_ : MeasurableSet (\u22c3 (b : \u2115), f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\nh\u2082 : (fun s => MeasurableSet s) s\u2082\nthis : extend m s\u2082 = extend m s\u2081 + extend m (s\u2082 \\ s\u2081)\n\u22a2 extend m s\u2081 \u2264 m s\u2082 h\u2082\n[PROOFSTEP]\nrw [\u2190 extend_eq m]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n Pairwise (Disjoint on f) \u2192\n m (\u22c3 (i : \u2115), f i) (_ : MeasurableSet (\u22c3 (b : \u2115), f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nhs : s\u2081 \u2286 s\u2082\nh\u2082 : (fun s => MeasurableSet s) s\u2082\nthis : extend m s\u2082 = extend m s\u2081 + extend m (s\u2082 \\ s\u2081)\n\u22a2 extend m s\u2081 \u2264 extend m s\u2082\n[PROOFSTEP]\nexact le_iff_exists_add.2 \u27e8_, this\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n Pairwise (Disjoint on f) \u2192\n m (\u22c3 (i : \u2115), f i) (_ : MeasurableSet (\u22c3 (b : \u2115), f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\n\u22a2 \u2200 (s : \u2115 \u2192 Set \u03b1), extend m (\u22c3 (i : \u2115), s i) \u2264 \u2211' (i : \u2115), extend m (s i)\n[PROOFSTEP]\nrefine' extend_iUnion_le_tsum_nat' MeasurableSet.iUnion _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n Pairwise (Disjoint on f) \u2192\n m (\u22c3 (i : \u2115), f i) (_ : MeasurableSet (\u22c3 (b : \u2115), f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\n\u22a2 \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n m (\u22c3 (i : \u2115), f i) (_ : MeasurableSet (\u22c3 (b : \u2115), f b)) \u2264 \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\n[PROOFSTEP]\nintro f h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n Pairwise (Disjoint on f) \u2192\n m (\u22c3 (i : \u2115), f i) (_ : MeasurableSet (\u22c3 (b : \u2115), f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\nf : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), MeasurableSet (f i)\n\u22a2 m (\u22c3 (i : \u2115), f i) (_ : MeasurableSet (\u22c3 (b : \u2115), f b)) \u2264 \u2211' (i : \u2115), m (f i) (_ : ?m.320579 (f i))\n[PROOFSTEP]\nsimp (config := { singlePass := true }) [iUnion_disjointed.symm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n Pairwise (Disjoint on f) \u2192\n m (\u22c3 (i : \u2115), f i) (_ : MeasurableSet (\u22c3 (b : \u2115), f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\nf : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), MeasurableSet (f i)\n\u22a2 m (\u22c3 (n : \u2115), disjointed (fun n => f n) n) (_ : MeasurableSet (\u22c3 (n : \u2115), disjointed (fun n => f n) n)) \u2264\n \u2211' (i : \u2115), m (f i) (_ : ?m.320579 (f i))\n[PROOFSTEP]\nrw [mU (MeasurableSet.disjointed h) (disjoint_disjointed _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n Pairwise (Disjoint on f) \u2192\n m (\u22c3 (i : \u2115), f i) (_ : MeasurableSet (\u22c3 (b : \u2115), f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\nf : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), MeasurableSet (f i)\n\u22a2 \u2211' (i : \u2115), m (disjointed (fun i => f i) i) (_ : MeasurableSet (disjointed (fun i => f i) i)) \u2264\n \u2211' (i : \u2115), m (f i) (_ : ?m.320579 (f i))\n[PROOFSTEP]\nrefine' ENNReal.tsum_le_tsum fun i => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n Pairwise (Disjoint on f) \u2192\n m (\u22c3 (i : \u2115), f i) (_ : MeasurableSet (\u22c3 (b : \u2115), f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\nf : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), MeasurableSet (f i)\ni : \u2115\n\u22a2 m (disjointed (fun i => f i) i) (_ : MeasurableSet (disjointed (fun i => f i) i)) \u2264 m (f i) (_ : ?m.320579 (f i))\n[PROOFSTEP]\nrw [\u2190 extend_eq m, \u2190 extend_eq m]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e\nm0 : m \u2205 (_ : MeasurableSet \u2205) = 0\nmU :\n \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),\n Pairwise (Disjoint on f) \u2192\n m (\u22c3 (i : \u2115), f i) (_ : MeasurableSet (\u22c3 (b : \u2115), f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i))\nf : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), MeasurableSet (f i)\ni : \u2115\n\u22a2 extend m (disjointed (fun i => f i) i) \u2264 extend m (f i)\n[PROOFSTEP]\nexact extend_mono m0 mU (MeasurableSet.disjointed h _) (disjointed_le f _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u22a2 m \u2264 trim m\n[PROOFSTEP]\napply le_ofFunction.mpr\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u22a2 \u2200 (s : Set \u03b1), \u2191m s \u2264 extend (fun s x => \u2191m s) s\n\u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : OuterMeasure \u03b1 \u22a2 extend (fun s x => \u2191m s) \u2205 = 0\n[PROOFSTEP]\nintro s\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191m s \u2264 extend (fun s x => \u2191m s) s\n\u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : OuterMeasure \u03b1 \u22a2 extend (fun s x => \u2191m s) \u2205 = 0\n[PROOFSTEP]\napply le_iInf\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2200 (i : (fun s => MeasurableSet s) s), \u2191m s \u2264 (fun s x => \u2191m s) s i\n\u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : OuterMeasure \u03b1 \u22a2 extend (fun s x => \u2191m s) \u2205 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u22a2 extend (fun s x => \u2191m s) \u2205 = 0\n[PROOFSTEP]\napply extend_empty\n[GOAL]\ncase m0\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u22a2 \u2191m \u2205 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase P0\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u22a2 MeasurableSet \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\nH : \u2200 {s : Set \u03b1}, MeasurableSet s \u2192 \u2191m\u2081 s = \u2191m\u2082 s\n\u22a2 trim m\u2081 = trim m\u2082\n[PROOFSTEP]\nunfold trim\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\nH : \u2200 {s : Set \u03b1}, MeasurableSet s \u2192 \u2191m\u2081 s = \u2191m\u2082 s\n\u22a2 inducedOuterMeasure (fun s x => \u2191m\u2081 s) (_ : MeasurableSet \u2205) (_ : \u2191m\u2081 \u2205 = 0) =\n inducedOuterMeasure (fun s x => \u2191m\u2082 s) (_ : MeasurableSet \u2205) (_ : \u2191m\u2082 \u2205 = 0)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_m\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\nH : \u2200 {s : Set \u03b1}, MeasurableSet s \u2192 \u2191m\u2081 s = \u2191m\u2082 s\n\u22a2 (fun s x => \u2191m\u2081 s) = fun s x => \u2191m\u2082 s\n[PROOFSTEP]\nfunext s hs\n[GOAL]\ncase e_m.h.h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\nH : \u2200 {s : Set \u03b1}, MeasurableSet s \u2192 \u2191m\u2081 s = \u2191m\u2082 s\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191m\u2081 s = \u2191m\u2082 s\n[PROOFSTEP]\nexact H hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\n\u22a2 m\u2081 \u2264 trim m\u2082 \u2194 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191m\u2081 s \u2264 \u2191m\u2082 s\n[PROOFSTEP]\nlet me := extend (fun s (_p : MeasurableSet s) => measureOf m\u2082 s)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\nme : Set \u03b1 \u2192 \u211d\u22650\u221e := extend fun s _p => \u2191m\u2082 s\n\u22a2 m\u2081 \u2264 trim m\u2082 \u2194 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191m\u2081 s \u2264 \u2191m\u2082 s\n[PROOFSTEP]\nhave me_empty : me \u2205 = 0 := by apply extend_empty; simp; simp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\nme : Set \u03b1 \u2192 \u211d\u22650\u221e := extend fun s _p => \u2191m\u2082 s\n\u22a2 me \u2205 = 0\n[PROOFSTEP]\napply extend_empty\n[GOAL]\ncase m0\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\nme : Set \u03b1 \u2192 \u211d\u22650\u221e := extend fun s _p => \u2191m\u2082 s\n\u22a2 \u2191m\u2082 \u2205 = 0\ncase P0\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\nme : Set \u03b1 \u2192 \u211d\u22650\u221e := extend fun s _p => \u2191m\u2082 s\n\u22a2 MeasurableSet \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\ncase P0\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\nme : Set \u03b1 \u2192 \u211d\u22650\u221e := extend fun s _p => \u2191m\u2082 s\n\u22a2 MeasurableSet \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\nme : Set \u03b1 \u2192 \u211d\u22650\u221e := extend fun s _p => \u2191m\u2082 s\nme_empty : me \u2205 = 0\n\u22a2 m\u2081 \u2264 trim m\u2082 \u2194 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191m\u2081 s \u2264 \u2191m\u2082 s\n[PROOFSTEP]\nhave : m\u2081 \u2264 OuterMeasure.ofFunction me me_empty \u2194 (\u2200 (s : Set \u03b1), measureOf m\u2081 s \u2264 me s) := le_ofFunction\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\nme : Set \u03b1 \u2192 \u211d\u22650\u221e := extend fun s _p => \u2191m\u2082 s\nme_empty : me \u2205 = 0\nthis : m\u2081 \u2264 OuterMeasure.ofFunction me me_empty \u2194 \u2200 (s : Set \u03b1), \u2191m\u2081 s \u2264 me s\n\u22a2 m\u2081 \u2264 trim m\u2082 \u2194 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191m\u2081 s \u2264 \u2191m\u2082 s\n[PROOFSTEP]\napply this.trans\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\nme : Set \u03b1 \u2192 \u211d\u22650\u221e := extend fun s _p => \u2191m\u2082 s\nme_empty : me \u2205 = 0\nthis : m\u2081 \u2264 OuterMeasure.ofFunction me me_empty \u2194 \u2200 (s : Set \u03b1), \u2191m\u2081 s \u2264 me s\n\u22a2 (\u2200 (s : Set \u03b1), \u2191m\u2081 s \u2264 me s) \u2194 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191m\u2081 s \u2264 \u2191m\u2082 s\n[PROOFSTEP]\napply forall_congr'\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\nme : Set \u03b1 \u2192 \u211d\u22650\u221e := extend fun s _p => \u2191m\u2082 s\nme_empty : me \u2205 = 0\nthis : m\u2081 \u2264 OuterMeasure.ofFunction me me_empty \u2194 \u2200 (s : Set \u03b1), \u2191m\u2081 s \u2264 me s\n\u22a2 \u2200 (a : Set \u03b1), \u2191m\u2081 a \u2264 me a \u2194 MeasurableSet a \u2192 \u2191m\u2081 a \u2264 \u2191m\u2082 a\n[PROOFSTEP]\nintro s\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\nme : Set \u03b1 \u2192 \u211d\u22650\u221e := extend fun s _p => \u2191m\u2082 s\nme_empty : me \u2205 = 0\nthis : m\u2081 \u2264 OuterMeasure.ofFunction me me_empty \u2194 \u2200 (s : Set \u03b1), \u2191m\u2081 s \u2264 me s\ns : Set \u03b1\n\u22a2 \u2191m\u2081 s \u2264 me s \u2194 MeasurableSet s \u2192 \u2191m\u2081 s \u2264 \u2191m\u2082 s\n[PROOFSTEP]\napply le_iInf_iff\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191(trim m\u2081) s \u2264 \u2191m\u2082 s \u2194 \u2191m\u2081 s \u2264 \u2191m\u2082 s\n[PROOFSTEP]\nrw [trim_eq _ hs]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 : OuterMeasure \u03b1\n\u22a2 trim m\u2081 = trim m\u2082 \u2194 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191m\u2081 s = \u2191m\u2082 s\n[PROOFSTEP]\nsimp only [le_antisymm_iff, trim_le_trim_iff, forall_and]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191(trim m) s = \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\n[PROOFSTEP]\nsimp (config := { singlePass := true }) only [iInf_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191(trim m) s = \u2a05 (t : Set \u03b1) (_ : MeasurableSet t) (_ : s \u2286 t), \u2191m t\n[PROOFSTEP]\nexact inducedOuterMeasure_eq_iInf MeasurableSet.iUnion (fun f _ => m.iUnion_nat f) (fun _ _ _ _ h => m.mono h) s\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191(trim m) s = \u2a05 (t : { t // s \u2286 t \u2227 MeasurableSet t }), \u2191m \u2191t\n[PROOFSTEP]\nsimp [iInf_subtype, iInf_and, trim_eq_iInf]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d : OuterMeasure \u03b1\n\u03b9 : Type u_2\nm : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191(sum fun i => trim (m i)) s \u2264 \u2191(trim (sum m)) s\n[PROOFSTEP]\nsimp [trim_eq_iInf]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d : OuterMeasure \u03b1\n\u03b9 : Type u_2\nm : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2200 (i : Set \u03b1),\n s \u2286 i \u2192\n MeasurableSet i \u2192 \u2211' (i : \u03b9), \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191(m i) t \u2264 \u2211' (i_3 : \u03b9), \u2191(m i_3) i\n[PROOFSTEP]\nexact fun t st ht => ENNReal.tsum_le_tsum fun i => iInf_le_of_le t <| iInf_le_of_le st <| iInf_le _ ht\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t = \u2191(trim m) s\n[PROOFSTEP]\nsimp only [trim_eq_iInf]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t = \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\n[PROOFSTEP]\nset ms := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), m t\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t = ms\n[PROOFSTEP]\nby_cases hs : ms = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : ms = \u22a4\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t = ms\n[PROOFSTEP]\nsimp only [hs]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : ms = \u22a4\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t = \u22a4\n[PROOFSTEP]\nsimp only [iInf_eq_top] at hs \n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u2200 (i : Set \u03b1), s \u2286 i \u2192 MeasurableSet i \u2192 \u2191m i = \u22a4\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t = \u22a4\n[PROOFSTEP]\nexact \u27e8univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u00acms = \u22a4\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t = ms\n[PROOFSTEP]\nhave : \u2200 r > ms, \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 m t < r :=\n by\n intro r hs\n have : \u2203 t, MeasurableSet t \u2227 s \u2286 t \u2227 measureOf m t < r := by simpa [iInf_lt_iff] using hs\n rcases this with \u27e8t, hmt, hin, hlt\u27e9\n exists t\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u00acms = \u22a4\n\u22a2 \u2200 (r : \u211d\u22650\u221e), r > ms \u2192 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\n[PROOFSTEP]\nintro r hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs\u271d : \u00acms = \u22a4\nr : \u211d\u22650\u221e\nhs : r > ms\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\n[PROOFSTEP]\nhave : \u2203 t, MeasurableSet t \u2227 s \u2286 t \u2227 measureOf m t < r := by simpa [iInf_lt_iff] using hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs\u271d : \u00acms = \u22a4\nr : \u211d\u22650\u221e\nhs : r > ms\n\u22a2 \u2203 t, MeasurableSet t \u2227 s \u2286 t \u2227 \u2191m t < r\n[PROOFSTEP]\nsimpa [iInf_lt_iff] using hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs\u271d : \u00acms = \u22a4\nr : \u211d\u22650\u221e\nhs : r > ms\nthis : \u2203 t, MeasurableSet t \u2227 s \u2286 t \u2227 \u2191m t < r\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\n[PROOFSTEP]\nrcases this with \u27e8t, hmt, hin, hlt\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs\u271d : \u00acms = \u22a4\nr : \u211d\u22650\u221e\nhs : r > ms\nt : Set \u03b1\nhmt : MeasurableSet t\nhin : s \u2286 t\nhlt : \u2191m t < r\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\n[PROOFSTEP]\nexists t\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u00acms = \u22a4\nthis : \u2200 (r : \u211d\u22650\u221e), r > ms \u2192 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t = ms\n[PROOFSTEP]\nhave : \u2200 n : \u2115, \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 m t < ms + (n : \u211d\u22650\u221e)\u207b\u00b9 :=\n by\n intro n\n refine' this _ (ENNReal.lt_add_right hs _)\n simp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u00acms = \u22a4\nthis : \u2200 (r : \u211d\u22650\u221e), r > ms \u2192 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\n\u22a2 \u2200 (n : \u2115), \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < ms + (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u00acms = \u22a4\nthis : \u2200 (r : \u211d\u22650\u221e), r > ms \u2192 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\nn : \u2115\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < ms + (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nrefine' this _ (ENNReal.lt_add_right hs _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u00acms = \u22a4\nthis : \u2200 (r : \u211d\u22650\u221e), r > ms \u2192 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\nn : \u2115\n\u22a2 (\u2191n)\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u00acms = \u22a4\nthis\u271d : \u2200 (r : \u211d\u22650\u221e), r > ms \u2192 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\nthis : \u2200 (n : \u2115), \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < ms + (\u2191n)\u207b\u00b9\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t = ms\n[PROOFSTEP]\nchoose t hsub hm hm' using this\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u00acms = \u22a4\nthis : \u2200 (r : \u211d\u22650\u221e), r > ms \u2192 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\nt : \u2115 \u2192 Set \u03b1\nhsub : \u2200 (n : \u2115), s \u2286 t n\nhm : \u2200 (n : \u2115), MeasurableSet (t n)\nhm' : \u2200 (n : \u2115), \u2191m (t n) < ms + (\u2191n)\u207b\u00b9\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t = ms\n[PROOFSTEP]\nrefine' \u27e8\u22c2 n, t n, subset_iInter hsub, MeasurableSet.iInter hm, _\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u00acms = \u22a4\nthis : \u2200 (r : \u211d\u22650\u221e), r > ms \u2192 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\nt : \u2115 \u2192 Set \u03b1\nhsub : \u2200 (n : \u2115), s \u2286 t n\nhm : \u2200 (n : \u2115), MeasurableSet (t n)\nhm' : \u2200 (n : \u2115), \u2191m (t n) < ms + (\u2191n)\u207b\u00b9\n\u22a2 \u2191m (\u22c2 (n : \u2115), t n) = ms\n[PROOFSTEP]\nhave : Tendsto (fun n : \u2115 => ms + (n : \u211d\u22650\u221e)\u207b\u00b9) atTop (\ud835\udcdd (ms + 0)) :=\n tendsto_const_nhds.add ENNReal.tendsto_inv_nat_nhds_zero\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u00acms = \u22a4\nthis\u271d : \u2200 (r : \u211d\u22650\u221e), r > ms \u2192 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\nt : \u2115 \u2192 Set \u03b1\nhsub : \u2200 (n : \u2115), s \u2286 t n\nhm : \u2200 (n : \u2115), MeasurableSet (t n)\nhm' : \u2200 (n : \u2115), \u2191m (t n) < ms + (\u2191n)\u207b\u00b9\nthis : Tendsto (fun n => ms + (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd (ms + 0))\n\u22a2 \u2191m (\u22c2 (n : \u2115), t n) = ms\n[PROOFSTEP]\nrw [add_zero] at this \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u00acms = \u22a4\nthis\u271d : \u2200 (r : \u211d\u22650\u221e), r > ms \u2192 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\nt : \u2115 \u2192 Set \u03b1\nhsub : \u2200 (n : \u2115), s \u2286 t n\nhm : \u2200 (n : \u2115), MeasurableSet (t n)\nhm' : \u2200 (n : \u2115), \u2191m (t n) < ms + (\u2191n)\u207b\u00b9\nthis : Tendsto (fun n => ms + (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd ms)\n\u22a2 \u2191m (\u22c2 (n : \u2115), t n) = ms\n[PROOFSTEP]\nrefine' le_antisymm (ge_of_tendsto' this fun n => _) _\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u00acms = \u22a4\nthis\u271d : \u2200 (r : \u211d\u22650\u221e), r > ms \u2192 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\nt : \u2115 \u2192 Set \u03b1\nhsub : \u2200 (n : \u2115), s \u2286 t n\nhm : \u2200 (n : \u2115), MeasurableSet (t n)\nhm' : \u2200 (n : \u2115), \u2191m (t n) < ms + (\u2191n)\u207b\u00b9\nthis : Tendsto (fun n => ms + (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd ms)\nn : \u2115\n\u22a2 \u2191m (\u22c2 (n : \u2115), t n) \u2264 ms + (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nexact le_trans (m.mono' <| iInter_subset t n) (hm' n).le\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u00acms = \u22a4\nthis\u271d : \u2200 (r : \u211d\u22650\u221e), r > ms \u2192 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\nt : \u2115 \u2192 Set \u03b1\nhsub : \u2200 (n : \u2115), s \u2286 t n\nhm : \u2200 (n : \u2115), MeasurableSet (t n)\nhm' : \u2200 (n : \u2115), \u2191m (t n) < ms + (\u2191n)\u207b\u00b9\nthis : Tendsto (fun n => ms + (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd ms)\n\u22a2 ms \u2264 \u2191m (\u22c2 (n : \u2115), t n)\n[PROOFSTEP]\nrefine' iInf_le_of_le (\u22c2 n, t n) _\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u00acms = \u22a4\nthis\u271d : \u2200 (r : \u211d\u22650\u221e), r > ms \u2192 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\nt : \u2115 \u2192 Set \u03b1\nhsub : \u2200 (n : \u2115), s \u2286 t n\nhm : \u2200 (n : \u2115), MeasurableSet (t n)\nhm' : \u2200 (n : \u2115), \u2191m (t n) < ms + (\u2191n)\u207b\u00b9\nthis : Tendsto (fun n => ms + (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd ms)\n\u22a2 \u2a05 (_ : s \u2286 \u22c2 (n : \u2115), t n) (_ : MeasurableSet (\u22c2 (n : \u2115), t n)), \u2191m (\u22c2 (n : \u2115), t n) \u2264 \u2191m (\u22c2 (n : \u2115), t n)\n[PROOFSTEP]\nrefine' iInf_le_of_le (subset_iInter hsub) _\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nms : \u211d\u22650\u221e := \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191m t\nhs : \u00acms = \u22a4\nthis\u271d : \u2200 (r : \u211d\u22650\u221e), r > ms \u2192 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t < r\nt : \u2115 \u2192 Set \u03b1\nhsub : \u2200 (n : \u2115), s \u2286 t n\nhm : \u2200 (n : \u2115), MeasurableSet (t n)\nhm' : \u2200 (n : \u2115), \u2191m (t n) < ms + (\u2191n)\u207b\u00b9\nthis : Tendsto (fun n => ms + (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd ms)\n\u22a2 \u2a05 (_ : MeasurableSet (\u22c2 (n : \u2115), t n)), \u2191m (\u22c2 (n : \u2115), t n) \u2264 \u2191m (\u22c2 (n : \u2115), t n)\n[PROOFSTEP]\nrefine' iInf_le _ (MeasurableSet.iInter hm)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nh : \u2191(trim m) s = 0\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t = 0\n[PROOFSTEP]\nrcases exists_measurable_superset_eq_trim m s with \u27e8t, hst, ht, hm\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm\u271d m : OuterMeasure \u03b1\ns : Set \u03b1\nh : \u2191(trim m) s = 0\nt : Set \u03b1\nhst : s \u2286 t\nht : MeasurableSet t\nhm : \u2191m t = \u2191(trim m) s\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191m t = 0\n[PROOFSTEP]\nexact \u27e8t, hst, ht, h \u25b8 hm\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2200 (i : \u03b9), \u2191(\u03bc i) t = \u2191(trim (\u03bc i)) s\n[PROOFSTEP]\nchoose t hst ht h\u03bct using fun i => (\u03bc i).exists_measurable_superset_eq_trim s\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\nhst : \u2200 (i : \u03b9), s \u2286 t i\nht : \u2200 (i : \u03b9), MeasurableSet (t i)\nh\u03bct : \u2200 (i : \u03b9), \u2191(\u03bc i) (t i) = \u2191(trim (\u03bc i)) s\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2200 (i : \u03b9), \u2191(\u03bc i) t = \u2191(trim (\u03bc i)) s\n[PROOFSTEP]\nreplace hst := subset_iInter hst\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\nht : \u2200 (i : \u03b9), MeasurableSet (t i)\nh\u03bct : \u2200 (i : \u03b9), \u2191(\u03bc i) (t i) = \u2191(trim (\u03bc i)) s\nhst : s \u2286 \u22c2 (i : \u03b9), t i\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2200 (i : \u03b9), \u2191(\u03bc i) t = \u2191(trim (\u03bc i)) s\n[PROOFSTEP]\nreplace ht := MeasurableSet.iInter ht\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\nh\u03bct : \u2200 (i : \u03b9), \u2191(\u03bc i) (t i) = \u2191(trim (\u03bc i)) s\nhst : s \u2286 \u22c2 (i : \u03b9), t i\nht : MeasurableSet (\u22c2 (b : \u03b9), t b)\n\u22a2 \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2200 (i : \u03b9), \u2191(\u03bc i) t = \u2191(trim (\u03bc i)) s\n[PROOFSTEP]\nrefine' \u27e8\u22c2 i, t i, hst, ht, fun i => le_antisymm _ _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\nh\u03bct : \u2200 (i : \u03b9), \u2191(\u03bc i) (t i) = \u2191(trim (\u03bc i)) s\nhst : s \u2286 \u22c2 (i : \u03b9), t i\nht : MeasurableSet (\u22c2 (b : \u03b9), t b)\ni : \u03b9\n\u22a2 \u2191(\u03bc i) (\u22c2 (i : \u03b9), t i) \u2264 \u2191(trim (\u03bc i)) s\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\nh\u03bct : \u2200 (i : \u03b9), \u2191(\u03bc i) (t i) = \u2191(trim (\u03bc i)) s\nhst : s \u2286 \u22c2 (i : \u03b9), t i\nht : MeasurableSet (\u22c2 (b : \u03b9), t b)\ni : \u03b9\n\u22a2 \u2191(trim (\u03bc i)) s \u2264 \u2191(\u03bc i) (\u22c2 (i : \u03b9), t i)\n[PROOFSTEP]\nexacts [h\u03bct i \u25b8 (\u03bc i).mono (iInter_subset _ _), (mono' _ hst).trans_eq ((\u03bc i).trim_eq ht)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 m\u2083 : OuterMeasure \u03b1\nop : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nh : \u2200 (s : Set \u03b1), \u2191m\u2081 s = op (\u2191m\u2082 s) (\u2191m\u2083 s)\ns : Set \u03b1\n\u22a2 \u2191(trim m\u2081) s = op (\u2191(trim m\u2082) s) (\u2191(trim m\u2083) s)\n[PROOFSTEP]\nrcases exists_measurable_superset_forall_eq_trim ![m\u2081, m\u2082, m\u2083] s with \u27e8t, _hst, _ht, htm\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 m\u2083 : OuterMeasure \u03b1\nop : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nh : \u2200 (s : Set \u03b1), \u2191m\u2081 s = op (\u2191m\u2082 s) (\u2191m\u2083 s)\ns t : Set \u03b1\n_hst : s \u2286 t\n_ht : MeasurableSet t\nhtm :\n \u2200 (i : Fin (Nat.succ (Nat.succ (Nat.succ 0)))),\n \u2191(Matrix.vecCons m\u2081 ![m\u2082, m\u2083] i) t = \u2191(trim (Matrix.vecCons m\u2081 ![m\u2082, m\u2083] i)) s\n\u22a2 \u2191(trim m\u2081) s = op (\u2191(trim m\u2082) s) (\u2191(trim m\u2083) s)\n[PROOFSTEP]\nsimp only [Fin.forall_fin_succ, Matrix.cons_val_zero, Matrix.cons_val_succ] at htm \n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm m\u2081 m\u2082 m\u2083 : OuterMeasure \u03b1\nop : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nh : \u2200 (s : Set \u03b1), \u2191m\u2081 s = op (\u2191m\u2082 s) (\u2191m\u2083 s)\ns t : Set \u03b1\n_hst : s \u2286 t\n_ht : MeasurableSet t\nhtm : \u2191m\u2081 t = \u2191(trim m\u2081) s \u2227 \u2191m\u2082 t = \u2191(trim m\u2082) s \u2227 \u2191m\u2083 t = \u2191(trim m\u2083) s \u2227 \u2200 (i : Fin 0), \u2191![] t = \u2191(trim ![]) s\n\u22a2 \u2191(trim m\u2081) s = op (\u2191(trim m\u2082) s) (\u2191(trim m\u2083) s)\n[PROOFSTEP]\nrw [\u2190 htm.1, \u2190 htm.2.1, \u2190 htm.2.2.1, h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\n\u22a2 trim (\u2a06 (i : \u03b9), \u03bc i) = \u2a06 (i : \u03b9), trim (\u03bc i)\n[PROOFSTEP]\nsimp_rw [\u2190 @iSup_plift_down _ \u03b9]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\n\u22a2 trim (\u2a06 (i : PLift \u03b9), \u03bc i.down) = \u2a06 (i : PLift \u03b9), trim (\u03bc i.down)\n[PROOFSTEP]\next1 s\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s = \u2191(\u2a06 (i : PLift \u03b9), trim (\u03bc i.down)) s\n[PROOFSTEP]\nobtain \u27e8t, _, _, h\u03bct\u27e9 :=\n exists_measurable_superset_forall_eq_trim (Option.elim' (\u2a06 i, \u03bc (PLift.down i)) (\u03bc \u2218 PLift.down)) s\n[GOAL]\ncase h.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns t : Set \u03b1\nleft\u271d\u00b9 : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct :\n \u2200 (i : Option (PLift \u03b9)),\n \u2191(Option.elim' (\u2a06 (i : PLift \u03b9), \u03bc i.down) (\u03bc \u2218 PLift.down) i) t =\n \u2191(trim (Option.elim' (\u2a06 (i : PLift \u03b9), \u03bc i.down) (\u03bc \u2218 PLift.down) i)) s\n\u22a2 \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s = \u2191(\u2a06 (i : PLift \u03b9), trim (\u03bc i.down)) s\n[PROOFSTEP]\nsimp only [Option.forall, Option.elim'] at h\u03bct \n[GOAL]\ncase h.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns t : Set \u03b1\nleft\u271d\u00b9 : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct :\n \u2191(\u2a06 (i : PLift \u03b9), \u03bc i.down) t = \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s \u2227\n \u2200 (x : PLift \u03b9), \u2191((\u03bc \u2218 PLift.down) x) t = \u2191(trim ((\u03bc \u2218 PLift.down) x)) s\n\u22a2 \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s = \u2191(\u2a06 (i : PLift \u03b9), trim (\u03bc i.down)) s\n[PROOFSTEP]\nsimp only [iSup_apply, \u2190 h\u03bct.1]\n[GOAL]\ncase h.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns t : Set \u03b1\nleft\u271d\u00b9 : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct :\n \u2191(\u2a06 (i : PLift \u03b9), \u03bc i.down) t = \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s \u2227\n \u2200 (x : PLift \u03b9), \u2191((\u03bc \u2218 PLift.down) x) t = \u2191(trim ((\u03bc \u2218 PLift.down) x)) s\n\u22a2 \u2a06 (i : PLift \u03b9), \u2191(\u03bc i.down) t = \u2a06 (i : PLift \u03b9), \u2191(trim (\u03bc i.down)) s\n[PROOFSTEP]\nexact iSup_congr h\u03bct.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm \u03bc : OuterMeasure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 trim (\u2191(restrict s) \u03bc) = \u2191(restrict s) (trim \u03bc)\n[PROOFSTEP]\nrefine' le_antisymm (fun t => _) (le_trim_iff.2 fun t ht => _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm \u03bc : OuterMeasure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191(trim (\u2191(restrict s) \u03bc)) t \u2264 \u2191(\u2191(restrict s) (trim \u03bc)) t\n[PROOFSTEP]\nrw [restrict_apply]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm \u03bc : OuterMeasure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191(trim (\u2191(restrict s) \u03bc)) t \u2264 \u2191(trim \u03bc) (t \u2229 s)\n[PROOFSTEP]\nrcases \u03bc.exists_measurable_superset_eq_trim (t \u2229 s) with \u27e8t', htt', ht', h\u03bct'\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm \u03bc : OuterMeasure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt t' : Set \u03b1\nhtt' : t \u2229 s \u2286 t'\nht' : MeasurableSet t'\nh\u03bct' : \u2191\u03bc t' = \u2191(trim \u03bc) (t \u2229 s)\n\u22a2 \u2191(trim (\u2191(restrict s) \u03bc)) t \u2264 \u2191(trim \u03bc) (t \u2229 s)\n[PROOFSTEP]\nrw [\u2190 h\u03bct']\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm \u03bc : OuterMeasure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt t' : Set \u03b1\nhtt' : t \u2229 s \u2286 t'\nht' : MeasurableSet t'\nh\u03bct' : \u2191\u03bc t' = \u2191(trim \u03bc) (t \u2229 s)\n\u22a2 \u2191(trim (\u2191(restrict s) \u03bc)) t \u2264 \u2191\u03bc t'\n[PROOFSTEP]\nrw [inter_subset] at htt' \n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm \u03bc : OuterMeasure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt t' : Set \u03b1\nhtt' : t \u2286 s\u1d9c \u222a t'\nht' : MeasurableSet t'\nh\u03bct' : \u2191\u03bc t' = \u2191(trim \u03bc) (t \u2229 s)\n\u22a2 \u2191(trim (\u2191(restrict s) \u03bc)) t \u2264 \u2191\u03bc t'\n[PROOFSTEP]\nrefine' (mono' _ htt').trans _\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm \u03bc : OuterMeasure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt t' : Set \u03b1\nhtt' : t \u2286 s\u1d9c \u222a t'\nht' : MeasurableSet t'\nh\u03bct' : \u2191\u03bc t' = \u2191(trim \u03bc) (t \u2229 s)\n\u22a2 \u2191(trim (\u2191(restrict s) \u03bc)) (s\u1d9c \u222a t') \u2264 \u2191\u03bc t'\n[PROOFSTEP]\nrw [trim_eq _ (hs.compl.union ht'), restrict_apply, union_inter_distrib_right, compl_inter_self, Set.empty_union]\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm \u03bc : OuterMeasure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt t' : Set \u03b1\nhtt' : t \u2286 s\u1d9c \u222a t'\nht' : MeasurableSet t'\nh\u03bct' : \u2191\u03bc t' = \u2191(trim \u03bc) (t \u2229 s)\n\u22a2 \u2191\u03bc (t' \u2229 s) \u2264 \u2191\u03bc t'\n[PROOFSTEP]\nexact \u03bc.mono' (inter_subset_left _ _)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nm \u03bc : OuterMeasure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191(\u2191(restrict s) (trim \u03bc)) t \u2264 \u2191(\u2191(restrict s) \u03bc) t\n[PROOFSTEP]\nrw [restrict_apply, trim_eq _ (ht.inter hs), restrict_apply]\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.OuterMeasure", "llama_tokens": 103892, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.4362083330004928}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u2081 : List (Sigma \u03b2)\nh\u2081 : NodupKeys l\u2081\nl\u2082 : List (Sigma \u03b2)\nnodupKeys\u271d : NodupKeys l\u2082\nH : { entries := l\u2081, nodupKeys := h\u2081 }.entries = { entries := l\u2082, nodupKeys := nodupKeys\u271d }.entries\n\u22a2 { entries := l\u2081, nodupKeys := h\u2081 } = { entries := l\u2082, nodupKeys := nodupKeys\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b2 a)\nxs ys : AList \u03b2\n\u22a2 Decidable (xs = ys)\n[PROOFSTEP]\nrw [ext_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b2 a)\nxs ys : AList \u03b2\n\u22a2 Decidable (xs.entries = ys.entries)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.entries \u2286 s\u2082.entries\n\u22a2 keys s\u2081 \u2286 keys s\u2082\n[PROOFSTEP]\nintro k hk\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.entries \u2286 s\u2082.entries\nk : \u03b1\nhk : k \u2208 keys s\u2081\n\u22a2 k \u2208 keys s\u2082\n[PROOFSTEP]\nletI : DecidableEq \u03b1 := Classical.decEq \u03b1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.entries \u2286 s\u2082.entries\nk : \u03b1\nhk : k \u2208 keys s\u2081\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\n\u22a2 k \u2208 keys s\u2082\n[PROOFSTEP]\nhave := h (mem_lookup_iff.1 (Option.get_mem (lookup_isSome.2 hk)))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.entries \u2286 s\u2082.entries\nk : \u03b1\nhk : k \u2208 keys s\u2081\nthis\u271d : DecidableEq \u03b1 := Classical.decEq \u03b1\nthis : { fst := k, snd := Option.get (lookup k s\u2081) (_ : Option.isSome (lookup k s\u2081) = true) } \u2208 s\u2082.entries\n\u22a2 k \u2208 keys s\u2082\n[PROOFSTEP]\nrw [\u2190 mem_lookup_iff, Option.mem_def] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.entries \u2286 s\u2082.entries\nk : \u03b1\nhk : k \u2208 keys s\u2081\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nthis\u271d : { fst := k, snd := Option.get (lookup k s\u2081) (_ : Option.isSome (lookup k s\u2081) = true) } \u2208 s\u2082.entries\nthis : lookup k s\u2082 = some (Option.get (lookup k s\u2081) (_ : Option.isSome (lookup k s\u2081) = true))\n\u22a2 k \u2208 keys s\u2082\n[PROOFSTEP]\nrw [\u2190 mem_keys, \u2190 lookup_isSome, this]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.entries \u2286 s\u2082.entries\nk : \u03b1\nhk : k \u2208 keys s\u2081\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nthis\u271d : { fst := k, snd := Option.get (lookup k s\u2081) (_ : Option.isSome (lookup k s\u2081) = true) } \u2208 s\u2082.entries\nthis : lookup k s\u2082 = some (Option.get (lookup k s\u2081) (_ : Option.isSome (lookup k s\u2081) = true))\n\u22a2 Option.isSome (some (Option.get (lookup k s\u2081) (_ : Option.isSome (lookup k s\u2081) = true))) = true\n[PROOFSTEP]\nexact Option.isSome_some\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a\ns : AList \u03b2\n\u22a2 a' \u2208 replace a b s \u2194 a' \u2208 s\n[PROOFSTEP]\nrw [mem_keys, keys_replace, \u2190 mem_keys]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\ns : AList \u03b2\n\u22a2 a' \u2208 erase a s \u2194 a' \u2260 a \u2227 a' \u2208 s\n[PROOFSTEP]\nrw [mem_keys, keys_erase, s.keys_nodup.mem_erase_iff, \u2190 mem_keys]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns : AList \u03b2\nh : \u00aca \u2208 s\n\u22a2 (insert a b s).entries = { fst := a, snd := b } :: s.entries\n[PROOFSTEP]\nrw [insert_entries, kerase_of_not_mem_keys h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns : AList \u03b2\n\u22a2 keys (insert a b s) = a :: List.erase (keys s) a\n[PROOFSTEP]\nsimp [insert, keys, keys_kerase]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns\u2081 s\u2082 : AList \u03b2\np : s\u2081.entries ~ s\u2082.entries\n\u22a2 (insert a b s\u2081).entries ~ (insert a b s\u2082).entries\n[PROOFSTEP]\nsimp only [insert_entries]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns\u2081 s\u2082 : AList \u03b2\np : s\u2081.entries ~ s\u2082.entries\n\u22a2 { fst := a, snd := b } :: kerase a s\u2081.entries ~ { fst := a, snd := b } :: kerase a s\u2082.entries\n[PROOFSTEP]\nexact p.kinsert s\u2081.nodupKeys\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns : AList \u03b2\n\u22a2 lookup a (insert a b s) = some b\n[PROOFSTEP]\nsimp only [lookup, insert, dlookup_kinsert]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : List (Sigma \u03b2)\n\u22a2 lookup a (toAList s) = dlookup a s\n[PROOFSTEP]\nrw [List.toAList, lookup, dlookup_dedupKeys]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb b' : \u03b2 a\ns : AList \u03b2\n\u22a2 insert a b' (insert a b s) = insert a b' s\n[PROOFSTEP]\next : 1\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb b' : \u03b2 a\ns : AList \u03b2\n\u22a2 (insert a b' (insert a b s)).entries = (insert a b' s).entries\n[PROOFSTEP]\nsimp only [AList.insert_entries, List.kerase_cons_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a\nb' : \u03b2 a'\ns : AList \u03b2\nh : a \u2260 a'\n\u22a2 (insert a' b' (insert a b s)).entries ~ (insert a b (insert a' b' s)).entries\n[PROOFSTEP]\nsimp only [insert_entries]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a\nb' : \u03b2 a'\ns : AList \u03b2\nh : a \u2260 a'\n\u22a2 { fst := a', snd := b' } :: kerase a' ({ fst := a, snd := b } :: kerase a s.entries) ~\n { fst := a, snd := b } :: kerase a ({ fst := a', snd := b' } :: kerase a' s.entries)\n[PROOFSTEP]\nrw [kerase_cons_ne, kerase_cons_ne, kerase_comm] <;> [apply Perm.swap; exact h; exact h.symm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a\nb' : \u03b2 a'\ns : AList \u03b2\nh : a \u2260 a'\n\u22a2 { fst := a', snd := b' } :: kerase a' ({ fst := a, snd := b } :: kerase a s.entries) ~\n { fst := a, snd := b } :: kerase a ({ fst := a', snd := b' } :: kerase a' s.entries)\n[PROOFSTEP]\nrw [kerase_cons_ne, kerase_cons_ne, kerase_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a\nb' : \u03b2 a'\ns : AList \u03b2\nh : a \u2260 a'\n\u22a2 { fst := a', snd := b' } :: { fst := a, snd := b } :: kerase a (kerase a' s.entries) ~\n { fst := a, snd := b } :: { fst := a', snd := b' } :: kerase a (kerase a' s.entries)\n[PROOFSTEP]\napply Perm.swap\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a\nb' : \u03b2 a'\ns : AList \u03b2\nh : a \u2260 a'\n\u22a2 a \u2260 { fst := a', snd := b' }.fst\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a\nb' : \u03b2 a'\ns : AList \u03b2\nh : a \u2260 a'\n\u22a2 a' \u2260 { fst := a, snd := b }.fst\n[PROOFSTEP]\nexact h.symm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb b' : \u03b2 a\n\u22a2 (insert a b (singleton a b')).entries = (singleton a b).entries\n[PROOFSTEP]\nsimp only [AList.insert_entries, List.kerase_cons_eq, and_self_iff, AList.singleton_entries, heq_iff_eq,\n eq_self_iff_true]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nc : Sigma \u03b2\nl : List (Sigma \u03b2)\nh : NodupKeys (c :: l)\n\u22a2 { entries := c :: l, nodupKeys := h } = insert c.fst c.snd { entries := l, nodupKeys := (_ : NodupKeys l) }\n[PROOFSTEP]\nsimpa [insert] using (kerase_of_not_mem_keys <| not_mem_keys_of_nodupKeys_cons h).symm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nC : AList \u03b2 \u2192 Sort u_1\nH0 : C \u2205\nIH : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 (l : AList \u03b2) \u2192 \u00aca \u2208 l \u2192 C l \u2192 C (insert a b l)\nc : Sigma \u03b2\nl : List (Sigma \u03b2)\nh : NodupKeys (c :: l)\n\u22a2 C { entries := c :: l, nodupKeys := h }\n[PROOFSTEP]\nrw [mk_cons_eq_insert]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nC : AList \u03b2 \u2192 Sort u_1\nH0 : C \u2205\nIH : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 (l : AList \u03b2) \u2192 \u00aca \u2208 l \u2192 C l \u2192 C (insert a b l)\nc : Sigma \u03b2\nl : List (Sigma \u03b2)\nh : NodupKeys (c :: l)\n\u22a2 C (insert c.fst c.snd { entries := l, nodupKeys := (_ : NodupKeys l) })\n[PROOFSTEP]\nrefine' IH _ _ _ _ (insertRec H0 IH _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nC : AList \u03b2 \u2192 Sort u_1\nH0 : C \u2205\nIH : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 (l : AList \u03b2) \u2192 \u00aca \u2208 l \u2192 C l \u2192 C (insert a b l)\nc : Sigma \u03b2\nl : List (Sigma \u03b2)\nh : NodupKeys (c :: l)\n\u22a2 \u00acc.fst \u2208 { entries := l, nodupKeys := (_ : NodupKeys l) }\n[PROOFSTEP]\nexact not_mem_keys_of_nodupKeys_cons h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nl : AList \u03b2\n\u22a2 True\n[PROOFSTEP]\ninduction l using AList.insertRec\n[GOAL]\ncase H0\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\n\u22a2 True\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase IH\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na\u271d\u00b2 : \u03b1\nb\u271d : \u03b2 a\u271d\u00b2\nl\u271d : AList \u03b2\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 l\u271d\na\u271d : True\n\u22a2 True\n[PROOFSTEP]\ntrivial\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nC : AList \u03b2 \u2192 Sort u_1\nH0 : C \u2205\nIH : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 (l : AList \u03b2) \u2192 \u00aca \u2208 l \u2192 C l \u2192 C (insert a b l)\n\u22a2 insertRec H0 IH \u2205 = H0\n[PROOFSTEP]\nchange @insertRec \u03b1 \u03b2 _ C H0 IH \u27e8[], _\u27e9 = H0\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nC : AList \u03b2 \u2192 Sort u_1\nH0 : C \u2205\nIH : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 (l : AList \u03b2) \u2192 \u00aca \u2208 l \u2192 C l \u2192 C (insert a b l)\n\u22a2 insertRec H0 IH { entries := [], nodupKeys := (_ : NodupKeys []) } = H0\n[PROOFSTEP]\nrw [insertRec]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nC : AList \u03b2 \u2192 Sort u_1\nH0 : C \u2205\nIH : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 (l : AList \u03b2) \u2192 \u00aca \u2208 l \u2192 C l \u2192 C (insert a b l)\nc : Sigma \u03b2\nl : AList \u03b2\nh : \u00acc.fst \u2208 l\n\u22a2 insertRec H0 IH (insert c.fst c.snd l) = IH c.fst c.snd l h (insertRec H0 IH l)\n[PROOFSTEP]\ncases' l with l hl\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nC : AList \u03b2 \u2192 Sort u_1\nH0 : C \u2205\nIH : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 (l : AList \u03b2) \u2192 \u00aca \u2208 l \u2192 C l \u2192 C (insert a b l)\nc : Sigma \u03b2\nl : List (Sigma \u03b2)\nhl : NodupKeys l\nh : \u00acc.fst \u2208 { entries := l, nodupKeys := hl }\n\u22a2 insertRec H0 IH (insert c.fst c.snd { entries := l, nodupKeys := hl }) =\n IH c.fst c.snd { entries := l, nodupKeys := hl } h (insertRec H0 IH { entries := l, nodupKeys := hl })\n[PROOFSTEP]\nsuffices\n HEq (@insertRec \u03b1 \u03b2 _ C H0 IH \u27e8c :: l, nodupKeys_cons.2 \u27e8h, hl\u27e9\u27e9)\n (IH c.1 c.2 \u27e8l, hl\u27e9 h (@insertRec \u03b1 \u03b2 _ C H0 IH \u27e8l, hl\u27e9))\n by\n cases c\n apply eq_of_heq\n convert this <;> rw [insert_of_neg h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nC : AList \u03b2 \u2192 Sort u_1\nH0 : C \u2205\nIH : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 (l : AList \u03b2) \u2192 \u00aca \u2208 l \u2192 C l \u2192 C (insert a b l)\nc : Sigma \u03b2\nl : List (Sigma \u03b2)\nhl : NodupKeys l\nh : \u00acc.fst \u2208 { entries := l, nodupKeys := hl }\nthis :\n HEq (insertRec H0 IH { entries := c :: l, nodupKeys := (_ : NodupKeys (c :: l)) })\n (IH c.fst c.snd { entries := l, nodupKeys := hl } h (insertRec H0 IH { entries := l, nodupKeys := hl }))\n\u22a2 insertRec H0 IH (insert c.fst c.snd { entries := l, nodupKeys := hl }) =\n IH c.fst c.snd { entries := l, nodupKeys := hl } h (insertRec H0 IH { entries := l, nodupKeys := hl })\n[PROOFSTEP]\ncases c\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nC : AList \u03b2 \u2192 Sort u_1\nH0 : C \u2205\nIH : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 (l : AList \u03b2) \u2192 \u00aca \u2208 l \u2192 C l \u2192 C (insert a b l)\nl : List (Sigma \u03b2)\nhl : NodupKeys l\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\nh : \u00ac{ fst := fst\u271d, snd := snd\u271d }.fst \u2208 { entries := l, nodupKeys := hl }\nthis :\n HEq\n (insertRec H0 IH\n { entries := { fst := fst\u271d, snd := snd\u271d } :: l,\n nodupKeys := (_ : NodupKeys ({ fst := fst\u271d, snd := snd\u271d } :: l)) })\n (IH { fst := fst\u271d, snd := snd\u271d }.fst { fst := fst\u271d, snd := snd\u271d }.snd { entries := l, nodupKeys := hl } h\n (insertRec H0 IH { entries := l, nodupKeys := hl }))\n\u22a2 insertRec H0 IH\n (insert { fst := fst\u271d, snd := snd\u271d }.fst { fst := fst\u271d, snd := snd\u271d }.snd { entries := l, nodupKeys := hl }) =\n IH { fst := fst\u271d, snd := snd\u271d }.fst { fst := fst\u271d, snd := snd\u271d }.snd { entries := l, nodupKeys := hl } h\n (insertRec H0 IH { entries := l, nodupKeys := hl })\n[PROOFSTEP]\napply eq_of_heq\n[GOAL]\ncase mk.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nC : AList \u03b2 \u2192 Sort u_1\nH0 : C \u2205\nIH : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 (l : AList \u03b2) \u2192 \u00aca \u2208 l \u2192 C l \u2192 C (insert a b l)\nl : List (Sigma \u03b2)\nhl : NodupKeys l\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\nh : \u00ac{ fst := fst\u271d, snd := snd\u271d }.fst \u2208 { entries := l, nodupKeys := hl }\nthis :\n HEq\n (insertRec H0 IH\n { entries := { fst := fst\u271d, snd := snd\u271d } :: l,\n nodupKeys := (_ : NodupKeys ({ fst := fst\u271d, snd := snd\u271d } :: l)) })\n (IH { fst := fst\u271d, snd := snd\u271d }.fst { fst := fst\u271d, snd := snd\u271d }.snd { entries := l, nodupKeys := hl } h\n (insertRec H0 IH { entries := l, nodupKeys := hl }))\n\u22a2 HEq\n (insertRec H0 IH\n (insert { fst := fst\u271d, snd := snd\u271d }.fst { fst := fst\u271d, snd := snd\u271d }.snd { entries := l, nodupKeys := hl }))\n (IH { fst := fst\u271d, snd := snd\u271d }.fst { fst := fst\u271d, snd := snd\u271d }.snd { entries := l, nodupKeys := hl } h\n (insertRec H0 IH { entries := l, nodupKeys := hl }))\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_1.h.e'_1.h.e'_3.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nC : AList \u03b2 \u2192 Sort u_1\nH0 : C \u2205\nIH : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 (l : AList \u03b2) \u2192 \u00aca \u2208 l \u2192 C l \u2192 C (insert a b l)\nl : List (Sigma \u03b2)\nhl : NodupKeys l\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\nh : \u00ac{ fst := fst\u271d, snd := snd\u271d }.fst \u2208 { entries := l, nodupKeys := hl }\nthis :\n HEq\n (insertRec H0 IH\n { entries := { fst := fst\u271d, snd := snd\u271d } :: l,\n nodupKeys := (_ : NodupKeys ({ fst := fst\u271d, snd := snd\u271d } :: l)) })\n (IH { fst := fst\u271d, snd := snd\u271d }.fst { fst := fst\u271d, snd := snd\u271d }.snd { entries := l, nodupKeys := hl } h\n (insertRec H0 IH { entries := l, nodupKeys := hl }))\n\u22a2 (insert { fst := fst\u271d, snd := snd\u271d }.fst { fst := fst\u271d, snd := snd\u271d }.snd { entries := l, nodupKeys := hl }).entries =\n { fst := fst\u271d, snd := snd\u271d } :: l\n[PROOFSTEP]\nrw [insert_of_neg h]\n[GOAL]\ncase h.e'_2.e'_7.h.e'_3.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nC : AList \u03b2 \u2192 Sort u_1\nH0 : C \u2205\nIH : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 (l : AList \u03b2) \u2192 \u00aca \u2208 l \u2192 C l \u2192 C (insert a b l)\nl : List (Sigma \u03b2)\nhl : NodupKeys l\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\nh : \u00ac{ fst := fst\u271d, snd := snd\u271d }.fst \u2208 { entries := l, nodupKeys := hl }\nthis :\n HEq\n (insertRec H0 IH\n { entries := { fst := fst\u271d, snd := snd\u271d } :: l,\n nodupKeys := (_ : NodupKeys ({ fst := fst\u271d, snd := snd\u271d } :: l)) })\n (IH { fst := fst\u271d, snd := snd\u271d }.fst { fst := fst\u271d, snd := snd\u271d }.snd { entries := l, nodupKeys := hl } h\n (insertRec H0 IH { entries := l, nodupKeys := hl }))\ne_1\u271d :\n C (insert { fst := fst\u271d, snd := snd\u271d }.fst { fst := fst\u271d, snd := snd\u271d }.snd { entries := l, nodupKeys := hl }) =\n C { entries := { fst := fst\u271d, snd := snd\u271d } :: l, nodupKeys := (_ : NodupKeys ({ fst := fst\u271d, snd := snd\u271d } :: l)) }\n\u22a2 (insert { fst := fst\u271d, snd := snd\u271d }.fst { fst := fst\u271d, snd := snd\u271d }.snd { entries := l, nodupKeys := hl }).entries =\n { fst := fst\u271d, snd := snd\u271d } :: l\n[PROOFSTEP]\nrw [insert_of_neg h]\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nC : AList \u03b2 \u2192 Sort u_1\nH0 : C \u2205\nIH : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 (l : AList \u03b2) \u2192 \u00aca \u2208 l \u2192 C l \u2192 C (insert a b l)\nc : Sigma \u03b2\nl : List (Sigma \u03b2)\nhl : NodupKeys l\nh : \u00acc.fst \u2208 { entries := l, nodupKeys := hl }\n\u22a2 HEq (insertRec H0 IH { entries := c :: l, nodupKeys := (_ : NodupKeys (c :: l)) })\n (IH c.fst c.snd { entries := l, nodupKeys := hl } h (insertRec H0 IH { entries := l, nodupKeys := hl }))\n[PROOFSTEP]\nrw [insertRec]\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\nC : AList \u03b2 \u2192 Sort u_1\nH0 : C \u2205\nIH : (a : \u03b1) \u2192 (b : \u03b2 a) \u2192 (l : AList \u03b2) \u2192 \u00aca \u2208 l \u2192 C l \u2192 C (insert a b l)\nc : Sigma \u03b2\nl : List (Sigma \u03b2)\nhl : NodupKeys l\nh : \u00acc.fst \u2208 { entries := l, nodupKeys := hl }\n\u22a2 HEq\n (Eq.mpr\n (_ :\n C { entries := c :: l, nodupKeys := (_ : NodupKeys (c :: l)) } =\n C (insert c.fst c.snd { entries := l, nodupKeys := (_ : NodupKeys l) }))\n (IH c.fst c.snd { entries := l, nodupKeys := (_ : NodupKeys l) }\n (_ : \u00acc.fst \u2208 List.keys { entries := l, nodupKeys := (_ : NodupKeys l) }.entries)\n (insertRec H0 IH { entries := l, nodupKeys := (_ : NodupKeys l) })))\n (IH c.fst c.snd { entries := l, nodupKeys := hl } h (insertRec H0 IH { entries := l, nodupKeys := hl }))\n[PROOFSTEP]\napply cast_heq\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : AList \u03b2\n\u22a2 NodupKeys (kextract a s.entries).snd\n[PROOFSTEP]\nrw [kextract_eq_dlookup_kerase]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : AList \u03b2\n\u22a2 NodupKeys (dlookup a s.entries, kerase a s.entries).snd\n[PROOFSTEP]\nexact s.nodupKeys.kerase _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : AList \u03b2\n\u22a2 extract a s = (lookup a s, erase a s)\n[PROOFSTEP]\nsimp [extract]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : AList \u03b2\n\u22a2 dlookup a s.entries = lookup a s \u2227\n { entries := kerase a s.entries, nodupKeys := (_ : NodupKeys (kerase a s.entries)) } = erase a s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : AList \u03b2\n\u22a2 dlookup a s.entries = lookup a s\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : AList \u03b2\n\u22a2 { entries := kerase a s.entries, nodupKeys := (_ : NodupKeys (kerase a s.entries)) } = erase a s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns : AList \u03b2\n\u22a2 (s \u222a \u2205).entries = s.entries\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 s\u2083 s\u2084 : AList \u03b2\np\u2081\u2082 : s\u2081.entries ~ s\u2082.entries\np\u2083\u2084 : s\u2083.entries ~ s\u2084.entries\n\u22a2 (s\u2081 \u222a s\u2083).entries ~ (s\u2082 \u222a s\u2084).entries\n[PROOFSTEP]\nsimp [p\u2081\u2082.kunion s\u2083.nodupKeys p\u2083\u2084]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns\u2081 s\u2082 : AList \u03b2\n\u22a2 insert a b (s\u2081 \u222a s\u2082) = insert a b s\u2081 \u222a s\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase a.a.a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns\u2081 s\u2082 : AList \u03b2\nn\u271d : \u2115\na\u271d : Sigma \u03b2\n\u22a2 a\u271d \u2208 get? (insert a b (s\u2081 \u222a s\u2082)).entries n\u271d \u2194 a\u271d \u2208 get? (insert a b s\u2081 \u222a s\u2082).entries n\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 s\u2083 : AList \u03b2\n\u22a2 \u2200 (x : \u03b1) (y : \u03b2 x), y \u2208 dlookup x (s\u2081 \u222a s\u2082 \u222a s\u2083).entries \u2194 y \u2208 dlookup x (s\u2081 \u222a (s\u2082 \u222a s\u2083)).entries\n[PROOFSTEP]\nsimp [not_or, or_assoc, and_or_left, and_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\n\u22a2 \u2200 (x : \u03b1) (y : \u03b2 x), y \u2208 dlookup x (s\u2081 \u222a s\u2082).entries \u2194 y \u2208 dlookup x (s\u2082 \u222a s\u2081).entries\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\n\u22a2 y\u271d \u2208 dlookup x\u271d (s\u2081 \u222a s\u2082).entries \u2194 y\u271d \u2208 dlookup x\u271d (s\u2082 \u222a s\u2081).entries\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d \u2228 \u00acx\u271d \u2208 List.keys s\u2081.entries \u2227 dlookup x\u271d s\u2082.entries = some y\u271d \u2194\n dlookup x\u271d s\u2082.entries = some y\u271d \u2228 \u00acx\u271d \u2208 List.keys s\u2082.entries \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d \u2228 \u00acx\u271d \u2208 List.keys s\u2081.entries \u2227 dlookup x\u271d s\u2082.entries = some y\u271d \u2192\n dlookup x\u271d s\u2082.entries = some y\u271d \u2228 \u00acx\u271d \u2208 List.keys s\u2082.entries \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\n\u22a2 dlookup x\u271d s\u2082.entries = some y\u271d \u2228 \u00acx\u271d \u2208 List.keys s\u2082.entries \u2227 dlookup x\u271d s\u2081.entries = some y\u271d \u2192\n dlookup x\u271d s\u2081.entries = some y\u271d \u2228 \u00acx\u271d \u2208 List.keys s\u2081.entries \u2227 dlookup x\u271d s\u2082.entries = some y\u271d\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d \u2228 \u00acx\u271d \u2208 List.keys s\u2081.entries \u2227 dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2082.entries = some y\u271d \u2228 \u00acx\u271d \u2208 List.keys s\u2082.entries \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n[PROOFSTEP]\ncases' h' with h' h'\n[GOAL]\ncase mp.inl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2082.entries = some y\u271d \u2228 \u00acx\u271d \u2208 List.keys s\u2082.entries \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n[PROOFSTEP]\nright\n[GOAL]\ncase mp.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 \u00acx\u271d \u2208 List.keys s\u2082.entries \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n[PROOFSTEP]\nrefine' \u27e8_, h'\u27e9\n[GOAL]\ncase mp.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 \u00acx\u271d \u2208 List.keys s\u2082.entries\n[PROOFSTEP]\napply h\n[GOAL]\ncase mp.inl.h.a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 x\u271d \u2208 keys s\u2081\n[PROOFSTEP]\nrw [keys, \u2190 List.dlookup_isSome, h']\n[GOAL]\ncase mp.inl.h.a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 Option.isSome (some y\u271d) = true\n[PROOFSTEP]\nexact rfl\n[GOAL]\ncase mp.inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : \u00acx\u271d \u2208 List.keys s\u2081.entries \u2227 dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2082.entries = some y\u271d \u2228 \u00acx\u271d \u2208 List.keys s\u2082.entries \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n[PROOFSTEP]\nleft\n[GOAL]\ncase mp.inr.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : \u00acx\u271d \u2208 List.keys s\u2081.entries \u2227 dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2082.entries = some y\u271d\n[PROOFSTEP]\nrw [h'.2]\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d \u2228 \u00acx\u271d \u2208 List.keys s\u2082.entries \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d \u2228 \u00acx\u271d \u2208 List.keys s\u2081.entries \u2227 dlookup x\u271d s\u2082.entries = some y\u271d\n[PROOFSTEP]\ncases' h' with h' h'\n[GOAL]\ncase mpr.inl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d \u2228 \u00acx\u271d \u2208 List.keys s\u2081.entries \u2227 dlookup x\u271d s\u2082.entries = some y\u271d\n[PROOFSTEP]\nright\n[GOAL]\ncase mpr.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 \u00acx\u271d \u2208 List.keys s\u2081.entries \u2227 dlookup x\u271d s\u2082.entries = some y\u271d\n[PROOFSTEP]\nrefine' \u27e8_, h'\u27e9\n[GOAL]\ncase mpr.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 \u00acx\u271d \u2208 List.keys s\u2081.entries\n[PROOFSTEP]\nintro h''\n[GOAL]\ncase mpr.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\nh'' : x\u271d \u2208 List.keys s\u2081.entries\n\u22a2 False\n[PROOFSTEP]\napply h _ h''\n[GOAL]\ncase mpr.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\nh'' : x\u271d \u2208 List.keys s\u2081.entries\n\u22a2 x\u271d \u2208 keys s\u2082\n[PROOFSTEP]\nrw [keys, \u2190 List.dlookup_isSome, h']\n[GOAL]\ncase mpr.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\nh'' : x\u271d \u2208 List.keys s\u2081.entries\n\u22a2 Option.isSome (some y\u271d) = true\n[PROOFSTEP]\nexact rfl\n[GOAL]\ncase mpr.inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : \u00acx\u271d \u2208 List.keys s\u2082.entries \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d \u2228 \u00acx\u271d \u2208 List.keys s\u2081.entries \u2227 dlookup x\u271d s\u2082.entries = some y\u271d\n[PROOFSTEP]\nleft\n[GOAL]\ncase mpr.inr.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : Disjoint s\u2081 s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : \u00acx\u271d \u2208 List.keys s\u2082.entries \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d\n[PROOFSTEP]\nrw [h'.2]\n", "meta": {"mathlib_filename": "Mathlib.Data.List.AList", "llama_tokens": 11930, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850933, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.4361090585197321}} {"text": "[GOAL]\nX Y : MonCat\n\u22a2 FunLike (\u2191X \u2192* \u2191Y) \u2191X fun x => \u2191Y\n[PROOFSTEP]\ninfer_instance\n -- porting note: added\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\n\u22a2 Group \u2191(of G)\n[PROOFSTEP]\nassumption\n[GOAL]\n\u22a2 ConcreteCategory CommMonCat\n[PROOFSTEP]\ndsimp only [CommMonCat]\n[GOAL]\n\u22a2 ConcreteCategory (Bundled CommMonoid)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX Y : CommMonCat\n\u22a2 FunLike (\u2191X \u2192* \u2191Y) \u2191X fun x => \u2191Y\n[PROOFSTEP]\ninfer_instance\n -- porting note: added\n[GOAL]\nR S : MonCat\ni : R \u27f6 S\nr : \u2191R\nh : r = 1\n\u22a2 \u2191i r = 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR S : CommMonCat\ni : R \u27f6 S\nr : \u2191R\nh : r = 1\n\u22a2 \u2191i r = 1\n[PROOFSTEP]\nsimp [h]\n -- We verify that when constructing a morphism in `CommMonCat`,\n -- when we construct the `toFun` field, the types are presented as `\u2191R`.\n[GOAL]\nR : CommMonCat\nx : \u2191R\n\u22a2 \u2191R\n[PROOFSTEP]\nmatch_target (R : Type u)\n[GOAL]\nR : CommMonCat\nx : \u2191R\n\u22a2 \u2191R\n[PROOFSTEP]\nguard_hyp x : (R : Type u)\n[GOAL]\nR : CommMonCat\nx : \u2191R\n\u22a2 \u2191R\n[PROOFSTEP]\nexact x * x\n[GOAL]\nR : CommMonCat\n\u22a2 (fun x => x * x) 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : CommMonCat\nx y : \u2191R\n\u22a2 OneHom.toFun { toFun := fun x => x * x, map_one' := (_ : 1 * 1 = 1) } (x * y) =\n OneHom.toFun { toFun := fun x => x * x, map_one' := (_ : 1 * 1 = 1) } x *\n OneHom.toFun { toFun := fun x => x * x, map_one' := (_ : 1 * 1 = 1) } y\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : CommMonCat\nx y : \u2191R\n\u22a2 x * y * (x * y) = x * x * (y * y)\n[PROOFSTEP]\nrw [mul_assoc x y (x * y), \u2190 mul_assoc y x y, mul_comm y x, mul_assoc, mul_assoc]\n[GOAL]\nX\u271d Y\u271d : Type u\nX Y : MonCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget MonCat).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nlet i := asIso ((forget MonCat).map f)\n[GOAL]\nX\u271d Y\u271d : Type u\nX Y : MonCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget MonCat).map f)\ni : (forget MonCat).obj X \u2245 (forget MonCat).obj Y := asIso ((forget MonCat).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nlet e : X \u2243* Y := MulEquiv.mk i.toEquiv (by aesop)\n[GOAL]\nX\u271d Y\u271d : Type u\nX Y : MonCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget MonCat).map f)\ni : (forget MonCat).obj X \u2245 (forget MonCat).obj Y := asIso ((forget MonCat).map f)\n\u22a2 \u2200 (x y : \u2191X), Equiv.toFun i.toEquiv (x * y) = Equiv.toFun i.toEquiv x * Equiv.toFun i.toEquiv y\n[PROOFSTEP]\naesop\n[GOAL]\nX\u271d Y\u271d : Type u\nX Y : MonCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget MonCat).map f)\ni : (forget MonCat).obj X \u2245 (forget MonCat).obj Y := asIso ((forget MonCat).map f)\ne : \u2191X \u2243* \u2191Y := { toEquiv := i.toEquiv, map_mul' := (_ : \u2200 (x_1 y : \u2191X), \u2191f (x_1 * y) = \u2191f x_1 * \u2191f y) }\n\u22a2 IsIso f\n[PROOFSTEP]\nexact IsIso.of_iso e.toMonCatIso\n[GOAL]\nX\u271d Y\u271d : Type u\nX Y : CommMonCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget CommMonCat).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nlet i := asIso ((forget CommMonCat).map f)\n[GOAL]\nX\u271d Y\u271d : Type u\nX Y : CommMonCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget CommMonCat).map f)\ni : (forget CommMonCat).obj X \u2245 (forget CommMonCat).obj Y := asIso ((forget CommMonCat).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nlet e : X \u2243* Y := MulEquiv.mk i.toEquiv (MonoidHom.map_mul (show MonoidHom X Y from f))\n[GOAL]\nX\u271d Y\u271d : Type u\nX Y : CommMonCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget CommMonCat).map f)\ni : (forget CommMonCat).obj X \u2245 (forget CommMonCat).obj Y := asIso ((forget CommMonCat).map f)\ne : \u2191X \u2243* \u2191Y :=\n { toEquiv := i.toEquiv,\n map_mul' :=\n (_ :\n \u2200 (a b : \u2191X),\n \u2191(let_fun this := f;\n this)\n (a * b) =\n \u2191(let_fun this := f;\n this)\n a *\n \u2191(let_fun this := f;\n this)\n b) }\n\u22a2 IsIso f\n[PROOFSTEP]\nexact IsIso.of_iso e.toCommMonCatIso\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.MonCat.Basic", "llama_tokens": 1705, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149868676283, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.4359599060253632}} {"text": "[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nsrc\u271d : Group { x // x \u2208 IsUnit.submonoid M } := inferInstanceAs (Group { x // x \u2208 IsUnit.submonoid M })\na b : { x // x \u2208 IsUnit.submonoid M }\n\u22a2 a * b = b * a\n[PROOFSTEP]\nconvert mul_comm a b\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\nS : Submonoid M\na _b : M\nx\u271d\u00b9 : a \u2208 {x | \u2203 y, x * \u2191y = 1}\nx\u271d : _b \u2208 {x | \u2203 y, x * \u2191y = 1}\na' : { x // x \u2208 S }\nha : a * \u2191a' = 1\nb' : { x // x \u2208 S }\nhb : _b * \u2191b' = 1\n\u22a2 a * _b * \u2191(b' * a') = 1\n[PROOFSTEP]\nsimp only [coe_mul, \u2190 mul_assoc, mul_assoc a, hb, mul_one, ha]\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\nS : Submonoid M\n\u22a2 leftInv (leftInv S) \u2264 S\n[PROOFSTEP]\nrintro x \u27e8\u27e8y, z, h\u2081\u27e9, h\u2082 : x * y = 1\u27e9\n[GOAL]\ncase intro.mk.intro\nM : Type u_1\ninst\u271d : Monoid M\nS : Submonoid M\nx y : M\nz : { x // x \u2208 S }\nh\u2081 : y * \u2191z = 1\nh\u2082 : x * y = 1\n\u22a2 x \u2208 S\n[PROOFSTEP]\nconvert z.prop\n[GOAL]\ncase h.e'_4\nM : Type u_1\ninst\u271d : Monoid M\nS : Submonoid M\nx y : M\nz : { x // x \u2208 S }\nh\u2081 : y * \u2191z = 1\nh\u2082 : x * y = 1\n\u22a2 x = \u2191z\n[PROOFSTEP]\nrw [\u2190 mul_one x, \u2190 h\u2081, \u2190 mul_assoc, h\u2082, one_mul]\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\n\u22a2 leftInv (leftInv S) = S\n[PROOFSTEP]\nrefine' le_antisymm S.leftInv_leftInv_le _\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\n\u22a2 S \u2264 leftInv (leftInv S)\n[PROOFSTEP]\nintro x hx\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nx : M\nhx : x \u2208 S\n\u22a2 x \u2208 leftInv (leftInv S)\n[PROOFSTEP]\nhave : x = ((hS hx).unit\u207b\u00b9\u207b\u00b9 : M\u02e3) := by\n rw [inv_inv (hS hx).unit]\n rfl\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nx : M\nhx : x \u2208 S\n\u22a2 x = \u2191(IsUnit.unit (_ : x \u2208 IsUnit.submonoid M))\u207b\u00b9\u207b\u00b9\n[PROOFSTEP]\nrw [inv_inv (hS hx).unit]\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nx : M\nhx : x \u2208 S\n\u22a2 x = \u2191(IsUnit.unit (_ : x \u2208 IsUnit.submonoid M))\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nx : M\nhx : x \u2208 S\nthis : x = \u2191(IsUnit.unit (_ : x \u2208 IsUnit.submonoid M))\u207b\u00b9\u207b\u00b9\n\u22a2 x \u2208 leftInv (leftInv S)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nx : M\nhx : x \u2208 S\nthis : x = \u2191(IsUnit.unit (_ : x \u2208 IsUnit.submonoid M))\u207b\u00b9\u207b\u00b9\n\u22a2 \u2191(IsUnit.unit (_ : x \u2208 IsUnit.submonoid M))\u207b\u00b9\u207b\u00b9 \u2208 leftInv (leftInv S)\n[PROOFSTEP]\nexact S.leftInv.unit_mem_leftInv _ (S.unit_mem_leftInv _ hx)\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nx : { x // x \u2208 leftInv S }\n\u22a2 \u2191(fromLeftInv S x) * \u2191x = 1\n[PROOFSTEP]\nrw [mul_comm, mul_fromLeftInv]\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\na : { x // x \u2208 leftInv S }\nb : M\n\u22a2 \u2191(fromLeftInv S a) = b \u2194 \u2191a * b = 1\n[PROOFSTEP]\nrw [\u2190 IsUnit.mul_right_inj (leftInv_le_isUnit _ a.prop), S.mul_fromLeftInv, eq_comm]\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nx y : { x // x \u2208 leftInv S }\n\u22a2 \u2191(OneHom.toFun { toFun := fromLeftInv S, map_one' := (_ : fromLeftInv S 1 = 1) } (x * y)) =\n \u2191(OneHom.toFun { toFun := fromLeftInv S, map_one' := (_ : fromLeftInv S 1 = 1) } x *\n OneHom.toFun { toFun := fromLeftInv S, map_one' := (_ : fromLeftInv S 1 = 1) } y)\n[PROOFSTEP]\nrw [fromLeftInv_eq_iff, mul_comm x, Submonoid.coe_mul, Submonoid.coe_mul, mul_assoc, \u2190 mul_assoc (x : M),\n mul_fromLeftInv, one_mul, mul_fromLeftInv]\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 S }\n\u22a2 { x // x \u2208 leftInv S }\n[PROOFSTEP]\nchoose x' hx using hS x.prop\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 S }\nx' : M\u02e3\nhx : \u2191x' = \u2191x\n\u22a2 { x // x \u2208 leftInv S }\n[PROOFSTEP]\nexact \u27e8x'.inv, x, hx \u25b8 x'.inv_val\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 leftInv S }\n\u22a2 \u2191((fun x =>\n (fun x' hx => { val := x'.inv, property := (_ : \u2203 y, x'.inv * \u2191y = 1) })\n (Classical.choose (_ : \u2191x \u2208 IsUnit.submonoid M))\n (_ : \u2191(Classical.choose (_ : \u2191x \u2208 IsUnit.submonoid M)) = \u2191x))\n (OneHom.toFun (\u2191src\u271d) x)) =\n \u2191x\n[PROOFSTEP]\ndsimp only\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 leftInv S }\n\u22a2 (Classical.choose (_ : \u2191(OneHom.toFun (\u2191(fromCommLeftInv S)) x) \u2208 IsUnit.submonoid M)).inv = \u2191x\n[PROOFSTEP]\ngeneralize_proofs h\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 leftInv S }\nh : \u2191(OneHom.toFun (\u2191(fromCommLeftInv S)) x) \u2208 IsUnit.submonoid M\n\u22a2 (Classical.choose h).inv = \u2191x\n[PROOFSTEP]\nrw [\u2190 h.choose.mul_left_inj]\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 leftInv S }\nh : \u2191(OneHom.toFun (\u2191(fromCommLeftInv S)) x) \u2208 IsUnit.submonoid M\n\u22a2 (Classical.choose h).inv * \u2191(Exists.choose h) = \u2191x * \u2191(Exists.choose h)\n[PROOFSTEP]\nconv => rhs; rw [h.choose_spec]\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 leftInv S }\nh : \u2191(OneHom.toFun (\u2191(fromCommLeftInv S)) x) \u2208 IsUnit.submonoid M\n| (Classical.choose h).inv * \u2191(Exists.choose h) = \u2191x * \u2191(Exists.choose h)\n[PROOFSTEP]\nrhs; rw [h.choose_spec]\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 leftInv S }\nh : \u2191(OneHom.toFun (\u2191(fromCommLeftInv S)) x) \u2208 IsUnit.submonoid M\n| (Classical.choose h).inv * \u2191(Exists.choose h) = \u2191x * \u2191(Exists.choose h)\n[PROOFSTEP]\nrhs; rw [h.choose_spec]\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 leftInv S }\nh : \u2191(OneHom.toFun (\u2191(fromCommLeftInv S)) x) \u2208 IsUnit.submonoid M\n| (Classical.choose h).inv * \u2191(Exists.choose h) = \u2191x * \u2191(Exists.choose h)\n[PROOFSTEP]\nrhs\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 leftInv S }\nh : \u2191(OneHom.toFun (\u2191(fromCommLeftInv S)) x) \u2208 IsUnit.submonoid M\n| \u2191x * \u2191(Exists.choose h)\n[PROOFSTEP]\nrw [h.choose_spec]\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 leftInv S }\nh : \u2191(OneHom.toFun (\u2191(fromCommLeftInv S)) x) \u2208 IsUnit.submonoid M\n\u22a2 (Classical.choose h).inv * \u2191(Exists.choose h) = \u2191x * \u2191(OneHom.toFun (\u2191(fromCommLeftInv S)) x)\n[PROOFSTEP]\nexact h.choose.inv_val.trans (S.mul_fromLeftInv x).symm\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 S }\n\u22a2 OneHom.toFun (\u2191src\u271d)\n ((fun x =>\n (fun x' hx => { val := x'.inv, property := (_ : \u2203 y, x'.inv * \u2191y = 1) })\n (Classical.choose (_ : \u2191x \u2208 IsUnit.submonoid M))\n (_ : \u2191(Classical.choose (_ : \u2191x \u2208 IsUnit.submonoid M)) = \u2191x))\n x) =\n x\n[PROOFSTEP]\ndsimp only [fromCommLeftInv]\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 S }\n\u22a2 fromLeftInv S\n { val := (Classical.choose (_ : \u2191x \u2208 IsUnit.submonoid M)).inv,\n property := (_ : \u2203 y, (Classical.choose (_ : \u2191x \u2208 IsUnit.submonoid M)).inv * \u2191y = 1) } =\n x\n[PROOFSTEP]\next\n[GOAL]\ncase a\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 S }\n\u22a2 \u2191(fromLeftInv S\n { val := (Classical.choose (_ : \u2191x \u2208 IsUnit.submonoid M)).inv,\n property := (_ : \u2203 y, (Classical.choose (_ : \u2191x \u2208 IsUnit.submonoid M)).inv * \u2191y = 1) }) =\n \u2191x\n[PROOFSTEP]\nrw [fromLeftInv_eq_iff]\n[GOAL]\ncase a\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 S }\n\u22a2 \u2191{ val := (Classical.choose (_ : \u2191x \u2208 IsUnit.submonoid M)).inv,\n property := (_ : \u2203 y, (Classical.choose (_ : \u2191x \u2208 IsUnit.submonoid M)).inv * \u2191y = 1) } *\n \u2191x =\n 1\n[PROOFSTEP]\nconvert (hS x.prop).choose.inv_val\n[GOAL]\ncase h.e'_2.h.e'_6\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nsrc\u271d : { x // x \u2208 leftInv S } \u2192* { x // x \u2208 S } := fromCommLeftInv S\nx : { x // x \u2208 S }\n\u22a2 \u2191x = \u2191(Exists.choose (_ : \u2191x \u2208 IsUnit.submonoid M))\n[PROOFSTEP]\nexact (hS x.prop).choose_spec.symm\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nx : { x // x \u2208 leftInv S }\n\u22a2 \u2191(\u2191(leftInvEquiv S hS) x) * \u2191x = 1\n[PROOFSTEP]\nsimpa only [leftInvEquiv_apply, fromCommLeftInv] using fromLeftInv_mul S x\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nx : { x // x \u2208 leftInv S }\n\u22a2 \u2191x * \u2191(\u2191(leftInvEquiv S hS) x) = 1\n[PROOFSTEP]\nsimp only [leftInvEquiv_apply, fromCommLeftInv, mul_fromLeftInv]\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nx : { x // x \u2208 S }\n\u22a2 \u2191(\u2191(MulEquiv.symm (leftInvEquiv S hS)) x) * \u2191x = 1\n[PROOFSTEP]\nconvert S.mul_leftInvEquiv hS ((S.leftInvEquiv hS).symm x)\n[GOAL]\ncase h.e'_2.h.e'_6.h.e'_3\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nx : { x // x \u2208 S }\n\u22a2 x = \u2191(leftInvEquiv S hS) (\u2191(MulEquiv.symm (leftInvEquiv S hS)) x)\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nx : { x // x \u2208 S }\n\u22a2 \u2191x * \u2191(\u2191(MulEquiv.symm (leftInvEquiv S hS)) x) = 1\n[PROOFSTEP]\nconvert S.leftInvEquiv_mul hS ((S.leftInvEquiv hS).symm x)\n[GOAL]\ncase h.e'_2.h.e'_5.h.e'_3\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nx : { x // x \u2208 S }\n\u22a2 x = \u2191(leftInvEquiv S hS) (\u2191(MulEquiv.symm (leftInvEquiv S hS)) x)\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\ninst\u271d : Group M\nS : Submonoid M\nx : { x // x \u2208 leftInv S }\n\u22a2 \u2191(fromLeftInv S x) = (\u2191x)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 mul_right_inj (x : M), mul_right_inv, mul_fromLeftInv]\n[GOAL]\nM : Type u_1\ninst\u271d : CommGroup M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nx : { x // x \u2208 S }\n\u22a2 \u2191(\u2191(MulEquiv.symm (leftInvEquiv S hS)) x) = (\u2191x)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 mul_right_inj (x : M), mul_right_inv, mul_leftInvEquiv_symm]\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Submonoid.Inverses", "llama_tokens": 5458, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743735019594, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4358806434281935}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\np : \u03b1 \u00d7 \u03b2 \u2192 Prop\n\u22a2 (\u2203 x, x \u2208 s \u00d7\u02e2 t \u2227 p x) \u2194 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 t \u2227 p (x, y)\n[PROOFSTEP]\nsimp [and_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 s \u00d7\u02e2 \u2205 = \u2205\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx\u271d : \u03b1 \u00d7 \u03b2\n\u22a2 x\u271d \u2208 s \u00d7\u02e2 \u2205 \u2194 x\u271d \u2208 \u2205\n[PROOFSTEP]\nexact and_false_iff _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 \u2205 \u00d7\u02e2 t = \u2205\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx\u271d : \u03b1 \u00d7 \u03b2\n\u22a2 x\u271d \u2208 \u2205 \u00d7\u02e2 t \u2194 x\u271d \u2208 \u2205\n[PROOFSTEP]\nexact false_and_iff _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 univ \u00d7\u02e2 univ = univ\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx\u271d : \u03b1 \u00d7 \u03b2\n\u22a2 x\u271d \u2208 univ \u00d7\u02e2 univ \u2194 x\u271d \u2208 univ\n[PROOFSTEP]\nexact true_and_iff _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nt : Set \u03b2\n\u22a2 univ \u00d7\u02e2 t = Prod.snd \u207b\u00b9' t\n[PROOFSTEP]\nsimp [prod_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ns : Set \u03b1\n\u22a2 s \u00d7\u02e2 univ = Prod.fst \u207b\u00b9' s\n[PROOFSTEP]\nsimp [prod_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 {a} \u00d7\u02e2 t = Prod.mk a '' t\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 (x, y) \u2208 {a} \u00d7\u02e2 t \u2194 (x, y) \u2208 Prod.mk a '' t\n[PROOFSTEP]\nsimp [and_left_comm, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 s \u00d7\u02e2 {b} = (fun a => (a, b)) '' s\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 (x, y) \u2208 s \u00d7\u02e2 {b} \u2194 (x, y) \u2208 (fun a => (a, b)) '' s\n[PROOFSTEP]\nsimp [and_left_comm, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 {a} \u00d7\u02e2 {b} = {(a, b)}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 (s\u2081 \u222a s\u2082) \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t \u222a s\u2082 \u00d7\u02e2 t\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 (x, y) \u2208 (s\u2081 \u222a s\u2082) \u00d7\u02e2 t \u2194 (x, y) \u2208 s\u2081 \u00d7\u02e2 t \u222a s\u2082 \u00d7\u02e2 t\n[PROOFSTEP]\nsimp [or_and_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 s \u00d7\u02e2 (t\u2081 \u222a t\u2082) = s \u00d7\u02e2 t\u2081 \u222a s \u00d7\u02e2 t\u2082\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 (x, y) \u2208 s \u00d7\u02e2 (t\u2081 \u222a t\u2082) \u2194 (x, y) \u2208 s \u00d7\u02e2 t\u2081 \u222a s \u00d7\u02e2 t\u2082\n[PROOFSTEP]\nsimp [and_or_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 (s\u2081 \u2229 s\u2082) \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t \u2229 s\u2082 \u00d7\u02e2 t\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 (x, y) \u2208 (s\u2081 \u2229 s\u2082) \u00d7\u02e2 t \u2194 (x, y) \u2208 s\u2081 \u00d7\u02e2 t \u2229 s\u2082 \u00d7\u02e2 t\n[PROOFSTEP]\nsimp only [\u2190 and_and_right, mem_inter_iff, mem_prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 s \u00d7\u02e2 (t\u2081 \u2229 t\u2082) = s \u00d7\u02e2 t\u2081 \u2229 s \u00d7\u02e2 t\u2082\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 (x, y) \u2208 s \u00d7\u02e2 (t\u2081 \u2229 t\u2082) \u2194 (x, y) \u2208 s \u00d7\u02e2 t\u2081 \u2229 s \u00d7\u02e2 t\u2082\n[PROOFSTEP]\nsimp only [\u2190 and_and_left, mem_inter_iff, mem_prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 s\u2081 \u00d7\u02e2 t\u2081 \u2229 s\u2082 \u00d7\u02e2 t\u2082 = (s\u2081 \u2229 s\u2082) \u00d7\u02e2 (t\u2081 \u2229 t\u2082)\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 (x, y) \u2208 s\u2081 \u00d7\u02e2 t\u2081 \u2229 s\u2082 \u00d7\u02e2 t\u2082 \u2194 (x, y) \u2208 (s\u2081 \u2229 s\u2082) \u00d7\u02e2 (t\u2081 \u2229 t\u2082)\n[PROOFSTEP]\nsimp [and_assoc, and_left_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 Disjoint (s\u2081 \u00d7\u02e2 t\u2081) (s\u2082 \u00d7\u02e2 t\u2082) \u2194 Disjoint s\u2081 s\u2082 \u2228 Disjoint t\u2081 t\u2082\n[PROOFSTEP]\nsimp_rw [disjoint_left, mem_prod, not_and_or, Prod.forall, and_imp, \u2190 @forall_or_right \u03b1, \u2190 @forall_or_left \u03b2, \u2190\n @forall_or_right (_ \u2208 s\u2081), \u2190 @forall_or_left (_ \u2208 t\u2081)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 insert a s \u00d7\u02e2 t = Prod.mk a '' t \u222a s \u00d7\u02e2 t\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 (x, y) \u2208 insert a s \u00d7\u02e2 t \u2194 (x, y) \u2208 Prod.mk a '' t \u222a s \u00d7\u02e2 t\n[PROOFSTEP]\nsimp (config := { contextual := true }) [image, iff_def, or_imp, Imp.swap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 s \u00d7\u02e2 insert b t = (fun a => (a, b)) '' s \u222a s \u00d7\u02e2 t\n[PROOFSTEP]\next\n \u27e8x, y\u27e9\n -- Porting note: was `simp (config := { contextual := true }) [image, iff_def, or_imp, Imp.swap]`\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 (x, y) \u2208 s \u00d7\u02e2 insert b t \u2194 (x, y) \u2208 (fun a => (a, b)) '' s \u222a s \u00d7\u02e2 t\n[PROOFSTEP]\nsimp [image, or_imp]\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 x \u2208 s \u2227 (y = b \u2228 y \u2208 t) \u2194 (\u2203 a, a \u2208 s \u2227 a = x \u2227 b = y) \u2228 x \u2208 s \u2227 y \u2208 t\n[PROOFSTEP]\nrefine \u27e8fun h => ?_, fun h => ?_\u27e9\n[GOAL]\ncase h.mk.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\nh : x \u2208 s \u2227 (y = b \u2228 y \u2208 t)\n\u22a2 (\u2203 a, a \u2208 s \u2227 a = x \u2227 b = y) \u2228 x \u2208 s \u2227 y \u2208 t\n[PROOFSTEP]\nobtain \u27e8hx, rfl | hy\u27e9 := h\n[GOAL]\ncase h.mk.refine_1.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na x : \u03b1\ny : \u03b2\nhx : x \u2208 s\n\u22a2 (\u2203 a, a \u2208 s \u2227 a = x \u2227 y = y) \u2228 x \u2208 s \u2227 y \u2208 t\n[PROOFSTEP]\nexact Or.inl \u27e8x, hx, rfl, rfl\u27e9\n[GOAL]\ncase h.mk.refine_1.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u22a2 (\u2203 a, a \u2208 s \u2227 a = x \u2227 b = y) \u2228 x \u2208 s \u2227 y \u2208 t\n[PROOFSTEP]\nexact Or.inr \u27e8hx, hy\u27e9\n[GOAL]\ncase h.mk.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\nh : (\u2203 a, a \u2208 s \u2227 a = x \u2227 b = y) \u2228 x \u2208 s \u2227 y \u2208 t\n\u22a2 x \u2208 s \u2227 (y = b \u2228 y \u2208 t)\n[PROOFSTEP]\nobtain \u27e8x, hx, rfl, rfl\u27e9 | \u27e8hx, hy\u27e9 := h\n[GOAL]\ncase h.mk.refine_2.inl.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 s \u2227 (b = b \u2228 b \u2208 t)\n[PROOFSTEP]\nexact \u27e8hx, Or.inl rfl\u27e9\n[GOAL]\ncase h.mk.refine_2.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u22a2 x \u2208 s \u2227 (y = b \u2228 y \u2208 t)\n[PROOFSTEP]\nexact \u27e8hx, Or.inr hy\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nhb : b \u2208 t\n\u22a2 (fun a => (a, b)) \u207b\u00b9' s \u00d7\u02e2 t = s\n[PROOFSTEP]\next a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na\u271d : \u03b1\nb : \u03b2\nhb : b \u2208 t\na : \u03b1\n\u22a2 a \u2208 (fun a => (a, b)) \u207b\u00b9' s \u00d7\u02e2 t \u2194 a \u2208 s\n[PROOFSTEP]\nsimp [hb]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nha : a \u2208 s\n\u22a2 Prod.mk a \u207b\u00b9' s \u00d7\u02e2 t = t\n[PROOFSTEP]\next b\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb\u271d : \u03b2\nha : a \u2208 s\nb : \u03b2\n\u22a2 b \u2208 Prod.mk a \u207b\u00b9' s \u00d7\u02e2 t \u2194 b \u2208 t\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nhb : \u00acb \u2208 t\n\u22a2 (fun a => (a, b)) \u207b\u00b9' s \u00d7\u02e2 t = \u2205\n[PROOFSTEP]\next a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na\u271d : \u03b1\nb : \u03b2\nhb : \u00acb \u2208 t\na : \u03b1\n\u22a2 a \u2208 (fun a => (a, b)) \u207b\u00b9' s \u00d7\u02e2 t \u2194 a \u2208 \u2205\n[PROOFSTEP]\nsimp [hb]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nha : \u00aca \u2208 s\n\u22a2 Prod.mk a \u207b\u00b9' s \u00d7\u02e2 t = \u2205\n[PROOFSTEP]\next b\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb\u271d : \u03b2\nha : \u00aca \u2208 s\nb : \u03b2\n\u22a2 b \u2208 Prod.mk a \u207b\u00b9' s \u00d7\u02e2 t \u2194 b \u2208 \u2205\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d : DecidablePred fun x => x \u2208 t\n\u22a2 (fun a => (a, b)) \u207b\u00b9' s \u00d7\u02e2 t = if b \u2208 t then s else \u2205\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d : DecidablePred fun x => x \u2208 t\nh : b \u2208 t\n\u22a2 (fun a => (a, b)) \u207b\u00b9' s \u00d7\u02e2 t = s\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d : DecidablePred fun x => x \u2208 t\nh : \u00acb \u2208 t\n\u22a2 (fun a => (a, b)) \u207b\u00b9' s \u00d7\u02e2 t = \u2205\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d : DecidablePred fun x => x \u2208 s\n\u22a2 Prod.mk a \u207b\u00b9' s \u00d7\u02e2 t = if a \u2208 s then t else \u2205\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d : DecidablePred fun x => x \u2208 s\nh : a \u2208 s\n\u22a2 Prod.mk a \u207b\u00b9' s \u00d7\u02e2 t = t\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d : DecidablePred fun x => x \u2208 s\nh : \u00aca \u2208 s\n\u22a2 Prod.mk a \u207b\u00b9' s \u00d7\u02e2 t = \u2205\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d : DecidablePred fun x => x \u2208 t\nf : \u03b3 \u2192 \u03b1\n\u22a2 (fun a => (f a, b)) \u207b\u00b9' s \u00d7\u02e2 t = if b \u2208 t then f \u207b\u00b9' s else \u2205\n[PROOFSTEP]\nrw [\u2190 mk_preimage_prod_left_eq_if, prod_preimage_left, preimage_preimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d : DecidablePred fun x => x \u2208 s\ng : \u03b4 \u2192 \u03b2\n\u22a2 (fun b => (a, g b)) \u207b\u00b9' s \u00d7\u02e2 t = if a \u2208 s then g \u207b\u00b9' t else \u2205\n[PROOFSTEP]\nrw [\u2190 mk_preimage_prod_right_eq_if, prod_preimage_right, preimage_preimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 Prod.swap \u207b\u00b9' s \u00d7\u02e2 t = t \u00d7\u02e2 s\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b2\ny : \u03b1\n\u22a2 (x, y) \u2208 Prod.swap \u207b\u00b9' s \u00d7\u02e2 t \u2194 (x, y) \u2208 t \u00d7\u02e2 s\n[PROOFSTEP]\nsimp [and_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 Prod.swap '' s \u00d7\u02e2 t = t \u00d7\u02e2 s\n[PROOFSTEP]\nrw [image_swap_eq_preimage_swap, preimage_swap_prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nm\u2081 : \u03b1 \u2192 \u03b3\nm\u2082 : \u03b2 \u2192 \u03b4\n\u22a2 \u2200 (x : \u03b3 \u00d7 \u03b4), x \u2208 (m\u2081 '' s) \u00d7\u02e2 (m\u2082 '' t) \u2194 x \u2208 (fun p => (m\u2081 p.fst, m\u2082 p.snd)) '' s \u00d7\u02e2 t\n[PROOFSTEP]\nsimp [-exists_and_right, exists_and_right.symm, and_left_comm, and_assoc, and_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nm\u2081 : \u03b1 \u2192 \u03b3\nm\u2082 : \u03b2 \u2192 \u03b4\n\u22a2 \u2200 (x : \u03b3 \u00d7 \u03b4), x \u2208 range m\u2081 \u00d7\u02e2 range m\u2082 \u2194 x \u2208 range fun p => (m\u2081 p.fst, m\u2082 p.snd)\n[PROOFSTEP]\nsimp [range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nm\u2081 : \u03b1 \u2192 \u03b3\n\u22a2 \u2200 (x : \u03b3 \u00d7 \u03b2), x \u2208 range m\u2081 \u00d7\u02e2 univ \u2194 x \u2208 range fun p => (m\u2081 p.fst, p.snd)\n[PROOFSTEP]\nsimp [range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nm\u2082 : \u03b2 \u2192 \u03b4\n\u22a2 \u2200 (x : \u03b1 \u00d7 \u03b4), x \u2208 univ \u00d7\u02e2 range m\u2082 \u2194 x \u2208 range fun p => (p.fst, m\u2082 p.snd)\n[PROOFSTEP]\nsimp [range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\n\u22a2 (range fun x => (f x, g x)) \u2286 range f \u00d7\u02e2 range g\n[PROOFSTEP]\nhave : (fun x => (f x, g x)) = Prod.map f g \u2218 fun x => (x, x) := funext fun x => rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\nthis : (fun x => (f x, g x)) = Prod.map f g \u2218 fun x => (x, x)\n\u22a2 (range fun x => (f x, g x)) \u2286 range f \u00d7\u02e2 range g\n[PROOFSTEP]\nrw [this, \u2190 range_prod_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\nthis : (fun x => (f x, g x)) = Prod.map f g \u2218 fun x => (x, x)\n\u22a2 range (Prod.map f g \u2218 fun x => (x, x)) \u2286 range (Prod.map f g)\n[PROOFSTEP]\napply range_comp_subset_range\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 s \u00d7\u02e2 t = \u2205 \u2194 s = \u2205 \u2228 t = \u2205\n[PROOFSTEP]\nsimp only [not_nonempty_iff_eq_empty.symm, prod_nonempty_iff, not_and_or]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nW : Set \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\n\u22a2 s \u00d7\u02e2 t \u2286 f \u207b\u00b9' W \u2194 \u2200 (a : \u03b1) (b : \u03b2), a \u2208 s \u2192 b \u2208 t \u2192 f (a, b) \u2208 W\n[PROOFSTEP]\nsimp [subset_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ns : Set \u03b1\n\u22a2 (fun x => (f x, g x)) '' s \u2286 (f '' s) \u00d7\u02e2 (g '' s)\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\n\u22a2 (fun x => (f x, g x)) x \u2208 (f '' s) \u00d7\u02e2 (g '' s)\n[PROOFSTEP]\nexact mk_mem_prod (mem_image_of_mem f hx) (mem_image_of_mem g hx)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nhb : b \u2208 t\n\u22a2 (fun a => (a, b)) '' s \u2286 s \u00d7\u02e2 t\n[PROOFSTEP]\nrintro _ \u27e8a, ha, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na\u271d : \u03b1\nb : \u03b2\nhb : b \u2208 t\na : \u03b1\nha : a \u2208 s\n\u22a2 (fun a => (a, b)) a \u2208 s \u00d7\u02e2 t\n[PROOFSTEP]\nexact \u27e8ha, hb\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nha : a \u2208 s\n\u22a2 Prod.mk a '' t \u2286 s \u00d7\u02e2 t\n[PROOFSTEP]\nrintro _ \u27e8b, hb, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb\u271d : \u03b2\nha : a \u2208 s\nb : \u03b2\nhb : b \u2208 t\n\u22a2 (a, b) \u2208 s \u00d7\u02e2 t\n[PROOFSTEP]\nexact \u27e8ha, hb\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 s \u00d7\u02e2 t \\ s\u2081 \u00d7\u02e2 t\u2081 = s \u00d7\u02e2 (t \\ t\u2081) \u222a (s \\ s\u2081) \u00d7\u02e2 t\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 x \u2208 s \u00d7\u02e2 t \\ s\u2081 \u00d7\u02e2 t\u2081 \u2194 x \u2208 s \u00d7\u02e2 (t \\ t\u2081) \u222a (s \\ s\u2081) \u00d7\u02e2 t\n[PROOFSTEP]\nby_cases h\u2081 : x.1 \u2208 s\u2081\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1 \u00d7 \u03b2\nh\u2081 : x.fst \u2208 s\u2081\n\u22a2 x \u2208 s \u00d7\u02e2 t \\ s\u2081 \u00d7\u02e2 t\u2081 \u2194 x \u2208 s \u00d7\u02e2 (t \\ t\u2081) \u222a (s \\ s\u2081) \u00d7\u02e2 t\n[PROOFSTEP]\nby_cases h\u2082 : x.2 \u2208 t\u2081\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1 \u00d7 \u03b2\nh\u2081 : \u00acx.fst \u2208 s\u2081\n\u22a2 x \u2208 s \u00d7\u02e2 t \\ s\u2081 \u00d7\u02e2 t\u2081 \u2194 x \u2208 s \u00d7\u02e2 (t \\ t\u2081) \u222a (s \\ s\u2081) \u00d7\u02e2 t\n[PROOFSTEP]\nby_cases h\u2082 : x.2 \u2208 t\u2081\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1 \u00d7 \u03b2\nh\u2081 : x.fst \u2208 s\u2081\nh\u2082 : x.snd \u2208 t\u2081\n\u22a2 x \u2208 s \u00d7\u02e2 t \\ s\u2081 \u00d7\u02e2 t\u2081 \u2194 x \u2208 s \u00d7\u02e2 (t \\ t\u2081) \u222a (s \\ s\u2081) \u00d7\u02e2 t\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1 \u00d7 \u03b2\nh\u2081 : x.fst \u2208 s\u2081\nh\u2082 : \u00acx.snd \u2208 t\u2081\n\u22a2 x \u2208 s \u00d7\u02e2 t \\ s\u2081 \u00d7\u02e2 t\u2081 \u2194 x \u2208 s \u00d7\u02e2 (t \\ t\u2081) \u222a (s \\ s\u2081) \u00d7\u02e2 t\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1 \u00d7 \u03b2\nh\u2081 : \u00acx.fst \u2208 s\u2081\nh\u2082 : x.snd \u2208 t\u2081\n\u22a2 x \u2208 s \u00d7\u02e2 t \\ s\u2081 \u00d7\u02e2 t\u2081 \u2194 x \u2208 s \u00d7\u02e2 (t \\ t\u2081) \u222a (s \\ s\u2081) \u00d7\u02e2 t\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1 \u00d7 \u03b2\nh\u2081 : \u00acx.fst \u2208 s\u2081\nh\u2082 : \u00acx.snd \u2208 t\u2081\n\u22a2 x \u2208 s \u00d7\u02e2 t \\ s\u2081 \u00d7\u02e2 t\u2081 \u2194 x \u2208 s \u00d7\u02e2 (t \\ t\u2081) \u222a (s \\ s\u2081) \u00d7\u02e2 t\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205\n[PROOFSTEP]\ncases' (s \u00d7\u02e2 t).eq_empty_or_nonempty with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : s \u00d7\u02e2 t = \u2205\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205\n[PROOFSTEP]\nsimp [h, prod_eq_empty_iff.1 h]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205\n[PROOFSTEP]\nhave st : s.Nonempty \u2227 t.Nonempty := by rwa [prod_nonempty_iff] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\n\u22a2 Set.Nonempty s \u2227 Set.Nonempty t\n[PROOFSTEP]\nrwa [prod_nonempty_iff] at h \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205\n[PROOFSTEP]\nrefine' \u27e8fun H => Or.inl \u27e8_, _\u27e9, _\u27e9\n[GOAL]\ncase inr.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081\n\u22a2 s \u2286 s\u2081\n[PROOFSTEP]\nhave := image_subset (Prod.fst : \u03b1 \u00d7 \u03b2 \u2192 \u03b1) H\n[GOAL]\ncase inr.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081\nthis : Prod.fst '' s \u00d7\u02e2 t \u2286 Prod.fst '' s\u2081 \u00d7\u02e2 t\u2081\n\u22a2 s \u2286 s\u2081\n[PROOFSTEP]\nrwa [fst_image_prod _ st.2, fst_image_prod _ (h.mono H).snd] at this \n[GOAL]\ncase inr.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081\n\u22a2 t \u2286 t\u2081\n[PROOFSTEP]\nhave := image_subset (Prod.snd : \u03b1 \u00d7 \u03b2 \u2192 \u03b2) H\n[GOAL]\ncase inr.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081\nthis : Prod.snd '' s \u00d7\u02e2 t \u2286 Prod.snd '' s\u2081 \u00d7\u02e2 t\u2081\n\u22a2 t \u2286 t\u2081\n[PROOFSTEP]\nrwa [snd_image_prod st.1, snd_image_prod (h.mono H).fst] at this \n[GOAL]\ncase inr.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\n\u22a2 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 \u2192 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081\n[PROOFSTEP]\nintro H\n[GOAL]\ncase inr.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081\n[PROOFSTEP]\nsimp only [st.1.ne_empty, st.2.ne_empty, or_false_iff] at H \n[GOAL]\ncase inr.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\nst : Set.Nonempty s \u2227 Set.Nonempty t\nH : s \u2286 s\u2081 \u2227 t \u2286 t\u2081\n\u22a2 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081\n[PROOFSTEP]\nexact prod_mono H.1 H.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\n\u22a2 s \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t\u2081 \u2194 s = s\u2081 \u2227 t = t\u2081\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\n\u22a2 s \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t\u2081 \u2192 s = s\u2081 \u2227 t = t\u2081\n[PROOFSTEP]\nintro heq\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\nheq : s \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t\u2081\n\u22a2 s = s\u2081 \u2227 t = t\u2081\n[PROOFSTEP]\nhave h\u2081 : (s\u2081 \u00d7\u02e2 t\u2081 : Set _).Nonempty := by rwa [\u2190 heq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\nheq : s \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t\u2081\n\u22a2 Set.Nonempty (s\u2081 \u00d7\u02e2 t\u2081)\n[PROOFSTEP]\nrwa [\u2190 heq]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\nheq : s \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t\u2081\nh\u2081 : Set.Nonempty (s\u2081 \u00d7\u02e2 t\u2081)\n\u22a2 s = s\u2081 \u2227 t = t\u2081\n[PROOFSTEP]\nrw [prod_nonempty_iff] at h h\u2081 \n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty s \u2227 Set.Nonempty t\nheq : s \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t\u2081\nh\u2081 : Set.Nonempty s\u2081 \u2227 Set.Nonempty t\u2081\n\u22a2 s = s\u2081 \u2227 t = t\u2081\n[PROOFSTEP]\nrw [\u2190 fst_image_prod s h.2, \u2190 fst_image_prod s\u2081 h\u2081.2, heq, eq_self_iff_true, true_and_iff, \u2190 snd_image_prod h.1 t, \u2190\n snd_image_prod h\u2081.1 t\u2081, heq]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\n\u22a2 s = s\u2081 \u2227 t = t\u2081 \u2192 s \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t\u2081\n[PROOFSTEP]\nrintro \u27e8rfl, rfl\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2082 : Set \u03b1\nt t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\n\u22a2 s \u00d7\u02e2 t = s \u00d7\u02e2 t\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 s \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t\u2081 \u2194 s = s\u2081 \u2227 t = t\u2081 \u2228 (s = \u2205 \u2228 t = \u2205) \u2227 (s\u2081 = \u2205 \u2228 t\u2081 = \u2205)\n[PROOFSTEP]\nsymm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 s = s\u2081 \u2227 t = t\u2081 \u2228 (s = \u2205 \u2228 t = \u2205) \u2227 (s\u2081 = \u2205 \u2228 t\u2081 = \u2205) \u2194 s \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t\u2081\n[PROOFSTEP]\ncases' eq_empty_or_nonempty (s \u00d7\u02e2 t) with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : s \u00d7\u02e2 t = \u2205\n\u22a2 s = s\u2081 \u2227 t = t\u2081 \u2228 (s = \u2205 \u2228 t = \u2205) \u2227 (s\u2081 = \u2205 \u2228 t\u2081 = \u2205) \u2194 s \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t\u2081\n[PROOFSTEP]\nsimp_rw [h, @eq_comm _ \u2205, prod_eq_empty_iff, prod_eq_empty_iff.mp h, true_and_iff, or_iff_right_iff_imp]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : s \u00d7\u02e2 t = \u2205\n\u22a2 s = s\u2081 \u2227 t = t\u2081 \u2192 s\u2081 = \u2205 \u2228 t\u2081 = \u2205\n[PROOFSTEP]\nrintro \u27e8rfl, rfl\u27e9\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2082 : Set \u03b1\nt t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : s \u00d7\u02e2 t = \u2205\n\u22a2 s = \u2205 \u2228 t = \u2205\n[PROOFSTEP]\nexact prod_eq_empty_iff.mp h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\n\u22a2 s = s\u2081 \u2227 t = t\u2081 \u2228 (s = \u2205 \u2228 t = \u2205) \u2227 (s\u2081 = \u2205 \u2228 t\u2081 = \u2205) \u2194 s \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t\u2081\n[PROOFSTEP]\nrw [prod_eq_prod_iff_of_nonempty h]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : Set.Nonempty (s \u00d7\u02e2 t)\n\u22a2 s = s\u2081 \u2227 t = t\u2081 \u2228 (s = \u2205 \u2228 t = \u2205) \u2227 (s\u2081 = \u2205 \u2228 t\u2081 = \u2205) \u2194 s = s\u2081 \u2227 t = t\u2081\n[PROOFSTEP]\nrw [nonempty_iff_ne_empty, Ne.def, prod_eq_empty_iff] at h \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nh : \u00ac(s = \u2205 \u2228 t = \u2205)\n\u22a2 s = s\u2081 \u2227 t = t\u2081 \u2228 (s = \u2205 \u2228 t = \u2205) \u2227 (s\u2081 = \u2205 \u2228 t\u2081 = \u2205) \u2194 s = s\u2081 \u2227 t = t\u2081\n[PROOFSTEP]\nsimp_rw [h, false_and_iff, or_false_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nht : Set.Nonempty t\n\u22a2 s \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t \u2194 s = s\u2081\n[PROOFSTEP]\nsimp_rw [prod_eq_prod_iff, ht.ne_empty, and_true_iff, or_iff_left_iff_imp, or_false_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nht : Set.Nonempty t\n\u22a2 s = \u2205 \u2227 s\u2081 = \u2205 \u2192 s = s\u2081\n[PROOFSTEP]\nrintro \u27e8rfl, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nht : Set.Nonempty t\n\u22a2 \u2205 = \u2205\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1\n\u22a2 (x, x) \u2208 diagonal \u03b1\n[PROOFSTEP]\nsimp [diagonal]\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\n\u22a2 (Prod.map (fun x => \u2191x) fun x => \u2191x) \u207b\u00b9' diagonal \u03b1 = diagonal \u2191s\n[PROOFSTEP]\next \u27e8\u27e8x, hx\u27e9, \u27e8y, hy\u27e9\u27e9\n[GOAL]\ncase h.mk.mk.mk\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\n\u22a2 ({ val := x, property := hx }, { val := y, property := hy }) \u2208 (Prod.map (fun x => \u2191x) fun x => \u2191x) \u207b\u00b9' diagonal \u03b1 \u2194\n ({ val := x, property := hx }, { val := y, property := hy }) \u2208 diagonal \u2191s\n[PROOFSTEP]\nsimp [Set.diagonal]\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\n\u22a2 (range fun x => (x, x)) = diagonal \u03b1\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\ns t : Set \u03b1\nx y : \u03b1\n\u22a2 ((x, y) \u2208 range fun x => (x, x)) \u2194 (x, y) \u2208 diagonal \u03b1\n[PROOFSTEP]\nsimp [diagonal, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t : Set \u03b1\ns : Set (\u03b1 \u00d7 \u03b1)\n\u22a2 diagonal \u03b1 \u2286 s \u2194 \u2200 (x : \u03b1), (x, x) \u2208 s\n[PROOFSTEP]\nrw [\u2190 range_diag, range_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\n\u22a2 (fun x => (x, x)) '' s = diagonal \u03b1 \u2229 s \u00d7\u02e2 s\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\n\u22a2 x \u2208 (fun x => (x, x)) '' s \u2194 x \u2208 diagonal \u03b1 \u2229 s \u00d7\u02e2 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\n\u22a2 x \u2208 (fun x => (x, x)) '' s \u2192 x \u2208 diagonal \u03b1 \u2229 s \u00d7\u02e2 s\n[PROOFSTEP]\nrintro \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\n\u22a2 (fun x => (x, x)) x \u2208 diagonal \u03b1 \u2229 s \u00d7\u02e2 s\n[PROOFSTEP]\nexact \u27e8rfl, hx, hx\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\n\u22a2 x \u2208 diagonal \u03b1 \u2229 s \u00d7\u02e2 s \u2192 x \u2208 (fun x => (x, x)) '' s\n[PROOFSTEP]\nobtain \u27e8x, y\u27e9 := x\n[GOAL]\ncase h.mpr.mk\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nx y : \u03b1\n\u22a2 (x, y) \u2208 diagonal \u03b1 \u2229 s \u00d7\u02e2 s \u2192 (x, y) \u2208 (fun x => (x, x)) '' s\n[PROOFSTEP]\nrintro \u27e8rfl : x = y, h2x\u27e9\n[GOAL]\ncase h.mpr.mk.intro\n\u03b1 : Type u_1\ns\u271d t s : Set \u03b1\nx : \u03b1\nh2x : (x, x) \u2208 s \u00d7\u02e2 s\n\u22a2 (x, x) \u2208 (fun x => (x, x)) '' s\n[PROOFSTEP]\nexact mem_image_of_mem _ h2x.1\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\n\u22a2 Set.Nonempty (offDiag s) \u2194 Set.Nontrivial s\n[PROOFSTEP]\nsimp [offDiag, Set.Nonempty, Set.Nontrivial]\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\n\u22a2 offDiag s = \u2205 \u2194 Set.Subsingleton s\n[PROOFSTEP]\nrw [\u2190 not_nonempty_iff_eq_empty, \u2190 not_nontrivial_iff, offDiag_nonempty.not]\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\n\u22a2 offDiag \u2205 = \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\na\u271d a : \u03b1\n\u22a2 offDiag {a} = \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\n\u22a2 \u2200 (x : \u03b1 \u00d7 \u03b1), x \u2208 offDiag univ \u2194 x \u2208 (diagonal \u03b1)\u1d9c\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nx : \u03b1 \u00d7 \u03b1\n\u22a2 x \u2208 offDiag (s \u2229 t) \u2194 x \u2208 offDiag s \u2229 offDiag t\n[PROOFSTEP]\nsimp only [mem_offDiag, mem_inter_iff]\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nx : \u03b1 \u00d7 \u03b1\n\u22a2 (x.fst \u2208 s \u2227 x.fst \u2208 t) \u2227 (x.snd \u2208 s \u2227 x.snd \u2208 t) \u2227 x.fst \u2260 x.snd \u2194\n (x.fst \u2208 s \u2227 x.snd \u2208 s \u2227 x.fst \u2260 x.snd) \u2227 x.fst \u2208 t \u2227 x.snd \u2208 t \u2227 x.fst \u2260 x.snd\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\n\u22a2 offDiag (s \u222a t) = offDiag s \u222a offDiag t \u222a s \u00d7\u02e2 t \u222a t \u00d7\u02e2 s\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\n\u22a2 x \u2208 offDiag (s \u222a t) \u2194 x \u2208 offDiag s \u222a offDiag t \u222a s \u00d7\u02e2 t \u222a t \u00d7\u02e2 s\n[PROOFSTEP]\nsimp only [mem_offDiag, mem_union, ne_eq, mem_prod]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\n\u22a2 (x.fst \u2208 s \u2228 x.fst \u2208 t) \u2227 (x.snd \u2208 s \u2228 x.snd \u2208 t) \u2227 \u00acx.fst = x.snd \u2194\n ((x.fst \u2208 s \u2227 x.snd \u2208 s \u2227 \u00acx.fst = x.snd \u2228 x.fst \u2208 t \u2227 x.snd \u2208 t \u2227 \u00acx.fst = x.snd) \u2228 x.fst \u2208 s \u2227 x.snd \u2208 t) \u2228\n x.fst \u2208 t \u2227 x.snd \u2208 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\n\u22a2 (x.fst \u2208 s \u2228 x.fst \u2208 t) \u2227 (x.snd \u2208 s \u2228 x.snd \u2208 t) \u2227 \u00acx.fst = x.snd \u2192\n ((x.fst \u2208 s \u2227 x.snd \u2208 s \u2227 \u00acx.fst = x.snd \u2228 x.fst \u2208 t \u2227 x.snd \u2208 t \u2227 \u00acx.fst = x.snd) \u2228 x.fst \u2208 s \u2227 x.snd \u2208 t) \u2228\n x.fst \u2208 t \u2227 x.snd \u2208 s\n[PROOFSTEP]\nrintro \u27e8h0 | h0, h1 | h1, h2\u27e9\n[GOAL]\ncase h.mp.intro.inl.intro.inl\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nh0 : x.fst \u2208 s\nh2 : \u00acx.fst = x.snd\nh1 : x.snd \u2208 s\n\u22a2 ((x.fst \u2208 s \u2227 x.snd \u2208 s \u2227 \u00acx.fst = x.snd \u2228 x.fst \u2208 t \u2227 x.snd \u2208 t \u2227 \u00acx.fst = x.snd) \u2228 x.fst \u2208 s \u2227 x.snd \u2208 t) \u2228\n x.fst \u2208 t \u2227 x.snd \u2208 s\n[PROOFSTEP]\nsimp [h0, h1, h2]\n[GOAL]\ncase h.mp.intro.inl.intro.inr\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nh0 : x.fst \u2208 s\nh2 : \u00acx.fst = x.snd\nh1 : x.snd \u2208 t\n\u22a2 ((x.fst \u2208 s \u2227 x.snd \u2208 s \u2227 \u00acx.fst = x.snd \u2228 x.fst \u2208 t \u2227 x.snd \u2208 t \u2227 \u00acx.fst = x.snd) \u2228 x.fst \u2208 s \u2227 x.snd \u2208 t) \u2228\n x.fst \u2208 t \u2227 x.snd \u2208 s\n[PROOFSTEP]\nsimp [h0, h1, h2]\n[GOAL]\ncase h.mp.intro.inr.intro.inl\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nh0 : x.fst \u2208 t\nh2 : \u00acx.fst = x.snd\nh1 : x.snd \u2208 s\n\u22a2 ((x.fst \u2208 s \u2227 x.snd \u2208 s \u2227 \u00acx.fst = x.snd \u2228 x.fst \u2208 t \u2227 x.snd \u2208 t \u2227 \u00acx.fst = x.snd) \u2228 x.fst \u2208 s \u2227 x.snd \u2208 t) \u2228\n x.fst \u2208 t \u2227 x.snd \u2208 s\n[PROOFSTEP]\nsimp [h0, h1, h2]\n[GOAL]\ncase h.mp.intro.inr.intro.inr\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nh0 : x.fst \u2208 t\nh2 : \u00acx.fst = x.snd\nh1 : x.snd \u2208 t\n\u22a2 ((x.fst \u2208 s \u2227 x.snd \u2208 s \u2227 \u00acx.fst = x.snd \u2228 x.fst \u2208 t \u2227 x.snd \u2208 t \u2227 \u00acx.fst = x.snd) \u2228 x.fst \u2208 s \u2227 x.snd \u2208 t) \u2228\n x.fst \u2208 t \u2227 x.snd \u2208 s\n[PROOFSTEP]\nsimp [h0, h1, h2]\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\n\u22a2 ((x.fst \u2208 s \u2227 x.snd \u2208 s \u2227 \u00acx.fst = x.snd \u2228 x.fst \u2208 t \u2227 x.snd \u2208 t \u2227 \u00acx.fst = x.snd) \u2228 x.fst \u2208 s \u2227 x.snd \u2208 t) \u2228\n x.fst \u2208 t \u2227 x.snd \u2208 s \u2192\n (x.fst \u2208 s \u2228 x.fst \u2208 t) \u2227 (x.snd \u2208 s \u2228 x.snd \u2208 t) \u2227 \u00acx.fst = x.snd\n[PROOFSTEP]\nrintro (((\u27e8h0, h1, h2\u27e9 | \u27e8h0, h1, h2\u27e9) | \u27e8h0, h1\u27e9) | \u27e8h0, h1\u27e9)\n[GOAL]\ncase h.mpr.inl.inl.inl.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nh0 : x.fst \u2208 s\nh1 : x.snd \u2208 s\nh2 : \u00acx.fst = x.snd\n\u22a2 (x.fst \u2208 s \u2228 x.fst \u2208 t) \u2227 (x.snd \u2208 s \u2228 x.snd \u2208 t) \u2227 \u00acx.fst = x.snd\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase h.mpr.inl.inl.inr.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nh0 : x.fst \u2208 t\nh1 : x.snd \u2208 t\nh2 : \u00acx.fst = x.snd\n\u22a2 (x.fst \u2208 s \u2228 x.fst \u2208 t) \u2227 (x.snd \u2208 s \u2228 x.snd \u2208 t) \u2227 \u00acx.fst = x.snd\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase h.mpr.inl.inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nh0 : x.fst \u2208 s\nh1 : x.snd \u2208 t\n\u22a2 (x.fst \u2208 s \u2228 x.fst \u2208 t) \u2227 (x.snd \u2208 s \u2228 x.snd \u2208 t) \u2227 \u00acx.fst = x.snd\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase h.mpr.inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nh0 : x.fst \u2208 t\nh1 : x.snd \u2208 s\n\u22a2 (x.fst \u2208 s \u2228 x.fst \u2208 t) \u2227 (x.snd \u2208 s \u2228 x.snd \u2208 t) \u2227 \u00acx.fst = x.snd\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase h.mpr.inl.inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nh0 : x.fst \u2208 s\nh1 : x.snd \u2208 t\n\u22a2 \u00acx.fst = x.snd\n[PROOFSTEP]\nrintro h3\n[GOAL]\ncase h.mpr.inl.inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nh0 : x.fst \u2208 s\nh1 : x.snd \u2208 t\nh3 : x.fst = x.snd\n\u22a2 False\n[PROOFSTEP]\nrw [h3] at h0 \n[GOAL]\ncase h.mpr.inl.inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nh0 : x.snd \u2208 s\nh1 : x.snd \u2208 t\nh3 : x.fst = x.snd\n\u22a2 False\n[PROOFSTEP]\nexact (Set.disjoint_left.mp h h0 h1)\n[GOAL]\ncase h.mpr.inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nh0 : x.fst \u2208 t\nh1 : x.snd \u2208 s\n\u22a2 \u00acx.fst = x.snd\n[PROOFSTEP]\nrintro h3\n[GOAL]\ncase h.mpr.inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nh0 : x.fst \u2208 t\nh1 : x.snd \u2208 s\nh3 : x.fst = x.snd\n\u22a2 False\n[PROOFSTEP]\nrw [h3] at h0 \n[GOAL]\ncase h.mpr.inr.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\nx\u271d : \u03b1 \u00d7 \u03b1\na : \u03b1\nh : Disjoint s t\nx : \u03b1 \u00d7 \u03b1\nh0 : x.snd \u2208 t\nh1 : x.snd \u2208 s\nh3 : x.fst = x.snd\n\u22a2 False\n[PROOFSTEP]\nexact (Set.disjoint_right.mp h h0 h1).elim\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 offDiag (insert a s) = offDiag s \u222a {a} \u00d7\u02e2 s \u222a s \u00d7\u02e2 {a}\n[PROOFSTEP]\nrw [insert_eq, union_comm, offDiag_union, offDiag_singleton, union_empty, union_right_comm]\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 Disjoint s {a}\n[PROOFSTEP]\nrw [disjoint_left]\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 \u2200 \u2983a_1 : \u03b1\u2984, a_1 \u2208 s \u2192 \u00aca_1 \u2208 {a}\n[PROOFSTEP]\nrintro b hb (rfl : b = a)\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\nb : \u03b1\nhb : b \u2208 s\nha : \u00acb \u2208 s\n\u22a2 False\n[PROOFSTEP]\nexact ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 f \u2208 pi univ t \u2194 \u2200 (i : \u03b9), f i \u2208 t i\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 pi \u2205 s = univ\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 x\u271d \u2208 pi \u2205 s \u2194 x\u271d \u2208 univ\n[PROOFSTEP]\nsimp [pi]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nx : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (x \u2208 pi s fun i => t i \u2229 t\u2081 i) \u2194 x \u2208 pi s t \u2229 pi s t\u2081\n[PROOFSTEP]\nsimp only [forall_and, mem_pi, mem_inter_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nhs : i \u2208 s\nht : t i = \u2205\n\u22a2 pi s t = \u2205\n[PROOFSTEP]\next f\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nhs : i \u2208 s\nht : t i = \u2205\nf : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 f \u2208 pi s t \u2194 f \u2208 \u2205\n[PROOFSTEP]\nsimp only [mem_empty_iff_false, not_forall, iff_false_iff, mem_pi, not_imp]\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nhs : i \u2208 s\nht : t i = \u2205\nf : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 \u2203 x, x \u2208 s \u2227 \u00acf x \u2208 t x\n[PROOFSTEP]\nexact \u27e8i, hs, by simp [ht]\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nhs : i \u2208 s\nht : t i = \u2205\nf : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 \u00acf i \u2208 t i\n[PROOFSTEP]\nsimp [ht]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 Set.Nonempty (pi s t) \u2194 \u2200 (i : \u03b9), \u2203 x, i \u2208 s \u2192 x \u2208 t i\n[PROOFSTEP]\nsimp [Classical.skolem, Set.Nonempty]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 Set.Nonempty (pi univ t) \u2194 \u2200 (i : \u03b9), Set.Nonempty (t i)\n[PROOFSTEP]\nsimp [Classical.skolem, Set.Nonempty]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 pi s t = \u2205 \u2194 \u2203 i, IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205\n[PROOFSTEP]\nrw [\u2190 not_nonempty_iff_eq_empty, pi_nonempty_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 (\u00ac\u2200 (i : \u03b9), \u2203 x, i \u2208 s \u2192 x \u2208 t i) \u2194 \u2203 i, IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205\n[PROOFSTEP]\npush_neg\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 (\u2203 i, \u2200 (x : \u03b1 i), i \u2208 s \u2227 \u00acx \u2208 t i) \u2194 \u2203 i, IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205\n[PROOFSTEP]\nrefine' exists_congr fun i => _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d i : \u03b9\n\u22a2 (\u2200 (x : \u03b1 i), i \u2208 s \u2227 \u00acx \u2208 t i) \u2194 IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205\n[PROOFSTEP]\ncases isEmpty_or_nonempty (\u03b1 i)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d i : \u03b9\nh\u271d : IsEmpty (\u03b1 i)\n\u22a2 (\u2200 (x : \u03b1 i), i \u2208 s \u2227 \u00acx \u2208 t i) \u2194 IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205\n[PROOFSTEP]\nsimp [*, forall_and, eq_empty_iff_forall_not_mem]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d i : \u03b9\nh\u271d : Nonempty (\u03b1 i)\n\u22a2 (\u2200 (x : \u03b1 i), i \u2208 s \u2227 \u00acx \u2208 t i) \u2194 IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205\n[PROOFSTEP]\nsimp [*, forall_and, eq_empty_iff_forall_not_mem]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 pi univ t = \u2205 \u2194 \u2203 i, t i = \u2205\n[PROOFSTEP]\nsimp [\u2190 not_nonempty_iff_eq_empty, univ_pi_nonempty_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 Disjoint (pi univ t\u2081) (pi univ t\u2082) \u2194 \u2203 i, Disjoint (t\u2081 i) (t\u2082 i)\n[PROOFSTEP]\nsimp only [disjoint_iff_inter_eq_empty, \u2190 pi_inter_distrib, univ_pi_eq_empty_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\n\u22a2 pi s t = \u2205 \u2194 \u2203 i, i \u2208 s \u2227 t i = \u2205\n[PROOFSTEP]\nsimp [pi_eq_empty_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\n\u22a2 Disjoint (pi s t\u2081) (pi s t\u2082) \u2194 \u2203 i, i \u2208 s \u2227 Disjoint (t\u2081 i) (t\u2082 i)\n[PROOFSTEP]\nsimp only [disjoint_iff_inter_eq_empty, \u2190 pi_inter_distrib, pi_eq_empty_iff']\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\n\u22a2 (range fun g i => f i (g i)) = pi univ fun i => range (f i)\n[PROOFSTEP]\nrefine Subset.antisymm ?_ fun x hx => ?_\n[GOAL]\ncase refine_1\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\n\u22a2 (range fun g i => f i (g i)) \u2286 pi univ fun i => range (f i)\n[PROOFSTEP]\nrintro _ \u27e8x, rfl\u27e9 i -\n[GOAL]\ncase refine_1.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\nx : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\n\u22a2 (fun g i => f i (g i)) x i \u2208 (fun i => range (f i)) i\n[PROOFSTEP]\nexact \u27e8x i, rfl\u27e9\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\nx : (i : \u03b9) \u2192 \u03b2 i\nhx : x \u2208 pi univ fun i => range (f i)\n\u22a2 x \u2208 range fun g i => f i (g i)\n[PROOFSTEP]\nchoose y hy using hx\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\nx : (i : \u03b9) \u2192 \u03b2 i\ny : (i : \u03b9) \u2192 i \u2208 univ \u2192 \u03b1 i\nhy : \u2200 (i : \u03b9) (a : i \u2208 univ), f i (y i a) = x i\n\u22a2 x \u2208 range fun g i => f i (g i)\n[PROOFSTEP]\nexact \u27e8fun i => y i trivial, funext fun i => hy i trivial\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d i : \u03b9\ns : Set \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 pi (insert i s) t = eval i \u207b\u00b9' t i \u2229 pi s t\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d i : \u03b9\ns : Set \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 x\u271d \u2208 pi (insert i s) t \u2194 x\u271d \u2208 eval i \u207b\u00b9' t i \u2229 pi s t\n[PROOFSTEP]\nsimp [pi, or_imp, forall_and]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d i : \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 pi {i} t = eval i \u207b\u00b9' t i\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d i : \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 x\u271d \u2208 pi {i} t \u2194 x\u271d \u2208 eval i \u207b\u00b9' t i\n[PROOFSTEP]\nsimp [pi]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nf g : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (g \u2208 pi univ fun i => {f i}) \u2194 g \u2208 {f}\n[PROOFSTEP]\nsimp [funext_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081\u271d t\u2082\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\np : \u03b9 \u2192 Prop\nh : DecidablePred p\ns : Set \u03b9\nt\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 (pi s fun i => if p i then t\u2081 i else t\u2082 i) = pi {i | i \u2208 s \u2227 p i} t\u2081 \u2229 pi {i | i \u2208 s \u2227 \u00acp i} t\u2082\n[PROOFSTEP]\next f\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081\u271d t\u2082\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\np : \u03b9 \u2192 Prop\nh : DecidablePred p\ns : Set \u03b9\nt\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (f \u2208 pi s fun i => if p i then t\u2081 i else t\u2082 i) \u2194 f \u2208 pi {i | i \u2208 s \u2227 p i} t\u2081 \u2229 pi {i | i \u2208 s \u2227 \u00acp i} t\u2082\n[PROOFSTEP]\nrefine' \u27e8fun h => _, _\u27e9\n[GOAL]\ncase h.refine'_1\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081\u271d t\u2082\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\np : \u03b9 \u2192 Prop\nh\u271d : DecidablePred p\ns : Set \u03b9\nt\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\nh : f \u2208 pi s fun i => if p i then t\u2081 i else t\u2082 i\n\u22a2 f \u2208 pi {i | i \u2208 s \u2227 p i} t\u2081 \u2229 pi {i | i \u2208 s \u2227 \u00acp i} t\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.refine'_1.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081\u271d t\u2082\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\np : \u03b9 \u2192 Prop\nh\u271d : DecidablePred p\ns : Set \u03b9\nt\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\nh : f \u2208 pi s fun i => if p i then t\u2081 i else t\u2082 i\n\u22a2 f \u2208 pi {i | i \u2208 s \u2227 p i} t\u2081\n[PROOFSTEP]\nrintro i \u27e8his, hpi\u27e9\n[GOAL]\ncase h.refine'_1.left.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081\u271d t\u2082\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\nh\u271d : DecidablePred p\ns : Set \u03b9\nt\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\nh : f \u2208 pi s fun i => if p i then t\u2081 i else t\u2082 i\ni : \u03b9\nhis : i \u2208 s\nhpi : p i\n\u22a2 f i \u2208 t\u2081 i\n[PROOFSTEP]\nsimpa [*] using h i\n[GOAL]\ncase h.refine'_1.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081\u271d t\u2082\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\np : \u03b9 \u2192 Prop\nh\u271d : DecidablePred p\ns : Set \u03b9\nt\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\nh : f \u2208 pi s fun i => if p i then t\u2081 i else t\u2082 i\n\u22a2 f \u2208 pi {i | i \u2208 s \u2227 \u00acp i} t\u2082\n[PROOFSTEP]\nrintro i \u27e8his, hpi\u27e9\n[GOAL]\ncase h.refine'_1.right.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081\u271d t\u2082\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\nh\u271d : DecidablePred p\ns : Set \u03b9\nt\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\nh : f \u2208 pi s fun i => if p i then t\u2081 i else t\u2082 i\ni : \u03b9\nhis : i \u2208 s\nhpi : \u00acp i\n\u22a2 f i \u2208 t\u2082 i\n[PROOFSTEP]\nsimpa [*] using h i\n[GOAL]\ncase h.refine'_2\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081\u271d t\u2082\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\np : \u03b9 \u2192 Prop\nh : DecidablePred p\ns : Set \u03b9\nt\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 f \u2208 pi {i | i \u2208 s \u2227 p i} t\u2081 \u2229 pi {i | i \u2208 s \u2227 \u00acp i} t\u2082 \u2192 f \u2208 pi s fun i => if p i then t\u2081 i else t\u2082 i\n[PROOFSTEP]\nrintro \u27e8ht\u2081, ht\u2082\u27e9 i his\n[GOAL]\ncase h.refine'_2.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081\u271d t\u2082\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\nh : DecidablePred p\ns : Set \u03b9\nt\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\nht\u2081 : f \u2208 pi {i | i \u2208 s \u2227 p i} t\u2081\nht\u2082 : f \u2208 pi {i | i \u2208 s \u2227 \u00acp i} t\u2082\ni : \u03b9\nhis : i \u2208 s\n\u22a2 f i \u2208 (fun i => if p i then t\u2081 i else t\u2082 i) i\n[PROOFSTEP]\nby_cases p i\n[GOAL]\ncase h.refine'_2.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081\u271d t\u2082\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\nh : DecidablePred p\ns : Set \u03b9\nt\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\nht\u2081 : f \u2208 pi {i | i \u2208 s \u2227 p i} t\u2081\nht\u2082 : f \u2208 pi {i | i \u2208 s \u2227 \u00acp i} t\u2082\ni : \u03b9\nhis : i \u2208 s\n\u22a2 f i \u2208 (fun i => if p i then t\u2081 i else t\u2082 i) i\n[PROOFSTEP]\nby_cases p i\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081\u271d t\u2082\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\nh\u271d : DecidablePred p\ns : Set \u03b9\nt\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\nht\u2081 : f \u2208 pi {i | i \u2208 s \u2227 p i} t\u2081\nht\u2082 : f \u2208 pi {i | i \u2208 s \u2227 \u00acp i} t\u2082\ni : \u03b9\nhis : i \u2208 s\nh : p i\n\u22a2 f i \u2208 (fun i => if p i then t\u2081 i else t\u2082 i) i\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081\u271d t\u2082\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\nh\u271d : DecidablePred p\ns : Set \u03b9\nt\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\nht\u2081 : f \u2208 pi {i | i \u2208 s \u2227 p i} t\u2081\nht\u2082 : f \u2208 pi {i | i \u2208 s \u2227 \u00acp i} t\u2082\ni : \u03b9\nhis : i \u2208 s\nh : \u00acp i\n\u22a2 f i \u2208 (fun i => if p i then t\u2081 i else t\u2082 i) i\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 pi (s\u2081 \u222a s\u2082) t = pi s\u2081 t \u2229 pi s\u2082 t\n[PROOFSTEP]\nsimp [pi, or_imp, forall_and, setOf_and]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ns : Set \u03b9\n\u22a2 pi s t \u2229 pi s\u1d9c t = pi univ t\n[PROOFSTEP]\nrw [\u2190 union_pi, union_compl_self]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nhi : \u00aci \u2208 s\nf : (j : \u03b9) \u2192 \u03b1 j\na : \u03b1 i\nt : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j)\nj : \u03b9\nhj : j \u2208 s\n\u22a2 t j (update f i a j) = t j (f j)\n[PROOFSTEP]\nrw [update_noteq]\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nhi : \u00aci \u2208 s\nf : (j : \u03b9) \u2192 \u03b1 j\na : \u03b1 i\nt : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j)\nj : \u03b9\nhj : j \u2208 s\n\u22a2 j \u2260 i\n[PROOFSTEP]\nexact fun h => hi (h \u25b8 hj)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\nf : (j : \u03b9) \u2192 \u03b1 j\na : \u03b1 i\nt : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j)\n\u22a2 (pi s fun j => t j (update f i a j)) = pi ({i} \u222a s \\ {i}) fun j => t j (update f i a j)\n[PROOFSTEP]\nrw [union_diff_self, union_eq_self_of_subset_left (singleton_subset_iff.2 hi)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\nf : (j : \u03b9) \u2192 \u03b1 j\na : \u03b1 i\nt : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j)\n\u22a2 (pi ({i} \u222a s \\ {i}) fun j => t j (update f i a j)) = {x | x i \u2208 t i a} \u2229 pi (s \\ {i}) fun j => t j (f j)\n[PROOFSTEP]\nrw [union_pi, singleton_pi', update_same, pi_update_of_not_mem]\n[GOAL]\ncase hi\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\nf : (j : \u03b9) \u2192 \u03b1 j\na : \u03b1 i\nt : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j)\n\u22a2 \u00aci \u2208 s \\ {i}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2\u271d : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\ninst\u271d : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type u_4\ni : \u03b9\nf : (j : \u03b9) \u2192 \u03b1 j\na : \u03b1 i\nt : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j)\n\u22a2 (pi univ fun j => t j (update f i a j)) = {x | x i \u2208 t i a} \u2229 pi {i}\u1d9c fun j => t j (f j)\n[PROOFSTEP]\nrw [compl_eq_univ_diff, \u2190 pi_update_of_mem (mem_univ _)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\ns : Set (\u03b1 i)\n\u22a2 pi univ (update (fun j => univ) i s) = eval i \u207b\u00b9' s\n[PROOFSTEP]\nrw [univ_pi_update i (fun j => (univ : Set (\u03b1 j))) s fun j t => t, pi_univ, inter_univ, preimage]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\nht : Set.Nonempty (pi s t)\ni : \u03b9\n\u22a2 t i \u2286 eval i '' pi s t\n[PROOFSTEP]\nclassical\nobtain \u27e8f, hf\u27e9 := ht\nrefine' fun y hy => \u27e8update f i y, fun j hj => _, update_same _ _ _\u27e9\nobtain rfl | hji := eq_or_ne j i <;> simp [*, hf _ hj]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\nht : Set.Nonempty (pi s t)\ni : \u03b9\n\u22a2 t i \u2286 eval i '' pi s t\n[PROOFSTEP]\nobtain \u27e8f, hf\u27e9 := ht\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d i : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nhf : f \u2208 pi s t\n\u22a2 t i \u2286 eval i '' pi s t\n[PROOFSTEP]\nrefine' fun y hy => \u27e8update f i y, fun j hj => _, update_same _ _ _\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d i : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nhf : f \u2208 pi s t\ny : \u03b1 i\nhy : y \u2208 t i\nj : \u03b9\nhj : j \u2208 s\n\u22a2 update f i y j \u2208 t j\n[PROOFSTEP]\nobtain rfl | hji := eq_or_ne j i\n[GOAL]\ncase intro.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nhf : f \u2208 pi s t\nj : \u03b9\nhj : j \u2208 s\ny : \u03b1 j\nhy : y \u2208 t j\n\u22a2 update f j y j \u2208 t j\n[PROOFSTEP]\nsimp [*, hf _ hj]\n[GOAL]\ncase intro.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d i : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nhf : f \u2208 pi s t\ny : \u03b1 i\nhy : y \u2208 t i\nj : \u03b9\nhj : j \u2208 s\nhji : j \u2260 i\n\u22a2 update f i y j \u2208 t j\n[PROOFSTEP]\nsimp [*, hf _ hj]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 pi s t\u2081 \u2286 pi s t\u2082 \u2194 (\u2200 (i : \u03b9), i \u2208 s \u2192 t\u2081 i \u2286 t\u2082 i) \u2228 pi s t\u2081 = \u2205\n[PROOFSTEP]\nrefine' \u27e8fun h => or_iff_not_imp_right.2 _, fun h => h.elim pi_mono fun h' => h'.symm \u25b8 empty_subset _\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nh : pi s t\u2081 \u2286 pi s t\u2082\n\u22a2 \u00acpi s t\u2081 = \u2205 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 t\u2081 i \u2286 t\u2082 i\n[PROOFSTEP]\nrw [\u2190 Ne.def, \u2190 nonempty_iff_ne_empty]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nh : pi s t\u2081 \u2286 pi s t\u2082\n\u22a2 Set.Nonempty (pi s t\u2081) \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 t\u2081 i \u2286 t\u2082 i\n[PROOFSTEP]\nintro hne i hi\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\nh : pi s t\u2081 \u2286 pi s t\u2082\nhne : Set.Nonempty (pi s t\u2081)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 t\u2081 i \u2286 t\u2082 i\n[PROOFSTEP]\nsimpa only [eval_image_pi hi hne, eval_image_pi hi (hne.mono h)] using image_subset (fun f : \u2200 i, \u03b1 i => f i) h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 pi univ t\u2081 \u2286 pi univ t\u2082 \u2194 (\u2200 (i : \u03b9), t\u2081 i \u2286 t\u2082 i) \u2228 \u2203 i, t\u2081 i = \u2205\n[PROOFSTEP]\nsimp [pi_subset_pi_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\ns : Set (\u03b1 i)\n\u22a2 eval i \u207b\u00b9' s = pi univ (update (fun i => univ) i s)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\ns : Set (\u03b1 i)\nx : (x : \u03b9) \u2192 \u03b1 x\n\u22a2 x \u2208 eval i \u207b\u00b9' s \u2194 x \u2208 pi univ (update (fun i => univ) i s)\n[PROOFSTEP]\nsimp [@forall_update_iff _ (fun i => Set (\u03b1 i)) _ _ _ _ fun i' y => x i' \u2208 y]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\ns : Set (\u03b1 i)\n\u22a2 eval i \u207b\u00b9' s = pi {i} (update (fun i => univ) i s)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\ns : Set (\u03b1 i)\nx\u271d : (x : \u03b9) \u2192 \u03b1 x\n\u22a2 x\u271d \u2208 eval i \u207b\u00b9' s \u2194 x\u271d \u2208 pi {i} (update (fun i => univ) i s)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nhi : i \u2208 s\nhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 f j \u2208 t j\n\u22a2 update f i \u207b\u00b9' pi s t = t i\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nhi : i \u2208 s\nhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 f j \u2208 t j\nx : \u03b1 i\n\u22a2 x \u2208 update f i \u207b\u00b9' pi s t \u2194 x \u2208 t i\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun hx j hj => _\u27e9\n[GOAL]\ncase h.refine'_1\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nhi : i \u2208 s\nhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 f j \u2208 t j\nx : \u03b1 i\nh : x \u2208 update f i \u207b\u00b9' pi s t\n\u22a2 x \u2208 t i\n[PROOFSTEP]\nconvert h i hi\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nhi : i \u2208 s\nhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 f j \u2208 t j\nx : \u03b1 i\nh : x \u2208 update f i \u207b\u00b9' pi s t\n\u22a2 x = update f i x i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.refine'_2\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nhi : i \u2208 s\nhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 f j \u2208 t j\nx : \u03b1 i\nhx : x \u2208 t i\nj : \u03b9\nhj : j \u2208 s\n\u22a2 update f i x j \u2208 t j\n[PROOFSTEP]\nobtain rfl | h := eq_or_ne j i\n[GOAL]\ncase h.refine'_2.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : DecidableEq \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nj : \u03b9\nhj hi : j \u2208 s\nhf : \u2200 (j_1 : \u03b9), j_1 \u2208 s \u2192 j_1 \u2260 j \u2192 f j_1 \u2208 t j_1\nx : \u03b1 j\nhx : x \u2208 t j\n\u22a2 update f j x j \u2208 t j\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase h.refine'_2.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nhi : i \u2208 s\nhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 f j \u2208 t j\nx : \u03b1 i\nhx : x \u2208 t i\nj : \u03b9\nhj : j \u2208 s\nh : j \u2260 i\n\u22a2 update f i x j \u2208 t j\n[PROOFSTEP]\nrw [update_noteq h]\n[GOAL]\ncase h.refine'_2.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i\nhi : i \u2208 s\nhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 f j \u2208 t j\nx : \u03b1 i\nhx : x \u2208 t i\nj : \u03b9\nhj : j \u2208 s\nh : j \u2260 i\n\u22a2 f j \u2208 t j\n[PROOFSTEP]\nexact hf j hj h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ns : Set \u03b9\ninst\u271d : DecidablePred fun x => x \u2208 s\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 (pi univ fun i => if i \u2208 s then t i else univ) = pi s t\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ns : Set \u03b9\ninst\u271d : DecidablePred fun x => x \u2208 s\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (x\u271d \u2208 pi univ fun i => if i \u2208 s then t i else univ) \u2194 x\u271d \u2208 pi s t\n[PROOFSTEP]\nsimp_rw [mem_univ_pi]\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ns : Set \u03b9\ninst\u271d : DecidablePred fun x => x \u2208 s\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (\u2200 (i : \u03b9), x\u271d i \u2208 if i \u2208 s then t i else univ) \u2194 x\u271d \u2208 pi s t\n[PROOFSTEP]\nrefine' forall_congr' fun i => _\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\ns : Set \u03b9\ninst\u271d : DecidablePred fun x => x \u2208 s\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\n\u22a2 (x\u271d i \u2208 if i \u2208 s then t i else univ) \u2194 i \u2208 s \u2192 x\u271d i \u2208 t i\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\ns : Set \u03b9\ninst\u271d : DecidablePred fun x => x \u2208 s\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\nh : i \u2208 s\n\u22a2 x\u271d i \u2208 t i \u2194 i \u2208 s \u2192 x\u271d i \u2208 t i\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\ns : Set \u03b9\ninst\u271d : DecidablePred fun x => x \u2208 s\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\nh : \u00aci \u2208 s\n\u22a2 x\u271d i \u2208 univ \u2194 i \u2208 s \u2192 x\u271d i \u2208 t i\n[PROOFSTEP]\nsimp [h]\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Prod", "llama_tokens": 32303, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4358806366777909}} {"text": "[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : StarRing R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : StarModule R A\nS\u2081 S\u2082 : StarSubalgebra R A\nh : S\u2081 \u2264 S\u2082\nhS\u2081 : IsClosed \u2191S\u2081\nsrc\u271d : Embedding \u2191(inclusion h) := embedding_inclusion h\n\u22a2 Subtype.val \u207b\u00b9' \u2191S\u2081 = range \u2191(inclusion h)\n[PROOFSTEP]\nconvert (Set.range_subtype_map id _).symm\n[GOAL]\ncase h.e'_2.h.e'_4\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : StarRing R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : StarModule R A\nS\u2081 S\u2082 : StarSubalgebra R A\nh : S\u2081 \u2264 S\u2082\nhS\u2081 : IsClosed \u2191S\u2081\nsrc\u271d : Embedding \u2191(inclusion h) := embedding_inclusion h\n\u22a2 \u2191S\u2081 = id '' {x | x \u2208 S\u2081}\n[PROOFSTEP]\nrw [Set.image_id]\n[GOAL]\ncase h.e'_2.h.e'_4\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : StarRing R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : StarModule R A\nS\u2081 S\u2082 : StarSubalgebra R A\nh : S\u2081 \u2264 S\u2082\nhS\u2081 : IsClosed \u2191S\u2081\nsrc\u271d : Embedding \u2191(inclusion h) := embedding_inclusion h\n\u22a2 \u2191S\u2081 = {x | x \u2208 S\u2081}\n[PROOFSTEP]\nrfl\n[GOAL]\ncase convert_4\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : StarRing R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : StarModule R A\nS\u2081 S\u2082 : StarSubalgebra R A\nh : S\u2081 \u2264 S\u2082\nhS\u2081 : IsClosed \u2191S\u2081\nsrc\u271d : Embedding \u2191(inclusion h) := embedding_inclusion h\n\u22a2 \u2200 (x : A), x \u2208 S\u2081 \u2192 id x \u2208 S\u2082\n[PROOFSTEP]\nintro _ h'\n[GOAL]\ncase convert_4\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : StarRing R\ninst\u271d\u2074 : TopologicalSpace A\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : StarModule R A\nS\u2081 S\u2082 : StarSubalgebra R A\nh : S\u2081 \u2264 S\u2082\nhS\u2081 : IsClosed \u2191S\u2081\nsrc\u271d : Embedding \u2191(inclusion h) := embedding_inclusion h\nx\u271d : A\nh' : x\u271d \u2208 S\u2081\n\u22a2 id x\u271d \u2208 S\u2082\n[PROOFSTEP]\napply h h'\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : Algebra R A\ninst\u271d\u2077 : StarRing A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSemiring A\ninst\u271d\u2074 : ContinuousStar A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : StarRing B\ns : Subalgebra R A\n\u22a2 Subalgebra.topologicalClosure (star s) = star (Subalgebra.topologicalClosure s)\n[PROOFSTEP]\nsuffices \u2200 t : Subalgebra R A, (star t).topologicalClosure \u2264 star t.topologicalClosure from\n le_antisymm (this s) (by simpa only [star_star] using Subalgebra.star_mono (this (star s)))\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : Algebra R A\ninst\u271d\u2077 : StarRing A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSemiring A\ninst\u271d\u2074 : ContinuousStar A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : StarRing B\ns : Subalgebra R A\nthis : \u2200 (t : Subalgebra R A), Subalgebra.topologicalClosure (star t) \u2264 star (Subalgebra.topologicalClosure t)\n\u22a2 star (Subalgebra.topologicalClosure s) \u2264 Subalgebra.topologicalClosure (star s)\n[PROOFSTEP]\nsimpa only [star_star] using Subalgebra.star_mono (this (star s))\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : Algebra R A\ninst\u271d\u2077 : StarRing A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSemiring A\ninst\u271d\u2074 : ContinuousStar A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : StarRing B\ns : Subalgebra R A\n\u22a2 \u2200 (t : Subalgebra R A), Subalgebra.topologicalClosure (star t) \u2264 star (Subalgebra.topologicalClosure t)\n[PROOFSTEP]\nexact fun t =>\n (star t).topologicalClosure_minimal (Subalgebra.star_mono subset_closure) (isClosed_closure.preimage continuous_star)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\n\u22a2 \u03c6 = \u03c8\n[PROOFSTEP]\nrw [FunLike.ext'_iff]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\n\u22a2 \u2191\u03c6 = \u2191\u03c8\n[PROOFSTEP]\nhave : Dense (Set.range <| inclusion (le_topologicalClosure S)) :=\n by\n refine' embedding_subtype_val.toInducing.dense_iff.2 fun x => _\n convert show \u2191x \u2208 closure (S : Set A) from x.prop\n rw [\u2190 Set.range_comp]\n exact\n Set.ext fun y =>\n \u27e8by\n rintro \u27e8y, rfl\u27e9\n exact y.prop, fun hy => \u27e8\u27e8y, hy\u27e9, rfl\u27e9\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\n\u22a2 Dense (range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))\n[PROOFSTEP]\nrefine' embedding_subtype_val.toInducing.dense_iff.2 fun x => _\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nx : { x // x \u2208 topologicalClosure S }\n\u22a2 \u2191x \u2208 closure (Subtype.val '' range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))\n[PROOFSTEP]\nconvert show \u2191x \u2208 closure (S : Set A) from x.prop\n[GOAL]\ncase h.e'_5.h.e'_3\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nx : { x // x \u2208 topologicalClosure S }\n\u22a2 Subtype.val '' range \u2191(inclusion (_ : S \u2264 topologicalClosure S)) = \u2191S\n[PROOFSTEP]\nrw [\u2190 Set.range_comp]\n[GOAL]\ncase h.e'_5.h.e'_3\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nx : { x // x \u2208 topologicalClosure S }\n\u22a2 range (Subtype.val \u2218 \u2191(inclusion (_ : S \u2264 topologicalClosure S))) = \u2191S\n[PROOFSTEP]\nexact\n Set.ext fun y =>\n \u27e8by\n rintro \u27e8y, rfl\u27e9\n exact y.prop, fun hy => \u27e8\u27e8y, hy\u27e9, rfl\u27e9\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nx : { x // x \u2208 topologicalClosure S }\ny : A\n\u22a2 y \u2208 range (Subtype.val \u2218 \u2191(inclusion (_ : S \u2264 topologicalClosure S))) \u2192 y \u2208 \u2191S\n[PROOFSTEP]\nrintro \u27e8y, rfl\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nx : { x // x \u2208 topologicalClosure S }\ny : { x // x \u2208 S }\n\u22a2 (Subtype.val \u2218 \u2191(inclusion (_ : S \u2264 topologicalClosure S))) y \u2208 \u2191S\n[PROOFSTEP]\nexact y.prop\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nthis : Dense (range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))\n\u22a2 \u2191\u03c6 = \u2191\u03c8\n[PROOFSTEP]\nrefine' Continuous.ext_on this h\u03c6 h\u03c8 _\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nthis : Dense (range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))\n\u22a2 EqOn (\u2191\u03c6) (\u2191\u03c8) (range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))\n[PROOFSTEP]\nrintro _ \u27e8x, rfl\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nthis : Dense (range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))\nx : { x // x \u2208 S }\n\u22a2 \u2191\u03c6 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x) = \u2191\u03c8 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x)\n[PROOFSTEP]\nsimpa only using FunLike.congr_fun h x\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : Algebra R A\ninst\u271d\u2079 : StarRing A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSemiring A\ninst\u271d\u2076 : ContinuousStar A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : StarRing B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\nS : StarSubalgebra R A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 topologicalClosure S } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2200 (x : { x // x \u2208 S }),\n \u2191\u03c6 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x) = \u2191\u03c8 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x)\n\u22a2 \u03c6 = \u03c8\n[PROOFSTEP]\nhave : (\u03c6 : S.topologicalClosure \u2192\u22c6\u2090[R] B) = (\u03c8 : S.topologicalClosure \u2192\u22c6\u2090[R] B) :=\n by\n refine StarAlgHom.ext_topologicalClosure (R := R) (A := A) (B := B) h\u03c6 h\u03c8 (StarAlgHom.ext ?_)\n simpa only [StarAlgHom.coe_comp, StarAlgHom.coe_coe] using h\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : Algebra R A\ninst\u271d\u2079 : StarRing A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSemiring A\ninst\u271d\u2076 : ContinuousStar A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : StarRing B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\nS : StarSubalgebra R A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 topologicalClosure S } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2200 (x : { x // x \u2208 S }),\n \u2191\u03c6 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x) = \u2191\u03c8 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x)\n\u22a2 \u2191\u03c6 = \u2191\u03c8\n[PROOFSTEP]\nrefine StarAlgHom.ext_topologicalClosure (R := R) (A := A) (B := B) h\u03c6 h\u03c8 (StarAlgHom.ext ?_)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : Algebra R A\ninst\u271d\u2079 : StarRing A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSemiring A\ninst\u271d\u2076 : ContinuousStar A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : StarRing B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\nS : StarSubalgebra R A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 topologicalClosure S } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2200 (x : { x // x \u2208 S }),\n \u2191\u03c6 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x) = \u2191\u03c8 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x)\n\u22a2 \u2200 (x : { x // x \u2208 S }),\n \u2191(StarAlgHom.comp (\u2191\u03c6) (inclusion (_ : S \u2264 topologicalClosure S))) x =\n \u2191(StarAlgHom.comp (\u2191\u03c8) (inclusion (_ : S \u2264 topologicalClosure S))) x\n[PROOFSTEP]\nsimpa only [StarAlgHom.coe_comp, StarAlgHom.coe_coe] using h\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : Algebra R A\ninst\u271d\u2079 : StarRing A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSemiring A\ninst\u271d\u2076 : ContinuousStar A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : StarRing B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\nS : StarSubalgebra R A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 topologicalClosure S } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2200 (x : { x // x \u2208 S }),\n \u2191\u03c6 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x) = \u2191\u03c8 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x)\nthis : \u2191\u03c6 = \u2191\u03c8\n\u22a2 \u03c6 = \u03c8\n[PROOFSTEP]\nrw [FunLike.ext'_iff, \u2190 StarAlgHom.coe_coe]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : Algebra R A\ninst\u271d\u2079 : StarRing A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSemiring A\ninst\u271d\u2076 : ContinuousStar A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : StarRing B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\nS : StarSubalgebra R A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 topologicalClosure S } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2200 (x : { x // x \u2208 S }),\n \u2191\u03c6 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x) = \u2191\u03c8 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x)\nthis : \u2191\u03c6 = \u2191\u03c8\n\u22a2 \u2191\u2191\u03c6 = \u2191\u03c8\n[PROOFSTEP]\napply congrArg _ this\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : TopologicalSemiring A\ninst\u271d\u2076 : ContinuousStar A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : Algebra R B\nx : A\n\u22a2 IsClosed (range Subtype.val)\n[PROOFSTEP]\nconvert elementalStarAlgebra.isClosed R x\n[GOAL]\ncase h.e'_3\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : TopologicalSemiring A\ninst\u271d\u2076 : ContinuousStar A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : Algebra R B\nx : A\n\u22a2 range Subtype.val = \u2191(elementalStarAlgebra R x)\n[PROOFSTEP]\nexact\n Set.ext fun y =>\n \u27e8by\n rintro \u27e8y, rfl\u27e9\n exact y.prop, fun hy => \u27e8\u27e8y, hy\u27e9, rfl\u27e9\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : TopologicalSemiring A\ninst\u271d\u2076 : ContinuousStar A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : Algebra R B\nx y : A\n\u22a2 y \u2208 range Subtype.val \u2192 y \u2208 \u2191(elementalStarAlgebra R x)\n[PROOFSTEP]\nrintro \u27e8y, rfl\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : TopologicalSemiring A\ninst\u271d\u2076 : ContinuousStar A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : Algebra R B\nx : A\ny : { x_1 // x_1 \u2208 \u2191(elementalStarAlgebra R x) }\n\u22a2 \u2191y \u2208 \u2191(elementalStarAlgebra R x)\n[PROOFSTEP]\nexact y.prop\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : StarRing A\ninst\u271d\u2079 : TopologicalSemiring A\ninst\u271d\u2078 : ContinuousStar A\ninst\u271d\u2077 : Algebra R A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : StarRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\na : A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 elementalStarAlgebra R a } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2191\u03c6 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) } =\n \u2191\u03c8 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) }\n\u22a2 \u03c6 = \u03c8\n[PROOFSTEP]\nrefine StarAlgHomClass.ext_topologicalClosure h\u03c6 h\u03c8 fun x => ?_\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : StarRing A\ninst\u271d\u2079 : TopologicalSemiring A\ninst\u271d\u2078 : ContinuousStar A\ninst\u271d\u2077 : Algebra R A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : StarRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\na : A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 elementalStarAlgebra R a } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2191\u03c6 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) } =\n \u2191\u03c8 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) }\nx : { x // x \u2208 adjoin R {a} }\n\u22a2 \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) x) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) x)\n[PROOFSTEP]\napply adjoin_induction' x ?_ ?_ ?_ ?_ ?_\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : StarRing A\ninst\u271d\u2079 : TopologicalSemiring A\ninst\u271d\u2078 : ContinuousStar A\ninst\u271d\u2077 : Algebra R A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : StarRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\na : A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 elementalStarAlgebra R a } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2191\u03c6 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) } =\n \u2191\u03c8 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) }\nx : { x // x \u2208 adjoin R {a} }\n\u22a2 \u2200 (x : A) (h : x \u2208 {a}),\n \u2191\u03c6\n (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a})))\n { val := x, property := (_ : x \u2208 \u2191(adjoin R {a})) }) =\n \u2191\u03c8\n (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a})))\n { val := x, property := (_ : x \u2208 \u2191(adjoin R {a})) })\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : StarRing A\ninst\u271d\u2079 : TopologicalSemiring A\ninst\u271d\u2078 : ContinuousStar A\ninst\u271d\u2077 : Algebra R A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : StarRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\na : A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 elementalStarAlgebra R a } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2191\u03c6 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) } =\n \u2191\u03c8 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) }\nx : { x // x \u2208 adjoin R {a} }\n\u22a2 \u2200 (r : R),\n \u2191\u03c6\n (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a})))\n (\u2191(algebraMap R { x // x \u2208 adjoin R {a} }) r)) =\n \u2191\u03c8\n (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a})))\n (\u2191(algebraMap R { x // x \u2208 adjoin R {a} }) r))\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : StarRing A\ninst\u271d\u2079 : TopologicalSemiring A\ninst\u271d\u2078 : ContinuousStar A\ninst\u271d\u2077 : Algebra R A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : StarRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\na : A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 elementalStarAlgebra R a } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2191\u03c6 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) } =\n \u2191\u03c8 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) }\nx : { x // x \u2208 adjoin R {a} }\n\u22a2 \u2200 (x y : { x // x \u2208 adjoin R {a} }),\n \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) x) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) x) \u2192\n \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) y) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) y) \u2192\n \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) (x + y)) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) (x + y))\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : StarRing A\ninst\u271d\u2079 : TopologicalSemiring A\ninst\u271d\u2078 : ContinuousStar A\ninst\u271d\u2077 : Algebra R A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : StarRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\na : A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 elementalStarAlgebra R a } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2191\u03c6 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) } =\n \u2191\u03c8 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) }\nx : { x // x \u2208 adjoin R {a} }\n\u22a2 \u2200 (x y : { x // x \u2208 adjoin R {a} }),\n \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) x) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) x) \u2192\n \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) y) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) y) \u2192\n \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) (x * y)) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) (x * y))\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : StarRing A\ninst\u271d\u2079 : TopologicalSemiring A\ninst\u271d\u2078 : ContinuousStar A\ninst\u271d\u2077 : Algebra R A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : StarRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\na : A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 elementalStarAlgebra R a } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2191\u03c6 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) } =\n \u2191\u03c8 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) }\nx : { x // x \u2208 adjoin R {a} }\n\u22a2 \u2200 (x : { x // x \u2208 adjoin R {a} }),\n \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) x) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) x) \u2192\n \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) (star x)) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) (star x))\n[PROOFSTEP]\nexacts [fun y hy => by simpa only [Set.mem_singleton_iff.mp hy] using h, fun r => by simp only [AlgHomClass.commutes],\n fun x y hx hy => by simp only [map_add, hx, hy], fun x y hx hy => by simp only [map_mul, hx, hy], fun x hx => by\n simp only [map_star, hx]]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : StarRing A\ninst\u271d\u2079 : TopologicalSemiring A\ninst\u271d\u2078 : ContinuousStar A\ninst\u271d\u2077 : Algebra R A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : StarRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\na : A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 elementalStarAlgebra R a } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2191\u03c6 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) } =\n \u2191\u03c8 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) }\nx : { x // x \u2208 adjoin R {a} }\ny : A\nhy : y \u2208 {a}\n\u22a2 \u2191\u03c6\n (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a})))\n { val := y, property := (_ : y \u2208 \u2191(adjoin R {a})) }) =\n \u2191\u03c8\n (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a})))\n { val := y, property := (_ : y \u2208 \u2191(adjoin R {a})) })\n[PROOFSTEP]\nsimpa only [Set.mem_singleton_iff.mp hy] using h\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : StarRing A\ninst\u271d\u2079 : TopologicalSemiring A\ninst\u271d\u2078 : ContinuousStar A\ninst\u271d\u2077 : Algebra R A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : StarRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\na : A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 elementalStarAlgebra R a } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2191\u03c6 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) } =\n \u2191\u03c8 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) }\nx : { x // x \u2208 adjoin R {a} }\nr : R\n\u22a2 \u2191\u03c6\n (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a})))\n (\u2191(algebraMap R { x // x \u2208 adjoin R {a} }) r)) =\n \u2191\u03c8\n (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a})))\n (\u2191(algebraMap R { x // x \u2208 adjoin R {a} }) r))\n[PROOFSTEP]\nsimp only [AlgHomClass.commutes]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : StarRing A\ninst\u271d\u2079 : TopologicalSemiring A\ninst\u271d\u2078 : ContinuousStar A\ninst\u271d\u2077 : Algebra R A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : StarRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\na : A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 elementalStarAlgebra R a } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2191\u03c6 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) } =\n \u2191\u03c8 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) }\nx\u271d x y : { x // x \u2208 adjoin R {a} }\nhx :\n \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) x) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) x)\nhy :\n \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) y) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) y)\n\u22a2 \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) (x + y)) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) (x + y))\n[PROOFSTEP]\nsimp only [map_add, hx, hy]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : StarRing A\ninst\u271d\u2079 : TopologicalSemiring A\ninst\u271d\u2078 : ContinuousStar A\ninst\u271d\u2077 : Algebra R A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : StarRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\na : A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 elementalStarAlgebra R a } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2191\u03c6 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) } =\n \u2191\u03c8 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) }\nx\u271d x y : { x // x \u2208 adjoin R {a} }\nhx :\n \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) x) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) x)\nhy :\n \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) y) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) y)\n\u22a2 \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) (x * y)) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) (x * y))\n[PROOFSTEP]\nsimp only [map_mul, hx, hy]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring R\ninst\u271d\u00b9\u00b3 : StarRing R\ninst\u271d\u00b9\u00b2 : TopologicalSpace A\ninst\u271d\u00b9\u00b9 : Semiring A\ninst\u271d\u00b9\u2070 : StarRing A\ninst\u271d\u2079 : TopologicalSemiring A\ninst\u271d\u2078 : ContinuousStar A\ninst\u271d\u2077 : Algebra R A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : StarRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : T2Space B\nF : Type u_4\na : A\ninst\u271d : StarAlgHomClass F R { x // x \u2208 elementalStarAlgebra R a } B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n \u2191\u03c6 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) } =\n \u2191\u03c8 { val := a, property := (_ : a \u2208 elementalStarAlgebra R a) }\nx\u271d x : { x // x \u2208 adjoin R {a} }\nhx :\n \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) x) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) x)\n\u22a2 \u2191\u03c6 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) (star x)) =\n \u2191\u03c8 (\u2191(StarSubalgebra.inclusion (_ : adjoin R {a} \u2264 topologicalClosure (adjoin R {a}))) (star x))\n[PROOFSTEP]\nsimp only [map_star, hx]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.StarSubalgebra", "llama_tokens": 14729, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743505760728, "lm_q2_score": 0.5888891307678321, "lm_q1q2_score": 0.4358806299273882}} {"text": "[GOAL]\nR : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type u_4\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup P\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R P\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : IsArtinian R M\ninst\u271d : IsArtinian R P\nf : M \u2192\u2097[R] N\ng : N \u2192\u2097[R] P\nhf : Function.Injective \u2191f\nhg : Function.Surjective \u2191g\nh : LinearMap.range f = LinearMap.ker g\n\u22a2 \u2200 (a : Submodule R N), Submodule.map f (Submodule.comap f a) = a \u2293 LinearMap.range f\n[PROOFSTEP]\nsimp [Submodule.map_comap_eq, inf_comm]\n[GOAL]\nR : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type u_4\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup P\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R P\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : IsArtinian R M\ninst\u271d : IsArtinian R P\nf : M \u2192\u2097[R] N\ng : N \u2192\u2097[R] P\nhf : Function.Injective \u2191f\nhg : Function.Surjective \u2191g\nh : LinearMap.range f = LinearMap.ker g\n\u22a2 \u2200 (a : Submodule R N), Submodule.comap g (Submodule.map g a) = a \u2294 LinearMap.range f\n[PROOFSTEP]\nsimp [Submodule.comap_map_eq, h]\n[GOAL]\nR\u271d : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type u_4\ninst\u271d\u2077 : Ring R\u271d\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup P\ninst\u271d\u2074 : AddCommGroup N\ninst\u271d\u00b3 : Module R\u271d M\ninst\u271d\u00b2 : Module R\u271d P\ninst\u271d\u00b9 : Module R\u271d N\nR : Type u_5\n\u03b9 : Type u_6\ninst\u271d : Finite \u03b9\n\u22a2 \u2200 {M : \u03b9 \u2192 Type u_7} [inst : Ring R] [inst_1 : (i : \u03b9) \u2192 AddCommGroup (M i)] [inst_2 : (i : \u03b9) \u2192 Module R (M i)]\n [inst_3 : \u2200 (i : \u03b9), IsArtinian R (M i)], IsArtinian R ((i : \u03b9) \u2192 M i)\n[PROOFSTEP]\napply Finite.induction_empty_option _ _ _ \u03b9\n[GOAL]\nR\u271d : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type u_4\ninst\u271d\u2077 : Ring R\u271d\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup P\ninst\u271d\u2074 : AddCommGroup N\ninst\u271d\u00b3 : Module R\u271d M\ninst\u271d\u00b2 : Module R\u271d P\ninst\u271d\u00b9 : Module R\u271d N\nR : Type u_5\n\u03b9 : Type u_6\ninst\u271d : Finite \u03b9\n\u22a2 \u2200 {\u03b1 \u03b2 : Type u_6},\n \u03b1 \u2243 \u03b2 \u2192\n (\u2200 {M : \u03b1 \u2192 Type u_7} [inst : Ring R] [inst_1 : (i : \u03b1) \u2192 AddCommGroup (M i)] [inst_2 : (i : \u03b1) \u2192 Module R (M i)]\n [inst_3 : \u2200 (i : \u03b1), IsArtinian R (M i)], IsArtinian R ((i : \u03b1) \u2192 M i)) \u2192\n \u2200 {M : \u03b2 \u2192 Type u_7} [inst : Ring R] [inst_1 : (i : \u03b2) \u2192 AddCommGroup (M i)] [inst_2 : (i : \u03b2) \u2192 Module R (M i)]\n [inst_3 : \u2200 (i : \u03b2), IsArtinian R (M i)], IsArtinian R ((i : \u03b2) \u2192 M i)\n[PROOFSTEP]\nintro \u03b1 \u03b2 e h\u03b1 M _ _ _ _\n[GOAL]\nR\u271d : Type u_1\nM\u271d : Type u_2\nP : Type u_3\nN : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u271d\ninst\u271d\u00b9\u2070 : AddCommGroup M\u271d\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : Module R\u271d M\u271d\ninst\u271d\u2076 : Module R\u271d P\ninst\u271d\u2075 : Module R\u271d N\nR : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : Finite \u03b9\n\u03b1 \u03b2 : Type u_6\ne : \u03b1 \u2243 \u03b2\nh\u03b1 :\n \u2200 {M : \u03b1 \u2192 Type u_7} [inst : Ring R] [inst_1 : (i : \u03b1) \u2192 AddCommGroup (M i)] [inst_2 : (i : \u03b1) \u2192 Module R (M i)]\n [inst_3 : \u2200 (i : \u03b1), IsArtinian R (M i)], IsArtinian R ((i : \u03b1) \u2192 M i)\nM : \u03b2 \u2192 Type u_7\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : (i : \u03b2) \u2192 AddCommGroup (M i)\ninst\u271d\u00b9 : (i : \u03b2) \u2192 Module R (M i)\ninst\u271d : \u2200 (i : \u03b2), IsArtinian R (M i)\n\u22a2 IsArtinian R ((i : \u03b2) \u2192 M i)\n[PROOFSTEP]\nhave := @h\u03b1\n[GOAL]\nR\u271d : Type u_1\nM\u271d : Type u_2\nP : Type u_3\nN : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u271d\ninst\u271d\u00b9\u2070 : AddCommGroup M\u271d\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : Module R\u271d M\u271d\ninst\u271d\u2076 : Module R\u271d P\ninst\u271d\u2075 : Module R\u271d N\nR : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : Finite \u03b9\n\u03b1 \u03b2 : Type u_6\ne : \u03b1 \u2243 \u03b2\nh\u03b1 :\n \u2200 {M : \u03b1 \u2192 Type u_7} [inst : Ring R] [inst_1 : (i : \u03b1) \u2192 AddCommGroup (M i)] [inst_2 : (i : \u03b1) \u2192 Module R (M i)]\n [inst_3 : \u2200 (i : \u03b1), IsArtinian R (M i)], IsArtinian R ((i : \u03b1) \u2192 M i)\nM : \u03b2 \u2192 Type u_7\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : (i : \u03b2) \u2192 AddCommGroup (M i)\ninst\u271d\u00b9 : (i : \u03b2) \u2192 Module R (M i)\ninst\u271d : \u2200 (i : \u03b2), IsArtinian R (M i)\nthis :\n \u2200 {M : \u03b1 \u2192 Type u_7} [inst : Ring R] [inst_1 : (i : \u03b1) \u2192 AddCommGroup (M i)] [inst_2 : (i : \u03b1) \u2192 Module R (M i)]\n [inst_3 : \u2200 (i : \u03b1), IsArtinian R (M i)], IsArtinian R ((i : \u03b1) \u2192 M i)\n\u22a2 IsArtinian R ((i : \u03b2) \u2192 M i)\n[PROOFSTEP]\nexact isArtinian_of_linearEquiv (LinearEquiv.piCongrLeft R M e)\n[GOAL]\nR\u271d : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type u_4\ninst\u271d\u2077 : Ring R\u271d\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup P\ninst\u271d\u2074 : AddCommGroup N\ninst\u271d\u00b3 : Module R\u271d M\ninst\u271d\u00b2 : Module R\u271d P\ninst\u271d\u00b9 : Module R\u271d N\nR : Type u_5\n\u03b9 : Type u_6\ninst\u271d : Finite \u03b9\n\u22a2 \u2200 {M : PEmpty \u2192 Type u_7} [inst : Ring R] [inst_1 : (i : PEmpty) \u2192 AddCommGroup (M i)]\n [inst_2 : (i : PEmpty) \u2192 Module R (M i)] [inst_3 : \u2200 (i : PEmpty), IsArtinian R (M i)],\n IsArtinian R ((i : PEmpty) \u2192 M i)\n[PROOFSTEP]\nintro M _ _ _ _\n[GOAL]\nR\u271d : Type u_1\nM\u271d : Type u_2\nP : Type u_3\nN : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u271d\ninst\u271d\u00b9\u2070 : AddCommGroup M\u271d\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : Module R\u271d M\u271d\ninst\u271d\u2076 : Module R\u271d P\ninst\u271d\u2075 : Module R\u271d N\nR : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2074 : Finite \u03b9\nM : PEmpty \u2192 Type u_7\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : (i : PEmpty) \u2192 AddCommGroup (M i)\ninst\u271d\u00b9 : (i : PEmpty) \u2192 Module R (M i)\ninst\u271d : \u2200 (i : PEmpty), IsArtinian R (M i)\n\u22a2 IsArtinian R ((i : PEmpty) \u2192 M i)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR\u271d : Type u_1\nM : Type u_2\nP : Type u_3\nN : Type u_4\ninst\u271d\u2077 : Ring R\u271d\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup P\ninst\u271d\u2074 : AddCommGroup N\ninst\u271d\u00b3 : Module R\u271d M\ninst\u271d\u00b2 : Module R\u271d P\ninst\u271d\u00b9 : Module R\u271d N\nR : Type u_5\n\u03b9 : Type u_6\ninst\u271d : Finite \u03b9\n\u22a2 \u2200 {\u03b1 : Type u_6} [inst : Fintype \u03b1],\n (\u2200 {M : \u03b1 \u2192 Type u_7} [inst : Ring R] [inst_1 : (i : \u03b1) \u2192 AddCommGroup (M i)] [inst_2 : (i : \u03b1) \u2192 Module R (M i)]\n [inst_3 : \u2200 (i : \u03b1), IsArtinian R (M i)], IsArtinian R ((i : \u03b1) \u2192 M i)) \u2192\n \u2200 {M : Option \u03b1 \u2192 Type u_7} [inst : Ring R] [inst_1 : (i : Option \u03b1) \u2192 AddCommGroup (M i)]\n [inst_2 : (i : Option \u03b1) \u2192 Module R (M i)] [inst_3 : \u2200 (i : Option \u03b1), IsArtinian R (M i)],\n IsArtinian R ((i : Option \u03b1) \u2192 M i)\n[PROOFSTEP]\nintro \u03b1 _ ih M _ _ _ _\n[GOAL]\nR\u271d : Type u_1\nM\u271d : Type u_2\nP : Type u_3\nN : Type u_4\ninst\u271d\u00b9\u00b2 : Ring R\u271d\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup N\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d P\ninst\u271d\u2076 : Module R\u271d N\nR : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : Finite \u03b9\n\u03b1 : Type u_6\ninst\u271d\u2074 : Fintype \u03b1\nih :\n \u2200 {M : \u03b1 \u2192 Type u_7} [inst : Ring R] [inst_1 : (i : \u03b1) \u2192 AddCommGroup (M i)] [inst_2 : (i : \u03b1) \u2192 Module R (M i)]\n [inst_3 : \u2200 (i : \u03b1), IsArtinian R (M i)], IsArtinian R ((i : \u03b1) \u2192 M i)\nM : Option \u03b1 \u2192 Type u_7\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : (i : Option \u03b1) \u2192 AddCommGroup (M i)\ninst\u271d\u00b9 : (i : Option \u03b1) \u2192 Module R (M i)\ninst\u271d : \u2200 (i : Option \u03b1), IsArtinian R (M i)\n\u22a2 IsArtinian R ((i : Option \u03b1) \u2192 M i)\n[PROOFSTEP]\nhave := @ih\n[GOAL]\nR\u271d : Type u_1\nM\u271d : Type u_2\nP : Type u_3\nN : Type u_4\ninst\u271d\u00b9\u00b2 : Ring R\u271d\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup N\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d P\ninst\u271d\u2076 : Module R\u271d N\nR : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : Finite \u03b9\n\u03b1 : Type u_6\ninst\u271d\u2074 : Fintype \u03b1\nih :\n \u2200 {M : \u03b1 \u2192 Type u_7} [inst : Ring R] [inst_1 : (i : \u03b1) \u2192 AddCommGroup (M i)] [inst_2 : (i : \u03b1) \u2192 Module R (M i)]\n [inst_3 : \u2200 (i : \u03b1), IsArtinian R (M i)], IsArtinian R ((i : \u03b1) \u2192 M i)\nM : Option \u03b1 \u2192 Type u_7\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : (i : Option \u03b1) \u2192 AddCommGroup (M i)\ninst\u271d\u00b9 : (i : Option \u03b1) \u2192 Module R (M i)\ninst\u271d : \u2200 (i : Option \u03b1), IsArtinian R (M i)\nthis :\n \u2200 {M : \u03b1 \u2192 Type u_7} [inst : Ring R] [inst_1 : (i : \u03b1) \u2192 AddCommGroup (M i)] [inst_2 : (i : \u03b1) \u2192 Module R (M i)]\n [inst_3 : \u2200 (i : \u03b1), IsArtinian R (M i)], IsArtinian R ((i : \u03b1) \u2192 M i)\n\u22a2 IsArtinian R ((i : Option \u03b1) \u2192 M i)\n[PROOFSTEP]\nexact isArtinian_of_linearEquiv (LinearEquiv.piOptionEquivProd R).symm\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\n\u22a2 Set.Finite s\n[PROOFSTEP]\nrefine'\n by_contradiction fun hf =>\n (RelEmbedding.wellFounded_iff_no_descending_seq.1 (wellFounded_submodule_lt (R := R) (M := M))).elim' _\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\n\u22a2 (fun x x_1 => x > x_1) \u21aar fun x x_1 => x < x_1\n[PROOFSTEP]\nhave f : \u2115 \u21aa s := Set.Infinite.natEmbedding s hf\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\n\u22a2 (fun x x_1 => x > x_1) \u21aar fun x x_1 => x < x_1\n[PROOFSTEP]\nhave : \u2200 n, (\u2191) \u2218 f '' {m | n \u2264 m} \u2286 s := by\n rintro n x \u27e8y, _, rfl\u27e9\n exact (f y).2\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\n\u22a2 \u2200 (n : \u2115), Subtype.val \u2218 \u2191f '' {m | n \u2264 m} \u2286 s\n[PROOFSTEP]\nrintro n x \u27e8y, _, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\nn y : \u2115\nleft\u271d : y \u2208 {m | n \u2264 m}\n\u22a2 (Subtype.val \u2218 \u2191f) y \u2208 s\n[PROOFSTEP]\nexact (f y).2\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\nthis : \u2200 (n : \u2115), Subtype.val \u2218 \u2191f '' {m | n \u2264 m} \u2286 s\n\u22a2 (fun x x_1 => x > x_1) \u21aar fun x x_1 => x < x_1\n[PROOFSTEP]\nhave : \u2200 a b : \u2115, a \u2264 b \u2194 span R (Subtype.val \u2218 f '' {m | b \u2264 m}) \u2264 span R (Subtype.val \u2218 f '' {m | a \u2264 m}) :=\n by\n intro a b\n rw [span_le_span_iff hs (this b) (this a), Set.image_subset_image_iff (Subtype.coe_injective.comp f.injective),\n Set.subset_def]\n simp only [Set.mem_setOf_eq]\n exact \u27e8fun hab x => le_trans hab, fun h => h _ le_rfl\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\nthis : \u2200 (n : \u2115), Subtype.val \u2218 \u2191f '' {m | n \u2264 m} \u2286 s\n\u22a2 \u2200 (a b : \u2115), a \u2264 b \u2194 span R (Subtype.val \u2218 \u2191f '' {m | b \u2264 m}) \u2264 span R (Subtype.val \u2218 \u2191f '' {m | a \u2264 m})\n[PROOFSTEP]\nintro a b\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\nthis : \u2200 (n : \u2115), Subtype.val \u2218 \u2191f '' {m | n \u2264 m} \u2286 s\na b : \u2115\n\u22a2 a \u2264 b \u2194 span R (Subtype.val \u2218 \u2191f '' {m | b \u2264 m}) \u2264 span R (Subtype.val \u2218 \u2191f '' {m | a \u2264 m})\n[PROOFSTEP]\nrw [span_le_span_iff hs (this b) (this a), Set.image_subset_image_iff (Subtype.coe_injective.comp f.injective),\n Set.subset_def]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\nthis : \u2200 (n : \u2115), Subtype.val \u2218 \u2191f '' {m | n \u2264 m} \u2286 s\na b : \u2115\n\u22a2 a \u2264 b \u2194 \u2200 (x : \u2115), x \u2208 {m | b \u2264 m} \u2192 x \u2208 {m | a \u2264 m}\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\nthis : \u2200 (n : \u2115), Subtype.val \u2218 \u2191f '' {m | n \u2264 m} \u2286 s\na b : \u2115\n\u22a2 a \u2264 b \u2194 \u2200 (x : \u2115), b \u2264 x \u2192 a \u2264 x\n[PROOFSTEP]\nexact \u27e8fun hab x => le_trans hab, fun h => h _ le_rfl\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\nthis\u271d : \u2200 (n : \u2115), Subtype.val \u2218 \u2191f '' {m | n \u2264 m} \u2286 s\nthis : \u2200 (a b : \u2115), a \u2264 b \u2194 span R (Subtype.val \u2218 \u2191f '' {m | b \u2264 m}) \u2264 span R (Subtype.val \u2218 \u2191f '' {m | a \u2264 m})\n\u22a2 (fun x x_1 => x > x_1) \u21aar fun x x_1 => x < x_1\n[PROOFSTEP]\nexact\n \u27e8\u27e8fun n => span R (Subtype.val \u2218 f '' {m | n \u2264 m}), fun x y =>\n by\n rw [le_antisymm_iff, \u2190 this y x, \u2190 this x y]\n exact fun \u27e8h\u2081, h\u2082\u27e9 => le_antisymm_iff.2 \u27e8h\u2082, h\u2081\u27e9\u27e9,\n by\n intro a b\n conv_rhs => rw [GT.gt, lt_iff_le_not_le, this, this, \u2190 lt_iff_le_not_le]\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\nthis\u271d : \u2200 (n : \u2115), Subtype.val \u2218 \u2191f '' {m | n \u2264 m} \u2286 s\nthis : \u2200 (a b : \u2115), a \u2264 b \u2194 span R (Subtype.val \u2218 \u2191f '' {m | b \u2264 m}) \u2264 span R (Subtype.val \u2218 \u2191f '' {m | a \u2264 m})\nx y : \u2115\n\u22a2 (fun n => span R (Subtype.val \u2218 \u2191f '' {m | n \u2264 m})) x = (fun n => span R (Subtype.val \u2218 \u2191f '' {m | n \u2264 m})) y \u2192 x = y\n[PROOFSTEP]\nrw [le_antisymm_iff, \u2190 this y x, \u2190 this x y]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\nthis\u271d : \u2200 (n : \u2115), Subtype.val \u2218 \u2191f '' {m | n \u2264 m} \u2286 s\nthis : \u2200 (a b : \u2115), a \u2264 b \u2194 span R (Subtype.val \u2218 \u2191f '' {m | b \u2264 m}) \u2264 span R (Subtype.val \u2218 \u2191f '' {m | a \u2264 m})\nx y : \u2115\n\u22a2 y \u2264 x \u2227 x \u2264 y \u2192 x = y\n[PROOFSTEP]\nexact fun \u27e8h\u2081, h\u2082\u27e9 => le_antisymm_iff.2 \u27e8h\u2082, h\u2081\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\nthis\u271d : \u2200 (n : \u2115), Subtype.val \u2218 \u2191f '' {m | n \u2264 m} \u2286 s\nthis : \u2200 (a b : \u2115), a \u2264 b \u2194 span R (Subtype.val \u2218 \u2191f '' {m | b \u2264 m}) \u2264 span R (Subtype.val \u2218 \u2191f '' {m | a \u2264 m})\n\u22a2 \u2200 {a b : \u2115},\n \u2191{ toFun := fun n => span R (Subtype.val \u2218 \u2191f '' {m | n \u2264 m}),\n inj' :=\n (_ :\n \u2200 (x y : \u2115),\n (fun n => span R (Subtype.val \u2218 \u2191f '' {m | n \u2264 m})) x =\n (fun n => span R (Subtype.val \u2218 \u2191f '' {m | n \u2264 m})) y \u2192\n x = y) }\n a <\n \u2191{ toFun := fun n => span R (Subtype.val \u2218 \u2191f '' {m | n \u2264 m}),\n inj' :=\n (_ :\n \u2200 (x y : \u2115),\n (fun n => span R (Subtype.val \u2218 \u2191f '' {m | n \u2264 m})) x =\n (fun n => span R (Subtype.val \u2218 \u2191f '' {m | n \u2264 m})) y \u2192\n x = y) }\n b \u2194\n a > b\n[PROOFSTEP]\nintro a b\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\nthis\u271d : \u2200 (n : \u2115), Subtype.val \u2218 \u2191f '' {m | n \u2264 m} \u2286 s\nthis : \u2200 (a b : \u2115), a \u2264 b \u2194 span R (Subtype.val \u2218 \u2191f '' {m | b \u2264 m}) \u2264 span R (Subtype.val \u2218 \u2191f '' {m | a \u2264 m})\na b : \u2115\n\u22a2 \u2191{ toFun := fun n => span R (Subtype.val \u2218 \u2191f '' {m | n \u2264 m}),\n inj' :=\n (_ :\n \u2200 (x y : \u2115),\n (fun n => span R (Subtype.val \u2218 \u2191f '' {m | n \u2264 m})) x =\n (fun n => span R (Subtype.val \u2218 \u2191f '' {m | n \u2264 m})) y \u2192\n x = y) }\n a <\n \u2191{ toFun := fun n => span R (Subtype.val \u2218 \u2191f '' {m | n \u2264 m}),\n inj' :=\n (_ :\n \u2200 (x y : \u2115),\n (fun n => span R (Subtype.val \u2218 \u2191f '' {m | n \u2264 m})) x =\n (fun n => span R (Subtype.val \u2218 \u2191f '' {m | n \u2264 m})) y \u2192\n x = y) }\n b \u2194\n a > b\n[PROOFSTEP]\nconv_rhs => rw [GT.gt, lt_iff_le_not_le, this, this, \u2190 lt_iff_le_not_le]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\nthis\u271d : \u2200 (n : \u2115), Subtype.val \u2218 \u2191f '' {m | n \u2264 m} \u2286 s\nthis : \u2200 (a b : \u2115), a \u2264 b \u2194 span R (Subtype.val \u2218 \u2191f '' {m | b \u2264 m}) \u2264 span R (Subtype.val \u2218 \u2191f '' {m | a \u2264 m})\na b : \u2115\n| a > b\n[PROOFSTEP]\nrw [GT.gt, lt_iff_le_not_le, this, this, \u2190 lt_iff_le_not_le]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\nthis\u271d : \u2200 (n : \u2115), Subtype.val \u2218 \u2191f '' {m | n \u2264 m} \u2286 s\nthis : \u2200 (a b : \u2115), a \u2264 b \u2194 span R (Subtype.val \u2218 \u2191f '' {m | b \u2264 m}) \u2264 span R (Subtype.val \u2218 \u2191f '' {m | a \u2264 m})\na b : \u2115\n| a > b\n[PROOFSTEP]\nrw [GT.gt, lt_iff_le_not_le, this, this, \u2190 lt_iff_le_not_le]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\nhf : \u00acSet.Finite s\nf : \u2115 \u21aa \u2191s\nthis\u271d : \u2200 (n : \u2115), Subtype.val \u2218 \u2191f '' {m | n \u2264 m} \u2286 s\nthis : \u2200 (a b : \u2115), a \u2264 b \u2194 span R (Subtype.val \u2218 \u2191f '' {m | b \u2264 m}) \u2264 span R (Subtype.val \u2218 \u2191f '' {m | a \u2264 m})\na b : \u2115\n| a > b\n[PROOFSTEP]\nrw [GT.gt, lt_iff_le_not_le, this, this, \u2190 lt_iff_le_not_le]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 (\u2200 (a : Set (Submodule R M)), Set.Nonempty a \u2192 \u2203 M', M' \u2208 a \u2227 \u2200 (I : Submodule R M), I \u2208 a \u2192 \u00acI < M') \u2194 IsArtinian R M\n[PROOFSTEP]\nrw [isArtinian_iff_wellFounded, WellFounded.wellFounded_iff_has_min]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 (\u2200 (f : \u2115 \u2192o (Submodule R M)\u1d52\u1d48), \u2203 n, \u2200 (m : \u2115), n \u2264 m \u2192 \u2191f n = \u2191f m) \u2194 IsArtinian R M\n[PROOFSTEP]\nrw [isArtinian_iff_wellFounded]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 (\u2200 (f : \u2115 \u2192o (Submodule R M)\u1d52\u1d48), \u2203 n, \u2200 (m : \u2115), n \u2264 m \u2192 \u2191f n = \u2191f m) \u2194 WellFounded fun x x_1 => x < x_1\n[PROOFSTEP]\nexact WellFounded.monotone_chain_condition.symm\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\n\u22a2 \u2203 n, n \u2260 0 \u2227 LinearMap.ker (f ^ n) \u2294 LinearMap.range (f ^ n) = \u22a4\n[PROOFSTEP]\nobtain \u27e8n, w\u27e9 := monotone_stabilizes (f.iterateRange.comp \u27e8fun n => n + 1, fun n m w => by linarith\u27e9)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn m : \u2115\nw : n \u2264 m\n\u22a2 (fun n => n + 1) n \u2264 (fun n => n + 1) m\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw :\n \u2200 (m : \u2115),\n n \u2264 m \u2192\n \u2191(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1,\n monotone' := (_ : \u2200 (n m : \u2115), n \u2264 m \u2192 (fun n => n + 1) n \u2264 (fun n => n + 1) m) })\n n =\n \u2191(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1,\n monotone' := (_ : \u2200 (n m : \u2115), n \u2264 m \u2192 (fun n => n + 1) n \u2264 (fun n => n + 1) m) })\n m\n\u22a2 \u2203 n, n \u2260 0 \u2227 LinearMap.ker (f ^ n) \u2294 LinearMap.range (f ^ n) = \u22a4\n[PROOFSTEP]\nspecialize w (n + 1 + n) (by linarith)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw :\n \u2200 (m : \u2115),\n n \u2264 m \u2192\n \u2191(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1,\n monotone' := (_ : \u2200 (n m : \u2115), n \u2264 m \u2192 (fun n => n + 1) n \u2264 (fun n => n + 1) m) })\n n =\n \u2191(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1,\n monotone' := (_ : \u2200 (n m : \u2115), n \u2264 m \u2192 (fun n => n + 1) n \u2264 (fun n => n + 1) m) })\n m\n\u22a2 n \u2264 n + 1 + n\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw :\n \u2191(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1, monotone' := (_ : \u2200 (n m : \u2115), n \u2264 m \u2192 (fun n => n + 1) n \u2264 (fun n => n + 1) m) })\n n =\n \u2191(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1, monotone' := (_ : \u2200 (n m : \u2115), n \u2264 m \u2192 (fun n => n + 1) n \u2264 (fun n => n + 1) m) })\n (n + 1 + n)\n\u22a2 \u2203 n, n \u2260 0 \u2227 LinearMap.ker (f ^ n) \u2294 LinearMap.range (f ^ n) = \u22a4\n[PROOFSTEP]\ndsimp at w \n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\n\u22a2 \u2203 n, n \u2260 0 \u2227 LinearMap.ker (f ^ n) \u2294 LinearMap.range (f ^ n) = \u22a4\n[PROOFSTEP]\nrefine' \u27e8n + 1, Nat.succ_ne_zero _, _\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\n\u22a2 LinearMap.ker (f ^ (n + 1)) \u2294 LinearMap.range (f ^ (n + 1)) = \u22a4\n[PROOFSTEP]\nsimp_rw [eq_top_iff', mem_sup]\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\n\u22a2 \u2200 (x : M), \u2203 y, y \u2208 LinearMap.ker (f ^ (n + 1)) \u2227 \u2203 z, z \u2208 LinearMap.range (f ^ (n + 1)) \u2227 y + z = x\n[PROOFSTEP]\nintro x\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx : M\n\u22a2 \u2203 y, y \u2208 LinearMap.ker (f ^ (n + 1)) \u2227 \u2203 z, z \u2208 LinearMap.range (f ^ (n + 1)) \u2227 y + z = x\n[PROOFSTEP]\nhave : (f ^ (n + 1)) x \u2208 LinearMap.range (f ^ (n + 1 + n + 1)) :=\n by\n rw [\u2190 w]\n exact mem_range_self _\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx : M\n\u22a2 \u2191(f ^ (n + 1)) x \u2208 LinearMap.range (f ^ (n + 1 + n + 1))\n[PROOFSTEP]\nrw [\u2190 w]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx : M\n\u22a2 \u2191(f ^ (n + 1)) x \u2208 LinearMap.range (f ^ (n + 1))\n[PROOFSTEP]\nexact mem_range_self _\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx : M\nthis : \u2191(f ^ (n + 1)) x \u2208 LinearMap.range (f ^ (n + 1 + n + 1))\n\u22a2 \u2203 y, y \u2208 LinearMap.ker (f ^ (n + 1)) \u2227 \u2203 z, z \u2208 LinearMap.range (f ^ (n + 1)) \u2227 y + z = x\n[PROOFSTEP]\nrcases this with \u27e8y, hy\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : \u2191(f ^ (n + 1 + n + 1)) y = \u2191(f ^ (n + 1)) x\n\u22a2 \u2203 y, y \u2208 LinearMap.ker (f ^ (n + 1)) \u2227 \u2203 z, z \u2208 LinearMap.range (f ^ (n + 1)) \u2227 y + z = x\n[PROOFSTEP]\nuse x - (f ^ (n + 1)) y\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : \u2191(f ^ (n + 1 + n + 1)) y = \u2191(f ^ (n + 1)) x\n\u22a2 x - \u2191(f ^ (n + 1)) y \u2208 LinearMap.ker (f ^ (n + 1)) \u2227\n \u2203 z, z \u2208 LinearMap.range (f ^ (n + 1)) \u2227 x - \u2191(f ^ (n + 1)) y + z = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : \u2191(f ^ (n + 1 + n + 1)) y = \u2191(f ^ (n + 1)) x\n\u22a2 x - \u2191(f ^ (n + 1)) y \u2208 LinearMap.ker (f ^ (n + 1))\n[PROOFSTEP]\nrw [LinearMap.mem_ker, LinearMap.map_sub, \u2190 hy, sub_eq_zero, pow_add]\n[GOAL]\ncase h.left\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : \u2191(f ^ (n + 1 + n + 1)) y = \u2191(f ^ (n + 1)) x\n\u22a2 \u2191(f ^ (n + 1 + n + 1)) y = \u2191(f ^ n * f ^ 1) (\u2191(f ^ n * f ^ 1) y)\n[PROOFSTEP]\nsimp [pow_add]\n[GOAL]\ncase h.right\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : \u2191(f ^ (n + 1 + n + 1)) y = \u2191(f ^ (n + 1)) x\n\u22a2 \u2203 z, z \u2208 LinearMap.range (f ^ (n + 1)) \u2227 x - \u2191(f ^ (n + 1)) y + z = x\n[PROOFSTEP]\nuse(f ^ (n + 1)) y\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\nn : \u2115\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : \u2191(f ^ (n + 1 + n + 1)) y = \u2191(f ^ (n + 1)) x\n\u22a2 \u2191(f ^ (n + 1)) y \u2208 LinearMap.range (f ^ (n + 1)) \u2227 x - \u2191(f ^ (n + 1)) y + \u2191(f ^ (n + 1)) y = x\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\ns : Injective \u2191f\n\u22a2 Surjective \u2191f\n[PROOFSTEP]\nobtain \u27e8n, ne, w\u27e9 := exists_endomorphism_iterate_ker_sup_range_eq_top f\n[GOAL]\ncase intro.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\ns : Injective \u2191f\nn : \u2115\nne : n \u2260 0\nw : LinearMap.ker (f ^ n) \u2294 LinearMap.range (f ^ n) = \u22a4\n\u22a2 Surjective \u2191f\n[PROOFSTEP]\nrw [LinearMap.ker_eq_bot.mpr (LinearMap.iterate_injective s n), bot_sup_eq, LinearMap.range_eq_top] at w \n[GOAL]\ncase intro.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\ns : Injective \u2191f\nn : \u2115\nne : n \u2260 0\nw : Surjective \u2191(f ^ n)\n\u22a2 Surjective \u2191f\n[PROOFSTEP]\nexact LinearMap.surjective_of_iterate_surjective ne w\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : \u2115 \u2192 Submodule R M\nh : \u2200 (n : \u2115), Disjoint (\u2191(partialSups (\u2191OrderDual.toDual \u2218 f)) n) (\u2191OrderDual.toDual (f (n + 1)))\n\u22a2 \u2203 n, \u2200 (m : \u2115), n \u2264 m \u2192 f m = \u22a4\n[PROOFSTEP]\nrsuffices \u27e8n, w\u27e9 : \u2203 n : \u2115, \u2200 m, n \u2264 m \u2192 OrderDual.toDual f (m + 1) = \u22a4\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : \u2115 \u2192 Submodule R M\nh : \u2200 (n : \u2115), Disjoint (\u2191(partialSups (\u2191OrderDual.toDual \u2218 f)) n) (\u2191OrderDual.toDual (f (n + 1)))\nn : \u2115\nw : \u2200 (m : \u2115), n \u2264 m \u2192 \u2191OrderDual.toDual f (m + 1) = \u22a4\n\u22a2 \u2203 n, \u2200 (m : \u2115), n \u2264 m \u2192 f m = \u22a4\n[PROOFSTEP]\nuse n + 1\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : \u2115 \u2192 Submodule R M\nh : \u2200 (n : \u2115), Disjoint (\u2191(partialSups (\u2191OrderDual.toDual \u2218 f)) n) (\u2191OrderDual.toDual (f (n + 1)))\nn : \u2115\nw : \u2200 (m : \u2115), n \u2264 m \u2192 \u2191OrderDual.toDual f (m + 1) = \u22a4\n\u22a2 \u2200 (m : \u2115), n + 1 \u2264 m \u2192 f m = \u22a4\n[PROOFSTEP]\nrintro (_ | m) p\n[GOAL]\ncase h.zero\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : \u2115 \u2192 Submodule R M\nh : \u2200 (n : \u2115), Disjoint (\u2191(partialSups (\u2191OrderDual.toDual \u2218 f)) n) (\u2191OrderDual.toDual (f (n + 1)))\nn : \u2115\nw : \u2200 (m : \u2115), n \u2264 m \u2192 \u2191OrderDual.toDual f (m + 1) = \u22a4\np : n + 1 \u2264 Nat.zero\n\u22a2 f Nat.zero = \u22a4\n[PROOFSTEP]\ncases p\n[GOAL]\ncase h.succ\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : \u2115 \u2192 Submodule R M\nh : \u2200 (n : \u2115), Disjoint (\u2191(partialSups (\u2191OrderDual.toDual \u2218 f)) n) (\u2191OrderDual.toDual (f (n + 1)))\nn : \u2115\nw : \u2200 (m : \u2115), n \u2264 m \u2192 \u2191OrderDual.toDual f (m + 1) = \u22a4\nm : \u2115\np : n + 1 \u2264 Nat.succ m\n\u22a2 f (Nat.succ m) = \u22a4\n[PROOFSTEP]\napply w\n[GOAL]\ncase h.succ.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : \u2115 \u2192 Submodule R M\nh : \u2200 (n : \u2115), Disjoint (\u2191(partialSups (\u2191OrderDual.toDual \u2218 f)) n) (\u2191OrderDual.toDual (f (n + 1)))\nn : \u2115\nw : \u2200 (m : \u2115), n \u2264 m \u2192 \u2191OrderDual.toDual f (m + 1) = \u22a4\nm : \u2115\np : n + 1 \u2264 Nat.succ m\n\u22a2 n \u2264 m\n[PROOFSTEP]\nexact Nat.succ_le_succ_iff.mp p\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : \u2115 \u2192 Submodule R M\nh : \u2200 (n : \u2115), Disjoint (\u2191(partialSups (\u2191OrderDual.toDual \u2218 f)) n) (\u2191OrderDual.toDual (f (n + 1)))\n\u22a2 \u2203 n, \u2200 (m : \u2115), n \u2264 m \u2192 \u2191OrderDual.toDual f (m + 1) = \u22a4\n[PROOFSTEP]\nobtain \u27e8n, w\u27e9 := monotone_stabilizes (partialSups (OrderDual.toDual \u2218 f))\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : \u2115 \u2192 Submodule R M\nh : \u2200 (n : \u2115), Disjoint (\u2191(partialSups (\u2191OrderDual.toDual \u2218 f)) n) (\u2191OrderDual.toDual (f (n + 1)))\nn : \u2115\nw : \u2200 (m : \u2115), n \u2264 m \u2192 \u2191(partialSups (\u2191OrderDual.toDual \u2218 f)) n = \u2191(partialSups (\u2191OrderDual.toDual \u2218 f)) m\n\u22a2 \u2203 n, \u2200 (m : \u2115), n \u2264 m \u2192 \u2191OrderDual.toDual f (m + 1) = \u22a4\n[PROOFSTEP]\nrefine' \u27e8n, fun m p => _\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : \u2115 \u2192 Submodule R M\nh : \u2200 (n : \u2115), Disjoint (\u2191(partialSups (\u2191OrderDual.toDual \u2218 f)) n) (\u2191OrderDual.toDual (f (n + 1)))\nn : \u2115\nw : \u2200 (m : \u2115), n \u2264 m \u2192 \u2191(partialSups (\u2191OrderDual.toDual \u2218 f)) n = \u2191(partialSups (\u2191OrderDual.toDual \u2218 f)) m\nm : \u2115\np : n \u2264 m\n\u22a2 \u2191OrderDual.toDual f (m + 1) = \u22a4\n[PROOFSTEP]\nexact (h m).eq_bot_of_ge (sup_eq_left.1 <| (w (m + 1) <| le_add_right p).symm.trans <| w m p)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nr : R\nn m : \u2115\nh : n \u2264 m\nx : M\nx\u271d : x \u2208 (fun n => LinearMap.range (r ^ n \u2022 LinearMap.id)) m\ny : M\nhy : \u2191(r ^ m \u2022 LinearMap.id) y = x\n\u22a2 \u2191(r ^ n \u2022 LinearMap.id) (r ^ (m - n) \u2022 y) = x\n[PROOFSTEP]\ndsimp at hy \u22a2\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nr : R\nn m : \u2115\nh : n \u2264 m\nx : M\nx\u271d : x \u2208 (fun n => LinearMap.range (r ^ n \u2022 LinearMap.id)) m\ny : M\nhy : r ^ m \u2022 y = x\n\u22a2 r ^ n \u2022 r ^ (m - n) \u2022 y = x\n[PROOFSTEP]\nrw [\u2190 smul_assoc, smul_eq_mul, \u2190 pow_add, \u2190 hy, add_tsub_cancel_of_le h]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nr : R\nx : M\n\u22a2 \u2203 n y, r ^ Nat.succ n \u2022 y = r ^ n \u2022 x\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := IsArtinian.range_smul_pow_stabilizes M r\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nr : R\nx : M\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 LinearMap.range (r ^ n \u2022 LinearMap.id) = LinearMap.range (r ^ m \u2022 LinearMap.id)\n\u22a2 \u2203 n y, r ^ Nat.succ n \u2022 y = r ^ n \u2022 x\n[PROOFSTEP]\nsimp_rw [SetLike.ext_iff] at hn \n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nr : R\nx : M\nn : \u2115\nhn :\n \u2200 (m : \u2115), n \u2264 m \u2192 \u2200 (x : M), x \u2208 LinearMap.range (r ^ n \u2022 LinearMap.id) \u2194 x \u2208 LinearMap.range (r ^ m \u2022 LinearMap.id)\n\u22a2 \u2203 n y, r ^ Nat.succ n \u2022 y = r ^ n \u2022 x\n[PROOFSTEP]\nexact \u27e8n, by simpa using hn n.succ n.le_succ (r ^ n \u2022 x)\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nr : R\nx : M\nn : \u2115\nhn :\n \u2200 (m : \u2115), n \u2264 m \u2192 \u2200 (x : M), x \u2208 LinearMap.range (r ^ n \u2022 LinearMap.id) \u2194 x \u2208 LinearMap.range (r ^ m \u2022 LinearMap.id)\n\u22a2 \u2203 y, r ^ Nat.succ n \u2022 y = r ^ n \u2022 x\n[PROOFSTEP]\nsimpa using hn n.succ n.le_succ (r ^ n \u2022 x)\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : Ring S\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Module S M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsScalarTower R S M\nh : IsArtinian R M\n\u22a2 IsArtinian S M\n[PROOFSTEP]\nrw [isArtinian_iff_wellFounded] at h \u22a2\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : Ring S\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Module S M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsScalarTower R S M\nh : WellFounded fun x x_1 => x < x_1\n\u22a2 WellFounded fun x x_1 => x < x_1\n[PROOFSTEP]\nrefine' (Submodule.restrictScalarsEmbedding R S M).wellFounded h\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nN : Submodule R M\ninst\u271d : IsArtinianRing R\nhN : FG N\n\u22a2 IsArtinian R { x // x \u2208 N }\n[PROOFSTEP]\nlet \u27e8s, hs\u27e9 := hN\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nN : Submodule R M\ninst\u271d : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R \u2191s = N\n\u22a2 IsArtinian R { x // x \u2208 N }\n[PROOFSTEP]\nhaveI := Classical.decEq M\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nN : Submodule R M\ninst\u271d : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R \u2191s = N\nthis : DecidableEq M\n\u22a2 IsArtinian R { x // x \u2208 N }\n[PROOFSTEP]\nhaveI := Classical.decEq R\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nN : Submodule R M\ninst\u271d : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R \u2191s = N\nthis\u271d : DecidableEq M\nthis : DecidableEq R\n\u22a2 IsArtinian R { x // x \u2208 N }\n[PROOFSTEP]\nhave : \u2200 x \u2208 s, x \u2208 N := fun x hx => hs \u25b8 Submodule.subset_span hx\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nN : Submodule R M\ninst\u271d : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R \u2191s = N\nthis\u271d\u00b9 : DecidableEq M\nthis\u271d : DecidableEq R\nthis : \u2200 (x : M), x \u2208 s \u2192 x \u2208 N\n\u22a2 IsArtinian R { x // x \u2208 N }\n[PROOFSTEP]\nrefine' @isArtinian_of_surjective _ ((\u2191s : Set M) \u2192\u2080 R) N _ _ _ _ _ _ _ isArtinian_finsupp\n[GOAL]\ncase refine'_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nN : Submodule R M\ninst\u271d : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R \u2191s = N\nthis\u271d\u00b9 : DecidableEq M\nthis\u271d : DecidableEq R\nthis : \u2200 (x : M), x \u2208 s \u2192 x \u2208 N\n\u22a2 (\u2191\u2191s \u2192\u2080 R) \u2192\u2097[R] { x // x \u2208 N }\n[PROOFSTEP]\nexact Finsupp.total (\u2191s : Set M) N R (fun i => \u27e8i, hs \u25b8 subset_span i.2\u27e9)\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nN : Submodule R M\ninst\u271d : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R \u2191s = N\nthis\u271d\u00b9 : DecidableEq M\nthis\u271d : DecidableEq R\nthis : \u2200 (x : M), x \u2208 s \u2192 x \u2208 N\n\u22a2 Surjective \u2191(Finsupp.total \u2191\u2191s { x // x \u2208 N } R fun i => { val := \u2191i, property := (_ : \u2191i \u2208 N) })\n[PROOFSTEP]\nrw [\u2190 LinearMap.range_eq_top, eq_top_iff, \u2190\n map_le_map_iff_of_injective (show Injective (Submodule.subtype N) from Subtype.val_injective), Submodule.map_top,\n range_subtype, \u2190 Submodule.map_top, \u2190 Submodule.map_comp, Submodule.map_top]\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nN : Submodule R M\ninst\u271d : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R \u2191s = N\nthis\u271d\u00b9 : DecidableEq M\nthis\u271d : DecidableEq R\nthis : \u2200 (x : M), x \u2208 s \u2192 x \u2208 N\n\u22a2 N \u2264\n LinearMap.range\n (LinearMap.comp (Submodule.subtype N)\n (Finsupp.total \u2191\u2191s { x // x \u2208 N } R fun i => { val := \u2191i, property := (_ : \u2191i \u2208 N) }))\n[PROOFSTEP]\nsubst N\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinianRing R\ns : Finset M\nthis\u271d\u00b9 : DecidableEq M\nthis\u271d : DecidableEq R\nhN : FG (span R \u2191s)\nthis : \u2200 (x : M), x \u2208 s \u2192 x \u2208 span R \u2191s\n\u22a2 span R \u2191s \u2264\n LinearMap.range\n (LinearMap.comp (Submodule.subtype (span R \u2191s))\n (Finsupp.total \u2191\u2191s { x // x \u2208 span R \u2191s } R fun i => { val := \u2191i, property := (_ : \u2191i \u2208 span R \u2191s) }))\n[PROOFSTEP]\nrefine span_le.2 (fun i hi => ?_)\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinianRing R\ns : Finset M\nthis\u271d\u00b9 : DecidableEq M\nthis\u271d : DecidableEq R\nhN : FG (span R \u2191s)\nthis : \u2200 (x : M), x \u2208 s \u2192 x \u2208 span R \u2191s\ni : M\nhi : i \u2208 \u2191s\n\u22a2 i \u2208\n \u2191(LinearMap.range\n (LinearMap.comp (Submodule.subtype (span R \u2191s))\n (Finsupp.total \u2191\u2191s { x // x \u2208 span R \u2191s } R fun i => { val := \u2191i, property := (_ : \u2191i \u2208 span R \u2191s) })))\n[PROOFSTEP]\nuse Finsupp.single \u27e8i, hi\u27e9 1\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinianRing R\ns : Finset M\nthis\u271d\u00b9 : DecidableEq M\nthis\u271d : DecidableEq R\nhN : FG (span R \u2191s)\nthis : \u2200 (x : M), x \u2208 s \u2192 x \u2208 span R \u2191s\ni : M\nhi : i \u2208 \u2191s\n\u22a2 \u2191(LinearMap.comp (Submodule.subtype (span R \u2191s))\n (Finsupp.total \u2191\u2191s { x // x \u2208 span R \u2191s } R fun i => { val := \u2191i, property := (_ : \u2191i \u2208 span R \u2191s) }))\n (Finsupp.single { val := i, property := hi } 1) =\n i\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\nS : Type u_2\ninst\u271d\u00b9 : Ring S\nF : Type u_3\ninst\u271d : RingHomClass F R S\nf : F\nhf : Surjective \u2191f\nH : IsArtinianRing R\n\u22a2 IsArtinianRing S\n[PROOFSTEP]\nrw [isArtinianRing_iff, isArtinian_iff_wellFounded] at H \u22a2\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\nS : Type u_2\ninst\u271d\u00b9 : Ring S\nF : Type u_3\ninst\u271d : RingHomClass F R S\nf : F\nhf : Surjective \u2191f\nH : WellFounded fun x x_1 => x < x_1\n\u22a2 WellFounded fun x x_1 => x < x_1\n[PROOFSTEP]\nexact (Ideal.orderEmbeddingOfSurjective f hf).wellFounded H\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\n\u22a2 IsNilpotent (Ideal.jacobson \u22a5)\n[PROOFSTEP]\nlet Jac := Ideal.jacobson (\u22a5 : Ideal R)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\n\u22a2 IsNilpotent (Ideal.jacobson \u22a5)\n[PROOFSTEP]\nlet f : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := \u27e8fun n => Jac ^ n, fun _ _ h => Ideal.pow_le_pow h\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\n\u22a2 IsNilpotent (Ideal.jacobson \u22a5)\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : \u2203 n, \u2200 m, n \u2264 m \u2192 Jac ^ n = Jac ^ m := IsArtinian.monotone_stabilizes f\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\n\u22a2 IsNilpotent (Ideal.jacobson \u22a5)\n[PROOFSTEP]\nrefine' \u27e8n, _\u27e9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\n\u22a2 Ideal.jacobson \u22a5 ^ n = 0\n[PROOFSTEP]\nlet J : Ideal R := annihilator (Jac ^ n)\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\n\u22a2 Ideal.jacobson \u22a5 ^ n = 0\n[PROOFSTEP]\nsuffices J = \u22a4 by\n have hJ : J \u2022 Jac ^ n = \u22a5 := annihilator_smul (Jac ^ n)\n simpa only [this, top_smul, Ideal.zero_eq_bot] using hJ\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nthis : J = \u22a4\n\u22a2 Ideal.jacobson \u22a5 ^ n = 0\n[PROOFSTEP]\nhave hJ : J \u2022 Jac ^ n = \u22a5 := annihilator_smul (Jac ^ n)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nthis : J = \u22a4\nhJ : J \u2022 Jac ^ n = \u22a5\n\u22a2 Ideal.jacobson \u22a5 ^ n = 0\n[PROOFSTEP]\nsimpa only [this, top_smul, Ideal.zero_eq_bot] using hJ\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\n\u22a2 J = \u22a4\n[PROOFSTEP]\nby_contra hJ\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : \u00acJ = \u22a4\n\u22a2 False\n[PROOFSTEP]\nchange J \u2260 \u22a4 at hJ \n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\n\u22a2 False\n[PROOFSTEP]\nrcases IsArtinian.set_has_minimal {J' : Ideal R | J < J'} \u27e8\u22a4, hJ.lt_top\u27e9 with\n \u27e8J', hJJ' : J < J', hJ' : \u2200 I, J < I \u2192 \u00acI < J'\u27e9\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nJ' : Submodule R R\nhJJ' : J < J'\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J'\n\u22a2 False\n[PROOFSTEP]\nrcases SetLike.exists_of_lt hJJ' with \u27e8x, hxJ', hxJ\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nJ' : Submodule R R\nhJJ' : J < J'\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J'\nx : R\nhxJ' : x \u2208 J'\nhxJ : \u00acx \u2208 J\n\u22a2 False\n[PROOFSTEP]\nobtain rfl : J \u2294 Ideal.span { x } = J' :=\n by\n apply eq_of_le_of_not_lt _ (hJ' (J \u2294 Ideal.span { x }) _)\n \u00b7 exact sup_le hJJ'.le (span_le.2 (singleton_subset_iff.2 hxJ'))\n \u00b7 rw [SetLike.lt_iff_le_and_exists]\n exact \u27e8le_sup_left, \u27e8x, mem_sup_right (mem_span_singleton_self x), hxJ\u27e9\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nJ' : Submodule R R\nhJJ' : J < J'\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J'\nx : R\nhxJ' : x \u2208 J'\nhxJ : \u00acx \u2208 J\n\u22a2 J \u2294 Ideal.span {x} = J'\n[PROOFSTEP]\napply eq_of_le_of_not_lt _ (hJ' (J \u2294 Ideal.span { x }) _)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nJ' : Submodule R R\nhJJ' : J < J'\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J'\nx : R\nhxJ' : x \u2208 J'\nhxJ : \u00acx \u2208 J\n\u22a2 J \u2294 Ideal.span {x} \u2264 J'\n[PROOFSTEP]\nexact sup_le hJJ'.le (span_le.2 (singleton_subset_iff.2 hxJ'))\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nJ' : Submodule R R\nhJJ' : J < J'\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J'\nx : R\nhxJ' : x \u2208 J'\nhxJ : \u00acx \u2208 J\n\u22a2 J < J \u2294 Ideal.span {x}\n[PROOFSTEP]\nrw [SetLike.lt_iff_le_and_exists]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nJ' : Submodule R R\nhJJ' : J < J'\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J'\nx : R\nhxJ' : x \u2208 J'\nhxJ : \u00acx \u2208 J\n\u22a2 J \u2264 J \u2294 Ideal.span {x} \u2227 \u2203 x_1, x_1 \u2208 J \u2294 Ideal.span {x} \u2227 \u00acx_1 \u2208 J\n[PROOFSTEP]\nexact \u27e8le_sup_left, \u27e8x, mem_sup_right (mem_span_singleton_self x), hxJ\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nx : R\nhxJ : \u00acx \u2208 J\nhJJ' : J < J \u2294 Ideal.span {x}\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J \u2294 Ideal.span {x}\nhxJ' : x \u2208 J \u2294 Ideal.span {x}\n\u22a2 False\n[PROOFSTEP]\nhave : J \u2294 Jac \u2022 Ideal.span { x } \u2264 J \u2294 Ideal.span { x } :=\n sup_le_sup_left (smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nx : R\nhxJ : \u00acx \u2208 J\nhJJ' : J < J \u2294 Ideal.span {x}\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J \u2294 Ideal.span {x}\nhxJ' : x \u2208 J \u2294 Ideal.span {x}\nthis : J \u2294 Jac \u2022 Ideal.span {x} \u2264 J \u2294 Ideal.span {x}\n\u22a2 False\n[PROOFSTEP]\nhave : Jac * Ideal.span { x } \u2264 J := by\n -- Need version 4 of Nakayama's lemma on Stacks\n by_contra H\n refine' H (smul_sup_le_of_le_smul_of_le_jacobson_bot (fg_span_singleton _) le_rfl (this.eq_of_not_lt (hJ' _ _)).ge)\n exact lt_of_le_of_ne le_sup_left fun h => H <| h.symm \u25b8 le_sup_right\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nx : R\nhxJ : \u00acx \u2208 J\nhJJ' : J < J \u2294 Ideal.span {x}\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J \u2294 Ideal.span {x}\nhxJ' : x \u2208 J \u2294 Ideal.span {x}\nthis : J \u2294 Jac \u2022 Ideal.span {x} \u2264 J \u2294 Ideal.span {x}\n\u22a2 Jac * Ideal.span {x} \u2264 J\n[PROOFSTEP]\nby_contra H\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nx : R\nhxJ : \u00acx \u2208 J\nhJJ' : J < J \u2294 Ideal.span {x}\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J \u2294 Ideal.span {x}\nhxJ' : x \u2208 J \u2294 Ideal.span {x}\nthis : J \u2294 Jac \u2022 Ideal.span {x} \u2264 J \u2294 Ideal.span {x}\nH : \u00acJac * Ideal.span {x} \u2264 J\n\u22a2 False\n[PROOFSTEP]\nrefine' H (smul_sup_le_of_le_smul_of_le_jacobson_bot (fg_span_singleton _) le_rfl (this.eq_of_not_lt (hJ' _ _)).ge)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nx : R\nhxJ : \u00acx \u2208 J\nhJJ' : J < J \u2294 Ideal.span {x}\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J \u2294 Ideal.span {x}\nhxJ' : x \u2208 J \u2294 Ideal.span {x}\nthis : J \u2294 Jac \u2022 Ideal.span {x} \u2264 J \u2294 Ideal.span {x}\nH : \u00acJac * Ideal.span {x} \u2264 J\n\u22a2 J < J \u2294 Jac \u2022 Ideal.span {x}\n[PROOFSTEP]\nexact lt_of_le_of_ne le_sup_left fun h => H <| h.symm \u25b8 le_sup_right\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nx : R\nhxJ : \u00acx \u2208 J\nhJJ' : J < J \u2294 Ideal.span {x}\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J \u2294 Ideal.span {x}\nhxJ' : x \u2208 J \u2294 Ideal.span {x}\nthis\u271d : J \u2294 Jac \u2022 Ideal.span {x} \u2264 J \u2294 Ideal.span {x}\nthis : Jac * Ideal.span {x} \u2264 J\n\u22a2 False\n[PROOFSTEP]\nhave : Ideal.span { x } * Jac ^ (n + 1) \u2264 \u22a5\n[GOAL]\ncase this\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nx : R\nhxJ : \u00acx \u2208 J\nhJJ' : J < J \u2294 Ideal.span {x}\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J \u2294 Ideal.span {x}\nhxJ' : x \u2208 J \u2294 Ideal.span {x}\nthis\u271d : J \u2294 Jac \u2022 Ideal.span {x} \u2264 J \u2294 Ideal.span {x}\nthis : Jac * Ideal.span {x} \u2264 J\n\u22a2 Ideal.span {x} * Jac ^ (n + 1) \u2264 \u22a5\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nx : R\nhxJ : \u00acx \u2208 J\nhJJ' : J < J \u2294 Ideal.span {x}\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J \u2294 Ideal.span {x}\nhxJ' : x \u2208 J \u2294 Ideal.span {x}\nthis\u271d\u00b9 : J \u2294 Jac \u2022 Ideal.span {x} \u2264 J \u2294 Ideal.span {x}\nthis\u271d : Jac * Ideal.span {x} \u2264 J\nthis : Ideal.span {x} * Jac ^ (n + 1) \u2264 \u22a5\n\u22a2 False\n[PROOFSTEP]\ncalc\n Ideal.span { x } * Jac ^ (n + 1) = Ideal.span { x } * Jac * Jac ^ n := by rw [pow_succ, \u2190 mul_assoc]\n _ \u2264 J * Jac ^ n := (mul_le_mul (by rwa [mul_comm]) le_rfl)\n _ = \u22a5 := by simp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nx : R\nhxJ : \u00acx \u2208 J\nhJJ' : J < J \u2294 Ideal.span {x}\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J \u2294 Ideal.span {x}\nhxJ' : x \u2208 J \u2294 Ideal.span {x}\nthis\u271d : J \u2294 Jac \u2022 Ideal.span {x} \u2264 J \u2294 Ideal.span {x}\nthis : Jac * Ideal.span {x} \u2264 J\n\u22a2 Ideal.span {x} * Jac ^ (n + 1) = Ideal.span {x} * Jac * Jac ^ n\n[PROOFSTEP]\nrw [pow_succ, \u2190 mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nx : R\nhxJ : \u00acx \u2208 J\nhJJ' : J < J \u2294 Ideal.span {x}\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J \u2294 Ideal.span {x}\nhxJ' : x \u2208 J \u2294 Ideal.span {x}\nthis\u271d : J \u2294 Jac \u2022 Ideal.span {x} \u2264 J \u2294 Ideal.span {x}\nthis : Jac * Ideal.span {x} \u2264 J\n\u22a2 Ideal.span {x} * Jac \u2264 J\n[PROOFSTEP]\nrwa [mul_comm]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nx : R\nhxJ : \u00acx \u2208 J\nhJJ' : J < J \u2294 Ideal.span {x}\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J \u2294 Ideal.span {x}\nhxJ' : x \u2208 J \u2294 Ideal.span {x}\nthis\u271d : J \u2294 Jac \u2022 Ideal.span {x} \u2264 J \u2294 Ideal.span {x}\nthis : Jac * Ideal.span {x} \u2264 J\n\u22a2 J * Jac ^ n = \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nx : R\nhxJ : \u00acx \u2208 J\nhJJ' : J < J \u2294 Ideal.span {x}\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J \u2294 Ideal.span {x}\nhxJ' : x \u2208 J \u2294 Ideal.span {x}\nthis\u271d\u00b9 : J \u2294 Jac \u2022 Ideal.span {x} \u2264 J \u2294 Ideal.span {x}\nthis\u271d : Jac * Ideal.span {x} \u2264 J\nthis : Ideal.span {x} * Jac ^ (n + 1) \u2264 \u22a5\n\u22a2 False\n[PROOFSTEP]\nrefine' hxJ (mem_annihilator.2 fun y hy => (mem_bot R).1 _)\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nx : R\nhxJ : \u00acx \u2208 J\nhJJ' : J < J \u2294 Ideal.span {x}\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J \u2294 Ideal.span {x}\nhxJ' : x \u2208 J \u2294 Ideal.span {x}\nthis\u271d\u00b9 : J \u2294 Jac \u2022 Ideal.span {x} \u2264 J \u2294 Ideal.span {x}\nthis\u271d : Jac * Ideal.span {x} \u2264 J\nthis : Ideal.span {x} * Jac ^ (n + 1) \u2264 \u22a5\ny : R\nhy : y \u2208 Jac ^ n\n\u22a2 x \u2022 y \u2208 \u22a5\n[PROOFSTEP]\nrefine' this (mul_mem_mul (mem_span_singleton_self x) _)\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsArtinianRing R\nJac : Ideal R := Ideal.jacobson \u22a5\nf : \u2115 \u2192o (Ideal R)\u1d52\u1d48 := { toFun := fun n => Jac ^ n, monotone' := (_ : \u2200 (x x_1 : \u2115), x \u2264 x_1 \u2192 Jac ^ x_1 \u2264 Jac ^ x) }\nn : \u2115\nhn : \u2200 (m : \u2115), n \u2264 m \u2192 Jac ^ n = Jac ^ m\nJ : Ideal R := annihilator (Jac ^ n)\nhJ : J \u2260 \u22a4\nx : R\nhxJ : \u00acx \u2208 J\nhJJ' : J < J \u2294 Ideal.span {x}\nhJ' : \u2200 (I : Ideal R), J < I \u2192 \u00acI < J \u2294 Ideal.span {x}\nhxJ' : x \u2208 J \u2294 Ideal.span {x}\nthis\u271d\u00b9 : J \u2294 Jac \u2022 Ideal.span {x} \u2264 J \u2294 Ideal.span {x}\nthis\u271d : Jac * Ideal.span {x} \u2264 J\nthis : Ideal.span {x} * Jac ^ (n + 1) \u2264 \u22a5\ny : R\nhy : y \u2208 Jac ^ n\n\u22a2 y \u2208 Jac ^ (n + 1)\n[PROOFSTEP]\nrwa [\u2190 hn (n + 1) (Nat.le_succ _)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\n\u22a2 \u2203 b, x * b = 1\n[PROOFSTEP]\nhave : IsArtinianRing (R \u29f8 p) := (@Ideal.Quotient.mk_surjective R _ p).isArtinianRing\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\n\u22a2 \u2203 b, x * b = 1\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : \u2203 (n : \u2115), Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\n\u22a2 \u2203 n, Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\n[PROOFSTEP]\nobtain \u27e8n, h\u27e9 :=\n IsArtinian.monotone_stabilizes (R := R \u29f8 p) (M := R \u29f8 p)\n \u27e8fun m => OrderDual.toDual (Ideal.span {x ^ m}), fun m n h y hy =>\n by\n dsimp [OrderDual.toDual] at *\n rw [Ideal.mem_span_singleton] at hy \u22a2\n obtain \u27e8z, rfl\u27e9 := hy\n exact dvd_mul_of_dvd_left (pow_dvd_pow _ h) _\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nm n : \u2115\nh : m \u2264 n\ny : R \u29f8 p\nhy : y \u2208 (fun m => \u2191OrderDual.toDual (Ideal.span {x ^ m})) n\n\u22a2 y \u2208 (fun m => \u2191OrderDual.toDual (Ideal.span {x ^ m})) m\n[PROOFSTEP]\ndsimp [OrderDual.toDual] at *\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : \u00acx = 0\nthis : IsArtinianRing (R \u29f8 p)\nm n : \u2115\nh : m \u2264 n\ny : R \u29f8 p\nhy : y \u2208 Ideal.span {x ^ n}\n\u22a2 y \u2208 Ideal.span {x ^ m}\n[PROOFSTEP]\nrw [Ideal.mem_span_singleton] at hy \u22a2\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : \u00acx = 0\nthis : IsArtinianRing (R \u29f8 p)\nm n : \u2115\nh : m \u2264 n\ny : R \u29f8 p\nhy : x ^ n \u2223 y\n\u22a2 x ^ m \u2223 y\n[PROOFSTEP]\nobtain \u27e8z, rfl\u27e9 := hy\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : \u00acx = 0\nthis : IsArtinianRing (R \u29f8 p)\nm n : \u2115\nh : m \u2264 n\nz : R \u29f8 p\n\u22a2 x ^ m \u2223 x ^ n * z\n[PROOFSTEP]\nexact dvd_mul_of_dvd_left (pow_dvd_pow _ h) _\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nn : \u2115\nh :\n \u2200 (m : \u2115),\n n \u2264 m \u2192\n \u2191{ toFun := fun m => \u2191OrderDual.toDual (Ideal.span {x ^ m}),\n monotone' :=\n (_ :\n \u2200 (m n : \u2115),\n m \u2264 n \u2192\n \u2200 (y : R \u29f8 p),\n y \u2208 (fun m => \u2191OrderDual.toDual (Ideal.span {x ^ m})) n \u2192\n y \u2208 (fun m => \u2191OrderDual.toDual (Ideal.span {x ^ m})) m) }\n n =\n \u2191{ toFun := fun m => \u2191OrderDual.toDual (Ideal.span {x ^ m}),\n monotone' :=\n (_ :\n \u2200 (m n : \u2115),\n m \u2264 n \u2192\n \u2200 (y : R \u29f8 p),\n y \u2208 (fun m => \u2191OrderDual.toDual (Ideal.span {x ^ m})) n \u2192\n y \u2208 (fun m => \u2191OrderDual.toDual (Ideal.span {x ^ m})) m) }\n m\n\u22a2 \u2203 n, Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\n[PROOFSTEP]\nexact \u27e8n, h (n + 1) <| by norm_num\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nn : \u2115\nh :\n \u2200 (m : \u2115),\n n \u2264 m \u2192\n \u2191{ toFun := fun m => \u2191OrderDual.toDual (Ideal.span {x ^ m}),\n monotone' :=\n (_ :\n \u2200 (m n : \u2115),\n m \u2264 n \u2192\n \u2200 (y : R \u29f8 p),\n y \u2208 (fun m => \u2191OrderDual.toDual (Ideal.span {x ^ m})) n \u2192\n y \u2208 (fun m => \u2191OrderDual.toDual (Ideal.span {x ^ m})) m) }\n n =\n \u2191{ toFun := fun m => \u2191OrderDual.toDual (Ideal.span {x ^ m}),\n monotone' :=\n (_ :\n \u2200 (m n : \u2115),\n m \u2264 n \u2192\n \u2200 (y : R \u29f8 p),\n y \u2208 (fun m => \u2191OrderDual.toDual (Ideal.span {x ^ m})) n \u2192\n y \u2208 (fun m => \u2191OrderDual.toDual (Ideal.span {x ^ m})) m) }\n m\n\u22a2 n \u2264 n + 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nn : \u2115\nhn : Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\n\u22a2 \u2203 b, x * b = 1\n[PROOFSTEP]\nhave H : x ^ n \u2208 Ideal.span {x ^ (n + 1)}\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nn : \u2115\nhn : Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\n\u22a2 x ^ n \u2208 Ideal.span {x ^ (n + 1)}\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nn : \u2115\nhn : Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\nH : x ^ n \u2208 Ideal.span {x ^ (n + 1)}\n\u22a2 \u2203 b, x * b = 1\n[PROOFSTEP]\n{rw [\u2190 hn]; refine Submodule.subset_span (Set.mem_singleton _)\n}\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nn : \u2115\nhn : Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\n\u22a2 x ^ n \u2208 Ideal.span {x ^ (n + 1)}\n[PROOFSTEP]\nrw [\u2190 hn]\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nn : \u2115\nhn : Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\n\u22a2 x ^ n \u2208 Ideal.span {x ^ n}\n[PROOFSTEP]\nrefine Submodule.subset_span (Set.mem_singleton _)\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nn : \u2115\nhn : Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\nH : x ^ n \u2208 Ideal.span {x ^ (n + 1)}\n\u22a2 \u2203 b, x * b = 1\n[PROOFSTEP]\nrw [Ideal.mem_span_singleton] at H \n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nn : \u2115\nhn : Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\nH : x ^ (n + 1) \u2223 x ^ n\n\u22a2 \u2203 b, x * b = 1\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := H\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nn : \u2115\nhn : Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\ny : R \u29f8 p\nhy : x ^ n = x ^ (n + 1) * y\n\u22a2 \u2203 b, x * b = 1\n[PROOFSTEP]\nrw [pow_add, mul_assoc, pow_one] at hy \n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nn : \u2115\nhn : Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\ny : R \u29f8 p\nhy : x ^ n = x ^ n * (x * y)\n\u22a2 \u2203 b, x * b = 1\n[PROOFSTEP]\nconv_lhs at hy => rw [\u2190 mul_one (x ^ n)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nn : \u2115\nhn : Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\ny : R \u29f8 p\nhy : x ^ n = x ^ n * (x * y)\n| x ^ n\n[PROOFSTEP]\nrw [\u2190 mul_one (x ^ n)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nn : \u2115\nhn : Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\ny : R \u29f8 p\nhy : x ^ n = x ^ n * (x * y)\n| x ^ n\n[PROOFSTEP]\nrw [\u2190 mul_one (x ^ n)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nn : \u2115\nhn : Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\ny : R \u29f8 p\nhy : x ^ n = x ^ n * (x * y)\n| x ^ n\n[PROOFSTEP]\nrw [\u2190 mul_one (x ^ n)]\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsArtinianRing R\np : Ideal R\ninst\u271d : Ideal.IsPrime p\nx : R \u29f8 p\nhx : x \u2260 0\nthis : IsArtinianRing (R \u29f8 p)\nn : \u2115\nhn : Ideal.span {x ^ n} = Ideal.span {x ^ (n + 1)}\ny : R \u29f8 p\nhy : x ^ n * 1 = x ^ n * (x * y)\n\u22a2 \u2203 b, x * b = 1\n[PROOFSTEP]\nexact \u27e8y, mul_left_cancel\u2080 (pow_ne_zero _ hx) hy.symm\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\n\u22a2 Surjective \u2191(algebraMap R L)\n[PROOFSTEP]\nintro r'\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nr' : L\n\u22a2 \u2203 a, \u2191(algebraMap R L) a = r'\n[PROOFSTEP]\nobtain \u27e8r\u2081, s, rfl\u27e9 := IsLocalization.mk'_surjective S r'\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nr\u2081 : R\ns : { x // x \u2208 S }\n\u22a2 \u2203 a, \u2191(algebraMap R L) a = IsLocalization.mk' L r\u2081 s\n[PROOFSTEP]\nobtain \u27e8r\u2082, h\u27e9 : \u2203 r : R, IsLocalization.mk' L 1 s = algebraMap R L r\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nr\u2081 : R\ns : { x // x \u2208 S }\n\u22a2 \u2203 r, IsLocalization.mk' L 1 s = \u2191(algebraMap R L) r\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nr\u2081 : R\ns : { x // x \u2208 S }\nr\u2082 : R\nh : IsLocalization.mk' L 1 s = \u2191(algebraMap R L) r\u2082\n\u22a2 \u2203 a, \u2191(algebraMap R L) a = IsLocalization.mk' L r\u2081 s\n[PROOFSTEP]\nswap\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nr\u2081 : R\ns : { x // x \u2208 S }\nr\u2082 : R\nh : IsLocalization.mk' L 1 s = \u2191(algebraMap R L) r\u2082\n\u22a2 \u2203 a, \u2191(algebraMap R L) a = IsLocalization.mk' L r\u2081 s\n[PROOFSTEP]\nexact \u27e8r\u2081 * r\u2082, by rw [IsLocalization.mk'_eq_mul_mk'_one, map_mul, h]\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nr\u2081 : R\ns : { x // x \u2208 S }\nr\u2082 : R\nh : IsLocalization.mk' L 1 s = \u2191(algebraMap R L) r\u2082\n\u22a2 \u2191(algebraMap R L) (r\u2081 * r\u2082) = IsLocalization.mk' L r\u2081 s\n[PROOFSTEP]\nrw [IsLocalization.mk'_eq_mul_mk'_one, map_mul, h]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nr\u2081 : R\ns : { x // x \u2208 S }\n\u22a2 \u2203 r, IsLocalization.mk' L 1 s = \u2191(algebraMap R L) r\n[PROOFSTEP]\nobtain \u27e8n, r, hr\u27e9 := IsArtinian.exists_pow_succ_smul_dvd (s : R) (1 : R)\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nr\u2081 : R\ns : { x // x \u2208 S }\nn : \u2115\nr : R\nhr : \u2191s ^ Nat.succ n \u2022 r = \u2191s ^ n \u2022 1\n\u22a2 \u2203 r, IsLocalization.mk' L 1 s = \u2191(algebraMap R L) r\n[PROOFSTEP]\nuse r\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nr\u2081 : R\ns : { x // x \u2208 S }\nn : \u2115\nr : R\nhr : \u2191s ^ Nat.succ n \u2022 r = \u2191s ^ n \u2022 1\n\u22a2 IsLocalization.mk' L 1 s = \u2191(algebraMap R L) r\n[PROOFSTEP]\nrw [smul_eq_mul, smul_eq_mul, pow_succ', mul_assoc] at hr \n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nr\u2081 : R\ns : { x // x \u2208 S }\nn : \u2115\nr : R\nhr : \u2191s ^ n * (\u2191s * r) = \u2191s ^ n * 1\n\u22a2 IsLocalization.mk' L 1 s = \u2191(algebraMap R L) r\n[PROOFSTEP]\napply_fun algebraMap R L at hr \n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nr\u2081 : R\ns : { x // x \u2208 S }\nn : \u2115\nr : R\nhr : \u2191(algebraMap R L) (\u2191s ^ n * (\u2191s * r)) = \u2191(algebraMap R L) (\u2191s ^ n * 1)\n\u22a2 IsLocalization.mk' L 1 s = \u2191(algebraMap R L) r\n[PROOFSTEP]\nsimp only [map_mul] at hr \n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst\u271d\u00b2 : CommRing L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsLocalization S L\nr\u2081 : R\ns : { x // x \u2208 S }\nn : \u2115\nr : R\nhr :\n \u2191(algebraMap R L) (\u2191s ^ n) * (\u2191(algebraMap R L) \u2191s * \u2191(algebraMap R L) r) =\n \u2191(algebraMap R L) (\u2191s ^ n) * \u2191(algebraMap R L) 1\n\u22a2 IsLocalization.mk' L 1 s = \u2191(algebraMap R L) r\n[PROOFSTEP]\nrw [\u2190 IsLocalization.mk'_one (M := S) L, IsLocalization.mk'_eq_iff_eq, mul_one, Submonoid.coe_one, \u2190\n (IsLocalization.map_units L (s ^ n)).mul_left_cancel hr, map_mul]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Artinian", "llama_tokens": 34952, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799928951399099, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.43549108874641274}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NormedCommRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nx\u271d\u00b9 : E\nx\u271d : x\u271d\u00b9 \u2208 s\n\u22a2 \u2016\u2191(\u2191B x\u271d\u00b9) 0\u2016 \u2264 1\n[PROOFSTEP]\nsimp only [map_zero, norm_zero, zero_le_one]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NormedCommRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\n\u22a2 polar B s = \u22c2 (x : E) (_ : x \u2208 s), {y | \u2016\u2191(\u2191B x) y\u2016 \u2264 1}\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NormedCommRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nx\u271d : F\n\u22a2 x\u271d \u2208 polar B s \u2194 x\u271d \u2208 \u22c2 (x : E) (_ : x \u2208 s), {y | \u2016\u2191(\u2191B x) y\u2016 \u2264 1}\n[PROOFSTEP]\nsimp only [polar_mem_iff, Set.mem_iInter, Set.mem_setOf_eq]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NormedCommRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\n\u22a2 polar B {0} = Set.univ\n[PROOFSTEP]\nrefine' Set.eq_univ_iff_forall.mpr fun y x hx => _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NormedCommRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ny : F\nx : E\nhx : x \u2208 {0}\n\u22a2 \u2016\u2191(\u2191B x) y\u2016 \u2264 1\n[PROOFSTEP]\nrw [Set.mem_singleton_iff.mp hx, map_zero, LinearMap.zero_apply, norm_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NormedCommRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ny : F\nx : E\nhx : x \u2208 {0}\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nexact zero_le_one\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NormedCommRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nx : E\nhx : x \u2208 s\ny : F\nhy : y \u2208 polar B s\n\u22a2 \u2016\u2191(\u2191(flip B) y) x\u2016 \u2264 1\n[PROOFSTEP]\nrw [B.flip_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NormedCommRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nx : E\nhx : x \u2208 s\ny : F\nhy : y \u2208 polar B s\n\u22a2 \u2016\u2191(\u2191B x) y\u2016 \u2264 1\n[PROOFSTEP]\nexact hy x hx\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NormedCommRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\n\u22a2 IsClosed (polar B s)\n[PROOFSTEP]\nrw [polar_eq_iInter]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NormedCommRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\n\u22a2 IsClosed (\u22c2 (x : E) (_ : x \u2208 s), {y | \u2016\u2191(\u2191B x) y\u2016 \u2264 1})\n[PROOFSTEP]\nrefine' isClosed_iInter fun x => isClosed_iInter fun _ => _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NormedCommRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nx : E\nx\u271d : x \u2208 s\n\u22a2 IsClosed {y | \u2016\u2191(\u2191B x) y\u2016 \u2264 1}\n[PROOFSTEP]\nexact isClosed_le (WeakBilin.eval_continuous B.flip x).norm continuous_const\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nh : SeparatingRight B\n\u22a2 polar B Set.univ = {0}\n[PROOFSTEP]\nrw [Set.eq_singleton_iff_unique_mem]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nh : SeparatingRight B\n\u22a2 0 \u2208 polar B Set.univ \u2227 \u2200 (x : F), x \u2208 polar B Set.univ \u2192 x = 0\n[PROOFSTEP]\nrefine' \u27e8by simp only [zero_mem_polar], fun y hy => h _ fun x => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nh : SeparatingRight B\n\u22a2 0 \u2208 polar B Set.univ\n[PROOFSTEP]\nsimp only [zero_mem_polar]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nh : SeparatingRight B\ny : F\nhy : y \u2208 polar B Set.univ\nx : E\n\u22a2 \u2191(\u2191B x) y = 0\n[PROOFSTEP]\nrefine' norm_le_zero_iff.mp (le_of_forall_le_of_dense fun \u03b5 h\u03b5 => _)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nh : SeparatingRight B\ny : F\nhy : y \u2208 polar B Set.univ\nx : E\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2016\u2191(\u2191B x) y\u2016 \u2264 \u03b5\n[PROOFSTEP]\nrcases NormedField.exists_norm_lt \ud835\udd5c h\u03b5 with \u27e8c, hc, hc\u03b5\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nh : SeparatingRight B\ny : F\nhy : y \u2208 polar B Set.univ\nx : E\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016\nhc\u03b5 : \u2016c\u2016 < \u03b5\n\u22a2 \u2016\u2191(\u2191B x) y\u2016 \u2264 \u03b5\n[PROOFSTEP]\ncalc\n \u2016B x y\u2016 = \u2016c\u2016 * \u2016B (c\u207b\u00b9 \u2022 x) y\u2016 := by\n rw [B.map_smul, LinearMap.smul_apply, Algebra.id.smul_eq_mul, norm_mul, norm_inv, mul_inv_cancel_left\u2080 hc.ne']\n _ \u2264 \u03b5 * 1 := by gcongr; exact hy _ trivial\n _ = \u03b5 := mul_one _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nh : SeparatingRight B\ny : F\nhy : y \u2208 polar B Set.univ\nx : E\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016\nhc\u03b5 : \u2016c\u2016 < \u03b5\n\u22a2 \u2016\u2191(\u2191B x) y\u2016 = \u2016c\u2016 * \u2016\u2191(\u2191B (c\u207b\u00b9 \u2022 x)) y\u2016\n[PROOFSTEP]\nrw [B.map_smul, LinearMap.smul_apply, Algebra.id.smul_eq_mul, norm_mul, norm_inv, mul_inv_cancel_left\u2080 hc.ne']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nh : SeparatingRight B\ny : F\nhy : y \u2208 polar B Set.univ\nx : E\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016\nhc\u03b5 : \u2016c\u2016 < \u03b5\n\u22a2 \u2016c\u2016 * \u2016\u2191(\u2191B (c\u207b\u00b9 \u2022 x)) y\u2016 \u2264 \u03b5 * 1\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2082\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nh : SeparatingRight B\ny : F\nhy : y \u2208 polar B Set.univ\nx : E\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016\nhc\u03b5 : \u2016c\u2016 < \u03b5\n\u22a2 \u2016\u2191(\u2191B (c\u207b\u00b9 \u2022 x)) y\u2016 \u2264 1\n[PROOFSTEP]\nexact hy _ trivial\n", "meta": {"mathlib_filename": "Mathlib.Analysis.LocallyConvex.Polar", "llama_tokens": 3728, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799928900257127, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.43549108589101837}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\na : \u03b1\n\u22a2 NoMaxOrder \u2191(Ici a)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\na : \u03b1\nh : sSup \u2205 \u2264 a\nsrc\u271d\u00b9 : OrderBot { x // a \u2264 x } := orderBot\nsrc\u271d : ConditionallyCompleteLinearOrder { x // a \u2264 x } := Nonneg.conditionallyCompleteLinearOrder\n\u22a2 \u2191(if ht : sSup (Subtype.val '' \u2205) \u2208 Ici a then { val := sSup (Subtype.val '' \u2205), property := ht } else default) = \u2191\u22a5\n[PROOFSTEP]\nrw [bot_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\na : \u03b1\nh : sSup \u2205 \u2264 a\nsrc\u271d\u00b9 : OrderBot { x // a \u2264 x } := orderBot\nsrc\u271d : ConditionallyCompleteLinearOrder { x // a \u2264 x } := Nonneg.conditionallyCompleteLinearOrder\n\u22a2 \u2191(if ht : sSup (Subtype.val '' \u2205) \u2208 Ici a then { val := sSup (Subtype.val '' \u2205), property := ht } else default) =\n \u2191{ val := a, property := (_ : a \u2264 a) }\n[PROOFSTEP]\ncases' h.lt_or_eq with h2 h2\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\na : \u03b1\nh : sSup \u2205 \u2264 a\nsrc\u271d\u00b9 : OrderBot { x // a \u2264 x } := orderBot\nsrc\u271d : ConditionallyCompleteLinearOrder { x // a \u2264 x } := Nonneg.conditionallyCompleteLinearOrder\nh2 : sSup \u2205 < a\n\u22a2 \u2191(if ht : sSup (Subtype.val '' \u2205) \u2208 Ici a then { val := sSup (Subtype.val '' \u2205), property := ht } else default) =\n \u2191{ val := a, property := (_ : a \u2264 a) }\n[PROOFSTEP]\nsimp [h2.not_le]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\na : \u03b1\nh : sSup \u2205 \u2264 a\nsrc\u271d\u00b9 : OrderBot { x // a \u2264 x } := orderBot\nsrc\u271d : ConditionallyCompleteLinearOrder { x // a \u2264 x } := Nonneg.conditionallyCompleteLinearOrder\nh2 : sSup \u2205 = a\n\u22a2 \u2191(if ht : sSup (Subtype.val '' \u2205) \u2208 Ici a then { val := sSup (Subtype.val '' \u2205), property := ht } else default) =\n \u2191{ val := a, property := (_ : a \u2264 a) }\n[PROOFSTEP]\nsimp [h2]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\nsrc\u271d\u00b9 : One { x // 0 \u2264 x } := one\nsrc\u271d : OrderedAddCommMonoid { x // 0 \u2264 x } := orderedAddCommMonoid\n\u22a2 NatCast.natCast 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\nsrc\u271d\u00b9 : One { x // 0 \u2264 x } := one\nsrc\u271d : OrderedAddCommMonoid { x // 0 \u2264 x } := orderedAddCommMonoid\nx\u271d : \u2115\n\u22a2 NatCast.natCast (x\u271d + 1) = NatCast.natCast x\u271d + 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\nsrc\u271d\u00b9 : One { x // 0 \u2264 x } := one\nsrc\u271d : OrderedAddCommMonoid { x // 0 \u2264 x } := orderedAddCommMonoid\nx\u271d : \u2115\n\u22a2 { val := \u2191x\u271d + 1, property := (_ : (fun x => 0 \u2264 x) (\u2191x\u271d + 1)) } = { val := \u2191x\u271d, property := (_ : 0 \u2264 \u2191x\u271d) } + 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\n\u22a2 MonoidWithZero { x // 0 \u2264 x }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCommSemiring \u03b1\n\u22a2 CommMonoidWithZero { x // 0 \u2264 x }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : OrderedCommRing \u03b1\ninst\u271d : NoZeroDivisors \u03b1\nsrc\u271d\u00b9 : CanonicallyOrderedAddMonoid { x // 0 \u2264 x } := canonicallyOrderedAddMonoid\nsrc\u271d : OrderedCommSemiring { x // 0 \u2264 x } := orderedCommSemiring\n\u22a2 \u2200 {a b : { x // 0 \u2264 x }}, a * b = 0 \u2192 a = 0 \u2228 b = 0\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : OrderedCommRing \u03b1\ninst\u271d : NoZeroDivisors \u03b1\nsrc\u271d\u00b9 : CanonicallyOrderedAddMonoid { x // 0 \u2264 x } := canonicallyOrderedAddMonoid\nsrc\u271d : OrderedCommSemiring { x // 0 \u2264 x } := orderedCommSemiring\na : \u03b1\nha : 0 \u2264 a\nb : \u03b1\nhb : 0 \u2264 b\n\u22a2 { val := a, property := ha } * { val := b, property := hb } = 0 \u2192\n { val := a, property := ha } = 0 \u2228 { val := b, property := hb } = 0\n[PROOFSTEP]\nsimp only [mk_mul_mk, mk_eq_zero, mul_eq_zero, imp_self]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\nh : 0 \u2264 a\n\u22a2 toNonneg a = { val := a, property := h }\n[PROOFSTEP]\nsimp [toNonneg, h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\nb : { x // 0 \u2264 x }\n\u22a2 toNonneg a \u2264 b \u2194 a \u2264 \u2191b\n[PROOFSTEP]\ncases' b with b hb\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nhb : 0 \u2264 b\n\u22a2 toNonneg a \u2264 { val := b, property := hb } \u2194 a \u2264 \u2191{ val := b, property := hb }\n[PROOFSTEP]\nsimp [toNonneg, hb]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LinearOrder \u03b1\na : { x // 0 \u2264 x }\nb : \u03b1\n\u22a2 a < toNonneg b \u2194 \u2191a < b\n[PROOFSTEP]\ncases' a with a ha\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LinearOrder \u03b1\nb a : \u03b1\nha : 0 \u2264 a\n\u22a2 { val := a, property := ha } < toNonneg b \u2194 \u2191{ val := a, property := ha } < b\n[PROOFSTEP]\nsimp [toNonneg, ha.not_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\n\u22a2 \u2200 (a b c : { x // 0 \u2264 x }), a - b \u2264 c \u2194 a \u2264 c + b\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9 \u27e8c, hc\u27e9\n[GOAL]\ncase mk.mk.mk\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na : \u03b1\nha : 0 \u2264 a\nb : \u03b1\nhb : 0 \u2264 b\nc : \u03b1\nhc : 0 \u2264 c\n\u22a2 { val := a, property := ha } - { val := b, property := hb } \u2264 { val := c, property := hc } \u2194\n { val := a, property := ha } \u2264 { val := c, property := hc } + { val := b, property := hb }\n[PROOFSTEP]\nsimp only [sub_le_iff_le_add, Subtype.mk_le_mk, mk_sub_mk, mk_add_mk, toNonneg_le, Subtype.coe_mk]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Nonneg.Ring", "llama_tokens": 2331, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321964553657, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.4354459257125089}} {"text": "[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\nh : Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)\n\u22a2 \u2191(Basis.det (OrthonormalBasis.toBasis e)) \u2191f = 1\n[PROOFSTEP]\napply (e.det_to_matrix_orthonormalBasis_real f).resolve_right\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\nh : Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)\n\u22a2 \u00ac\u2191(Basis.det (OrthonormalBasis.toBasis e)) \u2191f = -1\n[PROOFSTEP]\nhave : 0 < e.toBasis.det f := by\n rw [e.toBasis.orientation_eq_iff_det_pos] at h \n simpa using h\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\nh : Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)\n\u22a2 0 < \u2191(Basis.det (OrthonormalBasis.toBasis e)) \u2191f\n[PROOFSTEP]\nrw [e.toBasis.orientation_eq_iff_det_pos] at h \n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\nh : 0 < \u2191(Basis.det (OrthonormalBasis.toBasis e)) \u2191(OrthonormalBasis.toBasis f)\n\u22a2 0 < \u2191(Basis.det (OrthonormalBasis.toBasis e)) \u2191f\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\nh : Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)\nthis : 0 < \u2191(Basis.det (OrthonormalBasis.toBasis e)) \u2191f\n\u22a2 \u00ac\u2191(Basis.det (OrthonormalBasis.toBasis e)) \u2191f = -1\n[PROOFSTEP]\nlinarith\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\nh : Basis.orientation (OrthonormalBasis.toBasis e) \u2260 Basis.orientation (OrthonormalBasis.toBasis f)\n\u22a2 \u2191(Basis.det (OrthonormalBasis.toBasis e)) \u2191f = -1\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\nh : \u2191(Basis.det (OrthonormalBasis.toBasis e)) \u2191f \u2260 -1\n\u22a2 Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)\n[PROOFSTEP]\nsimp [e.toBasis.orientation_eq_iff_det_pos, (e.det_to_matrix_orthonormalBasis_real f).resolve_right h]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\n\u22a2 Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f) \u2194\n Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\n\u22a2 Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f) \u2192\n Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\nh : Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f)\n\u22a2 Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)\n[PROOFSTEP]\ndsimp [Basis.orientation]\n[GOAL]\ncase mp\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\nh : Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f)\n\u22a2 rayOfNeZero \u211d (Basis.det (OrthonormalBasis.toBasis e)) (_ : Basis.det (OrthonormalBasis.toBasis e) \u2260 0) =\n rayOfNeZero \u211d (Basis.det (OrthonormalBasis.toBasis f)) (_ : Basis.det (OrthonormalBasis.toBasis f) \u2260 0)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mpr\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\n\u22a2 Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f) \u2192\n Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\nh : Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)\n\u22a2 Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f)\n[PROOFSTEP]\nrw [e.toBasis.det.eq_smul_basis_det f.toBasis]\n[GOAL]\ncase mpr\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\nh : Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)\n\u22a2 \u2191(Basis.det (OrthonormalBasis.toBasis e)) \u2191(OrthonormalBasis.toBasis f) \u2022 Basis.det (OrthonormalBasis.toBasis f) =\n Basis.det (OrthonormalBasis.toBasis f)\n[PROOFSTEP]\nsimp [e.det_to_matrix_orthonormalBasis_of_same_orientation f h]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\nh : Basis.orientation (OrthonormalBasis.toBasis e) \u2260 Basis.orientation (OrthonormalBasis.toBasis f)\n\u22a2 Basis.det (OrthonormalBasis.toBasis e) = -Basis.det (OrthonormalBasis.toBasis f)\n[PROOFSTEP]\nrw [e.toBasis.det.eq_smul_basis_det f.toBasis]\n -- Porting note: added `neg_one_smul` with explicit type\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\nh : Basis.orientation (OrthonormalBasis.toBasis e) \u2260 Basis.orientation (OrthonormalBasis.toBasis f)\n\u22a2 \u2191(Basis.det (OrthonormalBasis.toBasis e)) \u2191(OrthonormalBasis.toBasis f) \u2022 Basis.det (OrthonormalBasis.toBasis f) =\n -Basis.det (OrthonormalBasis.toBasis f)\n[PROOFSTEP]\nsimp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h, neg_one_smul \u211d (M := AlternatingMap \u211d E \u211d \u03b9)]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\n\u22a2 Orthonormal \u211d \u2191(Basis.adjustToOrientation (OrthonormalBasis.toBasis e) x)\n[PROOFSTEP]\napply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\n\u22a2 \u2200 (i : \u03b9),\n \u2191(Basis.adjustToOrientation (OrthonormalBasis.toBasis e) x) i = \u2191e i \u2228\n \u2191(Basis.adjustToOrientation (OrthonormalBasis.toBasis e) x) i = -\u2191e i\n[PROOFSTEP]\nsimpa using e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\n\u22a2 Basis.orientation (OrthonormalBasis.toBasis (adjustToOrientation e x)) = x\n[PROOFSTEP]\nrw [e.toBasis_adjustToOrientation]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\n\u22a2 Basis.orientation (Basis.adjustToOrientation (OrthonormalBasis.toBasis e) x) = x\n[PROOFSTEP]\nexact e.toBasis.orientation_adjustToOrientation x\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\ni : \u03b9\n\u22a2 \u2191(adjustToOrientation e x) i = \u2191e i \u2228 \u2191(adjustToOrientation e x) i = -\u2191e i\n[PROOFSTEP]\nsimpa [\u2190 e.toBasis_adjustToOrientation] using e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x i\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\n\u22a2 Basis.det (OrthonormalBasis.toBasis (adjustToOrientation e x)) = Basis.det (OrthonormalBasis.toBasis e) \u2228\n Basis.det (OrthonormalBasis.toBasis (adjustToOrientation e x)) = -Basis.det (OrthonormalBasis.toBasis e)\n[PROOFSTEP]\nsimpa using e.toBasis.det_adjustToOrientation x\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \u211d E\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nne : Nonempty \u03b9\ne f : OrthonormalBasis \u03b9 \u211d E\nx : Orientation \u211d E \u03b9\nv : \u03b9 \u2192 E\n\u22a2 |\u2191(Basis.det (OrthonormalBasis.toBasis (adjustToOrientation e x))) v| = |\u2191(Basis.det (OrthonormalBasis.toBasis e)) v|\n[PROOFSTEP]\nsimp [toBasis_adjustToOrientation]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\nhn : 0 < n\nh : finrank \u211d E = n\nx : Orientation \u211d E (Fin n)\n\u22a2 OrthonormalBasis (Fin n) \u211d E\n[PROOFSTEP]\nhaveI := Fin.pos_iff_nonempty.1 hn\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\nhn : 0 < n\nh : finrank \u211d E = n\nx : Orientation \u211d E (Fin n)\nthis : Nonempty (Fin n)\n\u22a2 OrthonormalBasis (Fin n) \u211d E\n[PROOFSTEP]\nhaveI := finiteDimensional_of_finrank (h.symm \u25b8 hn : 0 < finrank \u211d E)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\nhn : 0 < n\nh : finrank \u211d E = n\nx : Orientation \u211d E (Fin n)\nthis\u271d : Nonempty (Fin n)\nthis : FiniteDimensional \u211d E\n\u22a2 OrthonormalBasis (Fin n) \u211d E\n[PROOFSTEP]\nexact ((@stdOrthonormalBasis _ _ _ _ _ this).reindex <| finCongr h).adjustToOrientation x\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\nhn : 0 < n\nh : finrank \u211d E = n\nx : Orientation \u211d E (Fin n)\n\u22a2 Basis.orientation (OrthonormalBasis.toBasis (Orientation.finOrthonormalBasis hn h x)) = x\n[PROOFSTEP]\nhaveI := Fin.pos_iff_nonempty.1 hn\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\nhn : 0 < n\nh : finrank \u211d E = n\nx : Orientation \u211d E (Fin n)\nthis : Nonempty (Fin n)\n\u22a2 Basis.orientation (OrthonormalBasis.toBasis (Orientation.finOrthonormalBasis hn h x)) = x\n[PROOFSTEP]\nhaveI := finiteDimensional_of_finrank (h.symm \u25b8 hn : 0 < finrank \u211d E)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\nhn : 0 < n\nh : finrank \u211d E = n\nx : Orientation \u211d E (Fin n)\nthis\u271d : Nonempty (Fin n)\nthis : FiniteDimensional \u211d E\n\u22a2 Basis.orientation (OrthonormalBasis.toBasis (Orientation.finOrthonormalBasis hn h x)) = x\n[PROOFSTEP]\nexact ((@stdOrthonormalBasis _ _ _ _ _ this).reindex <| finCongr h).orientation_adjustToOrientation x\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\n\u22a2 IsEmpty (Fin Nat.zero)\n[PROOFSTEP]\nsimp only [Nat.zero_eq]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\n\u22a2 IsEmpty (Fin 0)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\n\u22a2 AlternatingMap \u211d E \u211d (Fin n)\n[PROOFSTEP]\nclassical\ncases' n with n\n\u00b7 let opos : AlternatingMap \u211d E \u211d (Fin 0) := AlternatingMap.constOfIsEmpty \u211d E (Fin 0) (1 : \u211d)\n exact o.eq_or_eq_neg_of_isEmpty.by_cases (fun _ => opos) fun _ => -opos\n\u00b7 exact (o.finOrthonormalBasis n.succ_pos _i.out).toBasis.det\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\n\u22a2 AlternatingMap \u211d E \u211d (Fin n)\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\n\u22a2 AlternatingMap \u211d E \u211d (Fin Nat.zero)\n[PROOFSTEP]\nlet opos : AlternatingMap \u211d E \u211d (Fin 0) := AlternatingMap.constOfIsEmpty \u211d E (Fin 0) (1 : \u211d)\n[GOAL]\ncase zero\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nopos : AlternatingMap \u211d E \u211d (Fin 0) := AlternatingMap.constOfIsEmpty \u211d E (Fin 0) 1\n\u22a2 AlternatingMap \u211d E \u211d (Fin Nat.zero)\n[PROOFSTEP]\nexact o.eq_or_eq_neg_of_isEmpty.by_cases (fun _ => opos) fun _ => -opos\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\n\u22a2 AlternatingMap \u211d E \u211d (Fin (Nat.succ n))\n[PROOFSTEP]\nexact (o.finOrthonormalBasis n.succ_pos _i.out).toBasis.det\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i\u271d : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\n_i : Fact (finrank \u211d E = 0)\n\u22a2 volumeForm positiveOrientation = \u2191AlternatingMap.constLinearEquivOfIsEmpty 1\n[PROOFSTEP]\nsimp [volumeForm, Or.by_cases, if_pos]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i\u271d : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\n_i : Fact (finrank \u211d E = 0)\n\u22a2 volumeForm (-positiveOrientation) = -\u2191AlternatingMap.constLinearEquivOfIsEmpty 1\n[PROOFSTEP]\nsimp_rw [volumeForm, Or.by_cases, positiveOrientation]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i\u271d : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\n_i : Fact (finrank \u211d E = 0)\n\u22a2 Nat.rec (motive := fun t => 0 = t \u2192 AlternatingMap \u211d E \u211d (Fin 0))\n (fun h =>\n if hp :\n -rayOfNeZero \u211d (\u2191AlternatingMap.constLinearEquivOfIsEmpty 1)\n (_ :\n \u2191AlternatingMap.constLinearEquivOfIsEmpty 1 \u2260\n \u2191AlternatingMap.constLinearEquivOfIsEmpty AddMonoid.toZero.1) =\n rayOfNeZero \u211d (\u2191AlternatingMap.constLinearEquivOfIsEmpty 1)\n (_ :\n \u2191AlternatingMap.constLinearEquivOfIsEmpty 1 \u2260\n \u2191AlternatingMap.constLinearEquivOfIsEmpty AddMonoid.toZero.1) then\n AlternatingMap.constOfIsEmpty \u211d E (Fin 0) 1\n else -AlternatingMap.constOfIsEmpty \u211d E (Fin 0) 1)\n (fun n n_ih h =>\n Eq.rec (motive := fun x x_1 =>\n [_i : Fact (finrank \u211d E = x)] \u2192 Orientation \u211d E (Fin x) \u2192 AlternatingMap \u211d E \u211d (Fin x))\n (fun [Fact (finrank \u211d E = Nat.succ n)] o =>\n Basis.det\n (OrthonormalBasis.toBasis\n (Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o)))\n (_ : Nat.succ n = 0)\n (-rayOfNeZero \u211d (\u2191AlternatingMap.constLinearEquivOfIsEmpty 1)\n (_ :\n \u2191AlternatingMap.constLinearEquivOfIsEmpty 1 \u2260\n \u2191AlternatingMap.constLinearEquivOfIsEmpty AddMonoid.toZero.1)))\n 0 (_ : 0 = 0) =\n -\u2191AlternatingMap.constLinearEquivOfIsEmpty 1\n[PROOFSTEP]\napply if_neg\n[GOAL]\ncase hnc\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i\u271d : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\n_i : Fact (finrank \u211d E = 0)\n\u22a2 \u00ac-rayOfNeZero \u211d (\u2191AlternatingMap.constLinearEquivOfIsEmpty 1)\n (_ :\n \u2191AlternatingMap.constLinearEquivOfIsEmpty 1 \u2260\n \u2191AlternatingMap.constLinearEquivOfIsEmpty AddMonoid.toZero.1) =\n rayOfNeZero \u211d (\u2191AlternatingMap.constLinearEquivOfIsEmpty 1)\n (_ : \u2191AlternatingMap.constLinearEquivOfIsEmpty 1 \u2260 \u2191AlternatingMap.constLinearEquivOfIsEmpty AddMonoid.toZero.1)\n[PROOFSTEP]\nsimp only [neg_rayOfNeZero]\n[GOAL]\ncase hnc\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i\u271d : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\n_i : Fact (finrank \u211d E = 0)\n\u22a2 \u00acrayOfNeZero \u211d (-\u2191AlternatingMap.constLinearEquivOfIsEmpty 1) (_ : -\u2191AlternatingMap.constLinearEquivOfIsEmpty 1 \u2260 0) =\n rayOfNeZero \u211d (\u2191AlternatingMap.constLinearEquivOfIsEmpty 1)\n (_ : \u2191AlternatingMap.constLinearEquivOfIsEmpty 1 \u2260 \u2191AlternatingMap.constLinearEquivOfIsEmpty AddMonoid.toZero.1)\n[PROOFSTEP]\nrw [ray_eq_iff, SameRay.sameRay_comm]\n[GOAL]\ncase hnc\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i\u271d : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\n_i : Fact (finrank \u211d E = 0)\n\u22a2 \u00acSameRay \u211d (\u2191AlternatingMap.constLinearEquivOfIsEmpty 1) (-\u2191AlternatingMap.constLinearEquivOfIsEmpty 1)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase hnc\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i\u271d : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\n_i : Fact (finrank \u211d E = 0)\nh : SameRay \u211d (\u2191AlternatingMap.constLinearEquivOfIsEmpty 1) (-\u2191AlternatingMap.constLinearEquivOfIsEmpty 1)\n\u22a2 False\n[PROOFSTEP]\nsimpa using congr_arg AlternatingMap.constLinearEquivOfIsEmpty.symm (eq_zero_of_sameRay_self_neg h)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\nb : OrthonormalBasis (Fin n) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) = o\n\u22a2 volumeForm o = Basis.det (OrthonormalBasis.toBasis b)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nb : OrthonormalBasis (Fin Nat.zero) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) = o\n\u22a2 volumeForm o = Basis.det (OrthonormalBasis.toBasis b)\n[PROOFSTEP]\nclassical\nhave : o = positiveOrientation := hb.symm.trans b.toBasis.orientation_isEmpty\nsimp_rw [volumeForm, Or.by_cases, dif_pos this, Basis.det_isEmpty]\n[GOAL]\ncase zero\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nb : OrthonormalBasis (Fin Nat.zero) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) = o\n\u22a2 volumeForm o = Basis.det (OrthonormalBasis.toBasis b)\n[PROOFSTEP]\nhave : o = positiveOrientation := hb.symm.trans b.toBasis.orientation_isEmpty\n[GOAL]\ncase zero\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nb : OrthonormalBasis (Fin Nat.zero) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) = o\nthis : o = positiveOrientation\n\u22a2 volumeForm o = Basis.det (OrthonormalBasis.toBasis b)\n[PROOFSTEP]\nsimp_rw [volumeForm, Or.by_cases, dif_pos this, Basis.det_isEmpty]\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn\u271d : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n\u271d)\no : Orientation \u211d E (Fin (Nat.succ n\u271d))\nb : OrthonormalBasis (Fin (Nat.succ n\u271d)) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) = o\n\u22a2 volumeForm o = Basis.det (OrthonormalBasis.toBasis b)\n[PROOFSTEP]\nsimp_rw [volumeForm]\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn\u271d : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n\u271d)\no : Orientation \u211d E (Fin (Nat.succ n\u271d))\nb : OrthonormalBasis (Fin (Nat.succ n\u271d)) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) = o\n\u22a2 Basis.det\n (OrthonormalBasis.toBasis\n (Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n\u271d) (_ : finrank \u211d E = Nat.succ n\u271d) o)) =\n Basis.det (OrthonormalBasis.toBasis b)\n[PROOFSTEP]\nrw [same_orientation_iff_det_eq_det, hb]\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn\u271d : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n\u271d)\no : Orientation \u211d E (Fin (Nat.succ n\u271d))\nb : OrthonormalBasis (Fin (Nat.succ n\u271d)) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) = o\n\u22a2 Basis.orientation\n (OrthonormalBasis.toBasis\n (Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n\u271d) (_ : finrank \u211d E = Nat.succ n\u271d) o)) =\n o\n[PROOFSTEP]\nexact o.finOrthonormalBasis_orientation _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\nb : OrthonormalBasis (Fin n) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) \u2260 o\n\u22a2 volumeForm o = -Basis.det (OrthonormalBasis.toBasis b)\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nb : OrthonormalBasis (Fin Nat.zero) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) \u2260 o\n\u22a2 volumeForm o = -Basis.det (OrthonormalBasis.toBasis b)\n[PROOFSTEP]\nclassical\nhave : positiveOrientation \u2260 o := by rwa [b.toBasis.orientation_isEmpty] at hb \nsimp_rw [volumeForm, Or.by_cases, dif_neg this.symm, Basis.det_isEmpty]\n[GOAL]\ncase zero\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nb : OrthonormalBasis (Fin Nat.zero) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) \u2260 o\n\u22a2 volumeForm o = -Basis.det (OrthonormalBasis.toBasis b)\n[PROOFSTEP]\nhave : positiveOrientation \u2260 o := by rwa [b.toBasis.orientation_isEmpty] at hb \n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nb : OrthonormalBasis (Fin Nat.zero) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) \u2260 o\n\u22a2 positiveOrientation \u2260 o\n[PROOFSTEP]\nrwa [b.toBasis.orientation_isEmpty] at hb \n[GOAL]\ncase zero\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nb : OrthonormalBasis (Fin Nat.zero) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) \u2260 o\nthis : positiveOrientation \u2260 o\n\u22a2 volumeForm o = -Basis.det (OrthonormalBasis.toBasis b)\n[PROOFSTEP]\nsimp_rw [volumeForm, Or.by_cases, dif_neg this.symm, Basis.det_isEmpty]\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) \u2260 o\n\u22a2 volumeForm o = -Basis.det (OrthonormalBasis.toBasis b)\n[PROOFSTEP]\nlet e : OrthonormalBasis (Fin n.succ) \u211d E := o.finOrthonormalBasis n.succ_pos Fact.out\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) \u2260 o\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\n\u22a2 volumeForm o = -Basis.det (OrthonormalBasis.toBasis b)\n[PROOFSTEP]\nsimp_rw [volumeForm]\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) \u2260 o\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\n\u22a2 Basis.det\n (OrthonormalBasis.toBasis\n (Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o)) =\n -Basis.det (OrthonormalBasis.toBasis b)\n[PROOFSTEP]\napply e.det_eq_neg_det_of_opposite_orientation b\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) \u2260 o\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\n\u22a2 Basis.orientation (OrthonormalBasis.toBasis e) \u2260 Basis.orientation (OrthonormalBasis.toBasis b)\n[PROOFSTEP]\nconvert hb.symm\n[GOAL]\ncase h.e'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E\nhb : Basis.orientation (OrthonormalBasis.toBasis b) \u2260 o\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\n\u22a2 Basis.orientation (OrthonormalBasis.toBasis e) = o\n[PROOFSTEP]\nexact o.finOrthonormalBasis_orientation _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\n\u22a2 volumeForm (-o) = -volumeForm o\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\n\u22a2 volumeForm (-o) = -volumeForm o\n[PROOFSTEP]\nrefine' o.eq_or_eq_neg_of_isEmpty.elim _ _\n[GOAL]\ncase zero.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\n\u22a2 o = positiveOrientation \u2192 volumeForm (-o) = -volumeForm o\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase zero.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\n\u22a2 o = -positiveOrientation \u2192 volumeForm (-o) = -volumeForm o\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase zero.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\n\u22a2 volumeForm (-positiveOrientation) = -volumeForm positiveOrientation\n[PROOFSTEP]\nsimp [volumeForm_zero_neg]\n[GOAL]\ncase zero.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\n\u22a2 volumeForm (- -positiveOrientation) = -volumeForm (-positiveOrientation)\n[PROOFSTEP]\nrw [neg_neg (positiveOrientation (R := \u211d))]\n -- Porting note: added\n[GOAL]\ncase zero.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\n\u22a2 volumeForm positiveOrientation = -volumeForm (-positiveOrientation)\n[PROOFSTEP]\nsimp [volumeForm_zero_neg]\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\n\u22a2 volumeForm (-o) = -volumeForm o\n[PROOFSTEP]\nlet e : OrthonormalBasis (Fin n.succ) \u211d E := o.finOrthonormalBasis n.succ_pos Fact.out\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\n\u22a2 volumeForm (-o) = -volumeForm o\n[PROOFSTEP]\nhave h\u2081 : e.toBasis.orientation = o := o.finOrthonormalBasis_orientation _ _\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\nh\u2081 : Basis.orientation (OrthonormalBasis.toBasis e) = o\n\u22a2 volumeForm (-o) = -volumeForm o\n[PROOFSTEP]\nhave h\u2082 : e.toBasis.orientation \u2260 -o := by\n symm\n rw [e.toBasis.orientation_ne_iff_eq_neg, h\u2081]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\nh\u2081 : Basis.orientation (OrthonormalBasis.toBasis e) = o\n\u22a2 Basis.orientation (OrthonormalBasis.toBasis e) \u2260 -o\n[PROOFSTEP]\nsymm\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\nh\u2081 : Basis.orientation (OrthonormalBasis.toBasis e) = o\n\u22a2 -o \u2260 Basis.orientation (OrthonormalBasis.toBasis e)\n[PROOFSTEP]\nrw [e.toBasis.orientation_ne_iff_eq_neg, h\u2081]\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\nh\u2081 : Basis.orientation (OrthonormalBasis.toBasis e) = o\nh\u2082 : Basis.orientation (OrthonormalBasis.toBasis e) \u2260 -o\n\u22a2 volumeForm (-o) = -volumeForm o\n[PROOFSTEP]\nrw [o.volumeForm_robust e h\u2081, (-o).volumeForm_robust_neg e h\u2082]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\nb : OrthonormalBasis (Fin n) \u211d E\nv : Fin n \u2192 E\n\u22a2 |\u2191(volumeForm o) v| = |\u2191(Basis.det (OrthonormalBasis.toBasis b)) v|\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nb : OrthonormalBasis (Fin Nat.zero) \u211d E\nv : Fin Nat.zero \u2192 E\n\u22a2 |\u2191(volumeForm o) v| = |\u2191(Basis.det (OrthonormalBasis.toBasis b)) v|\n[PROOFSTEP]\nrefine' o.eq_or_eq_neg_of_isEmpty.elim _ _\n[GOAL]\ncase zero.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nb : OrthonormalBasis (Fin Nat.zero) \u211d E\nv : Fin Nat.zero \u2192 E\n\u22a2 o = positiveOrientation \u2192 |\u2191(volumeForm o) v| = |\u2191(Basis.det (OrthonormalBasis.toBasis b)) v|\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase zero.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nb : OrthonormalBasis (Fin Nat.zero) \u211d E\nv : Fin Nat.zero \u2192 E\n\u22a2 o = -positiveOrientation \u2192 |\u2191(volumeForm o) v| = |\u2191(Basis.det (OrthonormalBasis.toBasis b)) v|\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase zero.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\nb : OrthonormalBasis (Fin Nat.zero) \u211d E\nv : Fin Nat.zero \u2192 E\n\u22a2 |\u2191(volumeForm positiveOrientation) v| = |\u2191(Basis.det (OrthonormalBasis.toBasis b)) v|\n[PROOFSTEP]\nsimp\n[GOAL]\ncase zero.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\nb : OrthonormalBasis (Fin Nat.zero) \u211d E\nv : Fin Nat.zero \u2192 E\n\u22a2 |\u2191(volumeForm (-positiveOrientation)) v| = |\u2191(Basis.det (OrthonormalBasis.toBasis b)) v|\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn\u271d : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n\u271d)\no : Orientation \u211d E (Fin (Nat.succ n\u271d))\nb : OrthonormalBasis (Fin (Nat.succ n\u271d)) \u211d E\nv : Fin (Nat.succ n\u271d) \u2192 E\n\u22a2 |\u2191(volumeForm o) v| = |\u2191(Basis.det (OrthonormalBasis.toBasis b)) v|\n[PROOFSTEP]\nrw [o.volumeForm_robust (b.adjustToOrientation o) (b.orientation_adjustToOrientation o), b.abs_det_adjustToOrientation]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\nv : Fin n \u2192 E\n\u22a2 |\u2191(volumeForm o) v| \u2264 \u220f i : Fin n, \u2016v i\u2016\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nv : Fin Nat.zero \u2192 E\n\u22a2 |\u2191(volumeForm o) v| \u2264 \u220f i : Fin Nat.zero, \u2016v i\u2016\n[PROOFSTEP]\nrefine' o.eq_or_eq_neg_of_isEmpty.elim _ _\n[GOAL]\ncase zero.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nv : Fin Nat.zero \u2192 E\n\u22a2 o = positiveOrientation \u2192 |\u2191(volumeForm o) v| \u2264 \u220f i : Fin Nat.zero, \u2016v i\u2016\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase zero.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nv : Fin Nat.zero \u2192 E\n\u22a2 o = -positiveOrientation \u2192 |\u2191(volumeForm o) v| \u2264 \u220f i : Fin Nat.zero, \u2016v i\u2016\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase zero.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\nv : Fin Nat.zero \u2192 E\n\u22a2 |\u2191(volumeForm positiveOrientation) v| \u2264 \u220f i : Fin Nat.zero, \u2016v i\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase zero.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\nv : Fin Nat.zero \u2192 E\n\u22a2 |\u2191(volumeForm (-positiveOrientation)) v| \u2264 \u220f i : Fin Nat.zero, \u2016v i\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\n\u22a2 |\u2191(volumeForm o) v| \u2264 \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\nhaveI : FiniteDimensional \u211d E := fact_finiteDimensional_of_finrank_eq_succ n\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nthis : FiniteDimensional \u211d E\n\u22a2 |\u2191(volumeForm o) v| \u2264 \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\nhave : finrank \u211d E = Fintype.card (Fin n.succ) := by simpa using _i.out\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nthis : FiniteDimensional \u211d E\n\u22a2 finrank \u211d E = Fintype.card (Fin (Nat.succ n))\n[PROOFSTEP]\nsimpa using _i.out\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nthis\u271d : FiniteDimensional \u211d E\nthis : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\n\u22a2 |\u2191(volumeForm o) v| \u2264 \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\nlet b : OrthonormalBasis (Fin n.succ) \u211d E := gramSchmidtOrthonormalBasis this v\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nthis\u271d : FiniteDimensional \u211d E\nthis : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis this v\n\u22a2 |\u2191(volumeForm o) v| \u2264 \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\nhave hb : b.toBasis.det v = \u220f i, \u27eab i, v i\u27eb := gramSchmidtOrthonormalBasis_det this v\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nthis\u271d : FiniteDimensional \u211d E\nthis : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis this v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\n\u22a2 |\u2191(volumeForm o) v| \u2264 \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\nrw [o.volumeForm_robust' b, hb, Finset.abs_prod]\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nthis\u271d : FiniteDimensional \u211d E\nthis : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis this v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\n\u22a2 \u220f x : Fin (Nat.succ n), |inner (\u2191b x) (v x)| \u2264 \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\napply Finset.prod_le_prod\n[GOAL]\ncase succ.h0\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nthis\u271d : FiniteDimensional \u211d E\nthis : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis this v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\n\u22a2 \u2200 (i : Fin (Nat.succ n)), i \u2208 Finset.univ \u2192 0 \u2264 |inner (\u2191b i) (v i)|\n[PROOFSTEP]\nintro i _\n[GOAL]\ncase succ.h0\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nthis\u271d : FiniteDimensional \u211d E\nthis : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis this v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\ni : Fin (Nat.succ n)\na\u271d : i \u2208 Finset.univ\n\u22a2 0 \u2264 |inner (\u2191b i) (v i)|\n[PROOFSTEP]\npositivity\n[GOAL]\ncase succ.h1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nthis\u271d : FiniteDimensional \u211d E\nthis : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis this v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\n\u22a2 \u2200 (i : Fin (Nat.succ n)), i \u2208 Finset.univ \u2192 |inner (\u2191b i) (v i)| \u2264 \u2016v i\u2016\n[PROOFSTEP]\nintro i _\n[GOAL]\ncase succ.h1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nthis\u271d : FiniteDimensional \u211d E\nthis : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis this v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\ni : Fin (Nat.succ n)\na\u271d : i \u2208 Finset.univ\n\u22a2 |inner (\u2191b i) (v i)| \u2264 \u2016v i\u2016\n[PROOFSTEP]\nconvert abs_real_inner_le_norm (b i) (v i)\n[GOAL]\ncase h.e'_4\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nthis\u271d : FiniteDimensional \u211d E\nthis : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis this v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\ni : Fin (Nat.succ n)\na\u271d : i \u2208 Finset.univ\n\u22a2 \u2016v i\u2016 = \u2016\u2191b i\u2016 * \u2016v i\u2016\n[PROOFSTEP]\nsimp [b.orthonormal.1 i]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\nv : Fin n \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\n\u22a2 |\u2191(volumeForm o) v| = \u220f i : Fin n, \u2016v i\u2016\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nv : Fin Nat.zero \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\n\u22a2 |\u2191(volumeForm o) v| = \u220f i : Fin Nat.zero, \u2016v i\u2016\n[PROOFSTEP]\nrefine' o.eq_or_eq_neg_of_isEmpty.elim _ _\n[GOAL]\ncase zero.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nv : Fin Nat.zero \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\n\u22a2 o = positiveOrientation \u2192 |\u2191(volumeForm o) v| = \u220f i : Fin Nat.zero, \u2016v i\u2016\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase zero.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nv : Fin Nat.zero \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\n\u22a2 o = -positiveOrientation \u2192 |\u2191(volumeForm o) v| = \u220f i : Fin Nat.zero, \u2016v i\u2016\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase zero.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\nv : Fin Nat.zero \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\n\u22a2 |\u2191(volumeForm positiveOrientation) v| = \u220f i : Fin Nat.zero, \u2016v i\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase zero.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n_i : Fact (finrank \u211d E = Nat.zero)\nv : Fin Nat.zero \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\n\u22a2 |\u2191(volumeForm (-positiveOrientation)) v| = \u220f i : Fin Nat.zero, \u2016v i\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\n\u22a2 |\u2191(volumeForm o) v| = \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\nhaveI : FiniteDimensional \u211d E := fact_finiteDimensional_of_finrank_eq_succ n\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\n\u22a2 |\u2191(volumeForm o) v| = \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\nhave hdim : finrank \u211d E = Fintype.card (Fin n.succ) := by simpa using _i.out\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\n\u22a2 finrank \u211d E = Fintype.card (Fin (Nat.succ n))\n[PROOFSTEP]\nsimpa using _i.out\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\n\u22a2 |\u2191(volumeForm o) v| = \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\nlet b : OrthonormalBasis (Fin n.succ) \u211d E := gramSchmidtOrthonormalBasis hdim v\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\n\u22a2 |\u2191(volumeForm o) v| = \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\nhave hb : b.toBasis.det v = \u220f i, \u27eab i, v i\u27eb := gramSchmidtOrthonormalBasis_det hdim v\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\n\u22a2 |\u2191(volumeForm o) v| = \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\nrw [o.volumeForm_robust' b, hb, Finset.abs_prod]\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\n\u22a2 \u220f x : Fin (Nat.succ n), |inner (\u2191b x) (v x)| = \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\nby_cases h : \u2203 i, v i = 0\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\nh : \u2203 i, v i = 0\n\u22a2 \u220f x : Fin (Nat.succ n), |inner (\u2191b x) (v x)| = \u220f i : Fin (Nat.succ n), \u2016v i\u2016\ncase neg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\nh : \u00ac\u2203 i, v i = 0\n\u22a2 \u220f x : Fin (Nat.succ n), |inner (\u2191b x) (v x)| = \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := h\n[GOAL]\ncase pos.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\ni : Fin (Nat.succ n)\nhi : v i = 0\n\u22a2 \u220f x : Fin (Nat.succ n), |inner (\u2191b x) (v x)| = \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\nrw [Finset.prod_eq_zero (Finset.mem_univ i), Finset.prod_eq_zero (Finset.mem_univ i)]\n[GOAL]\ncase pos.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\ni : Fin (Nat.succ n)\nhi : v i = 0\n\u22a2 \u2016v i\u2016 = 0\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\ncase pos.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\ni : Fin (Nat.succ n)\nhi : v i = 0\n\u22a2 |inner (\u2191b i) (v i)| = 0\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\nh : \u00ac\u2203 i, v i = 0\n\u22a2 \u220f x : Fin (Nat.succ n), |inner (\u2191b x) (v x)| = \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\nh : \u2200 (i : Fin (Nat.succ n)), v i \u2260 0\n\u22a2 \u220f x : Fin (Nat.succ n), |inner (\u2191b x) (v x)| = \u220f i : Fin (Nat.succ n), \u2016v i\u2016\n[PROOFSTEP]\ncongr\n[GOAL]\ncase neg.e_f\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\nh : \u2200 (i : Fin (Nat.succ n)), v i \u2260 0\n\u22a2 (fun x => |inner (\u2191b x) (v x)|) = fun i => \u2016v i\u2016\n[PROOFSTEP]\next i\n[GOAL]\ncase neg.e_f.h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\nhb : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\nh : \u2200 (i : Fin (Nat.succ n)), v i \u2260 0\ni : Fin (Nat.succ n)\n\u22a2 |inner (\u2191b i) (v i)| = \u2016v i\u2016\n[PROOFSTEP]\nhave hb : b i = \u2016v i\u2016\u207b\u00b9 \u2022 v i := gramSchmidtOrthonormalBasis_apply_of_orthogonal hdim hv (h i)\n[GOAL]\ncase neg.e_f.h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\nhb\u271d : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\nh : \u2200 (i : Fin (Nat.succ n)), v i \u2260 0\ni : Fin (Nat.succ n)\nhb : \u2191b i = \u2016v i\u2016\u207b\u00b9 \u2022 v i\n\u22a2 |inner (\u2191b i) (v i)| = \u2016v i\u2016\n[PROOFSTEP]\nsimp only [hb, inner_smul_left, real_inner_self_eq_norm_mul_norm, IsROrC.conj_to_real]\n[GOAL]\ncase neg.e_f.h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\nhb\u271d : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\nh : \u2200 (i : Fin (Nat.succ n)), v i \u2260 0\ni : Fin (Nat.succ n)\nhb : \u2191b i = \u2016v i\u2016\u207b\u00b9 \u2022 v i\n\u22a2 |\u2016v i\u2016\u207b\u00b9 * (\u2016v i\u2016 * \u2016v i\u2016)| = \u2016v i\u2016\n[PROOFSTEP]\nrw [abs_of_nonneg]\n[GOAL]\ncase neg.e_f.h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\nhb\u271d : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\nh : \u2200 (i : Fin (Nat.succ n)), v i \u2260 0\ni : Fin (Nat.succ n)\nhb : \u2191b i = \u2016v i\u2016\u207b\u00b9 \u2022 v i\n\u22a2 \u2016v i\u2016\u207b\u00b9 * (\u2016v i\u2016 * \u2016v i\u2016) = \u2016v i\u2016\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase neg.e_f.h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nv : Fin (Nat.succ n) \u2192 E\nhv : Pairwise fun i j => inner (v i) (v j) = 0\nthis : FiniteDimensional \u211d E\nhdim : finrank \u211d E = Fintype.card (Fin (Nat.succ n))\nb : OrthonormalBasis (Fin (Nat.succ n)) \u211d E := gramSchmidtOrthonormalBasis hdim v\nhb\u271d : \u2191(Basis.det (OrthonormalBasis.toBasis b)) v = \u220f i : Fin (Nat.succ n), inner (\u2191b i) (v i)\nh : \u2200 (i : Fin (Nat.succ n)), v i \u2260 0\ni : Fin (Nat.succ n)\nhb : \u2191b i = \u2016v i\u2016\u207b\u00b9 \u2022 v i\n\u22a2 0 \u2264 \u2016v i\u2016\u207b\u00b9 * (\u2016v i\u2016 * \u2016v i\u2016)\n[PROOFSTEP]\npositivity\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\nv : OrthonormalBasis (Fin n) \u211d E\n\u22a2 |\u2191(volumeForm o) \u2191v| = 1\n[PROOFSTEP]\nsimpa [o.volumeForm_robust' v v] using congr_arg abs v.toBasis.det_self\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : Fact (finrank \u211d F = n)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx : Fin n \u2192 F\n\u22a2 \u2191(volumeForm (\u2191(map (Fin n) \u03c6.toLinearEquiv) o)) x = \u2191(volumeForm o) (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 x)\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\ninst\u271d : Fact (finrank \u211d F = Nat.zero)\nx : Fin Nat.zero \u2192 F\n\u22a2 \u2191(volumeForm (\u2191(map (Fin Nat.zero) \u03c6.toLinearEquiv) o)) x = \u2191(volumeForm o) (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 x)\n[PROOFSTEP]\nrefine' o.eq_or_eq_neg_of_isEmpty.elim _ _\n[GOAL]\ncase zero.refine'_1\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\ninst\u271d : Fact (finrank \u211d F = Nat.zero)\nx : Fin Nat.zero \u2192 F\n\u22a2 o = positiveOrientation \u2192\n \u2191(volumeForm (\u2191(map (Fin Nat.zero) \u03c6.toLinearEquiv) o)) x = \u2191(volumeForm o) (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 x)\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase zero.refine'_2\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\ninst\u271d : Fact (finrank \u211d F = Nat.zero)\nx : Fin Nat.zero \u2192 F\n\u22a2 o = -positiveOrientation \u2192\n \u2191(volumeForm (\u2191(map (Fin Nat.zero) \u03c6.toLinearEquiv) o)) x = \u2191(volumeForm o) (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 x)\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase zero.refine'_1\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\n_i : Fact (finrank \u211d E = Nat.zero)\ninst\u271d : Fact (finrank \u211d F = Nat.zero)\nx : Fin Nat.zero \u2192 F\n\u22a2 \u2191(volumeForm (\u2191(map (Fin Nat.zero) \u03c6.toLinearEquiv) positiveOrientation)) x =\n \u2191(volumeForm positiveOrientation) (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase zero.refine'_2\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\n_i : Fact (finrank \u211d E = Nat.zero)\ninst\u271d : Fact (finrank \u211d F = Nat.zero)\nx : Fin Nat.zero \u2192 F\n\u22a2 \u2191(volumeForm (\u2191(map (Fin Nat.zero) \u03c6.toLinearEquiv) (-positiveOrientation))) x =\n \u2191(volumeForm (-positiveOrientation)) (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\ninst\u271d : Fact (finrank \u211d F = Nat.succ n)\nx : Fin (Nat.succ n) \u2192 F\n\u22a2 \u2191(volumeForm (\u2191(map (Fin (Nat.succ n)) \u03c6.toLinearEquiv) o)) x = \u2191(volumeForm o) (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 x)\n[PROOFSTEP]\nlet e : OrthonormalBasis (Fin n.succ) \u211d E := o.finOrthonormalBasis n.succ_pos Fact.out\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\ninst\u271d : Fact (finrank \u211d F = Nat.succ n)\nx : Fin (Nat.succ n) \u2192 F\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\n\u22a2 \u2191(volumeForm (\u2191(map (Fin (Nat.succ n)) \u03c6.toLinearEquiv) o)) x = \u2191(volumeForm o) (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 x)\n[PROOFSTEP]\nhave he : e.toBasis.orientation = o := o.finOrthonormalBasis_orientation n.succ_pos Fact.out\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\ninst\u271d : Fact (finrank \u211d F = Nat.succ n)\nx : Fin (Nat.succ n) \u2192 F\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\nhe : Basis.orientation (OrthonormalBasis.toBasis e) = o\n\u22a2 \u2191(volumeForm (\u2191(map (Fin (Nat.succ n)) \u03c6.toLinearEquiv) o)) x = \u2191(volumeForm o) (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 x)\n[PROOFSTEP]\nhave he\u03c6 : (e.map \u03c6).toBasis.orientation = Orientation.map (Fin n.succ) \u03c6.toLinearEquiv o :=\n by\n rw [\u2190 he]\n exact e.toBasis.orientation_map \u03c6.toLinearEquiv\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\ninst\u271d : Fact (finrank \u211d F = Nat.succ n)\nx : Fin (Nat.succ n) \u2192 F\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\nhe : Basis.orientation (OrthonormalBasis.toBasis e) = o\n\u22a2 Basis.orientation (OrthonormalBasis.toBasis (OrthonormalBasis.map e \u03c6)) = \u2191(map (Fin (Nat.succ n)) \u03c6.toLinearEquiv) o\n[PROOFSTEP]\nrw [\u2190 he]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\ninst\u271d : Fact (finrank \u211d F = Nat.succ n)\nx : Fin (Nat.succ n) \u2192 F\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\nhe : Basis.orientation (OrthonormalBasis.toBasis e) = o\n\u22a2 Basis.orientation (OrthonormalBasis.toBasis (OrthonormalBasis.map e \u03c6)) =\n \u2191(map (Fin (Nat.succ n)) \u03c6.toLinearEquiv) (Basis.orientation (OrthonormalBasis.toBasis e))\n[PROOFSTEP]\nexact e.toBasis.orientation_map \u03c6.toLinearEquiv\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\ninst\u271d : Fact (finrank \u211d F = Nat.succ n)\nx : Fin (Nat.succ n) \u2192 F\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\nhe : Basis.orientation (OrthonormalBasis.toBasis e) = o\nhe\u03c6 :\n Basis.orientation (OrthonormalBasis.toBasis (OrthonormalBasis.map e \u03c6)) = \u2191(map (Fin (Nat.succ n)) \u03c6.toLinearEquiv) o\n\u22a2 \u2191(volumeForm (\u2191(map (Fin (Nat.succ n)) \u03c6.toLinearEquiv) o)) x = \u2191(volumeForm o) (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 x)\n[PROOFSTEP]\nrw [(Orientation.map (Fin n.succ) \u03c6.toLinearEquiv o).volumeForm_robust (e.map \u03c6) he\u03c6]\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\ninst\u271d : Fact (finrank \u211d F = Nat.succ n)\nx : Fin (Nat.succ n) \u2192 F\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\nhe : Basis.orientation (OrthonormalBasis.toBasis e) = o\nhe\u03c6 :\n Basis.orientation (OrthonormalBasis.toBasis (OrthonormalBasis.map e \u03c6)) = \u2191(map (Fin (Nat.succ n)) \u03c6.toLinearEquiv) o\n\u22a2 \u2191(Basis.det (OrthonormalBasis.toBasis (OrthonormalBasis.map e \u03c6))) x =\n \u2191(volumeForm o) (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 x)\n[PROOFSTEP]\nrw [o.volumeForm_robust e he]\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\ninst\u271d : Fact (finrank \u211d F = Nat.succ n)\nx : Fin (Nat.succ n) \u2192 F\ne : OrthonormalBasis (Fin (Nat.succ n)) \u211d E :=\n Orientation.finOrthonormalBasis (_ : 0 < Nat.succ n) (_ : finrank \u211d E = Nat.succ n) o\nhe : Basis.orientation (OrthonormalBasis.toBasis e) = o\nhe\u03c6 :\n Basis.orientation (OrthonormalBasis.toBasis (OrthonormalBasis.map e \u03c6)) = \u2191(map (Fin (Nat.succ n)) \u03c6.toLinearEquiv) o\n\u22a2 \u2191(Basis.det (OrthonormalBasis.toBasis (OrthonormalBasis.map e \u03c6))) x =\n \u2191(Basis.det (OrthonormalBasis.toBasis e)) (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 x)\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nn : \u2115\n_i : Fact (finrank \u211d E = n)\no : Orientation \u211d E (Fin n)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nx : Fin n \u2192 E\n\u22a2 \u2191(volumeForm o) (\u2191\u03c6 \u2218 x) = \u2191(volumeForm o) x\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nx : Fin Nat.zero \u2192 E\n\u22a2 \u2191(volumeForm o) (\u2191\u03c6 \u2218 x) = \u2191(volumeForm o) x\n[PROOFSTEP]\nrefine' o.eq_or_eq_neg_of_isEmpty.elim _ _\n[GOAL]\ncase zero.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nx : Fin Nat.zero \u2192 E\n\u22a2 o = positiveOrientation \u2192 \u2191(volumeForm o) (\u2191\u03c6 \u2218 x) = \u2191(volumeForm o) x\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase zero.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\n_i : Fact (finrank \u211d E = Nat.zero)\no : Orientation \u211d E (Fin Nat.zero)\nx : Fin Nat.zero \u2192 E\n\u22a2 o = -positiveOrientation \u2192 \u2191(volumeForm o) (\u2191\u03c6 \u2218 x) = \u2191(volumeForm o) x\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase zero.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\n_i : Fact (finrank \u211d E = Nat.zero)\nx : Fin Nat.zero \u2192 E\n\u22a2 \u2191(volumeForm positiveOrientation) (\u2191\u03c6 \u2218 x) = \u2191(volumeForm positiveOrientation) x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase zero.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\n_i : Fact (finrank \u211d E = Nat.zero)\nx : Fin Nat.zero \u2192 E\n\u22a2 \u2191(volumeForm (-positiveOrientation)) (\u2191\u03c6 \u2218 x) = \u2191(volumeForm (-positiveOrientation)) x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) \u2192 E\n\u22a2 \u2191(volumeForm o) (\u2191\u03c6 \u2218 x) = \u2191(volumeForm o) x\n[PROOFSTEP]\nhaveI : FiniteDimensional \u211d E := fact_finiteDimensional_of_finrank_eq_succ n\n[GOAL]\ncase succ\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) \u2192 E\nthis : FiniteDimensional \u211d E\n\u22a2 \u2191(volumeForm o) (\u2191\u03c6 \u2218 x) = \u2191(volumeForm o) x\n[PROOFSTEP]\nconvert o.volumeForm_map \u03c6 (\u03c6 \u2218 x)\n[GOAL]\ncase h.e'_2.h.e'_5.h.e'_6\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) \u2192 E\nthis : FiniteDimensional \u211d E\n\u22a2 o = \u2191(map (Fin (Nat.succ n)) \u03c6.toLinearEquiv) o\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_2.h.e'_5.h.e'_6\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) \u2192 E\nthis : FiniteDimensional \u211d E\n\u22a2 \u2191(map (Fin (Nat.succ n)) \u03c6.toLinearEquiv) o = o\n[PROOFSTEP]\nrwa [\u2190 o.map_eq_iff_det_pos \u03c6.toLinearEquiv] at h\u03c6 \n[GOAL]\ncase h.e'_2.h.e'_5.h.e'_6\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) \u2192 E\nthis : FiniteDimensional \u211d E\n\u22a2 Fintype.card (Fin (Nat.succ n)) = finrank \u211d E\n[PROOFSTEP]\nrw [_i.out, Fintype.card_fin]\n[GOAL]\ncase h.e'_3.h.e'_6\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) \u2192 E\nthis : FiniteDimensional \u211d E\n\u22a2 x = \u2191(LinearIsometryEquiv.symm \u03c6) \u2218 \u2191\u03c6 \u2218 x\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_3.h.e'_6.h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nn : \u2115\n_i : Fact (finrank \u211d E = Nat.succ n)\no : Orientation \u211d E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) \u2192 E\nthis : FiniteDimensional \u211d E\nx\u271d : Fin (Nat.succ n)\n\u22a2 x x\u271d = (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 \u2191\u03c6 \u2218 x) x\u271d\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Analysis.InnerProductSpace.Orientation", "llama_tokens": 31245, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.43544591824361073}} {"text": "[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u22a2 Finset.card (graph f) = n\n[PROOFSTEP]\nrw [graph, Finset.card_image_of_injective]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u22a2 Finset.card Finset.univ = n\n[PROOFSTEP]\nexact Finset.card_fin _\n[GOAL]\ncase H\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u22a2 Function.Injective fun i => (f i, i)\n[PROOFSTEP]\nintro _\n _\n -- Porting note: was `simp`\n[GOAL]\ncase H\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\na\u2081\u271d a\u2082\u271d : Fin n\n\u22a2 (fun i => (f i, i)) a\u2081\u271d = (fun i => (f i, i)) a\u2082\u271d \u2192 a\u2081\u271d = a\u2082\u271d\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase H\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\na\u2081\u271d a\u2082\u271d : Fin n\n\u22a2 (f a\u2081\u271d, a\u2081\u271d) = (f a\u2082\u271d, a\u2082\u271d) \u2192 a\u2081\u271d = a\u2082\u271d\n[PROOFSTEP]\nrw [Prod.ext_iff]\n[GOAL]\ncase H\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\na\u2081\u271d a\u2082\u271d : Fin n\n\u22a2 (f a\u2081\u271d, a\u2081\u271d).fst = (f a\u2082\u271d, a\u2082\u271d).fst \u2227 (f a\u2081\u271d, a\u2081\u271d).snd = (f a\u2082\u271d, a\u2082\u271d).snd \u2192 a\u2081\u271d = a\u2082\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\ni : Fin n\n\u22a2 (f i, i) \u2208 graph f\n[PROOFSTEP]\nsimp [graph]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\ni : Fin n\n\u22a2 (fun p => (\u2191p).snd)\n ((fun i => { val := (f i, i), property := (_ : (f i, i) \u2208 Finset.image (fun i => (f i, i)) Finset.univ) }) i) =\n i\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\nx\u271d : { x // x \u2208 graph f }\nx : \u03b1\ni : Fin n\nh : (x, i) \u2208 graph f\n\u22a2 (fun i => { val := (f i, i), property := (_ : (f i, i) \u2208 Finset.image (fun i => (f i, i)) Finset.univ) })\n ((fun p => (\u2191p).snd) { val := (x, i), property := h }) =\n { val := (x, i), property := h }\n[PROOFSTEP]\nsimp only [graph, Finset.mem_image, Finset.mem_univ, true_and] at h \n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\nx\u271d : { x // x \u2208 graph f }\nx : \u03b1\ni : Fin n\nh\u271d : (x, i) \u2208 graph f\nh : \u2203 a, (f a, a) = (x, i)\n\u22a2 (fun i => { val := (f i, i), property := (_ : (f i, i) \u2208 Finset.image (fun i => (f i, i)) Finset.univ) })\n ((fun p => (\u2191p).snd) { val := (x, i), property := h\u271d }) =\n { val := (x, i), property := h\u271d }\n[PROOFSTEP]\nobtain \u27e8i', hi'\u27e9 := h\n[GOAL]\ncase intro\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\nx\u271d : { x // x \u2208 graph f }\nx : \u03b1\ni : Fin n\nh : (x, i) \u2208 graph f\ni' : Fin n\nhi' : (f i', i') = (x, i)\n\u22a2 (fun i => { val := (f i, i), property := (_ : (f i, i) \u2208 Finset.image (fun i => (f i, i)) Finset.univ) })\n ((fun p => (\u2191p).snd) { val := (x, i), property := h }) =\n { val := (x, i), property := h }\n[PROOFSTEP]\nobtain \u27e8-, rfl\u27e9 := Prod.mk.inj_iff.mp hi'\n[GOAL]\ncase intro.intro\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\nx\u271d : { x // x \u2208 graph f }\nx : \u03b1\ni' : Fin n\nh : (x, i') \u2208 graph f\nhi' : (f i', i') = (x, i')\n\u22a2 (fun i => { val := (f i, i), property := (_ : (f i, i) \u2208 Finset.image (fun i => (f i, i)) Finset.univ) })\n ((fun p => (\u2191p).snd) { val := (x, i'), property := h }) =\n { val := (x, i'), property := h }\n[PROOFSTEP]\nsimpa\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u22a2 Finset.card (graph f) = n\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u22a2 graph.proj \u2218 (\u2191(graphEquiv\u2081 f) \u2218 \u2191(graphEquiv\u2081 f).symm) \u2218 \u2191(graphEquiv\u2082 f).toEquiv = graph.proj \u2218 \u2191(graphEquiv\u2082 f)\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u22a2 Monotone graph.proj\n[PROOFSTEP]\nrintro \u27e8\u27e8x, i\u27e9, hx\u27e9 \u27e8\u27e8y, j\u27e9, hy\u27e9 (_ | h)\n[GOAL]\ncase mk.mk.mk.mk.left\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\nx : \u03b1\ni : Fin n\nhx : (x, i) \u2208 graph f\ny : \u03b1\nj : Fin n\nhy : (y, j) \u2208 graph f\nh\u271d : x < y\n\u22a2 graph.proj { val := (x, i), property := hx } \u2264 graph.proj { val := (y, j), property := hy }\n[PROOFSTEP]\nexact le_of_lt \u2039_\u203a\n[GOAL]\ncase mk.mk.mk.mk.right\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\nx : \u03b1\ni : Fin n\nhx : (x, i) \u2208 graph f\nj : Fin n\nhy : (x, j) \u2208 graph f\nh\u271d : i \u2264 j\n\u22a2 graph.proj { val := (x, i), property := hx } \u2264 graph.proj { val := (x, j), property := hy }\n[PROOFSTEP]\nsimp [graph.proj]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u22a2 Monotone (f \u2218 \u2191(sort f))\n[PROOFSTEP]\nrw [self_comp_sort]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u22a2 Monotone (graph.proj \u2218 \u2191(graphEquiv\u2082 f))\n[PROOFSTEP]\nexact (monotone_proj f).comp (graphEquiv\u2082 f).monotone\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\n\u22a2 \u03c3 = sort f \u2194 StrictMono \u2191(\u03c3.trans (graphEquiv\u2081 f))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\n\u22a2 \u03c3 = sort f \u2192 StrictMono \u2191(\u03c3.trans (graphEquiv\u2081 f))\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\n\u22a2 StrictMono \u2191(\u03c3.trans (graphEquiv\u2081 f)) \u2192 \u03c3 = sort f\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\nh : \u03c3 = sort f\n\u22a2 StrictMono \u2191(\u03c3.trans (graphEquiv\u2081 f))\n[PROOFSTEP]\nrw [h, sort, Equiv.trans_assoc, Equiv.symm_trans_self]\n[GOAL]\ncase mp\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\nh : \u03c3 = sort f\n\u22a2 StrictMono \u2191((graphEquiv\u2082 f).trans (Equiv.refl { x // x \u2208 graph f }))\n[PROOFSTEP]\nexact (graphEquiv\u2082 f).strictMono\n[GOAL]\ncase mpr\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\nh : StrictMono \u2191(\u03c3.trans (graphEquiv\u2081 f))\n\u22a2 \u03c3 = sort f\n[PROOFSTEP]\nhave := Subsingleton.elim (graphEquiv\u2082 f) (h.orderIsoOfSurjective _ <| Equiv.surjective _)\n[GOAL]\ncase mpr\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\nh : StrictMono \u2191(\u03c3.trans (graphEquiv\u2081 f))\nthis :\n graphEquiv\u2082 f =\n StrictMono.orderIsoOfSurjective (\u2191(\u03c3.trans (graphEquiv\u2081 f))) h (_ : Function.Surjective \u2191(\u03c3.trans (graphEquiv\u2081 f)))\n\u22a2 \u03c3 = sort f\n[PROOFSTEP]\next1 x\n[GOAL]\ncase mpr.H\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\nh : StrictMono \u2191(\u03c3.trans (graphEquiv\u2081 f))\nthis :\n graphEquiv\u2082 f =\n StrictMono.orderIsoOfSurjective (\u2191(\u03c3.trans (graphEquiv\u2081 f))) h (_ : Function.Surjective \u2191(\u03c3.trans (graphEquiv\u2081 f)))\nx : Fin n\n\u22a2 \u2191\u03c3 x = \u2191(sort f) x\n[PROOFSTEP]\nexact (graphEquiv\u2081 f).apply_eq_iff_eq_symm_apply.1 (FunLike.congr_fun this x).symm\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\n\u22a2 \u03c3 = sort f \u2194 Monotone (f \u2218 \u2191\u03c3) \u2227 \u2200 (i j : Fin n), i < j \u2192 f (\u2191\u03c3 i) = f (\u2191\u03c3 j) \u2192 \u2191\u03c3 i < \u2191\u03c3 j\n[PROOFSTEP]\nrw [eq_sort_iff']\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\n\u22a2 StrictMono \u2191(\u03c3.trans (graphEquiv\u2081 f)) \u2194 Monotone (f \u2218 \u2191\u03c3) \u2227 \u2200 (i j : Fin n), i < j \u2192 f (\u2191\u03c3 i) = f (\u2191\u03c3 j) \u2192 \u2191\u03c3 i < \u2191\u03c3 j\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8(monotone_proj f).comp h.monotone, fun i j hij hfij => _\u27e9, fun h i j hij => _\u27e9\n[GOAL]\ncase refine'_1\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\nh : StrictMono \u2191(\u03c3.trans (graphEquiv\u2081 f))\ni j : Fin n\nhij : i < j\nhfij : f (\u2191\u03c3 i) = f (\u2191\u03c3 j)\n\u22a2 \u2191\u03c3 i < \u2191\u03c3 j\n[PROOFSTEP]\nexact (((Prod.Lex.lt_iff _ _).1 <| h hij).resolve_left hfij.not_lt).2\n[GOAL]\ncase refine'_2\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\nh : Monotone (f \u2218 \u2191\u03c3) \u2227 \u2200 (i j : Fin n), i < j \u2192 f (\u2191\u03c3 i) = f (\u2191\u03c3 j) \u2192 \u2191\u03c3 i < \u2191\u03c3 j\ni j : Fin n\nhij : i < j\n\u22a2 \u2191(\u03c3.trans (graphEquiv\u2081 f)) i < \u2191(\u03c3.trans (graphEquiv\u2081 f)) j\n[PROOFSTEP]\nobtain he | hl := (h.1 hij.le).eq_or_lt\n[GOAL]\ncase refine'_2.inl\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\nh : Monotone (f \u2218 \u2191\u03c3) \u2227 \u2200 (i j : Fin n), i < j \u2192 f (\u2191\u03c3 i) = f (\u2191\u03c3 j) \u2192 \u2191\u03c3 i < \u2191\u03c3 j\ni j : Fin n\nhij : i < j\nhe : (f \u2218 \u2191\u03c3) i = (f \u2218 \u2191\u03c3) j\n\u22a2 \u2191(\u03c3.trans (graphEquiv\u2081 f)) i < \u2191(\u03c3.trans (graphEquiv\u2081 f)) j\n[PROOFSTEP]\napply (Prod.Lex.lt_iff _ _).2\n[GOAL]\ncase refine'_2.inr\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\nh : Monotone (f \u2218 \u2191\u03c3) \u2227 \u2200 (i j : Fin n), i < j \u2192 f (\u2191\u03c3 i) = f (\u2191\u03c3 j) \u2192 \u2191\u03c3 i < \u2191\u03c3 j\ni j : Fin n\nhij : i < j\nhl : (f \u2218 \u2191\u03c3) i < (f \u2218 \u2191\u03c3) j\n\u22a2 \u2191(\u03c3.trans (graphEquiv\u2081 f)) i < \u2191(\u03c3.trans (graphEquiv\u2081 f)) j\n[PROOFSTEP]\napply (Prod.Lex.lt_iff _ _).2\n[GOAL]\ncase refine'_2.inl\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\nh : Monotone (f \u2218 \u2191\u03c3) \u2227 \u2200 (i j : Fin n), i < j \u2192 f (\u2191\u03c3 i) = f (\u2191\u03c3 j) \u2192 \u2191\u03c3 i < \u2191\u03c3 j\ni j : Fin n\nhij : i < j\nhe : (f \u2218 \u2191\u03c3) i = (f \u2218 \u2191\u03c3) j\n\u22a2 (f (\u2191\u03c3 i), \u2191\u03c3 i).fst < (f (\u2191\u03c3 j), \u2191\u03c3 j).fst \u2228\n (f (\u2191\u03c3 i), \u2191\u03c3 i).fst = (f (\u2191\u03c3 j), \u2191\u03c3 j).fst \u2227 (f (\u2191\u03c3 i), \u2191\u03c3 i).snd < (f (\u2191\u03c3 j), \u2191\u03c3 j).snd\ncase refine'_2.inr\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\nh : Monotone (f \u2218 \u2191\u03c3) \u2227 \u2200 (i j : Fin n), i < j \u2192 f (\u2191\u03c3 i) = f (\u2191\u03c3 j) \u2192 \u2191\u03c3 i < \u2191\u03c3 j\ni j : Fin n\nhij : i < j\nhl : (f \u2218 \u2191\u03c3) i < (f \u2218 \u2191\u03c3) j\n\u22a2 (f (\u2191\u03c3 i), \u2191\u03c3 i).fst < (f (\u2191\u03c3 j), \u2191\u03c3 j).fst \u2228\n (f (\u2191\u03c3 i), \u2191\u03c3 i).fst = (f (\u2191\u03c3 j), \u2191\u03c3 j).fst \u2227 (f (\u2191\u03c3 i), \u2191\u03c3 i).snd < (f (\u2191\u03c3 j), \u2191\u03c3 j).snd\n[PROOFSTEP]\nexacts [Or.inr \u27e8he, h.2 i j hij he\u27e9, Or.inl hl]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\n\u22a2 sort f = Equiv.refl (Fin n) \u2194 Monotone f\n[PROOFSTEP]\nrw [eq_comm, eq_sort_iff, Equiv.coe_refl, Function.comp.right_id]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\n\u22a2 (Monotone f \u2227 \u2200 (i j : Fin n), i < j \u2192 f (id i) = f (id j) \u2192 id i < id j) \u2194 Monotone f\n[PROOFSTEP]\nsimp only [id.def, and_iff_left_iff_imp]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\n\u22a2 Monotone f \u2192 \u2200 (i j : Fin n), i < j \u2192 f i = f j \u2192 i < j\n[PROOFSTEP]\nexact fun _ _ _ hij _ => hij\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\n\u22a2 (f \u2218 \u2191\u03c3) \u2218 \u2191(sort (f \u2218 \u2191\u03c3)) = f \u2218 \u2191(sort f)\n[PROOFSTEP]\nrw [Function.comp.assoc, \u2190 Equiv.Perm.coe_mul]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\n\u22a2 f \u2218 \u2191(\u03c3 * sort (f \u2218 \u2191\u03c3)) = f \u2218 \u2191(sort f)\n[PROOFSTEP]\nexact unique_monotone (monotone_sort (f \u2218 \u03c3)) (monotone_sort f)\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\nh : f \u2218 \u2191\u03c3 \u2260 f \u2218 \u2191(sort f)\n\u22a2 \u2203 i j, i < j \u2227 (f \u2218 \u2191\u03c3) j < (f \u2218 \u2191\u03c3) i\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf : Fin n \u2192 \u03b1\n\u03c3 : Equiv.Perm (Fin n)\nh : \u2200 (i j : Fin n), i < j \u2192 (f \u2218 \u2191\u03c3) i \u2264 (f \u2218 \u2191\u03c3) j\n\u22a2 f \u2218 \u2191\u03c3 = f \u2218 \u2191(sort f)\n[PROOFSTEP]\nexact comp_sort_eq_comp_iff_monotone.mpr (monotone_iff_forall_lt.mpr h)\n", "meta": {"mathlib_filename": "Mathlib.Data.Fin.Tuple.Sort", "llama_tokens": 5382, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.4354459182436106}} {"text": "[GOAL]\nd : \u2115\nh : d \u2260 0\nw : Nat.coprime (Int.natAbs 0) d\n\u22a2 mk' 0 d = 0\n[PROOFSTEP]\ncongr\n[GOAL]\nq : \u211a\n\u22a2 q.num = 0 \u2194 q = 0\n[PROOFSTEP]\ninduction q\n[GOAL]\ncase mk'\nnum\u271d : \u2124\nden\u271d : \u2115\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : Nat.coprime (Int.natAbs num\u271d) den\u271d\n\u22a2 (mk' num\u271d den\u271d).num = 0 \u2194 mk' num\u271d den\u271d = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk'.mp\nnum\u271d : \u2124\nden\u271d : \u2115\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : Nat.coprime (Int.natAbs num\u271d) den\u271d\n\u22a2 (mk' num\u271d den\u271d).num = 0 \u2192 mk' num\u271d den\u271d = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mk'.mp\nden\u271d : \u2115\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : Nat.coprime (Int.natAbs 0) den\u271d\n\u22a2 mk' 0 den\u271d = 0\n[PROOFSTEP]\nexact zero_mk _ _ _\n[GOAL]\ncase mk'.mpr\nnum\u271d : \u2124\nden\u271d : \u2115\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : Nat.coprime (Int.natAbs num\u271d) den\u271d\n\u22a2 mk' num\u271d den\u271d = 0 \u2192 (mk' num\u271d den\u271d).num = 0\n[PROOFSTEP]\nexact congr_arg num\n[GOAL]\na b : \u2124\nb0 : b \u2260 0\n\u22a2 a /. b = 0 \u2194 a = 0\n[PROOFSTEP]\nrw [\u2190 zero_divInt b, divInt_eq_iff b0 b0, zero_mul, mul_eq_zero, or_iff_left b0]\n[GOAL]\nC : \u211a \u2192 Sort u\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\nH : (n : \u2124) \u2192 (d : \u2115) \u2192 0 < d \u2192 Nat.coprime (Int.natAbs n) d \u2192 C (n /. \u2191d)\n\u22a2 C (mk' n d)\n[PROOFSTEP]\nrw [num_den']\n[GOAL]\nC : \u211a \u2192 Sort u\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\nH : (n : \u2124) \u2192 (d : \u2115) \u2192 0 < d \u2192 Nat.coprime (Int.natAbs n) d \u2192 C (n /. \u2191d)\n\u22a2 C (n /. \u2191d)\n[PROOFSTEP]\nexact H n d (Nat.pos_of_ne_zero h) c\n[GOAL]\nf : \u211a \u2192 \u211a \u2192 \u211a\nf\u2081 f\u2082 : \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124\nfv :\n \u2200 {n\u2081 : \u2124} {d\u2081 : \u2115} {h\u2081 : d\u2081 \u2260 0} {c\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081} {n\u2082 : \u2124} {d\u2082 : \u2115} {h\u2082 : d\u2082 \u2260 0}\n {c\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082}, f (mk' n\u2081 d\u2081) (mk' n\u2082 d\u2082) = f\u2081 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 /. f\u2082 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082\nf0 : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, d\u2081 \u2260 0 \u2192 d\u2082 \u2260 0 \u2192 f\u2082 n\u2081 d\u2081 n\u2082 d\u2082 \u2260 0\na b c d : \u2124\nb0 : b \u2260 0\nd0 : d \u2260 0\nH : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, a * d\u2081 = n\u2081 * b \u2192 c * d\u2082 = n\u2082 * d \u2192 f\u2081 n\u2081 d\u2081 n\u2082 d\u2082 * f\u2082 a b c d = f\u2081 a b c d * f\u2082 n\u2081 d\u2081 n\u2082 d\u2082\n\u22a2 f (a /. b) (c /. d) = f\u2081 a b c d /. f\u2082 a b c d\n[PROOFSTEP]\ngeneralize ha : a /. b = x\n[GOAL]\nf : \u211a \u2192 \u211a \u2192 \u211a\nf\u2081 f\u2082 : \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124\nfv :\n \u2200 {n\u2081 : \u2124} {d\u2081 : \u2115} {h\u2081 : d\u2081 \u2260 0} {c\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081} {n\u2082 : \u2124} {d\u2082 : \u2115} {h\u2082 : d\u2082 \u2260 0}\n {c\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082}, f (mk' n\u2081 d\u2081) (mk' n\u2082 d\u2082) = f\u2081 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 /. f\u2082 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082\nf0 : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, d\u2081 \u2260 0 \u2192 d\u2082 \u2260 0 \u2192 f\u2082 n\u2081 d\u2081 n\u2082 d\u2082 \u2260 0\na b c d : \u2124\nb0 : b \u2260 0\nd0 : d \u2260 0\nH : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, a * d\u2081 = n\u2081 * b \u2192 c * d\u2082 = n\u2082 * d \u2192 f\u2081 n\u2081 d\u2081 n\u2082 d\u2082 * f\u2082 a b c d = f\u2081 a b c d * f\u2082 n\u2081 d\u2081 n\u2082 d\u2082\nx : \u211a\nha : a /. b = x\n\u22a2 f x (c /. d) = f\u2081 a b c d /. f\u2082 a b c d\n[PROOFSTEP]\ncases' x with n\u2081 d\u2081 h\u2081 c\u2081\n[GOAL]\ncase mk'\nf : \u211a \u2192 \u211a \u2192 \u211a\nf\u2081 f\u2082 : \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124\nfv :\n \u2200 {n\u2081 : \u2124} {d\u2081 : \u2115} {h\u2081 : d\u2081 \u2260 0} {c\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081} {n\u2082 : \u2124} {d\u2082 : \u2115} {h\u2082 : d\u2082 \u2260 0}\n {c\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082}, f (mk' n\u2081 d\u2081) (mk' n\u2082 d\u2082) = f\u2081 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 /. f\u2082 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082\nf0 : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, d\u2081 \u2260 0 \u2192 d\u2082 \u2260 0 \u2192 f\u2082 n\u2081 d\u2081 n\u2082 d\u2082 \u2260 0\na b c d : \u2124\nb0 : b \u2260 0\nd0 : d \u2260 0\nH : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, a * d\u2081 = n\u2081 * b \u2192 c * d\u2082 = n\u2082 * d \u2192 f\u2081 n\u2081 d\u2081 n\u2082 d\u2082 * f\u2082 a b c d = f\u2081 a b c d * f\u2082 n\u2081 d\u2081 n\u2082 d\u2082\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nha : a /. b = mk' n\u2081 d\u2081\n\u22a2 f (mk' n\u2081 d\u2081) (c /. d) = f\u2081 a b c d /. f\u2082 a b c d\n[PROOFSTEP]\nrw [num_den'] at ha \n[GOAL]\ncase mk'\nf : \u211a \u2192 \u211a \u2192 \u211a\nf\u2081 f\u2082 : \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124\nfv :\n \u2200 {n\u2081 : \u2124} {d\u2081 : \u2115} {h\u2081 : d\u2081 \u2260 0} {c\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081} {n\u2082 : \u2124} {d\u2082 : \u2115} {h\u2082 : d\u2082 \u2260 0}\n {c\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082}, f (mk' n\u2081 d\u2081) (mk' n\u2082 d\u2082) = f\u2081 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 /. f\u2082 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082\nf0 : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, d\u2081 \u2260 0 \u2192 d\u2082 \u2260 0 \u2192 f\u2082 n\u2081 d\u2081 n\u2082 d\u2082 \u2260 0\na b c d : \u2124\nb0 : b \u2260 0\nd0 : d \u2260 0\nH : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, a * d\u2081 = n\u2081 * b \u2192 c * d\u2082 = n\u2082 * d \u2192 f\u2081 n\u2081 d\u2081 n\u2082 d\u2082 * f\u2082 a b c d = f\u2081 a b c d * f\u2082 n\u2081 d\u2081 n\u2082 d\u2082\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nha : a /. b = n\u2081 /. \u2191d\u2081\n\u22a2 f (mk' n\u2081 d\u2081) (c /. d) = f\u2081 a b c d /. f\u2082 a b c d\n[PROOFSTEP]\ngeneralize hc : c /. d = x\n[GOAL]\ncase mk'\nf : \u211a \u2192 \u211a \u2192 \u211a\nf\u2081 f\u2082 : \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124\nfv :\n \u2200 {n\u2081 : \u2124} {d\u2081 : \u2115} {h\u2081 : d\u2081 \u2260 0} {c\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081} {n\u2082 : \u2124} {d\u2082 : \u2115} {h\u2082 : d\u2082 \u2260 0}\n {c\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082}, f (mk' n\u2081 d\u2081) (mk' n\u2082 d\u2082) = f\u2081 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 /. f\u2082 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082\nf0 : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, d\u2081 \u2260 0 \u2192 d\u2082 \u2260 0 \u2192 f\u2082 n\u2081 d\u2081 n\u2082 d\u2082 \u2260 0\na b c d : \u2124\nb0 : b \u2260 0\nd0 : d \u2260 0\nH : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, a * d\u2081 = n\u2081 * b \u2192 c * d\u2082 = n\u2082 * d \u2192 f\u2081 n\u2081 d\u2081 n\u2082 d\u2082 * f\u2082 a b c d = f\u2081 a b c d * f\u2082 n\u2081 d\u2081 n\u2082 d\u2082\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nha : a /. b = n\u2081 /. \u2191d\u2081\nx : \u211a\nhc : c /. d = x\n\u22a2 f (mk' n\u2081 d\u2081) x = f\u2081 a b c d /. f\u2082 a b c d\n[PROOFSTEP]\ncases' x with n\u2082 d\u2082 h\u2082 c\u2082\n[GOAL]\ncase mk'.mk'\nf : \u211a \u2192 \u211a \u2192 \u211a\nf\u2081 f\u2082 : \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124\nfv :\n \u2200 {n\u2081 : \u2124} {d\u2081 : \u2115} {h\u2081 : d\u2081 \u2260 0} {c\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081} {n\u2082 : \u2124} {d\u2082 : \u2115} {h\u2082 : d\u2082 \u2260 0}\n {c\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082}, f (mk' n\u2081 d\u2081) (mk' n\u2082 d\u2082) = f\u2081 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 /. f\u2082 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082\nf0 : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, d\u2081 \u2260 0 \u2192 d\u2082 \u2260 0 \u2192 f\u2082 n\u2081 d\u2081 n\u2082 d\u2082 \u2260 0\na b c d : \u2124\nb0 : b \u2260 0\nd0 : d \u2260 0\nH : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, a * d\u2081 = n\u2081 * b \u2192 c * d\u2082 = n\u2082 * d \u2192 f\u2081 n\u2081 d\u2081 n\u2082 d\u2082 * f\u2082 a b c d = f\u2081 a b c d * f\u2082 n\u2081 d\u2081 n\u2082 d\u2082\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nha : a /. b = n\u2081 /. \u2191d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nhc : c /. d = mk' n\u2082 d\u2082\n\u22a2 f (mk' n\u2081 d\u2081) (mk' n\u2082 d\u2082) = f\u2081 a b c d /. f\u2082 a b c d\n[PROOFSTEP]\nrw [num_den'] at hc \n[GOAL]\ncase mk'.mk'\nf : \u211a \u2192 \u211a \u2192 \u211a\nf\u2081 f\u2082 : \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124\nfv :\n \u2200 {n\u2081 : \u2124} {d\u2081 : \u2115} {h\u2081 : d\u2081 \u2260 0} {c\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081} {n\u2082 : \u2124} {d\u2082 : \u2115} {h\u2082 : d\u2082 \u2260 0}\n {c\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082}, f (mk' n\u2081 d\u2081) (mk' n\u2082 d\u2082) = f\u2081 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 /. f\u2082 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082\nf0 : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, d\u2081 \u2260 0 \u2192 d\u2082 \u2260 0 \u2192 f\u2082 n\u2081 d\u2081 n\u2082 d\u2082 \u2260 0\na b c d : \u2124\nb0 : b \u2260 0\nd0 : d \u2260 0\nH : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, a * d\u2081 = n\u2081 * b \u2192 c * d\u2082 = n\u2082 * d \u2192 f\u2081 n\u2081 d\u2081 n\u2082 d\u2082 * f\u2082 a b c d = f\u2081 a b c d * f\u2082 n\u2081 d\u2081 n\u2082 d\u2082\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nha : a /. b = n\u2081 /. \u2191d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nhc : c /. d = n\u2082 /. \u2191d\u2082\n\u22a2 f (mk' n\u2081 d\u2081) (mk' n\u2082 d\u2082) = f\u2081 a b c d /. f\u2082 a b c d\n[PROOFSTEP]\nrw [fv]\n[GOAL]\ncase mk'.mk'\nf : \u211a \u2192 \u211a \u2192 \u211a\nf\u2081 f\u2082 : \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124\nfv :\n \u2200 {n\u2081 : \u2124} {d\u2081 : \u2115} {h\u2081 : d\u2081 \u2260 0} {c\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081} {n\u2082 : \u2124} {d\u2082 : \u2115} {h\u2082 : d\u2082 \u2260 0}\n {c\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082}, f (mk' n\u2081 d\u2081) (mk' n\u2082 d\u2082) = f\u2081 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 /. f\u2082 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082\nf0 : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, d\u2081 \u2260 0 \u2192 d\u2082 \u2260 0 \u2192 f\u2082 n\u2081 d\u2081 n\u2082 d\u2082 \u2260 0\na b c d : \u2124\nb0 : b \u2260 0\nd0 : d \u2260 0\nH : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, a * d\u2081 = n\u2081 * b \u2192 c * d\u2082 = n\u2082 * d \u2192 f\u2081 n\u2081 d\u2081 n\u2082 d\u2082 * f\u2082 a b c d = f\u2081 a b c d * f\u2082 n\u2081 d\u2081 n\u2082 d\u2082\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nha : a /. b = n\u2081 /. \u2191d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nhc : c /. d = n\u2082 /. \u2191d\u2082\n\u22a2 f\u2081 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 /. f\u2082 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 = f\u2081 a b c d /. f\u2082 a b c d\n[PROOFSTEP]\nhave d\u20810 := ne_of_gt (Int.ofNat_lt.2 $ Nat.pos_of_ne_zero h\u2081)\n[GOAL]\ncase mk'.mk'\nf : \u211a \u2192 \u211a \u2192 \u211a\nf\u2081 f\u2082 : \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124\nfv :\n \u2200 {n\u2081 : \u2124} {d\u2081 : \u2115} {h\u2081 : d\u2081 \u2260 0} {c\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081} {n\u2082 : \u2124} {d\u2082 : \u2115} {h\u2082 : d\u2082 \u2260 0}\n {c\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082}, f (mk' n\u2081 d\u2081) (mk' n\u2082 d\u2082) = f\u2081 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 /. f\u2082 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082\nf0 : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, d\u2081 \u2260 0 \u2192 d\u2082 \u2260 0 \u2192 f\u2082 n\u2081 d\u2081 n\u2082 d\u2082 \u2260 0\na b c d : \u2124\nb0 : b \u2260 0\nd0 : d \u2260 0\nH : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, a * d\u2081 = n\u2081 * b \u2192 c * d\u2082 = n\u2082 * d \u2192 f\u2081 n\u2081 d\u2081 n\u2082 d\u2082 * f\u2082 a b c d = f\u2081 a b c d * f\u2082 n\u2081 d\u2081 n\u2082 d\u2082\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nha : a /. b = n\u2081 /. \u2191d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nhc : c /. d = n\u2082 /. \u2191d\u2082\nd\u20810 : \u2191d\u2081 \u2260 \u21910\n\u22a2 f\u2081 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 /. f\u2082 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 = f\u2081 a b c d /. f\u2082 a b c d\n[PROOFSTEP]\nhave d\u20820 := ne_of_gt (Int.ofNat_lt.2 $ Nat.pos_of_ne_zero h\u2082)\n[GOAL]\ncase mk'.mk'\nf : \u211a \u2192 \u211a \u2192 \u211a\nf\u2081 f\u2082 : \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124 \u2192 \u2124\nfv :\n \u2200 {n\u2081 : \u2124} {d\u2081 : \u2115} {h\u2081 : d\u2081 \u2260 0} {c\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081} {n\u2082 : \u2124} {d\u2082 : \u2115} {h\u2082 : d\u2082 \u2260 0}\n {c\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082}, f (mk' n\u2081 d\u2081) (mk' n\u2082 d\u2082) = f\u2081 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 /. f\u2082 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082\nf0 : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, d\u2081 \u2260 0 \u2192 d\u2082 \u2260 0 \u2192 f\u2082 n\u2081 d\u2081 n\u2082 d\u2082 \u2260 0\na b c d : \u2124\nb0 : b \u2260 0\nd0 : d \u2260 0\nH : \u2200 {n\u2081 d\u2081 n\u2082 d\u2082 : \u2124}, a * d\u2081 = n\u2081 * b \u2192 c * d\u2082 = n\u2082 * d \u2192 f\u2081 n\u2081 d\u2081 n\u2082 d\u2082 * f\u2082 a b c d = f\u2081 a b c d * f\u2082 n\u2081 d\u2081 n\u2082 d\u2082\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nha : a /. b = n\u2081 /. \u2191d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nhc : c /. d = n\u2082 /. \u2191d\u2082\nd\u20810 : \u2191d\u2081 \u2260 \u21910\nd\u20820 : \u2191d\u2082 \u2260 \u21910\n\u22a2 f\u2081 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 /. f\u2082 n\u2081 (\u2191d\u2081) n\u2082 \u2191d\u2082 = f\u2081 a b c d /. f\u2082 a b c d\n[PROOFSTEP]\nexact (divInt_eq_iff (f0 d\u20810 d\u20820) (f0 b0 d0)).2 (H ((divInt_eq_iff b0 d\u20810).1 ha) ((divInt_eq_iff d0 d\u20820).1 hc))\n[GOAL]\na b c : \u211a\nn : \u2124\nd : \u2115\nh : d \u2260 0\n\u22a2 n /. \u2191d + 0 = n /. \u2191d\n[PROOFSTEP]\nrw [\u2190 zero_divInt d, add_def'', zero_mul, add_zero, divInt_mul_right]\n[GOAL]\na b c : \u211a\nn : \u2124\nd : \u2115\nh : d \u2260 0\n\u22a2 \u2191d \u2260 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase b0\na b c : \u211a\nn : \u2124\nd : \u2115\nh : d \u2260 0\n\u22a2 \u2191d \u2260 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase d0\na b c : \u211a\nn : \u2124\nd : \u2115\nh : d \u2260 0\n\u22a2 \u2191d \u2260 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\na b c : \u211a\nn : \u2124\nd : \u2115\nh : d \u2260 0\n\u22a2 0 + n /. \u2191d = n /. \u2191d\n[PROOFSTEP]\nrw [\u2190 zero_divInt d, add_def'', zero_mul, zero_add, divInt_mul_right]\n[GOAL]\na b c : \u211a\nn : \u2124\nd : \u2115\nh : d \u2260 0\n\u22a2 \u2191d \u2260 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase b0\na b c : \u211a\nn : \u2124\nd : \u2115\nh : d \u2260 0\n\u22a2 \u2191d \u2260 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase d0\na b c : \u211a\nn : \u2124\nd : \u2115\nh : d \u2260 0\n\u22a2 \u2191d \u2260 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\na b c : \u211a\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\n\u22a2 n\u2081 /. \u2191d\u2081 + n\u2082 /. \u2191d\u2082 = n\u2082 /. \u2191d\u2082 + n\u2081 /. \u2191d\u2081\n[PROOFSTEP]\nsimp [h\u2081, h\u2082, add_comm, mul_comm]\n[GOAL]\na b c : \u211a\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nn\u2083 : \u2124\nd\u2083 : \u2115\nh\u2083 : d\u2083 \u2260 0\n\u22a2 n\u2081 /. \u2191d\u2081 + n\u2082 /. \u2191d\u2082 + n\u2083 /. \u2191d\u2083 = n\u2081 /. \u2191d\u2081 + (n\u2082 /. \u2191d\u2082 + n\u2083 /. \u2191d\u2083)\n[PROOFSTEP]\nsimp [h\u2081, h\u2082, h\u2083]\n[GOAL]\na b c : \u211a\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nn\u2083 : \u2124\nd\u2083 : \u2115\nh\u2083 : d\u2083 \u2260 0\n\u22a2 ((n\u2081 * \u2191d\u2082 + n\u2082 * \u2191d\u2081) * \u2191d\u2083 + n\u2083 * (\u2191d\u2081 * \u2191d\u2082)) /. (\u2191d\u2081 * \u2191d\u2082 * \u2191d\u2083) =\n (n\u2081 * (\u2191d\u2082 * \u2191d\u2083) + (n\u2082 * \u2191d\u2083 + n\u2083 * \u2191d\u2082) * \u2191d\u2081) /. (\u2191d\u2081 * (\u2191d\u2082 * \u2191d\u2083))\n[PROOFSTEP]\nrw [mul_assoc, add_mul, add_mul, mul_assoc, add_assoc]\n[GOAL]\na b c : \u211a\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nn\u2083 : \u2124\nd\u2083 : \u2115\nh\u2083 : d\u2083 \u2260 0\n\u22a2 (n\u2081 * (\u2191d\u2082 * \u2191d\u2083) + (n\u2082 * \u2191d\u2081 * \u2191d\u2083 + n\u2083 * (\u2191d\u2081 * \u2191d\u2082))) /. (\u2191d\u2081 * (\u2191d\u2082 * \u2191d\u2083)) =\n (n\u2081 * (\u2191d\u2082 * \u2191d\u2083) + (n\u2082 * \u2191d\u2083 * \u2191d\u2081 + n\u2083 * \u2191d\u2082 * \u2191d\u2081)) /. (\u2191d\u2081 * (\u2191d\u2082 * \u2191d\u2083))\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_a\na b c : \u211a\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nn\u2083 : \u2124\nd\u2083 : \u2115\nh\u2083 : d\u2083 \u2260 0\n\u22a2 n\u2082 * \u2191d\u2081 * \u2191d\u2083 + n\u2083 * (\u2191d\u2081 * \u2191d\u2082) = n\u2082 * \u2191d\u2083 * \u2191d\u2081 + n\u2083 * \u2191d\u2082 * \u2191d\u2081\n[PROOFSTEP]\nac_rfl\n[GOAL]\na b c : \u211a\nn : \u2124\nd : \u2115\nh : d \u2260 0\n\u22a2 -(n /. \u2191d) + n /. \u2191d = 0\n[PROOFSTEP]\nsimp [h, mkRat_add_mkRat]\n[GOAL]\na b c : \u211a\n\u22a2 0 /. 1 = 0\n[PROOFSTEP]\nrw [divInt]\n[GOAL]\na b c : \u211a\n\u22a2 (match 0, 1 with\n | n, Int.ofNat d => inline (mkRat n d)\n | n, Int.negSucc d => normalize (-n) (Nat.succ d)) =\n 0\n[PROOFSTEP]\nsimp\n[GOAL]\na b c : \u211a\n\u22a2 1 /. 1 = 1\n[PROOFSTEP]\nrw [divInt]\n[GOAL]\na b c : \u211a\n\u22a2 (match 1, 1 with\n | n, Int.ofNat d => inline (mkRat n d)\n | n, Int.negSucc d => normalize (-n) (Nat.succ d)) =\n 1\n[PROOFSTEP]\nsimp\n[GOAL]\na b c : \u211a\n\u22a2 -1 /. 1 = -1\n[PROOFSTEP]\nrw [divInt]\n[GOAL]\na b c : \u211a\n\u22a2 (match -1, 1 with\n | n, Int.ofNat d => inline (mkRat n d)\n | n, Int.negSucc d => normalize (-n) (Nat.succ d)) =\n -1\n[PROOFSTEP]\nsimp\n[GOAL]\na b c : \u211a\nn : \u2124\n\u22a2 n /. 1 = \u2191n\n[PROOFSTEP]\nrw [divInt]\n[GOAL]\na b c : \u211a\nn : \u2124\n\u22a2 (match n, 1 with\n | n, Int.ofNat d => inline (mkRat n d)\n | n, Int.negSucc d => normalize (-n) (Nat.succ d)) =\n \u2191n\n[PROOFSTEP]\nsimp [mkRat, normalize]\n[GOAL]\na b c : \u211a\nn : \u2124\n\u22a2 mk' n 1 = \u2191n\n[PROOFSTEP]\nrfl\n[GOAL]\na b c : \u211a\nn : \u2124\n\u22a2 mkRat n 1 = \u2191n\n[PROOFSTEP]\nsimp [Rat.mkRat_eq, Rat.divInt_one]\n[GOAL]\na b c : \u211a\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nn\u2083 : \u2124\nd\u2083 : \u2115\nh\u2083 : d\u2083 \u2260 0\n\u22a2 n\u2081 /. \u2191d\u2081 * (n\u2082 /. \u2191d\u2082) * (n\u2083 /. \u2191d\u2083) = n\u2081 /. \u2191d\u2081 * (n\u2082 /. \u2191d\u2082 * (n\u2083 /. \u2191d\u2083))\n[PROOFSTEP]\nsimp [h\u2081, h\u2082, h\u2083, mul_ne_zero, mul_comm, mul_assoc, mul_left_comm]\n[GOAL]\na b c : \u211a\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nn\u2083 : \u2124\nd\u2083 : \u2115\nh\u2083 : d\u2083 \u2260 0\n\u22a2 (n\u2081 /. \u2191d\u2081 + n\u2082 /. \u2191d\u2082) * (n\u2083 /. \u2191d\u2083) = n\u2081 /. \u2191d\u2081 * (n\u2083 /. \u2191d\u2083) + n\u2082 /. \u2191d\u2082 * (n\u2083 /. \u2191d\u2083)\n[PROOFSTEP]\nsimp [h\u2081, h\u2082, h\u2083, mul_ne_zero]\n[GOAL]\na b c : \u211a\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nn\u2083 : \u2124\nd\u2083 : \u2115\nh\u2083 : d\u2083 \u2260 0\n\u22a2 (n\u2081 * \u2191d\u2082 + n\u2082 * \u2191d\u2081) * n\u2083 /. (\u2191d\u2081 * \u2191d\u2082 * \u2191d\u2083) =\n (n\u2081 * n\u2083 * (\u2191d\u2082 * \u2191d\u2083) + n\u2082 * n\u2083 * (\u2191d\u2081 * \u2191d\u2083)) /. (\u2191d\u2081 * \u2191d\u2083 * (\u2191d\u2082 * \u2191d\u2083))\n[PROOFSTEP]\nrw [\u2190 divInt_mul_right (Int.coe_nat_ne_zero.2 h\u2083), add_mul, add_mul]\n[GOAL]\na b c : \u211a\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nn\u2083 : \u2124\nd\u2083 : \u2115\nh\u2083 : d\u2083 \u2260 0\n\u22a2 (n\u2081 * \u2191d\u2082 * n\u2083 * \u2191d\u2083 + n\u2082 * \u2191d\u2081 * n\u2083 * \u2191d\u2083) /. (\u2191d\u2081 * \u2191d\u2082 * \u2191d\u2083 * \u2191d\u2083) =\n (n\u2081 * n\u2083 * (\u2191d\u2082 * \u2191d\u2083) + n\u2082 * n\u2083 * (\u2191d\u2081 * \u2191d\u2083)) /. (\u2191d\u2081 * \u2191d\u2083 * (\u2191d\u2082 * \u2191d\u2083))\n[PROOFSTEP]\nac_rfl\n[GOAL]\na b c : \u211a\n\u22a2 a * (b + c) = a * b + a * c\n[PROOFSTEP]\nrw [Rat.mul_comm, Rat.add_mul, Rat.mul_comm, Rat.mul_comm c a]\n[GOAL]\na b c : \u211a\n\u22a2 0 \u2260 1\n[PROOFSTEP]\nrw [ne_comm, \u2190 divInt_one_one, divInt_ne_zero one_ne_zero]\n[GOAL]\na b c : \u211a\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nexact one_ne_zero\n[GOAL]\na b c : \u211a\nn : \u2124\nd : \u2115\nh : d \u2260 0\na0 : n /. \u2191d \u2260 0\n\u22a2 n /. \u2191d * (n /. \u2191d)\u207b\u00b9 = 1\n[PROOFSTEP]\nhave n0 : n \u2260 0 := mt (by rintro rfl; simp) a0\n[GOAL]\na b c : \u211a\nn : \u2124\nd : \u2115\nh : d \u2260 0\na0 : n /. \u2191d \u2260 0\n\u22a2 n = 0 \u2192 n /. \u2191d = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\na b c : \u211a\nd : \u2115\nh : d \u2260 0\na0 : 0 /. \u2191d \u2260 0\n\u22a2 0 /. \u2191d = 0\n[PROOFSTEP]\nsimp\n[GOAL]\na b c : \u211a\nn : \u2124\nd : \u2115\nh : d \u2260 0\na0 : n /. \u2191d \u2260 0\nn0 : n \u2260 0\n\u22a2 n /. \u2191d * (n /. \u2191d)\u207b\u00b9 = 1\n[PROOFSTEP]\nsimpa [h, n0, mul_comm] using @divInt_mul_right 1 1 (n * d) (by simp [h, n0])\n[GOAL]\na b c : \u211a\nn : \u2124\nd : \u2115\nh : d \u2260 0\na0 : n /. \u2191d \u2260 0\nn0 : n \u2260 0\n\u22a2 n * \u2191d \u2260 0\n[PROOFSTEP]\nsimp [h, n0]\n[GOAL]\na b c : \u211a\nn : \u2115\n\u22a2 NatCast.natCast (n + 1) = NatCast.natCast n + 1\n[PROOFSTEP]\nsimp only [coe_int_eq_divInt, add_def'' one_ne_zero one_ne_zero, \u2190 divInt_one_one, Nat.cast_add, Nat.cast_one, mul_one]\n[GOAL]\na b c : \u211a\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\nchange Rat.inv 0 = 0\n[GOAL]\na b c : \u211a\n\u22a2 Rat.inv 0 = 0\n[PROOFSTEP]\nrw [Rat.inv_def]\n[GOAL]\na b c : \u211a\n\u22a2 \u21910.den /. 0.num = 0\n[PROOFSTEP]\nrfl\n[GOAL]\na b c : \u211a\n\u22a2 Nontrivial \u211a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b c : \u211a\n\u22a2 CommSemiring \u211a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b c : \u211a\n\u22a2 Semiring \u211a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b c : \u211a\n\u22a2 AddCommGroup \u211a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b c : \u211a\n\u22a2 AddGroup \u211a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b c : \u211a\n\u22a2 AddCommMonoid \u211a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b c : \u211a\n\u22a2 AddMonoid \u211a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b c : \u211a\n\u22a2 AddLeftCancelSemigroup \u211a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b c : \u211a\n\u22a2 AddRightCancelSemigroup \u211a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b c : \u211a\n\u22a2 AddCommSemigroup \u211a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b c : \u211a\n\u22a2 AddSemigroup \u211a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b c : \u211a\n\u22a2 CommMonoid \u211a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b c : \u211a\n\u22a2 Monoid \u211a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b c : \u211a\n\u22a2 CommSemigroup \u211a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b c : \u211a\n\u22a2 Semigroup \u211a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b c p q : \u211a\n\u22a2 p = q \u2194 p.num * \u2191q.den = q.num * \u2191p.den\n[PROOFSTEP]\nconv =>\n lhs\n rw [\u2190 @num_den p, \u2190 @num_den q]\n[GOAL]\na b c p q : \u211a\n| p = q \u2194 p.num * \u2191q.den = q.num * \u2191p.den\n[PROOFSTEP]\n lhs\n rw [\u2190 @num_den p, \u2190 @num_den q]\n[GOAL]\na b c p q : \u211a\n| p = q \u2194 p.num * \u2191q.den = q.num * \u2191p.den\n[PROOFSTEP]\n lhs\n rw [\u2190 @num_den p, \u2190 @num_den q]\n[GOAL]\na b c p q : \u211a\n| p = q \u2194 p.num * \u2191q.den = q.num * \u2191p.den\n[PROOFSTEP]\nlhs\n[GOAL]\na b c p q : \u211a\n| p = q\n[PROOFSTEP]\nrw [\u2190 @num_den p, \u2190 @num_den q]\n[GOAL]\na b c p q : \u211a\n\u22a2 p.num /. \u2191p.den = q.num /. \u2191q.den \u2194 p.num * \u2191q.den = q.num * \u2191p.den\n[PROOFSTEP]\napply Rat.divInt_eq_iff\n[GOAL]\ncase z\u2081\na b c p q : \u211a\n\u22a2 \u2191p.den \u2260 0\n[PROOFSTEP]\nrw [\u2190 Nat.cast_zero, Ne, Int.ofNat_inj]\n[GOAL]\ncase z\u2081\na b c p q : \u211a\n\u22a2 \u00acp.den = 0\n[PROOFSTEP]\napply den_nz\n[GOAL]\ncase z\u2082\na b c p q : \u211a\n\u22a2 \u2191q.den \u2260 0\n[PROOFSTEP]\nrw [\u2190 Nat.cast_zero, Ne, Int.ofNat_inj]\n[GOAL]\ncase z\u2082\na b c p q : \u211a\n\u22a2 \u00acq.den = 0\n[PROOFSTEP]\napply den_nz\n[GOAL]\na b c q : \u211a\nhq : q.num = 0\n\u22a2 q = 0\n[PROOFSTEP]\nhave : q = q.num /. q.den := num_den.symm\n[GOAL]\na b c q : \u211a\nhq : q.num = 0\nthis : q = q.num /. \u2191q.den\n\u22a2 q = 0\n[PROOFSTEP]\nsimpa [hq] using this\n[GOAL]\na b c q : \u211a\nx\u271d : q = 0\n\u22a2 q.num = 0\n[PROOFSTEP]\nsimp [*]\n[GOAL]\na b c q : \u211a\nn d : \u2124\nhq : q \u2260 0\nhqnd : q = n /. d\nthis : n = 0\n\u22a2 q = 0\n[PROOFSTEP]\nsimpa [this] using hqnd\n[GOAL]\na b c q : \u211a\nn d : \u2124\nhq : q \u2260 0\nhqnd : q = n /. d\nthis : d = 0\n\u22a2 q = 0\n[PROOFSTEP]\nsimpa [this] using hqnd\n[GOAL]\na b c q r : \u211a\n\u22a2 q * r = q.num * r.num /. \u2191(q.den * r.den)\n[PROOFSTEP]\nhave hq' : (\u2191q.den : \u2124) \u2260 0 := by have := den_nz q; simpa\n[GOAL]\na b c q r : \u211a\n\u22a2 \u2191q.den \u2260 0\n[PROOFSTEP]\nhave := den_nz q\n[GOAL]\na b c q r : \u211a\nthis : q.den \u2260 0\n\u22a2 \u2191q.den \u2260 0\n[PROOFSTEP]\nsimpa\n[GOAL]\na b c q r : \u211a\nhq' : \u2191q.den \u2260 0\n\u22a2 q * r = q.num * r.num /. \u2191(q.den * r.den)\n[PROOFSTEP]\nhave hr' : (\u2191r.den : \u2124) \u2260 0 := by have := den_nz r; simpa\n[GOAL]\na b c q r : \u211a\nhq' : \u2191q.den \u2260 0\n\u22a2 \u2191r.den \u2260 0\n[PROOFSTEP]\nhave := den_nz r\n[GOAL]\na b c q r : \u211a\nhq' : \u2191q.den \u2260 0\nthis : r.den \u2260 0\n\u22a2 \u2191r.den \u2260 0\n[PROOFSTEP]\nsimpa\n[GOAL]\na b c q r : \u211a\nhq' : \u2191q.den \u2260 0\nhr' : \u2191r.den \u2260 0\n\u22a2 q * r = q.num * r.num /. \u2191(q.den * r.den)\n[PROOFSTEP]\nsuffices q.num /. \u2191q.den * (r.num /. \u2191r.den) = q.num * r.num /. \u2191(q.den * r.den) by simpa [num_den] using this\n[GOAL]\na b c q r : \u211a\nhq' : \u2191q.den \u2260 0\nhr' : \u2191r.den \u2260 0\nthis : q.num /. \u2191q.den * (r.num /. \u2191r.den) = q.num * r.num /. \u2191(q.den * r.den)\n\u22a2 q * r = q.num * r.num /. \u2191(q.den * r.den)\n[PROOFSTEP]\nsimpa [num_den] using this\n[GOAL]\na b c q r : \u211a\nhq' : \u2191q.den \u2260 0\nhr' : \u2191r.den \u2260 0\n\u22a2 q.num /. \u2191q.den * (r.num /. \u2191r.den) = q.num * r.num /. \u2191(q.den * r.den)\n[PROOFSTEP]\nsimp [mul_def' hq' hr']\n[GOAL]\na b c q r : \u211a\nhr : r.num = 0\n\u22a2 q / r = q.num * \u2191r.den /. (\u2191q.den * r.num)\n[PROOFSTEP]\nhave hr' : r = 0 := zero_of_num_zero hr\n[GOAL]\na b c q r : \u211a\nhr : r.num = 0\nhr' : r = 0\n\u22a2 q / r = q.num * \u2191r.den /. (\u2191q.den * r.num)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\na b c q r : \u211a\nhr : \u00acr.num = 0\n\u22a2 q * r\u207b\u00b9 = q.num /. \u2191q.den * (r.num /. \u2191r.den)\u207b\u00b9\n[PROOFSTEP]\nsimp [num_den]\n[GOAL]\na b c q r : \u211a\nhr : \u00acr.num = 0\n\u22a2 q.num /. \u2191q.den * (r.num /. \u2191r.den)\u207b\u00b9 = q.num /. \u2191q.den * (\u2191r.den /. r.num)\n[PROOFSTEP]\nrw [inv_def']\n[GOAL]\na b c q r : \u211a\nhr : \u00acr.num = 0\n\u22a2 \u2191q.den \u2260 0\n[PROOFSTEP]\nsimpa using den_nz q\n[GOAL]\na\u271d b\u271d c\u271d : \u211a\na b c : \u2124\nh : c = 0\n\u22a2 (a + b) /. c = a /. c + b /. c\n[PROOFSTEP]\nsimp [h]\n[GOAL]\na\u271d b\u271d c\u271d : \u211a\na b c : \u2124\nh : \u00acc = 0\n\u22a2 (a + b) /. c = a /. c + b /. c\n[PROOFSTEP]\nrw [add_def'' h h, divInt_eq_iff h (mul_ne_zero h h)]\n[GOAL]\na\u271d b\u271d c\u271d : \u211a\na b c : \u2124\nh : \u00acc = 0\n\u22a2 (a + b) * (c * c) = (a * c + b * c) * c\n[PROOFSTEP]\nsimp [add_mul, mul_assoc]\n[GOAL]\na b c : \u211a\nn d : \u2124\n\u22a2 n /. d = \u2191n / \u2191d\n[PROOFSTEP]\nby_cases d0 : d = 0\n[GOAL]\ncase pos\na b c : \u211a\nn d : \u2124\nd0 : d = 0\n\u22a2 n /. d = \u2191n / \u2191d\n[PROOFSTEP]\nsimp [d0, div_zero]\n[GOAL]\ncase neg\na b c : \u211a\nn d : \u2124\nd0 : \u00acd = 0\n\u22a2 n /. d = \u2191n / \u2191d\n[PROOFSTEP]\nsimp [division_def, coe_int_eq_divInt, mul_def' one_ne_zero d0]\n[GOAL]\na b c : \u211a\nx : \u2124\nhx : x \u2260 0\nn d : \u2124\n\u22a2 n /. x * (x /. d) = n /. d\n[PROOFSTEP]\nby_cases hd : d = 0\n[GOAL]\ncase pos\na b c : \u211a\nx : \u2124\nhx : x \u2260 0\nn d : \u2124\nhd : d = 0\n\u22a2 n /. x * (x /. d) = n /. d\n[PROOFSTEP]\nrw [hd]\n[GOAL]\ncase pos\na b c : \u211a\nx : \u2124\nhx : x \u2260 0\nn d : \u2124\nhd : d = 0\n\u22a2 n /. x * (x /. 0) = n /. 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\na b c : \u211a\nx : \u2124\nhx : x \u2260 0\nn d : \u2124\nhd : \u00acd = 0\n\u22a2 n /. x * (x /. d) = n /. d\n[PROOFSTEP]\nrw [mul_def' hx hd, mul_comm x, divInt_mul_right hx]\n[GOAL]\na b c : \u211a\nx : \u2124\nhx : x \u2260 0\nn d : \u2124\n\u22a2 n /. x / (d /. x) = n /. d\n[PROOFSTEP]\nrw [div_eq_mul_inv, inv_def', divInt_mul_divInt_cancel hx]\n[GOAL]\na b c : \u211a\nx : \u2124\nhx : x \u2260 0\nn d : \u2124\n\u22a2 x /. n / (x /. d) = d /. n\n[PROOFSTEP]\nrw [div_eq_mul_inv, inv_def', mul_comm, divInt_mul_divInt_cancel hx]\n[GOAL]\na b c : \u211a\nn d : \u2124\n\u22a2 \u2191n / \u2191d = n /. d\n[PROOFSTEP]\nrepeat' rw [coe_int_eq_divInt]\n[GOAL]\na b c : \u211a\nn d : \u2124\n\u22a2 \u2191n / \u2191d = n /. d\n[PROOFSTEP]\nrw [coe_int_eq_divInt]\n[GOAL]\na b c : \u211a\nn d : \u2124\n\u22a2 n /. 1 / \u2191d = n /. d\n[PROOFSTEP]\nrw [coe_int_eq_divInt]\n[GOAL]\na b c : \u211a\nn d : \u2124\n\u22a2 n /. 1 / (d /. 1) = n /. d\n[PROOFSTEP]\nrw [coe_int_eq_divInt]\n[GOAL]\na b c : \u211a\nn d : \u2124\n\u22a2 n /. 1 / (d /. 1) = n /. d\n[PROOFSTEP]\nexact divInt_div_divInt_cancel_left one_ne_zero n d\n[GOAL]\na b c r : \u211a\n\u22a2 \u2191r.num / \u2191r.den = r\n[PROOFSTEP]\nrw [\u2190 Int.cast_ofNat]\n[GOAL]\na b c r : \u211a\n\u22a2 \u2191r.num / \u2191\u2191r.den = r\n[PROOFSTEP]\nerw [\u2190 divInt_eq_div, num_den]\n[GOAL]\na b c q : \u211a\nhq : q.den = 1\n\u22a2 \u2191q.num = q\n[PROOFSTEP]\nconv_rhs => rw [\u2190 @num_den q, hq]\n[GOAL]\na b c q : \u211a\nhq : q.den = 1\n| q\n[PROOFSTEP]\nrw [\u2190 @num_den q, hq]\n[GOAL]\na b c q : \u211a\nhq : q.den = 1\n| q\n[PROOFSTEP]\nrw [\u2190 @num_den q, hq]\n[GOAL]\na b c q : \u211a\nhq : q.den = 1\n| q\n[PROOFSTEP]\nrw [\u2190 @num_den q, hq]\n[GOAL]\na b c q : \u211a\nhq : q.den = 1\n\u22a2 \u2191q.num = q.num /. \u21911\n[PROOFSTEP]\nrw [coe_int_eq_divInt]\n[GOAL]\na b c q : \u211a\nhq : q.den = 1\n\u22a2 q.num /. 1 = q.num /. \u21911\n[PROOFSTEP]\nrfl\n[GOAL]\na b c : \u211a\nn : \u2115\n\u22a2 \u2191n = \u2191n /. 1\n[PROOFSTEP]\nrw [\u2190 Int.cast_ofNat, coe_int_eq_divInt]\n[GOAL]\na b c : \u211a\nn : \u2115\n\u22a2 (\u2191n).num = \u2191n\n[PROOFSTEP]\nrw [\u2190 Int.cast_ofNat, coe_int_num]\n[GOAL]\na b c : \u211a\nn : \u2115\n\u22a2 (\u2191n).den = 1\n[PROOFSTEP]\nrw [\u2190 Int.cast_ofNat, coe_int_den]\n[GOAL]\na b c : \u211a\nn : \u2124\nd : \u2115\n\u22a2 mkRat n d = \u2191n / \u2191d\n[PROOFSTEP]\nsimp [mkRat]\n[GOAL]\na b c : \u211a\nn : \u2124\nd : \u2115\n\u22a2 (if h : d = 0 then 0 else normalize n d) = \u2191n / \u2191d\n[PROOFSTEP]\nby_cases d = 0\n[GOAL]\na b c : \u211a\nn : \u2124\nd : \u2115\n\u22a2 (if h : d = 0 then 0 else normalize n d) = \u2191n / \u2191d\n[PROOFSTEP]\nby_cases d = 0\n[GOAL]\ncase pos\na b c : \u211a\nn : \u2124\nd : \u2115\nh : d = 0\n\u22a2 (if h : d = 0 then 0 else normalize n d) = \u2191n / \u2191d\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\na b c : \u211a\nn : \u2124\nd : \u2115\nh : \u00acd = 0\n\u22a2 (if h : d = 0 then 0 else normalize n d) = \u2191n / \u2191d\n[PROOFSTEP]\nsimp [h, HDiv.hDiv, Rat.div, Div.div]\n[GOAL]\ncase neg\na b c : \u211a\nn : \u2124\nd : \u2115\nh : \u00acd = 0\n\u22a2 normalize n d = \u2191n * Rat.inv \u2191d\n[PROOFSTEP]\nunfold Rat.inv\n[GOAL]\ncase neg\na b c : \u211a\nn : \u2124\nd : \u2115\nh : \u00acd = 0\n\u22a2 normalize n d =\n \u2191n *\n if h : (\u2191d).num < 0 then mk' (-\u2191(\u2191d).den) (Int.natAbs (\u2191d).num)\n else if h : (\u2191d).num > 0 then mk' (\u2191(\u2191d).den) (Int.natAbs (\u2191d).num) else \u2191d\n[PROOFSTEP]\nhave h\u2081 : 0 < d := Nat.pos_iff_ne_zero.2 h\n[GOAL]\ncase neg\na b c : \u211a\nn : \u2124\nd : \u2115\nh : \u00acd = 0\nh\u2081 : 0 < d\n\u22a2 normalize n d =\n \u2191n *\n if h : (\u2191d).num < 0 then mk' (-\u2191(\u2191d).den) (Int.natAbs (\u2191d).num)\n else if h : (\u2191d).num > 0 then mk' (\u2191(\u2191d).den) (Int.natAbs (\u2191d).num) else \u2191d\n[PROOFSTEP]\nhave h\u2082 : \u00ac(d : \u2124) < 0 := by simp\n[GOAL]\na b c : \u211a\nn : \u2124\nd : \u2115\nh : \u00acd = 0\nh\u2081 : 0 < d\n\u22a2 \u00ac\u2191d < 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\na b c : \u211a\nn : \u2124\nd : \u2115\nh : \u00acd = 0\nh\u2081 : 0 < d\nh\u2082 : \u00ac\u2191d < 0\n\u22a2 normalize n d =\n \u2191n *\n if h : (\u2191d).num < 0 then mk' (-\u2191(\u2191d).den) (Int.natAbs (\u2191d).num)\n else if h : (\u2191d).num > 0 then mk' (\u2191(\u2191d).den) (Int.natAbs (\u2191d).num) else \u2191d\n[PROOFSTEP]\nsimp [h, h\u2081, h\u2082, \u2190 Rat.normalize_eq_mk', Rat.normalize_eq_mkRat, \u2190 mkRat_one, Rat.mkRat_mul_mkRat]\n", "meta": {"mathlib_filename": "Mathlib.Data.Rat.Defs", "llama_tokens": 14447, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085708384735, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.4354304084102171}} {"text": "[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nh : Countable M\n\u22a2 Set.Countable (Quotient.mk' '' age L M)\n[PROOFSTEP]\nclassical\nrefine'\n (congr_arg _ (Set.ext <| forall_quotient_iff.2 fun N => _)).mp\n (countable_range fun s : Finset M => \u27e6\u27e8closure L (s : Set M), inferInstance\u27e9\u27e7)\nconstructor\n\u00b7 rintro \u27e8s, hs\u27e9\n use Bundled.of (closure L (s : Set M))\n exact \u27e8\u27e8(fg_iff_structure_fg _).1 (fg_closure s.finite_toSet), \u27e8Substructure.subtype _\u27e9\u27e9, hs\u27e9\n\u00b7 simp only [mem_range, Quotient.eq]\n rintro \u27e8P, \u27e8\u27e8s, hs\u27e9, \u27e8PM\u27e9\u27e9, hP2\u27e9\n have : P \u2248 N := by apply Quotient.eq'.mp; rw [hP2];\n rfl\n -- Porting note: added\n refine' \u27e8s.image PM, Setoid.trans (b := P) _ this\u27e9\n rw [\u2190 Embedding.coe_toHom, Finset.coe_image, closure_image PM.toHom, hs, \u2190 Hom.range_eq_map]\n exact \u27e8PM.equivRange.symm\u27e9\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nh : Countable M\n\u22a2 Set.Countable (Quotient.mk' '' age L M)\n[PROOFSTEP]\nrefine'\n (congr_arg _ (Set.ext <| forall_quotient_iff.2 fun N => _)).mp\n (countable_range fun s : Finset M => \u27e6\u27e8closure L (s : Set M), inferInstance\u27e9\u27e7)\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : Countable M\nN : Bundled (Structure L)\n\u22a2 (Quotient.mk equivSetoid N \u2208\n range fun s => Quotient.mk equivSetoid (Bundled.mk { x // x \u2208 LowerAdjoint.toFun (closure L) \u2191s })) \u2194\n Quotient.mk equivSetoid N \u2208 Quotient.mk' '' age L M\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : Countable M\nN : Bundled (Structure L)\n\u22a2 (Quotient.mk equivSetoid N \u2208\n range fun s => Quotient.mk equivSetoid (Bundled.mk { x // x \u2208 LowerAdjoint.toFun (closure L) \u2191s })) \u2192\n Quotient.mk equivSetoid N \u2208 Quotient.mk' '' age L M\n[PROOFSTEP]\nrintro \u27e8s, hs\u27e9\n[GOAL]\ncase mp.intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : Countable M\nN : Bundled (Structure L)\ns : Finset M\nhs :\n (fun s => Quotient.mk equivSetoid (Bundled.mk { x // x \u2208 LowerAdjoint.toFun (closure L) \u2191s })) s =\n Quotient.mk equivSetoid N\n\u22a2 Quotient.mk equivSetoid N \u2208 Quotient.mk' '' age L M\n[PROOFSTEP]\nuse Bundled.of (closure L (s : Set M))\n[GOAL]\ncase h\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : Countable M\nN : Bundled (Structure L)\ns : Finset M\nhs :\n (fun s => Quotient.mk equivSetoid (Bundled.mk { x // x \u2208 LowerAdjoint.toFun (closure L) \u2191s })) s =\n Quotient.mk equivSetoid N\n\u22a2 Bundled.of { x // x \u2208 LowerAdjoint.toFun (closure L) \u2191s } \u2208 age L M \u2227\n Quotient.mk' (Bundled.of { x // x \u2208 LowerAdjoint.toFun (closure L) \u2191s }) = Quotient.mk equivSetoid N\n[PROOFSTEP]\nexact \u27e8\u27e8(fg_iff_structure_fg _).1 (fg_closure s.finite_toSet), \u27e8Substructure.subtype _\u27e9\u27e9, hs\u27e9\n[GOAL]\ncase mpr\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : Countable M\nN : Bundled (Structure L)\n\u22a2 Quotient.mk equivSetoid N \u2208 Quotient.mk' '' age L M \u2192\n Quotient.mk equivSetoid N \u2208\n range fun s => Quotient.mk equivSetoid (Bundled.mk { x // x \u2208 LowerAdjoint.toFun (closure L) \u2191s })\n[PROOFSTEP]\nsimp only [mem_range, Quotient.eq]\n[GOAL]\ncase mpr\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : Countable M\nN : Bundled (Structure L)\n\u22a2 Quotient.mk equivSetoid N \u2208 Quotient.mk' '' age L M \u2192\n \u2203 y, Bundled.mk { x // x \u2208 LowerAdjoint.toFun (closure L) \u2191y } \u2248 N\n[PROOFSTEP]\nrintro \u27e8P, \u27e8\u27e8s, hs\u27e9, \u27e8PM\u27e9\u27e9, hP2\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.mk.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : Countable M\nN P : Bundled (Structure L)\nhP2 : Quotient.mk' P = Quotient.mk equivSetoid N\ns : Finset \u2191P\nhs : LowerAdjoint.toFun (closure L) \u2191s = \u22a4\nPM : \u2191P \u21aa[L] M\n\u22a2 \u2203 y, Bundled.mk { x // x \u2208 LowerAdjoint.toFun (closure L) \u2191y } \u2248 N\n[PROOFSTEP]\nhave : P \u2248 N := by apply Quotient.eq'.mp; rw [hP2];\n rfl\n -- Porting note: added\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : Countable M\nN P : Bundled (Structure L)\nhP2 : Quotient.mk' P = Quotient.mk equivSetoid N\ns : Finset \u2191P\nhs : LowerAdjoint.toFun (closure L) \u2191s = \u22a4\nPM : \u2191P \u21aa[L] M\n\u22a2 P \u2248 N\n[PROOFSTEP]\napply Quotient.eq'.mp\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : Countable M\nN P : Bundled (Structure L)\nhP2 : Quotient.mk' P = Quotient.mk equivSetoid N\ns : Finset \u2191P\nhs : LowerAdjoint.toFun (closure L) \u2191s = \u22a4\nPM : \u2191P \u21aa[L] M\n\u22a2 Quotient.mk' P = Quotient.mk' N\n[PROOFSTEP]\nrw [hP2]\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : Countable M\nN P : Bundled (Structure L)\nhP2 : Quotient.mk' P = Quotient.mk equivSetoid N\ns : Finset \u2191P\nhs : LowerAdjoint.toFun (closure L) \u2191s = \u22a4\nPM : \u2191P \u21aa[L] M\n\u22a2 Quotient.mk equivSetoid N = Quotient.mk' N\n[PROOFSTEP]\nrfl\n -- Porting note: added\n[GOAL]\ncase mpr.intro.intro.intro.mk.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : Countable M\nN P : Bundled (Structure L)\nhP2 : Quotient.mk' P = Quotient.mk equivSetoid N\ns : Finset \u2191P\nhs : LowerAdjoint.toFun (closure L) \u2191s = \u22a4\nPM : \u2191P \u21aa[L] M\nthis : P \u2248 N\n\u22a2 \u2203 y, Bundled.mk { x // x \u2208 LowerAdjoint.toFun (closure L) \u2191y } \u2248 N\n[PROOFSTEP]\nrefine' \u27e8s.image PM, Setoid.trans (b := P) _ this\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.mk.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : Countable M\nN P : Bundled (Structure L)\nhP2 : Quotient.mk' P = Quotient.mk equivSetoid N\ns : Finset \u2191P\nhs : LowerAdjoint.toFun (closure L) \u2191s = \u22a4\nPM : \u2191P \u21aa[L] M\nthis : P \u2248 N\n\u22a2 Bundled.mk { x // x \u2208 LowerAdjoint.toFun (closure L) \u2191(Finset.image (\u2191PM) s) } \u2248 P\n[PROOFSTEP]\nrw [\u2190 Embedding.coe_toHom, Finset.coe_image, closure_image PM.toHom, hs, \u2190 Hom.range_eq_map]\n[GOAL]\ncase mpr.intro.intro.intro.mk.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : Countable M\nN P : Bundled (Structure L)\nhP2 : Quotient.mk' P = Quotient.mk equivSetoid N\ns : Finset \u2191P\nhs : LowerAdjoint.toFun (closure L) \u2191s = \u22a4\nPM : \u2191P \u21aa[L] M\nthis : P \u2248 N\n\u22a2 Bundled.mk { x // x \u2208 Hom.range (Embedding.toHom PM) } \u2248 P\n[PROOFSTEP]\nexact \u27e8PM.equivRange.symm\u27e9\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u2076 : Structure L M\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\n\u22a2 age L (DirectLimit G f) = \u22c3 (i : \u03b9), age L (G i)\n[PROOFSTEP]\nclassical\next M\nsimp only [mem_iUnion]\nconstructor\n\u00b7 rintro \u27e8Mfg, \u27e8e\u27e9\u27e9\n obtain \u27e8s, hs\u27e9 := Mfg.range e.toHom\n let out := @Quotient.out _ (DirectLimit.setoid G f)\n obtain \u27e8i, hi\u27e9 := Finset.exists_le (s.image (Sigma.fst \u2218 out))\n have e' := (DirectLimit.of L \u03b9 G f i).equivRange.symm.toEmbedding\n refine' \u27e8i, Mfg, \u27e8e'.comp ((Substructure.inclusion _).comp e.equivRange.toEmbedding)\u27e9\u27e9\n rw [\u2190 hs, closure_le]\n intro x hx\n refine' \u27e8f (out x).1 i (hi (out x).1 (Finset.mem_image_of_mem _ hx)) (out x).2, _\u27e9\n rw [Embedding.coe_toHom, DirectLimit.of_apply, @Quotient.mk_eq_iff_out _ (_),\n DirectLimit.equiv_iff G f _ (hi (out x).1 (Finset.mem_image_of_mem _ hx)), DirectedSystem.map_self]\n rfl\n\u00b7 rintro \u27e8i, Mfg, \u27e8e\u27e9\u27e9\n exact \u27e8Mfg, \u27e8Embedding.comp (DirectLimit.of L \u03b9 G f i) e\u27e9\u27e9\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u2076 : Structure L M\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\n\u22a2 age L (DirectLimit G f) = \u22c3 (i : \u03b9), age L (G i)\n[PROOFSTEP]\next M\n[GOAL]\ncase h\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\n\u22a2 M \u2208 age L (DirectLimit G f) \u2194 M \u2208 \u22c3 (i : \u03b9), age L (G i)\n[PROOFSTEP]\nsimp only [mem_iUnion]\n[GOAL]\ncase h\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\n\u22a2 M \u2208 age L (DirectLimit G f) \u2194 \u2203 i, M \u2208 age L (G i)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\n\u22a2 M \u2208 age L (DirectLimit G f) \u2192 \u2203 i, M \u2208 age L (G i)\n[PROOFSTEP]\nrintro \u27e8Mfg, \u27e8e\u27e9\u27e9\n[GOAL]\ncase h.mp.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\n\u22a2 \u2203 i, M \u2208 age L (G i)\n[PROOFSTEP]\nobtain \u27e8s, hs\u27e9 := Mfg.range e.toHom\n[GOAL]\ncase h.mp.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\n\u22a2 \u2203 i, M \u2208 age L (G i)\n[PROOFSTEP]\nlet out := @Quotient.out _ (DirectLimit.setoid G f)\n[GOAL]\ncase h.mp.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\n\u22a2 \u2203 i, M \u2208 age L (G i)\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := Finset.exists_le (s.image (Sigma.fst \u2218 out))\n[GOAL]\ncase h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\n\u22a2 \u2203 i, M \u2208 age L (G i)\n[PROOFSTEP]\nhave e' := (DirectLimit.of L \u03b9 G f i).equivRange.symm.toEmbedding\n[GOAL]\ncase h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\n\u22a2 \u2203 i, M \u2208 age L (G i)\n[PROOFSTEP]\nrefine' \u27e8i, Mfg, \u27e8e'.comp ((Substructure.inclusion _).comp e.equivRange.toEmbedding)\u27e9\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\n\u22a2 Hom.range (Embedding.toHom e) \u2264 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i))\n[PROOFSTEP]\nrw [\u2190 hs, closure_le]\n[GOAL]\ncase h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\n\u22a2 \u2191s \u2286 \u2191(Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)))\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\nx : DirectLimit G f\nhx : x \u2208 \u2191s\n\u22a2 x \u2208 \u2191(Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)))\n[PROOFSTEP]\nrefine' \u27e8f (out x).1 i (hi (out x).1 (Finset.mem_image_of_mem _ hx)) (out x).2, _\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\nx : DirectLimit G f\nhx : x \u2208 \u2191s\n\u22a2 \u2191(Embedding.toHom (DirectLimit.of L \u03b9 G f i)) (\u2191(f (out x).fst i (_ : (out x).fst \u2264 i)) (out x).snd) = x\n[PROOFSTEP]\nrw [Embedding.coe_toHom, DirectLimit.of_apply, @Quotient.mk_eq_iff_out _ (_),\n DirectLimit.equiv_iff G f _ (hi (out x).1 (Finset.mem_image_of_mem _ hx)), DirectedSystem.map_self]\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\nx : DirectLimit G f\nhx : x \u2208 \u2191s\n\u22a2 (Structure.Sigma.mk f i (\u2191(f (out x).fst i (_ : (out x).fst \u2264 i)) (out x).snd)).fst \u2264 i\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\nx : DirectLimit G f\nhx : x \u2208 \u2191s\n\u22a2 (Structure.Sigma.mk f i (\u2191(f (out x).fst i (_ : (out x).fst \u2264 i)) (out x).snd)).fst \u2264 i\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.mpr\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\n\u22a2 (\u2203 i, M \u2208 age L (G i)) \u2192 M \u2208 age L (DirectLimit G f)\n[PROOFSTEP]\nrintro \u27e8i, Mfg, \u27e8e\u27e9\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\ni : \u03b9\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] G i\n\u22a2 M \u2208 age L (DirectLimit G f)\n[PROOFSTEP]\nexact \u27e8Mfg, \u27e8Embedding.comp (DirectLimit.of L \u03b9 G f i) e\u27e9\u27e9\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\n\u22a2 \u2203 M, Structure.CG L \u2191M \u2227 age L \u2191M = K\n[PROOFSTEP]\nobtain \u27e8F, hF\u27e9 := hc.exists_eq_range (hn.image _)\n[GOAL]\ncase intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : Quotient.mk' '' K = range F\n\u22a2 \u2203 M, Structure.CG L \u2191M \u2227 age L \u2191M = K\n[PROOFSTEP]\nsimp only [Set.ext_iff, forall_quotient_iff, mem_image, mem_range, Quotient.eq'] at hF \n[GOAL]\ncase intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF :\n \u2200 (a : Bundled (Structure L)),\n (\u2203 x, x \u2208 K \u2227 Quotient.mk' x = Quotient.mk equivSetoid a) \u2194 \u2203 y, F y = Quotient.mk equivSetoid a\n\u22a2 \u2203 M, Structure.CG L \u2191M \u2227 age L \u2191M = K\n[PROOFSTEP]\nsimp_rw [Quotient.eq_mk_iff_out] at hF \n[GOAL]\ncase intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\n\u22a2 \u2203 M, Structure.CG L \u2191M \u2227 age L \u2191M = K\n[PROOFSTEP]\nhave hF' : \u2200 n : \u2115, (F n).out \u2208 K := by\n intro n\n obtain \u27e8P, hP1, hP2\u27e9 :=\n (hF (F n).out).2\n \u27e8n, Setoid.refl _\u27e9\n -- Porting note: fix hP2 because `Quotient.out (Quotient.mk' x) \u2248 a` was not simplified\n -- to `x \u2248 a` in hF\n replace hP2 := Setoid.trans (Setoid.symm (Quotient.mk_out P)) hP2\n exact (h _ _ hP2).1 hP1\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\n\u22a2 \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\n[PROOFSTEP]\nintro n\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nn : \u2115\n\u22a2 Quotient.out (F n) \u2208 K\n[PROOFSTEP]\nobtain \u27e8P, hP1, hP2\u27e9 :=\n (hF (F n).out).2\n \u27e8n, Setoid.refl _\u27e9\n -- Porting note: fix hP2 because `Quotient.out (Quotient.mk' x) \u2248 a` was not simplified\n -- to `x \u2248 a` in hF\n[GOAL]\ncase intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nn : \u2115\nP : Bundled (Structure L)\nhP1 : P \u2208 K\nhP2 : Quotient.out (Quotient.mk' P) \u2248 Quotient.out (F n)\n\u22a2 Quotient.out (F n) \u2208 K\n[PROOFSTEP]\nreplace hP2 := Setoid.trans (Setoid.symm (Quotient.mk_out P)) hP2\n[GOAL]\ncase intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nn : \u2115\nP : Bundled (Structure L)\nhP1 : P \u2208 K\nhP2 : P \u2248 Quotient.out (F n)\n\u22a2 Quotient.out (F n) \u2208 K\n[PROOFSTEP]\nexact (h _ _ hP2).1 hP1\n[GOAL]\ncase intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\n\u22a2 \u2203 M, Structure.CG L \u2191M \u2227 age L \u2191M = K\n[PROOFSTEP]\nchoose P hPK hP hFP using fun (N : K) (n : \u2115) => jep N N.2 (F (n + 1)).out (hF' _)\n[GOAL]\ncase intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\n\u22a2 \u2203 M, Structure.CG L \u2191M \u2227 age L \u2191M = K\n[PROOFSTEP]\nlet G : \u2115 \u2192 K :=\n @Nat.rec (fun _ => K) \u27e8(F 0).out, hF' 0\u27e9 fun n N =>\n \u27e8P N n, hPK N n\u27e9\n -- Poting note: was\n -- let f : \u2200 i j, i \u2264 j \u2192 G i \u21aa[L] G j := DirectedSystem.natLeRec fun n => (hP _ n).some\n[GOAL]\ncase intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\n\u22a2 \u2203 M, Structure.CG L \u2191M \u2227 age L \u2191M = K\n[PROOFSTEP]\nlet f : \u2200 (i j : \u2115), i \u2264 j \u2192 (G i).val \u21aa[L] (G j).val :=\n by\n refine DirectedSystem.natLERec (G' := fun i => (G i).val) (L := L) ?_\n dsimp only\n exact (fun n => (hP _ n).some)\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\n\u22a2 (i j : \u2115) \u2192 i \u2264 j \u2192 \u2191\u2191(G i) \u21aa[L] \u2191\u2191(G j)\n[PROOFSTEP]\nrefine DirectedSystem.natLERec (G' := fun i => (G i).val) (L := L) ?_\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\n\u22a2 (n : \u2115) \u2192 (fun i => \u2191\u2191(G i)) n \u21aa[L] (fun i => \u2191\u2191(G i)) (n + 1)\n[PROOFSTEP]\ndsimp only\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\n\u22a2 (n : \u2115) \u2192\n \u2191\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n) \u21aa[L]\n \u2191\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) (n + 1))\n[PROOFSTEP]\nexact (fun n => (hP _ n).some)\n[GOAL]\ncase intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\nf : (i j : \u2115) \u2192 i \u2264 j \u2192 \u2191\u2191(G i) \u21aa[L] \u2191\u2191(G j) :=\n DirectedSystem.natLERec\n (id fun n =>\n Nonempty.some\n (_ :\n (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N))\n (\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n (P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)))\n\u22a2 \u2203 M, Structure.CG L \u2191M \u2227 age L \u2191M = K\n[PROOFSTEP]\nhave : DirectedSystem (fun n \u21a6 (G n).val) fun i j h \u21a6 \u2191(f i j h) := by dsimp; infer_instance\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\nf : (i j : \u2115) \u2192 i \u2264 j \u2192 \u2191\u2191(G i) \u21aa[L] \u2191\u2191(G j) :=\n DirectedSystem.natLERec\n (id fun n =>\n Nonempty.some\n (_ :\n (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N))\n (\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n (P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)))\n\u22a2 DirectedSystem (fun n => \u2191\u2191(G n)) fun i j h => \u2191(f i j h)\n[PROOFSTEP]\ndsimp\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\nf : (i j : \u2115) \u2192 i \u2264 j \u2192 \u2191\u2191(G i) \u21aa[L] \u2191\u2191(G j) :=\n DirectedSystem.natLERec\n (id fun n =>\n Nonempty.some\n (_ :\n (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N))\n (\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n (P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)))\n\u22a2 DirectedSystem\n (fun n =>\n \u2191\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n fun i j h =>\n \u2191(DirectedSystem.natLERec\n (fun n =>\n Nonempty.some\n (_ :\n (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N))\n (\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n (P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)))\n i j h)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\nf : (i j : \u2115) \u2192 i \u2264 j \u2192 \u2191\u2191(G i) \u21aa[L] \u2191\u2191(G j) :=\n DirectedSystem.natLERec\n (id fun n =>\n Nonempty.some\n (_ :\n (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N))\n (\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n (P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)))\nthis : DirectedSystem (fun n => \u2191\u2191(G n)) fun i j h => \u2191(f i j h)\n\u22a2 \u2203 M, Structure.CG L \u2191M \u2227 age L \u2191M = K\n[PROOFSTEP]\nrefine \u27e8Bundled.of (@DirectLimit L _ _ (fun n \u21a6 (G n).val) _ f _ _), ?_, ?_\u27e9\n[GOAL]\ncase intro.refine_1\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\nf : (i j : \u2115) \u2192 i \u2264 j \u2192 \u2191\u2191(G i) \u21aa[L] \u2191\u2191(G j) :=\n DirectedSystem.natLERec\n (id fun n =>\n Nonempty.some\n (_ :\n (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N))\n (\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n (P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)))\nthis : DirectedSystem (fun n => \u2191\u2191(G n)) fun i j h => \u2191(f i j h)\n\u22a2 Structure.CG L \u2191(Bundled.of (DirectLimit (fun n => \u2191\u2191(G n)) f))\n[PROOFSTEP]\nexact DirectLimit.cg _ (fun n => (fg _ (G n).2).cg)\n[GOAL]\ncase intro.refine_2\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\nf : (i j : \u2115) \u2192 i \u2264 j \u2192 \u2191\u2191(G i) \u21aa[L] \u2191\u2191(G j) :=\n DirectedSystem.natLERec\n (id fun n =>\n Nonempty.some\n (_ :\n (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N))\n (\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n (P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)))\nthis : DirectedSystem (fun n => \u2191\u2191(G n)) fun i j h => \u2191(f i j h)\n\u22a2 age L \u2191(Bundled.of (DirectLimit (fun n => \u2191\u2191(G n)) f)) = K\n[PROOFSTEP]\nrefine\n (age_directLimit (fun n \u21a6 (G n).val) f).trans\n (subset_antisymm (iUnion_subset fun n N hN => hp (G n).val (G n).2 hN) fun N KN => ?_)\n[GOAL]\ncase intro.refine_2\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\nf : (i j : \u2115) \u2192 i \u2264 j \u2192 \u2191\u2191(G i) \u21aa[L] \u2191\u2191(G j) :=\n DirectedSystem.natLERec\n (id fun n =>\n Nonempty.some\n (_ :\n (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N))\n (\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n (P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)))\nthis : DirectedSystem (fun n => \u2191\u2191(G n)) fun i j h => \u2191(f i j h)\nN : Bundled (Structure L)\nKN : N \u2208 K\n\u22a2 N \u2208 \u22c3 (i : \u2115), age L \u2191\u2191(G i)\n[PROOFSTEP]\nhave : Quotient.out (Quotient.mk' N) \u2248 N := Quotient.eq_mk_iff_out.mp rfl\n[GOAL]\ncase intro.refine_2\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\nf : (i j : \u2115) \u2192 i \u2264 j \u2192 \u2191\u2191(G i) \u21aa[L] \u2191\u2191(G j) :=\n DirectedSystem.natLERec\n (id fun n =>\n Nonempty.some\n (_ :\n (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N))\n (\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n (P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)))\nthis\u271d : DirectedSystem (fun n => \u2191\u2191(G n)) fun i j h => \u2191(f i j h)\nN : Bundled (Structure L)\nKN : N \u2208 K\nthis : Quotient.out (Quotient.mk' N) \u2248 N\n\u22a2 N \u2208 \u22c3 (i : \u2115), age L \u2191\u2191(G i)\n[PROOFSTEP]\nobtain \u27e8n, \u27e8e\u27e9\u27e9 := (hF N).1 \u27e8N, KN, this\u27e9\n[GOAL]\ncase intro.refine_2.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\nf : (i j : \u2115) \u2192 i \u2264 j \u2192 \u2191\u2191(G i) \u21aa[L] \u2191\u2191(G j) :=\n DirectedSystem.natLERec\n (id fun n =>\n Nonempty.some\n (_ :\n (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N))\n (\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n (P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)))\nthis\u271d : DirectedSystem (fun n => \u2191\u2191(G n)) fun i j h => \u2191(f i j h)\nN : Bundled (Structure L)\nKN : N \u2208 K\nthis : Quotient.out (Quotient.mk' N) \u2248 N\nn : \u2115\ne : \u2191(Quotient.out (F n)) \u2243[L] \u2191N\n\u22a2 N \u2208 \u22c3 (i : \u2115), age L \u2191\u2191(G i)\n[PROOFSTEP]\nrefine mem_iUnion_of_mem n \u27e8fg _ KN, \u27e8Embedding.comp ?_ e.symm.toEmbedding\u27e9\u27e9\n[GOAL]\ncase intro.refine_2.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\nf : (i j : \u2115) \u2192 i \u2264 j \u2192 \u2191\u2191(G i) \u21aa[L] \u2191\u2191(G j) :=\n DirectedSystem.natLERec\n (id fun n =>\n Nonempty.some\n (_ :\n (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N))\n (\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n (P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)))\nthis\u271d : DirectedSystem (fun n => \u2191\u2191(G n)) fun i j h => \u2191(f i j h)\nN : Bundled (Structure L)\nKN : N \u2208 K\nthis : Quotient.out (Quotient.mk' N) \u2248 N\nn : \u2115\ne : \u2191(Quotient.out (F n)) \u2243[L] \u2191N\n\u22a2 \u2191(Quotient.out (F n)) \u21aa[L] \u2191\u2191(G n)\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase intro.refine_2.intro.intro.zero\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\nf : (i j : \u2115) \u2192 i \u2264 j \u2192 \u2191\u2191(G i) \u21aa[L] \u2191\u2191(G j) :=\n DirectedSystem.natLERec\n (id fun n =>\n Nonempty.some\n (_ :\n (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N))\n (\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n (P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)))\nthis\u271d : DirectedSystem (fun n => \u2191\u2191(G n)) fun i j h => \u2191(f i j h)\nN : Bundled (Structure L)\nKN : N \u2208 K\nthis : Quotient.out (Quotient.mk' N) \u2248 N\ne : \u2191(Quotient.out (F Nat.zero)) \u2243[L] \u2191N\n\u22a2 \u2191(Quotient.out (F Nat.zero)) \u21aa[L] \u2191\u2191(G Nat.zero)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.refine_2.intro.intro.zero\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\nf : (i j : \u2115) \u2192 i \u2264 j \u2192 \u2191\u2191(G i) \u21aa[L] \u2191\u2191(G j) :=\n DirectedSystem.natLERec\n (id fun n =>\n Nonempty.some\n (_ :\n (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N))\n (\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n (P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)))\nthis\u271d : DirectedSystem (fun n => \u2191\u2191(G n)) fun i j h => \u2191(f i j h)\nN : Bundled (Structure L)\nKN : N \u2208 K\nthis : Quotient.out (Quotient.mk' N) \u2248 N\ne : \u2191(Quotient.out (F Nat.zero)) \u2243[L] \u2191N\n\u22a2 \u2191(Quotient.out (F 0)) \u21aa[L] \u2191(Quotient.out (F 0))\n[PROOFSTEP]\nexact Embedding.refl _ _\n[GOAL]\ncase intro.refine_2.intro.intro.succ\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\nf : (i j : \u2115) \u2192 i \u2264 j \u2192 \u2191\u2191(G i) \u21aa[L] \u2191\u2191(G j) :=\n DirectedSystem.natLERec\n (id fun n =>\n Nonempty.some\n (_ :\n (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N))\n (\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n (P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)))\nthis\u271d : DirectedSystem (fun n => \u2191\u2191(G n)) fun i j h => \u2191(f i j h)\nN : Bundled (Structure L)\nKN : N \u2208 K\nthis : Quotient.out (Quotient.mk' N) \u2248 N\nn : \u2115\ne : \u2191(Quotient.out (F (Nat.succ n))) \u2243[L] \u2191N\n\u22a2 \u2191(Quotient.out (F (Nat.succ n))) \u21aa[L] \u2191\u2191(G (Nat.succ n))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.refine_2.intro.intro.succ\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nhn : Set.Nonempty K\nh : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\nhc : Set.Countable (Quotient.mk' '' K)\nfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\nF : \u2115 \u2192 Quotient equivSetoid\nhF : \u2200 (a : Bundled (Structure L)), (\u2203 x, x \u2208 K \u2227 Quotient.out (Quotient.mk' x) \u2248 a) \u2194 \u2203 y, Quotient.out (F y) \u2248 a\nhF' : \u2200 (n : \u2115), Quotient.out (F n) \u2208 K\nP : \u2191K \u2192 \u2115 \u2192 Bundled (Structure L)\nhPK : \u2200 (N : \u2191K) (n : \u2115), P N n \u2208 K\nhP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (\u2191N) (P N n)\nhFP : \u2200 (N : \u2191K) (n : \u2115), (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N)) (Quotient.out (F (n + 1))) (P N n)\nG : \u2115 \u2192 \u2191K :=\n Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) } fun n N =>\n { val := P N n, property := (_ : P N n \u2208 K) }\nf : (i j : \u2115) \u2192 i \u2264 j \u2192 \u2191\u2191(G i) \u21aa[L] \u2191\u2191(G j) :=\n DirectedSystem.natLERec\n (id fun n =>\n Nonempty.some\n (_ :\n (fun M N => Nonempty (\u2191M \u21aa[L] \u2191N))\n (\u2191(Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n))\n (P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)))\nthis\u271d : DirectedSystem (fun n => \u2191\u2191(G n)) fun i j h => \u2191(f i j h)\nN : Bundled (Structure L)\nKN : N \u2208 K\nthis : Quotient.out (Quotient.mk' N) \u2248 N\nn : \u2115\ne : \u2191(Quotient.out (F (Nat.succ n))) \u2243[L] \u2191N\n\u22a2 \u2191(Quotient.out (F (Nat.succ n))) \u21aa[L]\n \u2191(P\n (Nat.rec { val := Quotient.out (F 0), property := (_ : Quotient.out (F 0) \u2208 K) }\n (fun n N => { val := P N n, property := (_ : P N n \u2208 K) }) n)\n n)\n[PROOFSTEP]\nexact (hFP _ n).some\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b2 : Structure L M\nN : Type w\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Countable ((l : \u2115) \u00d7 Functions L l)\n\u22a2 (\u2203 M, Countable \u2191M \u2227 age L \u2191M = K) \u2194\n Set.Nonempty K \u2227\n (\u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)) \u2227\n Set.Countable (Quotient.mk' '' K) \u2227\n (\u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M) \u2227 Hereditary K \u2227 JointEmbedding K\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b2 : Structure L M\nN : Type w\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Countable ((l : \u2115) \u00d7 Functions L l)\n\u22a2 (\u2203 M, Countable \u2191M \u2227 age L \u2191M = K) \u2192\n Set.Nonempty K \u2227\n (\u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)) \u2227\n Set.Countable (Quotient.mk' '' K) \u2227\n (\u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M) \u2227 Hereditary K \u2227 JointEmbedding K\n[PROOFSTEP]\nrintro \u27e8M, h1, h2, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.refl\nL : Language\nM\u271d : Type w\ninst\u271d\u00b2 : Structure L M\u271d\nN : Type w\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Countable ((l : \u2115) \u00d7 Functions L l)\nM : Bundled (Structure L)\nh1 : Countable \u2191M\n\u22a2 Set.Nonempty (age L \u2191M) \u2227\n (\u2200 (M_1 N : Bundled (Structure L)), Nonempty (\u2191M_1 \u2243[L] \u2191N) \u2192 (M_1 \u2208 age L \u2191M \u2194 N \u2208 age L \u2191M)) \u2227\n Set.Countable (Quotient.mk' '' age L \u2191M) \u2227\n (\u2200 (M_1 : Bundled (Structure L)), M_1 \u2208 age L \u2191M \u2192 Structure.FG L \u2191M_1) \u2227\n Hereditary (age L \u2191M) \u2227 JointEmbedding (age L \u2191M)\n[PROOFSTEP]\nrefine'\n \u27e8age.nonempty M, age.is_equiv_invariant L M, age.countable_quotient M, fun N hN => hN.1, age.hereditary M,\n age.jointEmbedding M\u27e9\n[GOAL]\ncase mpr\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b2 : Structure L M\nN : Type w\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Countable ((l : \u2115) \u00d7 Functions L l)\n\u22a2 Set.Nonempty K \u2227\n (\u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)) \u2227\n Set.Countable (Quotient.mk' '' K) \u2227\n (\u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M) \u2227 Hereditary K \u2227 JointEmbedding K \u2192\n \u2203 M, Countable \u2191M \u2227 age L \u2191M = K\n[PROOFSTEP]\nrintro \u27e8Kn, eqinv, cq, hfg, hp, jep\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b2 : Structure L M\nN : Type w\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Countable ((l : \u2115) \u00d7 Functions L l)\nKn : Set.Nonempty K\neqinv : \u2200 (M N : Bundled (Structure L)), Nonempty (\u2191M \u2243[L] \u2191N) \u2192 (M \u2208 K \u2194 N \u2208 K)\ncq : Set.Countable (Quotient.mk' '' K)\nhfg : \u2200 (M : Bundled (Structure L)), M \u2208 K \u2192 Structure.FG L \u2191M\nhp : Hereditary K\njep : JointEmbedding K\n\u22a2 \u2203 M, Countable \u2191M \u2227 age L \u2191M = K\n[PROOFSTEP]\nobtain \u27e8M, hM, rfl\u27e9 := exists_cg_is_age_of Kn eqinv cq hfg hp jep\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro.intro.intro\nL : Language\nM\u271d : Type w\ninst\u271d\u00b2 : Structure L M\u271d\nN : Type w\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Countable ((l : \u2115) \u00d7 Functions L l)\nM : Bundled (Structure L)\nhM : Structure.CG L \u2191M\nKn : Set.Nonempty (age L \u2191M)\neqinv : \u2200 (M_1 N : Bundled (Structure L)), Nonempty (\u2191M_1 \u2243[L] \u2191N) \u2192 (M_1 \u2208 age L \u2191M \u2194 N \u2208 age L \u2191M)\ncq : Set.Countable (Quotient.mk' '' age L \u2191M)\nhfg : \u2200 (M_1 : Bundled (Structure L)), M_1 \u2208 age L \u2191M \u2192 Structure.FG L \u2191M_1\nhp : Hereditary (age L \u2191M)\njep : JointEmbedding (age L \u2191M)\n\u22a2 \u2203 M_1, Countable \u2191M_1 \u2227 age L \u2191M_1 = age L \u2191M\n[PROOFSTEP]\nexact \u27e8M, Structure.cg_iff_countable.1 hM, rfl\u27e9\n[GOAL]\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN : Type w\ninst\u271d : Structure L N\nh : IsUltrahomogeneous L M\n\u22a2 Amalgamation (age L M)\n[PROOFSTEP]\nrintro N P Q NP NQ \u27e8Nfg, \u27e8-\u27e9\u27e9 \u27e8Pfg, \u27e8PM\u27e9\u27e9 \u27e8Qfg, \u27e8QM\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : IsUltrahomogeneous L M\nN P Q : Bundled (Structure L)\nNP : \u2191N \u21aa[L] \u2191P\nNQ : \u2191N \u21aa[L] \u2191Q\nNfg : Structure.FG L \u2191N\nPfg : Structure.FG L \u2191P\nPM : \u2191P \u21aa[L] M\nQfg : Structure.FG L \u2191Q\nQM : \u2191Q \u21aa[L] M\n\u22a2 \u2203 Q_1 NQ_1 PQ, Q_1 \u2208 age L M \u2227 Embedding.comp NQ_1 NP = Embedding.comp PQ NQ\n[PROOFSTEP]\nobtain \u27e8g, hg\u27e9 := h (PM.comp NP).toHom.range (Nfg.range _) ((QM.comp NQ).comp (PM.comp NP).equivRange.symm.toEmbedding)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : IsUltrahomogeneous L M\nN P Q : Bundled (Structure L)\nNP : \u2191N \u21aa[L] \u2191P\nNQ : \u2191N \u21aa[L] \u2191Q\nNfg : Structure.FG L \u2191N\nPfg : Structure.FG L \u2191P\nPM : \u2191P \u21aa[L] M\nQfg : Structure.FG L \u2191Q\nQM : \u2191Q \u21aa[L] M\ng : M \u2243[L] M\nhg :\n Embedding.comp (Embedding.comp QM NQ) (Equiv.toEmbedding (Equiv.symm (Embedding.equivRange (Embedding.comp PM NP)))) =\n Embedding.comp (Equiv.toEmbedding g) (subtype (Hom.range (Embedding.toHom (Embedding.comp PM NP))))\n\u22a2 \u2203 Q_1 NQ_1 PQ, Q_1 \u2208 age L M \u2227 Embedding.comp NQ_1 NP = Embedding.comp PQ NQ\n[PROOFSTEP]\nlet s := (g.toHom.comp PM.toHom).range \u2294 QM.toHom.range\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : IsUltrahomogeneous L M\nN P Q : Bundled (Structure L)\nNP : \u2191N \u21aa[L] \u2191P\nNQ : \u2191N \u21aa[L] \u2191Q\nNfg : Structure.FG L \u2191N\nPfg : Structure.FG L \u2191P\nPM : \u2191P \u21aa[L] M\nQfg : Structure.FG L \u2191Q\nQM : \u2191Q \u21aa[L] M\ng : M \u2243[L] M\nhg :\n Embedding.comp (Embedding.comp QM NQ) (Equiv.toEmbedding (Equiv.symm (Embedding.equivRange (Embedding.comp PM NP)))) =\n Embedding.comp (Equiv.toEmbedding g) (subtype (Hom.range (Embedding.toHom (Embedding.comp PM NP))))\ns : Substructure L M := Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)\n\u22a2 \u2203 Q_1 NQ_1 PQ, Q_1 \u2208 age L M \u2227 Embedding.comp NQ_1 NP = Embedding.comp PQ NQ\n[PROOFSTEP]\nrefine'\n \u27e8Bundled.of s, Embedding.comp (Substructure.inclusion le_sup_left) (g.toEmbedding.comp PM).equivRange.toEmbedding,\n Embedding.comp (Substructure.inclusion le_sup_right) QM.equivRange.toEmbedding,\n \u27e8(fg_iff_structure_fg _).1 (FG.sup (Pfg.range _) (Qfg.range _)), \u27e8Substructure.subtype _\u27e9\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : IsUltrahomogeneous L M\nN P Q : Bundled (Structure L)\nNP : \u2191N \u21aa[L] \u2191P\nNQ : \u2191N \u21aa[L] \u2191Q\nNfg : Structure.FG L \u2191N\nPfg : Structure.FG L \u2191P\nPM : \u2191P \u21aa[L] M\nQfg : Structure.FG L \u2191Q\nQM : \u2191Q \u21aa[L] M\ng : M \u2243[L] M\nhg :\n Embedding.comp (Embedding.comp QM NQ) (Equiv.toEmbedding (Equiv.symm (Embedding.equivRange (Embedding.comp PM NP)))) =\n Embedding.comp (Equiv.toEmbedding g) (subtype (Hom.range (Embedding.toHom (Embedding.comp PM NP))))\ns : Substructure L M := Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)\n\u22a2 Embedding.comp\n (Embedding.comp\n (Substructure.inclusion\n (_ :\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2264\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)))\n (Equiv.toEmbedding (Embedding.equivRange (Embedding.comp (Equiv.toEmbedding g) PM))))\n NP =\n Embedding.comp\n (Embedding.comp\n (Substructure.inclusion\n (_ :\n Hom.range (Embedding.toHom QM) \u2264\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)))\n (Equiv.toEmbedding (Embedding.equivRange QM)))\n NQ\n[PROOFSTEP]\next n\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.h\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : IsUltrahomogeneous L M\nN P Q : Bundled (Structure L)\nNP : \u2191N \u21aa[L] \u2191P\nNQ : \u2191N \u21aa[L] \u2191Q\nNfg : Structure.FG L \u2191N\nPfg : Structure.FG L \u2191P\nPM : \u2191P \u21aa[L] M\nQfg : Structure.FG L \u2191Q\nQM : \u2191Q \u21aa[L] M\ng : M \u2243[L] M\nhg :\n Embedding.comp (Embedding.comp QM NQ) (Equiv.toEmbedding (Equiv.symm (Embedding.equivRange (Embedding.comp PM NP)))) =\n Embedding.comp (Equiv.toEmbedding g) (subtype (Hom.range (Embedding.toHom (Embedding.comp PM NP))))\ns : Substructure L M := Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)\nn : \u2191N\n\u22a2 \u2191(Embedding.comp\n (Embedding.comp\n (Substructure.inclusion\n (_ :\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2264\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)))\n (Equiv.toEmbedding (Embedding.equivRange (Embedding.comp (Equiv.toEmbedding g) PM))))\n NP)\n n =\n \u2191(Embedding.comp\n (Embedding.comp\n (Substructure.inclusion\n (_ :\n Hom.range (Embedding.toHom QM) \u2264\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)))\n (Equiv.toEmbedding (Embedding.equivRange QM)))\n NQ)\n n\n[PROOFSTEP]\napply Subtype.ext\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.h.a\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : IsUltrahomogeneous L M\nN P Q : Bundled (Structure L)\nNP : \u2191N \u21aa[L] \u2191P\nNQ : \u2191N \u21aa[L] \u2191Q\nNfg : Structure.FG L \u2191N\nPfg : Structure.FG L \u2191P\nPM : \u2191P \u21aa[L] M\nQfg : Structure.FG L \u2191Q\nQM : \u2191Q \u21aa[L] M\ng : M \u2243[L] M\nhg :\n Embedding.comp (Embedding.comp QM NQ) (Equiv.toEmbedding (Equiv.symm (Embedding.equivRange (Embedding.comp PM NP)))) =\n Embedding.comp (Equiv.toEmbedding g) (subtype (Hom.range (Embedding.toHom (Embedding.comp PM NP))))\ns : Substructure L M := Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)\nn : \u2191N\n\u22a2 \u2191(\u2191(Embedding.comp\n (Embedding.comp\n (Substructure.inclusion\n (_ :\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2264\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)))\n (Equiv.toEmbedding (Embedding.equivRange (Embedding.comp (Equiv.toEmbedding g) PM))))\n NP)\n n) =\n \u2191(\u2191(Embedding.comp\n (Embedding.comp\n (Substructure.inclusion\n (_ :\n Hom.range (Embedding.toHom QM) \u2264\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)))\n (Equiv.toEmbedding (Embedding.equivRange QM)))\n NQ)\n n)\n[PROOFSTEP]\nhave hgn := (Embedding.ext_iff.1 hg) ((PM.comp NP).equivRange n)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.h.a\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : IsUltrahomogeneous L M\nN P Q : Bundled (Structure L)\nNP : \u2191N \u21aa[L] \u2191P\nNQ : \u2191N \u21aa[L] \u2191Q\nNfg : Structure.FG L \u2191N\nPfg : Structure.FG L \u2191P\nPM : \u2191P \u21aa[L] M\nQfg : Structure.FG L \u2191Q\nQM : \u2191Q \u21aa[L] M\ng : M \u2243[L] M\nhg :\n Embedding.comp (Embedding.comp QM NQ) (Equiv.toEmbedding (Equiv.symm (Embedding.equivRange (Embedding.comp PM NP)))) =\n Embedding.comp (Equiv.toEmbedding g) (subtype (Hom.range (Embedding.toHom (Embedding.comp PM NP))))\ns : Substructure L M := Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)\nn : \u2191N\nhgn :\n \u2191(Embedding.comp (Embedding.comp QM NQ)\n (Equiv.toEmbedding (Equiv.symm (Embedding.equivRange (Embedding.comp PM NP)))))\n (\u2191(Embedding.equivRange (Embedding.comp PM NP)) n) =\n \u2191(Embedding.comp (Equiv.toEmbedding g) (subtype (Hom.range (Embedding.toHom (Embedding.comp PM NP)))))\n (\u2191(Embedding.equivRange (Embedding.comp PM NP)) n)\n\u22a2 \u2191(\u2191(Embedding.comp\n (Embedding.comp\n (Substructure.inclusion\n (_ :\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2264\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)))\n (Equiv.toEmbedding (Embedding.equivRange (Embedding.comp (Equiv.toEmbedding g) PM))))\n NP)\n n) =\n \u2191(\u2191(Embedding.comp\n (Embedding.comp\n (Substructure.inclusion\n (_ :\n Hom.range (Embedding.toHom QM) \u2264\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)))\n (Equiv.toEmbedding (Embedding.equivRange QM)))\n NQ)\n n)\n[PROOFSTEP]\nsimp only [Embedding.comp_apply, Equiv.coe_toEmbedding, Equiv.symm_apply_apply, Substructure.coeSubtype,\n Embedding.equivRange_apply] at hgn \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.h.a\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : IsUltrahomogeneous L M\nN P Q : Bundled (Structure L)\nNP : \u2191N \u21aa[L] \u2191P\nNQ : \u2191N \u21aa[L] \u2191Q\nNfg : Structure.FG L \u2191N\nPfg : Structure.FG L \u2191P\nPM : \u2191P \u21aa[L] M\nQfg : Structure.FG L \u2191Q\nQM : \u2191Q \u21aa[L] M\ng : M \u2243[L] M\nhg :\n Embedding.comp (Embedding.comp QM NQ) (Equiv.toEmbedding (Equiv.symm (Embedding.equivRange (Embedding.comp PM NP)))) =\n Embedding.comp (Equiv.toEmbedding g) (subtype (Hom.range (Embedding.toHom (Embedding.comp PM NP))))\ns : Substructure L M := Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)\nn : \u2191N\nhgn : \u2191QM (\u2191NQ n) = \u2191g (\u2191PM (\u2191NP n))\n\u22a2 \u2191(\u2191(Embedding.comp\n (Embedding.comp\n (Substructure.inclusion\n (_ :\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2264\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)))\n (Equiv.toEmbedding (Embedding.equivRange (Embedding.comp (Equiv.toEmbedding g) PM))))\n NP)\n n) =\n \u2191(\u2191(Embedding.comp\n (Embedding.comp\n (Substructure.inclusion\n (_ :\n Hom.range (Embedding.toHom QM) \u2264\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)))\n (Equiv.toEmbedding (Embedding.equivRange QM)))\n NQ)\n n)\n[PROOFSTEP]\nsimp only [Embedding.comp_apply, Equiv.coe_toEmbedding]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.h.a\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : IsUltrahomogeneous L M\nN P Q : Bundled (Structure L)\nNP : \u2191N \u21aa[L] \u2191P\nNQ : \u2191N \u21aa[L] \u2191Q\nNfg : Structure.FG L \u2191N\nPfg : Structure.FG L \u2191P\nPM : \u2191P \u21aa[L] M\nQfg : Structure.FG L \u2191Q\nQM : \u2191Q \u21aa[L] M\ng : M \u2243[L] M\nhg :\n Embedding.comp (Embedding.comp QM NQ) (Equiv.toEmbedding (Equiv.symm (Embedding.equivRange (Embedding.comp PM NP)))) =\n Embedding.comp (Equiv.toEmbedding g) (subtype (Hom.range (Embedding.toHom (Embedding.comp PM NP))))\ns : Substructure L M := Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)\nn : \u2191N\nhgn : \u2191QM (\u2191NQ n) = \u2191g (\u2191PM (\u2191NP n))\n\u22a2 \u2191(\u2191(Substructure.inclusion\n (_ :\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2264\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)))\n (\u2191(Embedding.equivRange (Embedding.comp (Equiv.toEmbedding g) PM)) (\u2191NP n))) =\n \u2191(\u2191(Substructure.inclusion\n (_ :\n Hom.range (Embedding.toHom QM) \u2264\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)))\n (\u2191(Embedding.equivRange QM) (\u2191NQ n)))\n[PROOFSTEP]\nerw [Substructure.coe_inclusion, Substructure.coe_inclusion]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.h.a\nL : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u00b9 : Structure L M\nN\u271d : Type w\ninst\u271d : Structure L N\u271d\nh : IsUltrahomogeneous L M\nN P Q : Bundled (Structure L)\nNP : \u2191N \u21aa[L] \u2191P\nNQ : \u2191N \u21aa[L] \u2191Q\nNfg : Structure.FG L \u2191N\nPfg : Structure.FG L \u2191P\nPM : \u2191P \u21aa[L] M\nQfg : Structure.FG L \u2191Q\nQM : \u2191Q \u21aa[L] M\ng : M \u2243[L] M\nhg :\n Embedding.comp (Embedding.comp QM NQ) (Equiv.toEmbedding (Equiv.symm (Embedding.equivRange (Embedding.comp PM NP)))) =\n Embedding.comp (Equiv.toEmbedding g) (subtype (Hom.range (Embedding.toHom (Embedding.comp PM NP))))\ns : Substructure L M := Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM)\nn : \u2191N\nhgn : \u2191QM (\u2191NQ n) = \u2191g (\u2191PM (\u2191NP n))\n\u22a2 \u2191(Set.inclusion\n (_ :\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2264\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM))\n (\u2191(Embedding.equivRange (Embedding.comp (Equiv.toEmbedding g) PM)) (\u2191NP n))) =\n \u2191(Set.inclusion\n (_ :\n Hom.range (Embedding.toHom QM) \u2264\n Hom.range (Hom.comp (Equiv.toHom g) (Embedding.toHom PM)) \u2294 Hom.range (Embedding.toHom QM))\n (\u2191(Embedding.equivRange QM) (\u2191NQ n)))\n[PROOFSTEP]\nsimp only [Embedding.comp_apply, Equiv.coe_toEmbedding, Set.coe_inclusion, Embedding.equivRange_apply, hgn]\n", "meta": {"mathlib_filename": "Mathlib.ModelTheory.Fraisse", "llama_tokens": 35705, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.4352574392721748}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\n\u22a2 \u2191(hasseDeriv k) f = sum f fun i r => \u2191(monomial (i - k)) (\u2191(choose i k) * r)\n[PROOFSTEP]\ndsimp [hasseDeriv]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\n\u22a2 (sum f fun x x_1 => \u2191(monomial (x - k)) (choose x k \u2022 x_1)) = sum f fun i r => \u2191(monomial (i - k)) (\u2191(choose i k) * r)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\n\u22a2 (fun x x_1 => \u2191(monomial (x - k)) (choose x k \u2022 x_1)) = fun i r => \u2191(monomial (i - k)) (\u2191(choose i k) * r)\n[PROOFSTEP]\next\n[GOAL]\ncase e_f.h.h.a\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nx\u271d\u00b9 : \u2115\nx\u271d : R\nn\u271d : \u2115\n\u22a2 coeff (\u2191(monomial (x\u271d\u00b9 - k)) (choose x\u271d\u00b9 k \u2022 x\u271d)) n\u271d = coeff (\u2191(monomial (x\u271d\u00b9 - k)) (\u2191(choose x\u271d\u00b9 k) * x\u271d)) n\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f.h.h.a.e_a.h.e_6.h\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nx\u271d\u00b9 : \u2115\nx\u271d : R\nn\u271d : \u2115\n\u22a2 choose x\u271d\u00b9 k \u2022 x\u271d = \u2191(choose x\u271d\u00b9 k) * x\u271d\n[PROOFSTEP]\napply nsmul_eq_mul\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\n\u22a2 coeff (\u2191(hasseDeriv k) f) n = \u2191(choose (n + k) k) * coeff f (n + k)\n[PROOFSTEP]\nrw [hasseDeriv_apply, coeff_sum, sum_def, Finset.sum_eq_single (n + k), coeff_monomial]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\n\u22a2 (if n + k - k = n then \u2191(choose (n + k) k) * coeff f (n + k) else 0) = \u2191(choose (n + k) k) * coeff f (n + k)\n[PROOFSTEP]\nsimp only [if_true, add_tsub_cancel_right, eq_self_iff_true]\n[GOAL]\ncase h\u2080\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\n\u22a2 \u2200 (b : \u2115), b \u2208 support f \u2192 b \u2260 n + k \u2192 coeff (\u2191(monomial (b - k)) (\u2191(choose b k) * coeff f b)) n = 0\n[PROOFSTEP]\nintro i _hi hink\n[GOAL]\ncase h\u2080\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn i : \u2115\n_hi : i \u2208 support f\nhink : i \u2260 n + k\n\u22a2 coeff (\u2191(monomial (i - k)) (\u2191(choose i k) * coeff f i)) n = 0\n[PROOFSTEP]\nrw [coeff_monomial]\n[GOAL]\ncase h\u2080\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn i : \u2115\n_hi : i \u2208 support f\nhink : i \u2260 n + k\n\u22a2 (if i - k = n then \u2191(choose i k) * coeff f i else 0) = 0\n[PROOFSTEP]\nby_cases hik : i < k\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn i : \u2115\n_hi : i \u2208 support f\nhink : i \u2260 n + k\nhik : i < k\n\u22a2 (if i - k = n then \u2191(choose i k) * coeff f i else 0) = 0\n[PROOFSTEP]\nsimp only [Nat.choose_eq_zero_of_lt hik, ite_self, Nat.cast_zero, zero_mul]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn i : \u2115\n_hi : i \u2208 support f\nhink : i \u2260 n + k\nhik : \u00aci < k\n\u22a2 (if i - k = n then \u2191(choose i k) * coeff f i else 0) = 0\n[PROOFSTEP]\npush_neg at hik \n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn i : \u2115\n_hi : i \u2208 support f\nhink : i \u2260 n + k\nhik : k \u2264 i\n\u22a2 (if i - k = n then \u2191(choose i k) * coeff f i else 0) = 0\n[PROOFSTEP]\nrw [if_neg]\n[GOAL]\ncase neg.hnc\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn i : \u2115\n_hi : i \u2208 support f\nhink : i \u2260 n + k\nhik : k \u2264 i\n\u22a2 \u00aci - k = n\n[PROOFSTEP]\ncontrapose! hink\n[GOAL]\ncase neg.hnc\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn i : \u2115\n_hi : i \u2208 support f\nhik : k \u2264 i\nhink : i - k = n\n\u22a2 i = n + k\n[PROOFSTEP]\nexact (tsub_eq_iff_eq_add_of_le hik).mp hink\n[GOAL]\ncase h\u2081\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\n\u22a2 \u00acn + k \u2208 support f \u2192 coeff (\u2191(monomial (n + k - k)) (\u2191(choose (n + k) k) * coeff f (n + k))) n = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\u2081\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\nh : \u00acn + k \u2208 support f\n\u22a2 coeff (\u2191(monomial (n + k - k)) (\u2191(choose (n + k) k) * coeff f (n + k))) n = 0\n[PROOFSTEP]\nsimp only [not_mem_support_iff.mp h, monomial_zero_right, mul_zero, coeff_zero]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\n\u22a2 \u2191(hasseDeriv 0) f = f\n[PROOFSTEP]\nsimp only [hasseDeriv_apply, tsub_zero, Nat.choose_zero_right, Nat.cast_one, one_mul, sum_monomial_eq]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn : \u2115\nh : natDegree p < n\n\u22a2 \u2191(hasseDeriv n) p = 0\n[PROOFSTEP]\nrw [hasseDeriv_apply, sum_def]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn : \u2115\nh : natDegree p < n\n\u22a2 \u2211 n_1 in support p, \u2191(monomial (n_1 - n)) (\u2191(choose n_1 n) * coeff p n_1) = 0\n[PROOFSTEP]\nrefine' Finset.sum_eq_zero fun x hx => _\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn : \u2115\nh : natDegree p < n\nx : \u2115\nhx : x \u2208 support p\n\u22a2 \u2191(monomial (x - n)) (\u2191(choose x n) * coeff p x) = 0\n[PROOFSTEP]\nsimp [Nat.choose_eq_zero_of_lt ((le_natDegree_of_mem_supp _ hx).trans_lt h)]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\n\u22a2 \u2191(hasseDeriv 1) f = \u2191derivative f\n[PROOFSTEP]\nsimp only [hasseDeriv_apply, derivative_apply, \u2190 C_mul_X_pow_eq_monomial, Nat.choose_one_right,\n (Nat.cast_commute _ _).eq]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\nr : R\n\u22a2 \u2191(hasseDeriv k) (\u2191(monomial n) r) = \u2191(monomial (n - k)) (\u2191(choose n k) * r)\n[PROOFSTEP]\next i\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\nr : R\ni : \u2115\n\u22a2 coeff (\u2191(hasseDeriv k) (\u2191(monomial n) r)) i = coeff (\u2191(monomial (n - k)) (\u2191(choose n k) * r)) i\n[PROOFSTEP]\nsimp only [hasseDeriv_coeff, coeff_monomial]\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\nr : R\ni : \u2115\n\u22a2 (\u2191(choose (i + k) k) * if n = i + k then r else 0) = if n - k = i then \u2191(choose n k) * r else 0\n[PROOFSTEP]\nby_cases hnik : n = i + k\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\nr : R\ni : \u2115\nhnik : n = i + k\n\u22a2 (\u2191(choose (i + k) k) * if n = i + k then r else 0) = if n - k = i then \u2191(choose n k) * r else 0\n[PROOFSTEP]\nrw [if_pos hnik, if_pos, \u2190 hnik]\n[GOAL]\ncase pos.hc\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\nr : R\ni : \u2115\nhnik : n = i + k\n\u22a2 n - k = i\n[PROOFSTEP]\napply tsub_eq_of_eq_add_rev\n[GOAL]\ncase pos.hc.h\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\nr : R\ni : \u2115\nhnik : n = i + k\n\u22a2 n = k + i\n[PROOFSTEP]\nrwa [add_comm]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\nr : R\ni : \u2115\nhnik : \u00acn = i + k\n\u22a2 (\u2191(choose (i + k) k) * if n = i + k then r else 0) = if n - k = i then \u2191(choose n k) * r else 0\n[PROOFSTEP]\nrw [if_neg hnik, mul_zero]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\nr : R\ni : \u2115\nhnik : \u00acn = i + k\n\u22a2 0 = if n - k = i then \u2191(choose n k) * r else 0\n[PROOFSTEP]\nby_cases hkn : k \u2264 n\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\nr : R\ni : \u2115\nhnik : \u00acn = i + k\nhkn : k \u2264 n\n\u22a2 0 = if n - k = i then \u2191(choose n k) * r else 0\n[PROOFSTEP]\nrw [\u2190 tsub_eq_iff_eq_add_of_le hkn] at hnik \n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\nr : R\ni : \u2115\nhnik : \u00acn - k = i\nhkn : k \u2264 n\n\u22a2 0 = if n - k = i then \u2191(choose n k) * r else 0\n[PROOFSTEP]\nrw [if_neg hnik]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\nr : R\ni : \u2115\nhnik : \u00acn = i + k\nhkn : \u00ack \u2264 n\n\u22a2 0 = if n - k = i then \u2191(choose n k) * r else 0\n[PROOFSTEP]\npush_neg at hkn \n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nn : \u2115\nr : R\ni : \u2115\nhnik : \u00acn = i + k\nhkn : n < k\n\u22a2 0 = if n - k = i then \u2191(choose n k) * r else 0\n[PROOFSTEP]\nrw [Nat.choose_eq_zero_of_lt hkn, Nat.cast_zero, zero_mul, ite_self]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nr : R\nhk : 0 < k\n\u22a2 \u2191(hasseDeriv k) (\u2191C r) = 0\n[PROOFSTEP]\nrw [\u2190 monomial_zero_left, hasseDeriv_monomial, Nat.choose_eq_zero_of_lt hk, Nat.cast_zero, zero_mul,\n monomial_zero_right]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nhk : 0 < k\n\u22a2 \u2191(hasseDeriv k) 1 = 0\n[PROOFSTEP]\nrw [\u2190 C_1, hasseDeriv_C k _ hk]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nhk : 1 < k\n\u22a2 \u2191(hasseDeriv k) X = 0\n[PROOFSTEP]\nrw [\u2190 monomial_one_one_eq_X, hasseDeriv_monomial, Nat.choose_eq_zero_of_lt hk, Nat.cast_zero, zero_mul,\n monomial_zero_right]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\n\u22a2 \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\n\u22a2 \u2191(Nat.zero ! \u2022 hasseDeriv Nat.zero) = (\u2191derivative)^[Nat.zero]\n[PROOFSTEP]\nrw [hasseDeriv_zero, factorial_zero, iterate_zero, one_smul, LinearMap.id_coe]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk : \u2115\nih : \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\n\u22a2 \u2191((succ k)! \u2022 hasseDeriv (succ k)) = (\u2191derivative)^[succ k]\n[PROOFSTEP]\next f n : 2\n[GOAL]\ncase succ.h.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf\u271d : R[X]\nk : \u2115\nih : \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\nf : R[X]\nn : \u2115\n\u22a2 coeff (\u2191((succ k)! \u2022 hasseDeriv (succ k)) f) n = coeff ((\u2191derivative)^[succ k] f) n\n[PROOFSTEP]\nrw [iterate_succ_apply', \u2190 ih]\n[GOAL]\ncase succ.h.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf\u271d : R[X]\nk : \u2115\nih : \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\nf : R[X]\nn : \u2115\n\u22a2 coeff (\u2191((succ k)! \u2022 hasseDeriv (succ k)) f) n = coeff (\u2191derivative (\u2191(k ! \u2022 hasseDeriv k) f)) n\n[PROOFSTEP]\nsimp only [LinearMap.smul_apply, coeff_smul, LinearMap.map_smul_of_tower, coeff_derivative, hasseDeriv_coeff, \u2190\n @choose_symm_add _ k]\n[GOAL]\ncase succ.h.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf\u271d : R[X]\nk : \u2115\nih : \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\nf : R[X]\nn : \u2115\n\u22a2 (succ k)! \u2022 (\u2191(choose (n + succ k) (succ k)) * coeff f (n + succ k)) =\n k ! \u2022 (\u2191(choose (n + 1 + k) (n + 1)) * coeff f (n + 1 + k) * (\u2191n + 1))\n[PROOFSTEP]\nsimp only [nsmul_eq_mul, factorial_succ, mul_assoc, succ_eq_add_one, \u2190 add_assoc, add_right_comm n 1 k, \u2190 cast_succ]\n[GOAL]\ncase succ.h.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf\u271d : R[X]\nk : \u2115\nih : \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\nf : R[X]\nn : \u2115\n\u22a2 \u2191((k + 1) * k !) * (\u2191(choose (n + k + 1) (k + 1)) * coeff f (n + k + 1)) =\n \u2191k ! * (\u2191(choose (n + k + 1) (n + 1)) * (coeff f (n + k + 1) * \u2191(n + 1)))\n[PROOFSTEP]\nrw [\u2190 (cast_commute (n + 1) (f.coeff (n + k + 1))).eq]\n[GOAL]\ncase succ.h.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf\u271d : R[X]\nk : \u2115\nih : \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\nf : R[X]\nn : \u2115\n\u22a2 \u2191((k + 1) * k !) * (\u2191(choose (n + k + 1) (k + 1)) * coeff f (n + k + 1)) =\n \u2191k ! * (\u2191(choose (n + k + 1) (n + 1)) * (\u2191(n + 1) * coeff f (n + k + 1)))\n[PROOFSTEP]\nsimp only [\u2190 mul_assoc]\n[GOAL]\ncase succ.h.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf\u271d : R[X]\nk : \u2115\nih : \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\nf : R[X]\nn : \u2115\n\u22a2 \u2191((k + 1) * k !) * \u2191(choose (n + k + 1) (k + 1)) * coeff f (n + k + 1) =\n \u2191k ! * \u2191(choose (n + k + 1) (n + 1)) * \u2191(n + 1) * coeff f (n + k + 1)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase succ.h.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf\u271d : R[X]\nk : \u2115\nih : \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\nf : R[X]\nn : \u2115\n\u22a2 \u2191((k + 1) * k ! * choose (n + k + 1) (k + 1)) * coeff f (n + k + 1) =\n \u2191(k ! * choose (n + k + 1) (n + 1) * (n + 1)) * coeff f (n + k + 1)\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase succ.h.a.e_a.e_a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf\u271d : R[X]\nk : \u2115\nih : \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\nf : R[X]\nn : \u2115\n\u22a2 (k + 1) * k ! * choose (n + k + 1) (k + 1) = k ! * choose (n + k + 1) (n + 1) * (n + 1)\n[PROOFSTEP]\nrw [mul_comm (k + 1) _, mul_assoc, mul_assoc]\n[GOAL]\ncase succ.h.a.e_a.e_a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf\u271d : R[X]\nk : \u2115\nih : \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\nf : R[X]\nn : \u2115\n\u22a2 k ! * ((k + 1) * choose (n + k + 1) (k + 1)) = k ! * (choose (n + k + 1) (n + 1) * (n + 1))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase succ.h.a.e_a.e_a.e_a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf\u271d : R[X]\nk : \u2115\nih : \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\nf : R[X]\nn : \u2115\n\u22a2 (k + 1) * choose (n + k + 1) (k + 1) = choose (n + k + 1) (n + 1) * (n + 1)\n[PROOFSTEP]\nhave : n + k + 1 = n + (k + 1) := by apply add_assoc\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf\u271d : R[X]\nk : \u2115\nih : \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\nf : R[X]\nn : \u2115\n\u22a2 n + k + 1 = n + (k + 1)\n[PROOFSTEP]\napply add_assoc\n[GOAL]\ncase succ.h.a.e_a.e_a.e_a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf\u271d : R[X]\nk : \u2115\nih : \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\nf : R[X]\nn : \u2115\nthis : n + k + 1 = n + (k + 1)\n\u22a2 (k + 1) * choose (n + k + 1) (k + 1) = choose (n + k + 1) (n + 1) * (n + 1)\n[PROOFSTEP]\nrw [\u2190 choose_symm_of_eq_add this, choose_succ_right_eq, mul_comm]\n[GOAL]\ncase succ.h.a.e_a.e_a.e_a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf\u271d : R[X]\nk : \u2115\nih : \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\nf : R[X]\nn : \u2115\nthis : n + k + 1 = n + (k + 1)\n\u22a2 choose (n + k + 1) n * (k + 1) = choose (n + k + 1) n * (n + k + 1 - n)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase succ.h.a.e_a.e_a.e_a.e_a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf\u271d : R[X]\nk : \u2115\nih : \u2191(k ! \u2022 hasseDeriv k) = (\u2191derivative)^[k]\nf : R[X]\nn : \u2115\nthis : n + k + 1 = n + (k + 1)\n\u22a2 k + 1 = n + k + 1 - n\n[PROOFSTEP]\nrw [add_assoc, add_tsub_cancel_left]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l : \u2115\n\u22a2 LinearMap.comp (hasseDeriv k) (hasseDeriv l) = choose (k + l) k \u2022 hasseDeriv (k + l)\n[PROOFSTEP]\next i : 2\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\n\u22a2 \u2191(LinearMap.comp (LinearMap.comp (hasseDeriv k) (hasseDeriv l)) (monomial i)) 1 =\n \u2191(LinearMap.comp (choose (k + l) k \u2022 hasseDeriv (k + l)) (monomial i)) 1\n[PROOFSTEP]\nsimp only [LinearMap.smul_apply, comp_apply, LinearMap.coe_comp, smul_monomial, hasseDeriv_apply, mul_one,\n monomial_eq_zero_iff, sum_monomial_index, mul_zero, \u2190 tsub_add_eq_tsub_tsub, add_comm l k]\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\n\u22a2 \u2191(monomial (i - (k + l))) (\u2191(choose (i - l) k) * \u2191(choose i l)) =\n \u2191(monomial (i - (k + l))) (choose (k + l) k \u2022 \u2191(choose i (k + l)))\n[PROOFSTEP]\nrw_mod_cast [nsmul_eq_mul]\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\n\u22a2 \u2191(monomial (i - (k + l))) \u2191(choose (i - l) k * choose i l) =\n \u2191(monomial (i - (k + l))) (\u2191(choose (k + l) k) * \u2191(choose i (k + l)))\n[PROOFSTEP]\nrw [\u2190 Nat.cast_mul]\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\n\u22a2 \u2191(monomial (i - (k + l))) \u2191(choose (i - l) k * choose i l) =\n \u2191(monomial (i - (k + l))) \u2191(choose (k + l) k * choose i (k + l))\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase h.h.h.e_6.h.e_a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\n\u22a2 choose (i - l) k * choose i l = choose (k + l) k * choose i (k + l)\n[PROOFSTEP]\nby_cases hikl : i < k + l\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : i < k + l\n\u22a2 choose (i - l) k * choose i l = choose (k + l) k * choose i (k + l)\n[PROOFSTEP]\nrw [choose_eq_zero_of_lt hikl, mul_zero]\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : i < k + l\n\u22a2 choose (i - l) k * choose i l = 0\n[PROOFSTEP]\nby_cases hil : i < l\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : i < k + l\nhil : i < l\n\u22a2 choose (i - l) k * choose i l = 0\n[PROOFSTEP]\nrw [choose_eq_zero_of_lt hil, mul_zero]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : i < k + l\nhil : \u00aci < l\n\u22a2 choose (i - l) k * choose i l = 0\n[PROOFSTEP]\npush_neg at hil \n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : i < k + l\nhil : l \u2264 i\n\u22a2 choose (i - l) k * choose i l = 0\n[PROOFSTEP]\nrw [\u2190 tsub_lt_iff_right hil] at hikl \n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : i - l < k\nhil : l \u2264 i\n\u22a2 choose (i - l) k * choose i l = 0\n[PROOFSTEP]\nrw [choose_eq_zero_of_lt hikl, zero_mul]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : \u00aci < k + l\n\u22a2 choose (i - l) k * choose i l = choose (k + l) k * choose i (k + l)\n[PROOFSTEP]\npush_neg at hikl \n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : k + l \u2264 i\n\u22a2 choose (i - l) k * choose i l = choose (k + l) k * choose i (k + l)\n[PROOFSTEP]\napply @cast_injective \u211a\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : k + l \u2264 i\n\u22a2 \u2191(choose (i - l) k * choose i l) = \u2191(choose (k + l) k * choose i (k + l))\n[PROOFSTEP]\nhave h1 : l \u2264 i := le_of_add_le_right hikl\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : k + l \u2264 i\nh1 : l \u2264 i\n\u22a2 \u2191(choose (i - l) k * choose i l) = \u2191(choose (k + l) k * choose i (k + l))\n[PROOFSTEP]\nhave h2 : k \u2264 i - l := le_tsub_of_add_le_right hikl\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : k + l \u2264 i\nh1 : l \u2264 i\nh2 : k \u2264 i - l\n\u22a2 \u2191(choose (i - l) k * choose i l) = \u2191(choose (k + l) k * choose i (k + l))\n[PROOFSTEP]\nhave h3 : k \u2264 k + l := le_self_add\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : k + l \u2264 i\nh1 : l \u2264 i\nh2 : k \u2264 i - l\nh3 : k \u2264 k + l\n\u22a2 \u2191(choose (i - l) k * choose i l) = \u2191(choose (k + l) k * choose i (k + l))\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : k + l \u2264 i\nh1 : l \u2264 i\nh2 : k \u2264 i - l\nh3 : k \u2264 k + l\n\u22a2 \u2191(choose (i - l) k) * \u2191(choose i l) = \u2191(choose (k + l) k) * \u2191(choose i (k + l))\n[PROOFSTEP]\nrw [cast_choose \u211a h1, cast_choose \u211a h2, cast_choose \u211a h3, cast_choose \u211a hikl]\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : k + l \u2264 i\nh1 : l \u2264 i\nh2 : k \u2264 i - l\nh3 : k \u2264 k + l\n\u22a2 \u2191(i - l)! / (\u2191k ! * \u2191(i - l - k)!) * (\u2191i ! / (\u2191l ! * \u2191(i - l)!)) =\n \u2191(k + l)! / (\u2191k ! * \u2191(k + l - k)!) * (\u2191i ! / (\u2191(k + l)! * \u2191(i - (k + l))!))\n[PROOFSTEP]\nrw [show i - (k + l) = i - l - k by rw [add_comm]; apply tsub_add_eq_tsub_tsub]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : k + l \u2264 i\nh1 : l \u2264 i\nh2 : k \u2264 i - l\nh3 : k \u2264 k + l\n\u22a2 i - (k + l) = i - l - k\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : k + l \u2264 i\nh1 : l \u2264 i\nh2 : k \u2264 i - l\nh3 : k \u2264 k + l\n\u22a2 i - (l + k) = i - l - k\n[PROOFSTEP]\napply tsub_add_eq_tsub_tsub\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : k + l \u2264 i\nh1 : l \u2264 i\nh2 : k \u2264 i - l\nh3 : k \u2264 k + l\n\u22a2 \u2191(i - l)! / (\u2191k ! * \u2191(i - l - k)!) * (\u2191i ! / (\u2191l ! * \u2191(i - l)!)) =\n \u2191(k + l)! / (\u2191k ! * \u2191(k + l - k)!) * (\u2191i ! / (\u2191(k + l)! * \u2191(i - l - k)!))\n[PROOFSTEP]\nsimp only [add_tsub_cancel_left]\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : k + l \u2264 i\nh1 : l \u2264 i\nh2 : k \u2264 i - l\nh3 : k \u2264 k + l\n\u22a2 \u2191(i - l)! / (\u2191k ! * \u2191(i - l - k)!) * (\u2191i ! / (\u2191l ! * \u2191(i - l)!)) =\n \u2191(k + l)! / (\u2191k ! * \u2191l !) * (\u2191i ! / (\u2191(k + l)! * \u2191(i - l - k)!))\n[PROOFSTEP]\nhave H : \u2200 n : \u2115, (n ! : \u211a) \u2260 0 := by exact_mod_cast factorial_ne_zero\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : k + l \u2264 i\nh1 : l \u2264 i\nh2 : k \u2264 i - l\nh3 : k \u2264 k + l\n\u22a2 \u2200 (n : \u2115), \u2191n ! \u2260 0\n[PROOFSTEP]\nexact_mod_cast factorial_ne_zero\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : k + l \u2264 i\nh1 : l \u2264 i\nh2 : k \u2264 i - l\nh3 : k \u2264 k + l\nH : \u2200 (n : \u2115), \u2191n ! \u2260 0\n\u22a2 \u2191(i - l)! / (\u2191k ! * \u2191(i - l - k)!) * (\u2191i ! / (\u2191l ! * \u2191(i - l)!)) =\n \u2191(k + l)! / (\u2191k ! * \u2191l !) * (\u2191i ! / (\u2191(k + l)! * \u2191(i - l - k)!))\n[PROOFSTEP]\nfield_simp [H]\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Semiring R\nk\u271d : \u2115\nf : R[X]\nk l i : \u2115\nhikl : k + l \u2264 i\nh1 : l \u2264 i\nh2 : k \u2264 i - l\nh3 : k \u2264 k + l\nH : \u2200 (n : \u2115), \u2191n ! \u2260 0\n\u22a2 \u2191(i - l)! * \u2191i ! * (\u2191k ! * \u2191l ! * (\u2191(k + l)! * \u2191(i - l - k)!)) =\n \u2191(k + l)! * \u2191i ! * (\u2191k ! * \u2191(i - l - k)! * (\u2191l ! * \u2191(i - l)!))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn : \u2115\n\u22a2 natDegree (\u2191(hasseDeriv n) p) \u2264 natDegree p - n\n[PROOFSTEP]\nclassical\nrw [hasseDeriv_apply, sum_def]\nrefine' (natDegree_sum_le _ _).trans _\nsimp_rw [Function.comp, natDegree_monomial]\nrw [Finset.fold_ite, Finset.fold_const]\n\u00b7 simp only [ite_self, max_eq_right, zero_le', Finset.fold_max_le, true_and_iff, and_imp, tsub_le_iff_right,\n mem_support_iff, Ne.def, Finset.mem_filter]\n intro x hx hx'\n have hxp : x \u2264 p.natDegree := le_natDegree_of_ne_zero hx\n have hxn : n \u2264 x := by\n contrapose! hx'\n simp [Nat.choose_eq_zero_of_lt hx']\n rwa [tsub_add_cancel_of_le (hxn.trans hxp)]\n\u00b7 simp\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn : \u2115\n\u22a2 natDegree (\u2191(hasseDeriv n) p) \u2264 natDegree p - n\n[PROOFSTEP]\nrw [hasseDeriv_apply, sum_def]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn : \u2115\n\u22a2 natDegree (\u2211 n_1 in support p, \u2191(monomial (n_1 - n)) (\u2191(choose n_1 n) * coeff p n_1)) \u2264 natDegree p - n\n[PROOFSTEP]\nrefine' (natDegree_sum_le _ _).trans _\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn : \u2115\n\u22a2 Finset.fold max 0 (natDegree \u2218 fun n_1 => \u2191(monomial (n_1 - n)) (\u2191(choose n_1 n) * coeff p n_1)) (support p) \u2264\n natDegree p - n\n[PROOFSTEP]\nsimp_rw [Function.comp, natDegree_monomial]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn : \u2115\n\u22a2 Finset.fold max 0 (fun x => if \u2191(choose x n) * coeff p x = 0 then 0 else x - n) (support p) \u2264 natDegree p - n\n[PROOFSTEP]\nrw [Finset.fold_ite, Finset.fold_const]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn : \u2115\n\u22a2 max (if Finset.filter (fun x => \u2191(choose x n) * coeff p x = 0) (support p) = \u2205 then 0 else max 0 0)\n (Finset.fold max 0 (fun x => x - n) (Finset.filter (fun i => \u00ac\u2191(choose i n) * coeff p i = 0) (support p))) \u2264\n natDegree p - n\n[PROOFSTEP]\nsimp only [ite_self, max_eq_right, zero_le', Finset.fold_max_le, true_and_iff, and_imp, tsub_le_iff_right,\n mem_support_iff, Ne.def, Finset.mem_filter]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn : \u2115\n\u22a2 \u2200 (x : \u2115), \u00accoeff p x = 0 \u2192 \u00ac\u2191(choose x n) * coeff p x = 0 \u2192 x \u2264 natDegree p - n + n\n[PROOFSTEP]\nintro x hx hx'\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn x : \u2115\nhx : \u00accoeff p x = 0\nhx' : \u00ac\u2191(choose x n) * coeff p x = 0\n\u22a2 x \u2264 natDegree p - n + n\n[PROOFSTEP]\nhave hxp : x \u2264 p.natDegree := le_natDegree_of_ne_zero hx\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn x : \u2115\nhx : \u00accoeff p x = 0\nhx' : \u00ac\u2191(choose x n) * coeff p x = 0\nhxp : x \u2264 natDegree p\n\u22a2 x \u2264 natDegree p - n + n\n[PROOFSTEP]\nhave hxn : n \u2264 x := by\n contrapose! hx'\n simp [Nat.choose_eq_zero_of_lt hx']\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn x : \u2115\nhx : \u00accoeff p x = 0\nhx' : \u00ac\u2191(choose x n) * coeff p x = 0\nhxp : x \u2264 natDegree p\n\u22a2 n \u2264 x\n[PROOFSTEP]\ncontrapose! hx'\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn x : \u2115\nhx : \u00accoeff p x = 0\nhxp : x \u2264 natDegree p\nhx' : x < n\n\u22a2 \u2191(choose x n) * coeff p x = 0\n[PROOFSTEP]\nsimp [Nat.choose_eq_zero_of_lt hx']\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn x : \u2115\nhx : \u00accoeff p x = 0\nhx' : \u00ac\u2191(choose x n) * coeff p x = 0\nhxp : x \u2264 natDegree p\nhxn : n \u2264 x\n\u22a2 x \u2264 natDegree p - n + n\n[PROOFSTEP]\nrwa [tsub_add_cancel_of_le (hxn.trans hxp)]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf p : R[X]\nn : \u2115\n\u22a2 max 0 (max 0 0) = max 0 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nk : \u2115\nf : R[X]\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\n\u22a2 natDegree (\u2191(hasseDeriv n) p) = natDegree p - n\n[PROOFSTEP]\ncases' lt_or_le p.natDegree n with hn hn\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nk : \u2115\nf : R[X]\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\nhn : natDegree p < n\n\u22a2 natDegree (\u2191(hasseDeriv n) p) = natDegree p - n\n[PROOFSTEP]\nsimpa [hasseDeriv_eq_zero_of_lt_natDegree, hn] using (tsub_eq_zero_of_le hn.le).symm\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nk : \u2115\nf : R[X]\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\nhn : n \u2264 natDegree p\n\u22a2 natDegree (\u2191(hasseDeriv n) p) = natDegree p - n\n[PROOFSTEP]\nrefine' map_natDegree_eq_sub _ _\n[GOAL]\ncase inr.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nk : \u2115\nf : R[X]\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\nhn : n \u2264 natDegree p\n\u22a2 \u2200 (f : R[X]), natDegree f < n \u2192 \u2191(hasseDeriv n) f = 0\n[PROOFSTEP]\nexact fun h => hasseDeriv_eq_zero_of_lt_natDegree _ _\n[GOAL]\ncase inr.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nk : \u2115\nf : R[X]\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\nhn : n \u2264 natDegree p\n\u22a2 \u2200 (n_1 : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191(hasseDeriv n) (\u2191(monomial n_1) c)) = n_1 - n\n[PROOFSTEP]\nclassical\nsimp only [ite_eq_right_iff, Ne.def, natDegree_monomial, hasseDeriv_monomial]\nintro k c c0 hh\nrw [\u2190 nsmul_eq_mul, smul_eq_zero, Nat.choose_eq_zero_iff] at hh \nexact (tsub_eq_zero_of_le (Or.resolve_right hh c0).le).symm\n[GOAL]\ncase inr.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nk : \u2115\nf : R[X]\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\nhn : n \u2264 natDegree p\n\u22a2 \u2200 (n_1 : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191(hasseDeriv n) (\u2191(monomial n_1) c)) = n_1 - n\n[PROOFSTEP]\nsimp only [ite_eq_right_iff, Ne.def, natDegree_monomial, hasseDeriv_monomial]\n[GOAL]\ncase inr.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nk : \u2115\nf : R[X]\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\nhn : n \u2264 natDegree p\n\u22a2 \u2200 (n_1 : \u2115) (c : R), \u00acc = 0 \u2192 \u2191(choose n_1 n) * c = 0 \u2192 0 = n_1 - n\n[PROOFSTEP]\nintro k c c0 hh\n[GOAL]\ncase inr.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nk\u271d : \u2115\nf : R[X]\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\nhn : n \u2264 natDegree p\nk : \u2115\nc : R\nc0 : \u00acc = 0\nhh : \u2191(choose k n) * c = 0\n\u22a2 0 = k - n\n[PROOFSTEP]\nrw [\u2190 nsmul_eq_mul, smul_eq_zero, Nat.choose_eq_zero_iff] at hh \n[GOAL]\ncase inr.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nk\u271d : \u2115\nf : R[X]\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\nhn : n \u2264 natDegree p\nk : \u2115\nc : R\nc0 : \u00acc = 0\nhh : k < n \u2228 c = 0\n\u22a2 0 = k - n\n[PROOFSTEP]\nexact (tsub_eq_zero_of_le (Or.resolve_right hh c0).le).symm\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf\u271d f g : R[X]\n\u22a2 \u2191(hasseDeriv k) (f * g) = \u2211 ij in antidiagonal k, \u2191(hasseDeriv ij.fst) f * \u2191(hasseDeriv ij.snd) g\n[PROOFSTEP]\nlet D k := (@hasseDeriv R _ k).toAddMonoidHom\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf\u271d f g : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u22a2 \u2191(hasseDeriv k) (f * g) = \u2211 ij in antidiagonal k, \u2191(hasseDeriv ij.fst) f * \u2191(hasseDeriv ij.snd) g\n[PROOFSTEP]\nlet \u03a6 := @AddMonoidHom.mul R[X] _\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf\u271d f g : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\n\u22a2 \u2191(hasseDeriv k) (f * g) = \u2211 ij in antidiagonal k, \u2191(hasseDeriv ij.fst) f * \u2191(hasseDeriv ij.snd) g\n[PROOFSTEP]\nshow\n (compHom (D k)).comp \u03a6 f g = \u2211 ij : \u2115 \u00d7 \u2115 in antidiagonal k, ((compHom.comp ((compHom \u03a6) (D ij.1))).flip (D ij.2) f) g\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf\u271d f g : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\n\u22a2 \u2191(\u2191(AddMonoidHom.comp (\u2191compHom (D k)) \u03a6) f) g =\n \u2211 ij in antidiagonal k,\n \u2191(\u2191(\u2191(AddMonoidHom.flip (AddMonoidHom.comp compHom (\u2191(\u2191compHom \u03a6) (D ij.fst)))) (D ij.snd)) f) g\n[PROOFSTEP]\nsimp only [\u2190 finset_sum_apply]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf\u271d f g : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\n\u22a2 \u2191(\u2191(AddMonoidHom.comp (\u2191compHom (LinearMap.toAddMonoidHom (hasseDeriv k))) mul) f) g =\n \u2191(\u2191(\u2211 x in antidiagonal k,\n \u2191(AddMonoidHom.flip\n (AddMonoidHom.comp compHom (\u2191(\u2191compHom mul) (LinearMap.toAddMonoidHom (hasseDeriv x.fst)))))\n (LinearMap.toAddMonoidHom (hasseDeriv x.snd)))\n f)\n g\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_a\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf\u271d f g : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\n\u22a2 AddMonoidHom.comp (\u2191compHom (LinearMap.toAddMonoidHom (hasseDeriv k))) mul =\n \u2211 x in antidiagonal k,\n \u2191(AddMonoidHom.flip (AddMonoidHom.comp compHom (\u2191(\u2191compHom mul) (LinearMap.toAddMonoidHom (hasseDeriv x.fst)))))\n (LinearMap.toAddMonoidHom (hasseDeriv x.snd))\n[PROOFSTEP]\nclear f g\n[GOAL]\ncase e_a.e_a\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\n\u22a2 AddMonoidHom.comp (\u2191compHom (LinearMap.toAddMonoidHom (hasseDeriv k))) mul =\n \u2211 x in antidiagonal k,\n \u2191(AddMonoidHom.flip (AddMonoidHom.comp compHom (\u2191(\u2191compHom mul) (LinearMap.toAddMonoidHom (hasseDeriv x.fst)))))\n (LinearMap.toAddMonoidHom (hasseDeriv x.snd))\n[PROOFSTEP]\next m r n s : 4\n[GOAL]\ncase e_a.e_a.h.h.h.h\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\nm : \u2115\nr : R\nn : \u2115\ns : R\n\u22a2 \u2191(AddMonoidHom.comp\n (\u2191(AddMonoidHom.comp (AddMonoidHom.comp (\u2191compHom (LinearMap.toAddMonoidHom (hasseDeriv k))) mul)\n (LinearMap.toAddMonoidHom (monomial m)))\n r)\n (LinearMap.toAddMonoidHom (monomial n)))\n s =\n \u2191(AddMonoidHom.comp\n (\u2191(AddMonoidHom.comp\n (\u2211 x in antidiagonal k,\n \u2191(AddMonoidHom.flip\n (AddMonoidHom.comp compHom (\u2191(\u2191compHom mul) (LinearMap.toAddMonoidHom (hasseDeriv x.fst)))))\n (LinearMap.toAddMonoidHom (hasseDeriv x.snd)))\n (LinearMap.toAddMonoidHom (monomial m)))\n r)\n (LinearMap.toAddMonoidHom (monomial n)))\n s\n[PROOFSTEP]\nsimp only [finset_sum_apply, coe_mulLeft, coe_comp, flip_apply, Function.comp_apply, hasseDeriv_monomial,\n LinearMap.toAddMonoidHom_coe, compHom_apply_apply, coe_mul, monomial_mul_monomial]\n[GOAL]\ncase e_a.e_a.h.h.h.h\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\nm : \u2115\nr : R\nn : \u2115\ns : R\n\u22a2 \u2191(monomial (m + n - k)) (\u2191(choose (m + n) k) * (r * s)) =\n \u2211 x in antidiagonal k, \u2191(monomial (m - x.fst + (n - x.snd))) (\u2191(choose m x.fst) * r * (\u2191(choose n x.snd) * s))\n[PROOFSTEP]\nhave aux :\n \u2200 x : \u2115 \u00d7 \u2115,\n x \u2208 antidiagonal k \u2192\n monomial (m - x.1 + (n - x.2)) (\u2191(m.choose x.1) * r * (\u2191(n.choose x.2) * s)) =\n monomial (m + n - k) (\u2191(m.choose x.1) * \u2191(n.choose x.2) * (r * s)) :=\n by\n intro x hx\n rw [Finset.Nat.mem_antidiagonal] at hx \n subst hx\n by_cases hm : m < x.1\n \u00b7 simp only [Nat.choose_eq_zero_of_lt hm, Nat.cast_zero, zero_mul, monomial_zero_right]\n by_cases hn : n < x.2\n \u00b7 simp only [Nat.choose_eq_zero_of_lt hn, Nat.cast_zero, zero_mul, mul_zero, monomial_zero_right]\n push_neg at hm hn \n rw [tsub_add_eq_add_tsub hm, \u2190 add_tsub_assoc_of_le hn, \u2190 tsub_add_eq_tsub_tsub, add_comm x.2 x.1, mul_assoc, \u2190\n mul_assoc r, \u2190 (Nat.cast_commute _ r).eq, mul_assoc, mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\nm : \u2115\nr : R\nn : \u2115\ns : R\n\u22a2 \u2200 (x : \u2115 \u00d7 \u2115),\n x \u2208 antidiagonal k \u2192\n \u2191(monomial (m - x.fst + (n - x.snd))) (\u2191(choose m x.fst) * r * (\u2191(choose n x.snd) * s)) =\n \u2191(monomial (m + n - k)) (\u2191(choose m x.fst) * \u2191(choose n x.snd) * (r * s))\n[PROOFSTEP]\nintro x hx\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\nm : \u2115\nr : R\nn : \u2115\ns : R\nx : \u2115 \u00d7 \u2115\nhx : x \u2208 antidiagonal k\n\u22a2 \u2191(monomial (m - x.fst + (n - x.snd))) (\u2191(choose m x.fst) * r * (\u2191(choose n x.snd) * s)) =\n \u2191(monomial (m + n - k)) (\u2191(choose m x.fst) * \u2191(choose n x.snd) * (r * s))\n[PROOFSTEP]\nrw [Finset.Nat.mem_antidiagonal] at hx \n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\nm : \u2115\nr : R\nn : \u2115\ns : R\nx : \u2115 \u00d7 \u2115\nhx : x.fst + x.snd = k\n\u22a2 \u2191(monomial (m - x.fst + (n - x.snd))) (\u2191(choose m x.fst) * r * (\u2191(choose n x.snd) * s)) =\n \u2191(monomial (m + n - k)) (\u2191(choose m x.fst) * \u2191(choose n x.snd) * (r * s))\n[PROOFSTEP]\nsubst hx\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\nm : \u2115\nr : R\nn : \u2115\ns : R\nx : \u2115 \u00d7 \u2115\n\u22a2 \u2191(monomial (m - x.fst + (n - x.snd))) (\u2191(choose m x.fst) * r * (\u2191(choose n x.snd) * s)) =\n \u2191(monomial (m + n - (x.fst + x.snd))) (\u2191(choose m x.fst) * \u2191(choose n x.snd) * (r * s))\n[PROOFSTEP]\nby_cases hm : m < x.1\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\nm : \u2115\nr : R\nn : \u2115\ns : R\nx : \u2115 \u00d7 \u2115\nhm : m < x.fst\n\u22a2 \u2191(monomial (m - x.fst + (n - x.snd))) (\u2191(choose m x.fst) * r * (\u2191(choose n x.snd) * s)) =\n \u2191(monomial (m + n - (x.fst + x.snd))) (\u2191(choose m x.fst) * \u2191(choose n x.snd) * (r * s))\n[PROOFSTEP]\nsimp only [Nat.choose_eq_zero_of_lt hm, Nat.cast_zero, zero_mul, monomial_zero_right]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\nm : \u2115\nr : R\nn : \u2115\ns : R\nx : \u2115 \u00d7 \u2115\nhm : \u00acm < x.fst\n\u22a2 \u2191(monomial (m - x.fst + (n - x.snd))) (\u2191(choose m x.fst) * r * (\u2191(choose n x.snd) * s)) =\n \u2191(monomial (m + n - (x.fst + x.snd))) (\u2191(choose m x.fst) * \u2191(choose n x.snd) * (r * s))\n[PROOFSTEP]\nby_cases hn : n < x.2\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\nm : \u2115\nr : R\nn : \u2115\ns : R\nx : \u2115 \u00d7 \u2115\nhm : \u00acm < x.fst\nhn : n < x.snd\n\u22a2 \u2191(monomial (m - x.fst + (n - x.snd))) (\u2191(choose m x.fst) * r * (\u2191(choose n x.snd) * s)) =\n \u2191(monomial (m + n - (x.fst + x.snd))) (\u2191(choose m x.fst) * \u2191(choose n x.snd) * (r * s))\n[PROOFSTEP]\nsimp only [Nat.choose_eq_zero_of_lt hn, Nat.cast_zero, zero_mul, mul_zero, monomial_zero_right]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\nm : \u2115\nr : R\nn : \u2115\ns : R\nx : \u2115 \u00d7 \u2115\nhm : \u00acm < x.fst\nhn : \u00acn < x.snd\n\u22a2 \u2191(monomial (m - x.fst + (n - x.snd))) (\u2191(choose m x.fst) * r * (\u2191(choose n x.snd) * s)) =\n \u2191(monomial (m + n - (x.fst + x.snd))) (\u2191(choose m x.fst) * \u2191(choose n x.snd) * (r * s))\n[PROOFSTEP]\npush_neg at hm hn \n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\nm : \u2115\nr : R\nn : \u2115\ns : R\nx : \u2115 \u00d7 \u2115\nhm : x.fst \u2264 m\nhn : x.snd \u2264 n\n\u22a2 \u2191(monomial (m - x.fst + (n - x.snd))) (\u2191(choose m x.fst) * r * (\u2191(choose n x.snd) * s)) =\n \u2191(monomial (m + n - (x.fst + x.snd))) (\u2191(choose m x.fst) * \u2191(choose n x.snd) * (r * s))\n[PROOFSTEP]\nrw [tsub_add_eq_add_tsub hm, \u2190 add_tsub_assoc_of_le hn, \u2190 tsub_add_eq_tsub_tsub, add_comm x.2 x.1, mul_assoc, \u2190\n mul_assoc r, \u2190 (Nat.cast_commute _ r).eq, mul_assoc, mul_assoc]\n[GOAL]\ncase e_a.e_a.h.h.h.h\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\nm : \u2115\nr : R\nn : \u2115\ns : R\naux :\n \u2200 (x : \u2115 \u00d7 \u2115),\n x \u2208 antidiagonal k \u2192\n \u2191(monomial (m - x.fst + (n - x.snd))) (\u2191(choose m x.fst) * r * (\u2191(choose n x.snd) * s)) =\n \u2191(monomial (m + n - k)) (\u2191(choose m x.fst) * \u2191(choose n x.snd) * (r * s))\n\u22a2 \u2191(monomial (m + n - k)) (\u2191(choose (m + n) k) * (r * s)) =\n \u2211 x in antidiagonal k, \u2191(monomial (m - x.fst + (n - x.snd))) (\u2191(choose m x.fst) * r * (\u2191(choose n x.snd) * s))\n[PROOFSTEP]\nrw [Finset.sum_congr rfl aux]\n[GOAL]\ncase e_a.e_a.h.h.h.h\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\nm : \u2115\nr : R\nn : \u2115\ns : R\naux :\n \u2200 (x : \u2115 \u00d7 \u2115),\n x \u2208 antidiagonal k \u2192\n \u2191(monomial (m - x.fst + (n - x.snd))) (\u2191(choose m x.fst) * r * (\u2191(choose n x.snd) * s)) =\n \u2191(monomial (m + n - k)) (\u2191(choose m x.fst) * \u2191(choose n x.snd) * (r * s))\n\u22a2 \u2191(monomial (m + n - k)) (\u2191(choose (m + n) k) * (r * s)) =\n \u2211 x in antidiagonal k, \u2191(monomial (m + n - k)) (\u2191(choose m x.fst) * \u2191(choose n x.snd) * (r * s))\n[PROOFSTEP]\nrw [\u2190 LinearMap.map_sum, \u2190 Finset.sum_mul]\n[GOAL]\ncase e_a.e_a.h.h.h.h\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\nm : \u2115\nr : R\nn : \u2115\ns : R\naux :\n \u2200 (x : \u2115 \u00d7 \u2115),\n x \u2208 antidiagonal k \u2192\n \u2191(monomial (m - x.fst + (n - x.snd))) (\u2191(choose m x.fst) * r * (\u2191(choose n x.snd) * s)) =\n \u2191(monomial (m + n - k)) (\u2191(choose m x.fst) * \u2191(choose n x.snd) * (r * s))\n\u22a2 \u2191(monomial (m + n - k)) (\u2191(choose (m + n) k) * (r * s)) =\n \u2191(monomial (m + n - k)) ((\u2211 x in antidiagonal k, \u2191(choose m x.fst) * \u2191(choose n x.snd)) * (r * s))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_a.h.h.h.h.h.e_6.h.e_a\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nD : \u2115 \u2192 R[X] \u2192+ R[X] := fun k => LinearMap.toAddMonoidHom (hasseDeriv k)\n\u03a6 : R[X] \u2192+ R[X] \u2192+ R[X] := mul\nm : \u2115\nr : R\nn : \u2115\ns : R\naux :\n \u2200 (x : \u2115 \u00d7 \u2115),\n x \u2208 antidiagonal k \u2192\n \u2191(monomial (m - x.fst + (n - x.snd))) (\u2191(choose m x.fst) * r * (\u2191(choose n x.snd) * s)) =\n \u2191(monomial (m + n - k)) (\u2191(choose m x.fst) * \u2191(choose n x.snd) * (r * s))\n\u22a2 \u2191(choose (m + n) k) = \u2211 x in antidiagonal k, \u2191(choose m x.fst) * \u2191(choose n x.snd)\n[PROOFSTEP]\nrw_mod_cast [\u2190 Nat.add_choose_eq]\n", "meta": {"mathlib_filename": "Mathlib.Data.Polynomial.HasseDeriv", "llama_tokens": 20029, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.640635861701035, "lm_q1q2_score": 0.43510622287492323}} {"text": "[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nm : M\n\u22a2 \u2191(\u03b9 R) m \u2208 LinearMap.range (\u03b9 R) ^ 1\n[PROOFSTEP]\nsimpa only [pow_one] using LinearMap.mem_range_self _ m\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nm : M\n\u22a2 \u2191(\u03b9 R) m \u2208 LinearMap.range (\u03b9 R) ^ 1\n[PROOFSTEP]\nsimpa only [pow_one] using LinearMap.mem_range_self _ m\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nm : M\n\u22a2 \u2191(GradedAlgebra.\u03b9 R M) m * \u2191(GradedAlgebra.\u03b9 R M) m = 0\n[PROOFSTEP]\nrw [GradedAlgebra.\u03b9_apply, DirectSum.of_mul_of]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nm : M\n\u22a2 \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) (1 + 1))\n (GradedMonoid.GMul.mul { val := \u2191(\u03b9 R) m, property := (_ : \u2191(\u03b9 R) m \u2208 LinearMap.range (\u03b9 R) ^ 1) }\n { val := \u2191(\u03b9 R) m, property := (_ : \u2191(\u03b9 R) m \u2208 LinearMap.range (\u03b9 R) ^ 1) }) =\n 0\n[PROOFSTEP]\nrefine DFinsupp.single_eq_zero.mpr (Subtype.ext <| ExteriorAlgebra.\u03b9_sq_zero _)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 M \u2192\u2097[R] \u2a01 (i : \u2115), { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }\n[PROOFSTEP]\napply GradedAlgebra.\u03b9 R M\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }\n\u22a2 \u2191(lift\u03b9 R M) \u2191x = \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i) x\n[PROOFSTEP]\ncases' x with x hx\n[GOAL]\ncase mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\nhx : x \u2208 LinearMap.range (\u03b9 R) ^ i\n\u22a2 \u2191(lift\u03b9 R M) \u2191{ val := x, property := hx } =\n \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i) { val := x, property := hx }\n[PROOFSTEP]\ndsimp only [Subtype.coe_mk, DirectSum.lof_eq_of]\n -- Porting note: original statement was\n -- refine Submodule.pow_induction_on_left' _ (fun r => ?_) (fun x y i hx hy ihx ihy => ?_)\n -- (fun m hm i x hx ih => ?_) hx\n -- but it created invalid goals\n[GOAL]\ncase mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\nhx : x \u2208 LinearMap.range (\u03b9 R) ^ i\n\u22a2 \u2191(lift\u03b9 R M) x = \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i) { val := x, property := hx }\n[PROOFSTEP]\ninduction hx using Submodule.pow_induction_on_left' with\n| hr => simp_rw [AlgHom.commutes, DirectSum.algebraMap_apply]; rfl\n| hadd _ _ _ _ _ ihx ihy => simp_rw [AlgHom.map_add, ihx, ihy, \u2190 map_add]; rfl\n| hmul _ hm _ _ _ ih =>\n obtain \u27e8_, rfl\u27e9 := hm\n simp_rw [AlgHom.map_mul, ih, GradedAlgebra.lift\u03b9, lift_\u03b9_apply, GradedAlgebra.\u03b9_apply R M, DirectSum.of_mul_of]\n exact DirectSum.of_eq_of_gradedMonoid_eq (Sigma.subtype_ext (add_comm _ _) rfl)\n[GOAL]\ncase mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\nhx : x \u2208 LinearMap.range (\u03b9 R) ^ i\n\u22a2 \u2191(lift\u03b9 R M) x = \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i) { val := x, property := hx }\n[PROOFSTEP]\ninduction hx using Submodule.pow_induction_on_left' with\n| hr => simp_rw [AlgHom.commutes, DirectSum.algebraMap_apply]; rfl\n| hadd _ _ _ _ _ ihx ihy => simp_rw [AlgHom.map_add, ihx, ihy, \u2190 map_add]; rfl\n| hmul _ hm _ _ _ ih =>\n obtain \u27e8_, rfl\u27e9 := hm\n simp_rw [AlgHom.map_mul, ih, GradedAlgebra.lift\u03b9, lift_\u03b9_apply, GradedAlgebra.\u03b9_apply R M, DirectSum.of_mul_of]\n exact DirectSum.of_eq_of_gradedMonoid_eq (Sigma.subtype_ext (add_comm _ _) rfl)\n[GOAL]\ncase mk.hr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\nr\u271d : R\n\u22a2 \u2191(lift\u03b9 R M) (\u2191(algebraMap R (ExteriorAlgebra R M)) r\u271d) =\n \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) 0)\n { val := \u2191(algebraMap R (ExteriorAlgebra R M)) r\u271d,\n property := (_ : \u2191(algebraMap R (ExteriorAlgebra R M)) r\u271d \u2208 1) }\n[PROOFSTEP]\n\n| hr => simp_rw [AlgHom.commutes, DirectSum.algebraMap_apply]; rfl\n[GOAL]\ncase mk.hr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\nr\u271d : R\n\u22a2 \u2191(lift\u03b9 R M) (\u2191(algebraMap R (ExteriorAlgebra R M)) r\u271d) =\n \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) 0)\n { val := \u2191(algebraMap R (ExteriorAlgebra R M)) r\u271d,\n property := (_ : \u2191(algebraMap R (ExteriorAlgebra R M)) r\u271d \u2208 1) }\n[PROOFSTEP]\nsimp_rw [AlgHom.commutes, DirectSum.algebraMap_apply]\n[GOAL]\ncase mk.hr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\nr\u271d : R\n\u22a2 \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) 0) (\u2191DirectSum.GAlgebra.toFun r\u271d) =\n \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) 0)\n { val := \u2191(algebraMap R (ExteriorAlgebra R M)) r\u271d,\n property := (_ : \u2191(algebraMap R (ExteriorAlgebra R M)) r\u271d \u2208 1) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.hadd\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx x\u271d y\u271d : ExteriorAlgebra R M\ni\u271d : \u2115\nhx\u271d : x\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nhy\u271d : y\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nihx :\n \u2191(lift\u03b9 R M) x\u271d = \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i\u271d) { val := x\u271d, property := hx\u271d }\nihy :\n \u2191(lift\u03b9 R M) y\u271d = \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i\u271d) { val := y\u271d, property := hy\u271d }\n\u22a2 \u2191(lift\u03b9 R M) (x\u271d + y\u271d) =\n \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i\u271d)\n { val := x\u271d + y\u271d, property := (_ : x\u271d + y\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d) }\n[PROOFSTEP]\n\n| hadd _ _ _ _ _ ihx ihy => simp_rw [AlgHom.map_add, ihx, ihy, \u2190 map_add]; rfl\n[GOAL]\ncase mk.hadd\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx x\u271d y\u271d : ExteriorAlgebra R M\ni\u271d : \u2115\nhx\u271d : x\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nhy\u271d : y\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nihx :\n \u2191(lift\u03b9 R M) x\u271d = \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i\u271d) { val := x\u271d, property := hx\u271d }\nihy :\n \u2191(lift\u03b9 R M) y\u271d = \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i\u271d) { val := y\u271d, property := hy\u271d }\n\u22a2 \u2191(lift\u03b9 R M) (x\u271d + y\u271d) =\n \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i\u271d)\n { val := x\u271d + y\u271d, property := (_ : x\u271d + y\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d) }\n[PROOFSTEP]\nsimp_rw [AlgHom.map_add, ihx, ihy, \u2190 map_add]\n[GOAL]\ncase mk.hadd\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx x\u271d y\u271d : ExteriorAlgebra R M\ni\u271d : \u2115\nhx\u271d : x\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nhy\u271d : y\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nihx :\n \u2191(lift\u03b9 R M) x\u271d = \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i\u271d) { val := x\u271d, property := hx\u271d }\nihy :\n \u2191(lift\u03b9 R M) y\u271d = \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i\u271d) { val := y\u271d, property := hy\u271d }\n\u22a2 \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i\u271d)\n ({ val := x\u271d, property := hx\u271d } + { val := y\u271d, property := hy\u271d }) =\n \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i\u271d)\n { val := x\u271d + y\u271d, property := (_ : x\u271d + y\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.hmul\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx m\u271d : ExteriorAlgebra R M\nhm : m\u271d \u2208 LinearMap.range (\u03b9 R)\ni\u271d : \u2115\nx\u271d : ExteriorAlgebra R M\nhx\u271d : x\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nih :\n \u2191(lift\u03b9 R M) x\u271d = \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i\u271d) { val := x\u271d, property := hx\u271d }\n\u22a2 \u2191(lift\u03b9 R M) (m\u271d * x\u271d) =\n \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) (Nat.succ i\u271d))\n { val := m\u271d * x\u271d, property := (_ : m\u271d * x\u271d \u2208 LinearMap.range (\u03b9 R) * npowRec i\u271d (LinearMap.range (\u03b9 R))) }\n[PROOFSTEP]\n\n| hmul _ hm _ _ _ ih =>\n obtain \u27e8_, rfl\u27e9 := hm\n simp_rw [AlgHom.map_mul, ih, GradedAlgebra.lift\u03b9, lift_\u03b9_apply, GradedAlgebra.\u03b9_apply R M, DirectSum.of_mul_of]\n exact DirectSum.of_eq_of_gradedMonoid_eq (Sigma.subtype_ext (add_comm _ _) rfl)\n[GOAL]\ncase mk.hmul\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx m\u271d : ExteriorAlgebra R M\nhm : m\u271d \u2208 LinearMap.range (\u03b9 R)\ni\u271d : \u2115\nx\u271d : ExteriorAlgebra R M\nhx\u271d : x\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nih :\n \u2191(lift\u03b9 R M) x\u271d = \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i\u271d) { val := x\u271d, property := hx\u271d }\n\u22a2 \u2191(lift\u03b9 R M) (m\u271d * x\u271d) =\n \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) (Nat.succ i\u271d))\n { val := m\u271d * x\u271d, property := (_ : m\u271d * x\u271d \u2208 LinearMap.range (\u03b9 R) * npowRec i\u271d (LinearMap.range (\u03b9 R))) }\n[PROOFSTEP]\nobtain \u27e8_, rfl\u27e9 := hm\n[GOAL]\ncase mk.hmul.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\ni\u271d : \u2115\nx\u271d : ExteriorAlgebra R M\nhx\u271d : x\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nih :\n \u2191(lift\u03b9 R M) x\u271d = \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i\u271d) { val := x\u271d, property := hx\u271d }\nw\u271d : M\n\u22a2 \u2191(lift\u03b9 R M) (\u2191(\u03b9 R) w\u271d * x\u271d) =\n \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) (Nat.succ i\u271d))\n { val := \u2191(\u03b9 R) w\u271d * x\u271d,\n property := (_ : \u2191(\u03b9 R) w\u271d * x\u271d \u2208 LinearMap.range (\u03b9 R) * npowRec i\u271d (LinearMap.range (\u03b9 R))) }\n[PROOFSTEP]\nsimp_rw [AlgHom.map_mul, ih, GradedAlgebra.lift\u03b9, lift_\u03b9_apply, GradedAlgebra.\u03b9_apply R M, DirectSum.of_mul_of]\n[GOAL]\ncase mk.hmul.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\ni : \u2115\nx : ExteriorAlgebra R M\ni\u271d : \u2115\nx\u271d : ExteriorAlgebra R M\nhx\u271d : x\u271d \u2208 LinearMap.range (\u03b9 R) ^ i\u271d\nih :\n \u2191(lift\u03b9 R M) x\u271d = \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i\u271d) { val := x\u271d, property := hx\u271d }\nw\u271d : M\n\u22a2 \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) (1 + i\u271d))\n (GradedMonoid.GMul.mul { val := \u2191(\u03b9 R) w\u271d, property := (_ : \u2191(\u03b9 R) w\u271d \u2208 LinearMap.range (\u03b9 R) ^ 1) }\n { val := x\u271d, property := hx\u271d }) =\n \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) (Nat.succ i\u271d))\n { val := \u2191(\u03b9 R) w\u271d * x\u271d,\n property := (_ : \u2191(\u03b9 R) w\u271d * x\u271d \u2208 LinearMap.range (\u03b9 R) * npowRec i\u271d (LinearMap.range (\u03b9 R))) }\n[PROOFSTEP]\nexact DirectSum.of_eq_of_gradedMonoid_eq (Sigma.subtype_ext (add_comm _ _) rfl)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 ExteriorAlgebra R M \u2192\u2090[R] \u2a01 (i : \u2115), { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }\n[PROOFSTEP]\napply GradedAlgebra.lift\u03b9 R M\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 AlgHom.comp (DirectSum.coeAlgHom fun x => LinearMap.range (\u03b9 R) ^ x) (GradedAlgebra.lift\u03b9 R M) =\n AlgHom.id R (ExteriorAlgebra R M)\n[PROOFSTEP]\next m\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nm : M\n\u22a2 \u2191(LinearMap.comp\n (AlgHom.toLinearMap\n (AlgHom.comp (DirectSum.coeAlgHom fun x => LinearMap.range (\u03b9 R) ^ x) (GradedAlgebra.lift\u03b9 R M)))\n (CliffordAlgebra.\u03b9 0))\n m =\n \u2191(LinearMap.comp (AlgHom.toLinearMap (AlgHom.id R (ExteriorAlgebra R M))) (CliffordAlgebra.\u03b9 0)) m\n[PROOFSTEP]\ndsimp only [LinearMap.comp_apply, AlgHom.toLinearMap_apply, AlgHom.comp_apply, AlgHom.id_apply, GradedAlgebra.lift\u03b9]\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nm : M\n\u22a2 \u2191(DirectSum.coeAlgHom fun x => LinearMap.range (\u03b9 R) ^ x)\n (\u2191(\u2191(lift R)\n { val := GradedAlgebra.\u03b9 R M,\n property := (_ : \u2200 (m : M), \u2191(GradedAlgebra.\u03b9 R M) m * \u2191(GradedAlgebra.\u03b9 R M) m = 0) })\n (\u2191(CliffordAlgebra.\u03b9 0) m)) =\n \u2191(CliffordAlgebra.\u03b9 0) m\n[PROOFSTEP]\nrw [lift_\u03b9_apply, GradedAlgebra.\u03b9_apply R M, DirectSum.coeAlgHom_of, Subtype.coe_mk]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 \u2200 (i : \u2115) (x : { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }),\n \u2191(GradedAlgebra.lift\u03b9 R M) \u2191x = \u2191(DirectSum.of (fun i => { x // x \u2208 LinearMap.range (\u03b9 R) ^ i }) i) x\n[PROOFSTEP]\napply GradedAlgebra.lift\u03b9_eq R M\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.ExteriorAlgebra.Grading", "llama_tokens": 6096, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702880639791, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.43505870886959175}} {"text": "[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nX Y : D\nf : X \u27f6 Y\n\u22a2 (R\u2081 \u22d9 G).map f \u226b\n (fun X =>\n NatTrans.app adj\u2082.unit (G.obj (R\u2081.obj X)) \u226b\n R\u2082.map (NatTrans.app h (R\u2081.obj X) \u226b H.map (NatTrans.app adj\u2081.counit X)))\n Y =\n (fun X =>\n NatTrans.app adj\u2082.unit (G.obj (R\u2081.obj X)) \u226b\n R\u2082.map (NatTrans.app h (R\u2081.obj X) \u226b H.map (NatTrans.app adj\u2081.counit X)))\n X \u226b\n (H \u22d9 R\u2082).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nX Y : D\nf : X \u27f6 Y\n\u22a2 G.map (R\u2081.map f) \u226b\n NatTrans.app adj\u2082.unit (G.obj (R\u2081.obj Y)) \u226b\n R\u2082.map (NatTrans.app h (R\u2081.obj Y) \u226b H.map (NatTrans.app adj\u2081.counit Y)) =\n (NatTrans.app adj\u2082.unit (G.obj (R\u2081.obj X)) \u226b\n R\u2082.map (NatTrans.app h (R\u2081.obj X) \u226b H.map (NatTrans.app adj\u2081.counit X))) \u226b\n R\u2082.map (H.map f)\n[PROOFSTEP]\nrw [assoc, \u2190 R\u2082.map_comp, assoc, \u2190 H.map_comp, \u2190 adj\u2081.counit_naturality, H.map_comp, \u2190 Functor.comp_map L\u2081, \u2190\n h.naturality_assoc]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nX Y : D\nf : X \u27f6 Y\n\u22a2 G.map (R\u2081.map f) \u226b\n NatTrans.app adj\u2082.unit (G.obj (R\u2081.obj Y)) \u226b\n R\u2082.map (NatTrans.app h (R\u2081.obj Y) \u226b H.map (NatTrans.app adj\u2081.counit Y)) =\n NatTrans.app adj\u2082.unit (G.obj (R\u2081.obj X)) \u226b\n R\u2082.map ((G \u22d9 L\u2082).map (R\u2081.map f) \u226b NatTrans.app h (R\u2081.obj Y) \u226b H.map (NatTrans.app adj\u2081.counit Y))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : R\u2081 \u22d9 G \u27f6 H \u22d9 R\u2082\nX Y : C\nf : X \u27f6 Y\n\u22a2 (G \u22d9 L\u2082).map f \u226b\n (fun X =>\n L\u2082.map (G.map (NatTrans.app adj\u2081.unit X) \u226b NatTrans.app h (L\u2081.obj X)) \u226b\n NatTrans.app adj\u2082.counit (H.obj (L\u2081.obj X)))\n Y =\n (fun X =>\n L\u2082.map (G.map (NatTrans.app adj\u2081.unit X) \u226b NatTrans.app h (L\u2081.obj X)) \u226b\n NatTrans.app adj\u2082.counit (H.obj (L\u2081.obj X)))\n X \u226b\n (L\u2081 \u22d9 H).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : R\u2081 \u22d9 G \u27f6 H \u22d9 R\u2082\nX Y : C\nf : X \u27f6 Y\n\u22a2 L\u2082.map (G.map f) \u226b\n L\u2082.map (G.map (NatTrans.app adj\u2081.unit Y) \u226b NatTrans.app h (L\u2081.obj Y)) \u226b\n NatTrans.app adj\u2082.counit (H.obj (L\u2081.obj Y)) =\n (L\u2082.map (G.map (NatTrans.app adj\u2081.unit X) \u226b NatTrans.app h (L\u2081.obj X)) \u226b\n NatTrans.app adj\u2082.counit (H.obj (L\u2081.obj X))) \u226b\n H.map (L\u2081.map f)\n[PROOFSTEP]\nrw [\u2190 L\u2082.map_comp_assoc, \u2190 G.map_comp_assoc, \u2190 adj\u2081.unit_naturality, G.map_comp_assoc, \u2190 Functor.comp_map, h.naturality]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : R\u2081 \u22d9 G \u27f6 H \u22d9 R\u2082\nX Y : C\nf : X \u27f6 Y\n\u22a2 L\u2082.map (G.map (NatTrans.app adj\u2081.unit X) \u226b NatTrans.app h (L\u2081.obj X) \u226b (H \u22d9 R\u2082).map (L\u2081.map f)) \u226b\n NatTrans.app adj\u2082.counit (H.obj (L\u2081.obj Y)) =\n (L\u2082.map (G.map (NatTrans.app adj\u2081.unit X) \u226b NatTrans.app h (L\u2081.obj X)) \u226b\n NatTrans.app adj\u2082.counit (H.obj (L\u2081.obj X))) \u226b\n H.map (L\u2081.map f)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\n\u22a2 (fun h =>\n NatTrans.mk fun X =>\n L\u2082.map (G.map (NatTrans.app adj\u2081.unit X) \u226b NatTrans.app h (L\u2081.obj X)) \u226b\n NatTrans.app adj\u2082.counit (H.obj (L\u2081.obj X)))\n ((fun h =>\n NatTrans.mk fun X =>\n NatTrans.app adj\u2082.unit (G.obj (R\u2081.obj X)) \u226b\n R\u2082.map (NatTrans.app h (R\u2081.obj X) \u226b H.map (NatTrans.app adj\u2081.counit X)))\n h) =\n h\n[PROOFSTEP]\next X\n[GOAL]\ncase w.h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nX : C\n\u22a2 NatTrans.app\n ((fun h =>\n NatTrans.mk fun X =>\n L\u2082.map (G.map (NatTrans.app adj\u2081.unit X) \u226b NatTrans.app h (L\u2081.obj X)) \u226b\n NatTrans.app adj\u2082.counit (H.obj (L\u2081.obj X)))\n ((fun h =>\n NatTrans.mk fun X =>\n NatTrans.app adj\u2082.unit (G.obj (R\u2081.obj X)) \u226b\n R\u2082.map (NatTrans.app h (R\u2081.obj X) \u226b H.map (NatTrans.app adj\u2081.counit X)))\n h))\n X =\n NatTrans.app h X\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nX : C\n\u22a2 L\u2082.map\n (G.map (NatTrans.app adj\u2081.unit X) \u226b\n NatTrans.app adj\u2082.unit (G.obj (R\u2081.obj (L\u2081.obj X))) \u226b\n R\u2082.map (NatTrans.app h (R\u2081.obj (L\u2081.obj X)) \u226b H.map (NatTrans.app adj\u2081.counit (L\u2081.obj X)))) \u226b\n NatTrans.app adj\u2082.counit (H.obj (L\u2081.obj X)) =\n NatTrans.app h X\n[PROOFSTEP]\nsimp only [L\u2082.map_comp, assoc, adj\u2082.counit_naturality, adj\u2082.left_triangle_components_assoc, \u2190 Functor.comp_map G L\u2082,\n h.naturality_assoc, Functor.comp_map L\u2081, \u2190 H.map_comp, adj\u2081.left_triangle_components]\n[GOAL]\ncase w.h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nX : C\n\u22a2 NatTrans.app h X \u226b H.map (\ud835\udfd9 (L\u2081.obj X)) = NatTrans.app h X\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nX : C\n\u22a2 NatTrans.app h X \u226b H.map (\ud835\udfd9 (L\u2081.obj X)) = NatTrans.app h X\n[PROOFSTEP]\nsimp only [id_comp, \u2190 Functor.comp_map, \u2190 Functor.comp_obj, NatTrans.naturality_assoc]\n[GOAL]\ncase w.h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nX : C\n\u22a2 NatTrans.app h X \u226b H.map (\ud835\udfd9 (L\u2081.obj X)) = NatTrans.app h X\n[PROOFSTEP]\nsimp only [Functor.comp_obj, Functor.comp_map, \u2190 Functor.map_comp]\n[GOAL]\ncase w.h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nX : C\n\u22a2 NatTrans.app h X \u226b H.map (\ud835\udfd9 (L\u2081.obj X)) = NatTrans.app h X\n[PROOFSTEP]\nhave :\n Prefunctor.map L\u2081.toPrefunctor (NatTrans.app adj\u2081.unit X) \u226b\n NatTrans.app adj\u2081.counit (Prefunctor.obj L\u2081.toPrefunctor X) =\n \ud835\udfd9 _ :=\n by simp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nX : C\n\u22a2 L\u2081.map (NatTrans.app adj\u2081.unit X) \u226b NatTrans.app adj\u2081.counit (L\u2081.obj X) = \ud835\udfd9 (L\u2081.obj ((\ud835\udfed C).obj X))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase w.h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nX : C\nthis : L\u2081.map (NatTrans.app adj\u2081.unit X) \u226b NatTrans.app adj\u2081.counit (L\u2081.obj X) = \ud835\udfd9 (L\u2081.obj ((\ud835\udfed C).obj X))\n\u22a2 NatTrans.app h X \u226b H.map (\ud835\udfd9 (L\u2081.obj X)) = NatTrans.app h X\n[PROOFSTEP]\nsimp [this]\n -- See library note [dsimp, simp].\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : R\u2081 \u22d9 G \u27f6 H \u22d9 R\u2082\n\u22a2 (fun h =>\n NatTrans.mk fun X =>\n NatTrans.app adj\u2082.unit (G.obj (R\u2081.obj X)) \u226b\n R\u2082.map (NatTrans.app h (R\u2081.obj X) \u226b H.map (NatTrans.app adj\u2081.counit X)))\n ((fun h =>\n NatTrans.mk fun X =>\n L\u2082.map (G.map (NatTrans.app adj\u2081.unit X) \u226b NatTrans.app h (L\u2081.obj X)) \u226b\n NatTrans.app adj\u2082.counit (H.obj (L\u2081.obj X)))\n h) =\n h\n[PROOFSTEP]\next X\n[GOAL]\ncase w.h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : R\u2081 \u22d9 G \u27f6 H \u22d9 R\u2082\nX : D\n\u22a2 NatTrans.app\n ((fun h =>\n NatTrans.mk fun X =>\n NatTrans.app adj\u2082.unit (G.obj (R\u2081.obj X)) \u226b\n R\u2082.map (NatTrans.app h (R\u2081.obj X) \u226b H.map (NatTrans.app adj\u2081.counit X)))\n ((fun h =>\n NatTrans.mk fun X =>\n L\u2082.map (G.map (NatTrans.app adj\u2081.unit X) \u226b NatTrans.app h (L\u2081.obj X)) \u226b\n NatTrans.app adj\u2082.counit (H.obj (L\u2081.obj X)))\n h))\n X =\n NatTrans.app h X\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nh : R\u2081 \u22d9 G \u27f6 H \u22d9 R\u2082\nX : D\n\u22a2 NatTrans.app adj\u2082.unit (G.obj (R\u2081.obj X)) \u226b\n R\u2082.map\n ((L\u2082.map (G.map (NatTrans.app adj\u2081.unit (R\u2081.obj X)) \u226b NatTrans.app h (L\u2081.obj (R\u2081.obj X))) \u226b\n NatTrans.app adj\u2082.counit (H.obj (L\u2081.obj (R\u2081.obj X)))) \u226b\n H.map (NatTrans.app adj\u2081.counit X)) =\n NatTrans.app h X\n[PROOFSTEP]\nsimp [-Functor.comp_map, \u2190 Functor.comp_map H, Functor.comp_map R\u2081, -NatTrans.naturality, \u2190 h.naturality,\n -Functor.map_comp, \u2190 Functor.map_comp_assoc G, R\u2082.map_comp]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nf : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nY : D\n\u22a2 L\u2082.map (NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) f) Y) \u226b NatTrans.app adj\u2082.counit (H.obj Y) =\n NatTrans.app f (R\u2081.obj Y) \u226b H.map (NatTrans.app adj\u2081.counit Y)\n[PROOFSTEP]\nerw [Functor.map_comp]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nf : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nY : D\n\u22a2 (L\u2082.map (NatTrans.app adj\u2082.unit (G.obj (R\u2081.obj Y))) \u226b\n L\u2082.map (R\u2082.map (NatTrans.app f (R\u2081.obj Y) \u226b H.map (NatTrans.app adj\u2081.counit Y)))) \u226b\n NatTrans.app adj\u2082.counit (H.obj Y) =\n NatTrans.app f (R\u2081.obj Y) \u226b H.map (NatTrans.app adj\u2081.counit Y)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nf : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nX : C\n\u22a2 G.map (NatTrans.app adj\u2081.unit X) \u226b NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) f) (L\u2081.obj X) =\n NatTrans.app adj\u2082.unit (G.obj ((\ud835\udfed C).obj X)) \u226b R\u2082.map (NatTrans.app f ((\ud835\udfed C).obj X))\n[PROOFSTEP]\ndsimp [transferNatTrans]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nf : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nX : C\n\u22a2 G.map (NatTrans.app adj\u2081.unit X) \u226b\n NatTrans.app adj\u2082.unit (G.obj (R\u2081.obj (L\u2081.obj X))) \u226b\n R\u2082.map (NatTrans.app f (R\u2081.obj (L\u2081.obj X)) \u226b H.map (NatTrans.app adj\u2081.counit (L\u2081.obj X))) =\n NatTrans.app adj\u2082.unit (G.obj X) \u226b R\u2082.map (NatTrans.app f X)\n[PROOFSTEP]\nrw [\u2190 adj\u2082.unit_naturality_assoc, \u2190 R\u2082.map_comp, \u2190 Functor.comp_map G L\u2082, f.naturality_assoc, Functor.comp_map, \u2190\n H.map_comp]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nf : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nX : C\n\u22a2 NatTrans.app adj\u2082.unit (G.obj X) \u226b\n R\u2082.map (NatTrans.app f X \u226b H.map (L\u2081.map (NatTrans.app adj\u2081.unit X) \u226b NatTrans.app adj\u2081.counit (L\u2081.obj X))) =\n NatTrans.app adj\u2082.unit (G.obj X) \u226b R\u2082.map (NatTrans.app f X)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\nF : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : Category.{v\u2084, u\u2084} F\nG : C \u2964 E\nH : D \u2964 F\nL\u2081 : C \u2964 D\nR\u2081 : D \u2964 C\nL\u2082 : E \u2964 F\nR\u2082 : F \u2964 E\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nf : G \u22d9 L\u2082 \u27f6 L\u2081 \u22d9 H\nX : C\n\u22a2 NatTrans.app adj\u2082.unit (G.obj X) \u226b\n R\u2082.map (NatTrans.app f X \u226b H.map (L\u2081.map (NatTrans.app adj\u2081.unit X) \u226b NatTrans.app adj\u2081.counit (L\u2081.obj X))) =\n NatTrans.app adj\u2082.unit (G.obj X) \u226b R\u2082.map (NatTrans.app f X)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\nX : D\n\u22a2 L\u2082.map (NatTrans.app (\u2191(transferNatTransSelf adj\u2081 adj\u2082) f) X) \u226b NatTrans.app adj\u2082.counit X =\n NatTrans.app f (R\u2081.obj X) \u226b NatTrans.app adj\u2081.counit X\n[PROOFSTEP]\ndsimp [transferNatTransSelf]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\nX : D\n\u22a2 L\u2082.map\n (\ud835\udfd9 (R\u2081.obj X) \u226b\n NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv))\n X \u226b\n \ud835\udfd9 (R\u2082.obj X)) \u226b\n NatTrans.app adj\u2082.counit X =\n NatTrans.app f (R\u2081.obj X) \u226b NatTrans.app adj\u2081.counit X\n[PROOFSTEP]\nrw [id_comp, comp_id]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\nX : D\n\u22a2 L\u2082.map\n (NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv))\n X) \u226b\n NatTrans.app adj\u2082.counit X =\n NatTrans.app f (R\u2081.obj X) \u226b NatTrans.app adj\u2081.counit X\n[PROOFSTEP]\nhave := transferNatTrans_counit adj\u2081 adj\u2082 (L\u2082.leftUnitor.hom \u226b f \u226b L\u2081.rightUnitor.inv) X\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\nX : D\nthis :\n L\u2082.map\n (NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv))\n X) \u226b\n NatTrans.app adj\u2082.counit ((\ud835\udfed D).obj X) =\n NatTrans.app ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv) (R\u2081.obj X) \u226b\n (\ud835\udfed D).map (NatTrans.app adj\u2081.counit X)\n\u22a2 L\u2082.map\n (NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv))\n X) \u226b\n NatTrans.app adj\u2082.counit X =\n NatTrans.app f (R\u2081.obj X) \u226b NatTrans.app adj\u2081.counit X\n[PROOFSTEP]\ndsimp at this \n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\nX : D\nthis :\n L\u2082.map\n (NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv))\n X) \u226b\n NatTrans.app adj\u2082.counit X =\n (\ud835\udfd9 (L\u2082.obj (R\u2081.obj X)) \u226b NatTrans.app f (R\u2081.obj X) \u226b \ud835\udfd9 (L\u2081.obj (R\u2081.obj X))) \u226b NatTrans.app adj\u2081.counit X\n\u22a2 L\u2082.map\n (NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv))\n X) \u226b\n NatTrans.app adj\u2082.counit X =\n NatTrans.app f (R\u2081.obj X) \u226b NatTrans.app adj\u2081.counit X\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\nX : D\nthis :\n L\u2082.map\n (NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv))\n X) \u226b\n NatTrans.app adj\u2082.counit X =\n (\ud835\udfd9 (L\u2082.obj (R\u2081.obj X)) \u226b NatTrans.app f (R\u2081.obj X) \u226b \ud835\udfd9 (L\u2081.obj (R\u2081.obj X))) \u226b NatTrans.app adj\u2081.counit X\n\u22a2 (\ud835\udfd9 (L\u2082.obj (R\u2081.obj X)) \u226b NatTrans.app f (R\u2081.obj X) \u226b \ud835\udfd9 (L\u2081.obj (R\u2081.obj X))) \u226b NatTrans.app adj\u2081.counit X =\n NatTrans.app f (R\u2081.obj X) \u226b NatTrans.app adj\u2081.counit X\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\nX : C\n\u22a2 NatTrans.app adj\u2081.unit X \u226b NatTrans.app (\u2191(transferNatTransSelf adj\u2081 adj\u2082) f) (L\u2081.obj X) =\n NatTrans.app adj\u2082.unit X \u226b R\u2082.map (NatTrans.app f X)\n[PROOFSTEP]\ndsimp [transferNatTransSelf]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\nX : C\n\u22a2 NatTrans.app adj\u2081.unit X \u226b\n \ud835\udfd9 (R\u2081.obj (L\u2081.obj X)) \u226b\n NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv))\n (L\u2081.obj X) \u226b\n \ud835\udfd9 (R\u2082.obj (L\u2081.obj X)) =\n NatTrans.app adj\u2082.unit X \u226b R\u2082.map (NatTrans.app f X)\n[PROOFSTEP]\nrw [id_comp, comp_id]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\nX : C\n\u22a2 NatTrans.app adj\u2081.unit X \u226b\n NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv))\n (L\u2081.obj X) =\n NatTrans.app adj\u2082.unit X \u226b R\u2082.map (NatTrans.app f X)\n[PROOFSTEP]\nhave := unit_transferNatTrans adj\u2081 adj\u2082 (L\u2082.leftUnitor.hom \u226b f \u226b L\u2081.rightUnitor.inv) X\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\nX : C\nthis :\n (\ud835\udfed C).map (NatTrans.app adj\u2081.unit X) \u226b\n NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv))\n (L\u2081.obj X) =\n NatTrans.app adj\u2082.unit ((\ud835\udfed C).obj ((\ud835\udfed C).obj X)) \u226b\n R\u2082.map (NatTrans.app ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv) ((\ud835\udfed C).obj X))\n\u22a2 NatTrans.app adj\u2081.unit X \u226b\n NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv))\n (L\u2081.obj X) =\n NatTrans.app adj\u2082.unit X \u226b R\u2082.map (NatTrans.app f X)\n[PROOFSTEP]\ndsimp at this \n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\nX : C\nthis :\n NatTrans.app adj\u2081.unit X \u226b\n NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv))\n (L\u2081.obj X) =\n NatTrans.app adj\u2082.unit X \u226b R\u2082.map (\ud835\udfd9 (L\u2082.obj X) \u226b NatTrans.app f X \u226b \ud835\udfd9 (L\u2081.obj X))\n\u22a2 NatTrans.app adj\u2081.unit X \u226b\n NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv))\n (L\u2081.obj X) =\n NatTrans.app adj\u2082.unit X \u226b R\u2082.map (NatTrans.app f X)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\nX : C\nthis :\n NatTrans.app adj\u2081.unit X \u226b\n NatTrans.app (\u2191(transferNatTrans adj\u2081 adj\u2082) ((Functor.leftUnitor L\u2082).hom \u226b f \u226b (Functor.rightUnitor L\u2081).inv))\n (L\u2081.obj X) =\n NatTrans.app adj\u2082.unit X \u226b R\u2082.map (\ud835\udfd9 (L\u2082.obj X) \u226b NatTrans.app f X \u226b \ud835\udfd9 (L\u2081.obj X))\n\u22a2 NatTrans.app adj\u2082.unit X \u226b R\u2082.map (\ud835\udfd9 (L\u2082.obj X) \u226b NatTrans.app f X \u226b \ud835\udfd9 (L\u2081.obj X)) =\n NatTrans.app adj\u2082.unit X \u226b R\u2082.map (NatTrans.app f X)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\n\u22a2 \u2191(transferNatTransSelf adj\u2081 adj\u2081) (\ud835\udfd9 L\u2081) = \ud835\udfd9 R\u2081\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nx\u271d : D\n\u22a2 NatTrans.app (\u2191(transferNatTransSelf adj\u2081 adj\u2081) (\ud835\udfd9 L\u2081)) x\u271d = NatTrans.app (\ud835\udfd9 R\u2081) x\u271d\n[PROOFSTEP]\ndsimp [transferNatTransSelf, transferNatTrans]\n[GOAL]\ncase w.h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nx\u271d : D\n\u22a2 \ud835\udfd9 (R\u2081.obj x\u271d) \u226b\n (NatTrans.app adj\u2081.unit (R\u2081.obj x\u271d) \u226b\n R\u2081.map\n ((\ud835\udfd9 (L\u2081.obj (R\u2081.obj x\u271d)) \u226b \ud835\udfd9 (L\u2081.obj (R\u2081.obj x\u271d)) \u226b \ud835\udfd9 (L\u2081.obj (R\u2081.obj x\u271d))) \u226b\n NatTrans.app adj\u2081.counit x\u271d)) \u226b\n \ud835\udfd9 (R\u2081.obj x\u271d) =\n \ud835\udfd9 (R\u2081.obj x\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\n\u22a2 \u2191(transferNatTransSelf adj\u2081 adj\u2081).symm (\ud835\udfd9 R\u2081) = \ud835\udfd9 L\u2081\n[PROOFSTEP]\nrw [Equiv.symm_apply_eq]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\n\u22a2 \ud835\udfd9 R\u2081 = \u2191(transferNatTransSelf adj\u2081 adj\u2081) (\ud835\udfd9 L\u2081)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\ng : L\u2083 \u27f6 L\u2082\n\u22a2 \u2191(transferNatTransSelf adj\u2081 adj\u2082) f \u226b \u2191(transferNatTransSelf adj\u2082 adj\u2083) g = \u2191(transferNatTransSelf adj\u2081 adj\u2083) (g \u226b f)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\ng : L\u2083 \u27f6 L\u2082\nx\u271d : D\n\u22a2 NatTrans.app (\u2191(transferNatTransSelf adj\u2081 adj\u2082) f \u226b \u2191(transferNatTransSelf adj\u2082 adj\u2083) g) x\u271d =\n NatTrans.app (\u2191(transferNatTransSelf adj\u2081 adj\u2083) (g \u226b f)) x\u271d\n[PROOFSTEP]\ndsimp [transferNatTransSelf, transferNatTrans]\n[GOAL]\ncase w.h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\ng : L\u2083 \u27f6 L\u2082\nx\u271d : D\n\u22a2 (\ud835\udfd9 (R\u2081.obj x\u271d) \u226b\n (NatTrans.app adj\u2082.unit (R\u2081.obj x\u271d) \u226b\n R\u2082.map\n ((\ud835\udfd9 (L\u2082.obj (R\u2081.obj x\u271d)) \u226b NatTrans.app f (R\u2081.obj x\u271d) \u226b \ud835\udfd9 (L\u2081.obj (R\u2081.obj x\u271d))) \u226b\n NatTrans.app adj\u2081.counit x\u271d)) \u226b\n \ud835\udfd9 (R\u2082.obj x\u271d)) \u226b\n \ud835\udfd9 (R\u2082.obj x\u271d) \u226b\n (NatTrans.app adj\u2083.unit (R\u2082.obj x\u271d) \u226b\n R\u2083.map\n ((\ud835\udfd9 (L\u2083.obj (R\u2082.obj x\u271d)) \u226b NatTrans.app g (R\u2082.obj x\u271d) \u226b \ud835\udfd9 (L\u2082.obj (R\u2082.obj x\u271d))) \u226b\n NatTrans.app adj\u2082.counit x\u271d)) \u226b\n \ud835\udfd9 (R\u2083.obj x\u271d) =\n \ud835\udfd9 (R\u2081.obj x\u271d) \u226b\n (NatTrans.app adj\u2083.unit (R\u2081.obj x\u271d) \u226b\n R\u2083.map\n ((\ud835\udfd9 (L\u2083.obj (R\u2081.obj x\u271d)) \u226b\n (NatTrans.app g (R\u2081.obj x\u271d) \u226b NatTrans.app f (R\u2081.obj x\u271d)) \u226b \ud835\udfd9 (L\u2081.obj (R\u2081.obj x\u271d))) \u226b\n NatTrans.app adj\u2081.counit x\u271d)) \u226b\n \ud835\udfd9 (R\u2083.obj x\u271d)\n[PROOFSTEP]\nsimp only [id_comp, comp_id]\n[GOAL]\ncase w.h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\ng : L\u2083 \u27f6 L\u2082\nx\u271d : D\n\u22a2 (NatTrans.app adj\u2082.unit (R\u2081.obj x\u271d) \u226b R\u2082.map (NatTrans.app f (R\u2081.obj x\u271d) \u226b NatTrans.app adj\u2081.counit x\u271d)) \u226b\n NatTrans.app adj\u2083.unit (R\u2082.obj x\u271d) \u226b R\u2083.map (NatTrans.app g (R\u2082.obj x\u271d) \u226b NatTrans.app adj\u2082.counit x\u271d) =\n NatTrans.app adj\u2083.unit (R\u2081.obj x\u271d) \u226b\n R\u2083.map ((NatTrans.app g (R\u2081.obj x\u271d) \u226b NatTrans.app f (R\u2081.obj x\u271d)) \u226b NatTrans.app adj\u2081.counit x\u271d)\n[PROOFSTEP]\nrw [\u2190 adj\u2083.unit_naturality_assoc, \u2190 R\u2083.map_comp, g.naturality_assoc, L\u2082.map_comp, assoc, adj\u2082.counit_naturality,\n adj\u2082.left_triangle_components_assoc, assoc]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nL R : C \u2964 C\nadj : L \u22a3 R\nf : \ud835\udfed C \u27f6 L\nX : C\n\u22a2 NatTrans.app (\u2191(transferNatTransSelf adj Adjunction.id) f) X = NatTrans.app f (R.obj X) \u226b NatTrans.app adj.counit X\n[PROOFSTEP]\ndsimp [transferNatTransSelf, transferNatTrans, Adjunction.id]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nL R : C \u2964 C\nadj : L \u22a3 R\nf : \ud835\udfed C \u27f6 L\nX : C\n\u22a2 \ud835\udfd9 (R.obj X) \u226b\n (\ud835\udfd9 (R.obj X) \u226b (\ud835\udfd9 (R.obj X) \u226b NatTrans.app f (R.obj X) \u226b \ud835\udfd9 (L.obj (R.obj X))) \u226b NatTrans.app adj.counit X) \u226b \ud835\udfd9 X =\n NatTrans.app f (R.obj X) \u226b NatTrans.app adj.counit X\n[PROOFSTEP]\nsimp only [comp_id, id_comp]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nL R : C \u2964 C\nadj : L \u22a3 R\ng : R \u27f6 \ud835\udfed C\nX : C\n\u22a2 NatTrans.app (\u2191(transferNatTransSelf adj Adjunction.id).symm g) X = NatTrans.app adj.unit X \u226b NatTrans.app g (L.obj X)\n[PROOFSTEP]\ndsimp [transferNatTransSelf, transferNatTrans, Adjunction.id]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nL R : C \u2964 C\nadj : L \u22a3 R\ng : R \u27f6 \ud835\udfed C\nX : C\n\u22a2 \ud835\udfd9 X \u226b\n ((NatTrans.app adj.unit X \u226b \ud835\udfd9 (R.obj (L.obj X)) \u226b NatTrans.app g (L.obj X) \u226b \ud835\udfd9 (L.obj X)) \u226b \ud835\udfd9 (L.obj X)) \u226b\n \ud835\udfd9 (L.obj X) =\n NatTrans.app adj.unit X \u226b NatTrans.app g (L.obj X)\n[PROOFSTEP]\nsimp only [comp_id, id_comp]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : R\u2082 \u27f6 R\u2081\ng : R\u2083 \u27f6 R\u2082\n\u22a2 \u2191(transferNatTransSelf adj\u2082 adj\u2081).symm f \u226b \u2191(transferNatTransSelf adj\u2083 adj\u2082).symm g =\n \u2191(transferNatTransSelf adj\u2083 adj\u2081).symm (g \u226b f)\n[PROOFSTEP]\nrw [Equiv.eq_symm_apply, \u2190 transferNatTransSelf_comp _ adj\u2082]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : R\u2082 \u27f6 R\u2081\ng : R\u2083 \u27f6 R\u2082\n\u22a2 \u2191(transferNatTransSelf adj\u2083 adj\u2082) (\u2191(transferNatTransSelf adj\u2083 adj\u2082).symm g) \u226b\n \u2191(transferNatTransSelf adj\u2082 adj\u2081) (\u2191(transferNatTransSelf adj\u2082 adj\u2081).symm f) =\n g \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\ng : L\u2081 \u27f6 L\u2082\ngf : g \u226b f = \ud835\udfd9 L\u2081\n\u22a2 \u2191(transferNatTransSelf adj\u2081 adj\u2082) f \u226b \u2191(transferNatTransSelf adj\u2082 adj\u2081) g = \ud835\udfd9 R\u2081\n[PROOFSTEP]\nrw [transferNatTransSelf_comp, gf, transferNatTransSelf_id]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : R\u2081 \u27f6 R\u2082\ng : R\u2082 \u27f6 R\u2081\ngf : g \u226b f = \ud835\udfd9 R\u2082\n\u22a2 \u2191(transferNatTransSelf adj\u2081 adj\u2082).symm f \u226b \u2191(transferNatTransSelf adj\u2082 adj\u2081).symm g = \ud835\udfd9 L\u2082\n[PROOFSTEP]\nrw [transferNatTransSelf_symm_comp, gf, transferNatTransSelf_symm_id]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\ninst\u271d : IsIso f\n\u22a2 inv f \u226b f = \ud835\udfd9 L\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\ninst\u271d : IsIso f\n\u22a2 f \u226b inv f = \ud835\udfd9 L\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : R\u2081 \u27f6 R\u2082\ninst\u271d : IsIso f\n\u22a2 inv f \u226b f = \ud835\udfd9 R\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : R\u2081 \u27f6 R\u2082\ninst\u271d : IsIso f\n\u22a2 f \u226b inv f = \ud835\udfd9 R\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\ninst\u271d : IsIso (\u2191(transferNatTransSelf adj\u2081 adj\u2082) f)\n\u22a2 IsIso f\n[PROOFSTEP]\nsuffices IsIso ((transferNatTransSelf adj\u2081 adj\u2082).symm (transferNatTransSelf adj\u2081 adj\u2082 f)) by simpa using this\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\ninst\u271d : IsIso (\u2191(transferNatTransSelf adj\u2081 adj\u2082) f)\nthis : IsIso (\u2191(transferNatTransSelf adj\u2081 adj\u2082).symm (\u2191(transferNatTransSelf adj\u2081 adj\u2082) f))\n\u22a2 IsIso f\n[PROOFSTEP]\nsimpa using this\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : L\u2082 \u27f6 L\u2081\ninst\u271d : IsIso (\u2191(transferNatTransSelf adj\u2081 adj\u2082) f)\n\u22a2 IsIso (\u2191(transferNatTransSelf adj\u2081 adj\u2082).symm (\u2191(transferNatTransSelf adj\u2081 adj\u2082) f))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : R\u2081 \u27f6 R\u2082\ninst\u271d : IsIso (\u2191(transferNatTransSelf adj\u2081 adj\u2082).symm f)\n\u22a2 IsIso f\n[PROOFSTEP]\nsuffices IsIso ((transferNatTransSelf adj\u2081 adj\u2082) ((transferNatTransSelf adj\u2081 adj\u2082).symm f)) by simpa using this\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : R\u2081 \u27f6 R\u2082\ninst\u271d : IsIso (\u2191(transferNatTransSelf adj\u2081 adj\u2082).symm f)\nthis : IsIso (\u2191(transferNatTransSelf adj\u2081 adj\u2082) (\u2191(transferNatTransSelf adj\u2081 adj\u2082).symm f))\n\u22a2 IsIso f\n[PROOFSTEP]\nsimpa using this\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nL\u2081 L\u2082 L\u2083 : C \u2964 D\nR\u2081 R\u2082 R\u2083 : D \u2964 C\nadj\u2081 : L\u2081 \u22a3 R\u2081\nadj\u2082 : L\u2082 \u22a3 R\u2082\nadj\u2083 : L\u2083 \u22a3 R\u2083\nf : R\u2081 \u27f6 R\u2082\ninst\u271d : IsIso (\u2191(transferNatTransSelf adj\u2081 adj\u2082).symm f)\n\u22a2 IsIso (\u2191(transferNatTransSelf adj\u2081 adj\u2082) (\u2191(transferNatTransSelf adj\u2081 adj\u2082).symm f))\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Adjunction.Mates", "llama_tokens": 18011, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430520409023, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.43503718631525046}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\ninduction' f using MeasureTheory.SimpleFunc.induction with c s hs f\u2081 f\u2082 _ h\u2081 h\u2082 generalizing \u03b5\n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nlet f := SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)\n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nby_cases h : \u222b\u207b x, f x \u2202\u03bc = \u22a4\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun _ => c, fun x => _, lowerSemicontinuous_const, by simp only [_root_.top_add, le_top, h]\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), \u2191((fun x => c) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nsimp only [_root_.top_add, le_top, h]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nx : \u03b1\n\u22a2 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 (fun x => c) x\n[PROOFSTEP]\nsimp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,\n SimpleFunc.coe_piecewise]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nx : \u03b1\n\u22a2 Set.piecewise s (Function.const \u03b1 c) 0 x \u2264 c\n[PROOFSTEP]\nexact Set.indicator_le_self _ _ _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nby_cases hc : c = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : c = 0\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun _ => 0, _, lowerSemicontinuous_const, _\u27e9\n[GOAL]\ncase pos.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : c = 0\n\u22a2 \u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 (fun x => 0) x\n[PROOFSTEP]\nclassical simp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff, eq_self_iff_true,\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, le_zero_iff]\n[GOAL]\ncase pos.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : c = 0\n\u22a2 \u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 (fun x => 0) x\n[PROOFSTEP]\nsimp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff, eq_self_iff_true,\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, le_zero_iff]\n[GOAL]\ncase pos.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : c = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191((fun x => 0) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nsimp only [lintegral_const, zero_mul, zero_le, ENNReal.coe_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nhave : \u03bc s < \u03bc s + \u03b5 / c :=\n by\n have : (0 : \u211d\u22650\u221e) < \u03b5 / c := ENNReal.div_pos_iff.2 \u27e8\u03b50, ENNReal.coe_ne_top\u27e9\n simpa using ENNReal.add_lt_add_left ?aux this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\n\u22a2 \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n[PROOFSTEP]\nhave : (0 : \u211d\u22650\u221e) < \u03b5 / c := ENNReal.div_pos_iff.2 \u27e8\u03b50, ENNReal.coe_ne_top\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis : 0 < \u03b5 / \u2191c\n\u22a2 \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n[PROOFSTEP]\nsimpa using ENNReal.add_lt_add_left ?aux this\n[GOAL]\ncase aux\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis : 0 < \u03b5 / \u2191c\n\u22a2 \u2191\u2191\u03bc s \u2260 \u22a4\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\ncase aux =>\n classical simpa [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, Function.const_apply, lintegral_const,\n ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top, MeasurableSet.univ, ENNReal.mul_eq_top,\n SimpleFunc.const_zero, or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, false_and_iff, restrict_apply] using h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis : 0 < \u03b5 / \u2191c\n\u22a2 \u2191\u2191\u03bc s \u2260 \u22a4\n[PROOFSTEP]\ncase aux =>\n classical simpa [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, Function.const_apply, lintegral_const,\n ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top, MeasurableSet.univ, ENNReal.mul_eq_top,\n SimpleFunc.const_zero, or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, false_and_iff, restrict_apply] using h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis : 0 < \u03b5 / \u2191c\n\u22a2 \u2191\u2191\u03bc s \u2260 \u22a4\n[PROOFSTEP]\nclassical simpa [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, Function.const_apply, lintegral_const,\n ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top, MeasurableSet.univ, ENNReal.mul_eq_top,\n SimpleFunc.const_zero, or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, false_and_iff, restrict_apply] using h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis : 0 < \u03b5 / \u2191c\n\u22a2 \u2191\u2191\u03bc s \u2260 \u22a4\n[PROOFSTEP]\nsimpa [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, Function.const_apply, lintegral_const,\n ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top, MeasurableSet.univ, ENNReal.mul_eq_top,\n SimpleFunc.const_zero, or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, false_and_iff, restrict_apply] using h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nobtain \u27e8u, su, u_open, \u03bcu\u27e9 : \u2203 (u : _), u \u2287 s \u2227 IsOpen u \u2227 \u03bc u < \u03bc s + \u03b5 / c := s.exists_isOpen_lt_of_lt _ this\n[GOAL]\ncase neg.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrefine' \u27e8Set.indicator u fun _ => c, fun x => _, u_open.lowerSemicontinuous_indicator (zero_le _), _\u27e9\n[GOAL]\ncase neg.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nx : \u03b1\n\u22a2 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 Set.indicator u (fun x => c) x\n[PROOFSTEP]\nsimp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,\n SimpleFunc.coe_piecewise]\n[GOAL]\ncase neg.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nx : \u03b1\n\u22a2 Set.piecewise s (Function.const \u03b1 c) 0 x \u2264 Set.indicator u (fun x => c) x\n[PROOFSTEP]\nexact Set.indicator_le_indicator_of_subset su (fun x => zero_le _) _\n[GOAL]\ncase neg.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(Set.indicator u (fun x => c) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nsuffices (c : \u211d\u22650\u221e) * \u03bc u \u2264 c * \u03bc s + \u03b5 by\n classical simpa only [ENNReal.coe_indicator, u_open.measurableSet, lintegral_indicator, lintegral_const,\n MeasurableSet.univ, Measure.restrict_apply, Set.univ_inter, const_zero, coe_piecewise, coe_const, coe_zero,\n Set.piecewise_eq_indicator, Function.const_apply, hs]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis\u271d : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nthis : \u2191c * \u2191\u2191\u03bc u \u2264 \u2191c * \u2191\u2191\u03bc s + \u03b5\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(Set.indicator u (fun x => c) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nclassical simpa only [ENNReal.coe_indicator, u_open.measurableSet, lintegral_indicator, lintegral_const,\n MeasurableSet.univ, Measure.restrict_apply, Set.univ_inter, const_zero, coe_piecewise, coe_const, coe_zero,\n Set.piecewise_eq_indicator, Function.const_apply, hs]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis\u271d : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nthis : \u2191c * \u2191\u2191\u03bc u \u2264 \u2191c * \u2191\u2191\u03bc s + \u03b5\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(Set.indicator u (fun x => c) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nsimpa only [ENNReal.coe_indicator, u_open.measurableSet, lintegral_indicator, lintegral_const, MeasurableSet.univ,\n Measure.restrict_apply, Set.univ_inter, const_zero, coe_piecewise, coe_const, coe_zero, Set.piecewise_eq_indicator,\n Function.const_apply, hs]\n[GOAL]\ncase neg.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2191c * \u2191\u2191\u03bc u \u2264 \u2191c * \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\ncalc\n (c : \u211d\u22650\u221e) * \u03bc u \u2264 c * (\u03bc s + \u03b5 / c) := mul_le_mul_left' \u03bcu.le _\n _ = c * \u03bc s + \u03b5 := by\n simp_rw [mul_add]\n rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]\n simpa using hc\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2191c * (\u2191\u2191\u03bc s + \u03b5 / \u2191c) = \u2191c * \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nsimp_rw [mul_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2191c * \u2191\u2191\u03bc s + \u2191c * (\u03b5 / \u2191c) = \u2191c * \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nrw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2191c \u2260 0\n[PROOFSTEP]\nsimpa using hc\n[GOAL]\ncase h_add\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f\u2081 + f\u2082) x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrcases h\u2081 (ENNReal.half_pos \u03b50).ne' with \u27e8g\u2081, f\u2081_le_g\u2081, g\u2081cont, g\u2081int\u27e9\n[GOAL]\ncase h_add.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f\u2081 + f\u2082) x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrcases h\u2082 (ENNReal.half_pos \u03b50).ne' with \u27e8g\u2082, f\u2082_le_g\u2082, g\u2082cont, g\u2082int\u27e9\n[GOAL]\ncase h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f\u2081 + f\u2082) x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun x => g\u2081 x + g\u2082 x, fun x => add_le_add (f\u2081_le_g\u2081 x) (f\u2082_le_g\u2082 x), g\u2081cont.add g\u2082cont, _\u27e9\n[GOAL]\ncase h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191((fun x => g\u2081 x + g\u2082 x) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nsimp only [SimpleFunc.coe_add, ENNReal.coe_add, Pi.add_apply]\n[GOAL]\ncase h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) + \u2191(g\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) + \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrw [lintegral_add_left f\u2081.measurable.coe_nnreal_ennreal, lintegral_add_left g\u2081cont.measurable.coe_nnreal_ennreal]\n[GOAL]\ncase h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nconvert add_le_add g\u2081int g\u2082int using 1\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc + \u03b5 =\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5 / 2 + (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5 / 2)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 ENNReal.add_halves \u03b5]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5 / 2\n| \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrw [\u2190 ENNReal.add_halves \u03b5]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5 / 2\n| \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrw [\u2190 ENNReal.add_halves \u03b5]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5 / 2\n| \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrw [\u2190 ENNReal.add_halves \u03b5]\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc + (\u03b5 / 2 + \u03b5 / 2) =\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5 / 2 + (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5 / 2)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192 \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc + (\u03b5 / 2 + \u03b5 / 2) =\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u03b5 / 2 + (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc + \u03b5 / 2)\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrcases ENNReal.exists_pos_sum_of_countable' \u03b5pos \u2115 with \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nhave :\n \u2200 n,\n \u2203 g : \u03b1 \u2192 \u211d\u22650,\n (\u2200 x, SimpleFunc.eapproxDiff f n x \u2264 g x) \u2227\n LowerSemicontinuous g \u2227 (\u222b\u207b x, g x \u2202\u03bc) \u2264 (\u222b\u207b x, SimpleFunc.eapproxDiff f n x \u2202\u03bc) + \u03b4 n :=\n fun n => SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge \u03bc (SimpleFunc.eapproxDiff f n) (\u03b4pos n).ne'\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\nthis :\n \u2200 (n : \u2115),\n \u2203 g,\n (\u2200 (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g x) \u2227\n LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nchoose g f_le_g gcont hg using this\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun x => \u2211' n, g n x, fun x => _, _, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\nx : \u03b1\n\u22a2 f x \u2264 (fun x => \u2211' (n : \u2115), \u2191(g n x)) x\n[PROOFSTEP]\nrw [\u2190 SimpleFunc.tsum_eapproxDiff f hf]\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\nx : \u03b1\n\u22a2 \u2211' (n : \u2115), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2264 (fun x => \u2211' (n : \u2115), \u2191(g n x)) x\n[PROOFSTEP]\nexact ENNReal.tsum_le_tsum fun n => ENNReal.coe_le_coe.2 (f_le_g n x)\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 LowerSemicontinuous fun x => \u2211' (n : \u2115), \u2191(g n x)\n[PROOFSTEP]\nrefine' lowerSemicontinuous_tsum fun n => _\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\nn : \u2115\n\u22a2 LowerSemicontinuous fun x => \u2191(g n x)\n[PROOFSTEP]\nexact ENNReal.continuous_coe.comp_lowerSemicontinuous (gcont n) fun x y hxy => ENNReal.coe_le_coe.2 hxy\n[GOAL]\ncase intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u222b\u207b (x : \u03b1), (fun x => \u2211' (n : \u2115), \u2191(g n x)) x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\ncalc\n \u222b\u207b x, \u2211' n : \u2115, g n x \u2202\u03bc = \u2211' n, \u222b\u207b x, g n x \u2202\u03bc := by\n rw [lintegral_tsum fun n => (gcont n).measurable.coe_nnreal_ennreal.aemeasurable]\n _ \u2264 \u2211' n, ((\u222b\u207b x, SimpleFunc.eapproxDiff f n x \u2202\u03bc) + \u03b4 n) := (ENNReal.tsum_le_tsum hg)\n _ = \u2211' n, \u222b\u207b x, SimpleFunc.eapproxDiff f n x \u2202\u03bc + \u2211' n, \u03b4 n := ENNReal.tsum_add\n _ \u2264 (\u222b\u207b x : \u03b1, f x \u2202\u03bc) + \u03b5 := by\n refine' add_le_add _ h\u03b4.le\n rw [\u2190 lintegral_tsum]\n \u00b7 simp_rw [SimpleFunc.tsum_eapproxDiff f hf, le_refl]\n \u00b7 intro n; exact (SimpleFunc.measurable _).coe_nnreal_ennreal.aemeasurable\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u222b\u207b (x : \u03b1), \u2211' (n : \u2115), \u2191(g n x) \u2202\u03bc = \u2211' (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_tsum fun n => (gcont n).measurable.coe_nnreal_ennreal.aemeasurable]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u2211' (n : \u2115), \u03b4 n \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrefine' add_le_add _ h\u03b4.le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lintegral_tsum]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u222b\u207b (a : \u03b1), \u2211' (i : \u2115), \u2191(\u2191(SimpleFunc.eapproxDiff f i) a) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [SimpleFunc.tsum_eapproxDiff f hf, le_refl]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\n\u22a2 \u2200 (i : \u2115), AEMeasurable fun x => \u2191(\u2191(SimpleFunc.eapproxDiff f i) x)\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5\ng : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\nf_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x\ngcont : \u2200 (n : \u2115), LowerSemicontinuous (g n)\nhg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n\nn : \u2115\n\u22a2 AEMeasurable fun x => \u2191(\u2191(SimpleFunc.eapproxDiff f n) x)\n[PROOFSTEP]\nexact (SimpleFunc.measurable _).coe_nnreal_ennreal.aemeasurable\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nhave : \u03b5 / 2 \u2260 0 := (ENNReal.half_pos \u03b50).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrcases exists_pos_lintegral_lt_of_sigmaFinite \u03bc this with \u27e8w, wpos, wmeas, wint\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\nw : \u03b1 \u2192 \u211d\u22650\nwpos : \u2200 (x : \u03b1), 0 < w x\nwmeas : Measurable w\nwint : \u222b\u207b (x : \u03b1), \u2191(w x) \u2202\u03bc < \u03b5 / 2\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nlet f' x := ((f x + w x : \u211d\u22650) : \u211d\u22650\u221e)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\nw : \u03b1 \u2192 \u211d\u22650\nwpos : \u2200 (x : \u03b1), 0 < w x\nwmeas : Measurable w\nwint : \u222b\u207b (x : \u03b1), \u2191(w x) \u2202\u03bc < \u03b5 / 2\nf' : \u03b1 \u2192 \u211d\u22650\u221e := fun x => \u2191(f x + w x)\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrcases exists_le_lowerSemicontinuous_lintegral_ge \u03bc f' (fmeas.add wmeas).coe_nnreal_ennreal this with\n \u27e8g, le_g, gcont, gint\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\nw : \u03b1 \u2192 \u211d\u22650\nwpos : \u2200 (x : \u03b1), 0 < w x\nwmeas : Measurable w\nwint : \u222b\u207b (x : \u03b1), \u2191(w x) \u2202\u03bc < \u03b5 / 2\nf' : \u03b1 \u2192 \u211d\u22650\u221e := fun x => \u2191(f x + w x)\ng : \u03b1 \u2192 \u211d\u22650\u221e\nle_g : \u2200 (x : \u03b1), f' x \u2264 g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f' x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrefine' \u27e8g, fun x => _, gcont, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\nw : \u03b1 \u2192 \u211d\u22650\nwpos : \u2200 (x : \u03b1), 0 < w x\nwmeas : Measurable w\nwint : \u222b\u207b (x : \u03b1), \u2191(w x) \u2202\u03bc < \u03b5 / 2\nf' : \u03b1 \u2192 \u211d\u22650\u221e := fun x => \u2191(f x + w x)\ng : \u03b1 \u2192 \u211d\u22650\u221e\nle_g : \u2200 (x : \u03b1), f' x \u2264 g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f' x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\n\u22a2 \u2191(f x) < g x\n[PROOFSTEP]\ncalc\n (f x : \u211d\u22650\u221e) < f' x := by simpa only [\u2190 ENNReal.coe_lt_coe, add_zero] using add_lt_add_left (wpos x) (f x)\n _ \u2264 g x := le_g x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\nw : \u03b1 \u2192 \u211d\u22650\nwpos : \u2200 (x : \u03b1), 0 < w x\nwmeas : Measurable w\nwint : \u222b\u207b (x : \u03b1), \u2191(w x) \u2202\u03bc < \u03b5 / 2\nf' : \u03b1 \u2192 \u211d\u22650\u221e := fun x => \u2191(f x + w x)\ng : \u03b1 \u2192 \u211d\u22650\u221e\nle_g : \u2200 (x : \u03b1), f' x \u2264 g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f' x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\n\u22a2 \u2191(f x) < f' x\n[PROOFSTEP]\nsimpa only [\u2190 ENNReal.coe_lt_coe, add_zero] using add_lt_add_left (wpos x) (f x)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\nw : \u03b1 \u2192 \u211d\u22650\nwpos : \u2200 (x : \u03b1), 0 < w x\nwmeas : Measurable w\nwint : \u222b\u207b (x : \u03b1), \u2191(w x) \u2202\u03bc < \u03b5 / 2\nf' : \u03b1 \u2192 \u211d\u22650\u221e := fun x => \u2191(f x + w x)\ng : \u03b1 \u2192 \u211d\u22650\u221e\nle_g : \u2200 (x : \u03b1), f' x \u2264 g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f' x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\ncalc\n (\u222b\u207b x : \u03b1, g x \u2202\u03bc) \u2264 (\u222b\u207b x : \u03b1, f x + w x \u2202\u03bc) + \u03b5 / 2 := gint\n _ = ((\u222b\u207b x : \u03b1, f x \u2202\u03bc) + \u222b\u207b x : \u03b1, w x \u2202\u03bc) + \u03b5 / 2 := by rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]\n _ \u2264 (\u222b\u207b x : \u03b1, f x \u2202\u03bc) + \u03b5 / 2 + \u03b5 / 2 := (add_le_add_right (add_le_add_left wint.le _) _)\n _ = (\u222b\u207b x : \u03b1, f x \u2202\u03bc) + \u03b5 := by rw [add_assoc, ENNReal.add_halves]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\nw : \u03b1 \u2192 \u211d\u22650\nwpos : \u2200 (x : \u03b1), 0 < w x\nwmeas : Measurable w\nwint : \u222b\u207b (x : \u03b1), \u2191(w x) \u2202\u03bc < \u03b5 / 2\nf' : \u03b1 \u2192 \u211d\u22650\u221e := fun x => \u2191(f x + w x)\ng : \u03b1 \u2192 \u211d\u22650\u221e\nle_g : \u2200 (x : \u03b1), f' x \u2264 g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f' x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) + \u2191(w x) \u2202\u03bc + \u03b5 / 2 = \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(w x) \u2202\u03bc + \u03b5 / 2\n[PROOFSTEP]\nrw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : Measurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\nw : \u03b1 \u2192 \u211d\u22650\nwpos : \u2200 (x : \u03b1), 0 < w x\nwmeas : Measurable w\nwint : \u222b\u207b (x : \u03b1), \u2191(w x) \u2202\u03bc < \u03b5 / 2\nf' : \u03b1 \u2192 \u211d\u22650\u221e := fun x => \u2191(f x + w x)\ng : \u03b1 \u2192 \u211d\u22650\u221e\nle_g : \u2200 (x : \u03b1), f' x \u2264 g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f' x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2 + \u03b5 / 2 = \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrw [add_assoc, ENNReal.add_halves]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nhave : \u03b5 / 2 \u2260 0 := (ENNReal.half_pos \u03b50).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrcases exists_lt_lowerSemicontinuous_lintegral_ge \u03bc (fmeas.mk f) fmeas.measurable_mk this with\n \u27e8g0, f_lt_g0, g0_cont, g0_int\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrcases exists_measurable_superset_of_null fmeas.ae_eq_mk with \u27e8s, hs, smeas, \u03bcs\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrcases exists_le_lowerSemicontinuous_lintegral_ge \u03bc (s.indicator fun _x => \u221e) (measurable_const.indicator smeas)\n this with\n \u27e8g1, le_g1, g1_cont, g1_int\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun x => g0 x + g1 x, fun x => _, g0_cont.add g1_cont, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x\n[PROOFSTEP]\nby_cases h : x \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : x \u2208 s\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x\n[PROOFSTEP]\nhave := le_g1 x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis\u271d : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : x \u2208 s\nthis : Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x\n[PROOFSTEP]\nsimp only [h, Set.indicator_of_mem, top_le_iff] at this \n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis\u271d : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : x \u2208 s\nthis : g1 x = \u22a4\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x\n[PROOFSTEP]\nhave : f x = fmeas.mk f x := by rw [Set.compl_subset_comm] at hs ; exact hs h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 f x = AEMeasurable.mk f fmeas x\n[PROOFSTEP]\nrw [Set.compl_subset_comm] at hs \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : s\u1d9c \u2286 {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 f x = AEMeasurable.mk f fmeas x\n[PROOFSTEP]\nexact hs h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis\u271d : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : \u00acx \u2208 s\nthis : f x = AEMeasurable.mk f fmeas x\n\u22a2 \u2191(f x) < (fun x => g0 x + g1 x) x\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis\u271d : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\nx : \u03b1\nh : \u00acx \u2208 s\nthis : f x = AEMeasurable.mk f fmeas x\n\u22a2 \u2191(AEMeasurable.mk f fmeas x) < (fun x => g0 x + g1 x) x\n[PROOFSTEP]\nexact (f_lt_g0 x).trans_le le_self_add\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), (fun x => g0 x + g1 x) x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\ncalc\n \u222b\u207b x, g0 x + g1 x \u2202\u03bc = (\u222b\u207b x, g0 x \u2202\u03bc) + \u222b\u207b x, g1 x \u2202\u03bc := lintegral_add_left g0_cont.measurable _\n _ \u2264 (\u222b\u207b x, f x \u2202\u03bc) + \u03b5 / 2 + (0 + \u03b5 / 2) := by\n refine' add_le_add _ _\n \u00b7 convert g0_int using 2\n exact lintegral_congr_ae (fmeas.ae_eq_mk.fun_comp _)\n \u00b7 convert g1_int\n simp only [smeas, \u03bcs, lintegral_const, Set.univ_inter, MeasurableSet.univ, lintegral_indicator, mul_zero,\n restrict_apply]\n _ = (\u222b\u207b x, f x \u2202\u03bc) + \u03b5 := by simp only [add_assoc, ENNReal.add_halves, zero_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc + \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2 + (0 + \u03b5 / 2)\n[PROOFSTEP]\nrefine' add_le_add _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2\n[PROOFSTEP]\nconvert g0_int using 2\n[GOAL]\ncase h.e'_4.h.e'_5\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc = \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc\n[PROOFSTEP]\nexact lintegral_congr_ae (fmeas.ae_eq_mk.fun_comp _)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 0 + \u03b5 / 2\n[PROOFSTEP]\nconvert g1_int\n[GOAL]\ncase h.e'_4.h.e'_5\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 0 = \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc\n[PROOFSTEP]\nsimp only [smeas, \u03bcs, lintegral_const, Set.univ_inter, MeasurableSet.univ, lintegral_indicator, mul_zero,\n restrict_apply]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nthis : \u03b5 / 2 \u2260 0\ng0 : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g0 : \u2200 (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) < g0 x\ng0_cont : LowerSemicontinuous g0\ng0_int : \u222b\u207b (x : \u03b1), g0 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(AEMeasurable.mk f fmeas x) \u2202\u03bc + \u03b5 / 2\ns : Set \u03b1\nhs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}\u1d9c \u2286 s\nsmeas : MeasurableSet s\n\u03bcs : \u2191\u2191\u03bc s = 0\ng1 : \u03b1 \u2192 \u211d\u22650\u221e\nle_g1 : \u2200 (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2264 g1 x\ng1_cont : LowerSemicontinuous g1\ng1_int : \u222b\u207b (x : \u03b1), g1 x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), Set.indicator s (fun _x => \u22a4) x \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2 + (0 + \u03b5 / 2) = \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nsimp only [add_assoc, ENNReal.add_halves, zero_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nhave fmeas : AEMeasurable f \u03bc :=\n by\n convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable\n simp only [Real.toNNReal_coe]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nconvert fint.aestronglyMeasurable.real_toNNReal.aemeasurable\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nx\u271d : \u03b1\n\u22a2 f x\u271d = Real.toNNReal \u2191(f x\u271d)\n[PROOFSTEP]\nsimp only [Real.toNNReal_coe]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nfmeas : AEMeasurable f\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nlift \u03b5 to \u211d\u22650 using \u03b5pos.le\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5\n[PROOFSTEP]\nobtain \u27e8\u03b4, \u03b4pos, h\u03b4\u03b5\u27e9 : \u2203 \u03b4 : \u211d\u22650, 0 < \u03b4 \u2227 \u03b4 < \u03b5\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u03b4 < \u03b5\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5\n[PROOFSTEP]\nexact exists_between \u03b5pos\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5\n[PROOFSTEP]\nhave int_f_ne_top : (\u222b\u207b a : \u03b1, f a \u2202\u03bc) \u2260 \u221e := (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5\n[PROOFSTEP]\nrcases exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable \u03bc f fmeas (ENNReal.coe_ne_zero.2 \u03b4pos.ne') with\n \u27e8g, f_lt_g, gcont, gint\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5\n[PROOFSTEP]\nhave gint_ne : (\u222b\u207b x : \u03b1, g x \u2202\u03bc) \u2260 \u221e := ne_top_of_le_ne_top (by simpa) gint\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u2260 \u22a4\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5\n[PROOFSTEP]\nhave g_lt_top : \u2200\u1d50 x : \u03b1 \u2202\u03bc, g x < \u221e := ae_lt_top gcont.measurable gint_ne\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5\n[PROOFSTEP]\nhave Ig : (\u222b\u207b a : \u03b1, ENNReal.ofReal (g a).toReal \u2202\u03bc) = \u222b\u207b a : \u03b1, g a \u2202\u03bc :=\n by\n apply lintegral_congr_ae\n filter_upwards [g_lt_top] with _ hx\n simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n[PROOFSTEP]\napply lintegral_congr_ae\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (g a))) =\u1da0[ae \u03bc] fun a => g a\n[PROOFSTEP]\nfilter_upwards [g_lt_top] with _ hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\na\u271d : \u03b1\nhx : g a\u271d < \u22a4\n\u22a2 ENNReal.ofReal (ENNReal.toReal (g a\u271d)) = g a\u271d\n[PROOFSTEP]\nsimp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5\n[PROOFSTEP]\nrefine' \u27e8g, f_lt_g, gcont, g_lt_top, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 Integrable fun x => ENNReal.toReal (g x)\n[PROOFSTEP]\nrefine' \u27e8gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 HasFiniteIntegral fun x => ENNReal.toReal (g x)\n[PROOFSTEP]\nsimp only [hasFiniteIntegral_iff_norm, Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc < \u22a4\n[PROOFSTEP]\nconvert gint_ne.lt_top using 1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5\n[PROOFSTEP]\nrw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc) <\n ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) + \u2191\u03b5\n[PROOFSTEP]\ncalc\n ENNReal.toReal (\u222b\u207b a : \u03b1, ENNReal.ofReal (g a).toReal \u2202\u03bc) = ENNReal.toReal (\u222b\u207b a : \u03b1, g a \u2202\u03bc) := by congr 1\n _ \u2264 ENNReal.toReal ((\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b4) :=\n by\n apply ENNReal.toReal_mono _ gint\n simpa using int_f_ne_top\n _ = ENNReal.toReal (\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b4 := by\n rw [ENNReal.toReal_add int_f_ne_top ENNReal.coe_ne_top, ENNReal.coe_toReal]\n _ < ENNReal.toReal (\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b5 := (add_lt_add_left h\u03b4\u03b5 _)\n _ = (\u222b\u207b a : \u03b1, ENNReal.ofReal \u2191(f a) \u2202\u03bc).toReal + \u03b5 := by simp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc) = ENNReal.toReal (\u222b\u207b (a : \u03b1), g a \u2202\u03bc)\n[PROOFSTEP]\ncongr 1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), g a \u2202\u03bc) \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc + \u2191\u03b4)\n[PROOFSTEP]\napply ENNReal.toReal_mono _ gint\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u2260 \u22a4\n[PROOFSTEP]\nsimpa using int_f_ne_top\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc + \u2191\u03b4) = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc) + \u2191\u03b4\n[PROOFSTEP]\nrw [ENNReal.toReal_add int_f_ne_top ENNReal.coe_ne_top, ENNReal.coe_toReal]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc) + \u2191\u03b5 = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) + \u2191\u03b5\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.hf\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => \u2191(f x)\n[PROOFSTEP]\napply Filter.eventually_of_forall fun x => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => \u2191(f x)) x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.hfm\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 AEStronglyMeasurable (fun x => \u2191(f x)) \u03bc\n[PROOFSTEP]\nexact fmeas.coe_nnreal_real.aestronglyMeasurable\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.hf\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => ENNReal.toReal (g x)\n[PROOFSTEP]\napply Filter.eventually_of_forall fun x => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => ENNReal.toReal (g x)) x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.hfm\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (g x)) \u03bc\n[PROOFSTEP]\napply gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\ninduction' f using MeasureTheory.SimpleFunc.induction with c s hs f\u2081 f\u2082 _ h\u2081 h\u2082 generalizing \u03b5\n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nby_cases hc : c = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : c = 0\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun _ => 0, _, upperSemicontinuous_const, _\u27e9\n[GOAL]\ncase pos.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : c = 0\n\u22a2 \u2200 (x : \u03b1), (fun x => 0) x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x\n[PROOFSTEP]\nclassical simp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff, eq_self_iff_true,\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, le_zero_iff]\n[GOAL]\ncase pos.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : c = 0\n\u22a2 \u2200 (x : \u03b1), (fun x => 0) x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x\n[PROOFSTEP]\nsimp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff, eq_self_iff_true,\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, le_zero_iff]\n[GOAL]\ncase pos.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : c = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191((fun x => 0) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nclassical simp only [hc, Set.indicator_zero', lintegral_const, zero_mul, Pi.zero_apply, SimpleFunc.const_zero, zero_add,\n zero_le', SimpleFunc.coe_zero, Set.piecewise_eq_indicator, ENNReal.coe_zero, SimpleFunc.coe_piecewise, zero_le]\n[GOAL]\ncase pos.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : c = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191((fun x => 0) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nsimp only [hc, Set.indicator_zero', lintegral_const, zero_mul, Pi.zero_apply, SimpleFunc.const_zero, zero_add, zero_le',\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator, ENNReal.coe_zero, SimpleFunc.coe_piecewise, zero_le]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nhave \u03bcs_lt_top : \u03bc s < \u221e := by\n classical simpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, or_false_iff, lintegral_const,\n ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top, Measure.restrict_apply MeasurableSet.univ,\n ENNReal.mul_eq_top, SimpleFunc.const_zero, Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def,\n not_false_iff, SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, false_and_iff] using int_f\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u22a2 \u2191\u2191\u03bc s < \u22a4\n[PROOFSTEP]\nclassical simpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, or_false_iff, lintegral_const,\n ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top, Measure.restrict_apply MeasurableSet.univ,\n ENNReal.mul_eq_top, SimpleFunc.const_zero, Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def,\n not_false_iff, SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, false_and_iff] using int_f\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u22a2 \u2191\u2191\u03bc s < \u22a4\n[PROOFSTEP]\nsimpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, or_false_iff, lintegral_const,\n ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top, Measure.restrict_apply MeasurableSet.univ,\n ENNReal.mul_eq_top, SimpleFunc.const_zero, Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def,\n not_false_iff, SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, false_and_iff] using int_f\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nhave : (0 : \u211d\u22650\u221e) < \u03b5 / c := ENNReal.div_pos_iff.2 \u27e8\u03b50, ENNReal.coe_ne_top\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nobtain \u27e8F, Fs, F_closed, \u03bcF\u27e9 : \u2203 (F : _), F \u2286 s \u2227 IsClosed F \u2227 \u03bc s < \u03bc F + \u03b5 / c :=\n hs.exists_isClosed_lt_add \u03bcs_lt_top.ne this.ne'\n[GOAL]\ncase neg.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227\n UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrefine' \u27e8Set.indicator F fun _ => c, fun x => _, F_closed.upperSemicontinuous_indicator (zero_le _), _\u27e9\n[GOAL]\ncase neg.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\nx : \u03b1\n\u22a2 Set.indicator F (fun x => c) x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x\n[PROOFSTEP]\nsimp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,\n SimpleFunc.coe_piecewise]\n[GOAL]\ncase neg.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\nx : \u03b1\n\u22a2 Set.indicator F (fun x => c) x \u2264 Set.piecewise s (Function.const \u03b1 c) 0 x\n[PROOFSTEP]\nexact Set.indicator_le_indicator_of_subset Fs (fun x => zero_le _) _\n[GOAL]\ncase neg.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(Set.indicator F (fun x => c) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nsuffices (c : \u211d\u22650\u221e) * \u03bc s \u2264 c * \u03bc F + \u03b5 by\n classical simpa only [hs, F_closed.measurableSet, SimpleFunc.coe_const, Function.const_apply, lintegral_const,\n ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ, SimpleFunc.const_zero, lintegral_indicator,\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, Measure.restrict_apply]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis\u271d : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\nthis : \u2191c * \u2191\u2191\u03bc s \u2264 \u2191c * \u2191\u2191\u03bc F + \u03b5\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(Set.indicator F (fun x => c) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nclassical simpa only [hs, F_closed.measurableSet, SimpleFunc.coe_const, Function.const_apply, lintegral_const,\n ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ, SimpleFunc.const_zero, lintegral_indicator,\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, Measure.restrict_apply]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis\u271d : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\nthis : \u2191c * \u2191\u2191\u03bc s \u2264 \u2191c * \u2191\u2191\u03bc F + \u03b5\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(Set.indicator F (fun x => c) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nsimpa only [hs, F_closed.measurableSet, SimpleFunc.coe_const, Function.const_apply, lintegral_const,\n ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ, SimpleFunc.const_zero, lintegral_indicator,\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, Measure.restrict_apply]\n[GOAL]\ncase neg.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u2191c * \u2191\u2191\u03bc s \u2264 \u2191c * \u2191\u2191\u03bc F + \u03b5\n[PROOFSTEP]\ncalc\n (c : \u211d\u22650\u221e) * \u03bc s \u2264 c * (\u03bc F + \u03b5 / c) := mul_le_mul_left' \u03bcF.le _\n _ = c * \u03bc F + \u03b5 := by\n simp_rw [mul_add]\n rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]\n simpa using hc\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u2191c * (\u2191\u2191\u03bc F + \u03b5 / \u2191c) = \u2191c * \u2191\u2191\u03bc F + \u03b5\n[PROOFSTEP]\nsimp_rw [mul_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u2191c * \u2191\u2191\u03bc F + \u2191c * (\u03b5 / \u2191c) = \u2191c * \u2191\u2191\u03bc F + \u03b5\n[PROOFSTEP]\nrw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nhc : \u00acc = 0\n\u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4\nthis : 0 < \u03b5 / \u2191c\nF : Set \u03b1\nFs : F \u2286 s\nF_closed : IsClosed F\n\u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c\n\u22a2 \u2191c \u2260 0\n[PROOFSTEP]\nsimpa using hc\n[GOAL]\ncase h_add\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nhave A : ((\u222b\u207b x : \u03b1, f\u2081 x \u2202\u03bc) + \u222b\u207b x : \u03b1, f\u2082 x \u2202\u03bc) \u2260 \u22a4 := by rwa [\u2190 lintegral_add_left f\u2081.measurable.coe_nnreal_ennreal]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\n[PROOFSTEP]\nrwa [\u2190 lintegral_add_left f\u2081.measurable.coe_nnreal_ennreal]\n[GOAL]\ncase h_add\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrcases h\u2081 (ENNReal.add_ne_top.1 A).1 (ENNReal.half_pos \u03b50).ne' with \u27e8g\u2081, f\u2081_le_g\u2081, g\u2081cont, g\u2081int\u27e9\n[GOAL]\ncase h_add.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrcases h\u2082 (ENNReal.add_ne_top.1 A).2 (ENNReal.half_pos \u03b50).ne' with \u27e8g\u2082, f\u2082_le_g\u2082, g\u2082cont, g\u2082int\u27e9\n[GOAL]\ncase h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun x => g\u2081 x + g\u2082 x, fun x => add_le_add (f\u2081_le_g\u2081 x) (f\u2082_le_g\u2082 x), g\u2081cont.add g\u2082cont, _\u27e9\n[GOAL]\ncase h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191((fun x => g\u2081 x + g\u2082 x) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nsimp only [SimpleFunc.coe_add, ENNReal.coe_add, Pi.add_apply]\n[GOAL]\ncase h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) + \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) + \u2191(g\u2082 x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrw [lintegral_add_left f\u2081.measurable.coe_nnreal_ennreal, lintegral_add_left g\u2081cont.measurable.coe_nnreal_ennreal]\n[GOAL]\ncase h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nconvert add_le_add g\u2081int g\u2082int using 1\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + \u03b5 =\n \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 + (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 ENNReal.add_halves \u03b5]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n| \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrw [\u2190 ENNReal.add_halves \u03b5]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n| \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrw [\u2190 ENNReal.add_halves \u03b5]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n| \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrw [\u2190 ENNReal.add_halves \u03b5]\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + (\u03b5 / 2 + \u03b5 / 2) =\n \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 + (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\nint_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nh\u2082 :\n \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192\n \u2200 {\u03b5 : \u211d\u22650\u221e},\n \u03b5 \u2260 0 \u2192\n \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\nint_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nA : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x\ng\u2081cont : UpperSemicontinuous g\u2081\ng\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x\ng\u2082cont : UpperSemicontinuous g\u2082\ng\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + (\u03b5 / 2 + \u03b5 / 2) =\n \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 + (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2)\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), g x \u2264 f x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nobtain \u27e8fs, fs_le_f, int_fs\u27e9 : \u2203 fs : \u03b1 \u2192\u209b \u211d\u22650, (\u2200 x, fs x \u2264 f x) \u2227 (\u222b\u207b x, f x \u2202\u03bc) \u2264 (\u222b\u207b x, fs x \u2202\u03bc) + \u03b5 / 2 := by\n -- Porting note: need to name identifier (not `this`), because `conv_rhs at this` errors\n have aux := ENNReal.lt_add_right int_f (ENNReal.half_pos \u03b50).ne'\n conv_rhs at aux => rw [lintegral_eq_nnreal (fun x => (f x : \u211d\u22650\u221e)) \u03bc]\n erw [ENNReal.biSup_add] at aux <;> [skip; exact \u27e80, fun x => by simp\u27e9]\n simp only [lt_iSup_iff] at aux \n rcases aux with \u27e8fs, fs_le_f, int_fs\u27e9\n refine' \u27e8fs, fun x => by simpa only [ENNReal.coe_le_coe] using fs_le_f x, _\u27e9\n convert int_fs.le\n rw [\u2190 SimpleFunc.lintegral_eq_lintegral]\n rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 fs, (\u2200 (x : \u03b1), \u2191fs x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\n[PROOFSTEP]\nhave aux := ENNReal.lt_add_right int_f (ENNReal.half_pos \u03b50).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\naux : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 fs, (\u2200 (x : \u03b1), \u2191fs x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\n[PROOFSTEP]\nconv_rhs at aux => rw [lintegral_eq_nnreal (fun x => (f x : \u211d\u22650\u221e)) \u03bc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\naux : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2\n| \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2\n[PROOFSTEP]\nrw [lintegral_eq_nnreal (fun x => (f x : \u211d\u22650\u221e)) \u03bc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\naux : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2\n| \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2\n[PROOFSTEP]\nrw [lintegral_eq_nnreal (fun x => (f x : \u211d\u22650\u221e)) \u03bc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\naux : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2\n| \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5 / 2\n[PROOFSTEP]\nrw [lintegral_eq_nnreal (fun x => (f x : \u211d\u22650\u221e)) \u03bc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\naux :\n \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc <\n (\u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2191(f x)), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc) + \u03b5 / 2\n\u22a2 \u2203 fs, (\u2200 (x : \u03b1), \u2191fs x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\n[PROOFSTEP]\nerw [ENNReal.biSup_add] at aux <;> [skip; exact \u27e80, fun x => by simp\u27e9]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\naux :\n \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc <\n (\u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2191(f x)), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc) + \u03b5 / 2\n\u22a2 \u2203 fs, (\u2200 (x : \u03b1), \u2191fs x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\n[PROOFSTEP]\nerw [ENNReal.biSup_add] at aux \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\naux :\n \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc <\n \u2a06 (i : \u03b1 \u2192\u209b \u211d\u22650) (_ : i \u2208 fun \u03c6 => \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2191(f x)),\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc + \u03b5 / 2\n\u22a2 \u2203 fs, (\u2200 (x : \u03b1), \u2191fs x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\n[PROOFSTEP]\nskip\n[GOAL]\ncase hs\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\naux :\n \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc <\n (\u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2191(f x)), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc) + \u03b5 / 2\n\u22a2 Set.Nonempty fun \u03c6 => \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2191(f x)\n[PROOFSTEP]\nexact \u27e80, fun x => by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\naux :\n \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc <\n (\u2a06 (\u03c6 : \u03b1 \u2192\u209b \u211d\u22650) (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2191(f x)), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc) + \u03b5 / 2\nx : \u03b1\n\u22a2 \u2191(\u21910 x) \u2264 \u2191(f x)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\naux :\n \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc <\n \u2a06 (i : \u03b1 \u2192\u209b \u211d\u22650) (_ : i \u2208 fun \u03c6 => \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2191(f x)),\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc + \u03b5 / 2\n\u22a2 \u2203 fs, (\u2200 (x : \u03b1), \u2191fs x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\n[PROOFSTEP]\nsimp only [lt_iSup_iff] at aux \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\naux : \u2203 i i_1, \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc + \u03b5 / 2\n\u22a2 \u2203 fs, (\u2200 (x : \u03b1), \u2191fs x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\n[PROOFSTEP]\nrcases aux with \u27e8fs, fs_le_f, int_fs\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nfs : \u03b1 \u2192\u209b \u211d\u22650\nfs_le_f : fs \u2208 fun \u03c6 => \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2191(f x)\nint_fs : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) \u03bc + \u03b5 / 2\n\u22a2 \u2203 fs, (\u2200 (x : \u03b1), \u2191fs x \u2264 f x) \u2227 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\n[PROOFSTEP]\nrefine' \u27e8fs, fun x => by simpa only [ENNReal.coe_le_coe] using fs_le_f x, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nfs : \u03b1 \u2192\u209b \u211d\u22650\nfs_le_f : fs \u2208 fun \u03c6 => \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2191(f x)\nint_fs : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) \u03bc + \u03b5 / 2\nx : \u03b1\n\u22a2 \u2191fs x \u2264 f x\n[PROOFSTEP]\nsimpa only [ENNReal.coe_le_coe] using fs_le_f x\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nfs : \u03b1 \u2192\u209b \u211d\u22650\nfs_le_f : fs \u2208 fun \u03c6 => \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2191(f x)\nint_fs : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) \u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\n[PROOFSTEP]\nconvert int_fs.le\n[GOAL]\ncase h.e'_4.h.e'_5\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nfs : \u03b1 \u2192\u209b \u211d\u22650\nfs_le_f : fs \u2208 fun \u03c6 => \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2191(f x)\nint_fs : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) \u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc = SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) \u03bc\n[PROOFSTEP]\nrw [\u2190 SimpleFunc.lintegral_eq_lintegral]\n[GOAL]\ncase h.e'_4.h.e'_5\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nfs : \u03b1 \u2192\u209b \u211d\u22650\nfs_le_f : fs \u2208 fun \u03c6 => \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2191(f x)\nint_fs : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) \u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191(SimpleFunc.map ENNReal.some fs) a \u2202\u03bc\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nfs : \u03b1 \u2192\u209b \u211d\u22650\nfs_le_f : \u2200 (x : \u03b1), \u2191fs x \u2264 f x\nint_fs : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), g x \u2264 f x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nhave int_fs_lt_top : (\u222b\u207b x, fs x \u2202\u03bc) \u2260 \u221e :=\n by\n refine' ne_top_of_le_ne_top int_f (lintegral_mono fun x => _)\n simpa only [ENNReal.coe_le_coe] using fs_le_f x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nfs : \u03b1 \u2192\u209b \u211d\u22650\nfs_le_f : \u2200 (x : \u03b1), \u2191fs x \u2264 f x\nint_fs : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc \u2260 \u22a4\n[PROOFSTEP]\nrefine' ne_top_of_le_ne_top int_f (lintegral_mono fun x => _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nfs : \u03b1 \u2192\u209b \u211d\u22650\nfs_le_f : \u2200 (x : \u03b1), \u2191fs x \u2264 f x\nint_fs : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\nx : \u03b1\n\u22a2 \u2191(\u2191fs x) \u2264 \u2191(f x)\n[PROOFSTEP]\nsimpa only [ENNReal.coe_le_coe] using fs_le_f x\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nfs : \u03b1 \u2192\u209b \u211d\u22650\nfs_le_f : \u2200 (x : \u03b1), \u2191fs x \u2264 f x\nint_fs : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\nint_fs_lt_top : \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), g x \u2264 f x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nobtain \u27e8g, g_le_fs, gcont, gint\u27e9 :\n \u2203 g : \u03b1 \u2192 \u211d\u22650, (\u2200 x, g x \u2264 fs x) \u2227 UpperSemicontinuous g \u2227 (\u222b\u207b x, fs x \u2202\u03bc) \u2264 (\u222b\u207b x, g x \u2202\u03bc) + \u03b5 / 2 :=\n fs.exists_upperSemicontinuous_le_lintegral_le int_fs_lt_top (ENNReal.half_pos \u03b50).ne'\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nfs : \u03b1 \u2192\u209b \u211d\u22650\nfs_le_f : \u2200 (x : \u03b1), \u2191fs x \u2264 f x\nint_fs : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\nint_fs_lt_top : \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ng_le_fs : \u2200 (x : \u03b1), g x \u2264 \u2191fs x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), g x \u2264 f x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrefine' \u27e8g, fun x => (g_le_fs x).trans (fs_le_f x), gcont, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nfs : \u03b1 \u2192\u209b \u211d\u22650\nfs_le_f : \u2200 (x : \u03b1), \u2191fs x \u2264 f x\nint_fs : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\nint_fs_lt_top : \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ng_le_fs : \u2200 (x : \u03b1), g x \u2264 \u2191fs x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\ncalc\n (\u222b\u207b x, f x \u2202\u03bc) \u2264 (\u222b\u207b x, fs x \u2202\u03bc) + \u03b5 / 2 := int_fs\n _ \u2264 (\u222b\u207b x, g x \u2202\u03bc) + \u03b5 / 2 + \u03b5 / 2 := (add_le_add gint le_rfl)\n _ = (\u222b\u207b x, g x \u2202\u03bc) + \u03b5 := by rw [add_assoc, ENNReal.add_halves]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nint_f : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nfs : \u03b1 \u2192\u209b \u211d\u22650\nfs_le_f : \u2200 (x : \u03b1), \u2191fs x \u2264 f x\nint_fs : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc + \u03b5 / 2\nint_fs_lt_top : \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ng_le_fs : \u2200 (x : \u03b1), g x \u2264 \u2191fs x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(\u2191fs x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 / 2\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 / 2 + \u03b5 / 2 = \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrw [add_assoc, ENNReal.add_halves]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 f x) \u2227\n UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc\n[PROOFSTEP]\nlift \u03b5 to \u211d\u22650 using \u03b5pos.le\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 f x) \u2227\n UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc\n[PROOFSTEP]\nrw [NNReal.coe_pos, \u2190 ENNReal.coe_pos] at \u03b5pos \n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 f x) \u2227\n UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc\n[PROOFSTEP]\nhave If : (\u222b\u207b x, f x \u2202\u03bc) < \u221e := hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 f x) \u2227\n UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc\n[PROOFSTEP]\nrcases exists_upperSemicontinuous_le_lintegral_le f If.ne \u03b5pos.ne' with \u27e8g, gf, gcont, gint\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 f x) \u2227\n UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc\n[PROOFSTEP]\nhave Ig : (\u222b\u207b x, g x \u2202\u03bc) < \u221e := by\n refine' lt_of_le_of_lt (lintegral_mono fun x => _) If\n simpa using gf x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (lintegral_mono fun x => _) If\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nx : \u03b1\n\u22a2 \u2191(g x) \u2264 \u2191(f x)\n[PROOFSTEP]\nsimpa using gf x\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x \u2264 f x) \u2227\n UpperSemicontinuous g \u2227 (Integrable fun x => \u2191(g x)) \u2227 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc\n[PROOFSTEP]\nrefine' \u27e8g, gf, gcont, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 Integrable fun x => \u2191(g x)\n[PROOFSTEP]\nrefine' Integrable.mono fint gcont.measurable.coe_nnreal_real.aemeasurable.aestronglyMeasurable _\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016\u2191(g a)\u2016 \u2264 \u2016\u2191(f a)\u2016\n[PROOFSTEP]\nexact Filter.eventually_of_forall fun x => by simp [gf x]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\nx : \u03b1\n\u22a2 \u2016\u2191(g x)\u2016 \u2264 \u2016\u2191(f x)\u2016\n[PROOFSTEP]\nsimp [gf x]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc - \u2191\u03b5 \u2264 \u222b (x : \u03b1), \u2191(g x) \u2202\u03bc\n[PROOFSTEP]\nrw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) - \u2191\u03b5 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc)\n[PROOFSTEP]\nrw [sub_le_iff_le_add]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc) + \u2191\u03b5\n[PROOFSTEP]\nconvert ENNReal.toReal_mono _ gint\n[GOAL]\ncase h.e'_3.h.e'_1.h.e'_4.h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\nx\u271d : \u03b1\n\u22a2 ENNReal.ofReal \u2191(f x\u271d) = \u2191(f x\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc) + \u2191\u03b5 = ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5)\n[PROOFSTEP]\nrw [ENNReal.toReal_add Ig.ne ENNReal.coe_ne_top]\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc) + \u2191\u03b5 =\n ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) + ENNReal.toReal \u2191\u03b5\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5 \u2260 \u22a4\n[PROOFSTEP]\nsimpa using Ig.ne\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.hf\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => \u2191(g x)\n[PROOFSTEP]\napply Filter.eventually_of_forall\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.hf.hp\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => \u2191(g x)) x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.hfm\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 AEStronglyMeasurable (fun x => \u2191(g x)) \u03bc\n[PROOFSTEP]\nexact gcont.measurable.coe_nnreal_real.aemeasurable.aestronglyMeasurable\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.hf\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => \u2191(f x)\n[PROOFSTEP]\napply Filter.eventually_of_forall\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.hf.hp\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => \u2191(f x)) x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.hfm\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\nIf : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc < \u22a4\ng : \u03b1 \u2192 \u211d\u22650\ngf : \u2200 (x : \u03b1), g x \u2264 f x\ngcont : UpperSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5\nIg : \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc < \u22a4\n\u22a2 AEStronglyMeasurable (fun x => \u2191(f x)) \u03bc\n[PROOFSTEP]\nexact fint.aestronglyMeasurable\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (Integrable fun x => EReal.toReal (g x)) \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nlet \u03b4 : \u211d\u22650 := \u27e8\u03b5 / 2, (half_pos \u03b5pos).le\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (Integrable fun x => EReal.toReal (g x)) \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nhave \u03b4pos : 0 < \u03b4 := half_pos \u03b5pos\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (Integrable fun x => EReal.toReal (g x)) \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nlet fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (Integrable fun x => EReal.toReal (g x)) \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nhave int_fp : Integrable (fun x => (fp x : \u211d)) \u03bc := hf.real_toNNReal\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (Integrable fun x => EReal.toReal (g x)) \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrcases exists_lt_lowerSemicontinuous_integral_gt_nnreal fp int_fp \u03b4pos with\n \u27e8gp, fp_lt_gp, gpcont, gp_lt_top, gp_integrable, gpint\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (Integrable fun x => EReal.toReal (g x)) \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nlet fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (Integrable fun x => EReal.toReal (g x)) \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nhave int_fm : Integrable (fun x => (fm x : \u211d)) \u03bc := hf.neg.real_toNNReal\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (Integrable fun x => EReal.toReal (g x)) \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrcases exists_upperSemicontinuous_le_integral_le fm int_fm \u03b4pos with \u27e8gm, gm_le_fm, gmcont, gm_integrable, gmint\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (Integrable fun x => EReal.toReal (g x)) \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nlet g : \u03b1 \u2192 EReal := fun x => (gp x : EReal) - gm x\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (Integrable fun x => EReal.toReal (g x)) \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nhave ae_g : \u2200\u1d50 x \u2202\u03bc, (g x).toReal = (gp x : EReal).toReal - (gm x : EReal).toReal :=\n by\n filter_upwards [gp_lt_top] with _ hx\n rw [EReal.toReal_sub] <;> simp [hx.ne]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n[PROOFSTEP]\nfilter_upwards [gp_lt_top] with _ hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\na\u271d : \u03b1\nhx : gp a\u271d < \u22a4\n\u22a2 EReal.toReal (\u2191(gp a\u271d) - \u2191\u2191(gm a\u271d)) = EReal.toReal \u2191(gp a\u271d) - EReal.toReal \u2191\u2191(gm a\u271d)\n[PROOFSTEP]\nrw [EReal.toReal_sub]\n[GOAL]\ncase h.hx\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\na\u271d : \u03b1\nhx : gp a\u271d < \u22a4\n\u22a2 \u2191(gp a\u271d) \u2260 \u22a4\n[PROOFSTEP]\nsimp [hx.ne]\n[GOAL]\ncase h.h'x\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\na\u271d : \u03b1\nhx : gp a\u271d < \u22a4\n\u22a2 \u2191(gp a\u271d) \u2260 \u22a5\n[PROOFSTEP]\nsimp [hx.ne]\n[GOAL]\ncase h.hy\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\na\u271d : \u03b1\nhx : gp a\u271d < \u22a4\n\u22a2 \u2191\u2191(gm a\u271d) \u2260 \u22a4\n[PROOFSTEP]\nsimp [hx.ne]\n[GOAL]\ncase h.h'y\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\na\u271d : \u03b1\nhx : gp a\u271d < \u22a4\n\u22a2 \u2191\u2191(gm a\u271d) \u2260 \u22a5\n[PROOFSTEP]\nsimp [hx.ne]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n LowerSemicontinuous g \u2227\n (Integrable fun x => EReal.toReal (g x)) \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrefine' \u27e8g, ?lt, ?lsc, ?int, ?aelt, ?intlt\u27e9\n[GOAL]\ncase lt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x\ncase lsc\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g\ncase int\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 Integrable fun x => EReal.toReal (g x)\ncase aelt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ncase intlt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\ncase int =>\n show Integrable (fun x => EReal.toReal (g x)) \u03bc\n rw [integrable_congr ae_g]\n convert gp_integrable.sub gm_integrable\n simp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 Integrable fun x => EReal.toReal (g x)\n[PROOFSTEP]\ncase int =>\n show Integrable (fun x => EReal.toReal (g x)) \u03bc\n rw [integrable_congr ae_g]\n convert gp_integrable.sub gm_integrable\n simp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 Integrable fun x => EReal.toReal (g x)\n[PROOFSTEP]\nshow Integrable (fun x => EReal.toReal (g x)) \u03bc\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 Integrable fun x => EReal.toReal (g x)\n[PROOFSTEP]\nrw [integrable_congr ae_g]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 Integrable fun x => EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n[PROOFSTEP]\nconvert gp_integrable.sub gm_integrable\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx\u271d : \u03b1\n\u22a2 EReal.toReal \u2191(gp x\u271d) - EReal.toReal \u2191\u2191(gm x\u271d) = ((fun x => ENNReal.toReal (gp x)) - fun x => \u2191(gm x)) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase lt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x\ncase lsc\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g\ncase aelt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ncase intlt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\ncase intlt => show (\u222b x : \u03b1, (g x).toReal \u2202\u03bc) < (\u222b x : \u03b1, f x \u2202\u03bc) + \u03b5;\n exact\n calc\n (\u222b x : \u03b1, (g x).toReal \u2202\u03bc) = \u222b x : \u03b1, EReal.toReal (gp x) - EReal.toReal (gm x) \u2202\u03bc := integral_congr_ae ae_g\n _ = (\u222b x : \u03b1, EReal.toReal (gp x) \u2202\u03bc) - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc :=\n by\n simp only [EReal.toReal_coe_ennreal, ENNReal.coe_toReal]\n exact integral_sub gp_integrable gm_integrable\n _ < (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc :=\n by\n apply sub_lt_sub_right\n convert gpint\n simp only [EReal.toReal_coe_ennreal]\n _ \u2264 (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - ((\u222b x : \u03b1, \u2191(fm x) \u2202\u03bc) - \u03b4) := (sub_le_sub_left gmint _)\n _ = (\u222b x : \u03b1, f x \u2202\u03bc) + 2 * \u03b4 := by simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf]; ring\n _ = (\u222b x : \u03b1, f x \u2202\u03bc) + \u03b5 := by congr 1; field_simp [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\ncase intlt => show (\u222b x : \u03b1, (g x).toReal \u2202\u03bc) < (\u222b x : \u03b1, f x \u2202\u03bc) + \u03b5;\n exact\n calc\n (\u222b x : \u03b1, (g x).toReal \u2202\u03bc) = \u222b x : \u03b1, EReal.toReal (gp x) - EReal.toReal (gm x) \u2202\u03bc := integral_congr_ae ae_g\n _ = (\u222b x : \u03b1, EReal.toReal (gp x) \u2202\u03bc) - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc :=\n by\n simp only [EReal.toReal_coe_ennreal, ENNReal.coe_toReal]\n exact integral_sub gp_integrable gm_integrable\n _ < (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc :=\n by\n apply sub_lt_sub_right\n convert gpint\n simp only [EReal.toReal_coe_ennreal]\n _ \u2264 (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - ((\u222b x : \u03b1, \u2191(fm x) \u2202\u03bc) - \u03b4) := (sub_le_sub_left gmint _)\n _ = (\u222b x : \u03b1, f x \u2202\u03bc) + 2 * \u03b4 := by simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf]; ring\n _ = (\u222b x : \u03b1, f x \u2202\u03bc) + \u03b5 := by congr 1; field_simp [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nshow (\u222b x : \u03b1, (g x).toReal \u2202\u03bc) < (\u222b x : \u03b1, f x \u2202\u03bc) + \u03b5\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\nexact\n calc\n (\u222b x : \u03b1, (g x).toReal \u2202\u03bc) = \u222b x : \u03b1, EReal.toReal (gp x) - EReal.toReal (gm x) \u2202\u03bc := integral_congr_ae ae_g\n _ = (\u222b x : \u03b1, EReal.toReal (gp x) \u2202\u03bc) - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc :=\n by\n simp only [EReal.toReal_coe_ennreal, ENNReal.coe_toReal]\n exact integral_sub gp_integrable gm_integrable\n _ < (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc :=\n by\n apply sub_lt_sub_right\n convert gpint\n simp only [EReal.toReal_coe_ennreal]\n _ \u2264 (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - ((\u222b x : \u03b1, \u2191(fm x) \u2202\u03bc) - \u03b4) := (sub_le_sub_left gmint _)\n _ = (\u222b x : \u03b1, f x \u2202\u03bc) + 2 * \u03b4 := by simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf]; ring\n _ = (\u222b x : \u03b1, f x \u2202\u03bc) + \u03b5 := by congr 1; field_simp [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u2202\u03bc =\n \u222b (x : \u03b1), EReal.toReal \u2191(gp x) \u2202\u03bc - \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\n[PROOFSTEP]\nsimp only [EReal.toReal_coe_ennreal, ENNReal.coe_toReal]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), ENNReal.toReal (gp x) - \u2191(gm x) \u2202\u03bc = \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc - \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\n[PROOFSTEP]\nexact integral_sub gp_integrable gm_integrable\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal \u2191(gp x) \u2202\u03bc - \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + \u2191\u03b4 - \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\n[PROOFSTEP]\napply sub_lt_sub_right\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), EReal.toReal \u2191(gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + \u2191\u03b4\n[PROOFSTEP]\nconvert gpint\n[GOAL]\ncase h.e'_3.h.e'_7.h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx\u271d : \u03b1\n\u22a2 EReal.toReal \u2191(gp x\u271d) = ENNReal.toReal (gp x\u271d)\n[PROOFSTEP]\nsimp only [EReal.toReal_coe_ennreal]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + \u2191\u03b4 - (\u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - \u2191\u03b4) = \u222b (x : \u03b1), f x \u2202\u03bc + 2 * \u2191\u03b4\n[PROOFSTEP]\nsimp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), \u2191(Real.toNNReal (f x)) \u2202\u03bc + \u2191{ val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } -\n (\u222b (x : \u03b1), \u2191(Real.toNNReal (-f x)) \u2202\u03bc - \u2191{ val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }) =\n \u222b (x : \u03b1), \u2191(Real.toNNReal (f x)) \u2202\u03bc - \u222b (x : \u03b1), \u2191(Real.toNNReal (-f x)) \u2202\u03bc +\n 2 * \u2191{ val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n[PROOFSTEP]\nring\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u222b (x : \u03b1), f x \u2202\u03bc + 2 * \u2191\u03b4 = \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 2 * \u2191\u03b4 = \u03b5\n[PROOFSTEP]\nfield_simp [mul_comm]\n[GOAL]\ncase lt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x\ncase lsc\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g\ncase aelt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n[PROOFSTEP]\ncase aelt =>\n show \u2200\u1d50 x : \u03b1 \u2202\u03bc, g x < \u22a4\n filter_upwards [gp_lt_top] with ?_ hx\n simp only [sub_eq_add_neg, Ne.def, (EReal.add_lt_top _ _).ne, lt_top_iff_ne_top, lt_top_iff_ne_top.1 hx,\n EReal.coe_ennreal_eq_top_iff, not_false_iff, EReal.neg_eq_top_iff, EReal.coe_ennreal_ne_bot]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n[PROOFSTEP]\ncase aelt =>\n show \u2200\u1d50 x : \u03b1 \u2202\u03bc, g x < \u22a4\n filter_upwards [gp_lt_top] with ?_ hx\n simp only [sub_eq_add_neg, Ne.def, (EReal.add_lt_top _ _).ne, lt_top_iff_ne_top, lt_top_iff_ne_top.1 hx,\n EReal.coe_ennreal_eq_top_iff, not_false_iff, EReal.neg_eq_top_iff, EReal.coe_ennreal_ne_bot]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n[PROOFSTEP]\nshow \u2200\u1d50 x : \u03b1 \u2202\u03bc, g x < \u22a4\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n[PROOFSTEP]\nfilter_upwards [gp_lt_top] with ?_ hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx\u271d : \u03b1\nhx : gp x\u271d < \u22a4\n\u22a2 \u2191(gp x\u271d) - \u2191\u2191(gm x\u271d) < \u22a4\n[PROOFSTEP]\nsimp only [sub_eq_add_neg, Ne.def, (EReal.add_lt_top _ _).ne, lt_top_iff_ne_top, lt_top_iff_ne_top.1 hx,\n EReal.coe_ennreal_eq_top_iff, not_false_iff, EReal.neg_eq_top_iff, EReal.coe_ennreal_ne_bot]\n[GOAL]\ncase lt\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x\ncase lsc\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g\n[PROOFSTEP]\ncase lt =>\n show \u2200 x, (f x : EReal) < g x\n intro x\n rw [EReal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal (f x)]\n refine' EReal.sub_lt_sub_of_lt_of_le _ _ _ _\n \u00b7 simp only [EReal.coe_ennreal_lt_coe_ennreal_iff]; exact fp_lt_gp x\n \u00b7 simp only [ENNReal.coe_le_coe, EReal.coe_ennreal_le_coe_ennreal_iff]\n exact gm_le_fm x\n \u00b7 simp only [EReal.coe_ennreal_ne_bot, Ne.def, not_false_iff]\n \u00b7 simp only [EReal.coe_nnreal_ne_top, Ne.def, not_false_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x\n[PROOFSTEP]\ncase lt =>\n show \u2200 x, (f x : EReal) < g x\n intro x\n rw [EReal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal (f x)]\n refine' EReal.sub_lt_sub_of_lt_of_le _ _ _ _\n \u00b7 simp only [EReal.coe_ennreal_lt_coe_ennreal_iff]; exact fp_lt_gp x\n \u00b7 simp only [ENNReal.coe_le_coe, EReal.coe_ennreal_le_coe_ennreal_iff]\n exact gm_le_fm x\n \u00b7 simp only [EReal.coe_ennreal_ne_bot, Ne.def, not_false_iff]\n \u00b7 simp only [EReal.coe_nnreal_ne_top, Ne.def, not_false_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x\n[PROOFSTEP]\nshow \u2200 x, (f x : EReal) < g x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191(f x) < g x\n[PROOFSTEP]\nrw [EReal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal (f x)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191\u2191(Real.toNNReal (f x)) - \u2191\u2191(Real.toNNReal (-f x)) < g x\n[PROOFSTEP]\nrefine' EReal.sub_lt_sub_of_lt_of_le _ _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191\u2191(Real.toNNReal (f x)) < \u2191(gp x)\n[PROOFSTEP]\nsimp only [EReal.coe_ennreal_lt_coe_ennreal_iff]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191(Real.toNNReal (f x)) < gp x\n[PROOFSTEP]\nexact fp_lt_gp x\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191\u2191(gm x) \u2264 \u2191\u2191(Real.toNNReal (-f x))\n[PROOFSTEP]\nsimp only [ENNReal.coe_le_coe, EReal.coe_ennreal_le_coe_ennreal_iff]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 gm x \u2264 Real.toNNReal (-f x)\n[PROOFSTEP]\nexact gm_le_fm x\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191\u2191(gm x) \u2260 \u22a5\n[PROOFSTEP]\nsimp only [EReal.coe_ennreal_ne_bot, Ne.def, not_false_iff]\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 \u2191\u2191(Real.toNNReal (-f x)) \u2260 \u22a4\n[PROOFSTEP]\nsimp only [EReal.coe_nnreal_ne_top, Ne.def, not_false_iff]\n[GOAL]\ncase lsc\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g\n[PROOFSTEP]\ncase lsc =>\n show LowerSemicontinuous g\n apply LowerSemicontinuous.add'\n \u00b7\n exact\n continuous_coe_ennreal_ereal.comp_lowerSemicontinuous gpcont fun x y hxy =>\n EReal.coe_ennreal_le_coe_ennreal_iff.2 hxy\n \u00b7 apply continuous_neg.comp_upperSemicontinuous_antitone _ fun x y hxy => EReal.neg_le_neg_iff.2 hxy\n dsimp\n apply\n continuous_coe_ennreal_ereal.comp_upperSemicontinuous _ fun x y hxy => EReal.coe_ennreal_le_coe_ennreal_iff.2 hxy\n exact ENNReal.continuous_coe.comp_upperSemicontinuous gmcont fun x y hxy => ENNReal.coe_le_coe.2 hxy\n \u00b7 intro x\n exact EReal.continuousAt_add (by simp) (by simp)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g\n[PROOFSTEP]\ncase lsc =>\n show LowerSemicontinuous g\n apply LowerSemicontinuous.add'\n \u00b7\n exact\n continuous_coe_ennreal_ereal.comp_lowerSemicontinuous gpcont fun x y hxy =>\n EReal.coe_ennreal_le_coe_ennreal_iff.2 hxy\n \u00b7 apply continuous_neg.comp_upperSemicontinuous_antitone _ fun x y hxy => EReal.neg_le_neg_iff.2 hxy\n dsimp\n apply\n continuous_coe_ennreal_ereal.comp_upperSemicontinuous _ fun x y hxy => EReal.coe_ennreal_le_coe_ennreal_iff.2 hxy\n exact ENNReal.continuous_coe.comp_upperSemicontinuous gmcont fun x y hxy => ENNReal.coe_le_coe.2 hxy\n \u00b7 intro x\n exact EReal.continuousAt_add (by simp) (by simp)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g\n[PROOFSTEP]\nshow LowerSemicontinuous g\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous g\n[PROOFSTEP]\napply LowerSemicontinuous.add'\n[GOAL]\ncase hf\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous fun z => \u2191(gp z)\n[PROOFSTEP]\nexact\n continuous_coe_ennreal_ereal.comp_lowerSemicontinuous gpcont fun x y hxy => EReal.coe_ennreal_le_coe_ennreal_iff.2 hxy\n[GOAL]\ncase hg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 LowerSemicontinuous fun z => -\u2191\u2191(gm z)\n[PROOFSTEP]\napply continuous_neg.comp_upperSemicontinuous_antitone _ fun x y hxy => EReal.neg_le_neg_iff.2 hxy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 UpperSemicontinuous fun z => \u2191\u2191(gm z)\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 UpperSemicontinuous fun z => \u2191\u2191(gm z)\n[PROOFSTEP]\napply continuous_coe_ennreal_ereal.comp_upperSemicontinuous _ fun x y hxy => EReal.coe_ennreal_le_coe_ennreal_iff.2 hxy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 UpperSemicontinuous fun z => \u2191(gm z)\n[PROOFSTEP]\nexact ENNReal.continuous_coe.comp_upperSemicontinuous gmcont fun x y hxy => ENNReal.coe_le_coe.2 hxy\n[GOAL]\ncase hcont\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\n\u22a2 \u2200 (x : \u03b1), ContinuousAt (fun p => p.fst + p.snd) (\u2191(gp x), -\u2191\u2191(gm x))\n[PROOFSTEP]\nintro x\n[GOAL]\ncase hcont\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u2191(gp x), -\u2191\u2191(gm x))\n[PROOFSTEP]\nexact EReal.continuousAt_add (by simp) (by simp)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 (\u2191(gp x), -\u2191\u2191(gm x)).fst \u2260 \u22a4 \u2228 (\u2191(gp x), -\u2191\u2191(gm x)).snd \u2260 \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }\n\u03b4pos : 0 < \u03b4\nfp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x)\nint_fp : Integrable fun x => \u2191(fp x)\ngp : \u03b1 \u2192 \u211d\u22650\u221e\nfp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x\ngpcont : LowerSemicontinuous gp\ngp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4\ngp_integrable : Integrable fun x => ENNReal.toReal (gp x)\ngpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4\nfm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x)\nint_fm : Integrable fun x => \u2191(fm x)\ngm : \u03b1 \u2192 \u211d\u22650\ngm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x\ngmcont : UpperSemicontinuous gm\ngm_integrable : Integrable fun x => \u2191(gm x)\ngmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc\ng : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x)\nae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x)\nx : \u03b1\n\u22a2 (\u2191(gp x), -\u2191\u2191(gm x)).fst \u2260 \u22a5 \u2228 (\u2191(gp x), -\u2191\u2191(gm x)).snd \u2260 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x < \u2191(f x)) \u2227\n UpperSemicontinuous g \u2227\n (Integrable fun x => EReal.toReal (g x)) \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u22a5 < g x) \u2227 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrcases exists_lt_lowerSemicontinuous_integral_lt (fun x => -f x) hf.neg \u03b5pos with\n \u27e8g, g_lt_f, gcont, g_integrable, g_lt_top, gint\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 \u2203 g,\n (\u2200 (x : \u03b1), g x < \u2191(f x)) \u2227\n UpperSemicontinuous g \u2227\n (Integrable fun x => EReal.toReal (g x)) \u2227\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u22a5 < g x) \u2227 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun x => -g x, _, _, _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 \u2200 (x : \u03b1), (fun x => -g x) x < \u2191(f x)\n[PROOFSTEP]\nexact fun x => EReal.neg_lt_iff_neg_lt.1 (by simpa only [EReal.coe_neg] using g_lt_f x)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\nx : \u03b1\n\u22a2 -\u2191(f x) < g x\n[PROOFSTEP]\nsimpa only [EReal.coe_neg] using g_lt_f x\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 UpperSemicontinuous fun x => -g x\n[PROOFSTEP]\nexact continuous_neg.comp_lowerSemicontinuous_antitone gcont fun x y hxy => EReal.neg_le_neg_iff.2 hxy\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 Integrable fun x => EReal.toReal ((fun x => -g x) x)\n[PROOFSTEP]\nconvert g_integrable.neg\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\nx\u271d : \u03b1\n\u22a2 EReal.toReal ((fun x => -g x) x\u271d) = (-fun x => EReal.toReal (g x)) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u22a5 < (fun x => -g x) x\n[PROOFSTEP]\nsimpa [bot_lt_iff_ne_bot, lt_top_iff_ne_top] using g_lt_top\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_5\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5\n\u22a2 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal ((fun x => -g x) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nsimp_rw [integral_neg, lt_neg_add_iff_add_lt] at gint \n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_5\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (a : \u03b1), f a \u2202\u03bc + \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u03b5\n\u22a2 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal ((fun x => -g x) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nrw [add_comm] at gint \n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_5\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ng : \u03b1 \u2192 EReal\ng_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrable : Integrable fun x => EReal.toReal (g x)\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\ngint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc + \u222b (a : \u03b1), f a \u2202\u03bc < \u03b5\n\u22a2 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal ((fun x => -g x) x) \u2202\u03bc + \u03b5\n[PROOFSTEP]\nsimpa [integral_neg] using gint\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "llama_tokens": 138065, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430478583169, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.4350371841260001}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\ni : \u2115\n\u22a2 (head s :: tail s) i = s i\n[PROOFSTEP]\ncases i\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 (head s :: tail s) zero = s zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\nn\u271d : \u2115\n\u22a2 (head s :: tail s) (succ n\u271d) = s (succ n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn m : \u2115\ns : Stream' \u03b1\n\u22a2 drop n (drop m s) = drop (n + m) s\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn m : \u2115\ns : Stream' \u03b1\nn\u271d : \u2115\n\u22a2 nth (drop n (drop m s)) n\u271d = nth (drop (n + m) s) n\u271d\n[PROOFSTEP]\nsimp [Nat.add_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ni : \u2115\ns : Stream' \u03b1\n\u22a2 tail (drop i s) = drop (i + 1) s\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ni : \u2115\ns : Stream' \u03b1\nn\u271d : \u2115\n\u22a2 nth (tail (drop i s)) n\u271d = nth (drop (i + 1) s) n\u271d\n[PROOFSTEP]\nsimp [add_comm, add_assoc, add_left_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns : Stream' \u03b1\n\u22a2 tail (drop n s) = drop n (tail s)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : Stream' \u03b1\nn : \u2115\n\u22a2 head (drop n a) = nth a n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nx y : \u03b1\ns t : Stream' \u03b1\nh : x :: s = y :: t\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 nth_zero_cons x s, h, nth_zero_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nx y : \u03b1\ns t : Stream' \u03b1\nh : x :: s = y :: t\nn : \u2115\n\u22a2 nth s n = nth t n\n[PROOFSTEP]\nrw [\u2190 nth_succ_cons n _ x, h, nth_succ_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\ns : Stream' \u03b1\nb : \u03b1\nx\u271d : a \u2208 s\nn : \u2115\nh : (fun b => a = b) (nth s n)\n\u22a2 (fun b => a = b) (nth (b :: s) (succ n))\n[PROOFSTEP]\nrw [nth_succ, tail_cons, h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na b : \u03b1\ns : Stream' \u03b1\nx\u271d : a \u2208 b :: s\nn : \u2115\nh : (fun b => a = b) (nth (b :: s) n)\n\u22a2 a = b \u2228 a \u2208 s\n[PROOFSTEP]\ncases' n with n'\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na b : \u03b1\ns : Stream' \u03b1\nx\u271d : a \u2208 b :: s\nh : a = nth (b :: s) zero\n\u22a2 a = b \u2228 a \u2208 s\n[PROOFSTEP]\nleft\n[GOAL]\ncase zero.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na b : \u03b1\ns : Stream' \u03b1\nx\u271d : a \u2208 b :: s\nh : a = nth (b :: s) zero\n\u22a2 a = b\n[PROOFSTEP]\nexact h\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na b : \u03b1\ns : Stream' \u03b1\nx\u271d : a \u2208 b :: s\nn' : \u2115\nh : a = nth (b :: s) (succ n')\n\u22a2 a = b \u2228 a \u2208 s\n[PROOFSTEP]\nright\n[GOAL]\ncase succ.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na b : \u03b1\ns : Stream' \u03b1\nx\u271d : a \u2208 b :: s\nn' : \u2115\nh : a = nth (b :: s) (succ n')\n\u22a2 a \u2208 s\n[PROOFSTEP]\nrw [nth_succ, tail_cons] at h \n[GOAL]\ncase succ.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na b : \u03b1\ns : Stream' \u03b1\nx\u271d : a \u2208 b :: s\nn' : \u2115\nh : a = nth s n'\n\u22a2 a \u2208 s\n[PROOFSTEP]\nexact \u27e8n', h\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' \u03b1\n\u22a2 map f s = f (head s) :: map f (tail s)\n[PROOFSTEP]\nrw [\u2190 Stream'.eta (map f s), tail_map, head_map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Stream' \u03b1\n\u22a2 map f (a :: s) = f a :: map f s\n[PROOFSTEP]\nrw [\u2190 Stream'.eta (map f (a :: s)), map_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Stream' \u03b1\n\u22a2 head (f (head (a :: s)) :: map f (tail (a :: s))) :: tail (f (head (a :: s)) :: map f (tail (a :: s))) =\n f a :: map f s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Stream' \u03b1\nx\u271d : a \u2208 s\nn : \u2115\nh : (fun b => a = b) (nth s n)\n\u22a2 (fun b => f a = b) (nth (map f s) n)\n[PROOFSTEP]\nrw [nth_map, h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ns\u2081 : Stream' \u03b1\ns\u2082 : Stream' \u03b2\n\u22a2 zip f s\u2081 s\u2082 = f (head s\u2081) (head s\u2082) :: zip f (tail s\u2081) (tail s\u2082)\n[PROOFSTEP]\nrw [\u2190 Stream'.eta (zip f s\u2081 s\u2082)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ns\u2081 : Stream' \u03b1\ns\u2082 : Stream' \u03b2\n\u22a2 head (zip f s\u2081 s\u2082) :: tail (zip f s\u2081 s\u2082) = f (head s\u2081) (head s\u2082) :: zip f (tail s\u2081) (tail s\u2082)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\n\u22a2 const a = a :: const a\n[PROOFSTEP]\napply Stream'.ext\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\n\u22a2 \u2200 (n : \u2115), nth (const a) n = nth (a :: const a) n\n[PROOFSTEP]\nintro n\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nn : \u2115\n\u22a2 nth (const a) n = nth (a :: const a) n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase a.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\n\u22a2 nth (const a) zero = nth (a :: const a) zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nn\u271d : \u2115\n\u22a2 nth (const a) (succ n\u271d) = nth (a :: const a) (succ n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nthis : tail (a :: const a) = const a\n\u22a2 tail (const a) = const a\n[PROOFSTEP]\nrwa [\u2190 const_eq] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b1\na : \u03b1\n\u22a2 tail (iterate f a) = iterate f (f a)\n[PROOFSTEP]\next n\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b1\na : \u03b1\nn : \u2115\n\u22a2 nth (tail (iterate f a)) n = nth (iterate f (f a)) n\n[PROOFSTEP]\nrw [nth_tail]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b1\na : \u03b1\nn : \u2115\n\u22a2 nth (iterate f a) (n + 1) = nth (iterate f (f a)) n\n[PROOFSTEP]\ninduction' n with n' ih\n[GOAL]\ncase a.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b1\na : \u03b1\n\u22a2 nth (iterate f a) (zero + 1) = nth (iterate f (f a)) zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b1\na : \u03b1\nn' : \u2115\nih : nth (iterate f a) (n' + 1) = nth (iterate f (f a)) n'\n\u22a2 nth (iterate f a) (succ n' + 1) = nth (iterate f (f a)) (succ n')\n[PROOFSTEP]\nrw [nth_succ_iterate', ih, nth_succ_iterate']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b1\na : \u03b1\n\u22a2 iterate f a = a :: iterate f (f a)\n[PROOFSTEP]\nrw [\u2190 Stream'.eta (iterate f a)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b1\na : \u03b1\n\u22a2 head (iterate f a) :: tail (iterate f a) = a :: iterate f (f a)\n[PROOFSTEP]\nrw [tail_iterate]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b1\na : \u03b1\n\u22a2 head (iterate f a) :: iterate f (f a) = a :: iterate f (f a)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\nf : \u03b1 \u2192 \u03b1\na : \u03b1\n\u22a2 nth (iterate f a) (succ n) = nth (iterate f (f a)) n\n[PROOFSTEP]\nrw [nth_succ, tail_iterate]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d : Stream' \u03b1\nhh : head s\u2081\u271d = head s\u2082\u271d\nht\u2081 : s\u2081\u271d = tail s\u2081\u271d\nht\u2082 : s\u2082\u271d = tail s\u2082\u271d\ns\u2081 s\u2082 : Stream' \u03b1\nx\u271d : (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) s\u2081 s\u2082\nh\u2081 : head s\u2081 = head s\u2082\nh\u2082 : s\u2081 = tail s\u2081\nh\u2083 : s\u2082 = tail s\u2082\n\u22a2 head s\u2081 = head s\u2082 \u2227 (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) (tail s\u2081) (tail s\u2082)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d : Stream' \u03b1\nhh : head s\u2081\u271d = head s\u2082\u271d\nht\u2081 : s\u2081\u271d = tail s\u2081\u271d\nht\u2082 : s\u2082\u271d = tail s\u2082\u271d\ns\u2081 s\u2082 : Stream' \u03b1\nx\u271d : (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) s\u2081 s\u2082\nh\u2081 : head s\u2081 = head s\u2082\nh\u2082 : s\u2081 = tail s\u2081\nh\u2083 : s\u2082 = tail s\u2082\n\u22a2 head s\u2081 = head s\u2082\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d : Stream' \u03b1\nhh : head s\u2081\u271d = head s\u2082\u271d\nht\u2081 : s\u2081\u271d = tail s\u2081\u271d\nht\u2082 : s\u2082\u271d = tail s\u2082\u271d\ns\u2081 s\u2082 : Stream' \u03b1\nx\u271d : (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) s\u2081 s\u2082\nh\u2081 : head s\u2081 = head s\u2082\nh\u2082 : s\u2081 = tail s\u2081\nh\u2083 : s\u2082 = tail s\u2082\n\u22a2 (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) (tail s\u2081) (tail s\u2082)\n[PROOFSTEP]\nexact h\u2081\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d : Stream' \u03b1\nhh : head s\u2081\u271d = head s\u2082\u271d\nht\u2081 : s\u2081\u271d = tail s\u2081\u271d\nht\u2082 : s\u2082\u271d = tail s\u2082\u271d\ns\u2081 s\u2082 : Stream' \u03b1\nx\u271d : (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) s\u2081 s\u2082\nh\u2081 : head s\u2081 = head s\u2082\nh\u2082 : s\u2081 = tail s\u2081\nh\u2083 : s\u2082 = tail s\u2082\n\u22a2 (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) (tail s\u2081) (tail s\u2082)\n[PROOFSTEP]\nrw [\u2190 h\u2082, \u2190 h\u2083]\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d : Stream' \u03b1\nhh : head s\u2081\u271d = head s\u2082\u271d\nht\u2081 : s\u2081\u271d = tail s\u2081\u271d\nht\u2082 : s\u2082\u271d = tail s\u2082\u271d\ns\u2081 s\u2082 : Stream' \u03b1\nx\u271d : (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) s\u2081 s\u2082\nh\u2081 : head s\u2081 = head s\u2082\nh\u2082 : s\u2081 = tail s\u2081\nh\u2083 : s\u2082 = tail s\u2082\n\u22a2 (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) s\u2081 s\u2082\n[PROOFSTEP]\nrepeat' constructor\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d : Stream' \u03b1\nhh : head s\u2081\u271d = head s\u2082\u271d\nht\u2081 : s\u2081\u271d = tail s\u2081\u271d\nht\u2082 : s\u2082\u271d = tail s\u2082\u271d\ns\u2081 s\u2082 : Stream' \u03b1\nx\u271d : (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) s\u2081 s\u2082\nh\u2081 : head s\u2081 = head s\u2082\nh\u2082 : s\u2081 = tail s\u2081\nh\u2083 : s\u2082 = tail s\u2082\n\u22a2 (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) s\u2081 s\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase right.left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d : Stream' \u03b1\nhh : head s\u2081\u271d = head s\u2082\u271d\nht\u2081 : s\u2081\u271d = tail s\u2081\u271d\nht\u2082 : s\u2082\u271d = tail s\u2082\u271d\ns\u2081 s\u2082 : Stream' \u03b1\nx\u271d : (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) s\u2081 s\u2082\nh\u2081 : head s\u2081 = head s\u2082\nh\u2082 : s\u2081 = tail s\u2081\nh\u2083 : s\u2082 = tail s\u2082\n\u22a2 head s\u2081 = head s\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase right.right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d : Stream' \u03b1\nhh : head s\u2081\u271d = head s\u2082\u271d\nht\u2081 : s\u2081\u271d = tail s\u2081\u271d\nht\u2082 : s\u2082\u271d = tail s\u2082\u271d\ns\u2081 s\u2082 : Stream' \u03b1\nx\u271d : (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) s\u2081 s\u2082\nh\u2081 : head s\u2081 = head s\u2082\nh\u2082 : s\u2081 = tail s\u2081\nh\u2083 : s\u2082 = tail s\u2082\n\u22a2 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase right.right.left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d : Stream' \u03b1\nhh : head s\u2081\u271d = head s\u2082\u271d\nht\u2081 : s\u2081\u271d = tail s\u2081\u271d\nht\u2082 : s\u2082\u271d = tail s\u2082\u271d\ns\u2081 s\u2082 : Stream' \u03b1\nx\u271d : (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) s\u2081 s\u2082\nh\u2081 : head s\u2081 = head s\u2082\nh\u2082 : s\u2081 = tail s\u2081\nh\u2083 : s\u2082 = tail s\u2082\n\u22a2 s\u2081 = tail s\u2081\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase right.right.right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d : Stream' \u03b1\nhh : head s\u2081\u271d = head s\u2082\u271d\nht\u2081 : s\u2081\u271d = tail s\u2081\u271d\nht\u2082 : s\u2082\u271d = tail s\u2082\u271d\ns\u2081 s\u2082 : Stream' \u03b1\nx\u271d : (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) s\u2081 s\u2082\nh\u2081 : head s\u2081 = head s\u2082\nh\u2082 : s\u2081 = tail s\u2081\nh\u2083 : s\u2082 = tail s\u2082\n\u22a2 s\u2082 = tail s\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase right.left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d : Stream' \u03b1\nhh : head s\u2081\u271d = head s\u2082\u271d\nht\u2081 : s\u2081\u271d = tail s\u2081\u271d\nht\u2082 : s\u2082\u271d = tail s\u2082\u271d\ns\u2081 s\u2082 : Stream' \u03b1\nx\u271d : (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) s\u2081 s\u2082\nh\u2081 : head s\u2081 = head s\u2082\nh\u2082 : s\u2081 = tail s\u2081\nh\u2083 : s\u2082 = tail s\u2082\n\u22a2 head s\u2081 = head s\u2082\n[PROOFSTEP]\nassumption\n[GOAL]\ncase right.right.left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d : Stream' \u03b1\nhh : head s\u2081\u271d = head s\u2082\u271d\nht\u2081 : s\u2081\u271d = tail s\u2081\u271d\nht\u2082 : s\u2082\u271d = tail s\u2082\u271d\ns\u2081 s\u2082 : Stream' \u03b1\nx\u271d : (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) s\u2081 s\u2082\nh\u2081 : head s\u2081 = head s\u2082\nh\u2082 : s\u2081 = tail s\u2081\nh\u2083 : s\u2082 = tail s\u2082\n\u22a2 s\u2081 = tail s\u2081\n[PROOFSTEP]\nassumption\n[GOAL]\ncase right.right.right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d : Stream' \u03b1\nhh : head s\u2081\u271d = head s\u2082\u271d\nht\u2081 : s\u2081\u271d = tail s\u2081\u271d\nht\u2082 : s\u2082\u271d = tail s\u2082\u271d\ns\u2081 s\u2082 : Stream' \u03b1\nx\u271d : (fun s\u2081 s\u2082 => head s\u2081 = head s\u2082 \u2227 s\u2081 = tail s\u2081 \u2227 s\u2082 = tail s\u2082) s\u2081 s\u2082\nh\u2081 : head s\u2081 = head s\u2082\nh\u2082 : s\u2081 = tail s\u2081\nh\u2083 : s\u2082 = tail s\u2082\n\u22a2 s\u2082 = tail s\u2082\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b4 : Type w\na : \u03b1\n\u03b2 : Type u\nfr : Stream' \u03b1 \u2192 \u03b2\nch : fr (iterate id a) = fr (const a)\n\u22a2 fr (tail (iterate id a)) = fr (tail (const a))\n[PROOFSTEP]\nrw [tail_iterate, tail_const]\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b4 : Type w\na : \u03b1\n\u03b2 : Type u\nfr : Stream' \u03b1 \u2192 \u03b2\nch : fr (iterate id a) = fr (const a)\n\u22a2 fr (iterate id (id a)) = fr (const a)\n[PROOFSTEP]\nexact ch\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b1\na : \u03b1\n\u22a2 iterate f (f a) = map f (iterate f a)\n[PROOFSTEP]\nfunext n\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b1\na : \u03b1\nn : \u2115\n\u22a2 iterate f (f a) n = map f (iterate f a) n\n[PROOFSTEP]\ninduction' n with n' ih\n[GOAL]\ncase h.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b1\na : \u03b1\n\u22a2 iterate f (f a) zero = map f (iterate f a) zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b1\na : \u03b1\nn' : \u2115\nih : iterate f (f a) n' = map f (iterate f a) n'\n\u22a2 iterate f (f a) (succ n') = map f (iterate f a) (succ n')\n[PROOFSTEP]\nunfold map iterate nth\n[GOAL]\ncase h.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b1\na : \u03b1\nn' : \u2115\nih : iterate f (f a) n' = map f (iterate f a) n'\n\u22a2 f (iterate f (f a) n') = f (iterate f a (succ n'))\n[PROOFSTEP]\nrw [map, nth] at ih \n[GOAL]\ncase h.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b1\na : \u03b1\nn' : \u2115\nih : iterate f (f a) n' = f (iterate f a n')\n\u22a2 f (iterate f (f a) n') = f (iterate f a (succ n'))\n[PROOFSTEP]\nrw [iterate]\n[GOAL]\ncase h.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b1\na : \u03b1\nn' : \u2115\nih : iterate f (f a) n' = f (iterate f a n')\n\u22a2 f (iterate f (f a) n') = f (f (iterate f a n'))\n[PROOFSTEP]\nexact congrArg f ih\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b1\na : \u03b1\n\u22a2 corec f g a = f a :: corec f g (g a)\n[PROOFSTEP]\nrw [corec_def, map_eq, head_iterate, tail_iterate]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b1\na : \u03b1\n\u22a2 f a :: map f (iterate g (g a)) = f a :: corec f g (g a)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\n\u22a2 corec id id a = const a\n[PROOFSTEP]\nrw [corec_def, map_id, iterate_id]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ng : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\na : \u03b1\n\u22a2 unfolds g f a = g a :: unfolds g f (f a)\n[PROOFSTEP]\nunfold unfolds\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ng : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\na : \u03b1\n\u22a2 corec g f a = g a :: corec g f (f a)\n[PROOFSTEP]\nrw [corec_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\n\u22a2 \u2200 (n : \u2115) (s : Stream' \u03b1), nth (unfolds head tail s) n = nth s n\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\n\u22a2 \u2200 (s : Stream' \u03b1), nth (unfolds head tail s) n = nth s n\n[PROOFSTEP]\ninduction' n with n' ih\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\n\u22a2 \u2200 (s : Stream' \u03b1), nth (unfolds head tail s) zero = nth s zero\n[PROOFSTEP]\nintro s\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 nth (unfolds head tail s) zero = nth s zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn' : \u2115\nih : \u2200 (s : Stream' \u03b1), nth (unfolds head tail s) n' = nth s n'\n\u22a2 \u2200 (s : Stream' \u03b1), nth (unfolds head tail s) (succ n') = nth s (succ n')\n[PROOFSTEP]\nintro s\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn' : \u2115\nih : \u2200 (s : Stream' \u03b1), nth (unfolds head tail s) n' = nth s n'\ns : Stream' \u03b1\n\u22a2 nth (unfolds head tail s) (succ n') = nth s (succ n')\n[PROOFSTEP]\nrw [nth_succ, nth_succ, unfolds_eq, tail_cons, ih]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\n\u22a2 s\u2081 \u22c8 s\u2082 = head s\u2081 :: head s\u2082 :: (tail s\u2081 \u22c8 tail s\u2082)\n[PROOFSTEP]\nlet t := tail s\u2081 \u22c8 tail s\u2082\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nt : Stream' \u03b1 := tail s\u2081 \u22c8 tail s\u2082\n\u22a2 s\u2081 \u22c8 s\u2082 = head s\u2081 :: head s\u2082 :: (tail s\u2081 \u22c8 tail s\u2082)\n[PROOFSTEP]\nshow s\u2081 \u22c8 s\u2082 = head s\u2081 :: head s\u2082 :: t\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nt : Stream' \u03b1 := tail s\u2081 \u22c8 tail s\u2082\n\u22a2 s\u2081 \u22c8 s\u2082 = head s\u2081 :: head s\u2082 :: t\n[PROOFSTEP]\nunfold interleave\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nt : Stream' \u03b1 := tail s\u2081 \u22c8 tail s\u2082\n\u22a2 (corecOn (s\u2081, s\u2082)\n (fun x =>\n match x with\n | (s\u2081, snd) => head s\u2081)\n fun x =>\n match x with\n | (s\u2081, s\u2082) => (s\u2082, tail s\u2081)) =\n head s\u2081 :: head s\u2082 :: t\n[PROOFSTEP]\nunfold corecOn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nt : Stream' \u03b1 := tail s\u2081 \u22c8 tail s\u2082\n\u22a2 corec\n (fun x =>\n match x with\n | (s\u2081, snd) => head s\u2081)\n (fun x =>\n match x with\n | (s\u2081, s\u2082) => (s\u2082, tail s\u2081))\n (s\u2081, s\u2082) =\n head s\u2081 :: head s\u2082 :: t\n[PROOFSTEP]\nrw [corec_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nt : Stream' \u03b1 := tail s\u2081 \u22c8 tail s\u2082\n\u22a2 (match (s\u2081, s\u2082) with\n | (s\u2081, snd) => head s\u2081) ::\n corec\n (fun x =>\n match x with\n | (s\u2081, snd) => head s\u2081)\n (fun x =>\n match x with\n | (s\u2081, s\u2082) => (s\u2082, tail s\u2081))\n (match (s\u2081, s\u2082) with\n | (s\u2081, s\u2082) => (s\u2082, tail s\u2081)) =\n head s\u2081 :: head s\u2082 :: t\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nt : Stream' \u03b1 := tail s\u2081 \u22c8 tail s\u2082\n\u22a2 head s\u2081 :: corec (fun x => head x.fst) (fun x => (x.snd, tail x.fst)) (s\u2082, tail s\u2081) =\n head s\u2081 :: head s\u2082 :: (tail s\u2081 \u22c8 tail s\u2082)\n[PROOFSTEP]\nrw [corec_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nt : Stream' \u03b1 := tail s\u2081 \u22c8 tail s\u2082\n\u22a2 head s\u2081 ::\n head (s\u2082, tail s\u2081).fst ::\n corec (fun x => head x.fst) (fun x => (x.snd, tail x.fst)) ((s\u2082, tail s\u2081).snd, tail (s\u2082, tail s\u2081).fst) =\n head s\u2081 :: head s\u2082 :: (tail s\u2081 \u22c8 tail s\u2082)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\n\u22a2 tail (s\u2081 \u22c8 s\u2082) = s\u2082 \u22c8 tail s\u2081\n[PROOFSTEP]\nunfold interleave corecOn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\n\u22a2 tail\n (corec\n (fun x =>\n match x with\n | (s\u2081, snd) => head s\u2081)\n (fun x =>\n match x with\n | (s\u2081, s\u2082) => (s\u2082, tail s\u2081))\n (s\u2081, s\u2082)) =\n corec\n (fun x =>\n match x with\n | (s\u2081, snd) => head s\u2081)\n (fun x =>\n match x with\n | (s\u2081, s\u2082) => (s\u2082, tail s\u2081))\n (s\u2082, tail s\u2081)\n[PROOFSTEP]\nrw [corec_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\n\u22a2 tail\n ((match (s\u2081, s\u2082) with\n | (s\u2081, snd) => head s\u2081) ::\n corec\n (fun x =>\n match x with\n | (s\u2081, snd) => head s\u2081)\n (fun x =>\n match x with\n | (s\u2081, s\u2082) => (s\u2082, tail s\u2081))\n (match (s\u2081, s\u2082) with\n | (s\u2081, s\u2082) => (s\u2082, tail s\u2081))) =\n corec\n (fun x =>\n match x with\n | (s\u2081, snd) => head s\u2081)\n (fun x =>\n match x with\n | (s\u2081, s\u2082) => (s\u2082, tail s\u2081))\n (s\u2082, tail s\u2081)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\n\u22a2 tail s\u2081 \u22c8 tail s\u2082 = tail (tail (s\u2081 \u22c8 s\u2082))\n[PROOFSTEP]\nrw [interleave_eq s\u2081 s\u2082]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\n\u22a2 tail s\u2081 \u22c8 tail s\u2082 = tail (tail (head s\u2081 :: head s\u2082 :: (tail s\u2081 \u22c8 tail s\u2082)))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns\u2081 s\u2082 : Stream' \u03b1\n\u22a2 nth (s\u2081 \u22c8 s\u2082) (2 * (n + 1)) = nth s\u2081 (n + 1)\n[PROOFSTEP]\nchange nth (s\u2081 \u22c8 s\u2082) (succ (succ (2 * n))) = nth s\u2081 (succ n)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns\u2081 s\u2082 : Stream' \u03b1\n\u22a2 nth (s\u2081 \u22c8 s\u2082) (succ (succ (2 * n))) = nth s\u2081 (succ n)\n[PROOFSTEP]\nrw [nth_succ, nth_succ, interleave_eq, tail_cons, tail_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns\u2081 s\u2082 : Stream' \u03b1\n\u22a2 nth (tail s\u2081 \u22c8 tail s\u2082) (2 * n) = nth s\u2081 (succ n)\n[PROOFSTEP]\nhave : n < succ n := Nat.lt_succ_self n\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns\u2081 s\u2082 : Stream' \u03b1\nthis : n < succ n\n\u22a2 nth (tail s\u2081 \u22c8 tail s\u2082) (2 * n) = nth s\u2081 (succ n)\n[PROOFSTEP]\nrw [nth_interleave_left n (tail s\u2081) (tail s\u2082)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns\u2081 s\u2082 : Stream' \u03b1\nthis : n < succ n\n\u22a2 nth (tail s\u2081) n = nth s\u2081 (succ n)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns\u2081 s\u2082 : Stream' \u03b1\n\u22a2 nth (s\u2081 \u22c8 s\u2082) (2 * (n + 1) + 1) = nth s\u2082 (n + 1)\n[PROOFSTEP]\nchange nth (s\u2081 \u22c8 s\u2082) (succ (succ (2 * n + 1))) = nth s\u2082 (succ n)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns\u2081 s\u2082 : Stream' \u03b1\n\u22a2 nth (s\u2081 \u22c8 s\u2082) (succ (succ (2 * n + 1))) = nth s\u2082 (succ n)\n[PROOFSTEP]\nrw [nth_succ, nth_succ, interleave_eq, tail_cons, tail_cons, nth_interleave_right n (tail s\u2081) (tail s\u2082)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns\u2081 s\u2082 : Stream' \u03b1\n\u22a2 nth (tail s\u2082) n = nth s\u2082 (succ n)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\ns\u2081 s\u2082 : Stream' \u03b1\nx\u271d : a \u2208 s\u2081\nn : \u2115\nh : (fun b => a = b) (nth s\u2081 n)\n\u22a2 (fun b => a = b) (nth (s\u2081 \u22c8 s\u2082) (2 * n))\n[PROOFSTEP]\nrw [h, nth_interleave_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\ns\u2081 s\u2082 : Stream' \u03b1\nx\u271d : a \u2208 s\u2082\nn : \u2115\nh : (fun b => a = b) (nth s\u2082 n)\n\u22a2 (fun b => a = b) (nth (s\u2081 \u22c8 s\u2082) (2 * n + 1))\n[PROOFSTEP]\nrw [h, nth_interleave_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 tail (even s) = even (tail (tail s))\n[PROOFSTEP]\nunfold even\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 tail (corec (fun s => head s) (fun s => tail (tail s)) s) =\n corec (fun s => head s) (fun s => tail (tail s)) (tail (tail s))\n[PROOFSTEP]\nrw [corec_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 tail (head s :: corec (fun s => head s) (fun s => tail (tail s)) (tail (tail s))) =\n corec (fun s => head s) (fun s => tail (tail s)) (tail (tail s))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na\u2081 a\u2082 : \u03b1\ns : Stream' \u03b1\n\u22a2 even (a\u2081 :: a\u2082 :: s) = a\u2081 :: even s\n[PROOFSTEP]\nunfold even\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na\u2081 a\u2082 : \u03b1\ns : Stream' \u03b1\n\u22a2 corec (fun s => head s) (fun s => tail (tail s)) (a\u2081 :: a\u2082 :: s) =\n a\u2081 :: corec (fun s => head s) (fun s => tail (tail s)) s\n[PROOFSTEP]\nrw [corec_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na\u2081 a\u2082 : \u03b1\ns : Stream' \u03b1\n\u22a2 head (a\u2081 :: a\u2082 :: s) :: corec (fun s => head s) (fun s => tail (tail s)) (tail (tail (a\u2081 :: a\u2082 :: s))) =\n a\u2081 :: corec (fun s => head s) (fun s => tail (tail s)) s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d s\u2081' s\u2081 : Stream' \u03b1\nx\u271d : (fun s\u2081' s\u2081 => \u2203 s\u2082, s\u2081' = even (s\u2081 \u22c8 s\u2082)) s\u2081' s\u2081\ns\u2082 : Stream' \u03b1\nh\u2081 : s\u2081' = even (s\u2081 \u22c8 s\u2082)\n\u22a2 head s\u2081' = head s\u2081 \u2227 (fun s\u2081' s\u2081 => \u2203 s\u2082, s\u2081' = even (s\u2081 \u22c8 s\u2082)) (tail s\u2081') (tail s\u2081)\n[PROOFSTEP]\nrw [h\u2081]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d s\u2081' s\u2081 : Stream' \u03b1\nx\u271d : (fun s\u2081' s\u2081 => \u2203 s\u2082, s\u2081' = even (s\u2081 \u22c8 s\u2082)) s\u2081' s\u2081\ns\u2082 : Stream' \u03b1\nh\u2081 : s\u2081' = even (s\u2081 \u22c8 s\u2082)\n\u22a2 head (even (s\u2081 \u22c8 s\u2082)) = head s\u2081 \u2227 (fun s\u2081' s\u2081 => \u2203 s\u2082, s\u2081' = even (s\u2081 \u22c8 s\u2082)) (tail (even (s\u2081 \u22c8 s\u2082))) (tail s\u2081)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d s\u2081' s\u2081 : Stream' \u03b1\nx\u271d : (fun s\u2081' s\u2081 => \u2203 s\u2082, s\u2081' = even (s\u2081 \u22c8 s\u2082)) s\u2081' s\u2081\ns\u2082 : Stream' \u03b1\nh\u2081 : s\u2081' = even (s\u2081 \u22c8 s\u2082)\n\u22a2 head (even (s\u2081 \u22c8 s\u2082)) = head s\u2081\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d s\u2081' s\u2081 : Stream' \u03b1\nx\u271d : (fun s\u2081' s\u2081 => \u2203 s\u2082, s\u2081' = even (s\u2081 \u22c8 s\u2082)) s\u2081' s\u2081\ns\u2082 : Stream' \u03b1\nh\u2081 : s\u2081' = even (s\u2081 \u22c8 s\u2082)\n\u22a2 (fun s\u2081' s\u2081 => \u2203 s\u2082, s\u2081' = even (s\u2081 \u22c8 s\u2082)) (tail (even (s\u2081 \u22c8 s\u2082))) (tail s\u2081)\n[PROOFSTEP]\nexact \u27e8tail s\u2082, by rw [interleave_eq, even_cons_cons, tail_cons]\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081\u271d s\u2082\u271d s\u2081' s\u2081 : Stream' \u03b1\nx\u271d : (fun s\u2081' s\u2081 => \u2203 s\u2082, s\u2081' = even (s\u2081 \u22c8 s\u2082)) s\u2081' s\u2081\ns\u2082 : Stream' \u03b1\nh\u2081 : s\u2081' = even (s\u2081 \u22c8 s\u2082)\n\u22a2 tail (even (s\u2081 \u22c8 s\u2082)) = even (tail s\u2081 \u22c8 tail s\u2082)\n[PROOFSTEP]\nrw [interleave_eq, even_cons_cons, tail_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s' s : Stream' \u03b1\nh : s' = even s \u22c8 odd s\n\u22a2 head s' = head s \u2227 (fun s' s => s' = even s \u22c8 odd s) (tail s') (tail s)\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s' s : Stream' \u03b1\nh : s' = even s \u22c8 odd s\n\u22a2 head (even s \u22c8 odd s) = head s \u2227 (fun s' s => s' = even s \u22c8 odd s) (tail (even s \u22c8 odd s)) (tail s)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s' s : Stream' \u03b1\nh : s' = even s \u22c8 odd s\n\u22a2 head (even s \u22c8 odd s) = head s\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s' s : Stream' \u03b1\nh : s' = even s \u22c8 odd s\n\u22a2 (fun s' s => s' = even s \u22c8 odd s) (tail (even s \u22c8 odd s)) (tail s)\n[PROOFSTEP]\nsimp [odd_eq, odd_eq, tail_interleave, tail_even]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns : Stream' \u03b1\n\u22a2 nth (even s) (succ n) = nth s (2 * succ n)\n[PROOFSTEP]\nchange nth (even s) (succ n) = nth s (succ (succ (2 * n)))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns : Stream' \u03b1\n\u22a2 nth (even s) (succ n) = nth s (succ (succ (2 * n)))\n[PROOFSTEP]\nrw [nth_succ, nth_succ, tail_even, nth_even n]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns : Stream' \u03b1\n\u22a2 nth (tail (tail s)) (2 * n) = nth (tail s) (2 * n + 1)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns : Stream' \u03b1\n\u22a2 nth (odd s) n = nth s (2 * n + 1)\n[PROOFSTEP]\nrw [odd_eq, nth_even]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns : Stream' \u03b1\n\u22a2 nth (tail s) (2 * n) = nth s (2 * n + 1)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\ns : Stream' \u03b1\nx\u271d : a \u2208 even s\nn : \u2115\nh : (fun b => a = b) (nth (even s) n)\n\u22a2 (fun b => a = b) (nth s (2 * n))\n[PROOFSTEP]\nrw [h, nth_even]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\ns : Stream' \u03b1\nx\u271d : a \u2208 odd s\nn : \u2115\nh : (fun b => a = b) (nth (odd s) n)\n\u22a2 (fun b => a = b) (nth s (2 * n + 1))\n[PROOFSTEP]\nrw [h, nth_odd]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\ns : Stream' \u03b1\n\u22a2 a :: l\u2081 ++ l\u2082 ++\u209b s = a :: l\u2081 ++\u209b (l\u2082 ++\u209b s)\n[PROOFSTEP]\nrw [List.cons_append, cons_append_stream, cons_append_stream, append_append_stream l\u2081]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl : List \u03b1\ns : Stream' \u03b1\n\u22a2 map f (a :: l ++\u209b s) = List.map f (a :: l) ++\u209b map f s\n[PROOFSTEP]\nrw [cons_append_stream, List.map_cons, map_cons, cons_append_stream, map_append_stream f l]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 drop (List.length []) ([] ++\u209b s) = s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\ns : Stream' \u03b1\n\u22a2 drop (List.length (a :: l)) (a :: l ++\u209b s) = s\n[PROOFSTEP]\nrw [List.length_cons, drop_succ, cons_append_stream, tail_cons, drop_append_stream l s]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 [head s] ++\u209b tail s = s\n[PROOFSTEP]\nrw [cons_append_stream, nil_append_stream, Stream'.eta]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\nn : \u2115\n\u22a2 take (n + 1 + 1) s = take (n + 1) s ++ [nth s (n + 1)]\n[PROOFSTEP]\nrw [take_succ, take_succ' n, \u2190 List.cons_append, \u2190 take_succ, nth_tail]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns : Stream' \u03b1\n\u22a2 List.length (take n s) = n\n[PROOFSTEP]\ninduction n generalizing s\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 List.length (take zero s) = zero\n[PROOFSTEP]\nsimp [*, take_succ]\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn\u271d : \u2115\nn_ih\u271d : \u2200 (s : Stream' \u03b1), List.length (take n\u271d s) = n\u271d\ns : Stream' \u03b1\n\u22a2 List.length (take (succ n\u271d) s) = succ n\u271d\n[PROOFSTEP]\nsimp [*, take_succ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\nn : \u2115\n\u22a2 List.take 0 (take n s) = take (min n 0) s\n[PROOFSTEP]\nrw [min_zero, List.take_zero, take_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\nm : \u2115\n\u22a2 List.take m (take 0 s) = take (min 0 m) s\n[PROOFSTEP]\nrw [zero_min, take_zero, List.take_nil]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\nm n : \u2115\n\u22a2 List.take (m + 1) (take (n + 1) s) = take (min (n + 1) (m + 1)) s\n[PROOFSTEP]\nrw [take_succ, List.take_cons, Nat.min_succ_succ, take_succ, take_take]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\nk n : \u2115\nh : k + 1 < n + 1\n\u22a2 List.get? (take (n + 1) s) (k + 1) = some (nth s (k + 1))\n[PROOFSTEP]\nrw [take_succ, List.get?, get?_take (Nat.lt_of_succ_lt_succ h), nth_succ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\nxs : Stream' \u03b1\n\u22a2 List.dropLast (take n xs) = take (n - 1) xs\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nxs : Stream' \u03b1\n\u22a2 List.dropLast (take zero xs) = take (zero - 1) xs\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nxs : Stream' \u03b1\nn\u271d : \u2115\n\u22a2 List.dropLast (take (succ n\u271d) xs) = take (succ n\u271d - 1) xs\n[PROOFSTEP]\ncase zero => simp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nxs : Stream' \u03b1\n\u22a2 List.dropLast (take zero xs) = take (zero - 1) xs\n[PROOFSTEP]\ncase zero => simp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nxs : Stream' \u03b1\n\u22a2 List.dropLast (take zero xs) = take (zero - 1) xs\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nxs : Stream' \u03b1\nn\u271d : \u2115\n\u22a2 List.dropLast (take (succ n\u271d) xs) = take (succ n\u271d - 1) xs\n[PROOFSTEP]\ncase succ n => rw [take_succ', List.dropLast_concat, Nat.succ_sub_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nxs : Stream' \u03b1\nn : \u2115\n\u22a2 List.dropLast (take (succ n) xs) = take (succ n - 1) xs\n[PROOFSTEP]\ncase succ n => rw [take_succ', List.dropLast_concat, Nat.succ_sub_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nxs : Stream' \u03b1\nn : \u2115\n\u22a2 List.dropLast (take (succ n) xs) = take (succ n - 1) xs\n[PROOFSTEP]\nrw [take_succ', List.dropLast_concat, Nat.succ_sub_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\n\u22a2 \u2200 (n : \u2115) (s : Stream' \u03b1), take n s ++\u209b drop n s = s\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\n\u22a2 \u2200 (s : Stream' \u03b1), take n s ++\u209b drop n s = s\n[PROOFSTEP]\ninduction' n with n' ih\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\n\u22a2 \u2200 (s : Stream' \u03b1), take zero s ++\u209b drop zero s = s\n[PROOFSTEP]\nintro s\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 take zero s ++\u209b drop zero s = s\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn' : \u2115\nih : \u2200 (s : Stream' \u03b1), take n' s ++\u209b drop n' s = s\n\u22a2 \u2200 (s : Stream' \u03b1), take (succ n') s ++\u209b drop (succ n') s = s\n[PROOFSTEP]\nintro s\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn' : \u2115\nih : \u2200 (s : Stream' \u03b1), take n' s ++\u209b drop n' s = s\ns : Stream' \u03b1\n\u22a2 take (succ n') s ++\u209b drop (succ n') s = s\n[PROOFSTEP]\nrw [take_succ, drop_succ, cons_append_stream, ih (tail s), Stream'.eta]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\n\u22a2 (\u2200 (n : \u2115), take n s\u2081 = take n s\u2082) \u2192 s\u2081 = s\u2082\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nh : \u2200 (n : \u2115), take n s\u2081 = take n s\u2082\n\u22a2 s\u2081 = s\u2082\n[PROOFSTEP]\napply Stream'.ext\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nh : \u2200 (n : \u2115), take n s\u2081 = take n s\u2082\n\u22a2 \u2200 (n : \u2115), nth s\u2081 n = nth s\u2082 n\n[PROOFSTEP]\nintro n\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nh : \u2200 (n : \u2115), take n s\u2081 = take n s\u2082\nn : \u2115\n\u22a2 nth s\u2081 n = nth s\u2082 n\n[PROOFSTEP]\ninduction' n with n _\n[GOAL]\ncase a.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nh : \u2200 (n : \u2115), take n s\u2081 = take n s\u2082\n\u22a2 nth s\u2081 zero = nth s\u2082 zero\n[PROOFSTEP]\nhave aux := h 1\n[GOAL]\ncase a.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nh : \u2200 (n : \u2115), take n s\u2081 = take n s\u2082\naux : take 1 s\u2081 = take 1 s\u2082\n\u22a2 nth s\u2081 zero = nth s\u2082 zero\n[PROOFSTEP]\nsimp [take] at aux \n[GOAL]\ncase a.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nh : \u2200 (n : \u2115), take n s\u2081 = take n s\u2082\naux : head s\u2081 = head s\u2082\n\u22a2 nth s\u2081 zero = nth s\u2082 zero\n[PROOFSTEP]\nexact aux\n[GOAL]\ncase a.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nh : \u2200 (n : \u2115), take n s\u2081 = take n s\u2082\nn : \u2115\nn_ih\u271d : nth s\u2081 n = nth s\u2082 n\n\u22a2 nth s\u2081 (succ n) = nth s\u2082 (succ n)\n[PROOFSTEP]\nhave h\u2081 : some (nth s\u2081 (succ n)) = some (nth s\u2082 (succ n)) := by\n rw [\u2190 get?_take_succ, \u2190 get?_take_succ, h (succ (succ n))]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nh : \u2200 (n : \u2115), take n s\u2081 = take n s\u2082\nn : \u2115\nn_ih\u271d : nth s\u2081 n = nth s\u2082 n\n\u22a2 some (nth s\u2081 (succ n)) = some (nth s\u2082 (succ n))\n[PROOFSTEP]\nrw [\u2190 get?_take_succ, \u2190 get?_take_succ, h (succ (succ n))]\n[GOAL]\ncase a.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s\u2082 : Stream' \u03b1\nh : \u2200 (n : \u2115), take n s\u2081 = take n s\u2082\nn : \u2115\nn_ih\u271d : nth s\u2081 n = nth s\u2082 n\nh\u2081 : some (nth s\u2081 (succ n)) = some (nth s\u2082 (succ n))\n\u22a2 nth s\u2081 (succ n) = nth s\u2082 (succ n)\n[PROOFSTEP]\ninjection h\u2081\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\n\u22a2 \u2200 (l' : List \u03b1) (a' : \u03b1),\n corec Stream'.cycleF Stream'.cycleG (a', l', a, l) = a' :: l' ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\n[PROOFSTEP]\nintro l'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\nl' : List \u03b1\n\u22a2 \u2200 (a' : \u03b1),\n corec Stream'.cycleF Stream'.cycleG (a', l', a, l) = a' :: l' ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\n[PROOFSTEP]\ninduction' l' with a\u2081 l\u2081 ih\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\n\u22a2 \u2200 (a' : \u03b1),\n corec Stream'.cycleF Stream'.cycleG (a', [], a, l) = [a'] ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\n[PROOFSTEP]\nintros\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\na'\u271d : \u03b1\n\u22a2 corec Stream'.cycleF Stream'.cycleG (a'\u271d, [], a, l) = [a'\u271d] ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\n[PROOFSTEP]\nrw [corec_eq]\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\na'\u271d : \u03b1\n\u22a2 Stream'.cycleF (a'\u271d, [], a, l) :: corec Stream'.cycleF Stream'.cycleG (Stream'.cycleG (a'\u271d, [], a, l)) =\n [a'\u271d] ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\na\u2081 : \u03b1\nl\u2081 : List \u03b1\nih :\n \u2200 (a' : \u03b1),\n corec Stream'.cycleF Stream'.cycleG (a', l\u2081, a, l) = a' :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\n\u22a2 \u2200 (a' : \u03b1),\n corec Stream'.cycleF Stream'.cycleG (a', a\u2081 :: l\u2081, a, l) =\n a' :: a\u2081 :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\n[PROOFSTEP]\nintros\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\na\u2081 : \u03b1\nl\u2081 : List \u03b1\nih :\n \u2200 (a' : \u03b1),\n corec Stream'.cycleF Stream'.cycleG (a', l\u2081, a, l) = a' :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\na'\u271d : \u03b1\n\u22a2 corec Stream'.cycleF Stream'.cycleG (a'\u271d, a\u2081 :: l\u2081, a, l) =\n a'\u271d :: a\u2081 :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\n[PROOFSTEP]\nrw [corec_eq, Stream'.cycle_g_cons, ih a\u2081]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\na\u2081 : \u03b1\nl\u2081 : List \u03b1\nih :\n \u2200 (a' : \u03b1),\n corec Stream'.cycleF Stream'.cycleG (a', l\u2081, a, l) = a' :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\na'\u271d : \u03b1\n\u22a2 Stream'.cycleF (a'\u271d, a\u2081 :: l\u2081, a, l) :: (a\u2081 :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)) =\n a'\u271d :: a\u2081 :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nh : l \u2260 []\nainl : a \u2208 l\n\u22a2 a \u2208 cycle l h\n[PROOFSTEP]\nrw [cycle_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nh : l \u2260 []\nainl : a \u2208 l\n\u22a2 a \u2208 l ++\u209b cycle l h\n[PROOFSTEP]\nexact mem_append_stream_left _ ainl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\n\u22a2 [a] \u2260 []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b4 : Type w\na : \u03b1\n\u03b2 : Type u\nfr : Stream' \u03b1 \u2192 \u03b2\nch : fr (cycle [a] (_ : \u00ac[a] = [])) = fr (const a)\n\u22a2 fr (tail (cycle [a] (_ : \u00ac[a] = []))) = fr (tail (const a))\n[PROOFSTEP]\nrwa [cycle_eq, const_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 tails s = tail s :: tails (tail s)\n[PROOFSTEP]\nunfold tails\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 corec id tail (tail s) = tail s :: corec id tail (tail (tail s))\n[PROOFSTEP]\nrw [corec_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 id (tail s) :: corec id tail (tail (tail s)) = tail s :: corec id tail (tail (tail s))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\n\u22a2 \u2200 (n : \u2115) (s : Stream' \u03b1), nth (tails s) n = drop n (tail s)\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\n\u22a2 \u2200 (s : Stream' \u03b1), nth (tails s) n = drop n (tail s)\n[PROOFSTEP]\ninduction' n with n' ih\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\n\u22a2 \u2200 (s : Stream' \u03b1), nth (tails s) zero = drop zero (tail s)\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u271d : Stream' \u03b1\n\u22a2 nth (tails s\u271d) zero = drop zero (tail s\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn' : \u2115\nih : \u2200 (s : Stream' \u03b1), nth (tails s) n' = drop n' (tail s)\n\u22a2 \u2200 (s : Stream' \u03b1), nth (tails s) (succ n') = drop (succ n') (tail s)\n[PROOFSTEP]\nintro s\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn' : \u2115\nih : \u2200 (s : Stream' \u03b1), nth (tails s) n' = drop n' (tail s)\ns : Stream' \u03b1\n\u22a2 nth (tails s) (succ n') = drop (succ n') (tail s)\n[PROOFSTEP]\nrw [nth_succ, drop_succ, tails_eq, tail_cons, ih]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nl : List \u03b1\ns : Stream' \u03b1\n\u22a2 initsCore l s = l :: initsCore (l ++ [head s]) (tail s)\n[PROOFSTEP]\nunfold initsCore corecOn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nl : List \u03b1\ns : Stream' \u03b1\n\u22a2 corec\n (fun x =>\n match x with\n | (a, snd) => a)\n (fun p =>\n match p with\n | (l', s') => (l' ++ [head s'], tail s'))\n (l, s) =\n l ::\n corec\n (fun x =>\n match x with\n | (a, snd) => a)\n (fun p =>\n match p with\n | (l', s') => (l' ++ [head s'], tail s'))\n (l ++ [head s], tail s)\n[PROOFSTEP]\nrw [corec_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 tail (inits s) = initsCore [head s, head (tail s)] (tail (tail s))\n[PROOFSTEP]\nunfold inits\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 tail (initsCore [head s] (tail s)) = initsCore [head s, head (tail s)] (tail (tail s))\n[PROOFSTEP]\nrw [inits_core_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 tail ([head s] :: initsCore ([head s] ++ [head (tail s)]) (tail (tail s))) =\n initsCore [head s, head (tail s)] (tail (tail s))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\n\u22a2 \u2200 (a : \u03b1) (n : \u2115) (l : List \u03b1) (s : Stream' \u03b1), a :: nth (initsCore l s) n = nth (initsCore (a :: l) s) n\n[PROOFSTEP]\nintro a n\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nn : \u2115\n\u22a2 \u2200 (l : List \u03b1) (s : Stream' \u03b1), a :: nth (initsCore l s) n = nth (initsCore (a :: l) s) n\n[PROOFSTEP]\ninduction' n with n' ih\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\n\u22a2 \u2200 (l : List \u03b1) (s : Stream' \u03b1), a :: nth (initsCore l s) zero = nth (initsCore (a :: l) s) zero\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl\u271d : List \u03b1\ns\u271d : Stream' \u03b1\n\u22a2 a :: nth (initsCore l\u271d s\u271d) zero = nth (initsCore (a :: l\u271d) s\u271d) zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nn' : \u2115\nih : \u2200 (l : List \u03b1) (s : Stream' \u03b1), a :: nth (initsCore l s) n' = nth (initsCore (a :: l) s) n'\n\u22a2 \u2200 (l : List \u03b1) (s : Stream' \u03b1), a :: nth (initsCore l s) (succ n') = nth (initsCore (a :: l) s) (succ n')\n[PROOFSTEP]\nintro l s\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nn' : \u2115\nih : \u2200 (l : List \u03b1) (s : Stream' \u03b1), a :: nth (initsCore l s) n' = nth (initsCore (a :: l) s) n'\nl : List \u03b1\ns : Stream' \u03b1\n\u22a2 a :: nth (initsCore l s) (succ n') = nth (initsCore (a :: l) s) (succ n')\n[PROOFSTEP]\nrw [nth_succ, inits_core_eq, tail_cons, ih, inits_core_eq (a :: l) s]\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nn' : \u2115\nih : \u2200 (l : List \u03b1) (s : Stream' \u03b1), a :: nth (initsCore l s) n' = nth (initsCore (a :: l) s) n'\nl : List \u03b1\ns : Stream' \u03b1\n\u22a2 nth (initsCore (a :: (l ++ [head s])) (tail s)) n' =\n nth ((a :: l) :: initsCore (a :: l ++ [head s]) (tail s)) (succ n')\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\n\u22a2 \u2200 (n : \u2115) (s : Stream' \u03b1), nth (inits s) n = take (succ n) s\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\n\u22a2 \u2200 (s : Stream' \u03b1), nth (inits s) n = take (succ n) s\n[PROOFSTEP]\ninduction' n with n' ih\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\n\u22a2 \u2200 (s : Stream' \u03b1), nth (inits s) zero = take (succ zero) s\n[PROOFSTEP]\nintros\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u271d : Stream' \u03b1\n\u22a2 nth (inits s\u271d) zero = take (succ zero) s\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn' : \u2115\nih : \u2200 (s : Stream' \u03b1), nth (inits s) n' = take (succ n') s\n\u22a2 \u2200 (s : Stream' \u03b1), nth (inits s) (succ n') = take (succ (succ n')) s\n[PROOFSTEP]\nintros\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn' : \u2115\nih : \u2200 (s : Stream' \u03b1), nth (inits s) n' = take (succ n') s\ns\u271d : Stream' \u03b1\n\u22a2 nth (inits s\u271d) (succ n') = take (succ (succ n')) s\u271d\n[PROOFSTEP]\nrw [nth_succ, take_succ, \u2190 ih, tail_inits, inits_tail, cons_nth_inits_core]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 inits s = [head s] :: map (List.cons (head s)) (inits (tail s))\n[PROOFSTEP]\napply Stream'.ext\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 \u2200 (n : \u2115), nth (inits s) n = nth ([head s] :: map (List.cons (head s)) (inits (tail s))) n\n[PROOFSTEP]\nintro n\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\nn : \u2115\n\u22a2 nth (inits s) n = nth ([head s] :: map (List.cons (head s)) (inits (tail s))) n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase a.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 nth (inits s) zero = nth ([head s] :: map (List.cons (head s)) (inits (tail s))) zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\nn\u271d : \u2115\n\u22a2 nth (inits s) (succ n\u271d) = nth ([head s] :: map (List.cons (head s)) (inits (tail s))) (succ n\u271d)\n[PROOFSTEP]\nrw [nth_inits, nth_succ, tail_cons, nth_map, nth_inits]\n[GOAL]\ncase a.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\nn\u271d : \u2115\n\u22a2 take (succ (succ n\u271d)) s = head s :: take (succ n\u271d) (tail s)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 zip appendStream' (inits s) (tails s) = const s\n[PROOFSTEP]\napply Stream'.ext\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 \u2200 (n : \u2115), nth (zip appendStream' (inits s) (tails s)) n = nth (const s) n\n[PROOFSTEP]\nintro n\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\nn : \u2115\n\u22a2 nth (zip appendStream' (inits s) (tails s)) n = nth (const s) n\n[PROOFSTEP]\nrw [nth_zip, nth_inits, nth_tails, nth_const, take_succ, cons_append_stream, append_take_drop, Stream'.eta]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\n\u22a2 nats = 0 :: map succ nats\n[PROOFSTEP]\napply Stream'.ext\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\n\u22a2 \u2200 (n : \u2115), nth nats n = nth (0 :: map succ nats) n\n[PROOFSTEP]\nintro n\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\n\u22a2 nth nats n = nth (0 :: map succ nats) n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase a.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\n\u22a2 nth nats zero = nth (0 :: map succ nats) zero\ncase a.succ \u03b1 : Type u \u03b2 : Type v \u03b4 : Type w n\u271d : \u2115 \u22a2 nth nats (succ n\u271d) = nth (0 :: map succ nats) (succ n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn\u271d : \u2115\n\u22a2 nth nats (succ n\u271d) = nth (0 :: map succ nats) (succ n\u271d)\n[PROOFSTEP]\nrw [nth_succ]\n[GOAL]\ncase a.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn\u271d : \u2115\n\u22a2 nth (tail nats) n\u271d = nth (0 :: map succ nats) (succ n\u271d)\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Data.Stream.Init", "llama_tokens": 20635, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.6370307944803831, "lm_q1q2_score": 0.4348204883241249}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 lcm s f \u2223 a \u2194 \u2200 (b : \u03b2), b \u2208 s \u2192 f b \u2223 a\n[PROOFSTEP]\napply Iff.trans Multiset.lcm_dvd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200 (b : \u03b1), b \u2208 Multiset.map f s.val \u2192 b \u2223 a) \u2194 \u2200 (b : \u03b2), b \u2208 s \u2192 f b \u2223 a\n[PROOFSTEP]\nsimp only [Multiset.mem_map, and_imp, exists_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200 (b : \u03b1) (x : \u03b2), x \u2208 s.val \u2192 f x = b \u2192 b \u2223 a) \u2194 \u2200 (b : \u03b2), b \u2208 s \u2192 f b \u2223 a\n[PROOFSTEP]\nexact \u27e8fun k b hb \u21a6 k _ _ hb rfl, fun k a' b hb h \u21a6 h \u25b8 k _ hb\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\n\u22a2 lcm (insert b s) f = GCDMonoid.lcm (f b) (lcm s f)\n[PROOFSTEP]\nby_cases h : b \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\nh : b \u2208 s\n\u22a2 lcm (insert b s) f = GCDMonoid.lcm (f b) (lcm s f)\n[PROOFSTEP]\nrw [insert_eq_of_mem h, (lcm_eq_right_iff (f b) (s.lcm f) (Multiset.normalize_lcm (s.1.map f))).2 (dvd_lcm h)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\nh : \u00acb \u2208 s\n\u22a2 lcm (insert b s) f = GCDMonoid.lcm (f b) (lcm s f)\n[PROOFSTEP]\napply fold_insert h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2191normalize (lcm s f) = lcm s f\n[PROOFSTEP]\nsimp [lcm_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\n\u22a2 lcm (\u2205 \u222a s\u2082) f = GCDMonoid.lcm (lcm \u2205 f) (lcm s\u2082 f)\n[PROOFSTEP]\nrw [empty_union, lcm_empty, lcm_one_left, normalize_lcm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns\u271d s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b2\ns : Finset \u03b2\nx\u271d : \u00aca \u2208 s\nih : lcm (s \u222a s\u2082) f = GCDMonoid.lcm (lcm s f) (lcm s\u2082 f)\n\u22a2 lcm (insert a s \u222a s\u2082) f = GCDMonoid.lcm (lcm (insert a s) f) (lcm s\u2082 f)\n[PROOFSTEP]\nrw [insert_union, lcm_insert, lcm_insert, ih, lcm_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f g : \u03b2 \u2192 \u03b1\nhs : s\u2081 = s\u2082\nhfg : \u2200 (a : \u03b2), a \u2208 s\u2082 \u2192 f a = g a\n\u22a2 lcm s\u2081 f = lcm s\u2082 g\n[PROOFSTEP]\nsubst hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 : Finset \u03b2\nf\u271d f g : \u03b2 \u2192 \u03b1\nhfg : \u2200 (a : \u03b2), a \u2208 s\u2081 \u2192 f a = g a\n\u22a2 lcm s\u2081 f = lcm s\u2081 g\n[PROOFSTEP]\nexact Finset.fold_congr hfg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns\u271d s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\ng : \u03b3 \u2192 \u03b2\ns : Finset \u03b3\n\u22a2 lcm (image g s) f = lcm s (f \u2218 g)\n[PROOFSTEP]\nclassical induction' s using Finset.induction with c s _ ih <;> simp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns\u271d s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\ng : \u03b3 \u2192 \u03b2\ns : Finset \u03b3\n\u22a2 lcm (image g s) f = lcm s (f \u2218 g)\n[PROOFSTEP]\ninduction' s using Finset.induction with c s _ ih\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\ng : \u03b3 \u2192 \u03b2\n\u22a2 lcm (image g \u2205) f = lcm \u2205 (f \u2218 g)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns\u271d s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\ng : \u03b3 \u2192 \u03b2\nc : \u03b3\ns : Finset \u03b3\na\u271d : \u00acc \u2208 s\nih : lcm (image g s) f = lcm s (f \u2218 g)\n\u22a2 lcm (image g (insert c s)) f = lcm (insert c s) (f \u2218 g)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : Nontrivial \u03b1\n\u22a2 lcm s f = 0 \u2194 0 \u2208 f '' \u2191s\n[PROOFSTEP]\nsimp only [Multiset.mem_map, lcm_def, Multiset.lcm_eq_zero_iff, Set.mem_image, mem_coe, \u2190 Finset.mem_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 a \u2223 gcd s f \u2194 \u2200 (b : \u03b2), b \u2208 s \u2192 a \u2223 f b\n[PROOFSTEP]\napply Iff.trans Multiset.dvd_gcd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200 (b : \u03b1), b \u2208 Multiset.map f s.val \u2192 a \u2223 b) \u2194 \u2200 (b : \u03b2), b \u2208 s \u2192 a \u2223 f b\n[PROOFSTEP]\nsimp only [Multiset.mem_map, and_imp, exists_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200 (b : \u03b1) (x : \u03b2), x \u2208 s.val \u2192 f x = b \u2192 a \u2223 b) \u2194 \u2200 (b : \u03b2), b \u2208 s \u2192 a \u2223 f b\n[PROOFSTEP]\nexact \u27e8fun k b hb \u21a6 k _ _ hb rfl, fun k a' b hb h \u21a6 h \u25b8 k _ hb\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\n\u22a2 gcd (insert b s) f = GCDMonoid.gcd (f b) (gcd s f)\n[PROOFSTEP]\nby_cases h : b \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\nh : b \u2208 s\n\u22a2 gcd (insert b s) f = GCDMonoid.gcd (f b) (gcd s f)\n[PROOFSTEP]\nrw [insert_eq_of_mem h, (gcd_eq_right_iff (f b) (s.gcd f) (Multiset.normalize_gcd (s.1.map f))).2 (gcd_dvd h)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\nh : \u00acb \u2208 s\n\u22a2 gcd (insert b s) f = GCDMonoid.gcd (f b) (gcd s f)\n[PROOFSTEP]\napply fold_insert h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2191normalize (gcd s f) = gcd s f\n[PROOFSTEP]\nsimp [gcd_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\n\u22a2 gcd (\u2205 \u222a s\u2082) f = GCDMonoid.gcd (gcd \u2205 f) (gcd s\u2082 f)\n[PROOFSTEP]\nrw [empty_union, gcd_empty, gcd_zero_left, normalize_gcd]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns\u271d s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b2\ns : Finset \u03b2\nx\u271d : \u00aca \u2208 s\nih : gcd (s \u222a s\u2082) f = GCDMonoid.gcd (gcd s f) (gcd s\u2082 f)\n\u22a2 gcd (insert a s \u222a s\u2082) f = GCDMonoid.gcd (gcd (insert a s) f) (gcd s\u2082 f)\n[PROOFSTEP]\nrw [insert_union, gcd_insert, gcd_insert, ih, gcd_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f g : \u03b2 \u2192 \u03b1\nhs : s\u2081 = s\u2082\nhfg : \u2200 (a : \u03b2), a \u2208 s\u2082 \u2192 f a = g a\n\u22a2 gcd s\u2081 f = gcd s\u2082 g\n[PROOFSTEP]\nsubst hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 : Finset \u03b2\nf\u271d f g : \u03b2 \u2192 \u03b1\nhfg : \u2200 (a : \u03b2), a \u2208 s\u2081 \u2192 f a = g a\n\u22a2 gcd s\u2081 f = gcd s\u2081 g\n[PROOFSTEP]\nexact Finset.fold_congr hfg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns\u271d s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\ng : \u03b3 \u2192 \u03b2\ns : Finset \u03b3\n\u22a2 gcd (image g s) f = gcd s (f \u2218 g)\n[PROOFSTEP]\nclassical induction' s using Finset.induction with c s _ ih <;> simp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns\u271d s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\ng : \u03b3 \u2192 \u03b2\ns : Finset \u03b3\n\u22a2 gcd (image g s) f = gcd s (f \u2218 g)\n[PROOFSTEP]\ninduction' s using Finset.induction with c s _ ih\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\ng : \u03b3 \u2192 \u03b2\n\u22a2 gcd (image g \u2205) f = gcd \u2205 (f \u2218 g)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns\u271d s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b2\ng : \u03b3 \u2192 \u03b2\nc : \u03b3\ns : Finset \u03b3\na\u271d : \u00acc \u2208 s\nih : gcd (image g s) f = gcd s (f \u2218 g)\n\u22a2 gcd (image g (insert c s)) f = gcd (insert c s) (f \u2218 g)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 gcd s f = 0 \u2194 \u2200 (x : \u03b2), x \u2208 s \u2192 f x = 0\n[PROOFSTEP]\nrw [gcd_def, Multiset.gcd_eq_zero_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 (\u2200 (x : \u03b1), x \u2208 Multiset.map f s.val \u2192 x = 0) \u2194 \u2200 (x : \u03b2), x \u2208 s \u2192 f x = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 (\u2200 (x : \u03b1), x \u2208 Multiset.map f s.val \u2192 x = 0) \u2192 \u2200 (x : \u03b2), x \u2208 s \u2192 f x = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 (\u2200 (x : \u03b2), x \u2208 s \u2192 f x = 0) \u2192 \u2200 (x : \u03b1), x \u2208 Multiset.map f s.val \u2192 x = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (x : \u03b1), x \u2208 Multiset.map f s.val \u2192 x = 0\n\u22a2 \u2200 (x : \u03b2), x \u2208 s \u2192 f x = 0\n[PROOFSTEP]\nintro b bs\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (x : \u03b1), x \u2208 Multiset.map f s.val \u2192 x = 0\nb : \u03b2\nbs : b \u2208 s\n\u22a2 f b = 0\n[PROOFSTEP]\napply h (f b)\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (x : \u03b1), x \u2208 Multiset.map f s.val \u2192 x = 0\nb : \u03b2\nbs : b \u2208 s\n\u22a2 f b \u2208 Multiset.map f s.val\n[PROOFSTEP]\nsimp only [Multiset.mem_map, mem_def.1 bs]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (x : \u03b1), x \u2208 Multiset.map f s.val \u2192 x = 0\nb : \u03b2\nbs : b \u2208 s\n\u22a2 \u2203 a, a \u2208 s.val \u2227 f a = f b\n[PROOFSTEP]\nuse b\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (x : \u03b1), x \u2208 Multiset.map f s.val \u2192 x = 0\nb : \u03b2\nbs : b \u2208 s\n\u22a2 b \u2208 s.val \u2227 f b = f b\n[PROOFSTEP]\nsimp only [mem_def.1 bs, eq_self_iff_true, and_self]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (x : \u03b2), x \u2208 s \u2192 f x = 0\n\u22a2 \u2200 (x : \u03b1), x \u2208 Multiset.map f s.val \u2192 x = 0\n[PROOFSTEP]\nintro a as\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (x : \u03b2), x \u2208 s \u2192 f x = 0\na : \u03b1\nas : a \u2208 Multiset.map f s.val\n\u22a2 a = 0\n[PROOFSTEP]\nrw [Multiset.mem_map] at as \n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (x : \u03b2), x \u2208 s \u2192 f x = 0\na : \u03b1\nas : \u2203 a_1, a_1 \u2208 s.val \u2227 f a_1 = a\n\u22a2 a = 0\n[PROOFSTEP]\nrcases as with \u27e8b, \u27e8bs, rfl\u27e9\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (x : \u03b2), x \u2208 s \u2192 f x = 0\nb : \u03b2\nbs : b \u2208 s.val\n\u22a2 f b = 0\n[PROOFSTEP]\napply h b (mem_def.1 bs)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidablePred fun x => f x = 0\n\u22a2 gcd s f = gcd (filter (fun x => f x \u2260 0) s) f\n[PROOFSTEP]\nclassical\ntrans ((s.filter fun x \u21a6 f x = 0) \u222a s.filter fun x \u21a6 (f x \u2260 0)).gcd f\n\u00b7 rw [filter_union_filter_neg_eq]\nrw [gcd_union]\nrefine' Eq.trans (_ : _ = GCDMonoid.gcd (0 : \u03b1) _) (_ : GCDMonoid.gcd (0 : \u03b1) _ = _)\n\u00b7 exact (gcd (filter (fun x => (f x \u2260 0)) s) f)\n\u00b7 refine' congr (congr rfl <| s.induction_on _ _) (by simp)\n \u00b7 simp\n \u00b7 intro a s _ h\n rw [filter_insert]\n split_ifs with h1 <;> simp [h, h1]\nsimp only [gcd_zero_left, normalize_gcd]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidablePred fun x => f x = 0\n\u22a2 gcd s f = gcd (filter (fun x => f x \u2260 0) s) f\n[PROOFSTEP]\ntrans ((s.filter fun x \u21a6 f x = 0) \u222a s.filter fun x \u21a6 (f x \u2260 0)).gcd f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidablePred fun x => f x = 0\n\u22a2 gcd s f = gcd (filter (fun x => f x = 0) s \u222a filter (fun x => f x \u2260 0) s) f\n[PROOFSTEP]\nrw [filter_union_filter_neg_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidablePred fun x => f x = 0\n\u22a2 gcd (filter (fun x => f x = 0) s \u222a filter (fun x => f x \u2260 0) s) f = gcd (filter (fun x => f x \u2260 0) s) f\n[PROOFSTEP]\nrw [gcd_union]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidablePred fun x => f x = 0\n\u22a2 GCDMonoid.gcd (gcd (filter (fun x => f x = 0) s) f) (gcd (filter (fun x => f x \u2260 0) s) f) =\n gcd (filter (fun x => f x \u2260 0) s) f\n[PROOFSTEP]\nrefine' Eq.trans (_ : _ = GCDMonoid.gcd (0 : \u03b1) _) (_ : GCDMonoid.gcd (0 : \u03b1) _ = _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidablePred fun x => f x = 0\n\u22a2 \u03b1\n[PROOFSTEP]\nexact (gcd (filter (fun x => (f x \u2260 0)) s) f)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidablePred fun x => f x = 0\n\u22a2 GCDMonoid.gcd (gcd (filter (fun x => f x = 0) s) f) (gcd (filter (fun x => f x \u2260 0) s) f) =\n GCDMonoid.gcd 0 (gcd (filter (fun x => f x \u2260 0) s) f)\n[PROOFSTEP]\nrefine' congr (congr rfl <| s.induction_on _ _) (by simp)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidablePred fun x => f x = 0\n\u22a2 gcd (filter (fun x => f x \u2260 0) s) f = gcd (filter (fun x => f x \u2260 0) s) f\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidablePred fun x => f x = 0\n\u22a2 gcd (filter (fun x => f x = 0) \u2205) f = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidablePred fun x => f x = 0\n\u22a2 \u2200 \u2983a : \u03b2\u2984 {s : Finset \u03b2},\n \u00aca \u2208 s \u2192 gcd (filter (fun x => f x = 0) s) f = 0 \u2192 gcd (filter (fun x => f x = 0) (insert a s)) f = 0\n[PROOFSTEP]\nintro a s _ h\n[GOAL]\ncase refine'_2.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns\u271d s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidablePred fun x => f x = 0\na : \u03b2\ns : Finset \u03b2\na\u271d : \u00aca \u2208 s\nh : gcd (filter (fun x => f x = 0) s) f = 0\n\u22a2 gcd (filter (fun x => f x = 0) (insert a s)) f = 0\n[PROOFSTEP]\nrw [filter_insert]\n[GOAL]\ncase refine'_2.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns\u271d s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidablePred fun x => f x = 0\na : \u03b2\ns : Finset \u03b2\na\u271d : \u00aca \u2208 s\nh : gcd (filter (fun x => f x = 0) s) f = 0\n\u22a2 gcd (if f a = 0 then insert a (filter (fun x => f x = 0) s) else filter (fun x => f x = 0) s) f = 0\n[PROOFSTEP]\nsplit_ifs with h1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns\u271d s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidablePred fun x => f x = 0\na : \u03b2\ns : Finset \u03b2\na\u271d : \u00aca \u2208 s\nh : gcd (filter (fun x => f x = 0) s) f = 0\nh1 : f a = 0\n\u22a2 gcd (insert a (filter (fun x => f x = 0) s)) f = 0\n[PROOFSTEP]\nsimp [h, h1]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns\u271d s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidablePred fun x => f x = 0\na : \u03b2\ns : Finset \u03b2\na\u271d : \u00aca \u2208 s\nh : gcd (filter (fun x => f x = 0) s) f = 0\nh1 : \u00acf a = 0\n\u22a2 gcd (filter (fun x => f x = 0) s) f = 0\n[PROOFSTEP]\nsimp [h, h1]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\ninst\u271d : DecidablePred fun x => f x = 0\n\u22a2 GCDMonoid.gcd 0 (gcd (filter (fun x => f x \u2260 0) s) f) = gcd (filter (fun x => f x \u2260 0) s) f\n[PROOFSTEP]\nsimp only [gcd_zero_left, normalize_gcd]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (gcd s fun x => a * f x) = \u2191normalize a * gcd s f\n[PROOFSTEP]\nclassical\nrefine' s.induction_on _ _\n\u00b7 simp\n\u00b7 intro b t _ h\n rw [gcd_insert, gcd_insert, h, \u2190 gcd_mul_left]\n apply ((normalize_associated a).mul_right _).gcd_eq_right\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (gcd s fun x => a * f x) = \u2191normalize a * gcd s f\n[PROOFSTEP]\nrefine' s.induction_on _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (gcd \u2205 fun x => a * f x) = \u2191normalize a * gcd \u2205 f\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 \u2200 \u2983a_1 : \u03b2\u2984 {s : Finset \u03b2},\n \u00aca_1 \u2208 s \u2192\n (gcd s fun x => a * f x) = \u2191normalize a * gcd s f \u2192\n (gcd (insert a_1 s) fun x => a * f x) = \u2191normalize a * gcd (insert a_1 s) f\n[PROOFSTEP]\nintro b t _ h\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nb : \u03b2\nt : Finset \u03b2\na\u271d : \u00acb \u2208 t\nh : (gcd t fun x => a * f x) = \u2191normalize a * gcd t f\n\u22a2 (gcd (insert b t) fun x => a * f x) = \u2191normalize a * gcd (insert b t) f\n[PROOFSTEP]\nrw [gcd_insert, gcd_insert, h, \u2190 gcd_mul_left]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nb : \u03b2\nt : Finset \u03b2\na\u271d : \u00acb \u2208 t\nh : (gcd t fun x => a * f x) = \u2191normalize a * gcd t f\n\u22a2 GCDMonoid.gcd (a * f b) (\u2191normalize a * gcd t f) = GCDMonoid.gcd (a * f b) (a * gcd t f)\n[PROOFSTEP]\napply ((normalize_associated a).mul_right _).gcd_eq_right\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (gcd s fun x => f x * a) = gcd s f * \u2191normalize a\n[PROOFSTEP]\nclassical\nrefine' s.induction_on _ _\n\u00b7 simp\n\u00b7 intro b t _ h\n rw [gcd_insert, gcd_insert, h, \u2190 gcd_mul_right]\n apply ((normalize_associated a).mul_left _).gcd_eq_right\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (gcd s fun x => f x * a) = gcd s f * \u2191normalize a\n[PROOFSTEP]\nrefine' s.induction_on _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (gcd \u2205 fun x => f x * a) = gcd \u2205 f * \u2191normalize a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 \u2200 \u2983a_1 : \u03b2\u2984 {s : Finset \u03b2},\n \u00aca_1 \u2208 s \u2192\n (gcd s fun x => f x * a) = gcd s f * \u2191normalize a \u2192\n (gcd (insert a_1 s) fun x => f x * a) = gcd (insert a_1 s) f * \u2191normalize a\n[PROOFSTEP]\nintro b t _ h\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nb : \u03b2\nt : Finset \u03b2\na\u271d : \u00acb \u2208 t\nh : (gcd t fun x => f x * a) = gcd t f * \u2191normalize a\n\u22a2 (gcd (insert b t) fun x => f x * a) = gcd (insert b t) f * \u2191normalize a\n[PROOFSTEP]\nrw [gcd_insert, gcd_insert, h, \u2190 gcd_mul_right]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nb : \u03b2\nt : Finset \u03b2\na\u271d : \u00acb \u2208 t\nh : (gcd t fun x => f x * a) = gcd t f * \u2191normalize a\n\u22a2 GCDMonoid.gcd (f b * a) (gcd t f * \u2191normalize a) = GCDMonoid.gcd (f b * a) (gcd t f * a)\n[PROOFSTEP]\napply ((normalize_associated a).mul_left _).gcd_eq_right\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f g : \u03b2 \u2192 \u03b1\nhs : \u2203 x, x \u2208 s \u2227 f x \u2260 0\nhg : \u2200 (b : \u03b2), b \u2208 s \u2192 f b = gcd s f * g b\n\u22a2 gcd s f * gcd s g = gcd s f\n[PROOFSTEP]\nconv_lhs => rw [\u2190 normalize_gcd, \u2190 gcd_mul_left, \u2190 gcd_congr rfl hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f g : \u03b2 \u2192 \u03b1\nhs : \u2203 x, x \u2208 s \u2227 f x \u2260 0\nhg : \u2200 (b : \u03b2), b \u2208 s \u2192 f b = gcd s f * g b\n| gcd s f * gcd s g\n[PROOFSTEP]\nrw [\u2190 normalize_gcd, \u2190 gcd_mul_left, \u2190 gcd_congr rfl hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f g : \u03b2 \u2192 \u03b1\nhs : \u2203 x, x \u2208 s \u2227 f x \u2260 0\nhg : \u2200 (b : \u03b2), b \u2208 s \u2192 f b = gcd s f * g b\n| gcd s f * gcd s g\n[PROOFSTEP]\nrw [\u2190 normalize_gcd, \u2190 gcd_mul_left, \u2190 gcd_congr rfl hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f g : \u03b2 \u2192 \u03b1\nhs : \u2203 x, x \u2208 s \u2227 f x \u2260 0\nhg : \u2200 (b : \u03b2), b \u2208 s \u2192 f b = gcd s f * g b\n| gcd s f * gcd s g\n[PROOFSTEP]\nrw [\u2190 normalize_gcd, \u2190 gcd_mul_left, \u2190 gcd_congr rfl hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f g : \u03b2 \u2192 \u03b1\nhs : \u2203 x, x \u2208 s \u2227 f x \u2260 0\nhg : \u2200 (b : \u03b2), b \u2208 s \u2192 f b = gcd s f * g b\n\u22a2 \u00acgcd s f = 0\n[PROOFSTEP]\ncontrapose! hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f g : \u03b2 \u2192 \u03b1\nhg : \u2200 (b : \u03b2), b \u2208 s \u2192 f b = gcd s f * g b\nhs : gcd s f = 0\n\u22a2 \u2200 (x : \u03b2), x \u2208 s \u2192 f x = 0\n[PROOFSTEP]\nexact gcd_eq_zero_iff.1 hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f : \u03b2 \u2192 \u03b1\nhs : Finset.Nonempty s\n\u22a2 \u2203 g, (\u2200 (b : \u03b2), b \u2208 s \u2192 f b = gcd s f * g b) \u2227 gcd s g = 1\n[PROOFSTEP]\nclassical\nby_cases h : \u2200 x \u2208 s, f x = (0 : \u03b1)\n\u00b7 refine' \u27e8fun _ \u21a6 1, fun b hb \u21a6 by rw [h b hb, gcd_eq_zero_iff.2 h, mul_one], _\u27e9\n rw [gcd_eq_gcd_image, image_const hs, gcd_singleton, id, normalize_one]\n\u00b7 choose g' hg using @gcd_dvd _ _ _ _ s f\n push_neg at h \n refine' \u27e8fun b \u21a6 if hb : b \u2208 s then g' hb else 0, fun b hb \u21a6 _, extract_gcd' f _ h <| fun b hb \u21a6 _\u27e9\n simp only [hb, hg, dite_true]\n rw [dif_pos hb, hg hb]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f : \u03b2 \u2192 \u03b1\nhs : Finset.Nonempty s\n\u22a2 \u2203 g, (\u2200 (b : \u03b2), b \u2208 s \u2192 f b = gcd s f * g b) \u2227 gcd s g = 1\n[PROOFSTEP]\nby_cases h : \u2200 x \u2208 s, f x = (0 : \u03b1)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f : \u03b2 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u2200 (x : \u03b2), x \u2208 s \u2192 f x = 0\n\u22a2 \u2203 g, (\u2200 (b : \u03b2), b \u2208 s \u2192 f b = gcd s f * g b) \u2227 gcd s g = 1\n[PROOFSTEP]\nrefine' \u27e8fun _ \u21a6 1, fun b hb \u21a6 by rw [h b hb, gcd_eq_zero_iff.2 h, mul_one], _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f : \u03b2 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u2200 (x : \u03b2), x \u2208 s \u2192 f x = 0\nb : \u03b2\nhb : b \u2208 s\n\u22a2 f b = gcd s f * (fun x => 1) b\n[PROOFSTEP]\nrw [h b hb, gcd_eq_zero_iff.2 h, mul_one]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f : \u03b2 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u2200 (x : \u03b2), x \u2208 s \u2192 f x = 0\n\u22a2 (gcd s fun x => 1) = 1\n[PROOFSTEP]\nrw [gcd_eq_gcd_image, image_const hs, gcd_singleton, id, normalize_one]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f : \u03b2 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u00ac\u2200 (x : \u03b2), x \u2208 s \u2192 f x = 0\n\u22a2 \u2203 g, (\u2200 (b : \u03b2), b \u2208 s \u2192 f b = gcd s f * g b) \u2227 gcd s g = 1\n[PROOFSTEP]\nchoose g' hg using @gcd_dvd _ _ _ _ s f\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f : \u03b2 \u2192 \u03b1\nhs : Finset.Nonempty s\nh : \u00ac\u2200 (x : \u03b2), x \u2208 s \u2192 f x = 0\ng' : {b : \u03b2} \u2192 b \u2208 s \u2192 \u03b1\nhg : \u2200 {b : \u03b2} (hb : b \u2208 s), f b = gcd s f * g' hb\n\u22a2 \u2203 g, (\u2200 (b : \u03b2), b \u2208 s \u2192 f b = gcd s f * g b) \u2227 gcd s g = 1\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f : \u03b2 \u2192 \u03b1\nhs : Finset.Nonempty s\ng' : {b : \u03b2} \u2192 b \u2208 s \u2192 \u03b1\nhg : \u2200 {b : \u03b2} (hb : b \u2208 s), f b = gcd s f * g' hb\nh : \u2203 x, x \u2208 s \u2227 f x \u2260 0\n\u22a2 \u2203 g, (\u2200 (b : \u03b2), b \u2208 s \u2192 f b = gcd s f * g b) \u2227 gcd s g = 1\n[PROOFSTEP]\nrefine' \u27e8fun b \u21a6 if hb : b \u2208 s then g' hb else 0, fun b hb \u21a6 _, extract_gcd' f _ h <| fun b hb \u21a6 _\u27e9\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f : \u03b2 \u2192 \u03b1\nhs : Finset.Nonempty s\ng' : {b : \u03b2} \u2192 b \u2208 s \u2192 \u03b1\nhg : \u2200 {b : \u03b2} (hb : b \u2208 s), f b = gcd s f * g' hb\nh : \u2203 x, x \u2208 s \u2227 f x \u2260 0\nb : \u03b2\nhb : b \u2208 s\n\u22a2 f b = gcd s f * (fun b => if hb : b \u2208 s then g' hb else 0) b\ncase neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f : \u03b2 \u2192 \u03b1\nhs : Finset.Nonempty s\ng' : {b : \u03b2} \u2192 b \u2208 s \u2192 \u03b1\nhg : \u2200 {b : \u03b2} (hb : b \u2208 s), f b = gcd s f * g' hb\nh : \u2203 x, x \u2208 s \u2227 f x \u2260 0\nb : \u03b2\nhb : b \u2208 s\n\u22a2 f b = gcd s f * if hb : b \u2208 s then g' hb else 0\n[PROOFSTEP]\nsimp only [hb, hg, dite_true]\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns s\u2081 s\u2082 : Finset \u03b2\nf\u271d f : \u03b2 \u2192 \u03b1\nhs : Finset.Nonempty s\ng' : {b : \u03b2} \u2192 b \u2208 s \u2192 \u03b1\nhg : \u2200 {b : \u03b2} (hb : b \u2208 s), f b = gcd s f * g' hb\nh : \u2203 x, x \u2208 s \u2227 f x \u2260 0\nb : \u03b2\nhb : b \u2208 s\n\u22a2 f b = gcd s f * if hb : b \u2208 s then g' hb else 0\n[PROOFSTEP]\nrw [dif_pos hb, hg hb]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns : Finset \u03b2\nf g : \u03b2 \u2192 \u03b1\na : \u03b1\nh : \u2200 (x : \u03b2), x \u2208 s \u2192 a \u2223 f x - g x\n\u22a2 GCDMonoid.gcd a (gcd s f) = GCDMonoid.gcd a (gcd s g)\n[PROOFSTEP]\nclassical\nrevert h\nrefine' s.induction_on _ _\n\u00b7 simp\nintro b s _ hi h\nrw [gcd_insert, gcd_insert, gcd_comm (f b), \u2190 gcd_assoc, hi fun x hx \u21a6 h _ (mem_insert_of_mem hx), gcd_comm a,\n gcd_assoc, gcd_comm a (GCDMonoid.gcd _ _), gcd_comm (g b), gcd_assoc _ _ a, gcd_comm _ a]\nexact congr_arg _ (gcd_eq_of_dvd_sub_right (h _ (mem_insert_self _ _)))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns : Finset \u03b2\nf g : \u03b2 \u2192 \u03b1\na : \u03b1\nh : \u2200 (x : \u03b2), x \u2208 s \u2192 a \u2223 f x - g x\n\u22a2 GCDMonoid.gcd a (gcd s f) = GCDMonoid.gcd a (gcd s g)\n[PROOFSTEP]\nrevert h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns : Finset \u03b2\nf g : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200 (x : \u03b2), x \u2208 s \u2192 a \u2223 f x - g x) \u2192 GCDMonoid.gcd a (gcd s f) = GCDMonoid.gcd a (gcd s g)\n[PROOFSTEP]\nrefine' s.induction_on _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns : Finset \u03b2\nf g : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200 (x : \u03b2), x \u2208 \u2205 \u2192 a \u2223 f x - g x) \u2192 GCDMonoid.gcd a (gcd \u2205 f) = GCDMonoid.gcd a (gcd \u2205 g)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns : Finset \u03b2\nf g : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 \u2200 \u2983a_1 : \u03b2\u2984 {s : Finset \u03b2},\n \u00aca_1 \u2208 s \u2192\n ((\u2200 (x : \u03b2), x \u2208 s \u2192 a \u2223 f x - g x) \u2192 GCDMonoid.gcd a (gcd s f) = GCDMonoid.gcd a (gcd s g)) \u2192\n (\u2200 (x : \u03b2), x \u2208 insert a_1 s \u2192 a \u2223 f x - g x) \u2192\n GCDMonoid.gcd a (gcd (insert a_1 s) f) = GCDMonoid.gcd a (gcd (insert a_1 s) g)\n[PROOFSTEP]\nintro b s _ hi h\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns\u271d : Finset \u03b2\nf g : \u03b2 \u2192 \u03b1\na : \u03b1\nb : \u03b2\ns : Finset \u03b2\na\u271d : \u00acb \u2208 s\nhi : (\u2200 (x : \u03b2), x \u2208 s \u2192 a \u2223 f x - g x) \u2192 GCDMonoid.gcd a (gcd s f) = GCDMonoid.gcd a (gcd s g)\nh : \u2200 (x : \u03b2), x \u2208 insert b s \u2192 a \u2223 f x - g x\n\u22a2 GCDMonoid.gcd a (gcd (insert b s) f) = GCDMonoid.gcd a (gcd (insert b s) g)\n[PROOFSTEP]\nrw [gcd_insert, gcd_insert, gcd_comm (f b), \u2190 gcd_assoc, hi fun x hx \u21a6 h _ (mem_insert_of_mem hx), gcd_comm a,\n gcd_assoc, gcd_comm a (GCDMonoid.gcd _ _), gcd_comm (g b), gcd_assoc _ _ a, gcd_comm _ a]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\ns\u271d : Finset \u03b2\nf g : \u03b2 \u2192 \u03b1\na : \u03b1\nb : \u03b2\ns : Finset \u03b2\na\u271d : \u00acb \u2208 s\nhi : (\u2200 (x : \u03b2), x \u2208 s \u2192 a \u2223 f x - g x) \u2192 GCDMonoid.gcd a (gcd s f) = GCDMonoid.gcd a (gcd s g)\nh : \u2200 (x : \u03b2), x \u2208 insert b s \u2192 a \u2223 f x - g x\n\u22a2 GCDMonoid.gcd (gcd s g) (GCDMonoid.gcd a (f b)) = GCDMonoid.gcd (gcd s g) (GCDMonoid.gcd a (g b))\n[PROOFSTEP]\nexact congr_arg _ (gcd_eq_of_dvd_sub_right (h _ (mem_insert_self _ _)))\n", "meta": {"mathlib_filename": "Mathlib.Algebra.GCDMonoid.Finset", "llama_tokens": 16562, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154240079185319, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.43473682042632683}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\n\u22a2 \u2200 {s t : Multiset \u03b1}, a ::\u2098 s = a ::\u2098 t \u2194 s = t\n[PROOFSTEP]\nrintro \u27e8l\u2081\u27e9 \u27e8l\u2082\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns\u271d : Multiset \u03b1\nl\u2081 : List \u03b1\nt\u271d : Multiset \u03b1\nl\u2082 : List \u03b1\n\u22a2 a ::\u2098 Quot.mk Setoid.r l\u2081 = a ::\u2098 Quot.mk Setoid.r l\u2082 \u2194 Quot.mk Setoid.r l\u2081 = Quot.mk Setoid.r l\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Multiset \u03b1 \u2192 Prop\nempty : p 0\ncons : \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1}, p s \u2192 p (a ::\u2098 s)\n\u22a2 \u2200 (s : Multiset \u03b1), p s\n[PROOFSTEP]\nrintro \u27e8l\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Multiset \u03b1 \u2192 Prop\nempty : p 0\ncons : \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1}, p s \u2192 p (a ::\u2098 s)\ns\u271d : Multiset \u03b1\nl : List \u03b1\n\u22a2 p (Quot.mk Setoid.r l)\n[PROOFSTEP]\ninduction' l with _ _ ih <;> [exact empty; exact cons ih]\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Multiset \u03b1 \u2192 Prop\nempty : p 0\ncons : \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1}, p s \u2192 p (a ::\u2098 s)\ns\u271d : Multiset \u03b1\nl : List \u03b1\n\u22a2 p (Quot.mk Setoid.r l)\n[PROOFSTEP]\ninduction' l with _ _ ih\n[GOAL]\ncase mk.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Multiset \u03b1 \u2192 Prop\nempty : p 0\ncons : \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1}, p s \u2192 p (a ::\u2098 s)\ns\u271d : Multiset \u03b1\n\u22a2 p (Quot.mk Setoid.r [])\n[PROOFSTEP]\nexact empty\n[GOAL]\ncase mk.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Multiset \u03b1 \u2192 Prop\nempty : p 0\ncons : \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1}, p s \u2192 p (a ::\u2098 s)\ns\u271d : Multiset \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nih : p (Quot.mk Setoid.r tail\u271d)\n\u22a2 p (Quot.mk Setoid.r (head\u271d :: tail\u271d))\n[PROOFSTEP]\nexact cons ih\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Multiset \u03b1 \u2192 Sort u_4\nC_0 : C 0\nC_cons : (a : \u03b1) \u2192 (m : Multiset \u03b1) \u2192 C m \u2192 C (a ::\u2098 m)\nC_cons_heq :\n \u2200 (a a' : \u03b1) (m : Multiset \u03b1) (b : C m),\n HEq (C_cons a (a' ::\u2098 m) (C_cons a' m b)) (C_cons a' (a ::\u2098 m) (C_cons a m b))\nm : Multiset \u03b1\nl l' : List \u03b1\nh : l \u2248 l'\na\u271d : \u03b1\nl\u271d l'\u271d : List \u03b1\nb\u271d : C (Quotient.mk (isSetoid \u03b1) l\u271d)\nb'\u271d : C (Quotient.mk (isSetoid \u03b1) l'\u271d)\nhl : l\u271d ~ l'\u271d\nx\u271d : HEq b\u271d b'\u271d\n\u22a2 HEq (C_cons a\u271d (Quotient.mk (isSetoid \u03b1) l\u271d) b\u271d) (C_cons a\u271d (Quotient.mk (isSetoid \u03b1) l'\u271d) b'\u271d)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : Multiset \u03b1 \u2192 Sort u_4\nC_0 : C 0\nC_cons : (a : \u03b1) \u2192 (m : Multiset \u03b1) \u2192 C m \u2192 C (a ::\u2098 m)\nC_cons_heq :\n \u2200 (a a' : \u03b1) (m : Multiset \u03b1) (b : C m),\n HEq (C_cons a (a' ::\u2098 m) (C_cons a' m b)) (C_cons a' (a ::\u2098 m) (C_cons a m b))\nm : Multiset \u03b1\nl l' : List \u03b1\nh : l \u2248 l'\na\u271d : \u03b1\nl\u271d l'\u271d : List \u03b1\nb\u271d : C (Quotient.mk (isSetoid \u03b1) l\u271d)\nb'\u271d : C (Quotient.mk (isSetoid \u03b1) l'\u271d)\nhl : l\u271d ~ l'\u271d\nx\u271d : HEq b\u271d b'\u271d\n\u22a2 Quotient.mk (isSetoid \u03b1) l\u271d = Quotient.mk (isSetoid \u03b1) l'\u271d\n[PROOFSTEP]\nexact Quot.sound hl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Multiset \u03b1\nl : List \u03b1\nH : \u2200 (x : \u03b1), \u00acx \u2208 Quot.mk Setoid.r l\n\u22a2 Quot.mk Setoid.r l = 0\n[PROOFSTEP]\nrw [eq_nil_iff_forall_not_mem.mpr H]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Multiset \u03b1\nl : List \u03b1\nH : \u2200 (x : \u03b1), \u00acx \u2208 Quot.mk Setoid.r l\n\u22a2 Quot.mk Setoid.r [] = 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Multiset \u03b1\nl\u271d : List \u03b1\nhl : Quot.mk Setoid.r l\u271d \u2260 0\na : \u03b1\nl : List \u03b1\nx\u271d : Quot.mk Setoid.r (a :: l) \u2260 0\n\u22a2 a \u2208 Quot.mk Setoid.r (a :: l)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\n\u22a2 a ::\u2098 as = b ::\u2098 bs \u2194 a = b \u2227 as = bs \u2228 a \u2260 b \u2227 \u2203 cs, as = b ::\u2098 cs \u2227 bs = a ::\u2098 cs\n[PROOFSTEP]\nhaveI : DecidableEq \u03b1 := Classical.decEq \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis : DecidableEq \u03b1\n\u22a2 a ::\u2098 as = b ::\u2098 bs \u2194 a = b \u2227 as = bs \u2228 a \u2260 b \u2227 \u2203 cs, as = b ::\u2098 cs \u2227 bs = a ::\u2098 cs\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis : DecidableEq \u03b1\n\u22a2 a ::\u2098 as = b ::\u2098 bs \u2192 a = b \u2227 as = bs \u2228 a \u2260 b \u2227 \u2203 cs, as = b ::\u2098 cs \u2227 bs = a ::\u2098 cs\n[PROOFSTEP]\nintro eq\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis : DecidableEq \u03b1\neq : a ::\u2098 as = b ::\u2098 bs\n\u22a2 a = b \u2227 as = bs \u2228 a \u2260 b \u2227 \u2203 cs, as = b ::\u2098 cs \u2227 bs = a ::\u2098 cs\n[PROOFSTEP]\nby_cases h : a = b\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis : DecidableEq \u03b1\neq : a ::\u2098 as = b ::\u2098 bs\nh : a = b\n\u22a2 a = b \u2227 as = bs \u2228 a \u2260 b \u2227 \u2203 cs, as = b ::\u2098 cs \u2227 bs = a ::\u2098 cs\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nas bs : Multiset \u03b1\nthis : DecidableEq \u03b1\neq : a ::\u2098 as = a ::\u2098 bs\n\u22a2 a = a \u2227 as = bs \u2228 a \u2260 a \u2227 \u2203 cs, as = a ::\u2098 cs \u2227 bs = a ::\u2098 cs\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis : DecidableEq \u03b1\neq : a ::\u2098 as = b ::\u2098 bs\nh : \u00aca = b\n\u22a2 a = b \u2227 as = bs \u2228 a \u2260 b \u2227 \u2203 cs, as = b ::\u2098 cs \u2227 bs = a ::\u2098 cs\n[PROOFSTEP]\nhave : a \u2208 b ::\u2098 bs := eq \u25b8 mem_cons_self _ _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis\u271d : DecidableEq \u03b1\neq : a ::\u2098 as = b ::\u2098 bs\nh : \u00aca = b\nthis : a \u2208 b ::\u2098 bs\n\u22a2 a = b \u2227 as = bs \u2228 a \u2260 b \u2227 \u2203 cs, as = b ::\u2098 cs \u2227 bs = a ::\u2098 cs\n[PROOFSTEP]\nhave : a \u2208 bs := by simpa [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis\u271d : DecidableEq \u03b1\neq : a ::\u2098 as = b ::\u2098 bs\nh : \u00aca = b\nthis : a \u2208 b ::\u2098 bs\n\u22a2 a \u2208 bs\n[PROOFSTEP]\nsimpa [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis\u271d\u00b9 : DecidableEq \u03b1\neq : a ::\u2098 as = b ::\u2098 bs\nh : \u00aca = b\nthis\u271d : a \u2208 b ::\u2098 bs\nthis : a \u2208 bs\n\u22a2 a = b \u2227 as = bs \u2228 a \u2260 b \u2227 \u2203 cs, as = b ::\u2098 cs \u2227 bs = a ::\u2098 cs\n[PROOFSTEP]\nrcases exists_cons_of_mem this with \u27e8cs, hcs\u27e9\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis\u271d\u00b9 : DecidableEq \u03b1\neq : a ::\u2098 as = b ::\u2098 bs\nh : \u00aca = b\nthis\u271d : a \u2208 b ::\u2098 bs\nthis : a \u2208 bs\ncs : Multiset \u03b1\nhcs : bs = a ::\u2098 cs\n\u22a2 a = b \u2227 as = bs \u2228 a \u2260 b \u2227 \u2203 cs, as = b ::\u2098 cs \u2227 bs = a ::\u2098 cs\n[PROOFSTEP]\nsimp [h, hcs]\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis\u271d\u00b9 : DecidableEq \u03b1\neq : a ::\u2098 as = b ::\u2098 bs\nh : \u00aca = b\nthis\u271d : a \u2208 b ::\u2098 bs\nthis : a \u2208 bs\ncs : Multiset \u03b1\nhcs : bs = a ::\u2098 cs\n\u22a2 as = b ::\u2098 cs\n[PROOFSTEP]\nhave : a ::\u2098 as = b ::\u2098 a ::\u2098 cs := by simp [eq, hcs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis\u271d\u00b9 : DecidableEq \u03b1\neq : a ::\u2098 as = b ::\u2098 bs\nh : \u00aca = b\nthis\u271d : a \u2208 b ::\u2098 bs\nthis : a \u2208 bs\ncs : Multiset \u03b1\nhcs : bs = a ::\u2098 cs\n\u22a2 a ::\u2098 as = b ::\u2098 a ::\u2098 cs\n[PROOFSTEP]\nsimp [eq, hcs]\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis\u271d\u00b2 : DecidableEq \u03b1\neq : a ::\u2098 as = b ::\u2098 bs\nh : \u00aca = b\nthis\u271d\u00b9 : a \u2208 b ::\u2098 bs\nthis\u271d : a \u2208 bs\ncs : Multiset \u03b1\nhcs : bs = a ::\u2098 cs\nthis : a ::\u2098 as = b ::\u2098 a ::\u2098 cs\n\u22a2 as = b ::\u2098 cs\n[PROOFSTEP]\nhave : a ::\u2098 as = a ::\u2098 b ::\u2098 cs := by rwa [cons_swap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis\u271d\u00b2 : DecidableEq \u03b1\neq : a ::\u2098 as = b ::\u2098 bs\nh : \u00aca = b\nthis\u271d\u00b9 : a \u2208 b ::\u2098 bs\nthis\u271d : a \u2208 bs\ncs : Multiset \u03b1\nhcs : bs = a ::\u2098 cs\nthis : a ::\u2098 as = b ::\u2098 a ::\u2098 cs\n\u22a2 a ::\u2098 as = a ::\u2098 b ::\u2098 cs\n[PROOFSTEP]\nrwa [cons_swap]\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis\u271d\u00b3 : DecidableEq \u03b1\neq : a ::\u2098 as = b ::\u2098 bs\nh : \u00aca = b\nthis\u271d\u00b2 : a \u2208 b ::\u2098 bs\nthis\u271d\u00b9 : a \u2208 bs\ncs : Multiset \u03b1\nhcs : bs = a ::\u2098 cs\nthis\u271d : a ::\u2098 as = b ::\u2098 a ::\u2098 cs\nthis : a ::\u2098 as = a ::\u2098 b ::\u2098 cs\n\u22a2 as = b ::\u2098 cs\n[PROOFSTEP]\nsimpa using this\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis : DecidableEq \u03b1\n\u22a2 (a = b \u2227 as = bs \u2228 a \u2260 b \u2227 \u2203 cs, as = b ::\u2098 cs \u2227 bs = a ::\u2098 cs) \u2192 a ::\u2098 as = b ::\u2098 bs\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis : DecidableEq \u03b1\nh : a = b \u2227 as = bs \u2228 a \u2260 b \u2227 \u2203 cs, as = b ::\u2098 cs \u2227 bs = a ::\u2098 cs\n\u22a2 a ::\u2098 as = b ::\u2098 bs\n[PROOFSTEP]\nrcases h with (\u27e8eq\u2081, eq\u2082\u27e9 | \u27e8_, cs, eq\u2081, eq\u2082\u27e9)\n[GOAL]\ncase mpr.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis : DecidableEq \u03b1\neq\u2081 : a = b\neq\u2082 : as = bs\n\u22a2 a ::\u2098 as = b ::\u2098 bs\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase mpr.inr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nas bs : Multiset \u03b1\nthis : DecidableEq \u03b1\nleft\u271d : a \u2260 b\ncs : Multiset \u03b1\neq\u2081 : as = b ::\u2098 cs\neq\u2082 : bs = a ::\u2098 cs\n\u22a2 a ::\u2098 as = b ::\u2098 bs\n[PROOFSTEP]\nsimp [*, cons_swap a b]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\n\u22a2 b \u2208 {a} \u2194 b = a\n[PROOFSTEP]\nsimp only [\u2190 cons_zero, mem_cons, iff_self_iff, or_false_iff, not_mem_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\n\u22a2 a \u2208 {a}\n[PROOFSTEP]\nrw [\u2190 cons_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\n\u22a2 a \u2208 a ::\u2098 0\n[PROOFSTEP]\nexact mem_cons_self _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\n\u22a2 {a} = {b} \u2194 a = b\n[PROOFSTEP]\nsimp_rw [\u2190 cons_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\n\u22a2 a ::\u2098 0 = b ::\u2098 0 \u2194 a = b\n[PROOFSTEP]\nexact cons_inj_left _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl : List \u03b1\na : \u03b1\n\u22a2 \u2191l = {a} \u2194 l = [a]\n[PROOFSTEP]\nrw [\u2190 coe_singleton, coe_eq_coe, List.perm_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nm : Multiset \u03b1\n\u22a2 {a} = b ::\u2098 m \u2194 a = b \u2227 m = 0\n[PROOFSTEP]\nrw [\u2190 cons_zero, cons_eq_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\nm : Multiset \u03b1\n\u22a2 (a = b \u2227 0 = m \u2228 a \u2260 b \u2227 \u2203 cs, 0 = b ::\u2098 cs \u2227 m = a ::\u2098 cs) \u2194 a = b \u2227 m = 0\n[PROOFSTEP]\nsimp [eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns t : Multiset \u03b1\n\u22a2 a ::\u2098 s \u2286 t \u2194 a \u2208 t \u2227 s \u2286 t\n[PROOFSTEP]\nsimp [subset_iff, or_imp, forall_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Multiset \u03b1\n\u22a2 toList s = [] \u2194 s = 0\n[PROOFSTEP]\nrw [\u2190 coe_eq_zero, coe_toList]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns : Multiset \u03b1\n\u22a2 a \u2208 toList s \u2194 a \u2208 s\n[PROOFSTEP]\nrw [\u2190 mem_coe, coe_toList]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nm : Multiset \u03b1\n\u22a2 toList m = [a] \u2194 m = {a}\n[PROOFSTEP]\nrw [\u2190 perm_singleton, \u2190 coe_eq_coe, coe_toList, coe_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u22a2 \u2200 (a : Multiset \u03b1), a \u2264 a\n[PROOFSTEP]\nrintro \u27e8l\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d : Multiset \u03b1\nl : List \u03b1\n\u22a2 Quot.mk Setoid.r l \u2264 Quot.mk Setoid.r l\n[PROOFSTEP]\nexact Subperm.refl _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u22a2 \u2200 (a b c : Multiset \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n[PROOFSTEP]\nrintro \u27e8l\u2081\u27e9 \u27e8l\u2082\u27e9 \u27e8l\u2083\u27e9\n[GOAL]\ncase mk.mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d : Multiset \u03b1\nl\u2081 : List \u03b1\nb\u271d : Multiset \u03b1\nl\u2082 : List \u03b1\nc\u271d : Multiset \u03b1\nl\u2083 : List \u03b1\n\u22a2 Quot.mk Setoid.r l\u2081 \u2264 Quot.mk Setoid.r l\u2082 \u2192\n Quot.mk Setoid.r l\u2082 \u2264 Quot.mk Setoid.r l\u2083 \u2192 Quot.mk Setoid.r l\u2081 \u2264 Quot.mk Setoid.r l\u2083\n[PROOFSTEP]\nexact @Subperm.trans _ _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u22a2 \u2200 (a b : Multiset \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\nrintro \u27e8l\u2081\u27e9 \u27e8l\u2082\u27e9 h\u2081 h\u2082\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d : Multiset \u03b1\nl\u2081 : List \u03b1\nb\u271d : Multiset \u03b1\nl\u2082 : List \u03b1\nh\u2081 : Quot.mk Setoid.r l\u2081 \u2264 Quot.mk Setoid.r l\u2082\nh\u2082 : Quot.mk Setoid.r l\u2082 \u2264 Quot.mk Setoid.r l\u2081\n\u22a2 Quot.mk Setoid.r l\u2081 = Quot.mk Setoid.r l\u2082\n[PROOFSTEP]\nexact Quot.sound (Subperm.antisymm h\u2081 h\u2082)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\ns : Multiset \u03b1\na : \u03b1\nl : List \u03b1\nthis : l <+~ a :: l \u2227 \u00acl ~ a :: l\n\u22a2 Quot.mk Setoid.r l < a ::\u2098 Quot.mk Setoid.r l\n[PROOFSTEP]\nsimpa [lt_iff_le_and_ne]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Multiset \u03b1\na : \u03b1\nm : \u00aca \u2208 s\n\u22a2 s \u2264 a ::\u2098 t \u2194 s \u2264 t\n[PROOFSTEP]\nrefine' \u27e8_, fun h => le_trans h <| le_cons_self _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Multiset \u03b1\na : \u03b1\nm : \u00aca \u2208 s\n\u22a2 s \u2264 a ::\u2098 t \u2192 s \u2264 t\n[PROOFSTEP]\nsuffices \u2200 {t'} (_ : s \u2264 t') (_ : a \u2208 t'), a ::\u2098 s \u2264 t' by\n exact fun h => (cons_le_cons_iff a).1 (this h (mem_cons_self _ _))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Multiset \u03b1\na : \u03b1\nm : \u00aca \u2208 s\nthis : \u2200 {t' : Multiset \u03b1}, s \u2264 t' \u2192 a \u2208 t' \u2192 a ::\u2098 s \u2264 t'\n\u22a2 s \u2264 a ::\u2098 t \u2192 s \u2264 t\n[PROOFSTEP]\nexact fun h => (cons_le_cons_iff a).1 (this h (mem_cons_self _ _))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Multiset \u03b1\na : \u03b1\nm : \u00aca \u2208 s\n\u22a2 \u2200 {t' : Multiset \u03b1}, s \u2264 t' \u2192 a \u2208 t' \u2192 a ::\u2098 s \u2264 t'\n[PROOFSTEP]\nintrov h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Multiset \u03b1\na : \u03b1\nm : \u00aca \u2208 s\nt' : Multiset \u03b1\nh : s \u2264 t'\n\u22a2 a \u2208 t' \u2192 a ::\u2098 s \u2264 t'\n[PROOFSTEP]\nrevert m\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Multiset \u03b1\na : \u03b1\nt' : Multiset \u03b1\nh : s \u2264 t'\n\u22a2 \u00aca \u2208 s \u2192 a \u2208 t' \u2192 a ::\u2098 s \u2264 t'\n[PROOFSTEP]\nrefine' leInductionOn h _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Multiset \u03b1\na : \u03b1\nt' : Multiset \u03b1\nh : s \u2264 t'\n\u22a2 \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 <+ l\u2082 \u2192 \u00aca \u2208 \u2191l\u2081 \u2192 a \u2208 \u2191l\u2082 \u2192 a ::\u2098 \u2191l\u2081 \u2264 \u2191l\u2082\n[PROOFSTEP]\nintrov s m\u2081 m\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d t : Multiset \u03b1\na : \u03b1\nt' : Multiset \u03b1\nh : s\u271d \u2264 t'\nl\u2081 l\u2082 : List \u03b1\ns : l\u2081 <+ l\u2082\nm\u2081 : \u00aca \u2208 \u2191l\u2081\nm\u2082 : a \u2208 \u2191l\u2082\n\u22a2 a ::\u2098 \u2191l\u2081 \u2264 \u2191l\u2082\n[PROOFSTEP]\nrcases mem_split m\u2082 with \u27e8r\u2081, r\u2082, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d t : Multiset \u03b1\na : \u03b1\nt' : Multiset \u03b1\nh : s\u271d \u2264 t'\nl\u2081 : List \u03b1\nm\u2081 : \u00aca \u2208 \u2191l\u2081\nr\u2081 r\u2082 : List \u03b1\ns : l\u2081 <+ r\u2081 ++ a :: r\u2082\nm\u2082 : a \u2208 \u2191(r\u2081 ++ a :: r\u2082)\n\u22a2 a ::\u2098 \u2191l\u2081 \u2264 \u2191(r\u2081 ++ a :: r\u2082)\n[PROOFSTEP]\nexact perm_middle.subperm_left.2 ((subperm_cons _).2 <| ((sublist_or_mem_of_sublist s).resolve_right m\u2081).subperm)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Multiset \u03b1\n\u22a2 s \u2264 s + t\n[PROOFSTEP]\nsimpa using add_le_add_left (zero_le t) s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Multiset \u03b1\n\u22a2 s \u2264 t + s\n[PROOFSTEP]\nsimpa using add_le_add_right (zero_le t) s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns t : Multiset \u03b1\n\u22a2 a ::\u2098 s + t = a ::\u2098 (s + t)\n[PROOFSTEP]\nrw [\u2190 singleton_add, \u2190 singleton_add, add_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns t : Multiset \u03b1\n\u22a2 s + a ::\u2098 t = a ::\u2098 (s + t)\n[PROOFSTEP]\nrw [add_comm, cons_add, add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns : Multiset \u03b1\nn : \u2115\nh : a \u2208 n \u2022 s\n\u22a2 a \u2208 s\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns : Multiset \u03b1\nn : \u2115\nh\u271d : a \u2208 n \u2022 s\nh : a \u2208 zero \u2022 s\n\u22a2 a \u2208 s\n[PROOFSTEP]\nrw [zero_nsmul] at h \n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns : Multiset \u03b1\nn : \u2115\nh\u271d : a \u2208 n \u2022 s\nh : a \u2208 0\n\u22a2 a \u2208 s\n[PROOFSTEP]\nexact absurd h (not_mem_zero _)\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns : Multiset \u03b1\nn\u271d : \u2115\nh\u271d : a \u2208 n\u271d \u2022 s\nn : \u2115\nih : a \u2208 n \u2022 s \u2192 a \u2208 s\nh : a \u2208 succ n \u2022 s\n\u22a2 a \u2208 s\n[PROOFSTEP]\nrw [succ_nsmul, mem_add] at h \n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns : Multiset \u03b1\nn\u271d : \u2115\nh\u271d : a \u2208 n\u271d \u2022 s\nn : \u2115\nih : a \u2208 n \u2022 s \u2192 a \u2208 s\nh : a \u2208 s \u2228 a \u2208 n \u2022 s\n\u22a2 a \u2208 s\n[PROOFSTEP]\nexact h.elim id ih\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns : Multiset \u03b1\nn : \u2115\nh0 : n \u2260 0\n\u22a2 a \u2208 n \u2022 s \u2194 a \u2208 s\n[PROOFSTEP]\nrefine' \u27e8mem_of_mem_nsmul, fun h => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns : Multiset \u03b1\nn : \u2115\nh0 : n \u2260 0\nh : a \u2208 s\n\u22a2 a \u2208 n \u2022 s\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := exists_eq_succ_of_ne_zero h0\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns : Multiset \u03b1\nh : a \u2208 s\nn : \u2115\nh0 : succ n \u2260 0\n\u22a2 a \u2208 succ n \u2022 s\n[PROOFSTEP]\nrw [succ_nsmul, mem_add]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns : Multiset \u03b1\nh : a \u2208 s\nn : \u2115\nh0 : succ n \u2260 0\n\u22a2 a \u2208 s \u2228 a \u2208 n \u2022 s\n[PROOFSTEP]\nexact Or.inl h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Multiset \u03b1\nn : \u2115\na : \u03b1\n\u22a2 n \u2022 (a ::\u2098 s) = n \u2022 {a} + n \u2022 s\n[PROOFSTEP]\nrw [\u2190 singleton_add, nsmul_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Multiset \u03b1\n\u22a2 length (toList s) = \u2191card s\n[PROOFSTEP]\nrw [\u2190 coe_card, coe_toList]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Multiset \u03b1\nn : \u2115\n\u22a2 \u2191card (n \u2022 s) = n * \u2191card s\n[PROOFSTEP]\nrw [card.map_nsmul s n, Nat.nsmul_eq_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\n\u22a2 \u2191card {a} = 1\n[PROOFSTEP]\nsimp only [\u2190 cons_zero, card_zero, eq_self_iff_true, zero_add, card_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\n\u22a2 \u2191card {a, b} = 2\n[PROOFSTEP]\nrw [insert_eq_cons, card_cons, card_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Multiset \u03b1\ne : s = 0\n\u22a2 \u2191card s = 0\n[PROOFSTEP]\nsimp [e]\n[GOAL]\n\u03b1 : Type u_1\np : Multiset \u03b1 \u2192 Sort u_4\ns : Multiset \u03b1\nih : (s : Multiset \u03b1) \u2192 ((t : Multiset \u03b1) \u2192 t < s \u2192 p t) \u2192 p s\nt : Multiset \u03b1\n_h : t < s\n\u22a2 (invImage (fun a => PSigma.casesOn a fun s snd => \u2191card s) instWellFoundedRelation).1 { fst := t, snd := ih }\n { fst := s, snd := ih }\n[PROOFSTEP]\nexact card_lt_of_lt _h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Multiset \u03b1 \u2192 Sort u_4\ns : Multiset \u03b1\nH : (s : Multiset \u03b1) \u2192 ((t : Multiset \u03b1) \u2192 t < s \u2192 p t) \u2192 p s\n\u22a2 strongInductionOn s H = H s fun t _h => strongInductionOn t H\n[PROOFSTEP]\nrw [strongInductionOn]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\ns t : Multiset \u03b1\nht : \u2191card t \u2264 n\n_h : s < t\n\u22a2 (invImage (fun a => n - \u2191card a) instWellFoundedRelation).1 t s\n[PROOFSTEP]\nexact\n (tsub_lt_tsub_iff_left_of_le ht).2\n (card_lt_of_lt _h)\n -- Porting note: reorderd universes\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Multiset \u03b1 \u2192 Sort u_4\nn : \u2115\nH : (t\u2081 : Multiset \u03b1) \u2192 ({t\u2082 : Multiset \u03b1} \u2192 \u2191card t\u2082 \u2264 n \u2192 t\u2081 < t\u2082 \u2192 p t\u2082) \u2192 \u2191card t\u2081 \u2264 n \u2192 p t\u2081\ns : Multiset \u03b1\n\u22a2 strongDownwardInduction H s = H s fun {t\u2082} ht _hst => strongDownwardInduction H t\u2082 ht\n[PROOFSTEP]\nrw [strongDownwardInduction]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Multiset \u03b1 \u2192 Sort u_4\ns : Multiset \u03b1\nn : \u2115\nH : (t\u2081 : Multiset \u03b1) \u2192 ({t\u2082 : Multiset \u03b1} \u2192 \u2191card t\u2082 \u2264 n \u2192 t\u2081 < t\u2082 \u2192 p t\u2082) \u2192 \u2191card t\u2081 \u2264 n \u2192 p t\u2081\n\u22a2 (fun a => strongDownwardInductionOn s H a) = H s fun {t} ht _h => strongDownwardInductionOn t H ht\n[PROOFSTEP]\ndsimp only [strongDownwardInductionOn]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Multiset \u03b1 \u2192 Sort u_4\ns : Multiset \u03b1\nn : \u2115\nH : (t\u2081 : Multiset \u03b1) \u2192 ({t\u2082 : Multiset \u03b1} \u2192 \u2191card t\u2082 \u2264 n \u2192 t\u2081 < t\u2082 \u2192 p t\u2082) \u2192 \u2191card t\u2081 \u2264 n \u2192 p t\u2081\n\u22a2 (fun a => strongDownwardInduction H s a) = H s fun {t} ht _h => strongDownwardInduction H t ht\n[PROOFSTEP]\nrw [strongDownwardInduction]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nn : \u2115\n\u22a2 n \u2022 {a} = replicate n a\n[PROOFSTEP]\nrw [\u2190 replicate_one, nsmul_replicate, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nk n : \u2115\n\u22a2 replicate k a \u2264 replicate n a \u2194 List.replicate k a <+ List.replicate n a\n[PROOFSTEP]\nrw [\u2190 replicate_le_coe, coe_replicate]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Multiset \u03b1\nx : \u03b1\nn : \u2115\n\u22a2 m < replicate (n + 1) x \u2194 m \u2264 replicate n x\n[PROOFSTEP]\nrw [lt_iff_cons_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Multiset \u03b1\nx : \u03b1\nn : \u2115\n\u22a2 (\u2203 a, a ::\u2098 m \u2264 replicate (n + 1) x) \u2194 m \u2264 replicate n x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Multiset \u03b1\nx : \u03b1\nn : \u2115\n\u22a2 (\u2203 a, a ::\u2098 m \u2264 replicate (n + 1) x) \u2192 m \u2264 replicate n x\n[PROOFSTEP]\nrintro \u27e8x', hx'\u27e9\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Multiset \u03b1\nx : \u03b1\nn : \u2115\nx' : \u03b1\nhx' : x' ::\u2098 m \u2264 replicate (n + 1) x\n\u22a2 m \u2264 replicate n x\n[PROOFSTEP]\nhave := eq_of_mem_replicate (mem_of_le hx' (mem_cons_self _ _))\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Multiset \u03b1\nx : \u03b1\nn : \u2115\nx' : \u03b1\nhx' : x' ::\u2098 m \u2264 replicate (n + 1) x\nthis : x' = x\n\u22a2 m \u2264 replicate n x\n[PROOFSTEP]\nrwa [this, replicate_succ, cons_le_cons_iff] at hx' \n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Multiset \u03b1\nx : \u03b1\nn : \u2115\n\u22a2 m \u2264 replicate n x \u2192 \u2203 a, a ::\u2098 m \u2264 replicate (n + 1) x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Multiset \u03b1\nx : \u03b1\nn : \u2115\nh : m \u2264 replicate n x\n\u22a2 \u2203 a, a ::\u2098 m \u2264 replicate (n + 1) x\n[PROOFSTEP]\nrw [replicate_succ]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Multiset \u03b1\nx : \u03b1\nn : \u2115\nh : m \u2264 replicate n x\n\u22a2 \u2203 a, a ::\u2098 m \u2264 x ::\u2098 replicate n x\n[PROOFSTEP]\nexact \u27e8x, cons_le_cons _ h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Multiset \u03b1\na\u271d b : \u03b1\ns : Multiset \u03b1\na : \u03b1\nh : \u00aca \u2208 s\n\u22a2 s \u2264 a ::\u2098 erase s a\n[PROOFSTEP]\nrw [erase_of_not_mem h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Multiset \u03b1\na\u271d b : \u03b1\ns : Multiset \u03b1\na : \u03b1\nh : \u00aca \u2208 s\n\u22a2 s \u2264 a ::\u2098 s\n[PROOFSTEP]\napply le_cons_self\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\n\u22a2 s + {a} = t \u2194 a \u2208 t \u2227 s = erase t a\n[PROOFSTEP]\nrw [add_comm, singleton_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\n\u22a2 a ::\u2098 s = t \u2194 a \u2208 t \u2227 s = erase t a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\n\u22a2 a ::\u2098 s = t \u2192 a \u2208 t \u2227 s = erase t a\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Multiset \u03b1\na\u271d b : \u03b1\ns : Multiset \u03b1\na : \u03b1\n\u22a2 a \u2208 a ::\u2098 s \u2227 s = erase (a ::\u2098 s) a\n[PROOFSTEP]\nexact \u27e8s.mem_cons_self a, (s.erase_cons_head a).symm\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\n\u22a2 a \u2208 t \u2227 s = erase t a \u2192 a ::\u2098 s = t\n[PROOFSTEP]\nrintro \u27e8h, rfl\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t\u271d : Multiset \u03b1\na\u271d b : \u03b1\nt : Multiset \u03b1\na : \u03b1\nh : a \u2208 t\n\u22a2 a ::\u2098 erase t a = t\n[PROOFSTEP]\nexact cons_erase h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b a : \u03b1\ns t : Multiset \u03b1\nh : a \u2208 t\n\u22a2 erase (s + t) a = s + erase t a\n[PROOFSTEP]\nrw [add_comm, erase_add_left_pos s h, add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b a : \u03b1\ns t : Multiset \u03b1\nh : \u00aca \u2208 t\n\u22a2 erase (s + t) a = erase s a + t\n[PROOFSTEP]\nrw [add_comm, erase_add_right_neg s h, add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Multiset \u03b1\na\u271d b a : \u03b1\ns : Multiset \u03b1\nh : a \u2208 s\n\u22a2 erase s a < s\n[PROOFSTEP]\nsimpa [h] using lt_cons_self (s.erase a) a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\nh : s \u2264 a ::\u2098 t\nm : a \u2208 s\n\u22a2 erase s a \u2264 t\n[PROOFSTEP]\nrw [\u2190 cons_erase m] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\nh : a ::\u2098 erase s a \u2264 a ::\u2098 t\nm : a \u2208 s\n\u22a2 erase s a \u2264 t\n[PROOFSTEP]\nexact (cons_le_cons_iff _).1 h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Multiset \u03b1\na\u271d b a : \u03b1\ns : Multiset \u03b1\n\u22a2 \u2191card (erase s a) = if a \u2208 s then pred (\u2191card s) else \u2191card s\n[PROOFSTEP]\nby_cases h : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Multiset \u03b1\na\u271d b a : \u03b1\ns : Multiset \u03b1\nh : a \u2208 s\n\u22a2 \u2191card (erase s a) = if a \u2208 s then pred (\u2191card s) else \u2191card s\n[PROOFSTEP]\nrwa [card_erase_of_mem h, if_pos]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Multiset \u03b1\na\u271d b a : \u03b1\ns : Multiset \u03b1\nh : \u00aca \u2208 s\n\u22a2 \u2191card (erase s a) = if a \u2208 s then pred (\u2191card s) else \u2191card s\n[PROOFSTEP]\nrwa [erase_of_not_mem h, if_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\ns t : Multiset \u03b1\n\u22a2 s = t \u2192 (\u2200 (x : \u03b1), x \u2208 t \u2192 f x = g x) \u2192 map f s = map g t\n[PROOFSTEP]\nrintro rfl h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x = g x\n\u22a2 map f s = map g s\n[PROOFSTEP]\ninduction s using Quot.inductionOn\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\na\u271d : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 Quot.mk Setoid.r a\u271d \u2192 f x = g x\n\u22a2 map f (Quot.mk Setoid.r a\u271d) = map g (Quot.mk Setoid.r a\u271d)\n[PROOFSTEP]\nexact congr_arg _ (List.map_congr h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b2' : Type u_2\nm : Multiset \u03b1\nf : \u03b1 \u2192 \u03b2\nf' : \u03b1 \u2192 \u03b2'\nh : \u03b2 = \u03b2'\nhf : \u2200 (a : \u03b1), a \u2208 m \u2192 HEq (f a) (f' a)\n\u22a2 HEq (map f m) (map f' m)\n[PROOFSTEP]\nsubst h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Multiset \u03b1\nf f' : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 m \u2192 HEq (f a) (f' a)\n\u22a2 HEq (map f m) (map f' m)\n[PROOFSTEP]\nsimp at hf \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Multiset \u03b1\nf f' : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 m \u2192 f a = f' a\n\u22a2 HEq (map f m) (map f' m)\n[PROOFSTEP]\nsimp [map_congr rfl hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nt : \u03b1\n\u22a2 map f \u2218 cons t = cons (f t) \u2218 map f\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nt : \u03b1\nx\u271d : Multiset \u03b1\n\u22a2 (map f \u2218 cons t) x\u271d = (cons (f t) \u2218 map f) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nk : \u2115\na : \u03b1\n\u22a2 map f (replicate k a) = replicate k (f a)\n[PROOFSTEP]\nsimp only [\u2190 coe_replicate, coe_map, List.map_replicate]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nc : \u03b2 \u2192 \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : CanLift \u03b1 \u03b2 c p\n\u22a2 \u2200 (x : Multiset \u03b1), (\u2200 (x_1 : \u03b1), x_1 \u2208 x \u2192 p x_1) \u2192 \u2203 y, map c y = x\n[PROOFSTEP]\nrintro \u27e8l\u27e9 hl\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nc : \u03b2 \u2192 \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : CanLift \u03b1 \u03b2 c p\nx\u271d : Multiset \u03b1\nl : List \u03b1\nhl : \u2200 (x : \u03b1), x \u2208 Quot.mk Setoid.r l \u2192 p x\n\u22a2 \u2203 y, map c y = Quot.mk Setoid.r l\n[PROOFSTEP]\nlift l to List \u03b2 using hl\n[GOAL]\ncase mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nc : \u03b2 \u2192 \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : CanLift \u03b1 \u03b2 c p\nx\u271d : Multiset \u03b1\nl : List \u03b2\n\u22a2 \u2203 y, map c y = Quot.mk Setoid.r (List.map c l)\n[PROOFSTEP]\nexact \u27e8l, coe_map _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Multiset \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 map f s = 0 \u2194 s = 0\n[PROOFSTEP]\nrw [\u2190 Multiset.card_eq_zero, Multiset.card_map, Multiset.card_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nb : \u03b2\n\u22a2 map f s = {b} \u2194 \u2203 a, s = {a} \u2227 f a = b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nb : \u03b2\n\u22a2 map f s = {b} \u2192 \u2203 a, s = {a} \u2227 f a = b\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nb : \u03b2\nh : map f s = {b}\n\u22a2 \u2203 a, s = {a} \u2227 f a = b\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 : \u2203 a, s = { a } := by rw [\u2190 card_eq_one, \u2190 card_map, h, card_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nb : \u03b2\nh : map f s = {b}\n\u22a2 \u2203 a, s = {a}\n[PROOFSTEP]\nrw [\u2190 card_eq_one, \u2190 card_map, h, card_singleton]\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nb : \u03b2\nh : map f s = {b}\na : \u03b1\nha : s = {a}\n\u22a2 \u2203 a, s = {a} \u2227 f a = b\n[PROOFSTEP]\nrefine' \u27e8a, ha, _\u27e9\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nb : \u03b2\nh : map f s = {b}\na : \u03b1\nha : s = {a}\n\u22a2 f a = b\n[PROOFSTEP]\nrw [\u2190 mem_singleton, \u2190 h, ha, map_singleton, mem_singleton]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nb : \u03b2\n\u22a2 (\u2203 a, s = {a} \u2227 f a = b) \u2192 map f s = {b}\n[PROOFSTEP]\nrintro \u27e8a, rfl, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 map f {a} = {f a}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nb : \u03b2\n\u22a2 (\u2203 a, a \u2208 s \u2227 f a = b \u2227 map f (erase s a) = t) \u2194 map f s = b ::\u2098 t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nb : \u03b2\n\u22a2 (\u2203 a, a \u2208 s \u2227 f a = b \u2227 map f (erase s a) = t) \u2192 map f s = b ::\u2098 t\n[PROOFSTEP]\nrintro \u27e8a, ha, rfl, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\na : \u03b1\nha : a \u2208 s\n\u22a2 map f s = f a ::\u2098 map f (erase s a)\n[PROOFSTEP]\nrw [\u2190 map_cons, Multiset.cons_erase ha]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nb : \u03b2\n\u22a2 map f s = b ::\u2098 t \u2192 \u2203 a, a \u2208 s \u2227 f a = b \u2227 map f (erase s a) = t\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nb : \u03b2\nh : map f s = b ::\u2098 t\n\u22a2 \u2203 a, a \u2208 s \u2227 f a = b \u2227 map f (erase s a) = t\n[PROOFSTEP]\nhave : b \u2208 s.map f := by\n rw [h]\n exact mem_cons_self _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nb : \u03b2\nh : map f s = b ::\u2098 t\n\u22a2 b \u2208 map f s\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nb : \u03b2\nh : map f s = b ::\u2098 t\n\u22a2 b \u2208 b ::\u2098 t\n[PROOFSTEP]\nexact mem_cons_self _ _\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nb : \u03b2\nh : map f s = b ::\u2098 t\nthis : b \u2208 map f s\n\u22a2 \u2203 a, a \u2208 s \u2227 f a = b \u2227 map f (erase s a) = t\n[PROOFSTEP]\nobtain \u27e8a, h1, rfl\u27e9 := mem_map.mp this\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\na : \u03b1\nh1 : a \u2208 s\nh : map f s = f a ::\u2098 t\nthis : f a \u2208 map f s\n\u22a2 \u2203 a_1, a_1 \u2208 s \u2227 f a_1 = f a \u2227 map f (erase s a_1) = t\n[PROOFSTEP]\nobtain \u27e8u, rfl\u27e9 := exists_cons_of_mem h1\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nt : Multiset \u03b2\na : \u03b1\nu : Multiset \u03b1\nh1 : a \u2208 a ::\u2098 u\nh : map f (a ::\u2098 u) = f a ::\u2098 t\nthis : f a \u2208 map f (a ::\u2098 u)\n\u22a2 \u2203 a_1, a_1 \u2208 a ::\u2098 u \u2227 f a_1 = f a \u2227 map f (erase (a ::\u2098 u) a_1) = t\n[PROOFSTEP]\nrw [map_cons, cons_inj_right] at h \n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nt : Multiset \u03b2\na : \u03b1\nu : Multiset \u03b1\nh1 : a \u2208 a ::\u2098 u\nh : map f u = t\nthis : f a \u2208 map f (a ::\u2098 u)\n\u22a2 \u2203 a_1, a_1 \u2208 a ::\u2098 u \u2227 f a_1 = f a \u2227 map f (erase (a ::\u2098 u) a_1) = t\n[PROOFSTEP]\nrefine' \u27e8a, mem_cons_self _ _, rfl, _\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\nt : Multiset \u03b2\na : \u03b1\nu : Multiset \u03b1\nh1 : a \u2208 a ::\u2098 u\nh : map f u = t\nthis : f a \u2208 map f (a ::\u2098 u)\n\u22a2 map f (erase (a ::\u2098 u) a) = t\n[PROOFSTEP]\nrw [Multiset.erase_cons_head, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nb\u2081 b\u2082 : \u03b2\nl : List \u03b1\nh : b\u2081 \u2208 map (const \u03b1 b\u2082) \u2191l\n\u22a2 b\u2081 \u2208 replicate (?m.122527 h) b\u2082\n[PROOFSTEP]\nrwa [map_const] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns t : Multiset \u03b1\nh : s < t\n\u22a2 map f s < map f t\n[PROOFSTEP]\nrefine' (map_le_map h.le).lt_of_not_le fun H => h.ne <| eq_of_le_of_card_le h.le _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns t : Multiset \u03b1\nh : s < t\nH : map f t \u2264 map f s\n\u22a2 \u2191card t \u2264 \u2191card s\n[PROOFSTEP]\nrw [\u2190 s.card_map f, \u2190 t.card_map f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns t : Multiset \u03b1\nh : s < t\nH : map f t \u2264 map f s\n\u22a2 \u2191card (map f t) \u2264 \u2191card (map f s)\n[PROOFSTEP]\nexact card_le_of_le H\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nx : \u03b1\ns : Multiset \u03b1\n\u22a2 map f (erase s x) = erase (map f s) (f x)\n[PROOFSTEP]\ninduction' s using Multiset.induction_on with y s ih\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nx : \u03b1\n\u22a2 map f (erase 0 x) = erase (map f 0) (f x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nx y : \u03b1\ns : Multiset \u03b1\nih : map f (erase s x) = erase (map f s) (f x)\n\u22a2 map f (erase (y ::\u2098 s) x) = erase (map f (y ::\u2098 s)) (f x)\n[PROOFSTEP]\nby_cases hxy : y = x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nx y : \u03b1\ns : Multiset \u03b1\nih : map f (erase s x) = erase (map f s) (f x)\nhxy : y = x\n\u22a2 map f (erase (y ::\u2098 s) x) = erase (map f (y ::\u2098 s)) (f x)\n[PROOFSTEP]\ncases hxy\n[GOAL]\ncase pos.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nx : \u03b1\ns : Multiset \u03b1\nih : map f (erase s x) = erase (map f s) (f x)\n\u22a2 map f (erase (x ::\u2098 s) x) = erase (map f (x ::\u2098 s)) (f x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nx y : \u03b1\ns : Multiset \u03b1\nih : map f (erase s x) = erase (map f s) (f x)\nhxy : \u00acy = x\n\u22a2 map f (erase (y ::\u2098 s) x) = erase (map f (y ::\u2098 s)) (f x)\n[PROOFSTEP]\nrw [s.erase_cons_tail hxy, map_cons, map_cons, (s.map f).erase_cons_tail (hf.ne hxy), ih]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\n\u22a2 Surjective (map f)\n[PROOFSTEP]\nintro s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\ns : Multiset \u03b2\n\u22a2 \u2203 a, map f a = s\n[PROOFSTEP]\ninduction' s using Multiset.induction_on with x s ih\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\n\u22a2 \u2203 a, map f a = 0\n[PROOFSTEP]\nexact \u27e80, map_zero _\u27e9\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx : \u03b2\ns : Multiset \u03b2\nih : \u2203 a, map f a = s\n\u22a2 \u2203 a, map f a = x ::\u2098 s\n[PROOFSTEP]\nobtain \u27e8y, rfl\u27e9 := hf x\n[GOAL]\ncase cons.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\ns : Multiset \u03b2\nih : \u2203 a, map f a = s\ny : \u03b1\n\u22a2 \u2203 a, map f a = f y ::\u2098 s\n[PROOFSTEP]\nobtain \u27e8t, rfl\u27e9 := ih\n[GOAL]\ncase cons.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\ny : \u03b1\nt : Multiset \u03b1\n\u22a2 \u2203 a, map f a = f y ::\u2098 map f t\n[PROOFSTEP]\nexact \u27e8y ::\u2098 t, map_cons _ _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nH : LeftCommutative f\nx : \u03b2\nq : \u03b1 \u2192 Prop\np : \u03b2 \u2192 Prop\ns : Multiset \u03b1\nhpqf : \u2200 (a : \u03b1) (b : \u03b2), q a \u2192 p b \u2192 p (f a b)\npx : p x\nq_s : \u2200 (a : \u03b1), a \u2208 s \u2192 q a\n\u22a2 p (foldr f H x s)\n[PROOFSTEP]\nrevert s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nH : LeftCommutative f\nx : \u03b2\nq : \u03b1 \u2192 Prop\np : \u03b2 \u2192 Prop\nhpqf : \u2200 (a : \u03b1) (b : \u03b2), q a \u2192 p b \u2192 p (f a b)\npx : p x\n\u22a2 \u2200 (s : Multiset \u03b1), (\u2200 (a : \u03b1), a \u2208 s \u2192 q a) \u2192 p (foldr f H x s)\n[PROOFSTEP]\nrefine' Multiset.induction (by simp [px]) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nH : LeftCommutative f\nx : \u03b2\nq : \u03b1 \u2192 Prop\np : \u03b2 \u2192 Prop\nhpqf : \u2200 (a : \u03b1) (b : \u03b2), q a \u2192 p b \u2192 p (f a b)\npx : p x\n\u22a2 (\u2200 (a : \u03b1), a \u2208 0 \u2192 q a) \u2192 p (foldr f H x 0)\n[PROOFSTEP]\nsimp [px]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nH : LeftCommutative f\nx : \u03b2\nq : \u03b1 \u2192 Prop\np : \u03b2 \u2192 Prop\nhpqf : \u2200 (a : \u03b1) (b : \u03b2), q a \u2192 p b \u2192 p (f a b)\npx : p x\n\u22a2 \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1},\n ((\u2200 (a : \u03b1), a \u2208 s \u2192 q a) \u2192 p (foldr f H x s)) \u2192 (\u2200 (a_2 : \u03b1), a_2 \u2208 a ::\u2098 s \u2192 q a_2) \u2192 p (foldr f H x (a ::\u2098 s))\n[PROOFSTEP]\nintro a s hs hsa\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nH : LeftCommutative f\nx : \u03b2\nq : \u03b1 \u2192 Prop\np : \u03b2 \u2192 Prop\nhpqf : \u2200 (a : \u03b1) (b : \u03b2), q a \u2192 p b \u2192 p (f a b)\npx : p x\na : \u03b1\ns : Multiset \u03b1\nhs : (\u2200 (a : \u03b1), a \u2208 s \u2192 q a) \u2192 p (foldr f H x s)\nhsa : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 q a_1\n\u22a2 p (foldr f H x (a ::\u2098 s))\n[PROOFSTEP]\nrw [foldr_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nH : LeftCommutative f\nx : \u03b2\nq : \u03b1 \u2192 Prop\np : \u03b2 \u2192 Prop\nhpqf : \u2200 (a : \u03b1) (b : \u03b2), q a \u2192 p b \u2192 p (f a b)\npx : p x\na : \u03b1\ns : Multiset \u03b1\nhs : (\u2200 (a : \u03b1), a \u2208 s \u2192 q a) \u2192 p (foldr f H x s)\nhsa : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 q a_1\n\u22a2 p (f a (foldr f H x s))\n[PROOFSTEP]\nhave hps : \u2200 x : \u03b1, x \u2208 s \u2192 q x := fun x hxs => hsa x (mem_cons_of_mem hxs)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nH : LeftCommutative f\nx : \u03b2\nq : \u03b1 \u2192 Prop\np : \u03b2 \u2192 Prop\nhpqf : \u2200 (a : \u03b1) (b : \u03b2), q a \u2192 p b \u2192 p (f a b)\npx : p x\na : \u03b1\ns : Multiset \u03b1\nhs : (\u2200 (a : \u03b1), a \u2208 s \u2192 q a) \u2192 p (foldr f H x s)\nhsa : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 q a_1\nhps : \u2200 (x : \u03b1), x \u2208 s \u2192 q x\n\u22a2 p (f a (foldr f H x s))\n[PROOFSTEP]\nexact hpqf a (foldr f H x s) (hsa a (mem_cons_self a s)) (hs hps)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nH : RightCommutative f\nx : \u03b2\nq : \u03b1 \u2192 Prop\np : \u03b2 \u2192 Prop\ns : Multiset \u03b1\nhpqf : \u2200 (a : \u03b1) (b : \u03b2), q a \u2192 p b \u2192 p (f b a)\npx : p x\nq_s : \u2200 (a : \u03b1), a \u2208 s \u2192 q a\n\u22a2 p (foldl f H x s)\n[PROOFSTEP]\nrw [foldl_swap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nH : RightCommutative f\nx : \u03b2\nq : \u03b1 \u2192 Prop\np : \u03b2 \u2192 Prop\ns : Multiset \u03b1\nhpqf : \u2200 (a : \u03b1) (b : \u03b2), q a \u2192 p b \u2192 p (f b a)\npx : p x\nq_s : \u2200 (a : \u03b1), a \u2208 s \u2192 q a\n\u22a2 p (foldr (fun x y => f y x) (_ : \u2200 (_x _y : \u03b1) (_z : \u03b2), f (f _z _y) _x = f (f _z _x) _y) x s)\n[PROOFSTEP]\nexact foldr_induction' (fun x y => f y x) (fun x y z => (H _ _ _).symm) x q p s hpqf px q_s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\ns : Multiset \u03b1\nl\u2081 l\u2082 : List \u03b1\npp : l\u2081 ~ l\u2082\nH\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2082 \u2192 p a\nH\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2081 \u2192 p a\n\u22a2 \u2200 {s\u2082 : Multiset \u03b1} {e : \u2191l\u2081 = s\u2082} {H : \u2200 (a : \u03b1), a \u2208 s\u2082 \u2192 p a},\n Eq.ndrec (motive := fun s => (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 Multiset \u03b2) (fun x => \u2191(List.pmap f l\u2081 H\u2081)) e H =\n \u2191(List.pmap f l\u2081 H\u2081)\n[PROOFSTEP]\nintro s\u2082 e _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\ns : Multiset \u03b1\nl\u2081 l\u2082 : List \u03b1\npp : l\u2081 ~ l\u2082\nH\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2082 \u2192 p a\nH\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2081 \u2192 p a\ns\u2082 : Multiset \u03b1\ne : \u2191l\u2081 = s\u2082\nH\u271d : \u2200 (a : \u03b1), a \u2208 s\u2082 \u2192 p a\n\u22a2 Eq.ndrec (motive := fun s => (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 Multiset \u03b2) (fun x => \u2191(List.pmap f l\u2081 H\u2081)) e H\u271d =\n \u2191(List.pmap f l\u2081 H\u2081)\n[PROOFSTEP]\nsubst e\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\ns : Multiset \u03b1\nl\u2081 l\u2082 : List \u03b1\npp : l\u2081 ~ l\u2082\nH\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2082 \u2192 p a\nH\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2081 \u2192 p a\nH\u271d : \u2200 (a : \u03b1), a \u2208 \u2191l\u2081 \u2192 p a\n\u22a2 Eq.ndrec (motive := fun s => (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 Multiset \u03b2) (fun x => \u2191(List.pmap f l\u2081 H\u2081)) (_ : \u2191l\u2081 = \u2191l\u2081)\n H\u271d =\n \u2191(List.pmap f l\u2081 H\u2081)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\ns : Multiset \u03b1\nhx : x \u2208 s\n\u22a2 sizeOf x < sizeOf s\n[PROOFSTEP]\ninduction' s using Quot.inductionOn with l a b\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\ns : Multiset \u03b1\nhx\u271d : x \u2208 s\nl : List \u03b1\nhx : x \u2208 Quot.mk Setoid.r l\n\u22a2 sizeOf x < sizeOf (Quot.mk Setoid.r l)\n[PROOFSTEP]\nexact List.sizeOf_lt_sizeOf_of_mem hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nm : Multiset \u03b1\nl : List \u03b1\n\u22a2 List.pmap mk l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 (fun a_2 => a_2 \u2208 a ::\u2098 Quotient.mk (isSetoid \u03b1) l) x) =\n List.map (fun p => { val := \u2191p, property := (_ : \u2191p \u2208 a ::\u2098 Quotient.mk (isSetoid \u03b1) l) })\n (List.pmap mk l (_ : \u2200 (_a : \u03b1), _a \u2208 Quotient.mk (isSetoid \u03b1) l \u2192 _a \u2208 Quotient.mk (isSetoid \u03b1) l))\n[PROOFSTEP]\nrw [List.map_pmap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nm : Multiset \u03b1\nl : List \u03b1\n\u22a2 List.pmap mk l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 (fun a_2 => a_2 \u2208 a ::\u2098 Quotient.mk (isSetoid \u03b1) l) x) =\n List.pmap\n (fun a_1 h =>\n { val := \u2191{ val := a_1, property := h },\n property := (_ : \u2191{ val := a_1, property := h } \u2208 a ::\u2098 Quotient.mk (isSetoid \u03b1) l) })\n l (_ : \u2200 (_a : \u03b1), _a \u2208 Quotient.mk (isSetoid \u03b1) l \u2192 _a \u2208 Quotient.mk (isSetoid \u03b1) l)\n[PROOFSTEP]\nexact List.pmap_congr _ fun _ _ _ _ => Subtype.eq rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Multiset \u03b1\np : \u03b1 \u2192 Prop\nhp : (a : \u03b1) \u2192 Decidable (p a)\nl : List \u03b1\n\u22a2 (\u2200 (a : \u03b1), a \u2208 l \u2192 p a) \u2194 \u2200 (a : \u03b1), a \u2208 Quotient.mk (isSetoid \u03b1) l \u2192 p a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\nm : Multiset \u03b1\n\u03b2 : \u03b1 \u2192 Type u_4\nh : (a : \u03b1) \u2192 DecidableEq (\u03b2 a)\nf g : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a\n\u22a2 (\u2200 (a : \u03b1) (h : a \u2208 m), f a h = g a h) \u2194 f = g\n[PROOFSTEP]\nsimp [Function.funext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Multiset \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nl : List \u03b1\n\u22a2 (\u2203 a, a \u2208 l \u2227 p a) \u2194 \u2203 x, x \u2208 Quotient.mk (isSetoid \u03b1) l \u2227 p x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u : Multiset \u03b1\na b : \u03b1\n\u22a2 s - t \u2264 u \u2194 s \u2264 u + t\n[PROOFSTEP]\nrevert s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nt u : Multiset \u03b1\na b : \u03b1\n\u22a2 \u2200 {s : Multiset \u03b1}, s - t \u2264 u \u2194 s \u2264 u + t\n[PROOFSTEP]\nexact @(Multiset.induction_on t (by simp [Multiset.sub_zero]) fun a t IH s => by simp [IH, erase_le_iff_le_cons])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nt u : Multiset \u03b1\na b : \u03b1\n\u22a2 \u2200 {s : Multiset \u03b1}, s - 0 \u2264 u \u2194 s \u2264 u + 0\n[PROOFSTEP]\nsimp [Multiset.sub_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nt\u271d u : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s : Multiset \u03b1}, s - t \u2264 u \u2194 s \u2264 u + t\ns : Multiset \u03b1\n\u22a2 s - a ::\u2098 t \u2264 u \u2194 s \u2264 u + a ::\u2098 t\n[PROOFSTEP]\nsimp [IH, erase_le_iff_le_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na\u271d b a : \u03b1\ns t : Multiset \u03b1\nh : t \u2264 s\n\u22a2 a ::\u2098 s - t = a ::\u2098 (s - t)\n[PROOFSTEP]\nrw [\u2190 singleton_add, \u2190 singleton_add, add_tsub_assoc_of_le h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 Quotient.mk (isSetoid \u03b1) l\u2081 - Quotient.mk (isSetoid \u03b1) l\u2082 =\n foldl erase (_ : \u2200 (s : Multiset \u03b1) (a b : \u03b1), erase (erase s a) b = erase (erase s b) a)\n (Quotient.mk (isSetoid \u03b1) l\u2081) (Quotient.mk (isSetoid \u03b1) l\u2082)\n[PROOFSTEP]\nshow ofList (l\u2081.diff l\u2082) = foldl erase erase_comm l\u2081 l\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 \u2191(List.diff l\u2081 l\u2082) = foldl erase (_ : \u2200 (s : Multiset \u03b1) (a b : \u03b1), erase (erase s a) b = erase (erase s b) a) \u2191l\u2081 \u2191l\u2082\n[PROOFSTEP]\nrw [diff_eq_foldl l\u2081 l\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 \u2191(List.foldl List.erase l\u2081 l\u2082) =\n foldl erase (_ : \u2200 (s : Multiset \u03b1) (a b : \u03b1), erase (erase s a) b = erase (erase s b) a) \u2191l\u2081 \u2191l\u2082\n[PROOFSTEP]\nsymm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 foldl erase (_ : \u2200 (s : Multiset \u03b1) (a b : \u03b1), erase (erase s a) b = erase (erase s b) a) \u2191l\u2081 \u2191l\u2082 =\n \u2191(List.foldl List.erase l\u2081 l\u2082)\n[PROOFSTEP]\nexact foldl_hom _ _ _ _ _ fun x y => rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\nh : t \u2264 s\n\u22a2 \u2191card s = \u2191card t + \u2191card (s - t)\n[PROOFSTEP]\nrw [add_comm, \u2190 card_add, tsub_add_cancel_of_le h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u : Multiset \u03b1\na b : \u03b1\nh\u2081 : s \u2264 u\nh\u2082 : t \u2264 u\n\u22a2 s \u222a t \u2264 u\n[PROOFSTEP]\nrw [\u2190 eq_union_left h\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u : Multiset \u03b1\na b : \u03b1\nh\u2081 : s \u2264 u\nh\u2082 : t \u2264 u\n\u22a2 s \u222a t \u2264 u \u222a t\n[PROOFSTEP]\nexact union_le_union_right h\u2081 t\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\nfinj : Injective f\ns t : Multiset \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 List.map f (List.diff l\u2081 l\u2082 ++ l\u2082) = List.diff (List.map f l\u2081) (List.map f l\u2082) ++ List.map f l\u2082\n[PROOFSTEP]\nrw [List.map_append f, List.map_diff finj]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u : Multiset \u03b1\na b : \u03b1\n\u22a2 0 \u222a s = s\n[PROOFSTEP]\nsimp [union_def]\n --Porting note: new theorem\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u : Multiset \u03b1\na b : \u03b1\n\u22a2 s \u222a 0 = s\n[PROOFSTEP]\nsimp [union_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d\u00b9 t\u271d u : Multiset \u03b1\na\u271d b : \u03b1\ns\u271d : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nIH : \u2200 (t : Multiset \u03b1), s \u2229 t \u2264 t\nt : Multiset \u03b1\nh : a \u2208 t\n\u22a2 (a ::\u2098 s) \u2229 t \u2264 t\n[PROOFSTEP]\nsimpa [h] using cons_le_cons a (IH (t.erase a))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d\u00b9 t\u271d u : Multiset \u03b1\na\u271d b : \u03b1\ns\u271d : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nIH : \u2200 (t : Multiset \u03b1), s \u2229 t \u2264 t\nt : Multiset \u03b1\nh : \u00aca \u2208 t\n\u22a2 (a ::\u2098 s) \u2229 t \u2264 t\n[PROOFSTEP]\nsimp [h, IH]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u : Multiset \u03b1\na b : \u03b1\nh\u2081 : s \u2264 t\nh\u2082 : s \u2264 u\n\u22a2 s \u2264 t \u2229 u\n[PROOFSTEP]\nrevert s u\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nt : Multiset \u03b1\na b : \u03b1\n\u22a2 \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\n[PROOFSTEP]\nrefine @(Multiset.induction_on t ?_ fun a t IH => ?_)\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nt : Multiset \u03b1\na b : \u03b1\n\u22a2 \u2200 {s u : Multiset \u03b1}, s \u2264 0 \u2192 s \u2264 u \u2192 s \u2264 0 \u2229 u\n[PROOFSTEP]\nintros s u h\u2081 h\u2082\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\n\u22a2 \u2200 {s u : Multiset \u03b1}, s \u2264 a ::\u2098 t \u2192 s \u2264 u \u2192 s \u2264 (a ::\u2098 t) \u2229 u\n[PROOFSTEP]\nintros s u h\u2081 h\u2082\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nt : Multiset \u03b1\na b : \u03b1\ns u : Multiset \u03b1\nh\u2081 : s \u2264 0\nh\u2082 : s \u2264 u\n\u22a2 s \u2264 0 \u2229 u\n[PROOFSTEP]\nsimpa only [zero_inter, nonpos_iff_eq_zero] using h\u2081\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\ns u : Multiset \u03b1\nh\u2081 : s \u2264 a ::\u2098 t\nh\u2082 : s \u2264 u\n\u22a2 s \u2264 (a ::\u2098 t) \u2229 u\n[PROOFSTEP]\nby_cases h : a \u2208 u\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\ns u : Multiset \u03b1\nh\u2081 : s \u2264 a ::\u2098 t\nh\u2082 : s \u2264 u\nh : a \u2208 u\n\u22a2 s \u2264 (a ::\u2098 t) \u2229 u\n[PROOFSTEP]\nrw [cons_inter_of_pos _ h, \u2190 erase_le_iff_le_cons]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\ns u : Multiset \u03b1\nh\u2081 : s \u2264 a ::\u2098 t\nh\u2082 : s \u2264 u\nh : a \u2208 u\n\u22a2 erase s a \u2264 t \u2229 erase u a\n[PROOFSTEP]\nexact IH (erase_le_iff_le_cons.2 h\u2081) (erase_le_erase _ h\u2082)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\ns u : Multiset \u03b1\nh\u2081 : s \u2264 a ::\u2098 t\nh\u2082 : s \u2264 u\nh : \u00aca \u2208 u\n\u22a2 s \u2264 (a ::\u2098 t) \u2229 u\n[PROOFSTEP]\nrw [cons_inter_of_neg _ h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na\u271d b a : \u03b1\nt : Multiset \u03b1\nIH : \u2200 {s u : Multiset \u03b1}, s \u2264 t \u2192 s \u2264 u \u2192 s \u2264 t \u2229 u\ns u : Multiset \u03b1\nh\u2081 : s \u2264 a ::\u2098 t\nh\u2082 : s \u2264 u\nh : \u00aca \u2208 u\n\u22a2 s \u2264 t \u2229 u\n[PROOFSTEP]\nexact IH ((le_cons_of_not_mem <| mt (mem_of_le h\u2082) h).1 h\u2081) h\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u : Multiset \u03b1\na b : \u03b1\nx\u271d : a \u2208 s \u2227 a \u2208 t\nh\u2081 : a \u2208 s\nh\u2082 : a \u2208 t\n\u22a2 a \u2208 s \u2229 t\n[PROOFSTEP]\nrw [\u2190 cons_erase h\u2081, cons_inter_of_pos _ h\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u : Multiset \u03b1\na b : \u03b1\nx\u271d : a \u2208 s \u2227 a \u2208 t\nh\u2081 : a \u2208 s\nh\u2082 : a \u2208 t\n\u22a2 a \u2208 a ::\u2098 erase s a \u2229 erase t a\n[PROOFSTEP]\napply mem_cons_self\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u : Multiset \u03b1\na b : \u03b1\nh : s \u2264 t\n\u22a2 s \u222a t = t\n[PROOFSTEP]\nrw [union_comm, eq_union_left h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u\u271d : Multiset \u03b1\na b : \u03b1\ns t u : Multiset \u03b1\n\u22a2 s \u222a t + u = s + u \u222a (t + u)\n[PROOFSTEP]\nsimpa [(\u00b7 \u222a \u00b7), union, eq_comm, add_assoc] using\n show s + u - (t + u) = s - t by rw [add_comm t, tsub_add_eq_tsub_tsub, add_tsub_cancel_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u\u271d : Multiset \u03b1\na b : \u03b1\ns t u : Multiset \u03b1\n\u22a2 s + u - (t + u) = s - t\n[PROOFSTEP]\nrw [add_comm t, tsub_add_eq_tsub_tsub, add_tsub_cancel_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u\u271d : Multiset \u03b1\na b : \u03b1\ns t u : Multiset \u03b1\n\u22a2 s + (t \u222a u) = s + t \u222a (s + u)\n[PROOFSTEP]\nrw [add_comm, union_add_distrib, add_comm s, add_comm s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na\u271d b a : \u03b1\ns t : Multiset \u03b1\n\u22a2 a ::\u2098 (s \u222a t) = a ::\u2098 s \u222a a ::\u2098 t\n[PROOFSTEP]\nsimpa using add_union_distrib (a ::\u2098 0) s t\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u\u271d : Multiset \u03b1\na b : \u03b1\ns t u : Multiset \u03b1\n\u22a2 s \u2229 t + u = (s + u) \u2229 (t + u)\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u\u271d : Multiset \u03b1\na b : \u03b1\ns t u : Multiset \u03b1\nh : \u00acs \u2229 t + u = (s + u) \u2229 (t + u)\n\u22a2 False\n[PROOFSTEP]\ncases'\n lt_iff_cons_le.1\n (lt_of_le_of_ne (le_inter (add_le_add_right (inter_le_left s t) u) (add_le_add_right (inter_le_right s t) u))\n h) with\n a hl\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t u : Multiset \u03b1\nh : \u00acs \u2229 t + u = (s + u) \u2229 (t + u)\na : \u03b1\nhl : a ::\u2098 (s \u2229 t + u) \u2264 (s + u) \u2229 (t + u)\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 cons_add] at hl \n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t u : Multiset \u03b1\nh : \u00acs \u2229 t + u = (s + u) \u2229 (t + u)\na : \u03b1\nhl : a ::\u2098 s \u2229 t + u \u2264 (s + u) \u2229 (t + u)\n\u22a2 False\n[PROOFSTEP]\nexact\n not_le_of_lt (lt_cons_self (s \u2229 t) a)\n (le_inter (le_of_add_le_add_right (le_trans hl (inter_le_left _ _)))\n (le_of_add_le_add_right (le_trans hl (inter_le_right _ _))))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u\u271d : Multiset \u03b1\na b : \u03b1\ns t u : Multiset \u03b1\n\u22a2 s + t \u2229 u = (s + t) \u2229 (s + u)\n[PROOFSTEP]\nrw [add_comm, inter_add_distrib, add_comm s, add_comm s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na\u271d b a : \u03b1\ns t : Multiset \u03b1\n\u22a2 a ::\u2098 s \u2229 t = (a ::\u2098 s) \u2229 (a ::\u2098 t)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\n\u22a2 s \u222a t + s \u2229 t = s + t\n[PROOFSTEP]\napply _root_.le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\n\u22a2 s \u222a t + s \u2229 t \u2264 s + t\n[PROOFSTEP]\nrw [union_add_distrib]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\n\u22a2 s + s \u2229 t \u222a (t + s \u2229 t) \u2264 s + t\n[PROOFSTEP]\nrefine' union_le (add_le_add_left (inter_le_right _ _) _) _\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\n\u22a2 t + s \u2229 t \u2264 s + t\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\n\u22a2 s \u2229 t + t \u2264 s + t\n[PROOFSTEP]\nexact add_le_add_right (inter_le_left _ _) _\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\n\u22a2 s + t \u2264 s \u222a t + s \u2229 t\n[PROOFSTEP]\nrw [add_comm, add_inter_distrib]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\n\u22a2 t + s \u2264 (s \u222a t + s) \u2229 (s \u222a t + t)\n[PROOFSTEP]\nrefine' le_inter (add_le_add_right (le_union_right _ _) _) _\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\n\u22a2 t + s \u2264 s \u222a t + t\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\n\u22a2 s + t \u2264 s \u222a t + t\n[PROOFSTEP]\nexact add_le_add_right (le_union_left _ _) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\n\u22a2 s - t + s \u2229 t = s\n[PROOFSTEP]\nrw [inter_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\n\u22a2 s - t + t \u2229 s = s\n[PROOFSTEP]\nrevert s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t\u271d u : Multiset \u03b1\na b : \u03b1\nt : Multiset \u03b1\n\u22a2 \u2200 (s : Multiset \u03b1), s - t + t \u2229 s = s\n[PROOFSTEP]\nrefine' Multiset.induction_on t (by simp) fun a t IH s => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t\u271d u : Multiset \u03b1\na b : \u03b1\nt : Multiset \u03b1\n\u22a2 \u2200 (s : Multiset \u03b1), s - 0 + 0 \u2229 s = s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d\u00b9 u : Multiset \u03b1\na\u271d b : \u03b1\nt\u271d : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), s - t + t \u2229 s = s\ns : Multiset \u03b1\n\u22a2 s - a ::\u2098 t + (a ::\u2098 t) \u2229 s = s\n[PROOFSTEP]\nby_cases h : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d\u00b9 u : Multiset \u03b1\na\u271d b : \u03b1\nt\u271d : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), s - t + t \u2229 s = s\ns : Multiset \u03b1\nh : a \u2208 s\n\u22a2 s - a ::\u2098 t + (a ::\u2098 t) \u2229 s = s\n[PROOFSTEP]\nrw [cons_inter_of_pos _ h, sub_cons, add_cons, IH, cons_erase h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d\u00b9 u : Multiset \u03b1\na\u271d b : \u03b1\nt\u271d : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), s - t + t \u2229 s = s\ns : Multiset \u03b1\nh : \u00aca \u2208 s\n\u22a2 s - a ::\u2098 t + (a ::\u2098 t) \u2229 s = s\n[PROOFSTEP]\nrw [cons_inter_of_neg _ h, sub_cons, erase_of_not_mem h, IH]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\n\u22a2 s - s \u2229 t + ?m.192556 s t = s - t + ?m.192556 s t\n[PROOFSTEP]\nrw [sub_add_inter s t, tsub_add_cancel_of_le (inter_le_left s t)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\u271d\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns : Multiset \u03b1\n_l : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 Quot.mk Setoid.r _l \u2192 (p x \u2194 q x)\n\u22a2 \u2200 (x : \u03b1), x \u2208 _l \u2192 (decide (p x) = true \u2194 decide (q x) = true)\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\u271d\ns : Multiset \u03b1\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\nh : \u2200 (b : \u03b1), p b \u2192 q b\nl : List \u03b1\n\u22a2 \u2200 (a : \u03b1), decide (p a) = true \u2192 decide (q a) = true\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\nh : p a\n\u22a2 decide (p a) = true\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\nh : \u00acp a\n\u22a2 \u00acdecide (p a) = true\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\na : \u03b1\ns : Multiset \u03b1\n_l : List \u03b1\n\u22a2 a \u2208 filter p (Quot.mk Setoid.r _l) \u2194 a \u2208 Quot.mk Setoid.r _l \u2227 p a\n[PROOFSTEP]\nsimpa using List.mem_filter (p := (p \u00b7))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\n_l : List \u03b1\n\u22a2 List.filter (fun b => decide (p b)) _l = _l \u2194 \u2200 (a : \u03b1), a \u2208 Quot.mk Setoid.r _l \u2192 p a\n[PROOFSTEP]\nsimpa using List.filter_eq_self (p := (p \u00b7))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\n_l : List \u03b1\n\u22a2 List.filter (fun b => decide (p b)) _l = [] \u2194 \u2200 (a : \u03b1), a \u2208 Quot.mk Setoid.r _l \u2192 \u00acp a\n[PROOFSTEP]\nsimpa using List.filter_eq_nil (p := (p \u00b7))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\na : \u03b1\ns : Multiset \u03b1\n\u22a2 filter p (a ::\u2098 s) = (if p a then {a} else 0) + filter p s\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\na : \u03b1\ns : Multiset \u03b1\nh : p a\n\u22a2 filter p (a ::\u2098 s) = {a} + filter p s\n[PROOFSTEP]\nrw [filter_cons_of_pos _ h, singleton_add]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\na : \u03b1\ns : Multiset \u03b1\nh : \u00acp a\n\u22a2 filter p (a ::\u2098 s) = 0 + filter p s\n[PROOFSTEP]\nrw [filter_cons_of_neg _ h, zero_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\u271d\na : \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 filter p {a} = if p a then {a} else \u2205\n[PROOFSTEP]\nsimp only [singleton, filter_cons, filter_zero, add_zero, empty_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nn : \u2115\n\u22a2 filter p (n \u2022 s) = n \u2022 filter p s\n[PROOFSTEP]\nrefine' s.induction_on _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nn : \u2115\n\u22a2 filter p (n \u2022 0) = n \u2022 filter p 0\n[PROOFSTEP]\nsimp only [filter_zero, nsmul_zero]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nn : \u2115\n\u22a2 \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1}, filter p (n \u2022 s) = n \u2022 filter p s \u2192 filter p (n \u2022 (a ::\u2098 s)) = n \u2022 filter p (a ::\u2098 s)\n[PROOFSTEP]\nintro a ha ih\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nn : \u2115\na : \u03b1\nha : Multiset \u03b1\nih : filter p (n \u2022 ha) = n \u2022 filter p ha\n\u22a2 filter p (n \u2022 (a ::\u2098 ha)) = n \u2022 filter p (a ::\u2098 ha)\n[PROOFSTEP]\nrw [nsmul_cons, filter_add, ih, filter_cons, nsmul_add]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nn : \u2115\na : \u03b1\nha : Multiset \u03b1\nih : filter p (n \u2022 ha) = n \u2022 filter p ha\n\u22a2 filter p (n \u2022 {a}) + n \u2022 filter p ha = (n \u2022 if p a then {a} else 0) + n \u2022 filter p ha\n[PROOFSTEP]\ncongr\n[GOAL]\ncase refine'_2.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nn : \u2115\na : \u03b1\nha : Multiset \u03b1\nih : filter p (n \u2022 ha) = n \u2022 filter p ha\n\u22a2 filter p (n \u2022 {a}) = n \u2022 if p a then {a} else 0\n[PROOFSTEP]\nsplit_ifs with hp\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nn : \u2115\na : \u03b1\nha : Multiset \u03b1\nih : filter p (n \u2022 ha) = n \u2022 filter p ha\nhp : p a\n\u22a2 filter p (n \u2022 {a}) = n \u2022 {a}\n[PROOFSTEP]\nsimp only [filter_eq_self, nsmul_zero, filter_eq_nil]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nn : \u2115\na : \u03b1\nha : Multiset \u03b1\nih : filter p (n \u2022 ha) = n \u2022 filter p ha\nhp : p a\n\u22a2 \u2200 (a_1 : \u03b1), a_1 \u2208 n \u2022 {a} \u2192 p a_1\n[PROOFSTEP]\nintro b hb\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nn : \u2115\na : \u03b1\nha : Multiset \u03b1\nih : filter p (n \u2022 ha) = n \u2022 filter p ha\nhp : p a\nb : \u03b1\nhb : b \u2208 n \u2022 {a}\n\u22a2 p b\n[PROOFSTEP]\nrwa [mem_singleton.mp (mem_of_mem_nsmul hb)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nn : \u2115\na : \u03b1\nha : Multiset \u03b1\nih : filter p (n \u2022 ha) = n \u2022 filter p ha\nhp : \u00acp a\n\u22a2 filter p (n \u2022 {a}) = n \u2022 0\n[PROOFSTEP]\nsimp only [filter_eq_self, nsmul_zero, filter_eq_nil]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nn : \u2115\na : \u03b1\nha : Multiset \u03b1\nih : filter p (n \u2022 ha) = n \u2022 filter p ha\nhp : \u00acp a\n\u22a2 \u2200 (a_1 : \u03b1), a_1 \u2208 n \u2022 {a} \u2192 \u00acp a_1\n[PROOFSTEP]\nintro b hb\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nn : \u2115\na : \u03b1\nha : Multiset \u03b1\nih : filter p (n \u2022 ha) = n \u2022 filter p ha\nhp : \u00acp a\nb : \u03b1\nhb : b \u2208 n \u2022 {a}\n\u22a2 \u00acp b\n[PROOFSTEP]\nrwa [mem_singleton.mp (mem_of_mem_nsmul hb)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\n\u22a2 filter p (s - t) = filter p s - filter p t\n[PROOFSTEP]\nrevert s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\nt : Multiset \u03b1\n\u22a2 \u2200 (s : Multiset \u03b1), filter p (s - t) = filter p s - filter p t\n[PROOFSTEP]\nrefine' Multiset.induction_on t (by simp) fun a t IH s => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\nt : Multiset \u03b1\n\u22a2 \u2200 (s : Multiset \u03b1), filter p (s - 0) = filter p s - filter p 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), filter p (s - t) = filter p s - filter p t\ns : Multiset \u03b1\n\u22a2 filter p (s - a ::\u2098 t) = filter p s - filter p (a ::\u2098 t)\n[PROOFSTEP]\nrw [sub_cons, IH]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), filter p (s - t) = filter p s - filter p t\ns : Multiset \u03b1\n\u22a2 filter p (erase s a) - filter p t = filter p s - filter p (a ::\u2098 t)\n[PROOFSTEP]\nby_cases h : p a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), filter p (s - t) = filter p s - filter p t\ns : Multiset \u03b1\nh : p a\n\u22a2 filter p (erase s a) - filter p t = filter p s - filter p (a ::\u2098 t)\n[PROOFSTEP]\nrw [filter_cons_of_pos _ h, sub_cons]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), filter p (s - t) = filter p s - filter p t\ns : Multiset \u03b1\nh : p a\n\u22a2 filter p (erase s a) - filter p t = erase (filter p s) a - filter p t\n[PROOFSTEP]\ncongr\n[GOAL]\ncase pos.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), filter p (s - t) = filter p s - filter p t\ns : Multiset \u03b1\nh : p a\n\u22a2 filter p (erase s a) = erase (filter p s) a\n[PROOFSTEP]\nby_cases m : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), filter p (s - t) = filter p s - filter p t\ns : Multiset \u03b1\nh : p a\nm : a \u2208 s\n\u22a2 filter p (erase s a) = erase (filter p s) a\n[PROOFSTEP]\nrw [\u2190 cons_inj_right a, \u2190 filter_cons_of_pos _ h, cons_erase (mem_filter_of_mem m h), cons_erase m]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), filter p (s - t) = filter p s - filter p t\ns : Multiset \u03b1\nh : p a\nm : \u00aca \u2208 s\n\u22a2 filter p (erase s a) = erase (filter p s) a\n[PROOFSTEP]\nrw [erase_of_not_mem m, erase_of_not_mem (mt mem_of_mem_filter m)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), filter p (s - t) = filter p s - filter p t\ns : Multiset \u03b1\nh : \u00acp a\n\u22a2 filter p (erase s a) - filter p t = filter p s - filter p (a ::\u2098 t)\n[PROOFSTEP]\nrw [filter_cons_of_neg _ h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), filter p (s - t) = filter p s - filter p t\ns : Multiset \u03b1\nh : \u00acp a\n\u22a2 filter p (erase s a) - filter p t = filter p s - filter p t\n[PROOFSTEP]\nby_cases m : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), filter p (s - t) = filter p s - filter p t\ns : Multiset \u03b1\nh : \u00acp a\nm : a \u2208 s\n\u22a2 filter p (erase s a) - filter p t = filter p s - filter p t\n[PROOFSTEP]\nrw [(by rw [filter_cons_of_neg _ h] : filter p (erase s a) = filter p (a ::\u2098 erase s a)), cons_erase m]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), filter p (s - t) = filter p s - filter p t\ns : Multiset \u03b1\nh : \u00acp a\nm : a \u2208 s\n\u22a2 filter p (erase s a) = filter p (a ::\u2098 erase s a)\n[PROOFSTEP]\nrw [filter_cons_of_neg _ h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\nt\u271d : Multiset \u03b1\na : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), filter p (s - t) = filter p s - filter p t\ns : Multiset \u03b1\nh : \u00acp a\nm : \u00aca \u2208 s\n\u22a2 filter p (erase s a) - filter p t = filter p s - filter p t\n[PROOFSTEP]\nrw [erase_of_not_mem m]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\n\u22a2 filter p (s \u222a t) = filter p s \u222a filter p t\n[PROOFSTEP]\nsimp [(\u00b7 \u222a \u00b7), union]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\nq : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred q\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 filter p (filter q (Quot.mk Setoid.r l)) = filter (fun a => p a \u2227 q a) (Quot.mk Setoid.r l)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\nq : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred q\ns\u271d : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nIH : filter p s + filter q s = filter (fun a => p a \u2228 q a) s + filter (fun a => p a \u2227 q a) s\n\u22a2 filter p (a ::\u2098 s) + filter q (a ::\u2098 s) =\n filter (fun a => p a \u2228 q a) (a ::\u2098 s) + filter (fun a => p a \u2227 q a) (a ::\u2098 s)\n[PROOFSTEP]\nby_cases p a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\nq : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred q\ns\u271d : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nIH : filter p s + filter q s = filter (fun a => p a \u2228 q a) s + filter (fun a => p a \u2227 q a) s\n\u22a2 filter p (a ::\u2098 s) + filter q (a ::\u2098 s) =\n filter (fun a => p a \u2228 q a) (a ::\u2098 s) + filter (fun a => p a \u2227 q a) (a ::\u2098 s)\n[PROOFSTEP]\nby_cases p a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\nq : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred q\ns\u271d : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nIH : filter p s + filter q s = filter (fun a => p a \u2228 q a) s + filter (fun a => p a \u2227 q a) s\nh : p a\n\u22a2 filter p (a ::\u2098 s) + filter q (a ::\u2098 s) =\n filter (fun a => p a \u2228 q a) (a ::\u2098 s) + filter (fun a => p a \u2227 q a) (a ::\u2098 s)\n[PROOFSTEP]\nby_cases q a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\nq : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred q\ns\u271d : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nIH : filter p s + filter q s = filter (fun a => p a \u2228 q a) s + filter (fun a => p a \u2227 q a) s\nh : p a\n\u22a2 filter p (a ::\u2098 s) + filter q (a ::\u2098 s) =\n filter (fun a => p a \u2228 q a) (a ::\u2098 s) + filter (fun a => p a \u2227 q a) (a ::\u2098 s)\n[PROOFSTEP]\nby_cases q a\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\nq : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred q\ns\u271d : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nIH : filter p s + filter q s = filter (fun a => p a \u2228 q a) s + filter (fun a => p a \u2227 q a) s\nh : \u00acp a\n\u22a2 filter p (a ::\u2098 s) + filter q (a ::\u2098 s) =\n filter (fun a => p a \u2228 q a) (a ::\u2098 s) + filter (fun a => p a \u2227 q a) (a ::\u2098 s)\n[PROOFSTEP]\nby_cases q a\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\nq : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred q\ns\u271d : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nIH : filter p s + filter q s = filter (fun a => p a \u2228 q a) s + filter (fun a => p a \u2227 q a) s\nh : \u00acp a\n\u22a2 filter p (a ::\u2098 s) + filter q (a ::\u2098 s) =\n filter (fun a => p a \u2228 q a) (a ::\u2098 s) + filter (fun a => p a \u2227 q a) (a ::\u2098 s)\n[PROOFSTEP]\nby_cases q a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\nq : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred q\ns\u271d : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nIH : filter p s + filter q s = filter (fun a => p a \u2228 q a) s + filter (fun a => p a \u2227 q a) s\nh\u271d : p a\nh : q a\n\u22a2 filter p (a ::\u2098 s) + filter q (a ::\u2098 s) =\n filter (fun a => p a \u2228 q a) (a ::\u2098 s) + filter (fun a => p a \u2227 q a) (a ::\u2098 s)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\nq : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred q\ns\u271d : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nIH : filter p s + filter q s = filter (fun a => p a \u2228 q a) s + filter (fun a => p a \u2227 q a) s\nh\u271d : p a\nh : \u00acq a\n\u22a2 filter p (a ::\u2098 s) + filter q (a ::\u2098 s) =\n filter (fun a => p a \u2228 q a) (a ::\u2098 s) + filter (fun a => p a \u2227 q a) (a ::\u2098 s)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\nq : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred q\ns\u271d : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nIH : filter p s + filter q s = filter (fun a => p a \u2228 q a) s + filter (fun a => p a \u2227 q a) s\nh\u271d : \u00acp a\nh : q a\n\u22a2 filter p (a ::\u2098 s) + filter q (a ::\u2098 s) =\n filter (fun a => p a \u2228 q a) (a ::\u2098 s) + filter (fun a => p a \u2227 q a) (a ::\u2098 s)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\nq : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred q\ns\u271d : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nIH : filter p s + filter q s = filter (fun a => p a \u2228 q a) s + filter (fun a => p a \u2227 q a) s\nh\u271d : \u00acp a\nh : \u00acq a\n\u22a2 filter p (a ::\u2098 s) + filter q (a ::\u2098 s) =\n filter (fun a => p a \u2228 q a) (a ::\u2098 s) + filter (fun a => p a \u2227 q a) (a ::\u2098 s)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\n\u22a2 filter p s + filter (fun a => \u00acp a) s = s\n[PROOFSTEP]\nrw [filter_add_filter, filter_eq_self.2, filter_eq_nil.2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\n\u22a2 s + 0 = s\n[PROOFSTEP]\nsimp only [add_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2192 \u00ac(p a \u2227 \u00acp a)\n[PROOFSTEP]\nsimp [Decidable.em, -Bool.not_eq_true, -not_and, not_and_or, or_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2192 p a \u2228 \u00acp a\n[PROOFSTEP]\nsimp only [Bool.not_eq_true, decide_eq_true_eq, Bool.eq_false_or_eq_true, decide_True, implies_true, Decidable.em]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b2 \u2192 \u03b1\ns : Multiset \u03b2\nl : List \u03b2\n\u22a2 filter p (map f (Quot.mk Setoid.r l)) = map f (filter (p \u2218 f) (Quot.mk Setoid.r l))\n[PROOFSTEP]\nsimp [List.map_filter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b2 \u2192 \u03b1\ns : Multiset \u03b2\nl : List \u03b2\n\u22a2 List.map f (List.filter ((fun b => decide (p b)) \u2218 f) l) ~ List.map f (List.filter (fun b => decide (p (f b))) l)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 List.filterMap (Option.guard p) l = List.filter (fun b => decide (p b)) l\n[PROOFSTEP]\nrw [\u2190 List.filterMap_eq_filter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 List.filterMap (Option.guard p) l = List.filterMap (Option.guard fun x => decide (p x) = true) l\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 Option.guard p = Option.guard fun x => decide (p x) = true\n[PROOFSTEP]\nfunext a\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nl : List \u03b1\na : \u03b1\n\u22a2 Option.guard p a = Option.guard (fun x => decide (p x) = true) a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2192 Option \u03b2\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 List.filterMap f (List.filter (fun b => decide (p b)) l) = List.filterMap (fun x => if p x then f x else none) l\n[PROOFSTEP]\nsimpa using List.filterMap_filter p f l\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\na : \u03b1\ns : Multiset \u03b1\n\u22a2 \u2200 (a_1 : List \u03b1), p a \u2192 countp p (a ::\u2098 Quot.mk Setoid.r a_1) = countp p (Quot.mk Setoid.r a_1) + 1\n[PROOFSTEP]\nsimpa using List.countp_cons_of_pos (p \u00b7)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\na : \u03b1\ns : Multiset \u03b1\n\u22a2 \u2200 (a_1 : List \u03b1), \u00acp a \u2192 countp p (a ::\u2098 Quot.mk Setoid.r a_1) = countp p (Quot.mk Setoid.r a_1)\n[PROOFSTEP]\nsimpa using List.countp_cons_of_neg (p \u00b7)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nb : \u03b1\ns : Multiset \u03b1\n\u22a2 \u2200 (a : List \u03b1), countp p (b ::\u2098 Quot.mk Setoid.r a) = countp p (Quot.mk Setoid.r a) + if p b then 1 else 0\n[PROOFSTEP]\nsimp [List.countp_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns t : Multiset \u03b1\n\u22a2 countp p (s + t) = countp p s + countp p t\n[PROOFSTEP]\nsimp [countp_eq_card_filter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nn : \u2115\n\u22a2 countp p (n \u2022 s) = n * countp p s\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\n\u22a2 countp p (zero \u2022 s) = zero * countp p s\n[PROOFSTEP]\nsimp [*, succ_nsmul', succ_mul, zero_nsmul]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nn\u271d : \u2115\nn_ih\u271d : countp p (n\u271d \u2022 s) = n\u271d * countp p s\n\u22a2 countp p (succ n\u271d \u2022 s) = succ n\u271d * countp p s\n[PROOFSTEP]\nsimp [*, succ_nsmul', succ_mul, zero_nsmul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 \u2191card (Quot.mk Setoid.r l) = countp p (Quot.mk Setoid.r l) + countp (fun x => \u00acp x) (Quot.mk Setoid.r l)\n[PROOFSTEP]\nsimp [l.length_eq_countp_add_countp p]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\nh : t \u2264 s\n\u22a2 countp p (s - t) = countp p s - countp p t\n[PROOFSTEP]\nsimp [countp_eq_card_filter, h, filter_le_filter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns t : Multiset \u03b1\nh : s \u2264 t\n\u22a2 countp p s \u2264 countp p t\n[PROOFSTEP]\nsimpa [countp_eq_card_filter] using card_le_of_le (filter_le_filter p h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\nq : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred q\ns : Multiset \u03b1\n\u22a2 countp p (filter q s) = countp (fun a => p a \u2227 q a) s\n[PROOFSTEP]\nsimp [countp_eq_card_filter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\u271d\ns : Multiset \u03b1\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\nl : List \u03b1\n\u22a2 countp p (Quot.mk Setoid.r l) =\n countp p (filter q (Quot.mk Setoid.r l)) + countp p (filter (fun a => \u00acq a) (Quot.mk Setoid.r l))\n[PROOFSTEP]\nconvert l.countp_eq_countp_filter_add (p \u00b7) (q \u00b7)\n[GOAL]\ncase h.e'_3.h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\u271d\ns : Multiset \u03b1\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\nl : List \u03b1\n\u22a2 countp p (filter (fun a => \u00acq a) (Quot.mk Setoid.r l)) =\n List.countp (fun x => decide (p x)) (List.filter (fun a => decide \u00acdecide (q a) = true) l)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\u271d\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 countp p (map f s) = \u2191card (filter (fun a => p (f a)) s)\n[PROOFSTEP]\nrefine' Multiset.induction_on s _ fun a t IH => _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\u271d\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 countp p (map f 0) = \u2191card (filter (fun a => p (f a)) 0)\n[PROOFSTEP]\nrw [map_zero, countp_zero, filter_zero, card_zero]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\u271d\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\na : \u03b1\nt : Multiset \u03b1\nIH : countp p (map f t) = \u2191card (filter (fun a => p (f a)) t)\n\u22a2 countp p (map f (a ::\u2098 t)) = \u2191card (filter (fun a => p (f a)) (a ::\u2098 t))\n[PROOFSTEP]\nrw [map_cons, countp_cons, IH, filter_cons, card_add, apply_ite card, card_zero, card_singleton, add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\n_l : List \u03b1\n\u22a2 0 < countp p (Quot.mk Setoid.r _l) \u2194 \u2203 a, a \u2208 Quot.mk Setoid.r _l \u2227 p a\n[PROOFSTEP]\nsimpa using List.countp_pos (p \u00b7)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\n_l : List \u03b1\n\u22a2 countp p (Quot.mk Setoid.r _l) = 0 \u2194 \u2200 (a : \u03b1), a \u2208 Quot.mk Setoid.r _l \u2192 \u00acp a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Multiset \u03b1\n_l : List \u03b1\n\u22a2 countp p (Quot.mk Setoid.r _l) = \u2191card (Quot.mk Setoid.r _l) \u2194 \u2200 (a : \u03b1), a \u2208 Quot.mk Setoid.r _l \u2192 p a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\u271d\ns s' : Multiset \u03b1\nhs : s = s'\np p' : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred p'\nhp : \u2200 (x : \u03b1), x \u2208 s \u2192 p x = p' x\n\u22a2 countp p s = countp p' s'\n[PROOFSTEP]\nrevert hs hp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\u271d\ns s' : Multiset \u03b1\np p' : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred p'\n\u22a2 s = s' \u2192 (\u2200 (x : \u03b1), x \u2208 s \u2192 p x = p' x) \u2192 countp p s = countp p' s'\n[PROOFSTEP]\nexact\n Quot.induction_on\u2082 s s'\n (fun l l' hs hp => by\n simp only [quot_mk_to_coe'', coe_eq_coe] at hs \n apply hs.countp_congr\n simpa using hp)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\u271d\ns s' : Multiset \u03b1\np p' : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred p'\nl l' : List \u03b1\nhs : Quot.mk Setoid.r l = Quot.mk Setoid.r l'\nhp : \u2200 (x : \u03b1), x \u2208 Quot.mk Setoid.r l \u2192 p x = p' x\n\u22a2 countp p (Quot.mk Setoid.r l) = countp p' (Quot.mk Setoid.r l')\n[PROOFSTEP]\nsimp only [quot_mk_to_coe'', coe_eq_coe] at hs \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\u271d\ns s' : Multiset \u03b1\np p' : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred p'\nl l' : List \u03b1\nhp : \u2200 (x : \u03b1), x \u2208 Quot.mk Setoid.r l \u2192 p x = p' x\nhs : l ~ l'\n\u22a2 countp p (Quot.mk Setoid.r l) = countp p' (Quot.mk Setoid.r l')\n[PROOFSTEP]\napply hs.countp_congr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\u271d\ns s' : Multiset \u03b1\np p' : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred p'\nl l' : List \u03b1\nhp : \u2200 (x : \u03b1), x \u2208 Quot.mk Setoid.r l \u2192 p x = p' x\nhs : l ~ l'\n\u22a2 \u2200 (x : \u03b1), x \u2208 l \u2192 decide (p x) = decide (p' x)\n[PROOFSTEP]\nsimpa using hp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 count a \u2191l = List.count a l\n[PROOFSTEP]\nsimp_rw [count, List.count, coe_countp (a = \u00b7) l, @eq_comm _ a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 List.countp (fun b => decide (b = a)) l = List.countp (fun x => x == a) l\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\n\u22a2 count a {b} = if a = b then 1 else 0\n[PROOFSTEP]\nsimp only [count_cons, \u2190 cons_zero, count_zero, zero_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nn : \u2115\ns : Multiset \u03b1\n\u22a2 count a (n \u2022 s) = n * count a s\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\n\u22a2 count a (zero \u2022 s) = zero * count a s\n[PROOFSTEP]\nsimp [*, succ_nsmul', succ_mul, zero_nsmul]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nn\u271d : \u2115\nn_ih\u271d : count a (n\u271d \u2022 s) = n\u271d * count a s\n\u22a2 count a (succ n\u271d \u2022 s) = succ n\u271d * count a s\n[PROOFSTEP]\nsimp [*, succ_nsmul', succ_mul, zero_nsmul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\n\u22a2 0 < count a s \u2194 a \u2208 s\n[PROOFSTEP]\nsimp [count, countp_pos]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\n\u22a2 1 \u2264 count a s \u2194 a \u2208 s\n[PROOFSTEP]\nrw [succ_le_iff, count_pos]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\n\u22a2 count a s \u2260 0 \u2194 a \u2208 s\n[PROOFSTEP]\nsimp [Ne.def, count_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\n\u22a2 count a s = \u2191card s \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 a = x\n[PROOFSTEP]\nsimp [countp_eq_card, count, @eq_comm _ a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nn : \u2115\n\u22a2 count a (replicate n a) = n\n[PROOFSTEP]\nconvert List.count_replicate_self a n\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nn : \u2115\n\u22a2 count a (replicate n a) = List.count a (List.replicate n a)\n[PROOFSTEP]\nrw [\u2190 coe_count, coe_replicate]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 count a (replicate n b) = if a = b then n else 0\n[PROOFSTEP]\nconvert List.count_replicate a b n\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 count a (replicate n b) = List.count a (List.replicate n b)\n[PROOFSTEP]\nrw [\u2190 coe_count, coe_replicate]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 count a (erase (Quotient.mk (isSetoid \u03b1) l) a) = count a (Quotient.mk (isSetoid \u03b1) l) - 1\n[PROOFSTEP]\nconvert List.count_erase_self a l\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 count a (erase (Quotient.mk (isSetoid \u03b1) l) a) = List.count a (List.erase l a)\n[PROOFSTEP]\nrw [\u2190 coe_count]\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 count a (Quotient.mk (isSetoid \u03b1) l) = List.count a l\n[PROOFSTEP]\nrw [\u2190 coe_count]\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 count a (erase (Quotient.mk (isSetoid \u03b1) l) a) = count a \u2191(List.erase l a)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 count a (Quotient.mk (isSetoid \u03b1) l) = count a \u2191l\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab : a \u2260 b\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 count a (erase (Quotient.mk (isSetoid \u03b1) l) b) = count a (Quotient.mk (isSetoid \u03b1) l)\n[PROOFSTEP]\nconvert List.count_erase_of_ne ab l\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab : a \u2260 b\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 count a (erase (Quotient.mk (isSetoid \u03b1) l) b) = List.count a (List.erase l b)\n[PROOFSTEP]\nrw [\u2190 coe_count]\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab : a \u2260 b\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 count a (Quotient.mk (isSetoid \u03b1) l) = List.count a l\n[PROOFSTEP]\nrw [\u2190 coe_count]\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab : a \u2260 b\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 count a (erase (Quotient.mk (isSetoid \u03b1) l) b) = count a \u2191(List.erase l b)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab : a \u2260 b\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 count a (Quotient.mk (isSetoid \u03b1) l) = count a \u2191l\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns t : Multiset \u03b1\n\u22a2 count a (s - t) = count a s - count a t\n[PROOFSTEP]\nrevert s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nt : Multiset \u03b1\n\u22a2 \u2200 (s : Multiset \u03b1), count a (s - t) = count a s - count a t\n[PROOFSTEP]\nrefine' Multiset.induction_on t (by simp) fun b t IH s => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nt : Multiset \u03b1\n\u22a2 \u2200 (s : Multiset \u03b1), count a (s - 0) = count a s - count a 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nt\u271d : Multiset \u03b1\nb : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), count a (s - t) = count a s - count a t\ns : Multiset \u03b1\n\u22a2 count a (s - b ::\u2098 t) = count a s - count a (b ::\u2098 t)\n[PROOFSTEP]\nrw [sub_cons, IH]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nt\u271d : Multiset \u03b1\nb : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), count a (s - t) = count a s - count a t\ns : Multiset \u03b1\n\u22a2 count a (erase s b) - count a t = count a s - count a (b ::\u2098 t)\n[PROOFSTEP]\nrcases Decidable.eq_or_ne a b with rfl | ab\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nt\u271d t : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), count a (s - t) = count a s - count a t\ns : Multiset \u03b1\n\u22a2 count a (erase s a) - count a t = count a s - count a (a ::\u2098 t)\n[PROOFSTEP]\nrw [count_erase_self, count_cons_self, Nat.sub_sub, add_comm]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nt\u271d : Multiset \u03b1\nb : \u03b1\nt : Multiset \u03b1\nIH : \u2200 (s : Multiset \u03b1), count a (s - t) = count a s - count a t\ns : Multiset \u03b1\nab : a \u2260 b\n\u22a2 count a (erase s b) - count a t = count a s - count a (b ::\u2098 t)\n[PROOFSTEP]\nrw [count_erase_of_ne ab, count_cons_of_ne ab]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns t : Multiset \u03b1\n\u22a2 count a (s \u222a t) = max (count a s) (count a t)\n[PROOFSTEP]\nsimp [(\u00b7 \u222a \u00b7), union, tsub_add_eq_max]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns t : Multiset \u03b1\n\u22a2 count a (s \u2229 t) = min (count a s) (count a t)\n[PROOFSTEP]\napply @Nat.add_left_cancel (count a (s - t))\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns t : Multiset \u03b1\n\u22a2 count a (s - t) + count a (s \u2229 t) = count a (s - t) + min (count a s) (count a t)\n[PROOFSTEP]\nrw [\u2190 count_add, sub_add_inter, count_sub, tsub_add_min]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nn : \u2115\n_l : List \u03b1\n\u22a2 n \u2264 count a (Quot.mk Setoid.r _l) \u2194 replicate n a \u2264 Quot.mk Setoid.r _l\n[PROOFSTEP]\nsimp only [quot_mk_to_coe'', mem_coe, coe_count]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nn : \u2115\n_l : List \u03b1\n\u22a2 n \u2264 List.count a _l \u2194 replicate n a \u2264 \u2191_l\n[PROOFSTEP]\nexact le_count_iff_replicate_sublist.trans replicate_le_coe.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\na : \u03b1\ns : Multiset \u03b1\nh : p a\n_l : List \u03b1\n\u22a2 count a (filter p (Quot.mk Setoid.r _l)) = count a (Quot.mk Setoid.r _l)\n[PROOFSTEP]\nsimp only [quot_mk_to_coe'', coe_filter, mem_coe, coe_count, decide_eq_true_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\na : \u03b1\ns : Multiset \u03b1\nh : p a\n_l : List \u03b1\n\u22a2 List.count a (List.filter (fun b => decide (p b)) _l) = List.count a _l\n[PROOFSTEP]\napply count_filter\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\na : \u03b1\ns : Multiset \u03b1\nh : p a\n_l : List \u03b1\n\u22a2 decide (p a) = true\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\na : \u03b1\ns : Multiset \u03b1\n\u22a2 count a (filter p s) = if p a then count a s else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\na : \u03b1\ns : Multiset \u03b1\nh : p a\n\u22a2 count a (filter p s) = count a s\n[PROOFSTEP]\nexact count_filter_of_pos h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\na : \u03b1\ns : Multiset \u03b1\nh : \u00acp a\n\u22a2 count a (filter p s) = 0\n[PROOFSTEP]\nexact count_filter_of_neg h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\n_l\u2081 _l\u2082 : List \u03b1\n\u22a2 _l\u2081 \u2248 _l\u2082 \u2194 \u2200 (a : \u03b1), count a (Quotient.mk (isSetoid \u03b1) _l\u2081) = count a (Quotient.mk (isSetoid \u03b1) _l\u2082)\n[PROOFSTEP]\nsimp only [quot_mk_to_coe, coe_filter, mem_coe, coe_count, decide_eq_true_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\n_l\u2081 _l\u2082 : List \u03b1\n\u22a2 _l\u2081 \u2248 _l\u2082 \u2194 \u2200 (a : \u03b1), List.count a _l\u2081 = List.count a _l\u2082\n[PROOFSTEP]\napply perm_iff_count\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : List \u03b1\n\u22a2 \u2191s \u2229 \u2191t = \u2191(List.bagInter s t)\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : List \u03b1\na\u271d : \u03b1\n\u22a2 count a\u271d (\u2191s \u2229 \u2191t) = count a\u271d \u2191(List.bagInter s t)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\nal : \u2200 (a : \u03b1), count a s \u2264 count a t\n\u22a2 s \u2264 t\n[PROOFSTEP]\nrw [\u2190 (ext.2 fun a => by simp [max_eq_right (al a)] : s \u222a t = t)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\nal : \u2200 (a : \u03b1), count a s \u2264 count a t\na : \u03b1\n\u22a2 count a (s \u222a t) = count a t\n[PROOFSTEP]\nsimp [max_eq_right (al a)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\nal : \u2200 (a : \u03b1), count a s \u2264 count a t\n\u22a2 s \u2264 s \u222a t\n[PROOFSTEP]\napply le_union_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u : Multiset \u03b1\na : \u03b1\n\u22a2 count a (s \u2294 t \u2293 u) = count a ((s \u2294 t) \u2293 (s \u2294 u))\n[PROOFSTEP]\nsimp only [max_min_distrib_left, Multiset.count_inter, Multiset.sup_eq_union, Multiset.count_union,\n Multiset.inf_eq_inter]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\n\u22a2 count b (map f s) = \u2191card (filter (fun a => b = f a) s)\n[PROOFSTEP]\nsimp [Bool.beq_eq_decide_eq, eq_comm, count, countp_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Set.InjOn f {x | x \u2208 s}\nx : \u03b1\nH : x \u2208 s\n\u22a2 count (f x) (map f s) = count x s\n[PROOFSTEP]\nsuffices (filter (fun a : \u03b1 => f x = f a) s).count x = card (filter (fun a : \u03b1 => f x = f a) s)\n by\n rw [count, countp_map, \u2190 this]\n exact count_filter_of_pos $ rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Set.InjOn f {x | x \u2208 s}\nx : \u03b1\nH : x \u2208 s\nthis : count x (filter (fun a => f x = f a) s) = \u2191card (filter (fun a => f x = f a) s)\n\u22a2 count (f x) (map f s) = count x s\n[PROOFSTEP]\nrw [count, countp_map, \u2190 this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Set.InjOn f {x | x \u2208 s}\nx : \u03b1\nH : x \u2208 s\nthis : count x (filter (fun a => f x = f a) s) = \u2191card (filter (fun a => f x = f a) s)\n\u22a2 count x (filter (fun a => f x = f a) s) = count x s\n[PROOFSTEP]\nexact count_filter_of_pos $ rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Set.InjOn f {x | x \u2208 s}\nx : \u03b1\nH : x \u2208 s\n\u22a2 count x (filter (fun a => f x = f a) s) = \u2191card (filter (fun a => f x = f a) s)\n[PROOFSTEP]\nrw [eq_replicate_card.2 fun b hb => (hf H (mem_filter.1 hb).left _).symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Set.InjOn f {x | x \u2208 s}\nx : \u03b1\nH : x \u2208 s\n\u22a2 count x (replicate (\u2191card (filter (fun a => f x = f a) s)) x) =\n \u2191card (replicate (\u2191card (filter (fun a => f x = f a) s)) x)\n[PROOFSTEP]\nsimp only [count_replicate, eq_self_iff_true, if_true, card_replicate]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Set.InjOn f {x | x \u2208 s}\nx : \u03b1\nH : x \u2208 s\n\u22a2 \u2200 (b : \u03b1), b \u2208 filter (fun a => f x = f a) s \u2192 f x = f b\n[PROOFSTEP]\nsimp only [mem_filter, beq_iff_eq, and_imp, @eq_comm _ (f x), imp_self, implies_true]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\n\u22a2 count (f x) (map f s) = count x s\n[PROOFSTEP]\nby_cases H : x \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\nH : x \u2208 s\n\u22a2 count (f x) (map f s) = count x s\n[PROOFSTEP]\nexact count_map_eq_count f _ (Set.injOn_of_injective hf _) _ H\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\nH : \u00acx \u2208 s\n\u22a2 count (f x) (map f s) = count x s\n[PROOFSTEP]\nrw [count_eq_zero_of_not_mem H, count_eq_zero, mem_map]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\nH : \u00acx \u2208 s\n\u22a2 \u00ac\u2203 a, a \u2208 s \u2227 f a = f x\n[PROOFSTEP]\nrintro \u27e8k, hks, hkx\u27e9\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\nH : \u00acx \u2208 s\nk : \u03b1\nhks : k \u2208 s\nhkx : f k = f x\n\u22a2 False\n[PROOFSTEP]\nrw [hf hkx] at hks \n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\nH : \u00acx \u2208 s\nk : \u03b1\nhks : x \u2208 s\nhkx : f k = f x\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\nb : \u03b1\nl : List \u03b1\n\u22a2 filter (fun x => x = b) (Quotient.mk (isSetoid \u03b1) l) = replicate (count b (Quotient.mk (isSetoid \u03b1) l)) b\n[PROOFSTEP]\nsimp only [quot_mk_to_coe, coe_filter, mem_coe, coe_count]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\nb : \u03b1\nl : List \u03b1\n\u22a2 \u2191(List.filter (fun b_1 => decide (b_1 = b)) l) = replicate (List.count b l) b\n[PROOFSTEP]\nrw [List.filter_eq' l b, coe_replicate]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\nb : \u03b1\n\u22a2 filter (Eq b) s = replicate (count b s) b\n[PROOFSTEP]\nsimp_rw [\u2190 filter_eq', eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nn : \u2115\nx : \u03b1\ns : Multiset \u03b1\n\u22a2 replicate n x \u2229 s = replicate (min n (count x s)) x\n[PROOFSTEP]\next y\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nn : \u2115\nx : \u03b1\ns : Multiset \u03b1\ny : \u03b1\n\u22a2 count y (replicate n x \u2229 s) = count y (replicate (min n (count x s)) x)\n[PROOFSTEP]\nrw [count_inter, count_replicate, count_replicate]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nn : \u2115\nx : \u03b1\ns : Multiset \u03b1\ny : \u03b1\n\u22a2 min (if y = x then n else 0) (count y s) = if y = x then min n (count x s) else 0\n[PROOFSTEP]\nby_cases h : y = x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nn : \u2115\nx : \u03b1\ns : Multiset \u03b1\ny : \u03b1\nh : y = x\n\u22a2 min (if y = x then n else 0) (count y s) = if y = x then min n (count x s) else 0\n[PROOFSTEP]\nsimp only [h, if_true]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nn : \u2115\nx : \u03b1\ns : Multiset \u03b1\ny : \u03b1\nh : \u00acy = x\n\u22a2 min (if y = x then n else 0) (count y s) = if y = x then min n (count x s) else 0\n[PROOFSTEP]\nsimp only [h, if_false, zero_min]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\nn : \u2115\nx : \u03b1\n\u22a2 s \u2229 replicate n x = replicate (min (count x s) n) x\n[PROOFSTEP]\nrw [inter_comm, replicate_inter, min_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : AddZeroClass \u03b2\nf g : Multiset \u03b1 \u2192+ \u03b2\nh : \u2200 (x : \u03b1), \u2191f {x} = \u2191g {x}\n\u22a2 f = g\n[PROOFSTEP]\next s\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : AddZeroClass \u03b2\nf g : Multiset \u03b1 \u2192+ \u03b2\nh : \u2200 (x : \u03b1), \u2191f {x} = \u2191g {x}\ns : Multiset \u03b1\n\u22a2 \u2191f s = \u2191g s\n[PROOFSTEP]\ninduction' s using Multiset.induction_on with a s ih\n[GOAL]\ncase h.empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : AddZeroClass \u03b2\nf g : Multiset \u03b1 \u2192+ \u03b2\nh : \u2200 (x : \u03b1), \u2191f {x} = \u2191g {x}\n\u22a2 \u2191f 0 = \u2191g 0\n[PROOFSTEP]\nsimp only [_root_.map_zero]\n[GOAL]\ncase h.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : AddZeroClass \u03b2\nf g : Multiset \u03b1 \u2192+ \u03b2\nh : \u2200 (x : \u03b1), \u2191f {x} = \u2191g {x}\na : \u03b1\ns : Multiset \u03b1\nih : \u2191f s = \u2191g s\n\u22a2 \u2191f (a ::\u2098 s) = \u2191g (a ::\u2098 s)\n[PROOFSTEP]\nsimp only [\u2190 singleton_add, _root_.map_add, ih, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns t : Multiset \u03b1\n\u22a2 map f s \u2264 map f t \u2194 s \u2264 t\n[PROOFSTEP]\nclassical\nrefine' \u27e8fun h => le_iff_count.mpr fun a => _, map_le_map\u27e9\nsimpa [count_map_eq_count' f _ hf] using le_iff_count.mp h (f a)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns t : Multiset \u03b1\n\u22a2 map f s \u2264 map f t \u2194 s \u2264 t\n[PROOFSTEP]\nrefine' \u27e8fun h => le_iff_count.mpr fun a => _, map_le_map\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns t : Multiset \u03b1\nh : map f s \u2264 map f t\na : \u03b1\n\u22a2 count a s \u2264 count a t\n[PROOFSTEP]\nsimpa [count_map_eq_count' f _ hf] using le_iff_count.mp h (f a)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\na : \u03b1\n\u22a2 count a s = \u2191card (filter (fun x => a = x) s)\n[PROOFSTEP]\nrw [count, countp_eq_card_filter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Multiset \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 count True (map p s) = \u2191card (filter p s)\n[PROOFSTEP]\nsimp only [count_eq_card_filter_eq, map_filter, card_map, Function.comp.left_id, eq_true_eq_id, Function.comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Multiset \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 \u2191card (filter (fun x => True = p x) s) = \u2191card (filter p s)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e_6.h.e_p\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Multiset \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 (fun x => True = p x) = p\n[PROOFSTEP]\nfunext _\n[GOAL]\ncase h.e_6.h.e_p.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Multiset \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : \u03b1\n\u22a2 (True = p x\u271d) = p x\u271d\n[PROOFSTEP]\nsimp only [eq_iff_iff, true_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 (\u2200 (x : \u03b1), x \u2208 m \u2192 r x x) \u2192 Rel r m m\n[PROOFSTEP]\nrefine' m.induction_on _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 (\u2200 (x : \u03b1), x \u2208 0 \u2192 r x x) \u2192 Rel r 0 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u2200 (x : \u03b1), x \u2208 0 \u2192 r x x\n\u22a2 Rel r 0 0\n[PROOFSTEP]\napply Rel.zero\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1},\n ((\u2200 (x : \u03b1), x \u2208 s \u2192 r x x) \u2192 Rel r s s) \u2192 (\u2200 (x : \u03b1), x \u2208 a ::\u2098 s \u2192 r x x) \u2192 Rel r (a ::\u2098 s) (a ::\u2098 s)\n[PROOFSTEP]\nintro a m ih h\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm\u271d : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nm : Multiset \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 m \u2192 r x x) \u2192 Rel r m m\nh : \u2200 (x : \u03b1), x \u2208 a ::\u2098 m \u2192 r x x\n\u22a2 Rel r (a ::\u2098 m) (a ::\u2098 m)\n[PROOFSTEP]\nexact Rel.cons (h _ (mem_cons_self _ _)) (ih fun _ ha => h _ (mem_cons_of_mem ha))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns t : Multiset \u03b1\n\u22a2 Rel (fun x x_1 => x = x_1) s t \u2194 s = t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns t : Multiset \u03b1\n\u22a2 Rel (fun x x_1 => x = x_1) s t \u2192 s = t\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns t : Multiset \u03b1\nh : Rel (fun x x_1 => x = x_1) s t\n\u22a2 s = t\n[PROOFSTEP]\ninduction h\n[GOAL]\ncase mp.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns t : Multiset \u03b1\n\u22a2 0 = 0\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase mp.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns t : Multiset \u03b1\na\u271d\u00b2 b\u271d : \u03b1\nas\u271d bs\u271d : Multiset \u03b1\na\u271d\u00b9 : a\u271d\u00b2 = b\u271d\na\u271d : Rel (fun x x_1 => x = x_1) as\u271d bs\u271d\na_ih\u271d : as\u271d = bs\u271d\n\u22a2 a\u271d\u00b2 ::\u2098 as\u271d = b\u271d ::\u2098 bs\u271d\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns t : Multiset \u03b1\n\u22a2 s = t \u2192 Rel (fun x x_1 => x = x_1) s t\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns t : Multiset \u03b1\nh : s = t\n\u22a2 Rel (fun x x_1 => x = x_1) s t\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns : Multiset \u03b1\n\u22a2 Rel (fun x x_1 => x = x_1) s s\n[PROOFSTEP]\nexact rel_eq_refl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np\u271d : \u03b3 \u2192 \u03b4 \u2192 Prop\nr p : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\nhst : Rel r s t\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 r a b \u2192 p a b\n\u22a2 Rel p s t\n[PROOFSTEP]\ninduction hst\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np\u271d : \u03b3 \u2192 \u03b4 \u2192 Prop\nr p : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\nh : \u2200 (a : \u03b1), a \u2208 0 \u2192 \u2200 (b : \u03b2), b \u2208 0 \u2192 r a b \u2192 p a b\n\u22a2 Rel p 0 0\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np\u271d : \u03b3 \u2192 \u03b4 \u2192 Prop\nr p : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\na\u271d\u00b2 : \u03b1\nb\u271d : \u03b2\nas\u271d : Multiset \u03b1\nbs\u271d : Multiset \u03b2\na\u271d\u00b9 : r a\u271d\u00b2 b\u271d\na\u271d : Rel r as\u271d bs\u271d\na_ih\u271d : (\u2200 (a : \u03b1), a \u2208 as\u271d \u2192 \u2200 (b : \u03b2), b \u2208 bs\u271d \u2192 r a b \u2192 p a b) \u2192 Rel p as\u271d bs\u271d\nh : \u2200 (a : \u03b1), a \u2208 a\u271d\u00b2 ::\u2098 as\u271d \u2192 \u2200 (b : \u03b2), b \u2208 b\u271d ::\u2098 bs\u271d \u2192 r a b \u2192 p a b\n\u22a2 Rel p (a\u271d\u00b2 ::\u2098 as\u271d) (b\u271d ::\u2098 bs\u271d)\n[PROOFSTEP]\ncase zero => exact Rel.zero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np\u271d : \u03b3 \u2192 \u03b4 \u2192 Prop\nr p : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\nh : \u2200 (a : \u03b1), a \u2208 0 \u2192 \u2200 (b : \u03b2), b \u2208 0 \u2192 r a b \u2192 p a b\n\u22a2 Rel p 0 0\n[PROOFSTEP]\ncase zero => exact Rel.zero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np\u271d : \u03b3 \u2192 \u03b4 \u2192 Prop\nr p : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\nh : \u2200 (a : \u03b1), a \u2208 0 \u2192 \u2200 (b : \u03b2), b \u2208 0 \u2192 r a b \u2192 p a b\n\u22a2 Rel p 0 0\n[PROOFSTEP]\nexact Rel.zero\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np\u271d : \u03b3 \u2192 \u03b4 \u2192 Prop\nr p : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\na\u271d\u00b2 : \u03b1\nb\u271d : \u03b2\nas\u271d : Multiset \u03b1\nbs\u271d : Multiset \u03b2\na\u271d\u00b9 : r a\u271d\u00b2 b\u271d\na\u271d : Rel r as\u271d bs\u271d\na_ih\u271d : (\u2200 (a : \u03b1), a \u2208 as\u271d \u2192 \u2200 (b : \u03b2), b \u2208 bs\u271d \u2192 r a b \u2192 p a b) \u2192 Rel p as\u271d bs\u271d\nh : \u2200 (a : \u03b1), a \u2208 a\u271d\u00b2 ::\u2098 as\u271d \u2192 \u2200 (b : \u03b2), b \u2208 b\u271d ::\u2098 bs\u271d \u2192 r a b \u2192 p a b\n\u22a2 Rel p (a\u271d\u00b2 ::\u2098 as\u271d) (b\u271d ::\u2098 bs\u271d)\n[PROOFSTEP]\ncase cons a b s t hab _hst\n ih =>\n apply Rel.cons (h a (mem_cons_self _ _) b (mem_cons_self _ _) hab)\n exact ih fun a' ha' b' hb' h' => h a' (mem_cons_of_mem ha') b' (mem_cons_of_mem hb') h'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np\u271d : \u03b3 \u2192 \u03b4 \u2192 Prop\nr p : \u03b1 \u2192 \u03b2 \u2192 Prop\ns\u271d : Multiset \u03b1\nt\u271d : Multiset \u03b2\na : \u03b1\nb : \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nhab : r a b\n_hst : Rel r s t\nih : (\u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 r a b \u2192 p a b) \u2192 Rel p s t\nh : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 \u2200 (b_1 : \u03b2), b_1 \u2208 b ::\u2098 t \u2192 r a_1 b_1 \u2192 p a_1 b_1\n\u22a2 Rel p (a ::\u2098 s) (b ::\u2098 t)\n[PROOFSTEP]\ncase cons a b s t hab _hst\n ih =>\n apply Rel.cons (h a (mem_cons_self _ _) b (mem_cons_self _ _) hab)\n exact ih fun a' ha' b' hb' h' => h a' (mem_cons_of_mem ha') b' (mem_cons_of_mem hb') h'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np\u271d : \u03b3 \u2192 \u03b4 \u2192 Prop\nr p : \u03b1 \u2192 \u03b2 \u2192 Prop\ns\u271d : Multiset \u03b1\nt\u271d : Multiset \u03b2\na : \u03b1\nb : \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nhab : r a b\n_hst : Rel r s t\nih : (\u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 r a b \u2192 p a b) \u2192 Rel p s t\nh : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 \u2200 (b_1 : \u03b2), b_1 \u2208 b ::\u2098 t \u2192 r a_1 b_1 \u2192 p a_1 b_1\n\u22a2 Rel p (a ::\u2098 s) (b ::\u2098 t)\n[PROOFSTEP]\napply Rel.cons (h a (mem_cons_self _ _) b (mem_cons_self _ _) hab)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np\u271d : \u03b3 \u2192 \u03b4 \u2192 Prop\nr p : \u03b1 \u2192 \u03b2 \u2192 Prop\ns\u271d : Multiset \u03b1\nt\u271d : Multiset \u03b2\na : \u03b1\nb : \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nhab : r a b\n_hst : Rel r s t\nih : (\u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 r a b \u2192 p a b) \u2192 Rel p s t\nh : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 \u2200 (b_1 : \u03b2), b_1 \u2208 b ::\u2098 t \u2192 r a_1 b_1 \u2192 p a_1 b_1\n\u22a2 Rel p s t\n[PROOFSTEP]\nexact ih fun a' ha' b' hb' h' => h a' (mem_cons_of_mem ha') b' (mem_cons_of_mem hb') h'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\nu : Multiset \u03b1\nv : Multiset \u03b2\nhst : Rel r s t\nhuv : Rel r u v\n\u22a2 Rel r (s + u) (t + v)\n[PROOFSTEP]\ninduction hst\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\nu : Multiset \u03b1\nv : Multiset \u03b2\nhuv : Rel r u v\n\u22a2 Rel r (0 + u) (0 + v)\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\nu : Multiset \u03b1\nv : Multiset \u03b2\nhuv : Rel r u v\na\u271d\u00b2 : \u03b1\nb\u271d : \u03b2\nas\u271d : Multiset \u03b1\nbs\u271d : Multiset \u03b2\na\u271d\u00b9 : r a\u271d\u00b2 b\u271d\na\u271d : Rel r as\u271d bs\u271d\na_ih\u271d : Rel r (as\u271d + u) (bs\u271d + v)\n\u22a2 Rel r (a\u271d\u00b2 ::\u2098 as\u271d + u) (b\u271d ::\u2098 bs\u271d + v)\n[PROOFSTEP]\ncase zero => simpa using huv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\nu : Multiset \u03b1\nv : Multiset \u03b2\nhuv : Rel r u v\n\u22a2 Rel r (0 + u) (0 + v)\n[PROOFSTEP]\ncase zero => simpa using huv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\nu : Multiset \u03b1\nv : Multiset \u03b2\nhuv : Rel r u v\n\u22a2 Rel r (0 + u) (0 + v)\n[PROOFSTEP]\nsimpa using huv\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\nu : Multiset \u03b1\nv : Multiset \u03b2\nhuv : Rel r u v\na\u271d\u00b2 : \u03b1\nb\u271d : \u03b2\nas\u271d : Multiset \u03b1\nbs\u271d : Multiset \u03b2\na\u271d\u00b9 : r a\u271d\u00b2 b\u271d\na\u271d : Rel r as\u271d bs\u271d\na_ih\u271d : Rel r (as\u271d + u) (bs\u271d + v)\n\u22a2 Rel r (a\u271d\u00b2 ::\u2098 as\u271d + u) (b\u271d ::\u2098 bs\u271d + v)\n[PROOFSTEP]\ncase cons a b s t hab hst ih => simpa using ih.cons hab\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns\u271d : Multiset \u03b1\nt\u271d : Multiset \u03b2\nu : Multiset \u03b1\nv : Multiset \u03b2\nhuv : Rel r u v\na : \u03b1\nb : \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nhab : r a b\nhst : Rel r s t\nih : Rel r (s + u) (t + v)\n\u22a2 Rel r (a ::\u2098 s + u) (b ::\u2098 t + v)\n[PROOFSTEP]\ncase cons a b s t hab hst ih => simpa using ih.cons hab\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns\u271d : Multiset \u03b1\nt\u271d : Multiset \u03b2\nu : Multiset \u03b1\nv : Multiset \u03b2\nhuv : Rel r u v\na : \u03b1\nb : \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nhab : r a b\nhst : Rel r s t\nih : Rel r (s + u) (t + v)\n\u22a2 Rel r (a ::\u2098 s + u) (b ::\u2098 t + v)\n[PROOFSTEP]\nsimpa using ih.cons hab\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns t : Multiset \u03b1\n\u22a2 Rel (flip fun x x_1 => x = x_1) s t \u2194 s = t\n[PROOFSTEP]\nrw [rel_flip, rel_eq, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nb : Multiset \u03b2\n\u22a2 Rel r 0 b \u2194 b = 0\n[PROOFSTEP]\nrw [Rel_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nb : Multiset \u03b2\n\u22a2 (0 = 0 \u2227 b = 0 \u2228\n Exists fun {a} =>\n Exists fun {b_1} =>\n Exists fun {as} => Exists fun {bs} => r a b_1 \u2227 Rel r as bs \u2227 0 = a ::\u2098 as \u2227 b = b_1 ::\u2098 bs) \u2194\n b = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : Multiset \u03b1\n\u22a2 Rel r a 0 \u2194 a = 0\n[PROOFSTEP]\nrw [Rel_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : Multiset \u03b1\n\u22a2 (a = 0 \u2227 0 = 0 \u2228\n Exists fun {a_1} =>\n Exists fun {b} => Exists fun {as} => Exists fun {bs} => r a_1 b \u2227 Rel r as bs \u2227 a = a_1 ::\u2098 as \u2227 0 = b ::\u2098 bs) \u2194\n a = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nas : Multiset \u03b1\nbs : Multiset \u03b2\n\u22a2 Rel r (a ::\u2098 as) bs \u2194 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 bs = b ::\u2098 bs'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nas : Multiset \u03b1\nbs : Multiset \u03b2\n\u22a2 Rel r (a ::\u2098 as) bs \u2192 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 bs = b ::\u2098 bs'\n[PROOFSTEP]\ngeneralize hm : a ::\u2098 as = m\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nas : Multiset \u03b1\nbs : Multiset \u03b2\nm : Multiset \u03b1\nhm : a ::\u2098 as = m\n\u22a2 Rel r m bs \u2192 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 bs = b ::\u2098 bs'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nas : Multiset \u03b1\nbs : Multiset \u03b2\nm : Multiset \u03b1\nhm : a ::\u2098 as = m\nh : Rel r m bs\n\u22a2 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 bs = b ::\u2098 bs'\n[PROOFSTEP]\ninduction h generalizing as\n[GOAL]\ncase mp.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nbs : Multiset \u03b2\nm as : Multiset \u03b1\nhm : a ::\u2098 as = 0\n\u22a2 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 0 = b ::\u2098 bs'\ncase mp.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nbs : Multiset \u03b2\nm : Multiset \u03b1\na\u271d\u00b2 : \u03b1\nb\u271d : \u03b2\nas\u271d : Multiset \u03b1\nbs\u271d : Multiset \u03b2\na\u271d\u00b9 : r a\u271d\u00b2 b\u271d\na\u271d : Rel r as\u271d bs\u271d\na_ih\u271d : \u2200 {as : Multiset \u03b1}, a ::\u2098 as = as\u271d \u2192 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 bs\u271d = b ::\u2098 bs'\nas : Multiset \u03b1\nhm : a ::\u2098 as = a\u271d\u00b2 ::\u2098 as\u271d\n\u22a2 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 b\u271d ::\u2098 bs\u271d = b ::\u2098 bs'\n[PROOFSTEP]\ncase zero => simp at hm \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nbs : Multiset \u03b2\nm as : Multiset \u03b1\nhm : a ::\u2098 as = 0\n\u22a2 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 0 = b ::\u2098 bs'\n[PROOFSTEP]\ncase zero => simp at hm \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nbs : Multiset \u03b2\nm as : Multiset \u03b1\nhm : a ::\u2098 as = 0\n\u22a2 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 0 = b ::\u2098 bs'\n[PROOFSTEP]\nsimp at hm \n[GOAL]\ncase mp.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nbs : Multiset \u03b2\nm : Multiset \u03b1\na\u271d\u00b2 : \u03b1\nb\u271d : \u03b2\nas\u271d : Multiset \u03b1\nbs\u271d : Multiset \u03b2\na\u271d\u00b9 : r a\u271d\u00b2 b\u271d\na\u271d : Rel r as\u271d bs\u271d\na_ih\u271d : \u2200 {as : Multiset \u03b1}, a ::\u2098 as = as\u271d \u2192 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 bs\u271d = b ::\u2098 bs'\nas : Multiset \u03b1\nhm : a ::\u2098 as = a\u271d\u00b2 ::\u2098 as\u271d\n\u22a2 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 b\u271d ::\u2098 bs\u271d = b ::\u2098 bs'\n[PROOFSTEP]\ncase cons a' b as' bs ha'b h\n ih =>\n rcases cons_eq_cons.1 hm with (\u27e8eq\u2081, eq\u2082\u27e9 | \u27e8_h, cs, eq\u2081, eq\u2082\u27e9)\n \u00b7 subst eq\u2081\n subst eq\u2082\n exact \u27e8b, bs, ha'b, h, rfl\u27e9\n \u00b7 rcases ih eq\u2082.symm with \u27e8b', bs', h\u2081, h\u2082, eq\u27e9\n exact \u27e8b', b ::\u2098 bs', h\u2081, eq\u2081.symm \u25b8 Rel.cons ha'b h\u2082, eq.symm \u25b8 cons_swap _ _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nbs\u271d : Multiset \u03b2\nm : Multiset \u03b1\na' : \u03b1\nb : \u03b2\nas' : Multiset \u03b1\nbs : Multiset \u03b2\nha'b : r a' b\nh : Rel r as' bs\nih : \u2200 {as : Multiset \u03b1}, a ::\u2098 as = as' \u2192 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 bs = b ::\u2098 bs'\nas : Multiset \u03b1\nhm : a ::\u2098 as = a' ::\u2098 as'\n\u22a2 \u2203 b_1 bs', r a b_1 \u2227 Rel r as bs' \u2227 b ::\u2098 bs = b_1 ::\u2098 bs'\n[PROOFSTEP]\ncase cons a' b as' bs ha'b h\n ih =>\n rcases cons_eq_cons.1 hm with (\u27e8eq\u2081, eq\u2082\u27e9 | \u27e8_h, cs, eq\u2081, eq\u2082\u27e9)\n \u00b7 subst eq\u2081\n subst eq\u2082\n exact \u27e8b, bs, ha'b, h, rfl\u27e9\n \u00b7 rcases ih eq\u2082.symm with \u27e8b', bs', h\u2081, h\u2082, eq\u27e9\n exact \u27e8b', b ::\u2098 bs', h\u2081, eq\u2081.symm \u25b8 Rel.cons ha'b h\u2082, eq.symm \u25b8 cons_swap _ _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nbs\u271d : Multiset \u03b2\nm : Multiset \u03b1\na' : \u03b1\nb : \u03b2\nas' : Multiset \u03b1\nbs : Multiset \u03b2\nha'b : r a' b\nh : Rel r as' bs\nih : \u2200 {as : Multiset \u03b1}, a ::\u2098 as = as' \u2192 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 bs = b ::\u2098 bs'\nas : Multiset \u03b1\nhm : a ::\u2098 as = a' ::\u2098 as'\n\u22a2 \u2203 b_1 bs', r a b_1 \u2227 Rel r as bs' \u2227 b ::\u2098 bs = b_1 ::\u2098 bs'\n[PROOFSTEP]\nrcases cons_eq_cons.1 hm with (\u27e8eq\u2081, eq\u2082\u27e9 | \u27e8_h, cs, eq\u2081, eq\u2082\u27e9)\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nbs\u271d : Multiset \u03b2\nm : Multiset \u03b1\na' : \u03b1\nb : \u03b2\nas' : Multiset \u03b1\nbs : Multiset \u03b2\nha'b : r a' b\nh : Rel r as' bs\nih : \u2200 {as : Multiset \u03b1}, a ::\u2098 as = as' \u2192 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 bs = b ::\u2098 bs'\nas : Multiset \u03b1\nhm : a ::\u2098 as = a' ::\u2098 as'\neq\u2081 : a = a'\neq\u2082 : as = as'\n\u22a2 \u2203 b_1 bs', r a b_1 \u2227 Rel r as bs' \u2227 b ::\u2098 bs = b_1 ::\u2098 bs'\n[PROOFSTEP]\nsubst eq\u2081\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nbs\u271d : Multiset \u03b2\nm : Multiset \u03b1\nb : \u03b2\nas' : Multiset \u03b1\nbs : Multiset \u03b2\nh : Rel r as' bs\nih : \u2200 {as : Multiset \u03b1}, a ::\u2098 as = as' \u2192 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 bs = b ::\u2098 bs'\nas : Multiset \u03b1\neq\u2082 : as = as'\nha'b : r a b\nhm : a ::\u2098 as = a ::\u2098 as'\n\u22a2 \u2203 b_1 bs', r a b_1 \u2227 Rel r as bs' \u2227 b ::\u2098 bs = b_1 ::\u2098 bs'\n[PROOFSTEP]\nsubst eq\u2082\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nbs\u271d : Multiset \u03b2\nm : Multiset \u03b1\nb : \u03b2\nbs : Multiset \u03b2\nas : Multiset \u03b1\nha'b : r a b\nh : Rel r as bs\nih : \u2200 {as_1 : Multiset \u03b1}, a ::\u2098 as_1 = as \u2192 \u2203 b bs', r a b \u2227 Rel r as_1 bs' \u2227 bs = b ::\u2098 bs'\nhm : a ::\u2098 as = a ::\u2098 as\n\u22a2 \u2203 b_1 bs', r a b_1 \u2227 Rel r as bs' \u2227 b ::\u2098 bs = b_1 ::\u2098 bs'\n[PROOFSTEP]\nexact \u27e8b, bs, ha'b, h, rfl\u27e9\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nbs\u271d : Multiset \u03b2\nm : Multiset \u03b1\na' : \u03b1\nb : \u03b2\nas' : Multiset \u03b1\nbs : Multiset \u03b2\nha'b : r a' b\nh : Rel r as' bs\nih : \u2200 {as : Multiset \u03b1}, a ::\u2098 as = as' \u2192 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 bs = b ::\u2098 bs'\nas : Multiset \u03b1\nhm : a ::\u2098 as = a' ::\u2098 as'\n_h : a \u2260 a'\ncs : Multiset \u03b1\neq\u2081 : as = a' ::\u2098 cs\neq\u2082 : as' = a ::\u2098 cs\n\u22a2 \u2203 b_1 bs', r a b_1 \u2227 Rel r as bs' \u2227 b ::\u2098 bs = b_1 ::\u2098 bs'\n[PROOFSTEP]\nrcases ih eq\u2082.symm with \u27e8b', bs', h\u2081, h\u2082, eq\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nbs\u271d : Multiset \u03b2\nm : Multiset \u03b1\na' : \u03b1\nb : \u03b2\nas' : Multiset \u03b1\nbs : Multiset \u03b2\nha'b : r a' b\nh : Rel r as' bs\nih : \u2200 {as : Multiset \u03b1}, a ::\u2098 as = as' \u2192 \u2203 b bs', r a b \u2227 Rel r as bs' \u2227 bs = b ::\u2098 bs'\nas : Multiset \u03b1\nhm : a ::\u2098 as = a' ::\u2098 as'\n_h : a \u2260 a'\ncs : Multiset \u03b1\neq\u2081 : as = a' ::\u2098 cs\neq\u2082 : as' = a ::\u2098 cs\nb' : \u03b2\nbs' : Multiset \u03b2\nh\u2081 : r a b'\nh\u2082 : Rel r cs bs'\neq : bs = b' ::\u2098 bs'\n\u22a2 \u2203 b_1 bs', r a b_1 \u2227 Rel r as bs' \u2227 b ::\u2098 bs = b_1 ::\u2098 bs'\n[PROOFSTEP]\nexact \u27e8b', b ::\u2098 bs', h\u2081, eq\u2081.symm \u25b8 Rel.cons ha'b h\u2082, eq.symm \u25b8 cons_swap _ _ _\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nas : Multiset \u03b1\nbs : Multiset \u03b2\n\u22a2 (\u2203 b bs', r a b \u2227 Rel r as bs' \u2227 bs = b ::\u2098 bs') \u2192 Rel r (a ::\u2098 as) bs\n[PROOFSTEP]\nexact fun \u27e8b, bs', hab, h, Eq\u27e9 => Eq.symm \u25b8 Rel.cons hab h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas : Multiset \u03b1\nb : \u03b2\nbs : Multiset \u03b2\n\u22a2 Rel r as (b ::\u2098 bs) \u2194 \u2203 a as', r a b \u2227 Rel r as' bs \u2227 as = a ::\u2098 as'\n[PROOFSTEP]\nrw [\u2190 rel_flip, rel_cons_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas : Multiset \u03b1\nb : \u03b2\nbs : Multiset \u03b2\n\u22a2 (\u2203 b_1 bs', flip r b b_1 \u2227 Rel (flip r) bs bs' \u2227 as = b_1 ::\u2098 bs') \u2194 \u2203 a as', r a b \u2227 Rel r as' bs \u2227 as = a ::\u2098 as'\n[PROOFSTEP]\nrefine' exists\u2082_congr fun a as' => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas : Multiset \u03b1\nb : \u03b2\nbs : Multiset \u03b2\na : \u03b1\nas' : Multiset \u03b1\n\u22a2 flip r b a \u2227 Rel (flip r) bs as' \u2227 as = a ::\u2098 as' \u2194 r a b \u2227 Rel r as' bs \u2227 as = a ::\u2098 as'\n[PROOFSTEP]\nrw [rel_flip, flip]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas\u2080 as\u2081 : Multiset \u03b1\n\u22a2 \u2200 {bs : Multiset \u03b2}, Rel r (0 + as\u2081) bs \u2194 \u2203 bs\u2080 bs\u2081, Rel r 0 bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas\u2080 as\u2081 : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nih : \u2200 {bs : Multiset \u03b2}, Rel r (s + as\u2081) bs \u2194 \u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\nbs : Multiset \u03b2\n\u22a2 Rel r (a ::\u2098 s + as\u2081) bs \u2194 \u2203 bs\u2080 bs\u2081, Rel r (a ::\u2098 s) bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\n[PROOFSTEP]\nsimp only [ih, cons_add, rel_cons_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas\u2080 as\u2081 : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nih : \u2200 {bs : Multiset \u03b2}, Rel r (s + as\u2081) bs \u2194 \u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\nbs : Multiset \u03b2\n\u22a2 (\u2203 b bs', r a b \u2227 (\u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs' = bs\u2080 + bs\u2081) \u2227 bs = b ::\u2098 bs') \u2194\n \u2203 bs\u2080 bs\u2081, (\u2203 b bs', r a b \u2227 Rel r s bs' \u2227 bs\u2080 = b ::\u2098 bs') \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas\u2080 as\u2081 : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nih : \u2200 {bs : Multiset \u03b2}, Rel r (s + as\u2081) bs \u2194 \u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\nbs : Multiset \u03b2\n\u22a2 (\u2203 b bs', r a b \u2227 (\u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs' = bs\u2080 + bs\u2081) \u2227 bs = b ::\u2098 bs') \u2192\n \u2203 bs\u2080 bs\u2081, (\u2203 b bs', r a b \u2227 Rel r s bs' \u2227 bs\u2080 = b ::\u2098 bs') \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas\u2080 as\u2081 : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nih : \u2200 {bs : Multiset \u03b2}, Rel r (s + as\u2081) bs \u2194 \u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\nbs : Multiset \u03b2\nh : \u2203 b bs', r a b \u2227 (\u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs' = bs\u2080 + bs\u2081) \u2227 bs = b ::\u2098 bs'\n\u22a2 \u2203 bs\u2080 bs\u2081, (\u2203 b bs', r a b \u2227 Rel r s bs' \u2227 bs\u2080 = b ::\u2098 bs') \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\n[PROOFSTEP]\nrcases h with \u27e8b, bs', hab, h, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas\u2080 as\u2081 : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nih : \u2200 {bs : Multiset \u03b2}, Rel r (s + as\u2081) bs \u2194 \u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\nb : \u03b2\nbs' : Multiset \u03b2\nhab : r a b\nh : \u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs' = bs\u2080 + bs\u2081\n\u22a2 \u2203 bs\u2080 bs\u2081, (\u2203 b bs', r a b \u2227 Rel r s bs' \u2227 bs\u2080 = b ::\u2098 bs') \u2227 Rel r as\u2081 bs\u2081 \u2227 b ::\u2098 bs' = bs\u2080 + bs\u2081\n[PROOFSTEP]\nrcases h with \u27e8bs\u2080, bs\u2081, h\u2080, h\u2081, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas\u2080 as\u2081 : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nih : \u2200 {bs : Multiset \u03b2}, Rel r (s + as\u2081) bs \u2194 \u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\nb : \u03b2\nhab : r a b\nbs\u2080 bs\u2081 : Multiset \u03b2\nh\u2080 : Rel r s bs\u2080\nh\u2081 : Rel r as\u2081 bs\u2081\n\u22a2 \u2203 bs\u2080_1 bs\u2081_1,\n (\u2203 b bs', r a b \u2227 Rel r s bs' \u2227 bs\u2080_1 = b ::\u2098 bs') \u2227 Rel r as\u2081 bs\u2081_1 \u2227 b ::\u2098 (bs\u2080 + bs\u2081) = bs\u2080_1 + bs\u2081_1\n[PROOFSTEP]\nexact \u27e8b ::\u2098 bs\u2080, bs\u2081, \u27e8b, bs\u2080, hab, h\u2080, rfl\u27e9, h\u2081, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas\u2080 as\u2081 : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nih : \u2200 {bs : Multiset \u03b2}, Rel r (s + as\u2081) bs \u2194 \u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\nb : \u03b2\nhab : r a b\nbs\u2080 bs\u2081 : Multiset \u03b2\nh\u2080 : Rel r s bs\u2080\nh\u2081 : Rel r as\u2081 bs\u2081\n\u22a2 b ::\u2098 (bs\u2080 + bs\u2081) = b ::\u2098 bs\u2080 + bs\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas\u2080 as\u2081 : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nih : \u2200 {bs : Multiset \u03b2}, Rel r (s + as\u2081) bs \u2194 \u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\nbs : Multiset \u03b2\n\u22a2 (\u2203 bs\u2080 bs\u2081, (\u2203 b bs', r a b \u2227 Rel r s bs' \u2227 bs\u2080 = b ::\u2098 bs') \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081) \u2192\n \u2203 b bs', r a b \u2227 (\u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs' = bs\u2080 + bs\u2081) \u2227 bs = b ::\u2098 bs'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas\u2080 as\u2081 : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nih : \u2200 {bs : Multiset \u03b2}, Rel r (s + as\u2081) bs \u2194 \u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\nbs : Multiset \u03b2\nh : \u2203 bs\u2080 bs\u2081, (\u2203 b bs', r a b \u2227 Rel r s bs' \u2227 bs\u2080 = b ::\u2098 bs') \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\n\u22a2 \u2203 b bs', r a b \u2227 (\u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs' = bs\u2080 + bs\u2081) \u2227 bs = b ::\u2098 bs'\n[PROOFSTEP]\nrcases h with \u27e8bs\u2080, bs\u2081, h, h\u2081, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas\u2080 as\u2081 : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nih : \u2200 {bs : Multiset \u03b2}, Rel r (s + as\u2081) bs \u2194 \u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\nbs\u2080 bs\u2081 : Multiset \u03b2\nh : \u2203 b bs', r a b \u2227 Rel r s bs' \u2227 bs\u2080 = b ::\u2098 bs'\nh\u2081 : Rel r as\u2081 bs\u2081\n\u22a2 \u2203 b bs', r a b \u2227 (\u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs' = bs\u2080 + bs\u2081) \u2227 bs\u2080 + bs\u2081 = b ::\u2098 bs'\n[PROOFSTEP]\nrcases h with \u27e8b, bs, hab, h\u2080, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas\u2080 as\u2081 : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nih : \u2200 {bs : Multiset \u03b2}, Rel r (s + as\u2081) bs \u2194 \u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\nbs\u2081 : Multiset \u03b2\nh\u2081 : Rel r as\u2081 bs\u2081\nb : \u03b2\nbs : Multiset \u03b2\nhab : r a b\nh\u2080 : Rel r s bs\n\u22a2 \u2203 b_1 bs', r a b_1 \u2227 (\u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs' = bs\u2080 + bs\u2081) \u2227 b ::\u2098 bs + bs\u2081 = b_1 ::\u2098 bs'\n[PROOFSTEP]\nexact \u27e8b, bs + bs\u2081, hab, \u27e8bs, bs\u2081, h\u2080, h\u2081, rfl\u27e9, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas\u2080 as\u2081 : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nih : \u2200 {bs : Multiset \u03b2}, Rel r (s + as\u2081) bs \u2194 \u2203 bs\u2080 bs\u2081, Rel r s bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081\nbs\u2081 : Multiset \u03b2\nh\u2081 : Rel r as\u2081 bs\u2081\nb : \u03b2\nbs : Multiset \u03b2\nhab : r a b\nh\u2080 : Rel r s bs\n\u22a2 b ::\u2098 bs + bs\u2081 = b ::\u2098 (bs + bs\u2081)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas : Multiset \u03b1\nbs\u2080 bs\u2081 : Multiset \u03b2\n\u22a2 Rel r as (bs\u2080 + bs\u2081) \u2194 \u2203 as\u2080 as\u2081, Rel r as\u2080 bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 as = as\u2080 + as\u2081\n[PROOFSTEP]\nrw [\u2190 rel_flip, rel_add_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nas : Multiset \u03b1\nbs\u2080 bs\u2081 : Multiset \u03b2\n\u22a2 (\u2203 bs\u2080_1 bs\u2081_1, Rel (flip r) bs\u2080 bs\u2080_1 \u2227 Rel (flip r) bs\u2081 bs\u2081_1 \u2227 as = bs\u2080_1 + bs\u2081_1) \u2194\n \u2203 as\u2080 as\u2081, Rel r as\u2080 bs\u2080 \u2227 Rel r as\u2081 bs\u2081 \u2227 as = as\u2080 + as\u2081\n[PROOFSTEP]\nsimp [rel_flip]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns : Multiset \u03b3\nf : \u03b3 \u2192 \u03b1\n\u22a2 \u2200 {t : Multiset \u03b2}, Rel r (map f 0) t \u2194 Rel (fun a b => r (f a) b) 0 t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns : Multiset \u03b3\nf : \u03b3 \u2192 \u03b1\n\u22a2 \u2200 \u2983a : \u03b3\u2984 {s : Multiset \u03b3},\n (\u2200 {t : Multiset \u03b2}, Rel r (map f s) t \u2194 Rel (fun a b => r (f a) b) s t) \u2192\n \u2200 {t : Multiset \u03b2}, Rel r (map f (a ::\u2098 s)) t \u2194 Rel (fun a b => r (f a) b) (a ::\u2098 s) t\n[PROOFSTEP]\nsimp (config := { contextual := true }) [rel_cons_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b3\nf : \u03b3 \u2192 \u03b2\n\u22a2 Rel r s (map f t) \u2194 Rel (fun a b => r a (f b)) s t\n[PROOFSTEP]\nrw [\u2190 rel_flip, rel_map_left, \u2190 rel_flip]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b3\nf : \u03b3 \u2192 \u03b2\n\u22a2 Rel (flip fun a b => flip r (f a) b) s t \u2194 Rel (fun a b => r a (f b)) s t\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\nh : Rel r s t\n\u22a2 \u2191card s = \u2191card t\n[PROOFSTEP]\ninduction h\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\n\u22a2 \u2191card 0 = \u2191card 0\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\na\u271d\u00b2 : \u03b1\nb\u271d : \u03b2\nas\u271d : Multiset \u03b1\nbs\u271d : Multiset \u03b2\na\u271d\u00b9 : r a\u271d\u00b2 b\u271d\na\u271d : Rel r as\u271d bs\u271d\na_ih\u271d : \u2191card as\u271d = \u2191card bs\u271d\n\u22a2 \u2191card (a\u271d\u00b2 ::\u2098 as\u271d) = \u2191card (b\u271d ::\u2098 bs\u271d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\nh : Rel r s t\n\u22a2 \u2200 {a : \u03b1}, a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\n[PROOFSTEP]\ninduction' h with x y s t hxy _hst ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Multiset \u03b1\nt : Multiset \u03b2\n\u22a2 \u2200 {a : \u03b1}, a \u2208 0 \u2192 \u2203 b, b \u2208 0 \u2227 r a b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns\u271d : Multiset \u03b1\nt\u271d : Multiset \u03b2\nx : \u03b1\ny : \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nhxy : r x y\n_hst : Rel r s t\nih : \u2200 {a : \u03b1}, a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\n\u22a2 \u2200 {a : \u03b1}, a \u2208 x ::\u2098 s \u2192 \u2203 b, b \u2208 y ::\u2098 t \u2227 r a b\n[PROOFSTEP]\nintro a ha\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns\u271d : Multiset \u03b1\nt\u271d : Multiset \u03b2\nx : \u03b1\ny : \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nhxy : r x y\n_hst : Rel r s t\nih : \u2200 {a : \u03b1}, a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\na : \u03b1\nha : a \u2208 x ::\u2098 s\n\u22a2 \u2203 b, b \u2208 y ::\u2098 t \u2227 r a b\n[PROOFSTEP]\ncases' mem_cons.1 ha with ha ha\n[GOAL]\ncase cons.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns\u271d : Multiset \u03b1\nt\u271d : Multiset \u03b2\nx : \u03b1\ny : \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nhxy : r x y\n_hst : Rel r s t\nih : \u2200 {a : \u03b1}, a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\na : \u03b1\nha\u271d : a \u2208 x ::\u2098 s\nha : a = x\n\u22a2 \u2203 b, b \u2208 y ::\u2098 t \u2227 r a b\n[PROOFSTEP]\nexact \u27e8y, mem_cons_self _ _, ha.symm \u25b8 hxy\u27e9\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns\u271d : Multiset \u03b1\nt\u271d : Multiset \u03b2\nx : \u03b1\ny : \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nhxy : r x y\n_hst : Rel r s t\nih : \u2200 {a : \u03b1}, a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\na : \u03b1\nha\u271d : a \u2208 x ::\u2098 s\nha : a \u2208 s\n\u22a2 \u2203 b, b \u2208 y ::\u2098 t \u2227 r a b\n[PROOFSTEP]\nrcases ih ha with \u27e8b, hbt, hab\u27e9\n[GOAL]\ncase cons.inr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns\u271d : Multiset \u03b1\nt\u271d : Multiset \u03b2\nx : \u03b1\ny : \u03b2\ns : Multiset \u03b1\nt : Multiset \u03b2\nhxy : r x y\n_hst : Rel r s t\nih : \u2200 {a : \u03b1}, a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\na : \u03b1\nha\u271d : a \u2208 x ::\u2098 s\nha : a \u2208 s\nb : \u03b2\nhbt : b \u2208 t\nhab : r a b\n\u22a2 \u2203 b, b \u2208 y ::\u2098 t \u2227 r a b\n[PROOFSTEP]\nexact \u27e8b, mem_cons.2 (Or.inr hbt), hab\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm1 m2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 m2 \u2192 r a b\nhc : \u2191card m1 = \u2191card m2\n\u22a2 Rel r m1 m2\n[PROOFSTEP]\nrevert m1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 m2 \u2192 r a b) \u2192 \u2191card m1 = \u2191card m2 \u2192 Rel r m1 m2\n[PROOFSTEP]\nrefine' @(m2.induction_on _ _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 0 \u2192 r a b) \u2192 \u2191card m1 = \u2191card 0 \u2192 Rel r m1 0\n[PROOFSTEP]\nintro m _h hc\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nm : Multiset \u03b1\n_h : \u2200 (a b : \u03b1), a \u2208 m \u2192 b \u2208 0 \u2192 r a b\nhc : \u2191card m = \u2191card 0\n\u22a2 Rel r m 0\n[PROOFSTEP]\nrw [rel_zero_right, \u2190 card_eq_zero, hc, card_zero]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1},\n (\u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 s \u2192 r a b) \u2192 \u2191card m1 = \u2191card s \u2192 Rel r m1 s) \u2192\n \u2200 {m1 : Multiset \u03b1},\n (\u2200 (a_2 b : \u03b1), a_2 \u2208 m1 \u2192 b \u2208 a ::\u2098 s \u2192 r a_2 b) \u2192 \u2191card m1 = \u2191card (a ::\u2098 s) \u2192 Rel r m1 (a ::\u2098 s)\n[PROOFSTEP]\nintro a t ih m h hc\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 \u2191card m1 = \u2191card t \u2192 Rel r m1 t\nm : Multiset \u03b1\nh : \u2200 (a_1 b : \u03b1), a_1 \u2208 m \u2192 b \u2208 a ::\u2098 t \u2192 r a_1 b\nhc : \u2191card m = \u2191card (a ::\u2098 t)\n\u22a2 Rel r m (a ::\u2098 t)\n[PROOFSTEP]\nrw [card_cons] at hc \n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 \u2191card m1 = \u2191card t \u2192 Rel r m1 t\nm : Multiset \u03b1\nh : \u2200 (a_1 b : \u03b1), a_1 \u2208 m \u2192 b \u2208 a ::\u2098 t \u2192 r a_1 b\nhc : \u2191card m = \u2191card t + 1\n\u22a2 Rel r m (a ::\u2098 t)\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 := card_pos_iff_exists_mem.1 (show 0 < card m from hc.symm \u25b8 Nat.succ_pos _)\n[GOAL]\ncase refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 \u2191card m1 = \u2191card t \u2192 Rel r m1 t\nm : Multiset \u03b1\nh : \u2200 (a_1 b : \u03b1), a_1 \u2208 m \u2192 b \u2208 a ::\u2098 t \u2192 r a_1 b\nhc : \u2191card m = \u2191card t + 1\nb : \u03b1\nhb : b \u2208 m\n\u22a2 Rel r m (a ::\u2098 t)\n[PROOFSTEP]\nobtain \u27e8m', rfl\u27e9 := exists_cons_of_mem hb\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 \u2191card m1 = \u2191card t \u2192 Rel r m1 t\nb : \u03b1\nm' : Multiset \u03b1\nh : \u2200 (a_1 b_1 : \u03b1), a_1 \u2208 b ::\u2098 m' \u2192 b_1 \u2208 a ::\u2098 t \u2192 r a_1 b_1\nhc : \u2191card (b ::\u2098 m') = \u2191card t + 1\nhb : b \u2208 b ::\u2098 m'\n\u22a2 Rel r (b ::\u2098 m') (a ::\u2098 t)\n[PROOFSTEP]\nrefine' rel_cons_right.mpr \u27e8b, m', h _ _ hb (mem_cons_self _ _), ih _ _, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 \u2191card m1 = \u2191card t \u2192 Rel r m1 t\nb : \u03b1\nm' : Multiset \u03b1\nh : \u2200 (a_1 b_1 : \u03b1), a_1 \u2208 b ::\u2098 m' \u2192 b_1 \u2208 a ::\u2098 t \u2192 r a_1 b_1\nhc : \u2191card (b ::\u2098 m') = \u2191card t + 1\nhb : b \u2208 b ::\u2098 m'\n\u22a2 \u2200 (a b : \u03b1), a \u2208 m' \u2192 b \u2208 t \u2192 r a b\n[PROOFSTEP]\nexact fun _ _ ha hb => h _ _ (mem_cons_of_mem ha) (mem_cons_of_mem hb)\n[GOAL]\ncase refine'_2.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 \u2191card m1 = \u2191card t \u2192 Rel r m1 t\nb : \u03b1\nm' : Multiset \u03b1\nh : \u2200 (a_1 b_1 : \u03b1), a_1 \u2208 b ::\u2098 m' \u2192 b_1 \u2208 a ::\u2098 t \u2192 r a_1 b_1\nhc : \u2191card (b ::\u2098 m') = \u2191card t + 1\nhb : b \u2208 b ::\u2098 m'\n\u22a2 \u2191card m' = \u2191card t\n[PROOFSTEP]\nsimpa using hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm : Multiset \u03b1\na : \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nn : \u2115\nh : Rel r (replicate n a) m\nx : \u03b1\nhx : x \u2208 m\n\u22a2 r a x\n[PROOFSTEP]\nobtain \u27e8b, hb1, hb2\u27e9 := exists_mem_of_rel_of_mem (rel_flip.2 h) hx\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm : Multiset \u03b1\na : \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nn : \u2115\nh : Rel r (replicate n a) m\nx : \u03b1\nhx : x \u2208 m\nb : \u03b1\nhb1 : b \u2208 replicate n a\nhb2 : flip r x b\n\u22a2 r a x\n[PROOFSTEP]\nrwa [eq_of_mem_replicate hb1] at hb2 \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsTrans \u03b1 r\ns t u : Multiset \u03b1\nr1 : Rel r s t\nr2 : Rel r t u\n\u22a2 Rel r s u\n[PROOFSTEP]\ninduction' t using Multiset.induction_on with x t ih generalizing s u\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsTrans \u03b1 r\ns\u271d t u\u271d : Multiset \u03b1\nr1\u271d : Rel r s\u271d t\nr2\u271d : Rel r t u\u271d\ns u : Multiset \u03b1\nr1 : Rel r s 0\nr2 : Rel r 0 u\n\u22a2 Rel r s u\n[PROOFSTEP]\nrw [rel_zero_right.mp r1, rel_zero_left.mp r2, rel_zero_left]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsTrans \u03b1 r\ns\u271d t\u271d u\u271d : Multiset \u03b1\nr1\u271d : Rel r s\u271d t\u271d\nr2\u271d : Rel r t\u271d u\u271d\nx : \u03b1\nt : Multiset \u03b1\nih : \u2200 {s u : Multiset \u03b1}, Rel r s t \u2192 Rel r t u \u2192 Rel r s u\ns u : Multiset \u03b1\nr1 : Rel r s (x ::\u2098 t)\nr2 : Rel r (x ::\u2098 t) u\n\u22a2 Rel r s u\n[PROOFSTEP]\nobtain \u27e8a, as, ha1, ha2, rfl\u27e9 := rel_cons_right.mp r1\n[GOAL]\ncase cons.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsTrans \u03b1 r\ns t\u271d u\u271d : Multiset \u03b1\nr1\u271d : Rel r s t\u271d\nr2\u271d : Rel r t\u271d u\u271d\nx : \u03b1\nt : Multiset \u03b1\nih : \u2200 {s u : Multiset \u03b1}, Rel r s t \u2192 Rel r t u \u2192 Rel r s u\nu : Multiset \u03b1\nr2 : Rel r (x ::\u2098 t) u\na : \u03b1\nas : Multiset \u03b1\nha1 : r a x\nha2 : Rel r as t\nr1 : Rel r (a ::\u2098 as) (x ::\u2098 t)\n\u22a2 Rel r (a ::\u2098 as) u\n[PROOFSTEP]\nobtain \u27e8b, bs, hb1, hb2, rfl\u27e9 := rel_cons_left.mp r2\n[GOAL]\ncase cons.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsTrans \u03b1 r\ns t\u271d u : Multiset \u03b1\nr1\u271d : Rel r s t\u271d\nr2\u271d : Rel r t\u271d u\nx : \u03b1\nt : Multiset \u03b1\nih : \u2200 {s u : Multiset \u03b1}, Rel r s t \u2192 Rel r t u \u2192 Rel r s u\na : \u03b1\nas : Multiset \u03b1\nha1 : r a x\nha2 : Rel r as t\nr1 : Rel r (a ::\u2098 as) (x ::\u2098 t)\nb : \u03b1\nbs : Multiset \u03b1\nhb1 : r x b\nhb2 : Rel r t bs\nr2 : Rel r (x ::\u2098 t) (b ::\u2098 bs)\n\u22a2 Rel r (a ::\u2098 as) (b ::\u2098 bs)\n[PROOFSTEP]\nexact Multiset.Rel.cons (_root_.trans ha1 hb1) (ih ha2 hb2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : IsTrans \u03b1 r\ninst\u271d\u00b9 : IsSymm \u03b1 r\ns t : Multiset \u03b1\nx : \u03b1\ninst\u271d : DecidablePred (r x)\nh : Rel r s t\n\u22a2 countp (r x) s = countp (r x) t\n[PROOFSTEP]\ninduction' s using Multiset.induction_on with y s ih generalizing t\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : IsTrans \u03b1 r\ninst\u271d\u00b9 : IsSymm \u03b1 r\ns t\u271d : Multiset \u03b1\nx : \u03b1\ninst\u271d : DecidablePred (r x)\nh\u271d : Rel r s t\u271d\nt : Multiset \u03b1\nh : Rel r 0 t\n\u22a2 countp (r x) 0 = countp (r x) t\n[PROOFSTEP]\nrw [rel_zero_left.mp h]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : IsTrans \u03b1 r\ninst\u271d\u00b9 : IsSymm \u03b1 r\ns\u271d t\u271d : Multiset \u03b1\nx : \u03b1\ninst\u271d : DecidablePred (r x)\nh\u271d : Rel r s\u271d t\u271d\ny : \u03b1\ns : Multiset \u03b1\nih : \u2200 {t : Multiset \u03b1}, Rel r s t \u2192 countp (r x) s = countp (r x) t\nt : Multiset \u03b1\nh : Rel r (y ::\u2098 s) t\n\u22a2 countp (r x) (y ::\u2098 s) = countp (r x) t\n[PROOFSTEP]\nobtain \u27e8b, bs, hb1, hb2, rfl\u27e9 := rel_cons_left.mp h\n[GOAL]\ncase cons.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : IsTrans \u03b1 r\ninst\u271d\u00b9 : IsSymm \u03b1 r\ns\u271d t : Multiset \u03b1\nx : \u03b1\ninst\u271d : DecidablePred (r x)\nh\u271d : Rel r s\u271d t\ny : \u03b1\ns : Multiset \u03b1\nih : \u2200 {t : Multiset \u03b1}, Rel r s t \u2192 countp (r x) s = countp (r x) t\nb : \u03b1\nbs : Multiset \u03b1\nhb1 : r y b\nhb2 : Rel r s bs\nh : Rel r (y ::\u2098 s) (b ::\u2098 bs)\n\u22a2 countp (r x) (y ::\u2098 s) = countp (r x) (b ::\u2098 bs)\n[PROOFSTEP]\nrw [countp_cons, countp_cons, ih hb2]\n[GOAL]\ncase cons.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : IsTrans \u03b1 r\ninst\u271d\u00b9 : IsSymm \u03b1 r\ns\u271d t : Multiset \u03b1\nx : \u03b1\ninst\u271d : DecidablePred (r x)\nh\u271d : Rel r s\u271d t\ny : \u03b1\ns : Multiset \u03b1\nih : \u2200 {t : Multiset \u03b1}, Rel r s t \u2192 countp (r x) s = countp (r x) t\nb : \u03b1\nbs : Multiset \u03b1\nhb1 : r y b\nhb2 : Rel r s bs\nh : Rel r (y ::\u2098 s) (b ::\u2098 bs)\n\u22a2 (countp (r x) bs + if r x y then 1 else 0) = countp (r x) bs + if r x b then 1 else 0\n[PROOFSTEP]\nsimp only [decide_eq_true_eq, add_right_inj]\n[GOAL]\ncase cons.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : IsTrans \u03b1 r\ninst\u271d\u00b9 : IsSymm \u03b1 r\ns\u271d t : Multiset \u03b1\nx : \u03b1\ninst\u271d : DecidablePred (r x)\nh\u271d : Rel r s\u271d t\ny : \u03b1\ns : Multiset \u03b1\nih : \u2200 {t : Multiset \u03b1}, Rel r s t \u2192 countp (r x) s = countp (r x) t\nb : \u03b1\nbs : Multiset \u03b1\nhb1 : r y b\nhb2 : Rel r s bs\nh : Rel r (y ::\u2098 s) (b ::\u2098 bs)\n\u22a2 (if r x y then 1 else 0) = if r x b then 1 else 0\n[PROOFSTEP]\nrefine' (if_congr \u27e8fun h => _root_.trans h hb1, fun h => _root_.trans h (symm hb1)\u27e9 rfl rfl)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns t : Multiset \u03b1\n\u22a2 map f s = map f t \u2194 s = t\n[PROOFSTEP]\nrw [\u2190 rel_eq, \u2190 rel_eq, rel_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\ns t : Multiset \u03b1\n\u22a2 Rel (fun a b => f a = f b) s t \u2194 Rel (fun x x_1 => x = x_1) s t\n[PROOFSTEP]\nsimp only [hf.eq_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Multiset \u03b1\nhst : Rel r s t\na\u271d b\u271d : \u03b1\nas\u271d bs\u271d : Multiset \u03b1\nhab : r a\u271d b\u271d\n_hst : Rel r as\u271d bs\u271d\nih : map (Quot.mk r) as\u271d = map (Quot.mk r) bs\u271d\n\u22a2 map (Quot.mk r) (a\u271d ::\u2098 as\u271d) = map (Quot.mk r) (b\u271d ::\u2098 bs\u271d)\n[PROOFSTEP]\nsimp [ih, Quot.sound hab]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Multiset \u03b1\n\u22a2 Disjoint s t \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a \u2260 b\n[PROOFSTEP]\nsimp [disjoint_left, imp_not_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl : Multiset \u03b1\na : \u03b1\n\u22a2 Disjoint {a} l \u2194 \u00aca \u2208 l\n[PROOFSTEP]\nsimp [Disjoint]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl : Multiset \u03b1\na : \u03b1\n\u22a2 Disjoint l {a} \u2194 \u00aca \u2208 l\n[PROOFSTEP]\nrw [disjoint_comm, singleton_disjoint]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t u : Multiset \u03b1\n\u22a2 Disjoint (s + t) u \u2194 Disjoint s u \u2227 Disjoint t u\n[PROOFSTEP]\nsimp [Disjoint, or_imp, forall_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t u : Multiset \u03b1\n\u22a2 Disjoint s (t + u) \u2194 Disjoint s t \u2227 Disjoint s u\n[PROOFSTEP]\nrw [disjoint_comm, disjoint_add_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t u : Multiset \u03b1\n\u22a2 Disjoint t s \u2227 Disjoint u s \u2194 Disjoint s t \u2227 Disjoint s u\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns t : Multiset \u03b1\n\u22a2 Disjoint {a} t \u2227 Disjoint s t \u2194 \u00aca \u2208 t \u2227 Disjoint s t\n[PROOFSTEP]\nrw [singleton_disjoint]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns t : Multiset \u03b1\n\u22a2 Disjoint s (a ::\u2098 t) \u2194 \u00aca \u2208 s \u2227 Disjoint s t\n[PROOFSTEP]\nrw [disjoint_comm, disjoint_cons_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns t : Multiset \u03b1\n\u22a2 \u00aca \u2208 s \u2227 Disjoint t s \u2194 \u00aca \u2208 s \u2227 Disjoint s t\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\n\u22a2 s \u2229 t = 0 \u2194 Disjoint s t\n[PROOFSTEP]\nrw [\u2190 subset_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\n\u22a2 s \u2229 t \u2286 0 \u2194 Disjoint s t\n[PROOFSTEP]\nsimp [subset_iff, Disjoint]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u : Multiset \u03b1\n\u22a2 Disjoint (s \u222a t) u \u2194 Disjoint s u \u2227 Disjoint t u\n[PROOFSTEP]\nsimp [Disjoint, or_imp, forall_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u : Multiset \u03b1\n\u22a2 Disjoint s (t \u222a u) \u2194 Disjoint s t \u2227 Disjoint s u\n[PROOFSTEP]\nsimp [Disjoint, or_imp, forall_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\n\u22a2 s + t = s \u222a t \u2194 Disjoint s t\n[PROOFSTEP]\nsimp_rw [\u2190 inter_eq_zero_iff_disjoint, ext, count_add, count_union, count_inter, count_zero, Nat.min_eq_zero_iff,\n Nat.add_eq_max_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ns : Multiset \u03b1\nt : Multiset \u03b2\n\u22a2 Disjoint (map f s) (map g t) \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 f a \u2260 g b\n[PROOFSTEP]\nsimp [Disjoint, @eq_comm _ (f _) (g _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nl : List \u03b1\n\u22a2 \u2200 (a : List \u03b1), \u2191l = \u2191a \u2227 List.Pairwise r a \u2194 l ~ a \u2227 List.Pairwise r a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\nm : Multiset \u03b1\nh : Set.Pairwise {a | a \u2208 m} fun a\u2081 a\u2082 => r (f a\u2081) (f a\u2082)\nb\u2081 : \u03b2\nh\u2081 : b\u2081 \u2208 {b | b \u2208 map f m}\nb\u2082 : \u03b2\nh\u2082 : b\u2082 \u2208 {b | b \u2208 map f m}\nhn : b\u2081 \u2260 b\u2082\n\u22a2 r b\u2081 b\u2082\n[PROOFSTEP]\nobtain \u27e8\u27e8a\u2081, H\u2081, rfl\u27e9, a\u2082, H\u2082, rfl\u27e9 := Multiset.mem_map.1 h\u2081, Multiset.mem_map.1 h\u2082\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\nm : Multiset \u03b1\nh : Set.Pairwise {a | a \u2208 m} fun a\u2081 a\u2082 => r (f a\u2081) (f a\u2082)\na\u2081 : \u03b1\nH\u2081 : a\u2081 \u2208 m\nh\u2081 : f a\u2081 \u2208 {b | b \u2208 map f m}\na\u2082 : \u03b1\nH\u2082 : a\u2082 \u2208 m\nh\u2082 : f a\u2082 \u2208 {b | b \u2208 map f m}\nhn : f a\u2081 \u2260 f a\u2082\n\u22a2 r (f a\u2081) (f a\u2082)\n[PROOFSTEP]\nexact h H\u2081 H\u2082 (mt (congr_arg f) hn)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nl : Multiset \u03b1\n\u22a2 \u2200 (a b : List \u03b1) (p_1 : a \u2248 b),\n Eq.ndrec (motive := fun x => (\u2203! a, a \u2208 x \u2227 p a) \u2192 { a // a \u2208 x \u2227 p a })\n (fun ex_unique => List.chooseX p a (_ : \u2203 x, x \u2208 a \u2227 p x))\n (_ : Quotient.mk (isSetoid \u03b1) a = Quotient.mk (isSetoid \u03b1) b) =\n fun ex_unique => List.chooseX p b (_ : \u2203 x, x \u2208 b \u2227 p x)\n[PROOFSTEP]\nintros a b _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nl : Multiset \u03b1\na b : List \u03b1\np\u271d : a \u2248 b\n\u22a2 Eq.ndrec (motive := fun x => (\u2203! a, a \u2208 x \u2227 p a) \u2192 { a // a \u2208 x \u2227 p a })\n (fun ex_unique => List.chooseX p a (_ : \u2203 x, x \u2208 a \u2227 p x))\n (_ : Quotient.mk (isSetoid \u03b1) a = Quotient.mk (isSetoid \u03b1) b) =\n fun ex_unique => List.chooseX p b (_ : \u2203 x, x \u2208 b \u2227 p x)\n[PROOFSTEP]\nfunext hp\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nl : Multiset \u03b1\na b : List \u03b1\np\u271d : a \u2248 b\nhp : \u2203! a, a \u2208 Quotient.mk (isSetoid \u03b1) b \u2227 p a\n\u22a2 Eq.ndrec (motive := fun x => (\u2203! a, a \u2208 x \u2227 p a) \u2192 { a // a \u2208 x \u2227 p a })\n (fun ex_unique => List.chooseX p a (_ : \u2203 x, x \u2208 a \u2227 p x))\n (_ : Quotient.mk (isSetoid \u03b1) a = Quotient.mk (isSetoid \u03b1) b) hp =\n List.chooseX p b (_ : \u2203 x, x \u2208 b \u2227 p x)\n[PROOFSTEP]\nsuffices all_equal : \u2200 x y : { t // t \u2208 b \u2227 p t }, x = y\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nl : Multiset \u03b1\na b : List \u03b1\np\u271d : a \u2248 b\nhp : \u2203! a, a \u2208 Quotient.mk (isSetoid \u03b1) b \u2227 p a\nall_equal : \u2200 (x y : { t // t \u2208 b \u2227 p t }), x = y\n\u22a2 Eq.ndrec (motive := fun x => (\u2203! a, a \u2208 x \u2227 p a) \u2192 { a // a \u2208 x \u2227 p a })\n (fun ex_unique => List.chooseX p a (_ : \u2203 x, x \u2208 a \u2227 p x))\n (_ : Quotient.mk (isSetoid \u03b1) a = Quotient.mk (isSetoid \u03b1) b) hp =\n List.chooseX p b (_ : \u2203 x, x \u2208 b \u2227 p x)\n[PROOFSTEP]\napply all_equal\n[GOAL]\ncase all_equal\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nl : Multiset \u03b1\na b : List \u03b1\np\u271d : a \u2248 b\nhp : \u2203! a, a \u2208 Quotient.mk (isSetoid \u03b1) b \u2227 p a\n\u22a2 \u2200 (x y : { t // t \u2208 b \u2227 p t }), x = y\n[PROOFSTEP]\nrintro \u27e8x, px\u27e9 \u27e8y, py\u27e9\n[GOAL]\ncase all_equal.mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nl : Multiset \u03b1\na b : List \u03b1\np\u271d : a \u2248 b\nhp : \u2203! a, a \u2208 Quotient.mk (isSetoid \u03b1) b \u2227 p a\nx : \u03b1\npx : x \u2208 b \u2227 p x\ny : \u03b1\npy : y \u2208 b \u2227 p y\n\u22a2 { val := x, property := px } = { val := y, property := py }\n[PROOFSTEP]\nrcases hp with \u27e8z, \u27e8_z_mem_l, _pz\u27e9, z_unique\u27e9\n[GOAL]\ncase all_equal.mk.mk.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nl : Multiset \u03b1\na b : List \u03b1\np\u271d : a \u2248 b\nx : \u03b1\npx : x \u2208 b \u2227 p x\ny : \u03b1\npy : y \u2208 b \u2227 p y\nz : \u03b1\nz_unique : \u2200 (y : \u03b1), (fun a => a \u2208 Quotient.mk (isSetoid \u03b1) b \u2227 p a) y \u2192 y = z\n_z_mem_l : z \u2208 Quotient.mk (isSetoid \u03b1) b\n_pz : p z\n\u22a2 { val := x, property := px } = { val := y, property := py }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase all_equal.mk.mk.intro.intro.intro.e_val\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nl : Multiset \u03b1\na b : List \u03b1\np\u271d : a \u2248 b\nx : \u03b1\npx : x \u2208 b \u2227 p x\ny : \u03b1\npy : y \u2208 b \u2227 p y\nz : \u03b1\nz_unique : \u2200 (y : \u03b1), (fun a => a \u2208 Quotient.mk (isSetoid \u03b1) b \u2227 p a) y \u2192 y = z\n_z_mem_l : z \u2208 Quotient.mk (isSetoid \u03b1) b\n_pz : p z\n\u22a2 x = y\n[PROOFSTEP]\ncalc\n x = z := z_unique x px\n _ = y := (z_unique y py).symm\n", "meta": {"mathlib_filename": "Mathlib.Data.Multiset.Basic", "llama_tokens": 76270, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.4347368130522986}} {"text": "[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\n\u22a2 \u222b (x : \u211d) in x\u2080..x\u2080 + \u2191a, f x \u2264 \u2211 i in Finset.range a, f (x\u2080 + \u2191i)\n[PROOFSTEP]\nhave hint : \u2200 k : \u2115, k < a \u2192 IntervalIntegrable f volume (x\u2080 + k) (x\u2080 + (k + 1 : \u2115)) :=\n by\n intro k hk\n refine' (hf.mono _).intervalIntegrable\n rw [uIcc_of_le]\n \u00b7 apply Icc_subset_Icc\n \u00b7 simp only [le_add_iff_nonneg_right, Nat.cast_nonneg]\n \u00b7 simp only [add_le_add_iff_left, Nat.cast_le, Nat.succ_le_of_lt hk]\n \u00b7 simp only [add_le_add_iff_left, Nat.cast_le, Nat.le_succ]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\n\u22a2 \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n[PROOFSTEP]\nintro k hk\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n[PROOFSTEP]\nrefine' (hf.mono _).intervalIntegrable\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 uIcc (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1)) \u2286 Icc x\u2080 (x\u2080 + \u2191a)\n[PROOFSTEP]\nrw [uIcc_of_le]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 Icc (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1)) \u2286 Icc x\u2080 (x\u2080 + \u2191a)\n[PROOFSTEP]\napply Icc_subset_Icc\n[GOAL]\ncase h\u2081\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 x\u2080 \u2264 x\u2080 + \u2191k\n[PROOFSTEP]\nsimp only [le_add_iff_nonneg_right, Nat.cast_nonneg]\n[GOAL]\ncase h\u2082\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 x\u2080 + \u2191(k + 1) \u2264 x\u2080 + \u2191a\n[PROOFSTEP]\nsimp only [add_le_add_iff_left, Nat.cast_le, Nat.succ_le_of_lt hk]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 x\u2080 + \u2191k \u2264 x\u2080 + \u2191(k + 1)\n[PROOFSTEP]\nsimp only [add_le_add_iff_left, Nat.cast_le, Nat.le_succ]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n\u22a2 \u222b (x : \u211d) in x\u2080..x\u2080 + \u2191a, f x \u2264 \u2211 i in Finset.range a, f (x\u2080 + \u2191i)\n[PROOFSTEP]\ncalc\n \u222b x in x\u2080..x\u2080 + a, f x = \u2211 i in Finset.range a, \u222b x in x\u2080 + i..x\u2080 + (i + 1 : \u2115), f x :=\n by\n convert (intervalIntegral.sum_integral_adjacent_intervals hint).symm\n simp only [Nat.cast_zero, add_zero]\n _ \u2264 \u2211 i in Finset.range a, \u222b _ in x\u2080 + i..x\u2080 + (i + 1 : \u2115), f (x\u2080 + i) :=\n by\n apply Finset.sum_le_sum fun i hi => ?_\n have ia : i < a := Finset.mem_range.1 hi\n refine' intervalIntegral.integral_mono_on (by simp) (hint _ ia) (by simp) fun x hx => _\n apply hf _ _ hx.1\n \u00b7\n simp only [ia.le, mem_Icc, le_add_iff_nonneg_right, Nat.cast_nonneg, add_le_add_iff_left, Nat.cast_le,\n and_self_iff]\n \u00b7 refine' mem_Icc.2 \u27e8le_trans (by simp) hx.1, le_trans hx.2 _\u27e9\n simp only [add_le_add_iff_left, Nat.cast_le, Nat.succ_le_of_lt ia]\n _ = \u2211 i in Finset.range a, f (x\u2080 + i) := by simp\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n\u22a2 \u222b (x : \u211d) in x\u2080..x\u2080 + \u2191a, f x = \u2211 i in Finset.range a, \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f x\n[PROOFSTEP]\nconvert (intervalIntegral.sum_integral_adjacent_intervals hint).symm\n[GOAL]\ncase h.e'_2.h.e'_5\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n\u22a2 x\u2080 = x\u2080 + \u21910\n[PROOFSTEP]\nsimp only [Nat.cast_zero, add_zero]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n\u22a2 \u2211 i in Finset.range a, \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f x \u2264\n \u2211 i in Finset.range a, \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f (x\u2080 + \u2191i)\n[PROOFSTEP]\napply Finset.sum_le_sum fun i hi => ?_\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\n\u22a2 \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f x \u2264 \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f (x\u2080 + \u2191i)\n[PROOFSTEP]\nhave ia : i < a := Finset.mem_range.1 hi\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i < a\n\u22a2 \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f x \u2264 \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f (x\u2080 + \u2191i)\n[PROOFSTEP]\nrefine' intervalIntegral.integral_mono_on (by simp) (hint _ ia) (by simp) fun x hx => _\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i < a\n\u22a2 x\u2080 + \u2191i \u2264 x\u2080 + \u2191(i + 1)\n[PROOFSTEP]\nsimp\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i < a\n\u22a2 IntervalIntegrable (fun x => f (x\u2080 + \u2191i)) volume (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n[PROOFSTEP]\nsimp\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i < a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 f x \u2264 f (x\u2080 + \u2191i)\n[PROOFSTEP]\napply hf _ _ hx.1\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i < a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x\u2080 + \u2191i \u2208 Icc x\u2080 (x\u2080 + \u2191a)\n[PROOFSTEP]\nsimp only [ia.le, mem_Icc, le_add_iff_nonneg_right, Nat.cast_nonneg, add_le_add_iff_left, Nat.cast_le, and_self_iff]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i < a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x \u2208 Icc x\u2080 (x\u2080 + \u2191a)\n[PROOFSTEP]\nrefine' mem_Icc.2 \u27e8le_trans (by simp) hx.1, le_trans hx.2 _\u27e9\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i < a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x\u2080 \u2264 x\u2080 + \u2191i\n[PROOFSTEP]\nsimp\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i < a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x\u2080 + \u2191(i + 1) \u2264 x\u2080 + \u2191a\n[PROOFSTEP]\nsimp only [add_le_add_iff_left, Nat.cast_le, Nat.succ_le_of_lt ia]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n\u22a2 \u2211 i in Finset.range a, \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f (x\u2080 + \u2191i) = \u2211 i in Finset.range a, f (x\u2080 + \u2191i)\n[PROOFSTEP]\nsimp\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u222b (x : \u211d) in \u2191a..\u2191b, f x \u2264 \u2211 x in Finset.Ico a b, f \u2191x\n[PROOFSTEP]\nrw [(Nat.sub_add_cancel hab).symm, Nat.cast_add]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x \u2264 \u2211 x in Finset.Ico a (b - a + a), f \u2191x\n[PROOFSTEP]\nconv =>\n congr\n congr\n \u00b7skip\n \u00b7skip\n rw [add_comm]\n \u00b7skip\n \u00b7skip\n congr\n congr\n rw [\u2190 zero_add a]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x \u2264 \u2211 x in Finset.Ico a (b - a + a), f \u2191x\n[PROOFSTEP]\n congr\n congr\n \u00b7skip\n \u00b7skip\n rw [add_comm]\n \u00b7skip\n \u00b7skip\n congr\n congr\n rw [\u2190 zero_add a]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x \u2264 \u2211 x in Finset.Ico a (b - a + a), f \u2191x\n[PROOFSTEP]\n congr\n congr\n \u00b7skip\n \u00b7skip\n rw [add_comm]\n \u00b7skip\n \u00b7skip\n congr\n congr\n rw [\u2190 zero_add a]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x \u2264 \u2211 x in Finset.Ico a (b - a + a), f \u2191x\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x\ncase a x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | \u2211 x in Finset.Ico a (b - a + a), f \u2191x\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.f\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| fun x => f x\ncase a.a\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2191a\ncase a.b\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2191(b - a) + \u2191a\ncase a.\u03bc\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| volume\ncase a x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | \u2211 x in Finset.Ico a (b - a + a), f \u2191x\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a.f\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| fun x => f x\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.f\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| fun x => f x\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.f\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| fun x => f x\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2191a\ncase a.b\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2191(b - a) + \u2191a\ncase a.\u03bc\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| volume\ncase a x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | \u2211 x in Finset.Ico a (b - a + a), f \u2191x\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a.a\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2191a\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2191a\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2191a\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.b\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2191(b - a) + \u2191a\ncase a.\u03bc\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| volume\ncase a x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | \u2211 x in Finset.Ico a (b - a + a), f \u2191x\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\ncase a.b\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2191a + \u2191(b - a)\ncase a.\u03bc\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| volume\ncase a x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | \u2211 x in Finset.Ico a (b - a + a), f \u2191x\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a.b\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2191a + \u2191(b - a)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.b\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2191a + \u2191(b - a)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.b\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2191a + \u2191(b - a)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.\u03bc\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| volume\ncase a x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | \u2211 x in Finset.Ico a (b - a + a), f \u2191x\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a.\u03bc\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| volume\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.\u03bc\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| volume\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.\u03bc\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| volume\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2211 x in Finset.Ico a (b - a + a), f \u2191x\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.s\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| Finset.Ico a (b - a + a)\ncase a.f x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | fun x => f \u2191x\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.s.a\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| a\ncase a.s.b\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| b - a + a\ncase a.f x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | fun x => f \u2191x\n[PROOFSTEP]\nrw [\u2190 zero_add a]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u222b (x : \u211d) in \u2191a..\u2191a + \u2191(b - a), f x \u2264 \u2211 x in Finset.Ico (0 + a) (b - a + a), f \u2191x\n[PROOFSTEP]\nrw [\u2190 Finset.sum_Ico_add, Nat.Ico_zero_eq_range]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u222b (x : \u211d) in \u2191a..\u2191a + \u2191(b - a), f x \u2264 \u2211 x in Finset.range (b - a), f \u2191(a + x)\n[PROOFSTEP]\nconv =>\n rhs\n congr\n \u00b7skip\n ext\n rw [Nat.cast_add]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u222b (x : \u211d) in \u2191a..\u2191a + \u2191(b - a), f x \u2264 \u2211 x in Finset.range (b - a), f \u2191(a + x)\n[PROOFSTEP]\n rhs\n congr\n \u00b7skip\n ext\n rw [Nat.cast_add]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u222b (x : \u211d) in \u2191a..\u2191a + \u2191(b - a), f x \u2264 \u2211 x in Finset.range (b - a), f \u2191(a + x)\n[PROOFSTEP]\n rhs\n congr\n \u00b7skip\n ext\n rw [Nat.cast_add]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u222b (x : \u211d) in \u2191a..\u2191a + \u2191(b - a), f x \u2264 \u2211 x in Finset.range (b - a), f \u2191(a + x)\n[PROOFSTEP]\nrhs\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2211 x in Finset.range (b - a), f \u2191(a + x)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase s\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| Finset.range (b - a)\ncase f x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | fun x => f \u2191(a + x)\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase s\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| Finset.range (b - a)\n[PROOFSTEP]\nskip\n[GOAL]\ncase s\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| Finset.range (b - a)\n[PROOFSTEP]\nskip\n[GOAL]\ncase s\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| Finset.range (b - a)\n[PROOFSTEP]\nskip\n[GOAL]\ncase f\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| fun x => f \u2191(a + x)\n[PROOFSTEP]\next\n[GOAL]\ncase f.h\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\nx\u271d : \u2115\n| f \u2191(a + x\u271d)\n[PROOFSTEP]\nrw [Nat.cast_add]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u222b (x : \u211d) in \u2191a..\u2191a + \u2191(b - a), f x \u2264 \u2211 x in Finset.range (b - a), f (\u2191a + \u2191x)\n[PROOFSTEP]\napply AntitoneOn.integral_le_sum\n[GOAL]\ncase hf\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 AntitoneOn (fun x => f x) (Icc (\u2191a) (\u2191a + \u2191(b - a)))\n[PROOFSTEP]\nsimp only [hf, hab, Nat.cast_sub, add_sub_cancel'_right]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\n\u22a2 \u2211 i in Finset.range a, f (x\u2080 + \u2191(i + 1)) \u2264 \u222b (x : \u211d) in x\u2080..x\u2080 + \u2191a, f x\n[PROOFSTEP]\nhave hint : \u2200 k : \u2115, k < a \u2192 IntervalIntegrable f volume (x\u2080 + k) (x\u2080 + (k + 1 : \u2115)) :=\n by\n intro k hk\n refine' (hf.mono _).intervalIntegrable\n rw [uIcc_of_le]\n \u00b7 apply Icc_subset_Icc\n \u00b7 simp only [le_add_iff_nonneg_right, Nat.cast_nonneg]\n \u00b7 simp only [add_le_add_iff_left, Nat.cast_le, Nat.succ_le_of_lt hk]\n \u00b7 simp only [add_le_add_iff_left, Nat.cast_le, Nat.le_succ]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\n\u22a2 \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n[PROOFSTEP]\nintro k hk\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n[PROOFSTEP]\nrefine' (hf.mono _).intervalIntegrable\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 uIcc (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1)) \u2286 Icc x\u2080 (x\u2080 + \u2191a)\n[PROOFSTEP]\nrw [uIcc_of_le]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 Icc (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1)) \u2286 Icc x\u2080 (x\u2080 + \u2191a)\n[PROOFSTEP]\napply Icc_subset_Icc\n[GOAL]\ncase h\u2081\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 x\u2080 \u2264 x\u2080 + \u2191k\n[PROOFSTEP]\nsimp only [le_add_iff_nonneg_right, Nat.cast_nonneg]\n[GOAL]\ncase h\u2082\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 x\u2080 + \u2191(k + 1) \u2264 x\u2080 + \u2191a\n[PROOFSTEP]\nsimp only [add_le_add_iff_left, Nat.cast_le, Nat.succ_le_of_lt hk]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 x\u2080 + \u2191k \u2264 x\u2080 + \u2191(k + 1)\n[PROOFSTEP]\nsimp only [add_le_add_iff_left, Nat.cast_le, Nat.le_succ]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n\u22a2 \u2211 i in Finset.range a, f (x\u2080 + \u2191(i + 1)) \u2264 \u222b (x : \u211d) in x\u2080..x\u2080 + \u2191a, f x\n[PROOFSTEP]\ncalc\n (\u2211 i in Finset.range a, f (x\u2080 + (i + 1 : \u2115))) =\n \u2211 i in Finset.range a, \u222b _ in x\u2080 + i..x\u2080 + (i + 1 : \u2115), f (x\u2080 + (i + 1 : \u2115)) :=\n by simp\n _ \u2264 \u2211 i in Finset.range a, \u222b x in x\u2080 + i..x\u2080 + (i + 1 : \u2115), f x :=\n by\n apply Finset.sum_le_sum fun i hi => ?_\n have ia : i + 1 \u2264 a := Finset.mem_range.1 hi\n refine' intervalIntegral.integral_mono_on (by simp) (by simp) (hint _ ia) fun x hx => _\n apply hf _ _ hx.2\n \u00b7 refine' mem_Icc.2 \u27e8le_trans ((le_add_iff_nonneg_right _).2 (Nat.cast_nonneg _)) hx.1, le_trans hx.2 _\u27e9\n simp only [Nat.cast_le, add_le_add_iff_left, ia]\n \u00b7 refine' mem_Icc.2 \u27e8(le_add_iff_nonneg_right _).2 (Nat.cast_nonneg _), _\u27e9\n simp only [add_le_add_iff_left, Nat.cast_le, ia]\n _ = \u222b x in x\u2080..x\u2080 + a, f x :=\n by\n convert intervalIntegral.sum_integral_adjacent_intervals hint\n simp only [Nat.cast_zero, add_zero]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n\u22a2 \u2211 i in Finset.range a, f (x\u2080 + \u2191(i + 1)) =\n \u2211 i in Finset.range a, \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f (x\u2080 + \u2191(i + 1))\n[PROOFSTEP]\nsimp\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n\u22a2 \u2211 i in Finset.range a, \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f (x\u2080 + \u2191(i + 1)) \u2264\n \u2211 i in Finset.range a, \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f x\n[PROOFSTEP]\napply Finset.sum_le_sum fun i hi => ?_\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\n\u22a2 \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f (x\u2080 + \u2191(i + 1)) \u2264 \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f x\n[PROOFSTEP]\nhave ia : i + 1 \u2264 a := Finset.mem_range.1 hi\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\n\u22a2 \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f (x\u2080 + \u2191(i + 1)) \u2264 \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f x\n[PROOFSTEP]\nrefine' intervalIntegral.integral_mono_on (by simp) (by simp) (hint _ ia) fun x hx => _\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\n\u22a2 x\u2080 + \u2191i \u2264 x\u2080 + \u2191(i + 1)\n[PROOFSTEP]\nsimp\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\n\u22a2 IntervalIntegrable (fun x => f (x\u2080 + \u2191(i + 1))) volume (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n[PROOFSTEP]\nsimp\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 f (x\u2080 + \u2191(i + 1)) \u2264 f x\n[PROOFSTEP]\napply hf _ _ hx.2\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x \u2208 Icc x\u2080 (x\u2080 + \u2191a)\n[PROOFSTEP]\nrefine' mem_Icc.2 \u27e8le_trans ((le_add_iff_nonneg_right _).2 (Nat.cast_nonneg _)) hx.1, le_trans hx.2 _\u27e9\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x\u2080 + \u2191(i + 1) \u2264 x\u2080 + \u2191a\n[PROOFSTEP]\nsimp only [Nat.cast_le, add_le_add_iff_left, ia]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x\u2080 + \u2191(i + 1) \u2208 Icc x\u2080 (x\u2080 + \u2191a)\n[PROOFSTEP]\nrefine' mem_Icc.2 \u27e8(le_add_iff_nonneg_right _).2 (Nat.cast_nonneg _), _\u27e9\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x\u2080 + \u2191(i + 1) \u2264 x\u2080 + \u2191a\n[PROOFSTEP]\nsimp only [add_le_add_iff_left, Nat.cast_le, ia]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n\u22a2 \u2211 i in Finset.range a, \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f x = \u222b (x : \u211d) in x\u2080..x\u2080 + \u2191a, f x\n[PROOFSTEP]\nconvert intervalIntegral.sum_integral_adjacent_intervals hint\n[GOAL]\ncase h.e'_3.h.e'_5\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n\u22a2 x\u2080 = x\u2080 + \u21910\n[PROOFSTEP]\nsimp only [Nat.cast_zero, add_zero]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u2211 i in Finset.Ico a b, f \u2191(i + 1) \u2264 \u222b (x : \u211d) in \u2191a..\u2191b, f x\n[PROOFSTEP]\nrw [(Nat.sub_add_cancel hab).symm, Nat.cast_add]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u2211 i in Finset.Ico a (b - a + a), f \u2191(i + 1) \u2264 \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x\n[PROOFSTEP]\nconv =>\n congr\n congr\n congr\n rw [\u2190 zero_add a]\n \u00b7skip\n \u00b7skip\n \u00b7skip\n rw [add_comm]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2211 i in Finset.Ico a (b - a + a), f \u2191(i + 1) \u2264 \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x\n[PROOFSTEP]\n congr\n congr\n congr\n rw [\u2190 zero_add a]\n \u00b7skip\n \u00b7skip\n \u00b7skip\n rw [add_comm]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2211 i in Finset.Ico a (b - a + a), f \u2191(i + 1) \u2264 \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x\n[PROOFSTEP]\n congr\n congr\n congr\n rw [\u2190 zero_add a]\n \u00b7skip\n \u00b7skip\n \u00b7skip\n rw [add_comm]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2211 i in Finset.Ico a (b - a + a), f \u2191(i + 1) \u2264 \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2211 i in Finset.Ico a (b - a + a), f \u2191(i + 1)\ncase a x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.s\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| Finset.Ico a (b - a + a)\ncase a.f\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| fun i => f \u2191(i + 1)\ncase a x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.s.a\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| a\ncase a.s.b\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| b - a + a\ncase a.f\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| fun i => f \u2191(i + 1)\ncase a x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x\n[PROOFSTEP]\nrw [\u2190 zero_add a]\n[GOAL]\ncase a.s.a\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| 0 + a\ncase a.s.b\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| b - a + a\ncase a.f\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| fun i => f \u2191(i + 1)\ncase a x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a.s.a\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| 0 + a\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.s.a\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| 0 + a\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.s.a\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| 0 + a\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.s.b\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| b - a + a\ncase a.f\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| fun i => f \u2191(i + 1)\ncase a x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a.s.b\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| b - a + a\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.s.b\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| b - a + a\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.s.b\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| b - a + a\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.f\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| fun i => f \u2191(i + 1)\ncase a x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a.f\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| fun i => f \u2191(i + 1)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.f\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| fun i => f \u2191(i + 1)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.f\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| fun i => f \u2191(i + 1)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u2211 i in Finset.Ico (0 + a) (b - a + a), f \u2191(i + 1) \u2264 \u222b (x : \u211d) in \u2191a..\u2191a + \u2191(b - a), f x\n[PROOFSTEP]\nrw [\u2190 Finset.sum_Ico_add, Nat.Ico_zero_eq_range]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u2211 x in Finset.range (b - a), f \u2191(a + x + 1) \u2264 \u222b (x : \u211d) in \u2191a..\u2191a + \u2191(b - a), f x\n[PROOFSTEP]\nconv =>\n lhs\n congr\n congr\n \u00b7skip\n ext\n rw [add_assoc, Nat.cast_add]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2211 x in Finset.range (b - a), f \u2191(a + x + 1) \u2264 \u222b (x : \u211d) in \u2191a..\u2191a + \u2191(b - a), f x\n[PROOFSTEP]\n lhs\n congr\n congr\n \u00b7skip\n ext\n rw [add_assoc, Nat.cast_add]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2211 x in Finset.range (b - a), f \u2191(a + x + 1) \u2264 \u222b (x : \u211d) in \u2191a..\u2191a + \u2191(b - a), f x\n[PROOFSTEP]\n lhs\n congr\n congr\n \u00b7skip\n ext\n rw [add_assoc, Nat.cast_add]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2211 x in Finset.range (b - a), f \u2191(a + x + 1) \u2264 \u222b (x : \u211d) in \u2191a..\u2191a + \u2191(b - a), f x\n[PROOFSTEP]\nlhs\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| \u2211 x in Finset.range (b - a), f \u2191(a + x + 1)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase s\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| Finset.range (b - a)\ncase f x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | fun x => f \u2191(a + x + 1)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase s.n\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| b - a\ncase f x\u2080 : \u211d a b : \u2115 f : \u211d \u2192 \u211d hab : a \u2264 b hf : AntitoneOn f (Icc \u2191a \u2191b) | fun x => f \u2191(a + x + 1)\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase s.n\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| b - a\n[PROOFSTEP]\nskip\n[GOAL]\ncase s.n\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| b - a\n[PROOFSTEP]\nskip\n[GOAL]\ncase s.n\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| b - a\n[PROOFSTEP]\nskip\n[GOAL]\ncase f\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n| fun x => f \u2191(a + x + 1)\n[PROOFSTEP]\next\n[GOAL]\ncase f.h\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\nx\u271d : \u2115\n| f \u2191(a + x\u271d + 1)\n[PROOFSTEP]\nrw [add_assoc, Nat.cast_add]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u2211 x in Finset.range (b - a), f (\u2191a + \u2191(x + 1)) \u2264 \u222b (x : \u211d) in \u2191a..\u2191a + \u2191(b - a), f x\n[PROOFSTEP]\napply AntitoneOn.sum_le_integral\n[GOAL]\ncase hf\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 AntitoneOn f (Icc (\u2191a) (\u2191a + \u2191(b - a)))\n[PROOFSTEP]\nsimp only [hf, hab, Nat.cast_sub, add_sub_cancel'_right]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : MonotoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\n\u22a2 \u2211 i in Finset.range a, f (x\u2080 + \u2191i) \u2264 \u222b (x : \u211d) in x\u2080..x\u2080 + \u2191a, f x\n[PROOFSTEP]\nrw [\u2190 neg_le_neg_iff, \u2190 Finset.sum_neg_distrib, \u2190 intervalIntegral.integral_neg]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : MonotoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\n\u22a2 \u222b (x : \u211d) in x\u2080..x\u2080 + \u2191a, -f x \u2264 \u2211 x in Finset.range a, -f (x\u2080 + \u2191x)\n[PROOFSTEP]\nexact hf.neg.integral_le_sum\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : MonotoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u2211 x in Finset.Ico a b, f \u2191x \u2264 \u222b (x : \u211d) in \u2191a..\u2191b, f x\n[PROOFSTEP]\nrw [\u2190 neg_le_neg_iff, \u2190 Finset.sum_neg_distrib, \u2190 intervalIntegral.integral_neg]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : MonotoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u222b (x : \u211d) in \u2191a..\u2191b, -f x \u2264 \u2211 x in Finset.Ico a b, -f \u2191x\n[PROOFSTEP]\nexact hf.neg.integral_le_sum_Ico hab\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : MonotoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\n\u22a2 \u222b (x : \u211d) in x\u2080..x\u2080 + \u2191a, f x \u2264 \u2211 i in Finset.range a, f (x\u2080 + \u2191(i + 1))\n[PROOFSTEP]\nrw [\u2190 neg_le_neg_iff, \u2190 Finset.sum_neg_distrib, \u2190 intervalIntegral.integral_neg]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : MonotoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\n\u22a2 \u2211 x in Finset.range a, -f (x\u2080 + \u2191(x + 1)) \u2264 \u222b (x : \u211d) in x\u2080..x\u2080 + \u2191a, -f x\n[PROOFSTEP]\nexact hf.neg.sum_le_integral\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : MonotoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u222b (x : \u211d) in \u2191a..\u2191b, f x \u2264 \u2211 i in Finset.Ico a b, f \u2191(i + 1)\n[PROOFSTEP]\nrw [\u2190 neg_le_neg_iff, \u2190 Finset.sum_neg_distrib, \u2190 intervalIntegral.integral_neg]\n[GOAL]\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : MonotoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u2211 x in Finset.Ico a b, -f \u2191(x + 1) \u2264 \u222b (x : \u211d) in \u2191a..\u2191b, -f x\n[PROOFSTEP]\nexact hf.neg.sum_le_integral_Ico hab\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SumIntegralComparisons", "llama_tokens": 18350, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943822145998, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.43473027436533}} {"text": "[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM\u2082 : Type u_3\nM\u2081 : \u03b9 \u2192 Type u_4\ninst\u271d\u2079 : Finite \u03b9\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : Module.Finite R M\u2082\ninst\u271d\u2074 : Module.Free R M\u2082\nn : \u2115\nN : Fin n \u2192 Type u_5\ninst\u271d\u00b3 : (i : Fin n) \u2192 AddCommGroup (N i)\ninst\u271d\u00b2 : (i : Fin n) \u2192 Module R (N i)\ninst\u271d\u00b9 : \u2200 (i : Fin n), Module.Finite R (N i)\ninst\u271d : \u2200 (i : Fin n), Module.Free R (N i)\n\u22a2 Module.Free R (MultilinearMap R N M\u2082) \u2227 Module.Finite R (MultilinearMap R N M\u2082)\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b9 : Type u_1\nR : Type u_2\nM\u2082 : Type u_3\nM\u2081 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : Finite \u03b9\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2082\ninst\u271d\u00b9\u2070 : Module R M\u2082\ninst\u271d\u2079 : Module.Finite R M\u2082\ninst\u271d\u2078 : Module.Free R M\u2082\nn : \u2115\nN\u271d : Fin n \u2192 Type u_5\ninst\u271d\u2077 : (i : Fin n) \u2192 AddCommGroup (N\u271d i)\ninst\u271d\u2076 : (i : Fin n) \u2192 Module R (N\u271d i)\ninst\u271d\u2075 : \u2200 (i : Fin n), Module.Finite R (N\u271d i)\ninst\u271d\u2074 : \u2200 (i : Fin n), Module.Free R (N\u271d i)\nN : Fin Nat.zero \u2192 Type u_5\ninst\u271d\u00b3 : (i : Fin Nat.zero) \u2192 AddCommGroup (N i)\ninst\u271d\u00b2 : (i : Fin Nat.zero) \u2192 Module R (N i)\ninst\u271d\u00b9 : \u2200 (i : Fin Nat.zero), Module.Finite R (N i)\ninst\u271d : \u2200 (i : Fin Nat.zero), Module.Free R (N i)\n\u22a2 Module.Free R (MultilinearMap R N M\u2082) \u2227 Module.Finite R (MultilinearMap R N M\u2082)\n[PROOFSTEP]\nhaveI : IsEmpty (Fin Nat.zero) := inferInstanceAs (IsEmpty (Fin 0))\n[GOAL]\ncase zero\n\u03b9 : Type u_1\nR : Type u_2\nM\u2082 : Type u_3\nM\u2081 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : Finite \u03b9\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2082\ninst\u271d\u00b9\u2070 : Module R M\u2082\ninst\u271d\u2079 : Module.Finite R M\u2082\ninst\u271d\u2078 : Module.Free R M\u2082\nn : \u2115\nN\u271d : Fin n \u2192 Type u_5\ninst\u271d\u2077 : (i : Fin n) \u2192 AddCommGroup (N\u271d i)\ninst\u271d\u2076 : (i : Fin n) \u2192 Module R (N\u271d i)\ninst\u271d\u2075 : \u2200 (i : Fin n), Module.Finite R (N\u271d i)\ninst\u271d\u2074 : \u2200 (i : Fin n), Module.Free R (N\u271d i)\nN : Fin Nat.zero \u2192 Type u_5\ninst\u271d\u00b3 : (i : Fin Nat.zero) \u2192 AddCommGroup (N i)\ninst\u271d\u00b2 : (i : Fin Nat.zero) \u2192 Module R (N i)\ninst\u271d\u00b9 : \u2200 (i : Fin Nat.zero), Module.Finite R (N i)\ninst\u271d : \u2200 (i : Fin Nat.zero), Module.Free R (N i)\nthis : IsEmpty (Fin Nat.zero)\n\u22a2 Module.Free R (MultilinearMap R N M\u2082) \u2227 Module.Finite R (MultilinearMap R N M\u2082)\n[PROOFSTEP]\nexact \u27e8Module.Free.of_equiv (constLinearEquivOfIsEmpty R N M\u2082), Module.Finite.equiv (constLinearEquivOfIsEmpty R N M\u2082)\u27e9\n[GOAL]\ncase succ\n\u03b9 : Type u_1\nR : Type u_2\nM\u2082 : Type u_3\nM\u2081 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : Finite \u03b9\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2082\ninst\u271d\u00b9\u2070 : Module R M\u2082\ninst\u271d\u2079 : Module.Finite R M\u2082\ninst\u271d\u2078 : Module.Free R M\u2082\nn\u271d : \u2115\nN\u271d : Fin n\u271d \u2192 Type u_5\ninst\u271d\u2077 : (i : Fin n\u271d) \u2192 AddCommGroup (N\u271d i)\ninst\u271d\u2076 : (i : Fin n\u271d) \u2192 Module R (N\u271d i)\ninst\u271d\u2075 : \u2200 (i : Fin n\u271d), Module.Finite R (N\u271d i)\ninst\u271d\u2074 : \u2200 (i : Fin n\u271d), Module.Free R (N\u271d i)\nn : \u2115\nih :\n \u2200 (N : Fin n \u2192 Type u_5) [inst : (i : Fin n) \u2192 AddCommGroup (N i)] [inst_1 : (i : Fin n) \u2192 Module R (N i)]\n [inst_2 : \u2200 (i : Fin n), Module.Finite R (N i)] [inst_3 : \u2200 (i : Fin n), Module.Free R (N i)],\n Module.Free R (MultilinearMap R N M\u2082) \u2227 Module.Finite R (MultilinearMap R N M\u2082)\nN : Fin (Nat.succ n) \u2192 Type u_5\ninst\u271d\u00b3 : (i : Fin (Nat.succ n)) \u2192 AddCommGroup (N i)\ninst\u271d\u00b2 : (i : Fin (Nat.succ n)) \u2192 Module R (N i)\ninst\u271d\u00b9 : \u2200 (i : Fin (Nat.succ n)), Module.Finite R (N i)\ninst\u271d : \u2200 (i : Fin (Nat.succ n)), Module.Free R (N i)\n\u22a2 Module.Free R (MultilinearMap R N M\u2082) \u2227 Module.Finite R (MultilinearMap R N M\u2082)\n[PROOFSTEP]\nsuffices\n Module.Free R (N 0 \u2192\u2097[R] MultilinearMap R (fun i : Fin n => N i.succ) M\u2082) \u2227\n Module.Finite R (N 0 \u2192\u2097[R] MultilinearMap R (fun i : Fin n => N i.succ) M\u2082)\n by\n cases this\n exact\n \u27e8Module.Free.of_equiv (multilinearCurryLeftEquiv R N M\u2082), Module.Finite.equiv (multilinearCurryLeftEquiv R N M\u2082)\u27e9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM\u2082 : Type u_3\nM\u2081 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : Finite \u03b9\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2082\ninst\u271d\u00b9\u2070 : Module R M\u2082\ninst\u271d\u2079 : Module.Finite R M\u2082\ninst\u271d\u2078 : Module.Free R M\u2082\nn\u271d : \u2115\nN\u271d : Fin n\u271d \u2192 Type u_5\ninst\u271d\u2077 : (i : Fin n\u271d) \u2192 AddCommGroup (N\u271d i)\ninst\u271d\u2076 : (i : Fin n\u271d) \u2192 Module R (N\u271d i)\ninst\u271d\u2075 : \u2200 (i : Fin n\u271d), Module.Finite R (N\u271d i)\ninst\u271d\u2074 : \u2200 (i : Fin n\u271d), Module.Free R (N\u271d i)\nn : \u2115\nih :\n \u2200 (N : Fin n \u2192 Type u_5) [inst : (i : Fin n) \u2192 AddCommGroup (N i)] [inst_1 : (i : Fin n) \u2192 Module R (N i)]\n [inst_2 : \u2200 (i : Fin n), Module.Finite R (N i)] [inst_3 : \u2200 (i : Fin n), Module.Free R (N i)],\n Module.Free R (MultilinearMap R N M\u2082) \u2227 Module.Finite R (MultilinearMap R N M\u2082)\nN : Fin (Nat.succ n) \u2192 Type u_5\ninst\u271d\u00b3 : (i : Fin (Nat.succ n)) \u2192 AddCommGroup (N i)\ninst\u271d\u00b2 : (i : Fin (Nat.succ n)) \u2192 Module R (N i)\ninst\u271d\u00b9 : \u2200 (i : Fin (Nat.succ n)), Module.Finite R (N i)\ninst\u271d : \u2200 (i : Fin (Nat.succ n)), Module.Free R (N i)\nthis :\n Module.Free R (N 0 \u2192\u2097[R] MultilinearMap R (fun i => N (Fin.succ i)) M\u2082) \u2227\n Module.Finite R (N 0 \u2192\u2097[R] MultilinearMap R (fun i => N (Fin.succ i)) M\u2082)\n\u22a2 Module.Free R (MultilinearMap R N M\u2082) \u2227 Module.Finite R (MultilinearMap R N M\u2082)\n[PROOFSTEP]\ncases this\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\nM\u2082 : Type u_3\nM\u2081 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : Finite \u03b9\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2082\ninst\u271d\u00b9\u2070 : Module R M\u2082\ninst\u271d\u2079 : Module.Finite R M\u2082\ninst\u271d\u2078 : Module.Free R M\u2082\nn\u271d : \u2115\nN\u271d : Fin n\u271d \u2192 Type u_5\ninst\u271d\u2077 : (i : Fin n\u271d) \u2192 AddCommGroup (N\u271d i)\ninst\u271d\u2076 : (i : Fin n\u271d) \u2192 Module R (N\u271d i)\ninst\u271d\u2075 : \u2200 (i : Fin n\u271d), Module.Finite R (N\u271d i)\ninst\u271d\u2074 : \u2200 (i : Fin n\u271d), Module.Free R (N\u271d i)\nn : \u2115\nih :\n \u2200 (N : Fin n \u2192 Type u_5) [inst : (i : Fin n) \u2192 AddCommGroup (N i)] [inst_1 : (i : Fin n) \u2192 Module R (N i)]\n [inst_2 : \u2200 (i : Fin n), Module.Finite R (N i)] [inst_3 : \u2200 (i : Fin n), Module.Free R (N i)],\n Module.Free R (MultilinearMap R N M\u2082) \u2227 Module.Finite R (MultilinearMap R N M\u2082)\nN : Fin (Nat.succ n) \u2192 Type u_5\ninst\u271d\u00b3 : (i : Fin (Nat.succ n)) \u2192 AddCommGroup (N i)\ninst\u271d\u00b2 : (i : Fin (Nat.succ n)) \u2192 Module R (N i)\ninst\u271d\u00b9 : \u2200 (i : Fin (Nat.succ n)), Module.Finite R (N i)\ninst\u271d : \u2200 (i : Fin (Nat.succ n)), Module.Free R (N i)\nleft\u271d : Module.Free R (N 0 \u2192\u2097[R] MultilinearMap R (fun i => N (Fin.succ i)) M\u2082)\nright\u271d : Module.Finite R (N 0 \u2192\u2097[R] MultilinearMap R (fun i => N (Fin.succ i)) M\u2082)\n\u22a2 Module.Free R (MultilinearMap R N M\u2082) \u2227 Module.Finite R (MultilinearMap R N M\u2082)\n[PROOFSTEP]\nexact \u27e8Module.Free.of_equiv (multilinearCurryLeftEquiv R N M\u2082), Module.Finite.equiv (multilinearCurryLeftEquiv R N M\u2082)\u27e9\n[GOAL]\ncase succ\n\u03b9 : Type u_1\nR : Type u_2\nM\u2082 : Type u_3\nM\u2081 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : Finite \u03b9\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2082\ninst\u271d\u00b9\u2070 : Module R M\u2082\ninst\u271d\u2079 : Module.Finite R M\u2082\ninst\u271d\u2078 : Module.Free R M\u2082\nn\u271d : \u2115\nN\u271d : Fin n\u271d \u2192 Type u_5\ninst\u271d\u2077 : (i : Fin n\u271d) \u2192 AddCommGroup (N\u271d i)\ninst\u271d\u2076 : (i : Fin n\u271d) \u2192 Module R (N\u271d i)\ninst\u271d\u2075 : \u2200 (i : Fin n\u271d), Module.Finite R (N\u271d i)\ninst\u271d\u2074 : \u2200 (i : Fin n\u271d), Module.Free R (N\u271d i)\nn : \u2115\nih :\n \u2200 (N : Fin n \u2192 Type u_5) [inst : (i : Fin n) \u2192 AddCommGroup (N i)] [inst_1 : (i : Fin n) \u2192 Module R (N i)]\n [inst_2 : \u2200 (i : Fin n), Module.Finite R (N i)] [inst_3 : \u2200 (i : Fin n), Module.Free R (N i)],\n Module.Free R (MultilinearMap R N M\u2082) \u2227 Module.Finite R (MultilinearMap R N M\u2082)\nN : Fin (Nat.succ n) \u2192 Type u_5\ninst\u271d\u00b3 : (i : Fin (Nat.succ n)) \u2192 AddCommGroup (N i)\ninst\u271d\u00b2 : (i : Fin (Nat.succ n)) \u2192 Module R (N i)\ninst\u271d\u00b9 : \u2200 (i : Fin (Nat.succ n)), Module.Finite R (N i)\ninst\u271d : \u2200 (i : Fin (Nat.succ n)), Module.Free R (N i)\n\u22a2 Module.Free R (N 0 \u2192\u2097[R] MultilinearMap R (fun i => N (Fin.succ i)) M\u2082) \u2227\n Module.Finite R (N 0 \u2192\u2097[R] MultilinearMap R (fun i => N (Fin.succ i)) M\u2082)\n[PROOFSTEP]\ncases ih fun i => N i.succ\n[GOAL]\ncase succ.intro\n\u03b9 : Type u_1\nR : Type u_2\nM\u2082 : Type u_3\nM\u2081 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : Finite \u03b9\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2082\ninst\u271d\u00b9\u2070 : Module R M\u2082\ninst\u271d\u2079 : Module.Finite R M\u2082\ninst\u271d\u2078 : Module.Free R M\u2082\nn\u271d : \u2115\nN\u271d : Fin n\u271d \u2192 Type u_5\ninst\u271d\u2077 : (i : Fin n\u271d) \u2192 AddCommGroup (N\u271d i)\ninst\u271d\u2076 : (i : Fin n\u271d) \u2192 Module R (N\u271d i)\ninst\u271d\u2075 : \u2200 (i : Fin n\u271d), Module.Finite R (N\u271d i)\ninst\u271d\u2074 : \u2200 (i : Fin n\u271d), Module.Free R (N\u271d i)\nn : \u2115\nih :\n \u2200 (N : Fin n \u2192 Type u_5) [inst : (i : Fin n) \u2192 AddCommGroup (N i)] [inst_1 : (i : Fin n) \u2192 Module R (N i)]\n [inst_2 : \u2200 (i : Fin n), Module.Finite R (N i)] [inst_3 : \u2200 (i : Fin n), Module.Free R (N i)],\n Module.Free R (MultilinearMap R N M\u2082) \u2227 Module.Finite R (MultilinearMap R N M\u2082)\nN : Fin (Nat.succ n) \u2192 Type u_5\ninst\u271d\u00b3 : (i : Fin (Nat.succ n)) \u2192 AddCommGroup (N i)\ninst\u271d\u00b2 : (i : Fin (Nat.succ n)) \u2192 Module R (N i)\ninst\u271d\u00b9 : \u2200 (i : Fin (Nat.succ n)), Module.Finite R (N i)\ninst\u271d : \u2200 (i : Fin (Nat.succ n)), Module.Free R (N i)\nleft\u271d : Module.Free R (MultilinearMap R (fun i => N (Fin.succ i)) M\u2082)\nright\u271d : Module.Finite R (MultilinearMap R (fun i => N (Fin.succ i)) M\u2082)\n\u22a2 Module.Free R (N 0 \u2192\u2097[R] MultilinearMap R (fun i => N (Fin.succ i)) M\u2082) \u2227\n Module.Finite R (N 0 \u2192\u2097[R] MultilinearMap R (fun i => N (Fin.succ i)) M\u2082)\n[PROOFSTEP]\nexact \u27e8Module.Free.linearMap _ _ _, Module.Finite.linearMap _ _\u27e9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM\u2082 : Type u_3\nM\u2081 : \u03b9 \u2192 Type u_4\ninst\u271d\u2079 : Finite \u03b9\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : Module.Finite R M\u2082\ninst\u271d\u2074 : Module.Free R M\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (M\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), Module.Finite R (M\u2081 i)\ninst\u271d : \u2200 (i : \u03b9), Module.Free R (M\u2081 i)\n\u22a2 Module.Free R (MultilinearMap R M\u2081 M\u2082) \u2227 Module.Finite R (MultilinearMap R M\u2081 M\u2082)\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\nM\u2082 : Type u_3\nM\u2081 : \u03b9 \u2192 Type u_4\ninst\u271d\u2079 : Finite \u03b9\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : Module.Finite R M\u2082\ninst\u271d\u2074 : Module.Free R M\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (M\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), Module.Finite R (M\u2081 i)\ninst\u271d : \u2200 (i : \u03b9), Module.Free R (M\u2081 i)\nval\u271d : Fintype \u03b9\n\u22a2 Module.Free R (MultilinearMap R M\u2081 M\u2082) \u2227 Module.Finite R (MultilinearMap R M\u2081 M\u2082)\n[PROOFSTEP]\nhave := @free_and_finite_fin R M\u2082 _ _ _ _ _ (Fintype.card \u03b9) (fun x => M\u2081 ((Fintype.equivFin \u03b9).symm x))\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\nM\u2082 : Type u_3\nM\u2081 : \u03b9 \u2192 Type u_4\ninst\u271d\u2079 : Finite \u03b9\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : Module.Finite R M\u2082\ninst\u271d\u2074 : Module.Free R M\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (M\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), Module.Finite R (M\u2081 i)\ninst\u271d : \u2200 (i : \u03b9), Module.Free R (M\u2081 i)\nval\u271d : Fintype \u03b9\nthis :\n \u2200 [inst : (i : Fin (Fintype.card \u03b9)) \u2192 AddCommGroup (M\u2081 (\u2191(Fintype.equivFin \u03b9).symm i))]\n [inst_1 : (i : Fin (Fintype.card \u03b9)) \u2192 Module R (M\u2081 (\u2191(Fintype.equivFin \u03b9).symm i))]\n [inst_2 : \u2200 (i : Fin (Fintype.card \u03b9)), Module.Finite R (M\u2081 (\u2191(Fintype.equivFin \u03b9).symm i))]\n [inst_3 : \u2200 (i : Fin (Fintype.card \u03b9)), Module.Free R (M\u2081 (\u2191(Fintype.equivFin \u03b9).symm i))],\n Module.Free R (MultilinearMap R (fun x => M\u2081 (\u2191(Fintype.equivFin \u03b9).symm x)) M\u2082) \u2227\n Module.Finite R (MultilinearMap R (fun x => M\u2081 (\u2191(Fintype.equivFin \u03b9).symm x)) M\u2082)\n\u22a2 Module.Free R (MultilinearMap R M\u2081 M\u2082) \u2227 Module.Finite R (MultilinearMap R M\u2081 M\u2082)\n[PROOFSTEP]\ncases' this with l r\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nR : Type u_2\nM\u2082 : Type u_3\nM\u2081 : \u03b9 \u2192 Type u_4\ninst\u271d\u2079 : Finite \u03b9\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : Module.Finite R M\u2082\ninst\u271d\u2074 : Module.Free R M\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (M\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), Module.Finite R (M\u2081 i)\ninst\u271d : \u2200 (i : \u03b9), Module.Free R (M\u2081 i)\nval\u271d : Fintype \u03b9\nthis :\n \u2200 [inst : (i : Fin (Fintype.card \u03b9)) \u2192 AddCommGroup (M\u2081 (\u2191(Fintype.equivFin \u03b9).symm i))]\n [inst_1 : (i : Fin (Fintype.card \u03b9)) \u2192 Module R (M\u2081 (\u2191(Fintype.equivFin \u03b9).symm i))]\n [inst_2 : \u2200 (i : Fin (Fintype.card \u03b9)), Module.Finite R (M\u2081 (\u2191(Fintype.equivFin \u03b9).symm i))]\n [inst_3 : \u2200 (i : Fin (Fintype.card \u03b9)), Module.Free R (M\u2081 (\u2191(Fintype.equivFin \u03b9).symm i))],\n Module.Free R (MultilinearMap R (fun x => M\u2081 (\u2191(Fintype.equivFin \u03b9).symm x)) M\u2082) \u2227\n Module.Finite R (MultilinearMap R (fun x => M\u2081 (\u2191(Fintype.equivFin \u03b9).symm x)) M\u2082)\nl : Module.Free R (MultilinearMap R (fun x => M\u2081 (\u2191(Fintype.equivFin \u03b9).symm x)) M\u2082)\nr : Module.Finite R (MultilinearMap R (fun x => M\u2081 (\u2191(Fintype.equivFin \u03b9).symm x)) M\u2082)\n\u22a2 Module.Free R (MultilinearMap R M\u2081 M\u2082) \u2227 Module.Finite R (MultilinearMap R M\u2081 M\u2082)\n[PROOFSTEP]\nhave e := domDomCongrLinearEquiv' R M\u2081 M\u2082 (Fintype.equivFin \u03b9)\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nR : Type u_2\nM\u2082 : Type u_3\nM\u2081 : \u03b9 \u2192 Type u_4\ninst\u271d\u2079 : Finite \u03b9\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : Module.Finite R M\u2082\ninst\u271d\u2074 : Module.Free R M\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (M\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), Module.Finite R (M\u2081 i)\ninst\u271d : \u2200 (i : \u03b9), Module.Free R (M\u2081 i)\nval\u271d : Fintype \u03b9\nthis :\n \u2200 [inst : (i : Fin (Fintype.card \u03b9)) \u2192 AddCommGroup (M\u2081 (\u2191(Fintype.equivFin \u03b9).symm i))]\n [inst_1 : (i : Fin (Fintype.card \u03b9)) \u2192 Module R (M\u2081 (\u2191(Fintype.equivFin \u03b9).symm i))]\n [inst_2 : \u2200 (i : Fin (Fintype.card \u03b9)), Module.Finite R (M\u2081 (\u2191(Fintype.equivFin \u03b9).symm i))]\n [inst_3 : \u2200 (i : Fin (Fintype.card \u03b9)), Module.Free R (M\u2081 (\u2191(Fintype.equivFin \u03b9).symm i))],\n Module.Free R (MultilinearMap R (fun x => M\u2081 (\u2191(Fintype.equivFin \u03b9).symm x)) M\u2082) \u2227\n Module.Finite R (MultilinearMap R (fun x => M\u2081 (\u2191(Fintype.equivFin \u03b9).symm x)) M\u2082)\nl : Module.Free R (MultilinearMap R (fun x => M\u2081 (\u2191(Fintype.equivFin \u03b9).symm x)) M\u2082)\nr : Module.Finite R (MultilinearMap R (fun x => M\u2081 (\u2191(Fintype.equivFin \u03b9).symm x)) M\u2082)\ne : MultilinearMap R M\u2081 M\u2082 \u2243\u2097[R] MultilinearMap R (fun i => M\u2081 (\u2191(Fintype.equivFin \u03b9).symm i)) M\u2082\n\u22a2 Module.Free R (MultilinearMap R M\u2081 M\u2082) \u2227 Module.Finite R (MultilinearMap R M\u2081 M\u2082)\n[PROOFSTEP]\nexact \u27e8Module.Free.of_equiv e.symm, Module.Finite.equiv e.symm\u27e9\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Multilinear.FiniteDimensional", "llama_tokens": 6830, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744673038221, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.4343032204628459}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.1346, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\n\u22a2 N.map (NatTrans.app \u0393\u2082N\u2082.hom P) \u226b NatTrans.app N\u2082\u0393\u2082.hom (N.obj P) = \ud835\udfd9 (N.obj P)\n[PROOFSTEP]\nlet \u03b1 := N.mapIso (\u0393\u2082N\u2082.app P)\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.1346, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\n\u03b1 : N.obj ((\ud835\udfed (Karoubi (SimplicialObject C))).obj P) \u2245 N.obj ((N\u2082 \u22d9 \u0393\u2082).obj P) := N.mapIso (\u0393\u2082N\u2082.app P)\n\u22a2 N.map (NatTrans.app \u0393\u2082N\u2082.hom P) \u226b NatTrans.app N\u2082\u0393\u2082.hom (N.obj P) = \ud835\udfd9 (N.obj P)\n[PROOFSTEP]\nlet \u03b2 := N\u2082\u0393\u2082.app (N.obj P)\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.1346, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\n\u03b1 : N.obj ((\ud835\udfed (Karoubi (SimplicialObject C))).obj P) \u2245 N.obj ((N\u2082 \u22d9 \u0393\u2082).obj P) := N.mapIso (\u0393\u2082N\u2082.app P)\n\u03b2 : (\u0393\u2082 \u22d9 N\u2082).obj (N.obj P) \u2245 (\ud835\udfed (Karoubi (ChainComplex C \u2115))).obj (N.obj P) := N\u2082\u0393\u2082.app (N.obj P)\n\u22a2 N.map (NatTrans.app \u0393\u2082N\u2082.hom P) \u226b NatTrans.app N\u2082\u0393\u2082.hom (N.obj P) = \ud835\udfd9 (N.obj P)\n[PROOFSTEP]\nsymm\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.1346, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\n\u03b1 : N.obj ((\ud835\udfed (Karoubi (SimplicialObject C))).obj P) \u2245 N.obj ((N\u2082 \u22d9 \u0393\u2082).obj P) := N.mapIso (\u0393\u2082N\u2082.app P)\n\u03b2 : (\u0393\u2082 \u22d9 N\u2082).obj (N.obj P) \u2245 (\ud835\udfed (Karoubi (ChainComplex C \u2115))).obj (N.obj P) := N\u2082\u0393\u2082.app (N.obj P)\n\u22a2 \ud835\udfd9 (N.obj P) = N.map (NatTrans.app \u0393\u2082N\u2082.hom P) \u226b NatTrans.app N\u2082\u0393\u2082.hom (N.obj P)\n[PROOFSTEP]\nchange \ud835\udfd9 _ = \u03b1.hom \u226b \u03b2.hom\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.1346, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\n\u03b1 : N.obj ((\ud835\udfed (Karoubi (SimplicialObject C))).obj P) \u2245 N.obj ((N\u2082 \u22d9 \u0393\u2082).obj P) := N.mapIso (\u0393\u2082N\u2082.app P)\n\u03b2 : (\u0393\u2082 \u22d9 N\u2082).obj (N.obj P) \u2245 (\ud835\udfed (Karoubi (ChainComplex C \u2115))).obj (N.obj P) := N\u2082\u0393\u2082.app (N.obj P)\n\u22a2 \ud835\udfd9 (N.obj ((\ud835\udfed (Karoubi (SimplicialObject C))).obj P)) = \u03b1.hom \u226b \u03b2.hom\n[PROOFSTEP]\nrw [\u2190 Iso.inv_comp_eq, comp_id, \u2190 comp_id \u03b2.hom, \u2190 Iso.inv_comp_eq]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.1346, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\n\u03b1 : N.obj ((\ud835\udfed (Karoubi (SimplicialObject C))).obj P) \u2245 N.obj ((N\u2082 \u22d9 \u0393\u2082).obj P) := N.mapIso (\u0393\u2082N\u2082.app P)\n\u03b2 : (\u0393\u2082 \u22d9 N\u2082).obj (N.obj P) \u2245 (\ud835\udfed (Karoubi (ChainComplex C \u2115))).obj (N.obj P) := N\u2082\u0393\u2082.app (N.obj P)\n\u22a2 \u03b2.inv \u226b \u03b1.inv = \ud835\udfd9 ((\ud835\udfed (Karoubi (ChainComplex C \u2115))).obj (N.obj P))\n[PROOFSTEP]\nexact AlgebraicTopology.DoldKan.identity_N\u2082_objectwise P\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicTopology.DoldKan.EquivalenceAdditive", "llama_tokens": 1455, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754607093178, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.43418869443429864}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\nf : \u03b1 \u2192 \u03b2\n\u22a2 InjOn f (f \u207b\u00b9' \u2191\u2205)\n[PROOFSTEP]\nsimp [InjOn]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2191(preimage \u2205 f (_ : \u2200 (a : \u03b1), a \u2208 f \u207b\u00b9' \u2191\u2205 \u2192 \u2200 (a_2 : \u03b1), a_2 \u2208 f \u207b\u00b9' \u2191\u2205 \u2192 f a = f a_2 \u2192 a = a_2)) = \u2191\u2205\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u03b2\nhf : InjOn f (f \u207b\u00b9' \u2191univ)\n\u22a2 \u2191(preimage univ f hf) = \u2191univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns t : Finset \u03b2\nhs : InjOn f (f \u207b\u00b9' \u2191s)\nht : InjOn f (f \u207b\u00b9' \u2191t)\n\u22a2 \u2191(preimage (s \u2229 t) f\n (_ : \u2200 (x\u2081 : \u03b1), x\u2081 \u2208 f \u207b\u00b9' \u2191(s \u2229 t) \u2192 \u2200 (x\u2082 : \u03b1), x\u2082 \u2208 f \u207b\u00b9' \u2191(s \u2229 t) \u2192 f x\u2081 = f x\u2082 \u2192 x\u2081 = x\u2082)) =\n \u2191(preimage s f hs \u2229 preimage t f ht)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns t : Finset \u03b2\nhst : InjOn f (f \u207b\u00b9' \u2191(s \u222a t))\n\u22a2 \u2191(preimage (s \u222a t) f hst) =\n \u2191(preimage s f (_ : \u2200 (x\u2081 : \u03b1), x\u2081 \u2208 f \u207b\u00b9' \u2191s \u2192 \u2200 (x\u2082 : \u03b1), x\u2082 \u2208 f \u207b\u00b9' \u2191s \u2192 f x\u2081 = f x\u2082 \u2192 x\u2081 = x\u2082) \u222a\n preimage t f (_ : \u2200 (x\u2081 : \u03b1), x\u2081 \u2208 f \u207b\u00b9' \u2191t \u2192 \u2200 (x\u2082 : \u03b1), x\u2082 \u2208 f \u207b\u00b9' \u2191t \u2192 f x\u2081 = f x\u2082 \u2192 x\u2081 = x\u2082))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Finset \u03b2\nhf : Injective f\n\u22a2 \u2191(preimage s\u1d9c f (_ : InjOn f (f \u207b\u00b9' \u2191s\u1d9c))) = \u2191(preimage s f (_ : InjOn f (f \u207b\u00b9' \u2191s)))\u1d9c\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nhf : InjOn f (f \u207b\u00b9' \u2191t)\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2208 t) \u2194 s \u2286 preimage t f hf\n[PROOFSTEP]\nsimp only [subset_iff, mem_preimage]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\nf : \u03b1 \u21aa \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 map f s \u2286 t \u2194 s \u2286 preimage t \u2191f (_ : InjOn (\u2191f) (\u2191f \u207b\u00b9' \u2191t))\n[PROOFSTEP]\nclassical rw [map_eq_image, image_subset_iff_subset_preimage]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\nf : \u03b1 \u21aa \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 map f s \u2286 t \u2194 s \u2286 preimage t \u2191f (_ : InjOn (\u2191f) (\u2191f \u207b\u00b9' \u2191t))\n[PROOFSTEP]\nrw [map_eq_image, image_subset_iff_subset_preimage]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Finset \u03b2\ninst\u271d : (x : \u03b2) \u2192 Decidable (x \u2208 Set.range f)\nhf : InjOn f (f \u207b\u00b9' \u2191s)\n\u22a2 \u2191(image f (preimage s f hf)) = \u2191(filter (fun x => x \u2208 Set.range f) s)\n[PROOFSTEP]\nsimp only [coe_image, coe_preimage, coe_filter, Set.image_preimage_eq_inter_range, \u2190 Set.sep_mem_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Finset \u03b2\ninst\u271d : (x : \u03b2) \u2192 Decidable (x \u2208 Set.range f)\nhf : InjOn f (f \u207b\u00b9' \u2191s)\n\u22a2 {x | x \u2208 \u2191s \u2227 x \u2208 Set.range f} = {x | x \u2208 s \u2227 x \u2208 Set.range f}\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Finset \u03b2\nhf : BijOn f (f \u207b\u00b9' \u2191s) \u2191s\n\u22a2 \u2191(image f (preimage s f (_ : InjOn f (f \u207b\u00b9' \u2191s)))) = \u2191s\n[PROOFSTEP]\nsimpa using hf.image_eq\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\nf : \u03b1 \u21aa \u03b2\ns : Finset \u03b2\nt : Finset \u03b1\n\u22a2 s \u2286 map f t \u2194 \u2203 u x, s = map f u\n[PROOFSTEP]\nclassical\nrefine' \u27e8fun h => \u27e8_, preimage_subset h, _\u27e9, _\u27e9\n\u00b7 rw [map_eq_image, image_preimage, filter_true_of_mem]\n exact fun x hx \u21a6 coe_map_subset_range _ _ (h hx)\n\u00b7 rintro \u27e8u, hut, rfl\u27e9\n exact map_subset_map.2 hut\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\nf : \u03b1 \u21aa \u03b2\ns : Finset \u03b2\nt : Finset \u03b1\n\u22a2 s \u2286 map f t \u2194 \u2203 u x, s = map f u\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8_, preimage_subset h, _\u27e9, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\nf : \u03b1 \u21aa \u03b2\ns : Finset \u03b2\nt : Finset \u03b1\nh : s \u2286 map f t\n\u22a2 s = map f (preimage s \u2191f (_ : InjOn (\u2191f) (\u2191f \u207b\u00b9' \u2191s)))\n[PROOFSTEP]\nrw [map_eq_image, image_preimage, filter_true_of_mem]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\nf : \u03b1 \u21aa \u03b2\ns : Finset \u03b2\nt : Finset \u03b1\nh : s \u2286 map f t\n\u22a2 \u2200 (x : \u03b2), x \u2208 s \u2192 x \u2208 Set.range \u2191f\n[PROOFSTEP]\nexact fun x hx \u21a6 coe_map_subset_range _ _ (h hx)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\nf : \u03b1 \u21aa \u03b2\ns : Finset \u03b2\nt : Finset \u03b1\n\u22a2 (\u2203 u x, s = map f u) \u2192 s \u2286 map f t\n[PROOFSTEP]\nrintro \u27e8u, hut, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\nf : \u03b1 \u21aa \u03b2\nt u : Finset \u03b1\nhut : u \u2286 t\n\u22a2 map f u \u2286 map f t\n[PROOFSTEP]\nexact map_subset_map.2 hut\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Finset ((a : \u03b1) \u00d7 \u03b2 a)\nt : Finset \u03b1\n\u22a2 (Finset.sigma t fun a => preimage s (Sigma.mk a) (_ : InjOn (Sigma.mk a) (Sigma.mk a \u207b\u00b9' \u2191s))) =\n filter (fun a => a.fst \u2208 t) s\n[PROOFSTEP]\next x\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Finset ((a : \u03b1) \u00d7 \u03b2 a)\nt : Finset \u03b1\nx : (i : \u03b1) \u00d7 \u03b2 i\n\u22a2 (x \u2208 Finset.sigma t fun a => preimage s (Sigma.mk a) (_ : InjOn (Sigma.mk a) (Sigma.mk a \u207b\u00b9' \u2191s))) \u2194\n x \u2208 filter (fun a => a.fst \u2208 t) s\n[PROOFSTEP]\nsimp [and_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Finset ((a : \u03b1) \u00d7 \u03b2 a)\nt : Finset \u03b1\nht : image Sigma.fst s \u2286 t\n\u22a2 (Finset.sigma t fun a => preimage s (Sigma.mk a) (_ : InjOn (Sigma.mk a) (Sigma.mk a \u207b\u00b9' \u2191s))) = s\n[PROOFSTEP]\nrw [sigma_preimage_mk, filter_true_of_mem <| image_subset_iff.1 ht]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : CommMonoid \u03b2\nf : \u03b1 \u2192 \u03b3\ninst\u271d : DecidablePred fun x => x \u2208 Set.range f\ns : Finset \u03b3\nhf : InjOn f (f \u207b\u00b9' \u2191s)\ng : \u03b3 \u2192 \u03b2\n\u22a2 \u220f x in preimage s f hf, g (f x) = \u220f x in filter (fun x => x \u2208 Set.range f) s, g x\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b3\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : CommMonoid \u03b2\nf : \u03b1 \u2192 \u03b3\ninst\u271d : DecidablePred fun x => x \u2208 Set.range f\ns : Finset \u03b3\nhf : InjOn f (f \u207b\u00b9' \u2191s)\ng : \u03b3 \u2192 \u03b2\nthis : DecidableEq \u03b3\n\u22a2 \u220f x in preimage s f hf, g (f x) = \u220f x in filter (fun x => x \u2208 Set.range f) s, g x\n[PROOFSTEP]\ncalc\n (\u220f x in preimage s f hf, g (f x)) = \u220f x in image f (preimage s f hf), g x :=\n Eq.symm <| prod_image <| by simpa only [mem_preimage, InjOn] using hf\n _ = \u220f x in s.filter fun x => x \u2208 Set.range f, g x := by rw [image_preimage]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : CommMonoid \u03b2\nf : \u03b1 \u2192 \u03b3\ninst\u271d : DecidablePred fun x => x \u2208 Set.range f\ns : Finset \u03b3\nhf : InjOn f (f \u207b\u00b9' \u2191s)\ng : \u03b3 \u2192 \u03b2\nthis : DecidableEq \u03b3\n\u22a2 \u2200 (x : \u03b1), x \u2208 preimage s f hf \u2192 \u2200 (y : \u03b1), y \u2208 preimage s f hf \u2192 f x = f y \u2192 x = y\n[PROOFSTEP]\nsimpa only [mem_preimage, InjOn] using hf\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d\u00b9 : CommMonoid \u03b2\nf : \u03b1 \u2192 \u03b3\ninst\u271d : DecidablePred fun x => x \u2208 Set.range f\ns : Finset \u03b3\nhf : InjOn f (f \u207b\u00b9' \u2191s)\ng : \u03b3 \u2192 \u03b2\nthis : DecidableEq \u03b3\n\u22a2 \u220f x in image f (preimage s f hf), g x = \u220f x in filter (fun x => x \u2208 Set.range f) s, g x\n[PROOFSTEP]\nrw [image_preimage]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d : CommMonoid \u03b2\nf : \u03b1 \u2192 \u03b3\ns : Finset \u03b3\nhf : InjOn f (f \u207b\u00b9' \u2191s)\ng : \u03b3 \u2192 \u03b2\nhg : \u2200 (x : \u03b3), x \u2208 s \u2192 \u00acx \u2208 Set.range f \u2192 g x = 1\n\u22a2 \u220f x in preimage s f hf, g (f x) = \u220f x in s, g x\n[PROOFSTEP]\nclassical\nrw [prod_preimage', prod_filter_of_ne]\nexact fun x hx => Not.imp_symm (hg x hx)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d : CommMonoid \u03b2\nf : \u03b1 \u2192 \u03b3\ns : Finset \u03b3\nhf : InjOn f (f \u207b\u00b9' \u2191s)\ng : \u03b3 \u2192 \u03b2\nhg : \u2200 (x : \u03b3), x \u2208 s \u2192 \u00acx \u2208 Set.range f \u2192 g x = 1\n\u22a2 \u220f x in preimage s f hf, g (f x) = \u220f x in s, g x\n[PROOFSTEP]\nrw [prod_preimage', prod_filter_of_ne]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ninst\u271d : CommMonoid \u03b2\nf : \u03b1 \u2192 \u03b3\ns : Finset \u03b3\nhf : InjOn f (f \u207b\u00b9' \u2191s)\ng : \u03b3 \u2192 \u03b2\nhg : \u2200 (x : \u03b3), x \u2208 s \u2192 \u00acx \u2208 Set.range f \u2192 g x = 1\n\u22a2 \u2200 (x : \u03b3), x \u2208 s \u2192 g x \u2260 1 \u2192 x \u2208 Set.range f\n[PROOFSTEP]\nexact fun x hx => Not.imp_symm (hg x hx)\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.Preimage", "llama_tokens": 4148, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.43418868739439515}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ns : Set \u03b1\nt : \u03b9 \u2192 Set \u03b2\nh\u03b9 : Nonempty \u03b9\n\u22a2 s \u00d7\u02e2 \u22c2 (i : \u03b9), t i = \u22c2 (i : \u03b9), s \u00d7\u02e2 t i\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ns : Set \u03b1\nt : \u03b9 \u2192 Set \u03b2\nh\u03b9 : Nonempty \u03b9\nx : \u03b1 \u00d7 \u03b2\n\u22a2 x \u2208 s \u00d7\u02e2 \u22c2 (i : \u03b9), t i \u2194 x \u2208 \u22c2 (i : \u03b9), s \u00d7\u02e2 t i\n[PROOFSTEP]\nsimp only [mem_prod, mem_iInter]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ns : Set \u03b1\nt : \u03b9 \u2192 Set \u03b2\nh\u03b9 : Nonempty \u03b9\nx : \u03b1 \u00d7 \u03b2\n\u22a2 (x.fst \u2208 s \u2227 \u2200 (i : \u03b9), x.snd \u2208 t i) \u2194 \u2200 (i : \u03b9), x.fst \u2208 s \u2227 x.snd \u2208 t i\n[PROOFSTEP]\nexact \u27e8fun h i => \u27e8h.1, h.2 i\u27e9, fun h => \u27e8(h h\u03b9.some).1, fun i => (h i).2\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u22a2 \u22c3 (r : \u211a), Iic \u2191r = univ\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nx : \u211d\n\u22a2 x \u2208 \u22c3 (r : \u211a), Iic \u2191r \u2194 x \u2208 univ\n[PROOFSTEP]\nsimp only [mem_iUnion, mem_Iic, mem_univ, iff_true_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nx : \u211d\n\u22a2 \u2203 i, x \u2264 \u2191i\n[PROOFSTEP]\nobtain \u27e8r, hr\u27e9 := exists_rat_gt x\n[GOAL]\ncase h.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nx : \u211d\nr : \u211a\nhr : x < \u2191r\n\u22a2 \u2203 i, x \u2264 \u2191i\n[PROOFSTEP]\nexact \u27e8r, hr.le\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u22a2 \u22c2 (r : \u211a), Iic \u2191r = \u2205\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nx : \u211d\n\u22a2 x \u2208 \u22c2 (r : \u211a), Iic \u2191r \u2194 x \u2208 \u2205\n[PROOFSTEP]\nsimp only [mem_iInter, mem_Iic, mem_empty_iff_false, iff_false_iff, not_forall, not_le]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nx : \u211d\n\u22a2 \u2203 x_1, \u2191x_1 < x\n[PROOFSTEP]\nexact exists_rat_lt x\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderTopology \u03b1\n\u22a2 atBot \u2264 \ud835\udcdd \u22a5\n[PROOFSTEP]\ncases subsingleton_or_nontrivial \u03b1\n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderTopology \u03b1\nh\u271d : Subsingleton \u03b1\n\u22a2 atBot \u2264 \ud835\udcdd \u22a5\n[PROOFSTEP]\nsimp only [nhds_discrete, le_pure_iff, mem_atBot_sets, mem_singleton_iff, eq_iff_true_of_subsingleton, imp_true_iff,\n exists_const]\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderTopology \u03b1\nh\u271d : Nontrivial \u03b1\n\u22a2 atBot \u2264 \ud835\udcdd \u22a5\n[PROOFSTEP]\nhave h : atBot.HasBasis (fun _ : \u03b1 => True) Iic := @atBot_basis \u03b1 _ _\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderTopology \u03b1\nh\u271d : Nontrivial \u03b1\nh : HasBasis atBot (fun x => True) Iic\n\u22a2 atBot \u2264 \ud835\udcdd \u22a5\n[PROOFSTEP]\nhave h_nhds : (\ud835\udcdd \u22a5).HasBasis (fun a : \u03b1 => \u22a5 < a) fun a => Iio a := @nhds_bot_basis \u03b1 _ _ _ _ _\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderTopology \u03b1\nh\u271d : Nontrivial \u03b1\nh : HasBasis atBot (fun x => True) Iic\nh_nhds : HasBasis (\ud835\udcdd \u22a5) (fun a => \u22a5 < a) fun a => Iio a\n\u22a2 atBot \u2264 \ud835\udcdd \u22a5\n[PROOFSTEP]\nintro s\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderTopology \u03b1\nh\u271d : Nontrivial \u03b1\nh : HasBasis atBot (fun x => True) Iic\nh_nhds : HasBasis (\ud835\udcdd \u22a5) (fun a => \u22a5 < a) fun a => Iio a\ns : Set \u03b1\n\u22a2 s \u2208 \ud835\udcdd \u22a5 \u2192 s \u2208 atBot\n[PROOFSTEP]\nrw [h.mem_iff, h_nhds.mem_iff]\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderTopology \u03b1\nh\u271d : Nontrivial \u03b1\nh : HasBasis atBot (fun x => True) Iic\nh_nhds : HasBasis (\ud835\udcdd \u22a5) (fun a => \u22a5 < a) fun a => Iio a\ns : Set \u03b1\n\u22a2 (\u2203 i, \u22a5 < i \u2227 Iio i \u2286 s) \u2192 \u2203 i, True \u2227 Iic i \u2286 s\n[PROOFSTEP]\nrintro \u27e8a, ha_bot_lt, h_Iio_a_subset_s\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderTopology \u03b1\nh\u271d : Nontrivial \u03b1\nh : HasBasis atBot (fun x => True) Iic\nh_nhds : HasBasis (\ud835\udcdd \u22a5) (fun a => \u22a5 < a) fun a => Iio a\ns : Set \u03b1\na : \u03b1\nha_bot_lt : \u22a5 < a\nh_Iio_a_subset_s : Iio a \u2286 s\n\u22a2 \u2203 i, True \u2227 Iic i \u2286 s\n[PROOFSTEP]\nrefine' \u27e8\u22a5, trivial, _root_.trans _ h_Iio_a_subset_s\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderTopology \u03b1\nh\u271d : Nontrivial \u03b1\nh : HasBasis atBot (fun x => True) Iic\nh_nhds : HasBasis (\ud835\udcdd \u22a5) (fun a => \u22a5 < a) fun a => Iio a\ns : Set \u03b1\na : \u03b1\nha_bot_lt : \u22a5 < a\nh_Iio_a_subset_s : Iio a \u2286 s\n\u22a2 Iic \u22a5 \u2286 Iio a\n[PROOFSTEP]\nsimpa only [Iic_bot, singleton_subset_iff, mem_Iio]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\n\u22a2 ENNReal.ofReal (\u2a05 (i : \u03b1), f i) = \u2a05 (i : \u03b1), ENNReal.ofReal (f i)\n[PROOFSTEP]\nby_cases hf : BddBelow (range f)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : BddBelow (range f)\n\u22a2 ENNReal.ofReal (\u2a05 (i : \u03b1), f i) = \u2a05 (i : \u03b1), ENNReal.ofReal (f i)\n[PROOFSTEP]\nexact\n Monotone.map_ciInf_of_continuousAt ENNReal.continuous_ofReal.continuousAt\n (fun i j hij => ENNReal.ofReal_le_ofReal hij) hf\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\n\u22a2 ENNReal.ofReal (\u2a05 (i : \u03b1), f i) = \u2a05 (i : \u03b1), ENNReal.ofReal (f i)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\n\u22a2 \u2a05 (i : \u03b1), ENNReal.ofReal (f i) = ENNReal.ofReal (\u2a05 (i : \u03b1), f i)\n[PROOFSTEP]\nrw [Real.iInf_of_not_bddBelow hf, ENNReal.ofReal_zero, \u2190 ENNReal.bot_eq_zero, iInf_eq_bot]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\n\u22a2 \u2200 (b : \u211d\u22650\u221e), b > \u22a5 \u2192 \u2203 i, ENNReal.ofReal (f i) < b\n[PROOFSTEP]\nobtain \u27e8y, hy_mem, hy_neg\u27e9 := not_bddBelow_iff.mp hf 0\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\ny : \u211d\nhy_mem : y \u2208 range f\nhy_neg : y < 0\n\u22a2 \u2200 (b : \u211d\u22650\u221e), b > \u22a5 \u2192 \u2203 i, ENNReal.ofReal (f i) < b\n[PROOFSTEP]\nobtain \u27e8i, rfl\u27e9 := mem_range.mpr hy_mem\n[GOAL]\ncase neg.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\ni : \u03b1\nhy_mem : (fun y => f y) i \u2208 range f\nhy_neg : (fun y => f y) i < 0\n\u22a2 \u2200 (b : \u211d\u22650\u221e), b > \u22a5 \u2192 \u2203 i, ENNReal.ofReal (f i) < b\n[PROOFSTEP]\nrefine' fun x hx => \u27e8i, _\u27e9\n[GOAL]\ncase neg.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u211d\ninst\u271d : Nonempty \u03b1\nhf : \u00acBddBelow (range f)\ni : \u03b1\nhy_mem : (fun y => f y) i \u2208 range f\nhy_neg : (fun y => f y) i < 0\nx : \u211d\u22650\u221e\nhx : x > \u22a5\n\u22a2 ENNReal.ofReal (f i) < x\n[PROOFSTEP]\nrwa [ENNReal.ofReal_of_nonpos hy_neg.le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\n\u22a2 \u222b\u207b (a : \u03b1), \u2a05 (b : \u03b2), f b a \u2202\u03bc = \u2a05 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\ncases nonempty_encodable \u03b2\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\n\u22a2 \u222b\u207b (a : \u03b1), \u2a05 (b : \u03b2), f b a \u2202\u03bc = \u2a05 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b2\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n\u22a2 \u222b\u207b (a : \u03b1), \u2a05 (b : \u03b2), f b a \u2202\u03bc = \u2a05 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nconv =>\n lhs\n congr\n \u00b7skip\n \u00b7 ext x\n rw [WithTop.iInf_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n| \u222b\u207b (a : \u03b1), \u2a05 (b : \u03b2), f b a \u2202\u03bc = \u2a05 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\n lhs\n congr\n \u00b7skip\n \u00b7 ext x\n rw [WithTop.iInf_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n| \u222b\u207b (a : \u03b1), \u2a05 (b : \u03b2), f b a \u2202\u03bc = \u2a05 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\n lhs\n congr\n \u00b7skip\n \u00b7 ext x\n rw [WithTop.iInf_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n| \u222b\u207b (a : \u03b1), \u2a05 (b : \u03b2), f b a \u2202\u03bc = \u2a05 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nlhs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n| \u222b\u207b (a : \u03b1), \u2a05 (b : \u03b2), f b a \u2202\u03bc\n[PROOFSTEP]\ncongr\n[GOAL]\ncase \u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n| \u03bc\ncase f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n| fun a => \u2a05 (b : \u03b2), f b a\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase \u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n| \u03bc\n[PROOFSTEP]\nskip\n[GOAL]\ncase \u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n| \u03bc\n[PROOFSTEP]\nskip\n[GOAL]\ncase \u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n| \u03bc\n[PROOFSTEP]\nskip\n[GOAL]\ncase f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n| fun a => \u2a05 (b : \u03b2), f b a\n[PROOFSTEP]\n\u00b7 ext x\n rw [WithTop.iInf_empty]\n[GOAL]\ncase f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n| fun a => \u2a05 (b : \u03b2), f b a\n[PROOFSTEP]\n ext x\n rw [WithTop.iInf_empty]\n[GOAL]\ncase f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n| fun a => \u2a05 (b : \u03b2), f b a\n[PROOFSTEP]\n ext x\n rw [WithTop.iInf_empty]\n[GOAL]\ncase f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n| fun a => \u2a05 (b : \u03b2), f b a\n[PROOFSTEP]\next x\n[GOAL]\ncase f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\nx : \u03b1\n| \u2a05 (b : \u03b2), f b x\n[PROOFSTEP]\nrw [WithTop.iInf_empty]\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n\u22a2 \u222b\u207b (x : \u03b1), \u22a4 \u2202\u03bc = \u2a05 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nrw [WithTop.iInf_empty, lintegral_const]\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n\u22a2 \u22a4 * \u2191\u2191\u03bc univ = \u22a4\n[PROOFSTEP]\nrw [ENNReal.top_mul', if_neg]\n[GOAL]\ncase intro.inl.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : IsEmpty \u03b2\n\u22a2 \u00ac\u2191\u2191\u03bc univ = 0\n[PROOFSTEP]\nsimp only [Measure.measure_univ_eq_zero, h\u03bc, not_false_iff]\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\n\u22a2 \u222b\u207b (a : \u03b1), \u2a05 (b : \u03b2), f b a \u2202\u03bc = \u2a05 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\ninhabit \u03b2\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\n\u22a2 \u222b\u207b (a : \u03b1), \u2a05 (b : \u03b2), f b a \u2202\u03bc = \u2a05 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nhave : \u2200 a, \u2a05 b, f b a = \u2a05 n, f (h_directed.sequence f n) a :=\n by\n refine' fun a =>\n le_antisymm (le_iInf fun n => iInf_le _ _) (le_iInf fun b => iInf_le_of_le (Encodable.encode b + 1) _)\n exact h_directed.sequence_le b a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\n\u22a2 \u2200 (a : \u03b1), \u2a05 (b : \u03b2), f b a = \u2a05 (n : \u2115), f (Directed.sequence f h_directed n) a\n[PROOFSTEP]\nrefine' fun a => le_antisymm (le_iInf fun n => iInf_le _ _) (le_iInf fun b => iInf_le_of_le (Encodable.encode b + 1) _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 f (Directed.sequence f h_directed (Encodable.encode b + 1)) a \u2264 f b a\n[PROOFSTEP]\nexact h_directed.sequence_le b a\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\nthis : \u2200 (a : \u03b1), \u2a05 (b : \u03b2), f b a = \u2a05 (n : \u2115), f (Directed.sequence f h_directed n) a\n\u22a2 \u222b\u207b (a : \u03b1), \u2a05 (b : \u03b2), f b a \u2202\u03bc = \u2a05 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\ncalc\n \u222b\u207b a, \u2a05 b, f b a \u2202\u03bc\n _ = \u222b\u207b a, \u2a05 n, (f \u2218 h_directed.sequence f) n a \u2202\u03bc := by simp only [this, Function.comp_apply]\n _ = \u2a05 n, \u222b\u207b a, (f \u2218 h_directed.sequence f) n a \u2202\u03bc :=\n by\n rw [lintegral_iInf ?_ h_directed.sequence_anti]\n \u00b7 exact hf_int _\n \u00b7 exact (fun n => hf _)\n _ = \u2a05 b, \u222b\u207b a, f b a \u2202\u03bc := by\n refine' le_antisymm (le_iInf fun b => _) (le_iInf fun n => _)\n \u00b7 exact iInf_le_of_le (Encodable.encode b + 1) (lintegral_mono <| h_directed.sequence_le b)\n \u00b7 exact iInf_le (fun b => \u222b\u207b a, f b a \u2202\u03bc) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\nthis : \u2200 (a : \u03b1), \u2a05 (b : \u03b2), f b a = \u2a05 (n : \u2115), f (Directed.sequence f h_directed n) a\n\u22a2 \u222b\u207b (a : \u03b1), \u2a05 (b : \u03b2), f b a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2a05 (n : \u2115), (f \u2218 Directed.sequence f h_directed) n a \u2202\u03bc\n[PROOFSTEP]\nsimp only [this, Function.comp_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\nthis : \u2200 (a : \u03b1), \u2a05 (b : \u03b2), f b a = \u2a05 (n : \u2115), f (Directed.sequence f h_directed n) a\n\u22a2 \u222b\u207b (a : \u03b1), \u2a05 (n : \u2115), (f \u2218 Directed.sequence f h_directed) n a \u2202\u03bc =\n \u2a05 (n : \u2115), \u222b\u207b (a : \u03b1), (f \u2218 Directed.sequence f h_directed) n a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_iInf ?_ h_directed.sequence_anti]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\nthis : \u2200 (a : \u03b1), \u2a05 (b : \u03b2), f b a = \u2a05 (n : \u2115), f (Directed.sequence f h_directed n) a\n\u22a2 \u222b\u207b (a : \u03b1), (f \u2218 Directed.sequence f h_directed) 0 a \u2202\u03bc \u2260 \u22a4\n[PROOFSTEP]\nexact hf_int _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\nthis : \u2200 (a : \u03b1), \u2a05 (b : \u03b2), f b a = \u2a05 (n : \u2115), f (Directed.sequence f h_directed n) a\n\u22a2 \u2200 (n : \u2115), Measurable ((f \u2218 Directed.sequence f h_directed) n)\n[PROOFSTEP]\nexact (fun n => hf _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\nthis : \u2200 (a : \u03b1), \u2a05 (b : \u03b2), f b a = \u2a05 (n : \u2115), f (Directed.sequence f h_directed n) a\n\u22a2 \u2a05 (n : \u2115), \u222b\u207b (a : \u03b1), (f \u2218 Directed.sequence f h_directed) n a \u2202\u03bc = \u2a05 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nrefine' le_antisymm (le_iInf fun b => _) (le_iInf fun n => _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\nthis : \u2200 (a : \u03b1), \u2a05 (b : \u03b2), f b a = \u2a05 (n : \u2115), f (Directed.sequence f h_directed n) a\nb : \u03b2\n\u22a2 \u2a05 (n : \u2115), \u222b\u207b (a : \u03b1), (f \u2218 Directed.sequence f h_directed) n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f b a \u2202\u03bc\n[PROOFSTEP]\nexact iInf_le_of_le (Encodable.encode b + 1) (lintegral_mono <| h_directed.sequence_le b)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2260 0\nhf : \u2200 (b : \u03b2), Measurable (f b)\nhf_int : \u2200 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2260 \u22a4\nh_directed : Directed (fun x x_1 => x \u2265 x_1) f\nval\u271d : Encodable \u03b2\nh\u271d : Nonempty \u03b2\ninhabited_h : Inhabited \u03b2\nthis : \u2200 (a : \u03b1), \u2a05 (b : \u03b2), f b a = \u2a05 (n : \u2115), f (Directed.sequence f h_directed n) a\nn : \u2115\n\u22a2 \u2a05 (b : \u03b2), \u222b\u207b (a : \u03b1), f b a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), (f \u2218 Directed.sequence f h_directed) n a \u2202\u03bc\n[PROOFSTEP]\nexact iInf_le (fun b => \u222b\u207b a, f b a \u2202\u03bc) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(IicSnd \u03c1 r) s = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic r)\n[PROOFSTEP]\nrw [IicSnd, fst_apply hs, restrict_apply' (MeasurableSet.univ.prod (measurableSet_Iic : MeasurableSet (Iic r))), \u2190\n prod_univ, prod_inter_prod, inter_univ, univ_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr r' : \u211d\nh_le : r \u2264 r'\n\u22a2 IicSnd \u03c1 r \u2264 IicSnd \u03c1 r'\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr r' : \u211d\nh_le : r \u2264 r'\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(IicSnd \u03c1 r) s \u2264 \u2191\u2191(IicSnd \u03c1 r') s\n[PROOFSTEP]\nsimp_rw [IicSnd_apply \u03c1 _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr r' : \u211d\nh_le : r \u2264 r'\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic r) \u2264 \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic r')\n[PROOFSTEP]\nrefine' measure_mono (prod_subset_prod_iff.mpr (Or.inl \u27e8subset_rfl, Iic_subset_Iic.mpr _\u27e9))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr r' : \u211d\nh_le : r \u2264 r'\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 r \u2264 r'\n[PROOFSTEP]\nexact_mod_cast h_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211d\n\u22a2 IicSnd \u03c1 r \u2264 fst \u03c1\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(IicSnd \u03c1 r) s \u2264 \u2191\u2191(fst \u03c1) s\n[PROOFSTEP]\nsimp_rw [fst_apply hs, IicSnd_apply \u03c1 r hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic r) \u2264 \u2191\u2191\u03c1 (Prod.fst \u207b\u00b9' s)\n[PROOFSTEP]\nexact measure_mono (prod_subset_preimage_fst _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2a05 (r : { r' // t < r' }), \u2191\u2191(IicSnd \u03c1 \u2191\u2191r) s = \u2191\u2191(IicSnd \u03c1 \u2191t) s\n[PROOFSTEP]\nsimp_rw [\u03c1.IicSnd_apply _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2a05 (r : { r' // t < r' }), \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191\u2191r) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191t)\n[PROOFSTEP]\nrw [\u2190 measure_iInter_eq_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2191\u2191\u03c1 (\u22c2 (i : { r' // t < r' }), s \u00d7\u02e2 Iic \u2191\u2191i) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191t)\n[PROOFSTEP]\nrw [\u2190 prod_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 \u22c2 (i : { r' // t < r' }), Iic \u2191\u2191i) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191t)\n[PROOFSTEP]\ncongr with x : 1\n[GOAL]\ncase e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\n\u22a2 x \u2208 \u22c2 (i : { r' // t < r' }), Iic \u2191\u2191i \u2194 x \u2208 Iic \u2191t\n[PROOFSTEP]\nsimp only [mem_iInter, mem_Iic, Subtype.forall, Subtype.coe_mk]\n[GOAL]\ncase e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\n\u22a2 (\u2200 (a : \u211a), t < a \u2192 x \u2264 \u2191a) \u2194 x \u2264 \u2191t\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h a hta => h.trans _\u27e9\n[GOAL]\ncase e_a.e_a.h.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nh : \u2200 (a : \u211a), t < a \u2192 x \u2264 \u2191a\n\u22a2 x \u2264 \u2191t\n[PROOFSTEP]\nrefine' le_of_forall_lt_rat_imp_le fun q htq => h q _\n[GOAL]\ncase e_a.e_a.h.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nh : \u2200 (a : \u211a), t < a \u2192 x \u2264 \u2191a\nq : \u211a\nhtq : \u2191t < \u2191q\n\u22a2 t < q\n[PROOFSTEP]\nexact_mod_cast htq\n[GOAL]\ncase e_a.e_a.h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nh : x \u2264 \u2191t\na : \u211a\nhta : t < a\n\u22a2 \u2191t \u2264 \u2191a\n[PROOFSTEP]\nexact_mod_cast hta.le\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200 (i : { r' // t < r' }), MeasurableSet (s \u00d7\u02e2 Iic \u2191\u2191i)\n[PROOFSTEP]\nexact fun _ => hs.prod measurableSet_Iic\n[GOAL]\ncase hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 Directed (fun x x_1 => x \u2287 x_1) fun r => s \u00d7\u02e2 Iic \u2191\u2191r\n[PROOFSTEP]\nrefine' Monotone.directed_ge fun r r' hrr' => prod_subset_prod_iff.mpr (Or.inl \u27e8subset_rfl, _\u27e9)\n[GOAL]\ncase hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nr r' : { r' // t < r' }\nhrr' : r \u2264 r'\n\u22a2 Iic \u2191\u2191r \u2286 Iic \u2191\u2191r'\n[PROOFSTEP]\nrefine' Iic_subset_Iic.mpr _\n[GOAL]\ncase hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nr r' : { r' // t < r' }\nhrr' : r \u2264 r'\n\u22a2 \u2191\u2191r \u2264 \u2191\u2191r'\n[PROOFSTEP]\nexact_mod_cast hrr'\n[GOAL]\ncase hfin\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2203 i, \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191\u2191i) \u2260 \u22a4\n[PROOFSTEP]\nexact \u27e8\u27e8t + 1, lt_add_one _\u27e9, measure_ne_top \u03c1 _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Tendsto (fun r => \u2191\u2191(IicSnd \u03c1 \u2191r) s) atTop (\ud835\udcdd (\u2191\u2191(fst \u03c1) s))\n[PROOFSTEP]\nsimp_rw [\u03c1.IicSnd_apply _ hs, fst_apply hs, \u2190 prod_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (s \u00d7\u02e2 univ)))\n[PROOFSTEP]\nrw [\u2190 Real.iUnion_Iic_rat, prod_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c3 (i : \u211a), s \u00d7\u02e2 Iic \u2191i)))\n[PROOFSTEP]\nrefine' tendsto_measure_iUnion fun r q hr_le_q x => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u03b1\nhs : MeasurableSet s\nr q : \u211a\nhr_le_q : r \u2264 q\nx : \u03b1 \u00d7 \u211d\n\u22a2 x \u2208 s \u00d7\u02e2 Iic \u2191r \u2192 x \u2208 s \u00d7\u02e2 Iic \u2191q\n[PROOFSTEP]\nsimp only [mem_prod, mem_Iic, and_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u03b1\nhs : MeasurableSet s\nr q : \u211a\nhr_le_q : r \u2264 q\nx : \u03b1 \u00d7 \u211d\n\u22a2 x.fst \u2208 s \u2192 x.snd \u2264 \u2191r \u2192 x.fst \u2208 s \u2227 x.snd \u2264 \u2191q\n[PROOFSTEP]\nrefine' fun hxs hxr => \u27e8hxs, hxr.trans _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u03b1\nhs : MeasurableSet s\nr q : \u211a\nhr_le_q : r \u2264 q\nx : \u03b1 \u00d7 \u211d\nhxs : x.fst \u2208 s\nhxr : x.snd \u2264 \u2191r\n\u22a2 \u2191r \u2264 \u2191q\n[PROOFSTEP]\nexact_mod_cast hr_le_q\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Tendsto (fun r => \u2191\u2191(IicSnd \u03c1 \u2191r) s) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp_rw [\u03c1.IicSnd_apply _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nhave h_empty : \u03c1 (s \u00d7\u02e2 \u2205) = 0 := by simp only [prod_empty, measure_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\n[PROOFSTEP]\nsimp only [prod_empty, measure_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 h_empty, \u2190 Real.iInter_Iic_rat, prod_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (i : \u211a), s \u00d7\u02e2 Iic \u2191i)))\n[PROOFSTEP]\nsuffices h_neg : Tendsto (fun r : \u211a => \u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u03c1 (\u22c2 r : \u211a, s \u00d7\u02e2 Iic \u2191(-r))))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r))))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (i : \u211a), s \u00d7\u02e2 Iic \u2191i)))\n[PROOFSTEP]\nhave h_inter_eq : \u22c2 r : \u211a, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r : \u211a, s \u00d7\u02e2 Iic (r : \u211d) :=\n by\n ext1 x\n simp only [Rat.cast_eq_id, id.def, mem_iInter, mem_prod, mem_Iic]\n refine' \u27e8fun h i => \u27e8(h i).1, _\u27e9, fun h i => \u27e8(h i).1, _\u27e9\u27e9 <;> have h' := h (-i)\n \u00b7 rw [neg_neg] at h' ; exact h'.2\n \u00b7 exact h'.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r))))\n\u22a2 \u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191r\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\n\u22a2 x \u2208 \u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r) \u2194 x \u2208 \u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191r\n[PROOFSTEP]\nsimp only [Rat.cast_eq_id, id.def, mem_iInter, mem_prod, mem_Iic]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\n\u22a2 (\u2200 (i : \u211a), x.fst \u2208 s \u2227 x.snd \u2264 \u2191(-i)) \u2194 \u2200 (i : \u211a), x.fst \u2208 s \u2227 x.snd \u2264 \u2191i\n[PROOFSTEP]\nrefine' \u27e8fun h i => \u27e8(h i).1, _\u27e9, fun h i => \u27e8(h i).1, _\u27e9\u27e9\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\nh : \u2200 (i : \u211a), x.fst \u2208 s \u2227 x.snd \u2264 \u2191(-i)\ni : \u211a\n\u22a2 x.snd \u2264 \u2191i\n[PROOFSTEP]\nhave h' := h (-i)\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\nh : \u2200 (i : \u211a), x.fst \u2208 s \u2227 x.snd \u2264 \u2191i\ni : \u211a\n\u22a2 x.snd \u2264 \u2191(-i)\n[PROOFSTEP]\nhave h' := h (-i)\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\nh : \u2200 (i : \u211a), x.fst \u2208 s \u2227 x.snd \u2264 \u2191(-i)\ni : \u211a\nh' : x.fst \u2208 s \u2227 x.snd \u2264 \u2191(- -i)\n\u22a2 x.snd \u2264 \u2191i\n[PROOFSTEP]\nrw [neg_neg] at h' \n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\nh : \u2200 (i : \u211a), x.fst \u2208 s \u2227 x.snd \u2264 \u2191(-i)\ni : \u211a\nh' : x.fst \u2208 s \u2227 x.snd \u2264 \u2191i\n\u22a2 x.snd \u2264 \u2191i\n[PROOFSTEP]\nexact h'.2\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r))))\nx : \u03b1 \u00d7 \u211d\nh : \u2200 (i : \u211a), x.fst \u2208 s \u2227 x.snd \u2264 \u2191i\ni : \u211a\nh' : x.fst \u2208 s \u2227 x.snd \u2264 \u2191(-i)\n\u22a2 x.snd \u2264 \u2191(-i)\n[PROOFSTEP]\nexact h'.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r))))\nh_inter_eq : \u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191r\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (i : \u211a), s \u00d7\u02e2 Iic \u2191i)))\n[PROOFSTEP]\nrw [h_inter_eq] at h_neg \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191r)))\nh_inter_eq : \u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191r\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (i : \u211a), s \u00d7\u02e2 Iic \u2191i)))\n[PROOFSTEP]\nhave h_fun_eq : (fun r : \u211a => \u03c1 (s \u00d7\u02e2 Iic (r : \u211d))) = fun r : \u211a => \u03c1 (s \u00d7\u02e2 Iic \u2191(- -r)) := by simp_rw [neg_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191r)))\nh_inter_eq : \u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191r\n\u22a2 (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) = fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(- -r))\n[PROOFSTEP]\nsimp_rw [neg_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191r)))\nh_inter_eq : \u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191r\nh_fun_eq : (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) = fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(- -r))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (i : \u211a), s \u00d7\u02e2 Iic \u2191i)))\n[PROOFSTEP]\nrw [h_fun_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nh_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191r)))\nh_inter_eq : \u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191r\nh_fun_eq : (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) = fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(- -r))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(- -r))) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (i : \u211a), s \u00d7\u02e2 Iic \u2191i)))\n[PROOFSTEP]\nexact h_neg.comp tendsto_neg_atBot_atTop\n[GOAL]\ncase h_neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), s \u00d7\u02e2 Iic \u2191(-r))))\n[PROOFSTEP]\nrefine' tendsto_measure_iInter (fun q => hs.prod measurableSet_Iic) _ \u27e80, measure_ne_top \u03c1 _\u27e9\n[GOAL]\ncase h_neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\n\u22a2 Antitone fun r => s \u00d7\u02e2 Iic \u2191(-r)\n[PROOFSTEP]\nrefine' fun q r hqr => prod_subset_prod_iff.mpr (Or.inl \u27e8subset_rfl, fun x hx => _\u27e9)\n[GOAL]\ncase h_neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nq r : \u211a\nhqr : q \u2264 r\nx : \u211d\nhx : x \u2208 Iic \u2191(-r)\n\u22a2 x \u2208 Iic \u2191(-q)\n[PROOFSTEP]\nsimp only [Rat.cast_neg, mem_Iic] at hx \u22a2\n[GOAL]\ncase h_neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nq r : \u211a\nhqr : q \u2264 r\nx : \u211d\nhx : x \u2264 -\u2191r\n\u22a2 x \u2264 -\u2191q\n[PROOFSTEP]\nrefine' hx.trans (neg_le_neg _)\n[GOAL]\ncase h_neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\ns : Set \u03b1\nhs : MeasurableSet s\nh_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0\nq r : \u211a\nhqr : q \u2264 r\nx : \u211d\nhx : x \u2264 -\u2191r\n\u22a2 \u2191q \u2264 \u2191r\n[PROOFSTEP]\nexact_mod_cast hqr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s\n[PROOFSTEP]\nhave : \u2200 r, \u222b\u207b x in s, preCdf \u03c1 r x \u2202\u03c1.fst = \u222b\u207b x in s, (preCdf \u03c1 r * 1) x \u2202\u03c1.fst := by\n simp only [mul_one, eq_self_iff_true, forall_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n[PROOFSTEP]\nsimp only [mul_one, eq_self_iff_true, forall_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s\n[PROOFSTEP]\nrw [this, \u2190 set_lintegral_withDensity_eq_set_lintegral_mul _ measurable_preCdf _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (x : \u03b1) in s, OfNat.ofNat 1 x \u2202Measure.withDensity (Measure.fst \u03c1) (preCdf \u03c1 r) = \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s\n[PROOFSTEP]\nsimp only [withDensity_preCdf \u03c1 r, Pi.one_apply, lintegral_one, Measure.restrict_apply, MeasurableSet.univ, univ_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n\u22a2 Measurable 1\n[PROOFSTEP]\nrw [(_ : (1 : \u03b1 \u2192 \u211d\u22650\u221e) = fun _ => 1)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n\u22a2 Measurable fun x => 1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200 (r : \u211a), \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, (preCdf \u03c1 r * 1) x \u2202Measure.fst \u03c1\n\u22a2 1 = fun x => 1\n[PROOFSTEP]\nexacts [measurable_const, rfl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\n[PROOFSTEP]\nsimp_rw [Monotone, ae_all_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200 (i i_1 : \u211a), i \u2264 i_1 \u2192 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, preCdf \u03c1 i a \u2264 preCdf \u03c1 i_1 a\n[PROOFSTEP]\nrefine' fun r r' hrr' =>\n ae_le_of_forall_set_lintegral_le_of_sigmaFinite measurable_preCdf measurable_preCdf fun s hs _ => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr r' : \u211a\nhrr' : r \u2264 r'\ns : Set \u03b1\nhs : MeasurableSet s\nx\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 \u2264 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r' x \u2202Measure.fst \u03c1\n[PROOFSTEP]\nrw [set_lintegral_preCdf_fst \u03c1 r hs, set_lintegral_preCdf_fst \u03c1 r' hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr r' : \u211a\nhrr' : r \u2264 r'\ns : Set \u03b1\nhs : MeasurableSet s\nx\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4\n\u22a2 \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191r') s\n[PROOFSTEP]\nrefine' Measure.IicSnd_mono \u03c1 _ s hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr r' : \u211a\nhrr' : r \u2264 r'\ns : Set \u03b1\nhs : MeasurableSet s\nx\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4\n\u22a2 \u2191r \u2264 \u2191r'\n[PROOFSTEP]\nexact_mod_cast hrr'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 = \u2191\u2191(Measure.IicSnd \u03c1 \u2191t) s\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191t) s\n[PROOFSTEP]\nhave h : \u2200 q : Ioi t, \u222b\u207b x in s, \u2a05 r : Ioi t, preCdf \u03c1 r x \u2202\u03c1.fst \u2264 \u03c1.IicSnd q s :=\n by\n intro q\n rw [\u2190 set_lintegral_preCdf_fst \u03c1 _ hs]\n refine' set_lintegral_mono_ae _ measurable_preCdf _\n \u00b7 exact measurable_iInf fun _ => measurable_preCdf\n \u00b7 filter_upwards [monotone_preCdf _] with a _\n exact fun _ => iInf_le _ q\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2200 (q : \u2191(Ioi t)), \u222b\u207b (x : \u03b1) in s, \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191\u2191q) s\n[PROOFSTEP]\nintro q\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nq : \u2191(Ioi t)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191\u2191q) s\n[PROOFSTEP]\nrw [\u2190 set_lintegral_preCdf_fst \u03c1 _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nq : \u2191(Ioi t)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 (\u2191q) x \u2202Measure.fst \u03c1\n[PROOFSTEP]\nrefine' set_lintegral_mono_ae _ measurable_preCdf _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nq : \u2191(Ioi t)\n\u22a2 Measurable fun x => \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) x\n[PROOFSTEP]\nexact measurable_iInf fun _ => measurable_preCdf\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nq : \u2191(Ioi t)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, x \u2208 s \u2192 \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) x \u2264 preCdf \u03c1 (\u2191q) x\n[PROOFSTEP]\nfilter_upwards [monotone_preCdf _] with a _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nq : \u2191(Ioi t)\na : \u03b1\na\u271d : Monotone fun r => preCdf \u03c1 r a\n\u22a2 a \u2208 s \u2192 \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) a \u2264 preCdf \u03c1 (\u2191q) a\n[PROOFSTEP]\nexact fun _ => iInf_le _ q\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nh : \u2200 (q : \u2191(Ioi t)), \u222b\u207b (x : \u03b1) in s, \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191\u2191q) s\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191t) s\n[PROOFSTEP]\ncalc\n \u222b\u207b x in s, \u2a05 r : Ioi t, preCdf \u03c1 r x \u2202\u03c1.fst \u2264 \u2a05 q : Ioi t, \u03c1.IicSnd q s := le_iInf h\n _ = \u03c1.IicSnd t s := Measure.iInf_IicSnd_gt \u03c1 t hs\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.IicSnd \u03c1 \u2191t) s \u2264 \u222b\u207b (x : \u03b1) in s, \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1\n[PROOFSTEP]\nrw [(set_lintegral_preCdf_fst \u03c1 t hs).symm]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 t x \u2202Measure.fst \u03c1 \u2264 \u222b\u207b (x : \u03b1) in s, \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1\n[PROOFSTEP]\nrefine' set_lintegral_mono_ae measurable_preCdf _ _\n[GOAL]\ncase refine'_2.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Measurable fun x => \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) x\n[PROOFSTEP]\nexact measurable_iInf fun _ => measurable_preCdf\n[GOAL]\ncase refine'_2.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, x \u2208 s \u2192 preCdf \u03c1 t x \u2264 \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) x\n[PROOFSTEP]\nfilter_upwards [monotone_preCdf _] with a ha_mono\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\n\u22a2 a \u2208 s \u2192 preCdf \u03c1 t a \u2264 \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) a\n[PROOFSTEP]\nexact fun _ => le_iInf fun r => ha_mono (le_of_lt r.prop)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n[PROOFSTEP]\nrw [ae_all_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200 (i : \u211a), \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, preCdf \u03c1 i a \u2264 1\n[PROOFSTEP]\nrefine' fun r => ae_le_of_forall_set_lintegral_le_of_sigmaFinite measurable_preCdf measurable_const fun s hs _ => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nx\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 \u2264 \u222b\u207b (x : \u03b1) in s, 1 \u2202Measure.fst \u03c1\n[PROOFSTEP]\nrw [set_lintegral_preCdf_fst \u03c1 r hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nx\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4\n\u22a2 \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s \u2264 \u222b\u207b (x : \u03b1) in s, 1 \u2202Measure.fst \u03c1\n[PROOFSTEP]\nsimp only [Pi.one_apply, lintegral_one, Measure.restrict_apply, MeasurableSet.univ, univ_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nx\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4\n\u22a2 \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s \u2264 \u2191\u2191(Measure.fst \u03c1) s\n[PROOFSTEP]\nexact Measure.IicSnd_le_fst \u03c1 r s hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 r a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u2191\u2191\u03c1 univ))\n[PROOFSTEP]\nconvert \u03c1.tendsto_IicSnd_atTop MeasurableSet.univ\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx\u271d : \u211a\n\u22a2 \u222b\u207b (a : \u03b1), preCdf \u03c1 x\u271d a \u2202Measure.fst \u03c1 = \u2191\u2191(Measure.IicSnd \u03c1 \u2191x\u271d) univ\n[PROOFSTEP]\nrw [\u2190 set_lintegral_univ, set_lintegral_preCdf_fst \u03c1 _ MeasurableSet.univ]\n[GOAL]\ncase h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2191\u2191\u03c1 univ = \u2191\u2191(Measure.fst \u03c1) univ\n[PROOFSTEP]\nexact Measure.fst_univ.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 r a \u2202Measure.fst \u03c1) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert \u03c1.tendsto_IicSnd_atBot MeasurableSet.univ\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx\u271d : \u211a\n\u22a2 \u222b\u207b (a : \u03b1), preCdf \u03c1 x\u271d a \u2202Measure.fst \u03c1 = \u2191\u2191(Measure.IicSnd \u03c1 \u2191x\u271d) univ\n[PROOFSTEP]\nrw [\u2190 set_lintegral_univ, set_lintegral_preCdf_fst \u03c1 _ MeasurableSet.univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nhave h_mono := monotone_preCdf \u03c1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nhave h_le_one := preCdf_le_one \u03c1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nhave h_exists : \u2200\u1d50 a \u2202\u03c1.fst, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) :=\n by\n filter_upwards [h_mono, h_le_one] with a ha_mono ha_le_one\n have h_tendsto : Tendsto (fun r => preCdf \u03c1 r a) atTop atTop \u2228 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) :=\n tendsto_of_monotone ha_mono\n cases' h_tendsto with h_absurd h_tendsto\n \u00b7 rw [Monotone.tendsto_atTop_atTop_iff ha_mono] at h_absurd \n obtain \u27e8r, hr\u27e9 := h_absurd 2\n exact absurd (hr.trans (ha_le_one r)) ENNReal.one_lt_two.not_le\n \u00b7 exact h_tendsto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nfilter_upwards [h_mono, h_le_one] with a ha_mono ha_le_one\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nhave h_tendsto : Tendsto (fun r => preCdf \u03c1 r a) atTop atTop \u2228 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) :=\n tendsto_of_monotone ha_mono\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_tendsto : Tendsto (fun r => preCdf \u03c1 r a) atTop atTop \u2228 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n[PROOFSTEP]\ncases' h_tendsto with h_absurd h_tendsto\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_absurd : Tendsto (fun r => preCdf \u03c1 r a) atTop atTop\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nrw [Monotone.tendsto_atTop_atTop_iff ha_mono] at h_absurd \n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_absurd : \u2200 (b : \u211d\u22650\u221e), \u2203 a_1, b \u2264 preCdf \u03c1 a_1 a\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nobtain \u27e8r, hr\u27e9 := h_absurd 2\n[GOAL]\ncase h.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_absurd : \u2200 (b : \u211d\u22650\u221e), \u2203 a_1, b \u2264 preCdf \u03c1 a_1 a\nr : \u211a\nhr : 2 \u2264 preCdf \u03c1 r a\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nexact absurd (hr.trans (ha_le_one r)) ENNReal.one_lt_two.not_le\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\na : \u03b1\nha_mono : Monotone fun r => preCdf \u03c1 r a\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_tendsto : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nexact h_tendsto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nclassical\n -- let `F` be the pointwise limit of `preCdf` where it exists, and 0 elsewhere.\nlet F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then h.choose else 0\nhave h_tendsto_\u211a : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) :=\n by\n filter_upwards [h_exists] with a ha\n simp_rw [dif_pos ha]\n exact ha.choose_spec\nhave h_tendsto_\u2115 : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun n : \u2115 => preCdf \u03c1 n a) atTop (\ud835\udcdd (F a)) := by\n filter_upwards [h_tendsto_\u211a] with a ha using ha.comp tendsto_nat_cast_atTop_atTop\nhave hF_ae_meas : AEMeasurable F \u03c1.fst :=\n by\n refine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto_\u211a\n exact measurable_preCdf.aemeasurable\nhave hF_le_one : \u2200\u1d50 a \u2202\u03c1.fst, F a \u2264 1 := by\n filter_upwards [h_tendsto_\u211a, h_le_one] with a ha ha_le using le_of_tendsto' ha ha_le\nsuffices \u2200\u1d50 a \u2202\u03c1.fst, F a = 1 by\n filter_upwards [h_tendsto_\u211a, this] with a ha_tendsto ha_eq\n rwa [ha_eq] at ha_tendsto \nhave h_lintegral_eq : \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst :=\n by\n have h_lintegral : Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b a, F a \u2202\u03c1.fst)) :=\n by\n refine'\n lintegral_tendsto_of_tendsto_of_monotone\n (\n -- does this exist only for \u2115?fun _ => measurable_preCdf.aemeasurable)\n _ h_tendsto_\u2115\n filter_upwards [h_mono] with a ha\n refine' fun n m hnm => ha _\n exact_mod_cast hnm\n have h_lintegral' : Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b _, 1 \u2202\u03c1.fst)) :=\n by\n rw [lintegral_one, Measure.fst_univ]\n exact (tendsto_lintegral_preCdf_atTop \u03c1).comp tendsto_nat_cast_atTop_atTop\n exact tendsto_nhds_unique h_lintegral h_lintegral'\nhave : \u222b\u207b a, 1 - F a \u2202\u03c1.fst = 0 :=\n by\n rw [lintegral_sub' hF_ae_meas _ hF_le_one, h_lintegral_eq, tsub_self]\n calc\n \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := h_lintegral_eq\n _ = \u03c1.fst univ := lintegral_one\n _ = \u03c1 univ := Measure.fst_univ\n _ \u2260 \u221e := measure_ne_top \u03c1 _\nrw [lintegral_eq_zero_iff' (aemeasurable_const.sub hF_ae_meas)] at this \nfilter_upwards [this, hF_le_one] with ha h_one_sub_eq_zero h_le_one\nrw [Pi.zero_apply, tsub_eq_zero_iff_le] at h_one_sub_eq_zero \nexact le_antisymm h_le_one h_one_sub_eq_zero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nlet F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then h.choose else 0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nhave h_tendsto_\u211a : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) :=\n by\n filter_upwards [h_exists] with a ha\n simp_rw [dif_pos ha]\n exact ha.choose_spec\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\n[PROOFSTEP]\nfilter_upwards [h_exists] with a ha\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\na : \u03b1\nha : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atTop\n (\ud835\udcdd (if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0))\n[PROOFSTEP]\nsimp_rw [dif_pos ha]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\na : \u03b1\nha : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (Exists.choose ha))\n[PROOFSTEP]\nexact ha.choose_spec\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nhave h_tendsto_\u2115 : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun n : \u2115 => preCdf \u03c1 n a) atTop (\ud835\udcdd (F a)) := by\n filter_upwards [h_tendsto_\u211a] with a ha using ha.comp tendsto_nat_cast_atTop_atTop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\n[PROOFSTEP]\nfilter_upwards [h_tendsto_\u211a] with a ha using ha.comp tendsto_nat_cast_atTop_atTop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nhave hF_ae_meas : AEMeasurable F \u03c1.fst :=\n by\n refine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto_\u211a\n exact measurable_preCdf.aemeasurable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\n\u22a2 AEMeasurable F\n[PROOFSTEP]\nrefine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto_\u211a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nn : \u211a\n\u22a2 AEMeasurable fun x => preCdf \u03c1 n x\n[PROOFSTEP]\nexact measurable_preCdf.aemeasurable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nhave hF_le_one : \u2200\u1d50 a \u2202\u03c1.fst, F a \u2264 1 := by\n filter_upwards [h_tendsto_\u211a, h_le_one] with a ha ha_le using le_of_tendsto' ha ha_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\n[PROOFSTEP]\nfilter_upwards [h_tendsto_\u211a, h_le_one] with a ha ha_le using le_of_tendsto' ha ha_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nsuffices \u2200\u1d50 a \u2202\u03c1.fst, F a = 1 by\n filter_upwards [h_tendsto_\u211a, this] with a ha_tendsto ha_eq\n rwa [ha_eq] at ha_tendsto \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nfilter_upwards [h_tendsto_\u211a, this] with a ha_tendsto ha_eq\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1\na : \u03b1\nha_tendsto :\n Tendsto (fun r => preCdf \u03c1 r a) atTop\n (\ud835\udcdd (if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0))\nha_eq : (if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0) = 1\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nrwa [ha_eq] at ha_tendsto \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1\n[PROOFSTEP]\nhave h_lintegral_eq : \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst :=\n by\n have h_lintegral : Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b a, F a \u2202\u03c1.fst)) :=\n by\n refine'\n lintegral_tendsto_of_tendsto_of_monotone\n (\n -- does this exist only for \u2115?fun _ => measurable_preCdf.aemeasurable)\n _ h_tendsto_\u2115\n filter_upwards [h_mono] with a ha\n refine' fun n m hnm => ha _\n exact_mod_cast hnm\n have h_lintegral' : Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b _, 1 \u2202\u03c1.fst)) :=\n by\n rw [lintegral_one, Measure.fst_univ]\n exact (tendsto_lintegral_preCdf_atTop \u03c1).comp tendsto_nat_cast_atTop_atTop\n exact tendsto_nhds_unique h_lintegral h_lintegral'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\n[PROOFSTEP]\nhave h_lintegral : Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b a, F a \u2202\u03c1.fst)) :=\n by\n refine'\n lintegral_tendsto_of_tendsto_of_monotone\n (\n -- does this exist only for \u2115?fun _ => measurable_preCdf.aemeasurable)\n _ h_tendsto_\u2115\n filter_upwards [h_mono] with a ha\n refine' fun n m hnm => ha _\n exact_mod_cast hnm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n[PROOFSTEP]\nrefine'\n lintegral_tendsto_of_tendsto_of_monotone\n (\n -- does this exist only for \u2115?fun _ => measurable_preCdf.aemeasurable)\n _ h_tendsto_\u2115\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, Monotone fun n => preCdf \u03c1 (\u2191n) x\n[PROOFSTEP]\nfilter_upwards [h_mono] with a ha\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\n\u22a2 Monotone fun n => preCdf \u03c1 (\u2191n) a\n[PROOFSTEP]\nrefine' fun n m hnm => ha _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\nn m : \u2115\nhnm : n \u2264 m\n\u22a2 \u2191n \u2264 \u2191m\n[PROOFSTEP]\nexact_mod_cast hnm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\n[PROOFSTEP]\nhave h_lintegral' : Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b _, 1 \u2202\u03c1.fst)) :=\n by\n rw [lintegral_one, Measure.fst_univ]\n exact (tendsto_lintegral_preCdf_atTop \u03c1).comp tendsto_nat_cast_atTop_atTop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1))\n[PROOFSTEP]\nrw [lintegral_one, Measure.fst_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u2191\u2191\u03c1 univ))\n[PROOFSTEP]\nexact (tendsto_lintegral_preCdf_atTop \u03c1).comp tendsto_nat_cast_atTop_atTop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral' : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1))\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\n[PROOFSTEP]\nexact tendsto_nhds_unique h_lintegral h_lintegral'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1\n[PROOFSTEP]\nhave : \u222b\u207b a, 1 - F a \u2202\u03c1.fst = 0 :=\n by\n rw [lintegral_sub' hF_ae_meas _ hF_le_one, h_lintegral_eq, tsub_self]\n calc\n \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := h_lintegral_eq\n _ = \u03c1.fst univ := lintegral_one\n _ = \u03c1 univ := Measure.fst_univ\n _ \u2260 \u221e := measure_ne_top \u03c1 _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (a : \u03b1), 1 - F a \u2202Measure.fst \u03c1 = 0\n[PROOFSTEP]\nrw [lintegral_sub' hF_ae_meas _ hF_le_one, h_lintegral_eq, tsub_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 \u2260 \u22a4\n[PROOFSTEP]\ncalc\n \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := h_lintegral_eq\n _ = \u03c1.fst univ := lintegral_one\n _ = \u03c1 univ := Measure.fst_univ\n _ \u2260 \u221e := measure_ne_top \u03c1 _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\nthis : \u222b\u207b (a : \u03b1), 1 - F a \u2202Measure.fst \u03c1 = 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1\n[PROOFSTEP]\nrw [lintegral_eq_zero_iff' (aemeasurable_const.sub hF_ae_meas)] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\nthis : (fun a => 1 - F a) =\u1d50[Measure.fst \u03c1] 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1\n[PROOFSTEP]\nfilter_upwards [this, hF_le_one] with ha h_one_sub_eq_zero h_le_one\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one\u271d : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\nthis : (fun a => 1 - F a) =\u1d50[Measure.fst \u03c1] 0\nha : \u03b1\nh_one_sub_eq_zero :\n (1 - if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r ha) atTop (\ud835\udcdd l) then Exists.choose h else 0) = OfNat.ofNat 0 ha\nh_le_one : (if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r ha) atTop (\ud835\udcdd l) then Exists.choose h else 0) \u2264 1\n\u22a2 (if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r ha) atTop (\ud835\udcdd l) then Exists.choose h else 0) = 1\n[PROOFSTEP]\nrw [Pi.zero_apply, tsub_eq_zero_iff_le] at h_one_sub_eq_zero \n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a\nh_le_one\u271d : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a))\nh_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a))\nhF_ae_meas : AEMeasurable F\nhF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1\nthis : (fun a => 1 - F a) =\u1d50[Measure.fst \u03c1] 0\nha : \u03b1\nh_one_sub_eq_zero : 1 \u2264 if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r ha) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_le_one : (if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r ha) atTop (\ud835\udcdd l) then Exists.choose h else 0) \u2264 1\n\u22a2 (if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r ha) atTop (\ud835\udcdd l) then Exists.choose h else 0) = 1\n[PROOFSTEP]\nexact le_antisymm h_le_one h_one_sub_eq_zero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nsuffices \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n by\n filter_upwards [this] with a ha\n have h_eq_neg : (fun r : \u211a => preCdf \u03c1 r a) = fun r : \u211a => preCdf \u03c1 (- -r) a := by simp_rw [neg_neg]\n rw [h_eq_neg]\n exact ha.comp tendsto_neg_atBot_atTop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nfilter_upwards [this] with a ha\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\na : \u03b1\nha : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nhave h_eq_neg : (fun r : \u211a => preCdf \u03c1 r a) = fun r : \u211a => preCdf \u03c1 (- -r) a := by simp_rw [neg_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\na : \u03b1\nha : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n\u22a2 (fun r => preCdf \u03c1 r a) = fun r => preCdf \u03c1 (- -r) a\n[PROOFSTEP]\nsimp_rw [neg_neg]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\na : \u03b1\nha : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\nh_eq_neg : (fun r => preCdf \u03c1 r a) = fun r => preCdf \u03c1 (- -r) a\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [h_eq_neg]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\na : \u03b1\nha : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\nh_eq_neg : (fun r => preCdf \u03c1 r a) = fun r => preCdf \u03c1 (- -r) a\n\u22a2 Tendsto (fun r => preCdf \u03c1 (- -r) a) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nexact ha.comp tendsto_neg_atBot_atTop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave h_exists : \u2200\u1d50 a \u2202\u03c1.fst, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) :=\n by\n filter_upwards [monotone_preCdf \u03c1] with a ha\n have h_anti : Antitone fun r => preCdf \u03c1 (-r) a := fun p q hpq => ha (neg_le_neg hpq)\n have h_tendsto :\n Tendsto (fun r => preCdf \u03c1 (-r) a) atTop atBot \u2228 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) :=\n tendsto_of_antitone h_anti\n cases' h_tendsto with h_bot h_tendsto\n \u00b7 exact \u27e80, Tendsto.mono_right h_bot atBot_le_nhds_bot\u27e9\n \u00b7 exact h_tendsto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nfilter_upwards [monotone_preCdf \u03c1] with a ha\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nhave h_anti : Antitone fun r => preCdf \u03c1 (-r) a := fun p q hpq => ha (neg_le_neg hpq)\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\nh_anti : Antitone fun r => preCdf \u03c1 (-r) a\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nhave h_tendsto : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop atBot \u2228 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) :=\n tendsto_of_antitone h_anti\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\nh_anti : Antitone fun r => preCdf \u03c1 (-r) a\nh_tendsto : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop atBot \u2228 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n[PROOFSTEP]\ncases' h_tendsto with h_bot h_tendsto\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\nh_anti : Antitone fun r => preCdf \u03c1 (-r) a\nh_bot : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop atBot\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nexact \u27e80, Tendsto.mono_right h_bot atBot_le_nhds_bot\u27e9\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nha : Monotone fun r => preCdf \u03c1 r a\nh_anti : Antitone fun r => preCdf \u03c1 (-r) a\nh_tendsto : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nexact h_tendsto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nclassical\nlet F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then h.choose else 0\nhave h_tendsto : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) :=\n by\n filter_upwards [h_exists] with a ha\n simp_rw [dif_pos ha]\n exact ha.choose_spec\nsuffices h_lintegral_eq : \u222b\u207b a, F a \u2202\u03c1.fst = 0\n\u00b7 have hF_ae_meas : AEMeasurable F \u03c1.fst :=\n by\n refine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto\n exact measurable_preCdf.aemeasurable\n rw [lintegral_eq_zero_iff' hF_ae_meas] at h_lintegral_eq \n filter_upwards [h_tendsto, h_lintegral_eq] with a ha_tendsto ha_eq\n rwa [ha_eq] at ha_tendsto \nhave h_lintegral : Tendsto (fun r => \u222b\u207b a, preCdf \u03c1 (-r) a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b a, F a \u2202\u03c1.fst)) :=\n by\n refine'\n tendsto_lintegral_filter_of_dominated_convergence (fun _ => 1) (eventually_of_forall fun _ => measurable_preCdf)\n (eventually_of_forall fun _ => _) _ h_tendsto\n \u00b7 filter_upwards [preCdf_le_one \u03c1] with a ha using ha _\n \u00b7 rw [lintegral_one]\n exact measure_ne_top _ _\nhave h_lintegral' : Tendsto (fun r => \u222b\u207b a, preCdf \u03c1 (-r) a \u2202\u03c1.fst) atTop (\ud835\udcdd 0) :=\n by\n have h_lintegral_eq : (fun r => \u222b\u207b a, preCdf \u03c1 (-r) a \u2202\u03c1.fst) = fun r : \u211a => \u03c1 (univ \u00d7\u02e2 Iic (-r : \u211d)) :=\n by\n ext1 n\n rw [\u2190 set_lintegral_univ, set_lintegral_preCdf_fst \u03c1 _ MeasurableSet.univ, Measure.IicSnd_univ]\n norm_cast\n rw [h_lintegral_eq]\n have h_zero_eq_measure_iInter : (0 : \u211d\u22650\u221e) = \u03c1 (\u22c2 r : \u211a, univ \u00d7\u02e2 Iic (-r : \u211d)) :=\n by\n suffices \u22c2 r : \u211a, Iic (-(r : \u211d)) = \u2205 by rw [\u2190 prod_iInter, this, prod_empty, measure_empty]\n ext1 x\n simp only [mem_iInter, mem_Iic, mem_empty_iff_false, iff_false_iff, not_forall, not_le]\n simp_rw [neg_lt]\n exact exists_rat_gt _\n rw [h_zero_eq_measure_iInter]\n refine'\n tendsto_measure_iInter (fun n => MeasurableSet.univ.prod measurableSet_Iic) (fun i j hij x => _)\n \u27e80, measure_ne_top \u03c1 _\u27e9\n simp only [mem_prod, mem_univ, mem_Iic, true_and_iff]\n refine' fun hxj => hxj.trans (neg_le_neg _)\n exact_mod_cast hij\nexact tendsto_nhds_unique h_lintegral h_lintegral'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nlet F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then h.choose else 0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave h_tendsto : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) :=\n by\n filter_upwards [h_exists] with a ha\n simp_rw [dif_pos ha]\n exact ha.choose_spec\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\n[PROOFSTEP]\nfilter_upwards [h_exists] with a ha\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\na : \u03b1\nha : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun r => preCdf \u03c1 (-r) a) atTop\n (\ud835\udcdd (if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0))\n[PROOFSTEP]\nsimp_rw [dif_pos ha]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\na : \u03b1\nha : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (Exists.choose ha))\n[PROOFSTEP]\nexact ha.choose_spec\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsuffices h_lintegral_eq : \u222b\u207b a, F a \u2202\u03c1.fst = 0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave hF_ae_meas : AEMeasurable F \u03c1.fst :=\n by\n refine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto\n exact measurable_preCdf.aemeasurable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0\n\u22a2 AEMeasurable F\n[PROOFSTEP]\nrefine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0\nn : \u211a\n\u22a2 AEMeasurable fun x => preCdf \u03c1 (-n) x\n[PROOFSTEP]\nexact measurable_preCdf.aemeasurable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0\nhF_ae_meas : AEMeasurable F\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [lintegral_eq_zero_iff' hF_ae_meas] at h_lintegral_eq \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : F =\u1d50[Measure.fst \u03c1] 0\nhF_ae_meas : AEMeasurable F\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nfilter_upwards [h_tendsto, h_lintegral_eq] with a ha_tendsto ha_eq\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral_eq : F =\u1d50[Measure.fst \u03c1] 0\nhF_ae_meas : AEMeasurable F\na : \u03b1\nha_tendsto :\n Tendsto (fun r => preCdf \u03c1 (-r) a) atTop\n (\ud835\udcdd (if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0))\nha_eq : (if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0) = OfNat.ofNat 0 a\n\u22a2 Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrwa [ha_eq] at ha_tendsto \n[GOAL]\ncase h_lintegral_eq\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0\n[PROOFSTEP]\nhave h_lintegral : Tendsto (fun r => \u222b\u207b a, preCdf \u03c1 (-r) a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b a, F a \u2202\u03c1.fst)) :=\n by\n refine'\n tendsto_lintegral_filter_of_dominated_convergence (fun _ => 1) (eventually_of_forall fun _ => measurable_preCdf)\n (eventually_of_forall fun _ => _) _ h_tendsto\n \u00b7 filter_upwards [preCdf_le_one \u03c1] with a ha using ha _\n \u00b7 rw [lintegral_one]\n exact measure_ne_top _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n[PROOFSTEP]\nrefine'\n tendsto_lintegral_filter_of_dominated_convergence (fun _ => 1) (eventually_of_forall fun _ => measurable_preCdf)\n (eventually_of_forall fun _ => _) _ h_tendsto\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nx\u271d : \u211a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, preCdf \u03c1 (-x\u271d) a \u2264 (fun x => 1) a\n[PROOFSTEP]\nfilter_upwards [preCdf_le_one \u03c1] with a ha using ha _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\n\u22a2 \u222b\u207b (a : \u03b1), (fun x => 1) a \u2202Measure.fst \u03c1 \u2260 \u22a4\n[PROOFSTEP]\nrw [lintegral_one]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\n\u22a2 \u2191\u2191(Measure.fst \u03c1) univ \u2260 \u22a4\n[PROOFSTEP]\nexact measure_ne_top _ _\n[GOAL]\ncase h_lintegral_eq\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0\n[PROOFSTEP]\nhave h_lintegral' : Tendsto (fun r => \u222b\u207b a, preCdf \u03c1 (-r) a \u2202\u03c1.fst) atTop (\ud835\udcdd 0) :=\n by\n have h_lintegral_eq : (fun r => \u222b\u207b a, preCdf \u03c1 (-r) a \u2202\u03c1.fst) = fun r : \u211a => \u03c1 (univ \u00d7\u02e2 Iic (-r : \u211d)) :=\n by\n ext1 n\n rw [\u2190 set_lintegral_univ, set_lintegral_preCdf_fst \u03c1 _ MeasurableSet.univ, Measure.IicSnd_univ]\n norm_cast\n rw [h_lintegral_eq]\n have h_zero_eq_measure_iInter : (0 : \u211d\u22650\u221e) = \u03c1 (\u22c2 r : \u211a, univ \u00d7\u02e2 Iic (-r : \u211d)) :=\n by\n suffices \u22c2 r : \u211a, Iic (-(r : \u211d)) = \u2205 by rw [\u2190 prod_iInter, this, prod_empty, measure_empty]\n ext1 x\n simp only [mem_iInter, mem_Iic, mem_empty_iff_false, iff_false_iff, not_forall, not_le]\n simp_rw [neg_lt]\n exact exists_rat_gt _\n rw [h_zero_eq_measure_iInter]\n refine'\n tendsto_measure_iInter (fun n => MeasurableSet.univ.prod measurableSet_Iic) (fun i j hij x => _)\n \u27e80, measure_ne_top \u03c1 _\u27e9\n simp only [mem_prod, mem_univ, mem_Iic, true_and_iff]\n refine' fun hxj => hxj.trans (neg_le_neg _)\n exact_mod_cast hij\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave h_lintegral_eq : (fun r => \u222b\u207b a, preCdf \u03c1 (-r) a \u2202\u03c1.fst) = fun r : \u211a => \u03c1 (univ \u00d7\u02e2 Iic (-r : \u211d)) :=\n by\n ext1 n\n rw [\u2190 set_lintegral_univ, set_lintegral_preCdf_fst \u03c1 _ MeasurableSet.univ, Measure.IicSnd_univ]\n norm_cast\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\n\u22a2 (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nn : \u211a\n\u22a2 \u222b\u207b (a : \u03b1), preCdf \u03c1 (-n) a \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191n))\n[PROOFSTEP]\nrw [\u2190 set_lintegral_univ, set_lintegral_preCdf_fst \u03c1 _ MeasurableSet.univ, Measure.IicSnd_univ]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nn : \u211a\n\u22a2 \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic \u2191(-n)) = \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191n))\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [h_lintegral_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave h_zero_eq_measure_iInter : (0 : \u211d\u22650\u221e) = \u03c1 (\u22c2 r : \u211a, univ \u00d7\u02e2 Iic (-r : \u211d)) :=\n by\n suffices \u22c2 r : \u211a, Iic (-(r : \u211d)) = \u2205 by rw [\u2190 prod_iInter, this, prod_empty, measure_empty]\n ext1 x\n simp only [mem_iInter, mem_Iic, mem_empty_iff_false, iff_false_iff, not_forall, not_le]\n simp_rw [neg_lt]\n exact exists_rat_gt _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 0 = \u2191\u2191\u03c1 (\u22c2 (r : \u211a), univ \u00d7\u02e2 Iic (-\u2191r))\n[PROOFSTEP]\nsuffices \u22c2 r : \u211a, Iic (-(r : \u211d)) = \u2205 by rw [\u2190 prod_iInter, this, prod_empty, measure_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nthis : \u22c2 (r : \u211a), Iic (-\u2191r) = \u2205\n\u22a2 0 = \u2191\u2191\u03c1 (\u22c2 (r : \u211a), univ \u00d7\u02e2 Iic (-\u2191r))\n[PROOFSTEP]\nrw [\u2190 prod_iInter, this, prod_empty, measure_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 \u22c2 (r : \u211a), Iic (-\u2191r) = \u2205\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nx : \u211d\n\u22a2 x \u2208 \u22c2 (r : \u211a), Iic (-\u2191r) \u2194 x \u2208 \u2205\n[PROOFSTEP]\nsimp only [mem_iInter, mem_Iic, mem_empty_iff_false, iff_false_iff, not_forall, not_le]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nx : \u211d\n\u22a2 \u2203 x_1, -\u2191x_1 < x\n[PROOFSTEP]\nsimp_rw [neg_lt]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nx : \u211d\n\u22a2 \u2203 x_1, -x < \u2191x_1\n[PROOFSTEP]\nexact exists_rat_gt _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nh_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 (r : \u211a), univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [h_zero_eq_measure_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nh_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 (r : \u211a), univ \u00d7\u02e2 Iic (-\u2191r))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 (r : \u211a), univ \u00d7\u02e2 Iic (-\u2191r))))\n[PROOFSTEP]\nrefine'\n tendsto_measure_iInter (fun n => MeasurableSet.univ.prod measurableSet_Iic) (fun i j hij x => _)\n \u27e80, measure_ne_top \u03c1 _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nh_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 (r : \u211a), univ \u00d7\u02e2 Iic (-\u2191r))\ni j : \u211a\nhij : i \u2264 j\nx : \u03b1 \u00d7 \u211d\n\u22a2 x \u2208 univ \u00d7\u02e2 Iic (-\u2191j) \u2192 x \u2208 univ \u00d7\u02e2 Iic (-\u2191i)\n[PROOFSTEP]\nsimp only [mem_prod, mem_univ, mem_Iic, true_and_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nh_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 (r : \u211a), univ \u00d7\u02e2 Iic (-\u2191r))\ni j : \u211a\nhij : i \u2264 j\nx : \u03b1 \u00d7 \u211d\n\u22a2 x.snd \u2264 -\u2191j \u2192 x.snd \u2264 -\u2191i\n[PROOFSTEP]\nrefine' fun hxj => hxj.trans (neg_le_neg _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))\nh_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 (r : \u211a), univ \u00d7\u02e2 Iic (-\u2191r))\ni j : \u211a\nhij : i \u2264 j\nx : \u03b1 \u00d7 \u211d\nhxj : x.snd \u2264 -\u2191j\n\u22a2 \u2191i \u2264 \u2191j\n[PROOFSTEP]\nexact_mod_cast hij\n[GOAL]\ncase h_lintegral_eq\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nh_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l)\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0\nh_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a))\nh_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1))\nh_lintegral' : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd 0)\n\u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0\n[PROOFSTEP]\nexact tendsto_nhds_unique h_lintegral h_lintegral'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (t : \u211a), \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) a = preCdf \u03c1 t a\n[PROOFSTEP]\nrw [ae_all_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200 (i : \u211a), \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2a05 (r : \u2191(Ioi i)), preCdf \u03c1 (\u2191r) a = preCdf \u03c1 i a\n[PROOFSTEP]\nrefine' fun t => ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite _ measurable_preCdf _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\n\u22a2 Measurable fun a => \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) a\n[PROOFSTEP]\nexact measurable_iInf fun i => measurable_preCdf\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\n\u22a2 \u2200 (s : Set \u03b1),\n MeasurableSet s \u2192\n \u2191\u2191(Measure.fst \u03c1) s < \u22a4 \u2192\n \u222b\u207b (x : \u03b1) in s, \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 t x \u2202Measure.fst \u03c1\n[PROOFSTEP]\nintro s hs _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nt : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\na\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 t x \u2202Measure.fst \u03c1\n[PROOFSTEP]\nrw [set_lintegral_iInf_gt_preCdf \u03c1 t hs, set_lintegral_preCdf_fst \u03c1 t hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, HasCondCdf \u03c1 a\n[PROOFSTEP]\nfilter_upwards [monotone_preCdf \u03c1, preCdf_le_one \u03c1, tendsto_preCdf_atTop_one \u03c1, tendsto_preCdf_atBot_zero \u03c1,\n inf_gt_preCdf \u03c1] with a h1 h2 h3 h4 h5\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\na : \u03b1\nh1 : Monotone fun r => preCdf \u03c1 r a\nh2 : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\nh3 : Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\nh4 : Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)\nh5 : \u2200 (t : \u211a), \u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) a = preCdf \u03c1 t a\n\u22a2 HasCondCdf \u03c1 a\n[PROOFSTEP]\nexact \u27e8h1, h2, h3, h4, h5\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\n\u22a2 HasCondCdf \u03c1 a\n[PROOFSTEP]\nrw [condCdfSet, mem_compl_iff] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {b | \u00acHasCondCdf \u03c1 b}\n\u22a2 HasCondCdf \u03c1 a\n[PROOFSTEP]\nhave h_ss := subset_toMeasurable \u03c1.fst {b | \u00acHasCondCdf \u03c1 b}\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {b | \u00acHasCondCdf \u03c1 b}\nh_ss : {b | \u00acHasCondCdf \u03c1 b} \u2286 toMeasurable (Measure.fst \u03c1) {b | \u00acHasCondCdf \u03c1 b}\n\u22a2 HasCondCdf \u03c1 a\n[PROOFSTEP]\nby_contra ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {b | \u00acHasCondCdf \u03c1 b}\nh_ss : {b | \u00acHasCondCdf \u03c1 b} \u2286 toMeasurable (Measure.fst \u03c1) {b | \u00acHasCondCdf \u03c1 b}\nha : \u00acHasCondCdf \u03c1 a\n\u22a2 False\n[PROOFSTEP]\nexact h (h_ss ha)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 condCdfSet \u03c1\n[PROOFSTEP]\nsimp_rw [ae_iff, condCdfSet, not_mem_compl_iff, setOf_mem_eq, measure_toMeasurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 \u2191\u2191(Measure.fst \u03c1) {b | \u00acHasCondCdf \u03c1 b} = 0\n[PROOFSTEP]\nexact hasCondCdf_ae \u03c1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\nr : \u211a\n\u22a2 condCdfRat \u03c1 a r = if r < 0 then 0 else 1\n[PROOFSTEP]\nsimp only [condCdfRat, h, if_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\nr : \u211a\n\u22a2 condCdfRat \u03c1 a r = ENNReal.toReal (preCdf \u03c1 r a)\n[PROOFSTEP]\nsimp only [condCdfRat, h, if_true]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 Monotone (condCdfRat \u03c1 a)\n[PROOFSTEP]\nby_cases h : a \u2208 condCdfSet \u03c1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\n\u22a2 Monotone (condCdfRat \u03c1 a)\n[PROOFSTEP]\nsimp only [condCdfRat, h, if_true, forall_const, and_self_iff]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\n\u22a2 Monotone fun r => ENNReal.toReal (preCdf \u03c1 r a)\n[PROOFSTEP]\nintro r r' hrr'\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\nr r' : \u211a\nhrr' : r \u2264 r'\n\u22a2 (fun r => ENNReal.toReal (preCdf \u03c1 r a)) r \u2264 (fun r => ENNReal.toReal (preCdf \u03c1 r a)) r'\n[PROOFSTEP]\nhave h' := hasCondCdf_of_mem_condCdfSet h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\nr r' : \u211a\nhrr' : r \u2264 r'\nh' : HasCondCdf \u03c1 a\n\u22a2 (fun r => ENNReal.toReal (preCdf \u03c1 r a)) r \u2264 (fun r => ENNReal.toReal (preCdf \u03c1 r a)) r'\n[PROOFSTEP]\nhave h_ne_top : \u2200 r, preCdf \u03c1 r a \u2260 \u221e := fun r => ((h'.le_one r).trans_lt ENNReal.one_lt_top).ne\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\nr r' : \u211a\nhrr' : r \u2264 r'\nh' : HasCondCdf \u03c1 a\nh_ne_top : \u2200 (r : \u211a), preCdf \u03c1 r a \u2260 \u22a4\n\u22a2 (fun r => ENNReal.toReal (preCdf \u03c1 r a)) r \u2264 (fun r => ENNReal.toReal (preCdf \u03c1 r a)) r'\n[PROOFSTEP]\nrw [ENNReal.toReal_le_toReal (h_ne_top _) (h_ne_top _)]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\nr r' : \u211a\nhrr' : r \u2264 r'\nh' : HasCondCdf \u03c1 a\nh_ne_top : \u2200 (r : \u211a), preCdf \u03c1 r a \u2260 \u22a4\n\u22a2 preCdf \u03c1 r a \u2264 preCdf \u03c1 r' a\n[PROOFSTEP]\nexact h'.1 hrr'\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 Monotone (condCdfRat \u03c1 a)\n[PROOFSTEP]\nsimp only [condCdfRat, h, if_false]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 Monotone fun r => if r < 0 then 0 else 1\n[PROOFSTEP]\nintro x y hxy\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\nx y : \u211a\nhxy : x \u2264 y\n\u22a2 (fun r => if r < 0 then 0 else 1) x \u2264 (fun r => if r < 0 then 0 else 1) y\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\nx y : \u211a\nhxy : x \u2264 y\n\u22a2 (if x < 0 then 0 else 1) \u2264 if y < 0 then 0 else 1\n[PROOFSTEP]\nsplit_ifs with h_1 h_2 h_2\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\nx y : \u211a\nhxy : x \u2264 y\nh_1 : x < 0\nh_2 : y < 0\n\u22a2 0 \u2264 0\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\nx y : \u211a\nhxy : x \u2264 y\nh_1 : x < 0\nh_2 : \u00acy < 0\n\u22a2 0 \u2264 1\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\nx y : \u211a\nhxy : x \u2264 y\nh_1 : \u00acx < 0\nh_2 : y < 0\n\u22a2 1 \u2264 0\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\nx y : \u211a\nhxy : x \u2264 y\nh_1 : \u00acx < 0\nh_2 : \u00acy < 0\n\u22a2 1 \u2264 1\n[PROOFSTEP]\nexacts [le_rfl, zero_le_one, absurd (hxy.trans_lt h_2) h_1, le_rfl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nq : \u211a\n\u22a2 Measurable fun a => condCdfRat \u03c1 a q\n[PROOFSTEP]\nsimp_rw [condCdfRat, ite_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nq : \u211a\n\u22a2 Measurable fun a => if a \u2208 condCdfSet \u03c1 then ENNReal.toReal (preCdf \u03c1 q a) else if q < 0 then 0 else 1\n[PROOFSTEP]\nexact Measurable.ite (measurableSet_condCdfSet \u03c1) measurable_preCdf.ennreal_toReal measurable_const\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 0 \u2264 condCdfRat \u03c1 a r\n[PROOFSTEP]\nunfold condCdfRat\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 0 \u2264 ite (a \u2208 condCdfSet \u03c1) (fun r => ENNReal.toReal (preCdf \u03c1 r a)) (fun r => if r < 0 then 0 else 1) r\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh\u271d : a \u2208 condCdfSet \u03c1\n\u22a2 0 \u2264 (fun r => ENNReal.toReal (preCdf \u03c1 r a)) r\n[PROOFSTEP]\nexact ENNReal.toReal_nonneg\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh\u271d : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 0 \u2264 (fun r => if r < 0 then 0 else 1) r\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh\u271d : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 0 \u2264 if r < 0 then 0 else 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh\u271d\u00b9 : \u00aca \u2208 condCdfSet \u03c1\nh\u271d : r < 0\n\u22a2 0 \u2264 0\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh\u271d\u00b9 : \u00aca \u2208 condCdfSet \u03c1\nh\u271d : \u00acr < 0\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nexacts [le_rfl, zero_le_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 condCdfRat \u03c1 a r \u2264 1\n[PROOFSTEP]\nunfold condCdfRat\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 ite (a \u2208 condCdfSet \u03c1) (fun r => ENNReal.toReal (preCdf \u03c1 r a)) (fun r => if r < 0 then 0 else 1) r \u2264 1\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : a \u2208 condCdfSet \u03c1\n\u22a2 (fun r => ENNReal.toReal (preCdf \u03c1 r a)) r \u2264 1\n[PROOFSTEP]\nrefine' ENNReal.toReal_le_of_le_ofReal zero_le_one _\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : a \u2208 condCdfSet \u03c1\n\u22a2 preCdf \u03c1 r a \u2264 ENNReal.ofReal 1\n[PROOFSTEP]\nrw [ENNReal.ofReal_one]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : a \u2208 condCdfSet \u03c1\n\u22a2 preCdf \u03c1 r a \u2264 1\n[PROOFSTEP]\nexact (hasCondCdf_of_mem_condCdfSet h).le_one r\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 (fun r => if r < 0 then 0 else 1) r \u2264 1\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 (if r < 0 then 0 else 1) \u2264 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : \u00aca \u2208 condCdfSet \u03c1\nh\u271d : r < 0\n\u22a2 0 \u2264 1\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nh : \u00aca \u2208 condCdfSet \u03c1\nh\u271d : \u00acr < 0\n\u22a2 1 \u2264 1\n[PROOFSTEP]\nexacts [zero_le_one, le_rfl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 Tendsto (condCdfRat \u03c1 a) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nunfold condCdfRat\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 Tendsto (if a \u2208 condCdfSet \u03c1 then fun r => ENNReal.toReal (preCdf \u03c1 r a) else fun r => if r < 0 then 0 else 1) atBot\n (\ud835\udcdd 0)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\n\u22a2 Tendsto (fun r => ENNReal.toReal (preCdf \u03c1 r a)) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 ENNReal.zero_toReal, ENNReal.tendsto_toReal_iff]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nexact (hasCondCdf_of_mem_condCdfSet h).tendsto_atBot_zero\n[GOAL]\ncase pos.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\n\u22a2 \u2200 (i : \u211a), preCdf \u03c1 i a \u2260 \u22a4\n[PROOFSTEP]\nhave h' := hasCondCdf_of_mem_condCdfSet h\n[GOAL]\ncase pos.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\nh' : HasCondCdf \u03c1 a\n\u22a2 \u2200 (i : \u211a), preCdf \u03c1 i a \u2260 \u22a4\n[PROOFSTEP]\nexact fun r => ((h'.le_one r).trans_lt ENNReal.one_lt_top).ne\n[GOAL]\ncase pos.hx\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\n\u22a2 0 \u2260 \u22a4\n[PROOFSTEP]\nexact ENNReal.zero_ne_top\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 Tendsto (fun r => if r < 0 then 0 else 1) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' (tendsto_congr' _).mp tendsto_const_nhds\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 (fun x => 0) =\u1da0[atBot] fun r => if r < 0 then 0 else 1\n[PROOFSTEP]\nrw [EventuallyEq, eventually_atBot]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 \u2203 a, \u2200 (b : \u211a), b \u2264 a \u2192 0 = if b < 0 then 0 else 1\n[PROOFSTEP]\nrefine' \u27e8-1, fun q hq => (if_pos (hq.trans_lt _)).symm\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\nq : \u211a\nhq : q \u2264 -1\n\u22a2 -1 < 0\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 Tendsto (condCdfRat \u03c1 a) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nunfold condCdfRat\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 Tendsto (if a \u2208 condCdfSet \u03c1 then fun r => ENNReal.toReal (preCdf \u03c1 r a) else fun r => if r < 0 then 0 else 1) atTop\n (\ud835\udcdd 1)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\n\u22a2 Tendsto (fun r => ENNReal.toReal (preCdf \u03c1 r a)) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nhave h' := hasCondCdf_of_mem_condCdfSet h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\nh' : HasCondCdf \u03c1 a\n\u22a2 Tendsto (fun r => ENNReal.toReal (preCdf \u03c1 r a)) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nrw [\u2190 ENNReal.one_toReal, ENNReal.tendsto_toReal_iff]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\nh' : HasCondCdf \u03c1 a\n\u22a2 Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nexact h'.tendsto_atTop_one\n[GOAL]\ncase pos.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\nh' : HasCondCdf \u03c1 a\n\u22a2 \u2200 (i : \u211a), preCdf \u03c1 i a \u2260 \u22a4\n[PROOFSTEP]\nexact fun r => ((h'.le_one r).trans_lt ENNReal.one_lt_top).ne\n[GOAL]\ncase pos.hx\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : a \u2208 condCdfSet \u03c1\nh' : HasCondCdf \u03c1 a\n\u22a2 1 \u2260 \u22a4\n[PROOFSTEP]\nexact ENNReal.one_ne_top\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 Tendsto (fun r => if r < 0 then 0 else 1) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nrefine' (tendsto_congr' _).mp tendsto_const_nhds\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 (fun x => 1) =\u1da0[atTop] fun r => if r < 0 then 0 else 1\n[PROOFSTEP]\nrw [EventuallyEq, eventually_atTop]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 \u2203 a, \u2200 (b : \u211a), b \u2265 a \u2192 1 = if b < 0 then 0 else 1\n[PROOFSTEP]\nexact \u27e80, fun q hq => (if_neg (not_lt.mpr hq)).symm\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\n\u22a2 (fun a => condCdfRat \u03c1 a r) =\u1d50[Measure.fst \u03c1] fun a => ENNReal.toReal (preCdf \u03c1 r a)\n[PROOFSTEP]\nfilter_upwards [mem_condCdfSet_ae \u03c1] with a ha using condCdfRat_of_mem \u03c1 a ha r\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\n\u22a2 (fun a => ENNReal.ofReal (condCdfRat \u03c1 a r)) =\u1d50[Measure.fst \u03c1] preCdf \u03c1 r\n[PROOFSTEP]\nfilter_upwards [condCdfRat_ae_eq \u03c1 r, preCdf_le_one \u03c1] with a ha ha_le_one\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\na : \u03b1\nha : condCdfRat \u03c1 a r = ENNReal.toReal (preCdf \u03c1 r a)\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 ENNReal.ofReal (condCdfRat \u03c1 a r) = preCdf \u03c1 r a\n[PROOFSTEP]\nrw [ha, ENNReal.ofReal_toReal]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\na : \u03b1\nha : condCdfRat \u03c1 a r = ENNReal.toReal (preCdf \u03c1 r a)\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 preCdf \u03c1 r a \u2260 \u22a4\n[PROOFSTEP]\nexact ((ha_le_one r).trans_lt ENNReal.one_lt_top).ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\n\u22a2 \u2a05 (r : \u2191(Ioi t)), condCdfRat \u03c1 a \u2191r = condCdfRat \u03c1 a t\n[PROOFSTEP]\nby_cases ha : a \u2208 condCdfSet \u03c1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\n\u22a2 \u2a05 (r : \u2191(Ioi t)), condCdfRat \u03c1 a \u2191r = condCdfRat \u03c1 a t\n[PROOFSTEP]\nsimp_rw [condCdfRat_of_mem \u03c1 a ha]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\n\u22a2 \u2a05 (r : \u2191(Ioi t)), ENNReal.toReal (preCdf \u03c1 (\u2191r) a) = ENNReal.toReal (preCdf \u03c1 t a)\n[PROOFSTEP]\nhave ha' := hasCondCdf_of_mem_condCdfSet ha\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\nha' : HasCondCdf \u03c1 a\n\u22a2 \u2a05 (r : \u2191(Ioi t)), ENNReal.toReal (preCdf \u03c1 (\u2191r) a) = ENNReal.toReal (preCdf \u03c1 t a)\n[PROOFSTEP]\nrw [\u2190 ENNReal.toReal_iInf]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\nha' : HasCondCdf \u03c1 a\n\u22a2 ENNReal.toReal (\u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) a) = ENNReal.toReal (preCdf \u03c1 t a)\n[PROOFSTEP]\nsuffices \u2a05 i : \u21a5(Ioi t), preCdf \u03c1 (\u2191i) a = preCdf \u03c1 t a by rw [this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\nha' : HasCondCdf \u03c1 a\nthis : \u2a05 (i : \u2191(Ioi t)), preCdf \u03c1 (\u2191i) a = preCdf \u03c1 t a\n\u22a2 ENNReal.toReal (\u2a05 (r : \u2191(Ioi t)), preCdf \u03c1 (\u2191r) a) = ENNReal.toReal (preCdf \u03c1 t a)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\nha' : HasCondCdf \u03c1 a\n\u22a2 \u2a05 (i : \u2191(Ioi t)), preCdf \u03c1 (\u2191i) a = preCdf \u03c1 t a\n[PROOFSTEP]\nrw [\u2190 ha'.iInf_rat_gt_eq]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : a \u2208 condCdfSet \u03c1\nha' : HasCondCdf \u03c1 a\n\u22a2 \u2200 (i : \u2191(Ioi t)), preCdf \u03c1 (\u2191i) a \u2260 \u22a4\n[PROOFSTEP]\nexact fun r => ((ha'.le_one r).trans_lt ENNReal.one_lt_top).ne\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 \u2a05 (r : \u2191(Ioi t)), condCdfRat \u03c1 a \u2191r = condCdfRat \u03c1 a t\n[PROOFSTEP]\nsimp_rw [condCdfRat_of_not_mem \u03c1 a ha]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 (\u2a05 (r : \u2191(Ioi t)), if \u2191r < 0 then 0 else 1) = if t < 0 then 0 else 1\n[PROOFSTEP]\nhave h_bdd : BddBelow (range fun r : \u21a5(Ioi t) => ite ((r : \u211a) < 0) (0 : \u211d) 1) :=\n by\n refine' \u27e80, fun x hx => _\u27e9\n obtain \u27e8y, rfl\u27e9 := mem_range.mpr hx\n dsimp only\n split_ifs\n exacts [le_rfl, zero_le_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\n\u22a2 BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\n[PROOFSTEP]\nrefine' \u27e80, fun x hx => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nx : \u211d\nhx : x \u2208 range fun r => if \u2191r < 0 then 0 else 1\n\u22a2 0 \u2264 x\n[PROOFSTEP]\nobtain \u27e8y, rfl\u27e9 := mem_range.mpr hx\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\ny : \u2191(Ioi t)\nhx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1\n\u22a2 0 \u2264 (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\ny : \u2191(Ioi t)\nhx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1\n\u22a2 0 \u2264 if \u2191y < 0 then 0 else 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\ny : \u2191(Ioi t)\nhx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1\nh\u271d : \u2191y < 0\n\u22a2 0 \u2264 0\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\ny : \u2191(Ioi t)\nhx : (fun y => (fun r => if \u2191r < 0 then 0 else 1) y) y \u2208 range fun r => if \u2191r < 0 then 0 else 1\nh\u271d : \u00ac\u2191y < 0\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nexacts [le_rfl, zero_le_one]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\n\u22a2 (\u2a05 (r : \u2191(Ioi t)), if \u2191r < 0 then 0 else 1) = if t < 0 then 0 else 1\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\n\u22a2 (\u2a05 (r : \u2191(Ioi t)), if \u2191r < 0 then 0 else 1) = 0\n[PROOFSTEP]\nrefine' le_antisymm _ (le_ciInf fun x => _)\n[GOAL]\ncase pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\n\u22a2 (\u2a05 (r : \u2191(Ioi t)), if \u2191r < 0 then 0 else 1) \u2264 0\n[PROOFSTEP]\nobtain \u27e8q, htq, hq_neg\u27e9 : \u2203 q, t < q \u2227 q < 0 := by\n refine' \u27e8t / 2, _, _\u27e9\n \u00b7 linarith\n \u00b7 linarith\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\n\u22a2 \u2203 q, t < q \u2227 q < 0\n[PROOFSTEP]\nrefine' \u27e8t / 2, _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\n\u22a2 t < t / 2\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\n\u22a2 t / 2 < 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase pos.refine'_1.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nq : \u211a\nhtq : t < q\nhq_neg : q < 0\n\u22a2 (\u2a05 (r : \u2191(Ioi t)), if \u2191r < 0 then 0 else 1) \u2264 0\n[PROOFSTEP]\nrefine' (ciInf_le h_bdd \u27e8q, htq\u27e9).trans _\n[GOAL]\ncase pos.refine'_1.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nq : \u211a\nhtq : t < q\nhq_neg : q < 0\n\u22a2 (if \u2191{ val := q, property := htq } < 0 then 0 else 1) \u2264 0\n[PROOFSTEP]\nrw [if_pos]\n[GOAL]\ncase pos.refine'_1.intro.intro.hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nq : \u211a\nhtq : t < q\nhq_neg : q < 0\n\u22a2 \u2191{ val := q, property := htq } < 0\n[PROOFSTEP]\nrwa [Subtype.coe_mk]\n[GOAL]\ncase pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nx : \u2191(Ioi t)\n\u22a2 0 \u2264 if \u2191x < 0 then 0 else 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nx : \u2191(Ioi t)\nh\u271d : \u2191x < 0\n\u22a2 0 \u2264 0\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : t < 0\nx : \u2191(Ioi t)\nh\u271d : \u00ac\u2191x < 0\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nexacts [le_rfl, zero_le_one]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\n\u22a2 (\u2a05 (r : \u2191(Ioi t)), if \u2191r < 0 then 0 else 1) = 1\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\n\u22a2 (\u2a05 (r : \u2191(Ioi t)), if \u2191r < 0 then 0 else 1) \u2264 1\n[PROOFSTEP]\nrefine' (ciInf_le h_bdd \u27e8t + 1, lt_add_one t\u27e9).trans _\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\n\u22a2 (if \u2191{ val := t + 1, property := (_ : t < t + 1) } < 0 then 0 else 1) \u2264 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\nh\u271d : \u2191{ val := t + 1, property := (_ : t < t + 1) } < 0\n\u22a2 0 \u2264 1\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\nh\u271d : \u00ac\u2191{ val := t + 1, property := (_ : t < t + 1) } < 0\n\u22a2 1 \u2264 1\n[PROOFSTEP]\nexacts [zero_le_one, le_rfl]\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\n\u22a2 1 \u2264 \u2a05 (r : \u2191(Ioi t)), if \u2191r < 0 then 0 else 1\n[PROOFSTEP]\nrefine' le_ciInf fun x => _\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\nx : \u2191(Ioi t)\n\u22a2 1 \u2264 if \u2191x < 0 then 0 else 1\n[PROOFSTEP]\nrw [if_neg]\n[GOAL]\ncase neg.refine'_2.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : \u00act < 0\nx : \u2191(Ioi t)\n\u22a2 \u00ac\u2191x < 0\n[PROOFSTEP]\nrw [not_lt] at h \u22a2\n[GOAL]\ncase neg.refine'_2.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nt : \u211a\nha : \u00aca \u2208 condCdfSet \u03c1\nh_bdd : BddBelow (range fun r => if \u2191r < 0 then 0 else 1)\nh : 0 \u2264 t\nx : \u2191(Ioi t)\n\u22a2 0 \u2264 \u2191x\n[PROOFSTEP]\nexact h.trans (mem_Ioi.mp x.prop).le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 condCdf' \u03c1 a x = \u2a05 (r : { r // x < \u2191r }), condCdfRat \u03c1 a \u2191r\n[PROOFSTEP]\nrw [condCdf']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 condCdf' \u03c1 a \u2191r = condCdfRat \u03c1 a r\n[PROOFSTEP]\nrw [\u2190 inf_gt_condCdfRat \u03c1 a r, condCdf']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 \u2a05 (r_1 : { r' // \u2191r < \u2191r' }), condCdfRat \u03c1 a \u2191r_1 = \u2a05 (r_1 : \u2191(Ioi r)), condCdfRat \u03c1 a \u2191r_1\n[PROOFSTEP]\nrefine' Equiv.iInf_congr _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 { r' // \u2191r < \u2191r' } \u2243 \u2191(Ioi r)\n[PROOFSTEP]\nexact\n { toFun := fun t => \u27e8t.1, by exact_mod_cast t.2\u27e9\n invFun := fun t => \u27e8t.1, by exact_mod_cast t.2\u27e9\n left_inv := fun t => by simp only [Subtype.coe_eta]\n right_inv := fun t => by simp only [Subtype.coe_eta] }\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nt : { r' // \u2191r < \u2191r' }\n\u22a2 \u2191t \u2208 Ioi r\n[PROOFSTEP]\nexact_mod_cast t.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nt : \u2191(Ioi r)\n\u22a2 \u2191r < \u2191\u2191t\n[PROOFSTEP]\nexact_mod_cast t.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nt : { r' // \u2191r < \u2191r' }\n\u22a2 (fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) }) ((fun t => { val := \u2191t, property := (_ : r < \u2191t) }) t) = t\n[PROOFSTEP]\nsimp only [Subtype.coe_eta]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nt : \u2191(Ioi r)\n\u22a2 (fun t => { val := \u2191t, property := (_ : r < \u2191t) }) ((fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) }) t) = t\n[PROOFSTEP]\nsimp only [Subtype.coe_eta]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\n\u22a2 \u2200 (x : { r' // \u2191r < \u2191r' }),\n condCdfRat \u03c1 a\n \u2191(\u2191{ toFun := fun t => { val := \u2191t, property := (_ : r < \u2191t) },\n invFun := fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) },\n left_inv :=\n (_ :\n \u2200 (t : { r' // \u2191r < \u2191r' }),\n { val := \u2191t, property := (_ : \u2191r < \u2191\u2191((fun t => { val := \u2191t, property := (_ : r < \u2191t) }) t)) } =\n t),\n right_inv :=\n (_ :\n \u2200 (t : \u2191(Ioi r)),\n { val := \u2191t, property := (_ : r < \u2191((fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) }) t)) } =\n t) }\n x) =\n condCdfRat \u03c1 a \u2191x\n[PROOFSTEP]\nintro t\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211a\nt : { r' // \u2191r < \u2191r' }\n\u22a2 condCdfRat \u03c1 a\n \u2191(\u2191{ toFun := fun t => { val := \u2191t, property := (_ : r < \u2191t) },\n invFun := fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) },\n left_inv :=\n (_ :\n \u2200 (t : { r' // \u2191r < \u2191r' }),\n { val := \u2191t, property := (_ : \u2191r < \u2191\u2191((fun t => { val := \u2191t, property := (_ : r < \u2191t) }) t)) } = t),\n right_inv :=\n (_ :\n \u2200 (t : \u2191(Ioi r)),\n { val := \u2191t, property := (_ : r < \u2191((fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) }) t)) } =\n t) }\n t) =\n condCdfRat \u03c1 a \u2191t\n[PROOFSTEP]\nsimp only [Equiv.coe_fn_mk, Subtype.coe_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211d\n\u22a2 0 \u2264 condCdf' \u03c1 a r\n[PROOFSTEP]\nhave : Nonempty { r' : \u211a // r < \u2191r' } := by\n obtain \u27e8r, hrx\u27e9 := exists_rat_gt r\n exact \u27e8\u27e8r, hrx\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211d\n\u22a2 Nonempty { r' // r < \u2191r' }\n[PROOFSTEP]\nobtain \u27e8r, hrx\u27e9 := exists_rat_gt r\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr\u271d : \u211d\nr : \u211a\nhrx : r\u271d < \u2191r\n\u22a2 Nonempty { r' // r\u271d < \u2191r' }\n[PROOFSTEP]\nexact \u27e8\u27e8r, hrx\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211d\nthis : Nonempty { r' // r < \u2191r' }\n\u22a2 0 \u2264 condCdf' \u03c1 a r\n[PROOFSTEP]\nrw [condCdf'_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nr : \u211d\nthis : Nonempty { r' // r < \u2191r' }\n\u22a2 0 \u2264 \u2a05 (r_1 : { r' // r < \u2191r' }), condCdfRat \u03c1 a \u2191r_1\n[PROOFSTEP]\nexact le_ciInf fun r' => condCdfRat_nonneg \u03c1 a _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 BddBelow (range fun r => condCdfRat \u03c1 a \u2191r)\n[PROOFSTEP]\nrefine' \u27e80, fun z => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx z : \u211d\n\u22a2 (z \u2208 range fun r => condCdfRat \u03c1 a \u2191r) \u2192 0 \u2264 z\n[PROOFSTEP]\nrintro \u27e8u, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nu : { r' // x < \u2191r' }\n\u22a2 0 \u2264 (fun r => condCdfRat \u03c1 a \u2191r) u\n[PROOFSTEP]\nexact condCdfRat_nonneg \u03c1 a _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 Monotone (condCdf' \u03c1 a)\n[PROOFSTEP]\nintro x y hxy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx y : \u211d\nhxy : x \u2264 y\n\u22a2 condCdf' \u03c1 a x \u2264 condCdf' \u03c1 a y\n[PROOFSTEP]\nhave : Nonempty { r' : \u211a // y < \u2191r' } := by\n obtain \u27e8r, hrx\u27e9 := exists_rat_gt y\n exact \u27e8\u27e8r, hrx\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx y : \u211d\nhxy : x \u2264 y\n\u22a2 Nonempty { r' // y < \u2191r' }\n[PROOFSTEP]\nobtain \u27e8r, hrx\u27e9 := exists_rat_gt y\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx y : \u211d\nhxy : x \u2264 y\nr : \u211a\nhrx : y < \u2191r\n\u22a2 Nonempty { r' // y < \u2191r' }\n[PROOFSTEP]\nexact \u27e8\u27e8r, hrx\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx y : \u211d\nhxy : x \u2264 y\nthis : Nonempty { r' // y < \u2191r' }\n\u22a2 condCdf' \u03c1 a x \u2264 condCdf' \u03c1 a y\n[PROOFSTEP]\nsimp_rw [condCdf'_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx y : \u211d\nhxy : x \u2264 y\nthis : Nonempty { r' // y < \u2191r' }\n\u22a2 \u2a05 (r : { r' // x < \u2191r' }), condCdfRat \u03c1 a \u2191r \u2264 \u2a05 (r : { r' // y < \u2191r' }), condCdfRat \u03c1 a \u2191r\n[PROOFSTEP]\nrefine' le_ciInf fun r => (ciInf_le _ _).trans_eq _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx y : \u211d\nhxy : x \u2264 y\nthis : Nonempty { r' // y < \u2191r' }\nr : { r' // y < \u2191r' }\n\u22a2 BddBelow (range fun r => condCdfRat \u03c1 a \u2191r)\n[PROOFSTEP]\nexact bddBelow_range_condCdfRat_gt \u03c1 a x\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx y : \u211d\nhxy : x \u2264 y\nthis : Nonempty { r' // y < \u2191r' }\nr : { r' // y < \u2191r' }\n\u22a2 { r' // x < \u2191r' }\n[PROOFSTEP]\nexact \u27e8r.1, hxy.trans_lt r.prop\u27e9\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx y : \u211d\nhxy : x \u2264 y\nthis : Nonempty { r' // y < \u2191r' }\nr : { r' // y < \u2191r' }\n\u22a2 condCdfRat \u03c1 a \u2191{ val := \u2191r, property := (_ : x < \u2191\u2191r) } = condCdfRat \u03c1 a \u2191r\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ici x) x\n[PROOFSTEP]\nrw [\u2190 continuousWithinAt_Ioi_iff_Ici]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x\n[PROOFSTEP]\nconvert Monotone.tendsto_nhdsWithin_Ioi (monotone_condCdf' \u03c1 a) x\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (sInf (condCdf' \u03c1 a '' Ioi x)))\n[PROOFSTEP]\nrw [sInf_image']\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194\n Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 (a_1 : \u2191(Ioi x)), condCdf' \u03c1 a \u2191a_1))\n[PROOFSTEP]\nhave h' : \u2a05 r : Ioi x, condCdf' \u03c1 a r = \u2a05 r : { r' : \u211a // x < r' }, condCdf' \u03c1 a r :=\n by\n refine' Real.iInf_Ioi_eq_iInf_rat_gt x _ (monotone_condCdf' \u03c1 a)\n refine' \u27e80, fun z => _\u27e9\n rintro \u27e8u, -, rfl\u27e9\n exact condCdf'_nonneg \u03c1 a u\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 \u2a05 (r : \u2191(Ioi x)), condCdf' \u03c1 a \u2191r = \u2a05 (r : { r' // x < \u2191r' }), condCdf' \u03c1 a \u2191\u2191r\n[PROOFSTEP]\nrefine' Real.iInf_Ioi_eq_iInf_rat_gt x _ (monotone_condCdf' \u03c1 a)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 BddBelow (condCdf' \u03c1 a '' Ioi x)\n[PROOFSTEP]\nrefine' \u27e80, fun z => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx z : \u211d\n\u22a2 z \u2208 condCdf' \u03c1 a '' Ioi x \u2192 0 \u2264 z\n[PROOFSTEP]\nrintro \u27e8u, -, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx u : \u211d\n\u22a2 0 \u2264 condCdf' \u03c1 a u\n[PROOFSTEP]\nexact condCdf'_nonneg \u03c1 a u\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 (r : \u2191(Ioi x)), condCdf' \u03c1 a \u2191r = \u2a05 (r : { r' // x < \u2191r' }), condCdf' \u03c1 a \u2191\u2191r\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194\n Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 (a_1 : \u2191(Ioi x)), condCdf' \u03c1 a \u2191a_1))\n[PROOFSTEP]\nhave h'' : \u2a05 r : { r' : \u211a // x < r' }, condCdf' \u03c1 a r = \u2a05 r : { r' : \u211a // x < r' }, condCdfRat \u03c1 a r :=\n by\n congr with r\n exact condCdf'_eq_condCdfRat \u03c1 a r\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 (r : \u2191(Ioi x)), condCdf' \u03c1 a \u2191r = \u2a05 (r : { r' // x < \u2191r' }), condCdf' \u03c1 a \u2191\u2191r\n\u22a2 \u2a05 (r : { r' // x < \u2191r' }), condCdf' \u03c1 a \u2191\u2191r = \u2a05 (r : { r' // x < \u2191r' }), condCdfRat \u03c1 a \u2191r\n[PROOFSTEP]\ncongr with r\n[GOAL]\ncase e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 (r : \u2191(Ioi x)), condCdf' \u03c1 a \u2191r = \u2a05 (r : { r' // x < \u2191r' }), condCdf' \u03c1 a \u2191\u2191r\nr : { r' // x < \u2191r' }\n\u22a2 condCdf' \u03c1 a \u2191\u2191r = condCdfRat \u03c1 a \u2191r\n[PROOFSTEP]\nexact condCdf'_eq_condCdfRat \u03c1 a r\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 (r : \u2191(Ioi x)), condCdf' \u03c1 a \u2191r = \u2a05 (r : { r' // x < \u2191r' }), condCdf' \u03c1 a \u2191\u2191r\nh'' : \u2a05 (r : { r' // x < \u2191r' }), condCdf' \u03c1 a \u2191\u2191r = \u2a05 (r : { r' // x < \u2191r' }), condCdfRat \u03c1 a \u2191r\n\u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194\n Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 (a_1 : \u2191(Ioi x)), condCdf' \u03c1 a \u2191a_1))\n[PROOFSTEP]\nrw [h', h'', ContinuousWithinAt]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 (r : \u2191(Ioi x)), condCdf' \u03c1 a \u2191r = \u2a05 (r : { r' // x < \u2191r' }), condCdf' \u03c1 a \u2191\u2191r\nh'' : \u2a05 (r : { r' // x < \u2191r' }), condCdf' \u03c1 a \u2191\u2191r = \u2a05 (r : { r' // x < \u2191r' }), condCdfRat \u03c1 a \u2191r\n\u22a2 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (condCdf' \u03c1 a x)) \u2194\n Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 (r : { r' // x < \u2191r' }), condCdfRat \u03c1 a \u2191r))\n[PROOFSTEP]\ncongr!\n[GOAL]\ncase a.a.h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nh' : \u2a05 (r : \u2191(Ioi x)), condCdf' \u03c1 a \u2191r = \u2a05 (r : { r' // x < \u2191r' }), condCdf' \u03c1 a \u2191\u2191r\nh'' : \u2a05 (r : { r' // x < \u2191r' }), condCdf' \u03c1 a \u2191\u2191r = \u2a05 (r : { r' // x < \u2191r' }), condCdfRat \u03c1 a \u2191r\n\u22a2 condCdf' \u03c1 a x = \u2a05 (r : { r' // x < \u2191r' }), condCdfRat \u03c1 a \u2191r\n[PROOFSTEP]\nexact condCdf'_def'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 \u2191(condCdf \u03c1 a) x \u2264 1\n[PROOFSTEP]\nobtain \u27e8r, hrx\u27e9 := exists_rat_gt x\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nr : \u211a\nhrx : x < \u2191r\n\u22a2 \u2191(condCdf \u03c1 a) x \u2264 1\n[PROOFSTEP]\nrw [\u2190 StieltjesFunction.iInf_rat_gt_eq]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nr : \u211a\nhrx : x < \u2191r\n\u22a2 \u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r \u2264 1\n[PROOFSTEP]\nsimp_rw [condCdf_eq_condCdfRat]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nr : \u211a\nhrx : x < \u2191r\n\u22a2 \u2a05 (r : { r' // x < \u2191r' }), condCdfRat \u03c1 a \u2191r \u2264 1\n[PROOFSTEP]\nrefine' ciInf_le_of_le (bddBelow_range_condCdfRat_gt \u03c1 a x) _ (condCdfRat_le_one _ _ _)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\nr : \u211a\nhrx : x < \u2191r\n\u22a2 { r' // x < \u2191r' }\n[PROOFSTEP]\nexact \u27e8r, hrx\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nhave h_exists : \u2200 x : \u211d, \u2203 q : \u211a, x < q \u2227 \u2191q < x + 1 := fun x => exists_rat_btwn (lt_add_one x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nlet qs : \u211d \u2192 \u211a := fun x => (h_exists x).choose\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nhave hqs_tendsto : Tendsto qs atBot atBot := by\n rw [tendsto_atBot_atBot]\n refine' fun q => \u27e8q - 1, fun y hy => _\u27e9\n have h_le : \u2191(qs y) \u2264 (q : \u211d) - 1 + 1 := (h_exists y).choose_spec.2.le.trans (add_le_add hy le_rfl)\n rw [sub_add_cancel] at h_le \n exact_mod_cast h_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\n\u22a2 Tendsto qs atBot atBot\n[PROOFSTEP]\nrw [tendsto_atBot_atBot]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\n\u22a2 \u2200 (b : \u211a), \u2203 i, \u2200 (a : \u211d), a \u2264 i \u2192 qs a \u2264 b\n[PROOFSTEP]\nrefine' fun q => \u27e8q - 1, fun y hy => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nq : \u211a\ny : \u211d\nhy : y \u2264 \u2191q - 1\n\u22a2 qs y \u2264 q\n[PROOFSTEP]\nhave h_le : \u2191(qs y) \u2264 (q : \u211d) - 1 + 1 := (h_exists y).choose_spec.2.le.trans (add_le_add hy le_rfl)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nq : \u211a\ny : \u211d\nhy : y \u2264 \u2191q - 1\nh_le : \u2191(qs y) \u2264 \u2191q - 1 + 1\n\u22a2 qs y \u2264 q\n[PROOFSTEP]\nrw [sub_add_cancel] at h_le \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nq : \u211a\ny : \u211d\nhy : y \u2264 \u2191q - 1\nh_le : \u2191(qs y) \u2264 \u2191q\n\u22a2 qs y \u2264 q\n[PROOFSTEP]\nexact_mod_cast h_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nhqs_tendsto : Tendsto qs atBot atBot\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atBot (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine'\n tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds ((tendsto_condCdfRat_atBot \u03c1 a).comp hqs_tendsto)\n (condCdf_nonneg \u03c1 a) fun x => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nhqs_tendsto : Tendsto qs atBot atBot\nx : \u211d\n\u22a2 \u2191(condCdf \u03c1 a) x \u2264 (condCdfRat \u03c1 a \u2218 qs) x\n[PROOFSTEP]\nrw [Function.comp_apply, \u2190 condCdf_eq_condCdfRat]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x < \u2191q \u2227 \u2191q < x + 1\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x < \u2191q \u2227 \u2191q < x + 1)\nhqs_tendsto : Tendsto qs atBot atBot\nx : \u211d\n\u22a2 \u2191(condCdf \u03c1 a) x \u2264 \u2191(condCdf \u03c1 a) \u2191(qs x)\n[PROOFSTEP]\nexact (condCdf \u03c1 a).mono (h_exists x).choose_spec.1.le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nhave h_exists : \u2200 x : \u211d, \u2203 q : \u211a, x - 1 < q \u2227 \u2191q < x := fun x => exists_rat_btwn (sub_one_lt x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nlet qs : \u211d \u2192 \u211a := fun x => (h_exists x).choose\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nhave hqs_tendsto : Tendsto qs atTop atTop := by\n rw [tendsto_atTop_atTop]\n refine' fun q => \u27e8q + 1, fun y hy => _\u27e9\n have h_le : y - 1 \u2264 qs y := (h_exists y).choose_spec.1.le\n rw [sub_le_iff_le_add] at h_le \n exact_mod_cast le_of_add_le_add_right (hy.trans h_le)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\n\u22a2 Tendsto qs atTop atTop\n[PROOFSTEP]\nrw [tendsto_atTop_atTop]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\n\u22a2 \u2200 (b : \u211a), \u2203 i, \u2200 (a : \u211d), i \u2264 a \u2192 b \u2264 qs a\n[PROOFSTEP]\nrefine' fun q => \u27e8q + 1, fun y hy => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nq : \u211a\ny : \u211d\nhy : \u2191q + 1 \u2264 y\n\u22a2 q \u2264 qs y\n[PROOFSTEP]\nhave h_le : y - 1 \u2264 qs y := (h_exists y).choose_spec.1.le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nq : \u211a\ny : \u211d\nhy : \u2191q + 1 \u2264 y\nh_le : y - 1 \u2264 \u2191(qs y)\n\u22a2 q \u2264 qs y\n[PROOFSTEP]\nrw [sub_le_iff_le_add] at h_le \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nq : \u211a\ny : \u211d\nhy : \u2191q + 1 \u2264 y\nh_le : y \u2264 \u2191(qs y) + 1\n\u22a2 q \u2264 qs y\n[PROOFSTEP]\nexact_mod_cast le_of_add_le_add_right (hy.trans h_le)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nhqs_tendsto : Tendsto qs atTop atTop\n\u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nrefine'\n tendsto_of_tendsto_of_tendsto_of_le_of_le ((tendsto_condCdfRat_atTop \u03c1 a).comp hqs_tendsto) tendsto_const_nhds _\n (condCdf_le_one \u03c1 a)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nhqs_tendsto : Tendsto qs atTop atTop\n\u22a2 condCdfRat \u03c1 a \u2218 qs \u2264 \u2191(condCdf \u03c1 a)\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nhqs_tendsto : Tendsto qs atTop atTop\nx : \u211d\n\u22a2 (condCdfRat \u03c1 a \u2218 qs) x \u2264 \u2191(condCdf \u03c1 a) x\n[PROOFSTEP]\nrw [Function.comp_apply, \u2190 condCdf_eq_condCdfRat]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nh_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x\nqs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x)\nhqs_tendsto : Tendsto qs atTop atTop\nx : \u211d\n\u22a2 \u2191(condCdf \u03c1 a) \u2191(qs x) \u2264 \u2191(condCdf \u03c1 a) x\n[PROOFSTEP]\nexact (condCdf \u03c1 a).mono (le_of_lt (h_exists x).choose_spec.2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\n\u22a2 (fun a => \u2191(condCdf \u03c1 a) \u2191r) =\u1d50[Measure.fst \u03c1] fun a => ENNReal.toReal (preCdf \u03c1 r a)\n[PROOFSTEP]\nfilter_upwards [mem_condCdfSet_ae \u03c1] with a ha using (condCdf_eq_condCdfRat \u03c1 a r).trans (condCdfRat_of_mem \u03c1 a ha r)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\n\u22a2 (fun a => ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r)) =\u1d50[Measure.fst \u03c1] preCdf \u03c1 r\n[PROOFSTEP]\nfilter_upwards [condCdf_ae_eq \u03c1 r, preCdf_le_one \u03c1] with a ha ha_le_one\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\na : \u03b1\nha : \u2191(condCdf \u03c1 a) \u2191r = ENNReal.toReal (preCdf \u03c1 r a)\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) = preCdf \u03c1 r a\n[PROOFSTEP]\nrw [ha, ENNReal.ofReal_toReal]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\na : \u03b1\nha : \u2191(condCdf \u03c1 a) \u2191r = ENNReal.toReal (preCdf \u03c1 r a)\nha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1\n\u22a2 preCdf \u03c1 r a \u2260 \u22a4\n[PROOFSTEP]\nexact ((ha_le_one r).trans_lt ENNReal.one_lt_top).ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nx : \u211d\n\u22a2 Measurable fun a => \u2191(condCdf \u03c1 a) x\n[PROOFSTEP]\nhave : (fun a => condCdf \u03c1 a x) = fun a => \u2a05 r : { r' : \u211a // x < r' }, condCdfRat \u03c1 a \u2191r :=\n by\n ext1 a\n rw [\u2190 StieltjesFunction.iInf_rat_gt_eq]\n congr with q\n rw [condCdf_eq_condCdfRat]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nx : \u211d\n\u22a2 (fun a => \u2191(condCdf \u03c1 a) x) = fun a => \u2a05 (r : { r' // x < \u2191r' }), condCdfRat \u03c1 a \u2191r\n[PROOFSTEP]\next1 a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nx : \u211d\na : \u03b1\n\u22a2 \u2191(condCdf \u03c1 a) x = \u2a05 (r : { r' // x < \u2191r' }), condCdfRat \u03c1 a \u2191r\n[PROOFSTEP]\nrw [\u2190 StieltjesFunction.iInf_rat_gt_eq]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nx : \u211d\na : \u03b1\n\u22a2 \u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r = \u2a05 (r : { r' // x < \u2191r' }), condCdfRat \u03c1 a \u2191r\n[PROOFSTEP]\ncongr with q\n[GOAL]\ncase h.e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nx : \u211d\na : \u03b1\nq : { r' // x < \u2191r' }\n\u22a2 \u2191(condCdf \u03c1 a) \u2191\u2191q = condCdfRat \u03c1 a \u2191q\n[PROOFSTEP]\nrw [condCdf_eq_condCdfRat]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nx : \u211d\nthis : (fun a => \u2191(condCdf \u03c1 a) x) = fun a => \u2a05 (r : { r' // x < \u2191r' }), condCdfRat \u03c1 a \u2191r\n\u22a2 Measurable fun a => \u2191(condCdf \u03c1 a) x\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\nx : \u211d\nthis : (fun a => \u2191(condCdf \u03c1 a) x) = fun a => \u2a05 (r : { r' // x < \u2191r' }), condCdfRat \u03c1 a \u2191r\n\u22a2 Measurable fun a => \u2a05 (r : { r' // x < \u2191r' }), condCdfRat \u03c1 a \u2191r\n[PROOFSTEP]\nexact measurable_ciInf (fun q => measurable_condCdfRat \u03c1 q) fun a => bddBelow_range_condCdfRat_gt \u03c1 a _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)\n[PROOFSTEP]\nhave : \u2200\u1d50 a \u2202\u03c1.fst, a \u2208 s \u2192 ENNReal.ofReal (condCdf \u03c1 a r) = preCdf \u03c1 r a := by\n filter_upwards [ofReal_condCdf_ae_eq \u03c1 r] with a ha using fun _ => ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 s \u2192 ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) = preCdf \u03c1 r a\n[PROOFSTEP]\nfilter_upwards [ofReal_condCdf_ae_eq \u03c1 r] with a ha using fun _ => ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 s \u2192 ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) = preCdf \u03c1 r a\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)\n[PROOFSTEP]\nrw [set_lintegral_congr_fun hs this, set_lintegral_preCdf_fst \u03c1 r hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nr : \u211a\ns : Set \u03b1\nhs : MeasurableSet s\nthis : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 s \u2192 ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) = preCdf \u03c1 r a\n\u22a2 \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)\n[PROOFSTEP]\nexact \u03c1.IicSnd_apply r hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n[PROOFSTEP]\nby_cases h\u03c1_zero : \u03c1.fst.restrict s = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n[PROOFSTEP]\nrw [h\u03c1_zero, lintegral_zero_measure]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 0 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n[PROOFSTEP]\nrefine' le_antisymm (zero_le _) _\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) \u2264 0\n[PROOFSTEP]\ncalc\n \u03c1 (s \u00d7\u02e2 Iic x) \u2264 \u03c1 (Prod.fst \u207b\u00b9' s) := measure_mono (prod_subset_preimage_fst s (Iic x))\n _ = \u03c1.fst s := by rw [Measure.fst_apply hs]\n _ = \u03c1.fst.restrict s univ := by rw [Measure.restrict_apply_univ]\n _ = 0 := by simp only [h\u03c1_zero, Measure.coe_zero, Pi.zero_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 \u2191\u2191\u03c1 (Prod.fst \u207b\u00b9' s) = \u2191\u2191(Measure.fst \u03c1) s\n[PROOFSTEP]\nrw [Measure.fst_apply hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 \u2191\u2191(Measure.fst \u03c1) s = \u2191\u2191(Measure.restrict (Measure.fst \u03c1) s) univ\n[PROOFSTEP]\nrw [Measure.restrict_apply_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : Measure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 \u2191\u2191(Measure.restrict (Measure.fst \u03c1) s) univ = 0\n[PROOFSTEP]\nsimp only [h\u03c1_zero, Measure.coe_zero, Pi.zero_apply]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n[PROOFSTEP]\nhave h :\n \u222b\u207b a in s, ENNReal.ofReal (condCdf \u03c1 a x) \u2202\u03c1.fst =\n \u222b\u207b a in s, ENNReal.ofReal (\u2a05 r : { r' : \u211a // x < r' }, condCdf \u03c1 a r) \u2202\u03c1.fst :=\n by\n congr with a : 1\n rw [\u2190 (condCdf \u03c1 a).iInf_rat_gt_eq x]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\n[PROOFSTEP]\ncongr with a : 1\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\na : \u03b1\n\u22a2 ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) = ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r)\n[PROOFSTEP]\nrw [\u2190 (condCdf \u03c1 a).iInf_rat_gt_eq x]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n[PROOFSTEP]\nhave h_nonempty : Nonempty { r' : \u211a // x < \u2191r' } :=\n by\n obtain \u27e8r, hrx\u27e9 := exists_rat_gt x\n exact \u27e8\u27e8r, hrx\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\n\u22a2 Nonempty { r' // x < \u2191r' }\n[PROOFSTEP]\nobtain \u27e8r, hrx\u27e9 := exists_rat_gt x\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nr : \u211a\nhrx : x < \u2191r\n\u22a2 Nonempty { r' // x < \u2191r' }\n[PROOFSTEP]\nexact \u27e8\u27e8r, hrx\u27e9\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n[PROOFSTEP]\nsimp_rw [ENNReal.ofReal_cinfi]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2a05 (i : { r' // x < \u2191r' }), ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n[PROOFSTEP]\nhave h_coe : \u2200 b : { r' : \u211a // x < \u2191r' }, (b : \u211d) = ((b : \u211a) : \u211d) := fun _ => by congr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nx\u271d : { r' // x < \u2191r' }\n\u22a2 \u2191\u2191x\u271d = \u2191\u2191x\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2a05 (i : { r' // x < \u2191r' }), ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191i) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n[PROOFSTEP]\nrw [lintegral_iInf_directed_of_measurable h\u03c1_zero fun q : { r' : \u211a // x < \u2191r' } =>\n (measurable_condCdf \u03c1 q).ennreal_ofReal]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2a05 (b : { r' // x < \u2191r' }), \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\ncase neg.hf_int\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2200 (b : { r' // x < \u2191r' }), \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 \u2260 \u22a4\ncase neg.h_directed\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 Directed (fun x x_1 => x \u2265 x_1) fun q x_1 => ENNReal.ofReal (\u2191(condCdf \u03c1 x_1) \u2191\u2191q)\n[PROOFSTEP]\nrotate_left\n[GOAL]\ncase neg.hf_int\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2200 (b : { r' // x < \u2191r' }), \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 \u2260 \u22a4\n[PROOFSTEP]\nintro b\n[GOAL]\ncase neg.hf_int\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\nb : { r' // x < \u2191r' }\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 \u2260 \u22a4\n[PROOFSTEP]\nrw [set_lintegral_condCdf_rat \u03c1 _ hs]\n[GOAL]\ncase neg.hf_int\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\nb : { r' // x < \u2191r' }\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191\u2191b) \u2260 \u22a4\n[PROOFSTEP]\nexact measure_ne_top \u03c1 _\n[GOAL]\ncase neg.h_directed\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 Directed (fun x x_1 => x \u2265 x_1) fun q x_1 => ENNReal.ofReal (\u2191(condCdf \u03c1 x_1) \u2191\u2191q)\n[PROOFSTEP]\nrefine' Monotone.directed_ge fun i j hij a => ENNReal.ofReal_le_ofReal ((condCdf \u03c1 a).mono _)\n[GOAL]\ncase neg.h_directed\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ni j : { r' // x < \u2191r' }\nhij : i \u2264 j\na : \u03b1\n\u22a2 \u2191\u2191i \u2264 \u2191\u2191j\n[PROOFSTEP]\nrw [h_coe, h_coe]\n[GOAL]\ncase neg.h_directed\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ni j : { r' // x < \u2191r' }\nhij : i \u2264 j\na : \u03b1\n\u22a2 \u2191\u2191i \u2264 \u2191\u2191j\n[PROOFSTEP]\nexact_mod_cast hij\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2a05 (b : { r' // x < \u2191r' }), \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191\u2191b) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n[PROOFSTEP]\nsimp_rw [set_lintegral_condCdf_rat \u03c1 _ hs]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2a05 (b : { r' // x < \u2191r' }), \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191\u2191b) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n[PROOFSTEP]\nrw [\u2190 measure_iInter_eq_iInf]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2191\u2191\u03c1 (\u22c2 (i : { r' // x < \u2191r' }), s \u00d7\u02e2 Iic \u2191\u2191i) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n[PROOFSTEP]\nrw [\u2190 prod_iInter]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 \u22c2 (i : { r' // x < \u2191r' }), Iic \u2191\u2191i) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n[PROOFSTEP]\ncongr with y\n[GOAL]\ncase neg.e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ny : \u211d\n\u22a2 y \u2208 \u22c2 (i : { r' // x < \u2191r' }), Iic \u2191\u2191i \u2194 y \u2208 Iic x\n[PROOFSTEP]\nsimp only [mem_iInter, mem_Iic, Subtype.forall, Subtype.coe_mk]\n[GOAL]\ncase neg.e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ny : \u211d\n\u22a2 (\u2200 (a : \u211a), x < \u2191a \u2192 y \u2264 \u2191a) \u2194 y \u2264 x\n[PROOFSTEP]\nexact \u27e8le_of_forall_lt_rat_imp_le, fun hyx q hq => hyx.trans hq.le\u27e9\n[GOAL]\ncase neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2200 (i : { r' // x < \u2191r' }), MeasurableSet (s \u00d7\u02e2 Iic \u2191\u2191i)\n[PROOFSTEP]\nexact fun i => hs.prod measurableSet_Iic\n[GOAL]\ncase neg.hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 Directed (fun x x_1 => x \u2287 x_1) fun b => s \u00d7\u02e2 Iic \u2191\u2191b\n[PROOFSTEP]\nrefine' Monotone.directed_ge fun i j hij => _\n[GOAL]\ncase neg.hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ni j : { r' // x < \u2191r' }\nhij : i \u2264 j\n\u22a2 s \u00d7\u02e2 Iic \u2191\u2191i \u2264 s \u00d7\u02e2 Iic \u2191\u2191j\n[PROOFSTEP]\nrefine' prod_subset_prod_iff.mpr (Or.inl \u27e8subset_rfl, Iic_subset_Iic.mpr _\u27e9)\n[GOAL]\ncase neg.hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\ni j : { r' // x < \u2191r' }\nhij : i \u2264 j\n\u22a2 \u2191\u2191i \u2264 \u2191\u2191j\n[PROOFSTEP]\nexact_mod_cast hij\n[GOAL]\ncase neg.hfin\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03c1_zero : \u00acMeasure.restrict (Measure.fst \u03c1) s = 0\nh :\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 =\n \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2a05 (r : { r' // x < \u2191r' }), \u2191(condCdf \u03c1 a) \u2191\u2191r) \u2202Measure.fst \u03c1\nh_nonempty : Nonempty { r' // x < \u2191r' }\nh_coe : \u2200 (b : { r' // x < \u2191r' }), \u2191\u2191b = \u2191\u2191b\n\u22a2 \u2203 i, \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191\u2191i) \u2260 \u22a4\n[PROOFSTEP]\nexact \u27e8h_nonempty.some, measure_ne_top _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic x)\n[PROOFSTEP]\nrw [\u2190 set_lintegral_univ, set_lintegral_condCdf \u03c1 _ MeasurableSet.univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\n\u22a2 Integrable fun a => \u2191(condCdf \u03c1 a) x\n[PROOFSTEP]\nrefine' integrable_of_forall_fin_meas_le _ (measure_lt_top \u03c1.fst univ) _ fun t _ _ => _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\n\u22a2 AEStronglyMeasurable (fun a => \u2191(condCdf \u03c1 a) x) (Measure.fst \u03c1)\n[PROOFSTEP]\nexact (stronglyMeasurable_condCdf \u03c1 _).aestronglyMeasurable\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\n\u22a2 \u222b\u207b (x_1 : \u03b1) in t, \u2191\u2016\u2191(condCdf \u03c1 x_1) x\u2016\u208a \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.fst \u03c1) univ\n[PROOFSTEP]\nhave : \u2200 y, (\u2016condCdf \u03c1 y x\u2016\u208a : \u211d\u22650\u221e) \u2264 1 := by\n intro y\n rw [Real.nnnorm_of_nonneg (condCdf_nonneg _ _ _)]\n -- Porting note: was exact_mod_cast condCdf_le_one _ _ _\n simp only [ENNReal.coe_le_one_iff]\n exact condCdf_le_one _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\n\u22a2 \u2200 (y : \u03b1), \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1\n[PROOFSTEP]\nintro y\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\ny : \u03b1\n\u22a2 \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1\n[PROOFSTEP]\nrw [Real.nnnorm_of_nonneg (condCdf_nonneg _ _ _)]\n -- Porting note: was exact_mod_cast condCdf_le_one _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\ny : \u03b1\n\u22a2 \u2191{ val := \u2191(condCdf \u03c1 y) x, property := (_ : 0 \u2264 \u2191(condCdf \u03c1 y) x) } \u2264 1\n[PROOFSTEP]\nsimp only [ENNReal.coe_le_one_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\ny : \u03b1\n\u22a2 { val := \u2191(condCdf \u03c1 y) x, property := (_ : 0 \u2264 \u2191(condCdf \u03c1 y) x) } \u2264 1\n[PROOFSTEP]\nexact condCdf_le_one _ _ _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\nthis : \u2200 (y : \u03b1), \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1\n\u22a2 \u222b\u207b (x_1 : \u03b1) in t, \u2191\u2016\u2191(condCdf \u03c1 x_1) x\u2016\u208a \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.fst \u03c1) univ\n[PROOFSTEP]\nrefine' (set_lintegral_mono (measurable_condCdf _ _).ennnorm measurable_one fun y _ => this y).trans _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\nthis : \u2200 (y : \u03b1), \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1\n\u22a2 \u222b\u207b (x : \u03b1) in t, OfNat.ofNat 1 x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.fst \u03c1) univ\n[PROOFSTEP]\nsimp only [Pi.one_apply, lintegral_one, Measure.restrict_apply, MeasurableSet.univ, univ_inter]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\nt : Set \u03b1\nx\u271d\u00b9 : MeasurableSet t\nx\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4\nthis : \u2200 (y : \u03b1), \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1\n\u22a2 \u2191\u2191(Measure.fst \u03c1) t \u2264 \u2191\u2191(Measure.fst \u03c1) univ\n[PROOFSTEP]\nexact measure_mono (subset_univ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b (a : \u03b1) in s, \u2191(condCdf \u03c1 a) x \u2202Measure.fst \u03c1 = ENNReal.toReal (\u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x))\n[PROOFSTEP]\nhave h := set_lintegral_condCdf \u03c1 x hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh : \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\u22a2 \u222b (a : \u03b1) in s, \u2191(condCdf \u03c1 a) x \u2202Measure.fst \u03c1 = ENNReal.toReal (\u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x))\n[PROOFSTEP]\nrw [\u2190 ofReal_integral_eq_lintegral_ofReal] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh : ENNReal.ofReal (\u222b (x_1 : \u03b1) in s, \u2191(condCdf \u03c1 x_1) x \u2202Measure.fst \u03c1) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\u22a2 \u222b (a : \u03b1) in s, \u2191(condCdf \u03c1 a) x \u2202Measure.fst \u03c1 = ENNReal.toReal (\u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x))\n[PROOFSTEP]\nrw [\u2190 h, ENNReal.toReal_ofReal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh : ENNReal.ofReal (\u222b (x_1 : \u03b1) in s, \u2191(condCdf \u03c1 x_1) x \u2202Measure.fst \u03c1) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\u22a2 0 \u2264 \u222b (x_1 : \u03b1) in s, \u2191(condCdf \u03c1 x_1) x \u2202Measure.fst \u03c1\n[PROOFSTEP]\nexact integral_nonneg fun _ => condCdf_nonneg _ _ _\n[GOAL]\ncase hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh : \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\u22a2 Integrable fun a => \u2191(condCdf \u03c1 a) x\n[PROOFSTEP]\nexact (integrable_condCdf _ _).integrableOn\n[GOAL]\ncase f_nn\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\ns : Set \u03b1\nhs : MeasurableSet s\nh : \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)\n\u22a2 0 \u2264\u1d50[Measure.restrict (Measure.fst \u03c1) s] fun a => \u2191(condCdf \u03c1 a) x\n[PROOFSTEP]\nexact eventually_of_forall fun _ => condCdf_nonneg _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ninst\u271d : IsFiniteMeasure \u03c1\nx : \u211d\n\u22a2 \u222b (a : \u03b1), \u2191(condCdf \u03c1 a) x \u2202Measure.fst \u03c1 = ENNReal.toReal (\u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic x))\n[PROOFSTEP]\nrw [\u2190 set_integral_condCdf \u03c1 _ MeasurableSet.univ, Measure.restrict_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) (Iic x) = ENNReal.ofReal (\u2191(condCdf \u03c1 a) x)\n[PROOFSTEP]\nrw [\u2190 sub_zero (condCdf \u03c1 a x)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\nx : \u211d\n\u22a2 \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) (Iic x) = ENNReal.ofReal (\u2191(condCdf \u03c1 a) x - 0)\n[PROOFSTEP]\nexact (condCdf \u03c1 a).measure_Iic (tendsto_condCdf_atBot \u03c1 a) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ = 1\n[PROOFSTEP]\nrw [\u2190 ENNReal.ofReal_one, \u2190 sub_zero (1 : \u211d)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\na : \u03b1\n\u22a2 \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ = ENNReal.ofReal (1 - 0)\n[PROOFSTEP]\nexact StieltjesFunction.measure_univ _ (tendsto_condCdf_atBot \u03c1 a) (tendsto_condCdf_atTop \u03c1 a)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\n\u22a2 Measurable fun a => StieltjesFunction.measure (condCdf \u03c1 a)\n[PROOFSTEP]\nrw [Measure.measurable_measure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\n\u22a2 \u2200 (s : Set \u211d), MeasurableSet s \u2192 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s\n[PROOFSTEP]\nrefine' fun s hs =>\n ?_\n -- Porting note: supplied `C`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s\n[PROOFSTEP]\nrefine'\n MeasurableSpace.induction_on_inter (C := fun s => Measurable fun b \u21a6 StieltjesFunction.measure (condCdf \u03c1 b) s)\n (borel_eq_generateFrom_Iic \u211d) isPiSystem_Iic _ _ _ _ hs\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) \u2205\n[PROOFSTEP]\nsimp only [measure_empty, measurable_const]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set \u211d), t \u2208 range Iic \u2192 (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) t\n[PROOFSTEP]\nrintro S \u27e8u, rfl\u27e9\n[GOAL]\ncase refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u211d\nhs : MeasurableSet s\nu : \u211d\n\u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) (Iic u)\n[PROOFSTEP]\nsimp_rw [measure_condCdf_Iic \u03c1 _ u]\n[GOAL]\ncase refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u211d\nhs : MeasurableSet s\nu : \u211d\n\u22a2 Measurable fun b => ENNReal.ofReal (\u2191(condCdf \u03c1 b) u)\n[PROOFSTEP]\nexact (measurable_condCdf \u03c1 u).ennreal_ofReal\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set \u211d),\n MeasurableSet t \u2192\n (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) t \u2192\n (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) t\u1d9c\n[PROOFSTEP]\nintro t ht ht_cd_meas\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u211d\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\n\u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\u1d9c\n[PROOFSTEP]\nhave : (fun a => (condCdf \u03c1 a).measure t\u1d9c) = (fun a => (condCdf \u03c1 a).measure univ) - fun a => (condCdf \u03c1 a).measure t :=\n by\n ext1 a\n rw [measure_compl ht (measure_ne_top (condCdf \u03c1 a).measure _), Pi.sub_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u211d\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\n\u22a2 (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\u1d9c) =\n (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ) - fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\n[PROOFSTEP]\next1 a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u211d\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\na : \u03b1\n\u22a2 \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\u1d9c =\n ((fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ) - fun a =>\n \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t)\n a\n[PROOFSTEP]\nrw [measure_compl ht (measure_ne_top (condCdf \u03c1 a).measure _), Pi.sub_apply]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u211d\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\nthis :\n (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\u1d9c) =\n (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ) - fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\n\u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\u1d9c\n[PROOFSTEP]\nsimp_rw [this, measure_condCdf_univ \u03c1]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u211d\nhs : MeasurableSet s\nt : Set \u211d\nht : MeasurableSet t\nht_cd_meas : Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) t\nthis :\n (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\u1d9c) =\n (fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ) - fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t\n\u22a2 Measurable ((fun a => 1) - fun a => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) t)\n[PROOFSTEP]\nexact Measurable.sub measurable_const ht_cd_meas\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u211d\nhs : MeasurableSet s\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u211d),\n Pairwise (Disjoint on f) \u2192\n (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n (\u2200 (i : \u2115), (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (f i)) \u2192\n (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nintro f hf_disj hf_meas hf_cd_meas\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u211d\nhs : MeasurableSet s\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_cd_meas : \u2200 (i : \u2115), (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (f i)\n\u22a2 Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nsimp_rw [measure_iUnion hf_disj hf_meas]\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\n\u03c1 : Measure (\u03b1 \u00d7 \u211d)\ns : Set \u211d\nhs : MeasurableSet s\nf : \u2115 \u2192 Set \u211d\nhf_disj : Pairwise (Disjoint on f)\nhf_meas : \u2200 (i : \u2115), MeasurableSet (f i)\nhf_cd_meas : \u2200 (i : \u2115), (fun s => Measurable fun b => \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) s) (f i)\n\u22a2 Measurable fun b => \u2211' (i : \u2115), \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 b)) (f i)\n[PROOFSTEP]\nexact Measurable.ennreal_tsum hf_cd_meas\n", "meta": {"mathlib_filename": "Mathlib.Probability.Kernel.CondCdf", "llama_tokens": 111741, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.737158174177441, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4341044364297553}} {"text": "[GOAL]\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\ny : \u211d\n\u22a2 P \u22a5 \u2191y\n[PROOFSTEP]\nrcases lt_trichotomy y 0 with (hy | rfl | hy)\n[GOAL]\ncase inl\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\ny : \u211d\nhy : y < 0\n\u22a2 P \u22a5 \u2191y\ncase inr.inl\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\n\u22a2 P \u22a5 \u21910\ncase inr.inr\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\ny : \u211d\nhy : 0 < y\n\u22a2 P \u22a5 \u2191y\n[PROOFSTEP]\nexacts [bot_neg y hy, bot_zero, bot_pos y hy]\n[GOAL]\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\nx : \u211d\n\u22a2 P \u2191x \u22a5\n[PROOFSTEP]\nrcases lt_trichotomy x 0 with (hx | rfl | hx)\n[GOAL]\ncase inl\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\nx : \u211d\nhx : x < 0\n\u22a2 P \u2191x \u22a5\ncase inr.inl\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\n\u22a2 P \u21910 \u22a5\ncase inr.inr\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\nx : \u211d\nhx : 0 < x\n\u22a2 P \u2191x \u22a5\n[PROOFSTEP]\nexacts [neg_bot x hx, zero_bot, pos_bot x hx]\n[GOAL]\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\nx : \u211d\n\u22a2 P \u2191x \u22a4\n[PROOFSTEP]\nrcases lt_trichotomy x 0 with (hx | rfl | hx)\n[GOAL]\ncase inl\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\nx : \u211d\nhx : x < 0\n\u22a2 P \u2191x \u22a4\ncase inr.inl\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\n\u22a2 P \u21910 \u22a4\ncase inr.inr\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\nx : \u211d\nhx : 0 < x\n\u22a2 P \u2191x \u22a4\n[PROOFSTEP]\nexacts [neg_top x hx, zero_top, pos_top x hx]\n[GOAL]\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\ny : \u211d\n\u22a2 P \u22a4 \u2191y\n[PROOFSTEP]\nrcases lt_trichotomy y 0 with (hy | rfl | hy)\n[GOAL]\ncase inl\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\ny : \u211d\nhy : y < 0\n\u22a2 P \u22a4 \u2191y\ncase inr.inl\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\n\u22a2 P \u22a4 \u21910\ncase inr.inr\nP : EReal \u2192 EReal \u2192 Prop\ntop_top : P \u22a4 \u22a4\ntop_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a4 \u2191x\ntop_zero : P \u22a4 0\ntop_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a4 \u2191x\ntop_bot : P \u22a4 \u22a5\npos_top : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a4\npos_bot : \u2200 (x : \u211d), 0 < x \u2192 P \u2191x \u22a5\nzero_top : P 0 \u22a4\ncoe_coe : \u2200 (x y : \u211d), P \u2191x \u2191y\nzero_bot : P 0 \u22a5\nneg_top : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a4\nneg_bot : \u2200 (x : \u211d), x < 0 \u2192 P \u2191x \u22a5\nbot_top : P \u22a5 \u22a4\nbot_pos : \u2200 (x : \u211d), 0 < x \u2192 P \u22a5 \u2191x\nbot_zero : P \u22a5 0\nbot_neg : \u2200 (x : \u211d), x < 0 \u2192 P \u22a5 \u2191x\nbot_bot : P \u22a5 \u22a5\ny : \u211d\nhy : 0 < y\n\u22a2 P \u22a4 \u2191y\n[PROOFSTEP]\nexacts [top_neg y hy, top_zero, top_pos y hy]\n[GOAL]\nx y : EReal\n\u22a2 x * y = y * x\n[PROOFSTEP]\ninduction' x using EReal.rec with x\n[GOAL]\ncase h_bot\ny : EReal\n\u22a2 \u22a5 * y = y * \u22a5\n[PROOFSTEP]\ninduction' y using EReal.rec with y\n[GOAL]\ncase h_real\ny : EReal\nx : \u211d\n\u22a2 \u2191x * y = y * \u2191x\n[PROOFSTEP]\ninduction' y using EReal.rec with y\n[GOAL]\ncase h_top\ny : EReal\n\u22a2 \u22a4 * y = y * \u22a4\n[PROOFSTEP]\ninduction' y using EReal.rec with y\n[GOAL]\ncase h_bot.h_bot\n\u22a2 \u22a5 * \u22a5 = \u22a5 * \u22a5\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase h_bot.h_bot\n\u22a2 \u22a5 * \u22a5 = \u22a5 * \u22a5\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase h_bot.h_bot\n\u22a2 \u22a5 * \u22a5 = \u22a5 * \u22a5\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_bot.h_real\ny : \u211d\n\u22a2 \u22a5 * \u2191y = \u2191y * \u22a5\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase h_bot.h_real\ny : \u211d\n\u22a2 \u22a5 * \u2191y = \u2191y * \u22a5\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase h_bot.h_real\ny : \u211d\n\u22a2 \u22a5 * \u2191y = \u2191y * \u22a5\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_bot.h_top\n\u22a2 \u22a5 * \u22a4 = \u22a4 * \u22a5\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase h_bot.h_top\n\u22a2 \u22a5 * \u22a4 = \u22a4 * \u22a5\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase h_bot.h_top\n\u22a2 \u22a5 * \u22a4 = \u22a4 * \u22a5\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_real.h_bot\nx : \u211d\n\u22a2 \u2191x * \u22a5 = \u22a5 * \u2191x\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase h_real.h_bot\nx : \u211d\n\u22a2 \u2191x * \u22a5 = \u22a5 * \u2191x\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase h_real.h_bot\nx : \u211d\n\u22a2 \u2191x * \u22a5 = \u22a5 * \u2191x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_real.h_real\nx y : \u211d\n\u22a2 \u2191x * \u2191y = \u2191y * \u2191x\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase h_real.h_real\nx y : \u211d\n\u22a2 \u2191x * \u2191y = \u2191y * \u2191x\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase h_real.h_real\nx y : \u211d\n\u22a2 \u2191x * \u2191y = \u2191y * \u2191x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_real.h_top\nx : \u211d\n\u22a2 \u2191x * \u22a4 = \u22a4 * \u2191x\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase h_real.h_top\nx : \u211d\n\u22a2 \u2191x * \u22a4 = \u22a4 * \u2191x\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase h_real.h_top\nx : \u211d\n\u22a2 \u2191x * \u22a4 = \u22a4 * \u2191x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_top.h_bot\n\u22a2 \u22a4 * \u22a5 = \u22a5 * \u22a4\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase h_top.h_bot\n\u22a2 \u22a4 * \u22a5 = \u22a5 * \u22a4\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase h_top.h_bot\n\u22a2 \u22a4 * \u22a5 = \u22a5 * \u22a4\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_top.h_real\ny : \u211d\n\u22a2 \u22a4 * \u2191y = \u2191y * \u22a4\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase h_top.h_real\ny : \u211d\n\u22a2 \u22a4 * \u2191y = \u2191y * \u22a4\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase h_top.h_real\ny : \u211d\n\u22a2 \u22a4 * \u2191y = \u2191y * \u22a4\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_top.h_top\n\u22a2 \u22a4 * \u22a4 = \u22a4 * \u22a4\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase h_top.h_top\n\u22a2 \u22a4 * \u22a4 = \u22a4 * \u22a4\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase h_top.h_top\n\u22a2 \u22a4 * \u22a4 = \u22a4 * \u22a4\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_real.h_real\nx y : \u211d\n\u22a2 \u2191x * \u2191y = \u2191y * \u2191x\n[PROOFSTEP]\nrw [\u2190 coe_mul, \u2190 coe_mul, mul_comm]\n[GOAL]\nx : EReal\n\u22a2 x * 1 = x\n[PROOFSTEP]\nrw [EReal.mul_comm, EReal.one_mul]\n[GOAL]\nx : EReal\n\u22a2 x * 0 = 0\n[PROOFSTEP]\nrw [EReal.mul_comm, EReal.zero_mul]\n[GOAL]\nx : EReal\nhx : x \u2260 \u22a4 \u2227 x \u2260 \u22a5\n\u22a2 \u2203 y, \u2191y = x\n[PROOFSTEP]\ninduction x using EReal.rec\n[GOAL]\ncase h_bot\nhx : \u22a5 \u2260 \u22a4 \u2227 \u22a5 \u2260 \u22a5\n\u22a2 \u2203 y, \u2191y = \u22a5\n[PROOFSTEP]\nsimp at hx \n[GOAL]\ncase h_real\na\u271d : \u211d\nhx : \u2191a\u271d \u2260 \u22a4 \u2227 \u2191a\u271d \u2260 \u22a5\n\u22a2 \u2203 y, \u2191y = \u2191a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h_top\nhx : \u22a4 \u2260 \u22a4 \u2227 \u22a4 \u2260 \u22a5\n\u22a2 \u2203 y, \u2191y = \u22a4\n[PROOFSTEP]\nsimp at hx \n[GOAL]\n\u22a2 range Real.toEReal = {\u22a5, \u22a4}\u1d9c\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nx : EReal\n\u22a2 x \u2208 range Real.toEReal \u2194 x \u2208 {\u22a5, \u22a4}\u1d9c\n[PROOFSTEP]\ninduction x using EReal.rec\n[GOAL]\ncase h.h_bot\n\u22a2 \u22a5 \u2208 range Real.toEReal \u2194 \u22a5 \u2208 {\u22a5, \u22a4}\u1d9c\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.h_real\na\u271d : \u211d\n\u22a2 \u2191a\u271d \u2208 range Real.toEReal \u2194 \u2191a\u271d \u2208 {\u22a5, \u22a4}\u1d9c\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.h_top\n\u22a2 \u22a4 \u2208 range Real.toEReal \u2194 \u22a4 \u2208 {\u22a5, \u22a4}\u1d9c\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 range Real.toEReal = Ioo \u22a5 \u22a4\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nx : EReal\n\u22a2 x \u2208 range Real.toEReal \u2194 x \u2208 Ioo \u22a5 \u22a4\n[PROOFSTEP]\ninduction x using EReal.rec\n[GOAL]\ncase h.h_bot\n\u22a2 \u22a5 \u2208 range Real.toEReal \u2194 \u22a5 \u2208 Ioo \u22a5 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.h_real\na\u271d : \u211d\n\u22a2 \u2191a\u271d \u2208 range Real.toEReal \u2194 \u2191a\u271d \u2208 Ioo \u22a5 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.h_top\n\u22a2 \u22a4 \u2208 range Real.toEReal \u2194 \u22a4 \u2208 Ioo \u22a5 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nx y : EReal\nh : x \u2264 y\nhx : x \u2260 \u22a5\nhy : y \u2260 \u22a4\n\u22a2 toReal x \u2264 toReal y\n[PROOFSTEP]\nlift x to \u211d using \u27e8ne_top_of_le_ne_top hy h, hx\u27e9\n[GOAL]\ncase intro\ny : EReal\nhy : y \u2260 \u22a4\nx : \u211d\nh : \u2191x \u2264 y\nhx : \u2191x \u2260 \u22a5\n\u22a2 toReal \u2191x \u2264 toReal y\n[PROOFSTEP]\nlift y to \u211d using \u27e8hy, ne_bot_of_le_ne_bot hx h\u27e9\n[GOAL]\ncase intro.intro\nx : \u211d\nhx : \u2191x \u2260 \u22a5\ny : \u211d\nhy : \u2191y \u2260 \u22a4\nh : \u2191x \u2264 \u2191y\n\u22a2 toReal \u2191x \u2264 toReal \u2191y\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nx : EReal\nhx : x \u2260 \u22a4\nh'x : x \u2260 \u22a5\n\u22a2 \u2191(toReal x) = x\n[PROOFSTEP]\nlift x to \u211d using \u27e8hx, h'x\u27e9\n[GOAL]\ncase intro\nx : \u211d\nhx : \u2191x \u2260 \u22a4\nh'x : \u2191x \u2260 \u22a5\n\u22a2 \u2191(toReal \u2191x) = \u2191x\n[PROOFSTEP]\nrfl\n[GOAL]\nx : EReal\nh : x \u2260 \u22a4\n\u22a2 x \u2264 \u2191(toReal x)\n[PROOFSTEP]\nby_cases h' : x = \u22a5\n[GOAL]\ncase pos\nx : EReal\nh : x \u2260 \u22a4\nh' : x = \u22a5\n\u22a2 x \u2264 \u2191(toReal x)\n[PROOFSTEP]\nsimp only [h', bot_le]\n[GOAL]\ncase neg\nx : EReal\nh : x \u2260 \u22a4\nh' : \u00acx = \u22a5\n\u22a2 x \u2264 \u2191(toReal x)\n[PROOFSTEP]\nsimp only [le_refl, coe_toReal h h']\n[GOAL]\nx : EReal\nh : x \u2260 \u22a5\n\u22a2 \u2191(toReal x) \u2264 x\n[PROOFSTEP]\nby_cases h' : x = \u22a4\n[GOAL]\ncase pos\nx : EReal\nh : x \u2260 \u22a5\nh' : x = \u22a4\n\u22a2 \u2191(toReal x) \u2264 x\n[PROOFSTEP]\nsimp only [h', le_top]\n[GOAL]\ncase neg\nx : EReal\nh : x \u2260 \u22a5\nh' : \u00acx = \u22a4\n\u22a2 \u2191(toReal x) \u2264 x\n[PROOFSTEP]\nsimp only [le_refl, coe_toReal h' h]\n[GOAL]\nx : EReal\n\u22a2 x = \u22a4 \u2194 \u2200 (y : \u211d), \u2191y < x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nx : EReal\n\u22a2 x = \u22a4 \u2192 \u2200 (y : \u211d), \u2191y < x\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mp\n\u22a2 \u2200 (y : \u211d), \u2191y < \u22a4\n[PROOFSTEP]\nexact EReal.coe_lt_top\n[GOAL]\ncase mpr\nx : EReal\n\u22a2 (\u2200 (y : \u211d), \u2191y < x) \u2192 x = \u22a4\n[PROOFSTEP]\ncontrapose!\n[GOAL]\ncase mpr\nx : EReal\n\u22a2 x \u2260 \u22a4 \u2192 \u2203 y, x \u2264 \u2191y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nx : EReal\nh : x \u2260 \u22a4\n\u22a2 \u2203 y, x \u2264 \u2191y\n[PROOFSTEP]\nexact \u27e8x.toReal, le_coe_toReal h\u27e9\n[GOAL]\nx : EReal\n\u22a2 x = \u22a5 \u2194 \u2200 (y : \u211d), x < \u2191y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nx : EReal\n\u22a2 x = \u22a5 \u2192 \u2200 (y : \u211d), x < \u2191y\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mp\n\u22a2 \u2200 (y : \u211d), \u22a5 < \u2191y\n[PROOFSTEP]\nexact bot_lt_coe\n[GOAL]\ncase mpr\nx : EReal\n\u22a2 (\u2200 (y : \u211d), x < \u2191y) \u2192 x = \u22a5\n[PROOFSTEP]\ncontrapose!\n[GOAL]\ncase mpr\nx : EReal\n\u22a2 x \u2260 \u22a5 \u2192 \u2203 y, \u2191y \u2264 x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nx : EReal\nh : x \u2260 \u22a5\n\u22a2 \u2203 y, \u2191y \u2264 x\n[PROOFSTEP]\nexact \u27e8x.toReal, coe_toReal_le h\u27e9\n[GOAL]\nx : \u211d\u22650\u221e\n\u22a2 \u2191x = 0 \u2194 x = 0\n[PROOFSTEP]\nrw [\u2190 coe_ennreal_eq_coe_ennreal_iff, coe_ennreal_zero]\n[GOAL]\nx : \u211d\u22650\u221e\n\u22a2 \u2191x = 1 \u2194 x = 1\n[PROOFSTEP]\nrw [\u2190 coe_ennreal_eq_coe_ennreal_iff, coe_ennreal_one]\n[GOAL]\nx : \u211d\u22650\u221e\n\u22a2 0 < \u2191x \u2194 0 < x\n[PROOFSTEP]\nrw [\u2190 coe_ennreal_zero, coe_ennreal_lt_coe_ennreal_iff]\n[GOAL]\nx y : \u211d\u22650\u221e\n\u22a2 \u2191(x + y) = \u2191x + \u2191y\n[PROOFSTEP]\ncases x\n[GOAL]\ncase none\ny : \u211d\u22650\u221e\n\u22a2 \u2191(none + y) = \u2191none + \u2191y\n[PROOFSTEP]\ncases y\n[GOAL]\ncase some\ny : \u211d\u22650\u221e\nval\u271d : \u211d\u22650\n\u22a2 \u2191(Option.some val\u271d + y) = \u2191(Option.some val\u271d) + \u2191y\n[PROOFSTEP]\ncases y\n[GOAL]\ncase none.none\n\u22a2 \u2191(none + none) = \u2191none + \u2191none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase none.some\nval\u271d : \u211d\u22650\n\u22a2 \u2191(none + Option.some val\u271d) = \u2191none + \u2191(Option.some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.none\nval\u271d : \u211d\u22650\n\u22a2 \u2191(Option.some val\u271d + none) = \u2191(Option.some val\u271d) + \u2191none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u211d\u22650\n\u22a2 \u2191(Option.some val\u271d\u00b9 + Option.some val\u271d) = \u2191(Option.some val\u271d\u00b9) + \u2191(Option.some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nx : \u211d\u22650\n\u22a2 \u2191(\u22a4 * \u2191x) = \u22a4 * \u2191\u2191x\n[PROOFSTEP]\nrcases eq_or_ne x 0 with (rfl | h0)\n[GOAL]\ncase inl\n\u22a2 \u2191(\u22a4 * \u21910) = \u22a4 * \u2191\u21910\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nx : \u211d\u22650\nh0 : x \u2260 0\n\u22a2 \u2191(\u22a4 * \u2191x) = \u22a4 * \u2191\u2191x\n[PROOFSTEP]\nrw [ENNReal.top_mul (ENNReal.coe_ne_zero.2 h0)]\n[GOAL]\ncase inr\nx : \u211d\u22650\nh0 : x \u2260 0\n\u22a2 \u2191\u22a4 = \u22a4 * \u2191\u2191x\n[PROOFSTEP]\nexact Eq.symm <| if_pos <| NNReal.coe_pos.2 h0.bot_lt\n[GOAL]\nx : \u211d\u22650\n\u22a2 \u2191(\u2191x * \u22a4) = \u2191\u2191x * \u2191\u22a4\n[PROOFSTEP]\nrw [mul_comm, coe_ennreal_top_mul, EReal.mul_comm, coe_ennreal_top]\n[GOAL]\nx y : \u211d\u22650\n\u22a2 \u2191(\u2191x * \u2191y) = \u2191\u2191x * \u2191\u2191y\n[PROOFSTEP]\nsimp only [\u2190 ENNReal.coe_mul, coe_nnreal_eq_coe_real, NNReal.coe_mul, EReal.coe_mul]\n[GOAL]\na b : \u211d\nh : \u2191a < \u2191b\n\u22a2 \u2203 x, \u2191a < \u2191\u2191x \u2227 \u2191\u2191x < \u2191b\n[PROOFSTEP]\nsimp [exists_rat_btwn (EReal.coe_lt_coe_iff.1 h)]\n[GOAL]\na : \u211d\nx\u271d : \u2191a < \u22a4\nb : \u211a\nhab : a < \u2191b\n\u22a2 \u2191a < \u2191\u2191b\n[PROOFSTEP]\nsimpa using hab\n[GOAL]\na : \u211d\nx\u271d : \u22a5 < \u2191a\nb : \u211a\nhab : \u2191b < a\n\u22a2 \u2191\u2191b < \u2191a\n[PROOFSTEP]\nsimpa using hab\n[GOAL]\nx : \u211d\n\u22a2 \u2191x \u2208 {\u22a5, \u22a4}\u1d9c\n[PROOFSTEP]\nsimp\n[GOAL]\nx\u271d : \u2191{\u22a5, \u22a4}\u1d9c\nx : EReal\nhx : x \u2208 {\u22a5, \u22a4}\u1d9c\n\u22a2 (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 {\u22a5, \u22a4}\u1d9c) }) ((fun x => toReal \u2191x) { val := x, property := hx }) =\n { val := x, property := hx }\n[PROOFSTEP]\nlift x to \u211d\n[GOAL]\nx\u271d : \u2191{\u22a5, \u22a4}\u1d9c\nx : EReal\nhx : x \u2208 {\u22a5, \u22a4}\u1d9c\n\u22a2 x \u2260 \u22a4 \u2227 x \u2260 \u22a5\n[PROOFSTEP]\nsimpa [not_or, and_comm] using hx\n[GOAL]\ncase intro\nx\u271d : \u2191{\u22a5, \u22a4}\u1d9c\nx : \u211d\nhx : \u2191x \u2208 {\u22a5, \u22a4}\u1d9c\n\u22a2 (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 {\u22a5, \u22a4}\u1d9c) }) ((fun x => toReal \u2191x) { val := \u2191x, property := hx }) =\n { val := \u2191x, property := hx }\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\n\u22a2 (fun x => toReal \u2191x) ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 {\u22a5, \u22a4}\u1d9c) }) x) = x\n[PROOFSTEP]\nsimp\n[GOAL]\nx y : EReal\n\u22a2 \u22a5 < x + y \u2194 \u22a5 < x \u2227 \u22a5 < y\n[PROOFSTEP]\nsimp [bot_lt_iff_ne_bot, not_or]\n[GOAL]\nx y : EReal\nhx : x \u2260 \u22a4\nh'x : x \u2260 \u22a5\nhy : y \u2260 \u22a4\nh'y : y \u2260 \u22a5\n\u22a2 toReal (x + y) = toReal x + toReal y\n[PROOFSTEP]\nlift x to \u211d using \u27e8hx, h'x\u27e9\n[GOAL]\ncase intro\ny : EReal\nhy : y \u2260 \u22a4\nh'y : y \u2260 \u22a5\nx : \u211d\nhx : \u2191x \u2260 \u22a4\nh'x : \u2191x \u2260 \u22a5\n\u22a2 toReal (\u2191x + y) = toReal \u2191x + toReal y\n[PROOFSTEP]\nlift y to \u211d using \u27e8hy, h'y\u27e9\n[GOAL]\ncase intro.intro\nx : \u211d\nhx : \u2191x \u2260 \u22a4\nh'x : \u2191x \u2260 \u22a5\ny : \u211d\nhy : \u2191y \u2260 \u22a4\nh'y : \u2191y \u2260 \u22a5\n\u22a2 toReal (\u2191x + \u2191y) = toReal \u2191x + toReal \u2191y\n[PROOFSTEP]\nrfl\n[GOAL]\nx z : \u211d\nh : \u2191x + \u22a4 \u2264 \u2191x + \u2191z\n\u22a2 \u22a4 \u2264 \u2191z\n[PROOFSTEP]\nsimp only [coe_add_top, \u2190 coe_add, top_le_iff, coe_ne_top] at h \n[GOAL]\nx : \u211d\nx\u271d : EReal\nh : \u2191x + x\u271d \u2264 \u2191x + \u22a5\n\u22a2 x\u271d \u2264 \u22a5\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nx y z : \u211d\nh : \u2191x + \u2191y \u2264 \u2191x + \u2191z\n\u22a2 \u2191y \u2264 \u2191z\n[PROOFSTEP]\nsimpa only [\u2190 coe_add, EReal.coe_le_coe_iff, add_le_add_iff_left] using h\n[GOAL]\nx y : EReal\nh : x < y\nz : \u211d\n\u22a2 \u2191z + x < \u2191z + y\n[PROOFSTEP]\nsimpa [add_comm] using add_lt_add_right_coe h z\n[GOAL]\nx y z t : EReal\nh1 : x < y\nh2 : z < t\n\u22a2 x + z < y + t\n[PROOFSTEP]\nrcases eq_or_ne x \u22a5 with (rfl | hx)\n[GOAL]\ncase inl\ny z t : EReal\nh2 : z < t\nh1 : \u22a5 < y\n\u22a2 \u22a5 + z < y + t\n[PROOFSTEP]\nsimp [h1, bot_le.trans_lt h2]\n[GOAL]\ncase inr\nx y z t : EReal\nh1 : x < y\nh2 : z < t\nhx : x \u2260 \u22a5\n\u22a2 x + z < y + t\n[PROOFSTEP]\nlift x to \u211d using \u27e8h1.ne_top, hx\u27e9\n[GOAL]\ncase inr.intro\ny z t : EReal\nh2 : z < t\nx : \u211d\nh1 : \u2191x < y\nhx : \u2191x \u2260 \u22a5\n\u22a2 \u2191x + z < y + t\n[PROOFSTEP]\ncalc\n (x : EReal) + z < x + t := add_lt_add_left_coe h2 _\n _ \u2264 y + t := add_le_add_right h1.le _\n[GOAL]\nx y z t : EReal\nh : x < y\nh' : z \u2264 t\nhbot : t \u2260 \u22a5\nhtop : t = \u22a4 \u2192 z = \u22a4 \u2192 x = \u22a5\n\u22a2 x + z < y + t\n[PROOFSTEP]\nrcases h'.eq_or_lt with (rfl | hlt)\n[GOAL]\ncase inl\nx y z : EReal\nh : x < y\nh' : z \u2264 z\nhbot : z \u2260 \u22a5\nhtop : z = \u22a4 \u2192 z = \u22a4 \u2192 x = \u22a5\n\u22a2 x + z < y + z\n[PROOFSTEP]\nrcases eq_or_ne z \u22a4 with (rfl | hz)\n[GOAL]\ncase inl.inl\nx y : EReal\nh : x < y\nh' : \u22a4 \u2264 \u22a4\nhbot : \u22a4 \u2260 \u22a5\nhtop : \u22a4 = \u22a4 \u2192 \u22a4 = \u22a4 \u2192 x = \u22a5\n\u22a2 x + \u22a4 < y + \u22a4\n[PROOFSTEP]\nobtain rfl := htop rfl rfl\n[GOAL]\ncase inl.inl\ny : EReal\nh' : \u22a4 \u2264 \u22a4\nhbot : \u22a4 \u2260 \u22a5\nh : \u22a5 < y\nhtop : \u22a4 = \u22a4 \u2192 \u22a4 = \u22a4 \u2192 \u22a5 = \u22a5\n\u22a2 \u22a5 + \u22a4 < y + \u22a4\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase inl.inr\nx y z : EReal\nh : x < y\nh' : z \u2264 z\nhbot : z \u2260 \u22a5\nhtop : z = \u22a4 \u2192 z = \u22a4 \u2192 x = \u22a5\nhz : z \u2260 \u22a4\n\u22a2 x + z < y + z\n[PROOFSTEP]\nlift z to \u211d using \u27e8hz, hbot\u27e9\n[GOAL]\ncase inl.inr.intro\nx y : EReal\nh : x < y\nz : \u211d\nh' : \u2191z \u2264 \u2191z\nhbot : \u2191z \u2260 \u22a5\nhtop : \u2191z = \u22a4 \u2192 \u2191z = \u22a4 \u2192 x = \u22a5\nhz : \u2191z \u2260 \u22a4\n\u22a2 x + \u2191z < y + \u2191z\n[PROOFSTEP]\nexact add_lt_add_right_coe h z\n[GOAL]\ncase inr\nx y z t : EReal\nh : x < y\nh' : z \u2264 t\nhbot : t \u2260 \u22a5\nhtop : t = \u22a4 \u2192 z = \u22a4 \u2192 x = \u22a5\nhlt : z < t\n\u22a2 x + z < y + t\n[PROOFSTEP]\nexact add_lt_add h hlt\n[GOAL]\nx y : EReal\nhx : x \u2260 \u22a4\nhy : y \u2260 \u22a4\n\u22a2 x + y < \u22a4\n[PROOFSTEP]\nrw [\u2190 EReal.top_add_top]\n[GOAL]\nx y : EReal\nhx : x \u2260 \u22a4\nhy : y \u2260 \u22a4\n\u22a2 x + y < \u22a4 + \u22a4\n[PROOFSTEP]\nexact EReal.add_lt_add hx.lt_top hy.lt_top\n[GOAL]\n\u22a2 toReal (-\u22a4) = -toReal \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 toReal (-\u22a5) = -toReal \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\na b : EReal\n\u22a2 -a \u2264 b \u2194 -b \u2264 a\n[PROOFSTEP]\nrw [\u2190 neg_le_neg_iff, neg_neg]\n[GOAL]\na b : EReal\nh : a \u2264 -b\n\u22a2 b \u2264 -a\n[PROOFSTEP]\nrwa [\u2190 neg_neg b, EReal.neg_le, neg_neg]\n[GOAL]\na b : EReal\n\u22a2 -a < b \u2194 -b < a\n[PROOFSTEP]\nrw [\u2190 neg_lt_neg_iff, neg_neg]\n[GOAL]\nx y z t : EReal\nh : x < y\nh' : z \u2264 t\nhz : z \u2260 \u22a5\nht : t \u2260 \u22a4\n\u22a2 -t \u2260 \u22a5\n[PROOFSTEP]\nsimp [ht]\n[GOAL]\nx y z t : EReal\nh : x < y\nh' : z \u2264 t\nhz : z \u2260 \u22a5\nht : t \u2260 \u22a4\n\u22a2 -z \u2260 \u22a4\n[PROOFSTEP]\nsimp [hz]\n[GOAL]\nx : \u211d\n\u22a2 \u2191x = \u2191\u2191(Real.toNNReal x) - \u2191\u2191(Real.toNNReal (-x))\n[PROOFSTEP]\nrcases le_total 0 x with (h | h)\n[GOAL]\ncase inl\nx : \u211d\nh : 0 \u2264 x\n\u22a2 \u2191x = \u2191\u2191(Real.toNNReal x) - \u2191\u2191(Real.toNNReal (-x))\n[PROOFSTEP]\nlift x to \u211d\u22650 using h\n[GOAL]\ncase inl.intro\nx : \u211d\u22650\n\u22a2 \u2191\u2191x = \u2191\u2191(Real.toNNReal \u2191x) - \u2191\u2191(Real.toNNReal (-\u2191x))\n[PROOFSTEP]\nrw [Real.toNNReal_of_nonpos (neg_nonpos.mpr x.coe_nonneg), Real.toNNReal_coe, ENNReal.coe_zero, coe_ennreal_zero,\n sub_zero]\n[GOAL]\ncase inl.intro\nx : \u211d\u22650\n\u22a2 \u2191\u2191x = \u2191\u2191x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\nx : \u211d\nh : x \u2264 0\n\u22a2 \u2191x = \u2191\u2191(Real.toNNReal x) - \u2191\u2191(Real.toNNReal (-x))\n[PROOFSTEP]\nrw [Real.toNNReal_of_nonpos h, ENNReal.coe_zero, coe_ennreal_zero, coe_nnreal_eq_coe_real, Real.coe_toNNReal, zero_sub,\n coe_neg, neg_neg]\n[GOAL]\ncase inr.hr\nx : \u211d\nh : x \u2264 0\n\u22a2 0 \u2264 -x\n[PROOFSTEP]\nexact neg_nonneg.2 h\n[GOAL]\nx y : EReal\nhx : x \u2260 \u22a4\nh'x : x \u2260 \u22a5\nhy : y \u2260 \u22a4\nh'y : y \u2260 \u22a5\n\u22a2 toReal (x - y) = toReal x - toReal y\n[PROOFSTEP]\nlift x to \u211d using \u27e8hx, h'x\u27e9\n[GOAL]\ncase intro\ny : EReal\nhy : y \u2260 \u22a4\nh'y : y \u2260 \u22a5\nx : \u211d\nhx : \u2191x \u2260 \u22a4\nh'x : \u2191x \u2260 \u22a5\n\u22a2 toReal (\u2191x - y) = toReal \u2191x - toReal y\n[PROOFSTEP]\nlift y to \u211d using \u27e8hy, h'y\u27e9\n[GOAL]\ncase intro.intro\nx : \u211d\nhx : \u2191x \u2260 \u22a4\nh'x : \u2191x \u2260 \u22a5\ny : \u211d\nhy : \u2191y \u2260 \u22a4\nh'y : \u2191y \u2260 \u22a5\n\u22a2 toReal (\u2191x - \u2191y) = toReal \u2191x - toReal \u2191y\n[PROOFSTEP]\nrfl\n[GOAL]\nx : EReal\nh : 0 < x\n\u22a2 \u22a4 * x = \u22a4\n[PROOFSTEP]\nrw [EReal.mul_comm]\n[GOAL]\nx : EReal\nh : 0 < x\n\u22a2 x * \u22a4 = \u22a4\n[PROOFSTEP]\nexact mul_top_of_pos h\n[GOAL]\nx : EReal\nh : x < 0\n\u22a2 \u22a4 * x = \u22a5\n[PROOFSTEP]\nrw [EReal.mul_comm]\n[GOAL]\nx : EReal\nh : x < 0\n\u22a2 x * \u22a4 = \u22a5\n[PROOFSTEP]\nexact mul_top_of_neg h\n[GOAL]\nx : EReal\nh : 0 < x\n\u22a2 \u22a5 * x = \u22a5\n[PROOFSTEP]\nrw [EReal.mul_comm]\n[GOAL]\nx : EReal\nh : 0 < x\n\u22a2 x * \u22a5 = \u22a5\n[PROOFSTEP]\nexact mul_bot_of_pos h\n[GOAL]\nx : EReal\nh : x < 0\n\u22a2 \u22a5 * x = \u22a4\n[PROOFSTEP]\nrw [EReal.mul_comm]\n[GOAL]\nx : EReal\nh : x < 0\n\u22a2 x * \u22a5 = \u22a4\n[PROOFSTEP]\nexact mul_bot_of_neg h\n[GOAL]\nx y : EReal\n\u22a2 toReal (x * y) = toReal x * toReal y\n[PROOFSTEP]\ninduction x, y using induction\u2082_symm with\n| top_zero| zero_bot| top_top| top_bot| bot_bot => simp\n| symm h => rwa [mul_comm, EReal.mul_comm]\n| coe_coe => norm_cast\n| top_pos _ h => simp [top_mul_coe_of_pos h]\n| top_neg _ h => simp [top_mul_coe_of_neg h]\n| pos_bot _ h => simp [coe_mul_bot_of_pos h]\n| neg_bot _ h => simp [coe_mul_bot_of_neg h]\n[GOAL]\nx y : EReal\n\u22a2 toReal (x * y) = toReal x * toReal y\n[PROOFSTEP]\ninduction x, y using induction\u2082_symm with\n| top_zero => simp\n| zero_bot => simp\n| top_top => simp\n| top_bot => simp\n| bot_bot => simp\n| symm h => rwa [mul_comm, EReal.mul_comm]\n| coe_coe => norm_cast\n| top_pos _ h => simp [top_mul_coe_of_pos h]\n| top_neg _ h => simp [top_mul_coe_of_neg h]\n| pos_bot _ h => simp [coe_mul_bot_of_pos h]\n| neg_bot _ h => simp [coe_mul_bot_of_neg h]\n[GOAL]\nx y : EReal\n\u22a2 toReal (x * y) = toReal x * toReal y\n[PROOFSTEP]\ninduction x, y using induction\u2082_symm with\n| top_zero => simp\n| zero_bot => simp\n| top_top => simp\n| top_bot => simp\n| bot_bot => simp\n| symm h => rwa [mul_comm, EReal.mul_comm]\n| coe_coe => norm_cast\n| top_pos _ h => simp [top_mul_coe_of_pos h]\n| top_neg _ h => simp [top_mul_coe_of_neg h]\n| pos_bot _ h => simp [coe_mul_bot_of_pos h]\n| neg_bot _ h => simp [coe_mul_bot_of_neg h]\n[GOAL]\ncase top_zero\n\u22a2 toReal (\u22a4 * 0) = toReal \u22a4 * toReal 0\n[PROOFSTEP]\n\n| top_zero => simp\n[GOAL]\ncase top_zero\n\u22a2 toReal (\u22a4 * 0) = toReal \u22a4 * toReal 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase zero_bot\n\u22a2 toReal (0 * \u22a5) = toReal 0 * toReal \u22a5\n[PROOFSTEP]\n\n| zero_bot => simp\n[GOAL]\ncase zero_bot\n\u22a2 toReal (0 * \u22a5) = toReal 0 * toReal \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\ncase top_top\n\u22a2 toReal (\u22a4 * \u22a4) = toReal \u22a4 * toReal \u22a4\n[PROOFSTEP]\n\n| top_top => simp\n[GOAL]\ncase top_top\n\u22a2 toReal (\u22a4 * \u22a4) = toReal \u22a4 * toReal \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase top_bot\n\u22a2 toReal (\u22a4 * \u22a5) = toReal \u22a4 * toReal \u22a5\n[PROOFSTEP]\n\n| top_bot => simp\n[GOAL]\ncase top_bot\n\u22a2 toReal (\u22a4 * \u22a5) = toReal \u22a4 * toReal \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\ncase bot_bot\n\u22a2 toReal (\u22a5 * \u22a5) = toReal \u22a5 * toReal \u22a5\n[PROOFSTEP]\n\n| bot_bot => simp\n[GOAL]\ncase bot_bot\n\u22a2 toReal (\u22a5 * \u22a5) = toReal \u22a5 * toReal \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\ncase symm\nx\u271d y\u271d : EReal\nh : toReal (x\u271d * y\u271d) = toReal x\u271d * toReal y\u271d\n\u22a2 toReal (y\u271d * x\u271d) = toReal y\u271d * toReal x\u271d\n[PROOFSTEP]\n\n| symm h => rwa [mul_comm, EReal.mul_comm]\n[GOAL]\ncase symm\nx\u271d y\u271d : EReal\nh : toReal (x\u271d * y\u271d) = toReal x\u271d * toReal y\u271d\n\u22a2 toReal (y\u271d * x\u271d) = toReal y\u271d * toReal x\u271d\n[PROOFSTEP]\nrwa [mul_comm, EReal.mul_comm]\n[GOAL]\ncase coe_coe\nx\u271d y\u271d : \u211d\n\u22a2 toReal (\u2191x\u271d * \u2191y\u271d) = toReal \u2191x\u271d * toReal \u2191y\u271d\n[PROOFSTEP]\n\n| coe_coe => norm_cast\n[GOAL]\ncase coe_coe\nx\u271d y\u271d : \u211d\n\u22a2 toReal (\u2191x\u271d * \u2191y\u271d) = toReal \u2191x\u271d * toReal \u2191y\u271d\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase top_pos\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 toReal (\u22a4 * \u2191x\u271d) = toReal \u22a4 * toReal \u2191x\u271d\n[PROOFSTEP]\n\n| top_pos _ h => simp [top_mul_coe_of_pos h]\n[GOAL]\ncase top_pos\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 toReal (\u22a4 * \u2191x\u271d) = toReal \u22a4 * toReal \u2191x\u271d\n[PROOFSTEP]\nsimp [top_mul_coe_of_pos h]\n[GOAL]\ncase top_neg\nx\u271d : \u211d\nh : x\u271d < 0\n\u22a2 toReal (\u22a4 * \u2191x\u271d) = toReal \u22a4 * toReal \u2191x\u271d\n[PROOFSTEP]\n\n| top_neg _ h => simp [top_mul_coe_of_neg h]\n[GOAL]\ncase top_neg\nx\u271d : \u211d\nh : x\u271d < 0\n\u22a2 toReal (\u22a4 * \u2191x\u271d) = toReal \u22a4 * toReal \u2191x\u271d\n[PROOFSTEP]\nsimp [top_mul_coe_of_neg h]\n[GOAL]\ncase pos_bot\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 toReal (\u2191x\u271d * \u22a5) = toReal \u2191x\u271d * toReal \u22a5\n[PROOFSTEP]\n\n| pos_bot _ h => simp [coe_mul_bot_of_pos h]\n[GOAL]\ncase pos_bot\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 toReal (\u2191x\u271d * \u22a5) = toReal \u2191x\u271d * toReal \u22a5\n[PROOFSTEP]\nsimp [coe_mul_bot_of_pos h]\n[GOAL]\ncase neg_bot\nx\u271d : \u211d\nh : x\u271d < 0\n\u22a2 toReal (\u2191x\u271d * \u22a5) = toReal \u2191x\u271d * toReal \u22a5\n[PROOFSTEP]\n\n| neg_bot _ h => simp [coe_mul_bot_of_neg h]\n[GOAL]\ncase neg_bot\nx\u271d : \u211d\nh : x\u271d < 0\n\u22a2 toReal (\u2191x\u271d * \u22a5) = toReal \u2191x\u271d * toReal \u22a5\n[PROOFSTEP]\nsimp [coe_mul_bot_of_neg h]\n[GOAL]\nx y : EReal\n\u22a2 -x * y = -(x * y)\n[PROOFSTEP]\ninduction x, y using induction\u2082_neg_left with\n| top_zero| zero_top| zero_bot => simp only [zero_mul, mul_zero, neg_zero]\n| top_top| top_bot => rfl\n| neg_left h => rw [h, neg_neg, neg_neg]\n| coe_coe => norm_cast; exact neg_mul _ _\n| top_pos _ h => rw [top_mul_coe_of_pos h, neg_top, bot_mul_coe_of_pos h]\n| pos_top _ h => rw [coe_mul_top_of_pos h, neg_top, \u2190 coe_neg, coe_mul_top_of_neg (neg_neg_of_pos h)]\n| top_neg _ h => rw [top_mul_coe_of_neg h, neg_top, bot_mul_coe_of_neg h, neg_bot]\n| pos_bot _ h => rw [coe_mul_bot_of_pos h, neg_bot, \u2190 coe_neg, coe_mul_bot_of_neg (neg_neg_of_pos h)]\n[GOAL]\nx y : EReal\n\u22a2 -x * y = -(x * y)\n[PROOFSTEP]\ninduction x, y using induction\u2082_neg_left with\n| top_zero => simp only [zero_mul, mul_zero, neg_zero]\n| zero_top => simp only [zero_mul, mul_zero, neg_zero]\n| zero_bot => simp only [zero_mul, mul_zero, neg_zero]\n| top_top => rfl\n| top_bot => rfl\n| neg_left h => rw [h, neg_neg, neg_neg]\n| coe_coe => norm_cast; exact neg_mul _ _\n| top_pos _ h => rw [top_mul_coe_of_pos h, neg_top, bot_mul_coe_of_pos h]\n| pos_top _ h => rw [coe_mul_top_of_pos h, neg_top, \u2190 coe_neg, coe_mul_top_of_neg (neg_neg_of_pos h)]\n| top_neg _ h => rw [top_mul_coe_of_neg h, neg_top, bot_mul_coe_of_neg h, neg_bot]\n| pos_bot _ h => rw [coe_mul_bot_of_pos h, neg_bot, \u2190 coe_neg, coe_mul_bot_of_neg (neg_neg_of_pos h)]\n[GOAL]\nx y : EReal\n\u22a2 -x * y = -(x * y)\n[PROOFSTEP]\ninduction x, y using induction\u2082_neg_left with\n| top_zero => simp only [zero_mul, mul_zero, neg_zero]\n| zero_top => simp only [zero_mul, mul_zero, neg_zero]\n| zero_bot => simp only [zero_mul, mul_zero, neg_zero]\n| top_top => rfl\n| top_bot => rfl\n| neg_left h => rw [h, neg_neg, neg_neg]\n| coe_coe => norm_cast; exact neg_mul _ _\n| top_pos _ h => rw [top_mul_coe_of_pos h, neg_top, bot_mul_coe_of_pos h]\n| pos_top _ h => rw [coe_mul_top_of_pos h, neg_top, \u2190 coe_neg, coe_mul_top_of_neg (neg_neg_of_pos h)]\n| top_neg _ h => rw [top_mul_coe_of_neg h, neg_top, bot_mul_coe_of_neg h, neg_bot]\n| pos_bot _ h => rw [coe_mul_bot_of_pos h, neg_bot, \u2190 coe_neg, coe_mul_bot_of_neg (neg_neg_of_pos h)]\n[GOAL]\ncase top_zero\n\u22a2 -\u22a4 * 0 = -(\u22a4 * 0)\n[PROOFSTEP]\n\n| top_zero => simp only [zero_mul, mul_zero, neg_zero]\n[GOAL]\ncase top_zero\n\u22a2 -\u22a4 * 0 = -(\u22a4 * 0)\n[PROOFSTEP]\nsimp only [zero_mul, mul_zero, neg_zero]\n[GOAL]\ncase zero_top\n\u22a2 -0 * \u22a4 = -(0 * \u22a4)\n[PROOFSTEP]\n\n| zero_top => simp only [zero_mul, mul_zero, neg_zero]\n[GOAL]\ncase zero_top\n\u22a2 -0 * \u22a4 = -(0 * \u22a4)\n[PROOFSTEP]\nsimp only [zero_mul, mul_zero, neg_zero]\n[GOAL]\ncase zero_bot\n\u22a2 -0 * \u22a5 = -(0 * \u22a5)\n[PROOFSTEP]\n\n| zero_bot => simp only [zero_mul, mul_zero, neg_zero]\n[GOAL]\ncase zero_bot\n\u22a2 -0 * \u22a5 = -(0 * \u22a5)\n[PROOFSTEP]\nsimp only [zero_mul, mul_zero, neg_zero]\n[GOAL]\ncase top_top\n\u22a2 -\u22a4 * \u22a4 = -(\u22a4 * \u22a4)\n[PROOFSTEP]\n\n| top_top => rfl\n[GOAL]\ncase top_top\n\u22a2 -\u22a4 * \u22a4 = -(\u22a4 * \u22a4)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase top_bot\n\u22a2 -\u22a4 * \u22a5 = -(\u22a4 * \u22a5)\n[PROOFSTEP]\n\n| top_bot => rfl\n[GOAL]\ncase top_bot\n\u22a2 -\u22a4 * \u22a5 = -(\u22a4 * \u22a5)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg_left\nx\u271d y\u271d : EReal\nh : -x\u271d * y\u271d = -(x\u271d * y\u271d)\n\u22a2 - -x\u271d * y\u271d = -(-x\u271d * y\u271d)\n[PROOFSTEP]\n\n| neg_left h => rw [h, neg_neg, neg_neg]\n[GOAL]\ncase neg_left\nx\u271d y\u271d : EReal\nh : -x\u271d * y\u271d = -(x\u271d * y\u271d)\n\u22a2 - -x\u271d * y\u271d = -(-x\u271d * y\u271d)\n[PROOFSTEP]\nrw [h, neg_neg, neg_neg]\n[GOAL]\ncase coe_coe\nx\u271d y\u271d : \u211d\n\u22a2 -\u2191x\u271d * \u2191y\u271d = -(\u2191x\u271d * \u2191y\u271d)\n[PROOFSTEP]\n\n| coe_coe => norm_cast; exact neg_mul _ _\n[GOAL]\ncase coe_coe\nx\u271d y\u271d : \u211d\n\u22a2 -\u2191x\u271d * \u2191y\u271d = -(\u2191x\u271d * \u2191y\u271d)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase coe_coe\nx\u271d y\u271d : \u211d\n\u22a2 -x\u271d * y\u271d = -(x\u271d * y\u271d)\n[PROOFSTEP]\nexact neg_mul _ _\n[GOAL]\ncase top_pos\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 -\u22a4 * \u2191x\u271d = -(\u22a4 * \u2191x\u271d)\n[PROOFSTEP]\n\n| top_pos _ h => rw [top_mul_coe_of_pos h, neg_top, bot_mul_coe_of_pos h]\n[GOAL]\ncase top_pos\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 -\u22a4 * \u2191x\u271d = -(\u22a4 * \u2191x\u271d)\n[PROOFSTEP]\nrw [top_mul_coe_of_pos h, neg_top, bot_mul_coe_of_pos h]\n[GOAL]\ncase pos_top\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 -\u2191x\u271d * \u22a4 = -(\u2191x\u271d * \u22a4)\n[PROOFSTEP]\n\n| pos_top _ h => rw [coe_mul_top_of_pos h, neg_top, \u2190 coe_neg, coe_mul_top_of_neg (neg_neg_of_pos h)]\n[GOAL]\ncase pos_top\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 -\u2191x\u271d * \u22a4 = -(\u2191x\u271d * \u22a4)\n[PROOFSTEP]\nrw [coe_mul_top_of_pos h, neg_top, \u2190 coe_neg, coe_mul_top_of_neg (neg_neg_of_pos h)]\n[GOAL]\ncase top_neg\nx\u271d : \u211d\nh : x\u271d < 0\n\u22a2 -\u22a4 * \u2191x\u271d = -(\u22a4 * \u2191x\u271d)\n[PROOFSTEP]\n\n| top_neg _ h => rw [top_mul_coe_of_neg h, neg_top, bot_mul_coe_of_neg h, neg_bot]\n[GOAL]\ncase top_neg\nx\u271d : \u211d\nh : x\u271d < 0\n\u22a2 -\u22a4 * \u2191x\u271d = -(\u22a4 * \u2191x\u271d)\n[PROOFSTEP]\nrw [top_mul_coe_of_neg h, neg_top, bot_mul_coe_of_neg h, neg_bot]\n[GOAL]\ncase pos_bot\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 -\u2191x\u271d * \u22a5 = -(\u2191x\u271d * \u22a5)\n[PROOFSTEP]\n\n| pos_bot _ h => rw [coe_mul_bot_of_pos h, neg_bot, \u2190 coe_neg, coe_mul_bot_of_neg (neg_neg_of_pos h)]\n[GOAL]\ncase pos_bot\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 -\u2191x\u271d * \u22a5 = -(\u2191x\u271d * \u22a5)\n[PROOFSTEP]\nrw [coe_mul_bot_of_pos h, neg_bot, \u2190 coe_neg, coe_mul_bot_of_neg (neg_neg_of_pos h)]\n[GOAL]\nx y : EReal\n\u22a2 x * -y = -(x * y)\n[PROOFSTEP]\nrw [x.mul_comm, x.mul_comm]\n[GOAL]\nx y : EReal\n\u22a2 -y * x = -(y * x)\n[PROOFSTEP]\nexact y.neg_mul x\n[GOAL]\nx : EReal\n\u22a2 EReal.abs x = 0 \u2194 x = 0\n[PROOFSTEP]\ninduction x using EReal.rec\n[GOAL]\ncase h_bot\n\u22a2 EReal.abs \u22a5 = 0 \u2194 \u22a5 = 0\n[PROOFSTEP]\nsimp only [abs_bot, ENNReal.top_ne_zero, bot_ne_zero]\n[GOAL]\ncase h_real\na\u271d : \u211d\n\u22a2 EReal.abs \u2191a\u271d = 0 \u2194 \u2191a\u271d = 0\n[PROOFSTEP]\nsimp only [abs_def, coe_eq_zero, ENNReal.ofReal_eq_zero, abs_nonpos_iff]\n[GOAL]\ncase h_top\n\u22a2 EReal.abs \u22a4 = 0 \u2194 \u22a4 = 0\n[PROOFSTEP]\nsimp only [abs_top, ENNReal.top_ne_zero, top_ne_zero]\n[GOAL]\n\u22a2 EReal.abs 0 = 0\n[PROOFSTEP]\nrw [abs_eq_zero_iff]\n[GOAL]\nx : \u211d\n\u22a2 \u2191(EReal.abs \u2191x) = \u2191|x|\n[PROOFSTEP]\nrw [abs_def, \u2190 Real.coe_nnabs, ENNReal.ofReal_coe_nnreal]\n[GOAL]\nx : \u211d\n\u22a2 \u2191\u2191(\u2191Real.nnabs x) = \u2191\u2191(\u2191Real.nnabs x)\n[PROOFSTEP]\nrfl\n[GOAL]\nx : \u211d\n\u22a2 EReal.abs (-\u2191x) = EReal.abs \u2191x\n[PROOFSTEP]\nrw [abs_def, \u2190 coe_neg, abs_def, abs_neg]\n[GOAL]\nx y : EReal\n\u22a2 EReal.abs (x * y) = EReal.abs x * EReal.abs y\n[PROOFSTEP]\ninduction x, y using induction\u2082_symm_neg with\n| top_zero => simp only [zero_mul, mul_zero, abs_zero]\n| top_top => rfl\n| symm h => rwa [mul_comm, EReal.mul_comm]\n| coe_coe => simp only [\u2190 coe_mul, abs_def, _root_.abs_mul, ENNReal.ofReal_mul (abs_nonneg _)]\n| top_pos _ h =>\n rw [top_mul_coe_of_pos h, abs_top, ENNReal.top_mul]\n rw [Ne.def, abs_eq_zero_iff, coe_eq_zero]\n exact h.ne'\n| neg_left h => rwa [neg_mul, EReal.abs_neg, EReal.abs_neg]\n[GOAL]\nx y : EReal\n\u22a2 EReal.abs (x * y) = EReal.abs x * EReal.abs y\n[PROOFSTEP]\ninduction x, y using induction\u2082_symm_neg with\n| top_zero => simp only [zero_mul, mul_zero, abs_zero]\n| top_top => rfl\n| symm h => rwa [mul_comm, EReal.mul_comm]\n| coe_coe => simp only [\u2190 coe_mul, abs_def, _root_.abs_mul, ENNReal.ofReal_mul (abs_nonneg _)]\n| top_pos _ h =>\n rw [top_mul_coe_of_pos h, abs_top, ENNReal.top_mul]\n rw [Ne.def, abs_eq_zero_iff, coe_eq_zero]\n exact h.ne'\n| neg_left h => rwa [neg_mul, EReal.abs_neg, EReal.abs_neg]\n[GOAL]\ncase top_zero\n\u22a2 EReal.abs (\u22a4 * 0) = EReal.abs \u22a4 * EReal.abs 0\n[PROOFSTEP]\n\n| top_zero => simp only [zero_mul, mul_zero, abs_zero]\n[GOAL]\ncase top_zero\n\u22a2 EReal.abs (\u22a4 * 0) = EReal.abs \u22a4 * EReal.abs 0\n[PROOFSTEP]\nsimp only [zero_mul, mul_zero, abs_zero]\n[GOAL]\ncase top_top\n\u22a2 EReal.abs (\u22a4 * \u22a4) = EReal.abs \u22a4 * EReal.abs \u22a4\n[PROOFSTEP]\n\n| top_top => rfl\n[GOAL]\ncase top_top\n\u22a2 EReal.abs (\u22a4 * \u22a4) = EReal.abs \u22a4 * EReal.abs \u22a4\n[PROOFSTEP]\nrfl\n[GOAL]\ncase symm\nx\u271d y\u271d : EReal\nh : EReal.abs (x\u271d * y\u271d) = EReal.abs x\u271d * EReal.abs y\u271d\n\u22a2 EReal.abs (y\u271d * x\u271d) = EReal.abs y\u271d * EReal.abs x\u271d\n[PROOFSTEP]\n\n| symm h => rwa [mul_comm, EReal.mul_comm]\n[GOAL]\ncase symm\nx\u271d y\u271d : EReal\nh : EReal.abs (x\u271d * y\u271d) = EReal.abs x\u271d * EReal.abs y\u271d\n\u22a2 EReal.abs (y\u271d * x\u271d) = EReal.abs y\u271d * EReal.abs x\u271d\n[PROOFSTEP]\nrwa [mul_comm, EReal.mul_comm]\n[GOAL]\ncase coe_coe\nx\u271d y\u271d : \u211d\n\u22a2 EReal.abs (\u2191x\u271d * \u2191y\u271d) = EReal.abs \u2191x\u271d * EReal.abs \u2191y\u271d\n[PROOFSTEP]\n\n| coe_coe => simp only [\u2190 coe_mul, abs_def, _root_.abs_mul, ENNReal.ofReal_mul (abs_nonneg _)]\n[GOAL]\ncase coe_coe\nx\u271d y\u271d : \u211d\n\u22a2 EReal.abs (\u2191x\u271d * \u2191y\u271d) = EReal.abs \u2191x\u271d * EReal.abs \u2191y\u271d\n[PROOFSTEP]\nsimp only [\u2190 coe_mul, abs_def, _root_.abs_mul, ENNReal.ofReal_mul (abs_nonneg _)]\n[GOAL]\ncase top_pos\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 EReal.abs (\u22a4 * \u2191x\u271d) = EReal.abs \u22a4 * EReal.abs \u2191x\u271d\n[PROOFSTEP]\n\n| top_pos _ h =>\n rw [top_mul_coe_of_pos h, abs_top, ENNReal.top_mul]\n rw [Ne.def, abs_eq_zero_iff, coe_eq_zero]\n exact h.ne'\n[GOAL]\ncase top_pos\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 EReal.abs (\u22a4 * \u2191x\u271d) = EReal.abs \u22a4 * EReal.abs \u2191x\u271d\n[PROOFSTEP]\nrw [top_mul_coe_of_pos h, abs_top, ENNReal.top_mul]\n[GOAL]\ncase top_pos\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 EReal.abs \u2191x\u271d \u2260 0\n[PROOFSTEP]\nrw [Ne.def, abs_eq_zero_iff, coe_eq_zero]\n[GOAL]\ncase top_pos\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 \u00acx\u271d = 0\n[PROOFSTEP]\nexact h.ne'\n[GOAL]\ncase neg_left\nx\u271d y\u271d : EReal\nh : EReal.abs (x\u271d * y\u271d) = EReal.abs x\u271d * EReal.abs y\u271d\n\u22a2 EReal.abs (-x\u271d * y\u271d) = EReal.abs (-x\u271d) * EReal.abs y\u271d\n[PROOFSTEP]\n\n| neg_left h => rwa [neg_mul, EReal.abs_neg, EReal.abs_neg]\n[GOAL]\ncase neg_left\nx\u271d y\u271d : EReal\nh : EReal.abs (x\u271d * y\u271d) = EReal.abs x\u271d * EReal.abs y\u271d\n\u22a2 EReal.abs (-x\u271d * y\u271d) = EReal.abs (-x\u271d) * EReal.abs y\u271d\n[PROOFSTEP]\nrwa [neg_mul, EReal.abs_neg, EReal.abs_neg]\n[GOAL]\nx : \u211d\n\u22a2 \u2191sign \u2191x = \u2191sign x\n[PROOFSTEP]\nsimp only [sign, OrderHom.coe_mk, EReal.coe_pos, EReal.coe_neg']\n[GOAL]\nx : SignType\n\u22a2 \u2191\u2191x = \u2191x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase zero\n\u22a2 \u2191\u2191SignType.zero = \u2191SignType.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u22a2 \u2191\u2191SignType.neg = \u2191SignType.neg\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u22a2 \u2191\u2191SignType.pos = \u2191SignType.pos\n[PROOFSTEP]\nrfl\n[GOAL]\nx : \u211d\n\u22a2 \u2191sign (-\u2191x) = -\u2191sign \u2191x\n[PROOFSTEP]\nrw [\u2190 coe_neg, sign_coe, sign_coe, Left.sign_neg]\n[GOAL]\nx y : EReal\n\u22a2 \u2191sign (x * y) = \u2191sign x * \u2191sign y\n[PROOFSTEP]\ninduction x, y using induction\u2082_symm_neg with\n| top_zero => simp only [zero_mul, mul_zero, sign_zero]\n| top_top => rfl\n| symm h => rwa [mul_comm, EReal.mul_comm]\n| coe_coe => simp only [\u2190 coe_mul, sign_coe, _root_.sign_mul, ENNReal.ofReal_mul (abs_nonneg _)]\n| top_pos _ h => rw [top_mul_coe_of_pos h, sign_top, one_mul, sign_pos (EReal.coe_pos.2 h)]\n| neg_left h => rw [neg_mul, sign_neg, sign_neg, h, neg_mul]\n[GOAL]\nx y : EReal\n\u22a2 \u2191sign (x * y) = \u2191sign x * \u2191sign y\n[PROOFSTEP]\ninduction x, y using induction\u2082_symm_neg with\n| top_zero => simp only [zero_mul, mul_zero, sign_zero]\n| top_top => rfl\n| symm h => rwa [mul_comm, EReal.mul_comm]\n| coe_coe => simp only [\u2190 coe_mul, sign_coe, _root_.sign_mul, ENNReal.ofReal_mul (abs_nonneg _)]\n| top_pos _ h => rw [top_mul_coe_of_pos h, sign_top, one_mul, sign_pos (EReal.coe_pos.2 h)]\n| neg_left h => rw [neg_mul, sign_neg, sign_neg, h, neg_mul]\n[GOAL]\ncase top_zero\n\u22a2 \u2191sign (\u22a4 * 0) = \u2191sign \u22a4 * \u2191sign 0\n[PROOFSTEP]\n\n| top_zero => simp only [zero_mul, mul_zero, sign_zero]\n[GOAL]\ncase top_zero\n\u22a2 \u2191sign (\u22a4 * 0) = \u2191sign \u22a4 * \u2191sign 0\n[PROOFSTEP]\nsimp only [zero_mul, mul_zero, sign_zero]\n[GOAL]\ncase top_top\n\u22a2 \u2191sign (\u22a4 * \u22a4) = \u2191sign \u22a4 * \u2191sign \u22a4\n[PROOFSTEP]\n\n| top_top => rfl\n[GOAL]\ncase top_top\n\u22a2 \u2191sign (\u22a4 * \u22a4) = \u2191sign \u22a4 * \u2191sign \u22a4\n[PROOFSTEP]\nrfl\n[GOAL]\ncase symm\nx\u271d y\u271d : EReal\nh : \u2191sign (x\u271d * y\u271d) = \u2191sign x\u271d * \u2191sign y\u271d\n\u22a2 \u2191sign (y\u271d * x\u271d) = \u2191sign y\u271d * \u2191sign x\u271d\n[PROOFSTEP]\n\n| symm h => rwa [mul_comm, EReal.mul_comm]\n[GOAL]\ncase symm\nx\u271d y\u271d : EReal\nh : \u2191sign (x\u271d * y\u271d) = \u2191sign x\u271d * \u2191sign y\u271d\n\u22a2 \u2191sign (y\u271d * x\u271d) = \u2191sign y\u271d * \u2191sign x\u271d\n[PROOFSTEP]\nrwa [mul_comm, EReal.mul_comm]\n[GOAL]\ncase coe_coe\nx\u271d y\u271d : \u211d\n\u22a2 \u2191sign (\u2191x\u271d * \u2191y\u271d) = \u2191sign \u2191x\u271d * \u2191sign \u2191y\u271d\n[PROOFSTEP]\n\n| coe_coe => simp only [\u2190 coe_mul, sign_coe, _root_.sign_mul, ENNReal.ofReal_mul (abs_nonneg _)]\n[GOAL]\ncase coe_coe\nx\u271d y\u271d : \u211d\n\u22a2 \u2191sign (\u2191x\u271d * \u2191y\u271d) = \u2191sign \u2191x\u271d * \u2191sign \u2191y\u271d\n[PROOFSTEP]\nsimp only [\u2190 coe_mul, sign_coe, _root_.sign_mul, ENNReal.ofReal_mul (abs_nonneg _)]\n[GOAL]\ncase top_pos\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 \u2191sign (\u22a4 * \u2191x\u271d) = \u2191sign \u22a4 * \u2191sign \u2191x\u271d\n[PROOFSTEP]\n\n| top_pos _ h => rw [top_mul_coe_of_pos h, sign_top, one_mul, sign_pos (EReal.coe_pos.2 h)]\n[GOAL]\ncase top_pos\nx\u271d : \u211d\nh : 0 < x\u271d\n\u22a2 \u2191sign (\u22a4 * \u2191x\u271d) = \u2191sign \u22a4 * \u2191sign \u2191x\u271d\n[PROOFSTEP]\nrw [top_mul_coe_of_pos h, sign_top, one_mul, sign_pos (EReal.coe_pos.2 h)]\n[GOAL]\ncase neg_left\nx\u271d y\u271d : EReal\nh : \u2191sign (x\u271d * y\u271d) = \u2191sign x\u271d * \u2191sign y\u271d\n\u22a2 \u2191sign (-x\u271d * y\u271d) = \u2191sign (-x\u271d) * \u2191sign y\u271d\n[PROOFSTEP]\n\n| neg_left h => rw [neg_mul, sign_neg, sign_neg, h, neg_mul]\n[GOAL]\ncase neg_left\nx\u271d y\u271d : EReal\nh : \u2191sign (x\u271d * y\u271d) = \u2191sign x\u271d * \u2191sign y\u271d\n\u22a2 \u2191sign (-x\u271d * y\u271d) = \u2191sign (-x\u271d) * \u2191sign y\u271d\n[PROOFSTEP]\nrw [neg_mul, sign_neg, sign_neg, h, neg_mul]\n[GOAL]\n\u22a2 \u2191(\u2191sign \u22a5) * \u2191(EReal.abs \u22a5) = \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2191(\u2191sign \u22a4) * \u2191(EReal.abs \u22a4) = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\n\u22a2 \u2191(\u2191sign \u2191x) * \u2191(EReal.abs \u2191x) = \u2191x\n[PROOFSTEP]\nrw [sign_coe, coe_abs, \u2190 coe_coe_sign, \u2190 coe_mul, sign_mul_abs]\n[GOAL]\nx : EReal\n\u22a2 \u2191(EReal.abs x) * \u2191(\u2191sign x) = x\n[PROOFSTEP]\nrw [EReal.mul_comm, EReal.sign_mul_abs]\n[GOAL]\nx y : EReal\n\u22a2 EReal.abs x = EReal.abs y \u2227 \u2191sign x = \u2191sign y \u2194 x = y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nx y : EReal\n\u22a2 EReal.abs x = EReal.abs y \u2227 \u2191sign x = \u2191sign y \u2192 x = y\n[PROOFSTEP]\nrintro \u27e8habs, hsign\u27e9\n[GOAL]\ncase mp.intro\nx y : EReal\nhabs : EReal.abs x = EReal.abs y\nhsign : \u2191sign x = \u2191sign y\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 x.sign_mul_abs, \u2190 y.sign_mul_abs, habs, hsign]\n[GOAL]\ncase mpr\nx y : EReal\n\u22a2 x = y \u2192 EReal.abs x = EReal.abs y \u2227 \u2191sign x = \u2191sign y\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mpr\nx : EReal\n\u22a2 EReal.abs x = EReal.abs x \u2227 \u2191sign x = \u2191sign x\n[PROOFSTEP]\nexact \u27e8rfl, rfl\u27e9\n[GOAL]\nx y : EReal\n\u22a2 x \u2264 y \u2194\n \u2191sign x < \u2191sign y \u2228\n \u2191sign x = SignType.neg \u2227 \u2191sign y = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x \u2228\n \u2191sign x = SignType.zero \u2227 \u2191sign y = SignType.zero \u2228\n \u2191sign x = SignType.pos \u2227 \u2191sign y = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nx y : EReal\n\u22a2 x \u2264 y \u2192\n \u2191sign x < \u2191sign y \u2228\n \u2191sign x = SignType.neg \u2227 \u2191sign y = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x \u2228\n \u2191sign x = SignType.zero \u2227 \u2191sign y = SignType.zero \u2228\n \u2191sign x = SignType.pos \u2227 \u2191sign y = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nx y : EReal\nh : x \u2264 y\n\u22a2 \u2191sign x < \u2191sign y \u2228\n \u2191sign x = SignType.neg \u2227 \u2191sign y = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x \u2228\n \u2191sign x = SignType.zero \u2227 \u2191sign y = SignType.zero \u2228\n \u2191sign x = SignType.pos \u2227 \u2191sign y = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y\n[PROOFSTEP]\nrefine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_)\n[GOAL]\ncase mp\nx y : EReal\nh : x \u2264 y\nhs : \u2191sign x = \u2191sign y\n\u22a2 \u2191sign x = SignType.neg \u2227 \u2191sign y = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x \u2228\n \u2191sign x = SignType.zero \u2227 \u2191sign y = SignType.zero \u2228\n \u2191sign x = SignType.pos \u2227 \u2191sign y = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y\n[PROOFSTEP]\nrw [\u2190 x.sign_mul_abs, \u2190 y.sign_mul_abs] at h \n[GOAL]\ncase mp\nx y : EReal\nh : \u2191(\u2191sign x) * \u2191(EReal.abs x) \u2264 \u2191(\u2191sign y) * \u2191(EReal.abs y)\nhs : \u2191sign x = \u2191sign y\n\u22a2 \u2191sign x = SignType.neg \u2227 \u2191sign y = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x \u2228\n \u2191sign x = SignType.zero \u2227 \u2191sign y = SignType.zero \u2228\n \u2191sign x = SignType.pos \u2227 \u2191sign y = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y\n[PROOFSTEP]\ncases hy : sign y\n[GOAL]\ncase mp.zero\nx y : EReal\nh : \u2191(\u2191sign x) * \u2191(EReal.abs x) \u2264 \u2191(\u2191sign y) * \u2191(EReal.abs y)\nhs : \u2191sign x = \u2191sign y\nhy : \u2191sign y = SignType.zero\n\u22a2 \u2191sign x = SignType.neg \u2227 SignType.zero = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x \u2228\n \u2191sign x = SignType.zero \u2227 SignType.zero = SignType.zero \u2228\n \u2191sign x = SignType.pos \u2227 SignType.zero = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y\n[PROOFSTEP]\nrw [hs, hy] at h \u22a2\n[GOAL]\ncase mp.neg\nx y : EReal\nh : \u2191(\u2191sign x) * \u2191(EReal.abs x) \u2264 \u2191(\u2191sign y) * \u2191(EReal.abs y)\nhs : \u2191sign x = \u2191sign y\nhy : \u2191sign y = SignType.neg\n\u22a2 \u2191sign x = SignType.neg \u2227 SignType.neg = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x \u2228\n \u2191sign x = SignType.zero \u2227 SignType.neg = SignType.zero \u2228\n \u2191sign x = SignType.pos \u2227 SignType.neg = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y\n[PROOFSTEP]\nrw [hs, hy] at h \u22a2\n[GOAL]\ncase mp.pos\nx y : EReal\nh : \u2191(\u2191sign x) * \u2191(EReal.abs x) \u2264 \u2191(\u2191sign y) * \u2191(EReal.abs y)\nhs : \u2191sign x = \u2191sign y\nhy : \u2191sign y = SignType.pos\n\u22a2 \u2191sign x = SignType.neg \u2227 SignType.pos = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x \u2228\n \u2191sign x = SignType.zero \u2227 SignType.pos = SignType.zero \u2228\n \u2191sign x = SignType.pos \u2227 SignType.pos = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y\n[PROOFSTEP]\nrw [hs, hy] at h \u22a2\n[GOAL]\ncase mp.zero\nx y : EReal\nh : \u2191SignType.zero * \u2191(EReal.abs x) \u2264 \u2191SignType.zero * \u2191(EReal.abs y)\nhs : \u2191sign x = \u2191sign y\nhy : \u2191sign y = SignType.zero\n\u22a2 SignType.zero = SignType.neg \u2227 SignType.zero = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x \u2228\n SignType.zero = SignType.zero \u2227 SignType.zero = SignType.zero \u2228\n SignType.zero = SignType.pos \u2227 SignType.zero = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mp.neg\nx y : EReal\nh : \u2191SignType.neg * \u2191(EReal.abs x) \u2264 \u2191SignType.neg * \u2191(EReal.abs y)\nhs : \u2191sign x = \u2191sign y\nhy : \u2191sign y = SignType.neg\n\u22a2 SignType.neg = SignType.neg \u2227 SignType.neg = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x \u2228\n SignType.neg = SignType.zero \u2227 SignType.neg = SignType.zero \u2228\n SignType.neg = SignType.pos \u2227 SignType.neg = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y\n[PROOFSTEP]\nleft\n[GOAL]\ncase mp.neg.h\nx y : EReal\nh : \u2191SignType.neg * \u2191(EReal.abs x) \u2264 \u2191SignType.neg * \u2191(EReal.abs y)\nhs : \u2191sign x = \u2191sign y\nhy : \u2191sign y = SignType.neg\n\u22a2 SignType.neg = SignType.neg \u2227 SignType.neg = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase mp.pos\nx y : EReal\nh : \u2191SignType.pos * \u2191(EReal.abs x) \u2264 \u2191SignType.pos * \u2191(EReal.abs y)\nhs : \u2191sign x = \u2191sign y\nhy : \u2191sign y = SignType.pos\n\u22a2 SignType.pos = SignType.neg \u2227 SignType.pos = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x \u2228\n SignType.pos = SignType.zero \u2227 SignType.pos = SignType.zero \u2228\n SignType.pos = SignType.pos \u2227 SignType.pos = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y\n[PROOFSTEP]\nright\n[GOAL]\ncase mp.pos.h\nx y : EReal\nh : \u2191SignType.pos * \u2191(EReal.abs x) \u2264 \u2191SignType.pos * \u2191(EReal.abs y)\nhs : \u2191sign x = \u2191sign y\nhy : \u2191sign y = SignType.pos\n\u22a2 SignType.pos = SignType.zero \u2227 SignType.pos = SignType.zero \u2228\n SignType.pos = SignType.pos \u2227 SignType.pos = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y\n[PROOFSTEP]\nright\n[GOAL]\ncase mp.pos.h.h\nx y : EReal\nh : \u2191SignType.pos * \u2191(EReal.abs x) \u2264 \u2191SignType.pos * \u2191(EReal.abs y)\nhs : \u2191sign x = \u2191sign y\nhy : \u2191sign y = SignType.pos\n\u22a2 SignType.pos = SignType.pos \u2227 SignType.pos = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase mpr\nx y : EReal\n\u22a2 \u2191sign x < \u2191sign y \u2228\n \u2191sign x = SignType.neg \u2227 \u2191sign y = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x \u2228\n \u2191sign x = SignType.zero \u2227 \u2191sign y = SignType.zero \u2228\n \u2191sign x = SignType.pos \u2227 \u2191sign y = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y \u2192\n x \u2264 y\n[PROOFSTEP]\nrintro (h | h | h | h)\n[GOAL]\ncase mpr.inl\nx y : EReal\nh : \u2191sign x < \u2191sign y\n\u22a2 x \u2264 y\n[PROOFSTEP]\nexact (sign.monotone.reflect_lt h).le\n[GOAL]\ncase mpr.inr.inl\nx y : EReal\nh : \u2191sign x = SignType.neg \u2227 \u2191sign y = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x\n\u22a2 x \u2264 y\ncase mpr.inr.inr.inl\nx y : EReal\nh : \u2191sign x = SignType.zero \u2227 \u2191sign y = SignType.zero\n\u22a2 x \u2264 y\ncase mpr.inr.inr.inr x y : EReal h : \u2191sign x = SignType.pos \u2227 \u2191sign y = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y \u22a2 x \u2264 y\n[PROOFSTEP]\nall_goals rw [\u2190 x.sign_mul_abs, \u2190 y.sign_mul_abs]; simp [h]\n[GOAL]\ncase mpr.inr.inl\nx y : EReal\nh : \u2191sign x = SignType.neg \u2227 \u2191sign y = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x\n\u22a2 x \u2264 y\n[PROOFSTEP]\nrw [\u2190 x.sign_mul_abs, \u2190 y.sign_mul_abs]\n[GOAL]\ncase mpr.inr.inl\nx y : EReal\nh : \u2191sign x = SignType.neg \u2227 \u2191sign y = SignType.neg \u2227 EReal.abs y \u2264 EReal.abs x\n\u22a2 \u2191(\u2191sign x) * \u2191(EReal.abs x) \u2264 \u2191(\u2191sign y) * \u2191(EReal.abs y)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase mpr.inr.inr.inl\nx y : EReal\nh : \u2191sign x = SignType.zero \u2227 \u2191sign y = SignType.zero\n\u22a2 x \u2264 y\n[PROOFSTEP]\nrw [\u2190 x.sign_mul_abs, \u2190 y.sign_mul_abs]\n[GOAL]\ncase mpr.inr.inr.inl\nx y : EReal\nh : \u2191sign x = SignType.zero \u2227 \u2191sign y = SignType.zero\n\u22a2 \u2191(\u2191sign x) * \u2191(EReal.abs x) \u2264 \u2191(\u2191sign y) * \u2191(EReal.abs y)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase mpr.inr.inr.inr\nx y : EReal\nh : \u2191sign x = SignType.pos \u2227 \u2191sign y = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y\n\u22a2 x \u2264 y\n[PROOFSTEP]\nrw [\u2190 x.sign_mul_abs, \u2190 y.sign_mul_abs]\n[GOAL]\ncase mpr.inr.inr.inr\nx y : EReal\nh : \u2191sign x = SignType.pos \u2227 \u2191sign y = SignType.pos \u2227 EReal.abs x \u2264 EReal.abs y\n\u22a2 \u2191(\u2191sign x) * \u2191(EReal.abs x) \u2264 \u2191(\u2191sign y) * \u2191(EReal.abs y)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nsrc\u271d : MulZeroOneClass EReal := inferInstanceAs (MulZeroOneClass EReal)\nx y z : EReal\n\u22a2 x * y * z = x * (y * z)\n[PROOFSTEP]\nrw [\u2190 sign_eq_and_abs_eq_iff_eq]\n[GOAL]\nsrc\u271d : MulZeroOneClass EReal := inferInstanceAs (MulZeroOneClass EReal)\nx y z : EReal\n\u22a2 EReal.abs (x * y * z) = EReal.abs (x * (y * z)) \u2227 \u2191sign (x * y * z) = \u2191sign (x * (y * z))\n[PROOFSTEP]\nsimp only [mul_assoc, abs_mul, eq_self_iff_true, sign_mul, and_self_iff]\n[GOAL]\n\u22a2 Covariant { x // 0 < x } EReal (fun x y => \u2191x * y) fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nrintro \u27e8x, x0\u27e9 a b h\n[GOAL]\ncase mk\nx : EReal\nx0 : 0 < x\na b : EReal\nh : a \u2264 b\n\u22a2 (fun x y => \u2191x * y) { val := x, property := x0 } a \u2264 (fun x y => \u2191x * y) { val := x, property := x0 } b\n[PROOFSTEP]\nsimp only [le_iff_sign, EReal.sign_mul, sign_pos x0, one_mul, EReal.abs_mul] at h \u22a2\n[GOAL]\ncase mk\nx : EReal\nx0 : 0 < x\na b : EReal\nh :\n \u2191sign a < \u2191sign b \u2228\n \u2191sign a = SignType.neg \u2227 \u2191sign b = SignType.neg \u2227 EReal.abs b \u2264 EReal.abs a \u2228\n \u2191sign a = SignType.zero \u2227 \u2191sign b = SignType.zero \u2228\n \u2191sign a = SignType.pos \u2227 \u2191sign b = SignType.pos \u2227 EReal.abs a \u2264 EReal.abs b\n\u22a2 \u2191sign a < \u2191sign b \u2228\n \u2191sign a = SignType.neg \u2227 \u2191sign b = SignType.neg \u2227 EReal.abs x * EReal.abs b \u2264 EReal.abs x * EReal.abs a \u2228\n \u2191sign a = SignType.zero \u2227 \u2191sign b = SignType.zero \u2228\n \u2191sign a = SignType.pos \u2227 \u2191sign b = SignType.pos \u2227 EReal.abs x * EReal.abs a \u2264 EReal.abs x * EReal.abs b\n[PROOFSTEP]\nexact\n h.imp_right <|\n Or.imp (And.imp_right <| And.imp_right (mul_le_mul_left' \u00b7 _)) <|\n Or.imp_right <| And.imp_right <| And.imp_right (mul_le_mul_left' \u00b7 _)\n", "meta": {"mathlib_filename": "Mathlib.Data.Real.EReal", "llama_tokens": 27207, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7826624789529375, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.4339632286121159}} {"text": "[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr : R\nx y : (j : J) \u00d7 \u2191(F.obj j)\nh : Types.FilteredColimit.Rel (F \u22d9 forget (ModuleCat R)) x y\n\u22a2 colimitSMulAux F r x = colimitSMulAux F r y\n[PROOFSTEP]\napply M.mk_eq\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr : R\nx y : (j : J) \u00d7 \u2191(F.obj j)\nh : Types.FilteredColimit.Rel (F \u22d9 forget (ModuleCat R)) x y\n\u22a2 \u2203 k f g, \u2191(F.map f) { fst := x.fst, snd := r \u2022 x.snd }.snd = \u2191(F.map g) { fst := y.fst, snd := r \u2022 y.snd }.snd\n[PROOFSTEP]\nobtain \u27e8k, f, g, hfg\u27e9 := h\n[GOAL]\ncase h.intro.intro.intro\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr : R\nx y : (j : J) \u00d7 \u2191(F.obj j)\nk : J\nf : x.fst \u27f6 k\ng : y.fst \u27f6 k\nhfg : (F \u22d9 forget (ModuleCat R)).map f x.snd = (F \u22d9 forget (ModuleCat R)).map g y.snd\n\u22a2 \u2203 k f g, \u2191(F.map f) { fst := x.fst, snd := r \u2022 x.snd }.snd = \u2191(F.map g) { fst := y.fst, snd := r \u2022 y.snd }.snd\n[PROOFSTEP]\nuse k, f, g\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr : R\nx y : (j : J) \u00d7 \u2191(F.obj j)\nk : J\nf : x.fst \u27f6 k\ng : y.fst \u27f6 k\nhfg : (F \u22d9 forget (ModuleCat R)).map f x.snd = (F \u22d9 forget (ModuleCat R)).map g y.snd\n\u22a2 \u2191(F.map f) { fst := x.fst, snd := r \u2022 x.snd }.snd = \u2191(F.map g) { fst := y.fst, snd := r \u2022 y.snd }.snd\n[PROOFSTEP]\nsimp only [Functor.comp_obj, Functor.comp_map, forget_map] at hfg \n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr : R\nx y : (j : J) \u00d7 \u2191(F.obj j)\nk : J\nf : x.fst \u27f6 k\ng : y.fst \u27f6 k\nhfg : \u2191(F.map f) x.snd = \u2191(F.map g) y.snd\n\u22a2 \u2191(F.map f) { fst := x.fst, snd := r \u2022 x.snd }.snd = \u2191(F.map g) { fst := y.fst, snd := r \u2022 y.snd }.snd\n[PROOFSTEP]\nsimp [hfg]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr : R\nx : \u2191(M F)\n\u22a2 \u2191(M F)\n[PROOFSTEP]\nrefine' Quot.lift (colimitSMulAux F r) _ x\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr : R\nx : \u2191(M F)\n\u22a2 \u2200 (a b : (j : J) \u00d7 \u2191(F.obj j)),\n Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat)\n a b \u2192\n colimitSMulAux F r a = colimitSMulAux F r b\n[PROOFSTEP]\nintro x y h\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr : R\nx\u271d : \u2191(M F)\nx y : (j : J) \u00d7 \u2191(F.obj j)\nh :\n Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat)\n x y\n\u22a2 colimitSMulAux F r x = colimitSMulAux F r y\n[PROOFSTEP]\napply colimitSMulAux_eq_of_rel\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr : R\nx\u271d : \u2191(M F)\nx y : (j : J) \u00d7 \u2191(F.obj j)\nh :\n Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat)\n x y\n\u22a2 Types.FilteredColimit.Rel (F \u22d9 forget (ModuleCat R)) x y\n[PROOFSTEP]\napply Types.FilteredColimit.rel_of_quot_rel\n[GOAL]\ncase h.a\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr : R\nx\u271d : \u2191(M F)\nx y : (j : J) \u00d7 \u2191(F.obj j)\nh :\n Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat)\n x y\n\u22a2 Types.Quot.Rel (F \u22d9 forget (ModuleCat R)) x y\n[PROOFSTEP]\nexact h\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nx : \u2191(M F)\n\u22a2 1 \u2022 x = x\n[PROOFSTEP]\nrefine' Quot.inductionOn x _\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nx : \u2191(M F)\n\u22a2 \u2200\n (a :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j),\n 1 \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a =\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a\n[PROOFSTEP]\nclear x\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\n\u22a2 \u2200\n (a :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j),\n 1 \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a =\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a\n[PROOFSTEP]\nintro x\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nx :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 1 \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n x =\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n x\n[PROOFSTEP]\ncases' x with j x\n[GOAL]\ncase mk\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nj : J\nx :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 1 \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x } =\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x }\n[PROOFSTEP]\nerw [colimit_smul_mk_eq F 1 \u27e8j, x\u27e9]\n[GOAL]\ncase mk\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nj : J\nx :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 M.mk F { fst := { fst := j, snd := x }.fst, snd := 1 \u2022 { fst := j, snd := x }.snd } =\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nj : J\nx :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 M.mk F { fst := j, snd := x } =\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x }\n[PROOFSTEP]\nrfl\n -- Porting note: writing directly the `Module` instance makes things very slow.\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nx : \u2191(M F)\n\u22a2 1 \u2022 x = x\n[PROOFSTEP]\nrefine' Quot.inductionOn x _\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nx : \u2191(M F)\n\u22a2 \u2200\n (a :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j),\n 1 \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a =\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a\n[PROOFSTEP]\nclear x\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\n\u22a2 \u2200\n (a :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j),\n 1 \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a =\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a\n[PROOFSTEP]\nintro x\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nx :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 1 \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n x =\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n x\n[PROOFSTEP]\ncases' x with j x\n[GOAL]\ncase mk\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nj : J\nx :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 1 \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x } =\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x }\n[PROOFSTEP]\nerw [colimit_smul_mk_eq F 1 \u27e8j, x\u27e9, one_smul]\n[GOAL]\ncase mk\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nj : J\nx :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 M.mk F { fst := { fst := j, snd := x }.fst, snd := { fst := j, snd := x }.snd } =\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr s : R\nx : \u2191(M F)\n\u22a2 (r * s) \u2022 x = r \u2022 s \u2022 x\n[PROOFSTEP]\nrefine' Quot.inductionOn x _\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr s : R\nx : \u2191(M F)\n\u22a2 \u2200\n (a :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j),\n (r * s) \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a =\n r \u2022\n s \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a\n[PROOFSTEP]\nclear x\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr s : R\n\u22a2 \u2200\n (a :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j),\n (r * s) \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a =\n r \u2022\n s \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a\n[PROOFSTEP]\nintro x\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr s : R\nx :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 (r * s) \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n x =\n r \u2022\n s \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n x\n[PROOFSTEP]\ncases' x with j x\n[GOAL]\ncase mk\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr s : R\nj : J\nx :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 (r * s) \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x } =\n r \u2022\n s \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x }\n[PROOFSTEP]\nerw [colimit_smul_mk_eq F (r * s) \u27e8j, x\u27e9, colimit_smul_mk_eq F s \u27e8j, x\u27e9, colimit_smul_mk_eq F r \u27e8j, _\u27e9, mul_smul]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nsrc\u271d : MulAction R \u2191(M F) := colimitMulAction F\nr : R\n\u22a2 r \u2022 0 = 0\n[PROOFSTEP]\nerw [colimit_zero_eq _ (IsFiltered.Nonempty.some : J), colimit_smul_mk_eq, smul_zero]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nsrc\u271d : MulAction R \u2191(M F) := colimitMulAction F\nr : R\n\u22a2 M.mk F { fst := { fst := Nonempty.some (_ : Nonempty J), snd := 0 }.fst, snd := 0 } =\n AddMonCat.FilteredColimits.M.mk\n (((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat)\n { fst := Nonempty.some (_ : Nonempty J), snd := 0 }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nsrc\u271d : MulAction R \u2191(M F) := colimitMulAction F\nx : \u2191(M F)\n\u22a2 0 \u2022 x = 0\n[PROOFSTEP]\nrefine' Quot.inductionOn x _\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nsrc\u271d : MulAction R \u2191(M F) := colimitMulAction F\nx : \u2191(M F)\n\u22a2 \u2200\n (a :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j),\n 0 \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a =\n 0\n[PROOFSTEP]\nclear x\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nsrc\u271d : MulAction R \u2191(M F) := colimitMulAction F\n\u22a2 \u2200\n (a :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j),\n 0 \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a =\n 0\n[PROOFSTEP]\nintro x\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nsrc\u271d : MulAction R \u2191(M F) := colimitMulAction F\nx :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 0 \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n x =\n 0\n[PROOFSTEP]\ncases' x with j x\n[GOAL]\ncase mk\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nsrc\u271d : MulAction R \u2191(M F) := colimitMulAction F\nj : J\nx :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 0 \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x } =\n 0\n[PROOFSTEP]\nerw [colimit_smul_mk_eq, zero_smul, colimit_zero_eq _ j]\n[GOAL]\ncase mk\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nsrc\u271d : MulAction R \u2191(M F) := colimitMulAction F\nj : J\nx :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 M.mk F { fst := { fst := j, snd := x }.fst, snd := 0 } =\n AddMonCat.FilteredColimits.M.mk\n (((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat)\n { fst := j, snd := 0 }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr s : R\nx : \u2191(M F)\n\u22a2 (r + s) \u2022 x = r \u2022 x + s \u2022 x\n[PROOFSTEP]\nrefine' Quot.inductionOn x _\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr s : R\nx : \u2191(M F)\n\u22a2 \u2200\n (a :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j),\n (r + s) \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a =\n r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a +\n s \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a\n[PROOFSTEP]\nclear x\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr s : R\n\u22a2 \u2200\n (a :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j),\n (r + s) \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a =\n r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a +\n s \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a\n[PROOFSTEP]\nintro x\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr s : R\nx :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 (r + s) \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n x =\n r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n x +\n s \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n x\n[PROOFSTEP]\ncases' x with j x\n[GOAL]\ncase mk\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr s : R\nj : J\nx :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 (r + s) \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x } =\n r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x } +\n s \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x }\n[PROOFSTEP]\nerw [colimit_smul_mk_eq, _root_.add_smul, colimit_smul_mk_eq, colimit_smul_mk_eq,\n colimit_add_mk_eq _ \u27e8j, _\u27e9 \u27e8j, _\u27e9 j (\ud835\udfd9 j) (\ud835\udfd9 j), CategoryTheory.Functor.map_id, id_apply, id_apply]\n[GOAL]\ncase mk\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nr s : R\nj : J\nx :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 M.mk F { fst := { fst := j, snd := x }.fst, snd := r \u2022 { fst := j, snd := x }.snd + s \u2022 { fst := j, snd := x }.snd } =\n AddMonCat.FilteredColimits.M.mk\n (((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat)\n { fst := j, snd := SMul.smul r { fst := j, snd := x }.snd + SMul.smul s { fst := j, snd := x }.snd }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nsrc\u271d\u00b9 : MulAction R \u2191(M F) := colimitMulAction F\nsrc\u271d : SMulWithZero R \u2191(M F) := colimitSMulWithZero F\nr : R\nx y : \u2191(M F)\n\u22a2 r \u2022 (x + y) = r \u2022 x + r \u2022 y\n[PROOFSTEP]\nrefine' Quot.induction_on\u2082 x y _\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nsrc\u271d\u00b9 : MulAction R \u2191(M F) := colimitMulAction F\nsrc\u271d : SMulWithZero R \u2191(M F) := colimitSMulWithZero F\nr : R\nx y : \u2191(M F)\n\u22a2 \u2200\n (a b :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j),\n r \u2022\n (Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a +\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n b) =\n r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a +\n r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n b\n[PROOFSTEP]\nclear x y\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nsrc\u271d\u00b9 : MulAction R \u2191(M F) := colimitMulAction F\nsrc\u271d : SMulWithZero R \u2191(M F) := colimitSMulWithZero F\nr : R\n\u22a2 \u2200\n (a b :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j),\n r \u2022\n (Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a +\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n b) =\n r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a +\n r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n b\n[PROOFSTEP]\nintro x y\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nsrc\u271d\u00b9 : MulAction R \u2191(M F) := colimitMulAction F\nsrc\u271d : SMulWithZero R \u2191(M F) := colimitSMulWithZero F\nr : R\nx y :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 r \u2022\n (Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n x +\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n y) =\n r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n x +\n r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n y\n[PROOFSTEP]\ncases' x with i x\n[GOAL]\ncase mk\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nsrc\u271d\u00b9 : MulAction R \u2191(M F) := colimitMulAction F\nsrc\u271d : SMulWithZero R \u2191(M F) := colimitSMulWithZero F\nr : R\ny :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\ni : J\nx :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n i\n\u22a2 r \u2022\n (Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := i, snd := x } +\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n y) =\n r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := i, snd := x } +\n r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n y\n[PROOFSTEP]\ncases' y with j y\n[GOAL]\ncase mk.mk\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nsrc\u271d\u00b9 : MulAction R \u2191(M F) := colimitMulAction F\nsrc\u271d : SMulWithZero R \u2191(M F) := colimitSMulWithZero F\nr : R\ni : J\nx :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n i\nj : J\ny :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 r \u2022\n (Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := i, snd := x } +\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := y }) =\n r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := i, snd := x } +\n r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := y }\n[PROOFSTEP]\nerw [colimit_add_mk_eq _ \u27e8i, _\u27e9 \u27e8j, _\u27e9 (max' i j) (IsFiltered.leftToMax i j) (IsFiltered.rightToMax i j),\n colimit_smul_mk_eq, smul_add, colimit_smul_mk_eq, colimit_smul_mk_eq,\n colimit_add_mk_eq _ \u27e8i, _\u27e9 \u27e8j, _\u27e9 (max' i j) (IsFiltered.leftToMax i j) (IsFiltered.rightToMax i j),\n LinearMap.map_smul, LinearMap.map_smul]\n[GOAL]\ncase mk.mk\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nsrc\u271d\u00b9 : MulAction R \u2191(M F) := colimitMulAction F\nsrc\u271d : SMulWithZero R \u2191(M F) := colimitSMulWithZero F\nr : R\ni : J\nx :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n i\nj : J\ny :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 M.mk F\n {\n fst :=\n { fst := IsFiltered.max i j,\n snd :=\n \u2191((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat).map\n (IsFiltered.leftToMax i j))\n { fst := i, snd := x }.snd +\n \u2191((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat).map\n (IsFiltered.rightToMax i j))\n { fst := j, snd := y }.snd }.fst,\n snd :=\n r \u2022\n \u2191((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat).map\n (IsFiltered.leftToMax i j))\n { fst := i, snd := x }.snd +\n r \u2022\n \u2191((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat).map\n (IsFiltered.rightToMax i j))\n { fst := j, snd := y }.snd } =\n AddMonCat.FilteredColimits.M.mk\n (((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat)\n { fst := IsFiltered.max i j,\n snd :=\n r \u2022 \u2191(F.map (IsFiltered.leftToMax i j)) { fst := i, snd := x }.snd +\n r \u2022 \u2191(F.map (IsFiltered.rightToMax i j)) { fst := j, snd := y }.snd }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nj : J\nsrc\u271d : (F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat).obj j \u27f6\n ((Functor.const J).obj\n (AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat)).pt).obj\n j :=\n NatTrans.app (AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat)).\u03b9 j\nr : R\nx : \u2191(F.obj j)\n\u22a2 AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : \u2191((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat).obj j)),\n ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : \u2191((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat).obj j)),\n ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n x\n[PROOFSTEP]\nerw [colimit_smul_mk_eq F r \u27e8j, x\u27e9]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nj : J\nsrc\u271d : (F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat).obj j \u27f6\n ((Functor.const J).obj\n (AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat)).pt).obj\n j :=\n NatTrans.app (AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat)).\u03b9 j\nr : R\nx : \u2191(F.obj j)\n\u22a2 AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : \u2191((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat).obj j)),\n ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (r \u2022 x) =\n M.mk F { fst := { fst := j, snd := x }.fst, snd := r \u2022 { fst := j, snd := x }.snd }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nt : Cocone F\nsrc\u271d : (AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt \u27f6\n ((forget\u2082 (ModuleCat R) AddCommGroupCat).mapCocone t).pt :=\n IsColimit.desc\n (AddCommGroupCat.FilteredColimits.colimitCoconeIsColimit (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat))\n ((forget\u2082 (ModuleCat R) AddCommGroupCat).mapCocone t)\nr : R\nx : \u2191(colimit F)\n\u22a2 AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200\n (x y :\n \u2191(AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt),\n ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200\n (x y :\n \u2191(AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt),\n ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n x\n[PROOFSTEP]\nrefine' Quot.inductionOn x _\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nt : Cocone F\nsrc\u271d : (AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt \u27f6\n ((forget\u2082 (ModuleCat R) AddCommGroupCat).mapCocone t).pt :=\n IsColimit.desc\n (AddCommGroupCat.FilteredColimits.colimitCoconeIsColimit (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat))\n ((forget\u2082 (ModuleCat R) AddCommGroupCat).mapCocone t)\nr : R\nx : \u2191(colimit F)\n\u22a2 \u2200\n (a :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j),\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200\n (x y :\n \u2191(AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt),\n ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200\n (x y :\n \u2191(AddCommGroupCat.FilteredColimits.colimitCocone\n (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt),\n ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a)\n[PROOFSTEP]\nclear x\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nt : Cocone F\nsrc\u271d : (AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt \u27f6\n ((forget\u2082 (ModuleCat R) AddCommGroupCat).mapCocone t).pt :=\n IsColimit.desc\n (AddCommGroupCat.FilteredColimits.colimitCoconeIsColimit (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat))\n ((forget\u2082 (ModuleCat R) AddCommGroupCat).mapCocone t)\nr : R\n\u22a2 \u2200\n (a :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j),\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200\n (x y :\n \u2191(AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt),\n ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200\n (x y :\n \u2191(AddCommGroupCat.FilteredColimits.colimitCocone\n (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt),\n ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n a)\n[PROOFSTEP]\nintro x\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nt : Cocone F\nsrc\u271d : (AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt \u27f6\n ((forget\u2082 (ModuleCat R) AddCommGroupCat).mapCocone t).pt :=\n IsColimit.desc\n (AddCommGroupCat.FilteredColimits.colimitCoconeIsColimit (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat))\n ((forget\u2082 (ModuleCat R) AddCommGroupCat).mapCocone t)\nr : R\nx :\n (j : J) \u00d7\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200\n (x y :\n \u2191(AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt),\n ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200\n (x y :\n \u2191(AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt),\n ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n x)\n[PROOFSTEP]\ncases' x with j x\n[GOAL]\ncase mk\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nt : Cocone F\nsrc\u271d : (AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt \u27f6\n ((forget\u2082 (ModuleCat R) AddCommGroupCat).mapCocone t).pt :=\n IsColimit.desc\n (AddCommGroupCat.FilteredColimits.colimitCoconeIsColimit (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat))\n ((forget\u2082 (ModuleCat R) AddCommGroupCat).mapCocone t)\nr : R\nj : J\nx :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200\n (x y :\n \u2191(AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt),\n ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (r \u2022\n Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x }) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200\n (x y :\n \u2191(AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt),\n ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x })\n[PROOFSTEP]\nerw [colimit_smul_mk_eq]\n[GOAL]\ncase mk\nR : Type u\ninst\u271d\u00b2 : Ring R\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 ModuleCatMax R\nt : Cocone F\nsrc\u271d : (AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt \u27f6\n ((forget\u2082 (ModuleCat R) AddCommGroupCat).mapCocone t).pt :=\n IsColimit.desc\n (AddCommGroupCat.FilteredColimits.colimitCoconeIsColimit (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat))\n ((forget\u2082 (ModuleCat R) AddCommGroupCat).mapCocone t)\nr : R\nj : J\nx :\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat).obj\n j\n\u22a2 AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200\n (x y :\n \u2191(AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt),\n ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (M.mk F { fst := { fst := j, snd := x }.fst, snd := r \u2022 { fst := j, snd := x }.snd }) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200\n (x y :\n \u2191(AddCommGroupCat.FilteredColimits.colimitCocone (F \u22d9 forget\u2082 (ModuleCatMax R) AddCommGroupCat)).pt),\n ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (Quot.mk\n (Types.Quot.Rel\n ((((F \u22d9 forget\u2082 (ModuleCat R) AddCommGroupCat) \u22d9 forget\u2082 AddCommGroupCat AddGroupCat) \u22d9\n forget\u2082 AddGroupCat AddMonCat) \u22d9\n forget AddMonCat))\n { fst := j, snd := x })\n[PROOFSTEP]\nexact LinearMap.map_smul (t.\u03b9.app j) r x\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.ModuleCat.FilteredColimits", "llama_tokens": 20167, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389930307512, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4337329026990114}} {"text": "[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : Monoid G\n\u22a2 \u00acIsTorsion G \u2194 \u2203 g, \u00acIsOfFinOrder g\n[PROOFSTEP]\nrw [IsTorsion, not_forall]\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : Monoid G\ntG : IsTorsion G\nsrc\u271d : Monoid G := inst\u271d\ng : G\n\u22a2 g\u207b\u00b9 * g = 1\n[PROOFSTEP]\nerw [\u2190 pow_succ', tsub_add_cancel_of_le, pow_orderOf_eq_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : Monoid G\ntG : IsTorsion G\nsrc\u271d : Monoid G := inst\u271d\ng : G\n\u22a2 1 \u2264 orderOf g\n[PROOFSTEP]\nexact orderOf_pos' (tG g)\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Group H\nf : G \u2192* H\nhf : Function.Surjective \u2191f\ntG : IsTorsion G\nh : H\n\u22a2 IsOfFinOrder h\n[PROOFSTEP]\nobtain \u27e8g, hg\u27e9 := hf h\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Group H\nf : G \u2192* H\nhf : Function.Surjective \u2191f\ntG : IsTorsion G\nh : H\ng : G\nhg : \u2191f g = h\n\u22a2 IsOfFinOrder h\n[PROOFSTEP]\nrw [\u2190 hg]\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Group H\nf : G \u2192* H\nhf : Function.Surjective \u2191f\ntG : IsTorsion G\nh : H\ng : G\nhg : \u2191f g = h\n\u22a2 IsOfFinOrder (\u2191f g)\n[PROOFSTEP]\nexact f.isOfFinOrder (tG g)\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Group H\nf : G \u2192* H\nhN : N = MonoidHom.ker f\ntH : IsTorsion H\ntN : IsTorsion { x // x \u2208 N }\ng : G\n\u22a2 \u2203 n, 0 < n \u2227 g ^ n = 1\n[PROOFSTEP]\nobtain \u27e8ngn, ngnpos, hngn\u27e9 := (isOfFinOrder_iff_pow_eq_one _).mp (tH <| f g)\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Group H\nf : G \u2192* H\nhN : N = MonoidHom.ker f\ntH : IsTorsion H\ntN : IsTorsion { x // x \u2208 N }\ng : G\nngn : \u2115\nngnpos : 0 < ngn\nhngn : \u2191f g ^ ngn = 1\n\u22a2 \u2203 n, 0 < n \u2227 g ^ n = 1\n[PROOFSTEP]\nhave hmem := f.mem_ker.mpr ((f.map_pow g ngn).trans hngn)\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Group H\nf : G \u2192* H\nhN : N = MonoidHom.ker f\ntH : IsTorsion H\ntN : IsTorsion { x // x \u2208 N }\ng : G\nngn : \u2115\nngnpos : 0 < ngn\nhngn : \u2191f g ^ ngn = 1\nhmem : g ^ ngn \u2208 MonoidHom.ker f\n\u22a2 \u2203 n, 0 < n \u2227 g ^ n = 1\n[PROOFSTEP]\nlift g ^ ngn to N using hN.symm \u25b8 hmem with gn h\n[GOAL]\ncase intro.intro.intro\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Group H\nf : G \u2192* H\nhN : N = MonoidHom.ker f\ntH : IsTorsion H\ntN : IsTorsion { x // x \u2208 N }\ng : G\nngn : \u2115\nngnpos : 0 < ngn\nhngn : \u2191f g ^ ngn = 1\ngn : { x // x \u2208 N }\nh : \u2191gn = g ^ ngn\nhmem\u271d hmem : \u2191gn \u2208 MonoidHom.ker f\n\u22a2 \u2203 n, 0 < n \u2227 g ^ n = 1\n[PROOFSTEP]\nobtain \u27e8nn, nnpos, hnn\u27e9 := (isOfFinOrder_iff_pow_eq_one _).mp (tN gn)\n[GOAL]\ncase intro.intro.intro.intro.intro\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Group H\nf : G \u2192* H\nhN : N = MonoidHom.ker f\ntH : IsTorsion H\ntN : IsTorsion { x // x \u2208 N }\ng : G\nngn : \u2115\nngnpos : 0 < ngn\nhngn : \u2191f g ^ ngn = 1\ngn : { x // x \u2208 N }\nh : \u2191gn = g ^ ngn\nhmem\u271d hmem : \u2191gn \u2208 MonoidHom.ker f\nnn : \u2115\nnnpos : 0 < nn\nhnn : gn ^ nn = 1\n\u22a2 \u2203 n, 0 < n \u2227 g ^ n = 1\n[PROOFSTEP]\nexact \u27e8ngn * nn, mul_pos ngnpos nnpos, by rw [pow_mul, \u2190 h, \u2190 Subgroup.coe_pow, hnn, Subgroup.coe_one]\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Group H\nf : G \u2192* H\nhN : N = MonoidHom.ker f\ntH : IsTorsion H\ntN : IsTorsion { x // x \u2208 N }\ng : G\nngn : \u2115\nngnpos : 0 < ngn\nhngn : \u2191f g ^ ngn = 1\ngn : { x // x \u2208 N }\nh : \u2191gn = g ^ ngn\nhmem\u271d hmem : \u2191gn \u2208 MonoidHom.ker f\nnn : \u2115\nnnpos : 0 < nn\nhnn : gn ^ nn = 1\n\u22a2 g ^ (ngn * nn) = 1\n[PROOFSTEP]\nrw [pow_mul, \u2190 h, \u2190 Subgroup.coe_pow, hnn, Subgroup.coe_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Group H\nh : ExponentExists G\ng : G\n\u22a2 IsOfFinOrder g\n[PROOFSTEP]\nobtain \u27e8n, npos, hn\u27e9 := h\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Group H\ng : G\nn : \u2115\nnpos : 0 < n\nhn : \u2200 (g : G), g ^ n = 1\n\u22a2 IsOfFinOrder g\n[PROOFSTEP]\nexact (isOfFinOrder_iff_pow_eq_one g).mpr \u27e8n, npos, hn g\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nR : Type u_3\nM : Type u_4\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Module R M\ntR : IsTorsion R\nf : M\n\u22a2 \u2203 n, 0 < n \u2227 n \u2022 f = 0\n[PROOFSTEP]\nobtain \u27e8n, npos, hn\u27e9 := (isOfFinAddOrder_iff_nsmul_eq_zero _).mp (tR 1)\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\nR : Type u_3\nM : Type u_4\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Module R M\ntR : IsTorsion R\nf : M\nn : \u2115\nnpos : 0 < n\nhn : n \u2022 1 = 0\n\u22a2 \u2203 n, 0 < n \u2227 n \u2022 f = 0\n[PROOFSTEP]\nexact \u27e8n, npos, by simp only [nsmul_eq_smul_cast R _ f, \u2190 nsmul_one, hn, zero_smul]\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nR : Type u_3\nM : Type u_4\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Module R M\ntR : IsTorsion R\nf : M\nn : \u2115\nnpos : 0 < n\nhn : n \u2022 1 = 0\n\u22a2 n \u2022 f = 0\n[PROOFSTEP]\nsimp only [nsmul_eq_smul_cast R _ f, \u2190 nsmul_one, hn, zero_smul]\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : CommMonoid G\nx\u271d : { x // x \u2208 torsion G }\nx : G\nn : \u2115\nnpos : n > 0\nhn : Function.IsPeriodicPt ((fun x x_1 => x * x_1) x) n 1\n\u22a2 \u2191(((fun x x_1 => x * x_1)\n { val := x, property := (_ : \u2203 n, n > 0 \u2227 Function.IsPeriodicPt ((fun x x_1 => x * x_1) x) n 1) })^[n]\n 1) =\n \u21911\n[PROOFSTEP]\ndsimp\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : CommMonoid G\nx\u271d : { x // x \u2208 torsion G }\nx : G\nn : \u2115\nnpos : n > 0\nhn : Function.IsPeriodicPt ((fun x x_1 => x * x_1) x) n 1\n\u22a2 \u2191((fun x_1 => { val := x, property := (_ : \u2203 n, n > 0 \u2227 Function.IsPeriodicPt (fun x_2 => x * x_2) n 1) } * x_1)^[n]\n 1) =\n 1\n[PROOFSTEP]\nrw [mul_left_iterate]\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : CommMonoid G\nx\u271d : { x // x \u2208 torsion G }\nx : G\nn : \u2115\nnpos : n > 0\nhn : Function.IsPeriodicPt ((fun x x_1 => x * x_1) x) n 1\n\u22a2 \u2191((fun x_1 => { val := x, property := (_ : \u2203 n, n > 0 \u2227 Function.IsPeriodicPt (fun x_2 => x * x_2) n 1) } ^ n * x_1)\n 1) =\n 1\n[PROOFSTEP]\nchange _ * 1 = 1\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : CommMonoid G\nx\u271d : { x // x \u2208 torsion G }\nx : G\nn : \u2115\nnpos : n > 0\nhn : Function.IsPeriodicPt ((fun x x_1 => x * x_1) x) n 1\n\u22a2 \u2191({ val := x, property := (_ : \u2203 n, n > 0 \u2227 Function.IsPeriodicPt (fun x_1 => x * x_1) n 1) } ^ n) * 1 = 1\n[PROOFSTEP]\nrw [_root_.mul_one, SubmonoidClass.coe_pow, Subtype.coe_mk, (isPeriodicPt_mul_iff_pow_eq_one _).mp hn]\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : CommMonoid G\np : \u2115\nhp : Fact (Nat.Prime p)\na\u271d b\u271d : G\nhg\u2081 : a\u271d \u2208 {g | \u2203 n, orderOf g = p ^ n}\nhg\u2082 : b\u271d \u2208 {g | \u2203 n, orderOf g = p ^ n}\n\u22a2 \u2203 m, (a\u271d * b\u271d) ^ p ^ m = 1\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := exists_orderOf_eq_prime_pow_iff.mp hg\u2081\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\ninst\u271d : CommMonoid G\np : \u2115\nhp : Fact (Nat.Prime p)\na\u271d b\u271d : G\nhg\u2081 : a\u271d \u2208 {g | \u2203 n, orderOf g = p ^ n}\nhg\u2082 : b\u271d \u2208 {g | \u2203 n, orderOf g = p ^ n}\nm : \u2115\nhm : a\u271d ^ p ^ m = 1\n\u22a2 \u2203 m, (a\u271d * b\u271d) ^ p ^ m = 1\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := exists_orderOf_eq_prime_pow_iff.mp hg\u2082\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\ninst\u271d : CommMonoid G\np : \u2115\nhp : Fact (Nat.Prime p)\na\u271d b\u271d : G\nhg\u2081 : a\u271d \u2208 {g | \u2203 n, orderOf g = p ^ n}\nhg\u2082 : b\u271d \u2208 {g | \u2203 n, orderOf g = p ^ n}\nm : \u2115\nhm : a\u271d ^ p ^ m = 1\nn : \u2115\nhn : b\u271d ^ p ^ n = 1\n\u22a2 \u2203 m, (a\u271d * b\u271d) ^ p ^ m = 1\n[PROOFSTEP]\nexact \u27e8m + n, by rw [mul_pow, pow_add, pow_mul, hm, one_pow, Monoid.one_mul, mul_comm, pow_mul, hn, one_pow]\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : CommMonoid G\np : \u2115\nhp : Fact (Nat.Prime p)\na\u271d b\u271d : G\nhg\u2081 : a\u271d \u2208 {g | \u2203 n, orderOf g = p ^ n}\nhg\u2082 : b\u271d \u2208 {g | \u2203 n, orderOf g = p ^ n}\nm : \u2115\nhm : a\u271d ^ p ^ m = 1\nn : \u2115\nhn : b\u271d ^ p ^ n = 1\n\u22a2 (a\u271d * b\u271d) ^ p ^ (m + n) = 1\n[PROOFSTEP]\nrw [mul_pow, pow_add, pow_mul, hm, one_pow, Monoid.one_mul, mul_comm, pow_mul, hn, one_pow]\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : CommMonoid G\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 orderOf 1 = p ^ 0\n[PROOFSTEP]\nrw [pow_zero, orderOf_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : CommMonoid G\np : \u2115\nhp : Fact (Nat.Prime p)\ng : { x // x \u2208 primaryComponent G p }\n\u22a2 \u2203 n, orderOf g = p ^ n\n[PROOFSTEP]\nsimpa [primaryComponent] using g.property\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : CommMonoid G\np : \u2115\nhp : Fact (Nat.Prime p)\np' : \u2115\nhp' : Fact (Nat.Prime p')\nhne : p \u2260 p'\n\u22a2 \u2200 {x : G}, x \u2208 primaryComponent G p \u2192 x \u2208 primaryComponent G p' \u2192 x = 1\n[PROOFSTEP]\nrintro g \u27e8_ | n, hn\u27e9 \u27e8n', hn'\u27e9\n[GOAL]\ncase intro.zero.intro\nG : Type u_1\nH : Type u_2\ninst\u271d : CommMonoid G\np : \u2115\nhp : Fact (Nat.Prime p)\np' : \u2115\nhp' : Fact (Nat.Prime p')\nhne : p \u2260 p'\ng : G\nhn : orderOf g = p ^ Nat.zero\nn' : \u2115\nhn' : orderOf g = p' ^ n'\n\u22a2 g = 1\n[PROOFSTEP]\nrwa [pow_zero, orderOf_eq_one_iff] at hn \n[GOAL]\ncase intro.succ.intro\nG : Type u_1\nH : Type u_2\ninst\u271d : CommMonoid G\np : \u2115\nhp : Fact (Nat.Prime p)\np' : \u2115\nhp' : Fact (Nat.Prime p')\nhne : p \u2260 p'\ng : G\nn : \u2115\nhn : orderOf g = p ^ Nat.succ n\nn' : \u2115\nhn' : orderOf g = p' ^ n'\n\u22a2 g = 1\n[PROOFSTEP]\nexact absurd (eq_of_prime_pow_eq hp.out.prime hp'.out.prime n.succ_pos (hn.symm.trans hn')) hne\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : CommMonoid G\ntG : IsTorsion G\n\u22a2 torsion G = \u22a4\n[PROOFSTEP]\next\n[GOAL]\ncase h\nG : Type u_1\nH : Type u_2\ninst\u271d : CommMonoid G\ntG : IsTorsion G\nx\u271d : G\n\u22a2 x\u271d \u2208 torsion G \u2194 x\u271d \u2208 \u22a4\n[PROOFSTEP]\ntauto\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : Monoid G\n\u22a2 \u00acIsTorsionFree G \u2194 \u2203 g, g \u2260 1 \u2227 IsOfFinOrder g\n[PROOFSTEP]\nsimp_rw [IsTorsionFree, Ne.def, not_forall, Classical.not_not, exists_prop]\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : Group G\nhN : Nontrivial G\ntG : IsTorsion G\n\u22a2 \u2203 g, g \u2260 1 \u2227 IsOfFinOrder g\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := (nontrivial_iff_exists_ne (1 : G)).mp hN\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\ninst\u271d : Group G\nhN : Nontrivial G\ntG : IsTorsion G\nx : G\nhx : x \u2260 1\n\u22a2 \u2203 g, g \u2260 1 \u2227 IsOfFinOrder g\n[PROOFSTEP]\nexact \u27e8x, hx, tG x\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : Group G\nhN : Nontrivial G\ntfG : IsTorsionFree G\n\u22a2 \u2203 g, \u00acIsOfFinOrder g\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := (nontrivial_iff_exists_ne (1 : G)).mp hN\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\ninst\u271d : Group G\nhN : Nontrivial G\ntfG : IsTorsionFree G\nx : G\nhx : x \u2260 1\n\u22a2 \u2203 g, \u00acIsOfFinOrder g\n[PROOFSTEP]\nexact \u27e8x, (tfG x) hx\u27e9\n[GOAL]\nG : Type u_1\nH\u271d : Type u_2\ninst\u271d : Group G\ntG : IsTorsionFree G\nH : Subgroup G\nh : { x // x \u2208 H }\nhne : h \u2260 1\n\u22a2 \u2191h \u2260 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nG : Type u_1\nH : Type u_2\ninst\u271d : CommGroup G\ng : G \u29f8 torsion G\nhne : g \u2260 1\nhfin : IsOfFinOrder g\n\u22a2 g = 1\n[PROOFSTEP]\ninduction' g using QuotientGroup.induction_on' with g\n[GOAL]\ncase H\nG : Type u_1\nH : Type u_2\ninst\u271d : CommGroup G\ng\u271d : G \u29f8 torsion G\nhne\u271d : g\u271d \u2260 1\nhfin\u271d : IsOfFinOrder g\u271d\ng : G\nhne : \u2191g \u2260 1\nhfin : IsOfFinOrder \u2191g\n\u22a2 \u2191g = 1\n[PROOFSTEP]\nobtain \u27e8m, mpos, hm\u27e9 := (isOfFinOrder_iff_pow_eq_one _).mp hfin\n[GOAL]\ncase H.intro.intro\nG : Type u_1\nH : Type u_2\ninst\u271d : CommGroup G\ng\u271d : G \u29f8 torsion G\nhne\u271d : g\u271d \u2260 1\nhfin\u271d : IsOfFinOrder g\u271d\ng : G\nhne : \u2191g \u2260 1\nhfin : IsOfFinOrder \u2191g\nm : \u2115\nmpos : 0 < m\nhm : \u2191g ^ m = 1\n\u22a2 \u2191g = 1\n[PROOFSTEP]\nobtain \u27e8n, npos, hn\u27e9 := (isOfFinOrder_iff_pow_eq_one _).mp ((QuotientGroup.eq_one_iff _).mp hm)\n[GOAL]\ncase H.intro.intro.intro.intro\nG : Type u_1\nH : Type u_2\ninst\u271d : CommGroup G\ng\u271d : G \u29f8 torsion G\nhne\u271d : g\u271d \u2260 1\nhfin\u271d : IsOfFinOrder g\u271d\ng : G\nhne : \u2191g \u2260 1\nhfin : IsOfFinOrder \u2191g\nm : \u2115\nmpos : 0 < m\nhm : \u2191g ^ m = 1\nn : \u2115\nnpos : 0 < n\nhn : (fun x => x ^ m) g ^ n = 1\n\u22a2 \u2191g = 1\n[PROOFSTEP]\nexact\n (QuotientGroup.eq_one_iff g).mpr\n ((isOfFinOrder_iff_pow_eq_one _).mpr \u27e8m * n, mul_pos mpos npos, (pow_mul g m n).symm \u25b8 hn\u27e9)\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Torsion", "llama_tokens": 6120, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419958239132, "lm_q2_score": 0.6261241842048092, "lm_q1q2_score": 0.4336799045812385}} {"text": "[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nS : B\nl : List M\nhl : \u2200 (x : M), x \u2208 l \u2192 x \u2208 S\n\u22a2 List.prod l \u2208 S\n[PROOFSTEP]\nlift l to List S using hl\n[GOAL]\ncase intro\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nS : B\nl : List { x // x \u2208 S }\n\u22a2 List.prod (List.map Subtype.val l) \u2208 S\n[PROOFSTEP]\nrw [\u2190 coe_list_prod]\n[GOAL]\ncase intro\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nS : B\nl : List { x // x \u2208 S }\n\u22a2 \u2191(List.prod l) \u2208 S\n[PROOFSTEP]\nexact l.prod.coe_prop\n[GOAL]\nM\u271d : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2075 : Monoid M\u271d\ninst\u271d\u2074 : SetLike B M\u271d\ninst\u271d\u00b3 : SubmonoidClass B M\u271d\nS : B\nM : Type u_4\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nm : Multiset M\nhm : \u2200 (a : M), a \u2208 m \u2192 a \u2208 S\n\u22a2 Multiset.prod m \u2208 S\n[PROOFSTEP]\nlift m to Multiset S using hm\n[GOAL]\ncase intro\nM\u271d : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2075 : Monoid M\u271d\ninst\u271d\u2074 : SetLike B M\u271d\ninst\u271d\u00b3 : SubmonoidClass B M\u271d\nS : B\nM : Type u_4\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nm : Multiset { x // x \u2208 S }\n\u22a2 Multiset.prod (Multiset.map Subtype.val m) \u2208 S\n[PROOFSTEP]\nrw [\u2190 coe_multiset_prod]\n[GOAL]\ncase intro\nM\u271d : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2075 : Monoid M\u271d\ninst\u271d\u2074 : SetLike B M\u271d\ninst\u271d\u00b3 : SubmonoidClass B M\u271d\nS : B\nM : Type u_4\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nm : Multiset { x // x \u2208 S }\n\u22a2 \u2191(Multiset.prod m) \u2208 S\n[PROOFSTEP]\nexact m.prod.coe_prop\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nS : B\ns : Submonoid M\nl : List M\nhl : \u2200 (x : M), x \u2208 l \u2192 x \u2208 s\n\u22a2 List.prod l \u2208 s\n[PROOFSTEP]\nlift l to List s using hl\n[GOAL]\ncase intro\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nS : B\ns : Submonoid M\nl : List { x // x \u2208 s }\n\u22a2 List.prod (List.map Subtype.val l) \u2208 s\n[PROOFSTEP]\nrw [\u2190 coe_list_prod]\n[GOAL]\ncase intro\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nS : B\ns : Submonoid M\nl : List { x // x \u2208 s }\n\u22a2 \u2191(List.prod l) \u2208 s\n[PROOFSTEP]\nexact l.prod.coe_prop\n[GOAL]\nM\u271d : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Monoid M\u271d\ninst\u271d\u00b2 : SetLike B M\u271d\ninst\u271d\u00b9 : SubmonoidClass B M\u271d\nS\u271d : B\ns : Submonoid M\u271d\nM : Type u_4\ninst\u271d : CommMonoid M\nS : Submonoid M\nm : Multiset M\nhm : \u2200 (a : M), a \u2208 m \u2192 a \u2208 S\n\u22a2 Multiset.prod m \u2208 S\n[PROOFSTEP]\nlift m to Multiset S using hm\n[GOAL]\ncase intro\nM\u271d : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Monoid M\u271d\ninst\u271d\u00b2 : SetLike B M\u271d\ninst\u271d\u00b9 : SubmonoidClass B M\u271d\nS\u271d : B\ns : Submonoid M\u271d\nM : Type u_4\ninst\u271d : CommMonoid M\nS : Submonoid M\nm : Multiset { x // x \u2208 S }\n\u22a2 Multiset.prod (Multiset.map Subtype.val m) \u2208 S\n[PROOFSTEP]\nrw [\u2190 coe_multiset_prod]\n[GOAL]\ncase intro\nM\u271d : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b3 : Monoid M\u271d\ninst\u271d\u00b2 : SetLike B M\u271d\ninst\u271d\u00b9 : SubmonoidClass B M\u271d\nS\u271d : B\ns : Submonoid M\u271d\nM : Type u_4\ninst\u271d : CommMonoid M\nS : Submonoid M\nm : Multiset { x // x \u2208 S }\n\u22a2 \u2191(Multiset.prod m) \u2208 S\n[PROOFSTEP]\nexact m.prod.coe_prop\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nS\u271d : B\ns S : Submonoid M\nm : Multiset M\ncomm : Set.Pairwise {x | x \u2208 m} Commute\nh : \u2200 (x : M), x \u2208 m \u2192 x \u2208 S\n\u22a2 Multiset.noncommProd m comm \u2208 S\n[PROOFSTEP]\ninduction' m using Quotient.inductionOn with l\n[GOAL]\ncase h\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nS\u271d : B\ns S : Submonoid M\nm : Multiset M\ncomm\u271d : Set.Pairwise {x | x \u2208 m} Commute\nh\u271d : \u2200 (x : M), x \u2208 m \u2192 x \u2208 S\nl : List M\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid M) l} Commute\nh : \u2200 (x : M), x \u2208 Quotient.mk (List.isSetoid M) l \u2192 x \u2208 S\n\u22a2 Multiset.noncommProd (Quotient.mk (List.isSetoid M) l) comm \u2208 S\n[PROOFSTEP]\nsimp only [Multiset.quot_mk_to_coe, Multiset.noncommProd_coe]\n[GOAL]\ncase h\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nS\u271d : B\ns S : Submonoid M\nm : Multiset M\ncomm\u271d : Set.Pairwise {x | x \u2208 m} Commute\nh\u271d : \u2200 (x : M), x \u2208 m \u2192 x \u2208 S\nl : List M\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid M) l} Commute\nh : \u2200 (x : M), x \u2208 Quotient.mk (List.isSetoid M) l \u2192 x \u2208 S\n\u22a2 List.prod l \u2208 S\n[PROOFSTEP]\nexact Submonoid.list_prod_mem _ h\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nS\u271d : B\ns S : Submonoid M\n\u03b9 : Type u_4\nt : Finset \u03b9\nf : \u03b9 \u2192 M\ncomm : Set.Pairwise \u2191t fun a b => Commute (f a) (f b)\nh : \u2200 (c : \u03b9), c \u2208 t \u2192 f c \u2208 S\n\u22a2 Finset.noncommProd t f comm \u2208 S\n[PROOFSTEP]\napply multiset_noncommProd_mem\n[GOAL]\ncase h\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nS\u271d : B\ns S : Submonoid M\n\u03b9 : Type u_4\nt : Finset \u03b9\nf : \u03b9 \u2192 M\ncomm : Set.Pairwise \u2191t fun a b => Commute (f a) (f b)\nh : \u2200 (c : \u03b9), c \u2208 t \u2192 f c \u2208 S\n\u22a2 \u2200 (x : M), x \u2208 Multiset.map f t.val \u2192 x \u2208 S\n[PROOFSTEP]\nintro y\n[GOAL]\ncase h\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nS\u271d : B\ns S : Submonoid M\n\u03b9 : Type u_4\nt : Finset \u03b9\nf : \u03b9 \u2192 M\ncomm : Set.Pairwise \u2191t fun a b => Commute (f a) (f b)\nh : \u2200 (c : \u03b9), c \u2208 t \u2192 f c \u2208 S\ny : M\n\u22a2 y \u2208 Multiset.map f t.val \u2192 y \u2208 S\n[PROOFSTEP]\nrw [Multiset.mem_map]\n[GOAL]\ncase h\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nS\u271d : B\ns S : Submonoid M\n\u03b9 : Type u_4\nt : Finset \u03b9\nf : \u03b9 \u2192 M\ncomm : Set.Pairwise \u2191t fun a b => Commute (f a) (f b)\nh : \u2200 (c : \u03b9), c \u2208 t \u2192 f c \u2208 S\ny : M\n\u22a2 (\u2203 a, a \u2208 t.val \u2227 f a = y) \u2192 y \u2208 S\n[PROOFSTEP]\nrintro \u27e8x, \u27e8hx, rfl\u27e9\u27e9\n[GOAL]\ncase h.intro.intro\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nS\u271d : B\ns S : Submonoid M\n\u03b9 : Type u_4\nt : Finset \u03b9\nf : \u03b9 \u2192 M\ncomm : Set.Pairwise \u2191t fun a b => Commute (f a) (f b)\nh : \u2200 (c : \u03b9), c \u2208 t \u2192 f c \u2208 S\nx : \u03b9\nhx : x \u2208 t.val\n\u22a2 f x \u2208 S\n[PROOFSTEP]\nexact h x hx\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submonoid M\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : M\n\u22a2 x \u2208 \u2a06 (i : \u03b9), S i \u2194 \u2203 i, x \u2208 S i\n[PROOFSTEP]\nrefine' \u27e8_, fun \u27e8i, hi\u27e9 => (SetLike.le_def.1 <| le_iSup S i) hi\u27e9\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submonoid M\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : M\n\u22a2 x \u2208 \u2a06 (i : \u03b9), S i \u2192 \u2203 i, x \u2208 S i\n[PROOFSTEP]\nsuffices x \u2208 closure (\u22c3 i, (S i : Set M)) \u2192 \u2203 i, x \u2208 S i by simpa only [closure_iUnion, closure_eq (S _)] using this\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submonoid M\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : M\nthis : x \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i)) \u2192 \u2203 i, x \u2208 S i\n\u22a2 x \u2208 \u2a06 (i : \u03b9), S i \u2192 \u2203 i, x \u2208 S i\n[PROOFSTEP]\nsimpa only [closure_iUnion, closure_eq (S _)] using this\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submonoid M\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : M\n\u22a2 x \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i)) \u2192 \u2203 i, x \u2208 S i\n[PROOFSTEP]\nrefine' fun hx => closure_induction hx (fun _ => mem_iUnion.1) _ _\n[GOAL]\ncase refine'_1\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submonoid M\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : M\nhx : x \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\n\u22a2 \u2203 i, 1 \u2208 S i\n[PROOFSTEP]\nexact h\u03b9.elim fun i => \u27e8i, (S i).one_mem\u27e9\n[GOAL]\ncase refine'_2\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submonoid M\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : M\nhx : x \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\n\u22a2 \u2200 (x y : M), (\u2203 i, x \u2208 S i) \u2192 (\u2203 i, y \u2208 S i) \u2192 \u2203 i, x * y \u2208 S i\n[PROOFSTEP]\nrintro x y \u27e8i, hi\u27e9 \u27e8j, hj\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submonoid M\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx\u271d : M\nhx : x\u271d \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nx y : M\ni : \u03b9\nhi : x \u2208 S i\nj : \u03b9\nhj : y \u2208 S j\n\u22a2 \u2203 i, x * y \u2208 S i\n[PROOFSTEP]\nrcases hS i j with \u27e8k, hki, hkj\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Submonoid M\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx\u271d : M\nhx : x\u271d \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nx y : M\ni : \u03b9\nhi : x \u2208 S i\nj : \u03b9\nhj : y \u2208 S j\nk : \u03b9\nhki : S i \u2264 S k\nhkj : S j \u2264 S k\n\u22a2 \u2203 i, x * y \u2208 S i\n[PROOFSTEP]\nexact \u27e8k, (S k).mul_mem (hki hi) (hkj hj)\u27e9\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9 : MulOneClass M\n\u03b9 : Sort u_4\ninst\u271d : Nonempty \u03b9\nS : \u03b9 \u2192 Submonoid M\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : M\n\u22a2 x \u2208 \u2191(\u2a06 (i : \u03b9), S i) \u2194 x \u2208 \u22c3 (i : \u03b9), \u2191(S i)\n[PROOFSTEP]\nsimp [mem_iSup_of_directed hS]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\nS : Set (Submonoid M)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x \u2264 x_1) S\nx : M\n\u22a2 x \u2208 sSup S \u2194 \u2203 s, s \u2208 S \u2227 x \u2208 s\n[PROOFSTEP]\nhaveI : Nonempty S := Sne.to_subtype\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\nS : Set (Submonoid M)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x \u2264 x_1) S\nx : M\nthis : Nonempty \u2191S\n\u22a2 x \u2208 sSup S \u2194 \u2203 s, s \u2208 S \u2227 x \u2208 s\n[PROOFSTEP]\nsimp [sSup_eq_iSup', mem_iSup_of_directed hS.directed_val, SetCoe.exists, Subtype.coe_mk]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\nS : Set (Submonoid M)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x \u2264 x_1) S\nx : M\n\u22a2 x \u2208 \u2191(sSup S) \u2194 x \u2208 \u22c3 (s : Submonoid M) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\nsimp [mem_sSup_of_directedOn Sne hS]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\nS T : Submonoid M\n\u22a2 \u2200 {x : M}, x \u2208 S \u2192 x \u2208 S \u2294 T\n[PROOFSTEP]\nrw [\u2190 SetLike.le_def]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\nS T : Submonoid M\n\u22a2 S \u2264 S \u2294 T\n[PROOFSTEP]\nexact le_sup_left\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\nS T : Submonoid M\n\u22a2 \u2200 {x : M}, x \u2208 T \u2192 x \u2208 S \u2294 T\n[PROOFSTEP]\nrw [\u2190 SetLike.le_def]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\nS T : Submonoid M\n\u22a2 T \u2264 S \u2294 T\n[PROOFSTEP]\nexact le_sup_right\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nS : \u03b9 \u2192 Submonoid M\ni : \u03b9\n\u22a2 \u2200 {x : M}, x \u2208 S i \u2192 x \u2208 iSup S\n[PROOFSTEP]\nrw [\u2190 SetLike.le_def]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nS : \u03b9 \u2192 Submonoid M\ni : \u03b9\n\u22a2 S i \u2264 iSup S\n[PROOFSTEP]\nexact le_iSup _ _\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\nS : Set (Submonoid M)\ns : Submonoid M\nhs : s \u2208 S\n\u22a2 \u2200 {x : M}, x \u2208 s \u2192 x \u2208 sSup S\n[PROOFSTEP]\nrw [\u2190 SetLike.le_def]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\nS : Set (Submonoid M)\ns : Submonoid M\nhs : s \u2208 S\n\u22a2 s \u2264 sSup S\n[PROOFSTEP]\nexact le_sSup hs\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nS : \u03b9 \u2192 Submonoid M\nC : M \u2192 Prop\nx : M\nhx : x \u2208 \u2a06 (i : \u03b9), S i\nhp : \u2200 (i : \u03b9) (x : M), x \u2208 S i \u2192 C x\nh1 : C 1\nhmul : \u2200 (x y : M), C x \u2192 C y \u2192 C (x * y)\n\u22a2 C x\n[PROOFSTEP]\nrw [iSup_eq_closure] at hx \n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nS : \u03b9 \u2192 Submonoid M\nC : M \u2192 Prop\nx : M\nhx : x \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nhp : \u2200 (i : \u03b9) (x : M), x \u2208 S i \u2192 C x\nh1 : C 1\nhmul : \u2200 (x y : M), C x \u2192 C y \u2192 C (x * y)\n\u22a2 C x\n[PROOFSTEP]\nrefine closure_induction hx (fun x hx => ?_) h1 hmul\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nS : \u03b9 \u2192 Submonoid M\nC : M \u2192 Prop\nx\u271d : M\nhx\u271d : x\u271d \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nhp : \u2200 (i : \u03b9) (x : M), x \u2208 S i \u2192 C x\nh1 : C 1\nhmul : \u2200 (x y : M), C x \u2192 C y \u2192 C (x * y)\nx : M\nhx : x \u2208 \u22c3 (i : \u03b9), \u2191(S i)\n\u22a2 C x\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := Set.mem_iUnion.mp hx\n[GOAL]\ncase intro\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nS : \u03b9 \u2192 Submonoid M\nC : M \u2192 Prop\nx\u271d : M\nhx\u271d : x\u271d \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nhp : \u2200 (i : \u03b9) (x : M), x \u2208 S i \u2192 C x\nh1 : C 1\nhmul : \u2200 (x y : M), C x \u2192 C y \u2192 C (x * y)\nx : M\nhx : x \u2208 \u22c3 (i : \u03b9), \u2191(S i)\ni : \u03b9\nhi : x \u2208 \u2191(S i)\n\u22a2 C x\n[PROOFSTEP]\nexact hp _ _ hi\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nS : \u03b9 \u2192 Submonoid M\nC : (x : M) \u2192 x \u2208 \u2a06 (i : \u03b9), S i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : M) (hxS : x \u2208 S i), C x (_ : x \u2208 \u2a06 (i : \u03b9), S i)\nh1 : C 1 (_ : 1 \u2208 \u2a06 (i : \u03b9), S i)\nhmul :\n \u2200 (x y : M) (hx : x \u2208 \u2a06 (i : \u03b9), S i) (hy : y \u2208 \u2a06 (i : \u03b9), S i),\n C x hx \u2192 C y hy \u2192 C (x * y) (_ : x * y \u2208 \u2a06 (i : \u03b9), S i)\nx : M\nhx : x \u2208 \u2a06 (i : \u03b9), S i\n\u22a2 C x hx\n[PROOFSTEP]\nrefine' Exists.elim (_ : \u2203 Hx, C x Hx) fun (hx : x \u2208 \u2a06 i, S i) (hc : C x hx) => hc\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nS : \u03b9 \u2192 Submonoid M\nC : (x : M) \u2192 x \u2208 \u2a06 (i : \u03b9), S i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : M) (hxS : x \u2208 S i), C x (_ : x \u2208 \u2a06 (i : \u03b9), S i)\nh1 : C 1 (_ : 1 \u2208 \u2a06 (i : \u03b9), S i)\nhmul :\n \u2200 (x y : M) (hx : x \u2208 \u2a06 (i : \u03b9), S i) (hy : y \u2208 \u2a06 (i : \u03b9), S i),\n C x hx \u2192 C y hy \u2192 C (x * y) (_ : x * y \u2208 \u2a06 (i : \u03b9), S i)\nx : M\nhx : x \u2208 \u2a06 (i : \u03b9), S i\n\u22a2 \u2203 Hx, C x Hx\n[PROOFSTEP]\nrefine' @iSup_induction _ _ \u03b9 S (fun m => \u2203 hm, C m hm) _ hx (fun i x hx => _) _ fun x y => _\n[GOAL]\ncase refine'_1\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nS : \u03b9 \u2192 Submonoid M\nC : (x : M) \u2192 x \u2208 \u2a06 (i : \u03b9), S i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : M) (hxS : x \u2208 S i), C x (_ : x \u2208 \u2a06 (i : \u03b9), S i)\nh1 : C 1 (_ : 1 \u2208 \u2a06 (i : \u03b9), S i)\nhmul :\n \u2200 (x y : M) (hx : x \u2208 \u2a06 (i : \u03b9), S i) (hy : y \u2208 \u2a06 (i : \u03b9), S i),\n C x hx \u2192 C y hy \u2192 C (x * y) (_ : x * y \u2208 \u2a06 (i : \u03b9), S i)\nx\u271d : M\nhx\u271d : x\u271d \u2208 \u2a06 (i : \u03b9), S i\ni : \u03b9\nx : M\nhx : x \u2208 S i\n\u22a2 (fun m => \u2203 hm, C m hm) x\n[PROOFSTEP]\nexact \u27e8_, hp _ _ hx\u27e9\n[GOAL]\ncase refine'_2\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nS : \u03b9 \u2192 Submonoid M\nC : (x : M) \u2192 x \u2208 \u2a06 (i : \u03b9), S i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : M) (hxS : x \u2208 S i), C x (_ : x \u2208 \u2a06 (i : \u03b9), S i)\nh1 : C 1 (_ : 1 \u2208 \u2a06 (i : \u03b9), S i)\nhmul :\n \u2200 (x y : M) (hx : x \u2208 \u2a06 (i : \u03b9), S i) (hy : y \u2208 \u2a06 (i : \u03b9), S i),\n C x hx \u2192 C y hy \u2192 C (x * y) (_ : x * y \u2208 \u2a06 (i : \u03b9), S i)\nx : M\nhx : x \u2208 \u2a06 (i : \u03b9), S i\n\u22a2 (fun m => \u2203 hm, C m hm) 1\n[PROOFSTEP]\nexact \u27e8_, h1\u27e9\n[GOAL]\ncase refine'_3\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nS : \u03b9 \u2192 Submonoid M\nC : (x : M) \u2192 x \u2208 \u2a06 (i : \u03b9), S i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : M) (hxS : x \u2208 S i), C x (_ : x \u2208 \u2a06 (i : \u03b9), S i)\nh1 : C 1 (_ : 1 \u2208 \u2a06 (i : \u03b9), S i)\nhmul :\n \u2200 (x y : M) (hx : x \u2208 \u2a06 (i : \u03b9), S i) (hy : y \u2208 \u2a06 (i : \u03b9), S i),\n C x hx \u2192 C y hy \u2192 C (x * y) (_ : x * y \u2208 \u2a06 (i : \u03b9), S i)\nx\u271d : M\nhx : x\u271d \u2208 \u2a06 (i : \u03b9), S i\nx y : M\n\u22a2 (fun m => \u2203 hm, C m hm) x \u2192 (fun m => \u2203 hm, C m hm) y \u2192 (fun m => \u2203 hm, C m hm) (x * y)\n[PROOFSTEP]\nrintro \u27e8_, Cx\u27e9 \u27e8_, Cy\u27e9\n[GOAL]\ncase refine'_3.intro.intro\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : MulOneClass M\n\u03b9 : Sort u_4\nS : \u03b9 \u2192 Submonoid M\nC : (x : M) \u2192 x \u2208 \u2a06 (i : \u03b9), S i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : M) (hxS : x \u2208 S i), C x (_ : x \u2208 \u2a06 (i : \u03b9), S i)\nh1 : C 1 (_ : 1 \u2208 \u2a06 (i : \u03b9), S i)\nhmul :\n \u2200 (x y : M) (hx : x \u2208 \u2a06 (i : \u03b9), S i) (hy : y \u2208 \u2a06 (i : \u03b9), S i),\n C x hx \u2192 C y hy \u2192 C (x * y) (_ : x * y \u2208 \u2a06 (i : \u03b9), S i)\nx\u271d : M\nhx : x\u271d \u2208 \u2a06 (i : \u03b9), S i\nx y : M\nw\u271d\u00b9 : x \u2208 \u2a06 (i : \u03b9), S i\nCx : C x w\u271d\u00b9\nw\u271d : y \u2208 \u2a06 (i : \u03b9), S i\nCy : C y w\u271d\n\u22a2 \u2203 hm, C (x * y) hm\n[PROOFSTEP]\nrefine' \u27e8_, hmul _ _ _ _ Cx Cy\u27e9\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\nx y : M\n\u22a2 y \u2208 closure {x} \u2194 \u2203 n, x ^ n = y\n[PROOFSTEP]\nrw [closure_singleton_eq, mem_mrange]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\nx y : M\n\u22a2 (\u2203 x_1, \u2191(\u2191(powersHom M) x) x_1 = y) \u2194 \u2203 n, x ^ n = y\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\n\u22a2 closure {1} = \u22a5\n[PROOFSTEP]\nsimp [eq_bot_iff_forall, mem_closure_singleton]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\n\u03b1 : Type u_4\nf : \u03b1 \u2192 M\n\u22a2 mrange (\u2191FreeMonoid.lift f) = closure (Set.range f)\n[PROOFSTEP]\nrw [mrange_eq_map, \u2190 FreeMonoid.closure_range_of, map_mclosure, \u2190 Set.range_comp, FreeMonoid.lift_comp_of]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\n\u22a2 closure s = mrange (\u2191FreeMonoid.lift Subtype.val)\n[PROOFSTEP]\nrw [FreeMonoid.mrange_lift, Subtype.range_coe]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\n\u22a2 \u2191(closure s) = List.prod '' {l | \u2200 (x : M), x \u2208 l \u2192 x \u2208 s}\n[PROOFSTEP]\nrw [closure_eq_mrange, coe_mrange, \u2190 Set.range_list_map_coe, \u2190 Set.range_comp]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\n\u22a2 Set.range \u2191(\u2191FreeMonoid.lift Subtype.val) = Set.range (List.prod \u2218 List.map Subtype.val)\n[PROOFSTEP]\nexact congrArg _ (funext <| FreeMonoid.lift_apply _)\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\nx : M\nhx : x \u2208 closure s\n\u22a2 \u2203 l x_1, List.prod l = x\n[PROOFSTEP]\nrwa [\u2190 SetLike.mem_coe, closure_eq_image_prod, Set.mem_image_iff_bex] at hx \n[GOAL]\nM\u271d : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9 : Monoid M\u271d\nM : Type u_4\ninst\u271d : CommMonoid M\ns : Set M\nx : M\nhx : x \u2208 closure s\n\u22a2 \u2203 l x_1, Multiset.prod l = x\n[PROOFSTEP]\nobtain \u27e8l, h1, h2\u27e9 := exists_list_of_mem_closure hx\n[GOAL]\ncase intro.intro\nM\u271d : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9 : Monoid M\u271d\nM : Type u_4\ninst\u271d : CommMonoid M\ns : Set M\nx : M\nhx : x \u2208 closure s\nl : List M\nh1 : \u2200 (y : M), y \u2208 l \u2192 y \u2208 s\nh2 : List.prod l = x\n\u22a2 \u2203 l x_1, Multiset.prod l = x\n[PROOFSTEP]\nexact \u27e8l, h1, (Multiset.coe_prod l).trans h2\u27e9\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\np : M \u2192 Prop\nx : M\nh : x \u2208 closure s\nH1 : p 1\nHmul : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : M), p y \u2192 p (x * y)\n\u22a2 p x\n[PROOFSTEP]\nrw [closure_eq_mrange] at h \n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\np : M \u2192 Prop\nx : M\nh : x \u2208 mrange (\u2191FreeMonoid.lift Subtype.val)\nH1 : p 1\nHmul : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : M), p y \u2192 p (x * y)\n\u22a2 p x\n[PROOFSTEP]\nobtain \u27e8l, rfl\u27e9 := h\n[GOAL]\ncase intro\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\np : M \u2192 Prop\nH1 : p 1\nHmul : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : M), p y \u2192 p (x * y)\nl : FreeMonoid { x // x \u2208 s }\n\u22a2 p (\u2191(\u2191FreeMonoid.lift Subtype.val) l)\n[PROOFSTEP]\ninduction' l using FreeMonoid.recOn with x y ih\n[GOAL]\ncase intro.h0\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\np : M \u2192 Prop\nH1 : p 1\nHmul : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : M), p y \u2192 p (x * y)\n\u22a2 p (\u2191(\u2191FreeMonoid.lift Subtype.val) 1)\n[PROOFSTEP]\nexact H1\n[GOAL]\ncase intro.ih\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\np : M \u2192 Prop\nH1 : p 1\nHmul : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : M), p y \u2192 p (x * y)\nx : { x // x \u2208 s }\ny : FreeMonoid { x // x \u2208 s }\nih : p (\u2191(\u2191FreeMonoid.lift Subtype.val) y)\n\u22a2 p (\u2191(\u2191FreeMonoid.lift Subtype.val) (FreeMonoid.of x * y))\n[PROOFSTEP]\nsimpa only [map_mul, FreeMonoid.lift_eval_of] using Hmul _ x.prop _ ih\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\np : M \u2192 Prop\nhs : closure s = \u22a4\nx : M\nH1 : p 1\nHmul : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : M), p y \u2192 p (x * y)\n\u22a2 x \u2208 closure s\n[PROOFSTEP]\nrw [hs]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\np : M \u2192 Prop\nhs : closure s = \u22a4\nx : M\nH1 : p 1\nHmul : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : M), p y \u2192 p (x * y)\n\u22a2 x \u2208 \u22a4\n[PROOFSTEP]\nexact mem_top _\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\np : M \u2192 Prop\nhs : closure s = \u22a4\nx : M\nH1 : p 1\nHmul : \u2200 (x y : M), y \u2208 s \u2192 p x \u2192 p (x * y)\n\u22a2 x \u2208 closure s\n[PROOFSTEP]\nrw [hs]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\np : M \u2192 Prop\nhs : closure s = \u22a4\nx : M\nH1 : p 1\nHmul : \u2200 (x y : M), y \u2208 s \u2192 p x \u2192 p (x * y)\n\u22a2 x \u2208 \u22a4\n[PROOFSTEP]\nexact mem_top _\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\nn\u271d n : M\ni : \u2115\n\u22a2 (fun x x_1 => x ^ x_1) n\u271d i = n \u2194 \u2191(\u2191(powersHom M) n\u271d) i = n\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\nn\u271d n : M\ni : \u2115\n\u22a2 n\u271d ^ i = n \u2194 n\u271d ^ \u2191Multiplicative.toAdd i = n\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\nn : M\n\u22a2 powers n = closure {n}\n[PROOFSTEP]\next\n[GOAL]\ncase h\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\nn x\u271d : M\n\u22a2 x\u271d \u2208 powers n \u2194 x\u271d \u2208 closure {n}\n[PROOFSTEP]\nexact mem_closure_singleton.symm\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : DecidableEq M\nn : M\nh : Function.Injective fun m => n ^ m\nx\u271d\u00b9 x\u271d : Multiplicative \u2115\n\u22a2 Equiv.toFun\n { toFun := fun m => pow n (\u2191Multiplicative.toAdd m), invFun := fun m => \u2191Multiplicative.ofAdd (log m),\n left_inv := (_ : \u2200 (m : \u2115), log (pow n m) = m),\n right_inv := (_ : \u2200 (p : { x // x \u2208 powers n }), pow n (log p) = p) }\n (x\u271d\u00b9 * x\u271d) =\n Equiv.toFun\n { toFun := fun m => pow n (\u2191Multiplicative.toAdd m), invFun := fun m => \u2191Multiplicative.ofAdd (log m),\n left_inv := (_ : \u2200 (m : \u2115), log (pow n m) = m),\n right_inv := (_ : \u2200 (p : { x // x \u2208 powers n }), pow n (log p) = p) }\n x\u271d\u00b9 *\n Equiv.toFun\n { toFun := fun m => pow n (\u2191Multiplicative.toAdd m), invFun := fun m => \u2191Multiplicative.ofAdd (log m),\n left_inv := (_ : \u2200 (m : \u2115), log (pow n m) = m),\n right_inv := (_ : \u2200 (p : { x // x \u2208 powers n }), pow n (log p) = p) }\n x\u271d\n[PROOFSTEP]\nsimp only [pow, map_mul, ofAdd_add, toAdd_mul]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : Monoid M\nN : Type u_4\nF : Type u_5\ninst\u271d\u00b9 : Monoid N\ninst\u271d : MonoidHomClass F M N\nf : F\nm : M\n\u22a2 map f (powers m) = powers (\u2191f m)\n[PROOFSTEP]\nsimp only [powers_eq_closure, map_mclosure f, Set.image_singleton]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\nhcomm : \u2200 (a : M), a \u2208 s \u2192 \u2200 (b : M), b \u2208 s \u2192 a * b = b * a\nsrc\u271d : Monoid { x // x \u2208 closure s } := toMonoid (closure s)\nx y : { x // x \u2208 closure s }\n\u22a2 x * y = y * x\n[PROOFSTEP]\next\n[GOAL]\ncase a\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\nhcomm : \u2200 (a : M), a \u2208 s \u2192 \u2200 (b : M), b \u2208 s \u2192 a * b = b * a\nsrc\u271d : Monoid { x // x \u2208 closure s } := toMonoid (closure s)\nx y : { x // x \u2208 closure s }\n\u22a2 \u2191(x * y) = \u2191(y * x)\n[PROOFSTEP]\nsimp only [Submonoid.coe_mul]\n[GOAL]\ncase a\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\ns : Set M\nhcomm : \u2200 (a : M), a \u2208 s \u2192 \u2200 (b : M), b \u2208 s \u2192 a * b = b * a\nsrc\u271d : Monoid { x // x \u2208 closure s } := toMonoid (closure s)\nx y : { x // x \u2208 closure s }\n\u22a2 \u2191x * \u2191y = \u2191y * \u2191x\n[PROOFSTEP]\nexact\n closure_induction\u2082 x.prop y.prop hcomm Commute.one_left Commute.one_right (fun x y z => Commute.mul_left) fun x y z =>\n Commute.mul_right\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : MulAction M N\ninst\u271d\u00b9 : SMul N \u03b1\ninst\u271d : MulAction M \u03b1\ns : Set M\nhtop : Submonoid.closure s = \u22a4\nhs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), (x \u2022 y) \u2022 z = x \u2022 y \u2022 z\n\u22a2 IsScalarTower M N \u03b1\n[PROOFSTEP]\nrefine' \u27e8fun x => Submonoid.induction_of_closure_eq_top_left htop x _ _\u27e9\n[GOAL]\ncase refine'_1\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : MulAction M N\ninst\u271d\u00b9 : SMul N \u03b1\ninst\u271d : MulAction M \u03b1\ns : Set M\nhtop : Submonoid.closure s = \u22a4\nhs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), (x \u2022 y) \u2022 z = x \u2022 y \u2022 z\nx : M\n\u22a2 \u2200 (y : N) (z : \u03b1), (1 \u2022 y) \u2022 z = 1 \u2022 y \u2022 z\n[PROOFSTEP]\nintro y z\n[GOAL]\ncase refine'_1\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : MulAction M N\ninst\u271d\u00b9 : SMul N \u03b1\ninst\u271d : MulAction M \u03b1\ns : Set M\nhtop : Submonoid.closure s = \u22a4\nhs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), (x \u2022 y) \u2022 z = x \u2022 y \u2022 z\nx : M\ny : N\nz : \u03b1\n\u22a2 (1 \u2022 y) \u2022 z = 1 \u2022 y \u2022 z\n[PROOFSTEP]\nrw [one_smul, one_smul]\n[GOAL]\ncase refine'_2\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : MulAction M N\ninst\u271d\u00b9 : SMul N \u03b1\ninst\u271d : MulAction M \u03b1\ns : Set M\nhtop : Submonoid.closure s = \u22a4\nhs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), (x \u2022 y) \u2022 z = x \u2022 y \u2022 z\nx : M\n\u22a2 \u2200 (x : M),\n x \u2208 s \u2192\n \u2200 (y : M),\n (\u2200 (y_1 : N) (z : \u03b1), (y \u2022 y_1) \u2022 z = y \u2022 y_1 \u2022 z) \u2192\n \u2200 (y_1 : N) (z : \u03b1), ((x * y) \u2022 y_1) \u2022 z = (x * y) \u2022 y_1 \u2022 z\n[PROOFSTEP]\nclear x\n[GOAL]\ncase refine'_2\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : MulAction M N\ninst\u271d\u00b9 : SMul N \u03b1\ninst\u271d : MulAction M \u03b1\ns : Set M\nhtop : Submonoid.closure s = \u22a4\nhs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), (x \u2022 y) \u2022 z = x \u2022 y \u2022 z\n\u22a2 \u2200 (x : M),\n x \u2208 s \u2192\n \u2200 (y : M),\n (\u2200 (y_1 : N) (z : \u03b1), (y \u2022 y_1) \u2022 z = y \u2022 y_1 \u2022 z) \u2192\n \u2200 (y_1 : N) (z : \u03b1), ((x * y) \u2022 y_1) \u2022 z = (x * y) \u2022 y_1 \u2022 z\n[PROOFSTEP]\nintro x hx x' hx' y z\n[GOAL]\ncase refine'_2\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : MulAction M N\ninst\u271d\u00b9 : SMul N \u03b1\ninst\u271d : MulAction M \u03b1\ns : Set M\nhtop : Submonoid.closure s = \u22a4\nhs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), (x \u2022 y) \u2022 z = x \u2022 y \u2022 z\nx : M\nhx : x \u2208 s\nx' : M\nhx' : \u2200 (y : N) (z : \u03b1), (x' \u2022 y) \u2022 z = x' \u2022 y \u2022 z\ny : N\nz : \u03b1\n\u22a2 ((x * x') \u2022 y) \u2022 z = (x * x') \u2022 y \u2022 z\n[PROOFSTEP]\nrw [mul_smul, mul_smul, hs x hx, hx']\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SMul N \u03b1\ninst\u271d : MulAction M \u03b1\ns : Set M\nhtop : Submonoid.closure s = \u22a4\nhs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), x \u2022 y \u2022 z = y \u2022 x \u2022 z\n\u22a2 SMulCommClass M N \u03b1\n[PROOFSTEP]\nrefine' \u27e8fun x => Submonoid.induction_of_closure_eq_top_left htop x _ _\u27e9\n[GOAL]\ncase refine'_1\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SMul N \u03b1\ninst\u271d : MulAction M \u03b1\ns : Set M\nhtop : Submonoid.closure s = \u22a4\nhs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), x \u2022 y \u2022 z = y \u2022 x \u2022 z\nx : M\n\u22a2 \u2200 (n : N) (a : \u03b1), 1 \u2022 n \u2022 a = n \u2022 1 \u2022 a\n[PROOFSTEP]\nintro y z\n[GOAL]\ncase refine'_1\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SMul N \u03b1\ninst\u271d : MulAction M \u03b1\ns : Set M\nhtop : Submonoid.closure s = \u22a4\nhs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), x \u2022 y \u2022 z = y \u2022 x \u2022 z\nx : M\ny : N\nz : \u03b1\n\u22a2 1 \u2022 y \u2022 z = y \u2022 1 \u2022 z\n[PROOFSTEP]\nrw [one_smul, one_smul]\n[GOAL]\ncase refine'_2\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SMul N \u03b1\ninst\u271d : MulAction M \u03b1\ns : Set M\nhtop : Submonoid.closure s = \u22a4\nhs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), x \u2022 y \u2022 z = y \u2022 x \u2022 z\nx : M\n\u22a2 \u2200 (x : M),\n x \u2208 s \u2192 \u2200 (y : M), (\u2200 (n : N) (a : \u03b1), y \u2022 n \u2022 a = n \u2022 y \u2022 a) \u2192 \u2200 (n : N) (a : \u03b1), (x * y) \u2022 n \u2022 a = n \u2022 (x * y) \u2022 a\n[PROOFSTEP]\nclear x\n[GOAL]\ncase refine'_2\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SMul N \u03b1\ninst\u271d : MulAction M \u03b1\ns : Set M\nhtop : Submonoid.closure s = \u22a4\nhs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), x \u2022 y \u2022 z = y \u2022 x \u2022 z\n\u22a2 \u2200 (x : M),\n x \u2208 s \u2192 \u2200 (y : M), (\u2200 (n : N) (a : \u03b1), y \u2022 n \u2022 a = n \u2022 y \u2022 a) \u2192 \u2200 (n : N) (a : \u03b1), (x * y) \u2022 n \u2022 a = n \u2022 (x * y) \u2022 a\n[PROOFSTEP]\nintro x hx x' hx' y z\n[GOAL]\ncase refine'_2\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : SMul N \u03b1\ninst\u271d : MulAction M \u03b1\ns : Set M\nhtop : Submonoid.closure s = \u22a4\nhs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), x \u2022 y \u2022 z = y \u2022 x \u2022 z\nx : M\nhx : x \u2208 s\nx' : M\nhx' : \u2200 (n : N) (a : \u03b1), x' \u2022 n \u2022 a = n \u2022 x' \u2022 a\ny : N\nz : \u03b1\n\u22a2 (x * x') \u2022 y \u2022 z = y \u2022 (x * x') \u2022 z\n[PROOFSTEP]\nrw [mul_smul, mul_smul, hx', hs x hx]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\ninst\u271d : CommMonoid N\ns t : Submonoid N\n\u22a2 s \u2294 t = mrange (coprod (subtype s) (subtype t))\n[PROOFSTEP]\nrw [mrange_eq_map, \u2190 mrange_inl_sup_mrange_inr, map_sup, map_mrange, coprod_comp_inl, map_mrange, coprod_comp_inr,\n range_subtype, range_subtype]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\nN : Type u_4\ninst\u271d : CommMonoid N\ns t : Submonoid N\nx : N\n\u22a2 x \u2208 s \u2294 t \u2194 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x\n[PROOFSTEP]\nsimp only [ge_iff_le, sup_eq_range, mem_mrange, coprod_apply, coe_subtype, Prod.exists, Subtype.exists, exists_prop]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : AddMonoid A\nx y : A\n\u22a2 y \u2208 closure {x} \u2194 \u2203 n, n \u2022 x = y\n[PROOFSTEP]\nrw [closure_singleton_eq, AddMonoidHom.mem_mrange]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : AddMonoid A\nx y : A\n\u22a2 (\u2203 x_1, \u2191(\u2191(multiplesHom A) x) x_1 = y) \u2194 \u2203 n, n \u2022 x = y\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : AddMonoid A\n\u22a2 closure {0} = \u22a5\n[PROOFSTEP]\nsimp [eq_bot_iff_forall, mem_closure_singleton, nsmul_zero]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : AddMonoid A\nx n : A\ni : \u2115\n\u22a2 (fun i => i \u2022 x) i = n \u2194 \u2191(\u2191(multiplesHom A) x) i = n\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\na b : R\nha : a \u2208 AddSubmonoid.closure \u2191S\nhb : b \u2208 S\n\u22a2 a * b \u2208 AddSubmonoid.closure \u2191S\n[PROOFSTEP]\nrevert b\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\na : R\nha : a \u2208 AddSubmonoid.closure \u2191S\n\u22a2 \u2200 {b : R}, b \u2208 S \u2192 a * b \u2208 AddSubmonoid.closure \u2191S\n[PROOFSTEP]\nrefine' @AddSubmonoid.closure_induction _ _ _ (fun z => \u2200 (b : R), b \u2208 S \u2192 z * b \u2208 AddSubmonoid.closure S) _ ha _ _ _\n[GOAL]\ncase refine'_1\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\na : R\nha : a \u2208 AddSubmonoid.closure \u2191S\n\u22a2 \u2200 (x : R), x \u2208 \u2191S \u2192 (fun z => \u2200 (b : R), b \u2208 S \u2192 z * b \u2208 AddSubmonoid.closure \u2191S) x\n[PROOFSTEP]\nclear ha a\n[GOAL]\ncase refine'_2\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\na : R\nha : a \u2208 AddSubmonoid.closure \u2191S\n\u22a2 (fun z => \u2200 (b : R), b \u2208 S \u2192 z * b \u2208 AddSubmonoid.closure \u2191S) 0\n[PROOFSTEP]\nclear ha a\n[GOAL]\ncase refine'_3\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\na : R\nha : a \u2208 AddSubmonoid.closure \u2191S\n\u22a2 \u2200 (x y : R),\n (fun z => \u2200 (b : R), b \u2208 S \u2192 z * b \u2208 AddSubmonoid.closure \u2191S) x \u2192\n (fun z => \u2200 (b : R), b \u2208 S \u2192 z * b \u2208 AddSubmonoid.closure \u2191S) y \u2192\n (fun z => \u2200 (b : R), b \u2208 S \u2192 z * b \u2208 AddSubmonoid.closure \u2191S) (x + y)\n[PROOFSTEP]\nclear ha a\n[GOAL]\ncase refine'_1\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 \u2200 (x : R), x \u2208 \u2191S \u2192 (fun z => \u2200 (b : R), b \u2208 S \u2192 z * b \u2208 AddSubmonoid.closure \u2191S) x\n[PROOFSTEP]\nexact fun r hr b hb => AddSubmonoid.mem_closure.mpr fun y hy => hy (mul_mem hr hb)\n[GOAL]\ncase refine'_2\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 (fun z => \u2200 (b : R), b \u2208 S \u2192 z * b \u2208 AddSubmonoid.closure \u2191S) 0\n[PROOFSTEP]\nexact fun b _ => by simp only [zero_mul, (AddSubmonoid.closure (S : Set R)).zero_mem]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\nb : R\nx\u271d : b \u2208 S\n\u22a2 0 * b \u2208 AddSubmonoid.closure \u2191S\n[PROOFSTEP]\nsimp only [zero_mul, (AddSubmonoid.closure (S : Set R)).zero_mem]\n[GOAL]\ncase refine'_3\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 \u2200 (x y : R),\n (fun z => \u2200 (b : R), b \u2208 S \u2192 z * b \u2208 AddSubmonoid.closure \u2191S) x \u2192\n (fun z => \u2200 (b : R), b \u2208 S \u2192 z * b \u2208 AddSubmonoid.closure \u2191S) y \u2192\n (fun z => \u2200 (b : R), b \u2208 S \u2192 z * b \u2208 AddSubmonoid.closure \u2191S) (x + y)\n[PROOFSTEP]\nsimp_rw [add_mul]\n[GOAL]\ncase refine'_3\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 \u2200 (x y : R),\n (\u2200 (b : R), b \u2208 S \u2192 x * b \u2208 AddSubmonoid.closure \u2191S) \u2192\n (\u2200 (b : R), b \u2208 S \u2192 y * b \u2208 AddSubmonoid.closure \u2191S) \u2192 \u2200 (b : R), b \u2208 S \u2192 x * b + y * b \u2208 AddSubmonoid.closure \u2191S\n[PROOFSTEP]\nexact fun r s hr hs b hb => (AddSubmonoid.closure (S : Set R)).add_mem (hr _ hb) (hs _ hb)\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\na b : R\nha : a \u2208 AddSubmonoid.closure \u2191S\nhb : b \u2208 AddSubmonoid.closure \u2191S\n\u22a2 a * b \u2208 AddSubmonoid.closure \u2191S\n[PROOFSTEP]\nrevert a\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\nb : R\nhb : b \u2208 AddSubmonoid.closure \u2191S\n\u22a2 \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * b \u2208 AddSubmonoid.closure \u2191S\n[PROOFSTEP]\nrefine'\n @AddSubmonoid.closure_induction _ _ _\n (fun z => \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * z \u2208 AddSubmonoid.closure \u2191S) _ hb _ _ _\n[GOAL]\ncase refine'_1\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\nb : R\nhb : b \u2208 AddSubmonoid.closure \u2191S\n\u22a2 \u2200 (x : R), x \u2208 \u2191S \u2192 (fun z => \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * z \u2208 AddSubmonoid.closure \u2191S) x\n[PROOFSTEP]\nclear hb b\n[GOAL]\ncase refine'_2\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\nb : R\nhb : b \u2208 AddSubmonoid.closure \u2191S\n\u22a2 (fun z => \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * z \u2208 AddSubmonoid.closure \u2191S) 0\n[PROOFSTEP]\nclear hb b\n[GOAL]\ncase refine'_3\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\nb : R\nhb : b \u2208 AddSubmonoid.closure \u2191S\n\u22a2 \u2200 (x y : R),\n (fun z => \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * z \u2208 AddSubmonoid.closure \u2191S) x \u2192\n (fun z => \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * z \u2208 AddSubmonoid.closure \u2191S) y \u2192\n (fun z => \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * z \u2208 AddSubmonoid.closure \u2191S) (x + y)\n[PROOFSTEP]\nclear hb b\n[GOAL]\ncase refine'_1\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 \u2200 (x : R), x \u2208 \u2191S \u2192 (fun z => \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * z \u2208 AddSubmonoid.closure \u2191S) x\n[PROOFSTEP]\nexact fun r hr b hb => MulMemClass.mul_right_mem_add_closure hb hr\n[GOAL]\ncase refine'_2\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 (fun z => \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * z \u2208 AddSubmonoid.closure \u2191S) 0\n[PROOFSTEP]\nexact fun _ => by simp only [mul_zero, (AddSubmonoid.closure (S : Set R)).zero_mem]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\na\u271d : R\nx\u271d : a\u271d \u2208 AddSubmonoid.closure \u2191S\n\u22a2 a\u271d * 0 \u2208 AddSubmonoid.closure \u2191S\n[PROOFSTEP]\nsimp only [mul_zero, (AddSubmonoid.closure (S : Set R)).zero_mem]\n[GOAL]\ncase refine'_3\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 \u2200 (x y : R),\n (fun z => \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * z \u2208 AddSubmonoid.closure \u2191S) x \u2192\n (fun z => \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * z \u2208 AddSubmonoid.closure \u2191S) y \u2192\n (fun z => \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * z \u2208 AddSubmonoid.closure \u2191S) (x + y)\n[PROOFSTEP]\nsimp_rw [mul_add]\n[GOAL]\ncase refine'_3\nM : Type u_1\nA : Type u_2\nB : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : SetLike M R\ninst\u271d : MulMemClass M R\nS : M\n\u22a2 \u2200 (x y : R),\n (\u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * x \u2208 AddSubmonoid.closure \u2191S) \u2192\n (\u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * y \u2208 AddSubmonoid.closure \u2191S) \u2192\n \u2200 {a : R}, a \u2208 AddSubmonoid.closure \u2191S \u2192 a * x + a * y \u2208 AddSubmonoid.closure \u2191S\n[PROOFSTEP]\nexact fun r s hr hs b hb => (AddSubmonoid.closure (S : Set R)).add_mem (hr hb) (hs hb)\n[GOAL]\nM : Type u_1\nA\u271d : Type u_2\nB : Type u_3\nA : Type u_4\ninst\u271d : CommMonoid A\na b c : A\n\u22a2 c \u2208 closure {a, b} \u2194 \u2203 m n, a ^ m * b ^ n = c\n[PROOFSTEP]\nrw [\u2190 Set.singleton_union, Submonoid.closure_union, mem_sup]\n[GOAL]\nM : Type u_1\nA\u271d : Type u_2\nB : Type u_3\nA : Type u_4\ninst\u271d : CommMonoid A\na b c : A\n\u22a2 (\u2203 y, y \u2208 closure {a} \u2227 \u2203 z, z \u2208 closure {b} \u2227 y * z = c) \u2194 \u2203 m n, a ^ m * b ^ n = c\n[PROOFSTEP]\nsimp_rw [mem_closure_singleton, exists_exists_eq_and]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\nx : M\n\u22a2 \u2191Additive.ofMul '' \u2191(Submonoid.powers x) = \u2191(AddSubmonoid.multiples (\u2191Additive.ofMul x))\n[PROOFSTEP]\next\n[GOAL]\ncase h\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\nx : M\nx\u271d : Additive M\n\u22a2 x\u271d \u2208 \u2191Additive.ofMul '' \u2191(Submonoid.powers x) \u2194 x\u271d \u2208 \u2191(AddSubmonoid.multiples (\u2191Additive.ofMul x))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\nx : M\nx\u271d : Additive M\n\u22a2 x\u271d \u2208 \u2191Additive.ofMul '' \u2191(Submonoid.powers x) \u2192 x\u271d \u2208 \u2191(AddSubmonoid.multiples (\u2191Additive.ofMul x))\n[PROOFSTEP]\nrintro \u27e8y, \u27e8n, hy1\u27e9, hy2\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\nx : M\nx\u271d : Additive M\ny : M\nhy2 : \u2191Additive.ofMul y = x\u271d\nn : \u2115\nhy1 : (fun x x_1 => x ^ x_1) x n = y\n\u22a2 x\u271d \u2208 \u2191(AddSubmonoid.multiples (\u2191Additive.ofMul x))\n[PROOFSTEP]\nuse n\n[GOAL]\ncase h\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\nx : M\nx\u271d : Additive M\ny : M\nhy2 : \u2191Additive.ofMul y = x\u271d\nn : \u2115\nhy1 : (fun x x_1 => x ^ x_1) x n = y\n\u22a2 (fun i => i \u2022 \u2191Additive.ofMul x) n = x\u271d\n[PROOFSTEP]\nsimpa [\u2190 ofMul_pow, hy1]\n[GOAL]\ncase h.mpr\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\nx : M\nx\u271d : Additive M\n\u22a2 x\u271d \u2208 \u2191(AddSubmonoid.multiples (\u2191Additive.ofMul x)) \u2192 x\u271d \u2208 \u2191Additive.ofMul '' \u2191(Submonoid.powers x)\n[PROOFSTEP]\nrintro \u27e8n, hn\u27e9\n[GOAL]\ncase h.mpr.intro\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\nx : M\nx\u271d : Additive M\nn : \u2115\nhn : (fun i => i \u2022 \u2191Additive.ofMul x) n = x\u271d\n\u22a2 x\u271d \u2208 \u2191Additive.ofMul '' \u2191(Submonoid.powers x)\n[PROOFSTEP]\nrefine' \u27e8x ^ n, \u27e8n, rfl\u27e9, _\u27e9\n[GOAL]\ncase h.mpr.intro\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : Monoid M\nx : M\nx\u271d : Additive M\nn : \u2115\nhn : (fun i => i \u2022 \u2191Additive.ofMul x) n = x\u271d\n\u22a2 \u2191Additive.ofMul (x ^ n) = x\u271d\n[PROOFSTEP]\nrwa [ofMul_pow]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : AddMonoid A\nx : A\n\u22a2 \u2191Multiplicative.ofAdd '' \u2191(AddSubmonoid.multiples x) = \u2191(Submonoid.powers (\u2191Multiplicative.ofAdd x))\n[PROOFSTEP]\nsymm\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : AddMonoid A\nx : A\n\u22a2 \u2191(Submonoid.powers (\u2191Multiplicative.ofAdd x)) = \u2191Multiplicative.ofAdd '' \u2191(AddSubmonoid.multiples x)\n[PROOFSTEP]\nrw [Equiv.eq_image_iff_symm_image_eq]\n[GOAL]\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d : AddMonoid A\nx : A\n\u22a2 \u2191Multiplicative.ofAdd.symm '' \u2191(Submonoid.powers (\u2191Multiplicative.ofAdd x)) = \u2191(AddSubmonoid.multiples x)\n[PROOFSTEP]\nexact ofMul_image_powers_eq_multiples_ofMul\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Submonoid.Membership", "llama_tokens": 21056, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585903489892, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.4332740091093506}} {"text": "[GOAL]\nX Y : SemiNormedGroupCat\u2081\nf : X \u27f6 Y\n\u22a2 f \u226b\n mkHom (AddSubgroup.normedMk (NormedAddGroupHom.range \u2191f))\n (_ : \u2200 (m : \u2191Y), \u2016\u2191(AddSubgroup.normedMk (NormedAddGroupHom.range \u2191f)) m\u2016 \u2264 \u2016m\u2016) =\n 0 \u226b\n mkHom (AddSubgroup.normedMk (NormedAddGroupHom.range \u2191f))\n (_ : \u2200 (m : \u2191Y), \u2016\u2191(AddSubgroup.normedMk (NormedAddGroupHom.range \u2191f)) m\u2016 \u2264 \u2016m\u2016)\n[PROOFSTEP]\next x\n[GOAL]\ncase w.h\nX Y : SemiNormedGroupCat\u2081\nf : X \u27f6 Y\nx : \u2191X\n\u22a2 \u2191(f \u226b\n mkHom (AddSubgroup.normedMk (NormedAddGroupHom.range \u2191f))\n (_ : \u2200 (m : \u2191Y), \u2016\u2191(AddSubgroup.normedMk (NormedAddGroupHom.range \u2191f)) m\u2016 \u2264 \u2016m\u2016))\n x =\n \u2191(0 \u226b\n mkHom (AddSubgroup.normedMk (NormedAddGroupHom.range \u2191f))\n (_ : \u2200 (m : \u2191Y), \u2016\u2191(AddSubgroup.normedMk (NormedAddGroupHom.range \u2191f)) m\u2016 \u2264 \u2016m\u2016))\n x\n[PROOFSTEP]\nrw [Limits.zero_comp, comp_apply, SemiNormedGroupCat\u2081.mkHom_apply, SemiNormedGroupCat\u2081.zero_apply, \u2190\n NormedAddGroupHom.mem_ker, f.1.range.ker_normedMk, f.1.mem_range]\n[GOAL]\ncase w.h\nX Y : SemiNormedGroupCat\u2081\nf : X \u27f6 Y\nx : \u2191X\n\u22a2 \u2203 w, \u2191\u2191f w = \u2191f x\n[PROOFSTEP]\nuse x\n[GOAL]\ncase h\nX Y : SemiNormedGroupCat\u2081\nf : X \u27f6 Y\nx : \u2191X\n\u22a2 \u2191\u2191f x = \u2191f x\n[PROOFSTEP]\nrfl\n[GOAL]\nX Y : SemiNormedGroupCat\u2081\nf : X \u27f6 Y\ns : CokernelCofork f\n\u22a2 (cokernelCocone f).pt \u27f6 s.pt\n[PROOFSTEP]\nfconstructor\n -- The lift itself:\n[GOAL]\ncase val\nX Y : SemiNormedGroupCat\u2081\nf : X \u27f6 Y\ns : CokernelCofork f\n\u22a2 NormedAddGroupHom \u2191(cokernelCocone f).pt \u2191s.pt\n[PROOFSTEP]\napply NormedAddGroupHom.lift _ s.\u03c0.1\n[GOAL]\ncase val\nX Y : SemiNormedGroupCat\u2081\nf : X \u27f6 Y\ns : CokernelCofork f\n\u22a2 \u2200 (s_1 : \u2191((parallelPair f 0).obj WalkingParallelPair.one)), s_1 \u2208 NormedAddGroupHom.range \u2191f \u2192 \u2191\u2191(Cofork.\u03c0 s) s_1 = 0\n[PROOFSTEP]\nrintro _ \u27e8b, rfl\u27e9\n[GOAL]\ncase val.intro\nX Y : SemiNormedGroupCat\u2081\nf : X \u27f6 Y\ns : CokernelCofork f\nb : \u2191X\n\u22a2 \u2191\u2191(Cofork.\u03c0 s) (\u2191(NormedAddGroupHom.toAddMonoidHom \u2191f) b) = 0\n[PROOFSTEP]\nchange (f \u226b s.\u03c0) b = 0\n[GOAL]\ncase val.intro\nX Y : SemiNormedGroupCat\u2081\nf : X \u27f6 Y\ns : CokernelCofork f\nb : \u2191X\n\u22a2 \u2191(f \u226b Cofork.\u03c0 s) b = 0\n[PROOFSTEP]\nsimp\n -- The lift has norm at most one:\n[GOAL]\ncase property\nX Y : SemiNormedGroupCat\u2081\nf : X \u27f6 Y\ns : CokernelCofork f\n\u22a2 NormedAddGroupHom.NormNoninc\n (NormedAddGroupHom.lift (NormedAddGroupHom.range \u2191f) \u2191(Cofork.\u03c0 s)\n (_ :\n \u2200 (s_1 : \u2191((parallelPair f 0).obj WalkingParallelPair.one)),\n s_1 \u2208 NormedAddGroupHom.range \u2191f \u2192 \u2191\u2191(Cofork.\u03c0 s) s_1 = 0))\n[PROOFSTEP]\nexact NormedAddGroupHom.lift_normNoninc _ _ _ s.\u03c0.2\n[GOAL]\nX\u271d Y\u271d : SemiNormedGroupCat\u2081\nf : X\u271d \u27f6 Y\u271d\ns : CokernelCofork f\n\u22a2 Cofork.\u03c0 (cokernelCocone f) \u226b cokernelLift f s = Cofork.\u03c0 s\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nX\u271d Y\u271d : SemiNormedGroupCat\u2081\nf : X\u271d \u27f6 Y\u271d\ns : CokernelCofork f\nx\u271d : \u2191((parallelPair f 0).obj WalkingParallelPair.one)\n\u22a2 \u2191(Cofork.\u03c0 (cokernelCocone f) \u226b cokernelLift f s) x\u271d = \u2191(Cofork.\u03c0 s) x\u271d\n[PROOFSTEP]\napply NormedAddGroupHom.lift_mk f.1.range\n[GOAL]\ncase w.h.hf\nX\u271d Y\u271d : SemiNormedGroupCat\u2081\nf : X\u271d \u27f6 Y\u271d\ns : CokernelCofork f\nx\u271d : \u2191((parallelPair f 0).obj WalkingParallelPair.one)\n\u22a2 \u2200 (s_1 : \u2191Y\u271d), s_1 \u2208 NormedAddGroupHom.range \u2191f \u2192 \u2191\u2191(Cofork.\u03c0 s) s_1 = 0\n[PROOFSTEP]\nrintro _ \u27e8b, rfl\u27e9\n[GOAL]\ncase w.h.hf.intro\nX\u271d Y\u271d : SemiNormedGroupCat\u2081\nf : X\u271d \u27f6 Y\u271d\ns : CokernelCofork f\nx\u271d : \u2191((parallelPair f 0).obj WalkingParallelPair.one)\nb : \u2191X\u271d\n\u22a2 \u2191\u2191(Cofork.\u03c0 s) (\u2191(NormedAddGroupHom.toAddMonoidHom \u2191f) b) = 0\n[PROOFSTEP]\nchange (f \u226b s.\u03c0) b = 0\n[GOAL]\ncase w.h.hf.intro\nX\u271d Y\u271d : SemiNormedGroupCat\u2081\nf : X\u271d \u27f6 Y\u271d\ns : CokernelCofork f\nx\u271d : \u2191((parallelPair f 0).obj WalkingParallelPair.one)\nb : \u2191X\u271d\n\u22a2 \u2191(f \u226b Cofork.\u03c0 s) b = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 HasCokernels SemiNormedGroupCat\u2081\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nV W : SemiNormedGroupCat\nf g : V \u27f6 W\n\u22a2 NormedAddGroupHom.incl (NormedAddGroupHom.ker (f - g)) \u226b f =\n NormedAddGroupHom.incl (NormedAddGroupHom.ker (f - g)) \u226b g\n[PROOFSTEP]\nchange NormedAddGroupHom V W at f g \n[GOAL]\nV W : SemiNormedGroupCat\nf g : NormedAddGroupHom \u2191V \u2191W\n\u22a2 NormedAddGroupHom.incl (NormedAddGroupHom.ker (f - g)) \u226b f =\n NormedAddGroupHom.incl (NormedAddGroupHom.ker (f - g)) \u226b g\n[PROOFSTEP]\next v\n[GOAL]\ncase h\nV W : SemiNormedGroupCat\nf g : NormedAddGroupHom \u2191V \u2191W\nv : \u2191(of { x // x \u2208 NormedAddGroupHom.ker (f - g) })\n\u22a2 \u2191(NormedAddGroupHom.incl (NormedAddGroupHom.ker (f - g)) \u226b f) v =\n \u2191(NormedAddGroupHom.incl (NormedAddGroupHom.ker (f - g)) \u226b g) v\n[PROOFSTEP]\nhave : v.1 \u2208 (f - g).ker := v.2\n[GOAL]\ncase h\nV W : SemiNormedGroupCat\nf g : NormedAddGroupHom \u2191V \u2191W\nv : \u2191(of { x // x \u2208 NormedAddGroupHom.ker (f - g) })\nthis : \u2191v \u2208 NormedAddGroupHom.ker (f - g)\n\u22a2 \u2191(NormedAddGroupHom.incl (NormedAddGroupHom.ker (f - g)) \u226b f) v =\n \u2191(NormedAddGroupHom.incl (NormedAddGroupHom.ker (f - g)) \u226b g) v\n[PROOFSTEP]\nsimpa only [NormedAddGroupHom.incl_apply, Pi.zero_apply, coe_comp, NormedAddGroupHom.coe_zero,\n NormedAddGroupHom.mem_ker, NormedAddGroupHom.coe_sub, Pi.sub_apply, sub_eq_zero] using this\n[GOAL]\nV W : SemiNormedGroupCat\nf g : V \u27f6 W\nc : Fork f g\n\u22a2 \u2191(\u2191NormedAddGroupHom.compHom (f - g)) (Fork.\u03b9 c) = 0\n[PROOFSTEP]\nrw [AddMonoidHom.map_sub, AddMonoidHom.sub_apply, sub_eq_zero]\n[GOAL]\nV W : SemiNormedGroupCat\nf g : V \u27f6 W\nc : Fork f g\n\u22a2 \u2191(\u2191NormedAddGroupHom.compHom f) (Fork.\u03b9 c) = \u2191(\u2191NormedAddGroupHom.compHom g) (Fork.\u03b9 c)\n[PROOFSTEP]\nexact c.condition\n[GOAL]\nV W : SemiNormedGroupCat\nf g\u271d : V \u27f6 W\nc : Fork f g\u271d\ng : c.pt \u27f6 (fork f g\u271d).pt\nh : g \u226b Fork.\u03b9 (fork f g\u271d) = Fork.\u03b9 c\n\u22a2 g =\n (fun c =>\n NormedAddGroupHom.ker.lift (Fork.\u03b9 c) (f - g\u271d) (_ : \u2191(\u2191NormedAddGroupHom.compHom (f - g\u271d)) (Fork.\u03b9 c) = 0))\n c\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nV W : SemiNormedGroupCat\nf g\u271d : V \u27f6 W\nc : Fork f g\u271d\ng : c.pt \u27f6 (fork f g\u271d).pt\nh : g \u226b Fork.\u03b9 (fork f g\u271d) = Fork.\u03b9 c\nx : \u2191c.pt\n\u22a2 \u2191g x =\n \u2191((fun c =>\n NormedAddGroupHom.ker.lift (Fork.\u03b9 c) (f - g\u271d) (_ : \u2191(\u2191NormedAddGroupHom.compHom (f - g\u271d)) (Fork.\u03b9 c) = 0))\n c)\n x\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nV W : SemiNormedGroupCat\nf g\u271d : V \u27f6 W\nc : Fork f g\u271d\ng : c.pt \u27f6 (fork f g\u271d).pt\nh : g \u226b Fork.\u03b9 (fork f g\u271d) = Fork.\u03b9 c\nx : \u2191c.pt\n\u22a2 \u2191g x =\n \u2191(NormedAddGroupHom.ker.lift (Fork.\u03b9 c) (f - g\u271d)\n (_ : \u2191(\u2191NormedAddGroupHom.compHom f) (Fork.\u03b9 c) - \u2191(\u2191NormedAddGroupHom.compHom g\u271d) (Fork.\u03b9 c) = 0))\n x\n[PROOFSTEP]\nsimp_rw [\u2190 h]\n[GOAL]\ncase h\nV W : SemiNormedGroupCat\nf g\u271d : V \u27f6 W\nc : Fork f g\u271d\ng : c.pt \u27f6 (fork f g\u271d).pt\nh : g \u226b Fork.\u03b9 (fork f g\u271d) = Fork.\u03b9 c\nx : \u2191c.pt\n\u22a2 \u2191g x =\n \u2191(NormedAddGroupHom.ker.lift (g \u226b Fork.\u03b9 (fork f g\u271d)) (f - g\u271d)\n (_ : NormedAddGroupHom.comp (f - g\u271d) (g \u226b Fork.\u03b9 (fork f g\u271d)) = 0))\n x\n[PROOFSTEP]\nrfl\n[GOAL]\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\n\u22a2 f \u226b AddSubgroup.normedMk (NormedAddGroupHom.range f) = 0 \u226b AddSubgroup.normedMk (NormedAddGroupHom.range f)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\nx\u271d : \u2191X\n\u22a2 \u2191(f \u226b AddSubgroup.normedMk (NormedAddGroupHom.range f)) x\u271d =\n \u2191(0 \u226b AddSubgroup.normedMk (NormedAddGroupHom.range f)) x\u271d\n[PROOFSTEP]\nsimp only [comp_apply, Limits.zero_comp]\n -- porting note: `simp` not firing on the below\n[GOAL]\ncase h\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\nx\u271d : \u2191X\n\u22a2 \u2191(f \u226b AddSubgroup.normedMk (NormedAddGroupHom.range f)) x\u271d = \u21910 x\u271d\n[PROOFSTEP]\nrw [comp_apply, NormedAddGroupHom.zero_apply]\n -- porting note: Lean 3 didn't need this instance\n[GOAL]\ncase h\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\nx\u271d : \u2191X\n\u22a2 \u2191(AddSubgroup.normedMk (NormedAddGroupHom.range f)) (\u2191f x\u271d) = 0\n[PROOFSTEP]\nletI : SeminormedAddCommGroup ((forget SemiNormedGroupCat).obj Y) :=\n (inferInstance : SeminormedAddCommGroup Y)\n -- porting note: again simp doesn't seem to be firing in the below line\n[GOAL]\ncase h\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\nx\u271d : \u2191X\nthis : SeminormedAddCommGroup ((forget SemiNormedGroupCat).obj Y) := inferInstance\n\u22a2 \u2191(AddSubgroup.normedMk (NormedAddGroupHom.range f)) (\u2191f x\u271d) = 0\n[PROOFSTEP]\nrw [\u2190 NormedAddGroupHom.mem_ker, f.range.ker_normedMk, f.mem_range]\n[GOAL]\ncase h\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\nx\u271d : \u2191X\nthis : SeminormedAddCommGroup ((forget SemiNormedGroupCat).obj Y) := inferInstance\n\u22a2 \u2203 w, \u2191f w = \u2191f x\u271d\n[PROOFSTEP]\nsimp only [exists_apply_eq_apply]\n[GOAL]\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\ns : CokernelCofork f\n\u22a2 \u2200 (s_1 : \u2191Y), s_1 \u2208 NormedAddGroupHom.range f \u2192 \u2191(Cofork.\u03c0 s) s_1 = 0\n[PROOFSTEP]\nrintro _ \u27e8b, rfl\u27e9\n[GOAL]\ncase intro\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\ns : CokernelCofork f\nb : \u2191X\n\u22a2 \u2191(Cofork.\u03c0 s) (\u2191(NormedAddGroupHom.toAddMonoidHom f) b) = 0\n[PROOFSTEP]\nchange (f \u226b s.\u03c0) b = 0\n[GOAL]\ncase intro\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\ns : CokernelCofork f\nb : \u2191X\n\u22a2 \u2191(f \u226b Cofork.\u03c0 s) b = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\ns : CokernelCofork f\n\u22a2 Cofork.\u03c0 (cokernelCocone f) \u226b cokernelLift f s = Cofork.\u03c0 s\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\ns : CokernelCofork f\nx\u271d : \u2191((parallelPair f 0).obj WalkingParallelPair.one)\n\u22a2 \u2191(Cofork.\u03c0 (cokernelCocone f) \u226b cokernelLift f s) x\u271d = \u2191(Cofork.\u03c0 s) x\u271d\n[PROOFSTEP]\napply NormedAddGroupHom.lift_mk f.range\n[GOAL]\ncase h.hf\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\ns : CokernelCofork f\nx\u271d : \u2191((parallelPair f 0).obj WalkingParallelPair.one)\n\u22a2 \u2200 (s_1 : \u2191Y), s_1 \u2208 NormedAddGroupHom.range f \u2192 \u2191(Cofork.\u03c0 s) s_1 = 0\n[PROOFSTEP]\nrintro _ \u27e8b, rfl\u27e9\n[GOAL]\ncase h.hf.intro\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\ns : CokernelCofork f\nx\u271d : \u2191((parallelPair f 0).obj WalkingParallelPair.one)\nb : \u2191X\n\u22a2 \u2191(Cofork.\u03c0 s) (\u2191(NormedAddGroupHom.toAddMonoidHom f) b) = 0\n[PROOFSTEP]\nchange (f \u226b s.\u03c0) b = 0\n[GOAL]\ncase h.hf.intro\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\ns : CokernelCofork f\nx\u271d : \u2191((parallelPair f 0).obj WalkingParallelPair.one)\nb : \u2191X\n\u22a2 \u2191(f \u226b Cofork.\u03c0 s) b = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 HasCokernels SemiNormedGroupCat\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 f \u226b g = 0 \u226b g\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\n\u22a2 f \u226b explicitCokernel\u03c0 f = 0\n[PROOFSTEP]\nconvert (cokernelCocone f).w WalkingParallelPairHom.left\n[GOAL]\ncase h.e'_3.h\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\ne_1\u271d :\n (X \u27f6 explicitCokernel f) =\n ((parallelPair f 0).obj WalkingParallelPair.zero \u27f6\n ((Functor.const WalkingParallelPair).obj (cokernelCocone f).pt).obj WalkingParallelPair.one)\n\u22a2 0 = NatTrans.app (cokernelCocone f).\u03b9 WalkingParallelPair.zero\n[PROOFSTEP]\nsimp\n[GOAL]\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\nx : \u2191X\n\u22a2 \u2191(f \u226b explicitCokernel\u03c0 f) x = 0\n[PROOFSTEP]\nrw [comp_explicitCokernel\u03c0]\n[GOAL]\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\nx : \u2191X\n\u22a2 \u21910 x = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\ncond : f \u226b g = 0\nx : \u2191Y\n\u22a2 \u2191(explicitCokernel\u03c0 f \u226b explicitCokernelDesc cond) x = \u2191g x\n[PROOFSTEP]\nrw [explicitCokernel\u03c0_desc]\n[GOAL]\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\ne : explicitCokernel f \u27f6 Z\nhe : explicitCokernel\u03c0 f \u226b e = g\n\u22a2 e = explicitCokernelDesc w\n[PROOFSTEP]\napply (isColimitCokernelCocone f).uniq (Cofork.of\u03c0 g (by simp [w]))\n[GOAL]\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\ne : explicitCokernel f \u27f6 Z\nhe : explicitCokernel\u03c0 f \u226b e = g\n\u22a2 f \u226b g = 0 \u226b g\n[PROOFSTEP]\nsimp [w]\n[GOAL]\ncase x\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\ne : explicitCokernel f \u27f6 Z\nhe : explicitCokernel\u03c0 f \u226b e = g\n\u22a2 \u2200 (j : WalkingParallelPair),\n NatTrans.app (cokernelCocone f).\u03b9 j \u226b e = NatTrans.app (Cofork.of\u03c0 g (_ : f \u226b g = 0 \u226b g)).\u03b9 j\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase x.zero\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\ne : explicitCokernel f \u27f6 Z\nhe : explicitCokernel\u03c0 f \u226b e = g\n\u22a2 NatTrans.app (cokernelCocone f).\u03b9 WalkingParallelPair.zero \u226b e =\n NatTrans.app (Cofork.of\u03c0 g (_ : f \u226b g = 0 \u226b g)).\u03b9 WalkingParallelPair.zero\n[PROOFSTEP]\nconvert w.symm\n[GOAL]\ncase h.e'_2.h\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\ne : explicitCokernel f \u27f6 Z\nhe : explicitCokernel\u03c0 f \u226b e = g\ne_1\u271d : ((parallelPair f 0).obj WalkingParallelPair.zero \u27f6 (Cofork.of\u03c0 g (_ : f \u226b g = 0 \u226b g)).pt) = (X \u27f6 Z)\n\u22a2 NatTrans.app (cokernelCocone f).\u03b9 WalkingParallelPair.zero \u226b e = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase x.one\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\ne : explicitCokernel f \u27f6 Z\nhe : explicitCokernel\u03c0 f \u226b e = g\n\u22a2 NatTrans.app (cokernelCocone f).\u03b9 WalkingParallelPair.one \u226b e =\n NatTrans.app (Cofork.of\u03c0 g (_ : f \u226b g = 0 \u226b g)).\u03b9 WalkingParallelPair.one\n[PROOFSTEP]\nexact he\n[GOAL]\nX Y Z W : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nh : Z \u27f6 W\ncond' : f \u226b g = 0\n\u22a2 f \u226b g \u226b h = 0\n[PROOFSTEP]\nrw [\u2190 CategoryTheory.Category.assoc, cond', Limits.zero_comp]\n[GOAL]\nX Y Z W : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nh : Z \u27f6 W\ncond' : f \u226b g = 0\n\u22a2 explicitCokernelDesc cond' \u226b h = explicitCokernelDesc (_ : f \u226b g \u226b h = 0)\n[PROOFSTEP]\nrefine' explicitCokernelDesc_unique _ _ _\n[GOAL]\nX Y Z W : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nh : Z \u27f6 W\ncond' : f \u226b g = 0\n\u22a2 explicitCokernel\u03c0 f \u226b explicitCokernelDesc cond' \u226b h = g \u226b h\n[PROOFSTEP]\nrw [\u2190 CategoryTheory.Category.assoc, explicitCokernel\u03c0_desc]\n[GOAL]\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ne\u2081 e\u2082 : explicitCokernel f \u27f6 Z\nh : explicitCokernel\u03c0 f \u226b e\u2081 = explicitCokernel\u03c0 f \u226b e\u2082\n\u22a2 e\u2081 = e\u2082\n[PROOFSTEP]\nlet g : Y \u27f6 Z := explicitCokernel\u03c0 f \u226b e\u2082\n[GOAL]\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ne\u2081 e\u2082 : explicitCokernel f \u27f6 Z\nh : explicitCokernel\u03c0 f \u226b e\u2081 = explicitCokernel\u03c0 f \u226b e\u2082\ng : Y \u27f6 Z := explicitCokernel\u03c0 f \u226b e\u2082\n\u22a2 e\u2081 = e\u2082\n[PROOFSTEP]\nhave w : f \u226b g = 0 := by simp\n[GOAL]\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ne\u2081 e\u2082 : explicitCokernel f \u27f6 Z\nh : explicitCokernel\u03c0 f \u226b e\u2081 = explicitCokernel\u03c0 f \u226b e\u2082\ng : Y \u27f6 Z := explicitCokernel\u03c0 f \u226b e\u2082\n\u22a2 f \u226b g = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ne\u2081 e\u2082 : explicitCokernel f \u27f6 Z\nh : explicitCokernel\u03c0 f \u226b e\u2081 = explicitCokernel\u03c0 f \u226b e\u2082\ng : Y \u27f6 Z := explicitCokernel\u03c0 f \u226b e\u2082\nw : f \u226b g = 0\n\u22a2 e\u2081 = e\u2082\n[PROOFSTEP]\nhave : e\u2082 = explicitCokernelDesc w := by apply explicitCokernelDesc_unique; rfl\n[GOAL]\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ne\u2081 e\u2082 : explicitCokernel f \u27f6 Z\nh : explicitCokernel\u03c0 f \u226b e\u2081 = explicitCokernel\u03c0 f \u226b e\u2082\ng : Y \u27f6 Z := explicitCokernel\u03c0 f \u226b e\u2082\nw : f \u226b g = 0\n\u22a2 e\u2082 = explicitCokernelDesc w\n[PROOFSTEP]\napply explicitCokernelDesc_unique\n[GOAL]\ncase he\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ne\u2081 e\u2082 : explicitCokernel f \u27f6 Z\nh : explicitCokernel\u03c0 f \u226b e\u2081 = explicitCokernel\u03c0 f \u226b e\u2082\ng : Y \u27f6 Z := explicitCokernel\u03c0 f \u226b e\u2082\nw : f \u226b g = 0\n\u22a2 explicitCokernel\u03c0 f \u226b e\u2082 = g\n[PROOFSTEP]\nrfl\n[GOAL]\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ne\u2081 e\u2082 : explicitCokernel f \u27f6 Z\nh : explicitCokernel\u03c0 f \u226b e\u2081 = explicitCokernel\u03c0 f \u226b e\u2082\ng : Y \u27f6 Z := explicitCokernel\u03c0 f \u226b e\u2082\nw : f \u226b g = 0\nthis : e\u2082 = explicitCokernelDesc w\n\u22a2 e\u2081 = e\u2082\n[PROOFSTEP]\nrw [this]\n[GOAL]\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ne\u2081 e\u2082 : explicitCokernel f \u27f6 Z\nh : explicitCokernel\u03c0 f \u226b e\u2081 = explicitCokernel\u03c0 f \u226b e\u2082\ng : Y \u27f6 Z := explicitCokernel\u03c0 f \u226b e\u2082\nw : f \u226b g = 0\nthis : e\u2082 = explicitCokernelDesc w\n\u22a2 e\u2081 = explicitCokernelDesc w\n[PROOFSTEP]\napply explicitCokernelDesc_unique\n[GOAL]\ncase he\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ne\u2081 e\u2082 : explicitCokernel f \u27f6 Z\nh : explicitCokernel\u03c0 f \u226b e\u2081 = explicitCokernel\u03c0 f \u226b e\u2082\ng : Y \u27f6 Z := explicitCokernel\u03c0 f \u226b e\u2082\nw : f \u226b g = 0\nthis : e\u2082 = explicitCokernelDesc w\n\u22a2 explicitCokernel\u03c0 f \u226b e\u2081 = g\n[PROOFSTEP]\nexact h\n[GOAL]\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\n\u22a2 Epi (explicitCokernel\u03c0 f)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left_cancellation\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\n\u22a2 \u2200 {Z : SemiNormedGroupCat} (g h : explicitCokernel f \u27f6 Z), explicitCokernel\u03c0 f \u226b g = explicitCokernel\u03c0 f \u226b h \u2192 g = h\n[PROOFSTEP]\nintro Z g h H\n[GOAL]\ncase left_cancellation\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\nZ : SemiNormedGroupCat\ng h : explicitCokernel f \u27f6 Z\nH : explicitCokernel\u03c0 f \u226b g = explicitCokernel\u03c0 f \u226b h\n\u22a2 g = h\n[PROOFSTEP]\next x\n[GOAL]\ncase left_cancellation.h.h\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\nZ : SemiNormedGroupCat\ng h : explicitCokernel f \u27f6 Z\nH : explicitCokernel\u03c0 f \u226b g = explicitCokernel\u03c0 f \u226b h\nx : \u2191Y\n\u22a2 \u2191(explicitCokernel\u03c0 f \u226b g) x = \u2191(explicitCokernel\u03c0 f \u226b h) x\n[PROOFSTEP]\nrw [H]\n[GOAL]\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\ncond : f \u226b g = 0\nhg : NormedAddGroupHom.NormNoninc g\n\u22a2 NormedAddGroupHom.NormNoninc (explicitCokernelDesc cond)\n[PROOFSTEP]\nrefine' NormedAddGroupHom.NormNoninc.normNoninc_iff_norm_le_one.2 _\n[GOAL]\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\ncond : f \u226b g = 0\nhg : NormedAddGroupHom.NormNoninc g\n\u22a2 \u2016explicitCokernelDesc cond\u2016 \u2264 1\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_one]\n[GOAL]\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\ncond : f \u226b g = 0\nhg : NormedAddGroupHom.NormNoninc g\n\u22a2 \u2016explicitCokernelDesc cond\u2016 \u2264 \u21911\n[PROOFSTEP]\nexact explicitCokernelDesc_norm_le_of_norm_le cond 1 (NormedAddGroupHom.NormNoninc.normNoninc_iff_norm_le_one.1 hg)\n[GOAL]\nX Y Z W : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nh : Z \u27f6 W\ncond : f \u226b g = 0\ncond2 : g \u226b h = 0\n\u22a2 explicitCokernelDesc cond \u226b h = 0\n[PROOFSTEP]\nrw [\u2190 cancel_epi (explicitCokernel\u03c0 f), \u2190 Category.assoc, explicitCokernel\u03c0_desc]\n[GOAL]\nX Y Z W : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nh : Z \u27f6 W\ncond : f \u226b g = 0\ncond2 : g \u226b h = 0\n\u22a2 g \u226b h = explicitCokernel\u03c0 f \u226b 0\n[PROOFSTEP]\nsimp [cond2]\n[GOAL]\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\n\u22a2 explicitCokernel\u03c0 f \u226b (explicitCokernelIso f).hom = cokernel.\u03c0 f\n[PROOFSTEP]\nsimp [explicitCokernel\u03c0, explicitCokernelIso, IsColimit.coconePointUniqueUpToIso]\n[GOAL]\nX Y : SemiNormedGroupCat\nf : X \u27f6 Y\n\u22a2 cokernel.\u03c0 f \u226b (explicitCokernelIso f).inv = explicitCokernel\u03c0 f\n[PROOFSTEP]\nsimp [explicitCokernel\u03c0, explicitCokernelIso]\n[GOAL]\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 (explicitCokernelIso f).hom \u226b cokernel.desc f g w = explicitCokernelDesc w\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nX Y Z : SemiNormedGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 explicitCokernel\u03c0 f \u226b (explicitCokernelIso f).hom \u226b cokernel.desc f g w = explicitCokernel\u03c0 f \u226b explicitCokernelDesc w\n[PROOFSTEP]\nsimp [explicitCokernelDesc, explicitCokernel\u03c0, explicitCokernelIso, IsColimit.coconePointUniqueUpToIso]\n[GOAL]\nA B C D : SemiNormedGroupCat\nfab : A \u27f6 B\nfbd : B \u27f6 D\nfac : A \u27f6 C\nfcd : C \u27f6 D\nh : fab \u226b fbd = fac \u226b fcd\n\u22a2 fab \u226b fbd \u226b explicitCokernel\u03c0 fcd = 0\n[PROOFSTEP]\nsimp [reassoc_of% h]\n[GOAL]\nA B C D B' D' : SemiNormedGroupCat\nfab : A \u27f6 B\nfbd : B \u27f6 D\nfac : A \u27f6 C\nfcd : C \u27f6 D\nh : fab \u226b fbd = fac \u226b fcd\nfbb' : B \u27f6 B'\nfdd' : D \u27f6 D'\ncondb : fab \u226b fbb' = 0\ncondd : fcd \u226b fdd' = 0\ng : B' \u27f6 D'\nh' : fbb' \u226b g = fbd \u226b fdd'\n\u22a2 explicitCokernelDesc condb \u226b g = explicitCokernel.map h \u226b explicitCokernelDesc condd\n[PROOFSTEP]\ndelta explicitCokernel.map\n[GOAL]\nA B C D B' D' : SemiNormedGroupCat\nfab : A \u27f6 B\nfbd : B \u27f6 D\nfac : A \u27f6 C\nfcd : C \u27f6 D\nh : fab \u226b fbd = fac \u226b fcd\nfbb' : B \u27f6 B'\nfdd' : D \u27f6 D'\ncondb : fab \u226b fbb' = 0\ncondd : fcd \u226b fdd' = 0\ng : B' \u27f6 D'\nh' : fbb' \u226b g = fbd \u226b fdd'\n\u22a2 explicitCokernelDesc condb \u226b g =\n explicitCokernelDesc (_ : fab \u226b fbd \u226b explicitCokernel\u03c0 fcd = 0) \u226b explicitCokernelDesc condd\n[PROOFSTEP]\nsimp [\u2190 cancel_epi (explicitCokernel\u03c0 fab), \u2190 Category.assoc, explicitCokernel\u03c0_desc, h']\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Normed.Group.SemiNormedGroupCat.Kernels", "llama_tokens": 9360, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.43307687354066876}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG\u271d : SimpleGraph \u03b1\nH\u271d : SimpleGraph \u03b2\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 (fun x y => Adj G x.fst y.fst \u2227 x.snd = y.snd \u2228 Adj H x.snd y.snd \u2227 x.fst = y.fst) x y \u2192\n (fun x y => Adj G x.fst y.fst \u2227 x.snd = y.snd \u2228 Adj H x.snd y.snd \u2227 x.fst = y.fst) y x\n[PROOFSTEP]\nsimp [and_comm, or_comm, eq_comm, adj_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG\u271d : SimpleGraph \u03b1\nH\u271d : SimpleGraph \u03b2\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 \u00ac(fun x y => Adj G x.fst y.fst \u2227 x.snd = y.snd \u2228 Adj H x.snd y.snd \u2227 x.fst = y.fst) x x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\na\u2081 : \u03b1\nb : \u03b2\na\u2082 : \u03b1\n\u22a2 Adj (G \u25a1 H) (a\u2081, b) (a\u2082, b) \u2194 Adj G a\u2081 a\u2082\n[PROOFSTEP]\nsimp only [boxProd_adj, and_true, SimpleGraph.irrefl, false_and, or_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\na : \u03b1\nb\u2081 b\u2082 : \u03b2\n\u22a2 Adj (G \u25a1 H) (a, b\u2081) (a, b\u2082) \u2194 Adj H b\u2081 b\u2082\n[PROOFSTEP]\nsimp only [boxProd_adj, SimpleGraph.irrefl, false_and, and_true, false_or]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 neighborSet (G \u25a1 H) x = neighborSet G x.fst \u00d7\u02e2 {x.snd} \u222a {x.fst} \u00d7\u02e2 neighborSet H x.snd\n[PROOFSTEP]\next \u27e8a', b'\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nx : \u03b1 \u00d7 \u03b2\na' : \u03b1\nb' : \u03b2\n\u22a2 (a', b') \u2208 neighborSet (G \u25a1 H) x \u2194 (a', b') \u2208 neighborSet G x.fst \u00d7\u02e2 {x.snd} \u222a {x.fst} \u00d7\u02e2 neighborSet H x.snd\n[PROOFSTEP]\nsimp only [mem_neighborSet, Set.mem_union, boxProd_adj, Set.mem_prod, Set.mem_singleton_iff]\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nx : \u03b1 \u00d7 \u03b2\na' : \u03b1\nb' : \u03b2\n\u22a2 Adj G x.fst a' \u2227 x.snd = b' \u2228 Adj H x.snd b' \u2227 x.fst = a' \u2194 Adj G x.fst a' \u2227 b' = x.snd \u2228 a' = x.fst \u2227 Adj H x.snd b'\n[PROOFSTEP]\nsimp only [eq_comm, and_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nI : SimpleGraph \u03b3\nx y : (\u03b1 \u00d7 \u03b2) \u00d7 \u03b3\n\u22a2 Adj (G \u25a1 (H \u25a1 I)) (\u2191(Equiv.prodAssoc \u03b1 \u03b2 \u03b3) x) (\u2191(Equiv.prodAssoc \u03b1 \u03b2 \u03b3) y) \u2194 Adj (G \u25a1 H \u25a1 I) x y\n[PROOFSTEP]\nsimp only [boxProd_adj, Equiv.prodAssoc_apply, or_and_right, or_assoc, Prod.ext_iff, and_assoc,\n @and_comm (x.fst.fst = _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nb : \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableRel G.Adj\na\u2081 a\u2082 x z y : \u03b1\nh : Adj G x y\nw : Walk G y z\n\u22a2 ofBoxProdLeft (Walk.boxProdLeft H b (cons' x y z h w)) = cons' x y z h w\n[PROOFSTEP]\nrw [Walk.boxProdLeft, map_cons, ofBoxProdLeft, Or.by_cases, dif_pos, \u2190 Walk.boxProdLeft]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nb : \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableRel G.Adj\na\u2081 a\u2082 x z y : \u03b1\nh : Adj G x y\nw : Walk G y z\n\u22a2 cons (_ : Adj G (x, b).fst (\u2191(Embedding.toHom (boxProdLeft G H b)) y).fst) (ofBoxProdLeft (Walk.boxProdLeft H b w)) =\n cons' x y z h w\ncase hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nb : \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableRel G.Adj\na\u2081 a\u2082 x z y : \u03b1\nh : Adj G x y\nw : Walk G y z\n\u22a2 Adj G (x, b).fst (\u2191(Embedding.toHom (boxProdLeft G H b)) y).fst \u2227\n (x, b).snd = (\u2191(Embedding.toHom (boxProdLeft G H b)) y).snd\ncase hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nb : \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableRel G.Adj\na\u2081 a\u2082 x z y : \u03b1\nh : Adj G x y\nw : Walk G y z\n\u22a2 Adj G (x, b).fst (\u2191(Embedding.toHom (boxProdLeft G H b)) y).fst \u2227\n (x, b).snd = (\u2191(Embedding.toHom (boxProdLeft G H b)) y).snd\n[PROOFSTEP]\nsimp [ofBoxProdLeft_boxProdLeft]\n[GOAL]\ncase hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nb : \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableRel G.Adj\na\u2081 a\u2082 x z y : \u03b1\nh : Adj G x y\nw : Walk G y z\n\u22a2 Adj G (x, b).fst (\u2191(Embedding.toHom (boxProdLeft G H b)) y).fst \u2227\n (x, b).snd = (\u2191(Embedding.toHom (boxProdLeft G H b)) y).snd\ncase hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nb : \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableRel G.Adj\na\u2081 a\u2082 x z y : \u03b1\nh : Adj G x y\nw : Walk G y z\n\u22a2 Adj G (x, b).fst (\u2191(Embedding.toHom (boxProdLeft G H b)) y).fst \u2227\n (x, b).snd = (\u2191(Embedding.toHom (boxProdLeft G H b)) y).snd\n[PROOFSTEP]\nexact \u27e8h, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\na : \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\nb\u2081 b\u2082 x z y : \u03b1\nh : Adj G x y\nw : Walk G y z\n\u22a2 ofBoxProdRight (Walk.boxProdRight G a (cons' x y z h w)) = cons' x y z h w\n[PROOFSTEP]\nrw [Walk.boxProdRight, map_cons, ofBoxProdRight, Or.by_cases, dif_pos, \u2190 Walk.boxProdRight]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\na : \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\nb\u2081 b\u2082 x z y : \u03b1\nh : Adj G x y\nw : Walk G y z\n\u22a2 cons (_ : Adj G (a, x).snd (\u2191(Embedding.toHom (boxProdRight G G a)) y).snd)\n (ofBoxProdRight (Walk.boxProdRight G a w)) =\n cons' x y z h w\ncase hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\na : \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\nb\u2081 b\u2082 x z y : \u03b1\nh : Adj G x y\nw : Walk G y z\n\u22a2 Adj G (a, x).snd (\u2191(Embedding.toHom (boxProdRight G G a)) y).snd \u2227\n (a, x).fst = (\u2191(Embedding.toHom (boxProdRight G G a)) y).fst\ncase hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\na : \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\nb\u2081 b\u2082 x z y : \u03b1\nh : Adj G x y\nw : Walk G y z\n\u22a2 Adj G (a, x).snd (\u2191(Embedding.toHom (boxProdRight G G a)) y).snd \u2227\n (a, x).fst = (\u2191(Embedding.toHom (boxProdRight G G a)) y).fst\n[PROOFSTEP]\nsimp [ofBoxProdLeft_boxProdRight]\n[GOAL]\ncase hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\na : \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\nb\u2081 b\u2082 x z y : \u03b1\nh : Adj G x y\nw : Walk G y z\n\u22a2 Adj G (a, x).snd (\u2191(Embedding.toHom (boxProdRight G G a)) y).snd \u2227\n (a, x).fst = (\u2191(Embedding.toHom (boxProdRight G G a)) y).fst\ncase hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\na : \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\nb\u2081 b\u2082 x z y : \u03b1\nh : Adj G x y\nw : Walk G y z\n\u22a2 Adj G (a, x).snd (\u2191(Embedding.toHom (boxProdRight G G a)) y).snd \u2227\n (a, x).fst = (\u2191(Embedding.toHom (boxProdRight G G a)) y).fst\n[PROOFSTEP]\nexact \u27e8h, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Preconnected G\nhH : Preconnected H\n\u22a2 Preconnected (G \u25a1 H)\n[PROOFSTEP]\nrintro x y\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Preconnected G\nhH : Preconnected H\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 Reachable (G \u25a1 H) x y\n[PROOFSTEP]\nobtain \u27e8w\u2081\u27e9 := hG x.1 y.1\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Preconnected G\nhH : Preconnected H\nx y : \u03b1 \u00d7 \u03b2\nw\u2081 : Walk G x.fst y.fst\n\u22a2 Reachable (G \u25a1 H) x y\n[PROOFSTEP]\nobtain \u27e8w\u2082\u27e9 := hH x.2 y.2\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Preconnected G\nhH : Preconnected H\nx y : \u03b1 \u00d7 \u03b2\nw\u2081 : Walk G x.fst y.fst\nw\u2082 : Walk H x.snd y.snd\n\u22a2 Reachable (G \u25a1 H) x y\n[PROOFSTEP]\nrw [\u2190 @Prod.mk.eta _ _ x, \u2190 @Prod.mk.eta _ _ y]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Preconnected G\nhH : Preconnected H\nx y : \u03b1 \u00d7 \u03b2\nw\u2081 : Walk G x.fst y.fst\nw\u2082 : Walk H x.snd y.snd\n\u22a2 Reachable (G \u25a1 H) (x.fst, x.snd) (y.fst, y.snd)\n[PROOFSTEP]\nexact \u27e8(w\u2081.boxProdLeft _ _).append (w\u2082.boxProdRight _ _)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\ninst\u271d : Nonempty \u03b2\nh : Preconnected (G \u25a1 H)\n\u22a2 Preconnected G\n[PROOFSTEP]\nclassical\nrintro a\u2081 a\u2082\nobtain \u27e8w\u27e9 := h (a\u2081, Classical.arbitrary _) (a\u2082, Classical.arbitrary _)\nexact \u27e8w.ofBoxProdLeft\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\ninst\u271d : Nonempty \u03b2\nh : Preconnected (G \u25a1 H)\n\u22a2 Preconnected G\n[PROOFSTEP]\nrintro a\u2081 a\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\ninst\u271d : Nonempty \u03b2\nh : Preconnected (G \u25a1 H)\na\u2081 a\u2082 : \u03b1\n\u22a2 Reachable G a\u2081 a\u2082\n[PROOFSTEP]\nobtain \u27e8w\u27e9 := h (a\u2081, Classical.arbitrary _) (a\u2082, Classical.arbitrary _)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\ninst\u271d : Nonempty \u03b2\nh : Preconnected (G \u25a1 H)\na\u2081 a\u2082 : \u03b1\nw : Walk (G \u25a1 H) (a\u2081, Classical.arbitrary \u03b2) (a\u2082, Classical.arbitrary \u03b2)\n\u22a2 Reachable G a\u2081 a\u2082\n[PROOFSTEP]\nexact \u27e8w.ofBoxProdLeft\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\ninst\u271d : Nonempty \u03b1\nh : Preconnected (G \u25a1 H)\n\u22a2 Preconnected H\n[PROOFSTEP]\nclassical\nrintro b\u2081 b\u2082\nobtain \u27e8w\u27e9 := h (Classical.arbitrary _, b\u2081) (Classical.arbitrary _, b\u2082)\nexact \u27e8w.ofBoxProdRight\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\ninst\u271d : Nonempty \u03b1\nh : Preconnected (G \u25a1 H)\n\u22a2 Preconnected H\n[PROOFSTEP]\nrintro b\u2081 b\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\ninst\u271d : Nonempty \u03b1\nh : Preconnected (G \u25a1 H)\nb\u2081 b\u2082 : \u03b2\n\u22a2 Reachable H b\u2081 b\u2082\n[PROOFSTEP]\nobtain \u27e8w\u27e9 := h (Classical.arbitrary _, b\u2081) (Classical.arbitrary _, b\u2082)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\ninst\u271d : Nonempty \u03b1\nh : Preconnected (G \u25a1 H)\nb\u2081 b\u2082 : \u03b2\nw : Walk (G \u25a1 H) (Classical.arbitrary \u03b1, b\u2081) (Classical.arbitrary \u03b1, b\u2082)\n\u22a2 Reachable H b\u2081 b\u2082\n[PROOFSTEP]\nexact \u27e8w.ofBoxProdRight\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Connected G\nhH : Connected H\n\u22a2 Connected (G \u25a1 H)\n[PROOFSTEP]\nhaveI := hG.nonempty\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Connected G\nhH : Connected H\nthis : Nonempty \u03b1\n\u22a2 Connected (G \u25a1 H)\n[PROOFSTEP]\nhaveI := hH.nonempty\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Connected G\nhH : Connected H\nthis\u271d : Nonempty \u03b1\nthis : Nonempty \u03b2\n\u22a2 Connected (G \u25a1 H)\n[PROOFSTEP]\nexact \u27e8hG.preconnected.boxProd hH.preconnected\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nh : Connected (G \u25a1 H)\n\u22a2 Connected G\n[PROOFSTEP]\nhaveI := (nonempty_prod.1 h.nonempty).1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nh : Connected (G \u25a1 H)\nthis : Nonempty \u03b1\n\u22a2 Connected G\n[PROOFSTEP]\nhaveI := (nonempty_prod.1 h.nonempty).2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nh : Connected (G \u25a1 H)\nthis\u271d : Nonempty \u03b1\nthis : Nonempty \u03b2\n\u22a2 Connected G\n[PROOFSTEP]\nexact \u27e8h.preconnected.ofBoxProdLeft\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nh : Connected (G \u25a1 H)\n\u22a2 Connected H\n[PROOFSTEP]\nhaveI := (nonempty_prod.1 h.nonempty).1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nh : Connected (G \u25a1 H)\nthis : Nonempty \u03b1\n\u22a2 Connected H\n[PROOFSTEP]\nhaveI := (nonempty_prod.1 h.nonempty).2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nh : Connected (G \u25a1 H)\nthis\u271d : Nonempty \u03b1\nthis : Nonempty \u03b2\n\u22a2 Connected H\n[PROOFSTEP]\nexact \u27e8h.preconnected.ofBoxProdRight\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nx : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b9 : Fintype \u2191(neighborSet G x.fst)\ninst\u271d : Fintype \u2191(neighborSet H x.snd)\ny : \u03b1 \u00d7 \u03b2\n\u22a2 y \u2208\n Finset.disjUnion (neighborFinset G x.fst \u00d7\u02e2 {x.snd}) ({x.fst} \u00d7\u02e2 neighborFinset H x.snd)\n (_ : Disjoint (neighborFinset G x.fst \u00d7\u02e2 {x.snd}) ({x.fst} \u00d7\u02e2 neighborFinset H x.snd)) \u2194\n \u2191(Equiv.refl (\u03b1 \u00d7 \u03b2)) y \u2208 neighborSet (G \u25a1 H) x\n[PROOFSTEP]\nsimp_rw [Finset.mem_disjUnion, Finset.mem_product, Finset.mem_singleton, mem_neighborFinset, mem_neighborSet,\n Equiv.refl_apply, boxProd_adj]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nx : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b9 : Fintype \u2191(neighborSet G x.fst)\ninst\u271d : Fintype \u2191(neighborSet H x.snd)\ny : \u03b1 \u00d7 \u03b2\n\u22a2 Adj G x.fst y.fst \u2227 y.snd = x.snd \u2228 y.fst = x.fst \u2227 Adj H x.snd y.snd \u2194\n Adj G x.fst y.fst \u2227 x.snd = y.snd \u2228 Adj H x.snd y.snd \u2227 x.fst = y.fst\n[PROOFSTEP]\nsimp only [eq_comm, and_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nx : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b2 : Fintype \u2191(neighborSet G x.fst)\ninst\u271d\u00b9 : Fintype \u2191(neighborSet H x.snd)\ninst\u271d : Fintype \u2191(neighborSet (G \u25a1 H) x)\n\u22a2 neighborFinset (G \u25a1 H) x =\n Finset.disjUnion (neighborFinset G x.fst \u00d7\u02e2 {x.snd}) ({x.fst} \u00d7\u02e2 neighborFinset H x.snd)\n (_ : Disjoint (neighborFinset G x.fst \u00d7\u02e2 {x.snd}) ({x.fst} \u00d7\u02e2 neighborFinset H x.snd))\n[PROOFSTEP]\nletI : Fintype ((G \u25a1 H).neighborSet x) := SimpleGraph.boxProdFintypeNeighborSet _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nx : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b2 : Fintype \u2191(neighborSet G x.fst)\ninst\u271d\u00b9 : Fintype \u2191(neighborSet H x.snd)\ninst\u271d : Fintype \u2191(neighborSet (G \u25a1 H) x)\nthis : Fintype \u2191(neighborSet (G \u25a1 H) x) := boxProdFintypeNeighborSet x\n\u22a2 neighborFinset (G \u25a1 H) x =\n Finset.disjUnion (neighborFinset G x.fst \u00d7\u02e2 {x.snd}) ({x.fst} \u00d7\u02e2 neighborFinset H x.snd)\n (_ : Disjoint (neighborFinset G x.fst \u00d7\u02e2 {x.snd}) ({x.fst} \u00d7\u02e2 neighborFinset H x.snd))\n[PROOFSTEP]\nconvert_to (G \u25a1 H).neighborFinset x = _ using 2\n[GOAL]\ncase convert_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nx : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b2 : Fintype \u2191(neighborSet G x.fst)\ninst\u271d\u00b9 : Fintype \u2191(neighborSet H x.snd)\ninst\u271d : Fintype \u2191(neighborSet (G \u25a1 H) x)\nthis : Fintype \u2191(neighborSet (G \u25a1 H) x) := boxProdFintypeNeighborSet x\n\u22a2 neighborFinset (G \u25a1 H) x =\n Finset.disjUnion (neighborFinset G x.fst \u00d7\u02e2 {x.snd}) ({x.fst} \u00d7\u02e2 neighborFinset H x.snd)\n (_ : Disjoint (neighborFinset G x.fst \u00d7\u02e2 {x.snd}) ({x.fst} \u00d7\u02e2 neighborFinset H x.snd))\n[PROOFSTEP]\nexact Eq.trans (Finset.map_map _ _ _) Finset.attach_map_val\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nx : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b2 : Fintype \u2191(neighborSet G x.fst)\ninst\u271d\u00b9 : Fintype \u2191(neighborSet H x.snd)\ninst\u271d : Fintype \u2191(neighborSet (G \u25a1 H) x)\n\u22a2 degree (G \u25a1 H) x = degree G x.fst + degree H x.snd\n[PROOFSTEP]\nrw [degree, degree, degree, boxProd_neighborFinset, Finset.card_disjUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nx : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b2 : Fintype \u2191(neighborSet G x.fst)\ninst\u271d\u00b9 : Fintype \u2191(neighborSet H x.snd)\ninst\u271d : Fintype \u2191(neighborSet (G \u25a1 H) x)\n\u22a2 Finset.card (neighborFinset G x.fst \u00d7\u02e2 {x.snd}) + Finset.card ({x.fst} \u00d7\u02e2 neighborFinset H x.snd) =\n Finset.card (neighborFinset G x.fst) + Finset.card (neighborFinset H x.snd)\n[PROOFSTEP]\nsimp_rw [Finset.card_product, Finset.card_singleton, mul_one, one_mul]\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SimpleGraph.Prod", "llama_tokens": 7308, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390162, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.43307687354066865}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nn : \u2115\nr : R\n\u22a2 \u2191(opRingEquiv R) (op (\u2191(monomial n) r)) = \u2191(monomial n) (op r)\n[PROOFSTEP]\nsimp only [opRingEquiv, RingEquiv.coe_trans, Function.comp_apply, AddMonoidAlgebra.opRingEquiv_apply,\n RingEquiv.op_apply_apply, toFinsuppIso_apply, unop_op, toFinsupp_monomial, Finsupp.mapRange_single,\n toFinsuppIso_symm_apply, ofFinsupp_single]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nn : \u2115\n\u22a2 \u2191(opRingEquiv R) (op (\u2191C r * X ^ n)) = \u2191C (op r) * X ^ n\n[PROOFSTEP]\nsimp only [X_pow_mul, op_mul, op_pow, map_mul, map_pow, opRingEquiv_op_X, opRingEquiv_op_C]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nn : \u2115\nr : R\u1d50\u1d52\u1d56\n\u22a2 \u2191(opRingEquiv R) (\u2191(RingEquiv.symm (opRingEquiv R)) (\u2191(monomial n) r)) =\n \u2191(opRingEquiv R) (op (\u2191(monomial n) (unop r)))\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr : R\u1d50\u1d52\u1d56\nn : \u2115\n\u22a2 \u2191(RingEquiv.symm (opRingEquiv R)) (\u2191C r * X ^ n) = op (\u2191C (unop r) * X ^ n)\n[PROOFSTEP]\nrw [C_mul_X_pow_eq_monomial, opRingEquiv_symm_monomial, C_mul_X_pow_eq_monomial]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\np : R[X]\u1d50\u1d52\u1d56\nn : \u2115\n\u22a2 coeff (\u2191(opRingEquiv R) p) n = op (coeff (unop p) n)\n[PROOFSTEP]\ninduction' p using MulOpposite.rec' with p\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d : Semiring R\nn : \u2115\np : R[X]\n\u22a2 coeff (\u2191(opRingEquiv R) (op p)) n = op (coeff (unop (op p)) n)\n[PROOFSTEP]\ncases p\n[GOAL]\ncase h.ofFinsupp\nR : Type u_1\ninst\u271d : Semiring R\nn : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 coeff (\u2191(opRingEquiv R) (op { toFinsupp := toFinsupp\u271d })) n = op (coeff (unop (op { toFinsupp := toFinsupp\u271d })) n)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\np : R[X]\u1d50\u1d52\u1d56\n\u22a2 support (\u2191(opRingEquiv R) p) = support (unop p)\n[PROOFSTEP]\ninduction' p using MulOpposite.rec' with p\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d : Semiring R\np : R[X]\n\u22a2 support (\u2191(opRingEquiv R) (op p)) = support (unop (op p))\n[PROOFSTEP]\ncases p\n[GOAL]\ncase h.ofFinsupp\nR : Type u_1\ninst\u271d : Semiring R\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 support (\u2191(opRingEquiv R) (op { toFinsupp := toFinsupp\u271d })) = support (unop (op { toFinsupp := toFinsupp\u271d }))\n[PROOFSTEP]\nexact Finsupp.support_mapRange_of_injective (map_zero _) _ op_injective\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\np : R[X]\u1d50\u1d52\u1d56\n\u22a2 natDegree (\u2191(opRingEquiv R) p) = natDegree (unop p)\n[PROOFSTEP]\nby_cases p0 : p = 0\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : Semiring R\np : R[X]\u1d50\u1d52\u1d56\np0 : p = 0\n\u22a2 natDegree (\u2191(opRingEquiv R) p) = natDegree (unop p)\n[PROOFSTEP]\nsimp only [p0, _root_.map_zero, natDegree_zero, unop_zero]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\np : R[X]\u1d50\u1d52\u1d56\np0 : \u00acp = 0\n\u22a2 natDegree (\u2191(opRingEquiv R) p) = natDegree (unop p)\n[PROOFSTEP]\nsimp only [p0, natDegree_eq_support_max', Ne.def, AddEquivClass.map_eq_zero_iff, not_false_iff, support_opRingEquiv,\n unop_eq_zero_iff]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\np : R[X]\u1d50\u1d52\u1d56\n\u22a2 leadingCoeff (\u2191(opRingEquiv R) p) = op (leadingCoeff (unop p))\n[PROOFSTEP]\nrw [leadingCoeff, coeff_opRingEquiv, natDegree_opRingEquiv, leadingCoeff]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Polynomial.Opposites", "llama_tokens": 1557, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.712232184238947, "lm_q2_score": 0.6076631698328917, "lm_q1q2_score": 0.43279726673164265}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g\n\u22a2 List.prod l \u2208 alternatingGroup \u03b1 \u2194 Even (List.length l)\n[PROOFSTEP]\nrw [mem_alternatingGroup, sign_prod_list_swap hl, \u2190 Units.val_eq_one, Units.val_pow_eq_pow_val, Units.coe_neg_one,\n neg_one_pow_eq_one_iff_even]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g\n\u22a2 -1 \u2260 1\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u22a2 finRotate (bit1 n) \u2208 alternatingGroup (Fin (bit1 n))\n[PROOFSTEP]\nrw [mem_alternatingGroup, bit1, sign_finRotate, pow_bit0', Int.units_mul_self, one_pow]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\n\u22a2 2 * card { x // x \u2208 alternatingGroup \u03b1 } = card (Perm \u03b1)\n[PROOFSTEP]\nlet this := (QuotientGroup.quotientKerEquivOfSurjective _ (sign_surjective \u03b1)).toEquiv\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nthis : Perm \u03b1 \u29f8 MonoidHom.ker sign \u2243 \u2124\u02e3 :=\n (QuotientGroup.quotientKerEquivOfSurjective sign (_ : Function.Surjective \u2191sign)).toEquiv\n\u22a2 2 * card { x // x \u2208 alternatingGroup \u03b1 } = card (Perm \u03b1)\n[PROOFSTEP]\nrw [\u2190 Fintype.card_units_int, \u2190 Fintype.card_congr this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nontrivial \u03b1\nthis : Perm \u03b1 \u29f8 MonoidHom.ker sign \u2243 \u2124\u02e3 :=\n (QuotientGroup.quotientKerEquivOfSurjective sign (_ : Function.Surjective \u2191sign)).toEquiv\n\u22a2 card (Perm \u03b1 \u29f8 MonoidHom.ker sign) * card { x // x \u2208 alternatingGroup \u03b1 } = card (Perm \u03b1)\n[PROOFSTEP]\napply (Subgroup.card_eq_card_quotient_mul_card_subgroup _).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : { x // x \u2208 alternatingGroup \u03b1 }\nhc : IsConj \u2191\u03c3 \u2191\u03c4\nh\u03c3 : Finset.card (support \u2191\u03c3) + 2 \u2264 card \u03b1\n\u22a2 IsConj \u03c3 \u03c4\n[PROOFSTEP]\nobtain \u27e8\u03c3, h\u03c3\u27e9 := \u03c3\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c4 : { x // x \u2208 alternatingGroup \u03b1 }\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191\u03c4\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u22a2 IsConj { val := \u03c3, property := h\u03c3\u271d } \u03c4\n[PROOFSTEP]\nobtain \u27e8\u03c4, h\u03c4\u27e9 := \u03c4\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u22a2 IsConj { val := \u03c3, property := h\u03c3\u271d } { val := \u03c4, property := h\u03c4 }\n[PROOFSTEP]\nobtain \u27e8\u03c0, h\u03c0\u27e9 := isConj_iff.1 hc\n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u2191{ val := \u03c3, property := h\u03c3\u271d } * \u03c0\u207b\u00b9 = \u2191{ val := \u03c4, property := h\u03c4 }\n\u22a2 IsConj { val := \u03c3, property := h\u03c3\u271d } { val := \u03c4, property := h\u03c4 }\n[PROOFSTEP]\nrw [Subtype.coe_mk, Subtype.coe_mk] at h\u03c0 \n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\n\u22a2 IsConj { val := \u03c3, property := h\u03c3\u271d } { val := \u03c4, property := h\u03c4 }\n[PROOFSTEP]\ncases' Int.units_eq_one_or (Perm.sign \u03c0) with h h\n[GOAL]\ncase mk.mk.intro.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\nh : \u2191sign \u03c0 = 1\n\u22a2 IsConj { val := \u03c3, property := h\u03c3\u271d } { val := \u03c4, property := h\u03c4 }\n[PROOFSTEP]\nrw [isConj_iff]\n[GOAL]\ncase mk.mk.intro.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\nh : \u2191sign \u03c0 = 1\n\u22a2 \u2203 c, c * { val := \u03c3, property := h\u03c3\u271d } * c\u207b\u00b9 = { val := \u03c4, property := h\u03c4 }\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u03c0, mem_alternatingGroup.mp h\u27e9, Subtype.val_injective _\u27e9\n[GOAL]\ncase mk.mk.intro.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\nh : \u2191sign \u03c0 = 1\n\u22a2 \u2191({ val := \u03c0, property := (_ : \u2191sign \u03c0 = 1) } * { val := \u03c3, property := h\u03c3\u271d } *\n { val := \u03c0, property := (_ : \u2191sign \u03c0 = 1) }\u207b\u00b9) =\n \u2191{ val := \u03c4, property := h\u03c4 }\n[PROOFSTEP]\nsimpa only [Subtype.val, Subgroup.coe_mul, coe_inv, coe_mk] using h\u03c0\n[GOAL]\ncase mk.mk.intro.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\nh : \u2191sign \u03c0 = -1\n\u22a2 IsConj { val := \u03c3, property := h\u03c3\u271d } { val := \u03c4, property := h\u03c4 }\n[PROOFSTEP]\nhave h2 : 2 \u2264 \u03c3.support\u1d9c.card :=\n by\n rw [Finset.card_compl, le_tsub_iff_left \u03c3.support.card_le_univ]\n exact h\u03c3\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\nh : \u2191sign \u03c0 = -1\n\u22a2 2 \u2264 Finset.card (support \u03c3)\u1d9c\n[PROOFSTEP]\nrw [Finset.card_compl, le_tsub_iff_left \u03c3.support.card_le_univ]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\nh : \u2191sign \u03c0 = -1\n\u22a2 Finset.card (support \u03c3) + 2 \u2264 card \u03b1\n[PROOFSTEP]\nexact h\u03c3\n[GOAL]\ncase mk.mk.intro.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\nh : \u2191sign \u03c0 = -1\nh2 : 2 \u2264 Finset.card (support \u03c3)\u1d9c\n\u22a2 IsConj { val := \u03c3, property := h\u03c3\u271d } { val := \u03c4, property := h\u03c4 }\n[PROOFSTEP]\nobtain \u27e8a, ha, b, hb, ab\u27e9 := Finset.one_lt_card.1 h2\n[GOAL]\ncase mk.mk.intro.inr.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\nh : \u2191sign \u03c0 = -1\nh2 : 2 \u2264 Finset.card (support \u03c3)\u1d9c\na : \u03b1\nha : a \u2208 (support \u03c3)\u1d9c\nb : \u03b1\nhb : b \u2208 (support \u03c3)\u1d9c\nab : a \u2260 b\n\u22a2 IsConj { val := \u03c3, property := h\u03c3\u271d } { val := \u03c4, property := h\u03c4 }\n[PROOFSTEP]\nrefine' isConj_iff.2 \u27e8\u27e8\u03c0 * swap a b, _\u27e9, Subtype.val_injective _\u27e9\n[GOAL]\ncase mk.mk.intro.inr.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\nh : \u2191sign \u03c0 = -1\nh2 : 2 \u2264 Finset.card (support \u03c3)\u1d9c\na : \u03b1\nha : a \u2208 (support \u03c3)\u1d9c\nb : \u03b1\nhb : b \u2208 (support \u03c3)\u1d9c\nab : a \u2260 b\n\u22a2 \u03c0 * swap a b \u2208 alternatingGroup \u03b1\n[PROOFSTEP]\nrw [mem_alternatingGroup, MonoidHom.map_mul, h, sign_swap ab, Int.units_mul_self]\n[GOAL]\ncase mk.mk.intro.inr.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\nh : \u2191sign \u03c0 = -1\nh2 : 2 \u2264 Finset.card (support \u03c3)\u1d9c\na : \u03b1\nha : a \u2208 (support \u03c3)\u1d9c\nb : \u03b1\nhb : b \u2208 (support \u03c3)\u1d9c\nab : a \u2260 b\n\u22a2 \u2191({ val := \u03c0 * swap a b, property := (_ : \u03c0 * swap a b \u2208 alternatingGroup \u03b1) } * { val := \u03c3, property := h\u03c3\u271d } *\n { val := \u03c0 * swap a b, property := (_ : \u03c0 * swap a b \u2208 alternatingGroup \u03b1) }\u207b\u00b9) =\n \u2191{ val := \u03c4, property := h\u03c4 }\n[PROOFSTEP]\nsimp only [\u2190 h\u03c0, coe_mk, Subgroup.coe_mul, Subtype.val]\n[GOAL]\ncase mk.mk.intro.inr.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\nh : \u2191sign \u03c0 = -1\nh2 : 2 \u2264 Finset.card (support \u03c3)\u1d9c\na : \u03b1\nha : a \u2208 (support \u03c3)\u1d9c\nb : \u03b1\nhb : b \u2208 (support \u03c3)\u1d9c\nab : a \u2260 b\n\u22a2 \u03c0 * swap a b * \u03c3 * \u2191{ val := \u03c0 * swap a b, property := (_ : \u03c0 * swap a b \u2208 alternatingGroup \u03b1) }\u207b\u00b9 = \u03c0 * \u03c3 * \u03c0\u207b\u00b9\n[PROOFSTEP]\nhave hd : Disjoint (swap a b) \u03c3 :=\n by\n rw [disjoint_iff_disjoint_support, support_swap ab, Finset.disjoint_insert_left, Finset.disjoint_singleton_left]\n exact \u27e8Finset.mem_compl.1 ha, Finset.mem_compl.1 hb\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\nh : \u2191sign \u03c0 = -1\nh2 : 2 \u2264 Finset.card (support \u03c3)\u1d9c\na : \u03b1\nha : a \u2208 (support \u03c3)\u1d9c\nb : \u03b1\nhb : b \u2208 (support \u03c3)\u1d9c\nab : a \u2260 b\n\u22a2 Perm.Disjoint (swap a b) \u03c3\n[PROOFSTEP]\nrw [disjoint_iff_disjoint_support, support_swap ab, Finset.disjoint_insert_left, Finset.disjoint_singleton_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\nh : \u2191sign \u03c0 = -1\nh2 : 2 \u2264 Finset.card (support \u03c3)\u1d9c\na : \u03b1\nha : a \u2208 (support \u03c3)\u1d9c\nb : \u03b1\nhb : b \u2208 (support \u03c3)\u1d9c\nab : a \u2260 b\n\u22a2 \u00aca \u2208 support \u03c3 \u2227 \u00acb \u2208 support \u03c3\n[PROOFSTEP]\nexact \u27e8Finset.mem_compl.1 ha, Finset.mem_compl.1 hb\u27e9\n[GOAL]\ncase mk.mk.intro.inr.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\nh : \u2191sign \u03c0 = -1\nh2 : 2 \u2264 Finset.card (support \u03c3)\u1d9c\na : \u03b1\nha : a \u2208 (support \u03c3)\u1d9c\nb : \u03b1\nhb : b \u2208 (support \u03c3)\u1d9c\nab : a \u2260 b\nhd : Perm.Disjoint (swap a b) \u03c3\n\u22a2 \u03c0 * swap a b * \u03c3 * \u2191{ val := \u03c0 * swap a b, property := (_ : \u03c0 * swap a b \u2208 alternatingGroup \u03b1) }\u207b\u00b9 = \u03c0 * \u03c3 * \u03c0\u207b\u00b9\n[PROOFSTEP]\nrw [mul_assoc \u03c0 _ \u03c3, hd.commute.eq, coe_inv, coe_mk]\n[GOAL]\ncase mk.mk.intro.inr.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3\u271d : \u03c3 \u2208 alternatingGroup \u03b1\nh\u03c3 : Finset.card (support \u2191{ val := \u03c3, property := h\u03c3\u271d }) + 2 \u2264 card \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : \u03c4 \u2208 alternatingGroup \u03b1\nhc : IsConj \u2191{ val := \u03c3, property := h\u03c3\u271d } \u2191{ val := \u03c4, property := h\u03c4 }\n\u03c0 : Perm \u03b1\nh\u03c0 : \u03c0 * \u03c3 * \u03c0\u207b\u00b9 = \u03c4\nh : \u2191sign \u03c0 = -1\nh2 : 2 \u2264 Finset.card (support \u03c3)\u1d9c\na : \u03b1\nha : a \u2208 (support \u03c3)\u1d9c\nb : \u03b1\nhb : b \u2208 (support \u03c3)\u1d9c\nab : a \u2260 b\nhd : Perm.Disjoint (swap a b) \u03c3\n\u22a2 \u03c0 * (\u03c3 * swap a b) * (\u03c0 * swap a b)\u207b\u00b9 = \u03c0 * \u03c3 * \u03c0\u207b\u00b9\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh5 : 5 \u2264 card \u03b1\n\u03c3 \u03c4 : { x // x \u2208 alternatingGroup \u03b1 }\nh\u03c3 : IsThreeCycle \u2191\u03c3\nh\u03c4 : IsThreeCycle \u2191\u03c4\n\u22a2 Finset.card (support \u2191\u03c3) + 2 \u2264 card \u03b1\n[PROOFSTEP]\nrwa [h\u03c3.card_support]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 \u2208 alternatingGroup \u03b1\n\u22a2 \u03c3 \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nsuffices hind :\n \u2200 (n : \u2115) (l : List (Perm \u03b1)) (_ : \u2200 g, g \u2208 l \u2192 IsSwap g) (_ : l.length = 2 * n),\n l.prod \u2208 closure {\u03c3 : Perm \u03b1 | IsThreeCycle \u03c3}\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 \u2208 alternatingGroup \u03b1\nhind :\n \u2200 (n : \u2115) (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * n \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n\u22a2 \u03c3 \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nobtain \u27e8l, rfl, hl\u27e9 := truncSwapFactors \u03c3\n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nhind :\n \u2200 (n : \u2115) (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * n \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g\nh\u03c3 : List.prod l \u2208 alternatingGroup \u03b1\nx\u271d : Trunc { l_1 // List.prod l_1 = List.prod l \u2227 \u2200 (g : Perm \u03b1), g \u2208 l_1 \u2192 IsSwap g }\n\u22a2 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := (prod_list_swap_mem_alternatingGroup_iff_even_length hl).1 h\u03c3\n[GOAL]\ncase mk.mk.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nhind :\n \u2200 (n : \u2115) (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * n \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g\nh\u03c3 : List.prod l \u2208 alternatingGroup \u03b1\nx\u271d : Trunc { l_1 // List.prod l_1 = List.prod l \u2227 \u2200 (g : Perm \u03b1), g \u2208 l_1 \u2192 IsSwap g }\nn : \u2115\nhn : List.length l = n + n\n\u22a2 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nrw [\u2190 two_mul] at hn \n[GOAL]\ncase mk.mk.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nhind :\n \u2200 (n : \u2115) (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * n \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g\nh\u03c3 : List.prod l \u2208 alternatingGroup \u03b1\nx\u271d : Trunc { l_1 // List.prod l_1 = List.prod l \u2227 \u2200 (g : Perm \u03b1), g \u2208 l_1 \u2192 IsSwap g }\nn : \u2115\nhn : List.length l = 2 * n\n\u22a2 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nexact hind n l hl hn\n[GOAL]\ncase hind\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 \u2208 alternatingGroup \u03b1\n\u22a2 \u2200 (n : \u2115) (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * n \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nintro n\n[GOAL]\ncase hind\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 \u2208 alternatingGroup \u03b1\nn : \u2115\n\u22a2 \u2200 (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * n \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase hind.zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 \u2208 alternatingGroup \u03b1\n\u22a2 \u2200 (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * Nat.zero \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nintro l hl hn\n[GOAL]\ncase hind.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 \u2208 alternatingGroup \u03b1\nn : \u2115\nih :\n \u2200 (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * n \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n\u22a2 \u2200 (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * Nat.succ n \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nintro l hl hn\n[GOAL]\ncase hind.zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 \u2208 alternatingGroup \u03b1\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g\nhn : List.length l = 2 * Nat.zero\n\u22a2 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nsimp [List.length_eq_zero.1 hn, one_mem]\n[GOAL]\ncase hind.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 \u2208 alternatingGroup \u03b1\nn : \u2115\nih :\n \u2200 (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * n \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g\nhn : List.length l = 2 * Nat.succ n\n\u22a2 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nrw [Nat.mul_succ] at hn \n[GOAL]\ncase hind.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 \u2208 alternatingGroup \u03b1\nn : \u2115\nih :\n \u2200 (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * n \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g\nhn : List.length l = 2 * n + 2\n\u22a2 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nobtain \u27e8a, l, rfl\u27e9 := l.exists_of_length_succ hn\n[GOAL]\ncase hind.succ.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 \u2208 alternatingGroup \u03b1\nn : \u2115\nih :\n \u2200 (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * n \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\na : Perm \u03b1\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 a :: l \u2192 IsSwap g\nhn : List.length (a :: l) = 2 * n + 2\n\u22a2 List.prod (a :: l) \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nrw [List.length_cons, Nat.succ_inj'] at hn \n[GOAL]\ncase hind.succ.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 \u2208 alternatingGroup \u03b1\nn : \u2115\nih :\n \u2200 (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * n \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\na : Perm \u03b1\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 a :: l \u2192 IsSwap g\nhn : List.length l = 2 * n + 1\n\u22a2 List.prod (a :: l) \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nobtain \u27e8b, l, rfl\u27e9 := l.exists_of_length_succ hn\n[GOAL]\ncase hind.succ.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 \u2208 alternatingGroup \u03b1\nn : \u2115\nih :\n \u2200 (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * n \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\na b : Perm \u03b1\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 a :: b :: l \u2192 IsSwap g\nhn : List.length (b :: l) = 2 * n + 1\n\u22a2 List.prod (a :: b :: l) \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nrw [List.prod_cons, List.prod_cons, \u2190 mul_assoc]\n[GOAL]\ncase hind.succ.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 \u2208 alternatingGroup \u03b1\nn : \u2115\nih :\n \u2200 (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * n \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\na b : Perm \u03b1\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 a :: b :: l \u2192 IsSwap g\nhn : List.length (b :: l) = 2 * n + 1\n\u22a2 a * b * List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nrw [List.length_cons, Nat.succ_inj'] at hn \n[GOAL]\ncase hind.succ.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 \u2208 alternatingGroup \u03b1\nn : \u2115\nih :\n \u2200 (l : List (Perm \u03b1)),\n (\u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g) \u2192 List.length l = 2 * n \u2192 List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\na b : Perm \u03b1\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 a :: b :: l \u2192 IsSwap g\nhn : List.length l = 2 * n\n\u22a2 a * b * List.prod l \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nexact\n mul_mem\n (IsSwap.mul_mem_closure_three_cycles (hl a (List.mem_cons_self a _))\n (hl b (List.mem_cons_of_mem a (l.mem_cons_self b))))\n (ih _ (fun g hg => hl g (List.mem_cons_of_mem _ (List.mem_cons_of_mem _ hg))) hn)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh5 : 5 \u2264 card \u03b1\nf : Perm \u03b1\nhf : IsThreeCycle f\n\u22a2 \u22a4 \u2264 normalClosure {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }}\n[PROOFSTEP]\nhave hi : Function.Injective (alternatingGroup \u03b1).subtype := Subtype.coe_injective\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh5 : 5 \u2264 card \u03b1\nf : Perm \u03b1\nhf : IsThreeCycle f\nhi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1))\n\u22a2 \u22a4 \u2264 normalClosure {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }}\n[PROOFSTEP]\nrefine' eq_top_iff.1 (map_injective hi (le_antisymm (map_mono le_top) _))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh5 : 5 \u2264 card \u03b1\nf : Perm \u03b1\nhf : IsThreeCycle f\nhi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1))\n\u22a2 map (Subgroup.subtype (alternatingGroup \u03b1)) \u22a4 \u2264\n map (Subgroup.subtype (alternatingGroup \u03b1)) (normalClosure {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }})\n[PROOFSTEP]\nrw [\u2190 MonoidHom.range_eq_map, subtype_range, normalClosure, MonoidHom.map_closure]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh5 : 5 \u2264 card \u03b1\nf : Perm \u03b1\nhf : IsThreeCycle f\nhi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1))\n\u22a2 alternatingGroup \u03b1 \u2264\n closure\n (\u2191(Subgroup.subtype (alternatingGroup \u03b1)) ''\n Group.conjugatesOfSet {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }})\n[PROOFSTEP]\nrefine' (le_of_eq closure_three_cycles_eq_alternating.symm).trans (closure_mono _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh5 : 5 \u2264 card \u03b1\nf : Perm \u03b1\nhf : IsThreeCycle f\nhi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1))\n\u22a2 {\u03c3 | IsThreeCycle \u03c3} \u2286\n \u2191(Subgroup.subtype (alternatingGroup \u03b1)) ''\n Group.conjugatesOfSet {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }}\n[PROOFSTEP]\nintro g h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh5 : 5 \u2264 card \u03b1\nf : Perm \u03b1\nhf : IsThreeCycle f\nhi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1))\ng : Perm \u03b1\nh : g \u2208 {\u03c3 | IsThreeCycle \u03c3}\n\u22a2 g \u2208\n \u2191(Subgroup.subtype (alternatingGroup \u03b1)) ''\n Group.conjugatesOfSet {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }}\n[PROOFSTEP]\nobtain \u27e8c, rfl\u27e9 := isConj_iff.1 (isConj_iff_cycleType_eq.2 (hf.trans h.symm))\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh5 : 5 \u2264 card \u03b1\nf : Perm \u03b1\nhf : IsThreeCycle f\nhi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1))\nc : Perm \u03b1\nh : c * f * c\u207b\u00b9 \u2208 {\u03c3 | IsThreeCycle \u03c3}\n\u22a2 c * f * c\u207b\u00b9 \u2208\n \u2191(Subgroup.subtype (alternatingGroup \u03b1)) ''\n Group.conjugatesOfSet {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }}\n[PROOFSTEP]\nrefine' \u27e8\u27e8c * f * c\u207b\u00b9, h.mem_alternatingGroup\u27e9, _, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh5 : 5 \u2264 card \u03b1\nf : Perm \u03b1\nhf : IsThreeCycle f\nhi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1))\nc : Perm \u03b1\nh : c * f * c\u207b\u00b9 \u2208 {\u03c3 | IsThreeCycle \u03c3}\n\u22a2 { val := c * f * c\u207b\u00b9, property := (_ : c * f * c\u207b\u00b9 \u2208 alternatingGroup \u03b1) } \u2208\n Group.conjugatesOfSet {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }}\n[PROOFSTEP]\nrw [Group.mem_conjugatesOfSet_iff]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh5 : 5 \u2264 card \u03b1\nf : Perm \u03b1\nhf : IsThreeCycle f\nhi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1))\nc : Perm \u03b1\nh : c * f * c\u207b\u00b9 \u2208 {\u03c3 | IsThreeCycle \u03c3}\n\u22a2 \u2203 a,\n a \u2208 {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }} \u2227\n IsConj a { val := c * f * c\u207b\u00b9, property := (_ : c * f * c\u207b\u00b9 \u2208 alternatingGroup \u03b1) }\n[PROOFSTEP]\nexact \u27e8\u27e8f, hf.mem_alternatingGroup\u27e9, Set.mem_singleton _, isThreeCycle_isConj h5 hf h\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh : 3 \u2208 cycleType g\n\u22a2 IsThreeCycle (g * g)\n[PROOFSTEP]\nobtain \u27e8c, g', rfl, hd, _, h3\u27e9 := mem_cycleType_iff.1 h\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 cycleType (c * g')\nhd : Disjoint c g'\nleft\u271d : IsCycle c\nh3 : Finset.card (support c) = 3\n\u22a2 IsThreeCycle (c * g' * (c * g'))\n[PROOFSTEP]\nsimp only [mul_assoc]\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 cycleType (c * g')\nhd : Disjoint c g'\nleft\u271d : IsCycle c\nh3 : Finset.card (support c) = 3\n\u22a2 IsThreeCycle (c * (g' * (c * g')))\n[PROOFSTEP]\nrw [hd.commute.eq, \u2190 mul_assoc g']\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 cycleType (c * g')\nhd : Disjoint c g'\nleft\u271d : IsCycle c\nh3 : Finset.card (support c) = 3\n\u22a2 IsThreeCycle (c * (g' * g' * c))\n[PROOFSTEP]\nsuffices hg' : orderOf g' \u2223 2\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 cycleType (c * g')\nhd : Disjoint c g'\nleft\u271d : IsCycle c\nh3 : Finset.card (support c) = 3\nhg' : orderOf g' \u2223 2\n\u22a2 IsThreeCycle (c * (g' * g' * c))\n[PROOFSTEP]\nrw [\u2190 pow_two, orderOf_dvd_iff_pow_eq_one.1 hg', one_mul]\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 cycleType (c * g')\nhd : Disjoint c g'\nleft\u271d : IsCycle c\nh3 : Finset.card (support c) = 3\nhg' : orderOf g' \u2223 2\n\u22a2 IsThreeCycle (c * c)\n[PROOFSTEP]\nexact (card_support_eq_three_iff.1 h3).isThreeCycle_sq\n[GOAL]\ncase hg'\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 cycleType (c * g')\nhd : Disjoint c g'\nleft\u271d : IsCycle c\nh3 : Finset.card (support c) = 3\n\u22a2 orderOf g' \u2223 2\n[PROOFSTEP]\nrw [\u2190 lcm_cycleType, Multiset.lcm_dvd]\n[GOAL]\ncase hg'\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 cycleType (c * g')\nhd : Disjoint c g'\nleft\u271d : IsCycle c\nh3 : Finset.card (support c) = 3\n\u22a2 \u2200 (b : \u2115), b \u2208 cycleType g' \u2192 b \u2223 2\n[PROOFSTEP]\nintro n hn\n[GOAL]\ncase hg'\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 cycleType (c * g')\nhd : Disjoint c g'\nleft\u271d : IsCycle c\nh3 : Finset.card (support c) = 3\nn : \u2115\nhn : n \u2208 cycleType g'\n\u22a2 n \u2223 2\n[PROOFSTEP]\nrw [le_antisymm (two_le_of_mem_cycleType hn) (le_trans (le_card_support_of_mem_cycleType hn) _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 cycleType (c * g')\nhd : Disjoint c g'\nleft\u271d : IsCycle c\nh3 : Finset.card (support c) = 3\nn : \u2115\nhn : n \u2208 cycleType g'\n\u22a2 Finset.card (support g') \u2264 2\n[PROOFSTEP]\napply le_of_add_le_add_left\n[GOAL]\ncase bc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 cycleType (c * g')\nhd : Disjoint c g'\nleft\u271d : IsCycle c\nh3 : Finset.card (support c) = 3\nn : \u2115\nhn : n \u2208 cycleType g'\n\u22a2 ?a + Finset.card (support g') \u2264 ?a + 2\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 cycleType (c * g')\nhd : Disjoint c g'\nleft\u271d : IsCycle c\nh3 : Finset.card (support c) = 3\nn : \u2115\nhn : n \u2208 cycleType g'\n\u22a2 \u2115\n[PROOFSTEP]\nrw [\u2190 hd.card_support_mul, h3]\n[GOAL]\ncase bc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 cycleType (c * g')\nhd : Disjoint c g'\nleft\u271d : IsCycle c\nh3 : Finset.card (support c) = 3\nn : \u2115\nhn : n \u2208 cycleType g'\n\u22a2 Finset.card (support (c * g')) \u2264 3 + 2\n[PROOFSTEP]\nexact (c * g').support.card_le_univ\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh3 : 3 \u2264 card \u03b1\n\u22a2 Nontrivial { x // x \u2208 alternatingGroup \u03b1 }\n[PROOFSTEP]\nhaveI := Fintype.one_lt_card_iff_nontrivial.1 (lt_trans (by decide) h3)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh3 : 3 \u2264 card \u03b1\n\u22a2 1 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh3 : 3 \u2264 card \u03b1\nthis : Nontrivial \u03b1\n\u22a2 Nontrivial { x // x \u2208 alternatingGroup \u03b1 }\n[PROOFSTEP]\nrw [\u2190 Fintype.one_lt_card_iff_nontrivial]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh3 : 3 \u2264 card \u03b1\nthis : Nontrivial \u03b1\n\u22a2 1 < card { x // x \u2208 alternatingGroup \u03b1 }\n[PROOFSTEP]\nrefine' lt_of_mul_lt_mul_left _ (le_of_lt Nat.prime_two.pos)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh3 : 3 \u2264 card \u03b1\nthis : Nontrivial \u03b1\n\u22a2 2 * 1 < 2 * card { x // x \u2208 alternatingGroup \u03b1 }\n[PROOFSTEP]\nrw [two_mul_card_alternatingGroup, card_perm, \u2190 Nat.succ_le_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh3 : 3 \u2264 card \u03b1\nthis : Nontrivial \u03b1\n\u22a2 Nat.succ (2 * 1) \u2264 Nat.factorial (card \u03b1)\n[PROOFSTEP]\nexact le_trans h3 (card \u03b1).self_le_factorial\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u22a2 3 \u2264 card (Fin (n + 3))\n[PROOFSTEP]\nrw [card_fin]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u22a2 3 \u2264 n + 3\n[PROOFSTEP]\nexact le_add_left (le_refl 3)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 \u22a4 \u2264 normalClosure {{ val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }}\n[PROOFSTEP]\nhave h3 : IsThreeCycle (Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9) :=\n card_support_eq_three_iff.1 (by decide)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 Finset.card (support (Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9)) = 3\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh3 : IsThreeCycle (Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9)\n\u22a2 \u22a4 \u2264 normalClosure {{ val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }}\n[PROOFSTEP]\nrw [\u2190 h3.alternating_normalClosure (by rw [card_fin])]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh3 : IsThreeCycle (Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9)\n\u22a2 5 \u2264 card (Fin 5)\n[PROOFSTEP]\nrw [card_fin]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh3 : IsThreeCycle (Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9)\n\u22a2 normalClosure\n {{ val := Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9,\n property :=\n (_ :\n Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9 \u2208 alternatingGroup (Fin 5)) }} \u2264\n normalClosure {{ val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }}\n[PROOFSTEP]\nrefine' normalClosure_le_normal _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh3 : IsThreeCycle (Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9)\n\u22a2 {{ val := Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9,\n property :=\n (_ : Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9 \u2208 alternatingGroup (Fin 5)) }} \u2286\n \u2191(normalClosure {{ val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }})\n[PROOFSTEP]\nrw [Set.singleton_subset_iff, SetLike.mem_coe]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh3 : IsThreeCycle (Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9)\n\u22a2 { val := Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9,\n property :=\n (_ : Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9 \u2208 alternatingGroup (Fin 5)) } \u2208\n normalClosure {{ val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }}\n[PROOFSTEP]\nhave h : (\u27e8finRotate 5, finRotate_bit1_mem_alternatingGroup (n := 2)\u27e9 : alternatingGroup (Fin 5)) \u2208 normalClosure _ :=\n SetLike.mem_coe.1 (subset_normalClosure (Set.mem_singleton _))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nh3 : IsThreeCycle (Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9)\nh :\n { val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) } \u2208\n normalClosure {{ val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }}\n\u22a2 { val := Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9,\n property :=\n (_ : Fin.cycleRange 2 * finRotate 5 * (Fin.cycleRange 2)\u207b\u00b9 * (finRotate 5)\u207b\u00b9 \u2208 alternatingGroup (Fin 5)) } \u2208\n normalClosure {{ val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }}\n[PROOFSTEP]\nexact\n (mul_mem\n (Subgroup.normalClosure_normal.conj_mem _ h\n \u27e8Fin.cycleRange 2, Fin.isThreeCycle_cycleRange_two.mem_alternatingGroup\u27e9)\n (inv_mem h)\n --Porting note : added `: _` :\n _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 \u2191sign (swap 0 4 * swap 1 3) = 1\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 normalClosure {{ val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }} = \u22a4\n[PROOFSTEP]\nlet g1 := (\u27e8swap 0 2 * swap 0 1, mem_alternatingGroup.2 (by decide)\u27e9 : alternatingGroup (Fin 5))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 \u2191sign (swap 0 2 * swap 0 1) = 1\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng1 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 2 * swap 0 1, property := (_ : swap 0 2 * swap 0 1 \u2208 alternatingGroup (Fin 5)) }\n\u22a2 normalClosure {{ val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }} = \u22a4\n[PROOFSTEP]\nlet g2 := (\u27e8swap 0 4 * swap 1 3, mem_alternatingGroup.2 (by decide)\u27e9 : alternatingGroup (Fin 5))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng1 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 2 * swap 0 1, property := (_ : swap 0 2 * swap 0 1 \u2208 alternatingGroup (Fin 5)) }\n\u22a2 \u2191sign (swap 0 4 * swap 1 3) = 1\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng1 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 2 * swap 0 1, property := (_ : swap 0 2 * swap 0 1 \u2208 alternatingGroup (Fin 5)) }\ng2 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }\n\u22a2 normalClosure {{ val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }} = \u22a4\n[PROOFSTEP]\nhave h5 : g1 * g2 * g1\u207b\u00b9 * g2\u207b\u00b9 = \u27e8finRotate 5, finRotate_bit1_mem_alternatingGroup (n := 2)\u27e9 :=\n by\n rw [Subtype.ext_iff]\n simp only [Fin.val_mk, Subgroup.coe_mul, Subgroup.coe_inv, Fin.val_mk]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng1 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 2 * swap 0 1, property := (_ : swap 0 2 * swap 0 1 \u2208 alternatingGroup (Fin 5)) }\ng2 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }\n\u22a2 g1 * g2 * g1\u207b\u00b9 * g2\u207b\u00b9 = { val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }\n[PROOFSTEP]\nrw [Subtype.ext_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng1 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 2 * swap 0 1, property := (_ : swap 0 2 * swap 0 1 \u2208 alternatingGroup (Fin 5)) }\ng2 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }\n\u22a2 \u2191(g1 * g2 * g1\u207b\u00b9 * g2\u207b\u00b9) =\n \u2191{ val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }\n[PROOFSTEP]\nsimp only [Fin.val_mk, Subgroup.coe_mul, Subgroup.coe_inv, Fin.val_mk]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng1 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 2 * swap 0 1, property := (_ : swap 0 2 * swap 0 1 \u2208 alternatingGroup (Fin 5)) }\ng2 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }\nh5 :\n g1 * g2 * g1\u207b\u00b9 * g2\u207b\u00b9 = { val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }\n\u22a2 normalClosure {{ val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }} = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff, \u2190 normalClosure_finRotate_five]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng1 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 2 * swap 0 1, property := (_ : swap 0 2 * swap 0 1 \u2208 alternatingGroup (Fin 5)) }\ng2 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }\nh5 :\n g1 * g2 * g1\u207b\u00b9 * g2\u207b\u00b9 = { val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }\n\u22a2 normalClosure {{ val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }} \u2264\n normalClosure {{ val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }}\n[PROOFSTEP]\nrefine' normalClosure_le_normal _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng1 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 2 * swap 0 1, property := (_ : swap 0 2 * swap 0 1 \u2208 alternatingGroup (Fin 5)) }\ng2 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }\nh5 :\n g1 * g2 * g1\u207b\u00b9 * g2\u207b\u00b9 = { val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }\n\u22a2 {{ val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }} \u2286\n \u2191(normalClosure {{ val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }})\n[PROOFSTEP]\nrw [Set.singleton_subset_iff, SetLike.mem_coe, \u2190 h5]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng1 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 2 * swap 0 1, property := (_ : swap 0 2 * swap 0 1 \u2208 alternatingGroup (Fin 5)) }\ng2 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }\nh5 :\n g1 * g2 * g1\u207b\u00b9 * g2\u207b\u00b9 = { val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }\n\u22a2 g1 * g2 * g1\u207b\u00b9 * g2\u207b\u00b9 \u2208\n normalClosure {{ val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }}\n[PROOFSTEP]\nhave h : g2 \u2208 normalClosure { g2 } := SetLike.mem_coe.1 (subset_normalClosure (Set.mem_singleton _))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng1 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 2 * swap 0 1, property := (_ : swap 0 2 * swap 0 1 \u2208 alternatingGroup (Fin 5)) }\ng2 : { x // x \u2208 alternatingGroup (Fin 5) } :=\n { val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }\nh5 :\n g1 * g2 * g1\u207b\u00b9 * g2\u207b\u00b9 = { val := finRotate 5, property := (_ : finRotate (bit1 2) \u2208 alternatingGroup (Fin (bit1 2))) }\nh : g2 \u2208 normalClosure {g2}\n\u22a2 g1 * g2 * g1\u207b\u00b9 * g2\u207b\u00b9 \u2208\n normalClosure {{ val := swap 0 4 * swap 1 3, property := (_ : swap 0 4 * swap 1 3 \u2208 alternatingGroup (Fin 5)) }}\n[PROOFSTEP]\nexact mul_mem (Subgroup.normalClosure_normal.conj_mem _ h g1) (inv_mem h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nha : g \u2208 alternatingGroup (Fin 5)\nh1 : g \u2260 1\nh2 : \u2200 (n : \u2115), n \u2208 cycleType g \u2192 n = 2\n\u22a2 IsConj (swap 0 4 * swap 1 3) g\n[PROOFSTEP]\nhave h := g.support.card_le_univ\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nha : g \u2208 alternatingGroup (Fin 5)\nh1 : g \u2260 1\nh2 : \u2200 (n : \u2115), n \u2208 cycleType g \u2192 n = 2\nh : Finset.card (support g) \u2264 card (Fin 5)\n\u22a2 IsConj (swap 0 4 * swap 1 3) g\n[PROOFSTEP]\nrw [\u2190 Multiset.eq_replicate_card] at h2 \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nha : g \u2208 alternatingGroup (Fin 5)\nh1 : g \u2260 1\nh2 : cycleType g = Multiset.replicate (\u2191Multiset.card (cycleType g)) 2\nh : Finset.card (support g) \u2264 card (Fin 5)\n\u22a2 IsConj (swap 0 4 * swap 1 3) g\n[PROOFSTEP]\nrw [\u2190 sum_cycleType, h2, Multiset.sum_replicate, smul_eq_mul] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nha : g \u2208 alternatingGroup (Fin 5)\nh1 : g \u2260 1\nh2 : cycleType g = Multiset.replicate (\u2191Multiset.card (cycleType g)) 2\nh : \u2191Multiset.card (cycleType g) * 2 \u2264 card (Fin 5)\n\u22a2 IsConj (swap 0 4 * swap 1 3) g\n[PROOFSTEP]\nhave h : Multiset.card g.cycleType \u2264 3 :=\n le_of_mul_le_mul_right (le_trans h (by simp only [card_fin]; ring_nf)) (by simp)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nha : g \u2208 alternatingGroup (Fin 5)\nh1 : g \u2260 1\nh2 : cycleType g = Multiset.replicate (\u2191Multiset.card (cycleType g)) 2\nh : \u2191Multiset.card (cycleType g) * 2 \u2264 card (Fin 5)\n\u22a2 card (Fin 5) \u2264 3 * 2\n[PROOFSTEP]\nsimp only [card_fin]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nha : g \u2208 alternatingGroup (Fin 5)\nh1 : g \u2260 1\nh2 : cycleType g = Multiset.replicate (\u2191Multiset.card (cycleType g)) 2\nh : \u2191Multiset.card (cycleType g) * 2 \u2264 card (Fin 5)\n\u22a2 5 \u2264 3 * 2\n[PROOFSTEP]\nring_nf\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nha : g \u2208 alternatingGroup (Fin 5)\nh1 : g \u2260 1\nh2 : cycleType g = Multiset.replicate (\u2191Multiset.card (cycleType g)) 2\nh : \u2191Multiset.card (cycleType g) * 2 \u2264 card (Fin 5)\n\u22a2 0 < 2\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nha : g \u2208 alternatingGroup (Fin 5)\nh1 : g \u2260 1\nh2 : cycleType g = Multiset.replicate (\u2191Multiset.card (cycleType g)) 2\nh\u271d : \u2191Multiset.card (cycleType g) * 2 \u2264 card (Fin 5)\nh : \u2191Multiset.card (cycleType g) \u2264 3\n\u22a2 IsConj (swap 0 4 * swap 1 3) g\n[PROOFSTEP]\nrw [mem_alternatingGroup, sign_of_cycleType, h2] at ha \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nha :\n (-1) ^\n (Multiset.sum (Multiset.replicate (\u2191Multiset.card (cycleType g)) 2) +\n \u2191Multiset.card (Multiset.replicate (\u2191Multiset.card (cycleType g)) 2)) =\n 1\nh1 : g \u2260 1\nh2 : cycleType g = Multiset.replicate (\u2191Multiset.card (cycleType g)) 2\nh\u271d : \u2191Multiset.card (cycleType g) * 2 \u2264 card (Fin 5)\nh : \u2191Multiset.card (cycleType g) \u2264 3\n\u22a2 IsConj (swap 0 4 * swap 1 3) g\n[PROOFSTEP]\nnorm_num at ha \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh2 : cycleType g = Multiset.replicate (\u2191Multiset.card (cycleType g)) 2\nh\u271d : \u2191Multiset.card (cycleType g) * 2 \u2264 card (Fin 5)\nh : \u2191Multiset.card (cycleType g) \u2264 3\nha : (-1) ^ (\u2191Multiset.card (cycleType g) * 2 + \u2191Multiset.card (cycleType g)) = 1\n\u22a2 IsConj (swap 0 4 * swap 1 3) g\n[PROOFSTEP]\nrw [pow_add, pow_mul, Int.units_pow_two, one_mul, Units.ext_iff, Units.val_one, Units.val_pow_eq_pow_val,\n Units.coe_neg_one, neg_one_pow_eq_one_iff_even _] at ha \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh2 : cycleType g = Multiset.replicate (\u2191Multiset.card (cycleType g)) 2\nh\u271d : \u2191Multiset.card (cycleType g) * 2 \u2264 card (Fin 5)\nh : \u2191Multiset.card (cycleType g) \u2264 3\nha : Even (\u2191Multiset.card (cycleType g))\n\u22a2 IsConj (swap 0 4 * swap 1 3) g\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh2 : cycleType g = Multiset.replicate (\u2191Multiset.card (cycleType g)) 2\nh\u271d : \u2191Multiset.card (cycleType g) * 2 \u2264 card (Fin 5)\nh : \u2191Multiset.card (cycleType g) \u2264 3\nha : (-1) ^ \u2191Multiset.card (cycleType g) = 1\n\u22a2 -1 \u2260 1\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh2 : cycleType g = Multiset.replicate (\u2191Multiset.card (cycleType g)) 2\nh\u271d : \u2191Multiset.card (cycleType g) * 2 \u2264 card (Fin 5)\nh : \u2191Multiset.card (cycleType g) \u2264 3\nha : (-1) ^ \u2191Multiset.card (cycleType g) = 1\n\u22a2 -1 \u2260 1\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh2 : cycleType g = Multiset.replicate (\u2191Multiset.card (cycleType g)) 2\nh\u271d : \u2191Multiset.card (cycleType g) * 2 \u2264 card (Fin 5)\nh : \u2191Multiset.card (cycleType g) \u2264 3\nha : Even (\u2191Multiset.card (cycleType g))\n\u22a2 IsConj (swap 0 4 * swap 1 3) g\n[PROOFSTEP]\nrw [isConj_iff_cycleType_eq, h2]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh2 : cycleType g = Multiset.replicate (\u2191Multiset.card (cycleType g)) 2\nh\u271d : \u2191Multiset.card (cycleType g) * 2 \u2264 card (Fin 5)\nh : \u2191Multiset.card (cycleType g) \u2264 3\nha : Even (\u2191Multiset.card (cycleType g))\n\u22a2 cycleType (swap 0 4 * swap 1 3) = Multiset.replicate (\u2191Multiset.card (cycleType g)) 2\n[PROOFSTEP]\ninterval_cases h_1 : Multiset.card g.cycleType\n[GOAL]\ncase \u00ab0\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh_1 : \u2191Multiset.card (cycleType g) = 0\nh2 : cycleType g = Multiset.replicate 0 2\nh\u271d : 0 * 2 \u2264 card (Fin 5)\nh : 0 \u2264 3\nha : Even 0\n\u22a2 cycleType (swap 0 4 * swap 1 3) = Multiset.replicate 0 2\n[PROOFSTEP]\nexact (h1 (card_cycleType_eq_zero.1 h_1)).elim\n[GOAL]\ncase \u00ab1\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh_1 : \u2191Multiset.card (cycleType g) = 1\nh2 : cycleType g = Multiset.replicate 1 2\nh\u271d : 1 * 2 \u2264 card (Fin 5)\nh : 1 \u2264 3\nha : Even 1\n\u22a2 cycleType (swap 0 4 * swap 1 3) = Multiset.replicate 1 2\n[PROOFSTEP]\ncontrapose! ha\n[GOAL]\ncase \u00ab1\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh_1 : \u2191Multiset.card (cycleType g) = 1\nh2 : cycleType g = Multiset.replicate 1 2\nh\u271d : 1 * 2 \u2264 card (Fin 5)\nh : 1 \u2264 3\nha : cycleType (swap 0 4 * swap 1 3) \u2260 Multiset.replicate 1 2\n\u22a2 \u00acEven 1\n[PROOFSTEP]\nsimp [h_1]\n[GOAL]\ncase \u00ab2\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh_1 : \u2191Multiset.card (cycleType g) = 2\nh2 : cycleType g = Multiset.replicate 2 2\nh\u271d : 2 * 2 \u2264 card (Fin 5)\nh : 2 \u2264 3\nha : Even 2\n\u22a2 cycleType (swap 0 4 * swap 1 3) = Multiset.replicate 2 2\n[PROOFSTEP]\nhave h04 : (0 : Fin 5) \u2260 4 := by decide\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh_1 : \u2191Multiset.card (cycleType g) = 2\nh2 : cycleType g = Multiset.replicate 2 2\nh\u271d : 2 * 2 \u2264 card (Fin 5)\nh : 2 \u2264 3\nha : Even 2\n\u22a2 0 \u2260 4\n[PROOFSTEP]\ndecide\n[GOAL]\ncase \u00ab2\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh_1 : \u2191Multiset.card (cycleType g) = 2\nh2 : cycleType g = Multiset.replicate 2 2\nh\u271d : 2 * 2 \u2264 card (Fin 5)\nh : 2 \u2264 3\nha : Even 2\nh04 : 0 \u2260 4\n\u22a2 cycleType (swap 0 4 * swap 1 3) = Multiset.replicate 2 2\n[PROOFSTEP]\nhave h13 : (1 : Fin 5) \u2260 3 := by decide\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh_1 : \u2191Multiset.card (cycleType g) = 2\nh2 : cycleType g = Multiset.replicate 2 2\nh\u271d : 2 * 2 \u2264 card (Fin 5)\nh : 2 \u2264 3\nha : Even 2\nh04 : 0 \u2260 4\n\u22a2 1 \u2260 3\n[PROOFSTEP]\ndecide\n[GOAL]\ncase \u00ab2\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh_1 : \u2191Multiset.card (cycleType g) = 2\nh2 : cycleType g = Multiset.replicate 2 2\nh\u271d : 2 * 2 \u2264 card (Fin 5)\nh : 2 \u2264 3\nha : Even 2\nh04 : 0 \u2260 4\nh13 : 1 \u2260 3\n\u22a2 cycleType (swap 0 4 * swap 1 3) = Multiset.replicate 2 2\n[PROOFSTEP]\nrw [Disjoint.cycleType, (isCycle_swap h04).cycleType, (isCycle_swap h13).cycleType, card_support_swap h04,\n card_support_swap h13]\n[GOAL]\ncase \u00ab2\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh_1 : \u2191Multiset.card (cycleType g) = 2\nh2 : cycleType g = Multiset.replicate 2 2\nh\u271d : 2 * 2 \u2264 card (Fin 5)\nh : 2 \u2264 3\nha : Even 2\nh04 : 0 \u2260 4\nh13 : 1 \u2260 3\n\u22a2 \u2191[2] + \u2191[2] = Multiset.replicate 2 2\n[PROOFSTEP]\nrfl\n[GOAL]\ncase \u00ab2\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh_1 : \u2191Multiset.card (cycleType g) = 2\nh2 : cycleType g = Multiset.replicate 2 2\nh\u271d : 2 * 2 \u2264 card (Fin 5)\nh : 2 \u2264 3\nha : Even 2\nh04 : 0 \u2260 4\nh13 : 1 \u2260 3\n\u22a2 Perm.Disjoint (swap 0 4) (swap 1 3)\n[PROOFSTEP]\nrw [disjoint_iff_disjoint_support, support_swap h04, support_swap h13]\n[GOAL]\ncase \u00ab2\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh_1 : \u2191Multiset.card (cycleType g) = 2\nh2 : cycleType g = Multiset.replicate 2 2\nh\u271d : 2 * 2 \u2264 card (Fin 5)\nh : 2 \u2264 3\nha : Even 2\nh04 : 0 \u2260 4\nh13 : 1 \u2260 3\n\u22a2 _root_.Disjoint {0, 4} {1, 3}\n[PROOFSTEP]\ndecide\n[GOAL]\ncase \u00ab3\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh_1 : \u2191Multiset.card (cycleType g) = 3\nh2 : cycleType g = Multiset.replicate 3 2\nh\u271d : 3 * 2 \u2264 card (Fin 5)\nh : 3 \u2264 3\nha : Even 3\n\u22a2 cycleType (swap 0 4 * swap 1 3) = Multiset.replicate 3 2\n[PROOFSTEP]\ncontrapose! ha\n[GOAL]\ncase \u00ab3\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh1 : g \u2260 1\nh_1 : \u2191Multiset.card (cycleType g) = 3\nh2 : cycleType g = Multiset.replicate 3 2\nh\u271d : 3 * 2 \u2264 card (Fin 5)\nh : 3 \u2264 3\nha : cycleType (swap 0 4 * swap 1 3) \u2260 Multiset.replicate 3 2\n\u22a2 \u00acEven 3\n[PROOFSTEP]\nsimp [h_1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\n\u22a2 Normal H \u2192 H = \u22a5 \u2228 H = \u22a4\n[PROOFSTEP]\nintro Hn\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\n\u22a2 H = \u22a5 \u2228 H = \u22a4\n[PROOFSTEP]\nrefine' or_not.imp id fun Hb => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\nHb : \u00acH = \u22a5\n\u22a2 H = \u22a4\n[PROOFSTEP]\nrw [eq_bot_iff_forall] at Hb \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\nHb : \u00ac\u2200 (x : { x // x \u2208 alternatingGroup (Fin 5) }), x \u2208 H \u2192 x = 1\n\u22a2 H = \u22a4\n[PROOFSTEP]\npush_neg at Hb \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\nHb : \u2203 x, x \u2208 H \u2227 x \u2260 1\n\u22a2 H = \u22a4\n[PROOFSTEP]\nobtain \u27e8\u27e8g, gA\u27e9, gH, g1\u27e9 : \u2203 x : \u21a5(alternatingGroup (Fin 5)), x \u2208 H \u2227 x \u2260 1 := Hb\n[GOAL]\ncase intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : { val := g, property := gA } \u2208 H\ng1 : { val := g, property := gA } \u2260 1\n\u22a2 H = \u22a4\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, \u2190 Set.singleton_subset_iff] at gH \n[GOAL]\ncase intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\n\u22a2 H = \u22a4\n[PROOFSTEP]\nrefine'\n eq_top_iff.2\n (le_trans (ge_of_eq _) (normalClosure_le_normal gH))\n -- It suffices to show that the normal closure of `g` in $A_5$ is $A_5$.\n[GOAL]\ncase intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\n\u22a2 normalClosure {{ val := g, property := gA }} = \u22a4\n[PROOFSTEP]\nby_cases h2 :\n \u2200 n \u2208 g.cycleType,\n n =\n 2\n -- If the cycle decomposition of `g` consists entirely of swaps, then the cycle type is $(2,2)$.\n -- This means that it is conjugate to $(04)(13)$, whose normal closure is $A_5$.\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nh2 : \u2200 (n : \u2115), n \u2208 cycleType g \u2192 n = 2\n\u22a2 normalClosure {{ val := g, property := gA }} = \u22a4\n[PROOFSTEP]\nrw [Ne.def, Subtype.ext_iff] at g1 \n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : \u00ac\u2191{ val := g, property := gA } = \u21911\nh2 : \u2200 (n : \u2115), n \u2208 cycleType g \u2192 n = 2\n\u22a2 normalClosure {{ val := g, property := gA }} = \u22a4\n[PROOFSTEP]\nexact (isConj_swap_mul_swap_of_cycleType_two gA g1 h2).normalClosure_eq_top_of normalClosure_swap_mul_swap_five\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nh2 : \u00ac\u2200 (n : \u2115), n \u2208 cycleType g \u2192 n = 2\n\u22a2 normalClosure {{ val := g, property := gA }} = \u22a4\n[PROOFSTEP]\npush_neg at h2 \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nh2 : \u2203 n, n \u2208 cycleType g \u2227 n \u2260 2\n\u22a2 normalClosure {{ val := g, property := gA }} = \u22a4\n[PROOFSTEP]\nobtain \u27e8n, ng, n2\u27e9 : \u2203 n : \u2115, n \u2208 g.cycleType \u2227 n \u2260 2 := h2\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : n \u2208 cycleType g\nn2 : n \u2260 2\n\u22a2 normalClosure {{ val := g, property := gA }} = \u22a4\n[PROOFSTEP]\nhave n2' : 2 < n := lt_of_le_of_ne (two_le_of_mem_cycleType ng) n2.symm\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : n \u2208 cycleType g\nn2 : n \u2260 2\nn2' : 2 < n\n\u22a2 normalClosure {{ val := g, property := gA }} = \u22a4\n[PROOFSTEP]\nhave n5 : n \u2264 5 := le_trans ?_ g.support.card_le_univ\n[GOAL]\ncase neg.intro.intro.refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : n \u2208 cycleType g\nn2 : n \u2260 2\nn2' : 2 < n\nn5 : n \u2264 5\n\u22a2 normalClosure {{ val := g, property := gA }} = \u22a4\ncase neg.intro.intro.refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : n \u2208 cycleType g\nn2 : n \u2260 2\nn2' : 2 < n\n\u22a2 n \u2264 Finset.card (support g)\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg.intro.intro.refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : n \u2208 cycleType g\nn2 : n \u2260 2\nn2' : 2 < n\n\u22a2 n \u2264 Finset.card (support g)\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := Multiset.exists_cons_of_mem ng\n[GOAL]\ncase neg.intro.intro.refine_1.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : n \u2208 cycleType g\nn2 : n \u2260 2\nn2' : 2 < n\nm : Multiset \u2115\nhm : cycleType g = n ::\u2098 m\n\u22a2 n \u2264 Finset.card (support g)\n[PROOFSTEP]\nrw [\u2190 sum_cycleType, hm, Multiset.sum_cons]\n[GOAL]\ncase neg.intro.intro.refine_1.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : n \u2208 cycleType g\nn2 : n \u2260 2\nn2' : 2 < n\nm : Multiset \u2115\nhm : cycleType g = n ::\u2098 m\n\u22a2 n \u2264 n + Multiset.sum m\n[PROOFSTEP]\nexact le_add_right le_rfl\n[GOAL]\ncase neg.intro.intro.refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : n \u2208 cycleType g\nn2 : n \u2260 2\nn2' : 2 < n\nn5 : n \u2264 5\n\u22a2 normalClosure {{ val := g, property := gA }} = \u22a4\n[PROOFSTEP]\ninterval_cases n\n[GOAL]\ncase neg.intro.intro.refine_2.\u00ab3\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : 3 \u2208 cycleType g\nn2 : 3 \u2260 2\nn2' : 2 < 3\nn5 : 3 \u2264 5\n\u22a2 normalClosure {{ val := g, property := gA }} = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff, \u2190 (isThreeCycle_sq_of_three_mem_cycleType_five ng).alternating_normalClosure (by rw [card_fin])]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : 3 \u2208 cycleType g\nn2 : 3 \u2260 2\nn2' : 2 < 3\nn5 : 3 \u2264 5\n\u22a2 5 \u2264 card (Fin 5)\n[PROOFSTEP]\nrw [card_fin]\n[GOAL]\ncase neg.intro.intro.refine_2.\u00ab3\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : 3 \u2208 cycleType g\nn2 : 3 \u2260 2\nn2' : 2 < 3\nn5 : 3 \u2264 5\n\u22a2 normalClosure {{ val := g * g, property := (_ : g * g \u2208 alternatingGroup (Fin 5)) }} \u2264\n normalClosure {{ val := g, property := gA }}\n[PROOFSTEP]\nrefine' normalClosure_le_normal _\n[GOAL]\ncase neg.intro.intro.refine_2.\u00ab3\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : 3 \u2208 cycleType g\nn2 : 3 \u2260 2\nn2' : 2 < 3\nn5 : 3 \u2264 5\n\u22a2 {{ val := g * g, property := (_ : g * g \u2208 alternatingGroup (Fin 5)) }} \u2286\n \u2191(normalClosure {{ val := g, property := gA }})\n[PROOFSTEP]\nrw [Set.singleton_subset_iff, SetLike.mem_coe]\n[GOAL]\ncase neg.intro.intro.refine_2.\u00ab3\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : 3 \u2208 cycleType g\nn2 : 3 \u2260 2\nn2' : 2 < 3\nn5 : 3 \u2264 5\n\u22a2 { val := g * g, property := (_ : g * g \u2208 alternatingGroup (Fin 5)) } \u2208 normalClosure {{ val := g, property := gA }}\n[PROOFSTEP]\nhave h := SetLike.mem_coe.1 (subset_normalClosure (G := alternatingGroup (Fin 5)) (Set.mem_singleton \u27e8g, gA\u27e9))\n[GOAL]\ncase neg.intro.intro.refine_2.\u00ab3\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : 3 \u2208 cycleType g\nn2 : 3 \u2260 2\nn2' : 2 < 3\nn5 : 3 \u2264 5\nh : { val := g, property := gA } \u2208 normalClosure {{ val := g, property := gA }}\n\u22a2 { val := g * g, property := (_ : g * g \u2208 alternatingGroup (Fin 5)) } \u2208 normalClosure {{ val := g, property := gA }}\n[PROOFSTEP]\nexact mul_mem h h\n[GOAL]\ncase neg.intro.intro.refine_2.\u00ab4\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : 4 \u2208 cycleType g\nn2 : 4 \u2260 2\nn2' : 2 < 4\nn5 : 4 \u2264 5\n\u22a2 normalClosure {{ val := g, property := gA }} = \u22a4\n[PROOFSTEP]\nhave con := mem_alternatingGroup.1 gA\n[GOAL]\ncase neg.intro.intro.refine_2.\u00ab4\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : 4 \u2208 cycleType g\nn2 : 4 \u2260 2\nn2' : 2 < 4\nn5 : 4 \u2264 5\ncon : \u2191sign g = 1\n\u22a2 normalClosure {{ val := g, property := gA }} = \u22a4\n[PROOFSTEP]\ncontrapose! con\n[GOAL]\ncase neg.intro.intro.refine_2.\u00ab4\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : 4 \u2208 cycleType g\nn2 : 4 \u2260 2\nn2' : 2 < 4\nn5 : 4 \u2264 5\ncon : normalClosure {{ val := g, property := gA }} \u2260 \u22a4\n\u22a2 \u2191sign g \u2260 1\n[PROOFSTEP]\nrw [sign_of_cycleType, cycleType_of_card_le_mem_cycleType_add_two (by decide) ng]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : 4 \u2208 cycleType g\nn2 : 4 \u2260 2\nn2' : 2 < 4\nn5 : 4 \u2264 5\ncon : normalClosure {{ val := g, property := gA }} \u2260 \u22a4\n\u22a2 card (Fin 5) < 4 + 2\n[PROOFSTEP]\ndecide\n[GOAL]\ncase neg.intro.intro.refine_2.\u00ab4\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : 4 \u2208 cycleType g\nn2 : 4 \u2260 2\nn2' : 2 < 4\nn5 : 4 \u2264 5\ncon : normalClosure {{ val := g, property := gA }} \u2260 \u22a4\n\u22a2 (-1) ^ (Multiset.sum {4} + \u2191Multiset.card {4}) \u2260 1\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase neg.intro.intro.refine_2.\u00ab4\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : 4 \u2208 cycleType g\nn2 : 4 \u2260 2\nn2' : 2 < 4\nn5 : 4 \u2264 5\ncon : normalClosure {{ val := g, property := gA }} \u2260 \u22a4\n\u22a2 (-1) ^ (Multiset.sum {4} + \u2191Multiset.card {4}) \u2260 1\n[PROOFSTEP]\ndecide\n[GOAL]\ncase neg.intro.intro.refine_2.\u00ab5\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : 5 \u2208 cycleType g\nn2 : 5 \u2260 2\nn2' : 2 < 5\nn5 : 5 \u2264 5\n\u22a2 normalClosure {{ val := g, property := gA }} = \u22a4\n[PROOFSTEP]\nrefine' (isConj_iff_cycleType_eq.2 _).normalClosure_eq_top_of normalClosure_finRotate_five\n[GOAL]\ncase neg.intro.intro.refine_2.\u00ab5\u00bb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : 5 \u2208 cycleType g\nn2 : 5 \u2260 2\nn2' : 2 < 5\nn5 : 5 \u2264 5\n\u22a2 cycleType (finRotate 5) = cycleType g\n[PROOFSTEP]\nrw [cycleType_of_card_le_mem_cycleType_add_two (by decide) ng, cycleType_finRotate]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup { x // x \u2208 alternatingGroup (Fin 5) }\nHn : Normal H\ng : Perm (Fin 5)\ngA : g \u2208 alternatingGroup (Fin 5)\ngH : {{ val := g, property := gA }} \u2286 \u2191H\ng1 : { val := g, property := gA } \u2260 1\nn : \u2115\nng : 5 \u2208 cycleType g\nn2 : 5 \u2260 2\nn2' : 2 < 5\nn5 : 5 \u2264 5\n\u22a2 card (Fin 5) < 5 + 2\n[PROOFSTEP]\ndecide\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.SpecificGroups.Alternating", "llama_tokens": 30859, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321720225278, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.43279725930817453}} {"text": "[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableMul G\ng : G\n\u22a2 Measurable fun h => h / g\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv, measurable_mul_const]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nn : \u2115\n\u22a2 Measurable fun x => (x, n).fst ^ (x, n).snd\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\n\u22a2 Measurable fun x => (x, Nat.zero).fst ^ (x, Nat.zero).snd\n[PROOFSTEP]\nsimp only [Nat.zero_eq, pow_zero, \u2190 Pi.one_def, measurable_one]\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nn : \u2115\nih : Measurable fun x => (x, n).fst ^ (x, n).snd\n\u22a2 Measurable fun x => (x, Nat.succ n).fst ^ (x, Nat.succ n).snd\n[PROOFSTEP]\nsimp only [pow_succ]\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nn : \u2115\nih : Measurable fun x => (x, n).fst ^ (x, n).snd\n\u22a2 Measurable fun x => x * x ^ n\n[PROOFSTEP]\nexact measurable_id.mul ih\n[GOAL]\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_3\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : AddGroup E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : MeasurableSub\u2082 E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\n\u22a2 MeasurableSet {x | f x = g x}\n[PROOFSTEP]\nsuffices h_set_eq : {x : \u03b1 | f x = g x} = {x | (f - g) x = (0 : E)}\n[GOAL]\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_3\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : AddGroup E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : MeasurableSub\u2082 E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nh_set_eq : {x | f x = g x} = {x | (f - g) x = 0}\n\u22a2 MeasurableSet {x | f x = g x}\n[PROOFSTEP]\nrw [h_set_eq]\n[GOAL]\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_3\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : AddGroup E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : MeasurableSub\u2082 E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nh_set_eq : {x | f x = g x} = {x | (f - g) x = 0}\n\u22a2 MeasurableSet {x | (f - g) x = 0}\n[PROOFSTEP]\nexact (hf.sub hg) measurableSet_eq\n[GOAL]\ncase h_set_eq\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_3\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : AddGroup E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : MeasurableSub\u2082 E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\n\u22a2 {x | f x = g x} = {x | (f - g) x = 0}\n[PROOFSTEP]\next\n[GOAL]\ncase h_set_eq.h\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_3\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : AddGroup E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : MeasurableSub\u2082 E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 {x | f x = g x} \u2194 x\u271d \u2208 {x | (f - g) x = 0}\n[PROOFSTEP]\nsimp_rw [Set.mem_setOf_eq, Pi.sub_apply, sub_eq_zero]\n[GOAL]\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : Div G\nm : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : AddGroup E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : MeasurableSub\u2082 E\nf g : \u03b1 \u2192 E\nhf : AEMeasurable f\nhg : AEMeasurable g\n\u22a2 NullMeasurableSet {x | f x = g x}\n[PROOFSTEP]\napply (measurableSet_eq_fun hf.measurable_mk hg.measurable_mk).nullMeasurableSet.congr\n[GOAL]\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : Div G\nm : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : AddGroup E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : MeasurableSub\u2082 E\nf g : \u03b1 \u2192 E\nhf : AEMeasurable f\nhg : AEMeasurable g\n\u22a2 {x | AEMeasurable.mk f hf x = AEMeasurable.mk g hg x} =\u1d50[\u03bc] {x | f x = g x}\n[PROOFSTEP]\nfilter_upwards [hf.ae_eq_mk, hg.ae_eq_mk] with x hfx hgx\n[GOAL]\ncase h\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : Div G\nm : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : AddGroup E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : MeasurableSub\u2082 E\nf g : \u03b1 \u2192 E\nhf : AEMeasurable f\nhg : AEMeasurable g\nx : \u03b1\nhfx : f x = AEMeasurable.mk f hf x\nhgx : g x = AEMeasurable.mk g hg x\n\u22a2 setOf (fun x => AEMeasurable.mk f hf x = AEMeasurable.mk g hg x) x = setOf (fun x => f x = g x) x\n[PROOFSTEP]\nchange (hf.mk f x = hg.mk g x) = (f x = g x)\n[GOAL]\ncase h\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : Div G\nm : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : AddGroup E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : MeasurableSub\u2082 E\nf g : \u03b1 \u2192 E\nhf : AEMeasurable f\nhg : AEMeasurable g\nx : \u03b1\nhfx : f x = AEMeasurable.mk f hf x\nhgx : g x = AEMeasurable.mk g hg x\n\u22a2 (AEMeasurable.mk f hf x = AEMeasurable.mk g hg x) = (f x = g x)\n[PROOFSTEP]\nsimp only [hfx, hgx]\n[GOAL]\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_3\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\n\u22a2 MeasurableSet {x | f x = g x}\n[PROOFSTEP]\nhave : {x | f x = g x} = \u22c3 j, {x | f x = j} \u2229 {x | g x = j} :=\n by\n ext1 x\n simp only [Set.mem_setOf_eq, Set.mem_iUnion, Set.mem_inter_iff, exists_eq_right']\n[GOAL]\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_3\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\n\u22a2 {x | f x = g x} = \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_3\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nx : \u03b1\n\u22a2 x \u2208 {x | f x = g x} \u2194 x \u2208 \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq, Set.mem_iUnion, Set.mem_inter_iff, exists_eq_right']\n[GOAL]\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_3\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nthis : {x | f x = g x} = \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}\n\u22a2 MeasurableSet {x | f x = g x}\n[PROOFSTEP]\nrw [this]\n[GOAL]\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_3\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nthis : {x | f x = g x} = \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}\n\u22a2 MeasurableSet (\u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j})\n[PROOFSTEP]\nrefine' MeasurableSet.iUnion fun j => MeasurableSet.inter _ _\n[GOAL]\ncase refine'_1\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_3\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nthis : {x | f x = g x} = \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}\nj : E\n\u22a2 MeasurableSet {x | f x = j}\n[PROOFSTEP]\nexact hf (measurableSet_singleton j)\n[GOAL]\ncase refine'_2\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_3\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nthis : {x | f x = g x} = \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}\nj : E\n\u22a2 MeasurableSet {x | g x = j}\n[PROOFSTEP]\nexact hg (measurableSet_singleton j)\n[GOAL]\nG : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : Div G\nm\u271d : MeasurableSpace \u03b1\u271d\nf\u271d g\u271d : \u03b1\u271d \u2192 G\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_3\nE : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : AddGroup E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : MeasurableSub\u2082 E\nhm : m \u2264 m0\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nhfg : f =\u1d50[\u03bc] g\n\u22a2 f =\u1d50[Measure.trim \u03bc hm] g\n[PROOFSTEP]\nrwa [Filter.EventuallyEq, @ae_iff _ m, trim_measurableSet_eq hm _]\n[GOAL]\nG : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : Div G\nm\u271d : MeasurableSpace \u03b1\u271d\nf\u271d g\u271d : \u03b1\u271d \u2192 G\n\u03bc\u271d : Measure \u03b1\u271d\n\u03b1 : Type u_3\nE : Type u_4\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : AddGroup E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : MeasurableSub\u2082 E\nhm : m \u2264 m0\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nhfg : f =\u1d50[\u03bc] g\n\u22a2 MeasurableSet {a | \u00acf a = g a}\n[PROOFSTEP]\nexact @MeasurableSet.compl \u03b1 _ m (@measurableSet_eq_fun \u03b1 m E _ _ _ _ _ _ hf hg)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : MeasurableMul G\ninst\u271d : MeasurableInv G\nc : G\n\u22a2 Measurable fun x => c / x\n[PROOFSTEP]\nconvert measurable_inv.const_mul c using 1\n[GOAL]\ncase h.e'_5\nG : Type u_1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : MeasurableMul G\ninst\u271d : MeasurableInv G\nc : G\n\u22a2 (fun x => c / x) = fun x => c * x\u207b\u00b9\n[PROOFSTEP]\next1\n[GOAL]\ncase h.e'_5.h\nG : Type u_1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : MeasurableMul G\ninst\u271d : MeasurableInv G\nc x\u271d : G\n\u22a2 c / x\u271d = c * x\u271d\u207b\u00b9\n[PROOFSTEP]\napply div_eq_mul_inv\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : MeasurableMul G\ninst\u271d : MeasurableInv G\nc : G\n\u22a2 Measurable fun x => x / c\n[PROOFSTEP]\nconvert measurable_id.mul_const c\u207b\u00b9 using 1\n[GOAL]\ncase h.e'_5\nG : Type u_1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : MeasurableMul G\ninst\u271d : MeasurableInv G\nc : G\n\u22a2 (fun x => x / c) = fun x => id x * c\u207b\u00b9\n[PROOFSTEP]\next1\n[GOAL]\ncase h.e'_5.h\nG : Type u_1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : MeasurableMul G\ninst\u271d : MeasurableInv G\nc x\u271d : G\n\u22a2 x\u271d / c = id x\u271d * c\u207b\u00b9\n[PROOFSTEP]\napply div_eq_mul_inv\n[GOAL]\nG\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2075 : Inv G\u271d\ninst\u271d\u2074 : MeasurableSpace G\u271d\ninst\u271d\u00b3 : MeasurableInv G\u271d\nm : MeasurableSpace \u03b1\nf\u271d : \u03b1 \u2192 G\u271d\n\u03bc : Measure \u03b1\nG : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableInv G\nf : \u03b1 \u2192 G\nh : Measurable fun x => (f x)\u207b\u00b9\n\u22a2 Measurable f\n[PROOFSTEP]\nsimpa only [inv_inv] using h.inv\n[GOAL]\nG\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2075 : Inv G\u271d\ninst\u271d\u2074 : MeasurableSpace G\u271d\ninst\u271d\u00b3 : MeasurableInv G\u271d\nm : MeasurableSpace \u03b1\nf\u271d : \u03b1 \u2192 G\u271d\n\u03bc : Measure \u03b1\nG : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableInv G\nf : \u03b1 \u2192 G\nh : AEMeasurable fun x => (f x)\u207b\u00b9\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nsimpa only [inv_inv] using h.inv\n[GOAL]\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2075 : Inv G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : MeasurableInv G\nm : MeasurableSpace \u03b1\nf\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nG\u2080 : Type u_3\ninst\u271d\u00b2 : GroupWithZero G\u2080\ninst\u271d\u00b9 : MeasurableSpace G\u2080\ninst\u271d : MeasurableInv G\u2080\nf : \u03b1 \u2192 G\u2080\nh : Measurable fun x => (f x)\u207b\u00b9\n\u22a2 Measurable f\n[PROOFSTEP]\nsimpa only [inv_inv] using h.inv\n[GOAL]\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2075 : Inv G\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : MeasurableInv G\nm : MeasurableSpace \u03b1\nf\u271d : \u03b1 \u2192 G\n\u03bc : Measure \u03b1\nG\u2080 : Type u_3\ninst\u271d\u00b2 : GroupWithZero G\u2080\ninst\u271d\u00b9 : MeasurableSpace G\u2080\ninst\u271d : MeasurableInv G\u2080\nf : \u03b1 \u2192 G\u2080\nh : AEMeasurable fun x => (f x)\u207b\u00b9\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nsimpa only [inv_inv] using h.inv\n[GOAL]\nG : Type u\ninst\u271d\u00b3 : DivInvMonoid G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableMul\u2082 G\ninst\u271d : MeasurableInv G\nn : \u2124\n\u22a2 Measurable fun x => (x, n).fst ^ (x, n).snd\n[PROOFSTEP]\ncases' n with n n\n[GOAL]\ncase ofNat\nG : Type u\ninst\u271d\u00b3 : DivInvMonoid G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableMul\u2082 G\ninst\u271d : MeasurableInv G\nn : \u2115\n\u22a2 Measurable fun x => (x, Int.ofNat n).fst ^ (x, Int.ofNat n).snd\n[PROOFSTEP]\nsimp_rw [Int.ofNat_eq_coe, zpow_ofNat]\n[GOAL]\ncase ofNat\nG : Type u\ninst\u271d\u00b3 : DivInvMonoid G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableMul\u2082 G\ninst\u271d : MeasurableInv G\nn : \u2115\n\u22a2 Measurable fun x => x ^ n\n[PROOFSTEP]\nexact measurable_id.pow_const _\n[GOAL]\ncase negSucc\nG : Type u\ninst\u271d\u00b3 : DivInvMonoid G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableMul\u2082 G\ninst\u271d : MeasurableInv G\nn : \u2115\n\u22a2 Measurable fun x => (x, Int.negSucc n).fst ^ (x, Int.negSucc n).snd\n[PROOFSTEP]\nsimp_rw [zpow_negSucc]\n[GOAL]\ncase negSucc\nG : Type u\ninst\u271d\u00b3 : DivInvMonoid G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableMul\u2082 G\ninst\u271d : MeasurableInv G\nn : \u2115\n\u22a2 Measurable fun x => (x ^ (n + 1))\u207b\u00b9\n[PROOFSTEP]\nexact (measurable_id.pow_const (n + 1)).inv\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : MeasurableMul\u2082 G\ninst\u271d : MeasurableInv G\n\u22a2 Measurable fun p => p.fst / p.snd\n[PROOFSTEP]\nsimp only [div_eq_mul_inv]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : MeasurableMul\u2082 G\ninst\u271d : MeasurableInv G\n\u22a2 Measurable fun p => p.fst * p.snd\u207b\u00b9\n[PROOFSTEP]\nexact measurable_fst.mul measurable_snd.inv\n[GOAL]\nM : Type ?u.442912\n\u03b1 : Type ?u.442915\ninst\u271d\u2074 : MeasurableSpace M\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MulAction M \u03b1\ninst\u271d : MeasurableSMul M \u03b1\ns : Submonoid M\nc : { x // x \u2208 s }\n\u22a2 Measurable ((fun x x_1 => x \u2022 x_1) c)\n[PROOFSTEP]\nsimpa only using measurable_const_smul (c : M)\n[GOAL]\nM\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2076 : MeasurableSpace M\u271d\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : _root_.SMul M\u271d \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 M\u271d\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : MeasurableSMul M\u271d \u03b2\n\u03bc : Measure \u03b1\nM : Type u_4\ninst\u271d\u00b2 : AddMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableAdd\u2082 M\n\u22a2 Measurable (Function.uncurry fun x x_1 => x \u2022 x_1)\n[PROOFSTEP]\nsuffices Measurable fun p : M \u00d7 \u2115 => p.2 \u2022 p.1 by apply this.comp measurable_swap\n[GOAL]\nM\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2076 : MeasurableSpace M\u271d\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : _root_.SMul M\u271d \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 M\u271d\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : MeasurableSMul M\u271d \u03b2\n\u03bc : Measure \u03b1\nM : Type u_4\ninst\u271d\u00b2 : AddMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableAdd\u2082 M\nthis : Measurable fun p => p.snd \u2022 p.fst\n\u22a2 Measurable (Function.uncurry fun x x_1 => x \u2022 x_1)\n[PROOFSTEP]\napply this.comp measurable_swap\n[GOAL]\nM\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2076 : MeasurableSpace M\u271d\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : _root_.SMul M\u271d \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 M\u271d\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : MeasurableSMul M\u271d \u03b2\n\u03bc : Measure \u03b1\nM : Type u_4\ninst\u271d\u00b2 : AddMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableAdd\u2082 M\n\u22a2 Measurable fun p => p.snd \u2022 p.fst\n[PROOFSTEP]\nrefine' measurable_from_prod_countable fun n => _\n[GOAL]\nM\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2076 : MeasurableSpace M\u271d\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : _root_.SMul M\u271d \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 M\u271d\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : MeasurableSMul M\u271d \u03b2\n\u03bc : Measure \u03b1\nM : Type u_4\ninst\u271d\u00b2 : AddMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableAdd\u2082 M\nn : \u2115\n\u22a2 Measurable fun x => (x, n).snd \u2022 (x, n).fst\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nM\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2076 : MeasurableSpace M\u271d\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : _root_.SMul M\u271d \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 M\u271d\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : MeasurableSMul M\u271d \u03b2\n\u03bc : Measure \u03b1\nM : Type u_4\ninst\u271d\u00b2 : AddMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableAdd\u2082 M\n\u22a2 Measurable fun x => (x, Nat.zero).snd \u2022 (x, Nat.zero).fst\n[PROOFSTEP]\nsimp only [Nat.zero_eq, zero_smul, \u2190 Pi.zero_def, measurable_zero]\n[GOAL]\ncase succ\nM\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2076 : MeasurableSpace M\u271d\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : _root_.SMul M\u271d \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 M\u271d\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : MeasurableSMul M\u271d \u03b2\n\u03bc : Measure \u03b1\nM : Type u_4\ninst\u271d\u00b2 : AddMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableAdd\u2082 M\nn : \u2115\nih : Measurable fun x => (x, n).snd \u2022 (x, n).fst\n\u22a2 Measurable fun x => (x, Nat.succ n).snd \u2022 (x, Nat.succ n).fst\n[PROOFSTEP]\nsimp only [succ_nsmul]\n[GOAL]\ncase succ\nM\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2076 : MeasurableSpace M\u271d\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : _root_.SMul M\u271d \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 M\u271d\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : MeasurableSMul M\u271d \u03b2\n\u03bc : Measure \u03b1\nM : Type u_4\ninst\u271d\u00b2 : AddMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableAdd\u2082 M\nn : \u2115\nih : Measurable fun x => (x, n).snd \u2022 (x, n).fst\n\u22a2 Measurable fun x => x + n \u2022 x\n[PROOFSTEP]\nexact measurable_id.add ih\n[GOAL]\nM\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2077 : MeasurableSpace M\u271d\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : SMul M\u271d \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 M\u271d\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSMul M\u271d \u03b2\n\u03bc : Measure \u03b1\nM : Type u_4\ninst\u271d\u00b3 : SubNegMonoid M\ninst\u271d\u00b2 : MeasurableSpace M\ninst\u271d\u00b9 : MeasurableAdd\u2082 M\ninst\u271d : MeasurableNeg M\n\u22a2 Measurable (Function.uncurry fun x x_1 => x \u2022 x_1)\n[PROOFSTEP]\nsuffices Measurable fun p : M \u00d7 \u2124 => p.2 \u2022 p.1 by apply this.comp measurable_swap\n[GOAL]\nM\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2077 : MeasurableSpace M\u271d\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : SMul M\u271d \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 M\u271d\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSMul M\u271d \u03b2\n\u03bc : Measure \u03b1\nM : Type u_4\ninst\u271d\u00b3 : SubNegMonoid M\ninst\u271d\u00b2 : MeasurableSpace M\ninst\u271d\u00b9 : MeasurableAdd\u2082 M\ninst\u271d : MeasurableNeg M\nthis : Measurable fun p => p.snd \u2022 p.fst\n\u22a2 Measurable (Function.uncurry fun x x_1 => x \u2022 x_1)\n[PROOFSTEP]\napply this.comp measurable_swap\n[GOAL]\nM\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2077 : MeasurableSpace M\u271d\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : SMul M\u271d \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 M\u271d\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSMul M\u271d \u03b2\n\u03bc : Measure \u03b1\nM : Type u_4\ninst\u271d\u00b3 : SubNegMonoid M\ninst\u271d\u00b2 : MeasurableSpace M\ninst\u271d\u00b9 : MeasurableAdd\u2082 M\ninst\u271d : MeasurableNeg M\n\u22a2 Measurable fun p => p.snd \u2022 p.fst\n[PROOFSTEP]\nrefine' measurable_from_prod_countable fun n => _\n[GOAL]\nM\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2077 : MeasurableSpace M\u271d\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : SMul M\u271d \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 M\u271d\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSMul M\u271d \u03b2\n\u03bc : Measure \u03b1\nM : Type u_4\ninst\u271d\u00b3 : SubNegMonoid M\ninst\u271d\u00b2 : MeasurableSpace M\ninst\u271d\u00b9 : MeasurableAdd\u2082 M\ninst\u271d : MeasurableNeg M\nn : \u2124\n\u22a2 Measurable fun x => (x, n).snd \u2022 (x, n).fst\n[PROOFSTEP]\ninduction' n with n n ih\n[GOAL]\ncase ofNat\nM\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2077 : MeasurableSpace M\u271d\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : SMul M\u271d \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 M\u271d\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSMul M\u271d \u03b2\n\u03bc : Measure \u03b1\nM : Type u_4\ninst\u271d\u00b3 : SubNegMonoid M\ninst\u271d\u00b2 : MeasurableSpace M\ninst\u271d\u00b9 : MeasurableAdd\u2082 M\ninst\u271d : MeasurableNeg M\nn : \u2115\n\u22a2 Measurable fun x => (x, Int.ofNat n).snd \u2022 (x, Int.ofNat n).fst\n[PROOFSTEP]\nsimp only [Int.ofNat_eq_coe, ofNat_zsmul]\n[GOAL]\ncase ofNat\nM\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2077 : MeasurableSpace M\u271d\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : SMul M\u271d \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 M\u271d\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSMul M\u271d \u03b2\n\u03bc : Measure \u03b1\nM : Type u_4\ninst\u271d\u00b3 : SubNegMonoid M\ninst\u271d\u00b2 : MeasurableSpace M\ninst\u271d\u00b9 : MeasurableAdd\u2082 M\ninst\u271d : MeasurableNeg M\nn : \u2115\n\u22a2 Measurable fun x => n \u2022 x\n[PROOFSTEP]\nexact measurable_const_smul _\n[GOAL]\ncase negSucc\nM\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2077 : MeasurableSpace M\u271d\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : SMul M\u271d \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 M\u271d\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSMul M\u271d \u03b2\n\u03bc : Measure \u03b1\nM : Type u_4\ninst\u271d\u00b3 : SubNegMonoid M\ninst\u271d\u00b2 : MeasurableSpace M\ninst\u271d\u00b9 : MeasurableAdd\u2082 M\ninst\u271d : MeasurableNeg M\nn : \u2115\n\u22a2 Measurable fun x => (x, Int.negSucc n).snd \u2022 (x, Int.negSucc n).fst\n[PROOFSTEP]\nsimp only [negSucc_zsmul]\n[GOAL]\ncase negSucc\nM\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2077 : MeasurableSpace M\u271d\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : SMul M\u271d \u03b2\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 M\u271d\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u2074 : MeasurableSMul M\u271d \u03b2\n\u03bc : Measure \u03b1\nM : Type u_4\ninst\u271d\u00b3 : SubNegMonoid M\ninst\u271d\u00b2 : MeasurableSpace M\ninst\u271d\u00b9 : MeasurableAdd\u2082 M\ninst\u271d : MeasurableNeg M\nn : \u2115\n\u22a2 Measurable fun x => -((n + 1) \u2022 x)\n[PROOFSTEP]\nexact (measurable_const_smul _).neg\n[GOAL]\nM : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2079 : MeasurableSpace M\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : Monoid M\ninst\u271d\u2076 : MulAction M \u03b2\ninst\u271d\u2075 : MeasurableSMul M \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\nG : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MulAction G \u03b2\ninst\u271d : MeasurableSMul G \u03b2\nc : G\nh : Measurable fun x => c \u2022 f x\n\u22a2 Measurable f\n[PROOFSTEP]\nsimpa only [inv_smul_smul] using h.const_smul' c\u207b\u00b9\n[GOAL]\nM : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2079 : MeasurableSpace M\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : Monoid M\ninst\u271d\u2076 : MulAction M \u03b2\ninst\u271d\u2075 : MeasurableSMul M \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\nG : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MulAction G \u03b2\ninst\u271d : MeasurableSMul G \u03b2\nc : G\nh : AEMeasurable fun x => c \u2022 f x\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nsimpa only [inv_smul_smul] using h.const_smul' c\u207b\u00b9\n[GOAL]\nM : Type ?u.567803\n\u03b1 : Type ?u.567806\ninst\u271d\u2075 : MeasurableSpace M\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : SMul M \u03b1\ninst\u271d\u00b2 : SMul M\u1d50\u1d52\u1d56 \u03b1\ninst\u271d\u00b9 : IsCentralScalar M \u03b1\ninst\u271d : MeasurableSMul M \u03b1\nc : M\n\u22a2 Measurable fun x => MulOpposite.op c \u2022 x\n[PROOFSTEP]\nsimpa only [op_smul_eq_smul] using measurable_const_smul c\n[GOAL]\nM : Type ?u.567803\n\u03b1 : Type ?u.567806\ninst\u271d\u2075 : MeasurableSpace M\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : SMul M \u03b1\ninst\u271d\u00b2 : SMul M\u1d50\u1d52\u1d56 \u03b1\ninst\u271d\u00b9 : IsCentralScalar M \u03b1\ninst\u271d : MeasurableSMul M \u03b1\nx : \u03b1\n\u22a2 Measurable fun c => MulOpposite.op (unop c) \u2022 x\n[PROOFSTEP]\nsimpa only [op_smul_eq_smul] using (measurable_smul_const x).comp measurable_mul_unop\n[GOAL]\nM : Type ?u.570007\n\u03b1 : Type ?u.570010\ninst\u271d\u2075 : MeasurableSpace M\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : SMul M \u03b1\ninst\u271d\u00b2 : SMul M\u1d50\u1d52\u1d56 \u03b1\ninst\u271d\u00b9 : IsCentralScalar M \u03b1\ninst\u271d : MeasurableSMul\u2082 M \u03b1\n\u22a2 Measurable fun x => MulOpposite.op (unop x.fst) \u2022 x.snd\n[PROOFSTEP]\nsimp_rw [op_smul_eq_smul]\n[GOAL]\nM : Type ?u.570007\n\u03b1 : Type ?u.570010\ninst\u271d\u2075 : MeasurableSpace M\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : SMul M \u03b1\ninst\u271d\u00b2 : SMul M\u1d50\u1d52\u1d56 \u03b1\ninst\u271d\u00b9 : IsCentralScalar M \u03b1\ninst\u271d : MeasurableSMul\u2082 M \u03b1\n\u22a2 Measurable fun x => unop x.fst \u2022 x.snd\n[PROOFSTEP]\nrefine' (measurable_mul_unop.comp measurable_fst).smul measurable_snd\n[GOAL]\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl : List (\u03b1 \u2192 M)\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 Measurable f\n\u22a2 Measurable (prod l)\n[PROOFSTEP]\ninduction' l with f l ihl\n[GOAL]\ncase nil\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 Measurable f\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 [] \u2192 Measurable f\n\u22a2 Measurable (prod [])\n[PROOFSTEP]\nexact measurable_one\n[GOAL]\ncase cons\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 Measurable f\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 Measurable f) \u2192 Measurable (prod l)\nhl : \u2200 (f_1 : \u03b1 \u2192 M), f_1 \u2208 f :: l \u2192 Measurable f_1\n\u22a2 Measurable (prod (f :: l))\n[PROOFSTEP]\nrw [List.forall_mem_cons] at hl \n[GOAL]\ncase cons\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 Measurable f\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 Measurable f) \u2192 Measurable (prod l)\nhl : Measurable f \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 Measurable x\n\u22a2 Measurable (prod (f :: l))\n[PROOFSTEP]\nrw [List.prod_cons]\n[GOAL]\ncase cons\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 Measurable f\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 Measurable f) \u2192 Measurable (prod l)\nhl : Measurable f \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 Measurable x\n\u22a2 Measurable (f * prod l)\n[PROOFSTEP]\nexact hl.1.mul (ihl hl.2)\n[GOAL]\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl : List (\u03b1 \u2192 M)\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f\n\u22a2 AEMeasurable (prod l)\n[PROOFSTEP]\ninduction' l with f l ihl\n[GOAL]\ncase nil\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 [] \u2192 AEMeasurable f\n\u22a2 AEMeasurable (prod [])\n[PROOFSTEP]\nexact aemeasurable_one\n[GOAL]\ncase cons\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEMeasurable f\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f) \u2192 AEMeasurable (prod l)\nhl : \u2200 (f_1 : \u03b1 \u2192 M), f_1 \u2208 f :: l \u2192 AEMeasurable f_1\n\u22a2 AEMeasurable (prod (f :: l))\n[PROOFSTEP]\nrw [List.forall_mem_cons] at hl \n[GOAL]\ncase cons\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEMeasurable f\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f) \u2192 AEMeasurable (prod l)\nhl : AEMeasurable f \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 AEMeasurable x\n\u22a2 AEMeasurable (prod (f :: l))\n[PROOFSTEP]\nrw [List.prod_cons]\n[GOAL]\ncase cons\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl\u271d : List (\u03b1 \u2192 M)\nhl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEMeasurable f\nf : \u03b1 \u2192 M\nl : List (\u03b1 \u2192 M)\nihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f) \u2192 AEMeasurable (prod l)\nhl : AEMeasurable f \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 AEMeasurable x\n\u22a2 AEMeasurable (f * prod l)\n[PROOFSTEP]\nexact hl.1.mul (ihl hl.2)\n[GOAL]\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl : List (\u03b1 \u2192 M)\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 Measurable f\n\u22a2 Measurable fun x => prod (map (fun f => f x) l)\n[PROOFSTEP]\nsimpa only [\u2190 Pi.list_prod_apply] using l.measurable_prod' hl\n[GOAL]\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nl : List (\u03b1 \u2192 M)\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f\n\u22a2 AEMeasurable fun x => prod (map (fun f => f x) l)\n[PROOFSTEP]\nsimpa only [\u2190 Pi.list_prod_apply] using l.aemeasurable_prod' hl\n[GOAL]\nM : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 M\nl : Multiset (\u03b1 \u2192 M)\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 Measurable f\n\u22a2 Measurable (prod l)\n[PROOFSTEP]\nrcases l with \u27e8l\u27e9\n[GOAL]\ncase mk\nM : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 M\nl\u271d : Multiset (\u03b1 \u2192 M)\nl : List (\u03b1 \u2192 M)\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 Quot.mk Setoid.r l \u2192 Measurable f\n\u22a2 Measurable (prod (Quot.mk Setoid.r l))\n[PROOFSTEP]\nsimpa using l.measurable_prod' (by simpa using hl)\n[GOAL]\nM : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 M\nl\u271d : Multiset (\u03b1 \u2192 M)\nl : List (\u03b1 \u2192 M)\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 Quot.mk Setoid.r l \u2192 Measurable f\n\u22a2 \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 Measurable f\n[PROOFSTEP]\nsimpa using hl\n[GOAL]\nM : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 M\nl : Multiset (\u03b1 \u2192 M)\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f\n\u22a2 AEMeasurable (prod l)\n[PROOFSTEP]\nrcases l with \u27e8l\u27e9\n[GOAL]\ncase mk\nM : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 M\nl\u271d : Multiset (\u03b1 \u2192 M)\nl : List (\u03b1 \u2192 M)\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 Quot.mk Setoid.r l \u2192 AEMeasurable f\n\u22a2 AEMeasurable (prod (Quot.mk Setoid.r l))\n[PROOFSTEP]\nsimpa using l.aemeasurable_prod' (by simpa using hl)\n[GOAL]\nM : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 M\nl\u271d : Multiset (\u03b1 \u2192 M)\nl : List (\u03b1 \u2192 M)\nhl : \u2200 (f : \u03b1 \u2192 M), f \u2208 Quot.mk Setoid.r l \u2192 AEMeasurable f\n\u22a2 \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEMeasurable f\n[PROOFSTEP]\nsimpa using hl\n[GOAL]\nM : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 M\ns : Multiset (\u03b1 \u2192 M)\nhs : \u2200 (f : \u03b1 \u2192 M), f \u2208 s \u2192 Measurable f\n\u22a2 Measurable fun x => prod (map (fun f => f x) s)\n[PROOFSTEP]\nsimpa only [\u2190 Pi.multiset_prod_apply] using s.measurable_prod' hs\n[GOAL]\nM : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 M\ns : Multiset (\u03b1 \u2192 M)\nhs : \u2200 (f : \u03b1 \u2192 M), f \u2208 s \u2192 AEMeasurable f\n\u22a2 AEMeasurable fun x => prod (map (fun f => f x) s)\n[PROOFSTEP]\nsimpa only [\u2190 Pi.multiset_prod_apply] using s.aemeasurable_prod' hs\n[GOAL]\nM : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 M\ns : Finset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i)\n\u22a2 Measurable fun a => \u220f i in s, f i a\n[PROOFSTEP]\nsimpa only [\u2190 Finset.prod_apply] using s.measurable_prod' hf\n[GOAL]\nM : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : MeasurableSpace M\ninst\u271d : MeasurableMul\u2082 M\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 M\ns : Finset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 AEMeasurable (f i)\n\u22a2 AEMeasurable fun a => \u220f i in s, f i a\n[PROOFSTEP]\nsimpa only [\u2190 Finset.prod_apply] using s.aemeasurable_prod' hf\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Group.Arithmetic", "llama_tokens": 15389, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.4326849617162817}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c3 : Type u_2\nf : \u03c3 \u2192 R\ns : Finset \u03c3\nn : \u2115\n\u22a2 esymm (map f s.val) n = \u2211 t in Finset.powersetLen n s, Finset.prod t f\n[PROOFSTEP]\nsimp only [esymm, powersetLen_map, \u2190 Finset.map_val_val_powersetLen, map_map]\n[GOAL]\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c3 : Type u_2\nf : \u03c3 \u2192 R\ns : Finset \u03c3\nn : \u2115\n\u22a2 sum (map ((fun x => prod x) \u2218 (fun x => map f x) \u2218 Finset.val) (Finset.powersetLen n s).val) =\n \u2211 t in Finset.powersetLen n s, Finset.prod t f\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d : CommSemiring R\na\u271d b\u271d : MvPolynomial \u03c3 R\nha : a\u271d \u2208 setOf IsSymmetric\nhb : b\u271d \u2208 setOf IsSymmetric\ne : Perm \u03c3\n\u22a2 \u2191(rename \u2191e) (a\u271d * b\u271d) = a\u271d * b\u271d\n[PROOFSTEP]\nrw [AlgHom.map_mul, ha, hb]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d : CommSemiring R\na\u271d b\u271d : MvPolynomial \u03c3 R\nha :\n a\u271d \u2208\n {\n toSubsemigroup :=\n { carrier := setOf IsSymmetric,\n mul_mem' :=\n (_ :\n \u2200 {a b : MvPolynomial \u03c3 R},\n a \u2208 setOf IsSymmetric \u2192 b \u2208 setOf IsSymmetric \u2192 \u2200 (e : Perm \u03c3), \u2191(rename \u2191e) (a * b) = a * b) },\n one_mem' := (_ : 1 \u2208 setOf IsSymmetric) }.toSubsemigroup.carrier\nhb :\n b\u271d \u2208\n {\n toSubsemigroup :=\n { carrier := setOf IsSymmetric,\n mul_mem' :=\n (_ :\n \u2200 {a b : MvPolynomial \u03c3 R},\n a \u2208 setOf IsSymmetric \u2192 b \u2208 setOf IsSymmetric \u2192 \u2200 (e : Perm \u03c3), \u2191(rename \u2191e) (a * b) = a * b) },\n one_mem' := (_ : 1 \u2208 setOf IsSymmetric) }.toSubsemigroup.carrier\ne : Perm \u03c3\n\u22a2 \u2191(rename \u2191e) (a\u271d + b\u271d) = a\u271d + b\u271d\n[PROOFSTEP]\nrw [AlgHom.map_add, ha, hb]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nh\u03c6 : IsSymmetric \u03c6\nf : R \u2192+* S\ne : Perm \u03c3\n\u22a2 \u2191(rename \u2191e) (\u2191(MvPolynomial.map f) \u03c6) = \u2191(MvPolynomial.map f) \u03c6\n[PROOFSTEP]\nrw [\u2190 map_rename, h\u03c6]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\n\u22a2 esymm \u03c3 R = Multiset.esymm (Multiset.map X univ.val)\n[PROOFSTEP]\nrefine' funext fun n => (Finset.esymm_map_val X _ n).symm\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : Fintype \u03c4\ninst\u271d : Algebra R S\nf : \u03c3 \u2192 S\nn : \u2115\n\u22a2 \u2191(aeval f) (esymm \u03c3 R n) = Multiset.esymm (Multiset.map f univ.val) n\n[PROOFSTEP]\nsimp_rw [esymm, aeval_sum, aeval_prod, aeval_X, Finset.esymm_map_val]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\nn : \u2115\n\u22a2 esymm \u03c3 R n = \u2211 t in powersetLen n univ, \u2191(monomial (\u2211 i in t, Finsupp.single i 1)) 1\n[PROOFSTEP]\nsimp_rw [monomial_sum_one]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\nn : \u2115\n\u22a2 esymm \u03c3 R n = \u2211 x in powersetLen n univ, \u220f x in x, \u2191(monomial (Finsupp.single x 1)) 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\n\u22a2 esymm \u03c3 R 0 = 1\n[PROOFSTEP]\nsimp only [esymm, powersetLen_zero, sum_singleton, prod_empty]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\nn : \u2115\nf : R \u2192+* S\n\u22a2 \u2191(map f) (esymm \u03c3 R n) = esymm \u03c3 S n\n[PROOFSTEP]\nsimp_rw [esymm, map_sum, map_prod, map_X]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\nn : \u2115\ne : \u03c3 \u2243 \u03c4\n\u22a2 \u2191(rename \u2191e) (esymm \u03c3 R n) = \u2211 x in powersetLen n univ, \u220f i in x, X (\u2191e i)\n[PROOFSTEP]\nsimp_rw [esymm, map_sum, map_prod, rename_X]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\nn : \u2115\ne : \u03c3 \u2243 \u03c4\n\u22a2 \u2211 x in powersetLen n univ, \u220f i in x, X (\u2191e i) =\n \u2211 t in powersetLen n (Finset.map (Equiv.toEmbedding e) univ), \u220f i in t, X i\n[PROOFSTEP]\nsimp [Finset.powersetLen_map, -Finset.map_univ_equiv]\n --Porting note: Why did `mapEmbedding_apply` not work?\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\nn : \u2115\ne : \u03c3 \u2243 \u03c4\n\u22a2 \u2211 x in powersetLen n univ, \u220f x in x, X (\u2191e x) =\n \u2211 x in powersetLen n univ, \u220f i in \u2191(mapEmbedding (Equiv.toEmbedding e)) x, X i\n[PROOFSTEP]\ndsimp [mapEmbedding, OrderEmbedding.ofMapLEIff]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\nn : \u2115\ne : \u03c3 \u2243 \u03c4\n\u22a2 \u2211 x in powersetLen n univ, \u220f x in x, X (\u2191e x) =\n \u2211 x in powersetLen n univ, \u220f i in Finset.map (Equiv.toEmbedding e) x, X i\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\nn : \u2115\ne : \u03c3 \u2243 \u03c4\n\u22a2 \u2211 t in powersetLen n (Finset.map (Equiv.toEmbedding e) univ), \u220f i in t, X i = \u2211 t in powersetLen n univ, \u220f i in t, X i\n[PROOFSTEP]\nrw [Finset.map_univ_equiv]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\nn : \u2115\n\u22a2 IsSymmetric (esymm \u03c3 R n)\n[PROOFSTEP]\nintro\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\nn : \u2115\ne\u271d : Perm \u03c3\n\u22a2 \u2191(rename \u2191e\u271d) (esymm \u03c3 R n) = esymm \u03c3 R n\n[PROOFSTEP]\nrw [rename_esymm]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 support (esymm \u03c3 R n) =\n Finset.biUnion (powersetLen n univ) fun t => (Finsupp.single (\u2211 i in t, Finsupp.single i 1) 1).support\n[PROOFSTEP]\nrw [esymm_eq_sum_monomial]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 support (\u2211 t in powersetLen n univ, \u2191(monomial (\u2211 i in t, Finsupp.single i 1)) 1) =\n Finset.biUnion (powersetLen n univ) fun t => (Finsupp.single (\u2211 i in t, Finsupp.single i 1) 1).support\n[PROOFSTEP]\nsimp only [\u2190 single_eq_monomial]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 support (\u2211 x in powersetLen n univ, Finsupp.single (\u2211 i in x, Finsupp.single i 1) 1) =\n Finset.biUnion (powersetLen n univ) fun t => (Finsupp.single (\u2211 i in t, Finsupp.single i 1) 1).support\n[PROOFSTEP]\nrefine' Finsupp.support_sum_eq_biUnion (powersetLen n (univ : Finset \u03c3)) _\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 \u2200 (i\u2081 i\u2082 : Finset \u03c3),\n i\u2081 \u2260 i\u2082 \u2192\n Disjoint (Finsupp.single (\u2211 i in i\u2081, Finsupp.single i 1) 1).support\n (Finsupp.single (\u2211 i in i\u2082, Finsupp.single i 1) 1).support\n[PROOFSTEP]\nintro s t hst\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\n\u22a2 Disjoint (Finsupp.single (\u2211 i in s, Finsupp.single i 1) 1).support\n (Finsupp.single (\u2211 i in t, Finsupp.single i 1) 1).support\n[PROOFSTEP]\nrw [Finset.disjoint_left, Finsupp.support_single_ne_zero _ one_ne_zero]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\n\u22a2 \u2200 \u2983a : \u03c3 \u2192\u2080 \u2115\u2984, a \u2208 {\u2211 i in s, Finsupp.single i 1} \u2192 \u00aca \u2208 (Finsupp.single (\u2211 i in t, Finsupp.single i 1) 1).support\n[PROOFSTEP]\nrw [Finsupp.support_single_ne_zero _ one_ne_zero]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\n\u22a2 \u2200 \u2983a : \u03c3 \u2192\u2080 \u2115\u2984, a \u2208 {\u2211 i in s, Finsupp.single i 1} \u2192 \u00aca \u2208 {\u2211 i in t, Finsupp.single i 1}\n[PROOFSTEP]\nsimp only [one_ne_zero, mem_singleton, Finsupp.mem_support_iff]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\n\u22a2 \u2200 \u2983a : \u03c3 \u2192\u2080 \u2115\u2984, a = \u2211 i in s, Finsupp.single i 1 \u2192 \u00aca = \u2211 i in t, Finsupp.single i 1\n[PROOFSTEP]\nrintro a h rfl\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\n\u22a2 False\n[PROOFSTEP]\nhave := congr_arg Finsupp.support h\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\n\u22a2 False\n[PROOFSTEP]\nrw [Finsupp.support_sum_eq_biUnion, Finsupp.support_sum_eq_biUnion] at this \n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis\u271d\u00b9 : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nthis\u271d : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = (\u2211 i in s, Finsupp.single i 1).support\nthis : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = Finset.biUnion s fun i => (Finsupp.single i 1).support\n\u22a2 False\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis\u271d : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nthis : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = (\u2211 i in s, Finsupp.single i 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (Finsupp.single i\u2081 1).support (Finsupp.single i\u2082 1).support\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (Finsupp.single i\u2081 1).support (Finsupp.single i\u2082 1).support\n[PROOFSTEP]\nhave hsingle : \u2200 s : Finset \u03c3, \u2200 x : \u03c3, x \u2208 s \u2192 (Finsupp.single x 1).support = { x } :=\n by\n intros _ x _\n rw [Finsupp.support_single_ne_zero x one_ne_zero]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis\u271d\u00b9 : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nthis\u271d : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = (\u2211 i in s, Finsupp.single i 1).support\nthis : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = Finset.biUnion s fun i => (Finsupp.single i 1).support\n\u22a2 \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (Finsupp.single x 1).support = {x}\n[PROOFSTEP]\nintros _ x _\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis\u271d\u00b9 : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nthis\u271d : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = (\u2211 i in s, Finsupp.single i 1).support\nthis : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = Finset.biUnion s fun i => (Finsupp.single i 1).support\ns\u271d : Finset \u03c3\nx : \u03c3\na\u271d : x \u2208 s\u271d\n\u22a2 (Finsupp.single x 1).support = {x}\n[PROOFSTEP]\nrw [Finsupp.support_single_ne_zero x one_ne_zero]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis\u271d\u00b9 : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nthis\u271d : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = (\u2211 i in s, Finsupp.single i 1).support\nthis : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = Finset.biUnion s fun i => (Finsupp.single i 1).support\nhsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (Finsupp.single x 1).support = {x}\n\u22a2 False\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis\u271d : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nthis : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = (\u2211 i in s, Finsupp.single i 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (Finsupp.single i\u2081 1).support (Finsupp.single i\u2082 1).support\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (Finsupp.single i\u2081 1).support (Finsupp.single i\u2082 1).support\n[PROOFSTEP]\nhave hs := biUnion_congr (of_eq_true (eq_self s)) (hsingle s)\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis\u271d\u00b9 : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nthis\u271d : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = (\u2211 i in s, Finsupp.single i 1).support\nthis : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = Finset.biUnion s fun i => (Finsupp.single i 1).support\nhsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (Finsupp.single x 1).support = {x}\nhs : (Finset.biUnion s fun a => (Finsupp.single a 1).support) = Finset.biUnion s fun a => {a}\n\u22a2 False\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis\u271d : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nthis : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = (\u2211 i in s, Finsupp.single i 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (Finsupp.single i\u2081 1).support (Finsupp.single i\u2082 1).support\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (Finsupp.single i\u2081 1).support (Finsupp.single i\u2082 1).support\n[PROOFSTEP]\nhave ht := biUnion_congr (of_eq_true (eq_self t)) (hsingle t)\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis\u271d\u00b9 : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nthis\u271d : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = (\u2211 i in s, Finsupp.single i 1).support\nthis : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = Finset.biUnion s fun i => (Finsupp.single i 1).support\nhsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (Finsupp.single x 1).support = {x}\nhs : (Finset.biUnion s fun a => (Finsupp.single a 1).support) = Finset.biUnion s fun a => {a}\nht : (Finset.biUnion t fun a => (Finsupp.single a 1).support) = Finset.biUnion t fun a => {a}\n\u22a2 False\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis\u271d : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nthis : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = (\u2211 i in s, Finsupp.single i 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (Finsupp.single i\u2081 1).support (Finsupp.single i\u2082 1).support\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (Finsupp.single i\u2081 1).support (Finsupp.single i\u2082 1).support\n[PROOFSTEP]\nrw [hs, ht] at this \n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis\u271d\u00b9 : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nthis\u271d : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = (\u2211 i in s, Finsupp.single i 1).support\nthis : (Finset.biUnion t fun a => {a}) = Finset.biUnion s fun a => {a}\nhsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (Finsupp.single x 1).support = {x}\nhs : (Finset.biUnion s fun a => (Finsupp.single a 1).support) = Finset.biUnion s fun a => {a}\nht : (Finset.biUnion t fun a => (Finsupp.single a 1).support) = Finset.biUnion t fun a => {a}\n\u22a2 False\n[PROOFSTEP]\nsimp only [biUnion_singleton_eq_self] at this \n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis\u271d\u00b9 : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nthis\u271d : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = (\u2211 i in s, Finsupp.single i 1).support\nhsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (Finsupp.single x 1).support = {x}\nhs : (Finset.biUnion s fun a => (Finsupp.single a 1).support) = Finset.biUnion s fun a => {a}\nht : (Finset.biUnion t fun a => (Finsupp.single a 1).support) = Finset.biUnion t fun a => {a}\nthis : t = s\n\u22a2 False\n[PROOFSTEP]\nexact absurd this hst.symm\n[GOAL]\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis\u271d : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nthis : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = (\u2211 i in s, Finsupp.single i 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (Finsupp.single i\u2081 1).support (Finsupp.single i\u2082 1).support\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (Finsupp.single i\u2081 1).support (Finsupp.single i\u2082 1).support\n[PROOFSTEP]\nall_goals intro x y; simp [Finsupp.support_single_disjoint]\n[GOAL]\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis\u271d : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nthis : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = (\u2211 i in s, Finsupp.single i 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (Finsupp.single i\u2081 1).support (Finsupp.single i\u2082 1).support\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis\u271d : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nthis : (Finset.biUnion t fun i => (Finsupp.single i 1).support) = (\u2211 i in s, Finsupp.single i 1).support\nx y : \u03c3\n\u22a2 x \u2260 y \u2192 Disjoint (Finsupp.single x 1).support (Finsupp.single y 1).support\n[PROOFSTEP]\nsimp [Finsupp.support_single_disjoint]\n[GOAL]\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (Finsupp.single i\u2081 1).support (Finsupp.single i\u2082 1).support\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : \u2211 i in t, Finsupp.single i 1 = \u2211 i in s, Finsupp.single i 1\nthis : (\u2211 i in t, Finsupp.single i 1).support = (\u2211 i in s, Finsupp.single i 1).support\nx y : \u03c3\n\u22a2 x \u2260 y \u2192 Disjoint (Finsupp.single x 1).support (Finsupp.single y 1).support\n[PROOFSTEP]\nsimp [Finsupp.support_single_disjoint]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 support (esymm \u03c3 R n) = Finset.biUnion (powersetLen n univ) fun t => {\u2211 i in t, Finsupp.single i 1}\n[PROOFSTEP]\nrw [support_esymm'']\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 (Finset.biUnion (powersetLen n univ) fun t => (Finsupp.single (\u2211 i in t, Finsupp.single i 1) 1).support) =\n Finset.biUnion (powersetLen n univ) fun t => {\u2211 i in t, Finsupp.single i 1}\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_t\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 (fun t => (Finsupp.single (\u2211 i in t, Finsupp.single i 1) 1).support) = fun t => {\u2211 i in t, Finsupp.single i 1}\n[PROOFSTEP]\nfunext\n[GOAL]\ncase e_t.h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\nx\u271d : Finset \u03c3\n\u22a2 (Finsupp.single (\u2211 i in x\u271d, Finsupp.single i 1) 1).support = {\u2211 i in x\u271d, Finsupp.single i 1}\n[PROOFSTEP]\nexact Finsupp.support_single_ne_zero _ one_ne_zero\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 support (esymm \u03c3 R n) = image (fun t => \u2211 i in t, Finsupp.single i 1) (powersetLen n univ)\n[PROOFSTEP]\nrw [support_esymm']\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 (Finset.biUnion (powersetLen n univ) fun t => {\u2211 i in t, Finsupp.single i 1}) =\n image (fun t => \u2211 i in t, Finsupp.single i 1) (powersetLen n univ)\n[PROOFSTEP]\nexact biUnion_singleton\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : Fintype \u03c4\ninst\u271d : Nontrivial R\nn : \u2115\nhpos : 0 < n\nhn : n \u2264 Fintype.card \u03c3\n\u22a2 degrees (esymm \u03c3 R n) = univ.val\n[PROOFSTEP]\nclassical\nhave : (Finsupp.toMultiset \u2218 fun t : Finset \u03c3 => \u2211 i : \u03c3 in t, Finsupp.single i 1) = Finset.val :=\n by\n funext\n simp [Finsupp.toMultiset_sum_single]\nrw [degrees_def, support_esymm, sup_image, this]\nhave : ((powersetLen n univ).sup (fun (x : Finset \u03c3) => x)).val = sup (powersetLen n univ) val :=\n by\n refine' comp_sup_eq_sup_comp _ _ _\n \u00b7 intros\n simp only [union_val, sup_eq_union]\n congr\n \u00b7 rfl\nrw [\u2190 this]\nobtain \u27e8k, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero hpos.ne'\nsimpa using powersetLen_sup _ _ (Nat.lt_of_succ_le hn)\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : Fintype \u03c4\ninst\u271d : Nontrivial R\nn : \u2115\nhpos : 0 < n\nhn : n \u2264 Fintype.card \u03c3\n\u22a2 degrees (esymm \u03c3 R n) = univ.val\n[PROOFSTEP]\nhave : (Finsupp.toMultiset \u2218 fun t : Finset \u03c3 => \u2211 i : \u03c3 in t, Finsupp.single i 1) = Finset.val :=\n by\n funext\n simp [Finsupp.toMultiset_sum_single]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : Fintype \u03c4\ninst\u271d : Nontrivial R\nn : \u2115\nhpos : 0 < n\nhn : n \u2264 Fintype.card \u03c3\n\u22a2 (\u2191Finsupp.toMultiset \u2218 fun t => \u2211 i in t, Finsupp.single i 1) = val\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : Fintype \u03c4\ninst\u271d : Nontrivial R\nn : \u2115\nhpos : 0 < n\nhn : n \u2264 Fintype.card \u03c3\nx\u271d : Finset \u03c3\n\u22a2 (\u2191Finsupp.toMultiset \u2218 fun t => \u2211 i in t, Finsupp.single i 1) x\u271d = x\u271d.val\n[PROOFSTEP]\nsimp [Finsupp.toMultiset_sum_single]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : Fintype \u03c4\ninst\u271d : Nontrivial R\nn : \u2115\nhpos : 0 < n\nhn : n \u2264 Fintype.card \u03c3\nthis : (\u2191Finsupp.toMultiset \u2218 fun t => \u2211 i in t, Finsupp.single i 1) = val\n\u22a2 degrees (esymm \u03c3 R n) = univ.val\n[PROOFSTEP]\nrw [degrees_def, support_esymm, sup_image, this]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : Fintype \u03c4\ninst\u271d : Nontrivial R\nn : \u2115\nhpos : 0 < n\nhn : n \u2264 Fintype.card \u03c3\nthis : (\u2191Finsupp.toMultiset \u2218 fun t => \u2211 i in t, Finsupp.single i 1) = val\n\u22a2 sup (powersetLen n univ) val = univ.val\n[PROOFSTEP]\nhave : ((powersetLen n univ).sup (fun (x : Finset \u03c3) => x)).val = sup (powersetLen n univ) val :=\n by\n refine' comp_sup_eq_sup_comp _ _ _\n \u00b7 intros\n simp only [union_val, sup_eq_union]\n congr\n \u00b7 rfl\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : Fintype \u03c4\ninst\u271d : Nontrivial R\nn : \u2115\nhpos : 0 < n\nhn : n \u2264 Fintype.card \u03c3\nthis : (\u2191Finsupp.toMultiset \u2218 fun t => \u2211 i in t, Finsupp.single i 1) = val\n\u22a2 (sup (powersetLen n univ) fun x => x).val = sup (powersetLen n univ) val\n[PROOFSTEP]\nrefine' comp_sup_eq_sup_comp _ _ _\n[GOAL]\ncase refine'_1\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : Fintype \u03c4\ninst\u271d : Nontrivial R\nn : \u2115\nhpos : 0 < n\nhn : n \u2264 Fintype.card \u03c3\nthis : (\u2191Finsupp.toMultiset \u2218 fun t => \u2211 i in t, Finsupp.single i 1) = val\n\u22a2 \u2200 (x y : Finset \u03c3), (x \u2294 y).val = x.val \u2294 y.val\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : Fintype \u03c4\ninst\u271d : Nontrivial R\nn : \u2115\nhpos : 0 < n\nhn : n \u2264 Fintype.card \u03c3\nthis : (\u2191Finsupp.toMultiset \u2218 fun t => \u2211 i in t, Finsupp.single i 1) = val\nx\u271d y\u271d : Finset \u03c3\n\u22a2 (x\u271d \u2294 y\u271d).val = x\u271d.val \u2294 y\u271d.val\n[PROOFSTEP]\nsimp only [union_val, sup_eq_union]\n[GOAL]\ncase refine'_1\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : Fintype \u03c4\ninst\u271d : Nontrivial R\nn : \u2115\nhpos : 0 < n\nhn : n \u2264 Fintype.card \u03c3\nthis : (\u2191Finsupp.toMultiset \u2218 fun t => \u2211 i in t, Finsupp.single i 1) = val\nx\u271d y\u271d : Finset \u03c3\n\u22a2 x\u271d.val \u222a y\u271d.val = x\u271d.val \u2294 y\u271d.val\n[PROOFSTEP]\ncongr\n[GOAL]\ncase refine'_2\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : Fintype \u03c4\ninst\u271d : Nontrivial R\nn : \u2115\nhpos : 0 < n\nhn : n \u2264 Fintype.card \u03c3\nthis : (\u2191Finsupp.toMultiset \u2218 fun t => \u2211 i in t, Finsupp.single i 1) = val\n\u22a2 \u22a5.val = \u22a5\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : Fintype \u03c4\ninst\u271d : Nontrivial R\nn : \u2115\nhpos : 0 < n\nhn : n \u2264 Fintype.card \u03c3\nthis\u271d : (\u2191Finsupp.toMultiset \u2218 fun t => \u2211 i in t, Finsupp.single i 1) = val\nthis : (sup (powersetLen n univ) fun x => x).val = sup (powersetLen n univ) val\n\u22a2 sup (powersetLen n univ) val = univ.val\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : Fintype \u03c4\ninst\u271d : Nontrivial R\nn : \u2115\nhpos : 0 < n\nhn : n \u2264 Fintype.card \u03c3\nthis\u271d : (\u2191Finsupp.toMultiset \u2218 fun t => \u2211 i in t, Finsupp.single i 1) = val\nthis : (sup (powersetLen n univ) fun x => x).val = sup (powersetLen n univ) val\n\u22a2 (sup (powersetLen n univ) fun x => x).val = univ.val\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero hpos.ne'\n[GOAL]\ncase intro\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : Fintype \u03c4\ninst\u271d : Nontrivial R\nthis\u271d : (\u2191Finsupp.toMultiset \u2218 fun t => \u2211 i in t, Finsupp.single i 1) = val\nk : \u2115\nhpos : 0 < Nat.succ k\nhn : Nat.succ k \u2264 Fintype.card \u03c3\nthis : (sup (powersetLen (Nat.succ k) univ) fun x => x).val = sup (powersetLen (Nat.succ k) univ) val\n\u22a2 (sup (powersetLen (Nat.succ k) univ) fun x => x).val = univ.val\n[PROOFSTEP]\nsimpa using powersetLen_sup _ _ (Nat.lt_of_succ_le hn)\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\n\u22a2 psum \u03c3 R 0 = \u2191(Fintype.card \u03c3)\n[PROOFSTEP]\nsimp only [psum, _root_.pow_zero, \u2190 cast_card]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\n\u22a2 \u2191(card univ) = \u2191(Fintype.card \u03c3)\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\n\u22a2 psum \u03c3 R 1 = \u2211 i : \u03c3, X i\n[PROOFSTEP]\nsimp only [psum, _root_.pow_one]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Fintype \u03c4\nn : \u2115\ne : \u03c3 \u2243 \u03c4\n\u22a2 \u2191(rename \u2191e) (psum \u03c3 R n) = psum \u03c4 R n\n[PROOFSTEP]\nsimp_rw [psum, map_sum, map_pow, rename_X, e.sum_comp (X \u00b7 ^ n)]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.MvPolynomial.Symmetric", "llama_tokens": 16464, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.6370307875894139, "lm_q1q2_score": 0.4326577333761826}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d\u2075 : Ring R\nC : Type u_2\ninst\u271d\u2074 : Category.{?u.7905, u_2} C\ninst\u271d\u00b3 : Abelian C\ninst\u271d\u00b2 : Linear R C\ninst\u271d\u00b9 : EnoughProjectives C\nX Y : C\ninst\u271d : Projective X\nn : \u2115\nE : ((Functor.leftDerived ((linearYoneda R C).obj Y).rightOp (n + 1)).obj X).unop \u2245 0.unop :=\n (Iso.unop (Functor.leftDerivedObjProjectiveSucc ((linearYoneda R C).obj Y).rightOp n X)).symm\n\u22a2 0 \u226b 0 = \ud835\udfd9 0.unop\n[PROOFSTEP]\nlet Z : (ModuleCat R)\u1d52\u1d56 := 0\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : Ring R\nC : Type u_2\ninst\u271d\u2074 : Category.{?u.7905, u_2} C\ninst\u271d\u00b3 : Abelian C\ninst\u271d\u00b2 : Linear R C\ninst\u271d\u00b9 : EnoughProjectives C\nX Y : C\ninst\u271d : Projective X\nn : \u2115\nE : ((Functor.leftDerived ((linearYoneda R C).obj Y).rightOp (n + 1)).obj X).unop \u2245 0.unop :=\n (Iso.unop (Functor.leftDerivedObjProjectiveSucc ((linearYoneda R C).obj Y).rightOp n X)).symm\nZ : (ModuleCat R)\u1d52\u1d56 := 0\n\u22a2 0 \u226b 0 = \ud835\udfd9 0.unop\n[PROOFSTEP]\nrw [\u2190 (0 : 0 \u27f6 Z.unop).unop_op, \u2190 (0 : Z.unop \u27f6 0).unop_op, \u2190 unop_id, \u2190 unop_comp]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : Ring R\nC : Type u_2\ninst\u271d\u2074 : Category.{?u.10674, u_2} C\ninst\u271d\u00b3 : Abelian C\ninst\u271d\u00b2 : Linear R C\ninst\u271d\u00b9 : EnoughProjectives C\nX Y : C\ninst\u271d : Projective X\nn : \u2115\nE : ((Functor.leftDerived ((linearYoneda R C).obj Y).rightOp (n + 1)).obj X).unop \u2245 0.unop :=\n (Iso.unop (Functor.leftDerivedObjProjectiveSucc ((linearYoneda R C).obj Y).rightOp n X)).symm\nZ : (ModuleCat R)\u1d52\u1d56 := 0\n\u22a2 (0.op \u226b 0.op).unop = (\ud835\udfd9 0).unop\n[PROOFSTEP]\naesop\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Abelian.Ext", "llama_tokens": 747, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929002541068, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.43248302606972683}} {"text": "[GOAL]\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\n\u22a2 FunctionField Fq F \u2194 FiniteDimensional Fqt F\n[PROOFSTEP]\nlet e := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\n[GOAL]\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\n\u22a2 FunctionField Fq F \u2194 FiniteDimensional Fqt F\n[PROOFSTEP]\nhave : \u2200 (c) (x : F), e c \u2022 x = c \u2022 x := by\n intro c x\n rw [Algebra.smul_def, Algebra.smul_def]\n congr\n refine' congr_fun (f := fun c => algebraMap Fqt F (e c)) _ c\n refine' IsLocalization.ext (nonZeroDivisors Fq[X]) _ _ _ _ _ _ _ <;> intros <;>\n simp only [AlgEquiv.map_one, RingHom.map_one, AlgEquiv.map_mul, RingHom.map_mul, AlgEquiv.commutes, \u2190\n IsScalarTower.algebraMap_apply]\n[GOAL]\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\n\u22a2 \u2200 (c : RatFunc Fq) (x : F), \u2191e c \u2022 x = c \u2022 x\n[PROOFSTEP]\nintro c x\n[GOAL]\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nc : RatFunc Fq\nx : F\n\u22a2 \u2191e c \u2022 x = c \u2022 x\n[PROOFSTEP]\nrw [Algebra.smul_def, Algebra.smul_def]\n[GOAL]\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nc : RatFunc Fq\nx : F\n\u22a2 \u2191(algebraMap ((fun x => Fqt) c) F) (\u2191e c) * x = \u2191(algebraMap (RatFunc Fq) F) c * x\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nc : RatFunc Fq\nx : F\n\u22a2 \u2191(algebraMap ((fun x => Fqt) c) F) (\u2191e c) = \u2191(algebraMap (RatFunc Fq) F) c\n[PROOFSTEP]\nrefine' congr_fun (f := fun c => algebraMap Fqt F (e c)) _ c\n[GOAL]\ncase e_a\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nc : RatFunc Fq\nx : F\n\u22a2 (fun c => \u2191(algebraMap Fqt F) (\u2191e c)) = \u2191(algebraMap (RatFunc Fq) F)\n[PROOFSTEP]\nrefine' IsLocalization.ext (nonZeroDivisors Fq[X]) _ _ _ _ _ _ _\n[GOAL]\ncase e_a.refine'_1\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nc : RatFunc Fq\nx : F\n\u22a2 \u2191(algebraMap Fqt F) (\u2191e 1) = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase e_a.refine'_2\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nc : RatFunc Fq\nx : F\n\u22a2 \u2191(algebraMap (RatFunc Fq) F) 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase e_a.refine'_3\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nc : RatFunc Fq\nx : F\n\u22a2 \u2200 (a b : RatFunc Fq), \u2191(algebraMap Fqt F) (\u2191e (a * b)) = \u2191(algebraMap Fqt F) (\u2191e a) * \u2191(algebraMap Fqt F) (\u2191e b)\n[PROOFSTEP]\nintros\n[GOAL]\ncase e_a.refine'_4\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nc : RatFunc Fq\nx : F\n\u22a2 \u2200 (a b : RatFunc Fq),\n \u2191(algebraMap (RatFunc Fq) F) (a * b) = \u2191(algebraMap (RatFunc Fq) F) a * \u2191(algebraMap (RatFunc Fq) F) b\n[PROOFSTEP]\nintros\n[GOAL]\ncase e_a.refine'_5\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nc : RatFunc Fq\nx : F\n\u22a2 \u2200 (a : Fq[X]),\n \u2191(algebraMap Fqt F) (\u2191e (\u2191(algebraMap Fq[X] (RatFunc Fq)) a)) =\n \u2191(algebraMap (RatFunc Fq) F) (\u2191(algebraMap Fq[X] (RatFunc Fq)) a)\n[PROOFSTEP]\nintros\n[GOAL]\ncase e_a.refine'_1\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nc : RatFunc Fq\nx : F\n\u22a2 \u2191(algebraMap Fqt F) (\u2191e 1) = 1\n[PROOFSTEP]\nsimp only [AlgEquiv.map_one, RingHom.map_one, AlgEquiv.map_mul, RingHom.map_mul, AlgEquiv.commutes, \u2190\n IsScalarTower.algebraMap_apply]\n[GOAL]\ncase e_a.refine'_2\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nc : RatFunc Fq\nx : F\n\u22a2 \u2191(algebraMap (RatFunc Fq) F) 1 = 1\n[PROOFSTEP]\nsimp only [AlgEquiv.map_one, RingHom.map_one, AlgEquiv.map_mul, RingHom.map_mul, AlgEquiv.commutes, \u2190\n IsScalarTower.algebraMap_apply]\n[GOAL]\ncase e_a.refine'_3\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nc : RatFunc Fq\nx : F\na\u271d b\u271d : RatFunc Fq\n\u22a2 \u2191(algebraMap Fqt F) (\u2191e (a\u271d * b\u271d)) = \u2191(algebraMap Fqt F) (\u2191e a\u271d) * \u2191(algebraMap Fqt F) (\u2191e b\u271d)\n[PROOFSTEP]\nsimp only [AlgEquiv.map_one, RingHom.map_one, AlgEquiv.map_mul, RingHom.map_mul, AlgEquiv.commutes, \u2190\n IsScalarTower.algebraMap_apply]\n[GOAL]\ncase e_a.refine'_4\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nc : RatFunc Fq\nx : F\na\u271d b\u271d : RatFunc Fq\n\u22a2 \u2191(algebraMap (RatFunc Fq) F) (a\u271d * b\u271d) = \u2191(algebraMap (RatFunc Fq) F) a\u271d * \u2191(algebraMap (RatFunc Fq) F) b\u271d\n[PROOFSTEP]\nsimp only [AlgEquiv.map_one, RingHom.map_one, AlgEquiv.map_mul, RingHom.map_mul, AlgEquiv.commutes, \u2190\n IsScalarTower.algebraMap_apply]\n[GOAL]\ncase e_a.refine'_5\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nc : RatFunc Fq\nx : F\na\u271d : Fq[X]\n\u22a2 \u2191(algebraMap Fqt F) (\u2191e (\u2191(algebraMap Fq[X] (RatFunc Fq)) a\u271d)) =\n \u2191(algebraMap (RatFunc Fq) F) (\u2191(algebraMap Fq[X] (RatFunc Fq)) a\u271d)\n[PROOFSTEP]\nsimp only [AlgEquiv.map_one, RingHom.map_one, AlgEquiv.map_mul, RingHom.map_mul, AlgEquiv.commutes, \u2190\n IsScalarTower.algebraMap_apply]\n[GOAL]\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nthis : \u2200 (c : RatFunc Fq) (x : F), \u2191e c \u2022 x = c \u2022 x\n\u22a2 FunctionField Fq F \u2194 FiniteDimensional Fqt F\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nthis : \u2200 (c : RatFunc Fq) (x : F), \u2191e c \u2022 x = c \u2022 x\n\u22a2 FunctionField Fq F \u2192 FiniteDimensional Fqt F\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nthis : \u2200 (c : RatFunc Fq) (x : F), \u2191e c \u2022 x = c \u2022 x\n\u22a2 FiniteDimensional Fqt F \u2192 FunctionField Fq F\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nthis : \u2200 (c : RatFunc Fq) (x : F), \u2191e c \u2022 x = c \u2022 x\nh : FunctionField Fq F\n\u22a2 FiniteDimensional Fqt F\n[PROOFSTEP]\nlet b := FiniteDimensional.finBasis (RatFunc Fq) F\n[GOAL]\ncase mp\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nthis : \u2200 (c : RatFunc Fq) (x : F), \u2191e c \u2022 x = c \u2022 x\nh : FunctionField Fq F\nb : Basis (Fin (FiniteDimensional.finrank (RatFunc Fq) F)) (RatFunc Fq) F := FiniteDimensional.finBasis (RatFunc Fq) F\n\u22a2 FiniteDimensional Fqt F\n[PROOFSTEP]\nexact FiniteDimensional.of_fintype_basis (b.mapCoeffs e this)\n[GOAL]\ncase mpr\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nthis : \u2200 (c : RatFunc Fq) (x : F), \u2191e c \u2022 x = c \u2022 x\nh : FiniteDimensional Fqt F\n\u22a2 FunctionField Fq F\n[PROOFSTEP]\nlet b := FiniteDimensional.finBasis Fqt F\n[GOAL]\ncase mpr\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nthis : \u2200 (c : RatFunc Fq) (x : F), \u2191e c \u2022 x = c \u2022 x\nh : FiniteDimensional Fqt F\nb : Basis (Fin (FiniteDimensional.finrank Fqt F)) Fqt F := FiniteDimensional.finBasis Fqt F\n\u22a2 FunctionField Fq F\n[PROOFSTEP]\nrefine' FiniteDimensional.of_fintype_basis (b.mapCoeffs e.symm _)\n[GOAL]\ncase mpr\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nthis : \u2200 (c : RatFunc Fq) (x : F), \u2191e c \u2022 x = c \u2022 x\nh : FiniteDimensional Fqt F\nb : Basis (Fin (FiniteDimensional.finrank Fqt F)) Fqt F := FiniteDimensional.finBasis Fqt F\n\u22a2 \u2200 (c : Fqt) (x : F), \u2191\u2191(AlgEquiv.symm e) c \u2022 x = c \u2022 x\n[PROOFSTEP]\nintro c x\n[GOAL]\ncase mpr\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nthis : \u2200 (c : RatFunc Fq) (x : F), \u2191e c \u2022 x = c \u2022 x\nh : FiniteDimensional Fqt F\nb : Basis (Fin (FiniteDimensional.finrank Fqt F)) Fqt F := FiniteDimensional.finBasis Fqt F\nc : Fqt\nx : F\n\u22a2 \u2191\u2191(AlgEquiv.symm e) c \u2022 x = c \u2022 x\n[PROOFSTEP]\nconvert (this (e.symm c) x).symm\n[GOAL]\ncase h.e'_3.h.e'_5\nFq F : Type\ninst\u271d\u2079 : Field Fq\ninst\u271d\u2078 : Field F\nFqt : Type u_1\ninst\u271d\u2077 : Field Fqt\ninst\u271d\u2076 : Algebra Fq[X] Fqt\ninst\u271d\u2075 : IsFractionRing Fq[X] Fqt\ninst\u271d\u2074 : Algebra (RatFunc Fq) F\ninst\u271d\u00b3 : Algebra Fqt F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : IsScalarTower Fq[X] Fqt F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\ne : RatFunc Fq \u2243\u2090[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]\u2070 (RatFunc Fq) Fqt\nthis : \u2200 (c : RatFunc Fq) (x : F), \u2191e c \u2022 x = c \u2022 x\nh : FiniteDimensional Fqt F\nb : Basis (Fin (FiniteDimensional.finrank Fqt F)) Fqt F := FiniteDimensional.finBasis Fqt F\nc : Fqt\nx : F\n\u22a2 c = \u2191e (\u2191(AlgEquiv.symm e) c)\n[PROOFSTEP]\nsimp only [e.apply_symm_apply]\n[GOAL]\nFq F : Type\ninst\u271d\u2074 : Field Fq\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : Algebra (RatFunc Fq) F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\n\u22a2 Function.Injective \u2191(algebraMap Fq[X] F)\n[PROOFSTEP]\nrw [IsScalarTower.algebraMap_eq Fq[X] (RatFunc Fq) F]\n[GOAL]\nFq F : Type\ninst\u271d\u2074 : Field Fq\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : Algebra (RatFunc Fq) F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\n\u22a2 Function.Injective \u2191(RingHom.comp (algebraMap (RatFunc Fq) F) (algebraMap Fq[X] (RatFunc Fq)))\n[PROOFSTEP]\nexact (algebraMap (RatFunc Fq) F).injective.comp (IsFractionRing.injective Fq[X] (RatFunc Fq))\n[GOAL]\nFq F : Type\ninst\u271d\u2074 : Field Fq\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : Algebra (RatFunc Fq) F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\n\u22a2 Function.Injective \u2191(algebraMap Fq[X] { x // x \u2208 ringOfIntegers Fq F })\n[PROOFSTEP]\nhave hinj : Function.Injective (\u21d1(algebraMap Fq[X] F)) :=\n by\n rw [IsScalarTower.algebraMap_eq Fq[X] (RatFunc Fq) F]\n exact (algebraMap (RatFunc Fq) F).injective.comp (IsFractionRing.injective Fq[X] (RatFunc Fq))\n[GOAL]\nFq F : Type\ninst\u271d\u2074 : Field Fq\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : Algebra (RatFunc Fq) F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\n\u22a2 Function.Injective \u2191(algebraMap Fq[X] F)\n[PROOFSTEP]\nrw [IsScalarTower.algebraMap_eq Fq[X] (RatFunc Fq) F]\n[GOAL]\nFq F : Type\ninst\u271d\u2074 : Field Fq\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : Algebra (RatFunc Fq) F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\n\u22a2 Function.Injective \u2191(RingHom.comp (algebraMap (RatFunc Fq) F) (algebraMap Fq[X] (RatFunc Fq)))\n[PROOFSTEP]\nexact (algebraMap (RatFunc Fq) F).injective.comp (IsFractionRing.injective Fq[X] (RatFunc Fq))\n[GOAL]\nFq F : Type\ninst\u271d\u2074 : Field Fq\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : Algebra (RatFunc Fq) F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\nhinj : Function.Injective \u2191(algebraMap Fq[X] F)\n\u22a2 Function.Injective \u2191(algebraMap Fq[X] { x // x \u2208 ringOfIntegers Fq F })\n[PROOFSTEP]\nrw [injective_iff_map_eq_zero (algebraMap Fq[X] (\u21a5(ringOfIntegers Fq F)))]\n[GOAL]\nFq F : Type\ninst\u271d\u2074 : Field Fq\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : Algebra (RatFunc Fq) F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\nhinj : Function.Injective \u2191(algebraMap Fq[X] F)\n\u22a2 \u2200 (a : Fq[X]), \u2191(algebraMap Fq[X] { x // x \u2208 ringOfIntegers Fq F }) a = 0 \u2192 a = 0\n[PROOFSTEP]\nintro p hp\n[GOAL]\nFq F : Type\ninst\u271d\u2074 : Field Fq\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : Algebra (RatFunc Fq) F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\nhinj : Function.Injective \u2191(algebraMap Fq[X] F)\np : Fq[X]\nhp : \u2191(algebraMap Fq[X] { x // x \u2208 ringOfIntegers Fq F }) p = 0\n\u22a2 p = 0\n[PROOFSTEP]\nrw [\u2190 Subtype.coe_inj, Subalgebra.coe_zero] at hp \n[GOAL]\nFq F : Type\ninst\u271d\u2074 : Field Fq\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : Algebra (RatFunc Fq) F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\nhinj : Function.Injective \u2191(algebraMap Fq[X] F)\np : Fq[X]\nhp : \u2191(\u2191(algebraMap Fq[X] { x // x \u2208 ringOfIntegers Fq F }) p) = 0\n\u22a2 p = 0\n[PROOFSTEP]\nrw [injective_iff_map_eq_zero (algebraMap Fq[X] F)] at hinj \n[GOAL]\nFq F : Type\ninst\u271d\u2074 : Field Fq\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : Algebra (RatFunc Fq) F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\nhinj : \u2200 (a : Fq[X]), \u2191(algebraMap Fq[X] F) a = 0 \u2192 a = 0\np : Fq[X]\nhp : \u2191(\u2191(algebraMap Fq[X] { x // x \u2208 ringOfIntegers Fq F }) p) = 0\n\u22a2 p = 0\n[PROOFSTEP]\nexact hinj p hp\n[GOAL]\nFq F : Type\ninst\u271d\u2074 : Field Fq\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra Fq[X] F\ninst\u271d\u00b9 : Algebra (RatFunc Fq) F\ninst\u271d : IsScalarTower Fq[X] (RatFunc Fq) F\n\u22a2 \u00acIsField { x // x \u2208 ringOfIntegers Fq F }\n[PROOFSTEP]\nsimpa [\u2190 (IsIntegralClosure.isIntegral_algebra Fq[X] F).isField_iff_isField (algebraMap_injective Fq F)] using\n Polynomial.not_isField Fq\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\n\u22a2 \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree 1)) = 1\n[PROOFSTEP]\nrw [RatFunc.intDegree_one, ofAdd_zero, WithZero.coe_one]\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\n\u22a2 inftyValuationDef Fq (x * y) = inftyValuationDef Fq x * inftyValuationDef Fq y\n[PROOFSTEP]\nrw [inftyValuationDef, inftyValuationDef, inftyValuationDef]\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\n\u22a2 (if x * y = 0 then 0 else \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree (x * y)))) =\n (if x = 0 then 0 else \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree x))) *\n if y = 0 then 0 else \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree y))\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : x = 0\n\u22a2 (if x * y = 0 then 0 else \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree (x * y)))) =\n (if x = 0 then 0 else \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree x))) *\n if y = 0 then 0 else \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree y))\n[PROOFSTEP]\nrw [hx, zero_mul, if_pos (Eq.refl _), zero_mul]\n[GOAL]\ncase neg\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\n\u22a2 (if x * y = 0 then 0 else \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree (x * y)))) =\n (if x = 0 then 0 else \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree x))) *\n if y = 0 then 0 else \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree y))\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 (if x * y = 0 then 0 else \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree (x * y)))) =\n (if x = 0 then 0 else \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree x))) *\n if y = 0 then 0 else \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree y))\n[PROOFSTEP]\nrw [hy, mul_zero, if_pos (Eq.refl _), mul_zero]\n[GOAL]\ncase neg\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 (if x * y = 0 then 0 else \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree (x * y)))) =\n (if x = 0 then 0 else \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree x))) *\n if y = 0 then 0 else \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree y))\n[PROOFSTEP]\nrw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), \u2190 WithZero.coe_mul, WithZero.coe_inj, \u2190 ofAdd_add,\n RatFunc.intDegree_mul hx hy]\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\n\u22a2 inftyValuationDef Fq (x + y) \u2264 max (inftyValuationDef Fq x) (inftyValuationDef Fq y)\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : x = 0\n\u22a2 inftyValuationDef Fq (x + y) \u2264 max (inftyValuationDef Fq x) (inftyValuationDef Fq y)\n[PROOFSTEP]\nrw [hx, zero_add]\n[GOAL]\ncase pos\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : x = 0\n\u22a2 inftyValuationDef Fq y \u2264 max (inftyValuationDef Fq 0) (inftyValuationDef Fq y)\n[PROOFSTEP]\nconv_rhs => rw [inftyValuationDef, if_pos (Eq.refl _)]\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : x = 0\n| max (inftyValuationDef Fq 0) (inftyValuationDef Fq y)\n[PROOFSTEP]\nrw [inftyValuationDef, if_pos (Eq.refl _)]\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : x = 0\n| max (inftyValuationDef Fq 0) (inftyValuationDef Fq y)\n[PROOFSTEP]\nrw [inftyValuationDef, if_pos (Eq.refl _)]\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : x = 0\n| max (inftyValuationDef Fq 0) (inftyValuationDef Fq y)\n[PROOFSTEP]\nrw [inftyValuationDef, if_pos (Eq.refl _)]\n[GOAL]\ncase pos\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : x = 0\n\u22a2 inftyValuationDef Fq y \u2264 max 0 (inftyValuationDef Fq y)\n[PROOFSTEP]\nrw [max_eq_right (WithZero.zero_le (inftyValuationDef Fq y))]\n[GOAL]\ncase neg\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\n\u22a2 inftyValuationDef Fq (x + y) \u2264 max (inftyValuationDef Fq x) (inftyValuationDef Fq y)\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 inftyValuationDef Fq (x + y) \u2264 max (inftyValuationDef Fq x) (inftyValuationDef Fq y)\n[PROOFSTEP]\nrw [hy, add_zero]\n[GOAL]\ncase pos\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 inftyValuationDef Fq x \u2264 max (inftyValuationDef Fq x) (inftyValuationDef Fq 0)\n[PROOFSTEP]\nconv_rhs => rw [max_comm, inftyValuationDef, if_pos (Eq.refl _)]\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\nhy : y = 0\n| max (inftyValuationDef Fq x) (inftyValuationDef Fq 0)\n[PROOFSTEP]\nrw [max_comm, inftyValuationDef, if_pos (Eq.refl _)]\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\nhy : y = 0\n| max (inftyValuationDef Fq x) (inftyValuationDef Fq 0)\n[PROOFSTEP]\nrw [max_comm, inftyValuationDef, if_pos (Eq.refl _)]\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\nhy : y = 0\n| max (inftyValuationDef Fq x) (inftyValuationDef Fq 0)\n[PROOFSTEP]\nrw [max_comm, inftyValuationDef, if_pos (Eq.refl _)]\n[GOAL]\ncase pos\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 inftyValuationDef Fq x \u2264 max 0 (inftyValuationDef Fq x)\n[PROOFSTEP]\nrw [max_eq_right (WithZero.zero_le (inftyValuationDef Fq x))]\n[GOAL]\ncase neg\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 inftyValuationDef Fq (x + y) \u2264 max (inftyValuationDef Fq x) (inftyValuationDef Fq y)\n[PROOFSTEP]\nby_cases hxy : x + y = 0\n[GOAL]\ncase pos\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : x + y = 0\n\u22a2 inftyValuationDef Fq (x + y) \u2264 max (inftyValuationDef Fq x) (inftyValuationDef Fq y)\n[PROOFSTEP]\nrw [inftyValuationDef, if_pos hxy]\n[GOAL]\ncase pos\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : x + y = 0\n\u22a2 0 \u2264 max (inftyValuationDef Fq x) (inftyValuationDef Fq y)\n[PROOFSTEP]\nexact zero_le'\n[GOAL]\ncase neg\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : \u00acx + y = 0\n\u22a2 inftyValuationDef Fq (x + y) \u2264 max (inftyValuationDef Fq x) (inftyValuationDef Fq y)\n[PROOFSTEP]\nrw [inftyValuationDef, inftyValuationDef, inftyValuationDef, if_neg hx, if_neg hy, if_neg hxy]\n[GOAL]\ncase neg\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : \u00acx + y = 0\n\u22a2 \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree (x + y))) \u2264\n max \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree x)) \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree y))\n[PROOFSTEP]\nrw [le_max_iff, WithZero.coe_le_coe, Multiplicative.ofAdd_le, WithZero.coe_le_coe, Multiplicative.ofAdd_le, \u2190\n le_max_iff]\n[GOAL]\ncase neg\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : \u00acx + y = 0\n\u22a2 RatFunc.intDegree (x + y) \u2264 max (RatFunc.intDegree x) (RatFunc.intDegree y)\n[PROOFSTEP]\nexact RatFunc.intDegree_add_le hy hxy\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nx : RatFunc Fq\nhx : x \u2260 0\n\u22a2 inftyValuationDef Fq x = \u2191(\u2191Multiplicative.ofAdd (RatFunc.intDegree x))\n[PROOFSTEP]\nrw [inftyValuationDef, if_neg hx]\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nk : Fq\nhk : k \u2260 0\n\u22a2 inftyValuationDef Fq (\u2191RatFunc.C k) = \u2191(\u2191Multiplicative.ofAdd 0)\n[PROOFSTEP]\nhave hCk : RatFunc.C k \u2260 0 := (map_ne_zero _).mpr hk\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\nk : Fq\nhk : k \u2260 0\nhCk : \u2191RatFunc.C k \u2260 0\n\u22a2 inftyValuationDef Fq (\u2191RatFunc.C k) = \u2191(\u2191Multiplicative.ofAdd 0)\n[PROOFSTEP]\nrw [inftyValuationDef, if_neg hCk, RatFunc.intDegree_C]\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\n\u22a2 inftyValuationDef Fq RatFunc.X = \u2191(\u2191Multiplicative.ofAdd 1)\n[PROOFSTEP]\nrw [inftyValuationDef, if_neg RatFunc.X_ne_zero, RatFunc.intDegree_X]\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\np : Fq[X]\nhp : p \u2260 0\n\u22a2 inftyValuationDef Fq (\u2191(algebraMap Fq[X] (RatFunc Fq)) p) = \u2191(\u2191Multiplicative.ofAdd \u2191(Polynomial.natDegree p))\n[PROOFSTEP]\nhave hp' : algebraMap Fq[X] (RatFunc Fq) p \u2260 0 := by rw [Ne.def, RatFunc.algebraMap_eq_zero_iff]; exact hp\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\np : Fq[X]\nhp : p \u2260 0\n\u22a2 \u2191(algebraMap Fq[X] (RatFunc Fq)) p \u2260 0\n[PROOFSTEP]\nrw [Ne.def, RatFunc.algebraMap_eq_zero_iff]\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\np : Fq[X]\nhp : p \u2260 0\n\u22a2 \u00acp = 0\n[PROOFSTEP]\nexact hp\n[GOAL]\nFq F : Type\ninst\u271d\u00b2 : Field Fq\ninst\u271d\u00b9 : Field F\ninst\u271d : DecidableEq (RatFunc Fq)\np : Fq[X]\nhp : p \u2260 0\nhp' : \u2191(algebraMap Fq[X] (RatFunc Fq)) p \u2260 0\n\u22a2 inftyValuationDef Fq (\u2191(algebraMap Fq[X] (RatFunc Fq)) p) = \u2191(\u2191Multiplicative.ofAdd \u2191(Polynomial.natDegree p))\n[PROOFSTEP]\nrw [inftyValuationDef, if_neg hp', RatFunc.intDegree_polynomial]\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.FunctionField", "llama_tokens": 15303, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707281, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.432358836576842}} {"text": "[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\ns : Set A\n\u22a2 zeroLocus \ud835\udc9c \u2191(Ideal.span s) = zeroLocus \ud835\udc9c s\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\ns : Set A\nx : ProjectiveSpectrum \ud835\udc9c\n\u22a2 x \u2208 zeroLocus \ud835\udc9c \u2191(Ideal.span s) \u2194 x \u2208 zeroLocus \ud835\udc9c s\n[PROOFSTEP]\nexact (Submodule.gi _ _).gc s x.asHomogeneousIdeal.toIdeal\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 \u2191(vanishingIdeal t) = {f | \u2200 (x : ProjectiveSpectrum \ud835\udc9c), x \u2208 t \u2192 f \u2208 x.asHomogeneousIdeal}\n[PROOFSTEP]\next f\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nf : A\n\u22a2 f \u2208 \u2191(vanishingIdeal t) \u2194 f \u2208 {f | \u2200 (x : ProjectiveSpectrum \ud835\udc9c), x \u2208 t \u2192 f \u2208 x.asHomogeneousIdeal}\n[PROOFSTEP]\nrw [vanishingIdeal, SetLike.mem_coe, \u2190 HomogeneousIdeal.mem_iff, HomogeneousIdeal.toIdeal_iInf, Submodule.mem_iInf]\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nf : A\n\u22a2 (\u2200 (i : ProjectiveSpectrum \ud835\udc9c), f \u2208 HomogeneousIdeal.toIdeal (\u2a05 (_ : i \u2208 t), i.asHomogeneousIdeal)) \u2194\n f \u2208 {f | \u2200 (x : ProjectiveSpectrum \ud835\udc9c), x \u2208 t \u2192 f \u2208 x.asHomogeneousIdeal}\n[PROOFSTEP]\nrefine' forall_congr' fun x => _\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nf : A\nx : ProjectiveSpectrum \ud835\udc9c\n\u22a2 f \u2208 HomogeneousIdeal.toIdeal (\u2a05 (_ : x \u2208 t), x.asHomogeneousIdeal) \u2194 x \u2208 t \u2192 f \u2208 x.asHomogeneousIdeal\n[PROOFSTEP]\nrw [HomogeneousIdeal.toIdeal_iInf, Submodule.mem_iInf, HomogeneousIdeal.mem_iff]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nf : A\n\u22a2 f \u2208 vanishingIdeal t \u2194 \u2200 (x : ProjectiveSpectrum \ud835\udc9c), x \u2208 t \u2192 f \u2208 x.asHomogeneousIdeal\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, coe_vanishingIdeal, Set.mem_setOf_eq]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx : ProjectiveSpectrum \ud835\udc9c\n\u22a2 vanishingIdeal {x} = x.asHomogeneousIdeal\n[PROOFSTEP]\nsimp [vanishingIdeal]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\n\u22a2 GaloisConnection (fun s => zeroLocus \ud835\udc9c s) fun t => \u2191(vanishingIdeal t)\n[PROOFSTEP]\nhave ideal_gc : GaloisConnection Ideal.span _ := (Submodule.gi A _).gc\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nideal_gc : GaloisConnection Ideal.span SetLike.coe\n\u22a2 GaloisConnection (fun s => zeroLocus \ud835\udc9c s) fun t => \u2191(vanishingIdeal t)\n[PROOFSTEP]\nsimpa [zeroLocus_span, Function.comp] using GaloisConnection.compose ideal_gc (gc_ideal \ud835\udc9c)\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nI : HomogeneousIdeal \ud835\udc9c\nt : (Set (ProjectiveSpectrum \ud835\udc9c))\u1d52\u1d48\n\u22a2 (fun I => zeroLocus \ud835\udc9c \u2191I) I \u2264 t \u2194 I \u2264 (fun t => vanishingIdeal t) t\n[PROOFSTEP]\nsimpa [show I.toIdeal \u2264 (vanishingIdeal t).toIdeal \u2194 I \u2264 vanishingIdeal t from Iff.rfl] using\n subset_zeroLocus_iff_le_vanishingIdeal t I.toIdeal\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\n\u22a2 vanishingIdeal \u2205 = \u22a4\n[PROOFSTEP]\nsimpa using (gc_ideal _).u_top\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt t' : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 vanishingIdeal (t \u222a t') = vanishingIdeal t \u2293 vanishingIdeal t'\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt t' : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 HomogeneousIdeal.toIdeal (vanishingIdeal (t \u222a t')) = HomogeneousIdeal.toIdeal (vanishingIdeal t \u2293 vanishingIdeal t')\n[PROOFSTEP]\nexact (gc_ideal \ud835\udc9c).u_inf\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\ns : Set (Set A)\n\u22a2 zeroLocus \ud835\udc9c (\u22c3 (s' : Set A) (_ : s' \u2208 s), s') = \u22c2 (s' : Set A) (_ : s' \u2208 s), zeroLocus \ud835\udc9c s'\n[PROOFSTEP]\nsimp only [zeroLocus_iUnion]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\n\u03b3 : Sort u_3\nt : \u03b3 \u2192 Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 HomogeneousIdeal.toIdeal (vanishingIdeal (\u22c3 (i : \u03b3), t i)) =\n HomogeneousIdeal.toIdeal (\u2a05 (i : \u03b3), vanishingIdeal (t i))\n[PROOFSTEP]\nconvert (gc_ideal \ud835\udc9c).u_iInf\n[GOAL]\ncase h.e'_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\n\u03b3 : Sort u_3\nt : \u03b3 \u2192 Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 HomogeneousIdeal.toIdeal (\u2a05 (i : \u03b3), vanishingIdeal (t i)) =\n \u2a05 (i : \u03b3), HomogeneousIdeal.toIdeal (vanishingIdeal (t i))\n[PROOFSTEP]\nexact HomogeneousIdeal.toIdeal_iInf _\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\ns s' : Set A\n\u22a2 zeroLocus \ud835\udc9c s \u222a zeroLocus \ud835\udc9c s' = zeroLocus \ud835\udc9c \u2191(Ideal.span s \u2293 Ideal.span s')\n[PROOFSTEP]\nrw [zeroLocus_inf]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\ns s' : Set A\n\u22a2 zeroLocus \ud835\udc9c s \u222a zeroLocus \ud835\udc9c s' = zeroLocus \ud835\udc9c \u2191(Ideal.span s) \u222a zeroLocus \ud835\udc9c \u2191(Ideal.span s')\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf g : A\nx : ProjectiveSpectrum \ud835\udc9c\n\u22a2 x \u2208 zeroLocus \ud835\udc9c {f * g} \u2194 x \u2208 zeroLocus \ud835\udc9c {f} \u222a zeroLocus \ud835\udc9c {g}\n[PROOFSTEP]\nsimpa using x.isPrime.mul_mem_iff_mem_or_mem\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nn : \u2115\nhn : 0 < n\nx : ProjectiveSpectrum \ud835\udc9c\n\u22a2 x \u2208 zeroLocus \ud835\udc9c {f ^ n} \u2194 x \u2208 zeroLocus \ud835\udc9c {f}\n[PROOFSTEP]\nsimpa using x.isPrime.pow_mem_iff_mem n hn\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt t' : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 vanishingIdeal t \u2294 vanishingIdeal t' \u2264 vanishingIdeal (t \u2229 t')\n[PROOFSTEP]\nintro r\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt t' : Set (ProjectiveSpectrum \ud835\udc9c)\nr : A\n\u22a2 r \u2208 vanishingIdeal t \u2294 vanishingIdeal t' \u2192 r \u2208 vanishingIdeal (t \u2229 t')\n[PROOFSTEP]\nrw [\u2190 HomogeneousIdeal.mem_iff, HomogeneousIdeal.toIdeal_sup, mem_vanishingIdeal, Submodule.mem_sup]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt t' : Set (ProjectiveSpectrum \ud835\udc9c)\nr : A\n\u22a2 (\u2203 y,\n y \u2208 HomogeneousIdeal.toIdeal (vanishingIdeal t) \u2227\n \u2203 z, z \u2208 HomogeneousIdeal.toIdeal (vanishingIdeal t') \u2227 y + z = r) \u2192\n \u2200 (x : ProjectiveSpectrum \ud835\udc9c), x \u2208 t \u2229 t' \u2192 r \u2208 x.asHomogeneousIdeal\n[PROOFSTEP]\nrintro \u27e8f, hf, g, hg, rfl\u27e9 x \u27e8hxt, hxt'\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt t' : Set (ProjectiveSpectrum \ud835\udc9c)\nf : A\nhf : f \u2208 HomogeneousIdeal.toIdeal (vanishingIdeal t)\ng : A\nhg : g \u2208 HomogeneousIdeal.toIdeal (vanishingIdeal t')\nx : ProjectiveSpectrum \ud835\udc9c\nhxt : x \u2208 t\nhxt' : x \u2208 t'\n\u22a2 f + g \u2208 x.asHomogeneousIdeal\n[PROOFSTEP]\nerw [mem_vanishingIdeal] at hf hg \n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt t' : Set (ProjectiveSpectrum \ud835\udc9c)\nf : A\nhf : \u2200 (x : ProjectiveSpectrum \ud835\udc9c), x \u2208 t \u2192 f \u2208 x.asHomogeneousIdeal\ng : A\nhg : \u2200 (x : ProjectiveSpectrum \ud835\udc9c), x \u2208 t' \u2192 g \u2208 x.asHomogeneousIdeal\nx : ProjectiveSpectrum \ud835\udc9c\nhxt : x \u2208 t\nhxt' : x \u2208 t'\n\u22a2 f + g \u2208 x.asHomogeneousIdeal\n[PROOFSTEP]\napply Submodule.add_mem\n[GOAL]\ncase intro.intro.intro.intro.intro.h\u2081\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt t' : Set (ProjectiveSpectrum \ud835\udc9c)\nf : A\nhf : \u2200 (x : ProjectiveSpectrum \ud835\udc9c), x \u2208 t \u2192 f \u2208 x.asHomogeneousIdeal\ng : A\nhg : \u2200 (x : ProjectiveSpectrum \ud835\udc9c), x \u2208 t' \u2192 g \u2208 x.asHomogeneousIdeal\nx : ProjectiveSpectrum \ud835\udc9c\nhxt : x \u2208 t\nhxt' : x \u2208 t'\n\u22a2 f \u2208 HomogeneousIdeal.toIdeal x.asHomogeneousIdeal\n[PROOFSTEP]\nsolve_by_elim\n[GOAL]\ncase intro.intro.intro.intro.intro.h\u2082\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt t' : Set (ProjectiveSpectrum \ud835\udc9c)\nf : A\nhf : \u2200 (x : ProjectiveSpectrum \ud835\udc9c), x \u2208 t \u2192 f \u2208 x.asHomogeneousIdeal\ng : A\nhg : \u2200 (x : ProjectiveSpectrum \ud835\udc9c), x \u2208 t' \u2192 g \u2208 x.asHomogeneousIdeal\nx : ProjectiveSpectrum \ud835\udc9c\nhxt : x \u2208 t\nhxt' : x \u2208 t'\n\u22a2 g \u2208 HomogeneousIdeal.toIdeal x.asHomogeneousIdeal\n[PROOFSTEP]\nsolve_by_elim\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nI : ProjectiveSpectrum \ud835\udc9c\n\u22a2 I \u2208 (zeroLocus \ud835\udc9c {f})\u1d9c \u2194 \u00acf \u2208 I.asHomogeneousIdeal\n[PROOFSTEP]\nrw [Set.mem_compl_iff, mem_zeroLocus, Set.singleton_subset_iff]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nI : ProjectiveSpectrum \ud835\udc9c\n\u22a2 \u00acf \u2208 \u2191I.asHomogeneousIdeal \u2194 \u00acf \u2208 I.asHomogeneousIdeal\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\n\u22a2 zeroLocus \ud835\udc9c Set.univ = \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\n\u22a2 \u2200 (A_1 : Set (Set (ProjectiveSpectrum \ud835\udc9c))), A_1 \u2286 Set.range (zeroLocus \ud835\udc9c) \u2192 \u22c2\u2080 A_1 \u2208 Set.range (zeroLocus \ud835\udc9c)\n[PROOFSTEP]\nintro Zs h\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nZs : Set (Set (ProjectiveSpectrum \ud835\udc9c))\nh : Zs \u2286 Set.range (zeroLocus \ud835\udc9c)\n\u22a2 \u22c2\u2080 Zs \u2208 Set.range (zeroLocus \ud835\udc9c)\n[PROOFSTEP]\nrw [Set.sInter_eq_iInter]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nZs : Set (Set (ProjectiveSpectrum \ud835\udc9c))\nh : Zs \u2286 Set.range (zeroLocus \ud835\udc9c)\n\u22a2 \u22c2 (i : \u2191Zs), \u2191i \u2208 Set.range (zeroLocus \ud835\udc9c)\n[PROOFSTEP]\nlet f : Zs \u2192 Set _ := fun i => Classical.choose (h i.2)\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nZs : Set (Set (ProjectiveSpectrum \ud835\udc9c))\nh : Zs \u2286 Set.range (zeroLocus \ud835\udc9c)\nf : \u2191Zs \u2192 Set A := fun i => Classical.choose (_ : \u2191i \u2208 Set.range (zeroLocus \ud835\udc9c))\n\u22a2 \u22c2 (i : \u2191Zs), \u2191i \u2208 Set.range (zeroLocus \ud835\udc9c)\n[PROOFSTEP]\nhave H : (Set.iInter fun i \u21a6 zeroLocus \ud835\udc9c (f i)) \u2208 Set.range (zeroLocus \ud835\udc9c) := \u27e8_, zeroLocus_iUnion \ud835\udc9c _\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nZs : Set (Set (ProjectiveSpectrum \ud835\udc9c))\nh : Zs \u2286 Set.range (zeroLocus \ud835\udc9c)\nf : \u2191Zs \u2192 Set A := fun i => Classical.choose (_ : \u2191i \u2208 Set.range (zeroLocus \ud835\udc9c))\nH : \u22c2 (i : \u2191Zs), zeroLocus \ud835\udc9c (f i) \u2208 Set.range (zeroLocus \ud835\udc9c)\n\u22a2 \u22c2 (i : \u2191Zs), \u2191i \u2208 Set.range (zeroLocus \ud835\udc9c)\n[PROOFSTEP]\nconvert H using 2\n[GOAL]\ncase h.e'_4.h.e'_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nZs : Set (Set (ProjectiveSpectrum \ud835\udc9c))\nh : Zs \u2286 Set.range (zeroLocus \ud835\udc9c)\nf : \u2191Zs \u2192 Set A := fun i => Classical.choose (_ : \u2191i \u2208 Set.range (zeroLocus \ud835\udc9c))\nH : \u22c2 (i : \u2191Zs), zeroLocus \ud835\udc9c (f i) \u2208 Set.range (zeroLocus \ud835\udc9c)\n\u22a2 (fun i => \u2191i) = fun i => zeroLocus \ud835\udc9c (f i)\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h.e'_4.h.e'_3.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nZs : Set (Set (ProjectiveSpectrum \ud835\udc9c))\nh : Zs \u2286 Set.range (zeroLocus \ud835\udc9c)\nf : \u2191Zs \u2192 Set A := fun i => Classical.choose (_ : \u2191i \u2208 Set.range (zeroLocus \ud835\udc9c))\nH : \u22c2 (i : \u2191Zs), zeroLocus \ud835\udc9c (f i) \u2208 Set.range (zeroLocus \ud835\udc9c)\ni : \u2191Zs\n\u22a2 \u2191i = zeroLocus \ud835\udc9c (f i)\n[PROOFSTEP]\nexact (Classical.choose_spec (h i.2)).symm\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\n\u22a2 \u2200 (A_1 : Set (ProjectiveSpectrum \ud835\udc9c)),\n A_1 \u2208 Set.range (zeroLocus \ud835\udc9c) \u2192\n \u2200 (B : Set (ProjectiveSpectrum \ud835\udc9c)), B \u2208 Set.range (zeroLocus \ud835\udc9c) \u2192 A_1 \u222a B \u2208 Set.range (zeroLocus \ud835\udc9c)\n[PROOFSTEP]\nrintro _ \u27e8s, rfl\u27e9 _ \u27e8t, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\ns t : Set A\n\u22a2 zeroLocus \ud835\udc9c s \u222a zeroLocus \ud835\udc9c t \u2208 Set.range (zeroLocus \ud835\udc9c)\n[PROOFSTEP]\nexact \u27e8_, (union_zeroLocus \ud835\udc9c s t).symm\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nU : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 IsOpen U \u2194 \u2203 s, U\u1d9c = zeroLocus \ud835\udc9c s\n[PROOFSTEP]\nsimp only [@eq_comm _ U\u1d9c]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nU : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 IsOpen U \u2194 \u2203 s, zeroLocus \ud835\udc9c s = U\u1d9c\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nZ : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 IsClosed Z \u2194 \u2203 s, Z = zeroLocus \ud835\udc9c s\n[PROOFSTEP]\nrw [\u2190 isOpen_compl_iff, isOpen_iff, compl_compl]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\ns : Set A\n\u22a2 IsClosed (zeroLocus \ud835\udc9c s)\n[PROOFSTEP]\nrw [isClosed_iff_zeroLocus]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\ns : Set A\n\u22a2 \u2203 s_1, zeroLocus \ud835\udc9c s = zeroLocus \ud835\udc9c s_1\n[PROOFSTEP]\nexact \u27e8s, rfl\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t) = closure t\n[PROOFSTEP]\napply Set.Subset.antisymm\n[GOAL]\ncase h\u2081\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t) \u2286 closure t\n[PROOFSTEP]\nrintro x hx t' \u27e8ht', ht\u27e9\n[GOAL]\ncase h\u2081.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nx : ProjectiveSpectrum \ud835\udc9c\nhx : x \u2208 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)\nt' : Set (ProjectiveSpectrum \ud835\udc9c)\nht' : IsClosed t'\nht : t \u2286 t'\n\u22a2 x \u2208 t'\n[PROOFSTEP]\nobtain \u27e8fs, rfl\u27e9 : \u2203 s, t' = zeroLocus \ud835\udc9c s := by rwa [isClosed_iff_zeroLocus] at ht' \n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nx : ProjectiveSpectrum \ud835\udc9c\nhx : x \u2208 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)\nt' : Set (ProjectiveSpectrum \ud835\udc9c)\nht' : IsClosed t'\nht : t \u2286 t'\n\u22a2 \u2203 s, t' = zeroLocus \ud835\udc9c s\n[PROOFSTEP]\nrwa [isClosed_iff_zeroLocus] at ht' \n[GOAL]\ncase h\u2081.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nx : ProjectiveSpectrum \ud835\udc9c\nhx : x \u2208 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)\nfs : Set A\nht' : IsClosed (zeroLocus \ud835\udc9c fs)\nht : t \u2286 zeroLocus \ud835\udc9c fs\n\u22a2 x \u2208 zeroLocus \ud835\udc9c fs\n[PROOFSTEP]\nrw [subset_zeroLocus_iff_subset_vanishingIdeal] at ht \n[GOAL]\ncase h\u2081.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nx : ProjectiveSpectrum \ud835\udc9c\nhx : x \u2208 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)\nfs : Set A\nht' : IsClosed (zeroLocus \ud835\udc9c fs)\nht : fs \u2286 \u2191(vanishingIdeal t)\n\u22a2 x \u2208 zeroLocus \ud835\udc9c fs\n[PROOFSTEP]\nexact Set.Subset.trans ht hx\n[GOAL]\ncase h\u2082\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 closure t \u2286 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)\n[PROOFSTEP]\nrw [(isClosed_zeroLocus _ _).closure_subset_iff]\n[GOAL]\ncase h\u2082\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 t \u2286 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)\n[PROOFSTEP]\nexact subset_zeroLocus_vanishingIdeal \ud835\udc9c t\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 vanishingIdeal (closure t) = vanishingIdeal t\n[PROOFSTEP]\nhave := (gc_ideal \ud835\udc9c).u_l_u_eq_u t\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nthis :\n HomogeneousIdeal.toIdeal (vanishingIdeal (zeroLocus \ud835\udc9c \u2191(HomogeneousIdeal.toIdeal (vanishingIdeal t)))) =\n HomogeneousIdeal.toIdeal (vanishingIdeal t)\n\u22a2 vanishingIdeal (closure t) = vanishingIdeal t\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nthis :\n HomogeneousIdeal.toIdeal (vanishingIdeal (zeroLocus \ud835\udc9c \u2191(HomogeneousIdeal.toIdeal (vanishingIdeal t)))) =\n HomogeneousIdeal.toIdeal (vanishingIdeal t)\n\u22a2 HomogeneousIdeal.toIdeal (vanishingIdeal (closure t)) = HomogeneousIdeal.toIdeal (vanishingIdeal t)\n[PROOFSTEP]\nerw [zeroLocus_vanishingIdeal_eq_closure \ud835\udc9c t] at this \n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nthis : HomogeneousIdeal.toIdeal (vanishingIdeal (closure t)) = HomogeneousIdeal.toIdeal (vanishingIdeal t)\n\u22a2 HomogeneousIdeal.toIdeal (vanishingIdeal (closure t)) = HomogeneousIdeal.toIdeal (vanishingIdeal t)\n[PROOFSTEP]\nexact this\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nr : A\nx : ProjectiveSpectrum \ud835\udc9c\n\u22a2 x \u2208 \u2191(basicOpen \ud835\udc9c r) \u2194 x \u2208 (zeroLocus \ud835\udc9c {r})\u1d9c\n[PROOFSTEP]\nsimp only [Set.mem_compl_iff, mem_zeroLocus, Set.singleton_subset_iff]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nr : A\nx : ProjectiveSpectrum \ud835\udc9c\n\u22a2 x \u2208 \u2191(basicOpen \ud835\udc9c r) \u2194 \u00acr \u2208 \u2191x.asHomogeneousIdeal\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\n\u22a2 \u2191(basicOpen \ud835\udc9c 1) = \u2191\u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\n\u22a2 \u2191(basicOpen \ud835\udc9c 0) = \u2191\u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf g : A\n\u22a2 \u2191(basicOpen \ud835\udc9c (f * g)) = \u2191(basicOpen \ud835\udc9c f \u2293 basicOpen \ud835\udc9c g)\n[PROOFSTEP]\nsimp [zeroLocus_singleton_mul]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf g : A\n\u22a2 basicOpen \ud835\udc9c (f * g) \u2264 basicOpen \ud835\udc9c f\n[PROOFSTEP]\nrw [basicOpen_mul \ud835\udc9c f g]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf g : A\n\u22a2 basicOpen \ud835\udc9c f \u2293 basicOpen \ud835\udc9c g \u2264 basicOpen \ud835\udc9c f\n[PROOFSTEP]\nexact inf_le_left\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf g : A\n\u22a2 basicOpen \ud835\udc9c (f * g) \u2264 basicOpen \ud835\udc9c g\n[PROOFSTEP]\nrw [basicOpen_mul \ud835\udc9c f g]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf g : A\n\u22a2 basicOpen \ud835\udc9c f \u2293 basicOpen \ud835\udc9c g \u2264 basicOpen \ud835\udc9c g\n[PROOFSTEP]\nexact inf_le_right\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nn : \u2115\nhn : 0 < n\n\u22a2 \u2191(basicOpen \ud835\udc9c (f ^ n)) = \u2191(basicOpen \ud835\udc9c f)\n[PROOFSTEP]\nsimpa using zeroLocus_singleton_pow \ud835\udc9c f n hn\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nz : ProjectiveSpectrum \ud835\udc9c\n\u22a2 z \u2208 \u2191(basicOpen \ud835\udc9c f) \u2194 z \u2208 \u2191(\u2a06 (i : \u2115), basicOpen \ud835\udc9c (\u2191(GradedAlgebra.proj \ud835\udc9c i) f))\n[PROOFSTEP]\nerw [mem_coe_basicOpen, TopologicalSpace.Opens.mem_sSup]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nz : ProjectiveSpectrum \ud835\udc9c\n\u22a2 \u00acf \u2208 z.asHomogeneousIdeal \u2194 \u2203 u, (u \u2208 Set.range fun i => basicOpen \ud835\udc9c (\u2191(GradedAlgebra.proj \ud835\udc9c i) f)) \u2227 z \u2208 u\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nz : ProjectiveSpectrum \ud835\udc9c\n\u22a2 \u00acf \u2208 z.asHomogeneousIdeal \u2192 \u2203 u, (u \u2208 Set.range fun i => basicOpen \ud835\udc9c (\u2191(GradedAlgebra.proj \ud835\udc9c i) f)) \u2227 z \u2208 u\n[PROOFSTEP]\nintro hz\n[GOAL]\ncase mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nz : ProjectiveSpectrum \ud835\udc9c\n\u22a2 (\u2203 u, (u \u2208 Set.range fun i => basicOpen \ud835\udc9c (\u2191(GradedAlgebra.proj \ud835\udc9c i) f)) \u2227 z \u2208 u) \u2192 \u00acf \u2208 z.asHomogeneousIdeal\n[PROOFSTEP]\nintro hz\n[GOAL]\ncase mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nz : ProjectiveSpectrum \ud835\udc9c\nhz : \u00acf \u2208 z.asHomogeneousIdeal\n\u22a2 \u2203 u, (u \u2208 Set.range fun i => basicOpen \ud835\udc9c (\u2191(GradedAlgebra.proj \ud835\udc9c i) f)) \u2227 z \u2208 u\n[PROOFSTEP]\nrcases show \u2203 i, GradedAlgebra.proj \ud835\udc9c i f \u2209 z.asHomogeneousIdeal\n by\n contrapose! hz with H\n classical\n rw [\u2190 DirectSum.sum_support_decompose \ud835\udc9c f]\n apply Ideal.sum_mem _ fun i _ => H i with\n \u27e8i, hi\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nz : ProjectiveSpectrum \ud835\udc9c\nhz : \u00acf \u2208 z.asHomogeneousIdeal\n\u22a2 \u2203 i, \u00ac\u2191(GradedAlgebra.proj \ud835\udc9c i) f \u2208 z.asHomogeneousIdeal\n[PROOFSTEP]\ncontrapose! hz with H\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nz : ProjectiveSpectrum \ud835\udc9c\nH : \u2200 (i : \u2115), \u2191(GradedAlgebra.proj \ud835\udc9c i) f \u2208 z.asHomogeneousIdeal\n\u22a2 f \u2208 z.asHomogeneousIdeal\n[PROOFSTEP]\nclassical\nrw [\u2190 DirectSum.sum_support_decompose \ud835\udc9c f]\napply Ideal.sum_mem _ fun i _ => H i\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nz : ProjectiveSpectrum \ud835\udc9c\nH : \u2200 (i : \u2115), \u2191(GradedAlgebra.proj \ud835\udc9c i) f \u2208 z.asHomogeneousIdeal\n\u22a2 f \u2208 z.asHomogeneousIdeal\n[PROOFSTEP]\nrw [\u2190 DirectSum.sum_support_decompose \ud835\udc9c f]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nz : ProjectiveSpectrum \ud835\udc9c\nH : \u2200 (i : \u2115), \u2191(GradedAlgebra.proj \ud835\udc9c i) f \u2208 z.asHomogeneousIdeal\n\u22a2 \u2211 i in DFinsupp.support (\u2191(decompose \ud835\udc9c) f), \u2191(\u2191(\u2191(decompose \ud835\udc9c) f) i) \u2208 z.asHomogeneousIdeal\n[PROOFSTEP]\napply Ideal.sum_mem _ fun i _ => H i\n[GOAL]\ncase mp.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nz : ProjectiveSpectrum \ud835\udc9c\nhz : \u00acf \u2208 z.asHomogeneousIdeal\ni : \u2115\nhi : \u00ac\u2191(GradedAlgebra.proj \ud835\udc9c i) f \u2208 z.asHomogeneousIdeal\n\u22a2 \u2203 u, (u \u2208 Set.range fun i => basicOpen \ud835\udc9c (\u2191(GradedAlgebra.proj \ud835\udc9c i) f)) \u2227 z \u2208 u\n[PROOFSTEP]\nexact \u27e8basicOpen \ud835\udc9c (GradedAlgebra.proj \ud835\udc9c i f), \u27e8i, rfl\u27e9, by rwa [mem_basicOpen]\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nz : ProjectiveSpectrum \ud835\udc9c\nhz : \u00acf \u2208 z.asHomogeneousIdeal\ni : \u2115\nhi : \u00ac\u2191(GradedAlgebra.proj \ud835\udc9c i) f \u2208 z.asHomogeneousIdeal\n\u22a2 z \u2208 basicOpen \ud835\udc9c (\u2191(GradedAlgebra.proj \ud835\udc9c i) f)\n[PROOFSTEP]\nrwa [mem_basicOpen]\n[GOAL]\ncase mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nz : ProjectiveSpectrum \ud835\udc9c\nhz : \u2203 u, (u \u2208 Set.range fun i => basicOpen \ud835\udc9c (\u2191(GradedAlgebra.proj \ud835\udc9c i) f)) \u2227 z \u2208 u\n\u22a2 \u00acf \u2208 z.asHomogeneousIdeal\n[PROOFSTEP]\nobtain \u27e8_, \u27e8i, rfl\u27e9, hz\u27e9 := hz\n[GOAL]\ncase mpr.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nz : ProjectiveSpectrum \ud835\udc9c\ni : \u2115\nhz : z \u2208 (fun i => basicOpen \ud835\udc9c (\u2191(GradedAlgebra.proj \ud835\udc9c i) f)) i\n\u22a2 \u00acf \u2208 z.asHomogeneousIdeal\n[PROOFSTEP]\nexact fun rid => hz (z.1.2 i rid)\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\n\u22a2 IsTopologicalBasis (Set.range fun r => \u2191(basicOpen \ud835\udc9c r))\n[PROOFSTEP]\napply TopologicalSpace.isTopologicalBasis_of_open_of_nhds\n[GOAL]\ncase h_open\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\n\u22a2 \u2200 (u : Set (ProjectiveSpectrum \ud835\udc9c)), (u \u2208 Set.range fun r => \u2191(basicOpen \ud835\udc9c r)) \u2192 IsOpen u\n[PROOFSTEP]\nrintro _ \u27e8r, rfl\u27e9\n[GOAL]\ncase h_open.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nr : A\n\u22a2 IsOpen ((fun r => \u2191(basicOpen \ud835\udc9c r)) r)\n[PROOFSTEP]\nexact isOpen_basicOpen \ud835\udc9c\n[GOAL]\ncase h_nhds\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\n\u22a2 \u2200 (a : ProjectiveSpectrum \ud835\udc9c) (u : Set (ProjectiveSpectrum \ud835\udc9c)),\n a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, (v \u2208 Set.range fun r => \u2191(basicOpen \ud835\udc9c r)) \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nrintro p U hp \u27e8s, hs\u27e9\n[GOAL]\ncase h_nhds.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\np : ProjectiveSpectrum \ud835\udc9c\nU : Set (ProjectiveSpectrum \ud835\udc9c)\nhp : p \u2208 U\ns : Set A\nhs : zeroLocus \ud835\udc9c s = U\u1d9c\n\u22a2 \u2203 v, (v \u2208 Set.range fun r => \u2191(basicOpen \ud835\udc9c r)) \u2227 p \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nrw [\u2190 compl_compl U, Set.mem_compl_iff, \u2190 hs, mem_zeroLocus, Set.not_subset] at hp \n[GOAL]\ncase h_nhds.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\np : ProjectiveSpectrum \ud835\udc9c\nU : Set (ProjectiveSpectrum \ud835\udc9c)\ns : Set A\nhp : \u2203 a, a \u2208 s \u2227 \u00aca \u2208 \u2191p.asHomogeneousIdeal\nhs : zeroLocus \ud835\udc9c s = U\u1d9c\n\u22a2 \u2203 v, (v \u2208 Set.range fun r => \u2191(basicOpen \ud835\udc9c r)) \u2227 p \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nobtain \u27e8f, hfs, hfp\u27e9 := hp\n[GOAL]\ncase h_nhds.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\np : ProjectiveSpectrum \ud835\udc9c\nU : Set (ProjectiveSpectrum \ud835\udc9c)\ns : Set A\nhs : zeroLocus \ud835\udc9c s = U\u1d9c\nf : A\nhfs : f \u2208 s\nhfp : \u00acf \u2208 \u2191p.asHomogeneousIdeal\n\u22a2 \u2203 v, (v \u2208 Set.range fun r => \u2191(basicOpen \ud835\udc9c r)) \u2227 p \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nrefine' \u27e8basicOpen \ud835\udc9c f, \u27e8f, rfl\u27e9, hfp, _\u27e9\n[GOAL]\ncase h_nhds.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\np : ProjectiveSpectrum \ud835\udc9c\nU : Set (ProjectiveSpectrum \ud835\udc9c)\ns : Set A\nhs : zeroLocus \ud835\udc9c s = U\u1d9c\nf : A\nhfs : f \u2208 s\nhfp : \u00acf \u2208 \u2191p.asHomogeneousIdeal\n\u22a2 \u2191(basicOpen \ud835\udc9c f) \u2286 U\n[PROOFSTEP]\nrw [\u2190 Set.compl_subset_compl, \u2190 hs, basicOpen_eq_zeroLocus_compl, compl_compl]\n[GOAL]\ncase h_nhds.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\np : ProjectiveSpectrum \ud835\udc9c\nU : Set (ProjectiveSpectrum \ud835\udc9c)\ns : Set A\nhs : zeroLocus \ud835\udc9c s = U\u1d9c\nf : A\nhfs : f \u2208 s\nhfp : \u00acf \u2208 \u2191p.asHomogeneousIdeal\n\u22a2 zeroLocus \ud835\udc9c s \u2286 zeroLocus \ud835\udc9c {f}\n[PROOFSTEP]\nexact zeroLocus_anti_mono \ud835\udc9c (Set.singleton_subset_iff.mpr hfs)\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx\u271d\u00b9 x\u271d : ProjectiveSpectrum \ud835\udc9c\nasHomogeneousIdeal\u271d\u00b9 : HomogeneousIdeal \ud835\udc9c\nisPrime\u271d\u00b9 : Ideal.IsPrime (HomogeneousIdeal.toIdeal asHomogeneousIdeal\u271d\u00b9)\nnot_irrelevant_le\u271d\u00b9 : \u00acHomogeneousIdeal.irrelevant \ud835\udc9c \u2264 asHomogeneousIdeal\u271d\u00b9\nasHomogeneousIdeal\u271d : HomogeneousIdeal \ud835\udc9c\nisPrime\u271d : Ideal.IsPrime (HomogeneousIdeal.toIdeal asHomogeneousIdeal\u271d)\nnot_irrelevant_le\u271d : \u00acHomogeneousIdeal.irrelevant \ud835\udc9c \u2264 asHomogeneousIdeal\u271d\n\u22a2 { asHomogeneousIdeal := asHomogeneousIdeal\u271d\u00b9, isPrime := isPrime\u271d\u00b9,\n not_irrelevant_le := not_irrelevant_le\u271d\u00b9 }.asHomogeneousIdeal =\n { asHomogeneousIdeal := asHomogeneousIdeal\u271d, isPrime := isPrime\u271d,\n not_irrelevant_le := not_irrelevant_le\u271d }.asHomogeneousIdeal \u2192\n { asHomogeneousIdeal := asHomogeneousIdeal\u271d\u00b9, isPrime := isPrime\u271d\u00b9, not_irrelevant_le := not_irrelevant_le\u271d\u00b9 } =\n { asHomogeneousIdeal := asHomogeneousIdeal\u271d, isPrime := isPrime\u271d, not_irrelevant_le := not_irrelevant_le\u271d }\n[PROOFSTEP]\nsimp only [mk.injEq, imp_self]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx y : ProjectiveSpectrum \ud835\udc9c\n\u22a2 x \u2264 y \u2194 y \u2208 closure {x}\n[PROOFSTEP]\nrw [\u2190 as_ideal_le_as_ideal, \u2190 zeroLocus_vanishingIdeal_eq_closure, mem_zeroLocus, vanishingIdeal_singleton]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nx y : ProjectiveSpectrum \ud835\udc9c\n\u22a2 x.asHomogeneousIdeal \u2264 y.asHomogeneousIdeal \u2194 \u2191x.asHomogeneousIdeal \u2286 \u2191y.asHomogeneousIdeal\n[PROOFSTEP]\nsimp only [as_ideal_le_as_ideal, coe_subset_coe]\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology", "llama_tokens": 16949, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.4323492090936783}} {"text": "[GOAL]\nh : IsField \u2124\na : \u2124\n\u22a2 2 * a = 1 \u2192 1 = a + a\n[PROOFSTEP]\nrw [\u2190 two_mul]\n[GOAL]\nh : IsField \u2124\na : \u2124\n\u22a2 2 * a = 1 \u2192 1 = 2 * a\n[PROOFSTEP]\nexact Eq.symm\n[GOAL]\nK\u271d : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Field K\u271d\ninst\u271d\u00b9 : Field L\nnf : NumberField K\u271d\nK : Type u_3\ninst\u271d : Field K\nx : K\nhx : x \u2208 \ud835\udcde K\n\u22a2 IsIntegral \u2124 { val := x, property := hx }\n[PROOFSTEP]\nobtain \u27e8P, hPm, hP\u27e9 := hx\n[GOAL]\ncase intro.intro\nK\u271d : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Field K\u271d\ninst\u271d\u00b9 : Field L\nnf : NumberField K\u271d\nK : Type u_3\ninst\u271d : Field K\nx : K\nP : Polynomial \u2124\nhPm : Polynomial.Monic P\nhP : Polynomial.eval\u2082 (algebraMap \u2124 K) x P = 0\n\u22a2 IsIntegral \u2124 { val := x, property := (_ : \u2203 p, Polynomial.Monic p \u2227 Polynomial.eval\u2082 (algebraMap \u2124 K) x p = 0) }\n[PROOFSTEP]\nrefine' \u27e8P, hPm, _\u27e9\n[GOAL]\ncase intro.intro\nK\u271d : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Field K\u271d\ninst\u271d\u00b9 : Field L\nnf : NumberField K\u271d\nK : Type u_3\ninst\u271d : Field K\nx : K\nP : Polynomial \u2124\nhPm : Polynomial.Monic P\nhP : Polynomial.eval\u2082 (algebraMap \u2124 K) x P = 0\n\u22a2 Polynomial.eval\u2082 (algebraMap \u2124 { x // x \u2208 \ud835\udcde K })\n { val := x, property := (_ : \u2203 p, Polynomial.Monic p \u2227 Polynomial.eval\u2082 (algebraMap \u2124 K) x p = 0) } P =\n 0\n[PROOFSTEP]\nrw [\u2190 Polynomial.aeval_def, \u2190 Subalgebra.coe_eq_zero, Polynomial.aeval_subalgebra_coe, Polynomial.aeval_def,\n Subtype.coe_mk, hP]\n[GOAL]\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\nnf : NumberField K\ninst\u271d : Algebra K L\n\u22a2 \u2191((fun k => { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) }) 1) = \u21911\n[PROOFSTEP]\nsimp only [Subtype.coe_mk, Subalgebra.coe_one, map_one]\n[GOAL]\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\nnf : NumberField K\ninst\u271d : Algebra K L\nx y : { x // x \u2208 \ud835\udcde K }\n\u22a2 \u2191(OneHom.toFun\n { toFun := fun k => { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k => { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) }) 1 = 1) }\n (x * y)) =\n \u2191(OneHom.toFun\n { toFun := fun k => { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k => { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) }) 1 =\n 1) }\n x *\n OneHom.toFun\n { toFun := fun k => { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k => { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) }) 1 =\n 1) }\n y)\n[PROOFSTEP]\nsimp only [Subalgebra.coe_mul, map_mul, Subtype.coe_mk]\n[GOAL]\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\nnf : NumberField K\ninst\u271d : Algebra K L\n\u22a2 \u2191(OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k => { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : { x // x \u2208 \ud835\udcde K }),\n OneHom.toFun\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k =>\n { val := \u2191(algebraMap K L) \u2191k,\n property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k =>\n { val := \u2191(algebraMap K L) \u2191k,\n property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k =>\n { val := \u2191(algebraMap K L) \u2191k,\n property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) }\n y) })\n 0) =\n \u21910\n[PROOFSTEP]\nsimp only [Subtype.coe_mk, Subalgebra.coe_zero, map_zero]\n[GOAL]\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\nnf : NumberField K\ninst\u271d : Algebra K L\nx y : { x // x \u2208 \ud835\udcde K }\n\u22a2 \u2191(OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k => { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : { x // x \u2208 \ud835\udcde K }),\n OneHom.toFun\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k =>\n { val := \u2191(algebraMap K L) \u2191k,\n property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k =>\n { val := \u2191(algebraMap K L) \u2191k,\n property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k =>\n { val := \u2191(algebraMap K L) \u2191k,\n property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) }\n y) })\n (x + y)) =\n \u2191(OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k => { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : { x // x \u2208 \ud835\udcde K }),\n OneHom.toFun\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k =>\n { val := \u2191(algebraMap K L) \u2191k,\n property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k =>\n { val := \u2191(algebraMap K L) \u2191k,\n property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k =>\n { val := \u2191(algebraMap K L) \u2191k,\n property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) }\n y) })\n x +\n OneHom.toFun\n (\u2191{\n toOneHom :=\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k => { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : { x // x \u2208 \ud835\udcde K }),\n OneHom.toFun\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k =>\n { val := \u2191(algebraMap K L) \u2191k,\n property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k =>\n { val := \u2191(algebraMap K L) \u2191k,\n property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun k =>\n { val := \u2191(algebraMap K L) \u2191k, property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) },\n map_one' :=\n (_ :\n (fun k =>\n { val := \u2191(algebraMap K L) \u2191k,\n property := (_ : IsIntegral \u2124 (\u2191(algebraMap K L) \u2191k)) })\n 1 =\n 1) }\n y) })\n y)\n[PROOFSTEP]\nsimp only [map_add, Subalgebra.coe_add, Subtype.coe_mk]\n[GOAL]\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Field L\nnf : NumberField K\n\u22a2 \u00acIsField { x // x \u2208 \ud835\udcde K }\n[PROOFSTEP]\nhave h_inj : Function.Injective (algebraMap \u2124 (\ud835\udcde K)) := RingHom.injective_int (algebraMap \u2124 (\ud835\udcde K))\n[GOAL]\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Field L\nnf : NumberField K\nh_inj : Injective \u2191(algebraMap \u2124 { x // x \u2208 \ud835\udcde K })\n\u22a2 \u00acIsField { x // x \u2208 \ud835\udcde K }\n[PROOFSTEP]\nintro hf\n[GOAL]\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Field L\nnf : NumberField K\nh_inj : Injective \u2191(algebraMap \u2124 { x // x \u2208 \ud835\udcde K })\nhf : IsField { x // x \u2208 \ud835\udcde K }\n\u22a2 False\n[PROOFSTEP]\nexact Int.not_isField (((IsIntegralClosure.isIntegral_algebra \u2124 K).isField_iff_isField h_inj).mpr hf)\n[GOAL]\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Field L\nnf : NumberField K\nx : K\n\u22a2 x \u2208 Submodule.span \u2124 (Set.range \u2191(integralBasis K)) \u2194 x \u2208 \ud835\udcde K\n[PROOFSTEP]\nrw [integralBasis, Basis.localizationLocalization_span, Subalgebra.range_isScalarTower_toAlgHom,\n Subalgebra.mem_toSubmodule]\n[GOAL]\n\u22a2 FiniteDimensional \u211a \u211a\n[PROOFSTEP]\nconvert (inferInstance : FiniteDimensional \u211a \u211a)\n[GOAL]\nf : Polynomial \u211a\nhf : Fact (Irreducible f)\n\u22a2 FiniteDimensional \u211a (AdjoinRoot f)\n[PROOFSTEP]\nconvert (AdjoinRoot.powerBasis hf.out.ne_zero).finiteDimensional\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.NumberField.Basic", "llama_tokens": 4494, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7634837635542924, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.4321459614930072}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n : \u2115\nhqn : n \u2264 q\n\u22a2 HomologicalComplex.Hom.f (P (q + 1)) n = HomologicalComplex.Hom.f (P q) n\n[PROOFSTEP]\nrcases n with (_ | n)\n[GOAL]\ncase zero\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq : \u2115\nhqn : Nat.zero \u2264 q\n\u22a2 HomologicalComplex.Hom.f (P (q + 1)) Nat.zero = HomologicalComplex.Hom.f (P q) Nat.zero\n[PROOFSTEP]\nsimp only [Nat.zero_eq, P_f_0_eq]\n[GOAL]\ncase succ\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n : \u2115\nhqn : Nat.succ n \u2264 q\n\u22a2 HomologicalComplex.Hom.f (P (q + 1)) (Nat.succ n) = HomologicalComplex.Hom.f (P q) (Nat.succ n)\n[PROOFSTEP]\nsimp only [P_succ, add_right_eq_self, comp_add, HomologicalComplex.comp_f, HomologicalComplex.add_f_apply, comp_id]\n[GOAL]\ncase succ\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n : \u2115\nhqn : Nat.succ n \u2264 q\n\u22a2 HomologicalComplex.Hom.f (P q) (Nat.succ n) \u226b HomologicalComplex.Hom.f (H\u03c3 q) (Nat.succ n) = 0\n[PROOFSTEP]\nexact (HigherFacesVanish.of_P q n).comp_H\u03c3_eq_zero (Nat.succ_le_iff.mp hqn)\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n : \u2115\nhqn : n \u2264 q\n\u22a2 HomologicalComplex.Hom.f (Q (q + 1)) n = HomologicalComplex.Hom.f (Q q) n\n[PROOFSTEP]\nsimp only [Q, HomologicalComplex.sub_f_apply, P_is_eventually_constant hqn]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.41303, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nn : \u2115\n\u22a2 (fun n => HomologicalComplex.Hom.f (P n) n) (n + 1) \u226b AlternatingFaceMapComplex.objD X n =\n AlternatingFaceMapComplex.objD X n \u226b (fun n => HomologicalComplex.Hom.f (P n) n) n\n[PROOFSTEP]\nsimpa only [\u2190 P_is_eventually_constant (show n \u2264 n by rfl), AlternatingFaceMapComplex.obj_d_eq] using\n (P (n + 1) : K[X] \u27f6 _).comm (n + 1) n\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.41303, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nn : \u2115\n\u22a2 n \u2264 n\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\n\u22a2 HomologicalComplex.Hom.f QInfty 0 = 0\n[PROOFSTEP]\ndsimp [QInfty]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\n\u22a2 \ud835\udfd9 (X.obj (op [0])) - \ud835\udfd9 (X.obj (op [0])) = 0\n[PROOFSTEP]\nsimp only [sub_self]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f PInfty n \u226b HomologicalComplex.Hom.f PInfty n = HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nsimp only [PInfty_f, P_f_idem]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\n\u22a2 PInfty \u226b PInfty = PInfty\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f (PInfty \u226b PInfty) n = HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nexact PInfty_f_idem n\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\n\u22a2 QInfty \u226b QInfty = QInfty\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f (QInfty \u226b QInfty) n = HomologicalComplex.Hom.f QInfty n\n[PROOFSTEP]\nexact QInfty_f_idem n\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f PInfty n \u226b HomologicalComplex.Hom.f QInfty n = 0\n[PROOFSTEP]\ndsimp only [QInfty]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f PInfty n \u226b HomologicalComplex.Hom.f (\ud835\udfd9 K[X] - PInfty) n = 0\n[PROOFSTEP]\nsimp only [HomologicalComplex.sub_f_apply, HomologicalComplex.id_f, comp_sub, comp_id, PInfty_f_idem, sub_self]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\n\u22a2 PInfty \u226b QInfty = 0\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f (PInfty \u226b QInfty) n = HomologicalComplex.Hom.f 0 n\n[PROOFSTEP]\napply PInfty_f_comp_QInfty_f\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f QInfty n \u226b HomologicalComplex.Hom.f PInfty n = 0\n[PROOFSTEP]\ndsimp only [QInfty]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f (\ud835\udfd9 K[X] - PInfty) n \u226b HomologicalComplex.Hom.f PInfty n = 0\n[PROOFSTEP]\nsimp only [HomologicalComplex.sub_f_apply, HomologicalComplex.id_f, sub_comp, id_comp, PInfty_f_idem, sub_self]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\n\u22a2 QInfty \u226b PInfty = 0\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f (QInfty \u226b PInfty) n = HomologicalComplex.Hom.f 0 n\n[PROOFSTEP]\napply QInfty_f_comp_PInfty_f\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\n\u22a2 PInfty + QInfty = \ud835\udfd9 K[X]\n[PROOFSTEP]\ndsimp only [QInfty]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\n\u22a2 PInfty + (\ud835\udfd9 K[X] - PInfty) = \ud835\udfd9 K[X]\n[PROOFSTEP]\nsimp only [add_sub_cancel'_right]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.354986, u_1} C\ninst\u271d : Preadditive C\nX\u271d X Y : SimplicialObject C\nf : X \u27f6 Y\n\u22a2 (alternatingFaceMapComplex C).map f \u226b (fun x => PInfty) Y = (fun x => PInfty) X \u226b (alternatingFaceMapComplex C).map f\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.354986, u_1} C\ninst\u271d : Preadditive C\nX\u271d X Y : SimplicialObject C\nf : X \u27f6 Y\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f ((alternatingFaceMapComplex C).map f \u226b (fun x => PInfty) Y) n =\n HomologicalComplex.Hom.f ((fun x => PInfty) X \u226b (alternatingFaceMapComplex C).map f) n\n[PROOFSTEP]\nexact PInfty_f_naturality n f\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{u_4, u_1} C\ninst\u271d\u00b3 : Preadditive C\nX\u271d : SimplicialObject C\nD : Type u_2\ninst\u271d\u00b2 : Category.{u_3, u_2} D\ninst\u271d\u00b9 : Preadditive D\nG : C \u2964 D\ninst\u271d : Functor.Additive G\nX : SimplicialObject C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f PInfty n = G.map (HomologicalComplex.Hom.f PInfty n)\n[PROOFSTEP]\nsimp only [PInfty_f, map_P]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 (HomologicalComplex.Hom.f PInfty n).f = NatTrans.app Y.p (op [n]) \u226b HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nlet Y\u2081 := (karoubiFunctorCategoryEmbedding _ _).obj Y\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\n\u22a2 (HomologicalComplex.Hom.f PInfty n).f = NatTrans.app Y.p (op [n]) \u226b HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nlet Y\u2082 := Y.X\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\n\u22a2 (HomologicalComplex.Hom.f PInfty n).f = NatTrans.app Y.p (op [n]) \u226b HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nlet Y\u2083 := ((whiskering _ _).obj (toKaroubi C)).obj Y.X\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\n\u22a2 (HomologicalComplex.Hom.f PInfty n).f = NatTrans.app Y.p (op [n]) \u226b HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nlet Y\u2084 := (karoubiFunctorCategoryEmbedding _ _).obj ((toKaroubi _).obj Y.X)\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\n\u22a2 (HomologicalComplex.Hom.f PInfty n).f = NatTrans.app Y.p (op [n]) \u226b HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nlet P\u2081 : K[Y\u2081] \u27f6 _ := PInfty\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\n\u22a2 (HomologicalComplex.Hom.f PInfty n).f = NatTrans.app Y.p (op [n]) \u226b HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nlet P\u2082 : K[Y\u2082] \u27f6 _ := PInfty\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\n\u22a2 (HomologicalComplex.Hom.f PInfty n).f = NatTrans.app Y.p (op [n]) \u226b HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nlet P\u2083 : K[Y\u2083] \u27f6 _ := PInfty\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\n\u22a2 (HomologicalComplex.Hom.f PInfty n).f = NatTrans.app Y.p (op [n]) \u226b HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nlet P\u2084 : K[Y\u2084] \u27f6 _ := PInfty\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\n\u22a2 (HomologicalComplex.Hom.f PInfty n).f = NatTrans.app Y.p (op [n]) \u226b HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nchange (P\u2081.f n).f = Y.p.app (op [n]) \u226b P\u2082.f n\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\n\u22a2 (HomologicalComplex.Hom.f P\u2081 n).f = NatTrans.app Y.p (op [n]) \u226b HomologicalComplex.Hom.f P\u2082 n\n[PROOFSTEP]\nhave h\u2083\u2082 : (P\u2083.f n).f = P\u2082.f n := Karoubi.hom_ext_iff.mp (map_PInfty_f (toKaroubi C) Y\u2082 n)\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\nh\u2083\u2082 : (HomologicalComplex.Hom.f P\u2083 n).f = HomologicalComplex.Hom.f P\u2082 n\n\u22a2 (HomologicalComplex.Hom.f P\u2081 n).f = NatTrans.app Y.p (op [n]) \u226b HomologicalComplex.Hom.f P\u2082 n\n[PROOFSTEP]\nhave h\u2084\u2083 : P\u2084.f n = P\u2083.f n :=\n by\n have h := Functor.congr_obj (toKaroubi_comp_karoubiFunctorCategoryEmbedding _ _) Y\u2082\n simp only [\u2190 natTransPInfty_f_app]\n congr 1\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\nh\u2083\u2082 : (HomologicalComplex.Hom.f P\u2083 n).f = HomologicalComplex.Hom.f P\u2082 n\n\u22a2 HomologicalComplex.Hom.f P\u2084 n = HomologicalComplex.Hom.f P\u2083 n\n[PROOFSTEP]\nhave h := Functor.congr_obj (toKaroubi_comp_karoubiFunctorCategoryEmbedding _ _) Y\u2082\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\nh\u2083\u2082 : (HomologicalComplex.Hom.f P\u2083 n).f = HomologicalComplex.Hom.f P\u2082 n\nh :\n (toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u22d9 karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\u2082 =\n ((whiskeringRight SimplexCategory\u1d52\u1d56 C (Karoubi C)).obj (toKaroubi C)).obj Y\u2082\n\u22a2 HomologicalComplex.Hom.f P\u2084 n = HomologicalComplex.Hom.f P\u2083 n\n[PROOFSTEP]\nsimp only [\u2190 natTransPInfty_f_app]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\nh\u2083\u2082 : (HomologicalComplex.Hom.f P\u2083 n).f = HomologicalComplex.Hom.f P\u2082 n\nh :\n (toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u22d9 karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\u2082 =\n ((whiskeringRight SimplexCategory\u1d52\u1d56 C (Karoubi C)).obj (toKaroubi C)).obj Y\u2082\n\u22a2 NatTrans.app (natTransPInfty_f (Karoubi C) n)\n ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)) =\n NatTrans.app (natTransPInfty_f (Karoubi C) n) (((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X)\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\nh\u2083\u2082 : (HomologicalComplex.Hom.f P\u2083 n).f = HomologicalComplex.Hom.f P\u2082 n\nh\u2084\u2083 : HomologicalComplex.Hom.f P\u2084 n = HomologicalComplex.Hom.f P\u2083 n\n\u22a2 (HomologicalComplex.Hom.f P\u2081 n).f = NatTrans.app Y.p (op [n]) \u226b HomologicalComplex.Hom.f P\u2082 n\n[PROOFSTEP]\nhave h\u2081\u2084 :=\n Idempotents.natTrans_eq ((\ud835\udfd9 (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C)) \u25eb (natTransPInfty_f (Karoubi C) n))\n Y\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\nh\u2083\u2082 : (HomologicalComplex.Hom.f P\u2083 n).f = HomologicalComplex.Hom.f P\u2082 n\nh\u2084\u2083 : HomologicalComplex.Hom.f P\u2084 n = HomologicalComplex.Hom.f P\u2083 n\nh\u2081\u2084 :\n NatTrans.app (\ud835\udfd9 (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C) \u25eb natTransPInfty_f (Karoubi C) n) Y =\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n alternatingFaceMapComplex (Karoubi C) \u22d9 HomologicalComplex.eval (Karoubi C) (ComplexShape.down \u2115) n).map\n (Karoubi.decompId_i Y) \u226b\n NatTrans.app (\ud835\udfd9 (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C) \u25eb natTransPInfty_f (Karoubi C) n)\n (Karoubi.mk Y.X (\ud835\udfd9 Y.X)) \u226b\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n alternatingFaceMapComplex (Karoubi C) \u22d9 HomologicalComplex.eval (Karoubi C) (ComplexShape.down \u2115) n).map\n (Karoubi.decompId_p Y)\n\u22a2 (HomologicalComplex.Hom.f P\u2081 n).f = NatTrans.app Y.p (op [n]) \u226b HomologicalComplex.Hom.f P\u2082 n\n[PROOFSTEP]\ndsimp [natTransPInfty_f] at h\u2081\u2084 \n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\nh\u2083\u2082 : (HomologicalComplex.Hom.f P\u2083 n).f = HomologicalComplex.Hom.f P\u2082 n\nh\u2084\u2083 : HomologicalComplex.Hom.f P\u2084 n = HomologicalComplex.Hom.f P\u2083 n\nh\u2081\u2084 :\n (Karoubi.Hom.mk (NatTrans.app Y.p (op [n])) \u226b HomologicalComplex.Hom.f PInfty n) \u226b\n Karoubi.Hom.mk (NatTrans.app Y.p (op [n])) =\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)) (op [n]) \u226b\n ((Karoubi.Hom.mk (\ud835\udfd9 (Y.X.obj (op [n]))) \u226b HomologicalComplex.Hom.f PInfty n) \u226b\n Karoubi.Hom.mk (\ud835\udfd9 (Y.X.obj (op [n])))) \u226b\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)) (op [n])\n\u22a2 (HomologicalComplex.Hom.f P\u2081 n).f = NatTrans.app Y.p (op [n]) \u226b HomologicalComplex.Hom.f P\u2082 n\n[PROOFSTEP]\nerw [id_comp, id_comp, comp_id, comp_id] at h\u2081\u2084 \n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\nh\u2083\u2082 : (HomologicalComplex.Hom.f P\u2083 n).f = HomologicalComplex.Hom.f P\u2082 n\nh\u2084\u2083 : HomologicalComplex.Hom.f P\u2084 n = HomologicalComplex.Hom.f P\u2083 n\nh\u2081\u2084 :\n HomologicalComplex.Hom.f PInfty n =\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)) (op [n]) \u226b\n HomologicalComplex.Hom.f PInfty n \u226b\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)) (op [n])\n\u22a2 (HomologicalComplex.Hom.f P\u2081 n).f = NatTrans.app Y.p (op [n]) \u226b HomologicalComplex.Hom.f P\u2082 n\n[PROOFSTEP]\nrw [\u2190 h\u2083\u2082, \u2190 h\u2084\u2083, h\u2081\u2084]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\nh\u2083\u2082 : (HomologicalComplex.Hom.f P\u2083 n).f = HomologicalComplex.Hom.f P\u2082 n\nh\u2084\u2083 : HomologicalComplex.Hom.f P\u2084 n = HomologicalComplex.Hom.f P\u2083 n\nh\u2081\u2084 :\n HomologicalComplex.Hom.f PInfty n =\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)) (op [n]) \u226b\n HomologicalComplex.Hom.f PInfty n \u226b\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)) (op [n])\n\u22a2 (NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)) (op [n]) \u226b\n HomologicalComplex.Hom.f PInfty n \u226b\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)) (op [n])).f =\n NatTrans.app Y.p (op [n]) \u226b (HomologicalComplex.Hom.f P\u2084 n).f\n[PROOFSTEP]\nsimp only [KaroubiFunctorCategoryEmbedding.map_app_f, Karoubi.decompId_p_f, Karoubi.decompId_i_f, Karoubi.comp_f]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\nh\u2083\u2082 : (HomologicalComplex.Hom.f P\u2083 n).f = HomologicalComplex.Hom.f P\u2082 n\nh\u2084\u2083 : HomologicalComplex.Hom.f P\u2084 n = HomologicalComplex.Hom.f P\u2083 n\nh\u2081\u2084 :\n HomologicalComplex.Hom.f PInfty n =\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)) (op [n]) \u226b\n HomologicalComplex.Hom.f PInfty n \u226b\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)) (op [n])\n\u22a2 NatTrans.app Y.p (op [n]) \u226b (HomologicalComplex.Hom.f PInfty n).f \u226b NatTrans.app Y.p (op [n]) =\n NatTrans.app Y.p (op [n]) \u226b (HomologicalComplex.Hom.f PInfty n).f\n[PROOFSTEP]\nlet \u03c0 : Y\u2084 \u27f6 Y\u2084 := (toKaroubi _ \u22d9 karoubiFunctorCategoryEmbedding _ _).map Y.p\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\nh\u2083\u2082 : (HomologicalComplex.Hom.f P\u2083 n).f = HomologicalComplex.Hom.f P\u2082 n\nh\u2084\u2083 : HomologicalComplex.Hom.f P\u2084 n = HomologicalComplex.Hom.f P\u2083 n\nh\u2081\u2084 :\n HomologicalComplex.Hom.f PInfty n =\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)) (op [n]) \u226b\n HomologicalComplex.Hom.f PInfty n \u226b\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)) (op [n])\n\u03c0 : Y\u2084 \u27f6 Y\u2084 := (toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u22d9 karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).map Y.p\n\u22a2 NatTrans.app Y.p (op [n]) \u226b (HomologicalComplex.Hom.f PInfty n).f \u226b NatTrans.app Y.p (op [n]) =\n NatTrans.app Y.p (op [n]) \u226b (HomologicalComplex.Hom.f PInfty n).f\n[PROOFSTEP]\nhave eq := Karoubi.hom_ext_iff.mp (PInfty_f_naturality n \u03c0)\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\nh\u2083\u2082 : (HomologicalComplex.Hom.f P\u2083 n).f = HomologicalComplex.Hom.f P\u2082 n\nh\u2084\u2083 : HomologicalComplex.Hom.f P\u2084 n = HomologicalComplex.Hom.f P\u2083 n\nh\u2081\u2084 :\n HomologicalComplex.Hom.f PInfty n =\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)) (op [n]) \u226b\n HomologicalComplex.Hom.f PInfty n \u226b\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)) (op [n])\n\u03c0 : Y\u2084 \u27f6 Y\u2084 := (toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u22d9 karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).map Y.p\neq :\n (NatTrans.app \u03c0 (op [n]) \u226b HomologicalComplex.Hom.f PInfty n).f =\n (HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app \u03c0 (op [n])).f\n\u22a2 NatTrans.app Y.p (op [n]) \u226b (HomologicalComplex.Hom.f PInfty n).f \u226b NatTrans.app Y.p (op [n]) =\n NatTrans.app Y.p (op [n]) \u226b (HomologicalComplex.Hom.f PInfty n).f\n[PROOFSTEP]\nsimp only [Karoubi.comp_f] at eq \n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\nh\u2083\u2082 : (HomologicalComplex.Hom.f P\u2083 n).f = HomologicalComplex.Hom.f P\u2082 n\nh\u2084\u2083 : HomologicalComplex.Hom.f P\u2084 n = HomologicalComplex.Hom.f P\u2083 n\nh\u2081\u2084 :\n HomologicalComplex.Hom.f PInfty n =\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)) (op [n]) \u226b\n HomologicalComplex.Hom.f PInfty n \u226b\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)) (op [n])\n\u03c0 : Y\u2084 \u27f6 Y\u2084 := (toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u22d9 karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).map Y.p\neq :\n (NatTrans.app ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u22d9 karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).map Y.p)\n (op [n])).f \u226b\n (HomologicalComplex.Hom.f PInfty n).f =\n (HomologicalComplex.Hom.f PInfty n).f \u226b\n (NatTrans.app ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u22d9 karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).map Y.p)\n (op [n])).f\n\u22a2 NatTrans.app Y.p (op [n]) \u226b (HomologicalComplex.Hom.f PInfty n).f \u226b NatTrans.app Y.p (op [n]) =\n NatTrans.app Y.p (op [n]) \u226b (HomologicalComplex.Hom.f PInfty n).f\n[PROOFSTEP]\ndsimp at eq \n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nY : Karoubi (SimplicialObject C)\nn : \u2115\nY\u2081 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj Y\nY\u2082 : SimplicialObject C := Y.X\nY\u2083 : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X\nY\u2084 : SimplexCategory\u1d52\u1d56 \u2964 Karoubi C :=\n (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).obj ((toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C)).obj Y.X)\nP\u2081 : K[Y\u2081] \u27f6 K[Y\u2081] := PInfty\nP\u2082 : K[Y\u2082] \u27f6 K[Y\u2082] := PInfty\nP\u2083 : K[Y\u2083] \u27f6 K[Y\u2083] := PInfty\nP\u2084 : K[Y\u2084] \u27f6 K[Y\u2084] := PInfty\nh\u2083\u2082 : (HomologicalComplex.Hom.f P\u2083 n).f = HomologicalComplex.Hom.f P\u2082 n\nh\u2084\u2083 : HomologicalComplex.Hom.f P\u2084 n = HomologicalComplex.Hom.f P\u2083 n\nh\u2081\u2084 :\n HomologicalComplex.Hom.f PInfty n =\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)) (op [n]) \u226b\n HomologicalComplex.Hom.f PInfty n \u226b\n NatTrans.app (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)) (op [n])\n\u03c0 : Y\u2084 \u27f6 Y\u2084 := (toKaroubi (SimplexCategory\u1d52\u1d56 \u2964 C) \u22d9 karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C).map Y.p\neq :\n NatTrans.app Y.p (op [n]) \u226b (HomologicalComplex.Hom.f PInfty n).f =\n (HomologicalComplex.Hom.f PInfty n).f \u226b NatTrans.app Y.p (op [n])\n\u22a2 NatTrans.app Y.p (op [n]) \u226b (HomologicalComplex.Hom.f PInfty n).f \u226b NatTrans.app Y.p (op [n]) =\n NatTrans.app Y.p (op [n]) \u226b (HomologicalComplex.Hom.f PInfty n).f\n[PROOFSTEP]\nrw [\u2190 eq, app_idem_assoc Y (op [n])]\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicTopology.DoldKan.PInfty", "llama_tokens": 14603, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.43206732566506534}} {"text": "[GOAL]\na b : \u211d\nh : 0 < b\n\u22a2 IntegrableOn (fun x => exp (-b * x)) (Ioi a)\n[PROOFSTEP]\nhave : Tendsto (fun x => -exp (-b * x) / b) atTop (\ud835\udcdd (-0 / b)) :=\n by\n refine' Tendsto.div_const (Tendsto.neg _) _\n exact tendsto_exp_atBot.comp (tendsto_id.neg_const_mul_atTop (Right.neg_neg_iff.2 h))\n[GOAL]\na b : \u211d\nh : 0 < b\n\u22a2 Tendsto (fun x => -exp (-b * x) / b) atTop (\ud835\udcdd (-0 / b))\n[PROOFSTEP]\nrefine' Tendsto.div_const (Tendsto.neg _) _\n[GOAL]\na b : \u211d\nh : 0 < b\n\u22a2 Tendsto (fun x => exp (-b * x)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact tendsto_exp_atBot.comp (tendsto_id.neg_const_mul_atTop (Right.neg_neg_iff.2 h))\n[GOAL]\na b : \u211d\nh : 0 < b\nthis : Tendsto (fun x => -exp (-b * x) / b) atTop (\ud835\udcdd (-0 / b))\n\u22a2 IntegrableOn (fun x => exp (-b * x)) (Ioi a)\n[PROOFSTEP]\nrefine' integrableOn_Ioi_deriv_of_nonneg' (fun x _ => _) (fun x _ => (exp_pos _).le) this\n[GOAL]\na b : \u211d\nh : 0 < b\nthis : Tendsto (fun x => -exp (-b * x) / b) atTop (\ud835\udcdd (-0 / b))\nx : \u211d\nx\u271d : x \u2208 Ici a\n\u22a2 HasDerivAt (fun x => -exp (-b * x) / b) (exp (-b * x)) x\n[PROOFSTEP]\nsimpa [h.ne'] using ((hasDerivAt_id x).const_mul b).neg.exp.neg.div_const b\n[GOAL]\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\n\u22a2 IntegrableOn f (Ioi a)\n[PROOFSTEP]\ncases' h2.isBigOWith with c h3\n[GOAL]\ncase intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc : \u211d\nh3 : Asymptotics.IsBigOWith c atTop f fun x => exp (-b * x)\n\u22a2 IntegrableOn f (Ioi a)\n[PROOFSTEP]\nrw [Asymptotics.isBigOWith_iff, eventually_atTop] at h3 \n[GOAL]\ncase intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc : \u211d\nh3 : \u2203 a, \u2200 (b_1 : \u211d), b_1 \u2265 a \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\n\u22a2 IntegrableOn f (Ioi a)\n[PROOFSTEP]\ncases' h3 with r bdr\n[GOAL]\ncase intro.intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\n\u22a2 IntegrableOn f (Ioi a)\n[PROOFSTEP]\nlet v := max a r\n[GOAL]\ncase intro.intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\n\u22a2 IntegrableOn f (Ioi a)\n[PROOFSTEP]\nhave int_left : IntegrableOn f (Ioc a v) :=\n by\n rw [\u2190 intervalIntegrable_iff_integrable_Ioc_of_le (le_max_left a r)]\n have u : Icc a v \u2286 Ici a := Icc_subset_Ici_self\n exact (h1.mono u).intervalIntegrable_of_Icc (le_max_left a r)\n[GOAL]\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\n\u22a2 IntegrableOn f (Ioc a v)\n[PROOFSTEP]\nrw [\u2190 intervalIntegrable_iff_integrable_Ioc_of_le (le_max_left a r)]\n[GOAL]\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\n\u22a2 IntervalIntegrable f volume a (max a r)\n[PROOFSTEP]\nhave u : Icc a v \u2286 Ici a := Icc_subset_Ici_self\n[GOAL]\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\nu : Icc a v \u2286 Ici a\n\u22a2 IntervalIntegrable f volume a (max a r)\n[PROOFSTEP]\nexact (h1.mono u).intervalIntegrable_of_Icc (le_max_left a r)\n[GOAL]\ncase intro.intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\n\u22a2 IntegrableOn f (Ioi a)\n[PROOFSTEP]\nsuffices IntegrableOn f (Ioi v) by\n have t := integrableOn_union.mpr \u27e8int_left, this\u27e9\n simpa only [Ioc_union_Ioi_eq_Ioi, le_max_iff, le_refl, true_or_iff] using t\n[GOAL]\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : IntegrableOn f (Ioi v)\n\u22a2 IntegrableOn f (Ioi a)\n[PROOFSTEP]\nhave t := integrableOn_union.mpr \u27e8int_left, this\u27e9\n[GOAL]\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : IntegrableOn f (Ioi v)\nt : IntegrableOn f (Ioc a v \u222a Ioi v)\n\u22a2 IntegrableOn f (Ioi a)\n[PROOFSTEP]\nsimpa only [Ioc_union_Ioi_eq_Ioi, le_max_iff, le_refl, true_or_iff] using t\n[GOAL]\ncase intro.intro\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\n\u22a2 IntegrableOn f (Ioi v)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.left\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\n\u22a2 AEStronglyMeasurable f (Measure.restrict volume (Ioi v))\n[PROOFSTEP]\nexact (h1.mono <| Ioi_subset_Ici <| le_max_left a r).aestronglyMeasurable measurableSet_Ioi\n[GOAL]\ncase intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\n\u22a2 HasFiniteIntegral f\n[PROOFSTEP]\nhave : HasFiniteIntegral (fun x : \u211d => c * exp (-b * x)) (volume.restrict (Ioi v)) :=\n (exp_neg_integrableOn_Ioi v h0).hasFiniteIntegral.const_mul c\n[GOAL]\ncase intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * exp (-b * x)\n\u22a2 HasFiniteIntegral f\n[PROOFSTEP]\napply this.mono\n[GOAL]\ncase intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * exp (-b * x)\n\u22a2 \u2200\u1d50 (a : \u211d) \u2202Measure.restrict volume (Ioi v), \u2016f a\u2016 \u2264 \u2016c * exp (-b * a)\u2016\n[PROOFSTEP]\nrefine' (ae_restrict_iff' measurableSet_Ioi).mpr _\n[GOAL]\ncase intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * exp (-b * x)\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 Ioi v \u2192 \u2016f x\u2016 \u2264 \u2016c * exp (-b * x)\u2016\n[PROOFSTEP]\nrefine' ae_of_all _ fun x h1x => _\n[GOAL]\ncase intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * exp (-b * x)\nx : \u211d\nh1x : x \u2208 Ioi v\n\u22a2 \u2016f x\u2016 \u2264 \u2016c * exp (-b * x)\u2016\n[PROOFSTEP]\nrw [norm_mul, norm_eq_abs]\n[GOAL]\ncase intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * exp (-b * x)\nx : \u211d\nh1x : x \u2208 Ioi v\n\u22a2 |f x| \u2264 \u2016c\u2016 * \u2016exp (-b * x)\u2016\n[PROOFSTEP]\nrw [mem_Ioi] at h1x \n[GOAL]\ncase intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nbdr : \u2200 (b_1 : \u211d), b_1 \u2265 r \u2192 \u2016f b_1\u2016 \u2264 c * \u2016exp (-b * b_1)\u2016\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * exp (-b * x)\nx : \u211d\nh1x : v < x\n\u22a2 |f x| \u2264 \u2016c\u2016 * \u2016exp (-b * x)\u2016\n[PROOFSTEP]\nspecialize bdr x ((le_max_right a r).trans h1x.le)\n[GOAL]\ncase intro.intro.right\nf : \u211d \u2192 \u211d\na b : \u211d\nh0 : 0 < b\nh1 : ContinuousOn f (Ici a)\nh2 : f =O[atTop] fun x => exp (-b * x)\nc r : \u211d\nv : \u211d := max a r\nint_left : IntegrableOn f (Ioc a v)\nthis : HasFiniteIntegral fun x => c * exp (-b * x)\nx : \u211d\nh1x : v < x\nbdr : \u2016f x\u2016 \u2264 c * \u2016exp (-b * x)\u2016\n\u22a2 |f x| \u2264 \u2016c\u2016 * \u2016exp (-b * x)\u2016\n[PROOFSTEP]\nexact bdr.trans (mul_le_mul_of_nonneg_right (le_abs_self c) (norm_nonneg _))\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Integral.ExpDecay", "llama_tokens": 4620, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124812, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.43200586027441495}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : DecidableEq n\nA : Matrix n n \u03b1\nh : IsDiag A\ni j : n\n\u22a2 diagonal (diag A) i j = A i j\n[PROOFSTEP]\nobtain rfl | hij := Decidable.eq_or_ne i j\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : DecidableEq n\nA : Matrix n n \u03b1\nh : IsDiag A\ni : n\n\u22a2 diagonal (diag A) i i = A i i\n[PROOFSTEP]\nrw [diagonal_apply_eq, diag]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : DecidableEq n\nA : Matrix n n \u03b1\nh : IsDiag A\ni j : n\nhij : i \u2260 j\n\u22a2 diagonal (diag A) i j = A i j\n[PROOFSTEP]\nrw [diagonal_apply_ne _ hij, h hij]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : Zero \u03b2\nA : Matrix n n \u03b1\nha : IsDiag A\nf : \u03b1 \u2192 \u03b2\nhf : f 0 = 0\n\u22a2 IsDiag (Matrix.map A f)\n[PROOFSTEP]\nintro i j h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : Zero \u03b2\nA : Matrix n n \u03b1\nha : IsDiag A\nf : \u03b1 \u2192 \u03b2\nhf : f 0 = 0\ni j : n\nh : i \u2260 j\n\u22a2 Matrix.map A f i j = 0\n[PROOFSTEP]\nsimp [ha h, hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : AddGroup \u03b1\nA : Matrix n n \u03b1\nha : IsDiag A\n\u22a2 IsDiag (-A)\n[PROOFSTEP]\nintro i j h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : AddGroup \u03b1\nA : Matrix n n \u03b1\nha : IsDiag A\ni j : n\nh : i \u2260 j\n\u22a2 (-A) i j = 0\n[PROOFSTEP]\nsimp [ha h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : AddZeroClass \u03b1\nA B : Matrix n n \u03b1\nha : IsDiag A\nhb : IsDiag B\n\u22a2 IsDiag (A + B)\n[PROOFSTEP]\nintro i j h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : AddZeroClass \u03b1\nA B : Matrix n n \u03b1\nha : IsDiag A\nhb : IsDiag B\ni j : n\nh : i \u2260 j\n\u22a2 (A + B) i j = 0\n[PROOFSTEP]\nsimp [ha h, hb h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : AddGroup \u03b1\nA B : Matrix n n \u03b1\nha : IsDiag A\nhb : IsDiag B\n\u22a2 IsDiag (A - B)\n[PROOFSTEP]\nintro i j h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : AddGroup \u03b1\nA B : Matrix n n \u03b1\nha : IsDiag A\nhb : IsDiag B\ni j : n\nh : i \u2260 j\n\u22a2 (A - B) i j = 0\n[PROOFSTEP]\nsimp [ha h, hb h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b1\ninst\u271d : DistribMulAction R \u03b1\nk : R\nA : Matrix n n \u03b1\nha : IsDiag A\n\u22a2 IsDiag (k \u2022 A)\n[PROOFSTEP]\nintro i j h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b1\ninst\u271d : DistribMulAction R \u03b1\nk : R\nA : Matrix n n \u03b1\nha : IsDiag A\ni j : n\nh : i \u2260 j\n\u22a2 (k \u2022 A) i j = 0\n[PROOFSTEP]\nsimp [ha h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : StarRing \u03b1\nA : Matrix n n \u03b1\nha : IsDiag A\u1d34\n\u22a2 IsDiag A\n[PROOFSTEP]\nconvert ha.conjTranspose\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : StarRing \u03b1\nA : Matrix n n \u03b1\nha : IsDiag A\u1d34\n\u22a2 A = A\u1d34\u1d34\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : MulZeroClass \u03b1\nA : Matrix m m \u03b1\nB : Matrix n n \u03b1\nhA : IsDiag A\nhB : IsDiag B\n\u22a2 IsDiag (kroneckerMap (fun x x_1 => x * x_1) A B)\n[PROOFSTEP]\nrintro \u27e8a, b\u27e9 \u27e8c, d\u27e9 h\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : MulZeroClass \u03b1\nA : Matrix m m \u03b1\nB : Matrix n n \u03b1\nhA : IsDiag A\nhB : IsDiag B\na : m\nb : n\nc : m\nd : n\nh : (a, b) \u2260 (c, d)\n\u22a2 kroneckerMap (fun x x_1 => x * x_1) A B (a, b) (c, d) = 0\n[PROOFSTEP]\nsimp only [Prod.mk.inj_iff, Ne.def, not_and_or] at h \n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : MulZeroClass \u03b1\nA : Matrix m m \u03b1\nB : Matrix n n \u03b1\nhA : IsDiag A\nhB : IsDiag B\na : m\nb : n\nc : m\nd : n\nh : \u00aca = c \u2228 \u00acb = d\n\u22a2 kroneckerMap (fun x x_1 => x * x_1) A B (a, b) (c, d) = 0\n[PROOFSTEP]\ncases' h with hac hbd\n[GOAL]\ncase mk.mk.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : MulZeroClass \u03b1\nA : Matrix m m \u03b1\nB : Matrix n n \u03b1\nhA : IsDiag A\nhB : IsDiag B\na : m\nb : n\nc : m\nd : n\nhac : \u00aca = c\n\u22a2 kroneckerMap (fun x x_1 => x * x_1) A B (a, b) (c, d) = 0\n[PROOFSTEP]\nsimp [hA hac]\n[GOAL]\ncase mk.mk.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : MulZeroClass \u03b1\nA : Matrix m m \u03b1\nB : Matrix n n \u03b1\nhA : IsDiag A\nhB : IsDiag B\na : m\nb : n\nc : m\nd : n\nhbd : \u00acb = d\n\u22a2 kroneckerMap (fun x x_1 => x * x_1) A B (a, b) (c, d) = 0\n[PROOFSTEP]\nsimp [hB hbd]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix n n \u03b1\nh : IsDiag A\n\u22a2 IsSymm A\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix n n \u03b1\nh : IsDiag A\ni j : n\n\u22a2 A\u1d40 i j = A i j\n[PROOFSTEP]\nby_cases g : i = j\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix n n \u03b1\nh : IsDiag A\ni j : n\ng : i = j\n\u22a2 A\u1d40 i j = A i j\n[PROOFSTEP]\nrw [g, transpose_apply]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix n n \u03b1\nh : IsDiag A\ni j : n\ng : \u00aci = j\n\u22a2 A\u1d40 i j = A i j\n[PROOFSTEP]\nsimp [h g, h (Ne.symm g)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nD : Matrix n n \u03b1\nha : IsDiag A\nhd : IsDiag D\n\u22a2 IsDiag (Matrix.fromBlocks A 0 0 D)\n[PROOFSTEP]\nrintro (i | i) (j | j) hij\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nD : Matrix n n \u03b1\nha : IsDiag A\nhd : IsDiag D\ni j : m\nhij : Sum.inl i \u2260 Sum.inl j\n\u22a2 Matrix.fromBlocks A 0 0 D (Sum.inl i) (Sum.inl j) = 0\n[PROOFSTEP]\nexact ha (ne_of_apply_ne _ hij)\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nD : Matrix n n \u03b1\nha : IsDiag A\nhd : IsDiag D\ni : m\nj : n\nhij : Sum.inl i \u2260 Sum.inr j\n\u22a2 Matrix.fromBlocks A 0 0 D (Sum.inl i) (Sum.inr j) = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nD : Matrix n n \u03b1\nha : IsDiag A\nhd : IsDiag D\ni : n\nj : m\nhij : Sum.inr i \u2260 Sum.inl j\n\u22a2 Matrix.fromBlocks A 0 0 D (Sum.inr i) (Sum.inl j) = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nD : Matrix n n \u03b1\nha : IsDiag A\nhd : IsDiag D\ni j : n\nhij : Sum.inr i \u2260 Sum.inr j\n\u22a2 Matrix.fromBlocks A 0 0 D (Sum.inr i) (Sum.inr j) = 0\n[PROOFSTEP]\nexact hd (ne_of_apply_ne _ hij)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\n\u22a2 IsDiag (fromBlocks A B C D) \u2194 IsDiag A \u2227 B = 0 \u2227 C = 0 \u2227 IsDiag D\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\n\u22a2 IsDiag (fromBlocks A B C D) \u2192 IsDiag A \u2227 B = 0 \u2227 C = 0 \u2227 IsDiag D\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nh : IsDiag (fromBlocks A B C D)\n\u22a2 IsDiag A \u2227 B = 0 \u2227 C = 0 \u2227 IsDiag D\n[PROOFSTEP]\nrefine' \u27e8fun i j hij => _, ext fun i j => _, ext fun i j => _, fun i j hij => _\u27e9\n[GOAL]\ncase mp.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nh : IsDiag (fromBlocks A B C D)\ni j : m\nhij : i \u2260 j\n\u22a2 A i j = 0\n[PROOFSTEP]\nexact h (Sum.inl_injective.ne hij)\n[GOAL]\ncase mp.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nh : IsDiag (fromBlocks A B C D)\ni : m\nj : n\n\u22a2 B i j = OfNat.ofNat 0 i j\n[PROOFSTEP]\nexact h Sum.inl_ne_inr\n[GOAL]\ncase mp.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nh : IsDiag (fromBlocks A B C D)\ni : n\nj : m\n\u22a2 C i j = OfNat.ofNat 0 i j\n[PROOFSTEP]\nexact h Sum.inr_ne_inl\n[GOAL]\ncase mp.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nh : IsDiag (fromBlocks A B C D)\ni j : n\nhij : i \u2260 j\n\u22a2 D i j = 0\n[PROOFSTEP]\nexact h (Sum.inr_injective.ne hij)\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\n\u22a2 IsDiag A \u2227 B = 0 \u2227 C = 0 \u2227 IsDiag D \u2192 IsDiag (fromBlocks A B C D)\n[PROOFSTEP]\nrintro \u27e8ha, hb, hc, hd\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nha : IsDiag A\nhb : B = 0\nhc : C = 0\nhd : IsDiag D\n\u22a2 IsDiag (fromBlocks A B C D)\n[PROOFSTEP]\nconvert IsDiag.fromBlocks ha hd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nh : IsSymm (Matrix.fromBlocks A 0 C D)\nha : IsDiag A\nhd : IsDiag D\n\u22a2 IsDiag (Matrix.fromBlocks A 0 C D)\n[PROOFSTEP]\nrw [\u2190 (isSymm_fromBlocks_iff.1 h).2.1]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst\u271d : Zero \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nh : IsSymm (Matrix.fromBlocks A 0 C D)\nha : IsDiag A\nhd : IsDiag D\n\u22a2 IsDiag (Matrix.fromBlocks A 0 0\u1d40 D)\n[PROOFSTEP]\nexact ha.fromBlocks hd\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.IsDiag", "llama_tokens": 5123, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494421679929, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.4319935406406241}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : AddMonoidWithOne \u03b1\ninst\u271d : AddMonoidWithOne \u03b2\nn : \u2115\n\u22a2 (\u2191n).fst = \u2191n\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : AddMonoidWithOne \u03b1\ninst\u271d : AddMonoidWithOne \u03b2\n\u22a2 (\u2191Nat.zero).fst = \u2191Nat.zero\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : AddMonoidWithOne \u03b1\ninst\u271d : AddMonoidWithOne \u03b2\nn\u271d : \u2115\nn_ih\u271d : (\u2191n\u271d).fst = \u2191n\u271d\n\u22a2 (\u2191(Nat.succ n\u271d)).fst = \u2191(Nat.succ n\u271d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : AddMonoidWithOne \u03b1\ninst\u271d : AddMonoidWithOne \u03b2\nn : \u2115\n\u22a2 (\u2191n).snd = \u2191n\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : AddMonoidWithOne \u03b1\ninst\u271d : AddMonoidWithOne \u03b2\n\u22a2 (\u2191Nat.zero).snd = \u2191Nat.zero\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : AddMonoidWithOne \u03b1\ninst\u271d : AddMonoidWithOne \u03b2\nn\u271d : \u2115\nn_ih\u271d : (\u2191n\u271d).snd = \u2191n\u271d\n\u22a2 (\u2191(Nat.succ n\u271d)).snd = \u2191(Nat.succ n\u271d)\n[PROOFSTEP]\nsimp [*]\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Cast.Prod", "llama_tokens": 560, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056295505783, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.4315909201611935}} {"text": "[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\nhs : x \u2208 (chartAt H c).toLocalEquiv.source\nhd : dist (\u2191(extChartAt I c) x) (\u2191(extChartAt I c) c) \u2264 f.rIn\n\u22a2 \u2191f x = 1\n[PROOFSTEP]\nsimp only [f.eqOn_source hs, (\u00b7 \u2218 \u00b7), f.one_of_mem_closedBall hd]\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n\u22a2 support \u2191f = (chartAt H c).toLocalEquiv.source \u2229 \u2191(extChartAt I c) \u207b\u00b9' ball (\u2191(extChartAt I c) c) f.rOut\n[PROOFSTEP]\nrw [coe_def, support_indicator, support_comp_eq_preimage, \u2190 extChartAt_source I, \u2190\n (extChartAt I c).symm_image_target_inter_eq', \u2190 (extChartAt I c).symm_image_target_inter_eq', f.support_eq]\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n\u22a2 IsOpen (support \u2191f)\n[PROOFSTEP]\nrw [support_eq_inter_preimage]\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n\u22a2 IsOpen ((chartAt H c).toLocalEquiv.source \u2229 \u2191(extChartAt I c) \u207b\u00b9' ball (\u2191(extChartAt I c) c) f.rOut)\n[PROOFSTEP]\nexact isOpen_extChartAt_preimage I c isOpen_ball\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n\u22a2 support \u2191f = \u2191(LocalEquiv.symm (extChartAt I c)) '' (ball (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I)\n[PROOFSTEP]\nrw [f.support_eq_inter_preimage, \u2190 extChartAt_source I, \u2190 (extChartAt I c).symm_image_target_inter_eq', inter_comm,\n ball_inter_range_eq_ball_inter_target]\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n\u22a2 support \u2191f \u2286 (chartAt H c).toLocalEquiv.source\n[PROOFSTEP]\nrw [f.support_eq_inter_preimage, \u2190 extChartAt_source I]\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n\u22a2 (extChartAt I c).source \u2229 \u2191(extChartAt I c) \u207b\u00b9' ball (\u2191(extChartAt I c) c) f.rOut \u2286 (extChartAt I c).source\n[PROOFSTEP]\nexact inter_subset_left _ _\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhs : s \u2286 support \u2191f\n\u22a2 \u2191(extChartAt I c) '' s =\n closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I \u2229 \u2191(LocalEquiv.symm (extChartAt I c)) \u207b\u00b9' s\n[PROOFSTEP]\nrw [support_eq_inter_preimage, subset_inter_iff, \u2190 extChartAt_source I, \u2190 image_subset_iff] at hs \n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhs : s \u2286 (extChartAt I c).source \u2227 \u2191(extChartAt I c) '' s \u2286 ball (\u2191(extChartAt I c) c) f.rOut\n\u22a2 \u2191(extChartAt I c) '' s =\n closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I \u2229 \u2191(LocalEquiv.symm (extChartAt I c)) \u207b\u00b9' s\n[PROOFSTEP]\ncases' hs with hse hsf\n[GOAL]\ncase intro\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhse : s \u2286 (extChartAt I c).source\nhsf : \u2191(extChartAt I c) '' s \u2286 ball (\u2191(extChartAt I c) c) f.rOut\n\u22a2 \u2191(extChartAt I c) '' s =\n closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I \u2229 \u2191(LocalEquiv.symm (extChartAt I c)) \u207b\u00b9' s\n[PROOFSTEP]\napply Subset.antisymm\n[GOAL]\ncase intro.h\u2081\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhse : s \u2286 (extChartAt I c).source\nhsf : \u2191(extChartAt I c) '' s \u2286 ball (\u2191(extChartAt I c) c) f.rOut\n\u22a2 \u2191(extChartAt I c) '' s \u2286\n closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I \u2229 \u2191(LocalEquiv.symm (extChartAt I c)) \u207b\u00b9' s\n[PROOFSTEP]\nrefine' subset_inter (subset_inter (hsf.trans ball_subset_closedBall) _) _\n[GOAL]\ncase intro.h\u2081.refine'_1\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhse : s \u2286 (extChartAt I c).source\nhsf : \u2191(extChartAt I c) '' s \u2286 ball (\u2191(extChartAt I c) c) f.rOut\n\u22a2 \u2191(extChartAt I c) '' s \u2286 range \u2191I\n[PROOFSTEP]\nrintro _ \u27e8x, -, rfl\u27e9\n[GOAL]\ncase intro.h\u2081.refine'_1.intro.intro\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx\u271d : M\ns : Set M\nhse : s \u2286 (extChartAt I c).source\nhsf : \u2191(extChartAt I c) '' s \u2286 ball (\u2191(extChartAt I c) c) f.rOut\nx : M\n\u22a2 \u2191(extChartAt I c) x \u2208 range \u2191I\n[PROOFSTEP]\nexact mem_range_self _\n[GOAL]\ncase intro.h\u2081.refine'_2\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhse : s \u2286 (extChartAt I c).source\nhsf : \u2191(extChartAt I c) '' s \u2286 ball (\u2191(extChartAt I c) c) f.rOut\n\u22a2 \u2191(extChartAt I c) '' s \u2286 \u2191(LocalEquiv.symm (extChartAt I c)) \u207b\u00b9' s\n[PROOFSTEP]\nrw [(extChartAt I c).image_eq_target_inter_inv_preimage hse]\n[GOAL]\ncase intro.h\u2081.refine'_2\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhse : s \u2286 (extChartAt I c).source\nhsf : \u2191(extChartAt I c) '' s \u2286 ball (\u2191(extChartAt I c) c) f.rOut\n\u22a2 (extChartAt I c).target \u2229 \u2191(LocalEquiv.symm (extChartAt I c)) \u207b\u00b9' s \u2286 \u2191(LocalEquiv.symm (extChartAt I c)) \u207b\u00b9' s\n[PROOFSTEP]\nexact inter_subset_right _ _\n[GOAL]\ncase intro.h\u2082\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhse : s \u2286 (extChartAt I c).source\nhsf : \u2191(extChartAt I c) '' s \u2286 ball (\u2191(extChartAt I c) c) f.rOut\n\u22a2 closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I \u2229 \u2191(LocalEquiv.symm (extChartAt I c)) \u207b\u00b9' s \u2286\n \u2191(extChartAt I c) '' s\n[PROOFSTEP]\nrefine' Subset.trans (inter_subset_inter_left _ f.closedBall_subset) _\n[GOAL]\ncase intro.h\u2082\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhse : s \u2286 (extChartAt I c).source\nhsf : \u2191(extChartAt I c) '' s \u2286 ball (\u2191(extChartAt I c) c) f.rOut\n\u22a2 (extChartAt I c).target \u2229 \u2191(LocalEquiv.symm (extChartAt I c)) \u207b\u00b9' s \u2286 \u2191(extChartAt I c) '' s\n[PROOFSTEP]\nrw [(extChartAt I c).image_eq_target_inter_inv_preimage hse]\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n\u22a2 \u2191f x \u2208 Icc 0 1\n[PROOFSTEP]\nhave : f x = 0 \u2228 f x = _ := indicator_eq_zero_or_self _ _ _\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\nthis : \u2191f x = 0 \u2228 \u2191f x = (\u2191f.toContDiffBump \u2218 \u2191(extChartAt I c)) x\n\u22a2 \u2191f x \u2208 Icc 0 1\n[PROOFSTEP]\ncases' this with h h\n[GOAL]\ncase inl\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\nh : \u2191f x = 0\n\u22a2 \u2191f x \u2208 Icc 0 1\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase inr\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\nh : \u2191f x = (\u2191f.toContDiffBump \u2218 \u2191(extChartAt I c)) x\n\u22a2 \u2191f x \u2208 Icc 0 1\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase inl\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\nh : \u2191f x = 0\n\u22a2 0 \u2208 Icc 0 1\ncase inr\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\nh : \u2191f x = (\u2191f.toContDiffBump \u2218 \u2191(extChartAt I c)) x\n\u22a2 (\u2191f.toContDiffBump \u2218 \u2191(extChartAt I c)) x \u2208 Icc 0 1\n[PROOFSTEP]\nexacts [left_mem_Icc.2 zero_le_one, \u27e8f.nonneg, f.le_one\u27e9]\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\nhs : x \u2208 (chartAt H c).toLocalEquiv.source\nhd : dist (\u2191(extChartAt I c) x) (\u2191(extChartAt I c) c) < f.rIn\n\u22a2 \u2191f =\u1da0[\ud835\udcdd x] 1\n[PROOFSTEP]\nfilter_upwards [IsOpen.mem_nhds (isOpen_extChartAt_preimage I c isOpen_ball) \u27e8hs, hd\u27e9]\n[GOAL]\ncase h\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\nhs : x \u2208 (chartAt H c).toLocalEquiv.source\nhd : dist (\u2191(extChartAt I c) x) (\u2191(extChartAt I c) c) < f.rIn\n\u22a2 \u2200 (a : M),\n a \u2208 (chartAt H c).toLocalEquiv.source \u2229 \u2191(extChartAt I c) \u207b\u00b9' ball (\u2191(extChartAt I c) c) f.rIn \u2192\n \u2191f a = OfNat.ofNat 1 a\n[PROOFSTEP]\nrintro z \u27e8hzs, hzd\u27e9\n[GOAL]\ncase h.intro\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\nhs : x \u2208 (chartAt H c).toLocalEquiv.source\nhd : dist (\u2191(extChartAt I c) x) (\u2191(extChartAt I c) c) < f.rIn\nz : M\nhzs : z \u2208 (chartAt H c).toLocalEquiv.source\nhzd : z \u2208 \u2191(extChartAt I c) \u207b\u00b9' ball (\u2191(extChartAt I c) c) f.rIn\n\u22a2 \u2191f z = OfNat.ofNat 1 z\n[PROOFSTEP]\nexact f.one_of_dist_le hzs <| le_of_lt hzd\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n\u22a2 dist (\u2191(extChartAt I c) c) (\u2191(extChartAt I c) c) < f.rIn\n[PROOFSTEP]\nrw [dist_self]\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n\u22a2 0 < f.rIn\n[PROOFSTEP]\nexact f.rIn_pos\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx\u271d x : M\nhx : \u2191f x = OfNat.ofNat 1 x\n\u22a2 \u2191f x \u2260 0\n[PROOFSTEP]\nrw [hx]\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx\u271d x : M\nhx : \u2191f x = OfNat.ofNat 1 x\n\u22a2 OfNat.ofNat 1 x \u2260 0\n[PROOFSTEP]\nexact one_ne_zero\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n\u22a2 HasBasis (\ud835\udcdd[range \u2191I] \u2191(extChartAt I c) c) (fun x => True) fun f => closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I\n[PROOFSTEP]\nrefine'\n ((nhdsWithin_hasBasis nhds_basis_closedBall _).restrict_subset (extChartAt_target_mem_nhdsWithin _ _)).to_has_basis' _\n _\n[GOAL]\ncase refine'_1\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n\u22a2 \u2200 (i : \u211d),\n 0 < i \u2227 closedBall (\u2191(extChartAt I c) c) i \u2229 range \u2191I \u2286 (extChartAt I c).target \u2192\n \u2203 i', True \u2227 closedBall (\u2191(extChartAt I c) c) i'.rOut \u2229 range \u2191I \u2286 closedBall (\u2191(extChartAt I c) c) i \u2229 range \u2191I\n[PROOFSTEP]\nrintro R \u27e8hR0, hsub\u27e9\n[GOAL]\ncase refine'_1.intro\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\nR : \u211d\nhR0 : 0 < R\nhsub : closedBall (\u2191(extChartAt I c) c) R \u2229 range \u2191I \u2286 (extChartAt I c).target\n\u22a2 \u2203 i', True \u2227 closedBall (\u2191(extChartAt I c) c) i'.rOut \u2229 range \u2191I \u2286 closedBall (\u2191(extChartAt I c) c) R \u2229 range \u2191I\n[PROOFSTEP]\nexact \u27e8\u27e8\u27e8R / 2, R, half_pos hR0, half_lt_self hR0\u27e9, hsub\u27e9, trivial, Subset.rfl\u27e9\n[GOAL]\ncase refine'_2\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n\u22a2 \u2200 (i' : SmoothBumpFunction I c),\n True \u2192 closedBall (\u2191(extChartAt I c) c) i'.rOut \u2229 range \u2191I \u2208 \ud835\udcdd[range \u2191I] \u2191(extChartAt I c) c\n[PROOFSTEP]\nexact fun f _ => inter_mem (mem_nhdsWithin_of_mem_nhds <| closedBall_mem_nhds _ f.rOut_pos) self_mem_nhdsWithin\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhsc : IsClosed s\nhs : s \u2286 support \u2191f\n\u22a2 IsClosed (\u2191(extChartAt I c) '' s)\n[PROOFSTEP]\nrw [f.image_eq_inter_preimage_of_subset_support hs]\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhsc : IsClosed s\nhs : s \u2286 support \u2191f\n\u22a2 IsClosed (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I \u2229 \u2191(LocalEquiv.symm (extChartAt I c)) \u207b\u00b9' s)\n[PROOFSTEP]\nrefine' ContinuousOn.preimage_closed_of_closed ((continuousOn_extChartAt_symm _ _).mono f.closedBall_subset) _ hsc\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhsc : IsClosed s\nhs : s \u2286 support \u2191f\n\u22a2 IsClosed (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I)\n[PROOFSTEP]\nexact IsClosed.inter isClosed_ball I.closed_range\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhsc : IsClosed s\nhs : s \u2286 support \u2191f\n\u22a2 \u2203 r, r \u2208 Ioo 0 f.rOut \u2227 s \u2286 (chartAt H c).toLocalEquiv.source \u2229 \u2191(extChartAt I c) \u207b\u00b9' ball (\u2191(extChartAt I c) c) r\n[PROOFSTEP]\nset e := extChartAt I c\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhsc : IsClosed s\nhs : s \u2286 support \u2191f\ne : LocalEquiv M E := extChartAt I c\n\u22a2 \u2203 r, r \u2208 Ioo 0 f.rOut \u2227 s \u2286 (chartAt H c).toLocalEquiv.source \u2229 \u2191(extChartAt I c) \u207b\u00b9' ball (\u2191(extChartAt I c) c) r\n[PROOFSTEP]\nhave : IsClosed (e '' s) := f.isClosed_image_of_isClosed hsc hs\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhsc : IsClosed s\nhs : s \u2286 support \u2191f\ne : LocalEquiv M E := extChartAt I c\nthis : IsClosed (\u2191e '' s)\n\u22a2 \u2203 r, r \u2208 Ioo 0 f.rOut \u2227 s \u2286 (chartAt H c).toLocalEquiv.source \u2229 \u2191(extChartAt I c) \u207b\u00b9' ball (\u2191(extChartAt I c) c) r\n[PROOFSTEP]\nrw [support_eq_inter_preimage, subset_inter_iff, \u2190 image_subset_iff] at hs \n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhsc : IsClosed s\nhs : s \u2286 (chartAt H c).toLocalEquiv.source \u2227 \u2191(extChartAt I c) '' s \u2286 ball (\u2191(extChartAt I c) c) f.rOut\ne : LocalEquiv M E := extChartAt I c\nthis : IsClosed (\u2191e '' s)\n\u22a2 \u2203 r, r \u2208 Ioo 0 f.rOut \u2227 s \u2286 (chartAt H c).toLocalEquiv.source \u2229 \u2191(extChartAt I c) \u207b\u00b9' ball (\u2191(extChartAt I c) c) r\n[PROOFSTEP]\nrcases exists_pos_lt_subset_ball f.rOut_pos this hs.2 with \u27e8r, hrR, hr\u27e9\n[GOAL]\ncase intro.intro\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ns : Set M\nhsc : IsClosed s\nhs : s \u2286 (chartAt H c).toLocalEquiv.source \u2227 \u2191(extChartAt I c) '' s \u2286 ball (\u2191(extChartAt I c) c) f.rOut\ne : LocalEquiv M E := extChartAt I c\nthis : IsClosed (\u2191e '' s)\nr : \u211d\nhrR : r \u2208 Ioo 0 f.rOut\nhr : \u2191e '' s \u2286 ball (\u2191(extChartAt I c) c) r\n\u22a2 \u2203 r, r \u2208 Ioo 0 f.rOut \u2227 s \u2286 (chartAt H c).toLocalEquiv.source \u2229 \u2191(extChartAt I c) \u207b\u00b9' ball (\u2191(extChartAt I c) c) r\n[PROOFSTEP]\nexact \u27e8r, hrR, subset_inter hs.1 (image_subset_iff.1 hr)\u27e9\n[GOAL]\nE : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\nr : \u211d\nhr : r \u2208 Ioo 0 f.rOut\n\u22a2 support \u2191(updateRIn f r hr) = support \u2191f\n[PROOFSTEP]\nsimp only [support_eq_inter_preimage, updateRIn_rOut]\n[GOAL]\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ninst\u271d : T2Space M\n\u22a2 tsupport \u2191f \u2286 \u2191(LocalEquiv.symm (extChartAt I c)) '' (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I)\n[PROOFSTEP]\nrw [tsupport, support_eq_symm_image]\n[GOAL]\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ninst\u271d : T2Space M\n\u22a2 closure (\u2191(LocalEquiv.symm (extChartAt I c)) '' (ball (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I)) \u2286\n \u2191(LocalEquiv.symm (extChartAt I c)) '' (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I)\n[PROOFSTEP]\nexact\n closure_minimal (image_subset _ <| inter_subset_inter_left _ ball_subset_closedBall) f.isClosed_symm_image_closedBall\n[GOAL]\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ninst\u271d : T2Space M\n\u22a2 tsupport \u2191f \u2286 (chartAt H c).toLocalEquiv.source\n[PROOFSTEP]\nsimpa only [extChartAt_source] using f.tsupport_subset_extChartAt_source\n[GOAL]\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ninst\u271d : T2Space M\n\u22a2 HasBasis (\ud835\udcdd c) (fun x => True) fun f => tsupport \u2191f\n[PROOFSTEP]\nhave :\n (\ud835\udcdd c).HasBasis (fun _ : SmoothBumpFunction I c => True) fun f =>\n (extChartAt I c).symm '' (closedBall (extChartAt I c c) f.rOut \u2229 range I) :=\n by\n rw [\u2190 map_extChartAt_symm_nhdsWithin_range I c]\n exact nhdsWithin_range_basis.map _\n[GOAL]\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ninst\u271d : T2Space M\n\u22a2 HasBasis (\ud835\udcdd c) (fun x => True) fun f =>\n \u2191(LocalEquiv.symm (extChartAt I c)) '' (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I)\n[PROOFSTEP]\nrw [\u2190 map_extChartAt_symm_nhdsWithin_range I c]\n[GOAL]\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ninst\u271d : T2Space M\n\u22a2 HasBasis (map (\u2191(LocalEquiv.symm (extChartAt I c))) (\ud835\udcdd[range \u2191I] \u2191(extChartAt I c) c)) (fun x => True) fun f =>\n \u2191(LocalEquiv.symm (extChartAt I c)) '' (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I)\n[PROOFSTEP]\nexact nhdsWithin_range_basis.map _\n[GOAL]\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ninst\u271d : T2Space M\nthis :\n HasBasis (\ud835\udcdd c) (fun x => True) fun f =>\n \u2191(LocalEquiv.symm (extChartAt I c)) '' (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I)\n\u22a2 HasBasis (\ud835\udcdd c) (fun x => True) fun f => tsupport \u2191f\n[PROOFSTEP]\nrefine'\n this.to_has_basis' (fun f _ => \u27e8f, trivial, f.tsupport_subset_symm_image_closedBall\u27e9) fun f _ => f.tsupport_mem_nhds\n[GOAL]\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2075 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ninst\u271d\u00b9 : T2Space M\ninst\u271d : SmoothManifoldWithCorners I M\n\u22a2 Smooth I \ud835\udcd8(\u211d, \u211d) \u2191f\n[PROOFSTEP]\nrefine' contMDiff_of_support fun x hx => _\n[GOAL]\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2075 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx\u271d : M\ninst\u271d\u00b9 : T2Space M\ninst\u271d : SmoothManifoldWithCorners I M\nx : M\nhx : x \u2208 tsupport \u2191f\n\u22a2 ContMDiffAt I \ud835\udcd8(\u211d, \u211d) \u22a4 (\u2191f) x\n[PROOFSTEP]\nhave : x \u2208 (chartAt H c).source := f.tsupport_subset_chartAt_source hx\n[GOAL]\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2075 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx\u271d : M\ninst\u271d\u00b9 : T2Space M\ninst\u271d : SmoothManifoldWithCorners I M\nx : M\nhx : x \u2208 tsupport \u2191f\nthis : x \u2208 (chartAt H c).toLocalEquiv.source\n\u22a2 ContMDiffAt I \ud835\udcd8(\u211d, \u211d) \u22a4 (\u2191f) x\n[PROOFSTEP]\nrefine'\n ContMDiffAt.congr_of_eventuallyEq _ <|\n f.eqOn_source.eventuallyEq_of_mem <| IsOpen.mem_nhds (chartAt H c).open_source this\n[GOAL]\nE : Type uE\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2075 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx\u271d : M\ninst\u271d\u00b9 : T2Space M\ninst\u271d : SmoothManifoldWithCorners I M\nx : M\nhx : x \u2208 tsupport \u2191f\nthis : x \u2208 (chartAt H c).toLocalEquiv.source\n\u22a2 ContMDiffAt I \ud835\udcd8(\u211d, \u211d) \u22a4 (\u2191f.toContDiffBump \u2218 \u2191(extChartAt I c)) x\n[PROOFSTEP]\nexact f.contDiffAt.contMDiffAt.comp _ (contMDiffAt_extChartAt' this)\n[GOAL]\nE : Type uE\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ninst\u271d\u00b3 : T2Space M\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\ng : M \u2192 G\nhg : SmoothOn I \ud835\udcd8(\u211d, G) g (chartAt H c).toLocalEquiv.source\n\u22a2 Smooth I \ud835\udcd8(\u211d, G) fun x => \u2191f x \u2022 g x\n[PROOFSTEP]\nrefine contMDiff_of_support fun x hx => ?_\n[GOAL]\nE : Type uE\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx\u271d : M\ninst\u271d\u00b3 : T2Space M\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\ng : M \u2192 G\nhg : SmoothOn I \ud835\udcd8(\u211d, G) g (chartAt H c).toLocalEquiv.source\nx : M\nhx : x \u2208 tsupport fun x => \u2191f x \u2022 g x\n\u22a2 ContMDiffAt I \ud835\udcd8(\u211d, G) \u22a4 (fun x => \u2191f x \u2022 g x) x\n[PROOFSTEP]\nhave : x \u2208 (chartAt H c).source\n[GOAL]\ncase this\nE : Type uE\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx\u271d : M\ninst\u271d\u00b3 : T2Space M\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\ng : M \u2192 G\nhg : SmoothOn I \ud835\udcd8(\u211d, G) g (chartAt H c).toLocalEquiv.source\nx : M\nhx : x \u2208 tsupport fun x => \u2191f x \u2022 g x\n\u22a2 x \u2208 (chartAt H c).toLocalEquiv.source\n[PROOFSTEP]\nexact f.tsupport_subset_chartAt_source <| tsupport_smul_subset_left _ _ hx\n[GOAL]\nE : Type uE\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx\u271d : M\ninst\u271d\u00b3 : T2Space M\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\ng : M \u2192 G\nhg : SmoothOn I \ud835\udcd8(\u211d, G) g (chartAt H c).toLocalEquiv.source\nx : M\nhx : x \u2208 tsupport fun x => \u2191f x \u2022 g x\nthis : x \u2208 (chartAt H c).toLocalEquiv.source\n\u22a2 ContMDiffAt I \ud835\udcd8(\u211d, G) \u22a4 (fun x => \u2191f x \u2022 g x) x\n[PROOFSTEP]\nexact f.smoothAt.smul ((hg _ this).contMDiffAt <| IsOpen.mem_nhds (chartAt _ _).open_source this)\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Manifold.BumpFunction", "llama_tokens": 14921, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.689305616785446, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.43159091216863554}} {"text": "[GOAL]\nK : Type u_1\ninst\u271d : Field K\nx : (K \u2192+* \u2102) \u2192 \u2102\n\u03c6 : K \u2192+* \u2102\nhx : x \u2208 Submodule.span \u211d (Set.range \u2191(canonicalEmbedding K))\n\u22a2 \u2191(starRingEnd \u2102) (x \u03c6) = x (ComplexEmbedding.conjugate \u03c6)\n[PROOFSTEP]\nrefine Submodule.span_induction hx ?_ ?_ (fun _ _ hx hy => ?_) (fun a _ hx => ?_)\n[GOAL]\ncase refine_1\nK : Type u_1\ninst\u271d : Field K\nx : (K \u2192+* \u2102) \u2192 \u2102\n\u03c6 : K \u2192+* \u2102\nhx : x \u2208 Submodule.span \u211d (Set.range \u2191(canonicalEmbedding K))\n\u22a2 \u2200 (x : (K \u2192+* \u2102) \u2192 \u2102),\n x \u2208 Set.range \u2191(canonicalEmbedding K) \u2192 \u2191(starRingEnd \u2102) (x \u03c6) = x (ComplexEmbedding.conjugate \u03c6)\n[PROOFSTEP]\nrintro _ \u27e8x, rfl\u27e9\n[GOAL]\ncase refine_1.intro\nK : Type u_1\ninst\u271d : Field K\nx\u271d : (K \u2192+* \u2102) \u2192 \u2102\n\u03c6 : K \u2192+* \u2102\nhx : x\u271d \u2208 Submodule.span \u211d (Set.range \u2191(canonicalEmbedding K))\nx : K\n\u22a2 \u2191(starRingEnd \u2102) (\u2191(canonicalEmbedding K) x \u03c6) = \u2191(canonicalEmbedding K) x (ComplexEmbedding.conjugate \u03c6)\n[PROOFSTEP]\nrw [apply_at, apply_at, ComplexEmbedding.conjugate_coe_eq]\n[GOAL]\ncase refine_2\nK : Type u_1\ninst\u271d : Field K\nx : (K \u2192+* \u2102) \u2192 \u2102\n\u03c6 : K \u2192+* \u2102\nhx : x \u2208 Submodule.span \u211d (Set.range \u2191(canonicalEmbedding K))\n\u22a2 \u2191(starRingEnd \u2102) (OfNat.ofNat 0 \u03c6) = OfNat.ofNat 0 (ComplexEmbedding.conjugate \u03c6)\n[PROOFSTEP]\nrw [Pi.zero_apply, Pi.zero_apply, map_zero]\n[GOAL]\ncase refine_3\nK : Type u_1\ninst\u271d : Field K\nx : (K \u2192+* \u2102) \u2192 \u2102\n\u03c6 : K \u2192+* \u2102\nhx\u271d : x \u2208 Submodule.span \u211d (Set.range \u2191(canonicalEmbedding K))\nx\u271d\u00b9 x\u271d : (K \u2192+* \u2102) \u2192 \u2102\nhx : \u2191(starRingEnd \u2102) (x\u271d\u00b9 \u03c6) = x\u271d\u00b9 (ComplexEmbedding.conjugate \u03c6)\nhy : \u2191(starRingEnd \u2102) (x\u271d \u03c6) = x\u271d (ComplexEmbedding.conjugate \u03c6)\n\u22a2 \u2191(starRingEnd \u2102) ((x\u271d\u00b9 + x\u271d) \u03c6) = (x\u271d\u00b9 + x\u271d) (ComplexEmbedding.conjugate \u03c6)\n[PROOFSTEP]\nrw [Pi.add_apply, Pi.add_apply, map_add, hx, hy]\n[GOAL]\ncase refine_4\nK : Type u_1\ninst\u271d : Field K\nx : (K \u2192+* \u2102) \u2192 \u2102\n\u03c6 : K \u2192+* \u2102\nhx\u271d : x \u2208 Submodule.span \u211d (Set.range \u2191(canonicalEmbedding K))\na : \u211d\nx\u271d : (K \u2192+* \u2102) \u2192 \u2102\nhx : \u2191(starRingEnd \u2102) (x\u271d \u03c6) = x\u271d (ComplexEmbedding.conjugate \u03c6)\n\u22a2 \u2191(starRingEnd \u2102) ((a \u2022 x\u271d) \u03c6) = (a \u2022 x\u271d) (ComplexEmbedding.conjugate \u03c6)\n[PROOFSTEP]\nrw [Pi.smul_apply, Complex.real_smul, map_mul, Complex.conj_ofReal]\n[GOAL]\ncase refine_4\nK : Type u_1\ninst\u271d : Field K\nx : (K \u2192+* \u2102) \u2192 \u2102\n\u03c6 : K \u2192+* \u2102\nhx\u271d : x \u2208 Submodule.span \u211d (Set.range \u2191(canonicalEmbedding K))\na : \u211d\nx\u271d : (K \u2192+* \u2102) \u2192 \u2102\nhx : \u2191(starRingEnd \u2102) (x\u271d \u03c6) = x\u271d (ComplexEmbedding.conjugate \u03c6)\n\u22a2 \u2191a * \u2191(starRingEnd \u2102) (x\u271d \u03c6) = (a \u2022 x\u271d) (ComplexEmbedding.conjugate \u03c6)\n[PROOFSTEP]\nexact congrArg ((a : \u2102) * \u00b7) hx\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\n\u22a2 \u2016\u2191(canonicalEmbedding K) x\u2016\u208a = Finset.sup Finset.univ fun \u03c6 => \u2016\u2191\u03c6 x\u2016\u208a\n[PROOFSTEP]\nsimp_rw [Pi.nnnorm_def, apply_at]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\nr : \u211d\n\u22a2 \u2016\u2191(canonicalEmbedding K) x\u2016 \u2264 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\n[PROOFSTEP]\nobtain hr | hr := lt_or_le r 0\n[GOAL]\ncase inl\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\nr : \u211d\nhr : r < 0\n\u22a2 \u2016\u2191(canonicalEmbedding K) x\u2016 \u2264 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\n[PROOFSTEP]\nobtain \u27e8\u03c6\u27e9 := (inferInstance : Nonempty (K \u2192+* \u2102))\n[GOAL]\ncase inl.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\nr : \u211d\nhr : r < 0\n\u03c6 : K \u2192+* \u2102\n\u22a2 \u2016\u2191(canonicalEmbedding K) x\u2016 \u2264 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\n[PROOFSTEP]\nrefine iff_of_false ?_ ?_\n[GOAL]\ncase inl.intro.refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\nr : \u211d\nhr : r < 0\n\u03c6 : K \u2192+* \u2102\n\u22a2 \u00ac\u2016\u2191(canonicalEmbedding K) x\u2016 \u2264 r\ncase inl.intro.refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\nr : \u211d\nhr : r < 0\n\u03c6 : K \u2192+* \u2102\n\u22a2 \u00ac\u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\n[PROOFSTEP]\nexact (hr.trans_le (norm_nonneg _)).not_le\n[GOAL]\ncase inl.intro.refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\nr : \u211d\nhr : r < 0\n\u03c6 : K \u2192+* \u2102\n\u22a2 \u00ac\u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\n[PROOFSTEP]\nexact fun h => hr.not_le (le_trans (norm_nonneg _) (h \u03c6))\n[GOAL]\ncase inr\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 \u2016\u2191(canonicalEmbedding K) x\u2016 \u2264 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\n[PROOFSTEP]\nlift r to NNReal using hr\n[GOAL]\ncase inr.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\nr : NNReal\n\u22a2 \u2016\u2191(canonicalEmbedding K) x\u2016 \u2264 \u2191r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 \u2191r\n[PROOFSTEP]\nsimp_rw [\u2190 coe_nnnorm, nnnorm_eq, NNReal.coe_le_coe, Finset.sup_le_iff, Finset.mem_univ, forall_true_left]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\n\u22a2 Set.Finite (\u2191(integerLattice K) \u2229 Metric.closedBall 0 r)\n[PROOFSTEP]\nobtain hr | _ := lt_or_le r 0\n[GOAL]\ncase inl\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nhr : r < 0\n\u22a2 Set.Finite (\u2191(integerLattice K) \u2229 Metric.closedBall 0 r)\n[PROOFSTEP]\nsimp [Metric.closedBall_eq_empty.2 hr]\n[GOAL]\ncase inr\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\n\u22a2 Set.Finite (\u2191(integerLattice K) \u2229 Metric.closedBall 0 r)\n[PROOFSTEP]\nhave heq : \u2200 x, canonicalEmbedding K x \u2208 Metric.closedBall 0 r \u2194 \u2200 \u03c6 : K \u2192+* \u2102, \u2016\u03c6 x\u2016 \u2264 r := by intro x;\n rw [\u2190 norm_le_iff, mem_closedBall_zero_iff]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\n\u22a2 \u2200 (x : K), \u2191(canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\n[PROOFSTEP]\nintro x\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nx : K\n\u22a2 \u2191(canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\n[PROOFSTEP]\nrw [\u2190 norm_le_iff, mem_closedBall_zero_iff]\n[GOAL]\ncase inr\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), \u2191(canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\n\u22a2 Set.Finite (\u2191(integerLattice K) \u2229 Metric.closedBall 0 r)\n[PROOFSTEP]\nconvert (Embeddings.finite_of_norm_le K \u2102 r).image (canonicalEmbedding K)\n[GOAL]\ncase h.e'_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), \u2191(canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\n\u22a2 \u2191(integerLattice K) \u2229 Metric.closedBall 0 r =\n \u2191(canonicalEmbedding K) '' {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r}\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_2.h\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), \u2191(canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\nx\u271d : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 x\u271d \u2208 \u2191(integerLattice K) \u2229 Metric.closedBall 0 r \u2194\n x\u271d \u2208 \u2191(canonicalEmbedding K) '' {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.e'_2.h.mp\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), \u2191(canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\nx\u271d : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 x\u271d \u2208 \u2191(integerLattice K) \u2229 Metric.closedBall 0 r \u2192\n x\u271d \u2208 \u2191(canonicalEmbedding K) '' {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r}\n[PROOFSTEP]\nrintro \u27e8\u27e8_, \u27e8x, rfl\u27e9, rfl\u27e9, hx\u27e9\n[GOAL]\ncase h.e'_2.h.mp.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), \u2191(canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\nx : { x // x \u2208 \ud835\udcde K }\nhx : \u2191(canonicalEmbedding K) (\u2191(algebraMap { x // x \u2208 \ud835\udcde K } K) x) \u2208 Metric.closedBall 0 r\n\u22a2 \u2191(canonicalEmbedding K) (\u2191(algebraMap { x // x \u2208 \ud835\udcde K } K) x) \u2208\n \u2191(canonicalEmbedding K) '' {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r}\n[PROOFSTEP]\nexact \u27e8\u2191x, \u27e8SetLike.coe_mem x, fun \u03c6 => (heq x).mp hx \u03c6\u27e9, rfl\u27e9\n[GOAL]\ncase h.e'_2.h.mpr\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), \u2191(canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\nx\u271d : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 x\u271d \u2208 \u2191(canonicalEmbedding K) '' {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r} \u2192\n x\u271d \u2208 \u2191(integerLattice K) \u2229 Metric.closedBall 0 r\n[PROOFSTEP]\nrintro \u27e8x, \u27e8hx1, hx2\u27e9, rfl\u27e9\n[GOAL]\ncase h.e'_2.h.mpr.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), \u2191(canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\nx : K\nhx1 : IsIntegral \u2124 x\nhx2 : \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\n\u22a2 \u2191(canonicalEmbedding K) x \u2208 \u2191(integerLattice K) \u2229 Metric.closedBall 0 r\n[PROOFSTEP]\nexact \u27e8\u27e8x, \u27e8\u27e8x, hx1\u27e9, rfl\u27e9, rfl\u27e9, (heq x).mpr hx2\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u22a2 Basis (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }) \u2102 ((K \u2192+* \u2102) \u2192 \u2102)\n[PROOFSTEP]\nclassical\n -- Let `B` be the canonical basis of `(K \u2192+* \u2102) \u2192 \u2102`. We prove that the determinant of\n -- the image by `canonicalEmbedding` of the integral basis of `K` is nonzero. This\n -- will imply the result.\nlet B := Pi.basisFun \u2102 (K \u2192+* \u2102)\nlet e : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 (\ud835\udcde K) :=\n equivOfCardEq ((Embeddings.card K \u2102).trans (finrank_eq_card_basis (integralBasis K)))\nlet M := B.toMatrix (fun i => canonicalEmbedding K (integralBasis K (e i)))\nsuffices M.det \u2260 0 by\n rw [\u2190 isUnit_iff_ne_zero, \u2190 Basis.det_apply, \u2190 is_basis_iff_det] at this \n refine basisOfLinearIndependentOfCardEqFinrank ((linearIndependent_equiv e.symm).mpr this.1) ?_\n rw [\u2190 finrank_eq_card_chooseBasisIndex, RingOfIntegers.rank, finrank_fintype_fun_eq_card, Embeddings.card]\n -- In order to prove that the determinant is nonzero, we show that it is equal to the\n -- square of the discriminant of the integral basis and thus it is not zero\nlet N := Algebra.embeddingsMatrixReindex \u211a \u2102 (fun i => integralBasis K (e i)) RingHom.equivRatAlgHom\nrw [show M = N.transpose by {ext : 2; rfl\n }]\nrw [Matrix.det_transpose, \u2190 @pow_ne_zero_iff \u2102 _ _ _ 2 (by norm_num)]\nconvert (map_ne_zero_iff _ (algebraMap \u211a \u2102).injective).mpr (Algebra.discr_not_zero_of_basis \u211a (integralBasis K))\nrw [\u2190 Algebra.discr_reindex \u211a (integralBasis K) e.symm]\nexact\n (Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two \u211a \u2102 (fun i => integralBasis K (e i))\n RingHom.equivRatAlgHom).symm\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u22a2 Basis (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }) \u2102 ((K \u2192+* \u2102) \u2192 \u2102)\n[PROOFSTEP]\nlet B := Pi.basisFun \u2102 (K \u2192+* \u2102)\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\n\u22a2 Basis (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }) \u2102 ((K \u2192+* \u2102) \u2192 \u2102)\n[PROOFSTEP]\nlet e : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 (\ud835\udcde K) :=\n equivOfCardEq ((Embeddings.card K \u2102).trans (finrank_eq_card_basis (integralBasis K)))\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\ne : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K } :=\n equivOfCardEq (_ : card (K \u2192+* \u2102) = card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }))\n\u22a2 Basis (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }) \u2102 ((K \u2192+* \u2102) \u2192 \u2102)\n[PROOFSTEP]\nlet M := B.toMatrix (fun i => canonicalEmbedding K (integralBasis K (e i)))\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\ne : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K } :=\n equivOfCardEq (_ : card (K \u2192+* \u2102) = card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }))\nM : Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102 := Basis.toMatrix B fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))\n\u22a2 Basis (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }) \u2102 ((K \u2192+* \u2102) \u2192 \u2102)\n[PROOFSTEP]\nsuffices M.det \u2260 0 by\n rw [\u2190 isUnit_iff_ne_zero, \u2190 Basis.det_apply, \u2190 is_basis_iff_det] at this \n refine basisOfLinearIndependentOfCardEqFinrank ((linearIndependent_equiv e.symm).mpr this.1) ?_\n rw [\u2190 finrank_eq_card_chooseBasisIndex, RingOfIntegers.rank, finrank_fintype_fun_eq_card, Embeddings.card]\n -- In order to prove that the determinant is nonzero, we show that it is equal to the\n -- square of the discriminant of the integral basis and thus it is not zero\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\ne : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K } :=\n equivOfCardEq (_ : card (K \u2192+* \u2102) = card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }))\nM : Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102 := Basis.toMatrix B fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))\nthis : Matrix.det M \u2260 0\n\u22a2 Basis (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }) \u2102 ((K \u2192+* \u2102) \u2192 \u2102)\n[PROOFSTEP]\nrw [\u2190 isUnit_iff_ne_zero, \u2190 Basis.det_apply, \u2190 is_basis_iff_det] at this \n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\ne : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K } :=\n equivOfCardEq (_ : card (K \u2192+* \u2102) = card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }))\nM : Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102 := Basis.toMatrix B fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))\nthis :\n (LinearIndependent \u2102 fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))) \u2227\n Submodule.span \u2102 (Set.range fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))) = \u22a4\n\u22a2 Basis (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }) \u2102 ((K \u2192+* \u2102) \u2192 \u2102)\n[PROOFSTEP]\nrefine basisOfLinearIndependentOfCardEqFinrank ((linearIndependent_equiv e.symm).mpr this.1) ?_\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\ne : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K } :=\n equivOfCardEq (_ : card (K \u2192+* \u2102) = card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }))\nM : Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102 := Basis.toMatrix B fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))\nthis :\n (LinearIndependent \u2102 fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))) \u2227\n Submodule.span \u2102 (Set.range fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))) = \u22a4\n\u22a2 card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }) = finrank \u2102 ((K \u2192+* \u2102) \u2192 \u2102)\n[PROOFSTEP]\nrw [\u2190 finrank_eq_card_chooseBasisIndex, RingOfIntegers.rank, finrank_fintype_fun_eq_card, Embeddings.card]\n -- In order to prove that the determinant is nonzero, we show that it is equal to the\n -- square of the discriminant of the integral basis and thus it is not zero\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\ne : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K } :=\n equivOfCardEq (_ : card (K \u2192+* \u2102) = card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }))\nM : Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102 := Basis.toMatrix B fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))\n\u22a2 Matrix.det M \u2260 0\n[PROOFSTEP]\nlet N := Algebra.embeddingsMatrixReindex \u211a \u2102 (fun i => integralBasis K (e i)) RingHom.equivRatAlgHom\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\ne : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K } :=\n equivOfCardEq (_ : card (K \u2192+* \u2102) = card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }))\nM : Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102 := Basis.toMatrix B fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))\nN : (fun x => Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102) (Algebra.embeddingsMatrix \u211a \u2102 fun i => \u2191(integralBasis K) (\u2191e i)) :=\n Algebra.embeddingsMatrixReindex \u211a \u2102 (fun i => \u2191(integralBasis K) (\u2191e i)) RingHom.equivRatAlgHom\n\u22a2 Matrix.det M \u2260 0\n[PROOFSTEP]\nrw [show M = N.transpose by {ext : 2; rfl\n }]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\ne : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K } :=\n equivOfCardEq (_ : card (K \u2192+* \u2102) = card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }))\nM : Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102 := Basis.toMatrix B fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))\nN : (fun x => Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102) (Algebra.embeddingsMatrix \u211a \u2102 fun i => \u2191(integralBasis K) (\u2191e i)) :=\n Algebra.embeddingsMatrixReindex \u211a \u2102 (fun i => \u2191(integralBasis K) (\u2191e i)) RingHom.equivRatAlgHom\n\u22a2 M = Matrix.transpose N\n[PROOFSTEP]\n{ext : 2; rfl\n}\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\ne : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K } :=\n equivOfCardEq (_ : card (K \u2192+* \u2102) = card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }))\nM : Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102 := Basis.toMatrix B fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))\nN : (fun x => Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102) (Algebra.embeddingsMatrix \u211a \u2102 fun i => \u2191(integralBasis K) (\u2191e i)) :=\n Algebra.embeddingsMatrixReindex \u211a \u2102 (fun i => \u2191(integralBasis K) (\u2191e i)) RingHom.equivRatAlgHom\n\u22a2 M = Matrix.transpose N\n[PROOFSTEP]\next : 2\n[GOAL]\ncase a.h\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\ne : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K } :=\n equivOfCardEq (_ : card (K \u2192+* \u2102) = card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }))\nM : Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102 := Basis.toMatrix B fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))\nN : (fun x => Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102) (Algebra.embeddingsMatrix \u211a \u2102 fun i => \u2191(integralBasis K) (\u2191e i)) :=\n Algebra.embeddingsMatrixReindex \u211a \u2102 (fun i => \u2191(integralBasis K) (\u2191e i)) RingHom.equivRatAlgHom\ni\u271d x\u271d : K \u2192+* \u2102\n\u22a2 M i\u271d x\u271d = Matrix.transpose N i\u271d x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\ne : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K } :=\n equivOfCardEq (_ : card (K \u2192+* \u2102) = card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }))\nM : Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102 := Basis.toMatrix B fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))\nN : (fun x => Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102) (Algebra.embeddingsMatrix \u211a \u2102 fun i => \u2191(integralBasis K) (\u2191e i)) :=\n Algebra.embeddingsMatrixReindex \u211a \u2102 (fun i => \u2191(integralBasis K) (\u2191e i)) RingHom.equivRatAlgHom\n\u22a2 Matrix.det (Matrix.transpose N) \u2260 0\n[PROOFSTEP]\nrw [Matrix.det_transpose, \u2190 @pow_ne_zero_iff \u2102 _ _ _ 2 (by norm_num)]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\ne : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K } :=\n equivOfCardEq (_ : card (K \u2192+* \u2102) = card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }))\nM : Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102 := Basis.toMatrix B fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))\nN : (fun x => Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102) (Algebra.embeddingsMatrix \u211a \u2102 fun i => \u2191(integralBasis K) (\u2191e i)) :=\n Algebra.embeddingsMatrixReindex \u211a \u2102 (fun i => \u2191(integralBasis K) (\u2191e i)) RingHom.equivRatAlgHom\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\ne : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K } :=\n equivOfCardEq (_ : card (K \u2192+* \u2102) = card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }))\nM : Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102 := Basis.toMatrix B fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))\nN : (fun x => Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102) (Algebra.embeddingsMatrix \u211a \u2102 fun i => \u2191(integralBasis K) (\u2191e i)) :=\n Algebra.embeddingsMatrixReindex \u211a \u2102 (fun i => \u2191(integralBasis K) (\u2191e i)) RingHom.equivRatAlgHom\n\u22a2 Matrix.det N ^ 2 \u2260 0\n[PROOFSTEP]\nconvert (map_ne_zero_iff _ (algebraMap \u211a \u2102).injective).mpr (Algebra.discr_not_zero_of_basis \u211a (integralBasis K))\n[GOAL]\ncase h.e'_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\ne : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K } :=\n equivOfCardEq (_ : card (K \u2192+* \u2102) = card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }))\nM : Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102 := Basis.toMatrix B fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))\nN : (fun x => Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102) (Algebra.embeddingsMatrix \u211a \u2102 fun i => \u2191(integralBasis K) (\u2191e i)) :=\n Algebra.embeddingsMatrixReindex \u211a \u2102 (fun i => \u2191(integralBasis K) (\u2191e i)) RingHom.equivRatAlgHom\n\u22a2 Matrix.det N ^ 2 = \u2191(algebraMap \u211a \u2102) (Algebra.discr \u211a \u2191(integralBasis K))\n[PROOFSTEP]\nrw [\u2190 Algebra.discr_reindex \u211a (integralBasis K) e.symm]\n[GOAL]\ncase h.e'_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nB : Basis (K \u2192+* \u2102) \u2102 ((K \u2192+* \u2102) \u2192 \u2102) := Pi.basisFun \u2102 (K \u2192+* \u2102)\ne : (K \u2192+* \u2102) \u2243 Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K } :=\n equivOfCardEq (_ : card (K \u2192+* \u2102) = card (Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }))\nM : Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102 := Basis.toMatrix B fun i => \u2191(canonicalEmbedding K) (\u2191(integralBasis K) (\u2191e i))\nN : (fun x => Matrix (K \u2192+* \u2102) (K \u2192+* \u2102) \u2102) (Algebra.embeddingsMatrix \u211a \u2102 fun i => \u2191(integralBasis K) (\u2191e i)) :=\n Algebra.embeddingsMatrixReindex \u211a \u2102 (fun i => \u2191(integralBasis K) (\u2191e i)) RingHom.equivRatAlgHom\n\u22a2 Matrix.det N ^ 2 = \u2191(algebraMap \u211a \u2102) (Algebra.discr \u211a (\u2191(integralBasis K) \u2218 \u2191e.symm.symm))\n[PROOFSTEP]\nexact\n (Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two \u211a \u2102 (fun i => integralBasis K (e i))\n RingHom.equivRatAlgHom).symm\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\ni : Free.ChooseBasisIndex \u2124 { x // x \u2208 \ud835\udcde K }\n\u22a2 \u2191(latticeBasis K) i = \u2191(canonicalEmbedding K) (\u2191(integralBasis K) i)\n[PROOFSTEP]\nsimp only [latticeBasis, integralBasis_apply, coe_basisOfLinearIndependentOfCardEqFinrank, Function.comp_apply,\n Equiv.apply_symm_apply]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 x \u2208 Submodule.span \u2124 (Set.range \u2191(latticeBasis K)) \u2194 x \u2208 \u2191(canonicalEmbedding K) '' \u2191(\ud835\udcde K)\n[PROOFSTEP]\nrw [show Set.range (latticeBasis K) = (canonicalEmbedding K).toIntAlgHom.toLinearMap '' (Set.range (integralBasis K)) by\n rw [\u2190 Set.range_comp]; exact congrArg Set.range (funext (fun i => latticeBasis_apply K i))]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 Set.range \u2191(latticeBasis K) =\n \u2191(AlgHom.toLinearMap (RingHom.toIntAlgHom (canonicalEmbedding K))) '' Set.range \u2191(integralBasis K)\n[PROOFSTEP]\nrw [\u2190 Set.range_comp]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 Set.range \u2191(latticeBasis K) =\n Set.range (\u2191(AlgHom.toLinearMap (RingHom.toIntAlgHom (canonicalEmbedding K))) \u2218 \u2191(integralBasis K))\n[PROOFSTEP]\nexact congrArg Set.range (funext (fun i => latticeBasis_apply K i))\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 x \u2208\n Submodule.span \u2124\n (\u2191(AlgHom.toLinearMap (RingHom.toIntAlgHom (canonicalEmbedding K))) '' Set.range \u2191(integralBasis K)) \u2194\n x \u2208 \u2191(canonicalEmbedding K) '' \u2191(\ud835\udcde K)\n[PROOFSTEP]\nrw [\u2190 Submodule.map_span, \u2190 SetLike.mem_coe, Submodule.map_coe]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 x \u2208\n \u2191(AlgHom.toLinearMap (RingHom.toIntAlgHom (canonicalEmbedding K))) ''\n \u2191(Submodule.span \u2124 (Set.range \u2191(integralBasis K))) \u2194\n x \u2208 \u2191(canonicalEmbedding K) '' \u2191(\ud835\udcde K)\n[PROOFSTEP]\nrw [show (Submodule.span \u2124 (Set.range (integralBasis K)) : Set K) = \ud835\udcde K by ext; exact mem_span_integralBasis K]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 \u2191(Submodule.span \u2124 (Set.range \u2191(integralBasis K))) = \u2191(\ud835\udcde K)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : (K \u2192+* \u2102) \u2192 \u2102\nx\u271d : K\n\u22a2 x\u271d \u2208 \u2191(Submodule.span \u2124 (Set.range \u2191(integralBasis K))) \u2194 x\u271d \u2208 \u2191(\ud835\udcde K)\n[PROOFSTEP]\nexact mem_span_integralBasis K\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 x \u2208 \u2191(AlgHom.toLinearMap (RingHom.toIntAlgHom (canonicalEmbedding K))) '' \u2191(\ud835\udcde K) \u2194\n x \u2208 \u2191(canonicalEmbedding K) '' \u2191(\ud835\udcde K)\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.NumberField.CanonicalEmbedding", "llama_tokens": 12050, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.6548947155710233, "lm_q1q2_score": 0.43119414175732745}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF : H \u2192 \u211d \u2192 E\nF' : \u211d \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF_int : IntervalIntegrable (F x\u2080) \u03bc a b\nhF'_meas : AEStronglyMeasurable F' (Measure.restrict \u03bc (\u0399 a b))\nh_lip : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 a b \u2192 LipschitzOnWith (\u2191Real.nnabs (bound t)) (fun x => F x t) (ball x\u2080 \u03b5)\nbound_integrable : IntervalIntegrable bound \u03bc a b\nh_diff : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 a b \u2192 HasFDerivAt (fun x => F x t) (F' t) x\u2080\n\u22a2 IntervalIntegrable F' \u03bc a b \u2227 HasFDerivAt (fun x => \u222b (t : \u211d) in a..b, F x t \u2202\u03bc) (\u222b (t : \u211d) in a..b, F' t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nrw [\u2190 ae_restrict_iff' measurableSet_uIoc] at h_lip h_diff \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF : H \u2192 \u211d \u2192 E\nF' : \u211d \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF_int : IntervalIntegrable (F x\u2080) \u03bc a b\nhF'_meas : AEStronglyMeasurable F' (Measure.restrict \u03bc (\u0399 a b))\nh_lip : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), LipschitzOnWith (\u2191Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x\u2080 \u03b5)\nbound_integrable : IntervalIntegrable bound \u03bc a b\nh_diff : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), HasFDerivAt (fun x_1 => F x_1 x) (F' x) x\u2080\n\u22a2 IntervalIntegrable F' \u03bc a b \u2227 HasFDerivAt (fun x => \u222b (t : \u211d) in a..b, F x t \u2202\u03bc) (\u222b (t : \u211d) in a..b, F' t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nsimp only [intervalIntegrable_iff] at hF_int bound_integrable \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF : H \u2192 \u211d \u2192 E\nF' : \u211d \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF'_meas : AEStronglyMeasurable F' (Measure.restrict \u03bc (\u0399 a b))\nh_lip : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), LipschitzOnWith (\u2191Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x\u2080 \u03b5)\nh_diff : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), HasFDerivAt (fun x_1 => F x_1 x) (F' x) x\u2080\nhF_int : IntegrableOn (F x\u2080) (\u0399 a b)\nbound_integrable : IntegrableOn bound (\u0399 a b)\n\u22a2 IntegrableOn F' (\u0399 a b) \u2227 HasFDerivAt (fun x => \u222b (t : \u211d) in a..b, F x t \u2202\u03bc) (\u222b (t : \u211d) in a..b, F' t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nsimp only [intervalIntegral_eq_integral_uIoc]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF : H \u2192 \u211d \u2192 E\nF' : \u211d \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF'_meas : AEStronglyMeasurable F' (Measure.restrict \u03bc (\u0399 a b))\nh_lip : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), LipschitzOnWith (\u2191Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x\u2080 \u03b5)\nh_diff : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), HasFDerivAt (fun x_1 => F x_1 x) (F' x) x\u2080\nhF_int : IntegrableOn (F x\u2080) (\u0399 a b)\nbound_integrable : IntegrableOn bound (\u0399 a b)\n\u22a2 IntegrableOn F' (\u0399 a b) \u2227\n HasFDerivAt (fun x => (if a \u2264 b then 1 else -1) \u2022 \u222b (t : \u211d) in \u0399 a b, F x t \u2202\u03bc)\n ((if a \u2264 b then 1 else -1) \u2022 \u222b (t : \u211d) in \u0399 a b, F' t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave := hasFDerivAt_integral_of_dominated_loc_of_lip \u03b5_pos hF_meas hF_int hF'_meas h_lip bound_integrable h_diff\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF : H \u2192 \u211d \u2192 E\nF' : \u211d \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF'_meas : AEStronglyMeasurable F' (Measure.restrict \u03bc (\u0399 a b))\nh_lip : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), LipschitzOnWith (\u2191Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x\u2080 \u03b5)\nh_diff : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), HasFDerivAt (fun x_1 => F x_1 x) (F' x) x\u2080\nhF_int : IntegrableOn (F x\u2080) (\u0399 a b)\nbound_integrable : IntegrableOn bound (\u0399 a b)\nthis : Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u211d) in \u0399 a b, F x a \u2202\u03bc) (\u222b (a : \u211d) in \u0399 a b, F' a \u2202\u03bc) x\u2080\n\u22a2 IntegrableOn F' (\u0399 a b) \u2227\n HasFDerivAt (fun x => (if a \u2264 b then 1 else -1) \u2022 \u222b (t : \u211d) in \u0399 a b, F x t \u2202\u03bc)\n ((if a \u2264 b then 1 else -1) \u2022 \u222b (t : \u211d) in \u0399 a b, F' t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nexact \u27e8this.1, this.2.const_smul _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF : H \u2192 \u211d \u2192 E\nF' : H \u2192 \u211d \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF_int : IntervalIntegrable (F x\u2080) \u03bc a b\nhF'_meas : AEStronglyMeasurable (F' x\u2080) (Measure.restrict \u03bc (\u0399 a b))\nh_bound : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 a b \u2192 \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x t\u2016 \u2264 bound t\nbound_integrable : IntervalIntegrable bound \u03bc a b\nh_diff : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 a b \u2192 \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x => F x t) (F' x t) x\n\u22a2 HasFDerivAt (fun x => \u222b (t : \u211d) in a..b, F x t \u2202\u03bc) (\u222b (t : \u211d) in a..b, F' x\u2080 t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nrw [\u2190 ae_restrict_iff' measurableSet_uIoc] at h_bound h_diff \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF : H \u2192 \u211d \u2192 E\nF' : H \u2192 \u211d \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF_int : IntervalIntegrable (F x\u2080) \u03bc a b\nhF'_meas : AEStronglyMeasurable (F' x\u2080) (Measure.restrict \u03bc (\u0399 a b))\nh_bound : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2200 (x_1 : H), x_1 \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x_1 x\u2016 \u2264 bound x\nbound_integrable : IntervalIntegrable bound \u03bc a b\nh_diff :\n \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2200 (x_1 : H), x_1 \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1\n\u22a2 HasFDerivAt (fun x => \u222b (t : \u211d) in a..b, F x t \u2202\u03bc) (\u222b (t : \u211d) in a..b, F' x\u2080 t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nsimp only [intervalIntegrable_iff] at hF_int bound_integrable \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF : H \u2192 \u211d \u2192 E\nF' : H \u2192 \u211d \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF'_meas : AEStronglyMeasurable (F' x\u2080) (Measure.restrict \u03bc (\u0399 a b))\nh_bound : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2200 (x_1 : H), x_1 \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x_1 x\u2016 \u2264 bound x\nh_diff :\n \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2200 (x_1 : H), x_1 \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1\nhF_int : IntegrableOn (F x\u2080) (\u0399 a b)\nbound_integrable : IntegrableOn bound (\u0399 a b)\n\u22a2 HasFDerivAt (fun x => \u222b (t : \u211d) in a..b, F x t \u2202\u03bc) (\u222b (t : \u211d) in a..b, F' x\u2080 t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nsimp only [intervalIntegral_eq_integral_uIoc]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF : H \u2192 \u211d \u2192 E\nF' : H \u2192 \u211d \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF'_meas : AEStronglyMeasurable (F' x\u2080) (Measure.restrict \u03bc (\u0399 a b))\nh_bound : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2200 (x_1 : H), x_1 \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x_1 x\u2016 \u2264 bound x\nh_diff :\n \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2200 (x_1 : H), x_1 \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1\nhF_int : IntegrableOn (F x\u2080) (\u0399 a b)\nbound_integrable : IntegrableOn bound (\u0399 a b)\n\u22a2 HasFDerivAt (fun x => (if a \u2264 b then 1 else -1) \u2022 \u222b (t : \u211d) in \u0399 a b, F x t \u2202\u03bc)\n ((if a \u2264 b then 1 else -1) \u2022 \u222b (t : \u211d) in \u0399 a b, F' x\u2080 t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nexact\n (hasFDerivAt_integral_of_dominated_of_fderiv_le \u03b5_pos hF_meas hF_int hF'_meas h_bound bound_integrable\n h_diff).const_smul\n _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF : \ud835\udd5c \u2192 \u211d \u2192 E\nF' : \u211d \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF_int : IntervalIntegrable (F x\u2080) \u03bc a b\nhF'_meas : AEStronglyMeasurable F' (Measure.restrict \u03bc (\u0399 a b))\nh_lipsch : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 a b \u2192 LipschitzOnWith (\u2191Real.nnabs (bound t)) (fun x => F x t) (ball x\u2080 \u03b5)\nbound_integrable : IntervalIntegrable bound \u03bc a b\nh_diff : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 a b \u2192 HasDerivAt (fun x => F x t) (F' t) x\u2080\n\u22a2 IntervalIntegrable F' \u03bc a b \u2227 HasDerivAt (fun x => \u222b (t : \u211d) in a..b, F x t \u2202\u03bc) (\u222b (t : \u211d) in a..b, F' t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nrw [\u2190 ae_restrict_iff' measurableSet_uIoc] at h_lipsch h_diff \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF : \ud835\udd5c \u2192 \u211d \u2192 E\nF' : \u211d \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF_int : IntervalIntegrable (F x\u2080) \u03bc a b\nhF'_meas : AEStronglyMeasurable F' (Measure.restrict \u03bc (\u0399 a b))\nh_lipsch :\n \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), LipschitzOnWith (\u2191Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x\u2080 \u03b5)\nbound_integrable : IntervalIntegrable bound \u03bc a b\nh_diff : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x\u2080\n\u22a2 IntervalIntegrable F' \u03bc a b \u2227 HasDerivAt (fun x => \u222b (t : \u211d) in a..b, F x t \u2202\u03bc) (\u222b (t : \u211d) in a..b, F' t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nsimp only [intervalIntegrable_iff] at hF_int bound_integrable \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF : \ud835\udd5c \u2192 \u211d \u2192 E\nF' : \u211d \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF'_meas : AEStronglyMeasurable F' (Measure.restrict \u03bc (\u0399 a b))\nh_lipsch :\n \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), LipschitzOnWith (\u2191Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x\u2080 \u03b5)\nh_diff : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x\u2080\nhF_int : IntegrableOn (F x\u2080) (\u0399 a b)\nbound_integrable : IntegrableOn bound (\u0399 a b)\n\u22a2 IntegrableOn F' (\u0399 a b) \u2227 HasDerivAt (fun x => \u222b (t : \u211d) in a..b, F x t \u2202\u03bc) (\u222b (t : \u211d) in a..b, F' t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nsimp only [intervalIntegral_eq_integral_uIoc]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF : \ud835\udd5c \u2192 \u211d \u2192 E\nF' : \u211d \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF'_meas : AEStronglyMeasurable F' (Measure.restrict \u03bc (\u0399 a b))\nh_lipsch :\n \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), LipschitzOnWith (\u2191Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x\u2080 \u03b5)\nh_diff : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x\u2080\nhF_int : IntegrableOn (F x\u2080) (\u0399 a b)\nbound_integrable : IntegrableOn bound (\u0399 a b)\n\u22a2 IntegrableOn F' (\u0399 a b) \u2227\n HasDerivAt (fun x => (if a \u2264 b then 1 else -1) \u2022 \u222b (t : \u211d) in \u0399 a b, F x t \u2202\u03bc)\n ((if a \u2264 b then 1 else -1) \u2022 \u222b (t : \u211d) in \u0399 a b, F' t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave := hasDerivAt_integral_of_dominated_loc_of_lip \u03b5_pos hF_meas hF_int hF'_meas h_lipsch bound_integrable h_diff\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF : \ud835\udd5c \u2192 \u211d \u2192 E\nF' : \u211d \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF'_meas : AEStronglyMeasurable F' (Measure.restrict \u03bc (\u0399 a b))\nh_lipsch :\n \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), LipschitzOnWith (\u2191Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x\u2080 \u03b5)\nh_diff : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x\u2080\nhF_int : IntegrableOn (F x\u2080) (\u0399 a b)\nbound_integrable : IntegrableOn bound (\u0399 a b)\nthis : Integrable F' \u2227 HasDerivAt (fun x => \u222b (a : \u211d) in \u0399 a b, F x a \u2202\u03bc) (\u222b (a : \u211d) in \u0399 a b, F' a \u2202\u03bc) x\u2080\n\u22a2 IntegrableOn F' (\u0399 a b) \u2227\n HasDerivAt (fun x => (if a \u2264 b then 1 else -1) \u2022 \u222b (t : \u211d) in \u0399 a b, F x t \u2202\u03bc)\n ((if a \u2264 b then 1 else -1) \u2022 \u222b (t : \u211d) in \u0399 a b, F' t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nexact \u27e8this.1, this.2.const_smul _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF F' : \ud835\udd5c \u2192 \u211d \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF_int : IntervalIntegrable (F x\u2080) \u03bc a b\nhF'_meas : AEStronglyMeasurable (F' x\u2080) (Measure.restrict \u03bc (\u0399 a b))\nh_bound : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 a b \u2192 \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x t\u2016 \u2264 bound t\nbound_integrable : IntervalIntegrable bound \u03bc a b\nh_diff : \u2200\u1d50 (t : \u211d) \u2202\u03bc, t \u2208 \u0399 a b \u2192 \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x => F x t) (F' x t) x\n\u22a2 IntervalIntegrable (F' x\u2080) \u03bc a b \u2227\n HasDerivAt (fun x => \u222b (t : \u211d) in a..b, F x t \u2202\u03bc) (\u222b (t : \u211d) in a..b, F' x\u2080 t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nrw [\u2190 ae_restrict_iff' measurableSet_uIoc] at h_bound h_diff \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF F' : \ud835\udd5c \u2192 \u211d \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF_int : IntervalIntegrable (F x\u2080) \u03bc a b\nhF'_meas : AEStronglyMeasurable (F' x\u2080) (Measure.restrict \u03bc (\u0399 a b))\nh_bound : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2200 (x_1 : \ud835\udd5c), x_1 \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x_1 x\u2016 \u2264 bound x\nbound_integrable : IntervalIntegrable bound \u03bc a b\nh_diff :\n \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2200 (x_1 : \ud835\udd5c), x_1 \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1\n\u22a2 IntervalIntegrable (F' x\u2080) \u03bc a b \u2227\n HasDerivAt (fun x => \u222b (t : \u211d) in a..b, F x t \u2202\u03bc) (\u222b (t : \u211d) in a..b, F' x\u2080 t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nsimp only [intervalIntegrable_iff] at hF_int bound_integrable \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF F' : \ud835\udd5c \u2192 \u211d \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF'_meas : AEStronglyMeasurable (F' x\u2080) (Measure.restrict \u03bc (\u0399 a b))\nh_bound : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2200 (x_1 : \ud835\udd5c), x_1 \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x_1 x\u2016 \u2264 bound x\nh_diff :\n \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2200 (x_1 : \ud835\udd5c), x_1 \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1\nhF_int : IntegrableOn (F x\u2080) (\u0399 a b)\nbound_integrable : IntegrableOn bound (\u0399 a b)\n\u22a2 IntegrableOn (F' x\u2080) (\u0399 a b) \u2227 HasDerivAt (fun x => \u222b (t : \u211d) in a..b, F x t \u2202\u03bc) (\u222b (t : \u211d) in a..b, F' x\u2080 t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nsimp only [intervalIntegral_eq_integral_uIoc]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF F' : \ud835\udd5c \u2192 \u211d \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF'_meas : AEStronglyMeasurable (F' x\u2080) (Measure.restrict \u03bc (\u0399 a b))\nh_bound : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2200 (x_1 : \ud835\udd5c), x_1 \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x_1 x\u2016 \u2264 bound x\nh_diff :\n \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2200 (x_1 : \ud835\udd5c), x_1 \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1\nhF_int : IntegrableOn (F x\u2080) (\u0399 a b)\nbound_integrable : IntegrableOn bound (\u0399 a b)\n\u22a2 IntegrableOn (F' x\u2080) (\u0399 a b) \u2227\n HasDerivAt (fun x => (if a \u2264 b then 1 else -1) \u2022 \u222b (t : \u211d) in \u0399 a b, F x t \u2202\u03bc)\n ((if a \u2264 b then 1 else -1) \u2022 \u222b (t : \u211d) in \u0399 a b, F' x\u2080 t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave := hasDerivAt_integral_of_dominated_loc_of_deriv_le \u03b5_pos hF_meas hF_int hF'_meas h_bound bound_integrable h_diff\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : IsROrC \ud835\udd5c\n\u03bc : Measure \u211d\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\na b \u03b5 : \u211d\nbound : \u211d \u2192 \u211d\nF F' : \ud835\udd5c \u2192 \u211d \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (Measure.restrict \u03bc (\u0399 a b))\nhF'_meas : AEStronglyMeasurable (F' x\u2080) (Measure.restrict \u03bc (\u0399 a b))\nh_bound : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2200 (x_1 : \ud835\udd5c), x_1 \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x_1 x\u2016 \u2264 bound x\nh_diff :\n \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2200 (x_1 : \ud835\udd5c), x_1 \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1\nhF_int : IntegrableOn (F x\u2080) (\u0399 a b)\nbound_integrable : IntegrableOn bound (\u0399 a b)\nthis : Integrable (F' x\u2080) \u2227 HasDerivAt (fun n => \u222b (a : \u211d) in \u0399 a b, F n a \u2202\u03bc) (\u222b (a : \u211d) in \u0399 a b, F' x\u2080 a \u2202\u03bc) x\u2080\n\u22a2 IntegrableOn (F' x\u2080) (\u0399 a b) \u2227\n HasDerivAt (fun x => (if a \u2264 b then 1 else -1) \u2022 \u222b (t : \u211d) in \u0399 a b, F x t \u2202\u03bc)\n ((if a \u2264 b then 1 else -1) \u2022 \u222b (t : \u211d) in \u0399 a b, F' x\u2080 t \u2202\u03bc) x\u2080\n[PROOFSTEP]\nexact \u27e8this.1, this.2.const_smul _\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.ParametricIntervalIntegral", "llama_tokens": 10295, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7772998611746911, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.43098981543920656}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nF : C \u2964 D\nG H : D \u2964 E\n\u03b1 : G \u27f6 H\nX Y : C\nf : X \u27f6 Y\n\u22a2 (F \u22d9 G).map f \u226b (fun X => NatTrans.app \u03b1 (F.obj X)) Y = (fun X => NatTrans.app \u03b1 (F.obj X)) X \u226b (F \u22d9 H).map f\n[PROOFSTEP]\nrw [Functor.comp_map, Functor.comp_map, \u03b1.naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG H : C \u2964 D\n\u03b1 : G \u27f6 H\nF : D \u2964 E\nX Y : C\nf : X \u27f6 Y\n\u22a2 (G \u22d9 F).map f \u226b (fun X => F.map (NatTrans.app \u03b1 X)) Y = (fun X => F.map (NatTrans.app \u03b1 X)) X \u226b (H \u22d9 F).map f\n[PROOFSTEP]\nrw [Functor.comp_map, Functor.comp_map, \u2190 F.map_comp, \u2190 F.map_comp, \u03b1.naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nX\u271d Y\u271d : C \u2964 D\n\u03c4 : X\u271d \u27f6 Y\u271d\nH : D \u2964 E\nX Y : C\nf : X \u27f6 Y\n\u22a2 (((fun F => Functor.mk { obj := fun G => F \u22d9 G, map := fun {X Y} \u03b1 => whiskerLeft F \u03b1 }) X\u271d).obj H).map f \u226b\n (fun c => H.map (NatTrans.app \u03c4 c)) Y =\n (fun c => H.map (NatTrans.app \u03c4 c)) X \u226b\n (((fun F => Functor.mk { obj := fun G => F \u22d9 G, map := fun {X Y} \u03b1 => whiskerLeft F \u03b1 }) Y\u271d).obj H).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nX\u271d Y\u271d : C \u2964 D\n\u03c4 : X\u271d \u27f6 Y\u271d\nH : D \u2964 E\nX Y : C\nf : X \u27f6 Y\n\u22a2 H.map (X\u271d.map f) \u226b H.map (NatTrans.app \u03c4 Y) = H.map (NatTrans.app \u03c4 X) \u226b H.map (Y\u271d.map f)\n[PROOFSTEP]\nrw [\u2190 H.map_comp, \u2190 H.map_comp, \u2190 \u03c4.naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nX\u271d Y\u271d : C \u2964 D\n\u03c4 : X\u271d \u27f6 Y\u271d\nX Y : D \u2964 E\nf : X \u27f6 Y\n\u22a2 ((fun F => Functor.mk { obj := fun G => F \u22d9 G, map := fun {X Y} \u03b1 => whiskerLeft F \u03b1 }) X\u271d).map f \u226b\n (fun H => NatTrans.mk fun c => H.map (NatTrans.app \u03c4 c)) Y =\n (fun H => NatTrans.mk fun c => H.map (NatTrans.app \u03c4 c)) X \u226b\n ((fun F => Functor.mk { obj := fun G => F \u22d9 G, map := fun {X Y} \u03b1 => whiskerLeft F \u03b1 }) Y\u271d).map f\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nX\u271d Y\u271d : C \u2964 D\n\u03c4 : X\u271d \u27f6 Y\u271d\nX Y : D \u2964 E\nf : X \u27f6 Y\nx\u271d : C\n\u22a2 NatTrans.app\n (((fun F => Functor.mk { obj := fun G => F \u22d9 G, map := fun {X Y} \u03b1 => whiskerLeft F \u03b1 }) X\u271d).map f \u226b\n (fun H => NatTrans.mk fun c => H.map (NatTrans.app \u03c4 c)) Y)\n x\u271d =\n NatTrans.app\n ((fun H => NatTrans.mk fun c => H.map (NatTrans.app \u03c4 c)) X \u226b\n ((fun F => Functor.mk { obj := fun G => F \u22d9 G, map := fun {X Y} \u03b1 => whiskerLeft F \u03b1 }) Y\u271d).map f)\n x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nX\u271d Y\u271d : C \u2964 D\n\u03c4 : X\u271d \u27f6 Y\u271d\nX Y : D \u2964 E\nf : X \u27f6 Y\nx\u271d : C\n\u22a2 NatTrans.app f (X\u271d.obj x\u271d) \u226b Y.map (NatTrans.app \u03c4 x\u271d) = X.map (NatTrans.app \u03c4 x\u271d) \u226b NatTrans.app f (Y\u271d.obj x\u271d)\n[PROOFSTEP]\nrw [f.naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nX\u271d Y\u271d : D \u2964 E\n\u03c4 : X\u271d \u27f6 Y\u271d\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\n\u22a2 (((fun H => Functor.mk { obj := fun F => F \u22d9 H, map := fun {X Y} \u03b1 => whiskerRight \u03b1 H }) X\u271d).obj F).map f \u226b\n (fun c => NatTrans.app \u03c4 (F.obj c)) Y =\n (fun c => NatTrans.app \u03c4 (F.obj c)) X \u226b\n (((fun H => Functor.mk { obj := fun F => F \u22d9 H, map := fun {X Y} \u03b1 => whiskerRight \u03b1 H }) Y\u271d).obj F).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nX\u271d Y\u271d : D \u2964 E\n\u03c4 : X\u271d \u27f6 Y\u271d\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\n\u22a2 X\u271d.map (F.map f) \u226b NatTrans.app \u03c4 (F.obj Y) = NatTrans.app \u03c4 (F.obj X) \u226b Y\u271d.map (F.map f)\n[PROOFSTEP]\nrw [\u03c4.naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nX\u271d Y\u271d : D \u2964 E\n\u03c4 : X\u271d \u27f6 Y\u271d\nX Y : C \u2964 D\nf : X \u27f6 Y\n\u22a2 ((fun H => Functor.mk { obj := fun F => F \u22d9 H, map := fun {X Y} \u03b1 => whiskerRight \u03b1 H }) X\u271d).map f \u226b\n (fun F => NatTrans.mk fun c => NatTrans.app \u03c4 (F.obj c)) Y =\n (fun F => NatTrans.mk fun c => NatTrans.app \u03c4 (F.obj c)) X \u226b\n ((fun H => Functor.mk { obj := fun F => F \u22d9 H, map := fun {X Y} \u03b1 => whiskerRight \u03b1 H }) Y\u271d).map f\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nX\u271d Y\u271d : D \u2964 E\n\u03c4 : X\u271d \u27f6 Y\u271d\nX Y : C \u2964 D\nf : X \u27f6 Y\nx\u271d : C\n\u22a2 NatTrans.app\n (((fun H => Functor.mk { obj := fun F => F \u22d9 H, map := fun {X Y} \u03b1 => whiskerRight \u03b1 H }) X\u271d).map f \u226b\n (fun F => NatTrans.mk fun c => NatTrans.app \u03c4 (F.obj c)) Y)\n x\u271d =\n NatTrans.app\n ((fun F => NatTrans.mk fun c => NatTrans.app \u03c4 (F.obj c)) X \u226b\n ((fun H => Functor.mk { obj := fun F => F \u22d9 H, map := fun {X Y} \u03b1 => whiskerRight \u03b1 H }) Y\u271d).map f)\n x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nX\u271d Y\u271d : D \u2964 E\n\u03c4 : X\u271d \u27f6 Y\u271d\nX Y : C \u2964 D\nf : X \u27f6 Y\nx\u271d : C\n\u22a2 X\u271d.map (NatTrans.app f x\u271d) \u226b NatTrans.app \u03c4 (Y.obj x\u271d) = NatTrans.app \u03c4 (X.obj x\u271d) \u226b Y\u271d.map (NatTrans.app f x\u271d)\n[PROOFSTEP]\nrw [\u2190 NatTrans.naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF : D \u2964 E\ninst\u271d : Faithful F\nX\u271d Y\u271d : C \u2964 D\na\u2081\u271d a\u2082\u271d : X\u271d \u27f6 Y\u271d\nh\u03b1\u03b2 : ((whiskeringRight C D E).obj F).map a\u2081\u271d = ((whiskeringRight C D E).obj F).map a\u2082\u271d\n\u22a2 a\u2081\u271d = a\u2082\u271d\n[PROOFSTEP]\next X\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF : D \u2964 E\ninst\u271d : Faithful F\nX\u271d Y\u271d : C \u2964 D\na\u2081\u271d a\u2082\u271d : X\u271d \u27f6 Y\u271d\nh\u03b1\u03b2 : ((whiskeringRight C D E).obj F).map a\u2081\u271d = ((whiskeringRight C D E).obj F).map a\u2082\u271d\nX : C\n\u22a2 NatTrans.app a\u2081\u271d X = NatTrans.app a\u2082\u271d X\n[PROOFSTEP]\nexact (F.map_injective <| congr_fun (congr_arg NatTrans.app h\u03b1\u03b2) X)\n[GOAL]\nA : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} A\nB : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} B\nC : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} C\nD : Type u\u2084\ninst\u271d : Category.{v\u2084, u\u2084} D\nF : A \u2964 B\nG : B \u2964 C\n\u22a2 (associator F (\ud835\udfed B) G).hom \u226b whiskerLeft F (leftUnitor G).hom = whiskerRight (rightUnitor F).hom G\n[PROOFSTEP]\naesop_cat\n[GOAL]\nA : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} A\nB : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} B\nC : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} C\nD : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2084, u\u2084} D\nE : Type u\u2085\ninst\u271d : Category.{v\u2085, u\u2085} E\nF : A \u2964 B\nG : B \u2964 C\nH : C \u2964 D\nK : D \u2964 E\n\u22a2 whiskerRight (associator F G H).hom K \u226b (associator F (G \u22d9 H) K).hom \u226b whiskerLeft F (associator G H K).hom =\n (associator (F \u22d9 G) H K).hom \u226b (associator F G (H \u22d9 K)).hom\n[PROOFSTEP]\naesop_cat\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Whiskering", "llama_tokens": 3802, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390747, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.43094106024579265}} {"text": "[GOAL]\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\nE : Type u_4\ninst\u271d\u2076 : MeasurableSpace M\ninst\u271d\u2075 : MeasurableSpace N\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d : MeasurableSMul M \u03b1\nc : M\u1d48\u1d50\u1d43\n\u22a2 \u2200 (f g : { x // x \u2208 Lp E p }), c \u2022 (f + g) = c \u2022 f + c \u2022 g\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9, _\u27e9 \u27e8\u27e8\u27e9, _\u27e9\n[GOAL]\ncase mk.mk.mk.mk\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\nE : Type u_4\ninst\u271d\u2076 : MeasurableSpace M\ninst\u271d\u2075 : MeasurableSpace N\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d : MeasurableSMul M \u03b1\nc : M\u1d48\u1d50\u1d43\nval\u271d\u00b9 : \u03b1 \u2192\u2098[\u03bc] E\na\u271d\u00b9 : { f // AEStronglyMeasurable f \u03bc }\nproperty\u271d\u00b9 : Quot.mk Setoid.r a\u271d\u00b9 \u2208 Lp E p\nval\u271d : \u03b1 \u2192\u2098[\u03bc] E\na\u271d : { f // AEStronglyMeasurable f \u03bc }\nproperty\u271d : Quot.mk Setoid.r a\u271d \u2208 Lp E p\n\u22a2 c \u2022\n ({ val := Quot.mk Setoid.r a\u271d\u00b9, property := property\u271d\u00b9 } +\n { val := Quot.mk Setoid.r a\u271d, property := property\u271d }) =\n c \u2022 { val := Quot.mk Setoid.r a\u271d\u00b9, property := property\u271d\u00b9 } +\n c \u2022 { val := Quot.mk Setoid.r a\u271d, property := property\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\nE : Type u_4\ninst\u271d\u2076 : MeasurableSpace M\ninst\u271d\u2075 : MeasurableSpace N\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d : MeasurableSMul M \u03b1\nc : M\u1d48\u1d50\u1d43\nf : { x // x \u2208 Lp E p }\n\u22a2 c \u2022 -f = -(c \u2022 f)\n[PROOFSTEP]\nrcases f with \u27e8\u27e8_\u27e9, _\u27e9\n[GOAL]\ncase mk.mk\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\nE : Type u_4\ninst\u271d\u2076 : MeasurableSpace M\ninst\u271d\u2075 : MeasurableSpace N\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d : MeasurableSMul M \u03b1\nc : M\u1d48\u1d50\u1d43\nval\u271d : \u03b1 \u2192\u2098[\u03bc] E\na\u271d : { f // AEStronglyMeasurable f \u03bc }\nproperty\u271d : Quot.mk Setoid.r a\u271d \u2208 Lp E p\n\u22a2 c \u2022 -{ val := Quot.mk Setoid.r a\u271d, property := property\u271d } =\n -(c \u2022 { val := Quot.mk Setoid.r a\u271d, property := property\u271d })\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\nE : Type u_4\ninst\u271d\u2076 : MeasurableSpace M\ninst\u271d\u2075 : MeasurableSpace N\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d : MeasurableSMul M \u03b1\nc : M\u1d48\u1d50\u1d43\n\u22a2 \u2200 (f g : { x // x \u2208 Lp E p }), c \u2022 (f - g) = c \u2022 f - c \u2022 g\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9, _\u27e9 \u27e8\u27e8\u27e9, _\u27e9\n[GOAL]\ncase mk.mk.mk.mk\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\nE : Type u_4\ninst\u271d\u2076 : MeasurableSpace M\ninst\u271d\u2075 : MeasurableSpace N\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d : MeasurableSMul M \u03b1\nc : M\u1d48\u1d50\u1d43\nval\u271d\u00b9 : \u03b1 \u2192\u2098[\u03bc] E\na\u271d\u00b9 : { f // AEStronglyMeasurable f \u03bc }\nproperty\u271d\u00b9 : Quot.mk Setoid.r a\u271d\u00b9 \u2208 Lp E p\nval\u271d : \u03b1 \u2192\u2098[\u03bc] E\na\u271d : { f // AEStronglyMeasurable f \u03bc }\nproperty\u271d : Quot.mk Setoid.r a\u271d \u2208 Lp E p\n\u22a2 c \u2022\n ({ val := Quot.mk Setoid.r a\u271d\u00b9, property := property\u271d\u00b9 } -\n { val := Quot.mk Setoid.r a\u271d, property := property\u271d }) =\n c \u2022 { val := Quot.mk Setoid.r a\u271d\u00b9, property := property\u271d\u00b9 } -\n c \u2022 { val := Quot.mk Setoid.r a\u271d, property := property\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\nE : Type u_4\ninst\u271d\u2076 : MeasurableSpace M\ninst\u271d\u2075 : MeasurableSpace N\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d : MeasurableSMul M \u03b1\n\u22a2 \u2200 (a : M\u1d48\u1d50\u1d43) (x y : { x // x \u2208 Lp E p }), a \u2022 (x + y) = a \u2022 x + a \u2022 y\n[PROOFSTEP]\nrintro _ \u27e8\u27e8\u27e9, _\u27e9 \u27e8\u27e8\u27e9, _\u27e9\n[GOAL]\ncase mk.mk.mk.mk\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\nE : Type u_4\ninst\u271d\u2076 : MeasurableSpace M\ninst\u271d\u2075 : MeasurableSpace N\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d : MeasurableSMul M \u03b1\na\u271d\u00b2 : M\u1d48\u1d50\u1d43\nval\u271d\u00b9 : \u03b1 \u2192\u2098[\u03bc] E\na\u271d\u00b9 : { f // AEStronglyMeasurable f \u03bc }\nproperty\u271d\u00b9 : Quot.mk Setoid.r a\u271d\u00b9 \u2208 Lp E p\nval\u271d : \u03b1 \u2192\u2098[\u03bc] E\na\u271d : { f // AEStronglyMeasurable f \u03bc }\nproperty\u271d : Quot.mk Setoid.r a\u271d \u2208 Lp E p\n\u22a2 a\u271d\u00b2 \u2022\n ({ val := Quot.mk Setoid.r a\u271d\u00b9, property := property\u271d\u00b9 } +\n { val := Quot.mk Setoid.r a\u271d, property := property\u271d }) =\n a\u271d\u00b2 \u2022 { val := Quot.mk Setoid.r a\u271d\u00b9, property := property\u271d\u00b9 } +\n a\u271d\u00b2 \u2022 { val := Quot.mk Setoid.r a\u271d, property := property\u271d }\n[PROOFSTEP]\nrfl\n -- The next few lemmas follow from the `IsometricSMul` instance if `1 \u2264 p`\n[GOAL]\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\nE : Type u_4\ninst\u271d\u2076 : MeasurableSpace M\ninst\u271d\u2075 : MeasurableSpace N\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d : MeasurableSMul M \u03b1\nc : M\u1d48\u1d50\u1d43\nf g : { x // x \u2208 Lp E p }\n\u22a2 dist (c \u2022 f) (c \u2022 g) = dist f g\n[PROOFSTEP]\nsimp only [dist, \u2190 smul_Lp_sub, norm_smul_Lp]\n[GOAL]\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\nE : Type u_4\ninst\u271d\u2076 : MeasurableSpace M\ninst\u271d\u2075 : MeasurableSpace N\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d : MeasurableSMul M \u03b1\nc : M\u1d48\u1d50\u1d43\nf g : { x // x \u2208 Lp E p }\n\u22a2 edist (c \u2022 f) (c \u2022 g) = edist f g\n[PROOFSTEP]\nsimp only [Lp.edist_dist, dist_smul_Lp]\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.LpSpace.DomAct.Basic", "llama_tokens": 2797, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389930307512, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.4308918102596675}} {"text": "[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nI : LieIdeal R L\nh : LieModule.IsTrivial L { x // x \u2208 \u2191I }\nx y : { x // x \u2208 \u2191I }\n\u22a2 \u2045x, y\u2046 = 0\n[PROOFSTEP]\napply h.trivial\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\u2081\ninst\u271d\u00b2 : LieRing L\u2082\ninst\u271d\u00b9 : LieAlgebra R L\u2081\ninst\u271d : LieAlgebra R L\u2082\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nh\u2081 : Surjective \u2191f\nh\u2082 : IsLieAbelian L\u2081\nx y : L\u2082\n\u22a2 \u2045x, y\u2046 = 0\n[PROOFSTEP]\nobtain \u27e8u, rfl\u27e9 := h\u2081 x\n[GOAL]\ncase intro\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\u2081\ninst\u271d\u00b2 : LieRing L\u2082\ninst\u271d\u00b9 : LieAlgebra R L\u2081\ninst\u271d : LieAlgebra R L\u2082\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nh\u2081 : Surjective \u2191f\nh\u2082 : IsLieAbelian L\u2081\ny : L\u2082\nu : L\u2081\n\u22a2 \u2045\u2191f u, y\u2046 = 0\n[PROOFSTEP]\nobtain \u27e8v, rfl\u27e9 := h\u2081 y\n[GOAL]\ncase intro.intro\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\u2081\ninst\u271d\u00b2 : LieRing L\u2082\ninst\u271d\u00b9 : LieAlgebra R L\u2081\ninst\u271d : LieAlgebra R L\u2082\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nh\u2081 : Surjective \u2191f\nh\u2082 : IsLieAbelian L\u2081\nu v : L\u2081\n\u22a2 \u2045\u2191f u, \u2191f v\u2046 = 0\n[PROOFSTEP]\nrw [\u2190 LieHom.map_lie, trivial_lie_zero, LieHom.map_zero]\n[GOAL]\nA : Type v\ninst\u271d : Ring A\n\u22a2 (IsCommutative A fun x x_1 => x * x_1) \u2194 IsLieAbelian A\n[PROOFSTEP]\nhave h\u2081 : IsCommutative A (\u00b7 * \u00b7) \u2194 \u2200 a b : A, a * b = b * a := \u27e8fun h => h.1, fun h => \u27e8h\u27e9\u27e9\n[GOAL]\nA : Type v\ninst\u271d : Ring A\nh\u2081 : (IsCommutative A fun x x_1 => x * x_1) \u2194 \u2200 (a b : A), a * b = b * a\n\u22a2 (IsCommutative A fun x x_1 => x * x_1) \u2194 IsLieAbelian A\n[PROOFSTEP]\nhave h\u2082 : IsLieAbelian A \u2194 \u2200 a b : A, \u2045a, b\u2046 = 0 := \u27e8fun h => h.1, fun h => \u27e8h\u27e9\u27e9\n[GOAL]\nA : Type v\ninst\u271d : Ring A\nh\u2081 : (IsCommutative A fun x x_1 => x * x_1) \u2194 \u2200 (a b : A), a * b = b * a\nh\u2082 : IsLieAbelian A \u2194 \u2200 (a b : A), \u2045a, b\u2046 = 0\n\u22a2 (IsCommutative A fun x x_1 => x * x_1) \u2194 IsLieAbelian A\n[PROOFSTEP]\nsimp only [h\u2081, h\u2082, LieRing.of_associative_ring_bracket, sub_eq_zero]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx\u271d\u00b9 x\u271d : { x // x \u2208 \u2191\u22a5 }\nx : L\nhx : x \u2208 \u2191\u22a5\n\u22a2 \u2045{ val := x, property := hx }, x\u271d\u2046 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nx : L\n\u22a2 x \u2208 LieModule.ker R L M \u2194 \u2200 (m : M), \u2045x, m\u2046 = 0\n[PROOFSTEP]\nsimp only [LieModule.ker, LieHom.mem_ker, LinearMap.ext_iff, LinearMap.zero_apply, toEndomorphism_apply_apply]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nx y : M\nhx : x \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 = 0}\nhy : y \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 = 0}\nz : L\n\u22a2 \u2045z, x + y\u2046 = 0\n[PROOFSTEP]\nrw [lie_add, hx, hy, add_zero]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nc : R\nx : M\nhx :\n x \u2208\n {\n toAddSubsemigroup :=\n { carrier := {m | \u2200 (x : L), \u2045x, m\u2046 = 0},\n add_mem' :=\n (_ :\n \u2200 {x y : M},\n x \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 = 0} \u2192 y \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 = 0} \u2192 \u2200 (z : L), \u2045z, x + y\u2046 = 0) },\n zero_mem' := (_ : \u2200 (x : L), \u2045x, 0\u2046 = 0) }.toAddSubsemigroup.carrier\ny : L\n\u22a2 \u2045y, c \u2022 x\u2046 = 0\n[PROOFSTEP]\nrw [lie_smul, hx, smul_zero]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nx : L\nm : M\nhm :\n m \u2208\n {\n toAddSubmonoid :=\n {\n toAddSubsemigroup :=\n { carrier := {m | \u2200 (x : L), \u2045x, m\u2046 = 0},\n add_mem' :=\n (_ :\n \u2200 {x y : M},\n x \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 = 0} \u2192\n y \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 = 0} \u2192 \u2200 (z : L), \u2045z, x + y\u2046 = 0) },\n zero_mem' := (_ : \u2200 (x : L), \u2045x, 0\u2046 = 0) },\n smul_mem' :=\n (_ :\n \u2200 (c : R) (x : M),\n x \u2208\n {\n toAddSubsemigroup :=\n { carrier := {m | \u2200 (x : L), \u2045x, m\u2046 = 0},\n add_mem' :=\n (_ :\n \u2200 {x y : M},\n x \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 = 0} \u2192\n y \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 = 0} \u2192 \u2200 (z : L), \u2045z, x + y\u2046 = 0) },\n zero_mem' := (_ : \u2200 (x : L), \u2045x, 0\u2046 = 0) }.toAddSubsemigroup.carrier \u2192\n \u2200 (y : L), \u2045y, c \u2022 x\u2046 = 0) }.toAddSubmonoid.toAddSubsemigroup.carrier\ny : L\n\u22a2 \u2045y, \u2045x, m\u2046\u2046 = 0\n[PROOFSTEP]\nrw [hm, lie_zero]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nI : LieIdeal R L\n\u22a2 \u2045I, maxTrivSubmodule R L M\u2046 = \u22a5\n[PROOFSTEP]\nrw [\u2190 LieSubmodule.coe_toSubmodule_eq_iff, LieSubmodule.lieIdeal_oper_eq_linear_span, LieSubmodule.bot_coeSubmodule,\n Submodule.span_eq_bot]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nI : LieIdeal R L\n\u22a2 \u2200 (x : M), x \u2208 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2192 x = 0\n[PROOFSTEP]\nrintro m \u27e8\u27e8x, hx\u27e9, \u27e8\u27e8m, hm\u27e9, rfl\u27e9\u27e9\n[GOAL]\ncase intro.mk.intro.mk\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nI : LieIdeal R L\nx : L\nhx : x \u2208 I\nm : M\nhm : m \u2208 maxTrivSubmodule R L M\n\u22a2 \u2045\u2191{ val := x, property := hx }, \u2191{ val := m, property := hm }\u2046 = 0\n[PROOFSTEP]\nexact hm x\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN\u271d : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : LieRingModule L N\u271d\ninst\u271d : LieModule R L N\u271d\nN : LieSubmodule R L M\n\u22a2 N \u2264 maxTrivSubmodule R L M \u2194 \u2045\u22a4, N\u2046 = \u22a5\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h m hm => _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u\nL : Type v\nM : Type w\nN\u271d : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : LieRingModule L N\u271d\ninst\u271d : LieModule R L N\u271d\nN : LieSubmodule R L M\nh : N \u2264 maxTrivSubmodule R L M\n\u22a2 \u2045\u22a4, N\u2046 = \u22a5\n[PROOFSTEP]\nrw [\u2190 le_bot_iff, \u2190 ideal_oper_maxTrivSubmodule_eq_bot R L M \u22a4]\n[GOAL]\ncase refine'_1\nR : Type u\nL : Type v\nM : Type w\nN\u271d : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : LieRingModule L N\u271d\ninst\u271d : LieModule R L N\u271d\nN : LieSubmodule R L M\nh : N \u2264 maxTrivSubmodule R L M\n\u22a2 \u2045\u22a4, N\u2046 \u2264 \u2045\u22a4, maxTrivSubmodule R L M\u2046\n[PROOFSTEP]\nexact LieSubmodule.mono_lie_right _ _ \u22a4 h\n[GOAL]\ncase refine'_2\nR : Type u\nL : Type v\nM : Type w\nN\u271d : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : LieRingModule L N\u271d\ninst\u271d : LieModule R L N\u271d\nN : LieSubmodule R L M\nh : \u2045\u22a4, N\u2046 = \u22a5\nm : M\nhm : m \u2208 N\n\u22a2 m \u2208 maxTrivSubmodule R L M\n[PROOFSTEP]\nrw [mem_maxTrivSubmodule]\n[GOAL]\ncase refine'_2\nR : Type u\nL : Type v\nM : Type w\nN\u271d : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : LieRingModule L N\u271d\ninst\u271d : LieModule R L N\u271d\nN : LieSubmodule R L M\nh : \u2045\u22a4, N\u2046 = \u22a5\nm : M\nhm : m \u2208 N\n\u22a2 \u2200 (x : L), \u2045x, m\u2046 = 0\n[PROOFSTEP]\nrw [LieSubmodule.lie_eq_bot_iff] at h \n[GOAL]\ncase refine'_2\nR : Type u\nL : Type v\nM : Type w\nN\u271d : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : LieRingModule L N\u271d\ninst\u271d : LieModule R L N\u271d\nN : LieSubmodule R L M\nh : \u2200 (x : L), x \u2208 \u22a4 \u2192 \u2200 (m : M), m \u2208 N \u2192 \u2045x, m\u2046 = 0\nm : M\nhm : m \u2208 N\n\u22a2 \u2200 (x : L), \u2045x, m\u2046 = 0\n[PROOFSTEP]\nexact fun x => h x (LieSubmodule.mem_top x) m hm\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\n\u22a2 IsTrivial L M \u2194 maxTrivSubmodule R L M = \u22a4\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\n\u22a2 IsTrivial L M \u2192 maxTrivSubmodule R L M = \u22a4\n[PROOFSTEP]\nrintro \u27e8h\u27e9\n[GOAL]\ncase mp.mk\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nh : \u2200 (x : L) (m : M), \u2045x, m\u2046 = 0\n\u22a2 maxTrivSubmodule R L M = \u22a4\n[PROOFSTEP]\next\n[GOAL]\ncase mp.mk.h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nh : \u2200 (x : L) (m : M), \u2045x, m\u2046 = 0\nm\u271d : M\n\u22a2 m\u271d \u2208 maxTrivSubmodule R L M \u2194 m\u271d \u2208 \u22a4\n[PROOFSTEP]\nsimp only [mem_maxTrivSubmodule, h, forall_const, LieSubmodule.mem_top]\n[GOAL]\ncase mpr\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\n\u22a2 maxTrivSubmodule R L M = \u22a4 \u2192 IsTrivial L M\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nh : maxTrivSubmodule R L M = \u22a4\n\u22a2 IsTrivial L M\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.trivial\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nh : maxTrivSubmodule R L M = \u22a4\n\u22a2 \u2200 (x : L) (m : M), \u2045x, m\u2046 = 0\n[PROOFSTEP]\nintro x m\n[GOAL]\ncase mpr.trivial\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nh : maxTrivSubmodule R L M = \u22a4\nx : L\nm : M\n\u22a2 \u2045x, m\u2046 = 0\n[PROOFSTEP]\nrevert x\n[GOAL]\ncase mpr.trivial\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nh : maxTrivSubmodule R L M = \u22a4\nm : M\n\u22a2 \u2200 (x : L), \u2045x, m\u2046 = 0\n[PROOFSTEP]\nrw [\u2190 mem_maxTrivSubmodule R L M, h]\n[GOAL]\ncase mpr.trivial\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nh : maxTrivSubmodule R L M = \u22a4\nm : M\n\u22a2 m \u2208 \u22a4\n[PROOFSTEP]\nexact LieSubmodule.mem_top m\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : M \u2192\u2097\u2045R,L\u2046 N\nm n : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }\n\u22a2 (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) (m + n) =\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) m +\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) n\n[PROOFSTEP]\nsimp [Function.comp_apply]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : M \u2192\u2097\u2045R,L\u2046 N\nm n : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }\n\u22a2 { val := \u2191f \u2191m + \u2191f \u2191n, property := (_ : (fun x => x \u2208 \u2191(maxTrivSubmodule R L N)) (\u2191f \u2191m + \u2191f \u2191n)) } =\n { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) } +\n { val := \u2191f \u2191n, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191n\u2046 = 0) }\n[PROOFSTEP]\nrfl\n -- Porting note:\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : M \u2192\u2097\u2045R,L\u2046 N\nt : R\nm : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }\n\u22a2 AddHom.toFun\n { toFun := fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) (m + n) =\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) m +\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n { toFun := fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) (m + n) =\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) m +\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) n) }\n m\n[PROOFSTEP]\nsimp [Function.comp_apply]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : M \u2192\u2097\u2045R,L\u2046 N\nt : R\nm : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }\n\u22a2 { val := t \u2022 \u2191f \u2191m, property := (_ : (fun x => x \u2208 \u2191(maxTrivSubmodule R L N)) (t \u2022 \u2191f \u2191m)) } =\n t \u2022 { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }\n[PROOFSTEP]\nrfl\n -- these two were `by simpa`\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : M \u2192\u2097\u2045R,L\u2046 N\nx : L\nm : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) (m + n) =\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) m +\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n AddHom.toFun\n { toFun := fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) (m + n) =\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) m +\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n { toFun := fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) (m + n) =\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) m +\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) n) }\n m) }.toAddHom\n \u2045x, m\u2046 =\n \u2045x,\n AddHom.toFun\n {\n toAddHom :=\n { toFun := fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) (m + n) =\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) m +\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n AddHom.toFun\n { toFun := fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) (m + n) =\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) m +\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n { toFun := fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) (m + n) =\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) m +\n (fun m => { val := \u2191f \u2191m, property := (_ : \u2200 (x : L), \u2045x, \u2191f \u2191m\u2046 = 0) }) n) }\n m) }.toAddHom\n m\u2046\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\ne : M \u2243\u2097\u2045R,L\u2046 N\nsrc\u271d : { x // x \u2208 \u2191(maxTrivSubmodule R L M) } \u2192\u2097\u2045R,L\u2046 { x // x \u2208 \u2191(maxTrivSubmodule R L N) } :=\n maxTrivHom e.toLieModuleHom\nm : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }\n\u22a2 \u2191(maxTrivHom (LieModuleEquiv.symm e).toLieModuleHom)\n (AddHom.toFun\n (\u2191{\n toLinearMap :=\n {\n toAddHom :=\n { toFun := \u2191(maxTrivHom e.toLieModuleHom),\n map_add' :=\n (_ :\n \u2200 (x y : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n AddHom.toFun src\u271d.toAddHom (x + y) =\n AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) },\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }},\n AddHom.toFun src\u271d.toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun src\u271d.toAddHom m\u2046) }).toAddHom\n m) =\n m\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\ne : M \u2243\u2097\u2045R,L\u2046 N\nsrc\u271d : { x // x \u2208 \u2191(maxTrivSubmodule R L M) } \u2192\u2097\u2045R,L\u2046 { x // x \u2208 \u2191(maxTrivSubmodule R L N) } :=\n maxTrivHom e.toLieModuleHom\nm : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }\n\u22a2 \u2191(\u2191(maxTrivHom (LieModuleEquiv.symm e).toLieModuleHom)\n (AddHom.toFun\n (\u2191{\n toLinearMap :=\n {\n toAddHom :=\n { toFun := \u2191(maxTrivHom e.toLieModuleHom),\n map_add' :=\n (_ :\n \u2200 (x y : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n AddHom.toFun src\u271d.toAddHom (x + y) =\n AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) },\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }},\n AddHom.toFun src\u271d.toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun src\u271d.toAddHom m\u2046) }).toAddHom\n m)) =\n \u2191m\n[PROOFSTEP]\nsimp [LieModuleEquiv.coe_to_lieModuleHom]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\ne : M \u2243\u2097\u2045R,L\u2046 N\nsrc\u271d : { x // x \u2208 \u2191(maxTrivSubmodule R L M) } \u2192\u2097\u2045R,L\u2046 { x // x \u2208 \u2191(maxTrivSubmodule R L N) } :=\n maxTrivHom e.toLieModuleHom\nn : { x // x \u2208 \u2191(maxTrivSubmodule R L N) }\n\u22a2 AddHom.toFun\n (\u2191{\n toLinearMap :=\n {\n toAddHom :=\n { toFun := \u2191(maxTrivHom e.toLieModuleHom),\n map_add' :=\n (_ :\n \u2200 (x y : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n AddHom.toFun src\u271d.toAddHom (x + y) =\n AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) },\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }},\n AddHom.toFun src\u271d.toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun src\u271d.toAddHom m\u2046) }).toAddHom\n (\u2191(maxTrivHom (LieModuleEquiv.symm e).toLieModuleHom) n) =\n n\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\ne : M \u2243\u2097\u2045R,L\u2046 N\nsrc\u271d : { x // x \u2208 \u2191(maxTrivSubmodule R L M) } \u2192\u2097\u2045R,L\u2046 { x // x \u2208 \u2191(maxTrivSubmodule R L N) } :=\n maxTrivHom e.toLieModuleHom\nn : { x // x \u2208 \u2191(maxTrivSubmodule R L N) }\n\u22a2 \u2191(AddHom.toFun\n (\u2191{\n toLinearMap :=\n {\n toAddHom :=\n { toFun := \u2191(maxTrivHom e.toLieModuleHom),\n map_add' :=\n (_ :\n \u2200 (x y : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n AddHom.toFun src\u271d.toAddHom (x + y) =\n AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) },\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }},\n AddHom.toFun src\u271d.toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun src\u271d.toAddHom m\u2046) }).toAddHom\n (\u2191(maxTrivHom (LieModuleEquiv.symm e).toLieModuleHom) n)) =\n \u2191n\n[PROOFSTEP]\nsimp [LieModuleEquiv.coe_to_lieModuleHom]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\n\u22a2 maxTrivEquiv LieModuleEquiv.refl = LieModuleEquiv.refl\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nm\u271d : { x // x \u2208 \u2191(maxTrivSubmodule R L M) }\n\u22a2 \u2191(\u2191(maxTrivEquiv LieModuleEquiv.refl) m\u271d) = \u2191(\u2191LieModuleEquiv.refl m\u271d)\n[PROOFSTEP]\nsimp only [coe_maxTrivEquiv_apply, LieModuleEquiv.refl_apply]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }\nx : L\nm : M\n\u22a2 AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046\n[PROOFSTEP]\nhave hf : \u2045x, f.val\u2046 m = 0 := by rw [f.property x, LinearMap.zero_apply]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }\nx : L\nm : M\n\u22a2 \u2191\u2045x, \u2191f\u2046 m = 0\n[PROOFSTEP]\nrw [f.property x, LinearMap.zero_apply]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }\nx : L\nm : M\nhf : \u2191\u2045x, \u2191f\u2046 m = 0\n\u22a2 AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046\n[PROOFSTEP]\nrw [LieHom.lie_apply, sub_eq_zero, \u2190 LinearMap.toFun_eq_coe] at hf \n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }\nx : L\nm : M\nhf : \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046 = AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046\n\u22a2 AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046\n[PROOFSTEP]\nexact hf.symm\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf g : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }\n\u22a2 (fun f =>\n { toLinearMap := \u2191f,\n map_lie' := (_ : \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n (f + g) =\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ : \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n f +\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ : \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n g\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf g : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }\nm\u271d : M\n\u22a2 \u2191((fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ : \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n (f + g))\n m\u271d =\n \u2191((fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ : \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n f +\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ : \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n g)\n m\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nF : R\nG : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }\n\u22a2 AddHom.toFun\n {\n toFun := fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ : \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) },\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }),\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n (f + g) =\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n f +\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n g) }\n (F \u2022 G) =\n \u2191(RingHom.id R) F \u2022\n AddHom.toFun\n {\n toFun := fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ : \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) },\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }),\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n (f + g) =\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n f +\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n g) }\n G\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nF : R\nG : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }\nm\u271d : M\n\u22a2 \u2191(AddHom.toFun\n {\n toFun := fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ : \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) },\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }),\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n (f + g) =\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n f +\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n g) }\n (F \u2022 G))\n m\u271d =\n \u2191(\u2191(RingHom.id R) F \u2022\n AddHom.toFun\n {\n toFun := fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ : \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) },\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }),\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n (f + g) =\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n f +\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n g) }\n G)\n m\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nF : M \u2192\u2097\u2045R,L\u2046 N\nx : L\n\u22a2 \u2045x, \u2191F\u2046 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nF : M \u2192\u2097\u2045R,L\u2046 N\nx : L\nx\u271d : M\n\u22a2 \u2191\u2045x, \u2191F\u2046 x\u271d = \u21910 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }\n\u22a2 (fun F => { val := \u2191F, property := (_ : \u2200 (x : L), \u2045x, \u2191F\u2046 = 0) })\n (AddHom.toFun\n {\n toAddHom :=\n {\n toFun := fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ : \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) },\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }),\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n (f + g) =\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n f +\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n g) },\n map_smul' :=\n (_ :\n \u2200 (F : R) (G : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }),\n AddHom.toFun\n {\n toFun := fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) },\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }),\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n (f + g) =\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n f +\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n g) }\n (F \u2022 G) =\n \u2191(RingHom.id R) F \u2022\n AddHom.toFun\n {\n toFun := fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) },\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }),\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n (f + g) =\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 =\n \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n f +\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 =\n \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n g) }\n G) }.toAddHom\n f) =\n f\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nF : M \u2192\u2097\u2045R,L\u2046 N\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n {\n toFun := fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ : \u2200 {x : L} {m : M}, AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) },\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }),\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n (f + g) =\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n f +\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n g) },\n map_smul' :=\n (_ :\n \u2200 (F : R) (G : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }),\n AddHom.toFun\n {\n toFun := fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) },\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }),\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n (f + g) =\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n f +\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n g) }\n (F \u2022 G) =\n \u2191(RingHom.id R) F \u2022\n AddHom.toFun\n {\n toFun := fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) },\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 \u2191(maxTrivSubmodule R L (M \u2192\u2097[R] N)) }),\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n (f + g) =\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n f +\n (fun f =>\n { toLinearMap := \u2191f,\n map_lie' :=\n (_ :\n \u2200 {x : L} {m : M},\n AddHom.toFun (\u2191f).toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun (\u2191f).toAddHom m\u2046) })\n g) }\n G) }.toAddHom\n ((fun F => { val := \u2191F, property := (_ : \u2200 (x : L), \u2045x, \u2191F\u2046 = 0) }) F) =\n F\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : { x // x \u2208 maxTrivSubmodule R L (M \u2192\u2097[R] N) }\n\u22a2 \u2191(\u2191maxTrivLinearMapEquivLieModuleHom f) = \u2191\u2191f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : { x // x \u2208 maxTrivSubmodule R L (M \u2192\u2097[R] N) }\nx\u271d : M\n\u22a2 \u2191(\u2191maxTrivLinearMapEquivLieModuleHom f) x\u271d = \u2191\u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : { x // x \u2208 maxTrivSubmodule R L (M \u2192\u2097[R] N) }\n\u22a2 \u2191(\u2191maxTrivLinearMapEquivLieModuleHom f) = \u2191f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nf : { x // x \u2208 maxTrivSubmodule R L (M \u2192\u2097[R] N) }\nx\u271d : M\n\u22a2 \u2191\u2191(\u2191maxTrivLinearMapEquivLieModuleHom f) x\u271d = \u2191\u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\n\u22a2 LieModule.ker R L L = center R L\n[PROOFSTEP]\next y\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\ny : L\n\u22a2 y \u2208 LieModule.ker R L L \u2194 y \u2208 center R L\n[PROOFSTEP]\nsimp only [LieModule.mem_maxTrivSubmodule, LieModule.mem_ker, \u2190 lie_skew _ y, neg_eq_zero]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : LieRingModule L M\ninst\u271d\u00b9 : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\ninst\u271d : LieModule.IsTrivial L M\n\u22a2 \u2045I, N\u2046 = \u22a5\n[PROOFSTEP]\nsuffices : \u2045I, N\u2046 \u2264 \u22a5\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : LieRingModule L M\ninst\u271d\u00b9 : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\ninst\u271d : LieModule.IsTrivial L M\nthis : \u2045I, N\u2046 \u2264 \u22a5\n\u22a2 \u2045I, N\u2046 = \u22a5\ncase this\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : LieRingModule L M\ninst\u271d\u00b9 : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\ninst\u271d : LieModule.IsTrivial L M\n\u22a2 \u2045I, N\u2046 \u2264 \u22a5\n[PROOFSTEP]\nexact le_bot_iff.mp this\n[GOAL]\ncase this\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : LieRingModule L M\ninst\u271d\u00b9 : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\ninst\u271d : LieModule.IsTrivial L M\n\u22a2 \u2045I, N\u2046 \u2264 \u22a5\n[PROOFSTEP]\nrw [lieIdeal_oper_eq_span, LieSubmodule.lieSpan_le]\n[GOAL]\ncase this\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : LieRingModule L M\ninst\u271d\u00b9 : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\ninst\u271d : LieModule.IsTrivial L M\n\u22a2 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2286 \u2191\u22a5\n[PROOFSTEP]\nrintro m \u27e8x, n, h\u27e9\n[GOAL]\ncase this.intro.intro\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : LieRingModule L M\ninst\u271d\u00b9 : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\ninst\u271d : LieModule.IsTrivial L M\nm : M\nx : { x // x \u2208 I }\nn : { x // x \u2208 N }\nh : \u2045\u2191x, \u2191n\u2046 = m\n\u22a2 m \u2208 \u2191\u22a5\n[PROOFSTEP]\nrw [trivial_lie_zero] at h \n[GOAL]\ncase this.intro.intro\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : LieRingModule L M\ninst\u271d\u00b9 : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\ninst\u271d : LieModule.IsTrivial L M\nm : M\nx : { x // x \u2208 I }\nn : { x // x \u2208 N }\nh : 0 = m\n\u22a2 m \u2208 \u2191\u22a5\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\n\u22a2 IsLieAbelian { x // x \u2208 \u2191I } \u2194 \u2045I, I\u2046 = \u22a5\n[PROOFSTEP]\nsimp only [_root_.eq_bot_iff, lieIdeal_oper_eq_span, LieSubmodule.lieSpan_le, LieSubmodule.bot_coe,\n Set.subset_singleton_iff, Set.mem_setOf_eq, exists_imp]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\n\u22a2 IsLieAbelian { x // x \u2208 \u2191I } \u2194 \u2200 (y : L) (x : { x // x \u2208 I }) (x_1 : { x // x \u2208 I }), \u2045\u2191x, \u2191x_1\u2046 = y \u2192 y = 0\n[PROOFSTEP]\nrefine'\n \u27e8fun h z x y hz =>\n hz.symm.trans\n (((I : LieSubalgebra R L).coe_bracket x y).symm.trans ((coe_zero_iff_zero _ _).mpr (by apply h.trivial))),\n fun h => \u27e8fun x y => ((I : LieSubalgebra R L).coe_zero_iff_zero _).mp (h _ x y rfl)\u27e9\u27e9\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nh : IsLieAbelian { x // x \u2208 \u2191I }\nz : L\nx : { x // x \u2208 I }\ny : { x // x \u2208 I }\nhz : \u2045\u2191x, \u2191y\u2046 = z\n\u22a2 \u2045x, y\u2046 = 0\n[PROOFSTEP]\napply h.trivial\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Lie.Abelian", "llama_tokens": 25343, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389930307512, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.43089181025966744}} {"text": "[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Semiring M\nf g : AddMonoidAlgebra M \u03b9\n\u22a2 toDirectSum (f * g) = toDirectSum f * toDirectSum g\n[PROOFSTEP]\nlet to_hom : AddMonoidAlgebra M \u03b9 \u2192+ \u2a01 _ : \u03b9, M :=\n { toFun := toDirectSum\n map_zero' := toDirectSum_zero\n map_add' := toDirectSum_add }\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Semiring M\nf g : AddMonoidAlgebra M \u03b9\nto_hom : AddMonoidAlgebra M \u03b9 \u2192+ \u2a01 (x : \u03b9), M :=\n { toZeroHom := { toFun := toDirectSum, map_zero' := (_ : toDirectSum 0 = 0) },\n map_add' := (_ : \u2200 (f g : AddMonoidAlgebra M \u03b9), toDirectSum (f + g) = toDirectSum f + toDirectSum g) }\n\u22a2 toDirectSum (f * g) = toDirectSum f * toDirectSum g\n[PROOFSTEP]\nshow to_hom (f * g) = to_hom f * to_hom g\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Semiring M\nf g : AddMonoidAlgebra M \u03b9\nto_hom : AddMonoidAlgebra M \u03b9 \u2192+ \u2a01 (x : \u03b9), M :=\n { toZeroHom := { toFun := toDirectSum, map_zero' := (_ : toDirectSum 0 = 0) },\n map_add' := (_ : \u2200 (f g : AddMonoidAlgebra M \u03b9), toDirectSum (f + g) = toDirectSum f + toDirectSum g) }\n\u22a2 \u2191to_hom (f * g) = \u2191to_hom f * \u2191to_hom g\n[PROOFSTEP]\nlet _ : NonUnitalNonAssocSemiring (\u03b9 \u2192\u2080 M) := AddMonoidAlgebra.nonUnitalNonAssocSemiring\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Semiring M\nf g : AddMonoidAlgebra M \u03b9\nto_hom : AddMonoidAlgebra M \u03b9 \u2192+ \u2a01 (x : \u03b9), M :=\n { toZeroHom := { toFun := toDirectSum, map_zero' := (_ : toDirectSum 0 = 0) },\n map_add' := (_ : \u2200 (f g : AddMonoidAlgebra M \u03b9), toDirectSum (f + g) = toDirectSum f + toDirectSum g) }\nx\u271d : NonUnitalNonAssocSemiring (\u03b9 \u2192\u2080 M) := nonUnitalNonAssocSemiring\n\u22a2 \u2191to_hom (f * g) = \u2191to_hom f * \u2191to_hom g\n[PROOFSTEP]\nrevert f g\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Semiring M\nto_hom : AddMonoidAlgebra M \u03b9 \u2192+ \u2a01 (x : \u03b9), M :=\n { toZeroHom := { toFun := toDirectSum, map_zero' := (_ : toDirectSum 0 = 0) },\n map_add' := (_ : \u2200 (f g : AddMonoidAlgebra M \u03b9), toDirectSum (f + g) = toDirectSum f + toDirectSum g) }\nx\u271d : NonUnitalNonAssocSemiring (\u03b9 \u2192\u2080 M) := nonUnitalNonAssocSemiring\n\u22a2 \u2200 (f g : AddMonoidAlgebra M \u03b9), \u2191to_hom (f * g) = \u2191to_hom f * \u2191to_hom g\n[PROOFSTEP]\nrw [AddMonoidHom.map_mul_iff]\n -- porting note: does not find `addHom_ext'`, was `ext (xi xv yi yv) : 4`\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Semiring M\nto_hom : AddMonoidAlgebra M \u03b9 \u2192+ \u2a01 (x : \u03b9), M :=\n { toZeroHom := { toFun := toDirectSum, map_zero' := (_ : toDirectSum 0 = 0) },\n map_add' := (_ : \u2200 (f g : AddMonoidAlgebra M \u03b9), toDirectSum (f + g) = toDirectSum f + toDirectSum g) }\nx\u271d : NonUnitalNonAssocSemiring (\u03b9 \u2192\u2080 M) := nonUnitalNonAssocSemiring\n\u22a2 AddMonoidHom.compr\u2082 AddMonoidHom.mul to_hom = AddMonoidHom.compl\u2082 (AddMonoidHom.comp AddMonoidHom.mul to_hom) to_hom\n[PROOFSTEP]\nrefine Finsupp.addHom_ext' fun xi => AddMonoidHom.ext fun xv => ?_\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Semiring M\nto_hom : AddMonoidAlgebra M \u03b9 \u2192+ \u2a01 (x : \u03b9), M :=\n { toZeroHom := { toFun := toDirectSum, map_zero' := (_ : toDirectSum 0 = 0) },\n map_add' := (_ : \u2200 (f g : AddMonoidAlgebra M \u03b9), toDirectSum (f + g) = toDirectSum f + toDirectSum g) }\nx\u271d : NonUnitalNonAssocSemiring (\u03b9 \u2192\u2080 M) := nonUnitalNonAssocSemiring\nxi : \u03b9\nxv : M\n\u22a2 \u2191(AddMonoidHom.comp (AddMonoidHom.compr\u2082 AddMonoidHom.mul to_hom) (Finsupp.singleAddHom xi)) xv =\n \u2191(AddMonoidHom.comp (AddMonoidHom.compl\u2082 (AddMonoidHom.comp AddMonoidHom.mul to_hom) to_hom)\n (Finsupp.singleAddHom xi))\n xv\n[PROOFSTEP]\nrefine Finsupp.addHom_ext' fun yi => AddMonoidHom.ext fun yv => ?_\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Semiring M\nto_hom : AddMonoidAlgebra M \u03b9 \u2192+ \u2a01 (x : \u03b9), M :=\n { toZeroHom := { toFun := toDirectSum, map_zero' := (_ : toDirectSum 0 = 0) },\n map_add' := (_ : \u2200 (f g : AddMonoidAlgebra M \u03b9), toDirectSum (f + g) = toDirectSum f + toDirectSum g) }\nx\u271d : NonUnitalNonAssocSemiring (\u03b9 \u2192\u2080 M) := nonUnitalNonAssocSemiring\nxi : \u03b9\nxv : M\nyi : \u03b9\nyv : M\n\u22a2 \u2191(AddMonoidHom.comp (\u2191(AddMonoidHom.comp (AddMonoidHom.compr\u2082 AddMonoidHom.mul to_hom) (Finsupp.singleAddHom xi)) xv)\n (Finsupp.singleAddHom yi))\n yv =\n \u2191(AddMonoidHom.comp\n (\u2191(AddMonoidHom.comp (AddMonoidHom.compl\u2082 (AddMonoidHom.comp AddMonoidHom.mul to_hom) to_hom)\n (Finsupp.singleAddHom xi))\n xv)\n (Finsupp.singleAddHom yi))\n yv\n[PROOFSTEP]\ndsimp only [AddMonoidHom.comp_apply, AddMonoidHom.compl\u2082_apply, AddMonoidHom.compr\u2082_apply, AddMonoidHom.mul_apply,\n Finsupp.singleAddHom_apply]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Semiring M\nto_hom : AddMonoidAlgebra M \u03b9 \u2192+ \u2a01 (x : \u03b9), M :=\n { toZeroHom := { toFun := toDirectSum, map_zero' := (_ : toDirectSum 0 = 0) },\n map_add' := (_ : \u2200 (f g : AddMonoidAlgebra M \u03b9), toDirectSum (f + g) = toDirectSum f + toDirectSum g) }\nx\u271d : NonUnitalNonAssocSemiring (\u03b9 \u2192\u2080 M) := nonUnitalNonAssocSemiring\nxi : \u03b9\nxv : M\nyi : \u03b9\nyv : M\n\u22a2 \u2191{ toZeroHom := { toFun := toDirectSum, map_zero' := (_ : toDirectSum 0 = 0) },\n map_add' := (_ : \u2200 (f g : AddMonoidAlgebra M \u03b9), toDirectSum (f + g) = toDirectSum f + toDirectSum g) }\n (Finsupp.single xi xv * Finsupp.single yi yv) =\n \u2191{ toZeroHom := { toFun := toDirectSum, map_zero' := (_ : toDirectSum 0 = 0) },\n map_add' := (_ : \u2200 (f g : AddMonoidAlgebra M \u03b9), toDirectSum (f + g) = toDirectSum f + toDirectSum g) }\n (Finsupp.single xi xv) *\n \u2191{ toZeroHom := { toFun := toDirectSum, map_zero' := (_ : toDirectSum 0 = 0) },\n map_add' := (_ : \u2200 (f g : AddMonoidAlgebra M \u03b9), toDirectSum (f + g) = toDirectSum f + toDirectSum g) }\n (Finsupp.single yi yv)\n[PROOFSTEP]\nsimp only [AddMonoidHom.coe_mk, ZeroHom.coe_mk, toDirectSum_single]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Semiring M\nto_hom : AddMonoidAlgebra M \u03b9 \u2192+ \u2a01 (x : \u03b9), M :=\n { toZeroHom := { toFun := toDirectSum, map_zero' := (_ : toDirectSum 0 = 0) },\n map_add' := (_ : \u2200 (f g : AddMonoidAlgebra M \u03b9), toDirectSum (f + g) = toDirectSum f + toDirectSum g) }\nx\u271d : NonUnitalNonAssocSemiring (\u03b9 \u2192\u2080 M) := nonUnitalNonAssocSemiring\nxi : \u03b9\nxv : M\nyi : \u03b9\nyv : M\n\u22a2 toDirectSum (Finsupp.single xi xv * Finsupp.single yi yv) =\n \u2191(DirectSum.of (fun i => M) xi) xv * \u2191(DirectSum.of (fun i => M) yi) yv\n[PROOFSTEP]\nerw [AddMonoidAlgebra.single_mul_single, AddMonoidHom.coe_mk, ZeroHom.coe_mk, AddMonoidAlgebra.toDirectSum_single]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Semiring M\nto_hom : AddMonoidAlgebra M \u03b9 \u2192+ \u2a01 (x : \u03b9), M :=\n { toZeroHom := { toFun := toDirectSum, map_zero' := (_ : toDirectSum 0 = 0) },\n map_add' := (_ : \u2200 (f g : AddMonoidAlgebra M \u03b9), toDirectSum (f + g) = toDirectSum f + toDirectSum g) }\nx\u271d : NonUnitalNonAssocSemiring (\u03b9 \u2192\u2080 M) := nonUnitalNonAssocSemiring\nxi : \u03b9\nxv : M\nyi : \u03b9\nyv : M\n\u22a2 \u2191(DirectSum.of (fun i => M) (xi + yi)) (xv * yv) = DFinsupp.single xi xv * \u2191(DirectSum.of (fun i => M) yi) yv\n[PROOFSTEP]\nsimp only [AddMonoidHom.coe_comp, AddMonoidHom.coe_mul, AddMonoidHom.coe_mk, ZeroHom.coe_mk, Function.comp_apply,\n toDirectSum_single, AddMonoidHom.id_apply, Finsupp.singleAddHom_apply, AddMonoidHom.coe_mulLeft]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Semiring M\nto_hom : AddMonoidAlgebra M \u03b9 \u2192+ \u2a01 (x : \u03b9), M :=\n { toZeroHom := { toFun := toDirectSum, map_zero' := (_ : toDirectSum 0 = 0) },\n map_add' := (_ : \u2200 (f g : AddMonoidAlgebra M \u03b9), toDirectSum (f + g) = toDirectSum f + toDirectSum g) }\nx\u271d : NonUnitalNonAssocSemiring (\u03b9 \u2192\u2080 M) := nonUnitalNonAssocSemiring\nxi : \u03b9\nxv : M\nyi : \u03b9\nyv : M\n\u22a2 \u2191(DirectSum.of (fun i => M) (xi + yi)) (xv * yv) = DFinsupp.single xi xv * \u2191(DirectSum.of (fun i => M) yi) yv\n[PROOFSTEP]\nerw [DirectSum.of_mul_of, Mul.gMul_mul]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : Semiring M\ninst\u271d : (m : M) \u2192 Decidable (m \u2260 0)\nf g : \u2a01 (x : \u03b9), M\n\u22a2 toAddMonoidAlgebra (f * g) = toAddMonoidAlgebra f * toAddMonoidAlgebra g\n[PROOFSTEP]\napply_fun AddMonoidAlgebra.toDirectSum\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : Semiring M\ninst\u271d : (m : M) \u2192 Decidable (m \u2260 0)\nf g : \u2a01 (x : \u03b9), M\n\u22a2 AddMonoidAlgebra.toDirectSum (toAddMonoidAlgebra (f * g)) =\n AddMonoidAlgebra.toDirectSum (toAddMonoidAlgebra f * toAddMonoidAlgebra g)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inj\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : Semiring M\ninst\u271d : (m : M) \u2192 Decidable (m \u2260 0)\nf g : \u2a01 (x : \u03b9), M\n\u22a2 Function.Injective AddMonoidAlgebra.toDirectSum\n[PROOFSTEP]\napply Function.LeftInverse.injective\n[GOAL]\ncase inj.a\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : Semiring M\ninst\u271d : (m : M) \u2192 Decidable (m \u2260 0)\nf g : \u2a01 (x : \u03b9), M\n\u22a2 Function.LeftInverse ?inj.g AddMonoidAlgebra.toDirectSum\ncase inj.g\n\u03b9 : Type u_1\nR : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : Semiring M\ninst\u271d : (m : M) \u2192 Decidable (m \u2260 0)\nf g : \u2a01 (x : \u03b9), M\n\u22a2 (\u2a01 (x : \u03b9), M) \u2192 AddMonoidAlgebra M \u03b9\n[PROOFSTEP]\napply AddMonoidAlgebra.toDirectSum_toAddMonoidAlgebra\n", "meta": {"mathlib_filename": "Mathlib.Algebra.MonoidAlgebra.ToDirectSum", "llama_tokens": 4647, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936430859597, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.43079227892109545}} {"text": "[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nx : E\nhx : x \u2208 s\n\u22a2 p x 0 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)) (f x)\n[PROOFSTEP]\nrw [\u2190 h.zero_eq x hx]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nx : E\nhx : x \u2208 s\n\u22a2 p x 0 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)) (ContinuousMultilinearMap.uncurry0 (p x 0))\n[PROOFSTEP]\nexact (p x 0).uncurry0_curry0.symm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nh\u2081 : \u2200 (x : E), x \u2208 s \u2192 f\u2081 x = f x\n\u22a2 HasFTaylorSeriesUpToOn n f\u2081 p s\n[PROOFSTEP]\nrefine' \u27e8fun x hx => _, h.fderivWithin, h.cont\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nh\u2081 : \u2200 (x : E), x \u2208 s \u2192 f\u2081 x = f x\nx : E\nhx : x \u2208 s\n\u22a2 ContinuousMultilinearMap.uncurry0 (p x 0) = f\u2081 x\n[PROOFSTEP]\nrw [h\u2081 x hx]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nh\u2081 : \u2200 (x : E), x \u2208 s \u2192 f\u2081 x = f x\nx : E\nhx : x \u2208 s\n\u22a2 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\n[PROOFSTEP]\nexact h.zero_eq x hx\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\n\u22a2 ContinuousOn f s\n[PROOFSTEP]\nhave := (h.cont 0 bot_le).congr fun x hx => (h.zero_eq' hx).symm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nthis : ContinuousOn (fun x => \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)) (f x)) s\n\u22a2 ContinuousOn f s\n[PROOFSTEP]\nrwa [\u2190 (continuousMultilinearCurryFin0 \ud835\udd5c E F).symm.comp_continuousOn_iff]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpToOn 0 f p s \u2194 ContinuousOn f s \u2227 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\n[PROOFSTEP]\nrefine\n \u27e8fun H => \u27e8H.continuousOn, H.zero_eq\u27e9, fun H => \u27e8H.2, fun m hm => False.elim (not_le.2 hm bot_le), fun m hm \u21a6 ?_\u27e9\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContinuousOn f s \u2227 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nm : \u2115\nhm : \u2191m \u2264 0\n\u22a2 ContinuousOn (fun x => p x m) s\n[PROOFSTEP]\nobtain rfl : m = 0 := by exact_mod_cast hm.antisymm (zero_le _)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContinuousOn f s \u2227 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nm : \u2115\nhm : \u2191m \u2264 0\n\u22a2 m = 0\n[PROOFSTEP]\nexact_mod_cast hm.antisymm (zero_le _)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContinuousOn f s \u2227 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nhm : \u21910 \u2264 0\n\u22a2 ContinuousOn (fun x => p x 0) s\n[PROOFSTEP]\nhave : EqOn (p \u00b7 0) ((continuousMultilinearCurryFin0 \ud835\udd5c E F).symm \u2218 f) s := fun x hx \u21a6\n (continuousMultilinearCurryFin0 \ud835\udd5c E F).eq_symm_apply.2 (H.2 x hx)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContinuousOn f s \u2227 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nhm : \u21910 \u2264 0\nthis : EqOn (fun x => p x 0) (\u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)) \u2218 f) s\n\u22a2 ContinuousOn (fun x => p x 0) s\n[PROOFSTEP]\nrw [continuousOn_congr this, LinearIsometryEquiv.comp_continuousOn_iff]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContinuousOn f s \u2227 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nhm : \u21910 \u2264 0\nthis : EqOn (fun x => p x 0) (\u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)) \u2218 f) s\n\u22a2 ContinuousOn f s\n[PROOFSTEP]\nexact H.1\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpToOn \u22a4 f p s \u2194 \u2200 (n : \u2115), HasFTaylorSeriesUpToOn (\u2191n) f p s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpToOn \u22a4 f p s \u2192 \u2200 (n : \u2115), HasFTaylorSeriesUpToOn (\u2191n) f p s\n[PROOFSTEP]\nintro H n\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn \u22a4 f p s\nn : \u2115\n\u22a2 HasFTaylorSeriesUpToOn (\u2191n) f p s\n[PROOFSTEP]\nexact H.of_le le_top\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 (\u2200 (n : \u2115), HasFTaylorSeriesUpToOn (\u2191n) f p s) \u2192 HasFTaylorSeriesUpToOn \u22a4 f p s\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : \u2200 (n : \u2115), HasFTaylorSeriesUpToOn (\u2191n) f p s\n\u22a2 HasFTaylorSeriesUpToOn \u22a4 f p s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : \u2200 (n : \u2115), HasFTaylorSeriesUpToOn (\u2191n) f p s\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\n[PROOFSTEP]\nexact (H 0).zero_eq\n[GOAL]\ncase mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : \u2200 (n : \u2115), HasFTaylorSeriesUpToOn (\u2191n) f p s\n\u22a2 \u2200 (m : \u2115),\n \u2191m < \u22a4 \u2192\n \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x\n[PROOFSTEP]\nintro m _\n[GOAL]\ncase mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : \u2200 (n : \u2115), HasFTaylorSeriesUpToOn (\u2191n) f p s\nm : \u2115\na\u271d : \u2191m < \u22a4\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x\n[PROOFSTEP]\napply (H m.succ).fderivWithin m (WithTop.coe_lt_coe.2 (lt_add_one m))\n[GOAL]\ncase mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : \u2200 (n : \u2115), HasFTaylorSeriesUpToOn (\u2191n) f p s\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 \u22a4 \u2192 ContinuousOn (fun x => p x m) s\n[PROOFSTEP]\nintro m _\n[GOAL]\ncase mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : \u2200 (n : \u2115), HasFTaylorSeriesUpToOn (\u2191n) f p s\nm : \u2115\na\u271d : \u2191m \u2264 \u22a4\n\u22a2 ContinuousOn (fun x => p x m) s\n[PROOFSTEP]\napply (H m).cont m le_rfl\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nhn : 1 \u2264 n\nhx : x \u2208 s\n\u22a2 HasFDerivWithinAt f (\u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1)) s x\n[PROOFSTEP]\nhave A : \u2200 y \u2208 s, f y = (continuousMultilinearCurryFin0 \ud835\udd5c E F) (p y 0) := fun y hy \u21a6 (h.zero_eq y hy).symm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nhn : 1 \u2264 n\nhx : x \u2208 s\nA : \u2200 (y : E), y \u2208 s \u2192 f y = \u2191(continuousMultilinearCurryFin0 \ud835\udd5c E F) (p y 0)\n\u22a2 HasFDerivWithinAt f (\u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1)) s x\n[PROOFSTEP]\nsuffices H :\n HasFDerivWithinAt (continuousMultilinearCurryFin0 \ud835\udd5c E F \u2218 (p \u00b7 0)) (continuousMultilinearCurryFin1 \ud835\udd5c E F (p x 1)) s x\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nhn : 1 \u2264 n\nhx : x \u2208 s\nA : \u2200 (y : E), y \u2208 s \u2192 f y = \u2191(continuousMultilinearCurryFin0 \ud835\udd5c E F) (p y 0)\nH :\n HasFDerivWithinAt (\u2191(continuousMultilinearCurryFin0 \ud835\udd5c E F) \u2218 fun x => p x 0)\n (\u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1)) s x\n\u22a2 HasFDerivWithinAt f (\u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1)) s x\n[PROOFSTEP]\nexact H.congr A (A x hx)\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nhn : 1 \u2264 n\nhx : x \u2208 s\nA : \u2200 (y : E), y \u2208 s \u2192 f y = \u2191(continuousMultilinearCurryFin0 \ud835\udd5c E F) (p y 0)\n\u22a2 HasFDerivWithinAt (\u2191(continuousMultilinearCurryFin0 \ud835\udd5c E F) \u2218 fun x => p x 0)\n (\u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1)) s x\n[PROOFSTEP]\nrw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nhn : 1 \u2264 n\nhx : x \u2208 s\nA : \u2200 (y : E), y \u2208 s \u2192 f y = \u2191(continuousMultilinearCurryFin0 \ud835\udd5c E F) (p y 0)\n\u22a2 HasFDerivWithinAt (fun x => p x 0)\n (ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)).toLinearEquiv))\n (\u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1)))\n s x\n[PROOFSTEP]\nhave : ((0 : \u2115) : \u2115\u221e) < n := zero_lt_one.trans_le hn\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nhn : 1 \u2264 n\nhx : x \u2208 s\nA : \u2200 (y : E), y \u2208 s \u2192 f y = \u2191(continuousMultilinearCurryFin0 \ud835\udd5c E F) (p y 0)\nthis : \u21910 < n\n\u22a2 HasFDerivWithinAt (fun x => p x 0)\n (ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)).toLinearEquiv))\n (\u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1)))\n s x\n[PROOFSTEP]\nconvert h.fderivWithin _ this x hx\n[GOAL]\ncase h.e'_10.h.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nhn : 1 \u2264 n\nhx : x \u2208 s\nA : \u2200 (y : E), y \u2208 s \u2192 f y = \u2191(continuousMultilinearCurryFin0 \ud835\udd5c E F) (p y 0)\nthis : \u21910 < n\ne_7\u271d : ContinuousMultilinearMap.normedAddCommGroup' = ContinuousMultilinearMap.normedAddCommGroup\n\u22a2 ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)).toLinearEquiv))\n (\u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1)) =\n ContinuousMultilinearMap.curryLeft (p x (Nat.succ 0))\n[PROOFSTEP]\next y v\n[GOAL]\ncase h.e'_10.h.h.h.H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nhn : 1 \u2264 n\nhx : x \u2208 s\nA : \u2200 (y : E), y \u2208 s \u2192 f y = \u2191(continuousMultilinearCurryFin0 \ud835\udd5c E F) (p y 0)\nthis : \u21910 < n\ne_7\u271d : ContinuousMultilinearMap.normedAddCommGroup' = ContinuousMultilinearMap.normedAddCommGroup\ny : E\nv : Fin 0 \u2192 E\n\u22a2 \u2191(\u2191(ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)).toLinearEquiv))\n (\u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1)))\n y)\n v =\n \u2191(\u2191(ContinuousMultilinearMap.curryLeft (p x (Nat.succ 0))) y) v\n[PROOFSTEP]\nchange (p x 1) (snoc 0 y) = (p x 1) (cons y v)\n[GOAL]\ncase h.e'_10.h.h.h.H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nhn : 1 \u2264 n\nhx : x \u2208 s\nA : \u2200 (y : E), y \u2208 s \u2192 f y = \u2191(continuousMultilinearCurryFin0 \ud835\udd5c E F) (p y 0)\nthis : \u21910 < n\ne_7\u271d : ContinuousMultilinearMap.normedAddCommGroup' = ContinuousMultilinearMap.normedAddCommGroup\ny : E\nv : Fin 0 \u2192 E\n\u22a2 \u2191(p x 1) (snoc 0 y) = \u2191(p x 1) (cons y v)\n[PROOFSTEP]\ncongr with i\n[GOAL]\ncase h.e'_10.h.h.h.H.h.e_6.h.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nhn : 1 \u2264 n\nhx : x \u2208 s\nA : \u2200 (y : E), y \u2208 s \u2192 f y = \u2191(continuousMultilinearCurryFin0 \ud835\udd5c E F) (p y 0)\nthis : \u21910 < n\ne_7\u271d : ContinuousMultilinearMap.normedAddCommGroup' = ContinuousMultilinearMap.normedAddCommGroup\ny : E\nv : Fin 0 \u2192 E\ni : Fin (0 + 1)\n\u22a2 snoc 0 y i = cons y v i\n[PROOFSTEP]\nrw [Unique.eq_default (\u03b1 := Fin 1) i]\n[GOAL]\ncase h.e'_10.h.h.h.H.h.e_6.h.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nhn : 1 \u2264 n\nhx : x \u2208 s\nA : \u2200 (y : E), y \u2208 s \u2192 f y = \u2191(continuousMultilinearCurryFin0 \ud835\udd5c E F) (p y 0)\nthis : \u21910 < n\ne_7\u271d : ContinuousMultilinearMap.normedAddCommGroup' = ContinuousMultilinearMap.normedAddCommGroup\ny : E\nv : Fin 0 \u2192 E\ni : Fin (0 + 1)\n\u22a2 snoc 0 y default = cons y v default\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 HasFTaylorSeriesUpToOn (\u2191n + 1) f p s \u2194\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 HasFTaylorSeriesUpToOn (\u2191n + 1) f p s \u2192\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\n[PROOFSTEP]\nexact fun h \u21a6\n \u27e8h.of_le (WithTop.coe_le_coe.2 (Nat.le_succ n)), h.fderivWithin _ (WithTop.coe_lt_coe.2 (lt_add_one n)),\n h.cont (n + 1) le_rfl\u27e9\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s \u2192\n HasFTaylorSeriesUpToOn (\u2191n + 1) f p s\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\n\u22a2 HasFTaylorSeriesUpToOn (\u2191n + 1) f p s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\n[PROOFSTEP]\nexact h.1.zero_eq\n[GOAL]\ncase mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\n\u22a2 \u2200 (m : \u2115),\n \u2191m < \u2191n + 1 \u2192\n \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x\n[PROOFSTEP]\nintro m hm\n[GOAL]\ncase mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m < \u2191n + 1\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x\n[PROOFSTEP]\nby_cases h' : m < n\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m < \u2191n + 1\nh' : m < n\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x\n[PROOFSTEP]\nexact h.1.fderivWithin m (WithTop.coe_lt_coe.2 h')\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m < \u2191n + 1\nh' : \u00acm < n\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x\n[PROOFSTEP]\nhave : m = n := Nat.eq_of_lt_succ_of_not_lt (WithTop.coe_lt_coe.1 hm) h'\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m < \u2191n + 1\nh' : \u00acm < n\nthis : m = n\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m < \u2191n + 1\nh' : \u00acm < n\nthis : m = n\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x\n[PROOFSTEP]\nexact h.2.1\n[GOAL]\ncase mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 \u2191n + 1 \u2192 ContinuousOn (fun x => p x m) s\n[PROOFSTEP]\nintro m hm\n[GOAL]\ncase mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191n + 1\n\u22a2 ContinuousOn (fun x => p x m) s\n[PROOFSTEP]\nby_cases h' : m \u2264 n\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191n + 1\nh' : m \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) s\n[PROOFSTEP]\napply h.1.cont m (WithTop.coe_le_coe.2 h')\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191n + 1\nh' : \u00acm \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) s\n[PROOFSTEP]\nhave : m = n + 1 := le_antisymm (WithTop.coe_le_coe.1 hm) (not_le.1 h')\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191n + 1\nh' : \u00acm \u2264 n\nthis : m = n + 1\n\u22a2 ContinuousOn (fun x => p x m) s\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n (\u2200 (x : E),\n x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191n + 1\nh' : \u00acm \u2264 n\nthis : m = n + 1\n\u22a2 ContinuousOn (fun x => p x (n + 1)) s\n[PROOFSTEP]\nexact h.2.2\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\n\u22a2 HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\n\u22a2 \u2200 (x : E),\n x \u2208 s \u2192\n ContinuousMultilinearMap.uncurry0 (FormalMultilinearSeries.shift (p x) 0) =\n \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1)\n[PROOFSTEP]\nintro x _\n[GOAL]\ncase zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\nx : E\na\u271d : x \u2208 s\n\u22a2 ContinuousMultilinearMap.uncurry0 (FormalMultilinearSeries.shift (p x) 0) =\n \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\n\u22a2 \u2200 (m : \u2115),\n \u2191m < \u2191n \u2192\n \u2200 (x : E),\n x \u2208 s \u2192\n HasFDerivWithinAt (fun x => FormalMultilinearSeries.shift (p x) m)\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.shift (p x) (Nat.succ m))) s x\n[PROOFSTEP]\nintro m (hm : (m : \u2115\u221e) < n) x (hx : x \u2208 s)\n[GOAL]\ncase fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\nm : \u2115\nhm : \u2191m < \u2191n\nx : E\nhx : x \u2208 s\n\u22a2 HasFDerivWithinAt (fun x => FormalMultilinearSeries.shift (p x) m)\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.shift (p x) (Nat.succ m))) s x\n[PROOFSTEP]\nhave A : (m.succ : \u2115\u221e) < n.succ\n[GOAL]\ncase A\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\nm : \u2115\nhm : \u2191m < \u2191n\nx : E\nhx : x \u2208 s\n\u22a2 \u2191(Nat.succ m) < \u2191(Nat.succ n)\n[PROOFSTEP]\nrw [Nat.cast_lt] at hm \u22a2\n[GOAL]\ncase A\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\nm : \u2115\nhm : m < n\nx : E\nhx : x \u2208 s\n\u22a2 Nat.succ m < Nat.succ n\n[PROOFSTEP]\nexact Nat.succ_lt_succ hm\n[GOAL]\ncase fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\nm : \u2115\nhm : \u2191m < \u2191n\nx : E\nhx : x \u2208 s\nA : \u2191(Nat.succ m) < \u2191(Nat.succ n)\n\u22a2 HasFDerivWithinAt (fun x => FormalMultilinearSeries.shift (p x) m)\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.shift (p x) (Nat.succ m))) s x\n[PROOFSTEP]\nchange\n HasFDerivWithinAt ((continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F).symm \u2218 (p \u00b7 m.succ))\n (p x m.succ.succ).curryRight.curryLeft s x\n[GOAL]\ncase fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\nm : \u2115\nhm : \u2191m < \u2191n\nx : E\nhx : x \u2208 s\nA : \u2191(Nat.succ m) < \u2191(Nat.succ n)\n\u22a2 HasFDerivWithinAt\n (\u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F)) \u2218 fun x => p x (Nat.succ m))\n (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.curryRight (p x (Nat.succ (Nat.succ m))))) s x\n[PROOFSTEP]\nrw [((continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F).symm).comp_hasFDerivWithinAt_iff']\n[GOAL]\ncase fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\nm : \u2115\nhm : \u2191m < \u2191n\nx : E\nhx : x \u2208 s\nA : \u2191(Nat.succ m) < \u2191(Nat.succ n)\n\u22a2 HasFDerivWithinAt (fun x => p x (Nat.succ m))\n (ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm\n (LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.curryRight (p x (Nat.succ (Nat.succ m))))))\n s x\n[PROOFSTEP]\nconvert H.fderivWithin _ A x hx\n[GOAL]\ncase h.e'_10.h.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\nm : \u2115\nhm : \u2191m < \u2191n\nx : E\nhx : x \u2208 s\nA : \u2191(Nat.succ m) < \u2191(Nat.succ n)\ne_7\u271d : ContinuousMultilinearMap.normedAddCommGroup' = ContinuousMultilinearMap.normedAddCommGroup\n\u22a2 ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm\n (LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.curryRight (p x (Nat.succ (Nat.succ m))))) =\n ContinuousMultilinearMap.curryLeft (p x (Nat.succ (Nat.succ m)))\n[PROOFSTEP]\next y v\n[GOAL]\ncase h.e'_10.h.h.h.H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\nm : \u2115\nhm : \u2191m < \u2191n\nx : E\nhx : x \u2208 s\nA : \u2191(Nat.succ m) < \u2191(Nat.succ n)\ne_7\u271d : ContinuousMultilinearMap.normedAddCommGroup' = ContinuousMultilinearMap.normedAddCommGroup\ny : E\nv : Fin (Nat.succ m) \u2192 E\n\u22a2 \u2191(\u2191(ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm\n (LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.curryRight (p x (Nat.succ (Nat.succ m))))))\n y)\n v =\n \u2191(\u2191(ContinuousMultilinearMap.curryLeft (p x (Nat.succ (Nat.succ m)))) y) v\n[PROOFSTEP]\nchange p x (m + 2) (snoc (cons y (init v)) (v (last _))) = p x (m + 2) (cons y v)\n[GOAL]\ncase h.e'_10.h.h.h.H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\nm : \u2115\nhm : \u2191m < \u2191n\nx : E\nhx : x \u2208 s\nA : \u2191(Nat.succ m) < \u2191(Nat.succ n)\ne_7\u271d : ContinuousMultilinearMap.normedAddCommGroup' = ContinuousMultilinearMap.normedAddCommGroup\ny : E\nv : Fin (Nat.succ m) \u2192 E\n\u22a2 \u2191(p x (m + 2)) (snoc (cons y (init v)) (v (last m))) = \u2191(p x (m + 2)) (cons y v)\n[PROOFSTEP]\nrw [\u2190 cons_snoc_eq_snoc_cons, snoc_init_self]\n[GOAL]\ncase cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 \u2191n \u2192 ContinuousOn (fun x => FormalMultilinearSeries.shift (p x) m) s\n[PROOFSTEP]\nintro m (hm : (m : \u2115\u221e) \u2264 n)\n[GOAL]\ncase cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\nm : \u2115\nhm : \u2191m \u2264 \u2191n\n\u22a2 ContinuousOn (fun x => FormalMultilinearSeries.shift (p x) m) s\n[PROOFSTEP]\nsuffices A : ContinuousOn (p \u00b7 (m + 1)) s\n[GOAL]\ncase cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\nm : \u2115\nhm : \u2191m \u2264 \u2191n\nA : ContinuousOn (fun x => p x (m + 1)) s\n\u22a2 ContinuousOn (fun x => FormalMultilinearSeries.shift (p x) m) s\n[PROOFSTEP]\nexact ((continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F).symm).continuous.comp_continuousOn A\n[GOAL]\ncase A\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\nm : \u2115\nhm : \u2191m \u2264 \u2191n\n\u22a2 ContinuousOn (fun x => p x (m + 1)) s\n[PROOFSTEP]\nrefine H.cont _ ?_\n[GOAL]\ncase A\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\nm : \u2115\nhm : \u2191m \u2264 \u2191n\n\u22a2 \u2191(m + 1) \u2264 \u2191(n + 1)\n[PROOFSTEP]\nrw [Nat.cast_le] at hm \u22a2\n[GOAL]\ncase A\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\nm : \u2115\nhm : m \u2264 n\n\u22a2 m + 1 \u2264 n + 1\n[PROOFSTEP]\nexact Nat.succ_le_succ hm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s \u2194\n (\u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x) \u2227\n (\u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x) \u2227\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s \u2192\n (\u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x) \u2227\n (\u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x) \u2227\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\n\u22a2 (\u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x) \u2227\n (\u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x) \u2227\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\n[PROOFSTEP]\nrefine' \u27e8H.zero_eq, H.fderivWithin 0 (Nat.cast_lt.2 (Nat.succ_pos n)), _\u27e9\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nH : HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\n\u22a2 HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\n[PROOFSTEP]\nexact H.shift_of_succ\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 (\u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x) \u2227\n (\u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x) \u2227\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s \u2192\n HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\n[PROOFSTEP]\nrintro \u27e8Hzero_eq, Hfderiv_zero, Htaylor\u27e9\n[GOAL]\ncase mpr.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\n\u22a2 HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.intro.intro.zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\n[PROOFSTEP]\nexact Hzero_eq\n[GOAL]\ncase mpr.intro.intro.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\n\u22a2 \u2200 (m : \u2115),\n \u2191m < \u2191(n + 1) \u2192\n \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x\n[PROOFSTEP]\nintro m (hm : (m : \u2115\u221e) < n.succ) x (hx : x \u2208 s)\n[GOAL]\ncase mpr.intro.intro.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nm : \u2115\nhm : \u2191m < \u2191(Nat.succ n)\nx : E\nhx : x \u2208 s\n\u22a2 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x\n[PROOFSTEP]\ncases' m with m\n[GOAL]\ncase mpr.intro.intro.fderivWithin.zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nx : E\nhx : x \u2208 s\nhm : \u2191Nat.zero < \u2191(Nat.succ n)\n\u22a2 HasFDerivWithinAt (fun x => p x Nat.zero) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ Nat.zero))) s x\n[PROOFSTEP]\nexact Hfderiv_zero x hx\n[GOAL]\ncase mpr.intro.intro.fderivWithin.succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191(Nat.succ m) < \u2191(Nat.succ n)\n\u22a2 HasFDerivWithinAt (fun x => p x (Nat.succ m)) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ (Nat.succ m)))) s x\n[PROOFSTEP]\nhave A : (m : \u2115\u221e) < n := by\n rw [Nat.cast_lt] at hm \u22a2\n exact Nat.lt_of_succ_lt_succ hm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191(Nat.succ m) < \u2191(Nat.succ n)\n\u22a2 \u2191m < \u2191n\n[PROOFSTEP]\nrw [Nat.cast_lt] at hm \u22a2\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : Nat.succ m < Nat.succ n\n\u22a2 m < n\n[PROOFSTEP]\nexact Nat.lt_of_succ_lt_succ hm\n[GOAL]\ncase mpr.intro.intro.fderivWithin.succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191(Nat.succ m) < \u2191(Nat.succ n)\nA : \u2191m < \u2191n\n\u22a2 HasFDerivWithinAt (fun x => p x (Nat.succ m)) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ (Nat.succ m)))) s x\n[PROOFSTEP]\nhave :\n HasFDerivWithinAt ((continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F).symm \u2218 (p \u00b7 m.succ)) ((p x).shift m.succ).curryLeft\n s x :=\n Htaylor.fderivWithin _ A x hx\n[GOAL]\ncase mpr.intro.intro.fderivWithin.succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191(Nat.succ m) < \u2191(Nat.succ n)\nA : \u2191m < \u2191n\nthis :\n HasFDerivWithinAt\n (\u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F)) \u2218 fun x => p x (Nat.succ m))\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.shift (p x) (Nat.succ m))) s x\n\u22a2 HasFDerivWithinAt (fun x => p x (Nat.succ m)) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ (Nat.succ m)))) s x\n[PROOFSTEP]\nrw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff'] at this \n[GOAL]\ncase mpr.intro.intro.fderivWithin.succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191(Nat.succ m) < \u2191(Nat.succ n)\nA : \u2191m < \u2191n\nthis :\n HasFDerivWithinAt (fun x => p x (Nat.succ m))\n (ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm\n (LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.shift (p x) (Nat.succ m))))\n s x\n\u22a2 HasFDerivWithinAt (fun x => p x (Nat.succ m)) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ (Nat.succ m)))) s x\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_10.h.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191(Nat.succ m) < \u2191(Nat.succ n)\nA : \u2191m < \u2191n\nthis :\n HasFDerivWithinAt (fun x => p x (Nat.succ m))\n (ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm\n (LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.shift (p x) (Nat.succ m))))\n s x\ne_7\u271d : ContinuousMultilinearMap.normedAddCommGroup = ContinuousMultilinearMap.normedAddCommGroup'\n\u22a2 ContinuousMultilinearMap.curryLeft (p x (Nat.succ (Nat.succ m))) =\n ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm\n (LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.shift (p x) (Nat.succ m)))\n[PROOFSTEP]\next y v\n[GOAL]\ncase h.e'_10.h.h.h.H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191(Nat.succ m) < \u2191(Nat.succ n)\nA : \u2191m < \u2191n\nthis :\n HasFDerivWithinAt (fun x => p x (Nat.succ m))\n (ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm\n (LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.shift (p x) (Nat.succ m))))\n s x\ne_7\u271d : ContinuousMultilinearMap.normedAddCommGroup = ContinuousMultilinearMap.normedAddCommGroup'\ny : E\nv : Fin (Nat.succ m) \u2192 E\n\u22a2 \u2191(\u2191(ContinuousMultilinearMap.curryLeft (p x (Nat.succ (Nat.succ m)))) y) v =\n \u2191(\u2191(ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm\n (LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.shift (p x) (Nat.succ m))))\n y)\n v\n[PROOFSTEP]\nchange (p x (Nat.succ (Nat.succ m))) (cons y v) = (p x m.succ.succ) (snoc (cons y (init v)) (v (last _)))\n[GOAL]\ncase h.e'_10.h.h.h.H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191(Nat.succ m) < \u2191(Nat.succ n)\nA : \u2191m < \u2191n\nthis :\n HasFDerivWithinAt (fun x => p x (Nat.succ m))\n (ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm\n (LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.shift (p x) (Nat.succ m))))\n s x\ne_7\u271d : ContinuousMultilinearMap.normedAddCommGroup = ContinuousMultilinearMap.normedAddCommGroup'\ny : E\nv : Fin (Nat.succ m) \u2192 E\n\u22a2 \u2191(p x (Nat.succ (Nat.succ m))) (cons y v) = \u2191(p x (Nat.succ (Nat.succ m))) (snoc (cons y (init v)) (v (last m)))\n[PROOFSTEP]\nrw [\u2190 cons_snoc_eq_snoc_cons, snoc_init_self]\n[GOAL]\ncase mpr.intro.intro.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 \u2191(n + 1) \u2192 ContinuousOn (fun x => p x m) s\n[PROOFSTEP]\nintro m (hm : (m : \u2115\u221e) \u2264 n.succ)\n[GOAL]\ncase mpr.intro.intro.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191(Nat.succ n)\n\u22a2 ContinuousOn (fun x => p x m) s\n[PROOFSTEP]\ncases' m with m\n[GOAL]\ncase mpr.intro.intro.cont.zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nhm : \u2191Nat.zero \u2264 \u2191(Nat.succ n)\n\u22a2 ContinuousOn (fun x => p x Nat.zero) s\n[PROOFSTEP]\nhave : DifferentiableOn \ud835\udd5c (fun x => p x 0) s := fun x hx => (Hfderiv_zero x hx).differentiableWithinAt\n[GOAL]\ncase mpr.intro.intro.cont.zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nhm : \u2191Nat.zero \u2264 \u2191(Nat.succ n)\nthis : DifferentiableOn \ud835\udd5c (fun x => p x 0) s\n\u22a2 ContinuousOn (fun x => p x Nat.zero) s\n[PROOFSTEP]\nexact this.continuousOn\n[GOAL]\ncase mpr.intro.intro.cont.succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nm : \u2115\nhm : \u2191(Nat.succ m) \u2264 \u2191(Nat.succ n)\n\u22a2 ContinuousOn (fun x => p x (Nat.succ m)) s\n[PROOFSTEP]\nrefine (continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F).symm.comp_continuousOn_iff.mp ?_\n[GOAL]\ncase mpr.intro.intro.cont.succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nm : \u2115\nhm : \u2191(Nat.succ m) \u2264 \u2191(Nat.succ n)\n\u22a2 ContinuousOn (\u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F)) \u2218 fun x => p x (Nat.succ m))\n s\n[PROOFSTEP]\nrefine Htaylor.cont _ ?_\n[GOAL]\ncase mpr.intro.intro.cont.succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nm : \u2115\nhm : \u2191(Nat.succ m) \u2264 \u2191(Nat.succ n)\n\u22a2 \u2191m \u2264 \u2191n\n[PROOFSTEP]\nrw [Nat.cast_le] at hm \u22a2\n[GOAL]\ncase mpr.intro.intro.cont.succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nHzero_eq : \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\nHfderiv_zero : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) s x\nHtaylor :\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) s\nm : \u2115\nhm : Nat.succ m \u2264 Nat.succ n\n\u22a2 m \u2264 n\n[PROOFSTEP]\nexact Nat.lt_succ_iff.mp hm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 (\u2200 (m : \u2115), \u2191m \u2264 \u22a4 \u2192 ContDiffWithinAt \ud835\udd5c (\u2191m) f s x) \u2194 \u2200 (n : \u2115), ContDiffWithinAt \ud835\udd5c (\u2191n) f s x\n[PROOFSTEP]\nsimp only [forall_prop_of_true, le_top]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffWithinAt \ud835\udd5c n f s x\n\u22a2 ContinuousWithinAt f s x\n[PROOFSTEP]\nrcases h 0 bot_le with \u27e8u, hu, p, H\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffWithinAt \ud835\udd5c n f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn (\u21910) f p u\n\u22a2 ContinuousWithinAt f s x\n[PROOFSTEP]\nrw [mem_nhdsWithin_insert] at hu \n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffWithinAt \ud835\udd5c n f s x\nu : Set E\nhu : x \u2208 u \u2227 u \u2208 \ud835\udcdd[s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn (\u21910) f p u\n\u22a2 ContinuousWithinAt f s x\n[PROOFSTEP]\nexact (H.continuousOn.continuousWithinAt hu.1).mono_of_mem hu.2\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffWithinAt \ud835\udd5c n f s x\nt : Set E\nhst : s \u2208 \ud835\udcdd[t] x\n\u22a2 ContDiffWithinAt \ud835\udd5c n f t x\n[PROOFSTEP]\nintro m hm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffWithinAt \ud835\udd5c n f s x\nt : Set E\nhst : s \u2208 \ud835\udcdd[t] x\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x t] x \u2227 \u2203 p, HasFTaylorSeriesUpToOn (\u2191m) f p u\n[PROOFSTEP]\nrcases h m hm with \u27e8u, hu, p, H\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffWithinAt \ud835\udd5c n f s x\nt : Set E\nhst : s \u2208 \ud835\udcdd[t] x\nm : \u2115\nhm : \u2191m \u2264 n\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn (\u2191m) f p u\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x t] x \u2227 \u2203 p, HasFTaylorSeriesUpToOn (\u2191m) f p u\n[PROOFSTEP]\nexact \u27e8u, nhdsWithin_le_of_mem (insert_mem_nhdsWithin_insert hst) hu, p, H\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiffWithinAt \ud835\udd5c n f (insert x s) x \u2194 ContDiffWithinAt \ud835\udd5c n f s x\n[PROOFSTEP]\nsimp_rw [ContDiffWithinAt, insert_idem]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ny : E\n\u22a2 ContDiffWithinAt \ud835\udd5c n f (insert y s) x \u2194 ContDiffWithinAt \ud835\udd5c n f s x\n[PROOFSTEP]\nrcases eq_or_ne x y with (rfl | h)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiffWithinAt \ud835\udd5c n f (insert x s) x \u2194 ContDiffWithinAt \ud835\udd5c n f s x\n[PROOFSTEP]\nexact contDiffWithinAt_insert_self\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ny : E\nh : x \u2260 y\n\u22a2 ContDiffWithinAt \ud835\udd5c n f (insert y s) x \u2194 ContDiffWithinAt \ud835\udd5c n f s x\n[PROOFSTEP]\nsimp_rw [ContDiffWithinAt, insert_comm x y, nhdsWithin_insert_of_ne h]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhn : 1 \u2264 n\n\u22a2 DifferentiableWithinAt \ud835\udd5c f (insert x s) x\n[PROOFSTEP]\nrcases h 1 hn with \u27e8u, hu, p, H\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhn : 1 \u2264 n\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn (\u21911) f p u\n\u22a2 DifferentiableWithinAt \ud835\udd5c f (insert x s) x\n[PROOFSTEP]\nrcases mem_nhdsWithin.1 hu with \u27e8t, t_open, xt, tu\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhn : 1 \u2264 n\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn (\u21911) f p u\nt : Set E\nt_open : IsOpen t\nxt : x \u2208 t\ntu : t \u2229 insert x s \u2286 u\n\u22a2 DifferentiableWithinAt \ud835\udd5c f (insert x s) x\n[PROOFSTEP]\nrw [inter_comm] at tu \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhn : 1 \u2264 n\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn (\u21911) f p u\nt : Set E\nt_open : IsOpen t\nxt : x \u2208 t\ntu : insert x s \u2229 t \u2286 u\n\u22a2 DifferentiableWithinAt \ud835\udd5c f (insert x s) x\n[PROOFSTEP]\nhave := ((H.mono tu).differentiableOn le_rfl) x \u27e8mem_insert x s, xt\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhn : 1 \u2264 n\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn (\u21911) f p u\nt : Set E\nt_open : IsOpen t\nxt : x \u2208 t\ntu : insert x s \u2229 t \u2286 u\nthis : DifferentiableWithinAt \ud835\udd5c f (insert x s \u2229 t) x\n\u22a2 DifferentiableWithinAt \ud835\udd5c f (insert x s) x\n[PROOFSTEP]\nexact (differentiableWithinAt_inter (IsOpen.mem_nhds t_open xt)).1 this\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x \u2194\n \u2203 u,\n u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x \u2192\n \u2203 u,\n u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n[PROOFSTEP]\nrcases h n.succ le_rfl with \u27e8u, hu, p, Hp\u27e9\n[GOAL]\ncase mp.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ n)) f p u\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n[PROOFSTEP]\nrefine'\n \u27e8u, hu, fun y => (continuousMultilinearCurryFin1 \ud835\udd5c E F) (p y 1), fun y hy =>\n Hp.hasFDerivWithinAt (WithTop.coe_le_coe.2 (Nat.le_add_left 1 n)) hy, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ n)) f p u\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191n) (fun y => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p y 1)) u x\n[PROOFSTEP]\nintro m hm\n[GOAL]\ncase mp.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ n)) f p u\nm : \u2115\nhm : \u2191m \u2264 \u2191n\n\u22a2 \u2203 u_1,\n u_1 \u2208 \ud835\udcdd[insert x u] x \u2227\n \u2203 p_1, HasFTaylorSeriesUpToOn (\u2191m) (fun y => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p y 1)) p_1 u_1\n[PROOFSTEP]\nrefine' \u27e8u, _, fun y : E => (p y).shift, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ n)) f p u\nm : \u2115\nhm : \u2191m \u2264 \u2191n\n\u22a2 u \u2208 \ud835\udcdd[insert x u] x\n[PROOFSTEP]\nconvert @self_mem_nhdsWithin _ _ x u\n[GOAL]\ncase h.e'_5.h.e'_4\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ n)) f p u\nm : \u2115\nhm : \u2191m \u2264 \u2191n\n\u22a2 insert x u = u\n[PROOFSTEP]\nhave : x \u2208 insert x s := by simp\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ n)) f p u\nm : \u2115\nhm : \u2191m \u2264 \u2191n\n\u22a2 x \u2208 insert x s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_5.h.e'_4\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ n)) f p u\nm : \u2115\nhm : \u2191m \u2264 \u2191n\nthis : x \u2208 insert x s\n\u22a2 insert x u = u\n[PROOFSTEP]\nexact insert_eq_of_mem (mem_of_mem_nhdsWithin this hu)\n[GOAL]\ncase mp.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ n)) f p u\nm : \u2115\nhm : \u2191m \u2264 \u2191n\n\u22a2 HasFTaylorSeriesUpToOn (\u2191m) (fun y => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p y 1))\n (fun y => FormalMultilinearSeries.shift (p y)) u\n[PROOFSTEP]\nrw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp \n[GOAL]\ncase mp.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp :\n (\u2200 (x : E), x \u2208 u \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x) \u2227\n (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) u x) \u2227\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) u\nm : \u2115\nhm : \u2191m \u2264 \u2191n\n\u22a2 HasFTaylorSeriesUpToOn (\u2191m) (fun y => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p y 1))\n (fun y => FormalMultilinearSeries.shift (p y)) u\n[PROOFSTEP]\nexact Hp.2.2.of_le hm\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 (\u2203 u,\n u \u2208 \ud835\udcdd[insert x s] x \u2227\n \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x) \u2192\n ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\n[PROOFSTEP]\nrintro \u27e8u, hu, f', f'_eq_deriv, Hf'\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\n[PROOFSTEP]\nrw [contDiffWithinAt_nat]\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 p, HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p u\n[PROOFSTEP]\nrcases Hf' n le_rfl with \u27e8v, hv, p', Hp'\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 p, HasFTaylorSeriesUpToOn (\u2191(n + 1)) f p u\n[PROOFSTEP]\nrefine' \u27e8v \u2229 u, _, fun x => (p' x).unshift (f x), _\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\n\u22a2 v \u2229 u \u2208 \ud835\udcdd[insert x s] x\n[PROOFSTEP]\napply Filter.inter_mem _ hu\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\n\u22a2 v \u2208 \ud835\udcdd[insert x s] x\n[PROOFSTEP]\napply nhdsWithin_le_of_mem hu\n[GOAL]\ncase a\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\n\u22a2 v \u2208 \ud835\udcdd[u] x\n[PROOFSTEP]\nexact nhdsWithin_mono _ (subset_insert x u) hv\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\n\u22a2 HasFTaylorSeriesUpToOn (\u2191(n + 1)) f (fun x => FormalMultilinearSeries.unshift (p' x) (f x)) (v \u2229 u)\n[PROOFSTEP]\nrw [hasFTaylorSeriesUpToOn_succ_iff_right]\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\n\u22a2 (\u2200 (x : E), x \u2208 v \u2229 u \u2192 ContinuousMultilinearMap.uncurry0 (FormalMultilinearSeries.unshift (p' x) (f x) 0) = f x) \u2227\n (\u2200 (x : E),\n x \u2208 v \u2229 u \u2192\n HasFDerivWithinAt (fun y => FormalMultilinearSeries.unshift (p' y) (f y) 0)\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.unshift (p' x) (f x) 1)) (v \u2229 u) x) \u2227\n HasFTaylorSeriesUpToOn (\u2191n)\n (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (FormalMultilinearSeries.unshift (p' x) (f x) 1))\n (fun x => FormalMultilinearSeries.shift (FormalMultilinearSeries.unshift (p' x) (f x))) (v \u2229 u)\n[PROOFSTEP]\nrefine' \u27e8fun y _ => rfl, fun y hy => _, _\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\ny : E\nhy : y \u2208 v \u2229 u\n\u22a2 HasFDerivWithinAt (fun y => FormalMultilinearSeries.unshift (p' y) (f y) 0)\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.unshift (p' y) (f y) 1)) (v \u2229 u) y\n[PROOFSTEP]\nchange\n HasFDerivWithinAt (fun z => (continuousMultilinearCurryFin0 \ud835\udd5c E F).symm (f z))\n (FormalMultilinearSeries.unshift (p' y) (f y) 1).curryLeft (v \u2229 u) y\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\ny : E\nhy : y \u2208 v \u2229 u\n\u22a2 HasFDerivWithinAt (fun z => \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)) (f z))\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.unshift (p' y) (f y) 1)) (v \u2229 u) y\n[PROOFSTEP]\nerw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\ny : E\nhy : y \u2208 v \u2229 u\n\u22a2 HasFDerivWithinAt (fun z => f z)\n (ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm (LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.unshift (p' y) (f y) 1)))\n (v \u2229 u) y\n[PROOFSTEP]\nconvert (f'_eq_deriv y hy.2).mono (inter_subset_right v u)\n[GOAL]\ncase h.e'_10\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\ny : E\nhy : y \u2208 v \u2229 u\n\u22a2 ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm (LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.unshift (p' y) (f y) 1)) =\n f' y\n[PROOFSTEP]\nrw [\u2190 Hp'.zero_eq y hy.1]\n[GOAL]\ncase h.e'_10\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\ny : E\nhy : y \u2208 v \u2229 u\n\u22a2 ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm (LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.unshift (p' y) (f y) 1)) =\n ContinuousMultilinearMap.uncurry0 (p' y 0)\n[PROOFSTEP]\next z\n[GOAL]\ncase h.e'_10.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\ny : E\nhy : y \u2208 v \u2229 u\nz : E\n\u22a2 \u2191(ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm\n (LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.unshift (p' y) (f y) 1)))\n z =\n \u2191(ContinuousMultilinearMap.uncurry0 (p' y 0)) z\n[PROOFSTEP]\nchange ((p' y 0) (init (@cons 0 (fun _ => E) z 0))) (@cons 0 (fun _ => E) z 0 (last 0)) = ((p' y 0) 0) z\n[GOAL]\ncase h.e'_10.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\ny : E\nhy : y \u2208 v \u2229 u\nz : E\n\u22a2 \u2191(\u2191(p' y 0) (init (cons z 0))) (cons z 0 (last 0)) = \u2191(\u2191(p' y 0) 0) z\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e'_10.h.e_a.h.e_6.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\ny : E\nhy : y \u2208 v \u2229 u\nz : E\n\u22a2 init (cons z 0) = 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\n\u22a2 HasFTaylorSeriesUpToOn (\u2191n)\n (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (FormalMultilinearSeries.unshift (p' x) (f x) 1))\n (fun x => FormalMultilinearSeries.shift (FormalMultilinearSeries.unshift (p' x) (f x))) (v \u2229 u)\n[PROOFSTEP]\nconvert (Hp'.mono (inter_subset_left v u)).congr fun x hx => Hp'.zero_eq x hx.1 using 1\n[GOAL]\ncase h.e'_10\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\n\u22a2 (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (FormalMultilinearSeries.unshift (p' x) (f x) 1)) = fun x =>\n ContinuousMultilinearMap.uncurry0 (p' x 0)\n[PROOFSTEP]\next x y\n[GOAL]\ncase h.e'_10.h.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x\u271d s] x\u271d\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\u271d\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x\u271d u] x\u271d\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\nx y : E\n\u22a2 \u2191(\u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (FormalMultilinearSeries.unshift (p' x) (f x) 1)) y =\n \u2191(ContinuousMultilinearMap.uncurry0 (p' x 0)) y\n[PROOFSTEP]\nchange p' x 0 (init (@snoc 0 (fun _ : Fin 1 => E) 0 y)) y = p' x 0 0 y\n[GOAL]\ncase h.e'_10.h.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x\u271d s] x\u271d\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\u271d\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x\u271d u] x\u271d\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\nx y : E\n\u22a2 \u2191(\u2191(p' x 0) (init (snoc 0 y))) y = \u2191(\u2191(p' x 0) 0) y\n[PROOFSTEP]\nrw [init_snoc]\n[GOAL]\ncase h.e'_11\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nv : Set E\nhv : v \u2208 \ud835\udcdd[insert x u] x\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\n\u22a2 (fun x => FormalMultilinearSeries.shift (FormalMultilinearSeries.unshift (p' x) (f x))) = p'\n[PROOFSTEP]\next x k v y\n[GOAL]\ncase h.e'_11.h.h.H.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x\u271d s] x\u271d\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\u271d\nv\u271d : Set E\nhv : v\u271d \u2208 \ud835\udcdd[insert x\u271d u] x\u271d\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\u271d\nx : E\nk : \u2115\nv : Fin k \u2192 E\ny : E\n\u22a2 \u2191(\u2191(FormalMultilinearSeries.shift (FormalMultilinearSeries.unshift (p' x) (f x)) k) v) y = \u2191(\u2191(p' x k) v) y\n[PROOFSTEP]\nchange\n p' x k (init (@snoc k (fun _ : Fin k.succ => E) v y)) (@snoc k (fun _ : Fin k.succ => E) v y (last k)) = p' x k v y\n[GOAL]\ncase h.e'_11.h.h.H.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x\u271d s] x\u271d\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nf'_eq_deriv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nHf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\u271d\nv\u271d : Set E\nhv : v\u271d \u2208 \ud835\udcdd[insert x\u271d u] x\u271d\np' : E \u2192 FormalMultilinearSeries \ud835\udd5c E (E \u2192L[\ud835\udd5c] F)\nHp' : HasFTaylorSeriesUpToOn (\u2191n) f' p' v\u271d\nx : E\nk : \u2115\nv : Fin k \u2192 E\ny : E\n\u22a2 \u2191(\u2191(p' x k) (init (snoc v y))) (snoc v y (last k)) = \u2191(\u2191(p' x k) v) y\n[PROOFSTEP]\nrw [snoc_last, init_snoc]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x \u2194\n \u2203 u,\n u \u2208 \ud835\udcdd[insert x s] x \u2227\n u \u2286 insert x s \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) s x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' s x\n[PROOFSTEP]\nrefine' \u27e8fun hf => _, _\u27e9\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhf : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\n\u22a2 \u2203 u,\n u \u2208 \ud835\udcdd[insert x s] x \u2227\n u \u2286 insert x s \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) s x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' s x\n[PROOFSTEP]\nobtain \u27e8u, hu, f', huf', hf'\u27e9 := contDiffWithinAt_succ_iff_hasFDerivWithinAt.mp hf\n[GOAL]\ncase refine'_1.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhf : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhuf' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n\u22a2 \u2203 u,\n u \u2208 \ud835\udcdd[insert x s] x \u2227\n u \u2286 insert x s \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) s x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' s x\n[PROOFSTEP]\nobtain \u27e8w, hw, hxw, hwu\u27e9 := mem_nhdsWithin.mp hu\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhf : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhuf' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nw : Set E\nhw : IsOpen w\nhxw : x \u2208 w\nhwu : w \u2229 insert x s \u2286 u\n\u22a2 \u2203 u,\n u \u2208 \ud835\udcdd[insert x s] x \u2227\n u \u2286 insert x s \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) s x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' s x\n[PROOFSTEP]\nrw [inter_comm] at hwu \n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhf : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhuf' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nw : Set E\nhw : IsOpen w\nhxw : x \u2208 w\nhwu : insert x s \u2229 w \u2286 u\n\u22a2 \u2203 u,\n u \u2208 \ud835\udcdd[insert x s] x \u2227\n u \u2286 insert x s \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) s x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' s x\n[PROOFSTEP]\nrefine' \u27e8insert x s \u2229 w, inter_mem_nhdsWithin _ (hw.mem_nhds hxw), inter_subset_left _ _, f', fun y hy => _, _\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhf : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhuf' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nw : Set E\nhw : IsOpen w\nhxw : x \u2208 w\nhwu : insert x s \u2229 w \u2286 u\ny : E\nhy : y \u2208 insert x s \u2229 w\n\u22a2 HasFDerivWithinAt f (f' y) s y\n[PROOFSTEP]\nrefine' ((huf' y <| hwu hy).mono hwu).mono_of_mem _\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhf : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhuf' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nw : Set E\nhw : IsOpen w\nhxw : x \u2208 w\nhwu : insert x s \u2229 w \u2286 u\ny : E\nhy : y \u2208 insert x s \u2229 w\n\u22a2 insert x s \u2229 w \u2208 \ud835\udcdd[s] y\n[PROOFSTEP]\nrefine' mem_of_superset _ (inter_subset_inter_left _ (subset_insert _ _))\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhf : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhuf' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nw : Set E\nhw : IsOpen w\nhxw : x \u2208 w\nhwu : insert x s \u2229 w \u2286 u\ny : E\nhy : y \u2208 insert x s \u2229 w\n\u22a2 s \u2229 w \u2208 \ud835\udcdd[s] y\n[PROOFSTEP]\nrefine' inter_mem_nhdsWithin _ (hw.mem_nhds hy.2)\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhf : ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhuf' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nw : Set E\nhw : IsOpen w\nhxw : x \u2208 w\nhwu : insert x s \u2229 w \u2286 u\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191n) f' s x\n[PROOFSTEP]\nexact hf'.mono_of_mem (nhdsWithin_mono _ (subset_insert _ _) hu)\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 (\u2203 u,\n u \u2208 \ud835\udcdd[insert x s] x \u2227\n u \u2286 insert x s \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) s x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' s x) \u2192\n ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\n[PROOFSTEP]\nrw [\u2190 contDiffWithinAt_insert, contDiffWithinAt_succ_iff_hasFDerivWithinAt, insert_eq_of_mem (mem_insert _ _)]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 (\u2203 u,\n u \u2208 \ud835\udcdd[insert x s] x \u2227\n u \u2286 insert x s \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) s x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' s x) \u2192\n \u2203 u,\n u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n[PROOFSTEP]\nrintro \u27e8u, hu, hus, f', huf', hf'\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nhus : u \u2286 insert x s\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhuf' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) s x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' s x\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n[PROOFSTEP]\nrefine' \u27e8u, hu, f', fun y hy => (huf' y hy).insert'.mono hus, hf'.insert.mono hus\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np f' : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f f' s\n\u22a2 ContDiffOn \ud835\udd5c n f s\n[PROOFSTEP]\nintro x hx m hm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np f' : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f f' s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 p, HasFTaylorSeriesUpToOn (\u2191m) f p u\n[PROOFSTEP]\nuse s\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np f' : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f f' s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 s \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 p, HasFTaylorSeriesUpToOn (\u2191m) f p s\n[PROOFSTEP]\nsimp only [Set.insert_eq_of_mem hx, self_mem_nhdsWithin, true_and_iff]\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np f' : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f f' s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 \u2203 p, HasFTaylorSeriesUpToOn (\u2191m) f p s\n[PROOFSTEP]\nexact \u27e8f', hf.of_le hm\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nhm : \u2191m \u2264 n\nh : ContDiffWithinAt \ud835\udd5c n f s x\n\u22a2 \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 ContDiffOn \ud835\udd5c (\u2191m) f (insert x s \u2229 u)\n[PROOFSTEP]\nrcases h m hm with \u27e8t, ht, p, hp\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nhm : \u2191m \u2264 n\nh : ContDiffWithinAt \ud835\udd5c n f s x\nt : Set E\nht : t \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : HasFTaylorSeriesUpToOn (\u2191m) f p t\n\u22a2 \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 ContDiffOn \ud835\udd5c (\u2191m) f (insert x s \u2229 u)\n[PROOFSTEP]\nrcases mem_nhdsWithin.1 ht with \u27e8u, huo, hxu, hut\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nhm : \u2191m \u2264 n\nh : ContDiffWithinAt \ud835\udd5c n f s x\nt : Set E\nht : t \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : HasFTaylorSeriesUpToOn (\u2191m) f p t\nu : Set E\nhuo : IsOpen u\nhxu : x \u2208 u\nhut : u \u2229 insert x s \u2286 t\n\u22a2 \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 ContDiffOn \ud835\udd5c (\u2191m) f (insert x s \u2229 u)\n[PROOFSTEP]\nrw [inter_comm] at hut \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nhm : \u2191m \u2264 n\nh : ContDiffWithinAt \ud835\udd5c n f s x\nt : Set E\nht : t \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : HasFTaylorSeriesUpToOn (\u2191m) f p t\nu : Set E\nhuo : IsOpen u\nhxu : x \u2208 u\nhut : insert x s \u2229 u \u2286 t\n\u22a2 \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 ContDiffOn \ud835\udd5c (\u2191m) f (insert x s \u2229 u)\n[PROOFSTEP]\nexact \u27e8u, huo, hxu, (hp.mono hut).contDiffOn\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffWithinAt \ud835\udd5c (\u2191n) f s x\n\u22a2 \u2200\u1da0 (y : E) in \ud835\udcdd[insert x s] x, ContDiffWithinAt \ud835\udd5c (\u2191n) f s y\n[PROOFSTEP]\nrcases h.contDiffOn le_rfl with \u27e8u, hu, _, hd\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffWithinAt \ud835\udd5c (\u2191n) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nleft\u271d : u \u2286 insert x s\nhd : ContDiffOn \ud835\udd5c (\u2191n) f u\n\u22a2 \u2200\u1da0 (y : E) in \ud835\udcdd[insert x s] x, ContDiffWithinAt \ud835\udd5c (\u2191n) f s y\n[PROOFSTEP]\nhave : \u2200\u1da0 y : E in \ud835\udcdd[insert x s] x, u \u2208 \ud835\udcdd[insert x s] y \u2227 y \u2208 u := (eventually_nhdsWithin_nhdsWithin.2 hu).and hu\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffWithinAt \ud835\udd5c (\u2191n) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nleft\u271d : u \u2286 insert x s\nhd : ContDiffOn \ud835\udd5c (\u2191n) f u\nthis : \u2200\u1da0 (y : E) in \ud835\udcdd[insert x s] x, u \u2208 \ud835\udcdd[insert x s] y \u2227 y \u2208 u\n\u22a2 \u2200\u1da0 (y : E) in \ud835\udcdd[insert x s] x, ContDiffWithinAt \ud835\udd5c (\u2191n) f s y\n[PROOFSTEP]\nrefine' this.mono fun y hy => (hd y hy.2).mono_of_mem _\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffWithinAt \ud835\udd5c (\u2191n) f s x\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nleft\u271d : u \u2286 insert x s\nhd : ContDiffOn \ud835\udd5c (\u2191n) f u\nthis : \u2200\u1da0 (y : E) in \ud835\udcdd[insert x s] x, u \u2208 \ud835\udcdd[insert x s] y \u2227 y \u2208 u\ny : E\nhy : u \u2208 \ud835\udcdd[insert x s] y \u2227 y \u2208 u\n\u22a2 u \u2208 \ud835\udcdd[s] y\n[PROOFSTEP]\nexact nhdsWithin_mono y (subset_insert _ _) hy.1\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 (\u2200 (m : \u2115), \u2191m \u2264 \u22a4 \u2192 ContDiffOn \ud835\udd5c (\u2191m) f s) \u2194 \u2200 (n : \u2115), ContDiffOn \ud835\udd5c (\u2191n) f s\n[PROOFSTEP]\nsimp only [le_top, forall_prop_of_true]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 (\u2200 (n : \u2115\u221e), ContDiffOn \ud835\udd5c n f s) \u2194 \u2200 (n : \u2115), ContDiffOn \ud835\udd5c (\u2191n) f s\n[PROOFSTEP]\nrefine' \u27e8fun H n => H n, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 (\u2200 (n : \u2115), ContDiffOn \ud835\udd5c (\u2191n) f s) \u2192 \u2200 (n : \u2115\u221e), ContDiffOn \ud835\udd5c n f s\n[PROOFSTEP]\nrintro H (_ | n)\n[GOAL]\ncase none\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : \u2200 (n : \u2115), ContDiffOn \ud835\udd5c (\u2191n) f s\n\u22a2 ContDiffOn \ud835\udd5c none f s\ncase some\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : \u2200 (n : \u2115), ContDiffOn \ud835\udd5c (\u2191n) f s\nn : \u2115\n\u22a2 ContDiffOn \ud835\udd5c (some n) f s\n[PROOFSTEP]\nexacts [contDiffOn_top.2 H, H n]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : \u2200 (x : E), x \u2208 s \u2192 \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 ContDiffOn \ud835\udd5c n f (s \u2229 u)\n\u22a2 ContDiffOn \ud835\udd5c n f s\n[PROOFSTEP]\nintro x xs\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : \u2200 (x : E), x \u2208 s \u2192 \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 ContDiffOn \ud835\udd5c n f (s \u2229 u)\nx : E\nxs : x \u2208 s\n\u22a2 ContDiffWithinAt \ud835\udd5c n f s x\n[PROOFSTEP]\nrcases h x xs with \u27e8u, u_open, xu, hu\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : \u2200 (x : E), x \u2208 s \u2192 \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 ContDiffOn \ud835\udd5c n f (s \u2229 u)\nx : E\nxs : x \u2208 s\nu : Set E\nu_open : IsOpen u\nxu : x \u2208 u\nhu : ContDiffOn \ud835\udd5c n f (s \u2229 u)\n\u22a2 ContDiffWithinAt \ud835\udd5c n f s x\n[PROOFSTEP]\napply (contDiffWithinAt_inter _).1 (hu x \u27e8xs, xu\u27e9)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : \u2200 (x : E), x \u2208 s \u2192 \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 ContDiffOn \ud835\udd5c n f (s \u2229 u)\nx : E\nxs : x \u2208 s\nu : Set E\nu_open : IsOpen u\nxu : x \u2208 u\nhu : ContDiffOn \ud835\udd5c n f (s \u2229 u)\n\u22a2 u \u2208 \ud835\udcdd x\n[PROOFSTEP]\nexact IsOpen.mem_nhds u_open xu\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s \u2194\n \u2200 (x : E),\n x \u2208 s \u2192\n \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffOn \ud835\udd5c (\u2191n) f' u\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s \u2192\n \u2200 (x : E),\n x \u2208 s \u2192\n \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffOn \ud835\udd5c (\u2191n) f' u\n[PROOFSTEP]\nintro h x hx\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffOn \ud835\udd5c (\u2191n) f' u\n[PROOFSTEP]\nrcases(h x hx) n.succ le_rfl with \u27e8u, hu, p, Hp\u27e9\n[GOAL]\ncase mp.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ n)) f p u\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffOn \ud835\udd5c (\u2191n) f' u\n[PROOFSTEP]\nrefine'\n \u27e8u, hu, fun y => (continuousMultilinearCurryFin1 \ud835\udd5c E F) (p y 1), fun y hy =>\n Hp.hasFDerivWithinAt (WithTop.coe_le_coe.2 (Nat.le_add_left 1 n)) hy, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ n)) f p u\n\u22a2 ContDiffOn \ud835\udd5c (\u2191n) (fun y => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p y 1)) u\n[PROOFSTEP]\nrw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp \n[GOAL]\ncase mp.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp :\n (\u2200 (x : E), x \u2208 u \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x) \u2227\n (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) u x) \u2227\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) u\n\u22a2 ContDiffOn \ud835\udd5c (\u2191n) (fun y => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p y 1)) u\n[PROOFSTEP]\nintro z hz m hm\n[GOAL]\ncase mp.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp :\n (\u2200 (x : E), x \u2208 u \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x) \u2227\n (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) u x) \u2227\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) u\nz : E\nhz : z \u2208 u\nm : \u2115\nhm : \u2191m \u2264 \u2191n\n\u22a2 \u2203 u_1,\n u_1 \u2208 \ud835\udcdd[insert z u] z \u2227\n \u2203 p_1, HasFTaylorSeriesUpToOn (\u2191m) (fun y => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p y 1)) p_1 u_1\n[PROOFSTEP]\nrefine'\n \u27e8u, _, fun x : E => (p x).shift, Hp.2.2.of_le hm\u27e9\n -- Porting note: without the explicit arguments `convert` can not determine the type.\n[GOAL]\ncase mp.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp :\n (\u2200 (x : E), x \u2208 u \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x) \u2227\n (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) u x) \u2227\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) u\nz : E\nhz : z \u2208 u\nm : \u2115\nhm : \u2191m \u2264 \u2191n\n\u22a2 u \u2208 \ud835\udcdd[insert z u] z\n[PROOFSTEP]\nconvert @self_mem_nhdsWithin _ _ z u\n[GOAL]\ncase h.e'_5.h.e'_4\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp :\n (\u2200 (x : E), x \u2208 u \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x) \u2227\n (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) u x) \u2227\n HasFTaylorSeriesUpToOn (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1))\n (fun x => FormalMultilinearSeries.shift (p x)) u\nz : E\nhz : z \u2208 u\nm : \u2115\nhm : \u2191m \u2264 \u2191n\n\u22a2 insert z u = u\n[PROOFSTEP]\nexact insert_eq_of_mem hz\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 (\u2200 (x : E),\n x \u2208 s \u2192\n \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffOn \ud835\udd5c (\u2191n) f' u) \u2192\n ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\n[PROOFSTEP]\nintro h x hx\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n \u2200 (x : E),\n x \u2208 s \u2192\n \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffOn \ud835\udd5c (\u2191n) f' u\nx : E\nhx : x \u2208 s\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\n[PROOFSTEP]\nrw [contDiffWithinAt_succ_iff_hasFDerivWithinAt]\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n \u2200 (x : E),\n x \u2208 s \u2192\n \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffOn \ud835\udd5c (\u2191n) f' u\nx : E\nhx : x \u2208 s\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n[PROOFSTEP]\nrcases h x hx with \u27e8u, u_nhbd, f', hu, hf'\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n \u2200 (x : E),\n x \u2208 s \u2192\n \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffOn \ud835\udd5c (\u2191n) f' u\nx : E\nhx : x \u2208 s\nu : Set E\nu_nhbd : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhu : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffOn \ud835\udd5c (\u2191n) f' u\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n[PROOFSTEP]\nhave : x \u2208 u := mem_of_mem_nhdsWithin (mem_insert _ _) u_nhbd\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n \u2200 (x : E),\n x \u2208 s \u2192\n \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffOn \ud835\udd5c (\u2191n) f' u\nx : E\nhx : x \u2208 s\nu : Set E\nu_nhbd : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhu : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffOn \ud835\udd5c (\u2191n) f' u\nthis : x \u2208 u\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n[PROOFSTEP]\nexact \u27e8u, u_nhbd, f', hu, hf' x this\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 \u2016iteratedFDerivWithin \ud835\udd5c 0 f s x\u2016 = \u2016f x\u2016\n[PROOFSTEP]\nrw [iteratedFDerivWithin_zero_eq_comp, comp_apply, LinearIsometryEquiv.norm_map]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nn : \u2115\n\u22a2 fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f s) s =\n \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F)) \u2218\n iteratedFDerivWithin \ud835\udd5c (n + 1) f s\n[PROOFSTEP]\nrw [iteratedFDerivWithin_succ_eq_comp_left]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nn : \u2115\n\u22a2 fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f s) s =\n \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F)) \u2218\n \u2191(continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F) \u2218 fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f s) s\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ns : Set E\nn : \u2115\nx : E\n\u22a2 fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f s) s x =\n (\u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F)) \u2218\n \u2191(continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F) \u2218 fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f s) s)\n x\n[PROOFSTEP]\nsimp only [Function.comp_apply, LinearIsometryEquiv.symm_apply_apply]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 \u2016fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f s) s x\u2016 = \u2016iteratedFDerivWithin \ud835\udd5c (n + 1) f s x\u2016\n[PROOFSTEP]\nrw [iteratedFDerivWithin_succ_eq_comp_left, comp_apply, LinearIsometryEquiv.norm_map]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nm : Fin (n + 1) \u2192 E\n\u22a2 \u2191(iteratedFDerivWithin \ud835\udd5c (n + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last n))\n[PROOFSTEP]\ninduction' n with n IH generalizing x\n[GOAL]\ncase zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b9 n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm\u271d : Fin (n + 1) \u2192 E\nx : E\nhx : x \u2208 s\nm : Fin (Nat.zero + 1) \u2192 E\n\u22a2 \u2191(iteratedFDerivWithin \ud835\udd5c (Nat.zero + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c Nat.zero (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last Nat.zero))\n[PROOFSTEP]\nrw [iteratedFDerivWithin_succ_eq_comp_left, iteratedFDerivWithin_zero_eq_comp, iteratedFDerivWithin_zero_apply,\n Function.comp_apply, LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]\n[GOAL]\ncase zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b9 n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm\u271d : Fin (n + 1) \u2192 E\nx : E\nhx : x \u2208 s\nm : Fin (Nat.zero + 1) \u2192 E\n\u22a2 \u2191(\u2191(continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F)\n (ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)).toLinearEquiv))\n (fderivWithin \ud835\udd5c f s x)))\n m =\n \u2191(fderivWithin \ud835\udd5c f s x) (m (last Nat.zero))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b9 n\u271d\u00b9 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn\u271d : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm\u271d : Fin (n\u271d + 1) \u2192 E\nn : \u2115\nIH :\n \u2200 {x : E},\n x \u2208 s \u2192\n \u2200 (m : Fin (n + 1) \u2192 E),\n \u2191(iteratedFDerivWithin \ud835\udd5c (n + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last n))\nx : E\nhx : x \u2208 s\nm : Fin (Nat.succ n + 1) \u2192 E\n\u22a2 \u2191(iteratedFDerivWithin \ud835\udd5c (Nat.succ n + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c (Nat.succ n) (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last (Nat.succ n)))\n[PROOFSTEP]\nlet I := continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b9 n\u271d\u00b9 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn\u271d : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm\u271d : Fin (n\u271d + 1) \u2192 E\nn : \u2115\nIH :\n \u2200 {x : E},\n x \u2208 s \u2192\n \u2200 (m : Fin (n + 1) \u2192 E),\n \u2191(iteratedFDerivWithin \ud835\udd5c (n + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last n))\nx : E\nhx : x \u2208 s\nm : Fin (Nat.succ n + 1) \u2192 E\nI : ContinuousMultilinearMap \ud835\udd5c (fun i => E) (E \u2192L[\ud835\udd5c] F) \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F\n\u22a2 \u2191(iteratedFDerivWithin \ud835\udd5c (Nat.succ n + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c (Nat.succ n) (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last (Nat.succ n)))\n[PROOFSTEP]\nhave A :\n \u2200 y \u2208 s, iteratedFDerivWithin \ud835\udd5c n.succ f s y = (I \u2218 iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) y :=\n fun y hy \u21a6 by\n ext m\n rw [@IH y hy m]\n rfl\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b9 n\u271d\u00b9 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn\u271d : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm\u271d : Fin (n\u271d + 1) \u2192 E\nn : \u2115\nIH :\n \u2200 {x : E},\n x \u2208 s \u2192\n \u2200 (m : Fin (n + 1) \u2192 E),\n \u2191(iteratedFDerivWithin \ud835\udd5c (n + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last n))\nx : E\nhx : x \u2208 s\nm : Fin (Nat.succ n + 1) \u2192 E\nI : ContinuousMultilinearMap \ud835\udd5c (fun i => E) (E \u2192L[\ud835\udd5c] F) \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F\ny : E\nhy : y \u2208 s\n\u22a2 iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f s y = (\u2191I \u2218 iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) y\n[PROOFSTEP]\next m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b2 n\u271d\u00b9 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn\u271d : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm\u271d\u00b9 : Fin (n\u271d + 1) \u2192 E\nn : \u2115\nIH :\n \u2200 {x : E},\n x \u2208 s \u2192\n \u2200 (m : Fin (n + 1) \u2192 E),\n \u2191(iteratedFDerivWithin \ud835\udd5c (n + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last n))\nx : E\nhx : x \u2208 s\nm\u271d : Fin (Nat.succ n + 1) \u2192 E\nI : ContinuousMultilinearMap \ud835\udd5c (fun i => E) (E \u2192L[\ud835\udd5c] F) \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F\ny : E\nhy : y \u2208 s\nm : Fin (Nat.succ n) \u2192 E\n\u22a2 \u2191(iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f s y) m =\n \u2191((\u2191I \u2218 iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) y) m\n[PROOFSTEP]\nrw [@IH y hy m]\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b2 n\u271d\u00b9 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn\u271d : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm\u271d\u00b9 : Fin (n\u271d + 1) \u2192 E\nn : \u2115\nIH :\n \u2200 {x : E},\n x \u2208 s \u2192\n \u2200 (m : Fin (n + 1) \u2192 E),\n \u2191(iteratedFDerivWithin \ud835\udd5c (n + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last n))\nx : E\nhx : x \u2208 s\nm\u271d : Fin (Nat.succ n + 1) \u2192 E\nI : ContinuousMultilinearMap \ud835\udd5c (fun i => E) (E \u2192L[\ud835\udd5c] F) \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F\ny : E\nhy : y \u2208 s\nm : Fin (Nat.succ n) \u2192 E\n\u22a2 \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s y) (init m)) (m (last n)) =\n \u2191((\u2191I \u2218 iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) y) m\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b9 n\u271d\u00b9 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn\u271d : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm\u271d : Fin (n\u271d + 1) \u2192 E\nn : \u2115\nIH :\n \u2200 {x : E},\n x \u2208 s \u2192\n \u2200 (m : Fin (n + 1) \u2192 E),\n \u2191(iteratedFDerivWithin \ud835\udd5c (n + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last n))\nx : E\nhx : x \u2208 s\nm : Fin (Nat.succ n + 1) \u2192 E\nI : ContinuousMultilinearMap \ud835\udd5c (fun i => E) (E \u2192L[\ud835\udd5c] F) \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F\nA :\n \u2200 (y : E),\n y \u2208 s \u2192\n iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f s y = (\u2191I \u2218 iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) y\n\u22a2 \u2191(iteratedFDerivWithin \ud835\udd5c (Nat.succ n + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c (Nat.succ n) (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last (Nat.succ n)))\n[PROOFSTEP]\ncalc\n (iteratedFDerivWithin \ud835\udd5c (n + 2) f s x : (Fin (n + 2) \u2192 E) \u2192 F) m =\n (fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n.succ f s) s x : E \u2192 E[\u00d7n + 1]\u2192L[\ud835\udd5c] F) (m 0) (tail m) :=\n rfl\n _ =\n (fderivWithin \ud835\udd5c (I \u2218 iteratedFDerivWithin \ud835\udd5c n (fderivWithin \ud835\udd5c f s) s) s x : E \u2192 E[\u00d7n + 1]\u2192L[\ud835\udd5c] F) (m 0)\n (tail m) :=\n by rw [fderivWithin_congr A (A x hx)]\n _ =\n (I \u2218 fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n (fderivWithin \ud835\udd5c f s) s) s x : E \u2192 E[\u00d7n + 1]\u2192L[\ud835\udd5c] F) (m 0)\n (tail m) :=\n by simp only [LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]; rfl\n _ =\n (fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) s x : E \u2192 E[\u00d7n]\u2192L[\ud835\udd5c] E \u2192L[\ud835\udd5c] F) (m 0)\n (init (tail m)) ((tail m) (last n)) :=\n rfl\n _ = iteratedFDerivWithin \ud835\udd5c (Nat.succ n) (fun y => fderivWithin \ud835\udd5c f s y) s x (init m) (m (last (n + 1))) :=\n by\n rw [iteratedFDerivWithin_succ_apply_left, tail_init_eq_init_tail]\n rfl\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b9 n\u271d\u00b9 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn\u271d : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm\u271d : Fin (n\u271d + 1) \u2192 E\nn : \u2115\nIH :\n \u2200 {x : E},\n x \u2208 s \u2192\n \u2200 (m : Fin (n + 1) \u2192 E),\n \u2191(iteratedFDerivWithin \ud835\udd5c (n + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last n))\nx : E\nhx : x \u2208 s\nm : Fin (Nat.succ n + 1) \u2192 E\nI : ContinuousMultilinearMap \ud835\udd5c (fun i => E) (E \u2192L[\ud835\udd5c] F) \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F\nA :\n \u2200 (y : E),\n y \u2208 s \u2192\n iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f s y = (\u2191I \u2218 iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) y\n\u22a2 \u2191(\u2191(fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f s) s x) (m 0)) (tail m) =\n \u2191(\u2191(fderivWithin \ud835\udd5c (\u2191I \u2218 iteratedFDerivWithin \ud835\udd5c n (fderivWithin \ud835\udd5c f s) s) s x) (m 0)) (tail m)\n[PROOFSTEP]\nrw [fderivWithin_congr A (A x hx)]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b9 n\u271d\u00b9 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn\u271d : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm\u271d : Fin (n\u271d + 1) \u2192 E\nn : \u2115\nIH :\n \u2200 {x : E},\n x \u2208 s \u2192\n \u2200 (m : Fin (n + 1) \u2192 E),\n \u2191(iteratedFDerivWithin \ud835\udd5c (n + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last n))\nx : E\nhx : x \u2208 s\nm : Fin (Nat.succ n + 1) \u2192 E\nI : ContinuousMultilinearMap \ud835\udd5c (fun i => E) (E \u2192L[\ud835\udd5c] F) \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F\nA :\n \u2200 (y : E),\n y \u2208 s \u2192\n iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f s y = (\u2191I \u2218 iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) y\n\u22a2 \u2191(\u2191(fderivWithin \ud835\udd5c (\u2191I \u2218 iteratedFDerivWithin \ud835\udd5c n (fderivWithin \ud835\udd5c f s) s) s x) (m 0)) (tail m) =\n \u2191((\u2191I \u2218 \u2191(fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n (fderivWithin \ud835\udd5c f s) s) s x)) (m 0)) (tail m)\n[PROOFSTEP]\nsimp only [LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b9 n\u271d\u00b9 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn\u271d : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm\u271d : Fin (n\u271d + 1) \u2192 E\nn : \u2115\nIH :\n \u2200 {x : E},\n x \u2208 s \u2192\n \u2200 (m : Fin (n + 1) \u2192 E),\n \u2191(iteratedFDerivWithin \ud835\udd5c (n + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last n))\nx : E\nhx : x \u2208 s\nm : Fin (Nat.succ n + 1) \u2192 E\nI : ContinuousMultilinearMap \ud835\udd5c (fun i => E) (E \u2192L[\ud835\udd5c] F) \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F\nA :\n \u2200 (y : E),\n y \u2208 s \u2192\n iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f s y = (\u2191I \u2218 iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) y\n\u22a2 \u2191(\u2191(ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk (continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F).toLinearEquiv))\n (fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n (fderivWithin \ud835\udd5c f s) s) s x))\n (m 0))\n (tail m) =\n \u2191((\u2191(continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F) \u2218\n \u2191(fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n (fderivWithin \ud835\udd5c f s) s) s x))\n (m 0))\n (tail m)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b9 n\u271d\u00b9 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn\u271d : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm\u271d : Fin (n\u271d + 1) \u2192 E\nn : \u2115\nIH :\n \u2200 {x : E},\n x \u2208 s \u2192\n \u2200 (m : Fin (n + 1) \u2192 E),\n \u2191(iteratedFDerivWithin \ud835\udd5c (n + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last n))\nx : E\nhx : x \u2208 s\nm : Fin (Nat.succ n + 1) \u2192 E\nI : ContinuousMultilinearMap \ud835\udd5c (fun i => E) (E \u2192L[\ud835\udd5c] F) \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F\nA :\n \u2200 (y : E),\n y \u2208 s \u2192\n iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f s y = (\u2191I \u2218 iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) y\n\u22a2 \u2191(\u2191(\u2191(fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) s x) (m 0)) (init (tail m)))\n (tail m (last n)) =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c (Nat.succ n) (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last (n + 1)))\n[PROOFSTEP]\nrw [iteratedFDerivWithin_succ_apply_left, tail_init_eq_init_tail]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b9 n\u271d\u00b9 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn\u271d : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm\u271d : Fin (n\u271d + 1) \u2192 E\nn : \u2115\nIH :\n \u2200 {x : E},\n x \u2208 s \u2192\n \u2200 (m : Fin (n + 1) \u2192 E),\n \u2191(iteratedFDerivWithin \ud835\udd5c (n + 1) f s x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last n))\nx : E\nhx : x \u2208 s\nm : Fin (Nat.succ n + 1) \u2192 E\nI : ContinuousMultilinearMap \ud835\udd5c (fun i => E) (E \u2192L[\ud835\udd5c] F) \u2243\u2097\u1d62[\ud835\udd5c] ContinuousMultilinearMap \ud835\udd5c (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F\nA :\n \u2200 (y : E),\n y \u2208 s \u2192\n iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f s y = (\u2191I \u2218 iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) y\n\u22a2 \u2191(\u2191(\u2191(fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) s x) (m 0)) (init (tail m)))\n (tail m (last n)) =\n \u2191(\u2191(\u2191(fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) s x) (init m 0)) (init (tail m)))\n (m (last (n + 1)))\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\n\u22a2 iteratedFDerivWithin \ud835\udd5c (n + 1) f s x =\n (\u2191(continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F) \u2218 iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) x\n[PROOFSTEP]\next m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nm : Fin (n + 1) \u2192 E\n\u22a2 \u2191(iteratedFDerivWithin \ud835\udd5c (n + 1) f s x) m =\n \u2191((\u2191(continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F) \u2218 iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) x)\n m\n[PROOFSTEP]\nrw [iteratedFDerivWithin_succ_apply_right hs hx]\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nm : Fin (n + 1) \u2192 E\n\u22a2 \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s x) (init m)) (m (last n)) =\n \u2191((\u2191(continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F) \u2218 iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) x)\n m\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\n\u22a2 \u2016iteratedFDerivWithin \ud835\udd5c n (fderivWithin \ud835\udd5c f s) s x\u2016 = \u2016iteratedFDerivWithin \ud835\udd5c (n + 1) f s x\u2016\n[PROOFSTEP]\nrw [iteratedFDerivWithin_succ_eq_comp_right hs hx, comp_apply, LinearIsometryEquiv.norm_map]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : UniqueDiffWithinAt \ud835\udd5c s x\nm : Fin 1 \u2192 E\n\u22a2 \u2191(iteratedFDerivWithin \ud835\udd5c 1 f s x) m = \u2191(fderivWithin \ud835\udd5c f s x) (m 0)\n[PROOFSTEP]\nsimp only [iteratedFDerivWithin_succ_apply_left, iteratedFDerivWithin_zero_eq_comp,\n (continuousMultilinearCurryFin0 \ud835\udd5c E F).symm.comp_fderivWithin h]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : UniqueDiffWithinAt \ud835\udd5c s x\nm : Fin 1 \u2192 E\n\u22a2 \u2191(\u2191(ContinuousLinearMap.comp\n (\u2191(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)).toLinearEquiv))\n (fderivWithin \ud835\udd5c f s x))\n (m 0))\n (tail m) =\n \u2191(fderivWithin \ud835\udd5c f s x) (m 0)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\nn : \u2115\n\u22a2 iteratedFDerivWithin \ud835\udd5c n f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c n f t\n[PROOFSTEP]\ninduction' n with n ihn\n[GOAL]\ncase zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\n\u22a2 iteratedFDerivWithin \ud835\udd5c Nat.zero f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c Nat.zero f t\n[PROOFSTEP]\nexact h.mono fun y hy => FunLike.ext _ _ fun _ => hy\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\nn : \u2115\nihn : iteratedFDerivWithin \ud835\udd5c n f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c n f t\n\u22a2 iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f t\n[PROOFSTEP]\nhave : fderivWithin \ud835\udd5c _ t =\u1da0[\ud835\udcdd[s] x] fderivWithin \ud835\udd5c _ t := ihn.fderiv_within' ht\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\nn : \u2115\nihn : iteratedFDerivWithin \ud835\udd5c n f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c n f t\nthis : fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f\u2081 t) t =\u1da0[\ud835\udcdd[s] x] fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f t) t\n\u22a2 iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f t\n[PROOFSTEP]\napply this.mono\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\nn : \u2115\nihn : iteratedFDerivWithin \ud835\udd5c n f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c n f t\nthis : fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f\u2081 t) t =\u1da0[\ud835\udcdd[s] x] fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f t) t\n\u22a2 \u2200 (x : E),\n fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f\u2081 t) t x = fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f t) t x \u2192\n iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f\u2081 t x = iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f t x\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\nn : \u2115\nihn : iteratedFDerivWithin \ud835\udd5c n f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c n f t\nthis : fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f\u2081 t) t =\u1da0[\ud835\udcdd[s] x] fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f t) t\ny : E\nhy : fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f\u2081 t) t y = fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f t) t y\n\u22a2 iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f\u2081 t y = iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f t y\n[PROOFSTEP]\nsimp only [iteratedFDerivWithin_succ_eq_comp_left, hy, (\u00b7 \u2218 \u00b7)]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nn : \u2115\n\u22a2 f\u2081 =\u1da0[\ud835\udcdd[insert x s] x] f\n[PROOFSTEP]\nsimpa [EventuallyEq, hx]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ny : E\nh : s =\u1da0[\ud835\udcdd[{y}\u1d9c] x] t\nn : \u2115\n\u22a2 iteratedFDerivWithin \ud835\udd5c n f s =\u1da0[\ud835\udcdd x] iteratedFDerivWithin \ud835\udd5c n f t\n[PROOFSTEP]\ninduction' n with n ihn generalizing x\n[GOAL]\ncase zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ny : E\nh\u271d : s =\u1da0[\ud835\udcdd[{y}\u1d9c] x\u271d] t\nx : E\nh : s =\u1da0[\ud835\udcdd[{y}\u1d9c] x] t\n\u22a2 iteratedFDerivWithin \ud835\udd5c Nat.zero f s =\u1da0[\ud835\udcdd x] iteratedFDerivWithin \ud835\udd5c Nat.zero f t\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ny : E\nh\u271d : s =\u1da0[\ud835\udcdd[{y}\u1d9c] x\u271d] t\nn : \u2115\nihn : \u2200 {x : E}, s =\u1da0[\ud835\udcdd[{y}\u1d9c] x] t \u2192 iteratedFDerivWithin \ud835\udd5c n f s =\u1da0[\ud835\udcdd x] iteratedFDerivWithin \ud835\udd5c n f t\nx : E\nh : s =\u1da0[\ud835\udcdd[{y}\u1d9c] x] t\n\u22a2 iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f s =\u1da0[\ud835\udcdd x] iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f t\n[PROOFSTEP]\nrefine' (eventually_nhds_nhdsWithin.2 h).mono fun y hy => _\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ny\u271d : E\nh\u271d : s =\u1da0[\ud835\udcdd[{y\u271d}\u1d9c] x\u271d] t\nn : \u2115\nihn : \u2200 {x : E}, s =\u1da0[\ud835\udcdd[{y\u271d}\u1d9c] x] t \u2192 iteratedFDerivWithin \ud835\udd5c n f s =\u1da0[\ud835\udcdd x] iteratedFDerivWithin \ud835\udd5c n f t\nx : E\nh : s =\u1da0[\ud835\udcdd[{y\u271d}\u1d9c] x] t\ny : E\nhy : \u2200\u1da0 (x : E) in \ud835\udcdd[{y\u271d}\u1d9c] y, s x = t x\n\u22a2 iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f s y = iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f t y\n[PROOFSTEP]\nsimp only [iteratedFDerivWithin_succ_eq_comp_left, (\u00b7 \u2218 \u00b7)]\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ny\u271d : E\nh\u271d : s =\u1da0[\ud835\udcdd[{y\u271d}\u1d9c] x\u271d] t\nn : \u2115\nihn : \u2200 {x : E}, s =\u1da0[\ud835\udcdd[{y\u271d}\u1d9c] x] t \u2192 iteratedFDerivWithin \ud835\udd5c n f s =\u1da0[\ud835\udcdd x] iteratedFDerivWithin \ud835\udd5c n f t\nx : E\nh : s =\u1da0[\ud835\udcdd[{y\u271d}\u1d9c] x] t\ny : E\nhy : \u2200\u1da0 (x : E) in \ud835\udcdd[{y\u271d}\u1d9c] y, s x = t x\n\u22a2 \u2191(continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F) (fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f s) s y) =\n \u2191(continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F) (fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f t) t y)\n[PROOFSTEP]\nrw [(ihn hy).fderivWithin_eq_nhds, fderivWithin_congr_set' _ hy]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiffOn \ud835\udd5c 0 f s \u2194 ContinuousOn f s\n[PROOFSTEP]\nrefine' \u27e8fun H => H.continuousOn, fun H => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContinuousOn f s\n\u22a2 ContDiffOn \ud835\udd5c 0 f s\n[PROOFSTEP]\nintro x hx m hm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContinuousOn f s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 0\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 p, HasFTaylorSeriesUpToOn (\u2191m) f p u\n[PROOFSTEP]\nhave : (m : \u2115\u221e) = 0 := le_antisymm hm bot_le\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContinuousOn f s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 0\nthis : \u2191m = 0\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 p, HasFTaylorSeriesUpToOn (\u2191m) f p u\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContinuousOn f s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 0\nthis : \u2191m = 0\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 p, HasFTaylorSeriesUpToOn 0 f p u\n[PROOFSTEP]\nrefine' \u27e8insert x s, self_mem_nhdsWithin, ftaylorSeriesWithin \ud835\udd5c f s, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContinuousOn f s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 0\nthis : \u2191m = 0\n\u22a2 HasFTaylorSeriesUpToOn 0 f (ftaylorSeriesWithin \ud835\udd5c f s) (insert x s)\n[PROOFSTEP]\nrw [hasFTaylorSeriesUpToOn_zero_iff]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContinuousOn f s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 0\nthis : \u2191m = 0\n\u22a2 ContinuousOn f (insert x s) \u2227\n \u2200 (x_1 : E), x_1 \u2208 insert x s \u2192 ContinuousMultilinearMap.uncurry0 (ftaylorSeriesWithin \ud835\udd5c f s x_1 0) = f x_1\n[PROOFSTEP]\nexact \u27e8by rwa [insert_eq_of_mem hx], fun x _ => by simp [ftaylorSeriesWithin]\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContinuousOn f s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 0\nthis : \u2191m = 0\n\u22a2 ContinuousOn f (insert x s)\n[PROOFSTEP]\nrwa [insert_eq_of_mem hx]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d\u00b2 x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContinuousOn f s\nx\u271d\u00b9 : E\nhx : x\u271d\u00b9 \u2208 s\nm : \u2115\nhm : \u2191m \u2264 0\nthis : \u2191m = 0\nx : E\nx\u271d : x \u2208 insert x\u271d\u00b9 s\n\u22a2 ContinuousMultilinearMap.uncurry0 (ftaylorSeriesWithin \ud835\udd5c f s x 0) = f x\n[PROOFSTEP]\nsimp [ftaylorSeriesWithin]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\n\u22a2 ContDiffWithinAt \ud835\udd5c 0 f s x \u2194 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\n\u22a2 ContDiffWithinAt \ud835\udd5c 0 f s x \u2192 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nh : ContDiffWithinAt \ud835\udd5c 0 f s x\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u)\n[PROOFSTEP]\nobtain \u27e8u, H, p, hp\u27e9 := h 0 le_rfl\n[GOAL]\ncase mp.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nh : ContDiffWithinAt \ud835\udd5c 0 f s x\nu : Set E\nH : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : HasFTaylorSeriesUpToOn (\u21910) f p u\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u)\n[PROOFSTEP]\nrefine' \u27e8u, _, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nh : ContDiffWithinAt \ud835\udd5c 0 f s x\nu : Set E\nH : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : HasFTaylorSeriesUpToOn (\u21910) f p u\n\u22a2 u \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\nsimpa [hx] using H\n[GOAL]\ncase mp.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nh : ContDiffWithinAt \ud835\udd5c 0 f s x\nu : Set E\nH : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : HasFTaylorSeriesUpToOn (\u21910) f p u\n\u22a2 ContinuousOn f (s \u2229 u)\n[PROOFSTEP]\nsimp only [Nat.cast_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp \n[GOAL]\ncase mp.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nh : ContDiffWithinAt \ud835\udd5c 0 f s x\nu : Set E\nH : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : ContinuousOn f u \u2227 \u2200 (x : E), x \u2208 u \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\n\u22a2 ContinuousOn f (s \u2229 u)\n[PROOFSTEP]\nexact hp.1.mono (inter_subset_right s u)\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\n\u22a2 (\u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u)) \u2192 ContDiffWithinAt \ud835\udd5c 0 f s x\n[PROOFSTEP]\nrintro \u27e8u, H, hu\u27e9\n[GOAL]\ncase mpr.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nu : Set E\nH : u \u2208 \ud835\udcdd[s] x\nhu : ContinuousOn f (s \u2229 u)\n\u22a2 ContDiffWithinAt \ud835\udd5c 0 f s x\n[PROOFSTEP]\nrw [\u2190 contDiffWithinAt_inter' H]\n[GOAL]\ncase mpr.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nu : Set E\nH : u \u2208 \ud835\udcdd[s] x\nhu : ContinuousOn f (s \u2229 u)\n\u22a2 ContDiffWithinAt \ud835\udd5c 0 f (s \u2229 u) x\n[PROOFSTEP]\nhave h' : x \u2208 s \u2229 u := \u27e8hx, mem_of_mem_nhdsWithin hx H\u27e9\n[GOAL]\ncase mpr.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nu : Set E\nH : u \u2208 \ud835\udcdd[s] x\nhu : ContinuousOn f (s \u2229 u)\nh' : x \u2208 s \u2229 u\n\u22a2 ContDiffWithinAt \ud835\udd5c 0 f (s \u2229 u) x\n[PROOFSTEP]\nexact (contDiffOn_zero.mpr hu).contDiffWithinAt h'\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nm : \u2115\nhmn : \u2191m \u2264 n\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\n\u22a2 p x m = iteratedFDerivWithin \ud835\udd5c m f s x\n[PROOFSTEP]\ninduction' m with m IH generalizing x\n[GOAL]\ncase zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nm : \u2115\nhmn\u271d : \u2191m \u2264 n\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nx : E\nhmn : \u2191Nat.zero \u2264 n\nhx : x \u2208 s\n\u22a2 p x Nat.zero = iteratedFDerivWithin \ud835\udd5c Nat.zero f s x\n[PROOFSTEP]\nrw [Nat.zero_eq, h.zero_eq' hx, iteratedFDerivWithin_zero_eq_comp]\n[GOAL]\ncase zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nm : \u2115\nhmn\u271d : \u2191m \u2264 n\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nx : E\nhmn : \u2191Nat.zero \u2264 n\nhx : x \u2208 s\n\u22a2 \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)) (f x) =\n (\u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)) \u2218 f) x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b9 n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nm\u271d : \u2115\nhmn\u271d : \u2191m\u271d \u2264 n\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm : \u2115\nIH : \u2200 {x : E}, \u2191m \u2264 n \u2192 x \u2208 s \u2192 p x m = iteratedFDerivWithin \ud835\udd5c m f s x\nx : E\nhmn : \u2191(Nat.succ m) \u2264 n\nhx : x \u2208 s\n\u22a2 p x (Nat.succ m) = iteratedFDerivWithin \ud835\udd5c (Nat.succ m) f s x\n[PROOFSTEP]\nhave A : (m : \u2115\u221e) < n := lt_of_lt_of_le (WithTop.coe_lt_coe.2 (lt_add_one m)) hmn\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b9 n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nm\u271d : \u2115\nhmn\u271d : \u2191m\u271d \u2264 n\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm : \u2115\nIH : \u2200 {x : E}, \u2191m \u2264 n \u2192 x \u2208 s \u2192 p x m = iteratedFDerivWithin \ud835\udd5c m f s x\nx : E\nhmn : \u2191(Nat.succ m) \u2264 n\nhx : x \u2208 s\nA : \u2191m < n\n\u22a2 p x (Nat.succ m) = iteratedFDerivWithin \ud835\udd5c (Nat.succ m) f s x\n[PROOFSTEP]\nhave :\n HasFDerivWithinAt (fun y : E => iteratedFDerivWithin \ud835\udd5c m f s y)\n (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x :=\n (h.fderivWithin m A x hx).congr (fun y hy => (IH (le_of_lt A) hy).symm) (IH (le_of_lt A) hx).symm\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b9 n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nm\u271d : \u2115\nhmn\u271d : \u2191m\u271d \u2264 n\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm : \u2115\nIH : \u2200 {x : E}, \u2191m \u2264 n \u2192 x \u2208 s \u2192 p x m = iteratedFDerivWithin \ud835\udd5c m f s x\nx : E\nhmn : \u2191(Nat.succ m) \u2264 n\nhx : x \u2208 s\nA : \u2191m < n\nthis :\n HasFDerivWithinAt (fun y => iteratedFDerivWithin \ud835\udd5c m f s y) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s\n x\n\u22a2 p x (Nat.succ m) = iteratedFDerivWithin \ud835\udd5c (Nat.succ m) f s x\n[PROOFSTEP]\nrw [iteratedFDerivWithin_succ_eq_comp_left, Function.comp_apply, this.fderivWithin (hs x hx)]\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d\u00b9 n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p s\nm\u271d : \u2115\nhmn\u271d : \u2191m\u271d \u2264 n\nhs : UniqueDiffOn \ud835\udd5c s\nhx\u271d : x\u271d \u2208 s\nm : \u2115\nIH : \u2200 {x : E}, \u2191m \u2264 n \u2192 x \u2208 s \u2192 p x m = iteratedFDerivWithin \ud835\udd5c m f s x\nx : E\nhmn : \u2191(Nat.succ m) \u2264 n\nhx : x \u2208 s\nA : \u2191m < n\nthis :\n HasFDerivWithinAt (fun y => iteratedFDerivWithin \ud835\udd5c m f s y) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s\n x\n\u22a2 p x (Nat.succ m) =\n \u2191(continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m)))\n[PROOFSTEP]\nexact (ContinuousMultilinearMap.uncurry_curryLeft _).symm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\n\u22a2 HasFTaylorSeriesUpToOn n f (ftaylorSeriesWithin \ud835\udd5c f s) s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (ftaylorSeriesWithin \ud835\udd5c f s x 0) = f x\n[PROOFSTEP]\nintro x _\n[GOAL]\ncase zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nx : E\na\u271d : x \u2208 s\n\u22a2 ContinuousMultilinearMap.uncurry0 (ftaylorSeriesWithin \ud835\udd5c f s x 0) = f x\n[PROOFSTEP]\nsimp only [ftaylorSeriesWithin, ContinuousMultilinearMap.uncurry0_apply, iteratedFDerivWithin_zero_apply]\n[GOAL]\ncase fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\n\u22a2 \u2200 (m : \u2115),\n \u2191m < n \u2192\n \u2200 (x : E),\n x \u2208 s \u2192\n HasFDerivWithinAt (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m)\n (ContinuousMultilinearMap.curryLeft (ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m))) s x\n[PROOFSTEP]\nintro m hm x hx\n[GOAL]\ncase fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\n\u22a2 HasFDerivWithinAt (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m)\n (ContinuousMultilinearMap.curryLeft (ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m))) s x\n[PROOFSTEP]\nrcases(h x hx) m.succ (ENat.add_one_le_of_lt hm) with \u27e8u, hu, p, Hp\u27e9\n[GOAL]\ncase fderivWithin.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ m)) f p u\n\u22a2 HasFDerivWithinAt (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m)\n (ContinuousMultilinearMap.curryLeft (ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m))) s x\n[PROOFSTEP]\nrw [insert_eq_of_mem hx] at hu \n[GOAL]\ncase fderivWithin.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ m)) f p u\n\u22a2 HasFDerivWithinAt (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m)\n (ContinuousMultilinearMap.curryLeft (ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m))) s x\n[PROOFSTEP]\nrcases mem_nhdsWithin.1 hu with \u27e8o, o_open, xo, ho\u27e9\n[GOAL]\ncase fderivWithin.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ m)) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : o \u2229 s \u2286 u\n\u22a2 HasFDerivWithinAt (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m)\n (ContinuousMultilinearMap.curryLeft (ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m))) s x\n[PROOFSTEP]\nrw [inter_comm] at ho \n[GOAL]\ncase fderivWithin.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ m)) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\n\u22a2 HasFDerivWithinAt (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m)\n (ContinuousMultilinearMap.curryLeft (ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m))) s x\n[PROOFSTEP]\nhave : p x m.succ = ftaylorSeriesWithin \ud835\udd5c f s x m.succ :=\n by\n change p x m.succ = iteratedFDerivWithin \ud835\udd5c m.succ f s x\n rw [\u2190 iteratedFDerivWithin_inter_open o_open xo]\n exact (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn le_rfl (hs.inter o_open) \u27e8hx, xo\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ m)) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\n\u22a2 p x (Nat.succ m) = ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m)\n[PROOFSTEP]\nchange p x m.succ = iteratedFDerivWithin \ud835\udd5c m.succ f s x\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ m)) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\n\u22a2 p x (Nat.succ m) = iteratedFDerivWithin \ud835\udd5c (Nat.succ m) f s x\n[PROOFSTEP]\nrw [\u2190 iteratedFDerivWithin_inter_open o_open xo]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ m)) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\n\u22a2 p x (Nat.succ m) = iteratedFDerivWithin \ud835\udd5c (Nat.succ m) f (s \u2229 o) x\n[PROOFSTEP]\nexact (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn le_rfl (hs.inter o_open) \u27e8hx, xo\u27e9\n[GOAL]\ncase fderivWithin.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ m)) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\nthis : p x (Nat.succ m) = ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m)\n\u22a2 HasFDerivWithinAt (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m)\n (ContinuousMultilinearMap.curryLeft (ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m))) s x\n[PROOFSTEP]\nrw [\u2190 this, \u2190 hasFDerivWithinAt_inter (IsOpen.mem_nhds o_open xo)]\n[GOAL]\ncase fderivWithin.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ m)) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\nthis : p x (Nat.succ m) = ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m)\n\u22a2 HasFDerivWithinAt (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m)))\n (s \u2229 o) x\n[PROOFSTEP]\nhave A : \u2200 y \u2208 s \u2229 o, p y m = ftaylorSeriesWithin \ud835\udd5c f s y m :=\n by\n rintro y \u27e8hy, yo\u27e9\n change p y m = iteratedFDerivWithin \ud835\udd5c m f s y\n rw [\u2190 iteratedFDerivWithin_inter_open o_open yo]\n exact (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn (WithTop.coe_le_coe.2 (Nat.le_succ m)) (hs.inter o_open) \u27e8hy, yo\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ m)) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\nthis : p x (Nat.succ m) = ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m)\n\u22a2 \u2200 (y : E), y \u2208 s \u2229 o \u2192 p y m = ftaylorSeriesWithin \ud835\udd5c f s y m\n[PROOFSTEP]\nrintro y \u27e8hy, yo\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ m)) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\nthis : p x (Nat.succ m) = ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m)\ny : E\nhy : y \u2208 s\nyo : y \u2208 o\n\u22a2 p y m = ftaylorSeriesWithin \ud835\udd5c f s y m\n[PROOFSTEP]\nchange p y m = iteratedFDerivWithin \ud835\udd5c m f s y\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ m)) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\nthis : p x (Nat.succ m) = ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m)\ny : E\nhy : y \u2208 s\nyo : y \u2208 o\n\u22a2 p y m = iteratedFDerivWithin \ud835\udd5c m f s y\n[PROOFSTEP]\nrw [\u2190 iteratedFDerivWithin_inter_open o_open yo]\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ m)) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\nthis : p x (Nat.succ m) = ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m)\ny : E\nhy : y \u2208 s\nyo : y \u2208 o\n\u22a2 p y m = iteratedFDerivWithin \ud835\udd5c m f (s \u2229 o) y\n[PROOFSTEP]\nexact (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn (WithTop.coe_le_coe.2 (Nat.le_succ m)) (hs.inter o_open) \u27e8hy, yo\u27e9\n[GOAL]\ncase fderivWithin.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191(Nat.succ m)) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\nthis : p x (Nat.succ m) = ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m)\nA : \u2200 (y : E), y \u2208 s \u2229 o \u2192 p y m = ftaylorSeriesWithin \ud835\udd5c f s y m\n\u22a2 HasFDerivWithinAt (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m)))\n (s \u2229 o) x\n[PROOFSTEP]\nexact\n ((Hp.mono ho).fderivWithin m (WithTop.coe_lt_coe.2 (lt_add_one m)) x \u27e8hx, xo\u27e9).congr (fun y hy => (A y hy).symm)\n (A x \u27e8hx, xo\u27e9).symm\n[GOAL]\ncase cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m) s\n[PROOFSTEP]\nintro m hm\n[GOAL]\ncase cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m) s\n[PROOFSTEP]\napply continuousOn_of_locally_continuousOn\n[GOAL]\ncase cont.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 \u2203 t, IsOpen t \u2227 x \u2208 t \u2227 ContinuousOn (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m) (s \u2229 t)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase cont.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m \u2264 n\nx : E\nhx : x \u2208 s\n\u22a2 \u2203 t, IsOpen t \u2227 x \u2208 t \u2227 ContinuousOn (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m) (s \u2229 t)\n[PROOFSTEP]\nrcases h x hx m hm with \u27e8u, hu, p, Hp\u27e9\n[GOAL]\ncase cont.h.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m \u2264 n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191m) f p u\n\u22a2 \u2203 t, IsOpen t \u2227 x \u2208 t \u2227 ContinuousOn (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m) (s \u2229 t)\n[PROOFSTEP]\nrcases mem_nhdsWithin.1 hu with \u27e8o, o_open, xo, ho\u27e9\n[GOAL]\ncase cont.h.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m \u2264 n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191m) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : o \u2229 insert x s \u2286 u\n\u22a2 \u2203 t, IsOpen t \u2227 x \u2208 t \u2227 ContinuousOn (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m) (s \u2229 t)\n[PROOFSTEP]\nrw [insert_eq_of_mem hx] at ho \n[GOAL]\ncase cont.h.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m \u2264 n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191m) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : o \u2229 s \u2286 u\n\u22a2 \u2203 t, IsOpen t \u2227 x \u2208 t \u2227 ContinuousOn (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m) (s \u2229 t)\n[PROOFSTEP]\nrw [inter_comm] at ho \n[GOAL]\ncase cont.h.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m \u2264 n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191m) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\n\u22a2 \u2203 t, IsOpen t \u2227 x \u2208 t \u2227 ContinuousOn (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m) (s \u2229 t)\n[PROOFSTEP]\nrefine' \u27e8o, o_open, xo, _\u27e9\n[GOAL]\ncase cont.h.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m \u2264 n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191m) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\n\u22a2 ContinuousOn (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m) (s \u2229 o)\n[PROOFSTEP]\nhave A : \u2200 y \u2208 s \u2229 o, p y m = ftaylorSeriesWithin \ud835\udd5c f s y m :=\n by\n rintro y \u27e8hy, yo\u27e9\n change p y m = iteratedFDerivWithin \ud835\udd5c m f s y\n rw [\u2190 iteratedFDerivWithin_inter_open o_open yo]\n exact (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn le_rfl (hs.inter o_open) \u27e8hy, yo\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m \u2264 n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191m) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\n\u22a2 \u2200 (y : E), y \u2208 s \u2229 o \u2192 p y m = ftaylorSeriesWithin \ud835\udd5c f s y m\n[PROOFSTEP]\nrintro y \u27e8hy, yo\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m \u2264 n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191m) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\ny : E\nhy : y \u2208 s\nyo : y \u2208 o\n\u22a2 p y m = ftaylorSeriesWithin \ud835\udd5c f s y m\n[PROOFSTEP]\nchange p y m = iteratedFDerivWithin \ud835\udd5c m f s y\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m \u2264 n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191m) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\ny : E\nhy : y \u2208 s\nyo : y \u2208 o\n\u22a2 p y m = iteratedFDerivWithin \ud835\udd5c m f s y\n[PROOFSTEP]\nrw [\u2190 iteratedFDerivWithin_inter_open o_open yo]\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m \u2264 n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191m) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\ny : E\nhy : y \u2208 s\nyo : y \u2208 o\n\u22a2 p y m = iteratedFDerivWithin \ud835\udd5c m f (s \u2229 o) y\n[PROOFSTEP]\nexact (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn le_rfl (hs.inter o_open) \u27e8hy, yo\u27e9\n[GOAL]\ncase cont.h.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhm : \u2191m \u2264 n\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHp : HasFTaylorSeriesUpToOn (\u2191m) f p u\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\nA : \u2200 (y : E), y \u2208 s \u2229 o \u2192 p y m = ftaylorSeriesWithin \ud835\udd5c f s y m\n\u22a2 ContinuousOn (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m) (s \u2229 o)\n[PROOFSTEP]\nexact ((Hp.mono ho).cont m le_rfl).congr fun y hy => (A y hy).symm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHcont : \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nHdiff : \u2200 (m : \u2115), \u2191m < n \u2192 DifferentiableOn \ud835\udd5c (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\n\u22a2 ContDiffOn \ud835\udd5c n f s\n[PROOFSTEP]\nintro x hx m hm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHcont : \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nHdiff : \u2200 (m : \u2115), \u2191m < n \u2192 DifferentiableOn \ud835\udd5c (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x s] x \u2227 \u2203 p, HasFTaylorSeriesUpToOn (\u2191m) f p u\n[PROOFSTEP]\nrw [insert_eq_of_mem hx]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHcont : \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nHdiff : \u2200 (m : \u2115), \u2191m < n \u2192 DifferentiableOn \ud835\udd5c (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 \u2203 p, HasFTaylorSeriesUpToOn (\u2191m) f p u\n[PROOFSTEP]\nrefine' \u27e8s, self_mem_nhdsWithin, ftaylorSeriesWithin \ud835\udd5c f s, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHcont : \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nHdiff : \u2200 (m : \u2115), \u2191m < n \u2192 DifferentiableOn \ud835\udd5c (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 HasFTaylorSeriesUpToOn (\u2191m) f (ftaylorSeriesWithin \ud835\udd5c f s) s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHcont : \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nHdiff : \u2200 (m : \u2115), \u2191m < n \u2192 DifferentiableOn \ud835\udd5c (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (ftaylorSeriesWithin \ud835\udd5c f s x 0) = f x\n[PROOFSTEP]\nintro y _\n[GOAL]\ncase zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHcont : \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nHdiff : \u2200 (m : \u2115), \u2191m < n \u2192 DifferentiableOn \ud835\udd5c (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 n\ny : E\na\u271d : y \u2208 s\n\u22a2 ContinuousMultilinearMap.uncurry0 (ftaylorSeriesWithin \ud835\udd5c f s y 0) = f y\n[PROOFSTEP]\nsimp only [ftaylorSeriesWithin, ContinuousMultilinearMap.uncurry0_apply, iteratedFDerivWithin_zero_apply]\n[GOAL]\ncase fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHcont : \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nHdiff : \u2200 (m : \u2115), \u2191m < n \u2192 DifferentiableOn \ud835\udd5c (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 \u2200 (m_1 : \u2115),\n \u2191m_1 < \u2191m \u2192\n \u2200 (x : E),\n x \u2208 s \u2192\n HasFDerivWithinAt (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m_1)\n (ContinuousMultilinearMap.curryLeft (ftaylorSeriesWithin \ud835\udd5c f s x (Nat.succ m_1))) s x\n[PROOFSTEP]\nintro k hk y hy\n[GOAL]\ncase fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHcont : \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nHdiff : \u2200 (m : \u2115), \u2191m < n \u2192 DifferentiableOn \ud835\udd5c (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 n\nk : \u2115\nhk : \u2191k < \u2191m\ny : E\nhy : y \u2208 s\n\u22a2 HasFDerivWithinAt (fun x => ftaylorSeriesWithin \ud835\udd5c f s x k)\n (ContinuousMultilinearMap.curryLeft (ftaylorSeriesWithin \ud835\udd5c f s y (Nat.succ k))) s y\n[PROOFSTEP]\nconvert (Hdiff k (lt_of_lt_of_le hk hm) y hy).hasFDerivWithinAt\n[GOAL]\ncase cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHcont : \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nHdiff : \u2200 (m : \u2115), \u2191m < n \u2192 DifferentiableOn \ud835\udd5c (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 \u2200 (m_1 : \u2115), \u2191m_1 \u2264 \u2191m \u2192 ContinuousOn (fun x => ftaylorSeriesWithin \ud835\udd5c f s x m_1) s\n[PROOFSTEP]\nintro k hk\n[GOAL]\ncase cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nHcont : \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nHdiff : \u2200 (m : \u2115), \u2191m < n \u2192 DifferentiableOn \ud835\udd5c (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s\nx : E\nhx : x \u2208 s\nm : \u2115\nhm : \u2191m \u2264 n\nk : \u2115\nhk : \u2191k \u2264 \u2191m\n\u22a2 ContinuousOn (fun x => ftaylorSeriesWithin \ud835\udd5c f s x k) s\n[PROOFSTEP]\nexact Hcont k (le_trans hk hm)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhmn : \u2191m < n\nhs : UniqueDiffOn \ud835\udd5c (insert x s)\n\u22a2 DifferentiableWithinAt \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c m f s) s x\n[PROOFSTEP]\nrcases h.contDiffOn' (ENat.add_one_le_of_lt hmn) with \u27e8u, uo, xu, hu\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhmn : \u2191m < n\nhs : UniqueDiffOn \ud835\udd5c (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x \u2208 u\nhu : ContDiffOn \ud835\udd5c (\u2191(Add.add (\u2191m) 1)) f (insert x s \u2229 u)\n\u22a2 DifferentiableWithinAt \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c m f s) s x\n[PROOFSTEP]\nset t := insert x s \u2229 u\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhmn : \u2191m < n\nhs : UniqueDiffOn \ud835\udd5c (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x \u2208 u\nt : Set E := insert x s \u2229 u\nhu : ContDiffOn \ud835\udd5c (\u2191(Add.add (\u2191m) 1)) f t\n\u22a2 DifferentiableWithinAt \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c m f s) s x\n[PROOFSTEP]\nhave A : t =\u1da0[\ud835\udcdd[\u2260] x] s :=\n by\n simp only [set_eventuallyEq_iff_inf_principal, \u2190 nhdsWithin_inter']\n rw [\u2190 inter_assoc, nhdsWithin_inter_of_mem', \u2190 diff_eq_compl_inter, insert_diff_of_mem, diff_eq_compl_inter]\n exacts [rfl, mem_nhdsWithin_of_mem_nhds (uo.mem_nhds xu)]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhmn : \u2191m < n\nhs : UniqueDiffOn \ud835\udd5c (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x \u2208 u\nt : Set E := insert x s \u2229 u\nhu : ContDiffOn \ud835\udd5c (\u2191(Add.add (\u2191m) 1)) f t\n\u22a2 t =\u1da0[\ud835\udcdd[{x}\u1d9c] x] s\n[PROOFSTEP]\nsimp only [set_eventuallyEq_iff_inf_principal, \u2190 nhdsWithin_inter']\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhmn : \u2191m < n\nhs : UniqueDiffOn \ud835\udd5c (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x \u2208 u\nt : Set E := insert x s \u2229 u\nhu : ContDiffOn \ud835\udd5c (\u2191(Add.add (\u2191m) 1)) f t\n\u22a2 \ud835\udcdd[{x}\u1d9c \u2229 (insert x s \u2229 u)] x = \ud835\udcdd[{x}\u1d9c \u2229 s] x\n[PROOFSTEP]\nrw [\u2190 inter_assoc, nhdsWithin_inter_of_mem', \u2190 diff_eq_compl_inter, insert_diff_of_mem, diff_eq_compl_inter]\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhmn : \u2191m < n\nhs : UniqueDiffOn \ud835\udd5c (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x \u2208 u\nt : Set E := insert x s \u2229 u\nhu : ContDiffOn \ud835\udd5c (\u2191(Add.add (\u2191m) 1)) f t\n\u22a2 x \u2208 {x}\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhmn : \u2191m < n\nhs : UniqueDiffOn \ud835\udd5c (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x \u2208 u\nt : Set E := insert x s \u2229 u\nhu : ContDiffOn \ud835\udd5c (\u2191(Add.add (\u2191m) 1)) f t\n\u22a2 u \u2208 \ud835\udcdd[{x}\u1d9c \u2229 insert x s] x\n[PROOFSTEP]\nexacts [rfl, mem_nhdsWithin_of_mem_nhds (uo.mem_nhds xu)]\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhmn : \u2191m < n\nhs : UniqueDiffOn \ud835\udd5c (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x \u2208 u\nt : Set E := insert x s \u2229 u\nhu : ContDiffOn \ud835\udd5c (\u2191(Add.add (\u2191m) 1)) f t\nA : t =\u1da0[\ud835\udcdd[{x}\u1d9c] x] s\n\u22a2 DifferentiableWithinAt \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c m f s) s x\n[PROOFSTEP]\nhave B : iteratedFDerivWithin \ud835\udd5c m f s =\u1da0[\ud835\udcdd x] iteratedFDerivWithin \ud835\udd5c m f t :=\n iteratedFDerivWithin_eventually_congr_set' _ A.symm _\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhmn : \u2191m < n\nhs : UniqueDiffOn \ud835\udd5c (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x \u2208 u\nt : Set E := insert x s \u2229 u\nhu : ContDiffOn \ud835\udd5c (\u2191(Add.add (\u2191m) 1)) f t\nA : t =\u1da0[\ud835\udcdd[{x}\u1d9c] x] s\nB : iteratedFDerivWithin \ud835\udd5c m f s =\u1da0[\ud835\udcdd x] iteratedFDerivWithin \ud835\udd5c m f t\n\u22a2 DifferentiableWithinAt \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c m f s) s x\n[PROOFSTEP]\nhave C : DifferentiableWithinAt \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c m f t) t x :=\n hu.differentiableOn_iteratedFDerivWithin (Nat.cast_lt.2 m.lt_succ_self) (hs.inter uo) x \u27e8mem_insert _ _, xu\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhmn : \u2191m < n\nhs : UniqueDiffOn \ud835\udd5c (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x \u2208 u\nt : Set E := insert x s \u2229 u\nhu : ContDiffOn \ud835\udd5c (\u2191(Add.add (\u2191m) 1)) f t\nA : t =\u1da0[\ud835\udcdd[{x}\u1d9c] x] s\nB : iteratedFDerivWithin \ud835\udd5c m f s =\u1da0[\ud835\udcdd x] iteratedFDerivWithin \ud835\udd5c m f t\nC : DifferentiableWithinAt \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c m f t) t x\n\u22a2 DifferentiableWithinAt \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c m f s) s x\n[PROOFSTEP]\nrw [differentiableWithinAt_congr_set' _ A] at C \n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : \u2115\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhmn : \u2191m < n\nhs : UniqueDiffOn \ud835\udd5c (insert x s)\nu : Set E\nuo : IsOpen u\nxu : x \u2208 u\nt : Set E := insert x s \u2229 u\nhu : ContDiffOn \ud835\udd5c (\u2191(Add.add (\u2191m) 1)) f t\nA : t =\u1da0[\ud835\udcdd[{x}\u1d9c] x] s\nB : iteratedFDerivWithin \ud835\udd5c m f s =\u1da0[\ud835\udcdd x] iteratedFDerivWithin \ud835\udd5c m f t\nC : DifferentiableWithinAt \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c m f t) s x\n\u22a2 DifferentiableWithinAt \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c m f s) s x\n[PROOFSTEP]\nexact C.congr_of_eventuallyEq (B.filter_mono inf_le_left) B.self_of_nhds\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhf : DifferentiableOn \ud835\udd5c f s\nh : ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s\n\u22a2 ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhf : DifferentiableOn \ud835\udd5c f s\nh : ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s\nx : E\nhx : x \u2208 s\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x\n[PROOFSTEP]\nrw [contDiffWithinAt_succ_iff_hasFDerivWithinAt, insert_eq_of_mem hx]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhf : DifferentiableOn \ud835\udd5c f s\nh : ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s\nx : E\nhx : x \u2208 s\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n[PROOFSTEP]\nexact \u27e8s, self_mem_nhdsWithin, fderivWithin \ud835\udd5c f s, fun y hy => (hf y hy).hasFDerivWithinAt, h x hx\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\n\u22a2 ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s \u2194 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s\n[PROOFSTEP]\nrefine' \u27e8fun H => _, fun h => contDiffOn_succ_of_fderivWithin h.1 h.2\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nH : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\n\u22a2 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s\n[PROOFSTEP]\nrefine' \u27e8H.differentiableOn (WithTop.coe_le_coe.2 (Nat.le_add_left 1 n)), fun x hx => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nH : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s x\n[PROOFSTEP]\nrcases contDiffWithinAt_succ_iff_hasFDerivWithinAt.1 (H x hx) with \u27e8u, hu, f', hff', hf'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nH : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhff' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s x\n[PROOFSTEP]\nrcases mem_nhdsWithin.1 hu with \u27e8o, o_open, xo, ho\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nH : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhff' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : o \u2229 insert x s \u2286 u\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s x\n[PROOFSTEP]\nrw [inter_comm, insert_eq_of_mem hx] at ho \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nH : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhff' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s x\n[PROOFSTEP]\nhave := hf'.mono ho\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nH : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhff' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\nthis : ContDiffWithinAt \ud835\udd5c (\u2191n) f' (s \u2229 o) x\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s x\n[PROOFSTEP]\nrw [contDiffWithinAt_inter' (mem_nhdsWithin_of_mem_nhds (IsOpen.mem_nhds o_open xo))] at this \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nH : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhff' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\nthis : ContDiffWithinAt \ud835\udd5c (\u2191n) f' s x\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s x\n[PROOFSTEP]\napply this.congr_of_eventually_eq' _ hx\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nH : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhff' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\nthis : ContDiffWithinAt \ud835\udd5c (\u2191n) f' s x\n\u22a2 (fun y => fderivWithin \ud835\udd5c f s y) =\u1da0[\ud835\udcdd[s] x] f'\n[PROOFSTEP]\nhave : o \u2229 s \u2208 \ud835\udcdd[s] x := mem_nhdsWithin.2 \u27e8o, o_open, xo, Subset.refl _\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nH : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhff' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\nthis\u271d : ContDiffWithinAt \ud835\udd5c (\u2191n) f' s x\nthis : o \u2229 s \u2208 \ud835\udcdd[s] x\n\u22a2 (fun y => fderivWithin \ud835\udd5c f s y) =\u1da0[\ud835\udcdd[s] x] f'\n[PROOFSTEP]\nrw [inter_comm] at this \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nH : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhff' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\nthis\u271d : ContDiffWithinAt \ud835\udd5c (\u2191n) f' s x\nthis : s \u2229 o \u2208 \ud835\udcdd[s] x\n\u22a2 (fun y => fderivWithin \ud835\udd5c f s y) =\u1da0[\ud835\udcdd[s] x] f'\n[PROOFSTEP]\nrefine Filter.eventuallyEq_of_mem this fun y hy => ?_\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nH : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhff' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\nthis\u271d : ContDiffWithinAt \ud835\udd5c (\u2191n) f' s x\nthis : s \u2229 o \u2208 \ud835\udcdd[s] x\ny : E\nhy : y \u2208 s \u2229 o\n\u22a2 fderivWithin \ud835\udd5c f s y = f' y\n[PROOFSTEP]\nhave A : fderivWithin \ud835\udd5c f (s \u2229 o) y = f' y := ((hff' y (ho hy)).mono ho).fderivWithin (hs.inter o_open y hy)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nH : ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\nx : E\nhx : x \u2208 s\nu : Set E\nhu : u \u2208 \ud835\udcdd[insert x s] x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nhff' : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nhf' : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\no : Set E\no_open : IsOpen o\nxo : x \u2208 o\nho : s \u2229 o \u2286 u\nthis\u271d : ContDiffWithinAt \ud835\udd5c (\u2191n) f' s x\nthis : s \u2229 o \u2208 \ud835\udcdd[s] x\ny : E\nhy : y \u2208 s \u2229 o\nA : fderivWithin \ud835\udd5c f (s \u2229 o) y = f' y\n\u22a2 fderivWithin \ud835\udd5c f s y = f' y\n[PROOFSTEP]\nrwa [fderivWithin_inter (o_open.mem_nhds hy.2)] at A \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\n\u22a2 ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s \u2194 \u2203 f', ContDiffOn \ud835\udd5c (\u2191n) f' s \u2227 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n[PROOFSTEP]\nrw [contDiffOn_succ_iff_fderivWithin hs]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\n\u22a2 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s \u2194\n \u2203 f', ContDiffOn \ud835\udd5c (\u2191n) f' s \u2227 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8fderivWithin \ud835\udd5c f s, h.2, fun x hx => (h.1 x hx).hasFDerivWithinAt\u27e9, fun h => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nh : \u2203 f', ContDiffOn \ud835\udd5c (\u2191n) f' s \u2227 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s\n[PROOFSTEP]\nrcases h with \u27e8f', h1, h2\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nh1 : ContDiffOn \ud835\udd5c (\u2191n) f' s\nh2 : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s\n[PROOFSTEP]\nrefine' \u27e8fun x hx => (h2 x hx).differentiableWithinAt, fun x hx => _\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : UniqueDiffOn \ud835\udd5c s\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nh1 : ContDiffOn \ud835\udd5c (\u2191n) f' s\nh2 : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : x \u2208 s\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s x\n[PROOFSTEP]\nexact (h1 x hx).congr' (fun y hy => (h2 y hy).fderivWithin (hs y hy)) hx\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : IsOpen s\n\u22a2 ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s \u2194 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c f y) s\n[PROOFSTEP]\nrw [contDiffOn_succ_iff_fderivWithin hs.uniqueDiffOn]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : IsOpen s\n\u22a2 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s \u2194\n DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c f y) s\n[PROOFSTEP]\nexact Iff.rfl.and (contDiffOn_congr fun x hx \u21a6 fderivWithin_of_open hs hx)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : UniqueDiffOn \ud835\udd5c s\n\u22a2 ContDiffOn \ud835\udd5c \u22a4 f s \u2194 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c \u22a4 (fun y => fderivWithin \ud835\udd5c f s y) s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : UniqueDiffOn \ud835\udd5c s\n\u22a2 ContDiffOn \ud835\udd5c \u22a4 f s \u2192 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c \u22a4 (fun y => fderivWithin \ud835\udd5c f s y) s\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : UniqueDiffOn \ud835\udd5c s\nh : ContDiffOn \ud835\udd5c \u22a4 f s\n\u22a2 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c \u22a4 (fun y => fderivWithin \ud835\udd5c f s y) s\n[PROOFSTEP]\nrefine' \u27e8h.differentiableOn le_top, _\u27e9\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : UniqueDiffOn \ud835\udd5c s\nh : ContDiffOn \ud835\udd5c \u22a4 f s\n\u22a2 ContDiffOn \ud835\udd5c \u22a4 (fun y => fderivWithin \ud835\udd5c f s y) s\n[PROOFSTEP]\nrefine' contDiffOn_top.2 fun n => ((contDiffOn_succ_iff_fderivWithin hs).1 _).2\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : UniqueDiffOn \ud835\udd5c s\nh : ContDiffOn \ud835\udd5c \u22a4 f s\nn : \u2115\n\u22a2 ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s\n[PROOFSTEP]\nexact h.of_le le_top\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : UniqueDiffOn \ud835\udd5c s\n\u22a2 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c \u22a4 (fun y => fderivWithin \ud835\udd5c f s y) s \u2192 ContDiffOn \ud835\udd5c \u22a4 f s\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : UniqueDiffOn \ud835\udd5c s\nh : DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c \u22a4 (fun y => fderivWithin \ud835\udd5c f s y) s\n\u22a2 ContDiffOn \ud835\udd5c \u22a4 f s\n[PROOFSTEP]\nrefine' contDiffOn_top.2 fun n => _\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : UniqueDiffOn \ud835\udd5c s\nh : DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c \u22a4 (fun y => fderivWithin \ud835\udd5c f s y) s\nn : \u2115\n\u22a2 ContDiffOn \ud835\udd5c (\u2191n) f s\n[PROOFSTEP]\nhave A : (n : \u2115\u221e) \u2264 \u221e := le_top\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : UniqueDiffOn \ud835\udd5c s\nh : DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c \u22a4 (fun y => fderivWithin \ud835\udd5c f s y) s\nn : \u2115\nA : \u2191n \u2264 \u22a4\n\u22a2 ContDiffOn \ud835\udd5c (\u2191n) f s\n[PROOFSTEP]\napply ((contDiffOn_succ_iff_fderivWithin hs).2 \u27e8h.1, h.2.of_le A\u27e9).of_le\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : UniqueDiffOn \ud835\udd5c s\nh : DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c \u22a4 (fun y => fderivWithin \ud835\udd5c f s y) s\nn : \u2115\nA : \u2191n \u2264 \u22a4\n\u22a2 \u2191n \u2264 \u2191(n + 1)\n[PROOFSTEP]\nexact WithTop.coe_le_coe.2 (Nat.le_succ n)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : IsOpen s\n\u22a2 ContDiffOn \ud835\udd5c \u22a4 f s \u2194 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c \u22a4 (fun y => fderiv \ud835\udd5c f y) s\n[PROOFSTEP]\nrw [contDiffOn_top_iff_fderivWithin hs.uniqueDiffOn]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : IsOpen s\n\u22a2 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c \u22a4 (fun y => fderivWithin \ud835\udd5c f s y) s \u2194\n DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c \u22a4 (fun y => fderiv \ud835\udd5c f y) s\n[PROOFSTEP]\nexact Iff.rfl.and <| contDiffOn_congr fun x hx \u21a6 fderivWithin_of_open hs hx\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nhmn : m + 1 \u2264 n\n\u22a2 ContDiffOn \ud835\udd5c m (fun y => fderivWithin \ud835\udd5c f s y) s\n[PROOFSTEP]\ncases' m with m\n[GOAL]\ncase none\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nn : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nhmn : none + 1 \u2264 n\n\u22a2 ContDiffOn \ud835\udd5c none (fun y => fderivWithin \ud835\udd5c f s y) s\n[PROOFSTEP]\nchange \u221e + 1 \u2264 n at hmn \n[GOAL]\ncase none\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nn : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nhmn : \u22a4 + 1 \u2264 n\n\u22a2 ContDiffOn \ud835\udd5c none (fun y => fderivWithin \ud835\udd5c f s y) s\n[PROOFSTEP]\nhave : n = \u221e := by simpa using hmn\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nn : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nhmn : \u22a4 + 1 \u2264 n\n\u22a2 n = \u22a4\n[PROOFSTEP]\nsimpa using hmn\n[GOAL]\ncase none\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nn : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nhmn : \u22a4 + 1 \u2264 n\nthis : n = \u22a4\n\u22a2 ContDiffOn \ud835\udd5c none (fun y => fderivWithin \ud835\udd5c f s y) s\n[PROOFSTEP]\nrw [this] at hf \n[GOAL]\ncase none\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nn : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : ContDiffOn \ud835\udd5c \u22a4 f s\nhs : UniqueDiffOn \ud835\udd5c s\nhmn : \u22a4 + 1 \u2264 n\nthis : n = \u22a4\n\u22a2 ContDiffOn \ud835\udd5c none (fun y => fderivWithin \ud835\udd5c f s y) s\n[PROOFSTEP]\nexact ((contDiffOn_top_iff_fderivWithin hs).1 hf).2\n[GOAL]\ncase some\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nn : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhmn : some m + 1 \u2264 n\n\u22a2 ContDiffOn \ud835\udd5c (some m) (fun y => fderivWithin \ud835\udd5c f s y) s\n[PROOFSTEP]\nchange (m.succ : \u2115\u221e) \u2264 n at hmn \n[GOAL]\ncase some\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nn : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : ContDiffOn \ud835\udd5c n f s\nhs : UniqueDiffOn \ud835\udd5c s\nm : \u2115\nhmn : \u2191(Nat.succ m) \u2264 n\n\u22a2 ContDiffOn \ud835\udd5c (some m) (fun y => fderivWithin \ud835\udd5c f s y) s\n[PROOFSTEP]\nexact ((contDiffOn_succ_iff_fderivWithin hs).1 (hf.of_le hmn)).2\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpTo n f p\nx : E\n\u22a2 p x 0 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)) (f x)\n[PROOFSTEP]\nrw [\u2190 h.zero_eq x]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpTo n f p\nx : E\n\u22a2 p x 0 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)) (ContinuousMultilinearMap.uncurry0 (p x 0))\n[PROOFSTEP]\nexact (p x 0).uncurry0_curry0.symm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpToOn n f p univ \u2194 HasFTaylorSeriesUpTo n f p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpToOn n f p univ \u2192 HasFTaylorSeriesUpTo n f p\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 HasFTaylorSeriesUpTo n f p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 \u2200 (x : E), ContinuousMultilinearMap.uncurry0 (p x 0) = f x\n[PROOFSTEP]\nexact fun x => H.zero_eq x (mem_univ x)\n[GOAL]\ncase mp.fderiv\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 \u2200 (m : \u2115), \u2191m < n \u2192 \u2200 (x : E), HasFDerivAt (fun y => p y m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) x\n[PROOFSTEP]\nintro m hm x\n[GOAL]\ncase mp.fderiv\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m < n\nx : E\n\u22a2 HasFDerivAt (fun y => p y m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) x\n[PROOFSTEP]\nrw [\u2190 hasFDerivWithinAt_univ]\n[GOAL]\ncase mp.fderiv\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m < n\nx : E\n\u22a2 HasFDerivWithinAt (fun y => p y m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) univ x\n[PROOFSTEP]\nexact H.fderivWithin m hm x (mem_univ x)\n[GOAL]\ncase mp.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 Continuous fun x => p x m\n[PROOFSTEP]\nintro m hm\n[GOAL]\ncase mp.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 Continuous fun x => p x m\n[PROOFSTEP]\nrw [continuous_iff_continuousOn_univ]\n[GOAL]\ncase mp.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) univ\n[PROOFSTEP]\nexact H.cont m hm\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpTo n f p \u2192 HasFTaylorSeriesUpToOn n f p univ\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 HasFTaylorSeriesUpToOn n f p univ\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 \u2200 (x : E), x \u2208 univ \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\n[PROOFSTEP]\nexact fun x _ => H.zero_eq x\n[GOAL]\ncase mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 \u2200 (m : \u2115),\n \u2191m < n \u2192\n \u2200 (x : E),\n x \u2208 univ \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) univ x\n[PROOFSTEP]\nintro m hm x _\n[GOAL]\ncase mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m < n\nx : E\na\u271d : x \u2208 univ\n\u22a2 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) univ x\n[PROOFSTEP]\nrw [hasFDerivWithinAt_univ]\n[GOAL]\ncase mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m < n\nx : E\na\u271d : x \u2208 univ\n\u22a2 HasFDerivAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) x\n[PROOFSTEP]\nexact H.fderiv m hm x\n[GOAL]\ncase mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => p x m) univ\n[PROOFSTEP]\nintro m hm\n[GOAL]\ncase mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) univ\n[PROOFSTEP]\nrw [\u2190 continuous_iff_continuousOn_univ]\n[GOAL]\ncase mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 Continuous fun x => p x m\n[PROOFSTEP]\nexact H.cont m hm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpTo n f p\nhmn : m \u2264 n\n\u22a2 HasFTaylorSeriesUpTo m f p\n[PROOFSTEP]\nrw [\u2190 hasFTaylorSeriesUpToOn_univ_iff] at h \u22a2\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p univ\nhmn : m \u2264 n\n\u22a2 HasFTaylorSeriesUpToOn m f p univ\n[PROOFSTEP]\nexact h.of_le hmn\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpTo n f p\n\u22a2 Continuous f\n[PROOFSTEP]\nrw [\u2190 hasFTaylorSeriesUpToOn_univ_iff] at h \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 Continuous f\n[PROOFSTEP]\nrw [continuous_iff_continuousOn_univ]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 ContinuousOn f univ\n[PROOFSTEP]\nexact h.continuousOn\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpTo 0 f p \u2194 Continuous f \u2227 \u2200 (x : E), ContinuousMultilinearMap.uncurry0 (p x 0) = f x\n[PROOFSTEP]\nsimp [hasFTaylorSeriesUpToOn_univ_iff.symm, continuous_iff_continuousOn_univ, hasFTaylorSeriesUpToOn_zero_iff]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpTo \u22a4 f p \u2194 \u2200 (n : \u2115), HasFTaylorSeriesUpTo (\u2191n) f p\n[PROOFSTEP]\nsimp only [\u2190 hasFTaylorSeriesUpToOn_univ_iff, hasFTaylorSeriesUpToOn_top_iff]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpTo \u22a4 f p \u2194\n (\u2200 (x : E), ContinuousMultilinearMap.uncurry0 (p x 0) = f x) \u2227\n \u2200 (m : \u2115) (x : E), HasFDerivAt (fun y => p y m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) x\n[PROOFSTEP]\nsimp only [\u2190 hasFTaylorSeriesUpToOn_univ_iff, hasFTaylorSeriesUpToOn_top_iff', mem_univ, forall_true_left,\n hasFDerivWithinAt_univ]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpTo n f p\nhn : 1 \u2264 n\nx : E\n\u22a2 HasFDerivAt f (\u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1)) x\n[PROOFSTEP]\nrw [\u2190 hasFDerivWithinAt_univ]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpTo n f p\nhn : 1 \u2264 n\nx : E\n\u22a2 HasFDerivWithinAt f (\u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1)) univ x\n[PROOFSTEP]\nexact (hasFTaylorSeriesUpToOn_univ_iff.2 h).hasFDerivWithinAt hn (mem_univ _)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 HasFTaylorSeriesUpTo (\u2191(n + 1)) f p \u2194\n (\u2200 (x : E), ContinuousMultilinearMap.uncurry0 (p x 0) = f x) \u2227\n (\u2200 (x : E), HasFDerivAt (fun y => p y 0) (ContinuousMultilinearMap.curryLeft (p x 1)) x) \u2227\n HasFTaylorSeriesUpTo (\u2191n) (fun x => \u2191(continuousMultilinearCurryFin1 \ud835\udd5c E F) (p x 1)) fun x =>\n FormalMultilinearSeries.shift (p x)\n[PROOFSTEP]\nsimp only [hasFTaylorSeriesUpToOn_succ_iff_right, \u2190 hasFTaylorSeriesUpToOn_univ_iff, mem_univ, forall_true_left,\n hasFDerivWithinAt_univ]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiffAt \ud835\udd5c \u22a4 f x \u2194 \u2200 (n : \u2115), ContDiffAt \ud835\udd5c (\u2191n) f x\n[PROOFSTEP]\nsimp [\u2190 contDiffWithinAt_univ, contDiffWithinAt_top]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffWithinAt \ud835\udd5c n f s x\nhx : s \u2208 \ud835\udcdd x\n\u22a2 ContDiffAt \ud835\udd5c n f x\n[PROOFSTEP]\nrwa [ContDiffAt, \u2190 contDiffWithinAt_inter hx, univ_inter]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffAt \ud835\udd5c n f x\nhg : f\u2081 =\u1da0[\ud835\udcdd x] f\n\u22a2 f\u2081 =\u1da0[\ud835\udcdd[univ] x] f\n[PROOFSTEP]\nrwa [nhdsWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffAt \ud835\udd5c n f x\n\u22a2 ContinuousAt f x\n[PROOFSTEP]\nsimpa [continuousWithinAt_univ] using h.continuousWithinAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : ContDiffAt \ud835\udd5c n f x\nhn : 1 \u2264 n\n\u22a2 DifferentiableAt \ud835\udd5c f x\n[PROOFSTEP]\nsimpa [hn, differentiableWithinAt_univ] using h.differentiableWithinAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 ContDiffAt \ud835\udd5c (\u2191(n + 1)) f x \u2194\n \u2203 f', (\u2203 u, u \u2208 \ud835\udcdd x \u2227 \u2200 (x : E), x \u2208 u \u2192 HasFDerivAt f (f' x) x) \u2227 ContDiffAt \ud835\udd5c (\u2191n) f' x\n[PROOFSTEP]\nrw [\u2190 contDiffWithinAt_univ, contDiffWithinAt_succ_iff_hasFDerivWithinAt]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 (\u2203 u,\n u \u2208 \ud835\udcdd[insert x univ] x \u2227\n \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x) \u2194\n \u2203 f', (\u2203 u, u \u2208 \ud835\udcdd x \u2227 \u2200 (x : E), x \u2208 u \u2192 HasFDerivAt f (f' x) x) \u2227 ContDiffAt \ud835\udd5c (\u2191n) f' x\n[PROOFSTEP]\nsimp only [nhdsWithin_univ, exists_prop, mem_univ, insert_eq_of_mem]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 (\u2203 u, u \u2208 \ud835\udcdd x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x) \u2194\n \u2203 f', (\u2203 u, u \u2208 \ud835\udcdd x \u2227 \u2200 (x : E), x \u2208 u \u2192 HasFDerivAt f (f' x) x) \u2227 ContDiffAt \ud835\udd5c (\u2191n) f' x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 (\u2203 u, u \u2208 \ud835\udcdd x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x) \u2192\n \u2203 f', (\u2203 u, u \u2208 \ud835\udcdd x \u2227 \u2200 (x : E), x \u2208 u \u2192 HasFDerivAt f (f' x) x) \u2227 ContDiffAt \ud835\udd5c (\u2191n) f' x\n[PROOFSTEP]\nrintro \u27e8u, H, f', h_fderiv, h_cont_diff\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nH : u \u2208 \ud835\udcdd x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nh_fderiv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nh_cont_diff : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n\u22a2 \u2203 f', (\u2203 u, u \u2208 \ud835\udcdd x \u2227 \u2200 (x : E), x \u2208 u \u2192 HasFDerivAt f (f' x) x) \u2227 ContDiffAt \ud835\udd5c (\u2191n) f' x\n[PROOFSTEP]\nrcases mem_nhds_iff.mp H with \u27e8t, htu, ht, hxt\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nH : u \u2208 \ud835\udcdd x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nh_fderiv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nh_cont_diff : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nt : Set E\nhtu : t \u2286 u\nht : IsOpen t\nhxt : x \u2208 t\n\u22a2 \u2203 f', (\u2203 u, u \u2208 \ud835\udcdd x \u2227 \u2200 (x : E), x \u2208 u \u2192 HasFDerivAt f (f' x) x) \u2227 ContDiffAt \ud835\udd5c (\u2191n) f' x\n[PROOFSTEP]\nrefine' \u27e8f', \u27e8t, _\u27e9, h_cont_diff.contDiffAt H\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nH : u \u2208 \ud835\udcdd x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nh_fderiv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nh_cont_diff : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nt : Set E\nhtu : t \u2286 u\nht : IsOpen t\nhxt : x \u2208 t\n\u22a2 t \u2208 \ud835\udcdd x \u2227 \u2200 (x : E), x \u2208 t \u2192 HasFDerivAt f (f' x) x\n[PROOFSTEP]\nrefine' \u27e8mem_nhds_iff.mpr \u27e8t, Subset.rfl, ht, hxt\u27e9, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nH : u \u2208 \ud835\udcdd x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nh_fderiv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nh_cont_diff : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nt : Set E\nhtu : t \u2286 u\nht : IsOpen t\nhxt : x \u2208 t\n\u22a2 \u2200 (x : E), x \u2208 t \u2192 HasFDerivAt f (f' x) x\n[PROOFSTEP]\nintro y hyt\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nH : u \u2208 \ud835\udcdd x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nh_fderiv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nh_cont_diff : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nt : Set E\nhtu : t \u2286 u\nht : IsOpen t\nhxt : x \u2208 t\ny : E\nhyt : y \u2208 t\n\u22a2 HasFDerivAt f (f' y) y\n[PROOFSTEP]\nrefine' (h_fderiv y (htu hyt)).hasFDerivAt _\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t\u271d u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nu : Set E\nH : u \u2208 \ud835\udcdd x\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nh_fderiv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\nh_cont_diff : ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\nt : Set E\nhtu : t \u2286 u\nht : IsOpen t\nhxt : x \u2208 t\ny : E\nhyt : y \u2208 t\n\u22a2 u \u2208 \ud835\udcdd y\n[PROOFSTEP]\nexact mem_nhds_iff.mpr \u27e8t, htu, ht, hyt\u27e9\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 (\u2203 f', (\u2203 u, u \u2208 \ud835\udcdd x \u2227 \u2200 (x : E), x \u2208 u \u2192 HasFDerivAt f (f' x) x) \u2227 ContDiffAt \ud835\udd5c (\u2191n) f' x) \u2192\n \u2203 u, u \u2208 \ud835\udcdd x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n[PROOFSTEP]\nrintro \u27e8f', \u27e8u, H, h_fderiv\u27e9, h_cont_diff\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nh_cont_diff : ContDiffAt \ud835\udd5c (\u2191n) f' x\nu : Set E\nH : u \u2208 \ud835\udcdd x\nh_fderiv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivAt f (f' x) x\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x\n[PROOFSTEP]\nrefine' \u27e8u, H, f', _, h_cont_diff.contDiffWithinAt\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nh_cont_diff : ContDiffAt \ud835\udd5c (\u2191n) f' x\nu : Set E\nH : u \u2208 \ud835\udcdd x\nh_fderiv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivAt f (f' x) x\n\u22a2 \u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x\n[PROOFSTEP]\nintro x hxu\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nf' : E \u2192 E \u2192L[\ud835\udd5c] F\nh_cont_diff : ContDiffAt \ud835\udd5c (\u2191n) f' x\u271d\nu : Set E\nH : u \u2208 \ud835\udcdd x\u271d\nh_fderiv : \u2200 (x : E), x \u2208 u \u2192 HasFDerivAt f (f' x) x\nx : E\nhxu : x \u2208 u\n\u22a2 HasFDerivWithinAt f (f' x) u x\n[PROOFSTEP]\nexact (h_fderiv x hxu).hasFDerivWithinAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh : ContDiffAt \ud835\udd5c (\u2191n) f x\n\u22a2 \u2200\u1da0 (y : E) in \ud835\udcdd x, ContDiffAt \ud835\udd5c (\u2191n) f y\n[PROOFSTEP]\nsimpa [nhdsWithin_univ] using ContDiffWithinAt.eventually h\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiffOn \ud835\udd5c n f univ \u2194 ContDiff \ud835\udd5c n f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiffOn \ud835\udd5c n f univ \u2192 ContDiff \ud835\udd5c n f\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContDiffOn \ud835\udd5c n f univ\n\u22a2 ContDiff \ud835\udd5c n f\n[PROOFSTEP]\nuse ftaylorSeriesWithin \ud835\udd5c f univ\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContDiffOn \ud835\udd5c n f univ\n\u22a2 HasFTaylorSeriesUpTo n f (ftaylorSeriesWithin \ud835\udd5c f univ)\n[PROOFSTEP]\nrw [\u2190 hasFTaylorSeriesUpToOn_univ_iff]\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : ContDiffOn \ud835\udd5c n f univ\n\u22a2 HasFTaylorSeriesUpToOn n f (ftaylorSeriesWithin \ud835\udd5c f univ) univ\n[PROOFSTEP]\nexact H.ftaylorSeriesWithin uniqueDiffOn_univ\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiff \ud835\udd5c n f \u2192 ContDiffOn \ud835\udd5c n f univ\n[PROOFSTEP]\nrintro \u27e8p, hp\u27e9 x _ m hm\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : HasFTaylorSeriesUpTo n f p\nx : E\na\u271d : x \u2208 univ\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[insert x univ] x \u2227 \u2203 p, HasFTaylorSeriesUpToOn (\u2191m) f p u\n[PROOFSTEP]\nexact \u27e8univ, Filter.univ_sets _, p, (hp.hasFTaylorSeriesUpToOn univ).of_le hm\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiff \ud835\udd5c n f \u2194 \u2200 (x : E), ContDiffAt \ud835\udd5c n f x\n[PROOFSTEP]\nsimp [\u2190 contDiffOn_univ, ContDiffOn, ContDiffAt]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiff \ud835\udd5c \u22a4 f \u2194 \u2200 (n : \u2115), ContDiff \ud835\udd5c (\u2191n) f\n[PROOFSTEP]\nsimp [contDiffOn_univ.symm, contDiffOn_top]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 (\u2200 (n : \u2115\u221e), ContDiff \ud835\udd5c n f) \u2194 \u2200 (n : \u2115), ContDiff \ud835\udd5c (\u2191n) f\n[PROOFSTEP]\nsimp only [\u2190 contDiffOn_univ, contDiffOn_all_iff_nat]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiff \ud835\udd5c 0 f \u2194 Continuous f\n[PROOFSTEP]\nrw [\u2190 contDiffOn_univ, continuous_iff_continuousOn_univ]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiffOn \ud835\udd5c 0 f univ \u2194 ContinuousOn f univ\n[PROOFSTEP]\nexact contDiffOn_zero\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiffAt \ud835\udd5c 0 f x \u2194 \u2203 u, u \u2208 \ud835\udcdd x \u2227 ContinuousOn f u\n[PROOFSTEP]\nrw [\u2190 contDiffWithinAt_univ]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiffWithinAt \ud835\udd5c 0 f univ x \u2194 \u2203 u, u \u2208 \ud835\udcdd x \u2227 ContinuousOn f u\n[PROOFSTEP]\nsimp [contDiffWithinAt_zero, nhdsWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiffAt \ud835\udd5c 1 f x \u2194 \u2203 f' u, u \u2208 \ud835\udcdd x \u2227 ContinuousOn f' u \u2227 \u2200 (x : E), x \u2208 u \u2192 HasFDerivAt f (f' x) x\n[PROOFSTEP]\nsimp_rw [show (1 : \u2115\u221e) = (0 + 1 : \u2115) from (zero_add 1).symm, contDiffAt_succ_iff_hasFDerivAt,\n show ((0 : \u2115) : \u2115\u221e) = 0 from rfl, contDiffAt_zero, exists_mem_and_iff antitone_bforall antitone_continuousOn,\n and_comm]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiff \ud835\udd5c n f \u2194 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContDiff \ud835\udd5c (\u2191m) f\n[PROOFSTEP]\nsimp_rw [\u2190 contDiffOn_univ]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiffOn \ud835\udd5c n f univ \u2194 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContDiffOn \ud835\udd5c (\u2191m) f univ\n[PROOFSTEP]\nexact contDiffOn_iff_forall_nat_le\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 ContDiff \ud835\udd5c (\u2191(n + 1)) f \u2194 \u2203 f', ContDiff \ud835\udd5c (\u2191n) f' \u2227 \u2200 (x : E), HasFDerivAt f (f' x) x\n[PROOFSTEP]\nsimp only [\u2190 contDiffOn_univ, \u2190 hasFDerivWithinAt_univ, contDiffOn_succ_iff_has_fderiv_within uniqueDiffOn_univ,\n Set.mem_univ, forall_true_left]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 \u2016iteratedFDeriv \ud835\udd5c 0 f x\u2016 = \u2016f x\u2016\n[PROOFSTEP]\nrw [iteratedFDeriv_zero_eq_comp, comp_apply, LinearIsometryEquiv.norm_map]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f) =\n \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F)) \u2218 iteratedFDeriv \ud835\udd5c (n + 1) f\n[PROOFSTEP]\nrw [iteratedFDeriv_succ_eq_comp_left]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f) =\n \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F)) \u2218\n \u2191(continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F) \u2218 fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nx : E\n\u22a2 fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f) x =\n (\u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F)) \u2218\n \u2191(continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F) \u2218 fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f))\n x\n[PROOFSTEP]\nsimp only [Function.comp_apply, LinearIsometryEquiv.symm_apply_apply]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 tsupport (iteratedFDeriv \ud835\udd5c n f) \u2286 tsupport f\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 tsupport (iteratedFDeriv \ud835\udd5c Nat.zero f) \u2286 tsupport f\n[PROOFSTEP]\nrw [iteratedFDeriv_zero_eq_comp]\n[GOAL]\ncase zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 tsupport (\u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 \ud835\udd5c E F)) \u2218 f) \u2286 tsupport f\n[PROOFSTEP]\nexact closure_minimal ((support_comp_subset (LinearIsometryEquiv.map_zero _) _).trans subset_closure) isClosed_closure\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nIH : tsupport (iteratedFDeriv \ud835\udd5c n f) \u2286 tsupport f\n\u22a2 tsupport (iteratedFDeriv \ud835\udd5c (Nat.succ n) f) \u2286 tsupport f\n[PROOFSTEP]\nrw [iteratedFDeriv_succ_eq_comp_left]\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nIH : tsupport (iteratedFDeriv \ud835\udd5c n f) \u2286 tsupport f\n\u22a2 tsupport (\u2191(continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F) \u2218 fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f)) \u2286 tsupport f\n[PROOFSTEP]\nexact\n closure_minimal ((support_comp_subset (LinearIsometryEquiv.map_zero _) _).trans ((support_fderiv_subset \ud835\udd5c).trans IH))\n isClosed_closure\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 \u2016fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f) x\u2016 = \u2016iteratedFDeriv \ud835\udd5c (n + 1) f x\u2016\n[PROOFSTEP]\nrw [iteratedFDeriv_succ_eq_comp_left, comp_apply, LinearIsometryEquiv.norm_map]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 iteratedFDerivWithin \ud835\udd5c n f univ = iteratedFDeriv \ud835\udd5c n f\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 iteratedFDerivWithin \ud835\udd5c Nat.zero f univ = iteratedFDeriv \ud835\udd5c Nat.zero f\n[PROOFSTEP]\next x\n[GOAL]\ncase zero.h.H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d\u00b9 x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nx : E\nx\u271d : Fin Nat.zero \u2192 E\n\u22a2 \u2191(iteratedFDerivWithin \ud835\udd5c Nat.zero f univ x) x\u271d = \u2191(iteratedFDeriv \ud835\udd5c Nat.zero f x) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nIH : iteratedFDerivWithin \ud835\udd5c n f univ = iteratedFDeriv \ud835\udd5c n f\n\u22a2 iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f univ = iteratedFDeriv \ud835\udd5c (Nat.succ n) f\n[PROOFSTEP]\next x m\n[GOAL]\ncase succ.h.H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nIH : iteratedFDerivWithin \ud835\udd5c n f univ = iteratedFDeriv \ud835\udd5c n f\nx : E\nm : Fin (Nat.succ n) \u2192 E\n\u22a2 \u2191(iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f univ x) m = \u2191(iteratedFDeriv \ud835\udd5c (Nat.succ n) f x) m\n[PROOFSTEP]\nrw [iteratedFDeriv_succ_apply_left, iteratedFDerivWithin_succ_apply_left, IH, fderivWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhs : IsOpen s\n\u22a2 EqOn (iteratedFDerivWithin \ud835\udd5c n f s) (iteratedFDeriv \ud835\udd5c n f) s\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : IsOpen s\n\u22a2 EqOn (iteratedFDerivWithin \ud835\udd5c Nat.zero f s) (iteratedFDeriv \ud835\udd5c Nat.zero f) s\n[PROOFSTEP]\nintro x _\n[GOAL]\ncase zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : IsOpen s\nx : E\na\u271d : x \u2208 s\n\u22a2 iteratedFDerivWithin \ud835\udd5c Nat.zero f s x = iteratedFDeriv \ud835\udd5c Nat.zero f x\n[PROOFSTEP]\next1\n[GOAL]\ncase zero.H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d\u00b9 x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : IsOpen s\nx : E\na\u271d : x \u2208 s\nx\u271d : Fin Nat.zero \u2192 E\n\u22a2 \u2191(iteratedFDerivWithin \ud835\udd5c Nat.zero f s x) x\u271d = \u2191(iteratedFDeriv \ud835\udd5c Nat.zero f x) x\u271d\n[PROOFSTEP]\nsimp only [Nat.zero_eq, iteratedFDerivWithin_zero_apply, iteratedFDeriv_zero_apply]\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : IsOpen s\nn : \u2115\nIH : EqOn (iteratedFDerivWithin \ud835\udd5c n f s) (iteratedFDeriv \ud835\udd5c n f) s\n\u22a2 EqOn (iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f s) (iteratedFDeriv \ud835\udd5c (Nat.succ n) f) s\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : IsOpen s\nn : \u2115\nIH : EqOn (iteratedFDerivWithin \ud835\udd5c n f s) (iteratedFDeriv \ud835\udd5c n f) s\nx : E\nhx : x \u2208 s\n\u22a2 iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f s x = iteratedFDeriv \ud835\udd5c (Nat.succ n) f x\n[PROOFSTEP]\nrw [iteratedFDeriv_succ_eq_comp_left, iteratedFDerivWithin_succ_eq_comp_left]\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : IsOpen s\nn : \u2115\nIH : EqOn (iteratedFDerivWithin \ud835\udd5c n f s) (iteratedFDeriv \ud835\udd5c n f) s\nx : E\nhx : x \u2208 s\n\u22a2 (\u2191(continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F) \u2218 fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f s) s) x =\n (\u2191(continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F) \u2218 fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f)) x\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : IsOpen s\nn : \u2115\nIH : EqOn (iteratedFDerivWithin \ud835\udd5c n f s) (iteratedFDeriv \ud835\udd5c n f) s\nx : E\nhx : x \u2208 s\n\u22a2 \u2191(continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F) (fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f s) s x) =\n \u2191(continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun x => E) F) (fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f) x)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase succ.h.e_6.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : IsOpen s\nn : \u2115\nIH : EqOn (iteratedFDerivWithin \ud835\udd5c n f s) (iteratedFDeriv \ud835\udd5c n f) s\nx : E\nhx : x \u2208 s\n\u22a2 fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f s) s x = fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f) x\n[PROOFSTEP]\nrw [fderivWithin_of_open hs hx]\n[GOAL]\ncase succ.h.e_6.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : IsOpen s\nn : \u2115\nIH : EqOn (iteratedFDerivWithin \ud835\udd5c n f s) (iteratedFDeriv \ud835\udd5c n f) s\nx : E\nhx : x \u2208 s\n\u22a2 fderiv \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f s) x = fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f) x\n[PROOFSTEP]\napply Filter.EventuallyEq.fderiv_eq\n[GOAL]\ncase succ.h.e_6.h.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : IsOpen s\nn : \u2115\nIH : EqOn (iteratedFDerivWithin \ud835\udd5c n f s) (iteratedFDeriv \ud835\udd5c n f) s\nx : E\nhx : x \u2208 s\n\u22a2 iteratedFDerivWithin \ud835\udd5c n f s =\u1da0[\ud835\udcdd x] iteratedFDeriv \ud835\udd5c n f\n[PROOFSTEP]\nfilter_upwards [hs.mem_nhds hx]\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhs : IsOpen s\nn : \u2115\nIH : EqOn (iteratedFDerivWithin \ud835\udd5c n f s) (iteratedFDeriv \ud835\udd5c n f) s\nx : E\nhx : x \u2208 s\n\u22a2 \u2200 (a : E), a \u2208 s \u2192 iteratedFDerivWithin \ud835\udd5c n f s a = iteratedFDeriv \ud835\udd5c n f a\n[PROOFSTEP]\nexact IH\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ftaylorSeriesWithin \ud835\udd5c f univ = ftaylorSeries \ud835\udd5c f\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nx : E\n\u22a2 ftaylorSeriesWithin \ud835\udd5c f univ x = ftaylorSeries \ud835\udd5c f x\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nx : E\nn : \u2115\n\u22a2 ftaylorSeriesWithin \ud835\udd5c f univ x n = ftaylorSeries \ud835\udd5c f x n\n[PROOFSTEP]\nchange iteratedFDerivWithin \ud835\udd5c n f univ x = iteratedFDeriv \ud835\udd5c n f x\n[GOAL]\ncase h.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nx : E\nn : \u2115\n\u22a2 iteratedFDerivWithin \ud835\udd5c n f univ x = iteratedFDeriv \ud835\udd5c n f x\n[PROOFSTEP]\nrw [iteratedFDerivWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nm : Fin (n + 1) \u2192 E\n\u22a2 \u2191(iteratedFDeriv \ud835\udd5c (n + 1) f x) m = \u2191(\u2191(iteratedFDeriv \ud835\udd5c n (fun y => fderiv \ud835\udd5c f y) x) (init m)) (m (last n))\n[PROOFSTEP]\nrw [\u2190 iteratedFDerivWithin_univ, \u2190 iteratedFDerivWithin_univ, \u2190 fderivWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nm : Fin (n + 1) \u2192 E\n\u22a2 \u2191(iteratedFDerivWithin \ud835\udd5c (n + 1) f univ x) m =\n \u2191(\u2191(iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f univ y) univ x) (init m)) (m (last n))\n[PROOFSTEP]\nexact iteratedFDerivWithin_succ_apply_right uniqueDiffOn_univ (mem_univ _) _\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 iteratedFDeriv \ud835\udd5c (n + 1) f x =\n (\u2191(continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F) \u2218 iteratedFDeriv \ud835\udd5c n fun y => fderiv \ud835\udd5c f y) x\n[PROOFSTEP]\next m\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nm : Fin (n + 1) \u2192 E\n\u22a2 \u2191(iteratedFDeriv \ud835\udd5c (n + 1) f x) m =\n \u2191((\u2191(continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F) \u2218 iteratedFDeriv \ud835\udd5c n fun y => fderiv \ud835\udd5c f y) x) m\n[PROOFSTEP]\nrw [iteratedFDeriv_succ_apply_right]\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nm : Fin (n + 1) \u2192 E\n\u22a2 \u2191(\u2191(iteratedFDeriv \ud835\udd5c n (fun y => fderiv \ud835\udd5c f y) x) (init m)) (m (last n)) =\n \u2191((\u2191(continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F) \u2218 iteratedFDeriv \ud835\udd5c n fun y => fderiv \ud835\udd5c f y) x) m\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 \u2016iteratedFDeriv \ud835\udd5c n (fderiv \ud835\udd5c f) x\u2016 = \u2016iteratedFDeriv \ud835\udd5c (n + 1) f x\u2016\n[PROOFSTEP]\nrw [iteratedFDeriv_succ_eq_comp_right, comp_apply, LinearIsometryEquiv.norm_map]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : Fin 1 \u2192 E\n\u22a2 \u2191(iteratedFDeriv \ud835\udd5c 1 f x) m = \u2191(fderiv \ud835\udd5c f x) (m 0)\n[PROOFSTEP]\nrw [iteratedFDeriv_succ_apply_right, iteratedFDeriv_zero_apply]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nm : Fin 1 \u2192 E\n\u22a2 \u2191(fderiv \ud835\udd5c f x) (m (last 0)) = \u2191(fderiv \ud835\udd5c f x) (m 0)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiff \ud835\udd5c n f \u2194 HasFTaylorSeriesUpTo n f (ftaylorSeries \ud835\udd5c f)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiff \ud835\udd5c n f \u2192 HasFTaylorSeriesUpTo n f (ftaylorSeries \ud835\udd5c f)\n[PROOFSTEP]\nrw [\u2190 contDiffOn_univ, \u2190 hasFTaylorSeriesUpToOn_univ_iff, \u2190 ftaylorSeriesWithin_univ]\n[GOAL]\ncase mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiffOn \ud835\udd5c n f univ \u2192 HasFTaylorSeriesUpToOn n f (ftaylorSeriesWithin \ud835\udd5c f univ) univ\n[PROOFSTEP]\nexact fun h => ContDiffOn.ftaylorSeriesWithin h uniqueDiffOn_univ\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpTo n f (ftaylorSeries \ud835\udd5c f) \u2192 ContDiff \ud835\udd5c n f\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : HasFTaylorSeriesUpTo n f (ftaylorSeries \ud835\udd5c f)\n\u22a2 ContDiff \ud835\udd5c n f\n[PROOFSTEP]\nexact \u27e8ftaylorSeries \ud835\udd5c f, h\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiff \ud835\udd5c n f \u2194\n (\u2200 (m : \u2115), \u2191m \u2264 n \u2192 Continuous fun x => iteratedFDeriv \ud835\udd5c m f x) \u2227\n \u2200 (m : \u2115), \u2191m < n \u2192 Differentiable \ud835\udd5c fun x => iteratedFDeriv \ud835\udd5c m f x\n[PROOFSTEP]\nsimp [contDiffOn_univ.symm, continuous_iff_continuousOn_univ, differentiableOn_univ.symm, iteratedFDerivWithin_univ,\n contDiffOn_iff_continuousOn_differentiableOn uniqueDiffOn_univ]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 ContDiff \ud835\udd5c (\u2191(n + 1)) f \u2194 Differentiable \ud835\udd5c f \u2227 ContDiff \ud835\udd5c \u2191n fun y => fderiv \ud835\udd5c f y\n[PROOFSTEP]\nsimp only [\u2190 contDiffOn_univ, \u2190 differentiableOn_univ, \u2190 fderivWithin_univ,\n contDiffOn_succ_iff_fderivWithin uniqueDiffOn_univ]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiff \ud835\udd5c \u22a4 f \u2194 Differentiable \ud835\udd5c f \u2227 ContDiff \ud835\udd5c \u22a4 fun y => fderiv \ud835\udd5c f y\n[PROOFSTEP]\nsimp only [\u2190 contDiffOn_univ, \u2190 differentiableOn_univ, \u2190 fderivWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 ContDiffOn \ud835\udd5c \u22a4 f univ \u2194 DifferentiableOn \ud835\udd5c f univ \u2227 ContDiffOn \ud835\udd5c \u22a4 (fun y => fderivWithin \ud835\udd5c f univ y) univ\n[PROOFSTEP]\nrw [contDiffOn_top_iff_fderivWithin uniqueDiffOn_univ]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.ContDiffDef", "llama_tokens": 178880, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7879311956428947, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.43079204579682345}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d : Preorder \u03b9\nx : \u03b1\n\u22a2 x \u2208 notConvergentSeq f g n j \u2194 \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)\n[PROOFSTEP]\nsimp_rw [notConvergentSeq, Set.mem_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d : Preorder \u03b9\nx : \u03b1\n\u22a2 (\u2203 i i_1, x \u2208 {x | 1 / (\u2191n + 1) < dist (f i x) (g x)}) \u2194 \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (s \u2229 \u22c2 (j : \u03b9), notConvergentSeq f g n j) = 0\n[PROOFSTEP]\nsimp_rw [Metric.tendsto_atTop, ae_iff] at hfg \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\n\u22a2 \u2191\u2191\u03bc (s \u2229 \u22c2 (j : \u03b9), notConvergentSeq f g n j) = 0\n[PROOFSTEP]\nrw [\u2190 nonpos_iff_eq_zero, \u2190 hfg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\n\u22a2 \u2191\u2191\u03bc (s \u2229 \u22c2 (j : \u03b9), notConvergentSeq f g n j) \u2264\n \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)}\n[PROOFSTEP]\nrefine' measure_mono fun x => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\n\u22a2 x \u2208 s \u2229 \u22c2 (j : \u03b9), notConvergentSeq f g n j \u2192\n x \u2208 {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)}\n[PROOFSTEP]\nsimp only [Set.mem_inter_iff, Set.mem_iInter, ge_iff_le, mem_notConvergentSeq_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\n\u22a2 (x \u2208 s \u2227 \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)) \u2192\n x \u2208 {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), N \u2264 n \u2192 dist (f n a) (g a) < \u03b5)}\n[PROOFSTEP]\npush_neg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\n\u22a2 (x \u2208 s \u2227 \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)) \u2192\n x \u2208 {a | a \u2208 s \u2227 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (N : \u03b9), \u2203 n, N \u2264 n \u2227 \u03b5 \u2264 dist (f n a) (g a)}\n[PROOFSTEP]\nrintro \u27e8hmem, hx\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\nhmem : x \u2208 s\nhx : \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)\n\u22a2 x \u2208 {a | a \u2208 s \u2227 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (N : \u03b9), \u2203 n, N \u2264 n \u2227 \u03b5 \u2264 dist (f n a) (g a)}\n[PROOFSTEP]\nrefine' \u27e8hmem, 1 / (n + 1 : \u211d), Nat.one_div_pos_of_nat, fun N => _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\nhmem : x \u2208 s\nhx : \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)\nN : \u03b9\n\u22a2 \u2203 n_1, N \u2264 n_1 \u2227 1 / (\u2191n + 1) \u2264 dist (f n_1 x) (g x)\n[PROOFSTEP]\nobtain \u27e8n, hn\u2081, hn\u2082\u27e9 := hx N\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d\u00b9 : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn\u271d : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\nhmem : x \u2208 s\nhx : \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n\u271d + 1) < dist (f k x) (g x)\nN n : \u03b9\nhn\u2081 : N \u2264 n\nhn\u2082 : 1 / (\u2191n\u271d + 1) < dist (f n x) (g x)\n\u22a2 \u2203 n, N \u2264 n \u2227 1 / (\u2191n\u271d + 1) \u2264 dist (f n x) (g x)\n[PROOFSTEP]\nexact \u27e8n, hn\u2081, hn\u2082.le\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\n\u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\ncases' isEmpty_or_nonempty \u03b9 with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\nh : IsEmpty \u03b9\n\u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : (fun j => \u03bc (s \u2229 notConvergentSeq f g n j)) = fun j => 0 := by simp only [eq_iff_true_of_subsingleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\nh : IsEmpty \u03b9\n\u22a2 (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) = fun j => 0\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\nh : IsEmpty \u03b9\nthis : (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) = fun j => 0\n\u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\nh : IsEmpty \u03b9\nthis : (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) = fun j => 0\n\u22a2 Tendsto (fun j => 0) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact tendsto_const_nhds\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\nh : Nonempty \u03b9\n\u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 measure_inter_notConvergentSeq_eq_zero hfg n, Set.inter_iInter]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\nh : Nonempty \u03b9\n\u22a2 Tendsto (fun j => \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j)) atTop\n (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (i : \u03b9), s \u2229 notConvergentSeq (fun n x => f n x) (fun x => g x) n i)))\n[PROOFSTEP]\nrefine'\n tendsto_measure_iInter (fun n => hsm.inter <| notConvergentSeq_measurableSet hf hg)\n (fun k l hkl => Set.inter_subset_inter_right _ <| notConvergentSeq_antitone hkl)\n \u27e8h.some, (lt_of_le_of_lt (measure_mono <| Set.inter_subset_left _ _) (lt_top_iff_ne_top.2 hs)).ne\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\n\u22a2 \u2203 j, \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j) \u2264 ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)\n[PROOFSTEP]\nhave \u27e8N, hN\u27e9 :=\n (ENNReal.tendsto_atTop ENNReal.zero_ne_top).1 (measure_notConvergentSeq_tendsto_zero hf hg hsm hs hfg n)\n (ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n))\n (by\n rw [gt_iff_lt, ENNReal.ofReal_pos]\n exact mul_pos h\u03b5 (pow_pos (by norm_num) n))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\n\u22a2 ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n) > 0\n[PROOFSTEP]\nrw [gt_iff_lt, ENNReal.ofReal_pos]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\n\u22a2 0 < \u03b5 * 2\u207b\u00b9 ^ n\n[PROOFSTEP]\nexact mul_pos h\u03b5 (pow_pos (by norm_num) n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\n\u22a2 0 < 2\u207b\u00b9\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\nN : \u03b9\nhN :\n \u2200 (n_1 : \u03b9),\n n_1 \u2265 N \u2192\n \u2191\u2191\u03bc (s \u2229 notConvergentSeq (fun n => f n) g n n_1) \u2208\n Icc (0 - ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)) (0 + ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n))\n\u22a2 \u2203 j, \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j) \u2264 ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)\n[PROOFSTEP]\nrw [zero_add] at hN \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\nN : \u03b9\nhN :\n \u2200 (n_1 : \u03b9),\n n_1 \u2265 N \u2192\n \u2191\u2191\u03bc (s \u2229 notConvergentSeq (fun n => f n) g n n_1) \u2208\n Icc (0 - ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)) (ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n))\n\u22a2 \u2203 j, \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j) \u2264 ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)\n[PROOFSTEP]\nexact \u27e8N, (hN N le_rfl).2\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2191\u2191\u03bc (iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg) \u2264 ENNReal.ofReal \u03b5\n[PROOFSTEP]\nrefine'\n le_trans (measure_iUnion_le _)\n (le_trans (ENNReal.tsum_le_tsum <| notConvergentSeqLTIndex_spec (half_pos h\u03b5) hf hg hsm hs hfg) _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2211' (a : \u2115), ENNReal.ofReal (\u03b5 / 2 * 2\u207b\u00b9 ^ a) \u2264 ENNReal.ofReal \u03b5\n[PROOFSTEP]\nsimp_rw [ENNReal.ofReal_mul (half_pos h\u03b5).le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2211' (a : \u2115), ENNReal.ofReal (\u03b5 / 2) * ENNReal.ofReal (2\u207b\u00b9 ^ a) \u2264 ENNReal.ofReal \u03b5\n[PROOFSTEP]\nrw [ENNReal.tsum_mul_left, \u2190 ENNReal.ofReal_tsum_of_nonneg, inv_eq_one_div, tsum_geometric_two, \u2190\n ENNReal.ofReal_mul (half_pos h\u03b5).le, div_mul_cancel \u03b5 two_ne_zero]\n[GOAL]\ncase hf_nonneg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2200 (n : \u2115), 0 \u2264 2\u207b\u00b9 ^ n\n[PROOFSTEP]\nexact fun n => pow_nonneg (by norm_num) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\n\u22a2 0 \u2264 2\u207b\u00b9\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 Summable fun i => 2\u207b\u00b9 ^ i\n[PROOFSTEP]\nrw [inv_eq_one_div]\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 Summable fun i => (1 / 2) ^ i\n[PROOFSTEP]\nexact summable_geometric_two\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg \u2286 s\n[PROOFSTEP]\nrw [iUnionNotConvergentSeq, \u2190 Set.inter_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 s \u2229 \u22c3 (i : \u2115), notConvergentSeq (fun n => f n) g i (notConvergentSeqLTIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg i) \u2286 s\n[PROOFSTEP]\nexact Set.inter_subset_left _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 TendstoUniformlyOn f g atTop (s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg)\n[PROOFSTEP]\nrw [Metric.tendstoUniformlyOn_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2200 (\u03b5_1 : \u211d),\n \u03b5_1 > 0 \u2192\n \u2200\u1da0 (n : \u03b9) in atTop, \u2200 (x : \u03b1), x \u2208 s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg \u2192 dist (g x) (f n x) < \u03b5_1\n[PROOFSTEP]\nintro \u03b4 h\u03b4\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\n\u22a2 \u2200\u1da0 (n : \u03b9) in atTop, \u2200 (x : \u03b1), x \u2208 s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg \u2192 dist (g x) (f n x) < \u03b4\n[PROOFSTEP]\nobtain \u27e8N, hN\u27e9 := exists_nat_one_div_lt h\u03b4\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\n\u22a2 \u2200\u1da0 (n : \u03b9) in atTop, \u2200 (x : \u03b1), x \u2208 s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg \u2192 dist (g x) (f n x) < \u03b4\n[PROOFSTEP]\nrw [eventually_atTop]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\n\u22a2 \u2203 a, \u2200 (b : \u03b9), b \u2265 a \u2192 \u2200 (x : \u03b1), x \u2208 s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg \u2192 dist (g x) (f b x) < \u03b4\n[PROOFSTEP]\nrefine' \u27e8Egorov.notConvergentSeqLTIndex (half_pos h\u03b5) hf hg hsm hs hfg N, fun n hn x hx => _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLTIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhx : x \u2208 s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg\n\u22a2 dist (g x) (f n x) < \u03b4\n[PROOFSTEP]\nsimp only [Set.mem_diff, Egorov.iUnionNotConvergentSeq, not_exists, Set.mem_iUnion, Set.mem_inter_iff, not_and,\n exists_and_left] at hx \n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLTIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhx :\n x \u2208 s \u2227\n (x \u2208 s \u2192\n \u2200 (x_1 : \u2115),\n \u00acx \u2208 notConvergentSeq (fun n => f n) g x_1 (notConvergentSeqLTIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg x_1))\n\u22a2 dist (g x) (f n x) < \u03b4\n[PROOFSTEP]\nobtain \u27e8hxs, hx\u27e9 := hx\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLTIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhxs : x \u2208 s\nhx :\n x \u2208 s \u2192\n \u2200 (x_1 : \u2115),\n \u00acx \u2208 notConvergentSeq (fun n => f n) g x_1 (notConvergentSeqLTIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg x_1)\n\u22a2 dist (g x) (f n x) < \u03b4\n[PROOFSTEP]\nspecialize hx hxs N\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLTIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhxs : x \u2208 s\nhx : \u00acx \u2208 notConvergentSeq (fun n => f n) g N (notConvergentSeqLTIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N)\n\u22a2 dist (g x) (f n x) < \u03b4\n[PROOFSTEP]\nrw [Egorov.mem_notConvergentSeq_iff] at hx \n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLTIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhxs : x \u2208 s\nhx : \u00ac\u2203 k x_1, 1 / (\u2191N + 1) < dist (f k x) (g x)\n\u22a2 dist (g x) (f n x) < \u03b4\n[PROOFSTEP]\npush_neg at hx \n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLTIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhxs : x \u2208 s\nhx : \u2200 (k : \u03b9), notConvergentSeqLTIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N \u2264 k \u2192 dist (f k x) (g x) \u2264 1 / (\u2191N + 1)\n\u22a2 dist (g x) (f n x) < \u03b4\n[PROOFSTEP]\nrw [dist_comm]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLTIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhxs : x \u2208 s\nhx : \u2200 (k : \u03b9), notConvergentSeqLTIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N \u2264 k \u2192 dist (f k x) (g x) \u2264 1 / (\u2191N + 1)\n\u22a2 dist (f n x) (g x) < \u03b4\n[PROOFSTEP]\nexact lt_of_le_of_lt (hx n hn) hN\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : SemilatticeSup \u03b9\ninst\u271d\u00b3 : Nonempty \u03b9\ninst\u271d\u00b2 : Countable \u03b9\n\u03b3 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 t, MeasurableSet t \u2227 \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn f g atTop t\u1d9c\n[PROOFSTEP]\nhave \u27e8t, _, ht, htendsto\u27e9 :=\n tendstoUniformlyOn_of_ae_tendsto hf hg MeasurableSet.univ (measure_ne_top \u03bc Set.univ)\n (by filter_upwards [hfg] with _ htendsto _ using htendsto) h\u03b5\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : SemilatticeSup \u03b9\ninst\u271d\u00b3 : Nonempty \u03b9\ninst\u271d\u00b2 : Countable \u03b9\n\u03b3 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 univ \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n[PROOFSTEP]\nfilter_upwards [hfg] with _ htendsto _ using htendsto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : SemilatticeSup \u03b9\ninst\u271d\u00b3 : Nonempty \u03b9\ninst\u271d\u00b2 : Countable \u03b9\n\u03b3 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nt : Set \u03b1\nw\u271d : t \u2286 univ\nht : MeasurableSet t\nhtendsto : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn (fun n => f n) g atTop (univ \\ t)\n\u22a2 \u2203 t, MeasurableSet t \u2227 \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn f g atTop t\u1d9c\n[PROOFSTEP]\nrefine' \u27e8_, ht, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : SemilatticeSup \u03b9\ninst\u271d\u00b3 : Nonempty \u03b9\ninst\u271d\u00b2 : Countable \u03b9\n\u03b3 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nt : Set \u03b1\nw\u271d : t \u2286 univ\nht : MeasurableSet t\nhtendsto : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn (fun n => f n) g atTop (univ \\ t)\n\u22a2 \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn f g atTop t\u1d9c\n[PROOFSTEP]\nrwa [Set.compl_eq_univ_diff]\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.Egorov", "llama_tokens": 16019, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334527, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.4306453942318432}} {"text": "[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\nh_meas : NullMeasurableSet s\nh_exists : \u2200 (x : \u03b1), \u2203! g, g \u2022 x \u2208 s\na b : G\nhab : a \u2260 b\nx : \u03b1\nhxa : x \u2208 (fun g => g \u2022 s) a\nhxb : x \u2208 (fun g => g \u2022 s) b\n\u22a2 False\n[PROOFSTEP]\nrw [mem_smul_set_iff_inv_smul_mem] at hxa hxb \n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\nh_meas : NullMeasurableSet s\nh_exists : \u2200 (x : \u03b1), \u2203! g, g \u2022 x \u2208 s\na b : G\nhab : a \u2260 b\nx : \u03b1\nhxa : a\u207b\u00b9 \u2022 x \u2208 s\nhxb : b\u207b\u00b9 \u2022 x \u2208 s\n\u22a2 False\n[PROOFSTEP]\nexact hab (inv_injective <| (h_exists x).unique hxa hxb)\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : Group H\ninst\u271d\u2076 : MulAction G \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MulAction H \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Countable G\nh_meas : NullMeasurableSet s\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\nh_measure_univ_le : \u2191\u2191\u03bc univ \u2264 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 s)\naedisjoint : Pairwise (AEDisjoint \u03bc on fun g => g \u2022 s)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 g, g \u2022 x \u2208 s\n[PROOFSTEP]\nreplace h_meas : \u2200 g : G, NullMeasurableSet (g \u2022 s) \u03bc := fun g => by rw [\u2190 inv_inv g, \u2190 preimage_smul];\n exact h_meas.preimage (h_qmp g\u207b\u00b9)\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : Group H\ninst\u271d\u2076 : MulAction G \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MulAction H \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Countable G\nh_meas : NullMeasurableSet s\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\nh_measure_univ_le : \u2191\u2191\u03bc univ \u2264 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 s)\naedisjoint : Pairwise (AEDisjoint \u03bc on fun g => g \u2022 s)\ng : G\n\u22a2 NullMeasurableSet (g \u2022 s)\n[PROOFSTEP]\nrw [\u2190 inv_inv g, \u2190 preimage_smul]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : Group H\ninst\u271d\u2076 : MulAction G \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MulAction H \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Countable G\nh_meas : NullMeasurableSet s\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\nh_measure_univ_le : \u2191\u2191\u03bc univ \u2264 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 s)\naedisjoint : Pairwise (AEDisjoint \u03bc on fun g => g \u2022 s)\ng : G\n\u22a2 NullMeasurableSet ((fun x => g\u207b\u00b9 \u2022 x) \u207b\u00b9' s)\n[PROOFSTEP]\nexact h_meas.preimage (h_qmp g\u207b\u00b9)\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : Group H\ninst\u271d\u2076 : MulAction G \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MulAction H \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Countable G\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\nh_measure_univ_le : \u2191\u2191\u03bc univ \u2264 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 s)\naedisjoint : Pairwise (AEDisjoint \u03bc on fun g => g \u2022 s)\nh_meas : \u2200 (g : G), NullMeasurableSet (g \u2022 s)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 g, g \u2022 x \u2208 s\n[PROOFSTEP]\nhave h_meas' : NullMeasurableSet {a | \u2203 g : G, g \u2022 a \u2208 s} \u03bc := by rw [\u2190 iUnion_smul_eq_setOf_exists];\n exact .iUnion h_meas\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : Group H\ninst\u271d\u2076 : MulAction G \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MulAction H \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Countable G\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\nh_measure_univ_le : \u2191\u2191\u03bc univ \u2264 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 s)\naedisjoint : Pairwise (AEDisjoint \u03bc on fun g => g \u2022 s)\nh_meas : \u2200 (g : G), NullMeasurableSet (g \u2022 s)\n\u22a2 NullMeasurableSet {a | \u2203 g, g \u2022 a \u2208 s}\n[PROOFSTEP]\nrw [\u2190 iUnion_smul_eq_setOf_exists]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : Group H\ninst\u271d\u2076 : MulAction G \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MulAction H \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Countable G\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\nh_measure_univ_le : \u2191\u2191\u03bc univ \u2264 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 s)\naedisjoint : Pairwise (AEDisjoint \u03bc on fun g => g \u2022 s)\nh_meas : \u2200 (g : G), NullMeasurableSet (g \u2022 s)\n\u22a2 NullMeasurableSet (\u22c3 (g : G), g \u2022 s)\n[PROOFSTEP]\nexact .iUnion h_meas\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : Group H\ninst\u271d\u2076 : MulAction G \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MulAction H \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Countable G\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\nh_measure_univ_le : \u2191\u2191\u03bc univ \u2264 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 s)\naedisjoint : Pairwise (AEDisjoint \u03bc on fun g => g \u2022 s)\nh_meas : \u2200 (g : G), NullMeasurableSet (g \u2022 s)\nh_meas' : NullMeasurableSet {a | \u2203 g, g \u2022 a \u2208 s}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 g, g \u2022 x \u2208 s\n[PROOFSTEP]\nrw [ae_iff_measure_eq h_meas', \u2190 iUnion_smul_eq_setOf_exists]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : Group H\ninst\u271d\u2076 : MulAction G \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MulAction H \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Countable G\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\nh_measure_univ_le : \u2191\u2191\u03bc univ \u2264 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 s)\naedisjoint : Pairwise (AEDisjoint \u03bc on fun g => g \u2022 s)\nh_meas : \u2200 (g : G), NullMeasurableSet (g \u2022 s)\nh_meas' : NullMeasurableSet {a | \u2203 g, g \u2022 a \u2208 s}\n\u22a2 \u2191\u2191\u03bc (\u22c3 (g : G), g \u2022 s) = \u2191\u2191\u03bc univ\n[PROOFSTEP]\nrefine' le_antisymm (measure_mono <| subset_univ _) _\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : Group H\ninst\u271d\u2076 : MulAction G \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MulAction H \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Countable G\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\nh_measure_univ_le : \u2191\u2191\u03bc univ \u2264 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 s)\naedisjoint : Pairwise (AEDisjoint \u03bc on fun g => g \u2022 s)\nh_meas : \u2200 (g : G), NullMeasurableSet (g \u2022 s)\nh_meas' : NullMeasurableSet {a | \u2203 g, g \u2022 a \u2208 s}\n\u22a2 \u2191\u2191\u03bc univ \u2264 \u2191\u2191\u03bc (\u22c3 (g : G), g \u2022 s)\n[PROOFSTEP]\nrw [measure_iUnion\u2080 aedisjoint h_meas]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : Group H\ninst\u271d\u2076 : MulAction G \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MulAction H \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Countable G\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\nh_measure_univ_le : \u2191\u2191\u03bc univ \u2264 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 s)\naedisjoint : Pairwise (AEDisjoint \u03bc on fun g => g \u2022 s)\nh_meas : \u2200 (g : G), NullMeasurableSet (g \u2022 s)\nh_meas' : NullMeasurableSet {a | \u2203 g, g \u2022 a \u2208 s}\n\u22a2 \u2191\u2191\u03bc univ \u2264 \u2211' (i : G), \u2191\u2191\u03bc (i \u2022 s)\n[PROOFSTEP]\nexact h_measure_univ_le\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\ne : G \u2192 H\nhe : Bijective e\nhef : \u2200 (g : G), Semiconj f (fun x => e g \u2022 x) fun x => g \u2022 x\nx : \u03b2\nx\u271d : \u2203 g, g \u2022 f x \u2208 s\ng : G\nhg : g \u2022 f x \u2208 s\n\u22a2 e g \u2022 x \u2208 f \u207b\u00b9' s\n[PROOFSTEP]\nrwa [mem_preimage, hef g x]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\ne : G \u2192 H\nhe : Bijective e\nhef : \u2200 (g : G), Semiconj f (fun x => e g \u2022 x) fun x => g \u2022 x\na b : H\nhab : a \u2260 b\n\u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b\n[PROOFSTEP]\nlift e to G \u2243 H using he\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\na b : H\nhab : a \u2260 b\ne : G \u2243 H\nhef : \u2200 (g : G), Semiconj f (fun x => \u2191e g \u2022 x) fun x => g \u2022 x\n\u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b\n[PROOFSTEP]\nhave : (e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (e.symm b\u207b\u00b9)\u207b\u00b9 := by simp [hab]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\na b : H\nhab : a \u2260 b\ne : G \u2243 H\nhef : \u2200 (g : G), Semiconj f (fun x => \u2191e g \u2022 x) fun x => g \u2022 x\n\u22a2 (\u2191e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (\u2191e.symm b\u207b\u00b9)\u207b\u00b9\n[PROOFSTEP]\nsimp [hab]\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\na b : H\nhab : a \u2260 b\ne : G \u2243 H\nhef : \u2200 (g : G), Semiconj f (fun x => \u2191e g \u2022 x) fun x => g \u2022 x\nthis : (\u2191e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (\u2191e.symm b\u207b\u00b9)\u207b\u00b9\n\u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b\n[PROOFSTEP]\nhave := (h.aedisjoint this).preimage hf\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\na b : H\nhab : a \u2260 b\ne : G \u2243 H\nhef : \u2200 (g : G), Semiconj f (fun x => \u2191e g \u2022 x) fun x => g \u2022 x\nthis\u271d : (\u2191e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (\u2191e.symm b\u207b\u00b9)\u207b\u00b9\nthis : AEDisjoint \u03bd (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm a\u207b\u00b9)\u207b\u00b9) (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm b\u207b\u00b9)\u207b\u00b9)\n\u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b\n[PROOFSTEP]\nsimp only [Semiconj] at hef \n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b2 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\na b : H\nhab : a \u2260 b\ne : G \u2243 H\nhef : \u2200 (g : G) (x : \u03b2), f (\u2191e g \u2022 x) = g \u2022 f x\nthis\u271d : (\u2191e.symm a\u207b\u00b9)\u207b\u00b9 \u2260 (\u2191e.symm b\u207b\u00b9)\u207b\u00b9\nthis : AEDisjoint \u03bd (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm a\u207b\u00b9)\u207b\u00b9) (f \u207b\u00b9' (fun g => g \u2022 s) (\u2191e.symm b\u207b\u00b9)\u207b\u00b9)\n\u22a2 (AEDisjoint \u03bd on fun g => g \u2022 f \u207b\u00b9' s) a b\n[PROOFSTEP]\nsimpa only [onFun, \u2190 preimage_smul_inv, preimage_preimage, \u2190 hef, e.apply_symm_apply, inv_inv] using this\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b1 \u2243 \u03b2\nhf : QuasiMeasurePreserving \u2191f.symm\ne : H \u2243 G\nhef : \u2200 (g : H), Semiconj (\u2191f) (fun x => \u2191e g \u2022 x) fun x => g \u2022 x\n\u22a2 IsFundamentalDomain H (\u2191f '' s)\n[PROOFSTEP]\nrw [f.image_eq_preimage]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b1 \u2243 \u03b2\nhf : QuasiMeasurePreserving \u2191f.symm\ne : H \u2243 G\nhef : \u2200 (g : H), Semiconj (\u2191f) (fun x => \u2191e g \u2022 x) fun x => g \u2022 x\n\u22a2 IsFundamentalDomain H (\u2191f.symm \u207b\u00b9' s)\n[PROOFSTEP]\nrefine' h.preimage_of_equiv hf e.symm.bijective fun g x => _\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b1 \u2243 \u03b2\nhf : QuasiMeasurePreserving \u2191f.symm\ne : H \u2243 G\nhef : \u2200 (g : H), Semiconj (\u2191f) (fun x => \u2191e g \u2022 x) fun x => g \u2022 x\ng : G\nx : \u03b2\n\u22a2 \u2191f.symm ((fun x => \u2191e.symm g \u2022 x) x) = (fun x => g \u2022 x) (\u2191f.symm x)\n[PROOFSTEP]\nrcases f.surjective x with \u27e8x, rfl\u27e9\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MulAction H \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nh : IsFundamentalDomain G s\nf : \u03b1 \u2243 \u03b2\nhf : QuasiMeasurePreserving \u2191f.symm\ne : H \u2243 G\nhef : \u2200 (g : H), Semiconj (\u2191f) (fun x => \u2191e g \u2022 x) fun x => g \u2022 x\ng : G\nx : \u03b1\n\u22a2 \u2191f.symm ((fun x => \u2191e.symm g \u2022 x) (\u2191f x)) = (fun x => g \u2022 x) (\u2191f.symm (\u2191f x))\n[PROOFSTEP]\nrw [\u2190 hef _ _, f.symm_apply_apply, f.symm_apply_apply, e.apply_symm_apply]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : Group H\ninst\u271d\u2077 : MulAction G \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MulAction H \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nh : IsFundamentalDomain G s\ng g' : G\n\u22a2 (fun g' => g * g' * g\u207b\u00b9) ((fun g' => g\u207b\u00b9 * g' * g) g') = g'\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : Group H\ninst\u271d\u2077 : MulAction G \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MulAction H \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nh : IsFundamentalDomain G s\ng g' : G\n\u22a2 (fun g' => g\u207b\u00b9 * g' * g) ((fun g' => g * g' * g\u207b\u00b9) g') = g'\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : Group H\ninst\u271d\u2077 : MulAction G \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MulAction H \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nh : IsFundamentalDomain G s\ng g' : G\nx : \u03b1\n\u22a2 \u2191(MulAction.toPerm g)\n ((fun x =>\n \u2191{ toFun := fun g' => g\u207b\u00b9 * g' * g, invFun := fun g' => g * g' * g\u207b\u00b9,\n left_inv := (_ : \u2200 (g' : G), g * (g\u207b\u00b9 * g' * g) * g\u207b\u00b9 = g'),\n right_inv := (_ : \u2200 (g' : G), g\u207b\u00b9 * (g * g' * g\u207b\u00b9) * g = g') }\n g' \u2022\n x)\n x) =\n (fun x => g' \u2022 x) (\u2191(MulAction.toPerm g) x)\n[PROOFSTEP]\nsimp [smul_smul, mul_assoc]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nh : IsFundamentalDomain G s\nh\u03bd : \u03bd \u226a \u03bc\n\u22a2 (sum fun g => Measure.restrict \u03bd (g \u2022 s)) = \u03bd\n[PROOFSTEP]\nrw [\u2190 restrict_iUnion_ae (h.aedisjoint.mono fun i j h => h\u03bd h) fun g => (h.nullMeasurableSet_smul g).mono_ac h\u03bd,\n restrict_congr_set (h\u03bd h.iUnion_smul_ae_eq), restrict_univ]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nh : IsFundamentalDomain G s\nh\u03bd : \u03bd \u226a \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bd = \u2211' (g : G), \u222b\u207b (x : \u03b1) in g \u2022 s, f x \u2202\u03bd\n[PROOFSTEP]\nrw [\u2190 lintegral_sum_measure, h.sum_restrict_of_ac h\u03bd]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nh : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nt : Set \u03b1\n\u22a2 \u2211' (g : G), \u222b\u207b (x : \u03b1) in g \u2022 s, f x \u2202Measure.restrict \u03bc t = \u2211' (g : G), \u222b\u207b (x : \u03b1) in t \u2229 g \u2022 s, f x \u2202\u03bc\n[PROOFSTEP]\nsimp only [h.restrict_restrict, inter_comm]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nh : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nt : Set \u03b1\n\u22a2 \u2211' (g : G), \u222b\u207b (x : \u03b1) in t \u2229 g\u207b\u00b9 \u2022 s, f x \u2202\u03bc = \u2211' (g : G), \u222b\u207b (x : \u03b1) in g\u207b\u00b9 \u2022 (g \u2022 t \u2229 s), f x \u2202\u03bc\n[PROOFSTEP]\nsimp only [smul_set_inter, inv_smul_smul]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nh : IsFundamentalDomain G s\nh\u03bd : \u03bd \u226a \u03bc\nt : Set \u03b1\n\u22a2 \u2191\u2191\u03bd t = \u2211' (g : G), \u2191\u2191\u03bd (t \u2229 g \u2022 s)\n[PROOFSTEP]\nhave H : \u03bd.restrict t \u226a \u03bc := Measure.restrict_le_self.absolutelyContinuous.trans h\u03bd\n[GOAL]\nG : Type u_1\nH\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\u271d\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H\u271d \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nh : IsFundamentalDomain G s\nh\u03bd : \u03bd \u226a \u03bc\nt : Set \u03b1\nH : Measure.restrict \u03bd t \u226a \u03bc\n\u22a2 \u2191\u2191\u03bd t = \u2211' (g : G), \u2191\u2191\u03bd (t \u2229 g \u2022 s)\n[PROOFSTEP]\nsimpa only [set_lintegral_one, Pi.one_def, Measure.restrict_apply\u2080 ((h.nullMeasurableSet_smul _).mono_ac H),\n inter_comm] using h.lintegral_eq_tsum_of_ac H 1\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nh : IsFundamentalDomain G s\nt : Set \u03b1\n\u22a2 \u2191\u2191\u03bc t = \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 t \u2229 s)\n[PROOFSTEP]\nsimpa only [set_lintegral_one] using h.set_lintegral_eq_tsum' (fun _ => 1) t\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nh : IsFundamentalDomain G s\nt : Set \u03b1\nht : \u2200 (g : G), g \u2022 t = t\nhts : \u2191\u2191\u03bc (t \u2229 s) = 0\n\u22a2 \u2191\u2191\u03bc t = 0\n[PROOFSTEP]\nrw [measure_eq_tsum h]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nh : IsFundamentalDomain G s\nt : Set \u03b1\nht : \u2200 (g : G), g \u2022 t = t\nhts : \u2191\u2191\u03bc (t \u2229 s) = 0\n\u22a2 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 t \u2229 s) = 0\n[PROOFSTEP]\nsimp [ht, hts]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b9 : Group G\ninst\u271d\u00b9\u2070 : Group H\ninst\u271d\u2079 : MulAction G \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MulAction H \u03b2\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : MeasurableSMul G \u03b1\ninst\u271d\u00b2 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b9 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d : Finite G\nh : IsFundamentalDomain G s\nt : Set \u03b1\nht : \u2200 (g : G), g \u2022 t =\u1da0[ae \u03bc] t\n\u22a2 \u2191\u2191\u03bc t = Nat.card G \u2022 \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nhaveI : Fintype G := Fintype.ofFinite G\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b9 : Group G\ninst\u271d\u00b9\u2070 : Group H\ninst\u271d\u2079 : MulAction G \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MulAction H \u03b2\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : MeasurableSMul G \u03b1\ninst\u271d\u00b2 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b9 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d : Finite G\nh : IsFundamentalDomain G s\nt : Set \u03b1\nht : \u2200 (g : G), g \u2022 t =\u1da0[ae \u03bc] t\nthis : Fintype G\n\u22a2 \u2191\u2191\u03bc t = Nat.card G \u2022 \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nrw [h.measure_eq_tsum]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b9 : Group G\ninst\u271d\u00b9\u2070 : Group H\ninst\u271d\u2079 : MulAction G \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MulAction H \u03b2\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : MeasurableSMul G \u03b1\ninst\u271d\u00b2 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b9 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d : Finite G\nh : IsFundamentalDomain G s\nt : Set \u03b1\nht : \u2200 (g : G), g \u2022 t =\u1da0[ae \u03bc] t\nthis : Fintype G\n\u22a2 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 t \u2229 s) = Nat.card G \u2022 \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nreplace ht : \u2200 g : G, (g \u2022 t \u2229 s : Set \u03b1) =\u1d50[\u03bc] (t \u2229 s : Set \u03b1) := fun g => ae_eq_set_inter (ht g) (ae_eq_refl s)\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b9 : Group G\ninst\u271d\u00b9\u2070 : Group H\ninst\u271d\u2079 : MulAction G \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MulAction H \u03b2\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : MeasurableSMul G \u03b1\ninst\u271d\u00b2 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b9 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d : Finite G\nh : IsFundamentalDomain G s\nt : Set \u03b1\nthis : Fintype G\nht : \u2200 (g : G), g \u2022 t \u2229 s =\u1da0[ae \u03bc] t \u2229 s\n\u22a2 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 t \u2229 s) = Nat.card G \u2022 \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nsimp_rw [measure_congr (ht _), tsum_fintype, Finset.sum_const, Nat.card_eq_fintype_card, Finset.card_univ]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 \u2211' (g : G), \u222b\u207b (x : \u03b1) in s \u2229 g \u2022 t, f x \u2202\u03bc = \u2211' (g : G), \u222b\u207b (x : \u03b1) in g \u2022 t \u2229 s, f (g\u207b\u00b9 \u2022 x) \u2202\u03bc\n[PROOFSTEP]\nsimp only [hf, inter_comm]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nA : Set \u03b1\nhA\u2080 : MeasurableSet A\nhA : \u2200 (g : G), (fun x => g \u2022 x) \u207b\u00b9' A = A\n\u22a2 \u2191\u2191\u03bc (A \u2229 s) = \u2191\u2191\u03bc (A \u2229 t)\n[PROOFSTEP]\nhave : \u222b\u207b x in s, A.indicator 1 x \u2202\u03bc = \u222b\u207b x in t, A.indicator 1 x \u2202\u03bc :=\n by\n refine hs.set_lintegral_eq ht (Set.indicator A fun _ => 1) fun g x \u21a6 ?_\n convert (Set.indicator_comp_right (g \u2022 \u00b7 : \u03b1 \u2192 \u03b1) (g := fun _ \u21a6 (1 : \u211d\u22650\u221e))).symm\n rw [hA g]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nA : Set \u03b1\nhA\u2080 : MeasurableSet A\nhA : \u2200 (g : G), (fun x => g \u2022 x) \u207b\u00b9' A = A\n\u22a2 \u222b\u207b (x : \u03b1) in s, indicator A 1 x \u2202\u03bc = \u222b\u207b (x : \u03b1) in t, indicator A 1 x \u2202\u03bc\n[PROOFSTEP]\nrefine hs.set_lintegral_eq ht (Set.indicator A fun _ => 1) fun g x \u21a6 ?_\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nA : Set \u03b1\nhA\u2080 : MeasurableSet A\nhA : \u2200 (g : G), (fun x => g \u2022 x) \u207b\u00b9' A = A\ng : G\nx : \u03b1\n\u22a2 indicator A (fun x => 1) (g \u2022 x) = indicator A (fun x => 1) x\n[PROOFSTEP]\nconvert (Set.indicator_comp_right (g \u2022 \u00b7 : \u03b1 \u2192 \u03b1) (g := fun _ \u21a6 (1 : \u211d\u22650\u221e))).symm\n[GOAL]\ncase h.e'_3.h.e'_4\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nA : Set \u03b1\nhA\u2080 : MeasurableSet A\nhA : \u2200 (g : G), (fun x => g \u2022 x) \u207b\u00b9' A = A\ng : G\nx : \u03b1\n\u22a2 A = (fun x => g \u2022 x) \u207b\u00b9' A\n[PROOFSTEP]\nrw [hA g]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nA : Set \u03b1\nhA\u2080 : MeasurableSet A\nhA : \u2200 (g : G), (fun x => g \u2022 x) \u207b\u00b9' A = A\nthis : \u222b\u207b (x : \u03b1) in s, indicator A 1 x \u2202\u03bc = \u222b\u207b (x : \u03b1) in t, indicator A 1 x \u2202\u03bc\n\u22a2 \u2191\u2191\u03bc (A \u2229 s) = \u2191\u2191\u03bc (A \u2229 t)\n[PROOFSTEP]\nsimpa [Measure.restrict_apply hA\u2080, lintegral_indicator _ hA\u2080] using this\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\n\u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc t\n[PROOFSTEP]\nsimpa only [set_lintegral_one] using hs.set_lintegral_eq ht (fun _ => 1) fun _ _ => rfl\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc s) \u2194 AEStronglyMeasurable f (sum fun g => Measure.restrict \u03bc (g \u2022 t \u2229 s))\n[PROOFSTEP]\nsimp only [\u2190 ht.restrict_restrict, ht.sum_restrict_of_ac restrict_le_self.absolutelyContinuous]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 AEStronglyMeasurable f (sum fun g => Measure.restrict \u03bc (g \u2022 t \u2229 s)) \u2194\n \u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 (g\u207b\u00b9 \u2022 s \u2229 t)))\n[PROOFSTEP]\nsimp only [smul_set_inter, inter_comm, smul_inv_smul, aestronglyMeasurable_sum_measure_iff]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 (\u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g\u207b\u00b9 \u2022 (g\u207b\u00b9\u207b\u00b9 \u2022 s \u2229 t)))) \u2194\n \u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g\u207b\u00b9 \u2022 (g \u2022 s \u2229 t)))\n[PROOFSTEP]\nsimp only [inv_inv]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 (\u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g\u207b\u00b9 \u2022 (g \u2022 s \u2229 t)))) \u2194\n \u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 s \u2229 t))\n[PROOFSTEP]\nrefine' forall_congr' fun g => _\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\ng : G\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc (g\u207b\u00b9 \u2022 (g \u2022 s \u2229 t))) \u2194\n AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 s \u2229 t))\n[PROOFSTEP]\nhave he : MeasurableEmbedding ((\u00b7 \u2022 \u00b7) g\u207b\u00b9 : \u03b1 \u2192 \u03b1) := measurableEmbedding_const_smul _\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\ng : G\nhe : MeasurableEmbedding ((fun x x_1 => x \u2022 x_1) g\u207b\u00b9)\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc (g\u207b\u00b9 \u2022 (g \u2022 s \u2229 t))) \u2194\n AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 s \u2229 t))\n[PROOFSTEP]\nrw [\u2190 image_smul, \u2190 ((measurePreserving_smul g\u207b\u00b9 \u03bc).restrict_image_emb he _).aestronglyMeasurable_comp_iff he]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\ng : G\nhe : MeasurableEmbedding ((fun x x_1 => x \u2022 x_1) g\u207b\u00b9)\n\u22a2 AEStronglyMeasurable (f \u2218 fun x => g\u207b\u00b9 \u2022 x) (Measure.restrict \u03bc (g \u2022 s \u2229 t)) \u2194\n AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 s \u2229 t))\n[PROOFSTEP]\nsimp only [(\u00b7 \u2218 \u00b7), hf]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2\u271d : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\n\u03b2 : Type u_6\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 (\u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 s \u2229 t))) \u2194 AEStronglyMeasurable f (Measure.restrict \u03bc t)\n[PROOFSTEP]\nsimp only [\u2190 aestronglyMeasurable_sum_measure_iff, \u2190 hs.restrict_restrict,\n hs.sum_restrict_of_ac restrict_le_self.absolutelyContinuous]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 HasFiniteIntegral f \u2194 HasFiniteIntegral f\n[PROOFSTEP]\ndsimp only [HasFiniteIntegral]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4 \u2194 \u222b\u207b (a : \u03b1) in t, \u2191\u2016f a\u2016\u208a \u2202\u03bc < \u22a4\n[PROOFSTEP]\nrw [hs.set_lintegral_eq ht]\n[GOAL]\ncase hf\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 \u2200 (g : G) (x : \u03b1), \u2191\u2016f (g \u2022 x)\u2016\u208a = \u2191\u2016f x\u2016\u208a\n[PROOFSTEP]\nintro g x\n[GOAL]\ncase hf\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : Group H\ninst\u271d\u2078 : MulAction G \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MulAction H \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSMul G \u03b1\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d : Countable G\n\u03bd : Measure \u03b1\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\ng : G\nx : \u03b1\n\u22a2 \u2191\u2016f (g \u2022 x)\u2016\u208a = \u2191\u2016f x\u2016\u208a\n[PROOFSTEP]\nrw [hf]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nh : IsFundamentalDomain G s\nh\u03bd : \u03bd \u226a \u03bc\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 \u222b (x : \u03b1), f x \u2202\u03bd = \u2211' (g : G), \u222b (x : \u03b1) in g \u2022 s, f x \u2202\u03bd\n[PROOFSTEP]\nrw [\u2190 MeasureTheory.integral_sum_measure, h.sum_restrict_of_ac h\u03bd]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nh : IsFundamentalDomain G s\nh\u03bd : \u03bd \u226a \u03bc\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 Integrable fun x => f x\n[PROOFSTEP]\nrw [h.sum_restrict_of_ac h\u03bd]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nh : IsFundamentalDomain G s\nh\u03bd : \u03bd \u226a \u03bc\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 Integrable fun x => f x\n[PROOFSTEP]\nexact hf\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nh : IsFundamentalDomain G s\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 \u03bc \u226a \u03bc\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nh : IsFundamentalDomain G s\nf : \u03b1 \u2192 E\nt : Set \u03b1\nhf : IntegrableOn f t\n\u22a2 \u2211' (g : G), \u222b (x : \u03b1) in g \u2022 s, f x \u2202Measure.restrict \u03bc t = \u2211' (g : G), \u222b (x : \u03b1) in t \u2229 g \u2022 s, f x \u2202\u03bc\n[PROOFSTEP]\nsimp only [h.restrict_restrict, measure_smul, inter_comm]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t\u271d : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nh : IsFundamentalDomain G s\nf : \u03b1 \u2192 E\nt : Set \u03b1\nhf : IntegrableOn f t\n\u22a2 \u2211' (g : G), \u222b (x : \u03b1) in t \u2229 g\u207b\u00b9 \u2022 s, f x \u2202\u03bc = \u2211' (g : G), \u222b (x : \u03b1) in g\u207b\u00b9 \u2022 (g \u2022 t \u2229 s), f x \u2202\u03bc\n[PROOFSTEP]\nsimp only [smul_set_inter, inv_smul_smul]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc\n[PROOFSTEP]\nby_cases hfs : IntegrableOn f s \u03bc\n[GOAL]\ncase pos\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : IntegrableOn f s\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc\n[PROOFSTEP]\nhave hft : IntegrableOn f t \u03bc := by rwa [ht.integrableOn_iff hs hf]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : IntegrableOn f s\n\u22a2 IntegrableOn f t\n[PROOFSTEP]\nrwa [ht.integrableOn_iff hs hf]\n[GOAL]\ncase pos\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc\n[PROOFSTEP]\ncalc\n \u222b x in s, f x \u2202\u03bc = \u2211' g : G, \u222b x in s \u2229 g \u2022 t, f x \u2202\u03bc := ht.set_integral_eq_tsum hfs\n _ = \u2211' g : G, \u222b x in g \u2022 t \u2229 s, f (g\u207b\u00b9 \u2022 x) \u2202\u03bc := by simp only [hf, inter_comm]\n _ = \u222b x in t, f x \u2202\u03bc := (hs.set_integral_eq_tsum' hft).symm\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\n\u22a2 \u2211' (g : G), \u222b (x : \u03b1) in s \u2229 g \u2022 t, f x \u2202\u03bc = \u2211' (g : G), \u222b (x : \u03b1) in g \u2022 t \u2229 s, f (g\u207b\u00b9 \u2022 x) \u2202\u03bc\n[PROOFSTEP]\nsimp only [hf, inter_comm]\n[GOAL]\ncase neg\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : \u00acIntegrableOn f s\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc\n[PROOFSTEP]\nrw [integral_undef hfs, integral_undef]\n[GOAL]\ncase neg\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : \u00acIntegrableOn f s\n\u22a2 \u00acIntegrable fun x => f x\n[PROOFSTEP]\nrwa [hs.integrableOn_iff ht hf] at hfs \n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2191\u2191\u03bc s < \u2191\u2191\u03bc t\n\u22a2 \u2203 x, x \u2208 t \u2227 \u2203 y, y \u2208 t \u2227 \u2203 g, g \u2260 1 \u2227 g \u2022 x = y\n[PROOFSTEP]\ncontrapose! ht\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\n\u22a2 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc s\n[PROOFSTEP]\nrefine' hs.measure_le_of_pairwise_disjoint htm (Pairwise.aedisjoint fun g\u2081 g\u2082 hne => _)\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\n\u22a2 (Disjoint on fun g => g \u2022 t \u2229 s) g\u2081 g\u2082\n[PROOFSTEP]\ndsimp [Function.onFun]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\n\u22a2 Disjoint (g\u2081 \u2022 t \u2229 s) (g\u2082 \u2022 t \u2229 s)\n[PROOFSTEP]\nrefine' (Disjoint.inf_left _ _).inf_right _\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\n\u22a2 Disjoint (g\u2081 \u2022 t) (g\u2082 \u2022 t)\n[PROOFSTEP]\nrw [Set.disjoint_left]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\n\u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 g\u2081 \u2022 t \u2192 \u00aca \u2208 g\u2082 \u2022 t\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9 \u27e8y, hy, hxy : g\u2082 \u2022 y = g\u2081 \u2022 x\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\nx : \u03b1\nhx : x \u2208 t\ny : \u03b1\nhy : y \u2208 t\nhxy : g\u2082 \u2022 y = g\u2081 \u2022 x\n\u22a2 False\n[PROOFSTEP]\nrefine' ht x hx y hy (g\u2082\u207b\u00b9 * g\u2081) (mt inv_mul_eq_one.1 hne.symm) _\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nhtm : NullMeasurableSet t\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y\ng\u2081 g\u2082 : G\nhne : g\u2081 \u2260 g\u2082\nx : \u03b1\nhx : x \u2208 t\ny : \u03b1\nhy : y \u2208 t\nhxy : g\u2082 \u2022 y = g\u2081 \u2022 x\n\u22a2 (g\u2082\u207b\u00b9 * g\u2081) \u2022 x = y\n[PROOFSTEP]\nrw [mul_smul, \u2190 hxy, inv_smul_smul]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 essSup f (Measure.restrict \u03bc s) = essSup f \u03bc\n[PROOFSTEP]\nrefine' le_antisymm (essSup_mono_measure' Measure.restrict_le_self) _\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 essSup f \u03bc \u2264 essSup f (Measure.restrict \u03bc s)\n[PROOFSTEP]\nrw [essSup_eq_sInf (\u03bc.restrict s) f, essSup_eq_sInf \u03bc f]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 sInf {a | \u2191\u2191\u03bc {x | a < f x} = 0} \u2264 sInf {a | \u2191\u2191(Measure.restrict \u03bc s) {x | a < f x} = 0}\n[PROOFSTEP]\nrefine' sInf_le_sInf _\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\n\u22a2 {a | \u2191\u2191(Measure.restrict \u03bc s) {x | a < f x} = 0} \u2286 {a | \u2191\u2191\u03bc {x | a < f x} = 0}\n[PROOFSTEP]\nrintro a (ha : (\u03bc.restrict s) {x : \u03b1 | a < f x} = 0)\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191(Measure.restrict \u03bc s) {x | a < f x} = 0\n\u22a2 a \u2208 {a | \u2191\u2191\u03bc {x | a < f x} = 0}\n[PROOFSTEP]\nrw [Measure.restrict_apply\u2080' hs.nullMeasurableSet] at ha \n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0\n\u22a2 a \u2208 {a | \u2191\u2191\u03bc {x | a < f x} = 0}\n[PROOFSTEP]\nrefine' measure_zero_of_invariant hs _ _ ha\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0\n\u22a2 \u2200 (g : G), g \u2022 {x | a < f x} = {x | a < f x}\n[PROOFSTEP]\nintro \u03b3\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0\n\u03b3 : G\n\u22a2 \u03b3 \u2022 {x | a < f x} = {x | a < f x}\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0\n\u03b3 : G\nx : \u03b1\n\u22a2 x \u2208 \u03b3 \u2022 {x | a < f x} \u2194 x \u2208 {x | a < f x}\n[PROOFSTEP]\nrw [mem_smul_set_iff_inv_smul_mem]\n[GOAL]\ncase h\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x\na : \u211d\u22650\u221e\nha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0\n\u03b3 : G\nx : \u03b1\n\u22a2 \u03b3\u207b\u00b9 \u2022 x \u2208 {x | a < f x} \u2194 x \u2208 {x | a < f x}\n[PROOFSTEP]\nsimp only [mem_setOf_eq, hf \u03b3\u207b\u00b9 x]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 x \u2208 fundamentalFrontier G s \u2194 x \u2208 s \u2227 \u2203 g, g \u2260 1 \u2227 x \u2208 g \u2022 s\n[PROOFSTEP]\nsimp [fundamentalFrontier]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 x \u2208 fundamentalInterior G s \u2194 x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s\n[PROOFSTEP]\nsimp [fundamentalInterior]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G \u03b1\ns : Set \u03b1\nx : \u03b1\ninst\u271d\u00b2 : Group H\ninst\u271d\u00b9 : MulAction H \u03b1\ninst\u271d : SMulCommClass H G \u03b1\ng : H\n\u22a2 fundamentalFrontier G (g \u2022 s) = g \u2022 fundamentalFrontier G s\n[PROOFSTEP]\nsimp_rw [fundamentalFrontier, smul_set_inter, smul_set_Union, smul_comm g (_ : G) (_ : Set \u03b1)]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G \u03b1\ns : Set \u03b1\nx : \u03b1\ninst\u271d\u00b2 : Group H\ninst\u271d\u00b9 : MulAction H \u03b1\ninst\u271d : SMulCommClass H G \u03b1\ng : H\n\u22a2 fundamentalInterior G (g \u2022 s) = g \u2022 fundamentalInterior G s\n[PROOFSTEP]\nsimp_rw [fundamentalInterior, smul_set_sdiff, smul_set_Union, smul_comm g (_ : G) (_ : Set \u03b1)]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 Pairwise (Disjoint on fun g => g \u2022 fundamentalInterior G s)\n[PROOFSTEP]\nrefine' fun a b hab => disjoint_left.2 _\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx : \u03b1\na b : G\nhab : a \u2260 b\n\u22a2 \u2200 \u2983a_1 : \u03b1\u2984, a_1 \u2208 (fun g => g \u2022 fundamentalInterior G s) a \u2192 \u00aca_1 \u2208 (fun g => g \u2022 fundamentalInterior G s) b\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9 \u27e8y, hy, hxy\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\na b : G\nhab : a \u2260 b\nx : \u03b1\nhx : x \u2208 fundamentalInterior G s\ny : \u03b1\nhy : y \u2208 fundamentalInterior G s\nhxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x\n\u22a2 False\n[PROOFSTEP]\nrw [mem_fundamentalInterior] at hx hy \n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\na b : G\nhab : a \u2260 b\nx : \u03b1\nhx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s\ny : \u03b1\nhy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s\nhxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x\n\u22a2 False\n[PROOFSTEP]\nrefine' hx.2 (a\u207b\u00b9 * b) _ _\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\na b : G\nhab : a \u2260 b\nx : \u03b1\nhx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s\ny : \u03b1\nhy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s\nhxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x\n\u22a2 a\u207b\u00b9 * b \u2260 1\ncase intro.intro.intro.intro.refine'_2\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\na b : G\nhab : a \u2260 b\nx : \u03b1\nhx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s\ny : \u03b1\nhy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s\nhxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x\n\u22a2 x \u2208 (a\u207b\u00b9 * b) \u2022 s\n[PROOFSTEP]\nrwa [Ne.def, inv_mul_eq_iff_eq_mul, mul_one, eq_comm]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\na b : G\nhab : a \u2260 b\nx : \u03b1\nhx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s\ny : \u03b1\nhy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s\nhxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x\n\u22a2 x \u2208 (a\u207b\u00b9 * b) \u2022 s\n[PROOFSTEP]\nsimpa [mul_smul, \u2190 hxy, mem_inv_smul_set_iff] using hy.1\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : Countable G\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\n\u22a2 \u2191\u2191\u03bc (fundamentalFrontier G s) = 0\n[PROOFSTEP]\nsimpa only [fundamentalFrontier, iUnion\u2082_inter, measure_iUnion_null_iff', one_smul, measure_iUnion_null_iff,\n inter_comm s, Function.onFun] using fun g (hg : g \u2260 1) => hs.aedisjoint hg\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 g, g \u2022 x \u2208 fundamentalInterior G s\n[PROOFSTEP]\nsimp_rw [ae_iff, not_exists, \u2190 mem_inv_smul_set_iff, setOf_forall, \u2190 compl_setOf, setOf_mem_eq, \u2190 compl_iUnion]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : G), i\u207b\u00b9 \u2022 fundamentalInterior G s)\u1d9c = 0\n[PROOFSTEP]\nhave : ((\u22c3 g : G, g\u207b\u00b9 \u2022 s) \\ \u22c3 g : G, g\u207b\u00b9 \u2022 fundamentalFrontier G s) \u2286 \u22c3 g : G, g\u207b\u00b9 \u2022 fundamentalInterior G s := by\n simp_rw [diff_subset_iff, \u2190 iUnion_union_distrib, \u2190 smul_set_union (\u03b1 := G) (\u03b2 := \u03b1),\n fundamentalFrontier_union_fundamentalInterior];\n rfl\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\n\u22a2 (\u22c3 (g : G), g\u207b\u00b9 \u2022 s) \\ \u22c3 (g : G), g\u207b\u00b9 \u2022 fundamentalFrontier G s \u2286 \u22c3 (g : G), g\u207b\u00b9 \u2022 fundamentalInterior G s\n[PROOFSTEP]\nsimp_rw [diff_subset_iff, \u2190 iUnion_union_distrib, \u2190 smul_set_union (\u03b1 := G) (\u03b2 := \u03b1),\n fundamentalFrontier_union_fundamentalInterior]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\n\u22a2 \u22c3 (g : G), g\u207b\u00b9 \u2022 s \u2286 \u22c3 (g : G), g\u207b\u00b9 \u2022 s\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nthis : (\u22c3 (g : G), g\u207b\u00b9 \u2022 s) \\ \u22c3 (g : G), g\u207b\u00b9 \u2022 fundamentalFrontier G s \u2286 \u22c3 (g : G), g\u207b\u00b9 \u2022 fundamentalInterior G s\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : G), i\u207b\u00b9 \u2022 fundamentalInterior G s)\u1d9c = 0\n[PROOFSTEP]\nrefine' eq_bot_mono (\u03bc.mono <| compl_subset_compl.2 this) _\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nthis : (\u22c3 (g : G), g\u207b\u00b9 \u2022 s) \\ \u22c3 (g : G), g\u207b\u00b9 \u2022 fundamentalFrontier G s \u2286 \u22c3 (g : G), g\u207b\u00b9 \u2022 fundamentalInterior G s\n\u22a2 \u2191\u2191\u03bc ((\u22c3 (g : G), g\u207b\u00b9 \u2022 s) \\ \u22c3 (g : G), g\u207b\u00b9 \u2022 fundamentalFrontier G s)\u1d9c = \u22a5\n[PROOFSTEP]\nsimp only [iUnion_inv_smul, compl_sdiff, ENNReal.bot_eq_zero, himp_eq, sup_eq_union,\n @iUnion_smul_eq_setOf_exists _ _ _ _ s]\n[GOAL]\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u2076 : Countable G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsFundamentalDomain G s\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nthis : (\u22c3 (g : G), g\u207b\u00b9 \u2022 s) \\ \u22c3 (g : G), g\u207b\u00b9 \u2022 fundamentalFrontier G s \u2286 \u22c3 (g : G), g\u207b\u00b9 \u2022 fundamentalInterior G s\n\u22a2 \u2191\u2191\u03bc ((\u22c3 (g : G), g \u2022 fundamentalFrontier G s) \u222a {a | \u2203 g, g \u2022 a \u2208 s}\u1d9c) = 0\n[PROOFSTEP]\nexact measure_union_null (measure_iUnion_null fun _ => measure_smul_null hs.measure_fundamentalFrontier _) hs.ae_covers\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Group.FundamentalDomain", "llama_tokens": 33392, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585903489892, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.43051245781097286}} {"text": "[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Preadditive V\nc : ComplexShape \u03b9\n\u03b9\u271d : Type ?u.1725\nv : ComplexShape \u03b9\u271d\n\u22a2 Category.{?u.1722, max (max u v) ?u.1725} (HomotopyCategory V v)\n[PROOFSTEP]\ndsimp only [HomotopyCategory]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Preadditive V\nc : ComplexShape \u03b9\n\u03b9\u271d : Type ?u.1725\nv : ComplexShape \u03b9\u271d\n\u22a2 Category.{?u.1722, max (max u v) ?u.1725} (CategoryTheory.Quotient (homotopic V v))\n[PROOFSTEP]\ninfer_instance\n -- TODO the homotopy_category is preadditive\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Preadditive V\nc : ComplexShape \u03b9\nC D : HomologicalComplex V c\nf : C \u27f6 D\n\u22a2 Homotopy (Quot.out ((quotient V c).map f)) f\n[PROOFSTEP]\napply homotopyOfEq\n[GOAL]\ncase w\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Preadditive V\nc : ComplexShape \u03b9\nC D : HomologicalComplex V c\nf : C \u27f6 D\n\u22a2 (quotient V c).map (Quot.out ((quotient V c).map f)) = (quotient V c).map f\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Preadditive V\nc : ComplexShape \u03b9\nC D E : HomotopyCategory V c\nf : C \u27f6 D\ng : D \u27f6 E\n\u22a2 (quotient V c).map (Quot.out f \u226b Quot.out g) = f \u226b g\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Preadditive V\nc : ComplexShape \u03b9\nC D : HomologicalComplex V c\nf : HomotopyEquiv C D\n\u22a2 (quotient V c).map f.hom \u226b (quotient V c).map f.inv = \ud835\udfd9 ((quotient V c).obj C)\n[PROOFSTEP]\nrw [\u2190 (quotient V c).map_comp, \u2190 (quotient V c).map_id]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Preadditive V\nc : ComplexShape \u03b9\nC D : HomologicalComplex V c\nf : HomotopyEquiv C D\n\u22a2 (quotient V c).map (f.hom \u226b f.inv) = (quotient V c).map (\ud835\udfd9 C)\n[PROOFSTEP]\nexact eq_of_homotopy _ _ f.homotopyHomInvId\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Preadditive V\nc : ComplexShape \u03b9\nC D : HomologicalComplex V c\nf : HomotopyEquiv C D\n\u22a2 (quotient V c).map f.inv \u226b (quotient V c).map f.hom = \ud835\udfd9 ((quotient V c).obj D)\n[PROOFSTEP]\nrw [\u2190 (quotient V c).map_comp, \u2190 (quotient V c).map_id]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Preadditive V\nc : ComplexShape \u03b9\nC D : HomologicalComplex V c\nf : HomotopyEquiv C D\n\u22a2 (quotient V c).map (f.inv \u226b f.hom) = (quotient V c).map (\ud835\udfd9 D)\n[PROOFSTEP]\nexact eq_of_homotopy _ _ f.homotopyInvHomId\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Preadditive V\nc : ComplexShape \u03b9\nC D : HomologicalComplex V c\ni : (quotient V c).obj C \u2245 (quotient V c).obj D\n\u22a2 (quotient V c).map (Quot.out i.hom \u226b Quot.out i.inv) = (quotient V c).map (\ud835\udfd9 C)\n[PROOFSTEP]\nrw [quotient_map_out_comp_out, i.hom_inv_id, (quotient V c).map_id]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Preadditive V\nc : ComplexShape \u03b9\nC D : HomologicalComplex V c\ni : (quotient V c).obj C \u2245 (quotient V c).obj D\n\u22a2 (quotient V c).map (Quot.out i.inv \u226b Quot.out i.hom) = (quotient V c).map (\ud835\udfd9 D)\n[PROOFSTEP]\nrw [quotient_map_out_comp_out, i.inv_hom_id, (quotient V c).map_id]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : Preadditive V\nc\u271d : ComplexShape \u03b9\nW : Type u_2\ninst\u271d\u00b3 : Category.{?u.46796, u_2} W\ninst\u271d\u00b2 : Preadditive W\nF G : V \u2964 W\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nc : ComplexShape \u03b9\n\u22a2 \u2200 \u2983X Y : HomotopyCategory V c\u2984 (f : X \u27f6 Y),\n (Functor.mapHomotopyCategory F c).map f \u226b\n (fun C => (HomotopyCategory.quotient W c).map (app (mapHomologicalComplex \u03b1 c) C.as)) Y =\n (fun C => (HomotopyCategory.quotient W c).map (app (mapHomologicalComplex \u03b1 c) C.as)) X \u226b\n (Functor.mapHomotopyCategory G c).map f\n[PROOFSTEP]\nrintro \u27e8C\u27e9 \u27e8D\u27e9 \u27e8f : C \u27f6 D\u27e9\n[GOAL]\ncase mk.mk.mk\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : Preadditive V\nc\u271d : ComplexShape \u03b9\nW : Type u_2\ninst\u271d\u00b3 : Category.{?u.46796, u_2} W\ninst\u271d\u00b2 : Preadditive W\nF G : V \u2964 W\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nc : ComplexShape \u03b9\nC D : HomologicalComplex V c\nf\u271d : { as := C } \u27f6 { as := D }\nf : C \u27f6 D\n\u22a2 (Functor.mapHomotopyCategory F c).map (Quot.mk (Quotient.CompClosure (homotopic V c)) f) \u226b\n (fun C => (HomotopyCategory.quotient W c).map (app (mapHomologicalComplex \u03b1 c) C.as)) { as := D } =\n (fun C => (HomotopyCategory.quotient W c).map (app (mapHomologicalComplex \u03b1 c) C.as)) { as := C } \u226b\n (Functor.mapHomotopyCategory G c).map (Quot.mk (Quotient.CompClosure (homotopic V c)) f)\n[PROOFSTEP]\nsimp only [HomotopyCategory.quot_mk_eq_quotient_map, Functor.mapHomotopyCategory_map, \u2190 Functor.map_comp,\n NatTrans.naturality]\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2074 : Category.{v, u} V\ninst\u271d\u00b3 : Preadditive V\nc\u271d : ComplexShape \u03b9\nW : Type u_2\ninst\u271d\u00b2 : Category.{u_3, u_2} W\ninst\u271d\u00b9 : Preadditive W\nc : ComplexShape \u03b9\nF : V \u2964 W\ninst\u271d : Functor.Additive F\n\u22a2 mapHomotopyCategory (\ud835\udfd9 F) c = \ud835\udfd9 (Functor.mapHomotopyCategory F c)\n[PROOFSTEP]\naesop_cat\n[GOAL]\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u2076 : Category.{v, u} V\ninst\u271d\u2075 : Preadditive V\nc\u271d : ComplexShape \u03b9\nW : Type u_2\ninst\u271d\u2074 : Category.{u_3, u_2} W\ninst\u271d\u00b3 : Preadditive W\nc : ComplexShape \u03b9\nF G H : V \u2964 W\ninst\u271d\u00b2 : Functor.Additive F\ninst\u271d\u00b9 : Functor.Additive G\ninst\u271d : Functor.Additive H\n\u03b1 : F \u27f6 G\n\u03b2 : G \u27f6 H\n\u22a2 mapHomotopyCategory (\u03b1 \u226b \u03b2) c = mapHomotopyCategory \u03b1 c \u226b mapHomotopyCategory \u03b2 c\n[PROOFSTEP]\naesop_cat\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Homology.HomotopyCategory", "llama_tokens": 2540, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585786300049, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.43051245090979035}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\na b c d : \u03b1\nH : 0 \u2264 a\n\u22a2 0 \u2264 a ^ 0\n[PROOFSTEP]\nrw [pow_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\na b c d : \u03b1\nH : 0 \u2264 a\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nexact zero_le_one\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\na b c d : \u03b1\nH : 0 \u2264 a\nn : \u2115\n\u22a2 0 \u2264 a ^ (n + 1)\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\na b c d : \u03b1\nH : 0 \u2264 a\nn : \u2115\n\u22a2 0 \u2264 a * a ^ n\n[PROOFSTEP]\nexact mul_nonneg H (pow_nonneg H _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\na b c d : \u03b1\na2 : 2 \u2264 a\nb0 : 0 \u2264 b\n\u22a2 a + (a + a * b) \u2264 a * (2 + b)\n[PROOFSTEP]\nrw [mul_add, mul_two, add_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\na b c d : \u03b1\nh : 0 < a\n\u22a2 0 < bit1 a\n[PROOFSTEP]\nnontriviality\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : OrderedSemiring \u03b1\na b c d : \u03b1\nh : 0 < a\ninst\u271d : Nontrivial \u03b1\n\u22a2 0 < bit1 a\n[PROOFSTEP]\nexact bit1_pos h.le\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedRing \u03b1\na\u271d b\u271d c\u271d d : \u03b1\nsrc\u271d\u00b9 : OrderedRing \u03b1 := inst\u271d\nsrc\u271d : Semiring \u03b1 := Ring.toSemiring\na b c : \u03b1\nh : a \u2264 b\nhc : 0 \u2264 c\n\u22a2 c * a \u2264 c * b\n[PROOFSTEP]\nsimpa only [mul_sub, sub_nonneg] using OrderedRing.mul_nonneg _ _ hc (sub_nonneg.2 h)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedRing \u03b1\na\u271d b\u271d c\u271d d : \u03b1\nsrc\u271d\u00b9 : OrderedRing \u03b1 := inst\u271d\nsrc\u271d : Semiring \u03b1 := Ring.toSemiring\na b c : \u03b1\nh : a \u2264 b\nhc : 0 \u2264 c\n\u22a2 a * c \u2264 b * c\n[PROOFSTEP]\nsimpa only [sub_mul, sub_nonneg] using OrderedRing.mul_nonneg _ _ (sub_nonneg.2 h) hc\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedRing \u03b1\na b c d : \u03b1\nh : b \u2264 a\nhc : c \u2264 0\n\u22a2 c * a \u2264 c * b\n[PROOFSTEP]\nsimpa only [neg_mul, neg_le_neg_iff] using mul_le_mul_of_nonneg_left h (neg_nonneg.2 hc)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedRing \u03b1\na b c d : \u03b1\nh : b \u2264 a\nhc : c \u2264 0\n\u22a2 a * c \u2264 b * c\n[PROOFSTEP]\nsimpa only [mul_neg, neg_le_neg_iff] using mul_le_mul_of_nonneg_right h (neg_nonneg.2 hc)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedRing \u03b1\na b c d : \u03b1\nha : a \u2264 0\nhb : b \u2264 0\n\u22a2 0 \u2264 a * b\n[PROOFSTEP]\nsimpa only [zero_mul] using mul_le_mul_of_nonpos_right ha hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedRing \u03b1\na b c d : \u03b1\nhb : b \u2264 0\nh : a \u2264 1\n\u22a2 b \u2264 a * b\n[PROOFSTEP]\nsimpa only [one_mul] using mul_le_mul_of_nonpos_right h hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedRing \u03b1\na b c d : \u03b1\nhb : b \u2264 0\nh : 1 \u2264 a\n\u22a2 a * b \u2264 b\n[PROOFSTEP]\nsimpa only [one_mul] using mul_le_mul_of_nonpos_right h hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedRing \u03b1\na b c d : \u03b1\nha : a \u2264 0\nh : b \u2264 1\n\u22a2 a \u2264 a * b\n[PROOFSTEP]\nsimpa only [mul_one] using mul_le_mul_of_nonpos_left h ha\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedRing \u03b1\na b c d : \u03b1\nha : a \u2264 0\nh : 1 \u2264 b\n\u22a2 a * b \u2264 a\n[PROOFSTEP]\nsimpa only [mul_one] using mul_le_mul_of_nonpos_left h ha\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedRing \u03b1\na\u271d b\u271d c d a b : \u03b1\nh : a \u2264 b\n\u22a2 b = a + (b - a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedRing \u03b1\na\u271d b\u271d c\u271d d a b : \u03b1\nx\u271d : \u2203 c, c \u2265 0 \u2227 b = a + c\nc : \u03b1\nhc : c \u2265 0\nh : b = a + c\n\u22a2 a \u2264 b\n[PROOFSTEP]\nrw [h, le_add_iff_nonneg_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedRing \u03b1\na\u271d b\u271d c\u271d d a b : \u03b1\nx\u271d : \u2203 c, c \u2265 0 \u2227 b = a + c\nc : \u03b1\nhc : c \u2265 0\nh : b = a + c\n\u22a2 0 \u2264 c\n[PROOFSTEP]\nexact hc\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring \u03b1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : StrictOrderedSemiring \u03b1 := inst\u271d\u00b9\na b c : \u03b1\nhab : a \u2264 b\nhc : 0 \u2264 c\n\u22a2 c * a \u2264 c * b\n[PROOFSTEP]\nobtain rfl | hab := Decidable.eq_or_lt_of_le hab\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring \u03b1\na\u271d b c\u271d d : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : StrictOrderedSemiring \u03b1 := inst\u271d\u00b9\na c : \u03b1\nhc : 0 \u2264 c\nhab : a \u2264 a\n\u22a2 c * a \u2264 c * a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring \u03b1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : StrictOrderedSemiring \u03b1 := inst\u271d\u00b9\na b c : \u03b1\nhab\u271d : a \u2264 b\nhc : 0 \u2264 c\nhab : a < b\n\u22a2 c * a \u2264 c * b\n[PROOFSTEP]\nobtain rfl | hc := Decidable.eq_or_lt_of_le hc\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring \u03b1\na\u271d b\u271d c d : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : StrictOrderedSemiring \u03b1 := inst\u271d\u00b9\na b : \u03b1\nhab\u271d : a \u2264 b\nhab : a < b\nhc : 0 \u2264 0\n\u22a2 0 * a \u2264 0 * b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring \u03b1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : StrictOrderedSemiring \u03b1 := inst\u271d\u00b9\na b c : \u03b1\nhab\u271d : a \u2264 b\nhc\u271d : 0 \u2264 c\nhab : a < b\nhc : 0 < c\n\u22a2 c * a \u2264 c * b\n[PROOFSTEP]\nexact (mul_lt_mul_of_pos_left hab hc).le\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring \u03b1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : StrictOrderedSemiring \u03b1 := inst\u271d\u00b9\na b c : \u03b1\nhab : a \u2264 b\nhc : 0 \u2264 c\n\u22a2 a * c \u2264 b * c\n[PROOFSTEP]\nobtain rfl | hab := Decidable.eq_or_lt_of_le hab\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring \u03b1\na\u271d b c\u271d d : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : StrictOrderedSemiring \u03b1 := inst\u271d\u00b9\na c : \u03b1\nhc : 0 \u2264 c\nhab : a \u2264 a\n\u22a2 a * c \u2264 a * c\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring \u03b1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : StrictOrderedSemiring \u03b1 := inst\u271d\u00b9\na b c : \u03b1\nhab\u271d : a \u2264 b\nhc : 0 \u2264 c\nhab : a < b\n\u22a2 a * c \u2264 b * c\n[PROOFSTEP]\nobtain rfl | hc := Decidable.eq_or_lt_of_le hc\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring \u03b1\na\u271d b\u271d c d : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : StrictOrderedSemiring \u03b1 := inst\u271d\u00b9\na b : \u03b1\nhab\u271d : a \u2264 b\nhab : a < b\nhc : 0 \u2264 0\n\u22a2 a * 0 \u2264 b * 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring \u03b1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d : StrictOrderedSemiring \u03b1 := inst\u271d\u00b9\na b c : \u03b1\nhab\u271d : a \u2264 b\nhc\u271d : 0 \u2264 c\nhab : a < b\nhc : 0 < c\n\u22a2 a * c \u2264 b * c\n[PROOFSTEP]\nexact (mul_lt_mul_of_pos_right hab hc).le\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedSemiring \u03b1\na b c d : \u03b1\nH : 0 < a\n\u22a2 0 < a ^ 0\n[PROOFSTEP]\nnontriviality\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring \u03b1\na b c d : \u03b1\nH : 0 < a\ninst\u271d : Nontrivial \u03b1\n\u22a2 0 < a ^ 0\n[PROOFSTEP]\nrw [pow_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring \u03b1\na b c d : \u03b1\nH : 0 < a\ninst\u271d : Nontrivial \u03b1\n\u22a2 0 < 1\n[PROOFSTEP]\nexact zero_lt_one\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedSemiring \u03b1\na b c d : \u03b1\nH : 0 < a\nn : \u2115\n\u22a2 0 < a ^ (n + 1)\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedSemiring \u03b1\na b c d : \u03b1\nH : 0 < a\nn : \u2115\n\u22a2 0 < a * a ^ n\n[PROOFSTEP]\nexact mul_pos H (pow_pos H _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring \u03b1\na b c d : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nh1 : a < c\nh2 : b < d\nh3 : 0 \u2264 a\nh4 : 0 \u2264 b\nb0 : 0 = b\n\u22a2 a * b < c * d\n[PROOFSTEP]\nrw [\u2190 b0, mul_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedSemiring \u03b1\na b c d : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nh1 : a < c\nh2 : b < d\nh3 : 0 \u2264 a\nh4 : 0 \u2264 b\nb0 : 0 = b\n\u22a2 0 < c * d\n[PROOFSTEP]\nexact mul_pos (h3.trans_lt h1) (h4.trans_lt h2)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedSemiring \u03b1\na b c d : \u03b1\n\u22a2 a < c \u2192 b < d \u2192 0 \u2264 a \u2192 0 \u2264 b \u2192 a * b < c * d\n[PROOFSTEP]\nclassical exact Decidable.mul_lt_mul''\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedSemiring \u03b1\na b c d : \u03b1\n\u22a2 a < c \u2192 b < d \u2192 0 \u2264 a \u2192 0 \u2264 b \u2192 a * b < c * d\n[PROOFSTEP]\nexact Decidable.mul_lt_mul''\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedSemiring \u03b1\na b c d : \u03b1\nhn : 0 < a\nhm : 1 < b\n\u22a2 a < b * a\n[PROOFSTEP]\nconvert mul_lt_mul_of_pos_right hm hn\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedSemiring \u03b1\na b c d : \u03b1\nhn : 0 < a\nhm : 1 < b\n\u22a2 a = 1 * a\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedSemiring \u03b1\na b c d : \u03b1\nhn : 0 < a\nhm : 1 < b\n\u22a2 a < a * b\n[PROOFSTEP]\nconvert mul_lt_mul_of_pos_left hm hn\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedSemiring \u03b1\na b c d : \u03b1\nhn : 0 < a\nhm : 1 < b\n\u22a2 a = a * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedRing \u03b1\na\u271d b\u271d c\u271d : \u03b1\nsrc\u271d\u00b9 : StrictOrderedRing \u03b1 := inst\u271d\nsrc\u271d : Semiring \u03b1 := Ring.toSemiring\na b c : \u03b1\nh : a < b\nhc : 0 < c\n\u22a2 c * a < c * b\n[PROOFSTEP]\nsimpa only [mul_sub, sub_pos] using StrictOrderedRing.mul_pos _ _ hc (sub_pos.2 h)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedRing \u03b1\na\u271d b\u271d c\u271d : \u03b1\nsrc\u271d\u00b9 : StrictOrderedRing \u03b1 := inst\u271d\nsrc\u271d : Semiring \u03b1 := Ring.toSemiring\na b c : \u03b1\nh : a < b\nhc : 0 < c\n\u22a2 a * c < b * c\n[PROOFSTEP]\nsimpa only [sub_mul, sub_pos] using StrictOrderedRing.mul_pos _ _ (sub_pos.2 h) hc\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedRing \u03b1\na\u271d b\u271d c : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : StrictOrderedRing \u03b1 := inst\u271d\u00b9\nsrc\u271d : Semiring \u03b1 := Ring.toSemiring\na b : \u03b1\nha : 0 \u2264 a\nhb : 0 \u2264 b\n\u22a2 0 \u2264 a * b\n[PROOFSTEP]\nobtain ha | ha := Decidable.eq_or_lt_of_le ha\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedRing \u03b1\na\u271d b\u271d c : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : StrictOrderedRing \u03b1 := inst\u271d\u00b9\nsrc\u271d : Semiring \u03b1 := Ring.toSemiring\na b : \u03b1\nha\u271d : 0 \u2264 a\nhb : 0 \u2264 b\nha : 0 = a\n\u22a2 0 \u2264 a * b\n[PROOFSTEP]\nrw [\u2190 ha, zero_mul]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedRing \u03b1\na\u271d b\u271d c : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : StrictOrderedRing \u03b1 := inst\u271d\u00b9\nsrc\u271d : Semiring \u03b1 := Ring.toSemiring\na b : \u03b1\nha\u271d : 0 \u2264 a\nhb : 0 \u2264 b\nha : 0 < a\n\u22a2 0 \u2264 a * b\n[PROOFSTEP]\nobtain hb | hb := Decidable.eq_or_lt_of_le hb\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedRing \u03b1\na\u271d b\u271d c : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : StrictOrderedRing \u03b1 := inst\u271d\u00b9\nsrc\u271d : Semiring \u03b1 := Ring.toSemiring\na b : \u03b1\nha\u271d : 0 \u2264 a\nhb\u271d : 0 \u2264 b\nha : 0 < a\nhb : 0 = b\n\u22a2 0 \u2264 a * b\n[PROOFSTEP]\nrw [\u2190 hb, mul_zero]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u00b9 : StrictOrderedRing \u03b1\na\u271d b\u271d c : \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nsrc\u271d\u00b9 : StrictOrderedRing \u03b1 := inst\u271d\u00b9\nsrc\u271d : Semiring \u03b1 := Ring.toSemiring\na b : \u03b1\nha\u271d : 0 \u2264 a\nhb\u271d : 0 \u2264 b\nha : 0 < a\nhb : 0 < b\n\u22a2 0 \u2264 a * b\n[PROOFSTEP]\nexact (StrictOrderedRing.mul_pos _ _ ha hb).le\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedRing \u03b1\na b c : \u03b1\nh : b < a\nhc : c < 0\n\u22a2 c * a < c * b\n[PROOFSTEP]\nsimpa only [neg_mul, neg_lt_neg_iff] using mul_lt_mul_of_pos_left h (neg_pos_of_neg hc)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedRing \u03b1\na b c : \u03b1\nh : b < a\nhc : c < 0\n\u22a2 a * c < b * c\n[PROOFSTEP]\nsimpa only [mul_neg, neg_lt_neg_iff] using mul_lt_mul_of_pos_right h (neg_pos_of_neg hc)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedRing \u03b1\na\u271d b\u271d c a b : \u03b1\nha : a < 0\nhb : b < 0\n\u22a2 0 < a * b\n[PROOFSTEP]\nsimpa only [zero_mul] using mul_lt_mul_of_neg_right ha hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedRing \u03b1\na b c : \u03b1\nhb : b < 0\nh : a < 1\n\u22a2 b < a * b\n[PROOFSTEP]\nsimpa only [one_mul] using mul_lt_mul_of_neg_right h hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedRing \u03b1\na b c : \u03b1\nhb : b < 0\nh : 1 < a\n\u22a2 a * b < b\n[PROOFSTEP]\nsimpa only [one_mul] using mul_lt_mul_of_neg_right h hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedRing \u03b1\na b c : \u03b1\nha : a < 0\nh : b < 1\n\u22a2 a < a * b\n[PROOFSTEP]\nsimpa only [mul_one] using mul_lt_mul_of_neg_left h ha\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedRing \u03b1\na b c : \u03b1\nha : a < 0\nh : 1 < b\n\u22a2 a * b < a\n[PROOFSTEP]\nsimpa only [mul_one] using mul_lt_mul_of_neg_left h ha\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\nhab : 0 \u2264 a * b\n\u22a2 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\n[PROOFSTEP]\nrefine' Decidable.or_iff_not_and_not.2 _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\nhab : 0 \u2264 a * b\n\u22a2 \u00ac(\u00ac(0 \u2264 a \u2227 0 \u2264 b) \u2227 \u00ac(a \u2264 0 \u2227 b \u2264 0))\n[PROOFSTEP]\nsimp only [not_and, not_le]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\nhab : 0 \u2264 a * b\n\u22a2 (0 \u2264 a \u2192 b < 0) \u2192 \u00ac(a \u2264 0 \u2192 0 < b)\n[PROOFSTEP]\nintro ab nab\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\nhab : 0 \u2264 a * b\nab : 0 \u2264 a \u2192 b < 0\nnab : a \u2264 0 \u2192 0 < b\n\u22a2 False\n[PROOFSTEP]\napply\n not_lt_of_le hab\n _\n -- Porting note: for the middle case, we used to have `rfl`, but it is now rejected.\n -- https://github.com/leanprover/std4/issues/62\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\nhab : 0 \u2264 a * b\nab : 0 \u2264 a \u2192 b < 0\nnab : a \u2264 0 \u2192 0 < b\n\u22a2 a * b < 0\n[PROOFSTEP]\nrcases lt_trichotomy 0 a with (ha | ha | ha)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\nhab : 0 \u2264 a * b\nab : 0 \u2264 a \u2192 b < 0\nnab : a \u2264 0 \u2192 0 < b\nha : 0 < a\n\u22a2 a * b < 0\n[PROOFSTEP]\nexact mul_neg_of_pos_of_neg ha (ab ha.le)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\nhab : 0 \u2264 a * b\nab : 0 \u2264 a \u2192 b < 0\nnab : a \u2264 0 \u2192 0 < b\nha : 0 = a\n\u22a2 a * b < 0\n[PROOFSTEP]\nsubst ha\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\nb c d : \u03b1\nhab : 0 \u2264 0 * b\nab : 0 \u2264 0 \u2192 b < 0\nnab : 0 \u2264 0 \u2192 0 < b\n\u22a2 0 * b < 0\n[PROOFSTEP]\nexact ((ab le_rfl).asymm (nab le_rfl)).elim\n[GOAL]\ncase inr.inr\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\nhab : 0 \u2264 a * b\nab : 0 \u2264 a \u2192 b < 0\nnab : a \u2264 0 \u2192 0 < b\nha : a < 0\n\u22a2 a * b < 0\n[PROOFSTEP]\nexact mul_neg_of_neg_of_pos ha (nab ha.le)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\nh : 0 < c\n\u22a2 0 \u2264 c * b \u2194 0 \u2264 b\n[PROOFSTEP]\nsimpa using (mul_le_mul_left h : c * 0 \u2264 c * b \u2194 0 \u2264 b)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\nh : 0 < c\n\u22a2 0 \u2264 b * c \u2194 0 \u2264 b\n[PROOFSTEP]\nsimpa using (mul_le_mul_right h : 0 * c \u2264 b * c \u2194 0 \u2264 b)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\n\u22a2 bit0 a \u2264 bit0 b \u2194 a \u2264 b\n[PROOFSTEP]\nrw [bit0, bit0, \u2190 two_mul, \u2190 two_mul, mul_le_mul_left (zero_lt_two : 0 < (2 : \u03b1))]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\n\u22a2 bit0 a < bit0 b \u2194 a < b\n[PROOFSTEP]\nrw [bit0, bit0, \u2190 two_mul, \u2190 two_mul, mul_lt_mul_left (zero_lt_two : 0 < (2 : \u03b1))]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\n\u22a2 1 \u2264 bit1 a \u2194 0 \u2264 a\n[PROOFSTEP]\nrw [bit1, le_add_iff_nonneg_left, bit0, \u2190 two_mul, zero_le_mul_left (zero_lt_two : 0 < (2 : \u03b1))]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\n\u22a2 1 < bit1 a \u2194 0 < a\n[PROOFSTEP]\nrw [bit1, lt_add_iff_pos_left, bit0, \u2190 two_mul, zero_lt_mul_left (zero_lt_two : 0 < (2 : \u03b1))]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\n\u22a2 0 \u2264 bit0 a \u2194 0 \u2264 a\n[PROOFSTEP]\nrw [bit0, \u2190 two_mul, zero_le_mul_left (zero_lt_two : 0 < (2 : \u03b1))]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\na b c d : \u03b1\n\u22a2 0 < bit0 a \u2194 0 < a\n[PROOFSTEP]\nrw [bit0, \u2190 two_mul, zero_lt_mul_left (zero_lt_two : 0 < (2 : \u03b1))]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na b c : \u03b1\nsrc\u271d : LinearOrderedRing \u03b1 := inst\u271d\n\u22a2 \u2200 {a b : \u03b1}, a * b = 0 \u2192 a = 0 \u2228 b = 0\n[PROOFSTEP]\nintro a b hab\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c : \u03b1\nsrc\u271d : LinearOrderedRing \u03b1 := inst\u271d\na b : \u03b1\nhab : a * b = 0\n\u22a2 a = 0 \u2228 b = 0\n[PROOFSTEP]\nrefine' Decidable.or_iff_not_and_not.2 fun h => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c : \u03b1\nsrc\u271d : LinearOrderedRing \u03b1 := inst\u271d\na b : \u03b1\nhab : a * b = 0\nh : \u00aca = 0 \u2227 \u00acb = 0\n\u22a2 False\n[PROOFSTEP]\nrevert hab\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c : \u03b1\nsrc\u271d : LinearOrderedRing \u03b1 := inst\u271d\na b : \u03b1\nh : \u00aca = 0 \u2227 \u00acb = 0\n\u22a2 a * b = 0 \u2192 False\n[PROOFSTEP]\ncases' lt_or_gt_of_ne h.1 with ha ha\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c : \u03b1\nsrc\u271d : LinearOrderedRing \u03b1 := inst\u271d\na b : \u03b1\nh : \u00aca = 0 \u2227 \u00acb = 0\nha : a < 0\n\u22a2 a * b = 0 \u2192 False\n[PROOFSTEP]\ncases' lt_or_gt_of_ne h.2 with hb hb\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c : \u03b1\nsrc\u271d : LinearOrderedRing \u03b1 := inst\u271d\na b : \u03b1\nh : \u00aca = 0 \u2227 \u00acb = 0\nha : a > 0\n\u22a2 a * b = 0 \u2192 False\n[PROOFSTEP]\ncases' lt_or_gt_of_ne h.2 with hb hb\n[GOAL]\ncase inl.inl\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c : \u03b1\nsrc\u271d : LinearOrderedRing \u03b1 := inst\u271d\na b : \u03b1\nh : \u00aca = 0 \u2227 \u00acb = 0\nha : a < 0\nhb : b < 0\n\u22a2 a * b = 0 \u2192 False\ncase inl.inr\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c : \u03b1\nsrc\u271d : LinearOrderedRing \u03b1 := inst\u271d\na b : \u03b1\nh : \u00aca = 0 \u2227 \u00acb = 0\nha : a < 0\nhb : b > 0\n\u22a2 a * b = 0 \u2192 False\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c : \u03b1\nsrc\u271d : LinearOrderedRing \u03b1 := inst\u271d\na b : \u03b1\nh : \u00aca = 0 \u2227 \u00acb = 0\nha : a > 0\nhb : b < 0\n\u22a2 a * b = 0 \u2192 False\ncase inr.inr\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c : \u03b1\nsrc\u271d : LinearOrderedRing \u03b1 := inst\u271d\na b : \u03b1\nh : \u00aca = 0 \u2227 \u00acb = 0\nha : a > 0\nhb : b > 0\n\u22a2 a * b = 0 \u2192 False\n[PROOFSTEP]\nexacts [(mul_pos_of_neg_of_neg ha hb).ne.symm, (mul_neg_of_neg_of_pos ha hb).ne, (mul_neg_of_pos_of_neg ha hb).ne,\n (mul_pos ha hb).ne.symm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c\u271d : \u03b1\nsrc\u271d : Nontrivial \u03b1 := inferInstance\na b c : \u03b1\nha : a \u2260 0\nh : a * b = a * c\n\u22a2 b = c\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 mul_sub] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c\u271d : \u03b1\nsrc\u271d : Nontrivial \u03b1 := inferInstance\na b c : \u03b1\nha : a \u2260 0\nh\u271d : a * b = a * c\nh : a * (b - c) = 0\n\u22a2 b = c\n[PROOFSTEP]\nexact sub_eq_zero.1 ((eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_left ha)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c\u271d : \u03b1\nsrc\u271d : Nontrivial \u03b1 := inferInstance\na b c : \u03b1\nhb : b \u2260 0\nh : a * b = c * b\n\u22a2 a = c\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 sub_mul] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c\u271d : \u03b1\nsrc\u271d : Nontrivial \u03b1 := inferInstance\na b c : \u03b1\nhb : b \u2260 0\nh\u271d : a * b = c * b\nh : (a - c) * b = 0\n\u22a2 a = c\n[PROOFSTEP]\nexact sub_eq_zero.1 ((eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_right hb)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na b c : \u03b1\n\u22a2 a * b < 0 \u2194 0 < a \u2227 b < 0 \u2228 a < 0 \u2227 0 < b\n[PROOFSTEP]\nrw [\u2190 neg_pos, neg_mul_eq_mul_neg, mul_pos_iff, neg_pos, neg_lt_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c\u271d a b c : \u03b1\n\u22a2 0 \u2264 a * b \u2228 0 \u2264 b * c \u2228 0 \u2264 c * a\n[PROOFSTEP]\niterate 3 rw [mul_nonneg_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c\u271d a b c : \u03b1\n\u22a2 0 \u2264 a * b \u2228 0 \u2264 b * c \u2228 0 \u2264 c * a\n[PROOFSTEP]\nrw [mul_nonneg_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c\u271d a b c : \u03b1\n\u22a2 (0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0) \u2228 0 \u2264 b * c \u2228 0 \u2264 c * a\n[PROOFSTEP]\nrw [mul_nonneg_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c\u271d a b c : \u03b1\n\u22a2 (0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0) \u2228 (0 \u2264 b \u2227 0 \u2264 c \u2228 b \u2264 0 \u2227 c \u2264 0) \u2228 0 \u2264 c * a\n[PROOFSTEP]\nrw [mul_nonneg_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c\u271d a b c : \u03b1\n\u22a2 (0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0) \u2228 (0 \u2264 b \u2227 0 \u2264 c \u2228 b \u2264 0 \u2227 c \u2264 0) \u2228 0 \u2264 c \u2227 0 \u2264 a \u2228 c \u2264 0 \u2227 a \u2264 0\n[PROOFSTEP]\nhave or_a := le_total 0 a\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c\u271d a b c : \u03b1\nor_a : 0 \u2264 a \u2228 a \u2264 0\n\u22a2 (0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0) \u2228 (0 \u2264 b \u2227 0 \u2264 c \u2228 b \u2264 0 \u2227 c \u2264 0) \u2228 0 \u2264 c \u2227 0 \u2264 a \u2228 c \u2264 0 \u2227 a \u2264 0\n[PROOFSTEP]\nhave or_b := le_total 0 b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c\u271d a b c : \u03b1\nor_a : 0 \u2264 a \u2228 a \u2264 0\nor_b : 0 \u2264 b \u2228 b \u2264 0\n\u22a2 (0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0) \u2228 (0 \u2264 b \u2227 0 \u2264 c \u2228 b \u2264 0 \u2227 c \u2264 0) \u2228 0 \u2264 c \u2227 0 \u2264 a \u2228 c \u2264 0 \u2227 a \u2264 0\n[PROOFSTEP]\nhave or_c := le_total 0 c\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b\u271d c\u271d a b c : \u03b1\nor_a : 0 \u2264 a \u2228 a \u2264 0\nor_b : 0 \u2264 b \u2228 b \u2264 0\nor_c : 0 \u2264 c \u2228 c \u2264 0\n\u22a2 (0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0) \u2228 (0 \u2264 b \u2227 0 \u2264 c \u2228 b \u2264 0 \u2227 c \u2264 0) \u2228 0 \u2264 c \u2227 0 \u2264 a \u2228 c \u2264 0 \u2227 a \u2264 0\n[PROOFSTEP]\nexact\n Or.elim or_c\n (fun (h0 : 0 \u2264 c) =>\n Or.elim or_b\n (fun (h1 : 0 \u2264 b) =>\n Or.elim or_a (fun (h2 : 0 \u2264 a) => Or.inl (Or.inl \u27e8h2, h1\u27e9))\n (fun (_ : a \u2264 0) => Or.inr (Or.inl (Or.inl \u27e8h1, h0\u27e9))))\n (fun (h1 : b \u2264 0) =>\n Or.elim or_a (fun (h3 : 0 \u2264 a) => Or.inr (Or.inr (Or.inl \u27e8h0, h3\u27e9)))\n (fun (h3 : a \u2264 0) => Or.inl (Or.inr \u27e8h3, h1\u27e9))))\n (fun (h0 : c \u2264 0) =>\n Or.elim or_b\n (fun (h4 : 0 \u2264 b) =>\n Or.elim or_a (fun (h5 : 0 \u2264 a) => Or.inl (Or.inl \u27e8h5, h4\u27e9))\n (fun (h5 : a \u2264 0) => Or.inr (Or.inr (Or.inr \u27e8h0, h5\u27e9))))\n (fun (h4 : b \u2264 0) =>\n Or.elim or_a (fun (_ : 0 \u2264 a) => Or.inr (Or.inl (Or.inr \u27e8h4, h0\u27e9)))\n (fun (h6 : a \u2264 0) => Or.inl (Or.inr \u27e8h6, h4\u27e9))))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na b c : \u03b1\n\u22a2 a * b \u2264 0 \u2194 0 \u2264 a \u2227 b \u2264 0 \u2228 a \u2264 0 \u2227 0 \u2264 b\n[PROOFSTEP]\nrw [\u2190 neg_nonneg, neg_mul_eq_mul_neg, mul_nonneg_iff, neg_nonneg, neg_nonpos]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na b c : \u03b1\n\u22a2 -a \u2264 a \u2194 0 \u2264 a\n[PROOFSTEP]\nsimp [neg_le_iff_add_nonneg, \u2190 two_mul, mul_nonneg_iff, zero_le_one, (zero_lt_two' \u03b1).not_le]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na b c : \u03b1\n\u22a2 -a < a \u2194 0 < a\n[PROOFSTEP]\nsimp [neg_lt_iff_pos_add, \u2190 two_mul, mul_pos_iff, zero_lt_one, (zero_lt_two' \u03b1).not_lt]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na b c : \u03b1\n\u22a2 a \u2264 -a \u2194 - -a \u2264 -a\n[PROOFSTEP]\nrw [neg_neg]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na b c : \u03b1\n\u22a2 a < -a \u2194 - -a < -a\n[PROOFSTEP]\nrw [neg_neg]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na b c : \u03b1\nh : c * a < c * b\nhc : c \u2264 0\n\u22a2 -c * b < -c * a\n[PROOFSTEP]\nrwa [neg_mul, neg_mul, neg_lt_neg_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na b c : \u03b1\nh : a * c < b * c\nhc : c \u2264 0\n\u22a2 b * -c < a * -c\n[PROOFSTEP]\nrwa [mul_neg, mul_neg, neg_lt_neg_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b c a : \u03b1\n\u22a2 0 < a * a \u2194 a \u2260 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b c a : \u03b1\n\u22a2 0 < a * a \u2192 a \u2260 0\n[PROOFSTEP]\nrintro h rfl\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na b c : \u03b1\nh : 0 < 0 * 0\n\u22a2 False\n[PROOFSTEP]\nrw [mul_zero] at h \n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na b c : \u03b1\nh : 0 < 0\n\u22a2 False\n[PROOFSTEP]\nexact h.false\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b c a : \u03b1\n\u22a2 a \u2260 0 \u2192 0 < a * a\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b c a : \u03b1\nh : a \u2260 0\n\u22a2 0 < a * a\n[PROOFSTEP]\ncases' h.lt_or_lt with h h\n[GOAL]\ncase mpr.inl\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na\u271d b c a : \u03b1\nh\u271d : a \u2260 0\nh : a < 0\n\u22a2 0 < a * a\ncase mpr.inr \u03b1 : Type u \u03b2 : Type u_1 inst\u271d : LinearOrderedRing \u03b1 a\u271d b c a : \u03b1 h\u271d : a \u2260 0 h : 0 < a \u22a2 0 < a * a\n[PROOFSTEP]\nexacts [mul_pos_of_neg_of_neg h h, mul_pos h h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na b c x y : \u03b1\n\u22a2 x * x + y * y = 0 \u2194 x = 0 \u2227 y = 0\n[PROOFSTEP]\nrw [add_eq_zero_iff', mul_self_eq_zero, mul_self_eq_zero]\n[GOAL]\ncase ha\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na b c x y : \u03b1\n\u22a2 0 \u2264 x * x\n[PROOFSTEP]\napply mul_self_nonneg\n[GOAL]\ncase hb\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\na b c x y : \u03b1\n\u22a2 0 \u2264 y * y\n[PROOFSTEP]\napply mul_self_nonneg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedCommRing \u03b1\na b\u271d c\u271d d b c : \u03b1\nha : 0 \u2264 a\nhd : 0 \u2264 d\nba : b * a \u2264 max d b * max c a\ncd : c * d \u2264 max a c * max b d\n\u22a2 a * b \u2264 max a c * max d b\n[PROOFSTEP]\nsimpa [mul_comm, max_comm] using ba\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : LinearOrderedCommRing \u03b1\na b\u271d c\u271d d b c : \u03b1\nha : 0 \u2264 a\nhd : 0 \u2264 d\nba : b * a \u2264 max d b * max c a\ncd : c * d \u2264 max a c * max b d\n\u22a2 d * c \u2264 max a c * max d b\n[PROOFSTEP]\nsimpa [mul_comm, max_comm] using cd\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Ring.Defs", "llama_tokens": 13165, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432182679956, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.4304715784667369}} {"text": "[GOAL]\ns : \u211d\n\u22a2 (fun x => exp (-x) * x ^ s) =o[atTop] fun x => exp (-(1 / 2) * x)\n[PROOFSTEP]\nrefine' isLittleO_of_tendsto (fun x hx => _) _\n[GOAL]\ncase refine'_1\ns x : \u211d\nhx : exp (-(1 / 2) * x) = 0\n\u22a2 exp (-x) * x ^ s = 0\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase refine'_1.h\ns x : \u211d\nhx : exp (-(1 / 2) * x) = 0\n\u22a2 False\n[PROOFSTEP]\nexact (exp_pos (-(1 / 2) * x)).ne' hx\n[GOAL]\ncase refine'_2\ns : \u211d\n\u22a2 Tendsto (fun x => exp (-x) * x ^ s / exp (-(1 / 2) * x)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : (fun x : \u211d => exp (-x) * x ^ s / exp (-(1 / 2) * x)) = (fun x : \u211d => exp (1 / 2 * x) / x ^ s)\u207b\u00b9 :=\n by\n ext1 x\n field_simp [exp_ne_zero, exp_neg, \u2190 Real.exp_add]\n left\n ring\n[GOAL]\ns : \u211d\n\u22a2 (fun x => exp (-x) * x ^ s / exp (-(1 / 2) * x)) = (fun x => exp (1 / 2 * x) / x ^ s)\u207b\u00b9\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\ns x : \u211d\n\u22a2 exp (-x) * x ^ s / exp (-(1 / 2) * x) = (fun x => exp (1 / 2 * x) / x ^ s)\u207b\u00b9 x\n[PROOFSTEP]\nfield_simp [exp_ne_zero, exp_neg, \u2190 Real.exp_add]\n[GOAL]\ncase h\ns x : \u211d\n\u22a2 x = x * 2 + -x \u2228 x ^ s = 0\n[PROOFSTEP]\nleft\n[GOAL]\ncase h.h\ns x : \u211d\n\u22a2 x = x * 2 + -x\n[PROOFSTEP]\nring\n[GOAL]\ncase refine'_2\ns : \u211d\nthis : (fun x => exp (-x) * x ^ s / exp (-(1 / 2) * x)) = (fun x => exp (1 / 2 * x) / x ^ s)\u207b\u00b9\n\u22a2 Tendsto (fun x => exp (-x) * x ^ s / exp (-(1 / 2) * x)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase refine'_2\ns : \u211d\nthis : (fun x => exp (-x) * x ^ s / exp (-(1 / 2) * x)) = (fun x => exp (1 / 2 * x) / x ^ s)\u207b\u00b9\n\u22a2 Tendsto (fun x => exp (1 / 2 * x) / x ^ s)\u207b\u00b9 atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact (tendsto_exp_mul_div_rpow_atTop s (1 / 2) one_half_pos).inv_tendsto_atTop\n[GOAL]\ns : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => exp (-x) * x ^ (s - 1)) (Ioi 0)\n[PROOFSTEP]\nrw [\u2190 Ioc_union_Ioi_eq_Ioi (@zero_le_one \u211d _ _ _ _), integrableOn_union]\n[GOAL]\ns : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => exp (-x) * x ^ (s - 1)) (Ioc 0 1) \u2227 IntegrableOn (fun x => exp (-x) * x ^ (s - 1)) (Ioi 1)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\ns : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => exp (-x) * x ^ (s - 1)) (Ioc 0 1)\n[PROOFSTEP]\nrw [\u2190 integrableOn_Icc_iff_integrableOn_Ioc]\n[GOAL]\ncase left\ns : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => exp (-x) * x ^ (s - 1)) (Icc 0 1)\n[PROOFSTEP]\nrefine' IntegrableOn.continuousOn_mul continuousOn_id.neg.exp _ isCompact_Icc\n[GOAL]\ncase left\ns : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => x ^ (s - 1)) (Icc 0 1)\n[PROOFSTEP]\nrefine' (intervalIntegrable_iff_integrable_Icc_of_le zero_le_one).mp _\n[GOAL]\ncase left\ns : \u211d\nh : 0 < s\n\u22a2 IntervalIntegrable (fun x => x ^ (s - 1)) volume 0 1\n[PROOFSTEP]\nexact intervalIntegrable_rpow' (by linarith)\n[GOAL]\ns : \u211d\nh : 0 < s\n\u22a2 -1 < s - 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase right\ns : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => exp (-x) * x ^ (s - 1)) (Ioi 1)\n[PROOFSTEP]\nrefine' integrable_of_isBigO_exp_neg one_half_pos _ (Gamma_integrand_isLittleO _).isBigO\n[GOAL]\ncase right\ns : \u211d\nh : 0 < s\n\u22a2 ContinuousOn (fun x => exp (-x) * x ^ (s - 1)) (Ici 1)\n[PROOFSTEP]\nrefine' continuousOn_id.neg.exp.mul (continuousOn_id.rpow_const _)\n[GOAL]\ncase right\ns : \u211d\nh : 0 < s\n\u22a2 \u2200 (x : \u211d), x \u2208 Ici 1 \u2192 id x \u2260 0 \u2228 0 \u2264 s - 1\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase right\ns : \u211d\nh : 0 < s\nx : \u211d\nhx : x \u2208 Ici 1\n\u22a2 id x \u2260 0 \u2228 0 \u2264 s - 1\n[PROOFSTEP]\nexact Or.inl ((zero_lt_one : (0 : \u211d) < 1).trans_le hx).ne'\n[GOAL]\ns : \u2102\nhs : 0 < s.re\n\u22a2 IntegrableOn (fun x => \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) (Ioi 0)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\ns : \u2102\nhs : 0 < s.re\n\u22a2 AEStronglyMeasurable (fun x => \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) (Measure.restrict volume (Ioi 0))\n[PROOFSTEP]\nrefine' ContinuousOn.aestronglyMeasurable _ measurableSet_Ioi\n[GOAL]\ncase left\ns : \u2102\nhs : 0 < s.re\n\u22a2 ContinuousOn (fun x => \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) (Ioi 0)\n[PROOFSTEP]\napply (continuous_ofReal.comp continuous_neg.exp).continuousOn.mul\n[GOAL]\ncase left\ns : \u2102\nhs : 0 < s.re\n\u22a2 ContinuousOn (fun x => \u2191x ^ (s - 1)) (Ioi 0)\n[PROOFSTEP]\napply ContinuousAt.continuousOn\n[GOAL]\ncase left.hcont\ns : \u2102\nhs : 0 < s.re\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioi 0 \u2192 ContinuousAt (fun x => \u2191x ^ (s - 1)) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase left.hcont\ns : \u2102\nhs : 0 < s.re\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 ContinuousAt (fun x => \u2191x ^ (s - 1)) x\n[PROOFSTEP]\nhave : ContinuousAt (fun x : \u2102 => x ^ (s - 1)) \u2191x := by apply continuousAt_cpow_const; rw [ofReal_re]; exact Or.inl hx\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 ContinuousAt (fun x => x ^ (s - 1)) \u2191x\n[PROOFSTEP]\napply continuousAt_cpow_const\n[GOAL]\ncase ha\ns : \u2102\nhs : 0 < s.re\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 0 < (\u2191x).re \u2228 (\u2191x).im \u2260 0\n[PROOFSTEP]\nrw [ofReal_re]\n[GOAL]\ncase ha\ns : \u2102\nhs : 0 < s.re\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 0 < x \u2228 (\u2191x).im \u2260 0\n[PROOFSTEP]\nexact Or.inl hx\n[GOAL]\ncase left.hcont\ns : \u2102\nhs : 0 < s.re\nx : \u211d\nhx : x \u2208 Ioi 0\nthis : ContinuousAt (fun x => x ^ (s - 1)) \u2191x\n\u22a2 ContinuousAt (fun x => \u2191x ^ (s - 1)) x\n[PROOFSTEP]\nexact ContinuousAt.comp this continuous_ofReal.continuousAt\n[GOAL]\ncase right\ns : \u2102\nhs : 0 < s.re\n\u22a2 HasFiniteIntegral fun x => \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\n[PROOFSTEP]\nrw [\u2190 hasFiniteIntegral_norm_iff]\n[GOAL]\ncase right\ns : \u2102\nhs : 0 < s.re\n\u22a2 HasFiniteIntegral fun a => \u2016\u2191(Real.exp (-a)) * \u2191a ^ (s - 1)\u2016\n[PROOFSTEP]\nrefine' HasFiniteIntegral.congr (Real.GammaIntegral_convergent hs).2 _\n[GOAL]\ncase right\ns : \u2102\nhs : 0 < s.re\n\u22a2 (fun x => Real.exp (-x) * x ^ (s.re - 1)) =\u1d50[Measure.restrict volume (Ioi 0)] fun a =>\n \u2016\u2191(Real.exp (-a)) * \u2191a ^ (s - 1)\u2016\n[PROOFSTEP]\nrefine' (ae_restrict_iff' measurableSet_Ioi).mpr (ae_of_all _ fun x hx => _)\n[GOAL]\ncase right\ns : \u2102\nhs : 0 < s.re\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 (fun x => Real.exp (-x) * x ^ (s.re - 1)) x = (fun a => \u2016\u2191(Real.exp (-a)) * \u2191a ^ (s - 1)\u2016) x\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase right\ns : \u2102\nhs : 0 < s.re\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 Real.exp (-x) * x ^ (s.re - 1) = \u2016\u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\u2016\n[PROOFSTEP]\nrw [norm_eq_abs, map_mul, abs_of_nonneg <| le_of_lt <| exp_pos <| -x, abs_cpow_eq_rpow_re_of_pos hx _]\n[GOAL]\ncase right\ns : \u2102\nhs : 0 < s.re\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 Real.exp (-x) * x ^ (s.re - 1) = Real.exp (-x) * x ^ (s - 1).re\n[PROOFSTEP]\nsimp\n[GOAL]\ns : \u2102\n\u22a2 GammaIntegral (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaIntegral s)\n[PROOFSTEP]\nrw [GammaIntegral, GammaIntegral, \u2190 integral_conj]\n[GOAL]\ns : \u2102\n\u22a2 \u222b (x : \u211d) in Ioi 0, \u2191(Real.exp (-x)) * \u2191x ^ (\u2191(starRingEnd \u2102) s - 1) =\n \u222b (a : \u211d) in Ioi 0, \u2191(starRingEnd \u2102) (\u2191(Real.exp (-a)) * \u2191a ^ (s - 1))\n[PROOFSTEP]\nrefine' set_integral_congr measurableSet_Ioi fun x hx => _\n[GOAL]\ns : \u2102\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 \u2191(Real.exp (-x)) * \u2191x ^ (\u2191(starRingEnd \u2102) s - 1) = \u2191(starRingEnd \u2102) (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n[PROOFSTEP]\ndsimp only\n[GOAL]\ns : \u2102\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 \u2191(Real.exp (-x)) * \u2191x ^ (\u2191(starRingEnd \u2102) s - 1) = \u2191(starRingEnd \u2102) (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n[PROOFSTEP]\nrw [RingHom.map_mul, conj_ofReal, cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)),\n cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), \u2190 exp_conj, RingHom.map_mul, \u2190 ofReal_log (le_of_lt hx),\n conj_ofReal, RingHom.map_sub, RingHom.map_one]\n[GOAL]\ns : \u211d\n\u22a2 GammaIntegral \u2191s = \u2191(\u222b (x : \u211d) in Ioi 0, Real.exp (-x) * x ^ (s - 1))\n[PROOFSTEP]\nhave : \u2200 r : \u211d, Complex.ofReal' r = @IsROrC.ofReal \u2102 _ r := fun r => rfl\n[GOAL]\ns : \u211d\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\n\u22a2 GammaIntegral \u2191s = \u2191(\u222b (x : \u211d) in Ioi 0, Real.exp (-x) * x ^ (s - 1))\n[PROOFSTEP]\nrw [GammaIntegral]\n[GOAL]\ns : \u211d\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\n\u22a2 \u222b (x : \u211d) in Ioi 0, \u2191(Real.exp (-x)) * \u2191x ^ (\u2191s - 1) = \u2191(\u222b (x : \u211d) in Ioi 0, Real.exp (-x) * x ^ (s - 1))\n[PROOFSTEP]\nconv_rhs => rw [this, \u2190 _root_.integral_ofReal]\n[GOAL]\ns : \u211d\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\n| \u2191(\u222b (x : \u211d) in Ioi 0, Real.exp (-x) * x ^ (s - 1))\n[PROOFSTEP]\nrw [this, \u2190 _root_.integral_ofReal]\n[GOAL]\ns : \u211d\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\n| \u2191(\u222b (x : \u211d) in Ioi 0, Real.exp (-x) * x ^ (s - 1))\n[PROOFSTEP]\nrw [this, \u2190 _root_.integral_ofReal]\n[GOAL]\ns : \u211d\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\n| \u2191(\u222b (x : \u211d) in Ioi 0, Real.exp (-x) * x ^ (s - 1))\n[PROOFSTEP]\nrw [this, \u2190 _root_.integral_ofReal]\n[GOAL]\ns : \u211d\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\n\u22a2 \u222b (x : \u211d) in Ioi 0, \u2191(Real.exp (-x)) * \u2191x ^ (\u2191s - 1) = \u222b (a : \u211d) in Ioi 0, \u2191(Real.exp (-a) * a ^ (s - 1))\n[PROOFSTEP]\nrefine' set_integral_congr measurableSet_Ioi _\n[GOAL]\ns : \u211d\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\n\u22a2 EqOn (fun x => \u2191(Real.exp (-x)) * \u2191x ^ (\u2191s - 1)) (fun a => \u2191(Real.exp (-a) * a ^ (s - 1))) (Ioi 0)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ns : \u211d\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 (fun x => \u2191(Real.exp (-x)) * \u2191x ^ (\u2191s - 1)) x = (fun a => \u2191(Real.exp (-a) * a ^ (s - 1))) x\n[PROOFSTEP]\ndsimp only\n[GOAL]\ns : \u211d\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 \u2191(Real.exp (-x)) * \u2191x ^ (\u2191s - 1) = \u2191(Real.exp (-x) * x ^ (s - 1))\n[PROOFSTEP]\nconv_rhs => rw [\u2190 this]\n[GOAL]\ns : \u211d\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\nx : \u211d\nhx : x \u2208 Ioi 0\n| \u2191(Real.exp (-x) * x ^ (s - 1))\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\ns : \u211d\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\nx : \u211d\nhx : x \u2208 Ioi 0\n| \u2191(Real.exp (-x) * x ^ (s - 1))\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\ns : \u211d\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\nx : \u211d\nhx : x \u2208 Ioi 0\n| \u2191(Real.exp (-x) * x ^ (s - 1))\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\ns : \u211d\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 \u2191(Real.exp (-x)) * \u2191x ^ (\u2191s - 1) = \u2191(Real.exp (-x) * x ^ (s - 1))\n[PROOFSTEP]\nrw [ofReal_mul, ofReal_cpow (mem_Ioi.mp hx).le]\n[GOAL]\ns : \u211d\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 \u2191(Real.exp (-x)) * \u2191x ^ (\u2191s - 1) = \u2191(Real.exp (-x)) * \u2191x ^ \u2191(s - 1)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 GammaIntegral 1 = 1\n[PROOFSTEP]\nsimpa only [\u2190 ofReal_one, GammaIntegral_ofReal, ofReal_inj, sub_self, rpow_zero, mul_one] using\n integral_exp_neg_Ioi_zero\n[GOAL]\ns : \u2102\nX : \u211d\nhs : 0 < s.re\nhX : 0 \u2264 X\n\u22a2 IntervalIntegrable (fun x => \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) volume 0 X\n[PROOFSTEP]\nrw [intervalIntegrable_iff_integrable_Ioc_of_le hX]\n[GOAL]\ns : \u2102\nX : \u211d\nhs : 0 < s.re\nhX : 0 \u2264 X\n\u22a2 IntegrableOn (fun x => \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) (Ioc 0 X)\n[PROOFSTEP]\nexact IntegrableOn.mono_set (GammaIntegral_convergent hs) Ioc_subset_Ioi_self\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\n\u22a2 IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s)) volume 0 X\n[PROOFSTEP]\nconvert (Gamma_integrand_interval_integrable (s + 1) _ hX).neg\n[GOAL]\ncase h.e'_3.h\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nx\u271d : \u211d\n\u22a2 -(\u2191(Real.exp (-x\u271d)) * \u2191x\u271d ^ s) = (-fun x => \u2191(Real.exp (-x)) * \u2191x ^ (s + 1 - 1)) x\u271d\n[PROOFSTEP]\nsimp only [ofReal_exp, ofReal_neg, add_sub_cancel]\n[GOAL]\ncase h.e'_3.h\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nx\u271d : \u211d\n\u22a2 -(exp (-\u2191x\u271d) * \u2191x\u271d ^ s) = (-fun x => exp (-\u2191x) * \u2191x ^ s) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\n\u22a2 0 < (s + 1).re\n[PROOFSTEP]\nsimp only [add_re, one_re]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\n\u22a2 0 < s.re + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\n\u22a2 IntervalIntegrable (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 Y\n[PROOFSTEP]\nhave : (fun x => (-x).exp * (s * x ^ (s - 1)) : \u211d \u2192 \u2102) = (fun x => s * ((-x).exp * x ^ (s - 1)) : \u211d \u2192 \u2102) := by ext1;\n ring\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\n\u22a2 (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n[PROOFSTEP]\next1\n[GOAL]\ncase h\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nx\u271d : \u211d\n\u22a2 \u2191(Real.exp (-x\u271d)) * (s * \u2191x\u271d ^ (s - 1)) = s * (\u2191(Real.exp (-x\u271d)) * \u2191x\u271d ^ (s - 1))\n[PROOFSTEP]\nring\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n\u22a2 IntervalIntegrable (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 Y\n[PROOFSTEP]\nrw [this, intervalIntegrable_iff_integrable_Ioc_of_le hY]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n\u22a2 IntegrableOn (fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))) (Ioc 0 Y)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n\u22a2 AEStronglyMeasurable (fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))) (Measure.restrict volume (Ioc 0 Y))\n[PROOFSTEP]\nrefine' (continuousOn_const.mul _).aestronglyMeasurable measurableSet_Ioc\n[GOAL]\ncase left\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n\u22a2 ContinuousOn (fun x => \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) (Ioc 0 Y)\n[PROOFSTEP]\napply (continuous_ofReal.comp continuous_neg.exp).continuousOn.mul\n[GOAL]\ncase left\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n\u22a2 ContinuousOn (fun x => \u2191x ^ (s - 1)) (Ioc 0 Y)\n[PROOFSTEP]\napply ContinuousAt.continuousOn\n[GOAL]\ncase left.hcont\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioc 0 Y \u2192 ContinuousAt (fun x => \u2191x ^ (s - 1)) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase left.hcont\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\nx : \u211d\nhx : x \u2208 Ioc 0 Y\n\u22a2 ContinuousAt (fun x => \u2191x ^ (s - 1)) x\n[PROOFSTEP]\nrefine' (_ : ContinuousAt (fun x : \u2102 => x ^ (s - 1)) _).comp continuous_ofReal.continuousAt\n[GOAL]\ncase left.hcont\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\nx : \u211d\nhx : x \u2208 Ioc 0 Y\n\u22a2 ContinuousAt (fun x => x ^ (s - 1)) \u2191x\n[PROOFSTEP]\napply continuousAt_cpow_const\n[GOAL]\ncase left.hcont.ha\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\nx : \u211d\nhx : x \u2208 Ioc 0 Y\n\u22a2 0 < (\u2191x).re \u2228 (\u2191x).im \u2260 0\n[PROOFSTEP]\nrw [ofReal_re]\n[GOAL]\ncase left.hcont.ha\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\nx : \u211d\nhx : x \u2208 Ioc 0 Y\n\u22a2 0 < x \u2228 (\u2191x).im \u2260 0\n[PROOFSTEP]\nexact Or.inl hx.1\n[GOAL]\ncase right\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n\u22a2 HasFiniteIntegral fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n[PROOFSTEP]\nrw [\u2190 hasFiniteIntegral_norm_iff]\n[GOAL]\ncase right\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n\u22a2 HasFiniteIntegral fun a => \u2016s * (\u2191(Real.exp (-a)) * \u2191a ^ (s - 1))\u2016\n[PROOFSTEP]\nsimp_rw [norm_eq_abs, map_mul]\n[GOAL]\ncase right\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n\u22a2 HasFiniteIntegral fun a => \u2191abs s * (\u2191abs \u2191(Real.exp (-a)) * \u2191abs (\u2191a ^ (s - 1)))\n[PROOFSTEP]\nrefine' (((Real.GammaIntegral_convergent hs).mono_set Ioc_subset_Ioi_self).hasFiniteIntegral.congr _).const_mul _\n[GOAL]\ncase right\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n\u22a2 (fun x => Real.exp (-x) * x ^ (s.re - 1)) =\u1d50[Measure.restrict volume (Ioc 0 Y)] fun a =>\n \u2191abs \u2191(Real.exp (-a)) * \u2191abs (\u2191a ^ (s - 1))\n[PROOFSTEP]\nrw [EventuallyEq, ae_restrict_iff']\n[GOAL]\ncase right\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 Ioc 0 Y \u2192 Real.exp (-x) * x ^ (s.re - 1) = \u2191abs \u2191(Real.exp (-x)) * \u2191abs (\u2191x ^ (s - 1))\n[PROOFSTEP]\napply ae_of_all\n[GOAL]\ncase right.a\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioc 0 Y \u2192 Real.exp (-a) * a ^ (s.re - 1) = \u2191abs \u2191(Real.exp (-a)) * \u2191abs (\u2191a ^ (s - 1))\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase right.a\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\nx : \u211d\nhx : x \u2208 Ioc 0 Y\n\u22a2 Real.exp (-x) * x ^ (s.re - 1) = \u2191abs \u2191(Real.exp (-x)) * \u2191abs (\u2191x ^ (s - 1))\n[PROOFSTEP]\nrw [abs_of_nonneg (exp_pos _).le, abs_cpow_eq_rpow_re_of_pos hx.1]\n[GOAL]\ncase right.a\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\nx : \u211d\nhx : x \u2208 Ioc 0 Y\n\u22a2 Real.exp (-x) * x ^ (s.re - 1) = Real.exp (-x) * x ^ (s - 1).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\ns : \u2102\nhs : 0 < s.re\nY : \u211d\nhY : 0 \u2264 Y\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n\u22a2 MeasurableSet (Ioc 0 Y)\n[PROOFSTEP]\nexact measurableSet_Ioc\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\n\u22a2 partialGamma (s + 1) X = s * partialGamma s X - \u2191(Real.exp (-X)) * \u2191X ^ s\n[PROOFSTEP]\nrw [partialGamma, partialGamma, add_sub_cancel]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\n\u22a2 \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s =\n (s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) - \u2191(Real.exp (-X)) * \u2191X ^ s\n[PROOFSTEP]\nhave F_der_I :\n \u2200 x : \u211d,\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => (-x).exp * x ^ s : \u211d \u2192 \u2102) (-((-x).exp * x ^ s) + (-x).exp * (s * x ^ (s - 1))) x :=\n by\n intro x hx\n have d1 : HasDerivAt (fun y : \u211d => (-y).exp) (-(-x).exp) x := by simpa using (hasDerivAt_neg x).exp\n have d2 : HasDerivAt (fun y : \u211d => \u2191y ^ s) (s * x ^ (s - 1)) x :=\n by\n have t := @HasDerivAt.cpow_const _ _ _ s (hasDerivAt_id \u2191x) ?_\n simpa only [mul_one] using t.comp_ofReal\n simpa only [id.def, ofReal_re, ofReal_im, Ne.def, eq_self_iff_true, not_true, or_false_iff, mul_one] using hx.1\n simpa only [ofReal_neg, neg_mul] using d1.ofReal_comp.mul d2\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\n\u22a2 \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nx : \u211d\nhx : x \u2208 Ioo 0 X\n\u22a2 HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s) (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)))\n x\n[PROOFSTEP]\nhave d1 : HasDerivAt (fun y : \u211d => (-y).exp) (-(-x).exp) x := by simpa using (hasDerivAt_neg x).exp\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nx : \u211d\nhx : x \u2208 Ioo 0 X\n\u22a2 HasDerivAt (fun y => Real.exp (-y)) (-Real.exp (-x)) x\n[PROOFSTEP]\nsimpa using (hasDerivAt_neg x).exp\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nx : \u211d\nhx : x \u2208 Ioo 0 X\nd1 : HasDerivAt (fun y => Real.exp (-y)) (-Real.exp (-x)) x\n\u22a2 HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s) (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)))\n x\n[PROOFSTEP]\nhave d2 : HasDerivAt (fun y : \u211d => \u2191y ^ s) (s * x ^ (s - 1)) x :=\n by\n have t := @HasDerivAt.cpow_const _ _ _ s (hasDerivAt_id \u2191x) ?_\n simpa only [mul_one] using t.comp_ofReal\n simpa only [id.def, ofReal_re, ofReal_im, Ne.def, eq_self_iff_true, not_true, or_false_iff, mul_one] using hx.1\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nx : \u211d\nhx : x \u2208 Ioo 0 X\nd1 : HasDerivAt (fun y => Real.exp (-y)) (-Real.exp (-x)) x\n\u22a2 HasDerivAt (fun y => \u2191y ^ s) (s * \u2191x ^ (s - 1)) x\n[PROOFSTEP]\nhave t := @HasDerivAt.cpow_const _ _ _ s (hasDerivAt_id \u2191x) ?_\n[GOAL]\ncase refine_2\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nx : \u211d\nhx : x \u2208 Ioo 0 X\nd1 : HasDerivAt (fun y => Real.exp (-y)) (-Real.exp (-x)) x\nt : HasDerivAt (fun x => id x ^ s) (s * id \u2191x ^ (s - 1) * 1) \u2191x\n\u22a2 HasDerivAt (fun y => \u2191y ^ s) (s * \u2191x ^ (s - 1)) x\ncase refine_1\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nx : \u211d\nhx : x \u2208 Ioo 0 X\nd1 : HasDerivAt (fun y => Real.exp (-y)) (-Real.exp (-x)) x\n\u22a2 0 < (id \u2191x).re \u2228 (id \u2191x).im \u2260 0\n[PROOFSTEP]\nsimpa only [mul_one] using t.comp_ofReal\n[GOAL]\ncase refine_1\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nx : \u211d\nhx : x \u2208 Ioo 0 X\nd1 : HasDerivAt (fun y => Real.exp (-y)) (-Real.exp (-x)) x\n\u22a2 0 < (id \u2191x).re \u2228 (id \u2191x).im \u2260 0\n[PROOFSTEP]\nsimpa only [id.def, ofReal_re, ofReal_im, Ne.def, eq_self_iff_true, not_true, or_false_iff, mul_one] using hx.1\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nx : \u211d\nhx : x \u2208 Ioo 0 X\nd1 : HasDerivAt (fun y => Real.exp (-y)) (-Real.exp (-x)) x\nd2 : HasDerivAt (fun y => \u2191y ^ s) (s * \u2191x ^ (s - 1)) x\n\u22a2 HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s) (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)))\n x\n[PROOFSTEP]\nsimpa only [ofReal_neg, neg_mul] using d1.ofReal_comp.mul d2\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\n\u22a2 \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s =\n (s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) - \u2191(Real.exp (-X)) * \u2191X ^ s\n[PROOFSTEP]\nhave cont := (continuous_ofReal.comp continuous_neg.exp).mul (continuous_ofReal_cpow_const hs)\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\n\u22a2 \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s =\n (s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) - \u2191(Real.exp (-X)) * \u2191X ^ s\n[PROOFSTEP]\nhave der_ible := (Gamma_integrand_deriv_integrable_A hs hX).add (Gamma_integrand_deriv_integrable_B hs hX)\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\n\u22a2 \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s =\n (s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) - \u2191(Real.exp (-X)) * \u2191X ^ s\n[PROOFSTEP]\nhave int_eval := integral_eq_sub_of_hasDerivAt_of_le hX cont.continuousOn F_der_I der_ible\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\nint_eval :\n \u222b (y : \u211d) in 0 ..X, -(\u2191(Real.exp (-y)) * \u2191y ^ s) + \u2191(Real.exp (-y)) * (s * \u2191y ^ (s - 1)) =\n (ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s - (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s\n\u22a2 \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s =\n (s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) - \u2191(Real.exp (-X)) * \u2191X ^ s\n[PROOFSTEP]\napply_fun fun x : \u2102 => -x at int_eval \n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\nint_eval :\n -\u222b (y : \u211d) in 0 ..X, -(\u2191(Real.exp (-y)) * \u2191y ^ s) + \u2191(Real.exp (-y)) * (s * \u2191y ^ (s - 1)) =\n -((ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s - (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s)\n\u22a2 \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s =\n (s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) - \u2191(Real.exp (-X)) * \u2191X ^ s\n[PROOFSTEP]\nrw [intervalIntegral.integral_add (Gamma_integrand_deriv_integrable_A hs hX) (Gamma_integrand_deriv_integrable_B hs hX),\n intervalIntegral.integral_neg, neg_add, neg_neg] at int_eval \n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\nint_eval :\n (\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s) + -\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)) =\n -((ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s - (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s)\n\u22a2 \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s =\n (s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) - \u2191(Real.exp (-X)) * \u2191X ^ s\n[PROOFSTEP]\nrw [eq_sub_of_add_eq int_eval, sub_neg_eq_add, neg_sub, add_comm, add_sub]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\nint_eval :\n (\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s) + -\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)) =\n -((ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s - (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s)\n\u22a2 (\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) + (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s -\n (ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s =\n (s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) - \u2191(Real.exp (-X)) * \u2191X ^ s\n[PROOFSTEP]\nhave : (fun x => (-x).exp * (s * x ^ (s - 1)) : \u211d \u2192 \u2102) = (fun x => s * (-x).exp * x ^ (s - 1) : \u211d \u2192 \u2102) := by ext1; ring\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\nint_eval :\n (\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s) + -\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)) =\n -((ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s - (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s)\n\u22a2 (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\n[PROOFSTEP]\next1\n[GOAL]\ncase h\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\nint_eval :\n (\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s) + -\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)) =\n -((ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s - (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s)\nx\u271d : \u211d\n\u22a2 \u2191(Real.exp (-x\u271d)) * (s * \u2191x\u271d ^ (s - 1)) = s * \u2191(Real.exp (-x\u271d)) * \u2191x\u271d ^ (s - 1)\n[PROOFSTEP]\nring\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\nint_eval :\n (\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s) + -\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)) =\n -((ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s - (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s)\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\n\u22a2 (\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) + (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s -\n (ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s =\n (s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) - \u2191(Real.exp (-X)) * \u2191X ^ s\n[PROOFSTEP]\nrw [this]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\nint_eval :\n (\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s) + -\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)) =\n -((ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s - (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s)\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\n\u22a2 (\u222b (x : \u211d) in 0 ..X, s * \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) + (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s -\n (ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s =\n (s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) - \u2191(Real.exp (-X)) * \u2191X ^ s\n[PROOFSTEP]\nhave t := @integral_const_mul 0 X volume _ _ s fun x : \u211d => (-x).exp * x ^ (s - 1)\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\nint_eval :\n (\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s) + -\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)) =\n -((ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s - (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s)\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\nt : \u222b (x : \u211d) in 0 ..X, s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) = s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\n\u22a2 (\u222b (x : \u211d) in 0 ..X, s * \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) + (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s -\n (ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s =\n (s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) - \u2191(Real.exp (-X)) * \u2191X ^ s\n[PROOFSTEP]\nrw [\u2190 t, ofReal_zero, zero_cpow]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\nint_eval :\n (\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s) + -\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)) =\n -((ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s - (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s)\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\nt : \u222b (x : \u211d) in 0 ..X, s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) = s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\n\u22a2 (\u222b (x : \u211d) in 0 ..X, s * \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) + (ofReal' \u2218 fun y => Real.exp (-y)) 0 * 0 -\n (ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s =\n (\u222b (x : \u211d) in 0 ..X, s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))) - \u2191(Real.exp (-X)) * \u2191X ^ s\n[PROOFSTEP]\nrw [mul_zero, add_zero]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\nint_eval :\n (\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s) + -\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)) =\n -((ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s - (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s)\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\nt : \u222b (x : \u211d) in 0 ..X, s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) = s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\n\u22a2 (\u222b (x : \u211d) in 0 ..X, s * \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) - (ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s =\n (\u222b (x : \u211d) in 0 ..X, s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))) - \u2191(Real.exp (-X)) * \u2191X ^ s\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_f\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\nint_eval :\n (\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s) + -\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)) =\n -((ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s - (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s)\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\nt : \u222b (x : \u211d) in 0 ..X, s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) = s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\n\u22a2 (fun x => s * \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) = fun x => s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1))\n[PROOFSTEP]\next1\n[GOAL]\ncase e_a.e_f.h\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\nint_eval :\n (\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s) + -\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)) =\n -((ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s - (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s)\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\nt : \u222b (x : \u211d) in 0 ..X, s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) = s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\nx\u271d : \u211d\n\u22a2 s * \u2191(Real.exp (-x\u271d)) * \u2191x\u271d ^ (s - 1) = s * (\u2191(Real.exp (-x\u271d)) * \u2191x\u271d ^ (s - 1))\n[PROOFSTEP]\nring\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\nint_eval :\n (\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s) + -\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)) =\n -((ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s - (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s)\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\nt : \u222b (x : \u211d) in 0 ..X, s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) = s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\n\u22a2 s \u2260 0\n[PROOFSTEP]\ncontrapose! hs\n[GOAL]\ns : \u2102\nX : \u211d\nhX : 0 \u2264 X\nF_der_I :\n \u2200 (x : \u211d),\n x \u2208 Ioo 0 X \u2192\n HasDerivAt (fun x => \u2191(Real.exp (-x)) * \u2191x ^ s)\n (-(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) x\ncont : Continuous fun x => (ofReal' \u2218 fun y => Real.exp (-y)) x * \u2191x ^ s\nder_ible : IntervalIntegrable (fun x => -(\u2191(Real.exp (-x)) * \u2191x ^ s) + \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) volume 0 X\nint_eval :\n (\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ s) + -\u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1)) =\n -((ofReal' \u2218 fun y => Real.exp (-y)) X * \u2191X ^ s - (ofReal' \u2218 fun y => Real.exp (-y)) 0 * \u21910 ^ s)\nthis : (fun x => \u2191(Real.exp (-x)) * (s * \u2191x ^ (s - 1))) = fun x => s * \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\nt : \u222b (x : \u211d) in 0 ..X, s * (\u2191(Real.exp (-x)) * \u2191x ^ (s - 1)) = s * \u222b (x : \u211d) in 0 ..X, \u2191(Real.exp (-x)) * \u2191x ^ (s - 1)\nhs : s = 0\n\u22a2 s.re \u2264 0\n[PROOFSTEP]\nrw [hs, zero_re]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\n\u22a2 GammaIntegral (s + 1) = s * GammaIntegral s\n[PROOFSTEP]\nsuffices Tendsto (s + 1).partialGamma atTop (\ud835\udcdd <| s * GammaIntegral s)\n by\n refine' tendsto_nhds_unique _ this\n apply tendsto_partialGamma; rw [add_re, one_re]; linarith\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nthis : Tendsto (partialGamma (s + 1)) atTop (\ud835\udcdd (s * GammaIntegral s))\n\u22a2 GammaIntegral (s + 1) = s * GammaIntegral s\n[PROOFSTEP]\nrefine' tendsto_nhds_unique _ this\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nthis : Tendsto (partialGamma (s + 1)) atTop (\ud835\udcdd (s * GammaIntegral s))\n\u22a2 Tendsto (partialGamma (s + 1)) atTop (\ud835\udcdd (GammaIntegral (s + 1)))\n[PROOFSTEP]\napply tendsto_partialGamma\n[GOAL]\ncase hs\ns : \u2102\nhs : 0 < s.re\nthis : Tendsto (partialGamma (s + 1)) atTop (\ud835\udcdd (s * GammaIntegral s))\n\u22a2 0 < (s + 1).re\n[PROOFSTEP]\nrw [add_re, one_re]\n[GOAL]\ncase hs\ns : \u2102\nhs : 0 < s.re\nthis : Tendsto (partialGamma (s + 1)) atTop (\ud835\udcdd (s * GammaIntegral s))\n\u22a2 0 < s.re + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ns : \u2102\nhs : 0 < s.re\n\u22a2 Tendsto (partialGamma (s + 1)) atTop (\ud835\udcdd (s * GammaIntegral s))\n[PROOFSTEP]\nhave : (fun X : \u211d => s * partialGamma s X - X ^ s * (-X).exp) =\u1da0[atTop] (s + 1).partialGamma :=\n by\n apply eventuallyEq_of_mem (Ici_mem_atTop (0 : \u211d))\n intro X hX\n rw [partialGamma_add_one hs (mem_Ici.mp hX)]\n ring_nf\n[GOAL]\ns : \u2102\nhs : 0 < s.re\n\u22a2 (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) =\u1da0[atTop] partialGamma (s + 1)\n[PROOFSTEP]\napply eventuallyEq_of_mem (Ici_mem_atTop (0 : \u211d))\n[GOAL]\ns : \u2102\nhs : 0 < s.re\n\u22a2 EqOn (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) (partialGamma (s + 1)) (Ici 0)\n[PROOFSTEP]\nintro X hX\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : X \u2208 Ici 0\n\u22a2 (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) X = partialGamma (s + 1) X\n[PROOFSTEP]\nrw [partialGamma_add_one hs (mem_Ici.mp hX)]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nX : \u211d\nhX : X \u2208 Ici 0\n\u22a2 (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) X = s * partialGamma s X - \u2191(Real.exp (-X)) * \u2191X ^ s\n[PROOFSTEP]\nring_nf\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nthis : (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) =\u1da0[atTop] partialGamma (s + 1)\n\u22a2 Tendsto (partialGamma (s + 1)) atTop (\ud835\udcdd (s * GammaIntegral s))\n[PROOFSTEP]\nrefine' Tendsto.congr' this _\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nthis : (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) =\u1da0[atTop] partialGamma (s + 1)\n\u22a2 Tendsto (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) atTop (\ud835\udcdd (s * GammaIntegral s))\n[PROOFSTEP]\nsuffices Tendsto (fun X => -X ^ s * (-X).exp : \u211d \u2192 \u2102) atTop (\ud835\udcdd 0) by\n simpa using Tendsto.add (Tendsto.const_mul s (tendsto_partialGamma hs)) this\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nthis\u271d : (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) =\u1da0[atTop] partialGamma (s + 1)\nthis : Tendsto (fun X => -\u2191X ^ s * \u2191(Real.exp (-X))) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) atTop (\ud835\udcdd (s * GammaIntegral s))\n[PROOFSTEP]\nsimpa using Tendsto.add (Tendsto.const_mul s (tendsto_partialGamma hs)) this\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nthis : (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) =\u1da0[atTop] partialGamma (s + 1)\n\u22a2 Tendsto (fun X => -\u2191X ^ s * \u2191(Real.exp (-X))) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [tendsto_zero_iff_norm_tendsto_zero]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nthis : (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) =\u1da0[atTop] partialGamma (s + 1)\n\u22a2 Tendsto (fun e => \u2016-\u2191e ^ s * \u2191(Real.exp (-e))\u2016) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : (fun e : \u211d => \u2016-(e : \u2102) ^ s * (-e).exp\u2016) =\u1da0[atTop] fun e : \u211d => e ^ s.re * (-1 * e).exp :=\n by\n refine' eventuallyEq_of_mem (Ioi_mem_atTop 0) _\n intro x hx; dsimp only\n rw [norm_eq_abs, map_mul, abs.map_neg, abs_cpow_eq_rpow_re_of_pos hx, abs_of_nonneg (exp_pos (-x)).le, neg_mul,\n one_mul]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nthis : (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) =\u1da0[atTop] partialGamma (s + 1)\n\u22a2 (fun e => \u2016-\u2191e ^ s * \u2191(Real.exp (-e))\u2016) =\u1da0[atTop] fun e => e ^ s.re * Real.exp (-1 * e)\n[PROOFSTEP]\nrefine' eventuallyEq_of_mem (Ioi_mem_atTop 0) _\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nthis : (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) =\u1da0[atTop] partialGamma (s + 1)\n\u22a2 EqOn (fun e => \u2016-\u2191e ^ s * \u2191(Real.exp (-e))\u2016) (fun e => e ^ s.re * Real.exp (-1 * e)) (Ioi 0)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nthis : (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) =\u1da0[atTop] partialGamma (s + 1)\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 (fun e => \u2016-\u2191e ^ s * \u2191(Real.exp (-e))\u2016) x = (fun e => e ^ s.re * Real.exp (-1 * e)) x\n[PROOFSTEP]\ndsimp only\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nthis : (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) =\u1da0[atTop] partialGamma (s + 1)\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 \u2016-\u2191x ^ s * \u2191(Real.exp (-x))\u2016 = x ^ s.re * Real.exp (-1 * x)\n[PROOFSTEP]\nrw [norm_eq_abs, map_mul, abs.map_neg, abs_cpow_eq_rpow_re_of_pos hx, abs_of_nonneg (exp_pos (-x)).le, neg_mul, one_mul]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nthis\u271d : (fun X => s * partialGamma s X - \u2191X ^ s * \u2191(Real.exp (-X))) =\u1da0[atTop] partialGamma (s + 1)\nthis : (fun e => \u2016-\u2191e ^ s * \u2191(Real.exp (-e))\u2016) =\u1da0[atTop] fun e => e ^ s.re * Real.exp (-1 * e)\n\u22a2 Tendsto (fun e => \u2016-\u2191e ^ s * \u2191(Real.exp (-e))\u2016) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact (tendsto_congr' this).mpr (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_0 _ _ zero_lt_one)\n[GOAL]\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\n\u22a2 GammaAux n s = GammaAux n (s + 1) / s\n[PROOFSTEP]\ninduction' n with n hn generalizing s\n[GOAL]\ncase zero\ns\u271d : \u2102\nn : \u2115\nh1\u271d : -s\u271d.re < \u2191n\ns : \u2102\nh1 : -s.re < \u2191Nat.zero\n\u22a2 GammaAux Nat.zero s = GammaAux Nat.zero (s + 1) / s\n[PROOFSTEP]\nsimp only [Nat.zero_eq, CharP.cast_eq_zero, Left.neg_neg_iff] at h1 \n[GOAL]\ncase zero\ns\u271d : \u2102\nn : \u2115\nh1\u271d : -s\u271d.re < \u2191n\ns : \u2102\nh1 : 0 < s.re\n\u22a2 GammaAux Nat.zero s = GammaAux Nat.zero (s + 1) / s\n[PROOFSTEP]\ndsimp only [GammaAux]\n[GOAL]\ncase zero\ns\u271d : \u2102\nn : \u2115\nh1\u271d : -s\u271d.re < \u2191n\ns : \u2102\nh1 : 0 < s.re\n\u22a2 GammaIntegral s = GammaIntegral (s + 1) / s\n[PROOFSTEP]\nrw [GammaIntegral_add_one h1]\n[GOAL]\ncase zero\ns\u271d : \u2102\nn : \u2115\nh1\u271d : -s\u271d.re < \u2191n\ns : \u2102\nh1 : 0 < s.re\n\u22a2 GammaIntegral s = s * GammaIntegral s / s\n[PROOFSTEP]\nrw [mul_comm, mul_div_cancel]\n[GOAL]\ncase zero.h\ns\u271d : \u2102\nn : \u2115\nh1\u271d : -s\u271d.re < \u2191n\ns : \u2102\nh1 : 0 < s.re\n\u22a2 s \u2260 0\n[PROOFSTEP]\ncontrapose! h1\n[GOAL]\ncase zero.h\ns\u271d : \u2102\nn : \u2115\nh1\u271d : -s\u271d.re < \u2191n\ns : \u2102\nh1 : s = 0\n\u22a2 s.re \u2264 0\n[PROOFSTEP]\nrw [h1]\n[GOAL]\ncase zero.h\ns\u271d : \u2102\nn : \u2115\nh1\u271d : -s\u271d.re < \u2191n\ns : \u2102\nh1 : s = 0\n\u22a2 0.re \u2264 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : -s\u271d.re < \u2191n\u271d\nn : \u2115\nhn : \u2200 (s : \u2102), -s.re < \u2191n \u2192 GammaAux n s = GammaAux n (s + 1) / s\ns : \u2102\nh1 : -s.re < \u2191(Nat.succ n)\n\u22a2 GammaAux (Nat.succ n) s = GammaAux (Nat.succ n) (s + 1) / s\n[PROOFSTEP]\ndsimp only [GammaAux]\n[GOAL]\ncase succ\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : -s\u271d.re < \u2191n\u271d\nn : \u2115\nhn : \u2200 (s : \u2102), -s.re < \u2191n \u2192 GammaAux n s = GammaAux n (s + 1) / s\ns : \u2102\nh1 : -s.re < \u2191(Nat.succ n)\n\u22a2 GammaAux n (s + 1) / s = GammaAux n (s + 1 + 1) / (s + 1) / s\n[PROOFSTEP]\nhave hh1 : -(s + 1).re < n := by\n rw [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one] at h1 \n rw [add_re, one_re]; linarith\n[GOAL]\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : -s\u271d.re < \u2191n\u271d\nn : \u2115\nhn : \u2200 (s : \u2102), -s.re < \u2191n \u2192 GammaAux n s = GammaAux n (s + 1) / s\ns : \u2102\nh1 : -s.re < \u2191(Nat.succ n)\n\u22a2 -(s + 1).re < \u2191n\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one] at h1 \n[GOAL]\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : -s\u271d.re < \u2191n\u271d\nn : \u2115\nhn : \u2200 (s : \u2102), -s.re < \u2191n \u2192 GammaAux n s = GammaAux n (s + 1) / s\ns : \u2102\nh1 : -s.re < \u2191n + 1\n\u22a2 -(s + 1).re < \u2191n\n[PROOFSTEP]\nrw [add_re, one_re]\n[GOAL]\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : -s\u271d.re < \u2191n\u271d\nn : \u2115\nhn : \u2200 (s : \u2102), -s.re < \u2191n \u2192 GammaAux n s = GammaAux n (s + 1) / s\ns : \u2102\nh1 : -s.re < \u2191n + 1\n\u22a2 -(s.re + 1) < \u2191n\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase succ\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : -s\u271d.re < \u2191n\u271d\nn : \u2115\nhn : \u2200 (s : \u2102), -s.re < \u2191n \u2192 GammaAux n s = GammaAux n (s + 1) / s\ns : \u2102\nh1 : -s.re < \u2191(Nat.succ n)\nhh1 : -(s + 1).re < \u2191n\n\u22a2 GammaAux n (s + 1) / s = GammaAux n (s + 1 + 1) / (s + 1) / s\n[PROOFSTEP]\nrw [\u2190 hn (s + 1) hh1]\n[GOAL]\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\n\u22a2 GammaAux n s = GammaAux (n + 1) s\n[PROOFSTEP]\ncases' n with n n\n[GOAL]\ncase zero\ns : \u2102\nh1 : -s.re < \u2191Nat.zero\n\u22a2 GammaAux Nat.zero s = GammaAux (Nat.zero + 1) s\n[PROOFSTEP]\nsimp only [Nat.zero_eq, CharP.cast_eq_zero, Left.neg_neg_iff] at h1 \n[GOAL]\ncase zero\ns : \u2102\nh1 : 0 < s.re\n\u22a2 GammaAux Nat.zero s = GammaAux (Nat.zero + 1) s\n[PROOFSTEP]\ndsimp only [GammaAux]\n[GOAL]\ncase zero\ns : \u2102\nh1 : 0 < s.re\n\u22a2 GammaIntegral s = GammaIntegral (s + 1) / s\n[PROOFSTEP]\nrw [GammaIntegral_add_one h1, mul_div_cancel_left]\n[GOAL]\ncase zero.ha\ns : \u2102\nh1 : 0 < s.re\n\u22a2 s \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase zero.ha\nh1 : 0 < 0.re\n\u22a2 False\n[PROOFSTEP]\nrw [zero_re] at h1 \n[GOAL]\ncase zero.ha\nh1 : 0 < 0\n\u22a2 False\n[PROOFSTEP]\nexact h1.false\n[GOAL]\ncase succ\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191(Nat.succ n)\n\u22a2 GammaAux (Nat.succ n) s = GammaAux (Nat.succ n + 1) s\n[PROOFSTEP]\ndsimp only [GammaAux]\n[GOAL]\ncase succ\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191(Nat.succ n)\n\u22a2 GammaAux n (s + 1) / s = GammaAux n (s + 1 + 1) / (s + 1) / s\n[PROOFSTEP]\nhave : GammaAux n (s + 1 + 1) / (s + 1) = GammaAux n (s + 1) :=\n by\n have hh1 : -(s + 1).re < n :=\n by\n rw [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one] at h1 \n rw [add_re, one_re]; linarith\n rw [GammaAux_recurrence1 (s + 1) n hh1]\n[GOAL]\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191(Nat.succ n)\n\u22a2 GammaAux n (s + 1 + 1) / (s + 1) = GammaAux n (s + 1)\n[PROOFSTEP]\nhave hh1 : -(s + 1).re < n := by\n rw [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one] at h1 \n rw [add_re, one_re]; linarith\n[GOAL]\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191(Nat.succ n)\n\u22a2 -(s + 1).re < \u2191n\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one] at h1 \n[GOAL]\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n + 1\n\u22a2 -(s + 1).re < \u2191n\n[PROOFSTEP]\nrw [add_re, one_re]\n[GOAL]\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n + 1\n\u22a2 -(s.re + 1) < \u2191n\n[PROOFSTEP]\nlinarith\n[GOAL]\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191(Nat.succ n)\nhh1 : -(s + 1).re < \u2191n\n\u22a2 GammaAux n (s + 1 + 1) / (s + 1) = GammaAux n (s + 1)\n[PROOFSTEP]\nrw [GammaAux_recurrence1 (s + 1) n hh1]\n[GOAL]\ncase succ\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191(Nat.succ n)\nthis : GammaAux n (s + 1 + 1) / (s + 1) = GammaAux n (s + 1)\n\u22a2 GammaAux n (s + 1) / s = GammaAux n (s + 1 + 1) / (s + 1) / s\n[PROOFSTEP]\nrw [this]\n[GOAL]\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\n\u22a2 Gamma s = GammaAux n s\n[PROOFSTEP]\nhave u : \u2200 k : \u2115, GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s :=\n by\n intro k; induction' k with k hk\n \u00b7 simp [Gamma]\n \u00b7 rw [\u2190 hk, Nat.succ_eq_add_one, \u2190 add_assoc]\n refine' (GammaAux_recurrence2 s (\u230a1 - s.re\u230b\u208a + k) _).symm\n rw [Nat.cast_add]\n have i0 := Nat.sub_one_lt_floor (1 - s.re)\n simp only [sub_sub_cancel_left] at i0 \n refine' lt_add_of_lt_of_nonneg i0 _\n rw [\u2190 Nat.cast_zero, Nat.cast_le]; exact Nat.zero_le k\n[GOAL]\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\n\u22a2 \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\n[PROOFSTEP]\nintro k\n[GOAL]\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nk : \u2115\n\u22a2 GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\n[PROOFSTEP]\ninduction' k with k hk\n[GOAL]\ncase zero\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\n\u22a2 GammaAux (\u230a1 - s.re\u230b\u208a + Nat.zero) s = Gamma s\n[PROOFSTEP]\nsimp [Gamma]\n[GOAL]\ncase succ\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nk : \u2115\nhk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\n\u22a2 GammaAux (\u230a1 - s.re\u230b\u208a + Nat.succ k) s = Gamma s\n[PROOFSTEP]\nrw [\u2190 hk, Nat.succ_eq_add_one, \u2190 add_assoc]\n[GOAL]\ncase succ\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nk : \u2115\nhk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\n\u22a2 GammaAux (\u230a1 - s.re\u230b\u208a + k + 1) s = GammaAux (\u230a1 - s.re\u230b\u208a + k) s\n[PROOFSTEP]\nrefine' (GammaAux_recurrence2 s (\u230a1 - s.re\u230b\u208a + k) _).symm\n[GOAL]\ncase succ\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nk : \u2115\nhk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\n\u22a2 -s.re < \u2191(\u230a1 - s.re\u230b\u208a + k)\n[PROOFSTEP]\nrw [Nat.cast_add]\n[GOAL]\ncase succ\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nk : \u2115\nhk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\n\u22a2 -s.re < \u2191\u230a1 - s.re\u230b\u208a + \u2191k\n[PROOFSTEP]\nhave i0 := Nat.sub_one_lt_floor (1 - s.re)\n[GOAL]\ncase succ\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nk : \u2115\nhk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\ni0 : 1 - s.re - 1 < \u2191\u230a1 - s.re\u230b\u208a\n\u22a2 -s.re < \u2191\u230a1 - s.re\u230b\u208a + \u2191k\n[PROOFSTEP]\nsimp only [sub_sub_cancel_left] at i0 \n[GOAL]\ncase succ\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nk : \u2115\nhk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\ni0 : -s.re < \u2191\u230a1 - s.re\u230b\u208a\n\u22a2 -s.re < \u2191\u230a1 - s.re\u230b\u208a + \u2191k\n[PROOFSTEP]\nrefine' lt_add_of_lt_of_nonneg i0 _\n[GOAL]\ncase succ\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nk : \u2115\nhk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\ni0 : -s.re < \u2191\u230a1 - s.re\u230b\u208a\n\u22a2 0 \u2264 \u2191k\n[PROOFSTEP]\nrw [\u2190 Nat.cast_zero, Nat.cast_le]\n[GOAL]\ncase succ\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nk : \u2115\nhk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\ni0 : -s.re < \u2191\u230a1 - s.re\u230b\u208a\n\u22a2 0 \u2264 k\n[PROOFSTEP]\nexact Nat.zero_le k\n[GOAL]\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nu : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\n\u22a2 Gamma s = GammaAux n s\n[PROOFSTEP]\nconvert (u <| n - \u230a1 - s.re\u230b\u208a).symm\n[GOAL]\ncase h.e'_3.h.e'_1\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nu : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\n\u22a2 n = \u230a1 - s.re\u230b\u208a + (n - \u230a1 - s.re\u230b\u208a)\n[PROOFSTEP]\nrw [Nat.add_sub_of_le]\n[GOAL]\ncase h.e'_3.h.e'_1\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nu : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\n\u22a2 \u230a1 - s.re\u230b\u208a \u2264 n\n[PROOFSTEP]\nby_cases 0 \u2264 1 - s.re\n[GOAL]\ncase h.e'_3.h.e'_1\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nu : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\n\u22a2 \u230a1 - s.re\u230b\u208a \u2264 n\n[PROOFSTEP]\nby_cases 0 \u2264 1 - s.re\n[GOAL]\ncase pos\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nu : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\nh : 0 \u2264 1 - s.re\n\u22a2 \u230a1 - s.re\u230b\u208a \u2264 n\n[PROOFSTEP]\napply Nat.le_of_lt_succ\n[GOAL]\ncase pos.a\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nu : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\nh : 0 \u2264 1 - s.re\n\u22a2 \u230a1 - s.re\u230b\u208a < Nat.succ n\n[PROOFSTEP]\nexact_mod_cast lt_of_le_of_lt (Nat.floor_le h) (by linarith : 1 - s.re < n + 1)\n[GOAL]\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nu : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\nh : 0 \u2264 1 - s.re\n\u22a2 1 - s.re < \u2191n + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nu : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\nh : \u00ac0 \u2264 1 - s.re\n\u22a2 \u230a1 - s.re\u230b\u208a \u2264 n\n[PROOFSTEP]\nrw [Nat.floor_of_nonpos]\n[GOAL]\ncase neg\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nu : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\nh : \u00ac0 \u2264 1 - s.re\n\u22a2 0 \u2264 n\ncase neg\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nu : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\nh : \u00ac0 \u2264 1 - s.re\n\u22a2 1 - s.re \u2264 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\ns : \u2102\nn : \u2115\nh1 : -s.re < \u2191n\nu : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s\nh : \u00ac0 \u2264 1 - s.re\n\u22a2 1 - s.re \u2264 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ns : \u2102\nh2 : s \u2260 0\n\u22a2 Gamma (s + 1) = s * Gamma s\n[PROOFSTEP]\nlet n := \u230a1 - s.re\u230b\u208a\n[GOAL]\ns : \u2102\nh2 : s \u2260 0\nn : \u2115 := \u230a1 - s.re\u230b\u208a\n\u22a2 Gamma (s + 1) = s * Gamma s\n[PROOFSTEP]\nhave t1 : -s.re < n := by simpa only [sub_sub_cancel_left] using Nat.sub_one_lt_floor (1 - s.re)\n[GOAL]\ns : \u2102\nh2 : s \u2260 0\nn : \u2115 := \u230a1 - s.re\u230b\u208a\n\u22a2 -s.re < \u2191n\n[PROOFSTEP]\nsimpa only [sub_sub_cancel_left] using Nat.sub_one_lt_floor (1 - s.re)\n[GOAL]\ns : \u2102\nh2 : s \u2260 0\nn : \u2115 := \u230a1 - s.re\u230b\u208a\nt1 : -s.re < \u2191n\n\u22a2 Gamma (s + 1) = s * Gamma s\n[PROOFSTEP]\nhave t2 : -(s + 1).re < n := by rw [add_re, one_re]; linarith\n[GOAL]\ns : \u2102\nh2 : s \u2260 0\nn : \u2115 := \u230a1 - s.re\u230b\u208a\nt1 : -s.re < \u2191n\n\u22a2 -(s + 1).re < \u2191n\n[PROOFSTEP]\nrw [add_re, one_re]\n[GOAL]\ns : \u2102\nh2 : s \u2260 0\nn : \u2115 := \u230a1 - s.re\u230b\u208a\nt1 : -s.re < \u2191n\n\u22a2 -(s.re + 1) < \u2191n\n[PROOFSTEP]\nlinarith\n[GOAL]\ns : \u2102\nh2 : s \u2260 0\nn : \u2115 := \u230a1 - s.re\u230b\u208a\nt1 : -s.re < \u2191n\nt2 : -(s + 1).re < \u2191n\n\u22a2 Gamma (s + 1) = s * Gamma s\n[PROOFSTEP]\nrw [Gamma_eq_GammaAux s n t1, Gamma_eq_GammaAux (s + 1) n t2, GammaAux_recurrence1 s n t1]\n[GOAL]\ns : \u2102\nh2 : s \u2260 0\nn : \u2115 := \u230a1 - s.re\u230b\u208a\nt1 : -s.re < \u2191n\nt2 : -(s + 1).re < \u2191n\n\u22a2 GammaAux n (s + 1) = s * (GammaAux n (s + 1) / s)\n[PROOFSTEP]\nfield_simp\n[GOAL]\ns : \u2102\nh2 : s \u2260 0\nn : \u2115 := \u230a1 - s.re\u230b\u208a\nt1 : -s.re < \u2191n\nt2 : -(s + 1).re < \u2191n\n\u22a2 GammaAux \u230a1 - s.re\u230b\u208a (s + 1) * s = s * GammaAux \u230a1 - s.re\u230b\u208a (s + 1)\n[PROOFSTEP]\nring\n[GOAL]\ns : \u2102\nhs : 0 < s.re\n\u22a2 -s.re < \u21910\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ns : \u2102\nhs : 0 < s.re\n\u22a2 -s.re < 0\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u22a2 Gamma 1 = 1\n[PROOFSTEP]\nrw [Gamma_eq_integral]\n[GOAL]\n\u22a2 GammaIntegral 1 = 1\n\u22a2 0 < 1.re\n[PROOFSTEP]\nsimpa using GammaIntegral_one\n[GOAL]\n\u22a2 0 < 1.re\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 Gamma (\u2191n + 1) = \u2191n !\n[PROOFSTEP]\ninduction' n with n hn\n[GOAL]\ncase zero\n\u22a2 Gamma (\u2191Nat.zero + 1) = \u2191Nat.zero !\n[PROOFSTEP]\nsimpa using Gamma_one\n[GOAL]\ncase succ\nn : \u2115\nhn : Gamma (\u2191n + 1) = \u2191n !\n\u22a2 Gamma (\u2191(Nat.succ n) + 1) = \u2191(Nat.succ n)!\n[PROOFSTEP]\nrw [Gamma_add_one n.succ <| Nat.cast_ne_zero.mpr <| Nat.succ_ne_zero n]\n[GOAL]\ncase succ\nn : \u2115\nhn : Gamma (\u2191n + 1) = \u2191n !\n\u22a2 \u2191(Nat.succ n) * Gamma \u2191(Nat.succ n) = \u2191(Nat.succ n)!\n[PROOFSTEP]\nsimp only [Nat.cast_succ, Nat.factorial_succ, Nat.cast_mul]\n[GOAL]\ncase succ\nn : \u2115\nhn : Gamma (\u2191n + 1) = \u2191n !\n\u22a2 (\u2191n + 1) * Gamma (\u2191n + 1) = (\u2191n + 1) * \u2191n !\n[PROOFSTEP]\ncongr\n[GOAL]\n\u22a2 Gamma 0 = 0\n[PROOFSTEP]\nsimp_rw [Gamma, zero_re, sub_zero, Nat.floor_one, GammaAux, div_zero]\n[GOAL]\nn : \u2115\n\u22a2 Gamma (-\u2191n) = 0\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\u22a2 Gamma (-\u2191Nat.zero) = 0\n[PROOFSTEP]\nrw [Nat.cast_zero, neg_zero, Gamma_zero]\n[GOAL]\ncase succ\nn : \u2115\nIH : Gamma (-\u2191n) = 0\n\u22a2 Gamma (-\u2191(Nat.succ n)) = 0\n[PROOFSTEP]\nhave A : -(n.succ : \u2102) \u2260 0 := by\n rw [neg_ne_zero, Nat.cast_ne_zero]\n apply Nat.succ_ne_zero\n[GOAL]\nn : \u2115\nIH : Gamma (-\u2191n) = 0\n\u22a2 -\u2191(Nat.succ n) \u2260 0\n[PROOFSTEP]\nrw [neg_ne_zero, Nat.cast_ne_zero]\n[GOAL]\nn : \u2115\nIH : Gamma (-\u2191n) = 0\n\u22a2 Nat.succ n \u2260 0\n[PROOFSTEP]\napply Nat.succ_ne_zero\n[GOAL]\ncase succ\nn : \u2115\nIH : Gamma (-\u2191n) = 0\nA : -\u2191(Nat.succ n) \u2260 0\n\u22a2 Gamma (-\u2191(Nat.succ n)) = 0\n[PROOFSTEP]\nhave : -(n : \u2102) = -\u2191n.succ + 1 := by simp\n[GOAL]\nn : \u2115\nIH : Gamma (-\u2191n) = 0\nA : -\u2191(Nat.succ n) \u2260 0\n\u22a2 -\u2191n = -\u2191(Nat.succ n) + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nn : \u2115\nIH : Gamma (-\u2191n) = 0\nA : -\u2191(Nat.succ n) \u2260 0\nthis : -\u2191n = -\u2191(Nat.succ n) + 1\n\u22a2 Gamma (-\u2191(Nat.succ n)) = 0\n[PROOFSTEP]\nrw [this, Gamma_add_one _ A] at IH \n[GOAL]\ncase succ\nn : \u2115\nIH : -\u2191(Nat.succ n) * Gamma (-\u2191(Nat.succ n)) = 0\nA : -\u2191(Nat.succ n) \u2260 0\nthis : -\u2191n = -\u2191(Nat.succ n) + 1\n\u22a2 Gamma (-\u2191(Nat.succ n)) = 0\n[PROOFSTEP]\ncontrapose! IH\n[GOAL]\ncase succ\nn : \u2115\nA : -\u2191(Nat.succ n) \u2260 0\nthis : -\u2191n = -\u2191(Nat.succ n) + 1\nIH : Gamma (-\u2191(Nat.succ n)) \u2260 0\n\u22a2 -\u2191(Nat.succ n) * Gamma (-\u2191(Nat.succ n)) \u2260 0\n[PROOFSTEP]\nexact mul_ne_zero A IH\n[GOAL]\ns : \u2102\n\u22a2 Gamma (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (Gamma s)\n[PROOFSTEP]\nsuffices : \u2200 (n : \u2115) (s : \u2102), GammaAux n (conj s) = conj (GammaAux n s)\n[GOAL]\ns : \u2102\nthis : \u2200 (n : \u2115) (s : \u2102), GammaAux n (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaAux n s)\n\u22a2 Gamma (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (Gamma s)\ncase this s : \u2102 \u22a2 \u2200 (n : \u2115) (s : \u2102), GammaAux n (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaAux n s)\n[PROOFSTEP]\nexact this _ _\n[GOAL]\ncase this\ns : \u2102\n\u22a2 \u2200 (n : \u2115) (s : \u2102), GammaAux n (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaAux n s)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase this\ns : \u2102\nn : \u2115\n\u22a2 \u2200 (s : \u2102), GammaAux n (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaAux n s)\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase this.zero\ns : \u2102\n\u22a2 \u2200 (s : \u2102), GammaAux Nat.zero (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaAux Nat.zero s)\n[PROOFSTEP]\nrw [GammaAux]\n[GOAL]\ncase this.zero\ns : \u2102\n\u22a2 \u2200 (s : \u2102), GammaIntegral (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaIntegral s)\n[PROOFSTEP]\nexact GammaIntegral_conj\n[GOAL]\ncase this.succ\ns : \u2102\nn : \u2115\nIH : \u2200 (s : \u2102), GammaAux n (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaAux n s)\n\u22a2 \u2200 (s : \u2102), GammaAux (Nat.succ n) (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaAux (Nat.succ n) s)\n[PROOFSTEP]\nintro s\n[GOAL]\ncase this.succ\ns\u271d : \u2102\nn : \u2115\nIH : \u2200 (s : \u2102), GammaAux n (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaAux n s)\ns : \u2102\n\u22a2 GammaAux (Nat.succ n) (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaAux (Nat.succ n) s)\n[PROOFSTEP]\nrw [GammaAux]\n[GOAL]\ncase this.succ\ns\u271d : \u2102\nn : \u2115\nIH : \u2200 (s : \u2102), GammaAux n (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaAux n s)\ns : \u2102\n\u22a2 (fun s => GammaAux n (s + 1) / s) (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) ((fun s => GammaAux n (s + 1) / s) s)\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase this.succ\ns\u271d : \u2102\nn : \u2115\nIH : \u2200 (s : \u2102), GammaAux n (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaAux n s)\ns : \u2102\n\u22a2 GammaAux n (\u2191(starRingEnd \u2102) s + 1) / \u2191(starRingEnd \u2102) s = \u2191(starRingEnd \u2102) (GammaAux n (s + 1) / s)\n[PROOFSTEP]\nrw [div_eq_mul_inv _ s, RingHom.map_mul, conj_inv, \u2190 div_eq_mul_inv]\n[GOAL]\ncase this.succ\ns\u271d : \u2102\nn : \u2115\nIH : \u2200 (s : \u2102), GammaAux n (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaAux n s)\ns : \u2102\n\u22a2 GammaAux n (\u2191(starRingEnd \u2102) s + 1) / \u2191(starRingEnd \u2102) s = \u2191(starRingEnd \u2102) (GammaAux n (s + 1)) / \u2191(starRingEnd \u2102) s\n[PROOFSTEP]\nsuffices conj s + 1 = conj (s + 1) by rw [this, IH]\n[GOAL]\ns\u271d : \u2102\nn : \u2115\nIH : \u2200 (s : \u2102), GammaAux n (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaAux n s)\ns : \u2102\nthis : \u2191(starRingEnd \u2102) s + 1 = \u2191(starRingEnd \u2102) (s + 1)\n\u22a2 GammaAux n (\u2191(starRingEnd \u2102) s + 1) / \u2191(starRingEnd \u2102) s = \u2191(starRingEnd \u2102) (GammaAux n (s + 1)) / \u2191(starRingEnd \u2102) s\n[PROOFSTEP]\nrw [this, IH]\n[GOAL]\ncase this.succ\ns\u271d : \u2102\nn : \u2115\nIH : \u2200 (s : \u2102), GammaAux n (\u2191(starRingEnd \u2102) s) = \u2191(starRingEnd \u2102) (GammaAux n s)\ns : \u2102\n\u22a2 \u2191(starRingEnd \u2102) s + 1 = \u2191(starRingEnd \u2102) (s + 1)\n[PROOFSTEP]\nrw [RingHom.map_add, RingHom.map_one]\n[GOAL]\ns : \u2102\n\u22a2 GammaIntegral s = mellin (fun x => \u2191(Real.exp (-x))) s\n[PROOFSTEP]\nsimp only [mellin, GammaIntegral, smul_eq_mul, mul_comm]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\n\u22a2 HasDerivAt GammaIntegral (\u222b (t : \u211d) in Ioi 0, \u2191t ^ (s - 1) * (\u2191(Real.log t) * \u2191(Real.exp (-t)))) s\n[PROOFSTEP]\nrw [GammaIntegral_eq_mellin]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\n\u22a2 HasDerivAt (mellin fun x => \u2191(Real.exp (-x))) (\u222b (t : \u211d) in Ioi 0, \u2191t ^ (s - 1) * (\u2191(Real.log t) * \u2191(Real.exp (-t))))\n s\n[PROOFSTEP]\nconvert (mellin_hasDerivAt_of_isBigO_rpow (E := \u2102) _ _ (lt_add_one _) _ hs).2\n[GOAL]\ncase convert_2\ns : \u2102\nhs : 0 < s.re\n\u22a2 LocallyIntegrableOn (fun x => \u2191(Real.exp (-x))) (Ioi 0)\n[PROOFSTEP]\nrefine' (Continuous.continuousOn _).locallyIntegrableOn measurableSet_Ioi\n[GOAL]\ncase convert_2\ns : \u2102\nhs : 0 < s.re\n\u22a2 Continuous fun x => \u2191(Real.exp (-x))\n[PROOFSTEP]\nexact continuous_ofReal.comp (Real.continuous_exp.comp continuous_neg)\n[GOAL]\ncase convert_3\ns : \u2102\nhs : 0 < s.re\n\u22a2 (fun x => \u2191(Real.exp (-x))) =O[atTop] fun x => x ^ (-(s.re + 1))\n[PROOFSTEP]\nrw [\u2190 isBigO_norm_left]\n[GOAL]\ncase convert_3\ns : \u2102\nhs : 0 < s.re\n\u22a2 (fun x => \u2016\u2191(Real.exp (-x))\u2016) =O[atTop] fun x => x ^ (-(s.re + 1))\n[PROOFSTEP]\nsimp_rw [Complex.norm_eq_abs, abs_ofReal, \u2190 Real.norm_eq_abs, isBigO_norm_left]\n[GOAL]\ncase convert_3\ns : \u2102\nhs : 0 < s.re\n\u22a2 (fun x => Real.exp (-x)) =O[atTop] fun x => x ^ (-(s.re + 1))\n[PROOFSTEP]\nsimpa only [neg_one_mul] using (isLittleO_exp_neg_mul_rpow_atTop zero_lt_one _).isBigO\n[GOAL]\ncase convert_4\ns : \u2102\nhs : 0 < s.re\n\u22a2 (fun x => \u2191(Real.exp (-x))) =O[\ud835\udcdd[Ioi 0] 0] fun x => x ^ (-0)\n[PROOFSTEP]\nsimp_rw [neg_zero, rpow_zero]\n[GOAL]\ncase convert_4\ns : \u2102\nhs : 0 < s.re\n\u22a2 (fun x => \u2191(Real.exp (-x))) =O[\ud835\udcdd[Ioi 0] 0] fun x => 1\n[PROOFSTEP]\nrefine' isBigO_const_of_tendsto (_ : Tendsto _ _ (\ud835\udcdd (1 : \u2102))) one_ne_zero\n[GOAL]\ncase convert_4\ns : \u2102\nhs : 0 < s.re\n\u22a2 Tendsto (fun x => \u2191(Real.exp (-x))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\nrw [(by simp : (1 : \u2102) = Real.exp (-0))]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\n\u22a2 1 = \u2191(Real.exp (-0))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase convert_4\ns : \u2102\nhs : 0 < s.re\n\u22a2 Tendsto (fun x => \u2191(Real.exp (-x))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd \u2191(Real.exp (-0)))\n[PROOFSTEP]\nexact (continuous_ofReal.comp (Real.continuous_exp.comp continuous_neg)).continuousWithinAt\n[GOAL]\ns : \u2102\nn : \u2115\nh1 : 1 - s.re < \u2191n\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\n\u22a2 DifferentiableAt \u2102 (GammaAux n) s\n[PROOFSTEP]\ninduction' n with n hn generalizing s\n[GOAL]\ncase zero\ns\u271d : \u2102\nn : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\ns : \u2102\nh1 : 1 - s.re < \u2191Nat.zero\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\n\u22a2 DifferentiableAt \u2102 (GammaAux Nat.zero) s\n[PROOFSTEP]\nrefine' (hasDerivAt_GammaIntegral _).differentiableAt\n[GOAL]\ncase zero\ns\u271d : \u2102\nn : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\ns : \u2102\nh1 : 1 - s.re < \u2191Nat.zero\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\n\u22a2 0 < s.re\n[PROOFSTEP]\nrw [Nat.cast_zero] at h1 \n[GOAL]\ncase zero\ns\u271d : \u2102\nn : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\ns : \u2102\nh1 : 1 - s.re < 0\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\n\u22a2 0 < s.re\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase succ\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\nhn : \u2200 (s : \u2102), 1 - s.re < \u2191n \u2192 (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) s\ns : \u2102\nh1 : 1 - s.re < \u2191(Nat.succ n)\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\n\u22a2 DifferentiableAt \u2102 (GammaAux (Nat.succ n)) s\n[PROOFSTEP]\ndsimp only [GammaAux]\n[GOAL]\ncase succ\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\nhn : \u2200 (s : \u2102), 1 - s.re < \u2191n \u2192 (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) s\ns : \u2102\nh1 : 1 - s.re < \u2191(Nat.succ n)\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\n\u22a2 DifferentiableAt \u2102 (fun s => GammaAux n (s + 1) / s) s\n[PROOFSTEP]\nspecialize hn (s + 1)\n[GOAL]\ncase succ\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\ns : \u2102\nh1 : 1 - s.re < \u2191(Nat.succ n)\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\nhn : 1 - (s + 1).re < \u2191n \u2192 (\u2200 (m : \u2115), s + 1 \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) (s + 1)\n\u22a2 DifferentiableAt \u2102 (fun s => GammaAux n (s + 1) / s) s\n[PROOFSTEP]\nhave a : 1 - (s + 1).re < \u2191n := by rw [Nat.cast_succ] at h1 ; rw [Complex.add_re, Complex.one_re]; linarith\n[GOAL]\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\ns : \u2102\nh1 : 1 - s.re < \u2191(Nat.succ n)\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\nhn : 1 - (s + 1).re < \u2191n \u2192 (\u2200 (m : \u2115), s + 1 \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) (s + 1)\n\u22a2 1 - (s + 1).re < \u2191n\n[PROOFSTEP]\nrw [Nat.cast_succ] at h1 \n[GOAL]\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\ns : \u2102\nh1 : 1 - s.re < \u2191n + 1\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\nhn : 1 - (s + 1).re < \u2191n \u2192 (\u2200 (m : \u2115), s + 1 \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) (s + 1)\n\u22a2 1 - (s + 1).re < \u2191n\n[PROOFSTEP]\nrw [Complex.add_re, Complex.one_re]\n[GOAL]\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\ns : \u2102\nh1 : 1 - s.re < \u2191n + 1\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\nhn : 1 - (s + 1).re < \u2191n \u2192 (\u2200 (m : \u2115), s + 1 \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) (s + 1)\n\u22a2 1 - (s.re + 1) < \u2191n\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase succ\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\ns : \u2102\nh1 : 1 - s.re < \u2191(Nat.succ n)\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\nhn : 1 - (s + 1).re < \u2191n \u2192 (\u2200 (m : \u2115), s + 1 \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) (s + 1)\na : 1 - (s + 1).re < \u2191n\n\u22a2 DifferentiableAt \u2102 (fun s => GammaAux n (s + 1) / s) s\n[PROOFSTEP]\nhave b : \u2200 m : \u2115, s + 1 \u2260 -m := by\n intro m; have := h2 (1 + m)\n contrapose! this\n rw [\u2190 eq_sub_iff_add_eq] at this \n simpa using this\n[GOAL]\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\ns : \u2102\nh1 : 1 - s.re < \u2191(Nat.succ n)\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\nhn : 1 - (s + 1).re < \u2191n \u2192 (\u2200 (m : \u2115), s + 1 \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) (s + 1)\na : 1 - (s + 1).re < \u2191n\n\u22a2 \u2200 (m : \u2115), s + 1 \u2260 -\u2191m\n[PROOFSTEP]\nintro m\n[GOAL]\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\ns : \u2102\nh1 : 1 - s.re < \u2191(Nat.succ n)\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\nhn : 1 - (s + 1).re < \u2191n \u2192 (\u2200 (m : \u2115), s + 1 \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) (s + 1)\na : 1 - (s + 1).re < \u2191n\nm : \u2115\n\u22a2 s + 1 \u2260 -\u2191m\n[PROOFSTEP]\nhave := h2 (1 + m)\n[GOAL]\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\ns : \u2102\nh1 : 1 - s.re < \u2191(Nat.succ n)\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\nhn : 1 - (s + 1).re < \u2191n \u2192 (\u2200 (m : \u2115), s + 1 \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) (s + 1)\na : 1 - (s + 1).re < \u2191n\nm : \u2115\nthis : s \u2260 -\u2191(1 + m)\n\u22a2 s + 1 \u2260 -\u2191m\n[PROOFSTEP]\ncontrapose! this\n[GOAL]\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\ns : \u2102\nh1 : 1 - s.re < \u2191(Nat.succ n)\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\nhn : 1 - (s + 1).re < \u2191n \u2192 (\u2200 (m : \u2115), s + 1 \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) (s + 1)\na : 1 - (s + 1).re < \u2191n\nm : \u2115\nthis : s + 1 = -\u2191m\n\u22a2 s = -\u2191(1 + m)\n[PROOFSTEP]\nrw [\u2190 eq_sub_iff_add_eq] at this \n[GOAL]\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\ns : \u2102\nh1 : 1 - s.re < \u2191(Nat.succ n)\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\nhn : 1 - (s + 1).re < \u2191n \u2192 (\u2200 (m : \u2115), s + 1 \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) (s + 1)\na : 1 - (s + 1).re < \u2191n\nm : \u2115\nthis : s = -\u2191m - 1\n\u22a2 s = -\u2191(1 + m)\n[PROOFSTEP]\nsimpa using this\n[GOAL]\ncase succ\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\ns : \u2102\nh1 : 1 - s.re < \u2191(Nat.succ n)\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\nhn : 1 - (s + 1).re < \u2191n \u2192 (\u2200 (m : \u2115), s + 1 \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) (s + 1)\na : 1 - (s + 1).re < \u2191n\nb : \u2200 (m : \u2115), s + 1 \u2260 -\u2191m\n\u22a2 DifferentiableAt \u2102 (fun s => GammaAux n (s + 1) / s) s\n[PROOFSTEP]\nrefine' DifferentiableAt.div (DifferentiableAt.comp _ (hn a b) _) _ _\n[GOAL]\ncase succ.refine'_1\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\ns : \u2102\nh1 : 1 - s.re < \u2191(Nat.succ n)\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\nhn : 1 - (s + 1).re < \u2191n \u2192 (\u2200 (m : \u2115), s + 1 \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) (s + 1)\na : 1 - (s + 1).re < \u2191n\nb : \u2200 (m : \u2115), s + 1 \u2260 -\u2191m\n\u22a2 DifferentiableAt \u2102 (fun s => s + 1) s\n[PROOFSTEP]\nrw [differentiableAt_add_const_iff (1 : \u2102)]\n[GOAL]\ncase succ.refine'_1\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\ns : \u2102\nh1 : 1 - s.re < \u2191(Nat.succ n)\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\nhn : 1 - (s + 1).re < \u2191n \u2192 (\u2200 (m : \u2115), s + 1 \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) (s + 1)\na : 1 - (s + 1).re < \u2191n\nb : \u2200 (m : \u2115), s + 1 \u2260 -\u2191m\n\u22a2 DifferentiableAt \u2102 (fun s => s) s\n[PROOFSTEP]\nexact differentiableAt_id\n[GOAL]\ncase succ.refine'_2\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\ns : \u2102\nh1 : 1 - s.re < \u2191(Nat.succ n)\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\nhn : 1 - (s + 1).re < \u2191n \u2192 (\u2200 (m : \u2115), s + 1 \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) (s + 1)\na : 1 - (s + 1).re < \u2191n\nb : \u2200 (m : \u2115), s + 1 \u2260 -\u2191m\n\u22a2 DifferentiableAt \u2102 (fun s => s) s\n[PROOFSTEP]\nexact differentiableAt_id\n[GOAL]\ncase succ.refine'_3\ns\u271d : \u2102\nn\u271d : \u2115\nh1\u271d : 1 - s\u271d.re < \u2191n\u271d\nh2\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn : \u2115\ns : \u2102\nh1 : 1 - s.re < \u2191(Nat.succ n)\nh2 : \u2200 (m : \u2115), s \u2260 -\u2191m\nhn : 1 - (s + 1).re < \u2191n \u2192 (\u2200 (m : \u2115), s + 1 \u2260 -\u2191m) \u2192 DifferentiableAt \u2102 (GammaAux n) (s + 1)\na : 1 - (s + 1).re < \u2191n\nb : \u2200 (m : \u2115), s + 1 \u2260 -\u2191m\n\u22a2 s \u2260 0\n[PROOFSTEP]\nsimpa using h2 0\n[GOAL]\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\n\u22a2 DifferentiableAt \u2102 Gamma s\n[PROOFSTEP]\nlet n := \u230a1 - s.re\u230b\u208a + 1\n[GOAL]\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\n\u22a2 DifferentiableAt \u2102 Gamma s\n[PROOFSTEP]\nhave hn : 1 - s.re < n := by exact_mod_cast Nat.lt_floor_add_one (1 - s.re)\n[GOAL]\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\n\u22a2 1 - s.re < \u2191n\n[PROOFSTEP]\nexact_mod_cast Nat.lt_floor_add_one (1 - s.re)\n[GOAL]\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\n\u22a2 DifferentiableAt \u2102 Gamma s\n[PROOFSTEP]\napply (differentiableAt_GammaAux s n hn hs).congr_of_eventuallyEq\n[GOAL]\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\n\u22a2 Gamma =\u1da0[\ud835\udcdd s] GammaAux n\n[PROOFSTEP]\nlet S := {t : \u2102 | 1 - t.re < n}\n[GOAL]\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\nS : Set \u2102 := {t | 1 - t.re < \u2191n}\n\u22a2 Gamma =\u1da0[\ud835\udcdd s] GammaAux n\n[PROOFSTEP]\nhave : S \u2208 \ud835\udcdd s := by\n rw [mem_nhds_iff]; use S\n refine' \u27e8Subset.rfl, _, hn\u27e9\n have : S = re \u207b\u00b9' Ioi (1 - n : \u211d) := by ext; rw [preimage, Ioi, mem_setOf_eq, mem_setOf_eq, mem_setOf_eq];\n exact sub_lt_comm\n rw [this]\n refine' Continuous.isOpen_preimage continuous_re _ isOpen_Ioi\n[GOAL]\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\nS : Set \u2102 := {t | 1 - t.re < \u2191n}\n\u22a2 S \u2208 \ud835\udcdd s\n[PROOFSTEP]\nrw [mem_nhds_iff]\n[GOAL]\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\nS : Set \u2102 := {t | 1 - t.re < \u2191n}\n\u22a2 \u2203 t, t \u2286 S \u2227 IsOpen t \u2227 s \u2208 t\n[PROOFSTEP]\nuse S\n[GOAL]\ncase h\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\nS : Set \u2102 := {t | 1 - t.re < \u2191n}\n\u22a2 S \u2286 S \u2227 IsOpen S \u2227 s \u2208 S\n[PROOFSTEP]\nrefine' \u27e8Subset.rfl, _, hn\u27e9\n[GOAL]\ncase h\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\nS : Set \u2102 := {t | 1 - t.re < \u2191n}\n\u22a2 IsOpen S\n[PROOFSTEP]\nhave : S = re \u207b\u00b9' Ioi (1 - n : \u211d) := by ext; rw [preimage, Ioi, mem_setOf_eq, mem_setOf_eq, mem_setOf_eq];\n exact sub_lt_comm\n[GOAL]\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\nS : Set \u2102 := {t | 1 - t.re < \u2191n}\n\u22a2 S = re \u207b\u00b9' Ioi (1 - \u2191n)\n[PROOFSTEP]\next\n[GOAL]\ncase h\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\nS : Set \u2102 := {t | 1 - t.re < \u2191n}\nx\u271d : \u2102\n\u22a2 x\u271d \u2208 S \u2194 x\u271d \u2208 re \u207b\u00b9' Ioi (1 - \u2191n)\n[PROOFSTEP]\nrw [preimage, Ioi, mem_setOf_eq, mem_setOf_eq, mem_setOf_eq]\n[GOAL]\ncase h\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\nS : Set \u2102 := {t | 1 - t.re < \u2191n}\nx\u271d : \u2102\n\u22a2 1 - x\u271d.re < \u2191n \u2194 1 - \u2191n < x\u271d.re\n[PROOFSTEP]\nexact sub_lt_comm\n[GOAL]\ncase h\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\nS : Set \u2102 := {t | 1 - t.re < \u2191n}\nthis : S = re \u207b\u00b9' Ioi (1 - \u2191n)\n\u22a2 IsOpen S\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase h\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\nS : Set \u2102 := {t | 1 - t.re < \u2191n}\nthis : S = re \u207b\u00b9' Ioi (1 - \u2191n)\n\u22a2 IsOpen (re \u207b\u00b9' Ioi (1 - \u2191n))\n[PROOFSTEP]\nrefine' Continuous.isOpen_preimage continuous_re _ isOpen_Ioi\n[GOAL]\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\nS : Set \u2102 := {t | 1 - t.re < \u2191n}\nthis : S \u2208 \ud835\udcdd s\n\u22a2 Gamma =\u1da0[\ud835\udcdd s] GammaAux n\n[PROOFSTEP]\napply eventuallyEq_of_mem this\n[GOAL]\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\nS : Set \u2102 := {t | 1 - t.re < \u2191n}\nthis : S \u2208 \ud835\udcdd s\n\u22a2 EqOn Gamma (GammaAux n) S\n[PROOFSTEP]\nintro t ht\n[GOAL]\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\nS : Set \u2102 := {t | 1 - t.re < \u2191n}\nthis : S \u2208 \ud835\udcdd s\nt : \u2102\nht : t \u2208 S\n\u22a2 Gamma t = GammaAux n t\n[PROOFSTEP]\nrw [mem_setOf_eq] at ht \n[GOAL]\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\nS : Set \u2102 := {t | 1 - t.re < \u2191n}\nthis : S \u2208 \ud835\udcdd s\nt : \u2102\nht : 1 - t.re < \u2191n\n\u22a2 Gamma t = GammaAux n t\n[PROOFSTEP]\napply Gamma_eq_GammaAux\n[GOAL]\ncase h1\ns : \u2102\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115 := \u230a1 - s.re\u230b\u208a + 1\nhn : 1 - s.re < \u2191n\nS : Set \u2102 := {t | 1 - t.re < \u2191n}\nthis : S \u2208 \ud835\udcdd s\nt : \u2102\nht : 1 - t.re < \u2191n\n\u22a2 -t.re < \u2191n\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u22a2 Tendsto (fun z => z * Gamma z) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\nrw [show \ud835\udcdd (1 : \u2102) = \ud835\udcdd (Gamma (0 + 1)) by simp only [zero_add, Complex.Gamma_one]]\n[GOAL]\n\u22a2 \ud835\udcdd 1 = \ud835\udcdd (Gamma (0 + 1))\n[PROOFSTEP]\nsimp only [zero_add, Complex.Gamma_one]\n[GOAL]\n\u22a2 Tendsto (fun z => z * Gamma z) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (Gamma (0 + 1)))\n[PROOFSTEP]\nconvert (Tendsto.mono_left _ nhdsWithin_le_nhds).congr' (eventuallyEq_of_mem self_mem_nhdsWithin Complex.Gamma_add_one)\n[GOAL]\ncase convert_1\n\u22a2 Tendsto (fun x => Gamma (x + 1)) (\ud835\udcdd 0) (\ud835\udcdd (Gamma (0 + 1)))\n[PROOFSTEP]\nrefine' ContinuousAt.comp (g := Gamma) _ (continuous_id.add continuous_const).continuousAt\n[GOAL]\ncase convert_1\n\u22a2 ContinuousAt Gamma (0 + 1)\n[PROOFSTEP]\nrefine' (Complex.differentiableAt_Gamma _ fun m => _).continuousAt\n[GOAL]\ncase convert_1\nm : \u2115\n\u22a2 0 + 1 \u2260 -\u2191m\n[PROOFSTEP]\nrw [zero_add, \u2190 ofReal_nat_cast, \u2190 ofReal_neg, \u2190 ofReal_one, Ne.def, ofReal_inj]\n[GOAL]\ncase convert_1\nm : \u2115\n\u22a2 \u00ac1 = -\u2191m\n[PROOFSTEP]\nrefine' (lt_of_le_of_lt _ zero_lt_one).ne'\n[GOAL]\ncase convert_1\nm : \u2115\n\u22a2 -\u2191m \u2264 0\n[PROOFSTEP]\nexact neg_nonpos.mpr (Nat.cast_nonneg _)\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 Gamma s = \u222b (x : \u211d) in Ioi 0, exp (-x) * x ^ (s - 1)\n[PROOFSTEP]\nrw [Gamma, Complex.Gamma_eq_integral (by rwa [Complex.ofReal_re] : 0 < Complex.re s)]\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 0 < (\u2191s).re\n[PROOFSTEP]\nrwa [Complex.ofReal_re]\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 (Complex.GammaIntegral \u2191s).re = \u222b (x : \u211d) in Ioi 0, exp (-x) * x ^ (s - 1)\n[PROOFSTEP]\ndsimp only [Complex.GammaIntegral]\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 (\u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ (\u2191s - 1)).re = \u222b (x : \u211d) in Ioi 0, exp (-x) * x ^ (s - 1)\n[PROOFSTEP]\nsimp_rw [\u2190 Complex.ofReal_one, \u2190 Complex.ofReal_sub]\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 (\u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1)).re = \u222b (x : \u211d) in Ioi 0, exp (-x) * x ^ (s - 1)\n[PROOFSTEP]\nsuffices\n \u222b x : \u211d in Ioi 0, \u2191(exp (-x)) * (x : \u2102) ^ ((s - 1 : \u211d) : \u2102) = \u222b x : \u211d in Ioi 0, ((exp (-x) * x ^ (s - 1) : \u211d) : \u2102)\n by\n have cc : \u2200 r : \u211d, Complex.ofReal' r = @IsROrC.ofReal \u2102 _ r := fun r => rfl\n conv_lhs => rw [this]; enter [1, 2, x]; rw [cc]\n rw [_root_.integral_ofReal, \u2190 cc, Complex.ofReal_re]\n[GOAL]\ns : \u211d\nhs : 0 < s\nthis : \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1) = \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x) * x ^ (s - 1))\n\u22a2 (\u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1)).re = \u222b (x : \u211d) in Ioi 0, exp (-x) * x ^ (s - 1)\n[PROOFSTEP]\nhave cc : \u2200 r : \u211d, Complex.ofReal' r = @IsROrC.ofReal \u2102 _ r := fun r => rfl\n[GOAL]\ns : \u211d\nhs : 0 < s\nthis : \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1) = \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x) * x ^ (s - 1))\ncc : \u2200 (r : \u211d), \u2191r = \u2191r\n\u22a2 (\u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1)).re = \u222b (x : \u211d) in Ioi 0, exp (-x) * x ^ (s - 1)\n[PROOFSTEP]\nconv_lhs => rw [this]; enter [1, 2, x]; rw [cc]\n[GOAL]\ns : \u211d\nhs : 0 < s\nthis : \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1) = \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x) * x ^ (s - 1))\ncc : \u2200 (r : \u211d), \u2191r = \u2191r\n| (\u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1)).re\n[PROOFSTEP]\nrw [this]; enter [1, 2, x]; rw [cc]\n[GOAL]\ns : \u211d\nhs : 0 < s\nthis : \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1) = \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x) * x ^ (s - 1))\ncc : \u2200 (r : \u211d), \u2191r = \u2191r\n| (\u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1)).re\n[PROOFSTEP]\nrw [this]; enter [1, 2, x]; rw [cc]\n[GOAL]\ns : \u211d\nhs : 0 < s\nthis : \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1) = \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x) * x ^ (s - 1))\ncc : \u2200 (r : \u211d), \u2191r = \u2191r\n| (\u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1)).re\n[PROOFSTEP]\nrw [this]\n[GOAL]\ns : \u211d\nhs : 0 < s\nthis : \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1) = \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x) * x ^ (s - 1))\ncc : \u2200 (r : \u211d), \u2191r = \u2191r\n| (\u222b (x : \u211d) in Ioi 0, \u2191(exp (-x) * x ^ (s - 1))).re\n[PROOFSTEP]\nenter [1, 2, x]\n[GOAL]\ncase h\ns : \u211d\nhs : 0 < s\nthis : \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1) = \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x) * x ^ (s - 1))\ncc : \u2200 (r : \u211d), \u2191r = \u2191r\nx : \u211d\n| \u2191(exp (-x) * x ^ (s - 1))\n[PROOFSTEP]\nrw [cc]\n[GOAL]\ns : \u211d\nhs : 0 < s\nthis : \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1) = \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x) * x ^ (s - 1))\ncc : \u2200 (r : \u211d), \u2191r = \u2191r\n\u22a2 (\u222b (x : \u211d) in Ioi 0, \u2191(exp (-x) * x ^ (s - 1))).re = \u222b (x : \u211d) in Ioi 0, exp (-x) * x ^ (s - 1)\n[PROOFSTEP]\nrw [_root_.integral_ofReal, \u2190 cc, Complex.ofReal_re]\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1) = \u222b (x : \u211d) in Ioi 0, \u2191(exp (-x) * x ^ (s - 1))\n[PROOFSTEP]\nrefine' set_integral_congr measurableSet_Ioi fun x hx => _\n[GOAL]\ns : \u211d\nhs : 0 < s\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 \u2191(exp (-x)) * \u2191x ^ \u2191(s - 1) = \u2191(exp (-x) * x ^ (s - 1))\n[PROOFSTEP]\npush_cast\n[GOAL]\ns : \u211d\nhs : 0 < s\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 Complex.exp (-\u2191x) * \u2191x ^ (\u2191s - 1) = Complex.exp (-\u2191x) * \u2191(x ^ (s - 1))\n[PROOFSTEP]\nrw [Complex.ofReal_cpow (le_of_lt hx)]\n[GOAL]\ns : \u211d\nhs : 0 < s\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 Complex.exp (-\u2191x) * \u2191x ^ (\u2191s - 1) = Complex.exp (-\u2191x) * \u2191x ^ \u2191(s - 1)\n[PROOFSTEP]\npush_cast\n[GOAL]\ns : \u211d\nhs : 0 < s\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 Complex.exp (-\u2191x) * \u2191x ^ (\u2191s - 1) = Complex.exp (-\u2191x) * \u2191x ^ (\u2191s - 1)\n[PROOFSTEP]\nrfl\n[GOAL]\ns : \u211d\nhs : s \u2260 0\n\u22a2 Gamma (s + 1) = s * Gamma s\n[PROOFSTEP]\nsimp_rw [Gamma]\n[GOAL]\ns : \u211d\nhs : s \u2260 0\n\u22a2 (Complex.Gamma \u2191(s + 1)).re = s * (Complex.Gamma \u2191s).re\n[PROOFSTEP]\nrw [Complex.ofReal_add, Complex.ofReal_one, Complex.Gamma_add_one, Complex.ofReal_mul_re]\n[GOAL]\ncase h2\ns : \u211d\nhs : s \u2260 0\n\u22a2 \u2191s \u2260 0\n[PROOFSTEP]\nrwa [Complex.ofReal_ne_zero]\n[GOAL]\n\u22a2 Gamma 1 = 1\n[PROOFSTEP]\nrw [Gamma, Complex.ofReal_one, Complex.Gamma_one, Complex.one_re]\n[GOAL]\ns : \u211d\n\u22a2 Complex.Gamma \u2191s = \u2191(Gamma s)\n[PROOFSTEP]\nrw [Gamma, eq_comm, \u2190 Complex.conj_eq_iff_re, \u2190 Complex.Gamma_conj, Complex.conj_ofReal]\n[GOAL]\nn : \u2115\n\u22a2 Gamma (\u2191n + 1) = \u2191n !\n[PROOFSTEP]\nrw [Gamma, Complex.ofReal_add, Complex.ofReal_nat_cast, Complex.ofReal_one, Complex.Gamma_nat_eq_factorial, \u2190\n Complex.ofReal_nat_cast, Complex.ofReal_re]\n[GOAL]\n\u22a2 Gamma 0 = 0\n[PROOFSTEP]\nsimpa only [\u2190 Complex.ofReal_zero, Complex.Gamma_ofReal, Complex.ofReal_inj] using Complex.Gamma_zero\n[GOAL]\nn : \u2115\n\u22a2 Gamma (-\u2191n) = 0\n[PROOFSTEP]\nsimpa only [\u2190 Complex.ofReal_nat_cast, \u2190 Complex.ofReal_neg, Complex.Gamma_ofReal, Complex.ofReal_eq_zero] using\n Complex.Gamma_neg_nat_eq_zero n\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 0 < Gamma s\n[PROOFSTEP]\nrw [Gamma_eq_integral hs]\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 0 < \u222b (x : \u211d) in Ioi 0, exp (-x) * x ^ (s - 1)\n[PROOFSTEP]\nhave : (Function.support fun x : \u211d => exp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0 :=\n by\n rw [inter_eq_right_iff_subset]\n intro x hx\n rw [Function.mem_support]\n exact mul_ne_zero (exp_pos _).ne' (rpow_pos_of_pos hx _).ne'\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 (Function.support fun x => exp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n[PROOFSTEP]\nrw [inter_eq_right_iff_subset]\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 Ioi 0 \u2286 Function.support fun x => exp (-x) * x ^ (s - 1)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ns : \u211d\nhs : 0 < s\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 x \u2208 Function.support fun x => exp (-x) * x ^ (s - 1)\n[PROOFSTEP]\nrw [Function.mem_support]\n[GOAL]\ns : \u211d\nhs : 0 < s\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 exp (-x) * x ^ (s - 1) \u2260 0\n[PROOFSTEP]\nexact mul_ne_zero (exp_pos _).ne' (rpow_pos_of_pos hx _).ne'\n[GOAL]\ns : \u211d\nhs : 0 < s\nthis : (Function.support fun x => exp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 0 < \u222b (x : \u211d) in Ioi 0, exp (-x) * x ^ (s - 1)\n[PROOFSTEP]\nrw [set_integral_pos_iff_support_of_nonneg_ae]\n[GOAL]\ns : \u211d\nhs : 0 < s\nthis : (Function.support fun x => exp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 0 < \u2191\u2191volume ((Function.support fun x => exp (-x) * x ^ (s - 1)) \u2229 Ioi 0)\n[PROOFSTEP]\nrw [this, volume_Ioi, \u2190 ENNReal.ofReal_zero]\n[GOAL]\ns : \u211d\nhs : 0 < s\nthis : (Function.support fun x => exp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 ENNReal.ofReal 0 < \u22a4\n[PROOFSTEP]\nexact ENNReal.ofReal_lt_top\n[GOAL]\ncase hf\ns : \u211d\nhs : 0 < s\nthis : (Function.support fun x => exp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 0 \u2264\u1d50[Measure.restrict volume (Ioi 0)] fun x => exp (-x) * x ^ (s - 1)\n[PROOFSTEP]\nrefine' eventually_of_mem (self_mem_ae_restrict measurableSet_Ioi) _\n[GOAL]\ncase hf\ns : \u211d\nhs : 0 < s\nthis : (Function.support fun x => exp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioi 0 \u2192 OfNat.ofNat 0 x \u2264 (fun x => exp (-x) * x ^ (s - 1)) x\n[PROOFSTEP]\nexact fun x hx => (mul_pos (exp_pos _) (rpow_pos_of_pos hx _)).le\n[GOAL]\ncase hfi\ns : \u211d\nhs : 0 < s\nthis : (Function.support fun x => exp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 IntegrableOn (fun x => exp (-x) * x ^ (s - 1)) (Ioi 0)\n[PROOFSTEP]\nexact GammaIntegral_convergent hs\n[GOAL]\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\n\u22a2 Gamma s \u2260 0\n[PROOFSTEP]\nsuffices \u2200 {n : \u2115}, -(n : \u211d) < s \u2192 Gamma s \u2260 0 by\n apply this\n swap; exact \u230a-s\u230b\u208a + 1\n rw [neg_lt, Nat.cast_add, Nat.cast_one]\n exact Nat.lt_floor_add_one _\n[GOAL]\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nthis : \u2200 {n : \u2115}, -\u2191n < s \u2192 Gamma s \u2260 0\n\u22a2 Gamma s \u2260 0\n[PROOFSTEP]\napply this\n[GOAL]\ncase a\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nthis : \u2200 {n : \u2115}, -\u2191n < s \u2192 Gamma s \u2260 0\n\u22a2 -\u2191?n < s\ncase n s : \u211d hs : \u2200 (m : \u2115), s \u2260 -\u2191m this : \u2200 {n : \u2115}, -\u2191n < s \u2192 Gamma s \u2260 0 \u22a2 \u2115\n[PROOFSTEP]\nswap\n[GOAL]\ncase n\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nthis : \u2200 {n : \u2115}, -\u2191n < s \u2192 Gamma s \u2260 0\n\u22a2 \u2115\ncase a s : \u211d hs : \u2200 (m : \u2115), s \u2260 -\u2191m this : \u2200 {n : \u2115}, -\u2191n < s \u2192 Gamma s \u2260 0 \u22a2 -\u2191?n < s\n[PROOFSTEP]\nexact \u230a-s\u230b\u208a + 1\n[GOAL]\ncase a\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nthis : \u2200 {n : \u2115}, -\u2191n < s \u2192 Gamma s \u2260 0\n\u22a2 -\u2191(\u230a-s\u230b\u208a + 1) < s\n[PROOFSTEP]\nrw [neg_lt, Nat.cast_add, Nat.cast_one]\n[GOAL]\ncase a\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nthis : \u2200 {n : \u2115}, -\u2191n < s \u2192 Gamma s \u2260 0\n\u22a2 -s < \u2191\u230a-s\u230b\u208a + 1\n[PROOFSTEP]\nexact Nat.lt_floor_add_one _\n[GOAL]\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\n\u22a2 \u2200 {n : \u2115}, -\u2191n < s \u2192 Gamma s \u2260 0\n[PROOFSTEP]\nintro n\n[GOAL]\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nn : \u2115\n\u22a2 -\u2191n < s \u2192 Gamma s \u2260 0\n[PROOFSTEP]\ninduction' n with _ n_ih generalizing s\n[GOAL]\ncase zero\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\n\u22a2 -\u2191Nat.zero < s \u2192 Gamma s \u2260 0\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase zero\ns\u271d : \u211d\nhs\u271d\u00b9 : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\ns : \u211d\nhs\u271d : \u2200 (m : \u2115), s \u2260 -\u2191m\nhs : -\u2191Nat.zero < s\n\u22a2 Gamma s \u2260 0\n[PROOFSTEP]\nrefine' (Gamma_pos_of_pos _).ne'\n[GOAL]\ncase zero\ns\u271d : \u211d\nhs\u271d\u00b9 : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\ns : \u211d\nhs\u271d : \u2200 (m : \u2115), s \u2260 -\u2191m\nhs : -\u2191Nat.zero < s\n\u22a2 0 < s\n[PROOFSTEP]\nrwa [Nat.cast_zero, neg_zero] at hs \n[GOAL]\ncase succ\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn\u271d : \u2115\nn_ih : \u2200 {s : \u211d}, (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 -\u2191n\u271d < s \u2192 Gamma s \u2260 0\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\n\u22a2 -\u2191(Nat.succ n\u271d) < s \u2192 Gamma s \u2260 0\n[PROOFSTEP]\nintro hs'\n[GOAL]\ncase succ\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn\u271d : \u2115\nn_ih : \u2200 {s : \u211d}, (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 -\u2191n\u271d < s \u2192 Gamma s \u2260 0\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nhs' : -\u2191(Nat.succ n\u271d) < s\n\u22a2 Gamma s \u2260 0\n[PROOFSTEP]\nhave : Gamma (s + 1) \u2260 0 := by\n apply n_ih\n \u00b7 intro m\n specialize hs (1 + m)\n contrapose! hs\n rw [\u2190 eq_sub_iff_add_eq] at hs \n rw [hs]\n push_cast\n ring\n \u00b7 rw [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one, neg_add] at hs' \n linarith\n[GOAL]\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn\u271d : \u2115\nn_ih : \u2200 {s : \u211d}, (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 -\u2191n\u271d < s \u2192 Gamma s \u2260 0\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nhs' : -\u2191(Nat.succ n\u271d) < s\n\u22a2 Gamma (s + 1) \u2260 0\n[PROOFSTEP]\napply n_ih\n[GOAL]\ncase hs\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn\u271d : \u2115\nn_ih : \u2200 {s : \u211d}, (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 -\u2191n\u271d < s \u2192 Gamma s \u2260 0\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nhs' : -\u2191(Nat.succ n\u271d) < s\n\u22a2 \u2200 (m : \u2115), s + 1 \u2260 -\u2191m\n[PROOFSTEP]\nintro m\n[GOAL]\ncase hs\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn\u271d : \u2115\nn_ih : \u2200 {s : \u211d}, (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 -\u2191n\u271d < s \u2192 Gamma s \u2260 0\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nhs' : -\u2191(Nat.succ n\u271d) < s\nm : \u2115\n\u22a2 s + 1 \u2260 -\u2191m\n[PROOFSTEP]\nspecialize hs (1 + m)\n[GOAL]\ncase hs\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn\u271d : \u2115\nn_ih : \u2200 {s : \u211d}, (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 -\u2191n\u271d < s \u2192 Gamma s \u2260 0\ns : \u211d\nhs' : -\u2191(Nat.succ n\u271d) < s\nm : \u2115\nhs : s \u2260 -\u2191(1 + m)\n\u22a2 s + 1 \u2260 -\u2191m\n[PROOFSTEP]\ncontrapose! hs\n[GOAL]\ncase hs\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn\u271d : \u2115\nn_ih : \u2200 {s : \u211d}, (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 -\u2191n\u271d < s \u2192 Gamma s \u2260 0\ns : \u211d\nhs' : -\u2191(Nat.succ n\u271d) < s\nm : \u2115\nhs : s + 1 = -\u2191m\n\u22a2 s = -\u2191(1 + m)\n[PROOFSTEP]\nrw [\u2190 eq_sub_iff_add_eq] at hs \n[GOAL]\ncase hs\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn\u271d : \u2115\nn_ih : \u2200 {s : \u211d}, (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 -\u2191n\u271d < s \u2192 Gamma s \u2260 0\ns : \u211d\nhs' : -\u2191(Nat.succ n\u271d) < s\nm : \u2115\nhs : s = -\u2191m - 1\n\u22a2 s = -\u2191(1 + m)\n[PROOFSTEP]\nrw [hs]\n[GOAL]\ncase hs\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn\u271d : \u2115\nn_ih : \u2200 {s : \u211d}, (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 -\u2191n\u271d < s \u2192 Gamma s \u2260 0\ns : \u211d\nhs' : -\u2191(Nat.succ n\u271d) < s\nm : \u2115\nhs : s = -\u2191m - 1\n\u22a2 -\u2191m - 1 = -\u2191(1 + m)\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase hs\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn\u271d : \u2115\nn_ih : \u2200 {s : \u211d}, (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 -\u2191n\u271d < s \u2192 Gamma s \u2260 0\ns : \u211d\nhs' : -\u2191(Nat.succ n\u271d) < s\nm : \u2115\nhs : s = -\u2191m - 1\n\u22a2 -\u2191m - 1 = -(1 + \u2191m)\n[PROOFSTEP]\nring\n[GOAL]\ncase a\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn\u271d : \u2115\nn_ih : \u2200 {s : \u211d}, (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 -\u2191n\u271d < s \u2192 Gamma s \u2260 0\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nhs' : -\u2191(Nat.succ n\u271d) < s\n\u22a2 -\u2191n\u271d < s + 1\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one, neg_add] at hs' \n[GOAL]\ncase a\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn\u271d : \u2115\nn_ih : \u2200 {s : \u211d}, (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 -\u2191n\u271d < s \u2192 Gamma s \u2260 0\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nhs' : -\u2191n\u271d + -1 < s\n\u22a2 -\u2191n\u271d < s + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase succ\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn\u271d : \u2115\nn_ih : \u2200 {s : \u211d}, (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 -\u2191n\u271d < s \u2192 Gamma s \u2260 0\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nhs' : -\u2191(Nat.succ n\u271d) < s\nthis : Gamma (s + 1) \u2260 0\n\u22a2 Gamma s \u2260 0\n[PROOFSTEP]\nrw [Gamma_add_one, mul_ne_zero_iff] at this \n[GOAL]\ncase succ\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn\u271d : \u2115\nn_ih : \u2200 {s : \u211d}, (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 -\u2191n\u271d < s \u2192 Gamma s \u2260 0\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nhs' : -\u2191(Nat.succ n\u271d) < s\nthis : s \u2260 0 \u2227 Gamma s \u2260 0\n\u22a2 Gamma s \u2260 0\n[PROOFSTEP]\nexact this.2\n[GOAL]\ncase succ\ns\u271d : \u211d\nhs\u271d : \u2200 (m : \u2115), s\u271d \u2260 -\u2191m\nn\u271d : \u2115\nn_ih : \u2200 {s : \u211d}, (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 -\u2191n\u271d < s \u2192 Gamma s \u2260 0\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\nhs' : -\u2191(Nat.succ n\u271d) < s\nthis : Gamma (s + 1) \u2260 0\n\u22a2 s \u2260 0\n[PROOFSTEP]\nsimpa using hs 0\n[GOAL]\ns : \u211d\n\u22a2 Gamma s = 0 \u2192 \u2203 m, s = -\u2191m\n[PROOFSTEP]\ncontrapose!\n[GOAL]\ns : \u211d\n\u22a2 (\u2200 (m : \u2115), s \u2260 -\u2191m) \u2192 Gamma s \u2260 0\n[PROOFSTEP]\nexact Gamma_ne_zero\n[GOAL]\ns : \u211d\n\u22a2 (\u2203 m, s = -\u2191m) \u2192 Gamma s = 0\n[PROOFSTEP]\nrintro \u27e8m, rfl\u27e9\n[GOAL]\ncase intro\nm : \u2115\n\u22a2 Gamma (-\u2191m) = 0\n[PROOFSTEP]\nexact Gamma_neg_nat_eq_zero m\n[GOAL]\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\n\u22a2 DifferentiableAt \u211d Gamma s\n[PROOFSTEP]\nrefine' (Complex.differentiableAt_Gamma _ _).hasDerivAt.real_of_complex.differentiableAt\n[GOAL]\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\n\u22a2 \u2200 (m : \u2115), \u2191s \u2260 -\u2191m\n[PROOFSTEP]\nsimp_rw [\u2190 Complex.ofReal_nat_cast, \u2190 Complex.ofReal_neg, Ne.def, Complex.ofReal_inj]\n[GOAL]\ns : \u211d\nhs : \u2200 (m : \u2115), s \u2260 -\u2191m\n\u22a2 \u2200 (m : \u2115), \u00acs = -\u2191m\n[PROOFSTEP]\nexact hs\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "llama_tokens": 49468, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217431943271999, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.43047156418765276}} {"text": "[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : M\nhab : a \u2264 b\n\u22a2 a ^ 0 \u2264 b ^ 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : M\nhab : a \u2264 b\nk : \u2115\n\u22a2 a ^ (k + 1) \u2264 b ^ (k + 1)\n[PROOFSTEP]\nrw [pow_succ, pow_succ]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : M\nhab : a \u2264 b\nk : \u2115\n\u22a2 a * a ^ k \u2264 b * b ^ k\n[PROOFSTEP]\nexact mul_le_mul' hab (pow_le_pow_of_le_left' hab k)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx a : M\nH : 1 \u2264 a\n\u22a2 1 \u2264 a ^ 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx a : M\nH : 1 \u2264 a\nk : \u2115\n\u22a2 1 \u2264 a ^ (k + 1)\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx a : M\nH : 1 \u2264 a\nk : \u2115\n\u22a2 1 \u2264 a * a ^ k\n[PROOFSTEP]\nexact one_le_mul H (one_le_pow_of_one_le' H k)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx a : M\nn m : \u2115\nha : 1 \u2264 a\nh : n \u2264 m\nk : \u2115\nhk : n + k = m\n\u22a2 a ^ n * a ^ k = a ^ m\n[PROOFSTEP]\nrw [\u2190 hk, pow_add]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx a : M\nha : 1 < a\nk : \u2115\nhk : k \u2260 0\n\u22a2 1 < a ^ k\n[PROOFSTEP]\nrcases Nat.exists_eq_succ_of_ne_zero hk with \u27e8l, rfl\u27e9\n[GOAL]\ncase intro\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx a : M\nha : 1 < a\nl : \u2115\nhk : Nat.succ l \u2260 0\n\u22a2 1 < a ^ Nat.succ l\n[PROOFSTEP]\nclear hk\n[GOAL]\ncase intro\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx a : M\nha : 1 < a\nl : \u2115\n\u22a2 1 < a ^ Nat.succ l\n[PROOFSTEP]\ninduction' l with l IH\n[GOAL]\ncase intro.zero\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx a : M\nha : 1 < a\n\u22a2 1 < a ^ Nat.succ Nat.zero\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\ncase intro.zero\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx a : M\nha : 1 < a\n\u22a2 1 < a * a ^ Nat.zero\n[PROOFSTEP]\nsimpa using ha\n[GOAL]\ncase intro.succ\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx a : M\nha : 1 < a\nl : \u2115\nIH : 1 < a ^ Nat.succ l\n\u22a2 1 < a ^ Nat.succ (Nat.succ l)\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\ncase intro.succ\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx a : M\nha : 1 < a\nl : \u2115\nIH : 1 < a ^ Nat.succ l\n\u22a2 1 < a * a ^ (l + 1)\n[PROOFSTEP]\nexact one_lt_mul'' ha IH\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : M\nn m : \u2115\nha : 1 < a\nh : n < m\n\u22a2 a ^ n < a ^ m\n[PROOFSTEP]\nrcases Nat.le.dest h with \u27e8k, rfl\u27e9\n[GOAL]\ncase intro\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : M\nn : \u2115\nha : 1 < a\nk : \u2115\nh : n < Nat.succ n + k\n\u22a2 a ^ n < a ^ (Nat.succ n + k)\n[PROOFSTEP]\nclear h\n[GOAL]\ncase intro\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : M\nn : \u2115\nha : 1 < a\nk : \u2115\n\u22a2 a ^ n < a ^ (Nat.succ n + k)\n[PROOFSTEP]\nrw [pow_add, pow_succ', mul_assoc, \u2190 pow_succ]\n[GOAL]\ncase intro\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : M\nn : \u2115\nha : 1 < a\nk : \u2115\n\u22a2 a ^ n < a ^ n * a ^ (k + 1)\n[PROOFSTEP]\nexact lt_mul_of_one_lt_right' _ (one_lt_pow' ha k.succ_ne_zero)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\nhx : 1 \u2264 x\nn : \u2115\n\u22a2 1 \u2264 x ^ (n + 1)\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\nhx : 1 \u2264 x\nn : \u2115\n\u22a2 1 \u2264 x * x ^ n\n[PROOFSTEP]\nexact Left.one_le_mul hx <| Left.one_le_pow_of_le hx\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\nhx : x \u2264 1\nn : \u2115\n\u22a2 x ^ (n + 1) \u2264 1\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\nhx : x \u2264 1\nn : \u2115\n\u22a2 x * x ^ n \u2264 1\n[PROOFSTEP]\nexact Left.mul_le_one hx <| Left.pow_le_one_of_le hx\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\nhx : 1 \u2264 x\nn : \u2115\n\u22a2 1 \u2264 x ^ (n + 1)\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\nhx : 1 \u2264 x\nn : \u2115\n\u22a2 1 \u2264 x * x ^ n\n[PROOFSTEP]\nexact Right.one_le_mul hx <| Right.one_le_pow_of_le hx\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\nhx : x \u2264 1\nn : \u2115\n\u22a2 x ^ (n + 1) \u2264 1\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\nhx : x \u2264 1\nn : \u2115\n\u22a2 x * x ^ n \u2264 1\n[PROOFSTEP]\nexact Right.mul_le_one hx <| Right.pow_le_one_of_le hx\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Preorder M\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\nf : \u03b2 \u2192 M\nhf : StrictMono f\nx\u271d : 1 \u2260 0\n\u22a2 StrictMono fun a => f a ^ 1\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Preorder M\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\nf : \u03b2 \u2192 M\nhf : StrictMono f\nn : \u2115\nx\u271d : Nat.succ (Nat.succ n) \u2260 0\n\u22a2 StrictMono fun a => f a ^ Nat.succ (Nat.succ n)\n[PROOFSTEP]\nsimp_rw [pow_succ _ (n + 1)]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Preorder M\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\nf : \u03b2 \u2192 M\nhf : StrictMono f\nn : \u2115\nx\u271d : Nat.succ (Nat.succ n) \u2260 0\n\u22a2 StrictMono fun a => f a * f a ^ (n + 1)\n[PROOFSTEP]\nexact hf.mul' (StrictMono.pow_right' hf n.succ_ne_zero)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Preorder M\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nf : \u03b2 \u2192 M\nhf : Monotone f\n\u22a2 Monotone fun a => f a ^ 0\n[PROOFSTEP]\nsimpa using monotone_const\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Preorder M\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nf : \u03b2 \u2192 M\nhf : Monotone f\nn : \u2115\n\u22a2 Monotone fun a => f a ^ (n + 1)\n[PROOFSTEP]\nsimp_rw [pow_succ]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Preorder M\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nf : \u03b2 \u2192 M\nhf : Monotone f\nn : \u2115\n\u22a2 Monotone fun a => f a * f a ^ n\n[PROOFSTEP]\nexact hf.mul' (Monotone.pow_right hf _)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nn\u271d : \u2115\nx : M\nhn : 0 < n\u271d\nh : x < 1\nn : \u2115\nx\u271d : Nat.succ 0 \u2264 n\nih : x ^ n < 1\n\u22a2 x ^ (n + 1) < 1\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nn\u271d : \u2115\nx : M\nhn : 0 < n\u271d\nh : x < 1\nn : \u2115\nx\u271d : Nat.succ 0 \u2264 n\nih : x ^ n < 1\n\u22a2 x * x ^ n < 1\n[PROOFSTEP]\nexact mul_lt_one h ih\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\nn\u271d : \u2115\nx : M\nhn : 0 < n\u271d\nh : x < 1\nn : \u2115\nx\u271d : Nat.succ 0 \u2264 n\nih : x ^ n < 1\n\u22a2 x ^ (n + 1) < 1\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\nn\u271d : \u2115\nx : M\nhn : 0 < n\u271d\nh : x < 1\nn : \u2115\nx\u271d : Nat.succ 0 \u2264 n\nih : x ^ n < 1\n\u22a2 x * x ^ n < 1\n[PROOFSTEP]\nexact Right.mul_lt_one h ih\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : LinearOrder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\nn : \u2115\nhn : n \u2260 0\n\u22a2 x ^ n = 1 \u2194 x = 1\n[PROOFSTEP]\nsimp only [le_antisymm_iff]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : LinearOrder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\nn : \u2115\nhn : n \u2260 0\n\u22a2 x ^ n \u2264 1 \u2227 1 \u2264 x ^ n \u2194 x \u2264 1 \u2227 1 \u2264 x\n[PROOFSTEP]\nrw [pow_le_one_iff hn, one_le_pow_iff hn]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : LinearOrder M\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : M\nh : a * b < c * d\n\u22a2 min a b ^ 2 < max c d ^ 2\n[PROOFSTEP]\nsimp_rw [pow_two]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : LinearOrder M\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : M\nh : a * b < c * d\n\u22a2 min a b * min a b < max c d * max c d\n[PROOFSTEP]\nexact (mul_le_mul' inf_le_left inf_le_right).trans_lt (h.trans_le <| mul_le_mul' le_sup_left le_sup_right)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : LinearOrder M\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : M\nh : a * b < c ^ 2\n\u22a2 min a b < c\n[PROOFSTEP]\nsimpa using min_lt_max_of_mul_lt_mul (h.trans_eq <| pow_two _)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : LinearOrder M\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : M\nh : a ^ 2 < b * c\n\u22a2 a < max b c\n[PROOFSTEP]\nsimpa using min_lt_max_of_mul_lt_mul ((pow_two _).symm.trans_lt h)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : LinearOrder M\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c : M\nh : a * b \u2264 c ^ 2\n\u22a2 min a b \u2264 c\n[PROOFSTEP]\nsimpa using min_le_max_of_mul_le_mul (h.trans_eq <| pow_two _)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : LinearOrder M\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c : M\nh : a ^ 2 \u2264 b * c\n\u22a2 a \u2264 max b c\n[PROOFSTEP]\nsimpa using min_le_max_of_mul_le_mul ((pow_two _).symm.trans_le h)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : Preorder G\ninst\u271d : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : G\nH : 1 \u2264 x\nn : \u2124\nhn : 0 \u2264 n\n\u22a2 1 \u2264 x ^ n\n[PROOFSTEP]\nlift n to \u2115 using hn\n[GOAL]\ncase intro\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : Preorder G\ninst\u271d : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : G\nH : 1 \u2264 x\nn : \u2115\n\u22a2 1 \u2264 x ^ \u2191n\n[PROOFSTEP]\nrw [zpow_ofNat]\n[GOAL]\ncase intro\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : Preorder G\ninst\u271d : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : G\nH : 1 \u2264 x\nn : \u2115\n\u22a2 1 \u2264 x ^ n\n[PROOFSTEP]\napply one_le_pow_of_one_le' H\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nn\u271d m n : \u2115\n\u22a2 0 ^ (n + 1) \u2264 1\n[PROOFSTEP]\nrw [zero_pow n.succ_pos]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nn\u271d m n : \u2115\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nexact zero_le_one\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nn m : \u2115\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nhn : n \u2260 0\n\u22a2 x ^ n + y ^ n \u2264 (x + y) ^ n\n[PROOFSTEP]\nrcases Nat.exists_eq_succ_of_ne_zero hn with \u27e8k, rfl\u27e9\n[GOAL]\ncase intro\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nm : \u2115\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nk : \u2115\nhn : Nat.succ k \u2260 0\n\u22a2 x ^ Nat.succ k + y ^ Nat.succ k \u2264 (x + y) ^ Nat.succ k\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase intro.zero\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nm : \u2115\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nk : \u2115\nhn\u271d : Nat.succ k \u2260 0\nhn : Nat.succ Nat.zero \u2260 0\n\u22a2 x ^ Nat.succ Nat.zero + y ^ Nat.succ Nat.zero \u2264 (x + y) ^ Nat.succ Nat.zero\n[PROOFSTEP]\nhave eqn : Nat.succ Nat.zero = 1 := rfl\n[GOAL]\ncase intro.zero\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nm : \u2115\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nk : \u2115\nhn\u271d : Nat.succ k \u2260 0\nhn : Nat.succ Nat.zero \u2260 0\neqn : Nat.succ Nat.zero = 1\n\u22a2 x ^ Nat.succ Nat.zero + y ^ Nat.succ Nat.zero \u2264 (x + y) ^ Nat.succ Nat.zero\n[PROOFSTEP]\nrw [eqn]\n[GOAL]\ncase intro.zero\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nm : \u2115\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nk : \u2115\nhn\u271d : Nat.succ k \u2260 0\nhn : Nat.succ Nat.zero \u2260 0\neqn : Nat.succ Nat.zero = 1\n\u22a2 x ^ 1 + y ^ 1 \u2264 (x + y) ^ 1\n[PROOFSTEP]\nsimp only [pow_one, le_refl]\n[GOAL]\ncase intro.succ\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nm : \u2115\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nk\u271d : \u2115\nhn\u271d : Nat.succ k\u271d \u2260 0\nk : \u2115\nih : Nat.succ k \u2260 0 \u2192 x ^ Nat.succ k + y ^ Nat.succ k \u2264 (x + y) ^ Nat.succ k\nhn : Nat.succ (Nat.succ k) \u2260 0\n\u22a2 x ^ Nat.succ (Nat.succ k) + y ^ Nat.succ (Nat.succ k) \u2264 (x + y) ^ Nat.succ (Nat.succ k)\n[PROOFSTEP]\nlet n := k.succ\n[GOAL]\ncase intro.succ\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nm : \u2115\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nk\u271d : \u2115\nhn\u271d : Nat.succ k\u271d \u2260 0\nk : \u2115\nih : Nat.succ k \u2260 0 \u2192 x ^ Nat.succ k + y ^ Nat.succ k \u2264 (x + y) ^ Nat.succ k\nhn : Nat.succ (Nat.succ k) \u2260 0\nn : \u2115 := Nat.succ k\n\u22a2 x ^ Nat.succ (Nat.succ k) + y ^ Nat.succ (Nat.succ k) \u2264 (x + y) ^ Nat.succ (Nat.succ k)\n[PROOFSTEP]\nhave h1 := add_nonneg (mul_nonneg hx (pow_nonneg hy n)) (mul_nonneg hy (pow_nonneg hx n))\n[GOAL]\ncase intro.succ\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nm : \u2115\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nk\u271d : \u2115\nhn\u271d : Nat.succ k\u271d \u2260 0\nk : \u2115\nih : Nat.succ k \u2260 0 \u2192 x ^ Nat.succ k + y ^ Nat.succ k \u2264 (x + y) ^ Nat.succ k\nhn : Nat.succ (Nat.succ k) \u2260 0\nn : \u2115 := Nat.succ k\nh1 : 0 \u2264 x * y ^ n + y * x ^ n\n\u22a2 x ^ Nat.succ (Nat.succ k) + y ^ Nat.succ (Nat.succ k) \u2264 (x + y) ^ Nat.succ (Nat.succ k)\n[PROOFSTEP]\nhave h2 := add_nonneg hx hy\n[GOAL]\ncase intro.succ\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nm : \u2115\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nk\u271d : \u2115\nhn\u271d : Nat.succ k\u271d \u2260 0\nk : \u2115\nih : Nat.succ k \u2260 0 \u2192 x ^ Nat.succ k + y ^ Nat.succ k \u2264 (x + y) ^ Nat.succ k\nhn : Nat.succ (Nat.succ k) \u2260 0\nn : \u2115 := Nat.succ k\nh1 : 0 \u2264 x * y ^ n + y * x ^ n\nh2 : 0 \u2264 x + y\n\u22a2 x ^ Nat.succ (Nat.succ k) + y ^ Nat.succ (Nat.succ k) \u2264 (x + y) ^ Nat.succ (Nat.succ k)\n[PROOFSTEP]\ncalc\n x ^ n.succ + y ^ n.succ \u2264 x * x ^ n + y * y ^ n + (x * y ^ n + y * x ^ n) :=\n by\n rw [pow_succ _ n, pow_succ _ n]\n exact le_add_of_nonneg_right h1\n _ = (x + y) * (x ^ n + y ^ n) := by\n rw [add_mul, mul_add, mul_add, add_comm (y * x ^ n), \u2190 add_assoc, \u2190 add_assoc, add_assoc (x * x ^ n) (x * y ^ n),\n add_comm (x * y ^ n) (y * y ^ n), \u2190 add_assoc]\n _ \u2264 (x + y) ^ n.succ := by\n rw [pow_succ _ n]\n exact mul_le_mul_of_nonneg_left (ih (Nat.succ_ne_zero k)) h2\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nm : \u2115\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nk\u271d : \u2115\nhn\u271d : Nat.succ k\u271d \u2260 0\nk : \u2115\nih : Nat.succ k \u2260 0 \u2192 x ^ Nat.succ k + y ^ Nat.succ k \u2264 (x + y) ^ Nat.succ k\nhn : Nat.succ (Nat.succ k) \u2260 0\nn : \u2115 := Nat.succ k\nh1 : 0 \u2264 x * y ^ n + y * x ^ n\nh2 : 0 \u2264 x + y\n\u22a2 x ^ Nat.succ n + y ^ Nat.succ n \u2264 x * x ^ n + y * y ^ n + (x * y ^ n + y * x ^ n)\n[PROOFSTEP]\nrw [pow_succ _ n, pow_succ _ n]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nm : \u2115\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nk\u271d : \u2115\nhn\u271d : Nat.succ k\u271d \u2260 0\nk : \u2115\nih : Nat.succ k \u2260 0 \u2192 x ^ Nat.succ k + y ^ Nat.succ k \u2264 (x + y) ^ Nat.succ k\nhn : Nat.succ (Nat.succ k) \u2260 0\nn : \u2115 := Nat.succ k\nh1 : 0 \u2264 x * y ^ n + y * x ^ n\nh2 : 0 \u2264 x + y\n\u22a2 x * x ^ n + y * y ^ n \u2264 x * x ^ n + y * y ^ n + (x * y ^ n + y * x ^ n)\n[PROOFSTEP]\nexact le_add_of_nonneg_right h1\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nm : \u2115\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nk\u271d : \u2115\nhn\u271d : Nat.succ k\u271d \u2260 0\nk : \u2115\nih : Nat.succ k \u2260 0 \u2192 x ^ Nat.succ k + y ^ Nat.succ k \u2264 (x + y) ^ Nat.succ k\nhn : Nat.succ (Nat.succ k) \u2260 0\nn : \u2115 := Nat.succ k\nh1 : 0 \u2264 x * y ^ n + y * x ^ n\nh2 : 0 \u2264 x + y\n\u22a2 x * x ^ n + y * y ^ n + (x * y ^ n + y * x ^ n) = (x + y) * (x ^ n + y ^ n)\n[PROOFSTEP]\nrw [add_mul, mul_add, mul_add, add_comm (y * x ^ n), \u2190 add_assoc, \u2190 add_assoc, add_assoc (x * x ^ n) (x * y ^ n),\n add_comm (x * y ^ n) (y * y ^ n), \u2190 add_assoc]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nm : \u2115\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nk\u271d : \u2115\nhn\u271d : Nat.succ k\u271d \u2260 0\nk : \u2115\nih : Nat.succ k \u2260 0 \u2192 x ^ Nat.succ k + y ^ Nat.succ k \u2264 (x + y) ^ Nat.succ k\nhn : Nat.succ (Nat.succ k) \u2260 0\nn : \u2115 := Nat.succ k\nh1 : 0 \u2264 x * y ^ n + y * x ^ n\nh2 : 0 \u2264 x + y\n\u22a2 (x + y) * (x ^ n + y ^ n) \u2264 (x + y) ^ Nat.succ n\n[PROOFSTEP]\nrw [pow_succ _ n]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nm : \u2115\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nk\u271d : \u2115\nhn\u271d : Nat.succ k\u271d \u2260 0\nk : \u2115\nih : Nat.succ k \u2260 0 \u2192 x ^ Nat.succ k + y ^ Nat.succ k \u2264 (x + y) ^ Nat.succ k\nhn : Nat.succ (Nat.succ k) \u2260 0\nn : \u2115 := Nat.succ k\nh1 : 0 \u2264 x * y ^ n + y * x ^ n\nh2 : 0 \u2264 x + y\n\u22a2 (x + y) * (x ^ n + y ^ n) \u2264 (x + y) * (x + y) ^ n\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_left (ih (Nat.succ_ne_zero k)) h2\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nn\u271d m : \u2115\nh\u2080 : 0 \u2264 a\nh\u2081 : a < 1\nn : \u2115\nx\u271d : n + 1 \u2260 0\n\u22a2 a ^ (n + 1) < 1\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nn\u271d m : \u2115\nh\u2080 : 0 \u2264 a\nh\u2081 : a < 1\nn : \u2115\nx\u271d : n + 1 \u2260 0\n\u22a2 a * a ^ n < 1\n[PROOFSTEP]\nexact mul_lt_one_of_nonneg_of_lt_one_left h\u2080 h\u2081 (pow_le_one _ h\u2080 h\u2081.le)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nn m : \u2115\nH : 1 \u2264 a\n\u22a2 1 \u2264 a ^ 0\n[PROOFSTEP]\nrw [pow_zero]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nn\u271d m : \u2115\nH : 1 \u2264 a\nn : \u2115\n\u22a2 1 \u2264 a ^ (n + 1)\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nn\u271d m : \u2115\nH : 1 \u2264 a\nn : \u2115\n\u22a2 1 \u2264 a * a ^ n\n[PROOFSTEP]\nsimpa only [mul_one] using mul_le_mul H (one_le_pow_of_one_le H n) zero_le_one (le_trans zero_le_one H)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nn\u271d m : \u2115\nh : 1 \u2264 a\nn : \u2115\n\u22a2 a ^ n \u2264 a ^ (n + 1)\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nn\u271d m : \u2115\nh : 1 \u2264 a\nn : \u2115\n\u22a2 a ^ n \u2264 a * a ^ n\n[PROOFSTEP]\nexact le_mul_of_one_le_left (pow_nonneg (zero_le_one.trans h) _) h\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na\u271d x y : R\nn m : \u2115\na b : R\nha : 0 \u2264 a\nhab : a \u2264 b\n\u22a2 \u2200 (i : \u2115), a ^ i \u2264 b ^ i\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na\u271d x y : R\nn m : \u2115\na b : R\nha : 0 \u2264 a\nhab : a \u2264 b\ni : \u2115\n\u22a2 a ^ i \u2264 b ^ i\n[PROOFSTEP]\ninduction i with\n| zero => simp\n| succ k ih =>\n rw [pow_succ, pow_succ]\n apply mul_le_mul hab\n apply ih\n apply pow_nonneg ha\n apply le_trans ha hab\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na\u271d x y : R\nn m : \u2115\na b : R\nha : 0 \u2264 a\nhab : a \u2264 b\ni : \u2115\n\u22a2 a ^ i \u2264 b ^ i\n[PROOFSTEP]\ninduction i with\n| zero => simp\n| succ k ih =>\n rw [pow_succ, pow_succ]\n apply mul_le_mul hab\n apply ih\n apply pow_nonneg ha\n apply le_trans ha hab\n[GOAL]\ncase zero\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na\u271d x y : R\nn m : \u2115\na b : R\nha : 0 \u2264 a\nhab : a \u2264 b\n\u22a2 a ^ Nat.zero \u2264 b ^ Nat.zero\n[PROOFSTEP]\n\n| zero => simp\n[GOAL]\ncase zero\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na\u271d x y : R\nn m : \u2115\na b : R\nha : 0 \u2264 a\nhab : a \u2264 b\n\u22a2 a ^ Nat.zero \u2264 b ^ Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na\u271d x y : R\nn m : \u2115\na b : R\nha : 0 \u2264 a\nhab : a \u2264 b\nk : \u2115\nih : a ^ k \u2264 b ^ k\n\u22a2 a ^ Nat.succ k \u2264 b ^ Nat.succ k\n[PROOFSTEP]\n\n| succ k ih =>\n rw [pow_succ, pow_succ]\n apply mul_le_mul hab\n apply ih\n apply pow_nonneg ha\n apply le_trans ha hab\n[GOAL]\ncase succ\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na\u271d x y : R\nn m : \u2115\na b : R\nha : 0 \u2264 a\nhab : a \u2264 b\nk : \u2115\nih : a ^ k \u2264 b ^ k\n\u22a2 a ^ Nat.succ k \u2264 b ^ Nat.succ k\n[PROOFSTEP]\nrw [pow_succ, pow_succ]\n[GOAL]\ncase succ\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na\u271d x y : R\nn m : \u2115\na b : R\nha : 0 \u2264 a\nhab : a \u2264 b\nk : \u2115\nih : a ^ k \u2264 b ^ k\n\u22a2 a * a ^ k \u2264 b * b ^ k\n[PROOFSTEP]\napply mul_le_mul hab\n[GOAL]\ncase succ.h\u2082\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na\u271d x y : R\nn m : \u2115\na b : R\nha : 0 \u2264 a\nhab : a \u2264 b\nk : \u2115\nih : a ^ k \u2264 b ^ k\n\u22a2 a ^ k \u2264 b ^ k\ncase succ.c0\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na\u271d x y : R\nn m : \u2115\na b : R\nha : 0 \u2264 a\nhab : a \u2264 b\nk : \u2115\nih : a ^ k \u2264 b ^ k\n\u22a2 0 \u2264 a ^ k\ncase succ.b0\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na\u271d x y : R\nn m : \u2115\na b : R\nha : 0 \u2264 a\nhab : a \u2264 b\nk : \u2115\nih : a ^ k \u2264 b ^ k\n\u22a2 0 \u2264 b\n[PROOFSTEP]\napply ih\n[GOAL]\ncase succ.c0\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na\u271d x y : R\nn m : \u2115\na b : R\nha : 0 \u2264 a\nhab : a \u2264 b\nk : \u2115\nih : a ^ k \u2264 b ^ k\n\u22a2 0 \u2264 a ^ k\ncase succ.b0\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na\u271d x y : R\nn m : \u2115\na b : R\nha : 0 \u2264 a\nhab : a \u2264 b\nk : \u2115\nih : a ^ k \u2264 b ^ k\n\u22a2 0 \u2264 b\n[PROOFSTEP]\napply pow_nonneg ha\n[GOAL]\ncase succ.b0\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na\u271d x y : R\nn m : \u2115\na b : R\nha : 0 \u2264 a\nhab : a \u2264 b\nk : \u2115\nih : a ^ k \u2264 b ^ k\n\u22a2 0 \u2264 b\n[PROOFSTEP]\napply le_trans ha hab\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nn\u271d m : \u2115\nha : 1 < a\nn : \u2115\nx\u271d : n + 1 \u2260 0\n\u22a2 1 < a ^ (n + 1)\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : OrderedSemiring R\na x y : R\nn\u271d m : \u2115\nha : 1 < a\nn : \u2115\nx\u271d : n + 1 \u2260 0\n\u22a2 1 < a * a ^ n\n[PROOFSTEP]\nexact one_lt_mul_of_lt_of_le ha (one_le_pow_of_one_le ha.le _)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : StrictOrderedSemiring R\na x y : R\nn m : \u2115\nh : x < y\nhx : 0 \u2264 x\nhn : 0 < 0\n\u22a2 x ^ 0 < y ^ 0\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : StrictOrderedSemiring R\na x y : R\nn\u271d m : \u2115\nh : x < y\nhx : 0 \u2264 x\nn : \u2115\nx\u271d : 0 < n + 1\n\u22a2 x ^ (n + 1) < y ^ (n + 1)\n[PROOFSTEP]\nsimpa only [pow_succ'] using mul_lt_mul_of_le_of_le' (pow_le_pow_of_le_left hx h.le _) h (pow_pos (hx.trans_lt h) _) hx\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : StrictOrderedSemiring R\na x y : R\nn\u271d m : \u2115\nh : 1 < a\nthis : 0 < a\nn : \u2115\n\u22a2 a ^ n < a ^ (n + 1)\n[PROOFSTEP]\nsimpa only [one_mul, pow_succ] using mul_lt_mul h (le_refl (a ^ n)) (pow_pos this _) this.le\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : StrictOrderedSemiring R\na x y : R\nn\u271d m : \u2115\nh\u2080 : 0 < a\nh\u2081 : a < 1\nn : \u2115\n\u22a2 a ^ (n + 1) < a ^ n\n[PROOFSTEP]\nsimpa only [pow_succ, one_mul] using mul_lt_mul h\u2081 le_rfl (pow_pos h\u2080 n) zero_le_one\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : StrictOrderedSemiring R\na x y : R\nn m : \u2115\nha : 0 < a\n\u22a2 0 < a ^ 2\n[PROOFSTEP]\nrw [sq]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : StrictOrderedSemiring R\na x y : R\nn m : \u2115\nha : 0 < a\n\u22a2 0 < a * a\n[PROOFSTEP]\nexact mul_pos ha ha\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : StrictOrderedRing R\na : R\nha : a < 0\nn : \u2115\n\u22a2 0 < a ^ bit0 n\n[PROOFSTEP]\nrw [pow_bit0']\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : StrictOrderedRing R\na : R\nha : a < 0\nn : \u2115\n\u22a2 0 < (a * a) ^ n\n[PROOFSTEP]\nexact pow_pos (mul_pos_of_neg_of_neg ha ha) _\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : StrictOrderedRing R\na : R\nha : a < 0\nn : \u2115\n\u22a2 a ^ bit1 n < 0\n[PROOFSTEP]\nrw [bit1, pow_succ]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : StrictOrderedRing R\na : R\nha : a < 0\nn : \u2115\n\u22a2 a * a ^ bit0 n < 0\n[PROOFSTEP]\nexact mul_neg_of_neg_of_pos ha (pow_bit0_pos_of_neg ha n)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedSemiring R\na\u271d b a : R\nha : 0 \u2264 a\nn : \u2115\nhn : n \u2260 0\n\u22a2 a ^ n \u2264 1 \u2194 a \u2264 1\n[PROOFSTEP]\nrefine' \u27e8_, pow_le_one n ha\u27e9\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedSemiring R\na\u271d b a : R\nha : 0 \u2264 a\nn : \u2115\nhn : n \u2260 0\n\u22a2 a ^ n \u2264 1 \u2192 a \u2264 1\n[PROOFSTEP]\nrw [\u2190 not_lt, \u2190 not_lt]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedSemiring R\na\u271d b a : R\nha : 0 \u2264 a\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u00ac1 < a ^ n \u2192 \u00ac1 < a\n[PROOFSTEP]\nexact mt fun h => one_lt_pow h hn\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedSemiring R\na\u271d b a : R\nha : 0 \u2264 a\nn : \u2115\nhn : n \u2260 0\n\u22a2 1 \u2264 a ^ n \u2194 1 \u2264 a\n[PROOFSTEP]\nrefine' \u27e8_, fun h => one_le_pow_of_one_le h n\u27e9\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedSemiring R\na\u271d b a : R\nha : 0 \u2264 a\nn : \u2115\nhn : n \u2260 0\n\u22a2 1 \u2264 a ^ n \u2192 1 \u2264 a\n[PROOFSTEP]\nrw [\u2190 not_lt, \u2190 not_lt]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedSemiring R\na\u271d b a : R\nha : 0 \u2264 a\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u00aca ^ n < 1 \u2192 \u00aca < 1\n[PROOFSTEP]\nexact mt fun h => pow_lt_one ha h hn\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedSemiring R\na\u271d b\u271d a b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\n\u22a2 0 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedSemiring R\na b : R\nhb : 0 \u2264 b\n\u22a2 a * a < b * b \u2192 a < b\n[PROOFSTEP]\nsimp_rw [\u2190 sq]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedSemiring R\na b : R\nhb : 0 \u2264 b\n\u22a2 a ^ 2 < b ^ 2 \u2192 a < b\n[PROOFSTEP]\nexact lt_of_pow_lt_pow _ hb\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\nn : \u2115\n\u22a2 |(-1) ^ n| = 1\n[PROOFSTEP]\nrw [\u2190 pow_abs, abs_neg, abs_one, one_pow]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nh : 0 < n\n\u22a2 |a ^ n| = 1 \u2194 |a| = 1\n[PROOFSTEP]\nconvert pow_left_inj (abs_nonneg a) zero_le_one h\n[GOAL]\ncase h.e'_1.h.e'_2\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nh : 0 < n\n\u22a2 |a ^ n| = |a| ^ n\ncase h.e'_1.h.e'_3\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nh : 0 < n\n\u22a2 1 = 1 ^ n\n[PROOFSTEP]\nexacts [(pow_abs _ _).symm, (one_pow _).symm]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\n\u22a2 0 \u2264 a ^ bit0 n\n[PROOFSTEP]\nrw [pow_bit0]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\n\u22a2 0 \u2264 a ^ n * a ^ n\n[PROOFSTEP]\nexact mul_self_nonneg _\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhn : n \u2260 0\n\u22a2 0 < a ^ bit0 n \u2194 a \u2260 0\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => pow_bit0_pos h n\u27e9\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhn : n \u2260 0\nh : 0 < a ^ bit0 n\n\u22a2 a \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\nn : \u2115\nhn : n \u2260 0\nh : 0 < 0 ^ bit0 n\n\u22a2 False\n[PROOFSTEP]\nrw [zero_pow (Nat.zero_lt_bit0 hn)] at h \n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\nn : \u2115\nhn : n \u2260 0\nh : 0 < 0\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ h\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\nx\u271d y x : R\n\u22a2 |x| ^ 2 = x ^ 2\n[PROOFSTEP]\nsimpa only [sq] using abs_mul_abs_self x\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\nx\u271d y x : R\n\u22a2 |x ^ 2| = x ^ 2\n[PROOFSTEP]\nsimpa only [sq] using abs_mul_self x\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\nx y : R\n\u22a2 x ^ 2 < y ^ 2 \u2194 |x| < |y|\n[PROOFSTEP]\nsimpa only [sq_abs] using (@strictMonoOn_pow R _ _ two_pos).lt_iff_lt (abs_nonneg x) (abs_nonneg y)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\nx y : R\n\u22a2 x ^ 2 \u2264 y ^ 2 \u2194 |x| \u2264 |y|\n[PROOFSTEP]\nsimpa only [sq_abs] using (@strictMonoOn_pow R _ _ two_pos).le_iff_le (abs_nonneg x) (abs_nonneg y)\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\nx y : R\nh : x ^ 2 < y ^ 2\nhy : 0 \u2264 y\n\u22a2 |x| < y\n[PROOFSTEP]\nrwa [\u2190 abs_of_nonneg hy, \u2190 sq_lt_sq]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\nx y : R\nh : x ^ 2 \u2264 y ^ 2\nhy : 0 \u2264 y\n\u22a2 |x| \u2264 y\n[PROOFSTEP]\nrwa [\u2190 abs_of_nonneg hy, \u2190 sq_le_sq]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\nx\u271d y\u271d x y : R\n\u22a2 x ^ 2 = y ^ 2 \u2194 |x| = |y|\n[PROOFSTEP]\nsimp only [le_antisymm_iff, sq_le_sq]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\nx\u271d y x : R\n\u22a2 x ^ 2 \u2264 1 \u2194 |x| \u2264 1\n[PROOFSTEP]\nsimpa only [one_pow, abs_one] using @sq_le_sq _ _ x 1\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\nx\u271d y x : R\n\u22a2 x ^ 2 < 1 \u2194 |x| < 1\n[PROOFSTEP]\nsimpa only [one_pow, abs_one] using @sq_lt_sq _ _ x 1\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\nx\u271d y x : R\n\u22a2 1 \u2264 x ^ 2 \u2194 1 \u2264 |x|\n[PROOFSTEP]\nsimpa only [one_pow, abs_one] using @sq_le_sq _ _ 1 x\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\nx\u271d y x : R\n\u22a2 1 < x ^ 2 \u2194 1 < |x|\n[PROOFSTEP]\nsimpa only [one_pow, abs_one] using @sq_lt_sq _ _ 1 x\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b9 : LinearOrderedCommMonoidWithZero M\ninst\u271d : NoZeroDivisors M\na : M\nn : \u2115\nhn : 0 < n\n\u22a2 0 < a ^ n \u2194 0 < a\n[PROOFSTEP]\nsimp_rw [zero_lt_iff, pow_ne_zero_iff hn]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedCommGroupWithZero M\na : M\nm n : \u2115\nha : 1 < a\n\u22a2 a ^ n < a ^ Nat.succ n\n[PROOFSTEP]\nrw [\u2190 one_mul (a ^ n), pow_succ]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedCommGroupWithZero M\na : M\nm n : \u2115\nha : 1 < a\n\u22a2 1 * a ^ n < a * a ^ n\n[PROOFSTEP]\nexact mul_lt_right\u2080 _ ha (pow_ne_zero _ (zero_lt_one.trans ha).ne')\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedCommGroupWithZero M\na : M\nm n : \u2115\nha : 1 < a\nhmn : m < n\n\u22a2 a ^ m < a ^ n\n[PROOFSTEP]\ninduction' hmn with n _ ih\n[GOAL]\ncase refl\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedCommGroupWithZero M\na : M\nm n : \u2115\nha : 1 < a\n\u22a2 a ^ m < a ^ Nat.succ m\ncase step\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedCommGroupWithZero M\na : M\nm n\u271d : \u2115\nha : 1 < a\nn : \u2115\na\u271d : Nat.le (Nat.succ m) n\nih : a ^ m < a ^ n\n\u22a2 a ^ m < a ^ Nat.succ n\n[PROOFSTEP]\nexacts [pow_lt_pow_succ ha, lt_trans ih (pow_lt_pow_succ ha)]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : LinearOrder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nf : R \u2192* M\n\u22a2 \u2191f (-1) ^ Nat.succ 1 = 1\n[PROOFSTEP]\nrw [\u2190 map_pow, neg_one_sq, map_one]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : LinearOrder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nf : R \u2192* M\nx : R\n\u22a2 \u2191f (-x) = \u2191f x\n[PROOFSTEP]\nrw [\u2190 neg_one_mul, map_mul, map_neg_one, one_mul]\n[GOAL]\n\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : LinearOrder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nf : R \u2192* M\nx y : R\n\u22a2 \u2191f (x - y) = \u2191f (y - x)\n[PROOFSTEP]\nrw [\u2190 map_neg, neg_sub]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.GroupPower.Order", "llama_tokens": 20699, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743735019595, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4302656509324074}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_int : IntegrableOn f s\ng_int : IntegrableOn g s\nhs : MeasurableSet s\nhfg : f \u2264\u1da0[ae (Measure.restrict \u03bc s)] g\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = ENNReal.ofReal (\u222b (y : \u03b1) in s, (g - f) y \u2202\u03bc)\n[PROOFSTEP]\nhave h : g - f =\u1d50[\u03bc.restrict s] fun x => Real.toNNReal (g x - f x) :=\n hfg.mono fun x hx => (Real.coe_toNNReal _ <| sub_nonneg.2 hx).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_int : IntegrableOn f s\ng_int : IntegrableOn g s\nhs : MeasurableSet s\nhfg : f \u2264\u1da0[ae (Measure.restrict \u03bc s)] g\nh : g - f =\u1da0[ae (Measure.restrict \u03bc s)] fun x => \u2191(Real.toNNReal (g x - f x))\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = ENNReal.ofReal (\u222b (y : \u03b1) in s, (g - f) y \u2202\u03bc)\n[PROOFSTEP]\nrw [volume_regionBetween_eq_lintegral f_int.aemeasurable g_int.aemeasurable hs, integral_congr_ae h, lintegral_congr_ae,\n lintegral_coe_eq_integral _ ((integrable_congr h).mp (g_int.sub f_int))]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_int : IntegrableOn f s\ng_int : IntegrableOn g s\nhs : MeasurableSet s\nhfg : f \u2264\u1da0[ae (Measure.restrict \u03bc s)] g\nh : g - f =\u1da0[ae (Measure.restrict \u03bc s)] fun x => \u2191(Real.toNNReal (g x - f x))\n\u22a2 (fun y => ENNReal.ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun a => \u2191(Real.toNNReal (g a - f a))\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_int : IntegrableOn f s\ng_int : IntegrableOn g s\nhs : MeasurableSet s\nhfg : f \u2264\u1da0[ae (Measure.restrict \u03bc s)] g\nh : g - f =\u1da0[ae (Measure.restrict \u03bc s)] fun x => \u2191(Real.toNNReal (g x - f x))\n\u22a2 (fun y => ENNReal.ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun a => \u2191(Real.toNNReal (g a - f a))\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n\u22a2 Integrable \u2191f\n[PROOFSTEP]\nrefine'\n @integrable_of_summable_norm_restrict \u211d \u2124 E _ volume _ _ _ _ _ _ _ _\n (summable_of_nonneg_of_le\n (fun n : \u2124 =>\n mul_nonneg (norm_nonneg (f.restrict (\u27e8Icc (n : \u211d) ((n : \u211d) + 1), isCompact_Icc\u27e9 : Compacts \u211d)))\n ENNReal.toReal_nonneg)\n (fun n => _) hf)\n _\n -- porting note: `refine` was able to find that on its own before\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n\u22a2 \u2124 \u2192 Compacts \u211d\n[PROOFSTEP]\nintro n\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\nn : \u2124\n\u22a2 Compacts \u211d\n[PROOFSTEP]\nexact \u27e8Icc (n : \u211d) ((n : \u211d) + 1), isCompact_Icc\u27e9\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\nn : \u2124\n\u22a2 \u2016ContinuousMap.restrict (\u2191{ carrier := Icc (\u2191n) (\u2191n + 1), isCompact' := (_ : IsCompact (Icc (\u2191n) (\u2191n + 1))) }) f\u2016 *\n ENNReal.toReal (\u2191\u2191volume \u2191{ carrier := Icc (\u2191n) (\u2191n + 1), isCompact' := (_ : IsCompact (Icc (\u2191n) (\u2191n + 1))) }) \u2264\n \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n[PROOFSTEP]\nsimp only [Compacts.coe_mk, Real.volume_Icc, add_sub_cancel', ENNReal.toReal_ofReal zero_le_one, mul_one,\n norm_le _ (norm_nonneg _)]\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\nn : \u2124\n\u22a2 \u2200 (x : \u2191(Icc (\u2191n) (\u2191n + 1))),\n \u2016\u2191(ContinuousMap.restrict (Icc (\u2191n) (\u2191n + 1)) f) x\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n[PROOFSTEP]\nintro x\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\nn : \u2124\nx : \u2191(Icc (\u2191n) (\u2191n + 1))\n\u22a2 \u2016\u2191(ContinuousMap.restrict (Icc (\u2191n) (\u2191n + 1)) f) x\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n[PROOFSTEP]\nhave :=\n ((f.comp <| ContinuousMap.addRight n).restrict (Icc 0 1)).norm_coe_le_norm\n \u27e8x - n, \u27e8sub_nonneg.mpr x.2.1, sub_le_iff_le_add'.mpr x.2.2\u27e9\u27e9\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\nn : \u2124\nx : \u2191(Icc (\u2191n) (\u2191n + 1))\nthis :\n \u2016\u2191(ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n)))\n { val := \u2191x - \u2191n, property := (_ : 0 \u2264 \u2191x - \u2191n \u2227 \u2191x - \u2191n \u2264 1) }\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n\u22a2 \u2016\u2191(ContinuousMap.restrict (Icc (\u2191n) (\u2191n + 1)) f) x\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n[PROOFSTEP]\nsimpa only [ContinuousMap.restrict_apply, comp_apply, coe_addRight, Subtype.coe_mk, sub_add_cancel] using this\n[GOAL]\ncase refine'_3\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nhf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016\n\u22a2 \u22c3 (i : \u2124), \u2191{ carrier := Icc (\u2191i) (\u2191i + 1), isCompact' := (_ : IsCompact (Icc (\u2191i) (\u2191i + 1))) } = univ\n[PROOFSTEP]\nexact iUnion_Icc_int_cast \u211d\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x\n[PROOFSTEP]\nhave A : MeasurableEmbedding fun x : \u211d => -x := (Homeomorph.neg \u211d).closedEmbedding.measurableEmbedding\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x\n[PROOFSTEP]\nhave := MeasurableEmbedding.set_integral_map (\u03bc := volume) A f (Ici (-c))\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\nthis : \u222b (y : \u211d) in Ici (-c), f y \u2202Measure.map (fun x => -x) volume = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x)\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x\n[PROOFSTEP]\nrw [Measure.map_neg_eq_self (volume : Measure \u211d)] at this \n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\nthis : \u222b (y : \u211d) in Ici (-c), f y = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x)\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x\n[PROOFSTEP]\nsimp_rw [\u2190 integral_Ici_eq_integral_Ioi, this, neg_preimage, preimage_neg_Ici, neg_neg]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\n\u22a2 \u222b (x : \u211d) in Ioi c, f (-x) = \u222b (x : \u211d) in Iic (-c), f x\n[PROOFSTEP]\nrw [\u2190 neg_neg c, \u2190 integral_comp_neg_Iic]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\n\u22a2 \u222b (x : \u211d) in Iic (-c), f (- -x) = \u222b (x : \u211d) in Iic (- - -c), f x\n[PROOFSTEP]\nsimp only [neg_neg]\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "llama_tokens": 3558, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.43026564426896313}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nC : \u211d\n\u22a2 Tendsto (fun n => C / \u2191n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa only [mul_zero] using tendsto_const_nhds.mul tendsto_inverse_atTop_nhds_0_nat\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 NNReal.tendsto_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u22a2 Tendsto (fun a => \u2191(\u2191a)\u207b\u00b9) atTop (\ud835\udcdd \u21910)\n[PROOFSTEP]\nexact _root_.tendsto_inverse_atTop_nhds_0_nat\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nC : \u211d\u22650\n\u22a2 Tendsto (fun n => C / \u2191n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa using tendsto_const_nhds.mul NNReal.tendsto_inverse_atTop_nhds_0_nat\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nthis : Tendsto (fun n => 1 / \u2191(n + 1)) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => 1 / (\u2191n + 1)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : Semiring \ud835\udd5c\ninst\u271d\u00b3 : Algebra \u211d\u22650 \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSemiring \ud835\udd5c\ninst\u271d : ContinuousSMul \u211d\u22650 \ud835\udd5c\n\u22a2 Tendsto (\u2191(algebraMap \u211d\u22650 \ud835\udd5c) \u2218 fun n => (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert (continuous_algebraMap \u211d\u22650 \ud835\udd5c).continuousAt.tendsto.comp tendsto_inverse_atTop_nhds_0_nat\n[GOAL]\ncase h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2074 : Semiring \ud835\udd5c\ninst\u271d\u00b3 : Algebra \u211d\u22650 \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSemiring \ud835\udd5c\ninst\u271d : ContinuousSMul \u211d\u22650 \ud835\udd5c\n\u22a2 0 = \u2191(algebraMap \u211d\u22650 \ud835\udd5c) 0\n[PROOFSTEP]\nrw [map_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b2 : Algebra \u211d \ud835\udd5c\ninst\u271d\u00b9 : ContinuousSMul \u211d \ud835\udd5c\ninst\u271d : TopologicalDivisionRing \ud835\udd5c\nx : \ud835\udd5c\n\u22a2 Tendsto (fun n => \u2191n / (\u2191n + x)) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nrefine' Tendsto.congr' ((eventually_ne_atTop 0).mp (eventually_of_forall fun n hn => _)) _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b2 : Algebra \u211d \ud835\udd5c\ninst\u271d\u00b9 : ContinuousSMul \u211d \ud835\udd5c\ninst\u271d : TopologicalDivisionRing \ud835\udd5c\nx : \ud835\udd5c\n\u22a2 \u2115 \u2192 \ud835\udd5c\n[PROOFSTEP]\nexact fun n : \u2115 => 1 / (1 + x / n)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b2 : Algebra \u211d \ud835\udd5c\ninst\u271d\u00b9 : ContinuousSMul \u211d \ud835\udd5c\ninst\u271d : TopologicalDivisionRing \ud835\udd5c\nx : \ud835\udd5c\nn : \u2115\nhn : n \u2260 0\n\u22a2 1 / (1 + x / \u2191n) = (fun n => \u2191n / (\u2191n + x)) n\n[PROOFSTEP]\nfield_simp [Nat.cast_ne_zero.mpr hn]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b2 : Algebra \u211d \ud835\udd5c\ninst\u271d\u00b9 : ContinuousSMul \u211d \ud835\udd5c\ninst\u271d : TopologicalDivisionRing \ud835\udd5c\nx : \ud835\udd5c\n\u22a2 Tendsto (fun n => 1 / (1 + x / \u2191n)) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nhave : \ud835\udcdd (1 : \ud835\udd5c) = \ud835\udcdd (1 / (1 + x * (0 : \ud835\udd5c))) := by rw [mul_zero, add_zero, div_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b2 : Algebra \u211d \ud835\udd5c\ninst\u271d\u00b9 : ContinuousSMul \u211d \ud835\udd5c\ninst\u271d : TopologicalDivisionRing \ud835\udd5c\nx : \ud835\udd5c\n\u22a2 \ud835\udcdd 1 = \ud835\udcdd (1 / (1 + x * 0))\n[PROOFSTEP]\nrw [mul_zero, add_zero, div_one]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b2 : Algebra \u211d \ud835\udd5c\ninst\u271d\u00b9 : ContinuousSMul \u211d \ud835\udd5c\ninst\u271d : TopologicalDivisionRing \ud835\udd5c\nx : \ud835\udd5c\nthis : \ud835\udcdd 1 = \ud835\udcdd (1 / (1 + x * 0))\n\u22a2 Tendsto (fun n => 1 / (1 + x / \u2191n)) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b2 : Algebra \u211d \ud835\udd5c\ninst\u271d\u00b9 : ContinuousSMul \u211d \ud835\udd5c\ninst\u271d : TopologicalDivisionRing \ud835\udd5c\nx : \ud835\udd5c\nthis : \ud835\udcdd 1 = \ud835\udcdd (1 / (1 + x * 0))\n\u22a2 Tendsto (fun n => 1 / (1 + x / \u2191n)) atTop (\ud835\udcdd (1 / (1 + x * 0)))\n[PROOFSTEP]\nrefine' tendsto_const_nhds.div (tendsto_const_nhds.add _) (by simp)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b2 : Algebra \u211d \ud835\udd5c\ninst\u271d\u00b9 : ContinuousSMul \u211d \ud835\udd5c\ninst\u271d : TopologicalDivisionRing \ud835\udd5c\nx : \ud835\udd5c\nthis : \ud835\udcdd 1 = \ud835\udcdd (1 / (1 + x * 0))\n\u22a2 1 + x * 0 \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b2 : Algebra \u211d \ud835\udd5c\ninst\u271d\u00b9 : ContinuousSMul \u211d \ud835\udd5c\ninst\u271d : TopologicalDivisionRing \ud835\udd5c\nx : \ud835\udd5c\nthis : \ud835\udcdd 1 = \ud835\udcdd (1 / (1 + x * 0))\n\u22a2 Tendsto (fun n => x / \u2191n) atTop (\ud835\udcdd (x * 0))\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b2 : Algebra \u211d \ud835\udd5c\ninst\u271d\u00b9 : ContinuousSMul \u211d \ud835\udd5c\ninst\u271d : TopologicalDivisionRing \ud835\udd5c\nx : \ud835\udd5c\nthis : \ud835\udcdd 1 = \ud835\udcdd (1 / (1 + x * 0))\n\u22a2 Tendsto (fun n => x * (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd (x * 0))\n[PROOFSTEP]\nrefine' tendsto_const_nhds.mul _\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b2 : Algebra \u211d \ud835\udd5c\ninst\u271d\u00b9 : ContinuousSMul \u211d \ud835\udd5c\ninst\u271d : TopologicalDivisionRing \ud835\udd5c\nx : \ud835\udd5c\nthis : \ud835\udcdd 1 = \ud835\udcdd (1 / (1 + x * 0))\n\u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave := ((continuous_algebraMap \u211d \ud835\udd5c).tendsto _).comp tendsto_inverse_atTop_nhds_0_nat\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b2 : Algebra \u211d \ud835\udd5c\ninst\u271d\u00b9 : ContinuousSMul \u211d \ud835\udd5c\ninst\u271d : TopologicalDivisionRing \ud835\udd5c\nx : \ud835\udd5c\nthis\u271d : \ud835\udcdd 1 = \ud835\udcdd (1 / (1 + x * 0))\nthis : Tendsto (\u2191(algebraMap \u211d \ud835\udd5c) \u2218 fun n => (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd (\u2191(algebraMap \u211d \ud835\udd5c) 0))\n\u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [map_zero, Filter.tendsto_atTop'] at this \n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b2 : Algebra \u211d \ud835\udd5c\ninst\u271d\u00b9 : ContinuousSMul \u211d \ud835\udd5c\ninst\u271d : TopologicalDivisionRing \ud835\udd5c\nx : \ud835\udd5c\nthis\u271d : \ud835\udcdd 1 = \ud835\udcdd (1 / (1 + x * 0))\nthis : \u2200 (s : Set \ud835\udd5c), s \u2208 \ud835\udcdd 0 \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 (\u2191(algebraMap \u211d \ud835\udd5c) \u2218 fun n => (\u2191n)\u207b\u00b9) b \u2208 s\n\u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' Iff.mpr tendsto_atTop' _\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b2 : Algebra \u211d \ud835\udd5c\ninst\u271d\u00b9 : ContinuousSMul \u211d \ud835\udd5c\ninst\u271d : TopologicalDivisionRing \ud835\udd5c\nx : \ud835\udd5c\nthis\u271d : \ud835\udcdd 1 = \ud835\udcdd (1 / (1 + x * 0))\nthis : \u2200 (s : Set \ud835\udd5c), s \u2208 \ud835\udcdd 0 \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 (\u2191(algebraMap \u211d \ud835\udd5c) \u2218 fun n => (\u2191n)\u207b\u00b9) b \u2208 s\n\u22a2 \u2200 (s : Set \ud835\udd5c), s \u2208 \ud835\udcdd 0 \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 (\u2191b)\u207b\u00b9 \u2208 s\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b2 : Algebra \u211d \ud835\udd5c\ninst\u271d\u00b9 : ContinuousSMul \u211d \ud835\udd5c\ninst\u271d : TopologicalDivisionRing \ud835\udd5c\nx : \ud835\udd5c\nthis\u271d : \ud835\udcdd 1 = \ud835\udcdd (1 / (1 + x * 0))\nthis : \u2200 (s : Set \ud835\udd5c), s \u2208 \ud835\udcdd 0 \u2192 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 (\u2191(algebraMap \u211d \ud835\udd5c) \u2218 fun n => (\u2191n)\u207b\u00b9) b \u2208 s\ns\u271d : Set \ud835\udd5c\na\u271d : s\u271d \u2208 \ud835\udcdd 0\n\u22a2 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 (\u2191b)\u207b\u00b9 \u2208 s\u271d\n[PROOFSTEP]\nsimp_all only [comp_apply, map_inv\u2080, map_natCast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nh\u2081 : 0 \u2264 r\nh\u2082 : r < 1\nhr : 0 = r\n\u22a2 Tendsto (fun n => r ^ (n + 1)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp [_root_.pow_succ, \u2190 hr, tendsto_const_nhds]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nh\u2081 : 0 \u2264 r\nh\u2082 : r < 1\nhr : 0 < r\nthis : Tendsto (fun n => r\u207b\u00b9 ^ n) atTop atTop\nn : \u2115\n\u22a2 ((fun r => r\u207b\u00b9) \u2218 fun n => r\u207b\u00b9 ^ n) n = r ^ n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\n\u22a2 Tendsto (fun n => r ^ n) atTop (\ud835\udcdd 0) \u2194 |r| < 1\n[PROOFSTEP]\nrw [tendsto_zero_iff_abs_tendsto_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\n\u22a2 Tendsto (abs \u2218 fun n => r ^ n) atTop (\ud835\udcdd 0) \u2194 |r| < 1\n[PROOFSTEP]\nrefine \u27e8fun h \u21a6 by_contra (fun hr_le \u21a6 ?_), fun h \u21a6 ?_\u27e9\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nh : Tendsto (abs \u2218 fun n => r ^ n) atTop (\ud835\udcdd 0)\nhr_le : \u00ac|r| < 1\n\u22a2 False\n[PROOFSTEP]\nby_cases hr : 1 = |r|\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nh : Tendsto (abs \u2218 fun n => r ^ n) atTop (\ud835\udcdd 0)\nhr_le : \u00ac|r| < 1\nhr : 1 = |r|\n\u22a2 False\n[PROOFSTEP]\nreplace h : Tendsto (fun n : \u2115 \u21a6 |r| ^ n) atTop (\ud835\udcdd 0) := by simpa only [\u2190 abs_pow, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nh : Tendsto (abs \u2218 fun n => r ^ n) atTop (\ud835\udcdd 0)\nhr_le : \u00ac|r| < 1\nhr : 1 = |r|\n\u22a2 Tendsto (fun n => |r| ^ n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa only [\u2190 abs_pow, h]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nhr_le : \u00ac|r| < 1\nhr : 1 = |r|\nh : Tendsto (fun n => |r| ^ n) atTop (\ud835\udcdd 0)\n\u22a2 False\n[PROOFSTEP]\nsimp only [hr.symm, one_pow] at h \n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nhr_le : \u00ac|r| < 1\nhr : 1 = |r|\nh : Tendsto (fun n => 1) atTop (\ud835\udcdd 0)\n\u22a2 False\n[PROOFSTEP]\nexact zero_ne_one <| tendsto_nhds_unique h tendsto_const_nhds\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nh : Tendsto (abs \u2218 fun n => r ^ n) atTop (\ud835\udcdd 0)\nhr_le : \u00ac|r| < 1\nhr : \u00ac1 = |r|\n\u22a2 False\n[PROOFSTEP]\napply @not_tendsto_nhds_of_tendsto_atTop \ud835\udd5c \u2115 _ _ _ _ atTop _ (fun n \u21a6 |r| ^ n) _ 0 _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nh : Tendsto (abs \u2218 fun n => r ^ n) atTop (\ud835\udcdd 0)\nhr_le : \u00ac|r| < 1\nhr : \u00ac1 = |r|\n\u22a2 Tendsto (fun n => |r| ^ n) atTop atTop\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nh : Tendsto (abs \u2218 fun n => r ^ n) atTop (\ud835\udcdd 0)\nhr_le : \u00ac|r| < 1\nhr : \u00ac1 = |r|\n\u22a2 Tendsto (fun n => |r| ^ n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine (pow_strictMono_right $ lt_of_le_of_ne (le_of_not_lt hr_le) hr).monotone.tendsto_atTop_atTop (fun b \u21a6 ?_)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nh : Tendsto (abs \u2218 fun n => r ^ n) atTop (\ud835\udcdd 0)\nhr_le : \u00ac|r| < 1\nhr : \u00ac1 = |r|\nb : \ud835\udd5c\n\u22a2 \u2203 a, b \u2264 |r| ^ a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nh : Tendsto (abs \u2218 fun n => r ^ n) atTop (\ud835\udcdd 0)\nhr_le : \u00ac|r| < 1\nhr : \u00ac1 = |r|\n\u22a2 Tendsto (fun n => |r| ^ n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := (pow_unbounded_of_one_lt b (lt_of_le_of_ne (le_of_not_lt hr_le) hr))\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nh : Tendsto (abs \u2218 fun n => r ^ n) atTop (\ud835\udcdd 0)\nhr_le : \u00ac|r| < 1\nhr : \u00ac1 = |r|\nb : \ud835\udd5c\nn : \u2115\nhn : b < |r| ^ n\n\u22a2 \u2203 a, b \u2264 |r| ^ a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nh : Tendsto (abs \u2218 fun n => r ^ n) atTop (\ud835\udcdd 0)\nhr_le : \u00ac|r| < 1\nhr : \u00ac1 = |r|\n\u22a2 Tendsto (fun n => |r| ^ n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexacts [\u27e8n, le_of_lt hn\u27e9, by simpa only [\u2190 abs_pow]]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nh : Tendsto (abs \u2218 fun n => r ^ n) atTop (\ud835\udcdd 0)\nhr_le : \u00ac|r| < 1\nhr : \u00ac1 = |r|\n\u22a2 Tendsto (fun n => |r| ^ n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa only [\u2190 abs_pow]\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : Archimedean \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nr : \ud835\udd5c\nh : |r| < 1\n\u22a2 Tendsto (abs \u2218 fun n => r ^ n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa only [\u2190 abs_pow] using (tendsto_pow_atTop_nhds_0_of_lt_1 (abs_nonneg r)) h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nu : \u2115 \u2192 \u211d\nc : \u211d\nhc : 0 \u2264 c\nn : \u2115\nhn : 0 < n\nh : \u2200 (k : \u2115), k < n \u2192 c * u k < u (k + 1)\n\u22a2 c ^ n * u 0 < u n\n[PROOFSTEP]\napply (monotone_mul_left_of_nonneg hc).seq_pos_lt_seq_of_le_of_lt hn _ _ h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nu : \u2115 \u2192 \u211d\nc : \u211d\nhc : 0 \u2264 c\nn : \u2115\nhn : 0 < n\nh : \u2200 (k : \u2115), k < n \u2192 c * u k < u (k + 1)\n\u22a2 c ^ 0 * u 0 \u2264 u 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nu : \u2115 \u2192 \u211d\nc : \u211d\nhc : 0 \u2264 c\nn : \u2115\nhn : 0 < n\nh : \u2200 (k : \u2115), k < n \u2192 c * u k < u (k + 1)\n\u22a2 \u2200 (k : \u2115), k < n \u2192 c ^ (k + 1) * u 0 \u2264 c * (c ^ k * u 0)\n[PROOFSTEP]\nsimp [_root_.pow_succ, mul_assoc, le_refl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nu : \u2115 \u2192 \u211d\nc : \u211d\nhc : 0 \u2264 c\nn : \u2115\nh : \u2200 (k : \u2115), k < n \u2192 c * u k \u2264 u (k + 1)\n\u22a2 c ^ n * u 0 \u2264 u n\n[PROOFSTEP]\napply (monotone_mul_left_of_nonneg hc).seq_le_seq n _ _ h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nu : \u2115 \u2192 \u211d\nc : \u211d\nhc : 0 \u2264 c\nn : \u2115\nh : \u2200 (k : \u2115), k < n \u2192 c * u k \u2264 u (k + 1)\n\u22a2 c ^ 0 * u 0 \u2264 u 0\n[PROOFSTEP]\nsimp [_root_.pow_succ, mul_assoc, le_refl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nu : \u2115 \u2192 \u211d\nc : \u211d\nhc : 0 \u2264 c\nn : \u2115\nh : \u2200 (k : \u2115), k < n \u2192 c * u k \u2264 u (k + 1)\n\u22a2 \u2200 (k : \u2115), k < n \u2192 c ^ (k + 1) * u 0 \u2264 c * (c ^ k * u 0)\n[PROOFSTEP]\nsimp [_root_.pow_succ, mul_assoc, le_refl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nu : \u2115 \u2192 \u211d\nc : \u211d\nhc : 0 \u2264 c\nn : \u2115\nhn : 0 < n\nh : \u2200 (k : \u2115), k < n \u2192 u (k + 1) < c * u k\n\u22a2 u n < c ^ n * u 0\n[PROOFSTEP]\napply (monotone_mul_left_of_nonneg hc).seq_pos_lt_seq_of_lt_of_le hn _ h _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nu : \u2115 \u2192 \u211d\nc : \u211d\nhc : 0 \u2264 c\nn : \u2115\nhn : 0 < n\nh : \u2200 (k : \u2115), k < n \u2192 u (k + 1) < c * u k\n\u22a2 u 0 \u2264 c ^ 0 * u 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nu : \u2115 \u2192 \u211d\nc : \u211d\nhc : 0 \u2264 c\nn : \u2115\nhn : 0 < n\nh : \u2200 (k : \u2115), k < n \u2192 u (k + 1) < c * u k\n\u22a2 \u2200 (k : \u2115), k < n \u2192 c * (c ^ k * u 0) \u2264 c ^ (k + 1) * u 0\n[PROOFSTEP]\nsimp [_root_.pow_succ, mul_assoc, le_refl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nu : \u2115 \u2192 \u211d\nc : \u211d\nhc : 0 \u2264 c\nn : \u2115\nh : \u2200 (k : \u2115), k < n \u2192 u (k + 1) \u2264 c * u k\n\u22a2 u n \u2264 c ^ n * u 0\n[PROOFSTEP]\napply (monotone_mul_left_of_nonneg hc).seq_le_seq n _ h _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nu : \u2115 \u2192 \u211d\nc : \u211d\nhc : 0 \u2264 c\nn : \u2115\nh : \u2200 (k : \u2115), k < n \u2192 u (k + 1) \u2264 c * u k\n\u22a2 u 0 \u2264 c ^ 0 * u 0\n[PROOFSTEP]\nsimp [_root_.pow_succ, mul_assoc, le_refl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nu : \u2115 \u2192 \u211d\nc : \u211d\nhc : 0 \u2264 c\nn : \u2115\nh : \u2200 (k : \u2115), k < n \u2192 u (k + 1) \u2264 c * u k\n\u22a2 \u2200 (k : \u2115), k < n \u2192 c * (c ^ k * u 0) \u2264 c ^ (k + 1) * u 0\n[PROOFSTEP]\nsimp [_root_.pow_succ, mul_assoc, le_refl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\nhr : r < 1\n\u22a2 Tendsto (fun a => \u2191(r ^ a)) atTop (\ud835\udcdd \u21910)\n[PROOFSTEP]\nsimp only [NNReal.coe_pow, NNReal.coe_zero, _root_.tendsto_pow_atTop_nhds_0_of_lt_1 r.coe_nonneg hr]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\u221e\nhr : r < 1\n\u22a2 Tendsto (fun n => r ^ n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrcases ENNReal.lt_iff_exists_coe.1 hr with \u27e8r, rfl, hr'\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\nhr' : \u2191r < 1\nhr : \u2191r < 1\n\u22a2 Tendsto (fun n => \u2191r ^ n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 ENNReal.coe_zero]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\nhr' : \u2191r < 1\nhr : \u2191r < 1\n\u22a2 Tendsto (fun n => \u2191r ^ n) atTop (\ud835\udcdd \u21910)\n[PROOFSTEP]\nnorm_cast at *\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\nhr' hr : r < 1\n\u22a2 Tendsto (fun a => r ^ a) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\napply NNReal.tendsto_pow_atTop_nhds_0_of_lt_1 hr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\nh\u2081 : 0 \u2264 r\nh\u2082 : r < 1\nthis\u271d : r \u2260 1\nthis : Tendsto (fun n => (r ^ n - 1) * (r - 1)\u207b\u00b9) atTop (\ud835\udcdd ((0 - 1) * (r - 1)\u207b\u00b9))\n\u22a2 Tendsto (fun n => \u2211 i in Finset.range n, r ^ i) atTop (\ud835\udcdd (1 - r)\u207b\u00b9)\n[PROOFSTEP]\nsimp_all [neg_inv, geom_sum_eq, div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u22a2 HasSum (fun n => (1 / 2) ^ n) 2\n[PROOFSTEP]\nconvert hasSum_geometric_of_lt_1 _ _\n[GOAL]\ncase h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u22a2 2 = (1 - 1 / 2)\u207b\u00b9\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase convert_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u22a2 0 \u2264 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase convert_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u22a2 1 / 2 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, (1 / 2) ^ i \u2264 2\n[PROOFSTEP]\nhave : \u2200 i, 0 \u2264 (1 / (2 : \u211d)) ^ i := by\n intro i\n apply pow_nonneg\n norm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn : \u2115\n\u22a2 \u2200 (i : \u2115), 0 \u2264 (1 / 2) ^ i\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn i : \u2115\n\u22a2 0 \u2264 (1 / 2) ^ i\n[PROOFSTEP]\napply pow_nonneg\n[GOAL]\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn i : \u2115\n\u22a2 0 \u2264 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn : \u2115\nthis : \u2200 (i : \u2115), 0 \u2264 (1 / 2) ^ i\n\u22a2 \u2211 i in Finset.range n, (1 / 2) ^ i \u2264 2\n[PROOFSTEP]\nconvert sum_le_tsum (range n) (fun i _ => this i) summable_geometric_two\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn : \u2115\nthis : \u2200 (i : \u2115), 0 \u2264 (1 / 2) ^ i\n\u22a2 2 = \u2211' (i : \u2115), (1 / 2) ^ i\n[PROOFSTEP]\nexact tsum_geometric_two.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn : \u2115\n\u22a2 (\u2211' (i : \u2115), if n \u2264 i then 2\u207b\u00b9 ^ i else 0) = 2 * 2\u207b\u00b9 ^ n\n[PROOFSTEP]\nhave A : Summable fun i : \u2115 => ite (n \u2264 i) ((2\u207b\u00b9 : \u211d) ^ i) 0 := by\n apply summable_of_nonneg_of_le _ _ summable_geometric_two <;>\n \u00b7 intro i\n by_cases hi : n \u2264 i <;> simp [hi]; apply pow_nonneg; exact zero_le_two\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn : \u2115\n\u22a2 Summable fun i => if n \u2264 i then 2\u207b\u00b9 ^ i else 0\n[PROOFSTEP]\napply summable_of_nonneg_of_le _ _ summable_geometric_two\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn : \u2115\n\u22a2 \u2200 (b : \u2115), 0 \u2264 if n \u2264 b then 2\u207b\u00b9 ^ b else 0\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn i : \u2115\n\u22a2 0 \u2264 if n \u2264 i then 2\u207b\u00b9 ^ i else 0\n[PROOFSTEP]\nby_cases hi : n \u2264 i\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn i : \u2115\nhi : n \u2264 i\n\u22a2 0 \u2264 if n \u2264 i then 2\u207b\u00b9 ^ i else 0\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn i : \u2115\nhi : \u00acn \u2264 i\n\u22a2 0 \u2264 if n \u2264 i then 2\u207b\u00b9 ^ i else 0\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn i : \u2115\nhi : n \u2264 i\n\u22a2 0 \u2264 2 ^ i\n[PROOFSTEP]\napply pow_nonneg\n[GOAL]\ncase pos.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn i : \u2115\nhi : n \u2264 i\n\u22a2 0 \u2264 2\n[PROOFSTEP]\nexact zero_le_two\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn : \u2115\n\u22a2 \u2200 (b : \u2115), (if n \u2264 b then 2\u207b\u00b9 ^ b else 0) \u2264 (1 / 2) ^ b\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn i : \u2115\n\u22a2 (if n \u2264 i then 2\u207b\u00b9 ^ i else 0) \u2264 (1 / 2) ^ i\n[PROOFSTEP]\nby_cases hi : n \u2264 i\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn i : \u2115\nhi : n \u2264 i\n\u22a2 (if n \u2264 i then 2\u207b\u00b9 ^ i else 0) \u2264 (1 / 2) ^ i\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn i : \u2115\nhi : \u00acn \u2264 i\n\u22a2 (if n \u2264 i then 2\u207b\u00b9 ^ i else 0) \u2264 (1 / 2) ^ i\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn i : \u2115\nhi : \u00acn \u2264 i\n\u22a2 0 \u2264 2 ^ i\n[PROOFSTEP]\napply pow_nonneg\n[GOAL]\ncase neg.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn i : \u2115\nhi : \u00acn \u2264 i\n\u22a2 0 \u2264 2\n[PROOFSTEP]\nexact zero_le_two\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn : \u2115\nA : Summable fun i => if n \u2264 i then 2\u207b\u00b9 ^ i else 0\n\u22a2 (\u2211' (i : \u2115), if n \u2264 i then 2\u207b\u00b9 ^ i else 0) = 2 * 2\u207b\u00b9 ^ n\n[PROOFSTEP]\nhave B : ((Finset.range n).sum fun i : \u2115 => ite (n \u2264 i) ((2\u207b\u00b9 : \u211d) ^ i) 0) = 0 :=\n Finset.sum_eq_zero fun i hi => ite_eq_right_iff.2 fun h => (lt_irrefl _ ((Finset.mem_range.1 hi).trans_le h)).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn : \u2115\nA : Summable fun i => if n \u2264 i then 2\u207b\u00b9 ^ i else 0\nB : (\u2211 i in Finset.range n, if n \u2264 i then 2\u207b\u00b9 ^ i else 0) = 0\n\u22a2 (\u2211' (i : \u2115), if n \u2264 i then 2\u207b\u00b9 ^ i else 0) = 2 * 2\u207b\u00b9 ^ n\n[PROOFSTEP]\nsimp only [\u2190 _root_.sum_add_tsum_nat_add n A, B, if_true, zero_add, zero_le', le_add_iff_nonneg_left, pow_add,\n _root_.tsum_mul_right, tsum_geometric_inv_two]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\na : \u211d\n\u22a2 HasSum (fun n => a / 2 / 2 ^ n) a\n[PROOFSTEP]\nconvert HasSum.mul_left (a / 2) (hasSum_geometric_of_lt_1 (le_of_lt one_half_pos) one_half_lt_one) using 1\n[GOAL]\ncase h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\na : \u211d\n\u22a2 (fun n => a / 2 / 2 ^ n) = fun i => a / 2 * (1 / 2) ^ i\n[PROOFSTEP]\nfunext n\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\na : \u211d\nn : \u2115\n\u22a2 a / 2 / 2 ^ n = a / 2 * (1 / 2) ^ n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\na : \u211d\nn : \u2115\n\u22a2 a / 2 / 2 ^ n = a / 2 * (2 ^ n)\u207b\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\na : \u211d\n\u22a2 a = a / 2 * (1 - 1 / 2)\u207b\u00b9\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\nhr : r < 1\n\u22a2 HasSum (fun n => r ^ n) (1 - r)\u207b\u00b9\n[PROOFSTEP]\napply NNReal.hasSum_coe.1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\nhr : r < 1\n\u22a2 HasSum (fun a => \u2191(r ^ a)) \u2191(1 - r)\u207b\u00b9\n[PROOFSTEP]\npush_cast\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\nhr : r < 1\n\u22a2 HasSum (fun a => \u2191r ^ a) (\u2191(1 - r))\u207b\u00b9\n[PROOFSTEP]\nrw [NNReal.coe_sub (le_of_lt hr)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\nhr : r < 1\n\u22a2 HasSum (fun a => \u2191r ^ a) (\u21911 - \u2191r)\u207b\u00b9\n[PROOFSTEP]\nexact hasSum_geometric_of_lt_1 r.coe_nonneg hr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\u221e\n\u22a2 \u2211' (n : \u2115), r ^ n = (1 - r)\u207b\u00b9\n[PROOFSTEP]\ncases' lt_or_le r 1 with hr hr\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\u221e\nhr : r < 1\n\u22a2 \u2211' (n : \u2115), r ^ n = (1 - r)\u207b\u00b9\n[PROOFSTEP]\nrcases ENNReal.lt_iff_exists_coe.1 hr with \u27e8r, rfl, hr'\u27e9\n[GOAL]\ncase inl.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\nhr' : \u2191r < 1\nhr : \u2191r < 1\n\u22a2 \u2211' (n : \u2115), \u2191r ^ n = (1 - \u2191r)\u207b\u00b9\n[PROOFSTEP]\nnorm_cast at *\n[GOAL]\ncase inl.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\nhr' hr : r < 1\n\u22a2 \u2211' (n : \u2115), \u2191(r ^ n) = (1 - \u2191r)\u207b\u00b9\n[PROOFSTEP]\nconvert ENNReal.tsum_coe_eq (NNReal.hasSum_geometric hr)\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\nhr' hr : r < 1\n\u22a2 (1 - \u2191r)\u207b\u00b9 = \u2191(1 - r)\u207b\u00b9\n[PROOFSTEP]\nrw [ENNReal.coe_inv <| ne_of_gt <| tsub_pos_iff_lt.2 hr, coe_sub, coe_one]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\u221e\nhr : 1 \u2264 r\n\u22a2 \u2211' (n : \u2115), r ^ n = (1 - r)\u207b\u00b9\n[PROOFSTEP]\nrw [tsub_eq_zero_iff_le.mpr hr, ENNReal.inv_zero, ENNReal.tsum_eq_iSup_nat, iSup_eq_top]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\u221e\nhr : 1 \u2264 r\n\u22a2 \u2200 (b : \u211d\u22650\u221e), b < \u22a4 \u2192 \u2203 i, b < \u2211 a in Finset.range i, r ^ a\n[PROOFSTEP]\nrefine' fun a ha => (ENNReal.exists_nat_gt (lt_top_iff_ne_top.1 ha)).imp fun n hn => lt_of_lt_of_le hn _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\u221e\nhr : 1 \u2264 r\na : \u211d\u22650\u221e\nha : a < \u22a4\nn : \u2115\nhn : a < \u2191n\n\u22a2 \u2191n \u2264 \u2211 a in Finset.range n, r ^ a\n[PROOFSTEP]\ncalc\n (n : \u211d\u22650\u221e) = \u2211 i in range n, 1 := by rw [sum_const, nsmul_one, card_range]\n _ \u2264 \u2211 i in range n, r ^ i := by gcongr; apply one_le_pow_of_one_le' hr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\u221e\nhr : 1 \u2264 r\na : \u211d\u22650\u221e\nha : a < \u22a4\nn : \u2115\nhn : a < \u2191n\n\u22a2 \u2191n = \u2211 i in Finset.range n, 1\n[PROOFSTEP]\nrw [sum_const, nsmul_one, card_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\u221e\nhr : 1 \u2264 r\na : \u211d\u22650\u221e\nha : a < \u22a4\nn : \u2115\nhn : a < \u2191n\n\u22a2 \u2211 i in Finset.range n, 1 \u2264 \u2211 i in Finset.range n, r ^ i\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nr : \u211d\u22650\u221e\nhr : 1 \u2264 r\na : \u211d\u22650\u221e\nha : a < \u22a4\nn : \u2115\nhn : a < \u2191n\ni\u271d : \u2115\na\u271d : i\u271d \u2208 Finset.range n\n\u22a2 1 \u2264 r ^ i\u271d\n[PROOFSTEP]\napply one_le_pow_of_one_le' hr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\nr C : \u211d\u22650\u221e\nhr : r < 1\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C * r ^ n\n\u22a2 CauchySeq f\n[PROOFSTEP]\nrefine' cauchySeq_of_edist_le_of_tsum_ne_top _ hu _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\nr C : \u211d\u22650\u221e\nhr : r < 1\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C * r ^ n\n\u22a2 \u2211' (n : \u2115), C * r ^ n \u2260 \u22a4\n[PROOFSTEP]\nrw [ENNReal.tsum_mul_left, ENNReal.tsum_geometric]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\nr C : \u211d\u22650\u221e\nhr : r < 1\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C * r ^ n\n\u22a2 C * (1 - r)\u207b\u00b9 \u2260 \u22a4\n[PROOFSTEP]\nrefine' ENNReal.mul_ne_top hC (ENNReal.inv_ne_top.2 _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\nr C : \u211d\u22650\u221e\nhr : r < 1\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C * r ^ n\n\u22a2 1 - r \u2260 0\n[PROOFSTEP]\nexact (tsub_pos_iff_lt.2 hr).ne'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\nr C : \u211d\u22650\u221e\nhr : r < 1\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C * r ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\n\u22a2 edist (f n) a \u2264 C * r ^ n / (1 - r)\n[PROOFSTEP]\nconvert edist_le_tsum_of_edist_le_of_tendsto _ hu ha _\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\nr C : \u211d\u22650\u221e\nhr : r < 1\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C * r ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\n\u22a2 C * r ^ n / (1 - r) = \u2211' (m : \u2115), C * r ^ (n + m)\n[PROOFSTEP]\nsimp only [pow_add, ENNReal.tsum_mul_left, ENNReal.tsum_geometric, div_eq_mul_inv, mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\nr C : \u211d\u22650\u221e\nhr : r < 1\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C * r ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\n\u22a2 edist (f 0) a \u2264 C / (1 - r)\n[PROOFSTEP]\nsimpa only [_root_.pow_zero, mul_one] using edist_le_of_edist_le_geometric_of_tendsto r C hu ha 0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\nC : \u211d\u22650\u221e\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C / 2 ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\n\u22a2 CauchySeq f\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, ENNReal.inv_pow] at hu \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\nC : \u211d\u22650\u221e\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C * 2\u207b\u00b9 ^ n\n\u22a2 CauchySeq f\n[PROOFSTEP]\nrefine' cauchySeq_of_edist_le_geometric 2\u207b\u00b9 C _ hC hu\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\nC : \u211d\u22650\u221e\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C * 2\u207b\u00b9 ^ n\n\u22a2 2\u207b\u00b9 < 1\n[PROOFSTEP]\nsimp [ENNReal.one_lt_two]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\nC : \u211d\u22650\u221e\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C / 2 ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\n\u22a2 edist (f n) a \u2264 2 * C / 2 ^ n\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, ENNReal.inv_pow] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\nC : \u211d\u22650\u221e\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C * 2\u207b\u00b9 ^ n\n\u22a2 edist (f n) a \u2264 2 * C * 2\u207b\u00b9 ^ n\n[PROOFSTEP]\nrw [mul_assoc, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\nC : \u211d\u22650\u221e\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C * 2\u207b\u00b9 ^ n\n\u22a2 edist (f n) a \u2264 C * 2\u207b\u00b9 ^ n * 2\n[PROOFSTEP]\nconvert edist_le_of_edist_le_geometric_of_tendsto 2\u207b\u00b9 C hu ha n using 1\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\nC : \u211d\u22650\u221e\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C * 2\u207b\u00b9 ^ n\n\u22a2 C * 2\u207b\u00b9 ^ n * 2 = C * 2\u207b\u00b9 ^ n / (1 - 2\u207b\u00b9)\n[PROOFSTEP]\nrw [ENNReal.one_sub_inv_two, div_eq_mul_inv, inv_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\nC : \u211d\u22650\u221e\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C / 2 ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\n\u22a2 edist (f 0) a \u2264 2 * C\n[PROOFSTEP]\nsimpa only [_root_.pow_zero, div_eq_mul_inv, inv_one, mul_one] using\n edist_le_of_edist_le_geometric_two_of_tendsto C hu ha 0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nr C : \u211d\nhr : r < 1\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C * r ^ n\n\u22a2 HasSum (fun n => C * r ^ n) (C / (1 - r))\n[PROOFSTEP]\nrcases sign_cases_of_C_mul_pow_nonneg fun n => dist_nonneg.trans (hu n) with (rfl | \u27e8_, r\u2080\u27e9)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nr : \u211d\nhr : r < 1\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 0 * r ^ n\n\u22a2 HasSum (fun n => 0 * r ^ n) (0 / (1 - r))\n[PROOFSTEP]\nsimp [hasSum_zero]\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nr C : \u211d\nhr : r < 1\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C * r ^ n\nleft\u271d : 0 < C\nr\u2080 : 0 \u2264 r\n\u22a2 HasSum (fun n => C * r ^ n) (C / (1 - r))\n[PROOFSTEP]\nrefine' HasSum.mul_left C _\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nr C : \u211d\nhr : r < 1\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C * r ^ n\nleft\u271d : 0 < C\nr\u2080 : 0 \u2264 r\n\u22a2 HasSum (fun n => r ^ n) (1 - r)\u207b\u00b9\n[PROOFSTEP]\nsimpa using hasSum_geometric_of_lt_1 r\u2080 hr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nr C : \u211d\nhr : r < 1\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C * r ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\n\u22a2 dist (f n) a \u2264 C * r ^ n / (1 - r)\n[PROOFSTEP]\nhave := aux_hasSum_of_le_geometric hr hu\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nr C : \u211d\nhr : r < 1\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C * r ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\nthis : HasSum (fun n => C * r ^ n) (C / (1 - r))\n\u22a2 dist (f n) a \u2264 C * r ^ n / (1 - r)\n[PROOFSTEP]\nconvert dist_le_tsum_of_dist_le_of_tendsto _ hu \u27e8_, this\u27e9 ha n\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nr C : \u211d\nhr : r < 1\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C * r ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\nthis : HasSum (fun n => C * r ^ n) (C / (1 - r))\n\u22a2 C * r ^ n / (1 - r) = \u2211' (m : \u2115), C * r ^ (n + m)\n[PROOFSTEP]\nsimp only [pow_add, mul_left_comm C, mul_div_right_comm]\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nr C : \u211d\nhr : r < 1\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C * r ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\nthis : HasSum (fun n => C * r ^ n) (C / (1 - r))\n\u22a2 C / (1 - r) * r ^ n = \u2211' (m : \u2115), r ^ n * (C * r ^ m)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nr C : \u211d\nhr : r < 1\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C * r ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\nthis : HasSum (fun n => C * r ^ n) (C / (1 - r))\n\u22a2 r ^ n * (C / (1 - r)) = \u2211' (m : \u2115), r ^ n * (C * r ^ m)\n[PROOFSTEP]\nexact (this.mul_left _).tsum_eq.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nr C : \u211d\nhr : r < 1\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C * r ^ n\nhu\u2082 : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C / 2 / 2 ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\n\u22a2 dist (f n) a \u2264 C / 2 ^ n\n[PROOFSTEP]\nconvert dist_le_tsum_of_dist_le_of_tendsto _ hu\u2082 (summable_geometric_two' C) ha n\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nr C : \u211d\nhr : r < 1\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C * r ^ n\nhu\u2082 : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C / 2 / 2 ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\n\u22a2 C / 2 ^ n = \u2211' (m : \u2115), C / 2 / 2 ^ (n + m)\n[PROOFSTEP]\nsimp only [add_comm n, pow_add, \u2190 div_div]\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nr C : \u211d\nhr : r < 1\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C * r ^ n\nhu\u2082 : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C / 2 / 2 ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\n\u22a2 C / 2 ^ n = \u2211' (m : \u2115), C / 2 / 2 ^ m / 2 ^ n\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nr C : \u211d\nhr : r < 1\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C * r ^ n\nhu\u2082 : \u2200 (n : \u2115), dist (f n) (f (n + 1)) \u2264 C / 2 / 2 ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\n\u22a2 \u2211' (m : \u2115), C / 2 / 2 ^ m / 2 ^ n = C / 2 ^ n\n[PROOFSTEP]\nexact ((hasSum_geometric_two' C).div_const _).tsum_eq\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : \u211d\nf : \u2115 \u2192 \u2115\nhm : 1 < m\nfi : \u2200 (i : \u2115), i \u2264 f i\n\u22a2 Summable fun i => 1 / m ^ f i\n[PROOFSTEP]\nrefine'\n summable_of_nonneg_of_le (fun a => one_div_nonneg.mpr (pow_nonneg (zero_le_one.trans hm.le) _)) (fun a => _)\n (summable_geometric_of_lt_1 (one_div_nonneg.mpr (zero_le_one.trans hm.le))\n ((one_div_lt (zero_lt_one.trans hm) zero_lt_one).mpr (one_div_one.le.trans_lt hm)))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : \u211d\nf : \u2115 \u2192 \u2115\nhm : 1 < m\nfi : \u2200 (i : \u2115), i \u2264 f i\na : \u2115\n\u22a2 1 / m ^ f a \u2264 (1 / m) ^ a\n[PROOFSTEP]\nrw [div_pow, one_pow]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : \u211d\nf : \u2115 \u2192 \u2115\nhm : 1 < m\nfi : \u2200 (i : \u2115), i \u2264 f i\na : \u2115\n\u22a2 1 / m ^ f a \u2264 1 / m ^ a\n[PROOFSTEP]\nrefine' (one_div_le_one_div _ _).mpr (pow_le_pow hm.le (fi a))\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : \u211d\nf : \u2115 \u2192 \u2115\nhm : 1 < m\nfi : \u2200 (i : \u2115), i \u2264 f i\na : \u2115\n\u22a2 0 < m ^ f a\n[PROOFSTEP]\nexact pow_pos (zero_lt_one.trans hm) _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : \u211d\nf : \u2115 \u2192 \u2115\nhm : 1 < m\nfi : \u2200 (i : \u2115), i \u2264 f i\na : \u2115\n\u22a2 0 < m ^ a\n[PROOFSTEP]\nexact pow_pos (zero_lt_one.trans hm) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b9 : Type ?u.561991\ninst\u271d : Encodable \u03b9\n\u22a2 { \u03b5' // (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2203 c, HasSum \u03b5' c \u2227 c \u2264 \u03b5 }\n[PROOFSTEP]\nlet f n := \u03b5 / 2 / 2 ^ n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b9 : Type ?u.561991\ninst\u271d : Encodable \u03b9\nf : \u2115 \u2192 \u211d := fun n => \u03b5 / 2 / 2 ^ n\n\u22a2 { \u03b5' // (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2203 c, HasSum \u03b5' c \u2227 c \u2264 \u03b5 }\n[PROOFSTEP]\nhave hf : HasSum f \u03b5 := hasSum_geometric_two' _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b9 : Type ?u.561991\ninst\u271d : Encodable \u03b9\nf : \u2115 \u2192 \u211d := fun n => \u03b5 / 2 / 2 ^ n\nhf : HasSum f \u03b5\n\u22a2 { \u03b5' // (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2203 c, HasSum \u03b5' c \u2227 c \u2264 \u03b5 }\n[PROOFSTEP]\nhave f0 : \u2200 n, 0 < f n := fun n => div_pos (half_pos h\u03b5) (pow_pos zero_lt_two _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b9 : Type ?u.561991\ninst\u271d : Encodable \u03b9\nf : \u2115 \u2192 \u211d := fun n => \u03b5 / 2 / 2 ^ n\nhf : HasSum f \u03b5\nf0 : \u2200 (n : \u2115), 0 < f n\n\u22a2 { \u03b5' // (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2203 c, HasSum \u03b5' c \u2227 c \u2264 \u03b5 }\n[PROOFSTEP]\nrefine' \u27e8f \u2218 Encodable.encode, fun i => f0 _, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b9 : Type ?u.561991\ninst\u271d : Encodable \u03b9\nf : \u2115 \u2192 \u211d := fun n => \u03b5 / 2 / 2 ^ n\nhf : HasSum f \u03b5\nf0 : \u2200 (n : \u2115), 0 < f n\n\u22a2 \u2203 c, HasSum (f \u2218 Encodable.encode) c \u2227 c \u2264 \u03b5\n[PROOFSTEP]\nrcases hf.summable.comp_injective (@Encodable.encode_injective \u03b9 _) with \u27e8c, hg\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b9 : Type ?u.561991\ninst\u271d : Encodable \u03b9\nf : \u2115 \u2192 \u211d := fun n => \u03b5 / 2 / 2 ^ n\nhf : HasSum f \u03b5\nf0 : \u2200 (n : \u2115), 0 < f n\nc : \u211d\nhg : HasSum (f \u2218 Encodable.encode) c\n\u22a2 \u2203 c, HasSum (f \u2218 Encodable.encode) c \u2227 c \u2264 \u03b5\n[PROOFSTEP]\nrefine' \u27e8c, hg, hasSum_le_inj _ (@Encodable.encode_injective \u03b9 _) _ _ hg hf\u27e9\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b9 : Type ?u.561991\ninst\u271d : Encodable \u03b9\nf : \u2115 \u2192 \u211d := fun n => \u03b5 / 2 / 2 ^ n\nhf : HasSum f \u03b5\nf0 : \u2200 (n : \u2115), 0 < f n\nc : \u211d\nhg : HasSum (f \u2218 Encodable.encode) c\n\u22a2 \u2200 (c : \u2115), \u00acc \u2208 Set.range Encodable.encode \u2192 0 \u2264 f c\n[PROOFSTEP]\nintro i _\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b9 : Type ?u.561991\ninst\u271d : Encodable \u03b9\nf : \u2115 \u2192 \u211d := fun n => \u03b5 / 2 / 2 ^ n\nhf : HasSum f \u03b5\nf0 : \u2200 (n : \u2115), 0 < f n\nc : \u211d\nhg : HasSum (f \u2218 Encodable.encode) c\ni : \u2115\na\u271d : \u00aci \u2208 Set.range Encodable.encode\n\u22a2 0 \u2264 f i\n[PROOFSTEP]\nexact le_of_lt (f0 _)\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b9 : Type ?u.561991\ninst\u271d : Encodable \u03b9\nf : \u2115 \u2192 \u211d := fun n => \u03b5 / 2 / 2 ^ n\nhf : HasSum f \u03b5\nf0 : \u2200 (n : \u2115), 0 < f n\nc : \u211d\nhg : HasSum (f \u2218 Encodable.encode) c\n\u22a2 \u2200 (i : \u03b9), (f \u2218 Encodable.encode) i \u2264 f (Encodable.encode i)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b9 : Type ?u.561991\ninst\u271d : Encodable \u03b9\nf : \u2115 \u2192 \u211d := fun n => \u03b5 / 2 / 2 ^ n\nhf : HasSum f \u03b5\nf0 : \u2200 (n : \u2115), 0 < f n\nc : \u211d\nhg : HasSum (f \u2218 Encodable.encode) c\nn : \u03b9\n\u22a2 (f \u2218 Encodable.encode) n \u2264 f (Encodable.encode n)\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2203 c, HasSum (fun i => \u03b5' \u2191i) c \u2227 c \u2264 \u03b5\n[PROOFSTEP]\nhaveI := hs.toEncodable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nthis : Encodable \u2191s\n\u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2203 c, HasSum (fun i => \u03b5' \u2191i) c \u2227 c \u2264 \u03b5\n[PROOFSTEP]\nrcases posSumOfEncodable h\u03b5 s with \u27e8f, hf0, \u27e8c, hfc, hc\u03b5\u27e9\u27e9\n[GOAL]\ncase mk.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nthis : Encodable \u2191s\nf : \u2191s \u2192 \u211d\nhf0 : \u2200 (i : \u2191s), 0 < f i\nc : \u211d\nhfc : HasSum f c\nhc\u03b5 : c \u2264 \u03b5\n\u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2203 c, HasSum (fun i => \u03b5' \u2191i) c \u2227 c \u2264 \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun i => if h : i \u2208 s then f \u27e8i, h\u27e9 else 1, fun i => _, \u27e8c, _, hc\u03b5\u27e9\u27e9\n[GOAL]\ncase mk.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nthis : Encodable \u2191s\nf : \u2191s \u2192 \u211d\nhf0 : \u2200 (i : \u2191s), 0 < f i\nc : \u211d\nhfc : HasSum f c\nhc\u03b5 : c \u2264 \u03b5\ni : \u03b9\n\u22a2 0 < (fun i => if h : i \u2208 s then f { val := i, property := h } else 1) i\n[PROOFSTEP]\nconv_rhs => simp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nthis : Encodable \u2191s\nf : \u2191s \u2192 \u211d\nhf0 : \u2200 (i : \u2191s), 0 < f i\nc : \u211d\nhfc : HasSum f c\nhc\u03b5 : c \u2264 \u03b5\ni : \u03b9\n| (fun i => if h : i \u2208 s then f { val := i, property := h } else 1) i\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nthis : Encodable \u2191s\nf : \u2191s \u2192 \u211d\nhf0 : \u2200 (i : \u2191s), 0 < f i\nc : \u211d\nhfc : HasSum f c\nhc\u03b5 : c \u2264 \u03b5\ni : \u03b9\n| (fun i => if h : i \u2208 s then f { val := i, property := h } else 1) i\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nthis : Encodable \u2191s\nf : \u2191s \u2192 \u211d\nhf0 : \u2200 (i : \u2191s), 0 < f i\nc : \u211d\nhfc : HasSum f c\nhc\u03b5 : c \u2264 \u03b5\ni : \u03b9\n| (fun i => if h : i \u2208 s then f { val := i, property := h } else 1) i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nthis : Encodable \u2191s\nf : \u2191s \u2192 \u211d\nhf0 : \u2200 (i : \u2191s), 0 < f i\nc : \u211d\nhfc : HasSum f c\nhc\u03b5 : c \u2264 \u03b5\ni : \u03b9\n\u22a2 0 < if h : i \u2208 s then f { val := i, property := h } else 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nthis : Encodable \u2191s\nf : \u2191s \u2192 \u211d\nhf0 : \u2200 (i : \u2191s), 0 < f i\nc : \u211d\nhfc : HasSum f c\nhc\u03b5 : c \u2264 \u03b5\ni : \u03b9\nh\u271d : i \u2208 s\n\u22a2 0 < f { val := i, property := h\u271d }\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nthis : Encodable \u2191s\nf : \u2191s \u2192 \u211d\nhf0 : \u2200 (i : \u2191s), 0 < f i\nc : \u211d\nhfc : HasSum f c\nhc\u03b5 : c \u2264 \u03b5\ni : \u03b9\nh\u271d : \u00aci \u2208 s\n\u22a2 0 < 1\n[PROOFSTEP]\nexacts [hf0 _, zero_lt_one]\n[GOAL]\ncase mk.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nthis : Encodable \u2191s\nf : \u2191s \u2192 \u211d\nhf0 : \u2200 (i : \u2191s), 0 < f i\nc : \u211d\nhfc : HasSum f c\nhc\u03b5 : c \u2264 \u03b5\n\u22a2 HasSum (fun i => (fun i => if h : i \u2208 s then f { val := i, property := h } else 1) \u2191i) c\n[PROOFSTEP]\nsimpa only [Subtype.coe_prop, dif_pos, Subtype.coe_eta]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2200 (t : Finset \u03b9), \u2191t \u2286 s \u2192 \u2211 i in t, \u03b5' i \u2264 \u03b5\n[PROOFSTEP]\nrcases hs.exists_pos_hasSum_le h\u03b5 with \u27e8\u03b5', hpos, c, h\u03b5'c, hc\u03b5\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b5' : \u03b9 \u2192 \u211d\nhpos : \u2200 (i : \u03b9), 0 < \u03b5' i\nc : \u211d\nh\u03b5'c : HasSum (fun i => \u03b5' \u2191i) c\nhc\u03b5 : c \u2264 \u03b5\n\u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2200 (t : Finset \u03b9), \u2191t \u2286 s \u2192 \u2211 i in t, \u03b5' i \u2264 \u03b5\n[PROOFSTEP]\nrefine' \u27e8\u03b5', hpos, fun t ht => _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b5' : \u03b9 \u2192 \u211d\nhpos : \u2200 (i : \u03b9), 0 < \u03b5' i\nc : \u211d\nh\u03b5'c : HasSum (fun i => \u03b5' \u2191i) c\nhc\u03b5 : c \u2264 \u03b5\nt : Finset \u03b9\nht : \u2191t \u2286 s\n\u22a2 \u2211 i in t, \u03b5' i \u2264 \u03b5\n[PROOFSTEP]\nrw [\u2190 sum_subtype_of_mem _ ht]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b5' : \u03b9 \u2192 \u211d\nhpos : \u2200 (i : \u03b9), 0 < \u03b5' i\nc : \u211d\nh\u03b5'c : HasSum (fun i => \u03b5' \u2191i) c\nhc\u03b5 : c \u2264 \u03b5\nt : Finset \u03b9\nht : \u2191t \u2286 s\n\u22a2 \u2211 x in Finset.subtype (fun x => x \u2208 s) t, \u03b5' \u2191x \u2264 \u03b5\n[PROOFSTEP]\nrefine' (sum_le_hasSum _ _ h\u03b5'c).trans hc\u03b5\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\ns : Set \u03b9\nhs : Set.Countable s\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b5' : \u03b9 \u2192 \u211d\nhpos : \u2200 (i : \u03b9), 0 < \u03b5' i\nc : \u211d\nh\u03b5'c : HasSum (fun i => \u03b5' \u2191i) c\nhc\u03b5 : c \u2264 \u03b5\nt : Finset \u03b9\nht : \u2191t \u2286 s\n\u22a2 \u2200 (i : \u2191s), \u00aci \u2208 Finset.subtype (fun x => x \u2208 s) t \u2192 0 \u2264 \u03b5' \u2191i\n[PROOFSTEP]\nexact fun _ _ => (hpos _).le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\n\u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2203 c, HasSum \u03b5' c \u2227 c < \u03b5\n[PROOFSTEP]\ncases nonempty_encodable \u03b9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\nval\u271d : Encodable \u03b9\n\u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2203 c, HasSum \u03b5' c \u2227 c < \u03b5\n[PROOFSTEP]\nobtain \u27e8a, a0, a\u03b5\u27e9 := exists_between (pos_iff_ne_zero.2 h\u03b5)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\nval\u271d : Encodable \u03b9\na : \u211d\u22650\na0 : 0 < a\na\u03b5 : a < \u03b5\n\u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2203 c, HasSum \u03b5' c \u2227 c < \u03b5\n[PROOFSTEP]\nobtain \u27e8\u03b5', h\u03b5', c, hc, hc\u03b5\u27e9 := posSumOfEncodable a0 \u03b9\n[GOAL]\ncase intro.intro.intro.mk.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\nval\u271d : Encodable \u03b9\na : \u211d\u22650\na0 : 0 < a\na\u03b5 : a < \u03b5\n\u03b5' : \u03b9 \u2192 \u211d\nh\u03b5' : \u2200 (i : \u03b9), 0 < \u03b5' i\nc : \u211d\nhc : HasSum \u03b5' c\nhc\u03b5 : c \u2264 (fun a => \u2191a) a\n\u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2203 c, HasSum \u03b5' c \u2227 c < \u03b5\n[PROOFSTEP]\nexact\n \u27e8fun i => \u27e8\u03b5' i, (h\u03b5' i).le\u27e9, fun i => NNReal.coe_lt_coe.1 <| h\u03b5' i,\n \u27e8c, hasSum_le (fun i => (h\u03b5' i).le) hasSum_zero hc\u27e9, NNReal.hasSum_coe.1 hc,\n a\u03b5.trans_le' <| NNReal.coe_le_coe.1 hc\u03b5\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\n\u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2211' (i : \u03b9), \u2191(\u03b5' i) < \u03b5\n[PROOFSTEP]\nrcases exists_between (pos_iff_ne_zero.2 h\u03b5) with \u27e8r, h0r, hr\u03b5\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\nr : \u211d\u22650\u221e\nh0r : 0 < r\nhr\u03b5 : r < \u03b5\n\u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2211' (i : \u03b9), \u2191(\u03b5' i) < \u03b5\n[PROOFSTEP]\nrcases lt_iff_exists_coe.1 hr\u03b5 with \u27e8x, rfl, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\nx : \u211d\u22650\nright\u271d : \u2191x < \u03b5\nh0r : 0 < \u2191x\nhr\u03b5 : \u2191x < \u03b5\n\u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2211' (i : \u03b9), \u2191(\u03b5' i) < \u03b5\n[PROOFSTEP]\nrcases NNReal.exists_pos_sum_of_countable (coe_pos.1 h0r).ne' \u03b9 with \u27e8\u03b5', hp, c, hc, hcr\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\nx : \u211d\u22650\nright\u271d : \u2191x < \u03b5\nh0r : 0 < \u2191x\nhr\u03b5 : \u2191x < \u03b5\n\u03b5' : \u03b9 \u2192 \u211d\u22650\nhp : \u2200 (i : \u03b9), 0 < \u03b5' i\nc : \u211d\u22650\nhc : HasSum \u03b5' c\nhcr : c < x\n\u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2211' (i : \u03b9), \u2191(\u03b5' i) < \u03b5\n[PROOFSTEP]\nexact \u27e8\u03b5', hp, (ENNReal.tsum_coe_eq hc).symm \u25b8 lt_trans (coe_lt_coe.2 hcr) hr\u03b5\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\nw : \u03b9 \u2192 \u211d\u22650\u221e\nhw : \u2200 (i : \u03b9), w i \u2260 \u22a4\n\u22a2 \u2203 \u03b4, (\u2200 (i : \u03b9), 0 < \u03b4 i) \u2227 \u2211' (i : \u03b9), w i * \u2191(\u03b4 i) < \u03b5\n[PROOFSTEP]\nlift w to \u03b9 \u2192 \u211d\u22650 using hw\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\nw : \u03b9 \u2192 \u211d\u22650\n\u22a2 \u2203 \u03b4, (\u2200 (i : \u03b9), 0 < \u03b4 i) \u2227 \u2211' (i : \u03b9), (fun i => \u2191(w i)) i * \u2191(\u03b4 i) < \u03b5\n[PROOFSTEP]\nrcases exists_pos_sum_of_countable h\u03b5 \u03b9 with \u27e8\u03b4', Hpos, Hsum\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\nw \u03b4' : \u03b9 \u2192 \u211d\u22650\nHpos : \u2200 (i : \u03b9), 0 < \u03b4' i\nHsum : \u2211' (i : \u03b9), \u2191(\u03b4' i) < \u03b5\n\u22a2 \u2203 \u03b4, (\u2200 (i : \u03b9), 0 < \u03b4 i) \u2227 \u2211' (i : \u03b9), (fun i => \u2191(w i)) i * \u2191(\u03b4 i) < \u03b5\n[PROOFSTEP]\nhave : \u2200 i, 0 < max 1 (w i) := fun i => zero_lt_one.trans_le (le_max_left _ _)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\nw \u03b4' : \u03b9 \u2192 \u211d\u22650\nHpos : \u2200 (i : \u03b9), 0 < \u03b4' i\nHsum : \u2211' (i : \u03b9), \u2191(\u03b4' i) < \u03b5\nthis : \u2200 (i : \u03b9), 0 < max 1 (w i)\n\u22a2 \u2203 \u03b4, (\u2200 (i : \u03b9), 0 < \u03b4 i) \u2227 \u2211' (i : \u03b9), (fun i => \u2191(w i)) i * \u2191(\u03b4 i) < \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun i => \u03b4' i / max 1 (w i), fun i => div_pos (Hpos _) (this i), _\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\nw \u03b4' : \u03b9 \u2192 \u211d\u22650\nHpos : \u2200 (i : \u03b9), 0 < \u03b4' i\nHsum : \u2211' (i : \u03b9), \u2191(\u03b4' i) < \u03b5\nthis : \u2200 (i : \u03b9), 0 < max 1 (w i)\n\u22a2 \u2211' (i : \u03b9), (fun i => \u2191(w i)) i * \u2191((fun i => \u03b4' i / max 1 (w i)) i) < \u03b5\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (ENNReal.tsum_le_tsum fun i => _) Hsum\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\nw \u03b4' : \u03b9 \u2192 \u211d\u22650\nHpos : \u2200 (i : \u03b9), 0 < \u03b4' i\nHsum : \u2211' (i : \u03b9), \u2191(\u03b4' i) < \u03b5\nthis : \u2200 (i : \u03b9), 0 < max 1 (w i)\ni : \u03b9\n\u22a2 (fun i => \u2191(w i)) i * \u2191((fun i => \u03b4' i / max 1 (w i)) i) \u2264 \u2191(\u03b4' i)\n[PROOFSTEP]\nrw [coe_div (this i).ne']\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\nw \u03b4' : \u03b9 \u2192 \u211d\u22650\nHpos : \u2200 (i : \u03b9), 0 < \u03b4' i\nHsum : \u2211' (i : \u03b9), \u2191(\u03b4' i) < \u03b5\nthis : \u2200 (i : \u03b9), 0 < max 1 (w i)\ni : \u03b9\n\u22a2 (fun i => \u2191(w i)) i * (\u2191(\u03b4' i) / \u2191(max 1 (w i))) \u2264 \u2191(\u03b4' i)\n[PROOFSTEP]\nrefine' mul_le_of_le_div' (mul_le_mul_left' (ENNReal.inv_le_inv.2 _) _)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b9 : Type u_4\ninst\u271d : Countable \u03b9\nw \u03b4' : \u03b9 \u2192 \u211d\u22650\nHpos : \u2200 (i : \u03b9), 0 < \u03b4' i\nHsum : \u2211' (i : \u03b9), \u2191(\u03b4' i) < \u03b5\nthis : \u2200 (i : \u03b9), 0 < max 1 (w i)\ni : \u03b9\n\u22a2 (fun i => \u2191(w i)) i \u2264 \u2191(max 1 (w i))\n[PROOFSTEP]\nexact coe_le_coe.2 (le_max_right _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn : \u2115\n\u22a2 0 \u2264 \u2191n !\n[PROOFSTEP]\nexact_mod_cast n.factorial_pos.le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn : \u2115\n\u22a2 0 \u2264 \u2191n\n[PROOFSTEP]\nexact_mod_cast n.zero_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u22a2 \u2200\u1da0 (b : \u2115) in atTop, \u2191b ! / \u2191b ^ b \u2264 1 / \u2191b\n[PROOFSTEP]\nrefine' (eventually_gt_atTop 0).mono fun n hn => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nn : \u2115\nhn : 0 < n\n\u22a2 \u2191n ! / \u2191n ^ n \u2264 1 / \u2191n\n[PROOFSTEP]\nrcases Nat.exists_eq_succ_of_ne_zero hn.ne.symm with \u27e8k, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\n\u22a2 \u2191(succ k)! / \u2191(succ k) ^ succ k \u2264 1 / \u2191(succ k)\n[PROOFSTEP]\nrw [\u2190 prod_range_add_one_eq_factorial, pow_eq_prod_const, div_eq_mul_inv, \u2190 inv_eq_one_div, prod_natCast, Nat.cast_succ,\n \u2190 prod_inv_distrib, \u2190 prod_mul_distrib, Finset.prod_range_succ']\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\n\u22a2 (\u220f k_1 in Finset.range k, \u2191(k_1 + 1 + 1) * (\u2191k + 1)\u207b\u00b9) * (\u2191(0 + 1) * (\u2191k + 1)\u207b\u00b9) \u2264 (\u2191k + 1)\u207b\u00b9\n[PROOFSTEP]\nsimp only [prod_range_succ', one_mul, Nat.cast_add, zero_add, Nat.cast_one]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\n\u22a2 (\u220f x in Finset.range k, (\u2191x + 1 + 1) * (\u2191k + 1)\u207b\u00b9) * (\u2191k + 1)\u207b\u00b9 \u2264 (\u2191k + 1)\u207b\u00b9\n[PROOFSTEP]\nrefine' mul_le_of_le_one_left (inv_nonneg.mpr <| by exact_mod_cast hn.le) (prod_le_one _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\n\u22a2 0 \u2264 \u2191k + 1\n[PROOFSTEP]\nexact_mod_cast hn.le\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\n\u22a2 \u2200 (i : \u2115), i \u2208 Finset.range k \u2192 0 \u2264 (\u2191i + 1 + 1) * (\u2191k + 1)\u207b\u00b9\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\n\u22a2 \u2200 (i : \u2115), i \u2208 Finset.range k \u2192 (\u2191i + 1 + 1) * (\u2191k + 1)\u207b\u00b9 \u2264 1\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\nx : \u2115\nhx : x \u2208 Finset.range k\n\u22a2 0 \u2264 (\u2191x + 1 + 1) * (\u2191k + 1)\u207b\u00b9\n[PROOFSTEP]\nrw [Finset.mem_range] at hx \n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\nx : \u2115\nhx : x \u2208 Finset.range k\n\u22a2 (\u2191x + 1 + 1) * (\u2191k + 1)\u207b\u00b9 \u2264 1\n[PROOFSTEP]\nrw [Finset.mem_range] at hx \n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\nx : \u2115\nhx : x < k\n\u22a2 0 \u2264 (\u2191x + 1 + 1) * (\u2191k + 1)\u207b\u00b9\n[PROOFSTEP]\nrefine' mul_nonneg _ (inv_nonneg.mpr _)\n[GOAL]\ncase intro.refine'_1.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\nx : \u2115\nhx : x < k\n\u22a2 0 \u2264 \u2191x + 1 + 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.refine'_1.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\nx : \u2115\nhx : x < k\n\u22a2 0 \u2264 \u2191k + 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.refine'_1.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\nx : \u2115\nhx : x < k\n\u22a2 0 \u2264 x + 1 + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase intro.refine'_1.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\nx : \u2115\nhx : x < k\n\u22a2 0 \u2264 k + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\nx : \u2115\nhx : x < k\n\u22a2 (\u2191x + 1 + 1) * (\u2191k + 1)\u207b\u00b9 \u2264 1\n[PROOFSTEP]\nrefine' (div_le_one <| by exact_mod_cast hn).mpr _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\nx : \u2115\nhx : x < k\n\u22a2 0 < \u2191k + 1\n[PROOFSTEP]\nexact_mod_cast hn\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\nx : \u2115\nhx : x < k\n\u22a2 \u2191x + 1 + 1 \u2264 \u2191k + 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nk : \u2115\nhn : 0 < succ k\nx : \u2115\nhx : x < k\n\u22a2 x + 1 + 1 \u2264 k + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\nx : \u2115\n\u22a2 x \u2264 \u230amax 0 (\u2191x + 1)\u230b\u208a\n[PROOFSTEP]\nsimp [Nat.le_floor_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\n\u22a2 Tendsto (fun x => \u2191\u230aa * x\u230b\u208a / x) atTop (\ud835\udcdd a)\n[PROOFSTEP]\nhave A : Tendsto (fun x : R => a - x\u207b\u00b9) atTop (\ud835\udcdd (a - 0)) := tendsto_const_nhds.sub tendsto_inv_atTop_zero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a - x\u207b\u00b9) atTop (\ud835\udcdd (a - 0))\n\u22a2 Tendsto (fun x => \u2191\u230aa * x\u230b\u208a / x) atTop (\ud835\udcdd a)\n[PROOFSTEP]\nrw [sub_zero] at A \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a - x\u207b\u00b9) atTop (\ud835\udcdd a)\n\u22a2 Tendsto (fun x => \u2191\u230aa * x\u230b\u208a / x) atTop (\ud835\udcdd a)\n[PROOFSTEP]\napply tendsto_of_tendsto_of_tendsto_of_le_of_le' A tendsto_const_nhds\n[GOAL]\ncase hgf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a - x\u207b\u00b9) atTop (\ud835\udcdd a)\n\u22a2 \u2200\u1da0 (b : R) in atTop, a - b\u207b\u00b9 \u2264 \u2191\u230aa * b\u230b\u208a / b\n[PROOFSTEP]\nrefine' eventually_atTop.2 \u27e81, fun x hx => _\u27e9\n[GOAL]\ncase hgf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a - x\u207b\u00b9) atTop (\ud835\udcdd a)\nx : R\nhx : x \u2265 1\n\u22a2 a - x\u207b\u00b9 \u2264 \u2191\u230aa * x\u230b\u208a / x\n[PROOFSTEP]\nsimp only [le_div_iff (zero_lt_one.trans_le hx), _root_.sub_mul, inv_mul_cancel (zero_lt_one.trans_le hx).ne']\n[GOAL]\ncase hgf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a - x\u207b\u00b9) atTop (\ud835\udcdd a)\nx : R\nhx : x \u2265 1\n\u22a2 a * x - 1 \u2264 \u2191\u230aa * x\u230b\u208a\n[PROOFSTEP]\nhave := Nat.lt_floor_add_one (a * x)\n[GOAL]\ncase hgf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a - x\u207b\u00b9) atTop (\ud835\udcdd a)\nx : R\nhx : x \u2265 1\nthis : a * x < \u2191\u230aa * x\u230b\u208a + 1\n\u22a2 a * x - 1 \u2264 \u2191\u230aa * x\u230b\u208a\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hfh\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a - x\u207b\u00b9) atTop (\ud835\udcdd a)\n\u22a2 \u2200\u1da0 (b : R) in atTop, \u2191\u230aa * b\u230b\u208a / b \u2264 a\n[PROOFSTEP]\nrefine' eventually_atTop.2 \u27e81, fun x hx => _\u27e9\n[GOAL]\ncase hfh\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a - x\u207b\u00b9) atTop (\ud835\udcdd a)\nx : R\nhx : x \u2265 1\n\u22a2 \u2191\u230aa * x\u230b\u208a / x \u2264 a\n[PROOFSTEP]\nrw [div_le_iff (zero_lt_one.trans_le hx)]\n[GOAL]\ncase hfh\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a - x\u207b\u00b9) atTop (\ud835\udcdd a)\nx : R\nhx : x \u2265 1\n\u22a2 \u2191\u230aa * x\u230b\u208a \u2264 a * x\n[PROOFSTEP]\nsimp [Nat.floor_le (mul_nonneg ha (zero_le_one.trans hx))]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\n\u22a2 Tendsto (fun x => \u2191\u230ax\u230b\u208a / x) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nsimpa using tendsto_nat_floor_mul_div_atTop (zero_le_one' R)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\n\u22a2 Tendsto (fun x => \u2191\u2308a * x\u2309\u208a / x) atTop (\ud835\udcdd a)\n[PROOFSTEP]\nhave A : Tendsto (fun x : R => a + x\u207b\u00b9) atTop (\ud835\udcdd (a + 0)) := tendsto_const_nhds.add tendsto_inv_atTop_zero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a + x\u207b\u00b9) atTop (\ud835\udcdd (a + 0))\n\u22a2 Tendsto (fun x => \u2191\u2308a * x\u2309\u208a / x) atTop (\ud835\udcdd a)\n[PROOFSTEP]\nrw [add_zero] at A \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a + x\u207b\u00b9) atTop (\ud835\udcdd a)\n\u22a2 Tendsto (fun x => \u2191\u2308a * x\u2309\u208a / x) atTop (\ud835\udcdd a)\n[PROOFSTEP]\napply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds A\n[GOAL]\ncase hgf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a + x\u207b\u00b9) atTop (\ud835\udcdd a)\n\u22a2 \u2200\u1da0 (b : R) in atTop, a \u2264 \u2191\u2308a * b\u2309\u208a / b\n[PROOFSTEP]\nrefine' eventually_atTop.2 \u27e81, fun x hx => _\u27e9\n[GOAL]\ncase hgf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a + x\u207b\u00b9) atTop (\ud835\udcdd a)\nx : R\nhx : x \u2265 1\n\u22a2 a \u2264 \u2191\u2308a * x\u2309\u208a / x\n[PROOFSTEP]\nrw [le_div_iff (zero_lt_one.trans_le hx)]\n[GOAL]\ncase hgf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a + x\u207b\u00b9) atTop (\ud835\udcdd a)\nx : R\nhx : x \u2265 1\n\u22a2 a * x \u2264 \u2191\u2308a * x\u2309\u208a\n[PROOFSTEP]\nexact Nat.le_ceil _\n[GOAL]\ncase hfh\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a + x\u207b\u00b9) atTop (\ud835\udcdd a)\n\u22a2 \u2200\u1da0 (b : R) in atTop, \u2191\u2308a * b\u2309\u208a / b \u2264 a + b\u207b\u00b9\n[PROOFSTEP]\nrefine' eventually_atTop.2 \u27e81, fun x hx => _\u27e9\n[GOAL]\ncase hfh\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\na : R\nha : 0 \u2264 a\nA : Tendsto (fun x => a + x\u207b\u00b9) atTop (\ud835\udcdd a)\nx : R\nhx : x \u2265 1\n\u22a2 \u2191\u2308a * x\u2309\u208a / x \u2264 a + x\u207b\u00b9\n[PROOFSTEP]\nsimp [div_le_iff (zero_lt_one.trans_le hx), inv_mul_cancel (zero_lt_one.trans_le hx).ne',\n (Nat.ceil_lt_add_one (mul_nonneg ha (zero_le_one.trans hx))).le, add_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : LinearOrderedField R\ninst\u271d\u00b9 : OrderTopology R\ninst\u271d : FloorRing R\n\u22a2 Tendsto (fun x => \u2191\u2308x\u2309\u208a / x) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nsimpa using tendsto_nat_ceil_mul_div_atTop (zero_le_one' R)\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecificLimits.Basic", "llama_tokens": 32977, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4302656442689631}} {"text": "[GOAL]\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nk : \u2115\nh : \u2200 (i : \u2115), i < k + 1 \u2192 coeff x i = 0\n\u22a2 \u2191verschiebung (shift x (Nat.succ k)) = shift x k\n[PROOFSTEP]\next \u27e8j\u27e9\n[GOAL]\ncase h.zero\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nk : \u2115\nh : \u2200 (i : \u2115), i < k + 1 \u2192 coeff x i = 0\n\u22a2 coeff (\u2191verschiebung (shift x (Nat.succ k))) Nat.zero = coeff (shift x k) Nat.zero\n[PROOFSTEP]\nrw [verschiebung_coeff_zero, shift_coeff, h]\n[GOAL]\ncase h.zero.a\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nk : \u2115\nh : \u2200 (i : \u2115), i < k + 1 \u2192 coeff x i = 0\n\u22a2 k + Nat.zero < k + 1\n[PROOFSTEP]\napply Nat.lt_succ_self\n[GOAL]\ncase h.succ\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nk : \u2115\nh : \u2200 (i : \u2115), i < k + 1 \u2192 coeff x i = 0\nn\u271d : \u2115\n\u22a2 coeff (\u2191verschiebung (shift x (Nat.succ k))) (Nat.succ n\u271d) = coeff (shift x k) (Nat.succ n\u271d)\n[PROOFSTEP]\nsimp only [verschiebung_coeff_succ, shift]\n[GOAL]\ncase h.succ\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nk : \u2115\nh : \u2200 (i : \u2115), i < k + 1 \u2192 coeff x i = 0\nn\u271d : \u2115\n\u22a2 coeff x (Nat.succ k + n\u271d) = coeff x (k + Nat.succ n\u271d)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.succ.e_a\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nk : \u2115\nh : \u2200 (i : \u2115), i < k + 1 \u2192 coeff x i = 0\nn\u271d : \u2115\n\u22a2 Nat.succ k + n\u271d = k + Nat.succ n\u271d\n[PROOFSTEP]\nrw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]\n[GOAL]\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nn : \u2115\nh : \u2200 (i : \u2115), i < n \u2192 coeff x i = 0\n\u22a2 x = (\u2191verschiebung)^[n] (shift x n)\n[PROOFSTEP]\ninduction' n with k ih\n[GOAL]\ncase zero\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nn : \u2115\nh\u271d : \u2200 (i : \u2115), i < n \u2192 coeff x i = 0\nh : \u2200 (i : \u2115), i < Nat.zero \u2192 coeff x i = 0\n\u22a2 x = (\u2191verschiebung)^[Nat.zero] (shift x Nat.zero)\n[PROOFSTEP]\ncases x\n[GOAL]\ncase zero.mk'\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nn : \u2115\ncoeff\u271d : \u2115 \u2192 R\nh\u271d : \u2200 (i : \u2115), i < n \u2192 coeff { coeff := coeff\u271d } i = 0\nh : \u2200 (i : \u2115), i < Nat.zero \u2192 coeff { coeff := coeff\u271d } i = 0\n\u22a2 { coeff := coeff\u271d } = (\u2191verschiebung)^[Nat.zero] (shift { coeff := coeff\u271d } Nat.zero)\n[PROOFSTEP]\nsimp [shift]\n[GOAL]\ncase succ\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nn : \u2115\nh\u271d : \u2200 (i : \u2115), i < n \u2192 coeff x i = 0\nk : \u2115\nih : (\u2200 (i : \u2115), i < k \u2192 coeff x i = 0) \u2192 x = (\u2191verschiebung)^[k] (shift x k)\nh : \u2200 (i : \u2115), i < Nat.succ k \u2192 coeff x i = 0\n\u22a2 x = (\u2191verschiebung)^[Nat.succ k] (shift x (Nat.succ k))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase succ\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nn : \u2115\nh\u271d : \u2200 (i : \u2115), i < n \u2192 coeff x i = 0\nk : \u2115\nih : (\u2200 (i : \u2115), i < k \u2192 coeff x i = 0) \u2192 x = (\u2191verschiebung)^[k] (shift x k)\nh : \u2200 (i : \u2115), i < Nat.succ k \u2192 coeff x i = 0\n\u22a2 x = (\u2191verschiebung)^[k] (\u2191verschiebung (shift x (Nat.succ k)))\n[PROOFSTEP]\nrw [verschiebung_shift]\n[GOAL]\ncase succ\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nn : \u2115\nh\u271d : \u2200 (i : \u2115), i < n \u2192 coeff x i = 0\nk : \u2115\nih : (\u2200 (i : \u2115), i < k \u2192 coeff x i = 0) \u2192 x = (\u2191verschiebung)^[k] (shift x k)\nh : \u2200 (i : \u2115), i < Nat.succ k \u2192 coeff x i = 0\n\u22a2 x = (\u2191verschiebung)^[k] (shift x k)\n[PROOFSTEP]\nexact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))\n[GOAL]\ncase succ.h\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nn : \u2115\nh\u271d : \u2200 (i : \u2115), i < n \u2192 coeff x i = 0\nk : \u2115\nih : (\u2200 (i : \u2115), i < k \u2192 coeff x i = 0) \u2192 x = (\u2191verschiebung)^[k] (shift x k)\nh : \u2200 (i : \u2115), i < Nat.succ k \u2192 coeff x i = 0\n\u22a2 \u2200 (i : \u2115), i < k + 1 \u2192 coeff x i = 0\n[PROOFSTEP]\nexact h\n[GOAL]\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nhx : x \u2260 0\n\u22a2 \u2203 n x', coeff x' 0 \u2260 0 \u2227 x = (\u2191verschiebung)^[n] x'\n[PROOFSTEP]\nhave hex : \u2203 k : \u2115, x.coeff k \u2260 0 := by\n by_contra' hall\n apply hx\n ext i\n simp only [hall, zero_coeff]\n[GOAL]\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nhx : x \u2260 0\n\u22a2 \u2203 k, coeff x k \u2260 0\n[PROOFSTEP]\nby_contra' hall\n[GOAL]\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nhx : x \u2260 0\nhall : \u2200 (k : \u2115), coeff x k = 0\n\u22a2 False\n[PROOFSTEP]\napply hx\n[GOAL]\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nhx : x \u2260 0\nhall : \u2200 (k : \u2115), coeff x k = 0\n\u22a2 x = 0\n[PROOFSTEP]\next i\n[GOAL]\ncase h\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nhx : x \u2260 0\nhall : \u2200 (k : \u2115), coeff x k = 0\ni : \u2115\n\u22a2 coeff x i = coeff 0 i\n[PROOFSTEP]\nsimp only [hall, zero_coeff]\n[GOAL]\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nhx : x \u2260 0\nhex : \u2203 k, coeff x k \u2260 0\n\u22a2 \u2203 n x', coeff x' 0 \u2260 0 \u2227 x = (\u2191verschiebung)^[n] x'\n[PROOFSTEP]\nlet n := Nat.find hex\n[GOAL]\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nhx : x \u2260 0\nhex : \u2203 k, coeff x k \u2260 0\nn : \u2115 := Nat.find hex\n\u22a2 \u2203 n x', coeff x' 0 \u2260 0 \u2227 x = (\u2191verschiebung)^[n] x'\n[PROOFSTEP]\nuse n, x.shift n\n[GOAL]\ncase h\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nhx : x \u2260 0\nhex : \u2203 k, coeff x k \u2260 0\nn : \u2115 := Nat.find hex\n\u22a2 coeff (shift x n) 0 \u2260 0 \u2227 x = (\u2191verschiebung)^[n] (shift x n)\n[PROOFSTEP]\nrefine' \u27e8Nat.find_spec hex, eq_iterate_verschiebung fun i hi => not_not.mp _\u27e9\n[GOAL]\ncase h\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nhx : x \u2260 0\nhex : \u2203 k, coeff x k \u2260 0\nn : \u2115 := Nat.find hex\ni : \u2115\nhi : i < n\n\u22a2 \u00ac\u00accoeff x i = 0\n[PROOFSTEP]\nexact Nat.find_min hex hi\n[GOAL]\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nx y : \ud835\udd4e R\n\u22a2 x * y = 0 \u2192 x = 0 \u2228 y = 0\n[PROOFSTEP]\ncontrapose!\n[GOAL]\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nx y : \ud835\udd4e R\n\u22a2 x \u2260 0 \u2227 y \u2260 0 \u2192 x * y \u2260 0\n[PROOFSTEP]\nrintro \u27e8ha, hb\u27e9\n[GOAL]\ncase intro\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nx y : \ud835\udd4e R\nha : x \u2260 0\nhb : y \u2260 0\n\u22a2 x * y \u2260 0\n[PROOFSTEP]\nrcases verschiebung_nonzero ha with \u27e8na, wa, hwa0, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\ny : \ud835\udd4e R\nhb : y \u2260 0\nna : \u2115\nwa : \ud835\udd4e R\nhwa0 : coeff wa 0 \u2260 0\nha : (\u2191verschiebung)^[na] wa \u2260 0\n\u22a2 (\u2191verschiebung)^[na] wa * y \u2260 0\n[PROOFSTEP]\nrcases verschiebung_nonzero hb with \u27e8nb, wb, hwb0, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nna : \u2115\nwa : \ud835\udd4e R\nhwa0 : coeff wa 0 \u2260 0\nha : (\u2191verschiebung)^[na] wa \u2260 0\nnb : \u2115\nwb : \ud835\udd4e R\nhwb0 : coeff wb 0 \u2260 0\nhb : (\u2191verschiebung)^[nb] wb \u2260 0\n\u22a2 (\u2191verschiebung)^[na] wa * (\u2191verschiebung)^[nb] wb \u2260 0\n[PROOFSTEP]\nrefine' ne_of_apply_ne (fun x => x.coeff (na + nb)) _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nna : \u2115\nwa : \ud835\udd4e R\nhwa0 : coeff wa 0 \u2260 0\nha : (\u2191verschiebung)^[na] wa \u2260 0\nnb : \u2115\nwb : \ud835\udd4e R\nhwb0 : coeff wb 0 \u2260 0\nhb : (\u2191verschiebung)^[nb] wb \u2260 0\n\u22a2 (fun x => coeff x (na + nb)) ((\u2191verschiebung)^[na] wa * (\u2191verschiebung)^[nb] wb) \u2260 (fun x => coeff x (na + nb)) 0\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nna : \u2115\nwa : \ud835\udd4e R\nhwa0 : coeff wa 0 \u2260 0\nha : (\u2191verschiebung)^[na] wa \u2260 0\nnb : \u2115\nwb : \ud835\udd4e R\nhwb0 : coeff wb 0 \u2260 0\nhb : (\u2191verschiebung)^[nb] wb \u2260 0\n\u22a2 coeff ((\u2191verschiebung)^[na] wa * (\u2191verschiebung)^[nb] wb) (na + nb) \u2260 coeff 0 (na + nb)\n[PROOFSTEP]\nrw [iterate_verschiebung_mul_coeff, zero_coeff]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nna : \u2115\nwa : \ud835\udd4e R\nhwa0 : coeff wa 0 \u2260 0\nha : (\u2191verschiebung)^[na] wa \u2260 0\nnb : \u2115\nwb : \ud835\udd4e R\nhwb0 : coeff wb 0 \u2260 0\nhb : (\u2191verschiebung)^[nb] wb \u2260 0\n\u22a2 coeff wa 0 ^ p ^ nb * coeff wb 0 ^ p ^ na \u2260 0\n[PROOFSTEP]\nexact mul_ne_zero (pow_ne_zero _ hwa0) (pow_ne_zero _ hwb0)\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.WittVector.Domain", "llama_tokens": 4613, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149868676283, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.43019030685880155}} {"text": "[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\nf g : \u03b1 \u2192\u2099+* \u03b2\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\ng : \u03b1 \u2192\u2099+* \u03b2\ntoMulHom\u271d : \u03b1 \u2192\u2099* \u03b2\nmap_zero'\u271d : MulHom.toFun toMulHom\u271d 0 = 0\nmap_add'\u271d : \u2200 (x y : \u03b1), MulHom.toFun toMulHom\u271d (x + y) = MulHom.toFun toMulHom\u271d x + MulHom.toFun toMulHom\u271d y\nh : (fun f => f.toFun) { toMulHom := toMulHom\u271d, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d } = (fun f => f.toFun) g\n\u22a2 { toMulHom := toMulHom\u271d, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\ntoMulHom\u271d\u00b9 : \u03b1 \u2192\u2099* \u03b2\nmap_zero'\u271d\u00b9 : MulHom.toFun toMulHom\u271d\u00b9 0 = 0\nmap_add'\u271d\u00b9 : \u2200 (x y : \u03b1), MulHom.toFun toMulHom\u271d\u00b9 (x + y) = MulHom.toFun toMulHom\u271d\u00b9 x + MulHom.toFun toMulHom\u271d\u00b9 y\ntoMulHom\u271d : \u03b1 \u2192\u2099* \u03b2\nmap_zero'\u271d : MulHom.toFun toMulHom\u271d 0 = 0\nmap_add'\u271d : \u2200 (x y : \u03b1), MulHom.toFun toMulHom\u271d (x + y) = MulHom.toFun toMulHom\u271d x + MulHom.toFun toMulHom\u271d y\nh :\n (fun f => f.toFun) { toMulHom := toMulHom\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9, map_add' := map_add'\u271d\u00b9 } =\n (fun f => f.toFun) { toMulHom := toMulHom\u271d, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d }\n\u22a2 { toMulHom := toMulHom\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9, map_add' := map_add'\u271d\u00b9 } =\n { toMulHom := toMulHom\u271d, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toMulHom\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\ntoMulHom\u271d\u00b9 : \u03b1 \u2192\u2099* \u03b2\nmap_zero'\u271d\u00b9 : MulHom.toFun toMulHom\u271d\u00b9 0 = 0\nmap_add'\u271d\u00b9 : \u2200 (x y : \u03b1), MulHom.toFun toMulHom\u271d\u00b9 (x + y) = MulHom.toFun toMulHom\u271d\u00b9 x + MulHom.toFun toMulHom\u271d\u00b9 y\ntoMulHom\u271d : \u03b1 \u2192\u2099* \u03b2\nmap_zero'\u271d : MulHom.toFun toMulHom\u271d 0 = 0\nmap_add'\u271d : \u2200 (x y : \u03b1), MulHom.toFun toMulHom\u271d (x + y) = MulHom.toFun toMulHom\u271d x + MulHom.toFun toMulHom\u271d y\nh :\n (fun f => f.toFun) { toMulHom := toMulHom\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9, map_add' := map_add'\u271d\u00b9 } =\n (fun f => f.toFun) { toMulHom := toMulHom\u271d, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d }\n\u22a2 toMulHom\u271d\u00b9 = toMulHom\u271d\n[PROOFSTEP]\napply FunLike.coe_injective'\n[GOAL]\ncase mk.mk.e_toMulHom.a\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d : NonUnitalNonAssocSemiring \u03b2\ntoMulHom\u271d\u00b9 : \u03b1 \u2192\u2099* \u03b2\nmap_zero'\u271d\u00b9 : MulHom.toFun toMulHom\u271d\u00b9 0 = 0\nmap_add'\u271d\u00b9 : \u2200 (x y : \u03b1), MulHom.toFun toMulHom\u271d\u00b9 (x + y) = MulHom.toFun toMulHom\u271d\u00b9 x + MulHom.toFun toMulHom\u271d\u00b9 y\ntoMulHom\u271d : \u03b1 \u2192\u2099* \u03b2\nmap_zero'\u271d : MulHom.toFun toMulHom\u271d 0 = 0\nmap_add'\u271d : \u2200 (x y : \u03b1), MulHom.toFun toMulHom\u271d (x + y) = MulHom.toFun toMulHom\u271d x + MulHom.toFun toMulHom\u271d y\nh :\n (fun f => f.toFun) { toMulHom := toMulHom\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9, map_add' := map_add'\u271d\u00b9 } =\n (fun f => f.toFun) { toMulHom := toMulHom\u271d, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d }\n\u22a2 \u2191toMulHom\u271d\u00b9 = \u2191toMulHom\u271d\n[PROOFSTEP]\nexact h\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\n\u22a2 \u03b1 \u2192\u2099+* \u03b1\n[PROOFSTEP]\nrefine' { toFun := id .. }\n[GOAL]\ncase refine'_1\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\n\u22a2 \u2200 (x y : \u03b1), id (x * y) = id x * id y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\n\u22a2 MulHom.toFun { toFun := id, map_mul' := (_ : \u2200 (x y : \u03b1), id (x * y) = id x * id y) } 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\n\u22a2 \u2200 (x y : \u03b1),\n MulHom.toFun { toFun := id, map_mul' := (_ : \u2200 (x y : \u03b1), id (x * y) = id x * id y) } (x + y) =\n MulHom.toFun { toFun := id, map_mul' := (_ : \u2200 (x y : \u03b1), id (x * y) = id x * id y) } x +\n MulHom.toFun { toFun := id, map_mul' := (_ : \u2200 (x y : \u03b1), id (x * y) = id x * id y) } y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 id (x\u271d * y\u271d) = id x\u271d * id y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\n\u22a2 MulHom.toFun { toFun := id, map_mul' := (_ : \u2200 (x y : \u03b1), id (x * y) = id (x * y)) } 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_3\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 MulHom.toFun { toFun := id, map_mul' := (_ : \u2200 (x y : \u03b1), id (x * y) = id (x * y)) } (x\u271d + y\u271d) =\n MulHom.toFun { toFun := id, map_mul' := (_ : \u2200 (x y : \u03b1), id (x * y) = id (x * y)) } x\u271d +\n MulHom.toFun { toFun := id, map_mul' := (_ : \u2200 (x y : \u03b1), id (x * y) = id (x * y)) } y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b3\ng\u271d : \u03b2 \u2192\u2099+* \u03b3\nf : \u03b1 \u2192\u2099+* \u03b2\ng : \u03b2 \u2192\u2099+* \u03b3\n\u22a2 comp g 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase a\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b3\ng\u271d : \u03b2 \u2192\u2099+* \u03b3\nf : \u03b1 \u2192\u2099+* \u03b2\ng : \u03b2 \u2192\u2099+* \u03b3\nx\u271d : \u03b1\n\u22a2 \u2191(comp g 0) x\u271d = \u21910 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b3\ng : \u03b2 \u2192\u2099+* \u03b3\nf\u271d f : \u03b1 \u2192\u2099+* \u03b2\n\u22a2 comp 0 f = 0\n[PROOFSTEP]\next\n[GOAL]\ncase a\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b3\ng : \u03b2 \u2192\u2099+* \u03b3\nf\u271d f : \u03b1 \u2192\u2099+* \u03b2\nx\u271d : \u03b1\n\u22a2 \u2191(comp 0 f) x\u271d = \u21910 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b2\ninst\u271d : NonUnitalNonAssocSemiring \u03b3\ng\u271d : \u03b2 \u2192\u2099+* \u03b3\nf : \u03b1 \u2192\u2099+* \u03b2\ng : \u03b2 \u2192\u2099+* \u03b3\nf\u2081 f\u2082 : \u03b1 \u2192\u2099+* \u03b2\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\nx : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 x) = \u2191g (\u2191f\u2082 x)\n[PROOFSTEP]\nrw [\u2190 comp_apply, h, comp_apply]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : NonAssocSemiring \u03b1\ninst\u271d\u00b9 : NonAssocSemiring \u03b2\ninst\u271d : RingHomClass F \u03b1 \u03b2\nf : F\na : \u03b1\n\u22a2 \u2191f (bit1 a) = bit1 (\u2191f a)\n[PROOFSTEP]\nsimp [bit1]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\nf g : \u03b1 \u2192+* \u03b2\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\ng : \u03b1 \u2192+* \u03b2\ntoMonoidHom\u271d : \u03b1 \u2192* \u03b2\nmap_zero'\u271d : OneHom.toFun (\u2191toMonoidHom\u271d) 0 = 0\nmap_add'\u271d :\n \u2200 (x y : \u03b1), OneHom.toFun (\u2191toMonoidHom\u271d) (x + y) = OneHom.toFun (\u2191toMonoidHom\u271d) x + OneHom.toFun (\u2191toMonoidHom\u271d) y\nh :\n (fun f => f.toFun) { toMonoidHom := toMonoidHom\u271d, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d } =\n (fun f => f.toFun) g\n\u22a2 { toMonoidHom := toMonoidHom\u271d, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\ntoMonoidHom\u271d\u00b9 : \u03b1 \u2192* \u03b2\nmap_zero'\u271d\u00b9 : OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) 0 = 0\nmap_add'\u271d\u00b9 :\n \u2200 (x y : \u03b1), OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) (x + y) = OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) x + OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) y\ntoMonoidHom\u271d : \u03b1 \u2192* \u03b2\nmap_zero'\u271d : OneHom.toFun (\u2191toMonoidHom\u271d) 0 = 0\nmap_add'\u271d :\n \u2200 (x y : \u03b1), OneHom.toFun (\u2191toMonoidHom\u271d) (x + y) = OneHom.toFun (\u2191toMonoidHom\u271d) x + OneHom.toFun (\u2191toMonoidHom\u271d) y\nh :\n (fun f => f.toFun) { toMonoidHom := toMonoidHom\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9, map_add' := map_add'\u271d\u00b9 } =\n (fun f => f.toFun) { toMonoidHom := toMonoidHom\u271d, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d }\n\u22a2 { toMonoidHom := toMonoidHom\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9, map_add' := map_add'\u271d\u00b9 } =\n { toMonoidHom := toMonoidHom\u271d, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toMonoidHom\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\ntoMonoidHom\u271d\u00b9 : \u03b1 \u2192* \u03b2\nmap_zero'\u271d\u00b9 : OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) 0 = 0\nmap_add'\u271d\u00b9 :\n \u2200 (x y : \u03b1), OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) (x + y) = OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) x + OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) y\ntoMonoidHom\u271d : \u03b1 \u2192* \u03b2\nmap_zero'\u271d : OneHom.toFun (\u2191toMonoidHom\u271d) 0 = 0\nmap_add'\u271d :\n \u2200 (x y : \u03b1), OneHom.toFun (\u2191toMonoidHom\u271d) (x + y) = OneHom.toFun (\u2191toMonoidHom\u271d) x + OneHom.toFun (\u2191toMonoidHom\u271d) y\nh :\n (fun f => f.toFun) { toMonoidHom := toMonoidHom\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9, map_add' := map_add'\u271d\u00b9 } =\n (fun f => f.toFun) { toMonoidHom := toMonoidHom\u271d, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d }\n\u22a2 toMonoidHom\u271d\u00b9 = toMonoidHom\u271d\n[PROOFSTEP]\napply FunLike.coe_injective'\n[GOAL]\ncase mk.mk.e_toMonoidHom.a\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\ntoMonoidHom\u271d\u00b9 : \u03b1 \u2192* \u03b2\nmap_zero'\u271d\u00b9 : OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) 0 = 0\nmap_add'\u271d\u00b9 :\n \u2200 (x y : \u03b1), OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) (x + y) = OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) x + OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) y\ntoMonoidHom\u271d : \u03b1 \u2192* \u03b2\nmap_zero'\u271d : OneHom.toFun (\u2191toMonoidHom\u271d) 0 = 0\nmap_add'\u271d :\n \u2200 (x y : \u03b1), OneHom.toFun (\u2191toMonoidHom\u271d) (x + y) = OneHom.toFun (\u2191toMonoidHom\u271d) x + OneHom.toFun (\u2191toMonoidHom\u271d) y\nh :\n (fun f => f.toFun) { toMonoidHom := toMonoidHom\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9, map_add' := map_add'\u271d\u00b9 } =\n (fun f => f.toFun) { toMonoidHom := toMonoidHom\u271d, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d }\n\u22a2 \u2191toMonoidHom\u271d\u00b9 = \u2191toMonoidHom\u271d\n[PROOFSTEP]\nexact h\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\nf : \u03b1 \u2192+* \u03b2\n\u22a2 \u2191(toMonoidWithZeroHom f) = \u2191f\n[PROOFSTEP]\nrfl\n[GOAL]\nF\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\nf\u271d : \u03b1 \u2192+* \u03b2\nx y : \u03b1\nF : Type u_5\ninst\u271d\u00b9 : RingHomClass F \u03b1 \u03b2\nf : F\np : Prop\ninst\u271d : Decidable p\n\u22a2 \u2191f (if p then 0 else 1) = if p then 0 else 1\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nF\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\nf\u271d : \u03b1 \u2192+* \u03b2\nx y : \u03b1\nF : Type u_5\ninst\u271d\u00b9 : RingHomClass F \u03b1 \u03b2\nf : F\np : Prop\ninst\u271d : Decidable p\nh : p\n\u22a2 \u2191f (if p then 0 else 1) = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nF\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\nf\u271d : \u03b1 \u2192+* \u03b2\nx y : \u03b1\nF : Type u_5\ninst\u271d\u00b9 : RingHomClass F \u03b1 \u03b2\nf : F\np : Prop\ninst\u271d : Decidable p\nh : \u00acp\n\u22a2 \u2191f (if p then 0 else 1) = 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nF\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\nf\u271d : \u03b1 \u2192+* \u03b2\nx y : \u03b1\nF : Type u_5\ninst\u271d\u00b9 : RingHomClass F \u03b1 \u03b2\nf : F\np : Prop\ninst\u271d : Decidable p\n\u22a2 \u2191f (if p then 1 else 0) = if p then 1 else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nF\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\nf\u271d : \u03b1 \u2192+* \u03b2\nx y : \u03b1\nF : Type u_5\ninst\u271d\u00b9 : RingHomClass F \u03b1 \u03b2\nf : F\np : Prop\ninst\u271d : Decidable p\nh : p\n\u22a2 \u2191f (if p then 1 else 0) = 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nF\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\nf\u271d : \u03b1 \u2192+* \u03b2\nx y : \u03b1\nF : Type u_5\ninst\u271d\u00b9 : RingHomClass F \u03b1 \u03b2\nf : F\np : Prop\ninst\u271d : Decidable p\nh : \u00acp\n\u22a2 \u2191f (if p then 1 else 0) = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\nf : \u03b1 \u2192+* \u03b2\nx y : \u03b1\n\u22a2 0 = 1 \u2194 \u2191f 1 = 0\n[PROOFSTEP]\nrw [map_one, eq_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b2 : NonAssocSemiring \u03b1\nx\u271d\u00b9 : NonAssocSemiring \u03b2\nf : \u03b1 \u2192+* \u03b2\nx\u271d y : \u03b1\nh : \u2191f 1 = 0\nx : \u03b1\n\u22a2 \u2191f x = 0\n[PROOFSTEP]\nrw [\u2190 mul_one x, map_mul, h, mul_zero]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b3 : NonAssocSemiring \u03b1\nx\u271d\u00b2 : NonAssocSemiring \u03b2\nf : \u03b1 \u2192+* \u03b2\nx\u271d\u00b9 y\u271d : \u03b1\nh : \u2200 (x : \u03b1), \u2191f x = 0\ny : \u03b2\nx\u271d : y \u2208 Set.range \u2191f\nx : \u03b1\nhx : \u2191f x = y\n\u22a2 y \u2208 {0}\n[PROOFSTEP]\nsimp [\u2190 hx, h x]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\nf : \u03b1 \u2192+* \u03b2\nx y\u271d : \u03b1\nh : \u2200 (x : \u03b1), \u2191f x = 0\ny : \u03b2\nhy : y \u2208 {0}\n\u22a2 \u2191f 0 = y\n[PROOFSTEP]\nsimpa using hy.symm\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\nx\u271d : NonAssocSemiring \u03b2\nf : \u03b1 \u2192+* \u03b2\nx y : \u03b1\ninst\u271d : Nontrivial \u03b2\nh : 1 = 0\n\u22a2 \u2191f 1 = 0\n[PROOFSTEP]\nrw [h, map_zero]\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\u271d\nx\u271d : NonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonAssocSemiring \u03b1\n\u22a2 \u03b1 \u2192+* \u03b1\n[PROOFSTEP]\nrefine' { toFun := _root_.id .. }\n[GOAL]\ncase refine'_1\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\u271d\nx\u271d : NonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonAssocSemiring \u03b1\n\u22a2 _root_.id 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\u271d\nx\u271d : NonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonAssocSemiring \u03b1\n\u22a2 \u2200 (x y : \u03b1),\n OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } (x * y) =\n OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } x *\n OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\u271d\nx\u271d : NonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonAssocSemiring \u03b1\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := _root_.id, map_one' := ?refine'_1 },\n map_mul' :=\n (_ :\n \u2200 (x y : \u03b1),\n OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } (x * y) =\n OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } x *\n OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } y) })\n 0 =\n 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_4\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\u271d\nx\u271d : NonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonAssocSemiring \u03b1\n\u22a2 \u2200 (x y : \u03b1),\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := _root_.id, map_one' := ?refine'_1 },\n map_mul' :=\n (_ :\n \u2200 (x y : \u03b1),\n OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } (x * y) =\n OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } x *\n OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } y) })\n (x + y) =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := _root_.id, map_one' := ?refine'_1 },\n map_mul' :=\n (_ :\n \u2200 (x y : \u03b1),\n OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } (x * y) =\n OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } x *\n OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } y) })\n x +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := _root_.id, map_one' := ?refine'_1 },\n map_mul' :=\n (_ :\n \u2200 (x y : \u03b1),\n OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } (x * y) =\n OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } x *\n OneHom.toFun { toFun := _root_.id, map_one' := ?refine'_1 } y) })\n y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\u271d\nx\u271d : NonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonAssocSemiring \u03b1\n\u22a2 _root_.id 1 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b2 : NonAssocSemiring \u03b1\u271d\nx\u271d\u00b9 : NonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonAssocSemiring \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x\u271d * y\u271d) =\n OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } x\u271d *\n OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_3\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b9 : NonAssocSemiring \u03b1\u271d\nx\u271d : NonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonAssocSemiring \u03b1\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u03b1),\n OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y) =\n OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y)) })\n 0 =\n 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_4\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b2 : NonAssocSemiring \u03b1\u271d\nx\u271d\u00b9 : NonAssocSemiring \u03b2\n\u03b1 : Type u_5\ninst\u271d : NonAssocSemiring \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u03b1),\n OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y) =\n OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y)) })\n (x\u271d + y\u271d) =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u03b1),\n OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y) =\n OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y)) })\n x\u271d +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u03b1),\n OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y) =\n OneHom.toFun { toFun := _root_.id, map_one' := (_ : _root_.id 1 = _root_.id 1) } (x * y)) })\n y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b2 : NonAssocSemiring \u03b1\nx\u271d\u00b9 : NonAssocSemiring \u03b2\nx\u271d : NonAssocSemiring \u03b3\ng : \u03b2 \u2192+* \u03b3\nf : \u03b1 \u2192+* \u03b2\nsrc\u271d : \u03b1 \u2192\u2099+* \u03b3 := NonUnitalRingHom.comp (toNonUnitalRingHom g) (toNonUnitalRingHom f)\n\u22a2 (\u2191g \u2218 \u2191f) 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b2 : NonAssocSemiring \u03b1\nx\u271d\u00b9 : NonAssocSemiring \u03b2\nx\u271d : NonAssocSemiring \u03b3\ng : \u03b2 \u2192+* \u03b3\nf\u2081 f\u2082 : \u03b1 \u2192+* \u03b2\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\nx : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 x) = \u2191g (\u2191f\u2082 x)\n[PROOFSTEP]\nrw [\u2190 comp_apply, h, comp_apply]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Ring \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : Ring \u03b2\nf : \u03b2 \u2192+* \u03b1\nhf : Injective \u2191f\n\u22a2 IsDomain \u03b2\n[PROOFSTEP]\nhaveI := pullback_nonzero f f.map_zero f.map_one\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Ring \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : Ring \u03b2\nf : \u03b2 \u2192+* \u03b1\nhf : Injective \u2191f\nthis : Nontrivial \u03b2\n\u22a2 IsDomain \u03b2\n[PROOFSTEP]\nhaveI := IsRightCancelMulZero.to_noZeroDivisors \u03b1\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Ring \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : Ring \u03b2\nf : \u03b2 \u2192+* \u03b1\nhf : Injective \u2191f\nthis\u271d : Nontrivial \u03b2\nthis : NoZeroDivisors \u03b1\n\u22a2 IsDomain \u03b2\n[PROOFSTEP]\nhaveI := hf.noZeroDivisors f f.map_zero f.map_mul\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Ring \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : Ring \u03b2\nf : \u03b2 \u2192+* \u03b1\nhf : Injective \u2191f\nthis\u271d\u00b9 : Nontrivial \u03b2\nthis\u271d : NoZeroDivisors \u03b1\nthis : NoZeroDivisors \u03b2\n\u22a2 IsDomain \u03b2\n[PROOFSTEP]\nexact NoZeroDivisors.to_isDomain \u03b2\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : CommRing \u03b2\nf : \u03b2 \u2192+ \u03b1\nh : \u2200 (x : \u03b2), \u2191f (x * x) = \u2191f x * \u2191f x\nh_two : 2 \u2260 0\nh_one : \u2191f 1 = 1\nx y : \u03b2\n\u22a2 OneHom.toFun { toFun := f.toFun, map_one' := h_one } (x * y) =\n OneHom.toFun { toFun := f.toFun, map_one' := h_one } x * OneHom.toFun { toFun := f.toFun, map_one' := h_one } y\n[PROOFSTEP]\nhave hxy := h (x + y)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : CommRing \u03b2\nf : \u03b2 \u2192+ \u03b1\nh : \u2200 (x : \u03b2), \u2191f (x * x) = \u2191f x * \u2191f x\nh_two : 2 \u2260 0\nh_one : \u2191f 1 = 1\nx y : \u03b2\nhxy : \u2191f ((x + y) * (x + y)) = \u2191f (x + y) * \u2191f (x + y)\n\u22a2 OneHom.toFun { toFun := f.toFun, map_one' := h_one } (x * y) =\n OneHom.toFun { toFun := f.toFun, map_one' := h_one } x * OneHom.toFun { toFun := f.toFun, map_one' := h_one } y\n[PROOFSTEP]\nrw [mul_add, add_mul, add_mul, f.map_add, f.map_add, f.map_add, f.map_add, h x, h y, add_mul, mul_add, mul_add, \u2190\n sub_eq_zero, add_comm (f x * f x + f (y * x)), \u2190 sub_sub, \u2190 sub_sub, \u2190 sub_sub, mul_comm y x, mul_comm (f y) (f x)] at\n hxy \n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : CommRing \u03b2\nf : \u03b2 \u2192+ \u03b1\nh : \u2200 (x : \u03b2), \u2191f (x * x) = \u2191f x * \u2191f x\nh_two : 2 \u2260 0\nh_one : \u2191f 1 = 1\nx y : \u03b2\nhxy\u271d : \u2191f x * \u2191f x + \u2191f (y * x) + (\u2191f (x * y) + \u2191f y * \u2191f y) = \u2191f x * \u2191f x + \u2191f x * \u2191f y + (\u2191f y * \u2191f x + \u2191f y * \u2191f y)\nhxy : \u2191f (x * y) + \u2191f y * \u2191f y + (\u2191f x * \u2191f x + \u2191f (x * y)) - \u2191f x * \u2191f x - \u2191f x * \u2191f y - \u2191f x * \u2191f y - \u2191f y * \u2191f y = 0\n\u22a2 OneHom.toFun { toFun := f.toFun, map_one' := h_one } (x * y) =\n OneHom.toFun { toFun := f.toFun, map_one' := h_one } x * OneHom.toFun { toFun := f.toFun, map_one' := h_one } y\n[PROOFSTEP]\nsimp only [add_assoc, add_sub_assoc, add_sub_cancel'_right] at hxy \n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : CommRing \u03b2\nf : \u03b2 \u2192+ \u03b1\nh : \u2200 (x : \u03b2), \u2191f (x * x) = \u2191f x * \u2191f x\nh_two : 2 \u2260 0\nh_one : \u2191f 1 = 1\nx y : \u03b2\nhxy\u271d : \u2191f x * \u2191f x + \u2191f (y * x) + (\u2191f (x * y) + \u2191f y * \u2191f y) = \u2191f x * \u2191f x + \u2191f x * \u2191f y + (\u2191f y * \u2191f x + \u2191f y * \u2191f y)\nhxy : \u2191f (x * y) + (\u2191f (x * y) - \u2191f x * \u2191f y - \u2191f x * \u2191f y) = 0\n\u22a2 OneHom.toFun { toFun := f.toFun, map_one' := h_one } (x * y) =\n OneHom.toFun { toFun := f.toFun, map_one' := h_one } x * OneHom.toFun { toFun := f.toFun, map_one' := h_one } y\n[PROOFSTEP]\nrw [sub_sub, \u2190 two_mul, \u2190 add_sub_assoc, \u2190 two_mul, \u2190 mul_sub, mul_eq_zero (M\u2080 := \u03b1), sub_eq_zero,\n or_iff_not_imp_left] at hxy \n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : CommRing \u03b2\nf : \u03b2 \u2192+ \u03b1\nh : \u2200 (x : \u03b2), \u2191f (x * x) = \u2191f x * \u2191f x\nh_two : 2 \u2260 0\nh_one : \u2191f 1 = 1\nx y : \u03b2\nhxy\u271d : \u2191f x * \u2191f x + \u2191f (y * x) + (\u2191f (x * y) + \u2191f y * \u2191f y) = \u2191f x * \u2191f x + \u2191f x * \u2191f y + (\u2191f y * \u2191f x + \u2191f y * \u2191f y)\nhxy : \u00ac2 = 0 \u2192 \u2191f (x * y) = \u2191f x * \u2191f y\n\u22a2 OneHom.toFun { toFun := f.toFun, map_one' := h_one } (x * y) =\n OneHom.toFun { toFun := f.toFun, map_one' := h_one } x * OneHom.toFun { toFun := f.toFun, map_one' := h_one } y\n[PROOFSTEP]\nexact hxy h_two\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : CommRing \u03b2\nf : \u03b2 \u2192+ \u03b1\nh : \u2200 (x : \u03b2), \u2191f (x * x) = \u2191f x * \u2191f x\nh_two : 2 \u2260 0\nh_one : \u2191f 1 = 1\n\u22a2 \u2191(mkRingHomOfMulSelfOfTwoNeZero f h h_two h_one) = f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : CommRing \u03b2\nf : \u03b2 \u2192+ \u03b1\nh : \u2200 (x : \u03b2), \u2191f (x * x) = \u2191f x * \u2191f x\nh_two : 2 \u2260 0\nh_one : \u2191f 1 = 1\nx\u271d : \u03b2\n\u22a2 \u2191\u2191(mkRingHomOfMulSelfOfTwoNeZero f h h_two h_one) x\u271d = \u2191f x\u271d\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Hom.Ring", "llama_tokens": 12953, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149868676283, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.43019030685880155}} {"text": "[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 negY W x\u2081 (negY W x\u2081 y\u2081) = y\u2081\n[PROOFSTEP]\nsimp only [negY]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 -(-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) - W.a\u2081 * x\u2081 - W.a\u2083 = y\u2081\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 negY (baseChange W A) (\u2191(algebraMap R A) x\u2081) (\u2191(algebraMap R A) y\u2081) = \u2191(algebraMap R A) (negY W x\u2081 y\u2081)\n[PROOFSTEP]\nsimp only [negY]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 -\u2191(algebraMap R A) y\u2081 - (baseChange W A).a\u2081 * \u2191(algebraMap R A) x\u2081 - (baseChange W A).a\u2083 =\n \u2191(algebraMap R A) (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)\n[PROOFSTEP]\nmap_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 -\u2191(algebraMap R A) y\u2081 - (baseChange W A).a\u2081 * \u2191(algebraMap R A) x\u2081 - (baseChange W A).a\u2083 =\n -\u2191(algebraMap R A) y\u2081 - \u2191(algebraMap R A) W.a\u2081 * \u2191(algebraMap R A) x\u2081 - \u2191(algebraMap R A) W.a\u2083\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081\u271d x\u2082 y\u2081\u271d y\u2082 L : R\nx\u2081 y\u2081 : A\n\u22a2 negY (baseChange W B) (\u2191(algebraMap A B) x\u2081) (\u2191(algebraMap A B) y\u2081) = \u2191(algebraMap A B) (negY (baseChange W A) x\u2081 y\u2081)\n[PROOFSTEP]\nrw [\u2190 baseChange_negY, baseChange_baseChange]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 eval x\u2081 (eval (\u2191C y\u2081) (negPolynomial W)) = negY W x\u2081 y\u2081\n[PROOFSTEP]\nrw [negY, sub_sub, negPolynomial]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 eval x\u2081 (eval (\u2191C y\u2081) (-Y - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083))) = -y\u2081 - (W.a\u2081 * x\u2081 + W.a\u2083)\n[PROOFSTEP]\neval_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 XYIdeal W x\u2081 (\u2191C y\u2081) = XYIdeal W x\u2081 (linePolynomial x\u2081 y\u2081 L)\n[PROOFSTEP]\nsimp only [XYIdeal, XClass, YClass, linePolynomial]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 span {\u2191(CoordinateRing.mk W) (\u2191C (Y - \u2191C x\u2081)), \u2191(CoordinateRing.mk W) (Y - \u2191C (\u2191C y\u2081))} =\n span {\u2191(CoordinateRing.mk W) (\u2191C (Y - \u2191C x\u2081)), \u2191(CoordinateRing.mk W) (Y - \u2191C (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081))}\n[PROOFSTEP]\nrw [\u2190 span_pair_add_mul_right <| CoordinateRing.mk W <| C <| C <| -L, \u2190 _root_.map_mul, \u2190 map_add]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 span\n {\u2191(CoordinateRing.mk W) (\u2191C (Y - \u2191C x\u2081)),\n \u2191(CoordinateRing.mk W) (Y - \u2191C (\u2191C y\u2081) + \u2191C (Y - \u2191C x\u2081) * \u2191C (\u2191C (-L)))} =\n span {\u2191(CoordinateRing.mk W) (\u2191C (Y - \u2191C x\u2081)), \u2191(CoordinateRing.mk W) (Y - \u2191C (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081))}\n[PROOFSTEP]\napply congr_arg (_ \u2218 _ \u2218 _ \u2218 _)\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 Y - \u2191C (\u2191C y\u2081) + \u2191C (Y - \u2191C x\u2081) * \u2191C (\u2191C (-L)) = Y - \u2191C (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081)\n[PROOFSTEP]\nC_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 Y - \u2191C (\u2191C y\u2081) + (\u2191C Y - \u2191C (\u2191C x\u2081)) * -\u2191C (\u2191C L) = Y - (\u2191C (\u2191C L) * (\u2191C Y - \u2191C (\u2191C x\u2081)) + \u2191C (\u2191C y\u2081))\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 \u2191C (addPolynomial W x\u2081 y\u2081 L) =\n (Y - \u2191C (linePolynomial x\u2081 y\u2081 L)) * (negPolynomial W - \u2191C (linePolynomial x\u2081 y\u2081 L)) + WeierstrassCurve.polynomial W\n[PROOFSTEP]\nrw [addPolynomial, linePolynomial, WeierstrassCurve.polynomial, negPolynomial]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 \u2191C\n (eval (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081)\n (Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086))) =\n (Y - \u2191C (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081)) * (-Y - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - \u2191C (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081)) +\n (Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086))\n[PROOFSTEP]\neval_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 \u2191C\n ((\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081) ^ 2 + (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081) -\n (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n (Y - \u2191C (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081)) * (-Y - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - \u2191C (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081)) +\n (Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086))\n[PROOFSTEP]\nC_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 (\u2191C (\u2191C L) * (\u2191C Y - \u2191C (\u2191C x\u2081)) + \u2191C (\u2191C y\u2081)) ^ 2 +\n (\u2191C (\u2191C W.a\u2081) * \u2191C Y + \u2191C (\u2191C W.a\u2083)) * (\u2191C (\u2191C L) * (\u2191C Y - \u2191C (\u2191C x\u2081)) + \u2191C (\u2191C y\u2081)) -\n (\u2191C Y ^ 3 + \u2191C (\u2191C W.a\u2082) * \u2191C Y ^ 2 + \u2191C (\u2191C W.a\u2084) * \u2191C Y + \u2191C (\u2191C W.a\u2086)) =\n (Y - (\u2191C (\u2191C L) * (\u2191C Y - \u2191C (\u2191C x\u2081)) + \u2191C (\u2191C y\u2081))) *\n (-Y - (\u2191C (\u2191C W.a\u2081) * \u2191C Y + \u2191C (\u2191C W.a\u2083)) - (\u2191C (\u2191C L) * (\u2191C Y - \u2191C (\u2191C x\u2081)) + \u2191C (\u2191C y\u2081))) +\n (Y ^ 2 + (\u2191C (\u2191C W.a\u2081) * \u2191C Y + \u2191C (\u2191C W.a\u2083)) * Y -\n (\u2191C Y ^ 3 + \u2191C (\u2191C W.a\u2082) * \u2191C Y ^ 2 + \u2191C (\u2191C W.a\u2084) * \u2191C Y + \u2191C (\u2191C W.a\u2086)))\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 \u2191C (addPolynomial W x\u2081 y\u2081 L) - (Y - \u2191C (linePolynomial x\u2081 y\u2081 L)) * (negPolynomial W - \u2191C (linePolynomial x\u2081 y\u2081 L)) =\n WeierstrassCurve.polynomial W * 1\n[PROOFSTEP]\nrw [W.C_addPolynomial, add_sub_cancel', mul_one]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 addPolynomial W x\u2081 y\u2081 L =\n -Cubic.toPoly\n { a := 1, b := -L ^ 2 - W.a\u2081 * L + W.a\u2082,\n c := 2 * x\u2081 * L ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * L + (-W.a\u2081 * y\u2081 + W.a\u2084),\n d := -x\u2081 ^ 2 * L ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * L - (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }\n[PROOFSTEP]\nrw [addPolynomial, linePolynomial, WeierstrassCurve.polynomial, Cubic.toPoly]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 eval (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081)\n (Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n -(\u2191C\n { a := 1, b := -L ^ 2 - W.a\u2081 * L + W.a\u2082,\n c := 2 * x\u2081 * L ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * L + (-W.a\u2081 * y\u2081 + W.a\u2084),\n d := -x\u2081 ^ 2 * L ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * L - (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.a *\n Y ^ 3 +\n \u2191C\n { a := 1, b := -L ^ 2 - W.a\u2081 * L + W.a\u2082,\n c := 2 * x\u2081 * L ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * L + (-W.a\u2081 * y\u2081 + W.a\u2084),\n d := -x\u2081 ^ 2 * L ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * L - (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.b *\n Y ^ 2 +\n \u2191C\n { a := 1, b := -L ^ 2 - W.a\u2081 * L + W.a\u2082,\n c := 2 * x\u2081 * L ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * L + (-W.a\u2081 * y\u2081 + W.a\u2084),\n d := -x\u2081 ^ 2 * L ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * L - (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.c *\n Y +\n \u2191C\n { a := 1, b := -L ^ 2 - W.a\u2081 * L + W.a\u2082,\n c := 2 * x\u2081 * L ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * L + (-W.a\u2081 * y\u2081 + W.a\u2084),\n d := -x\u2081 ^ 2 * L ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * L - (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.d)\n[PROOFSTEP]\neval_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081) ^ 2 + (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081) -\n (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086) =\n -(\u2191C 1 * Y ^ 3 + \u2191C (-L ^ 2 - W.a\u2081 * L + W.a\u2082) * Y ^ 2 +\n \u2191C (2 * x\u2081 * L ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * L + (-W.a\u2081 * y\u2081 + W.a\u2084)) * Y +\n \u2191C (-x\u2081 ^ 2 * L ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * L - (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086)))\n[PROOFSTEP]\nC_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081) ^ 2 + (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081) -\n (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086) =\n -(1 * Y ^ 3 + (-\u2191C L ^ 2 - \u2191C W.a\u2081 * \u2191C L + \u2191C W.a\u2082) * Y ^ 2 +\n (2 * \u2191C x\u2081 * \u2191C L ^ 2 + (\u2191C W.a\u2081 * \u2191C x\u2081 - 2 * \u2191C y\u2081 - \u2191C W.a\u2083) * \u2191C L + (-\u2191C W.a\u2081 * \u2191C y\u2081 + \u2191C W.a\u2084)) * Y +\n (-\u2191C x\u2081 ^ 2 * \u2191C L ^ 2 + (2 * \u2191C x\u2081 * \u2191C y\u2081 + \u2191C W.a\u2083 * \u2191C x\u2081) * \u2191C L -\n (\u2191C y\u2081 ^ 2 + \u2191C W.a\u2083 * \u2191C y\u2081 - \u2191C W.a\u2086)))\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 addX (baseChange W A) (\u2191(algebraMap R A) x\u2081) (\u2191(algebraMap R A) x\u2082) (\u2191(algebraMap R A) L) =\n \u2191(algebraMap R A) (addX W x\u2081 x\u2082 L)\n[PROOFSTEP]\nsimp only [addX]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 \u2191(algebraMap R A) L ^ 2 + (baseChange W A).a\u2081 * \u2191(algebraMap R A) L - (baseChange W A).a\u2082 - \u2191(algebraMap R A) x\u2081 -\n \u2191(algebraMap R A) x\u2082 =\n \u2191(algebraMap R A) (L ^ 2 + W.a\u2081 * L - W.a\u2082 - x\u2081 - x\u2082)\n[PROOFSTEP]\nmap_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 \u2191(algebraMap R A) L ^ 2 + (baseChange W A).a\u2081 * \u2191(algebraMap R A) L - (baseChange W A).a\u2082 - \u2191(algebraMap R A) x\u2081 -\n \u2191(algebraMap R A) x\u2082 =\n \u2191(algebraMap R A) L ^ 2 + \u2191(algebraMap R A) W.a\u2081 * \u2191(algebraMap R A) L - \u2191(algebraMap R A) W.a\u2082 -\n \u2191(algebraMap R A) x\u2081 -\n \u2191(algebraMap R A) x\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081\u271d x\u2082\u271d y\u2081 y\u2082 L\u271d : R\nx\u2081 x\u2082 L : A\n\u22a2 addX (baseChange W B) (\u2191(algebraMap A B) x\u2081) (\u2191(algebraMap A B) x\u2082) (\u2191(algebraMap A B) L) =\n \u2191(algebraMap A B) (addX (baseChange W A) x\u2081 x\u2082 L)\n[PROOFSTEP]\nrw [\u2190 baseChange_addX, baseChange_baseChange]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 addY' (baseChange W A) (\u2191(algebraMap R A) x\u2081) (\u2191(algebraMap R A) x\u2082) (\u2191(algebraMap R A) y\u2081) (\u2191(algebraMap R A) L) =\n \u2191(algebraMap R A) (addY' W x\u2081 x\u2082 y\u2081 L)\n[PROOFSTEP]\nsimp only [addY', baseChange_addX]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 \u2191(algebraMap R A) L * (\u2191(algebraMap R A) (addX W x\u2081 x\u2082 L) - \u2191(algebraMap R A) x\u2081) + \u2191(algebraMap R A) y\u2081 =\n \u2191(algebraMap R A) (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081)\n[PROOFSTEP]\nmap_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082 L\u271d : R\nx\u2081 x\u2082 y\u2081 L : A\n\u22a2 addY' (baseChange W B) (\u2191(algebraMap A B) x\u2081) (\u2191(algebraMap A B) x\u2082) (\u2191(algebraMap A B) y\u2081) (\u2191(algebraMap A B) L) =\n \u2191(algebraMap A B) (addY' (baseChange W A) x\u2081 x\u2082 y\u2081 L)\n[PROOFSTEP]\nrw [\u2190 baseChange_addY', baseChange_baseChange]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 addY (baseChange W A) (\u2191(algebraMap R A) x\u2081) (\u2191(algebraMap R A) x\u2082) (\u2191(algebraMap R A) y\u2081) (\u2191(algebraMap R A) L) =\n \u2191(algebraMap R A) (addY W x\u2081 x\u2082 y\u2081 L)\n[PROOFSTEP]\nsimp only [addY, baseChange_addY', baseChange_addX, baseChange_negY]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082 L\u271d : R\nx\u2081 x\u2082 y\u2081 L : A\n\u22a2 addY (baseChange W B) (\u2191(algebraMap A B) x\u2081) (\u2191(algebraMap A B) x\u2082) (\u2191(algebraMap A B) y\u2081) (\u2191(algebraMap A B) L) =\n \u2191(algebraMap A B) (addY (baseChange W A) x\u2081 x\u2082 y\u2081 L)\n[PROOFSTEP]\nrw [\u2190 baseChange_addY, baseChange_baseChange]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 XYIdeal W (addX W x\u2081 x\u2082 L) (\u2191C (addY W x\u2081 x\u2082 y\u2081 L)) =\n span {\u2191(CoordinateRing.mk W) (negPolynomial W - \u2191C (linePolynomial x\u2081 y\u2081 L))} \u2294 XIdeal W (addX W x\u2081 x\u2082 L)\n[PROOFSTEP]\nsimp only [XYIdeal, XIdeal, XClass, YClass, addY, addY', negY, negPolynomial, linePolynomial]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 span\n {\u2191(CoordinateRing.mk W) (\u2191C (Y - \u2191C (addX W x\u2081 x\u2082 L))),\n \u2191(CoordinateRing.mk W) (Y - \u2191C (\u2191C (-(L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) - W.a\u2081 * addX W x\u2081 x\u2082 L - W.a\u2083)))} =\n span {\u2191(CoordinateRing.mk W) (-Y - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - \u2191C (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081))} \u2294\n span {\u2191(CoordinateRing.mk W) (\u2191C (Y - \u2191C (addX W x\u2081 x\u2082 L)))}\n[PROOFSTEP]\nrw [sub_sub <| -Y, neg_sub_left Y, map_neg, span_singleton_neg, sup_comm, \u2190 span_insert, \u2190\n span_pair_add_mul_right <| CoordinateRing.mk W <| C <| C <| W.a\u2081 + L, \u2190 _root_.map_mul, \u2190 map_add]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 span\n {\u2191(CoordinateRing.mk W) (\u2191C (Y - \u2191C (addX W x\u2081 x\u2082 L))),\n \u2191(CoordinateRing.mk W)\n (Y - \u2191C (\u2191C (-(L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) - W.a\u2081 * addX W x\u2081 x\u2082 L - W.a\u2083)) +\n \u2191C (Y - \u2191C (addX W x\u2081 x\u2082 L)) * \u2191C (\u2191C (W.a\u2081 + L)))} =\n span\n {\u2191(CoordinateRing.mk W) (\u2191C (Y - \u2191C (addX W x\u2081 x\u2082 L))),\n \u2191(CoordinateRing.mk W) (\u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) + \u2191C (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081) + Y)}\n[PROOFSTEP]\napply congr_arg (_ \u2218 _ \u2218 _ \u2218 _)\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 Y - \u2191C (\u2191C (-(L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) - W.a\u2081 * addX W x\u2081 x\u2082 L - W.a\u2083)) +\n \u2191C (Y - \u2191C (addX W x\u2081 x\u2082 L)) * \u2191C (\u2191C (W.a\u2081 + L)) =\n \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) + \u2191C (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081) + Y\n[PROOFSTEP]\nC_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 Y -\n (-(\u2191C (\u2191C L) * (\u2191C (\u2191C (addX W x\u2081 x\u2082 L)) - \u2191C (\u2191C x\u2081)) + \u2191C (\u2191C y\u2081)) - \u2191C (\u2191C W.a\u2081) * \u2191C (\u2191C (addX W x\u2081 x\u2082 L)) -\n \u2191C (\u2191C W.a\u2083)) +\n (\u2191C Y - \u2191C (\u2191C (addX W x\u2081 x\u2082 L))) * (\u2191C (\u2191C W.a\u2081) + \u2191C (\u2191C L)) =\n \u2191C (\u2191C W.a\u2081) * \u2191C Y + \u2191C (\u2191C W.a\u2083) + (\u2191C (\u2191C L) * (\u2191C Y - \u2191C (\u2191C x\u2081)) + \u2191C (\u2191C y\u2081)) + Y\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 WeierstrassCurve.equation W (addX W x\u2081 x\u2082 L) (addY' W x\u2081 x\u2082 y\u2081 L) \u2194\n eval (addX W x\u2081 x\u2082 L) (addPolynomial W x\u2081 y\u2081 L) = 0\n[PROOFSTEP]\nrw [WeierstrassCurve.equation, addY', addPolynomial, linePolynomial, WeierstrassCurve.polynomial]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 eval (addX W x\u2081 x\u2082 L)\n (eval (\u2191C (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081))\n (Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086))) =\n 0 \u2194\n eval (addX W x\u2081 x\u2082 L)\n (eval (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081)\n (Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086))) =\n 0\n[PROOFSTEP]\neval_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\nhx' : WeierstrassCurve.equation W (addX W x\u2081 x\u2082 L) (addY' W x\u2081 x\u2082 y\u2081 L)\nhx : eval (addX W x\u2081 x\u2082 L) (\u2191derivative (addPolynomial W x\u2081 y\u2081 L)) \u2260 0\n\u22a2 WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2082 L) (addY' W x\u2081 x\u2082 y\u2081 L)\n[PROOFSTEP]\nrw [WeierstrassCurve.nonsingular, and_iff_right hx', addY', WeierstrassCurve.polynomialX, WeierstrassCurve.polynomialY]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\nhx' : WeierstrassCurve.equation W (addX W x\u2081 x\u2082 L) (addY' W x\u2081 x\u2082 y\u2081 L)\nhx : eval (addX W x\u2081 x\u2082 L) (\u2191derivative (addPolynomial W x\u2081 y\u2081 L)) \u2260 0\n\u22a2 eval (addX W x\u2081 x\u2082 L)\n (eval (\u2191C (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081))\n (\u2191C (\u2191C W.a\u2081) * Y - \u2191C (\u2191C 3 * Y ^ 2 + \u2191C (2 * W.a\u2082) * Y + \u2191C W.a\u2084))) \u2260\n 0 \u2228\n eval (addX W x\u2081 x\u2082 L) (eval (\u2191C (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081)) (\u2191C (\u2191C 2) * Y + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083))) \u2260 0\n[PROOFSTEP]\neval_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\nhx' : WeierstrassCurve.equation W (addX W x\u2081 x\u2082 L) (addY' W x\u2081 x\u2082 y\u2081 L)\nhx : eval (addX W x\u2081 x\u2082 L) (\u2191derivative (addPolynomial W x\u2081 y\u2081 L)) \u2260 0\n\u22a2 W.a\u2081 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) - (3 * addX W x\u2081 x\u2082 L ^ 2 + 2 * W.a\u2082 * addX W x\u2081 x\u2082 L + W.a\u2084) \u2260 0 \u2228\n 2 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) + (W.a\u2081 * addX W x\u2081 x\u2082 L + W.a\u2083) \u2260 0\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\nhx' : WeierstrassCurve.equation W (addX W x\u2081 x\u2082 L) (addY' W x\u2081 x\u2082 y\u2081 L)\nhx :\n W.a\u2081 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) - (3 * addX W x\u2081 x\u2082 L ^ 2 + 2 * W.a\u2082 * addX W x\u2081 x\u2082 L + W.a\u2084) = 0 \u2227\n 2 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) + (W.a\u2081 * addX W x\u2081 x\u2082 L + W.a\u2083) = 0\n\u22a2 eval (addX W x\u2081 x\u2082 L) (\u2191derivative (addPolynomial W x\u2081 y\u2081 L)) = 0\n[PROOFSTEP]\nrw [addPolynomial, linePolynomial, WeierstrassCurve.polynomial]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\nhx' : WeierstrassCurve.equation W (addX W x\u2081 x\u2082 L) (addY' W x\u2081 x\u2082 y\u2081 L)\nhx :\n W.a\u2081 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) - (3 * addX W x\u2081 x\u2082 L ^ 2 + 2 * W.a\u2082 * addX W x\u2081 x\u2082 L + W.a\u2084) = 0 \u2227\n 2 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) + (W.a\u2081 * addX W x\u2081 x\u2082 L + W.a\u2083) = 0\n\u22a2 eval (addX W x\u2081 x\u2082 L)\n (\u2191derivative\n (eval (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081)\n (Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)))) =\n 0\n[PROOFSTEP]\neval_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\nhx' : WeierstrassCurve.equation W (addX W x\u2081 x\u2082 L) (addY' W x\u2081 x\u2082 y\u2081 L)\nhx :\n W.a\u2081 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) - (3 * addX W x\u2081 x\u2082 L ^ 2 + 2 * W.a\u2082 * addX W x\u2081 x\u2082 L + W.a\u2084) = 0 \u2227\n 2 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) + (W.a\u2081 * addX W x\u2081 x\u2082 L + W.a\u2083) = 0\n\u22a2 eval (addX W x\u2081 x\u2082 L)\n (\u2191derivative\n ((\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081) ^ 2 + (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081) -\n (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086))) =\n 0\n[PROOFSTEP]\nderivative_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\nhx' : WeierstrassCurve.equation W (addX W x\u2081 x\u2082 L) (addY' W x\u2081 x\u2082 y\u2081 L)\nhx :\n W.a\u2081 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) - (3 * addX W x\u2081 x\u2082 L ^ 2 + 2 * W.a\u2082 * addX W x\u2081 x\u2082 L + W.a\u2084) = 0 \u2227\n 2 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) + (W.a\u2081 * addX W x\u2081 x\u2082 L + W.a\u2083) = 0\n\u22a2 eval (addX W x\u2081 x\u2082 L)\n (\u2191C 2 * (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081) * (0 * (Y - \u2191C x\u2081) + \u2191C L * (1 - 0) + 0) +\n ((0 * Y + \u2191C W.a\u2081 * 1 + 0) * (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081) +\n (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * (0 * (Y - \u2191C x\u2081) + \u2191C L * (1 - 0) + 0)) -\n (\u2191C \u21913 * Y ^ (3 - 1) + (0 * Y ^ 2 + \u2191C W.a\u2082 * (\u2191C 2 * Y * 1)) + (0 * Y + \u2191C W.a\u2084 * 1) + 0)) =\n 0\n[PROOFSTEP]\nsimp only [zero_add, add_zero, sub_zero, zero_mul, mul_one]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\nhx' : WeierstrassCurve.equation W (addX W x\u2081 x\u2082 L) (addY' W x\u2081 x\u2082 y\u2081 L)\nhx :\n W.a\u2081 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) - (3 * addX W x\u2081 x\u2082 L ^ 2 + 2 * W.a\u2082 * addX W x\u2081 x\u2082 L + W.a\u2084) = 0 \u2227\n 2 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) + (W.a\u2081 * addX W x\u2081 x\u2082 L + W.a\u2083) = 0\n\u22a2 eval (addX W x\u2081 x\u2082 L)\n (\u2191C 2 * (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081) * \u2191C L +\n (\u2191C W.a\u2081 * (\u2191C L * (Y - \u2191C x\u2081) + \u2191C y\u2081) + (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * \u2191C L) -\n (\u2191C \u21913 * Y ^ (3 - 1) + \u2191C W.a\u2082 * (\u2191C 2 * Y) + \u2191C W.a\u2084)) =\n 0\n[PROOFSTEP]\neval_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\nhx' : WeierstrassCurve.equation W (addX W x\u2081 x\u2082 L) (addY' W x\u2081 x\u2082 y\u2081 L)\nhx :\n W.a\u2081 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) - (3 * addX W x\u2081 x\u2082 L ^ 2 + 2 * W.a\u2082 * addX W x\u2081 x\u2082 L + W.a\u2084) = 0 \u2227\n 2 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) + (W.a\u2081 * addX W x\u2081 x\u2082 L + W.a\u2083) = 0\n\u22a2 2 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) * L +\n (W.a\u2081 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) + (W.a\u2081 * addX W x\u2081 x\u2082 L + W.a\u2083) * L) -\n (\u21913 * addX W x\u2081 x\u2082 L ^ (3 - 1) + W.a\u2082 * (2 * addX W x\u2081 x\u2082 L) + W.a\u2084) =\n 0\n[PROOFSTEP]\nlinear_combination (norm := (norm_num1; ring1)) hx.left + L * hx.right\n[GOAL]\ncase a\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\nhx' : WeierstrassCurve.equation W (addX W x\u2081 x\u2082 L) (addY' W x\u2081 x\u2082 y\u2081 L)\nhx :\n W.a\u2081 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) - (3 * addX W x\u2081 x\u2082 L ^ 2 + 2 * W.a\u2082 * addX W x\u2081 x\u2082 L + W.a\u2084) = 0 \u2227\n 2 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) + (W.a\u2081 * addX W x\u2081 x\u2082 L + W.a\u2083) = 0\n\u22a2 2 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) * L +\n (W.a\u2081 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) + (W.a\u2081 * addX W x\u2081 x\u2082 L + W.a\u2083) * L) -\n (\u21913 * addX W x\u2081 x\u2082 L ^ (3 - 1) + W.a\u2082 * (2 * addX W x\u2081 x\u2082 L) + W.a\u2084) -\n 0 -\n (W.a\u2081 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) - (3 * addX W x\u2081 x\u2082 L ^ 2 + 2 * W.a\u2082 * addX W x\u2081 x\u2082 L + W.a\u2084) +\n L * (2 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) + (W.a\u2081 * addX W x\u2081 x\u2082 L + W.a\u2083)) -\n (0 + L * 0)) =\n 0\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase a\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\nhx' : WeierstrassCurve.equation W (addX W x\u2081 x\u2082 L) (addY' W x\u2081 x\u2082 y\u2081 L)\nhx :\n W.a\u2081 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) - (3 * addX W x\u2081 x\u2082 L ^ 2 + 2 * W.a\u2082 * addX W x\u2081 x\u2082 L + W.a\u2084) = 0 \u2227\n 2 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) + (W.a\u2081 * addX W x\u2081 x\u2082 L + W.a\u2083) = 0\n\u22a2 2 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) * L +\n (W.a\u2081 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) + (W.a\u2081 * addX W x\u2081 x\u2082 L + W.a\u2083) * L) -\n (3 * addX W x\u2081 x\u2082 L ^ 2 + W.a\u2082 * (2 * addX W x\u2081 x\u2082 L) + W.a\u2084) -\n 0 -\n (W.a\u2081 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) - (3 * addX W x\u2081 x\u2082 L ^ 2 + 2 * W.a\u2082 * addX W x\u2081 x\u2082 L + W.a\u2084) +\n L * (2 * (L * (addX W x\u2081 x\u2082 L - x\u2081) + y\u2081) + (W.a\u2081 * addX W x\u2081 x\u2082 L + W.a\u2083)) -\n (0 + L * 0)) =\n 0\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 WeierstrassCurve.equation W x\u2081 (negY W x\u2081 y\u2081) \u2194 WeierstrassCurve.equation W x\u2081 y\u2081\n[PROOFSTEP]\nrw [equation_iff, equation_iff, negY]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) ^ 2 + W.a\u2081 * x\u2081 * (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) + W.a\u2083 * (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) =\n x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086 \u2194\n y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\n[PROOFSTEP]\ncongr! 1\n[GOAL]\ncase a.h.e'_2\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) ^ 2 + W.a\u2081 * x\u2081 * (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) + W.a\u2083 * (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) =\n y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 WeierstrassCurve.nonsingular W x\u2081 (negY W x\u2081 y\u2081) \u2194 WeierstrassCurve.nonsingular W x\u2081 y\u2081\n[PROOFSTEP]\nrw [nonsingular_iff, equation_neg_iff, \u2190 negY, negY_negY, \u2190 @ne_comm _ y\u2081, nonsingular_iff]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 WeierstrassCurve.equation W x\u2081 y\u2081 \u2227 (W.a\u2081 * negY W x\u2081 y\u2081 \u2260 3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 \u2228 y\u2081 \u2260 negY W x\u2081 y\u2081) \u2194\n WeierstrassCurve.equation W x\u2081 y\u2081 \u2227 (W.a\u2081 * y\u2081 \u2260 3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 \u2228 y\u2081 \u2260 -y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)\n[PROOFSTEP]\nexact\n and_congr_right' <|\n (iff_congr not_and_or.symm not_and_or.symm).mpr <| not_congr <| and_congr_left fun h => by rw [\u2190 h]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\nh : y\u2081 = negY W x\u2081 y\u2081\n\u22a2 W.a\u2081 * negY W x\u2081 y\u2081 = 3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 \u2194 W.a\u2081 * y\u2081 = 3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 \u2200 (x : Point W), - -x = x\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase zero\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L : R\n\u22a2 - -zero = zero\n[PROOFSTEP]\nsimp [zero_def]\n[GOAL]\ncase some\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L x\u271d y\u271d : R\nh\u271d : WeierstrassCurve.nonsingular W x\u271d y\u271d\n\u22a2 - -some h\u271d = some h\u271d\n[PROOFSTEP]\nsimp [zero_def]\n[GOAL]\ncase some\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u2081 x\u2082 y\u2081 y\u2082 L x\u271d y\u271d : R\nh\u271d : WeierstrassCurve.nonsingular W x\u271d y\u271d\n\u22a2 W.a\u2083 - (-y\u271d - W.a\u2081 * x\u271d) - W.a\u2081 * x\u271d - W.a\u2083 = y\u271d\n[PROOFSTEP]\nring1\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : y\u2081 = negY W x\u2082 y\u2082\n\u22a2 slope W x\u2081 x\u2082 y\u2081 y\u2082 = 0\n[PROOFSTEP]\nrw [slope, if_pos hx, if_pos hy]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 slope W x\u2081 x\u2082 y\u2081 y\u2082 = (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)\n[PROOFSTEP]\nrw [slope, if_pos hx, if_neg hy]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 \u2260 x\u2082\n\u22a2 slope W x\u2081 x\u2082 y\u2081 y\u2082 = (y\u2081 - y\u2082) / (x\u2081 - x\u2082)\n[PROOFSTEP]\nrw [slope, if_neg hx]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 slope W x\u2081 x\u2082 y\u2081 y\u2082 =\n -eval x\u2081 (eval (\u2191C y\u2081) (WeierstrassCurve.polynomialX W)) / eval x\u2081 (eval (\u2191C y\u2081) (WeierstrassCurve.polynomialY W))\n[PROOFSTEP]\nrw [slope_of_Y_ne hx hy, eval_polynomialX, neg_sub]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081) =\n (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / eval x\u2081 (eval (\u2191C y\u2081) (WeierstrassCurve.polynomialY W))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 y\u2081 - negY W x\u2081 y\u2081 = eval x\u2081 (eval (\u2191C y\u2081) (WeierstrassCurve.polynomialY W))\n[PROOFSTEP]\nrw [negY, eval_polynomialY]\n[GOAL]\ncase e_a\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) = 2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083\n[PROOFSTEP]\nring1\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 slope (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2081) (\u2191(algebraMap F K) y\u2082) =\n \u2191(algebraMap F K) (slope W x\u2081 x\u2082 y\u2081 y\u2082)\n[PROOFSTEP]\nby_cases hx : x\u2081 = x\u2082\n[GOAL]\ncase pos\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\n\u22a2 slope (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2081) (\u2191(algebraMap F K) y\u2082) =\n \u2191(algebraMap F K) (slope W x\u2081 x\u2082 y\u2081 y\u2082)\n[PROOFSTEP]\nby_cases hy : y\u2081 = W.negY x\u2082 y\u2082\n[GOAL]\ncase pos\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : y\u2081 = negY W x\u2082 y\u2082\n\u22a2 slope (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2081) (\u2191(algebraMap F K) y\u2082) =\n \u2191(algebraMap F K) (slope W x\u2081 x\u2082 y\u2081 y\u2082)\n[PROOFSTEP]\nrw [slope_of_Y_eq hx hy, slope_of_Y_eq <| congr_arg _ hx, map_zero]\n[GOAL]\ncase pos\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : y\u2081 = negY W x\u2082 y\u2082\n\u22a2 \u2191(algebraMap F K) y\u2081 = negY (baseChange W K) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2082)\n[PROOFSTEP]\nrw [hy, baseChange_negY]\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : \u00acy\u2081 = negY W x\u2082 y\u2082\n\u22a2 slope (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2081) (\u2191(algebraMap F K) y\u2082) =\n \u2191(algebraMap F K) (slope W x\u2081 x\u2082 y\u2081 y\u2082)\n[PROOFSTEP]\nrw [slope_of_Y_ne hx hy, slope_of_Y_ne <| congr_arg _ hx]\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : \u00acy\u2081 = negY W x\u2082 y\u2082\n\u22a2 (3 * \u2191(algebraMap F K) x\u2081 ^ 2 + 2 * (baseChange W K).a\u2082 * \u2191(algebraMap F K) x\u2081 + (baseChange W K).a\u2084 -\n (baseChange W K).a\u2081 * \u2191(algebraMap F K) y\u2081) /\n (\u2191(algebraMap F K) y\u2081 - negY (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) y\u2081)) =\n \u2191(algebraMap F K) ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081))\n[PROOFSTEP]\nmap_simp\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : \u00acy\u2081 = negY W x\u2082 y\u2082\n\u22a2 (3 * \u2191(algebraMap F K) x\u2081 ^ 2 + 2 * (baseChange W K).a\u2082 * \u2191(algebraMap F K) x\u2081 + (baseChange W K).a\u2084 -\n (baseChange W K).a\u2081 * \u2191(algebraMap F K) y\u2081) /\n (\u2191(algebraMap F K) y\u2081 - negY (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) y\u2081)) =\n (3 * \u2191(algebraMap F K) x\u2081 ^ 2 + 2 * \u2191(algebraMap F K) W.a\u2082 * \u2191(algebraMap F K) x\u2081 + \u2191(algebraMap F K) W.a\u2084 -\n \u2191(algebraMap F K) W.a\u2081 * \u2191(algebraMap F K) y\u2081) /\n (\u2191(algebraMap F K) y\u2081 - \u2191(algebraMap F K) (negY W x\u2081 y\u2081))\n[PROOFSTEP]\nsimp only [baseChange_negY]\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : \u00acy\u2081 = negY W x\u2082 y\u2082\n\u22a2 (3 * \u2191(algebraMap F K) x\u2081 ^ 2 + 2 * (baseChange W K).a\u2082 * \u2191(algebraMap F K) x\u2081 + (baseChange W K).a\u2084 -\n (baseChange W K).a\u2081 * \u2191(algebraMap F K) y\u2081) /\n (\u2191(algebraMap F K) y\u2081 - \u2191(algebraMap F K) (negY W x\u2081 y\u2081)) =\n (3 * \u2191(algebraMap F K) x\u2081 ^ 2 + 2 * \u2191(algebraMap F K) W.a\u2082 * \u2191(algebraMap F K) x\u2081 + \u2191(algebraMap F K) W.a\u2084 -\n \u2191(algebraMap F K) W.a\u2081 * \u2191(algebraMap F K) y\u2081) /\n (\u2191(algebraMap F K) y\u2081 - \u2191(algebraMap F K) (negY W x\u2081 y\u2081))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : \u00acy\u2081 = negY W x\u2082 y\u2082\n\u22a2 \u2191(algebraMap F K) y\u2081 \u2260 negY (baseChange W K) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2082)\n[PROOFSTEP]\nrw [baseChange_negY]\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : \u00acy\u2081 = negY W x\u2082 y\u2082\n\u22a2 \u2191(algebraMap F K) y\u2081 \u2260 \u2191(algebraMap F K) (negY W x\u2082 y\u2082)\n[PROOFSTEP]\ncontrapose! hy\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : \u2191(algebraMap F K) y\u2081 = \u2191(algebraMap F K) (negY W x\u2082 y\u2082)\n\u22a2 y\u2081 = negY W x\u2082 y\u2082\n[PROOFSTEP]\nexact NoZeroSMulDivisors.algebraMap_injective F K hy\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 slope (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2081) (\u2191(algebraMap F K) y\u2082) =\n \u2191(algebraMap F K) (slope W x\u2081 x\u2082 y\u2081 y\u2082)\n[PROOFSTEP]\nrw [slope_of_X_ne hx, slope_of_X_ne]\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 (\u2191(algebraMap F K) y\u2081 - \u2191(algebraMap F K) y\u2082) / (\u2191(algebraMap F K) x\u2081 - \u2191(algebraMap F K) x\u2082) =\n \u2191(algebraMap F K) ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))\n[PROOFSTEP]\nmap_simp\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 \u2191(algebraMap F K) x\u2081 \u2260 \u2191(algebraMap F K) x\u2082\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : \u2191(algebraMap F K) x\u2081 = \u2191(algebraMap F K) x\u2082\n\u22a2 x\u2081 = x\u2082\n[PROOFSTEP]\nexact NoZeroSMulDivisors.algebraMap_injective F K hx\n[GOAL]\nF\u271d : Type u\ninst\u271d\u2079 : Field F\u271d\nW\u271d : WeierstrassCurve F\u271d\nK\u271d : Type v\ninst\u271d\u2078 : Field K\u271d\ninst\u271d\u2077 : Algebra F\u271d K\u271d\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\u271d\nh\u2081 : WeierstrassCurve.nonsingular W\u271d x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W\u271d x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W\u271d x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W\u271d x\u2082\u271d y\u2082\u271d\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\n\u22a2 slope (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2081) (\u2191(algebraMap F K) y\u2082) =\n \u2191(algebraMap F K) (slope (baseChange W F) x\u2081 x\u2082 y\u2081 y\u2082)\n[PROOFSTEP]\nrw [\u2190 baseChange_slope, baseChange_baseChange]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\n\u22a2 y\u2081 = y\u2082 \u2228 y\u2081 = negY W x\u2082 y\u2082\n[PROOFSTEP]\nrw [equation_iff] at h\u2081' h\u2082' \n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhx : x\u2081 = x\u2082\n\u22a2 y\u2081 = y\u2082 \u2228 y\u2081 = negY W x\u2082 y\u2082\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 @sub_eq_zero _ _ y\u2081, \u2190 mul_eq_zero, negY]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhx : x\u2081 = x\u2082\n\u22a2 (y\u2081 - y\u2082) * (y\u2081 - (-y\u2082 - W.a\u2081 * x\u2082 - W.a\u2083)) = 0\n[PROOFSTEP]\nlinear_combination (norm := (rw [hx]; ring1)) h\u2081' - h\u2082'\n[GOAL]\ncase a\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhx : x\u2081 = x\u2082\n\u22a2 (y\u2081 - y\u2082) * (y\u2081 - (-y\u2082 - W.a\u2081 * x\u2082 - W.a\u2083)) - 0 -\n (y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 - (y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082) -\n (x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086 - (x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086))) =\n 0\n[PROOFSTEP]\nrw [hx]\n[GOAL]\ncase a\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhx : x\u2081 = x\u2082\n\u22a2 (y\u2081 - y\u2082) * (y\u2081 - (-y\u2082 - W.a\u2081 * x\u2082 - W.a\u2083)) - 0 -\n (y\u2081 ^ 2 + W.a\u2081 * x\u2082 * y\u2081 + W.a\u2083 * y\u2081 - (y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082) -\n (x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086 - (x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086))) =\n 0\n[PROOFSTEP]\nring1\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 XYIdeal W x\u2082 (\u2191C y\u2082) = XYIdeal W x\u2082 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nclear h\u2082\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 XYIdeal W x\u2082 (\u2191C y\u2082) = XYIdeal W x\u2082 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nhave hy\u2082 : y\u2082 = (linePolynomial x\u2081 y\u2081 <| W.slope x\u2081 x\u2082 y\u2081 y\u2082).eval x\u2082 :=\n by\n by_cases hx : x\u2081 = x\u2082\n \u00b7 rcases hx, Y_eq_of_Y_ne h\u2081' h\u2082' hx <| hxy hx with \u27e8rfl, rfl\u27e9\n field_simp [linePolynomial, sub_ne_zero_of_ne (hxy rfl)]\n \u00b7 field_simp [linePolynomial, slope_of_X_ne hx, sub_ne_zero_of_ne hx]\n ring1\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 y\u2082 = eval x\u2082 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nby_cases hx : x\u2081 = x\u2082\n[GOAL]\ncase pos\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\n\u22a2 y\u2082 = eval x\u2082 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nrcases hx, Y_eq_of_Y_ne h\u2081' h\u2082' hx <| hxy hx with \u27e8rfl, rfl\u27e9\n[GOAL]\ncase pos\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' h\u2082' : WeierstrassCurve.equation W x\u2081 y\u2081\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 negY W x\u2081 y\u2081\n\u22a2 y\u2081 = eval x\u2081 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2081 y\u2081 y\u2081))\n[PROOFSTEP]\nfield_simp [linePolynomial, sub_ne_zero_of_ne (hxy rfl)]\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 y\u2082 = eval x\u2082 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nfield_simp [linePolynomial, slope_of_X_ne hx, sub_ne_zero_of_ne hx]\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 y\u2082 * (x\u2081 - x\u2082) = (y\u2081 - y\u2082) * (x\u2082 - x\u2081) + y\u2081 * (x\u2081 - x\u2082)\n[PROOFSTEP]\nring1\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhy\u2082 : y\u2082 = eval x\u2082 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n\u22a2 XYIdeal W x\u2082 (\u2191C y\u2082) = XYIdeal W x\u2082 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nnth_rw 1 [hy\u2082]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhy\u2082 : y\u2082 = eval x\u2082 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n\u22a2 XYIdeal W x\u2082 (\u2191C (eval x\u2082 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))) =\n XYIdeal W x\u2082 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nsimp only [XYIdeal, XClass, YClass, linePolynomial]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhy\u2082 : y\u2082 = eval x\u2082 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n\u22a2 span\n {\u2191(CoordinateRing.mk W) (\u2191C (Y - \u2191C x\u2082)),\n \u2191(CoordinateRing.mk W) (Y - \u2191C (\u2191C (eval x\u2082 (\u2191C (slope W x\u2081 x\u2082 y\u2081 y\u2082) * (Y - \u2191C x\u2081) + \u2191C y\u2081))))} =\n span\n {\u2191(CoordinateRing.mk W) (\u2191C (Y - \u2191C x\u2082)),\n \u2191(CoordinateRing.mk W) (Y - \u2191C (\u2191C (slope W x\u2081 x\u2082 y\u2081 y\u2082) * (Y - \u2191C x\u2081) + \u2191C y\u2081))}\n[PROOFSTEP]\nrw [\u2190 span_pair_add_mul_right <| CoordinateRing.mk W <| C <| C <| -W.slope x\u2081 x\u2082 y\u2081 y\u2082, \u2190 _root_.map_mul, \u2190 map_add]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhy\u2082 : y\u2082 = eval x\u2082 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n\u22a2 span\n {\u2191(CoordinateRing.mk W) (\u2191C (Y - \u2191C x\u2082)),\n \u2191(CoordinateRing.mk W)\n (Y - \u2191C (\u2191C (eval x\u2082 (\u2191C (slope W x\u2081 x\u2082 y\u2081 y\u2082) * (Y - \u2191C x\u2081) + \u2191C y\u2081))) +\n \u2191C (Y - \u2191C x\u2082) * \u2191C (\u2191C (-slope W x\u2081 x\u2082 y\u2081 y\u2082)))} =\n span\n {\u2191(CoordinateRing.mk W) (\u2191C (Y - \u2191C x\u2082)),\n \u2191(CoordinateRing.mk W) (Y - \u2191C (\u2191C (slope W x\u2081 x\u2082 y\u2081 y\u2082) * (Y - \u2191C x\u2081) + \u2191C y\u2081))}\n[PROOFSTEP]\napply congr_arg (_ \u2218 _ \u2218 _ \u2218 _)\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhy\u2082 : y\u2082 = eval x\u2082 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n\u22a2 Y - \u2191C (\u2191C (eval x\u2082 (\u2191C (slope W x\u2081 x\u2082 y\u2081 y\u2082) * (Y - \u2191C x\u2081) + \u2191C y\u2081))) +\n \u2191C (Y - \u2191C x\u2082) * \u2191C (\u2191C (-slope W x\u2081 x\u2082 y\u2081 y\u2082)) =\n Y - \u2191C (\u2191C (slope W x\u2081 x\u2082 y\u2081 y\u2082) * (Y - \u2191C x\u2081) + \u2191C y\u2081)\n[PROOFSTEP]\neval_simp\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhy\u2082 : y\u2082 = eval x\u2082 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n\u22a2 Y - \u2191C (\u2191C (slope W x\u2081 x\u2082 y\u2081 y\u2082 * (x\u2082 - x\u2081) + y\u2081)) + \u2191C (Y - \u2191C x\u2082) * \u2191C (\u2191C (-slope W x\u2081 x\u2082 y\u2081 y\u2082)) =\n Y - \u2191C (\u2191C (slope W x\u2081 x\u2082 y\u2081 y\u2082) * (Y - \u2191C x\u2081) + \u2191C y\u2081)\n[PROOFSTEP]\nC_simp\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhy\u2082 : y\u2082 = eval x\u2082 (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n\u22a2 Y - (\u2191C (\u2191C (slope W x\u2081 x\u2082 y\u2081 y\u2082)) * (\u2191C (\u2191C x\u2082) - \u2191C (\u2191C x\u2081)) + \u2191C (\u2191C y\u2081)) +\n (\u2191C Y - \u2191C (\u2191C x\u2082)) * -\u2191C (\u2191C (slope W x\u2081 x\u2082 y\u2081 y\u2082)) =\n Y - (\u2191C (\u2191C (slope W x\u2081 x\u2082 y\u2081 y\u2082)) * (\u2191C Y - \u2191C (\u2191C x\u2081)) + \u2191C (\u2191C y\u2081))\n[PROOFSTEP]\nring1\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 addPolynomial W x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082) =\n -((Y - \u2191C x\u2081) * (Y - \u2191C x\u2082) * (Y - \u2191C (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))))\n[PROOFSTEP]\nclear h\u2082\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 addPolynomial W x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082) =\n -((Y - \u2191C x\u2081) * (Y - \u2191C x\u2082) * (Y - \u2191C (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))))\n[PROOFSTEP]\nrw [addPolynomial_eq, neg_inj, Cubic.prod_X_sub_C_eq, Cubic.toPoly_injective]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 { a := 1, b := -slope W x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 - W.a\u2081 * slope W x\u2081 x\u2082 y\u2081 y\u2082 + W.a\u2082,\n c := 2 * x\u2081 * slope W x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * slope W x\u2081 x\u2082 y\u2081 y\u2082 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * slope W x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * slope W x\u2081 x\u2082 y\u2081 y\u2082 -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n { a := 1, b := -(x\u2081 + x\u2082 + addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)),\n c := x\u2081 * x\u2082 + x\u2081 * addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) + x\u2082 * addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082),\n d := -(x\u2081 * x\u2082 * addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) }\n[PROOFSTEP]\nby_cases hx : x\u2081 = x\u2082\n[GOAL]\ncase pos\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\n\u22a2 { a := 1, b := -slope W x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 - W.a\u2081 * slope W x\u2081 x\u2082 y\u2081 y\u2082 + W.a\u2082,\n c := 2 * x\u2081 * slope W x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * slope W x\u2081 x\u2082 y\u2081 y\u2082 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * slope W x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * slope W x\u2081 x\u2082 y\u2081 y\u2082 -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n { a := 1, b := -(x\u2081 + x\u2082 + addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)),\n c := x\u2081 * x\u2082 + x\u2081 * addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) + x\u2082 * addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082),\n d := -(x\u2081 * x\u2082 * addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) }\n[PROOFSTEP]\nrcases hx, Y_eq_of_Y_ne h\u2081' h\u2082' hx (hxy hx) with \u27e8rfl, rfl\u27e9\n[GOAL]\ncase pos\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' h\u2082' : WeierstrassCurve.equation W x\u2081 y\u2081\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 negY W x\u2081 y\u2081\n\u22a2 { a := 1, b := -slope W x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 - W.a\u2081 * slope W x\u2081 x\u2081 y\u2081 y\u2081 + W.a\u2082,\n c := 2 * x\u2081 * slope W x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * slope W x\u2081 x\u2081 y\u2081 y\u2081 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * slope W x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * slope W x\u2081 x\u2081 y\u2081 y\u2081 -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n { a := 1, b := -(x\u2081 + x\u2081 + addX W x\u2081 x\u2081 (slope W x\u2081 x\u2081 y\u2081 y\u2081)),\n c := x\u2081 * x\u2081 + x\u2081 * addX W x\u2081 x\u2081 (slope W x\u2081 x\u2081 y\u2081 y\u2081) + x\u2081 * addX W x\u2081 x\u2081 (slope W x\u2081 x\u2081 y\u2081 y\u2081),\n d := -(x\u2081 * x\u2081 * addX W x\u2081 x\u2081 (slope W x\u2081 x\u2081 y\u2081 y\u2081)) }\n[PROOFSTEP]\nrw [equation_iff] at h\u2081' h\u2082' \n[GOAL]\ncase pos\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' h\u2082' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 negY W x\u2081 y\u2081\n\u22a2 { a := 1, b := -slope W x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 - W.a\u2081 * slope W x\u2081 x\u2081 y\u2081 y\u2081 + W.a\u2082,\n c := 2 * x\u2081 * slope W x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * slope W x\u2081 x\u2081 y\u2081 y\u2081 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * slope W x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * slope W x\u2081 x\u2081 y\u2081 y\u2081 -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n { a := 1, b := -(x\u2081 + x\u2081 + addX W x\u2081 x\u2081 (slope W x\u2081 x\u2081 y\u2081 y\u2081)),\n c := x\u2081 * x\u2081 + x\u2081 * addX W x\u2081 x\u2081 (slope W x\u2081 x\u2081 y\u2081 y\u2081) + x\u2081 * addX W x\u2081 x\u2081 (slope W x\u2081 x\u2081 y\u2081 y\u2081),\n d := -(x\u2081 * x\u2081 * addX W x\u2081 x\u2081 (slope W x\u2081 x\u2081 y\u2081 y\u2081)) }\n[PROOFSTEP]\nrw [slope_of_Y_ne rfl <| hxy rfl]\n[GOAL]\ncase pos\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' h\u2082' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 negY W x\u2081 y\u2081\n\u22a2 { a := 1,\n b :=\n -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 -\n W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n W.a\u2082,\n c :=\n 2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 +\n (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 +\n (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n { a := 1, b := -(x\u2081 + x\u2081 + addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081))),\n c :=\n x\u2081 * x\u2081 + x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)),\n d := -(x\u2081 * x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081))) }\n[PROOFSTEP]\nrw [negY, \u2190 sub_ne_zero] at hxy \n[GOAL]\ncase pos\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' h\u2082' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy\u271d : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 -y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n b :=\n -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 -\n W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n W.a\u2082,\n c :=\n 2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 +\n (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 +\n (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n { a := 1, b := -(x\u2081 + x\u2081 + addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081))),\n c :=\n x\u2081 * x\u2081 + x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)),\n d := -(x\u2081 * x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081))) }\n[PROOFSTEP]\next\n[GOAL]\ncase pos.a\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' h\u2082' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy\u271d : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 -y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n b :=\n -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 -\n W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n W.a\u2082,\n c :=\n 2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 +\n (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 +\n (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.a =\n { a := 1, b := -(x\u2081 + x\u2081 + addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081))),\n c :=\n x\u2081 * x\u2081 + x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)),\n d := -(x\u2081 * x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081))) }.a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.b\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' h\u2082' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy\u271d : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 -y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n b :=\n -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 -\n W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n W.a\u2082,\n c :=\n 2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 +\n (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 +\n (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.b =\n { a := 1, b := -(x\u2081 + x\u2081 + addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081))),\n c :=\n x\u2081 * x\u2081 + x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)),\n d := -(x\u2081 * x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081))) }.b\n[PROOFSTEP]\nsimp only [addX]\n[GOAL]\ncase pos.b\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' h\u2082' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy\u271d : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 -y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 -\n W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n W.a\u2082 =\n -(x\u2081 + x\u2081 +\n (((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 +\n W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) -\n W.a\u2082 -\n x\u2081 -\n x\u2081))\n[PROOFSTEP]\nring1\n[GOAL]\ncase pos.c\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' h\u2082' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy\u271d : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 -y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n b :=\n -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 -\n W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n W.a\u2082,\n c :=\n 2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 +\n (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 +\n (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.c =\n { a := 1, b := -(x\u2081 + x\u2081 + addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081))),\n c :=\n x\u2081 * x\u2081 + x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)),\n d := -(x\u2081 * x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081))) }.c\n[PROOFSTEP]\nfield_simp [hxy rfl]\n[GOAL]\ncase pos.c\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' h\u2082' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy\u271d : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 -y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 2 * x\u2081 * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) +\n (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) *\n (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 +\n (-(W.a\u2081 * y\u2081) + W.a\u2084) * ((y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083))) =\n x\u2081 * x\u2081 * ((y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083))) +\n x\u2081 *\n ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) +\n W.a\u2081 * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 -\n (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * W.a\u2082 -\n (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081 -\n (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081) +\n x\u2081 *\n ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) +\n W.a\u2081 * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 -\n (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * W.a\u2082 -\n (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081 -\n (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081)\n[PROOFSTEP]\nring1\n[GOAL]\ncase pos.d\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' h\u2082' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy\u271d : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 -y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n b :=\n -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 -\n W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n W.a\u2082,\n c :=\n 2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 +\n (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 +\n (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.d =\n { a := 1, b := -(x\u2081 + x\u2081 + addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081))),\n c :=\n x\u2081 * x\u2081 + x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)),\n d := -(x\u2081 * x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081))) }.d\n[PROOFSTEP]\nlinear_combination (norm := (field_simp [hxy rfl]; ring1)) -h\u2081'\n[GOAL]\ncase a\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' h\u2082' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy\u271d : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 -y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n b :=\n -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 -\n W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n W.a\u2082,\n c :=\n 2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 +\n (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) *\n ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) ^ 2 +\n (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.d -\n { a := 1,\n b := -(x\u2081 + x\u2081 + addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081))),\n c :=\n x\u2081 * x\u2081 + x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)) +\n x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081)),\n d := -(x\u2081 * x\u2081 * addX W x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - negY W x\u2081 y\u2081))) }.d -\n (-(y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081) - -(x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086)) =\n 0\n[PROOFSTEP]\nfield_simp [hxy rfl]\n[GOAL]\ncase a\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' h\u2082' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy\u271d : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 -y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 -(x\u2081 ^ 2 * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083))) +\n (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) *\n (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 -\n (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) +\n x\u2081 * x\u2081 *\n ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) +\n W.a\u2081 * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 -\n (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * W.a\u2082 -\n (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081 -\n (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081) -\n (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) *\n (-(W.a\u2083 * y\u2081) + (-(W.a\u2081 * x\u2081 * y\u2081) + -y\u2081 ^ 2) - (-W.a\u2086 + (-(W.a\u2084 * x\u2081) + (-(W.a\u2082 * x\u2081 ^ 2) + -x\u2081 ^ 3)))) =\n 0\n[PROOFSTEP]\nring1\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 { a := 1, b := -slope W x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 - W.a\u2081 * slope W x\u2081 x\u2082 y\u2081 y\u2082 + W.a\u2082,\n c := 2 * x\u2081 * slope W x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * slope W x\u2081 x\u2082 y\u2081 y\u2082 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * slope W x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * slope W x\u2081 x\u2082 y\u2081 y\u2082 -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n { a := 1, b := -(x\u2081 + x\u2082 + addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)),\n c := x\u2081 * x\u2082 + x\u2081 * addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) + x\u2082 * addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082),\n d := -(x\u2081 * x\u2082 * addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) }\n[PROOFSTEP]\nrw [equation_iff] at h\u2081' h\u2082' \n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 { a := 1, b := -slope W x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 - W.a\u2081 * slope W x\u2081 x\u2082 y\u2081 y\u2082 + W.a\u2082,\n c := 2 * x\u2081 * slope W x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * slope W x\u2081 x\u2082 y\u2081 y\u2082 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * slope W x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * slope W x\u2081 x\u2082 y\u2081 y\u2082 -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n { a := 1, b := -(x\u2081 + x\u2082 + addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)),\n c := x\u2081 * x\u2082 + x\u2081 * addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) + x\u2082 * addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082),\n d := -(x\u2081 * x\u2082 * addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) }\n[PROOFSTEP]\nrw [slope_of_X_ne hx]\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n c :=\n 2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n { a := 1, b := -(x\u2081 + x\u2082 + addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n c := x\u2081 * x\u2082 + x\u2081 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n d := -(x\u2081 * x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero] at hx \n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u271d : \u00acx\u2081 = x\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n c :=\n 2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n { a := 1, b := -(x\u2081 + x\u2082 + addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n c := x\u2081 * x\u2082 + x\u2081 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n d := -(x\u2081 * x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }\n[PROOFSTEP]\next\n[GOAL]\ncase neg.a\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u271d : \u00acx\u2081 = x\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n c :=\n 2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.a =\n { a := 1, b := -(x\u2081 + x\u2082 + addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n c := x\u2081 * x\u2082 + x\u2081 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n d := -(x\u2081 * x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg.b\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u271d : \u00acx\u2081 = x\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n c :=\n 2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.b =\n { a := 1, b := -(x\u2081 + x\u2082 + addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n c := x\u2081 * x\u2082 + x\u2081 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n d := -(x\u2081 * x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.b\n[PROOFSTEP]\nsimp only [addX]\n[GOAL]\ncase neg.b\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u271d : \u00acx\u2081 = x\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082 =\n -(x\u2081 + x\u2082 + (((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) - W.a\u2082 - x\u2081 - x\u2082))\n[PROOFSTEP]\nring1\n[GOAL]\ncase neg.c\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u271d : \u00acx\u2081 = x\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n c :=\n 2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.c =\n { a := 1, b := -(x\u2081 + x\u2082 + addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n c := x\u2081 * x\u2082 + x\u2081 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n d := -(x\u2081 * x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.c\n[PROOFSTEP]\napply mul_right_injective\u2080 hx\n[GOAL]\ncase neg.c.a\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u271d : \u00acx\u2081 = x\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (fun x x_1 => x * x_1) (x\u2081 - x\u2082)\n { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n c :=\n 2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.c =\n (fun x x_1 => x * x_1) (x\u2081 - x\u2082)\n { a := 1, b := -(x\u2081 + x\u2082 + addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n c := x\u2081 * x\u2082 + x\u2081 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n d := -(x\u2081 * x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.c\n[PROOFSTEP]\nlinear_combination (norm := (field_simp [hx]; ring1)) h\u2082' - h\u2081'\n[GOAL]\ncase a\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u271d : \u00acx\u2081 = x\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (fun x x_1 => x * x_1) (x\u2081 - x\u2082)\n { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n c :=\n 2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.c -\n (fun x x_1 => x * x_1) (x\u2081 - x\u2082)\n { a := 1, b := -(x\u2081 + x\u2082 + addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n c := x\u2081 * x\u2082 + x\u2081 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n d := -(x\u2081 * x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.c -\n (y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 - (y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081) -\n (x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086 - (x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086))) =\n 0\n[PROOFSTEP]\nfield_simp [hx]\n[GOAL]\ncase a\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u271d : \u00acx\u2081 = x\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (x\u2081 - x\u2082) *\n (2 * x\u2081 * (y\u2081 - y\u2082) ^ 2 * (x\u2081 - x\u2082) + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * (y\u2081 - y\u2082) * (x\u2081 - x\u2082) ^ 2 +\n (-(W.a\u2081 * y\u2081) + W.a\u2084) * ((x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082))) -\n (x\u2081 - x\u2082) *\n (x\u2081 * x\u2082 * ((x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082)) +\n x\u2081 *\n ((y\u2081 - y\u2082) ^ 2 * (x\u2081 - x\u2082) + W.a\u2081 * (y\u2081 - y\u2082) * (x\u2081 - x\u2082) ^ 2 - (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * W.a\u2082 -\n (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2081 -\n (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2082) +\n x\u2082 *\n ((y\u2081 - y\u2082) ^ 2 * (x\u2081 - x\u2082) + W.a\u2081 * (y\u2081 - y\u2082) * (x\u2081 - x\u2082) ^ 2 - (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * W.a\u2082 -\n (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2081 -\n (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2082)) -\n (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) *\n (y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 - (y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081) -\n (x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 - (x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081))) =\n 0\n[PROOFSTEP]\nring1\n[GOAL]\ncase neg.d\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u271d : \u00acx\u2081 = x\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n c :=\n 2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.d =\n { a := 1, b := -(x\u2081 + x\u2082 + addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n c := x\u2081 * x\u2082 + x\u2081 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n d := -(x\u2081 * x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.d\n[PROOFSTEP]\napply mul_right_injective\u2080 hx\n[GOAL]\ncase neg.d.a\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u271d : \u00acx\u2081 = x\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (fun x x_1 => x * x_1) (x\u2081 - x\u2082)\n { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n c :=\n 2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.d =\n (fun x x_1 => x * x_1) (x\u2081 - x\u2082)\n { a := 1, b := -(x\u2081 + x\u2082 + addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n c := x\u2081 * x\u2082 + x\u2081 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n d := -(x\u2081 * x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.d\n[PROOFSTEP]\nlinear_combination (norm := (field_simp [hx]; ring1)) x\u2082 * h\u2081' - x\u2081 * h\u2082'\n[GOAL]\ncase a\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u271d : \u00acx\u2081 = x\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (fun x x_1 => x * x_1) (x\u2081 - x\u2082)\n { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n c :=\n 2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n (-W.a\u2081 * y\u2081 + W.a\u2084),\n d :=\n -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.d -\n (fun x x_1 => x * x_1) (x\u2081 - x\u2082)\n { a := 1, b := -(x\u2081 + x\u2082 + addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n c := x\u2081 * x\u2082 + x\u2081 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n d := -(x\u2081 * x\u2082 * addX W x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.d -\n (x\u2082 * (y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081) - x\u2081 * (y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082) -\n (x\u2082 * (x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086) - x\u2081 * (x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086))) =\n 0\n[PROOFSTEP]\nfield_simp [hx]\n[GOAL]\ncase a\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082' : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u271d : \u00acx\u2081 = x\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (x\u2081 - x\u2082) *\n (-(x\u2081 ^ 2 * (y\u2081 - y\u2082) ^ 2 * (x\u2081 - x\u2082)) + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * (y\u2081 - y\u2082) * (x\u2081 - x\u2082) ^ 2 -\n (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086)) +\n (x\u2081 - x\u2082) *\n (x\u2081 * x\u2082 *\n ((y\u2081 - y\u2082) ^ 2 * (x\u2081 - x\u2082) + W.a\u2081 * (y\u2081 - y\u2082) * (x\u2081 - x\u2082) ^ 2 - (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * W.a\u2082 -\n (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2081 -\n (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2082)) -\n (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) *\n (x\u2082 * (y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081) - x\u2081 * (y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082) -\n (x\u2082 * (x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086) - x\u2081 * (x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086))) =\n 0\n[PROOFSTEP]\nring1\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 \u2191(mk W) (\u2191C (addPolynomial W x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) =\n -(XClass W x\u2081 * XClass W x\u2082 * XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))\n[PROOFSTEP]\nsimp only [addPolynomial_slope h\u2081' h\u2082' hxy, C_neg, mk, map_neg, neg_inj, _root_.map_mul]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 \u2191(AdjoinRoot.mk (WeierstrassCurve.polynomial W)) (\u2191C (Y - \u2191C x\u2081)) *\n \u2191(AdjoinRoot.mk (WeierstrassCurve.polynomial W)) (\u2191C (Y - \u2191C x\u2082)) *\n \u2191(AdjoinRoot.mk (WeierstrassCurve.polynomial W)) (\u2191C (Y - \u2191C (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))) =\n XClass W x\u2081 * XClass W x\u2082 * XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 \u2191derivative (addPolynomial W x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) =\n -((Y - \u2191C x\u2081) * (Y - \u2191C x\u2082) + (Y - \u2191C x\u2081) * (Y - \u2191C (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) +\n (Y - \u2191C x\u2082) * (Y - \u2191C (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))))\n[PROOFSTEP]\nrw [addPolynomial_slope h\u2081' h\u2082' hxy]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 \u2191derivative (-((Y - \u2191C x\u2081) * (Y - \u2191C x\u2082) * (Y - \u2191C (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))))) =\n -((Y - \u2191C x\u2081) * (Y - \u2191C x\u2082) + (Y - \u2191C x\u2081) * (Y - \u2191C (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) +\n (Y - \u2191C x\u2082) * (Y - \u2191C (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))))\n[PROOFSTEP]\nderivative_simp\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 -(((1 - 0) * (Y - \u2191C x\u2082) + (Y - \u2191C x\u2081) * (1 - 0)) * (Y - \u2191C (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) +\n (Y - \u2191C x\u2081) * (Y - \u2191C x\u2082) * (1 - 0)) =\n -((Y - \u2191C x\u2081) * (Y - \u2191C x\u2082) + (Y - \u2191C x\u2081) * (Y - \u2191C (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) +\n (Y - \u2191C x\u2082) * (Y - \u2191C (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))))\n[PROOFSTEP]\nring1\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 WeierstrassCurve.equation W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (addY' W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nrw [equation_add_iff, addPolynomial_slope h\u2081' h\u2082' hxy]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 eval (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n (-((Y - \u2191C x\u2081) * (Y - \u2191C x\u2082) * (Y - \u2191C (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))))) =\n 0\n[PROOFSTEP]\neval_simp\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 -((addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) - x\u2081) * (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) - x\u2082) *\n (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) - addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) =\n 0\n[PROOFSTEP]\nrw [neg_eq_zero, sub_self, mul_zero]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (addY' W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nclear h\u2082'\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (addY' W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nby_cases hx\u2081 : W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) = x\u2081\n[GOAL]\ncase pos\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u2081 : addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2081\n\u22a2 WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (addY' W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nrwa [addY', hx\u2081, sub_self, mul_zero, zero_add]\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u2081 : \u00acaddX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2081\n\u22a2 WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (addY' W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nby_cases hx\u2082 : W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) = x\u2082\n[GOAL]\ncase pos\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u2081 : \u00acaddX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2081\nhx\u2082 : addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2082\n\u22a2 WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (addY' W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nby_cases hx : x\u2081 = x\u2082\n[GOAL]\ncase pos\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u2081 : \u00acaddX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2081\nhx\u2082 : addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2082\nhx : x\u2081 = x\u2082\n\u22a2 WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (addY' W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nsubst hx\n[GOAL]\ncase pos\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2081 y\u2082\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 negY W x\u2081 y\u2082\nhx\u2081 : \u00acaddX W x\u2081 x\u2081 (slope W x\u2081 x\u2081 y\u2081 y\u2082) = x\u2081\nhx\u2082 : addX W x\u2081 x\u2081 (slope W x\u2081 x\u2081 y\u2081 y\u2082) = x\u2081\n\u22a2 WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2081 (slope W x\u2081 x\u2081 y\u2081 y\u2082)) (addY' W x\u2081 x\u2081 y\u2081 (slope W x\u2081 x\u2081 y\u2081 y\u2082))\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u2081 : \u00acaddX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2081\nhx\u2082 : addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (addY' W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\nrwa [addY', \u2190 neg_sub, mul_neg, hx\u2082, slope_of_X_ne hx, div_mul_cancel _ <| sub_ne_zero_of_ne hx, neg_sub,\n sub_add_cancel]\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u2081 : \u00acaddX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2081\nhx\u2082 : \u00acaddX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2082\n\u22a2 WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (addY' W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n[PROOFSTEP]\napply W.nonsingular_add_of_eval_derivative_ne_zero _ _ _ _ (equation_add' h\u2081.1 h\u2082.1 hxy)\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u2081 : \u00acaddX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2081\nhx\u2082 : \u00acaddX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2082\n\u22a2 eval (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (\u2191derivative (addPolynomial W x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) \u2260 0\n[PROOFSTEP]\nrw [derivative_addPolynomial_slope h\u2081.left h\u2082.left hxy]\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u2081 : \u00acaddX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2081\nhx\u2082 : \u00acaddX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2082\n\u22a2 eval (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))\n (-((Y - \u2191C x\u2081) * (Y - \u2191C x\u2082) + (Y - \u2191C x\u2081) * (Y - \u2191C (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) +\n (Y - \u2191C x\u2082) * (Y - \u2191C (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))))) \u2260\n 0\n[PROOFSTEP]\neval_simp\n[GOAL]\ncase neg\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nhx\u2081 : \u00acaddX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2081\nhx\u2082 : \u00acaddX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) = x\u2082\n\u22a2 -((addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) - x\u2081) * (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) - x\u2082) +\n (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) - x\u2081) *\n (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) - addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) +\n (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) - x\u2082) *\n (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082) - addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) \u2260\n 0\n[PROOFSTEP]\nsimpa only [neg_ne_zero, sub_self, mul_zero, add_zero] using mul_ne_zero (sub_ne_zero_of_ne hx\u2081) (sub_ne_zero_of_ne hx\u2082)\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 \u2200 (a : Point W), 0 + a = a\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase zero\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 0 + zero = zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nx\u271d y\u271d : F\nh\u271d : WeierstrassCurve.nonsingular W x\u271d y\u271d\n\u22a2 0 + some h\u271d = some h\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 \u2200 (a : Point W), a + 0 = a\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase zero\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 zero + 0 = zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nx\u271d y\u271d : F\nh\u271d : WeierstrassCurve.nonsingular W x\u271d y\u271d\n\u22a2 some h\u271d + 0 = some h\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : y\u2081 = negY W x\u2082 y\u2082\n\u22a2 some h\u2081 + some h\u2082 = 0\n[PROOFSTEP]\nsimp only [\u2190 add_def, add, dif_pos hx, dif_pos hy]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 some h\u2081 + some h\u2082 =\n some\n (_ : WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (addY W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))\n[PROOFSTEP]\nsimp only [\u2190 add_def, add, dif_pos hx, dif_neg hy]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nW : WeierstrassCurve F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhx : x\u2081 \u2260 x\u2082\n\u22a2 some h\u2081 + some h\u2082 =\n some\n (_ : WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (addY W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))\n[PROOFSTEP]\nsimp only [\u2190 add_def, add, dif_neg hx]\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 XYIdeal W x\u2081 (\u2191C (negY W x\u2081 y\u2081)) * XYIdeal W x\u2081 (\u2191C y\u2081) = XIdeal W x\u2081\n[PROOFSTEP]\nhave Y_rw :\n (Y - C (C y\u2081)) * (Y - C (C (W.negY x\u2081 y\u2081))) -\n C (X - C x\u2081) * (C (X ^ 2 + C (x\u2081 + W.a\u2082) * X + C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - C (C W.a\u2081) * Y) =\n W.polynomial * 1 :=\n by\n linear_combination (norm := (rw [negY, WeierstrassCurve.polynomial]; C_simp; ring1))\n congr_arg C (congr_arg C ((W.equation_iff _ _).mp h\u2081.left).symm)\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\n[PROOFSTEP]\nlinear_combination (norm := (rw [negY, WeierstrassCurve.polynomial]; C_simp; ring1))\n congr_arg C (congr_arg C ((W.equation_iff _ _).mp h\u2081.left).symm)\n[GOAL]\ncase a\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) -\n WeierstrassCurve.polynomial W * 1 -\n (\u2191C (\u2191C (x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086)) - \u2191C (\u2191C (y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081))) =\n 0\n[PROOFSTEP]\nrw [negY, WeierstrassCurve.polynomial]\n[GOAL]\ncase a\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) -\n (Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) * 1 -\n (\u2191C (\u2191C (x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086)) - \u2191C (\u2191C (y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081))) =\n 0\n[PROOFSTEP]\nC_simp\n[GOAL]\ncase a\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 (Y - \u2191C (\u2191C y\u2081)) * (Y - (-\u2191C (\u2191C y\u2081) - \u2191C (\u2191C W.a\u2081) * \u2191C (\u2191C x\u2081) - \u2191C (\u2191C W.a\u2083))) -\n (\u2191C Y - \u2191C (\u2191C x\u2081)) *\n (\u2191C Y ^ 2 + (\u2191C (\u2191C x\u2081) + \u2191C (\u2191C W.a\u2082)) * \u2191C Y +\n (\u2191C (\u2191C x\u2081) ^ 2 + \u2191C (\u2191C W.a\u2082) * \u2191C (\u2191C x\u2081) + \u2191C (\u2191C W.a\u2084)) -\n \u2191C (\u2191C W.a\u2081) * Y) -\n (Y ^ 2 + (\u2191C (\u2191C W.a\u2081) * \u2191C Y + \u2191C (\u2191C W.a\u2083)) * Y -\n (\u2191C Y ^ 3 + \u2191C (\u2191C W.a\u2082) * \u2191C Y ^ 2 + \u2191C (\u2191C W.a\u2084) * \u2191C Y + \u2191C (\u2191C W.a\u2086))) *\n 1 -\n (\u2191C (\u2191C x\u2081) ^ 3 + \u2191C (\u2191C W.a\u2082) * \u2191C (\u2191C x\u2081) ^ 2 + \u2191C (\u2191C W.a\u2084) * \u2191C (\u2191C x\u2081) + \u2191C (\u2191C W.a\u2086) -\n (\u2191C (\u2191C y\u2081) ^ 2 + \u2191C (\u2191C W.a\u2081) * \u2191C (\u2191C x\u2081) * \u2191C (\u2191C y\u2081) + \u2191C (\u2191C W.a\u2083) * \u2191C (\u2191C y\u2081))) =\n 0\n[PROOFSTEP]\nring1\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\n\u22a2 XYIdeal W x\u2081 (\u2191C (negY W x\u2081 y\u2081)) * XYIdeal W x\u2081 (\u2191C y\u2081) = XIdeal W x\u2081\n[PROOFSTEP]\nsimp_rw [XYIdeal, XClass, YClass, span_pair_mul_span_pair, mul_comm, \u2190 _root_.map_mul,\n AdjoinRoot.mk_eq_mk.mpr \u27e81, Y_rw\u27e9, _root_.map_mul, span_insert, \u2190 span_singleton_mul_span_singleton, \u2190 mul_sup, \u2190\n span_insert]\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\n\u22a2 span {\u2191(CoordinateRing.mk W) (\u2191C (Y - \u2191C x\u2081))} *\n span\n {\u2191(CoordinateRing.mk W) (\u2191C (Y - \u2191C x\u2081)), \u2191(CoordinateRing.mk W) (Y - \u2191C (\u2191C y\u2081)),\n \u2191(CoordinateRing.mk W) (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))),\n \u2191(AdjoinRoot.mk (WeierstrassCurve.polynomial W))\n (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y)} =\n XIdeal W x\u2081\n[PROOFSTEP]\nconvert mul_top (_ : Ideal W.CoordinateRing) using 2\n[GOAL]\ncase h.e'_2.h.e'_6\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\n\u22a2 span\n {\u2191(CoordinateRing.mk W) (\u2191C (Y - \u2191C x\u2081)), \u2191(CoordinateRing.mk W) (Y - \u2191C (\u2191C y\u2081)),\n \u2191(CoordinateRing.mk W) (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))),\n \u2191(AdjoinRoot.mk (WeierstrassCurve.polynomial W))\n (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y)} =\n \u22a4\n[PROOFSTEP]\nsimp_rw [\u2190 @Set.image_singleton _ _ <| CoordinateRing.mk W, \u2190 Set.image_insert_eq, \u2190 map_span]\n[GOAL]\ncase h.e'_2.h.e'_6\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\n\u22a2 Ideal.map (CoordinateRing.mk W)\n (span\n {\u2191C (Y - \u2191C x\u2081), Y - \u2191C (\u2191C y\u2081), Y - \u2191C (\u2191C (negY W x\u2081 y\u2081)),\n \u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y}) =\n \u22a4\n[PROOFSTEP]\nconvert map_top (R := F[X][Y]) (CoordinateRing.mk W) using 1\n[GOAL]\ncase h.e'_2\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\n\u22a2 Ideal.map (CoordinateRing.mk W)\n (span\n {\u2191C (Y - \u2191C x\u2081), Y - \u2191C (\u2191C y\u2081), Y - \u2191C (\u2191C (negY W x\u2081 y\u2081)),\n \u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y}) =\n Ideal.map (CoordinateRing.mk W) \u22a4\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h.e'_2.h\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\n\u22a2 span\n {\u2191C (Y - \u2191C x\u2081), Y - \u2191C (\u2191C y\u2081), Y - \u2191C (\u2191C (negY W x\u2081 y\u2081)),\n \u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y} =\n \u22a4\n[PROOFSTEP]\nsimp_rw [eq_top_iff_one, mem_span_insert', mem_span_singleton']\n[GOAL]\ncase h.e'_2.h\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\n\u22a2 \u2203 a a_1 a_2 a_3,\n a_3 * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n 1 + a * \u2191C (Y - \u2191C x\u2081) + a_1 * (Y - \u2191C (\u2191C y\u2081)) + a_2 * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081)))\n[PROOFSTEP]\nrcases((W.nonsingular_iff' _ _).mp h\u2081).right with hx | hy\n[GOAL]\ncase h.e'_2.h.inl\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\nhx : W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084) \u2260 0\n\u22a2 \u2203 a a_1 a_2 a_3,\n a_3 * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n 1 + a * \u2191C (Y - \u2191C x\u2081) + a_1 * (Y - \u2191C (\u2191C y\u2081)) + a_2 * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081)))\n[PROOFSTEP]\nlet W_X := W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084)\n[GOAL]\ncase h.e'_2.h.inl\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\nhx : W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084) \u2260 0\nW_X : F := W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084)\n\u22a2 \u2203 a a_1 a_2 a_3,\n a_3 * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n 1 + a * \u2191C (Y - \u2191C x\u2081) + a_1 * (Y - \u2191C (\u2191C y\u2081)) + a_2 * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081)))\n[PROOFSTEP]\nrefine \u27e8C (C W_X\u207b\u00b9 * -(X + C (2 * x\u2081 + W.a\u2082))), C (C <| W_X\u207b\u00b9 * W.a\u2081), 0, C (C <| W_X\u207b\u00b9 * -1), ?_\u27e9\n[GOAL]\ncase h.e'_2.h.inl\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\nhx : W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084) \u2260 0\nW_X : F := W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084)\n\u22a2 \u2191C (\u2191C (W_X\u207b\u00b9 * -1)) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n 1 + \u2191C (\u2191C W_X\u207b\u00b9 * -(Y + \u2191C (2 * x\u2081 + W.a\u2082))) * \u2191C (Y - \u2191C x\u2081) + \u2191C (\u2191C (W_X\u207b\u00b9 * W.a\u2081)) * (Y - \u2191C (\u2191C y\u2081)) +\n 0 * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081)))\n[PROOFSTEP]\nrw [\u2190 mul_right_inj' <| C_ne_zero.mpr <| C_ne_zero.mpr hx]\n[GOAL]\ncase h.e'_2.h.inl\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\nhx : W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084) \u2260 0\nW_X : F := W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084)\n\u22a2 \u2191C (\u2191C (W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084))) *\n (\u2191C (\u2191C (W_X\u207b\u00b9 * -1)) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y)) =\n \u2191C (\u2191C (W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084))) *\n (1 + \u2191C (\u2191C W_X\u207b\u00b9 * -(Y + \u2191C (2 * x\u2081 + W.a\u2082))) * \u2191C (Y - \u2191C x\u2081) + \u2191C (\u2191C (W_X\u207b\u00b9 * W.a\u2081)) * (Y - \u2191C (\u2191C y\u2081)) +\n 0 * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))))\n[PROOFSTEP]\nsimp only [mul_add, \u2190 mul_assoc, \u2190 C_mul, mul_inv_cancel hx]\n[GOAL]\ncase h.e'_2.h.inl\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\nhx : W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084) \u2260 0\nW_X : F := W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084)\n\u22a2 \u2191C (\u2191C (1 * -1)) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n \u2191C (\u2191C (W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084))) * 1 +\n \u2191C (\u2191C 1 * -(Y + \u2191C (2 * x\u2081 + W.a\u2082)) * (Y - \u2191C x\u2081)) +\n \u2191C (\u2191C (1 * W.a\u2081)) * (Y - \u2191C (\u2191C y\u2081)) +\n \u2191C (\u2191C (W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084))) * 0 * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081)))\n[PROOFSTEP]\nC_simp\n[GOAL]\ncase h.e'_2.h.inl\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\nhx : W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084) \u2260 0\nW_X : F := W.a\u2081 * y\u2081 - (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084)\n\u22a2 1 * -1 *\n (\u2191C Y ^ 2 + (\u2191C (\u2191C x\u2081) + \u2191C (\u2191C W.a\u2082)) * \u2191C Y + (\u2191C (\u2191C x\u2081) ^ 2 + \u2191C (\u2191C W.a\u2082) * \u2191C (\u2191C x\u2081) + \u2191C (\u2191C W.a\u2084)) -\n \u2191C (\u2191C W.a\u2081) * Y) =\n (\u2191C (\u2191C W.a\u2081) * \u2191C (\u2191C y\u2081) - (3 * \u2191C (\u2191C x\u2081) ^ 2 + 2 * \u2191C (\u2191C W.a\u2082) * \u2191C (\u2191C x\u2081) + \u2191C (\u2191C W.a\u2084))) * 1 +\n 1 * -(\u2191C Y + (2 * \u2191C (\u2191C x\u2081) + \u2191C (\u2191C W.a\u2082))) * (\u2191C Y - \u2191C (\u2191C x\u2081)) +\n 1 * \u2191C (\u2191C W.a\u2081) * (Y - \u2191C (\u2191C y\u2081)) +\n (\u2191C (\u2191C W.a\u2081) * \u2191C (\u2191C y\u2081) - (3 * \u2191C (\u2191C x\u2081) ^ 2 + 2 * \u2191C (\u2191C W.a\u2082) * \u2191C (\u2191C x\u2081) + \u2191C (\u2191C W.a\u2084))) * 0 *\n (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081)))\n[PROOFSTEP]\nring1\n[GOAL]\ncase h.e'_2.h.inr\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\nhy : 2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083 \u2260 0\n\u22a2 \u2203 a a_1 a_2 a_3,\n a_3 * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n 1 + a * \u2191C (Y - \u2191C x\u2081) + a_1 * (Y - \u2191C (\u2191C y\u2081)) + a_2 * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081)))\n[PROOFSTEP]\nlet W_Y := 2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083\n[GOAL]\ncase h.e'_2.h.inr\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\nhy : 2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083 \u2260 0\nW_Y : F := 2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083\n\u22a2 \u2203 a a_1 a_2 a_3,\n a_3 * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n 1 + a * \u2191C (Y - \u2191C x\u2081) + a_1 * (Y - \u2191C (\u2191C y\u2081)) + a_2 * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081)))\n[PROOFSTEP]\nrefine \u27e80, C (C W_Y\u207b\u00b9), C (C <| W_Y\u207b\u00b9 * -1), 0, ?_\u27e9\n[GOAL]\ncase h.e'_2.h.inr\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\nhy : 2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083 \u2260 0\nW_Y : F := 2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083\n\u22a2 0 * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n 1 + 0 * \u2191C (Y - \u2191C x\u2081) + \u2191C (\u2191C W_Y\u207b\u00b9) * (Y - \u2191C (\u2191C y\u2081)) + \u2191C (\u2191C (W_Y\u207b\u00b9 * -1)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081)))\n[PROOFSTEP]\nrw [negY, \u2190 mul_right_inj' <| C_ne_zero.mpr <| C_ne_zero.mpr hy]\n[GOAL]\ncase h.e'_2.h.inr\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\nhy : 2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083 \u2260 0\nW_Y : F := 2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083\n\u22a2 \u2191C (\u2191C (2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083)) *\n (0 * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y)) =\n \u2191C (\u2191C (2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083)) *\n (1 + 0 * \u2191C (Y - \u2191C x\u2081) + \u2191C (\u2191C W_Y\u207b\u00b9) * (Y - \u2191C (\u2191C y\u2081)) +\n \u2191C (\u2191C (W_Y\u207b\u00b9 * -1)) * (Y - \u2191C (\u2191C (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083))))\n[PROOFSTEP]\nsimp only [mul_add, \u2190 mul_assoc, \u2190 C_mul, mul_inv_cancel hy]\n[GOAL]\ncase h.e'_2.h.inr\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\nhy : 2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083 \u2260 0\nW_Y : F := 2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083\n\u22a2 \u2191C (\u2191C (2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083)) * 0 *\n (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n \u2191C (\u2191C (2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083)) * 1 + \u2191C (\u2191C (2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083)) * 0 * \u2191C (Y - \u2191C x\u2081) +\n \u2191C (\u2191C 1) * (Y - \u2191C (\u2191C y\u2081)) +\n \u2191C (\u2191C (1 * -1)) * (Y - \u2191C (\u2191C (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)))\n[PROOFSTEP]\nC_simp\n[GOAL]\ncase h.e'_2.h.inr\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nY_rw :\n (Y - \u2191C (\u2191C y\u2081)) * (Y - \u2191C (\u2191C (negY W x\u2081 y\u2081))) -\n \u2191C (Y - \u2191C x\u2081) * (\u2191C (Y ^ 2 + \u2191C (x\u2081 + W.a\u2082) * Y + \u2191C (x\u2081 ^ 2 + W.a\u2082 * x\u2081 + W.a\u2084)) - \u2191C (\u2191C W.a\u2081) * Y) =\n WeierstrassCurve.polynomial W * 1\nhy : 2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083 \u2260 0\nW_Y : F := 2 * y\u2081 + W.a\u2081 * x\u2081 + W.a\u2083\n\u22a2 (2 * \u2191C (\u2191C y\u2081) + \u2191C (\u2191C W.a\u2081) * \u2191C (\u2191C x\u2081) + \u2191C (\u2191C W.a\u2083)) * 0 *\n (\u2191C Y ^ 2 + (\u2191C (\u2191C x\u2081) + \u2191C (\u2191C W.a\u2082)) * \u2191C Y + (\u2191C (\u2191C x\u2081) ^ 2 + \u2191C (\u2191C W.a\u2082) * \u2191C (\u2191C x\u2081) + \u2191C (\u2191C W.a\u2084)) -\n \u2191C (\u2191C W.a\u2081) * Y) =\n (2 * \u2191C (\u2191C y\u2081) + \u2191C (\u2191C W.a\u2081) * \u2191C (\u2191C x\u2081) + \u2191C (\u2191C W.a\u2083)) * 1 +\n (2 * \u2191C (\u2191C y\u2081) + \u2191C (\u2191C W.a\u2081) * \u2191C (\u2191C x\u2081) + \u2191C (\u2191C W.a\u2083)) * 0 * (\u2191C Y - \u2191C (\u2191C x\u2081)) +\n 1 * (Y - \u2191C (\u2191C y\u2081)) +\n 1 * -1 * (Y - (-\u2191C (\u2191C y\u2081) - \u2191C (\u2191C W.a\u2081) * \u2191C (\u2191C x\u2081) - \u2191C (\u2191C W.a\u2083)))\n[PROOFSTEP]\nring1\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 \u2191(XYIdeal W x\u2081 (\u2191C y\u2081)) * (\u2191(XYIdeal W x\u2081 (\u2191C (negY W x\u2081 y\u2081))) * (\u2191(XIdeal W x\u2081))\u207b\u00b9) = 1\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 FractionalIdeal.coeIdeal_mul, mul_comm <| XYIdeal W _ _, XYIdeal_neg_mul h\u2081, XIdeal,\n FractionalIdeal.coe_ideal_span_singleton_mul_inv W.FunctionField <| XClass_ne_zero W x\u2081]\n -- porting note: increased `maxHeartbeats` for `ring1`\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 XIdeal W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) * (XYIdeal W x\u2081 (\u2191C y\u2081) * XYIdeal W x\u2082 (\u2191C y\u2082)) =\n YIdeal W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) *\n XYIdeal W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (\u2191C (addY W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))\n[PROOFSTEP]\nclear h\u2082\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 XIdeal W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) * (XYIdeal W x\u2081 (\u2191C y\u2081) * XYIdeal W x\u2082 (\u2191C y\u2082)) =\n YIdeal W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) *\n XYIdeal W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (\u2191C (addY W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))\n[PROOFSTEP]\nhave sup_rw : \u2200 a b c d : Ideal W.CoordinateRing, a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c := fun _ _ c _ => by\n rw [\u2190 sup_assoc, @sup_comm _ _ c, sup_sup_sup_comm, \u2190 sup_assoc]\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nx\u271d\u00b2 x\u271d\u00b9 c x\u271d : Ideal (CoordinateRing W)\n\u22a2 x\u271d\u00b2 \u2294 (x\u271d\u00b9 \u2294 (c \u2294 x\u271d)) = x\u271d\u00b2 \u2294 x\u271d \u2294 x\u271d\u00b9 \u2294 c\n[PROOFSTEP]\nrw [\u2190 sup_assoc, @sup_comm _ _ c, sup_sup_sup_comm, \u2190 sup_assoc]\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\n\u22a2 XIdeal W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) * (XYIdeal W x\u2081 (\u2191C y\u2081) * XYIdeal W x\u2082 (\u2191C y\u2082)) =\n YIdeal W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) *\n XYIdeal W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (\u2191C (addY W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))\n[PROOFSTEP]\nrw [XYIdeal_add_eq, XIdeal, mul_comm, W.XYIdeal_eq\u2081 x\u2081 y\u2081 <| W.slope x\u2081 x\u2082 y\u2081 y\u2082, XYIdeal, XYIdeal_eq\u2082 h\u2081' h\u2082' hxy,\n XYIdeal, span_pair_mul_span_pair]\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\n\u22a2 span\n {XClass W x\u2081 * XClass W x\u2082, XClass W x\u2081 * YClass W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)),\n YClass W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) * XClass W x\u2082,\n YClass W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) *\n YClass W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))} *\n span {XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))} =\n YIdeal W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) *\n (span {\u2191(CoordinateRing.mk W) (negPolynomial W - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))} \u2294\n span {XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))})\n[PROOFSTEP]\nsimp_rw [span_insert, sup_rw, sup_mul, span_singleton_mul_span_singleton]\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\n\u22a2 span {XClass W x\u2081 * XClass W x\u2082 * XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))} \u2294\n span\n {YClass W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) *\n YClass W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) *\n XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))} \u2294\n span\n {XClass W x\u2081 * YClass W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) *\n XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))} \u2294\n span\n {YClass W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) * XClass W x\u2082 *\n XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))} =\n YIdeal W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) *\n (span {\u2191(CoordinateRing.mk W) (negPolynomial W - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))} \u2294\n span {XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))})\n[PROOFSTEP]\nrw [\u2190 neg_eq_iff_eq_neg.mpr <| CoordinateRing.C_addPolynomial_slope h\u2081' h\u2082' hxy, span_singleton_neg,\n CoordinateRing.C_addPolynomial, _root_.map_mul, YClass]\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\n\u22a2 span\n {\u2191(CoordinateRing.mk W) (Y - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) *\n \u2191(CoordinateRing.mk W) (negPolynomial W - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))} \u2294\n span\n {\u2191(CoordinateRing.mk W) (Y - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) *\n \u2191(CoordinateRing.mk W) (Y - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) *\n XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))} \u2294\n span\n {XClass W x\u2081 * \u2191(CoordinateRing.mk W) (Y - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) *\n XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))} \u2294\n span\n {\u2191(CoordinateRing.mk W) (Y - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) * XClass W x\u2082 *\n XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))} =\n YIdeal W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) *\n (span {\u2191(CoordinateRing.mk W) (negPolynomial W - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))} \u2294\n span {XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))})\n[PROOFSTEP]\nsimp_rw [mul_comm <| XClass W x\u2081, mul_assoc, \u2190 span_singleton_mul_span_singleton, \u2190 mul_sup]\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\n\u22a2 span {\u2191(CoordinateRing.mk W) (Y - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))} *\n (span {\u2191(CoordinateRing.mk W) (negPolynomial W - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))} \u2294\n span {\u2191(CoordinateRing.mk W) (Y - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))} *\n span {XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))} \u2294\n span {XClass W x\u2081} * span {XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))} \u2294\n span {XClass W x\u2082} * span {XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))}) =\n YIdeal W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) *\n (span {\u2191(CoordinateRing.mk W) (negPolynomial W - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))} \u2294\n span {XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))})\n[PROOFSTEP]\nrw [span_singleton_mul_span_singleton, \u2190 span_insert, \u2190\n span_pair_add_mul_right <| -(XClass W <| W.addX x\u2081 x\u2082 <| W.slope x\u2081 x\u2082 y\u2081 y\u2082), mul_neg, \u2190 sub_eq_add_neg, \u2190 sub_mul, \u2190\n map_sub <| CoordinateRing.mk W, sub_sub_sub_cancel_right, span_insert, \u2190 span_singleton_mul_span_singleton, \u2190 sup_rw,\n \u2190 sup_mul, \u2190 sup_mul]\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\n\u22a2 span {\u2191(CoordinateRing.mk W) (Y - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))} *\n (span {\u2191(CoordinateRing.mk W) (negPolynomial W - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))} \u2294\n (span {XClass W x\u2081} \u2294 (span {XClass W x\u2082} \u2294 span {\u2191(CoordinateRing.mk W) (Y - negPolynomial W)})) *\n span {XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))}) =\n YIdeal W (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) *\n (span {\u2191(CoordinateRing.mk W) (negPolynomial W - \u2191C (linePolynomial x\u2081 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082)))} \u2294\n span {XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))})\n[PROOFSTEP]\napply congr_arg (_ \u2218 _)\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\n\u22a2 (span {XClass W x\u2081} \u2294 (span {XClass W x\u2082} \u2294 span {\u2191(CoordinateRing.mk W) (Y - negPolynomial W)})) *\n span {XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))} =\n span {XClass W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082))}\n[PROOFSTEP]\nconvert top_mul (_ : Ideal W.CoordinateRing)\n[GOAL]\ncase h.e'_2.h.e'_5\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\n\u22a2 span {XClass W x\u2081} \u2294 (span {XClass W x\u2082} \u2294 span {\u2191(CoordinateRing.mk W) (Y - negPolynomial W)}) = \u22a4\n[PROOFSTEP]\nsimp_rw [XClass, \u2190 @Set.image_singleton _ _ <| CoordinateRing.mk W, \u2190 map_span, \u2190 Ideal.map_sup, eq_top_iff_one,\n mem_map_iff_of_surjective _ <| AdjoinRoot.mk_surjective W.monic_polynomial, \u2190 span_insert, mem_span_insert',\n mem_span_singleton']\n[GOAL]\ncase h.e'_2.h.e'_5\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\n\u22a2 \u2203 x,\n (\u2203 a a_1 a_2, a_2 * (Y - negPolynomial W) = x + a * \u2191C (Y - \u2191C x\u2081) + a_1 * \u2191C (Y - \u2191C x\u2082)) \u2227\n \u2191(AdjoinRoot.mk (WeierstrassCurve.polynomial W)) x = 1\n[PROOFSTEP]\nby_cases hx : x\u2081 = x\u2082\n[GOAL]\ncase pos\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nhx : x\u2081 = x\u2082\n\u22a2 \u2203 x,\n (\u2203 a a_1 a_2, a_2 * (Y - negPolynomial W) = x + a * \u2191C (Y - \u2191C x\u2081) + a_1 * \u2191C (Y - \u2191C x\u2082)) \u2227\n \u2191(AdjoinRoot.mk (WeierstrassCurve.polynomial W)) x = 1\n[PROOFSTEP]\nrcases hx, Y_eq_of_Y_ne h\u2081' h\u2082' hx (hxy hx) with \u27e8rfl, rfl\u27e9\n[GOAL]\ncase pos\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nh\u2082' : WeierstrassCurve.equation W x\u2081 y\u2081\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 negY W x\u2081 y\u2081\n\u22a2 \u2203 x,\n (\u2203 a a_1 a_2, a_2 * (Y - negPolynomial W) = x + a * \u2191C (Y - \u2191C x\u2081) + a_1 * \u2191C (Y - \u2191C x\u2081)) \u2227\n \u2191(AdjoinRoot.mk (WeierstrassCurve.polynomial W)) x = 1\n[PROOFSTEP]\nlet y := (y\u2081 - W.negY x\u2081 y\u2081) ^ 2\n[GOAL]\ncase pos\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nh\u2082' : WeierstrassCurve.equation W x\u2081 y\u2081\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 negY W x\u2081 y\u2081\ny : F := (y\u2081 - negY W x\u2081 y\u2081) ^ 2\n\u22a2 \u2203 x,\n (\u2203 a a_1 a_2, a_2 * (Y - negPolynomial W) = x + a * \u2191C (Y - \u2191C x\u2081) + a_1 * \u2191C (Y - \u2191C x\u2081)) \u2227\n \u2191(AdjoinRoot.mk (WeierstrassCurve.polynomial W)) x = 1\n[PROOFSTEP]\nreplace hxy := pow_ne_zero 2 (sub_ne_zero_of_ne <| hxy rfl)\n[GOAL]\ncase pos\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nh\u2082' : WeierstrassCurve.equation W x\u2081 y\u2081\ny : F := (y\u2081 - negY W x\u2081 y\u2081) ^ 2\nhxy : (y\u2081 - negY W x\u2081 y\u2081) ^ 2 \u2260 0\n\u22a2 \u2203 x,\n (\u2203 a a_1 a_2, a_2 * (Y - negPolynomial W) = x + a * \u2191C (Y - \u2191C x\u2081) + a_1 * \u2191C (Y - \u2191C x\u2081)) \u2227\n \u2191(AdjoinRoot.mk (WeierstrassCurve.polynomial W)) x = 1\n[PROOFSTEP]\nrefine\n \u27e81 + C (C <| y\u207b\u00b9 * 4) * W.polynomial,\n \u27e8C <| C y\u207b\u00b9 * (C 4 * X ^ 2 + C (4 * x\u2081 + W.b\u2082) * X + C (4 * x\u2081 ^ 2 + W.b\u2082 * x\u2081 + 2 * W.b\u2084)), 0,\n C (C y\u207b\u00b9) * (Y - W.negPolynomial), ?_\u27e9,\n by rw [map_add, map_one, _root_.map_mul <| CoordinateRing.mk W, AdjoinRoot.mk_self, mul_zero, add_zero]\u27e9\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nh\u2082' : WeierstrassCurve.equation W x\u2081 y\u2081\ny : F := (y\u2081 - negY W x\u2081 y\u2081) ^ 2\nhxy : (y\u2081 - negY W x\u2081 y\u2081) ^ 2 \u2260 0\n\u22a2 \u2191(AdjoinRoot.mk (WeierstrassCurve.polynomial W)) (1 + \u2191C (\u2191C (y\u207b\u00b9 * 4)) * WeierstrassCurve.polynomial W) = 1\n[PROOFSTEP]\nrw [map_add, map_one, _root_.map_mul <| CoordinateRing.mk W, AdjoinRoot.mk_self, mul_zero, add_zero]\n[GOAL]\ncase pos\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nh\u2082' : WeierstrassCurve.equation W x\u2081 y\u2081\ny : F := (y\u2081 - negY W x\u2081 y\u2081) ^ 2\nhxy : (y\u2081 - negY W x\u2081 y\u2081) ^ 2 \u2260 0\n\u22a2 \u2191C (\u2191C y\u207b\u00b9) * (Y - negPolynomial W) * (Y - negPolynomial W) =\n 1 + \u2191C (\u2191C (y\u207b\u00b9 * 4)) * WeierstrassCurve.polynomial W +\n \u2191C (\u2191C y\u207b\u00b9 * (\u2191C 4 * Y ^ 2 + \u2191C (4 * x\u2081 + b\u2082 W) * Y + \u2191C (4 * x\u2081 ^ 2 + b\u2082 W * x\u2081 + 2 * b\u2084 W))) *\n \u2191C (Y - \u2191C x\u2081) +\n 0 * \u2191C (Y - \u2191C x\u2081)\n[PROOFSTEP]\nrw [WeierstrassCurve.polynomial, negPolynomial, \u2190 mul_right_inj' <| C_ne_zero.mpr <| C_ne_zero.mpr hxy]\n[GOAL]\ncase pos\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nh\u2082' : WeierstrassCurve.equation W x\u2081 y\u2081\ny : F := (y\u2081 - negY W x\u2081 y\u2081) ^ 2\nhxy : (y\u2081 - negY W x\u2081 y\u2081) ^ 2 \u2260 0\n\u22a2 \u2191C (\u2191C ((y\u2081 - negY W x\u2081 y\u2081) ^ 2)) *\n (\u2191C (\u2191C y\u207b\u00b9) * (Y - (-Y - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083))) * (Y - (-Y - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)))) =\n \u2191C (\u2191C ((y\u2081 - negY W x\u2081 y\u2081) ^ 2)) *\n (1 +\n \u2191C (\u2191C (y\u207b\u00b9 * 4)) *\n (Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) +\n \u2191C (\u2191C y\u207b\u00b9 * (\u2191C 4 * Y ^ 2 + \u2191C (4 * x\u2081 + b\u2082 W) * Y + \u2191C (4 * x\u2081 ^ 2 + b\u2082 W * x\u2081 + 2 * b\u2084 W))) *\n \u2191C (Y - \u2191C x\u2081) +\n 0 * \u2191C (Y - \u2191C x\u2081))\n[PROOFSTEP]\nsimp only [mul_add, \u2190 mul_assoc, \u2190 C_mul, mul_inv_cancel hxy]\n[GOAL]\ncase pos\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nh\u2082' : WeierstrassCurve.equation W x\u2081 y\u2081\ny : F := (y\u2081 - negY W x\u2081 y\u2081) ^ 2\nhxy : (y\u2081 - negY W x\u2081 y\u2081) ^ 2 \u2260 0\n\u22a2 \u2191C (\u2191C 1) * (Y - (-Y - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083))) * (Y - (-Y - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083))) =\n \u2191C (\u2191C ((y\u2081 - negY W x\u2081 y\u2081) ^ 2)) * 1 +\n \u2191C (\u2191C (1 * 4)) *\n (Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) +\n \u2191C\n ((\u2191C (1 * 4) * Y ^ 2 + \u2191C (1 * 4 * x\u2081 + 1 * b\u2082 W) * Y + \u2191C (1 * 4 * x\u2081 ^ 2 + 1 * b\u2082 W * x\u2081 + 1 * 2 * b\u2084 W)) *\n (Y - \u2191C x\u2081)) +\n \u2191C (\u2191C ((y\u2081 - negY W x\u2081 y\u2081) ^ 2)) * 0 * \u2191C (Y - \u2191C x\u2081)\n[PROOFSTEP]\nlinear_combination (norm := (rw [b\u2082, b\u2084, negY]; C_simp; ring1))\n -4 * congr_arg C (congr_arg C <| (W.equation_iff _ _).mp h\u2081')\n[GOAL]\ncase a\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nh\u2082' : WeierstrassCurve.equation W x\u2081 y\u2081\ny : F := (y\u2081 - negY W x\u2081 y\u2081) ^ 2\nhxy : (y\u2081 - negY W x\u2081 y\u2081) ^ 2 \u2260 0\n\u22a2 \u2191C (\u2191C 1) * (Y - (-Y - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083))) * (Y - (-Y - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083))) -\n (\u2191C (\u2191C ((y\u2081 - negY W x\u2081 y\u2081) ^ 2)) * 1 +\n \u2191C (\u2191C (1 * 4)) *\n (Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) +\n \u2191C\n ((\u2191C (1 * 4) * Y ^ 2 + \u2191C (1 * 4 * x\u2081 + 1 * b\u2082 W) * Y +\n \u2191C (1 * 4 * x\u2081 ^ 2 + 1 * b\u2082 W * x\u2081 + 1 * 2 * b\u2084 W)) *\n (Y - \u2191C x\u2081)) +\n \u2191C (\u2191C ((y\u2081 - negY W x\u2081 y\u2081) ^ 2)) * 0 * \u2191C (Y - \u2191C x\u2081)) -\n (-4 * \u2191C (\u2191C (y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081)) - -4 * \u2191C (\u2191C (x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086))) =\n 0\n[PROOFSTEP]\nrw [b\u2082, b\u2084, negY]\n[GOAL]\ncase a\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nh\u2082' : WeierstrassCurve.equation W x\u2081 y\u2081\ny : F := (y\u2081 - negY W x\u2081 y\u2081) ^ 2\nhxy : (y\u2081 - negY W x\u2081 y\u2081) ^ 2 \u2260 0\n\u22a2 \u2191C (\u2191C 1) * (Y - (-Y - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083))) * (Y - (-Y - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083))) -\n (\u2191C (\u2191C ((y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2)) * 1 +\n \u2191C (\u2191C (1 * 4)) *\n (Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) +\n \u2191C\n ((\u2191C (1 * 4) * Y ^ 2 + \u2191C (1 * 4 * x\u2081 + 1 * (W.a\u2081 ^ 2 + 4 * W.a\u2082)) * Y +\n \u2191C (1 * 4 * x\u2081 ^ 2 + 1 * (W.a\u2081 ^ 2 + 4 * W.a\u2082) * x\u2081 + 1 * 2 * (2 * W.a\u2084 + W.a\u2081 * W.a\u2083))) *\n (Y - \u2191C x\u2081)) +\n \u2191C (\u2191C ((y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2)) * 0 * \u2191C (Y - \u2191C x\u2081)) -\n (-4 * \u2191C (\u2191C (y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081)) - -4 * \u2191C (\u2191C (x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086))) =\n 0\n[PROOFSTEP]\nC_simp\n[GOAL]\ncase a\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nh\u2082' : WeierstrassCurve.equation W x\u2081 y\u2081\ny : F := (y\u2081 - negY W x\u2081 y\u2081) ^ 2\nhxy : (y\u2081 - negY W x\u2081 y\u2081) ^ 2 \u2260 0\n\u22a2 1 * (Y - (-Y - (\u2191C (\u2191C W.a\u2081) * \u2191C Y + \u2191C (\u2191C W.a\u2083)))) * (Y - (-Y - (\u2191C (\u2191C W.a\u2081) * \u2191C Y + \u2191C (\u2191C W.a\u2083)))) -\n ((\u2191C (\u2191C y\u2081) - (-\u2191C (\u2191C y\u2081) - \u2191C (\u2191C W.a\u2081) * \u2191C (\u2191C x\u2081) - \u2191C (\u2191C W.a\u2083))) ^ 2 * 1 +\n 1 * 4 *\n (Y ^ 2 + (\u2191C (\u2191C W.a\u2081) * \u2191C Y + \u2191C (\u2191C W.a\u2083)) * Y -\n (\u2191C Y ^ 3 + \u2191C (\u2191C W.a\u2082) * \u2191C Y ^ 2 + \u2191C (\u2191C W.a\u2084) * \u2191C Y + \u2191C (\u2191C W.a\u2086))) +\n (1 * 4 * \u2191C Y ^ 2 + (1 * 4 * \u2191C (\u2191C x\u2081) + 1 * (\u2191C (\u2191C W.a\u2081) ^ 2 + 4 * \u2191C (\u2191C W.a\u2082))) * \u2191C Y +\n (1 * 4 * \u2191C (\u2191C x\u2081) ^ 2 + 1 * (\u2191C (\u2191C W.a\u2081) ^ 2 + 4 * \u2191C (\u2191C W.a\u2082)) * \u2191C (\u2191C x\u2081) +\n 1 * 2 * (2 * \u2191C (\u2191C W.a\u2084) + \u2191C (\u2191C W.a\u2081) * \u2191C (\u2191C W.a\u2083)))) *\n (\u2191C Y - \u2191C (\u2191C x\u2081)) +\n (\u2191C (\u2191C y\u2081) - (-\u2191C (\u2191C y\u2081) - \u2191C (\u2191C W.a\u2081) * \u2191C (\u2191C x\u2081) - \u2191C (\u2191C W.a\u2083))) ^ 2 * 0 * (\u2191C Y - \u2191C (\u2191C x\u2081))) -\n (-4 * (\u2191C (\u2191C y\u2081) ^ 2 + \u2191C (\u2191C W.a\u2081) * \u2191C (\u2191C x\u2081) * \u2191C (\u2191C y\u2081) + \u2191C (\u2191C W.a\u2083) * \u2191C (\u2191C y\u2081)) -\n -4 * (\u2191C (\u2191C x\u2081) ^ 3 + \u2191C (\u2191C W.a\u2082) * \u2191C (\u2191C x\u2081) ^ 2 + \u2191C (\u2191C W.a\u2084) * \u2191C (\u2191C x\u2081) + \u2191C (\u2191C W.a\u2086))) =\n 0\n[PROOFSTEP]\nring1\n[GOAL]\ncase neg\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nhx : \u00acx\u2081 = x\u2082\n\u22a2 \u2203 x,\n (\u2203 a a_1 a_2, a_2 * (Y - negPolynomial W) = x + a * \u2191C (Y - \u2191C x\u2081) + a_1 * \u2191C (Y - \u2191C x\u2082)) \u2227\n \u2191(AdjoinRoot.mk (WeierstrassCurve.polynomial W)) x = 1\n[PROOFSTEP]\nreplace hx := sub_ne_zero_of_ne hx\n[GOAL]\ncase neg\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nhx : x\u2081 - x\u2082 \u2260 0\n\u22a2 \u2203 x,\n (\u2203 a a_1 a_2, a_2 * (Y - negPolynomial W) = x + a * \u2191C (Y - \u2191C x\u2081) + a_1 * \u2191C (Y - \u2191C x\u2082)) \u2227\n \u2191(AdjoinRoot.mk (WeierstrassCurve.polynomial W)) x = 1\n[PROOFSTEP]\nrefine \u27e8_, \u27e8\u27e8C <| C (x\u2081 - x\u2082)\u207b\u00b9, C <| C <| (x\u2081 - x\u2082)\u207b\u00b9 * -1, 0, ?_\u27e9, map_one _\u27e9\u27e9\n[GOAL]\ncase neg\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nhx : x\u2081 - x\u2082 \u2260 0\n\u22a2 0 * (Y - negPolynomial W) = 1 + \u2191C (\u2191C (x\u2081 - x\u2082)\u207b\u00b9) * \u2191C (Y - \u2191C x\u2081) + \u2191C (\u2191C ((x\u2081 - x\u2082)\u207b\u00b9 * -1)) * \u2191C (Y - \u2191C x\u2082)\n[PROOFSTEP]\nrw [\u2190 mul_right_inj' <| C_ne_zero.mpr <| C_ne_zero.mpr hx]\n[GOAL]\ncase neg\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nhx : x\u2081 - x\u2082 \u2260 0\n\u22a2 \u2191C (\u2191C (x\u2081 - x\u2082)) * (0 * (Y - negPolynomial W)) =\n \u2191C (\u2191C (x\u2081 - x\u2082)) * (1 + \u2191C (\u2191C (x\u2081 - x\u2082)\u207b\u00b9) * \u2191C (Y - \u2191C x\u2081) + \u2191C (\u2191C ((x\u2081 - x\u2082)\u207b\u00b9 * -1)) * \u2191C (Y - \u2191C x\u2082))\n[PROOFSTEP]\nsimp only [\u2190 mul_assoc, mul_add, \u2190 C_mul, mul_inv_cancel hx]\n[GOAL]\ncase neg\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nhx : x\u2081 - x\u2082 \u2260 0\n\u22a2 \u2191C (\u2191C (x\u2081 - x\u2082)) * 0 * (Y - negPolynomial W) =\n \u2191C (\u2191C (x\u2081 - x\u2082)) * 1 + \u2191C (\u2191C 1 * (Y - \u2191C x\u2081)) + \u2191C (\u2191C (1 * -1) * (Y - \u2191C x\u2082))\n[PROOFSTEP]\nC_simp\n[GOAL]\ncase neg\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\nsup_rw : \u2200 (a b c d : Ideal (CoordinateRing W)), a \u2294 (b \u2294 (c \u2294 d)) = a \u2294 d \u2294 b \u2294 c\nhx : x\u2081 - x\u2082 \u2260 0\n\u22a2 (\u2191C (\u2191C x\u2081) - \u2191C (\u2191C x\u2082)) * 0 * (Y - negPolynomial W) =\n (\u2191C (\u2191C x\u2081) - \u2191C (\u2191C x\u2082)) * 1 + 1 * (\u2191C Y - \u2191C (\u2191C x\u2081)) + 1 * -1 * (\u2191C Y - \u2191C (\u2191C x\u2082))\n[PROOFSTEP]\nring1\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 \u2191ClassGroup.mk (XYIdeal' (_ : WeierstrassCurve.nonsingular W x\u2081 (negY W x\u2081 y\u2081))) * \u2191ClassGroup.mk (XYIdeal' h\u2081) = 1\n[PROOFSTEP]\nrw [\u2190 _root_.map_mul]\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 \u2191ClassGroup.mk (XYIdeal' (_ : WeierstrassCurve.nonsingular W x\u2081 (negY W x\u2081 y\u2081)) * XYIdeal' h\u2081) = 1\n[PROOFSTEP]\nexact\n (ClassGroup.mk_eq_one_of_coe_ideal <| by\n exact\n (FractionalIdeal.coeIdeal_mul _ _).symm.trans <| FractionalIdeal.coeIdeal_inj.mpr <| XYIdeal_neg_mul h\u2081).mpr\n \u27e8_, XClass_ne_zero W _, rfl\u27e9\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 \u2191(XYIdeal' (_ : WeierstrassCurve.nonsingular W x\u2081 (negY W x\u2081 y\u2081)) * XYIdeal' h\u2081) = \u2191(span {XClass W ?m.4456306})\n[PROOFSTEP]\nexact (FractionalIdeal.coeIdeal_mul _ _).symm.trans <| FractionalIdeal.coeIdeal_inj.mpr <| XYIdeal_neg_mul h\u2081\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 \u2191ClassGroup.mk (XYIdeal' h\u2081) * \u2191ClassGroup.mk (XYIdeal' h\u2082) =\n \u2191ClassGroup.mk\n (XYIdeal'\n (_ :\n WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (addY W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))))\n[PROOFSTEP]\nrw [\u2190 _root_.map_mul]\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 \u2191ClassGroup.mk (XYIdeal' h\u2081 * XYIdeal' h\u2082) =\n \u2191ClassGroup.mk\n (XYIdeal'\n (_ :\n WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (addY W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))))\n[PROOFSTEP]\nexact\n (ClassGroup.mk_eq_mk_of_coe_ideal (by exact (FractionalIdeal.coeIdeal_mul _ _).symm) <| XYIdeal'_eq _).mpr\n \u27e8_, _, XClass_ne_zero W _, YClass_ne_zero W _, XYIdeal_mul_XYIdeal h\u2081.left h\u2082.left hxy\u27e9\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 negY W x\u2082 y\u2082\n\u22a2 \u2191(XYIdeal' h\u2081 * XYIdeal' h\u2082) = \u2191(XYIdeal W x\u2081 (\u2191C y\u2081) * XYIdeal W x\u2082 (\u2191C y\u2082))\n[PROOFSTEP]\nexact (FractionalIdeal.coeIdeal_mul _ _).symm\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 \u2200 (x y : Point W),\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (x + y) =\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } x +\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } y\n[PROOFSTEP]\nrintro (_ | @\u27e8x\u2081, y\u2081, h\u2081\u27e9) (_ | @\u27e8x\u2082, y\u2082, h\u2082\u27e9)\n[GOAL]\ncase zero.zero\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (zero + zero) =\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } zero +\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } zero\ncase zero.some\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082\u271d y\u2081 y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082\u271d : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2082 y\u2082 : F\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\n\u22a2 ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (zero + some h\u2082) =\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } zero +\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (some h\u2082)\ncase some.zero\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082 y\u2081\u271d y\u2082 : F\nh\u2081\u271d : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\n\u22a2 ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (some h\u2081 + zero) =\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (some h\u2081) +\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } zero\ncase some.some\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081\u271d : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082\u271d : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\n\u22a2 ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (some h\u2081 + some h\u2082) =\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (some h\u2081) +\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (some h\u2082)\n[PROOFSTEP]\nany_goals simp only [zero_def, toClassFun, _root_.zero_add, _root_.add_zero]\n[GOAL]\ncase zero.zero\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (zero + zero) =\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } zero +\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } zero\n[PROOFSTEP]\nsimp only [zero_def, toClassFun, _root_.zero_add, _root_.add_zero]\n[GOAL]\ncase zero.some\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082\u271d y\u2081 y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082\u271d : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2082 y\u2082 : F\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\n\u22a2 ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (zero + some h\u2082) =\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } zero +\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (some h\u2082)\n[PROOFSTEP]\nsimp only [zero_def, toClassFun, _root_.zero_add, _root_.add_zero]\n[GOAL]\ncase some.zero\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082 y\u2081\u271d y\u2082 : F\nh\u2081\u271d : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\n\u22a2 ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (some h\u2081 + zero) =\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (some h\u2081) +\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } zero\n[PROOFSTEP]\nsimp only [zero_def, toClassFun, _root_.zero_add, _root_.add_zero]\n[GOAL]\ncase some.some\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081\u271d : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082\u271d : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\n\u22a2 ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (some h\u2081 + some h\u2082) =\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (some h\u2081) +\n ZeroHom.toFun { toFun := toClassFun, map_zero' := (_ : toClassFun 0 = toClassFun 0) } (some h\u2082)\n[PROOFSTEP]\nsimp only [zero_def, toClassFun, _root_.zero_add, _root_.add_zero]\n[GOAL]\ncase some.some\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081\u271d : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082\u271d : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\n\u22a2 (match some h\u2081 + some h\u2082 with\n | zero => 0\n | some h => \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h))) =\n \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h\u2081)) + \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h\u2082))\n[PROOFSTEP]\nby_cases hx : x\u2081 = x\u2082\n[GOAL]\ncase pos\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081\u271d : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082\u271d : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\n\u22a2 (match some h\u2081 + some h\u2082 with\n | zero => 0\n | some h => \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h))) =\n \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h\u2081)) + \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h\u2082))\n[PROOFSTEP]\nby_cases hy : y\u2081 = W.negY x\u2082 y\u2082\n[GOAL]\ncase pos\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081\u271d : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082\u271d : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : y\u2081 = negY W x\u2082 y\u2082\n\u22a2 (match some h\u2081 + some h\u2082 with\n | zero => 0\n | some h => \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h))) =\n \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h\u2081)) + \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h\u2082))\n[PROOFSTEP]\nsubsts hx hy\n[GOAL]\ncase pos\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082 y\u2081 y\u2082\u271d : F\nh\u2081\u271d : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\nh\u2082\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\u271d\nx\u2081 y\u2082 : F\nh\u2082 : WeierstrassCurve.nonsingular W x\u2081 y\u2082\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 (negY W x\u2081 y\u2082)\n\u22a2 (match some h\u2081 + some h\u2082 with\n | zero => 0\n | some h => \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h))) =\n \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h\u2081)) + \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h\u2082))\n[PROOFSTEP]\nsimpa only [some_add_some_of_Y_eq rfl rfl] using (mk_XYIdeal'_mul_mk_XYIdeal'_of_Y_eq h\u2082).symm\n[GOAL]\ncase neg\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081\u271d : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082\u271d : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : \u00acy\u2081 = negY W x\u2082 y\u2082\n\u22a2 (match some h\u2081 + some h\u2082 with\n | zero => 0\n | some h => \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h))) =\n \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h\u2081)) + \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h\u2082))\n[PROOFSTEP]\nsimpa only [some_add_some_of_Y_ne hx hy] using (mk_XYIdeal'_mul_mk_XYIdeal' h\u2081 h\u2082 fun _ => hy).symm\n[GOAL]\ncase neg\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081\u271d : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082\u271d : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 (match some h\u2081 + some h\u2082 with\n | zero => 0\n | some h => \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h))) =\n \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h\u2081)) + \u2191Additive.ofMul (\u2191ClassGroup.mk (XYIdeal' h\u2082))\n[PROOFSTEP]\nsimpa only [some_add_some_of_X_ne hx] using (mk_XYIdeal'_mul_mk_XYIdeal' h\u2081 h\u2082 fun h => (hx h).elim).symm\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nP Q : Point W\n\u22a2 P + Q = 0 \u2194 P = -Q\n[PROOFSTEP]\nrcases P, Q with \u27e8_ | @\u27e8x\u2081, y\u2081, _\u27e9, _ | @\u27e8x\u2082, y\u2082, _\u27e9\u27e9\n[GOAL]\ncase zero.zero\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 zero + zero = 0 \u2194 zero = -zero\ncase zero.some\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082\u271d y\u2081 y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\n\u22a2 zero + some h\u271d = 0 \u2194 zero = -some h\u271d\ncase some.zero\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082 y\u2081\u271d y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nx\u2081 y\u2081 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2081 y\u2081\n\u22a2 some h\u271d + zero = 0 \u2194 some h\u271d = -zero\ncase some.some\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\n\u22a2 some h\u271d\u00b9 + some h\u271d = 0 \u2194 some h\u271d\u00b9 = -some h\u271d\n[PROOFSTEP]\nany_goals rfl\n[GOAL]\ncase zero.zero\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 zero + zero = 0 \u2194 zero = -zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase zero.some\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082\u271d y\u2081 y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\n\u22a2 zero + some h\u271d = 0 \u2194 zero = -some h\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.zero\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082 y\u2081\u271d y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nx\u2081 y\u2081 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2081 y\u2081\n\u22a2 some h\u271d + zero = 0 \u2194 some h\u271d = -zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\n\u22a2 some h\u271d\u00b9 + some h\u271d = 0 \u2194 some h\u271d\u00b9 = -some h\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase zero.some\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082\u271d y\u2081 y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\n\u22a2 zero + some h\u271d = 0 \u2194 zero = -some h\u271d\n[PROOFSTEP]\nrw [zero_def, zero_add, \u2190 neg_eq_iff_eq_neg, neg_zero, eq_comm]\n[GOAL]\ncase some.some\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\n\u22a2 some h\u271d\u00b9 + some h\u271d = 0 \u2194 some h\u271d\u00b9 = -some h\u271d\n[PROOFSTEP]\nrw [neg_some, some.injEq]\n[GOAL]\ncase some.some\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\n\u22a2 some h\u271d\u00b9 + some h\u271d = 0 \u2194 x\u2081 = x\u2082 \u2227 y\u2081 = negY W x\u2082 y\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase some.some.mp\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\n\u22a2 some h\u271d\u00b9 + some h\u271d = 0 \u2192 x\u2081 = x\u2082 \u2227 y\u2081 = negY W x\u2082 y\u2082\n[PROOFSTEP]\nintro h\n[GOAL]\ncase some.some.mp\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh : some h\u271d\u00b9 + some h\u271d = 0\n\u22a2 x\u2081 = x\u2082 \u2227 y\u2081 = negY W x\u2082 y\u2082\n[PROOFSTEP]\nby_cases hx : x\u2081 = x\u2082\n[GOAL]\ncase pos\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh : some h\u271d\u00b9 + some h\u271d = 0\nhx : x\u2081 = x\u2082\n\u22a2 x\u2081 = x\u2082 \u2227 y\u2081 = negY W x\u2082 y\u2082\n[PROOFSTEP]\nby_cases hy : y\u2081 = W.negY x\u2082 y\u2082\n[GOAL]\ncase pos\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh : some h\u271d\u00b9 + some h\u271d = 0\nhx : x\u2081 = x\u2082\nhy : y\u2081 = negY W x\u2082 y\u2082\n\u22a2 x\u2081 = x\u2082 \u2227 y\u2081 = negY W x\u2082 y\u2082\n[PROOFSTEP]\nexact \u27e8hx, hy\u27e9\n[GOAL]\ncase neg\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh : some h\u271d\u00b9 + some h\u271d = 0\nhx : x\u2081 = x\u2082\nhy : \u00acy\u2081 = negY W x\u2082 y\u2082\n\u22a2 x\u2081 = x\u2082 \u2227 y\u2081 = negY W x\u2082 y\u2082\n[PROOFSTEP]\nrw [some_add_some_of_Y_ne hx hy] at h \n[GOAL]\ncase neg\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : \u00acy\u2081 = negY W x\u2082 y\u2082\nh :\n some\n (_ :\n WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (addY W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) =\n 0\n\u22a2 x\u2081 = x\u2082 \u2227 y\u2081 = negY W x\u2082 y\u2082\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh : some h\u271d\u00b9 + some h\u271d = 0\nhx : \u00acx\u2081 = x\u2082\n\u22a2 x\u2081 = x\u2082 \u2227 y\u2081 = negY W x\u2082 y\u2082\n[PROOFSTEP]\nrw [some_add_some_of_X_ne hx] at h \n[GOAL]\ncase neg\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\nh :\n some\n (_ :\n WeierstrassCurve.nonsingular W (addX W x\u2081 x\u2082 (slope W x\u2081 x\u2082 y\u2081 y\u2082)) (addY W x\u2081 x\u2082 y\u2081 (slope W x\u2081 x\u2082 y\u2081 y\u2082))) =\n 0\n\u22a2 x\u2081 = x\u2082 \u2227 y\u2081 = negY W x\u2082 y\u2082\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase some.some.mpr\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081\u271d x\u2082\u271d y\u2081\u271d y\u2082\u271d : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081\u271d y\u2081\u271d\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082\u271d y\u2082\u271d\nh\u2081' : WeierstrassCurve.equation W x\u2081\u271d y\u2081\u271d\nh\u2082' : WeierstrassCurve.equation W x\u2082\u271d y\u2082\u271d\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular W x\u2082 y\u2082\n\u22a2 x\u2081 = x\u2082 \u2227 y\u2081 = negY W x\u2082 y\u2082 \u2192 some h\u271d\u00b9 + some h\u271d = 0\n[PROOFSTEP]\nexact fun \u27e8hx, hy\u27e9 => some_add_some_of_Y_eq hx hy\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nP : Point W\n\u22a2 -P + P = 0\n[PROOFSTEP]\nrw [add_eq_zero]\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nP Q : Point W\n\u22a2 -P + Q = 0 \u2194 P = Q\n[PROOFSTEP]\nrw [add_eq_zero, neg_inj]\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nP : Point W\n\u22a2 \u2191toClass P = 0 \u2194 P = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nP : Point W\n\u22a2 \u2191toClass P = 0 \u2192 P = 0\n[PROOFSTEP]\nintro hP\n[GOAL]\ncase mp\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nP : Point W\nhP : \u2191toClass P = 0\n\u22a2 P = 0\n[PROOFSTEP]\nrcases P with (_ | @\u27e8_, _, \u27e8h, _\u27e9\u27e9)\n[GOAL]\ncase mp.zero\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nhP : \u2191toClass zero = 0\n\u22a2 zero = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mp.some.intro\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nx\u271d y\u271d : F\nh : WeierstrassCurve.equation W x\u271d y\u271d\nright\u271d :\n eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialX W)) \u2260 0 \u2228\n eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialY W)) \u2260 0\nhP :\n \u2191toClass\n (some\n (_ :\n WeierstrassCurve.equation W x\u271d y\u271d \u2227\n (eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialX W)) \u2260 0 \u2228\n eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialY W)) \u2260 0))) =\n 0\n\u22a2 some\n (_ :\n WeierstrassCurve.equation W x\u271d y\u271d \u2227\n (eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialX W)) \u2260 0 \u2228\n eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialY W)) \u2260 0)) =\n 0\n[PROOFSTEP]\nrcases(ClassGroup.mk_eq_one_of_coe_ideal <| by rfl).mp hP with \u27e8p, h0, hp\u27e9\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nx\u271d y\u271d : F\nh : WeierstrassCurve.equation W x\u271d y\u271d\nright\u271d :\n eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialX W)) \u2260 0 \u2228\n eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialY W)) \u2260 0\nhP :\n \u2191toClass\n (some\n (_ :\n WeierstrassCurve.equation W x\u271d y\u271d \u2227\n (eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialX W)) \u2260 0 \u2228\n eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialY W)) \u2260 0))) =\n 0\n\u22a2 \u2191(XYIdeal'\n (_ :\n WeierstrassCurve.equation W x\u271d y\u271d \u2227\n (eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialX W)) \u2260 0 \u2228\n eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialY W)) \u2260 0))) =\n \u2191?m.4705169\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mp.some.intro.intro.intro\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nx\u271d y\u271d : F\nh : WeierstrassCurve.equation W x\u271d y\u271d\nright\u271d :\n eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialX W)) \u2260 0 \u2228\n eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialY W)) \u2260 0\nhP :\n \u2191toClass\n (some\n (_ :\n WeierstrassCurve.equation W x\u271d y\u271d \u2227\n (eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialX W)) \u2260 0 \u2228\n eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialY W)) \u2260 0))) =\n 0\np : CoordinateRing W\nh0 : p \u2260 0\nhp : XYIdeal W x\u271d (\u2191C y\u271d) = span {p}\n\u22a2 some\n (_ :\n WeierstrassCurve.equation W x\u271d y\u271d \u2227\n (eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialX W)) \u2260 0 \u2228\n eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialY W)) \u2260 0)) =\n 0\n[PROOFSTEP]\napply (p.natDegree_norm_ne_one _).elim\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nx\u271d y\u271d : F\nh : WeierstrassCurve.equation W x\u271d y\u271d\nright\u271d :\n eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialX W)) \u2260 0 \u2228\n eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialY W)) \u2260 0\nhP :\n \u2191toClass\n (some\n (_ :\n WeierstrassCurve.equation W x\u271d y\u271d \u2227\n (eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialX W)) \u2260 0 \u2228\n eval x\u271d (eval (\u2191C y\u271d) (WeierstrassCurve.polynomialY W)) \u2260 0))) =\n 0\np : CoordinateRing W\nh0 : p \u2260 0\nhp : XYIdeal W x\u271d (\u2191C y\u271d) = span {p}\n\u22a2 natDegree (\u2191(Algebra.norm F[X]) p) = 1\n[PROOFSTEP]\nrw [\u2190 finrank_quotient_span_eq_natDegree_norm (CoordinateRing.basis W) h0, \u2190\n (quotientEquivAlgOfEq F hp).toLinearEquiv.finrank_eq, (quotientXYIdealEquiv W h).toLinearEquiv.finrank_eq,\n FiniteDimensional.finrank_self]\n[GOAL]\ncase mpr\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nP : Point W\n\u22a2 P = 0 \u2192 \u2191toClass P = 0\n[PROOFSTEP]\nexact congr_arg toClass\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\n\u22a2 Function.Injective \u2191toClass\n[PROOFSTEP]\nrintro (_ | h) _ hP\n[GOAL]\ncase zero\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\na\u2082\u271d : Point W\nhP : \u2191toClass zero = \u2191toClass a\u2082\u271d\n\u22a2 zero = a\u2082\u271d\ncase some\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nx\u271d y\u271d : F\nh : WeierstrassCurve.nonsingular W x\u271d y\u271d\na\u2082\u271d : Point W\nhP : \u2191toClass (some h) = \u2191toClass a\u2082\u271d\n\u22a2 some h = a\u2082\u271d\n[PROOFSTEP]\nall_goals rw [\u2190 neg_add_eq_zero, \u2190 toClass_eq_zero, map_add, \u2190 hP]\n[GOAL]\ncase zero\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\na\u2082\u271d : Point W\nhP : \u2191toClass zero = \u2191toClass a\u2082\u271d\n\u22a2 zero = a\u2082\u271d\n[PROOFSTEP]\nrw [\u2190 neg_add_eq_zero, \u2190 toClass_eq_zero, map_add, \u2190 hP]\n[GOAL]\ncase some\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nx\u271d y\u271d : F\nh : WeierstrassCurve.nonsingular W x\u271d y\u271d\na\u2082\u271d : Point W\nhP : \u2191toClass (some h) = \u2191toClass a\u2082\u271d\n\u22a2 some h = a\u2082\u271d\n[PROOFSTEP]\nrw [\u2190 neg_add_eq_zero, \u2190 toClass_eq_zero, map_add, \u2190 hP]\n[GOAL]\ncase zero\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\na\u2082\u271d : Point W\nhP : \u2191toClass zero = \u2191toClass a\u2082\u271d\n\u22a2 \u2191toClass (-zero) + \u2191toClass zero = 0\n[PROOFSTEP]\nexact zero_add 0\n[GOAL]\ncase some\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nx\u271d y\u271d : F\nh : WeierstrassCurve.nonsingular W x\u271d y\u271d\na\u2082\u271d : Point W\nhP : \u2191toClass (some h) = \u2191toClass a\u2082\u271d\n\u22a2 \u2191toClass (-some h) + \u2191toClass (some h) = 0\n[PROOFSTEP]\nexact mk_XYIdeal'_mul_mk_XYIdeal'_of_Y_eq h\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nP Q : Point W\n\u22a2 \u2191toClass (P + Q) = \u2191toClass (Q + P)\n[PROOFSTEP]\nsimp only [map_add, _root_.add_comm]\n[GOAL]\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : WeierstrassCurve.nonsingular W x\u2081 y\u2081\nh\u2082 : WeierstrassCurve.nonsingular W x\u2082 y\u2082\nh\u2081' : WeierstrassCurve.equation W x\u2081 y\u2081\nh\u2082' : WeierstrassCurve.equation W x\u2082 y\u2082\nP Q R : Point W\n\u22a2 \u2191toClass (P + Q + R) = \u2191toClass (P + (Q + R))\n[PROOFSTEP]\nsimp only [map_add, _root_.add_assoc]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\n\u22a2 \u2200 (x y : W\u27eeF\u27ef),\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (x + y) =\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) } x +\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) } y\n[PROOFSTEP]\nrintro (_ | @\u27e8x\u2081, y\u2081, _\u27e9) (_ | @\u27e8x\u2082, y\u2082, _\u27e9)\n[GOAL]\ncase zero.zero\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\n\u22a2 ZeroHom.toFun { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (zero + zero) =\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) } zero +\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) } zero\ncase zero.some\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\n\u22a2 ZeroHom.toFun { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (zero + some h\u271d) =\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) } zero +\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d)\ncase some.zero\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\n\u22a2 ZeroHom.toFun { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d + zero) =\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d) +\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) } zero\ncase some.some\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\n\u22a2 ZeroHom.toFun { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d\u00b9 + some h\u271d) =\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d\u00b9) +\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d)\n[PROOFSTEP]\nany_goals rfl\n[GOAL]\ncase zero.zero\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\n\u22a2 ZeroHom.toFun { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (zero + zero) =\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) } zero +\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) } zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase zero.some\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\n\u22a2 ZeroHom.toFun { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (zero + some h\u271d) =\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) } zero +\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.zero\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\n\u22a2 ZeroHom.toFun { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d + zero) =\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d) +\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) } zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\n\u22a2 ZeroHom.toFun { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d\u00b9 + some h\u271d) =\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d\u00b9) +\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\n\u22a2 ZeroHom.toFun { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d\u00b9 + some h\u271d) =\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d\u00b9) +\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d)\n[PROOFSTEP]\nby_cases hx : x\u2081 = x\u2082\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\nhx : x\u2081 = x\u2082\n\u22a2 ZeroHom.toFun { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d\u00b9 + some h\u271d) =\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d\u00b9) +\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d)\n[PROOFSTEP]\nby_cases hy : y\u2081 = (W.baseChange F).negY x\u2082 y\u2082\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : y\u2081 = negY (baseChange W F) x\u2082 y\u2082\n\u22a2 ZeroHom.toFun { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d\u00b9 + some h\u271d) =\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d\u00b9) +\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d)\n[PROOFSTEP]\nsimp only [some_add_some_of_Y_eq hx hy, ofBaseChangeFun]\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : y\u2081 = negY (baseChange W F) x\u2082 y\u2082\n\u22a2 0 =\n some (_ : WeierstrassCurve.nonsingular (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) y\u2081)) +\n some (_ : WeierstrassCurve.nonsingular (baseChange W K) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2082))\n[PROOFSTEP]\nrw [some_add_some_of_Y_eq <| congr_arg _ hx]\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : y\u2081 = negY (baseChange W F) x\u2082 y\u2082\n\u22a2 \u2191(algebraMap F K) y\u2081 = negY (baseChange W K) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2082)\n[PROOFSTEP]\nrw [hy, baseChange_negY_of_baseChange]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : \u00acy\u2081 = negY (baseChange W F) x\u2082 y\u2082\n\u22a2 ZeroHom.toFun { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d\u00b9 + some h\u271d) =\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d\u00b9) +\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d)\n[PROOFSTEP]\nsimp only [some_add_some_of_Y_ne hx hy, ofBaseChangeFun]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : \u00acy\u2081 = negY (baseChange W F) x\u2082 y\u2082\n\u22a2 some\n (_ :\n WeierstrassCurve.nonsingular (baseChange W K)\n (\u2191(algebraMap F K) (addX (baseChange W F) x\u2081 x\u2082 (slope (baseChange W F) x\u2081 x\u2082 y\u2081 y\u2082)))\n (\u2191(algebraMap F K) (addY (baseChange W F) x\u2081 x\u2082 y\u2081 (slope (baseChange W F) x\u2081 x\u2082 y\u2081 y\u2082)))) =\n some (_ : WeierstrassCurve.nonsingular (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) y\u2081)) +\n some (_ : WeierstrassCurve.nonsingular (baseChange W K) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2082))\n[PROOFSTEP]\nrw [some_add_some_of_Y_ne <| congr_arg _ hx]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : \u00acy\u2081 = negY (baseChange W F) x\u2082 y\u2082\n\u22a2 some\n (_ :\n WeierstrassCurve.nonsingular (baseChange W K)\n (\u2191(algebraMap F K) (addX (baseChange W F) x\u2081 x\u2082 (slope (baseChange W F) x\u2081 x\u2082 y\u2081 y\u2082)))\n (\u2191(algebraMap F K) (addY (baseChange W F) x\u2081 x\u2082 y\u2081 (slope (baseChange W F) x\u2081 x\u2082 y\u2081 y\u2082)))) =\n some\n (_ :\n WeierstrassCurve.nonsingular (baseChange W K)\n (addX (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) x\u2082)\n (slope (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2081)\n (\u2191(algebraMap F K) y\u2082)))\n (addY (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2081)\n (slope (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2081)\n (\u2191(algebraMap F K) y\u2082))))\n[PROOFSTEP]\nsimp only [baseChange_addX_of_baseChange, baseChange_addY_of_baseChange, baseChange_slope_of_baseChange]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : \u00acy\u2081 = negY (baseChange W F) x\u2082 y\u2082\n\u22a2 \u2191(algebraMap F K) y\u2081 \u2260 negY (baseChange W K) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2082)\n[PROOFSTEP]\nrw [baseChange_negY_of_baseChange]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : \u00acy\u2081 = negY (baseChange W F) x\u2082 y\u2082\n\u22a2 \u2191(algebraMap F K) y\u2081 \u2260 \u2191(algebraMap F K) (negY (baseChange W F) x\u2082 y\u2082)\n[PROOFSTEP]\ncontrapose! hy\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\nhx : x\u2081 = x\u2082\nhy : \u2191(algebraMap F K) y\u2081 = \u2191(algebraMap F K) (negY (baseChange W F) x\u2082 y\u2082)\n\u22a2 y\u2081 = negY (baseChange W F) x\u2082 y\u2082\n[PROOFSTEP]\nexact NoZeroSMulDivisors.algebraMap_injective F K hy\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 ZeroHom.toFun { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d\u00b9 + some h\u271d) =\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d\u00b9) +\n ZeroHom.toFun\n { toFun := ofBaseChangeFun W F K, map_zero' := (_ : ofBaseChangeFun W F K 0 = ofBaseChangeFun W F K 0) }\n (some h\u271d)\n[PROOFSTEP]\nsimp only [some_add_some_of_X_ne hx, ofBaseChangeFun]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 some\n (_ :\n WeierstrassCurve.nonsingular (baseChange W K)\n (\u2191(algebraMap F K) (addX (baseChange W F) x\u2081 x\u2082 (slope (baseChange W F) x\u2081 x\u2082 y\u2081 y\u2082)))\n (\u2191(algebraMap F K) (addY (baseChange W F) x\u2081 x\u2082 y\u2081 (slope (baseChange W F) x\u2081 x\u2082 y\u2081 y\u2082)))) =\n some (_ : WeierstrassCurve.nonsingular (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) y\u2081)) +\n some (_ : WeierstrassCurve.nonsingular (baseChange W K) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2082))\n[PROOFSTEP]\nrw [some_add_some_of_X_ne]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 some\n (_ :\n WeierstrassCurve.nonsingular (baseChange W K)\n (\u2191(algebraMap F K) (addX (baseChange W F) x\u2081 x\u2082 (slope (baseChange W F) x\u2081 x\u2082 y\u2081 y\u2082)))\n (\u2191(algebraMap F K) (addY (baseChange W F) x\u2081 x\u2082 y\u2081 (slope (baseChange W F) x\u2081 x\u2082 y\u2081 y\u2082)))) =\n some\n (_ :\n WeierstrassCurve.nonsingular (baseChange W K)\n (addX (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) x\u2082)\n (slope (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2081)\n (\u2191(algebraMap F K) y\u2082)))\n (addY (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2081)\n (slope (baseChange W K) (\u2191(algebraMap F K) x\u2081) (\u2191(algebraMap F K) x\u2082) (\u2191(algebraMap F K) y\u2081)\n (\u2191(algebraMap F K) y\u2082))))\n[PROOFSTEP]\nsimp only [baseChange_addX_of_baseChange, baseChange_addY_of_baseChange, baseChange_slope_of_baseChange]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 \u2191(algebraMap F K) x\u2081 \u2260 \u2191(algebraMap F K) x\u2082\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u2081 y\u2081 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u2081 y\u2081\nx\u2082 y\u2082 : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u2082 y\u2082\nhx : \u2191(algebraMap F K) x\u2081 = \u2191(algebraMap F K) x\u2082\n\u22a2 x\u2081 = x\u2082\n[PROOFSTEP]\nexact NoZeroSMulDivisors.algebraMap_injective F K hx\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\n\u22a2 Function.Injective \u2191(ofBaseChange W F K)\n[PROOFSTEP]\nrintro (_ | _) (_ | _) h\n[GOAL]\ncase zero.zero\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nh : \u2191(ofBaseChange W F K) zero = \u2191(ofBaseChange W F K) zero\n\u22a2 zero = zero\ncase zero.some\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u271d y\u271d : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u271d y\u271d\nh : \u2191(ofBaseChange W F K) zero = \u2191(ofBaseChange W F K) (some h\u271d)\n\u22a2 zero = some h\u271d\ncase some.zero\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u271d y\u271d : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u271d y\u271d\nh : \u2191(ofBaseChange W F K) (some h\u271d) = \u2191(ofBaseChange W F K) zero\n\u22a2 some h\u271d = zero\ncase some.some\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u271d\u00b9 y\u271d\u00b9 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u271d\u00b9 y\u271d\u00b9\nx\u271d y\u271d : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u271d y\u271d\nh : \u2191(ofBaseChange W F K) (some h\u271d\u00b9) = \u2191(ofBaseChange W F K) (some h\u271d)\n\u22a2 some h\u271d\u00b9 = some h\u271d\n[PROOFSTEP]\nany_goals contradiction\n[GOAL]\ncase zero.zero\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nh : \u2191(ofBaseChange W F K) zero = \u2191(ofBaseChange W F K) zero\n\u22a2 zero = zero\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase zero.some\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u271d y\u271d : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u271d y\u271d\nh : \u2191(ofBaseChange W F K) zero = \u2191(ofBaseChange W F K) (some h\u271d)\n\u22a2 zero = some h\u271d\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase some.zero\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u271d y\u271d : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u271d y\u271d\nh : \u2191(ofBaseChange W F K) (some h\u271d) = \u2191(ofBaseChange W F K) zero\n\u22a2 some h\u271d = zero\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase some.some\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u271d\u00b9 y\u271d\u00b9 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u271d\u00b9 y\u271d\u00b9\nx\u271d y\u271d : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u271d y\u271d\nh : \u2191(ofBaseChange W F K) (some h\u271d\u00b9) = \u2191(ofBaseChange W F K) (some h\u271d)\n\u22a2 some h\u271d\u00b9 = some h\u271d\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase zero.zero\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nh : \u2191(ofBaseChange W F K) zero = \u2191(ofBaseChange W F K) zero\n\u22a2 zero = zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u271d\u00b9 y\u271d\u00b9 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u271d\u00b9 y\u271d\u00b9\nx\u271d y\u271d : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u271d y\u271d\nh : \u2191(ofBaseChange W F K) (some h\u271d\u00b9) = \u2191(ofBaseChange W F K) (some h\u271d)\n\u22a2 some h\u271d\u00b9 = some h\u271d\n[PROOFSTEP]\nrw [some.injEq]\n[GOAL]\ncase some.some\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nF : Type v\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R F\nK : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : IsScalarTower R F K\nx\u271d\u00b9 y\u271d\u00b9 : F\nh\u271d\u00b9 : WeierstrassCurve.nonsingular (baseChange W F) x\u271d\u00b9 y\u271d\u00b9\nx\u271d y\u271d : F\nh\u271d : WeierstrassCurve.nonsingular (baseChange W F) x\u271d y\u271d\nh : \u2191(ofBaseChange W F K) (some h\u271d\u00b9) = \u2191(ofBaseChange W F K) (some h\u271d)\n\u22a2 x\u271d\u00b9 = x\u271d \u2227 y\u271d\u00b9 = y\u271d\n[PROOFSTEP]\nexact\n \u27e8NoZeroSMulDivisors.algebraMap_injective F K (some.inj h).left,\n NoZeroSMulDivisors.algebraMap_injective F K (some.inj h).right\u27e9\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.EllipticCurve.Point", "llama_tokens": 110355, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149868676283, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.43019030685880155}} {"text": "[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : LE \u03b1\n\u22a2 OrderTop \u03b1 \u2295' NoTopOrder \u03b1\n[PROOFSTEP]\nby_cases H : \u2200 a : \u03b1, \u2203 b, \u00acb \u2264 a\n[GOAL]\ncase pos\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : LE \u03b1\nH : \u2200 (a : \u03b1), \u2203 b, \u00acb \u2264 a\n\u22a2 OrderTop \u03b1 \u2295' NoTopOrder \u03b1\n[PROOFSTEP]\nexact PSum.inr \u27e8H\u27e9\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : LE \u03b1\nH : \u00ac\u2200 (a : \u03b1), \u2203 b, \u00acb \u2264 a\n\u22a2 OrderTop \u03b1 \u2295' NoTopOrder \u03b1\n[PROOFSTEP]\npush_neg at H \n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : LE \u03b1\nH : \u2203 a, \u2200 (b : \u03b1), b \u2264 a\n\u22a2 OrderTop \u03b1 \u2295' NoTopOrder \u03b1\n[PROOFSTEP]\nletI : Top \u03b1 := \u27e8Classical.choose H\u27e9\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : LE \u03b1\nH : \u2203 a, \u2200 (b : \u03b1), b \u2264 a\nthis : Top \u03b1 := { top := Classical.choose H }\n\u22a2 OrderTop \u03b1 \u2295' NoTopOrder \u03b1\n[PROOFSTEP]\nexact PSum.inl \u27e8Classical.choose_spec H\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : OrderTop \u03b2\nf : \u03b1 \u2192 \u03b2\nH : StrictMono f\na : \u03b1\nh_top : f a = \u22a4\nx : \u03b1\np : \u03b2\n\u22a2 p \u2264 f a\n[PROOFSTEP]\nrw [h_top]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : OrderTop \u03b2\nf : \u03b1 \u2192 \u03b2\nH : StrictMono f\na : \u03b1\nh_top : f a = \u22a4\nx : \u03b1\np : \u03b2\n\u22a2 p \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\nhA : PartialOrder \u03b1\nA : OrderTop \u03b1\nhB : PartialOrder \u03b1\nB : OrderTop \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 \u22a4 = \u22a4\n[PROOFSTEP]\ncases PartialOrder.ext H\n[GOAL]\ncase refl\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\nhA : PartialOrder \u03b1\nA B : OrderTop \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 \u22a4 = \u22a4\n[PROOFSTEP]\napply top_unique\n[GOAL]\ncase refl.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\nhA : PartialOrder \u03b1\nA B : OrderTop \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 \u22a4 \u2264 \u22a4\n[PROOFSTEP]\nexact @le_top _ _ A _\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\nA B : OrderTop \u03b1\n\u22a2 A = B\n[PROOFSTEP]\nrcases A with \u27e8ha\u27e9\n[GOAL]\ncase mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\nB : OrderTop \u03b1\ntoTop\u271d : Top \u03b1\nha : \u2200 (a : \u03b1), a \u2264 \u22a4\n\u22a2 mk ha = B\n[PROOFSTEP]\nrcases B with \u27e8hb\u27e9\n[GOAL]\ncase mk.mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\ntoTop\u271d\u00b9 : Top \u03b1\nha : \u2200 (a : \u03b1), a \u2264 \u22a4\ntoTop\u271d : Top \u03b1\nhb : \u2200 (a : \u03b1), a \u2264 \u22a4\n\u22a2 mk ha = mk hb\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toTop\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\ntoTop\u271d\u00b9 : Top \u03b1\nha : \u2200 (a : \u03b1), a \u2264 \u22a4\ntoTop\u271d : Top \u03b1\nhb : \u2200 (a : \u03b1), a \u2264 \u22a4\n\u22a2 toTop\u271d\u00b9 = toTop\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase mk.mk.e_toTop.top\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\ntoTop\u271d\u00b9 : Top \u03b1\nha : \u2200 (a : \u03b1), a \u2264 \u22a4\ntoTop\u271d : Top \u03b1\nhb : \u2200 (a : \u03b1), a \u2264 \u22a4\n\u22a2 \u22a4 = \u22a4\n[PROOFSTEP]\nexact le_antisymm (hb _) (ha _)\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : LE \u03b1\n\u22a2 OrderBot \u03b1 \u2295' NoBotOrder \u03b1\n[PROOFSTEP]\nby_cases H : \u2200 a : \u03b1, \u2203 b, \u00aca \u2264 b\n[GOAL]\ncase pos\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : LE \u03b1\nH : \u2200 (a : \u03b1), \u2203 b, \u00aca \u2264 b\n\u22a2 OrderBot \u03b1 \u2295' NoBotOrder \u03b1\n[PROOFSTEP]\nexact PSum.inr \u27e8H\u27e9\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : LE \u03b1\nH : \u00ac\u2200 (a : \u03b1), \u2203 b, \u00aca \u2264 b\n\u22a2 OrderBot \u03b1 \u2295' NoBotOrder \u03b1\n[PROOFSTEP]\npush_neg at H \n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : LE \u03b1\nH : \u2203 a, \u2200 (b : \u03b1), a \u2264 b\n\u22a2 OrderBot \u03b1 \u2295' NoBotOrder \u03b1\n[PROOFSTEP]\nletI : Bot \u03b1 := \u27e8Classical.choose H\u27e9\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : LE \u03b1\nH : \u2203 a, \u2200 (b : \u03b1), a \u2264 b\nthis : Bot \u03b1 := { bot := Classical.choose H }\n\u22a2 OrderBot \u03b1 \u2295' NoBotOrder \u03b1\n[PROOFSTEP]\nexact PSum.inl \u27e8Classical.choose_spec H\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : OrderBot \u03b2\nf : \u03b1 \u2192 \u03b2\nH : StrictMono f\na : \u03b1\nh_bot : f a = \u22a5\nx : \u03b1\np : \u03b2\n\u22a2 f a \u2264 p\n[PROOFSTEP]\nrw [h_bot]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : OrderBot \u03b2\nf : \u03b1 \u2192 \u03b2\nH : StrictMono f\na : \u03b1\nh_bot : f a = \u22a5\nx : \u03b1\np : \u03b2\n\u22a2 \u22a5 \u2264 p\n[PROOFSTEP]\nexact bot_le\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\nhA : PartialOrder \u03b1\nA : OrderBot \u03b1\nhB : PartialOrder \u03b1\nB : OrderBot \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 \u22a5 = \u22a5\n[PROOFSTEP]\ncases PartialOrder.ext H\n[GOAL]\ncase refl\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\nhA : PartialOrder \u03b1\nA B : OrderBot \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 \u22a5 = \u22a5\n[PROOFSTEP]\napply bot_unique\n[GOAL]\ncase refl.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\nhA : PartialOrder \u03b1\nA B : OrderBot \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 \u22a5 \u2264 \u22a5\n[PROOFSTEP]\nexact @bot_le _ _ A _\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\nA B : OrderBot \u03b1\n\u22a2 A = B\n[PROOFSTEP]\nrcases A with \u27e8ha\u27e9\n[GOAL]\ncase mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\nB : OrderBot \u03b1\ntoBot\u271d : Bot \u03b1\nha : \u2200 (a : \u03b1), \u22a5 \u2264 a\n\u22a2 mk ha = B\n[PROOFSTEP]\nrcases B with \u27e8hb\u27e9\n[GOAL]\ncase mk.mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\ntoBot\u271d\u00b9 : Bot \u03b1\nha : \u2200 (a : \u03b1), \u22a5 \u2264 a\ntoBot\u271d : Bot \u03b1\nhb : \u2200 (a : \u03b1), \u22a5 \u2264 a\n\u22a2 mk ha = mk hb\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toBot\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\ntoBot\u271d\u00b9 : Bot \u03b1\nha : \u2200 (a : \u03b1), \u22a5 \u2264 a\ntoBot\u271d : Bot \u03b1\nhb : \u2200 (a : \u03b1), \u22a5 \u2264 a\n\u22a2 toBot\u271d\u00b9 = toBot\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase mk.mk.e_toBot.bot\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\ntoBot\u271d\u00b9 : Bot \u03b1\nha : \u2200 (a : \u03b1), \u22a5 \u2264 a\ntoBot\u271d : Bot \u03b1\nhb : \u2200 (a : \u03b1), \u22a5 \u2264 a\n\u22a2 \u22a5 = \u22a5\n[PROOFSTEP]\nexact le_antisymm (ha _) (hb _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : OrderBot \u03b1\na b : \u03b1\n\u22a2 a \u2294 b = \u22a5 \u2194 a = \u22a5 \u2227 b = \u22a5\n[PROOFSTEP]\nrw [eq_bot_iff, sup_le_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : OrderBot \u03b1\na b : \u03b1\n\u22a2 a \u2264 \u22a5 \u2227 b \u2264 \u22a5 \u2194 a = \u22a5 \u2227 b = \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\nA B : BoundedOrder \u03b1\n\u22a2 A = B\n[PROOFSTEP]\nhave ht : @BoundedOrder.toOrderTop \u03b1 _ A = @BoundedOrder.toOrderTop \u03b1 _ B := OrderTop.ext\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\nA B : BoundedOrder \u03b1\nht : toOrderTop = toOrderTop\n\u22a2 A = B\n[PROOFSTEP]\nhave hb : @BoundedOrder.toOrderBot \u03b1 _ A = @BoundedOrder.toOrderBot \u03b1 _ B := OrderBot.ext\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\nA B : BoundedOrder \u03b1\nht : toOrderTop = toOrderTop\nhb : toOrderBot = toOrderBot\n\u22a2 A = B\n[PROOFSTEP]\ncases A\n[GOAL]\ncase mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\nB : BoundedOrder \u03b1\ntoOrderTop\u271d : OrderTop \u03b1\ntoOrderBot\u271d : OrderBot \u03b1\nht : toOrderTop = toOrderTop\nhb : toOrderBot = toOrderBot\n\u22a2 mk = B\n[PROOFSTEP]\ncases B\n[GOAL]\ncase mk.mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\ntoOrderTop\u271d\u00b9 : OrderTop \u03b1\ntoOrderBot\u271d\u00b9 : OrderBot \u03b1\ntoOrderTop\u271d : OrderTop \u03b1\ntoOrderBot\u271d : OrderBot \u03b1\nht : toOrderTop = toOrderTop\nhb : toOrderBot = toOrderBot\n\u22a2 mk = mk\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b3 : LE \u03b1\ninst\u271d\u00b2 : Top \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : OrderTop \u03b2\nf : \u03b1 \u2192 \u03b2\nmap_le : \u2200 (a b : \u03b1), f a \u2264 f b \u2192 a \u2264 b\nmap_top : f \u22a4 = \u22a4\na : \u03b1\n\u22a2 f a \u2264 f \u22a4\n[PROOFSTEP]\nrw [map_top]\n -- Porting note: lean3 didn't need the type annotation\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b3 : LE \u03b1\ninst\u271d\u00b2 : Top \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : OrderTop \u03b2\nf : \u03b1 \u2192 \u03b2\nmap_le : \u2200 (a b : \u03b1), f a \u2264 f b \u2192 a \u2264 b\nmap_top : f \u22a4 = \u22a4\na : \u03b1\n\u22a2 f a \u2264 \u22a4\n[PROOFSTEP]\nexact @le_top \u03b2 _ _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b3 : LE \u03b1\ninst\u271d\u00b2 : Bot \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : OrderBot \u03b2\nf : \u03b1 \u2192 \u03b2\nmap_le : \u2200 (a b : \u03b1), f a \u2264 f b \u2192 a \u2264 b\nmap_bot : f \u22a5 = \u22a5\na : \u03b1\n\u22a2 f \u22a5 \u2264 f a\n[PROOFSTEP]\nrw [map_bot]\n -- Porting note: lean3 didn't need the type annotation\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b3 : LE \u03b1\ninst\u271d\u00b2 : Bot \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : OrderBot \u03b2\nf : \u03b1 \u2192 \u03b2\nmap_le : \u2200 (a b : \u03b1), f a \u2264 f b \u2192 a \u2264 b\nmap_bot : f \u22a5 = \u22a5\na : \u03b1\n\u22a2 \u22a5 \u2264 f a\n[PROOFSTEP]\nexact @bot_le \u03b2 _ _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderBot (Subtype p)\nhbot : p \u22a5\nx : { x // p x }\n\u22a2 \u2191x = \u22a5 \u2194 x = \u22a5\n[PROOFSTEP]\nrw [\u2190 coe_bot hbot, ext_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : OrderTop (Subtype p)\nhtop : p \u22a4\nx : { x // p x }\n\u22a2 \u2191x = \u22a4 \u2194 x = \u22a4\n[PROOFSTEP]\nrw [\u2190 coe_top htop, ext_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\na b : \u03b1\n\u22a2 min a b = \u22a5 \u2194 a = \u22a5 \u2228 b = \u22a5\n[PROOFSTEP]\nsimp only [\u2190 inf_eq_min, \u2190 le_bot_iff, inf_le_iff]\n", "meta": {"mathlib_filename": "Mathlib.Order.BoundedOrder", "llama_tokens": 5010, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149868676283, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.43019030685880155}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : Mul \u03b1\ninst\u271d : MulOneClass \u03b2\nf : \u03b1 \u2192\u2099* \u03b2\nx y : WithOne \u03b1\n\u22a2 OneHom.toFun\n { toFun := fun x => Option.casesOn x 1 \u2191f,\n map_one' := (_ : (fun x => Option.casesOn x 1 \u2191f) 1 = (fun x => Option.casesOn x 1 \u2191f) 1) }\n (1 * y) =\n OneHom.toFun\n { toFun := fun x => Option.casesOn x 1 \u2191f,\n map_one' := (_ : (fun x => Option.casesOn x 1 \u2191f) 1 = (fun x => Option.casesOn x 1 \u2191f) 1) }\n 1 *\n OneHom.toFun\n { toFun := fun x => Option.casesOn x 1 \u2191f,\n map_one' := (_ : (fun x => Option.casesOn x 1 \u2191f) 1 = (fun x => Option.casesOn x 1 \u2191f) 1) }\n y\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : Mul \u03b1\ninst\u271d : MulOneClass \u03b2\nf : \u03b1 \u2192\u2099* \u03b2\nx y : WithOne \u03b1\n\u22a2 OneHom.toFun\n { toFun := fun x => Option.casesOn x 1 \u2191f,\n map_one' := (_ : (fun x => Option.casesOn x 1 \u2191f) 1 = (fun x => Option.casesOn x 1 \u2191f) 1) }\n y =\n OneHom.toFun\n { toFun := fun x => Option.casesOn x 1 \u2191f,\n map_one' := (_ : (fun x => Option.casesOn x 1 \u2191f) 1 = (fun x => Option.casesOn x 1 \u2191f) 1) }\n 1 *\n OneHom.toFun\n { toFun := fun x => Option.casesOn x 1 \u2191f,\n map_one' := (_ : (fun x => Option.casesOn x 1 \u2191f) 1 = (fun x => Option.casesOn x 1 \u2191f) 1) }\n y\n[PROOFSTEP]\nexact (one_mul _).symm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : Mul \u03b1\ninst\u271d : MulOneClass \u03b2\nf : \u03b1 \u2192\u2099* \u03b2\nx\u271d y : WithOne \u03b1\nx : \u03b1\n\u22a2 OneHom.toFun\n { toFun := fun x => Option.casesOn x 1 \u2191f,\n map_one' := (_ : (fun x => Option.casesOn x 1 \u2191f) 1 = (fun x => Option.casesOn x 1 \u2191f) 1) }\n (\u2191x * 1) =\n OneHom.toFun\n { toFun := fun x => Option.casesOn x 1 \u2191f,\n map_one' := (_ : (fun x => Option.casesOn x 1 \u2191f) 1 = (fun x => Option.casesOn x 1 \u2191f) 1) }\n \u2191x *\n OneHom.toFun\n { toFun := fun x => Option.casesOn x 1 \u2191f,\n map_one' := (_ : (fun x => Option.casesOn x 1 \u2191f) 1 = (fun x => Option.casesOn x 1 \u2191f) 1) }\n 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : Mul \u03b1\ninst\u271d : MulOneClass \u03b2\nf : \u03b1 \u2192\u2099* \u03b2\nx\u271d y : WithOne \u03b1\nx : \u03b1\n\u22a2 OneHom.toFun\n { toFun := fun x => Option.casesOn x 1 \u2191f,\n map_one' := (_ : (fun x => Option.casesOn x 1 \u2191f) 1 = (fun x => Option.casesOn x 1 \u2191f) 1) }\n \u2191x =\n OneHom.toFun\n { toFun := fun x => Option.casesOn x 1 \u2191f,\n map_one' := (_ : (fun x => Option.casesOn x 1 \u2191f) 1 = (fun x => Option.casesOn x 1 \u2191f) 1) }\n \u2191x *\n OneHom.toFun\n { toFun := fun x => Option.casesOn x 1 \u2191f,\n map_one' := (_ : (fun x => Option.casesOn x 1 \u2191f) 1 = (fun x => Option.casesOn x 1 \u2191f) 1) }\n 1\n[PROOFSTEP]\nexact (mul_one _).symm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : Mul \u03b2\ninst\u271d : Mul \u03b3\n\u22a2 map (MulHom.id \u03b1) = MonoidHom.id (WithOne \u03b1)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : Mul \u03b2\ninst\u271d : Mul \u03b3\nx : WithOne \u03b1\n\u22a2 \u2191(map (MulHom.id \u03b1)) x = \u2191(MonoidHom.id (WithOne \u03b1)) x\n[PROOFSTEP]\ninduction x using WithOne.cases_on\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : Mul \u03b2\ninst\u271d : Mul \u03b3\n\u22a2 \u2191(map (MulHom.id \u03b1)) 1 = \u2191(MonoidHom.id (WithOne \u03b1)) 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : Mul \u03b2\ninst\u271d : Mul \u03b3\na\u271d : \u03b1\n\u22a2 \u2191(map (MulHom.id \u03b1)) \u2191a\u271d = \u2191(MonoidHom.id (WithOne \u03b1)) \u2191a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : Mul \u03b2\ninst\u271d : Mul \u03b3\nf : \u03b1 \u2192\u2099* \u03b2\ng : \u03b2 \u2192\u2099* \u03b3\nx : WithOne \u03b1\n\u22a2 \u2191(map g) (\u2191(map f) x) = \u2191(map (MulHom.comp g f)) x\n[PROOFSTEP]\ninduction x using WithOne.cases_on\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : Mul \u03b2\ninst\u271d : Mul \u03b3\nf : \u03b1 \u2192\u2099* \u03b2\ng : \u03b2 \u2192\u2099* \u03b3\n\u22a2 \u2191(map g) (\u2191(map f) 1) = \u2191(map (MulHom.comp g f)) 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : Mul \u03b2\ninst\u271d : Mul \u03b3\nf : \u03b1 \u2192\u2099* \u03b2\ng : \u03b2 \u2192\u2099* \u03b3\na\u271d : \u03b1\n\u22a2 \u2191(map g) (\u2191(map f) \u2191a\u271d) = \u2191(map (MulHom.comp g f)) \u2191a\u271d\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Group.WithOne.Basic", "llama_tokens": 2011, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.4301396890660718}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nh : \u00acAffineIndependent \ud835\udd5c Subtype.val\nx : E\nm : x \u2208 \u2191(convexHull \ud835\udd5c) \u2191t\n\u22a2 \u2203 y, x \u2208 \u2191(convexHull \ud835\udd5c) \u2191(erase t \u2191y)\n[PROOFSTEP]\nsimp only [Finset.convexHull_eq, mem_setOf_eq] at m \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nh : \u00acAffineIndependent \ud835\udd5c Subtype.val\nx : E\nm : \u2203 w x_1 x_2, centerMass t w id = x\n\u22a2 \u2203 y w x_1 x_2, centerMass (erase t \u2191y) w id = x\n[PROOFSTEP]\nobtain \u27e8f, fpos, fsum, rfl\u27e9 := m\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nh : \u00acAffineIndependent \ud835\udd5c Subtype.val\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\n\u22a2 \u2203 y w x x, centerMass (erase t \u2191y) w id = centerMass t f id\n[PROOFSTEP]\nobtain \u27e8g, gcombo, gsum, gpos\u27e9 := exists_nontrivial_relation_sum_zero_of_not_affine_ind h\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nh : \u00acAffineIndependent \ud835\udd5c Subtype.val\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 x, x \u2208 t \u2227 g x \u2260 0\n\u22a2 \u2203 y w x x, centerMass (erase t \u2191y) w id = centerMass t f id\n[PROOFSTEP]\nreplace gpos := exists_pos_of_sum_zero_of_exists_nonzero g gsum gpos\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nh : \u00acAffineIndependent \ud835\udd5c Subtype.val\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\n\u22a2 \u2203 y w x x, centerMass (erase t \u2191y) w id = centerMass t f id\n[PROOFSTEP]\nclear h\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\n\u22a2 \u2203 y w x x, centerMass (erase t \u2191y) w id = centerMass t f id\n[PROOFSTEP]\nlet s := @Finset.filter _ (fun z => 0 < g z) (fun _ => LinearOrder.decidableLT _ _) t\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\n\u22a2 \u2203 y w x x, centerMass (erase t \u2191y) w id = centerMass t f id\n[PROOFSTEP]\nobtain \u27e8i\u2080, mem, w\u27e9 : \u2203 i\u2080 \u2208 s, \u2200 i \u2208 s, f i\u2080 / g i\u2080 \u2264 f i / g i :=\n by\n apply s.exists_min_image fun z => f z / g z\n obtain \u27e8x, hx, hgx\u27e9 : \u2203 x \u2208 t, 0 < g x := gpos\n exact \u27e8x, mem_filter.mpr \u27e8hx, hgx\u27e9\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\n\u22a2 \u2203 i\u2080, i\u2080 \u2208 s \u2227 \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\n[PROOFSTEP]\napply s.exists_min_image fun z => f z / g z\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\n\u22a2 Finset.Nonempty s\n[PROOFSTEP]\nobtain \u27e8x, hx, hgx\u27e9 : \u2203 x \u2208 t, 0 < g x := gpos\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ns : Finset E := filter (fun z => 0 < g z) t\nx : E\nhx : x \u2208 t\nhgx : 0 < g x\n\u22a2 Finset.Nonempty s\n[PROOFSTEP]\nexact \u27e8x, mem_filter.mpr \u27e8hx, hgx\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\n\u22a2 \u2203 y w x x, centerMass (erase t \u2191y) w id = centerMass t f id\n[PROOFSTEP]\nhave hg : 0 < g i\u2080 := by\n rw [mem_filter] at mem \n exact mem.2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\n\u22a2 0 < g i\u2080\n[PROOFSTEP]\nrw [mem_filter] at mem \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 t \u2227 0 < g i\u2080\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\n\u22a2 0 < g i\u2080\n[PROOFSTEP]\nexact mem.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\n\u22a2 \u2203 y w x x, centerMass (erase t \u2191y) w id = centerMass t f id\n[PROOFSTEP]\nhave hi\u2080 : i\u2080 \u2208 t := filter_subset _ _ mem\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\n\u22a2 \u2203 y w x x, centerMass (erase t \u2191y) w id = centerMass t f id\n[PROOFSTEP]\nlet k : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\n\u22a2 \u2203 y w x x, centerMass (erase t \u2191y) w id = centerMass t f id\n[PROOFSTEP]\nhave hk : k i\u2080 = 0 := by field_simp [ne_of_gt hg]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\n\u22a2 k i\u2080 = 0\n[PROOFSTEP]\nfield_simp [ne_of_gt hg]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\n\u22a2 \u2203 y w x x, centerMass (erase t \u2191y) w id = centerMass t f id\n[PROOFSTEP]\nhave ksum : \u2211 e in t.erase i\u2080, k e = 1 := by\n calc\n \u2211 e in t.erase i\u2080, k e = \u2211 e in t, k e := by\n conv_rhs => rw [\u2190 insert_erase hi\u2080, sum_insert (not_mem_erase i\u2080 t), hk, zero_add]\n _ = \u2211 e in t, (f e - f i\u2080 / g i\u2080 * g e) := rfl\n _ = 1 := by rw [sum_sub_distrib, fsum, \u2190 mul_sum, gsum, mul_zero, sub_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\n\u22a2 \u2211 e in erase t i\u2080, k e = 1\n[PROOFSTEP]\ncalc\n \u2211 e in t.erase i\u2080, k e = \u2211 e in t, k e := by\n conv_rhs => rw [\u2190 insert_erase hi\u2080, sum_insert (not_mem_erase i\u2080 t), hk, zero_add]\n _ = \u2211 e in t, (f e - f i\u2080 / g i\u2080 * g e) := rfl\n _ = 1 := by rw [sum_sub_distrib, fsum, \u2190 mul_sum, gsum, mul_zero, sub_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\n\u22a2 \u2211 e in erase t i\u2080, k e = \u2211 e in t, k e\n[PROOFSTEP]\nconv_rhs => rw [\u2190 insert_erase hi\u2080, sum_insert (not_mem_erase i\u2080 t), hk, zero_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\n| \u2211 e in t, k e\n[PROOFSTEP]\nrw [\u2190 insert_erase hi\u2080, sum_insert (not_mem_erase i\u2080 t), hk, zero_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\n| \u2211 e in t, k e\n[PROOFSTEP]\nrw [\u2190 insert_erase hi\u2080, sum_insert (not_mem_erase i\u2080 t), hk, zero_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\n| \u2211 e in t, k e\n[PROOFSTEP]\nrw [\u2190 insert_erase hi\u2080, sum_insert (not_mem_erase i\u2080 t), hk, zero_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\n\u22a2 \u2211 e in t, (f e - f i\u2080 / g i\u2080 * g e) = 1\n[PROOFSTEP]\nrw [sum_sub_distrib, fsum, \u2190 mul_sum, gsum, mul_zero, sub_zero]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\n\u22a2 \u2203 y w x x, centerMass (erase t \u2191y) w id = centerMass t f id\n[PROOFSTEP]\nrefine' \u27e8\u27e8i\u2080, hi\u2080\u27e9, k, _, by convert ksum, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\n\u22a2 \u2211 y in erase t \u2191{ val := i\u2080, property := hi\u2080 }, k y = 1\n[PROOFSTEP]\nconvert ksum\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\n\u22a2 \u2200 (y : E), y \u2208 erase t \u2191{ val := i\u2080, property := hi\u2080 } \u2192 0 \u2264 k y\n[PROOFSTEP]\nsimp only [and_imp, sub_nonneg, mem_erase, Ne.def, Subtype.coe_mk]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\n\u22a2 \u2200 (y : E), \u00acy = i\u2080 \u2192 y \u2208 t \u2192 f i\u2080 / g i\u2080 * g y \u2264 f y\n[PROOFSTEP]\nintro e _ het\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\ne : E\na\u271d : \u00ace = i\u2080\nhet : e \u2208 t\n\u22a2 f i\u2080 / g i\u2080 * g e \u2264 f e\n[PROOFSTEP]\nby_cases hes : e \u2208 s\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\ne : E\na\u271d : \u00ace = i\u2080\nhet : e \u2208 t\nhes : e \u2208 s\n\u22a2 f i\u2080 / g i\u2080 * g e \u2264 f e\n[PROOFSTEP]\nhave hge : 0 < g e := by\n rw [mem_filter] at hes \n exact hes.2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\ne : E\na\u271d : \u00ace = i\u2080\nhet : e \u2208 t\nhes : e \u2208 s\n\u22a2 0 < g e\n[PROOFSTEP]\nrw [mem_filter] at hes \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\ne : E\na\u271d : \u00ace = i\u2080\nhet : e \u2208 t\nhes : e \u2208 t \u2227 0 < g e\n\u22a2 0 < g e\n[PROOFSTEP]\nexact hes.2\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\ne : E\na\u271d : \u00ace = i\u2080\nhet : e \u2208 t\nhes : e \u2208 s\nhge : 0 < g e\n\u22a2 f i\u2080 / g i\u2080 * g e \u2264 f e\n[PROOFSTEP]\nrw [\u2190 le_div_iff hge]\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\ne : E\na\u271d : \u00ace = i\u2080\nhet : e \u2208 t\nhes : e \u2208 s\nhge : 0 < g e\n\u22a2 f i\u2080 / g i\u2080 \u2264 f e / g e\n[PROOFSTEP]\nexact w _ hes\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\ne : E\na\u271d : \u00ace = i\u2080\nhet : e \u2208 t\nhes : \u00ace \u2208 s\n\u22a2 f i\u2080 / g i\u2080 * g e \u2264 f e\n[PROOFSTEP]\ncalc\n _ \u2264 0 := by\n apply mul_nonpos_of_nonneg_of_nonpos\n \u00b7 apply div_nonneg (fpos i\u2080 (mem_of_subset (filter_subset _ t) mem)) (le_of_lt hg)\n \u00b7 simpa only [mem_filter, het, true_and_iff, not_lt] using hes\n _ \u2264 f e := fpos e het\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\ne : E\na\u271d : \u00ace = i\u2080\nhet : e \u2208 t\nhes : \u00ace \u2208 s\n\u22a2 f i\u2080 / g i\u2080 * g e \u2264 0\n[PROOFSTEP]\napply mul_nonpos_of_nonneg_of_nonpos\n[GOAL]\ncase ha\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\ne : E\na\u271d : \u00ace = i\u2080\nhet : e \u2208 t\nhes : \u00ace \u2208 s\n\u22a2 0 \u2264 f i\u2080 / g i\u2080\n[PROOFSTEP]\napply div_nonneg (fpos i\u2080 (mem_of_subset (filter_subset _ t) mem)) (le_of_lt hg)\n[GOAL]\ncase hb\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\ne : E\na\u271d : \u00ace = i\u2080\nhet : e \u2208 t\nhes : \u00ace \u2208 s\n\u22a2 g e \u2264 0\n[PROOFSTEP]\nsimpa only [mem_filter, het, true_and_iff, not_lt] using hes\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\n\u22a2 centerMass (erase t \u2191{ val := i\u2080, property := hi\u2080 }) k id = centerMass t f id\n[PROOFSTEP]\nrw [Subtype.coe_mk, centerMass_eq_of_sum_1 _ id ksum]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\n\u22a2 \u2211 i in erase t i\u2080, k i \u2022 id i = centerMass t f id\n[PROOFSTEP]\ncalc\n \u2211 e in t.erase i\u2080, k e \u2022 e = \u2211 e in t, k e \u2022 e := sum_erase _ (by rw [hk, zero_smul])\n _ = \u2211 e in t, (f e - f i\u2080 / g i\u2080 * g e) \u2022 e := rfl\n _ = t.centerMass f id := by\n simp only [sub_smul, mul_smul, sum_sub_distrib, \u2190 smul_sum, gcombo, smul_zero, sub_zero, centerMass, fsum, inv_one,\n one_smul, id.def]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\n\u22a2 k i\u2080 \u2022 i\u2080 = 0\n[PROOFSTEP]\nrw [hk, zero_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : DecidableEq E\nt : Finset E\nf : E \u2192 \ud835\udd5c\nfpos : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 f y\nfsum : \u2211 y in t, f y = 1\ng : E \u2192 \ud835\udd5c\ngcombo : \u2211 e in t, g e \u2022 e = 0\ngsum : \u2211 e in t, g e = 0\ngpos : \u2203 i, i \u2208 t \u2227 0 < g i\ns : Finset E := filter (fun z => 0 < g z) t\ni\u2080 : E\nmem : i\u2080 \u2208 s\nw : \u2200 (i : E), i \u2208 s \u2192 f i\u2080 / g i\u2080 \u2264 f i / g i\nhg : 0 < g i\u2080\nhi\u2080 : i\u2080 \u2208 t\nk : E \u2192 \ud835\udd5c := fun z => f z - f i\u2080 / g i\u2080 * g z\nhk : k i\u2080 = 0\nksum : \u2211 e in erase t i\u2080, k e = 1\n\u22a2 \u2211 e in t, (f e - f i\u2080 / g i\u2080 * g e) \u2022 e = centerMass t f id\n[PROOFSTEP]\nsimp only [sub_smul, mul_smul, sum_sub_distrib, \u2190 smul_sum, gcombo, smul_zero, sub_zero, centerMass, fsum, inv_one,\n one_smul, id.def]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\n\u22a2 Set.Nonempty {t | \u2191t \u2286 s \u2227 x \u2208 \u2191(convexHull \ud835\udd5c) \u2191t}\n[PROOFSTEP]\nsimpa only [convexHull_eq_union_convexHull_finite_subsets s, exists_prop, mem_iUnion] using hx\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\n\u22a2 Finset.Nonempty (minCardFinsetOfMemConvexHull hx)\n[PROOFSTEP]\nrw [\u2190 Finset.coe_nonempty, \u2190 @convexHull_nonempty_iff \ud835\udd5c]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\n\u22a2 Set.Nonempty (\u2191(convexHull \ud835\udd5c) \u2191(minCardFinsetOfMemConvexHull hx))\n[PROOFSTEP]\nexact \u27e8x, mem_minCardFinsetOfMemConvexHull hx\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : x \u2208 \u2191(convexHull \ud835\udd5c) \u2191t\n\u22a2 t \u2208 {t | \u2191t \u2286 s \u2227 x \u2208 \u2191(convexHull \ud835\udd5c) \u2191t}\n[PROOFSTEP]\nexact \u27e8ht\u2081, ht\u2082\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\n\u22a2 AffineIndependent \ud835\udd5c Subtype.val\n[PROOFSTEP]\nlet k := (minCardFinsetOfMemConvexHull hx).card - 1\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\nk : \u2115 := card (minCardFinsetOfMemConvexHull hx) - 1\n\u22a2 AffineIndependent \ud835\udd5c Subtype.val\n[PROOFSTEP]\nhave hk : (minCardFinsetOfMemConvexHull hx).card = k + 1 :=\n (Nat.succ_pred_eq_of_pos (Finset.card_pos.mpr (minCardFinsetOfMemConvexHull_nonempty hx))).symm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\nk : \u2115 := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\n\u22a2 AffineIndependent \ud835\udd5c Subtype.val\n[PROOFSTEP]\nclassical\nby_contra h\nobtain \u27e8p, hp\u27e9 := mem_convexHull_erase h (mem_minCardFinsetOfMemConvexHull hx)\nhave contra :=\n minCardFinsetOfMemConvexHull_card_le_card hx\n (Set.Subset.trans (Finset.erase_subset (p : E) (minCardFinsetOfMemConvexHull hx))\n (minCardFinsetOfMemConvexHull_subseteq hx))\n hp\nrw [\u2190 not_lt] at contra \napply contra\nerw [card_erase_of_mem p.2, hk]\nexact lt_add_one _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\nk : \u2115 := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\n\u22a2 AffineIndependent \ud835\udd5c Subtype.val\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\nk : \u2115 := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : \u00acAffineIndependent \ud835\udd5c Subtype.val\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := mem_convexHull_erase h (mem_minCardFinsetOfMemConvexHull hx)\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\nk : \u2115 := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : \u00acAffineIndependent \ud835\udd5c Subtype.val\np : \u2191\u2191(minCardFinsetOfMemConvexHull hx)\nhp : x \u2208 \u2191(convexHull \ud835\udd5c) \u2191(erase (minCardFinsetOfMemConvexHull hx) \u2191p)\n\u22a2 False\n[PROOFSTEP]\nhave contra :=\n minCardFinsetOfMemConvexHull_card_le_card hx\n (Set.Subset.trans (Finset.erase_subset (p : E) (minCardFinsetOfMemConvexHull hx))\n (minCardFinsetOfMemConvexHull_subseteq hx))\n hp\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\nk : \u2115 := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : \u00acAffineIndependent \ud835\udd5c Subtype.val\np : \u2191\u2191(minCardFinsetOfMemConvexHull hx)\nhp : x \u2208 \u2191(convexHull \ud835\udd5c) \u2191(erase (minCardFinsetOfMemConvexHull hx) \u2191p)\ncontra : card (minCardFinsetOfMemConvexHull hx) \u2264 card (erase (minCardFinsetOfMemConvexHull hx) \u2191p)\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 not_lt] at contra \n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\nk : \u2115 := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : \u00acAffineIndependent \ud835\udd5c Subtype.val\np : \u2191\u2191(minCardFinsetOfMemConvexHull hx)\nhp : x \u2208 \u2191(convexHull \ud835\udd5c) \u2191(erase (minCardFinsetOfMemConvexHull hx) \u2191p)\ncontra : \u00accard (erase (minCardFinsetOfMemConvexHull hx) \u2191p) < card (minCardFinsetOfMemConvexHull hx)\n\u22a2 False\n[PROOFSTEP]\napply contra\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\nk : \u2115 := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : \u00acAffineIndependent \ud835\udd5c Subtype.val\np : \u2191\u2191(minCardFinsetOfMemConvexHull hx)\nhp : x \u2208 \u2191(convexHull \ud835\udd5c) \u2191(erase (minCardFinsetOfMemConvexHull hx) \u2191p)\ncontra : \u00accard (erase (minCardFinsetOfMemConvexHull hx) \u2191p) < card (minCardFinsetOfMemConvexHull hx)\n\u22a2 card (erase (minCardFinsetOfMemConvexHull hx) \u2191p) < card (minCardFinsetOfMemConvexHull hx)\n[PROOFSTEP]\nerw [card_erase_of_mem p.2, hk]\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\nk : \u2115 := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : \u00acAffineIndependent \ud835\udd5c Subtype.val\np : \u2191\u2191(minCardFinsetOfMemConvexHull hx)\nhp : x \u2208 \u2191(convexHull \ud835\udd5c) \u2191(erase (minCardFinsetOfMemConvexHull hx) \u2191p)\ncontra : \u00accard (erase (minCardFinsetOfMemConvexHull hx) \u2191p) < card (minCardFinsetOfMemConvexHull hx)\n\u22a2 k + 1 - 1 < k + 1\n[PROOFSTEP]\nexact lt_add_one _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\n\u22a2 \u2191(convexHull \ud835\udd5c) s = \u22c3 (t : Finset E) (_ : \u2191t \u2286 s) (_ : AffineIndependent \ud835\udd5c Subtype.val), \u2191(convexHull \ud835\udd5c) \u2191t\n[PROOFSTEP]\napply Set.Subset.antisymm\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\n\u22a2 \u2191(convexHull \ud835\udd5c) s \u2286 \u22c3 (t : Finset E) (_ : \u2191t \u2286 s) (_ : AffineIndependent \ud835\udd5c Subtype.val), \u2191(convexHull \ud835\udd5c) \u2191t\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\n\u22a2 x \u2208 \u22c3 (t : Finset E) (_ : \u2191t \u2286 s) (_ : AffineIndependent \ud835\udd5c Subtype.val), \u2191(convexHull \ud835\udd5c) \u2191t\n[PROOFSTEP]\nsimp only [exists_prop, Set.mem_iUnion]\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\n\u22a2 \u2203 i, \u2191i \u2286 s \u2227 AffineIndependent \ud835\udd5c Subtype.val \u2227 x \u2208 \u2191(convexHull \ud835\udd5c) \u2191i\n[PROOFSTEP]\nexact\n \u27e8Caratheodory.minCardFinsetOfMemConvexHull hx, Caratheodory.minCardFinsetOfMemConvexHull_subseteq hx,\n Caratheodory.affineIndependent_minCardFinsetOfMemConvexHull hx, Caratheodory.mem_minCardFinsetOfMemConvexHull hx\u27e9\n[GOAL]\ncase h\u2082\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\n\u22a2 \u22c3 (t : Finset E) (_ : \u2191t \u2286 s) (_ : AffineIndependent \ud835\udd5c Subtype.val), \u2191(convexHull \ud835\udd5c) \u2191t \u2286 \u2191(convexHull \ud835\udd5c) s\n[PROOFSTEP]\niterate 3 convert Set.iUnion_subset _; intro\n[GOAL]\ncase h\u2082\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\n\u22a2 \u22c3 (t : Finset E) (_ : \u2191t \u2286 s) (_ : AffineIndependent \ud835\udd5c Subtype.val), \u2191(convexHull \ud835\udd5c) \u2191t \u2286 \u2191(convexHull \ud835\udd5c) s\n[PROOFSTEP]\nconvert Set.iUnion_subset _\n[GOAL]\ncase h\u2082.convert_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\n\u22a2 \u2200 (i : Finset E), \u22c3 (_ : \u2191i \u2286 s) (_ : AffineIndependent \ud835\udd5c Subtype.val), \u2191(convexHull \ud835\udd5c) \u2191i \u2286 \u2191(convexHull \ud835\udd5c) s\n[PROOFSTEP]\nintro\n[GOAL]\ncase h\u2082.convert_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\ni\u271d : Finset E\n\u22a2 \u22c3 (_ : \u2191i\u271d \u2286 s) (_ : AffineIndependent \ud835\udd5c Subtype.val), \u2191(convexHull \ud835\udd5c) \u2191i\u271d \u2286 \u2191(convexHull \ud835\udd5c) s\n[PROOFSTEP]\nconvert Set.iUnion_subset _\n[GOAL]\ncase h\u2082.convert_5.convert_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\ni\u271d : Finset E\n\u22a2 \u2191i\u271d \u2286 s \u2192 \u22c3 (_ : AffineIndependent \ud835\udd5c Subtype.val), \u2191(convexHull \ud835\udd5c) \u2191i\u271d \u2286 \u2191(convexHull \ud835\udd5c) s\n[PROOFSTEP]\nintro\n[GOAL]\ncase h\u2082.convert_5.convert_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\ni\u271d\u00b9 : Finset E\ni\u271d : \u2191i\u271d\u00b9 \u2286 s\n\u22a2 \u22c3 (_ : AffineIndependent \ud835\udd5c Subtype.val), \u2191(convexHull \ud835\udd5c) \u2191i\u271d\u00b9 \u2286 \u2191(convexHull \ud835\udd5c) s\n[PROOFSTEP]\nconvert Set.iUnion_subset _\n[GOAL]\ncase h\u2082.convert_5.convert_5.convert_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\ni\u271d\u00b9 : Finset E\ni\u271d : \u2191i\u271d\u00b9 \u2286 s\n\u22a2 AffineIndependent \ud835\udd5c Subtype.val \u2192 \u2191(convexHull \ud835\udd5c) \u2191i\u271d\u00b9 \u2286 \u2191(convexHull \ud835\udd5c) s\n[PROOFSTEP]\nintro\n[GOAL]\ncase h\u2082.convert_5.convert_5.convert_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\ni\u271d\u00b2 : Finset E\ni\u271d\u00b9 : \u2191i\u271d\u00b2 \u2286 s\ni\u271d : AffineIndependent \ud835\udd5c Subtype.val\n\u22a2 \u2191(convexHull \ud835\udd5c) \u2191i\u271d\u00b2 \u2286 \u2191(convexHull \ud835\udd5c) s\n[PROOFSTEP]\nexact convexHull_mono \u2039_\u203a\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c) s\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x\n[PROOFSTEP]\nrw [convexHull_eq_union] at hx \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u22c3 (t : Finset E) (_ : \u2191t \u2286 s) (_ : AffineIndependent \ud835\udd5c Subtype.val), \u2191(convexHull \ud835\udd5c) \u2191t\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x\n[PROOFSTEP]\nsimp only [exists_prop, Set.mem_iUnion] at hx \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : \u2203 i, \u2191i \u2286 s \u2227 AffineIndependent \ud835\udd5c Subtype.val \u2227 x \u2208 \u2191(convexHull \ud835\udd5c) \u2191i\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x\n[PROOFSTEP]\nobtain \u27e8t, ht\u2081, ht\u2082, ht\u2083\u27e9 := hx\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nht\u2083 : x \u2208 \u2191(convexHull \ud835\udd5c) \u2191t\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x\n[PROOFSTEP]\nsimp only [t.convexHull_eq, exists_prop, Set.mem_setOf_eq] at ht\u2083 \n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nht\u2083 : \u2203 w, (\u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y) \u2227 \u2211 y in t, w y = 1 \u2227 centerMass t w id = x\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x\n[PROOFSTEP]\nobtain \u27e8w, hw\u2081, hw\u2082, hw\u2083\u27e9 := ht\u2083\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x\n[PROOFSTEP]\nlet t' := t.filter fun i => w i \u2260 0\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x\n[PROOFSTEP]\nrefine' \u27e8t', t'.fintypeCoeSort, ((\u2191) : t' \u2192 E), w \u2218 ((\u2191) : t' \u2192 E), _, _, _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 Set.range Subtype.val \u2286 s\n[PROOFSTEP]\nrw [Subtype.range_coe_subtype]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 {x | x \u2208 t'} \u2286 s\n[PROOFSTEP]\nexact Subset.trans (Finset.filter_subset _ t) ht\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 AffineIndependent \ud835\udd5c Subtype.val\n[PROOFSTEP]\nexact ht\u2082.comp_embedding \u27e8_, inclusion_injective (Finset.filter_subset (fun i => w i \u2260 0) t)\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_3\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2200 (i : { x // x \u2208 t' }), 0 < (w \u2218 Subtype.val) i\n[PROOFSTEP]\nexact fun i => (hw\u2081 _ (Finset.mem_filter.mp i.2).1).lt_of_ne (Finset.mem_filter.mp i.property).2.symm\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_4\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2211 i : { x // x \u2208 t' }, (w \u2218 Subtype.val) i = 1\n[PROOFSTEP]\nerw [Finset.sum_attach, Finset.sum_filter_ne_zero, hw\u2082]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2211 i : { x // x \u2208 t' }, (w \u2218 Subtype.val) i \u2022 \u2191i = x\n[PROOFSTEP]\nchange (\u2211 i : t' in t'.attach, (fun e => w e \u2022 e) \u2191i) = x\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2211 i in attach t', (fun e => w e \u2022 e) \u2191i = x\n[PROOFSTEP]\nerw [Finset.sum_attach (f := fun e => w e \u2022 e), Finset.sum_filter_of_ne]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2211 x in t, w x \u2022 x = x\n[PROOFSTEP]\nrw [t.centerMass_eq_of_sum_1 id hw\u2082] at hw\u2083 \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : \u2211 i in t, w i \u2022 id i = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2211 x in t, w x \u2022 x = x\n[PROOFSTEP]\nexact hw\u2083\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2200 (x : E), x \u2208 t \u2192 w x \u2022 x \u2260 0 \u2192 w x \u2260 0\n[PROOFSTEP]\nintro e _ hwe contra\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\ne : E\na\u271d : e \u2208 t\nhwe : w e \u2022 e \u2260 0\ncontra : w e = 0\n\u22a2 False\n[PROOFSTEP]\napply hwe\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\ne : E\na\u271d : e \u2208 t\nhwe : w e \u2022 e \u2260 0\ncontra : w e = 0\n\u22a2 w e \u2022 e = 0\n[PROOFSTEP]\nrw [contra, zero_smul]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.Caratheodory", "llama_tokens": 23345, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.43013968906607175}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\ns : Finset (Perm \u03b1)\nh1 : \u2200 (f : Perm \u03b1), f \u2208 s \u2192 IsCycle f\nh2 : Set.Pairwise (\u2191s) Disjoint\nh0 : Finset.noncommProd s id (_ : Set.Pairwise \u2191s fun a b => Commute (id a) (id b)) = \u03c3\n\u22a2 cycleType \u03c3 = map (Finset.card \u2218 support) s.val\n[PROOFSTEP]\nrw [cycleType_def]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\ns : Finset (Perm \u03b1)\nh1 : \u2200 (f : Perm \u03b1), f \u2208 s \u2192 IsCycle f\nh2 : Set.Pairwise (\u2191s) Disjoint\nh0 : Finset.noncommProd s id (_ : Set.Pairwise \u2191s fun a b => Commute (id a) (id b)) = \u03c3\n\u22a2 map (Finset.card \u2218 support) (cycleFactorsFinset \u03c3).val = map (Finset.card \u2218 support) s.val\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s.e_self\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\ns : Finset (Perm \u03b1)\nh1 : \u2200 (f : Perm \u03b1), f \u2208 s \u2192 IsCycle f\nh2 : Set.Pairwise (\u2191s) Disjoint\nh0 : Finset.noncommProd s id (_ : Set.Pairwise \u2191s fun a b => Commute (id a) (id b)) = \u03c3\n\u22a2 cycleFactorsFinset \u03c3 = s\n[PROOFSTEP]\nrw [cycleFactorsFinset_eq_finset]\n[GOAL]\ncase e_s.e_self\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\ns : Finset (Perm \u03b1)\nh1 : \u2200 (f : Perm \u03b1), f \u2208 s \u2192 IsCycle f\nh2 : Set.Pairwise (\u2191s) Disjoint\nh0 : Finset.noncommProd s id (_ : Set.Pairwise \u2191s fun a b => Commute (id a) (id b)) = \u03c3\n\u22a2 (\u2200 (f : Perm \u03b1), f \u2208 s \u2192 IsCycle f) \u2227\n \u2203 h, Finset.noncommProd s id (_ : Set.Pairwise \u2191s fun a b => Commute (id a) (id b)) = \u03c3\n[PROOFSTEP]\nexact \u27e8h1, h2, h0\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nl : List (Perm \u03b1)\nh0 : List.prod l = \u03c3\nh1 : \u2200 (\u03c3 : Perm \u03b1), \u03c3 \u2208 l \u2192 IsCycle \u03c3\nh2 : List.Pairwise Disjoint l\n\u22a2 cycleType \u03c3 = \u2191(List.map (Finset.card \u2218 support) l)\n[PROOFSTEP]\nhave hl : l.Nodup := nodup_of_pairwise_disjoint_cycles h1 h2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nl : List (Perm \u03b1)\nh0 : List.prod l = \u03c3\nh1 : \u2200 (\u03c3 : Perm \u03b1), \u03c3 \u2208 l \u2192 IsCycle \u03c3\nh2 : List.Pairwise Disjoint l\nhl : List.Nodup l\n\u22a2 cycleType \u03c3 = \u2191(List.map (Finset.card \u2218 support) l)\n[PROOFSTEP]\nrw [cycleType_eq' l.toFinset]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nl : List (Perm \u03b1)\nh0 : List.prod l = \u03c3\nh1 : \u2200 (\u03c3 : Perm \u03b1), \u03c3 \u2208 l \u2192 IsCycle \u03c3\nh2 : List.Pairwise Disjoint l\nhl : List.Nodup l\n\u22a2 map (Finset.card \u2218 support) (List.toFinset l).val = \u2191(List.map (Finset.card \u2218 support) l)\n[PROOFSTEP]\nsimp [List.dedup_eq_self.mpr hl, (\u00b7 \u2218 \u00b7)]\n[GOAL]\ncase h1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nl : List (Perm \u03b1)\nh0 : List.prod l = \u03c3\nh1 : \u2200 (\u03c3 : Perm \u03b1), \u03c3 \u2208 l \u2192 IsCycle \u03c3\nh2 : List.Pairwise Disjoint l\nhl : List.Nodup l\n\u22a2 \u2200 (f : Perm \u03b1), f \u2208 List.toFinset l \u2192 IsCycle f\n[PROOFSTEP]\nsimpa using h1\n[GOAL]\ncase h2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nl : List (Perm \u03b1)\nh0 : List.prod l = \u03c3\nh1 : \u2200 (\u03c3 : Perm \u03b1), \u03c3 \u2208 l \u2192 IsCycle \u03c3\nh2 : List.Pairwise Disjoint l\nhl : List.Nodup l\n\u22a2 Set.Pairwise (\u2191(List.toFinset l)) Disjoint\n[PROOFSTEP]\nsimpa [hl] using h2\n[GOAL]\ncase h0\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nl : List (Perm \u03b1)\nh0 : List.prod l = \u03c3\nh1 : \u2200 (\u03c3 : Perm \u03b1), \u03c3 \u2208 l \u2192 IsCycle \u03c3\nh2 : List.Pairwise Disjoint l\nhl : List.Nodup l\n\u22a2 Finset.noncommProd (List.toFinset l) id (_ : Set.Pairwise \u2191(List.toFinset l) fun a b => Commute (id a) (id b)) = \u03c3\n[PROOFSTEP]\nsimp [hl, h0]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 cycleType \u03c3 = 0 \u2194 \u03c3 = 1\n[PROOFSTEP]\nsimp [cycleType_def, cycleFactorsFinset_eq_empty_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 \u2191card (cycleType \u03c3) = 0 \u2194 \u03c3 = 1\n[PROOFSTEP]\nrw [card_eq_zero, cycleType_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nn : \u2115\nh : n \u2208 cycleType \u03c3\n\u22a2 2 \u2264 n\n[PROOFSTEP]\nsimp only [cycleType_def, \u2190 Finset.mem_def, Function.comp_apply, Multiset.mem_map, mem_cycleFactorsFinset_iff] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nn : \u2115\nh : \u2203 a, (IsCycle a \u2227 \u2200 (a_1 : \u03b1), a_1 \u2208 support a \u2192 \u2191a a_1 = \u2191\u03c3 a_1) \u2227 Finset.card (support a) = n\n\u22a2 2 \u2264 n\n[PROOFSTEP]\nobtain \u27e8_, \u27e8hc, -\u27e9, rfl\u27e9 := h\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 w\u271d : Perm \u03b1\nhc : IsCycle w\u271d\n\u22a2 2 \u2264 Finset.card (support w\u271d)\n[PROOFSTEP]\nexact hc.two_le_card_support\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 \u2191card (cycleType \u03c3) = 1 \u2194 IsCycle \u03c3\n[PROOFSTEP]\nrw [card_eq_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 (\u2203 a, cycleType \u03c3 = {a}) \u2194 IsCycle \u03c3\n[PROOFSTEP]\nsimp_rw [cycleType_def, Multiset.map_eq_singleton, \u2190 Finset.singleton_val, Finset.val_inj,\n cycleFactorsFinset_eq_singleton_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 (\u2203 a a_1, (IsCycle \u03c3 \u2227 \u03c3 = a_1) \u2227 (Finset.card \u2218 support) a_1 = a) \u2194 IsCycle \u03c3\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 (\u2203 a a_1, (IsCycle \u03c3 \u2227 \u03c3 = a_1) \u2227 (Finset.card \u2218 support) a_1 = a) \u2192 IsCycle \u03c3\n[PROOFSTEP]\nrintro \u27e8_, _, \u27e8h, -\u27e9, -\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nw\u271d\u00b9 : \u2115\nw\u271d : Perm \u03b1\nh : IsCycle \u03c3\n\u22a2 IsCycle \u03c3\n[PROOFSTEP]\nexact h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 IsCycle \u03c3 \u2192 \u2203 a a_1, (IsCycle \u03c3 \u2227 \u03c3 = a_1) \u2227 (Finset.card \u2218 support) a_1 = a\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh : IsCycle \u03c3\n\u22a2 \u2203 a a_1, (IsCycle \u03c3 \u2227 \u03c3 = a_1) \u2227 (Finset.card \u2218 support) a_1 = a\n[PROOFSTEP]\nuse \u03c3.support.card, \u03c3\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh : IsCycle \u03c3\n\u22a2 (IsCycle \u03c3 \u2227 \u03c3 = \u03c3) \u2227 (Finset.card \u2218 support) \u03c3 = Finset.card (support \u03c3)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\nh : Disjoint \u03c3 \u03c4\n\u22a2 Perm.cycleType (\u03c3 * \u03c4) = Perm.cycleType \u03c3 + Perm.cycleType \u03c4\n[PROOFSTEP]\nrw [cycleType_def, cycleType_def, cycleType_def, h.cycleFactorsFinset_mul_eq_union, \u2190 Multiset.map_add,\n Finset.union_val, Multiset.add_eq_union_iff_disjoint.mpr _]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\nh : Disjoint \u03c3 \u03c4\n\u22a2 Multiset.Disjoint (cycleFactorsFinset \u03c3).val (cycleFactorsFinset \u03c4).val\n[PROOFSTEP]\nexact Finset.disjoint_val.2 h.disjoint_cycleFactorsFinset\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3\u271d \u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u22a2 (fun \u03c4 => cycleType \u03c4\u207b\u00b9 = cycleType \u03c4) \u03c3\n[PROOFSTEP]\nsimp only [h\u03c3.cycleType, h\u03c3.inv.cycleType, support_inv]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3\u271d \u03c3 \u03c4 : Perm \u03b1\nh\u03c3\u03c4 : Disjoint \u03c3 \u03c4\nx\u271d : IsCycle \u03c3\nh\u03c3 : (fun \u03c4 => cycleType \u03c4\u207b\u00b9 = cycleType \u03c4) \u03c3\nh\u03c4 : (fun \u03c4 => cycleType \u03c4\u207b\u00b9 = cycleType \u03c4) \u03c4\n\u22a2 (fun \u03c4 => cycleType \u03c4\u207b\u00b9 = cycleType \u03c4) (\u03c3 * \u03c4)\n[PROOFSTEP]\nsimp only [mul_inv_rev, h\u03c3\u03c4.cycleType, h\u03c3\u03c4.symm.inv_left.inv_right.cycleType, h\u03c3, h\u03c4, add_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\n\u22a2 cycleType (\u03c4 * \u03c3 * \u03c4\u207b\u00b9) = cycleType \u03c3\n[PROOFSTEP]\ninduction \u03c3 using cycle_induction_on with\n| base_one => simp\n| base_cycles \u03c3 h\u03c3 => rw [h\u03c3.cycleType, h\u03c3.conj.cycleType, card_support_conj]\n| induction_disjoint \u03c3 \u03c0 hd _ h\u03c3 h\u03c0 => rw [\u2190 conj_mul, hd.cycleType, (hd.conj _).cycleType, h\u03c3, h\u03c0]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\n\u22a2 cycleType (\u03c4 * \u03c3 * \u03c4\u207b\u00b9) = cycleType \u03c3\n[PROOFSTEP]\ninduction \u03c3 using cycle_induction_on with\n| base_one => simp\n| base_cycles \u03c3 h\u03c3 => rw [h\u03c3.cycleType, h\u03c3.conj.cycleType, card_support_conj]\n| induction_disjoint \u03c3 \u03c0 hd _ h\u03c3 h\u03c0 => rw [\u2190 conj_mul, hd.cycleType, (hd.conj _).cycleType, h\u03c3, h\u03c0]\n[GOAL]\ncase base_one\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c4 : Perm \u03b1\n\u22a2 cycleType (\u03c4 * 1 * \u03c4\u207b\u00b9) = cycleType 1\n[PROOFSTEP]\n\n| base_one => simp\n[GOAL]\ncase base_one\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c4 : Perm \u03b1\n\u22a2 cycleType (\u03c4 * 1 * \u03c4\u207b\u00b9) = cycleType 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c4 \u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u22a2 cycleType (\u03c4 * \u03c3 * \u03c4\u207b\u00b9) = cycleType \u03c3\n[PROOFSTEP]\n\n| base_cycles \u03c3 h\u03c3 => rw [h\u03c3.cycleType, h\u03c3.conj.cycleType, card_support_conj]\n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c4 \u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u22a2 cycleType (\u03c4 * \u03c3 * \u03c4\u207b\u00b9) = cycleType \u03c3\n[PROOFSTEP]\nrw [h\u03c3.cycleType, h\u03c3.conj.cycleType, card_support_conj]\n[GOAL]\ncase induction_disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c4 \u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\na\u271d : IsCycle \u03c3\nh\u03c3 : cycleType (\u03c4 * \u03c3 * \u03c4\u207b\u00b9) = cycleType \u03c3\nh\u03c0 : cycleType (\u03c4 * \u03c0 * \u03c4\u207b\u00b9) = cycleType \u03c0\n\u22a2 cycleType (\u03c4 * (\u03c3 * \u03c0) * \u03c4\u207b\u00b9) = cycleType (\u03c3 * \u03c0)\n[PROOFSTEP]\n\n| induction_disjoint \u03c3 \u03c0 hd _ h\u03c3 h\u03c0 => rw [\u2190 conj_mul, hd.cycleType, (hd.conj _).cycleType, h\u03c3, h\u03c0]\n[GOAL]\ncase induction_disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c4 \u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\na\u271d : IsCycle \u03c3\nh\u03c3 : cycleType (\u03c4 * \u03c3 * \u03c4\u207b\u00b9) = cycleType \u03c3\nh\u03c0 : cycleType (\u03c4 * \u03c0 * \u03c4\u207b\u00b9) = cycleType \u03c0\n\u22a2 cycleType (\u03c4 * (\u03c3 * \u03c0) * \u03c4\u207b\u00b9) = cycleType (\u03c3 * \u03c0)\n[PROOFSTEP]\nrw [\u2190 conj_mul, hd.cycleType, (hd.conj _).cycleType, h\u03c3, h\u03c0]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 sum (cycleType \u03c3) = Finset.card (support \u03c3)\n[PROOFSTEP]\ninduction \u03c3 using cycle_induction_on with\n| base_one => simp\n| base_cycles \u03c3 h\u03c3 => rw [h\u03c3.cycleType, coe_sum, List.sum_singleton]\n| induction_disjoint \u03c3 \u03c4 hd _ h\u03c3 h\u03c4 => rw [hd.cycleType, sum_add, h\u03c3, h\u03c4, hd.card_support_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 sum (cycleType \u03c3) = Finset.card (support \u03c3)\n[PROOFSTEP]\ninduction \u03c3 using cycle_induction_on with\n| base_one => simp\n| base_cycles \u03c3 h\u03c3 => rw [h\u03c3.cycleType, coe_sum, List.sum_singleton]\n| induction_disjoint \u03c3 \u03c4 hd _ h\u03c3 h\u03c4 => rw [hd.cycleType, sum_add, h\u03c3, h\u03c4, hd.card_support_mul]\n[GOAL]\ncase base_one\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 sum (cycleType 1) = Finset.card (support 1)\n[PROOFSTEP]\n\n| base_one => simp\n[GOAL]\ncase base_one\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 sum (cycleType 1) = Finset.card (support 1)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u22a2 sum (cycleType \u03c3) = Finset.card (support \u03c3)\n[PROOFSTEP]\n\n| base_cycles \u03c3 h\u03c3 => rw [h\u03c3.cycleType, coe_sum, List.sum_singleton]\n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u22a2 sum (cycleType \u03c3) = Finset.card (support \u03c3)\n[PROOFSTEP]\nrw [h\u03c3.cycleType, coe_sum, List.sum_singleton]\n[GOAL]\ncase induction_disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c4\na\u271d : IsCycle \u03c3\nh\u03c3 : sum (cycleType \u03c3) = Finset.card (support \u03c3)\nh\u03c4 : sum (cycleType \u03c4) = Finset.card (support \u03c4)\n\u22a2 sum (cycleType (\u03c3 * \u03c4)) = Finset.card (support (\u03c3 * \u03c4))\n[PROOFSTEP]\n\n| induction_disjoint \u03c3 \u03c4 hd _ h\u03c3 h\u03c4 => rw [hd.cycleType, sum_add, h\u03c3, h\u03c4, hd.card_support_mul]\n[GOAL]\ncase induction_disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c4\na\u271d : IsCycle \u03c3\nh\u03c3 : sum (cycleType \u03c3) = Finset.card (support \u03c3)\nh\u03c4 : sum (cycleType \u03c4) = Finset.card (support \u03c4)\n\u22a2 sum (cycleType (\u03c3 * \u03c4)) = Finset.card (support (\u03c3 * \u03c4))\n[PROOFSTEP]\nrw [hd.cycleType, sum_add, h\u03c3, h\u03c4, hd.card_support_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 \u2191sign \u03c3 = prod (map (fun n => -(-1) ^ n) (cycleType \u03c3))\n[PROOFSTEP]\ninduction \u03c3 using cycle_induction_on with\n| base_one => simp\n| base_cycles \u03c3 h\u03c3 => simp [h\u03c3.cycleType, h\u03c3.sign]\n| induction_disjoint \u03c3 \u03c4 hd _ h\u03c3 h\u03c4 => simp [h\u03c3, h\u03c4, hd.cycleType]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 \u2191sign \u03c3 = prod (map (fun n => -(-1) ^ n) (cycleType \u03c3))\n[PROOFSTEP]\ninduction \u03c3 using cycle_induction_on with\n| base_one => simp\n| base_cycles \u03c3 h\u03c3 => simp [h\u03c3.cycleType, h\u03c3.sign]\n| induction_disjoint \u03c3 \u03c4 hd _ h\u03c3 h\u03c4 => simp [h\u03c3, h\u03c4, hd.cycleType]\n[GOAL]\ncase base_one\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 \u2191sign 1 = prod (map (fun n => -(-1) ^ n) (cycleType 1))\n[PROOFSTEP]\n\n| base_one => simp\n[GOAL]\ncase base_one\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 \u2191sign 1 = prod (map (fun n => -(-1) ^ n) (cycleType 1))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u22a2 \u2191sign \u03c3 = prod (map (fun n => -(-1) ^ n) (cycleType \u03c3))\n[PROOFSTEP]\n\n| base_cycles \u03c3 h\u03c3 => simp [h\u03c3.cycleType, h\u03c3.sign]\n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u22a2 \u2191sign \u03c3 = prod (map (fun n => -(-1) ^ n) (cycleType \u03c3))\n[PROOFSTEP]\nsimp [h\u03c3.cycleType, h\u03c3.sign]\n[GOAL]\ncase induction_disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c4\na\u271d : IsCycle \u03c3\nh\u03c3 : \u2191sign \u03c3 = prod (map (fun n => -(-1) ^ n) (cycleType \u03c3))\nh\u03c4 : \u2191sign \u03c4 = prod (map (fun n => -(-1) ^ n) (cycleType \u03c4))\n\u22a2 \u2191sign (\u03c3 * \u03c4) = prod (map (fun n => -(-1) ^ n) (cycleType (\u03c3 * \u03c4)))\n[PROOFSTEP]\n\n| induction_disjoint \u03c3 \u03c4 hd _ h\u03c3 h\u03c4 => simp [h\u03c3, h\u03c4, hd.cycleType]\n[GOAL]\ncase induction_disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c4\na\u271d : IsCycle \u03c3\nh\u03c3 : \u2191sign \u03c3 = prod (map (fun n => -(-1) ^ n) (cycleType \u03c3))\nh\u03c4 : \u2191sign \u03c4 = prod (map (fun n => -(-1) ^ n) (cycleType \u03c4))\n\u22a2 \u2191sign (\u03c3 * \u03c4) = prod (map (fun n => -(-1) ^ n) (cycleType (\u03c3 * \u03c4)))\n[PROOFSTEP]\nsimp [h\u03c3, h\u03c4, hd.cycleType]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\n\u22a2 \u2191sign f = (-1) ^ (sum (cycleType f) + \u2191card (cycleType f))\n[PROOFSTEP]\nrw [sign_of_cycleType']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\n\u22a2 prod (map (fun n => -(-1) ^ n) (cycleType f)) = (-1) ^ (sum (cycleType f) + \u2191card (cycleType f))\n[PROOFSTEP]\ninduction' f.cycleType using Multiset.induction_on with a s ihs\n[GOAL]\ncase empty\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\n\u22a2 prod (map (fun n => -(-1) ^ n) 0) = (-1) ^ (sum 0 + \u2191card 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\na : \u2115\ns : Multiset \u2115\nihs : prod (map (fun n => -(-1) ^ n) s) = (-1) ^ (sum s + \u2191card s)\n\u22a2 prod (map (fun n => -(-1) ^ n) (a ::\u2098 s)) = (-1) ^ (sum (a ::\u2098 s) + \u2191card (a ::\u2098 s))\n[PROOFSTEP]\nrw [Multiset.map_cons, Multiset.prod_cons, Multiset.sum_cons, Multiset.card_cons, ihs]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\na : \u2115\ns : Multiset \u2115\nihs : prod (map (fun n => -(-1) ^ n) s) = (-1) ^ (sum s + \u2191card s)\n\u22a2 -(-1) ^ a * (-1) ^ (sum s + \u2191card s) = (-1) ^ (a + sum s + (\u2191card s + 1))\n[PROOFSTEP]\nsimp only [pow_add, pow_one, mul_neg_one, neg_mul, mul_neg, mul_assoc, mul_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 Multiset.lcm (cycleType \u03c3) = orderOf \u03c3\n[PROOFSTEP]\ninduction \u03c3 using cycle_induction_on with\n| base_one => simp\n| base_cycles \u03c3 h\u03c3 => simp [h\u03c3.cycleType, h\u03c3.orderOf]\n| induction_disjoint \u03c3 \u03c4 hd _ h\u03c3 h\u03c4 => simp [hd.cycleType, hd.orderOf, lcm_eq_nat_lcm, h\u03c3, h\u03c4]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 Multiset.lcm (cycleType \u03c3) = orderOf \u03c3\n[PROOFSTEP]\ninduction \u03c3 using cycle_induction_on with\n| base_one => simp\n| base_cycles \u03c3 h\u03c3 => simp [h\u03c3.cycleType, h\u03c3.orderOf]\n| induction_disjoint \u03c3 \u03c4 hd _ h\u03c3 h\u03c4 => simp [hd.cycleType, hd.orderOf, lcm_eq_nat_lcm, h\u03c3, h\u03c4]\n[GOAL]\ncase base_one\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 Multiset.lcm (cycleType 1) = orderOf 1\n[PROOFSTEP]\n\n| base_one => simp\n[GOAL]\ncase base_one\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 Multiset.lcm (cycleType 1) = orderOf 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u22a2 Multiset.lcm (cycleType \u03c3) = orderOf \u03c3\n[PROOFSTEP]\n\n| base_cycles \u03c3 h\u03c3 => simp [h\u03c3.cycleType, h\u03c3.orderOf]\n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u22a2 Multiset.lcm (cycleType \u03c3) = orderOf \u03c3\n[PROOFSTEP]\nsimp [h\u03c3.cycleType, h\u03c3.orderOf]\n[GOAL]\ncase induction_disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c4\na\u271d : IsCycle \u03c3\nh\u03c3 : Multiset.lcm (cycleType \u03c3) = orderOf \u03c3\nh\u03c4 : Multiset.lcm (cycleType \u03c4) = orderOf \u03c4\n\u22a2 Multiset.lcm (cycleType (\u03c3 * \u03c4)) = orderOf (\u03c3 * \u03c4)\n[PROOFSTEP]\n\n| induction_disjoint \u03c3 \u03c4 hd _ h\u03c3 h\u03c4 => simp [hd.cycleType, hd.orderOf, lcm_eq_nat_lcm, h\u03c3, h\u03c4]\n[GOAL]\ncase induction_disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c4\na\u271d : IsCycle \u03c3\nh\u03c3 : Multiset.lcm (cycleType \u03c3) = orderOf \u03c3\nh\u03c4 : Multiset.lcm (cycleType \u03c4) = orderOf \u03c4\n\u22a2 Multiset.lcm (cycleType (\u03c3 * \u03c4)) = orderOf (\u03c3 * \u03c4)\n[PROOFSTEP]\nsimp [hd.cycleType, hd.orderOf, lcm_eq_nat_lcm, h\u03c3, h\u03c4]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nn : \u2115\nh : n \u2208 cycleType \u03c3\n\u22a2 n \u2223 orderOf \u03c3\n[PROOFSTEP]\nrw [\u2190 lcm_cycleType]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nn : \u2115\nh : n \u2208 cycleType \u03c3\n\u22a2 n \u2223 Multiset.lcm (cycleType \u03c3)\n[PROOFSTEP]\nexact dvd_lcm h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx : \u03b1\n\u22a2 orderOf (cycleOf f x) \u2223 orderOf f\n[PROOFSTEP]\nby_cases hx : f x = x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u2191f x = x\n\u22a2 orderOf (cycleOf f x) \u2223 orderOf f\n[PROOFSTEP]\nrw [\u2190 cycleOf_eq_one_iff] at hx \n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx\u271d : \u2191f x = x\nhx : cycleOf f x = 1\n\u22a2 orderOf (cycleOf f x) \u2223 orderOf f\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00ac\u2191f x = x\n\u22a2 orderOf (cycleOf f x) \u2223 orderOf f\n[PROOFSTEP]\nrefine dvd_of_mem_cycleType ?_\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00ac\u2191f x = x\n\u22a2 orderOf (cycleOf f x) \u2208 cycleType f\n[PROOFSTEP]\nrw [cycleType, Multiset.mem_map]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00ac\u2191f x = x\n\u22a2 \u2203 a, a \u2208 (cycleFactorsFinset f).val \u2227 (Finset.card \u2218 support) a = orderOf (cycleOf f x)\n[PROOFSTEP]\nrefine' \u27e8f.cycleOf x, _, _\u27e9\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00ac\u2191f x = x\n\u22a2 cycleOf f x \u2208 (cycleFactorsFinset f).val\n[PROOFSTEP]\nrwa [\u2190 Finset.mem_def, cycleOf_mem_cycleFactorsFinset_iff, mem_support]\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00ac\u2191f x = x\n\u22a2 (Finset.card \u2218 support) (cycleOf f x) = orderOf (cycleOf f x)\n[PROOFSTEP]\nsimp [(isCycle_cycleOf _ hx).orderOf]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ 2 = 1\nn : \u2115\nhn : n \u2208 cycleType \u03c3\n\u22a2 2 \u2223 n\n[PROOFSTEP]\nrw [le_antisymm (Nat.le_of_dvd zero_lt_two <| (dvd_of_mem_cycleType hn).trans <| orderOf_dvd_of_pow_eq_one h\u03c3)\n (two_le_of_mem_cycleType hn)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : Nat.Prime (orderOf \u03c3)\n\u22a2 \u2203 n, cycleType \u03c3 = replicate (n + 1) (orderOf \u03c3)\n[PROOFSTEP]\nrefine \u27e8Multiset.card \u03c3.cycleType - 1, eq_replicate.2 \u27e8?_, fun n hn \u21a6 ?_\u27e9\u27e9\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : Nat.Prime (orderOf \u03c3)\n\u22a2 \u2191card (cycleType \u03c3) = \u2191card (cycleType \u03c3) - 1 + 1\n[PROOFSTEP]\nrw [tsub_add_cancel_of_le]\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : Nat.Prime (orderOf \u03c3)\n\u22a2 1 \u2264 \u2191card (cycleType \u03c3)\n[PROOFSTEP]\nrw [Nat.succ_le_iff, card_cycleType_pos, Ne.def, \u2190 orderOf_eq_one_iff]\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : Nat.Prime (orderOf \u03c3)\n\u22a2 \u00acorderOf \u03c3 = 1\n[PROOFSTEP]\nexact h\u03c3.ne_one\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : Nat.Prime (orderOf \u03c3)\nn : \u2115\nhn : n \u2208 cycleType \u03c3\n\u22a2 n = orderOf \u03c3\n[PROOFSTEP]\nexact (h\u03c3.eq_one_or_self_of_dvd n (dvd_of_mem_cycleType hn)).resolve_left (one_lt_of_mem_cycleType hn).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh1 : Nat.Prime (orderOf \u03c3)\nh2 : Finset.card (support \u03c3) < 2 * orderOf \u03c3\n\u22a2 IsCycle \u03c3\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := cycleType_prime_order h1\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh1 : Nat.Prime (orderOf \u03c3)\nh2 : Finset.card (support \u03c3) < 2 * orderOf \u03c3\nn : \u2115\nhn : cycleType \u03c3 = replicate (n + 1) (orderOf \u03c3)\n\u22a2 IsCycle \u03c3\n[PROOFSTEP]\nrw [\u2190 \u03c3.sum_cycleType, hn, Multiset.sum_replicate, nsmul_eq_mul, Nat.cast_id, mul_lt_mul_right (orderOf_pos \u03c3),\n Nat.succ_lt_succ_iff, Nat.lt_succ_iff, le_zero_iff] at h2 \n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh1 : Nat.Prime (orderOf \u03c3)\nn : \u2115\nh2 : n = 0\nhn : cycleType \u03c3 = replicate (n + 1) (orderOf \u03c3)\n\u22a2 IsCycle \u03c3\n[PROOFSTEP]\nrw [\u2190 card_cycleType_eq_one, hn, card_replicate, h2]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf g : Perm \u03b1\nhf : f \u2208 cycleFactorsFinset g\n\u22a2 cycleType f \u2264 cycleType g\n[PROOFSTEP]\nhave hf' := mem_cycleFactorsFinset_iff.1 hf\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf g : Perm \u03b1\nhf : f \u2208 cycleFactorsFinset g\nhf' : IsCycle f \u2227 \u2200 (a : \u03b1), a \u2208 support f \u2192 \u2191f a = \u2191g a\n\u22a2 cycleType f \u2264 cycleType g\n[PROOFSTEP]\nrw [cycleType_def, cycleType_def, hf'.left.cycleFactorsFinset_eq_singleton]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf g : Perm \u03b1\nhf : f \u2208 cycleFactorsFinset g\nhf' : IsCycle f \u2227 \u2200 (a : \u03b1), a \u2208 support f \u2192 \u2191f a = \u2191g a\n\u22a2 map (Finset.card \u2218 support) {f}.val \u2264 map (Finset.card \u2218 support) (cycleFactorsFinset g).val\n[PROOFSTEP]\nrefine map_le_map ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf g : Perm \u03b1\nhf : f \u2208 cycleFactorsFinset g\nhf' : IsCycle f \u2227 \u2200 (a : \u03b1), a \u2208 support f \u2192 \u2191f a = \u2191g a\n\u22a2 {f}.val \u2264 (cycleFactorsFinset g).val\n[PROOFSTEP]\nsimpa only [Finset.singleton_val, singleton_le, Finset.mem_val] using hf\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nf g : Perm \u03b1\nhf : f \u2208 cycleFactorsFinset g\n\u22a2 cycleType (g * f\u207b\u00b9) + cycleType f = cycleType g - cycleType f + cycleType f\n[PROOFSTEP]\nrw [\u2190 (disjoint_mul_inv_of_mem_cycleFactorsFinset hf).cycleType, inv_mul_cancel_right,\n tsub_add_cancel_of_le (cycleType_le_of_mem_cycleFactorsFinset hf)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\nh : cycleType \u03c3 = cycleType \u03c4\n\u22a2 IsConj \u03c3 \u03c4\n[PROOFSTEP]\ninduction \u03c3 using cycle_induction_on generalizing \u03c4 with\n| base_one =>\n rw [cycleType_one, eq_comm, cycleType_eq_zero] at h \n rw [h]\n| base_cycles \u03c3 h\u03c3 =>\n have h\u03c4 := card_cycleType_eq_one.2 h\u03c3\n rw [h, card_cycleType_eq_one] at h\u03c4 \n apply h\u03c3.isConj h\u03c4\n rw [h\u03c3.cycleType, h\u03c4.cycleType, coe_eq_coe, List.singleton_perm] at h \n exact List.singleton_injective h\n| induction_disjoint \u03c3 \u03c0 hd hc h\u03c3 h\u03c0 =>\n rw [hd.cycleType] at h \n have h' : \u03c3.support.card \u2208 \u03c4.cycleType := by simp [\u2190 h, hc.cycleType]\n obtain \u27e8\u03c3', h\u03c3'l, h\u03c3'\u27e9 := Multiset.mem_map.mp h'\n have key : IsConj (\u03c3' * \u03c4 * \u03c3'\u207b\u00b9) \u03c4 := (isConj_iff.2 \u27e8\u03c3', rfl\u27e9).symm\n refine' IsConj.trans _ key\n rw [mul_assoc]\n have hs : \u03c3.cycleType = \u03c3'.cycleType :=\n by\n rw [\u2190 Finset.mem_def, mem_cycleFactorsFinset_iff] at h\u03c3'l \n rw [hc.cycleType, \u2190 h\u03c3', h\u03c3'l.left.cycleType]; rfl\n refine' hd.isConj_mul (h\u03c3 hs) (h\u03c0 _) _\n \u00b7 rw [cycleType_mul_inv_mem_cycleFactorsFinset_eq_sub, \u2190 h, add_comm, hs, add_tsub_cancel_right]\n rwa [Finset.mem_def]\n \u00b7 exact (disjoint_mul_inv_of_mem_cycleFactorsFinset h\u03c3'l).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\nh : cycleType \u03c3 = cycleType \u03c4\n\u22a2 IsConj \u03c3 \u03c4\n[PROOFSTEP]\ninduction \u03c3 using cycle_induction_on generalizing \u03c4 with\n| base_one =>\n rw [cycleType_one, eq_comm, cycleType_eq_zero] at h \n rw [h]\n| base_cycles \u03c3 h\u03c3 =>\n have h\u03c4 := card_cycleType_eq_one.2 h\u03c3\n rw [h, card_cycleType_eq_one] at h\u03c4 \n apply h\u03c3.isConj h\u03c4\n rw [h\u03c3.cycleType, h\u03c4.cycleType, coe_eq_coe, List.singleton_perm] at h \n exact List.singleton_injective h\n| induction_disjoint \u03c3 \u03c0 hd hc h\u03c3 h\u03c0 =>\n rw [hd.cycleType] at h \n have h' : \u03c3.support.card \u2208 \u03c4.cycleType := by simp [\u2190 h, hc.cycleType]\n obtain \u27e8\u03c3', h\u03c3'l, h\u03c3'\u27e9 := Multiset.mem_map.mp h'\n have key : IsConj (\u03c3' * \u03c4 * \u03c3'\u207b\u00b9) \u03c4 := (isConj_iff.2 \u27e8\u03c3', rfl\u27e9).symm\n refine' IsConj.trans _ key\n rw [mul_assoc]\n have hs : \u03c3.cycleType = \u03c3'.cycleType :=\n by\n rw [\u2190 Finset.mem_def, mem_cycleFactorsFinset_iff] at h\u03c3'l \n rw [hc.cycleType, \u2190 h\u03c3', h\u03c3'l.left.cycleType]; rfl\n refine' hd.isConj_mul (h\u03c3 hs) (h\u03c0 _) _\n \u00b7 rw [cycleType_mul_inv_mem_cycleFactorsFinset_eq_sub, \u2190 h, add_comm, hs, add_tsub_cancel_right]\n rwa [Finset.mem_def]\n \u00b7 exact (disjoint_mul_inv_of_mem_cycleFactorsFinset h\u03c3'l).symm\n[GOAL]\ncase base_one\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c4 : Perm \u03b1\nh : cycleType 1 = cycleType \u03c4\n\u22a2 IsConj 1 \u03c4\n[PROOFSTEP]\n\n| base_one =>\n rw [cycleType_one, eq_comm, cycleType_eq_zero] at h \n rw [h]\n[GOAL]\ncase base_one\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c4 : Perm \u03b1\nh : cycleType 1 = cycleType \u03c4\n\u22a2 IsConj 1 \u03c4\n[PROOFSTEP]\nrw [cycleType_one, eq_comm, cycleType_eq_zero] at h \n[GOAL]\ncase base_one\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c4 : Perm \u03b1\nh : \u03c4 = 1\n\u22a2 IsConj 1 \u03c4\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 = cycleType \u03c4\n\u22a2 IsConj \u03c3 \u03c4\n[PROOFSTEP]\n\n| base_cycles \u03c3 h\u03c3 =>\n have h\u03c4 := card_cycleType_eq_one.2 h\u03c3\n rw [h, card_cycleType_eq_one] at h\u03c4 \n apply h\u03c3.isConj h\u03c4\n rw [h\u03c3.cycleType, h\u03c4.cycleType, coe_eq_coe, List.singleton_perm] at h \n exact List.singleton_injective h\n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 = cycleType \u03c4\n\u22a2 IsConj \u03c3 \u03c4\n[PROOFSTEP]\nhave h\u03c4 := card_cycleType_eq_one.2 h\u03c3\n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 = cycleType \u03c4\nh\u03c4 : \u2191card (cycleType \u03c3) = 1\n\u22a2 IsConj \u03c3 \u03c4\n[PROOFSTEP]\nrw [h, card_cycleType_eq_one] at h\u03c4 \n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 = cycleType \u03c4\nh\u03c4 : IsCycle \u03c4\n\u22a2 IsConj \u03c3 \u03c4\n[PROOFSTEP]\napply h\u03c3.isConj h\u03c4\n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 = cycleType \u03c4\nh\u03c4 : IsCycle \u03c4\n\u22a2 Finset.card (support \u03c3) = Finset.card (support \u03c4)\n[PROOFSTEP]\nrw [h\u03c3.cycleType, h\u03c4.cycleType, coe_eq_coe, List.singleton_perm] at h \n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u03c4 : Perm \u03b1\nh : [Finset.card (support \u03c3)] = [Finset.card (support \u03c4)]\nh\u03c4 : IsCycle \u03c4\n\u22a2 Finset.card (support \u03c3) = Finset.card (support \u03c4)\n[PROOFSTEP]\nexact List.singleton_injective h\n[GOAL]\ncase induction_disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType (\u03c3 * \u03c0) = cycleType \u03c4\n\u22a2 IsConj (\u03c3 * \u03c0) \u03c4\n[PROOFSTEP]\n\n| induction_disjoint \u03c3 \u03c0 hd hc h\u03c3 h\u03c0 =>\n rw [hd.cycleType] at h \n have h' : \u03c3.support.card \u2208 \u03c4.cycleType := by simp [\u2190 h, hc.cycleType]\n obtain \u27e8\u03c3', h\u03c3'l, h\u03c3'\u27e9 := Multiset.mem_map.mp h'\n have key : IsConj (\u03c3' * \u03c4 * \u03c3'\u207b\u00b9) \u03c4 := (isConj_iff.2 \u27e8\u03c3', rfl\u27e9).symm\n refine' IsConj.trans _ key\n rw [mul_assoc]\n have hs : \u03c3.cycleType = \u03c3'.cycleType :=\n by\n rw [\u2190 Finset.mem_def, mem_cycleFactorsFinset_iff] at h\u03c3'l \n rw [hc.cycleType, \u2190 h\u03c3', h\u03c3'l.left.cycleType]; rfl\n refine' hd.isConj_mul (h\u03c3 hs) (h\u03c0 _) _\n \u00b7 rw [cycleType_mul_inv_mem_cycleFactorsFinset_eq_sub, \u2190 h, add_comm, hs, add_tsub_cancel_right]\n rwa [Finset.mem_def]\n \u00b7 exact (disjoint_mul_inv_of_mem_cycleFactorsFinset h\u03c3'l).symm\n[GOAL]\ncase induction_disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType (\u03c3 * \u03c0) = cycleType \u03c4\n\u22a2 IsConj (\u03c3 * \u03c0) \u03c4\n[PROOFSTEP]\nrw [hd.cycleType] at h \n[GOAL]\ncase induction_disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 + cycleType \u03c0 = cycleType \u03c4\n\u22a2 IsConj (\u03c3 * \u03c0) \u03c4\n[PROOFSTEP]\nhave h' : \u03c3.support.card \u2208 \u03c4.cycleType := by simp [\u2190 h, hc.cycleType]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 + cycleType \u03c0 = cycleType \u03c4\n\u22a2 Finset.card (support \u03c3) \u2208 cycleType \u03c4\n[PROOFSTEP]\nsimp [\u2190 h, hc.cycleType]\n[GOAL]\ncase induction_disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 + cycleType \u03c0 = cycleType \u03c4\nh' : Finset.card (support \u03c3) \u2208 cycleType \u03c4\n\u22a2 IsConj (\u03c3 * \u03c0) \u03c4\n[PROOFSTEP]\nobtain \u27e8\u03c3', h\u03c3'l, h\u03c3'\u27e9 := Multiset.mem_map.mp h'\n[GOAL]\ncase induction_disjoint.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 + cycleType \u03c0 = cycleType \u03c4\nh' : Finset.card (support \u03c3) \u2208 cycleType \u03c4\n\u03c3' : Perm \u03b1\nh\u03c3'l : \u03c3' \u2208 (cycleFactorsFinset \u03c4).val\nh\u03c3' : (Finset.card \u2218 support) \u03c3' = Finset.card (support \u03c3)\n\u22a2 IsConj (\u03c3 * \u03c0) \u03c4\n[PROOFSTEP]\nhave key : IsConj (\u03c3' * \u03c4 * \u03c3'\u207b\u00b9) \u03c4 := (isConj_iff.2 \u27e8\u03c3', rfl\u27e9).symm\n[GOAL]\ncase induction_disjoint.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 + cycleType \u03c0 = cycleType \u03c4\nh' : Finset.card (support \u03c3) \u2208 cycleType \u03c4\n\u03c3' : Perm \u03b1\nh\u03c3'l : \u03c3' \u2208 (cycleFactorsFinset \u03c4).val\nh\u03c3' : (Finset.card \u2218 support) \u03c3' = Finset.card (support \u03c3)\nkey : IsConj (\u03c3' * \u03c4 * \u03c3'\u207b\u00b9) \u03c4\n\u22a2 IsConj (\u03c3 * \u03c0) \u03c4\n[PROOFSTEP]\nrefine' IsConj.trans _ key\n[GOAL]\ncase induction_disjoint.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 + cycleType \u03c0 = cycleType \u03c4\nh' : Finset.card (support \u03c3) \u2208 cycleType \u03c4\n\u03c3' : Perm \u03b1\nh\u03c3'l : \u03c3' \u2208 (cycleFactorsFinset \u03c4).val\nh\u03c3' : (Finset.card \u2218 support) \u03c3' = Finset.card (support \u03c3)\nkey : IsConj (\u03c3' * \u03c4 * \u03c3'\u207b\u00b9) \u03c4\n\u22a2 IsConj (\u03c3 * \u03c0) (\u03c3' * \u03c4 * \u03c3'\u207b\u00b9)\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\ncase induction_disjoint.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 + cycleType \u03c0 = cycleType \u03c4\nh' : Finset.card (support \u03c3) \u2208 cycleType \u03c4\n\u03c3' : Perm \u03b1\nh\u03c3'l : \u03c3' \u2208 (cycleFactorsFinset \u03c4).val\nh\u03c3' : (Finset.card \u2218 support) \u03c3' = Finset.card (support \u03c3)\nkey : IsConj (\u03c3' * \u03c4 * \u03c3'\u207b\u00b9) \u03c4\n\u22a2 IsConj (\u03c3 * \u03c0) (\u03c3' * (\u03c4 * \u03c3'\u207b\u00b9))\n[PROOFSTEP]\nhave hs : \u03c3.cycleType = \u03c3'.cycleType :=\n by\n rw [\u2190 Finset.mem_def, mem_cycleFactorsFinset_iff] at h\u03c3'l \n rw [hc.cycleType, \u2190 h\u03c3', h\u03c3'l.left.cycleType]; rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 + cycleType \u03c0 = cycleType \u03c4\nh' : Finset.card (support \u03c3) \u2208 cycleType \u03c4\n\u03c3' : Perm \u03b1\nh\u03c3'l : \u03c3' \u2208 (cycleFactorsFinset \u03c4).val\nh\u03c3' : (Finset.card \u2218 support) \u03c3' = Finset.card (support \u03c3)\nkey : IsConj (\u03c3' * \u03c4 * \u03c3'\u207b\u00b9) \u03c4\n\u22a2 cycleType \u03c3 = cycleType \u03c3'\n[PROOFSTEP]\nrw [\u2190 Finset.mem_def, mem_cycleFactorsFinset_iff] at h\u03c3'l \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 + cycleType \u03c0 = cycleType \u03c4\nh' : Finset.card (support \u03c3) \u2208 cycleType \u03c4\n\u03c3' : Perm \u03b1\nh\u03c3'l : IsCycle \u03c3' \u2227 \u2200 (a : \u03b1), a \u2208 support \u03c3' \u2192 \u2191\u03c3' a = \u2191\u03c4 a\nh\u03c3' : (Finset.card \u2218 support) \u03c3' = Finset.card (support \u03c3)\nkey : IsConj (\u03c3' * \u03c4 * \u03c3'\u207b\u00b9) \u03c4\n\u22a2 cycleType \u03c3 = cycleType \u03c3'\n[PROOFSTEP]\nrw [hc.cycleType, \u2190 h\u03c3', h\u03c3'l.left.cycleType]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 + cycleType \u03c0 = cycleType \u03c4\nh' : Finset.card (support \u03c3) \u2208 cycleType \u03c4\n\u03c3' : Perm \u03b1\nh\u03c3'l : IsCycle \u03c3' \u2227 \u2200 (a : \u03b1), a \u2208 support \u03c3' \u2192 \u2191\u03c3' a = \u2191\u03c4 a\nh\u03c3' : (Finset.card \u2218 support) \u03c3' = Finset.card (support \u03c3)\nkey : IsConj (\u03c3' * \u03c4 * \u03c3'\u207b\u00b9) \u03c4\n\u22a2 \u2191[(Finset.card \u2218 support) \u03c3'] = \u2191[Finset.card (support \u03c3')]\n[PROOFSTEP]\nrfl\n[GOAL]\ncase induction_disjoint.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 + cycleType \u03c0 = cycleType \u03c4\nh' : Finset.card (support \u03c3) \u2208 cycleType \u03c4\n\u03c3' : Perm \u03b1\nh\u03c3'l : \u03c3' \u2208 (cycleFactorsFinset \u03c4).val\nh\u03c3' : (Finset.card \u2218 support) \u03c3' = Finset.card (support \u03c3)\nkey : IsConj (\u03c3' * \u03c4 * \u03c3'\u207b\u00b9) \u03c4\nhs : cycleType \u03c3 = cycleType \u03c3'\n\u22a2 IsConj (\u03c3 * \u03c0) (\u03c3' * (\u03c4 * \u03c3'\u207b\u00b9))\n[PROOFSTEP]\nrefine' hd.isConj_mul (h\u03c3 hs) (h\u03c0 _) _\n[GOAL]\ncase induction_disjoint.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 + cycleType \u03c0 = cycleType \u03c4\nh' : Finset.card (support \u03c3) \u2208 cycleType \u03c4\n\u03c3' : Perm \u03b1\nh\u03c3'l : \u03c3' \u2208 (cycleFactorsFinset \u03c4).val\nh\u03c3' : (Finset.card \u2218 support) \u03c3' = Finset.card (support \u03c3)\nkey : IsConj (\u03c3' * \u03c4 * \u03c3'\u207b\u00b9) \u03c4\nhs : cycleType \u03c3 = cycleType \u03c3'\n\u22a2 cycleType \u03c0 = cycleType (\u03c4 * \u03c3'\u207b\u00b9)\n[PROOFSTEP]\nrw [cycleType_mul_inv_mem_cycleFactorsFinset_eq_sub, \u2190 h, add_comm, hs, add_tsub_cancel_right]\n[GOAL]\ncase induction_disjoint.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 + cycleType \u03c0 = cycleType \u03c4\nh' : Finset.card (support \u03c3) \u2208 cycleType \u03c4\n\u03c3' : Perm \u03b1\nh\u03c3'l : \u03c3' \u2208 (cycleFactorsFinset \u03c4).val\nh\u03c3' : (Finset.card \u2218 support) \u03c3' = Finset.card (support \u03c3)\nkey : IsConj (\u03c3' * \u03c4 * \u03c3'\u207b\u00b9) \u03c4\nhs : cycleType \u03c3 = cycleType \u03c3'\n\u22a2 \u03c3' \u2208 cycleFactorsFinset \u03c4\n[PROOFSTEP]\nrwa [Finset.mem_def]\n[GOAL]\ncase induction_disjoint.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c0\nhc : IsCycle \u03c3\nh\u03c3 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c3 = cycleType \u03c4 \u2192 IsConj \u03c3 \u03c4\nh\u03c0 : \u2200 {\u03c4 : Perm \u03b1}, cycleType \u03c0 = cycleType \u03c4 \u2192 IsConj \u03c0 \u03c4\n\u03c4 : Perm \u03b1\nh : cycleType \u03c3 + cycleType \u03c0 = cycleType \u03c4\nh' : Finset.card (support \u03c3) \u2208 cycleType \u03c4\n\u03c3' : Perm \u03b1\nh\u03c3'l : \u03c3' \u2208 (cycleFactorsFinset \u03c4).val\nh\u03c3' : (Finset.card \u2218 support) \u03c3' = Finset.card (support \u03c3)\nkey : IsConj (\u03c3' * \u03c4 * \u03c3'\u207b\u00b9) \u03c4\nhs : cycleType \u03c3 = cycleType \u03c3'\n\u22a2 Disjoint \u03c3' (\u03c4 * \u03c3'\u207b\u00b9)\n[PROOFSTEP]\nexact (disjoint_mul_inv_of_mem_cycleFactorsFinset h\u03c3'l).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\nh : IsConj \u03c3 \u03c4\n\u22a2 cycleType \u03c3 = cycleType \u03c4\n[PROOFSTEP]\nobtain \u27e8\u03c0, rfl\u27e9 := isConj_iff.1 h\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c0 : Perm \u03b1\nh : IsConj \u03c3 (\u03c0 * \u03c3 * \u03c0\u207b\u00b9)\n\u22a2 cycleType \u03c3 = cycleType (\u03c0 * \u03c3 * \u03c0\u207b\u00b9)\n[PROOFSTEP]\nrw [cycleType_conj]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : Fintype \u03b1\ninst\u271d\u00b3 : DecidableEq \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\n\u22a2 cycleType (extendDomain g f) = cycleType g\n[PROOFSTEP]\ninduction g using cycle_induction_on with\n| base_one => rw [extendDomain_one, cycleType_one, cycleType_one]\n| base_cycles \u03c3 h\u03c3 => rw [(h\u03c3.extendDomain f).cycleType, h\u03c3.cycleType, card_support_extend_domain]\n| induction_disjoint \u03c3 \u03c4 hd _ h\u03c3 h\u03c4 => rw [hd.cycleType, \u2190 extendDomain_mul, (hd.extendDomain f).cycleType, h\u03c3, h\u03c4]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : Fintype \u03b1\ninst\u271d\u00b3 : DecidableEq \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\n\u22a2 cycleType (extendDomain g f) = cycleType g\n[PROOFSTEP]\ninduction g using cycle_induction_on with\n| base_one => rw [extendDomain_one, cycleType_one, cycleType_one]\n| base_cycles \u03c3 h\u03c3 => rw [(h\u03c3.extendDomain f).cycleType, h\u03c3.cycleType, card_support_extend_domain]\n| induction_disjoint \u03c3 \u03c4 hd _ h\u03c3 h\u03c4 => rw [hd.cycleType, \u2190 extendDomain_mul, (hd.extendDomain f).cycleType, h\u03c3, h\u03c4]\n[GOAL]\ncase base_one\n\u03b1 : Type u_1\ninst\u271d\u2074 : Fintype \u03b1\ninst\u271d\u00b3 : DecidableEq \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\n\u22a2 cycleType (extendDomain 1 f) = cycleType 1\n[PROOFSTEP]\n\n| base_one => rw [extendDomain_one, cycleType_one, cycleType_one]\n[GOAL]\ncase base_one\n\u03b1 : Type u_1\ninst\u271d\u2074 : Fintype \u03b1\ninst\u271d\u00b3 : DecidableEq \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\n\u22a2 cycleType (extendDomain 1 f) = cycleType 1\n[PROOFSTEP]\nrw [extendDomain_one, cycleType_one, cycleType_one]\n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u2074 : Fintype \u03b1\ninst\u271d\u00b3 : DecidableEq \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\n\u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u22a2 cycleType (extendDomain \u03c3 f) = cycleType \u03c3\n[PROOFSTEP]\n\n| base_cycles \u03c3 h\u03c3 => rw [(h\u03c3.extendDomain f).cycleType, h\u03c3.cycleType, card_support_extend_domain]\n[GOAL]\ncase base_cycles\n\u03b1 : Type u_1\ninst\u271d\u2074 : Fintype \u03b1\ninst\u271d\u00b3 : DecidableEq \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\n\u03c3 : Perm \u03b1\nh\u03c3 : IsCycle \u03c3\n\u22a2 cycleType (extendDomain \u03c3 f) = cycleType \u03c3\n[PROOFSTEP]\nrw [(h\u03c3.extendDomain f).cycleType, h\u03c3.cycleType, card_support_extend_domain]\n[GOAL]\ncase induction_disjoint\n\u03b1 : Type u_1\ninst\u271d\u2074 : Fintype \u03b1\ninst\u271d\u00b3 : DecidableEq \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\n\u03c3 \u03c4 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c4\na\u271d : IsCycle \u03c3\nh\u03c3 : cycleType (extendDomain \u03c3 f) = cycleType \u03c3\nh\u03c4 : cycleType (extendDomain \u03c4 f) = cycleType \u03c4\n\u22a2 cycleType (extendDomain (\u03c3 * \u03c4) f) = cycleType (\u03c3 * \u03c4)\n[PROOFSTEP]\n\n| induction_disjoint \u03c3 \u03c4 hd _ h\u03c3 h\u03c4 => rw [hd.cycleType, \u2190 extendDomain_mul, (hd.extendDomain f).cycleType, h\u03c3, h\u03c4]\n[GOAL]\ncase induction_disjoint\n\u03b1 : Type u_1\ninst\u271d\u2074 : Fintype \u03b1\ninst\u271d\u00b3 : DecidableEq \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\n\u03c3 \u03c4 : Perm \u03b1\nhd : Disjoint \u03c3 \u03c4\na\u271d : IsCycle \u03c3\nh\u03c3 : cycleType (extendDomain \u03c3 f) = cycleType \u03c3\nh\u03c4 : cycleType (extendDomain \u03c4 f) = cycleType \u03c4\n\u22a2 cycleType (extendDomain (\u03c3 * \u03c4) f) = cycleType (\u03c3 * \u03c4)\n[PROOFSTEP]\nrw [hd.cycleType, \u2190 extendDomain_mul, (hd.extendDomain f).cycleType, h\u03c3, h\u03c4]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u03c3 : Perm \u03b1\n\u22a2 n \u2208 cycleType \u03c3 \u2194 \u2203 c \u03c4, \u03c3 = c * \u03c4 \u2227 Disjoint c \u03c4 \u2227 IsCycle c \u2227 Finset.card (support c) = n\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u03c3 : Perm \u03b1\n\u22a2 n \u2208 cycleType \u03c3 \u2192 \u2203 c \u03c4, \u03c3 = c * \u03c4 \u2227 Disjoint c \u03c4 \u2227 IsCycle c \u2227 Finset.card (support c) = n\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u03c3 : Perm \u03b1\nh : n \u2208 cycleType \u03c3\n\u22a2 \u2203 c \u03c4, \u03c3 = c * \u03c4 \u2227 Disjoint c \u03c4 \u2227 IsCycle c \u2227 Finset.card (support c) = n\n[PROOFSTEP]\nobtain \u27e8l, rfl, hlc, hld\u27e9 := truncCycleFactors \u03c3\n[GOAL]\ncase mp.mk.mk.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\nl : List (Perm \u03b1)\nh : n \u2208 cycleType (List.prod l)\nx\u271d : Trunc { l_1 // List.prod l_1 = List.prod l \u2227 (\u2200 (g : Perm \u03b1), g \u2208 l_1 \u2192 IsCycle g) \u2227 List.Pairwise Disjoint l_1 }\nhlc : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsCycle g\nhld : List.Pairwise Disjoint l\n\u22a2 \u2203 c \u03c4, List.prod l = c * \u03c4 \u2227 Disjoint c \u03c4 \u2227 IsCycle c \u2227 Finset.card (support c) = n\n[PROOFSTEP]\nrw [cycleType_eq _ rfl hlc hld, Multiset.mem_coe, List.mem_map] at h \n[GOAL]\ncase mp.mk.mk.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\nl : List (Perm \u03b1)\nh : \u2203 a, a \u2208 l \u2227 (Finset.card \u2218 support) a = n\nx\u271d : Trunc { l_1 // List.prod l_1 = List.prod l \u2227 (\u2200 (g : Perm \u03b1), g \u2208 l_1 \u2192 IsCycle g) \u2227 List.Pairwise Disjoint l_1 }\nhlc : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsCycle g\nhld : List.Pairwise Disjoint l\n\u22a2 \u2203 c \u03c4, List.prod l = c * \u03c4 \u2227 Disjoint c \u03c4 \u2227 IsCycle c \u2227 Finset.card (support c) = n\n[PROOFSTEP]\nobtain \u27e8c, cl, rfl\u27e9 := h\n[GOAL]\ncase mp.mk.mk.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List (Perm \u03b1)\nx\u271d : Trunc { l_1 // List.prod l_1 = List.prod l \u2227 (\u2200 (g : Perm \u03b1), g \u2208 l_1 \u2192 IsCycle g) \u2227 List.Pairwise Disjoint l_1 }\nhlc : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsCycle g\nhld : List.Pairwise Disjoint l\nc : Perm \u03b1\ncl : c \u2208 l\n\u22a2 \u2203 c_1 \u03c4, List.prod l = c_1 * \u03c4 \u2227 Disjoint c_1 \u03c4 \u2227 IsCycle c_1 \u2227 Finset.card (support c_1) = (Finset.card \u2218 support) c\n[PROOFSTEP]\nrw [(List.perm_cons_erase cl).pairwise_iff Disjoint.symmetric] at hld \n[GOAL]\ncase mp.mk.mk.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List (Perm \u03b1)\nx\u271d : Trunc { l_1 // List.prod l_1 = List.prod l \u2227 (\u2200 (g : Perm \u03b1), g \u2208 l_1 \u2192 IsCycle g) \u2227 List.Pairwise Disjoint l_1 }\nhlc : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsCycle g\nc : Perm \u03b1\nhld : List.Pairwise Disjoint (c :: List.erase l c)\ncl : c \u2208 l\n\u22a2 \u2203 c_1 \u03c4, List.prod l = c_1 * \u03c4 \u2227 Disjoint c_1 \u03c4 \u2227 IsCycle c_1 \u2227 Finset.card (support c_1) = (Finset.card \u2218 support) c\n[PROOFSTEP]\nrefine' \u27e8c, (l.erase c).prod, _, _, hlc _ cl, rfl\u27e9\n[GOAL]\ncase mp.mk.mk.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List (Perm \u03b1)\nx\u271d : Trunc { l_1 // List.prod l_1 = List.prod l \u2227 (\u2200 (g : Perm \u03b1), g \u2208 l_1 \u2192 IsCycle g) \u2227 List.Pairwise Disjoint l_1 }\nhlc : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsCycle g\nc : Perm \u03b1\nhld : List.Pairwise Disjoint (c :: List.erase l c)\ncl : c \u2208 l\n\u22a2 List.prod l = c * List.prod (List.erase l c)\n[PROOFSTEP]\nrw [\u2190 List.prod_cons, (List.perm_cons_erase cl).symm.prod_eq' (hld.imp Disjoint.commute)]\n[GOAL]\ncase mp.mk.mk.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List (Perm \u03b1)\nx\u271d : Trunc { l_1 // List.prod l_1 = List.prod l \u2227 (\u2200 (g : Perm \u03b1), g \u2208 l_1 \u2192 IsCycle g) \u2227 List.Pairwise Disjoint l_1 }\nhlc : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsCycle g\nc : Perm \u03b1\nhld : List.Pairwise Disjoint (c :: List.erase l c)\ncl : c \u2208 l\n\u22a2 Disjoint c (List.prod (List.erase l c))\n[PROOFSTEP]\nexact disjoint_prod_right _ fun g => List.rel_of_pairwise_cons hld\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u03c3 : Perm \u03b1\n\u22a2 (\u2203 c \u03c4, \u03c3 = c * \u03c4 \u2227 Disjoint c \u03c4 \u2227 IsCycle c \u2227 Finset.card (support c) = n) \u2192 n \u2208 cycleType \u03c3\n[PROOFSTEP]\nrintro \u27e8c, t, rfl, hd, hc, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc t : Perm \u03b1\nhd : Disjoint c t\nhc : IsCycle c\n\u22a2 Finset.card (support c) \u2208 cycleType (c * t)\n[PROOFSTEP]\nsimp [hd.cycleType, hc.cycleType]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\ng : Perm \u03b1\nhn2 : Fintype.card \u03b1 < n + 2\nhng : n \u2208 cycleType g\n\u22a2 cycleType g = {n}\n[PROOFSTEP]\nobtain \u27e8c, g', rfl, hd, hc, rfl\u27e9 := mem_cycleType_iff.1 hng\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm \u03b1\nhd : Disjoint c g'\nhc : IsCycle c\nhn2 : Fintype.card \u03b1 < Finset.card (support c) + 2\nhng : Finset.card (support c) \u2208 cycleType (c * g')\n\u22a2 cycleType (c * g') = {Finset.card (support c)}\n[PROOFSTEP]\nby_cases g'1 : g' = 1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm \u03b1\nhd : Disjoint c g'\nhc : IsCycle c\nhn2 : Fintype.card \u03b1 < Finset.card (support c) + 2\nhng : Finset.card (support c) \u2208 cycleType (c * g')\ng'1 : g' = 1\n\u22a2 cycleType (c * g') = {Finset.card (support c)}\n[PROOFSTEP]\nrw [hd.cycleType, hc.cycleType, coe_singleton, g'1, cycleType_one, add_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm \u03b1\nhd : Disjoint c g'\nhc : IsCycle c\nhn2 : Fintype.card \u03b1 < Finset.card (support c) + 2\nhng : Finset.card (support c) \u2208 cycleType (c * g')\ng'1 : \u00acg' = 1\n\u22a2 cycleType (c * g') = {Finset.card (support c)}\n[PROOFSTEP]\ncontrapose! hn2\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm \u03b1\nhd : Disjoint c g'\nhc : IsCycle c\nhng : Finset.card (support c) \u2208 cycleType (c * g')\ng'1 : \u00acg' = 1\nhn2 : cycleType (c * g') \u2260 {Finset.card (support c)}\n\u22a2 Finset.card (support c) + 2 \u2264 Fintype.card \u03b1\n[PROOFSTEP]\napply le_trans _ (c * g').support.card_le_univ\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm \u03b1\nhd : Disjoint c g'\nhc : IsCycle c\nhng : Finset.card (support c) \u2208 cycleType (c * g')\ng'1 : \u00acg' = 1\nhn2 : cycleType (c * g') \u2260 {Finset.card (support c)}\n\u22a2 Finset.card (support c) + 2 \u2264 Finset.card (support (c * g'))\n[PROOFSTEP]\nrw [hd.card_support_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm \u03b1\nhd : Disjoint c g'\nhc : IsCycle c\nhng : Finset.card (support c) \u2208 cycleType (c * g')\ng'1 : \u00acg' = 1\nhn2 : cycleType (c * g') \u2260 {Finset.card (support c)}\n\u22a2 Finset.card (support c) + 2 \u2264 Finset.card (support c) + Finset.card (support g')\n[PROOFSTEP]\nexact add_le_add_left (two_le_card_support_of_ne_one g'1) _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\n\u22a2 Finset.card (support \u03c3)\u1d9c \u2261 Fintype.card \u03b1 [MOD p]\n[PROOFSTEP]\nrw [Nat.modEq_iff_dvd', \u2190 Finset.card_compl, compl_compl, \u2190 sum_cycleType]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\n\u22a2 p \u2223 sum (cycleType \u03c3)\n[PROOFSTEP]\nrefine Multiset.dvd_sum fun k hk => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\nk : \u2115\nhk : k \u2208 cycleType \u03c3\n\u22a2 p \u2223 k\n[PROOFSTEP]\nobtain \u27e8m, -, hm\u27e9 := (Nat.dvd_prime_pow hp.out).mp (orderOf_dvd_of_pow_eq_one h\u03c3)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\nk : \u2115\nhk : k \u2208 cycleType \u03c3\nm : \u2115\nhm : orderOf \u03c3 = p ^ m\n\u22a2 p \u2223 k\n[PROOFSTEP]\nobtain \u27e8l, -, rfl\u27e9 := (Nat.dvd_prime_pow hp.out).mp ((congr_arg _ hm).mp (dvd_of_mem_cycleType hk))\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\nm : \u2115\nhm : orderOf \u03c3 = p ^ m\nl : \u2115\nhk : p ^ l \u2208 cycleType \u03c3\n\u22a2 p \u2223 p ^ l\n[PROOFSTEP]\nexact dvd_pow_self _ fun h => (one_lt_of_mem_cycleType hk).ne <| by rw [h, pow_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\nm : \u2115\nhm : orderOf \u03c3 = p ^ m\nl : \u2115\nhk : p ^ l \u2208 cycleType \u03c3\nh : l = 0\n\u22a2 1 = p ^ l\n[PROOFSTEP]\nrw [h, pow_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\n\u22a2 Finset.card (support \u03c3)\u1d9c \u2264 Fintype.card \u03b1\n[PROOFSTEP]\nexact Finset.card_le_univ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\nh\u03b1 : \u00acp \u2223 Fintype.card \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\n\u22a2 \u2203 a, \u2191\u03c3 a = a\n[PROOFSTEP]\nclassical\ncontrapose! h\u03b1\nsimp_rw [\u2190 mem_support, \u2190 Finset.eq_univ_iff_forall] at h\u03b1 \nexact\n Nat.modEq_zero_iff_dvd.1\n ((congr_arg _ (Finset.card_eq_zero.2 (compl_eq_bot.2 h\u03b1))).mp (card_compl_support_modEq h\u03c3).symm)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\nh\u03b1 : \u00acp \u2223 Fintype.card \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\n\u22a2 \u2203 a, \u2191\u03c3 a = a\n[PROOFSTEP]\ncontrapose! h\u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\nh\u03b1 : \u2200 (a : \u03b1), \u2191\u03c3 a \u2260 a\n\u22a2 p \u2223 Fintype.card \u03b1\n[PROOFSTEP]\nsimp_rw [\u2190 mem_support, \u2190 Finset.eq_univ_iff_forall] at h\u03b1 \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\nh\u03b1 : support \u03c3 = Finset.univ\n\u22a2 p \u2223 Fintype.card \u03b1\n[PROOFSTEP]\nexact\n Nat.modEq_zero_iff_dvd.1\n ((congr_arg _ (Finset.card_eq_zero.2 (compl_eq_bot.2 h\u03b1))).mp (card_compl_support_modEq h\u03c3).symm)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\nh\u03b1 : p \u2223 Fintype.card \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\na : \u03b1\nha : \u2191\u03c3 a = a\n\u22a2 \u2203 b, \u2191\u03c3 b = b \u2227 b \u2260 a\n[PROOFSTEP]\nclassical\nhave h : \u2200 b : \u03b1, b \u2208 \u03c3.support\u1d9c \u2194 \u03c3 b = b := fun b => by rw [Finset.mem_compl, mem_support, Classical.not_not]\nobtain \u27e8b, hb1, hb2\u27e9 :=\n Finset.exists_ne_of_one_lt_card\n (hp.out.one_lt.trans_le\n (Nat.le_of_dvd (Finset.card_pos.mpr \u27e8a, (h a).mpr ha\u27e9)\n (Nat.modEq_zero_iff_dvd.mp ((card_compl_support_modEq h\u03c3).trans (Nat.modEq_zero_iff_dvd.mpr h\u03b1)))))\n a\nexact \u27e8b, (h b).mp hb1, hb2\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\nh\u03b1 : p \u2223 Fintype.card \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\na : \u03b1\nha : \u2191\u03c3 a = a\n\u22a2 \u2203 b, \u2191\u03c3 b = b \u2227 b \u2260 a\n[PROOFSTEP]\nhave h : \u2200 b : \u03b1, b \u2208 \u03c3.support\u1d9c \u2194 \u03c3 b = b := fun b => by rw [Finset.mem_compl, mem_support, Classical.not_not]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\nh\u03b1 : p \u2223 Fintype.card \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\na : \u03b1\nha : \u2191\u03c3 a = a\nb : \u03b1\n\u22a2 b \u2208 (support \u03c3)\u1d9c \u2194 \u2191\u03c3 b = b\n[PROOFSTEP]\nrw [Finset.mem_compl, mem_support, Classical.not_not]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\nh\u03b1 : p \u2223 Fintype.card \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\na : \u03b1\nha : \u2191\u03c3 a = a\nh : \u2200 (b : \u03b1), b \u2208 (support \u03c3)\u1d9c \u2194 \u2191\u03c3 b = b\n\u22a2 \u2203 b, \u2191\u03c3 b = b \u2227 b \u2260 a\n[PROOFSTEP]\nobtain \u27e8b, hb1, hb2\u27e9 :=\n Finset.exists_ne_of_one_lt_card\n (hp.out.one_lt.trans_le\n (Nat.le_of_dvd (Finset.card_pos.mpr \u27e8a, (h a).mpr ha\u27e9)\n (Nat.modEq_zero_iff_dvd.mp ((card_compl_support_modEq h\u03c3).trans (Nat.modEq_zero_iff_dvd.mpr h\u03b1)))))\n a\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\nh\u03b1 : p \u2223 Fintype.card \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\na : \u03b1\nha : \u2191\u03c3 a = a\nh : \u2200 (b : \u03b1), b \u2208 (support \u03c3)\u1d9c \u2194 \u2191\u03c3 b = b\nb : \u03b1\nhb1 : b \u2208 (support \u03c3)\u1d9c\nhb2 : b \u2260 a\n\u22a2 \u2203 b, \u2191\u03c3 b = b \u2227 b \u2260 a\n[PROOFSTEP]\nexact \u27e8b, (h b).mp hb1, hb2\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : Nat.Prime (orderOf \u03c3)\nh2 : Fintype.card \u03b1 < 2 * orderOf \u03c3\n\u22a2 IsCycle \u03c3\n[PROOFSTEP]\nclassical exact isCycle_of_prime_order h1 (lt_of_le_of_lt \u03c3.support.card_le_univ h2)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : Nat.Prime (orderOf \u03c3)\nh2 : Fintype.card \u03b1 < 2 * orderOf \u03c3\n\u22a2 IsCycle \u03c3\n[PROOFSTEP]\nexact isCycle_of_prime_order h1 (lt_of_le_of_lt \u03c3.support.card_le_univ h2)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : Nat.Prime (Fintype.card \u03b1)\nh2 : orderOf \u03c3 = Fintype.card \u03b1\n\u22a2 Fintype.card \u03b1 < 2 * orderOf \u03c3\n[PROOFSTEP]\nrw [\u2190 one_mul (Fintype.card \u03b1), \u2190 h2, mul_lt_mul_right (orderOf_pos \u03c3)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : Nat.Prime (Fintype.card \u03b1)\nh2 : orderOf \u03c3 = Fintype.card \u03b1\n\u22a2 1 < 2\n[PROOFSTEP]\nexact one_lt_two\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nG : Type u_2\ninst\u271d : Group G\nn : \u2115\n\u22a2 vectorsProdEqOne G 1 = {1 ::\u1d65 Vector.nil}\n[PROOFSTEP]\nsimp_rw [Set.eq_singleton_iff_unique_mem, mem_iff, Vector.toList_singleton, List.prod_singleton, Vector.head_cons,\n true_and]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nG : Type u_2\ninst\u271d : Group G\nn : \u2115\n\u22a2 \u2200 (x : Vector G 1), Vector.head x = 1 \u2192 x = 1 ::\u1d65 Vector.nil\n[PROOFSTEP]\nexact fun v hv => v.cons_head_tail.symm.trans (congr_arg\u2082 Vector.cons hv v.tail.eq_nil)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nG : Type u_2\ninst\u271d : Group G\nn : \u2115\n\u22a2 Unique \u2191(vectorsProdEqOne G 0)\n[PROOFSTEP]\nrw [zero_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nG : Type u_2\ninst\u271d : Group G\nn : \u2115\n\u22a2 Unique \u2191{Vector.nil}\n[PROOFSTEP]\nexact Set.uniqueSingleton Vector.nil\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nG : Type u_2\ninst\u271d : Group G\nn : \u2115\n\u22a2 Unique \u2191(vectorsProdEqOne G 1)\n[PROOFSTEP]\nrw [one_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nG : Type u_2\ninst\u271d : Group G\nn : \u2115\n\u22a2 Unique \u2191{1 ::\u1d65 Vector.nil}\n[PROOFSTEP]\nexact Set.uniqueSingleton (Vector.nil.cons 1)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nG : Type u_2\ninst\u271d : Group G\nn : \u2115\nv : Vector G n\n\u22a2 (List.prod (Vector.toList v))\u207b\u00b9 ::\u1d65 v \u2208 vectorsProdEqOne G (n + 1)\n[PROOFSTEP]\nrw [mem_iff, Vector.toList_cons, List.prod_cons, inv_mul_self]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nG : Type u_2\ninst\u271d : Group G\nn : \u2115\nv : \u2191(vectorsProdEqOne G (n + 1))\n\u22a2 Vector.head \u2191v * List.prod (Vector.toList (Vector.tail \u2191v)) = 1\n[PROOFSTEP]\nrw [\u2190 List.prod_cons, \u2190 Vector.toList_cons, v.1.cons_head_tail]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nG : Type u_2\ninst\u271d : Group G\nn : \u2115\nv : \u2191(vectorsProdEqOne G (n + 1))\n\u22a2 List.prod (Vector.toList \u2191v) = 1\n[PROOFSTEP]\nexact v.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\n\u22a2 \u2203 x, orderOf x = p\n[PROOFSTEP]\nhave hp' : p - 1 \u2260 0 := mt tsub_eq_zero_iff_le.mp (not_le_of_lt hp.out.one_lt)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\n\u22a2 \u2203 x, orderOf x = p\n[PROOFSTEP]\nhave Scard :=\n calc\n p \u2223 Fintype.card G ^ (p - 1) := hdvd.trans (dvd_pow (dvd_refl _) hp')\n _ = Fintype.card (vectorsProdEqOne G p) := (VectorsProdEqOne.card G p).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\n\u22a2 \u2203 x, orderOf x = p\n[PROOFSTEP]\nlet f : \u2115 \u2192 vectorsProdEqOne G p \u2192 vectorsProdEqOne G p := fun k v => VectorsProdEqOne.rotate v k\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\n\u22a2 \u2203 x, orderOf x = p\n[PROOFSTEP]\nhave hf1 : \u2200 v, f 0 v = v := VectorsProdEqOne.rotate_zero\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f 0 v = v\n\u22a2 \u2203 x, orderOf x = p\n[PROOFSTEP]\nhave hf2 : \u2200 j k v, f k (f j v) = f (j + k) v := fun j k v => VectorsProdEqOne.rotate_rotate v j k\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f 0 v = v\nhf2 : \u2200 (j k : \u2115) (v : \u2191(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\n\u22a2 \u2203 x, orderOf x = p\n[PROOFSTEP]\nhave hf3 : \u2200 v, f p v = v := VectorsProdEqOne.rotate_length\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f 0 v = v\nhf2 : \u2200 (j k : \u2115) (v : \u2191(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f p v = v\n\u22a2 \u2203 x, orderOf x = p\n[PROOFSTEP]\nlet \u03c3 :=\n Equiv.mk (f 1) (f (p - 1)) (fun s => by rw [hf2, add_tsub_cancel_of_le hp.out.one_lt.le, hf3]) fun s => by\n rw [hf2, tsub_add_cancel_of_le hp.out.one_lt.le, hf3]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f 0 v = v\nhf2 : \u2200 (j k : \u2115) (v : \u2191(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f p v = v\ns : \u2191(vectorsProdEqOne G p)\n\u22a2 f (p - 1) (f 1 s) = s\n[PROOFSTEP]\nrw [hf2, add_tsub_cancel_of_le hp.out.one_lt.le, hf3]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f 0 v = v\nhf2 : \u2200 (j k : \u2115) (v : \u2191(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f p v = v\ns : \u2191(vectorsProdEqOne G p)\n\u22a2 f 1 (f (p - 1) s) = s\n[PROOFSTEP]\nrw [hf2, tsub_add_cancel_of_le hp.out.one_lt.le, hf3]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f 0 v = v\nhf2 : \u2200 (j k : \u2115) (v : \u2191(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f p v = v\n\u03c3 : \u2191(vectorsProdEqOne G p) \u2243 \u2191(vectorsProdEqOne G p) :=\n { toFun := f 1, invFun := f (p - 1), left_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f (p - 1) (f 1 s) = s),\n right_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f 1 (f (p - 1) s) = s) }\n\u22a2 \u2203 x, orderOf x = p\n[PROOFSTEP]\nhave h\u03c3 : \u2200 k v, (\u03c3 ^ k) v = f k v := fun k v =>\n Nat.rec (hf1 v).symm (fun k hk => Eq.trans (congr_arg \u03c3 hk) (hf2 k 1 v)) k\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f 0 v = v\nhf2 : \u2200 (j k : \u2115) (v : \u2191(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f p v = v\n\u03c3 : \u2191(vectorsProdEqOne G p) \u2243 \u2191(vectorsProdEqOne G p) :=\n { toFun := f 1, invFun := f (p - 1), left_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f (p - 1) (f 1 s) = s),\n right_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f 1 (f (p - 1) s) = s) }\nh\u03c3 : \u2200 (k : \u2115) (v : \u2191(vectorsProdEqOne G p)), \u2191(\u03c3 ^ k) v = f k v\n\u22a2 \u2203 x, orderOf x = p\n[PROOFSTEP]\nreplace h\u03c3 : \u03c3 ^ p ^ 1 = 1 := Perm.ext fun v => by rw [pow_one, h\u03c3, hf3, one_apply]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f 0 v = v\nhf2 : \u2200 (j k : \u2115) (v : \u2191(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f p v = v\n\u03c3 : \u2191(vectorsProdEqOne G p) \u2243 \u2191(vectorsProdEqOne G p) :=\n { toFun := f 1, invFun := f (p - 1), left_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f (p - 1) (f 1 s) = s),\n right_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f 1 (f (p - 1) s) = s) }\nh\u03c3 : \u2200 (k : \u2115) (v : \u2191(vectorsProdEqOne G p)), \u2191(\u03c3 ^ k) v = f k v\nv : \u2191(vectorsProdEqOne G p)\n\u22a2 \u2191(\u03c3 ^ p ^ 1) v = \u21911 v\n[PROOFSTEP]\nrw [pow_one, h\u03c3, hf3, one_apply]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f 0 v = v\nhf2 : \u2200 (j k : \u2115) (v : \u2191(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f p v = v\n\u03c3 : \u2191(vectorsProdEqOne G p) \u2243 \u2191(vectorsProdEqOne G p) :=\n { toFun := f 1, invFun := f (p - 1), left_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f (p - 1) (f 1 s) = s),\n right_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f 1 (f (p - 1) s) = s) }\nh\u03c3 : \u03c3 ^ p ^ 1 = 1\n\u22a2 \u2203 x, orderOf x = p\n[PROOFSTEP]\nlet v\u2080 : vectorsProdEqOne G p := \u27e8Vector.replicate p 1, (List.prod_replicate p 1).trans (one_pow p)\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f 0 v = v\nhf2 : \u2200 (j k : \u2115) (v : \u2191(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f p v = v\n\u03c3 : \u2191(vectorsProdEqOne G p) \u2243 \u2191(vectorsProdEqOne G p) :=\n { toFun := f 1, invFun := f (p - 1), left_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f (p - 1) (f 1 s) = s),\n right_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f 1 (f (p - 1) s) = s) }\nh\u03c3 : \u03c3 ^ p ^ 1 = 1\nv\u2080 : \u2191(vectorsProdEqOne G p) := { val := Vector.replicate p 1, property := (_ : List.prod (List.replicate p 1) = 1) }\n\u22a2 \u2203 x, orderOf x = p\n[PROOFSTEP]\nhave hv\u2080 : \u03c3 v\u2080 = v\u2080 := Subtype.ext (Subtype.ext (List.rotate_replicate (1 : G) p 1))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f 0 v = v\nhf2 : \u2200 (j k : \u2115) (v : \u2191(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f p v = v\n\u03c3 : \u2191(vectorsProdEqOne G p) \u2243 \u2191(vectorsProdEqOne G p) :=\n { toFun := f 1, invFun := f (p - 1), left_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f (p - 1) (f 1 s) = s),\n right_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f 1 (f (p - 1) s) = s) }\nh\u03c3 : \u03c3 ^ p ^ 1 = 1\nv\u2080 : \u2191(vectorsProdEqOne G p) := { val := Vector.replicate p 1, property := (_ : List.prod (List.replicate p 1) = 1) }\nhv\u2080 : \u2191\u03c3 v\u2080 = v\u2080\n\u22a2 \u2203 x, orderOf x = p\n[PROOFSTEP]\nobtain \u27e8v, hv1, hv2\u27e9 := exists_fixed_point_of_prime' Scard h\u03c3 hv\u2080\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f 0 v = v\nhf2 : \u2200 (j k : \u2115) (v : \u2191(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f p v = v\n\u03c3 : \u2191(vectorsProdEqOne G p) \u2243 \u2191(vectorsProdEqOne G p) :=\n { toFun := f 1, invFun := f (p - 1), left_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f (p - 1) (f 1 s) = s),\n right_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f 1 (f (p - 1) s) = s) }\nh\u03c3 : \u03c3 ^ p ^ 1 = 1\nv\u2080 : \u2191(vectorsProdEqOne G p) := { val := Vector.replicate p 1, property := (_ : List.prod (List.replicate p 1) = 1) }\nhv\u2080 : \u2191\u03c3 v\u2080 = v\u2080\nv : \u2191(vectorsProdEqOne G p)\nhv1 : \u2191\u03c3 v = v\nhv2 : v \u2260 v\u2080\n\u22a2 \u2203 x, orderOf x = p\n[PROOFSTEP]\nrefine'\n Exists.imp (fun g hg => orderOf_eq_prime _ fun hg' => hv2 _)\n (List.rotate_one_eq_self_iff_eq_replicate.mp (Subtype.ext_iff.mp (Subtype.ext_iff.mp hv1)))\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f 0 v = v\nhf2 : \u2200 (j k : \u2115) (v : \u2191(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f p v = v\n\u03c3 : \u2191(vectorsProdEqOne G p) \u2243 \u2191(vectorsProdEqOne G p) :=\n { toFun := f 1, invFun := f (p - 1), left_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f (p - 1) (f 1 s) = s),\n right_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f 1 (f (p - 1) s) = s) }\nh\u03c3 : \u03c3 ^ p ^ 1 = 1\nv\u2080 : \u2191(vectorsProdEqOne G p) := { val := Vector.replicate p 1, property := (_ : List.prod (List.replicate p 1) = 1) }\nhv\u2080 : \u2191\u03c3 v\u2080 = v\u2080\nv : \u2191(vectorsProdEqOne G p)\nhv1 : \u2191\u03c3 v = v\nhv2 : v \u2260 v\u2080\ng : G\nhg : \u2191\u2191v = List.replicate (List.length \u2191\u2191v) g\n\u22a2 g ^ p = 1\n[PROOFSTEP]\nrw [\u2190 List.prod_replicate, \u2190 v.1.2, \u2190 hg, show v.val.val.prod = 1 from v.2]\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f 0 v = v\nhf2 : \u2200 (j k : \u2115) (v : \u2191(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f p v = v\n\u03c3 : \u2191(vectorsProdEqOne G p) \u2243 \u2191(vectorsProdEqOne G p) :=\n { toFun := f 1, invFun := f (p - 1), left_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f (p - 1) (f 1 s) = s),\n right_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f 1 (f (p - 1) s) = s) }\nh\u03c3 : \u03c3 ^ p ^ 1 = 1\nv\u2080 : \u2191(vectorsProdEqOne G p) := { val := Vector.replicate p 1, property := (_ : List.prod (List.replicate p 1) = 1) }\nhv\u2080 : \u2191\u03c3 v\u2080 = v\u2080\nv : \u2191(vectorsProdEqOne G p)\nhv1 : \u2191\u03c3 v = v\nhv2 : v \u2260 v\u2080\ng : G\nhg : \u2191\u2191v = List.replicate (List.length \u2191\u2191v) g\nhg' : g = 1\n\u22a2 v = v\u2080\n[PROOFSTEP]\nrw [Subtype.ext_iff_val, Subtype.ext_iff_val, hg, hg', v.1.2]\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\nhp' : p - 1 \u2260 0\nScard : p \u2223 Fintype.card \u2191(vectorsProdEqOne G p)\nf : \u2115 \u2192 \u2191(vectorsProdEqOne G p) \u2192 \u2191(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f 0 v = v\nhf2 : \u2200 (j k : \u2115) (v : \u2191(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : \u2200 (v : \u2191(vectorsProdEqOne G p)), f p v = v\n\u03c3 : \u2191(vectorsProdEqOne G p) \u2243 \u2191(vectorsProdEqOne G p) :=\n { toFun := f 1, invFun := f (p - 1), left_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f (p - 1) (f 1 s) = s),\n right_inv := (_ : \u2200 (s : \u2191(vectorsProdEqOne G p)), f 1 (f (p - 1) s) = s) }\nh\u03c3 : \u03c3 ^ p ^ 1 = 1\nv\u2080 : \u2191(vectorsProdEqOne G p) := { val := Vector.replicate p 1, property := (_ : List.prod (List.replicate p 1) = 1) }\nhv\u2080 : \u2191\u03c3 v\u2080 = v\u2080\nv : \u2191(vectorsProdEqOne G p)\nhv1 : \u2191\u03c3 v = v\nhv2 : v \u2260 v\u2080\ng : G\nhg : \u2191\u2191v = List.replicate (List.length \u2191\u2191v) g\nhg' : g = 1\n\u22a2 List.replicate p 1 = \u2191\u2191v\u2080\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\n\u22a2 p \u2223 Fintype.card (Multiplicative G)\n[PROOFSTEP]\nconvert hdvd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup (Perm \u03b1)\nd : DecidablePred fun x => x \u2208 H\n\u03c4 : Perm \u03b1\nh0 : Nat.Prime (Fintype.card \u03b1)\nh1 : Fintype.card \u03b1 \u2223 Fintype.card { x // x \u2208 H }\nh2 : \u03c4 \u2208 H\nh3 : IsSwap \u03c4\n\u22a2 H = \u22a4\n[PROOFSTEP]\nhaveI : Fact (Fintype.card \u03b1).Prime := \u27e8h0\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup (Perm \u03b1)\nd : DecidablePred fun x => x \u2208 H\n\u03c4 : Perm \u03b1\nh0 : Nat.Prime (Fintype.card \u03b1)\nh1 : Fintype.card \u03b1 \u2223 Fintype.card { x // x \u2208 H }\nh2 : \u03c4 \u2208 H\nh3 : IsSwap \u03c4\nthis : Fact (Nat.Prime (Fintype.card \u03b1))\n\u22a2 H = \u22a4\n[PROOFSTEP]\nobtain \u27e8\u03c3, h\u03c3\u27e9 := exists_prime_orderOf_dvd_card (Fintype.card \u03b1) h1\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup (Perm \u03b1)\nd : DecidablePred fun x => x \u2208 H\n\u03c4 : Perm \u03b1\nh0 : Nat.Prime (Fintype.card \u03b1)\nh1 : Fintype.card \u03b1 \u2223 Fintype.card { x // x \u2208 H }\nh2 : \u03c4 \u2208 H\nh3 : IsSwap \u03c4\nthis : Fact (Nat.Prime (Fintype.card \u03b1))\n\u03c3 : { x // x \u2208 H }\nh\u03c3 : orderOf \u03c3 = Fintype.card \u03b1\n\u22a2 H = \u22a4\n[PROOFSTEP]\nhave h\u03c31 : orderOf (\u03c3 : Perm \u03b1) = Fintype.card \u03b1 := (orderOf_subgroup \u03c3).trans h\u03c3\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup (Perm \u03b1)\nd : DecidablePred fun x => x \u2208 H\n\u03c4 : Perm \u03b1\nh0 : Nat.Prime (Fintype.card \u03b1)\nh1 : Fintype.card \u03b1 \u2223 Fintype.card { x // x \u2208 H }\nh2 : \u03c4 \u2208 H\nh3 : IsSwap \u03c4\nthis : Fact (Nat.Prime (Fintype.card \u03b1))\n\u03c3 : { x // x \u2208 H }\nh\u03c3 : orderOf \u03c3 = Fintype.card \u03b1\nh\u03c31 : orderOf \u2191\u03c3 = Fintype.card \u03b1\n\u22a2 H = \u22a4\n[PROOFSTEP]\nhave h\u03c32 : IsCycle \u2191\u03c3 := isCycle_of_prime_order'' h0 h\u03c31\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup (Perm \u03b1)\nd : DecidablePred fun x => x \u2208 H\n\u03c4 : Perm \u03b1\nh0 : Nat.Prime (Fintype.card \u03b1)\nh1 : Fintype.card \u03b1 \u2223 Fintype.card { x // x \u2208 H }\nh2 : \u03c4 \u2208 H\nh3 : IsSwap \u03c4\nthis : Fact (Nat.Prime (Fintype.card \u03b1))\n\u03c3 : { x // x \u2208 H }\nh\u03c3 : orderOf \u03c3 = Fintype.card \u03b1\nh\u03c31 : orderOf \u2191\u03c3 = Fintype.card \u03b1\nh\u03c32 : IsCycle \u2191\u03c3\n\u22a2 H = \u22a4\n[PROOFSTEP]\nhave h\u03c33 : (\u03c3 : Perm \u03b1).support = \u22a4 := Finset.eq_univ_of_card (\u03c3 : Perm \u03b1).support (h\u03c32.orderOf.symm.trans h\u03c31)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup (Perm \u03b1)\nd : DecidablePred fun x => x \u2208 H\n\u03c4 : Perm \u03b1\nh0 : Nat.Prime (Fintype.card \u03b1)\nh1 : Fintype.card \u03b1 \u2223 Fintype.card { x // x \u2208 H }\nh2 : \u03c4 \u2208 H\nh3 : IsSwap \u03c4\nthis : Fact (Nat.Prime (Fintype.card \u03b1))\n\u03c3 : { x // x \u2208 H }\nh\u03c3 : orderOf \u03c3 = Fintype.card \u03b1\nh\u03c31 : orderOf \u2191\u03c3 = Fintype.card \u03b1\nh\u03c32 : IsCycle \u2191\u03c3\nh\u03c33 : support \u2191\u03c3 = \u22a4\n\u22a2 H = \u22a4\n[PROOFSTEP]\nhave h\u03c34 : Subgroup.closure {\u2191\u03c3, \u03c4} = \u22a4 := closure_prime_cycle_swap h0 h\u03c32 h\u03c33 h3\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup (Perm \u03b1)\nd : DecidablePred fun x => x \u2208 H\n\u03c4 : Perm \u03b1\nh0 : Nat.Prime (Fintype.card \u03b1)\nh1 : Fintype.card \u03b1 \u2223 Fintype.card { x // x \u2208 H }\nh2 : \u03c4 \u2208 H\nh3 : IsSwap \u03c4\nthis : Fact (Nat.Prime (Fintype.card \u03b1))\n\u03c3 : { x // x \u2208 H }\nh\u03c3 : orderOf \u03c3 = Fintype.card \u03b1\nh\u03c31 : orderOf \u2191\u03c3 = Fintype.card \u03b1\nh\u03c32 : IsCycle \u2191\u03c3\nh\u03c33 : support \u2191\u03c3 = \u22a4\nh\u03c34 : Subgroup.closure {\u2191\u03c3, \u03c4} = \u22a4\n\u22a2 H = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff, \u2190 h\u03c34, Subgroup.closure_le, Set.insert_subset_iff, Set.singleton_subset_iff]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nH : Subgroup (Perm \u03b1)\nd : DecidablePred fun x => x \u2208 H\n\u03c4 : Perm \u03b1\nh0 : Nat.Prime (Fintype.card \u03b1)\nh1 : Fintype.card \u03b1 \u2223 Fintype.card { x // x \u2208 H }\nh2 : \u03c4 \u2208 H\nh3 : IsSwap \u03c4\nthis : Fact (Nat.Prime (Fintype.card \u03b1))\n\u03c3 : { x // x \u2208 H }\nh\u03c3 : orderOf \u03c3 = Fintype.card \u03b1\nh\u03c31 : orderOf \u2191\u03c3 = Fintype.card \u03b1\nh\u03c32 : IsCycle \u2191\u03c3\nh\u03c33 : support \u2191\u03c3 = \u22a4\nh\u03c34 : Subgroup.closure {\u2191\u03c3, \u03c4} = \u22a4\n\u22a2 \u2191\u03c3 \u2208 \u2191H \u2227 \u03c4 \u2208 \u2191H\n[PROOFSTEP]\nexact \u27e8Subtype.mem \u03c3, h2\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nn : \u2115\nhn : n \u2208 cycleType \u03c3 + replicate (Fintype.card \u03b1 - Finset.card (support \u03c3)) 1\n\u22a2 0 < n\n[PROOFSTEP]\ncases' mem_add.mp hn with hn hn\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nn : \u2115\nhn\u271d : n \u2208 cycleType \u03c3 + replicate (Fintype.card \u03b1 - Finset.card (support \u03c3)) 1\nhn : n \u2208 cycleType \u03c3\n\u22a2 0 < n\n[PROOFSTEP]\nexact zero_lt_one.trans (one_lt_of_mem_cycleType hn)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nn : \u2115\nhn\u271d : n \u2208 cycleType \u03c3 + replicate (Fintype.card \u03b1 - Finset.card (support \u03c3)) 1\nhn : n \u2208 replicate (Fintype.card \u03b1 - Finset.card (support \u03c3)) 1\n\u22a2 0 < n\n[PROOFSTEP]\nexact lt_of_lt_of_le zero_lt_one (ge_of_eq (Multiset.eq_of_mem_replicate hn))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 sum (cycleType \u03c3 + replicate (Fintype.card \u03b1 - Finset.card (support \u03c3)) 1) = Fintype.card \u03b1\n[PROOFSTEP]\nrw [sum_add, sum_cycleType, Multiset.sum_replicate, nsmul_eq_mul, Nat.cast_id, mul_one,\n add_tsub_cancel_of_le \u03c3.support.card_le_univ]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 filter (fun n => 2 \u2264 n) (partition \u03c3).parts = cycleType \u03c3\n[PROOFSTEP]\nrw [parts_partition, filter_add, Multiset.filter_eq_self.2 fun _ => two_le_of_mem_cycleType,\n Multiset.filter_eq_nil.2 fun a h => ?_, add_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\na : \u2115\nh : a \u2208 replicate (Fintype.card \u03b1 - Finset.card (support \u03c3)) 1\n\u22a2 \u00ac2 \u2264 a\n[PROOFSTEP]\nrw [Multiset.eq_of_mem_replicate h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\na : \u2115\nh : a \u2208 replicate (Fintype.card \u03b1 - Finset.card (support \u03c3)) 1\n\u22a2 \u00ac2 \u2264 1\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\n\u22a2 IsConj \u03c3 \u03c4 \u2194 partition \u03c3 = partition \u03c4\n[PROOFSTEP]\nrw [isConj_iff_cycleType_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\n\u22a2 cycleType \u03c3 = cycleType \u03c4 \u2194 partition \u03c3 = partition \u03c4\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\nh : cycleType \u03c3 = cycleType \u03c4\n\u22a2 partition \u03c3 = partition \u03c4\n[PROOFSTEP]\nrw [Nat.Partition.ext_iff, parts_partition, parts_partition, \u2190 sum_cycleType, \u2190 sum_cycleType, h]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\nh : partition \u03c3 = partition \u03c4\n\u22a2 cycleType \u03c3 = cycleType \u03c4\n[PROOFSTEP]\nrw [\u2190 filter_parts_partition_eq_cycleType, \u2190 filter_parts_partition_eq_cycleType, h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh : IsThreeCycle \u03c3\n\u22a2 Finset.card (support \u03c3) = 3\n[PROOFSTEP]\nrw [\u2190 sum_cycleType, h.cycleType, Multiset.sum_singleton]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 Finset.card (support \u03c3) = 3 \u2194 IsThreeCycle \u03c3\n[PROOFSTEP]\nrefine' \u27e8fun h => _, IsThreeCycle.card_support\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh : Finset.card (support \u03c3) = 3\n\u22a2 IsThreeCycle \u03c3\n[PROOFSTEP]\nby_cases h0 : \u03c3.cycleType = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh : Finset.card (support \u03c3) = 3\nh0 : Perm.cycleType \u03c3 = 0\n\u22a2 IsThreeCycle \u03c3\n[PROOFSTEP]\nrw [\u2190 sum_cycleType, h0, sum_zero] at h \n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh : 0 = 3\nh0 : Perm.cycleType \u03c3 = 0\n\u22a2 IsThreeCycle \u03c3\n[PROOFSTEP]\nexact (ne_of_lt zero_lt_three h).elim\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh : Finset.card (support \u03c3) = 3\nh0 : \u00acPerm.cycleType \u03c3 = 0\n\u22a2 IsThreeCycle \u03c3\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := exists_mem_of_ne_zero h0\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh : Finset.card (support \u03c3) = 3\nh0 : \u00acPerm.cycleType \u03c3 = 0\nn : \u2115\nhn : n \u2208 Perm.cycleType \u03c3\n\u22a2 IsThreeCycle \u03c3\n[PROOFSTEP]\nby_cases h1 : \u03c3.cycleType.erase n = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh : Finset.card (support \u03c3) = 3\nh0 : \u00acPerm.cycleType \u03c3 = 0\nn : \u2115\nhn : n \u2208 Perm.cycleType \u03c3\nh1 : erase (Perm.cycleType \u03c3) n = 0\n\u22a2 IsThreeCycle \u03c3\n[PROOFSTEP]\nrw [\u2190 sum_cycleType, \u2190 cons_erase hn, h1, cons_zero, Multiset.sum_singleton] at h \n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh0 : \u00acPerm.cycleType \u03c3 = 0\nn : \u2115\nh : n = 3\nhn : n \u2208 Perm.cycleType \u03c3\nh1 : erase (Perm.cycleType \u03c3) n = 0\n\u22a2 IsThreeCycle \u03c3\n[PROOFSTEP]\nrw [IsThreeCycle, \u2190 cons_erase hn, h1, h, \u2190 cons_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh : Finset.card (support \u03c3) = 3\nh0 : \u00acPerm.cycleType \u03c3 = 0\nn : \u2115\nhn : n \u2208 Perm.cycleType \u03c3\nh1 : \u00acerase (Perm.cycleType \u03c3) n = 0\n\u22a2 IsThreeCycle \u03c3\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := exists_mem_of_ne_zero h1\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh : Finset.card (support \u03c3) = 3\nh0 : \u00acPerm.cycleType \u03c3 = 0\nn : \u2115\nhn : n \u2208 Perm.cycleType \u03c3\nh1 : \u00acerase (Perm.cycleType \u03c3) n = 0\nm : \u2115\nhm : m \u2208 erase (Perm.cycleType \u03c3) n\n\u22a2 IsThreeCycle \u03c3\n[PROOFSTEP]\nrw [\u2190 sum_cycleType, \u2190 cons_erase hn, \u2190 cons_erase hm, Multiset.sum_cons, Multiset.sum_cons] at h \n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh0 : \u00acPerm.cycleType \u03c3 = 0\nn : \u2115\nhn : n \u2208 Perm.cycleType \u03c3\nh1 : \u00acerase (Perm.cycleType \u03c3) n = 0\nm : \u2115\nh : n + (m + sum (erase (erase (Perm.cycleType \u03c3) n) m)) = 3\nhm : m \u2208 erase (Perm.cycleType \u03c3) n\n\u22a2 IsThreeCycle \u03c3\n[PROOFSTEP]\nhave : \u2200 {k}, 2 \u2264 m \u2192 2 \u2264 n \u2192 n + (m + k) = 3 \u2192 False :=\n by\n intros\n linarith\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh0 : \u00acPerm.cycleType \u03c3 = 0\nn : \u2115\nhn : n \u2208 Perm.cycleType \u03c3\nh1 : \u00acerase (Perm.cycleType \u03c3) n = 0\nm : \u2115\nh : n + (m + sum (erase (erase (Perm.cycleType \u03c3) n) m)) = 3\nhm : m \u2208 erase (Perm.cycleType \u03c3) n\n\u22a2 \u2200 {k : \u2115}, 2 \u2264 m \u2192 2 \u2264 n \u2192 n + (m + k) = 3 \u2192 False\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh0 : \u00acPerm.cycleType \u03c3 = 0\nn : \u2115\nhn : n \u2208 Perm.cycleType \u03c3\nh1 : \u00acerase (Perm.cycleType \u03c3) n = 0\nm : \u2115\nh : n + (m + sum (erase (erase (Perm.cycleType \u03c3) n) m)) = 3\nhm : m \u2208 erase (Perm.cycleType \u03c3) n\nk\u271d : \u2115\na\u271d\u00b2 : 2 \u2264 m\na\u271d\u00b9 : 2 \u2264 n\na\u271d : n + (m + k\u271d) = 3\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh0 : \u00acPerm.cycleType \u03c3 = 0\nn : \u2115\nhn : n \u2208 Perm.cycleType \u03c3\nh1 : \u00acerase (Perm.cycleType \u03c3) n = 0\nm : \u2115\nh : n + (m + sum (erase (erase (Perm.cycleType \u03c3) n) m)) = 3\nhm : m \u2208 erase (Perm.cycleType \u03c3) n\nthis : \u2200 {k : \u2115}, 2 \u2264 m \u2192 2 \u2264 n \u2192 n + (m + k) = 3 \u2192 False\n\u22a2 IsThreeCycle \u03c3\n[PROOFSTEP]\ncases this (two_le_of_mem_cycleType (mem_of_mem_erase hm)) (two_le_of_mem_cycleType hn) h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh : IsThreeCycle \u03c3\n\u22a2 IsCycle \u03c3\n[PROOFSTEP]\nrw [\u2190 card_cycleType_eq_one, h.cycleType, card_singleton]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh : IsThreeCycle \u03c3\n\u22a2 \u2191Perm.sign \u03c3 = 1\n[PROOFSTEP]\nrw [Equiv.Perm.sign_of_cycleType, h.cycleType]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 : Perm \u03b1\nh : IsThreeCycle \u03c3\n\u22a2 (-1) ^ (sum {3} + \u2191card {3}) = 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 f : Perm \u03b1\nh : IsThreeCycle f\n\u22a2 IsThreeCycle f\u207b\u00b9\n[PROOFSTEP]\nrwa [IsThreeCycle, cycleType_inv]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 f : Perm \u03b1\n\u22a2 IsThreeCycle f\u207b\u00b9 \u2192 IsThreeCycle f\n[PROOFSTEP]\nrw [\u2190 inv_inv f]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 f : Perm \u03b1\n\u22a2 IsThreeCycle f\u207b\u00b9\u207b\u00b9\u207b\u00b9 \u2192 IsThreeCycle f\u207b\u00b9\u207b\u00b9\n[PROOFSTEP]\napply inv\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 g : Perm \u03b1\nht : IsThreeCycle g\n\u22a2 _root_.orderOf g = 3\n[PROOFSTEP]\nrw [\u2190 lcm_cycleType, ht.cycleType, Multiset.lcm_singleton, normalize_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 g : Perm \u03b1\nht : IsThreeCycle g\n\u22a2 IsThreeCycle (g * g)\n[PROOFSTEP]\nrw [\u2190 pow_two, \u2190 card_support_eq_three_iff, support_pow_coprime, ht.card_support]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 g : Perm \u03b1\nht : IsThreeCycle g\n\u22a2 Nat.coprime 2 (_root_.orderOf g)\n[PROOFSTEP]\nrw [ht.orderOf]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 g : Perm \u03b1\nht : IsThreeCycle g\n\u22a2 Nat.coprime 2 3\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nab : a \u2260 b\nac : a \u2260 c\nbc : b \u2260 c\n\u22a2 IsThreeCycle (swap a b * swap a c)\n[PROOFSTEP]\nsuffices h : support (swap a b * swap a c) = { a, b, c }\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nab : a \u2260 b\nac : a \u2260 c\nbc : b \u2260 c\nh : support (swap a b * swap a c) = {a, b, c}\n\u22a2 IsThreeCycle (swap a b * swap a c)\n[PROOFSTEP]\nrw [\u2190 card_support_eq_three_iff, h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nab : a \u2260 b\nac : a \u2260 c\nbc : b \u2260 c\nh : support (swap a b * swap a c) = {a, b, c}\n\u22a2 Finset.card {a, b, c} = 3\n[PROOFSTEP]\nsimp [ab, ac, bc]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nab : a \u2260 b\nac : a \u2260 c\nbc : b \u2260 c\n\u22a2 support (swap a b * swap a c) = {a, b, c}\n[PROOFSTEP]\napply le_antisymm ((support_mul_le _ _).trans fun x => _) fun x hx => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nab : a \u2260 b\nac : a \u2260 c\nbc : b \u2260 c\n\u22a2 \u2200 (x : \u03b1), x \u2208 support (swap a b) \u2294 support (swap a c) \u2192 x \u2208 {a, b, c}\n[PROOFSTEP]\nsimp [ab, ac, bc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nab : a \u2260 b\nac : a \u2260 c\nbc : b \u2260 c\nx : \u03b1\nhx : x \u2208 {a, b, c}\n\u22a2 x \u2208 support (swap a b * swap a c)\n[PROOFSTEP]\nsimp only [Finset.mem_insert, Finset.mem_singleton] at hx \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nab : a \u2260 b\nac : a \u2260 c\nbc : b \u2260 c\nx : \u03b1\nhx : x = a \u2228 x = b \u2228 x = c\n\u22a2 x \u2208 support (swap a b * swap a c)\n[PROOFSTEP]\nrw [mem_support]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nab : a \u2260 b\nac : a \u2260 c\nbc : b \u2260 c\nx : \u03b1\nhx : x = a \u2228 x = b \u2228 x = c\n\u22a2 \u2191(swap a b * swap a c) x \u2260 x\n[PROOFSTEP]\nsimp only [Perm.coe_mul, Function.comp_apply, Ne.def]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nab : a \u2260 b\nac : a \u2260 c\nbc : b \u2260 c\nx : \u03b1\nhx : x = a \u2228 x = b \u2228 x = c\n\u22a2 \u00ac\u2191(swap a b) (\u2191(swap a c) x) = x\n[PROOFSTEP]\nobtain rfl | rfl | rfl := hx\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nb c : \u03b1\nbc : b \u2260 c\nx : \u03b1\nab : x \u2260 b\nac : x \u2260 c\n\u22a2 \u00ac\u2191(swap x b) (\u2191(swap x c) x) = x\n[PROOFSTEP]\nrw [swap_apply_left, swap_apply_of_ne_of_ne ac.symm bc.symm]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nb c : \u03b1\nbc : b \u2260 c\nx : \u03b1\nab : x \u2260 b\nac : x \u2260 c\n\u22a2 \u00acc = x\n[PROOFSTEP]\nexact ac.symm\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na c : \u03b1\nac : a \u2260 c\nx : \u03b1\nab : a \u2260 x\nbc : x \u2260 c\n\u22a2 \u00ac\u2191(swap a x) (\u2191(swap a c) x) = x\n[PROOFSTEP]\nrw [swap_apply_of_ne_of_ne ab.symm bc, swap_apply_right]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na c : \u03b1\nac : a \u2260 c\nx : \u03b1\nab : a \u2260 x\nbc : x \u2260 c\n\u22a2 \u00aca = x\n[PROOFSTEP]\nexact ab\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab : a \u2260 b\nx : \u03b1\nac : a \u2260 x\nbc : b \u2260 x\n\u22a2 \u00ac\u2191(swap a b) (\u2191(swap a x) x) = x\n[PROOFSTEP]\nrw [swap_apply_right, swap_apply_left]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab : a \u2260 b\nx : \u03b1\nac : a \u2260 x\nbc : b \u2260 x\n\u22a2 \u00acb = x\n[PROOFSTEP]\nexact bc\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nab : a \u2260 b\nac : a \u2260 c\n\u22a2 swap a b * swap a c \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nby_cases bc : b = c\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nab : a \u2260 b\nac : a \u2260 c\nbc : b = c\n\u22a2 swap a b * swap a c \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nsubst bc\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab ac : a \u2260 b\n\u22a2 swap a b * swap a b \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nsimp [one_mem]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nab : a \u2260 b\nac : a \u2260 c\nbc : \u00acb = c\n\u22a2 swap a b * swap a c \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nexact subset_closure (isThreeCycle_swap_mul_swap_same ab ac bc)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c3 \u03c4 : Perm \u03b1\nh\u03c3 : IsSwap \u03c3\nh\u03c4 : IsSwap \u03c4\n\u22a2 \u03c3 * \u03c4 \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nobtain \u27e8a, b, ab, rfl\u27e9 := h\u03c3\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u03c4 : Perm \u03b1\nh\u03c4 : IsSwap \u03c4\na b : \u03b1\nab : a \u2260 b\n\u22a2 swap a b * \u03c4 \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nobtain \u27e8c, d, cd, rfl\u27e9 := h\u03c4\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab : a \u2260 b\nc d : \u03b1\ncd : c \u2260 d\n\u22a2 swap a b * swap c d \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nby_cases ac : a = c\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab : a \u2260 b\nc d : \u03b1\ncd : c \u2260 d\nac : a = c\n\u22a2 swap a b * swap c d \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nsubst ac\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab : a \u2260 b\nd : \u03b1\ncd : a \u2260 d\n\u22a2 swap a b * swap a d \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nexact swap_mul_swap_same_mem_closure_three_cycles ab cd\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab : a \u2260 b\nc d : \u03b1\ncd : c \u2260 d\nac : \u00aca = c\n\u22a2 swap a b * swap c d \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nhave h' : swap a b * swap c d = swap a b * swap a c * (swap c a * swap c d) := by simp [swap_comm c a, mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab : a \u2260 b\nc d : \u03b1\ncd : c \u2260 d\nac : \u00aca = c\n\u22a2 swap a b * swap c d = swap a b * swap a c * (swap c a * swap c d)\n[PROOFSTEP]\nsimp [swap_comm c a, mul_assoc]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab : a \u2260 b\nc d : \u03b1\ncd : c \u2260 d\nac : \u00aca = c\nh' : swap a b * swap c d = swap a b * swap a c * (swap c a * swap c d)\n\u22a2 swap a b * swap c d \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nrw [h']\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab : a \u2260 b\nc d : \u03b1\ncd : c \u2260 d\nac : \u00aca = c\nh' : swap a b * swap c d = swap a b * swap a c * (swap c a * swap c d)\n\u22a2 swap a b * swap a c * (swap c a * swap c d) \u2208 closure {\u03c3 | IsThreeCycle \u03c3}\n[PROOFSTEP]\nexact\n mul_mem (swap_mul_swap_same_mem_closure_three_cycles ab ac)\n (swap_mul_swap_same_mem_closure_three_cycles (Ne.symm ac) cd)\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Perm.Cycle.Type", "llama_tokens": 44254, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737214979745, "lm_q2_score": 0.6297745935070806, "lm_q1q2_score": 0.42986758799500213}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\u271d\ninst\u271d : Preorder \u03b3\u271d\n\u03b2 \u03b3 : Type ?u.55\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\n\u22a2 Monotone fun x => \u2191f x >>= \u2191g x\n[PROOFSTEP]\nintro x y h a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\u271d\ninst\u271d : Preorder \u03b3\u271d\n\u03b2 \u03b3 : Type ?u.55\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx y : \u03b1\nh : x \u2264 y\na : \u03b3\n\u22a2 a \u2208 (fun x => \u2191f x >>= \u2191g x) x \u2192 a \u2208 (fun x => \u2191f x >>= \u2191g x) y\n[PROOFSTEP]\nsimp only [and_imp, exists_prop, Part.bind_eq_bind, Part.mem_bind_iff, exists_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\u271d\ninst\u271d : Preorder \u03b3\u271d\n\u03b2 \u03b3 : Type ?u.55\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx y : \u03b1\nh : x \u2264 y\na : \u03b3\n\u22a2 \u2200 (x_1 : \u03b2), x_1 \u2208 \u2191f x \u2192 a \u2208 \u2191g x x_1 \u2192 \u2203 a_3, a_3 \u2208 \u2191f y \u2227 a \u2208 \u2191g y a_3\n[PROOFSTEP]\nintro b hb ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\u271d\ninst\u271d : Preorder \u03b3\u271d\n\u03b2 \u03b3 : Type ?u.55\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx y : \u03b1\nh : x \u2264 y\na : \u03b3\nb : \u03b2\nhb : b \u2208 \u2191f x\nha : a \u2208 \u2191g x b\n\u22a2 \u2203 a_1, a_1 \u2208 \u2191f y \u2227 a \u2208 \u2191g y a_1\n[PROOFSTEP]\nrefine' \u27e8b, f.monotone h _ hb, g.monotone h _ _ ha\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nc c' : Chain \u03b1\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\nb : \u03b2\nh : \u2203 a, a \u2208 c \u2227 \u2191f a = b\n\u22a2 b \u2208 map c f\n[PROOFSTEP]\nrcases h with \u27e8w, h, h'\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nc c' : Chain \u03b1\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\nb : \u03b2\nw : \u03b1\nh : w \u2208 c\nh' : \u2191f w = b\n\u22a2 b \u2208 map c f\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nc c' : Chain \u03b1\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\nw : \u03b1\nh : w \u2208 c\n\u22a2 \u2191f w \u2208 map c f\n[PROOFSTEP]\napply mem_map c _ h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nc c' : Chain \u03b1\nf : \u03b1 \u2192o \u03b2\ng\u271d : \u03b2 \u2192o \u03b3\ng : \u03b1 \u2192o \u03b2\nh : f \u2264 g\ni : \u2115\n\u22a2 \u2203 j, \u2191(map c f) i \u2264 \u2191(map c g) j\n[PROOFSTEP]\nsimp [mem_map_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nc c' : Chain \u03b1\nf : \u03b1 \u2192o \u03b2\ng\u271d : \u03b2 \u2192o \u03b3\ng : \u03b1 \u2192o \u03b2\nh : f \u2264 g\ni : \u2115\n\u22a2 \u2203 j, \u2191f (\u2191c i) \u2264 \u2191g (\u2191c j)\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nc c' : Chain \u03b1\nf : \u03b1 \u2192o \u03b2\ng\u271d : \u03b2 \u2192o \u03b3\ng : \u03b1 \u2192o \u03b2\nh : f \u2264 g\ni : \u2115\n\u22a2 \u2203 j, \u2191f (\u2191c i) \u2264 \u2191g (\u2191c j)\n[PROOFSTEP]\nexists i\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nc c' : Chain \u03b1\nf : \u03b1 \u2192o \u03b2\ng\u271d : \u03b2 \u2192o \u03b3\ng : \u03b1 \u2192o \u03b2\nh : f \u2264 g\ni : \u2115\n\u22a2 \u2191f (\u2191c i) \u2264 \u2191g (\u2191c i)\n[PROOFSTEP]\napply h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\nf : \u03b2 \u2192o \u03b1\n\u03c9Sup\u2080 : Chain \u03b2 \u2192 \u03b2\nh : \u2200 (x y : \u03b2), \u2191f x \u2264 \u2191f y \u2192 x \u2264 y\nh' : \u2200 (c : Chain \u03b2), \u2191f (\u03c9Sup\u2080 c) = \u03c9Sup (Chain.map c f)\nc : Chain \u03b2\ni : \u2115\n\u22a2 \u2191f (\u2191c i) \u2264 \u2191f (\u03c9Sup\u2080 c)\n[PROOFSTEP]\nrw [h']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\nf : \u03b2 \u2192o \u03b1\n\u03c9Sup\u2080 : Chain \u03b2 \u2192 \u03b2\nh : \u2200 (x y : \u03b2), \u2191f x \u2264 \u2191f y \u2192 x \u2264 y\nh' : \u2200 (c : Chain \u03b2), \u2191f (\u03c9Sup\u2080 c) = \u03c9Sup (Chain.map c f)\nc : Chain \u03b2\ni : \u2115\n\u22a2 \u2191f (\u2191c i) \u2264 \u03c9Sup (Chain.map c f)\n[PROOFSTEP]\napply le_\u03c9Sup (c.map f)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\nf : \u03b2 \u2192o \u03b1\n\u03c9Sup\u2080 : Chain \u03b2 \u2192 \u03b2\nh : \u2200 (x y : \u03b2), \u2191f x \u2264 \u2191f y \u2192 x \u2264 y\nh' : \u2200 (c : Chain \u03b2), \u2191f (\u03c9Sup\u2080 c) = \u03c9Sup (Chain.map c f)\nc : Chain \u03b2\nx : \u03b2\nhx : \u2200 (i : \u2115), \u2191c i \u2264 x\n\u22a2 \u2191f (\u03c9Sup\u2080 c) \u2264 \u2191f x\n[PROOFSTEP]\nrw [h']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\nf : \u03b2 \u2192o \u03b1\n\u03c9Sup\u2080 : Chain \u03b2 \u2192 \u03b2\nh : \u2200 (x y : \u03b2), \u2191f x \u2264 \u2191f y \u2192 x \u2264 y\nh' : \u2200 (c : Chain \u03b2), \u2191f (\u03c9Sup\u2080 c) = \u03c9Sup (Chain.map c f)\nc : Chain \u03b2\nx : \u03b2\nhx : \u2200 (i : \u2115), \u2191c i \u2264 x\n\u22a2 \u03c9Sup (Chain.map c f) \u2264 \u2191f x\n[PROOFSTEP]\napply \u03c9Sup_le\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\nf : \u03b2 \u2192o \u03b1\n\u03c9Sup\u2080 : Chain \u03b2 \u2192 \u03b2\nh : \u2200 (x y : \u03b2), \u2191f x \u2264 \u2191f y \u2192 x \u2264 y\nh' : \u2200 (c : Chain \u03b2), \u2191f (\u03c9Sup\u2080 c) = \u03c9Sup (Chain.map c f)\nc : Chain \u03b2\nx : \u03b2\nhx : \u2200 (i : \u2115), \u2191c i \u2264 x\n\u22a2 \u2200 (i : \u2115), \u2191(Chain.map c f) i \u2264 \u2191f x\n[PROOFSTEP]\nintro i\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\nf : \u03b2 \u2192o \u03b1\n\u03c9Sup\u2080 : Chain \u03b2 \u2192 \u03b2\nh : \u2200 (x y : \u03b2), \u2191f x \u2264 \u2191f y \u2192 x \u2264 y\nh' : \u2200 (c : Chain \u03b2), \u2191f (\u03c9Sup\u2080 c) = \u03c9Sup (Chain.map c f)\nc : Chain \u03b2\nx : \u03b2\nhx : \u2200 (i : \u2115), \u2191c i \u2264 x\ni : \u2115\n\u22a2 \u2191(Chain.map c f) i \u2264 \u2191f x\n[PROOFSTEP]\napply f.monotone (hx i)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc : Chain \u03b1\nx : \u03b1\nh : \u2200 (i : \u2115), \u2191c i \u2264 x \u2228 x \u2264 \u2191c i\nthis : \u00ac\u2200 (i : \u2115), \u2191c i \u2264 x\n\u22a2 \u2203 i, \u00ac\u2191c i \u2264 x\n[PROOFSTEP]\nsimp only [not_forall] at this \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc : Chain \u03b1\nx : \u03b1\nh : \u2200 (i : \u2115), \u2191c i \u2264 x \u2228 x \u2264 \u2191c i\nthis : \u2203 x_1, \u00ac\u2191c x_1 \u2264 x\n\u22a2 \u2203 i, \u00ac\u2191c i \u2264 x\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc\u2080 c\u2081 : Chain \u03b1\nh : c\u2080 \u2264 c\u2081\ni : \u2115\n\u22a2 \u2191c\u2080 i \u2264 \u03c9Sup c\u2081\n[PROOFSTEP]\nobtain \u27e8_, h\u27e9 := h i\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc\u2080 c\u2081 : Chain \u03b1\nh\u271d : c\u2080 \u2264 c\u2081\ni w\u271d : \u2115\nh : \u2191c\u2080 i \u2264 \u2191c\u2081 w\u271d\n\u22a2 \u2191c\u2080 i \u2264 \u03c9Sup c\u2081\n[PROOFSTEP]\nexact le_trans h (le_\u03c9Sup _ _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc : Chain \u03b1\nx : \u03b1\n\u22a2 \u03c9Sup c \u2264 x \u2194 \u2200 (i : \u2115), \u2191c i \u2264 x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc : Chain \u03b1\nx : \u03b1\n\u22a2 \u03c9Sup c \u2264 x \u2192 \u2200 (i : \u2115), \u2191c i \u2264 x\n[PROOFSTEP]\nintros\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc : Chain \u03b1\nx : \u03b1\n\u22a2 (\u2200 (i : \u2115), \u2191c i \u2264 x) \u2192 \u03c9Sup c \u2264 x\n[PROOFSTEP]\nintros\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc : Chain \u03b1\nx : \u03b1\na\u271d : \u03c9Sup c \u2264 x\ni\u271d : \u2115\n\u22a2 \u2191c i\u271d \u2264 x\n[PROOFSTEP]\ntrans \u03c9Sup c\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc : Chain \u03b1\nx : \u03b1\na\u271d : \u03c9Sup c \u2264 x\ni\u271d : \u2115\n\u22a2 \u2191c i\u271d \u2264 \u03c9Sup c\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc : Chain \u03b1\nx : \u03b1\na\u271d : \u03c9Sup c \u2264 x\ni\u271d : \u2115\n\u22a2 \u03c9Sup c \u2264 x\n[PROOFSTEP]\nexact le_\u03c9Sup _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc : Chain \u03b1\nx : \u03b1\na\u271d : \u03c9Sup c \u2264 x\ni\u271d : \u2115\n\u22a2 \u03c9Sup c \u2264 x\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc : Chain \u03b1\nx : \u03b1\na\u271d : \u2200 (i : \u2115), \u2191c i \u2264 x\n\u22a2 \u03c9Sup c \u2264 x\n[PROOFSTEP]\nexact \u03c9Sup_le _ _ \u2039_\u203a\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\n\u22a2 Continuous OrderHom.id\n[PROOFSTEP]\nintro c\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\nc : Chain \u03b1\n\u22a2 \u2191OrderHom.id (\u03c9Sup c) = \u03c9Sup (map c OrderHom.id)\n[PROOFSTEP]\nrw [c.map_id]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\nc : Chain \u03b1\n\u22a2 \u2191OrderHom.id (\u03c9Sup c) = \u03c9Sup c\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\nhfc : Continuous f\nhgc : Continuous g\n\u22a2 Continuous (OrderHom.comp g f)\n[PROOFSTEP]\ndsimp [Continuous] at *\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\nhfc : \u2200 (c : Chain \u03b1), \u2191f (\u03c9Sup c) = \u03c9Sup (map c f)\nhgc : \u2200 (c : Chain \u03b2), \u2191g (\u03c9Sup c) = \u03c9Sup (map c g)\n\u22a2 \u2200 (c : Chain \u03b1), \u2191g (\u2191f (\u03c9Sup c)) = \u03c9Sup (map c (OrderHom.comp g f))\n[PROOFSTEP]\nintro\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\nhfc : \u2200 (c : Chain \u03b1), \u2191f (\u03c9Sup c) = \u03c9Sup (map c f)\nhgc : \u2200 (c : Chain \u03b2), \u2191g (\u03c9Sup c) = \u03c9Sup (map c g)\nc\u271d : Chain \u03b1\n\u22a2 \u2191g (\u2191f (\u03c9Sup c\u271d)) = \u03c9Sup (map c\u271d (OrderHom.comp g f))\n[PROOFSTEP]\nrw [hfc, hgc, Chain.map_comp]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\nx : \u03b2\nc : Chain \u03b1\nz : \u03b2\n\u22a2 \u2191(\u2191(OrderHom.const \u03b1) x) (\u03c9Sup c) \u2264 z \u2194 \u03c9Sup (map c (\u2191(OrderHom.const \u03b1) x)) \u2264 z\n[PROOFSTEP]\nrw [\u03c9Sup_le_iff, Chain.map_coe, OrderHom.const_coe_coe]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\nx : \u03b2\nc : Chain \u03b1\nz : \u03b2\n\u22a2 Function.const \u03b1 x (\u03c9Sup c) \u2264 z \u2194 \u2200 (i : \u2115), (Function.const \u03b1 x \u2218 \u2191c) i \u2264 z\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na b : \u03b1\nha : some a \u2208 c\nhb : some b \u2208 c\n\u22a2 a = b\n[PROOFSTEP]\ncases' ha with i ha\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na b : \u03b1\nhb : some b \u2208 c\ni : \u2115\nha : some a = \u2191c i\n\u22a2 a = b\n[PROOFSTEP]\nreplace ha := ha.symm\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na b : \u03b1\nhb : some b \u2208 c\ni : \u2115\nha : \u2191c i = some a\n\u22a2 a = b\n[PROOFSTEP]\ncases' hb with j hb\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na b : \u03b1\ni : \u2115\nha : \u2191c i = some a\nj : \u2115\nhb : some b = \u2191c j\n\u22a2 a = b\n[PROOFSTEP]\nreplace hb := hb.symm\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na b : \u03b1\ni : \u2115\nha : \u2191c i = some a\nj : \u2115\nhb : \u2191c j = some b\n\u22a2 a = b\n[PROOFSTEP]\nrw [eq_some_iff] at ha hb \n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na b : \u03b1\ni : \u2115\nha : a \u2208 \u2191c i\nj : \u2115\nhb : b \u2208 \u2191c j\n\u22a2 a = b\n[PROOFSTEP]\ncases' le_total i j with hij hji\n[GOAL]\ncase intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na b : \u03b1\ni : \u2115\nha : a \u2208 \u2191c i\nj : \u2115\nhb : b \u2208 \u2191c j\nhij : i \u2264 j\n\u22a2 a = b\n[PROOFSTEP]\nhave := c.monotone hij _ ha\n[GOAL]\ncase intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na b : \u03b1\ni : \u2115\nha : a \u2208 \u2191c i\nj : \u2115\nhb : b \u2208 \u2191c j\nhij : i \u2264 j\nthis : a \u2208 \u2191c j\n\u22a2 a = b\n[PROOFSTEP]\napply mem_unique this hb\n[GOAL]\ncase intro.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na b : \u03b1\ni : \u2115\nha : a \u2208 \u2191c i\nj : \u2115\nhb : b \u2208 \u2191c j\nhji : j \u2264 i\n\u22a2 a = b\n[PROOFSTEP]\nhave := c.monotone hji _ hb\n[GOAL]\ncase intro.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na b : \u03b1\ni : \u2115\nha : a \u2208 \u2191c i\nj : \u2115\nhb : b \u2208 \u2191c j\nhji : j \u2264 i\nthis : b \u2208 \u2191c i\n\u22a2 a = b\n[PROOFSTEP]\napply Eq.symm\n[GOAL]\ncase intro.intro.inr.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na b : \u03b1\ni : \u2115\nha : a \u2208 \u2191c i\nj : \u2115\nhb : b \u2208 \u2191c j\nhji : j \u2264 i\nthis : b \u2208 \u2191c i\n\u22a2 b = a\n[PROOFSTEP]\napply mem_unique this ha\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na : \u03b1\nh : a \u2208 Part.\u03c9Sup c\n\u22a2 some a \u2208 c\n[PROOFSTEP]\nsimp only [Part.\u03c9Sup] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na : \u03b1\nh : a \u2208 if h : \u2203 a, some a \u2208 c then some (choose h) else none\n\u22a2 some a \u2208 c\n[PROOFSTEP]\nsplit_ifs at h with h_1\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na : \u03b1\nh_1 : \u2203 a, some a \u2208 c\nh : a \u2208 some (choose h_1)\n\u22a2 some a \u2208 c\n[PROOFSTEP]\nhave h' := Classical.choose_spec h_1\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na : \u03b1\nh_1 : \u2203 a, some a \u2208 c\nh : a \u2208 some (choose h_1)\nh' : some (choose h_1) \u2208 c\n\u22a2 some a \u2208 c\n[PROOFSTEP]\nrw [\u2190 eq_some_iff] at h \n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na : \u03b1\nh_1 : \u2203 a, some a \u2208 c\nh : some (choose h_1) = some a\nh' : some (choose h_1) \u2208 c\n\u22a2 some a \u2208 c\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na : \u03b1\nh_1 : \u2203 a, some a \u2208 c\nh : some (choose h_1) = some a\nh' : some (choose h_1) \u2208 c\n\u22a2 some (choose h_1) \u2208 c\n[PROOFSTEP]\nexact h'\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\na : \u03b1\nh_1 : \u00ac\u2203 a, some a \u2208 c\nh : a \u2208 none\n\u22a2 some a \u2208 c\n[PROOFSTEP]\nrcases h with \u27e8\u27e8\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\ni : \u2115\n\u22a2 \u2191c i \u2264 Part.\u03c9Sup c\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\ni : \u2115\nx : \u03b1\nhx : x \u2208 \u2191c i\n\u22a2 x \u2208 Part.\u03c9Sup c\n[PROOFSTEP]\nrw [\u2190 eq_some_iff] at hx \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\ni : \u2115\nx : \u03b1\nhx : \u2191c i = some x\n\u22a2 Part.\u03c9Sup c = some x\n[PROOFSTEP]\nrw [\u03c9Sup_eq_some, \u2190 hx]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\ni : \u2115\nx : \u03b1\nhx : \u2191c i = some x\n\u22a2 some x \u2208 c\n[PROOFSTEP]\nrw [\u2190 hx]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\ni : \u2115\nx : \u03b1\nhx : \u2191c i = some x\n\u22a2 \u2191c i \u2208 c\n[PROOFSTEP]\nexact \u27e8i, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u22a2 \u2200 (c : Chain (Part \u03b1)) (x : Part \u03b1), (\u2200 (i : \u2115), \u2191c i \u2264 x) \u2192 Part.\u03c9Sup c \u2264 x\n[PROOFSTEP]\nrintro c x hx a ha\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\nx : Part \u03b1\nhx : \u2200 (i : \u2115), \u2191c i \u2264 x\na : \u03b1\nha : a \u2208 Part.\u03c9Sup c\n\u22a2 a \u2208 x\n[PROOFSTEP]\nreplace ha := mem_chain_of_mem_\u03c9Sup ha\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\nx : Part \u03b1\nhx : \u2200 (i : \u2115), \u2191c i \u2264 x\na : \u03b1\nha : some a \u2208 c\n\u22a2 a \u2208 x\n[PROOFSTEP]\ncases' ha with i ha\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\nx : Part \u03b1\nhx : \u2200 (i : \u2115), \u2191c i \u2264 x\na : \u03b1\ni : \u2115\nha : some a = \u2191c i\n\u22a2 a \u2208 x\n[PROOFSTEP]\napply hx i\n[GOAL]\ncase intro.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\nx : Part \u03b1\nhx : \u2200 (i : \u2115), \u2191c i \u2264 x\na : \u03b1\ni : \u2115\nha : some a = \u2191c i\n\u22a2 a \u2208 \u2191c i\n[PROOFSTEP]\nrw [\u2190 ha]\n[GOAL]\ncase intro.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\nx : Part \u03b1\nhx : \u2200 (i : \u2115), \u2191c i \u2264 x\na : \u03b1\ni : \u2115\nha : some a = \u2191c i\n\u22a2 a \u2208 some a\n[PROOFSTEP]\napply mem_some\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\n\u22a2 x \u2208 \u03c9Sup c \u2194 some x \u2208 c\n[PROOFSTEP]\nsimp [OmegaCompletePartialOrder.\u03c9Sup, Part.\u03c9Sup]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\n\u22a2 (x \u2208 if h : \u2203 a, some a \u2208 c then some (choose h) else none) \u2194 some x \u2208 c\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\n\u22a2 (x \u2208 if h : \u2203 a, some a \u2208 c then some (choose h) else none) \u2192 some x \u2208 c\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\nh : \u2203 a, some a \u2208 c\n\u22a2 x \u2208 some (choose h) \u2192 some x \u2208 c\ncase neg \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 x : \u03b1 c : Chain (Part \u03b1) h : \u00ac\u2203 a, some a \u2208 c \u22a2 x \u2208 none \u2192 some x \u2208 c\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\nh : \u00ac\u2203 a, some a \u2208 c\n\u22a2 x \u2208 none \u2192 some x \u2208 c\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\nh : \u2203 a, some a \u2208 c\n\u22a2 x \u2208 some (choose h) \u2192 some x \u2208 c\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\nh : \u2203 a, some a \u2208 c\n\u22a2 x \u2208 some (choose h) \u2192 some x \u2208 c\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\nh : \u2203 a, some a \u2208 c\nh' : x \u2208 some (choose h)\n\u22a2 some x \u2208 c\n[PROOFSTEP]\nhave hh := Classical.choose_spec h\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\nh : \u2203 a, some a \u2208 c\nh' : x \u2208 some (choose h)\nhh : some (choose h) \u2208 c\n\u22a2 some x \u2208 c\n[PROOFSTEP]\nsimp only [mem_some_iff] at h' \n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\nh : \u2203 a, some a \u2208 c\nhh : some (choose h) \u2208 c\nh' : x = choose h\n\u22a2 some x \u2208 c\n[PROOFSTEP]\nsubst x\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nc : Chain (Part \u03b1)\nh : \u2203 a, some a \u2208 c\nhh : some (choose h) \u2208 c\n\u22a2 some (choose h) \u2208 c\n[PROOFSTEP]\nexact hh\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\n\u22a2 some x \u2208 c \u2192 x \u2208 if h : \u2203 a, some a \u2208 c then some (choose h) else none\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\nh : some x \u2208 c\n\u22a2 x \u2208 if h : \u2203 a, some a \u2208 c then some (choose h) else none\n[PROOFSTEP]\nhave h' : \u2203 a : \u03b1, some a \u2208 c := \u27e8_, h\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\nh : some x \u2208 c\nh' : \u2203 a, some a \u2208 c\n\u22a2 x \u2208 if h : \u2203 a, some a \u2208 c then some (choose h) else none\n[PROOFSTEP]\nrw [dif_pos h']\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\nh : some x \u2208 c\nh' : \u2203 a, some a \u2208 c\n\u22a2 x \u2208 some (choose h')\n[PROOFSTEP]\nhave hh := Classical.choose_spec h'\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\nh : some x \u2208 c\nh' : \u2203 a, some a \u2208 c\nhh : some (choose h') \u2208 c\n\u22a2 x \u2208 some (choose h')\n[PROOFSTEP]\nrw [eq_of_chain hh h]\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nx : \u03b1\nc : Chain (Part \u03b1)\nh : some x \u2208 c\nh' : \u2203 a, some a \u2208 c\nhh : some (choose h') \u2208 c\n\u22a2 x \u2208 some x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\n\u03b3 : Type u_3\ninst\u271d : (a : \u03b1) \u2192 OmegaCompletePartialOrder (\u03b2 a)\nc : Chain ((a : \u03b1) \u2192 \u03b2 a)\nf : (a : \u03b1) \u2192 \u03b2 a\nhf : \u2200 (i : \u2115), \u2191c i \u2264 f\na : \u03b1\n\u22a2 \u2200 (i : \u2115), \u2191(map c (evalOrderHom a)) i \u2264 f a\n[PROOFSTEP]\nrintro i\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\n\u03b3 : Type u_3\ninst\u271d : (a : \u03b1) \u2192 OmegaCompletePartialOrder (\u03b2 a)\nc : Chain ((a : \u03b1) \u2192 \u03b2 a)\nf : (a : \u03b1) \u2192 \u03b2 a\nhf : \u2200 (i : \u2115), \u2191c i \u2264 f\na : \u03b1\ni : \u2115\n\u22a2 \u2191(map c (evalOrderHom a)) i \u2264 f a\n[PROOFSTEP]\napply hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : (x : \u03b1) \u2192 OmegaCompletePartialOrder (\u03b2 x)\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b3 \u2192 (x : \u03b1) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b1), Continuous' fun g => f g x\n\u22a2 Continuous { toFun := f, monotone' := (_ : \u2200 (x y : \u03b3), x \u2264 y \u2192 \u2200 (a : \u03b1), (fun g => f g a) x \u2264 (fun g => f g a) y) }\n[PROOFSTEP]\nintro c\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : (x : \u03b1) \u2192 OmegaCompletePartialOrder (\u03b2 x)\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b3 \u2192 (x : \u03b1) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b1), Continuous' fun g => f g x\nc : Chain \u03b3\n\u22a2 \u2191{ toFun := f, monotone' := (_ : \u2200 (x y : \u03b3), x \u2264 y \u2192 \u2200 (a : \u03b1), (fun g => f g a) x \u2264 (fun g => f g a) y) } (\u03c9Sup c) =\n \u03c9Sup\n (map c { toFun := f, monotone' := (_ : \u2200 (x y : \u03b3), x \u2264 y \u2192 \u2200 (a : \u03b1), (fun g => f g a) x \u2264 (fun g => f g a) y) })\n[PROOFSTEP]\next a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : (x : \u03b1) \u2192 OmegaCompletePartialOrder (\u03b2 x)\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b3 \u2192 (x : \u03b1) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b1), Continuous' fun g => f g x\nc : Chain \u03b3\na : \u03b1\n\u22a2 \u2191{ toFun := f, monotone' := (_ : \u2200 (x y : \u03b3), x \u2264 y \u2192 \u2200 (a : \u03b1), (fun g => f g a) x \u2264 (fun g => f g a) y) } (\u03c9Sup c)\n a =\n \u03c9Sup\n (map c { toFun := f, monotone' := (_ : \u2200 (x y : \u03b3), x \u2264 y \u2192 \u2200 (a : \u03b1), (fun g => f g a) x \u2264 (fun g => f g a) y) })\n a\n[PROOFSTEP]\napply (hf a).to_bundled _ c\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nc\u2080 : Chain \u03b1\nc\u2081 : Chain \u03b2\n\u22a2 \u03c9Sup (Chain.zip c\u2080 c\u2081) = (\u03c9Sup c\u2080, \u03c9Sup c\u2081)\n[PROOFSTEP]\napply eq_of_forall_ge_iff\n[GOAL]\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nc\u2080 : Chain \u03b1\nc\u2081 : Chain \u03b2\n\u22a2 \u2200 (c : \u03b1 \u00d7 \u03b2), \u03c9Sup (Chain.zip c\u2080 c\u2081) \u2264 c \u2194 (\u03c9Sup c\u2080, \u03c9Sup c\u2081) \u2264 c\n[PROOFSTEP]\nrintro \u27e8z\u2081, z\u2082\u27e9\n[GOAL]\ncase H.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nc\u2080 : Chain \u03b1\nc\u2081 : Chain \u03b2\nz\u2081 : \u03b1\nz\u2082 : \u03b2\n\u22a2 \u03c9Sup (Chain.zip c\u2080 c\u2081) \u2264 (z\u2081, z\u2082) \u2194 (\u03c9Sup c\u2080, \u03c9Sup c\u2081) \u2264 (z\u2081, z\u2082)\n[PROOFSTEP]\nsimp [\u03c9Sup_le_iff, forall_and]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nx\u271d : Chain \u03b1\ni : \u2115\nc : \u2115 \u2192 \u03b1\nmonotone'\u271d : Monotone c\n\u22a2 \u2191{ toFun := c, monotone' := monotone'\u271d } i \u2264 (fun c => \u2a06 (i : \u2115), \u2191c i) { toFun := c, monotone' := monotone'\u271d }\n[PROOFSTEP]\nsimp only [OrderHom.coe_mk]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nx\u271d : Chain \u03b1\ni : \u2115\nc : \u2115 \u2192 \u03b1\nmonotone'\u271d : Monotone c\n\u22a2 \u2191{ toFun := c, monotone' := monotone'\u271d } i \u2264 \u2a06 (i : \u2115), \u2191{ toFun := c, monotone' := monotone'\u271d } i\n[PROOFSTEP]\napply le_iSup_of_le i\n[GOAL]\n\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nx\u271d : Chain \u03b1\ni : \u2115\nc : \u2115 \u2192 \u03b1\nmonotone'\u271d : Monotone c\n\u22a2 \u2191{ toFun := c, monotone' := monotone'\u271d } i \u2264 \u2191{ toFun := c, monotone' := monotone'\u271d } i\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nx\u271d : Chain \u03b1\ns : \u03b1\nc : \u2115 \u2192 \u03b1\nmonotone'\u271d : Monotone c\nhs : \u2200 (i : \u2115), \u2191{ toFun := c, monotone' := monotone'\u271d } i \u2264 s\n\u22a2 (fun c => \u2a06 (i : \u2115), \u2191c i) { toFun := c, monotone' := monotone'\u271d } \u2264 s\n[PROOFSTEP]\nsimp only [iSup_le_iff, OrderHom.coe_mk] at hs \u22a2\n[GOAL]\n\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nx\u271d : Chain \u03b1\ns : \u03b1\nc : \u2115 \u2192 \u03b1\nmonotone'\u271d : Monotone c\nhs : \u2200 (i : \u2115), \u2191{ toFun := c, monotone' := monotone'\u271d } i \u2264 s\n\u22a2 \u2200 (i : \u2115), \u2191{ toFun := c, monotone' := monotone'\u271d } i \u2264 s\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u\ninst\u271d : CompleteLattice \u03b1\nx\u271d : Chain \u03b1\ns : \u03b1\nc : \u2115 \u2192 \u03b1\nmonotone'\u271d : Monotone c\nhs : \u2200 (i : \u2115), \u2191{ toFun := c, monotone' := monotone'\u271d } i \u2264 s\ni : \u2115\n\u22a2 \u2191{ toFun := c, monotone' := monotone'\u271d } i \u2264 s\n[PROOFSTEP]\napply hs i\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\ns : Set (\u03b1 \u2192o \u03b2)\nhs : \u2200 (f : \u03b1 \u2192o \u03b2), f \u2208 s \u2192 Continuous f\n\u22a2 Continuous (sSup s)\n[PROOFSTEP]\nintro c\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\ns : Set (\u03b1 \u2192o \u03b2)\nhs : \u2200 (f : \u03b1 \u2192o \u03b2), f \u2208 s \u2192 Continuous f\nc : Chain \u03b1\n\u22a2 \u2191(sSup s) (\u03c9Sup c) = \u03c9Sup (Chain.map c (sSup s))\n[PROOFSTEP]\napply eq_of_forall_ge_iff\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\ns : Set (\u03b1 \u2192o \u03b2)\nhs : \u2200 (f : \u03b1 \u2192o \u03b2), f \u2208 s \u2192 Continuous f\nc : Chain \u03b1\n\u22a2 \u2200 (c_1 : \u03b2), \u2191(sSup s) (\u03c9Sup c) \u2264 c_1 \u2194 \u03c9Sup (Chain.map c (sSup s)) \u2264 c_1\n[PROOFSTEP]\nintro z\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\ns : Set (\u03b1 \u2192o \u03b2)\nhs : \u2200 (f : \u03b1 \u2192o \u03b2), f \u2208 s \u2192 Continuous f\nc : Chain \u03b1\nz : \u03b2\n\u22a2 \u2191(sSup s) (\u03c9Sup c) \u2264 z \u2194 \u03c9Sup (Chain.map c (sSup s)) \u2264 z\n[PROOFSTEP]\nsuffices (\u2200 f \u2208 s, \u2200 (n), (f : _) (c n) \u2264 z) \u2194 \u2200 (n), \u2200 f \u2208 s, (f : _) (c n) \u2264 z by\n simpa (config := { contextual := true }) [\u03c9Sup_le_iff, hs _ _ _] using this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\ns : Set (\u03b1 \u2192o \u03b2)\nhs : \u2200 (f : \u03b1 \u2192o \u03b2), f \u2208 s \u2192 Continuous f\nc : Chain \u03b1\nz : \u03b2\nthis : (\u2200 (f : \u03b1 \u2192o \u03b2), f \u2208 s \u2192 \u2200 (n : \u2115), \u2191f (\u2191c n) \u2264 z) \u2194 \u2200 (n : \u2115) (f : \u03b1 \u2192o \u03b2), f \u2208 s \u2192 \u2191f (\u2191c n) \u2264 z\n\u22a2 \u2191(sSup s) (\u03c9Sup c) \u2264 z \u2194 \u03c9Sup (Chain.map c (sSup s)) \u2264 z\n[PROOFSTEP]\nsimpa (config := { contextual := true }) [\u03c9Sup_le_iff, hs _ _ _] using this\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\ns : Set (\u03b1 \u2192o \u03b2)\nhs : \u2200 (f : \u03b1 \u2192o \u03b2), f \u2208 s \u2192 Continuous f\nc : Chain \u03b1\nz : \u03b2\n\u22a2 (\u2200 (f : \u03b1 \u2192o \u03b2), f \u2208 s \u2192 \u2200 (n : \u2115), \u2191f (\u2191c n) \u2264 z) \u2194 \u2200 (n : \u2115) (f : \u03b1 \u2192o \u03b2), f \u2208 s \u2192 \u2191f (\u2191c n) \u2264 z\n[PROOFSTEP]\nexact \u27e8fun H n f hf => H f hf n, fun H f hf n => H n f hf\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\ns : Set (\u03b1 \u2192 \u03b2)\nhc : \u2200 (f : \u03b1 \u2192 \u03b2), f \u2208 s \u2192 Continuous' f\n\u22a2 Continuous' (sSup s)\n[PROOFSTEP]\nlift s to Set (\u03b1 \u2192o \u03b2) using fun f hf => (hc f hf).to_monotone\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\ns : Set (\u03b1 \u2192o \u03b2)\nhc : \u2200 (f : \u03b1 \u2192 \u03b2), f \u2208 (fun x x_1 => x '' x_1) FunLike.coe s \u2192 Continuous' f\n\u22a2 Continuous' (sSup ((fun x x_1 => x '' x_1) FunLike.coe s))\n[PROOFSTEP]\nsimp only [Set.ball_image_iff, continuous'_coe] at hc \n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\ns : Set (\u03b1 \u2192o \u03b2)\nhc : \u2200 (x : \u03b1 \u2192o \u03b2), x \u2208 s \u2192 Continuous x\n\u22a2 Continuous' (sSup ((fun x x_1 => x '' x_1) FunLike.coe s))\n[PROOFSTEP]\nrw [sSup_image]\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\ns : Set (\u03b1 \u2192o \u03b2)\nhc : \u2200 (x : \u03b1 \u2192o \u03b2), x \u2208 s \u2192 Continuous x\n\u22a2 Continuous' (\u2a06 (a : \u03b1 \u2192o \u03b2) (_ : a \u2208 s), \u2191a)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\ns : Set (\u03b1 \u2192o \u03b2)\nhc : \u2200 (x : \u03b1 \u2192o \u03b2), x \u2208 s \u2192 Continuous x\n\u22a2 Continuous (\u2a06 (i : \u03b1 \u2192o \u03b2) (_ : i \u2208 s), i)\n[PROOFSTEP]\nexact iSup_continuous fun f => iSup_continuous fun hf => hc f hf\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\nf g : \u03b1 \u2192o \u03b2\nhf : Continuous f\nhg : Continuous g\n\u22a2 Continuous (f \u2294 g)\n[PROOFSTEP]\nrw [\u2190 sSup_pair]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\nf g : \u03b1 \u2192o \u03b2\nhf : Continuous f\nhg : Continuous g\n\u22a2 Continuous (sSup {f, g})\n[PROOFSTEP]\napply sSup_continuous\n[GOAL]\ncase hs\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\nf g : \u03b1 \u2192o \u03b2\nhf : Continuous f\nhg : Continuous g\n\u22a2 \u2200 (f_1 : \u03b1 \u2192o \u03b2), f_1 \u2208 {f, g} \u2192 Continuous f_1\n[PROOFSTEP]\nrintro f (rfl | rfl | _)\n[GOAL]\ncase hs.inl\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\ng : \u03b1 \u2192o \u03b2\nhg : Continuous g\nf : \u03b1 \u2192o \u03b2\nhf : Continuous f\n\u22a2 Continuous f\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hs.inr.refl\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\nf g : \u03b1 \u2192o \u03b2\nhf : Continuous f\nhg : Continuous g\n\u22a2 Continuous g\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u22a2 Continuous \u22a4\n[PROOFSTEP]\nintro c\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\nc : Chain \u03b1\n\u22a2 \u2191\u22a4 (\u03c9Sup c) = \u03c9Sup (Chain.map c \u22a4)\n[PROOFSTEP]\napply eq_of_forall_ge_iff\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\nc : Chain \u03b1\n\u22a2 \u2200 (c_1 : \u03b2), \u2191\u22a4 (\u03c9Sup c) \u2264 c_1 \u2194 \u03c9Sup (Chain.map c \u22a4) \u2264 c_1\n[PROOFSTEP]\nintro z\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\nc : Chain \u03b1\nz : \u03b2\n\u22a2 \u2191\u22a4 (\u03c9Sup c) \u2264 z \u2194 \u03c9Sup (Chain.map c \u22a4) \u2264 z\n[PROOFSTEP]\nsimp only [OrderHom.instTopOrderHom_top, OrderHom.const_coe_coe, Function.const, top_le_iff, \u03c9Sup_le_iff, Chain.map_coe,\n Function.comp, forall_const]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u22a2 Continuous \u22a5\n[PROOFSTEP]\nrw [\u2190 sSup_empty]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u22a2 Continuous (sSup \u2205)\n[PROOFSTEP]\nexact sSup_continuous _ fun f hf => hf.elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLinearOrder \u03b2\nf g : \u03b1 \u2192o \u03b2\nhf : Continuous f\nhg : Continuous g\n\u22a2 Continuous (f \u2293 g)\n[PROOFSTEP]\nrefine' fun c => eq_of_forall_ge_iff fun z => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLinearOrder \u03b2\nf g : \u03b1 \u2192o \u03b2\nhf : Continuous f\nhg : Continuous g\nc : Chain \u03b1\nz : \u03b2\n\u22a2 \u2191(f \u2293 g) (\u03c9Sup c) \u2264 z \u2194 \u03c9Sup (Chain.map c (f \u2293 g)) \u2264 z\n[PROOFSTEP]\nsimp only [inf_le_iff, hf c, hg c, \u03c9Sup_le_iff, \u2190 forall_or_left, \u2190 forall_or_right, Chain.map_coe, OrderHom.coe_inf,\n ge_iff_le, Pi.inf_apply, Function.comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLinearOrder \u03b2\nf g : \u03b1 \u2192o \u03b2\nhf : Continuous f\nhg : Continuous g\nc : Chain \u03b1\nz : \u03b2\n\u22a2 (\u2200 (x x_1 : \u2115), \u2191f (\u2191c x_1) \u2264 z \u2228 \u2191g (\u2191c x) \u2264 z) \u2194 \u2200 (i : \u2115), \u2191f (\u2191c i) \u2264 z \u2228 \u2191g (\u2191c i) \u2264 z\n[PROOFSTEP]\nexact\n \u27e8\u03bb h _ => h _ _, \u03bb h i j =>\n (h (max j i)).imp (le_trans $ f.mono $ c.mono $ le_max_left _ _) (le_trans $ g.mono $ c.mono $ le_max_right _ _)\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u22a2 Function.Injective fun f => f.toFun\n[PROOFSTEP]\nrintro \u27e8\u27e9 \u27e8\u27e9 h\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\ntoOrderHom\u271d\u00b9 : \u03b1 \u2192o \u03b2\ncont\u271d\u00b9 : Continuous toOrderHom\u271d\u00b9\ntoOrderHom\u271d : \u03b1 \u2192o \u03b2\ncont\u271d : Continuous toOrderHom\u271d\nh :\n (fun f => f.toFun) { toOrderHom := toOrderHom\u271d\u00b9, cont := cont\u271d\u00b9 } =\n (fun f => f.toFun) { toOrderHom := toOrderHom\u271d, cont := cont\u271d }\n\u22a2 { toOrderHom := toOrderHom\u271d\u00b9, cont := cont\u271d\u00b9 } = { toOrderHom := toOrderHom\u271d, cont := cont\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toOrderHom\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\ntoOrderHom\u271d\u00b9 : \u03b1 \u2192o \u03b2\ncont\u271d\u00b9 : Continuous toOrderHom\u271d\u00b9\ntoOrderHom\u271d : \u03b1 \u2192o \u03b2\ncont\u271d : Continuous toOrderHom\u271d\nh :\n (fun f => f.toFun) { toOrderHom := toOrderHom\u271d\u00b9, cont := cont\u271d\u00b9 } =\n (fun f => f.toFun) { toOrderHom := toOrderHom\u271d, cont := cont\u271d }\n\u22a2 toOrderHom\u271d\u00b9 = toOrderHom\u271d\n[PROOFSTEP]\nexact FunLike.ext' h\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u22a2 Function.Injective fun f => f.toFun\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9\u27e9 \u27e8\u27e8\u27e9\u27e9 h\n[GOAL]\ncase mk.mk.mk.mk\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmonotone'\u271d\u00b9 : Monotone toFun\u271d\u00b9\ncont\u271d\u00b9 : Continuous { toFun := toFun\u271d\u00b9, monotone' := monotone'\u271d\u00b9 }\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmonotone'\u271d : Monotone toFun\u271d\ncont\u271d : Continuous { toFun := toFun\u271d, monotone' := monotone'\u271d }\nh :\n (fun f => f.toFun) { toOrderHom := { toFun := toFun\u271d\u00b9, monotone' := monotone'\u271d\u00b9 }, cont := cont\u271d\u00b9 } =\n (fun f => f.toFun) { toOrderHom := { toFun := toFun\u271d, monotone' := monotone'\u271d }, cont := cont\u271d }\n\u22a2 { toOrderHom := { toFun := toFun\u271d\u00b9, monotone' := monotone'\u271d\u00b9 }, cont := cont\u271d\u00b9 } =\n { toOrderHom := { toFun := toFun\u271d, monotone' := monotone'\u271d }, cont := cont\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\np : Prop\nhp : Decidable p\nf g : \u03b1 \u2192 \u03b2\nhf : Continuous' f\nhg : Continuous' g\n\u22a2 Continuous' fun x => if p then f x else g x\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\np : Prop\nhp : Decidable p\nf g : \u03b1 \u2192 \u03b2\nhf : Continuous' f\nhg : Continuous' g\nh\u271d : p\n\u22a2 Continuous' fun x => f x\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\np : Prop\nhp : Decidable p\nf g : \u03b1 \u2192 \u03b2\nhf : Continuous' f\nhg : Continuous' g\nh\u271d : \u00acp\n\u22a2 Continuous' fun x => g x\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\n\u22a2 \u03c9Sup (Chain.map c (OrderHom.bind f g)) = \u03c9Sup (Chain.map c f) >>= \u03c9Sup (Chain.map c g)\n[PROOFSTEP]\napply eq_of_forall_ge_iff\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\n\u22a2 \u2200 (c_1 : Part \u03b3), \u03c9Sup (Chain.map c (OrderHom.bind f g)) \u2264 c_1 \u2194 \u03c9Sup (Chain.map c f) >>= \u03c9Sup (Chain.map c g) \u2264 c_1\n[PROOFSTEP]\nintro x\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\n\u22a2 \u03c9Sup (Chain.map c (OrderHom.bind f g)) \u2264 x \u2194 \u03c9Sup (Chain.map c f) >>= \u03c9Sup (Chain.map c g) \u2264 x\n[PROOFSTEP]\nsimp only [\u03c9Sup_le_iff, Part.bind_le, Chain.mem_map_iff, and_imp, OrderHom.bind_coe, exists_imp]\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\n\u22a2 (\u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x) \u2194\n \u2200 (a : \u03b2), a \u2208 \u03c9Sup (Chain.map c f) \u2192 \u03c9Sup (Chain.map c g) a \u2264 x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase H.mp\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\n\u22a2 (\u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x) \u2192\n \u2200 (a : \u03b2), a \u2208 \u03c9Sup (Chain.map c f) \u2192 \u03c9Sup (Chain.map c g) a \u2264 x\n[PROOFSTEP]\nintro h'''\n[GOAL]\ncase H.mpr\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\n\u22a2 (\u2200 (a : \u03b2), a \u2208 \u03c9Sup (Chain.map c f) \u2192 \u03c9Sup (Chain.map c g) a \u2264 x) \u2192\n \u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x\n[PROOFSTEP]\nintro h'''\n[GOAL]\ncase H.mp\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x\n\u22a2 \u2200 (a : \u03b2), a \u2208 \u03c9Sup (Chain.map c f) \u2192 \u03c9Sup (Chain.map c g) a \u2264 x\n[PROOFSTEP]\nintro b hb\n[GOAL]\ncase H.mp\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x\nb : \u03b2\nhb : b \u2208 \u03c9Sup (Chain.map c f)\n\u22a2 \u03c9Sup (Chain.map c g) b \u2264 x\n[PROOFSTEP]\napply \u03c9Sup_le _ _ _\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x\nb : \u03b2\nhb : b \u2208 \u03c9Sup (Chain.map c f)\n\u22a2 \u2200 (i : \u2115), \u2191(Chain.map (Chain.map c g) (Pi.evalOrderHom b)) i \u2264 x\n[PROOFSTEP]\nrintro i y hy\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x\nb : \u03b2\nhb : b \u2208 \u03c9Sup (Chain.map c f)\ni : \u2115\ny : \u03b3\nhy : y \u2208 \u2191(Chain.map (Chain.map c g) (Pi.evalOrderHom b)) i\n\u22a2 y \u2208 x\n[PROOFSTEP]\nsimp only [Part.mem_\u03c9Sup] at hb \n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x\nb : \u03b2\ni : \u2115\ny : \u03b3\nhy : y \u2208 \u2191(Chain.map (Chain.map c g) (Pi.evalOrderHom b)) i\nhb : Part.some b \u2208 Chain.map c f\n\u22a2 y \u2208 x\n[PROOFSTEP]\nrcases hb with \u27e8j, hb\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x\nb : \u03b2\ni : \u2115\ny : \u03b3\nhy : y \u2208 \u2191(Chain.map (Chain.map c g) (Pi.evalOrderHom b)) i\nj : \u2115\nhb : Part.some b = \u2191(Chain.map c f) j\n\u22a2 y \u2208 x\n[PROOFSTEP]\nreplace hb := hb.symm\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x\nb : \u03b2\ni : \u2115\ny : \u03b3\nhy : y \u2208 \u2191(Chain.map (Chain.map c g) (Pi.evalOrderHom b)) i\nj : \u2115\nhb : \u2191(Chain.map c f) j = Part.some b\n\u22a2 y \u2208 x\n[PROOFSTEP]\nsimp only [Part.eq_some_iff, Chain.map_coe, Function.comp_apply, OrderHom.apply_coe] at hy hb \n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x\nb : \u03b2\ni : \u2115\ny : \u03b3\nhy : y \u2208 \u2191(Pi.evalOrderHom b) (\u2191g (\u2191c i))\nj : \u2115\nhb : b \u2208 \u2191(Chain.map c f) j\n\u22a2 y \u2208 x\n[PROOFSTEP]\nreplace hb : b \u2208 f (c (max i j)) := f.mono (c.mono (le_max_right i j)) _ hb\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x\nb : \u03b2\ni : \u2115\ny : \u03b3\nhy : y \u2208 \u2191(Pi.evalOrderHom b) (\u2191g (\u2191c i))\nj : \u2115\nhb : b \u2208 \u2191f (\u2191c (max i j))\n\u22a2 y \u2208 x\n[PROOFSTEP]\nreplace hy : y \u2208 g (c (max i j)) b := g.mono (c.mono (le_max_left i j)) _ _ hy\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x\nb : \u03b2\ni : \u2115\ny : \u03b3\nj : \u2115\nhb : b \u2208 \u2191f (\u2191c (max i j))\nhy : y \u2208 \u2191g (\u2191c (max i j)) b\n\u22a2 y \u2208 x\n[PROOFSTEP]\napply h''' (max i j)\n[GOAL]\ncase intro.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x\nb : \u03b2\ni : \u2115\ny : \u03b3\nj : \u2115\nhb : b \u2208 \u2191f (\u2191c (max i j))\nhy : y \u2208 \u2191g (\u2191c (max i j)) b\n\u22a2 y \u2208 \u2191(Chain.map c (OrderHom.bind f g)) (max i j)\n[PROOFSTEP]\nsimp only [exists_prop, Part.bind_eq_bind, Part.mem_bind_iff, Chain.map_coe, Function.comp_apply, OrderHom.bind_coe]\n[GOAL]\ncase intro.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x\nb : \u03b2\ni : \u2115\ny : \u03b3\nj : \u2115\nhb : b \u2208 \u2191f (\u2191c (max i j))\nhy : y \u2208 \u2191g (\u2191c (max i j)) b\n\u22a2 \u2203 a, a \u2208 \u2191f (\u2191c (max i j)) \u2227 y \u2208 \u2191g (\u2191c (max i j)) a\n[PROOFSTEP]\nexact \u27e8_, hb, hy\u27e9\n[GOAL]\ncase H.mpr\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (a : \u03b2), a \u2208 \u03c9Sup (Chain.map c f) \u2192 \u03c9Sup (Chain.map c g) a \u2264 x\n\u22a2 \u2200 (i : \u2115), \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x\n[PROOFSTEP]\nintro i\n[GOAL]\ncase H.mpr\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (a : \u03b2), a \u2208 \u03c9Sup (Chain.map c f) \u2192 \u03c9Sup (Chain.map c g) a \u2264 x\ni : \u2115\n\u22a2 \u2191(Chain.map c (OrderHom.bind f g)) i \u2264 x\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase H.mpr\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (a : \u03b2), a \u2208 \u03c9Sup (Chain.map c f) \u2192 \u03c9Sup (Chain.map c g) a \u2264 x\ni : \u2115\ny : \u03b3\nhy : y \u2208 \u2191(Chain.map c (OrderHom.bind f g)) i\n\u22a2 y \u2208 x\n[PROOFSTEP]\nsimp only [exists_prop, Part.bind_eq_bind, Part.mem_bind_iff, Chain.map_coe, Function.comp_apply, OrderHom.bind_coe] at\n hy \n[GOAL]\ncase H.mpr\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (a : \u03b2), a \u2208 \u03c9Sup (Chain.map c f) \u2192 \u03c9Sup (Chain.map c g) a \u2264 x\ni : \u2115\ny : \u03b3\nhy : \u2203 a, a \u2208 \u2191f (\u2191c i) \u2227 y \u2208 \u2191g (\u2191c i) a\n\u22a2 y \u2208 x\n[PROOFSTEP]\nrcases hy with \u27e8b, hb\u2080, hb\u2081\u27e9\n[GOAL]\ncase H.mpr.intro.intro\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (a : \u03b2), a \u2208 \u03c9Sup (Chain.map c f) \u2192 \u03c9Sup (Chain.map c g) a \u2264 x\ni : \u2115\ny : \u03b3\nb : \u03b2\nhb\u2080 : b \u2208 \u2191f (\u2191c i)\nhb\u2081 : y \u2208 \u2191g (\u2191c i) b\n\u22a2 y \u2208 x\n[PROOFSTEP]\napply h''' b _\n[GOAL]\ncase H.mpr.intro.intro.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (a : \u03b2), a \u2208 \u03c9Sup (Chain.map c f) \u2192 \u03c9Sup (Chain.map c g) a \u2264 x\ni : \u2115\ny : \u03b3\nb : \u03b2\nhb\u2080 : b \u2208 \u2191f (\u2191c i)\nhb\u2081 : y \u2208 \u2191g (\u2191c i) b\n\u22a2 y \u2208 \u03c9Sup (Chain.map c g) b\n[PROOFSTEP]\napply le_\u03c9Sup (c.map g) _ _ _ hb\u2081\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nc : Chain \u03b1\nf : \u03b1 \u2192o Part \u03b2\ng : \u03b1 \u2192o \u03b2 \u2192 Part \u03b3\nx : Part \u03b3\nh''' : \u2200 (a : \u03b2), a \u2208 \u03c9Sup (Chain.map c f) \u2192 \u03c9Sup (Chain.map c g) a \u2264 x\ni : \u2115\ny : \u03b3\nb : \u03b2\nhb\u2080 : b \u2208 \u2191f (\u2191c i)\nhb\u2081 : y \u2208 \u2191g (\u2191c i) b\n\u22a2 b \u2208 \u03c9Sup (Chain.map c f)\n[PROOFSTEP]\napply le_\u03c9Sup (c.map f) i _ hb\u2080\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b1 \u2192 Part \u03b2\ng : \u03b1 \u2192 \u03b2 \u2192 Part \u03b3\nhf : Monotone f\nhf' : Continuous { toFun := f, monotone' := hf }\nhg : Monotone g\nhg' : Continuous { toFun := g, monotone' := hg }\n\u22a2 Continuous (OrderHom.bind { toFun := f, monotone' := hf } { toFun := g, monotone' := hg })\n[PROOFSTEP]\nintro c\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b1 \u2192 Part \u03b2\ng : \u03b1 \u2192 \u03b2 \u2192 Part \u03b3\nhf : Monotone f\nhf' : Continuous { toFun := f, monotone' := hf }\nhg : Monotone g\nhg' : Continuous { toFun := g, monotone' := hg }\nc : Chain \u03b1\n\u22a2 \u2191(OrderHom.bind { toFun := f, monotone' := hf } { toFun := g, monotone' := hg }) (\u03c9Sup c) =\n \u03c9Sup (Chain.map c (OrderHom.bind { toFun := f, monotone' := hf } { toFun := g, monotone' := hg }))\n[PROOFSTEP]\nrw [\u03c9Sup_bind, \u2190 hf', \u2190 hg']\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b1 \u2192 Part \u03b2\ng : \u03b1 \u2192 \u03b2 \u2192 Part \u03b3\nhf : Monotone f\nhf' : Continuous { toFun := f, monotone' := hf }\nhg : Monotone g\nhg' : Continuous { toFun := g, monotone' := hg }\nc : Chain \u03b1\n\u22a2 \u2191(OrderHom.bind { toFun := f, monotone' := hf } { toFun := g, monotone' := hg }) (\u03c9Sup c) =\n \u2191{ toFun := f, monotone' := hf } (\u03c9Sup c) >>= \u2191{ toFun := g, monotone' := hg } (\u03c9Sup c)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 Part \u03b2\nhg : Continuous' g\n\u22a2 Continuous' fun x => f <$> g x\n[PROOFSTEP]\nsimp only [map_eq_bind_pure_comp]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 Part \u03b2\nhg : Continuous' g\n\u22a2 Continuous' fun x => g x >>= pure \u2218 f\n[PROOFSTEP]\napply bind_continuous' _ _ hg\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 Part \u03b2\nhg : Continuous' g\n\u22a2 Continuous' fun x => pure \u2218 f\n[PROOFSTEP]\napply const_continuous'\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b1 \u2192 Part (\u03b2 \u2192 \u03b3)\ng : \u03b1 \u2192 Part \u03b2\nhf : Continuous' f\nhg : Continuous' g\n\u22a2 Continuous' fun x => Seq.seq (f x) fun x_1 => g x\n[PROOFSTEP]\nsimp only [seq_eq_bind_map]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b1 \u2192 Part (\u03b2 \u2192 \u03b3)\ng : \u03b1 \u2192 Part \u03b2\nhf : Continuous' f\nhg : Continuous' g\n\u22a2 Continuous' fun x => do\n let x_1 \u2190 f x\n x_1 <$> g x\n[PROOFSTEP]\napply bind_continuous' _ _ hf\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b1 \u2192 Part (\u03b2 \u2192 \u03b3)\ng : \u03b1 \u2192 Part \u03b2\nhf : Continuous' f\nhg : Continuous' g\n\u22a2 Continuous' fun x x_1 => x_1 <$> g x\n[PROOFSTEP]\napply Pi.OmegaCompletePartialOrder.flip\u2082_continuous'\n[GOAL]\ncase hf\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b1 \u2192 Part (\u03b2 \u2192 \u03b3)\ng : \u03b1 \u2192 Part \u03b2\nhf : Continuous' f\nhg : Continuous' g\n\u22a2 \u2200 (x : \u03b2 \u2192 \u03b3), Continuous' fun g_1 => x <$> g g_1\n[PROOFSTEP]\nintro\n[GOAL]\ncase hf\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b1 \u2192 Part (\u03b2 \u2192 \u03b3)\ng : \u03b1 \u2192 Part \u03b2\nhf : Continuous' f\nhg : Continuous' g\nx\u271d : \u03b2 \u2192 \u03b3\n\u22a2 Continuous' fun g_1 => x\u271d <$> g g_1\n[PROOFSTEP]\napply map_continuous' _ _ hg\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192\ud835\udc84 \u03b2\nh : f = \u2191g\n\u22a2 Continuous (OrderHom.copy g.toOrderHom f h)\n[PROOFSTEP]\nrw [OrderHom.copy_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192\ud835\udc84 \u03b2\nh : f = \u2191g\n\u22a2 Continuous g.toOrderHom\n[PROOFSTEP]\nexact g.cont\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc\u2080 : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc\u2081 : Chain \u03b1\nz : \u03b2\n\u22a2 (\u2200 (i j : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 j) \u2264 z) \u2194 \u2200 (i : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 i) \u2264 z\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc\u2080 : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc\u2081 : Chain \u03b1\nz : \u03b2\n\u22a2 (\u2200 (i j : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 j) \u2264 z) \u2192 \u2200 (i : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 i) \u2264 z\n[PROOFSTEP]\nintrov h\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc\u2080 : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc\u2081 : Chain \u03b1\nz : \u03b2\n\u22a2 (\u2200 (i : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 i) \u2264 z) \u2192 \u2200 (i j : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 j) \u2264 z\n[PROOFSTEP]\nintrov h\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc\u2080 : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc\u2081 : Chain \u03b1\nz : \u03b2\nh : \u2200 (i j : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 j) \u2264 z\ni : \u2115\n\u22a2 \u2191(\u2191c\u2080 i) (\u2191c\u2081 i) \u2264 z\n[PROOFSTEP]\napply h\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc\u2080 : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc\u2081 : Chain \u03b1\nz : \u03b2\nh : \u2200 (i : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 i) \u2264 z\ni j : \u2115\n\u22a2 \u2191(\u2191c\u2080 i) (\u2191c\u2081 j) \u2264 z\n[PROOFSTEP]\napply le_trans _ (h (max i j))\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc\u2080 : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc\u2081 : Chain \u03b1\nz : \u03b2\nh : \u2200 (i : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 i) \u2264 z\ni j : \u2115\n\u22a2 \u2191(\u2191c\u2080 i) (\u2191c\u2081 j) \u2264 \u2191(\u2191c\u2080 (max i j)) (\u2191c\u2081 (max i j))\n[PROOFSTEP]\ntrans c\u2080 i (c\u2081 (max i j))\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc\u2080 : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc\u2081 : Chain \u03b1\nz : \u03b2\nh : \u2200 (i : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 i) \u2264 z\ni j : \u2115\n\u22a2 \u2191(\u2191c\u2080 i) (\u2191c\u2081 j) \u2264 \u2191(\u2191c\u2080 i) (\u2191c\u2081 (max i j))\n[PROOFSTEP]\napply (c\u2080 i).monotone\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc\u2080 : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc\u2081 : Chain \u03b1\nz : \u03b2\nh : \u2200 (i : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 i) \u2264 z\ni j : \u2115\n\u22a2 \u2191c\u2081 j \u2264 \u2191c\u2081 (max i j)\n[PROOFSTEP]\napply c\u2081.monotone\n[GOAL]\ncase a.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc\u2080 : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc\u2081 : Chain \u03b1\nz : \u03b2\nh : \u2200 (i : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 i) \u2264 z\ni j : \u2115\n\u22a2 j \u2264 max i j\n[PROOFSTEP]\napply le_max_right\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc\u2080 : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc\u2081 : Chain \u03b1\nz : \u03b2\nh : \u2200 (i : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 i) \u2264 z\ni j : \u2115\n\u22a2 \u2191(\u2191c\u2080 i) (\u2191c\u2081 (max i j)) \u2264 \u2191(\u2191c\u2080 (max i j)) (\u2191c\u2081 (max i j))\n[PROOFSTEP]\napply c\u2080.monotone\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc\u2080 : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc\u2081 : Chain \u03b1\nz : \u03b2\nh : \u2200 (i : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 i) \u2264 z\ni j : \u2115\n\u22a2 i \u2264 max i j\n[PROOFSTEP]\napply le_max_left\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc\u2080 : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc\u2081 : Chain \u03b1\nz : \u03b2\n\u22a2 (\u2200 (j i : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 j) \u2264 z) \u2194 \u2200 (i : \u2115), \u2191(\u2191c\u2080 i) (\u2191c\u2081 i) \u2264 z\n[PROOFSTEP]\nrw [forall_swap, forall_forall_merge]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc' : Chain \u03b1\n\u22a2 \u2191(\u03c9Sup (Chain.map c toMono)) (\u03c9Sup c') = \u03c9Sup (Chain.map c' (\u03c9Sup (Chain.map c toMono)))\n[PROOFSTEP]\napply eq_of_forall_ge_iff\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc' : Chain \u03b1\n\u22a2 \u2200 (c_1 : \u03b2), \u2191(\u03c9Sup (Chain.map c toMono)) (\u03c9Sup c') \u2264 c_1 \u2194 \u03c9Sup (Chain.map c' (\u03c9Sup (Chain.map c toMono))) \u2264 c_1\n[PROOFSTEP]\nintro z\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc' : Chain \u03b1\nz : \u03b2\n\u22a2 \u2191(\u03c9Sup (Chain.map c toMono)) (\u03c9Sup c') \u2264 z \u2194 \u03c9Sup (Chain.map c' (\u03c9Sup (Chain.map c toMono))) \u2264 z\n[PROOFSTEP]\nsimp only [\u03c9Sup_le_iff, (c _).continuous, Chain.map_coe, OrderHom.apply_coe, toMono_coe,\n OrderHom.omegaCompletePartialOrder_\u03c9Sup_coe, forall_forall_merge, OrderHomClass.coe_coe, forall_forall_merge',\n (\u00b7 \u2218 \u00b7), Function.eval]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nx y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1\nh : x \u2264 y\n\u22a2 (fun f => \u2191f.fst f.snd) x \u2264 (fun f => \u2191f.fst f.snd) y\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nx y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1\nh : x \u2264 y\n\u22a2 \u2191x.fst x.snd \u2264 \u2191y.fst y.snd\n[PROOFSTEP]\ntrans y.fst x.snd <;> [apply h.1; apply y.1.monotone h.2]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nx y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1\nh : x \u2264 y\n\u22a2 \u2191x.fst x.snd \u2264 \u2191y.fst y.snd\n[PROOFSTEP]\ntrans y.fst x.snd\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nx y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1\nh : x \u2264 y\n\u22a2 \u2191x.fst x.snd \u2264 \u2191y.fst x.snd\n[PROOFSTEP]\napply h.1\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nx y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1\nh : x \u2264 y\n\u22a2 \u2191y.fst x.snd \u2264 \u2191y.fst y.snd\n[PROOFSTEP]\napply y.1.monotone h.2\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u22a2 Continuous\n { toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 (fun f => \u2191f.fst f.snd) x \u2264 (fun f => \u2191f.fst f.snd) y) }\n[PROOFSTEP]\nintro c\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\n\u22a2 \u2191{ toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 (fun f => \u2191f.fst f.snd) x \u2264 (fun f => \u2191f.fst f.snd) y) }\n (\u03c9Sup c) =\n \u03c9Sup\n (Chain.map c\n { toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 (fun f => \u2191f.fst f.snd) x \u2264 (fun f => \u2191f.fst f.snd) y) })\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\n\u22a2 \u2191{ toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 (fun f => \u2191f.fst f.snd) x \u2264 (fun f => \u2191f.fst f.snd) y) }\n (\u03c9Sup c) \u2264\n \u03c9Sup\n (Chain.map c\n { toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 (fun f => \u2191f.fst f.snd) x \u2264 (fun f => \u2191f.fst f.snd) y) })\n[PROOFSTEP]\napply \u03c9Sup_le\n[GOAL]\ncase a.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\n\u22a2 \u2200 (i : \u2115),\n \u2191(Chain.map (Chain.map (Chain.map c OrderHom.fst) toMono) (OrderHom.apply (\u03c9Sup c).snd)) i \u2264\n \u03c9Sup\n (Chain.map c\n { toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 (fun f => \u2191f.fst f.snd) x \u2264 (fun f => \u2191f.fst f.snd) y) })\n[PROOFSTEP]\nintro i\n[GOAL]\ncase a.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\ni : \u2115\n\u22a2 \u2191(Chain.map (Chain.map (Chain.map c OrderHom.fst) toMono) (OrderHom.apply (\u03c9Sup c).snd)) i \u2264\n \u03c9Sup\n (Chain.map c\n { toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 (fun f => \u2191f.fst f.snd) x \u2264 (fun f => \u2191f.fst f.snd) y) })\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\ni : \u2115\n\u22a2 \u2191(\u2191c i).fst (\u03c9Sup (Chain.map c OrderHom.snd)) \u2264\n \u03c9Sup\n (Chain.map c\n { toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 \u2191x.fst x.snd \u2264 \u2191y.fst y.snd) })\n[PROOFSTEP]\nrw [(c _).fst.continuous]\n[GOAL]\ncase a.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\ni : \u2115\n\u22a2 \u03c9Sup (Chain.map (Chain.map c OrderHom.snd) \u2191(\u2191c i).fst) \u2264\n \u03c9Sup\n (Chain.map c\n { toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 \u2191x.fst x.snd \u2264 \u2191y.fst y.snd) })\n[PROOFSTEP]\napply \u03c9Sup_le\n[GOAL]\ncase a.a.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\ni : \u2115\n\u22a2 \u2200 (i_1 : \u2115),\n \u2191(Chain.map (Chain.map c OrderHom.snd) \u2191(\u2191c i).fst) i_1 \u2264\n \u03c9Sup\n (Chain.map c\n { toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 \u2191x.fst x.snd \u2264 \u2191y.fst y.snd) })\n[PROOFSTEP]\nintro j\n[GOAL]\ncase a.a.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\ni j : \u2115\n\u22a2 \u2191(Chain.map (Chain.map c OrderHom.snd) \u2191(\u2191c i).fst) j \u2264\n \u03c9Sup\n (Chain.map c\n { toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 \u2191x.fst x.snd \u2264 \u2191y.fst y.snd) })\n[PROOFSTEP]\napply le_\u03c9Sup_of_le (max i j)\n[GOAL]\ncase a.a.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\ni j : \u2115\n\u22a2 \u2191(Chain.map (Chain.map c OrderHom.snd) \u2191(\u2191c i).fst) j \u2264\n \u2191(Chain.map c\n { toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 \u2191x.fst x.snd \u2264 \u2191y.fst y.snd) })\n (max i j)\n[PROOFSTEP]\napply apply_mono\n[GOAL]\ncase a.a.a.h\u2081\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\ni j : \u2115\n\u22a2 (\u2191c i).fst \u2264 (\u2191c (max i j)).fst\ncase a.a.a.h\u2082\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\ni j : \u2115\n\u22a2 \u2191(Chain.map c OrderHom.snd) j \u2264 (\u2191c (max i j)).snd\n[PROOFSTEP]\nexact monotone_fst (OrderHom.mono _ (le_max_left _ _))\n[GOAL]\ncase a.a.a.h\u2082\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\ni j : \u2115\n\u22a2 \u2191(Chain.map c OrderHom.snd) j \u2264 (\u2191c (max i j)).snd\n[PROOFSTEP]\nexact monotone_snd (OrderHom.mono _ (le_max_right _ _))\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\n\u22a2 \u03c9Sup\n (Chain.map c\n { toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 (fun f => \u2191f.fst f.snd) x \u2264 (fun f => \u2191f.fst f.snd) y) }) \u2264\n \u2191{ toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 (fun f => \u2191f.fst f.snd) x \u2264 (fun f => \u2191f.fst f.snd) y) }\n (\u03c9Sup c)\n[PROOFSTEP]\napply \u03c9Sup_le\n[GOAL]\ncase a.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\n\u22a2 \u2200 (i : \u2115),\n \u2191(Chain.map c\n { toFun := fun f => \u2191f.fst f.snd,\n monotone' :=\n (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 (fun f => \u2191f.fst f.snd) x \u2264 (fun f => \u2191f.fst f.snd) y) })\n i \u2264\n \u2191{ toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 (fun f => \u2191f.fst f.snd) x \u2264 (fun f => \u2191f.fst f.snd) y) }\n (\u03c9Sup c)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase a.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\ni : \u2115\n\u22a2 \u2191(Chain.map c\n { toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 (fun f => \u2191f.fst f.snd) x \u2264 (fun f => \u2191f.fst f.snd) y) })\n i \u2264\n \u2191{ toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 (fun f => \u2191f.fst f.snd) x \u2264 (fun f => \u2191f.fst f.snd) y) }\n (\u03c9Sup c)\n[PROOFSTEP]\napply le_\u03c9Sup_of_le i\n[GOAL]\ncase a.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\ni : \u2115\n\u22a2 \u2191(Chain.map c\n { toFun := fun f => \u2191f.fst f.snd,\n monotone' := (_ : \u2200 (x y : (\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1), x \u2264 y \u2192 (fun f => \u2191f.fst f.snd) x \u2264 (fun f => \u2191f.fst f.snd) y) })\n i \u2264\n \u2191(Chain.map (Chain.map (Chain.map c OrderHom.fst) toMono) (OrderHom.apply (\u03c9Sup c).snd)) i\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\ni : \u2115\n\u22a2 \u2191(\u2191c i).fst (\u2191c i).snd \u2264 \u2191(\u2191c i).fst (\u03c9Sup (Chain.map c OrderHom.snd))\n[PROOFSTEP]\napply OrderHom.mono _\n[GOAL]\ncase a.a.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\ni : \u2115\n\u22a2 (\u2191c i).snd \u2264 \u03c9Sup (Chain.map c OrderHom.snd)\n[PROOFSTEP]\napply le_\u03c9Sup_of_le i\n[GOAL]\ncase a.a.a\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc : Chain ((\u03b1 \u2192\ud835\udc84 \u03b2) \u00d7 \u03b1)\ni : \u2115\n\u22a2 (\u2191c i).snd \u2264 \u2191(Chain.map c OrderHom.snd) i\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\nc\u2080 : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)\nc\u2081 : Chain \u03b1\n\u22a2 \u2191(\u03c9Sup c\u2080) (\u03c9Sup c\u2081) = \u2191Prod.apply (\u03c9Sup (Chain.zip c\u2080 c\u2081))\n[PROOFSTEP]\nsimp [Prod.apply_apply, Prod.\u03c9Sup_zip]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\u271d\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b1 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192\ud835\udc84 \u03b3\n\u22a2 Continuous { toFun := fun x y => \u2191(f y) x, monotone' := (_ : \u2200 (x y : \u03b2), x \u2264 y \u2192 \u2200 (a : \u03b1), \u2191(f a) x \u2264 \u2191(f a) y) }\n[PROOFSTEP]\nintro _\n[GOAL]\n\u03b1\u271d : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\u271d\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b1 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192\ud835\udc84 \u03b3\nc\u271d : Chain \u03b2\n\u22a2 \u2191{ toFun := fun x y => \u2191(f y) x, monotone' := (_ : \u2200 (x y : \u03b2), x \u2264 y \u2192 \u2200 (a : \u03b1), \u2191(f a) x \u2264 \u2191(f a) y) } (\u03c9Sup c\u271d) =\n \u03c9Sup\n (Chain.map c\u271d\n { toFun := fun x y => \u2191(f y) x, monotone' := (_ : \u2200 (x y : \u03b2), x \u2264 y \u2192 \u2200 (a : \u03b1), \u2191(f a) x \u2264 \u2191(f a) y) })\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1\u271d : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\u271d\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b1 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192\ud835\udc84 \u03b3\nc\u271d : Chain \u03b2\nx : \u03b1\n\u22a2 \u2191{ toFun := fun x y => \u2191(f y) x, monotone' := (_ : \u2200 (x y : \u03b2), x \u2264 y \u2192 \u2200 (a : \u03b1), \u2191(f a) x \u2264 \u2191(f a) y) } (\u03c9Sup c\u271d)\n x =\n \u03c9Sup\n (Chain.map c\u271d\n { toFun := fun x y => \u2191(f y) x, monotone' := (_ : \u2200 (x y : \u03b2), x \u2264 y \u2192 \u2200 (a : \u03b1), \u2191(f a) x \u2264 \u2191(f a) y) })\n x\n[PROOFSTEP]\nchange f _ _ = _\n[GOAL]\ncase h\n\u03b1\u271d : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\u271d\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b1 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192\ud835\udc84 \u03b3\nc\u271d : Chain \u03b2\nx : \u03b1\n\u22a2 \u2191(f x) (\u03c9Sup c\u271d) =\n \u03c9Sup\n (Chain.map c\u271d\n { toFun := fun x y => \u2191(f y) x, monotone' := (_ : \u2200 (x y : \u03b2), x \u2264 y \u2192 \u2200 (a : \u03b1), \u2191(f a) x \u2264 \u2191(f a) y) })\n x\n[PROOFSTEP]\nrw [(f _).continuous]\n[GOAL]\ncase h\n\u03b1\u271d : Type u\n\u03b1' : Type u_1\n\u03b2 : Type v\n\u03b2' : Type u_2\n\u03b3 : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\u271d\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b1 : Type u_5\nf : \u03b1 \u2192 \u03b2 \u2192\ud835\udc84 \u03b3\nc\u271d : Chain \u03b2\nx : \u03b1\n\u22a2 \u03c9Sup (Chain.map c\u271d \u2191(f x)) =\n \u03c9Sup\n (Chain.map c\u271d\n { toFun := fun x y => \u2191(f y) x, monotone' := (_ : \u2200 (x y : \u03b2), x \u2264 y \u2192 \u2200 (a : \u03b1), \u2191(f a) x \u2264 \u2191(f a) y) })\n x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b1 \u2192\ud835\udc84 Part \u03b2\ng : \u03b1 \u2192\ud835\udc84 \u03b2 \u2192 Part \u03b3\nc : Chain \u03b1\n\u22a2 \u2191(OrderHom.bind (\u2191f) g.toOrderHom) (\u03c9Sup c) = \u03c9Sup (Chain.map c (OrderHom.bind (\u2191f) g.toOrderHom))\n[PROOFSTEP]\nrw [\u03c9Sup_bind, \u2190 f.continuous, g.toOrderHom_eq_coe, \u2190 g.continuous]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b1 \u2192\ud835\udc84 Part \u03b2\ng : \u03b1 \u2192\ud835\udc84 \u03b2 \u2192 Part \u03b3\nc : Chain \u03b1\n\u22a2 \u2191(OrderHom.bind \u2191f \u2191g) (\u03c9Sup c) = \u2191f (\u03c9Sup c) >>= \u2191g (\u03c9Sup c)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192\ud835\udc84 Part \u03b2\n\u22a2 (fun x => f <$> \u2191g x) = \u2191(bind g (const (pure \u2218 f)))\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192\ud835\udc84 Part \u03b2\nx\u271d : \u03b1\n\u22a2 f <$> \u2191g x\u271d = \u2191(bind g (const (pure \u2218 f))) x\u271d\n[PROOFSTEP]\nsimp only [map_eq_bind_pure_comp, bind, coe_mk, OrderHom.bind_coe, coe_apply, coe_toOrderHom, const_apply]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b1 \u2192\ud835\udc84 Part (\u03b2 \u2192 \u03b3)\ng : \u03b1 \u2192\ud835\udc84 Part \u03b2\n\u22a2 (fun x => Seq.seq (\u2191f x) fun x_1 => \u2191g x) = \u2191(bind f (flip (_root_.flip map g)))\n[PROOFSTEP]\next\n[GOAL]\ncase h.H\n\u03b1 : Type u\n\u03b1' : Type u_1\n\u03b2\u271d : Type v\n\u03b2' : Type u_2\n\u03b3\u271d : Type u_3\n\u03c6 : Type u_4\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b1 \u2192\ud835\udc84 Part (\u03b2 \u2192 \u03b3)\ng : \u03b1 \u2192\ud835\udc84 Part \u03b2\nx\u271d : \u03b1\na\u271d : \u03b3\n\u22a2 (a\u271d \u2208 Seq.seq (\u2191f x\u271d) fun x => \u2191g x\u271d) \u2194 a\u271d \u2208 \u2191(bind f (flip (_root_.flip map g))) x\u271d\n[PROOFSTEP]\nsimp only [seq_eq_bind_map, Part.bind_eq_bind, Part.mem_bind_iff, flip_apply, _root_.flip, map_apply, bind_apply]\n", "meta": {"mathlib_filename": "Mathlib.Order.OmegaCompletePartialOrder", "llama_tokens": 41193, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872243177518, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4297351658896287}} {"text": "[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\n\u22a2 sInf (a \u2022 s) = a \u2022 sInf s\n[PROOFSTEP]\nobtain rfl | hs := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\n\u22a2 sInf (a \u2022 \u2205) = a \u2022 sInf \u2205\n[PROOFSTEP]\nrw [smul_set_empty, Real.sInf_empty, smul_zero]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : Set.Nonempty s\n\u22a2 sInf (a \u2022 s) = a \u2022 sInf s\n[PROOFSTEP]\nobtain rfl | ha' := ha.eq_or_lt\n[GOAL]\ncase inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : Set.Nonempty s\nha : 0 \u2264 0\n\u22a2 sInf (0 \u2022 s) = 0 \u2022 sInf s\n[PROOFSTEP]\nrw [zero_smul_set hs, zero_smul]\n[GOAL]\ncase inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : Set.Nonempty s\nha : 0 \u2264 0\n\u22a2 sInf 0 = 0\n[PROOFSTEP]\nexact csInf_singleton 0\n[GOAL]\ncase inr.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : Set.Nonempty s\nha' : 0 < a\n\u22a2 sInf (a \u2022 s) = a \u2022 sInf s\n[PROOFSTEP]\nby_cases h : BddBelow s\n[GOAL]\ncase pos\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : Set.Nonempty s\nha' : 0 < a\nh : BddBelow s\n\u22a2 sInf (a \u2022 s) = a \u2022 sInf s\n[PROOFSTEP]\nexact ((OrderIso.smulLeft \u211d ha').map_csInf' hs h).symm\n[GOAL]\ncase neg\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : Set.Nonempty s\nha' : 0 < a\nh : \u00acBddBelow s\n\u22a2 sInf (a \u2022 s) = a \u2022 sInf s\n[PROOFSTEP]\nrw [Real.sInf_of_not_bddBelow (mt (bddBelow_smul_iff_of_pos ha').1 h), Real.sInf_of_not_bddBelow h, smul_zero]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\n\u22a2 sSup (a \u2022 s) = a \u2022 sSup s\n[PROOFSTEP]\nobtain rfl | hs := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\n\u22a2 sSup (a \u2022 \u2205) = a \u2022 sSup \u2205\n[PROOFSTEP]\nrw [smul_set_empty, Real.sSup_empty, smul_zero]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : Set.Nonempty s\n\u22a2 sSup (a \u2022 s) = a \u2022 sSup s\n[PROOFSTEP]\nobtain rfl | ha' := ha.eq_or_lt\n[GOAL]\ncase inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : Set.Nonempty s\nha : 0 \u2264 0\n\u22a2 sSup (0 \u2022 s) = 0 \u2022 sSup s\n[PROOFSTEP]\nrw [zero_smul_set hs, zero_smul]\n[GOAL]\ncase inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : Set.Nonempty s\nha : 0 \u2264 0\n\u22a2 sSup 0 = 0\n[PROOFSTEP]\nexact csSup_singleton 0\n[GOAL]\ncase inr.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : Set.Nonempty s\nha' : 0 < a\n\u22a2 sSup (a \u2022 s) = a \u2022 sSup s\n[PROOFSTEP]\nby_cases h : BddAbove s\n[GOAL]\ncase pos\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : Set.Nonempty s\nha' : 0 < a\nh : BddAbove s\n\u22a2 sSup (a \u2022 s) = a \u2022 sSup s\n[PROOFSTEP]\nexact ((OrderIso.smulLeft \u211d ha').map_csSup' hs h).symm\n[GOAL]\ncase neg\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : Set.Nonempty s\nha' : 0 < a\nh : \u00acBddAbove s\n\u22a2 sSup (a \u2022 s) = a \u2022 sSup s\n[PROOFSTEP]\nrw [Real.sSup_of_not_bddAbove (mt (bddAbove_smul_iff_of_pos ha').1 h), Real.sSup_of_not_bddAbove h, smul_zero]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\n\u22a2 sInf (a \u2022 s) = a \u2022 sSup s\n[PROOFSTEP]\nobtain rfl | hs := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\n\u22a2 sInf (a \u2022 \u2205) = a \u2022 sSup \u2205\n[PROOFSTEP]\nrw [smul_set_empty, Real.sInf_empty, Real.sSup_empty, smul_zero]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : Set.Nonempty s\n\u22a2 sInf (a \u2022 s) = a \u2022 sSup s\n[PROOFSTEP]\nobtain rfl | ha' := ha.eq_or_lt\n[GOAL]\ncase inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : Set.Nonempty s\nha : 0 \u2264 0\n\u22a2 sInf (0 \u2022 s) = 0 \u2022 sSup s\n[PROOFSTEP]\nrw [zero_smul_set hs, zero_smul]\n[GOAL]\ncase inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : Set.Nonempty s\nha : 0 \u2264 0\n\u22a2 sInf 0 = 0\n[PROOFSTEP]\nexact csInf_singleton 0\n[GOAL]\ncase inr.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : Set.Nonempty s\nha' : a < 0\n\u22a2 sInf (a \u2022 s) = a \u2022 sSup s\n[PROOFSTEP]\nby_cases h : BddAbove s\n[GOAL]\ncase pos\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : Set.Nonempty s\nha' : a < 0\nh : BddAbove s\n\u22a2 sInf (a \u2022 s) = a \u2022 sSup s\n[PROOFSTEP]\nexact ((OrderIso.smulLeftDual \u211d ha').map_csSup' hs h).symm\n[GOAL]\ncase neg\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : Set.Nonempty s\nha' : a < 0\nh : \u00acBddAbove s\n\u22a2 sInf (a \u2022 s) = a \u2022 sSup s\n[PROOFSTEP]\nrw [Real.sInf_of_not_bddBelow (mt (bddBelow_smul_iff_of_neg ha').1 h), Real.sSup_of_not_bddAbove h, smul_zero]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\n\u22a2 sSup (a \u2022 s) = a \u2022 sInf s\n[PROOFSTEP]\nobtain rfl | hs := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\n\u22a2 sSup (a \u2022 \u2205) = a \u2022 sInf \u2205\n[PROOFSTEP]\nrw [smul_set_empty, Real.sSup_empty, Real.sInf_empty, smul_zero]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : Set.Nonempty s\n\u22a2 sSup (a \u2022 s) = a \u2022 sInf s\n[PROOFSTEP]\nobtain rfl | ha' := ha.eq_or_lt\n[GOAL]\ncase inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : Set.Nonempty s\nha : 0 \u2264 0\n\u22a2 sSup (0 \u2022 s) = 0 \u2022 sInf s\n[PROOFSTEP]\nrw [zero_smul_set hs, zero_smul]\n[GOAL]\ncase inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : Set.Nonempty s\nha : 0 \u2264 0\n\u22a2 sSup 0 = 0\n[PROOFSTEP]\nexact csSup_singleton 0\n[GOAL]\ncase inr.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : Set.Nonempty s\nha' : a < 0\n\u22a2 sSup (a \u2022 s) = a \u2022 sInf s\n[PROOFSTEP]\nby_cases h : BddBelow s\n[GOAL]\ncase pos\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : Set.Nonempty s\nha' : a < 0\nh : BddBelow s\n\u22a2 sSup (a \u2022 s) = a \u2022 sInf s\n[PROOFSTEP]\nexact ((OrderIso.smulLeftDual \u211d ha').map_csInf' hs h).symm\n[GOAL]\ncase neg\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : Set.Nonempty s\nha' : a < 0\nh : \u00acBddBelow s\n\u22a2 sSup (a \u2022 s) = a \u2022 sInf s\n[PROOFSTEP]\nrw [Real.sSup_of_not_bddAbove (mt (bddAbove_smul_iff_of_neg ha').1 h), Real.sInf_of_not_bddBelow h, smul_zero]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d : LinearOrderedField \u03b1\nr : \u211d\nha : 0 \u2264 r\nf : \u03b9 \u2192 \u211d\n\u22a2 (\u2a05 (i : \u03b9), f i) * r = \u2a05 (i : \u03b9), f i * r\n[PROOFSTEP]\nsimp only [Real.mul_iInf_of_nonneg ha, mul_comm]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d : LinearOrderedField \u03b1\nr : \u211d\nha : 0 \u2264 r\nf : \u03b9 \u2192 \u211d\n\u22a2 (\u2a06 (i : \u03b9), f i) * r = \u2a06 (i : \u03b9), f i * r\n[PROOFSTEP]\nsimp only [Real.mul_iSup_of_nonneg ha, mul_comm]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d : LinearOrderedField \u03b1\nr : \u211d\nha : r \u2264 0\nf : \u03b9 \u2192 \u211d\n\u22a2 (\u2a05 (i : \u03b9), f i) * r = \u2a06 (i : \u03b9), f i * r\n[PROOFSTEP]\nsimp only [Real.mul_iInf_of_nonpos ha, mul_comm]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d : LinearOrderedField \u03b1\nr : \u211d\nha : r \u2264 0\nf : \u03b9 \u2192 \u211d\n\u22a2 (\u2a06 (i : \u03b9), f i) * r = \u2a05 (i : \u03b9), f i * r\n[PROOFSTEP]\nsimp only [Real.mul_iSup_of_nonpos ha, mul_comm]\n", "meta": {"mathlib_filename": "Mathlib.Data.Real.Pointwise", "llama_tokens": 4747, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147275, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4297351594626956}} {"text": "[GOAL]\n\u22a2 ConcreteCategory BoolRingCat\n[PROOFSTEP]\ndsimp [BoolRingCat]\n[GOAL]\n\u22a2 ConcreteCategory (Bundled BooleanRing)\n[PROOFSTEP]\ninfer_instance\n -- Porting note: disabled `simps`\n -- Invalid simp lemma BoolRingCat.hasForgetToCommRing_forget\u2082_obj_str_add.\n -- The given definition is not a constructor application:\n -- inferInstance.1\n -- @[simps]\n[GOAL]\n\u03b1 \u03b2 : BoolRingCat\ne : \u2191\u03b1 \u2243+* \u2191\u03b2\n\u22a2 \u2191e \u226b \u2191(RingEquiv.symm e) = \ud835\udfd9 \u03b1\n[PROOFSTEP]\next\n[GOAL]\ncase w\n\u03b1 \u03b2 : BoolRingCat\ne : \u2191\u03b1 \u2243+* \u2191\u03b2\nx\u271d : (forget BoolRingCat).obj \u03b1\n\u22a2 \u2191(\u2191e \u226b \u2191(RingEquiv.symm e)) x\u271d = \u2191(\ud835\udfd9 \u03b1) x\u271d\n[PROOFSTEP]\nexact e.symm_apply_apply _\n[GOAL]\n\u03b1 \u03b2 : BoolRingCat\ne : \u2191\u03b1 \u2243+* \u2191\u03b2\n\u22a2 \u2191(RingEquiv.symm e) \u226b \u2191e = \ud835\udfd9 \u03b2\n[PROOFSTEP]\next\n[GOAL]\ncase w\n\u03b1 \u03b2 : BoolRingCat\ne : \u2191\u03b1 \u2243+* \u2191\u03b2\nx\u271d : (forget BoolRingCat).obj \u03b2\n\u22a2 \u2191(\u2191(RingEquiv.symm e) \u226b \u2191e) x\u271d = \u2191(\ud835\udfd9 \u03b2) x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.BoolRingCat", "llama_tokens": 467, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702761768248, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.4296064019894863}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b9 r\nx y : \u03a0\u2080 (i : \u03b9), \u03b1 i\nhlt : x < y\n\u22a2 \u2203 i, (\u2200 (j : \u03b9), r j i \u2192 \u2191x j \u2264 \u2191y j \u2227 \u2191y j \u2264 \u2191x j) \u2227 \u2191x i < \u2191y i\n[PROOFSTEP]\nobtain \u27e8hle, j, hlt\u27e9 := Pi.lt_def.1 hlt\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b9 r\nx y : \u03a0\u2080 (i : \u03b9), \u03b1 i\nhlt\u271d : x < y\nhle : (fun i => \u2191x i) \u2264 fun i => \u2191y i\nj : \u03b9\nhlt : \u2191x j < \u2191y j\n\u22a2 \u2203 i, (\u2200 (j : \u03b9), r j i \u2192 \u2191x j \u2264 \u2191y j \u2227 \u2191y j \u2264 \u2191x j) \u2227 \u2191x i < \u2191y i\n[PROOFSTEP]\nclassical\nhave : (x.neLocus y : Set \u03b9).WellFoundedOn r := (x.neLocus y).finite_toSet.wellFoundedOn\nobtain \u27e8i, hi, hl\u27e9 := this.has_min {i | x i < y i} \u27e8\u27e8j, mem_neLocus.2 hlt.ne\u27e9, hlt\u27e9\nrefine' \u27e8i, fun k hk \u21a6 \u27e8hle k, _\u27e9, hi\u27e9\nexact of_not_not fun h \u21a6 hl \u27e8k, mem_neLocus.2 (ne_of_not_le h).symm\u27e9 ((hle k).lt_of_not_le h) hk\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b9 r\nx y : \u03a0\u2080 (i : \u03b9), \u03b1 i\nhlt\u271d : x < y\nhle : (fun i => \u2191x i) \u2264 fun i => \u2191y i\nj : \u03b9\nhlt : \u2191x j < \u2191y j\n\u22a2 \u2203 i, (\u2200 (j : \u03b9), r j i \u2192 \u2191x j \u2264 \u2191y j \u2227 \u2191y j \u2264 \u2191x j) \u2227 \u2191x i < \u2191y i\n[PROOFSTEP]\nhave : (x.neLocus y : Set \u03b9).WellFoundedOn r := (x.neLocus y).finite_toSet.wellFoundedOn\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b9 r\nx y : \u03a0\u2080 (i : \u03b9), \u03b1 i\nhlt\u271d : x < y\nhle : (fun i => \u2191x i) \u2264 fun i => \u2191y i\nj : \u03b9\nhlt : \u2191x j < \u2191y j\nthis : Set.WellFoundedOn (\u2191(neLocus x y)) r\n\u22a2 \u2203 i, (\u2200 (j : \u03b9), r j i \u2192 \u2191x j \u2264 \u2191y j \u2227 \u2191y j \u2264 \u2191x j) \u2227 \u2191x i < \u2191y i\n[PROOFSTEP]\nobtain \u27e8i, hi, hl\u27e9 := this.has_min {i | x i < y i} \u27e8\u27e8j, mem_neLocus.2 hlt.ne\u27e9, hlt\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b9 r\nx y : \u03a0\u2080 (i : \u03b9), \u03b1 i\nhlt\u271d : x < y\nhle : (fun i => \u2191x i) \u2264 fun i => \u2191y i\nj : \u03b9\nhlt : \u2191x j < \u2191y j\nthis : Set.WellFoundedOn (\u2191(neLocus x y)) r\ni : \u2191\u2191(neLocus x y)\nhi : i \u2208 {i | \u2191x \u2191i < \u2191y \u2191i}\nhl : \u2200 (x_1 : \u2191\u2191(neLocus x y)), x_1 \u2208 {i | \u2191x \u2191i < \u2191y \u2191i} \u2192 \u00acr \u2191x_1 \u2191i\n\u22a2 \u2203 i, (\u2200 (j : \u03b9), r j i \u2192 \u2191x j \u2264 \u2191y j \u2227 \u2191y j \u2264 \u2191x j) \u2227 \u2191x i < \u2191y i\n[PROOFSTEP]\nrefine' \u27e8i, fun k hk \u21a6 \u27e8hle k, _\u27e9, hi\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b9 r\nx y : \u03a0\u2080 (i : \u03b9), \u03b1 i\nhlt\u271d : x < y\nhle : (fun i => \u2191x i) \u2264 fun i => \u2191y i\nj : \u03b9\nhlt : \u2191x j < \u2191y j\nthis : Set.WellFoundedOn (\u2191(neLocus x y)) r\ni : \u2191\u2191(neLocus x y)\nhi : i \u2208 {i | \u2191x \u2191i < \u2191y \u2191i}\nhl : \u2200 (x_1 : \u2191\u2191(neLocus x y)), x_1 \u2208 {i | \u2191x \u2191i < \u2191y \u2191i} \u2192 \u00acr \u2191x_1 \u2191i\nk : \u03b9\nhk : r k \u2191i\n\u22a2 \u2191y k \u2264 \u2191x k\n[PROOFSTEP]\nexact of_not_not fun h \u21a6 hl \u27e8k, mem_neLocus.2 (ne_of_not_le h).symm\u27e9 ((hle k).lt_of_not_le h) hk\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b9 r\nx y : \u03a0\u2080 (i : \u03b9), \u03b1 i\nhlt : x < y\n\u22a2 Pi.Lex r (fun {i} x x_1 => x < x_1) \u2191x \u2191y\n[PROOFSTEP]\nsimp_rw [Pi.Lex, le_antisymm_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b9 r\nx y : \u03a0\u2080 (i : \u03b9), \u03b1 i\nhlt : x < y\n\u22a2 \u2203 i, (\u2200 (j : \u03b9), r j i \u2192 \u2191x j \u2264 \u2191y j \u2227 \u2191y j \u2264 \u2191x j) \u2227 \u2191x i < \u2191y i\n[PROOFSTEP]\nexact lex_lt_of_lt_of_preorder r hlt\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nP : Lex (\u03a0\u2080 (i : \u03b9), \u03b1 i) \u2192 Lex (\u03a0\u2080 (i : \u03b9), \u03b1 i) \u2192 Sort u_3\nh_lt : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex f < \u2191toLex g \u2192 P (\u2191toLex f) (\u2191toLex g)\nh_eq : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex f = \u2191toLex g \u2192 P (\u2191toLex f) (\u2191toLex g)\nh_gt : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex g < \u2191toLex f \u2192 P (\u2191toLex f) (\u2191toLex g)\nf g : \u03a0\u2080 (i : \u03b9), \u03b1 i\nwit : \u03b9\nh : Finset.min (neLocus f g) = \u2191wit\n\u22a2 P (\u2191toLex f) (\u2191toLex g)\n[PROOFSTEP]\napply (mem_neLocus.mp <| Finset.mem_of_min h).lt_or_lt.by_cases\n[GOAL]\ncase h\u2081\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nP : Lex (\u03a0\u2080 (i : \u03b9), \u03b1 i) \u2192 Lex (\u03a0\u2080 (i : \u03b9), \u03b1 i) \u2192 Sort u_3\nh_lt : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex f < \u2191toLex g \u2192 P (\u2191toLex f) (\u2191toLex g)\nh_eq : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex f = \u2191toLex g \u2192 P (\u2191toLex f) (\u2191toLex g)\nh_gt : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex g < \u2191toLex f \u2192 P (\u2191toLex f) (\u2191toLex g)\nf g : \u03a0\u2080 (i : \u03b9), \u03b1 i\nwit : \u03b9\nh : Finset.min (neLocus f g) = \u2191wit\n\u22a2 \u2191f wit < \u2191g wit \u2192 P (\u2191toLex f) (\u2191toLex g)\n[PROOFSTEP]\nintro hwit\n[GOAL]\ncase h\u2082\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nP : Lex (\u03a0\u2080 (i : \u03b9), \u03b1 i) \u2192 Lex (\u03a0\u2080 (i : \u03b9), \u03b1 i) \u2192 Sort u_3\nh_lt : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex f < \u2191toLex g \u2192 P (\u2191toLex f) (\u2191toLex g)\nh_eq : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex f = \u2191toLex g \u2192 P (\u2191toLex f) (\u2191toLex g)\nh_gt : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex g < \u2191toLex f \u2192 P (\u2191toLex f) (\u2191toLex g)\nf g : \u03a0\u2080 (i : \u03b9), \u03b1 i\nwit : \u03b9\nh : Finset.min (neLocus f g) = \u2191wit\n\u22a2 \u2191g wit < \u2191f wit \u2192 P (\u2191toLex f) (\u2191toLex g)\n[PROOFSTEP]\nintro hwit\n[GOAL]\ncase h\u2081\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nP : Lex (\u03a0\u2080 (i : \u03b9), \u03b1 i) \u2192 Lex (\u03a0\u2080 (i : \u03b9), \u03b1 i) \u2192 Sort u_3\nh_lt : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex f < \u2191toLex g \u2192 P (\u2191toLex f) (\u2191toLex g)\nh_eq : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex f = \u2191toLex g \u2192 P (\u2191toLex f) (\u2191toLex g)\nh_gt : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex g < \u2191toLex f \u2192 P (\u2191toLex f) (\u2191toLex g)\nf g : \u03a0\u2080 (i : \u03b9), \u03b1 i\nwit : \u03b9\nh : Finset.min (neLocus f g) = \u2191wit\nhwit : \u2191f wit < \u2191g wit\n\u22a2 P (\u2191toLex f) (\u2191toLex g)\n[PROOFSTEP]\nexact h_lt \u27e8wit, fun j hj \u21a6 not_mem_neLocus.mp (Finset.not_mem_of_lt_min hj h), hwit\u27e9\n[GOAL]\ncase h\u2082\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nP : Lex (\u03a0\u2080 (i : \u03b9), \u03b1 i) \u2192 Lex (\u03a0\u2080 (i : \u03b9), \u03b1 i) \u2192 Sort u_3\nh_lt : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex f < \u2191toLex g \u2192 P (\u2191toLex f) (\u2191toLex g)\nh_eq : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex f = \u2191toLex g \u2192 P (\u2191toLex f) (\u2191toLex g)\nh_gt : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex g < \u2191toLex f \u2192 P (\u2191toLex f) (\u2191toLex g)\nf g : \u03a0\u2080 (i : \u03b9), \u03b1 i\nwit : \u03b9\nh : Finset.min (neLocus f g) = \u2191wit\nhwit : \u2191g wit < \u2191f wit\n\u22a2 P (\u2191toLex f) (\u2191toLex g)\n[PROOFSTEP]\nexact h_gt \u27e8wit, fun j hj \u21a6 not_mem_neLocus.mp (Finset.not_mem_of_lt_min hj <| by rwa [neLocus_comm]), hwit\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nP : Lex (\u03a0\u2080 (i : \u03b9), \u03b1 i) \u2192 Lex (\u03a0\u2080 (i : \u03b9), \u03b1 i) \u2192 Sort u_3\nh_lt : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex f < \u2191toLex g \u2192 P (\u2191toLex f) (\u2191toLex g)\nh_eq : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex f = \u2191toLex g \u2192 P (\u2191toLex f) (\u2191toLex g)\nh_gt : {f g : \u03a0\u2080 (i : \u03b9), \u03b1 i} \u2192 \u2191toLex g < \u2191toLex f \u2192 P (\u2191toLex f) (\u2191toLex g)\nf g : \u03a0\u2080 (i : \u03b9), \u03b1 i\nwit : \u03b9\nh : Finset.min (neLocus f g) = \u2191wit\nhwit : \u2191g wit < \u2191f wit\nj : \u03b9\nhj : (fun x x_1 => x < x_1) j wit\n\u22a2 Finset.min (neLocus (\u2191ofLex (\u2191toLex g)) (\u2191ofLex (\u2191toLex f))) = \u2191wit\n[PROOFSTEP]\nrwa [neLocus_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\n\u22a2 Monotone \u2191toLex\n[PROOFSTEP]\nintro a b h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\na b : \u03a0\u2080 (i : \u03b9), \u03b1 i\nh : a \u2264 b\n\u22a2 \u2191toLex a \u2264 \u2191toLex b\n[PROOFSTEP]\nrefine' le_of_lt_or_eq (or_iff_not_imp_right.2 fun hne \u21a6 _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\na b : \u03a0\u2080 (i : \u03b9), \u03b1 i\nh : a \u2264 b\nhne : \u00ac\u2191toLex a = \u2191toLex b\n\u22a2 \u2191toLex a < \u2191toLex b\n[PROOFSTEP]\nclassical exact\n \u27e8Finset.min' _ (nonempty_neLocus_iff.2 hne), fun j hj \u21a6 not_mem_neLocus.1 fun h \u21a6 (Finset.min'_le _ _ h).not_lt hj,\n (h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b1 i)\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\na b : \u03a0\u2080 (i : \u03b9), \u03b1 i\nh : a \u2264 b\nhne : \u00ac\u2191toLex a = \u2191toLex b\n\u22a2 \u2191toLex a < \u2191toLex b\n[PROOFSTEP]\nexact\n \u27e8Finset.min' _ (nonempty_neLocus_iff.2 hne), fun j hj \u21a6 not_mem_neLocus.1 fun h \u21a6 (Finset.min'_le _ _ h).not_lt hj,\n (h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Data.DFinsupp.Lex", "llama_tokens": 5082, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195269001831, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.42953418251242714}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03bc \u03bd : Measure \u03b1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u22a2 Measure.bind (\u03bc + \u03bd) \u2191\u03ba = Measure.bind \u03bc \u2191\u03ba + Measure.bind \u03bd \u2191\u03ba\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03bc \u03bd : Measure \u03b1\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.bind (\u03bc + \u03bd) \u2191\u03ba) s = \u2191\u2191(Measure.bind \u03bc \u2191\u03ba + Measure.bind \u03bd \u2191\u03ba) s\n[PROOFSTEP]\nrw [Measure.bind_apply hs (kernel.measurable _), lintegral_add_measure, Measure.coe_add, Pi.add_apply,\n Measure.bind_apply hs (kernel.measurable _), Measure.bind_apply hs (kernel.measurable _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bc : Measure \u03b1\nr : \u211d\u22650\u221e\n\u22a2 Measure.bind (r \u2022 \u03bc) \u2191\u03ba = r \u2022 Measure.bind \u03bc \u2191\u03ba\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bc : Measure \u03b1\nr : \u211d\u22650\u221e\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.bind (r \u2022 \u03bc) \u2191\u03ba) s = \u2191\u2191(r \u2022 Measure.bind \u03bc \u2191\u03ba) s\n[PROOFSTEP]\nrw [Measure.bind_apply hs (kernel.measurable _), lintegral_smul_measure, Measure.coe_smul, Pi.smul_apply,\n Measure.bind_apply hs (kernel.measurable _), smul_eq_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bc : Measure \u03b1\n\u22a2 const \u03b1 (Measure.bind \u03bc \u2191\u03ba) = \u03ba \u2218\u2096 const \u03b1 \u03bc\n[PROOFSTEP]\next a s hs\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(const \u03b1 (Measure.bind \u03bc \u2191\u03ba)) a) s = \u2191\u2191(\u2191(\u03ba \u2218\u2096 const \u03b1 \u03bc) a) s\n[PROOFSTEP]\nsimp_rw [comp_apply' _ _ _ hs, const_apply, Measure.bind_apply hs (kernel.measurable _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\n\u03bc : Measure \u03b1\na : \u03b1\n\u22a2 \u2191(\u03ba \u2218\u2096 const \u03b1 \u03bc) a = Measure.bind \u03bc \u2191\u03ba\n[PROOFSTEP]\nrw [\u2190 const_apply (\u03bc.bind \u03ba) a, const_bind_eq_comp_const \u03ba \u03bc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b1 }\n\u03bc : Measure \u03b1\nh\u03ba : Invariant \u03ba \u03bc\n\u22a2 \u03ba \u2218\u2096 const \u03b1 \u03bc = const \u03b1 \u03bc\n[PROOFSTEP]\nrw [\u2190 const_bind_eq_comp_const \u03ba \u03bc, h\u03ba.def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b1 }\n\u03bc : Measure \u03b1\ninst\u271d : IsSFiniteKernel \u03ba\nh\u03ba : Invariant \u03ba \u03bc\nh\u03b7 : Invariant \u03b7 \u03bc\n\u22a2 Invariant (\u03ba \u2218\u2096 \u03b7) \u03bc\n[PROOFSTEP]\ncases' isEmpty_or_nonempty \u03b1 with _ h\u03b1\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b1 }\n\u03bc : Measure \u03b1\ninst\u271d : IsSFiniteKernel \u03ba\nh\u03ba : Invariant \u03ba \u03bc\nh\u03b7 : Invariant \u03b7 \u03bc\nh\u271d : IsEmpty \u03b1\n\u22a2 Invariant (\u03ba \u2218\u2096 \u03b7) \u03bc\n[PROOFSTEP]\nexact Subsingleton.elim _ _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\n\u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b1 }\n\u03bc : Measure \u03b1\ninst\u271d : IsSFiniteKernel \u03ba\nh\u03ba : Invariant \u03ba \u03bc\nh\u03b7 : Invariant \u03b7 \u03bc\nh\u03b1 : Nonempty \u03b1\n\u22a2 Invariant (\u03ba \u2218\u2096 \u03b7) \u03bc\n[PROOFSTEP]\nsimp_rw [Invariant, \u2190 comp_const_apply_eq_bind (\u03ba \u2218\u2096 \u03b7) \u03bc h\u03b1.some, comp_assoc, h\u03b7.comp_const, h\u03ba.comp_const,\n const_apply]\n", "meta": {"mathlib_filename": "Mathlib.Probability.Kernel.Invariance", "llama_tokens": 1772, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.42951421278873386}} {"text": "[GOAL]\n\u03b1 : Type u_1\nk : \u2115\nA : Finset \u2115\nh\u2081 : \u2200 {x : \u2115}, x \u2208 A \u2192 x < k\n\u22a2 Finset.sum A (Nat.pow 2) < 2 ^ k\n[PROOFSTEP]\napply lt_of_le_of_lt (sum_le_sum_of_subset fun t => mem_range.2 \u2218 h\u2081)\n[GOAL]\n\u03b1 : Type u_1\nk : \u2115\nA : Finset \u2115\nh\u2081 : \u2200 {x : \u2115}, x \u2208 A \u2192 x < k\n\u22a2 \u2211 x in range k, Nat.pow 2 x < 2 ^ k\n[PROOFSTEP]\nhave z := geom_sum_mul_add 1 k\n[GOAL]\n\u03b1 : Type u_1\nk : \u2115\nA : Finset \u2115\nh\u2081 : \u2200 {x : \u2115}, x \u2208 A \u2192 x < k\nz : (\u2211 i in range k, (1 + 1) ^ i) * 1 + 1 = (1 + 1) ^ k\n\u22a2 \u2211 x in range k, Nat.pow 2 x < 2 ^ k\n[PROOFSTEP]\nrw [mul_one, one_add_one_eq_two] at z \n[GOAL]\n\u03b1 : Type u_1\nk : \u2115\nA : Finset \u2115\nh\u2081 : \u2200 {x : \u2115}, x \u2208 A \u2192 x < k\nz : \u2211 i in range k, 2 ^ i + 1 = 2 ^ k\n\u22a2 \u2211 x in range k, Nat.pow 2 x < 2 ^ k\n[PROOFSTEP]\nrw [\u2190 z]\n[GOAL]\n\u03b1 : Type u_1\nk : \u2115\nA : Finset \u2115\nh\u2081 : \u2200 {x : \u2115}, x \u2208 A \u2192 x < k\nz : \u2211 i in range k, 2 ^ i + 1 = 2 ^ k\n\u22a2 \u2211 x in range k, Nat.pow 2 x < \u2211 i in range k, 2 ^ i + 1\n[PROOFSTEP]\napply Nat.lt_succ_self\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\n\u22a2 toColex (image f A) < toColex (image f B) \u2194 toColex A < toColex B\n[PROOFSTEP]\nsimp only [Colex.lt_def, not_exists, mem_image, exists_prop, not_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\n\u22a2 (\u2203 k,\n (\u2200 {x : \u03b2}, k < x \u2192 ((\u2203 a, a \u2208 A \u2227 f a = x) \u2194 \u2203 a, a \u2208 B \u2227 f a = x)) \u2227\n (\u2200 (x : \u03b1), x \u2208 A \u2192 \u00acf x = k) \u2227 \u2203 a, a \u2208 B \u2227 f a = k) \u2194\n \u2203 k, (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\n\u22a2 (\u2203 k,\n (\u2200 {x : \u03b2}, k < x \u2192 ((\u2203 a, a \u2208 A \u2227 f a = x) \u2194 \u2203 a, a \u2208 B \u2227 f a = x)) \u2227\n (\u2200 (x : \u03b1), x \u2208 A \u2192 \u00acf x = k) \u2227 \u2203 a, a \u2208 B \u2227 f a = k) \u2192\n \u2203 k, (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B\n[PROOFSTEP]\nrintro \u27e8k, z, q, k', _, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk' : \u03b1\nleft\u271d : k' \u2208 B\nz : \u2200 {x : \u03b2}, f k' < x \u2192 ((\u2203 a, a \u2208 A \u2227 f a = x) \u2194 \u2203 a, a \u2208 B \u2227 f a = x)\nq : \u2200 (x : \u03b1), x \u2208 A \u2192 \u00acf x = f k'\n\u22a2 \u2203 k, (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B\n[PROOFSTEP]\nexact \u27e8k', @fun x hx => by simpa [h\u2081.injective.eq_iff] using z (h\u2081 hx), fun t => q _ t rfl, \u2039k' \u2208 B\u203a\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk' : \u03b1\nleft\u271d : k' \u2208 B\nz : \u2200 {x : \u03b2}, f k' < x \u2192 ((\u2203 a, a \u2208 A \u2227 f a = x) \u2194 \u2203 a, a \u2208 B \u2227 f a = x)\nq : \u2200 (x : \u03b1), x \u2208 A \u2192 \u00acf x = f k'\nx : \u03b1\nhx : k' < x\n\u22a2 x \u2208 A \u2194 x \u2208 B\n[PROOFSTEP]\nsimpa [h\u2081.injective.eq_iff] using z (h\u2081 hx)\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\n\u22a2 (\u2203 k, (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B) \u2192\n \u2203 k,\n (\u2200 {x : \u03b2}, k < x \u2192 ((\u2203 a, a \u2208 A \u2227 f a = x) \u2194 \u2203 a, a \u2208 B \u2227 f a = x)) \u2227\n (\u2200 (x : \u03b1), x \u2208 A \u2192 \u00acf x = k) \u2227 \u2203 a, a \u2208 B \u2227 f a = k\n[PROOFSTEP]\nrintro \u27e8k, z, ka, _\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\n\u22a2 \u2203 k,\n (\u2200 {x : \u03b2}, k < x \u2192 ((\u2203 a, a \u2208 A \u2227 f a = x) \u2194 \u2203 a, a \u2208 B \u2227 f a = x)) \u2227\n (\u2200 (x : \u03b1), x \u2208 A \u2192 \u00acf x = k) \u2227 \u2203 a, a \u2208 B \u2227 f a = k\n[PROOFSTEP]\nrefine' \u27e8f k, @fun x hx => _, _, k, \u2039k \u2208 B\u203a, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx : \u03b2\nhx : f k < x\n\u22a2 (\u2203 a, a \u2208 A \u2227 f a = x) \u2194 \u2203 a, a \u2208 B \u2227 f a = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.intro.intro.intro.refine'_1.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx : \u03b2\nhx : f k < x\n\u22a2 (\u2203 a, a \u2208 A \u2227 f a = x) \u2192 \u2203 a, a \u2208 B \u2227 f a = x\ncase mpr.intro.intro.intro.refine'_1.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx : \u03b2\nhx : f k < x\n\u22a2 (\u2203 a, a \u2208 B \u2227 f a = x) \u2192 \u2203 a, a \u2208 A \u2227 f a = x\n[PROOFSTEP]\nany_goals\n rintro \u27e8x', hx', rfl\u27e9\n refine' \u27e8x', _, rfl\u27e9\n first\n | rwa [\u2190 z _]\n | rwa [z _]\n rwa [StrictMono.lt_iff_lt h\u2081] at hx \n[GOAL]\ncase mpr.intro.intro.intro.refine'_1.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx : \u03b2\nhx : f k < x\n\u22a2 (\u2203 a, a \u2208 A \u2227 f a = x) \u2192 \u2203 a, a \u2208 B \u2227 f a = x\n[PROOFSTEP]\nrintro \u27e8x', hx', rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.refine'_1.mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx' : \u03b1\nhx' : x' \u2208 A\nhx : f k < f x'\n\u22a2 \u2203 a, a \u2208 B \u2227 f a = f x'\n[PROOFSTEP]\nrefine' \u27e8x', _, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.refine'_1.mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx' : \u03b1\nhx' : x' \u2208 A\nhx : f k < f x'\n\u22a2 x' \u2208 B\n[PROOFSTEP]\nfirst\n| rwa [\u2190 z _]\n| rwa [z _]\n[GOAL]\ncase mpr.intro.intro.intro.refine'_1.mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx' : \u03b1\nhx' : x' \u2208 A\nhx : f k < f x'\n\u22a2 x' \u2208 B\n[PROOFSTEP]\nrwa [\u2190 z _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx' : \u03b1\nhx' : x' \u2208 A\nhx : f k < f x'\n\u22a2 k < x'\n[PROOFSTEP]\nrwa [StrictMono.lt_iff_lt h\u2081] at hx \n[GOAL]\ncase mpr.intro.intro.intro.refine'_1.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx : \u03b2\nhx : f k < x\n\u22a2 (\u2203 a, a \u2208 B \u2227 f a = x) \u2192 \u2203 a, a \u2208 A \u2227 f a = x\n[PROOFSTEP]\nrintro \u27e8x', hx', rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.refine'_1.mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx' : \u03b1\nhx' : x' \u2208 B\nhx : f k < f x'\n\u22a2 \u2203 a, a \u2208 A \u2227 f a = f x'\n[PROOFSTEP]\nrefine' \u27e8x', _, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.refine'_1.mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx' : \u03b1\nhx' : x' \u2208 B\nhx : f k < f x'\n\u22a2 x' \u2208 A\n[PROOFSTEP]\nfirst\n| rwa [\u2190 z _]\n| rwa [z _]\n[GOAL]\ncase mpr.intro.intro.intro.refine'_1.mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx' : \u03b1\nhx' : x' \u2208 B\nhx : f k < f x'\n\u22a2 x' \u2208 A\n[PROOFSTEP]\nrwa [\u2190 z _]\n[GOAL]\ncase mpr.intro.intro.intro.refine'_1.mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx' : \u03b1\nhx' : x' \u2208 B\nhx : f k < f x'\n\u22a2 x' \u2208 A\n[PROOFSTEP]\nrwa [z _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx' : \u03b1\nhx' : x' \u2208 B\nhx : f k < f x'\n\u22a2 k < x'\n[PROOFSTEP]\nrwa [StrictMono.lt_iff_lt h\u2081] at hx \n[GOAL]\ncase mpr.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\n\u22a2 \u2200 (x : \u03b1), x \u2208 A \u2192 \u00acf x = f k\n[PROOFSTEP]\nsimp only [h\u2081.injective, Function.Injective.eq_iff]\n[GOAL]\ncase mpr.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nka : \u00ack \u2208 A\nright\u271d : k \u2208 B\n\u22a2 \u2200 (x : \u03b1), x \u2208 A \u2192 \u00acx = k\n[PROOFSTEP]\nexact fun x hx => ne_of_mem_of_not_mem hx ka\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nA B : Finset (Fin n)\n\u22a2 toColex (image (fun i => \u2191i) A) < toColex (image (fun i => \u2191i) B) \u2194 toColex A < toColex B\n[PROOFSTEP]\nrefine' Colex.hom_lt_iff _ _ _\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nA B : Finset (Fin n)\n\u22a2 StrictMono fun i => \u2191i\n[PROOFSTEP]\nexact (fun x y k => k)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\na b c : Colex \u03b1\n\u22a2 a < b \u2192 b < c \u2192 a < c\n[PROOFSTEP]\nrintro \u27e8k\u2081, k\u2081z, notinA, inB\u27e9 \u27e8k\u2082, k\u2082z, notinB, inC\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\na b c : Colex \u03b1\nk\u2081 : \u03b1\nk\u2081z : \u2200 {x : \u03b1}, k\u2081 < x \u2192 (x \u2208 a \u2194 x \u2208 b)\nnotinA : \u00ack\u2081 \u2208 a\ninB : k\u2081 \u2208 b\nk\u2082 : \u03b1\nk\u2082z : \u2200 {x : \u03b1}, k\u2082 < x \u2192 (x \u2208 b \u2194 x \u2208 c)\nnotinB : \u00ack\u2082 \u2208 b\ninC : k\u2082 \u2208 c\n\u22a2 a < c\n[PROOFSTEP]\ncases' lt_or_gt_of_ne (ne_of_mem_of_not_mem inB notinB) with h h\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\na b c : Colex \u03b1\nk\u2081 : \u03b1\nk\u2081z : \u2200 {x : \u03b1}, k\u2081 < x \u2192 (x \u2208 a \u2194 x \u2208 b)\nnotinA : \u00ack\u2081 \u2208 a\ninB : k\u2081 \u2208 b\nk\u2082 : \u03b1\nk\u2082z : \u2200 {x : \u03b1}, k\u2082 < x \u2192 (x \u2208 b \u2194 x \u2208 c)\nnotinB : \u00ack\u2082 \u2208 b\ninC : k\u2082 \u2208 c\nh : k\u2081 < k\u2082\n\u22a2 a < c\n[PROOFSTEP]\nrefine' \u27e8k\u2082, @fun x hx => _, _, inC\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.inl.refine'_1\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\na b c : Colex \u03b1\nk\u2081 : \u03b1\nk\u2081z : \u2200 {x : \u03b1}, k\u2081 < x \u2192 (x \u2208 a \u2194 x \u2208 b)\nnotinA : \u00ack\u2081 \u2208 a\ninB : k\u2081 \u2208 b\nk\u2082 : \u03b1\nk\u2082z : \u2200 {x : \u03b1}, k\u2082 < x \u2192 (x \u2208 b \u2194 x \u2208 c)\nnotinB : \u00ack\u2082 \u2208 b\ninC : k\u2082 \u2208 c\nh : k\u2081 < k\u2082\nx : \u03b1\nhx : k\u2082 < x\n\u22a2 x \u2208 a \u2194 x \u2208 c\ncase intro.intro.intro.intro.intro.intro.inl.refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\na b c : Colex \u03b1\nk\u2081 : \u03b1\nk\u2081z : \u2200 {x : \u03b1}, k\u2081 < x \u2192 (x \u2208 a \u2194 x \u2208 b)\nnotinA : \u00ack\u2081 \u2208 a\ninB : k\u2081 \u2208 b\nk\u2082 : \u03b1\nk\u2082z : \u2200 {x : \u03b1}, k\u2082 < x \u2192 (x \u2208 b \u2194 x \u2208 c)\nnotinB : \u00ack\u2082 \u2208 b\ninC : k\u2082 \u2208 c\nh : k\u2081 < k\u2082\n\u22a2 \u00ack\u2082 \u2208 a\n[PROOFSTEP]\nrw [\u2190 k\u2082z hx]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.inl.refine'_1\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\na b c : Colex \u03b1\nk\u2081 : \u03b1\nk\u2081z : \u2200 {x : \u03b1}, k\u2081 < x \u2192 (x \u2208 a \u2194 x \u2208 b)\nnotinA : \u00ack\u2081 \u2208 a\ninB : k\u2081 \u2208 b\nk\u2082 : \u03b1\nk\u2082z : \u2200 {x : \u03b1}, k\u2082 < x \u2192 (x \u2208 b \u2194 x \u2208 c)\nnotinB : \u00ack\u2082 \u2208 b\ninC : k\u2082 \u2208 c\nh : k\u2081 < k\u2082\nx : \u03b1\nhx : k\u2082 < x\n\u22a2 x \u2208 a \u2194 x \u2208 b\ncase intro.intro.intro.intro.intro.intro.inl.refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\na b c : Colex \u03b1\nk\u2081 : \u03b1\nk\u2081z : \u2200 {x : \u03b1}, k\u2081 < x \u2192 (x \u2208 a \u2194 x \u2208 b)\nnotinA : \u00ack\u2081 \u2208 a\ninB : k\u2081 \u2208 b\nk\u2082 : \u03b1\nk\u2082z : \u2200 {x : \u03b1}, k\u2082 < x \u2192 (x \u2208 b \u2194 x \u2208 c)\nnotinB : \u00ack\u2082 \u2208 b\ninC : k\u2082 \u2208 c\nh : k\u2081 < k\u2082\n\u22a2 \u00ack\u2082 \u2208 a\n[PROOFSTEP]\napply k\u2081z (Trans.trans h hx)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.inl.refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\na b c : Colex \u03b1\nk\u2081 : \u03b1\nk\u2081z : \u2200 {x : \u03b1}, k\u2081 < x \u2192 (x \u2208 a \u2194 x \u2208 b)\nnotinA : \u00ack\u2081 \u2208 a\ninB : k\u2081 \u2208 b\nk\u2082 : \u03b1\nk\u2082z : \u2200 {x : \u03b1}, k\u2082 < x \u2192 (x \u2208 b \u2194 x \u2208 c)\nnotinB : \u00ack\u2082 \u2208 b\ninC : k\u2082 \u2208 c\nh : k\u2081 < k\u2082\n\u22a2 \u00ack\u2082 \u2208 a\n[PROOFSTEP]\nrwa [k\u2081z h]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\na b c : Colex \u03b1\nk\u2081 : \u03b1\nk\u2081z : \u2200 {x : \u03b1}, k\u2081 < x \u2192 (x \u2208 a \u2194 x \u2208 b)\nnotinA : \u00ack\u2081 \u2208 a\ninB : k\u2081 \u2208 b\nk\u2082 : \u03b1\nk\u2082z : \u2200 {x : \u03b1}, k\u2082 < x \u2192 (x \u2208 b \u2194 x \u2208 c)\nnotinB : \u00ack\u2082 \u2208 b\ninC : k\u2082 \u2208 c\nh : k\u2081 > k\u2082\n\u22a2 a < c\n[PROOFSTEP]\nrefine' \u27e8k\u2081, @fun x hx => _, notinA, by rwa [\u2190 k\u2082z h]\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\na b c : Colex \u03b1\nk\u2081 : \u03b1\nk\u2081z : \u2200 {x : \u03b1}, k\u2081 < x \u2192 (x \u2208 a \u2194 x \u2208 b)\nnotinA : \u00ack\u2081 \u2208 a\ninB : k\u2081 \u2208 b\nk\u2082 : \u03b1\nk\u2082z : \u2200 {x : \u03b1}, k\u2082 < x \u2192 (x \u2208 b \u2194 x \u2208 c)\nnotinB : \u00ack\u2082 \u2208 b\ninC : k\u2082 \u2208 c\nh : k\u2081 > k\u2082\n\u22a2 k\u2081 \u2208 c\n[PROOFSTEP]\nrwa [\u2190 k\u2082z h]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\na b c : Colex \u03b1\nk\u2081 : \u03b1\nk\u2081z : \u2200 {x : \u03b1}, k\u2081 < x \u2192 (x \u2208 a \u2194 x \u2208 b)\nnotinA : \u00ack\u2081 \u2208 a\ninB : k\u2081 \u2208 b\nk\u2082 : \u03b1\nk\u2082z : \u2200 {x : \u03b1}, k\u2082 < x \u2192 (x \u2208 b \u2194 x \u2208 c)\nnotinB : \u00ack\u2082 \u2208 b\ninC : k\u2082 \u2208 c\nh : k\u2081 > k\u2082\nx : \u03b1\nhx : k\u2081 < x\n\u22a2 x \u2208 a \u2194 x \u2208 c\n[PROOFSTEP]\nrw [k\u2081z hx]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\na b c : Colex \u03b1\nk\u2081 : \u03b1\nk\u2081z : \u2200 {x : \u03b1}, k\u2081 < x \u2192 (x \u2208 a \u2194 x \u2208 b)\nnotinA : \u00ack\u2081 \u2208 a\ninB : k\u2081 \u2208 b\nk\u2082 : \u03b1\nk\u2082z : \u2200 {x : \u03b1}, k\u2082 < x \u2192 (x \u2208 b \u2194 x \u2208 c)\nnotinB : \u00ack\u2082 \u2208 b\ninC : k\u2082 \u2208 c\nh : k\u2081 > k\u2082\nx : \u03b1\nhx : k\u2081 < x\n\u22a2 x \u2208 b \u2194 x \u2208 c\n[PROOFSTEP]\napply k\u2082z (Trans.trans h hx)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\n\u22a2 A < B \u2228 A = B \u2228 B < A\n[PROOFSTEP]\nby_cases h\u2081 : A = B\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : A = B\n\u22a2 A < B \u2228 A = B \u2228 B < A\n[PROOFSTEP]\ntauto\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\n\u22a2 A < B \u2228 A = B \u2228 B < A\n[PROOFSTEP]\nhave h : Finset.Nonempty (A \\ B \u222a B \\ A) := by\n rw [nonempty_iff_ne_empty]\n intro a\n simp only [union_eq_empty_iff, sdiff_eq_empty_iff_subset] at a \n apply h\u2081 (Subset.antisymm a.1 a.2)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\n\u22a2 Finset.Nonempty (A \\ B \u222a B \\ A)\n[PROOFSTEP]\nrw [nonempty_iff_ne_empty]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\n\u22a2 A \\ B \u222a B \\ A \u2260 \u2205\n[PROOFSTEP]\nintro a\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\na : A \\ B \u222a B \\ A = \u2205\n\u22a2 False\n[PROOFSTEP]\nsimp only [union_eq_empty_iff, sdiff_eq_empty_iff_subset] at a \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\na : A \u2286 B \u2227 B \u2286 A\n\u22a2 False\n[PROOFSTEP]\napply h\u2081 (Subset.antisymm a.1 a.2)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\n\u22a2 A < B \u2228 A = B \u2228 B < A\n[PROOFSTEP]\nrcases exists_max_image (A \\ B \u222a B \\ A) id h with \u27e8k, \u27e8hk, z\u27e9\u27e9\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nhk : k \u2208 A \\ B \u222a B \\ A\nz : \u2200 (x' : \u03b1), x' \u2208 A \\ B \u222a B \\ A \u2192 id x' \u2264 id k\n\u22a2 A < B \u2228 A = B \u2228 B < A\n[PROOFSTEP]\nsimp only [mem_union, mem_sdiff] at hk \n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nz : \u2200 (x' : \u03b1), x' \u2208 A \\ B \u222a B \\ A \u2192 id x' \u2264 id k\nhk : k \u2208 A \u2227 \u00ack \u2208 B \u2228 k \u2208 B \u2227 \u00ack \u2208 A\n\u22a2 A < B \u2228 A = B \u2228 B < A\n[PROOFSTEP]\ncases' hk with hk hk\n[GOAL]\ncase neg.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nz : \u2200 (x' : \u03b1), x' \u2208 A \\ B \u222a B \\ A \u2192 id x' \u2264 id k\nhk : k \u2208 A \u2227 \u00ack \u2208 B\n\u22a2 A < B \u2228 A = B \u2228 B < A\n[PROOFSTEP]\nright\n[GOAL]\ncase neg.intro.intro.inl.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nz : \u2200 (x' : \u03b1), x' \u2208 A \\ B \u222a B \\ A \u2192 id x' \u2264 id k\nhk : k \u2208 A \u2227 \u00ack \u2208 B\n\u22a2 A = B \u2228 B < A\n[PROOFSTEP]\nright\n[GOAL]\ncase neg.intro.intro.inl.h.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nz : \u2200 (x' : \u03b1), x' \u2208 A \\ B \u222a B \\ A \u2192 id x' \u2264 id k\nhk : k \u2208 A \u2227 \u00ack \u2208 B\n\u22a2 B < A\n[PROOFSTEP]\nrefine' \u27e8k, @fun t th => _, hk.2, hk.1\u27e9\n[GOAL]\ncase neg.intro.intro.inl.h.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nz : \u2200 (x' : \u03b1), x' \u2208 A \\ B \u222a B \\ A \u2192 id x' \u2264 id k\nhk : k \u2208 A \u2227 \u00ack \u2208 B\nt : \u03b1\nth : k < t\n\u22a2 t \u2208 B \u2194 t \u2208 A\n[PROOFSTEP]\nspecialize z t\n[GOAL]\ncase neg.intro.intro.inl.h.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nhk : k \u2208 A \u2227 \u00ack \u2208 B\nt : \u03b1\nth : k < t\nz : t \u2208 A \\ B \u222a B \\ A \u2192 id t \u2264 id k\n\u22a2 t \u2208 B \u2194 t \u2208 A\n[PROOFSTEP]\nby_contra h\u2082\n[GOAL]\ncase neg.intro.intro.inl.h.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nhk : k \u2208 A \u2227 \u00ack \u2208 B\nt : \u03b1\nth : k < t\nz : t \u2208 A \\ B \u222a B \\ A \u2192 id t \u2264 id k\nh\u2082 : \u00ac(t \u2208 B \u2194 t \u2208 A)\n\u22a2 False\n[PROOFSTEP]\nsimp only [mem_union, mem_sdiff, id.def] at z \n[GOAL]\ncase neg.intro.intro.inl.h.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nhk : k \u2208 A \u2227 \u00ack \u2208 B\nt : \u03b1\nth : k < t\nh\u2082 : \u00ac(t \u2208 B \u2194 t \u2208 A)\nz : t \u2208 A \u2227 \u00act \u2208 B \u2228 t \u2208 B \u2227 \u00act \u2208 A \u2192 t \u2264 k\n\u22a2 False\n[PROOFSTEP]\nrw [not_iff, iff_iff_and_or_not_and_not, not_not, and_comm] at h\u2082 \n[GOAL]\ncase neg.intro.intro.inl.h.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nhk : k \u2208 A \u2227 \u00ack \u2208 B\nt : \u03b1\nth : k < t\nh\u2082 : t \u2208 A \u2227 \u00act \u2208 B \u2228 t \u2208 B \u2227 \u00act \u2208 A\nz : t \u2208 A \u2227 \u00act \u2208 B \u2228 t \u2208 B \u2227 \u00act \u2208 A \u2192 t \u2264 k\n\u22a2 False\n[PROOFSTEP]\napply not_le_of_lt th (z h\u2082)\n[GOAL]\ncase neg.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nz : \u2200 (x' : \u03b1), x' \u2208 A \\ B \u222a B \\ A \u2192 id x' \u2264 id k\nhk : k \u2208 B \u2227 \u00ack \u2208 A\n\u22a2 A < B \u2228 A = B \u2228 B < A\n[PROOFSTEP]\nleft\n[GOAL]\ncase neg.intro.intro.inr.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nz : \u2200 (x' : \u03b1), x' \u2208 A \\ B \u222a B \\ A \u2192 id x' \u2264 id k\nhk : k \u2208 B \u2227 \u00ack \u2208 A\n\u22a2 A < B\n[PROOFSTEP]\nrefine' \u27e8k, @fun t th => _, hk.2, hk.1\u27e9\n[GOAL]\ncase neg.intro.intro.inr.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nz : \u2200 (x' : \u03b1), x' \u2208 A \\ B \u222a B \\ A \u2192 id x' \u2264 id k\nhk : k \u2208 B \u2227 \u00ack \u2208 A\nt : \u03b1\nth : k < t\n\u22a2 t \u2208 A \u2194 t \u2208 B\n[PROOFSTEP]\nspecialize z t\n[GOAL]\ncase neg.intro.intro.inr.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nhk : k \u2208 B \u2227 \u00ack \u2208 A\nt : \u03b1\nth : k < t\nz : t \u2208 A \\ B \u222a B \\ A \u2192 id t \u2264 id k\n\u22a2 t \u2208 A \u2194 t \u2208 B\n[PROOFSTEP]\nby_contra h\u2083\n[GOAL]\ncase neg.intro.intro.inr.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nhk : k \u2208 B \u2227 \u00ack \u2208 A\nt : \u03b1\nth : k < t\nz : t \u2208 A \\ B \u222a B \\ A \u2192 id t \u2264 id k\nh\u2083 : \u00ac(t \u2208 A \u2194 t \u2208 B)\n\u22a2 False\n[PROOFSTEP]\nsimp only [mem_union, mem_sdiff, id.def] at z \n[GOAL]\ncase neg.intro.intro.inr.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nhk : k \u2208 B \u2227 \u00ack \u2208 A\nt : \u03b1\nth : k < t\nh\u2083 : \u00ac(t \u2208 A \u2194 t \u2208 B)\nz : t \u2208 A \u2227 \u00act \u2208 B \u2228 t \u2208 B \u2227 \u00act \u2208 A \u2192 t \u2264 k\n\u22a2 False\n[PROOFSTEP]\nrw [not_iff, iff_iff_and_or_not_and_not, not_not, and_comm, or_comm] at h\u2083 \n[GOAL]\ncase neg.intro.intro.inr.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Colex \u03b1\nh\u2081 : \u00acA = B\nh : Finset.Nonempty (A \\ B \u222a B \\ A)\nk : \u03b1\nhk : k \u2208 B \u2227 \u00ack \u2208 A\nt : \u03b1\nth : k < t\nh\u2083 : t \u2208 A \u2227 \u00act \u2208 B \u2228 t \u2208 B \u2227 \u00act \u2208 A\nz : t \u2208 A \u2227 \u00act \u2208 B \u2228 t \u2208 B \u2227 \u00act \u2208 A \u2192 t \u2264 k\n\u22a2 False\n[PROOFSTEP]\napply not_le_of_lt th (z h\u2083)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA\u271d B\u271d : Colex \u03b1\nA B : Finset \u03b1\n\u22a2 toColex A < toColex B \u2194 \u2203 k, k \u2208 B \u2227 (\u2200 (x : \u03b1), x \u2208 A \u222a B \u2192 k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A\n[PROOFSTEP]\nrw [Colex.lt_def]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA\u271d B\u271d : Colex \u03b1\nA B : Finset \u03b1\n\u22a2 (\u2203 k, (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B) \u2194\n \u2203 k, k \u2208 B \u2227 (\u2200 (x : \u03b1), x \u2208 A \u222a B \u2192 k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A\n[PROOFSTEP]\napply exists_congr\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA\u271d B\u271d : Colex \u03b1\nA B : Finset \u03b1\n\u22a2 \u2200 (a : \u03b1),\n (\u2200 {x : \u03b1}, a < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00aca \u2208 A \u2227 a \u2208 B \u2194\n a \u2208 B \u2227 (\u2200 (x : \u03b1), x \u2208 A \u222a B \u2192 a < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00aca \u2208 A\n[PROOFSTEP]\nsimp only [mem_union, exists_prop, or_imp, and_comm (a := _ \u2208 B), and_assoc]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA\u271d B\u271d : Colex \u03b1\nA B : Finset \u03b1\n\u22a2 \u2200 (a : \u03b1),\n (\u2200 {x : \u03b1}, a < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00aca \u2208 A \u2227 a \u2208 B \u2194\n (\u2200 (x : \u03b1), (x \u2208 A \u2192 a < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 (x \u2208 B \u2192 a < x \u2192 (x \u2208 A \u2194 x \u2208 B))) \u2227 \u00aca \u2208 A \u2227 a \u2208 B\n[PROOFSTEP]\nintro k\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA\u271d B\u271d : Colex \u03b1\nA B : Finset \u03b1\nk : \u03b1\n\u22a2 (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B \u2194\n (\u2200 (x : \u03b1), (x \u2208 A \u2192 k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 (x \u2208 B \u2192 k < x \u2192 (x \u2208 A \u2194 x \u2208 B))) \u2227 \u00ack \u2208 A \u2227 k \u2208 B\n[PROOFSTEP]\nrefine' and_congr_left' (forall_congr' _)\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA\u271d B\u271d : Colex \u03b1\nA B : Finset \u03b1\nk : \u03b1\n\u22a2 \u2200 (a : \u03b1), k < a \u2192 (a \u2208 A \u2194 a \u2208 B) \u2194 (a \u2208 A \u2192 k < a \u2192 (a \u2208 A \u2194 a \u2208 B)) \u2227 (a \u2208 B \u2192 k < a \u2192 (a \u2208 A \u2194 a \u2208 B))\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d\u00b9 : LT (Colex \u03b1) := instLT\nsrc\u271d : LE (Colex \u03b1) := instLE\nA B : Colex \u03b1\n\u22a2 A < B \u2194 A \u2264 B \u2227 \u00acB \u2264 A\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d\u00b9 : LT (Colex \u03b1) := instLT\nsrc\u271d : LE (Colex \u03b1) := instLE\nA B : Colex \u03b1\n\u22a2 A < B \u2192 A \u2264 B \u2227 \u00acB \u2264 A\n[PROOFSTEP]\nintro t\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d\u00b9 : LT (Colex \u03b1) := instLT\nsrc\u271d : LE (Colex \u03b1) := instLE\nA B : Colex \u03b1\nt : A < B\n\u22a2 A \u2264 B \u2227 \u00acB \u2264 A\n[PROOFSTEP]\nrefine' \u27e8Or.inl t, _\u27e9\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d\u00b9 : LT (Colex \u03b1) := instLT\nsrc\u271d : LE (Colex \u03b1) := instLE\nA B : Colex \u03b1\nt : A < B\n\u22a2 \u00acB \u2264 A\n[PROOFSTEP]\nrintro (i | rfl)\n[GOAL]\ncase mp.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d\u00b9 : LT (Colex \u03b1) := instLT\nsrc\u271d : LE (Colex \u03b1) := instLE\nA B : Colex \u03b1\nt : A < B\ni : B < A\n\u22a2 False\n[PROOFSTEP]\napply asymm_of _ t i\n[GOAL]\ncase mp.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d\u00b9 : LT (Colex \u03b1) := instLT\nsrc\u271d : LE (Colex \u03b1) := instLE\nB : Colex \u03b1\nt : B < B\n\u22a2 False\n[PROOFSTEP]\napply irrefl _ t\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d\u00b9 : LT (Colex \u03b1) := instLT\nsrc\u271d : LE (Colex \u03b1) := instLE\nA B : Colex \u03b1\n\u22a2 A \u2264 B \u2227 \u00acB \u2264 A \u2192 A < B\n[PROOFSTEP]\nrintro \u27e8h\u2081 | rfl, h\u2082\u27e9\n[GOAL]\ncase mpr.intro.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d\u00b9 : LT (Colex \u03b1) := instLT\nsrc\u271d : LE (Colex \u03b1) := instLE\nA B : Colex \u03b1\nh\u2082 : \u00acB \u2264 A\nh\u2081 : A < B\n\u22a2 A < B\n[PROOFSTEP]\napply h\u2081\n[GOAL]\ncase mpr.intro.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d\u00b9 : LT (Colex \u03b1) := instLT\nsrc\u271d : LE (Colex \u03b1) := instLE\nA : Colex \u03b1\nh\u2082 : \u00acA \u2264 A\n\u22a2 A < A\n[PROOFSTEP]\napply h\u2082.elim (Or.inr rfl)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d\u00b9 : LT (Colex \u03b1) := instLT\nsrc\u271d : LE (Colex \u03b1) := instLE\n\u22a2 DecidableRel fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nletI : DecidableEq (Finset.Colex \u03b1) := inferInstanceAs (DecidableEq (Finset \u03b1))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d\u00b9 : LT (Colex \u03b1) := instLT\nsrc\u271d : LE (Colex \u03b1) := instLE\nthis : DecidableEq (Colex \u03b1) := inferInstanceAs (DecidableEq (Finset \u03b1))\n\u22a2 DecidableRel fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nexact fun A B => inferInstanceAs (Decidable (A < B \u2228 A = B))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMono f\nA B : Finset \u03b1\n\u22a2 toColex (image f A) \u2264 toColex (image f B) \u2194 toColex A \u2264 toColex B\n[PROOFSTEP]\nrw [le_iff_le_iff_lt_iff_lt, hom_lt_iff h\u2081]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Finset \u03b1\nt : \u03b1\nh\u2081 : toColex A < toColex B\nh\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t\n\u22a2 \u2200 (x : \u03b1), x \u2208 A \u2192 x < t\n[PROOFSTEP]\nrw [Colex.lt_def] at h\u2081 \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Finset \u03b1\nt : \u03b1\nh\u2081 : \u2203 k, (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B\nh\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t\n\u22a2 \u2200 (x : \u03b1), x \u2208 A \u2192 x < t\n[PROOFSTEP]\nrcases h\u2081 with \u27e8k, z, _, _\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Finset \u03b1\nt : \u03b1\nh\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nleft\u271d : \u00ack \u2208 A\nright\u271d : k \u2208 B\n\u22a2 \u2200 (x : \u03b1), x \u2208 A \u2192 x < t\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Finset \u03b1\nt : \u03b1\nh\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nleft\u271d : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx : \u03b1\nhx : x \u2208 A\n\u22a2 x < t\n[PROOFSTEP]\napply lt_of_not_ge\n[GOAL]\ncase intro.intro.intro.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Finset \u03b1\nt : \u03b1\nh\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nleft\u271d : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx : \u03b1\nhx : x \u2208 A\n\u22a2 \u00acx \u2265 t\n[PROOFSTEP]\nintro a\n[GOAL]\ncase intro.intro.intro.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Finset \u03b1\nt : \u03b1\nh\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nleft\u271d : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx : \u03b1\nhx : x \u2208 A\na : x \u2265 t\n\u22a2 False\n[PROOFSTEP]\nrefine' not_lt_of_ge a (h\u2082 x _)\n[GOAL]\ncase intro.intro.intro.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Finset \u03b1\nt : \u03b1\nh\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nleft\u271d : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx : \u03b1\nhx : x \u2208 A\na : x \u2265 t\n\u22a2 x \u2208 B\n[PROOFSTEP]\nrwa [\u2190 z]\n[GOAL]\ncase intro.intro.intro.h\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Finset \u03b1\nt : \u03b1\nh\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nleft\u271d : \u00ack \u2208 A\nright\u271d : k \u2208 B\nx : \u03b1\nhx : x \u2208 A\na : x \u2265 t\n\u22a2 k < x\n[PROOFSTEP]\napply lt_of_lt_of_le (h\u2082 k \u2039_\u203a) a\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\n\u22a2 toColex s < toColex {r} \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x < r\n[PROOFSTEP]\nsimp only [lt_def, mem_singleton, \u2190 and_assoc, exists_eq_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\n\u22a2 (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x < r\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\n\u22a2 (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 x < r\n[PROOFSTEP]\nintro t x hx\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nt : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x < r\n[PROOFSTEP]\nrw [\u2190 not_le]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nt : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u00acr \u2264 x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nt : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\nx : \u03b1\nhx : x \u2208 s\nh : r \u2264 x\n\u22a2 False\n[PROOFSTEP]\nrcases lt_or_eq_of_le h with (h\u2081 | rfl)\n[GOAL]\ncase mp.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nt : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\nx : \u03b1\nhx : x \u2208 s\nh : r \u2264 x\nh\u2081 : r < x\n\u22a2 False\n[PROOFSTEP]\nexact ne_of_irrefl h\u2081 ((t.1 h\u2081).1 hx).symm\n[GOAL]\ncase mp.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nt : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\nhx : r \u2208 s\nh : r \u2264 r\n\u22a2 False\n[PROOFSTEP]\nexact t.2 hx\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 x < r) \u2192 (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\n[PROOFSTEP]\nexact fun h => \u27e8fun {z} hz => \u27e8fun i => (asymm hz (h _ i)).elim, fun i => (hz.ne' i).elim\u27e9, by simpa using h r\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 x < r\n\u22a2 \u00acr \u2208 s\n[PROOFSTEP]\nsimpa using h r\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\n\u22a2 toColex {r} \u2264 toColex s \u2194 \u2203 x, x \u2208 s \u2227 r \u2264 x\n[PROOFSTEP]\nsimp only [\u2190 not_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\n\u22a2 \u00actoColex s < toColex {r} \u2194 \u2203 x, x \u2208 s \u2227 \u00acx < r\n[PROOFSTEP]\nsimp [lt_singleton_iff_mem_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr s : \u03b1\n\u22a2 toColex {r} < toColex {s} \u2194 r < s\n[PROOFSTEP]\nsimp [lt_singleton_iff_mem_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr s : \u03b1\n\u22a2 toColex {r} \u2264 toColex {s} \u2194 r \u2264 s\n[PROOFSTEP]\nrw [le_iff_le_iff_lt_iff_lt, singleton_lt_iff_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\n\u22a2 toColex (A \\ B) < toColex (B \\ A) \u2194 toColex A < toColex B\n[PROOFSTEP]\nrw [Colex.lt_def, Colex.lt_def]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\n\u22a2 (\u2203 k, (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)) \u2227 \u00ack \u2208 A \\ B \u2227 k \u2208 B \\ A) \u2194\n \u2203 k, (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B\n[PROOFSTEP]\napply exists_congr\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\n\u22a2 \u2200 (a : \u03b1),\n (\u2200 {x : \u03b1}, a < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)) \u2227 \u00aca \u2208 A \\ B \u2227 a \u2208 B \\ A \u2194\n (\u2200 {x : \u03b1}, a < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00aca \u2208 A \u2227 a \u2208 B\n[PROOFSTEP]\nintro k\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nk : \u03b1\n\u22a2 (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)) \u2227 \u00ack \u2208 A \\ B \u2227 k \u2208 B \\ A \u2194\n (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B\n[PROOFSTEP]\nsimp only [mem_sdiff, not_and, not_not]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nk : \u03b1\n\u22a2 (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2227 \u00acx \u2208 B \u2194 x \u2208 B \u2227 \u00acx \u2208 A)) \u2227 (k \u2208 A \u2192 k \u2208 B) \u2227 k \u2208 B \u2227 \u00ack \u2208 A \u2194\n (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nk : \u03b1\n\u22a2 (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2227 \u00acx \u2208 B \u2194 x \u2208 B \u2227 \u00acx \u2208 A)) \u2227 (k \u2208 A \u2192 k \u2208 B) \u2227 k \u2208 B \u2227 \u00ack \u2208 A \u2192\n (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B\n[PROOFSTEP]\nrintro \u27e8z, kAB, kB, kA\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2227 \u00acx \u2208 B \u2194 x \u2208 B \u2227 \u00acx \u2208 A)\nkAB : k \u2208 A \u2192 k \u2208 B\nkB : k \u2208 B\nkA : \u00ack \u2208 A\n\u22a2 (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B\n[PROOFSTEP]\nrefine' \u27e8_, kA, kB\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2227 \u00acx \u2208 B \u2194 x \u2208 B \u2227 \u00acx \u2208 A)\nkAB : k \u2208 A \u2192 k \u2208 B\nkB : k \u2208 B\nkA : \u00ack \u2208 A\n\u22a2 \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h.mp.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2227 \u00acx \u2208 B \u2194 x \u2208 B \u2227 \u00acx \u2208 A)\nkAB : k \u2208 A \u2192 k \u2208 B\nkB : k \u2208 B\nkA : \u00ack \u2208 A\nx : \u03b1\nhx : k < x\n\u22a2 x \u2208 A \u2194 x \u2208 B\n[PROOFSTEP]\nspecialize z hx\n[GOAL]\ncase h.mp.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nk : \u03b1\nkAB : k \u2208 A \u2192 k \u2208 B\nkB : k \u2208 B\nkA : \u00ack \u2208 A\nx : \u03b1\nhx : k < x\nz : x \u2208 A \u2227 \u00acx \u2208 B \u2194 x \u2208 B \u2227 \u00acx \u2208 A\n\u22a2 x \u2208 A \u2194 x \u2208 B\n[PROOFSTEP]\ntauto\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nk : \u03b1\n\u22a2 (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B \u2192\n (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2227 \u00acx \u2208 B \u2194 x \u2208 B \u2227 \u00acx \u2208 A)) \u2227 (k \u2208 A \u2192 k \u2208 B) \u2227 k \u2208 B \u2227 \u00ack \u2208 A\n[PROOFSTEP]\nrintro \u27e8z, kA, kB\u27e9\n[GOAL]\ncase h.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nkA : \u00ack \u2208 A\nkB : k \u2208 B\n\u22a2 (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2227 \u00acx \u2208 B \u2194 x \u2208 B \u2227 \u00acx \u2208 A)) \u2227 (k \u2208 A \u2192 k \u2208 B) \u2227 k \u2208 B \u2227 \u00ack \u2208 A\n[PROOFSTEP]\nrefine' \u27e8_, fun _ => kB, kB, kA\u27e9\n[GOAL]\ncase h.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nkA : \u00ack \u2208 A\nkB : k \u2208 B\n\u22a2 \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2227 \u00acx \u2208 B \u2194 x \u2208 B \u2227 \u00acx \u2208 A)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : DecidableEq \u03b1\nA B : Finset \u03b1\nk : \u03b1\nz : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)\nkA : \u00ack \u2208 A\nkB : k \u2208 B\nx : \u03b1\nhx : k < x\n\u22a2 x \u2208 A \u2227 \u00acx \u2208 B \u2194 x \u2208 B \u2227 \u00acx \u2208 A\n[PROOFSTEP]\nrw [z hx]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Finset \u03b1\n\u22a2 toColex (A \\ B) \u2264 toColex (B \\ A) \u2194 toColex A \u2264 toColex B\n[PROOFSTEP]\nrw [le_iff_le_iff_lt_iff_lt, sdiff_lt_sdiff_iff_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA : Finset \u03b1\nhA : Finset.Nonempty A\n\u22a2 toColex \u2205 < toColex A\n[PROOFSTEP]\nrw [Colex.lt_def]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA : Finset \u03b1\nhA : Finset.Nonempty A\n\u22a2 \u2203 k, (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 \u2205 \u2194 x \u2208 A)) \u2227 \u00ack \u2208 \u2205 \u2227 k \u2208 A\n[PROOFSTEP]\nrefine' \u27e8max' _ hA, _, by simp, max'_mem _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA : Finset \u03b1\nhA : Finset.Nonempty A\n\u22a2 \u00acmax' A hA \u2208 \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA : Finset \u03b1\nhA : Finset.Nonempty A\n\u22a2 \u2200 {x : \u03b1}, max' A hA < x \u2192 (x \u2208 \u2205 \u2194 x \u2208 A)\n[PROOFSTEP]\nsimp only [false_iff_iff, not_mem_empty]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA : Finset \u03b1\nhA : Finset.Nonempty A\n\u22a2 \u2200 {x : \u03b1}, max' A hA < x \u2192 \u00acx \u2208 A\n[PROOFSTEP]\nintro x hx t\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA : Finset \u03b1\nhA : Finset.Nonempty A\nx : \u03b1\nhx : max' A hA < x\nt : x \u2208 A\n\u22a2 False\n[PROOFSTEP]\napply not_le_of_lt hx (le_max' _ _ t)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Finset \u03b1\nh : A \u2282 B\n\u22a2 toColex A < toColex B\n[PROOFSTEP]\nrw [\u2190 sdiff_lt_sdiff_iff_lt, sdiff_eq_empty_iff_subset.2 h.1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Finset \u03b1\nh : A \u2282 B\n\u22a2 toColex \u2205 < toColex (B \\ A)\n[PROOFSTEP]\nexact empty_toColex_lt (by simpa [Finset.Nonempty] using exists_of_ssubset h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Finset \u03b1\nh : A \u2282 B\n\u22a2 Finset.Nonempty (B \\ A)\n[PROOFSTEP]\nsimpa [Finset.Nonempty] using exists_of_ssubset h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA : Finset \u03b1\n\u22a2 toColex \u2205 \u2264 toColex A\n[PROOFSTEP]\nrcases A.eq_empty_or_nonempty with (rfl | hA)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\n\u22a2 toColex \u2205 \u2264 toColex \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA : Finset \u03b1\nhA : Finset.Nonempty A\n\u22a2 toColex \u2205 \u2264 toColex A\n[PROOFSTEP]\napply (empty_toColex_lt hA).le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Finset \u03b1\nh : A \u2286 B\n\u22a2 toColex A \u2264 toColex B\n[PROOFSTEP]\nrw [\u2190 sdiff_le_sdiff_iff_le, sdiff_eq_empty_iff_subset.2 h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nA B : Finset \u03b1\nh : A \u2286 B\n\u22a2 toColex \u2205 \u2264 toColex (B \\ A)\n[PROOFSTEP]\napply empty_toColex_le\n[GOAL]\n\u03b1 : Type u_1\nA B : Finset \u2115\n\u22a2 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i \u2194 toColex A < toColex B\n[PROOFSTEP]\nhave z : \u2200 A B : Finset \u2115, A.toColex < B.toColex \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i :=\n by\n intro A B\n rw [\u2190 sdiff_lt_sdiff_iff_lt, Colex.lt_def]\n rintro \u27e8k, z, kA, kB\u27e9\n rw [\u2190 sdiff_union_inter A B]\n conv_rhs => rw [\u2190 sdiff_union_inter B A]\n rw [sum_union (disjoint_sdiff_inter _ _), sum_union (disjoint_sdiff_inter _ _), inter_comm, add_lt_add_iff_right]\n apply lt_of_lt_of_le (@Nat.sum_two_pow_lt k (A \\ B) _)\n \u00b7 apply single_le_sum (fun _ _ => Nat.zero_le _) kB\n intro x hx\n apply lt_of_le_of_ne (le_of_not_lt _)\n \u00b7 apply ne_of_mem_of_not_mem hx kA\n intro kx\n have := (z kx).1 hx\n rw [mem_sdiff] at this hx \n exact hx.2 this.1\n[GOAL]\n\u03b1 : Type u_1\nA B : Finset \u2115\n\u22a2 \u2200 (A B : Finset \u2115), toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i\n[PROOFSTEP]\nintro A B\n[GOAL]\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\n\u22a2 toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i\n[PROOFSTEP]\nrw [\u2190 sdiff_lt_sdiff_iff_lt, Colex.lt_def]\n[GOAL]\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\n\u22a2 (\u2203 k, (\u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)) \u2227 \u00ack \u2208 A \\ B \u2227 k \u2208 B \\ A) \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i\n[PROOFSTEP]\nrintro \u27e8k, z, kA, kB\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i\n[PROOFSTEP]\nrw [\u2190 sdiff_union_inter A B]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 \u2211 i in A \\ B \u222a A \u2229 B, 2 ^ i < \u2211 i in B, 2 ^ i\n[PROOFSTEP]\nconv_rhs => rw [\u2190 sdiff_union_inter B A]\n[GOAL]\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n| \u2211 i in B, 2 ^ i\n[PROOFSTEP]\nrw [\u2190 sdiff_union_inter B A]\n[GOAL]\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n| \u2211 i in B, 2 ^ i\n[PROOFSTEP]\nrw [\u2190 sdiff_union_inter B A]\n[GOAL]\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n| \u2211 i in B, 2 ^ i\n[PROOFSTEP]\nrw [\u2190 sdiff_union_inter B A]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 \u2211 i in A \\ B \u222a A \u2229 B, 2 ^ i < \u2211 i in B \\ A \u222a B \u2229 A, 2 ^ i\n[PROOFSTEP]\nrw [sum_union (disjoint_sdiff_inter _ _), sum_union (disjoint_sdiff_inter _ _), inter_comm, add_lt_add_iff_right]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 \u2211 x in A \\ B, 2 ^ x < \u2211 x in B \\ A, 2 ^ x\n[PROOFSTEP]\napply lt_of_lt_of_le (@Nat.sum_two_pow_lt k (A \\ B) _)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 2 ^ k \u2264 \u2211 x in B \\ A, 2 ^ x\n[PROOFSTEP]\napply single_le_sum (fun _ _ => Nat.zero_le _) kB\n[GOAL]\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 \u2200 {x : \u2115}, x \u2208 A \\ B \u2192 x < k\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \\ B\n\u22a2 x < k\n[PROOFSTEP]\napply lt_of_le_of_ne (le_of_not_lt _)\n[GOAL]\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \\ B\n\u22a2 x \u2260 k\n[PROOFSTEP]\napply ne_of_mem_of_not_mem hx kA\n[GOAL]\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \\ B\n\u22a2 \u00ack < x\n[PROOFSTEP]\nintro kx\n[GOAL]\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \\ B\nkx : k < x\n\u22a2 False\n[PROOFSTEP]\nhave := (z kx).1 hx\n[GOAL]\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \\ B\nkx : k < x\nthis : x \u2208 B \\ A\n\u22a2 False\n[PROOFSTEP]\nrw [mem_sdiff] at this hx \n[GOAL]\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \u2227 \u00acx \u2208 B\nkx : k < x\nthis : x \u2208 B \u2227 \u00acx \u2208 A\n\u22a2 False\n[PROOFSTEP]\nexact hx.2 this.1\n[GOAL]\n\u03b1 : Type u_1\nA B : Finset \u2115\nz : \u2200 (A B : Finset \u2115), toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i\n\u22a2 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i \u2194 toColex A < toColex B\n[PROOFSTEP]\nrefine' \u27e8fun h => (lt_trichotomy A B).resolve_right fun h\u2081 => h\u2081.elim _ (not_lt_of_gt h \u2218 z _ _), z A B\u27e9\n[GOAL]\n\u03b1 : Type u_1\nA B : Finset \u2115\nz : \u2200 (A B : Finset \u2115), toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i\nh : \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i\nh\u2081 : A = B \u2228 B < A\n\u22a2 A = B \u2192 False\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\nA : Finset \u2115\nz : \u2200 (A B : Finset \u2115), toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i\nh : \u2211 i in A, 2 ^ i < \u2211 i in A, 2 ^ i\nh\u2081 : A = A \u2228 A < A\n\u22a2 False\n[PROOFSTEP]\napply irrefl _ h\n[GOAL]\n\u03b1 : Type u_1\nA B : Finset \u2115\n\u22a2 \u2211 i in A, 2 ^ i \u2264 \u2211 i in B, 2 ^ i \u2194 toColex A \u2264 toColex B\n[PROOFSTEP]\nrw [le_iff_le_iff_lt_iff_lt, sum_two_pow_lt_iff_lt]\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.Colex", "llama_tokens": 22429, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.42951421278873386}} {"text": "[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf' : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u22a2 LinearMap.range (ContinuousLinearMap.prod f' \u03c6') \u2260 \u22a4\n[PROOFSTEP]\nintro htop\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf' : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nhtop : LinearMap.range (ContinuousLinearMap.prod f' \u03c6') = \u22a4\n\u22a2 False\n[PROOFSTEP]\nset f\u03c6 := fun x => (f x, \u03c6 x)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf' : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nhtop : LinearMap.range (ContinuousLinearMap.prod f' \u03c6') = \u22a4\nf\u03c6 : E \u2192 F \u00d7 \u211d := fun x => (f x, \u03c6 x)\n\u22a2 False\n[PROOFSTEP]\nhave A : map \u03c6 (\ud835\udcdd[f \u207b\u00b9' {f x\u2080}] x\u2080) = \ud835\udcdd (\u03c6 x\u2080) :=\n by\n change map (Prod.snd \u2218 f\u03c6) (\ud835\udcdd[f\u03c6 \u207b\u00b9' {p | p.1 = f x\u2080}] x\u2080) = \ud835\udcdd (\u03c6 x\u2080)\n rw [\u2190 map_map, nhdsWithin, map_inf_principal_preimage, (hf'.prod h\u03c6').map_nhds_eq_of_surj htop]\n exact map_snd_nhdsWithin _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf' : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nhtop : LinearMap.range (ContinuousLinearMap.prod f' \u03c6') = \u22a4\nf\u03c6 : E \u2192 F \u00d7 \u211d := fun x => (f x, \u03c6 x)\n\u22a2 map \u03c6 (\ud835\udcdd[f \u207b\u00b9' {f x\u2080}] x\u2080) = \ud835\udcdd (\u03c6 x\u2080)\n[PROOFSTEP]\nchange map (Prod.snd \u2218 f\u03c6) (\ud835\udcdd[f\u03c6 \u207b\u00b9' {p | p.1 = f x\u2080}] x\u2080) = \ud835\udcdd (\u03c6 x\u2080)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf' : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nhtop : LinearMap.range (ContinuousLinearMap.prod f' \u03c6') = \u22a4\nf\u03c6 : E \u2192 F \u00d7 \u211d := fun x => (f x, \u03c6 x)\n\u22a2 map (Prod.snd \u2218 f\u03c6) (\ud835\udcdd[f\u03c6 \u207b\u00b9' {p | p.fst = f x\u2080}] x\u2080) = \ud835\udcdd (\u03c6 x\u2080)\n[PROOFSTEP]\nrw [\u2190 map_map, nhdsWithin, map_inf_principal_preimage, (hf'.prod h\u03c6').map_nhds_eq_of_surj htop]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf' : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nhtop : LinearMap.range (ContinuousLinearMap.prod f' \u03c6') = \u22a4\nf\u03c6 : E \u2192 F \u00d7 \u211d := fun x => (f x, \u03c6 x)\n\u22a2 map Prod.snd (\ud835\udcdd (f x\u2080, \u03c6 x\u2080) \u2293 \ud835\udcdf {p | p.fst = f x\u2080}) = \ud835\udcdd (\u03c6 x\u2080)\n[PROOFSTEP]\nexact map_snd_nhdsWithin _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf' : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nhtop : LinearMap.range (ContinuousLinearMap.prod f' \u03c6') = \u22a4\nf\u03c6 : E \u2192 F \u00d7 \u211d := fun x => (f x, \u03c6 x)\nA : map \u03c6 (\ud835\udcdd[f \u207b\u00b9' {f x\u2080}] x\u2080) = \ud835\udcdd (\u03c6 x\u2080)\n\u22a2 False\n[PROOFSTEP]\nexact hextr.not_nhds_le_map A.ge\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf' : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u22a2 \u2203 \u039b \u039b\u2080, (\u039b, \u039b\u2080) \u2260 0 \u2227 \u2200 (x : E), \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\n[PROOFSTEP]\nrcases Submodule.exists_le_ker_of_lt_top _ (lt_top_iff_ne_top.2 <| hextr.range_ne_top_of_hasStrictFDerivAt hf' h\u03c6') with\n \u27e8\u039b', h0, h\u039b'\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf' : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u039b' : F \u00d7 \u211d \u2192\u2097[\u211d] \u211d\nh0 : \u039b' \u2260 0\nh\u039b' : LinearMap.range (ContinuousLinearMap.prod f' \u03c6') \u2264 LinearMap.ker \u039b'\n\u22a2 \u2203 \u039b \u039b\u2080, (\u039b, \u039b\u2080) \u2260 0 \u2227 \u2200 (x : E), \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\n[PROOFSTEP]\nset e : ((F \u2192\u2097[\u211d] \u211d) \u00d7 \u211d) \u2243\u2097[\u211d] F \u00d7 \u211d \u2192\u2097[\u211d] \u211d :=\n ((LinearEquiv.refl \u211d (F \u2192\u2097[\u211d] \u211d)).prod (LinearMap.ringLmapEquivSelf \u211d \u211d \u211d).symm).trans (LinearMap.coprodEquiv \u211d)\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf' : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u039b' : F \u00d7 \u211d \u2192\u2097[\u211d] \u211d\nh0 : \u039b' \u2260 0\nh\u039b' : LinearMap.range (ContinuousLinearMap.prod f' \u03c6') \u2264 LinearMap.ker \u039b'\ne : ((F \u2192\u2097[\u211d] \u211d) \u00d7 \u211d) \u2243\u2097[\u211d] F \u00d7 \u211d \u2192\u2097[\u211d] \u211d :=\n LinearEquiv.trans\n (LinearEquiv.prod (LinearEquiv.refl \u211d (F \u2192\u2097[\u211d] \u211d)) (LinearEquiv.symm (LinearMap.ringLmapEquivSelf \u211d \u211d \u211d)))\n (LinearMap.coprodEquiv \u211d)\n\u22a2 \u2203 \u039b \u039b\u2080, (\u039b, \u039b\u2080) \u2260 0 \u2227 \u2200 (x : E), \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\n[PROOFSTEP]\nrcases e.surjective \u039b' with \u27e8\u27e8\u039b, \u039b\u2080\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.mk\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf' : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\ne : ((F \u2192\u2097[\u211d] \u211d) \u00d7 \u211d) \u2243\u2097[\u211d] F \u00d7 \u211d \u2192\u2097[\u211d] \u211d :=\n LinearEquiv.trans\n (LinearEquiv.prod (LinearEquiv.refl \u211d (F \u2192\u2097[\u211d] \u211d)) (LinearEquiv.symm (LinearMap.ringLmapEquivSelf \u211d \u211d \u211d)))\n (LinearMap.coprodEquiv \u211d)\n\u039b : F \u2192\u2097[\u211d] \u211d\n\u039b\u2080 : \u211d\nh0 : \u2191e (\u039b, \u039b\u2080) \u2260 0\nh\u039b' : LinearMap.range (ContinuousLinearMap.prod f' \u03c6') \u2264 LinearMap.ker (\u2191e (\u039b, \u039b\u2080))\n\u22a2 \u2203 \u039b \u039b\u2080, (\u039b, \u039b\u2080) \u2260 0 \u2227 \u2200 (x : E), \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\n[PROOFSTEP]\nrefine' \u27e8\u039b, \u039b\u2080, e.map_ne_zero_iff.1 h0, fun x => _\u27e9\n[GOAL]\ncase intro.intro.intro.mk\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf' : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\ne : ((F \u2192\u2097[\u211d] \u211d) \u00d7 \u211d) \u2243\u2097[\u211d] F \u00d7 \u211d \u2192\u2097[\u211d] \u211d :=\n LinearEquiv.trans\n (LinearEquiv.prod (LinearEquiv.refl \u211d (F \u2192\u2097[\u211d] \u211d)) (LinearEquiv.symm (LinearMap.ringLmapEquivSelf \u211d \u211d \u211d)))\n (LinearMap.coprodEquiv \u211d)\n\u039b : F \u2192\u2097[\u211d] \u211d\n\u039b\u2080 : \u211d\nh0 : \u2191e (\u039b, \u039b\u2080) \u2260 0\nh\u039b' : LinearMap.range (ContinuousLinearMap.prod f' \u03c6') \u2264 LinearMap.ker (\u2191e (\u039b, \u039b\u2080))\nx : E\n\u22a2 \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\n[PROOFSTEP]\nconvert LinearMap.congr_fun (LinearMap.range_le_ker_iff.1 h\u039b') x using 1\n -- squeezed `simp [mul_comm]` to speed up elaboration\n[GOAL]\ncase h.e'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf' : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\ne : ((F \u2192\u2097[\u211d] \u211d) \u00d7 \u211d) \u2243\u2097[\u211d] F \u00d7 \u211d \u2192\u2097[\u211d] \u211d :=\n LinearEquiv.trans\n (LinearEquiv.prod (LinearEquiv.refl \u211d (F \u2192\u2097[\u211d] \u211d)) (LinearEquiv.symm (LinearMap.ringLmapEquivSelf \u211d \u211d \u211d)))\n (LinearMap.coprodEquiv \u211d)\n\u039b : F \u2192\u2097[\u211d] \u211d\n\u039b\u2080 : \u211d\nh0 : \u2191e (\u039b, \u039b\u2080) \u2260 0\nh\u039b' : LinearMap.range (ContinuousLinearMap.prod f' \u03c6') \u2264 LinearMap.ker (\u2191e (\u039b, \u039b\u2080))\nx : E\n\u22a2 \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = \u2191(LinearMap.comp (\u2191e (\u039b, \u039b\u2080)) \u2191(ContinuousLinearMap.prod f' \u03c6')) x\n[PROOFSTEP]\nsimp only [smul_eq_mul, LinearEquiv.trans_apply, LinearEquiv.prod_apply, LinearEquiv.refl_apply,\n LinearMap.ringLmapEquivSelf_symm_apply, LinearMap.coprodEquiv_apply, ContinuousLinearMap.coe_prod,\n LinearMap.coprod_comp_prod, LinearMap.add_apply, LinearMap.coe_comp, ContinuousLinearMap.coe_coe, Function.comp_apply,\n LinearMap.coe_smulRight, LinearMap.one_apply, mul_comm]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nf : E \u2192 \u211d\nf' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u22a2 \u2203 a b, (a, b) \u2260 0 \u2227 a \u2022 f' + b \u2022 \u03c6' = 0\n[PROOFSTEP]\nobtain \u27e8\u039b, \u039b\u2080, h\u039b, hf\u039b\u27e9 := hextr.exists_linear_map_of_hasStrictFDerivAt hf' h\u03c6'\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nf : E \u2192 \u211d\nf' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u039b : Module.Dual \u211d \u211d\n\u039b\u2080 : \u211d\nh\u039b : (\u039b, \u039b\u2080) \u2260 0\nhf\u039b : \u2200 (x : E), \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\n\u22a2 \u2203 a b, (a, b) \u2260 0 \u2227 a \u2022 f' + b \u2022 \u03c6' = 0\n[PROOFSTEP]\nrefine' \u27e8\u039b 1, \u039b\u2080, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nf : E \u2192 \u211d\nf' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u039b : Module.Dual \u211d \u211d\n\u039b\u2080 : \u211d\nh\u039b : (\u039b, \u039b\u2080) \u2260 0\nhf\u039b : \u2200 (x : E), \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\n\u22a2 (\u2191\u039b 1, \u039b\u2080) \u2260 0\n[PROOFSTEP]\ncontrapose! h\u039b\n[GOAL]\ncase intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nf : E \u2192 \u211d\nf' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u039b : Module.Dual \u211d \u211d\n\u039b\u2080 : \u211d\nhf\u039b : \u2200 (x : E), \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\nh\u039b : (\u2191\u039b 1, \u039b\u2080) = 0\n\u22a2 (\u039b, \u039b\u2080) = 0\n[PROOFSTEP]\nsimp only [Prod.mk_eq_zero] at h\u039b \u22a2\n[GOAL]\ncase intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nf : E \u2192 \u211d\nf' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u039b : Module.Dual \u211d \u211d\n\u039b\u2080 : \u211d\nhf\u039b : \u2200 (x : E), \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\nh\u039b : \u2191\u039b 1 = 0 \u2227 \u039b\u2080 = 0\n\u22a2 \u039b = 0 \u2227 \u039b\u2080 = 0\n[PROOFSTEP]\nrefine' \u27e8LinearMap.ext fun x => _, h\u039b.2\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nf : E \u2192 \u211d\nf' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u039b : Module.Dual \u211d \u211d\n\u039b\u2080 : \u211d\nhf\u039b : \u2200 (x : E), \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\nh\u039b : \u2191\u039b 1 = 0 \u2227 \u039b\u2080 = 0\nx : \u211d\n\u22a2 \u2191\u039b x = \u21910 x\n[PROOFSTEP]\nsimpa [h\u039b.1] using \u039b.map_smul x 1\n[GOAL]\ncase intro.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nf : E \u2192 \u211d\nf' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u039b : Module.Dual \u211d \u211d\n\u039b\u2080 : \u211d\nh\u039b : (\u039b, \u039b\u2080) \u2260 0\nhf\u039b : \u2200 (x : E), \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\n\u22a2 \u2191\u039b 1 \u2022 f' + \u039b\u2080 \u2022 \u03c6' = 0\n[PROOFSTEP]\next x\n[GOAL]\ncase intro.intro.intro.refine'_2.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nf : E \u2192 \u211d\nf' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u039b : Module.Dual \u211d \u211d\n\u039b\u2080 : \u211d\nh\u039b : (\u039b, \u039b\u2080) \u2260 0\nhf\u039b : \u2200 (x : E), \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\nx : E\n\u22a2 \u2191(\u2191\u039b 1 \u2022 f' + \u039b\u2080 \u2022 \u03c6') x = \u21910 x\n[PROOFSTEP]\nhave H\u2081 : \u039b (f' x) = f' x * \u039b 1 := by simpa only [mul_one, Algebra.id.smul_eq_mul] using \u039b.map_smul (f' x) 1\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nf : E \u2192 \u211d\nf' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u039b : Module.Dual \u211d \u211d\n\u039b\u2080 : \u211d\nh\u039b : (\u039b, \u039b\u2080) \u2260 0\nhf\u039b : \u2200 (x : E), \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\nx : E\n\u22a2 \u2191\u039b (\u2191f' x) = \u2191f' x * \u2191\u039b 1\n[PROOFSTEP]\nsimpa only [mul_one, Algebra.id.smul_eq_mul] using \u039b.map_smul (f' x) 1\n[GOAL]\ncase intro.intro.intro.refine'_2.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nf : E \u2192 \u211d\nf' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u039b : Module.Dual \u211d \u211d\n\u039b\u2080 : \u211d\nh\u039b : (\u039b, \u039b\u2080) \u2260 0\nhf\u039b : \u2200 (x : E), \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\nx : E\nH\u2081 : \u2191\u039b (\u2191f' x) = \u2191f' x * \u2191\u039b 1\n\u22a2 \u2191(\u2191\u039b 1 \u2022 f' + \u039b\u2080 \u2022 \u03c6') x = \u21910 x\n[PROOFSTEP]\nhave H\u2082 : f' x * \u039b 1 + \u039b\u2080 * \u03c6' x = 0 := by simpa only [Algebra.id.smul_eq_mul, H\u2081] using hf\u039b x\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nf : E \u2192 \u211d\nf' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u039b : Module.Dual \u211d \u211d\n\u039b\u2080 : \u211d\nh\u039b : (\u039b, \u039b\u2080) \u2260 0\nhf\u039b : \u2200 (x : E), \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\nx : E\nH\u2081 : \u2191\u039b (\u2191f' x) = \u2191f' x * \u2191\u039b 1\n\u22a2 \u2191f' x * \u2191\u039b 1 + \u039b\u2080 * \u2191\u03c6' x = 0\n[PROOFSTEP]\nsimpa only [Algebra.id.smul_eq_mul, H\u2081] using hf\u039b x\n[GOAL]\ncase intro.intro.intro.refine'_2.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\nf : E \u2192 \u211d\nf' : E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | f x = f x\u2080} x\u2080\nhf' : HasStrictFDerivAt f f' x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u039b : Module.Dual \u211d \u211d\n\u039b\u2080 : \u211d\nh\u039b : (\u039b, \u039b\u2080) \u2260 0\nhf\u039b : \u2200 (x : E), \u2191\u039b (\u2191f' x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\nx : E\nH\u2081 : \u2191\u039b (\u2191f' x) = \u2191f' x * \u2191\u039b 1\nH\u2082 : \u2191f' x * \u2191\u039b 1 + \u039b\u2080 * \u2191\u03c6' x = 0\n\u22a2 \u2191(\u2191\u039b 1 \u2022 f' + \u039b\u2080 \u2022 \u03c6') x = \u21910 x\n[PROOFSTEP]\nsimpa [mul_comm] using H\u2082\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | \u2200 (i : \u03b9), f i x = f i x\u2080} x\u2080\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u22a2 \u2203 \u039b \u039b\u2080, (\u039b, \u039b\u2080) \u2260 0 \u2227 \u2211 i : \u03b9, \u039b i \u2022 f' i + \u039b\u2080 \u2022 \u03c6' = 0\n[PROOFSTEP]\nletI := Classical.decEq \u03b9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | \u2200 (i : \u03b9), f i x = f i x\u2080} x\u2080\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 \u2203 \u039b \u039b\u2080, (\u039b, \u039b\u2080) \u2260 0 \u2227 \u2211 i : \u03b9, \u039b i \u2022 f' i + \u039b\u2080 \u2022 \u03c6' = 0\n[PROOFSTEP]\nreplace hextr : IsLocalExtrOn \u03c6 {x | (fun i => f i x) = fun i => f i x\u2080} x\u2080\n[GOAL]\ncase hextr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | \u2200 (i : \u03b9), f i x = f i x\u2080} x\u2080\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 IsLocalExtrOn \u03c6 {x | (fun i => f i x) = fun i => f i x\u2080} x\u2080\n[PROOFSTEP]\nsimpa only [Function.funext_iff] using hextr\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\nhextr : IsLocalExtrOn \u03c6 {x | (fun i => f i x) = fun i => f i x\u2080} x\u2080\n\u22a2 \u2203 \u039b \u039b\u2080, (\u039b, \u039b\u2080) \u2260 0 \u2227 \u2211 i : \u03b9, \u039b i \u2022 f' i + \u039b\u2080 \u2022 \u03c6' = 0\n[PROOFSTEP]\nrcases hextr.exists_linear_map_of_hasStrictFDerivAt (hasStrictFDerivAt_pi.2 fun i => hf' i) h\u03c6' with \u27e8\u039b, \u039b\u2080, h0, hsum\u27e9\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\nhextr : IsLocalExtrOn \u03c6 {x | (fun i => f i x) = fun i => f i x\u2080} x\u2080\n\u039b : Module.Dual \u211d (\u03b9 \u2192 \u211d)\n\u039b\u2080 : \u211d\nh0 : (\u039b, \u039b\u2080) \u2260 0\nhsum : \u2200 (x : E), \u2191\u039b (\u2191(ContinuousLinearMap.pi fun i => f' i) x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\n\u22a2 \u2203 \u039b \u039b\u2080, (\u039b, \u039b\u2080) \u2260 0 \u2227 \u2211 i : \u03b9, \u039b i \u2022 f' i + \u039b\u2080 \u2022 \u03c6' = 0\n[PROOFSTEP]\nrcases(LinearEquiv.piRing \u211d \u211d \u03b9 \u211d).symm.surjective \u039b with \u27e8\u039b, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\nhextr : IsLocalExtrOn \u03c6 {x | (fun i => f i x) = fun i => f i x\u2080} x\u2080\n\u039b\u2080 : \u211d\n\u039b : \u03b9 \u2192 \u211d\nh0 : (\u2191(LinearEquiv.symm (LinearEquiv.piRing \u211d \u211d \u03b9 \u211d)) \u039b, \u039b\u2080) \u2260 0\nhsum :\n \u2200 (x : E),\n \u2191(\u2191(LinearEquiv.symm (LinearEquiv.piRing \u211d \u211d \u03b9 \u211d)) \u039b) (\u2191(ContinuousLinearMap.pi fun i => f' i) x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\n\u22a2 \u2203 \u039b \u039b\u2080, (\u039b, \u039b\u2080) \u2260 0 \u2227 \u2211 i : \u03b9, \u039b i \u2022 f' i + \u039b\u2080 \u2022 \u03c6' = 0\n[PROOFSTEP]\nrefine' \u27e8\u039b, \u039b\u2080, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\nhextr : IsLocalExtrOn \u03c6 {x | (fun i => f i x) = fun i => f i x\u2080} x\u2080\n\u039b\u2080 : \u211d\n\u039b : \u03b9 \u2192 \u211d\nh0 : (\u2191(LinearEquiv.symm (LinearEquiv.piRing \u211d \u211d \u03b9 \u211d)) \u039b, \u039b\u2080) \u2260 0\nhsum :\n \u2200 (x : E),\n \u2191(\u2191(LinearEquiv.symm (LinearEquiv.piRing \u211d \u211d \u03b9 \u211d)) \u039b) (\u2191(ContinuousLinearMap.pi fun i => f' i) x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\n\u22a2 (\u039b, \u039b\u2080) \u2260 0\n[PROOFSTEP]\nsimpa only [Ne.def, Prod.ext_iff, LinearEquiv.map_eq_zero_iff, Prod.fst_zero] using h0\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\nhextr : IsLocalExtrOn \u03c6 {x | (fun i => f i x) = fun i => f i x\u2080} x\u2080\n\u039b\u2080 : \u211d\n\u039b : \u03b9 \u2192 \u211d\nh0 : (\u2191(LinearEquiv.symm (LinearEquiv.piRing \u211d \u211d \u03b9 \u211d)) \u039b, \u039b\u2080) \u2260 0\nhsum :\n \u2200 (x : E),\n \u2191(\u2191(LinearEquiv.symm (LinearEquiv.piRing \u211d \u211d \u03b9 \u211d)) \u039b) (\u2191(ContinuousLinearMap.pi fun i => f' i) x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\n\u22a2 \u2211 i : \u03b9, \u039b i \u2022 f' i + \u039b\u2080 \u2022 \u03c6' = 0\n[PROOFSTEP]\next x\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\nhextr : IsLocalExtrOn \u03c6 {x | (fun i => f i x) = fun i => f i x\u2080} x\u2080\n\u039b\u2080 : \u211d\n\u039b : \u03b9 \u2192 \u211d\nh0 : (\u2191(LinearEquiv.symm (LinearEquiv.piRing \u211d \u211d \u03b9 \u211d)) \u039b, \u039b\u2080) \u2260 0\nhsum :\n \u2200 (x : E),\n \u2191(\u2191(LinearEquiv.symm (LinearEquiv.piRing \u211d \u211d \u03b9 \u211d)) \u039b) (\u2191(ContinuousLinearMap.pi fun i => f' i) x) + \u039b\u2080 \u2022 \u2191\u03c6' x = 0\nx : E\n\u22a2 \u2191(\u2211 i : \u03b9, \u039b i \u2022 f' i + \u039b\u2080 \u2022 \u03c6') x = \u21910 x\n[PROOFSTEP]\nsimpa [mul_comm] using hsum x\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | \u2200 (i : \u03b9), f i x = f i x\u2080} x\u2080\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\n\u22a2 \u00acLinearIndependent \u211d (Option.elim' \u03c6' f')\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | \u2200 (i : \u03b9), f i x = f i x\u2080} x\u2080\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nval\u271d : Fintype \u03b9\n\u22a2 \u00acLinearIndependent \u211d (Option.elim' \u03c6' f')\n[PROOFSTEP]\nrw [Fintype.linearIndependent_iff]\n[GOAL]\ncase intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | \u2200 (i : \u03b9), f i x = f i x\u2080} x\u2080\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nval\u271d : Fintype \u03b9\n\u22a2 \u00ac\u2200 (g : Option \u03b9 \u2192 \u211d), \u2211 i : Option \u03b9, g i \u2022 Option.elim' \u03c6' f' i = 0 \u2192 \u2200 (i : Option \u03b9), g i = 0\n[PROOFSTEP]\npush_neg\n[GOAL]\ncase intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | \u2200 (i : \u03b9), f i x = f i x\u2080} x\u2080\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 g, \u2211 i : Option \u03b9, g i \u2022 Option.elim' \u03c6' f' i = 0 \u2227 \u2203 i, g i \u2260 0\n[PROOFSTEP]\nrcases hextr.exists_multipliers_of_hasStrictFDerivAt hf' h\u03c6' with \u27e8\u039b, \u039b\u2080, h\u039b, h\u039bf\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | \u2200 (i : \u03b9), f i x = f i x\u2080} x\u2080\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nval\u271d : Fintype \u03b9\n\u039b : \u03b9 \u2192 \u211d\n\u039b\u2080 : \u211d\nh\u039b : (\u039b, \u039b\u2080) \u2260 0\nh\u039bf : \u2211 i : \u03b9, \u039b i \u2022 f' i + \u039b\u2080 \u2022 \u03c6' = 0\n\u22a2 \u2203 g, \u2211 i : Option \u03b9, g i \u2022 Option.elim' \u03c6' f' i = 0 \u2227 \u2203 i, g i \u2260 0\n[PROOFSTEP]\nrefine' \u27e8Option.elim' \u039b\u2080 \u039b, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | \u2200 (i : \u03b9), f i x = f i x\u2080} x\u2080\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nval\u271d : Fintype \u03b9\n\u039b : \u03b9 \u2192 \u211d\n\u039b\u2080 : \u211d\nh\u039b : (\u039b, \u039b\u2080) \u2260 0\nh\u039bf : \u2211 i : \u03b9, \u039b i \u2022 f' i + \u039b\u2080 \u2022 \u03c6' = 0\n\u22a2 \u2211 i : Option \u03b9, Option.elim' \u039b\u2080 \u039b i \u2022 Option.elim' \u03c6' f' i = 0\n[PROOFSTEP]\nsimpa [add_comm] using h\u039bf\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf\u271d : E \u2192 F\n\u03c6 : E \u2192 \u211d\nx\u2080 : E\nf'\u271d : E \u2192L[\u211d] F\n\u03c6' : E \u2192L[\u211d] \u211d\n\u03b9 : Type u_3\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E \u2192 \u211d\nf' : \u03b9 \u2192 E \u2192L[\u211d] \u211d\nhextr : IsLocalExtrOn \u03c6 {x | \u2200 (i : \u03b9), f i x = f i x\u2080} x\u2080\nhf' : \u2200 (i : \u03b9), HasStrictFDerivAt (f i) (f' i) x\u2080\nh\u03c6' : HasStrictFDerivAt \u03c6 \u03c6' x\u2080\nval\u271d : Fintype \u03b9\n\u039b : \u03b9 \u2192 \u211d\n\u039b\u2080 : \u211d\nh\u039b : (\u039b, \u039b\u2080) \u2260 0\nh\u039bf : \u2211 i : \u03b9, \u039b i \u2022 f' i + \u039b\u2080 \u2022 \u03c6' = 0\n\u22a2 \u2203 i, Option.elim' \u039b\u2080 \u039b i \u2260 0\n[PROOFSTEP]\nsimpa only [Function.funext_iff, not_and_or, or_comm, Option.exists, Prod.mk_eq_zero, Ne.def, not_forall] using h\u039b\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.LagrangeMultipliers", "llama_tokens": 15422, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833945721305, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.42948361493903536}} {"text": "[GOAL]\nI : Type v\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nJ : Type v\u2081\ninst\u271d : SmallCategory J\nF : J \u2964 ((i : I) \u2192 C i)\nc : (i : I) \u2192 Cone (F \u22d9 Pi.eval C i)\nj j' : J\nf : j \u27f6 j'\n\u22a2 ((Functor.const J).obj fun i => (c i).pt).map f \u226b (fun j i => NatTrans.app (c i).\u03c0 j) j' =\n (fun j i => NatTrans.app (c i).\u03c0 j) j \u226b F.map f\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h\nI : Type v\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nJ : Type v\u2081\ninst\u271d : SmallCategory J\nF : J \u2964 ((i : I) \u2192 C i)\nc : (i : I) \u2192 Cone (F \u22d9 Pi.eval C i)\nj j' : J\nf : j \u27f6 j'\ni : I\n\u22a2 (((Functor.const J).obj fun i => (c i).pt).map f \u226b (fun j i => NatTrans.app (c i).\u03c0 j) j') i =\n ((fun j i => NatTrans.app (c i).\u03c0 j) j \u226b F.map f) i\n[PROOFSTEP]\nexact (c i).\u03c0.naturality f\n[GOAL]\nI : Type v\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nJ : Type v\u2081\ninst\u271d : SmallCategory J\nF : J \u2964 ((i : I) \u2192 C i)\nc : (i : I) \u2192 Cocone (F \u22d9 Pi.eval C i)\nj j' : J\nf : j \u27f6 j'\n\u22a2 F.map f \u226b (fun j i => NatTrans.app (c i).\u03b9 j) j' =\n (fun j i => NatTrans.app (c i).\u03b9 j) j \u226b ((Functor.const J).obj fun i => (c i).pt).map f\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h\nI : Type v\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nJ : Type v\u2081\ninst\u271d : SmallCategory J\nF : J \u2964 ((i : I) \u2192 C i)\nc : (i : I) \u2192 Cocone (F \u22d9 Pi.eval C i)\nj j' : J\nf : j \u27f6 j'\ni : I\n\u22a2 (F.map f \u226b (fun j i => NatTrans.app (c i).\u03b9 j) j') i =\n ((fun j i => NatTrans.app (c i).\u03b9 j) j \u226b ((Functor.const J).obj fun i => (c i).pt).map f) i\n[PROOFSTEP]\nexact (c i).\u03b9.naturality f\n[GOAL]\nI : Type v\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nJ : Type v\u2081\ninst\u271d : SmallCategory J\nF : J \u2964 ((i : I) \u2192 C i)\nc : (i : I) \u2192 Cone (F \u22d9 Pi.eval C i)\nP : (i : I) \u2192 IsLimit (c i)\ns : Cone F\nj : J\n\u22a2 (fun s i => IsLimit.lift (P i) (coneCompEval s i)) s \u226b NatTrans.app (coneOfConeCompEval c).\u03c0 j = NatTrans.app s.\u03c0 j\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h\nI : Type v\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nJ : Type v\u2081\ninst\u271d : SmallCategory J\nF : J \u2964 ((i : I) \u2192 C i)\nc : (i : I) \u2192 Cone (F \u22d9 Pi.eval C i)\nP : (i : I) \u2192 IsLimit (c i)\ns : Cone F\nj : J\ni : I\n\u22a2 ((fun s i => IsLimit.lift (P i) (coneCompEval s i)) s \u226b NatTrans.app (coneOfConeCompEval c).\u03c0 j) i =\n NatTrans.app s.\u03c0 j i\n[PROOFSTEP]\nexact (P i).fac (coneCompEval s i) j\n[GOAL]\nI : Type v\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nJ : Type v\u2081\ninst\u271d : SmallCategory J\nF : J \u2964 ((i : I) \u2192 C i)\nc : (i : I) \u2192 Cone (F \u22d9 Pi.eval C i)\nP : (i : I) \u2192 IsLimit (c i)\ns : Cone F\nm : s.pt \u27f6 (coneOfConeCompEval c).pt\nw : \u2200 (j : J), m \u226b NatTrans.app (coneOfConeCompEval c).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m = (fun s i => IsLimit.lift (P i) (coneCompEval s i)) s\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h\nI : Type v\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nJ : Type v\u2081\ninst\u271d : SmallCategory J\nF : J \u2964 ((i : I) \u2192 C i)\nc : (i : I) \u2192 Cone (F \u22d9 Pi.eval C i)\nP : (i : I) \u2192 IsLimit (c i)\ns : Cone F\nm : s.pt \u27f6 (coneOfConeCompEval c).pt\nw : \u2200 (j : J), m \u226b NatTrans.app (coneOfConeCompEval c).\u03c0 j = NatTrans.app s.\u03c0 j\ni : I\n\u22a2 m i = (fun s i => IsLimit.lift (P i) (coneCompEval s i)) s i\n[PROOFSTEP]\nexact (P i).uniq (coneCompEval s i) (m i) fun j => congr_fun (w j) i\n[GOAL]\nI : Type v\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nJ : Type v\u2081\ninst\u271d : SmallCategory J\nF : J \u2964 ((i : I) \u2192 C i)\nc : (i : I) \u2192 Cocone (F \u22d9 Pi.eval C i)\nP : (i : I) \u2192 IsColimit (c i)\ns : Cocone F\nj : J\n\u22a2 NatTrans.app (coconeOfCoconeCompEval c).\u03b9 j \u226b (fun s i => IsColimit.desc (P i) (coconeCompEval s i)) s =\n NatTrans.app s.\u03b9 j\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h\nI : Type v\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nJ : Type v\u2081\ninst\u271d : SmallCategory J\nF : J \u2964 ((i : I) \u2192 C i)\nc : (i : I) \u2192 Cocone (F \u22d9 Pi.eval C i)\nP : (i : I) \u2192 IsColimit (c i)\ns : Cocone F\nj : J\ni : I\n\u22a2 (NatTrans.app (coconeOfCoconeCompEval c).\u03b9 j \u226b (fun s i => IsColimit.desc (P i) (coconeCompEval s i)) s) i =\n NatTrans.app s.\u03b9 j i\n[PROOFSTEP]\nexact (P i).fac (coconeCompEval s i) j\n[GOAL]\nI : Type v\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nJ : Type v\u2081\ninst\u271d : SmallCategory J\nF : J \u2964 ((i : I) \u2192 C i)\nc : (i : I) \u2192 Cocone (F \u22d9 Pi.eval C i)\nP : (i : I) \u2192 IsColimit (c i)\ns : Cocone F\nm : (coconeOfCoconeCompEval c).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (coconeOfCoconeCompEval c).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n\u22a2 m = (fun s i => IsColimit.desc (P i) (coconeCompEval s i)) s\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h\nI : Type v\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nJ : Type v\u2081\ninst\u271d : SmallCategory J\nF : J \u2964 ((i : I) \u2192 C i)\nc : (i : I) \u2192 Cocone (F \u22d9 Pi.eval C i)\nP : (i : I) \u2192 IsColimit (c i)\ns : Cocone F\nm : (coconeOfCoconeCompEval c).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (coconeOfCoconeCompEval c).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\ni : I\n\u22a2 m i = (fun s i => IsColimit.desc (P i) (coconeCompEval s i)) s i\n[PROOFSTEP]\nexact (P i).uniq (coconeCompEval s i) (m i) fun j => congr_fun (w j) i\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Pi", "llama_tokens": 2586, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833789613196, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.42948360628330173}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g\u271d : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ng : \u03ba \u2192 \u03b1\ne : \u03b9 \u2192 \u03ba\nhe : Injective e\nhs : \u2200 (c : \u03ba), \u00acc \u2208 Set.range e \u2192 0 \u2264 g c\nh : \u2200 (i : \u03b9), f i \u2264 g (e i)\nhf : HasSum f a\u2081\nhg : HasSum g a\u2082\n\u22a2 a\u2081 \u2264 a\u2082\n[PROOFSTEP]\nrw [\u2190 hasSum_extend_zero he] at hf \n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g\u271d : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ng : \u03ba \u2192 \u03b1\ne : \u03b9 \u2192 \u03ba\nhe : Injective e\nhs : \u2200 (c : \u03ba), \u00acc \u2208 Set.range e \u2192 0 \u2264 g c\nh : \u2200 (i : \u03b9), f i \u2264 g (e i)\nhf : HasSum (extend e f 0) a\u2081\nhg : HasSum g a\u2082\n\u22a2 a\u2081 \u2264 a\u2082\n[PROOFSTEP]\nrefine hasSum_le (fun c => ?_) hf hg\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g\u271d : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ng : \u03ba \u2192 \u03b1\ne : \u03b9 \u2192 \u03ba\nhe : Injective e\nhs : \u2200 (c : \u03ba), \u00acc \u2208 Set.range e \u2192 0 \u2264 g c\nh : \u2200 (i : \u03b9), f i \u2264 g (e i)\nhf : HasSum (extend e f 0) a\u2081\nhg : HasSum g a\u2082\nc : \u03ba\n\u22a2 extend e f 0 c \u2264 g c\n[PROOFSTEP]\nobtain \u27e8i, rfl\u27e9 | h := em (c \u2208 Set.range e)\n[GOAL]\ncase inl.intro\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g\u271d : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ng : \u03ba \u2192 \u03b1\ne : \u03b9 \u2192 \u03ba\nhe : Injective e\nhs : \u2200 (c : \u03ba), \u00acc \u2208 Set.range e \u2192 0 \u2264 g c\nh : \u2200 (i : \u03b9), f i \u2264 g (e i)\nhf : HasSum (extend e f 0) a\u2081\nhg : HasSum g a\u2082\ni : \u03b9\n\u22a2 extend e f 0 (e i) \u2264 g (e i)\n[PROOFSTEP]\nrw [he.extend_apply]\n[GOAL]\ncase inl.intro\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g\u271d : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ng : \u03ba \u2192 \u03b1\ne : \u03b9 \u2192 \u03ba\nhe : Injective e\nhs : \u2200 (c : \u03ba), \u00acc \u2208 Set.range e \u2192 0 \u2264 g c\nh : \u2200 (i : \u03b9), f i \u2264 g (e i)\nhf : HasSum (extend e f 0) a\u2081\nhg : HasSum g a\u2082\ni : \u03b9\n\u22a2 f i \u2264 g (e i)\n[PROOFSTEP]\nexact h _\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g\u271d : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ng : \u03ba \u2192 \u03b1\ne : \u03b9 \u2192 \u03ba\nhe : Injective e\nhs : \u2200 (c : \u03ba), \u00acc \u2208 Set.range e \u2192 0 \u2264 g c\nh\u271d : \u2200 (i : \u03b9), f i \u2264 g (e i)\nhf : HasSum (extend e f 0) a\u2081\nhg : HasSum g a\u2082\nc : \u03ba\nh : \u00acc \u2208 Set.range e\n\u22a2 extend e f 0 c \u2264 g c\n[PROOFSTEP]\nrw [extend_apply' _ _ _ h]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g\u271d : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ng : \u03ba \u2192 \u03b1\ne : \u03b9 \u2192 \u03ba\nhe : Injective e\nhs : \u2200 (c : \u03ba), \u00acc \u2208 Set.range e \u2192 0 \u2264 g c\nh\u271d : \u2200 (i : \u03b9), f i \u2264 g (e i)\nhf : HasSum (extend e f 0) a\u2081\nhg : HasSum g a\u2082\nc : \u03ba\nh : \u00acc \u2208 Set.range e\n\u22a2 OfNat.ofNat 0 c \u2264 g c\n[PROOFSTEP]\nexact hs _ h\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nhf : HasSum f a\ni : \u03b9\nhb : \u2200 (j : \u03b9), j \u2260 i \u2192 0 \u2264 f j\n\u22a2 \u2200 (i_1 : \u03b9), \u00aci_1 \u2208 {i} \u2192 0 \u2264 f i_1\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nha\u2082 : 0 \u2264 a\u2082\nh : \u2200 (s : Finset \u03b9), \u2211 i in s, f i \u2264 a\u2082\n\u22a2 \u2211' (i : \u03b9), f i \u2264 a\u2082\n[PROOFSTEP]\nby_cases hf : Summable f\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nha\u2082 : 0 \u2264 a\u2082\nh : \u2200 (s : Finset \u03b9), \u2211 i in s, f i \u2264 a\u2082\nhf : Summable f\n\u22a2 \u2211' (i : \u03b9), f i \u2264 a\u2082\n[PROOFSTEP]\nexact tsum_le_of_sum_le hf h\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nha\u2082 : 0 \u2264 a\u2082\nh : \u2200 (s : Finset \u03b9), \u2211 i in s, f i \u2264 a\u2082\nhf : \u00acSummable f\n\u22a2 \u2211' (i : \u03b9), f i \u2264 a\u2082\n[PROOFSTEP]\nrw [tsum_eq_zero_of_not_summable hf]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nha\u2082 : 0 \u2264 a\u2082\nh : \u2200 (s : Finset \u03b9), \u2211 i in s, f i \u2264 a\u2082\nhf : \u00acSummable f\n\u22a2 0 \u2264 a\u2082\n[PROOFSTEP]\nexact ha\u2082\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 (i : \u03b9), 0 \u2264 g i\n\u22a2 0 \u2264 \u2211' (i : \u03b9), g i\n[PROOFSTEP]\nby_cases hg : Summable g\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 (i : \u03b9), 0 \u2264 g i\nhg : Summable g\n\u22a2 0 \u2264 \u2211' (i : \u03b9), g i\n[PROOFSTEP]\nexact hg.hasSum.nonneg h\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 (i : \u03b9), 0 \u2264 g i\nhg : \u00acSummable g\n\u22a2 0 \u2264 \u2211' (i : \u03b9), g i\n[PROOFSTEP]\nrw [tsum_eq_zero_of_not_summable hg]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 (i : \u03b9), f i \u2264 0\n\u22a2 \u2211' (i : \u03b9), f i \u2264 0\n[PROOFSTEP]\nby_cases hf : Summable f\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 (i : \u03b9), f i \u2264 0\nhf : Summable f\n\u22a2 \u2211' (i : \u03b9), f i \u2264 0\n[PROOFSTEP]\nexact hf.hasSum.nonpos h\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 (i : \u03b9), f i \u2264 0\nhf : \u00acSummable f\n\u22a2 \u2211' (i : \u03b9), f i \u2264 0\n[PROOFSTEP]\nrw [tsum_eq_zero_of_not_summable hf]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nhf : \u2200 (i : \u03b9), 0 \u2264 f i\n\u22a2 HasSum f 0 \u2194 f = 0\n[PROOFSTEP]\nrefine' \u27e8fun hf' => _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nhf : \u2200 (i : \u03b9), 0 \u2264 f i\nhf' : HasSum f 0\n\u22a2 f = 0\n[PROOFSTEP]\next i\n[GOAL]\ncase refine'_1.h\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nhf : \u2200 (i : \u03b9), 0 \u2264 f i\nhf' : HasSum f 0\ni : \u03b9\n\u22a2 f i = OfNat.ofNat 0 i\n[PROOFSTEP]\nexact (hf i).antisymm' (le_hasSum hf' _ fun j _ => hf j)\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nhf : \u2200 (i : \u03b9), 0 \u2264 f i\n\u22a2 f = 0 \u2192 HasSum f 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\ng : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nhf : \u2200 (i : \u03b9), 0 \u2264 OfNat.ofNat 0 i\n\u22a2 HasSum 0 0\n[PROOFSTEP]\nexact hasSum_zero\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na\u2081 a\u2082 : \u03b1\ni : \u03b9\nh : f \u2264 g\nhi : f i < g i\nhf : HasSum f a\u2081\nhg : HasSum g a\u2082\n\u22a2 a\u2081 < a\u2082\n[PROOFSTEP]\nhave : update f i 0 \u2264 update g i 0 := update_le_update_iff.mpr \u27e8rfl.le, fun i _ => h i\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na\u2081 a\u2082 : \u03b1\ni : \u03b9\nh : f \u2264 g\nhi : f i < g i\nhf : HasSum f a\u2081\nhg : HasSum g a\u2082\nthis : update f i 0 \u2264 update g i 0\n\u22a2 a\u2081 < a\u2082\n[PROOFSTEP]\nhave : 0 - f i + a\u2081 \u2264 0 - g i + a\u2082 := hasSum_le this (hf.update i 0) (hg.update i 0)\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na\u2081 a\u2082 : \u03b1\ni : \u03b9\nh : f \u2264 g\nhi : f i < g i\nhf : HasSum f a\u2081\nhg : HasSum g a\u2082\nthis\u271d : update f i 0 \u2264 update g i 0\nthis : 0 - f i + a\u2081 \u2264 0 - g i + a\u2082\n\u22a2 a\u2081 < a\u2082\n[PROOFSTEP]\nsimpa only [zero_sub, add_neg_cancel_left] using add_lt_add_of_lt_of_le hi this\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na\u2081 a\u2082 : \u03b1\ni\u271d : \u03b9\nhsum : Summable g\nhg : \u2200 (i : \u03b9), 0 \u2264 g i\ni : \u03b9\nhi : 0 < g i\n\u22a2 0 < \u2211' (i : \u03b9), g i\n[PROOFSTEP]\nrw [\u2190 tsum_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na\u2081 a\u2082 : \u03b1\ni\u271d : \u03b9\nhsum : Summable g\nhg : \u2200 (i : \u03b9), 0 \u2264 g i\ni : \u03b9\nhi : 0 < g i\n\u22a2 \u2211' (x : ?m.47496), 0 < \u2211' (i : \u03b9), g i\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalAddGroup \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na\u2081 a\u2082 : \u03b1\ni\u271d : \u03b9\nhsum : Summable g\nhg : \u2200 (i : \u03b9), 0 \u2264 g i\ni : \u03b9\nhi : 0 < g i\n\u22a2 Type ?u.47494\n[PROOFSTEP]\nexact tsum_lt_tsum hg hi summable_zero hsum\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf : \u03b9 \u2192 \u03b1\na : \u03b1\nhf : Summable f\n\u22a2 \u2211' (i : \u03b9), f i = 0 \u2194 \u2200 (x : \u03b9), f x = 0\n[PROOFSTEP]\nrw [\u2190 hasSum_zero_iff, hf.hasSum_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf : \u03b9 \u2192 \u03b1\na : \u03b1\nhf : Summable f\n\u22a2 \u2211' (i : \u03b9), f i \u2260 0 \u2194 \u2203 x, f x \u2260 0\n[PROOFSTEP]\nrw [Ne.def, tsum_eq_zero_iff hf, not_forall]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformAddGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u03b9 \u2192 \u03b1\ns : Set \u03b9 := {x | 0 \u2264 f x}\nh1 : \u2200 (x : \u2191s), |f \u2191x| = f \u2191x\nh2 : \u2200 (x : \u2191s\u1d9c), |f \u2191x| = -f \u2191x\n\u22a2 ((Summable fun x => |f \u2191x|) \u2227 Summable fun x => |f \u2191x|) \u2194 (Summable fun x => f \u2191x) \u2227 Summable fun x => -f \u2191x\n[PROOFSTEP]\nsimp only [h1, h2]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformAddGroup \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u03b9 \u2192 \u03b1\ns : Set \u03b9 := {x | 0 \u2264 f x}\nh1 : \u2200 (x : \u2191s), |f \u2191x| = f \u2191x\nh2 : \u2200 (x : \u2191s\u1d9c), |f \u2191x| = -f \u2191x\n\u22a2 ((Summable fun x => f \u2191x) \u2227 Summable fun x => -f \u2191x) \u2194 Summable f\n[PROOFSTEP]\nsimp only [summable_neg_iff, summable_subtype_and_compl]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Archimedean \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nb : \u03b1\nhb : 0 < b\nhf : Summable fun x => b\n\u22a2 Finite \u03b9\n[PROOFSTEP]\nhave H : \u2200 s : Finset \u03b9, s.card \u2022 b \u2264 \u2211' _ : \u03b9, b := fun s => by\n simpa using sum_le_hasSum s (fun a _ => hb.le) hf.hasSum\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Archimedean \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nb : \u03b1\nhb : 0 < b\nhf : Summable fun x => b\ns : Finset \u03b9\n\u22a2 card s \u2022 b \u2264 \u2211' (x : \u03b9), b\n[PROOFSTEP]\nsimpa using sum_le_hasSum s (fun a _ => hb.le) hf.hasSum\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Archimedean \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nb : \u03b1\nhb : 0 < b\nhf : Summable fun x => b\nH : \u2200 (s : Finset \u03b9), card s \u2022 b \u2264 \u2211' (x : \u03b9), b\n\u22a2 Finite \u03b9\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := Archimedean.arch (\u2211' _ : \u03b9, b) hb\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Archimedean \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nb : \u03b1\nhb : 0 < b\nhf : Summable fun x => b\nH : \u2200 (s : Finset \u03b9), card s \u2022 b \u2264 \u2211' (x : \u03b9), b\nn : \u2115\nhn : \u2211' (x : \u03b9), b \u2264 n \u2022 b\n\u22a2 Finite \u03b9\n[PROOFSTEP]\nhave : \u2200 s : Finset \u03b9, s.card \u2264 n := fun s => by simpa [nsmul_le_nsmul_iff hb] using (H s).trans hn\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Archimedean \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nb : \u03b1\nhb : 0 < b\nhf : Summable fun x => b\nH : \u2200 (s : Finset \u03b9), card s \u2022 b \u2264 \u2211' (x : \u03b9), b\nn : \u2115\nhn : \u2211' (x : \u03b9), b \u2264 n \u2022 b\ns : Finset \u03b9\n\u22a2 card s \u2264 n\n[PROOFSTEP]\nsimpa [nsmul_le_nsmul_iff hb] using (H s).trans hn\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Archimedean \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nb : \u03b1\nhb : 0 < b\nhf : Summable fun x => b\nH : \u2200 (s : Finset \u03b9), card s \u2022 b \u2264 \u2211' (x : \u03b9), b\nn : \u2115\nhn : \u2211' (x : \u03b9), b \u2264 n \u2022 b\nthis : \u2200 (s : Finset \u03b9), card s \u2264 n\n\u22a2 Finite \u03b9\n[PROOFSTEP]\nhave : Fintype \u03b9 := fintypeOfFinsetCardLe n this\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Archimedean \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nb : \u03b1\nhb : 0 < b\nhf : Summable fun x => b\nH : \u2200 (s : Finset \u03b9), card s \u2022 b \u2264 \u2211' (x : \u03b9), b\nn : \u2115\nhn : \u2211' (x : \u03b9), b \u2264 n \u2022 b\nthis\u271d : \u2200 (s : Finset \u03b9), card s \u2264 n\nthis : Fintype \u03b9\n\u22a2 Finite \u03b9\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.InfiniteSum.Order", "llama_tokens": 6992, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707281, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.42939825716643687}} {"text": "[GOAL]\n\u22a2 1 = \u2016\u2191reClm 1\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u20161\u2016 \u2264 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 1 = \u2016\u2191imClm I\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2016I\u2016 \u2264 1\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Complex.OperatorNorm", "llama_tokens": 103, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.4293891542046626}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf : Filter \u03b1\ng : Filter \u03b2\n\u22a2 \u2200 {x y : Set \u03b3},\n x \u2208 {s | \u2203 u v, u \u2208 f \u2227 v \u2208 g \u2227 image2 m u v \u2286 s} \u2192\n y \u2208 {s | \u2203 u v, u \u2208 f \u2227 v \u2208 g \u2227 image2 m u v \u2286 s} \u2192 x \u2229 y \u2208 {s | \u2203 u v, u \u2208 f \u2227 v \u2208 g \u2227 image2 m u v \u2286 s}\n[PROOFSTEP]\nsimp only [exists_prop, Set.mem_setOf_eq, subset_inter_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf : Filter \u03b1\ng : Filter \u03b2\n\u22a2 \u2200 {x y : Set \u03b3},\n (\u2203 u v, u \u2208 f \u2227 v \u2208 g \u2227 image2 m u v \u2286 x) \u2192\n (\u2203 u v, u \u2208 f \u2227 v \u2208 g \u2227 image2 m u v \u2286 y) \u2192 \u2203 u v, u \u2208 f \u2227 v \u2208 g \u2227 image2 m u v \u2286 x \u2227 image2 m u v \u2286 y\n[PROOFSTEP]\nrintro _ _ \u27e8s\u2081, s\u2082, hs\u2081, hs\u2082, hs\u27e9 \u27e8t\u2081, t\u2082, ht\u2081, ht\u2082, ht\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081\u271d s\u2082\u271d : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf : Filter \u03b1\ng : Filter \u03b2\nx\u271d y\u271d : Set \u03b3\ns\u2081 : Set \u03b1\ns\u2082 : Set \u03b2\nhs\u2081 : s\u2081 \u2208 f\nhs\u2082 : s\u2082 \u2208 g\nhs : image2 m s\u2081 s\u2082 \u2286 x\u271d\nt\u2081 : Set \u03b1\nt\u2082 : Set \u03b2\nht\u2081 : t\u2081 \u2208 f\nht\u2082 : t\u2082 \u2208 g\nht : image2 m t\u2081 t\u2082 \u2286 y\u271d\n\u22a2 \u2203 u v, u \u2208 f \u2227 v \u2208 g \u2227 image2 m u v \u2286 x\u271d \u2227 image2 m u v \u2286 y\u271d\n[PROOFSTEP]\nexact\n \u27e8s\u2081 \u2229 t\u2081, s\u2082 \u2229 t\u2082, inter_sets f hs\u2081 ht\u2081, inter_sets g hs\u2082 ht\u2082,\n (image2_subset (inter_subset_left _ _) <| inter_subset_left _ _).trans hs,\n (image2_subset (inter_subset_right _ _) <| inter_subset_right _ _).trans ht\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf : Filter \u03b1\ng : Filter \u03b2\n\u22a2 map (fun p => m p.fst p.snd) (f \u00d7\u02e2 g) = map\u2082 m f g\n[PROOFSTEP]\next s\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf : Filter \u03b1\ng : Filter \u03b2\ns : Set \u03b3\n\u22a2 s \u2208 map (fun p => m p.fst p.snd) (f \u00d7\u02e2 g) \u2194 s \u2208 map\u2082 m f g\n[PROOFSTEP]\nsimp [mem_prod_iff, prod_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nf : Filter \u03b1\ng : Filter \u03b2\n\u22a2 map\u2082 Prod.mk f g = f \u00d7\u02e2 g\n[PROOFSTEP]\nsimp only [\u2190 map_prod_eq_map\u2082, map_id']\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\n\u22a2 map\u2082 m f g = \u22a5 \u2194 f = \u22a5 \u2228 g = \u22a5\n[PROOFSTEP]\nsimp only [\u2190 empty_mem_iff_bot, mem_map\u2082_iff, subset_empty_iff, image2_eq_empty_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\n\u22a2 (\u2203 s t, s \u2208 f \u2227 t \u2208 g \u2227 (s = \u2205 \u2228 t = \u2205)) \u2194 \u2205 \u2208 f \u2228 \u2205 \u2208 g\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\n\u22a2 (\u2203 s t, s \u2208 f \u2227 t \u2208 g \u2227 (s = \u2205 \u2228 t = \u2205)) \u2192 \u2205 \u2208 f \u2228 \u2205 \u2208 g\n[PROOFSTEP]\nrintro \u27e8s, t, hs, ht, rfl | rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nt : Set \u03b2\nht : t \u2208 g\nhs : \u2205 \u2208 f\n\u22a2 \u2205 \u2208 f \u2228 \u2205 \u2208 g\n[PROOFSTEP]\nexact Or.inl hs\n[GOAL]\ncase mp.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\ns : Set \u03b1\nhs : s \u2208 f\nht : \u2205 \u2208 g\n\u22a2 \u2205 \u2208 f \u2228 \u2205 \u2208 g\n[PROOFSTEP]\nexact Or.inr ht\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\n\u22a2 \u2205 \u2208 f \u2228 \u2205 \u2208 g \u2192 \u2203 s t, s \u2208 f \u2227 t \u2208 g \u2227 (s = \u2205 \u2228 t = \u2205)\n[PROOFSTEP]\nrintro (h | h)\n[GOAL]\ncase mpr.inl\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nh : \u2205 \u2208 f\n\u22a2 \u2203 s t, s \u2208 f \u2227 t \u2208 g \u2227 (s = \u2205 \u2228 t = \u2205)\n[PROOFSTEP]\nexact \u27e8_, _, h, univ_mem, Or.inl rfl\u27e9\n[GOAL]\ncase mpr.inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nh : \u2205 \u2208 g\n\u22a2 \u2203 s t, s \u2208 f \u2227 t \u2208 g \u2227 (s = \u2205 \u2228 t = \u2205)\n[PROOFSTEP]\nexact \u27e8_, _, univ_mem, h, Or.inr rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\n\u22a2 NeBot (map\u2082 m f g) \u2194 NeBot f \u2227 NeBot g\n[PROOFSTEP]\nsimp_rw [neBot_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\n\u22a2 map\u2082 m f g \u2260 \u22a5 \u2194 f \u2260 \u22a5 \u2227 g \u2260 \u22a5\n[PROOFSTEP]\nexact map\u2082_eq_bot_iff.not.trans not_or\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\n\u22a2 map\u2082 m (f\u2081 \u2294 f\u2082) g = map\u2082 m f\u2081 g \u2294 map\u2082 m f\u2082 g\n[PROOFSTEP]\next u\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\n\u22a2 u \u2208 map\u2082 m (f\u2081 \u2294 f\u2082) g \u2194 u \u2208 map\u2082 m f\u2081 g \u2294 map\u2082 m f\u2082 g\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\n\u22a2 u \u2208 map\u2082 m (f\u2081 \u2294 f\u2082) g \u2192 u \u2208 map\u2082 m f\u2081 g \u2294 map\u2082 m f\u2082 g\n[PROOFSTEP]\nrintro \u27e8s, t, \u27e8h\u2081, h\u2082\u27e9, ht, hu\u27e9\n[GOAL]\ncase a.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081\u271d h\u2082\u271d : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\ns : Set \u03b1\nt : Set \u03b2\nh\u2081 : s \u2208 f\u2081.sets\nh\u2082 : s \u2208 f\u2082.sets\nht : t \u2208 g\nhu : image2 m s t \u2286 u\n\u22a2 u \u2208 map\u2082 m f\u2081 g \u2294 map\u2082 m f\u2082 g\n[PROOFSTEP]\nexact \u27e8mem_of_superset (image2_mem_map\u2082 h\u2081 ht) hu, mem_of_superset (image2_mem_map\u2082 h\u2082 ht) hu\u27e9\n[GOAL]\ncase a.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\n\u22a2 u \u2208 map\u2082 m f\u2081 g \u2294 map\u2082 m f\u2082 g \u2192 u \u2208 map\u2082 m (f\u2081 \u2294 f\u2082) g\n[PROOFSTEP]\nrintro \u27e8\u27e8s\u2081, t\u2081, hs\u2081, ht\u2081, hu\u2081\u27e9, s\u2082, t\u2082, hs\u2082, ht\u2082, hu\u2082\u27e9\n[GOAL]\ncase a.mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081\u271d s\u2082\u271d : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\ns\u2081 : Set \u03b1\nt\u2081 : Set \u03b2\nhs\u2081 : s\u2081 \u2208 f\u2081\nht\u2081 : t\u2081 \u2208 g\nhu\u2081 : image2 m s\u2081 t\u2081 \u2286 u\ns\u2082 : Set \u03b1\nt\u2082 : Set \u03b2\nhs\u2082 : s\u2082 \u2208 f\u2082\nht\u2082 : t\u2082 \u2208 g\nhu\u2082 : image2 m s\u2082 t\u2082 \u2286 u\n\u22a2 u \u2208 map\u2082 m (f\u2081 \u2294 f\u2082) g\n[PROOFSTEP]\nrefine' \u27e8s\u2081 \u222a s\u2082, t\u2081 \u2229 t\u2082, union_mem_sup hs\u2081 hs\u2082, inter_mem ht\u2081 ht\u2082, _\u27e9\n[GOAL]\ncase a.mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081\u271d s\u2082\u271d : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\ns\u2081 : Set \u03b1\nt\u2081 : Set \u03b2\nhs\u2081 : s\u2081 \u2208 f\u2081\nht\u2081 : t\u2081 \u2208 g\nhu\u2081 : image2 m s\u2081 t\u2081 \u2286 u\ns\u2082 : Set \u03b1\nt\u2082 : Set \u03b2\nhs\u2082 : s\u2082 \u2208 f\u2082\nht\u2082 : t\u2082 \u2208 g\nhu\u2082 : image2 m s\u2082 t\u2082 \u2286 u\n\u22a2 image2 m (s\u2081 \u222a s\u2082) (t\u2081 \u2229 t\u2082) \u2286 u\n[PROOFSTEP]\nrw [image2_union_left]\n[GOAL]\ncase a.mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081\u271d s\u2082\u271d : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\ns\u2081 : Set \u03b1\nt\u2081 : Set \u03b2\nhs\u2081 : s\u2081 \u2208 f\u2081\nht\u2081 : t\u2081 \u2208 g\nhu\u2081 : image2 m s\u2081 t\u2081 \u2286 u\ns\u2082 : Set \u03b1\nt\u2082 : Set \u03b2\nhs\u2082 : s\u2082 \u2208 f\u2082\nht\u2082 : t\u2082 \u2208 g\nhu\u2082 : image2 m s\u2082 t\u2082 \u2286 u\n\u22a2 image2 m s\u2081 (t\u2081 \u2229 t\u2082) \u222a image2 m s\u2082 (t\u2081 \u2229 t\u2082) \u2286 u\n[PROOFSTEP]\nexact\n union_subset ((image2_subset_left <| inter_subset_left _ _).trans hu\u2081)\n ((image2_subset_left <| inter_subset_right _ _).trans hu\u2082)\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\n\u22a2 map\u2082 m f (g\u2081 \u2294 g\u2082) = map\u2082 m f g\u2081 \u2294 map\u2082 m f g\u2082\n[PROOFSTEP]\next u\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\n\u22a2 u \u2208 map\u2082 m f (g\u2081 \u2294 g\u2082) \u2194 u \u2208 map\u2082 m f g\u2081 \u2294 map\u2082 m f g\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\n\u22a2 u \u2208 map\u2082 m f (g\u2081 \u2294 g\u2082) \u2192 u \u2208 map\u2082 m f g\u2081 \u2294 map\u2082 m f g\u2082\n[PROOFSTEP]\nrintro \u27e8s, t, hs, \u27e8h\u2081, h\u2082\u27e9, hu\u27e9\n[GOAL]\ncase a.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081\u271d h\u2082\u271d : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhs : s \u2208 f\nhu : image2 m s t \u2286 u\nh\u2081 : t \u2208 g\u2081.sets\nh\u2082 : t \u2208 g\u2082.sets\n\u22a2 u \u2208 map\u2082 m f g\u2081 \u2294 map\u2082 m f g\u2082\n[PROOFSTEP]\nexact \u27e8mem_of_superset (image2_mem_map\u2082 hs h\u2081) hu, mem_of_superset (image2_mem_map\u2082 hs h\u2082) hu\u27e9\n[GOAL]\ncase a.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\n\u22a2 u \u2208 map\u2082 m f g\u2081 \u2294 map\u2082 m f g\u2082 \u2192 u \u2208 map\u2082 m f (g\u2081 \u2294 g\u2082)\n[PROOFSTEP]\nrintro \u27e8\u27e8s\u2081, t\u2081, hs\u2081, ht\u2081, hu\u2081\u27e9, s\u2082, t\u2082, hs\u2082, ht\u2082, hu\u2082\u27e9\n[GOAL]\ncase a.mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081\u271d s\u2082\u271d : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\ns\u2081 : Set \u03b1\nt\u2081 : Set \u03b2\nhs\u2081 : s\u2081 \u2208 f\nht\u2081 : t\u2081 \u2208 g\u2081\nhu\u2081 : image2 m s\u2081 t\u2081 \u2286 u\ns\u2082 : Set \u03b1\nt\u2082 : Set \u03b2\nhs\u2082 : s\u2082 \u2208 f\nht\u2082 : t\u2082 \u2208 g\u2082\nhu\u2082 : image2 m s\u2082 t\u2082 \u2286 u\n\u22a2 u \u2208 map\u2082 m f (g\u2081 \u2294 g\u2082)\n[PROOFSTEP]\nrefine' \u27e8s\u2081 \u2229 s\u2082, t\u2081 \u222a t\u2082, inter_mem hs\u2081 hs\u2082, union_mem_sup ht\u2081 ht\u2082, _\u27e9\n[GOAL]\ncase a.mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081\u271d s\u2082\u271d : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\ns\u2081 : Set \u03b1\nt\u2081 : Set \u03b2\nhs\u2081 : s\u2081 \u2208 f\nht\u2081 : t\u2081 \u2208 g\u2081\nhu\u2081 : image2 m s\u2081 t\u2081 \u2286 u\ns\u2082 : Set \u03b1\nt\u2082 : Set \u03b2\nhs\u2082 : s\u2082 \u2208 f\nht\u2082 : t\u2082 \u2208 g\u2082\nhu\u2082 : image2 m s\u2082 t\u2082 \u2286 u\n\u22a2 image2 m (s\u2081 \u2229 s\u2082) (t\u2081 \u222a t\u2082) \u2286 u\n[PROOFSTEP]\nrw [image2_union_right]\n[GOAL]\ncase a.mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081\u271d s\u2082\u271d : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\ns\u2081 : Set \u03b1\nt\u2081 : Set \u03b2\nhs\u2081 : s\u2081 \u2208 f\nht\u2081 : t\u2081 \u2208 g\u2081\nhu\u2081 : image2 m s\u2081 t\u2081 \u2286 u\ns\u2082 : Set \u03b1\nt\u2082 : Set \u03b2\nhs\u2082 : s\u2082 \u2208 f\nht\u2082 : t\u2082 \u2208 g\u2082\nhu\u2082 : image2 m s\u2082 t\u2082 \u2286 u\n\u22a2 image2 m (s\u2081 \u2229 s\u2082) t\u2081 \u222a image2 m (s\u2081 \u2229 s\u2082) t\u2082 \u2286 u\n[PROOFSTEP]\nexact\n union_subset ((image2_subset_right <| inter_subset_left _ _).trans hu\u2081)\n ((image2_subset_right <| inter_subset_right _ _).trans hu\u2082)\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\nh : u \u2208 map (fun b => m a b) g\n\u22a2 image2 m {a} ((fun b => m a b) \u207b\u00b9' u) \u2286 u\n[PROOFSTEP]\nrw [image2_singleton_left, image_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\nh : u \u2208 map (fun a => m a b) f\n\u22a2 image2 m ((fun a => m a b) \u207b\u00b9' u) {b} \u2286 u\n[PROOFSTEP]\nrw [image2_singleton_right, image_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\n\u22a2 map\u2082 m (pure a) (pure b) = pure (m a b)\n[PROOFSTEP]\nrw [map\u2082_pure_right, map_pure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf : Filter \u03b1\ng : Filter \u03b2\n\u22a2 map\u2082 m f g = map\u2082 (fun a b => m b a) g f\n[PROOFSTEP]\next u\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf : Filter \u03b1\ng : Filter \u03b2\nu : Set \u03b3\n\u22a2 u \u2208 map\u2082 m f g \u2194 u \u2208 map\u2082 (fun a b => m b a) g f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf : Filter \u03b1\ng : Filter \u03b2\nu : Set \u03b3\n\u22a2 u \u2208 map\u2082 m f g \u2192 u \u2208 map\u2082 (fun a b => m b a) g f\n[PROOFSTEP]\nrintro \u27e8s, t, hs, ht, hu\u27e9\n[GOAL]\ncase a.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf : Filter \u03b1\ng : Filter \u03b2\nu : Set \u03b3\n\u22a2 u \u2208 map\u2082 (fun a b => m b a) g f \u2192 u \u2208 map\u2082 m f g\n[PROOFSTEP]\nrintro \u27e8s, t, hs, ht, hu\u27e9\n[GOAL]\ncase a.mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf : Filter \u03b1\ng : Filter \u03b2\nu : Set \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhs : s \u2208 f\nht : t \u2208 g\nhu : image2 m s t \u2286 u\n\u22a2 u \u2208 map\u2082 (fun a b => m b a) g f\n[PROOFSTEP]\nrefine' \u27e8t, s, ht, hs, by rwa [image2_swap]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf : Filter \u03b1\ng : Filter \u03b2\nu : Set \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhs : s \u2208 f\nht : t \u2208 g\nhu : image2 m s t \u2286 u\n\u22a2 image2 (fun a b => m b a) t s \u2286 u\n[PROOFSTEP]\nrwa [image2_swap]\n[GOAL]\ncase a.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf : Filter \u03b1\ng : Filter \u03b2\nu : Set \u03b3\ns : Set \u03b2\nt : Set \u03b1\nhs : s \u2208 g\nht : t \u2208 f\nhu : image2 (fun a b => m b a) s t \u2286 u\n\u22a2 u \u2208 map\u2082 m f g\n[PROOFSTEP]\nrefine' \u27e8t, s, ht, hs, by rwa [image2_swap]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf : Filter \u03b1\ng : Filter \u03b2\nu : Set \u03b3\ns : Set \u03b2\nt : Set \u03b1\nhs : s \u2208 g\nht : t \u2208 f\nhu : image2 (fun a b => m b a) s t \u2286 u\n\u22a2 image2 m t s \u2286 u\n[PROOFSTEP]\nrwa [image2_swap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nh : NeBot g\n\u22a2 map\u2082 (fun x x_1 => x) f g = f\n[PROOFSTEP]\next u\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nh : NeBot g\nu : Set \u03b1\n\u22a2 u \u2208 map\u2082 (fun x x_1 => x) f g \u2194 u \u2208 f\n[PROOFSTEP]\nrefine' \u27e8_, fun hu => \u27e8_, _, hu, univ_mem, (image2_left <| h.nonempty_of_mem univ_mem).subset\u27e9\u27e9\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nh : NeBot g\nu : Set \u03b1\n\u22a2 u \u2208 map\u2082 (fun x x_1 => x) f g \u2192 u \u2208 f\n[PROOFSTEP]\nrintro \u27e8s, t, hs, ht, hu\u27e9\n[GOAL]\ncase a.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nh : NeBot g\nu s : Set \u03b1\nt : Set \u03b2\nhs : s \u2208 f\nht : t \u2208 g\nhu : image2 (fun x x_1 => x) s t \u2286 u\n\u22a2 u \u2208 f\n[PROOFSTEP]\nrw [image2_left (h.nonempty_of_mem ht)] at hu \n[GOAL]\ncase a.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nh : NeBot g\nu s : Set \u03b1\nt : Set \u03b2\nhs : s \u2208 f\nht : t \u2208 g\nhu : s \u2286 u\n\u22a2 u \u2208 f\n[PROOFSTEP]\nexact mem_of_superset hs hu\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nh : NeBot f\n\u22a2 map\u2082 (fun x y => y) f g = g\n[PROOFSTEP]\nrw [map\u2082_swap, map\u2082_left h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\nf : Filter \u03b1\ng : Filter \u03b2\nh : Filter \u03b3\n\u22a2 \u2200 {x y : Set \u03b4},\n x \u2208 {s | \u2203 u v w, u \u2208 f \u2227 v \u2208 g \u2227 w \u2208 h \u2227 image3 m u v w \u2286 s} \u2192\n y \u2208 {s | \u2203 u v w, u \u2208 f \u2227 v \u2208 g \u2227 w \u2208 h \u2227 image3 m u v w \u2286 s} \u2192\n x \u2229 y \u2208 {s | \u2203 u v w, u \u2208 f \u2227 v \u2208 g \u2227 w \u2208 h \u2227 image3 m u v w \u2286 s}\n[PROOFSTEP]\nsimp only [exists_prop, mem_setOf_eq, subset_inter_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\nf : Filter \u03b1\ng : Filter \u03b2\nh : Filter \u03b3\n\u22a2 \u2200 {x y : Set \u03b4},\n (\u2203 u v w, u \u2208 f \u2227 v \u2208 g \u2227 w \u2208 h \u2227 image3 m u v w \u2286 x) \u2192\n (\u2203 u v w, u \u2208 f \u2227 v \u2208 g \u2227 w \u2208 h \u2227 image3 m u v w \u2286 y) \u2192\n \u2203 u v w, u \u2208 f \u2227 v \u2208 g \u2227 w \u2208 h \u2227 image3 m u v w \u2286 x \u2227 image3 m u v w \u2286 y\n[PROOFSTEP]\nrintro _ _ \u27e8s\u2081, s\u2082, s\u2083, hs\u2081, hs\u2082, hs\u2083, hs\u27e9 \u27e8t\u2081, t\u2082, t\u2083, ht\u2081, ht\u2082, ht\u2083, ht\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng\u271d g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns s\u2081\u271d s\u2082\u271d : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\nf : Filter \u03b1\ng : Filter \u03b2\nh : Filter \u03b3\nx\u271d y\u271d : Set \u03b4\ns\u2081 : Set \u03b1\ns\u2082 : Set \u03b2\ns\u2083 : Set \u03b3\nhs\u2081 : s\u2081 \u2208 f\nhs\u2082 : s\u2082 \u2208 g\nhs\u2083 : s\u2083 \u2208 h\nhs : image3 m s\u2081 s\u2082 s\u2083 \u2286 x\u271d\nt\u2081 : Set \u03b1\nt\u2082 : Set \u03b2\nt\u2083 : Set \u03b3\nht\u2081 : t\u2081 \u2208 f\nht\u2082 : t\u2082 \u2208 g\nht\u2083 : t\u2083 \u2208 h\nht : image3 m t\u2081 t\u2082 t\u2083 \u2286 y\u271d\n\u22a2 \u2203 u v w, u \u2208 f \u2227 v \u2208 g \u2227 w \u2208 h \u2227 image3 m u v w \u2286 x\u271d \u2227 image3 m u v w \u2286 y\u271d\n[PROOFSTEP]\nexact\n \u27e8s\u2081 \u2229 t\u2081, s\u2082 \u2229 t\u2082, s\u2083 \u2229 t\u2083, inter_mem hs\u2081 ht\u2081, inter_mem hs\u2082 ht\u2082, inter_mem hs\u2083 ht\u2083,\n (image3_mono (inter_subset_left _ _) (inter_subset_left _ _) <| inter_subset_left _ _).trans hs,\n (image3_mono (inter_subset_right _ _) (inter_subset_right _ _) <| inter_subset_right _ _).trans ht\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\n\u22a2 map\u2082 m (map\u2082 n f g) h = map\u2083 (fun a b c => m (n a b) c) f g h\n[PROOFSTEP]\next w\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nw : Set \u03b5\n\u22a2 w \u2208 map\u2082 m (map\u2082 n f g) h \u2194 w \u2208 map\u2083 (fun a b c => m (n a b) c) f g h\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nw : Set \u03b5\n\u22a2 w \u2208 map\u2082 m (map\u2082 n f g) h \u2192 w \u2208 map\u2083 (fun a b c => m (n a b) c) f g h\n[PROOFSTEP]\nrintro \u27e8s, t, \u27e8u, v, hu, hv, hs\u27e9, ht, hw\u27e9\n[GOAL]\ncase a.mp.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nw : Set \u03b5\ns : Set \u03b4\nt : Set \u03b3\nu : Set \u03b1\nv : Set \u03b2\nhu : u \u2208 f\nhv : v \u2208 g\nhs : image2 n u v \u2286 s\nht : t \u2208 h\nhw : image2 m s t \u2286 w\n\u22a2 w \u2208 map\u2083 (fun a b c => m (n a b) c) f g h\n[PROOFSTEP]\nrefine' \u27e8u, v, t, hu, hv, ht, _\u27e9\n[GOAL]\ncase a.mp.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nw : Set \u03b5\ns : Set \u03b4\nt : Set \u03b3\nu : Set \u03b1\nv : Set \u03b2\nhu : u \u2208 f\nhv : v \u2208 g\nhs : image2 n u v \u2286 s\nht : t \u2208 h\nhw : image2 m s t \u2286 w\n\u22a2 image3 (fun a b c => m (n a b) c) u v t \u2286 w\n[PROOFSTEP]\nrw [\u2190 image2_image2_left]\n[GOAL]\ncase a.mp.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nw : Set \u03b5\ns : Set \u03b4\nt : Set \u03b3\nu : Set \u03b1\nv : Set \u03b2\nhu : u \u2208 f\nhv : v \u2208 g\nhs : image2 n u v \u2286 s\nht : t \u2208 h\nhw : image2 m s t \u2286 w\n\u22a2 image2 m (image2 (fun a b => n a b) u v) t \u2286 w\n[PROOFSTEP]\nexact (image2_subset_right hs).trans hw\n[GOAL]\ncase a.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nw : Set \u03b5\n\u22a2 w \u2208 map\u2083 (fun a b c => m (n a b) c) f g h \u2192 w \u2208 map\u2082 m (map\u2082 n f g) h\n[PROOFSTEP]\nrintro \u27e8s, t, u, hs, ht, hu, hw\u27e9\n[GOAL]\ncase a.mpr.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nw : Set \u03b5\ns : Set \u03b1\nt : Set \u03b2\nu : Set \u03b3\nhs : s \u2208 f\nht : t \u2208 g\nhu : u \u2208 h\nhw : image3 (fun a b c => m (n a b) c) s t u \u2286 w\n\u22a2 w \u2208 map\u2082 m (map\u2082 n f g) h\n[PROOFSTEP]\nexact \u27e8_, u, image2_mem_map\u2082 hs ht, hu, by rwa [image2_image2_left]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nw : Set \u03b5\ns : Set \u03b1\nt : Set \u03b2\nu : Set \u03b3\nhs : s \u2208 f\nht : t \u2208 g\nhu : u \u2208 h\nhw : image3 (fun a b c => m (n a b) c) s t u \u2286 w\n\u22a2 image2 m (image2 n s t) u \u2286 w\n[PROOFSTEP]\nrwa [image2_image2_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\n\u22a2 map\u2082 m f (map\u2082 n g h) = map\u2083 (fun a b c => m a (n b c)) f g h\n[PROOFSTEP]\next w\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nw : Set \u03b5\n\u22a2 w \u2208 map\u2082 m f (map\u2082 n g h) \u2194 w \u2208 map\u2083 (fun a b c => m a (n b c)) f g h\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nw : Set \u03b5\n\u22a2 w \u2208 map\u2082 m f (map\u2082 n g h) \u2192 w \u2208 map\u2083 (fun a b c => m a (n b c)) f g h\n[PROOFSTEP]\nrintro \u27e8s, t, hs, \u27e8u, v, hu, hv, ht\u27e9, hw\u27e9\n[GOAL]\ncase a.mp.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nw : Set \u03b5\ns : Set \u03b1\nt : Set \u03b4\nhs : s \u2208 f\nhw : image2 m s t \u2286 w\nu : Set \u03b2\nv : Set \u03b3\nhu : u \u2208 g\nhv : v \u2208 h\nht : image2 n u v \u2286 t\n\u22a2 w \u2208 map\u2083 (fun a b c => m a (n b c)) f g h\n[PROOFSTEP]\nrefine' \u27e8s, u, v, hs, hu, hv, _\u27e9\n[GOAL]\ncase a.mp.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nw : Set \u03b5\ns : Set \u03b1\nt : Set \u03b4\nhs : s \u2208 f\nhw : image2 m s t \u2286 w\nu : Set \u03b2\nv : Set \u03b3\nhu : u \u2208 g\nhv : v \u2208 h\nht : image2 n u v \u2286 t\n\u22a2 image3 (fun a b c => m a (n b c)) s u v \u2286 w\n[PROOFSTEP]\nrw [\u2190 image2_image2_right]\n[GOAL]\ncase a.mp.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nw : Set \u03b5\ns : Set \u03b1\nt : Set \u03b4\nhs : s \u2208 f\nhw : image2 m s t \u2286 w\nu : Set \u03b2\nv : Set \u03b3\nhu : u \u2208 g\nhv : v \u2208 h\nht : image2 n u v \u2286 t\n\u22a2 image2 (fun a => m a) s (image2 (fun b c => n b c) u v) \u2286 w\n[PROOFSTEP]\nexact (image2_subset_left ht).trans hw\n[GOAL]\ncase a.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nw : Set \u03b5\n\u22a2 w \u2208 map\u2083 (fun a b c => m a (n b c)) f g h \u2192 w \u2208 map\u2082 m f (map\u2082 n g h)\n[PROOFSTEP]\nrintro \u27e8s, t, u, hs, ht, hu, hw\u27e9\n[GOAL]\ncase a.mpr.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nw : Set \u03b5\ns : Set \u03b1\nt : Set \u03b2\nu : Set \u03b3\nhs : s \u2208 f\nht : t \u2208 g\nhu : u \u2208 h\nhw : image3 (fun a b c => m a (n b c)) s t u \u2286 w\n\u22a2 w \u2208 map\u2082 m f (map\u2082 n g h)\n[PROOFSTEP]\nexact \u27e8s, _, hs, image2_mem_map\u2082 ht hu, by rwa [image2_image2_right]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nw : Set \u03b5\ns : Set \u03b1\nt : Set \u03b2\nu : Set \u03b3\nhs : s \u2208 f\nht : t \u2208 g\nhu : u \u2208 h\nhw : image3 (fun a b c => m a (n b c)) s t u \u2286 w\n\u22a2 image2 m s (image2 n t u) \u2286 w\n[PROOFSTEP]\nrwa [image2_image2_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn : \u03b3 \u2192 \u03b4\n\u22a2 map n (map\u2082 m f g) = map\u2082 (fun a b => n (m a b)) f g\n[PROOFSTEP]\nrw [\u2190 map_prod_eq_map\u2082, \u2190 map_prod_eq_map\u2082, map_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn : \u03b3 \u2192 \u03b4\n\u22a2 map (n \u2218 fun p => m p.fst p.snd) (f \u00d7\u02e2 g) = map (fun p => n (m p.fst p.snd)) (f \u00d7\u02e2 g)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b3 \u2192 \u03b2 \u2192 \u03b4\nn : \u03b1 \u2192 \u03b3\n\u22a2 map\u2082 m (map n f) g = map\u2082 (fun a b => m (n a) b) f g\n[PROOFSTEP]\nrw [\u2190 map_prod_eq_map\u2082, \u2190 map_prod_eq_map\u2082, \u2190 @map_id _ g, prod_map_map_eq, map_map, map_id]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b3 \u2192 \u03b2 \u2192 \u03b4\nn : \u03b1 \u2192 \u03b3\n\u22a2 map ((fun p => m p.fst p.snd) \u2218 fun p => (n p.fst, id p.snd)) (f \u00d7\u02e2 g) = map (fun p => m (n p.fst) p.snd) (f \u00d7\u02e2 g)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nn : \u03b2 \u2192 \u03b3\n\u22a2 map\u2082 m f (map n g) = map\u2082 (fun a b => m a (n b)) f g\n[PROOFSTEP]\nrw [map\u2082_swap, map\u2082_map_left, map\u2082_swap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm' : \u03b1 \u2192 \u03b5' \u2192 \u03b5\nn' : \u03b2 \u2192 \u03b3 \u2192 \u03b5'\nh : Filter \u03b3\nh_assoc : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = m' a (n' b c)\n\u22a2 map\u2082 m (map\u2082 n f g) h = map\u2082 m' f (map\u2082 n' g h)\n[PROOFSTEP]\nsimp only [map\u2082_map\u2082_left, map\u2082_map\u2082_right, h_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nn : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh_comm : \u2200 (a : \u03b1) (b : \u03b2), m a b = n b a\n\u22a2 map\u2082 (fun a b => m b a) g f = map\u2082 n g f\n[PROOFSTEP]\nsimp_rw [h_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nm' : \u03b1 \u2192 \u03b3 \u2192 \u03b4'\nn' : \u03b2 \u2192 \u03b4' \u2192 \u03b5\nh_left_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m a (n b c) = n' b (m' a c)\n\u22a2 map\u2082 m f (map\u2082 n g h) = map\u2082 n' g (map\u2082 m' f h)\n[PROOFSTEP]\nrw [map\u2082_swap m', map\u2082_swap m]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nm' : \u03b1 \u2192 \u03b3 \u2192 \u03b4'\nn' : \u03b2 \u2192 \u03b4' \u2192 \u03b5\nh_left_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m a (n b c) = n' b (m' a c)\n\u22a2 map\u2082 (fun a b => m b a) (map\u2082 n g h) f = map\u2082 n' g (map\u2082 (fun a b => m' b a) h f)\n[PROOFSTEP]\nexact map\u2082_assoc fun _ _ _ => h_left_comm _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm' : \u03b1 \u2192 \u03b3 \u2192 \u03b4'\nn' : \u03b4' \u2192 \u03b2 \u2192 \u03b5\nh_right_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m' a c) b\n\u22a2 map\u2082 m (map\u2082 n f g) h = map\u2082 n' (map\u2082 m' f h) g\n[PROOFSTEP]\nrw [map\u2082_swap n, map\u2082_swap n']\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm' : \u03b1 \u2192 \u03b3 \u2192 \u03b4'\nn' : \u03b4' \u2192 \u03b2 \u2192 \u03b5\nh_right_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m' a c) b\n\u22a2 map\u2082 m (map\u2082 (fun a b => n b a) g f) h = map\u2082 (fun a b => n' b a) g (map\u2082 m' f h)\n[PROOFSTEP]\nexact map\u2082_assoc fun _ _ _ => h_right_comm _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nn : \u03b3 \u2192 \u03b4\nm' : \u03b1' \u2192 \u03b2' \u2192 \u03b4\nn\u2081 : \u03b1 \u2192 \u03b1'\nn\u2082 : \u03b2 \u2192 \u03b2'\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2), n (m a b) = m' (n\u2081 a) (n\u2082 b)\n\u22a2 map n (map\u2082 m f g) = map\u2082 m' (map n\u2081 f) (map n\u2082 g)\n[PROOFSTEP]\nsimp_rw [map_map\u2082, map\u2082_map_left, map\u2082_map_right, h_distrib]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03b2'\nm\u2082 : \u03b1 \u2192 \u03b3 \u2192 \u03b3'\nn' : \u03b2' \u2192 \u03b3' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m a (n b c) = n' (m\u2081 a b) (m\u2082 a c)\n\u22a2 map\u2082 m f (map\u2082 n g h) \u2264 map\u2082 n' (map\u2082 m\u2081 f g) (map\u2082 m\u2082 f h)\n[PROOFSTEP]\nrintro s \u27e8t\u2081, t\u2082, \u27e8u\u2081, v, hu\u2081, hv, ht\u2081\u27e9, \u27e8u\u2082, w, hu\u2082, hw, ht\u2082\u27e9, hs\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03b2'\nm\u2082 : \u03b1 \u2192 \u03b3 \u2192 \u03b3'\nn' : \u03b2' \u2192 \u03b3' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m a (n b c) = n' (m\u2081 a b) (m\u2082 a c)\ns : Set \u03b5\nt\u2081 : Set \u03b2'\nt\u2082 : Set \u03b3'\nu\u2081 : Set \u03b1\nv : Set \u03b2\nhu\u2081 : u\u2081 \u2208 f\nhv : v \u2208 g\nht\u2081 : image2 m\u2081 u\u2081 v \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nu\u2082 : Set \u03b1\nw : Set \u03b3\nhu\u2082 : u\u2082 \u2208 f\nhw : w \u2208 h\nht\u2082 : image2 m\u2082 u\u2082 w \u2286 t\u2082\n\u22a2 s \u2208 map\u2082 m f (map\u2082 n g h)\n[PROOFSTEP]\nrefine' \u27e8u\u2081 \u2229 u\u2082, _, inter_mem hu\u2081 hu\u2082, image2_mem_map\u2082 hv hw, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03b2'\nm\u2082 : \u03b1 \u2192 \u03b3 \u2192 \u03b3'\nn' : \u03b2' \u2192 \u03b3' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m a (n b c) = n' (m\u2081 a b) (m\u2082 a c)\ns : Set \u03b5\nt\u2081 : Set \u03b2'\nt\u2082 : Set \u03b3'\nu\u2081 : Set \u03b1\nv : Set \u03b2\nhu\u2081 : u\u2081 \u2208 f\nhv : v \u2208 g\nht\u2081 : image2 m\u2081 u\u2081 v \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nu\u2082 : Set \u03b1\nw : Set \u03b3\nhu\u2082 : u\u2082 \u2208 f\nhw : w \u2208 h\nht\u2082 : image2 m\u2082 u\u2082 w \u2286 t\u2082\n\u22a2 image2 m (u\u2081 \u2229 u\u2082) (image2 n v w) \u2286 s\n[PROOFSTEP]\nrefine' (image2_distrib_subset_left h_distrib).trans ((image2_subset _ _).trans hs)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03b2'\nm\u2082 : \u03b1 \u2192 \u03b3 \u2192 \u03b3'\nn' : \u03b2' \u2192 \u03b3' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m a (n b c) = n' (m\u2081 a b) (m\u2082 a c)\ns : Set \u03b5\nt\u2081 : Set \u03b2'\nt\u2082 : Set \u03b3'\nu\u2081 : Set \u03b1\nv : Set \u03b2\nhu\u2081 : u\u2081 \u2208 f\nhv : v \u2208 g\nht\u2081 : image2 m\u2081 u\u2081 v \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nu\u2082 : Set \u03b1\nw : Set \u03b3\nhu\u2082 : u\u2082 \u2208 f\nhw : w \u2208 h\nht\u2082 : image2 m\u2082 u\u2082 w \u2286 t\u2082\n\u22a2 image2 (fun a b => m\u2081 a b) (u\u2081 \u2229 u\u2082) v \u2286 t\u2081\n[PROOFSTEP]\nexact (image2_subset_right <| inter_subset_left _ _).trans ht\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b1 \u2192 \u03b4 \u2192 \u03b5\nn : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b2 \u2192 \u03b2'\nm\u2082 : \u03b1 \u2192 \u03b3 \u2192 \u03b3'\nn' : \u03b2' \u2192 \u03b3' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m a (n b c) = n' (m\u2081 a b) (m\u2082 a c)\ns : Set \u03b5\nt\u2081 : Set \u03b2'\nt\u2082 : Set \u03b3'\nu\u2081 : Set \u03b1\nv : Set \u03b2\nhu\u2081 : u\u2081 \u2208 f\nhv : v \u2208 g\nht\u2081 : image2 m\u2081 u\u2081 v \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nu\u2082 : Set \u03b1\nw : Set \u03b3\nhu\u2082 : u\u2082 \u2208 f\nhw : w \u2208 h\nht\u2082 : image2 m\u2082 u\u2082 w \u2286 t\u2082\n\u22a2 image2 (fun a c => m\u2082 a c) (u\u2081 \u2229 u\u2082) w \u2286 t\u2082\n[PROOFSTEP]\nexact (image2_subset_right <| inter_subset_right _ _).trans ht\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nm\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\nn' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m\u2081 a c) (m\u2082 b c)\n\u22a2 map\u2082 m (map\u2082 n f g) h \u2264 map\u2082 n' (map\u2082 m\u2081 f h) (map\u2082 m\u2082 g h)\n[PROOFSTEP]\nrintro s \u27e8t\u2081, t\u2082, \u27e8u, w\u2081, hu, hw\u2081, ht\u2081\u27e9, \u27e8v, w\u2082, hv, hw\u2082, ht\u2082\u27e9, hs\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nm\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\nn' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m\u2081 a c) (m\u2082 b c)\ns : Set \u03b5\nt\u2081 : Set \u03b1'\nt\u2082 : Set \u03b2'\nu : Set \u03b1\nw\u2081 : Set \u03b3\nhu : u \u2208 f\nhw\u2081 : w\u2081 \u2208 h\nht\u2081 : image2 m\u2081 u w\u2081 \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nv : Set \u03b2\nw\u2082 : Set \u03b3\nhv : v \u2208 g\nhw\u2082 : w\u2082 \u2208 h\nht\u2082 : image2 m\u2082 v w\u2082 \u2286 t\u2082\n\u22a2 s \u2208 map\u2082 m (map\u2082 n f g) h\n[PROOFSTEP]\nrefine' \u27e8_, w\u2081 \u2229 w\u2082, image2_mem_map\u2082 hu hv, inter_mem hw\u2081 hw\u2082, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nm\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\nn' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m\u2081 a c) (m\u2082 b c)\ns : Set \u03b5\nt\u2081 : Set \u03b1'\nt\u2082 : Set \u03b2'\nu : Set \u03b1\nw\u2081 : Set \u03b3\nhu : u \u2208 f\nhw\u2081 : w\u2081 \u2208 h\nht\u2081 : image2 m\u2081 u w\u2081 \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nv : Set \u03b2\nw\u2082 : Set \u03b3\nhv : v \u2208 g\nhw\u2082 : w\u2082 \u2208 h\nht\u2082 : image2 m\u2082 v w\u2082 \u2286 t\u2082\n\u22a2 image2 m (image2 n u v) (w\u2081 \u2229 w\u2082) \u2286 s\n[PROOFSTEP]\nrefine' (image2_distrib_subset_right h_distrib).trans ((image2_subset _ _).trans hs)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nm\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\nn' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m\u2081 a c) (m\u2082 b c)\ns : Set \u03b5\nt\u2081 : Set \u03b1'\nt\u2082 : Set \u03b2'\nu : Set \u03b1\nw\u2081 : Set \u03b3\nhu : u \u2208 f\nhw\u2081 : w\u2081 \u2208 h\nht\u2081 : image2 m\u2081 u w\u2081 \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nv : Set \u03b2\nw\u2082 : Set \u03b3\nhv : v \u2208 g\nhw\u2082 : w\u2082 \u2208 h\nht\u2082 : image2 m\u2082 v w\u2082 \u2286 t\u2082\n\u22a2 image2 (fun a c => m\u2081 a c) u (w\u2081 \u2229 w\u2082) \u2286 t\u2081\n[PROOFSTEP]\nexact (image2_subset_left <| inter_subset_left _ _).trans ht\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nm\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\nn' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m\u2081 a c) (m\u2082 b c)\ns : Set \u03b5\nt\u2081 : Set \u03b1'\nt\u2082 : Set \u03b2'\nu : Set \u03b1\nw\u2081 : Set \u03b3\nhu : u \u2208 f\nhw\u2081 : w\u2081 \u2208 h\nht\u2081 : image2 m\u2081 u w\u2081 \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nv : Set \u03b2\nw\u2082 : Set \u03b3\nhv : v \u2208 g\nhw\u2082 : w\u2082 \u2208 h\nht\u2082 : image2 m\u2082 v w\u2082 \u2286 t\u2082\n\u22a2 image2 (fun b c => m\u2082 b c) v (w\u2081 \u2229 w\u2082) \u2286 t\u2082\n[PROOFSTEP]\nexact (image2_subset_left <| inter_subset_right _ _).trans ht\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nn : \u03b3 \u2192 \u03b4\nm' : \u03b2' \u2192 \u03b1' \u2192 \u03b4\nn\u2081 : \u03b2 \u2192 \u03b2'\nn\u2082 : \u03b1 \u2192 \u03b1'\nh_antidistrib : \u2200 (a : \u03b1) (b : \u03b2), n (m a b) = m' (n\u2081 b) (n\u2082 a)\n\u22a2 map n (map\u2082 m f g) = map\u2082 m' (map n\u2081 g) (map n\u2082 f)\n[PROOFSTEP]\nrw [map\u2082_swap m]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nn : \u03b3 \u2192 \u03b4\nm' : \u03b2' \u2192 \u03b1' \u2192 \u03b4\nn\u2081 : \u03b2 \u2192 \u03b2'\nn\u2082 : \u03b1 \u2192 \u03b1'\nh_antidistrib : \u2200 (a : \u03b1) (b : \u03b2), n (m a b) = m' (n\u2081 b) (n\u2082 a)\n\u22a2 map n (map\u2082 (fun a b => m b a) g f) = map\u2082 m' (map n\u2081 g) (map n\u2082 f)\n[PROOFSTEP]\nexact map_map\u2082_distrib fun _ _ => h_antidistrib _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na\u271d : \u03b1\nb : \u03b2\nc : \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\na : \u03b1\nh : \u2200 (b : \u03b2), f a b = b\nl : Filter \u03b2\n\u22a2 map\u2082 f (pure a) l = l\n[PROOFSTEP]\nrw [map\u2082_pure_left, show f a = id from funext h, map_id]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb\u271d : \u03b2\nc : \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\nb : \u03b2\nh : \u2200 (a : \u03b1), f a b = a\nl : Filter \u03b1\n\u22a2 map\u2082 f l (pure b) = l\n[PROOFSTEP]\nrw [map\u2082_pure_right, funext h, map_id']\n", "meta": {"mathlib_filename": "Mathlib.Order.Filter.NAry", "llama_tokens": 29215, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.4293891542046626}} {"text": "[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : ConditionallyCompleteLinearOrderedField \u03b1\n\u22a2 \u2200 (x : \u03b1), \u2203 n, x < \u2191n\n[PROOFSTEP]\nby_contra' h\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : ConditionallyCompleteLinearOrderedField \u03b1\nh : \u2203 x, \u2200 (n : \u2115), \u2191n \u2264 x\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8x, h\u27e9 := h\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : ConditionallyCompleteLinearOrderedField \u03b1\nx : \u03b1\nh : \u2200 (n : \u2115), \u2191n \u2264 x\n\u22a2 False\n[PROOFSTEP]\nhave :=\n csSup_le _ _ (range_nonempty Nat.cast)\n (forall_range_iff.2 fun m => le_sub_iff_add_le.2 <| le_csSup _ _ \u27e8x, forall_range_iff.2 h\u27e9 \u27e8m + 1, Nat.cast_succ m\u27e9)\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : ConditionallyCompleteLinearOrderedField \u03b1\nx : \u03b1\nh : \u2200 (n : \u2115), \u2191n \u2264 x\nthis : sSup (range Nat.cast) \u2264 sSup (range fun i => \u2191i) - 1\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : DivisionRing \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb : \u03b2\nq : \u211a\na : \u03b1\n\u22a2 cutMap \u03b1 a = Iio a \u2229 range Rat.cast\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : DivisionRing \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb : \u03b2\nq : \u211a\na x\u271d : \u03b1\n\u22a2 x\u271d \u2208 cutMap \u03b1 a \u2194 x\u271d \u2208 Iio a \u2229 range Rat.cast\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : DivisionRing \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb : \u03b2\nq : \u211a\na x\u271d : \u03b1\n\u22a2 x\u271d \u2208 cutMap \u03b1 a \u2192 x\u271d \u2208 Iio a \u2229 range Rat.cast\n[PROOFSTEP]\nrintro \u27e8q, h, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : DivisionRing \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb : \u03b2\nq\u271d : \u211a\na : \u03b1\nq : \u211a\nh : q \u2208 {t | \u2191t < a}\n\u22a2 \u2191q \u2208 Iio a \u2229 range Rat.cast\n[PROOFSTEP]\nexact \u27e8h, q, rfl\u27e9\n[GOAL]\ncase h.mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : DivisionRing \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb : \u03b2\nq : \u211a\na x\u271d : \u03b1\n\u22a2 x\u271d \u2208 Iio a \u2229 range Rat.cast \u2192 x\u271d \u2208 cutMap \u03b1 a\n[PROOFSTEP]\nrintro \u27e8h, q, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : DivisionRing \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb : \u03b2\nq\u271d : \u211a\na : \u03b1\nq : \u211a\nh : \u2191q \u2208 Iio a\n\u22a2 \u2191q \u2208 cutMap \u03b1 a\n[PROOFSTEP]\nexact \u27e8q, h, rfl\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : LinearOrderedField \u03b2\na a\u2081 a\u2082 : \u03b1\nb : \u03b2\nq\u271d q : \u211a\n\u22a2 cutMap \u03b2 \u2191q = Rat.cast '' {r | \u2191r < \u2191q}\n[PROOFSTEP]\nsimp_rw [cutMap, Rat.cast_lt]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb : \u03b2\nq : \u211a\ninst\u271d : Archimedean \u03b1\na : \u03b1\n\u22a2 BddAbove (cutMap \u03b2 a)\n[PROOFSTEP]\nobtain \u27e8q, hq\u27e9 := exists_rat_gt a\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb : \u03b2\nq\u271d : \u211a\ninst\u271d : Archimedean \u03b1\na : \u03b1\nq : \u211a\nhq : a < \u2191q\n\u22a2 BddAbove (cutMap \u03b2 a)\n[PROOFSTEP]\nexact \u27e8q, ball_image_iff.2 fun r hr => by exact_mod_cast (hq.trans' hr).le\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb : \u03b2\nq\u271d : \u211a\ninst\u271d : Archimedean \u03b1\na : \u03b1\nq : \u211a\nhq : a < \u2191q\nr : \u211a\nhr : r \u2208 {t | \u2191t < a}\n\u22a2 \u2191r \u2264 \u2191q\n[PROOFSTEP]\nexact_mod_cast (hq.trans' hr).le\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d : \u03b2\nq : \u211a\ninst\u271d : Archimedean \u03b1\na b : \u03b1\n\u22a2 cutMap \u03b2 (a + b) = cutMap \u03b2 a + cutMap \u03b2 b\n[PROOFSTEP]\nrefine (image_subset_iff.2 fun q hq => ?_).antisymm ?_\n[GOAL]\ncase refine_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d : \u03b2\nq\u271d : \u211a\ninst\u271d : Archimedean \u03b1\na b : \u03b1\nq : \u211a\nhq : q \u2208 {t | \u2191t < a + b}\n\u22a2 q \u2208 Rat.cast \u207b\u00b9' (cutMap \u03b2 a + cutMap \u03b2 b)\n[PROOFSTEP]\nrw [mem_setOf_eq, \u2190 sub_lt_iff_lt_add] at hq \n[GOAL]\ncase refine_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d : \u03b2\nq\u271d : \u211a\ninst\u271d : Archimedean \u03b1\na b : \u03b1\nq : \u211a\nhq : \u2191q - b < a\n\u22a2 q \u2208 Rat.cast \u207b\u00b9' (cutMap \u03b2 a + cutMap \u03b2 b)\n[PROOFSTEP]\nobtain \u27e8q\u2081, hq\u2081q, hq\u2081ab\u27e9 := exists_rat_btwn hq\n[GOAL]\ncase refine_1.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d : \u03b2\nq\u271d : \u211a\ninst\u271d : Archimedean \u03b1\na b : \u03b1\nq : \u211a\nhq : \u2191q - b < a\nq\u2081 : \u211a\nhq\u2081q : \u2191q - b < \u2191q\u2081\nhq\u2081ab : \u2191q\u2081 < a\n\u22a2 q \u2208 Rat.cast \u207b\u00b9' (cutMap \u03b2 a + cutMap \u03b2 b)\n[PROOFSTEP]\nrefine \u27e8q\u2081, q - q\u2081, by rwa [coe_mem_cutMap_iff], ?_, add_sub_cancel'_right _ _\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d : \u03b2\nq\u271d : \u211a\ninst\u271d : Archimedean \u03b1\na b : \u03b1\nq : \u211a\nhq : \u2191q - b < a\nq\u2081 : \u211a\nhq\u2081q : \u2191q - b < \u2191q\u2081\nhq\u2081ab : \u2191q\u2081 < a\n\u22a2 \u2191q\u2081 \u2208 cutMap \u03b2 a\n[PROOFSTEP]\nrwa [coe_mem_cutMap_iff]\n[GOAL]\ncase refine_1.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d : \u03b2\nq\u271d : \u211a\ninst\u271d : Archimedean \u03b1\na b : \u03b1\nq : \u211a\nhq : \u2191q - b < a\nq\u2081 : \u211a\nhq\u2081q : \u2191q - b < \u2191q\u2081\nhq\u2081ab : \u2191q\u2081 < a\n\u22a2 \u2191q - \u2191q\u2081 \u2208 cutMap \u03b2 b\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase refine_1.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d : \u03b2\nq\u271d : \u211a\ninst\u271d : Archimedean \u03b1\na b : \u03b1\nq : \u211a\nhq : \u2191q - b < a\nq\u2081 : \u211a\nhq\u2081q : \u2191q - b < \u2191q\u2081\nhq\u2081ab : \u2191q\u2081 < a\n\u22a2 \u2191(q - q\u2081) \u2208 cutMap \u03b2 b\n[PROOFSTEP]\nrw [coe_mem_cutMap_iff]\n[GOAL]\ncase refine_1.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d : \u03b2\nq\u271d : \u211a\ninst\u271d : Archimedean \u03b1\na b : \u03b1\nq : \u211a\nhq : \u2191q - b < a\nq\u2081 : \u211a\nhq\u2081q : \u2191q - b < \u2191q\u2081\nhq\u2081ab : \u2191q\u2081 < a\n\u22a2 \u2191(q - q\u2081) < b\n[PROOFSTEP]\nexact_mod_cast sub_lt_comm.mp hq\u2081q\n[GOAL]\ncase refine_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d : \u03b2\nq : \u211a\ninst\u271d : Archimedean \u03b1\na b : \u03b1\n\u22a2 cutMap \u03b2 a + cutMap \u03b2 b \u2286 Rat.cast '' {t | \u2191t < a + b}\n[PROOFSTEP]\nrintro _ \u27e8_, _, \u27e8qa, ha, rfl\u27e9, \u27e8qb, hb, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase refine_2.intro.intro.intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d : \u03b2\nq : \u211a\ninst\u271d : Archimedean \u03b1\na b : \u03b1\nqa : \u211a\nha : qa \u2208 {t | \u2191t < a}\nqb : \u211a\nhb : qb \u2208 {t | \u2191t < b}\n\u22a2 (fun x x_1 => x + x_1) \u2191qa \u2191qb \u2208 Rat.cast '' {t | \u2191t < a + b}\n[PROOFSTEP]\nrefine' \u27e8qa + qb, _, by norm_cast\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d : \u03b2\nq : \u211a\ninst\u271d : Archimedean \u03b1\na b : \u03b1\nqa : \u211a\nha : qa \u2208 {t | \u2191t < a}\nqb : \u211a\nhb : qb \u2208 {t | \u2191t < b}\n\u22a2 \u2191(qa + qb) = (fun x x_1 => x + x_1) \u2191qa \u2191qb\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase refine_2.intro.intro.intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d : \u03b2\nq : \u211a\ninst\u271d : Archimedean \u03b1\na b : \u03b1\nqa : \u211a\nha : qa \u2208 {t | \u2191t < a}\nqb : \u211a\nhb : qb \u2208 {t | \u2191t < b}\n\u22a2 qa + qb \u2208 {t | \u2191t < a + b}\n[PROOFSTEP]\nrw [mem_setOf_eq, cast_add]\n[GOAL]\ncase refine_2.intro.intro.intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : LinearOrderedField \u03b2\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d : \u03b2\nq : \u211a\ninst\u271d : Archimedean \u03b1\na b : \u03b1\nqa : \u211a\nha : qa \u2208 {t | \u2191t < a}\nqb : \u211a\nhb : qb \u2208 {t | \u2191t < b}\n\u22a2 \u2191qa + \u2191qb < a + b\n[PROOFSTEP]\nexact add_lt_add ha hb\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\nq : \u211a\n\u22a2 inducedMap \u03b1 \u03b2 \u2191q = \u2191q\n[PROOFSTEP]\nrefine csSup_eq_of_forall_le_of_forall_lt_exists_gt (cutMap_nonempty \u03b2 (q : \u03b1)) (fun x h => ?_) fun w h => ?_\n[GOAL]\ncase refine_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\nq : \u211a\nx : \u03b2\nh : x \u2208 cutMap \u03b2 \u2191q\n\u22a2 x \u2264 \u2191q\n[PROOFSTEP]\nrw [cutMap_coe] at h \n[GOAL]\ncase refine_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\nq : \u211a\nx : \u03b2\nh : x \u2208 Rat.cast '' {r | \u2191r < \u2191q}\n\u22a2 x \u2264 \u2191q\n[PROOFSTEP]\nobtain \u27e8r, h, rfl\u27e9 := h\n[GOAL]\ncase refine_1.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\nq r : \u211a\nh : r \u2208 {r | \u2191r < \u2191q}\n\u22a2 \u2191r \u2264 \u2191q\n[PROOFSTEP]\nexact le_of_lt h\n[GOAL]\ncase refine_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\nq : \u211a\nw : \u03b2\nh : w < \u2191q\n\u22a2 \u2203 a, a \u2208 cutMap \u03b2 \u2191q \u2227 w < a\n[PROOFSTEP]\nobtain \u27e8q', hwq, hq\u27e9 := exists_rat_btwn h\n[GOAL]\ncase refine_2.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\nq : \u211a\nw : \u03b2\nh : w < \u2191q\nq' : \u211a\nhwq : w < \u2191q'\nhq : \u2191q' < \u2191q\n\u22a2 \u2203 a, a \u2208 cutMap \u03b2 \u2191q \u2227 w < a\n[PROOFSTEP]\nrw [cutMap_coe]\n[GOAL]\ncase refine_2.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\nq : \u211a\nw : \u03b2\nh : w < \u2191q\nq' : \u211a\nhwq : w < \u2191q'\nhq : \u2191q' < \u2191q\n\u22a2 \u2203 a, a \u2208 Rat.cast '' {r | \u2191r < \u2191q} \u2227 w < a\n[PROOFSTEP]\nexact \u27e8q', \u27e8_, hq, rfl\u27e9, hwq\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\n\u22a2 inducedMap \u03b1 \u03b2 0 = 0\n[PROOFSTEP]\nexact_mod_cast inducedMap_rat \u03b1 \u03b2 0\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\n\u22a2 inducedMap \u03b1 \u03b2 1 = 1\n[PROOFSTEP]\nexact_mod_cast inducedMap_rat \u03b1 \u03b2 1\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\n\u22a2 \u2191q < inducedMap \u03b1 \u03b2 a \u2194 \u2191q < a\n[PROOFSTEP]\nrefine \u27e8fun h => ?_, fun hq => ?_\u27e9\n[GOAL]\ncase refine_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nh : \u2191q < inducedMap \u03b1 \u03b2 a\n\u22a2 \u2191q < a\n[PROOFSTEP]\nrw [\u2190 inducedMap_rat \u03b1] at h \n[GOAL]\ncase refine_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nh : inducedMap \u03b1 \u03b2 \u2191q < inducedMap \u03b1 \u03b2 a\n\u22a2 \u2191q < a\n[PROOFSTEP]\nexact (inducedMap_mono \u03b1 \u03b2).reflect_lt h\n[GOAL]\ncase refine_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nhq : \u2191q < a\n\u22a2 \u2191q < inducedMap \u03b1 \u03b2 a\n[PROOFSTEP]\nobtain \u27e8q', hq, hqa\u27e9 := exists_rat_btwn hq\n[GOAL]\ncase refine_2.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nhq\u271d : \u2191q < a\nq' : \u211a\nhq : \u2191q < \u2191q'\nhqa : \u2191q' < a\n\u22a2 \u2191q < inducedMap \u03b1 \u03b2 a\n[PROOFSTEP]\napply lt_csSup_of_lt (cutMap_bddAbove \u03b2 a) (coe_mem_cutMap_iff.mpr hqa)\n[GOAL]\ncase refine_2.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nhq\u271d : \u2191q < a\nq' : \u211a\nhq : \u2191q < \u2191q'\nhqa : \u2191q' < a\n\u22a2 \u2191q < \u2191q'\n[PROOFSTEP]\nexact_mod_cast hq\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq\u271d : \u211a\nx\u271d : \u2203 q, b < \u2191q \u2227 \u2191q < a\nq : \u211a\nhbq : b < \u2191q\nhqa : \u2191q < a\n\u22a2 \u2191q < inducedMap \u03b1 \u03b2 a\n[PROOFSTEP]\nrwa [coe_lt_inducedMap_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na\u271d : \u03b1\nb : \u03b2\nq\u271d : \u211a\na : \u03b1\nq : \u211a\n\u22a2 \u2191q < inducedMap \u03b2 \u03b3 (inducedMap \u03b1 \u03b2 a) \u2194 \u2191q < inducedMap \u03b1 \u03b3 a\n[PROOFSTEP]\nrw [coe_lt_inducedMap_iff, coe_lt_inducedMap_iff, Iff.comm, coe_lt_inducedMap_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb\u271d : \u03b2\nq : \u211a\nb : \u03b2\n\u22a2 inducedMap \u03b3 \u03b2 (inducedMap \u03b2 \u03b3 b) = b\n[PROOFSTEP]\nrw [inducedMap_inducedMap, inducedMap_self]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nx y : \u03b1\n\u22a2 inducedMap \u03b1 \u03b2 (x + y) = inducedMap \u03b1 \u03b2 x + inducedMap \u03b1 \u03b2 y\n[PROOFSTEP]\nrw [inducedMap, cutMap_add]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nx y : \u03b1\n\u22a2 sSup (cutMap \u03b2 x + cutMap \u03b2 y) = inducedMap \u03b1 \u03b2 x + inducedMap \u03b1 \u03b2 y\n[PROOFSTEP]\nexact csSup_add (cutMap_nonempty \u03b2 x) (cutMap_bddAbove \u03b2 x) (cutMap_nonempty \u03b2 y) (cutMap_bddAbove \u03b2 y)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb\u271d : \u03b2\nq : \u211a\nha : 0 < a\nb : \u03b2\nhb : b \u2208 cutMap \u03b2 (a * a)\n\u22a2 b \u2264 inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\n[PROOFSTEP]\nobtain \u27e8q, hb, rfl\u27e9 := hb\n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq\u271d : \u211a\nha : 0 < a\nq : \u211a\nhb : q \u2208 {t | \u2191t < a * a}\n\u22a2 \u2191q \u2264 inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\n[PROOFSTEP]\nobtain \u27e8q', hq', hqq', hqa\u27e9 := exists_rat_pow_btwn two_ne_zero hb (mul_self_pos.2 ha.ne')\n[GOAL]\ncase intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq\u271d : \u211a\nha : 0 < a\nq : \u211a\nhb : q \u2208 {t | \u2191t < a * a}\nq' : \u211a\nhq' : 0 < q'\nhqq' : \u2191q < \u2191q' ^ 2\nhqa : \u2191q' ^ 2 < a * a\n\u22a2 \u2191q \u2264 inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\n[PROOFSTEP]\ntrans (q' : \u03b2) ^ 2\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq\u271d : \u211a\nha : 0 < a\nq : \u211a\nhb : q \u2208 {t | \u2191t < a * a}\nq' : \u211a\nhq' : 0 < q'\nhqq' : \u2191q < \u2191q' ^ 2\nhqa : \u2191q' ^ 2 < a * a\n\u22a2 \u2191q \u2264 \u2191q' ^ 2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq\u271d : \u211a\nha : 0 < a\nq : \u211a\nhb : q \u2208 {t | \u2191t < a * a}\nq' : \u211a\nhq' : 0 < q'\nhqq' : \u2191q < \u2191q' ^ 2\nhqa : \u2191q' ^ 2 < a * a\n\u22a2 \u2191q' ^ 2 \u2264 inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\n[PROOFSTEP]\nexact_mod_cast hqq'.le\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq\u271d : \u211a\nha : 0 < a\nq : \u211a\nhb : q \u2208 {t | \u2191t < a * a}\nq' : \u211a\nhq' : 0 < q'\nhqq' : \u2191q < \u2191q' ^ 2\nhqa : \u2191q' ^ 2 < a * a\n\u22a2 \u2191q' ^ 2 \u2264 inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\n[PROOFSTEP]\nrw [pow_two] at hqa \u22a2\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq\u271d : \u211a\nha : 0 < a\nq : \u211a\nhb : q \u2208 {t | \u2191t < a * a}\nq' : \u211a\nhq' : 0 < q'\nhqq' : \u2191q < \u2191q' ^ 2\nhqa : \u2191q' * \u2191q' < a * a\n\u22a2 \u2191q' * \u2191q' \u2264 inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\n[PROOFSTEP]\nexact\n mul_self_le_mul_self (by exact_mod_cast hq'.le)\n (le_csSup (cutMap_bddAbove \u03b2 a) <| coe_mem_cutMap_iff.2 <| lt_of_mul_self_lt_mul_self ha.le hqa)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq\u271d : \u211a\nha : 0 < a\nq : \u211a\nhb : q \u2208 {t | \u2191t < a * a}\nq' : \u211a\nhq' : 0 < q'\nhqq' : \u2191q < \u2191q' ^ 2\nhqa : \u2191q' * \u2191q' < a * a\n\u22a2 0 \u2264 \u2191q'\n[PROOFSTEP]\nexact_mod_cast hq'.le\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb\u271d : \u03b2\nq : \u211a\nha : 0 < a\nb : \u03b2\nhba : b < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\n\u22a2 \u2203 c, c \u2208 cutMap \u03b2 (a * a) \u2227 b < c\n[PROOFSTEP]\nobtain hb | hb := lt_or_le b 0\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb\u271d : \u03b2\nq : \u211a\nha : 0 < a\nb : \u03b2\nhba : b < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\nhb : b < 0\n\u22a2 \u2203 c, c \u2208 cutMap \u03b2 (a * a) \u2227 b < c\n[PROOFSTEP]\nrefine \u27e80, ?_, hb\u27e9\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb\u271d : \u03b2\nq : \u211a\nha : 0 < a\nb : \u03b2\nhba : b < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\nhb : b < 0\n\u22a2 0 \u2208 cutMap \u03b2 (a * a)\n[PROOFSTEP]\nrw [\u2190 Rat.cast_zero, coe_mem_cutMap_iff, Rat.cast_zero]\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb\u271d : \u03b2\nq : \u211a\nha : 0 < a\nb : \u03b2\nhba : b < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\nhb : b < 0\n\u22a2 0 < a * a\n[PROOFSTEP]\nexact mul_self_pos.2 ha.ne'\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb\u271d : \u03b2\nq : \u211a\nha : 0 < a\nb : \u03b2\nhba : b < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\nhb : 0 \u2264 b\n\u22a2 \u2203 c, c \u2208 cutMap \u03b2 (a * a) \u2227 b < c\n[PROOFSTEP]\nobtain \u27e8q, hq, hbq, hqa\u27e9 := exists_rat_pow_btwn two_ne_zero hba (hb.trans_lt hba)\n[GOAL]\ncase inr.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb\u271d : \u03b2\nq\u271d : \u211a\nha : 0 < a\nb : \u03b2\nhba : b < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\nhb : 0 \u2264 b\nq : \u211a\nhq : 0 < q\nhbq : b < \u2191q ^ 2\nhqa : \u2191q ^ 2 < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\n\u22a2 \u2203 c, c \u2208 cutMap \u03b2 (a * a) \u2227 b < c\n[PROOFSTEP]\nrw [\u2190 cast_pow] at hbq \n[GOAL]\ncase inr.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb\u271d : \u03b2\nq\u271d : \u211a\nha : 0 < a\nb : \u03b2\nhba : b < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\nhb : 0 \u2264 b\nq : \u211a\nhq : 0 < q\nhbq : b < \u2191(q ^ 2)\nhqa : \u2191q ^ 2 < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\n\u22a2 \u2203 c, c \u2208 cutMap \u03b2 (a * a) \u2227 b < c\n[PROOFSTEP]\nrefine \u27e8(q ^ 2 : \u211a), coe_mem_cutMap_iff.2 ?_, hbq\u27e9\n[GOAL]\ncase inr.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb\u271d : \u03b2\nq\u271d : \u211a\nha : 0 < a\nb : \u03b2\nhba : b < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\nhb : 0 \u2264 b\nq : \u211a\nhq : 0 < q\nhbq : b < \u2191(q ^ 2)\nhqa : \u2191q ^ 2 < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\n\u22a2 \u2191(q ^ 2) < a * a\n[PROOFSTEP]\nrw [pow_two] at hqa \u22a2\n[GOAL]\ncase inr.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb\u271d : \u03b2\nq\u271d : \u211a\nha : 0 < a\nb : \u03b2\nhba : b < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\nhb : 0 \u2264 b\nq : \u211a\nhq : 0 < q\nhbq : b < \u2191(q ^ 2)\nhqa : \u2191q * \u2191q < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\n\u22a2 \u2191(q * q) < a * a\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase inr.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb\u271d : \u03b2\nq\u271d : \u211a\nha : 0 < a\nb : \u03b2\nhba : b < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\nhb : 0 \u2264 b\nq : \u211a\nhq : 0 < q\nhbq : b < \u2191(q ^ 2)\nhqa : \u2191q * \u2191q < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\n\u22a2 \u2191q * \u2191q < a * a\n[PROOFSTEP]\nobtain \u27e8q', hq', hqa'\u27e9 := lt_inducedMap_iff.1 (lt_of_mul_self_lt_mul_self (inducedMap_nonneg ha.le) hqa)\n[GOAL]\ncase inr.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb\u271d : \u03b2\nq\u271d : \u211a\nha : 0 < a\nb : \u03b2\nhba : b < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\nhb : 0 \u2264 b\nq : \u211a\nhq : 0 < q\nhbq : b < \u2191(q ^ 2)\nhqa : \u2191q * \u2191q < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\nq' : \u211a\nhq' : \u2191q < \u2191q'\nhqa' : \u2191q' < a\n\u22a2 \u2191q * \u2191q < a * a\n[PROOFSTEP]\nexact mul_self_lt_mul_self (by exact_mod_cast hq.le) (hqa'.trans' <| by assumption_mod_cast)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb\u271d : \u03b2\nq\u271d : \u211a\nha : 0 < a\nb : \u03b2\nhba : b < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\nhb : 0 \u2264 b\nq : \u211a\nhq : 0 < q\nhbq : b < \u2191(q ^ 2)\nhqa : \u2191q * \u2191q < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\nq' : \u211a\nhq' : \u2191q < \u2191q'\nhqa' : \u2191q' < a\n\u22a2 0 \u2264 \u2191q\n[PROOFSTEP]\nexact_mod_cast hq.le\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb\u271d : \u03b2\nq\u271d : \u211a\nha : 0 < a\nb : \u03b2\nhba : b < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\nhb : 0 \u2264 b\nq : \u211a\nhq : 0 < q\nhbq : b < \u2191(q ^ 2)\nhqa : \u2191q * \u2191q < inducedMap \u03b1 \u03b2 a * inducedMap \u03b1 \u03b2 a\nq' : \u211a\nhq' : \u2191q < \u2191q'\nhqa' : \u2191q' < a\n\u22a2 \u2191q < \u2191q'\n[PROOFSTEP]\nassumption_mod_cast\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\n\u22a2 \u2200 (x : \u03b1), \u2191(inducedAddHom \u03b1 \u03b2) (x * x) = \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x\n[PROOFSTEP]\nsuffices : \u2200 x, 0 < x \u2192 inducedAddHom \u03b1 \u03b2 (x * x) = inducedAddHom \u03b1 \u03b2 x * inducedAddHom \u03b1 \u03b2 x\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nthis : \u2200 (x : \u03b1), 0 < x \u2192 \u2191(inducedAddHom \u03b1 \u03b2) (x * x) = \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x\n\u22a2 \u2200 (x : \u03b1), \u2191(inducedAddHom \u03b1 \u03b2) (x * x) = \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x\n[PROOFSTEP]\nintro x\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nthis : \u2200 (x : \u03b1), 0 < x \u2192 \u2191(inducedAddHom \u03b1 \u03b2) (x * x) = \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x\nx : \u03b1\n\u22a2 \u2191(inducedAddHom \u03b1 \u03b2) (x * x) = \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x\n[PROOFSTEP]\nobtain h | rfl | h := lt_trichotomy x 0\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nthis : \u2200 (x : \u03b1), 0 < x \u2192 \u2191(inducedAddHom \u03b1 \u03b2) (x * x) = \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x\nx : \u03b1\nh : x < 0\n\u22a2 \u2191(inducedAddHom \u03b1 \u03b2) (x * x) = \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x\n[PROOFSTEP]\nconvert this (-x) (neg_pos.2 h) using 1\n[GOAL]\ncase h.e'_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nthis : \u2200 (x : \u03b1), 0 < x \u2192 \u2191(inducedAddHom \u03b1 \u03b2) (x * x) = \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x\nx : \u03b1\nh : x < 0\n\u22a2 \u2191(inducedAddHom \u03b1 \u03b2) (x * x) = \u2191(inducedAddHom \u03b1 \u03b2) (-x * -x)\n[PROOFSTEP]\nrw [neg_mul, mul_neg, neg_neg]\n[GOAL]\ncase h.e'_3\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nthis : \u2200 (x : \u03b1), 0 < x \u2192 \u2191(inducedAddHom \u03b1 \u03b2) (x * x) = \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x\nx : \u03b1\nh : x < 0\n\u22a2 \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x = \u2191(inducedAddHom \u03b1 \u03b2) (-x) * \u2191(inducedAddHom \u03b1 \u03b2) (-x)\n[PROOFSTEP]\nsimp_rw [AddMonoidHom.map_neg, neg_mul, mul_neg, neg_neg]\n[GOAL]\ncase inr.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nthis : \u2200 (x : \u03b1), 0 < x \u2192 \u2191(inducedAddHom \u03b1 \u03b2) (x * x) = \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x\n\u22a2 \u2191(inducedAddHom \u03b1 \u03b2) (0 * 0) = \u2191(inducedAddHom \u03b1 \u03b2) 0 * \u2191(inducedAddHom \u03b1 \u03b2) 0\n[PROOFSTEP]\nsimp only [mul_zero, AddMonoidHom.map_zero]\n[GOAL]\ncase inr.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nthis : \u2200 (x : \u03b1), 0 < x \u2192 \u2191(inducedAddHom \u03b1 \u03b2) (x * x) = \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x\nx : \u03b1\nh : 0 < x\n\u22a2 \u2191(inducedAddHom \u03b1 \u03b2) (x * x) = \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x\n[PROOFSTEP]\nexact this x h\n[GOAL]\ncase this\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\n\u22a2 \u2200 (x : \u03b1), 0 < x \u2192 \u2191(inducedAddHom \u03b1 \u03b2) (x * x) = \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x\n[PROOFSTEP]\nrefine fun x hx => csSup_eq_of_forall_le_of_forall_lt_exists_gt (cutMap_nonempty \u03b2 _) ?_ ?_\n[GOAL]\ncase this.refine_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nx : \u03b1\nhx : 0 < x\n\u22a2 \u2200 (a : (fun x => \u03b2) (x * x)), a \u2208 cutMap \u03b2 (x * x) \u2192 a \u2264 \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x\n[PROOFSTEP]\nexact le_inducedMap_mul_self_of_mem_cutMap hx\n[GOAL]\ncase this.refine_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nx : \u03b1\nhx : 0 < x\n\u22a2 \u2200 (w : (fun x => \u03b2) (x * x)), w < \u2191(inducedAddHom \u03b1 \u03b2) x * \u2191(inducedAddHom \u03b1 \u03b2) x \u2192 \u2203 a, a \u2208 cutMap \u03b2 (x * x) \u2227 w < a\n[PROOFSTEP]\nexact exists_mem_cutMap_mul_self_of_lt_inducedMap_mul_self hx\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nsrc\u271d : \u03b2 \u2192+*o \u03b3 := inducedOrderRingHom \u03b2 \u03b3\n\u22a2 \u2200 {a b : \u03b2},\n Equiv.toFun\n {\n toEquiv :=\n { toFun := src\u271d.toFun, invFun := inducedMap \u03b3 \u03b2,\n left_inv := (_ : \u2200 (b : \u03b2), inducedMap \u03b3 \u03b2 (inducedMap \u03b2 \u03b3 b) = b),\n right_inv := (_ : \u2200 (b : \u03b3), inducedMap \u03b2 \u03b3 (inducedMap \u03b3 \u03b2 b) = b) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u03b2),\n OneHom.toFun (\u2191\u2191src\u271d.toRingHom) (x * y) =\n OneHom.toFun (\u2191\u2191src\u271d.toRingHom) x * OneHom.toFun (\u2191\u2191src\u271d.toRingHom) y),\n map_add' :=\n (_ :\n \u2200 (x y : \u03b2),\n OneHom.toFun (\u2191\u2191src\u271d.toRingHom) (x + y) =\n OneHom.toFun (\u2191\u2191src\u271d.toRingHom) x + OneHom.toFun (\u2191\u2191src\u271d.toRingHom) y) }.toEquiv\n a \u2264\n Equiv.toFun\n {\n toEquiv :=\n { toFun := src\u271d.toFun, invFun := inducedMap \u03b3 \u03b2,\n left_inv := (_ : \u2200 (b : \u03b2), inducedMap \u03b3 \u03b2 (inducedMap \u03b2 \u03b3 b) = b),\n right_inv := (_ : \u2200 (b : \u03b3), inducedMap \u03b2 \u03b3 (inducedMap \u03b3 \u03b2 b) = b) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u03b2),\n OneHom.toFun (\u2191\u2191src\u271d.toRingHom) (x * y) =\n OneHom.toFun (\u2191\u2191src\u271d.toRingHom) x * OneHom.toFun (\u2191\u2191src\u271d.toRingHom) y),\n map_add' :=\n (_ :\n \u2200 (x y : \u03b2),\n OneHom.toFun (\u2191\u2191src\u271d.toRingHom) (x + y) =\n OneHom.toFun (\u2191\u2191src\u271d.toRingHom) x + OneHom.toFun (\u2191\u2191src\u271d.toRingHom) y) }.toEquiv\n b \u2194\n a \u2264 b\n[PROOFSTEP]\ndsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nsrc\u271d : \u03b2 \u2192+*o \u03b3 := inducedOrderRingHom \u03b2 \u03b3\n\u22a2 \u2200 {a b : \u03b2}, \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom a \u2264 \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom b \u2194 a \u2264 b\n[PROOFSTEP]\nrefine \u27e8fun h => ?_, fun h => inducedMap_mono _ _ h\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nsrc\u271d : \u03b2 \u2192+*o \u03b3 := inducedOrderRingHom \u03b2 \u03b3\na\u271d b\u271d : \u03b2\nh : \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom a\u271d \u2264 \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom b\u271d\n\u22a2 a\u271d \u2264 b\u271d\n[PROOFSTEP]\nconvert inducedMap_mono \u03b3 \u03b2 h\n[GOAL]\ncase h.e'_3\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nsrc\u271d : \u03b2 \u2192+*o \u03b3 := inducedOrderRingHom \u03b2 \u03b3\na\u271d b\u271d : \u03b2\nh : \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom a\u271d \u2264 \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom b\u271d\n\u22a2 a\u271d = inducedMap \u03b3 \u03b2 (\u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom a\u271d)\n[PROOFSTEP]\nrw [inducedOrderRingHom, AddMonoidHom.coe_fn_mkRingHomOfMulSelfOfTwoNeZero, inducedAddHom]\n[GOAL]\ncase h.e'_3\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nsrc\u271d : \u03b2 \u2192+*o \u03b3 := inducedOrderRingHom \u03b2 \u03b3\na\u271d b\u271d : \u03b2\nh : \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom a\u271d \u2264 \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom b\u271d\n\u22a2 a\u271d =\n inducedMap \u03b3 \u03b2\n (\u2191{ toZeroHom := { toFun := inducedMap \u03b2 \u03b3, map_zero' := (_ : inducedMap \u03b2 \u03b3 0 = 0) },\n map_add' := (_ : \u2200 (x y : \u03b2), inducedMap \u03b2 \u03b3 (x + y) = inducedMap \u03b2 \u03b3 x + inducedMap \u03b2 \u03b3 y) }\n a\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.e'_3\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nsrc\u271d : \u03b2 \u2192+*o \u03b3 := inducedOrderRingHom \u03b2 \u03b3\na\u271d b\u271d : \u03b2\nh : \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom a\u271d \u2264 \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom b\u271d\n\u22a2 a\u271d = inducedMap \u03b3 \u03b2 (inducedMap \u03b2 \u03b3 a\u271d)\n[PROOFSTEP]\nrw [inducedMap_inv_self \u03b2 \u03b3 _]\n[GOAL]\ncase h.e'_4\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nsrc\u271d : \u03b2 \u2192+*o \u03b3 := inducedOrderRingHom \u03b2 \u03b3\na\u271d b\u271d : \u03b2\nh : \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom a\u271d \u2264 \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom b\u271d\n\u22a2 b\u271d = inducedMap \u03b3 \u03b2 (\u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom b\u271d)\n[PROOFSTEP]\nrw [inducedOrderRingHom, AddMonoidHom.coe_fn_mkRingHomOfMulSelfOfTwoNeZero, inducedAddHom]\n[GOAL]\ncase h.e'_4\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nsrc\u271d : \u03b2 \u2192+*o \u03b3 := inducedOrderRingHom \u03b2 \u03b3\na\u271d b\u271d : \u03b2\nh : \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom a\u271d \u2264 \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom b\u271d\n\u22a2 b\u271d =\n inducedMap \u03b3 \u03b2\n (\u2191{ toZeroHom := { toFun := inducedMap \u03b2 \u03b3, map_zero' := (_ : inducedMap \u03b2 \u03b3 0 = 0) },\n map_add' := (_ : \u2200 (x y : \u03b2), inducedMap \u03b2 \u03b3 (x + y) = inducedMap \u03b2 \u03b3 x + inducedMap \u03b2 \u03b3 y) }\n b\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.e'_4\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrderedField \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderedField \u03b3\ninst\u271d : Archimedean \u03b1\na : \u03b1\nb : \u03b2\nq : \u211a\nsrc\u271d : \u03b2 \u2192+*o \u03b3 := inducedOrderRingHom \u03b2 \u03b3\na\u271d b\u271d : \u03b2\nh : \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom a\u271d \u2264 \u2191(inducedOrderRingHom \u03b2 \u03b3).toRingHom b\u271d\n\u22a2 b\u271d = inducedMap \u03b3 \u03b2 (inducedMap \u03b2 \u03b3 b\u271d)\n[PROOFSTEP]\nrw [inducedMap_inv_self \u03b2 \u03b3 _]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nR : Type u_5\nS : Type u_6\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : LinearOrderedRing S\nhR : \u2200 (r : R), 0 \u2264 r \u2192 \u2203 s, s ^ 2 = r\nf : R \u2192+* S\nr : R\nh : 0 \u2264 r\n\u22a2 0 \u2264 \u2191f r\n[PROOFSTEP]\nobtain \u27e8s, rfl\u27e9 := hR r h\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nR : Type u_5\nS : Type u_6\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : LinearOrderedRing S\nhR : \u2200 (r : R), 0 \u2264 r \u2192 \u2203 s, s ^ 2 = r\nf : R \u2192+* S\ns : R\nh : 0 \u2264 s ^ 2\n\u22a2 0 \u2264 \u2191f (s ^ 2)\n[PROOFSTEP]\nrw [map_pow]\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nR : Type u_5\nS : Type u_6\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : LinearOrderedRing S\nhR : \u2200 (r : R), 0 \u2264 r \u2192 \u2203 s, s ^ 2 = r\nf : R \u2192+* S\ns : R\nh : 0 \u2264 s ^ 2\n\u22a2 0 \u2264 \u2191f s ^ 2\n[PROOFSTEP]\napply sq_nonneg\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.CompleteField", "llama_tokens": 18288, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.4293891542046626}} {"text": "[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\n\u22a2 fourierCoeff (Periodic.lift (_ : Periodic (\u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight (n \u2022 1)))) 1))\n m =\n \ud835\udcd5 \u2191f \u2191m\n[PROOFSTEP]\nlet e : C(\u211d, \u2102) := (fourier (-m)).comp \u27e8((\u2191) : \u211d \u2192 UnitAddCircle), continuous_quotient_mk'\u27e9\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\n\u22a2 fourierCoeff (Periodic.lift (_ : Periodic (\u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight (n \u2022 1)))) 1))\n m =\n \ud835\udcd5 \u2191f \u2191m\n[PROOFSTEP]\nhave neK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016(e * g).restrict K\u2016 = \u2016g.restrict K\u2016 :=\n by\n have : \u2200 x : \u211d, \u2016e x\u2016 = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m \u2022 x))\n intro K g\n simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\n\u22a2 \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\n[PROOFSTEP]\nhave : \u2200 x : \u211d, \u2016e x\u2016 = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m \u2022 x))\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nthis : \u2200 (x : \u211d), \u2016\u2191e x\u2016 = 1\n\u22a2 \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\n[PROOFSTEP]\nintro K g\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nthis : \u2200 (x : \u211d), \u2016\u2191e x\u2016 = 1\nK : Compacts \u211d\ng : C(\u211d, \u2102)\n\u22a2 \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\n[PROOFSTEP]\nsimp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\n\u22a2 fourierCoeff (Periodic.lift (_ : Periodic (\u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight (n \u2022 1)))) 1))\n m =\n \ud835\udcd5 \u2191f \u2191m\n[PROOFSTEP]\nhave eadd : \u2200 (n : \u2124), e.comp (ContinuousMap.addRight n) = e :=\n by\n intro n; ext1 x\n have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))\n simpa only [mul_one] using this.int_mul n x\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\n\u22a2 \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\n[PROOFSTEP]\nintro n\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\nn : \u2124\n\u22a2 ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\nn : \u2124\nx : \u211d\n\u22a2 \u2191(ContinuousMap.comp e (ContinuousMap.addRight \u2191n)) x = \u2191e x\n[PROOFSTEP]\nhave : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))\n[GOAL]\ncase h\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\nn : \u2124\nx : \u211d\nthis : Periodic (\u2191e) 1\n\u22a2 \u2191(ContinuousMap.comp e (ContinuousMap.addRight \u2191n)) x = \u2191e x\n[PROOFSTEP]\nsimpa only [mul_one] using this.int_mul n x\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\n\u22a2 fourierCoeff (Periodic.lift (_ : Periodic (\u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight (n \u2022 1)))) 1))\n m =\n \ud835\udcd5 \u2191f \u2191m\n[PROOFSTEP]\ncalc\n fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =\n \u222b x in (0 : \u211d)..1, e x * (\u2211' n : \u2124, f.comp (ContinuousMap.addRight n)) x :=\n by\n simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply, coe_mk, Periodic.lift_coe,\n zsmul_one, smul_eq_mul]\n -- Transform sum in C(\u211d, \u2102) evaluated at x into pointwise sum of values.\n _ = \u222b x in (0 : \u211d)..1, \u2211' n : \u2124, (e * f.comp (ContinuousMap.addRight n)) x := by\n simp_rw [coe_mul, Pi.mul_apply, \u2190 ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]\n -- Swap sum and integral.\n _ = \u2211' n : \u2124, \u222b x in (0 : \u211d)..1, (e * f.comp (ContinuousMap.addRight n)) x :=\n by\n refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm\n convert hf \u27e8uIcc 0 1, isCompact_uIcc\u27e9 using 1\n exact funext fun n => neK _ _\n _ = \u2211' n : \u2124, \u222b x in (0 : \u211d)..1, (e * f).comp (ContinuousMap.addRight n) x :=\n by\n simp only [ContinuousMap.comp_apply, mul_comp] at eadd \u22a2\n simp_rw [eadd]\n -- Rearrange sum of interval integrals into an integral over `\u211d`.\n _ = \u222b x, e x * f x :=\n by\n suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq\n apply integrable_of_summable_norm_Icc\n convert hf \u27e8Icc 0 1, isCompact_Icc\u27e9 using 1\n simp_rw [mul_comp] at eadd \u22a2\n simp_rw [eadd]\n exact\n funext fun n =>\n neK \u27e8Icc 0 1, isCompact_Icc\u27e9\n _\n -- Minor tidying to finish\n _ = \ud835\udcd5 f m := by\n rw [fourierIntegral_eq_integral_exp_smul]\n congr 1 with x : 1\n rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]\n congr 2\n push_cast\n ring\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\n\u22a2 fourierCoeff (Periodic.lift (_ : Periodic (\u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight (n \u2022 1)))) 1))\n m =\n \u222b (x : \u211d) in 0 ..1, \u2191e x * \u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight \u2191n)) x\n[PROOFSTEP]\nsimp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply, coe_mk, Periodic.lift_coe,\n zsmul_one, smul_eq_mul]\n -- Transform sum in C(\u211d, \u2102) evaluated at x into pointwise sum of values.\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\n\u22a2 \u222b (x : \u211d) in 0 ..1, \u2191e x * \u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight \u2191n)) x =\n \u222b (x : \u211d) in 0 ..1, \u2211' (n : \u2124), \u2191(e * ContinuousMap.comp f (ContinuousMap.addRight \u2191n)) x\n[PROOFSTEP]\nsimp_rw [coe_mul, Pi.mul_apply, \u2190 ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]\n -- Swap sum and integral.\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\n\u22a2 \u222b (x : \u211d) in 0 ..1, \u2211' (n : \u2124), \u2191(e * ContinuousMap.comp f (ContinuousMap.addRight \u2191n)) x =\n \u2211' (n : \u2124), \u222b (x : \u211d) in 0 ..1, \u2191(e * ContinuousMap.comp f (ContinuousMap.addRight \u2191n)) x\n[PROOFSTEP]\nrefine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\n\u22a2 Summable fun i =>\n \u2016ContinuousMap.restrict (\u2191{ carrier := uIcc 0 1, isCompact' := (_ : IsCompact (uIcc 0 1)) })\n (e * ContinuousMap.comp f (ContinuousMap.addRight \u2191i))\u2016\n[PROOFSTEP]\nconvert hf \u27e8uIcc 0 1, isCompact_uIcc\u27e9 using 1\n[GOAL]\ncase h.e'_5\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\n\u22a2 (fun i =>\n \u2016ContinuousMap.restrict (\u2191{ carrier := uIcc 0 1, isCompact' := (_ : IsCompact (uIcc 0 1)) })\n (e * ContinuousMap.comp f (ContinuousMap.addRight \u2191i))\u2016) =\n fun n =>\n \u2016ContinuousMap.restrict (\u2191{ carrier := uIcc 0 1, isCompact' := (_ : IsCompact (uIcc 0 1)) })\n (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\n[PROOFSTEP]\nexact funext fun n => neK _ _\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\n\u22a2 \u2211' (n : \u2124), \u222b (x : \u211d) in 0 ..1, \u2191(e * ContinuousMap.comp f (ContinuousMap.addRight \u2191n)) x =\n \u2211' (n : \u2124), \u222b (x : \u211d) in 0 ..1, \u2191(ContinuousMap.comp (e * f) (ContinuousMap.addRight \u2191n)) x\n[PROOFSTEP]\nsimp only [ContinuousMap.comp_apply, mul_comp] at eadd \u22a2\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd :\n \u2200 (n : \u2124),\n ContinuousMap.comp (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk))\n (ContinuousMap.addRight \u2191n) =\n ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\n\u22a2 \u2211' (n : \u2124),\n \u222b (x : \u211d) in 0 ..1,\n \u2191(ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) *\n ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\n x =\n \u2211' (n : \u2124),\n \u222b (x : \u211d) in 0 ..1,\n \u2191(ContinuousMap.comp (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk))\n (ContinuousMap.addRight \u2191n) *\n ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\n x\n[PROOFSTEP]\nsimp_rw [eadd]\n -- Rearrange sum of interval integrals into an integral over `\u211d`.\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\n\u22a2 \u2211' (n : \u2124), \u222b (x : \u211d) in 0 ..1, \u2191(ContinuousMap.comp (e * f) (ContinuousMap.addRight \u2191n)) x = \u222b (x : \u211d), \u2191e x * \u2191f x\n[PROOFSTEP]\nsuffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\n\u22a2 Integrable \u2191(e * f)\n[PROOFSTEP]\napply integrable_of_summable_norm_Icc\n[GOAL]\ncase hf\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\n\u22a2 Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (ContinuousMap.comp (e * f) (ContinuousMap.addRight \u2191n))\u2016\n[PROOFSTEP]\nconvert hf \u27e8Icc 0 1, isCompact_Icc\u27e9 using 1\n[GOAL]\ncase h.e'_5\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\n\u22a2 (fun n => \u2016ContinuousMap.restrict (Icc 0 1) (ContinuousMap.comp (e * f) (ContinuousMap.addRight \u2191n))\u2016) = fun n =>\n \u2016ContinuousMap.restrict (\u2191{ carrier := Icc 0 1, isCompact' := (_ : IsCompact (Icc 0 1)) })\n (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\n[PROOFSTEP]\nsimp_rw [mul_comp] at eadd \u22a2\n[GOAL]\ncase h.e'_5\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd :\n \u2200 (n : \u2124),\n ContinuousMap.comp (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk))\n (ContinuousMap.addRight \u2191n) =\n ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\n\u22a2 (fun n =>\n \u2016ContinuousMap.restrict (Icc 0 1)\n (ContinuousMap.comp (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk))\n (ContinuousMap.addRight \u2191n) *\n ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016) =\n fun n =>\n \u2016ContinuousMap.restrict (\u2191{ carrier := Icc 0 1, isCompact' := (_ : IsCompact (Icc 0 1)) })\n (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\n[PROOFSTEP]\nsimp_rw [eadd]\n[GOAL]\ncase h.e'_5\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd :\n \u2200 (n : \u2124),\n ContinuousMap.comp (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk))\n (ContinuousMap.addRight \u2191n) =\n ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\n\u22a2 (fun n =>\n \u2016ContinuousMap.restrict (Icc 0 1)\n (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) *\n ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016) =\n fun n =>\n \u2016ContinuousMap.restrict (\u2191{ carrier := Icc 0 1, isCompact' := (_ : IsCompact (Icc 0 1)) })\n (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\n[PROOFSTEP]\nexact\n funext fun n =>\n neK \u27e8Icc 0 1, isCompact_Icc\u27e9\n _\n -- Minor tidying to finish\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\n\u22a2 \u222b (x : \u211d), \u2191e x * \u2191f x = \ud835\udcd5 \u2191f \u2191m\n[PROOFSTEP]\nrw [fourierIntegral_eq_integral_exp_smul]\n[GOAL]\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\n\u22a2 \u222b (x : \u211d), \u2191e x * \u2191f x = \u222b (v : \u211d), Complex.exp (\u2191(-2 * \u03c0 * v * \u2191m) * I) \u2022 \u2191f v\n[PROOFSTEP]\ncongr 1 with x : 1\n[GOAL]\ncase e_f.h\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\nx : \u211d\n\u22a2 \u2191e x * \u2191f x = Complex.exp (\u2191(-2 * \u03c0 * x * \u2191m) * I) \u2022 \u2191f x\n[PROOFSTEP]\nrw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]\n[GOAL]\ncase e_f.h\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\nx : \u211d\n\u22a2 Complex.exp (2 * \u2191\u03c0 * I * \u2191(-m) * \u2191x / \u21911) * \u2191f x = Complex.exp (\u2191(-2 * \u03c0 * x * \u2191m) * I) * \u2191f x\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_f.h.e_a.e_z\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\nx : \u211d\n\u22a2 2 * \u2191\u03c0 * I * \u2191(-m) * \u2191x / \u21911 = \u2191(-2 * \u03c0 * x * \u2191m) * I\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase e_f.h.e_a.e_z\nf : C(\u211d, \u2102)\nhf :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nm : \u2124\ne : C(\u211d, \u2102) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)\nneK : \u2200 (K : Compacts \u211d) (g : C(\u211d, \u2102)), \u2016ContinuousMap.restrict (\u2191K) (e * g)\u2016 = \u2016ContinuousMap.restrict (\u2191K) g\u2016\neadd : \u2200 (n : \u2124), ContinuousMap.comp e (ContinuousMap.addRight \u2191n) = e\nx : \u211d\n\u22a2 2 * \u2191\u03c0 * I * -\u2191m * \u2191x / 1 = -2 * \u2191\u03c0 * \u2191x * \u2191m * I\n[PROOFSTEP]\nring\n[GOAL]\nf : C(\u211d, \u2102)\nh_norm :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nh_sum : Summable fun n => \ud835\udcd5 \u2191f \u2191n\n\u22a2 \u2211' (n : \u2124), \u2191f \u2191n = \u2211' (n : \u2124), \ud835\udcd5 \u2191f \u2191n\n[PROOFSTEP]\nlet F : C(UnitAddCircle, \u2102) :=\n \u27e8(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)\u27e9\n[GOAL]\nf : C(\u211d, \u2102)\nh_norm :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nh_sum : Summable fun n => \ud835\udcd5 \u2191f \u2191n\nF : C(UnitAddCircle, \u2102) :=\n ContinuousMap.mk\n (Periodic.lift (_ : Periodic (\u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight (n \u2022 1)))) 1))\n\u22a2 \u2211' (n : \u2124), \u2191f \u2191n = \u2211' (n : \u2124), \ud835\udcd5 \u2191f \u2191n\n[PROOFSTEP]\nhave : Summable (fourierCoeff F) := by\n convert h_sum\n exact Real.fourierCoeff_tsum_comp_add h_norm _\n[GOAL]\nf : C(\u211d, \u2102)\nh_norm :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nh_sum : Summable fun n => \ud835\udcd5 \u2191f \u2191n\nF : C(UnitAddCircle, \u2102) :=\n ContinuousMap.mk\n (Periodic.lift (_ : Periodic (\u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight (n \u2022 1)))) 1))\n\u22a2 Summable (fourierCoeff \u2191F)\n[PROOFSTEP]\nconvert h_sum\n[GOAL]\ncase h.e'_5.h\nf : C(\u211d, \u2102)\nh_norm :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nh_sum : Summable fun n => \ud835\udcd5 \u2191f \u2191n\nF : C(UnitAddCircle, \u2102) :=\n ContinuousMap.mk\n (Periodic.lift (_ : Periodic (\u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight (n \u2022 1)))) 1))\nx\u271d : \u2124\n\u22a2 fourierCoeff (\u2191F) x\u271d = \ud835\udcd5 \u2191f \u2191x\u271d\n[PROOFSTEP]\nexact Real.fourierCoeff_tsum_comp_add h_norm _\n[GOAL]\nf : C(\u211d, \u2102)\nh_norm :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nh_sum : Summable fun n => \ud835\udcd5 \u2191f \u2191n\nF : C(UnitAddCircle, \u2102) :=\n ContinuousMap.mk\n (Periodic.lift (_ : Periodic (\u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight (n \u2022 1)))) 1))\nthis : Summable (fourierCoeff \u2191F)\n\u22a2 \u2211' (n : \u2124), \u2191f \u2191n = \u2211' (n : \u2124), \ud835\udcd5 \u2191f \u2191n\n[PROOFSTEP]\nconvert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1\n[GOAL]\ncase h.e'_2\nf : C(\u211d, \u2102)\nh_norm :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nh_sum : Summable fun n => \ud835\udcd5 \u2191f \u2191n\nF : C(UnitAddCircle, \u2102) :=\n ContinuousMap.mk\n (Periodic.lift (_ : Periodic (\u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight (n \u2022 1)))) 1))\nthis : Summable (fourierCoeff \u2191F)\n\u22a2 \u2211' (n : \u2124), \u2191f \u2191n = \u2191F 0\n[PROOFSTEP]\nhave := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq\n[GOAL]\ncase h.e'_2\nf : C(\u211d, \u2102)\nh_norm :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nh_sum : Summable fun n => \ud835\udcd5 \u2191f \u2191n\nF : C(UnitAddCircle, \u2102) :=\n ContinuousMap.mk\n (Periodic.lift (_ : Periodic (\u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight (n \u2022 1)))) 1))\nthis\u271d : Summable (fourierCoeff \u2191F)\nthis :\n \u2211' (b : \u2124), \u2191(ContinuousMap.comp f (ContinuousMap.addRight \u2191b)) 0 =\n \u2191(\u2211' (b : \u2124), ContinuousMap.comp f (ContinuousMap.addRight \u2191b)) 0\n\u22a2 \u2211' (n : \u2124), \u2191f \u2191n = \u2191F 0\n[PROOFSTEP]\nsimpa only [coe_mk, \u2190 QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using\n this\n[GOAL]\ncase h.e'_3\nf : C(\u211d, \u2102)\nh_norm :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nh_sum : Summable fun n => \ud835\udcd5 \u2191f \u2191n\nF : C(UnitAddCircle, \u2102) :=\n ContinuousMap.mk\n (Periodic.lift (_ : Periodic (\u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight (n \u2022 1)))) 1))\nthis : Summable (fourierCoeff \u2191F)\n\u22a2 \u2211' (n : \u2124), \ud835\udcd5 \u2191f \u2191n = \u2211' (b : \u2124), fourierCoeff (\u2191F) b \u2022 \u2191(fourier b) 0\n[PROOFSTEP]\ncongr 1 with n : 1\n[GOAL]\ncase h.e'_3.e_f.h\nf : C(\u211d, \u2102)\nh_norm :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nh_sum : Summable fun n => \ud835\udcd5 \u2191f \u2191n\nF : C(UnitAddCircle, \u2102) :=\n ContinuousMap.mk\n (Periodic.lift (_ : Periodic (\u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight (n \u2022 1)))) 1))\nthis : Summable (fourierCoeff \u2191F)\nn : \u2124\n\u22a2 \ud835\udcd5 \u2191f \u2191n = fourierCoeff (\u2191F) n \u2022 \u2191(fourier n) 0\n[PROOFSTEP]\nrw [\u2190 Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]\n[GOAL]\ncase h.e'_3.e_f.h\nf : C(\u211d, \u2102)\nh_norm :\n \u2200 (K : Compacts \u211d), Summable fun n => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight \u2191n))\u2016\nh_sum : Summable fun n => \ud835\udcd5 \u2191f \u2191n\nF : C(UnitAddCircle, \u2102) :=\n ContinuousMap.mk\n (Periodic.lift (_ : Periodic (\u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight (n \u2022 1)))) 1))\nthis : Summable (fourierCoeff \u2191F)\nn : \u2124\n\u22a2 fourierCoeff (Periodic.lift (_ : Periodic (\u2191(\u2211' (n : \u2124), ContinuousMap.comp f (ContinuousMap.addRight (n \u2022 1)))) 1))\n n =\n fourierCoeff (\u2191F) n\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\n\u22a2 (fun x => \u2016ContinuousMap.restrict (Icc (x + R) (x + S)) f\u2016) =O[atTop] fun x => |x| ^ (-b)\n[PROOFSTEP]\nhave claim : \u2200 x : \u211d, max 0 (-2 * R) < x \u2192 \u2200 y : \u211d, x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b) :=\n by\n intro x hx y hy\n rw [max_lt_iff] at hx \n have hxR : 0 < x + R := by\n rcases le_or_lt 0 R with (h | h)\n \u00b7 exact add_pos_of_pos_of_nonneg hx.1 h\n \u00b7 rw [\u2190 sub_lt_iff_lt_add, zero_sub]\n refine' lt_trans _ hx.2\n rwa [neg_mul, neg_lt_neg_iff, two_mul, add_lt_iff_neg_left]\n have hy' : 0 < y := hxR.trans_le hy\n have : y ^ (-b) \u2264 (x + R) ^ (-b) :=\n by\n rw [rpow_neg hy'.le, rpow_neg hxR.le, inv_le_inv (rpow_pos_of_pos hy' _) (rpow_pos_of_pos hxR _)]\n exact rpow_le_rpow hxR.le hy hb.le\n refine' this.trans _\n rw [\u2190 mul_rpow one_half_pos.le hx.1.le, rpow_neg (mul_pos one_half_pos hx.1).le, rpow_neg hxR.le]\n refine' inv_le_inv_of_le (rpow_pos_of_pos (mul_pos one_half_pos hx.1) _) _\n exact rpow_le_rpow (mul_pos one_half_pos hx.1).le (by linarith) hb.le\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\n\u22a2 \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\n[PROOFSTEP]\nintro x hx y hy\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : max 0 (-2 * R) < x\ny : \u211d\nhy : x + R \u2264 y\n\u22a2 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\n[PROOFSTEP]\nrw [max_lt_iff] at hx \n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : 0 < x \u2227 -2 * R < x\ny : \u211d\nhy : x + R \u2264 y\n\u22a2 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\n[PROOFSTEP]\nhave hxR : 0 < x + R := by\n rcases le_or_lt 0 R with (h | h)\n \u00b7 exact add_pos_of_pos_of_nonneg hx.1 h\n \u00b7 rw [\u2190 sub_lt_iff_lt_add, zero_sub]\n refine' lt_trans _ hx.2\n rwa [neg_mul, neg_lt_neg_iff, two_mul, add_lt_iff_neg_left]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : 0 < x \u2227 -2 * R < x\ny : \u211d\nhy : x + R \u2264 y\n\u22a2 0 < x + R\n[PROOFSTEP]\nrcases le_or_lt 0 R with (h | h)\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : 0 < x \u2227 -2 * R < x\ny : \u211d\nhy : x + R \u2264 y\nh : 0 \u2264 R\n\u22a2 0 < x + R\n[PROOFSTEP]\nexact add_pos_of_pos_of_nonneg hx.1 h\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : 0 < x \u2227 -2 * R < x\ny : \u211d\nhy : x + R \u2264 y\nh : R < 0\n\u22a2 0 < x + R\n[PROOFSTEP]\nrw [\u2190 sub_lt_iff_lt_add, zero_sub]\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : 0 < x \u2227 -2 * R < x\ny : \u211d\nhy : x + R \u2264 y\nh : R < 0\n\u22a2 -R < x\n[PROOFSTEP]\nrefine' lt_trans _ hx.2\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : 0 < x \u2227 -2 * R < x\ny : \u211d\nhy : x + R \u2264 y\nh : R < 0\n\u22a2 -R < -2 * R\n[PROOFSTEP]\nrwa [neg_mul, neg_lt_neg_iff, two_mul, add_lt_iff_neg_left]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : 0 < x \u2227 -2 * R < x\ny : \u211d\nhy : x + R \u2264 y\nhxR : 0 < x + R\n\u22a2 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\n[PROOFSTEP]\nhave hy' : 0 < y := hxR.trans_le hy\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : 0 < x \u2227 -2 * R < x\ny : \u211d\nhy : x + R \u2264 y\nhxR : 0 < x + R\nhy' : 0 < y\n\u22a2 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\n[PROOFSTEP]\nhave : y ^ (-b) \u2264 (x + R) ^ (-b) :=\n by\n rw [rpow_neg hy'.le, rpow_neg hxR.le, inv_le_inv (rpow_pos_of_pos hy' _) (rpow_pos_of_pos hxR _)]\n exact rpow_le_rpow hxR.le hy hb.le\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : 0 < x \u2227 -2 * R < x\ny : \u211d\nhy : x + R \u2264 y\nhxR : 0 < x + R\nhy' : 0 < y\n\u22a2 y ^ (-b) \u2264 (x + R) ^ (-b)\n[PROOFSTEP]\nrw [rpow_neg hy'.le, rpow_neg hxR.le, inv_le_inv (rpow_pos_of_pos hy' _) (rpow_pos_of_pos hxR _)]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : 0 < x \u2227 -2 * R < x\ny : \u211d\nhy : x + R \u2264 y\nhxR : 0 < x + R\nhy' : 0 < y\n\u22a2 (x + R) ^ b \u2264 y ^ b\n[PROOFSTEP]\nexact rpow_le_rpow hxR.le hy hb.le\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : 0 < x \u2227 -2 * R < x\ny : \u211d\nhy : x + R \u2264 y\nhxR : 0 < x + R\nhy' : 0 < y\nthis : y ^ (-b) \u2264 (x + R) ^ (-b)\n\u22a2 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\n[PROOFSTEP]\nrefine' this.trans _\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : 0 < x \u2227 -2 * R < x\ny : \u211d\nhy : x + R \u2264 y\nhxR : 0 < x + R\nhy' : 0 < y\nthis : y ^ (-b) \u2264 (x + R) ^ (-b)\n\u22a2 (x + R) ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\n[PROOFSTEP]\nrw [\u2190 mul_rpow one_half_pos.le hx.1.le, rpow_neg (mul_pos one_half_pos hx.1).le, rpow_neg hxR.le]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : 0 < x \u2227 -2 * R < x\ny : \u211d\nhy : x + R \u2264 y\nhxR : 0 < x + R\nhy' : 0 < y\nthis : y ^ (-b) \u2264 (x + R) ^ (-b)\n\u22a2 ((x + R) ^ b)\u207b\u00b9 \u2264 ((1 / 2 * x) ^ b)\u207b\u00b9\n[PROOFSTEP]\nrefine' inv_le_inv_of_le (rpow_pos_of_pos (mul_pos one_half_pos hx.1) _) _\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : 0 < x \u2227 -2 * R < x\ny : \u211d\nhy : x + R \u2264 y\nhxR : 0 < x + R\nhy' : 0 < y\nthis : y ^ (-b) \u2264 (x + R) ^ (-b)\n\u22a2 (1 / 2 * x) ^ b \u2264 (x + R) ^ b\n[PROOFSTEP]\nexact rpow_le_rpow (mul_pos one_half_pos hx.1).le (by linarith) hb.le\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S x : \u211d\nhx : 0 < x \u2227 -2 * R < x\ny : \u211d\nhy : x + R \u2264 y\nhxR : 0 < x + R\nhy' : 0 < y\nthis : y ^ (-b) \u2264 (x + R) ^ (-b)\n\u22a2 1 / 2 * x \u2264 x + R\n[PROOFSTEP]\nlinarith\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\n\u22a2 (fun x => \u2016ContinuousMap.restrict (Icc (x + R) (x + S)) f\u2016) =O[atTop] fun x => |x| ^ (-b)\n[PROOFSTEP]\nobtain \u27e8c, hc, hc'\u27e9 := hf.exists_pos\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nhc' : IsBigOWith c atTop \u2191f fun x => |x| ^ (-b)\n\u22a2 (fun x => \u2016ContinuousMap.restrict (Icc (x + R) (x + S)) f\u2016) =O[atTop] fun x => |x| ^ (-b)\n[PROOFSTEP]\nsimp only [IsBigO, IsBigOWith, eventually_atTop] at hc' \u22a2\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nhc' : \u2203 a, \u2200 (b_1 : \u211d), b_1 \u2265 a \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\n\u22a2 \u2203 c a, \u2200 (b_1 : \u211d), b_1 \u2265 a \u2192 \u2016\u2016ContinuousMap.restrict (Icc (b_1 + R) (b_1 + S)) f\u2016\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\n[PROOFSTEP]\nobtain \u27e8d, hd\u27e9 := hc'\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\n\u22a2 \u2203 c a, \u2200 (b_1 : \u211d), b_1 \u2265 a \u2192 \u2016\u2016ContinuousMap.restrict (Icc (b_1 + R) (b_1 + S)) f\u2016\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\n[PROOFSTEP]\nrefine' \u27e8c * (1 / 2) ^ (-b), \u27e8max (1 + max 0 (-2 * R)) (d - R), fun x hx => _\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\nx : \u211d\nhx : x \u2265 max (1 + max 0 (-2 * R)) (d - R)\n\u22a2 \u2016\u2016ContinuousMap.restrict (Icc (x + R) (x + S)) f\u2016\u2016 \u2264 c * (1 / 2) ^ (-b) * \u2016|x| ^ (-b)\u2016\n[PROOFSTEP]\nrw [ge_iff_le, max_le_iff] at hx \n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\nx : \u211d\nhx : 1 + max 0 (-2 * R) \u2264 x \u2227 d - R \u2264 x\n\u22a2 \u2016\u2016ContinuousMap.restrict (Icc (x + R) (x + S)) f\u2016\u2016 \u2264 c * (1 / 2) ^ (-b) * \u2016|x| ^ (-b)\u2016\n[PROOFSTEP]\nhave hx' : max 0 (-2 * R) < x := by linarith\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\nx : \u211d\nhx : 1 + max 0 (-2 * R) \u2264 x \u2227 d - R \u2264 x\n\u22a2 max 0 (-2 * R) < x\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\nx : \u211d\nhx : 1 + max 0 (-2 * R) \u2264 x \u2227 d - R \u2264 x\nhx' : max 0 (-2 * R) < x\n\u22a2 \u2016\u2016ContinuousMap.restrict (Icc (x + R) (x + S)) f\u2016\u2016 \u2264 c * (1 / 2) ^ (-b) * \u2016|x| ^ (-b)\u2016\n[PROOFSTEP]\nrw [max_lt_iff] at hx' \n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\nx : \u211d\nhx : 1 + max 0 (-2 * R) \u2264 x \u2227 d - R \u2264 x\nhx' : 0 < x \u2227 -2 * R < x\n\u22a2 \u2016\u2016ContinuousMap.restrict (Icc (x + R) (x + S)) f\u2016\u2016 \u2264 c * (1 / 2) ^ (-b) * \u2016|x| ^ (-b)\u2016\n[PROOFSTEP]\nrw [norm_norm,\n ContinuousMap.norm_le _ (mul_nonneg (mul_nonneg hc.le <| rpow_nonneg_of_nonneg one_half_pos.le _) (norm_nonneg _))]\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\nx : \u211d\nhx : 1 + max 0 (-2 * R) \u2264 x \u2227 d - R \u2264 x\nhx' : 0 < x \u2227 -2 * R < x\n\u22a2 \u2200 (x_1 : \u2191(Icc (x + R) (x + S))),\n \u2016\u2191(ContinuousMap.restrict (Icc (x + R) (x + S)) f) x_1\u2016 \u2264 c * (1 / 2) ^ (-b) * \u2016|x| ^ (-b)\u2016\n[PROOFSTEP]\nrefine' fun y => (hd y.1 (by linarith [hx.1, y.2.1])).trans _\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\nx : \u211d\nhx : 1 + max 0 (-2 * R) \u2264 x \u2227 d - R \u2264 x\nhx' : 0 < x \u2227 -2 * R < x\ny : \u2191(Icc (x + R) (x + S))\n\u22a2 \u2191y \u2265 d\n[PROOFSTEP]\nlinarith [hx.1, y.2.1]\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\nx : \u211d\nhx : 1 + max 0 (-2 * R) \u2264 x \u2227 d - R \u2264 x\nhx' : 0 < x \u2227 -2 * R < x\ny : \u2191(Icc (x + R) (x + S))\n\u22a2 c * \u2016|\u2191y| ^ (-b)\u2016 \u2264 c * (1 / 2) ^ (-b) * \u2016|x| ^ (-b)\u2016\n[PROOFSTEP]\nhave A : \u2200 x : \u211d, 0 \u2264 |x| ^ (-b) := fun x => by positivity\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\nx\u271d : \u211d\nhx : 1 + max 0 (-2 * R) \u2264 x\u271d \u2227 d - R \u2264 x\u271d\nhx' : 0 < x\u271d \u2227 -2 * R < x\u271d\ny : \u2191(Icc (x\u271d + R) (x\u271d + S))\nx : \u211d\n\u22a2 0 \u2264 |x| ^ (-b)\n[PROOFSTEP]\npositivity\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\nx : \u211d\nhx : 1 + max 0 (-2 * R) \u2264 x \u2227 d - R \u2264 x\nhx' : 0 < x \u2227 -2 * R < x\ny : \u2191(Icc (x + R) (x + S))\nA : \u2200 (x : \u211d), 0 \u2264 |x| ^ (-b)\n\u22a2 c * \u2016|\u2191y| ^ (-b)\u2016 \u2264 c * (1 / 2) ^ (-b) * \u2016|x| ^ (-b)\u2016\n[PROOFSTEP]\nrw [mul_assoc, mul_le_mul_left hc, norm_of_nonneg (A _), norm_of_nonneg (A _)]\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\nx : \u211d\nhx : 1 + max 0 (-2 * R) \u2264 x \u2227 d - R \u2264 x\nhx' : 0 < x \u2227 -2 * R < x\ny : \u2191(Icc (x + R) (x + S))\nA : \u2200 (x : \u211d), 0 \u2264 |x| ^ (-b)\n\u22a2 |\u2191y| ^ (-b) \u2264 (1 / 2) ^ (-b) * |x| ^ (-b)\n[PROOFSTEP]\nconvert claim x (by linarith only [hx.1]) y.1 y.2.1\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\nx : \u211d\nhx : 1 + max 0 (-2 * R) \u2264 x \u2227 d - R \u2264 x\nhx' : 0 < x \u2227 -2 * R < x\ny : \u2191(Icc (x + R) (x + S))\nA : \u2200 (x : \u211d), 0 \u2264 |x| ^ (-b)\n\u22a2 max 0 (-2 * R) < x\n[PROOFSTEP]\nlinarith only [hx.1]\n[GOAL]\ncase h.e'_3.h.e'_5\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\nx : \u211d\nhx : 1 + max 0 (-2 * R) \u2264 x \u2227 d - R \u2264 x\nhx' : 0 < x \u2227 -2 * R < x\ny : \u2191(Icc (x + R) (x + S))\nA : \u2200 (x : \u211d), 0 \u2264 |x| ^ (-b)\n\u22a2 |\u2191y| = \u2191y\n[PROOFSTEP]\napply abs_of_nonneg\n[GOAL]\ncase h.e'_3.h.e'_5.h\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\nx : \u211d\nhx : 1 + max 0 (-2 * R) \u2264 x \u2227 d - R \u2264 x\nhx' : 0 < x \u2227 -2 * R < x\ny : \u2191(Icc (x + R) (x + S))\nA : \u2200 (x : \u211d), 0 \u2264 |x| ^ (-b)\n\u22a2 0 \u2264 \u2191y\n[PROOFSTEP]\nlinarith [y.2.1]\n[GOAL]\ncase h.e'_4.h.e'_6.h.e'_5\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atTop] fun x => |x| ^ (-b)\nR S : \u211d\nclaim : \u2200 (x : \u211d), max 0 (-2 * R) < x \u2192 \u2200 (y : \u211d), x + R \u2264 y \u2192 y ^ (-b) \u2264 (1 / 2) ^ (-b) * x ^ (-b)\nc : \u211d\nhc : 0 < c\nd : \u211d\nhd : \u2200 (b_1 : \u211d), b_1 \u2265 d \u2192 \u2016\u2191f b_1\u2016 \u2264 c * \u2016|b_1| ^ (-b)\u2016\nx : \u211d\nhx : 1 + max 0 (-2 * R) \u2264 x \u2227 d - R \u2264 x\nhx' : 0 < x \u2227 -2 * R < x\ny : \u2191(Icc (x + R) (x + S))\nA : \u2200 (x : \u211d), 0 \u2264 |x| ^ (-b)\n\u22a2 |x| = x\n[PROOFSTEP]\nexact abs_of_pos hx'.1\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\n\u22a2 (fun x => \u2016ContinuousMap.restrict (Icc (x + R) (x + S)) f\u2016) =O[atBot] fun x => |x| ^ (-b)\n[PROOFSTEP]\nhave h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : \u211d => |x| ^ (-b) :=\n by\n convert hf.comp_tendsto tendsto_neg_atTop_atBot using 1\n ext1 x; simp only [Function.comp_apply, abs_neg]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\n\u22a2 \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\n[PROOFSTEP]\nconvert hf.comp_tendsto tendsto_neg_atTop_atBot using 1\n[GOAL]\ncase h.e'_8\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\n\u22a2 (fun x => |x| ^ (-b)) = (fun x => |x| ^ (-b)) \u2218 Neg.neg\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h.e'_8.h\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S x : \u211d\n\u22a2 |x| ^ (-b) = ((fun x => |x| ^ (-b)) \u2218 Neg.neg) x\n[PROOFSTEP]\nsimp only [Function.comp_apply, abs_neg]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\n\u22a2 (fun x => \u2016ContinuousMap.restrict (Icc (x + R) (x + S)) f\u2016) =O[atBot] fun x => |x| ^ (-b)\n[PROOFSTEP]\nhave h2 := (isBigO_norm_Icc_restrict_atTop hb h1 (-S) (-R)).comp_tendsto tendsto_neg_atBot_atTop\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n ((fun x => |x| ^ (-b)) \u2218 Neg.neg)\n\u22a2 (fun x => \u2016ContinuousMap.restrict (Icc (x + R) (x + S)) f\u2016) =O[atBot] fun x => |x| ^ (-b)\n[PROOFSTEP]\nhave : (fun x : \u211d => |x| ^ (-b)) \u2218 Neg.neg = fun x : \u211d => |x| ^ (-b) := by ext1 x;\n simp only [Function.comp_apply, abs_neg]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n ((fun x => |x| ^ (-b)) \u2218 Neg.neg)\n\u22a2 (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n ((fun x => |x| ^ (-b)) \u2218 Neg.neg)\nx : \u211d\n\u22a2 ((fun x => |x| ^ (-b)) \u2218 Neg.neg) x = |x| ^ (-b)\n[PROOFSTEP]\nsimp only [Function.comp_apply, abs_neg]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n ((fun x => |x| ^ (-b)) \u2218 Neg.neg)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\n\u22a2 (fun x => \u2016ContinuousMap.restrict (Icc (x + R) (x + S)) f\u2016) =O[atBot] fun x => |x| ^ (-b)\n[PROOFSTEP]\nrw [this] at h2 \n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\n\u22a2 (fun x => \u2016ContinuousMap.restrict (Icc (x + R) (x + S)) f\u2016) =O[atBot] fun x => |x| ^ (-b)\n[PROOFSTEP]\nrefine' (isBigO_of_le _ fun x => _).trans h2\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\nx : \u211d\n\u22a2 \u2016\u2016ContinuousMap.restrict (Icc (x + R) (x + S)) f\u2016\u2016 \u2264\n \u2016((fun x =>\n \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg)\n x\u2016\n[PROOFSTEP]\nrw [norm_norm, Function.comp_apply, norm_norm, ContinuousMap.norm_le _ (norm_nonneg _)]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\nx : \u211d\n\u22a2 \u2200 (x_1 : \u2191(Icc (x + R) (x + S))),\n \u2016\u2191(ContinuousMap.restrict (Icc (x + R) (x + S)) f) x_1\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (-x + -S) (-x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase mk\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\nx\u271d x : \u211d\nhx : x \u2208 Icc (x\u271d + R) (x\u271d + S)\n\u22a2 \u2016\u2191(ContinuousMap.restrict (Icc (x\u271d + R) (x\u271d + S)) f) { val := x, property := hx }\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (-x\u271d + -S) (-x\u271d + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016\n[PROOFSTEP]\nrw [ContinuousMap.restrict_apply_mk]\n[GOAL]\ncase mk\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\nx\u271d x : \u211d\nhx : x \u2208 Icc (x\u271d + R) (x\u271d + S)\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016ContinuousMap.restrict (Icc (-x\u271d + -S) (-x\u271d + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016\n[PROOFSTEP]\nrefine' (le_of_eq _).trans (ContinuousMap.norm_coe_le_norm _ \u27e8-x, _\u27e9)\n[GOAL]\ncase mk.refine'_1\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\nx\u271d x : \u211d\nhx : x \u2208 Icc (x\u271d + R) (x\u271d + S)\n\u22a2 \u2016\u2191f x\u2016 =\n \u2016\u2191(ContinuousMap.restrict (Icc (-x\u271d + -S) (-x\u271d + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a)))\n { val := -x, property := ?mk.refine'_2 }\u2016\ncase mk.refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\nx\u271d x : \u211d\nhx : x \u2208 Icc (x\u271d + R) (x\u271d + S)\n\u22a2 -x \u2208 Icc (-x\u271d + -S) (-x\u271d + -R)\n[PROOFSTEP]\nrw [ContinuousMap.restrict_apply_mk, ContinuousMap.comp_apply, ContinuousMap.coe_mk, ContinuousMap.coe_mk, neg_neg]\n[GOAL]\ncase mk.refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\nx\u271d x : \u211d\nhx : x \u2208 Icc (x\u271d + R) (x\u271d + S)\n\u22a2 -x \u2208 Icc (-x\u271d + -S) (-x\u271d + -R)\ncase mk.refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\nx\u271d x : \u211d\nhx : x \u2208 Icc (x\u271d + R) (x\u271d + S)\n\u22a2 -x \u2208 Icc (-x\u271d + -S) (-x\u271d + -R)\ncase mk.refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\nx\u271d x : \u211d\nhx : x \u2208 Icc (x\u271d + R) (x\u271d + S)\n\u22a2 -x \u2208 Icc (-x\u271d + -S) (-x\u271d + -R)\ncase mk.refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\nx\u271d x : \u211d\nhx : x \u2208 Icc (x\u271d + R) (x\u271d + S)\n\u22a2 -x \u2208 Icc (-x\u271d + -S) (-x\u271d + -R)\ncase mk.refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\nx\u271d x : \u211d\nhx : x \u2208 Icc (x\u271d + R) (x\u271d + S)\n\u22a2 -x \u2208 Icc (-x\u271d + -S) (-x\u271d + -R)\ncase mk.refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\nx\u271d x : \u211d\nhx : x \u2208 Icc (x\u271d + R) (x\u271d + S)\n\u22a2 -x \u2208 Icc (-x\u271d + -S) (-x\u271d + -R)\n[PROOFSTEP]\nexact \u27e8by linarith [hx.2], by linarith [hx.1]\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\nx\u271d x : \u211d\nhx : x \u2208 Icc (x\u271d + R) (x\u271d + S)\n\u22a2 -x\u271d + -S \u2264 -x\n[PROOFSTEP]\nlinarith [hx.2]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[atBot] fun x => |x| ^ (-b)\nR S : \u211d\nh1 : \u2191(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n ((fun x => \u2016ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a => -a))\u2016) \u2218\n Neg.neg) =O[atBot]\n fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) \u2218 Neg.neg = fun x => |x| ^ (-b)\nx\u271d x : \u211d\nhx : x \u2208 Icc (x\u271d + R) (x\u271d + S)\n\u22a2 -x \u2264 -x\u271d + -R\n[PROOFSTEP]\nlinarith [hx.1]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[cocompact \u211d] fun x => |x| ^ (-b)\nK : Compacts \u211d\n\u22a2 (fun x => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016) =O[cocompact \u211d] fun x =>\n |x| ^ (-b)\n[PROOFSTEP]\nobtain \u27e8r, hr\u27e9 := K.isCompact.bounded.subset_ball 0\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[cocompact \u211d] fun x => |x| ^ (-b)\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Metric.closedBall 0 r\n\u22a2 (fun x => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016) =O[cocompact \u211d] fun x =>\n |x| ^ (-b)\n[PROOFSTEP]\nrw [closedBall_eq_Icc, zero_add, zero_sub] at hr \n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[cocompact \u211d] fun x => |x| ^ (-b)\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\n\u22a2 (fun x => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016) =O[cocompact \u211d] fun x =>\n |x| ^ (-b)\n[PROOFSTEP]\nhave : \u2200 x : \u211d, \u2016(f.comp (ContinuousMap.addRight x)).restrict K\u2016 \u2264 \u2016f.restrict (Icc (x - r) (x + r))\u2016 :=\n by\n intro x\n rw [ContinuousMap.norm_le _ (norm_nonneg _)]\n rintro \u27e8y, hy\u27e9\n refine' (le_of_eq _).trans (ContinuousMap.norm_coe_le_norm _ \u27e8y + x, _\u27e9)\n \u00b7 simp_rw [ContinuousMap.restrict_apply, ContinuousMap.comp_apply, ContinuousMap.coe_addRight]\n \u00b7 exact \u27e8by linarith [(hr hy).1], by linarith [(hr hy).2]\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[cocompact \u211d] fun x => |x| ^ (-b)\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\n\u22a2 \u2200 (x : \u211d),\n \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (x - r) (x + r)) f\u2016\n[PROOFSTEP]\nintro x\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[cocompact \u211d] fun x => |x| ^ (-b)\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\nx : \u211d\n\u22a2 \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (x - r) (x + r)) f\u2016\n[PROOFSTEP]\nrw [ContinuousMap.norm_le _ (norm_nonneg _)]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[cocompact \u211d] fun x => |x| ^ (-b)\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\nx : \u211d\n\u22a2 \u2200 (x_1 : \u2191\u2191K),\n \u2016\u2191(ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))) x_1\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (x - r) (x + r)) f\u2016\n[PROOFSTEP]\nrintro \u27e8y, hy\u27e9\n[GOAL]\ncase mk\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[cocompact \u211d] fun x => |x| ^ (-b)\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\nx y : \u211d\nhy : y \u2208 \u2191K\n\u22a2 \u2016\u2191(ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))) { val := y, property := hy }\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (x - r) (x + r)) f\u2016\n[PROOFSTEP]\nrefine' (le_of_eq _).trans (ContinuousMap.norm_coe_le_norm _ \u27e8y + x, _\u27e9)\n[GOAL]\ncase mk.refine'_1\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[cocompact \u211d] fun x => |x| ^ (-b)\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\nx y : \u211d\nhy : y \u2208 \u2191K\n\u22a2 \u2016\u2191(ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))) { val := y, property := hy }\u2016 =\n \u2016\u2191(ContinuousMap.restrict (Icc (x - r) (x + r)) f) { val := y + x, property := ?mk.refine'_2 }\u2016\n[PROOFSTEP]\nsimp_rw [ContinuousMap.restrict_apply, ContinuousMap.comp_apply, ContinuousMap.coe_addRight]\n[GOAL]\ncase mk.refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[cocompact \u211d] fun x => |x| ^ (-b)\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\nx y : \u211d\nhy : y \u2208 \u2191K\n\u22a2 y + x \u2208 Icc (x - r) (x + r)\n[PROOFSTEP]\nexact \u27e8by linarith [(hr hy).1], by linarith [(hr hy).2]\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[cocompact \u211d] fun x => |x| ^ (-b)\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\nx y : \u211d\nhy : y \u2208 \u2191K\n\u22a2 x - r \u2264 y + x\n[PROOFSTEP]\nlinarith [(hr hy).1]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[cocompact \u211d] fun x => |x| ^ (-b)\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\nx y : \u211d\nhy : y \u2208 \u2191K\n\u22a2 y + x \u2264 x + r\n[PROOFSTEP]\nlinarith [(hr hy).2]\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nhf : \u2191f =O[cocompact \u211d] fun x => |x| ^ (-b)\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\nthis :\n \u2200 (x : \u211d),\n \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (x - r) (x + r)) f\u2016\n\u22a2 (fun x => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016) =O[cocompact \u211d] fun x =>\n |x| ^ (-b)\n[PROOFSTEP]\nsimp_rw [cocompact_eq, isBigO_sup] at hf \u22a2\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\nthis :\n \u2200 (x : \u211d),\n \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (x - r) (x + r)) f\u2016\nhf : (\u2191f =O[atBot] fun x => |x| ^ (-b)) \u2227 \u2191f =O[atTop] fun x => |x| ^ (-b)\n\u22a2 ((fun x => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016) =O[atBot] fun x =>\n |x| ^ (-b)) \u2227\n (fun x => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016) =O[atTop] fun x =>\n |x| ^ (-b)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.left\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\nthis :\n \u2200 (x : \u211d),\n \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (x - r) (x + r)) f\u2016\nhf : (\u2191f =O[atBot] fun x => |x| ^ (-b)) \u2227 \u2191f =O[atTop] fun x => |x| ^ (-b)\n\u22a2 (fun x => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016) =O[atBot] fun x =>\n |x| ^ (-b)\n[PROOFSTEP]\nrefine' (isBigO_of_le atBot _).trans (isBigO_norm_Icc_restrict_atBot hb hf.1 (-r) r)\n[GOAL]\ncase intro.left\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\nthis :\n \u2200 (x : \u211d),\n \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (x - r) (x + r)) f\u2016\nhf : (\u2191f =O[atBot] fun x => |x| ^ (-b)) \u2227 \u2191f =O[atTop] fun x => |x| ^ (-b)\n\u22a2 \u2200 (x : \u211d),\n \u2016\u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016\u2016 \u2264\n \u2016\u2016ContinuousMap.restrict (Icc (x + -r) (x + r)) f\u2016\u2016\n[PROOFSTEP]\nsimp_rw [norm_norm]\n[GOAL]\ncase intro.left\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\nthis :\n \u2200 (x : \u211d),\n \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (x - r) (x + r)) f\u2016\nhf : (\u2191f =O[atBot] fun x => |x| ^ (-b)) \u2227 \u2191f =O[atTop] fun x => |x| ^ (-b)\n\u22a2 \u2200 (x : \u211d),\n \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (x + -r) (x + r)) f\u2016\n[PROOFSTEP]\nexact this\n[GOAL]\ncase intro.right\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\nthis :\n \u2200 (x : \u211d),\n \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (x - r) (x + r)) f\u2016\nhf : (\u2191f =O[atBot] fun x => |x| ^ (-b)) \u2227 \u2191f =O[atTop] fun x => |x| ^ (-b)\n\u22a2 (fun x => \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016) =O[atTop] fun x =>\n |x| ^ (-b)\n[PROOFSTEP]\nrefine' (isBigO_of_le atTop _).trans (isBigO_norm_Icc_restrict_atTop hb hf.2 (-r) r)\n[GOAL]\ncase intro.right\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\nthis :\n \u2200 (x : \u211d),\n \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (x - r) (x + r)) f\u2016\nhf : (\u2191f =O[atBot] fun x => |x| ^ (-b)) \u2227 \u2191f =O[atTop] fun x => |x| ^ (-b)\n\u22a2 \u2200 (x : \u211d),\n \u2016\u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016\u2016 \u2264\n \u2016\u2016ContinuousMap.restrict (Icc (x + -r) (x + r)) f\u2016\u2016\n[PROOFSTEP]\nsimp_rw [norm_norm]\n[GOAL]\ncase intro.right\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : C(\u211d, E)\nb : \u211d\nhb : 0 < b\nK : Compacts \u211d\nr : \u211d\nhr : \u2191K \u2286 Icc (-r) r\nthis :\n \u2200 (x : \u211d),\n \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (x - r) (x + r)) f\u2016\nhf : (\u2191f =O[atBot] fun x => |x| ^ (-b)) \u2227 \u2191f =O[atTop] fun x => |x| ^ (-b)\n\u22a2 \u2200 (x : \u211d),\n \u2016ContinuousMap.restrict (\u2191K) (ContinuousMap.comp f (ContinuousMap.addRight x))\u2016 \u2264\n \u2016ContinuousMap.restrict (Icc (x + -r) (x + r)) f\u2016\n[PROOFSTEP]\nexact this\n[GOAL]\nf g : SchwartzMap \u211d \u2102\nhfg : \ud835\udcd5 \u2191f = \u2191g\n\u22a2 \u2211' (n : \u2124), \u2191f \u2191n = \u2211' (n : \u2124), \u2191g \u2191n\n[PROOFSTEP]\nsimp_rw [\u2190 hfg]\n[GOAL]\nf g : SchwartzMap \u211d \u2102\nhfg : \ud835\udcd5 \u2191f = \u2191g\n\u22a2 \u2211' (n : \u2124), \u2191f \u2191n = \u2211' (n : \u2124), \ud835\udcd5 \u2191f \u2191n\n[PROOFSTEP]\nrw [Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay f.continuous one_lt_two (f.isBigO_cocompact_rpow (-2))]\n[GOAL]\nf g : SchwartzMap \u211d \u2102\nhfg : \ud835\udcd5 \u2191f = \u2191g\n\u22a2 \ud835\udcd5 \u2191f =O[cocompact \u211d] fun x => |x| ^ (-2)\n[PROOFSTEP]\nrw [hfg]\n[GOAL]\nf g : SchwartzMap \u211d \u2102\nhfg : \ud835\udcd5 \u2191f = \u2191g\n\u22a2 \u2191g =O[cocompact \u211d] fun x => |x| ^ (-2)\n[PROOFSTEP]\nexact g.isBigO_cocompact_rpow (-2)\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Fourier.PoissonSummation", "llama_tokens": 34100, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7905303186696747, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.4291498842404264}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\nm : Type u_6\nn : Type u_7\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM N : Matrix n n A\n\u22a2 det (M * N) = det (N * M)\n[PROOFSTEP]\nrw [det_mul, det_mul, mul_comm]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\u271d\ninst\u271d\u2079 : Module R M\u271d\nM'\u271d : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\u271d\ninst\u271d\u2077 : Module R M'\u271d\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2074 : CommRing A\nm : Type u_6\nn : Type u_7\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nM : Matrix n m A\nN M' : Matrix m n A\nhMM' : M * M' = 1\nhM'M : M' * M = 1\n\u22a2 det (M * N) = det (N * M)\n[PROOFSTEP]\nnontriviality A\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\u271d\ninst\u271d\u2079 : Module R M\u271d\nM'\u271d : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\u271d\ninst\u271d\u2077 : Module R M'\u271d\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2074 : CommRing A\nm : Type u_6\nn : Type u_7\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nM : Matrix n m A\nN M' : Matrix m n A\nhMM' : M * M' = 1\nhM'M : M' * M = 1\n\u271d : Nontrivial A\n\u22a2 det (M * N) = det (N * M)\n[PROOFSTEP]\nlet e := indexEquivOfInv hMM' hM'M\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\u271d\ninst\u271d\u2079 : Module R M\u271d\nM'\u271d : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\u271d\ninst\u271d\u2077 : Module R M'\u271d\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2074 : CommRing A\nm : Type u_6\nn : Type u_7\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nM : Matrix n m A\nN M' : Matrix m n A\nhMM' : M * M' = 1\nhM'M : M' * M = 1\n\u271d : Nontrivial A\ne : n \u2243 m := indexEquivOfInv hMM' hM'M\n\u22a2 det (M * N) = det (N * M)\n[PROOFSTEP]\nrw [\u2190 det_submatrix_equiv_self e, \u2190 submatrix_mul_equiv _ _ _ (Equiv.refl n) _, det_comm, submatrix_mul_equiv,\n Equiv.coe_refl, submatrix_id_id]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\u271d\ninst\u271d\u2079 : Module R M\u271d\nM'\u271d : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\u271d\ninst\u271d\u2077 : Module R M'\u271d\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2074 : CommRing A\nm : Type u_6\nn : Type u_7\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nM : Matrix m n A\nM' : Matrix n m A\nN : Matrix n n A\nhMM' : M * M' = 1\nhM'M : M' * M = 1\n\u22a2 det (M * N * M') = det N\n[PROOFSTEP]\nrw [\u2190 det_comm' hM'M hMM', \u2190 Matrix.mul_assoc, hM'M, Matrix.one_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq \u03ba\nb : Basis \u03b9 A M\nc : Basis \u03ba A M\nf : M \u2192\u2097[A] M\n\u22a2 det (\u2191(toMatrix b b) f) = det (\u2191(toMatrix c c) f)\n[PROOFSTEP]\nrw [\u2190 linearMap_toMatrix_mul_basis_toMatrix c b c, \u2190 basis_toMatrix_mul_linearMap_toMatrix b c b,\n Matrix.det_conj_of_mul_eq_one]\n[GOAL]\ncase hMM'\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq \u03ba\nb : Basis \u03b9 A M\nc : Basis \u03ba A M\nf : M \u2192\u2097[A] M\n\u22a2 Basis.toMatrix c \u2191b * Basis.toMatrix b \u2191c = 1\n[PROOFSTEP]\nrw [Basis.toMatrix_mul_toMatrix, Basis.toMatrix_self]\n[GOAL]\ncase hM'M\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq \u03ba\nb : Basis \u03b9 A M\nc : Basis \u03ba A M\nf : M \u2192\u2097[A] M\n\u22a2 Basis.toMatrix b \u2191c * Basis.toMatrix c \u2191b = 1\n[PROOFSTEP]\nrw [Basis.toMatrix_mul_toMatrix, Basis.toMatrix_self]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nM' : Type u_3\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Module A M\n\u03ba : Type u_6\ninst\u271d : Fintype \u03ba\nb : Basis \u03b9 A M\nf : M \u2192\u2097[A] M\n\u22a2 \u2191(detAux (Trunc.mk b)) f = det (\u2191(toMatrix b b) f)\n[PROOFSTEP]\nrw [detAux]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nM' : Type u_3\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Module A M\n\u03ba : Type u_6\ninst\u271d : Fintype \u03ba\nb : Basis \u03b9 A M\nf : M \u2192\u2097[A] M\n\u22a2 \u2191(Trunc.lift (fun b => MonoidHom.comp detMonoidHom \u2191(toMatrixAlgEquiv b))\n (_ :\n \u2200 (b c : Basis \u03b9 A M),\n (fun b => MonoidHom.comp detMonoidHom \u2191(toMatrixAlgEquiv b)) b =\n (fun b => MonoidHom.comp detMonoidHom \u2191(toMatrixAlgEquiv b)) c)\n (Trunc.mk b))\n f =\n det (\u2191(toMatrix b b) f)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\u03b9' : Type u_7\ninst\u271d\u00b9 : Fintype \u03b9'\ninst\u271d : DecidableEq \u03b9'\ntb : Trunc (Basis \u03b9 A M)\nb' : Basis \u03b9' A M\nf : M \u2192\u2097[A] M\n\u22a2 \u2191(detAux tb) f = det (\u2191(toMatrix b' b') f)\n[PROOFSTEP]\ninduction tb using Trunc.induction_on with\n| h b => rw [detAux_def', det_toMatrix_eq_det_toMatrix b b']\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\u03b9' : Type u_7\ninst\u271d\u00b9 : Fintype \u03b9'\ninst\u271d : DecidableEq \u03b9'\ntb : Trunc (Basis \u03b9 A M)\nb' : Basis \u03b9' A M\nf : M \u2192\u2097[A] M\n\u22a2 \u2191(detAux tb) f = det (\u2191(toMatrix b' b') f)\n[PROOFSTEP]\ninduction tb using Trunc.induction_on with\n| h b => rw [detAux_def', det_toMatrix_eq_det_toMatrix b b']\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\u03b9' : Type u_7\ninst\u271d\u00b9 : Fintype \u03b9'\ninst\u271d : DecidableEq \u03b9'\nb' : Basis \u03b9' A M\nf : M \u2192\u2097[A] M\nb : Basis \u03b9 A M\n\u22a2 \u2191(detAux (Trunc.mk b)) f = det (\u2191(toMatrix b' b') f)\n[PROOFSTEP]\n\n| h b => rw [detAux_def', det_toMatrix_eq_det_toMatrix b b']\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\u03b9' : Type u_7\ninst\u271d\u00b9 : Fintype \u03b9'\ninst\u271d : DecidableEq \u03b9'\nb' : Basis \u03b9' A M\nf : M \u2192\u2097[A] M\nb : Basis \u03b9 A M\n\u22a2 \u2191(detAux (Trunc.mk b)) f = det (\u2191(toMatrix b' b') f)\n[PROOFSTEP]\nrw [detAux_def', det_toMatrix_eq_det_toMatrix b b']\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq M\n\u22a2 \u2191LinearMap.det =\n \u2191(if H : \u2203 s, Nonempty (Basis { x // x \u2208 s } A M) then\n detAux (Trunc.mk (Nonempty.some (_ : Nonempty (Basis { x // x \u2208 Exists.choose H } A M))))\n else 1)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq M\nx\u271d : M \u2192\u2097[A] M\n\u22a2 \u2191LinearMap.det x\u271d =\n \u2191(if H : \u2203 s, Nonempty (Basis { x // x \u2208 s } A M) then\n detAux (Trunc.mk (Nonempty.some (_ : Nonempty (Basis { x // x \u2208 Exists.choose H } A M))))\n else 1)\n x\u271d\n[PROOFSTEP]\nrw [LinearMap.det_def]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq M\nx\u271d : M \u2192\u2097[A] M\n\u22a2 \u2191(if H : \u2203 s, Nonempty (Basis { x // x \u2208 s } A M) then\n detAux (Trunc.mk (Nonempty.some (_ : Nonempty (Basis { x // x \u2208 Exists.choose H } A M))))\n else 1)\n x\u271d =\n \u2191(if H : \u2203 s, Nonempty (Basis { x // x \u2208 s } A M) then\n detAux (Trunc.mk (Nonempty.some (_ : Nonempty (Basis { x // x \u2208 Exists.choose H } A M))))\n else 1)\n x\u271d\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq M\nx\u271d : M \u2192\u2097[A] M\nh\u271d : \u2203 s, Nonempty (Basis { x // x \u2208 s } A M)\n\u22a2 \u2191(detAux (Trunc.mk (Nonempty.some (_ : Nonempty (Basis { x // x \u2208 Exists.choose h\u271d } A M))))) x\u271d =\n \u2191(detAux (Trunc.mk (Nonempty.some (_ : Nonempty (Basis { x // x \u2208 Exists.choose h\u271d } A M))))) x\u271d\n[PROOFSTEP]\ncongr\n -- use the correct `DecidableEq` instance\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq M\nx\u271d : M \u2192\u2097[A] M\nh\u271d : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } A M)\n\u22a2 \u21911 x\u271d = \u21911 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq M\ns : Finset M\nb : Basis { x // x \u2208 s } A M\nf : M \u2192\u2097[A] M\n\u22a2 \u2191LinearMap.det f = det (\u2191(toMatrix b b) f)\n[PROOFSTEP]\nhave : \u2203 s : Finset M, Nonempty (Basis s A M) := \u27e8s, \u27e8b\u27e9\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq M\ns : Finset M\nb : Basis { x // x \u2208 s } A M\nf : M \u2192\u2097[A] M\nthis : \u2203 s, Nonempty (Basis { x // x \u2208 s } A M)\n\u22a2 \u2191LinearMap.det f = det (\u2191(toMatrix b b) f)\n[PROOFSTEP]\nrw [LinearMap.coe_det, dif_pos, detAux_def'' _ b]\n[GOAL]\ncase hc\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq M\ns : Finset M\nb : Basis { x // x \u2208 s } A M\nf : M \u2192\u2097[A] M\nthis : \u2203 s, Nonempty (Basis { x // x \u2208 s } A M)\n\u22a2 \u2203 s, Nonempty (Basis { x // x \u2208 s } A M)\n[PROOFSTEP]\nassumption\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nM' : Type u_3\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Module A M\n\u03ba : Type u_6\ninst\u271d : Fintype \u03ba\nb : Basis \u03b9 A M\nf : M \u2192\u2097[A] M\n\u22a2 det (\u2191(toMatrix b b) f) = \u2191LinearMap.det f\n[PROOFSTEP]\nhaveI := Classical.decEq M\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nM' : Type u_3\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Module A M\n\u03ba : Type u_6\ninst\u271d : Fintype \u03ba\nb : Basis \u03b9 A M\nf : M \u2192\u2097[A] M\nthis : DecidableEq M\n\u22a2 det (\u2191(toMatrix b b) f) = \u2191LinearMap.det f\n[PROOFSTEP]\nrw [det_eq_det_toMatrix_of_finset b.reindexFinsetRange]\n -- Porting note: moved out of `rw` due to error\n -- typeclass instance problem is stuck, it is often due to metavariables `DecidableEq ?m.628881`\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nM' : Type u_3\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Module A M\n\u03ba : Type u_6\ninst\u271d : Fintype \u03ba\nb : Basis \u03b9 A M\nf : M \u2192\u2097[A] M\nthis : DecidableEq M\n\u22a2 det (\u2191(toMatrix b b) f) = det (\u2191(toMatrix (Basis.reindexFinsetRange b) (Basis.reindexFinsetRange b)) f)\n[PROOFSTEP]\napply det_toMatrix_eq_det_toMatrix b\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9\u271d : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\u271d\ninst\u271d\u2075 : Fintype \u03b9\u271d\ne : Basis \u03b9\u271d R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\u03b9 : Type u_7\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf : (\u03b9 \u2192 A) \u2192\u2097[A] \u03b9 \u2192 A\n\u22a2 det (\u2191toMatrix' f) = \u2191LinearMap.det f\n[PROOFSTEP]\nsimp [\u2190 toMatrix_eq_toMatrix']\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nM' : Type u_3\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Module A M\n\u03ba : Type u_6\ninst\u271d : Fintype \u03ba\nb : Basis \u03b9 R M\nf : Matrix \u03b9 \u03b9 R\n\u22a2 \u2191LinearMap.det (\u2191(toLin b b) f) = det f\n[PROOFSTEP]\nrw [\u2190 LinearMap.det_toMatrix b, LinearMap.toMatrix_toLin]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nM' : Type u_3\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Module A M\n\u03ba : Type u_6\ninst\u271d : Fintype \u03ba\nf : Matrix \u03b9 \u03b9 R\n\u22a2 \u2191LinearMap.det (\u2191toLin' f) = det f\n[PROOFSTEP]\nsimp only [\u2190 toLin_eq_toLin', det_toLin]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq M\nP : A \u2192 Prop\nf : M \u2192\u2097[A] M\nhb : \u2200 (s : Finset M) (b : Basis { x // x \u2208 s } A M), P (det (\u2191(toMatrix b b) f))\nh1 : P 1\n\u22a2 P (\u2191LinearMap.det f)\n[PROOFSTEP]\nrw [LinearMap.det_def]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq M\nP : A \u2192 Prop\nf : M \u2192\u2097[A] M\nhb : \u2200 (s : Finset M) (b : Basis { x // x \u2208 s } A M), P (det (\u2191(toMatrix b b) f))\nh1 : P 1\n\u22a2 P\n (\u2191(if H : \u2203 s, Nonempty (Basis { x // x \u2208 s } A M) then\n detAux (Trunc.mk (Nonempty.some (_ : Nonempty (Basis { x // x \u2208 Exists.choose H } A M))))\n else 1)\n f)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq M\nP : A \u2192 Prop\nf : M \u2192\u2097[A] M\nhb : \u2200 (s : Finset M) (b : Basis { x // x \u2208 s } A M), P (det (\u2191(toMatrix b b) f))\nh1 : P 1\nh : \u2203 s, Nonempty (Basis { x // x \u2208 s } A M)\n\u22a2 P (\u2191(detAux (Trunc.mk (Nonempty.some (_ : Nonempty (Basis { x // x \u2208 Exists.choose h } A M))))) f)\n[PROOFSTEP]\nconvert hb _ h.choose_spec.some\n[GOAL]\ncase h.e'_1\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq M\nP : A \u2192 Prop\nf : M \u2192\u2097[A] M\nhb : \u2200 (s : Finset M) (b : Basis { x // x \u2208 s } A M), P (det (\u2191(toMatrix b b) f))\nh1 : P 1\nh : \u2203 s, Nonempty (Basis { x // x \u2208 s } A M)\n\u22a2 \u2191(detAux (Trunc.mk (Nonempty.some (_ : Nonempty (Basis { x // x \u2208 Exists.choose h } A M))))) f =\n det\n (\u2191(toMatrix (Nonempty.some (_ : Nonempty (Basis { x // x \u2208 Exists.choose h } A M)))\n (Nonempty.some (_ : Nonempty (Basis { x // x \u2208 Exists.choose h } A M))))\n f)\n[PROOFSTEP]\nconvert detAux_def'' (Trunc.mk h.choose_spec.some) h.choose_spec.some f\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq M\nP : A \u2192 Prop\nf : M \u2192\u2097[A] M\nhb : \u2200 (s : Finset M) (b : Basis { x // x \u2208 s } A M), P (det (\u2191(toMatrix b b) f))\nh1 : P 1\nh : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } A M)\n\u22a2 P (\u21911 f)\n[PROOFSTEP]\nexact h1\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\nc : \ud835\udd5c\nf : M \u2192\u2097[\ud835\udd5c] M\n\u22a2 \u2191LinearMap.det (c \u2022 f) = c ^ FiniteDimensional.finrank \ud835\udd5c M * \u2191LinearMap.det f\n[PROOFSTEP]\nby_cases H : \u2203 s : Finset M, Nonempty (Basis s \ud835\udd5c M)\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\nc : \ud835\udd5c\nf : M \u2192\u2097[\ud835\udd5c] M\nH : \u2203 s, Nonempty (Basis { x // x \u2208 s } \ud835\udd5c M)\n\u22a2 \u2191LinearMap.det (c \u2022 f) = c ^ FiniteDimensional.finrank \ud835\udd5c M * \u2191LinearMap.det f\n[PROOFSTEP]\nhave : FiniteDimensional \ud835\udd5c M := by\n rcases H with \u27e8s, \u27e8hs\u27e9\u27e9\n exact FiniteDimensional.of_fintype_basis hs\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\nc : \ud835\udd5c\nf : M \u2192\u2097[\ud835\udd5c] M\nH : \u2203 s, Nonempty (Basis { x // x \u2208 s } \ud835\udd5c M)\n\u22a2 FiniteDimensional \ud835\udd5c M\n[PROOFSTEP]\nrcases H with \u27e8s, \u27e8hs\u27e9\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\nc : \ud835\udd5c\nf : M \u2192\u2097[\ud835\udd5c] M\ns : Finset M\nhs : Basis { x // x \u2208 s } \ud835\udd5c M\n\u22a2 FiniteDimensional \ud835\udd5c M\n[PROOFSTEP]\nexact FiniteDimensional.of_fintype_basis hs\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\nc : \ud835\udd5c\nf : M \u2192\u2097[\ud835\udd5c] M\nH : \u2203 s, Nonempty (Basis { x // x \u2208 s } \ud835\udd5c M)\nthis : FiniteDimensional \ud835\udd5c M\n\u22a2 \u2191LinearMap.det (c \u2022 f) = c ^ FiniteDimensional.finrank \ud835\udd5c M * \u2191LinearMap.det f\n[PROOFSTEP]\nsimp only [\u2190 det_toMatrix (FiniteDimensional.finBasis \ud835\udd5c M), LinearEquiv.map_smul, Fintype.card_fin, Matrix.det_smul]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\nc : \ud835\udd5c\nf : M \u2192\u2097[\ud835\udd5c] M\nH : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } \ud835\udd5c M)\n\u22a2 \u2191LinearMap.det (c \u2022 f) = c ^ FiniteDimensional.finrank \ud835\udd5c M * \u2191LinearMap.det f\n[PROOFSTEP]\nclassical\nhave : FiniteDimensional.finrank \ud835\udd5c M = 0 := finrank_eq_zero_of_not_exists_basis H\nsimp [coe_det, H, this]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\nc : \ud835\udd5c\nf : M \u2192\u2097[\ud835\udd5c] M\nH : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } \ud835\udd5c M)\n\u22a2 \u2191LinearMap.det (c \u2022 f) = c ^ FiniteDimensional.finrank \ud835\udd5c M * \u2191LinearMap.det f\n[PROOFSTEP]\nhave : FiniteDimensional.finrank \ud835\udd5c M = 0 := finrank_eq_zero_of_not_exists_basis H\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\nc : \ud835\udd5c\nf : M \u2192\u2097[\ud835\udd5c] M\nH : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } \ud835\udd5c M)\nthis : FiniteDimensional.finrank \ud835\udd5c M = 0\n\u22a2 \u2191LinearMap.det (c \u2022 f) = c ^ FiniteDimensional.finrank \ud835\udd5c M * \u2191LinearMap.det f\n[PROOFSTEP]\nsimp [coe_det, H, this]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9\u271d : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\u271d\ninst\u271d\u2075 : Fintype \u03b9\u271d\ne : Basis \u03b9\u271d R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\u03b9 : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Nonempty \u03b9\nb : Basis \u03b9 A M\n\u22a2 \u2191LinearMap.det 0 = 0\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9\u271d : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\u271d\ninst\u271d\u2075 : Fintype \u03b9\u271d\ne : Basis \u03b9\u271d R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\u03b9 : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Nonempty \u03b9\nb : Basis \u03b9 A M\nthis : DecidableEq \u03b9\n\u22a2 \u2191LinearMap.det 0 = 0\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9\u271d : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\u271d\ninst\u271d\u2075 : Fintype \u03b9\u271d\ne : Basis \u03b9\u271d R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\u03b9 : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Nonempty \u03b9\nb : Basis \u03b9 A M\nthis : DecidableEq \u03b9\nval\u271d : Fintype \u03b9\n\u22a2 \u2191LinearMap.det 0 = 0\n[PROOFSTEP]\nrwa [\u2190 det_toMatrix b, LinearEquiv.map_zero, det_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\n\u22a2 \u2191LinearMap.det 0 = 0 ^ FiniteDimensional.finrank \ud835\udd5c M\n[PROOFSTEP]\nsimp only [\u2190 zero_smul \ud835\udd5c (1 : M \u2192\u2097[\ud835\udd5c] M), det_smul, mul_one, MonoidHom.map_one]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : Subsingleton M\nf : M \u2192\u2097[R] M\n\u22a2 \u2191LinearMap.det f = 1\n[PROOFSTEP]\nhave b : Basis (Fin 0) R M := Basis.empty M\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : Subsingleton M\nf : M \u2192\u2097[R] M\nb : Basis (Fin 0) R M\n\u22a2 \u2191LinearMap.det f = 1\n[PROOFSTEP]\nrw [\u2190 f.det_toMatrix b]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : Subsingleton M\nf : M \u2192\u2097[R] M\nb : Basis (Fin 0) R M\n\u22a2 det (\u2191(toMatrix b b) f) = 1\n[PROOFSTEP]\nexact Matrix.det_isEmpty\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\nh : FiniteDimensional.finrank \ud835\udd5c M = 0\nf : M \u2192\u2097[\ud835\udd5c] M\n\u22a2 \u2191LinearMap.det f = 1\n[PROOFSTEP]\nclassical\nrefine' @LinearMap.det_cases M _ \ud835\udd5c _ _ _ (fun t => t = 1) f _ rfl\nintro s b\nhave : IsEmpty s := by\n rw [\u2190 Fintype.card_eq_zero_iff]\n exact (FiniteDimensional.finrank_eq_card_basis b).symm.trans h\nexact Matrix.det_isEmpty\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\nh : FiniteDimensional.finrank \ud835\udd5c M = 0\nf : M \u2192\u2097[\ud835\udd5c] M\n\u22a2 \u2191LinearMap.det f = 1\n[PROOFSTEP]\nrefine' @LinearMap.det_cases M _ \ud835\udd5c _ _ _ (fun t => t = 1) f _ rfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\nh : FiniteDimensional.finrank \ud835\udd5c M = 0\nf : M \u2192\u2097[\ud835\udd5c] M\n\u22a2 \u2200 (s : Finset M) (b : Basis { x // x \u2208 s } \ud835\udd5c M), (fun t => t = 1) (det (\u2191(toMatrix b b) f))\n[PROOFSTEP]\nintro s b\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\nh : FiniteDimensional.finrank \ud835\udd5c M = 0\nf : M \u2192\u2097[\ud835\udd5c] M\ns : Finset M\nb : Basis { x // x \u2208 s } \ud835\udd5c M\n\u22a2 det (\u2191(toMatrix b b) f) = 1\n[PROOFSTEP]\nhave : IsEmpty s := by\n rw [\u2190 Fintype.card_eq_zero_iff]\n exact (FiniteDimensional.finrank_eq_card_basis b).symm.trans h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\nh : FiniteDimensional.finrank \ud835\udd5c M = 0\nf : M \u2192\u2097[\ud835\udd5c] M\ns : Finset M\nb : Basis { x // x \u2208 s } \ud835\udd5c M\n\u22a2 IsEmpty { x // x \u2208 s }\n[PROOFSTEP]\nrw [\u2190 Fintype.card_eq_zero_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\nh : FiniteDimensional.finrank \ud835\udd5c M = 0\nf : M \u2192\u2097[\ud835\udd5c] M\ns : Finset M\nb : Basis { x // x \u2208 s } \ud835\udd5c M\n\u22a2 Fintype.card { x // x \u2208 s } = 0\n[PROOFSTEP]\nexact (FiniteDimensional.finrank_eq_card_basis b).symm.trans h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u271d\ninst\u271d\u00b9\u2070 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2077 : DecidableEq \u03b9\ninst\u271d\u2076 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Module A M\u271d\n\u03ba : Type u_6\ninst\u271d\u00b3 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b2 : Field \ud835\udd5c\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module \ud835\udd5c M\nh : FiniteDimensional.finrank \ud835\udd5c M = 0\nf : M \u2192\u2097[\ud835\udd5c] M\ns : Finset M\nb : Basis { x // x \u2208 s } \ud835\udd5c M\nthis : IsEmpty { x // x \u2208 s }\n\u22a2 det (\u2191(toMatrix b b) f) = 1\n[PROOFSTEP]\nexact Matrix.det_isEmpty\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\nN : Type u_7\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module A N\nf : M \u2192\u2097[A] M\ne : M \u2243\u2097[A] N\n\u22a2 \u2191LinearMap.det (comp (\u2191e) (comp f \u2191(LinearEquiv.symm e))) = \u2191LinearMap.det f\n[PROOFSTEP]\nclassical\nby_cases H : \u2203 s : Finset M, Nonempty (Basis s A M)\n\u00b7 rcases H with \u27e8s, \u27e8b\u27e9\u27e9\n rw [\u2190 det_toMatrix b f, \u2190 det_toMatrix (b.map e), toMatrix_comp (b.map e) b (b.map e), toMatrix_comp (b.map e) b b, \u2190\n Matrix.mul_assoc, Matrix.det_conj_of_mul_eq_one]\n \u00b7 rw [\u2190 toMatrix_comp, LinearEquiv.comp_coe, e.symm_trans_self, LinearEquiv.refl_toLinearMap, toMatrix_id]\n \u00b7 rw [\u2190 toMatrix_comp, LinearEquiv.comp_coe, e.self_trans_symm, LinearEquiv.refl_toLinearMap, toMatrix_id]\n\u00b7 have H' : \u00ac\u2203 t : Finset N, Nonempty (Basis t A N) :=\n by\n contrapose! H\n rcases H with \u27e8s, \u27e8b\u27e9\u27e9\n exact \u27e8_, \u27e8(b.map e.symm).reindexFinsetRange\u27e9\u27e9\n simp only [coe_det, H, H', MonoidHom.one_apply, dif_neg]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\nN : Type u_7\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module A N\nf : M \u2192\u2097[A] M\ne : M \u2243\u2097[A] N\n\u22a2 \u2191LinearMap.det (comp (\u2191e) (comp f \u2191(LinearEquiv.symm e))) = \u2191LinearMap.det f\n[PROOFSTEP]\nby_cases H : \u2203 s : Finset M, Nonempty (Basis s A M)\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\nN : Type u_7\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module A N\nf : M \u2192\u2097[A] M\ne : M \u2243\u2097[A] N\nH : \u2203 s, Nonempty (Basis { x // x \u2208 s } A M)\n\u22a2 \u2191LinearMap.det (comp (\u2191e) (comp f \u2191(LinearEquiv.symm e))) = \u2191LinearMap.det f\n[PROOFSTEP]\nrcases H with \u27e8s, \u27e8b\u27e9\u27e9\n[GOAL]\ncase pos.intro.intro\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\nN : Type u_7\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module A N\nf : M \u2192\u2097[A] M\ne : M \u2243\u2097[A] N\ns : Finset M\nb : Basis { x // x \u2208 s } A M\n\u22a2 \u2191LinearMap.det (comp (\u2191e) (comp f \u2191(LinearEquiv.symm e))) = \u2191LinearMap.det f\n[PROOFSTEP]\nrw [\u2190 det_toMatrix b f, \u2190 det_toMatrix (b.map e), toMatrix_comp (b.map e) b (b.map e), toMatrix_comp (b.map e) b b, \u2190\n Matrix.mul_assoc, Matrix.det_conj_of_mul_eq_one]\n[GOAL]\ncase pos.intro.intro.hMM'\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\nN : Type u_7\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module A N\nf : M \u2192\u2097[A] M\ne : M \u2243\u2097[A] N\ns : Finset M\nb : Basis { x // x \u2208 s } A M\n\u22a2 \u2191(toMatrix b (Basis.map b e)) \u2191e * \u2191(toMatrix (Basis.map b e) b) \u2191(LinearEquiv.symm e) = 1\n[PROOFSTEP]\nrw [\u2190 toMatrix_comp, LinearEquiv.comp_coe, e.symm_trans_self, LinearEquiv.refl_toLinearMap, toMatrix_id]\n[GOAL]\ncase pos.intro.intro.hM'M\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\nN : Type u_7\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module A N\nf : M \u2192\u2097[A] M\ne : M \u2243\u2097[A] N\ns : Finset M\nb : Basis { x // x \u2208 s } A M\n\u22a2 \u2191(toMatrix (Basis.map b e) b) \u2191(LinearEquiv.symm e) * \u2191(toMatrix b (Basis.map b e)) \u2191e = 1\n[PROOFSTEP]\nrw [\u2190 toMatrix_comp, LinearEquiv.comp_coe, e.self_trans_symm, LinearEquiv.refl_toLinearMap, toMatrix_id]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\nN : Type u_7\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module A N\nf : M \u2192\u2097[A] M\ne : M \u2243\u2097[A] N\nH : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } A M)\n\u22a2 \u2191LinearMap.det (comp (\u2191e) (comp f \u2191(LinearEquiv.symm e))) = \u2191LinearMap.det f\n[PROOFSTEP]\nhave H' : \u00ac\u2203 t : Finset N, Nonempty (Basis t A N) := by\n contrapose! H\n rcases H with \u27e8s, \u27e8b\u27e9\u27e9\n exact \u27e8_, \u27e8(b.map e.symm).reindexFinsetRange\u27e9\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\nN : Type u_7\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module A N\nf : M \u2192\u2097[A] M\ne : M \u2243\u2097[A] N\nH : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } A M)\n\u22a2 \u00ac\u2203 t, Nonempty (Basis { x // x \u2208 t } A N)\n[PROOFSTEP]\ncontrapose! H\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\nN : Type u_7\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module A N\nf : M \u2192\u2097[A] M\ne : M \u2243\u2097[A] N\nH : \u2203 t, Nonempty (Basis { x // x \u2208 t } A N)\n\u22a2 \u2203 s, Nonempty (Basis { x // x \u2208 s } A M)\n[PROOFSTEP]\nrcases H with \u27e8s, \u27e8b\u27e9\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\nN : Type u_7\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module A N\nf : M \u2192\u2097[A] M\ne : M \u2243\u2097[A] N\ns : Finset N\nb : Basis { x // x \u2208 s } A N\n\u22a2 \u2203 s, Nonempty (Basis { x // x \u2208 s } A M)\n[PROOFSTEP]\nexact \u27e8_, \u27e8(b.map e.symm).reindexFinsetRange\u27e9\u27e9\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\nN : Type u_7\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module A N\nf : M \u2192\u2097[A] M\ne : M \u2243\u2097[A] N\nH : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } A M)\nH' : \u00ac\u2203 t, Nonempty (Basis { x // x \u2208 t } A N)\n\u22a2 \u2191LinearMap.det (comp (\u2191e) (comp f \u2191(LinearEquiv.symm e))) = \u2191LinearMap.det f\n[PROOFSTEP]\nsimp only [coe_det, H, H', MonoidHom.one_apply, dif_neg]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA\u271d : Type u_5\ninst\u271d\u2074 : CommRing A\u271d\ninst\u271d\u00b3 : Module A\u271d M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\nA : Type u_7\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nf : M \u2192\u2097[A] M\nhf : IsUnit f\n\u22a2 IsUnit (\u2191LinearMap.det f)\n[PROOFSTEP]\nobtain \u27e8g, hg\u27e9 : \u2203 g, f.comp g = 1 := hf.exists_right_inv\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA\u271d : Type u_5\ninst\u271d\u2074 : CommRing A\u271d\ninst\u271d\u00b3 : Module A\u271d M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\nA : Type u_7\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nf : M \u2192\u2097[A] M\nhf : IsUnit f\ng : M \u2192\u2097[A] M\nhg : comp f g = 1\n\u22a2 IsUnit (\u2191LinearMap.det f)\n[PROOFSTEP]\nhave : LinearMap.det f * LinearMap.det g = 1 := by simp only [\u2190 LinearMap.det_comp, hg, MonoidHom.map_one]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA\u271d : Type u_5\ninst\u271d\u2074 : CommRing A\u271d\ninst\u271d\u00b3 : Module A\u271d M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\nA : Type u_7\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nf : M \u2192\u2097[A] M\nhf : IsUnit f\ng : M \u2192\u2097[A] M\nhg : comp f g = 1\n\u22a2 \u2191LinearMap.det f * \u2191LinearMap.det g = 1\n[PROOFSTEP]\nsimp only [\u2190 LinearMap.det_comp, hg, MonoidHom.map_one]\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA\u271d : Type u_5\ninst\u271d\u2074 : CommRing A\u271d\ninst\u271d\u00b3 : Module A\u271d M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\nA : Type u_7\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nf : M \u2192\u2097[A] M\nhf : IsUnit f\ng : M \u2192\u2097[A] M\nhg : comp f g = 1\nthis : \u2191LinearMap.det f * \u2191LinearMap.det g = 1\n\u22a2 IsUnit (\u2191LinearMap.det f)\n[PROOFSTEP]\nexact isUnit_of_mul_eq_one _ _ this\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nhf : \u2191LinearMap.det f \u2260 1\n\u22a2 FiniteDimensional \ud835\udd5c M\n[PROOFSTEP]\nby_cases H : \u2203 s : Finset M, Nonempty (Basis s \ud835\udd5c M)\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nhf : \u2191LinearMap.det f \u2260 1\nH : \u2203 s, Nonempty (Basis { x // x \u2208 s } \ud835\udd5c M)\n\u22a2 FiniteDimensional \ud835\udd5c M\n[PROOFSTEP]\nrcases H with \u27e8s, \u27e8hs\u27e9\u27e9\n[GOAL]\ncase pos.intro.intro\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nhf : \u2191LinearMap.det f \u2260 1\ns : Finset M\nhs : Basis { x // x \u2208 s } \ud835\udd5c M\n\u22a2 FiniteDimensional \ud835\udd5c M\n[PROOFSTEP]\nexact FiniteDimensional.of_fintype_basis hs\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nhf : \u2191LinearMap.det f \u2260 1\nH : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } \ud835\udd5c M)\n\u22a2 FiniteDimensional \ud835\udd5c M\n[PROOFSTEP]\nclassical simp [LinearMap.coe_det, H] at hf \n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nhf : \u2191LinearMap.det f \u2260 1\nH : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } \ud835\udd5c M)\n\u22a2 FiniteDimensional \ud835\udd5c M\n[PROOFSTEP]\nsimp [LinearMap.coe_det, H] at hf \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nhf : \u2191LinearMap.det f = 0\n\u22a2 range f < \u22a4\n[PROOFSTEP]\nhave : FiniteDimensional \ud835\udd5c M := by simp [f.finiteDimensional_of_det_ne_one, hf]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nhf : \u2191LinearMap.det f = 0\n\u22a2 FiniteDimensional \ud835\udd5c M\n[PROOFSTEP]\nsimp [f.finiteDimensional_of_det_ne_one, hf]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nhf : \u2191LinearMap.det f = 0\nthis : FiniteDimensional \ud835\udd5c M\n\u22a2 range f < \u22a4\n[PROOFSTEP]\ncontrapose hf\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nthis : FiniteDimensional \ud835\udd5c M\nhf : \u00acrange f < \u22a4\n\u22a2 \u00ac\u2191LinearMap.det f = 0\n[PROOFSTEP]\nsimp only [lt_top_iff_ne_top, Classical.not_not, \u2190 isUnit_iff_range_eq_top] at hf \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nthis : FiniteDimensional \ud835\udd5c M\nhf : IsUnit f\n\u22a2 \u00ac\u2191LinearMap.det f = 0\n[PROOFSTEP]\nexact isUnit_iff_ne_zero.1 (f.isUnit_det hf)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nhf : \u2191LinearMap.det f = 0\n\u22a2 \u22a5 < ker f\n[PROOFSTEP]\nhave : FiniteDimensional \ud835\udd5c M := by simp [f.finiteDimensional_of_det_ne_one, hf]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nhf : \u2191LinearMap.det f = 0\n\u22a2 FiniteDimensional \ud835\udd5c M\n[PROOFSTEP]\nsimp [f.finiteDimensional_of_det_ne_one, hf]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nhf : \u2191LinearMap.det f = 0\nthis : FiniteDimensional \ud835\udd5c M\n\u22a2 \u22a5 < ker f\n[PROOFSTEP]\ncontrapose hf\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nthis : FiniteDimensional \ud835\udd5c M\nhf : \u00ac\u22a5 < ker f\n\u22a2 \u00ac\u2191LinearMap.det f = 0\n[PROOFSTEP]\nsimp only [bot_lt_iff_ne_bot, Classical.not_not, \u2190 isUnit_iff_ker_eq_bot] at hf \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : Module R M\nM' : Type u_3\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Module A M\n\u03ba : Type u_6\ninst\u271d\u00b2 : Fintype \u03ba\n\ud835\udd5c : Type u_7\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nthis : FiniteDimensional \ud835\udd5c M\nhf : IsUnit f\n\u22a2 \u00ac\u2191LinearMap.det f = 0\n[PROOFSTEP]\nexact isUnit_iff_ne_zero.1 (f.isUnit_det hf)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nf : M \u2243\u2097[R] M\ne : M \u2243\u2097[R] M'\n\u22a2 \u2191LinearEquiv.det (trans (trans (symm e) f) e) = \u2191LinearEquiv.det f\n[PROOFSTEP]\nrw [\u2190 Units.eq_iff, coe_det, coe_det, \u2190 comp_coe, \u2190 comp_coe, LinearMap.det_conj]\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nf : M \u2243\u2097[A] M\n\u22a2 \u2191LinearMap.det \u2191f * \u2191LinearMap.det \u2191(symm f) = 1\n[PROOFSTEP]\nsimp [\u2190 LinearMap.det_comp]\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nf : M \u2243\u2097[A] M\n\u22a2 \u2191LinearMap.det \u2191(symm f) * \u2191LinearMap.det \u2191f = 1\n[PROOFSTEP]\nsimp [\u2190 LinearMap.det_comp]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nf : M \u2243\u2097[R] M'\nv : Basis \u03b9 R M\nv' : Basis \u03b9 R M'\n\u22a2 IsUnit (det (\u2191(toMatrix v v') \u2191f))\n[PROOFSTEP]\napply isUnit_det_of_left_inverse\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nf : M \u2243\u2097[R] M'\nv : Basis \u03b9 R M\nv' : Basis \u03b9 R M'\n\u22a2 ?B * \u2191(toMatrix v v') \u2191f = 1\ncase B\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nf : M \u2243\u2097[R] M'\nv : Basis \u03b9 R M\nv' : Basis \u03b9 R M'\n\u22a2 Matrix \u03b9 \u03b9 R\n[PROOFSTEP]\nsimpa using (LinearMap.toMatrix_comp v v' v f.symm f).symm\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\n\ud835\udd5c : Type u_5\ninst\u271d\u00b9 : Field \ud835\udd5c\ninst\u271d : Module \ud835\udd5c M\nf : M \u2243\u2097[\ud835\udd5c] M\n\u22a2 \u2191LinearMap.det \u2191(symm f) = (\u2191LinearMap.det \u2191f)\u207b\u00b9\n[PROOFSTEP]\nfield_simp [IsUnit.ne_zero f.isUnit_det']\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nf : M \u2192\u2097[R] M'\nv : Basis \u03b9 R M\nv' : Basis \u03b9 R M'\nh : IsUnit (det (\u2191(toMatrix v v') f))\nx : M\n\u22a2 \u2191(\u2191(toLin v' v) (\u2191(toMatrix v v') f)\u207b\u00b9) (\u2191f x) = \u2191(\u2191(toLin v v) ((\u2191(toMatrix v v') f)\u207b\u00b9 * \u2191(toMatrix v v') f)) x\n[PROOFSTEP]\nrw [toLin_mul v v' v, toLin_toMatrix, LinearMap.comp_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nf : M \u2192\u2097[R] M'\nv : Basis \u03b9 R M\nv' : Basis \u03b9 R M'\nh : IsUnit (det (\u2191(toMatrix v v') f))\nx : M\n\u22a2 \u2191(\u2191(toLin v v) ((\u2191(toMatrix v v') f)\u207b\u00b9 * \u2191(toMatrix v v') f)) x = x\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nf : M \u2192\u2097[R] M'\nv : Basis \u03b9 R M\nv' : Basis \u03b9 R M'\nh : IsUnit (det (\u2191(toMatrix v v') f))\nx : M'\n\u22a2 \u2191f (\u2191(\u2191(toLin v' v) (\u2191(toMatrix v v') f)\u207b\u00b9) x) = \u2191(\u2191(toLin v' v') (\u2191(toMatrix v v') f * (\u2191(toMatrix v v') f)\u207b\u00b9)) x\n[PROOFSTEP]\nrw [toLin_mul v' v v', LinearMap.comp_apply, toLin_toMatrix v v']\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nf : M \u2192\u2097[R] M'\nv : Basis \u03b9 R M\nv' : Basis \u03b9 R M'\nh : IsUnit (det (\u2191(toMatrix v v') f))\nx : M'\n\u22a2 \u2191(\u2191(toLin v' v') (\u2191(toMatrix v v') f * (\u2191(toMatrix v v') f)\u207b\u00b9)) x = x\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nf : M \u2192\u2097[R] M'\nv : Basis \u03b9 R M\nv' : Basis \u03b9 R M'\nh : IsUnit (det (\u2191(toMatrix v v') f))\n\u22a2 \u2191(ofIsUnitDet h) = f\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nf : M \u2192\u2097[R] M'\nv : Basis \u03b9 R M\nv' : Basis \u03b9 R M'\nh : IsUnit (det (\u2191(toMatrix v v') f))\nx : M\n\u22a2 \u2191\u2191(ofIsUnitDet h) x = \u2191f x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : Field \ud835\udd5c\nM : Type u_6\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module \ud835\udd5c M\ninst\u271d : FiniteDimensional \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nhf : \u2191LinearMap.det f \u2260 0\n\u22a2 IsUnit (det (\u2191(toMatrix (FiniteDimensional.finBasis \ud835\udd5c M) (FiniteDimensional.finBasis \ud835\udd5c M)) f))\n[PROOFSTEP]\nrw [LinearMap.det_toMatrix]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\n\ud835\udd5c : Type u_5\ninst\u271d\u00b3 : Field \ud835\udd5c\nM : Type u_6\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module \ud835\udd5c M\ninst\u271d : FiniteDimensional \ud835\udd5c M\nf : M \u2192\u2097[\ud835\udd5c] M\nhf : \u2191LinearMap.det f \u2260 0\n\u22a2 IsUnit (\u2191LinearMap.det f)\n[PROOFSTEP]\nexact isUnit_iff_ne_zero.2 hf\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), \u2191f x = \u2191f' (\u2191e x)\n\u22a2 Associated (\u2191LinearMap.det f) (\u2191LinearMap.det f')\n[PROOFSTEP]\nsuffices Associated (LinearMap.det (f' \u2218\u2097 \u2191e)) (LinearMap.det f')\n by\n convert this using 2\n ext x\n exact h x\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), \u2191f x = \u2191f' (\u2191e x)\nthis : Associated (\u2191LinearMap.det (comp f' \u2191e)) (\u2191LinearMap.det f')\n\u22a2 Associated (\u2191LinearMap.det f) (\u2191LinearMap.det f')\n[PROOFSTEP]\nconvert this using 2\n[GOAL]\ncase h.e'_3.h.e'_6\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), \u2191f x = \u2191f' (\u2191e x)\nthis : Associated (\u2191LinearMap.det (comp f' \u2191e)) (\u2191LinearMap.det f')\n\u22a2 f = comp f' \u2191e\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_3.h.e'_6.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), \u2191f x = \u2191f' (\u2191e x)\nthis : Associated (\u2191LinearMap.det (comp f' \u2191e)) (\u2191LinearMap.det f')\nx : M\n\u22a2 \u2191f x = \u2191(comp f' \u2191e) x\n[PROOFSTEP]\nexact h x\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), \u2191f x = \u2191f' (\u2191e x)\n\u22a2 Associated (\u2191LinearMap.det (comp f' \u2191e)) (\u2191LinearMap.det f')\n[PROOFSTEP]\nrw [\u2190 mul_one (LinearMap.det f'), LinearMap.det_comp]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), \u2191f x = \u2191f' (\u2191e x)\n\u22a2 Associated (\u2191LinearMap.det f' * \u2191LinearMap.det \u2191e) (\u2191LinearMap.det f' * 1)\n[PROOFSTEP]\nexact Associated.mul_left _ (associated_one_iff_isUnit.mpr e.isUnit_det')\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nN : Type u_5\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : N \u2192\u2097[R] M\ne e' : M \u2243\u2097[R] N\n\u22a2 Associated (\u2191LinearMap.det (comp f \u2191e)) (\u2191LinearMap.det (comp f \u2191e'))\n[PROOFSTEP]\nrefine' LinearMap.associated_det_of_eq_comp (e.trans e'.symm) _ _ _\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nN : Type u_5\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : N \u2192\u2097[R] M\ne e' : M \u2243\u2097[R] N\n\u22a2 \u2200 (x : M), \u2191(comp f \u2191e) x = \u2191(comp f \u2191e') (\u2191(LinearEquiv.trans e (LinearEquiv.symm e')) x)\n[PROOFSTEP]\nintro x\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nN : Type u_5\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : N \u2192\u2097[R] M\ne e' : M \u2243\u2097[R] N\nx : M\n\u22a2 \u2191(comp f \u2191e) x = \u2191(comp f \u2191e') (\u2191(LinearEquiv.trans e (LinearEquiv.symm e')) x)\n[PROOFSTEP]\nsimp only [LinearMap.comp_apply, LinearEquiv.coe_coe, LinearEquiv.trans_apply, LinearEquiv.apply_symm_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\n\u22a2 \u2200 [inst : DecidableEq \u03b9] (m : \u03b9 \u2192 M) (i : \u03b9) (x y : M),\n (fun v => Matrix.det (toMatrix e v)) (update m i (x + y)) =\n (fun v => Matrix.det (toMatrix e v)) (update m i x) + (fun v => Matrix.det (toMatrix e v)) (update m i y)\n[PROOFSTEP]\nintro inst v i x y\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\ninst : DecidableEq \u03b9\nv : \u03b9 \u2192 M\ni : \u03b9\nx y : M\n\u22a2 (fun v => Matrix.det (toMatrix e v)) (update v i (x + y)) =\n (fun v => Matrix.det (toMatrix e v)) (update v i x) + (fun v => Matrix.det (toMatrix e v)) (update v i y)\n[PROOFSTEP]\ncases Subsingleton.elim inst \u2039_\u203a\n[GOAL]\ncase refl\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\ninst : DecidableEq \u03b9\nv : \u03b9 \u2192 M\ni : \u03b9\nx y : M\n\u22a2 (fun v => Matrix.det (toMatrix e v)) (update v i (x + y)) =\n (fun v => Matrix.det (toMatrix e v)) (update v i x) + (fun v => Matrix.det (toMatrix e v)) (update v i y)\n[PROOFSTEP]\nsimp only [e.toMatrix_update, LinearEquiv.map_add, Finsupp.coe_add]\n -- Porting note: was `exact det_update_column_add _ _ _ _`\n[GOAL]\ncase refl\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\ninst : DecidableEq \u03b9\nv : \u03b9 \u2192 M\ni : \u03b9\nx y : M\n\u22a2 Matrix.det (updateColumn (toMatrix e v) i (\u2191(\u2191e.repr x) + \u2191(\u2191e.repr y))) =\n Matrix.det (updateColumn (toMatrix e v) i \u2191(\u2191e.repr x)) + Matrix.det (updateColumn (toMatrix e v) i \u2191(\u2191e.repr y))\n[PROOFSTEP]\nconvert det_updateColumn_add (e.toMatrix v) i (e.repr x) (e.repr y)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\n\u22a2 \u2200 [inst : DecidableEq \u03b9] (m : \u03b9 \u2192 M) (i : \u03b9) (c : R) (x : M),\n (fun v => Matrix.det (toMatrix e v)) (update m i (c \u2022 x)) = c \u2022 (fun v => Matrix.det (toMatrix e v)) (update m i x)\n[PROOFSTEP]\nintro inst u i c x\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\ninst : DecidableEq \u03b9\nu : \u03b9 \u2192 M\ni : \u03b9\nc : R\nx : M\n\u22a2 (fun v => Matrix.det (toMatrix e v)) (update u i (c \u2022 x)) = c \u2022 (fun v => Matrix.det (toMatrix e v)) (update u i x)\n[PROOFSTEP]\ncases Subsingleton.elim inst \u2039_\u203a\n[GOAL]\ncase refl\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\ninst : DecidableEq \u03b9\nu : \u03b9 \u2192 M\ni : \u03b9\nc : R\nx : M\n\u22a2 (fun v => Matrix.det (toMatrix e v)) (update u i (c \u2022 x)) = c \u2022 (fun v => Matrix.det (toMatrix e v)) (update u i x)\n[PROOFSTEP]\nsimp only [e.toMatrix_update, Algebra.id.smul_eq_mul, LinearEquiv.map_smul]\n -- Porting note: was `apply det_update_column_smul`\n[GOAL]\ncase refl\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\ninst : DecidableEq \u03b9\nu : \u03b9 \u2192 M\ni : \u03b9\nc : R\nx : M\n\u22a2 Matrix.det (updateColumn (toMatrix e u) i \u2191(c \u2022 \u2191e.repr x)) =\n c * Matrix.det (updateColumn (toMatrix e u) i \u2191(\u2191e.repr x))\n[PROOFSTEP]\nconvert det_updateColumn_smul (e.toMatrix u) i c (e.repr x)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\n\u22a2 \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9),\n v i = v j \u2192\n i \u2260 j \u2192\n MultilinearMap.toFun\n { toFun := fun v => Matrix.det (toMatrix e v),\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : \u03b9 \u2192 M) (i : \u03b9) (x y : M),\n (fun v => Matrix.det (toMatrix e v)) (update v i (x + y)) =\n (fun v => Matrix.det (toMatrix e v)) (update v i x) +\n (fun v => Matrix.det (toMatrix e v)) (update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (u : \u03b9 \u2192 M) (i : \u03b9) (c : R) (x : M),\n (fun v => Matrix.det (toMatrix e v)) (update u i (c \u2022 x)) =\n c \u2022 (fun v => Matrix.det (toMatrix e v)) (update u i x)) }\n v =\n 0\n[PROOFSTEP]\nintro v i j h hij\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\ni j : \u03b9\nh : v i = v j\nhij : i \u2260 j\n\u22a2 MultilinearMap.toFun\n { toFun := fun v => Matrix.det (toMatrix e v),\n map_add' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (v : \u03b9 \u2192 M) (i : \u03b9) (x y : M),\n (fun v => Matrix.det (toMatrix e v)) (update v i (x + y)) =\n (fun v => Matrix.det (toMatrix e v)) (update v i x) +\n (fun v => Matrix.det (toMatrix e v)) (update v i y)),\n map_smul' :=\n (_ :\n \u2200 [inst : DecidableEq \u03b9] (u : \u03b9 \u2192 M) (i : \u03b9) (c : R) (x : M),\n (fun v => Matrix.det (toMatrix e v)) (update u i (c \u2022 x)) =\n c \u2022 (fun v => Matrix.det (toMatrix e v)) (update u i x)) }\n v =\n 0\n[PROOFSTEP]\nsimp only\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\ni j : \u03b9\nh : v i = v j\nhij : i \u2260 j\n\u22a2 Matrix.det (toMatrix e v) = 0\n[PROOFSTEP]\nrw [\u2190 Function.update_eq_self i v, h, \u2190 det_transpose, e.toMatrix_update, \u2190 updateRow_transpose, \u2190\n e.toMatrix_transpose_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\ni j : \u03b9\nh : v i = v j\nhij : i \u2260 j\n\u22a2 Matrix.det (updateRow (toMatrix e v)\u1d40 i ((toMatrix e v)\u1d40 j)) = 0\n[PROOFSTEP]\napply det_zero_of_row_eq hij\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\ni j : \u03b9\nh : v i = v j\nhij : i \u2260 j\n\u22a2 updateRow (toMatrix e v)\u1d40 i ((toMatrix e v)\u1d40 j) i = updateRow (toMatrix e v)\u1d40 i ((toMatrix e v)\u1d40 j) j\n[PROOFSTEP]\nrw [updateRow_ne hij.symm, updateRow_self]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\n\u22a2 \u2191(det e) \u2191e = 1\n[PROOFSTEP]\nsimp [e.det_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommRing R\nM : Type u_2\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : Module R M\nM' : Type u_3\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\ne : Basis \u03b9 R M\ninst\u271d : IsEmpty \u03b9\n\u22a2 det e = AlternatingMap.constOfIsEmpty R M \u03b9 1\n[PROOFSTEP]\next v\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u2077 : CommRing R\nM : Type u_2\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : Module R M\nM' : Type u_3\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\ne : Basis \u03b9 R M\ninst\u271d : IsEmpty \u03b9\nv : \u03b9 \u2192 M\n\u22a2 \u2191(det e) v = \u2191(AlternatingMap.constOfIsEmpty R M \u03b9 1) v\n[PROOFSTEP]\nexact Matrix.det_isEmpty\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommRing R\nM : Type u_2\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : Module R M\nM' : Type u_3\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\ne : Basis \u03b9 R M\ninst\u271d : Nontrivial R\nh : det e = 0\n\u22a2 False\n[PROOFSTEP]\nsimpa [h] using e.det_self\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\n\u22a2 LinearIndependent R v \u2227 span R (Set.range v) = \u22a4 \u2194 IsUnit (\u2191(Basis.det e) v)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\n\u22a2 LinearIndependent R v \u2227 span R (Set.range v) = \u22a4 \u2192 IsUnit (\u2191(Basis.det e) v)\n[PROOFSTEP]\nrintro \u27e8hli, hspan\u27e9\n[GOAL]\ncase mp.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\nhli : LinearIndependent R v\nhspan : span R (Set.range v) = \u22a4\n\u22a2 IsUnit (\u2191(Basis.det e) v)\n[PROOFSTEP]\nset v' := Basis.mk hli hspan.ge\n[GOAL]\ncase mp.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\nhli : LinearIndependent R v\nhspan : span R (Set.range v) = \u22a4\nv' : Basis \u03b9 R M := Basis.mk hli (_ : \u22a4 \u2264 span R (Set.range v))\n\u22a2 IsUnit (\u2191(Basis.det e) v)\n[PROOFSTEP]\nrw [e.det_apply]\n[GOAL]\ncase mp.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\nhli : LinearIndependent R v\nhspan : span R (Set.range v) = \u22a4\nv' : Basis \u03b9 R M := Basis.mk hli (_ : \u22a4 \u2264 span R (Set.range v))\n\u22a2 IsUnit (det (Basis.toMatrix e v))\n[PROOFSTEP]\nconvert LinearEquiv.isUnit_det (LinearEquiv.refl R M) v' e using 2\n[GOAL]\ncase h.e'_3.h.e'_6\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\nhli : LinearIndependent R v\nhspan : span R (Set.range v) = \u22a4\nv' : Basis \u03b9 R M := Basis.mk hli (_ : \u22a4 \u2264 span R (Set.range v))\n\u22a2 Basis.toMatrix e v = \u2191(toMatrix v' e) \u2191(LinearEquiv.refl R M)\n[PROOFSTEP]\next i j\n[GOAL]\ncase h.e'_3.h.e'_6.a.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\nhli : LinearIndependent R v\nhspan : span R (Set.range v) = \u22a4\nv' : Basis \u03b9 R M := Basis.mk hli (_ : \u22a4 \u2264 span R (Set.range v))\ni j : \u03b9\n\u22a2 Basis.toMatrix e v i j = \u2191(toMatrix v' e) (\u2191(LinearEquiv.refl R M)) i j\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\n\u22a2 IsUnit (\u2191(Basis.det e) v) \u2192 LinearIndependent R v \u2227 span R (Set.range v) = \u22a4\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\nh : IsUnit (\u2191(Basis.det e) v)\n\u22a2 LinearIndependent R v \u2227 span R (Set.range v) = \u22a4\n[PROOFSTEP]\nrw [Basis.det_apply, Basis.toMatrix_eq_toMatrix_constr] at h \n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\nh : IsUnit (det (\u2191(toMatrix e e) (\u2191(Basis.constr e \u2115) v)))\n\u22a2 LinearIndependent R v \u2227 span R (Set.range v) = \u22a4\n[PROOFSTEP]\nset v' := Basis.map e (LinearEquiv.ofIsUnitDet h) with v'_def\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\nh : IsUnit (det (\u2191(toMatrix e e) (\u2191(Basis.constr e \u2115) v)))\nv' : Basis \u03b9 R M := Basis.map e (LinearEquiv.ofIsUnitDet h)\nv'_def : v' = Basis.map e (LinearEquiv.ofIsUnitDet h)\n\u22a2 LinearIndependent R v \u2227 span R (Set.range v) = \u22a4\n[PROOFSTEP]\nhave : \u21d1v' = v := by\n ext i\n rw [v'_def, Basis.map_apply, LinearEquiv.ofIsUnitDet_apply, e.constr_basis]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\nh : IsUnit (det (\u2191(toMatrix e e) (\u2191(Basis.constr e \u2115) v)))\nv' : Basis \u03b9 R M := Basis.map e (LinearEquiv.ofIsUnitDet h)\nv'_def : v' = Basis.map e (LinearEquiv.ofIsUnitDet h)\n\u22a2 \u2191v' = v\n[PROOFSTEP]\next i\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\nh : IsUnit (det (\u2191(toMatrix e e) (\u2191(Basis.constr e \u2115) v)))\nv' : Basis \u03b9 R M := Basis.map e (LinearEquiv.ofIsUnitDet h)\nv'_def : v' = Basis.map e (LinearEquiv.ofIsUnitDet h)\ni : \u03b9\n\u22a2 \u2191v' i = v i\n[PROOFSTEP]\nrw [v'_def, Basis.map_apply, LinearEquiv.ofIsUnitDet_apply, e.constr_basis]\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\nh : IsUnit (det (\u2191(toMatrix e e) (\u2191(Basis.constr e \u2115) v)))\nv' : Basis \u03b9 R M := Basis.map e (LinearEquiv.ofIsUnitDet h)\nv'_def : v' = Basis.map e (LinearEquiv.ofIsUnitDet h)\nthis : \u2191v' = v\n\u22a2 LinearIndependent R v \u2227 span R (Set.range v) = \u22a4\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nv : \u03b9 \u2192 M\nh : IsUnit (det (\u2191(toMatrix e e) (\u2191(Basis.constr e \u2115) v)))\nv' : Basis \u03b9 R M := Basis.map e (LinearEquiv.ofIsUnitDet h)\nv'_def : v' = Basis.map e (LinearEquiv.ofIsUnitDet h)\nthis : \u2191v' = v\n\u22a2 LinearIndependent R \u2191v' \u2227 span R (Set.range \u2191v') = \u22a4\n[PROOFSTEP]\nexact \u27e8v'.linearIndependent, v'.span_eq\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nf : AlternatingMap R M R \u03b9\n\u22a2 f = \u2191f \u2191e \u2022 Basis.det e\n[PROOFSTEP]\nrefine' Basis.ext_alternating e fun i h => _\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nf : AlternatingMap R M R \u03b9\ni : \u03b9 \u2192 \u03b9\nh : Injective i\n\u22a2 (\u2191f fun i_1 => \u2191e (i i_1)) = \u2191(\u2191f \u2191e \u2022 Basis.det e) fun i_1 => \u2191e (i i_1)\n[PROOFSTEP]\nlet \u03c3 : Equiv.Perm \u03b9 := Equiv.ofBijective i (Finite.injective_iff_bijective.1 h)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nf : AlternatingMap R M R \u03b9\ni : \u03b9 \u2192 \u03b9\nh : Injective i\n\u03c3 : Equiv.Perm \u03b9 := Equiv.ofBijective i (_ : Bijective i)\n\u22a2 (\u2191f fun i_1 => \u2191e (i i_1)) = \u2191(\u2191f \u2191e \u2022 Basis.det e) fun i_1 => \u2191e (i i_1)\n[PROOFSTEP]\nchange f (e \u2218 \u03c3) = (f e \u2022 e.det) (e \u2218 \u03c3)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne : Basis \u03b9 R M\nf : AlternatingMap R M R \u03b9\ni : \u03b9 \u2192 \u03b9\nh : Injective i\n\u03c3 : Equiv.Perm \u03b9 := Equiv.ofBijective i (_ : Bijective i)\n\u22a2 \u2191f (\u2191e \u2218 \u2191\u03c3) = \u2191(\u2191f \u2191e \u2022 Basis.det e) (\u2191e \u2218 \u2191\u03c3)\n[PROOFSTEP]\nsimp [AlternatingMap.map_perm, Basis.det_self]\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommRing R\nM : Type u_2\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : Module R M\nM' : Type u_3\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R M'\n\u03b9\u271d : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\u271d\ninst\u271d\u00b9 : Fintype \u03b9\u271d\ne\u271d : Basis \u03b9\u271d R M\n\u03b9 : Type u_5\ninst\u271d : Finite \u03b9\ne : Basis \u03b9 R M\nf : AlternatingMap R M R \u03b9\nh : \u2191f \u2191e = 0\n\u22a2 f = 0\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2077 : CommRing R\nM : Type u_2\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : Module R M\nM' : Type u_3\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R M'\n\u03b9\u271d : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\u271d\ninst\u271d\u00b9 : Fintype \u03b9\u271d\ne\u271d : Basis \u03b9\u271d R M\n\u03b9 : Type u_5\ninst\u271d : Finite \u03b9\ne : Basis \u03b9 R M\nf : AlternatingMap R M R \u03b9\nh : \u2191f \u2191e = 0\nval\u271d : Fintype \u03b9\n\u22a2 f = 0\n[PROOFSTEP]\nletI := Classical.decEq \u03b9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2077 : CommRing R\nM : Type u_2\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : Module R M\nM' : Type u_3\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R M'\n\u03b9\u271d : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\u271d\ninst\u271d\u00b9 : Fintype \u03b9\u271d\ne\u271d : Basis \u03b9\u271d R M\n\u03b9 : Type u_5\ninst\u271d : Finite \u03b9\ne : Basis \u03b9 R M\nf : AlternatingMap R M R \u03b9\nh : \u2191f \u2191e = 0\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 f = 0\n[PROOFSTEP]\nsimpa [h] using f.eq_smul_basis_det e\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 A M\nf : M \u2192\u2097[A] M\nv : \u03b9 \u2192 M\n\u22a2 \u2191(det e) (\u2191f \u2218 v) = \u2191LinearMap.det f * \u2191(det e) v\n[PROOFSTEP]\nrw [Basis.det_apply, Basis.det_apply, \u2190 f.det_toMatrix e, \u2190 Matrix.det_mul, e.toMatrix_eq_toMatrix_constr (f \u2218 v),\n e.toMatrix_eq_toMatrix_constr v, \u2190 toMatrix_comp, e.constr_comp]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nM' : Type u_3\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Module A M\ninst\u271d : Module A M'\nb : Basis \u03b9 A M\nb' : Basis \u03b9 A M'\nf : M \u2192\u2097[A] M'\n\u22a2 \u2191(det b') (\u2191f \u2218 \u2191b) = \u2191LinearMap.det (LinearMap.comp f \u2191(Basis.equiv b' b (Equiv.refl \u03b9)))\n[PROOFSTEP]\nrw [Basis.det_apply, \u2190 LinearMap.det_toMatrix b', LinearMap.toMatrix_comp _ b, Matrix.det_mul,\n LinearMap.toMatrix_basis_equiv, Matrix.det_one, mul_one]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nM' : Type u_3\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Module A M\ninst\u271d : Module A M'\nb : Basis \u03b9 A M\nb' : Basis \u03b9 A M'\nf : M \u2192\u2097[A] M'\n\u22a2 Matrix.det (toMatrix b' (\u2191f \u2218 \u2191b)) = Matrix.det (\u2191(LinearMap.toMatrix b b') f)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_M\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nM' : Type u_3\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Module A M\ninst\u271d : Module A M'\nb : Basis \u03b9 A M\nb' : Basis \u03b9 A M'\nf : M \u2192\u2097[A] M'\n\u22a2 toMatrix b' (\u2191f \u2218 \u2191b) = \u2191(LinearMap.toMatrix b b') f\n[PROOFSTEP]\next i j\n[GOAL]\ncase e_M.a.h\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nM' : Type u_3\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Module A M\ninst\u271d : Module A M'\nb : Basis \u03b9 A M\nb' : Basis \u03b9 A M'\nf : M \u2192\u2097[A] M'\ni j : \u03b9\n\u22a2 toMatrix b' (\u2191f \u2218 \u2191b) i j = \u2191(LinearMap.toMatrix b b') f i j\n[PROOFSTEP]\nrw [Basis.toMatrix_apply, LinearMap.toMatrix_apply, Function.comp_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03b9' : Type u_6\ninst\u271d\u00b9 : Fintype \u03b9'\ninst\u271d : DecidableEq \u03b9'\nb : Basis \u03b9 R M\nv : \u03b9' \u2192 M\ne : \u03b9 \u2243 \u03b9'\n\u22a2 \u2191(det (reindex b e)) v = \u2191(det b) (v \u2218 \u2191e)\n[PROOFSTEP]\nrw [Basis.det_apply, Basis.toMatrix_reindex', det_reindexAlgEquiv, Basis.det_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nM' : Type u_3\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Module A M\n\u03b9' : Type u_6\ninst\u271d\u00b9 : Fintype \u03b9'\ninst\u271d : DecidableEq \u03b9'\nb : Basis \u03b9 R M\nv : \u03b9 \u2192 M\ne : \u03b9' \u2243 \u03b9\n\u22a2 \u2191(det (reindex b e.symm)) (v \u2218 \u2191e) = \u2191(det b) v\n[PROOFSTEP]\nrw [Basis.det_reindex, Function.comp.assoc, e.self_comp_symm, Function.comp.right_id]\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nb : Basis \u03b9 R M\nf : M \u2243\u2097[R] M'\nv : \u03b9 \u2192 M'\n\u22a2 \u2191(det (Basis.map b f)) v = \u2191(det b) (\u2191(LinearEquiv.symm f) \u2218 v)\n[PROOFSTEP]\nrw [Basis.det_apply, Basis.toMatrix_map, Basis.det_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\n\u22a2 Basis.det (basisFun R \u03b9) = detRowAlternating\n[PROOFSTEP]\next M\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM\u271d : Type u_2\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\u271d\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\u271d\nM : \u03b9 \u2192 \u03b9 \u2192 R\n\u22a2 \u2191(Basis.det (basisFun R \u03b9)) M = \u2191detRowAlternating M\n[PROOFSTEP]\nrw [Basis.det_apply, Basis.coePiBasisFun.toMatrix_eq_transpose, det_transpose]\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nv : \u03b9 \u2192 M\nhli : LinearIndependent R v\nhsp : \u22a4 \u2264 span R (Set.range v)\ni : \u03b9\n\u22a2 \u2191(det e) v \u2022 coord (Basis.mk hli hsp) i = MultilinearMap.toLinearMap (\u2191(det e)) v i\n[PROOFSTEP]\napply (Basis.mk hli hsp).ext\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nv : \u03b9 \u2192 M\nhli : LinearIndependent R v\nhsp : \u22a4 \u2264 span R (Set.range v)\ni : \u03b9\n\u22a2 \u2200 (i_1 : \u03b9),\n \u2191(\u2191(det e) v \u2022 coord (Basis.mk hli hsp) i) (\u2191(Basis.mk hli hsp) i_1) =\n \u2191(MultilinearMap.toLinearMap (\u2191(det e)) v i) (\u2191(Basis.mk hli hsp) i_1)\n[PROOFSTEP]\nintro k\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nv : \u03b9 \u2192 M\nhli : LinearIndependent R v\nhsp : \u22a4 \u2264 span R (Set.range v)\ni k : \u03b9\n\u22a2 \u2191(\u2191(det e) v \u2022 coord (Basis.mk hli hsp) i) (\u2191(Basis.mk hli hsp) k) =\n \u2191(MultilinearMap.toLinearMap (\u2191(det e)) v i) (\u2191(Basis.mk hli hsp) k)\n[PROOFSTEP]\nrcases eq_or_ne k i with (rfl | hik)\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nv : \u03b9 \u2192 M\nhli : LinearIndependent R v\nhsp : \u22a4 \u2264 span R (Set.range v)\nk : \u03b9\n\u22a2 \u2191(\u2191(det e) v \u2022 coord (Basis.mk hli hsp) k) (\u2191(Basis.mk hli hsp) k) =\n \u2191(MultilinearMap.toLinearMap (\u2191(det e)) v k) (\u2191(Basis.mk hli hsp) k)\n[PROOFSTEP]\nsimp only [Algebra.id.smul_eq_mul, Basis.coe_mk, LinearMap.smul_apply, LinearMap.coe_mk,\n MultilinearMap.toLinearMap_apply]\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nv : \u03b9 \u2192 M\nhli : LinearIndependent R v\nhsp : \u22a4 \u2264 span R (Set.range v)\ni k : \u03b9\nhik : k \u2260 i\n\u22a2 \u2191(\u2191(det e) v \u2022 coord (Basis.mk hli hsp) i) (\u2191(Basis.mk hli hsp) k) =\n \u2191(MultilinearMap.toLinearMap (\u2191(det e)) v i) (\u2191(Basis.mk hli hsp) k)\n[PROOFSTEP]\nsimp only [Algebra.id.smul_eq_mul, Basis.coe_mk, LinearMap.smul_apply, LinearMap.coe_mk,\n MultilinearMap.toLinearMap_apply]\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nv : \u03b9 \u2192 M\nhli : LinearIndependent R v\nhsp : \u22a4 \u2264 span R (Set.range v)\nk : \u03b9\n\u22a2 \u2191(det e) v * \u2191(coord (Basis.mk hli hsp) k) (v k) = \u2191\u2191(det e) (update v k (v k))\n[PROOFSTEP]\nrw [Basis.mk_coord_apply_eq, mul_one, update_eq_self]\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nv : \u03b9 \u2192 M\nhli : LinearIndependent R v\nhsp : \u22a4 \u2264 span R (Set.range v)\nk : \u03b9\n\u22a2 \u2191(det e) v = \u2191\u2191(det e) v\n[PROOFSTEP]\ncongr\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nv : \u03b9 \u2192 M\nhli : LinearIndependent R v\nhsp : \u22a4 \u2264 span R (Set.range v)\ni k : \u03b9\nhik : k \u2260 i\n\u22a2 \u2191(det e) v * \u2191(coord (Basis.mk hli hsp) i) (v k) = \u2191\u2191(det e) (update v i (v k))\n[PROOFSTEP]\nrw [Basis.mk_coord_apply_ne hik, mul_zero, eq_comm]\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nv : \u03b9 \u2192 M\nhli : LinearIndependent R v\nhsp : \u22a4 \u2264 span R (Set.range v)\ni k : \u03b9\nhik : k \u2260 i\n\u22a2 \u2191\u2191(det e) (update v i (v k)) = 0\n[PROOFSTEP]\nexact e.det.map_eq_zero_of_eq _ (by simp [hik, Function.update_apply]) hik\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nv : \u03b9 \u2192 M\nhli : LinearIndependent R v\nhsp : \u22a4 \u2264 span R (Set.range v)\ni k : \u03b9\nhik : k \u2260 i\n\u22a2 update v i (v k) k = update v i (v k) i\n[PROOFSTEP]\nsimp [hik, Function.update_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\n\u22a2 det (unitsSMul e w) = \u2191(\u220f i : \u03b9, w i)\u207b\u00b9 \u2022 det e\n[PROOFSTEP]\next f\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\nf : \u03b9 \u2192 M\n\u22a2 \u2191(det (unitsSMul e w)) f = \u2191(\u2191(\u220f i : \u03b9, w i)\u207b\u00b9 \u2022 det e) f\n[PROOFSTEP]\nchange (Matrix.det fun i j => (e.unitsSMul w).repr (f j) i) = (\u2191(\u220f i, w i)\u207b\u00b9 : R) \u2022 Matrix.det fun i j => e.repr (f j) i\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\nf : \u03b9 \u2192 M\n\u22a2 (Matrix.det fun i j => \u2191(\u2191(unitsSMul e w).repr (f j)) i) =\n \u2191(\u220f i : \u03b9, w i)\u207b\u00b9 \u2022 Matrix.det fun i j => \u2191(\u2191e.repr (f j)) i\n[PROOFSTEP]\nsimp only [e.repr_unitsSMul]\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\nf : \u03b9 \u2192 M\n\u22a2 (Matrix.det fun i j => (w i)\u207b\u00b9 \u2022 \u2191(\u2191e.repr (f j)) i) = \u2191(\u220f i : \u03b9, w i)\u207b\u00b9 \u2022 Matrix.det fun i j => \u2191(\u2191e.repr (f j)) i\n[PROOFSTEP]\nconvert Matrix.det_mul_column (fun i => (\u2191(w i)\u207b\u00b9 : R)) fun i j => e.repr (f j) i\n[GOAL]\ncase h.e'_3.h.e'_1\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\nf : \u03b9 \u2192 M\n\u22a2 \u2191(\u220f i : \u03b9, w i)\u207b\u00b9 = \u220f i : \u03b9, \u2191(w i)\u207b\u00b9\n[PROOFSTEP]\nsimp only [\u2190 Finset.prod_inv_distrib]\n -- Porting note: was `simp [\u2190 Finset.prod_inv_distrib]`\n[GOAL]\ncase h.e'_3.h.e'_1\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\nf : \u03b9 \u2192 M\n\u22a2 \u2191(\u220f x : \u03b9, (w x)\u207b\u00b9) = \u220f i : \u03b9, \u2191(w i)\u207b\u00b9\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nw : \u03b9 \u2192 R\u02e3\n\u22a2 \u2191(det e) \u2191(unitsSMul e w) = \u220f i : \u03b9, \u2191(w i)\n[PROOFSTEP]\nsimp [Basis.det_apply]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Determinant", "llama_tokens": 47443, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7905303186696747, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.4291498842404264}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn : \u2115\n\u22a2 (x + y) ^ p ^ n =\n x ^ p ^ n + y ^ p ^ n + \u2191p * \u2211 k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191(Nat.choose (p ^ n) k / p)\n[PROOFSTEP]\ntrans x ^ p ^ n + y ^ p ^ n + \u2211 k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * (p ^ n).choose k\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn : \u2115\n\u22a2 (x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n + \u2211 k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191(Nat.choose (p ^ n) k)\n[PROOFSTEP]\nsimp_rw [h.add_pow, \u2190 Nat.Ico_zero_eq_range, Nat.Ico_succ_right, Icc_eq_cons_Ico (zero_le _), Finset.sum_cons,\n Ico_eq_cons_Ioo (pow_pos hp.pos _), Finset.sum_cons, tsub_self, tsub_zero, pow_zero, Nat.choose_zero_right,\n Nat.choose_self, Nat.cast_one, mul_one, one_mul, \u2190 add_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn : \u2115\n\u22a2 x ^ p ^ n + y ^ p ^ n + \u2211 k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191(Nat.choose (p ^ n) k) =\n x ^ p ^ n + y ^ p ^ n + \u2191p * \u2211 k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191(Nat.choose (p ^ n) k / p)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn : \u2115\n\u22a2 \u2211 k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191(Nat.choose (p ^ n) k) =\n \u2191p * \u2211 k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191(Nat.choose (p ^ n) k / p)\n[PROOFSTEP]\nsimp_rw [Finset.mul_sum, Nat.cast_comm, mul_assoc _ _ (p : R), \u2190 Nat.cast_mul]\n[GOAL]\ncase e_a\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn : \u2115\n\u22a2 \u2211 k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191(Nat.choose (p ^ n) k) =\n \u2211 x_1 in Ioo 0 (p ^ n), x ^ x_1 * y ^ (p ^ n - x_1) * \u2191(Nat.choose (p ^ n) x_1 / p * p)\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun i hi => _\n[GOAL]\ncase e_a\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn i : \u2115\nhi : i \u2208 Ioo 0 (p ^ n)\n\u22a2 x ^ i * y ^ (p ^ n - i) * \u2191(Nat.choose (p ^ n) i) = x ^ i * y ^ (p ^ n - i) * \u2191(Nat.choose (p ^ n) i / p * p)\n[PROOFSTEP]\nrw [mem_Ioo] at hi \n[GOAL]\ncase e_a\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn i : \u2115\nhi : 0 < i \u2227 i < p ^ n\n\u22a2 x ^ i * y ^ (p ^ n - i) * \u2191(Nat.choose (p ^ n) i) = x ^ i * y ^ (p ^ n - i) * \u2191(Nat.choose (p ^ n) i / p * p)\n[PROOFSTEP]\nrw [Nat.div_mul_cancel (hp.dvd_choose_pow hi.1.ne' hi.2.ne)]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\n\u22a2 (x + y) ^ p = x ^ p + y ^ p + \u2191p * \u2211 k in Ioo 0 p, x ^ k * y ^ (p - k) * \u2191(Nat.choose p k / p)\n[PROOFSTEP]\nsimpa using h.add_pow_prime_pow_eq hp 1\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\ninst\u271d : Fintype R\n\u22a2 \u2191(Fintype.card R) = 0\n[PROOFSTEP]\nrw [\u2190 nsmul_one, card_nsmul_eq_zero]\n[GOAL]\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nn : \u2115\n\u22a2 \u2191n = 0 \u2194 addOrderOf 1 \u2223 n\n[PROOFSTEP]\nrw [\u2190 Nat.smul_one_eq_coe, addOrderOf_dvd_iff_nsmul_eq_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\np : \u2115\ninst\u271d : CharP R p\na : \u2124\n\u22a2 \u2191a = 0 \u2194 \u2191p \u2223 a\n[PROOFSTEP]\nrcases lt_trichotomy a 0 with (h | rfl | h)\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\np : \u2115\ninst\u271d : CharP R p\na : \u2124\nh : a < 0\n\u22a2 \u2191a = 0 \u2194 \u2191p \u2223 a\n[PROOFSTEP]\nrw [\u2190 neg_eq_zero, \u2190 Int.cast_neg, \u2190 dvd_neg]\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\np : \u2115\ninst\u271d : CharP R p\na : \u2124\nh : a < 0\n\u22a2 \u2191(-a) = 0 \u2194 \u2191p \u2223 -a\n[PROOFSTEP]\nlift -a to \u2115 using neg_nonneg.mpr (le_of_lt h) with b\n[GOAL]\ncase inl.intro\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\np : \u2115\ninst\u271d : CharP R p\na : \u2124\nh : a < 0\nb : \u2115\n\u22a2 \u2191\u2191b = 0 \u2194 \u2191p \u2223 \u2191b\n[PROOFSTEP]\nrw [Int.cast_ofNat, CharP.cast_eq_zero_iff R p, Int.coe_nat_dvd]\n[GOAL]\ncase inr.inl\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\np : \u2115\ninst\u271d : CharP R p\n\u22a2 \u21910 = 0 \u2194 \u2191p \u2223 0\n[PROOFSTEP]\nsimp only [Int.cast_zero, eq_self_iff_true, dvd_zero]\n[GOAL]\ncase inr.inr\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\np : \u2115\ninst\u271d : CharP R p\na : \u2124\nh : 0 < a\n\u22a2 \u2191a = 0 \u2194 \u2191p \u2223 a\n[PROOFSTEP]\nlift a to \u2115 using le_of_lt h with b\n[GOAL]\ncase inr.inr.intro\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\np : \u2115\ninst\u271d : CharP R p\na : \u2124\nb : \u2115\nh\u271d h : 0 < \u2191b\n\u22a2 \u2191\u2191b = 0 \u2194 \u2191p \u2223 \u2191b\n[PROOFSTEP]\nrw [Int.cast_ofNat, CharP.cast_eq_zero_iff R p, Int.coe_nat_dvd]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\np : \u2115\ninst\u271d : CharP R p\na b : \u2124\n\u22a2 \u2191a = \u2191b \u2194 a \u2261 b [ZMOD \u2191p]\n[PROOFSTEP]\nrw [eq_comm, \u2190 sub_eq_zero, \u2190 Int.cast_sub, CharP.int_cast_eq_zero_iff R p, Int.modEq_iff_dvd]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\np : \u2115\ninst\u271d : CharP R p\na b : \u2115\n\u22a2 \u2191a = \u2191b \u2194 a \u2261 b [MOD p]\n[PROOFSTEP]\nrw [\u2190 Int.cast_ofNat, \u2190 Int.cast_ofNat b]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\np : \u2115\ninst\u271d : CharP R p\na b : \u2115\n\u22a2 \u2191\u2191a = \u2191\u2191b \u2194 a \u2261 b [MOD p]\n[PROOFSTEP]\nexact (CharP.intCast_eq_intCast _ _).trans Int.coe_nat_modEq_iff\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : AddMonoidWithOne R\ninst\u271d : CharZero R\nx : \u2115\n\u22a2 \u2191x = 0 \u2194 0 \u2223 x\n[PROOFSTEP]\nrw [zero_dvd_iff, \u2190 Nat.cast_zero, Nat.cast_inj]\n[GOAL]\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nthis : DecidableEq R := Classical.decEq R\nH : \u2200 (p : \u2115), \u2191p = 0 \u2192 p = 0\nx : \u2115\n\u22a2 \u2191x = 0 \u2194 0 \u2223 x\n[PROOFSTEP]\nrw [zero_dvd_iff]\n[GOAL]\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nthis : DecidableEq R := Classical.decEq R\nH : \u2200 (p : \u2115), \u2191p = 0 \u2192 p = 0\nx : \u2115\n\u22a2 \u2191x = 0 \u2194 x = 0\n[PROOFSTEP]\nexact \u27e8H x, by rintro rfl; simp\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nthis : DecidableEq R := Classical.decEq R\nH : \u2200 (p : \u2115), \u2191p = 0 \u2192 p = 0\nx : \u2115\n\u22a2 x = 0 \u2192 \u2191x = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nthis : DecidableEq R := Classical.decEq R\nH : \u2200 (p : \u2115), \u2191p = 0 \u2192 p = 0\n\u22a2 \u21910 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nthis : DecidableEq R := Classical.decEq R\nH : \u00ac\u2200 (p : \u2115), \u2191p = 0 \u2192 p = 0\nx : \u2115\nH1 : \u2191x = 0\nH2 : \u00acx % Nat.find (_ : \u2203 x, \u00ac(\u2191x = 0 \u2192 x = 0)) = 0\n\u22a2 \u2191(x % Nat.find (_ : \u2203 x, \u00ac(\u2191x = 0 \u2192 x = 0))) = 0\n[PROOFSTEP]\nrwa [\u2190 Nat.mod_add_div x (Nat.find (not_forall.1 H)), Nat.cast_add, Nat.cast_mul,\n of_not_not (not_not_of_not_imp <| Nat.find_spec (not_forall.1 H)), zero_mul, add_zero] at H1 \n[GOAL]\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nthis : DecidableEq R := Classical.decEq R\nH : \u00ac\u2200 (p : \u2115), \u2191p = 0 \u2192 p = 0\nx : \u2115\nH1 : Nat.find (_ : \u2203 x, \u00ac(\u2191x = 0 \u2192 x = 0)) \u2223 x\n\u22a2 \u2191x = 0\n[PROOFSTEP]\nrw [\u2190 Nat.mul_div_cancel' H1, Nat.cast_mul, of_not_not (not_not_of_not_imp <| Nat.find_spec (not_forall.1 H)), zero_mul]\n[GOAL]\nR : Type u_1\ninst\u271d : NonAssocSemiring R\n\u22a2 \u2200 (x : \u2115), \u2191x = 0 \u2194 ringChar R \u2223 x\n[PROOFSTEP]\nletI : CharP R (ringChar R) := (Classical.choose_spec (CharP.exists_unique R)).1\n[GOAL]\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nthis : CharP R (ringChar R) := (Classical.choose_spec (CharP.exists_unique R)).left\n\u22a2 \u2200 (x : \u2115), \u2191x = 0 \u2194 ringChar R \u2223 x\n[PROOFSTEP]\nexact CharP.cast_eq_zero_iff R (ringChar R)\n[GOAL]\nR : Type u_1\ninst\u271d : NonAssocSemiring R\n\u22a2 \u2191(ringChar R) = 0\n[PROOFSTEP]\nrw [ringChar.spec]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\np : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\n\u22a2 (x + y) ^ p = x ^ p + y ^ p\n[PROOFSTEP]\nlet \u27e8r, hr\u27e9 := h.exists_add_pow_prime_eq hp.out\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\np : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nr : R\nhr : (x + y) ^ p = x ^ p + y ^ p + \u2191p * r\n\u22a2 (x + y) ^ p = x ^ p + y ^ p\n[PROOFSTEP]\nsimp [hr]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\n\u22a2 (x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n\n[PROOFSTEP]\nlet \u27e8r, hr\u27e9 := h.exists_add_pow_prime_pow_eq hp.out n\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nr : R\nhr : (x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n + \u2191p * r\n\u22a2 (x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n\n[PROOFSTEP]\nsimp [hr]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\n\u22a2 (x - y) ^ p = x ^ p - y ^ p\n[PROOFSTEP]\nrw [eq_sub_iff_add_eq, \u2190 add_pow_char_of_commute _ _ _ (Commute.sub_left h rfl)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\n\u22a2 (x - y + y) ^ p = x ^ p\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nn : \u2115\nx y : R\nh : Commute x y\n\u22a2 (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n[PROOFSTEP]\ninduction n with\n| zero => simp\n| succ n n_ih =>\n rw [pow_succ', pow_mul, pow_mul, pow_mul, n_ih]\n apply sub_pow_char_of_commute; apply Commute.pow_pow h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nn : \u2115\nx y : R\nh : Commute x y\n\u22a2 (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n[PROOFSTEP]\ninduction n with\n| zero => simp\n| succ n n_ih =>\n rw [pow_succ', pow_mul, pow_mul, pow_mul, n_ih]\n apply sub_pow_char_of_commute; apply Commute.pow_pow h\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\n\u22a2 (x - y) ^ p ^ Nat.zero = x ^ p ^ Nat.zero - y ^ p ^ Nat.zero\n[PROOFSTEP]\n\n| zero => simp\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\n\u22a2 (x - y) ^ p ^ Nat.zero = x ^ p ^ Nat.zero - y ^ p ^ Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nn : \u2115\nn_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n\u22a2 (x - y) ^ p ^ Nat.succ n = x ^ p ^ Nat.succ n - y ^ p ^ Nat.succ n\n[PROOFSTEP]\n\n| succ n n_ih =>\n rw [pow_succ', pow_mul, pow_mul, pow_mul, n_ih]\n apply sub_pow_char_of_commute; apply Commute.pow_pow h\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nn : \u2115\nn_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n\u22a2 (x - y) ^ p ^ Nat.succ n = x ^ p ^ Nat.succ n - y ^ p ^ Nat.succ n\n[PROOFSTEP]\nrw [pow_succ', pow_mul, pow_mul, pow_mul, n_ih]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nn : \u2115\nn_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n\u22a2 (x ^ p ^ n - y ^ p ^ n) ^ p = (x ^ p ^ n) ^ p - (y ^ p ^ n) ^ p\n[PROOFSTEP]\napply sub_pow_char_of_commute\n[GOAL]\ncase succ.h\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nn : \u2115\nn_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n\u22a2 Commute (x ^ p ^ n) (y ^ p ^ n)\n[PROOFSTEP]\napply Commute.pow_pow h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (2 < p)\n\u22a2 -1 \u2260 1\n[PROOFSTEP]\nsuffices (2 : R) \u2260 0 by\n intro h\n symm at h \n rw [\u2190 sub_eq_zero, sub_neg_eq_add] at h \n norm_num at h \n exact this h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (2 < p)\nthis : 2 \u2260 0\n\u22a2 -1 \u2260 1\n[PROOFSTEP]\nintro h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (2 < p)\nthis : 2 \u2260 0\nh : -1 = 1\n\u22a2 False\n[PROOFSTEP]\nsymm at h \n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (2 < p)\nthis : 2 \u2260 0\nh : 1 = -1\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, sub_neg_eq_add] at h \n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (2 < p)\nthis : 2 \u2260 0\nh\u271d : 1 = -1\nh : 1 + 1 = 0\n\u22a2 False\n[PROOFSTEP]\nnorm_num at h \n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (2 < p)\nthis : 2 \u2260 0\nh\u271d : 1 = -1\nh : 2 = 0\n\u22a2 False\n[PROOFSTEP]\nexact this h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (2 < p)\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (2 < p)\nh : 2 = 0\n\u22a2 False\n[PROOFSTEP]\nrw [show (2 : R) = (2 : \u2115) by norm_cast] at h \n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (2 < p)\nh : 2 = 0\n\u22a2 2 = \u21912\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (2 < p)\nh : \u21912 = 0\n\u22a2 False\n[PROOFSTEP]\nhave := (CharP.cast_eq_zero_iff R p 2).mp h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (2 < p)\nh : \u21912 = 0\nthis : p \u2223 2\n\u22a2 False\n[PROOFSTEP]\nhave := Nat.le_of_dvd (by decide) this\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (2 < p)\nh : \u21912 = 0\nthis : p \u2223 2\n\u22a2 0 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (2 < p)\nh : \u21912 = 0\nthis\u271d : p \u2223 2\nthis : p \u2264 2\n\u22a2 False\n[PROOFSTEP]\nrw [fact_iff] at *\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : 2 < p\nh : \u21912 = 0\nthis\u271d : p \u2223 2\nthis : p \u2264 2\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 (-1) ^ p = -1\n[PROOFSTEP]\nrw [eq_neg_iff_add_eq_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 (-1) ^ p + 1 = 0\n[PROOFSTEP]\nnth_rw 2 [\u2190 one_pow p]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 (-1) ^ p + 1 ^ p = 0\n[PROOFSTEP]\nrw [\u2190 add_pow_char, add_left_neg, zero_pow (Fact.out (p := Nat.Prime p)).pos]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\np n : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 (-1) ^ p ^ n = -1\n[PROOFSTEP]\nrw [eq_neg_iff_add_eq_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\np n : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 (-1) ^ p ^ n + 1 = 0\n[PROOFSTEP]\nnth_rw 2 [\u2190 one_pow (p ^ n)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\np n : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 (-1) ^ p ^ n + 1 ^ p ^ n = 0\n[PROOFSTEP]\nrw [\u2190 add_pow_char_pow, add_left_neg, zero_pow (pow_pos (Fact.out (p := Nat.Prime p)).pos _)]\n[GOAL]\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : Semiring L\ninst\u271d : Nontrivial L\nf : K \u2192+* L\np : \u2115\n\u22a2 CharP K p \u2194 CharP L p\n[PROOFSTEP]\nsimp only [charP_iff, \u2190 f.injective.eq_iff, map_natCast f, f.map_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nS : Type v\ninst\u271d\u00b3 : CommSemiring S\nf : R \u2192* S\ng : R \u2192+* S\np : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CharP R p\ninst\u271d : CharP S p\nx y : R\nn : \u2115\n\u22a2 (\u2191(frobenius R p))^[n] x = x ^ p ^ n\n[PROOFSTEP]\ninduction n with\n| zero => simp\n| succ n n_ih => rw [Function.iterate_succ', pow_succ', pow_mul, Function.comp_apply, frobenius_def, n_ih]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nS : Type v\ninst\u271d\u00b3 : CommSemiring S\nf : R \u2192* S\ng : R \u2192+* S\np : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CharP R p\ninst\u271d : CharP S p\nx y : R\nn : \u2115\n\u22a2 (\u2191(frobenius R p))^[n] x = x ^ p ^ n\n[PROOFSTEP]\ninduction n with\n| zero => simp\n| succ n n_ih => rw [Function.iterate_succ', pow_succ', pow_mul, Function.comp_apply, frobenius_def, n_ih]\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nS : Type v\ninst\u271d\u00b3 : CommSemiring S\nf : R \u2192* S\ng : R \u2192+* S\np : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CharP R p\ninst\u271d : CharP S p\nx y : R\n\u22a2 (\u2191(frobenius R p))^[Nat.zero] x = x ^ p ^ Nat.zero\n[PROOFSTEP]\n\n| zero => simp\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nS : Type v\ninst\u271d\u00b3 : CommSemiring S\nf : R \u2192* S\ng : R \u2192+* S\np : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CharP R p\ninst\u271d : CharP S p\nx y : R\n\u22a2 (\u2191(frobenius R p))^[Nat.zero] x = x ^ p ^ Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nS : Type v\ninst\u271d\u00b3 : CommSemiring S\nf : R \u2192* S\ng : R \u2192+* S\np : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CharP R p\ninst\u271d : CharP S p\nx y : R\nn : \u2115\nn_ih : (\u2191(frobenius R p))^[n] x = x ^ p ^ n\n\u22a2 (\u2191(frobenius R p))^[Nat.succ n] x = x ^ p ^ Nat.succ n\n[PROOFSTEP]\n\n| succ n n_ih => rw [Function.iterate_succ', pow_succ', pow_mul, Function.comp_apply, frobenius_def, n_ih]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nS : Type v\ninst\u271d\u00b3 : CommSemiring S\nf : R \u2192* S\ng : R \u2192+* S\np : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : CharP R p\ninst\u271d : CharP S p\nx y : R\nn : \u2115\nn_ih : (\u2191(frobenius R p))^[n] x = x ^ p ^ n\n\u22a2 (\u2191(frobenius R p))^[Nat.succ n] x = x ^ p ^ Nat.succ n\n[PROOFSTEP]\nrw [Function.iterate_succ', pow_succ', pow_mul, Function.comp_apply, frobenius_def, n_ih]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsReduced R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx h : R\nH : \u2191(frobenius R p) x = \u2191(frobenius R p) h\n\u22a2 x = h\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero] at H \u22a2\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsReduced R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx h : R\nH\u271d : \u2191(frobenius R p) x = \u2191(frobenius R p) h\nH : \u2191(frobenius R p) x - \u2191(frobenius R p) h = 0\n\u22a2 x - h = 0\n[PROOFSTEP]\nrw [\u2190 frobenius_sub] at H \n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsReduced R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx h : R\nH\u271d : \u2191(frobenius R p) x = \u2191(frobenius R p) h\nH : \u2191(frobenius R p) (x - h) = 0\n\u22a2 x - h = 0\n[PROOFSTEP]\nexact IsReduced.eq_zero _ \u27e8_, H\u27e9\n[GOAL]\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Finite R\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsReduced R\ninst\u271d : CharP R 2\na : R\n\u22a2 IsSquare a\n[PROOFSTEP]\ncases nonempty_fintype R\n[GOAL]\ncase intro\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Finite R\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsReduced R\ninst\u271d : CharP R 2\na : R\nval\u271d : Fintype R\n\u22a2 IsSquare a\n[PROOFSTEP]\nexact\n Exists.imp (fun b h => pow_two b \u25b8 Eq.symm h)\n (((Fintype.bijective_iff_injective_and_card _).mpr \u27e8frobenius_inj R 2, rfl\u27e9).surjective a)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocRing R\np : \u2115\ninst\u271d : CharP R p\nk : \u2115\n\u22a2 \u2191k = \u2191(k % p + p * (k / p))\n[PROOFSTEP]\nrw [Nat.mod_add_div]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocRing R\np : \u2115\ninst\u271d : CharP R p\nk : \u2115\n\u22a2 \u2191(k % p + p * (k / p)) = \u2191(k % p)\n[PROOFSTEP]\nsimp [cast_eq_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\np : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : Finite R\n\u22a2 p \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Finite R\ninst\u271d : CharP R 0\n\u22a2 False\n[PROOFSTEP]\nhaveI : CharZero R := charP_to_charZero R\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Finite R\ninst\u271d : CharP R 0\nthis : CharZero R\n\u22a2 False\n[PROOFSTEP]\ncases nonempty_fintype R\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Finite R\ninst\u271d : CharP R 0\nthis : CharZero R\nval\u271d : Fintype R\n\u22a2 False\n[PROOFSTEP]\nexact absurd Nat.cast_injective (not_injective_infinite_finite ((\u2191) : \u2115 \u2192 R))\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 inst\u271d : NonAssocRing R\n\u22a2 ringChar R = 0 \u2194 CharZero R\n[PROOFSTEP]\nrw [ringChar.eq_iff, charP_zero_iff_charZero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsReduced R\np k m : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx : R\n\u22a2 x ^ (p ^ k * m) = 1 \u2194 x ^ m = 1\n[PROOFSTEP]\ninduction' k with k hk\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsReduced R\np m : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx : R\n\u22a2 x ^ (p ^ Nat.zero * m) = 1 \u2194 x ^ m = 1\n[PROOFSTEP]\nrw [pow_zero, one_mul]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsReduced R\np m : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx : R\nk : \u2115\nhk : x ^ (p ^ k * m) = 1 \u2194 x ^ m = 1\n\u22a2 x ^ (p ^ Nat.succ k * m) = 1 \u2194 x ^ m = 1\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase succ.refine'_1\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsReduced R\np m : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx : R\nk : \u2115\nhk : x ^ (p ^ k * m) = 1 \u2194 x ^ m = 1\nh : x ^ (p ^ Nat.succ k * m) = 1\n\u22a2 x ^ m = 1\n[PROOFSTEP]\nrw [pow_succ, mul_assoc, pow_mul', \u2190 frobenius_def, \u2190 frobenius_one p] at h \n[GOAL]\ncase succ.refine'_1\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsReduced R\np m : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx : R\nk : \u2115\nhk : x ^ (p ^ k * m) = 1 \u2194 x ^ m = 1\nh\u271d : \u2191(frobenius R p) (x ^ (p ^ k * m)) = 1\nh : \u2191(frobenius R p) (x ^ (p ^ k * m)) = \u2191(frobenius R p) 1\n\u22a2 x ^ m = 1\n[PROOFSTEP]\nexact hk.1 (frobenius_inj R p h)\n[GOAL]\ncase succ.refine'_2\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsReduced R\np m : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx : R\nk : \u2115\nhk : x ^ (p ^ k * m) = 1 \u2194 x ^ m = 1\nh : x ^ m = 1\n\u22a2 x ^ (p ^ Nat.succ k * m) = 1\n[PROOFSTEP]\nrw [pow_mul', h, one_pow]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : Nontrivial R\np : \u2115\nhc : CharP R p\nhp : p = 1\n\u22a2 1 = 0\n[PROOFSTEP]\nsimpa using (cast_eq_zero_iff R p 1).mpr (hp \u25b8 dvd_refl p)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : NoZeroDivisors R\np : \u2115\nhc : CharP R p\nhp : 2 \u2264 p\nd : \u2115\nhdvd : \u2203 e, p = d * e\ne : \u2115\nhmul : p = d * e\nthis\u271d\u00b3 : \u2191p = 0\nthis\u271d\u00b2 : \u2191d * \u2191e = 0\nhe : \u2191e = 0\nthis\u271d\u00b9 : p \u2223 e\nthis\u271d : e \u2223 p\nthis : e = p\nh\u2080 : 0 < p\n\u22a2 d * p = 1 * p\n[PROOFSTEP]\nrw [\u2039e = p\u203a] at hmul \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : NoZeroDivisors R\np : \u2115\nhc : CharP R p\nhp : 2 \u2264 p\nd : \u2115\nhdvd : \u2203 e, p = d * e\ne : \u2115\nhmul : p = d * p\nthis\u271d\u00b3 : \u2191p = 0\nthis\u271d\u00b2 : \u2191d * \u2191e = 0\nhe : \u2191e = 0\nthis\u271d\u00b9 : p \u2223 e\nthis\u271d : e \u2223 p\nthis : e = p\nh\u2080 : 0 < p\n\u22a2 d * p = 1 * p\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : NoZeroDivisors R\np : \u2115\nhc : CharP R p\nhp : 2 \u2264 p\nd : \u2115\nhdvd : \u2203 e, p = d * e\ne : \u2115\nhmul : p = d * p\nthis\u271d\u00b3 : \u2191p = 0\nthis\u271d\u00b2 : \u2191d * \u2191e = 0\nhe : \u2191e = 0\nthis\u271d\u00b9 : p \u2223 e\nthis\u271d : e \u2223 p\nthis : e = p\nh\u2080 : 0 < p\n\u22a2 d * p = p\n[PROOFSTEP]\nexact Eq.symm hmul\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : CharP R 1\nr : R\n\u22a2 r = 1 * r\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : CharP R 1\nr : R\n\u22a2 1 * r = \u21911 * r\n[PROOFSTEP]\nrw [Nat.cast_one]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : CharP R 1\nr : R\n\u22a2 \u21911 * r = 0 * r\n[PROOFSTEP]\nrw [CharP.cast_eq_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : CharP R 1\nr : R\n\u22a2 0 * r = 0\n[PROOFSTEP]\nrw [zero_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : CharP R 1\nthis : \u2200 (r : R), r = 0\na b : R\n\u22a2 a = b\n[PROOFSTEP]\nrw [this a, this b]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : Nontrivial R\n\u22a2 ringChar R \u2260 1\n[PROOFSTEP]\nintro h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : Nontrivial R\nh : ringChar R = 1\n\u22a2 False\n[PROOFSTEP]\napply zero_ne_one' R\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : Nontrivial R\nh : ringChar R = 1\n\u22a2 0 = 1\n[PROOFSTEP]\nsymm\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : Nontrivial R\nh : ringChar R = 1\n\u22a2 1 = 0\n[PROOFSTEP]\nrw [\u2190 Nat.cast_one, ringChar.spec, h]\n[GOAL]\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nv : \u2115\nhv : v \u2260 1\nhr : CharP R v\nh : \u21911 = 0\n\u22a2 v = 1\n[PROOFSTEP]\nrwa [CharP.cast_eq_zero_iff _ v, Nat.dvd_one] at h \n[GOAL]\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : Nontrivial R\nhR : ringChar R \u2260 2\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nrw [Ne.def, (by norm_cast : (2 : R) = (2 : \u2115)), ringChar.spec, Nat.dvd_prime Nat.prime_two]\n[GOAL]\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : Nontrivial R\nhR : ringChar R \u2260 2\n\u22a2 2 = \u21912\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : NonAssocSemiring R\ninst\u271d : Nontrivial R\nhR : ringChar R \u2260 2\n\u22a2 \u00ac(ringChar R = 1 \u2228 ringChar R = 2)\n[PROOFSTEP]\nexact mt (or_iff_left hR).mp CharP.ringChar_ne_one\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Fintype R\nn : \u2115\nhn : Fintype.card R = n\nhR : \u2200 (i : \u2115), i < n \u2192 \u2191i = 0 \u2192 i = 0\n\u22a2 \u2200 (x : \u2115), \u2191x = 0 \u2194 n \u2223 x\n[PROOFSTEP]\nhave H : (n : R) = 0 := by rw [\u2190 hn, CharP.cast_card_eq_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Fintype R\nn : \u2115\nhn : Fintype.card R = n\nhR : \u2200 (i : \u2115), i < n \u2192 \u2191i = 0 \u2192 i = 0\n\u22a2 \u2191n = 0\n[PROOFSTEP]\nrw [\u2190 hn, CharP.cast_card_eq_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Fintype R\nn : \u2115\nhn : Fintype.card R = n\nhR : \u2200 (i : \u2115), i < n \u2192 \u2191i = 0 \u2192 i = 0\nH : \u2191n = 0\n\u22a2 \u2200 (x : \u2115), \u2191x = 0 \u2194 n \u2223 x\n[PROOFSTEP]\nintro k\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Fintype R\nn : \u2115\nhn : Fintype.card R = n\nhR : \u2200 (i : \u2115), i < n \u2192 \u2191i = 0 \u2192 i = 0\nH : \u2191n = 0\nk : \u2115\n\u22a2 \u2191k = 0 \u2194 n \u2223 k\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Fintype R\nn : \u2115\nhn : Fintype.card R = n\nhR : \u2200 (i : \u2115), i < n \u2192 \u2191i = 0 \u2192 i = 0\nH : \u2191n = 0\nk : \u2115\n\u22a2 \u2191k = 0 \u2192 n \u2223 k\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Fintype R\nn : \u2115\nhn : Fintype.card R = n\nhR : \u2200 (i : \u2115), i < n \u2192 \u2191i = 0 \u2192 i = 0\nH : \u2191n = 0\nk : \u2115\nh : \u2191k = 0\n\u22a2 n \u2223 k\n[PROOFSTEP]\nrw [\u2190 Nat.mod_add_div k n, Nat.cast_add, Nat.cast_mul, H, zero_mul, add_zero] at h \n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Fintype R\nn : \u2115\nhn : Fintype.card R = n\nhR : \u2200 (i : \u2115), i < n \u2192 \u2191i = 0 \u2192 i = 0\nH : \u2191n = 0\nk : \u2115\nh : \u2191(k % n) = 0\n\u22a2 n \u2223 k\n[PROOFSTEP]\nrw [Nat.dvd_iff_mod_eq_zero]\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Fintype R\nn : \u2115\nhn : Fintype.card R = n\nhR : \u2200 (i : \u2115), i < n \u2192 \u2191i = 0 \u2192 i = 0\nH : \u2191n = 0\nk : \u2115\nh : \u2191(k % n) = 0\n\u22a2 k % n = 0\n[PROOFSTEP]\napply hR _ (Nat.mod_lt _ _) h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Fintype R\nn : \u2115\nhn : Fintype.card R = n\nhR : \u2200 (i : \u2115), i < n \u2192 \u2191i = 0 \u2192 i = 0\nH : \u2191n = 0\nk : \u2115\nh : \u2191(k % n) = 0\n\u22a2 n > 0\n[PROOFSTEP]\nrw [\u2190 hn, gt_iff_lt, Fintype.card_pos_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Fintype R\nn : \u2115\nhn : Fintype.card R = n\nhR : \u2200 (i : \u2115), i < n \u2192 \u2191i = 0 \u2192 i = 0\nH : \u2191n = 0\nk : \u2115\nh : \u2191(k % n) = 0\n\u22a2 Nonempty R\n[PROOFSTEP]\nexact \u27e80\u27e9\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Fintype R\nn : \u2115\nhn : Fintype.card R = n\nhR : \u2200 (i : \u2115), i < n \u2192 \u2191i = 0 \u2192 i = 0\nH : \u2191n = 0\nk : \u2115\n\u22a2 n \u2223 k \u2192 \u2191k = 0\n[PROOFSTEP]\nrintro \u27e8k, rfl\u27e9\n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Fintype R\nn : \u2115\nhn : Fintype.card R = n\nhR : \u2200 (i : \u2115), i < n \u2192 \u2191i = 0 \u2192 i = 0\nH : \u2191n = 0\nk : \u2115\n\u22a2 \u2191(n * k) = 0\n[PROOFSTEP]\nrw [Nat.cast_mul, H, zero_mul]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b3 : NonAssocRing R\u271d\ninst\u271d\u00b2 : Fintype R\u271d\nn\u271d : \u2115\nR : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Fintype R\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nhn : Fintype.card R = p ^ n\nhR : \u2200 (i : \u2115), i \u2264 n \u2192 \u2191p ^ i = 0 \u2192 i = n\n\u22a2 CharP R (p ^ n)\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 := CharP.exists R\n[GOAL]\ncase intro\nR\u271d : Type u_1\ninst\u271d\u00b3 : NonAssocRing R\u271d\ninst\u271d\u00b2 : Fintype R\u271d\nn\u271d : \u2115\nR : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Fintype R\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nhn : Fintype.card R = p ^ n\nhR : \u2200 (i : \u2115), i \u2264 n \u2192 \u2191p ^ i = 0 \u2192 i = n\nc : \u2115\nhc : CharP R c\n\u22a2 CharP R (p ^ n)\n[PROOFSTEP]\nhave hcpn : c \u2223 p ^ n := by rw [\u2190 CharP.cast_eq_zero_iff R c, \u2190 hn, CharP.cast_card_eq_zero]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b3 : NonAssocRing R\u271d\ninst\u271d\u00b2 : Fintype R\u271d\nn\u271d : \u2115\nR : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Fintype R\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nhn : Fintype.card R = p ^ n\nhR : \u2200 (i : \u2115), i \u2264 n \u2192 \u2191p ^ i = 0 \u2192 i = n\nc : \u2115\nhc : CharP R c\n\u22a2 c \u2223 p ^ n\n[PROOFSTEP]\nrw [\u2190 CharP.cast_eq_zero_iff R c, \u2190 hn, CharP.cast_card_eq_zero]\n[GOAL]\ncase intro\nR\u271d : Type u_1\ninst\u271d\u00b3 : NonAssocRing R\u271d\ninst\u271d\u00b2 : Fintype R\u271d\nn\u271d : \u2115\nR : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Fintype R\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nhn : Fintype.card R = p ^ n\nhR : \u2200 (i : \u2115), i \u2264 n \u2192 \u2191p ^ i = 0 \u2192 i = n\nc : \u2115\nhc : CharP R c\nhcpn : c \u2223 p ^ n\n\u22a2 CharP R (p ^ n)\n[PROOFSTEP]\nobtain \u27e8i, hi, hc\u27e9 : \u2203 i \u2264 n, c = p ^ i := by rwa [Nat.dvd_prime_pow hp.1] at hcpn \n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b3 : NonAssocRing R\u271d\ninst\u271d\u00b2 : Fintype R\u271d\nn\u271d : \u2115\nR : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Fintype R\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nhn : Fintype.card R = p ^ n\nhR : \u2200 (i : \u2115), i \u2264 n \u2192 \u2191p ^ i = 0 \u2192 i = n\nc : \u2115\nhc : CharP R c\nhcpn : c \u2223 p ^ n\n\u22a2 \u2203 i, i \u2264 n \u2227 c = p ^ i\n[PROOFSTEP]\nrwa [Nat.dvd_prime_pow hp.1] at hcpn \n[GOAL]\ncase intro.intro.intro\nR\u271d : Type u_1\ninst\u271d\u00b3 : NonAssocRing R\u271d\ninst\u271d\u00b2 : Fintype R\u271d\nn\u271d : \u2115\nR : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Fintype R\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nhn : Fintype.card R = p ^ n\nhR : \u2200 (i : \u2115), i \u2264 n \u2192 \u2191p ^ i = 0 \u2192 i = n\nc : \u2115\nhc\u271d : CharP R c\nhcpn : c \u2223 p ^ n\ni : \u2115\nhi : i \u2264 n\nhc : c = p ^ i\n\u22a2 CharP R (p ^ n)\n[PROOFSTEP]\nobtain rfl : i = n := by\n apply hR i hi\n rw [\u2190 Nat.cast_pow, \u2190 hc, CharP.cast_eq_zero]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b3 : NonAssocRing R\u271d\ninst\u271d\u00b2 : Fintype R\u271d\nn\u271d : \u2115\nR : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Fintype R\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nhn : Fintype.card R = p ^ n\nhR : \u2200 (i : \u2115), i \u2264 n \u2192 \u2191p ^ i = 0 \u2192 i = n\nc : \u2115\nhc\u271d : CharP R c\nhcpn : c \u2223 p ^ n\ni : \u2115\nhi : i \u2264 n\nhc : c = p ^ i\n\u22a2 i = n\n[PROOFSTEP]\napply hR i hi\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b3 : NonAssocRing R\u271d\ninst\u271d\u00b2 : Fintype R\u271d\nn\u271d : \u2115\nR : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Fintype R\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nhn : Fintype.card R = p ^ n\nhR : \u2200 (i : \u2115), i \u2264 n \u2192 \u2191p ^ i = 0 \u2192 i = n\nc : \u2115\nhc\u271d : CharP R c\nhcpn : c \u2223 p ^ n\ni : \u2115\nhi : i \u2264 n\nhc : c = p ^ i\n\u22a2 \u2191p ^ i = 0\n[PROOFSTEP]\nrw [\u2190 Nat.cast_pow, \u2190 hc, CharP.cast_eq_zero]\n[GOAL]\ncase intro.intro.intro\nR\u271d : Type u_1\ninst\u271d\u00b3 : NonAssocRing R\u271d\ninst\u271d\u00b2 : Fintype R\u271d\nn : \u2115\nR : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Fintype R\np : \u2115\nhp : Fact (Nat.Prime p)\nc : \u2115\nhc\u271d : CharP R c\ni : \u2115\nhc : c = p ^ i\nhn : Fintype.card R = p ^ i\nhR : \u2200 (i_1 : \u2115), i_1 \u2264 i \u2192 \u2191p ^ i_1 = 0 \u2192 i_1 = i\nhcpn : c \u2223 p ^ i\nhi : i \u2264 i\n\u22a2 CharP R (p ^ i)\n[PROOFSTEP]\nrwa [\u2190 hc]\n[GOAL]\nR : Type u_1\nS : Type v\ninst\u271d\u00b3 : AddMonoidWithOne R\ninst\u271d\u00b2 : AddMonoidWithOne S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : CharP S q\n\u22a2 \u2200 (x : \u2115), \u2191x = 0 \u2194 lcm p q \u2223 x\n[PROOFSTEP]\nsimp [Prod.ext_iff, CharP.cast_eq_zero_iff R p, CharP.cast_eq_zero_iff S q, Nat.lcm_dvd_iff]\n[GOAL]\nR : Type u_1\nS : Type v\ninst\u271d\u00b3 : AddMonoidWithOne R\ninst\u271d\u00b2 : AddMonoidWithOne S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : CharP S p\n\u22a2 CharP (R \u00d7 S) p\n[PROOFSTEP]\nconvert Nat.lcm.charP R S p p\n[GOAL]\ncase h.e'_3\nR : Type u_1\nS : Type v\ninst\u271d\u00b3 : AddMonoidWithOne R\ninst\u271d\u00b2 : AddMonoidWithOne S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : CharP S p\n\u22a2 p = Nat.lcm p p\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Nontrivial R\nhR : ringChar R \u2260 2\n\u22a2 Set.InjOn Int.cast {0, 1, -1}\n[PROOFSTEP]\nrintro _ (rfl | rfl | rfl) _ (rfl | rfl | rfl) h\n[GOAL]\ncase inl.inl\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Nontrivial R\nhR : ringChar R \u2260 2\nh : \u21910 = \u21910\n\u22a2 0 = 0\n[PROOFSTEP]\nsimp only [cast_neg, cast_one, cast_zero, neg_eq_zero, one_ne_zero, zero_ne_one, zero_eq_neg] at h \u22a2\n[GOAL]\ncase inl.inr.inl\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Nontrivial R\nhR : ringChar R \u2260 2\nh : \u21910 = \u21911\n\u22a2 0 = 1\n[PROOFSTEP]\nsimp only [cast_neg, cast_one, cast_zero, neg_eq_zero, one_ne_zero, zero_ne_one, zero_eq_neg] at h \u22a2\n[GOAL]\ncase inl.inr.inr\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Nontrivial R\nhR : ringChar R \u2260 2\nh : \u21910 = \u2191(-1)\n\u22a2 0 = -1\n[PROOFSTEP]\nsimp only [cast_neg, cast_one, cast_zero, neg_eq_zero, one_ne_zero, zero_ne_one, zero_eq_neg] at h \u22a2\n[GOAL]\ncase inr.inl.inl\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Nontrivial R\nhR : ringChar R \u2260 2\nh : \u21911 = \u21910\n\u22a2 1 = 0\n[PROOFSTEP]\nsimp only [cast_neg, cast_one, cast_zero, neg_eq_zero, one_ne_zero, zero_ne_one, zero_eq_neg] at h \u22a2\n[GOAL]\ncase inr.inl.inr.inl\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Nontrivial R\nhR : ringChar R \u2260 2\nh : \u21911 = \u21911\n\u22a2 1 = 1\n[PROOFSTEP]\nsimp only [cast_neg, cast_one, cast_zero, neg_eq_zero, one_ne_zero, zero_ne_one, zero_eq_neg] at h \u22a2\n[GOAL]\ncase inr.inl.inr.inr\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Nontrivial R\nhR : ringChar R \u2260 2\nh : \u21911 = \u2191(-1)\n\u22a2 1 = -1\n[PROOFSTEP]\nsimp only [cast_neg, cast_one, cast_zero, neg_eq_zero, one_ne_zero, zero_ne_one, zero_eq_neg] at h \u22a2\n[GOAL]\ncase inr.inr.inl\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Nontrivial R\nhR : ringChar R \u2260 2\nh : \u2191(-1) = \u21910\n\u22a2 -1 = 0\n[PROOFSTEP]\nsimp only [cast_neg, cast_one, cast_zero, neg_eq_zero, one_ne_zero, zero_ne_one, zero_eq_neg] at h \u22a2\n[GOAL]\ncase inr.inr.inr.inl\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Nontrivial R\nhR : ringChar R \u2260 2\nh : \u2191(-1) = \u21911\n\u22a2 -1 = 1\n[PROOFSTEP]\nsimp only [cast_neg, cast_one, cast_zero, neg_eq_zero, one_ne_zero, zero_ne_one, zero_eq_neg] at h \u22a2\n[GOAL]\ncase inr.inr.inr.inr\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Nontrivial R\nhR : ringChar R \u2260 2\nh : \u2191(-1) = \u2191(-1)\n\u22a2 -1 = -1\n[PROOFSTEP]\nsimp only [cast_neg, cast_one, cast_zero, neg_eq_zero, one_ne_zero, zero_ne_one, zero_eq_neg] at h \u22a2\n[GOAL]\ncase inr.inl.inr.inr\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Nontrivial R\nhR : ringChar R \u2260 2\nh : 1 = -1\n\u22a2 False\n[PROOFSTEP]\nexact (Ring.neg_one_ne_one_of_char_ne_two hR).symm h\n[GOAL]\ncase inr.inr.inr.inl\nR\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : NonAssocRing R\ninst\u271d : Nontrivial R\nhR : ringChar R \u2260 2\nh : -1 = 1\n\u22a2 False\n[PROOFSTEP]\nexact (Ring.neg_one_ne_one_of_char_ne_two hR) h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : AddMonoidWithOne R\nr : R\nn p\u271d : \u2115\na : \u2115+\np : \u2115\ninst\u271d : CharP R p\nk : \u2115\nh : NeZero \u2191k\n\u22a2 \u00acp \u2223 k\n[PROOFSTEP]\nrwa [\u2190 CharP.cast_eq_zero_iff R p k, \u2190 Ne.def, \u2190 neZero_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\n\u22a2 CharZero R \u2194 \u2200 (p : \u2115), Nat.Prime p \u2192 \u2191p \u2260 0\n[PROOFSTEP]\nrefine \u27e8fun h p hp => by simp [hp.ne_zero], fun h => ?_\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : CharZero R\np : \u2115\nhp : Nat.Prime p\n\u22a2 \u2191p \u2260 0\n[PROOFSTEP]\nsimp [hp.ne_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 \u2191p \u2260 0\n\u22a2 CharZero R\n[PROOFSTEP]\nlet p := ringChar R\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 \u2191p \u2260 0\np : \u2115 := ringChar R\n\u22a2 CharZero R\n[PROOFSTEP]\ncases CharP.char_is_prime_or_zero R p with\n| inl hp => simpa using h p hp\n| inr h =>\n haveI : CharP R 0 := h \u25b8 inferInstance\n exact CharP.charP_to_charZero R\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 \u2191p \u2260 0\np : \u2115 := ringChar R\nx\u271d : Nat.Prime p \u2228 p = 0\n\u22a2 CharZero R\n[PROOFSTEP]\ncases CharP.char_is_prime_or_zero R p with\n| inl hp => simpa using h p hp\n| inr h =>\n haveI : CharP R 0 := h \u25b8 inferInstance\n exact CharP.charP_to_charZero R\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 \u2191p \u2260 0\np : \u2115 := ringChar R\nhp : Nat.Prime p\n\u22a2 CharZero R\n[PROOFSTEP]\n\n| inl hp => simpa using h p hp\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 \u2191p \u2260 0\np : \u2115 := ringChar R\nhp : Nat.Prime p\n\u22a2 CharZero R\n[PROOFSTEP]\nsimpa using h p hp\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh\u271d : \u2200 (p : \u2115), Nat.Prime p \u2192 \u2191p \u2260 0\np : \u2115 := ringChar R\nh : p = 0\n\u22a2 CharZero R\n[PROOFSTEP]\n\n| inr h =>\n haveI : CharP R 0 := h \u25b8 inferInstance\n exact CharP.charP_to_charZero R\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh\u271d : \u2200 (p : \u2115), Nat.Prime p \u2192 \u2191p \u2260 0\np : \u2115 := ringChar R\nh : p = 0\n\u22a2 CharZero R\n[PROOFSTEP]\nhaveI : CharP R 0 := h \u25b8 inferInstance\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh\u271d : \u2200 (p : \u2115), Nat.Prime p \u2192 \u2191p \u2260 0\np : \u2115 := ringChar R\nh : p = 0\nthis : CharP R 0\n\u22a2 CharZero R\n[PROOFSTEP]\nexact CharP.charP_to_charZero R\n", "meta": {"mathlib_filename": "Mathlib.Algebra.CharP.Basic", "llama_tokens": 19346, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.42892566000872273}} {"text": "[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\u271d\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\ns : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 Ioo x y \u2286 s\n\u22a2 OrdConnected s\n[PROOFSTEP]\nrw [ordConnected_iff]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\u271d\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\ns : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 Ioo x y \u2286 s\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x \u2264 y \u2192 Icc x y \u2286 s\n[PROOFSTEP]\nintro x hx y hy hxy\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\u271d\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\ns : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 Ioo x y \u2286 s\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2264 y\n\u22a2 Icc x y \u2286 s\n[PROOFSTEP]\nrcases eq_or_lt_of_le hxy with (rfl | hxy')\n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\u271d\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\ns : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 Ioo x y \u2286 s\nx : \u03b1\nhx hy : x \u2208 s\nhxy : x \u2264 x\n\u22a2 Icc x x \u2286 s\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\u271d\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\ns : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 Ioo x y \u2286 s\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2264 y\nhxy' : x < y\n\u22a2 Icc x y \u2286 s\n[PROOFSTEP]\nrw [\u2190 Ioc_insert_left hxy, \u2190 Ioo_insert_right hxy']\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\u271d\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\ns : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x < y \u2192 Ioo x y \u2286 s\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2264 y\nhxy' : x < y\n\u22a2 insert x (insert y (Ioo x y)) \u2286 s\n[PROOFSTEP]\nexact insert_subset_iff.2 \u27e8hx, insert_subset_iff.2 \u27e8hy, hs x hx y hy hxy'\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns\u271d t s : Set \u03b1\nh : OrdConnected (\u2191ofDual \u207b\u00b9' s)\n\u22a2 OrdConnected s\n[PROOFSTEP]\nsimpa only [ordConnected_def] using h.dual\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\u271d\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\na : \u03b1\n\u22a2 OrdConnected {a}\n[PROOFSTEP]\nrw [\u2190 Icc_self]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\u271d\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\na : \u03b1\n\u22a2 OrdConnected (Icc a a)\n[PROOFSTEP]\nexact ordConnected_Icc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\nE : Type u_3\ninst\u271d : OrderIsoClass E \u03b1 \u03b2\ne : E\ns : Set \u03b1\nhs : OrdConnected s\n\u22a2 OrdConnected (\u2191e '' s)\n[PROOFSTEP]\nerw [(e : \u03b1 \u2243o \u03b2).image_eq_preimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns\u271d t : Set \u03b1\nE : Type u_3\ninst\u271d : OrderIsoClass E \u03b1 \u03b2\ne : E\ns : Set \u03b1\nhs : OrdConnected s\n\u22a2 OrdConnected (\u2191(OrderIso.symm \u2191e) \u207b\u00b9' s)\n[PROOFSTEP]\napply ordConnected_preimage (e : \u03b1 \u2243o \u03b2).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\nE : Type u_3\ninst\u271d : OrderIsoClass E \u03b1 \u03b2\ne : E\n\u22a2 OrdConnected (range \u2191e)\n[PROOFSTEP]\nsimp_rw [\u2190 image_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns t : Set \u03b1\nE : Type u_3\ninst\u271d : OrderIsoClass E \u03b1 \u03b2\ne : E\n\u22a2 OrdConnected (\u2191e '' univ)\n[PROOFSTEP]\nexact ordConnected_image (e : \u03b1 \u2243o \u03b2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns\u271d t s : Set \u03b1\n\u22a2 OrdConnected (\u2191ofDual \u207b\u00b9' s) \u2194 OrdConnected s\n[PROOFSTEP]\nsimp_rw [ordConnected_def, toDual.surjective.forall, dual_Icc, Subtype.forall']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns\u271d t s : Set \u03b1\n\u22a2 (\u2200 (x x_1 : { a // \u2191toDual a \u2208 \u2191ofDual \u207b\u00b9' s }), \u2191ofDual \u207b\u00b9' Icc \u2191x_1 \u2191x \u2286 \u2191ofDual \u207b\u00b9' s) \u2194\n \u2200 (x x_1 : { a // a \u2208 s }), Icc \u2191x \u2191x_1 \u2286 s\n[PROOFSTEP]\nexact forall_swap\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\ns : Set \u03b1\nhs : IsAntichain (fun x x_1 => x \u2264 x_1) s\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nz : \u03b1\nhz : z \u2208 Icc x y\n\u22a2 z \u2208 s\n[PROOFSTEP]\nobtain rfl := hs.eq hx hy (hz.1.trans hz.2)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\ns : Set \u03b1\nhs : IsAntichain (fun x x_1 => x \u2264 x_1) s\nx : \u03b1\nhx : x \u2208 s\nz : \u03b1\nhy : x \u2208 s\nhz : z \u2208 Icc x x\n\u22a2 z \u2208 s\n[PROOFSTEP]\nrw [Icc_self, mem_singleton_iff] at hz \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\ns : Set \u03b1\nhs : IsAntichain (fun x x_1 => x \u2264 x_1) s\nx : \u03b1\nhx : x \u2208 s\nz : \u03b1\nhy : x \u2208 s\nhz : z = x\n\u22a2 z \u2208 s\n[PROOFSTEP]\nrwa [hz]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\nx : \u03b1\nh : \u2200 (y : \u03b1), y \u2208 s \u2192 [[x, y]] \u2286 s\ny : \u03b1\nhy : y \u2208 s\nz : \u03b1\nhz : z \u2208 s\n\u22a2 [[y, x]] \u222a [[x, z]] = [[x, y]] \u222a [[x, z]]\n[PROOFSTEP]\nrw [uIcc_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\n\u22a2 OrdConnected s \u2194 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 [[y, x]] \u2286 s\n[PROOFSTEP]\nsimp_rw [ordConnected_iff_uIcc_subset_left hx, uIcc_comm]\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Intervals.OrdConnected", "llama_tokens": 2683, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943603346811, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.4289256538165029}} {"text": "[GOAL]\nR : Type u\ninst\u271d : Ring R\n\u03b9 : Type v\nZ : \u03b9 \u2192 ModuleCatMax R\ns : Cone (Discrete.functor Z)\nj : Discrete \u03b9\n\u22a2 (fun s => LinearMap.pi fun j => NatTrans.app s.\u03c0 { as := j }) s \u226b NatTrans.app (productCone Z).\u03c0 j =\n NatTrans.app s.\u03c0 j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase mk\nR : Type u\ninst\u271d : Ring R\n\u03b9 : Type v\nZ : \u03b9 \u2192 ModuleCatMax R\ns : Cone (Discrete.functor Z)\nas\u271d : \u03b9\n\u22a2 (fun s => LinearMap.pi fun j => NatTrans.app s.\u03c0 { as := j }) s \u226b NatTrans.app (productCone Z).\u03c0 { as := as\u271d } =\n NatTrans.app s.\u03c0 { as := as\u271d }\n[PROOFSTEP]\naesop\n[GOAL]\nR : Type u\ninst\u271d : Ring R\n\u03b9 : Type v\nZ : \u03b9 \u2192 ModuleCatMax R\ns : Cone (Discrete.functor Z)\nm : s.pt \u27f6 (productCone Z).pt\nw : \u2200 (j : Discrete \u03b9), m \u226b NatTrans.app (productCone Z).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m = (fun s => LinearMap.pi fun j => NatTrans.app s.\u03c0 { as := j }) s\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\ninst\u271d : Ring R\n\u03b9 : Type v\nZ : \u03b9 \u2192 ModuleCatMax R\ns : Cone (Discrete.functor Z)\nm : s.pt \u27f6 (productCone Z).pt\nw : \u2200 (j : Discrete \u03b9), m \u226b NatTrans.app (productCone Z).\u03c0 j = NatTrans.app s.\u03c0 j\nx : \u2191s.pt\n\u22a2 \u2191m x = \u2191((fun s => LinearMap.pi fun j => NatTrans.app s.\u03c0 { as := j }) s) x\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h.h\nR : Type u\ninst\u271d : Ring R\n\u03b9 : Type v\nZ : \u03b9 \u2192 ModuleCatMax R\ns : Cone (Discrete.functor Z)\nm : s.pt \u27f6 (productCone Z).pt\nw : \u2200 (j : Discrete \u03b9), m \u226b NatTrans.app (productCone Z).\u03c0 j = NatTrans.app s.\u03c0 j\nx : \u2191s.pt\ni : \u03b9\n\u22a2 \u2191m x i = \u2191((fun s => LinearMap.pi fun j => NatTrans.app s.\u03c0 { as := j }) s) x i\n[PROOFSTEP]\nexact LinearMap.congr_fun (w \u27e8i\u27e9) x\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.ModuleCat.Products", "llama_tokens": 715, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.4288795672360339}} {"text": "[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsBezout R\nx y : R\n\u22a2 Submodule.IsPrincipal (Ideal.span {x, y})\n[PROOFSTEP]\nclassical exact isPrincipal_of_FG (Ideal.span { x, y }) \u27e8{ x, y }, by simp\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsBezout R\nx y : R\n\u22a2 Submodule.IsPrincipal (Ideal.span {x, y})\n[PROOFSTEP]\nexact isPrincipal_of_FG (Ideal.span { x, y }) \u27e8{ x, y }, by simp\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsBezout R\nx y : R\n\u22a2 Ideal.span \u2191{x, y} = Ideal.span {x, y}\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 IsBezout R \u2194 \u2200 (x y : R), Submodule.IsPrincipal (Ideal.span {x, y})\n[PROOFSTEP]\nclassical\nconstructor\n\u00b7 intro H x y; infer_instance\n\u00b7 intro H\n constructor\n apply Submodule.fg_induction\n \u00b7 exact fun _ => \u27e8\u27e8_, rfl\u27e9\u27e9\n \u00b7 rintro _ _ \u27e8\u27e8x, rfl\u27e9\u27e9 \u27e8\u27e8y, rfl\u27e9\u27e9; rw [\u2190 Submodule.span_insert]; exact H _ _\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 IsBezout R \u2194 \u2200 (x y : R), Submodule.IsPrincipal (Ideal.span {x, y})\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\ninst\u271d : CommRing R\n\u22a2 IsBezout R \u2192 \u2200 (x y : R), Submodule.IsPrincipal (Ideal.span {x, y})\n[PROOFSTEP]\nintro H x y\n[GOAL]\ncase mp\nR : Type u\ninst\u271d : CommRing R\nH : IsBezout R\nx y : R\n\u22a2 Submodule.IsPrincipal (Ideal.span {x, y})\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase mpr\nR : Type u\ninst\u271d : CommRing R\n\u22a2 (\u2200 (x y : R), Submodule.IsPrincipal (Ideal.span {x, y})) \u2192 IsBezout R\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\nR : Type u\ninst\u271d : CommRing R\nH : \u2200 (x y : R), Submodule.IsPrincipal (Ideal.span {x, y})\n\u22a2 IsBezout R\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.isPrincipal_of_FG\nR : Type u\ninst\u271d : CommRing R\nH : \u2200 (x y : R), Submodule.IsPrincipal (Ideal.span {x, y})\n\u22a2 \u2200 (I : Ideal R), Ideal.FG I \u2192 Submodule.IsPrincipal I\n[PROOFSTEP]\napply Submodule.fg_induction\n[GOAL]\ncase mpr.isPrincipal_of_FG.h\u2081\nR : Type u\ninst\u271d : CommRing R\nH : \u2200 (x y : R), Submodule.IsPrincipal (Ideal.span {x, y})\n\u22a2 \u2200 (x : R), Submodule.IsPrincipal (Submodule.span R {x})\n[PROOFSTEP]\nexact fun _ => \u27e8\u27e8_, rfl\u27e9\u27e9\n[GOAL]\ncase mpr.isPrincipal_of_FG.h\u2082\nR : Type u\ninst\u271d : CommRing R\nH : \u2200 (x y : R), Submodule.IsPrincipal (Ideal.span {x, y})\n\u22a2 \u2200 (M\u2081 M\u2082 : Submodule R R), Submodule.IsPrincipal M\u2081 \u2192 Submodule.IsPrincipal M\u2082 \u2192 Submodule.IsPrincipal (M\u2081 \u2294 M\u2082)\n[PROOFSTEP]\nrintro _ _ \u27e8\u27e8x, rfl\u27e9\u27e9 \u27e8\u27e8y, rfl\u27e9\u27e9\n[GOAL]\ncase mpr.isPrincipal_of_FG.h\u2082.mk.intro.mk.intro\nR : Type u\ninst\u271d : CommRing R\nH : \u2200 (x y : R), Submodule.IsPrincipal (Ideal.span {x, y})\nx y : R\n\u22a2 Submodule.IsPrincipal (Submodule.span R {x} \u2294 Submodule.span R {y})\n[PROOFSTEP]\nrw [\u2190 Submodule.span_insert]\n[GOAL]\ncase mpr.isPrincipal_of_FG.h\u2082.mk.intro.mk.intro\nR : Type u\ninst\u271d : CommRing R\nH : \u2200 (x y : R), Submodule.IsPrincipal (Ideal.span {x, y})\nx y : R\n\u22a2 Submodule.IsPrincipal (Submodule.span R {x, y})\n[PROOFSTEP]\nexact H _ _\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsBezout R\nx y : R\n\u22a2 x \u2208 {x, y}\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsBezout R\nx y : R\n\u22a2 y \u2208 {x, y}\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsBezout R\nx y z : R\nhx : z \u2223 x\nhy : z \u2223 y\n\u22a2 z \u2223 gcd x y\n[PROOFSTEP]\nrw [\u2190 Ideal.span_singleton_le_span_singleton] at hx hy \u22a2\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsBezout R\nx y z : R\nhx : Ideal.span {x} \u2264 Ideal.span {z}\nhy : Ideal.span {y} \u2264 Ideal.span {z}\n\u22a2 Ideal.span {gcd x y} \u2264 Ideal.span {z}\n[PROOFSTEP]\nrw [span_gcd, Ideal.span_insert, sup_le_iff]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsBezout R\nx y z : R\nhx : Ideal.span {x} \u2264 Ideal.span {z}\nhy : Ideal.span {y} \u2264 Ideal.span {z}\n\u22a2 Ideal.span {x} \u2264 Ideal.span {z} \u2227 Ideal.span {y} \u2264 Ideal.span {z}\n[PROOFSTEP]\nexact \u27e8hx, hy\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsBezout R\nx y : R\n\u22a2 gcd x y \u2208 Ideal.span {x, y}\n[PROOFSTEP]\nrw [\u2190 span_gcd]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsBezout R\nx y : R\n\u22a2 gcd x y \u2208 Ideal.span {gcd x y}\n[PROOFSTEP]\napply Ideal.subset_span\n[GOAL]\ncase a\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsBezout R\nx y : R\n\u22a2 gcd x y \u2208 {gcd x y}\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsBezout R\n\u22a2 IsIntegrallyClosed R\n[PROOFSTEP]\nclassical exact GCDMonoid.toIsIntegrallyClosed\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsBezout R\n\u22a2 IsIntegrallyClosed R\n[PROOFSTEP]\nexact GCDMonoid.toIsIntegrallyClosed\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nS : Type v\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\ninst\u271d : IsBezout R\n\u22a2 IsBezout S\n[PROOFSTEP]\nrw [iff_span_pair_isPrincipal]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nS : Type v\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\ninst\u271d : IsBezout R\n\u22a2 \u2200 (x y : S), Submodule.IsPrincipal (Ideal.span {x, y})\n[PROOFSTEP]\nintro x y\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nS : Type v\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\ninst\u271d : IsBezout R\nx y : S\n\u22a2 Submodule.IsPrincipal (Ideal.span {x, y})\n[PROOFSTEP]\nobtain \u27e8\u27e8x, rfl\u27e9, \u27e8y, rfl\u27e9\u27e9 := hf x, hf y\n[GOAL]\ncase intro.intro\nR : Type u\ninst\u271d\u00b2 : CommRing R\nS : Type v\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\ninst\u271d : IsBezout R\nx y : R\n\u22a2 Submodule.IsPrincipal (Ideal.span {\u2191f x, \u2191f y})\n[PROOFSTEP]\nuse f (gcd x y)\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b2 : CommRing R\nS : Type v\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\ninst\u271d : IsBezout R\nx y : R\n\u22a2 Ideal.span {\u2191f x, \u2191f y} = Submodule.span S {\u2191f (gcd x y)}\n[PROOFSTEP]\ntrans Ideal.map f (Ideal.span {gcd x y})\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nS : Type v\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\ninst\u271d : IsBezout R\nx y : R\n\u22a2 Ideal.span {\u2191f x, \u2191f y} = Ideal.map f (Ideal.span {gcd x y})\n[PROOFSTEP]\nrw [span_gcd, Ideal.map_span, Set.image_insert_eq, Set.image_singleton]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nS : Type v\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\ninst\u271d : IsBezout R\nx y : R\n\u22a2 Ideal.map f (Ideal.span {gcd x y}) = Submodule.span S {\u2191f (gcd x y)}\n[PROOFSTEP]\nrw [Ideal.map_span, Set.image_singleton]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nS : Type v\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\ninst\u271d : IsBezout R\nx y : R\n\u22a2 Ideal.span {\u2191f (gcd x y)} = Submodule.span S {\u2191f (gcd x y)}\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\n\u22a2 List.TFAE [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R]\n[PROOFSTEP]\nclassical\ntfae_have 1 \u2192 2\n\u00b7 intro H; exact \u27e8fun I => isPrincipal_of_FG _ (IsNoetherian.noetherian _)\u27e9\ntfae_have 2 \u2192 3\n\u00b7 intro; infer_instance\ntfae_have 3 \u2192 4\n\u00b7 intro; infer_instance\ntfae_have 4 \u2192 1\n\u00b7 rintro \u27e8h\u27e9\n rw [isNoetherianRing_iff, isNoetherian_iff_fg_wellFounded]\n apply RelEmbedding.wellFounded _ h\n have : \u2200 I : { J : Ideal R // J.FG }, \u2203 x : R, (I : Ideal R) = Ideal.span { x } := fun \u27e8I, hI\u27e9 =>\n (IsBezout.isPrincipal_of_FG I hI).1\n choose f hf using this\n exact\n { toFun := f\n inj' := fun x y e => by ext1; rw [hf, hf, e]\n map_rel_iff' := by\n dsimp\n intro a b\n rw [\u2190 Ideal.span_singleton_lt_span_singleton, \u2190 hf, \u2190 hf]\n rfl }\ntfae_finish\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\n\u22a2 List.TFAE [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R]\n[PROOFSTEP]\ntfae_have 1 \u2192 2\n[GOAL]\ncase tfae_1_to_2\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\n\u22a2 IsNoetherianRing R \u2192 IsPrincipalIdealRing R\n[PROOFSTEP]\nintro H\n[GOAL]\ncase tfae_1_to_2\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\nH : IsNoetherianRing R\n\u22a2 IsPrincipalIdealRing R\n[PROOFSTEP]\nexact \u27e8fun I => isPrincipal_of_FG _ (IsNoetherian.noetherian _)\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\n\u22a2 List.TFAE [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R]\n[PROOFSTEP]\ntfae_have 2 \u2192 3\n[GOAL]\ncase tfae_2_to_3\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\n\u22a2 IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\n[PROOFSTEP]\nintro\n[GOAL]\ncase tfae_2_to_3\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\n\u271d : IsPrincipalIdealRing R\n\u22a2 UniqueFactorizationMonoid R\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\n\u22a2 List.TFAE [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R]\n[PROOFSTEP]\ntfae_have 3 \u2192 4\n[GOAL]\ncase tfae_3_to_4\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\n\u22a2 UniqueFactorizationMonoid R \u2192 WfDvdMonoid R\n[PROOFSTEP]\nintro\n[GOAL]\ncase tfae_3_to_4\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\n\u271d : UniqueFactorizationMonoid R\n\u22a2 WfDvdMonoid R\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R \u2192 WfDvdMonoid R\n\u22a2 List.TFAE [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R]\n[PROOFSTEP]\ntfae_have 4 \u2192 1\n[GOAL]\ncase tfae_4_to_1\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R \u2192 WfDvdMonoid R\n\u22a2 WfDvdMonoid R \u2192 IsNoetherianRing R\n[PROOFSTEP]\nrintro \u27e8h\u27e9\n[GOAL]\ncase tfae_4_to_1.mk\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R \u2192 WfDvdMonoid R\nh : WellFounded DvdNotUnit\n\u22a2 IsNoetherianRing R\n[PROOFSTEP]\nrw [isNoetherianRing_iff, isNoetherian_iff_fg_wellFounded]\n[GOAL]\ncase tfae_4_to_1.mk\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R \u2192 WfDvdMonoid R\nh : WellFounded DvdNotUnit\n\u22a2 WellFounded fun x x_1 => x > x_1\n[PROOFSTEP]\napply RelEmbedding.wellFounded _ h\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R \u2192 WfDvdMonoid R\nh : WellFounded DvdNotUnit\n\u22a2 (fun x x_1 => x > x_1) \u21aar DvdNotUnit\n[PROOFSTEP]\nhave : \u2200 I : { J : Ideal R // J.FG }, \u2203 x : R, (I : Ideal R) = Ideal.span { x } := fun \u27e8I, hI\u27e9 =>\n (IsBezout.isPrincipal_of_FG I hI).1\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R \u2192 WfDvdMonoid R\nh : WellFounded DvdNotUnit\nthis : \u2200 (I : { J // Ideal.FG J }), \u2203 x, \u2191I = Ideal.span {x}\n\u22a2 (fun x x_1 => x > x_1) \u21aar DvdNotUnit\n[PROOFSTEP]\nchoose f hf using this\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R \u2192 WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } \u2192 R\nhf : \u2200 (I : { J // Ideal.FG J }), \u2191I = Ideal.span {f I}\n\u22a2 (fun x x_1 => x > x_1) \u21aar DvdNotUnit\n[PROOFSTEP]\nexact\n { toFun := f\n inj' := fun x y e => by ext1; rw [hf, hf, e]\n map_rel_iff' := by\n dsimp\n intro a b\n rw [\u2190 Ideal.span_singleton_lt_span_singleton, \u2190 hf, \u2190 hf]\n rfl }\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R \u2192 WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } \u2192 R\nhf : \u2200 (I : { J // Ideal.FG J }), \u2191I = Ideal.span {f I}\nx y : { N // Submodule.FG N }\ne : f x = f y\n\u22a2 x = y\n[PROOFSTEP]\next1\n[GOAL]\ncase a\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R \u2192 WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } \u2192 R\nhf : \u2200 (I : { J // Ideal.FG J }), \u2191I = Ideal.span {f I}\nx y : { N // Submodule.FG N }\ne : f x = f y\n\u22a2 \u2191x = \u2191y\n[PROOFSTEP]\nrw [hf, hf, e]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R \u2192 WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } \u2192 R\nhf : \u2200 (I : { J // Ideal.FG J }), \u2191I = Ideal.span {f I}\n\u22a2 \u2200 {a b : { N // Submodule.FG N }},\n DvdNotUnit (\u2191{ toFun := f, inj' := (_ : \u2200 (x y : { N // Submodule.FG N }), f x = f y \u2192 x = y) } a)\n (\u2191{ toFun := f, inj' := (_ : \u2200 (x y : { N // Submodule.FG N }), f x = f y \u2192 x = y) } b) \u2194\n a > b\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R \u2192 WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } \u2192 R\nhf : \u2200 (I : { J // Ideal.FG J }), \u2191I = Ideal.span {f I}\n\u22a2 \u2200 {a b : { N // Submodule.FG N }}, DvdNotUnit (f a) (f b) \u2194 a > b\n[PROOFSTEP]\nintro a b\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R \u2192 WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } \u2192 R\nhf : \u2200 (I : { J // Ideal.FG J }), \u2191I = Ideal.span {f I}\na b : { N // Submodule.FG N }\n\u22a2 DvdNotUnit (f a) (f b) \u2194 a > b\n[PROOFSTEP]\nrw [\u2190 Ideal.span_singleton_lt_span_singleton, \u2190 hf, \u2190 hf]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R \u2192 WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // Ideal.FG J } \u2192 R\nhf : \u2200 (I : { J // Ideal.FG J }), \u2191I = Ideal.span {f I}\na b : { N // Submodule.FG N }\n\u22a2 \u2191b < \u2191a \u2194 a > b\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsBezout R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R \u2192 IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R \u2192 UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R \u2192 WfDvdMonoid R\ntfae_4_to_1 : WfDvdMonoid R \u2192 IsNoetherianRing R\n\u22a2 List.TFAE [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R]\n[PROOFSTEP]\ntfae_finish\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Bezout", "llama_tokens": 7754, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.4288795672360339}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u2191toDFinsupp (replicate n a) = DFinsupp.single a n\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\na : \u03b1\nn : \u2115\ni : \u03b1\n\u22a2 \u2191(\u2191toDFinsupp (replicate n a)) i = \u2191(DFinsupp.single a n) i\n[PROOFSTEP]\ndsimp [toDFinsupp]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\na : \u03b1\nn : \u2115\ni : \u03b1\n\u22a2 count i (replicate n a) = \u2191(DFinsupp.single a n) i\n[PROOFSTEP]\nsimp [count_replicate, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\na : \u03b1\n\u22a2 \u2191toDFinsupp {a} = DFinsupp.single a 1\n[PROOFSTEP]\nrw [\u2190 replicate_one, toDFinsupp_replicate]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\n\u22a2 AddMonoidHom.comp DFinsupp.toMultiset toDFinsupp = AddMonoidHom.id (Multiset \u03b1)\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\nx\u271d a\u271d : \u03b1\n\u22a2 count a\u271d (\u2191(AddMonoidHom.comp DFinsupp.toMultiset toDFinsupp) {x\u271d}) = count a\u271d (\u2191(AddMonoidHom.id (Multiset \u03b1)) {x\u271d})\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\n\u22a2 AddMonoidHom.comp toDFinsupp DFinsupp.toMultiset = AddMonoidHom.id (\u03a0\u2080 (x : \u03b1), \u2115)\n[PROOFSTEP]\next\n[GOAL]\ncase H.a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\nx\u271d i\u271d : \u03b1\n\u22a2 \u2191(\u2191(AddMonoidHom.comp (AddMonoidHom.comp toDFinsupp DFinsupp.toMultiset) (DFinsupp.singleAddHom (fun i => \u2115) x\u271d)) 1)\n i\u271d =\n \u2191(\u2191(AddMonoidHom.comp (AddMonoidHom.id (\u03a0\u2080 (x : \u03b1), \u2115)) (DFinsupp.singleAddHom (fun i => \u2115) x\u271d)) 1) i\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\n\u22a2 \u2191toDFinsupp s \u2264 \u2191toDFinsupp t \u2194 s \u2264 t\n[PROOFSTEP]\nsimp [Multiset.le_iff_count, DFinsupp.le_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d s t : Multiset \u03b1\n\u22a2 \u2191toDFinsupp (s \u2229 t) = \u2191toDFinsupp s \u2293 \u2191toDFinsupp t\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d s t : Multiset \u03b1\ni : \u03b1\n\u22a2 \u2191(\u2191toDFinsupp (s \u2229 t)) i = \u2191(\u2191toDFinsupp s \u2293 \u2191toDFinsupp t) i\n[PROOFSTEP]\nsimp [inf_eq_min]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d s t : Multiset \u03b1\n\u22a2 \u2191toDFinsupp (s \u222a t) = \u2191toDFinsupp s \u2294 \u2191toDFinsupp t\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d s t : Multiset \u03b1\ni : \u03b1\n\u22a2 \u2191(\u2191toDFinsupp (s \u222a t)) i = \u2191(\u2191toDFinsupp s \u2294 \u2191toDFinsupp t) i\n[PROOFSTEP]\nsimp [sup_eq_max]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\nf g : \u03a0\u2080 (_a : \u03b1), \u2115\n\u22a2 \u2191toMultiset f \u2264 \u2191toMultiset g \u2194 f \u2264 g\n[PROOFSTEP]\nsimp_rw [\u2190 Multiset.toDFinsupp_le_toDFinsupp, toMultiset_toDFinsupp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\nf g : \u03a0\u2080 (_a : \u03b1), \u2115\n\u22a2 \u2191toMultiset f < \u2191toMultiset g \u2194 f < g\n[PROOFSTEP]\nsimp_rw [\u2190 Multiset.toDFinsupp_lt_toDFinsupp, toMultiset_toDFinsupp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\nf g : \u03a0\u2080 (_a : \u03b1), \u2115\n\u22a2 \u2191Multiset.toDFinsupp (\u2191toMultiset (f \u2293 g)) = \u2191Multiset.toDFinsupp (\u2191toMultiset f \u2229 \u2191toMultiset g)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : DecidableEq \u03b1\nf g : \u03a0\u2080 (_a : \u03b1), \u2115\n\u22a2 \u2191Multiset.toDFinsupp (\u2191toMultiset (f \u2294 g)) = \u2191Multiset.toDFinsupp (\u2191toMultiset f \u222a \u2191toMultiset g)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.DFinsupp.Multiset", "llama_tokens": 1799, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.4288795672360339}} {"text": "[GOAL]\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nn : \u2115\nhnil : IsNilpotent r\n\u22a2 IsNilpotent (\u2191C r * X ^ n)\n[PROOFSTEP]\nrefine' Commute.isNilpotent_mul_left (commute_X_pow _ _).symm _\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nn : \u2115\nhnil : IsNilpotent r\n\u22a2 IsNilpotent (\u2191C r)\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := hnil\n[GOAL]\ncase intro\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nn m : \u2115\nhm : r ^ m = 0\n\u22a2 IsNilpotent (\u2191C r)\n[PROOFSTEP]\nrefine' \u27e8m, _\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nn m : \u2115\nhm : r ^ m = 0\n\u22a2 \u2191C r ^ m = 0\n[PROOFSTEP]\nrw [\u2190 C_pow, hm, C_0]\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nn : \u2115\nhnil : IsNilpotent r\n\u22a2 IsNilpotent (X ^ n * \u2191C r)\n[PROOFSTEP]\nrw [commute_X_pow]\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nn : \u2115\nhnil : IsNilpotent r\n\u22a2 IsNilpotent (\u2191C r * X ^ n)\n[PROOFSTEP]\nexact isNilpotent_C_mul_pow_X_of_isNilpotent n hnil\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nn k : \u2115\n\u22a2 \u2191(monomial n) r ^ k = 0 \u2194 r ^ k = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nk : \u2115\n\u22a2 \u2191C r ^ k = 0 \u2194 r ^ k = 0\n[PROOFSTEP]\nsimpa only [\u2190 C_pow] using C_eq_zero\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\n\u22a2 IsNilpotent (X * P) \u2194 IsNilpotent P\n[PROOFSTEP]\nrefine' \u27e8fun h \u21a6 _, _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nh : IsNilpotent (X * P)\n\u22a2 IsNilpotent P\n[PROOFSTEP]\nrwa [Commute.isNilpotent_mul_right_iff (commute_X P) (by simp)] at h \n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nh : IsNilpotent (X * P)\n\u22a2 X \u2208 nonZeroDivisorsRight R[X]\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\n\u22a2 IsNilpotent P \u2192 IsNilpotent (X * P)\n[PROOFSTEP]\nrintro \u27e8k, hk\u27e9\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nk : \u2115\nhk : P ^ k = 0\n\u22a2 IsNilpotent (X * P)\n[PROOFSTEP]\nexact \u27e8k, by simp [(commute_X P).mul_pow, hk]\u27e9\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\nk : \u2115\nhk : P ^ k = 0\n\u22a2 (X * P) ^ k = 0\n[PROOFSTEP]\nsimp [(commute_X P).mul_pow, hk]\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\n\u22a2 IsNilpotent (P * X) \u2194 IsNilpotent P\n[PROOFSTEP]\nrw [\u2190 commute_X P]\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : Semiring R\nP : R[X]\n\u22a2 IsNilpotent (X * P) \u2194 IsNilpotent P\n[PROOFSTEP]\nexact isNilpotent_X_mul_iff\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\n\u22a2 IsNilpotent P \u2194 \u2200 (i : \u2115), IsNilpotent (coeff P i)\n[PROOFSTEP]\nrefine' \u27e8P.recOnHorner (by simp) (fun p r hp\u2080 _ hp hpr i \u21a6 _) (fun p _ hnp hpX i \u21a6 _), fun h \u21a6 _\u27e9\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\n\u22a2 IsNilpotent 0 \u2192 \u2200 (i : \u2115), IsNilpotent (coeff 0 i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_1\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nh : \u2200 (i : \u2115), IsNilpotent (coeff P i)\n\u22a2 IsNilpotent P\n[PROOFSTEP]\nrw [\u2190 sum_monomial_eq P]\n[GOAL]\ncase refine'_1\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nh : \u2200 (i : \u2115), IsNilpotent (coeff P i)\n\u22a2 IsNilpotent (sum P fun n a => \u2191(monomial n) a)\n[PROOFSTEP]\nexact isNilpotent_sum (fun i _ \u21a6 by simpa only [isNilpotent_monomial_iff] using h i)\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nh : \u2200 (i : \u2115), IsNilpotent (coeff P i)\ni : \u2115\nx\u271d : i \u2208 support P\n\u22a2 IsNilpotent ((fun n a => \u2191(monomial n) a) i (coeff P i))\n[PROOFSTEP]\nsimpa only [isNilpotent_monomial_iff] using h i\n[GOAL]\ncase refine'_2\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : coeff p 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (coeff p i)\nhpr : IsNilpotent (p + \u2191C r)\ni : \u2115\n\u22a2 IsNilpotent (coeff (p + \u2191C r) i)\n[PROOFSTEP]\nhave hr : IsNilpotent (C r) := by\n obtain \u27e8k, hk\u27e9 := hpr\n replace hp : eval 0 p = 0 := by rwa [coeff_zero_eq_aeval_zero] at hp\u2080 \n refine' isNilpotent_C_iff.mpr \u27e8k, _\u27e9\n simpa [coeff_zero_eq_aeval_zero, hp] using congr_arg (fun q \u21a6 coeff q 0) hk\n[GOAL]\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : coeff p 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (coeff p i)\nhpr : IsNilpotent (p + \u2191C r)\ni : \u2115\n\u22a2 IsNilpotent (\u2191C r)\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := hpr\n[GOAL]\ncase intro\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : coeff p 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (coeff p i)\ni k : \u2115\nhk : (p + \u2191C r) ^ k = 0\n\u22a2 IsNilpotent (\u2191C r)\n[PROOFSTEP]\nreplace hp : eval 0 p = 0 := by rwa [coeff_zero_eq_aeval_zero] at hp\u2080 \n[GOAL]\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : coeff p 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (coeff p i)\ni k : \u2115\nhk : (p + \u2191C r) ^ k = 0\n\u22a2 eval 0 p = 0\n[PROOFSTEP]\nrwa [coeff_zero_eq_aeval_zero] at hp\u2080 \n[GOAL]\ncase intro\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : coeff p 0 = 0\nx\u271d : r \u2260 0\ni k : \u2115\nhk : (p + \u2191C r) ^ k = 0\nhp : eval 0 p = 0\n\u22a2 IsNilpotent (\u2191C r)\n[PROOFSTEP]\nrefine' isNilpotent_C_iff.mpr \u27e8k, _\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : coeff p 0 = 0\nx\u271d : r \u2260 0\ni k : \u2115\nhk : (p + \u2191C r) ^ k = 0\nhp : eval 0 p = 0\n\u22a2 r ^ k = 0\n[PROOFSTEP]\nsimpa [coeff_zero_eq_aeval_zero, hp] using congr_arg (fun q \u21a6 coeff q 0) hk\n[GOAL]\ncase refine'_2\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : coeff p 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (coeff p i)\nhpr : IsNilpotent (p + \u2191C r)\ni : \u2115\nhr : IsNilpotent (\u2191C r)\n\u22a2 IsNilpotent (coeff (p + \u2191C r) i)\n[PROOFSTEP]\ncases' i with i\n[GOAL]\ncase refine'_2.zero\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : coeff p 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (coeff p i)\nhpr : IsNilpotent (p + \u2191C r)\nhr : IsNilpotent (\u2191C r)\n\u22a2 IsNilpotent (coeff (p + \u2191C r) Nat.zero)\ncase refine'_2.succ\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : coeff p 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (coeff p i)\nhpr : IsNilpotent (p + \u2191C r)\nhr : IsNilpotent (\u2191C r)\ni : \u2115\n\u22a2 IsNilpotent (coeff (p + \u2191C r) (Nat.succ i))\n[PROOFSTEP]\nsimpa [hp\u2080] using hr\n[GOAL]\ncase refine'_2.succ\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : coeff p 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (coeff p i)\nhpr : IsNilpotent (p + \u2191C r)\nhr : IsNilpotent (\u2191C r)\ni : \u2115\n\u22a2 IsNilpotent (coeff (p + \u2191C r) (Nat.succ i))\n[PROOFSTEP]\nsimp only [coeff_add, coeff_C_succ, add_zero]\n[GOAL]\ncase refine'_2.succ\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : coeff p 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (coeff p i)\nhpr : IsNilpotent (p + \u2191C r)\nhr : IsNilpotent (\u2191C r)\ni : \u2115\n\u22a2 IsNilpotent (coeff p (Nat.succ i))\n[PROOFSTEP]\napply hp\n[GOAL]\ncase refine'_2.succ.a\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : coeff p 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (coeff p i)\nhpr : IsNilpotent (p + \u2191C r)\nhr : IsNilpotent (\u2191C r)\ni : \u2115\n\u22a2 IsNilpotent p\n[PROOFSTEP]\nsimpa using Commute.isNilpotent_sub (Commute.all _ _) hpr hr\n[GOAL]\ncase refine'_3\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP p : R[X]\nx\u271d : p \u2260 0\nhnp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (coeff p i)\nhpX : IsNilpotent (p * X)\ni : \u2115\n\u22a2 IsNilpotent (coeff (p * X) i)\n[PROOFSTEP]\ncases' i with i\n[GOAL]\ncase refine'_3.zero\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP p : R[X]\nx\u271d : p \u2260 0\nhnp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (coeff p i)\nhpX : IsNilpotent (p * X)\n\u22a2 IsNilpotent (coeff (p * X) Nat.zero)\ncase refine'_3.succ\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP p : R[X]\nx\u271d : p \u2260 0\nhnp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (coeff p i)\nhpX : IsNilpotent (p * X)\ni : \u2115\n\u22a2 IsNilpotent (coeff (p * X) (Nat.succ i))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3.succ\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP p : R[X]\nx\u271d : p \u2260 0\nhnp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (coeff p i)\nhpX : IsNilpotent (p * X)\ni : \u2115\n\u22a2 IsNilpotent (coeff (p * X) (Nat.succ i))\n[PROOFSTEP]\nsimpa using hnp (isNilpotent_mul_X_iff.mp hpX) i\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\n\u22a2 IsNilpotent (reverse P) \u2194 IsNilpotent P\n[PROOFSTEP]\nsimp only [Polynomial.isNilpotent_iff, coeff_reverse]\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\n\u22a2 (\u2200 (i : \u2115), IsNilpotent (coeff P (\u2191(revAt (natDegree P)) i))) \u2194 \u2200 (i : \u2115), IsNilpotent (coeff P i)\n[PROOFSTEP]\nrefine' \u27e8fun h i \u21a6 _, fun h i \u21a6 _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nh : \u2200 (i : \u2115), IsNilpotent (coeff P (\u2191(revAt (natDegree P)) i))\ni : \u2115\n\u22a2 IsNilpotent (coeff P i)\n[PROOFSTEP]\ncases' le_or_lt i P.natDegree with hi hi\n[GOAL]\ncase refine'_2\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nh : \u2200 (i : \u2115), IsNilpotent (coeff P i)\ni : \u2115\n\u22a2 IsNilpotent (coeff P (\u2191(revAt (natDegree P)) i))\n[PROOFSTEP]\ncases' le_or_lt i P.natDegree with hi hi\n[GOAL]\ncase refine'_1.inl\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nh : \u2200 (i : \u2115), IsNilpotent (coeff P (\u2191(revAt (natDegree P)) i))\ni : \u2115\nhi : i \u2264 natDegree P\n\u22a2 IsNilpotent (coeff P i)\n[PROOFSTEP]\nsimpa [tsub_tsub_cancel_of_le hi] using h (P.natDegree - i)\n[GOAL]\ncase refine'_1.inr\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nh : \u2200 (i : \u2115), IsNilpotent (coeff P (\u2191(revAt (natDegree P)) i))\ni : \u2115\nhi : natDegree P < i\n\u22a2 IsNilpotent (coeff P i)\n[PROOFSTEP]\nsimp [coeff_eq_zero_of_natDegree_lt hi]\n[GOAL]\ncase refine'_2.inl\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nh : \u2200 (i : \u2115), IsNilpotent (coeff P i)\ni : \u2115\nhi : i \u2264 natDegree P\n\u22a2 IsNilpotent (coeff P (\u2191(revAt (natDegree P)) i))\n[PROOFSTEP]\nsimpa only [hi, revAt_le] using h (P.natDegree - i)\n[GOAL]\ncase refine'_2.inr\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nh : \u2200 (i : \u2115), IsNilpotent (coeff P i)\ni : \u2115\nhi : natDegree P < i\n\u22a2 IsNilpotent (coeff P (\u2191(revAt (natDegree P)) i))\n[PROOFSTEP]\nsimpa only [revAt_eq_self_of_lt hi] using h i\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\n\u22a2 IsUnit P\n[PROOFSTEP]\ninduction' h : P.natDegree using Nat.strong_induction_on with k hind generalizing P\n[GOAL]\ncase h\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\n\u22a2 IsUnit P\n[PROOFSTEP]\nby_cases hdeg : P.natDegree = 0\n[GOAL]\ncase pos\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : natDegree P = 0\n\u22a2 IsUnit P\ncase neg\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : \u00acnatDegree P = 0\n\u22a2 IsUnit P\n[PROOFSTEP]\n{ rw [eq_C_of_natDegree_eq_zero hdeg]\n exact hunit.map C\n}\n[GOAL]\ncase pos\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : natDegree P = 0\n\u22a2 IsUnit P\n[PROOFSTEP]\nrw [eq_C_of_natDegree_eq_zero hdeg]\n[GOAL]\ncase pos\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : natDegree P = 0\n\u22a2 IsUnit (\u2191C (coeff P 0))\n[PROOFSTEP]\nexact hunit.map C\n[GOAL]\ncase neg\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : \u00acnatDegree P = 0\n\u22a2 IsUnit P\n[PROOFSTEP]\nset P\u2081 := P.eraseLead with hP\u2081\n[GOAL]\ncase neg\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : \u00acnatDegree P = 0\nP\u2081 : R[X] := eraseLead P\nhP\u2081 : P\u2081 = eraseLead P\n\u22a2 IsUnit P\n[PROOFSTEP]\nsuffices IsUnit P\u2081 by\n rw [\u2190 eraseLead_add_monomial_natDegree_leadingCoeff P, \u2190 C_mul_X_pow_eq_monomial]\n obtain \u27e8Q, hQ\u27e9 := this\n rw [\u2190 hP\u2081, \u2190 hQ]\n refine'\n Commute.IsNilpotent.add_isUnit (isNilpotent_C_mul_pow_X_of_isNilpotent _ (hnil _ hdeg))\n ((Commute.all _ _).mul_left (Commute.all _ _))\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : \u00acnatDegree P = 0\nP\u2081 : R[X] := eraseLead P\nhP\u2081 : P\u2081 = eraseLead P\nthis : IsUnit P\u2081\n\u22a2 IsUnit P\n[PROOFSTEP]\nrw [\u2190 eraseLead_add_monomial_natDegree_leadingCoeff P, \u2190 C_mul_X_pow_eq_monomial]\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : \u00acnatDegree P = 0\nP\u2081 : R[X] := eraseLead P\nhP\u2081 : P\u2081 = eraseLead P\nthis : IsUnit P\u2081\n\u22a2 IsUnit (eraseLead P + \u2191C (leadingCoeff P) * X ^ natDegree P)\n[PROOFSTEP]\nobtain \u27e8Q, hQ\u27e9 := this\n[GOAL]\ncase intro\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : \u00acnatDegree P = 0\nP\u2081 : R[X] := eraseLead P\nhP\u2081 : P\u2081 = eraseLead P\nQ : R[X]\u02e3\nhQ : \u2191Q = P\u2081\n\u22a2 IsUnit (eraseLead P + \u2191C (leadingCoeff P) * X ^ natDegree P)\n[PROOFSTEP]\nrw [\u2190 hP\u2081, \u2190 hQ]\n[GOAL]\ncase intro\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : \u00acnatDegree P = 0\nP\u2081 : R[X] := eraseLead P\nhP\u2081 : P\u2081 = eraseLead P\nQ : R[X]\u02e3\nhQ : \u2191Q = P\u2081\n\u22a2 IsUnit (\u2191Q + \u2191C (leadingCoeff P) * X ^ natDegree P)\n[PROOFSTEP]\nrefine'\n Commute.IsNilpotent.add_isUnit (isNilpotent_C_mul_pow_X_of_isNilpotent _ (hnil _ hdeg))\n ((Commute.all _ _).mul_left (Commute.all _ _))\n[GOAL]\ncase neg\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : \u00acnatDegree P = 0\nP\u2081 : R[X] := eraseLead P\nhP\u2081 : P\u2081 = eraseLead P\n\u22a2 IsUnit P\u2081\n[PROOFSTEP]\nhave hdeg\u2082 := lt_of_le_of_lt P.eraseLead_natDegree_le (Nat.sub_lt (Nat.pos_of_ne_zero hdeg) zero_lt_one)\n[GOAL]\ncase neg\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : \u00acnatDegree P = 0\nP\u2081 : R[X] := eraseLead P\nhP\u2081 : P\u2081 = eraseLead P\nhdeg\u2082 : natDegree (eraseLead P) < natDegree P\n\u22a2 IsUnit P\u2081\n[PROOFSTEP]\nrefine' hind P\u2081.natDegree _ _ (fun i hi => _) rfl\n[GOAL]\ncase neg.refine'_1\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : \u00acnatDegree P = 0\nP\u2081 : R[X] := eraseLead P\nhP\u2081 : P\u2081 = eraseLead P\nhdeg\u2082 : natDegree (eraseLead P) < natDegree P\n\u22a2 natDegree P\u2081 < k\n[PROOFSTEP]\nsimp_rw [\u2190 h, hdeg\u2082]\n[GOAL]\ncase neg.refine'_2\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : \u00acnatDegree P = 0\nP\u2081 : R[X] := eraseLead P\nhP\u2081 : P\u2081 = eraseLead P\nhdeg\u2082 : natDegree (eraseLead P) < natDegree P\n\u22a2 IsUnit (coeff P\u2081 0)\n[PROOFSTEP]\nsimp_rw [eraseLead_coeff_of_ne _ (Ne.symm hdeg), hunit]\n[GOAL]\ncase neg.refine'_3\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : \u00acnatDegree P = 0\nP\u2081 : R[X] := eraseLead P\nhP\u2081 : P\u2081 = eraseLead P\nhdeg\u2082 : natDegree (eraseLead P) < natDegree P\ni : \u2115\nhi : i \u2260 0\n\u22a2 IsNilpotent (coeff P\u2081 i)\n[PROOFSTEP]\nby_cases H : i \u2264 P\u2081.natDegree\n[GOAL]\ncase pos\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : \u00acnatDegree P = 0\nP\u2081 : R[X] := eraseLead P\nhP\u2081 : P\u2081 = eraseLead P\nhdeg\u2082 : natDegree (eraseLead P) < natDegree P\ni : \u2115\nhi : i \u2260 0\nH : i \u2264 natDegree P\u2081\n\u22a2 IsNilpotent (coeff P\u2081 i)\ncase neg\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : \u00acnatDegree P = 0\nP\u2081 : R[X] := eraseLead P\nhP\u2081 : P\u2081 = eraseLead P\nhdeg\u2082 : natDegree (eraseLead P) < natDegree P\ni : \u2115\nhi : i \u2260 0\nH : \u00aci \u2264 natDegree P\u2081\n\u22a2 IsNilpotent (coeff P\u2081 i)\n[PROOFSTEP]\nsimp_rw [eraseLead_coeff_of_ne _ (ne_of_lt (lt_of_le_of_lt H hdeg\u2082)), hnil i hi]\n[GOAL]\ncase neg\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP\u271d : R[X]\nhunit\u271d : IsUnit (coeff P\u271d 0)\nhnil\u271d : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P\u271d i)\nx\u271d : \u2115\nh\u271d : natDegree P\u271d = x\u271d\nk : \u2115\nhind :\n \u2200 (m : \u2115),\n m < k \u2192 \u2200 {P : R[X]}, IsUnit (coeff P 0) \u2192 (\u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)) \u2192 natDegree P = m \u2192 IsUnit P\nP : R[X]\nhunit : IsUnit (coeff P 0)\nhnil : \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\nh : natDegree P = k\nhdeg : \u00acnatDegree P = 0\nP\u2081 : R[X] := eraseLead P\nhP\u2081 : P\u2081 = eraseLead P\nhdeg\u2082 : natDegree (eraseLead P) < natDegree P\ni : \u2115\nhi : i \u2260 0\nH : \u00aci \u2264 natDegree P\u2081\n\u22a2 IsNilpotent (coeff P\u2081 i)\n[PROOFSTEP]\nsimp_rw [coeff_eq_zero_of_natDegree_lt (lt_of_not_ge H), IsNilpotent.zero]\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\n\u22a2 IsUnit (coeff P 0) \u2227 \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\n[PROOFSTEP]\nobtain \u27e8Q, hQ\u27e9 := IsUnit.exists_right_inv hunit\n[GOAL]\ncase intro\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\n\u22a2 IsUnit (coeff P 0) \u2227 \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.left\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\n\u22a2 IsUnit (coeff P 0)\n[PROOFSTEP]\nrefine' isUnit_of_mul_eq_one _ (Q.coeff 0) _\n[GOAL]\ncase intro.left\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\n\u22a2 coeff P 0 * coeff Q 0 = 1\n[PROOFSTEP]\nhave h := (mul_coeff_zero P Q).symm\n[GOAL]\ncase intro.left\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\nh : coeff P 0 * coeff Q 0 = coeff (P * Q) 0\n\u22a2 coeff P 0 * coeff Q 0 = 1\n[PROOFSTEP]\nrwa [hQ, coeff_one_zero] at h \n[GOAL]\ncase intro.right\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\n\u22a2 \u2200 (i : \u2115), i \u2260 0 \u2192 IsNilpotent (coeff P i)\n[PROOFSTEP]\nintros n hn\n[GOAL]\ncase intro.right\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\nn : \u2115\nhn : n \u2260 0\n\u22a2 IsNilpotent (coeff P n)\n[PROOFSTEP]\nrw [nilpotent_iff_mem_prime]\n[GOAL]\ncase intro.right\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u2200 (J : Ideal R), Ideal.IsPrime J \u2192 coeff P n \u2208 J\n[PROOFSTEP]\nintros I hI\n[GOAL]\ncase intro.right\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\nn : \u2115\nhn : n \u2260 0\nI : Ideal R\nhI : Ideal.IsPrime I\n\u22a2 coeff P n \u2208 I\n[PROOFSTEP]\nlet f := mapRingHom (Ideal.Quotient.mk I)\n[GOAL]\ncase intro.right\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\nn : \u2115\nhn : n \u2260 0\nI : Ideal R\nhI : Ideal.IsPrime I\nf : R[X] \u2192+* (R \u29f8 I)[X] := mapRingHom (Ideal.Quotient.mk I)\n\u22a2 coeff P n \u2208 I\n[PROOFSTEP]\nhave hPQ : degree (f P) = 0 \u2227 degree (f Q) = 0 := by\n rw [\u2190 Nat.WithBot.add_eq_zero_iff, \u2190 degree_mul, \u2190 _root_.map_mul, hQ, map_one, degree_one]\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\nn : \u2115\nhn : n \u2260 0\nI : Ideal R\nhI : Ideal.IsPrime I\nf : R[X] \u2192+* (R \u29f8 I)[X] := mapRingHom (Ideal.Quotient.mk I)\n\u22a2 degree (\u2191f P) = 0 \u2227 degree (\u2191f Q) = 0\n[PROOFSTEP]\nrw [\u2190 Nat.WithBot.add_eq_zero_iff, \u2190 degree_mul, \u2190 _root_.map_mul, hQ, map_one, degree_one]\n[GOAL]\ncase intro.right\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\nn : \u2115\nhn : n \u2260 0\nI : Ideal R\nhI : Ideal.IsPrime I\nf : R[X] \u2192+* (R \u29f8 I)[X] := mapRingHom (Ideal.Quotient.mk I)\nhPQ : degree (\u2191f P) = 0 \u2227 degree (\u2191f Q) = 0\n\u22a2 coeff P n \u2208 I\n[PROOFSTEP]\nhave hcoeff : (f P).coeff n = 0 := by\n refine' coeff_eq_zero_of_degree_lt _\n rw [hPQ.1]\n exact (@WithBot.coe_pos _ _ _ n).2 (Ne.bot_lt hn)\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\nn : \u2115\nhn : n \u2260 0\nI : Ideal R\nhI : Ideal.IsPrime I\nf : R[X] \u2192+* (R \u29f8 I)[X] := mapRingHom (Ideal.Quotient.mk I)\nhPQ : degree (\u2191f P) = 0 \u2227 degree (\u2191f Q) = 0\n\u22a2 coeff (\u2191f P) n = 0\n[PROOFSTEP]\nrefine' coeff_eq_zero_of_degree_lt _\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\nn : \u2115\nhn : n \u2260 0\nI : Ideal R\nhI : Ideal.IsPrime I\nf : R[X] \u2192+* (R \u29f8 I)[X] := mapRingHom (Ideal.Quotient.mk I)\nhPQ : degree (\u2191f P) = 0 \u2227 degree (\u2191f Q) = 0\n\u22a2 degree (\u2191f P) < \u2191n\n[PROOFSTEP]\nrw [hPQ.1]\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\nn : \u2115\nhn : n \u2260 0\nI : Ideal R\nhI : Ideal.IsPrime I\nf : R[X] \u2192+* (R \u29f8 I)[X] := mapRingHom (Ideal.Quotient.mk I)\nhPQ : degree (\u2191f P) = 0 \u2227 degree (\u2191f Q) = 0\n\u22a2 0 < \u2191n\n[PROOFSTEP]\nexact (@WithBot.coe_pos _ _ _ n).2 (Ne.bot_lt hn)\n[GOAL]\ncase intro.right\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\nn : \u2115\nhn : n \u2260 0\nI : Ideal R\nhI : Ideal.IsPrime I\nf : R[X] \u2192+* (R \u29f8 I)[X] := mapRingHom (Ideal.Quotient.mk I)\nhPQ : degree (\u2191f P) = 0 \u2227 degree (\u2191f Q) = 0\nhcoeff : coeff (\u2191f P) n = 0\n\u22a2 coeff P n \u2208 I\n[PROOFSTEP]\nrw [coe_mapRingHom, coeff_map, \u2190 RingHom.mem_ker, Ideal.mk_ker] at hcoeff \n[GOAL]\ncase intro.right\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\nn : \u2115\nhn : n \u2260 0\nI : Ideal R\nhI : Ideal.IsPrime I\nf : R[X] \u2192+* (R \u29f8 I)[X] := mapRingHom (Ideal.Quotient.mk I)\nhPQ : degree (\u2191f P) = 0 \u2227 degree (\u2191f Q) = 0\nhcoeff : coeff P n \u2208 I\n\u22a2 coeff P n \u2208 I\n[PROOFSTEP]\nexact hcoeff\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\n\u22a2 IsUnit (\u2191C r + X * P) \u2194 IsUnit r \u2227 IsNilpotent P\n[PROOFSTEP]\nhave : \u2200 i, coeff (C r + X * P) (i + 1) = coeff P i := by simp\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\n\u22a2 \u2200 (i : \u2115), coeff (\u2191C r + X * P) (i + 1) = coeff P i\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nthis : \u2200 (i : \u2115), coeff (\u2191C r + X * P) (i + 1) = coeff P i\n\u22a2 IsUnit (\u2191C r + X * P) \u2194 IsUnit r \u2227 IsNilpotent P\n[PROOFSTEP]\nsimp_rw [isUnit_iff_coeff_isUnit_isNilpotent, Nat.forall_ne_zero_iff, this]\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nthis : \u2200 (i : \u2115), coeff (\u2191C r + X * P) (i + 1) = coeff P i\n\u22a2 (IsUnit (coeff (\u2191C r + X * P) 0) \u2227 \u2200 (i : \u2115), IsNilpotent (coeff P i)) \u2194 IsUnit r \u2227 IsNilpotent P\n[PROOFSTEP]\nsimp only [coeff_add, coeff_C_zero, mul_coeff_zero, coeff_X_zero, zero_mul, add_zero, and_congr_right_iff, \u2190\n Polynomial.isNilpotent_iff]\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\n\u22a2 IsUnit P \u2194 IsUnit (eval 0 P) \u2227 IsNilpotent (P /\u2098 X)\n[PROOFSTEP]\nsuffices : P = C (eval 0 P) + X * (P /\u2098 X)\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nthis : P = \u2191C (eval 0 P) + X * (P /\u2098 X)\n\u22a2 IsUnit P \u2194 IsUnit (eval 0 P) \u2227 IsNilpotent (P /\u2098 X)\n[PROOFSTEP]\nconv_lhs => rw [this]; simp\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nthis : P = \u2191C (eval 0 P) + X * (P /\u2098 X)\n| IsUnit P\n[PROOFSTEP]\nrw [this]; simp\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nthis : P = \u2191C (eval 0 P) + X * (P /\u2098 X)\n| IsUnit P\n[PROOFSTEP]\nrw [this]; simp\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nthis : P = \u2191C (eval 0 P) + X * (P /\u2098 X)\n| IsUnit P\n[PROOFSTEP]\nrw [this]\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nthis : P = \u2191C (eval 0 P) + X * (P /\u2098 X)\n| IsUnit (\u2191C (eval 0 P) + X * (P /\u2098 X))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase this\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\n\u22a2 P = \u2191C (eval 0 P) + X * (P /\u2098 X)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 modByMonic_add_div P monic_X]\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\n| P\n[PROOFSTEP]\nrw [\u2190 modByMonic_add_div P monic_X]\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\n| P\n[PROOFSTEP]\nrw [\u2190 modByMonic_add_div P monic_X]\n[GOAL]\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\n| P\n[PROOFSTEP]\nrw [\u2190 modByMonic_add_div P monic_X]\n[GOAL]\ncase this\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\n\u22a2 P %\u2098 X + X * (P /\u2098 X) = \u2191C (eval 0 P) + X * (P /\u2098 X)\n[PROOFSTEP]\nsimp [modByMonic_X]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Polynomial.Nilpotent", "llama_tokens": 16082, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850154599563, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.42887955971495345}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\n\u22a2 \u2203 P, IsEquipartition P \u2227 l \u2264 Finset.card P.parts \u2227 Finset.card P.parts \u2264 bound \u03b5 l \u2227 IsUniform P G \u03b5\n[PROOFSTEP]\nobtain h\u03b1 | h\u03b1 :=\n le_total (card \u03b1)\n (bound \u03b5 l)\n -- If `card \u03b1 \u2264 bound \u03b5 l`, then the partition into singletons is acceptable.\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : Fintype.card \u03b1 \u2264 bound \u03b5 l\n\u22a2 \u2203 P, IsEquipartition P \u2227 l \u2264 Finset.card P.parts \u2227 Finset.card P.parts \u2264 bound \u03b5 l \u2227 IsUniform P G \u03b5\n[PROOFSTEP]\nrefine' \u27e8\u22a5, bot_isEquipartition _, _\u27e9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : Fintype.card \u03b1 \u2264 bound \u03b5 l\n\u22a2 l \u2264 Finset.card \u22a5.parts \u2227 Finset.card \u22a5.parts \u2264 bound \u03b5 l \u2227 IsUniform \u22a5 G \u03b5\n[PROOFSTEP]\nrw [card_bot, card_univ]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : Fintype.card \u03b1 \u2264 bound \u03b5 l\n\u22a2 l \u2264 Fintype.card \u03b1 \u2227 Fintype.card \u03b1 \u2264 bound \u03b5 l \u2227 IsUniform \u22a5 G \u03b5\n[PROOFSTEP]\nexact\n \u27e8hl, h\u03b1, botIsUniform _ h\u03b5\u27e9\n -- Else, let's start from a dummy equipartition of size `initialBound \u03b5 l`.\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\n\u22a2 \u2203 P, IsEquipartition P \u2227 l \u2264 Finset.card P.parts \u2227 Finset.card P.parts \u2264 bound \u03b5 l \u2227 IsUniform P G \u03b5\n[PROOFSTEP]\nlet t := initialBound \u03b5 l\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\n\u22a2 \u2203 P, IsEquipartition P \u2227 l \u2264 Finset.card P.parts \u2227 Finset.card P.parts \u2264 bound \u03b5 l \u2227 IsUniform P G \u03b5\n[PROOFSTEP]\nhave ht\u03b1 : t \u2264 (univ : Finset \u03b1).card := (initialBound_le_bound _ _).trans (by rwa [Finset.card_univ])\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\n\u22a2 bound \u03b5 l \u2264 Finset.card univ\n[PROOFSTEP]\nrwa [Finset.card_univ]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\n\u22a2 \u2203 P, IsEquipartition P \u2227 l \u2264 Finset.card P.parts \u2227 Finset.card P.parts \u2264 bound \u03b5 l \u2227 IsUniform P G \u03b5\n[PROOFSTEP]\nobtain \u27e8dum, hdum\u2081, hdum\u2082\u27e9 := exists_equipartition_card_eq (univ : Finset \u03b1) (initialBound_pos _ _).ne' ht\u03b1\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\n\u22a2 \u2203 P, IsEquipartition P \u2227 l \u2264 Finset.card P.parts \u2227 Finset.card P.parts \u2264 bound \u03b5 l \u2227 IsUniform P G \u03b5\n[PROOFSTEP]\nobtain h\u03b5\u2081 | h\u03b5\u2081 := le_total 1 \u03b5\n[GOAL]\ncase inr.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : 1 \u2264 \u03b5\n\u22a2 \u2203 P, IsEquipartition P \u2227 l \u2264 Finset.card P.parts \u2227 Finset.card P.parts \u2264 bound \u03b5 l \u2227 IsUniform P G \u03b5\n[PROOFSTEP]\nexact\n \u27e8dum, hdum\u2081, (le_initialBound \u03b5 l).trans hdum\u2082.ge, hdum\u2082.le.trans (initialBound_le_bound \u03b5 l),\n (dum.isUniformOne G).mono h\u03b5\u2081\u27e9\n -- Else, set up the induction on energy. We phrase it through the existence for each `i` of an\n -- equipartition of size bounded by `stepBound^[i] (initialBound \u03b5 l)` and which is either\n -- `\u03b5`-uniform or has energy at least `\u03b5 ^ 5 / 4 * i`.\n[GOAL]\ncase inr.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 \u2203 P, IsEquipartition P \u2227 l \u2264 Finset.card P.parts \u2227 Finset.card P.parts \u2264 bound \u03b5 l \u2227 IsUniform P G \u03b5\n[PROOFSTEP]\nhave : Nonempty \u03b1 := by\n rw [\u2190 Fintype.card_pos_iff]\n exact (bound_pos _ _).trans_le h\u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 Nonempty \u03b1\n[PROOFSTEP]\nrw [\u2190 Fintype.card_pos_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 0 < Fintype.card \u03b1\n[PROOFSTEP]\nexact (bound_pos _ _).trans_le h\u03b1\n[GOAL]\ncase inr.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\n\u22a2 \u2203 P, IsEquipartition P \u2227 l \u2264 Finset.card P.parts \u2227 Finset.card P.parts \u2264 bound \u03b5 l \u2227 IsUniform P G \u03b5\n[PROOFSTEP]\nsuffices h :\n \u2200 i,\n \u2203 P : Finpartition (univ : Finset \u03b1),\n P.IsEquipartition \u2227\n t \u2264 P.parts.card \u2227\n P.parts.card \u2264 stepBound^[i] t \u2227\n (P.IsUniform G \u03b5 \u2228 \u03b5 ^ 5 / 4 * i \u2264 P.energy G)\n -- For `i > 4 / \u03b5 ^ 5` we know that the partition we get can't have energy `\u2265 \u03b5 ^ 5 / 4 * i > 1`,\n -- so it must instead be `\u03b5`-uniform and we won.\n[GOAL]\ncase inr.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\nh :\n \u2200 (i : \u2115),\n \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G))\n\u22a2 \u2203 P, IsEquipartition P \u2227 l \u2264 Finset.card P.parts \u2227 Finset.card P.parts \u2264 bound \u03b5 l \u2227 IsUniform P G \u03b5\n[PROOFSTEP]\nobtain \u27e8P, hP\u2081, hP\u2082, hP\u2083, hP\u2084\u27e9 := h (\u230a4 / \u03b5 ^ 5\u230b\u208a + 1)\n[GOAL]\ncase inr.intro.intro.inr.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\nh :\n \u2200 (i : \u2115),\n \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G))\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\n\u22a2 \u2203 P, IsEquipartition P \u2227 l \u2264 Finset.card P.parts \u2227 Finset.card P.parts \u2264 bound \u03b5 l \u2227 IsUniform P G \u03b5\n[PROOFSTEP]\nrefine' \u27e8P, hP\u2081, (le_initialBound _ _).trans hP\u2082, hP\u2083.trans _, hP\u2084.resolve_right fun hPenergy => lt_irrefl (1 : \u211d) _\u27e9\n[GOAL]\ncase inr.intro.intro.inr.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\nh :\n \u2200 (i : \u2115),\n \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G))\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\n\u22a2 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1] t \u2264 bound \u03b5 l\n[PROOFSTEP]\nrw [iterate_succ_apply']\n[GOAL]\ncase inr.intro.intro.inr.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\nh :\n \u2200 (i : \u2115),\n \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G))\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\n\u22a2 stepBound (stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a] t) \u2264 bound \u03b5 l\n[PROOFSTEP]\nexact mul_le_mul_left' (pow_le_pow_of_le_left (by norm_num) (by norm_num) _) _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\nh :\n \u2200 (i : \u2115),\n \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G))\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\n\u22a2 0 \u2264 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\nh :\n \u2200 (i : \u2115),\n \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G))\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\n\u22a2 4 \u2264 16\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr.intro.intro.inr.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\nh :\n \u2200 (i : \u2115),\n \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G))\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\nhPenergy : \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\n\u22a2 1 < 1\n[PROOFSTEP]\ncalc\n (1 : \u211d) = \u03b5 ^ 5 / \u21914 * (\u21914 / \u03b5 ^ 5) := by rw [mul_comm, div_mul_div_cancel 4 (pow_pos h\u03b5 5).ne']; norm_num\n _ < \u03b5 ^ 5 / 4 * (\u230a4 / \u03b5 ^ 5\u230b\u208a + 1) := ((mul_lt_mul_left <| by positivity).2 (Nat.lt_floor_add_one _))\n _ \u2264 (P.energy G : \u211d) := by rwa [\u2190 Nat.cast_add_one]\n _ \u2264 1 := by exact_mod_cast P.energy_le_one G\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\nh :\n \u2200 (i : \u2115),\n \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G))\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\nhPenergy : \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\n\u22a2 1 = \u03b5 ^ 5 / 4 * (4 / \u03b5 ^ 5)\n[PROOFSTEP]\nrw [mul_comm, div_mul_div_cancel 4 (pow_pos h\u03b5 5).ne']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\nh :\n \u2200 (i : \u2115),\n \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G))\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\nhPenergy : \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\n\u22a2 1 = 4 / 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\nh :\n \u2200 (i : \u2115),\n \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G))\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\nhPenergy : \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\n\u22a2 0 < \u03b5 ^ 5 / 4\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\nh :\n \u2200 (i : \u2115),\n \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G))\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\nhPenergy : \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\n\u22a2 \u03b5 ^ 5 / \u21914 * (\u2191\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\n[PROOFSTEP]\nrwa [\u2190 Nat.cast_add_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\nh :\n \u2200 (i : \u2115),\n \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G))\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\nhPenergy : \u03b5 ^ 5 / 4 * \u2191(\u230a\u21914 / \u03b5 ^ 5\u230b\u208a + 1) \u2264 \u2191(energy P G)\n\u22a2 \u2191(energy P G) \u2264 1\n[PROOFSTEP]\nexact_mod_cast P.energy_le_one G\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\n\u22a2 \u2200 (i : \u2115),\n \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G))\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\n\u22a2 \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G))\n[PROOFSTEP]\ninduction' i with i ih\n[GOAL]\ncase h.zero\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\n\u22a2 \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[Nat.zero] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191Nat.zero \u2264 \u2191(energy P G))\n[PROOFSTEP]\nrefine' \u27e8dum, hdum\u2081, hdum\u2082.ge, hdum\u2082.le, Or.inr _\u27e9\n[GOAL]\ncase h.zero\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\n\u22a2 \u03b5 ^ 5 / 4 * \u2191Nat.zero \u2264 \u2191(energy dum G)\n[PROOFSTEP]\nrw [Nat.cast_zero, mul_zero]\n[GOAL]\ncase h.zero\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\n\u22a2 0 \u2264 \u2191(energy dum G)\n[PROOFSTEP]\nexact_mod_cast dum.energy_nonneg G\n[GOAL]\ncase h.succ\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nih :\n \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G))\n\u22a2 \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[Nat.succ i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(Nat.succ i) \u2264 \u2191(energy P G))\n[PROOFSTEP]\nobtain \u27e8P, hP\u2081, hP\u2082, hP\u2083, hP\u2084\u27e9 := ih\n[GOAL]\ncase h.succ.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\n\u22a2 \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[Nat.succ i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(Nat.succ i) \u2264 \u2191(energy P G))\n[PROOFSTEP]\nby_cases huniform : P.IsUniform G \u03b5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nhuniform : IsUniform P G \u03b5\n\u22a2 \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[Nat.succ i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(Nat.succ i) \u2264 \u2191(energy P G))\n[PROOFSTEP]\nrefine' \u27e8P, hP\u2081, hP\u2082, _, Or.inl huniform\u27e9\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nhuniform : IsUniform P G \u03b5\n\u22a2 Finset.card P.parts \u2264 stepBound^[Nat.succ i] t\n[PROOFSTEP]\nrw [iterate_succ_apply']\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nhuniform : IsUniform P G \u03b5\n\u22a2 Finset.card P.parts \u2264 stepBound (stepBound^[i] t)\n[PROOFSTEP]\nexact\n hP\u2083.trans\n (le_stepBound _)\n -- Else, `P` must instead have energy at least `\u03b5 ^ 5 / 4 * i`.\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhP\u2084 : IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nhuniform : \u00acIsUniform P G \u03b5\n\u22a2 \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[Nat.succ i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(Nat.succ i) \u2264 \u2191(energy P G))\n[PROOFSTEP]\nreplace hP\u2084 := hP\u2084.resolve_left huniform\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\n\u22a2 \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[Nat.succ i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(Nat.succ i) \u2264 \u2191(energy P G))\n[PROOFSTEP]\nhave h\u03b5l' : \u2191100 < \u21914 ^ P.parts.card * \u03b5 ^ 5 :=\n (hundred_lt_pow_initialBound_mul h\u03b5 l).trans_le\n (mul_le_mul_of_nonneg_right (pow_le_pow (by norm_num) hP\u2082) <| by positivity)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\n\u22a2 1 \u2264 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\n\u22a2 0 \u2264 \u03b5 ^ 5\n[PROOFSTEP]\npositivity\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\n\u22a2 \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[Nat.succ i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(Nat.succ i) \u2264 \u2191(energy P G))\n[PROOFSTEP]\nhave hi : (i : \u211d) \u2264 4 / \u03b5 ^ 5 :=\n by\n have hi : \u03b5 ^ 5 / 4 * \u2191i \u2264 1 := hP\u2084.trans (by exact_mod_cast P.energy_le_one G)\n rw [div_mul_eq_mul_div, div_le_iff (show (0 : \u211d) < 4 by norm_num)] at hi \n norm_num at hi \n rwa [le_div_iff' (pow_pos h\u03b5 _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\n\u22a2 \u2191i \u2264 \u21914 / \u03b5 ^ 5\n[PROOFSTEP]\nhave hi : \u03b5 ^ 5 / 4 * \u2191i \u2264 1 := hP\u2084.trans (by exact_mod_cast P.energy_le_one G)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\n\u22a2 \u2191(energy P G) \u2264 1\n[PROOFSTEP]\nexact_mod_cast P.energy_le_one G\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhi : \u03b5 ^ 5 / 4 * \u2191i \u2264 1\n\u22a2 \u2191i \u2264 \u21914 / \u03b5 ^ 5\n[PROOFSTEP]\nrw [div_mul_eq_mul_div, div_le_iff (show (0 : \u211d) < 4 by norm_num)] at hi \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhi : \u03b5 ^ 5 * \u2191i / 4 \u2264 1\n\u22a2 0 < 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhi : \u03b5 ^ 5 * \u2191i \u2264 1 * 4\n\u22a2 \u2191i \u2264 \u21914 / \u03b5 ^ 5\n[PROOFSTEP]\nnorm_num at hi \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhi : \u03b5 ^ 5 * \u2191i \u2264 4\n\u22a2 \u2191i \u2264 \u21914 / \u03b5 ^ 5\n[PROOFSTEP]\nrwa [le_div_iff' (pow_pos h\u03b5 _)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhi : \u2191i \u2264 \u21914 / \u03b5 ^ 5\n\u22a2 \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[Nat.succ i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(Nat.succ i) \u2264 \u2191(energy P G))\n[PROOFSTEP]\nhave hsize : P.parts.card \u2264 stepBound^[\u230a4 / \u03b5 ^ 5\u230b\u208a] t :=\n hP\u2083.trans (monotone_iterate_of_id_le le_stepBound (Nat.le_floor hi) _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhi : \u2191i \u2264 \u21914 / \u03b5 ^ 5\nhsize : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a] t\n\u22a2 \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[Nat.succ i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(Nat.succ i) \u2264 \u2191(energy P G))\n[PROOFSTEP]\nhave hP\u03b1 : P.parts.card * 16 ^ P.parts.card \u2264 card \u03b1 :=\n (Nat.mul_le_mul hsize (Nat.pow_le_pow_of_le_right (by norm_num) hsize)).trans h\u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhi : \u2191i \u2264 \u21914 / \u03b5 ^ 5\nhsize : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a] t\n\u22a2 16 > 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhi : \u2191i \u2264 \u21914 / \u03b5 ^ 5\nhsize : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a] t\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\n\u22a2 \u2203 P,\n IsEquipartition P \u2227\n t \u2264 Finset.card P.parts \u2227\n Finset.card P.parts \u2264 stepBound^[Nat.succ i] t \u2227 (IsUniform P G \u03b5 \u2228 \u03b5 ^ 5 / 4 * \u2191(Nat.succ i) \u2264 \u2191(energy P G))\n[PROOFSTEP]\nrefine'\n \u27e8increment hP\u2081 G \u03b5, increment_isEquipartition hP\u2081 G \u03b5, _, _,\n Or.inr <| le_trans _ <| energy_increment hP\u2081 ((seven_le_initialBound \u03b5 l).trans hP\u2082) h\u03b5l' hP\u03b1 huniform h\u03b5\u2081\u27e9\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhi : \u2191i \u2264 \u21914 / \u03b5 ^ 5\nhsize : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a] t\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\n\u22a2 t \u2264 Finset.card (increment hP\u2081 G \u03b5).parts\n[PROOFSTEP]\nrw [card_increment hP\u03b1 huniform]\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhi : \u2191i \u2264 \u21914 / \u03b5 ^ 5\nhsize : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a] t\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\n\u22a2 t \u2264 stepBound (Finset.card P.parts)\n[PROOFSTEP]\nexact hP\u2082.trans (le_stepBound _)\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhi : \u2191i \u2264 \u21914 / \u03b5 ^ 5\nhsize : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a] t\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\n\u22a2 Finset.card (increment hP\u2081 G \u03b5).parts \u2264 stepBound^[Nat.succ i] t\n[PROOFSTEP]\nrw [card_increment hP\u03b1 huniform, iterate_succ_apply']\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhi : \u2191i \u2264 \u21914 / \u03b5 ^ 5\nhsize : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a] t\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\n\u22a2 stepBound (Finset.card P.parts) \u2264 stepBound (stepBound^[i] t)\n[PROOFSTEP]\nexact stepBound_mono hP\u2083\n[GOAL]\ncase neg.refine'_3\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhi : \u2191i \u2264 \u21914 / \u03b5 ^ 5\nhsize : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a] t\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\n\u22a2 \u03b5 ^ 5 / 4 * \u2191(Nat.succ i) \u2264 \u2191(energy P G) + \u03b5 ^ 5 / 4\n[PROOFSTEP]\nrw [Nat.cast_succ, mul_add, mul_one]\n[GOAL]\ncase neg.refine'_3\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nl : \u2115\nh\u03b5 : 0 < \u03b5\nhl : l \u2264 Fintype.card \u03b1\nh\u03b1 : bound \u03b5 l \u2264 Fintype.card \u03b1\nt : \u2115 := initialBound \u03b5 l\nht\u03b1 : t \u2264 Finset.card univ\ndum : Finpartition univ\nhdum\u2081 : IsEquipartition dum\nhdum\u2082 : Finset.card dum.parts = initialBound \u03b5 l\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : Nonempty \u03b1\ni : \u2115\nP : Finpartition univ\nhP\u2081 : IsEquipartition P\nhP\u2082 : t \u2264 Finset.card P.parts\nhP\u2083 : Finset.card P.parts \u2264 stepBound^[i] t\nhuniform : \u00acIsUniform P G \u03b5\nhP\u2084 : \u03b5 ^ 5 / 4 * \u2191i \u2264 \u2191(energy P G)\nh\u03b5l' : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhi : \u2191i \u2264 \u21914 / \u03b5 ^ 5\nhsize : Finset.card P.parts \u2264 stepBound^[\u230a\u21914 / \u03b5 ^ 5\u230b\u208a] t\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\n\u22a2 \u03b5 ^ 5 / 4 * \u2191i + \u03b5 ^ 5 / 4 \u2264 \u2191(energy P G) + \u03b5 ^ 5 / 4\n[PROOFSTEP]\nexact add_le_add_right hP\u2084 _\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma", "llama_tokens": 19545, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982315512489, "lm_q2_score": 0.6619228691808012, "lm_q1q2_score": 0.42879246407865174}} {"text": "[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type u_2\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_3\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nf : G \u2192* H\nemb : OpenEmbedding \u2191f\nU : Set H\nhU : U \u2208 nhds 1\n\u22a2 \u2191f \u207b\u00b9' U \u2208 nhds 1\n[PROOFSTEP]\napply emb.continuous.tendsto\n[GOAL]\ncase a\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type u_2\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_3\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nf : G \u2192* H\nemb : OpenEmbedding \u2191f\nU : Set H\nhU : U \u2208 nhds 1\n\u22a2 U \u2208 nhds (\u2191f 1)\n[PROOFSTEP]\nrwa [f.map_one]\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type u_2\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_3\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nhU : U \u2208 nhds 1\n\u22a2 \u2203 V W, \u2191V \u00d7\u02e2 \u2191W \u2286 U\n[PROOFSTEP]\nerw [nhds_prod_eq, Filter.mem_prod_iff] at hU \n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type u_2\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_3\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nhU : \u2203 t\u2081, t\u2081 \u2208 nhds 1 \u2227 \u2203 t\u2082, t\u2082 \u2208 nhds 1 \u2227 t\u2081 \u00d7\u02e2 t\u2082 \u2286 U\n\u22a2 \u2203 V W, \u2191V \u00d7\u02e2 \u2191W \u2286 U\n[PROOFSTEP]\nrcases hU with \u27e8U\u2081, hU\u2081, U\u2082, hU\u2082, h\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type u_2\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_3\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\n\u22a2 \u2203 V W, \u2191V \u00d7\u02e2 \u2191W \u2286 U\n[PROOFSTEP]\ncases' is_nonarchimedean _ hU\u2081 with V hV\n[GOAL]\ncase intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type u_2\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_3\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\n\u22a2 \u2203 V W, \u2191V \u00d7\u02e2 \u2191W \u2286 U\n[PROOFSTEP]\ncases' is_nonarchimedean _ hU\u2082 with W hW\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type u_2\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_3\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\nW : OpenSubgroup K\nhW : \u2191W \u2286 U\u2082\n\u22a2 \u2203 V W, \u2191V \u00d7\u02e2 \u2191W \u2286 U\n[PROOFSTEP]\nuse V\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type u_2\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_3\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\nW : OpenSubgroup K\nhW : \u2191W \u2286 U\u2082\n\u22a2 \u2203 W, \u2191V \u00d7\u02e2 \u2191W \u2286 U\n[PROOFSTEP]\nuse W\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type u_2\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_3\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\nW : OpenSubgroup K\nhW : \u2191W \u2286 U\u2082\n\u22a2 \u2191V \u00d7\u02e2 \u2191W \u2286 U\n[PROOFSTEP]\nrw [Set.prod_subset_iff]\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type u_2\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_3\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\nW : OpenSubgroup K\nhW : \u2191W \u2286 U\u2082\n\u22a2 \u2200 (x : G), x \u2208 \u2191V \u2192 \u2200 (y : K), y \u2208 \u2191W \u2192 (x, y) \u2208 U\n[PROOFSTEP]\nintro x hX y hY\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type u_2\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_3\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\nW : OpenSubgroup K\nhW : \u2191W \u2286 U\u2082\nx : G\nhX : x \u2208 \u2191V\ny : K\nhY : y \u2208 \u2191W\n\u22a2 (x, y) \u2208 U\n[PROOFSTEP]\nexact Set.Subset.trans (Set.prod_mono hV hW) h (Set.mem_sep hX hY)\n[GOAL]\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type u_2\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_3\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 G)\nhU : U \u2208 nhds 1\nV W : OpenSubgroup G\nh : \u2191V \u00d7\u02e2 \u2191W \u2286 U\n\u22a2 \u2191(V \u2293 W) \u00d7\u02e2 \u2191(V \u2293 W) \u2286 U\n[PROOFSTEP]\nrefine' Set.Subset.trans (Set.prod_mono _ _) \u2039_\u203a\n[GOAL]\ncase refine'_1\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type u_2\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_3\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 G)\nhU : U \u2208 nhds 1\nV W : OpenSubgroup G\nh : \u2191V \u00d7\u02e2 \u2191W \u2286 U\n\u22a2 \u2191(V \u2293 W) \u2286 \u2191V\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type u_2\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_3\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 G)\nhU : U \u2208 nhds 1\nV W : OpenSubgroup G\nh : \u2191V \u00d7\u02e2 \u2191W \u2286 U\n\u22a2 \u2191(V \u2293 W) \u2286 \u2191W\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : NonarchimedeanRing R\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : TopologicalSpace S\ninst\u271d : NonarchimedeanRing S\nU : OpenAddSubgroup R\n\u22a2 \u2203 V, \u2191V * \u2191V \u2286 \u2191U\n[PROOFSTEP]\nlet \u27e8V, H\u27e9 :=\n prod_self_subset\n (IsOpen.mem_nhds (IsOpen.preimage continuous_mul U.isOpen)\n (by simpa only [Set.mem_preimage, SetLike.mem_coe, Prod.snd_zero, mul_zero] using U.zero_mem))\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : NonarchimedeanRing R\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : TopologicalSpace S\ninst\u271d : NonarchimedeanRing S\nU : OpenAddSubgroup R\n\u22a2 0 \u2208 (fun p => p.fst * p.snd) \u207b\u00b9' \u2191U\n[PROOFSTEP]\nsimpa only [Set.mem_preimage, SetLike.mem_coe, Prod.snd_zero, mul_zero] using U.zero_mem\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : NonarchimedeanRing R\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : TopologicalSpace S\ninst\u271d : NonarchimedeanRing S\nU V : OpenAddSubgroup R\nH : \u2191V \u00d7\u02e2 \u2191V \u2286 (fun p => p.fst * p.snd) \u207b\u00b9' \u2191U\n\u22a2 \u2203 V, \u2191V * \u2191V \u2286 \u2191U\n[PROOFSTEP]\nuse V\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : NonarchimedeanRing R\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : TopologicalSpace S\ninst\u271d : NonarchimedeanRing S\nU V : OpenAddSubgroup R\nH : \u2191V \u00d7\u02e2 \u2191V \u2286 (fun p => p.fst * p.snd) \u207b\u00b9' \u2191U\n\u22a2 \u2191V * \u2191V \u2286 \u2191U\n[PROOFSTEP]\nrintro v \u27e8a, b, ha, hb, hv\u27e9\n[GOAL]\ncase h.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : NonarchimedeanRing R\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : TopologicalSpace S\ninst\u271d : NonarchimedeanRing S\nU V : OpenAddSubgroup R\nH : \u2191V \u00d7\u02e2 \u2191V \u2286 (fun p => p.fst * p.snd) \u207b\u00b9' \u2191U\nv a b : R\nha : a \u2208 \u2191V\nhb : b \u2208 \u2191V\nhv : (fun x x_1 => x * x_1) a b = v\n\u22a2 v \u2208 \u2191U\n[PROOFSTEP]\nhave hy := H (Set.mk_mem_prod ha hb)\n[GOAL]\ncase h.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : NonarchimedeanRing R\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : TopologicalSpace S\ninst\u271d : NonarchimedeanRing S\nU V : OpenAddSubgroup R\nH : \u2191V \u00d7\u02e2 \u2191V \u2286 (fun p => p.fst * p.snd) \u207b\u00b9' \u2191U\nv a b : R\nha : a \u2208 \u2191V\nhb : b \u2208 \u2191V\nhv : (fun x x_1 => x * x_1) a b = v\nhy : (a, b) \u2208 (fun p => p.fst * p.snd) \u207b\u00b9' \u2191U\n\u22a2 v \u2208 \u2191U\n[PROOFSTEP]\nsimp only [Set.mem_preimage, SetLike.mem_coe, hv] at hy \n[GOAL]\ncase h.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : NonarchimedeanRing R\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : TopologicalSpace S\ninst\u271d : NonarchimedeanRing S\nU V : OpenAddSubgroup R\nH : \u2191V \u00d7\u02e2 \u2191V \u2286 (fun p => p.fst * p.snd) \u207b\u00b9' \u2191U\nv a b : R\nha : a \u2208 \u2191V\nhb : b \u2208 \u2191V\nhv : (fun x x_1 => x * x_1) a b = v\nhy : v \u2208 U\n\u22a2 v \u2208 \u2191U\n[PROOFSTEP]\nrw [SetLike.mem_coe]\n[GOAL]\ncase h.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : NonarchimedeanRing R\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : TopologicalSpace S\ninst\u271d : NonarchimedeanRing S\nU V : OpenAddSubgroup R\nH : \u2191V \u00d7\u02e2 \u2191V \u2286 (fun p => p.fst * p.snd) \u207b\u00b9' \u2191U\nv a b : R\nha : a \u2208 \u2191V\nhb : b \u2208 \u2191V\nhv : (fun x x_1 => x * x_1) a b = v\nhy : v \u2208 U\n\u22a2 v \u2208 U\n[PROOFSTEP]\nexact hy\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Nonarchimedean.Basic", "llama_tokens": 4909, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228625116081, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.4287924507567795}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf\u271d : \ud835\udd5c \u2192 E\na\u271d b\u271d : \ud835\udd5c\ns : Set \ud835\udd5c\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ng : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\nH : a = b \u2192 DifferentiableAt \ud835\udd5c g a\n\u22a2 dslope (\u2191f \u2218 g) a b = \u2191f (dslope g a b)\n[PROOFSTEP]\nrcases eq_or_ne b a with (rfl | hne)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf\u271d : \ud835\udd5c \u2192 E\na b\u271d : \ud835\udd5c\ns : Set \ud835\udd5c\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ng : \ud835\udd5c \u2192 E\nb : \ud835\udd5c\nH : b = b \u2192 DifferentiableAt \ud835\udd5c g b\n\u22a2 dslope (\u2191f \u2218 g) b b = \u2191f (dslope g b b)\n[PROOFSTEP]\nsimp only [dslope_same]\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf\u271d : \ud835\udd5c \u2192 E\na b\u271d : \ud835\udd5c\ns : Set \ud835\udd5c\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ng : \ud835\udd5c \u2192 E\nb : \ud835\udd5c\nH : b = b \u2192 DifferentiableAt \ud835\udd5c g b\n\u22a2 deriv (\u2191f \u2218 g) b = \u2191f (deriv g b)\n[PROOFSTEP]\nexact (f.hasFDerivAt.comp_hasDerivAt b (H rfl).hasDerivAt).deriv\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf\u271d : \ud835\udd5c \u2192 E\na\u271d b\u271d : \ud835\udd5c\ns : Set \ud835\udd5c\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ng : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\nH : a = b \u2192 DifferentiableAt \ud835\udd5c g a\nhne : b \u2260 a\n\u22a2 dslope (\u2191f \u2218 g) a b = \u2191f (dslope g a b)\n[PROOFSTEP]\nsimpa only [dslope_of_ne _ hne] using f.toLinearMap.slope_comp g a b\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf\u271d : \ud835\udd5c \u2192 E\na\u271d b\u271d : \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\n\u22a2 (b - a) \u2022 dslope f a b = f b - f a\n[PROOFSTEP]\nrcases eq_or_ne b a with (rfl | hne)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf\u271d : \ud835\udd5c \u2192 E\na b\u271d : \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 E\nb : \ud835\udd5c\n\u22a2 (b - b) \u2022 dslope f b b = f b - f b\n[PROOFSTEP]\nsimp [dslope_of_ne, *]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf\u271d : \ud835\udd5c \u2192 E\na\u271d b\u271d : \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\nhne : b \u2260 a\n\u22a2 (b - a) \u2022 dslope f a b = f b - f a\n[PROOFSTEP]\nsimp [dslope_of_ne, *]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf\u271d : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 E\nh : b \u2260 a\n\u22a2 dslope (fun x => (x - a) \u2022 f x) a b = f b\n[PROOFSTEP]\nrw [dslope_of_ne _ h, slope_sub_smul _ h.symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\n\u22a2 ContinuousAt (dslope f a) a \u2194 DifferentiableAt \ud835\udd5c f a\n[PROOFSTEP]\nsimp only [dslope, continuousAt_update_same, \u2190 hasDerivAt_deriv_iff, hasDerivAt_iff_tendsto_slope]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : ContinuousWithinAt (dslope f a) s b\n\u22a2 ContinuousWithinAt f s b\n[PROOFSTEP]\nhave : ContinuousWithinAt (fun x => (x - a) \u2022 dslope f a x + f a) s b :=\n ((continuousWithinAt_id.sub continuousWithinAt_const).smul h).add continuousWithinAt_const\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : ContinuousWithinAt (dslope f a) s b\nthis : ContinuousWithinAt (fun x => (x - a) \u2022 dslope f a x + f a) s b\n\u22a2 ContinuousWithinAt f s b\n[PROOFSTEP]\nsimpa only [sub_smul_dslope, sub_add_cancel] using this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : b \u2260 a\n\u22a2 ContinuousWithinAt (dslope f a) s b \u2194 ContinuousWithinAt f s b\n[PROOFSTEP]\nrefine' \u27e8ContinuousWithinAt.of_dslope, fun hc => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : b \u2260 a\nhc : ContinuousWithinAt f s b\n\u22a2 ContinuousWithinAt (dslope f a) s b\n[PROOFSTEP]\nsimp only [dslope, continuousWithinAt_update_of_ne h]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : b \u2260 a\nhc : ContinuousWithinAt f s b\n\u22a2 ContinuousWithinAt (slope f a) s b\n[PROOFSTEP]\nexact\n ((continuousWithinAt_id.sub continuousWithinAt_const).inv\u2080 (sub_ne_zero.2 h)).smul (hc.sub continuousWithinAt_const)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : b \u2260 a\n\u22a2 ContinuousAt (dslope f a) b \u2194 ContinuousAt f b\n[PROOFSTEP]\nsimp only [\u2190 continuousWithinAt_univ, continuousWithinAt_dslope_of_ne h]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : s \u2208 \ud835\udcdd a\n\u22a2 ContinuousOn (dslope f a) s \u2194 ContinuousOn f s \u2227 DifferentiableAt \ud835\udd5c f a\n[PROOFSTEP]\nrefine' \u27e8fun hc => \u27e8hc.of_dslope, continuousAt_dslope_same.1 <| hc.continuousAt h\u27e9, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : s \u2208 \ud835\udcdd a\n\u22a2 ContinuousOn f s \u2227 DifferentiableAt \ud835\udd5c f a \u2192 ContinuousOn (dslope f a) s\n[PROOFSTEP]\nrintro \u27e8hc, hd\u27e9 x hx\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : s \u2208 \ud835\udcdd a\nhc : ContinuousOn f s\nhd : DifferentiableAt \ud835\udd5c f a\nx : \ud835\udd5c\nhx : x \u2208 s\n\u22a2 ContinuousWithinAt (dslope f a) s x\n[PROOFSTEP]\nrcases eq_or_ne x a with (rfl | hne)\n[GOAL]\ncase intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\nb : \ud835\udd5c\ns : Set \ud835\udd5c\nhc : ContinuousOn f s\nx : \ud835\udd5c\nhx : x \u2208 s\nh : s \u2208 \ud835\udcdd x\nhd : DifferentiableAt \ud835\udd5c f x\n\u22a2 ContinuousWithinAt (dslope f x) s x\ncase intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : s \u2208 \ud835\udcdd a\nhc : ContinuousOn f s\nhd : DifferentiableAt \ud835\udd5c f a\nx : \ud835\udd5c\nhx : x \u2208 s\nhne : x \u2260 a\n\u22a2 ContinuousWithinAt (dslope f a) s x\n[PROOFSTEP]\nexacts [(continuousAt_dslope_same.2 hd).continuousWithinAt, (continuousWithinAt_dslope_of_ne hne).2 (hc x hx)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : DifferentiableWithinAt \ud835\udd5c (dslope f a) s b\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s b\n[PROOFSTEP]\nsimpa only [id, sub_smul_dslope f a, sub_add_cancel] using\n ((differentiableWithinAt_id.sub_const a).smul h).add_const (f a)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : b \u2260 a\n\u22a2 DifferentiableWithinAt \ud835\udd5c (dslope f a) s b \u2194 DifferentiableWithinAt \ud835\udd5c f s b\n[PROOFSTEP]\nrefine' \u27e8DifferentiableWithinAt.of_dslope, fun hd => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : b \u2260 a\nhd : DifferentiableWithinAt \ud835\udd5c f s b\n\u22a2 DifferentiableWithinAt \ud835\udd5c (dslope f a) s b\n[PROOFSTEP]\nrefine'\n (((differentiableWithinAt_id.sub_const a).inv (sub_ne_zero.2 h)).smul (hd.sub_const (f a))).congr_of_eventuallyEq _\n (dslope_of_ne _ h)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : b \u2260 a\nhd : DifferentiableWithinAt \ud835\udd5c f s b\n\u22a2 dslope f a =\u1da0[\ud835\udcdd[s] b] fun y => (id y - a)\u207b\u00b9 \u2022 (f y - f a)\n[PROOFSTEP]\nrefine' (eqOn_dslope_slope _ _).eventuallyEq_of_mem _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : b \u2260 a\nhd : DifferentiableWithinAt \ud835\udd5c f s b\n\u22a2 {a}\u1d9c \u2208 \ud835\udcdd[s] b\n[PROOFSTEP]\nexact mem_nhdsWithin_of_mem_nhds (isOpen_ne.mem_nhds h)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\na b : \ud835\udd5c\ns : Set \ud835\udd5c\nh : b \u2260 a\n\u22a2 DifferentiableAt \ud835\udd5c (dslope f a) b \u2194 DifferentiableAt \ud835\udd5c f b\n[PROOFSTEP]\nsimp only [\u2190 differentiableWithinAt_univ, differentiableWithinAt_dslope_of_ne h]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.Dslope", "llama_tokens": 4419, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754607093178, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4286968362774222}} {"text": "[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u03b2\n\u22a2 \u2200 (x : \u03b1 \u2295 \u03b2), x \u2208 disjSum univ univ\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u03b2\nval\u271d : \u03b1\n\u22a2 Sum.inl val\u271d \u2208 disjSum univ univ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u03b2\nval\u271d : \u03b2\n\u22a2 Sum.inr val\u271d \u2208 disjSum univ univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : \u03b1\nh : Fintype { b // b \u2260 a }\n\u22a2 Function.Bijective (Sum.elim Subtype.val Subtype.val)\n[PROOFSTEP]\nclassical exact (Equiv.sumCompl (\u00b7 = a)).bijective\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : \u03b1\nh : Fintype { b // b \u2260 a }\n\u22a2 Function.Bijective (Sum.elim Subtype.val Subtype.val)\n[PROOFSTEP]\nexact (Equiv.sumCompl (\u00b7 = a)).bijective\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\n\u22a2 image (fun i => b \u2191i) univ = erase (image b univ) k\n[PROOFSTEP]\napply subset_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\n\u22a2 image (fun i => b \u2191i) univ \u2286 erase (image b univ) k\n[PROOFSTEP]\nrw [image_subset_iff]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (x : { a // b a \u2260 k }), x \u2208 univ \u2192 b \u2191x \u2208 erase (image b univ) k\n[PROOFSTEP]\nintro i _\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\ni : { a // b a \u2260 k }\na\u271d : i \u2208 univ\n\u22a2 b \u2191i \u2208 erase (image b univ) k\n[PROOFSTEP]\napply mem_erase_of_ne_of_mem i.2 (mem_image_of_mem _ (mem_univ _))\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\n\u22a2 erase (image b univ) k \u2286 image (fun i => b \u2191i) univ\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\ni : \u03b2\nhi : i \u2208 erase (image b univ) k\n\u22a2 i \u2208 image (fun i => b \u2191i) univ\n[PROOFSTEP]\nrw [mem_image]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\ni : \u03b2\nhi : i \u2208 erase (image b univ) k\n\u22a2 \u2203 a, a \u2208 univ \u2227 b \u2191a = i\n[PROOFSTEP]\nrcases mem_image.1 (erase_subset _ _ hi) with \u27e8a, _, ha\u27e9\n[GOAL]\ncase a.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\ni : \u03b2\nhi : i \u2208 erase (image b univ) k\na : \u03b1\nleft\u271d : a \u2208 univ\nha : b a = i\n\u22a2 \u2203 a, a \u2208 univ \u2227 b \u2191a = i\n[PROOFSTEP]\nsubst ha\n[GOAL]\ncase a.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\na : \u03b1\nleft\u271d : a \u2208 univ\nhi : b a \u2208 erase (image b univ) k\n\u22a2 \u2203 a_1, a_1 \u2208 univ \u2227 b \u2191a_1 = b a\n[PROOFSTEP]\nexact \u27e8\u27e8a, ne_of_mem_erase hi\u27e9, mem_univ _, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\n\u22a2 image (fun i => b \u2191i) univ \u2282 image b univ\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\n\u22a2 image (fun i => b \u2191i) univ \u2286 image b univ\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\nx : \u03b2\nhx : x \u2208 image (fun i => b \u2191i) univ\n\u22a2 x \u2208 image b univ\n[PROOFSTEP]\nrcases mem_image.1 hx with \u27e8y, _, hy\u27e9\n[GOAL]\ncase left.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\nx : \u03b2\nhx : x \u2208 image (fun i => b \u2191i) univ\ny : { a // p (b a) }\nleft\u271d : y \u2208 univ\nhy : b \u2191y = x\n\u22a2 x \u2208 image b univ\n[PROOFSTEP]\nexact hy \u25b8 mem_image_of_mem b (mem_univ (y : \u03b1))\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\n\u22a2 \u00acimage b univ \u2286 image (fun i => b \u2191i) univ\n[PROOFSTEP]\nintro h\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\nh : image b univ \u2286 image (fun i => b \u2191i) univ\n\u22a2 False\n[PROOFSTEP]\nrw [mem_image] at hk \n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : \u2203 a, a \u2208 univ \u2227 b a = k\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\nh : image b univ \u2286 image (fun i => b \u2191i) univ\n\u22a2 False\n[PROOFSTEP]\nrcases hk with \u27e8k', _, hk'\u27e9\n[GOAL]\ncase right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\nh : image b univ \u2286 image (fun i => b \u2191i) univ\nk' : \u03b1\nleft\u271d : k' \u2208 univ\nhk' : b k' = k\n\u22a2 False\n[PROOFSTEP]\nsubst hk'\n[GOAL]\ncase right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nb : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nh : image b univ \u2286 image (fun i => b \u2191i) univ\nk' : \u03b1\nleft\u271d : k' \u2208 univ\nhp : \u00acp (b k')\n\u22a2 False\n[PROOFSTEP]\nhave := h (mem_image_of_mem b (mem_univ k'))\n[GOAL]\ncase right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nb : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nh : image b univ \u2286 image (fun i => b \u2191i) univ\nk' : \u03b1\nleft\u271d : k' \u2208 univ\nhp : \u00acp (b k')\nthis : b k' \u2208 image (fun i => b \u2191i) univ\n\u22a2 False\n[PROOFSTEP]\nrw [mem_image] at this \n[GOAL]\ncase right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nb : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nh : image b univ \u2286 image (fun i => b \u2191i) univ\nk' : \u03b1\nleft\u271d : k' \u2208 univ\nhp : \u00acp (b k')\nthis : \u2203 a, a \u2208 univ \u2227 b \u2191a = b k'\n\u22a2 False\n[PROOFSTEP]\nrcases this with \u27e8j, _, hj'\u27e9\n[GOAL]\ncase right.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nb : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nh : image b univ \u2286 image (fun i => b \u2191i) univ\nk' : \u03b1\nleft\u271d\u00b9 : k' \u2208 univ\nhp : \u00acp (b k')\nj : { a // p (b a) }\nleft\u271d : j \u2208 univ\nhj' : b \u2191j = b k'\n\u22a2 False\n[PROOFSTEP]\nexact hp (hj' \u25b8 j.2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nhfst : image f s \u2286 t\nhfs : Set.InjOn f \u2191s\n\u22a2 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\n[PROOFSTEP]\nclassical\ninduction' s using Finset.induction with a s has H generalizing f\n\u00b7 obtain \u27e8e\u27e9 : Nonempty (\u03b1 \u2243 \u21a5t) := by rwa [\u2190 Fintype.card_eq, Fintype.card_coe]\n use e\n simp\nhave hfst' : Finset.image f s \u2286 t := (Finset.image_mono _ (s.subset_insert a)).trans hfst\nhave hfs' : Set.InjOn f s := hfs.mono (s.subset_insert a)\nobtain \u27e8g', hg'\u27e9 := H hfst' hfs'\nhave hfat : f a \u2208 t := hfst (mem_image_of_mem _ (s.mem_insert_self a))\nuse g'.trans (Equiv.swap (\u27e8f a, hfat\u27e9 : t) (g' a))\nsimp_rw [mem_insert]\nrintro i (rfl | hi)\n\u00b7 simp\nrw [Equiv.trans_apply, Equiv.swap_apply_of_ne_of_ne, hg' _ hi]\n\u00b7\n exact\n ne_of_apply_ne Subtype.val\n (ne_of_eq_of_ne (hg' _ hi) <| hfs.ne (subset_insert _ _ hi) (mem_insert_self _ _) <| ne_of_mem_of_not_mem hi has)\n\u00b7 exact g'.injective.ne (ne_of_mem_of_not_mem hi has)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nhfst : image f s \u2286 t\nhfs : Set.InjOn f \u2191s\n\u22a2 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\n[PROOFSTEP]\ninduction' s using Finset.induction with a s has H generalizing f\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nhfst\u271d : image f\u271d s \u2286 t\nhfs\u271d : Set.InjOn f\u271d \u2191s\nf : \u03b1 \u2192 \u03b2\nhfst : image f \u2205 \u2286 t\nhfs : Set.InjOn f \u2191\u2205\n\u22a2 \u2203 g, \u2200 (i : \u03b1), i \u2208 \u2205 \u2192 \u2191(\u2191g i) = f i\n[PROOFSTEP]\nobtain \u27e8e\u27e9 : Nonempty (\u03b1 \u2243 \u21a5t) := by rwa [\u2190 Fintype.card_eq, Fintype.card_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nhfst\u271d : image f\u271d s \u2286 t\nhfs\u271d : Set.InjOn f\u271d \u2191s\nf : \u03b1 \u2192 \u03b2\nhfst : image f \u2205 \u2286 t\nhfs : Set.InjOn f \u2191\u2205\n\u22a2 Nonempty (\u03b1 \u2243 { x // x \u2208 t })\n[PROOFSTEP]\nrwa [\u2190 Fintype.card_eq, Fintype.card_coe]\n[GOAL]\ncase empty.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nhfst\u271d : image f\u271d s \u2286 t\nhfs\u271d : Set.InjOn f\u271d \u2191s\nf : \u03b1 \u2192 \u03b2\nhfst : image f \u2205 \u2286 t\nhfs : Set.InjOn f \u2191\u2205\ne : \u03b1 \u2243 { x // x \u2208 t }\n\u22a2 \u2203 g, \u2200 (i : \u03b1), i \u2208 \u2205 \u2192 \u2191(\u2191g i) = f i\n[PROOFSTEP]\nuse e\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nhfst\u271d : image f\u271d s \u2286 t\nhfs\u271d : Set.InjOn f\u271d \u2191s\nf : \u03b1 \u2192 \u03b2\nhfst : image f \u2205 \u2286 t\nhfs : Set.InjOn f \u2191\u2205\ne : \u03b1 \u2243 { x // x \u2208 t }\n\u22a2 \u2200 (i : \u03b1), i \u2208 \u2205 \u2192 \u2191(\u2191e i) = f i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nhfst\u271d : image f\u271d s\u271d \u2286 t\nhfs\u271d : Set.InjOn f\u271d \u2191s\u271d\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nH : \u2200 {f : \u03b1 \u2192 \u03b2}, image f s \u2286 t \u2192 Set.InjOn f \u2191s \u2192 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\nf : \u03b1 \u2192 \u03b2\nhfst : image f (insert a s) \u2286 t\nhfs : Set.InjOn f \u2191(insert a s)\n\u22a2 \u2203 g, \u2200 (i : \u03b1), i \u2208 insert a s \u2192 \u2191(\u2191g i) = f i\n[PROOFSTEP]\nhave hfst' : Finset.image f s \u2286 t := (Finset.image_mono _ (s.subset_insert a)).trans hfst\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nhfst\u271d : image f\u271d s\u271d \u2286 t\nhfs\u271d : Set.InjOn f\u271d \u2191s\u271d\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nH : \u2200 {f : \u03b1 \u2192 \u03b2}, image f s \u2286 t \u2192 Set.InjOn f \u2191s \u2192 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\nf : \u03b1 \u2192 \u03b2\nhfst : image f (insert a s) \u2286 t\nhfs : Set.InjOn f \u2191(insert a s)\nhfst' : image f s \u2286 t\n\u22a2 \u2203 g, \u2200 (i : \u03b1), i \u2208 insert a s \u2192 \u2191(\u2191g i) = f i\n[PROOFSTEP]\nhave hfs' : Set.InjOn f s := hfs.mono (s.subset_insert a)\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nhfst\u271d : image f\u271d s\u271d \u2286 t\nhfs\u271d : Set.InjOn f\u271d \u2191s\u271d\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nH : \u2200 {f : \u03b1 \u2192 \u03b2}, image f s \u2286 t \u2192 Set.InjOn f \u2191s \u2192 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\nf : \u03b1 \u2192 \u03b2\nhfst : image f (insert a s) \u2286 t\nhfs : Set.InjOn f \u2191(insert a s)\nhfst' : image f s \u2286 t\nhfs' : Set.InjOn f \u2191s\n\u22a2 \u2203 g, \u2200 (i : \u03b1), i \u2208 insert a s \u2192 \u2191(\u2191g i) = f i\n[PROOFSTEP]\nobtain \u27e8g', hg'\u27e9 := H hfst' hfs'\n[GOAL]\ncase insert.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nhfst\u271d : image f\u271d s\u271d \u2286 t\nhfs\u271d : Set.InjOn f\u271d \u2191s\u271d\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nH : \u2200 {f : \u03b1 \u2192 \u03b2}, image f s \u2286 t \u2192 Set.InjOn f \u2191s \u2192 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\nf : \u03b1 \u2192 \u03b2\nhfst : image f (insert a s) \u2286 t\nhfs : Set.InjOn f \u2191(insert a s)\nhfst' : image f s \u2286 t\nhfs' : Set.InjOn f \u2191s\ng' : \u03b1 \u2243 { x // x \u2208 t }\nhg' : \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g' i) = f i\n\u22a2 \u2203 g, \u2200 (i : \u03b1), i \u2208 insert a s \u2192 \u2191(\u2191g i) = f i\n[PROOFSTEP]\nhave hfat : f a \u2208 t := hfst (mem_image_of_mem _ (s.mem_insert_self a))\n[GOAL]\ncase insert.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nhfst\u271d : image f\u271d s\u271d \u2286 t\nhfs\u271d : Set.InjOn f\u271d \u2191s\u271d\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nH : \u2200 {f : \u03b1 \u2192 \u03b2}, image f s \u2286 t \u2192 Set.InjOn f \u2191s \u2192 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\nf : \u03b1 \u2192 \u03b2\nhfst : image f (insert a s) \u2286 t\nhfs : Set.InjOn f \u2191(insert a s)\nhfst' : image f s \u2286 t\nhfs' : Set.InjOn f \u2191s\ng' : \u03b1 \u2243 { x // x \u2208 t }\nhg' : \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g' i) = f i\nhfat : f a \u2208 t\n\u22a2 \u2203 g, \u2200 (i : \u03b1), i \u2208 insert a s \u2192 \u2191(\u2191g i) = f i\n[PROOFSTEP]\nuse g'.trans (Equiv.swap (\u27e8f a, hfat\u27e9 : t) (g' a))\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nhfst\u271d : image f\u271d s\u271d \u2286 t\nhfs\u271d : Set.InjOn f\u271d \u2191s\u271d\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nH : \u2200 {f : \u03b1 \u2192 \u03b2}, image f s \u2286 t \u2192 Set.InjOn f \u2191s \u2192 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\nf : \u03b1 \u2192 \u03b2\nhfst : image f (insert a s) \u2286 t\nhfs : Set.InjOn f \u2191(insert a s)\nhfst' : image f s \u2286 t\nhfs' : Set.InjOn f \u2191s\ng' : \u03b1 \u2243 { x // x \u2208 t }\nhg' : \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g' i) = f i\nhfat : f a \u2208 t\n\u22a2 \u2200 (i : \u03b1), i \u2208 insert a s \u2192 \u2191(\u2191(g'.trans (Equiv.swap { val := f a, property := hfat } (\u2191g' a))) i) = f i\n[PROOFSTEP]\nsimp_rw [mem_insert]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nhfst\u271d : image f\u271d s\u271d \u2286 t\nhfs\u271d : Set.InjOn f\u271d \u2191s\u271d\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nH : \u2200 {f : \u03b1 \u2192 \u03b2}, image f s \u2286 t \u2192 Set.InjOn f \u2191s \u2192 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\nf : \u03b1 \u2192 \u03b2\nhfst : image f (insert a s) \u2286 t\nhfs : Set.InjOn f \u2191(insert a s)\nhfst' : image f s \u2286 t\nhfs' : Set.InjOn f \u2191s\ng' : \u03b1 \u2243 { x // x \u2208 t }\nhg' : \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g' i) = f i\nhfat : f a \u2208 t\n\u22a2 \u2200 (i : \u03b1), i = a \u2228 i \u2208 s \u2192 \u2191(\u2191(g'.trans (Equiv.swap { val := f a, property := hfat } (\u2191g' a))) i) = f i\n[PROOFSTEP]\nrintro i (rfl | hi)\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nhfst\u271d : image f\u271d s\u271d \u2286 t\nhfs\u271d : Set.InjOn f\u271d \u2191s\u271d\ns : Finset \u03b1\nH : \u2200 {f : \u03b1 \u2192 \u03b2}, image f s \u2286 t \u2192 Set.InjOn f \u2191s \u2192 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\nf : \u03b1 \u2192 \u03b2\nhfst' : image f s \u2286 t\nhfs' : Set.InjOn f \u2191s\ng' : \u03b1 \u2243 { x // x \u2208 t }\nhg' : \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g' i) = f i\ni : \u03b1\nhas : \u00aci \u2208 s\nhfst : image f (insert i s) \u2286 t\nhfs : Set.InjOn f \u2191(insert i s)\nhfat : f i \u2208 t\n\u22a2 \u2191(\u2191(g'.trans (Equiv.swap { val := f i, property := hfat } (\u2191g' i))) i) = f i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nhfst\u271d : image f\u271d s\u271d \u2286 t\nhfs\u271d : Set.InjOn f\u271d \u2191s\u271d\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nH : \u2200 {f : \u03b1 \u2192 \u03b2}, image f s \u2286 t \u2192 Set.InjOn f \u2191s \u2192 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\nf : \u03b1 \u2192 \u03b2\nhfst : image f (insert a s) \u2286 t\nhfs : Set.InjOn f \u2191(insert a s)\nhfst' : image f s \u2286 t\nhfs' : Set.InjOn f \u2191s\ng' : \u03b1 \u2243 { x // x \u2208 t }\nhg' : \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g' i) = f i\nhfat : f a \u2208 t\ni : \u03b1\nhi : i \u2208 s\n\u22a2 \u2191(\u2191(g'.trans (Equiv.swap { val := f a, property := hfat } (\u2191g' a))) i) = f i\n[PROOFSTEP]\nrw [Equiv.trans_apply, Equiv.swap_apply_of_ne_of_ne, hg' _ hi]\n[GOAL]\ncase h.inr.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nhfst\u271d : image f\u271d s\u271d \u2286 t\nhfs\u271d : Set.InjOn f\u271d \u2191s\u271d\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nH : \u2200 {f : \u03b1 \u2192 \u03b2}, image f s \u2286 t \u2192 Set.InjOn f \u2191s \u2192 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\nf : \u03b1 \u2192 \u03b2\nhfst : image f (insert a s) \u2286 t\nhfs : Set.InjOn f \u2191(insert a s)\nhfst' : image f s \u2286 t\nhfs' : Set.InjOn f \u2191s\ng' : \u03b1 \u2243 { x // x \u2208 t }\nhg' : \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g' i) = f i\nhfat : f a \u2208 t\ni : \u03b1\nhi : i \u2208 s\n\u22a2 \u2191g' i \u2260 { val := f a, property := hfat }\n[PROOFSTEP]\nexact\n ne_of_apply_ne Subtype.val\n (ne_of_eq_of_ne (hg' _ hi) <| hfs.ne (subset_insert _ _ hi) (mem_insert_self _ _) <| ne_of_mem_of_not_mem hi has)\n[GOAL]\ncase h.inr.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nhfst\u271d : image f\u271d s\u271d \u2286 t\nhfs\u271d : Set.InjOn f\u271d \u2191s\u271d\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nH : \u2200 {f : \u03b1 \u2192 \u03b2}, image f s \u2286 t \u2192 Set.InjOn f \u2191s \u2192 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\nf : \u03b1 \u2192 \u03b2\nhfst : image f (insert a s) \u2286 t\nhfs : Set.InjOn f \u2191(insert a s)\nhfst' : image f s \u2286 t\nhfs' : Set.InjOn f \u2191s\ng' : \u03b1 \u2243 { x // x \u2208 t }\nhg' : \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g' i) = f i\nhfat : f a \u2208 t\ni : \u03b1\nhi : i \u2208 s\n\u22a2 \u2191g' i \u2260 \u2191g' a\n[PROOFSTEP]\nexact g'.injective.ne (ne_of_mem_of_not_mem hi has)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b1\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhfst : MapsTo f s \u2191t\nhfs : InjOn f s\n\u22a2 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\n[PROOFSTEP]\nclassical\nlet s' : Finset \u03b1 := s.toFinset\nhave hfst' : s'.image f \u2286 t := by simpa [\u2190 Finset.coe_subset] using hfst\nhave hfs' : Set.InjOn f s' := by simpa using hfs\nobtain \u27e8g, hg\u27e9 := Finset.exists_equiv_extend_of_card_eq h\u03b1t hfst' hfs'\nrefine' \u27e8g, fun i hi => _\u27e9\napply hg\nsimpa using hi\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b1\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhfst : MapsTo f s \u2191t\nhfs : InjOn f s\n\u22a2 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\n[PROOFSTEP]\nlet s' : Finset \u03b1 := s.toFinset\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b1\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhfst : MapsTo f s \u2191t\nhfs : InjOn f s\ns' : Finset \u03b1 := toFinset s\n\u22a2 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\n[PROOFSTEP]\nhave hfst' : s'.image f \u2286 t := by simpa [\u2190 Finset.coe_subset] using hfst\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b1\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhfst : MapsTo f s \u2191t\nhfs : InjOn f s\ns' : Finset \u03b1 := toFinset s\n\u22a2 Finset.image f s' \u2286 t\n[PROOFSTEP]\nsimpa [\u2190 Finset.coe_subset] using hfst\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b1\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhfst : MapsTo f s \u2191t\nhfs : InjOn f s\ns' : Finset \u03b1 := toFinset s\nhfst' : Finset.image f s' \u2286 t\n\u22a2 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\n[PROOFSTEP]\nhave hfs' : Set.InjOn f s' := by simpa using hfs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b1\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhfst : MapsTo f s \u2191t\nhfs : InjOn f s\ns' : Finset \u03b1 := toFinset s\nhfst' : Finset.image f s' \u2286 t\n\u22a2 InjOn f \u2191s'\n[PROOFSTEP]\nsimpa using hfs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b1\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhfst : MapsTo f s \u2191t\nhfs : InjOn f s\ns' : Finset \u03b1 := toFinset s\nhfst' : Finset.image f s' \u2286 t\nhfs' : InjOn f \u2191s'\n\u22a2 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\n[PROOFSTEP]\nobtain \u27e8g, hg\u27e9 := Finset.exists_equiv_extend_of_card_eq h\u03b1t hfst' hfs'\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b1\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhfst : MapsTo f s \u2191t\nhfs : InjOn f s\ns' : Finset \u03b1 := toFinset s\nhfst' : Finset.image f s' \u2286 t\nhfs' : InjOn f \u2191s'\ng : \u03b1 \u2243 { x // x \u2208 t }\nhg : \u2200 (i : \u03b1), i \u2208 s' \u2192 \u2191(\u2191g i) = f i\n\u22a2 \u2203 g, \u2200 (i : \u03b1), i \u2208 s \u2192 \u2191(\u2191g i) = f i\n[PROOFSTEP]\nrefine' \u27e8g, fun i hi => _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b1\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhfst : MapsTo f s \u2191t\nhfs : InjOn f s\ns' : Finset \u03b1 := toFinset s\nhfst' : Finset.image f s' \u2286 t\nhfs' : InjOn f \u2191s'\ng : \u03b1 \u2243 { x // x \u2208 t }\nhg : \u2200 (i : \u03b1), i \u2208 s' \u2192 \u2191(\u2191g i) = f i\ni : \u03b1\nhi : i \u2208 s\n\u22a2 \u2191(\u2191g i) = f i\n[PROOFSTEP]\napply hg\n[GOAL]\ncase intro.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b1\nt : Finset \u03b2\nh\u03b1t : Fintype.card \u03b1 = card t\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhfst : MapsTo f s \u2191t\nhfs : InjOn f s\ns' : Finset \u03b1 := toFinset s\nhfst' : Finset.image f s' \u2286 t\nhfs' : InjOn f \u2191s'\ng : \u03b1 \u2243 { x // x \u2208 t }\nhg : \u2200 (i : \u03b1), i \u2208 s' \u2192 \u2191(\u2191g i) = f i\ni : \u03b1\nhi : i \u2208 s\n\u22a2 i \u2208 s'\n[PROOFSTEP]\nsimpa using hi\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : Fintype { x // p x }\ninst\u271d\u00b9 : Fintype { x // q x }\ninst\u271d : Fintype { x // p x \u2228 q x }\n\u22a2 card { x // p x \u2228 q x } \u2264 card { x // p x } + card { x // q x }\n[PROOFSTEP]\nclassical\nconvert Fintype.card_le_of_embedding (subtypeOrLeftEmbedding p q)\nrw [Fintype.card_sum]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : Fintype { x // p x }\ninst\u271d\u00b9 : Fintype { x // q x }\ninst\u271d : Fintype { x // p x \u2228 q x }\n\u22a2 card { x // p x \u2228 q x } \u2264 card { x // p x } + card { x // q x }\n[PROOFSTEP]\nconvert Fintype.card_le_of_embedding (subtypeOrLeftEmbedding p q)\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : Fintype { x // p x }\ninst\u271d\u00b9 : Fintype { x // q x }\ninst\u271d : Fintype { x // p x \u2228 q x }\n\u22a2 card { x // p x } + card { x // q x } = card ({ x // p x } \u2295 { x // q x })\n[PROOFSTEP]\nrw [Fintype.card_sum]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\np q : \u03b1 \u2192 Prop\nh : Disjoint p q\ninst\u271d\u00b2 : Fintype { x // p x }\ninst\u271d\u00b9 : Fintype { x // q x }\ninst\u271d : Fintype { x // p x \u2228 q x }\n\u22a2 card { x // p x \u2228 q x } = card { x // p x } + card { x // q x }\n[PROOFSTEP]\nclassical\nconvert Fintype.card_congr (subtypeOrEquiv p q h)\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\np q : \u03b1 \u2192 Prop\nh : Disjoint p q\ninst\u271d\u00b2 : Fintype { x // p x }\ninst\u271d\u00b9 : Fintype { x // q x }\ninst\u271d : Fintype { x // p x \u2228 q x }\n\u22a2 card { x // p x \u2228 q x } = card { x // p x } + card { x // q x }\n[PROOFSTEP]\nconvert Fintype.card_congr (subtypeOrEquiv p q h)\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\np q : \u03b1 \u2192 Prop\nh : Disjoint p q\ninst\u271d\u00b2 : Fintype { x // p x }\ninst\u271d\u00b9 : Fintype { x // q x }\ninst\u271d : Fintype { x // p x \u2228 q x }\n\u22a2 card { x // p x } + card { x // q x } = card ({ x // p x } \u2295 { x // q x })\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u22a2 Infinite (\u03b1 \u2295 \u03b2) \u2194 Infinite \u03b1 \u2228 Infinite \u03b2\n[PROOFSTEP]\nrefine' \u27e8fun H => _, fun H => H.elim (@Sum.infinite_of_left \u03b1 \u03b2) (@Sum.infinite_of_right \u03b1 \u03b2)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nH : Infinite (\u03b1 \u2295 \u03b2)\n\u22a2 Infinite \u03b1 \u2228 Infinite \u03b2\n[PROOFSTEP]\ncontrapose! H\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nH : \u00acInfinite \u03b1 \u2227 \u00acInfinite \u03b2\n\u22a2 \u00acInfinite (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\nhaveI := fintypeOfNotInfinite H.1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nH : \u00acInfinite \u03b1 \u2227 \u00acInfinite \u03b2\nthis : Fintype \u03b1\n\u22a2 \u00acInfinite (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\nhaveI := fintypeOfNotInfinite H.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nH : \u00acInfinite \u03b1 \u2227 \u00acInfinite \u03b2\nthis\u271d : Fintype \u03b1\nthis : Fintype \u03b2\n\u22a2 \u00acInfinite (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\nexact Infinite.false\n", "meta": {"mathlib_filename": "Mathlib.Data.Fintype.Sum", "llama_tokens": 11911, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410572017153, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.4286741966149066}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\n\u22a2 affineSpan k (range id) = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\nf g : AffineBasis \u03b9 k P\nh : f.toFun = g.toFun\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ng : AffineBasis \u03b9 k P\ntoFun\u271d : \u03b9 \u2192 P\nind'\u271d : AffineIndependent k toFun\u271d\ntot'\u271d : affineSpan k (range toFun\u271d) = \u22a4\nh : { toFun := toFun\u271d, ind' := ind'\u271d, tot' := tot'\u271d }.toFun = g.toFun\n\u22a2 { toFun := toFun\u271d, ind' := ind'\u271d, tot' := tot'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ntoFun\u271d\u00b9 : \u03b9 \u2192 P\nind'\u271d\u00b9 : AffineIndependent k toFun\u271d\u00b9\ntot'\u271d\u00b9 : affineSpan k (range toFun\u271d\u00b9) = \u22a4\ntoFun\u271d : \u03b9 \u2192 P\nind'\u271d : AffineIndependent k toFun\u271d\ntot'\u271d : affineSpan k (range toFun\u271d) = \u22a4\nh : { toFun := toFun\u271d\u00b9, ind' := ind'\u271d\u00b9, tot' := tot'\u271d\u00b9 }.toFun = { toFun := toFun\u271d, ind' := ind'\u271d, tot' := tot'\u271d }.toFun\n\u22a2 { toFun := toFun\u271d\u00b9, ind' := ind'\u271d\u00b9, tot' := tot'\u271d\u00b9 } = { toFun := toFun\u271d, ind' := ind'\u271d, tot' := tot'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\nh\u03b9 : IsEmpty \u03b9\n\u22a2 False\n[PROOFSTEP]\nsimpa only [@range_eq_empty _ _ h\u03b9, AffineSubspace.span_empty, bot_ne_top] using b.tot\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne\u271d e : \u03b9 \u2243 \u03b9'\n\u22a2 affineSpan k (range (\u2191b \u2218 \u2191e.symm)) = \u22a4\n[PROOFSTEP]\nrw [e.symm.surjective.range_comp]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne\u271d e : \u03b9 \u2243 \u03b9'\n\u22a2 affineSpan k (range \u2191b) = \u22a4\n[PROOFSTEP]\nexact b.3\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\n\u22a2 \u22a4 \u2264 Submodule.span k (range fun i_1 => \u2191b \u2191i_1 -\u1d65 \u2191b i)\n[PROOFSTEP]\nsuffices Submodule.span k (range fun j : { x // x \u2260 i } => b \u2191j -\u1d65 b i) = vectorSpan k (range b) by\n rw [this, \u2190 direction_affineSpan, b.tot, AffineSubspace.direction_top]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\nthis : Submodule.span k (range fun j => \u2191b \u2191j -\u1d65 \u2191b i) = vectorSpan k (range \u2191b)\n\u22a2 \u22a4 \u2264 Submodule.span k (range fun i_1 => \u2191b \u2191i_1 -\u1d65 \u2191b i)\n[PROOFSTEP]\nrw [this, \u2190 direction_affineSpan, b.tot, AffineSubspace.direction_top]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\n\u22a2 Submodule.span k (range fun j => \u2191b \u2191j -\u1d65 \u2191b i) = vectorSpan k (range \u2191b)\n[PROOFSTEP]\nconv_rhs => rw [\u2190 image_univ]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\n| vectorSpan k (range \u2191b)\n[PROOFSTEP]\nrw [\u2190 image_univ]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\n| vectorSpan k (range \u2191b)\n[PROOFSTEP]\nrw [\u2190 image_univ]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\n| vectorSpan k (range \u2191b)\n[PROOFSTEP]\nrw [\u2190 image_univ]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\n\u22a2 Submodule.span k (range fun j => \u2191b \u2191j -\u1d65 \u2191b i) = vectorSpan k (\u2191b '' univ)\n[PROOFSTEP]\nrw [vectorSpan_image_eq_span_vsub_set_right_ne k b (mem_univ i)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\n\u22a2 Submodule.span k (range fun j => \u2191b \u2191j -\u1d65 \u2191b i) = Submodule.span k ((fun x => x -\u1d65 \u2191b i) '' (\u2191b '' (univ \\ {i})))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\n\u22a2 (range fun j => \u2191b \u2191j -\u1d65 \u2191b i) = (fun x => x -\u1d65 \u2191b i) '' (\u2191b '' (univ \\ {i}))\n[PROOFSTEP]\next v\n[GOAL]\ncase e_s.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\nv : V\n\u22a2 (v \u2208 range fun j => \u2191b \u2191j -\u1d65 \u2191b i) \u2194 v \u2208 (fun x => x -\u1d65 \u2191b i) '' (\u2191b '' (univ \\ {i}))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\nj : { j // j \u2260 i }\n\u22a2 \u2191(basisOf b i) j = \u2191b \u2191j -\u1d65 \u2191b i\n[PROOFSTEP]\nsimp [basisOf]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9'\n\u22a2 basisOf (reindex b e) i =\n Basis.reindex (basisOf b (\u2191e.symm i)) (Equiv.subtypeEquiv e (_ : \u2200 (x : \u03b9), \u00acx = \u2191e.symm i \u2194 \u00ac\u2191e x = i))\n[PROOFSTEP]\next j\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9'\nj : { j // j \u2260 i }\n\u22a2 \u2191(basisOf (reindex b e) i) j =\n \u2191(Basis.reindex (basisOf b (\u2191e.symm i)) (Equiv.subtypeEquiv e (_ : \u2200 (x : \u03b9), \u00acx = \u2191e.symm i \u2194 \u00ac\u2191e x = i))) j\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\nq : P\nv : V\n\u22a2 (fun q => 1 - \u2191(Basis.sumCoords (basisOf b i)) (q -\u1d65 \u2191b i)) (v +\u1d65 q) =\n \u2191(-Basis.sumCoords (basisOf b i)) v +\u1d65 (fun q => 1 - \u2191(Basis.sumCoords (basisOf b i)) (q -\u1d65 \u2191b i)) q\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\nq : P\nv : V\n\u22a2 1 - \u2191(Basis.sumCoords (basisOf b i)) (v +\u1d65 q -\u1d65 \u2191b i) =\n \u2191(-Basis.sumCoords (basisOf b i)) v +\u1d65 (1 - \u2191(Basis.sumCoords (basisOf b i)) (q -\u1d65 \u2191b i))\n[PROOFSTEP]\nrw [vadd_vsub_assoc, LinearMap.map_add, vadd_eq_add, LinearMap.neg_apply, sub_add_eq_sub_sub_swap, add_comm,\n sub_eq_add_neg]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9'\n\u22a2 coord (reindex b e) i = coord b (\u2191e.symm i)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9'\np\u271d : P\n\u22a2 \u2191(coord (reindex b e) i) p\u271d = \u2191(coord b (\u2191e.symm i)) p\u271d\n[PROOFSTEP]\nclassical simp [AffineBasis.coord]\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9'\np\u271d : P\n\u22a2 \u2191(coord (reindex b e) i) p\u271d = \u2191(coord b (\u2191e.symm i)) p\u271d\n[PROOFSTEP]\nsimp [AffineBasis.coord]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\n\u22a2 \u2191(coord b i) (\u2191b i) = 1\n[PROOFSTEP]\nsimp only [coord, Basis.coe_sumCoords, LinearEquiv.map_zero, LinearEquiv.coe_coe, sub_zero, AffineMap.coe_mk,\n Finsupp.sum_zero_index, vsub_self]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\nh : i \u2260 j\n\u22a2 \u2191(coord b i) (\u2191b j) = 0\n[PROOFSTEP]\nrw [coord, AffineMap.coe_mk, \u2190 @Subtype.coe_mk _ (fun j => j \u2260 i) j h.symm, \u2190 b.basisOf_apply,\n Basis.sumCoords_self_apply, sub_self]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : DecidableEq \u03b9\ni j : \u03b9\n\u22a2 \u2191(coord b i) (\u2191b j) = if i = j then 1 else 0\n[PROOFSTEP]\ncases' eq_or_ne i j with h h\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : DecidableEq \u03b9\ni j : \u03b9\nh : i = j\n\u22a2 \u2191(coord b i) (\u2191b j) = if i = j then 1 else 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : DecidableEq \u03b9\ni j : \u03b9\nh : i \u2260 j\n\u22a2 \u2191(coord b i) (\u2191b j) = if i = j then 1 else 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\nhi : i \u2208 s\nw : \u03b9 \u2192 k\nhw : Finset.sum s w = 1\n\u22a2 \u2191(coord b i) (\u2191(Finset.affineCombination k s \u2191b) w) = w i\n[PROOFSTEP]\nclassical simp only [coord_apply, hi, Finset.affineCombination_eq_linear_combination, if_true, mul_boole, hw,\n Function.comp_apply, smul_eq_mul, s.sum_ite_eq, s.map_affineCombination b w hw]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\nhi : i \u2208 s\nw : \u03b9 \u2192 k\nhw : Finset.sum s w = 1\n\u22a2 \u2191(coord b i) (\u2191(Finset.affineCombination k s \u2191b) w) = w i\n[PROOFSTEP]\nsimp only [coord_apply, hi, Finset.affineCombination_eq_linear_combination, if_true, mul_boole, hw, Function.comp_apply,\n smul_eq_mul, s.sum_ite_eq, s.map_affineCombination b w hw]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\nhi : \u00aci \u2208 s\nw : \u03b9 \u2192 k\nhw : Finset.sum s w = 1\n\u22a2 \u2191(coord b i) (\u2191(Finset.affineCombination k s \u2191b) w) = 0\n[PROOFSTEP]\nclassical simp only [coord_apply, hi, Finset.affineCombination_eq_linear_combination, if_false, mul_boole, hw,\n Function.comp_apply, smul_eq_mul, s.sum_ite_eq, s.map_affineCombination b w hw]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\nhi : \u00aci \u2208 s\nw : \u03b9 \u2192 k\nhw : Finset.sum s w = 1\n\u22a2 \u2191(coord b i) (\u2191(Finset.affineCombination k s \u2191b) w) = 0\n[PROOFSTEP]\nsimp only [coord_apply, hi, Finset.affineCombination_eq_linear_combination, if_false, mul_boole, hw,\n Function.comp_apply, smul_eq_mul, s.sum_ite_eq, s.map_affineCombination b w hw]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fintype \u03b9\nq : P\n\u22a2 \u2211 i : \u03b9, \u2191(coord b i) q = 1\n[PROOFSTEP]\nhave hq : q \u2208 affineSpan k (range b) := by\n rw [b.tot]\n exact AffineSubspace.mem_top k V q\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fintype \u03b9\nq : P\n\u22a2 q \u2208 affineSpan k (range \u2191b)\n[PROOFSTEP]\nrw [b.tot]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fintype \u03b9\nq : P\n\u22a2 q \u2208 \u22a4\n[PROOFSTEP]\nexact AffineSubspace.mem_top k V q\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fintype \u03b9\nq : P\nhq : q \u2208 affineSpan k (range \u2191b)\n\u22a2 \u2211 i : \u03b9, \u2191(coord b i) q = 1\n[PROOFSTEP]\nobtain \u27e8w, hw, rfl\u27e9 := eq_affineCombination_of_mem_affineSpan_of_fintype hq\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fintype \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i : \u03b9, w i = 1\nhq : \u2191(Finset.affineCombination k Finset.univ \u2191b) w \u2208 affineSpan k (range \u2191b)\n\u22a2 \u2211 i : \u03b9, \u2191(coord b i) (\u2191(Finset.affineCombination k Finset.univ \u2191b) w) = 1\n[PROOFSTEP]\nconvert hw\n[GOAL]\ncase h.e'_2.a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fintype \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i : \u03b9, w i = 1\nhq : \u2191(Finset.affineCombination k Finset.univ \u2191b) w \u2208 affineSpan k (range \u2191b)\nx\u271d : \u03b9\na\u271d : x\u271d \u2208 Finset.univ\n\u22a2 \u2191(coord b x\u271d) (\u2191(Finset.affineCombination k Finset.univ \u2191b) w) = w x\u271d\n[PROOFSTEP]\nexact b.coord_apply_combination_of_mem (Finset.mem_univ _) hw\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fintype \u03b9\nq : P\n\u22a2 (\u2191(Finset.affineCombination k Finset.univ \u2191b) fun i => \u2191(coord b i) q) = q\n[PROOFSTEP]\nhave hq : q \u2208 affineSpan k (range b) := by\n rw [b.tot]\n exact AffineSubspace.mem_top k V q\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fintype \u03b9\nq : P\n\u22a2 q \u2208 affineSpan k (range \u2191b)\n[PROOFSTEP]\nrw [b.tot]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fintype \u03b9\nq : P\n\u22a2 q \u2208 \u22a4\n[PROOFSTEP]\nexact AffineSubspace.mem_top k V q\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fintype \u03b9\nq : P\nhq : q \u2208 affineSpan k (range \u2191b)\n\u22a2 (\u2191(Finset.affineCombination k Finset.univ \u2191b) fun i => \u2191(coord b i) q) = q\n[PROOFSTEP]\nobtain \u27e8w, hw, rfl\u27e9 := eq_affineCombination_of_mem_affineSpan_of_fintype hq\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fintype \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i : \u03b9, w i = 1\nhq : \u2191(Finset.affineCombination k Finset.univ \u2191b) w \u2208 affineSpan k (range \u2191b)\n\u22a2 (\u2191(Finset.affineCombination k Finset.univ \u2191b) fun i =>\n \u2191(coord b i) (\u2191(Finset.affineCombination k Finset.univ \u2191b) w)) =\n \u2191(Finset.affineCombination k Finset.univ \u2191b) w\n[PROOFSTEP]\ncongr\n[GOAL]\ncase intro.intro.h.e_6.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fintype \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i : \u03b9, w i = 1\nhq : \u2191(Finset.affineCombination k Finset.univ \u2191b) w \u2208 affineSpan k (range \u2191b)\n\u22a2 (fun i => \u2191(coord b i) (\u2191(Finset.affineCombination k Finset.univ \u2191b) w)) = w\n[PROOFSTEP]\next i\n[GOAL]\ncase intro.intro.h.e_6.h.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fintype \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i : \u03b9, w i = 1\nhq : \u2191(Finset.affineCombination k Finset.univ \u2191b) w \u2208 affineSpan k (range \u2191b)\ni : \u03b9\n\u22a2 \u2191(coord b i) (\u2191(Finset.affineCombination k Finset.univ \u2191b) w) = w i\n[PROOFSTEP]\nexact b.coord_apply_combination_of_mem (Finset.mem_univ i) hw\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb\u271d : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fintype \u03b9\nb : AffineBasis \u03b9 k V\nv : V\n\u22a2 \u2211 i : \u03b9, \u2191(coord b i) v \u2022 \u2191b i = v\n[PROOFSTEP]\nhave hb := b.affineCombination_coord_eq_self v\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb\u271d : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Fintype \u03b9\nb : AffineBasis \u03b9 k V\nv : V\nhb : (\u2191(Finset.affineCombination k Finset.univ \u2191b) fun i => \u2191(coord b i) v) = v\n\u22a2 \u2211 i : \u03b9, \u2191(coord b i) v \u2022 \u2191b i = v\n[PROOFSTEP]\nrwa [Finset.univ.affineCombination_eq_linear_combination _ _ (b.sum_coord_apply_eq_one v)] at hb \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Finite \u03b9\nq\u2081 q\u2082 : P\nh : \u2200 (i : \u03b9), \u2191(coord b i) q\u2081 = \u2191(coord b i) q\u2082\n\u22a2 q\u2081 = q\u2082\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Finite \u03b9\nq\u2081 q\u2082 : P\nh : \u2200 (i : \u03b9), \u2191(coord b i) q\u2081 = \u2191(coord b i) q\u2082\nval\u271d : Fintype \u03b9\n\u22a2 q\u2081 = q\u2082\n[PROOFSTEP]\nrw [\u2190 b.affineCombination_coord_eq_self q\u2081, \u2190 b.affineCombination_coord_eq_self q\u2082]\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Finite \u03b9\nq\u2081 q\u2082 : P\nh : \u2200 (i : \u03b9), \u2191(coord b i) q\u2081 = \u2191(coord b i) q\u2082\nval\u271d : Fintype \u03b9\n\u22a2 (\u2191(Finset.affineCombination k Finset.univ \u2191b) fun i => \u2191(coord b i) q\u2081) =\n \u2191(Finset.affineCombination k Finset.univ \u2191b) fun i => \u2191(coord b i) q\u2082\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Subsingleton \u03b9\ni : \u03b9\n\u22a2 \u2191(coord b i) = 1\n[PROOFSTEP]\next q\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Subsingleton \u03b9\ni : \u03b9\nq : P\n\u22a2 \u2191(coord b i) q = OfNat.ofNat 1 q\n[PROOFSTEP]\nhave hp : (range b).Subsingleton := by\n rw [\u2190 image_univ]\n apply Subsingleton.image\n apply subsingleton_of_subsingleton\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Subsingleton \u03b9\ni : \u03b9\nq : P\n\u22a2 Set.Subsingleton (range \u2191b)\n[PROOFSTEP]\nrw [\u2190 image_univ]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Subsingleton \u03b9\ni : \u03b9\nq : P\n\u22a2 Set.Subsingleton (\u2191b '' univ)\n[PROOFSTEP]\napply Subsingleton.image\n[GOAL]\ncase hs\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Subsingleton \u03b9\ni : \u03b9\nq : P\n\u22a2 Set.Subsingleton univ\n[PROOFSTEP]\napply subsingleton_of_subsingleton\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Subsingleton \u03b9\ni : \u03b9\nq : P\nhp : Set.Subsingleton (range \u2191b)\n\u22a2 \u2191(coord b i) q = OfNat.ofNat 1 q\n[PROOFSTEP]\nhaveI := AffineSubspace.subsingleton_of_subsingleton_span_eq_top hp b.tot\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Subsingleton \u03b9\ni : \u03b9\nq : P\nhp : Set.Subsingleton (range \u2191b)\nthis : Subsingleton P\n\u22a2 \u2191(coord b i) q = OfNat.ofNat 1 q\n[PROOFSTEP]\nlet s : Finset \u03b9 := { i }\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Subsingleton \u03b9\ni : \u03b9\nq : P\nhp : Set.Subsingleton (range \u2191b)\nthis : Subsingleton P\ns : Finset \u03b9 := {i}\n\u22a2 \u2191(coord b i) q = OfNat.ofNat 1 q\n[PROOFSTEP]\nhave hi : i \u2208 s := by simp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Subsingleton \u03b9\ni : \u03b9\nq : P\nhp : Set.Subsingleton (range \u2191b)\nthis : Subsingleton P\ns : Finset \u03b9 := {i}\n\u22a2 i \u2208 s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Subsingleton \u03b9\ni : \u03b9\nq : P\nhp : Set.Subsingleton (range \u2191b)\nthis : Subsingleton P\ns : Finset \u03b9 := {i}\nhi : i \u2208 s\n\u22a2 \u2191(coord b i) q = OfNat.ofNat 1 q\n[PROOFSTEP]\nhave hw : s.sum (Function.const \u03b9 (1 : k)) = 1 := by simp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Subsingleton \u03b9\ni : \u03b9\nq : P\nhp : Set.Subsingleton (range \u2191b)\nthis : Subsingleton P\ns : Finset \u03b9 := {i}\nhi : i \u2208 s\n\u22a2 Finset.sum s (Function.const \u03b9 1) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Subsingleton \u03b9\ni : \u03b9\nq : P\nhp : Set.Subsingleton (range \u2191b)\nthis : Subsingleton P\ns : Finset \u03b9 := {i}\nhi : i \u2208 s\nhw : Finset.sum s (Function.const \u03b9 1) = 1\n\u22a2 \u2191(coord b i) q = OfNat.ofNat 1 q\n[PROOFSTEP]\nhave hq : q = s.affineCombination k b (Function.const \u03b9 (1 : k)) := by simp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Subsingleton \u03b9\ni : \u03b9\nq : P\nhp : Set.Subsingleton (range \u2191b)\nthis : Subsingleton P\ns : Finset \u03b9 := {i}\nhi : i \u2208 s\nhw : Finset.sum s (Function.const \u03b9 1) = 1\n\u22a2 q = \u2191(Finset.affineCombination k s \u2191b) (Function.const \u03b9 1)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Subsingleton \u03b9\ni : \u03b9\nq : P\nhp : Set.Subsingleton (range \u2191b)\nthis : Subsingleton P\ns : Finset \u03b9 := {i}\nhi : i \u2208 s\nhw : Finset.sum s (Function.const \u03b9 1) = 1\nhq : q = \u2191(Finset.affineCombination k s \u2191b) (Function.const \u03b9 1)\n\u22a2 \u2191(coord b i) q = OfNat.ofNat 1 q\n[PROOFSTEP]\nrw [Pi.one_apply, hq, b.coord_apply_combination_of_mem hi hw, Function.const_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Nontrivial \u03b9\ni : \u03b9\n\u22a2 Function.Surjective \u2191(coord b i)\n[PROOFSTEP]\nclassical\nintro x\nobtain \u27e8j, hij\u27e9 := exists_ne i\nlet s : Finset \u03b9 := { i, j }\nhave hi : i \u2208 s := by simp\nhave _ : j \u2208 s := by simp\nlet w : \u03b9 \u2192 k := fun j' => if j' = i then x else 1 - x\nhave hw : s.sum w = 1 := by\n -- Porting note: previously this subgoal worked just by:\n -- simp [hij, Finset.sum_ite, Finset.filter_insert, Finset.filter_eq']\n -- I'm not sure why `simp` can not successfully use `Finset.filter_eq'`.\n simp [Finset.sum_ite, Finset.filter_insert, hij]\n erw [Finset.filter_eq']\n simp [hij.symm]\nuse s.affineCombination k b w\nsimp [b.coord_apply_combination_of_mem hi hw]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Nontrivial \u03b9\ni : \u03b9\n\u22a2 Function.Surjective \u2191(coord b i)\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Nontrivial \u03b9\ni : \u03b9\nx : k\n\u22a2 \u2203 a, \u2191(coord b i) a = x\n[PROOFSTEP]\nobtain \u27e8j, hij\u27e9 := exists_ne i\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Nontrivial \u03b9\ni : \u03b9\nx : k\nj : \u03b9\nhij : j \u2260 i\n\u22a2 \u2203 a, \u2191(coord b i) a = x\n[PROOFSTEP]\nlet s : Finset \u03b9 := { i, j }\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Nontrivial \u03b9\ni : \u03b9\nx : k\nj : \u03b9\nhij : j \u2260 i\ns : Finset \u03b9 := {i, j}\n\u22a2 \u2203 a, \u2191(coord b i) a = x\n[PROOFSTEP]\nhave hi : i \u2208 s := by simp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Nontrivial \u03b9\ni : \u03b9\nx : k\nj : \u03b9\nhij : j \u2260 i\ns : Finset \u03b9 := {i, j}\n\u22a2 i \u2208 s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Nontrivial \u03b9\ni : \u03b9\nx : k\nj : \u03b9\nhij : j \u2260 i\ns : Finset \u03b9 := {i, j}\nhi : i \u2208 s\n\u22a2 \u2203 a, \u2191(coord b i) a = x\n[PROOFSTEP]\nhave _ : j \u2208 s := by simp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Nontrivial \u03b9\ni : \u03b9\nx : k\nj : \u03b9\nhij : j \u2260 i\ns : Finset \u03b9 := {i, j}\nhi : i \u2208 s\n\u22a2 j \u2208 s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Nontrivial \u03b9\ni : \u03b9\nx : k\nj : \u03b9\nhij : j \u2260 i\ns : Finset \u03b9 := {i, j}\nhi : i \u2208 s\nx\u271d : j \u2208 s\n\u22a2 \u2203 a, \u2191(coord b i) a = x\n[PROOFSTEP]\nlet w : \u03b9 \u2192 k := fun j' => if j' = i then x else 1 - x\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Nontrivial \u03b9\ni : \u03b9\nx : k\nj : \u03b9\nhij : j \u2260 i\ns : Finset \u03b9 := {i, j}\nhi : i \u2208 s\nx\u271d : j \u2208 s\nw : \u03b9 \u2192 k := fun j' => if j' = i then x else 1 - x\n\u22a2 \u2203 a, \u2191(coord b i) a = x\n[PROOFSTEP]\nhave hw : s.sum w = 1 := by\n -- Porting note: previously this subgoal worked just by:\n -- simp [hij, Finset.sum_ite, Finset.filter_insert, Finset.filter_eq']\n -- I'm not sure why `simp` can not successfully use `Finset.filter_eq'`.\n simp [Finset.sum_ite, Finset.filter_insert, hij]\n erw [Finset.filter_eq']\n simp [hij.symm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Nontrivial \u03b9\ni : \u03b9\nx : k\nj : \u03b9\nhij : j \u2260 i\ns : Finset \u03b9 := {i, j}\nhi : i \u2208 s\nx\u271d : j \u2208 s\nw : \u03b9 \u2192 k := fun j' => if j' = i then x else 1 - x\n\u22a2 Finset.sum s w = 1\n[PROOFSTEP]\nsimp [Finset.sum_ite, Finset.filter_insert, hij]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Nontrivial \u03b9\ni : \u03b9\nx : k\nj : \u03b9\nhij : j \u2260 i\ns : Finset \u03b9 := {i, j}\nhi : i \u2208 s\nx\u271d : j \u2208 s\nw : \u03b9 \u2192 k := fun j' => if j' = i then x else 1 - x\n\u22a2 \u2191(Finset.card (insert i (Finset.filter (fun x => x = i) {j}))) * x + (1 - x) = 1\n[PROOFSTEP]\nerw [Finset.filter_eq']\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Nontrivial \u03b9\ni : \u03b9\nx : k\nj : \u03b9\nhij : j \u2260 i\ns : Finset \u03b9 := {i, j}\nhi : i \u2208 s\nx\u271d : j \u2208 s\nw : \u03b9 \u2192 k := fun j' => if j' = i then x else 1 - x\n\u22a2 \u2191(Finset.card (insert i (if i \u2208 {j} then {i} else \u2205))) * x + (1 - x) = 1\n[PROOFSTEP]\nsimp [hij.symm]\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Nontrivial \u03b9\ni : \u03b9\nx : k\nj : \u03b9\nhij : j \u2260 i\ns : Finset \u03b9 := {i, j}\nhi : i \u2208 s\nx\u271d : j \u2208 s\nw : \u03b9 \u2192 k := fun j' => if j' = i then x else 1 - x\nhw : Finset.sum s w = 1\n\u22a2 \u2203 a, \u2191(coord b i) a = x\n[PROOFSTEP]\nuse s.affineCombination k b w\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ns\u271d : Finset \u03b9\ni\u271d j\u271d : \u03b9\ne : \u03b9 \u2243 \u03b9'\ninst\u271d : Nontrivial \u03b9\ni : \u03b9\nx : k\nj : \u03b9\nhij : j \u2260 i\ns : Finset \u03b9 := {i, j}\nhi : i \u2208 s\nx\u271d : j \u2208 s\nw : \u03b9 \u2192 k := fun j' => if j' = i then x else 1 - x\nhw : Finset.sum s w = 1\n\u22a2 \u2191(coord b i) (\u2191(Finset.affineCombination k s \u2191b) w) = x\n[PROOFSTEP]\nsimp [b.coord_apply_combination_of_mem hi hw]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\nv w : V\n\u22a2 (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) (v + w) =\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) v + (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) w\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\nv w : V\ni : \u03b9\n\u22a2 (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) (v + w) i =\n ((fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) v + (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) w) i\n[PROOFSTEP]\nsimp only [LinearMap.map_add, Pi.add_apply, neg_add]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\nt : k\nv : V\n\u22a2 AddHom.toFun\n { toFun := fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v,\n map_add' :=\n (_ :\n \u2200 (v w : V),\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) (v + w) =\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) v +\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) w) }\n (t \u2022 v) =\n \u2191(RingHom.id k) t \u2022\n AddHom.toFun\n { toFun := fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v,\n map_add' :=\n (_ :\n \u2200 (v w : V),\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) (v + w) =\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) v +\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) w) }\n v\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\nt : k\nv : V\ni : \u03b9\n\u22a2 AddHom.toFun\n { toFun := fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v,\n map_add' :=\n (_ :\n \u2200 (v w : V),\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) (v + w) =\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) v +\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) w) }\n (t \u2022 v) i =\n (\u2191(RingHom.id k) t \u2022\n AddHom.toFun\n { toFun := fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v,\n map_add' :=\n (_ :\n \u2200 (v w : V),\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) (v + w) =\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) v +\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) w) }\n v)\n i\n[PROOFSTEP]\nsimp only [LinearMap.map_smul, Pi.smul_apply, smul_neg, RingHom.id_apply, mul_neg]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\np : P\nv : V\n\u22a2 (fun q i => \u2191(coord b i) q) (v +\u1d65 p) =\n \u2191{\n toAddHom :=\n { toFun := fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v,\n map_add' :=\n (_ :\n \u2200 (v w : V),\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) (v + w) =\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) v +\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) w) },\n map_smul' :=\n (_ :\n \u2200 (t : k) (v : V),\n AddHom.toFun\n { toFun := fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v,\n map_add' :=\n (_ :\n \u2200 (v w : V),\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) (v + w) =\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) v +\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) w) }\n (t \u2022 v) =\n \u2191(RingHom.id k) t \u2022\n AddHom.toFun\n { toFun := fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v,\n map_add' :=\n (_ :\n \u2200 (v w : V),\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) (v + w) =\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) v +\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) w) }\n v) }\n v +\u1d65\n (fun q i => \u2191(coord b i) q) p\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\np : P\nv : V\ni : \u03b9\n\u22a2 (fun q i => \u2191(coord b i) q) (v +\u1d65 p) i =\n (\u2191{\n toAddHom :=\n { toFun := fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v,\n map_add' :=\n (_ :\n \u2200 (v w : V),\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) (v + w) =\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) v +\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) w) },\n map_smul' :=\n (_ :\n \u2200 (t : k) (v : V),\n AddHom.toFun\n { toFun := fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v,\n map_add' :=\n (_ :\n \u2200 (v w : V),\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) (v + w) =\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) v +\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) w) }\n (t \u2022 v) =\n \u2191(RingHom.id k) t \u2022\n AddHom.toFun\n { toFun := fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v,\n map_add' :=\n (_ :\n \u2200 (v w : V),\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) (v + w) =\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) v +\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) w) }\n v) }\n v +\u1d65\n (fun q i => \u2191(coord b i) q) p)\n i\n[PROOFSTEP]\nrw [LinearMap.coe_mk, Pi.vadd_apply']\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\np : P\nv : V\ni : \u03b9\n\u22a2 (fun q i => \u2191(coord b i) q) (v +\u1d65 p) i =\n \u2191{ toFun := fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v,\n map_add' :=\n (_ :\n \u2200 (v w : V),\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) (v + w) =\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) v +\n (fun v i => -\u2191(Basis.sumCoords (basisOf b i)) v) w) }\n v i +\u1d65\n (fun q i => \u2191(coord b i) q) p i\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\np : P\nv : V\ni : \u03b9\n\u22a2 \u2191(coord b i) (v +\u1d65 p) = (-Finsupp.sum (\u2191(basisOf b i).repr v) fun x => id) + \u2191(coord b i) p\n[PROOFSTEP]\nrw [AffineMap.map_vadd, linear_eq_sumCoords, LinearMap.neg_apply]\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni\u271d j : \u03b9\ne : \u03b9 \u2243 \u03b9'\np : P\nv : V\ni : \u03b9\n\u22a2 -\u2191(Basis.sumCoords (basisOf b i)) v +\u1d65 \u2191(coord b i) p =\n (-Finsupp.sum (\u2191(basisOf b i).repr v) fun x => id) + \u2191(coord b i) p\n[PROOFSTEP]\nsimp only [ne_eq, Basis.coe_sumCoords, vadd_eq_add]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : DivisionRing k\ninst\u271d\u00b9 : Module k V\ninst\u271d : CharZero k\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191(coord b i) (Finset.centroid k s \u2191b) = (\u2191(Finset.card s))\u207b\u00b9\n[PROOFSTEP]\nrw [Finset.centroid, b.coord_apply_combination_of_mem hi (s.sum_centroidWeights_eq_one_of_nonempty _ \u27e8i, hi\u27e9),\n Finset.centroidWeights, Function.const_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : DivisionRing k\ninst\u271d : Module k V\nt : Set P\nht : affineSpan k t = \u22a4\n\u22a2 \u2203 s x b, \u2191b = Subtype.val\n[PROOFSTEP]\nobtain \u27e8s, hst, h_tot, h_ind\u27e9 := exists_affineIndependent k V t\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : DivisionRing k\ninst\u271d : Module k V\nt : Set P\nht : affineSpan k t = \u22a4\ns : Set P\nhst : s \u2286 t\nh_tot : affineSpan k s = affineSpan k t\nh_ind : AffineIndependent k Subtype.val\n\u22a2 \u2203 s x b, \u2191b = Subtype.val\n[PROOFSTEP]\nrefine' \u27e8s, hst, \u27e8(\u2191), h_ind, _\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : DivisionRing k\ninst\u271d : Module k V\nt : Set P\nht : affineSpan k t = \u22a4\ns : Set P\nhst : s \u2286 t\nh_tot : affineSpan k s = affineSpan k t\nh_ind : AffineIndependent k Subtype.val\n\u22a2 affineSpan k (range Subtype.val) = \u22a4\n[PROOFSTEP]\nrw [Subtype.range_coe, h_tot, ht]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.AffineSpace.Basis", "llama_tokens": 22063, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943925708562, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.4285935144491532}} {"text": "[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\nH\u271d H : LieSubalgebra R L\ninst\u271d : IsCartanSubalgebra H\n\u22a2 LieSubmodule.normalizer (toLieSubmodule H) = toLieSubmodule H\n[PROOFSTEP]\nrw [\u2190 LieSubmodule.coe_toSubmodule_eq_iff, coe_normalizer_eq_normalizer, IsCartanSubalgebra.self_normalizing,\n coe_toLieSubmodule]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\nH\u271d H : LieSubalgebra R L\ninst\u271d : IsCartanSubalgebra H\nk : \u2115\n\u22a2 LieSubmodule.ucs k (toLieSubmodule H) = toLieSubmodule H\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\nR : Type u\nL : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\nH\u271d H : LieSubalgebra R L\ninst\u271d : IsCartanSubalgebra H\n\u22a2 LieSubmodule.ucs Nat.zero (toLieSubmodule H) = toLieSubmodule H\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u\nL : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\nH\u271d H : LieSubalgebra R L\ninst\u271d : IsCartanSubalgebra H\nk : \u2115\nih : LieSubmodule.ucs k (toLieSubmodule H) = toLieSubmodule H\n\u22a2 LieSubmodule.ucs (Nat.succ k) (toLieSubmodule H) = toLieSubmodule H\n[PROOFSTEP]\nsimp [ih]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\n\u22a2 IsCartanSubalgebra H \u2194 LieSubmodule.IsUcsLimit (toLieSubmodule H)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\n\u22a2 IsCartanSubalgebra H \u2192 LieSubmodule.IsUcsLimit (toLieSubmodule H)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\nh : IsCartanSubalgebra H\n\u22a2 LieSubmodule.IsUcsLimit (toLieSubmodule H)\n[PROOFSTEP]\nhave h\u2081 : LieAlgebra.IsNilpotent R H := by infer_instance\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\nh : IsCartanSubalgebra H\n\u22a2 LieAlgebra.IsNilpotent R { x // x \u2208 H }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase mp\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\nh : IsCartanSubalgebra H\nh\u2081 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\n\u22a2 LieSubmodule.IsUcsLimit (toLieSubmodule H)\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h\u2081\n[GOAL]\ncase mp.intro\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\nh : IsCartanSubalgebra H\nh\u2081 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nk : \u2115\nhk : toLieSubmodule H \u2264 LieSubmodule.ucs k \u22a5\n\u22a2 LieSubmodule.IsUcsLimit (toLieSubmodule H)\n[PROOFSTEP]\nreplace hk : H.toLieSubmodule = LieSubmodule.ucs k \u22a5 :=\n le_antisymm hk (LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_self_of_isCartanSubalgebra k)\n[GOAL]\ncase mp.intro\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\nh : IsCartanSubalgebra H\nh\u2081 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nk : \u2115\nhk : toLieSubmodule H = LieSubmodule.ucs k \u22a5\n\u22a2 LieSubmodule.IsUcsLimit (toLieSubmodule H)\n[PROOFSTEP]\nrefine' \u27e8k, fun l hl => _\u27e9\n[GOAL]\ncase mp.intro\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\nh : IsCartanSubalgebra H\nh\u2081 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nk : \u2115\nhk : toLieSubmodule H = LieSubmodule.ucs k \u22a5\nl : \u2115\nhl : k \u2264 l\n\u22a2 LieSubmodule.ucs l \u22a5 = toLieSubmodule H\n[PROOFSTEP]\nrw [\u2190 Nat.sub_add_cancel hl, LieSubmodule.ucs_add, \u2190 hk, LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra]\n[GOAL]\ncase mpr\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\n\u22a2 LieSubmodule.IsUcsLimit (toLieSubmodule H) \u2192 IsCartanSubalgebra H\n[PROOFSTEP]\nrintro \u27e8k, hk\u27e9\n[GOAL]\ncase mpr.intro\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\nk : \u2115\nhk : \u2200 (l : \u2115), k \u2264 l \u2192 LieSubmodule.ucs l \u22a5 = toLieSubmodule H\n\u22a2 IsCartanSubalgebra H\n[PROOFSTEP]\nexact\n { nilpotent := by\n dsimp only [LieAlgebra.IsNilpotent]\n erw [H.toLieSubmodule.isNilpotent_iff_exists_lcs_eq_bot]\n use k\n rw [_root_.eq_bot_iff, LieSubmodule.lcs_le_iff, hk k (le_refl k)]\n self_normalizing := by\n have hk' := hk (k + 1) k.le_succ\n rw [LieSubmodule.ucs_succ, hk k (le_refl k)] at hk' \n rw [\u2190 LieSubalgebra.coe_to_submodule_eq_iff, \u2190 LieSubalgebra.coe_normalizer_eq_normalizer, hk',\n LieSubalgebra.coe_toLieSubmodule] }\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\nk : \u2115\nhk : \u2200 (l : \u2115), k \u2264 l \u2192 LieSubmodule.ucs l \u22a5 = toLieSubmodule H\n\u22a2 LieAlgebra.IsNilpotent R { x // x \u2208 H }\n[PROOFSTEP]\ndsimp only [LieAlgebra.IsNilpotent]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\nk : \u2115\nhk : \u2200 (l : \u2115), k \u2264 l \u2192 LieSubmodule.ucs l \u22a5 = toLieSubmodule H\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 H } { x // x \u2208 H }\n[PROOFSTEP]\nerw [H.toLieSubmodule.isNilpotent_iff_exists_lcs_eq_bot]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\nk : \u2115\nhk : \u2200 (l : \u2115), k \u2264 l \u2192 LieSubmodule.ucs l \u22a5 = toLieSubmodule H\n\u22a2 \u2203 k, LieSubmodule.lcs k (toLieSubmodule H) = \u22a5\n[PROOFSTEP]\nuse k\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\nk : \u2115\nhk : \u2200 (l : \u2115), k \u2264 l \u2192 LieSubmodule.ucs l \u22a5 = toLieSubmodule H\n\u22a2 LieSubmodule.lcs k (toLieSubmodule H) = \u22a5\n[PROOFSTEP]\nrw [_root_.eq_bot_iff, LieSubmodule.lcs_le_iff, hk k (le_refl k)]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\nk : \u2115\nhk : \u2200 (l : \u2115), k \u2264 l \u2192 LieSubmodule.ucs l \u22a5 = toLieSubmodule H\n\u22a2 normalizer H = H\n[PROOFSTEP]\nhave hk' := hk (k + 1) k.le_succ\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\nk : \u2115\nhk : \u2200 (l : \u2115), k \u2264 l \u2192 LieSubmodule.ucs l \u22a5 = toLieSubmodule H\nhk' : LieSubmodule.ucs (k + 1) \u22a5 = toLieSubmodule H\n\u22a2 normalizer H = H\n[PROOFSTEP]\nrw [LieSubmodule.ucs_succ, hk k (le_refl k)] at hk' \n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nH : LieSubalgebra R L\nk : \u2115\nhk : \u2200 (l : \u2115), k \u2264 l \u2192 LieSubmodule.ucs l \u22a5 = toLieSubmodule H\nhk' : LieSubmodule.normalizer (toLieSubmodule H) = toLieSubmodule H\n\u22a2 normalizer H = H\n[PROOFSTEP]\nrw [\u2190 LieSubalgebra.coe_to_submodule_eq_iff, \u2190 LieSubalgebra.coe_normalizer_eq_normalizer, hk',\n LieSubalgebra.coe_toLieSubmodule]\n[GOAL]\nR\u271d : Type u\nL\u271d : Type v\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : LieRing L\u271d\ninst\u271d\u00b3 : LieAlgebra R\u271d L\u271d\nH : LieSubalgebra R\u271d L\u271d\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nI : LieIdeal R L\n\u22a2 LieSubalgebra.normalizer (\u2191R L I) = \u22a4\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR\u271d : Type u\nL\u271d : Type v\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : LieRing L\u271d\ninst\u271d\u00b3 : LieAlgebra R\u271d L\u271d\nH : LieSubalgebra R\u271d L\u271d\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nI : LieIdeal R L\nx : L\n\u22a2 x \u2208 LieSubalgebra.normalizer (\u2191R L I) \u2194 x \u2208 \u22a4\n[PROOFSTEP]\nsimpa only [LieSubalgebra.mem_normalizer_iff, LieSubalgebra.mem_top, iff_true_iff] using fun y hy => I.lie_mem hy\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LieRing L\ninst\u271d\u00b9 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d : IsNilpotent R L\n\u22a2 LieSubalgebra.normalizer \u22a4 = \u22a4\n[PROOFSTEP]\nrw [\u2190 top_coe_lieSubalgebra, normalizer_eq_top, top_coe_lieSubalgebra]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Lie.CartanSubalgebra", "llama_tokens": 3620, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943925708561, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.42859351444915317}} {"text": "[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\n\u22a2 Monotone fun s => posTangentConeAt s a\n[PROOFSTEP]\nrintro s t hst y \u27e8c, d, hd, hc, hcd\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns t : Set E\nhst : s \u2264 t\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, a + d n \u2208 s\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\n\u22a2 y \u2208 (fun s => posTangentConeAt s a) t\n[PROOFSTEP]\nexact \u27e8c, d, mem_of_superset hd fun h hn => hst hn, hc, hcd\u27e9\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nx y : E\nh : segment \u211d x y \u2286 s\n\u22a2 y - x \u2208 posTangentConeAt s x\n[PROOFSTEP]\nlet c := fun n : \u2115 => (2 : \u211d) ^ n\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nx y : E\nh : segment \u211d x y \u2286 s\nc : \u2115 \u2192 \u211d := fun n => 2 ^ n\n\u22a2 y - x \u2208 posTangentConeAt s x\n[PROOFSTEP]\nlet d := fun n : \u2115 => (c n)\u207b\u00b9 \u2022 (y - x)\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nx y : E\nh : segment \u211d x y \u2286 s\nc : \u2115 \u2192 \u211d := fun n => 2 ^ n\nd : \u2115 \u2192 E := fun n => (c n)\u207b\u00b9 \u2022 (y - x)\n\u22a2 y - x \u2208 posTangentConeAt s x\n[PROOFSTEP]\nrefine' \u27e8c, d, Filter.univ_mem' fun n => h _, tendsto_pow_atTop_atTop_of_one_lt one_lt_two, _\u27e9\n[GOAL]\ncase refine'_1\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nx y : E\nh : segment \u211d x y \u2286 s\nc : \u2115 \u2192 \u211d := fun n => 2 ^ n\nd : \u2115 \u2192 E := fun n => (c n)\u207b\u00b9 \u2022 (y - x)\nn : \u2115\n\u22a2 x + d n \u2208 segment \u211d x y\ncase refine'_2\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nx y : E\nh : segment \u211d x y \u2286 s\nc : \u2115 \u2192 \u211d := fun n => 2 ^ n\nd : \u2115 \u2192 E := fun n => (c n)\u207b\u00b9 \u2022 (y - x)\n\u22a2 Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd (y - x))\n[PROOFSTEP]\nshow x + d n \u2208 segment \u211d x y\n[GOAL]\ncase refine'_1\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nx y : E\nh : segment \u211d x y \u2286 s\nc : \u2115 \u2192 \u211d := fun n => 2 ^ n\nd : \u2115 \u2192 E := fun n => (c n)\u207b\u00b9 \u2022 (y - x)\nn : \u2115\n\u22a2 x + d n \u2208 segment \u211d x y\n[PROOFSTEP]\nrw [segment_eq_image']\n[GOAL]\ncase refine'_1\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nx y : E\nh : segment \u211d x y \u2286 s\nc : \u2115 \u2192 \u211d := fun n => 2 ^ n\nd : \u2115 \u2192 E := fun n => (c n)\u207b\u00b9 \u2022 (y - x)\nn : \u2115\n\u22a2 x + d n \u2208 (fun \u03b8 => x + \u03b8 \u2022 (y - x)) '' Icc 0 1\n[PROOFSTEP]\nrefine' \u27e8(c n)\u207b\u00b9, \u27e8_, _\u27e9, rfl\u27e9\n[GOAL]\ncase refine'_1.refine'_1\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nx y : E\nh : segment \u211d x y \u2286 s\nc : \u2115 \u2192 \u211d := fun n => 2 ^ n\nd : \u2115 \u2192 E := fun n => (c n)\u207b\u00b9 \u2022 (y - x)\nn : \u2115\n\u22a2 0 \u2264 (c n)\u207b\u00b9\ncase refine'_1.refine'_2\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nx y : E\nh : segment \u211d x y \u2286 s\nc : \u2115 \u2192 \u211d := fun n => 2 ^ n\nd : \u2115 \u2192 E := fun n => (c n)\u207b\u00b9 \u2022 (y - x)\nn : \u2115\n\u22a2 (c n)\u207b\u00b9 \u2264 1\n[PROOFSTEP]\nexacts [inv_nonneg.2 (pow_nonneg zero_le_two _), inv_le_one (one_le_pow_of_one_le one_le_two _)]\n[GOAL]\ncase refine'_2\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nx y : E\nh : segment \u211d x y \u2286 s\nc : \u2115 \u2192 \u211d := fun n => 2 ^ n\nd : \u2115 \u2192 E := fun n => (c n)\u207b\u00b9 \u2022 (y - x)\n\u22a2 Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd (y - x))\n[PROOFSTEP]\nshow Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd (y - x))\n[GOAL]\ncase refine'_2\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nx y : E\nh : segment \u211d x y \u2286 s\nc : \u2115 \u2192 \u211d := fun n => 2 ^ n\nd : \u2115 \u2192 E := fun n => (c n)\u207b\u00b9 \u2022 (y - x)\n\u22a2 Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd (y - x))\n[PROOFSTEP]\nexact tendsto_const_nhds.congr fun n \u21a6 (smul_inv_smul\u2080 (pow_ne_zero _ two_ne_zero) _).symm\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nx y : E\nh : segment \u211d x (x + y) \u2286 s\n\u22a2 y \u2208 posTangentConeAt s x\n[PROOFSTEP]\nsimpa only [add_sub_cancel'] using mem_posTangentConeAt_of_segment_subset h\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nhy : y \u2208 posTangentConeAt s a\n\u22a2 \u2191f' y \u2264 0\n[PROOFSTEP]\nrcases hy with \u27e8c, d, hd, hc, hcd\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, a + d n \u2208 s\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\n\u22a2 \u2191f' y \u2264 0\n[PROOFSTEP]\nhave hc' : Tendsto (\u2016c \u00b7\u2016) atTop atTop := tendsto_abs_atTop_atTop.comp hc\n[GOAL]\ncase intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, a + d n \u2208 s\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\n\u22a2 \u2191f' y \u2264 0\n[PROOFSTEP]\nsuffices : \u2200\u1da0 n in atTop, c n \u2022 (f (a + d n) - f a) \u2264 0\n[GOAL]\ncase intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, a + d n \u2208 s\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\nthis : \u2200\u1da0 (n : \u2115) in atTop, c n \u2022 (f (a + d n) - f a) \u2264 0\n\u22a2 \u2191f' y \u2264 0\n[PROOFSTEP]\nexact le_of_tendsto (hf.lim atTop hd hc' hcd) this\n[GOAL]\ncase this\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, a + d n \u2208 s\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, c n \u2022 (f (a + d n) - f a) \u2264 0\n[PROOFSTEP]\nreplace hd : Tendsto (fun n => a + d n) atTop (\ud835\udcdd[s] (a + 0))\n[GOAL]\ncase hd\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, a + d n \u2208 s\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\n\u22a2 Tendsto (fun n => a + d n) atTop (\ud835\udcdd[s] (a + 0))\n[PROOFSTEP]\nexact tendsto_nhdsWithin_iff.2 \u27e8tendsto_const_nhds.add (tangentConeAt.lim_zero _ hc' hcd), hd\u27e9\n[GOAL]\ncase this\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\nhd : Tendsto (fun n => a + d n) atTop (\ud835\udcdd[s] (a + 0))\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, c n \u2022 (f (a + d n) - f a) \u2264 0\n[PROOFSTEP]\nrw [add_zero] at hd \n[GOAL]\ncase this\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\nhd : Tendsto (fun n => a + d n) atTop (\ud835\udcdd[s] a)\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, c n \u2022 (f (a + d n) - f a) \u2264 0\n[PROOFSTEP]\nfilter_upwards [hd.eventually h, hc.eventually_ge_atTop 0] with n hfn hcn\n[GOAL]\ncase h\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nc : \u2115 \u2192 \u211d\nd : \u2115 \u2192 E\nhc : Tendsto c atTop atTop\nhcd : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nhc' : Tendsto (fun x => \u2016c x\u2016) atTop atTop\nhd : Tendsto (fun n => a + d n) atTop (\ud835\udcdd[s] a)\nn : \u2115\nhfn : f (a + d n) \u2264 f a\nhcn : 0 \u2264 c n\n\u22a2 c n \u2022 (f (a + d n) - f a) \u2264 0\n[PROOFSTEP]\nexact mul_nonpos_of_nonneg_of_nonpos hcn (sub_nonpos.2 hfn)\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\ny : E\nhy : y \u2208 posTangentConeAt s a\nhf : \u00acDifferentiableWithinAt \u211d f s a\n\u22a2 \u2191(fderivWithin \u211d f s a) y \u2264 0\n[PROOFSTEP]\nrw [fderivWithin_zero_of_not_differentiableWithinAt hf]\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\ny : E\nhy : y \u2208 posTangentConeAt s a\nhf : \u00acDifferentiableWithinAt \u211d f s a\n\u22a2 \u21910 y \u2264 0\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nhy : y \u2208 posTangentConeAt s a\nhy' : -y \u2208 posTangentConeAt s a\n\u22a2 0 \u2264 \u2191f' y\n[PROOFSTEP]\nsimpa using h.hasFDerivWithinAt_nonpos hf hy'\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\ny : E\nhy : y \u2208 posTangentConeAt s a\nhy' : -y \u2208 posTangentConeAt s a\nhf : \u00acDifferentiableWithinAt \u211d f s a\n\u22a2 \u2191(fderivWithin \u211d f s a) y = 0\n[PROOFSTEP]\nrw [fderivWithin_zero_of_not_differentiableWithinAt hf]\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMaxOn f s a\ny : E\nhy : y \u2208 posTangentConeAt s a\nhy' : -y \u2208 posTangentConeAt s a\nhf : \u00acDifferentiableWithinAt \u211d f s a\n\u22a2 \u21910 y = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMinOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nhy : y \u2208 posTangentConeAt s a\n\u22a2 0 \u2264 \u2191f' y\n[PROOFSTEP]\nsimpa using h.neg.hasFDerivWithinAt_nonpos hf.neg hy\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMinOn f s a\ny : E\nhy : y \u2208 posTangentConeAt s a\nhf : \u00acDifferentiableWithinAt \u211d f s a\n\u22a2 0 \u2264 \u2191(fderivWithin \u211d f s a) y\n[PROOFSTEP]\nrw [fderivWithin_zero_of_not_differentiableWithinAt hf]\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMinOn f s a\ny : E\nhy : y \u2208 posTangentConeAt s a\nhf : \u00acDifferentiableWithinAt \u211d f s a\n\u22a2 0 \u2264 \u21910 y\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMinOn f s a\nhf : HasFDerivWithinAt f f' s a\ny : E\nhy : y \u2208 posTangentConeAt s a\nhy' : -y \u2208 posTangentConeAt s a\n\u22a2 \u2191f' y = 0\n[PROOFSTEP]\nsimpa using h.neg.hasFDerivWithinAt_eq_zero hf.neg hy hy'\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMinOn f s a\ny : E\nhy : y \u2208 posTangentConeAt s a\nhy' : -y \u2208 posTangentConeAt s a\nhf : \u00acDifferentiableWithinAt \u211d f s a\n\u22a2 \u2191(fderivWithin \u211d f s a) y = 0\n[PROOFSTEP]\nrw [fderivWithin_zero_of_not_differentiableWithinAt hf]\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\ns : Set E\nh : IsLocalMinOn f s a\ny : E\nhy : y \u2208 posTangentConeAt s a\nhy' : -y \u2208 posTangentConeAt s a\nhf : \u00acDifferentiableWithinAt \u211d f s a\n\u22a2 \u21910 y = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\nh : IsLocalMin f a\nhf : HasFDerivAt f f' a\n\u22a2 f' = 0\n[PROOFSTEP]\next y\n[GOAL]\ncase h\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\nh : IsLocalMin f a\nhf : HasFDerivAt f f' a\ny : E\n\u22a2 \u2191f' y = \u21910 y\n[PROOFSTEP]\napply (h.on univ).hasFDerivWithinAt_eq_zero hf.hasFDerivWithinAt\n[GOAL]\ncase h.hy\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\nh : IsLocalMin f a\nhf : HasFDerivAt f f' a\ny : E\n\u22a2 y \u2208 posTangentConeAt univ a\n[PROOFSTEP]\nrw [posTangentConeAt_univ]\n[GOAL]\ncase h.hy'\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\nh : IsLocalMin f a\nhf : HasFDerivAt f f' a\ny : E\n\u22a2 -y \u2208 posTangentConeAt univ a\n[PROOFSTEP]\nrw [posTangentConeAt_univ]\n[GOAL]\ncase h.hy\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\nh : IsLocalMin f a\nhf : HasFDerivAt f f' a\ny : E\n\u22a2 y \u2208 univ\n[PROOFSTEP]\napply mem_univ\n[GOAL]\ncase h.hy'\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : E \u2192 \u211d\na : E\nf' : E \u2192L[\u211d] \u211d\nh : IsLocalMin f a\nhf : HasFDerivAt f f' a\ny : E\n\u22a2 -y \u2208 univ\n[PROOFSTEP]\napply mem_univ\n[GOAL]\nf : \u211d \u2192 \u211d\nf' a b : \u211d\nh : IsLocalMin f a\nhf : HasDerivAt f f' a\n\u22a2 f' = 0\n[PROOFSTEP]\nsimpa using FunLike.congr_fun (h.hasFDerivAt_eq_zero (hasDerivAt_iff_hasFDerivAt.1 hf)) 1\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.LocalExtr.Basic", "llama_tokens": 7036, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943925708562, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.4285935144491531}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192 Finset \u03b1\n\u22a2 f \u2208 Finset.finsupp s t \u2194 f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i\n[PROOFSTEP]\nrefine' mem_map.trans \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192 Finset \u03b1\n\u22a2 (\u2203 a, a \u2208 pi s t \u2227 \u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } a = f) \u2192\n f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i\n[PROOFSTEP]\nrintro \u27e8f, hf, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Finset \u03b1\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\nhf : f \u2208 pi s t\n\u22a2 (\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f).support \u2286 s \u2227\n \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f) i \u2208 t i\n[PROOFSTEP]\nrefine' \u27e8support_indicator_subset _ _, fun i hi => _\u27e9\n[GOAL]\ncase refine'_1.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Finset \u03b1\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\nhf : f \u2208 pi s t\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f) i \u2208 t i\n[PROOFSTEP]\nconvert mem_pi.1 hf i hi\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Finset \u03b1\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\nhf : f \u2208 pi s t\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } f) i = f i hi\n[PROOFSTEP]\nexact indicator_of_mem hi _\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192 Finset \u03b1\n\u22a2 (f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i) \u2192\n \u2203 a, a \u2208 pi s t \u2227 \u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } a = f\n[PROOFSTEP]\nrefine' fun h => \u27e8fun i _ => f i, mem_pi.2 h.2, _\u27e9\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i\n\u22a2 (\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } fun i x => \u2191f i) = f\n[PROOFSTEP]\next i\n[GOAL]\ncase refine'_2.h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : f.support \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i\ni : \u03b9\n\u22a2 \u2191(\u2191{ toFun := indicator s, inj' := (_ : Function.Injective fun f => indicator s f) } fun i x => \u2191f i) i = \u2191f i\n[PROOFSTEP]\nexact ite_eq_left_iff.2 fun hi => (not_mem_support_iff.1 fun H => hi <| h.1 H).symm\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192\u2080 Finset \u03b1\nht : t.support \u2286 s\n\u22a2 f \u2208 Finset.finsupp s \u2191t \u2194 \u2200 (i : \u03b9), \u2191f i \u2208 \u2191t i\n[PROOFSTEP]\nrefine'\n mem_finsupp_iff.trans\n (forall_and.symm.trans <|\n forall_congr' fun i =>\n \u27e8fun h => _, fun h => \u27e8fun hi => ht <| mem_support_iff.2 fun H => mem_support_iff.1 hi _, fun _ => h\u27e9\u27e9)\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192\u2080 Finset \u03b1\nht : t.support \u2286 s\ni : \u03b9\nh : (i \u2208 f.support \u2192 i \u2208 s) \u2227 (i \u2208 s \u2192 \u2191f i \u2208 \u2191t i)\n\u22a2 \u2191f i \u2208 \u2191t i\n[PROOFSTEP]\nby_cases hi : i \u2208 s\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192\u2080 Finset \u03b1\nht : t.support \u2286 s\ni : \u03b9\nh : (i \u2208 f.support \u2192 i \u2208 s) \u2227 (i \u2208 s \u2192 \u2191f i \u2208 \u2191t i)\nhi : i \u2208 s\n\u22a2 \u2191f i \u2208 \u2191t i\n[PROOFSTEP]\nexact h.2 hi\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192\u2080 Finset \u03b1\nht : t.support \u2286 s\ni : \u03b9\nh : (i \u2208 f.support \u2192 i \u2208 s) \u2227 (i \u2208 s \u2192 \u2191f i \u2208 \u2191t i)\nhi : \u00aci \u2208 s\n\u22a2 \u2191f i \u2208 \u2191t i\n[PROOFSTEP]\nrw [not_mem_support_iff.1 (mt h.1 hi), not_mem_support_iff.1 fun H => hi <| ht H]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192\u2080 Finset \u03b1\nht : t.support \u2286 s\ni : \u03b9\nh : (i \u2208 f.support \u2192 i \u2208 s) \u2227 (i \u2208 s \u2192 \u2191f i \u2208 \u2191t i)\nhi : \u00aci \u2208 s\n\u22a2 0 \u2208 0\n[PROOFSTEP]\nexact zero_mem_zero\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192\u2080 \u03b1\nt : \u03b9 \u2192\u2080 Finset \u03b1\nht : t.support \u2286 s\ni : \u03b9\nh : \u2191f i \u2208 \u2191t i\nhi : i \u2208 f.support\nH : \u2191t i = 0\n\u22a2 \u2191f i = 0\n[PROOFSTEP]\nrwa [H, mem_zero] at h \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf\u271d : \u03b9 \u2192\u2080 \u03b1\nf : \u03b9 \u2192\u2080 Finset \u03b1\n\u22a2 card (pi f) = prod f fun i => \u2191(card (\u2191f i))\n[PROOFSTEP]\nrw [pi, card_finsupp]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf\u271d : \u03b9 \u2192\u2080 \u03b1\nf : \u03b9 \u2192\u2080 Finset \u03b1\n\u22a2 \u220f i in f.support, card (\u2191f i) = prod f fun i => \u2191(card (\u2191f i))\n[PROOFSTEP]\nexact Finset.prod_congr rfl fun i _ => by simp only [Pi.nat_apply, Nat.cast_id]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf\u271d : \u03b9 \u2192\u2080 \u03b1\nf : \u03b9 \u2192\u2080 Finset \u03b1\ni : \u03b9\nx\u271d : i \u2208 f.support\n\u22a2 card (\u2191f i) = (fun i => \u2191(card (\u2191f i))) i (\u2191f i)\n[PROOFSTEP]\nsimp only [Pi.nat_apply, Nat.cast_id]\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.Finsupp", "llama_tokens": 2483, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419958239132, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.42859331460512695}} {"text": "[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ns : Set \u03b1\na : \u03b1\ninst\u271d : MulHomClass F \u03b1 \u03b2\nf : F\nhf : InjOn (\u2191f) (s * s)\nh : MulSalemSpencer s\n\u22a2 MulSalemSpencer (\u2191f '' s)\n[PROOFSTEP]\nrintro _ _ _ \u27e8a, ha, rfl\u27e9 \u27e8b, hb, rfl\u27e9 \u27e8c, hc, rfl\u27e9 habc\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ns : Set \u03b1\na\u271d : \u03b1\ninst\u271d : MulHomClass F \u03b1 \u03b2\nf : F\nhf : InjOn (\u2191f) (s * s)\nh : MulSalemSpencer s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 s\nhabc : \u2191f a * \u2191f b = \u2191f c * \u2191f c\n\u22a2 \u2191f a = \u2191f b\n[PROOFSTEP]\nrw [h ha hb hc (hf (mul_mem_mul ha hb) (mul_mem_mul hc hc) <| by rwa [map_mul, map_mul])]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ns : Set \u03b1\na\u271d : \u03b1\ninst\u271d : MulHomClass F \u03b1 \u03b2\nf : F\nhf : InjOn (\u2191f) (s * s)\nh : MulSalemSpencer s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 s\nhabc : \u2191f a * \u2191f b = \u2191f c * \u2191f c\n\u22a2 \u2191f (a * b) = \u2191f (c * c)\n[PROOFSTEP]\nrwa [map_mul, map_mul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 MulSalemSpencer (insert a s) \u2194\n MulSalemSpencer s \u2227\n (\u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 a * b = c * c \u2192 a = b) \u2227 \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 b * c = a * a \u2192 b = c\n[PROOFSTEP]\nrefine'\n \u27e8fun hs =>\n \u27e8hs.mono (subset_insert _ _), fun b c hb hc => hs (Or.inl rfl) (Or.inr hb) (Or.inr hc), fun b c hb hc =>\n hs (Or.inr hb) (Or.inr hc) (Or.inl rfl)\u27e9,\n _\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 (MulSalemSpencer s \u2227\n (\u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 a * b = c * c \u2192 a = b) \u2227 \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 b * c = a * a \u2192 b = c) \u2192\n MulSalemSpencer (insert a s)\n[PROOFSTEP]\nrintro \u27e8hs, ha, ha'\u27e9 b c d hb hc hd h\n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nha : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 a * b = c * c \u2192 a = b\nha' : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 b * c = a * a \u2192 b = c\nb c d : \u03b1\nhb : b \u2208 insert a s\nhc : c \u2208 insert a s\nhd : d \u2208 insert a s\nh : b * c = d * d\n\u22a2 b = c\n[PROOFSTEP]\nrw [mem_insert_iff] at hb hc hd \n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nha : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 a * b = c * c \u2192 a = b\nha' : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 b * c = a * a \u2192 b = c\nb c d : \u03b1\nhb : b = a \u2228 b \u2208 s\nhc : c = a \u2228 c \u2208 s\nhd : d = a \u2228 d \u2208 s\nh : b * c = d * d\n\u22a2 b = c\n[PROOFSTEP]\nobtain rfl | hb := hb\n[GOAL]\ncase intro.intro.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\nhs : MulSalemSpencer s\nb c d : \u03b1\nh : b * c = d * d\nha : \u2200 \u2983b_1 c : \u03b1\u2984, b_1 \u2208 s \u2192 c \u2208 s \u2192 b * b_1 = c * c \u2192 b = b_1\nha' : \u2200 \u2983b_1 c : \u03b1\u2984, b_1 \u2208 s \u2192 c \u2208 s \u2192 b_1 * c = b * b \u2192 b_1 = c\nhc : c = b \u2228 c \u2208 s\nhd : d = b \u2228 d \u2208 s\n\u22a2 b = c\n[PROOFSTEP]\nobtain rfl | hc := hc\n[GOAL]\ncase intro.intro.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nha : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 a * b = c * c \u2192 a = b\nha' : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 b * c = a * a \u2192 b = c\nb c d : \u03b1\nhc : c = a \u2228 c \u2208 s\nhd : d = a \u2228 d \u2208 s\nh : b * c = d * d\nhb : b \u2208 s\n\u22a2 b = c\n[PROOFSTEP]\nobtain rfl | hc := hc\n[GOAL]\ncase intro.intro.inl.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\nhs : MulSalemSpencer s\nc d : \u03b1\nh : c * c = d * d\nha : \u2200 \u2983b c_1 : \u03b1\u2984, b \u2208 s \u2192 c_1 \u2208 s \u2192 c * b = c_1 * c_1 \u2192 c = b\nha' : \u2200 \u2983b c_1 : \u03b1\u2984, b \u2208 s \u2192 c_1 \u2208 s \u2192 b * c_1 = c * c \u2192 b = c_1\nhd : d = c \u2228 d \u2208 s\n\u22a2 c = c\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro.inl.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\nhs : MulSalemSpencer s\nb c d : \u03b1\nh : b * c = d * d\nha : \u2200 \u2983b_1 c : \u03b1\u2984, b_1 \u2208 s \u2192 c \u2208 s \u2192 b * b_1 = c * c \u2192 b = b_1\nha' : \u2200 \u2983b_1 c : \u03b1\u2984, b_1 \u2208 s \u2192 c \u2208 s \u2192 b_1 * c = b * b \u2192 b_1 = c\nhd : d = b \u2228 d \u2208 s\nhc : c \u2208 s\n\u22a2 b = c\ncase intro.intro.inr.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\nhs : MulSalemSpencer s\nb c d : \u03b1\nh : b * c = d * d\nhb : b \u2208 s\nha : \u2200 \u2983b c_1 : \u03b1\u2984, b \u2208 s \u2192 c_1 \u2208 s \u2192 c * b = c_1 * c_1 \u2192 c = b\nha' : \u2200 \u2983b c_1 : \u03b1\u2984, b \u2208 s \u2192 c_1 \u2208 s \u2192 b * c_1 = c * c \u2192 b = c_1\nhd : d = c \u2228 d \u2208 s\n\u22a2 b = c\ncase intro.intro.inr.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nha : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 a * b = c * c \u2192 a = b\nha' : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 b * c = a * a \u2192 b = c\nb c d : \u03b1\nhd : d = a \u2228 d \u2208 s\nh : b * c = d * d\nhb : b \u2208 s\nhc : c \u2208 s\n\u22a2 b = c\n[PROOFSTEP]\nall_goals obtain rfl | hd := hd\n[GOAL]\ncase intro.intro.inl.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\nhs : MulSalemSpencer s\nb c d : \u03b1\nh : b * c = d * d\nha : \u2200 \u2983b_1 c : \u03b1\u2984, b_1 \u2208 s \u2192 c \u2208 s \u2192 b * b_1 = c * c \u2192 b = b_1\nha' : \u2200 \u2983b_1 c : \u03b1\u2984, b_1 \u2208 s \u2192 c \u2208 s \u2192 b_1 * c = b * b \u2192 b_1 = c\nhd : d = b \u2228 d \u2208 s\nhc : c \u2208 s\n\u22a2 b = c\n[PROOFSTEP]\nobtain rfl | hd := hd\n[GOAL]\ncase intro.intro.inr.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\nhs : MulSalemSpencer s\nb c d : \u03b1\nh : b * c = d * d\nhb : b \u2208 s\nha : \u2200 \u2983b c_1 : \u03b1\u2984, b \u2208 s \u2192 c_1 \u2208 s \u2192 c * b = c_1 * c_1 \u2192 c = b\nha' : \u2200 \u2983b c_1 : \u03b1\u2984, b \u2208 s \u2192 c_1 \u2208 s \u2192 b * c_1 = c * c \u2192 b = c_1\nhd : d = c \u2228 d \u2208 s\n\u22a2 b = c\n[PROOFSTEP]\nobtain rfl | hd := hd\n[GOAL]\ncase intro.intro.inr.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nha : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 a * b = c * c \u2192 a = b\nha' : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 b * c = a * a \u2192 b = c\nb c d : \u03b1\nhd : d = a \u2228 d \u2208 s\nh : b * c = d * d\nhb : b \u2208 s\nhc : c \u2208 s\n\u22a2 b = c\n[PROOFSTEP]\nobtain rfl | hd := hd\n[GOAL]\ncase intro.intro.inl.inr.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\nhs : MulSalemSpencer s\nc d : \u03b1\nhc : c \u2208 s\nh : d * c = d * d\nha : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 d * b = c * c \u2192 d = b\nha' : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 b * c = d * d \u2192 b = c\n\u22a2 d = c\n[PROOFSTEP]\nexact (mul_left_cancel h).symm\n[GOAL]\ncase intro.intro.inl.inr.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\nhs : MulSalemSpencer s\nb c d : \u03b1\nh : b * c = d * d\nha : \u2200 \u2983b_1 c : \u03b1\u2984, b_1 \u2208 s \u2192 c \u2208 s \u2192 b * b_1 = c * c \u2192 b = b_1\nha' : \u2200 \u2983b_1 c : \u03b1\u2984, b_1 \u2208 s \u2192 c \u2208 s \u2192 b_1 * c = b * b \u2192 b_1 = c\nhc : c \u2208 s\nhd : d \u2208 s\n\u22a2 b = c\n[PROOFSTEP]\nexact ha hc hd h\n[GOAL]\ncase intro.intro.inr.inl.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\nhs : MulSalemSpencer s\nb d : \u03b1\nhb : b \u2208 s\nh : b * d = d * d\nha : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 d * b = c * c \u2192 d = b\nha' : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 b * c = d * d \u2192 b = c\n\u22a2 b = d\n[PROOFSTEP]\nexact mul_right_cancel h\n[GOAL]\ncase intro.intro.inr.inl.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\nhs : MulSalemSpencer s\nb c d : \u03b1\nh : b * c = d * d\nhb : b \u2208 s\nha : \u2200 \u2983b c_1 : \u03b1\u2984, b \u2208 s \u2192 c_1 \u2208 s \u2192 c * b = c_1 * c_1 \u2192 c = b\nha' : \u2200 \u2983b c_1 : \u03b1\u2984, b \u2208 s \u2192 c_1 \u2208 s \u2192 b * c_1 = c * c \u2192 b = c_1\nhd : d \u2208 s\n\u22a2 b = c\n[PROOFSTEP]\nexact (ha hb hd <| (mul_comm _ _).trans h).symm\n[GOAL]\ncase intro.intro.inr.inr.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\nhs : MulSalemSpencer s\nb c d : \u03b1\nh : b * c = d * d\nhb : b \u2208 s\nhc : c \u2208 s\nha : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 d * b = c * c \u2192 d = b\nha' : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 b * c = d * d \u2192 b = c\n\u22a2 b = c\n[PROOFSTEP]\nexact ha' hb hc h\n[GOAL]\ncase intro.intro.inr.inr.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nha : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 a * b = c * c \u2192 a = b\nha' : \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 b * c = a * a \u2192 b = c\nb c d : \u03b1\nh : b * c = d * d\nhb : b \u2208 s\nhc : c \u2208 s\nhd : d \u2208 s\n\u22a2 b = c\n[PROOFSTEP]\nexact hs hb hc hd h\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na\u271d a b : \u03b1\n\u22a2 MulSalemSpencer {a, b}\n[PROOFSTEP]\nrw [mulSalemSpencer_insert]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na\u271d a b : \u03b1\n\u22a2 MulSalemSpencer {b} \u2227\n (\u2200 \u2983b_1 c : \u03b1\u2984, b_1 \u2208 {b} \u2192 c \u2208 {b} \u2192 a * b_1 = c * c \u2192 a = b_1) \u2227\n \u2200 \u2983b_1 c : \u03b1\u2984, b_1 \u2208 {b} \u2192 c \u2208 {b} \u2192 b_1 * c = a * a \u2192 b_1 = c\n[PROOFSTEP]\nrefine' \u27e8mulSalemSpencer_singleton _, _, _\u27e9\n[GOAL]\ncase refine'_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na\u271d a b : \u03b1\n\u22a2 \u2200 \u2983b_1 c : \u03b1\u2984, b_1 \u2208 {b} \u2192 c \u2208 {b} \u2192 a * b_1 = c * c \u2192 a = b_1\n[PROOFSTEP]\nrintro c d (rfl : c = b) (rfl : d = c)\n[GOAL]\ncase refine'_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na\u271d a d : \u03b1\n\u22a2 a * d = d * d \u2192 a = d\n[PROOFSTEP]\nexact mul_right_cancel\n[GOAL]\ncase refine'_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na\u271d a b : \u03b1\n\u22a2 \u2200 \u2983b_1 c : \u03b1\u2984, b_1 \u2208 {b} \u2192 c \u2208 {b} \u2192 b_1 * c = a * a \u2192 b_1 = c\n[PROOFSTEP]\nrintro c d (rfl : c = b) (rfl : d = c) _\n[GOAL]\ncase refine'_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na\u271d\u00b9 a d : \u03b1\na\u271d : d * d = a * a\n\u22a2 d = d\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\n\u22a2 MulSalemSpencer ((fun x x_1 => x * x_1) a '' s)\n[PROOFSTEP]\nrintro _ _ _ \u27e8b, hb, rfl\u27e9 \u27e8c, hc, rfl\u27e9 \u27e8d, hd, rfl\u27e9 h\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nb : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 s\nd : \u03b1\nhd : d \u2208 s\nh : (fun x x_1 => x * x_1) a b * (fun x x_1 => x * x_1) a c = (fun x x_1 => x * x_1) a d * (fun x x_1 => x * x_1) a d\n\u22a2 (fun x x_1 => x * x_1) a b = (fun x x_1 => x * x_1) a c\n[PROOFSTEP]\nrw [mul_mul_mul_comm, mul_mul_mul_comm a d] at h \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nb : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 s\nd : \u03b1\nhd : d \u2208 s\nh : a * a * (b * c) = a * a * (d * d)\n\u22a2 (fun x x_1 => x * x_1) a b = (fun x x_1 => x * x_1) a c\n[PROOFSTEP]\nrw [hs hb hc hd (mul_left_cancel h)]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\n\u22a2 MulSalemSpencer ((fun x => x * a) '' s)\n[PROOFSTEP]\nrintro _ _ _ \u27e8b, hb, rfl\u27e9 \u27e8c, hc, rfl\u27e9 \u27e8d, hd, rfl\u27e9 h\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nb : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 s\nd : \u03b1\nhd : d \u2208 s\nh : (fun x => x * a) b * (fun x => x * a) c = (fun x => x * a) d * (fun x => x * a) d\n\u22a2 (fun x => x * a) b = (fun x => x * a) c\n[PROOFSTEP]\nrw [mul_mul_mul_comm, mul_mul_mul_comm d] at h \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nb : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 s\nd : \u03b1\nhd : d \u2208 s\nh : b * c * (a * a) = d * d * (a * a)\n\u22a2 (fun x => x * a) b = (fun x => x * a) c\n[PROOFSTEP]\nrw [hs hb hc hd (mul_right_cancel h)]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer ((fun x x_1 => x * x_1) a '' s)\nb c d : \u03b1\nhb : b \u2208 s\nhc : c \u2208 s\nhd : d \u2208 s\nh : b * c = d * d\n\u22a2 a * b * (a * c) = (fun x x_1 => x * x_1) a d * (fun x x_1 => x * x_1) a d\n[PROOFSTEP]\nrw [mul_mul_mul_comm, h, mul_mul_mul_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : CancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer ((fun x => x * a) '' s)\nb c d : \u03b1\nhb : b \u2208 s\nhc : c \u2208 s\nhd : d \u2208 s\nh : b * c = d * d\n\u22a2 b * a * (c * a) = d * a * (d * a)\n[PROOFSTEP]\nrw [mul_mul_mul_comm, h, mul_mul_mul_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : \u2200 (i : \u03b1), i \u2208 s \u2192 i < a\n\u22a2 MulSalemSpencer (insert a s) \u2194 MulSalemSpencer s \u2227 \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 a * b = c * c \u2192 a = b\n[PROOFSTEP]\nrefine' mulSalemSpencer_insert.trans _\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : \u2200 (i : \u03b1), i \u2208 s \u2192 i < a\n\u22a2 (MulSalemSpencer s \u2227\n (\u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 a * b = c * c \u2192 a = b) \u2227 \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 b * c = a * a \u2192 b = c) \u2194\n MulSalemSpencer s \u2227 \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 a * b = c * c \u2192 a = b\n[PROOFSTEP]\nrw [\u2190 and_assoc]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns : Set \u03b1\na : \u03b1\nhs : \u2200 (i : \u03b1), i \u2208 s \u2192 i < a\n\u22a2 ((MulSalemSpencer s \u2227 \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 a * b = c * c \u2192 a = b) \u2227\n \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 b * c = a * a \u2192 b = c) \u2194\n MulSalemSpencer s \u2227 \u2200 \u2983b c : \u03b1\u2984, b \u2208 s \u2192 c \u2208 s \u2192 a * b = c * c \u2192 a = b\n[PROOFSTEP]\nexact and_iff_left fun b c hb hc h => ((mul_lt_mul_of_lt_of_lt (hs _ hb) (hs _ hc)).ne h).elim\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NoZeroDivisors \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nha : a \u2260 0\n\u22a2 MulSalemSpencer ((fun x x_1 => x * x_1) a '' s)\n[PROOFSTEP]\nrintro _ _ _ \u27e8b, hb, rfl\u27e9 \u27e8c, hc, rfl\u27e9 \u27e8d, hd, rfl\u27e9 h\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NoZeroDivisors \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nha : a \u2260 0\nb : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 s\nd : \u03b1\nhd : d \u2208 s\nh : (fun x x_1 => x * x_1) a b * (fun x x_1 => x * x_1) a c = (fun x x_1 => x * x_1) a d * (fun x x_1 => x * x_1) a d\n\u22a2 (fun x x_1 => x * x_1) a b = (fun x x_1 => x * x_1) a c\n[PROOFSTEP]\nrw [mul_mul_mul_comm, mul_mul_mul_comm a d] at h \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NoZeroDivisors \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nha : a \u2260 0\nb : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 s\nd : \u03b1\nhd : d \u2208 s\nh : a * a * (b * c) = a * a * (d * d)\n\u22a2 (fun x x_1 => x * x_1) a b = (fun x x_1 => x * x_1) a c\n[PROOFSTEP]\nrw [hs hb hc hd (mul_left_cancel\u2080 (mul_ne_zero ha ha) h)]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NoZeroDivisors \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nha : a \u2260 0\n\u22a2 MulSalemSpencer ((fun x => x * a) '' s)\n[PROOFSTEP]\nrintro _ _ _ \u27e8b, hb, rfl\u27e9 \u27e8c, hc, rfl\u27e9 \u27e8d, hd, rfl\u27e9 h\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NoZeroDivisors \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nha : a \u2260 0\nb : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 s\nd : \u03b1\nhd : d \u2208 s\nh : (fun x => x * a) b * (fun x => x * a) c = (fun x => x * a) d * (fun x => x * a) d\n\u22a2 (fun x => x * a) b = (fun x => x * a) c\n[PROOFSTEP]\nrw [mul_mul_mul_comm, mul_mul_mul_comm d] at h \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NoZeroDivisors \u03b1\ns : Set \u03b1\na : \u03b1\nhs : MulSalemSpencer s\nha : a \u2260 0\nb : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 s\nd : \u03b1\nhd : d \u2208 s\nh : b * c * (a * a) = d * d * (a * a)\n\u22a2 (fun x => x * a) b = (fun x => x * a) c\n[PROOFSTEP]\nrw [hs hb hc hd (mul_right_cancel\u2080 (mul_ne_zero ha ha) h)]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NoZeroDivisors \u03b1\ns : Set \u03b1\na : \u03b1\nha : a \u2260 0\nhs : MulSalemSpencer ((fun x x_1 => x * x_1) a '' s)\nb c d : \u03b1\nhb : b \u2208 s\nhc : c \u2208 s\nhd : d \u2208 s\nh : b * c = d * d\n\u22a2 a * b * (a * c) = (fun x x_1 => x * x_1) a d * (fun x x_1 => x * x_1) a d\n[PROOFSTEP]\nrw [mul_mul_mul_comm, h, mul_mul_mul_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NoZeroDivisors \u03b1\ns : Set \u03b1\na : \u03b1\nha : a \u2260 0\nhs : MulSalemSpencer ((fun x => x * a) '' s)\nb c d : \u03b1\nhb : b \u2208 s\nhc : c \u2208 s\nhd : d \u2208 s\nh : b * c = d * d\n\u22a2 b * a * (c * a) = d * a * (d * a)\n[PROOFSTEP]\nrw [mul_mul_mul_comm, h, mul_mul_mul_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Set \u2115\n\u22a2 AddSalemSpencer s \u2194 \u2200 \u2983a b c : \u2115\u2984, a \u2208 s \u2192 b \u2208 s \u2192 c \u2208 s \u2192 a + b = c + c \u2192 a = c\n[PROOFSTEP]\nrefine' forall\u2084_congr fun a b c _ => forall\u2083_congr fun _ _ habc => \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Set \u2115\na b c : \u2115\nx\u271d\u00b2 : a \u2208 s\nx\u271d\u00b9 : b \u2208 s\nx\u271d : c \u2208 s\nhabc : a + b = c + c\n\u22a2 a = b \u2192 a = c\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Set \u2115\na c : \u2115\nx\u271d\u00b2 : a \u2208 s\nx\u271d\u00b9 : c \u2208 s\nx\u271d : a \u2208 s\nhabc : a + a = c + c\n\u22a2 a = c\n[PROOFSTEP]\nsimp_rw [\u2190 two_mul] at habc \n[GOAL]\ncase refine'_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Set \u2115\na c : \u2115\nx\u271d\u00b2 : a \u2208 s\nx\u271d\u00b9 : c \u2208 s\nx\u271d : a \u2208 s\nhabc : 2 * a = 2 * c\n\u22a2 a = c\n[PROOFSTEP]\nexact mul_left_cancel\u2080 two_ne_zero habc\n[GOAL]\ncase refine'_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Set \u2115\na b c : \u2115\nx\u271d\u00b2 : a \u2208 s\nx\u271d\u00b9 : b \u2208 s\nx\u271d : c \u2208 s\nhabc : a + b = c + c\n\u22a2 a = c \u2192 a = b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Set \u2115\na b : \u2115\nx\u271d\u00b2 : a \u2208 s\nx\u271d\u00b9 : b \u2208 s\nx\u271d : a \u2208 s\nhabc : a + b = a + a\n\u22a2 a = b\n[PROOFSTEP]\nexact (add_left_cancel habc).symm\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs\u2080 : IsClosed s\nhs\u2081 : StrictConvex \ud835\udd5c s\n\u22a2 AddSalemSpencer (frontier s)\n[PROOFSTEP]\nintro a b c ha hb hc habc\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs\u2080 : IsClosed s\nhs\u2081 : StrictConvex \ud835\udd5c s\na b c : E\nha : a \u2208 frontier s\nhb : b \u2208 frontier s\nhc : c \u2208 frontier s\nhabc : a + b = c + c\n\u22a2 a = b\n[PROOFSTEP]\nobtain rfl : (1 / 2 : \ud835\udd5c) \u2022 a + (1 / 2 : \ud835\udd5c) \u2022 b = c := by\n rwa [\u2190 smul_add, one_div, inv_smul_eq_iff\u2080 (show (2 : \ud835\udd5c) \u2260 0 by norm_num), two_smul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs\u2080 : IsClosed s\nhs\u2081 : StrictConvex \ud835\udd5c s\na b c : E\nha : a \u2208 frontier s\nhb : b \u2208 frontier s\nhc : c \u2208 frontier s\nhabc : a + b = c + c\n\u22a2 (1 / 2) \u2022 a + (1 / 2) \u2022 b = c\n[PROOFSTEP]\nrwa [\u2190 smul_add, one_div, inv_smul_eq_iff\u2080 (show (2 : \ud835\udd5c) \u2260 0 by norm_num), two_smul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs\u2080 : IsClosed s\nhs\u2081 : StrictConvex \ud835\udd5c s\na b c : E\nha : a \u2208 frontier s\nhb : b \u2208 frontier s\nhc : c \u2208 frontier s\nhabc : a + b = c + c\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs\u2080 : IsClosed s\nhs\u2081 : StrictConvex \ud835\udd5c s\na b : E\nha : a \u2208 frontier s\nhb : b \u2208 frontier s\nhc : (1 / 2) \u2022 a + (1 / 2) \u2022 b \u2208 frontier s\nhabc : a + b = (1 / 2) \u2022 a + (1 / 2) \u2022 b + ((1 / 2) \u2022 a + (1 / 2) \u2022 b)\n\u22a2 a = b\n[PROOFSTEP]\nexact hs\u2081.eq (hs\u2080.frontier_subset ha) (hs\u2080.frontier_subset hb) one_half_pos one_half_pos (add_halves _) hc.2\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx : E\nr : \u211d\n\u22a2 AddSalemSpencer (sphere x r)\n[PROOFSTEP]\nobtain rfl | hr := eq_or_ne r 0\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx : E\n\u22a2 AddSalemSpencer (sphere x 0)\n[PROOFSTEP]\nrw [sphere_zero]\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx : E\n\u22a2 AddSalemSpencer {x}\n[PROOFSTEP]\nexact addSalemSpencer_singleton _\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx : E\nr : \u211d\nhr : r \u2260 0\n\u22a2 AddSalemSpencer (sphere x r)\n[PROOFSTEP]\nconvert addSalemSpencer_frontier isClosed_ball (strictConvex_closedBall \u211d x r)\n[GOAL]\ncase h.e'_3\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx : E\nr : \u211d\nhr : r \u2260 0\n\u22a2 sphere x r = frontier (closedBall x r)\n[PROOFSTEP]\nexact (frontier_closedBall _ hr).symm\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns t : Finset \u03b1\n\u22a2 Monotone fun s => Nat.findGreatest (fun m => \u2203 t x, card t = m \u2227 MulSalemSpencer \u2191t) (card s)\n[PROOFSTEP]\nrintro t u htu\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns t\u271d t u : Finset \u03b1\nhtu : t \u2264 u\n\u22a2 (fun s => Nat.findGreatest (fun m => \u2203 t x, card t = m \u2227 MulSalemSpencer \u2191t) (card s)) t \u2264\n (fun s => Nat.findGreatest (fun m => \u2203 t x, card t = m \u2227 MulSalemSpencer \u2191t) (card s)) u\n[PROOFSTEP]\nrefine' Nat.findGreatest_mono (fun m => _) (card_le_of_subset htu)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns t\u271d t u : Finset \u03b1\nhtu : t \u2264 u\nm : \u2115\n\u22a2 (\u2203 t_1 x, card t_1 = m \u2227 MulSalemSpencer \u2191t_1) \u2264 \u2203 t x, card t = m \u2227 MulSalemSpencer \u2191t\n[PROOFSTEP]\nrintro \u27e8v, hvt, hv\u27e9\n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns t\u271d t u : Finset \u03b1\nhtu : t \u2264 u\nm : \u2115\nv : Finset \u03b1\nhvt : v \u2286 t\nhv : card v = m \u2227 MulSalemSpencer \u2191v\n\u22a2 \u2203 t x, card t = m \u2227 MulSalemSpencer \u2191t\n[PROOFSTEP]\nexact \u27e8v, hvt.trans htu, hv\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns t : Finset \u03b1\n\u22a2 MulSalemSpencer \u2191\u2205\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns t : Finset \u03b1\n\u22a2 MulSalemSpencer \u2205\n[PROOFSTEP]\nexact mulSalemSpencer_empty\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns t : Finset \u03b1\nn : \u2115\na : \u03b1\n\u22a2 \u2191mulRothNumber {a} = 1\n[PROOFSTEP]\nrefine' MulSalemSpencer.roth_number_eq _\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns t : Finset \u03b1\nn : \u2115\na : \u03b1\n\u22a2 MulSalemSpencer \u2191{a}\n[PROOFSTEP]\nrw [coe_singleton]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns t : Finset \u03b1\nn : \u2115\na : \u03b1\n\u22a2 MulSalemSpencer {a}\n[PROOFSTEP]\nexact mulSalemSpencer_singleton a\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns\u271d t\u271d : Finset \u03b1\nn : \u2115\ns t u : Finset \u03b1\nhus : u \u2286 s \u222a t\nhcard : card u = \u2191mulRothNumber (s \u222a t)\nhu : MulSalemSpencer \u2191u\n\u22a2 card u = card (u \u2229 s \u222a u \u2229 t)\n[PROOFSTEP]\nrw [\u2190 inter_distrib_left, (inter_eq_left_iff_subset _ _).2 hus]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns\u271d t\u271d : Finset \u03b1\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 \u2191mulRothNumber s * \u2191mulRothNumber t \u2264 \u2191mulRothNumber (s \u00d7\u02e2 t)\n[PROOFSTEP]\nobtain \u27e8u, hus, hucard, hu\u27e9 := mulRothNumber_spec s\n[GOAL]\ncase intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns\u271d t\u271d : Finset \u03b1\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\nu : Finset \u03b1\nhus : u \u2286 s\nhucard : card u = \u2191mulRothNumber s\nhu : MulSalemSpencer \u2191u\n\u22a2 \u2191mulRothNumber s * \u2191mulRothNumber t \u2264 \u2191mulRothNumber (s \u00d7\u02e2 t)\n[PROOFSTEP]\nobtain \u27e8v, hvt, hvcard, hv\u27e9 := mulRothNumber_spec t\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns\u271d t\u271d : Finset \u03b1\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\nu : Finset \u03b1\nhus : u \u2286 s\nhucard : card u = \u2191mulRothNumber s\nhu : MulSalemSpencer \u2191u\nv : Finset \u03b2\nhvt : v \u2286 t\nhvcard : card v = \u2191mulRothNumber t\nhv : MulSalemSpencer \u2191v\n\u22a2 \u2191mulRothNumber s * \u2191mulRothNumber t \u2264 \u2191mulRothNumber (s \u00d7\u02e2 t)\n[PROOFSTEP]\nrw [\u2190 hucard, \u2190 hvcard, \u2190 card_product]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns\u271d t\u271d : Finset \u03b1\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\nu : Finset \u03b1\nhus : u \u2286 s\nhucard : card u = \u2191mulRothNumber s\nhu : MulSalemSpencer \u2191u\nv : Finset \u03b2\nhvt : v \u2286 t\nhvcard : card v = \u2191mulRothNumber t\nhv : MulSalemSpencer \u2191v\n\u22a2 card (u \u00d7\u02e2 v) \u2264 \u2191mulRothNumber (s \u00d7\u02e2 t)\n[PROOFSTEP]\nrefine' MulSalemSpencer.le_mulRothNumber _ (product_subset_product hus hvt)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns\u271d t\u271d : Finset \u03b1\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\nu : Finset \u03b1\nhus : u \u2286 s\nhucard : card u = \u2191mulRothNumber s\nhu : MulSalemSpencer \u2191u\nv : Finset \u03b2\nhvt : v \u2286 t\nhvcard : card v = \u2191mulRothNumber t\nhv : MulSalemSpencer \u2191v\n\u22a2 MulSalemSpencer \u2191(u \u00d7\u02e2 v)\n[PROOFSTEP]\nrw [coe_product]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns\u271d t\u271d : Finset \u03b1\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\nu : Finset \u03b1\nhus : u \u2286 s\nhucard : card u = \u2191mulRothNumber s\nhu : MulSalemSpencer \u2191u\nv : Finset \u03b2\nhvt : v \u2286 t\nhvcard : card v = \u2191mulRothNumber t\nhv : MulSalemSpencer \u2191v\n\u22a2 MulSalemSpencer (\u2191u \u00d7\u02e2 \u2191v)\n[PROOFSTEP]\nexact hu.prod hv\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns t : Finset \u03b1\nn : \u2115\nh : \u2200 (t : Finset \u03b1), t \u2208 powersetLen n s \u2192 \u00acMulSalemSpencer \u2191t\n\u22a2 \u2191mulRothNumber s < n\n[PROOFSTEP]\nobtain \u27e8t, hts, hcard, ht\u27e9 := mulRothNumber_spec s\n[GOAL]\ncase intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns t\u271d : Finset \u03b1\nn : \u2115\nh : \u2200 (t : Finset \u03b1), t \u2208 powersetLen n s \u2192 \u00acMulSalemSpencer \u2191t\nt : Finset \u03b1\nhts : t \u2286 s\nhcard : card t = \u2191mulRothNumber s\nht : MulSalemSpencer \u2191t\n\u22a2 \u2191mulRothNumber s < n\n[PROOFSTEP]\nrw [\u2190 hcard, \u2190 not_le]\n[GOAL]\ncase intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns t\u271d : Finset \u03b1\nn : \u2115\nh : \u2200 (t : Finset \u03b1), t \u2208 powersetLen n s \u2192 \u00acMulSalemSpencer \u2191t\nt : Finset \u03b1\nhts : t \u2286 s\nhcard : card t = \u2191mulRothNumber s\nht : MulSalemSpencer \u2191t\n\u22a2 \u00acn \u2264 card t\n[PROOFSTEP]\nintro hn\n[GOAL]\ncase intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns t\u271d : Finset \u03b1\nn : \u2115\nh : \u2200 (t : Finset \u03b1), t \u2208 powersetLen n s \u2192 \u00acMulSalemSpencer \u2191t\nt : Finset \u03b1\nhts : t \u2286 s\nhcard : card t = \u2191mulRothNumber s\nht : MulSalemSpencer \u2191t\nhn : n \u2264 card t\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8u, hut, rfl\u27e9 := exists_smaller_set t n hn\n[GOAL]\ncase intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : Monoid \u03b2\ns t\u271d t : Finset \u03b1\nhts : t \u2286 s\nhcard : card t = \u2191mulRothNumber s\nht : MulSalemSpencer \u2191t\nu : Finset \u03b1\nhut : u \u2286 t\nh : \u2200 (t : Finset \u03b1), t \u2208 powersetLen (card u) s \u2192 \u00acMulSalemSpencer \u2191t\nhn : card u \u2264 card t\n\u22a2 False\n[PROOFSTEP]\nexact h _ (mem_powersetLen.2 \u27e8hut.trans hts, rfl\u27e9) (ht.mono hut)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CancelCommMonoid \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 \u2191mulRothNumber (map (mulLeftEmbedding a) s) = \u2191mulRothNumber s\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CancelCommMonoid \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 \u2191mulRothNumber (map (mulLeftEmbedding a) s) \u2264 \u2191mulRothNumber s\n[PROOFSTEP]\nobtain \u27e8u, hus, hcard, hu\u27e9 := mulRothNumber_spec (s.map <| mulLeftEmbedding a)\n[GOAL]\ncase refine'_1.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CancelCommMonoid \u03b1\ns : Finset \u03b1\na : \u03b1\nu : Finset \u03b1\nhus : u \u2286 map (mulLeftEmbedding a) s\nhcard : card u = \u2191mulRothNumber (map (mulLeftEmbedding a) s)\nhu : MulSalemSpencer \u2191u\n\u22a2 \u2191mulRothNumber (map (mulLeftEmbedding a) s) \u2264 \u2191mulRothNumber s\n[PROOFSTEP]\nrw [subset_map_iff] at hus \n[GOAL]\ncase refine'_1.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CancelCommMonoid \u03b1\ns : Finset \u03b1\na : \u03b1\nu : Finset \u03b1\nhus : \u2203 u_1 x, u = map (mulLeftEmbedding a) u_1\nhcard : card u = \u2191mulRothNumber (map (mulLeftEmbedding a) s)\nhu : MulSalemSpencer \u2191u\n\u22a2 \u2191mulRothNumber (map (mulLeftEmbedding a) s) \u2264 \u2191mulRothNumber s\n[PROOFSTEP]\nobtain \u27e8u, hus, rfl\u27e9 := hus\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CancelCommMonoid \u03b1\ns : Finset \u03b1\na : \u03b1\nu : Finset \u03b1\nhus : u \u2286 s\nhcard : card (map (mulLeftEmbedding a) u) = \u2191mulRothNumber (map (mulLeftEmbedding a) s)\nhu : MulSalemSpencer \u2191(map (mulLeftEmbedding a) u)\n\u22a2 \u2191mulRothNumber (map (mulLeftEmbedding a) s) \u2264 \u2191mulRothNumber s\n[PROOFSTEP]\nrw [coe_map] at hu \n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CancelCommMonoid \u03b1\ns : Finset \u03b1\na : \u03b1\nu : Finset \u03b1\nhus : u \u2286 s\nhcard : card (map (mulLeftEmbedding a) u) = \u2191mulRothNumber (map (mulLeftEmbedding a) s)\nhu : MulSalemSpencer (\u2191(mulLeftEmbedding a) '' \u2191u)\n\u22a2 \u2191mulRothNumber (map (mulLeftEmbedding a) s) \u2264 \u2191mulRothNumber s\n[PROOFSTEP]\nrw [\u2190 hcard, card_map]\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CancelCommMonoid \u03b1\ns : Finset \u03b1\na : \u03b1\nu : Finset \u03b1\nhus : u \u2286 s\nhcard : card (map (mulLeftEmbedding a) u) = \u2191mulRothNumber (map (mulLeftEmbedding a) s)\nhu : MulSalemSpencer (\u2191(mulLeftEmbedding a) '' \u2191u)\n\u22a2 card u \u2264 \u2191mulRothNumber s\n[PROOFSTEP]\nexact (mulSalemSpencer_mul_left_iff.1 hu).le_mulRothNumber hus\n[GOAL]\ncase refine'_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CancelCommMonoid \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 \u2191mulRothNumber s \u2264 \u2191mulRothNumber (map (mulLeftEmbedding a) s)\n[PROOFSTEP]\nobtain \u27e8u, hus, hcard, hu\u27e9 := mulRothNumber_spec s\n[GOAL]\ncase refine'_2.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CancelCommMonoid \u03b1\ns : Finset \u03b1\na : \u03b1\nu : Finset \u03b1\nhus : u \u2286 s\nhcard : card u = \u2191mulRothNumber s\nhu : MulSalemSpencer \u2191u\n\u22a2 \u2191mulRothNumber s \u2264 \u2191mulRothNumber (map (mulLeftEmbedding a) s)\n[PROOFSTEP]\nhave h : MulSalemSpencer (u.map <| mulLeftEmbedding a : Set \u03b1) :=\n by\n rw [coe_map]\n exact hu.mul_left\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CancelCommMonoid \u03b1\ns : Finset \u03b1\na : \u03b1\nu : Finset \u03b1\nhus : u \u2286 s\nhcard : card u = \u2191mulRothNumber s\nhu : MulSalemSpencer \u2191u\n\u22a2 MulSalemSpencer \u2191(map (mulLeftEmbedding a) u)\n[PROOFSTEP]\nrw [coe_map]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CancelCommMonoid \u03b1\ns : Finset \u03b1\na : \u03b1\nu : Finset \u03b1\nhus : u \u2286 s\nhcard : card u = \u2191mulRothNumber s\nhu : MulSalemSpencer \u2191u\n\u22a2 MulSalemSpencer (\u2191(mulLeftEmbedding a) '' \u2191u)\n[PROOFSTEP]\nexact hu.mul_left\n[GOAL]\ncase refine'_2.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CancelCommMonoid \u03b1\ns : Finset \u03b1\na : \u03b1\nu : Finset \u03b1\nhus : u \u2286 s\nhcard : card u = \u2191mulRothNumber s\nhu : MulSalemSpencer \u2191u\nh : MulSalemSpencer \u2191(map (mulLeftEmbedding a) u)\n\u22a2 \u2191mulRothNumber s \u2264 \u2191mulRothNumber (map (mulLeftEmbedding a) s)\n[PROOFSTEP]\nconvert h.le_mulRothNumber (map_subset_map.2 hus) using 1\n[GOAL]\ncase h.e'_3\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CancelCommMonoid \u03b1\ns : Finset \u03b1\na : \u03b1\nu : Finset \u03b1\nhus : u \u2286 s\nhcard : card u = \u2191mulRothNumber s\nhu : MulSalemSpencer \u2191u\nh : MulSalemSpencer \u2191(map (mulLeftEmbedding a) u)\n\u22a2 \u2191mulRothNumber s = card (map (mulLeftEmbedding a) u)\n[PROOFSTEP]\nrw [card_map, hcard]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CancelCommMonoid \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 \u2191mulRothNumber (map (mulRightEmbedding a) s) = \u2191mulRothNumber s\n[PROOFSTEP]\nrw [\u2190 mul_left_embedding_eq_mul_right_embedding, mulRothNumber_map_mul_left s a]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Finset \u2115\nk n M N : \u2115\n\u22a2 \u2191rothNumberNat (M + N) \u2264 \u2191rothNumberNat M + \u2191rothNumberNat N\n[PROOFSTEP]\nsimp_rw [rothNumberNat_def]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Finset \u2115\nk n M N : \u2115\n\u22a2 \u2191addRothNumber (range (M + N)) \u2264 \u2191addRothNumber (range M) + \u2191addRothNumber (range N)\n[PROOFSTEP]\nrw [range_add_eq_union, \u2190 addRothNumber_map_add_left (range N) M]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Finset \u2115\nk n M N : \u2115\n\u22a2 \u2191addRothNumber (range M \u222a map (addLeftEmbedding M) (range N)) \u2264\n \u2191addRothNumber (range M) + \u2191addRothNumber (map (addLeftEmbedding M) (range N))\n[PROOFSTEP]\nexact addRothNumber_union_le _ _\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Finset \u2115\nk n a b : \u2115\n\u22a2 \u2191addRothNumber (Ico a b) = \u2191rothNumberNat (b - a)\n[PROOFSTEP]\nobtain h | h := le_total b a\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Finset \u2115\nk n a b : \u2115\nh : b \u2264 a\n\u22a2 \u2191addRothNumber (Ico a b) = \u2191rothNumberNat (b - a)\n[PROOFSTEP]\nrw [tsub_eq_zero_of_le h, Ico_eq_empty_of_le h, rothNumberNat_zero, addRothNumber_empty]\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Finset \u2115\nk n a b : \u2115\nh : a \u2264 b\n\u22a2 \u2191addRothNumber (Ico a b) = \u2191rothNumberNat (b - a)\n[PROOFSTEP]\nconvert addRothNumber_map_add_left _ a\n[GOAL]\ncase h.e'_2.h.e'_6\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Finset \u2115\nk n a b : \u2115\nh : a \u2264 b\n\u22a2 Ico a b = map (addLeftEmbedding a) (range (b - a))\n[PROOFSTEP]\nrw [range_eq_Ico, map_eq_image]\n[GOAL]\ncase h.e'_2.h.e'_6\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Finset \u2115\nk n a b : \u2115\nh : a \u2264 b\n\u22a2 Ico a b = image (\u2191(addLeftEmbedding a)) (Ico 0 (b - a))\n[PROOFSTEP]\nconvert (image_add_left_Ico 0 (b - a) _).symm\n[GOAL]\ncase h.e'_2.h.e'_5\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Finset \u2115\nk n a b : \u2115\nh : a \u2264 b\n\u22a2 b = a + (b - a)\n[PROOFSTEP]\nexact (add_tsub_cancel_of_le h).symm\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\ud835\udd5c : Type u_4\nE : Type u_5\ns : Finset \u2115\nk n : \u2115\n\u22a2 \u2200 (x : \u2115), \u2016\u2191(\u2191rothNumberNat x)\u2016 \u2264 \u2016\u2191x\u2016\n[PROOFSTEP]\nsimpa only [Real.norm_coe_nat, Nat.cast_le] using rothNumberNat_le\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.Additive.SalemSpencer", "llama_tokens": 19835, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125848754471, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.42853636538808737}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nP : Karoubi (Karoubi C)\n\u22a2 P.p.f \u226b P.p.f = P.p.f\n[PROOFSTEP]\nsimpa only [hom_ext_iff, comp_f] using P.idem\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nP Q : Karoubi (Karoubi C)\nf : P \u27f6 Q\n\u22a2 P.p.f \u226b f.f.f = f.f.f \u226b Q.p.f\n[PROOFSTEP]\nsimpa only [hom_ext_iff, comp_f] using p_comm f\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.1759, u_1} C\nP : Karoubi (Karoubi C)\n\u22a2 P.p.f \u226b P.p.f = P.p.f\n[PROOFSTEP]\nsimpa only [hom_ext_iff] using P.idem\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.1759, u_1} C\nX\u271d Y\u271d : Karoubi (Karoubi C)\nf : X\u271d \u27f6 Y\u271d\n\u22a2 f.f.f = ((fun P => Karoubi.mk P.X.X P.p.f) X\u271d).p \u226b f.f.f \u226b ((fun P => Karoubi.mk P.X.X P.p.f) Y\u271d).p\n[PROOFSTEP]\nsimpa only [hom_ext_iff] using f.comm\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.7379, u_1} C\n\u22a2 \u2200 (X : Karoubi C), (\ud835\udfed (Karoubi C)).obj X = (toKaroubi (Karoubi C) \u22d9 inverse C).obj X\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.7379, u_1} C\n\u22a2 \u2200 (X Y : Karoubi C) (f : X \u27f6 Y),\n (\ud835\udfed (Karoubi C)).map f =\n eqToHom (_ : (\ud835\udfed (Karoubi C)).obj X = (toKaroubi (Karoubi C) \u22d9 inverse C).obj X) \u226b\n (toKaroubi (Karoubi C) \u22d9 inverse C).map f \u226b\n eqToHom (_ : (toKaroubi (Karoubi C) \u22d9 inverse C).obj Y = (\ud835\udfed (Karoubi C)).obj Y)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.11252, u_1} C\nP : Karoubi (Karoubi C)\n\u22a2 P.p.f = ((inverse C \u22d9 toKaroubi (Karoubi C)).obj P).X.p \u226b P.p.f \u226b ((\ud835\udfed (Karoubi (Karoubi C))).obj P).X.p\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.11252, u_1} C\nP : Karoubi (Karoubi C)\n\u22a2 Hom.mk P.p.f = ((inverse C \u22d9 toKaroubi (Karoubi C)).obj P).p \u226b Hom.mk P.p.f \u226b ((\ud835\udfed (Karoubi (Karoubi C))).obj P).p\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.11252, u_1} C\nP : Karoubi (Karoubi C)\n\u22a2 P.p.f = ((\ud835\udfed (Karoubi (Karoubi C))).obj P).X.p \u226b P.p.f \u226b ((inverse C \u22d9 toKaroubi (Karoubi C)).obj P).X.p\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.11252, u_1} C\nP : Karoubi (Karoubi C)\n\u22a2 Hom.mk P.p.f = ((\ud835\udfed (Karoubi (Karoubi C))).obj P).p \u226b Hom.mk P.p.f \u226b ((inverse C \u22d9 toKaroubi (Karoubi C)).obj P).p\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Idempotents.KaroubiKaroubi", "llama_tokens": 1204, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4285363590538048}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b2\ninst\u271d : CompleteLattice \u03b1\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2a06 (i : \u2115) (b : \u03b2) (_ : b \u2208 decode\u2082 \u03b2 i), f b = \u2a06 (b : \u03b2), f b\n[PROOFSTEP]\nrw [iSup_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b2\ninst\u271d : CompleteLattice \u03b1\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2a06 (j : \u03b2) (i : \u2115) (_ : j \u2208 decode\u2082 \u03b2 i), f j = \u2a06 (b : \u03b2), f b\n[PROOFSTEP]\nsimp only [mem_decode\u2082, iSup_iSup_eq_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nC : Set \u03b1 \u2192 Prop\nH0 : C \u2205\nH1 : \u2200 (b : \u03b2), C (f b)\nn : \u2115\n\u22a2 C (\u22c3 (b : \u03b2) (_ : b \u2208 none), f b)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nC : Set \u03b1 \u2192 Prop\nH0 : C \u2205\nH1 : \u2200 (b : \u03b2), C (f b)\nn : \u2115\n\u22a2 C \u2205\n[PROOFSTEP]\napply H0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nC : Set \u03b1 \u2192 Prop\nH0 : C \u2205\nH1 : \u2200 (b : \u03b2), C (f b)\nn : \u2115\nb : \u03b2\n\u22a2 C (\u22c3 (b_1 : \u03b2) (_ : b_1 \u2208 some b), f b_1)\n[PROOFSTEP]\nconvert H1 b\n[GOAL]\ncase h.e'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nC : Set \u03b1 \u2192 Prop\nH0 : C \u2205\nH1 : \u2200 (b : \u03b2), C (f b)\nn : \u2115\nb : \u03b2\n\u22a2 \u22c3 (b_1 : \u03b2) (_ : b_1 \u2208 some b), f b_1 = f b\n[PROOFSTEP]\nsimp [ext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\n\u22a2 Pairwise (Disjoint on fun i => \u22c3 (b : \u03b2) (_ : b \u2208 decode\u2082 \u03b2 i), f b)\n[PROOFSTEP]\nrintro i j ij\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\ni j : \u2115\nij : i \u2260 j\n\u22a2 (Disjoint on fun i => \u22c3 (b : \u03b2) (_ : b \u2208 decode\u2082 \u03b2 i), f b) i j\n[PROOFSTEP]\nrefine' disjoint_left.mpr fun x => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\ni j : \u2115\nij : i \u2260 j\nx : \u03b1\n\u22a2 x \u2208 (fun i => \u22c3 (b : \u03b2) (_ : b \u2208 decode\u2082 \u03b2 i), f b) i \u2192 \u00acx \u2208 (fun i => \u22c3 (b : \u03b2) (_ : b \u2208 decode\u2082 \u03b2 i), f b) j\n[PROOFSTEP]\nsuffices \u2200 a, encode a = i \u2192 x \u2208 f a \u2192 \u2200 b, encode b = j \u2192 x \u2209 f b by simpa [decode\u2082_eq_some]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\ni j : \u2115\nij : i \u2260 j\nx : \u03b1\nthis : \u2200 (a : \u03b2), encode a = i \u2192 x \u2208 f a \u2192 \u2200 (b : \u03b2), encode b = j \u2192 \u00acx \u2208 f b\n\u22a2 x \u2208 (fun i => \u22c3 (b : \u03b2) (_ : b \u2208 decode\u2082 \u03b2 i), f b) i \u2192 \u00acx \u2208 (fun i => \u22c3 (b : \u03b2) (_ : b \u2208 decode\u2082 \u03b2 i), f b) j\n[PROOFSTEP]\nsimpa [decode\u2082_eq_some]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\ni j : \u2115\nij : i \u2260 j\nx : \u03b1\n\u22a2 \u2200 (a : \u03b2), encode a = i \u2192 x \u2208 f a \u2192 \u2200 (b : \u03b2), encode b = j \u2192 \u00acx \u2208 f b\n[PROOFSTEP]\nrintro a rfl ha b rfl hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nx : \u03b1\na : \u03b2\nha : x \u2208 f a\nb : \u03b2\nij : encode a \u2260 encode b\nhb : x \u2208 f b\n\u22a2 False\n[PROOFSTEP]\nexact (hd (mt (congr_arg encode) ij)).le_bot \u27e8ha, hb\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Logic.Encodable.Lattice", "llama_tokens": 1533, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.752012562644147, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.42853635271952206}} {"text": "[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\ns' : Finset \u2191{x' | x' \u2260 x}\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => erase (t \u2191x') a)\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b9\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\ns' : Finset \u2191{x' | x' \u2260 x}\nthis : DecidableEq \u03b9\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => erase (t \u2191x') a)\n[PROOFSTEP]\nspecialize ha (s'.image fun z => z.1)\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\ns' : Finset \u2191{x' | x' \u2260 x}\nthis : DecidableEq \u03b9\nha :\n Finset.Nonempty (image (fun z => \u2191z) s') \u2192\n image (fun z => \u2191z) s' \u2260 univ \u2192 card (image (fun z => \u2191z) s') < card (Finset.biUnion (image (fun z => \u2191z) s') t)\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => erase (t \u2191x') a)\n[PROOFSTEP]\nrw [Nonempty.image_iff, Finset.card_image_of_injective s' Subtype.coe_injective] at ha \n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\ns' : Finset \u2191{x' | x' \u2260 x}\nthis : DecidableEq \u03b9\nha : Finset.Nonempty s' \u2192 image (fun z => \u2191z) s' \u2260 univ \u2192 card s' < card (Finset.biUnion (image (fun z => \u2191z) s') t)\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => erase (t \u2191x') a)\n[PROOFSTEP]\nby_cases he : s'.Nonempty\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\ns' : Finset \u2191{x' | x' \u2260 x}\nthis : DecidableEq \u03b9\nha : Finset.Nonempty s' \u2192 image (fun z => \u2191z) s' \u2260 univ \u2192 card s' < card (Finset.biUnion (image (fun z => \u2191z) s') t)\nhe : Finset.Nonempty s'\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => erase (t \u2191x') a)\n[PROOFSTEP]\nhave ha' : s'.card < (s'.biUnion fun x => t x).card :=\n by\n convert ha he fun h => by simpa [\u2190 h] using mem_univ x using 2\n ext x\n simp only [mem_image, mem_biUnion, exists_prop, SetCoe.exists, exists_and_right, exists_eq_right, Subtype.coe_mk]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\ns' : Finset \u2191{x' | x' \u2260 x}\nthis : DecidableEq \u03b9\nha : Finset.Nonempty s' \u2192 image (fun z => \u2191z) s' \u2260 univ \u2192 card s' < card (Finset.biUnion (image (fun z => \u2191z) s') t)\nhe : Finset.Nonempty s'\n\u22a2 card s' < card (Finset.biUnion s' fun x_1 => t \u2191x_1)\n[PROOFSTEP]\nconvert ha he fun h => by simpa [\u2190 h] using mem_univ x using 2\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\ns' : Finset \u2191{x' | x' \u2260 x}\nthis : DecidableEq \u03b9\nha : Finset.Nonempty s' \u2192 image (fun z => \u2191z) s' \u2260 univ \u2192 card s' < card (Finset.biUnion (image (fun z => \u2191z) s') t)\nhe : Finset.Nonempty s'\nh : image (fun z => \u2191z) s' = univ\n\u22a2 False\n[PROOFSTEP]\nsimpa [\u2190 h] using mem_univ x\n[GOAL]\ncase h.e'_4.h.e'_2\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\ns' : Finset \u2191{x' | x' \u2260 x}\nthis : DecidableEq \u03b9\nha : Finset.Nonempty s' \u2192 image (fun z => \u2191z) s' \u2260 univ \u2192 card s' < card (Finset.biUnion (image (fun z => \u2191z) s') t)\nhe : Finset.Nonempty s'\n\u22a2 (Finset.biUnion s' fun x_1 => t \u2191x_1) = Finset.biUnion (image (fun z => \u2191z) s') t\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_4.h.e'_2.a\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx\u271d : \u03b9\na : \u03b1\ns' : Finset \u2191{x' | x' \u2260 x\u271d}\nthis : DecidableEq \u03b9\nha : Finset.Nonempty s' \u2192 image (fun z => \u2191z) s' \u2260 univ \u2192 card s' < card (Finset.biUnion (image (fun z => \u2191z) s') t)\nhe : Finset.Nonempty s'\nx : \u03b1\n\u22a2 (x \u2208 Finset.biUnion s' fun x => t \u2191x) \u2194 x \u2208 Finset.biUnion (image (fun z => \u2191z) s') t\n[PROOFSTEP]\nsimp only [mem_image, mem_biUnion, exists_prop, SetCoe.exists, exists_and_right, exists_eq_right, Subtype.coe_mk]\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\ns' : Finset \u2191{x' | x' \u2260 x}\nthis : DecidableEq \u03b9\nha : Finset.Nonempty s' \u2192 image (fun z => \u2191z) s' \u2260 univ \u2192 card s' < card (Finset.biUnion (image (fun z => \u2191z) s') t)\nhe : Finset.Nonempty s'\nha' : card s' < card (Finset.biUnion s' fun x_1 => t \u2191x_1)\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => erase (t \u2191x') a)\n[PROOFSTEP]\nrw [\u2190 erase_biUnion]\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\ns' : Finset \u2191{x' | x' \u2260 x}\nthis : DecidableEq \u03b9\nha : Finset.Nonempty s' \u2192 image (fun z => \u2191z) s' \u2260 univ \u2192 card s' < card (Finset.biUnion (image (fun z => \u2191z) s') t)\nhe : Finset.Nonempty s'\nha' : card s' < card (Finset.biUnion s' fun x_1 => t \u2191x_1)\n\u22a2 card s' \u2264 card (erase (Finset.biUnion s' fun x' => t \u2191x') a)\n[PROOFSTEP]\nby_cases hb : a \u2208 s'.biUnion fun x => t x\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\ns' : Finset \u2191{x' | x' \u2260 x}\nthis : DecidableEq \u03b9\nha : Finset.Nonempty s' \u2192 image (fun z => \u2191z) s' \u2260 univ \u2192 card s' < card (Finset.biUnion (image (fun z => \u2191z) s') t)\nhe : Finset.Nonempty s'\nha' : card s' < card (Finset.biUnion s' fun x_1 => t \u2191x_1)\nhb : a \u2208 Finset.biUnion s' fun x_1 => t \u2191x_1\n\u22a2 card s' \u2264 card (erase (Finset.biUnion s' fun x' => t \u2191x') a)\n[PROOFSTEP]\nrw [card_erase_of_mem hb]\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\ns' : Finset \u2191{x' | x' \u2260 x}\nthis : DecidableEq \u03b9\nha : Finset.Nonempty s' \u2192 image (fun z => \u2191z) s' \u2260 univ \u2192 card s' < card (Finset.biUnion (image (fun z => \u2191z) s') t)\nhe : Finset.Nonempty s'\nha' : card s' < card (Finset.biUnion s' fun x_1 => t \u2191x_1)\nhb : a \u2208 Finset.biUnion s' fun x_1 => t \u2191x_1\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x_1 => t \u2191x_1) - 1\n[PROOFSTEP]\nexact Nat.le_pred_of_lt ha'\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\ns' : Finset \u2191{x' | x' \u2260 x}\nthis : DecidableEq \u03b9\nha : Finset.Nonempty s' \u2192 image (fun z => \u2191z) s' \u2260 univ \u2192 card s' < card (Finset.biUnion (image (fun z => \u2191z) s') t)\nhe : Finset.Nonempty s'\nha' : card s' < card (Finset.biUnion s' fun x_1 => t \u2191x_1)\nhb : \u00aca \u2208 Finset.biUnion s' fun x_1 => t \u2191x_1\n\u22a2 card s' \u2264 card (erase (Finset.biUnion s' fun x' => t \u2191x') a)\n[PROOFSTEP]\nrw [erase_eq_of_not_mem hb]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\ns' : Finset \u2191{x' | x' \u2260 x}\nthis : DecidableEq \u03b9\nha : Finset.Nonempty s' \u2192 image (fun z => \u2191z) s' \u2260 univ \u2192 card s' < card (Finset.biUnion (image (fun z => \u2191z) s') t)\nhe : Finset.Nonempty s'\nha' : card s' < card (Finset.biUnion s' fun x_1 => t \u2191x_1)\nhb : \u00aca \u2208 Finset.biUnion s' fun x_1 => t \u2191x_1\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x_1 => t \u2191x_1)\n[PROOFSTEP]\nexact Nat.le_of_lt ha'\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\ns' : Finset \u2191{x' | x' \u2260 x}\nthis : DecidableEq \u03b9\nha : Finset.Nonempty s' \u2192 image (fun z => \u2191z) s' \u2260 univ \u2192 card s' < card (Finset.biUnion (image (fun z => \u2191z) s') t)\nhe : \u00acFinset.Nonempty s'\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => erase (t \u2191x') a)\n[PROOFSTEP]\nrw [nonempty_iff_ne_empty, not_not] at he \n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\ns' : Finset \u2191{x' | x' \u2260 x}\nthis : DecidableEq \u03b9\nha : Finset.Nonempty s' \u2192 image (fun z => \u2191z) s' \u2260 univ \u2192 card s' < card (Finset.biUnion (image (fun z => \u2191z) s') t)\nhe : s' = \u2205\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => erase (t \u2191x') a)\n[PROOFSTEP]\nsubst s'\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nx : \u03b9\na : \u03b1\nthis : DecidableEq \u03b9\nha : Finset.Nonempty \u2205 \u2192 image (fun z => \u2191z) \u2205 \u2260 univ \u2192 card \u2205 < card (Finset.biUnion (image (fun z => \u2191z) \u2205) t)\n\u22a2 card \u2205 \u2264 card (Finset.biUnion \u2205 fun x' => erase (t \u2191x') a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nhaveI : Nonempty \u03b9 := Fintype.card_pos_iff.mp (hn.symm \u25b8 Nat.succ_pos _)\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis : Nonempty \u03b9\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b9\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nlet x := Classical.arbitrary \u03b9\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nhave tx_ne : (t x).Nonempty := by\n rw [\u2190 Finset.card_pos]\n calc\n 0 < 1 := Nat.one_pos\n _ \u2264 (Finset.biUnion { x } t).card := (ht { x })\n _ = (t x).card := by rw [Finset.singleton_biUnion]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\n\u22a2 Finset.Nonempty (t x)\n[PROOFSTEP]\nrw [\u2190 Finset.card_pos]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\n\u22a2 0 < card (t x)\n[PROOFSTEP]\ncalc\n 0 < 1 := Nat.one_pos\n _ \u2264 (Finset.biUnion { x } t).card := (ht { x })\n _ = (t x).card := by rw [Finset.singleton_biUnion]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\n\u22a2 card (Finset.biUnion {x} t) = card (t x)\n[PROOFSTEP]\nrw [Finset.singleton_biUnion]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ntx_ne : Finset.Nonempty (t x)\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nchoose y hy using tx_ne\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nlet \u03b9' := {x' : \u03b9 | x' \u2260 x}\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nlet t' : \u03b9' \u2192 Finset \u03b1 := fun x' => (t x').erase y\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nhave card_\u03b9' : Fintype.card \u03b9' = n :=\n calc\n Fintype.card \u03b9' = Fintype.card \u03b9 - 1 := Set.card_ne_eq _\n _ = n := by rw [hn, Nat.add_succ_sub_one, add_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\n\u22a2 Fintype.card \u03b9 - 1 = n\n[PROOFSTEP]\nrw [hn, Nat.add_succ_sub_one, add_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nrcases ih t' card_\u03b9'.le (hall_cond_of_erase y ha) with\n \u27e8f', hfinj, hfr\u27e9\n -- Extend the resulting function.\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nrefine' \u27e8fun z => if h : z = x then y else f' \u27e8z, h\u27e9, _, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\n\u22a2 Function.Injective fun z => if h : z = x then y else f' { val := z, property := h }\n[PROOFSTEP]\nrintro z\u2081 z\u2082\n[GOAL]\ncase intro.intro.refine'_1\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\nz\u2081 z\u2082 : \u03b9\n\u22a2 (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2081 =\n (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2082 \u2192\n z\u2081 = z\u2082\n[PROOFSTEP]\nhave key : \u2200 {x}, y \u2260 f' x := by\n intro x h\n simpa [\u2190 h] using hfr x\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\nz\u2081 z\u2082 : \u03b9\n\u22a2 \u2200 {x : \u2191\u03b9'}, y \u2260 f' x\n[PROOFSTEP]\nintro x h\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx\u271d : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\u271d\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x\u271d}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\nz\u2081 z\u2082 : \u03b9\nx : \u2191\u03b9'\nh : y = f' x\n\u22a2 False\n[PROOFSTEP]\nsimpa [\u2190 h] using hfr x\n[GOAL]\ncase intro.intro.refine'_1\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\nz\u2081 z\u2082 : \u03b9\nkey : \u2200 {x : \u2191\u03b9'}, y \u2260 f' x\n\u22a2 (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2081 =\n (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2082 \u2192\n z\u2081 = z\u2082\n[PROOFSTEP]\nby_cases h\u2081 : z\u2081 = x\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\nz\u2081 z\u2082 : \u03b9\nkey : \u2200 {x : \u2191\u03b9'}, y \u2260 f' x\nh\u2081 : z\u2081 = x\n\u22a2 (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2081 =\n (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2082 \u2192\n z\u2081 = z\u2082\n[PROOFSTEP]\nby_cases h\u2082 : z\u2082 = x\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\nz\u2081 z\u2082 : \u03b9\nkey : \u2200 {x : \u2191\u03b9'}, y \u2260 f' x\nh\u2081 : \u00acz\u2081 = x\n\u22a2 (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2081 =\n (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2082 \u2192\n z\u2081 = z\u2082\n[PROOFSTEP]\nby_cases h\u2082 : z\u2082 = x\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\nz\u2081 z\u2082 : \u03b9\nkey : \u2200 {x : \u2191\u03b9'}, y \u2260 f' x\nh\u2081 : z\u2081 = x\nh\u2082 : z\u2082 = x\n\u22a2 (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2081 =\n (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2082 \u2192\n z\u2081 = z\u2082\n[PROOFSTEP]\nsimp [h\u2081, h\u2082, hfinj.eq_iff, key, key.symm]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\nz\u2081 z\u2082 : \u03b9\nkey : \u2200 {x : \u2191\u03b9'}, y \u2260 f' x\nh\u2081 : z\u2081 = x\nh\u2082 : \u00acz\u2082 = x\n\u22a2 (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2081 =\n (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2082 \u2192\n z\u2081 = z\u2082\n[PROOFSTEP]\nsimp [h\u2081, h\u2082, hfinj.eq_iff, key, key.symm]\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\nz\u2081 z\u2082 : \u03b9\nkey : \u2200 {x : \u2191\u03b9'}, y \u2260 f' x\nh\u2081 : \u00acz\u2081 = x\nh\u2082 : z\u2082 = x\n\u22a2 (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2081 =\n (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2082 \u2192\n z\u2081 = z\u2082\n[PROOFSTEP]\nsimp [h\u2081, h\u2082, hfinj.eq_iff, key, key.symm]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\nz\u2081 z\u2082 : \u03b9\nkey : \u2200 {x : \u2191\u03b9'}, y \u2260 f' x\nh\u2081 : \u00acz\u2081 = x\nh\u2082 : \u00acz\u2082 = x\n\u22a2 (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2081 =\n (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2082 \u2192\n z\u2081 = z\u2082\n[PROOFSTEP]\nsimp [h\u2081, h\u2082, hfinj.eq_iff, key, key.symm]\n[GOAL]\ncase intro.intro.refine'_2\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\n\u22a2 \u2200 (x_1 : \u03b9), (fun z => if h : z = x then y else f' { val := z, property := h }) x_1 \u2208 t x_1\n[PROOFSTEP]\nintro z\n[GOAL]\ncase intro.intro.refine'_2\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\nz : \u03b9\n\u22a2 (fun z => if h : z = x then y else f' { val := z, property := h }) z \u2208 t z\n[PROOFSTEP]\nsimp only [ne_eq, Set.mem_setOf_eq]\n[GOAL]\ncase intro.intro.refine'_2\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\nz : \u03b9\n\u22a2 (if h : z = Classical.arbitrary \u03b9 then y else f' { val := z, property := h }) \u2208 t z\n[PROOFSTEP]\nsplit_ifs with hz\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\nz : \u03b9\nhz : z = Classical.arbitrary \u03b9\n\u22a2 y \u2208 t z\n[PROOFSTEP]\nrwa [hz]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nhfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x\nz : \u03b9\nhz : \u00acz = Classical.arbitrary \u03b9\n\u22a2 f' { val := z, property := hz } \u2208 t z\n[PROOFSTEP]\nspecialize hfr \u27e8z, hz\u27e9\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nz : \u03b9\nhz : \u00acz = Classical.arbitrary \u03b9\nhfr : f' { val := z, property := hz } \u2208 t' { val := z, property := hz }\n\u22a2 f' { val := z, property := hz } \u2208 t z\n[PROOFSTEP]\nrw [mem_erase] at hfr \n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\nthis\u271d : Nonempty \u03b9\nthis : DecidableEq \u03b9\nx : \u03b9 := Classical.arbitrary \u03b9\ny : \u03b1\nhy : y \u2208 t x\n\u03b9' : Set \u03b9 := {x' | x' \u2260 x}\nt' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y\ncard_\u03b9' : Fintype.card \u2191\u03b9' = n\nf' : \u2191\u03b9' \u2192 \u03b1\nhfinj : Function.Injective f'\nz : \u03b9\nhz : \u00acz = Classical.arbitrary \u03b9\nhfr : f' { val := z, property := hz } \u2260 y \u2227 f' { val := z, property := hz } \u2208 t \u2191{ val := z, property := hz }\n\u22a2 f' { val := z, property := hz } \u2208 t z\n[PROOFSTEP]\nexact hfr.2\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191\u2191s\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun a' => t \u2191a')\n[PROOFSTEP]\nclassical\nrw [\u2190 card_image_of_injective s' Subtype.coe_injective]\nconvert ht (s'.image fun z => z.1) using 1\napply congr_arg\next y\nsimp\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191\u2191s\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun a' => t \u2191a')\n[PROOFSTEP]\nrw [\u2190 card_image_of_injective s' Subtype.coe_injective]\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191\u2191s\n\u22a2 card (image (fun a => \u2191a) s') \u2264 card (Finset.biUnion s' fun a' => t \u2191a')\n[PROOFSTEP]\nconvert ht (s'.image fun z => z.1) using 1\n[GOAL]\ncase h.e'_4\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191\u2191s\n\u22a2 card (Finset.biUnion s' fun a' => t \u2191a') = card (Finset.biUnion (image (fun z => \u2191z) s') t)\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h.e'_4.h\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191\u2191s\n\u22a2 (Finset.biUnion s' fun a' => t \u2191a') = Finset.biUnion (image (fun z => \u2191z) s') t\n[PROOFSTEP]\next y\n[GOAL]\ncase h.e'_4.h.a\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191\u2191s\ny : \u03b1\n\u22a2 (y \u2208 Finset.biUnion s' fun a' => t \u2191a') \u2194 y \u2208 Finset.biUnion (image (fun z => \u2191z) s') t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s)\u1d9c\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b9\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis : DecidableEq \u03b9\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)\n[PROOFSTEP]\nhave disj : Disjoint s (s'.image fun z => z.1) :=\n by\n simp only [disjoint_left, not_exists, mem_image, exists_prop, SetCoe.exists, exists_and_right, exists_eq_right,\n Subtype.coe_mk]\n intro x hx hc _\n exact absurd hx hc\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis : DecidableEq \u03b9\n\u22a2 Disjoint s (image (fun z => \u2191z) s')\n[PROOFSTEP]\nsimp only [disjoint_left, not_exists, mem_image, exists_prop, SetCoe.exists, exists_and_right, exists_eq_right,\n Subtype.coe_mk]\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis : DecidableEq \u03b9\n\u22a2 \u2200 \u2983a : \u03b9\u2984, a \u2208 s \u2192 \u2200 (x : a \u2208 (\u2191s)\u1d9c), \u00ac{ val := a, property := (_ : a \u2208 (\u2191s)\u1d9c) } \u2208 s'\n[PROOFSTEP]\nintro x hx hc _\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis : DecidableEq \u03b9\nx : \u03b9\nhx : x \u2208 s\nhc : x \u2208 (\u2191s)\u1d9c\na\u271d : { val := x, property := (_ : x \u2208 (\u2191s)\u1d9c) } \u2208 s'\n\u22a2 False\n[PROOFSTEP]\nexact absurd hx hc\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)\n[PROOFSTEP]\nhave : s'.card = (s \u222a s'.image fun z => z.1).card - s.card := by\n simp [disj, card_image_of_injective _ Subtype.coe_injective]\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\n\u22a2 card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n[PROOFSTEP]\nsimp [disj, card_image_of_injective _ Subtype.coe_injective]\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)\n[PROOFSTEP]\nrw [this, hus]\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 card (s \u222a image (fun z => \u2191z) s') - card (Finset.biUnion s t) \u2264\n card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)\n[PROOFSTEP]\nrefine' (tsub_le_tsub_right (ht _) _).trans _\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 card (Finset.biUnion (s \u222a image (fun z => \u2191z) s') t) - card (Finset.biUnion s t) \u2264\n card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)\n[PROOFSTEP]\nrw [\u2190 card_sdiff]\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 card (Finset.biUnion (s \u222a image (fun z => \u2191z) s') t \\ Finset.biUnion s t) \u2264\n card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)\n[PROOFSTEP]\nrefine' (card_le_of_subset _).trans le_rfl\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 Finset.biUnion (s \u222a image (fun z => \u2191z) s') t \\ Finset.biUnion s t \u2286\n Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t\n[PROOFSTEP]\nintro t\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d\u00b9 : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt\u271d : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t\u271d)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t\u271d)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\nt : \u03b1\n\u22a2 t \u2208 Finset.biUnion (s \u222a image (fun z => \u2191z) s') t\u271d \\ Finset.biUnion s t\u271d \u2192\n t \u2208 Finset.biUnion s' fun x' => t\u271d \u2191x' \\ Finset.biUnion s t\u271d\n[PROOFSTEP]\nsimp only [mem_biUnion, mem_sdiff, not_exists, mem_image, and_imp, mem_union, exists_and_right, exists_imp]\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d\u00b9 : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt\u271d : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t\u271d)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t\u271d)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\nt : \u03b1\n\u22a2 \u2200 (x : \u03b9),\n (x \u2208 s \u2228 \u2203 a, a \u2208 s' \u2227 \u2191a = x) \u2192\n t \u2208 t\u271d x \u2192 (\u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)) \u2192 \u2203 a, a \u2208 s' \u2227 t \u2208 t\u271d \u2191a \u2227 \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)\n[PROOFSTEP]\nrintro x (hx | \u27e8x', hx', rfl\u27e9) rat hs\n[GOAL]\ncase inl\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d\u00b9 : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt\u271d : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t\u271d)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t\u271d)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\nt : \u03b1\nx : \u03b9\nhx : x \u2208 s\nrat : t \u2208 t\u271d x\nhs : \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)\n\u22a2 \u2203 a, a \u2208 s' \u2227 t \u2208 t\u271d \u2191a \u2227 \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)\n[PROOFSTEP]\nexact False.elim <| (hs x) <| And.intro hx rat\n[GOAL]\ncase inr.intro.intro\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d\u00b9 : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt\u271d : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t\u271d)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t\u271d)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\nt : \u03b1\nx' : \u2191(\u2191s)\u1d9c\nhx' : x' \u2208 s'\nrat : t \u2208 t\u271d \u2191x'\nhs : \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)\n\u22a2 \u2203 a, a \u2208 s' \u2227 t \u2208 t\u271d \u2191a \u2227 \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)\n[PROOFSTEP]\nuse x', hx', rat, hs\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 Finset.biUnion s t \u2286 Finset.biUnion (s \u222a image (fun z => \u2191z) s') t\n[PROOFSTEP]\napply biUnion_subset_biUnion_of_subset_left\n[GOAL]\ncase h\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s)\u1d9c\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 s \u2286 s \u222a image (fun z => \u2191z) s'\n[PROOFSTEP]\napply subset_union_left\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b9\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = n + 1\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nrw [Nat.add_one] at hn \n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nhave card_\u03b9'_le : Fintype.card s \u2264 n := by\n apply Nat.le_of_lt_succ\n calc\n Fintype.card s = s.card := Fintype.card_coe _\n _ < Fintype.card \u03b9 := ((card_lt_iff_ne_univ _).mpr hns)\n _ = n.succ := hn\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\n\u22a2 Fintype.card { x // x \u2208 s } \u2264 n\n[PROOFSTEP]\napply Nat.le_of_lt_succ\n[GOAL]\ncase a\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\n\u22a2 Fintype.card { x // x \u2208 s } < Nat.succ n\n[PROOFSTEP]\ncalc\n Fintype.card s = s.card := Fintype.card_coe _\n _ < Fintype.card \u03b9 := ((card_lt_iff_ne_univ _).mpr hns)\n _ = n.succ := hn\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nlet t' : s \u2192 Finset \u03b1 := fun x' => t x'\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nrcases ih t' card_\u03b9'_le (hall_cond_of_restrict ht) with\n \u27e8f', hf', hsf'\u27e9\n -- Restrict to `s\u1d9c` in the domain and `(s.biUnion t)\u1d9c` in the codomain.\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nset \u03b9'' := (s : Set \u03b9)\u1d9c\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nlet t'' : \u03b9'' \u2192 Finset \u03b1 := fun a'' => t a'' \\ s.biUnion t\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nhave card_\u03b9''_le : Fintype.card \u03b9'' \u2264 n :=\n by\n simp_rw [\u2190 Nat.lt_succ_iff, \u2190 hn, \u2190 Finset.coe_compl, coe_sort_coe]\n rwa [Fintype.card_coe, card_compl_lt_iff_nonempty]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\n\u22a2 Fintype.card \u2191\u03b9'' \u2264 n\n[PROOFSTEP]\nsimp_rw [\u2190 Nat.lt_succ_iff, \u2190 hn, \u2190 Finset.coe_compl, coe_sort_coe]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\n\u22a2 Fintype.card { x // x \u2208 s\u1d9c } < Fintype.card \u03b9\n[PROOFSTEP]\nrwa [Fintype.card_coe, card_compl_lt_iff_nonempty]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nrcases ih t'' card_\u03b9''_le (hall_cond_of_compl hus ht) with\n \u27e8f'', hf'', hsf''\u27e9\n -- Put them together\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nhave f'_mem_biUnion : \u2200 (x') (hx' : x' \u2208 s), f' \u27e8x', hx'\u27e9 \u2208 s.biUnion t :=\n by\n intro x' hx'\n rw [mem_biUnion]\n exact \u27e8x', hx', hsf' _\u27e9\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\n\u22a2 \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\n[PROOFSTEP]\nintro x' hx'\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nx' : \u03b9\nhx' : x' \u2208 s\n\u22a2 f' { val := x', property := hx' } \u2208 Finset.biUnion s t\n[PROOFSTEP]\nrw [mem_biUnion]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nx' : \u03b9\nhx' : x' \u2208 s\n\u22a2 \u2203 a, a \u2208 s \u2227 f' { val := x', property := hx' } \u2208 t a\n[PROOFSTEP]\nexact \u27e8x', hx', hsf' _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nhave f''_not_mem_biUnion : \u2200 (x'') (hx'' : \u00acx'' \u2208 s), \u00acf'' \u27e8x'', hx''\u27e9 \u2208 s.biUnion t :=\n by\n intro x'' hx''\n have h := hsf'' \u27e8x'', hx''\u27e9\n rw [mem_sdiff] at h \n exact h.2\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\n\u22a2 \u2200 (x'' : \u03b9) (hx'' : \u00acx'' \u2208 s), \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\n[PROOFSTEP]\nintro x'' hx''\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\nx'' : \u03b9\nhx'' : \u00acx'' \u2208 s\n\u22a2 \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\n[PROOFSTEP]\nhave h := hsf'' \u27e8x'', hx''\u27e9\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\nx'' : \u03b9\nhx'' : \u00acx'' \u2208 s\nh : f'' { val := x'', property := hx'' } \u2208 t'' { val := x'', property := hx'' }\n\u22a2 \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\n[PROOFSTEP]\nrw [mem_sdiff] at h \n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\nx'' : \u03b9\nhx'' : \u00acx'' \u2208 s\nh :\n f'' { val := x'', property := hx'' } \u2208 t \u2191{ val := x'', property := hx'' } \u2227\n \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\n\u22a2 \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\n[PROOFSTEP]\nexact h.2\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\nf''_not_mem_biUnion : \u2200 (x'' : \u03b9) (hx'' : \u00acx'' \u2208 s), \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nhave im_disj : \u2200 (x' x'' : \u03b9) (hx' : x' \u2208 s) (hx'' : \u00acx'' \u2208 s), f' \u27e8x', hx'\u27e9 \u2260 f'' \u27e8x'', hx''\u27e9 :=\n by\n intro x x' hx' hx'' h\n apply f''_not_mem_biUnion x' hx''\n rw [\u2190 h]\n apply f'_mem_biUnion x\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\nf''_not_mem_biUnion : \u2200 (x'' : \u03b9) (hx'' : \u00acx'' \u2208 s), \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\n\u22a2 \u2200 (x' x'' : \u03b9) (hx' : x' \u2208 s) (hx'' : \u00acx'' \u2208 s),\n f' { val := x', property := hx' } \u2260 f'' { val := x'', property := hx'' }\n[PROOFSTEP]\nintro x x' hx' hx'' h\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\nf''_not_mem_biUnion : \u2200 (x'' : \u03b9) (hx'' : \u00acx'' \u2208 s), \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\nx x' : \u03b9\nhx' : x \u2208 s\nhx'' : \u00acx' \u2208 s\nh : f' { val := x, property := hx' } = f'' { val := x', property := hx'' }\n\u22a2 False\n[PROOFSTEP]\napply f''_not_mem_biUnion x' hx''\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\nf''_not_mem_biUnion : \u2200 (x'' : \u03b9) (hx'' : \u00acx'' \u2208 s), \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\nx x' : \u03b9\nhx' : x \u2208 s\nhx'' : \u00acx' \u2208 s\nh : f' { val := x, property := hx' } = f'' { val := x', property := hx'' }\n\u22a2 f'' { val := x', property := hx'' } \u2208 Finset.biUnion s t\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\nf''_not_mem_biUnion : \u2200 (x'' : \u03b9) (hx'' : \u00acx'' \u2208 s), \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\nx x' : \u03b9\nhx' : x \u2208 s\nhx'' : \u00acx' \u2208 s\nh : f' { val := x, property := hx' } = f'' { val := x', property := hx'' }\n\u22a2 f' { val := x, property := hx' } \u2208 Finset.biUnion s t\n[PROOFSTEP]\napply f'_mem_biUnion x\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\nf''_not_mem_biUnion : \u2200 (x'' : \u03b9) (hx'' : \u00acx'' \u2208 s), \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\nim_disj :\n \u2200 (x' x'' : \u03b9) (hx' : x' \u2208 s) (hx'' : \u00acx'' \u2208 s),\n f' { val := x', property := hx' } \u2260 f'' { val := x'', property := hx'' }\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nrefine' \u27e8fun x => if h : x \u2208 s then f' \u27e8x, h\u27e9 else f'' \u27e8x, h\u27e9, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\nf''_not_mem_biUnion : \u2200 (x'' : \u03b9) (hx'' : \u00acx'' \u2208 s), \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\nim_disj :\n \u2200 (x' x'' : \u03b9) (hx' : x' \u2208 s) (hx'' : \u00acx'' \u2208 s),\n f' { val := x', property := hx' } \u2260 f'' { val := x'', property := hx'' }\n\u22a2 Function.Injective fun x => if h : x \u2208 s then f' { val := x, property := h } else f'' { val := x, property := h }\n[PROOFSTEP]\nrefine' hf'.dite _ hf'' (@fun x x' => im_disj x x' _ _)\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\nf''_not_mem_biUnion : \u2200 (x'' : \u03b9) (hx'' : \u00acx'' \u2208 s), \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\nim_disj :\n \u2200 (x' x'' : \u03b9) (hx' : x' \u2208 s) (hx'' : \u00acx'' \u2208 s),\n f' { val := x', property := hx' } \u2260 f'' { val := x'', property := hx'' }\n\u22a2 \u2200 (x : \u03b9), (fun x => if h : x \u2208 s then f' { val := x, property := h } else f'' { val := x, property := h }) x \u2208 t x\n[PROOFSTEP]\nintro x\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\nf''_not_mem_biUnion : \u2200 (x'' : \u03b9) (hx'' : \u00acx'' \u2208 s), \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\nim_disj :\n \u2200 (x' x'' : \u03b9) (hx' : x' \u2208 s) (hx'' : \u00acx'' \u2208 s),\n f' { val := x', property := hx' } \u2260 f'' { val := x'', property := hx'' }\nx : \u03b9\n\u22a2 (fun x => if h : x \u2208 s then f' { val := x, property := h } else f'' { val := x, property := h }) x \u2208 t x\n[PROOFSTEP]\nsimp only [of_eq_true]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\nf''_not_mem_biUnion : \u2200 (x'' : \u03b9) (hx'' : \u00acx'' \u2208 s), \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\nim_disj :\n \u2200 (x' x'' : \u03b9) (hx' : x' \u2208 s) (hx'' : \u00acx'' \u2208 s),\n f' { val := x', property := hx' } \u2260 f'' { val := x'', property := hx'' }\nx : \u03b9\n\u22a2 (if h : x \u2208 s then f' { val := x, property := h } else f'' { val := x, property := h }) \u2208 t x\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\nf''_not_mem_biUnion : \u2200 (x'' : \u03b9) (hx'' : \u00acx'' \u2208 s), \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\nim_disj :\n \u2200 (x' x'' : \u03b9) (hx' : x' \u2208 s) (hx'' : \u00acx'' \u2208 s),\n f' { val := x', property := hx' } \u2260 f'' { val := x'', property := hx'' }\nx : \u03b9\nh : x \u2208 s\n\u22a2 f' { val := x, property := h } \u2208 t x\n[PROOFSTEP]\nexact hsf' \u27e8x, h\u27e9\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\nn : \u2115\nhn : Fintype.card \u03b9 = Nat.succ n\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nih :\n \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nhs : Finset.Nonempty s\nhns : s \u2260 univ\nhus : card s = card (Finset.biUnion s t)\nthis : DecidableEq \u03b9\ncard_\u03b9'_le : Fintype.card { x // x \u2208 s } \u2264 n\nt' : { x // x \u2208 s } \u2192 Finset \u03b1 := fun x' => t \u2191x'\nf' : { x // x \u2208 s } \u2192 \u03b1\nhf' : Function.Injective f'\nhsf' : \u2200 (x : { x // x \u2208 s }), f' x \u2208 t' x\n\u03b9'' : Set \u03b9 := (\u2191s)\u1d9c\nt'' : \u2191\u03b9'' \u2192 Finset \u03b1 := fun a'' => t \u2191a'' \\ Finset.biUnion s t\ncard_\u03b9''_le : Fintype.card \u2191\u03b9'' \u2264 n\nf'' : \u2191\u03b9'' \u2192 \u03b1\nhf'' : Function.Injective f''\nhsf'' : \u2200 (x : \u2191\u03b9''), f'' x \u2208 t'' x\nf'_mem_biUnion : \u2200 (x' : \u03b9) (hx' : x' \u2208 s), f' { val := x', property := hx' } \u2208 Finset.biUnion s t\nf''_not_mem_biUnion : \u2200 (x'' : \u03b9) (hx'' : \u00acx'' \u2208 s), \u00acf'' { val := x'', property := hx'' } \u2208 Finset.biUnion s t\nim_disj :\n \u2200 (x' x'' : \u03b9) (hx' : x' \u2208 s) (hx'' : \u00acx'' \u2208 s),\n f' { val := x', property := hx' } \u2260 f'' { val := x'', property := hx'' }\nx : \u03b9\nh : \u00acx \u2208 s\n\u22a2 f'' { val := x, property := h } \u2208 t x\n[PROOFSTEP]\nexact sdiff_subset _ _ (hsf'' \u27e8x, h\u27e9)\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Finite \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Finite \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\ninduction' hn : Fintype.card \u03b9 using Nat.strong_induction_on with n ih generalizing \u03b9\n[GOAL]\ncase intro.h\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d\u00b9 : Finite \u03b9\u271d\nht\u271d : \u2200 (s : Finset \u03b9\u271d), card s \u2264 card (Finset.biUnion s t\u271d)\nval\u271d\u00b9 : Fintype \u03b9\u271d\nx\u271d : \u2115\nhn\u271d : Fintype.card \u03b9\u271d = x\u271d\nn : \u2115\nih :\n \u2200 (m : \u2115),\n m < n \u2192\n \u2200 {\u03b9 : Type u} {t : \u03b9 \u2192 Finset \u03b1} [inst : Finite \u03b9],\n (\u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)) \u2192\n \u2200 (val : Fintype \u03b9), Fintype.card \u03b9 = m \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Finite \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nval\u271d : Fintype \u03b9\nhn : Fintype.card \u03b9 = n\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nrcases n with (_ | _)\n[GOAL]\ncase intro.h.zero\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d\u00b9 : Finite \u03b9\u271d\nht\u271d : \u2200 (s : Finset \u03b9\u271d), card s \u2264 card (Finset.biUnion s t\u271d)\nval\u271d\u00b9 : Fintype \u03b9\u271d\nx\u271d : \u2115\nhn\u271d : Fintype.card \u03b9\u271d = x\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Finite \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nval\u271d : Fintype \u03b9\nih :\n \u2200 (m : \u2115),\n m < Nat.zero \u2192\n \u2200 {\u03b9 : Type u} {t : \u03b9 \u2192 Finset \u03b1} [inst : Finite \u03b9],\n (\u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)) \u2192\n \u2200 (val : Fintype \u03b9), Fintype.card \u03b9 = m \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\nhn : Fintype.card \u03b9 = Nat.zero\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nrw [Fintype.card_eq_zero_iff] at hn \n[GOAL]\ncase intro.h.zero\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d\u00b9 : Finite \u03b9\u271d\nht\u271d : \u2200 (s : Finset \u03b9\u271d), card s \u2264 card (Finset.biUnion s t\u271d)\nval\u271d\u00b9 : Fintype \u03b9\u271d\nx\u271d : \u2115\nhn\u271d : Fintype.card \u03b9\u271d = x\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Finite \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nval\u271d : Fintype \u03b9\nih :\n \u2200 (m : \u2115),\n m < Nat.zero \u2192\n \u2200 {\u03b9 : Type u} {t : \u03b9 \u2192 Finset \u03b1} [inst : Finite \u03b9],\n (\u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)) \u2192\n \u2200 (val : Fintype \u03b9), Fintype.card \u03b9 = m \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\nhn : IsEmpty \u03b9\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nexact \u27e8isEmptyElim, isEmptyElim, isEmptyElim\u27e9\n[GOAL]\ncase intro.h.succ\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d\u00b9 : Finite \u03b9\u271d\nht\u271d : \u2200 (s : Finset \u03b9\u271d), card s \u2264 card (Finset.biUnion s t\u271d)\nval\u271d\u00b9 : Fintype \u03b9\u271d\nx\u271d : \u2115\nhn\u271d : Fintype.card \u03b9\u271d = x\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Finite \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nval\u271d : Fintype \u03b9\nn\u271d : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ n\u271d \u2192\n \u2200 {\u03b9 : Type u} {t : \u03b9 \u2192 Finset \u03b1} [inst : Finite \u03b9],\n (\u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)) \u2192\n \u2200 (val : Fintype \u03b9), Fintype.card \u03b9 = m \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\nhn : Fintype.card \u03b9 = Nat.succ n\u271d\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nhave ih' :\n \u2200 (\u03b9' : Type u) [Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 _ \u2192\n (\u2200 s' : Finset \u03b9', s'.card \u2264 (s'.biUnion t').card) \u2192 \u2203 f : \u03b9' \u2192 \u03b1, Function.Injective f \u2227 \u2200 x, f x \u2208 t' x :=\n by\n intro \u03b9' _ _ h\u03b9' ht'\n exact ih _ (Nat.lt_succ_of_le h\u03b9') ht' _ rfl\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d\u00b9 : Finite \u03b9\u271d\nht\u271d : \u2200 (s : Finset \u03b9\u271d), card s \u2264 card (Finset.biUnion s t\u271d)\nval\u271d\u00b9 : Fintype \u03b9\u271d\nx\u271d : \u2115\nhn\u271d : Fintype.card \u03b9\u271d = x\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Finite \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nval\u271d : Fintype \u03b9\nn\u271d : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ n\u271d \u2192\n \u2200 {\u03b9 : Type u} {t : \u03b9 \u2192 Finset \u03b1} [inst : Finite \u03b9],\n (\u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)) \u2192\n \u2200 (val : Fintype \u03b9), Fintype.card \u03b9 = m \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\nhn : Fintype.card \u03b9 = Nat.succ n\u271d\n\u22a2 \u2200 (\u03b9' : Type u) [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 ?m.46100 \u03b9' t' \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\n[PROOFSTEP]\nintro \u03b9' _ _ h\u03b9' ht'\n[GOAL]\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b3 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d\u00b2 : Finite \u03b9\u271d\nht\u271d : \u2200 (s : Finset \u03b9\u271d), card s \u2264 card (Finset.biUnion s t\u271d)\nval\u271d\u00b9 : Fintype \u03b9\u271d\nx\u271d : \u2115\nhn\u271d : Fintype.card \u03b9\u271d = x\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d\u00b9 : Finite \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nval\u271d : Fintype \u03b9\nn\u271d : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ n\u271d \u2192\n \u2200 {\u03b9 : Type u} {t : \u03b9 \u2192 Finset \u03b1} [inst : Finite \u03b9],\n (\u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)) \u2192\n \u2200 (val : Fintype \u03b9), Fintype.card \u03b9 = m \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\nhn : Fintype.card \u03b9 = Nat.succ n\u271d\n\u03b9' : Type u\ninst\u271d : Fintype \u03b9'\nt'\u271d : \u03b9' \u2192 Finset \u03b1\nh\u03b9' : Fintype.card \u03b9' \u2264 ?m.46100 \u03b9' t'\u271d\nht' : \u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t'\u271d)\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t'\u271d x\n[PROOFSTEP]\nexact ih _ (Nat.lt_succ_of_le h\u03b9') ht' _ rfl\n[GOAL]\ncase intro.h.succ\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d\u00b9 : Finite \u03b9\u271d\nht\u271d : \u2200 (s : Finset \u03b9\u271d), card s \u2264 card (Finset.biUnion s t\u271d)\nval\u271d\u00b9 : Fintype \u03b9\u271d\nx\u271d : \u2115\nhn\u271d : Fintype.card \u03b9\u271d = x\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Finite \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nval\u271d : Fintype \u03b9\nn\u271d : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ n\u271d \u2192\n \u2200 {\u03b9 : Type u} {t : \u03b9 \u2192 Finset \u03b1} [inst : Finite \u03b9],\n (\u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)) \u2192\n \u2200 (val : Fintype \u03b9), Fintype.card \u03b9 = m \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\nhn : Fintype.card \u03b9 = Nat.succ n\u271d\nih' :\n \u2200 (\u03b9' : Type u) [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n\u271d \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nby_cases h : \u2200 s : Finset \u03b9, s.Nonempty \u2192 s \u2260 univ \u2192 s.card < (s.biUnion t).card\n[GOAL]\ncase pos\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d\u00b9 : Finite \u03b9\u271d\nht\u271d : \u2200 (s : Finset \u03b9\u271d), card s \u2264 card (Finset.biUnion s t\u271d)\nval\u271d\u00b9 : Fintype \u03b9\u271d\nx\u271d : \u2115\nhn\u271d : Fintype.card \u03b9\u271d = x\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Finite \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nval\u271d : Fintype \u03b9\nn\u271d : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ n\u271d \u2192\n \u2200 {\u03b9 : Type u} {t : \u03b9 \u2192 Finset \u03b1} [inst : Finite \u03b9],\n (\u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)) \u2192\n \u2200 (val : Fintype \u03b9), Fintype.card \u03b9 = m \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\nhn : Fintype.card \u03b9 = Nat.succ n\u271d\nih' :\n \u2200 (\u03b9' : Type u) [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n\u271d \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nh : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nrefine' hall_hard_inductive_step_A hn ht (@fun \u03b9' => ih' \u03b9') h\n[GOAL]\ncase neg\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d\u00b9 : Finite \u03b9\u271d\nht\u271d : \u2200 (s : Finset \u03b9\u271d), card s \u2264 card (Finset.biUnion s t\u271d)\nval\u271d\u00b9 : Fintype \u03b9\u271d\nx\u271d : \u2115\nhn\u271d : Fintype.card \u03b9\u271d = x\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Finite \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nval\u271d : Fintype \u03b9\nn\u271d : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ n\u271d \u2192\n \u2200 {\u03b9 : Type u} {t : \u03b9 \u2192 Finset \u03b1} [inst : Finite \u03b9],\n (\u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)) \u2192\n \u2200 (val : Fintype \u03b9), Fintype.card \u03b9 = m \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\nhn : Fintype.card \u03b9 = Nat.succ n\u271d\nih' :\n \u2200 (\u03b9' : Type u) [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n\u271d \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nh : \u00ac\u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t)\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d\u00b9 : Finite \u03b9\u271d\nht\u271d : \u2200 (s : Finset \u03b9\u271d), card s \u2264 card (Finset.biUnion s t\u271d)\nval\u271d\u00b9 : Fintype \u03b9\u271d\nx\u271d : \u2115\nhn\u271d : Fintype.card \u03b9\u271d = x\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Finite \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nval\u271d : Fintype \u03b9\nn\u271d : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ n\u271d \u2192\n \u2200 {\u03b9 : Type u} {t : \u03b9 \u2192 Finset \u03b1} [inst : Finite \u03b9],\n (\u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)) \u2192\n \u2200 (val : Fintype \u03b9), Fintype.card \u03b9 = m \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\nhn : Fintype.card \u03b9 = Nat.succ n\u271d\nih' :\n \u2200 (\u03b9' : Type u) [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n\u271d \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\nh : \u2203 s, Finset.Nonempty s \u2227 s \u2260 univ \u2227 card (Finset.biUnion s t) \u2264 card s\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nrcases h with \u27e8s, sne, snu, sle\u27e9\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d\u00b9 : Finite \u03b9\u271d\nht\u271d : \u2200 (s : Finset \u03b9\u271d), card s \u2264 card (Finset.biUnion s t\u271d)\nval\u271d\u00b9 : Fintype \u03b9\u271d\nx\u271d : \u2115\nhn\u271d : Fintype.card \u03b9\u271d = x\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ninst\u271d : Finite \u03b9\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nval\u271d : Fintype \u03b9\nn\u271d : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ n\u271d \u2192\n \u2200 {\u03b9 : Type u} {t : \u03b9 \u2192 Finset \u03b1} [inst : Finite \u03b9],\n (\u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)) \u2192\n \u2200 (val : Fintype \u03b9), Fintype.card \u03b9 = m \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\nhn : Fintype.card \u03b9 = Nat.succ n\u271d\nih' :\n \u2200 (\u03b9' : Type u) [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1),\n Fintype.card \u03b9' \u2264 n\u271d \u2192\n (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x\ns : Finset \u03b9\nsne : Finset.Nonempty s\nsnu : s \u2260 univ\nsle : card (Finset.biUnion s t) \u2264 card s\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nexact hall_hard_inductive_step_B hn ht (@fun \u03b9' => ih' \u03b9') s sne snu (Nat.le_antisymm (ht _) sle)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\n\u22a2 (\u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)) \u2194 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\n\u22a2 (\u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nexact HallMarriageTheorem.hall_hard_inductive\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\n\u22a2 (\u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x) \u2192 \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\n[PROOFSTEP]\nrintro \u27e8f, hf\u2081, hf\u2082\u27e9 s\n[GOAL]\ncase mpr.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nf : \u03b9 \u2192 \u03b1\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (x : \u03b9), f x \u2208 t x\ns : Finset \u03b9\n\u22a2 card s \u2264 card (Finset.biUnion s t)\n[PROOFSTEP]\nrw [\u2190 card_image_of_injective s hf\u2081]\n[GOAL]\ncase mpr.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nf : \u03b9 \u2192 \u03b1\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (x : \u03b9), f x \u2208 t x\ns : Finset \u03b9\n\u22a2 card (image f s) \u2264 card (Finset.biUnion s t)\n[PROOFSTEP]\napply card_le_of_subset\n[GOAL]\ncase mpr.intro.intro.a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nf : \u03b9 \u2192 \u03b1\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (x : \u03b9), f x \u2208 t x\ns : Finset \u03b9\n\u22a2 image f s \u2286 Finset.biUnion s t\n[PROOFSTEP]\nintro\n[GOAL]\ncase mpr.intro.intro.a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nf : \u03b9 \u2192 \u03b1\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (x : \u03b9), f x \u2208 t x\ns : Finset \u03b9\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 image f s \u2192 a\u271d \u2208 Finset.biUnion s t\n[PROOFSTEP]\nrw [mem_image, mem_biUnion]\n[GOAL]\ncase mpr.intro.intro.a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nf : \u03b9 \u2192 \u03b1\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (x : \u03b9), f x \u2208 t x\ns : Finset \u03b9\na\u271d : \u03b1\n\u22a2 (\u2203 a, a \u2208 s \u2227 f a = a\u271d) \u2192 \u2203 a, a \u2208 s \u2227 a\u271d \u2208 t a\n[PROOFSTEP]\nrintro \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.a.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nf : \u03b9 \u2192 \u03b1\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (x : \u03b9), f x \u2208 t x\ns : Finset \u03b9\nx : \u03b9\nhx : x \u2208 s\n\u22a2 \u2203 a, a \u2208 s \u2227 f x \u2208 t a\n[PROOFSTEP]\nexact \u27e8x, hx, hf\u2082 x\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.Hall.Finite", "llama_tokens": 49031, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581626286834, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.42851231822973695}} {"text": "[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\n\u22a2 ContravariantClass (WithZero \u03b1) (WithZero \u03b1) (fun x x_1 => x * x_1) fun x x_1 => x < x_1\n[PROOFSTEP]\nrefine \u27e8fun a b c h => ?_\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c : WithZero \u03b1\nh : a * b < a * c\n\u22a2 b < c\n[PROOFSTEP]\nhave := ((zero_le _).trans_lt h).ne'\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c : WithZero \u03b1\nh : a * b < a * c\nthis : a * c \u2260 0\n\u22a2 b < c\n[PROOFSTEP]\ninduction a using WithZero.recZeroCoe\n[GOAL]\ncase h\u2081\n\u03b1 : Type u\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nb c : WithZero \u03b1\nh : 0 * b < 0 * c\nthis : 0 * c \u2260 0\n\u22a2 b < c\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h\u2081.h\n\u03b1 : Type u\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nb c : WithZero \u03b1\nh : 0 * b < 0 * c\nthis : 0 * c \u2260 0\n\u22a2 False\n[PROOFSTEP]\nexact left_ne_zero_of_mul this rfl\n[GOAL]\ncase h\u2082\n\u03b1 : Type u\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nb c : WithZero \u03b1\na\u271d : \u03b1\nh : \u2191a\u271d * b < \u2191a\u271d * c\nthis : \u2191a\u271d * c \u2260 0\n\u22a2 b < c\n[PROOFSTEP]\ninduction c using WithZero.recZeroCoe\n[GOAL]\ncase h\u2082.h\u2081\n\u03b1 : Type u\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nb : WithZero \u03b1\na\u271d : \u03b1\nh : \u2191a\u271d * b < \u2191a\u271d * 0\nthis : \u2191a\u271d * 0 \u2260 0\n\u22a2 b < 0\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h\u2082.h\u2081.h\n\u03b1 : Type u\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nb : WithZero \u03b1\na\u271d : \u03b1\nh : \u2191a\u271d * b < \u2191a\u271d * 0\nthis : \u2191a\u271d * 0 \u2260 0\n\u22a2 False\n[PROOFSTEP]\nexact right_ne_zero_of_mul this rfl\n[GOAL]\ncase h\u2082.h\u2082\n\u03b1 : Type u\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nb : WithZero \u03b1\na\u271d\u00b9 a\u271d : \u03b1\nh : \u2191a\u271d\u00b9 * b < \u2191a\u271d\u00b9 * \u2191a\u271d\nthis : \u2191a\u271d\u00b9 * \u2191a\u271d \u2260 0\n\u22a2 b < \u2191a\u271d\n[PROOFSTEP]\ninduction b using WithZero.recZeroCoe\n[GOAL]\ncase h\u2082.h\u2082.h\u2081\n\u03b1 : Type u\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u271d\u00b9 a\u271d : \u03b1\nthis : \u2191a\u271d\u00b9 * \u2191a\u271d \u2260 0\nh : \u2191a\u271d\u00b9 * 0 < \u2191a\u271d\u00b9 * \u2191a\u271d\n\u22a2 0 < \u2191a\u271d\ncase h\u2082.h\u2082.h\u2082\n\u03b1 : Type u\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u271d\u00b2 a\u271d\u00b9 : \u03b1\nthis : \u2191a\u271d\u00b2 * \u2191a\u271d\u00b9 \u2260 0\na\u271d : \u03b1\nh : \u2191a\u271d\u00b2 * \u2191a\u271d < \u2191a\u271d\u00b2 * \u2191a\u271d\u00b9\n\u22a2 \u2191a\u271d < \u2191a\u271d\u00b9\n[PROOFSTEP]\nexacts [zero_lt_coe _, coe_lt_coe.mpr (lt_of_mul_lt_mul_left' <| coe_lt_coe.mp h)]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Monoid.WithZero.Basic", "llama_tokens": 1578, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.42819922561362195}} {"text": "[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 R S\na : S\n\u22a2 \u2191(RingHom.mapMatrix (algebraMap R R\u2098)) (\u2191(leftMulMatrix b) a) =\n \u2191(leftMulMatrix (Basis.localizationLocalization R\u2098 M S\u2098 b)) (\u2191(algebraMap S S\u2098) a)\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 R S\na : S\ni j : \u03b9\n\u22a2 \u2191(RingHom.mapMatrix (algebraMap R R\u2098)) (\u2191(leftMulMatrix b) a) i j =\n \u2191(leftMulMatrix (Basis.localizationLocalization R\u2098 M S\u2098 b)) (\u2191(algebraMap S S\u2098) a) i j\n[PROOFSTEP]\nsimp only [Matrix.map_apply, RingHom.mapMatrix_apply, leftMulMatrix_eq_repr_mul, \u2190 map_mul,\n Basis.localizationLocalization_apply, Basis.localizationLocalization_repr_algebraMap]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\na : S\n\u22a2 \u2191(norm R\u2098) (\u2191(algebraMap S S\u2098) a) = \u2191(algebraMap R R\u2098) (\u2191(norm R) a)\n[PROOFSTEP]\ncases subsingleton_or_nontrivial R\n[GOAL]\ncase inl\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\na : S\nh\u271d : Subsingleton R\n\u22a2 \u2191(norm R\u2098) (\u2191(algebraMap S S\u2098) a) = \u2191(algebraMap R R\u2098) (\u2191(norm R) a)\n[PROOFSTEP]\nhaveI : Subsingleton R\u2098 := Module.subsingleton R R\u2098\n[GOAL]\ncase inl\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\na : S\nh\u271d : Subsingleton R\nthis : Subsingleton R\u2098\n\u22a2 \u2191(norm R\u2098) (\u2191(algebraMap S S\u2098) a) = \u2191(algebraMap R R\u2098) (\u2191(norm R) a)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\na : S\nh\u271d : Nontrivial R\n\u22a2 \u2191(norm R\u2098) (\u2191(algebraMap S S\u2098) a) = \u2191(algebraMap R R\u2098) (\u2191(norm R) a)\n[PROOFSTEP]\nlet b := Module.Free.chooseBasis R S\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\na : S\nh\u271d : Nontrivial R\nb : Basis (Module.Free.ChooseBasisIndex R S) R S := Module.Free.chooseBasis R S\n\u22a2 \u2191(norm R\u2098) (\u2191(algebraMap S S\u2098) a) = \u2191(algebraMap R R\u2098) (\u2191(norm R) a)\n[PROOFSTEP]\nletI := Classical.decEq (Module.Free.ChooseBasisIndex R S)\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\na : S\nh\u271d : Nontrivial R\nb : Basis (Module.Free.ChooseBasisIndex R S) R S := Module.Free.chooseBasis R S\nthis : DecidableEq (Module.Free.ChooseBasisIndex R S) := Classical.decEq (Module.Free.ChooseBasisIndex R S)\n\u22a2 \u2191(norm R\u2098) (\u2191(algebraMap S S\u2098) a) = \u2191(algebraMap R R\u2098) (\u2191(norm R) a)\n[PROOFSTEP]\nrw [Algebra.norm_eq_matrix_det (b.localizationLocalization R\u2098 M S\u2098), Algebra.norm_eq_matrix_det b, RingHom.map_det, \u2190\n Algebra.map_leftMulMatrix_localization]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\na : S\n\u22a2 \u2191(trace R\u2098 S\u2098) (\u2191(algebraMap S S\u2098) a) = \u2191(algebraMap R R\u2098) (\u2191(trace R S) a)\n[PROOFSTEP]\ncases subsingleton_or_nontrivial R\n[GOAL]\ncase inl\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\na : S\nh\u271d : Subsingleton R\n\u22a2 \u2191(trace R\u2098 S\u2098) (\u2191(algebraMap S S\u2098) a) = \u2191(algebraMap R R\u2098) (\u2191(trace R S) a)\n[PROOFSTEP]\nhaveI : Subsingleton R\u2098 := Module.subsingleton R R\u2098\n[GOAL]\ncase inl\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\na : S\nh\u271d : Subsingleton R\nthis : Subsingleton R\u2098\n\u22a2 \u2191(trace R\u2098 S\u2098) (\u2191(algebraMap S S\u2098) a) = \u2191(algebraMap R R\u2098) (\u2191(trace R S) a)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\na : S\nh\u271d : Nontrivial R\n\u22a2 \u2191(trace R\u2098 S\u2098) (\u2191(algebraMap S S\u2098) a) = \u2191(algebraMap R R\u2098) (\u2191(trace R S) a)\n[PROOFSTEP]\nlet b := Module.Free.chooseBasis R S\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\na : S\nh\u271d : Nontrivial R\nb : Basis (Module.Free.ChooseBasisIndex R S) R S := Module.Free.chooseBasis R S\n\u22a2 \u2191(trace R\u2098 S\u2098) (\u2191(algebraMap S S\u2098) a) = \u2191(algebraMap R R\u2098) (\u2191(trace R S) a)\n[PROOFSTEP]\nletI := Classical.decEq (Module.Free.ChooseBasisIndex R S)\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\na : S\nh\u271d : Nontrivial R\nb : Basis (Module.Free.ChooseBasisIndex R S) R S := Module.Free.chooseBasis R S\nthis : DecidableEq (Module.Free.ChooseBasisIndex R S) := Classical.decEq (Module.Free.ChooseBasisIndex R S)\n\u22a2 \u2191(trace R\u2098 S\u2098) (\u2191(algebraMap S S\u2098) a) = \u2191(algebraMap R R\u2098) (\u2191(trace R S) a)\n[PROOFSTEP]\nrw [Algebra.trace_eq_matrix_trace (b.localizationLocalization R\u2098 M S\u2098), Algebra.trace_eq_matrix_trace b, \u2190\n Algebra.map_leftMulMatrix_localization]\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\nR\u2098 : Type u_3\nS\u2098 : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u2098\ninst\u271d\u00b9\u2070 : Algebra R R\u2098\ninst\u271d\u2079 : CommRing S\u2098\ninst\u271d\u2078 : Algebra S S\u2098\nM : Submonoid R\ninst\u271d\u2077 : IsLocalization M R\u2098\ninst\u271d\u2076 : IsLocalization (algebraMapSubmonoid S M) S\u2098\ninst\u271d\u2075 : Algebra R\u2098 S\u2098\ninst\u271d\u2074 : Algebra R S\u2098\ninst\u271d\u00b3 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b2 : IsScalarTower R S S\u2098\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\na : S\nh\u271d : Nontrivial R\nb : Basis (Module.Free.ChooseBasisIndex R S) R S := Module.Free.chooseBasis R S\nthis : DecidableEq (Module.Free.ChooseBasisIndex R S) := Classical.decEq (Module.Free.ChooseBasisIndex R S)\n\u22a2 Matrix.trace (\u2191(RingHom.mapMatrix (algebraMap R R\u2098)) (\u2191(leftMulMatrix b) a)) =\n \u2191(algebraMap R R\u2098) (Matrix.trace (\u2191(leftMulMatrix b) a))\n[PROOFSTEP]\nexact (AddMonoidHom.map_trace (algebraMap R R\u2098).toAddMonoidHom _).symm\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2\u00b9 : CommRing R\ninst\u271d\u00b2\u2070 : CommRing S\ninst\u271d\u00b9\u2079 : Algebra R S\nR\u2098 : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2078 : CommRing R\u2098\ninst\u271d\u00b9\u2077 : Algebra R R\u2098\ninst\u271d\u00b9\u2076 : CommRing S\u2098\u271d\ninst\u271d\u00b9\u2075 : Algebra S S\u2098\u271d\nM : Submonoid R\ninst\u271d\u00b9\u2074 : IsLocalization M R\u2098\ninst\u271d\u00b9\u00b3 : IsLocalization (algebraMapSubmonoid S M) S\u2098\u271d\ninst\u271d\u00b9\u00b2 : Algebra R\u2098 S\u2098\u271d\ninst\u271d\u00b9\u00b9 : Algebra R S\u2098\u271d\ninst\u271d\u00b9\u2070 : IsScalarTower R R\u2098 S\u2098\u271d\ninst\u271d\u2079 : IsScalarTower R S S\u2098\u271d\nS\u2098 : Type u_5\ninst\u271d\u2078 : CommRing S\u2098\ninst\u271d\u2077 : Algebra S S\u2098\ninst\u271d\u2076 : Algebra R\u2098 S\u2098\ninst\u271d\u2075 : Algebra R S\u2098\ninst\u271d\u2074 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b3 : IsScalarTower R S S\u2098\ninst\u271d\u00b2 : IsLocalization (algebraMapSubmonoid S M) S\u2098\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 R S\n\u22a2 traceMatrix R\u2098 \u2191(Basis.localizationLocalization R\u2098 M S\u2098 b) = \u2191(RingHom.mapMatrix (algebraMap R R\u2098)) (traceMatrix R \u2191b)\n[PROOFSTEP]\nhave : Module.Finite R S := Module.Finite.of_basis b\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2\u00b9 : CommRing R\ninst\u271d\u00b2\u2070 : CommRing S\ninst\u271d\u00b9\u2079 : Algebra R S\nR\u2098 : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2078 : CommRing R\u2098\ninst\u271d\u00b9\u2077 : Algebra R R\u2098\ninst\u271d\u00b9\u2076 : CommRing S\u2098\u271d\ninst\u271d\u00b9\u2075 : Algebra S S\u2098\u271d\nM : Submonoid R\ninst\u271d\u00b9\u2074 : IsLocalization M R\u2098\ninst\u271d\u00b9\u00b3 : IsLocalization (algebraMapSubmonoid S M) S\u2098\u271d\ninst\u271d\u00b9\u00b2 : Algebra R\u2098 S\u2098\u271d\ninst\u271d\u00b9\u00b9 : Algebra R S\u2098\u271d\ninst\u271d\u00b9\u2070 : IsScalarTower R R\u2098 S\u2098\u271d\ninst\u271d\u2079 : IsScalarTower R S S\u2098\u271d\nS\u2098 : Type u_5\ninst\u271d\u2078 : CommRing S\u2098\ninst\u271d\u2077 : Algebra S S\u2098\ninst\u271d\u2076 : Algebra R\u2098 S\u2098\ninst\u271d\u2075 : Algebra R S\u2098\ninst\u271d\u2074 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b3 : IsScalarTower R S S\u2098\ninst\u271d\u00b2 : IsLocalization (algebraMapSubmonoid S M) S\u2098\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 R S\nthis : Module.Finite R S\n\u22a2 traceMatrix R\u2098 \u2191(Basis.localizationLocalization R\u2098 M S\u2098 b) = \u2191(RingHom.mapMatrix (algebraMap R R\u2098)) (traceMatrix R \u2191b)\n[PROOFSTEP]\nhave : Module.Free R S := Module.Free.of_basis b\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2\u00b9 : CommRing R\ninst\u271d\u00b2\u2070 : CommRing S\ninst\u271d\u00b9\u2079 : Algebra R S\nR\u2098 : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2078 : CommRing R\u2098\ninst\u271d\u00b9\u2077 : Algebra R R\u2098\ninst\u271d\u00b9\u2076 : CommRing S\u2098\u271d\ninst\u271d\u00b9\u2075 : Algebra S S\u2098\u271d\nM : Submonoid R\ninst\u271d\u00b9\u2074 : IsLocalization M R\u2098\ninst\u271d\u00b9\u00b3 : IsLocalization (algebraMapSubmonoid S M) S\u2098\u271d\ninst\u271d\u00b9\u00b2 : Algebra R\u2098 S\u2098\u271d\ninst\u271d\u00b9\u00b9 : Algebra R S\u2098\u271d\ninst\u271d\u00b9\u2070 : IsScalarTower R R\u2098 S\u2098\u271d\ninst\u271d\u2079 : IsScalarTower R S S\u2098\u271d\nS\u2098 : Type u_5\ninst\u271d\u2078 : CommRing S\u2098\ninst\u271d\u2077 : Algebra S S\u2098\ninst\u271d\u2076 : Algebra R\u2098 S\u2098\ninst\u271d\u2075 : Algebra R S\u2098\ninst\u271d\u2074 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b3 : IsScalarTower R S S\u2098\ninst\u271d\u00b2 : IsLocalization (algebraMapSubmonoid S M) S\u2098\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 R S\nthis\u271d : Module.Finite R S\nthis : Module.Free R S\n\u22a2 traceMatrix R\u2098 \u2191(Basis.localizationLocalization R\u2098 M S\u2098 b) = \u2191(RingHom.mapMatrix (algebraMap R R\u2098)) (traceMatrix R \u2191b)\n[PROOFSTEP]\next i j : 2\n[GOAL]\ncase a.h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2\u00b9 : CommRing R\ninst\u271d\u00b2\u2070 : CommRing S\ninst\u271d\u00b9\u2079 : Algebra R S\nR\u2098 : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2078 : CommRing R\u2098\ninst\u271d\u00b9\u2077 : Algebra R R\u2098\ninst\u271d\u00b9\u2076 : CommRing S\u2098\u271d\ninst\u271d\u00b9\u2075 : Algebra S S\u2098\u271d\nM : Submonoid R\ninst\u271d\u00b9\u2074 : IsLocalization M R\u2098\ninst\u271d\u00b9\u00b3 : IsLocalization (algebraMapSubmonoid S M) S\u2098\u271d\ninst\u271d\u00b9\u00b2 : Algebra R\u2098 S\u2098\u271d\ninst\u271d\u00b9\u00b9 : Algebra R S\u2098\u271d\ninst\u271d\u00b9\u2070 : IsScalarTower R R\u2098 S\u2098\u271d\ninst\u271d\u2079 : IsScalarTower R S S\u2098\u271d\nS\u2098 : Type u_5\ninst\u271d\u2078 : CommRing S\u2098\ninst\u271d\u2077 : Algebra S S\u2098\ninst\u271d\u2076 : Algebra R\u2098 S\u2098\ninst\u271d\u2075 : Algebra R S\u2098\ninst\u271d\u2074 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b3 : IsScalarTower R S S\u2098\ninst\u271d\u00b2 : IsLocalization (algebraMapSubmonoid S M) S\u2098\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 R S\nthis\u271d : Module.Finite R S\nthis : Module.Free R S\ni j : \u03b9\n\u22a2 traceMatrix R\u2098 (\u2191(Basis.localizationLocalization R\u2098 M S\u2098 b)) i j =\n \u2191(RingHom.mapMatrix (algebraMap R R\u2098)) (traceMatrix R \u2191b) i j\n[PROOFSTEP]\nsimp_rw [RingHom.mapMatrix_apply, Matrix.map_apply, traceMatrix_apply, traceForm_apply,\n Basis.localizationLocalization_apply, \u2190 map_mul]\n[GOAL]\ncase a.h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2\u00b9 : CommRing R\ninst\u271d\u00b2\u2070 : CommRing S\ninst\u271d\u00b9\u2079 : Algebra R S\nR\u2098 : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2078 : CommRing R\u2098\ninst\u271d\u00b9\u2077 : Algebra R R\u2098\ninst\u271d\u00b9\u2076 : CommRing S\u2098\u271d\ninst\u271d\u00b9\u2075 : Algebra S S\u2098\u271d\nM : Submonoid R\ninst\u271d\u00b9\u2074 : IsLocalization M R\u2098\ninst\u271d\u00b9\u00b3 : IsLocalization (algebraMapSubmonoid S M) S\u2098\u271d\ninst\u271d\u00b9\u00b2 : Algebra R\u2098 S\u2098\u271d\ninst\u271d\u00b9\u00b9 : Algebra R S\u2098\u271d\ninst\u271d\u00b9\u2070 : IsScalarTower R R\u2098 S\u2098\u271d\ninst\u271d\u2079 : IsScalarTower R S S\u2098\u271d\nS\u2098 : Type u_5\ninst\u271d\u2078 : CommRing S\u2098\ninst\u271d\u2077 : Algebra S S\u2098\ninst\u271d\u2076 : Algebra R\u2098 S\u2098\ninst\u271d\u2075 : Algebra R S\u2098\ninst\u271d\u2074 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b3 : IsScalarTower R S S\u2098\ninst\u271d\u00b2 : IsLocalization (algebraMapSubmonoid S M) S\u2098\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 R S\nthis\u271d : Module.Finite R S\nthis : Module.Free R S\ni j : \u03b9\n\u22a2 \u2191(trace R\u2098 S\u2098) (\u2191(algebraMap S S\u2098) (\u2191b i * \u2191b j)) = \u2191(algebraMap R R\u2098) (\u2191(trace R S) (\u2191b i * \u2191b j))\n[PROOFSTEP]\nexact Algebra.trace_localization R M _\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2\u00b9 : CommRing R\ninst\u271d\u00b2\u2070 : CommRing S\ninst\u271d\u00b9\u2079 : Algebra R S\nR\u2098 : Type u_3\nS\u2098\u271d : Type u_4\ninst\u271d\u00b9\u2078 : CommRing R\u2098\ninst\u271d\u00b9\u2077 : Algebra R R\u2098\ninst\u271d\u00b9\u2076 : CommRing S\u2098\u271d\ninst\u271d\u00b9\u2075 : Algebra S S\u2098\u271d\nM : Submonoid R\ninst\u271d\u00b9\u2074 : IsLocalization M R\u2098\ninst\u271d\u00b9\u00b3 : IsLocalization (algebraMapSubmonoid S M) S\u2098\u271d\ninst\u271d\u00b9\u00b2 : Algebra R\u2098 S\u2098\u271d\ninst\u271d\u00b9\u00b9 : Algebra R S\u2098\u271d\ninst\u271d\u00b9\u2070 : IsScalarTower R R\u2098 S\u2098\u271d\ninst\u271d\u2079 : IsScalarTower R S S\u2098\u271d\nS\u2098 : Type u_5\ninst\u271d\u2078 : CommRing S\u2098\ninst\u271d\u2077 : Algebra S S\u2098\ninst\u271d\u2076 : Algebra R\u2098 S\u2098\ninst\u271d\u2075 : Algebra R S\u2098\ninst\u271d\u2074 : IsScalarTower R R\u2098 S\u2098\ninst\u271d\u00b3 : IsScalarTower R S S\u2098\ninst\u271d\u00b2 : IsLocalization (algebraMapSubmonoid S M) S\u2098\n\u03b9 : Type u_6\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 R S\n\u22a2 discr R\u2098 \u2191(Basis.localizationLocalization R\u2098 M S\u2098 b) = \u2191(algebraMap R R\u2098) (discr R \u2191b)\n[PROOFSTEP]\nrw [Algebra.discr_def, Algebra.discr_def, RingHom.map_det, Algebra.traceMatrix_localizationLocalization]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Localization.NormTrace", "llama_tokens": 9229, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936324115011, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.4278424205626497}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nas : List \u03b1\n\u22a2 \ud835\udcdd as = traverse \ud835\udcdd as\n[PROOFSTEP]\nrefine' nhds_mkOfNhds _ _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nas : List \u03b1\n\u22a2 pure \u2264 traverse \ud835\udcdd\n[PROOFSTEP]\nintro l\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nas l : List \u03b1\n\u22a2 pure l \u2264 traverse \ud835\udcdd l\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase refine'_1.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nas : List \u03b1\n\u22a2 pure [] \u2264 traverse \ud835\udcdd []\ncase refine'_1.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nas : List \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : pure tail\u271d \u2264 traverse \ud835\udcdd tail\u271d\n\u22a2 pure (head\u271d :: tail\u271d) \u2264 traverse \ud835\udcdd (head\u271d :: tail\u271d)\n[PROOFSTEP]\ncase nil => exact le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nas : List \u03b1\n\u22a2 pure [] \u2264 traverse \ud835\udcdd []\n[PROOFSTEP]\ncase nil => exact le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nas : List \u03b1\n\u22a2 pure [] \u2264 traverse \ud835\udcdd []\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase refine'_1.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nas : List \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : pure tail\u271d \u2264 traverse \ud835\udcdd tail\u271d\n\u22a2 pure (head\u271d :: tail\u271d) \u2264 traverse \ud835\udcdd (head\u271d :: tail\u271d)\n[PROOFSTEP]\ncase cons a l\n ih =>\n suffices List.cons <$> pure a <*> pure l \u2264 List.cons <$> \ud835\udcdd a <*> traverse \ud835\udcdd l by simpa only [functor_norm] using this\n exact Filter.seq_mono (Filter.map_mono <| pure_le_nhds a) ih\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nas : List \u03b1\na : \u03b1\nl : List \u03b1\nih : pure l \u2264 traverse \ud835\udcdd l\n\u22a2 pure (a :: l) \u2264 traverse \ud835\udcdd (a :: l)\n[PROOFSTEP]\ncase cons a l\n ih =>\n suffices List.cons <$> pure a <*> pure l \u2264 List.cons <$> \ud835\udcdd a <*> traverse \ud835\udcdd l by simpa only [functor_norm] using this\n exact Filter.seq_mono (Filter.map_mono <| pure_le_nhds a) ih\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nas : List \u03b1\na : \u03b1\nl : List \u03b1\nih : pure l \u2264 traverse \ud835\udcdd l\n\u22a2 pure (a :: l) \u2264 traverse \ud835\udcdd (a :: l)\n[PROOFSTEP]\nsuffices List.cons <$> pure a <*> pure l \u2264 List.cons <$> \ud835\udcdd a <*> traverse \ud835\udcdd l by simpa only [functor_norm] using this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nas : List \u03b1\na : \u03b1\nl : List \u03b1\nih : pure l \u2264 traverse \ud835\udcdd l\nthis : (Seq.seq (List.cons <$> pure a) fun x => pure l) \u2264 Seq.seq (List.cons <$> \ud835\udcdd a) fun x => traverse \ud835\udcdd l\n\u22a2 pure (a :: l) \u2264 traverse \ud835\udcdd (a :: l)\n[PROOFSTEP]\nsimpa only [functor_norm] using this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nas : List \u03b1\na : \u03b1\nl : List \u03b1\nih : pure l \u2264 traverse \ud835\udcdd l\n\u22a2 (Seq.seq (List.cons <$> pure a) fun x => pure l) \u2264 Seq.seq (List.cons <$> \ud835\udcdd a) fun x => traverse \ud835\udcdd l\n[PROOFSTEP]\nexact Filter.seq_mono (Filter.map_mono <| pure_le_nhds a) ih\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nas : List \u03b1\n\u22a2 \u2200 (a : List \u03b1) (s : Set (List \u03b1)),\n s \u2208 traverse \ud835\udcdd a \u2192 \u2203 t, t \u2208 traverse \ud835\udcdd a \u2227 t \u2286 s \u2227 \u2200 (a' : List \u03b1), a' \u2208 t \u2192 s \u2208 traverse \ud835\udcdd a'\n[PROOFSTEP]\nintro l s hs\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nas l : List \u03b1\ns : Set (List \u03b1)\nhs : s \u2208 traverse \ud835\udcdd l\n\u22a2 \u2203 t, t \u2208 traverse \ud835\udcdd l \u2227 t \u2286 s \u2227 \u2200 (a' : List \u03b1), a' \u2208 t \u2192 s \u2208 traverse \ud835\udcdd a'\n[PROOFSTEP]\nrcases(mem_traverse_iff _ _).1 hs with \u27e8u, hu, hus\u27e9\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nas l : List \u03b1\ns : Set (List \u03b1)\nhs : s \u2208 traverse \ud835\udcdd l\nu : List (Set \u03b1)\nhu : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\nhus : sequence u \u2286 s\n\u22a2 \u2203 t, t \u2208 traverse \ud835\udcdd l \u2227 t \u2286 s \u2227 \u2200 (a' : List \u03b1), a' \u2208 t \u2192 s \u2208 traverse \ud835\udcdd a'\n[PROOFSTEP]\nclear as hs\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu : List (Set \u03b1)\nhu : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\nhus : sequence u \u2286 s\n\u22a2 \u2203 t, t \u2208 traverse \ud835\udcdd l \u2227 t \u2286 s \u2227 \u2200 (a' : List \u03b1), a' \u2208 t \u2192 s \u2208 traverse \ud835\udcdd a'\n[PROOFSTEP]\nhave : \u2203 v : List (Set \u03b1), l.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) v \u2227 sequence v \u2286 s\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu : List (Set \u03b1)\nhu : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\nhus : sequence u \u2286 s\n\u22a2 \u2203 v, List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l v \u2227 sequence v \u2286 s\ncase refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu : List (Set \u03b1)\nhu : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\nhus : sequence u \u2286 s\nthis : \u2203 v, List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l v \u2227 sequence v \u2286 s\n\u22a2 \u2203 t, t \u2208 traverse \ud835\udcdd l \u2227 t \u2286 s \u2227 \u2200 (a' : List \u03b1), a' \u2208 t \u2192 s \u2208 traverse \ud835\udcdd a'\n[PROOFSTEP]\ninduction hu generalizing s\n[GOAL]\ncase this.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\nu : List (Set \u03b1)\ns : Set (List \u03b1)\nhus : sequence [] \u2286 s\n\u22a2 \u2203 v, List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) [] v \u2227 sequence v \u2286 s\ncase this.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\nu : List (Set \u03b1)\na\u271d\u00b2 : \u03b1\nb\u271d : Set \u03b1\nl\u2081\u271d : List \u03b1\nl\u2082\u271d : List (Set \u03b1)\na\u271d\u00b9 : b\u271d \u2208 \ud835\udcdd a\u271d\u00b2\na\u271d : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l\u2081\u271d l\u2082\u271d\na_ih\u271d : \u2200 (s : Set (List \u03b1)), sequence l\u2082\u271d \u2286 s \u2192 \u2203 v, List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l\u2081\u271d v \u2227 sequence v \u2286 s\ns : Set (List \u03b1)\nhus : sequence (b\u271d :: l\u2082\u271d) \u2286 s\n\u22a2 \u2203 v, List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) (a\u271d\u00b2 :: l\u2081\u271d) v \u2227 sequence v \u2286 s\ncase refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu : List (Set \u03b1)\nhu : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\nhus : sequence u \u2286 s\nthis : \u2203 v, List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l v \u2227 sequence v \u2286 s\n\u22a2 \u2203 t, t \u2208 traverse \ud835\udcdd l \u2227 t \u2286 s \u2227 \u2200 (a' : List \u03b1), a' \u2208 t \u2192 s \u2208 traverse \ud835\udcdd a'\n[PROOFSTEP]\ncase nil _hs =>\n exists []\n simp only [List.forall\u2082_nil_left_iff, exists_eq_left]\n exact\n \u27e8trivial, hus\u27e9\n -- porting note -- renamed reordered variables based on previous types\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\n_hs : TopologicalSpace \u03b2\nl : List \u03b1\nu : List (Set \u03b1)\ns : Set (List \u03b1)\nhus : sequence [] \u2286 s\n\u22a2 \u2203 v, List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) [] v \u2227 sequence v \u2286 s\n[PROOFSTEP]\ncase nil _hs =>\n exists []\n simp only [List.forall\u2082_nil_left_iff, exists_eq_left]\n exact\n \u27e8trivial, hus\u27e9\n -- porting note -- renamed reordered variables based on previous types\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\n_hs : TopologicalSpace \u03b2\nl : List \u03b1\nu : List (Set \u03b1)\ns : Set (List \u03b1)\nhus : sequence [] \u2286 s\n\u22a2 \u2203 v, List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) [] v \u2227 sequence v \u2286 s\n[PROOFSTEP]\nexists []\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\n_hs : TopologicalSpace \u03b2\nl : List \u03b1\nu : List (Set \u03b1)\ns : Set (List \u03b1)\nhus : sequence [] \u2286 s\n\u22a2 List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) [] [] \u2227 sequence [] \u2286 s\n[PROOFSTEP]\nsimp only [List.forall\u2082_nil_left_iff, exists_eq_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\n_hs : TopologicalSpace \u03b2\nl : List \u03b1\nu : List (Set \u03b1)\ns : Set (List \u03b1)\nhus : sequence [] \u2286 s\n\u22a2 True \u2227 sequence [] \u2286 s\n[PROOFSTEP]\nexact\n \u27e8trivial, hus\u27e9\n -- porting note -- renamed reordered variables based on previous types\n[GOAL]\ncase this.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\nu : List (Set \u03b1)\na\u271d\u00b2 : \u03b1\nb\u271d : Set \u03b1\nl\u2081\u271d : List \u03b1\nl\u2082\u271d : List (Set \u03b1)\na\u271d\u00b9 : b\u271d \u2208 \ud835\udcdd a\u271d\u00b2\na\u271d : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l\u2081\u271d l\u2082\u271d\na_ih\u271d : \u2200 (s : Set (List \u03b1)), sequence l\u2082\u271d \u2286 s \u2192 \u2203 v, List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l\u2081\u271d v \u2227 sequence v \u2286 s\ns : Set (List \u03b1)\nhus : sequence (b\u271d :: l\u2082\u271d) \u2286 s\n\u22a2 \u2203 v, List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) (a\u271d\u00b2 :: l\u2081\u271d) v \u2227 sequence v \u2286 s\ncase refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu : List (Set \u03b1)\nhu : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\nhus : sequence u \u2286 s\nthis : \u2203 v, List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l v \u2227 sequence v \u2286 s\n\u22a2 \u2203 t, t \u2208 traverse \ud835\udcdd l \u2227 t \u2286 s \u2227 \u2200 (a' : List \u03b1), a' \u2208 t \u2192 s \u2208 traverse \ud835\udcdd a'\n[PROOFSTEP]\ncase cons a s as ss hts h ht _ ih =>\n rcases mem_nhds_iff.1 ht with \u27e8u, hut, hu\u27e9\n rcases ih _ Subset.rfl with \u27e8v, hv, hvss\u27e9\n exact \u27e8u :: v, List.Forall\u2082.cons hu hv, Subset.trans (Set.seq_mono (Set.image_subset _ hut) hvss) hus\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : TopologicalSpace \u03b1\ns\u271d : TopologicalSpace \u03b2\nl : List \u03b1\nu : List (Set \u03b1)\nas : \u03b1\nss : Set \u03b1\nhts : List \u03b1\nh : List (Set \u03b1)\nht : ss \u2208 \ud835\udcdd as\na\u271d : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) hts h\nih : \u2200 (s : Set (List \u03b1)), sequence h \u2286 s \u2192 \u2203 v, List.Forall\u2082 (fun a_1 s => IsOpen s \u2227 a_1 \u2208 s) hts v \u2227 sequence v \u2286 s\ns : Set (List \u03b1)\nhus : sequence (ss :: h) \u2286 s\n\u22a2 \u2203 v, List.Forall\u2082 (fun a_1 s => IsOpen s \u2227 a_1 \u2208 s) (as :: hts) v \u2227 sequence v \u2286 s\n[PROOFSTEP]\ncase cons a s as ss hts h ht _ ih =>\n rcases mem_nhds_iff.1 ht with \u27e8u, hut, hu\u27e9\n rcases ih _ Subset.rfl with \u27e8v, hv, hvss\u27e9\n exact \u27e8u :: v, List.Forall\u2082.cons hu hv, Subset.trans (Set.seq_mono (Set.image_subset _ hut) hvss) hus\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : TopologicalSpace \u03b1\ns\u271d : TopologicalSpace \u03b2\nl : List \u03b1\nu : List (Set \u03b1)\nas : \u03b1\nss : Set \u03b1\nhts : List \u03b1\nh : List (Set \u03b1)\nht : ss \u2208 \ud835\udcdd as\na\u271d : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) hts h\nih : \u2200 (s : Set (List \u03b1)), sequence h \u2286 s \u2192 \u2203 v, List.Forall\u2082 (fun a_1 s => IsOpen s \u2227 a_1 \u2208 s) hts v \u2227 sequence v \u2286 s\ns : Set (List \u03b1)\nhus : sequence (ss :: h) \u2286 s\n\u22a2 \u2203 v, List.Forall\u2082 (fun a_1 s => IsOpen s \u2227 a_1 \u2208 s) (as :: hts) v \u2227 sequence v \u2286 s\n[PROOFSTEP]\nrcases mem_nhds_iff.1 ht with \u27e8u, hut, hu\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : TopologicalSpace \u03b1\ns\u271d : TopologicalSpace \u03b2\nl : List \u03b1\nu\u271d : List (Set \u03b1)\nas : \u03b1\nss : Set \u03b1\nhts : List \u03b1\nh : List (Set \u03b1)\nht : ss \u2208 \ud835\udcdd as\na\u271d : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) hts h\nih : \u2200 (s : Set (List \u03b1)), sequence h \u2286 s \u2192 \u2203 v, List.Forall\u2082 (fun a_1 s => IsOpen s \u2227 a_1 \u2208 s) hts v \u2227 sequence v \u2286 s\ns : Set (List \u03b1)\nhus : sequence (ss :: h) \u2286 s\nu : Set \u03b1\nhut : u \u2286 ss\nhu : IsOpen u \u2227 as \u2208 u\n\u22a2 \u2203 v, List.Forall\u2082 (fun a_1 s => IsOpen s \u2227 a_1 \u2208 s) (as :: hts) v \u2227 sequence v \u2286 s\n[PROOFSTEP]\nrcases ih _ Subset.rfl with \u27e8v, hv, hvss\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : TopologicalSpace \u03b1\ns\u271d : TopologicalSpace \u03b2\nl : List \u03b1\nu\u271d : List (Set \u03b1)\nas : \u03b1\nss : Set \u03b1\nhts : List \u03b1\nh : List (Set \u03b1)\nht : ss \u2208 \ud835\udcdd as\na\u271d : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) hts h\nih : \u2200 (s : Set (List \u03b1)), sequence h \u2286 s \u2192 \u2203 v, List.Forall\u2082 (fun a_1 s => IsOpen s \u2227 a_1 \u2208 s) hts v \u2227 sequence v \u2286 s\ns : Set (List \u03b1)\nhus : sequence (ss :: h) \u2286 s\nu : Set \u03b1\nhut : u \u2286 ss\nhu : IsOpen u \u2227 as \u2208 u\nv : List (Set \u03b1)\nhv : List.Forall\u2082 (fun a_1 s => IsOpen s \u2227 a_1 \u2208 s) hts v\nhvss : sequence v \u2286 sequence h\n\u22a2 \u2203 v, List.Forall\u2082 (fun a_1 s => IsOpen s \u2227 a_1 \u2208 s) (as :: hts) v \u2227 sequence v \u2286 s\n[PROOFSTEP]\nexact \u27e8u :: v, List.Forall\u2082.cons hu hv, Subset.trans (Set.seq_mono (Set.image_subset _ hut) hvss) hus\u27e9\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu : List (Set \u03b1)\nhu : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\nhus : sequence u \u2286 s\nthis : \u2203 v, List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l v \u2227 sequence v \u2286 s\n\u22a2 \u2203 t, t \u2208 traverse \ud835\udcdd l \u2227 t \u2286 s \u2227 \u2200 (a' : List \u03b1), a' \u2208 t \u2192 s \u2208 traverse \ud835\udcdd a'\n[PROOFSTEP]\nrcases this with \u27e8v, hv, hvs\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu : List (Set \u03b1)\nhu : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\nhus : sequence u \u2286 s\nv : List (Set \u03b1)\nhv : List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l v\nhvs : sequence v \u2286 s\n\u22a2 \u2203 t, t \u2208 traverse \ud835\udcdd l \u2227 t \u2286 s \u2227 \u2200 (a' : List \u03b1), a' \u2208 t \u2192 s \u2208 traverse \ud835\udcdd a'\n[PROOFSTEP]\nrefine' \u27e8sequence v, mem_traverse _ _ _, hvs, _\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu : List (Set \u03b1)\nhu : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\nhus : sequence u \u2286 s\nv : List (Set \u03b1)\nhv : List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l v\nhvs : sequence v \u2286 s\n\u22a2 List.Forall\u2082 (fun b c => id c \u2208 \ud835\udcdd b) l v\n[PROOFSTEP]\nexact hv.imp fun a s \u27e8hs, ha\u27e9 => IsOpen.mem_nhds hs ha\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu : List (Set \u03b1)\nhu : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\nhus : sequence u \u2286 s\nv : List (Set \u03b1)\nhv : List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l v\nhvs : sequence v \u2286 s\n\u22a2 \u2200 (a' : List \u03b1), a' \u2208 sequence v \u2192 s \u2208 traverse \ud835\udcdd a'\n[PROOFSTEP]\nintro u hu\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu\u271d : List (Set \u03b1)\nhu\u271d : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\u271d\nhus : sequence u\u271d \u2286 s\nv : List (Set \u03b1)\nhv : List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l v\nhvs : sequence v \u2286 s\nu : List \u03b1\nhu : u \u2208 sequence v\n\u22a2 s \u2208 traverse \ud835\udcdd u\n[PROOFSTEP]\nhave hu := (List.mem_traverse _ _).1 hu\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu\u271d : List (Set \u03b1)\nhu\u271d\u00b9 : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\u271d\nhus : sequence u\u271d \u2286 s\nv : List (Set \u03b1)\nhv : List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l v\nhvs : sequence v \u2286 s\nu : List \u03b1\nhu\u271d : u \u2208 sequence v\nhu : List.Forall\u2082 (fun b a => b \u2208 id a) u v\n\u22a2 s \u2208 traverse \ud835\udcdd u\n[PROOFSTEP]\nhave : List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) u v :=\n by\n refine' List.Forall\u2082.flip _\n replace hv := hv.flip\n simp only [List.forall\u2082_and_left, flip] at hv \u22a2\n exact \u27e8hv.1, hu.flip\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu\u271d : List (Set \u03b1)\nhu\u271d\u00b9 : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\u271d\nhus : sequence u\u271d \u2286 s\nv : List (Set \u03b1)\nhv : List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l v\nhvs : sequence v \u2286 s\nu : List \u03b1\nhu\u271d : u \u2208 sequence v\nhu : List.Forall\u2082 (fun b a => b \u2208 id a) u v\n\u22a2 List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) u v\n[PROOFSTEP]\nrefine' List.Forall\u2082.flip _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu\u271d : List (Set \u03b1)\nhu\u271d\u00b9 : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\u271d\nhus : sequence u\u271d \u2286 s\nv : List (Set \u03b1)\nhv : List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l v\nhvs : sequence v \u2286 s\nu : List \u03b1\nhu\u271d : u \u2208 sequence v\nhu : List.Forall\u2082 (fun b a => b \u2208 id a) u v\n\u22a2 List.Forall\u2082 (flip fun a s => IsOpen s \u2227 a \u2208 s) v u\n[PROOFSTEP]\nreplace hv := hv.flip\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu\u271d : List (Set \u03b1)\nhu\u271d\u00b9 : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\u271d\nhus : sequence u\u271d \u2286 s\nv : List (Set \u03b1)\nhvs : sequence v \u2286 s\nu : List \u03b1\nhu\u271d : u \u2208 sequence v\nhu : List.Forall\u2082 (fun b a => b \u2208 id a) u v\nhv : List.Forall\u2082 (fun s a => IsOpen s \u2227 a \u2208 s) v l\n\u22a2 List.Forall\u2082 (flip fun a s => IsOpen s \u2227 a \u2208 s) v u\n[PROOFSTEP]\nsimp only [List.forall\u2082_and_left, flip] at hv \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu\u271d : List (Set \u03b1)\nhu\u271d\u00b9 : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\u271d\nhus : sequence u\u271d \u2286 s\nv : List (Set \u03b1)\nhvs : sequence v \u2286 s\nu : List \u03b1\nhu\u271d : u \u2208 sequence v\nhu : List.Forall\u2082 (fun b a => b \u2208 id a) u v\nhv : (\u2200 (a : Set \u03b1), a \u2208 v \u2192 IsOpen a) \u2227 List.Forall\u2082 (fun a b => b \u2208 a) v l\n\u22a2 (\u2200 (a : Set \u03b1), a \u2208 v \u2192 IsOpen a) \u2227 List.Forall\u2082 (fun a b => b \u2208 a) v u\n[PROOFSTEP]\nexact \u27e8hv.1, hu.flip\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu\u271d : List (Set \u03b1)\nhu\u271d\u00b9 : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\u271d\nhus : sequence u\u271d \u2286 s\nv : List (Set \u03b1)\nhv : List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l v\nhvs : sequence v \u2286 s\nu : List \u03b1\nhu\u271d : u \u2208 sequence v\nhu : List.Forall\u2082 (fun b a => b \u2208 id a) u v\nthis : List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) u v\n\u22a2 s \u2208 traverse \ud835\udcdd u\n[PROOFSTEP]\nrefine' mem_of_superset _ hvs\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\ns : Set (List \u03b1)\nu\u271d : List (Set \u03b1)\nhu\u271d\u00b9 : List.Forall\u2082 (fun b s => s \u2208 \ud835\udcdd b) l u\u271d\nhus : sequence u\u271d \u2286 s\nv : List (Set \u03b1)\nhv : List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) l v\nhvs : sequence v \u2286 s\nu : List \u03b1\nhu\u271d : u \u2208 sequence v\nhu : List.Forall\u2082 (fun b a => b \u2208 id a) u v\nthis : List.Forall\u2082 (fun a s => IsOpen s \u2227 a \u2208 s) u v\n\u22a2 sequence v \u2208 traverse \ud835\udcdd u\n[PROOFSTEP]\nexact mem_traverse _ _ (this.imp fun a s \u27e8hs, ha\u27e9 => IsOpen.mem_nhds hs ha)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\n\u22a2 \ud835\udcdd [] = pure []\n[PROOFSTEP]\nrw [nhds_list, List.traverse_nil _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 \ud835\udcdd (a :: l) = Seq.seq (List.cons <$> \ud835\udcdd a) fun x => \ud835\udcdd l\n[PROOFSTEP]\nrw [nhds_list, List.traverse_cons _, \u2190 nhds_list]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 Tendsto (fun p => p.fst :: p.snd) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (a :: l))\n[PROOFSTEP]\nrw [nhds_cons, Tendsto, Filter.map_prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 Filter.seq (Filter.map (fun a b => (a, b).fst :: (a, b).snd) (\ud835\udcdd a)) (\ud835\udcdd l) \u2264 Seq.seq (cons <$> \ud835\udcdd a) fun x => \ud835\udcdd l\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\n\u03b2 : Type u_3\nf : List \u03b1 \u2192 \u03b2\nb : Filter \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 Tendsto f (\ud835\udcdd (a :: l)) b \u2194 Tendsto (fun p => f (p.fst :: p.snd)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) b\n[PROOFSTEP]\nhave : \ud835\udcdd (a :: l) = (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l).map fun p : \u03b1 \u00d7 List \u03b1 => p.1 :: p.2 :=\n by\n simp only [nhds_cons, Filter.prod_eq, (Filter.map_def _ _).symm, (Filter.seq_eq_filter_seq _ _).symm]\n simp [-Filter.map_def, (\u00b7 \u2218 \u00b7), functor_norm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\n\u03b2 : Type u_3\nf : List \u03b1 \u2192 \u03b2\nb : Filter \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 \ud835\udcdd (a :: l) = Filter.map (fun p => p.fst :: p.snd) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l)\n[PROOFSTEP]\nsimp only [nhds_cons, Filter.prod_eq, (Filter.map_def _ _).symm, (Filter.seq_eq_filter_seq _ _).symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\n\u03b2 : Type u_3\nf : List \u03b1 \u2192 \u03b2\nb : Filter \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 (Seq.seq (cons <$> \ud835\udcdd a) fun x => \ud835\udcdd l) = (fun p => p.fst :: p.snd) <$> Seq.seq (Prod.mk <$> \ud835\udcdd a) fun x => \ud835\udcdd l\n[PROOFSTEP]\nsimp [-Filter.map_def, (\u00b7 \u2218 \u00b7), functor_norm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\n\u03b2 : Type u_3\nf : List \u03b1 \u2192 \u03b2\nb : Filter \u03b2\na : \u03b1\nl : List \u03b1\nthis : \ud835\udcdd (a :: l) = Filter.map (fun p => p.fst :: p.snd) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l)\n\u22a2 Tendsto f (\ud835\udcdd (a :: l)) b \u2194 Tendsto (fun p => f (p.fst :: p.snd)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) b\n[PROOFSTEP]\nrw [this, Filter.tendsto_map'_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\n\u03b2 : Type u_3\nf : List \u03b1 \u2192 \u03b2\nb : Filter \u03b2\na : \u03b1\nl : List \u03b1\nthis : \ud835\udcdd (a :: l) = Filter.map (fun p => p.fst :: p.snd) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l)\n\u22a2 Tendsto (f \u2218 fun p => p.fst :: p.snd) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) b \u2194 Tendsto (fun p => f (p.fst :: p.snd)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\n\u03b2 : Type u_3\nf : List \u03b1 \u2192 \u03b2\nr : List \u03b1 \u2192 Filter \u03b2\nh_nil : Tendsto f (pure []) (r [])\nh_cons : \u2200 (l : List \u03b1) (a : \u03b1), Tendsto f (\ud835\udcdd l) (r l) \u2192 Tendsto (fun p => f (p.fst :: p.snd)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (r (a :: l))\n\u22a2 Tendsto f (\ud835\udcdd []) (r [])\n[PROOFSTEP]\nrwa [nhds_nil]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\n\u03b2 : Type u_3\nf : List \u03b1 \u2192 \u03b2\nr : List \u03b1 \u2192 Filter \u03b2\nh_nil : Tendsto f (pure []) (r [])\nh_cons : \u2200 (l : List \u03b1) (a : \u03b1), Tendsto f (\ud835\udcdd l) (r l) \u2192 Tendsto (fun p => f (p.fst :: p.snd)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (r (a :: l))\na : \u03b1\nl : List \u03b1\n\u22a2 Tendsto f (\ud835\udcdd (a :: l)) (r (a :: l))\n[PROOFSTEP]\nrw [tendsto_cons_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\n\u03b2 : Type u_3\nf : List \u03b1 \u2192 \u03b2\nr : List \u03b1 \u2192 Filter \u03b2\nh_nil : Tendsto f (pure []) (r [])\nh_cons : \u2200 (l : List \u03b1) (a : \u03b1), Tendsto f (\ud835\udcdd l) (r l) \u2192 Tendsto (fun p => f (p.fst :: p.snd)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (r (a :: l))\na : \u03b1\nl : List \u03b1\n\u22a2 Tendsto (fun p => f (p.fst :: p.snd)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (r (a :: l))\n[PROOFSTEP]\nexact h_cons l a (@tendsto_nhds _ _ _ h_nil h_cons l)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\n\u22a2 \u2200 (l : List \u03b1), ContinuousAt length l\n[PROOFSTEP]\nsimp only [ContinuousAt, nhds_discrete]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\n\u22a2 \u2200 (l : List \u03b1), Tendsto length (\ud835\udcdd l) (pure (length l))\n[PROOFSTEP]\nrefine' tendsto_nhds _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\n\u22a2 Tendsto length (pure []) (pure (length []))\n[PROOFSTEP]\nexact tendsto_pure_pure _ _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\n\u22a2 \u2200 (l : List \u03b1) (a : \u03b1),\n Tendsto length (\ud835\udcdd l) (pure (length l)) \u2192\n Tendsto (fun p => length (p.fst :: p.snd)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (pure (length (a :: l)))\n[PROOFSTEP]\nintro l a ih\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\na : \u03b1\nih : Tendsto length (\ud835\udcdd l) (pure (length l))\n\u22a2 Tendsto (fun p => length (p.fst :: p.snd)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (pure (length (a :: l)))\n[PROOFSTEP]\ndsimp only [List.length]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\na : \u03b1\nih : Tendsto length (\ud835\udcdd l) (pure (length l))\n\u22a2 Tendsto (fun p => length p.snd + 1) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (pure (length l + 1))\n[PROOFSTEP]\nrefine' Tendsto.comp (tendsto_pure_pure (fun x => x + 1) _) _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl : List \u03b1\na : \u03b1\nih : Tendsto length (\ud835\udcdd l) (pure (length l))\n\u22a2 Tendsto (fun p => Nat.add (length p.snd) 0) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (pure (length l))\n[PROOFSTEP]\nrefine' Tendsto.comp ih tendsto_snd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na : \u03b1\nn : \u2115\n\u22a2 Tendsto (fun p => insertNth (n + 1) p.fst p.snd) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd []) (\ud835\udcdd (insertNth (n + 1) a []))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na : \u03b1\nn : \u2115\na' : \u03b1\nl : List \u03b1\n\u22a2 Tendsto (fun p => insertNth (n + 1) p.fst p.snd) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd (a' :: l)) (\ud835\udcdd (insertNth (n + 1) a (a' :: l)))\n[PROOFSTEP]\nhave : \ud835\udcdd a \u00d7\u02e2 \ud835\udcdd (a' :: l) = (\ud835\udcdd a \u00d7\u02e2 (\ud835\udcdd a' \u00d7\u02e2 \ud835\udcdd l)).map fun p : \u03b1 \u00d7 \u03b1 \u00d7 List \u03b1 => (p.1, p.2.1 :: p.2.2) :=\n by\n simp only [nhds_cons, Filter.prod_eq, \u2190 Filter.map_def, \u2190 Filter.seq_eq_filter_seq]\n simp [-Filter.map_def, (\u00b7 \u2218 \u00b7), functor_norm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na : \u03b1\nn : \u2115\na' : \u03b1\nl : List \u03b1\n\u22a2 \ud835\udcdd a \u00d7\u02e2 \ud835\udcdd (a' :: l) = Filter.map (fun p => (p.fst, p.snd.fst :: p.snd.snd)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd a' \u00d7\u02e2 \ud835\udcdd l)\n[PROOFSTEP]\nsimp only [nhds_cons, Filter.prod_eq, \u2190 Filter.map_def, \u2190 Filter.seq_eq_filter_seq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na : \u03b1\nn : \u2115\na' : \u03b1\nl : List \u03b1\n\u22a2 (Seq.seq (Prod.mk <$> \ud835\udcdd a) fun x => Seq.seq (cons <$> \ud835\udcdd a') fun x => \ud835\udcdd l) =\n (fun p => (p.fst, p.snd.fst :: p.snd.snd)) <$>\n Seq.seq (Prod.mk <$> \ud835\udcdd a) fun x => Seq.seq (Prod.mk <$> \ud835\udcdd a') fun x => \ud835\udcdd l\n[PROOFSTEP]\nsimp [-Filter.map_def, (\u00b7 \u2218 \u00b7), functor_norm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na : \u03b1\nn : \u2115\na' : \u03b1\nl : List \u03b1\nthis : \ud835\udcdd a \u00d7\u02e2 \ud835\udcdd (a' :: l) = Filter.map (fun p => (p.fst, p.snd.fst :: p.snd.snd)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd a' \u00d7\u02e2 \ud835\udcdd l)\n\u22a2 Tendsto (fun p => insertNth (n + 1) p.fst p.snd) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd (a' :: l)) (\ud835\udcdd (insertNth (n + 1) a (a' :: l)))\n[PROOFSTEP]\nrw [this, tendsto_map'_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na : \u03b1\nn : \u2115\na' : \u03b1\nl : List \u03b1\nthis : \ud835\udcdd a \u00d7\u02e2 \ud835\udcdd (a' :: l) = Filter.map (fun p => (p.fst, p.snd.fst :: p.snd.snd)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd a' \u00d7\u02e2 \ud835\udcdd l)\n\u22a2 Tendsto ((fun p => insertNth (n + 1) p.fst p.snd) \u2218 fun p => (p.fst, p.snd.fst :: p.snd.snd)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd a' \u00d7\u02e2 \ud835\udcdd l)\n (\ud835\udcdd (insertNth (n + 1) a (a' :: l)))\n[PROOFSTEP]\nexact\n (tendsto_fst.comp tendsto_snd).cons\n ((@tendsto_insertNth' _ n l).comp <| tendsto_fst.prod_mk <| tendsto_snd.comp tendsto_snd)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\nx\u271d : \u03b1 \u00d7 List \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 ContinuousAt (fun p => insertNth n p.fst p.snd) (a, l)\n[PROOFSTEP]\nrw [ContinuousAt, nhds_prod_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\nx\u271d : \u03b1 \u00d7 List \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 Tendsto (fun p => insertNth n p.fst p.snd) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (insertNth n (a, l).fst (a, l).snd))\n[PROOFSTEP]\nexact tendsto_insertNth'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nx\u271d : \u2115\n\u22a2 Tendsto (fun l => removeNth l x\u271d) (\ud835\udcdd []) (\ud835\udcdd (removeNth [] x\u271d))\n[PROOFSTEP]\nrw [nhds_nil]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nx\u271d : \u2115\n\u22a2 Tendsto (fun l => removeNth l x\u271d) (pure []) (\ud835\udcdd (removeNth [] x\u271d))\n[PROOFSTEP]\nexact tendsto_pure_nhds _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 Tendsto (fun l => removeNth l 0) (\ud835\udcdd (a :: l)) (\ud835\udcdd (removeNth (a :: l) 0))\n[PROOFSTEP]\nrw [tendsto_cons_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 Tendsto (fun p => removeNth (p.fst :: p.snd) 0) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (removeNth (a :: l) 0))\n[PROOFSTEP]\nexact tendsto_snd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\na : \u03b1\nl : List \u03b1\n\u22a2 Tendsto (fun l => removeNth l (n + 1)) (\ud835\udcdd (a :: l)) (\ud835\udcdd (removeNth (a :: l) (n + 1)))\n[PROOFSTEP]\nrw [tendsto_cons_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\na : \u03b1\nl : List \u03b1\n\u22a2 Tendsto (fun p => removeNth (p.fst :: p.snd) (n + 1)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (removeNth (a :: l) (n + 1)))\n[PROOFSTEP]\ndsimp [removeNth]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\na : \u03b1\nl : List \u03b1\n\u22a2 Tendsto (fun p => p.fst :: removeNth p.snd n) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (a :: removeNth l n))\n[PROOFSTEP]\nexact tendsto_fst.cons ((@tendsto_removeNth n l).comp tendsto_snd)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : ContinuousMul \u03b1\nl : List \u03b1\n\u22a2 Tendsto prod (\ud835\udcdd l) (\ud835\udcdd (prod l))\n[PROOFSTEP]\ninduction' l with x l ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : ContinuousMul \u03b1\n\u22a2 Tendsto prod (\ud835\udcdd []) (\ud835\udcdd (prod []))\n[PROOFSTEP]\nsimp (config := { contextual := true }) [nhds_nil, mem_of_mem_nhds, tendsto_pure_left]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : ContinuousMul \u03b1\nx : \u03b1\nl : List \u03b1\nih : Tendsto prod (\ud835\udcdd l) (\ud835\udcdd (prod l))\n\u22a2 Tendsto prod (\ud835\udcdd (x :: l)) (\ud835\udcdd (prod (x :: l)))\n[PROOFSTEP]\nsimp_rw [tendsto_cons_iff, prod_cons]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : ContinuousMul \u03b1\nx : \u03b1\nl : List \u03b1\nih : Tendsto prod (\ud835\udcdd l) (\ud835\udcdd (prod l))\n\u22a2 Tendsto (fun p => p.fst * prod p.snd) (\ud835\udcdd x \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (x * prod l))\n[PROOFSTEP]\nhave := continuous_iff_continuousAt.mp continuous_mul (x, l.prod)\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : ContinuousMul \u03b1\nx : \u03b1\nl : List \u03b1\nih : Tendsto prod (\ud835\udcdd l) (\ud835\udcdd (prod l))\nthis : ContinuousAt (fun p => p.fst * p.snd) (x, prod l)\n\u22a2 Tendsto (fun p => p.fst * prod p.snd) (\ud835\udcdd x \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (x * prod l))\n[PROOFSTEP]\nrw [ContinuousAt, nhds_prod_eq] at this \n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : ContinuousMul \u03b1\nx : \u03b1\nl : List \u03b1\nih : Tendsto prod (\ud835\udcdd l) (\ud835\udcdd (prod l))\nthis : Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd x \u00d7\u02e2 \ud835\udcdd (prod l)) (\ud835\udcdd ((x, prod l).fst * (x, prod l).snd))\n\u22a2 Tendsto (fun p => p.fst * prod p.snd) (\ud835\udcdd x \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (x * prod l))\n[PROOFSTEP]\nexact this.comp (tendsto_id.prod_map ih)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\n\u22a2 TopologicalSpace (Vector \u03b1 n)\n[PROOFSTEP]\nunfold Vector\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\n\u22a2 TopologicalSpace { l // List.length l = n }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\na : \u03b1\nl : Vector \u03b1 n\n\u22a2 Tendsto (fun p => p.fst ::\u1d65 p.snd) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (a ::\u1d65 l))\n[PROOFSTEP]\nrw [tendsto_subtype_rng, cons_val]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\na : \u03b1\nl : Vector \u03b1 n\n\u22a2 Tendsto (fun x => \u2191(x.fst ::\u1d65 x.snd)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (a :: \u2191l))\n[PROOFSTEP]\nexact tendsto_fst.cons (Tendsto.comp continuousAt_subtype_val tendsto_snd)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\ni : Fin (n + 1)\na : \u03b1\nl : List \u03b1\nhl : List.length l = n\n\u22a2 Tendsto (fun p => insertNth p.fst i p.snd) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd { val := l, property := hl })\n (\ud835\udcdd (insertNth a i { val := l, property := hl }))\n[PROOFSTEP]\nrw [insertNth, tendsto_subtype_rng]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\ni : Fin (n + 1)\na : \u03b1\nl : List \u03b1\nhl : List.length l = n\n\u22a2 Tendsto (fun x => \u2191(insertNth x.fst i x.snd)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd { val := l, property := hl })\n (\ud835\udcdd\n \u2191{ val := List.insertNth (\u2191i) a \u2191{ val := l, property := hl },\n property := (_ : List.length (List.insertNth (\u2191i) a \u2191{ val := l, property := hl }) = n + 1) })\n[PROOFSTEP]\nsimp [insertNth_val]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\ni : Fin (n + 1)\na : \u03b1\nl : List \u03b1\nhl : List.length l = n\n\u22a2 Tendsto (fun x => List.insertNth (\u2191i) x.fst \u2191x.snd) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd { val := l, property := hl })\n (\ud835\udcdd (List.insertNth (\u2191i) a l))\n[PROOFSTEP]\nexact List.tendsto_insertNth tendsto_fst (Tendsto.comp continuousAt_subtype_val tendsto_snd : _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\ni : Fin (n + 1)\nx\u271d : \u03b1 \u00d7 Vector \u03b1 n\na : \u03b1\nl : Vector \u03b1 n\n\u22a2 ContinuousAt (fun p => insertNth p.fst i p.snd) (a, l)\n[PROOFSTEP]\nrw [ContinuousAt, nhds_prod_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\ni : Fin (n + 1)\nx\u271d : \u03b1 \u00d7 Vector \u03b1 n\na : \u03b1\nl : Vector \u03b1 n\n\u22a2 Tendsto (fun p => insertNth p.fst i p.snd) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (insertNth (a, l).fst i (a, l).snd))\n[PROOFSTEP]\nexact tendsto_insertNth\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\ni : Fin (n + 1)\nl : List \u03b1\nhl : List.length l = n + 1\n\u22a2 ContinuousAt (removeNth i) { val := l, property := hl }\n[PROOFSTEP]\nrw [ContinuousAt, removeNth, tendsto_subtype_rng]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\ni : Fin (n + 1)\nl : List \u03b1\nhl : List.length l = n + 1\n\u22a2 Tendsto (fun x => \u2191(removeNth i x)) (\ud835\udcdd { val := l, property := hl })\n (\ud835\udcdd\n \u2191(match { val := l, property := hl } with\n | { val := l, property := p } =>\n { val := List.removeNth l \u2191i, property := (_ : List.length (List.removeNth l \u2191i) = n + 1 - 1) }))\n[PROOFSTEP]\nsimp only [Vector.removeNth_val]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nn : \u2115\ni : Fin (n + 1)\nl : List \u03b1\nhl : List.length l = n + 1\n\u22a2 Tendsto (fun x => List.removeNth \u2191x \u2191i) (\ud835\udcdd { val := l, property := hl }) (\ud835\udcdd (List.removeNth l \u2191i))\n[PROOFSTEP]\nexact Tendsto.comp List.tendsto_removeNth continuousAt_subtype_val\n", "meta": {"mathlib_filename": "Mathlib.Topology.List", "llama_tokens": 15851, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432182679957, "lm_q2_score": 0.5926665999540697, "lm_q1q2_score": 0.427753099210801}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : ArtinianObject X\nh : \u00acIsZero X\n\u22a2 \u2203 Y, Simple (underlying.obj Y)\n[PROOFSTEP]\nhaveI : Nontrivial (Subobject X) := nontrivial_of_not_isZero h\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : ArtinianObject X\nh : \u00acIsZero X\nthis : Nontrivial (Subobject X)\n\u22a2 \u2203 Y, Simple (underlying.obj Y)\n[PROOFSTEP]\nhaveI := isAtomic_of_orderBot_wellFounded_lt (ArtinianObject.subobject_lt_wellFounded X)\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : ArtinianObject X\nh : \u00acIsZero X\nthis\u271d : Nontrivial (Subobject X)\nthis : IsAtomic (Subobject X)\n\u22a2 \u2203 Y, Simple (underlying.obj Y)\n[PROOFSTEP]\nobtain \u27e8Y, s\u27e9 := (IsAtomic.eq_bot_or_exists_atom_le (\u22a4 : Subobject X)).resolve_left top_ne_bot\n[GOAL]\ncase intro\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : ArtinianObject X\nh : \u00acIsZero X\nthis\u271d : Nontrivial (Subobject X)\nthis : IsAtomic (Subobject X)\nY : Subobject X\ns : IsAtom Y \u2227 Y \u2264 \u22a4\n\u22a2 \u2203 Y, Simple (underlying.obj Y)\n[PROOFSTEP]\nexact \u27e8Y, (subobject_simple_iff_isAtom _).mpr s.1\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{?u.7023, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : ArtinianObject X\nh : \u00acIsZero X\n\u22a2 Mono (simpleSubobjectArrow h)\n[PROOFSTEP]\ndsimp only [simpleSubobjectArrow]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{?u.7023, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : ArtinianObject X\nh : \u00acIsZero X\n\u22a2 Mono (arrow (Exists.choose (_ : \u2203 Y, Simple (underlying.obj Y))))\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Noetherian", "llama_tokens": 836, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217431943271999, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.427753085021891}} {"text": "[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nC : HomologicalComplex (HomologicalComplex V c) c'\ni : \u03b9\nj j' : \u03b9'\nw : \u00acComplexShape.Rel c' j j'\n\u22a2 (fun j j' => Hom.f (d C j j') i) j j' = 0\n[PROOFSTEP]\nsimp_all only [shape, zero_f]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nC : HomologicalComplex (HomologicalComplex V c) c'\ni i' : \u03b9\nw : \u00acComplexShape.Rel c i i'\n\u22a2 (fun i i' => Hom.mk fun j => d (X C j) i i') i i' = 0\n[PROOFSTEP]\next j\n[GOAL]\ncase h\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nC : HomologicalComplex (HomologicalComplex V c) c'\ni i' : \u03b9\nw : \u00acComplexShape.Rel c i i'\nj : \u03b9'\n\u22a2 Hom.f ((fun i i' => Hom.mk fun j => d (X C j) i i') i i') j = Hom.f 0 j\n[PROOFSTEP]\nexact (C.X j).shape i i' w\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nC : HomologicalComplex (HomologicalComplex V c) c'\ni j : \u03b9'\nx\u271d : ComplexShape.Rel c' i j\n\u22a2 (fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) i \u226b d ((flip V c c' \u22d9 flip V c' c).obj C) i j =\n d ((\ud835\udfed (HomologicalComplex (HomologicalComplex V c) c')).obj C) i j \u226b (fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) j\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nC : HomologicalComplex (HomologicalComplex V c) c'\ni j : \u03b9'\nx\u271d : ComplexShape.Rel c' i j\ni\u271d : \u03b9\n\u22a2 Hom.f ((fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) i \u226b d ((flip V c c' \u22d9 flip V c' c).obj C) i j) i\u271d =\n Hom.f\n (d ((\ud835\udfed (HomologicalComplex (HomologicalComplex V c) c')).obj C) i j \u226b\n (fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) j)\n i\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nC : HomologicalComplex (HomologicalComplex V c) c'\ni j : \u03b9'\nx\u271d : ComplexShape.Rel c' i j\ni\u271d : \u03b9\n\u22a2 \ud835\udfd9 (X (X C i) i\u271d) \u226b Hom.f (d C i j) i\u271d = Hom.f (d C i j) i\u271d \u226b \ud835\udfd9 (X (X C j) i\u271d)\n[PROOFSTEP]\nsimp only [Category.id_comp, Category.comp_id]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nC : HomologicalComplex (HomologicalComplex V c) c'\ni j : \u03b9'\nx\u271d : ComplexShape.Rel c' i j\n\u22a2 (fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) i \u226b d ((\ud835\udfed (HomologicalComplex (HomologicalComplex V c) c')).obj C) i j =\n d ((flip V c c' \u22d9 flip V c' c).obj C) i j \u226b (fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) j\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nC : HomologicalComplex (HomologicalComplex V c) c'\ni j : \u03b9'\nx\u271d : ComplexShape.Rel c' i j\ni\u271d : \u03b9\n\u22a2 Hom.f\n ((fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) i \u226b\n d ((\ud835\udfed (HomologicalComplex (HomologicalComplex V c) c')).obj C) i j)\n i\u271d =\n Hom.f (d ((flip V c c' \u22d9 flip V c' c).obj C) i j \u226b (fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) j) i\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nC : HomologicalComplex (HomologicalComplex V c) c'\ni j : \u03b9'\nx\u271d : ComplexShape.Rel c' i j\ni\u271d : \u03b9\n\u22a2 \ud835\udfd9 (X (X C i) i\u271d) \u226b Hom.f (d C i j) i\u271d = Hom.f (d C i j) i\u271d \u226b \ud835\udfd9 (X (X C j) i\u271d)\n[PROOFSTEP]\nsimp only [Category.id_comp, Category.comp_id]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nX Y : HomologicalComplex (HomologicalComplex V c) c'\nf : X \u27f6 Y\n\u22a2 (\ud835\udfed (HomologicalComplex (HomologicalComplex V c) c')).map f \u226b\n ((fun C =>\n Iso.mk (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j))\n (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j)))\n Y).hom =\n ((fun C =>\n Iso.mk (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j))\n (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j)))\n X).hom \u226b\n (flip V c c' \u22d9 flip V c' c).map f\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nX Y : HomologicalComplex (HomologicalComplex V c) c'\nf : X \u27f6 Y\ni\u271d\u00b9 : \u03b9'\ni\u271d : \u03b9\n\u22a2 Hom.f\n (Hom.f\n ((\ud835\udfed (HomologicalComplex (HomologicalComplex V c) c')).map f \u226b\n ((fun C =>\n Iso.mk (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j))\n (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j)))\n Y).hom)\n i\u271d\u00b9)\n i\u271d =\n Hom.f\n (Hom.f\n (((fun C =>\n Iso.mk (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j))\n (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j)))\n X).hom \u226b\n (flip V c c' \u22d9 flip V c' c).map f)\n i\u271d\u00b9)\n i\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.h\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nX Y : HomologicalComplex (HomologicalComplex V c) c'\nf : X \u27f6 Y\ni\u271d\u00b9 : \u03b9'\ni\u271d : \u03b9\n\u22a2 Hom.f (Hom.f f i\u271d\u00b9) i\u271d \u226b \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X Y i\u271d\u00b9) i\u271d) =\n \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X X i\u271d\u00b9) i\u271d) \u226b Hom.f (Hom.f f i\u271d\u00b9) i\u271d\n[PROOFSTEP]\nsimp only [Category.id_comp, Category.comp_id]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nC : HomologicalComplex (HomologicalComplex V c') c\ni j : \u03b9\nx\u271d : ComplexShape.Rel c i j\n\u22a2 (fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) i \u226b d ((\ud835\udfed (HomologicalComplex (HomologicalComplex V c') c)).obj C) i j =\n d ((flip V c' c \u22d9 flip V c c').obj C) i j \u226b (fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) j\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nC : HomologicalComplex (HomologicalComplex V c') c\ni j : \u03b9\nx\u271d : ComplexShape.Rel c i j\ni\u271d : \u03b9'\n\u22a2 Hom.f\n ((fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) i \u226b\n d ((\ud835\udfed (HomologicalComplex (HomologicalComplex V c') c)).obj C) i j)\n i\u271d =\n Hom.f (d ((flip V c' c \u22d9 flip V c c').obj C) i j \u226b (fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) j) i\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nC : HomologicalComplex (HomologicalComplex V c') c\ni j : \u03b9\nx\u271d : ComplexShape.Rel c i j\ni\u271d : \u03b9'\n\u22a2 \ud835\udfd9 (X (X C i) i\u271d) \u226b Hom.f (d C i j) i\u271d = Hom.f (d C i j) i\u271d \u226b \ud835\udfd9 (X (X C j) i\u271d)\n[PROOFSTEP]\nsimp only [Category.id_comp, Category.comp_id]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nC : HomologicalComplex (HomologicalComplex V c') c\ni j : \u03b9\nx\u271d : ComplexShape.Rel c i j\n\u22a2 (fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) i \u226b d ((flip V c' c \u22d9 flip V c c').obj C) i j =\n d ((\ud835\udfed (HomologicalComplex (HomologicalComplex V c') c)).obj C) i j \u226b (fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) j\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nC : HomologicalComplex (HomologicalComplex V c') c\ni j : \u03b9\nx\u271d : ComplexShape.Rel c i j\ni\u271d : \u03b9'\n\u22a2 Hom.f ((fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) i \u226b d ((flip V c' c \u22d9 flip V c c').obj C) i j) i\u271d =\n Hom.f\n (d ((\ud835\udfed (HomologicalComplex (HomologicalComplex V c') c)).obj C) i j \u226b\n (fun i => Hom.mk fun j => \ud835\udfd9 (X (X C i) j)) j)\n i\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nC : HomologicalComplex (HomologicalComplex V c') c\ni j : \u03b9\nx\u271d : ComplexShape.Rel c i j\ni\u271d : \u03b9'\n\u22a2 \ud835\udfd9 (X (X C i) i\u271d) \u226b Hom.f (d C i j) i\u271d = Hom.f (d C i j) i\u271d \u226b \ud835\udfd9 (X (X C j) i\u271d)\n[PROOFSTEP]\nsimp only [Category.id_comp, Category.comp_id]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nX Y : HomologicalComplex (HomologicalComplex V c') c\nf : X \u27f6 Y\n\u22a2 (flip V c' c \u22d9 flip V c c').map f \u226b\n ((fun C =>\n Iso.mk (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j))\n (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j)))\n Y).hom =\n ((fun C =>\n Iso.mk (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j))\n (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j)))\n X).hom \u226b\n (\ud835\udfed (HomologicalComplex (HomologicalComplex V c') c)).map f\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nX Y : HomologicalComplex (HomologicalComplex V c') c\nf : X \u27f6 Y\ni\u271d\u00b9 : \u03b9\ni\u271d : \u03b9'\n\u22a2 Hom.f\n (Hom.f\n ((flip V c' c \u22d9 flip V c c').map f \u226b\n ((fun C =>\n Iso.mk (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j))\n (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j)))\n Y).hom)\n i\u271d\u00b9)\n i\u271d =\n Hom.f\n (Hom.f\n (((fun C =>\n Iso.mk (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j))\n (Hom.mk fun i => Hom.mk fun j => \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X C i) j)))\n X).hom \u226b\n (\ud835\udfed (HomologicalComplex (HomologicalComplex V c') c)).map f)\n i\u271d\u00b9)\n i\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.h\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : HasZeroMorphisms V\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u03b9' : Type u_2\nc' : ComplexShape \u03b9'\nX Y : HomologicalComplex (HomologicalComplex V c') c\nf : X \u27f6 Y\ni\u271d\u00b9 : \u03b9\ni\u271d : \u03b9'\n\u22a2 Hom.f (Hom.f f i\u271d\u00b9) i\u271d \u226b \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X Y i\u271d\u00b9) i\u271d) =\n \ud835\udfd9 (HomologicalComplex.X (HomologicalComplex.X X i\u271d\u00b9) i\u271d) \u226b Hom.f (Hom.f f i\u271d\u00b9) i\u271d\n[PROOFSTEP]\nsimp only [Category.id_comp, Category.comp_id]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Homology.Flip", "llama_tokens": 5222, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506635289836, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.4276199284191845}} {"text": "[GOAL]\nE : Type u_1\nL : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : AddCommGroup L\ninst\u271d\u2074 : Countable L\ninst\u271d\u00b3 : AddAction L E\ninst\u271d\u00b2 : MeasurableSpace L\ninst\u271d\u00b9 : MeasurableVAdd L E\ninst\u271d : VAddInvariantMeasure L E \u03bc\nfund : IsAddFundamentalDomain L F\nhS : NullMeasurableSet s\nh : \u2191\u2191\u03bc F < \u2191\u2191\u03bc s\n\u22a2 \u2203 x y, x \u2260 y \u2227 \u00acDisjoint (x +\u1d65 s) (y +\u1d65 s)\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\nE : Type u_1\nL : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : AddCommGroup L\ninst\u271d\u2074 : Countable L\ninst\u271d\u00b3 : AddAction L E\ninst\u271d\u00b2 : MeasurableSpace L\ninst\u271d\u00b9 : MeasurableVAdd L E\ninst\u271d : VAddInvariantMeasure L E \u03bc\nfund : IsAddFundamentalDomain L F\nhS : NullMeasurableSet s\nh : \u2200 (x y : L), x \u2260 y \u2192 Disjoint (x +\u1d65 s) (y +\u1d65 s)\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F\n[PROOFSTEP]\nexact\n ((fund.measure_eq_tsum _).trans\n (measure_iUnion\u2080 (Pairwise.mono h fun i j hij => (hij.mono inf_le_left inf_le_left).aedisjoint) fun _ =>\n (hS.vadd _).inter fund.nullMeasurableSet).symm).trans_le\n (measure_mono <| Set.iUnion_subset fun _ => Set.inter_subset_right _ _)\n[GOAL]\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\n\u22a2 \u2203 x x_1, \u2191x \u2208 s\n[PROOFSTEP]\nhave h_vol : \u03bc F < \u03bc ((2\u207b\u00b9 : \u211d) \u2022 s) :=\n by\n rw [addHaar_smul_of_nonneg \u03bc (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9) s, \u2190\n mul_lt_mul_right (pow_ne_zero (finrank \u211d E) (two_ne_zero' _)) (pow_ne_top two_ne_top), mul_right_comm,\n ofReal_pow (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9), \u2190 ofReal_inv_of_pos zero_lt_two]\n norm_num\n rwa [\u2190 mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul]\n[GOAL]\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\n\u22a2 \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\n[PROOFSTEP]\nrw [addHaar_smul_of_nonneg \u03bc (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9) s, \u2190\n mul_lt_mul_right (pow_ne_zero (finrank \u211d E) (two_ne_zero' _)) (pow_ne_top two_ne_top), mul_right_comm,\n ofReal_pow (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9), \u2190 ofReal_inv_of_pos zero_lt_two]\n[GOAL]\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\n\u22a2 0 \u2264 2\u207b\u00b9\n[PROOFSTEP]\nnorm_num\n[GOAL]\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\n\u22a2 0 \u2264 2\u207b\u00b9\n[PROOFSTEP]\nnorm_num\n[GOAL]\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\n\u22a2 \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < (ENNReal.ofReal 2)\u207b\u00b9 ^ finrank \u211d E * 2 ^ finrank \u211d E * \u2191\u2191\u03bc s\n[PROOFSTEP]\nnorm_num\n[GOAL]\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\n\u22a2 \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < 2\u207b\u00b9 ^ finrank \u211d E * 2 ^ finrank \u211d E * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrwa [\u2190 mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul]\n[GOAL]\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\n\u22a2 \u2203 x x_1, \u2191x \u2208 s\n[PROOFSTEP]\nobtain \u27e8x, y, hxy, h\u27e9 := exists_pair_mem_lattice_not_disjoint_vadd fund ((h_conv.smul _).nullMeasurableSet _) h_vol\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\n\u22a2 \u2203 x x_1, \u2191x \u2208 s\n[PROOFSTEP]\nobtain \u27e8_, \u27e8v, hv, rfl\u27e9, w, hw, hvw\u27e9 := Set.not_disjoint_iff.mp h\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : v \u2208 2\u207b\u00b9 \u2022 s\nw : E\nhw : w \u2208 2\u207b\u00b9 \u2022 s\nhvw : (fun x => y +\u1d65 x) w = (fun x_1 => x +\u1d65 x_1) v\n\u22a2 \u2203 x x_1, \u2191x \u2208 s\n[PROOFSTEP]\nrefine' \u27e8x - y, sub_ne_zero.2 hxy, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : v \u2208 2\u207b\u00b9 \u2022 s\nw : E\nhw : w \u2208 2\u207b\u00b9 \u2022 s\nhvw : (fun x => y +\u1d65 x) w = (fun x_1 => x +\u1d65 x_1) v\n\u22a2 \u2191(x - y) \u2208 s\n[PROOFSTEP]\nrw [Set.mem_inv_smul_set_iff\u2080 (two_ne_zero' \u211d)] at hv hw \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : 2 \u2022 v \u2208 s\nw : E\nhw : 2 \u2022 w \u2208 s\nhvw : (fun x => y +\u1d65 x) w = (fun x_1 => x +\u1d65 x_1) v\n\u22a2 \u2191(x - y) \u2208 s\n[PROOFSTEP]\nsimp_rw [AddSubgroup.vadd_def, vadd_eq_add, add_comm _ w, \u2190 sub_eq_sub_iff_add_eq_add, \u2190 AddSubgroup.coe_sub] at hvw \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : 2 \u2022 v \u2208 s\nw : E\nhw : 2 \u2022 w \u2208 s\nhvw : w - v = \u2191(x - y)\n\u22a2 \u2191(x - y) \u2208 s\n[PROOFSTEP]\nrw [\u2190 hvw, \u2190 inv_smul_smul\u2080 (two_ne_zero' \u211d) (_ - _), smul_sub, sub_eq_add_neg, smul_add]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : 2 \u2022 v \u2208 s\nw : E\nhw : 2 \u2022 w \u2208 s\nhvw : w - v = \u2191(x - y)\n\u22a2 2\u207b\u00b9 \u2022 2 \u2022 w + 2\u207b\u00b9 \u2022 -(2 \u2022 v) \u2208 s\n[PROOFSTEP]\nrefine' h_conv hw (h_symm _ hv) _ _ _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : 2 \u2022 v \u2208 s\nw : E\nhw : 2 \u2022 w \u2208 s\nhvw : w - v = \u2191(x - y)\n\u22a2 0 \u2264 2\u207b\u00b9\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : 2 \u2022 v \u2208 s\nw : E\nhw : 2 \u2022 w \u2208 s\nhvw : w - v = \u2191(x - y)\n\u22a2 0 \u2264 2\u207b\u00b9\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\nE : Type u_1\nL\u271d : Type u_2\ninst\u271d\u2076 : MeasurableSpace E\n\u03bc : Measure E\nF s : Set E\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\nL : AddSubgroup E\ninst\u271d : Countable { x // x \u2208 L }\nfund : IsAddFundamentalDomain { x // x \u2208 L } F\nh\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s\nh_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s\nh_conv : Convex \u211d s\nh_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s)\nx y : { x // x \u2208 L }\nhxy : x \u2260 y\nh : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s)\nv : E\nhv : 2 \u2022 v \u2208 s\nw : E\nhw : 2 \u2022 w \u2208 s\nhvw : w - v = \u2191(x - y)\n\u22a2 2\u207b\u00b9 + 2\u207b\u00b9 = 1\n[PROOFSTEP]\nnorm_num\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Group.GeometryOfNumbers", "llama_tokens": 6232, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506526772883, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.42761992231861645}} {"text": "[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\ntoSetoid\u271d\u00b9 : Setoid X\nisOpen_setOf_rel\u271d\u00b9 : \u2200 (x : X), IsOpen (setOf (Setoid.Rel toSetoid\u271d\u00b9 x))\ntoSetoid\u271d : Setoid X\nisOpen_setOf_rel\u271d : \u2200 (x : X), IsOpen (setOf (Setoid.Rel toSetoid\u271d x))\nx\u271d :\n { toSetoid := toSetoid\u271d\u00b9, isOpen_setOf_rel := isOpen_setOf_rel\u271d\u00b9 }.toSetoid =\n { toSetoid := toSetoid\u271d, isOpen_setOf_rel := isOpen_setOf_rel\u271d }.toSetoid\n\u22a2 { toSetoid := toSetoid\u271d\u00b9, isOpen_setOf_rel := isOpen_setOf_rel\u271d\u00b9 } =\n { toSetoid := toSetoid\u271d, isOpen_setOf_rel := isOpen_setOf_rel\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nA : Set X\nh : IsClopen A\nx : X\n\u22a2 IsOpen (setOf (Setoid.Rel { r := fun x y => x \u2208 A \u2194 y \u2208 A, iseqv := (_ : Equivalence fun x y => x \u2208 A \u2194 y \u2208 A) } x))\n[PROOFSTEP]\nby_cases hx : x \u2208 A\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nA : Set X\nh : IsClopen A\nx : X\nhx : x \u2208 A\n\u22a2 IsOpen (setOf (Setoid.Rel { r := fun x y => x \u2208 A \u2194 y \u2208 A, iseqv := (_ : Equivalence fun x y => x \u2208 A \u2194 y \u2208 A) } x))\n[PROOFSTEP]\nsimp [Setoid.Rel, hx, h.1, h.2, \u2190 compl_setOf]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nA : Set X\nh : IsClopen A\nx : X\nhx : \u00acx \u2208 A\n\u22a2 IsOpen (setOf (Setoid.Rel { r := fun x y => x \u2208 A \u2194 y \u2208 A, iseqv := (_ : Equivalence fun x y => x \u2208 A \u2194 y \u2208 A) } x))\n[PROOFSTEP]\nsimp [Setoid.Rel, hx, h.1, h.2, \u2190 compl_setOf]\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nx : X\n\u22a2 IsOpen {proj S x}\n[PROOFSTEP]\nrw [\u2190 S.proj_quotientMap.isOpen_preimage, fiber_eq]\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nx : X\n\u22a2 IsOpen (setOf (Setoid.Rel S.toSetoid x))\n[PROOFSTEP]\nexact S.isOpen_setOf_rel _\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nx : X\n\u22a2 IsClopen (setOf (Setoid.Rel S.toSetoid x))\n[PROOFSTEP]\nrw [\u2190 fiber_eq]\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nx : X\n\u22a2 IsClopen (proj S \u207b\u00b9' {proj S x})\n[PROOFSTEP]\napply isClopen_preimage\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS a : DiscreteQuotient X\n\u22a2 a \u2264 \u22a4\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\n\u22a2 \u2200 (a b : Quotient \u22a4.toSetoid), a = b\n[PROOFSTEP]\nrintro \u27e8_\u27e9 \u27e8_\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\na\u271d\u00b2 : Quotient \u22a4.toSetoid\na\u271d\u00b9 : X\nb\u271d : Quotient \u22a4.toSetoid\na\u271d : X\n\u22a2 Quot.mk Setoid.r a\u271d\u00b9 = Quot.mk Setoid.r a\u271d\n[PROOFSTEP]\nexact Quotient.sound trivial\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\ng : C(Y, Z)\nf : C(X, Y)\nA B : DiscreteQuotient Y\nh : A \u2264 B\n\u22a2 comap f A \u2264 comap f B\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS A B C : DiscreteQuotient X\n\u22a2 ofLE (_ : A \u2264 A) = id\n[PROOFSTEP]\next \u27e8\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS A B C : DiscreteQuotient X\nx\u271d : Quotient A.toSetoid\na\u271d : X\n\u22a2 ofLE (_ : A \u2264 A) (Quot.mk Setoid.r a\u271d) = id (Quot.mk Setoid.r a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS A B C : DiscreteQuotient X\na : Quotient A.toSetoid\n\u22a2 ofLE (_ : A \u2264 A) a = a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS A B C : DiscreteQuotient X\nh\u2081 : A \u2264 B\nh\u2082 : B \u2264 C\nx : Quotient A.toSetoid\n\u22a2 ofLE h\u2082 (ofLE h\u2081 x) = ofLE (_ : A \u2264 C) x\n[PROOFSTEP]\nrcases x with \u27e8\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS A B C : DiscreteQuotient X\nh\u2081 : A \u2264 B\nh\u2082 : B \u2264 C\nx : Quotient A.toSetoid\na\u271d : X\n\u22a2 ofLE h\u2082 (ofLE h\u2081 (Quot.mk Setoid.r a\u271d)) = ofLE (_ : A \u2264 C) (Quot.mk Setoid.r a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d : LocallyConnectedSpace X\nx : X\n\u22a2 IsOpen (setOf (Setoid.Rel (connectedComponentSetoid X) x))\n[PROOFSTEP]\nconvert isOpen_connectedComponent (x := x)\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d : LocallyConnectedSpace X\nx : X\n\u22a2 setOf (Setoid.Rel (connectedComponentSetoid X) x) = connectedComponent x\n[PROOFSTEP]\next y\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d : LocallyConnectedSpace X\nx y : X\n\u22a2 y \u2208 setOf (Setoid.Rel (connectedComponentSetoid X) x) \u2194 y \u2208 connectedComponent x\n[PROOFSTEP]\nsimpa only [connectedComponentSetoid, \u2190 connectedComponent_eq_iff_mem] using eq_comm\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d : DiscreteTopology X\nx y : X\n\u22a2 proj \u22a5 x = proj \u22a5 y \u2194 x = y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\n\u22a2 LEComap g B C \u2192 LEComap f A B \u2192 LEComap (ContinuousMap.comp g f) A C\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\n\u22a2 map (ContinuousMap.id X) (_ : LEComap (ContinuousMap.id X) A A) = id\n[PROOFSTEP]\next \u27e8\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\nx\u271d : Quotient A.toSetoid\na\u271d : X\n\u22a2 map (ContinuousMap.id X) (_ : LEComap (ContinuousMap.id X) A A) (Quot.mk Setoid.r a\u271d) = id (Quot.mk Setoid.r a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\nh1 : LEComap g B C\nh2 : LEComap f A B\n\u22a2 map (ContinuousMap.comp g f) (_ : LEComap (ContinuousMap.comp g f) A C) = map g h1 \u2218 map f h2\n[PROOFSTEP]\next \u27e8\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\nh1 : LEComap g B C\nh2 : LEComap f A B\nx\u271d : Quotient A.toSetoid\na\u271d : X\n\u22a2 map (ContinuousMap.comp g f) (_ : LEComap (ContinuousMap.comp g f) A C) (Quot.mk Setoid.r a\u271d) =\n (map g h1 \u2218 map f h2) (Quot.mk Setoid.r a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\ncond : LEComap f A B\nh : B \u2264 B'\na : Quotient A.toSetoid\n\u22a2 ofLE h (map f cond a) = map f (_ : LEComap f A B') a\n[PROOFSTEP]\nrcases a with \u27e8\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\ncond : LEComap f A B\nh : B \u2264 B'\na : Quotient A.toSetoid\na\u271d : X\n\u22a2 ofLE h (map f cond (Quot.mk Setoid.r a\u271d)) = map f (_ : LEComap f A B') (Quot.mk Setoid.r a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\ncond : LEComap f A B\nh : A' \u2264 A\nc : Quotient A'.toSetoid\n\u22a2 map f cond (ofLE h c) = map f (_ : LEComap f A' B) c\n[PROOFSTEP]\nrcases c with \u27e8\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS : DiscreteQuotient X\nf : C(X, Y)\nA A' : DiscreteQuotient X\nB B' : DiscreteQuotient Y\ng : C(Y, Z)\nC : DiscreteQuotient Z\ncond : LEComap f A B\nh : A' \u2264 A\nc : Quotient A'.toSetoid\na\u271d : X\n\u22a2 map f cond (ofLE h (Quot.mk Setoid.r a\u271d)) = map f (_ : LEComap f A' B) (Quot.mk Setoid.r a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d\u00b9 : T2Space X\ninst\u271d : CompactSpace X\ndisc : TotallyDisconnectedSpace X\nx y : X\nh : \u2200 (Q : DiscreteQuotient X), proj Q x = proj Q y\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 mem_singleton_iff, \u2190 connectedComponent_eq_singleton, connectedComponent_eq_iInter_clopen, mem_iInter]\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d\u00b9 : T2Space X\ninst\u271d : CompactSpace X\ndisc : TotallyDisconnectedSpace X\nx y : X\nh : \u2200 (Q : DiscreteQuotient X), proj Q x = proj Q y\n\u22a2 \u2200 (i : { Z // IsClopen Z \u2227 y \u2208 Z }), x \u2208 \u2191i\n[PROOFSTEP]\nrintro \u27e8U, hU1, hU2\u27e9\n[GOAL]\ncase mk.intro\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d\u00b9 : T2Space X\ninst\u271d : CompactSpace X\ndisc : TotallyDisconnectedSpace X\nx y : X\nh : \u2200 (Q : DiscreteQuotient X), proj Q x = proj Q y\nU : Set X\nhU1 : IsClopen U\nhU2 : y \u2208 U\n\u22a2 x \u2208 \u2191{ val := U, property := (_ : IsClopen U \u2227 y \u2208 U) }\n[PROOFSTEP]\nexact (Quotient.exact' (h (ofClopen hU1))).mpr hU2\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS A B : DiscreteQuotient X\nh : A \u2264 B\na : Quotient A.toSetoid\n\u22a2 proj A \u207b\u00b9' {a} \u2286 proj B \u207b\u00b9' {ofLE h a}\n[PROOFSTEP]\nrcases A.proj_surjective a with \u27e8a, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS A B : DiscreteQuotient X\nh : A \u2264 B\na : X\n\u22a2 proj A \u207b\u00b9' {proj A a} \u2286 proj B \u207b\u00b9' {ofLE h (proj A a)}\n[PROOFSTEP]\nrw [fiber_eq, ofLE_proj, fiber_eq]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nS A B : DiscreteQuotient X\nh : A \u2264 B\na : X\n\u22a2 setOf (Setoid.Rel A.toSetoid a) \u2286 setOf (Setoid.Rel B.toSetoid a)\n[PROOFSTEP]\nexact fun _ h' => h h'\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d : CompactSpace X\nQs : (Q : DiscreteQuotient X) \u2192 Quotient Q.toSetoid\ncompat : \u2200 (A B : DiscreteQuotient X) (h : A \u2264 B), ofLE h (Qs A) = Qs B\n\u22a2 \u2203 x, \u2200 (Q : DiscreteQuotient X), proj Q x = Qs Q\n[PROOFSTEP]\nhave H\u2081 : \u2200 Q\u2081 Q\u2082, Q\u2081 \u2264 Q\u2082 \u2192 proj Q\u2081 \u207b\u00b9' {Qs Q\u2081} \u2286 proj Q\u2082 \u207b\u00b9' {Qs Q\u2082} := fun _ _ h =>\n by\n rw [\u2190 compat _ _ h]\n exact fiber_subset_ofLE _ _\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d : CompactSpace X\nQs : (Q : DiscreteQuotient X) \u2192 Quotient Q.toSetoid\ncompat : \u2200 (A B : DiscreteQuotient X) (h : A \u2264 B), ofLE h (Qs A) = Qs B\nx\u271d\u00b9 x\u271d : DiscreteQuotient X\nh : x\u271d\u00b9 \u2264 x\u271d\n\u22a2 proj x\u271d\u00b9 \u207b\u00b9' {Qs x\u271d\u00b9} \u2286 proj x\u271d \u207b\u00b9' {Qs x\u271d}\n[PROOFSTEP]\nrw [\u2190 compat _ _ h]\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d : CompactSpace X\nQs : (Q : DiscreteQuotient X) \u2192 Quotient Q.toSetoid\ncompat : \u2200 (A B : DiscreteQuotient X) (h : A \u2264 B), ofLE h (Qs A) = Qs B\nx\u271d\u00b9 x\u271d : DiscreteQuotient X\nh : x\u271d\u00b9 \u2264 x\u271d\n\u22a2 proj x\u271d\u00b9 \u207b\u00b9' {Qs x\u271d\u00b9} \u2286 proj x\u271d \u207b\u00b9' {ofLE h (Qs x\u271d\u00b9)}\n[PROOFSTEP]\nexact fiber_subset_ofLE _ _\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d : CompactSpace X\nQs : (Q : DiscreteQuotient X) \u2192 Quotient Q.toSetoid\ncompat : \u2200 (A B : DiscreteQuotient X) (h : A \u2264 B), ofLE h (Qs A) = Qs B\nH\u2081 : \u2200 (Q\u2081 Q\u2082 : DiscreteQuotient X), Q\u2081 \u2264 Q\u2082 \u2192 proj Q\u2081 \u207b\u00b9' {Qs Q\u2081} \u2286 proj Q\u2082 \u207b\u00b9' {Qs Q\u2082}\n\u22a2 \u2203 x, \u2200 (Q : DiscreteQuotient X), proj Q x = Qs Q\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 : Set.Nonempty (\u22c2 Q, proj Q \u207b\u00b9' {Qs Q}) :=\n IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed (fun Q : DiscreteQuotient X => Q.proj \u207b\u00b9' {Qs _})\n (directed_of_inf H\u2081) (fun Q => (singleton_nonempty _).preimage Q.proj_surjective)\n (fun Q => (Q.isClosed_preimage {Qs _}).isCompact) fun Q => Q.isClosed_preimage _\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d : CompactSpace X\nQs : (Q : DiscreteQuotient X) \u2192 Quotient Q.toSetoid\ncompat : \u2200 (A B : DiscreteQuotient X) (h : A \u2264 B), ofLE h (Qs A) = Qs B\nH\u2081 : \u2200 (Q\u2081 Q\u2082 : DiscreteQuotient X), Q\u2081 \u2264 Q\u2082 \u2192 proj Q\u2081 \u207b\u00b9' {Qs Q\u2081} \u2286 proj Q\u2082 \u207b\u00b9' {Qs Q\u2082}\nx : X\nhx : x \u2208 \u22c2 (Q : DiscreteQuotient X), proj Q \u207b\u00b9' {Qs Q}\n\u22a2 \u2203 x, \u2200 (Q : DiscreteQuotient X), proj Q x = Qs Q\n[PROOFSTEP]\nexact \u27e8x, mem_iInter.1 hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d : CompactSpace X\n\u22a2 Finite (Quotient S.toSetoid)\n[PROOFSTEP]\nhave : CompactSpace S := Quotient.compactSpace\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d : CompactSpace X\nthis : CompactSpace (Quotient S.toSetoid)\n\u22a2 Finite (Quotient S.toSetoid)\n[PROOFSTEP]\nrwa [\u2190 isCompact_univ_iff, isCompact_iff_finite, finite_univ_iff] at this \n", "meta": {"mathlib_filename": "Mathlib.Topology.DiscreteQuotient", "llama_tokens": 7607, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544085240401, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.4275115401180621}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\np : Pmf \u03b1\nb : \u03b2\n\u22a2 \u2191(map f p) b = \u2211' (a : \u03b1), if b = f a then \u2191p a else 0\n[PROOFSTEP]\nsimp [map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\np : Pmf \u03b1\nb\u271d b : \u03b2\n\u22a2 b \u2208 support (map f p) \u2194 b \u2208 f '' support p\n[PROOFSTEP]\nsimp [map, @eq_comm \u03b2 b]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\np : Pmf \u03b1\nb : \u03b2\n\u22a2 b \u2208 support (map f p) \u2194 \u2203 a, a \u2208 support p \u2227 f a = b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\np : Pmf \u03b1\nb : \u03b2\ng : \u03b2 \u2192 \u03b3\n\u22a2 map g (map f p) = map (g \u2218 f) p\n[PROOFSTEP]\nsimp [map, Function.comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\np : Pmf \u03b1\nb : \u03b2\n\u22a2 map (Function.const \u03b1 b) p = pure b\n[PROOFSTEP]\nsimp only [map, Function.comp, bind_const, Function.const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\np : Pmf \u03b1\nb : \u03b2\ns : Set \u03b2\n\u22a2 \u2191(toOuterMeasure (map f p)) s = \u2191(toOuterMeasure p) (f \u207b\u00b9' s)\n[PROOFSTEP]\nsimp [map, Set.indicator, toOuterMeasure_apply p (f \u207b\u00b9' s)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\np : Pmf \u03b1\nb : \u03b2\ns : Set \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nhf : Measurable f\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(toMeasure (map f p)) s = \u2191\u2191(toMeasure p) (f \u207b\u00b9' s)\n[PROOFSTEP]\nrw [toMeasure_apply_eq_toOuterMeasure_apply _ s hs,\n toMeasure_apply_eq_toOuterMeasure_apply _ (f \u207b\u00b9' s) (measurableSet_preimage hf hs)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\np : Pmf \u03b1\nb : \u03b2\ns : Set \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nhf : Measurable f\nhs : MeasurableSet s\n\u22a2 \u2191(toOuterMeasure (map f p)) s = \u2191(toOuterMeasure p) (f \u207b\u00b9' s)\n[PROOFSTEP]\nexact toOuterMeasure_map_apply f p s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nq : Pmf (\u03b1 \u2192 \u03b2)\np : Pmf \u03b1\nb : \u03b2\n\u22a2 \u2191(seq q p) b = \u2211' (f : \u03b1 \u2192 \u03b2) (a : \u03b1), if b = f a then \u2191q f * \u2191p a else 0\n[PROOFSTEP]\nsimp only [seq, mul_boole, bind_apply, pure_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nq : Pmf (\u03b1 \u2192 \u03b2)\np : Pmf \u03b1\nb : \u03b2\n\u22a2 (\u2211' (a : \u03b1 \u2192 \u03b2), \u2191q a * \u2211' (a_1 : \u03b1), if b = a a_1 then \u2191p a_1 else 0) =\n \u2211' (f : \u03b1 \u2192 \u03b2) (a : \u03b1), if b = f a then \u2191q f * \u2191p a else 0\n[PROOFSTEP]\nrefine' tsum_congr fun f => ENNReal.tsum_mul_left.symm.trans (tsum_congr fun a => _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nq : Pmf (\u03b1 \u2192 \u03b2)\np : Pmf \u03b1\nb : \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 (\u2191q f * if b = f a then \u2191p a else 0) = if b = f a then \u2191q f * \u2191p a else 0\n[PROOFSTEP]\nsimpa only [mul_zero] using mul_ite (b = f a) (q f) (p a) 0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nq : Pmf (\u03b1 \u2192 \u03b2)\np : Pmf \u03b1\nb\u271d b : \u03b2\n\u22a2 b \u2208 support (seq q p) \u2194 b \u2208 \u22c3 (f : \u03b1 \u2192 \u03b2) (_ : f \u2208 support q), f '' support p\n[PROOFSTEP]\nsimp [-mem_support_iff, seq, @eq_comm \u03b2 b]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nq : Pmf (\u03b1 \u2192 \u03b2)\np : Pmf \u03b1\nb : \u03b2\n\u22a2 b \u2208 support (seq q p) \u2194 \u2203 f, f \u2208 support q \u2227 b \u2208 f '' support p\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Finset \u03b1\nh : \u2211 a in s, f a = 1\nh' : \u2200 (a : \u03b1), \u00aca \u2208 s \u2192 f a = 0\na : \u03b1\n\u22a2 a \u2208 support (ofFinset f s h h') \u2194 a \u2208 \u2191s \u2229 Function.support f\n[PROOFSTEP]\nsimpa [mem_support_iff] using mt (h' a)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u211d\u22650\u221e\ns : Finset \u03b1\nh : \u2211 a in s, f a = 1\nh' : \u2200 (a : \u03b1), \u00aca \u2208 s \u2192 f a = 0\na : \u03b1\n\u22a2 a \u2208 support (ofFinset f s h h') \u2194 a \u2208 s \u2227 f a \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf0 : tsum f \u2260 0\nhf : tsum f \u2260 \u22a4\na : \u03b1\n\u22a2 a \u2208 support (normalize f hf0 hf) \u2194 a \u2208 Function.support f\n[PROOFSTEP]\nsimp [hf, mem_support_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf0 : tsum f \u2260 0\nhf : tsum f \u2260 \u22a4\na : \u03b1\n\u22a2 a \u2208 support (normalize f hf0 hf) \u2194 f a \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns : Set \u03b1\nh : \u2203 a, a \u2208 s \u2227 a \u2208 support p\n\u22a2 tsum (Set.indicator s \u2191p) \u2260 0\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns : Set \u03b1\nh : \u2203 a, a \u2208 s \u2227 a \u2208 support p\na : \u03b1\n\u22a2 \u2191(filter p s h) a = Set.indicator s (\u2191p) a * (\u2211' (a' : \u03b1), Set.indicator s (\u2191p) a')\u207b\u00b9\n[PROOFSTEP]\nrw [filter, normalize_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns : Set \u03b1\nh : \u2203 a, a \u2208 s \u2227 a \u2208 support p\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 \u2191(filter p s h) a = 0\n[PROOFSTEP]\nrw [filter_apply, Set.indicator_apply_eq_zero.mpr fun ha' => absurd ha' ha, zero_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns : Set \u03b1\nh : \u2203 a, a \u2208 s \u2227 a \u2208 support p\na : \u03b1\n\u22a2 \u2191(filter p s h) a = 0 \u2194 \u00aca \u2208 s \u2228 \u00aca \u2208 support p\n[PROOFSTEP]\nerw [apply_eq_zero_iff, support_filter, Set.mem_inter_iff, not_and_or]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : Pmf \u03b1\ns : Set \u03b1\nh : \u2203 a, a \u2208 s \u2227 a \u2208 support p\na : \u03b1\n\u22a2 \u2191(filter p s h) a \u2260 0 \u2194 a \u2208 s \u2227 a \u2208 support p\n[PROOFSTEP]\nrw [Ne.def, filter_apply_eq_zero_iff, not_or, Classical.not_not, Classical.not_not]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\n\u22a2 \u2211 a : Bool, (fun b => bif b then p else 1 - p) a = 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\nb : Bool\n\u22a2 support (bernoulli p h) = {b | bif b then p \u2260 0 else p \u2260 1}\n[PROOFSTEP]\nrefine' Set.ext fun b => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\nb\u271d b : Bool\n\u22a2 b \u2208 support (bernoulli p h) \u2194 b \u2208 {b | bif b then p \u2260 0 else p \u2260 1}\n[PROOFSTEP]\ninduction b\n[GOAL]\ncase false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\nb : Bool\n\u22a2 false \u2208 support (bernoulli p h) \u2194 false \u2208 {b | bif b then p \u2260 0 else p \u2260 1}\n[PROOFSTEP]\nsimp_rw [mem_support_iff, bernoulli_apply, Bool.cond_false, Ne.def, tsub_eq_zero_iff_le, not_le]\n[GOAL]\ncase false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\nb : Bool\n\u22a2 p < 1 \u2194 false \u2208 {b | bif b then \u00acp = 0 else \u00acp = 1}\n[PROOFSTEP]\nexact \u27e8ne_of_lt, lt_of_le_of_ne h\u27e9\n[GOAL]\ncase true\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\nb : Bool\n\u22a2 true \u2208 support (bernoulli p h) \u2194 true \u2208 {b | bif b then p \u2260 0 else p \u2260 1}\n[PROOFSTEP]\nsimp only [mem_support_iff, bernoulli_apply, Bool.cond_true, Set.mem_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\nb : Bool\n\u22a2 b \u2208 support (bernoulli p h) \u2194 bif b then p \u2260 0 else p \u2260 1\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Probability.ProbabilityMassFunction.Constructions", "llama_tokens": 3316, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321720225278, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.427473458738533}} {"text": "[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn\u271d : \u2115\ninst\u271d : Monoid G\nn : \u2115\nA : Rep k G\nf g : (Fin n \u2192 G) \u2192 CoeSort.coe A\n\u22a2 (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j => (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n (f + g) =\n (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j => (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n f +\n (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j => (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n g\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn\u271d : \u2115\ninst\u271d : Monoid G\nn : \u2115\nA : Rep k G\nf g : (Fin n \u2192 G) \u2192 CoeSort.coe A\nx : Fin (n + 1) \u2192 G\n\u22a2 (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j => (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n (f + g) x =\n ((fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j => (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n f +\n (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j => (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n g)\n x\n[PROOFSTEP]\nsimp_rw [Pi.add_apply, map_add, smul_add, Finset.sum_add_distrib, add_add_add_comm]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn\u271d : \u2115\ninst\u271d : Monoid G\nn : \u2115\nA : Rep k G\nr : k\nf : (Fin n \u2192 G) \u2192 CoeSort.coe A\n\u22a2 AddHom.toFun\n {\n toFun := fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j => (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g),\n map_add' :=\n (_ :\n \u2200 (f g : (Fin n \u2192 G) \u2192 CoeSort.coe A),\n (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j => (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n (f + g) =\n (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j =>\n (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n f +\n (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j =>\n (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n g) }\n (r \u2022 f) =\n \u2191(RingHom.id k) r \u2022\n AddHom.toFun\n {\n toFun := fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j => (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g),\n map_add' :=\n (_ :\n \u2200 (f g : (Fin n \u2192 G) \u2192 CoeSort.coe A),\n (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j =>\n (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n (f + g) =\n (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j =>\n (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n f +\n (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j =>\n (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n g) }\n f\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn\u271d : \u2115\ninst\u271d : Monoid G\nn : \u2115\nA : Rep k G\nr : k\nf : (Fin n \u2192 G) \u2192 CoeSort.coe A\nx : Fin (n + 1) \u2192 G\n\u22a2 AddHom.toFun\n {\n toFun := fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j => (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g),\n map_add' :=\n (_ :\n \u2200 (f g : (Fin n \u2192 G) \u2192 CoeSort.coe A),\n (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j => (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n (f + g) =\n (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j =>\n (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n f +\n (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j =>\n (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n g) }\n (r \u2022 f) x =\n (\u2191(RingHom.id k) r \u2022\n AddHom.toFun\n {\n toFun := fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j => (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g),\n map_add' :=\n (_ :\n \u2200 (f g : (Fin n \u2192 G) \u2192 CoeSort.coe A),\n (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j =>\n (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n (f + g) =\n (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j =>\n (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n f +\n (fun f g =>\n \u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j =>\n (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g))\n g) }\n f)\n x\n[PROOFSTEP]\nsimp_rw [Pi.smul_apply, RingHom.id_apply, map_smul, smul_add, Finset.smul_sum, \u2190 smul_assoc, smul_eq_mul, mul_comm r]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\n\u22a2 d n A =\n (LinearEquiv.toModuleIso (diagonalHomEquiv n A)).inv \u226b\n HomologicalComplex.d (linearYonedaObjResolution A) n (n + 1) \u226b\n (LinearEquiv.toModuleIso (diagonalHomEquiv (n + 1) A)).hom\n[PROOFSTEP]\next f g\n[GOAL]\ncase h.h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\nf : (Fin n \u2192 G) \u2192 CoeSort.coe A\ng : Fin (n + 1) \u2192 G\n\u22a2 \u2191(d n A) f g =\n \u2191((LinearEquiv.toModuleIso (diagonalHomEquiv n A)).inv \u226b\n HomologicalComplex.d (linearYonedaObjResolution A) n (n + 1) \u226b\n (LinearEquiv.toModuleIso (diagonalHomEquiv (n + 1) A)).hom)\n f g\n[PROOFSTEP]\nchange d n A f g = diagonalHomEquiv (n + 1) A ((resolution k G).d (n + 1) n \u226b (diagonalHomEquiv n A).symm f) g\n[GOAL]\ncase h.h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\nf : (Fin n \u2192 G) \u2192 CoeSort.coe A\ng : Fin (n + 1) \u2192 G\n\u22a2 \u2191(d n A) f g =\n \u2191(diagonalHomEquiv (n + 1) A)\n (HomologicalComplex.d (resolution k G) (n + 1) n \u226b \u2191(LinearEquiv.symm (diagonalHomEquiv n A)) f) g\n[PROOFSTEP]\nrw [diagonalHomEquiv_apply, Action.comp_hom, ModuleCat.comp_def, LinearMap.comp_apply, Resolution.d_eq]\n[GOAL]\ncase h.h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\nf : (Fin n \u2192 G) \u2192 CoeSort.coe A\ng : Fin (n + 1) \u2192 G\n\u22a2 \u2191(d n A) f g =\n \u2191(\u2191(LinearEquiv.symm (diagonalHomEquiv n A)) f).hom\n (\u2191(Resolution.d k G (n + 1)) (Finsupp.single (Fin.partialProd g) 1))\n[PROOFSTEP]\nerw [Resolution.d_of (Fin.partialProd g)]\n[GOAL]\ncase h.h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\nf : (Fin n \u2192 G) \u2192 CoeSort.coe A\ng : Fin (n + 1) \u2192 G\n\u22a2 \u2191(d n A) f g =\n \u2191(\u2191(LinearEquiv.symm (diagonalHomEquiv n A)) f).hom\n (Finset.sum Finset.univ fun p => Finsupp.single (Fin.partialProd g \u2218 Fin.succAbove p) ((-1) ^ \u2191p))\n[PROOFSTEP]\nrw [LinearMap.map_sum]\n[GOAL]\ncase h.h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\nf : (Fin n \u2192 G) \u2192 CoeSort.coe A\ng : Fin (n + 1) \u2192 G\n\u22a2 \u2191(d n A) f g =\n Finset.sum Finset.univ fun i =>\n \u2191(\u2191(LinearEquiv.symm (diagonalHomEquiv n A)) f).hom\n (Finsupp.single (Fin.partialProd g \u2218 Fin.succAbove i) ((-1) ^ \u2191i))\n[PROOFSTEP]\nsimp only [\u2190 Finsupp.smul_single_one _ ((-1 : k) ^ _)]\n[GOAL]\ncase h.h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\nf : (Fin n \u2192 G) \u2192 CoeSort.coe A\ng : Fin (n + 1) \u2192 G\n\u22a2 \u2191(d n A) f g =\n Finset.sum Finset.univ fun x =>\n \u2191(\u2191(LinearEquiv.symm (diagonalHomEquiv n A)) f).hom\n ((-1) ^ \u2191x \u2022 Finsupp.single (Fin.partialProd g \u2218 Fin.succAbove x) 1)\n[PROOFSTEP]\nrw [d_apply, @Fin.sum_univ_succ _ _ (n + 1), Fin.val_zero, pow_zero, one_smul, Fin.succAbove_zero,\n diagonalHomEquiv_symm_apply f (Fin.partialProd g \u2218 @Fin.succ (n + 1))]\n[GOAL]\ncase h.h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\nf : (Fin n \u2192 G) \u2192 CoeSort.coe A\ng : Fin (n + 1) \u2192 G\n\u22a2 (\u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j => (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g)) =\n \u2191(\u2191(\u03c1 A) ((Fin.partialProd g \u2218 Fin.succ) 0))\n (f fun i => ((Fin.partialProd g \u2218 Fin.succ) (Fin.castSucc i))\u207b\u00b9 * (Fin.partialProd g \u2218 Fin.succ) (Fin.succ i)) +\n Finset.sum Finset.univ fun i =>\n \u2191(\u2191(LinearEquiv.symm (diagonalHomEquiv n A)) f).hom\n ((-1) ^ \u2191(Fin.succ i) \u2022 Finsupp.single (Fin.partialProd g \u2218 Fin.succAbove (Fin.succ i)) 1)\n[PROOFSTEP]\nsimp_rw [Function.comp_apply, Fin.partialProd_succ, Fin.castSucc_zero, Fin.partialProd_zero, one_mul]\n[GOAL]\ncase h.h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\nf : (Fin n \u2192 G) \u2192 CoeSort.coe A\ng : Fin (n + 1) \u2192 G\n\u22a2 (\u2191(\u2191(\u03c1 A) (g 0)) (f fun i => g (Fin.succ i)) +\n Finset.sum Finset.univ fun j => (-1) ^ (\u2191j + 1) \u2022 f (Fin.contractNth j (fun x x_1 => x * x_1) g)) =\n \u2191(\u2191(\u03c1 A) (g 0))\n (f fun i =>\n (Fin.partialProd g (Fin.castSucc (Fin.castSucc i)) * g (Fin.castSucc i))\u207b\u00b9 *\n (Fin.partialProd g (Fin.castSucc (Fin.succ i)) * g (Fin.succ i))) +\n Finset.sum Finset.univ fun i =>\n \u2191(\u2191(LinearEquiv.symm (diagonalHomEquiv n A)) f).hom\n ((-1) ^ \u2191(Fin.succ i) \u2022 Finsupp.single (Fin.partialProd g \u2218 Fin.succAbove (Fin.succ i)) 1)\n[PROOFSTEP]\nrcongr x\n[GOAL]\ncase h.h.e_a.h.e_6.h.e_a.h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\nf : (Fin n \u2192 G) \u2192 CoeSort.coe A\ng : Fin (n + 1) \u2192 G\nx : Fin n\n\u22a2 g (Fin.succ x) =\n (Fin.partialProd g (Fin.castSucc (Fin.castSucc x)) * g (Fin.castSucc x))\u207b\u00b9 *\n (Fin.partialProd g (Fin.castSucc (Fin.succ x)) * g (Fin.succ x))\n[PROOFSTEP]\nhave := Fin.partialProd_right_inv g (Fin.castSucc x)\n[GOAL]\ncase h.h.e_a.h.e_6.h.e_a.h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\nf : (Fin n \u2192 G) \u2192 CoeSort.coe A\ng : Fin (n + 1) \u2192 G\nx : Fin n\nthis :\n (Fin.partialProd g (Fin.castSucc (Fin.castSucc x)))\u207b\u00b9 * Fin.partialProd g (Fin.succ (Fin.castSucc x)) =\n g (Fin.castSucc x)\n\u22a2 g (Fin.succ x) =\n (Fin.partialProd g (Fin.castSucc (Fin.castSucc x)) * g (Fin.castSucc x))\u207b\u00b9 *\n (Fin.partialProd g (Fin.castSucc (Fin.succ x)) * g (Fin.succ x))\n[PROOFSTEP]\nsimp only [mul_inv_rev, Fin.castSucc_fin_succ] at this \u22a2\n[GOAL]\ncase h.h.e_a.h.e_6.h.e_a.h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\nf : (Fin n \u2192 G) \u2192 CoeSort.coe A\ng : Fin (n + 1) \u2192 G\nx : Fin n\nthis :\n (Fin.partialProd g (Fin.castSucc (Fin.castSucc x)))\u207b\u00b9 * Fin.partialProd g (Fin.succ (Fin.castSucc x)) =\n g (Fin.castSucc x)\n\u22a2 g (Fin.succ x) =\n (g (Fin.castSucc x))\u207b\u00b9 * (Fin.partialProd g (Fin.castSucc (Fin.castSucc x)))\u207b\u00b9 *\n (Fin.partialProd g (Fin.succ (Fin.castSucc x)) * g (Fin.succ x))\n[PROOFSTEP]\nrw [mul_assoc, \u2190 mul_assoc _ _ (g x.succ), this, inv_mul_cancel_left]\n[GOAL]\ncase h.h.e_a.e_f.h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\nf : (Fin n \u2192 G) \u2192 CoeSort.coe A\ng : Fin (n + 1) \u2192 G\nx : Fin (n + 1)\n\u22a2 (-1) ^ (\u2191x + 1) \u2022 f (Fin.contractNth x (fun x x_1 => x * x_1) g) =\n \u2191(\u2191(LinearEquiv.symm (diagonalHomEquiv n A)) f).hom\n ((-1) ^ \u2191(Fin.succ x) \u2022 Finsupp.single (Fin.partialProd g \u2218 Fin.succAbove (Fin.succ x)) 1)\n[PROOFSTEP]\nrw [map_smul, diagonalHomEquiv_symm_partialProd_succ, Fin.val_succ]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn\u271d : \u2115\ninst\u271d : Group G\nA : Rep k G\nn : \u2115\n\u22a2 (fun n => d n A) n \u226b (fun n => d n A) (n + 1) = 0\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn\u271d : \u2115\ninst\u271d : Group G\nA : Rep k G\nn : \u2115\nx : \u2191((fun n => ModuleCat.of k ((Fin n \u2192 G) \u2192 CoeSort.coe A)) n)\n\u22a2 \u2191((fun n => d n A) n \u226b (fun n => d n A) (n + 1)) x = \u21910 x\n[PROOFSTEP]\nhave := LinearMap.ext_iff.1 ((linearYonedaObjResolution A).d_comp_d n (n + 1) (n + 2))\n[GOAL]\ncase h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn\u271d : \u2115\ninst\u271d : Group G\nA : Rep k G\nn : \u2115\nx : \u2191((fun n => ModuleCat.of k ((Fin n \u2192 G) \u2192 CoeSort.coe A)) n)\nthis :\n \u2200 (x : \u2191(HomologicalComplex.X (linearYonedaObjResolution A) n)),\n \u2191(HomologicalComplex.d (linearYonedaObjResolution A) n (n + 1) \u226b\n HomologicalComplex.d (linearYonedaObjResolution A) (n + 1) (n + 2))\n x =\n \u21910 x\n\u22a2 \u2191((fun n => d n A) n \u226b (fun n => d n A) (n + 1)) x = \u21910 x\n[PROOFSTEP]\nsimp only [ModuleCat.comp_def, LinearMap.comp_apply] at this \n[GOAL]\ncase h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn\u271d : \u2115\ninst\u271d : Group G\nA : Rep k G\nn : \u2115\nx : \u2191((fun n => ModuleCat.of k ((Fin n \u2192 G) \u2192 CoeSort.coe A)) n)\nthis :\n \u2200 (x : \u2191(HomologicalComplex.X (linearYonedaObjResolution A) n)),\n \u2191(HomologicalComplex.d (linearYonedaObjResolution A) (n + 1) (n + 2))\n (\u2191(HomologicalComplex.d (linearYonedaObjResolution A) n (n + 1)) x) =\n \u21910 x\n\u22a2 \u2191((fun n => d n A) n \u226b (fun n => d n A) (n + 1)) x = \u21910 x\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn\u271d : \u2115\ninst\u271d : Group G\nA : Rep k G\nn : \u2115\nx : \u2191((fun n => ModuleCat.of k ((Fin n \u2192 G) \u2192 CoeSort.coe A)) n)\nthis :\n \u2200 (x : \u2191(HomologicalComplex.X (linearYonedaObjResolution A) n)),\n \u2191(HomologicalComplex.d (linearYonedaObjResolution A) (n + 1) (n + 2))\n (\u2191(HomologicalComplex.d (linearYonedaObjResolution A) n (n + 1)) x) =\n \u21910 x\n\u22a2 \u2191(d n A \u226b d (n + 1) A) x = \u21910 x\n[PROOFSTEP]\nsimp only [d_eq, LinearEquiv.toModuleIso_inv, LinearEquiv.toModuleIso_hom, ModuleCat.coe_comp, Function.comp_apply]\n /- Porting note: I can see I need to rewrite `LinearEquiv.coe_coe` twice to at\n least reduce the need for `symm_apply_apply` to be an `erw`. However, even `erw` refuses to\n rewrite the second `coe_coe`... -/\n[GOAL]\ncase h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn\u271d : \u2115\ninst\u271d : Group G\nA : Rep k G\nn : \u2115\nx : \u2191((fun n => ModuleCat.of k ((Fin n \u2192 G) \u2192 CoeSort.coe A)) n)\nthis :\n \u2200 (x : \u2191(HomologicalComplex.X (linearYonedaObjResolution A) n)),\n \u2191(HomologicalComplex.d (linearYonedaObjResolution A) (n + 1) (n + 2))\n (\u2191(HomologicalComplex.d (linearYonedaObjResolution A) n (n + 1)) x) =\n \u21910 x\n\u22a2 \u2191\u2191(Rep.diagonalHomEquiv (n + 1 + 1) A)\n (\u2191(HomologicalComplex.d (linearYonedaObjResolution A) (n + 1) (n + 1 + 1))\n (\u2191\u2191(LinearEquiv.symm (Rep.diagonalHomEquiv (n + 1) A))\n (\u2191\u2191(Rep.diagonalHomEquiv (n + 1) A)\n (\u2191(HomologicalComplex.d (linearYonedaObjResolution A) n (n + 1))\n (\u2191\u2191(LinearEquiv.symm (Rep.diagonalHomEquiv n A)) x))))) =\n \u21910 x\n[PROOFSTEP]\nerw [LinearEquiv.symm_apply_apply, this]\n[GOAL]\ncase h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn\u271d : \u2115\ninst\u271d : Group G\nA : Rep k G\nn : \u2115\nx : \u2191((fun n => ModuleCat.of k ((Fin n \u2192 G) \u2192 CoeSort.coe A)) n)\nthis :\n \u2200 (x : \u2191(HomologicalComplex.X (linearYonedaObjResolution A) n)),\n \u2191(HomologicalComplex.d (linearYonedaObjResolution A) (n + 1) (n + 2))\n (\u2191(HomologicalComplex.d (linearYonedaObjResolution A) n (n + 1)) x) =\n \u21910 x\n\u22a2 \u2191\u2191(Rep.diagonalHomEquiv (n + 1 + 1) A) (\u21910 (\u2191\u2191(LinearEquiv.symm (Rep.diagonalHomEquiv n A)) x)) = \u21910 x\n[PROOFSTEP]\nexact map_zero _\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\n\u22a2 inhomogeneousCochains A \u2245 linearYonedaObjResolution A\n[PROOFSTEP]\nrefine' HomologicalComplex.Hom.isoOfComponents (fun i => (Rep.diagonalHomEquiv i A).toModuleIso.symm) _\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\n\u22a2 \u2200 (i j : \u2115),\n ComplexShape.Rel (ComplexShape.up \u2115) i j \u2192\n ((fun i => (LinearEquiv.toModuleIso (Rep.diagonalHomEquiv i A)).symm) i).hom \u226b\n HomologicalComplex.d (linearYonedaObjResolution A) i j =\n HomologicalComplex.d (inhomogeneousCochains A) i j \u226b\n ((fun i => (LinearEquiv.toModuleIso (Rep.diagonalHomEquiv i A)).symm) j).hom\n[PROOFSTEP]\nrintro i j (h : i + 1 = j)\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\ni j : \u2115\nh : i + 1 = j\n\u22a2 ((fun i => (LinearEquiv.toModuleIso (Rep.diagonalHomEquiv i A)).symm) i).hom \u226b\n HomologicalComplex.d (linearYonedaObjResolution A) i j =\n HomologicalComplex.d (inhomogeneousCochains A) i j \u226b\n ((fun i => (LinearEquiv.toModuleIso (Rep.diagonalHomEquiv i A)).symm) j).hom\n[PROOFSTEP]\nsubst h\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\nn : \u2115\ninst\u271d : Group G\nA : Rep k G\ni : \u2115\n\u22a2 ((fun i => (LinearEquiv.toModuleIso (Rep.diagonalHomEquiv i A)).symm) i).hom \u226b\n HomologicalComplex.d (linearYonedaObjResolution A) i (i + 1) =\n HomologicalComplex.d (inhomogeneousCochains A) i (i + 1) \u226b\n ((fun i => (LinearEquiv.toModuleIso (Rep.diagonalHomEquiv i A)).symm) (i + 1)).hom\n[PROOFSTEP]\nsimp only [CochainComplex.of_d, d_eq, Category.assoc, Iso.symm_hom, Iso.hom_inv_id, Category.comp_id]\n", "meta": {"mathlib_filename": "Mathlib.RepresentationTheory.GroupCohomology.Basic", "llama_tokens": 8162, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8104789040926008, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.4273789179785919}} {"text": "[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' \u03b9'' : Finset \u03b9\nh : \u03b9' \u2286 \u03b9''\nf : \u2191(hallMatchingsOn t \u03b9'')\n\u22a2 \u2191(hallMatchingsOn t \u03b9')\n[PROOFSTEP]\nrefine' \u27e8fun i => f.val \u27e8i, h i.property\u27e9, _\u27e9\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' \u03b9'' : Finset \u03b9\nh : \u03b9' \u2286 \u03b9''\nf : \u2191(hallMatchingsOn t \u03b9'')\n\u22a2 (fun i => \u2191f { val := \u2191i, property := (_ : \u2191i \u2208 \u03b9'') }) \u2208 hallMatchingsOn t \u03b9'\n[PROOFSTEP]\ncases' f.property with hinj hc\n[GOAL]\ncase intro\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' \u03b9'' : Finset \u03b9\nh : \u03b9' \u2286 \u03b9''\nf : \u2191(hallMatchingsOn t \u03b9'')\nhinj : Function.Injective \u2191f\nhc : \u2200 (x : { x // x \u2208 \u03b9'' }), \u2191f x \u2208 t \u2191x\n\u22a2 (fun i => \u2191f { val := \u2191i, property := (_ : \u2191i \u2208 \u03b9'') }) \u2208 hallMatchingsOn t \u03b9'\n[PROOFSTEP]\nrefine' \u27e8_, fun i => hc \u27e8i, h i.property\u27e9\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' \u03b9'' : Finset \u03b9\nh : \u03b9' \u2286 \u03b9''\nf : \u2191(hallMatchingsOn t \u03b9'')\nhinj : Function.Injective \u2191f\nhc : \u2200 (x : { x // x \u2208 \u03b9'' }), \u2191f x \u2208 t \u2191x\n\u22a2 Function.Injective fun i => \u2191f { val := \u2191i, property := (_ : \u2191i \u2208 \u03b9'') }\n[PROOFSTEP]\nrintro \u27e8i, hi\u27e9 \u27e8j, hj\u27e9 hh\n[GOAL]\ncase intro.mk.mk\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' \u03b9'' : Finset \u03b9\nh : \u03b9' \u2286 \u03b9''\nf : \u2191(hallMatchingsOn t \u03b9'')\nhinj : Function.Injective \u2191f\nhc : \u2200 (x : { x // x \u2208 \u03b9'' }), \u2191f x \u2208 t \u2191x\ni : \u03b9\nhi : i \u2208 \u03b9'\nj : \u03b9\nhj : j \u2208 \u03b9'\nhh :\n (fun i => \u2191f { val := \u2191i, property := (_ : \u2191i \u2208 \u03b9'') }) { val := i, property := hi } =\n (fun i => \u2191f { val := \u2191i, property := (_ : \u2191i \u2208 \u03b9'') }) { val := j, property := hj }\n\u22a2 { val := i, property := hi } = { val := j, property := hj }\n[PROOFSTEP]\nsimpa only [Subtype.mk_eq_mk] using hinj hh\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\n\u03b9' : Finset \u03b9\n\u22a2 Nonempty \u2191(hallMatchingsOn t \u03b9')\n[PROOFSTEP]\nclassical\nrefine' \u27e8Classical.indefiniteDescription _ _\u27e9\napply (all_card_le_biUnion_card_iff_existsInjective' fun i : \u03b9' => t i).mp\nintro s'\nconvert h (s'.image (\u2191)) using 1\nsimp only [card_image_of_injective s' Subtype.coe_injective]\nrw [image_biUnion]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\n\u03b9' : Finset \u03b9\n\u22a2 Nonempty \u2191(hallMatchingsOn t \u03b9')\n[PROOFSTEP]\nrefine' \u27e8Classical.indefiniteDescription _ _\u27e9\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\n\u03b9' : Finset \u03b9\n\u22a2 \u2203 x, x \u2208 hallMatchingsOn t \u03b9'\n[PROOFSTEP]\napply (all_card_le_biUnion_card_iff_existsInjective' fun i : \u03b9' => t i).mp\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\n\u03b9' : Finset \u03b9\n\u22a2 \u2200 (s : Finset { x // x \u2208 \u03b9' }), card s \u2264 card (Finset.biUnion s fun i => t \u2191i)\n[PROOFSTEP]\nintro s'\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\n\u03b9' : Finset \u03b9\ns' : Finset { x // x \u2208 \u03b9' }\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun i => t \u2191i)\n[PROOFSTEP]\nconvert h (s'.image (\u2191)) using 1\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\n\u03b9' : Finset \u03b9\ns' : Finset { x // x \u2208 \u03b9' }\n\u22a2 card s' = card (image Subtype.val s')\ncase h.e'_4\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\n\u03b9' : Finset \u03b9\ns' : Finset { x // x \u2208 \u03b9' }\n\u22a2 card (Finset.biUnion s' fun i => t \u2191i) = card (Finset.biUnion (image Subtype.val s') t)\n[PROOFSTEP]\nsimp only [card_image_of_injective s' Subtype.coe_injective]\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\n\u03b9' : Finset \u03b9\ns' : Finset { x // x \u2208 \u03b9' }\n\u22a2 card (Finset.biUnion s' fun i => t \u2191i) = card (Finset.biUnion (image Subtype.val s') t)\n[PROOFSTEP]\nrw [image_biUnion]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' : Finset \u03b9\n\u22a2 Finite \u2191(hallMatchingsOn t \u03b9')\n[PROOFSTEP]\nclassical\nrw [hallMatchingsOn]\nlet g : hallMatchingsOn t \u03b9' \u2192 \u03b9' \u2192 \u03b9'.biUnion t := by\n rintro f i\n refine' \u27e8f.val i, _\u27e9\n rw [mem_biUnion]\n exact \u27e8i, i.property, f.property.2 i\u27e9\napply Finite.of_injective g\nintro f f' h\next a\nrw [Function.funext_iff] at h \nsimpa using h a\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' : Finset \u03b9\n\u22a2 Finite \u2191(hallMatchingsOn t \u03b9')\n[PROOFSTEP]\nrw [hallMatchingsOn]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' : Finset \u03b9\n\u22a2 Finite \u2191{f | Function.Injective f \u2227 \u2200 (x : { x // x \u2208 \u03b9' }), f x \u2208 t \u2191x}\n[PROOFSTEP]\nlet g : hallMatchingsOn t \u03b9' \u2192 \u03b9' \u2192 \u03b9'.biUnion t := by\n rintro f i\n refine' \u27e8f.val i, _\u27e9\n rw [mem_biUnion]\n exact \u27e8i, i.property, f.property.2 i\u27e9\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' : Finset \u03b9\n\u22a2 \u2191(hallMatchingsOn t \u03b9') \u2192 { x // x \u2208 \u03b9' } \u2192 { x // x \u2208 Finset.biUnion \u03b9' t }\n[PROOFSTEP]\nrintro f i\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' : Finset \u03b9\nf : \u2191(hallMatchingsOn t \u03b9')\ni : { x // x \u2208 \u03b9' }\n\u22a2 { x // x \u2208 Finset.biUnion \u03b9' t }\n[PROOFSTEP]\nrefine' \u27e8f.val i, _\u27e9\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' : Finset \u03b9\nf : \u2191(hallMatchingsOn t \u03b9')\ni : { x // x \u2208 \u03b9' }\n\u22a2 \u2191f i \u2208 Finset.biUnion \u03b9' t\n[PROOFSTEP]\nrw [mem_biUnion]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' : Finset \u03b9\nf : \u2191(hallMatchingsOn t \u03b9')\ni : { x // x \u2208 \u03b9' }\n\u22a2 \u2203 a, a \u2208 \u03b9' \u2227 \u2191f i \u2208 t a\n[PROOFSTEP]\nexact \u27e8i, i.property, f.property.2 i\u27e9\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' : Finset \u03b9\ng : \u2191(hallMatchingsOn t \u03b9') \u2192 { x // x \u2208 \u03b9' } \u2192 { x // x \u2208 Finset.biUnion \u03b9' t } :=\n fun f i => { val := \u2191f i, property := (_ : \u2191f i \u2208 Finset.biUnion \u03b9' t) }\n\u22a2 Finite \u2191{f | Function.Injective f \u2227 \u2200 (x : { x // x \u2208 \u03b9' }), f x \u2208 t \u2191x}\n[PROOFSTEP]\napply Finite.of_injective g\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' : Finset \u03b9\ng : \u2191(hallMatchingsOn t \u03b9') \u2192 { x // x \u2208 \u03b9' } \u2192 { x // x \u2208 Finset.biUnion \u03b9' t } :=\n fun f i => { val := \u2191f i, property := (_ : \u2191f i \u2208 Finset.biUnion \u03b9' t) }\n\u22a2 Function.Injective g\n[PROOFSTEP]\nintro f f' h\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' : Finset \u03b9\ng : \u2191(hallMatchingsOn t \u03b9') \u2192 { x // x \u2208 \u03b9' } \u2192 { x // x \u2208 Finset.biUnion \u03b9' t } :=\n fun f i => { val := \u2191f i, property := (_ : \u2191f i \u2208 Finset.biUnion \u03b9' t) }\nf f' : \u2191(hallMatchingsOn t \u03b9')\nh : g f = g f'\n\u22a2 f = f'\n[PROOFSTEP]\next a\n[GOAL]\ncase a.h\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' : Finset \u03b9\ng : \u2191(hallMatchingsOn t \u03b9') \u2192 { x // x \u2208 \u03b9' } \u2192 { x // x \u2208 Finset.biUnion \u03b9' t } :=\n fun f i => { val := \u2191f i, property := (_ : \u2191f i \u2208 Finset.biUnion \u03b9' t) }\nf f' : \u2191(hallMatchingsOn t \u03b9')\nh : g f = g f'\na : { x // x \u2208 \u03b9' }\n\u22a2 \u2191f a = \u2191f' a\n[PROOFSTEP]\nrw [Function.funext_iff] at h \n[GOAL]\ncase a.h\n\u03b9 : Type u\n\u03b1 : Type v\nt : \u03b9 \u2192 Finset \u03b1\n\u03b9' : Finset \u03b9\ng : \u2191(hallMatchingsOn t \u03b9') \u2192 { x // x \u2208 \u03b9' } \u2192 { x // x \u2208 Finset.biUnion \u03b9' t } :=\n fun f i => { val := \u2191f i, property := (_ : \u2191f i \u2208 Finset.biUnion \u03b9' t) }\nf f' : \u2191(hallMatchingsOn t \u03b9')\nh : \u2200 (a : { x // x \u2208 \u03b9' }), g f a = g f' a\na : { x // x \u2208 \u03b9' }\n\u22a2 \u2191f a = \u2191f' a\n[PROOFSTEP]\nsimpa using h a\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\n\u22a2 (\u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)) \u2194 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\n\u22a2 (\u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nhaveI : \u2200 \u03b9' : (Finset \u03b9)\u1d52\u1d56, Nonempty ((hallMatchingsFunctor t).obj \u03b9') := fun \u03b9' =>\n hallMatchingsOn.nonempty t h \u03b9'.unop\n[GOAL]\ncase mp\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nclassical\nhaveI : \u2200 \u03b9' : (Finset \u03b9)\u1d52\u1d56, Finite ((hallMatchingsFunctor t).obj \u03b9') :=\n by\n intro \u03b9'\n rw [hallMatchingsFunctor]\n infer_instance\n -- Apply the compactness argument\nobtain \u27e8u, hu\u27e9 :=\n nonempty_sections_of_finite_inverse_system\n (hallMatchingsFunctor t)\n -- Interpret the resulting section of the inverse limit\nrefine' \u27e8_, _, _\u27e9\n\u00b7\n -- Build the matching function from the sectionexact fun i =>\n (u (Opposite.op ({ i } : Finset \u03b9))).val \u27e8i, by simp only [Opposite.unop_op, mem_singleton]\u27e9\n\u00b7\n -- Show that it is injective\n intro i i'\n have subi : ({ i } : Finset \u03b9) \u2286 { i, i' } := by simp\n have subi' : ({ i' } : Finset \u03b9) \u2286 { i, i' } := by simp\n rw [\u2190 Finset.le_iff_subset] at subi subi' \n simp only\n rw [\u2190 hu (CategoryTheory.homOfLE subi).op, \u2190 hu (CategoryTheory.homOfLE subi').op]\n let uii' := u (Opposite.op ({ i, i' } : Finset \u03b9))\n exact fun h => Subtype.mk_eq_mk.mp (uii'.property.1 h)\n\u00b7\n -- Show that it maps each index to the corresponding finite set\n intro i\n apply (u (Opposite.op ({ i } : Finset \u03b9))).property.2\n[GOAL]\ncase mp\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nhaveI : \u2200 \u03b9' : (Finset \u03b9)\u1d52\u1d56, Finite ((hallMatchingsFunctor t).obj \u03b9') :=\n by\n intro \u03b9'\n rw [hallMatchingsFunctor]\n infer_instance\n -- Apply the compactness argument\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\n\u22a2 \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\n[PROOFSTEP]\nintro \u03b9'\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\n\u03b9' : (Finset \u03b9)\u1d52\u1d56\n\u22a2 Finite ((hallMatchingsFunctor t).obj \u03b9')\n[PROOFSTEP]\nrw [hallMatchingsFunctor]\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\n\u03b9' : (Finset \u03b9)\u1d52\u1d56\n\u22a2 Finite\n ((CategoryTheory.Functor.mk\n { obj := fun \u03b9' => \u2191(hallMatchingsOn t \u03b9'.unop),\n map := fun {\u03b9' \u03b9''} g f => hallMatchingsOn.restrict t (_ : \u03b9''.unop \u2264 \u03b9'.unop) f }).obj\n \u03b9')\n[PROOFSTEP]\ninfer_instance\n -- Apply the compactness argument\n[GOAL]\ncase mp\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nobtain \u27e8u, hu\u27e9 :=\n nonempty_sections_of_finite_inverse_system\n (hallMatchingsFunctor t)\n -- Interpret the resulting section of the inverse limit\n[GOAL]\ncase mp.intro\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\nu : (j : (Finset \u03b9)\u1d52\u1d56) \u2192 (hallMatchingsFunctor t).obj j\nhu : u \u2208 Functor.sections (hallMatchingsFunctor t)\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x\n[PROOFSTEP]\nrefine' \u27e8_, _, _\u27e9\n[GOAL]\ncase mp.intro.refine'_1\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\nu : (j : (Finset \u03b9)\u1d52\u1d56) \u2192 (hallMatchingsFunctor t).obj j\nhu : u \u2208 Functor.sections (hallMatchingsFunctor t)\n\u22a2 \u03b9 \u2192 \u03b1\n[PROOFSTEP]\nexact fun i => (u (Opposite.op ({ i } : Finset \u03b9))).val \u27e8i, by simp only [Opposite.unop_op, mem_singleton]\u27e9\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\nu : (j : (Finset \u03b9)\u1d52\u1d56) \u2192 (hallMatchingsFunctor t).obj j\nhu : u \u2208 Functor.sections (hallMatchingsFunctor t)\ni : \u03b9\n\u22a2 i \u2208 (Opposite.op {i}).unop\n[PROOFSTEP]\nsimp only [Opposite.unop_op, mem_singleton]\n[GOAL]\ncase mp.intro.refine'_2\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\nu : (j : (Finset \u03b9)\u1d52\u1d56) \u2192 (hallMatchingsFunctor t).obj j\nhu : u \u2208 Functor.sections (hallMatchingsFunctor t)\n\u22a2 Function.Injective fun i => \u2191(u (Opposite.op {i})) { val := i, property := (_ : i \u2208 {i}) }\n[PROOFSTEP]\nintro i i'\n[GOAL]\ncase mp.intro.refine'_2\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\nu : (j : (Finset \u03b9)\u1d52\u1d56) \u2192 (hallMatchingsFunctor t).obj j\nhu : u \u2208 Functor.sections (hallMatchingsFunctor t)\ni i' : \u03b9\n\u22a2 (fun i => \u2191(u (Opposite.op {i})) { val := i, property := (_ : i \u2208 {i}) }) i =\n (fun i => \u2191(u (Opposite.op {i})) { val := i, property := (_ : i \u2208 {i}) }) i' \u2192\n i = i'\n[PROOFSTEP]\nhave subi : ({ i } : Finset \u03b9) \u2286 { i, i' } := by simp\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\nu : (j : (Finset \u03b9)\u1d52\u1d56) \u2192 (hallMatchingsFunctor t).obj j\nhu : u \u2208 Functor.sections (hallMatchingsFunctor t)\ni i' : \u03b9\n\u22a2 {i} \u2286 {i, i'}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mp.intro.refine'_2\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\nu : (j : (Finset \u03b9)\u1d52\u1d56) \u2192 (hallMatchingsFunctor t).obj j\nhu : u \u2208 Functor.sections (hallMatchingsFunctor t)\ni i' : \u03b9\nsubi : {i} \u2286 {i, i'}\n\u22a2 (fun i => \u2191(u (Opposite.op {i})) { val := i, property := (_ : i \u2208 {i}) }) i =\n (fun i => \u2191(u (Opposite.op {i})) { val := i, property := (_ : i \u2208 {i}) }) i' \u2192\n i = i'\n[PROOFSTEP]\nhave subi' : ({ i' } : Finset \u03b9) \u2286 { i, i' } := by simp\n[GOAL]\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\nu : (j : (Finset \u03b9)\u1d52\u1d56) \u2192 (hallMatchingsFunctor t).obj j\nhu : u \u2208 Functor.sections (hallMatchingsFunctor t)\ni i' : \u03b9\nsubi : {i} \u2286 {i, i'}\n\u22a2 {i'} \u2286 {i, i'}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mp.intro.refine'_2\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\nu : (j : (Finset \u03b9)\u1d52\u1d56) \u2192 (hallMatchingsFunctor t).obj j\nhu : u \u2208 Functor.sections (hallMatchingsFunctor t)\ni i' : \u03b9\nsubi : {i} \u2286 {i, i'}\nsubi' : {i'} \u2286 {i, i'}\n\u22a2 (fun i => \u2191(u (Opposite.op {i})) { val := i, property := (_ : i \u2208 {i}) }) i =\n (fun i => \u2191(u (Opposite.op {i})) { val := i, property := (_ : i \u2208 {i}) }) i' \u2192\n i = i'\n[PROOFSTEP]\nrw [\u2190 Finset.le_iff_subset] at subi subi' \n[GOAL]\ncase mp.intro.refine'_2\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\nu : (j : (Finset \u03b9)\u1d52\u1d56) \u2192 (hallMatchingsFunctor t).obj j\nhu : u \u2208 Functor.sections (hallMatchingsFunctor t)\ni i' : \u03b9\nsubi : {i} \u2264 {i, i'}\nsubi' : {i'} \u2264 {i, i'}\n\u22a2 (fun i => \u2191(u (Opposite.op {i})) { val := i, property := (_ : i \u2208 {i}) }) i =\n (fun i => \u2191(u (Opposite.op {i})) { val := i, property := (_ : i \u2208 {i}) }) i' \u2192\n i = i'\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase mp.intro.refine'_2\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\nu : (j : (Finset \u03b9)\u1d52\u1d56) \u2192 (hallMatchingsFunctor t).obj j\nhu : u \u2208 Functor.sections (hallMatchingsFunctor t)\ni i' : \u03b9\nsubi : {i} \u2264 {i, i'}\nsubi' : {i'} \u2264 {i, i'}\n\u22a2 \u2191(u (Opposite.op {i})) { val := i, property := (_ : i \u2208 {i}) } =\n \u2191(u (Opposite.op {i'})) { val := i', property := (_ : i' \u2208 {i'}) } \u2192\n i = i'\n[PROOFSTEP]\nrw [\u2190 hu (CategoryTheory.homOfLE subi).op, \u2190 hu (CategoryTheory.homOfLE subi').op]\n[GOAL]\ncase mp.intro.refine'_2\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\nu : (j : (Finset \u03b9)\u1d52\u1d56) \u2192 (hallMatchingsFunctor t).obj j\nhu : u \u2208 Functor.sections (hallMatchingsFunctor t)\ni i' : \u03b9\nsubi : {i} \u2264 {i, i'}\nsubi' : {i'} \u2264 {i, i'}\n\u22a2 \u2191((hallMatchingsFunctor t).map (homOfLE subi).op (u (Opposite.op {i, i'}))) { val := i, property := (_ : i \u2208 {i}) } =\n \u2191((hallMatchingsFunctor t).map (homOfLE subi').op (u (Opposite.op {i, i'})))\n { val := i', property := (_ : i' \u2208 {i'}) } \u2192\n i = i'\n[PROOFSTEP]\nlet uii' := u (Opposite.op ({ i, i' } : Finset \u03b9))\n[GOAL]\ncase mp.intro.refine'_2\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\nu : (j : (Finset \u03b9)\u1d52\u1d56) \u2192 (hallMatchingsFunctor t).obj j\nhu : u \u2208 Functor.sections (hallMatchingsFunctor t)\ni i' : \u03b9\nsubi : {i} \u2264 {i, i'}\nsubi' : {i'} \u2264 {i, i'}\nuii' : (hallMatchingsFunctor t).obj (Opposite.op {i, i'}) := u (Opposite.op {i, i'})\n\u22a2 \u2191((hallMatchingsFunctor t).map (homOfLE subi).op (u (Opposite.op {i, i'}))) { val := i, property := (_ : i \u2208 {i}) } =\n \u2191((hallMatchingsFunctor t).map (homOfLE subi').op (u (Opposite.op {i, i'})))\n { val := i', property := (_ : i' \u2208 {i'}) } \u2192\n i = i'\n[PROOFSTEP]\nexact fun h => Subtype.mk_eq_mk.mp (uii'.property.1 h)\n[GOAL]\ncase mp.intro.refine'_3\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\nu : (j : (Finset \u03b9)\u1d52\u1d56) \u2192 (hallMatchingsFunctor t).obj j\nhu : u \u2208 Functor.sections (hallMatchingsFunctor t)\n\u22a2 \u2200 (x : \u03b9), \u2191(u (Opposite.op {x})) { val := x, property := (_ : x \u2208 {x}) } \u2208 t x\n[PROOFSTEP]\nintro i\n[GOAL]\ncase mp.intro.refine'_3\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nh : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\nthis\u271d : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Nonempty ((hallMatchingsFunctor t).obj \u03b9')\nthis : \u2200 (\u03b9' : (Finset \u03b9)\u1d52\u1d56), Finite ((hallMatchingsFunctor t).obj \u03b9')\nu : (j : (Finset \u03b9)\u1d52\u1d56) \u2192 (hallMatchingsFunctor t).obj j\nhu : u \u2208 Functor.sections (hallMatchingsFunctor t)\ni : \u03b9\n\u22a2 \u2191(u (Opposite.op {i})) { val := i, property := (_ : i \u2208 {i}) } \u2208 t i\n[PROOFSTEP]\napply (u (Opposite.op ({ i } : Finset \u03b9))).property.2\n[GOAL]\ncase mpr\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\n\u22a2 (\u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x) \u2192 \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\n[PROOFSTEP]\nrintro \u27e8f, hf\u2081, hf\u2082\u27e9 s\n[GOAL]\ncase mpr.intro.intro\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nf : \u03b9 \u2192 \u03b1\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (x : \u03b9), f x \u2208 t x\ns : Finset \u03b9\n\u22a2 card s \u2264 card (Finset.biUnion s t)\n[PROOFSTEP]\nrw [\u2190 Finset.card_image_of_injective s hf\u2081]\n[GOAL]\ncase mpr.intro.intro\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nf : \u03b9 \u2192 \u03b1\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (x : \u03b9), f x \u2208 t x\ns : Finset \u03b9\n\u22a2 card (image f s) \u2264 card (Finset.biUnion s t)\n[PROOFSTEP]\napply Finset.card_le_of_subset\n[GOAL]\ncase mpr.intro.intro.a\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nf : \u03b9 \u2192 \u03b1\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (x : \u03b9), f x \u2208 t x\ns : Finset \u03b9\n\u22a2 image f s \u2286 Finset.biUnion s t\n[PROOFSTEP]\nintro\n[GOAL]\ncase mpr.intro.intro.a\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nf : \u03b9 \u2192 \u03b1\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (x : \u03b9), f x \u2208 t x\ns : Finset \u03b9\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 image f s \u2192 a\u271d \u2208 Finset.biUnion s t\n[PROOFSTEP]\nrw [Finset.mem_image, Finset.mem_biUnion]\n[GOAL]\ncase mpr.intro.intro.a\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nf : \u03b9 \u2192 \u03b1\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (x : \u03b9), f x \u2208 t x\ns : Finset \u03b9\na\u271d : \u03b1\n\u22a2 (\u2203 a, a \u2208 s \u2227 f a = a\u271d) \u2192 \u2203 a, a \u2208 s \u2227 a\u271d \u2208 t a\n[PROOFSTEP]\nrintro \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.a.intro.intro\n\u03b9 : Type u\n\u03b1 : Type v\ninst\u271d : DecidableEq \u03b1\nt : \u03b9 \u2192 Finset \u03b1\nf : \u03b9 \u2192 \u03b1\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (x : \u03b9), f x \u2208 t x\ns : Finset \u03b9\nx : \u03b9\nhx : x \u2208 s\n\u22a2 \u2203 a, a \u2208 s \u2227 f x \u2208 t a\n[PROOFSTEP]\nexact \u27e8x, hx, hf\u2082 x\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\nA : Finset \u03b1\n\u22a2 Fintype \u2191(Rel.image r \u2191A)\n[PROOFSTEP]\nhave h : Rel.image r A = (A.biUnion fun a => (Rel.image r { a }).toFinset : Set \u03b2) :=\n by\n ext\n -- Porting note: added `Set.mem_toFinset`\n simp [Rel.image, (Set.mem_toFinset)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\nA : Finset \u03b1\n\u22a2 Rel.image r \u2191A = \u2191(Finset.biUnion A fun a => Set.toFinset (Rel.image r {a}))\n[PROOFSTEP]\next\n -- Porting note: added `Set.mem_toFinset`\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\nA : Finset \u03b1\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 Rel.image r \u2191A \u2194 x\u271d \u2208 \u2191(Finset.biUnion A fun a => Set.toFinset (Rel.image r {a}))\n[PROOFSTEP]\nsimp [Rel.image, (Set.mem_toFinset)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\nA : Finset \u03b1\nh : Rel.image r \u2191A = \u2191(Finset.biUnion A fun a => Set.toFinset (Rel.image r {a}))\n\u22a2 Fintype \u2191(Rel.image r \u2191A)\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\nA : Finset \u03b1\nh : Rel.image r \u2191A = \u2191(Finset.biUnion A fun a => Set.toFinset (Rel.image r {a}))\n\u22a2 Fintype \u2191\u2191(Finset.biUnion A fun a => Set.toFinset (Rel.image r {a}))\n[PROOFSTEP]\napply FinsetCoe.fintype\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\n\u22a2 (\u2200 (A : Finset \u03b1), Finset.card A \u2264 card \u2191(Rel.image r \u2191A)) \u2194 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b1), r x (f x)\n[PROOFSTEP]\nlet r' a := (Rel.image r { a }).toFinset\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => Set.toFinset (Rel.image r {a})\n\u22a2 (\u2200 (A : Finset \u03b1), Finset.card A \u2264 card \u2191(Rel.image r \u2191A)) \u2194 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b1), r x (f x)\n[PROOFSTEP]\nhave h : \u2200 A : Finset \u03b1, Fintype.card (Rel.image r A) = (A.biUnion r').card :=\n by\n intro A\n rw [\u2190 Set.toFinset_card]\n apply congr_arg\n ext b\n simp [Rel.image, (Set.mem_toFinset)]\n -- Porting note: added `Set.mem_toFinset`\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => Set.toFinset (Rel.image r {a})\n\u22a2 \u2200 (A : Finset \u03b1), card \u2191(Rel.image r \u2191A) = Finset.card (Finset.biUnion A r')\n[PROOFSTEP]\nintro A\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => Set.toFinset (Rel.image r {a})\nA : Finset \u03b1\n\u22a2 card \u2191(Rel.image r \u2191A) = Finset.card (Finset.biUnion A r')\n[PROOFSTEP]\nrw [\u2190 Set.toFinset_card]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => Set.toFinset (Rel.image r {a})\nA : Finset \u03b1\n\u22a2 Finset.card (Set.toFinset (Rel.image r \u2191A)) = Finset.card (Finset.biUnion A r')\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => Set.toFinset (Rel.image r {a})\nA : Finset \u03b1\n\u22a2 Set.toFinset (Rel.image r \u2191A) = Finset.biUnion A r'\n[PROOFSTEP]\next b\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => Set.toFinset (Rel.image r {a})\nA : Finset \u03b1\nb : \u03b2\n\u22a2 b \u2208 Set.toFinset (Rel.image r \u2191A) \u2194 b \u2208 Finset.biUnion A r'\n[PROOFSTEP]\nsimp [Rel.image, (Set.mem_toFinset)]\n -- Porting note: added `Set.mem_toFinset`\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => Set.toFinset (Rel.image r {a})\nh : \u2200 (A : Finset \u03b1), card \u2191(Rel.image r \u2191A) = Finset.card (Finset.biUnion A r')\n\u22a2 (\u2200 (A : Finset \u03b1), Finset.card A \u2264 card \u2191(Rel.image r \u2191A)) \u2194 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b1), r x (f x)\n[PROOFSTEP]\nhave h' : \u2200 (f : \u03b1 \u2192 \u03b2) (x), r x (f x) \u2194 f x \u2208 r' x := by simp [Rel.image, (Set.mem_toFinset)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => Set.toFinset (Rel.image r {a})\nh : \u2200 (A : Finset \u03b1), card \u2191(Rel.image r \u2191A) = Finset.card (Finset.biUnion A r')\n\u22a2 \u2200 (f : \u03b1 \u2192 \u03b2) (x : \u03b1), r x (f x) \u2194 f x \u2208 r' x\n[PROOFSTEP]\nsimp [Rel.image, (Set.mem_toFinset)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => Set.toFinset (Rel.image r {a})\nh : \u2200 (A : Finset \u03b1), card \u2191(Rel.image r \u2191A) = Finset.card (Finset.biUnion A r')\nh' : \u2200 (f : \u03b1 \u2192 \u03b2) (x : \u03b1), r x (f x) \u2194 f x \u2208 r' x\n\u22a2 (\u2200 (A : Finset \u03b1), Finset.card A \u2264 card \u2191(Rel.image r \u2191A)) \u2194 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b1), r x (f x)\n[PROOFSTEP]\nsimp only [h, h']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 Fintype \u2191(Rel.image r {a})\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => Set.toFinset (Rel.image r {a})\nh : \u2200 (A : Finset \u03b1), card \u2191(Rel.image r \u2191A) = Finset.card (Finset.biUnion A r')\nh' : \u2200 (f : \u03b1 \u2192 \u03b2) (x : \u03b1), r x (f x) \u2194 f x \u2208 r' x\n\u22a2 (\u2200 (A : Finset \u03b1), Finset.card A \u2264 Finset.card (Finset.biUnion A fun a => Set.toFinset (Rel.image r {a}))) \u2194\n \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b1), f x \u2208 Set.toFinset (Rel.image r {x})\n[PROOFSTEP]\napply Finset.all_card_le_biUnion_card_iff_exists_injective\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Fintype \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\n\u22a2 (\u2200 (A : Finset \u03b1), Finset.card A \u2264 Finset.card (filter (fun b => \u2203 a, a \u2208 A \u2227 r a b) univ)) \u2194\n \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b1), r x (f x)\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Fintype \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\nthis : DecidableEq \u03b2\n\u22a2 (\u2200 (A : Finset \u03b1), Finset.card A \u2264 Finset.card (filter (fun b => \u2203 a, a \u2208 A \u2227 r a b) univ)) \u2194\n \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b1), r x (f x)\n[PROOFSTEP]\nlet r' a := univ.filter fun b => r a b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Fintype \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\nthis : DecidableEq \u03b2\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => filter (fun b => r a b) univ\n\u22a2 (\u2200 (A : Finset \u03b1), Finset.card A \u2264 Finset.card (filter (fun b => \u2203 a, a \u2208 A \u2227 r a b) univ)) \u2194\n \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b1), r x (f x)\n[PROOFSTEP]\nhave h : \u2200 A : Finset \u03b1, (univ.filter fun b : \u03b2 => \u2203 a \u2208 A, r a b) = A.biUnion r' :=\n by\n intro A\n ext b\n simp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Fintype \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\nthis : DecidableEq \u03b2\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => filter (fun b => r a b) univ\n\u22a2 \u2200 (A : Finset \u03b1), filter (fun b => \u2203 a, a \u2208 A \u2227 r a b) univ = Finset.biUnion A r'\n[PROOFSTEP]\nintro A\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Fintype \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\nthis : DecidableEq \u03b2\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => filter (fun b => r a b) univ\nA : Finset \u03b1\n\u22a2 filter (fun b => \u2203 a, a \u2208 A \u2227 r a b) univ = Finset.biUnion A r'\n[PROOFSTEP]\next b\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Fintype \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\nthis : DecidableEq \u03b2\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => filter (fun b => r a b) univ\nA : Finset \u03b1\nb : \u03b2\n\u22a2 b \u2208 filter (fun b => \u2203 a, a \u2208 A \u2227 r a b) univ \u2194 b \u2208 Finset.biUnion A r'\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Fintype \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\nthis : DecidableEq \u03b2\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => filter (fun b => r a b) univ\nh : \u2200 (A : Finset \u03b1), filter (fun b => \u2203 a, a \u2208 A \u2227 r a b) univ = Finset.biUnion A r'\n\u22a2 (\u2200 (A : Finset \u03b1), Finset.card A \u2264 Finset.card (filter (fun b => \u2203 a, a \u2208 A \u2227 r a b) univ)) \u2194\n \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b1), r x (f x)\n[PROOFSTEP]\nhave h' : \u2200 (f : \u03b1 \u2192 \u03b2) (x), r x (f x) \u2194 f x \u2208 r' x := by simp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Fintype \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\nthis : DecidableEq \u03b2\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => filter (fun b => r a b) univ\nh : \u2200 (A : Finset \u03b1), filter (fun b => \u2203 a, a \u2208 A \u2227 r a b) univ = Finset.biUnion A r'\n\u22a2 \u2200 (f : \u03b1 \u2192 \u03b2) (x : \u03b1), r x (f x) \u2194 f x \u2208 r' x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Fintype \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\nthis : DecidableEq \u03b2\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => filter (fun b => r a b) univ\nh : \u2200 (A : Finset \u03b1), filter (fun b => \u2203 a, a \u2208 A \u2227 r a b) univ = Finset.biUnion A r'\nh' : \u2200 (f : \u03b1 \u2192 \u03b2) (x : \u03b1), r x (f x) \u2194 f x \u2208 r' x\n\u22a2 (\u2200 (A : Finset \u03b1), Finset.card A \u2264 Finset.card (filter (fun b => \u2203 a, a \u2208 A \u2227 r a b) univ)) \u2194\n \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b1), r x (f x)\n[PROOFSTEP]\nsimp_rw [h, h']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Fintype \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d : (a : \u03b1) \u2192 DecidablePred (r a)\nthis : DecidableEq \u03b2\nr' : \u03b1 \u2192 Finset \u03b2 := fun a => filter (fun b => r a b) univ\nh : \u2200 (A : Finset \u03b1), filter (fun b => \u2203 a, a \u2208 A \u2227 r a b) univ = Finset.biUnion A r'\nh' : \u2200 (f : \u03b1 \u2192 \u03b2) (x : \u03b1), r x (f x) \u2194 f x \u2208 r' x\n\u22a2 (\u2200 (A : Finset \u03b1), Finset.card A \u2264 Finset.card (Finset.biUnion A fun a => filter (fun b => r a b) univ)) \u2194\n \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b1), f x \u2208 filter (fun b => r x b) univ\n[PROOFSTEP]\napply Finset.all_card_le_biUnion_card_iff_exists_injective\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.Hall.Basic", "llama_tokens": 15370, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737473266735, "lm_q2_score": 0.6261241842048093, "lm_q1q2_score": 0.4273759307045331}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\n\u22a2 \u2203 b, \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\n[PROOFSTEP]\nby_cases h : \u2203 b, (a, b) \u2208 f.val\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u2203 b, (a, b) \u2208 \u2191f\n\u22a2 \u2203 b, \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\n[PROOFSTEP]\ncases' h with b hb\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nhb : (a, b) \u2208 \u2191f\n\u22a2 \u2203 b, \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\n[PROOFSTEP]\nexact \u27e8b, fun p hp \u21a6 f.prop _ hp _ hb\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\n\u22a2 \u2203 b, \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\n[PROOFSTEP]\nhave :\n \u2200 x \u2208 (f.val.filter fun p : \u03b1 \u00d7 \u03b2 \u21a6 p.fst < a).image Prod.snd,\n \u2200 y \u2208 (f.val.filter fun p : \u03b1 \u00d7 \u03b2 \u21a6 a < p.fst).image Prod.snd, x < y :=\n by\n intro x hx y hy\n rw [Finset.mem_image] at hx hy \n rcases hx with \u27e8p, hp1, rfl\u27e9\n rcases hy with \u27e8q, hq1, rfl\u27e9\n rw [Finset.mem_filter] at hp1 hq1 \n rw [\u2190 lt_iff_lt_of_cmp_eq_cmp (f.prop _ hp1.1 _ hq1.1)]\n exact lt_trans hp1.right hq1.right\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\n\u22a2 \u2200 (x : \u03b2),\n x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 x < y\n[PROOFSTEP]\nintro x hx y hy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\nx : \u03b2\nhx : x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f)\ny : \u03b2\nhy : y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f)\n\u22a2 x < y\n[PROOFSTEP]\nrw [Finset.mem_image] at hx hy \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\nx : \u03b2\nhx : \u2203 a_1, a_1 \u2208 Finset.filter (fun p => p.fst < a) \u2191f \u2227 a_1.snd = x\ny : \u03b2\nhy : \u2203 a_1, a_1 \u2208 Finset.filter (fun p => a < p.fst) \u2191f \u2227 a_1.snd = y\n\u22a2 x < y\n[PROOFSTEP]\nrcases hx with \u27e8p, hp1, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\ny : \u03b2\nhy : \u2203 a_1, a_1 \u2208 Finset.filter (fun p => a < p.fst) \u2191f \u2227 a_1.snd = y\np : \u03b1 \u00d7 \u03b2\nhp1 : p \u2208 Finset.filter (fun p => p.fst < a) \u2191f\n\u22a2 p.snd < y\n[PROOFSTEP]\nrcases hy with \u27e8q, hq1, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\np : \u03b1 \u00d7 \u03b2\nhp1 : p \u2208 Finset.filter (fun p => p.fst < a) \u2191f\nq : \u03b1 \u00d7 \u03b2\nhq1 : q \u2208 Finset.filter (fun p => a < p.fst) \u2191f\n\u22a2 p.snd < q.snd\n[PROOFSTEP]\nrw [Finset.mem_filter] at hp1 hq1 \n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\np : \u03b1 \u00d7 \u03b2\nhp1 : p \u2208 \u2191f \u2227 p.fst < a\nq : \u03b1 \u00d7 \u03b2\nhq1 : q \u2208 \u2191f \u2227 a < q.fst\n\u22a2 p.snd < q.snd\n[PROOFSTEP]\nrw [\u2190 lt_iff_lt_of_cmp_eq_cmp (f.prop _ hp1.1 _ hq1.1)]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\np : \u03b1 \u00d7 \u03b2\nhp1 : p \u2208 \u2191f \u2227 p.fst < a\nq : \u03b1 \u00d7 \u03b2\nhq1 : q \u2208 \u2191f \u2227 a < q.fst\n\u22a2 p.fst < q.fst\n[PROOFSTEP]\nexact lt_trans hp1.right hq1.right\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\nthis :\n \u2200 (x : \u03b2),\n x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 x < y\n\u22a2 \u2203 b, \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\n[PROOFSTEP]\ncases' exists_between_finsets _ _ this with b hb\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\nthis :\n \u2200 (x : \u03b2),\n x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 x < y\nb : \u03b2\nhb :\n (\u2200 (x : \u03b2), x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192 x < b) \u2227\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 b < y\n\u22a2 \u2203 b, \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\n[PROOFSTEP]\nuse b\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\nthis :\n \u2200 (x : \u03b2),\n x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 x < y\nb : \u03b2\nhb :\n (\u2200 (x : \u03b2), x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192 x < b) \u2227\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 b < y\n\u22a2 \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\n[PROOFSTEP]\nrintro \u27e8p1, p2\u27e9 hp\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\nthis :\n \u2200 (x : \u03b2),\n x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 x < y\nb : \u03b2\nhb :\n (\u2200 (x : \u03b2), x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192 x < b) \u2227\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 b < y\np1 : \u03b1\np2 : \u03b2\nhp : (p1, p2) \u2208 \u2191f\n\u22a2 cmp (p1, p2).fst a = cmp (p1, p2).snd b\n[PROOFSTEP]\nhave : p1 \u2260 a := fun he \u21a6 h \u27e8p2, he \u25b8 hp\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\nthis\u271d :\n \u2200 (x : \u03b2),\n x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 x < y\nb : \u03b2\nhb :\n (\u2200 (x : \u03b2), x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192 x < b) \u2227\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 b < y\np1 : \u03b1\np2 : \u03b2\nhp : (p1, p2) \u2208 \u2191f\nthis : p1 \u2260 a\n\u22a2 cmp (p1, p2).fst a = cmp (p1, p2).snd b\n[PROOFSTEP]\ncases' lt_or_gt_of_ne this with hl hr\n[GOAL]\ncase h.mk.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\nthis\u271d :\n \u2200 (x : \u03b2),\n x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 x < y\nb : \u03b2\nhb :\n (\u2200 (x : \u03b2), x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192 x < b) \u2227\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 b < y\np1 : \u03b1\np2 : \u03b2\nhp : (p1, p2) \u2208 \u2191f\nthis : p1 \u2260 a\nhl : p1 < a\n\u22a2 cmp (p1, p2).fst a = cmp (p1, p2).snd b\n[PROOFSTEP]\nhave : p1 < a \u2227 p2 < b := \u27e8hl, hb.1 _ (Finset.mem_image.mpr \u27e8(p1, p2), Finset.mem_filter.mpr \u27e8hp, hl\u27e9, rfl\u27e9)\u27e9\n[GOAL]\ncase h.mk.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\nthis\u271d\u00b9 :\n \u2200 (x : \u03b2),\n x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 x < y\nb : \u03b2\nhb :\n (\u2200 (x : \u03b2), x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192 x < b) \u2227\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 b < y\np1 : \u03b1\np2 : \u03b2\nhp : (p1, p2) \u2208 \u2191f\nthis\u271d : p1 \u2260 a\nhl : p1 < a\nthis : p1 < a \u2227 p2 < b\n\u22a2 cmp (p1, p2).fst a = cmp (p1, p2).snd b\n[PROOFSTEP]\nrw [\u2190 cmp_eq_lt_iff, \u2190 cmp_eq_lt_iff] at this \n[GOAL]\ncase h.mk.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\nthis\u271d\u00b9 :\n \u2200 (x : \u03b2),\n x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 x < y\nb : \u03b2\nhb :\n (\u2200 (x : \u03b2), x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192 x < b) \u2227\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 b < y\np1 : \u03b1\np2 : \u03b2\nhp : (p1, p2) \u2208 \u2191f\nthis\u271d : p1 \u2260 a\nhl : p1 < a\nthis : cmp p1 a = Ordering.lt \u2227 cmp p2 b = Ordering.lt\n\u22a2 cmp (p1, p2).fst a = cmp (p1, p2).snd b\n[PROOFSTEP]\nexact this.1.trans this.2.symm\n[GOAL]\ncase h.mk.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\nthis\u271d :\n \u2200 (x : \u03b2),\n x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 x < y\nb : \u03b2\nhb :\n (\u2200 (x : \u03b2), x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192 x < b) \u2227\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 b < y\np1 : \u03b1\np2 : \u03b2\nhp : (p1, p2) \u2208 \u2191f\nthis : p1 \u2260 a\nhr : p1 > a\n\u22a2 cmp (p1, p2).fst a = cmp (p1, p2).snd b\n[PROOFSTEP]\nhave : a < p1 \u2227 b < p2 := \u27e8hr, hb.2 _ (Finset.mem_image.mpr \u27e8(p1, p2), Finset.mem_filter.mpr \u27e8hp, hr\u27e9, rfl\u27e9)\u27e9\n[GOAL]\ncase h.mk.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\nthis\u271d\u00b9 :\n \u2200 (x : \u03b2),\n x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 x < y\nb : \u03b2\nhb :\n (\u2200 (x : \u03b2), x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192 x < b) \u2227\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 b < y\np1 : \u03b1\np2 : \u03b2\nhp : (p1, p2) \u2208 \u2191f\nthis\u271d : p1 \u2260 a\nhr : p1 > a\nthis : a < p1 \u2227 b < p2\n\u22a2 cmp (p1, p2).fst a = cmp (p1, p2).snd b\n[PROOFSTEP]\nrw [\u2190 cmp_eq_gt_iff, \u2190 cmp_eq_gt_iff] at this \n[GOAL]\ncase h.mk.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nf : PartialIso \u03b1 \u03b2\na : \u03b1\nh : \u00ac\u2203 b, (a, b) \u2208 \u2191f\nthis\u271d\u00b9 :\n \u2200 (x : \u03b2),\n x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 x < y\nb : \u03b2\nhb :\n (\u2200 (x : \u03b2), x \u2208 Finset.image Prod.snd (Finset.filter (fun p => p.fst < a) \u2191f) \u2192 x < b) \u2227\n \u2200 (y : \u03b2), y \u2208 Finset.image Prod.snd (Finset.filter (fun p => a < p.fst) \u2191f) \u2192 b < y\np1 : \u03b1\np2 : \u03b2\nhp : (p1, p2) \u2208 \u2191f\nthis\u271d : p1 \u2260 a\nhr : p1 > a\nthis : cmp p1 a = Ordering.gt \u2227 cmp p2 b = Ordering.gt\n\u22a2 cmp (p1, p2).fst a = cmp (p1, p2).snd b\n[PROOFSTEP]\nexact this.1.trans this.2.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : Finset (\u03b1 \u00d7 \u03b2)\nhf : \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 f \u2192 \u2200 (q : \u03b1 \u00d7 \u03b2), q \u2208 f \u2192 cmp p.fst q.fst = cmp p.snd q.snd\np : \u03b2 \u00d7 \u03b1\nhp : p \u2208 Finset.image (\u2191(Equiv.prodComm \u03b1 \u03b2)) f\nq : \u03b2 \u00d7 \u03b1\nhq : q \u2208 Finset.image (\u2191(Equiv.prodComm \u03b1 \u03b2)) f\n\u22a2 \u2191(Equiv.prodComm \u03b1 \u03b2).symm p \u2208 f\n[PROOFSTEP]\nrw [\u2190 Finset.mem_coe, Finset.coe_image, Equiv.image_eq_preimage] at hp \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : Finset (\u03b1 \u00d7 \u03b2)\nhf : \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 f \u2192 \u2200 (q : \u03b1 \u00d7 \u03b2), q \u2208 f \u2192 cmp p.fst q.fst = cmp p.snd q.snd\np : \u03b2 \u00d7 \u03b1\nhp : p \u2208 \u2191(Equiv.prodComm \u03b1 \u03b2).symm \u207b\u00b9' \u2191f\nq : \u03b2 \u00d7 \u03b1\nhq : q \u2208 Finset.image (\u2191(Equiv.prodComm \u03b1 \u03b2)) f\n\u22a2 \u2191(Equiv.prodComm \u03b1 \u03b2).symm p \u2208 f\n[PROOFSTEP]\nrwa [\u2190 Finset.mem_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : Finset (\u03b1 \u00d7 \u03b2)\nhf : \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 f \u2192 \u2200 (q : \u03b1 \u00d7 \u03b2), q \u2208 f \u2192 cmp p.fst q.fst = cmp p.snd q.snd\np : \u03b2 \u00d7 \u03b1\nhp : p \u2208 Finset.image (\u2191(Equiv.prodComm \u03b1 \u03b2)) f\nq : \u03b2 \u00d7 \u03b1\nhq : q \u2208 Finset.image (\u2191(Equiv.prodComm \u03b1 \u03b2)) f\n\u22a2 \u2191(Equiv.prodComm \u03b1 \u03b2).symm q \u2208 f\n[PROOFSTEP]\nrw [\u2190 Finset.mem_coe, Finset.coe_image, Equiv.image_eq_preimage] at hq \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : Finset (\u03b1 \u00d7 \u03b2)\nhf : \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 f \u2192 \u2200 (q : \u03b1 \u00d7 \u03b2), q \u2208 f \u2192 cmp p.fst q.fst = cmp p.snd q.snd\np : \u03b2 \u00d7 \u03b1\nhp : p \u2208 Finset.image (\u2191(Equiv.prodComm \u03b1 \u03b2)) f\nq : \u03b2 \u00d7 \u03b1\nhq : q \u2208 \u2191(Equiv.prodComm \u03b1 \u03b2).symm \u207b\u00b9' \u2191f\n\u22a2 \u2191(Equiv.prodComm \u03b1 \u03b2).symm q \u2208 f\n[PROOFSTEP]\nrwa [\u2190 Finset.mem_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\na : \u03b1\nf : PartialIso \u03b1 \u03b2\n\u22a2 \u2203 y, y \u2208 {f | \u2203 b, (a, b) \u2208 \u2191f} \u2227 f \u2264 y\n[PROOFSTEP]\ncases' exists_across f a with b a_b\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\na : \u03b1\nf : PartialIso \u03b1 \u03b2\nb : \u03b2\na_b : \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\n\u22a2 \u2203 y, y \u2208 {f | \u2203 b, (a, b) \u2208 \u2191f} \u2227 f \u2264 y\n[PROOFSTEP]\nrefine \u27e8\u27e8insert (a, b) f.val, fun p hp q hq \u21a6 ?_\u27e9, \u27e8b, Finset.mem_insert_self _ _\u27e9, Finset.subset_insert _ _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\na : \u03b1\nf : PartialIso \u03b1 \u03b2\nb : \u03b2\na_b : \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\np : \u03b1 \u00d7 \u03b2\nhp : p \u2208 insert (a, b) \u2191f\nq : \u03b1 \u00d7 \u03b2\nhq : q \u2208 insert (a, b) \u2191f\n\u22a2 cmp p.fst q.fst = cmp p.snd q.snd\n[PROOFSTEP]\nrw [Finset.mem_insert] at hp hq \n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\na : \u03b1\nf : PartialIso \u03b1 \u03b2\nb : \u03b2\na_b : \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\np : \u03b1 \u00d7 \u03b2\nhp : p = (a, b) \u2228 p \u2208 \u2191f\nq : \u03b1 \u00d7 \u03b2\nhq : q = (a, b) \u2228 q \u2208 \u2191f\n\u22a2 cmp p.fst q.fst = cmp p.snd q.snd\n[PROOFSTEP]\nrcases hp with (rfl | pf)\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\na : \u03b1\nf : PartialIso \u03b1 \u03b2\nb : \u03b2\na_b : \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\nq : \u03b1 \u00d7 \u03b2\nhq : q = (a, b) \u2228 q \u2208 \u2191f\n\u22a2 cmp (a, b).fst q.fst = cmp (a, b).snd q.snd\n[PROOFSTEP]\nrcases hq with (rfl | qf)\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\na : \u03b1\nf : PartialIso \u03b1 \u03b2\nb : \u03b2\na_b : \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\np q : \u03b1 \u00d7 \u03b2\nhq : q = (a, b) \u2228 q \u2208 \u2191f\npf : p \u2208 \u2191f\n\u22a2 cmp p.fst q.fst = cmp p.snd q.snd\n[PROOFSTEP]\nrcases hq with (rfl | qf)\n[GOAL]\ncase intro.inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\na : \u03b1\nf : PartialIso \u03b1 \u03b2\nb : \u03b2\na_b : \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\n\u22a2 cmp (a, b).fst (a, b).fst = cmp (a, b).snd (a, b).snd\n[PROOFSTEP]\nsimp only [cmp_self_eq_eq]\n[GOAL]\ncase intro.inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\na : \u03b1\nf : PartialIso \u03b1 \u03b2\nb : \u03b2\na_b : \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\nq : \u03b1 \u00d7 \u03b2\nqf : q \u2208 \u2191f\n\u22a2 cmp (a, b).fst q.fst = cmp (a, b).snd q.snd\n[PROOFSTEP]\nrw [cmp_eq_cmp_symm]\n[GOAL]\ncase intro.inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\na : \u03b1\nf : PartialIso \u03b1 \u03b2\nb : \u03b2\na_b : \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\nq : \u03b1 \u00d7 \u03b2\nqf : q \u2208 \u2191f\n\u22a2 cmp q.fst (a, b).fst = cmp q.snd (a, b).snd\n[PROOFSTEP]\nexact a_b _ qf\n[GOAL]\ncase intro.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\na : \u03b1\nf : PartialIso \u03b1 \u03b2\nb : \u03b2\na_b : \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\np : \u03b1 \u00d7 \u03b2\npf : p \u2208 \u2191f\n\u22a2 cmp p.fst (a, b).fst = cmp p.snd (a, b).snd\n[PROOFSTEP]\nexact a_b _ pf\n[GOAL]\ncase intro.inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\na : \u03b1\nf : PartialIso \u03b1 \u03b2\nb : \u03b2\na_b : \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 \u2191f \u2192 cmp p.fst a = cmp p.snd b\np q : \u03b1 \u00d7 \u03b2\npf : p \u2208 \u2191f\nqf : q \u2208 \u2191f\n\u22a2 cmp p.fst q.fst = cmp p.snd q.snd\n[PROOFSTEP]\nexact f.prop _ pf _ qf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b1\ninst\u271d\u00b2 : NoMinOrder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : Nonempty \u03b1\nb : \u03b2\nf : PartialIso \u03b1 \u03b2\n\u22a2 \u2203 y, y \u2208 {f | \u2203 a, (a, b) \u2208 \u2191f} \u2227 f \u2264 y\n[PROOFSTEP]\nrcases(definedAtLeft \u03b1 b).mem_gt f.comm with \u27e8f', \u27e8a, ha\u27e9, hl\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b1\ninst\u271d\u00b2 : NoMinOrder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : Nonempty \u03b1\nb : \u03b2\nf : PartialIso \u03b1 \u03b2\nf' : PartialIso \u03b2 \u03b1\nhl : PartialIso.comm f \u2264 f'\na : \u03b1\nha : (b, a) \u2208 \u2191f'\n\u22a2 \u2203 y, y \u2208 {f | \u2203 a, (a, b) \u2208 \u2191f} \u2227 f \u2264 y\n[PROOFSTEP]\nrefine' \u27e8f'.comm, \u27e8a, _\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b1\ninst\u271d\u00b2 : NoMinOrder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : Nonempty \u03b1\nb : \u03b2\nf : PartialIso \u03b1 \u03b2\nf' : PartialIso \u03b2 \u03b1\nhl : PartialIso.comm f \u2264 f'\na : \u03b1\nha : (b, a) \u2208 \u2191f'\n\u22a2 (a, b) \u2208 \u2191(PartialIso.comm f')\n[PROOFSTEP]\nchange (a, b) \u2208 f'.val.image _\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b1\ninst\u271d\u00b2 : NoMinOrder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : Nonempty \u03b1\nb : \u03b2\nf : PartialIso \u03b1 \u03b2\nf' : PartialIso \u03b2 \u03b1\nhl : PartialIso.comm f \u2264 f'\na : \u03b1\nha : (b, a) \u2208 \u2191f'\n\u22a2 (a, b) \u2208 Finset.image \u2191(Equiv.prodComm \u03b2 \u03b1) \u2191f'\n[PROOFSTEP]\nrwa [\u2190 Finset.mem_coe, Finset.coe_image, Equiv.image_eq_preimage]\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b1\ninst\u271d\u00b2 : NoMinOrder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : Nonempty \u03b1\nb : \u03b2\nf : PartialIso \u03b1 \u03b2\nf' : PartialIso \u03b2 \u03b1\nhl : PartialIso.comm f \u2264 f'\na : \u03b1\nha : (b, a) \u2208 \u2191f'\n\u22a2 f \u2264 PartialIso.comm f'\n[PROOFSTEP]\nchange _ \u2286 f'.val.image _\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : LinearOrder \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b1\ninst\u271d\u00b2 : NoMinOrder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : Nonempty \u03b1\nb : \u03b2\nf : PartialIso \u03b1 \u03b2\nf' : PartialIso \u03b2 \u03b1\nhl : PartialIso.comm f \u2264 f'\na : \u03b1\nha : (b, a) \u2208 \u2191f'\n\u22a2 (fun a => \u2191a) f \u2286 Finset.image \u2191(Equiv.prodComm \u03b2 \u03b1) \u2191f'\n[PROOFSTEP]\nrwa [\u2190 Finset.coe_subset, Finset.coe_image, \u2190 Equiv.subset_image, \u2190 Finset.coe_image, Finset.coe_subset]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Encodable \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b2\ninst\u271d : Nontrivial \u03b2\n\u22a2 Nonempty (\u03b1 \u21aao \u03b2)\n[PROOFSTEP]\nrcases exists_pair_lt \u03b2 with \u27e8x, y, hxy\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Encodable \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b2\ninst\u271d : Nontrivial \u03b2\nx y : \u03b2\nhxy : x < y\n\u22a2 Nonempty (\u03b1 \u21aao \u03b2)\n[PROOFSTEP]\ncases' exists_between hxy with a ha\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Encodable \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b2\ninst\u271d : Nontrivial \u03b2\nx y : \u03b2\nhxy : x < y\na : \u03b2\nha : x < a \u2227 a < y\n\u22a2 Nonempty (\u03b1 \u21aao \u03b2)\n[PROOFSTEP]\nhaveI : Nonempty (Set.Ioo x y) := \u27e8\u27e8a, ha\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Encodable \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b2\ninst\u271d : Nontrivial \u03b2\nx y : \u03b2\nhxy : x < y\na : \u03b2\nha : x < a \u2227 a < y\nthis : Nonempty \u2191(Set.Ioo x y)\n\u22a2 Nonempty (\u03b1 \u21aao \u03b2)\n[PROOFSTEP]\nlet our_ideal : Ideal (PartialIso \u03b1 _) := idealOfCofinals default (definedAtLeft (Set.Ioo x y))\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Encodable \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b2\ninst\u271d : Nontrivial \u03b2\nx y : \u03b2\nhxy : x < y\na : \u03b2\nha : x < a \u2227 a < y\nthis : Nonempty \u2191(Set.Ioo x y)\nour_ideal : Ideal (PartialIso \u03b1 \u2191(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft \u2191(Set.Ioo x y))\n\u22a2 Nonempty (\u03b1 \u21aao \u03b2)\n[PROOFSTEP]\nlet F a := funOfIdeal a our_ideal (cofinal_meets_idealOfCofinals _ _ a)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Encodable \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b2\ninst\u271d : Nontrivial \u03b2\nx y : \u03b2\nhxy : x < y\na : \u03b2\nha : x < a \u2227 a < y\nthis : Nonempty \u2191(Set.Ioo x y)\nour_ideal : Ideal (PartialIso \u03b1 \u2191(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft \u2191(Set.Ioo x y))\nF : (a : \u03b1) \u2192 { b // \u2203 f, f \u2208 our_ideal \u2227 (a, b) \u2208 \u2191f } :=\n fun a =>\n funOfIdeal a our_ideal\n (_ : \u2203 x_1, x_1 \u2208 definedAtLeft (\u2191(Set.Ioo x y)) a \u2227 x_1 \u2208 idealOfCofinals default (definedAtLeft \u2191(Set.Ioo x y)))\n\u22a2 Nonempty (\u03b1 \u21aao \u03b2)\n[PROOFSTEP]\nrefine \u27e8RelEmbedding.trans (OrderEmbedding.ofStrictMono (fun a \u21a6 (F a).val) fun a\u2081 a\u2082 \u21a6 ?_) (OrderEmbedding.subtype _)\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Encodable \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b2\ninst\u271d : Nontrivial \u03b2\nx y : \u03b2\nhxy : x < y\na : \u03b2\nha : x < a \u2227 a < y\nthis : Nonempty \u2191(Set.Ioo x y)\nour_ideal : Ideal (PartialIso \u03b1 \u2191(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft \u2191(Set.Ioo x y))\nF : (a : \u03b1) \u2192 { b // \u2203 f, f \u2208 our_ideal \u2227 (a, b) \u2208 \u2191f } :=\n fun a =>\n funOfIdeal a our_ideal\n (_ : \u2203 x_1, x_1 \u2208 definedAtLeft (\u2191(Set.Ioo x y)) a \u2227 x_1 \u2208 idealOfCofinals default (definedAtLeft \u2191(Set.Ioo x y)))\na\u2081 a\u2082 : \u03b1\n\u22a2 a\u2081 < a\u2082 \u2192 (fun a => \u2191(F a)) a\u2081 < (fun a => \u2191(F a)) a\u2082\n[PROOFSTEP]\nrcases(F a\u2081).prop with \u27e8f, hf, ha\u2081\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Encodable \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b2\ninst\u271d : Nontrivial \u03b2\nx y : \u03b2\nhxy : x < y\na : \u03b2\nha : x < a \u2227 a < y\nthis : Nonempty \u2191(Set.Ioo x y)\nour_ideal : Ideal (PartialIso \u03b1 \u2191(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft \u2191(Set.Ioo x y))\nF : (a : \u03b1) \u2192 { b // \u2203 f, f \u2208 our_ideal \u2227 (a, b) \u2208 \u2191f } :=\n fun a =>\n funOfIdeal a our_ideal\n (_ : \u2203 x_1, x_1 \u2208 definedAtLeft (\u2191(Set.Ioo x y)) a \u2227 x_1 \u2208 idealOfCofinals default (definedAtLeft \u2191(Set.Ioo x y)))\na\u2081 a\u2082 : \u03b1\nf : PartialIso \u03b1 \u2191(Set.Ioo x y)\nhf : f \u2208 our_ideal\nha\u2081 : (a\u2081, \u2191(F a\u2081)) \u2208 \u2191f\n\u22a2 a\u2081 < a\u2082 \u2192 (fun a => \u2191(F a)) a\u2081 < (fun a => \u2191(F a)) a\u2082\n[PROOFSTEP]\nrcases(F a\u2082).prop with \u27e8g, hg, ha\u2082\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Encodable \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b2\ninst\u271d : Nontrivial \u03b2\nx y : \u03b2\nhxy : x < y\na : \u03b2\nha : x < a \u2227 a < y\nthis : Nonempty \u2191(Set.Ioo x y)\nour_ideal : Ideal (PartialIso \u03b1 \u2191(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft \u2191(Set.Ioo x y))\nF : (a : \u03b1) \u2192 { b // \u2203 f, f \u2208 our_ideal \u2227 (a, b) \u2208 \u2191f } :=\n fun a =>\n funOfIdeal a our_ideal\n (_ : \u2203 x_1, x_1 \u2208 definedAtLeft (\u2191(Set.Ioo x y)) a \u2227 x_1 \u2208 idealOfCofinals default (definedAtLeft \u2191(Set.Ioo x y)))\na\u2081 a\u2082 : \u03b1\nf : PartialIso \u03b1 \u2191(Set.Ioo x y)\nhf : f \u2208 our_ideal\nha\u2081 : (a\u2081, \u2191(F a\u2081)) \u2208 \u2191f\ng : PartialIso \u03b1 \u2191(Set.Ioo x y)\nhg : g \u2208 our_ideal\nha\u2082 : (a\u2082, \u2191(F a\u2082)) \u2208 \u2191g\n\u22a2 a\u2081 < a\u2082 \u2192 (fun a => \u2191(F a)) a\u2081 < (fun a => \u2191(F a)) a\u2082\n[PROOFSTEP]\nrcases our_ideal.directed _ hf _ hg with \u27e8m, _hm, fm, gm\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Encodable \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b2\ninst\u271d : Nontrivial \u03b2\nx y : \u03b2\nhxy : x < y\na : \u03b2\nha : x < a \u2227 a < y\nthis : Nonempty \u2191(Set.Ioo x y)\nour_ideal : Ideal (PartialIso \u03b1 \u2191(Set.Ioo x y)) := idealOfCofinals default (definedAtLeft \u2191(Set.Ioo x y))\nF : (a : \u03b1) \u2192 { b // \u2203 f, f \u2208 our_ideal \u2227 (a, b) \u2208 \u2191f } :=\n fun a =>\n funOfIdeal a our_ideal\n (_ : \u2203 x_1, x_1 \u2208 definedAtLeft (\u2191(Set.Ioo x y)) a \u2227 x_1 \u2208 idealOfCofinals default (definedAtLeft \u2191(Set.Ioo x y)))\na\u2081 a\u2082 : \u03b1\nf : PartialIso \u03b1 \u2191(Set.Ioo x y)\nhf : f \u2208 our_ideal\nha\u2081 : (a\u2081, \u2191(F a\u2081)) \u2208 \u2191f\ng : PartialIso \u03b1 \u2191(Set.Ioo x y)\nhg : g \u2208 our_ideal\nha\u2082 : (a\u2082, \u2191(F a\u2082)) \u2208 \u2191g\nm : PartialIso \u03b1 \u2191(Set.Ioo x y)\n_hm : m \u2208 \u2191our_ideal\nfm : f \u2264 m\ngm : g \u2264 m\n\u22a2 a\u2081 < a\u2082 \u2192 (fun a => \u2191(F a)) a\u2081 < (fun a => \u2191(F a)) a\u2082\n[PROOFSTEP]\nexact (lt_iff_lt_of_cmp_eq_cmp <| m.prop (a\u2081, _) (fm ha\u2081) (a\u2082, _) (gm ha\u2082)).mp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9\u00b9 : LinearOrder \u03b1\ninst\u271d\u00b9\u2070 : LinearOrder \u03b2\ninst\u271d\u2079 : Encodable \u03b1\ninst\u271d\u2078 : DenselyOrdered \u03b1\ninst\u271d\u2077 : NoMinOrder \u03b1\ninst\u271d\u2076 : NoMaxOrder \u03b1\ninst\u271d\u2075 : Nonempty \u03b1\ninst\u271d\u2074 : Encodable \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nto_cofinal : \u03b1 \u2295 \u03b2 \u2192 Cofinal (PartialIso \u03b1 \u03b2) := fun p => Sum.recOn p (definedAtLeft \u03b2) (definedAtRight \u03b1)\nour_ideal : Ideal (PartialIso \u03b1 \u03b2) := idealOfCofinals default to_cofinal\nF : (a : \u03b1) \u2192 { b // \u2203 f, f \u2208 our_ideal \u2227 (a, b) \u2208 \u2191f } :=\n fun a => funOfIdeal a our_ideal (_ : \u2203 x, x \u2208 to_cofinal (Sum.inl a) \u2227 x \u2208 idealOfCofinals default to_cofinal)\nG : (b : \u03b2) \u2192 { a // \u2203 f, f \u2208 our_ideal \u2227 (a, b) \u2208 \u2191f } :=\n fun b => invOfIdeal b our_ideal (_ : \u2203 x, x \u2208 to_cofinal (Sum.inr b) \u2227 x \u2208 idealOfCofinals default to_cofinal)\na : \u03b1\nb : \u03b2\n\u22a2 cmp a ((fun b => \u2191(G b)) b) = cmp ((fun a => \u2191(F a)) a) b\n[PROOFSTEP]\nrcases(F a).prop with \u27e8f, hf, ha\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9\u00b9 : LinearOrder \u03b1\ninst\u271d\u00b9\u2070 : LinearOrder \u03b2\ninst\u271d\u2079 : Encodable \u03b1\ninst\u271d\u2078 : DenselyOrdered \u03b1\ninst\u271d\u2077 : NoMinOrder \u03b1\ninst\u271d\u2076 : NoMaxOrder \u03b1\ninst\u271d\u2075 : Nonempty \u03b1\ninst\u271d\u2074 : Encodable \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nto_cofinal : \u03b1 \u2295 \u03b2 \u2192 Cofinal (PartialIso \u03b1 \u03b2) := fun p => Sum.recOn p (definedAtLeft \u03b2) (definedAtRight \u03b1)\nour_ideal : Ideal (PartialIso \u03b1 \u03b2) := idealOfCofinals default to_cofinal\nF : (a : \u03b1) \u2192 { b // \u2203 f, f \u2208 our_ideal \u2227 (a, b) \u2208 \u2191f } :=\n fun a => funOfIdeal a our_ideal (_ : \u2203 x, x \u2208 to_cofinal (Sum.inl a) \u2227 x \u2208 idealOfCofinals default to_cofinal)\nG : (b : \u03b2) \u2192 { a // \u2203 f, f \u2208 our_ideal \u2227 (a, b) \u2208 \u2191f } :=\n fun b => invOfIdeal b our_ideal (_ : \u2203 x, x \u2208 to_cofinal (Sum.inr b) \u2227 x \u2208 idealOfCofinals default to_cofinal)\na : \u03b1\nb : \u03b2\nf : PartialIso \u03b1 \u03b2\nhf : f \u2208 our_ideal\nha : (a, \u2191(F a)) \u2208 \u2191f\n\u22a2 cmp a ((fun b => \u2191(G b)) b) = cmp ((fun a => \u2191(F a)) a) b\n[PROOFSTEP]\nrcases(G b).prop with \u27e8g, hg, hb\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9\u00b9 : LinearOrder \u03b1\ninst\u271d\u00b9\u2070 : LinearOrder \u03b2\ninst\u271d\u2079 : Encodable \u03b1\ninst\u271d\u2078 : DenselyOrdered \u03b1\ninst\u271d\u2077 : NoMinOrder \u03b1\ninst\u271d\u2076 : NoMaxOrder \u03b1\ninst\u271d\u2075 : Nonempty \u03b1\ninst\u271d\u2074 : Encodable \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nto_cofinal : \u03b1 \u2295 \u03b2 \u2192 Cofinal (PartialIso \u03b1 \u03b2) := fun p => Sum.recOn p (definedAtLeft \u03b2) (definedAtRight \u03b1)\nour_ideal : Ideal (PartialIso \u03b1 \u03b2) := idealOfCofinals default to_cofinal\nF : (a : \u03b1) \u2192 { b // \u2203 f, f \u2208 our_ideal \u2227 (a, b) \u2208 \u2191f } :=\n fun a => funOfIdeal a our_ideal (_ : \u2203 x, x \u2208 to_cofinal (Sum.inl a) \u2227 x \u2208 idealOfCofinals default to_cofinal)\nG : (b : \u03b2) \u2192 { a // \u2203 f, f \u2208 our_ideal \u2227 (a, b) \u2208 \u2191f } :=\n fun b => invOfIdeal b our_ideal (_ : \u2203 x, x \u2208 to_cofinal (Sum.inr b) \u2227 x \u2208 idealOfCofinals default to_cofinal)\na : \u03b1\nb : \u03b2\nf : PartialIso \u03b1 \u03b2\nhf : f \u2208 our_ideal\nha : (a, \u2191(F a)) \u2208 \u2191f\ng : PartialIso \u03b1 \u03b2\nhg : g \u2208 our_ideal\nhb : (\u2191(G b), b) \u2208 \u2191g\n\u22a2 cmp a ((fun b => \u2191(G b)) b) = cmp ((fun a => \u2191(F a)) a) b\n[PROOFSTEP]\nrcases our_ideal.directed _ hf _ hg with \u27e8m, _, fm, gm\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9\u00b9 : LinearOrder \u03b1\ninst\u271d\u00b9\u2070 : LinearOrder \u03b2\ninst\u271d\u2079 : Encodable \u03b1\ninst\u271d\u2078 : DenselyOrdered \u03b1\ninst\u271d\u2077 : NoMinOrder \u03b1\ninst\u271d\u2076 : NoMaxOrder \u03b1\ninst\u271d\u2075 : Nonempty \u03b1\ninst\u271d\u2074 : Encodable \u03b2\ninst\u271d\u00b3 : DenselyOrdered \u03b2\ninst\u271d\u00b2 : NoMinOrder \u03b2\ninst\u271d\u00b9 : NoMaxOrder \u03b2\ninst\u271d : Nonempty \u03b2\nto_cofinal : \u03b1 \u2295 \u03b2 \u2192 Cofinal (PartialIso \u03b1 \u03b2) := fun p => Sum.recOn p (definedAtLeft \u03b2) (definedAtRight \u03b1)\nour_ideal : Ideal (PartialIso \u03b1 \u03b2) := idealOfCofinals default to_cofinal\nF : (a : \u03b1) \u2192 { b // \u2203 f, f \u2208 our_ideal \u2227 (a, b) \u2208 \u2191f } :=\n fun a => funOfIdeal a our_ideal (_ : \u2203 x, x \u2208 to_cofinal (Sum.inl a) \u2227 x \u2208 idealOfCofinals default to_cofinal)\nG : (b : \u03b2) \u2192 { a // \u2203 f, f \u2208 our_ideal \u2227 (a, b) \u2208 \u2191f } :=\n fun b => invOfIdeal b our_ideal (_ : \u2203 x, x \u2208 to_cofinal (Sum.inr b) \u2227 x \u2208 idealOfCofinals default to_cofinal)\na : \u03b1\nb : \u03b2\nf : PartialIso \u03b1 \u03b2\nhf : f \u2208 our_ideal\nha : (a, \u2191(F a)) \u2208 \u2191f\ng : PartialIso \u03b1 \u03b2\nhg : g \u2208 our_ideal\nhb : (\u2191(G b), b) \u2208 \u2191g\nm : PartialIso \u03b1 \u03b2\nleft\u271d : m \u2208 \u2191our_ideal\nfm : f \u2264 m\ngm : g \u2264 m\n\u22a2 cmp a ((fun b => \u2191(G b)) b) = cmp ((fun a => \u2191(F a)) a) b\n[PROOFSTEP]\nexact m.prop (a, _) (fm ha) (_, b) (gm hb)\n", "meta": {"mathlib_filename": "Mathlib.Order.CountableDenseLinearOrder", "llama_tokens": 16694, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149758396752, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.42729586882237325}} {"text": "[GOAL]\np : Prop\nas ps as\u2081 : List Prop\n\u22a2 as = List.reverseAux as\u2081 ps \u2192 implies (mk as) p ps (List.length as\u2081) \u2192 p\n[PROOFSTEP]\ninduction ps generalizing as\u2081 with\n| nil => exact fun _ \u21a6 id\n| cons a as ih =>\n refine fun e H \u21a6 @ih (a :: as\u2081) e (H ?_)\n subst e; clear ih H\n suffices \u2200 n n', n' = List.length as\u2081 + n \u2192 \u2200 bs, mk (as\u2081.reverseAux bs) n' \u2194 mk bs n from this 0 _ rfl (a :: as)\n induction as\u2081 with simp\n | cons b as\u2081 ih => exact fun n bs \u21a6 ih (n + 1) _ (Nat.succ_add ..) _\n[GOAL]\np : Prop\nas ps as\u2081 : List Prop\n\u22a2 as = List.reverseAux as\u2081 ps \u2192 implies (mk as) p ps (List.length as\u2081) \u2192 p\n[PROOFSTEP]\ninduction ps generalizing as\u2081 with\n| nil => exact fun _ \u21a6 id\n| cons a as ih =>\n refine fun e H \u21a6 @ih (a :: as\u2081) e (H ?_)\n subst e; clear ih H\n suffices \u2200 n n', n' = List.length as\u2081 + n \u2192 \u2200 bs, mk (as\u2081.reverseAux bs) n' \u2194 mk bs n from this 0 _ rfl (a :: as)\n induction as\u2081 with simp\n | cons b as\u2081 ih => exact fun n bs \u21a6 ih (n + 1) _ (Nat.succ_add ..) _\n[GOAL]\ncase nil\np : Prop\nas as\u2081 : List Prop\n\u22a2 as = List.reverseAux as\u2081 [] \u2192 implies (mk as) p [] (List.length as\u2081) \u2192 p\n[PROOFSTEP]\n\n| nil => exact fun _ \u21a6 id\n[GOAL]\ncase nil\np : Prop\nas as\u2081 : List Prop\n\u22a2 as = List.reverseAux as\u2081 [] \u2192 implies (mk as) p [] (List.length as\u2081) \u2192 p\n[PROOFSTEP]\nexact fun _ \u21a6 id\n[GOAL]\ncase cons\np : Prop\nas\u271d : List Prop\na : Prop\nas : List Prop\nih : \u2200 (as\u2081 : List Prop), as\u271d = List.reverseAux as\u2081 as \u2192 implies (mk as\u271d) p as (List.length as\u2081) \u2192 p\nas\u2081 : List Prop\n\u22a2 as\u271d = List.reverseAux as\u2081 (a :: as) \u2192 implies (mk as\u271d) p (a :: as) (List.length as\u2081) \u2192 p\n[PROOFSTEP]\n\n| cons a as ih =>\n refine fun e H \u21a6 @ih (a :: as\u2081) e (H ?_)\n subst e; clear ih H\n suffices \u2200 n n', n' = List.length as\u2081 + n \u2192 \u2200 bs, mk (as\u2081.reverseAux bs) n' \u2194 mk bs n from this 0 _ rfl (a :: as)\n induction as\u2081 with simp\n | cons b as\u2081 ih => exact fun n bs \u21a6 ih (n + 1) _ (Nat.succ_add ..) _\n[GOAL]\ncase cons\np : Prop\nas\u271d : List Prop\na : Prop\nas : List Prop\nih : \u2200 (as\u2081 : List Prop), as\u271d = List.reverseAux as\u2081 as \u2192 implies (mk as\u271d) p as (List.length as\u2081) \u2192 p\nas\u2081 : List Prop\n\u22a2 as\u271d = List.reverseAux as\u2081 (a :: as) \u2192 implies (mk as\u271d) p (a :: as) (List.length as\u2081) \u2192 p\n[PROOFSTEP]\nrefine fun e H \u21a6 @ih (a :: as\u2081) e (H ?_)\n[GOAL]\ncase cons\np : Prop\nas\u271d : List Prop\na : Prop\nas : List Prop\nih : \u2200 (as\u2081 : List Prop), as\u271d = List.reverseAux as\u2081 as \u2192 implies (mk as\u271d) p as (List.length as\u2081) \u2192 p\nas\u2081 : List Prop\ne : as\u271d = List.reverseAux as\u2081 (a :: as)\nH : implies (mk as\u271d) p (a :: as) (List.length as\u2081)\n\u22a2 mk as\u271d (List.length as\u2081) \u2194 a\n[PROOFSTEP]\nsubst e\n[GOAL]\ncase cons\np a : Prop\nas as\u2081 : List Prop\nih :\n \u2200 (as\u2081_1 : List Prop),\n List.reverseAux as\u2081 (a :: as) = List.reverseAux as\u2081_1 as \u2192\n implies (mk (List.reverseAux as\u2081 (a :: as))) p as (List.length as\u2081_1) \u2192 p\nH : implies (mk (List.reverseAux as\u2081 (a :: as))) p (a :: as) (List.length as\u2081)\n\u22a2 mk (List.reverseAux as\u2081 (a :: as)) (List.length as\u2081) \u2194 a\n[PROOFSTEP]\nclear ih H\n[GOAL]\ncase cons\np a : Prop\nas as\u2081 : List Prop\n\u22a2 mk (List.reverseAux as\u2081 (a :: as)) (List.length as\u2081) \u2194 a\n[PROOFSTEP]\nsuffices \u2200 n n', n' = List.length as\u2081 + n \u2192 \u2200 bs, mk (as\u2081.reverseAux bs) n' \u2194 mk bs n from this 0 _ rfl (a :: as)\n[GOAL]\ncase cons\np a : Prop\nas as\u2081 : List Prop\n\u22a2 \u2200 (n n' : \u2115), n' = List.length as\u2081 + n \u2192 \u2200 (bs : List Prop), mk (List.reverseAux as\u2081 bs) n' \u2194 mk bs n\n[PROOFSTEP]\ninduction as\u2081 with simp\n| cons b as\u2081 ih => exact fun n bs \u21a6 ih (n + 1) _ (Nat.succ_add ..) _\n[GOAL]\ncase cons\np a : Prop\nas as\u2081 : List Prop\n\u22a2 \u2200 (n n' : \u2115), n' = List.length as\u2081 + n \u2192 \u2200 (bs : List Prop), mk (List.reverseAux as\u2081 bs) n' \u2194 mk bs n\n[PROOFSTEP]\ninduction as\u2081 with simp\n| cons b as\u2081 ih => exact fun n bs \u21a6 ih (n + 1) _ (Nat.succ_add ..) _\n[GOAL]\ncase cons.cons\np a : Prop\nas : List Prop\nb : Prop\nas\u2081 : List Prop\nih : \u2200 (n n' : \u2115), n' = List.length as\u2081 + n \u2192 \u2200 (bs : List Prop), mk (List.reverseAux as\u2081 bs) n' \u2194 mk bs n\n\u22a2 \u2200 (n n' : \u2115), n' = List.length (b :: as\u2081) + n \u2192 \u2200 (bs : List Prop), mk (List.reverseAux (b :: as\u2081) bs) n' \u2194 mk bs n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.cons\np a : Prop\nas : List Prop\nb : Prop\nas\u2081 : List Prop\nih : \u2200 (n n' : \u2115), n' = List.length as\u2081 + n \u2192 \u2200 (bs : List Prop), mk (List.reverseAux as\u2081 bs) n' \u2194 mk bs n\n\u22a2 \u2200 (n : \u2115) (bs : List Prop), mk (List.reverseAux as\u2081 (b :: bs)) (Nat.succ (List.length as\u2081) + n) \u2194 mk bs n\n[PROOFSTEP]\n\n| cons b as\u2081 ih => exact fun n bs \u21a6 ih (n + 1) _ (Nat.succ_add ..) _\n[GOAL]\ncase cons.cons\np a : Prop\nas : List Prop\nb : Prop\nas\u2081 : List Prop\nih : \u2200 (n n' : \u2115), n' = List.length as\u2081 + n \u2192 \u2200 (bs : List Prop), mk (List.reverseAux as\u2081 bs) n' \u2194 mk bs n\n\u22a2 \u2200 (n : \u2115) (bs : List Prop), mk (List.reverseAux as\u2081 (b :: bs)) (Nat.succ (List.length as\u2081) + n) \u2194 mk bs n\n[PROOFSTEP]\nexact fun n bs \u21a6 ih (n + 1) _ (Nat.succ_add ..) _\n[GOAL]\ncase cons.nil\np a : Prop\nas : List Prop\n\u22a2 \u2200 (n n' : \u2115), n' = List.length [] + n \u2192 \u2200 (bs : List Prop), mk (List.reverseAux [] bs) n' \u2194 mk bs n\n[PROOFSTEP]\nsimp\n[GOAL]\nv : Valuation\nf\u2081 : Fmla\na : Prop\nf\u2082 : Fmla\nb : Prop\nh\u2081 : reify v f\u2081 a\nh\u2082 : reify v f\u2082 b\n\u22a2 reify v (and f\u2081 f\u2082) (a \u2228 b)\n[PROOFSTEP]\nrefine \u27e8fun H \u21a6 by_contra fun hn \u21a6 H \u27e8fun c h \u21a6 by_contra fun hn' \u21a6 ?_\u27e9\u27e9\n[GOAL]\nv : Valuation\nf\u2081 : Fmla\na : Prop\nf\u2082 : Fmla\nb : Prop\nh\u2081 : reify v f\u2081 a\nh\u2082 : reify v f\u2082 b\nH : \u00acValuation.satisfies_fmla v (and f\u2081 f\u2082)\nhn : \u00ac(a \u2228 b)\nc : Clause\nh : c \u2208 and f\u2081 f\u2082\nhn' : \u00acValuation.satisfies v c\n\u22a2 False\n[PROOFSTEP]\nrcases List.mem_append.1 h with h | h\n[GOAL]\ncase inl\nv : Valuation\nf\u2081 : Fmla\na : Prop\nf\u2082 : Fmla\nb : Prop\nh\u2081 : reify v f\u2081 a\nh\u2082 : reify v f\u2082 b\nH : \u00acValuation.satisfies_fmla v (and f\u2081 f\u2082)\nhn : \u00ac(a \u2228 b)\nc : Clause\nh\u271d : c \u2208 and f\u2081 f\u2082\nhn' : \u00acValuation.satisfies v c\nh : c \u2208 f\u2081\n\u22a2 False\n[PROOFSTEP]\nexact hn $ Or.inl $ h\u2081.1 fun Hc \u21a6 hn' $ Hc.1 _ h\n[GOAL]\ncase inr\nv : Valuation\nf\u2081 : Fmla\na : Prop\nf\u2082 : Fmla\nb : Prop\nh\u2081 : reify v f\u2081 a\nh\u2082 : reify v f\u2082 b\nH : \u00acValuation.satisfies_fmla v (and f\u2081 f\u2082)\nhn : \u00ac(a \u2228 b)\nc : Clause\nh\u271d : c \u2208 and f\u2081 f\u2082\nhn' : \u00acValuation.satisfies v c\nh : c \u2208 f\u2082\n\u22a2 False\n[PROOFSTEP]\nexact hn $ Or.inr $ h\u2082.1 fun Hc \u21a6 hn' $ Hc.1 _ h\n", "meta": {"mathlib_filename": "Mathlib.Tactic.Sat.FromLRAT", "llama_tokens": 2804, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.42697563900267615}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nx : Multiset \u03b1 \u00d7 Multiset \u03b1\nl : List \u03b1\n\u22a2 x \u2208 antidiagonal (Quotient.mk (isSetoid \u03b1) l) \u2194 x.fst + x.snd = Quotient.mk (isSetoid \u03b1) l\n[PROOFSTEP]\ndsimp only [quot_mk_to_coe, antidiagonal_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nx : Multiset \u03b1 \u00d7 Multiset \u03b1\nl : List \u03b1\n\u22a2 x \u2208 \u2191(revzip (powersetAux l)) \u2194 x.fst + x.snd = \u2191l\n[PROOFSTEP]\nrefine' \u27e8fun h => revzip_powersetAux h, fun h \u21a6 _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nx : Multiset \u03b1 \u00d7 Multiset \u03b1\nl : List \u03b1\nh : x.fst + x.snd = \u2191l\n\u22a2 x \u2208 \u2191(revzip (powersetAux l))\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nx : Multiset \u03b1 \u00d7 Multiset \u03b1\nl : List \u03b1\nh : x.fst + x.snd = \u2191l\nthis : DecidableEq \u03b1\n\u22a2 x \u2208 \u2191(revzip (powersetAux l))\n[PROOFSTEP]\nsimp only [revzip_powersetAux_lemma l revzip_powersetAux, h.symm, ge_iff_le, mem_coe, List.mem_map, mem_powersetAux]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nx : Multiset \u03b1 \u00d7 Multiset \u03b1\nl : List \u03b1\nh : x.fst + x.snd = \u2191l\nthis : DecidableEq \u03b1\n\u22a2 \u2203 a, a \u2264 x.fst + x.snd \u2227 (a, x.fst + x.snd - a) = x\n[PROOFSTEP]\ncases' x with x\u2081 x\u2082\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nl : List \u03b1\nthis : DecidableEq \u03b1\nx\u2081 x\u2082 : Multiset \u03b1\nh : (x\u2081, x\u2082).fst + (x\u2081, x\u2082).snd = \u2191l\n\u22a2 \u2203 a, a \u2264 (x\u2081, x\u2082).fst + (x\u2081, x\u2082).snd \u2227 (a, (x\u2081, x\u2082).fst + (x\u2081, x\u2082).snd - a) = (x\u2081, x\u2082)\n[PROOFSTEP]\nexact \u27e8x\u2081, le_add_right _ _, by rw [add_tsub_cancel_left x\u2081 x\u2082]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nl : List \u03b1\nthis : DecidableEq \u03b1\nx\u2081 x\u2082 : Multiset \u03b1\nh : (x\u2081, x\u2082).fst + (x\u2081, x\u2082).snd = \u2191l\n\u22a2 (x\u2081, (x\u2081, x\u2082).fst + (x\u2081, x\u2082).snd - x\u2081) = (x\u2081, x\u2082)\n[PROOFSTEP]\nrw [add_tsub_cancel_left x\u2081 x\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 map Prod.fst (antidiagonal (Quotient.mk (isSetoid \u03b1) l)) = powerset (Quotient.mk (isSetoid \u03b1) l)\n[PROOFSTEP]\nsimp [powersetAux']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 map Prod.snd (antidiagonal (Quotient.mk (isSetoid \u03b1) l)) = powerset (Quotient.mk (isSetoid \u03b1) l)\n[PROOFSTEP]\nsimp [powersetAux']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 antidiagonal (a ::\u2098 Quotient.mk (isSetoid \u03b1) l) =\n map (Prod.map id (cons a)) (antidiagonal (Quotient.mk (isSetoid \u03b1) l)) +\n map (Prod.map (cons a) id) (antidiagonal (Quotient.mk (isSetoid \u03b1) l))\n[PROOFSTEP]\nsimp only [revzip, reverse_append, quot_mk_to_coe, coe_eq_coe, powersetAux'_cons, cons_coe, coe_map, antidiagonal_coe',\n coe_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 zip (powersetAux' l ++ List.map (cons a) (powersetAux' l))\n (reverse (List.map (cons a) (powersetAux' l)) ++ reverse (powersetAux' l)) ~\n List.map (fun x => Prod.map id (cons a) x) (zip (powersetAux' l) (reverse (powersetAux' l))) ++\n List.map (fun x => Prod.map (cons a) id x) (zip (powersetAux' l) (reverse (powersetAux' l)))\n[PROOFSTEP]\nrw [\u2190 zip_map, \u2190 zip_map, zip_append, (_ : _ ++ _ = _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 zip (powersetAux' l) (reverse (List.map (cons a) (powersetAux' l))) ++\n zip (List.map (cons a) (powersetAux' l)) (reverse (powersetAux' l)) =\n zip (List.map id (powersetAux' l)) (List.map (cons a) (reverse (powersetAux' l))) ++\n zip (List.map (cons a) (powersetAux' l)) (List.map id (reverse (powersetAux' l)))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 powersetAux' l = List.map id (powersetAux' l)\ncase e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 reverse (List.map (cons a) (powersetAux' l)) = List.map (cons a) (reverse (powersetAux' l))\ncase e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 reverse (powersetAux' l) = List.map id (reverse (powersetAux' l))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 reverse (List.map (cons a) (powersetAux' l)) = List.map (cons a) (reverse (powersetAux' l))\ncase e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 reverse (powersetAux' l) = List.map id (reverse (powersetAux' l))\n[PROOFSTEP]\nrw [map_reverse]\n[GOAL]\ncase e_a.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 reverse (powersetAux' l) = List.map id (reverse (powersetAux' l))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 length (powersetAux' l) = length (reverse (List.map (cons a) (powersetAux' l)))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\n\u22a2 antidiagonal s = map (fun t => (s - t, t)) (powerset s)\n[PROOFSTEP]\ninduction' s using Multiset.induction_on with a s hs\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\n\u22a2 antidiagonal 0 = map (fun t => (0 - t, t)) (powerset 0)\n[PROOFSTEP]\nsimp only [antidiagonal_zero, powerset_zero, zero_tsub, map_singleton]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nhs : antidiagonal s = map (fun t => (s - t, t)) (powerset s)\n\u22a2 antidiagonal (a ::\u2098 s) = map (fun t => (a ::\u2098 s - t, t)) (powerset (a ::\u2098 s))\n[PROOFSTEP]\nsimp_rw [antidiagonal_cons, powerset_cons, map_add, hs, map_map, Function.comp, Prod.map_mk, id.def, sub_cons,\n erase_cons_head]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nhs : antidiagonal s = map (fun t => (s - t, t)) (powerset s)\n\u22a2 map (fun x => (s - x, a ::\u2098 x)) (powerset s) + map (fun x => (a ::\u2098 (s - x), x)) (powerset s) =\n map (fun x => (a ::\u2098 s - x, x)) (powerset s) + map (fun x => (s - x, a ::\u2098 x)) (powerset s)\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nhs : antidiagonal s = map (fun t => (s - t, t)) (powerset s)\n\u22a2 map (fun x => (a ::\u2098 (s - x), x)) (powerset s) + map (fun x => (s - x, a ::\u2098 x)) (powerset s) =\n map (fun x => (a ::\u2098 s - x, x)) (powerset s) + map (fun x => (s - x, a ::\u2098 x)) (powerset s)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase cons.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nhs : antidiagonal s = map (fun t => (s - t, t)) (powerset s)\n\u22a2 map (fun x => (a ::\u2098 (s - x), x)) (powerset s) = map (fun x => (a ::\u2098 s - x, x)) (powerset s)\n[PROOFSTEP]\nrefine' Multiset.map_congr rfl fun x hx \u21a6 _\n[GOAL]\ncase cons.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nhs : antidiagonal s = map (fun t => (s - t, t)) (powerset s)\nx : Multiset \u03b1\nhx : x \u2208 powerset s\n\u22a2 (a ::\u2098 (s - x), x) = (a ::\u2098 s - x, x)\n[PROOFSTEP]\nrw [cons_sub_of_le _ (mem_powerset.mp hx)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\n\u22a2 \u2191card (antidiagonal s) = 2 ^ \u2191card s\n[PROOFSTEP]\nhave := card_powerset s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nthis : \u2191card (powerset s) = 2 ^ \u2191card s\n\u22a2 \u2191card (antidiagonal s) = 2 ^ \u2191card s\n[PROOFSTEP]\nrwa [\u2190 antidiagonal_map_fst, card_map] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommSemiring \u03b2\ns : Multiset \u03b1\nf g : \u03b1 \u2192 \u03b2\n\u22a2 prod (map (fun a => f a + g a) s) = sum (map (fun p => prod (map f p.fst) * prod (map g p.snd)) (antidiagonal s))\n[PROOFSTEP]\nrefine' s.induction_on _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommSemiring \u03b2\ns : Multiset \u03b1\nf g : \u03b1 \u2192 \u03b2\n\u22a2 prod (map (fun a => f a + g a) 0) = sum (map (fun p => prod (map f p.fst) * prod (map g p.snd)) (antidiagonal 0))\n[PROOFSTEP]\nsimp only [map_zero, prod_zero, antidiagonal_zero, map_singleton, mul_one, sum_singleton]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommSemiring \u03b2\ns : Multiset \u03b1\nf g : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1},\n prod (map (fun a => f a + g a) s) = sum (map (fun p => prod (map f p.fst) * prod (map g p.snd)) (antidiagonal s)) \u2192\n prod (map (fun a => f a + g a) (a ::\u2098 s)) =\n sum (map (fun p => prod (map f p.fst) * prod (map g p.snd)) (antidiagonal (a ::\u2098 s)))\n[PROOFSTEP]\nintro a s ih\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommSemiring \u03b2\ns\u271d : Multiset \u03b1\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Multiset \u03b1\nih : prod (map (fun a => f a + g a) s) = sum (map (fun p => prod (map f p.fst) * prod (map g p.snd)) (antidiagonal s))\n\u22a2 prod (map (fun a => f a + g a) (a ::\u2098 s)) =\n sum (map (fun p => prod (map f p.fst) * prod (map g p.snd)) (antidiagonal (a ::\u2098 s)))\n[PROOFSTEP]\nsimp only [map_cons, prod_cons, ih, sum_map_mul_left.symm, add_mul, mul_left_comm (f a), mul_left_comm (g a),\n sum_map_add, antidiagonal_cons, Prod_map, id_eq, map_add, map_map, Function.comp_apply, mul_assoc, sum_add]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommSemiring \u03b2\ns\u271d : Multiset \u03b1\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Multiset \u03b1\nih : prod (map (fun a => f a + g a) s) = sum (map (fun p => prod (map f p.fst) * prod (map g p.snd)) (antidiagonal s))\n\u22a2 sum (map (fun i => prod (map f i.fst) * (f a * prod (map g i.snd))) (antidiagonal s)) +\n sum (map (fun i => prod (map f i.fst) * (g a * prod (map g i.snd))) (antidiagonal s)) =\n sum (map (fun x => prod (map f x.fst) * (g a * prod (map g x.snd))) (antidiagonal s)) +\n sum (map (fun i => prod (map f i.fst) * (f a * prod (map g i.snd))) (antidiagonal s))\n[PROOFSTEP]\nexact add_comm _ _\n", "meta": {"mathlib_filename": "Mathlib.Data.Multiset.Antidiagonal", "llama_tokens": 4384, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494421679929, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.4269756310101741}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := hs\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\n\u22a2 Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\n[PROOFSTEP]\nexact \u27e8\u27e8fun _ => x, fun i j _ => le_rfl, fun _ => hx\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf f' : \u03b1 \u2192 E\ns : Set \u03b1\nh : \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 edist (f x) (f' x) = 0\n\u22a2 eVariationOn f s = eVariationOn f' s\n[PROOFSTEP]\ndsimp only [eVariationOn]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf f' : \u03b1 \u2192 E\ns : Set \u03b1\nh : \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 edist (f x) (f' x) = 0\n\u22a2 \u2a06 (p : \u2115 \u00d7 { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }),\n \u2211 i in Finset.range p.fst, edist (f (\u2191p.snd (i + 1))) (f (\u2191p.snd i)) =\n \u2a06 (p : \u2115 \u00d7 { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }),\n \u2211 i in Finset.range p.fst, edist (f' (\u2191p.snd (i + 1))) (f' (\u2191p.snd i))\n[PROOFSTEP]\ncongr 1 with p : 1\n[GOAL]\ncase e_s.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf f' : \u03b1 \u2192 E\ns : Set \u03b1\nh : \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 edist (f x) (f' x) = 0\np : \u2115 \u00d7 { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\n\u22a2 \u2211 i in Finset.range p.fst, edist (f (\u2191p.snd (i + 1))) (f (\u2191p.snd i)) =\n \u2211 i in Finset.range p.fst, edist (f' (\u2191p.snd (i + 1))) (f' (\u2191p.snd i))\n[PROOFSTEP]\ncongr 1 with i : 1\n[GOAL]\ncase e_s.h.e_f.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf f' : \u03b1 \u2192 E\ns : Set \u03b1\nh : \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 edist (f x) (f' x) = 0\np : \u2115 \u00d7 { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\ni : \u2115\n\u22a2 edist (f (\u2191p.snd (i + 1))) (f (\u2191p.snd i)) = edist (f' (\u2191p.snd (i + 1))) (f' (\u2191p.snd i))\n[PROOFSTEP]\nrw [edist_congr_right (h <| p.snd.prop.2 (i + 1)), edist_congr_left (h <| p.snd.prop.2 i)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf f' : \u03b1 \u2192 E\ns : Set \u03b1\nh : EqOn f f' s\nx : \u03b1\nxs : x \u2208 s\n\u22a2 edist (f x) (f' x) = 0\n[PROOFSTEP]\nrw [h xs, edist_self]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nm n : \u2115\nu : \u2115 \u2192 \u03b1\nhu : MonotoneOn u (Icc m n)\nus : \u2200 (i : \u2115), i \u2208 Icc m n \u2192 u i \u2208 s\n\u22a2 \u2211 i in Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) \u2264 eVariationOn f s\n[PROOFSTEP]\ncases' le_total n m with hnm hmn\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nm n : \u2115\nu : \u2115 \u2192 \u03b1\nhu : MonotoneOn u (Icc m n)\nus : \u2200 (i : \u2115), i \u2208 Icc m n \u2192 u i \u2208 s\nhnm : n \u2264 m\n\u22a2 \u2211 i in Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) \u2264 eVariationOn f s\n[PROOFSTEP]\nsimp [Finset.Ico_eq_empty_of_le hnm]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nm n : \u2115\nu : \u2115 \u2192 \u03b1\nhu : MonotoneOn u (Icc m n)\nus : \u2200 (i : \u2115), i \u2208 Icc m n \u2192 u i \u2208 s\nhmn : m \u2264 n\n\u22a2 \u2211 i in Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) \u2264 eVariationOn f s\n[PROOFSTEP]\nlet \u03c0 := projIcc m n hmn\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nm n : \u2115\nu : \u2115 \u2192 \u03b1\nhu : MonotoneOn u (Icc m n)\nus : \u2200 (i : \u2115), i \u2208 Icc m n \u2192 u i \u2208 s\nhmn : m \u2264 n\n\u03c0 : \u2115 \u2192 \u2191(Icc m n) := projIcc m n hmn\n\u22a2 \u2211 i in Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) \u2264 eVariationOn f s\n[PROOFSTEP]\nlet v i := u (\u03c0 i)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nm n : \u2115\nu : \u2115 \u2192 \u03b1\nhu : MonotoneOn u (Icc m n)\nus : \u2200 (i : \u2115), i \u2208 Icc m n \u2192 u i \u2208 s\nhmn : m \u2264 n\n\u03c0 : \u2115 \u2192 \u2191(Icc m n) := projIcc m n hmn\nv : \u2115 \u2192 \u03b1 := fun i => u \u2191(\u03c0 i)\n\u22a2 \u2211 i in Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) \u2264 eVariationOn f s\n[PROOFSTEP]\ncalc\n \u2211 i in Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) = \u2211 i in Finset.Ico m n, edist (f (v (i + 1))) (f (v i)) :=\n Finset.sum_congr rfl fun i hi \u21a6 by\n rw [Finset.mem_Ico] at hi \n simp only [projIcc_of_mem hmn \u27e8hi.1, hi.2.le\u27e9, projIcc_of_mem hmn \u27e8hi.1.trans i.le_succ, hi.2\u27e9]\n _ \u2264 \u2211 i in Finset.range n, edist (f (v (i + 1))) (f (v i)) :=\n (Finset.sum_mono_set _ (Nat.Iio_eq_range \u25b8 Finset.Ico_subset_Iio_self))\n _ \u2264 eVariationOn f s := sum_le _ _ (fun i j h \u21a6 hu (\u03c0 i).2 (\u03c0 j).2 (monotone_projIcc hmn h)) fun i \u21a6 us _ (\u03c0 i).2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nm n : \u2115\nu : \u2115 \u2192 \u03b1\nhu : MonotoneOn u (Icc m n)\nus : \u2200 (i : \u2115), i \u2208 Icc m n \u2192 u i \u2208 s\nhmn : m \u2264 n\n\u03c0 : \u2115 \u2192 \u2191(Icc m n) := projIcc m n hmn\nv : \u2115 \u2192 \u03b1 := fun i => u \u2191(\u03c0 i)\ni : \u2115\nhi : i \u2208 Finset.Ico m n\n\u22a2 edist (f (u (i + 1))) (f (u i)) = edist (f (v (i + 1))) (f (v i))\n[PROOFSTEP]\nrw [Finset.mem_Ico] at hi \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nm n : \u2115\nu : \u2115 \u2192 \u03b1\nhu : MonotoneOn u (Icc m n)\nus : \u2200 (i : \u2115), i \u2208 Icc m n \u2192 u i \u2208 s\nhmn : m \u2264 n\n\u03c0 : \u2115 \u2192 \u2191(Icc m n) := projIcc m n hmn\nv : \u2115 \u2192 \u03b1 := fun i => u \u2191(\u03c0 i)\ni : \u2115\nhi : m \u2264 i \u2227 i < n\n\u22a2 edist (f (u (i + 1))) (f (u i)) = edist (f (v (i + 1))) (f (v i))\n[PROOFSTEP]\nsimp only [projIcc_of_mem hmn \u27e8hi.1, hi.2.le\u27e9, projIcc_of_mem hmn \u27e8hi.1.trans i.le_succ, hi.2\u27e9]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : MonotoneOn u (Iic n)\nus : \u2200 (i : \u2115), i \u2264 n \u2192 u i \u2208 s\n\u22a2 \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 eVariationOn f s\n[PROOFSTEP]\nsimpa using sum_le_of_monotoneOn_Icc f (m := 0) (hu.mono Icc_subset_Iic_self) fun i hi \u21a6 us i hi.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nhst : t \u2286 s\n\u22a2 eVariationOn f t \u2264 eVariationOn f s\n[PROOFSTEP]\napply iSup_le _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nhst : t \u2286 s\n\u22a2 \u2200 (i : \u2115 \u00d7 { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }),\n \u2211 i_1 in Finset.range i.fst, edist (f (\u2191i.snd (i_1 + 1))) (f (\u2191i.snd i_1)) \u2264 eVariationOn f s\n[PROOFSTEP]\nrintro \u27e8n, \u27e8u, hu, ut\u27e9\u27e9\n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nhst : t \u2286 s\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nut : \u2200 (i : \u2115), u i \u2208 t\n\u22a2 \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).snd i)) \u2264\n eVariationOn f s\n[PROOFSTEP]\nexact sum_le f n hu fun i => hst (ut i)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 edist (f x) (f y) \u2264 eVariationOn f s\n[PROOFSTEP]\nwlog hxy : y \u2264 x generalizing x y\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\nthis : \u2200 {x y : \u03b1}, x \u2208 s \u2192 y \u2208 s \u2192 y \u2264 x \u2192 edist (f x) (f y) \u2264 eVariationOn f s\nhxy : \u00acy \u2264 x\n\u22a2 edist (f x) (f y) \u2264 eVariationOn f s\n[PROOFSTEP]\nrw [edist_comm]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\nthis : \u2200 {x y : \u03b1}, x \u2208 s \u2192 y \u2208 s \u2192 y \u2264 x \u2192 edist (f x) (f y) \u2264 eVariationOn f s\nhxy : \u00acy \u2264 x\n\u22a2 edist (f y) (f x) \u2264 eVariationOn f s\n[PROOFSTEP]\nexact this hy hx (le_of_not_le hxy)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\nhxy : y \u2264 x\n\u22a2 edist (f x) (f y) \u2264 eVariationOn f s\n[PROOFSTEP]\nlet u : \u2115 \u2192 \u03b1 := fun n => if n = 0 then y else x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\nhxy : y \u2264 x\nu : \u2115 \u2192 \u03b1 := fun n => if n = 0 then y else x\n\u22a2 edist (f x) (f y) \u2264 eVariationOn f s\n[PROOFSTEP]\nhave hu : Monotone u :=\n monotone_nat_of_le_succ fun\n | 0 => hxy\n | (_ + 1) => le_rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\nhxy : y \u2264 x\nu : \u2115 \u2192 \u03b1 := fun n => if n = 0 then y else x\nhu : Monotone u\n\u22a2 edist (f x) (f y) \u2264 eVariationOn f s\n[PROOFSTEP]\nhave us : \u2200 i, u i \u2208 s := fun\n | 0 => hy\n | (_ + 1) => hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\nhxy : y \u2264 x\nu : \u2115 \u2192 \u03b1 := fun n => if n = 0 then y else x\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\n\u22a2 edist (f x) (f y) \u2264 eVariationOn f s\n[PROOFSTEP]\nsimpa only [Finset.sum_range_one] using sum_le f 1 hu us\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\n\u22a2 eVariationOn f s = 0 \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 edist (f x) (f y) = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\n\u22a2 eVariationOn f s = 0 \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 edist (f x) (f y) = 0\n[PROOFSTEP]\nrintro h x xs y ys\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nh : eVariationOn f s = 0\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nys : y \u2208 s\n\u22a2 edist (f x) (f y) = 0\n[PROOFSTEP]\nrw [\u2190 le_zero_iff, \u2190 h]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nh : eVariationOn f s = 0\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nys : y \u2208 s\n\u22a2 edist (f x) (f y) \u2264 eVariationOn f s\n[PROOFSTEP]\nexact edist_le f xs ys\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 edist (f x) (f y) = 0) \u2192 eVariationOn f s = 0\n[PROOFSTEP]\nrintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 edist (f x) (f y) = 0\n\u22a2 eVariationOn f s = 0\n[PROOFSTEP]\ndsimp only [eVariationOn]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 edist (f x) (f y) = 0\n\u22a2 \u2a06 (p : \u2115 \u00d7 { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }),\n \u2211 i in Finset.range p.fst, edist (f (\u2191p.snd (i + 1))) (f (\u2191p.snd i)) =\n 0\n[PROOFSTEP]\nrw [ENNReal.iSup_eq_zero]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 edist (f x) (f y) = 0\n\u22a2 \u2200 (i : \u2115 \u00d7 { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }),\n \u2211 i_1 in Finset.range i.fst, edist (f (\u2191i.snd (i_1 + 1))) (f (\u2191i.snd i_1)) = 0\n[PROOFSTEP]\nrintro \u27e8n, u, um, us\u27e9\n[GOAL]\ncase mpr.mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 edist (f x) (f y) = 0\nn : \u2115\nu : \u2115 \u2192 \u03b1\num : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\n\u22a2 \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd i)) =\n 0\n[PROOFSTEP]\nexact Finset.sum_eq_zero fun i _ => h _ (us i.succ) _ (us i)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : Set.Subsingleton (f '' s)\n\u22a2 eVariationOn f s = 0\n[PROOFSTEP]\nrw [eq_zero_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : Set.Subsingleton (f '' s)\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 edist (f x) (f y) = 0\n[PROOFSTEP]\nrintro x xs y ys\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : Set.Subsingleton (f '' s)\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nys : y \u2208 s\n\u22a2 edist (f x) (f y) = 0\n[PROOFSTEP]\nrw [hf \u27e8x, xs, rfl\u27e9 \u27e8y, ys, rfl\u27e9, edist_self]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\n\u03b9 : Type u_3\nF : \u03b9 \u2192 \u03b1 \u2192 E\np : Filter \u03b9\nf : \u03b1 \u2192 E\ns : Set \u03b1\nFfs : \u2200 (x : \u03b1), x \u2208 s \u2192 Tendsto (fun i => F i x) p (\ud835\udcdd (f x))\nv : \u211d\u22650\u221e\nhv : v < eVariationOn f s\n\u22a2 \u2200\u1da0 (n : \u03b9) in p, v < eVariationOn (F n) s\n[PROOFSTEP]\nobtain \u27e8\u27e8n, \u27e8u, um, us\u27e9\u27e9, hlt\u27e9 :\n \u2203 p : \u2115 \u00d7 { u : \u2115 \u2192 \u03b1 // Monotone u \u2227 \u2200 i, u i \u2208 s },\n v < \u2211 i in Finset.range p.1, edist (f ((p.2 : \u2115 \u2192 \u03b1) (i + 1))) (f ((p.2 : \u2115 \u2192 \u03b1) i)) :=\n lt_iSup_iff.mp hv\n[GOAL]\ncase intro.mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\n\u03b9 : Type u_3\nF : \u03b9 \u2192 \u03b1 \u2192 E\np : Filter \u03b9\nf : \u03b1 \u2192 E\ns : Set \u03b1\nFfs : \u2200 (x : \u03b1), x \u2208 s \u2192 Tendsto (fun i => F i x) p (\ud835\udcdd (f x))\nv : \u211d\u22650\u221e\nhv : v < eVariationOn f s\nn : \u2115\nu : \u2115 \u2192 \u03b1\num : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nhlt :\n v <\n \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd i))\n\u22a2 \u2200\u1da0 (n : \u03b9) in p, v < eVariationOn (F n) s\n[PROOFSTEP]\nhave :\n Tendsto (fun j => \u2211 i : \u2115 in Finset.range n, edist (F j (u (i + 1))) (F j (u i))) p\n (\ud835\udcdd (\u2211 i : \u2115 in Finset.range n, edist (f (u (i + 1))) (f (u i)))) :=\n by\n apply tendsto_finset_sum\n exact fun i _ => Tendsto.edist (Ffs (u i.succ) (us i.succ)) (Ffs (u i) (us i))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\n\u03b9 : Type u_3\nF : \u03b9 \u2192 \u03b1 \u2192 E\np : Filter \u03b9\nf : \u03b1 \u2192 E\ns : Set \u03b1\nFfs : \u2200 (x : \u03b1), x \u2208 s \u2192 Tendsto (fun i => F i x) p (\ud835\udcdd (f x))\nv : \u211d\u22650\u221e\nhv : v < eVariationOn f s\nn : \u2115\nu : \u2115 \u2192 \u03b1\num : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nhlt :\n v <\n \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd i))\n\u22a2 Tendsto (fun j => \u2211 i in Finset.range n, edist (F j (u (i + 1))) (F j (u i))) p\n (\ud835\udcdd (\u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i))))\n[PROOFSTEP]\napply tendsto_finset_sum\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\n\u03b9 : Type u_3\nF : \u03b9 \u2192 \u03b1 \u2192 E\np : Filter \u03b9\nf : \u03b1 \u2192 E\ns : Set \u03b1\nFfs : \u2200 (x : \u03b1), x \u2208 s \u2192 Tendsto (fun i => F i x) p (\ud835\udcdd (f x))\nv : \u211d\u22650\u221e\nhv : v < eVariationOn f s\nn : \u2115\nu : \u2115 \u2192 \u03b1\num : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nhlt :\n v <\n \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd i))\n\u22a2 \u2200 (i : \u2115),\n i \u2208 Finset.range n \u2192 Tendsto (fun b => edist (F b (u (i + 1))) (F b (u i))) p (\ud835\udcdd (edist (f (u (i + 1))) (f (u i))))\n[PROOFSTEP]\nexact fun i _ => Tendsto.edist (Ffs (u i.succ) (us i.succ)) (Ffs (u i) (us i))\n[GOAL]\ncase intro.mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\n\u03b9 : Type u_3\nF : \u03b9 \u2192 \u03b1 \u2192 E\np : Filter \u03b9\nf : \u03b1 \u2192 E\ns : Set \u03b1\nFfs : \u2200 (x : \u03b1), x \u2208 s \u2192 Tendsto (fun i => F i x) p (\ud835\udcdd (f x))\nv : \u211d\u22650\u221e\nhv : v < eVariationOn f s\nn : \u2115\nu : \u2115 \u2192 \u03b1\num : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nhlt :\n v <\n \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd i))\nthis :\n Tendsto (fun j => \u2211 i in Finset.range n, edist (F j (u (i + 1))) (F j (u i))) p\n (\ud835\udcdd (\u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i))))\n\u22a2 \u2200\u1da0 (n : \u03b9) in p, v < eVariationOn (F n) s\n[PROOFSTEP]\nexact (eventually_gt_of_tendsto_gt hlt this).mono fun i h => h.trans_le (sum_le (F i) n um us)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\ns : Set \u03b1\nf : \u03b1 \u2192\u1d64[singleton '' s] E\n\u22a2 LowerSemicontinuousAt (fun f => eVariationOn f s) f\n[PROOFSTEP]\napply @lowerSemicontinuous_aux _ _ _ _ (UniformOnFun \u03b1 E (s.image singleton)) id (\ud835\udcdd f) f s _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\ns : Set \u03b1\nf : \u03b1 \u2192\u1d64[singleton '' s] E\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 Tendsto (fun i => id i x) (\ud835\udcdd f) (\ud835\udcdd (f x))\n[PROOFSTEP]\nsimpa only [UniformOnFun.tendsto_iff_tendstoUniformlyOn, mem_image, forall_exists_index, and_imp,\n forall_apply_eq_imp_iff\u2082, tendstoUniformlyOn_singleton_iff_tendsto] using @tendsto_id _ (\ud835\udcdd f)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\ns : Set \u03b1\nf : \u03b1 \u2192\u1d64[{s}] E\n\u22a2 LowerSemicontinuousAt (fun f => eVariationOn f s) f\n[PROOFSTEP]\napply @lowerSemicontinuous_aux _ _ _ _ (UniformOnFun \u03b1 E { s }) id (\ud835\udcdd f) f s _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\ns : Set \u03b1\nf : \u03b1 \u2192\u1d64[{s}] E\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 Tendsto (fun i => id i x) (\ud835\udcdd f) (\ud835\udcdd (f x))\n[PROOFSTEP]\nhave := @tendsto_id _ (\ud835\udcdd f)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\ns : Set \u03b1\nf : \u03b1 \u2192\u1d64[{s}] E\nthis : Tendsto id (\ud835\udcdd f) (\ud835\udcdd f)\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 Tendsto (fun i => id i x) (\ud835\udcdd f) (\ud835\udcdd (f x))\n[PROOFSTEP]\nrw [UniformOnFun.tendsto_iff_tendstoUniformlyOn] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\ns : Set \u03b1\nf : \u03b1 \u2192\u1d64[{s}] E\nthis : \u2200 (s_1 : Set \u03b1), s_1 \u2208 {s} \u2192 TendstoUniformlyOn id f (\ud835\udcdd f) s_1\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 Tendsto (fun i => id i x) (\ud835\udcdd f) (\ud835\udcdd (f x))\n[PROOFSTEP]\nsimp_rw [\u2190 tendstoUniformlyOn_singleton_iff_tendsto]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\ns : Set \u03b1\nf : \u03b1 \u2192\u1d64[{s}] E\nthis : \u2200 (s_1 : Set \u03b1), s_1 \u2208 {s} \u2192 TendstoUniformlyOn id f (\ud835\udcdd f) s_1\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 TendstoUniformlyOn (fun n => id n) f (\ud835\udcdd f) {x}\n[PROOFSTEP]\nexact fun x xs => (this s rfl).mono (singleton_subset_iff.mpr xs)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE\u271d : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\u271d\nE : Type u_3\ninst\u271d : PseudoMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nh : BoundedVariationOn f s\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 dist (f x) (f y) \u2264 ENNReal.toReal (eVariationOn f s)\n[PROOFSTEP]\nrw [\u2190 ENNReal.ofReal_le_ofReal_iff ENNReal.toReal_nonneg, ENNReal.ofReal_toReal h, \u2190 edist_dist]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE\u271d : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\u271d\nE : Type u_3\ninst\u271d : PseudoMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nh : BoundedVariationOn f s\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 edist (f x) (f y) \u2264 eVariationOn f s\n[PROOFSTEP]\nexact edist_le f hx hy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nh : BoundedVariationOn f s\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 f x - f y \u2264 ENNReal.toReal (eVariationOn f s)\n[PROOFSTEP]\napply (le_abs_self _).trans\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nh : BoundedVariationOn f s\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 |f x - f y| \u2264 ENNReal.toReal (eVariationOn f s)\n[PROOFSTEP]\nrw [\u2190 Real.dist_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nh : BoundedVariationOn f s\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 dist (f x) (f y) \u2264 ENNReal.toReal (eVariationOn f s)\n[PROOFSTEP]\nexact h.dist_le hx hy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\n\u22a2 \u2203 v m,\n Monotone v \u2227\n (\u2200 (i : \u2115), v i \u2208 s) \u2227\n x \u2208 v '' Iio m \u2227\n \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264\n \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n[PROOFSTEP]\nrcases le_or_lt (u n) x with (h | h)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\n\u22a2 \u2203 v m,\n Monotone v \u2227\n (\u2200 (i : \u2115), v i \u2208 s) \u2227\n x \u2208 v '' Iio m \u2227\n \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264\n \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n[PROOFSTEP]\nlet v i := if i \u2264 n then u i else x\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\n\u22a2 \u2203 v m,\n Monotone v \u2227\n (\u2200 (i : \u2115), v i \u2208 s) \u2227\n x \u2208 v '' Iio m \u2227\n \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264\n \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n[PROOFSTEP]\nhave vs : \u2200 i, v i \u2208 s := fun i \u21a6 by\n simp only\n split_ifs\n \u00b7 exact us i\n \u00b7 exact hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\ni : \u2115\n\u22a2 v i \u2208 s\n[PROOFSTEP]\nsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\ni : \u2115\n\u22a2 (if i \u2264 n then u i else x) \u2208 s\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\ni : \u2115\nh\u271d : i \u2264 n\n\u22a2 u i \u2208 s\n[PROOFSTEP]\nexact us i\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\ni : \u2115\nh\u271d : \u00aci \u2264 n\n\u22a2 x \u2208 s\n[PROOFSTEP]\nexact hx\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\n\u22a2 \u2203 v m,\n Monotone v \u2227\n (\u2200 (i : \u2115), v i \u2208 s) \u2227\n x \u2208 v '' Iio m \u2227\n \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264\n \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n[PROOFSTEP]\nhave hv : Monotone v := by\n refine monotone_nat_of_le_succ fun i => ?_\n simp only\n rcases lt_trichotomy i n with (hi | rfl | hi)\n \u00b7 have : i + 1 \u2264 n := Nat.succ_le_of_lt hi\n simp only [hi.le, this, if_true]\n exact hu (Nat.le_succ i)\n \u00b7 simp only [le_refl, if_true, add_le_iff_nonpos_right, le_zero_iff, Nat.one_ne_zero, if_false, h]\n \u00b7 have A : \u00aci \u2264 n := hi.not_le\n have B : \u00aci + 1 \u2264 n := fun h => A (i.le_succ.trans h)\n simp only [A, B, if_false, le_rfl]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\n\u22a2 Monotone v\n[PROOFSTEP]\nrefine monotone_nat_of_le_succ fun i => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\ni : \u2115\n\u22a2 v i \u2264 v (i + 1)\n[PROOFSTEP]\nsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\ni : \u2115\n\u22a2 (if i \u2264 n then u i else x) \u2264 if i + 1 \u2264 n then u (i + 1) else x\n[PROOFSTEP]\nrcases lt_trichotomy i n with (hi | rfl | hi)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\ni : \u2115\nhi : i < n\n\u22a2 (if i \u2264 n then u i else x) \u2264 if i + 1 \u2264 n then u (i + 1) else x\n[PROOFSTEP]\nhave : i + 1 \u2264 n := Nat.succ_le_of_lt hi\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\ni : \u2115\nhi : i < n\nthis : i + 1 \u2264 n\n\u22a2 (if i \u2264 n then u i else x) \u2264 if i + 1 \u2264 n then u (i + 1) else x\n[PROOFSTEP]\nsimp only [hi.le, this, if_true]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\ni : \u2115\nhi : i < n\nthis : i + 1 \u2264 n\n\u22a2 u i \u2264 u (i + 1)\n[PROOFSTEP]\nexact hu (Nat.le_succ i)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\ni : \u2115\nh : u i \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i_1 => if i_1 \u2264 i then u i_1 else x\nvs : \u2200 (i : \u2115), v i \u2208 s\n\u22a2 (if i \u2264 i then u i else x) \u2264 if i + 1 \u2264 i then u (i + 1) else x\n[PROOFSTEP]\nsimp only [le_refl, if_true, add_le_iff_nonpos_right, le_zero_iff, Nat.one_ne_zero, if_false, h]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\ni : \u2115\nhi : n < i\n\u22a2 (if i \u2264 n then u i else x) \u2264 if i + 1 \u2264 n then u (i + 1) else x\n[PROOFSTEP]\nhave A : \u00aci \u2264 n := hi.not_le\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\ni : \u2115\nhi : n < i\nA : \u00aci \u2264 n\n\u22a2 (if i \u2264 n then u i else x) \u2264 if i + 1 \u2264 n then u (i + 1) else x\n[PROOFSTEP]\nhave B : \u00aci + 1 \u2264 n := fun h => A (i.le_succ.trans h)\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\ni : \u2115\nhi : n < i\nA : \u00aci \u2264 n\nB : \u00aci + 1 \u2264 n\n\u22a2 (if i \u2264 n then u i else x) \u2264 if i + 1 \u2264 n then u (i + 1) else x\n[PROOFSTEP]\nsimp only [A, B, if_false, le_rfl]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\nhv : Monotone v\n\u22a2 \u2203 v m,\n Monotone v \u2227\n (\u2200 (i : \u2115), v i \u2208 s) \u2227\n x \u2208 v '' Iio m \u2227\n \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264\n \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n[PROOFSTEP]\nrefine' \u27e8v, n + 2, hv, vs, (mem_image _ _ _).2 \u27e8n + 1, _, _\u27e9, _\u27e9\n[GOAL]\ncase inl.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\nhv : Monotone v\n\u22a2 n + 1 \u2208 Iio (n + 2)\n[PROOFSTEP]\nrw [mem_Iio]\n[GOAL]\ncase inl.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\nhv : Monotone v\n\u22a2 n + 1 < n + 2\n[PROOFSTEP]\nexact Nat.lt_succ_self (n + 1)\n[GOAL]\ncase inl.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\nhv : Monotone v\n\u22a2 v (n + 1) = x\n[PROOFSTEP]\nhave : \u00acn + 1 \u2264 n := Nat.not_succ_le_self n\n[GOAL]\ncase inl.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\nhv : Monotone v\nthis : \u00acn + 1 \u2264 n\n\u22a2 v (n + 1) = x\n[PROOFSTEP]\nsimp only [this, ite_eq_right_iff, IsEmpty.forall_iff]\n[GOAL]\ncase inl.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\nhv : Monotone v\n\u22a2 \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range (n + 2), edist (f (v (j + 1))) (f (v j))\n[PROOFSTEP]\ncalc\n (\u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i))) = \u2211 i in Finset.range n, edist (f (v (i + 1))) (f (v i)) :=\n by\n apply Finset.sum_congr rfl fun i hi => ?_\n simp only [Finset.mem_range] at hi \n have : i + 1 \u2264 n := Nat.succ_le_of_lt hi\n simp only [hi.le, this, if_true]\n _ \u2264 \u2211 j in Finset.range (n + 2), edist (f (v (j + 1))) (f (v j)) :=\n Finset.sum_le_sum_of_subset (Finset.range_mono (Nat.le_add_right n 2))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\nhv : Monotone v\n\u22a2 \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) = \u2211 i in Finset.range n, edist (f (v (i + 1))) (f (v i))\n[PROOFSTEP]\napply Finset.sum_congr rfl fun i hi => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\nhv : Monotone v\ni : \u2115\nhi : i \u2208 Finset.range n\n\u22a2 edist (f (u (i + 1))) (f (u i)) = edist (f (v (i + 1))) (f (v i))\n[PROOFSTEP]\nsimp only [Finset.mem_range] at hi \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\nhv : Monotone v\ni : \u2115\nhi : i < n\n\u22a2 edist (f (u (i + 1))) (f (u i)) = edist (f (v (i + 1))) (f (v i))\n[PROOFSTEP]\nhave : i + 1 \u2264 n := Nat.succ_le_of_lt hi\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : u n \u2264 x\nv : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else x\nvs : \u2200 (i : \u2115), v i \u2208 s\nhv : Monotone v\ni : \u2115\nhi : i < n\nthis : i + 1 \u2264 n\n\u22a2 edist (f (u (i + 1))) (f (u i)) = edist (f (v (i + 1))) (f (v i))\n[PROOFSTEP]\nsimp only [hi.le, this, if_true]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\n\u22a2 \u2203 v m,\n Monotone v \u2227\n (\u2200 (i : \u2115), v i \u2208 s) \u2227\n x \u2208 v '' Iio m \u2227\n \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264\n \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n[PROOFSTEP]\nhave exists_N : \u2203 N, N \u2264 n \u2227 x < u N := \u27e8n, le_rfl, h\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\n\u22a2 \u2203 v m,\n Monotone v \u2227\n (\u2200 (i : \u2115), v i \u2208 s) \u2227\n x \u2208 v '' Iio m \u2227\n \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264\n \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n[PROOFSTEP]\nlet N := Nat.find exists_N\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\n\u22a2 \u2203 v m,\n Monotone v \u2227\n (\u2200 (i : \u2115), v i \u2208 s) \u2227\n x \u2208 v '' Iio m \u2227\n \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264\n \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n[PROOFSTEP]\nhave hN : N \u2264 n \u2227 x < u N := Nat.find_spec exists_N\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\n\u22a2 \u2203 v m,\n Monotone v \u2227\n (\u2200 (i : \u2115), v i \u2208 s) \u2227\n x \u2208 v '' Iio m \u2227\n \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264\n \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n[PROOFSTEP]\nlet w : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\n\u22a2 \u2203 v m,\n Monotone v \u2227\n (\u2200 (i : \u2115), v i \u2208 s) \u2227\n x \u2208 v '' Iio m \u2227\n \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264\n \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n[PROOFSTEP]\nhave ws : \u2200 i, w i \u2208 s := by\n dsimp only\n intro i\n split_ifs\n exacts [us _, hx, us _]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\n\u22a2 \u2200 (i : \u2115), w i \u2208 s\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\n\u22a2 \u2200 (i : \u2115), (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)) \u2208 s\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\ni : \u2115\n\u22a2 (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)) \u2208 s\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\ni : \u2115\nh\u271d : i < Nat.find exists_N\n\u22a2 u i \u2208 s\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\ni : \u2115\nh\u271d\u00b9 : \u00aci < Nat.find exists_N\nh\u271d : i = Nat.find exists_N\n\u22a2 x \u2208 s\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\ni : \u2115\nh\u271d\u00b9 : \u00aci < Nat.find exists_N\nh\u271d : \u00aci = Nat.find exists_N\n\u22a2 u (i - 1) \u2208 s\n[PROOFSTEP]\nexacts [us _, hx, us _]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\n\u22a2 \u2203 v m,\n Monotone v \u2227\n (\u2200 (i : \u2115), v i \u2208 s) \u2227\n x \u2208 v '' Iio m \u2227\n \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264\n \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n[PROOFSTEP]\nhave hw : Monotone w := by\n apply monotone_nat_of_le_succ fun i => ?_\n dsimp only\n rcases lt_trichotomy (i + 1) N with (hi | hi | hi)\n \u00b7 have : i < N := Nat.lt_of_le_of_lt (Nat.le_succ i) hi\n simp only [hi, this, if_true]\n exact hu (Nat.le_succ _)\n \u00b7 have A : i < N := hi \u25b8 i.lt_succ_self\n have B : \u00aci + 1 < N := by rw [\u2190 hi]; exact fun h => h.ne rfl\n rw [if_pos A, if_neg B, if_pos hi]\n have T := Nat.find_min exists_N A\n push_neg at T \n exact T (A.le.trans hN.1)\n \u00b7 have A : \u00aci < N := (Nat.lt_succ_iff.mp hi).not_lt\n have B : \u00aci + 1 < N := hi.not_lt\n have C : \u00aci + 1 = N := hi.ne.symm\n have D : i + 1 - 1 = i := Nat.pred_succ i\n rw [if_neg A, if_neg B, if_neg C, D]\n split_ifs\n \u00b7 exact hN.2.le.trans (hu (le_of_not_lt A))\n \u00b7 exact hu (Nat.pred_le _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\n\u22a2 Monotone w\n[PROOFSTEP]\napply monotone_nat_of_le_succ fun i => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\n\u22a2 w i \u2264 w (i + 1)\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\n\u22a2 (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)) \u2264\n if i + 1 < Nat.find exists_N then u (i + 1) else if i + 1 = Nat.find exists_N then x else u (i + 1 - 1)\n[PROOFSTEP]\nrcases lt_trichotomy (i + 1) N with (hi | hi | hi)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : i + 1 < N\n\u22a2 (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)) \u2264\n if i + 1 < Nat.find exists_N then u (i + 1) else if i + 1 = Nat.find exists_N then x else u (i + 1 - 1)\n[PROOFSTEP]\nhave : i < N := Nat.lt_of_le_of_lt (Nat.le_succ i) hi\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : i + 1 < N\nthis : i < N\n\u22a2 (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)) \u2264\n if i + 1 < Nat.find exists_N then u (i + 1) else if i + 1 = Nat.find exists_N then x else u (i + 1 - 1)\n[PROOFSTEP]\nsimp only [hi, this, if_true]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : i + 1 < N\nthis : i < N\n\u22a2 u i \u2264 u (i + 1)\n[PROOFSTEP]\nexact hu (Nat.le_succ _)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : i + 1 = N\n\u22a2 (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)) \u2264\n if i + 1 < Nat.find exists_N then u (i + 1) else if i + 1 = Nat.find exists_N then x else u (i + 1 - 1)\n[PROOFSTEP]\nhave A : i < N := hi \u25b8 i.lt_succ_self\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : i + 1 = N\nA : i < N\n\u22a2 (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)) \u2264\n if i + 1 < Nat.find exists_N then u (i + 1) else if i + 1 = Nat.find exists_N then x else u (i + 1 - 1)\n[PROOFSTEP]\nhave B : \u00aci + 1 < N := by rw [\u2190 hi]; exact fun h => h.ne rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : i + 1 = N\nA : i < N\n\u22a2 \u00aci + 1 < N\n[PROOFSTEP]\nrw [\u2190 hi]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : i + 1 = N\nA : i < N\n\u22a2 \u00aci + 1 < i + 1\n[PROOFSTEP]\nexact fun h => h.ne rfl\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : i + 1 = N\nA : i < N\nB : \u00aci + 1 < N\n\u22a2 (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)) \u2264\n if i + 1 < Nat.find exists_N then u (i + 1) else if i + 1 = Nat.find exists_N then x else u (i + 1 - 1)\n[PROOFSTEP]\nrw [if_pos A, if_neg B, if_pos hi]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : i + 1 = N\nA : i < N\nB : \u00aci + 1 < N\n\u22a2 u i \u2264 x\n[PROOFSTEP]\nhave T := Nat.find_min exists_N A\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : i + 1 = N\nA : i < N\nB : \u00aci + 1 < N\nT : \u00ac(i \u2264 n \u2227 x < u i)\n\u22a2 u i \u2264 x\n[PROOFSTEP]\npush_neg at T \n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : i + 1 = N\nA : i < N\nB : \u00aci + 1 < N\nT : i \u2264 n \u2192 u i \u2264 x\n\u22a2 u i \u2264 x\n[PROOFSTEP]\nexact T (A.le.trans hN.1)\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : N < i + 1\n\u22a2 (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)) \u2264\n if i + 1 < Nat.find exists_N then u (i + 1) else if i + 1 = Nat.find exists_N then x else u (i + 1 - 1)\n[PROOFSTEP]\nhave A : \u00aci < N := (Nat.lt_succ_iff.mp hi).not_lt\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : N < i + 1\nA : \u00aci < N\n\u22a2 (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)) \u2264\n if i + 1 < Nat.find exists_N then u (i + 1) else if i + 1 = Nat.find exists_N then x else u (i + 1 - 1)\n[PROOFSTEP]\nhave B : \u00aci + 1 < N := hi.not_lt\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : N < i + 1\nA : \u00aci < N\nB : \u00aci + 1 < N\n\u22a2 (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)) \u2264\n if i + 1 < Nat.find exists_N then u (i + 1) else if i + 1 = Nat.find exists_N then x else u (i + 1 - 1)\n[PROOFSTEP]\nhave C : \u00aci + 1 = N := hi.ne.symm\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : N < i + 1\nA : \u00aci < N\nB : \u00aci + 1 < N\nC : \u00aci + 1 = N\n\u22a2 (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)) \u2264\n if i + 1 < Nat.find exists_N then u (i + 1) else if i + 1 = Nat.find exists_N then x else u (i + 1 - 1)\n[PROOFSTEP]\nhave D : i + 1 - 1 = i := Nat.pred_succ i\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : N < i + 1\nA : \u00aci < N\nB : \u00aci + 1 < N\nC : \u00aci + 1 = N\nD : i + 1 - 1 = i\n\u22a2 (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)) \u2264\n if i + 1 < Nat.find exists_N then u (i + 1) else if i + 1 = Nat.find exists_N then x else u (i + 1 - 1)\n[PROOFSTEP]\nrw [if_neg A, if_neg B, if_neg C, D]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : N < i + 1\nA : \u00aci < N\nB : \u00aci + 1 < N\nC : \u00aci + 1 = N\nD : i + 1 - 1 = i\n\u22a2 (if i = Nat.find exists_N then x else u (i - 1)) \u2264 u i\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : N < i + 1\nA : \u00aci < N\nB : \u00aci + 1 < N\nC : \u00aci + 1 = N\nD : i + 1 - 1 = i\nh\u271d : i = Nat.find exists_N\n\u22a2 x \u2264 u i\n[PROOFSTEP]\nexact hN.2.le.trans (hu (le_of_not_lt A))\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\ni : \u2115\nhi : N < i + 1\nA : \u00aci < N\nB : \u00aci + 1 < N\nC : \u00aci + 1 = N\nD : i + 1 - 1 = i\nh\u271d : \u00aci = Nat.find exists_N\n\u22a2 u (i - 1) \u2264 u i\n[PROOFSTEP]\nexact hu (Nat.pred_le _)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\n\u22a2 \u2203 v m,\n Monotone v \u2227\n (\u2200 (i : \u2115), v i \u2208 s) \u2227\n x \u2208 v '' Iio m \u2227\n \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264\n \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n[PROOFSTEP]\nrefine' \u27e8w, n + 1, hw, ws, (mem_image _ _ _).2 \u27e8N, hN.1.trans_lt (Nat.lt_succ_self n), _\u27e9, _\u27e9\n[GOAL]\ncase inr.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\n\u22a2 w N = x\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase inr.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\n\u22a2 (if Nat.find exists_N < Nat.find exists_N then u (Nat.find exists_N)\n else if Nat.find exists_N = Nat.find exists_N then x else u (Nat.find exists_N - 1)) =\n x\n[PROOFSTEP]\nrw [if_neg (lt_irrefl N), if_pos rfl]\n[GOAL]\ncase inr.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\n\u22a2 \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range (n + 1), edist (f (w (j + 1))) (f (w j))\n[PROOFSTEP]\nrcases eq_or_lt_of_le (zero_le N) with (Npos | Npos)\n[GOAL]\ncase inr.refine'_2.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 = N\n\u22a2 \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range (n + 1), edist (f (w (j + 1))) (f (w j))\n[PROOFSTEP]\ncalc\n (\u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i))) =\n \u2211 i in Finset.range n, edist (f (w (1 + i + 1))) (f (w (1 + i))) :=\n by\n apply Finset.sum_congr rfl fun i _hi => ?_\n dsimp only\n simp only [\u2190 Npos, Nat.not_lt_zero, Nat.add_succ_sub_one, add_zero, if_false, add_eq_zero_iff, Nat.one_ne_zero,\n false_and_iff, Nat.succ_add_sub_one, zero_add]\n rw [add_comm 1 i]\n _ = \u2211 i in Finset.Ico 1 (n + 1), edist (f (w (i + 1))) (f (w i)) :=\n by\n rw [Finset.range_eq_Ico]\n exact Finset.sum_Ico_add (fun i => edist (f (w (i + 1))) (f (w i))) 0 n 1\n _ \u2264 \u2211 j in Finset.range (n + 1), edist (f (w (j + 1))) (f (w j)) :=\n by\n apply Finset.sum_le_sum_of_subset _\n rw [Finset.range_eq_Ico]\n exact Finset.Ico_subset_Ico zero_le_one le_rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 = N\n\u22a2 \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) =\n \u2211 i in Finset.range n, edist (f (w (1 + i + 1))) (f (w (1 + i)))\n[PROOFSTEP]\napply Finset.sum_congr rfl fun i _hi => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 = N\ni : \u2115\n_hi : i \u2208 Finset.range n\n\u22a2 edist (f (u (i + 1))) (f (u i)) = edist (f (w (1 + i + 1))) (f (w (1 + i)))\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 = N\ni : \u2115\n_hi : i \u2208 Finset.range n\n\u22a2 edist (f (u (i + 1))) (f (u i)) =\n edist\n (f\n (if 1 + i + 1 < Nat.find exists_N then u (1 + i + 1)\n else if 1 + i + 1 = Nat.find exists_N then x else u (1 + i + 1 - 1)))\n (f (if 1 + i < Nat.find exists_N then u (1 + i) else if 1 + i = Nat.find exists_N then x else u (1 + i - 1)))\n[PROOFSTEP]\nsimp only [\u2190 Npos, Nat.not_lt_zero, Nat.add_succ_sub_one, add_zero, if_false, add_eq_zero_iff, Nat.one_ne_zero,\n false_and_iff, Nat.succ_add_sub_one, zero_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 = N\ni : \u2115\n_hi : i \u2208 Finset.range n\n\u22a2 edist (f (u (i + 1))) (f (u i)) = edist (f (u (1 + i))) (f (u i))\n[PROOFSTEP]\nrw [add_comm 1 i]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 = N\n\u22a2 \u2211 i in Finset.range n, edist (f (w (1 + i + 1))) (f (w (1 + i))) =\n \u2211 i in Finset.Ico 1 (n + 1), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\nrw [Finset.range_eq_Ico]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 = N\n\u22a2 \u2211 i in Finset.Ico 0 n, edist (f (w (1 + i + 1))) (f (w (1 + i))) =\n \u2211 i in Finset.Ico 1 (n + 1), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\nexact Finset.sum_Ico_add (fun i => edist (f (w (i + 1))) (f (w i))) 0 n 1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 = N\n\u22a2 \u2211 i in Finset.Ico 1 (n + 1), edist (f (w (i + 1))) (f (w i)) \u2264\n \u2211 j in Finset.range (n + 1), edist (f (w (j + 1))) (f (w j))\n[PROOFSTEP]\napply Finset.sum_le_sum_of_subset _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 = N\n\u22a2 Finset.Ico 1 (n + 1) \u2286 Finset.range (n + 1)\n[PROOFSTEP]\nrw [Finset.range_eq_Ico]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 = N\n\u22a2 Finset.Ico 1 (n + 1) \u2286 Finset.Ico 0 (n + 1)\n[PROOFSTEP]\nexact Finset.Ico_subset_Ico zero_le_one le_rfl\n[GOAL]\ncase inr.refine'_2.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range (n + 1), edist (f (w (j + 1))) (f (w j))\n[PROOFSTEP]\ncalc\n (\u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i))) =\n ((\u2211 i in Finset.Ico 0 (N - 1), edist (f (u (i + 1))) (f (u i))) +\n \u2211 i in Finset.Ico (N - 1) N, edist (f (u (i + 1))) (f (u i))) +\n \u2211 i in Finset.Ico N n, edist (f (u (i + 1))) (f (u i)) :=\n by\n rw [Finset.sum_Ico_consecutive, Finset.sum_Ico_consecutive, Finset.range_eq_Ico]\n \u00b7 exact zero_le _\n \u00b7 exact hN.1\n \u00b7 exact zero_le _\n \u00b7 exact Nat.pred_le _\n _ =\n (\u2211 i in Finset.Ico 0 (N - 1), edist (f (w (i + 1))) (f (w i))) + edist (f (u N)) (f (u (N - 1))) +\n \u2211 i in Finset.Ico N n, edist (f (w (1 + i + 1))) (f (w (1 + i))) :=\n by\n congr 1; congr 1\n \u00b7 apply Finset.sum_congr rfl fun i hi => ?_\n simp only [Finset.mem_Ico, zero_le', true_and_iff] at hi \n dsimp only\n have A : i + 1 < N := Nat.lt_pred_iff.1 hi\n have B : i < N := Nat.lt_of_succ_lt A\n rw [if_pos A, if_pos B]\n \u00b7 have A : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos\n have : Finset.Ico (N - 1) N = {N - 1} := by rw [\u2190 Nat.Ico_succ_singleton, A]\n simp only [this, A, Finset.sum_singleton]\n \u00b7 apply Finset.sum_congr rfl fun i hi => ?_\n rw [Finset.mem_Ico] at hi \n dsimp only\n have A : \u00ac1 + i + 1 < N := fun h => by\n rw [add_assoc, add_comm] at h \n exact hi.left.not_lt (i.lt_succ_self.trans (i.succ.lt_succ_self.trans h))\n have B : \u00ac1 + i + 1 = N := fun h => by\n rw [\u2190 h, add_assoc, add_comm] at hi \n exact Nat.not_succ_le_self i (i.succ.le_succ.trans hi.left)\n have C : \u00ac1 + i < N := fun h => by\n rw [add_comm] at h \n exact hi.left.not_lt (i.lt_succ_self.trans h)\n have D : \u00ac1 + i = N := fun h => by\n rw [\u2190 h, add_comm, Nat.succ_le_iff] at hi \n exact hi.left.ne rfl\n rw [if_neg A, if_neg B, if_neg C, if_neg D]\n congr 3 <;> \u00b7 rw [add_comm, Nat.sub_one]; apply Nat.pred_succ\n _ =\n (\u2211 i in Finset.Ico 0 (N - 1), edist (f (w (i + 1))) (f (w i))) + edist (f (w (N + 1))) (f (w (N - 1))) +\n \u2211 i in Finset.Ico (N + 1) (n + 1), edist (f (w (i + 1))) (f (w i)) :=\n by\n congr 1; congr 1\n \u00b7 dsimp only\n have A : \u00acN + 1 < N := Nat.not_succ_lt_self\n have B : N - 1 < N := Nat.pred_lt Npos.ne'\n simp only [A, not_and, not_lt, Nat.succ_ne_self, Nat.add_succ_sub_one, add_zero, if_false, B, if_true]\n \u00b7 exact Finset.sum_Ico_add (fun i => edist (f (w (i + 1))) (f (w i))) N n 1\n _ \u2264\n ((\u2211 i in Finset.Ico 0 (N - 1), edist (f (w (i + 1))) (f (w i))) +\n \u2211 i in Finset.Ico (N - 1) (N + 1), edist (f (w (i + 1))) (f (w i))) +\n \u2211 i in Finset.Ico (N + 1) (n + 1), edist (f (w (i + 1))) (f (w i)) :=\n by\n refine' add_le_add (add_le_add le_rfl _) le_rfl\n have A : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos\n have B : N - 1 + 1 < N + 1 := A.symm \u25b8 N.lt_succ_self\n have C : N - 1 < N + 1 := lt_of_le_of_lt N.pred_le N.lt_succ_self\n rw [Finset.sum_eq_sum_Ico_succ_bot C, Finset.sum_eq_sum_Ico_succ_bot B, A, Finset.Ico_self, Finset.sum_empty,\n add_zero, add_comm (edist _ _)]\n exact edist_triangle _ _ _\n _ = \u2211 j in Finset.range (n + 1), edist (f (w (j + 1))) (f (w j)) :=\n by\n rw [Finset.sum_Ico_consecutive, Finset.sum_Ico_consecutive, Finset.range_eq_Ico]\n \u00b7 exact zero_le _\n \u00b7 exact Nat.succ_le_succ hN.left\n \u00b7 exact zero_le _\n \u00b7 exact N.pred_le.trans N.le_succ\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) =\n \u2211 i in Finset.Ico 0 (N - 1), edist (f (u (i + 1))) (f (u i)) +\n \u2211 i in Finset.Ico (N - 1) N, edist (f (u (i + 1))) (f (u i)) +\n \u2211 i in Finset.Ico N n, edist (f (u (i + 1))) (f (u i))\n[PROOFSTEP]\nrw [Finset.sum_Ico_consecutive, Finset.sum_Ico_consecutive, Finset.range_eq_Ico]\n[GOAL]\ncase hmn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 0 \u2264 N\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase hnk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 N \u2264 n\n[PROOFSTEP]\nexact hN.1\n[GOAL]\ncase hmn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 0 \u2264 N - 1\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase hnk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 N - 1 \u2264 N\n[PROOFSTEP]\nexact Nat.pred_le _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 \u2211 i in Finset.Ico 0 (N - 1), edist (f (u (i + 1))) (f (u i)) +\n \u2211 i in Finset.Ico (N - 1) N, edist (f (u (i + 1))) (f (u i)) +\n \u2211 i in Finset.Ico N n, edist (f (u (i + 1))) (f (u i)) =\n \u2211 i in Finset.Ico 0 (N - 1), edist (f (w (i + 1))) (f (w i)) + edist (f (u N)) (f (u (N - 1))) +\n \u2211 i in Finset.Ico N n, edist (f (w (1 + i + 1))) (f (w (1 + i)))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 \u2211 i in Finset.Ico 0 (N - 1), edist (f (u (i + 1))) (f (u i)) +\n \u2211 i in Finset.Ico (N - 1) N, edist (f (u (i + 1))) (f (u i)) =\n \u2211 i in Finset.Ico 0 (N - 1), edist (f (w (i + 1))) (f (w i)) + edist (f (u N)) (f (u (N - 1)))\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 \u2211 i in Finset.Ico N n, edist (f (u (i + 1))) (f (u i)) =\n \u2211 i in Finset.Ico N n, edist (f (w (1 + i + 1))) (f (w (1 + i)))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 \u2211 i in Finset.Ico 0 (N - 1), edist (f (u (i + 1))) (f (u i)) =\n \u2211 i in Finset.Ico 0 (N - 1), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\napply Finset.sum_congr rfl fun i hi => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : i \u2208 Finset.Ico 0 (N - 1)\n\u22a2 edist (f (u (i + 1))) (f (u i)) = edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\nsimp only [Finset.mem_Ico, zero_le', true_and_iff] at hi \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : i < Nat.find exists_N - 1\n\u22a2 edist (f (u (i + 1))) (f (u i)) = edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : i < Nat.find exists_N - 1\n\u22a2 edist (f (u (i + 1))) (f (u i)) =\n edist (f (if i + 1 < Nat.find exists_N then u (i + 1) else if i + 1 = Nat.find exists_N then x else u (i + 1 - 1)))\n (f (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)))\n[PROOFSTEP]\nhave A : i + 1 < N := Nat.lt_pred_iff.1 hi\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : i < Nat.find exists_N - 1\nA : i + 1 < N\n\u22a2 edist (f (u (i + 1))) (f (u i)) =\n edist (f (if i + 1 < Nat.find exists_N then u (i + 1) else if i + 1 = Nat.find exists_N then x else u (i + 1 - 1)))\n (f (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)))\n[PROOFSTEP]\nhave B : i < N := Nat.lt_of_succ_lt A\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : i < Nat.find exists_N - 1\nA : i + 1 < N\nB : i < N\n\u22a2 edist (f (u (i + 1))) (f (u i)) =\n edist (f (if i + 1 < Nat.find exists_N then u (i + 1) else if i + 1 = Nat.find exists_N then x else u (i + 1 - 1)))\n (f (if i < Nat.find exists_N then u i else if i = Nat.find exists_N then x else u (i - 1)))\n[PROOFSTEP]\nrw [if_pos A, if_pos B]\n[GOAL]\ncase e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 \u2211 i in Finset.Ico (N - 1) N, edist (f (u (i + 1))) (f (u i)) = edist (f (u N)) (f (u (N - 1)))\n[PROOFSTEP]\nhave A : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos\n[GOAL]\ncase e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\nA : N - 1 + 1 = N\n\u22a2 \u2211 i in Finset.Ico (N - 1) N, edist (f (u (i + 1))) (f (u i)) = edist (f (u N)) (f (u (N - 1)))\n[PROOFSTEP]\nhave : Finset.Ico (N - 1) N = {N - 1} := by rw [\u2190 Nat.Ico_succ_singleton, A]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\nA : N - 1 + 1 = N\n\u22a2 Finset.Ico (N - 1) N = {N - 1}\n[PROOFSTEP]\nrw [\u2190 Nat.Ico_succ_singleton, A]\n[GOAL]\ncase e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\nA : N - 1 + 1 = N\nthis : Finset.Ico (N - 1) N = {N - 1}\n\u22a2 \u2211 i in Finset.Ico (N - 1) N, edist (f (u (i + 1))) (f (u i)) = edist (f (u N)) (f (u (N - 1)))\n[PROOFSTEP]\nsimp only [this, A, Finset.sum_singleton]\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 \u2211 i in Finset.Ico N n, edist (f (u (i + 1))) (f (u i)) =\n \u2211 i in Finset.Ico N n, edist (f (w (1 + i + 1))) (f (w (1 + i)))\n[PROOFSTEP]\napply Finset.sum_congr rfl fun i hi => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : i \u2208 Finset.Ico N n\n\u22a2 edist (f (u (i + 1))) (f (u i)) = edist (f (w (1 + i + 1))) (f (w (1 + i)))\n[PROOFSTEP]\nrw [Finset.mem_Ico] at hi \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\n\u22a2 edist (f (u (i + 1))) (f (u i)) = edist (f (w (1 + i + 1))) (f (w (1 + i)))\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\n\u22a2 edist (f (u (i + 1))) (f (u i)) =\n edist\n (f\n (if 1 + i + 1 < Nat.find exists_N then u (1 + i + 1)\n else if 1 + i + 1 = Nat.find exists_N then x else u (1 + i + 1 - 1)))\n (f (if 1 + i < Nat.find exists_N then u (1 + i) else if 1 + i = Nat.find exists_N then x else u (1 + i - 1)))\n[PROOFSTEP]\nhave A : \u00ac1 + i + 1 < N := fun h => by\n rw [add_assoc, add_comm] at h \n exact hi.left.not_lt (i.lt_succ_self.trans (i.succ.lt_succ_self.trans h))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh\u271d : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\nh : 1 + i + 1 < N\n\u22a2 False\n[PROOFSTEP]\nrw [add_assoc, add_comm] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh\u271d : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\nh : i + 1 + 1 < N\n\u22a2 False\n[PROOFSTEP]\nexact hi.left.not_lt (i.lt_succ_self.trans (i.succ.lt_succ_self.trans h))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\nA : \u00ac1 + i + 1 < N\n\u22a2 edist (f (u (i + 1))) (f (u i)) =\n edist\n (f\n (if 1 + i + 1 < Nat.find exists_N then u (1 + i + 1)\n else if 1 + i + 1 = Nat.find exists_N then x else u (1 + i + 1 - 1)))\n (f (if 1 + i < Nat.find exists_N then u (1 + i) else if 1 + i = Nat.find exists_N then x else u (1 + i - 1)))\n[PROOFSTEP]\nhave B : \u00ac1 + i + 1 = N := fun h => by\n rw [\u2190 h, add_assoc, add_comm] at hi \n exact Nat.not_succ_le_self i (i.succ.le_succ.trans hi.left)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh\u271d : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\nA : \u00ac1 + i + 1 < N\nh : 1 + i + 1 = N\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 h, add_assoc, add_comm] at hi \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh\u271d : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : i + 1 + 1 \u2264 i \u2227 i < n\nA : \u00ac1 + i + 1 < N\nh : 1 + i + 1 = N\n\u22a2 False\n[PROOFSTEP]\nexact Nat.not_succ_le_self i (i.succ.le_succ.trans hi.left)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\nA : \u00ac1 + i + 1 < N\nB : \u00ac1 + i + 1 = N\n\u22a2 edist (f (u (i + 1))) (f (u i)) =\n edist\n (f\n (if 1 + i + 1 < Nat.find exists_N then u (1 + i + 1)\n else if 1 + i + 1 = Nat.find exists_N then x else u (1 + i + 1 - 1)))\n (f (if 1 + i < Nat.find exists_N then u (1 + i) else if 1 + i = Nat.find exists_N then x else u (1 + i - 1)))\n[PROOFSTEP]\nhave C : \u00ac1 + i < N := fun h => by\n rw [add_comm] at h \n exact hi.left.not_lt (i.lt_succ_self.trans h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh\u271d : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\nA : \u00ac1 + i + 1 < N\nB : \u00ac1 + i + 1 = N\nh : 1 + i < N\n\u22a2 False\n[PROOFSTEP]\nrw [add_comm] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh\u271d : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\nA : \u00ac1 + i + 1 < N\nB : \u00ac1 + i + 1 = N\nh : i + 1 < N\n\u22a2 False\n[PROOFSTEP]\nexact hi.left.not_lt (i.lt_succ_self.trans h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\nA : \u00ac1 + i + 1 < N\nB : \u00ac1 + i + 1 = N\nC : \u00ac1 + i < N\n\u22a2 edist (f (u (i + 1))) (f (u i)) =\n edist\n (f\n (if 1 + i + 1 < Nat.find exists_N then u (1 + i + 1)\n else if 1 + i + 1 = Nat.find exists_N then x else u (1 + i + 1 - 1)))\n (f (if 1 + i < Nat.find exists_N then u (1 + i) else if 1 + i = Nat.find exists_N then x else u (1 + i - 1)))\n[PROOFSTEP]\nhave D : \u00ac1 + i = N := fun h => by\n rw [\u2190 h, add_comm, Nat.succ_le_iff] at hi \n exact hi.left.ne rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh\u271d : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\nA : \u00ac1 + i + 1 < N\nB : \u00ac1 + i + 1 = N\nC : \u00ac1 + i < N\nh : 1 + i = N\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 h, add_comm, Nat.succ_le_iff] at hi \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh\u271d : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : i < i \u2227 i < n\nA : \u00ac1 + i + 1 < N\nB : \u00ac1 + i + 1 = N\nC : \u00ac1 + i < N\nh : 1 + i = N\n\u22a2 False\n[PROOFSTEP]\nexact hi.left.ne rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\nA : \u00ac1 + i + 1 < N\nB : \u00ac1 + i + 1 = N\nC : \u00ac1 + i < N\nD : \u00ac1 + i = N\n\u22a2 edist (f (u (i + 1))) (f (u i)) =\n edist\n (f\n (if 1 + i + 1 < Nat.find exists_N then u (1 + i + 1)\n else if 1 + i + 1 = Nat.find exists_N then x else u (1 + i + 1 - 1)))\n (f (if 1 + i < Nat.find exists_N then u (1 + i) else if 1 + i = Nat.find exists_N then x else u (1 + i - 1)))\n[PROOFSTEP]\nrw [if_neg A, if_neg B, if_neg C, if_neg D]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\nA : \u00ac1 + i + 1 < N\nB : \u00ac1 + i + 1 = N\nC : \u00ac1 + i < N\nD : \u00ac1 + i = N\n\u22a2 edist (f (u (i + 1))) (f (u i)) = edist (f (u (1 + i + 1 - 1))) (f (u (1 + i - 1)))\n[PROOFSTEP]\ncongr 3\n[GOAL]\ncase e_a.e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\nA : \u00ac1 + i + 1 < N\nB : \u00ac1 + i + 1 = N\nC : \u00ac1 + i < N\nD : \u00ac1 + i = N\n\u22a2 i + 1 = 1 + i + 1 - 1\n[PROOFSTEP]\nrw [add_comm, Nat.sub_one]\n[GOAL]\ncase e_a.e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\nA : \u00ac1 + i + 1 < N\nB : \u00ac1 + i + 1 = N\nC : \u00ac1 + i < N\nD : \u00ac1 + i = N\n\u22a2 1 + i = Nat.pred (1 + i + 1)\n[PROOFSTEP]\napply Nat.pred_succ\n[GOAL]\ncase e_a.e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\nA : \u00ac1 + i + 1 < N\nB : \u00ac1 + i + 1 = N\nC : \u00ac1 + i < N\nD : \u00ac1 + i = N\n\u22a2 i = 1 + i - 1\n[PROOFSTEP]\nrw [add_comm, Nat.sub_one]\n[GOAL]\ncase e_a.e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\ni : \u2115\nhi : N \u2264 i \u2227 i < n\nA : \u00ac1 + i + 1 < N\nB : \u00ac1 + i + 1 = N\nC : \u00ac1 + i < N\nD : \u00ac1 + i = N\n\u22a2 i = Nat.pred (i + 1)\n[PROOFSTEP]\napply Nat.pred_succ\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 \u2211 i in Finset.Ico 0 (N - 1), edist (f (w (i + 1))) (f (w i)) + edist (f (u N)) (f (u (N - 1))) +\n \u2211 i in Finset.Ico N n, edist (f (w (1 + i + 1))) (f (w (1 + i))) =\n \u2211 i in Finset.Ico 0 (N - 1), edist (f (w (i + 1))) (f (w i)) + edist (f (w (N + 1))) (f (w (N - 1))) +\n \u2211 i in Finset.Ico (N + 1) (n + 1), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 \u2211 i in Finset.Ico 0 (N - 1), edist (f (w (i + 1))) (f (w i)) + edist (f (u N)) (f (u (N - 1))) =\n \u2211 i in Finset.Ico 0 (N - 1), edist (f (w (i + 1))) (f (w i)) + edist (f (w (N + 1))) (f (w (N - 1)))\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 \u2211 i in Finset.Ico N n, edist (f (w (1 + i + 1))) (f (w (1 + i))) =\n \u2211 i in Finset.Ico (N + 1) (n + 1), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 edist (f (u N)) (f (u (N - 1))) = edist (f (w (N + 1))) (f (w (N - 1)))\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 edist (f (u (Nat.find exists_N))) (f (u (Nat.find exists_N - 1))) =\n edist\n (f\n (if Nat.find exists_N + 1 < Nat.find exists_N then u (Nat.find exists_N + 1)\n else if Nat.find exists_N + 1 = Nat.find exists_N then x else u (Nat.find exists_N + 1 - 1)))\n (f\n (if Nat.find exists_N - 1 < Nat.find exists_N then u (Nat.find exists_N - 1)\n else if Nat.find exists_N - 1 = Nat.find exists_N then x else u (Nat.find exists_N - 1 - 1)))\n[PROOFSTEP]\nhave A : \u00acN + 1 < N := Nat.not_succ_lt_self\n[GOAL]\ncase e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\nA : \u00acN + 1 < N\n\u22a2 edist (f (u (Nat.find exists_N))) (f (u (Nat.find exists_N - 1))) =\n edist\n (f\n (if Nat.find exists_N + 1 < Nat.find exists_N then u (Nat.find exists_N + 1)\n else if Nat.find exists_N + 1 = Nat.find exists_N then x else u (Nat.find exists_N + 1 - 1)))\n (f\n (if Nat.find exists_N - 1 < Nat.find exists_N then u (Nat.find exists_N - 1)\n else if Nat.find exists_N - 1 = Nat.find exists_N then x else u (Nat.find exists_N - 1 - 1)))\n[PROOFSTEP]\nhave B : N - 1 < N := Nat.pred_lt Npos.ne'\n[GOAL]\ncase e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\nA : \u00acN + 1 < N\nB : N - 1 < N\n\u22a2 edist (f (u (Nat.find exists_N))) (f (u (Nat.find exists_N - 1))) =\n edist\n (f\n (if Nat.find exists_N + 1 < Nat.find exists_N then u (Nat.find exists_N + 1)\n else if Nat.find exists_N + 1 = Nat.find exists_N then x else u (Nat.find exists_N + 1 - 1)))\n (f\n (if Nat.find exists_N - 1 < Nat.find exists_N then u (Nat.find exists_N - 1)\n else if Nat.find exists_N - 1 = Nat.find exists_N then x else u (Nat.find exists_N - 1 - 1)))\n[PROOFSTEP]\nsimp only [A, not_and, not_lt, Nat.succ_ne_self, Nat.add_succ_sub_one, add_zero, if_false, B, if_true]\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 \u2211 i in Finset.Ico N n, edist (f (w (1 + i + 1))) (f (w (1 + i))) =\n \u2211 i in Finset.Ico (N + 1) (n + 1), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\nexact Finset.sum_Ico_add (fun i => edist (f (w (i + 1))) (f (w i))) N n 1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 \u2211 i in Finset.Ico 0 (N - 1), edist (f (w (i + 1))) (f (w i)) + edist (f (w (N + 1))) (f (w (N - 1))) +\n \u2211 i in Finset.Ico (N + 1) (n + 1), edist (f (w (i + 1))) (f (w i)) \u2264\n \u2211 i in Finset.Ico 0 (N - 1), edist (f (w (i + 1))) (f (w i)) +\n \u2211 i in Finset.Ico (N - 1) (N + 1), edist (f (w (i + 1))) (f (w i)) +\n \u2211 i in Finset.Ico (N + 1) (n + 1), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\nrefine' add_le_add (add_le_add le_rfl _) le_rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 edist (f (w (N + 1))) (f (w (N - 1))) \u2264 \u2211 i in Finset.Ico (N - 1) (N + 1), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\nhave A : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\nA : N - 1 + 1 = N\n\u22a2 edist (f (w (N + 1))) (f (w (N - 1))) \u2264 \u2211 i in Finset.Ico (N - 1) (N + 1), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\nhave B : N - 1 + 1 < N + 1 := A.symm \u25b8 N.lt_succ_self\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\nA : N - 1 + 1 = N\nB : N - 1 + 1 < N + 1\n\u22a2 edist (f (w (N + 1))) (f (w (N - 1))) \u2264 \u2211 i in Finset.Ico (N - 1) (N + 1), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\nhave C : N - 1 < N + 1 := lt_of_le_of_lt N.pred_le N.lt_succ_self\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\nA : N - 1 + 1 = N\nB : N - 1 + 1 < N + 1\nC : N - 1 < N + 1\n\u22a2 edist (f (w (N + 1))) (f (w (N - 1))) \u2264 \u2211 i in Finset.Ico (N - 1) (N + 1), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\nrw [Finset.sum_eq_sum_Ico_succ_bot C, Finset.sum_eq_sum_Ico_succ_bot B, A, Finset.Ico_self, Finset.sum_empty, add_zero,\n add_comm (edist _ _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\nA : N - 1 + 1 = N\nB : N - 1 + 1 < N + 1\nC : N - 1 < N + 1\n\u22a2 edist (f (w (N + 1))) (f (w (N - 1))) \u2264 edist (f (w (N + 1))) (f (w N)) + edist (f (w N)) (f (w (N - 1)))\n[PROOFSTEP]\nexact edist_triangle _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 \u2211 i in Finset.Ico 0 (N - 1), edist (f (w (i + 1))) (f (w i)) +\n \u2211 i in Finset.Ico (N - 1) (N + 1), edist (f (w (i + 1))) (f (w i)) +\n \u2211 i in Finset.Ico (N + 1) (n + 1), edist (f (w (i + 1))) (f (w i)) =\n \u2211 j in Finset.range (n + 1), edist (f (w (j + 1))) (f (w j))\n[PROOFSTEP]\nrw [Finset.sum_Ico_consecutive, Finset.sum_Ico_consecutive, Finset.range_eq_Ico]\n[GOAL]\ncase hmn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 0 \u2264 N + 1\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase hnk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 N + 1 \u2264 n + 1\n[PROOFSTEP]\nexact Nat.succ_le_succ hN.left\n[GOAL]\ncase hmn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 0 \u2264 N - 1\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase hnk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nn : \u2115\nh : x < u n\nexists_N : \u2203 N, N \u2264 n \u2227 x < u N\nN : \u2115 := Nat.find exists_N\nhN : N \u2264 n \u2227 x < u N\nw : \u2115 \u2192 \u03b1 := fun i => if i < N then u i else if i = N then x else u (i - 1)\nws : \u2200 (i : \u2115), w i \u2208 s\nhw : Monotone w\nNpos : 0 < N\n\u22a2 N - 1 \u2264 N + 1\n[PROOFSTEP]\nexact N.pred_le.trans N.le_succ\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\n\u22a2 eVariationOn f s + eVariationOn f t \u2264 eVariationOn f (s \u222a t)\n[PROOFSTEP]\nby_cases hs : s = \u2205\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : s = \u2205\n\u22a2 eVariationOn f s + eVariationOn f t \u2264 eVariationOn f (s \u222a t)\n[PROOFSTEP]\nsimp [hs]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\n\u22a2 eVariationOn f s + eVariationOn f t \u2264 eVariationOn f (s \u222a t)\n[PROOFSTEP]\nhave : Nonempty { u // Monotone u \u2227 \u2200 i : \u2115, u i \u2208 s } := nonempty_monotone_mem (nonempty_iff_ne_empty.2 hs)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\n\u22a2 eVariationOn f s + eVariationOn f t \u2264 eVariationOn f (s \u222a t)\n[PROOFSTEP]\nby_cases ht : t = \u2205\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : t = \u2205\n\u22a2 eVariationOn f s + eVariationOn f t \u2264 eVariationOn f (s \u222a t)\n[PROOFSTEP]\nsimp [ht]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\n\u22a2 eVariationOn f s + eVariationOn f t \u2264 eVariationOn f (s \u222a t)\n[PROOFSTEP]\nhave : Nonempty { u // Monotone u \u2227 \u2200 i : \u2115, u i \u2208 t } := nonempty_monotone_mem (nonempty_iff_ne_empty.2 ht)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\n\u22a2 eVariationOn f s + eVariationOn f t \u2264 eVariationOn f (s \u222a t)\n[PROOFSTEP]\nrefine'\n ENNReal.iSup_add_iSup_le\n _\n /- We start from two sequences `u` and `v` along `s` and `t` respectively, and we build a new\n sequence `w` along `s \u222a t` by juxtaposing them. Its variation is larger than the sum of the\n variations. -/\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\n\u22a2 \u2200 (i : \u2115 \u00d7 { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }) (j : \u2115 \u00d7 { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }),\n \u2211 i_1 in Finset.range i.fst, edist (f (\u2191i.snd (i_1 + 1))) (f (\u2191i.snd i_1)) +\n \u2211 i in Finset.range j.fst, edist (f (\u2191j.snd (i + 1))) (f (\u2191j.snd i)) \u2264\n eVariationOn f (s \u222a t)\n[PROOFSTEP]\nrintro \u27e8n, \u27e8u, hu, us\u27e9\u27e9 \u27e8m, \u27e8v, hv, vt\u27e9\u27e9\n[GOAL]\ncase neg.mk.mk.intro.mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\n\u22a2 \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd i)) +\n \u2211 i in Finset.range (m, { val := v, property := (_ : Monotone v \u2227 \u2200 (i : \u2115), v i \u2208 t) }).fst,\n edist (f (\u2191(m, { val := v, property := (_ : Monotone v \u2227 \u2200 (i : \u2115), v i \u2208 t) }).snd (i + 1)))\n (f (\u2191(m, { val := v, property := (_ : Monotone v \u2227 \u2200 (i : \u2115), v i \u2208 t) }).snd i)) \u2264\n eVariationOn f (s \u222a t)\n[PROOFSTEP]\nlet w i := if i \u2264 n then u i else v (i - (n + 1))\n[GOAL]\ncase neg.mk.mk.intro.mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\n\u22a2 \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd i)) +\n \u2211 i in Finset.range (m, { val := v, property := (_ : Monotone v \u2227 \u2200 (i : \u2115), v i \u2208 t) }).fst,\n edist (f (\u2191(m, { val := v, property := (_ : Monotone v \u2227 \u2200 (i : \u2115), v i \u2208 t) }).snd (i + 1)))\n (f (\u2191(m, { val := v, property := (_ : Monotone v \u2227 \u2200 (i : \u2115), v i \u2208 t) }).snd i)) \u2264\n eVariationOn f (s \u222a t)\n[PROOFSTEP]\nhave wst : \u2200 i, w i \u2208 s \u222a t := by\n intro i\n by_cases hi : i \u2264 n\n \u00b7 simp [hi, us]\n \u00b7 simp [hi, vt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\n\u22a2 \u2200 (i : \u2115), w i \u2208 s \u222a t\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\ni : \u2115\n\u22a2 w i \u2208 s \u222a t\n[PROOFSTEP]\nby_cases hi : i \u2264 n\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\ni : \u2115\nhi : i \u2264 n\n\u22a2 w i \u2208 s \u222a t\n[PROOFSTEP]\nsimp [hi, us]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\ni : \u2115\nhi : \u00aci \u2264 n\n\u22a2 w i \u2208 s \u222a t\n[PROOFSTEP]\nsimp [hi, vt]\n[GOAL]\ncase neg.mk.mk.intro.mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\n\u22a2 \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd i)) +\n \u2211 i in Finset.range (m, { val := v, property := (_ : Monotone v \u2227 \u2200 (i : \u2115), v i \u2208 t) }).fst,\n edist (f (\u2191(m, { val := v, property := (_ : Monotone v \u2227 \u2200 (i : \u2115), v i \u2208 t) }).snd (i + 1)))\n (f (\u2191(m, { val := v, property := (_ : Monotone v \u2227 \u2200 (i : \u2115), v i \u2208 t) }).snd i)) \u2264\n eVariationOn f (s \u222a t)\n[PROOFSTEP]\nhave hw : Monotone w := by\n intro i j hij\n dsimp only\n split_ifs with h_1 h_2 h_2\n \u00b7 exact hu hij\n \u00b7 apply h _ (us _) _ (vt _)\n \u00b7 exfalso; exact h_1 (hij.trans h_2)\n \u00b7 apply hv (tsub_le_tsub hij le_rfl)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\n\u22a2 Monotone w\n[PROOFSTEP]\nintro i j hij\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\ni j : \u2115\nhij : i \u2264 j\n\u22a2 w i \u2264 w j\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\ni j : \u2115\nhij : i \u2264 j\n\u22a2 (if i \u2264 n then u i else v (i - (n + 1))) \u2264 if j \u2264 n then u j else v (j - (n + 1))\n[PROOFSTEP]\nsplit_ifs with h_1 h_2 h_2\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\ni j : \u2115\nhij : i \u2264 j\nh_1 : i \u2264 n\nh_2 : j \u2264 n\n\u22a2 u i \u2264 u j\n[PROOFSTEP]\nexact hu hij\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\ni j : \u2115\nhij : i \u2264 j\nh_1 : i \u2264 n\nh_2 : \u00acj \u2264 n\n\u22a2 u i \u2264 v (j - (n + 1))\n[PROOFSTEP]\napply h _ (us _) _ (vt _)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\ni j : \u2115\nhij : i \u2264 j\nh_1 : \u00aci \u2264 n\nh_2 : j \u2264 n\n\u22a2 v (i - (n + 1)) \u2264 u j\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\ni j : \u2115\nhij : i \u2264 j\nh_1 : \u00aci \u2264 n\nh_2 : j \u2264 n\n\u22a2 False\n[PROOFSTEP]\nexact h_1 (hij.trans h_2)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\ni j : \u2115\nhij : i \u2264 j\nh_1 : \u00aci \u2264 n\nh_2 : \u00acj \u2264 n\n\u22a2 v (i - (n + 1)) \u2264 v (j - (n + 1))\n[PROOFSTEP]\napply hv (tsub_le_tsub hij le_rfl)\n[GOAL]\ncase neg.mk.mk.intro.mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd i)) +\n \u2211 i in Finset.range (m, { val := v, property := (_ : Monotone v \u2227 \u2200 (i : \u2115), v i \u2208 t) }).fst,\n edist (f (\u2191(m, { val := v, property := (_ : Monotone v \u2227 \u2200 (i : \u2115), v i \u2208 t) }).snd (i + 1)))\n (f (\u2191(m, { val := v, property := (_ : Monotone v \u2227 \u2200 (i : \u2115), v i \u2208 t) }).snd i)) \u2264\n eVariationOn f (s \u222a t)\n[PROOFSTEP]\ncalc\n ((\u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i))) +\n \u2211 i : \u2115 in Finset.range m, edist (f (v (i + 1))) (f (v i))) =\n (\u2211 i in Finset.range n, edist (f (w (i + 1))) (f (w i))) +\n \u2211 i : \u2115 in Finset.range m, edist (f (w (n + 1 + i + 1))) (f (w (n + 1 + i))) :=\n by\n dsimp only\n congr 1\n \u00b7 refine Finset.sum_congr rfl fun i hi => ?_\n simp only [Finset.mem_range] at hi \n have : i + 1 \u2264 n := Nat.succ_le_of_lt hi\n simp [hi.le, this]\n \u00b7 refine Finset.sum_congr rfl fun i hi => ?_\n simp only [Finset.mem_range] at hi \n have B : \u00acn + 1 + i \u2264 n := by linarith\n have A : \u00acn + 1 + i + 1 \u2264 n := fun h => B ((n + 1 + i).le_succ.trans h)\n have C : n + 1 + i - n = i + 1 := by\n rw [tsub_eq_iff_eq_add_of_le]\n \u00b7 abel\n \u00b7 exact n.le_succ.trans (n.succ.le_add_right i)\n simp only [A, B, C, Nat.succ_sub_succ_eq_sub, if_false, add_tsub_cancel_left]\n _ =\n (\u2211 i in Finset.range n, edist (f (w (i + 1))) (f (w i))) +\n \u2211 i : \u2115 in Finset.Ico (n + 1) (n + 1 + m), edist (f (w (i + 1))) (f (w i)) :=\n by\n congr 1\n rw [Finset.range_eq_Ico]\n convert Finset.sum_Ico_add (fun i : \u2115 => edist (f (w (i + 1))) (f (w i))) 0 m (n + 1) using 3 <;> abel\n _ \u2264 \u2211 i in Finset.range (n + 1 + m), edist (f (w (i + 1))) (f (w i)) :=\n by\n rw [\u2190 Finset.sum_union]\n \u00b7 apply Finset.sum_le_sum_of_subset _\n rintro i hi\n simp only [Finset.mem_union, Finset.mem_range, Finset.mem_Ico] at hi \u22a2\n cases' hi with hi hi\n \u00b7 exact lt_of_lt_of_le hi (n.le_succ.trans (n.succ.le_add_right m))\n \u00b7 exact hi.2\n \u00b7 refine Finset.disjoint_left.2 fun i hi h'i => ?_\n simp only [Finset.mem_Ico, Finset.mem_range] at hi h'i \n exact hi.not_lt (Nat.lt_of_succ_le h'i.left)\n _ \u2264 eVariationOn f (s \u222a t) := sum_le f _ hw wst\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) + \u2211 i in Finset.range m, edist (f (v (i + 1))) (f (v i)) =\n \u2211 i in Finset.range n, edist (f (w (i + 1))) (f (w i)) +\n \u2211 i in Finset.range m, edist (f (w (n + 1 + i + 1))) (f (w (n + 1 + i)))\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) + \u2211 i in Finset.range m, edist (f (v (i + 1))) (f (v i)) =\n \u2211 i in Finset.range n,\n edist (f (if i + 1 \u2264 n then u (i + 1) else v (i + 1 - (n + 1)))) (f (if i \u2264 n then u i else v (i - (n + 1)))) +\n \u2211 i in Finset.range m,\n edist (f (if n + 1 + i + 1 \u2264 n then u (n + 1 + i + 1) else v (n + 1 + i + 1 - (n + 1))))\n (f (if n + 1 + i \u2264 n then u (n + 1 + i) else v (n + 1 + i - (n + 1))))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) =\n \u2211 i in Finset.range n,\n edist (f (if i + 1 \u2264 n then u (i + 1) else v (i + 1 - (n + 1)))) (f (if i \u2264 n then u i else v (i - (n + 1))))\n[PROOFSTEP]\nrefine Finset.sum_congr rfl fun i hi => ?_\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i \u2208 Finset.range n\n\u22a2 edist (f (u (i + 1))) (f (u i)) =\n edist (f (if i + 1 \u2264 n then u (i + 1) else v (i + 1 - (n + 1)))) (f (if i \u2264 n then u i else v (i - (n + 1))))\n[PROOFSTEP]\nsimp only [Finset.mem_range] at hi \n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i < n\n\u22a2 edist (f (u (i + 1))) (f (u i)) =\n edist (f (if i + 1 \u2264 n then u (i + 1) else v (i + 1 - (n + 1)))) (f (if i \u2264 n then u i else v (i - (n + 1))))\n[PROOFSTEP]\nhave : i + 1 \u2264 n := Nat.succ_le_of_lt hi\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d\u00b9 : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i < n\nthis : i + 1 \u2264 n\n\u22a2 edist (f (u (i + 1))) (f (u i)) =\n edist (f (if i + 1 \u2264 n then u (i + 1) else v (i + 1 - (n + 1)))) (f (if i \u2264 n then u i else v (i - (n + 1))))\n[PROOFSTEP]\nsimp [hi.le, this]\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 \u2211 i in Finset.range m, edist (f (v (i + 1))) (f (v i)) =\n \u2211 i in Finset.range m,\n edist (f (if n + 1 + i + 1 \u2264 n then u (n + 1 + i + 1) else v (n + 1 + i + 1 - (n + 1))))\n (f (if n + 1 + i \u2264 n then u (n + 1 + i) else v (n + 1 + i - (n + 1))))\n[PROOFSTEP]\nrefine Finset.sum_congr rfl fun i hi => ?_\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i \u2208 Finset.range m\n\u22a2 edist (f (v (i + 1))) (f (v i)) =\n edist (f (if n + 1 + i + 1 \u2264 n then u (n + 1 + i + 1) else v (n + 1 + i + 1 - (n + 1))))\n (f (if n + 1 + i \u2264 n then u (n + 1 + i) else v (n + 1 + i - (n + 1))))\n[PROOFSTEP]\nsimp only [Finset.mem_range] at hi \n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i < m\n\u22a2 edist (f (v (i + 1))) (f (v i)) =\n edist (f (if n + 1 + i + 1 \u2264 n then u (n + 1 + i + 1) else v (n + 1 + i + 1 - (n + 1))))\n (f (if n + 1 + i \u2264 n then u (n + 1 + i) else v (n + 1 + i - (n + 1))))\n[PROOFSTEP]\nhave B : \u00acn + 1 + i \u2264 n := by linarith\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i < m\n\u22a2 \u00acn + 1 + i \u2264 n\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i < m\nB : \u00acn + 1 + i \u2264 n\n\u22a2 edist (f (v (i + 1))) (f (v i)) =\n edist (f (if n + 1 + i + 1 \u2264 n then u (n + 1 + i + 1) else v (n + 1 + i + 1 - (n + 1))))\n (f (if n + 1 + i \u2264 n then u (n + 1 + i) else v (n + 1 + i - (n + 1))))\n[PROOFSTEP]\nhave A : \u00acn + 1 + i + 1 \u2264 n := fun h => B ((n + 1 + i).le_succ.trans h)\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i < m\nB : \u00acn + 1 + i \u2264 n\nA : \u00acn + 1 + i + 1 \u2264 n\n\u22a2 edist (f (v (i + 1))) (f (v i)) =\n edist (f (if n + 1 + i + 1 \u2264 n then u (n + 1 + i + 1) else v (n + 1 + i + 1 - (n + 1))))\n (f (if n + 1 + i \u2264 n then u (n + 1 + i) else v (n + 1 + i - (n + 1))))\n[PROOFSTEP]\nhave C : n + 1 + i - n = i + 1 := by\n rw [tsub_eq_iff_eq_add_of_le]\n \u00b7 abel\n \u00b7 exact n.le_succ.trans (n.succ.le_add_right i)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i < m\nB : \u00acn + 1 + i \u2264 n\nA : \u00acn + 1 + i + 1 \u2264 n\n\u22a2 n + 1 + i - n = i + 1\n[PROOFSTEP]\nrw [tsub_eq_iff_eq_add_of_le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i < m\nB : \u00acn + 1 + i \u2264 n\nA : \u00acn + 1 + i + 1 \u2264 n\n\u22a2 n + 1 + i = i + 1 + n\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i < m\nB : \u00acn + 1 + i \u2264 n\nA : \u00acn + 1 + i + 1 \u2264 n\n\u22a2 n + 1 + i = i + 1 + n\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i < m\nB : \u00acn + 1 + i \u2264 n\nA : \u00acn + 1 + i + 1 \u2264 n\n\u22a2 n \u2264 n + 1 + i\n[PROOFSTEP]\nexact n.le_succ.trans (n.succ.le_add_right i)\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i < m\nB : \u00acn + 1 + i \u2264 n\nA : \u00acn + 1 + i + 1 \u2264 n\nC : n + 1 + i - n = i + 1\n\u22a2 edist (f (v (i + 1))) (f (v i)) =\n edist (f (if n + 1 + i + 1 \u2264 n then u (n + 1 + i + 1) else v (n + 1 + i + 1 - (n + 1))))\n (f (if n + 1 + i \u2264 n then u (n + 1 + i) else v (n + 1 + i - (n + 1))))\n[PROOFSTEP]\nsimp only [A, B, C, Nat.succ_sub_succ_eq_sub, if_false, add_tsub_cancel_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 \u2211 i in Finset.range n, edist (f (w (i + 1))) (f (w i)) +\n \u2211 i in Finset.range m, edist (f (w (n + 1 + i + 1))) (f (w (n + 1 + i))) =\n \u2211 i in Finset.range n, edist (f (w (i + 1))) (f (w i)) +\n \u2211 i in Finset.Ico (n + 1) (n + 1 + m), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 \u2211 i in Finset.range m, edist (f (w (n + 1 + i + 1))) (f (w (n + 1 + i))) =\n \u2211 i in Finset.Ico (n + 1) (n + 1 + m), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\nrw [Finset.range_eq_Ico]\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 \u2211 i in Finset.Ico 0 m, edist (f (w (n + 1 + i + 1))) (f (w (n + 1 + i))) =\n \u2211 i in Finset.Ico (n + 1) (n + 1 + m), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\nconvert Finset.sum_Ico_add (fun i : \u2115 => edist (f (w (i + 1))) (f (w i))) 0 m (n + 1) using 3\n[GOAL]\ncase h.e'_3.h.h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 n + 1 = 0 + (n + 1)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_3.h.h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 n + 1 = 0 + (n + 1)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_3.h.h.e'_5\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 n + 1 + m = m + (n + 1)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_3.h.h.e'_5\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 n + 1 + m = m + (n + 1)\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 \u2211 i in Finset.range n, edist (f (w (i + 1))) (f (w i)) +\n \u2211 i in Finset.Ico (n + 1) (n + 1 + m), edist (f (w (i + 1))) (f (w i)) \u2264\n \u2211 i in Finset.range (n + 1 + m), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\nrw [\u2190 Finset.sum_union]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 \u2211 x in Finset.range n \u222a Finset.Ico (n + 1) (n + 1 + m), edist (f (w (x + 1))) (f (w x)) \u2264\n \u2211 i in Finset.range (n + 1 + m), edist (f (w (i + 1))) (f (w i))\n[PROOFSTEP]\napply Finset.sum_le_sum_of_subset _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 Finset.range n \u222a Finset.Ico (n + 1) (n + 1 + m) \u2286 Finset.range (n + 1 + m)\n[PROOFSTEP]\nrintro i hi\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i \u2208 Finset.range n \u222a Finset.Ico (n + 1) (n + 1 + m)\n\u22a2 i \u2208 Finset.range (n + 1 + m)\n[PROOFSTEP]\nsimp only [Finset.mem_union, Finset.mem_range, Finset.mem_Ico] at hi \u22a2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i < n \u2228 n + 1 \u2264 i \u2227 i < n + 1 + m\n\u22a2 i < n + 1 + m\n[PROOFSTEP]\ncases' hi with hi hi\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i < n\n\u22a2 i < n + 1 + m\n[PROOFSTEP]\nexact lt_of_lt_of_le hi (n.le_succ.trans (n.succ.le_add_right m))\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : n + 1 \u2264 i \u2227 i < n + 1 + m\n\u22a2 i < n + 1 + m\n[PROOFSTEP]\nexact hi.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\n\u22a2 Disjoint (Finset.range n) (Finset.Ico (n + 1) (n + 1 + m))\n[PROOFSTEP]\nrefine Finset.disjoint_left.2 fun i hi h'i => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i \u2208 Finset.range n\nh'i : i \u2208 Finset.Ico (n + 1) (n + 1 + m)\n\u22a2 False\n[PROOFSTEP]\nsimp only [Finset.mem_Ico, Finset.mem_range] at hi h'i \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 x \u2264 y\nhs : \u00acs = \u2205\nthis\u271d : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }\nht : \u00act = \u2205\nthis : Nonempty { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\nm : \u2115\nv : \u2115 \u2192 \u03b1\nhv : Monotone v\nvt : \u2200 (i : \u2115), v i \u2208 t\nw : \u2115 \u2192 \u03b1 := fun i => if i \u2264 n then u i else v (i - (n + 1))\nwst : \u2200 (i : \u2115), w i \u2208 s \u222a t\nhw : Monotone w\ni : \u2115\nhi : i < n\nh'i : n + 1 \u2264 i \u2227 i < n + 1 + m\n\u22a2 False\n[PROOFSTEP]\nexact hi.not_lt (Nat.lt_of_succ_le h'i.left)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\n\u22a2 eVariationOn f (s \u222a t) = eVariationOn f s + eVariationOn f t\n[PROOFSTEP]\nclassical\napply le_antisymm _ (eVariationOn.add_le_union f fun a ha b hb => le_trans (hs.2 ha) (ht.2 hb))\napply iSup_le _\nrintro \u27e8n, \u27e8u, hu, ust\u27e9\u27e9\nobtain \u27e8v, m, hv, vst, xv, huv\u27e9 :\n \u2203 (v : \u2115 \u2192 \u03b1) (m : \u2115),\n Monotone v \u2227\n (\u2200 i, v i \u2208 s \u222a t) \u2227\n x \u2208 v '' Iio m \u2227\n (\u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i))) \u2264\n \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nexact eVariationOn.add_point f (mem_union_left t hs.1) u hu ust n\nobtain \u27e8N, hN, Nx\u27e9 : \u2203 N, N < m \u2227 v N = x\nexact xv\ncalc\n (\u2211 j in Finset.range n, edist (f (u (j + 1))) (f (u j))) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j)) :=\n huv\n _ =\n (\u2211 j in Finset.Ico 0 N, edist (f (v (j + 1))) (f (v j))) +\n \u2211 j in Finset.Ico N m, edist (f (v (j + 1))) (f (v j)) :=\n by rw [Finset.range_eq_Ico, Finset.sum_Ico_consecutive _ (zero_le _) hN.le]\n _ \u2264 eVariationOn f s + eVariationOn f t := by\n refine' add_le_add _ _\n \u00b7 apply sum_le_of_monotoneOn_Icc _ (hv.monotoneOn _) fun i hi => ?_\n rcases vst i with (h | h); \u00b7 exact h\n have : v i = x := by\n apply le_antisymm\n \u00b7 rw [\u2190 Nx]; exact hv hi.2\n \u00b7 exact ht.2 h\n rw [this]\n exact hs.1\n \u00b7 apply sum_le_of_monotoneOn_Icc _ (hv.monotoneOn _) fun i hi => ?_\n rcases vst i with (h | h); swap; \u00b7 exact h\n have : v i = x := by\n apply le_antisymm\n \u00b7 exact hs.2 h\n \u00b7 rw [\u2190 Nx]; exact hv hi.1\n rw [this]\n exact ht.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\n\u22a2 eVariationOn f (s \u222a t) = eVariationOn f s + eVariationOn f t\n[PROOFSTEP]\napply le_antisymm _ (eVariationOn.add_le_union f fun a ha b hb => le_trans (hs.2 ha) (ht.2 hb))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\n\u22a2 eVariationOn f (s \u222a t) \u2264 eVariationOn f s + eVariationOn f t\n[PROOFSTEP]\napply iSup_le _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\n\u22a2 \u2200 (i : \u2115 \u00d7 { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t }),\n \u2211 i_1 in Finset.range i.fst, edist (f (\u2191i.snd (i_1 + 1))) (f (\u2191i.snd i_1)) \u2264 eVariationOn f s + eVariationOn f t\n[PROOFSTEP]\nrintro \u27e8n, \u27e8u, hu, ust\u27e9\u27e9\n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\n\u22a2 \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t) }).snd i)) \u2264\n eVariationOn f s + eVariationOn f t\n[PROOFSTEP]\nobtain \u27e8v, m, hv, vst, xv, huv\u27e9 :\n \u2203 (v : \u2115 \u2192 \u03b1) (m : \u2115),\n Monotone v \u2227\n (\u2200 i, v i \u2208 s \u222a t) \u2227\n x \u2208 v '' Iio m \u2227\n (\u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i))) \u2264\n \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\n\u22a2 \u2203 v m,\n Monotone v \u2227\n (\u2200 (i : \u2115), v i \u2208 s \u222a t) \u2227\n x \u2208 v '' Iio m \u2227\n \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264\n \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\ncase mk.mk.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n\u22a2 \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t) }).snd i)) \u2264\n eVariationOn f s + eVariationOn f t\n[PROOFSTEP]\nexact eVariationOn.add_point f (mem_union_left t hs.1) u hu ust n\n[GOAL]\ncase mk.mk.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n\u22a2 \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t) }).snd i)) \u2264\n eVariationOn f s + eVariationOn f t\n[PROOFSTEP]\nobtain \u27e8N, hN, Nx\u27e9 : \u2203 N, N < m \u2227 v N = x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\n\u22a2 \u2203 N, N < m \u2227 v N = x\ncase mk.mk.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\n\u22a2 \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t) }).snd i)) \u2264\n eVariationOn f s + eVariationOn f t\n[PROOFSTEP]\nexact xv\n[GOAL]\ncase mk.mk.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\n\u22a2 \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u222a t) }).snd i)) \u2264\n eVariationOn f s + eVariationOn f t\n[PROOFSTEP]\ncalc\n (\u2211 j in Finset.range n, edist (f (u (j + 1))) (f (u j))) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j)) :=\n huv\n _ =\n (\u2211 j in Finset.Ico 0 N, edist (f (v (j + 1))) (f (v j))) +\n \u2211 j in Finset.Ico N m, edist (f (v (j + 1))) (f (v j)) :=\n by rw [Finset.range_eq_Ico, Finset.sum_Ico_consecutive _ (zero_le _) hN.le]\n _ \u2264 eVariationOn f s + eVariationOn f t := by\n refine' add_le_add _ _\n \u00b7 apply sum_le_of_monotoneOn_Icc _ (hv.monotoneOn _) fun i hi => ?_\n rcases vst i with (h | h); \u00b7 exact h\n have : v i = x := by\n apply le_antisymm\n \u00b7 rw [\u2190 Nx]; exact hv hi.2\n \u00b7 exact ht.2 h\n rw [this]\n exact hs.1\n \u00b7 apply sum_le_of_monotoneOn_Icc _ (hv.monotoneOn _) fun i hi => ?_\n rcases vst i with (h | h); swap; \u00b7 exact h\n have : v i = x := by\n apply le_antisymm\n \u00b7 exact hs.2 h\n \u00b7 rw [\u2190 Nx]; exact hv hi.1\n rw [this]\n exact ht.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\n\u22a2 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j)) =\n \u2211 j in Finset.Ico 0 N, edist (f (v (j + 1))) (f (v j)) + \u2211 j in Finset.Ico N m, edist (f (v (j + 1))) (f (v j))\n[PROOFSTEP]\nrw [Finset.range_eq_Ico, Finset.sum_Ico_consecutive _ (zero_le _) hN.le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\n\u22a2 \u2211 j in Finset.Ico 0 N, edist (f (v (j + 1))) (f (v j)) + \u2211 j in Finset.Ico N m, edist (f (v (j + 1))) (f (v j)) \u2264\n eVariationOn f s + eVariationOn f t\n[PROOFSTEP]\nrefine' add_le_add _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\n\u22a2 \u2211 j in Finset.Ico 0 N, edist (f (v (j + 1))) (f (v j)) \u2264 eVariationOn f s\n[PROOFSTEP]\napply sum_le_of_monotoneOn_Icc _ (hv.monotoneOn _) fun i hi => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc 0 N\n\u22a2 v i \u2208 s\n[PROOFSTEP]\nrcases vst i with (h | h)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc 0 N\nh : v i \u2208 s\n\u22a2 v i \u2208 s\n[PROOFSTEP]\nexact h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc 0 N\nh : v i \u2208 t\n\u22a2 v i \u2208 s\n[PROOFSTEP]\nhave : v i = x := by\n apply le_antisymm\n \u00b7 rw [\u2190 Nx]; exact hv hi.2\n \u00b7 exact ht.2 h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc 0 N\nh : v i \u2208 t\n\u22a2 v i = x\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc 0 N\nh : v i \u2208 t\n\u22a2 v i \u2264 x\n[PROOFSTEP]\nrw [\u2190 Nx]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc 0 N\nh : v i \u2208 t\n\u22a2 v i \u2264 v N\n[PROOFSTEP]\nexact hv hi.2\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc 0 N\nh : v i \u2208 t\n\u22a2 x \u2264 v i\n[PROOFSTEP]\nexact ht.2 h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc 0 N\nh : v i \u2208 t\nthis : v i = x\n\u22a2 v i \u2208 s\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc 0 N\nh : v i \u2208 t\nthis : v i = x\n\u22a2 x \u2208 s\n[PROOFSTEP]\nexact hs.1\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\n\u22a2 \u2211 j in Finset.Ico N m, edist (f (v (j + 1))) (f (v j)) \u2264 eVariationOn f t\n[PROOFSTEP]\napply sum_le_of_monotoneOn_Icc _ (hv.monotoneOn _) fun i hi => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc N m\n\u22a2 v i \u2208 t\n[PROOFSTEP]\nrcases vst i with (h | h)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc N m\nh : v i \u2208 s\n\u22a2 v i \u2208 t\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc N m\nh : v i \u2208 t\n\u22a2 v i \u2208 t\n[PROOFSTEP]\nswap\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc N m\nh : v i \u2208 t\n\u22a2 v i \u2208 t\n[PROOFSTEP]\nexact h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc N m\nh : v i \u2208 s\n\u22a2 v i \u2208 t\n[PROOFSTEP]\nhave : v i = x := by\n apply le_antisymm\n \u00b7 exact hs.2 h\n \u00b7 rw [\u2190 Nx]; exact hv hi.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc N m\nh : v i \u2208 s\n\u22a2 v i = x\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc N m\nh : v i \u2208 s\n\u22a2 v i \u2264 x\n[PROOFSTEP]\nexact hs.2 h\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc N m\nh : v i \u2208 s\n\u22a2 x \u2264 v i\n[PROOFSTEP]\nrw [\u2190 Nx]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc N m\nh : v i \u2208 s\n\u22a2 v N \u2264 v i\n[PROOFSTEP]\nexact hv hi.1\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc N m\nh : v i \u2208 s\nthis : v i = x\n\u22a2 v i \u2208 t\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nx : \u03b1\nhs : IsGreatest s x\nht : IsLeast t x\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nust : \u2200 (i : \u2115), u i \u2208 s \u222a t\nv : \u2115 \u2192 \u03b1\nm : \u2115\nhv : Monotone v\nvst : \u2200 (i : \u2115), v i \u2208 s \u222a t\nxv : x \u2208 v '' Iio m\nhuv : \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) \u2264 \u2211 j in Finset.range m, edist (f (v (j + 1))) (f (v j))\nN : \u2115\nhN : N < m\nNx : v N = x\ni : \u2115\nhi : i \u2208 Icc N m\nh : v i \u2208 s\nthis : v i = x\n\u22a2 x \u2208 t\n[PROOFSTEP]\nexact ht.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\na b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\nhb : b \u2208 s\n\u22a2 eVariationOn f (s \u2229 Icc a b) + eVariationOn f (s \u2229 Icc b c) = eVariationOn f (s \u2229 Icc a c)\n[PROOFSTEP]\nhave A : IsGreatest (s \u2229 Icc a b) b := \u27e8\u27e8hb, hab, le_rfl\u27e9, (inter_subset_right _ _).trans Icc_subset_Iic_self\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\na b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\nhb : b \u2208 s\nA : IsGreatest (s \u2229 Icc a b) b\n\u22a2 eVariationOn f (s \u2229 Icc a b) + eVariationOn f (s \u2229 Icc b c) = eVariationOn f (s \u2229 Icc a c)\n[PROOFSTEP]\nhave B : IsLeast (s \u2229 Icc b c) b := \u27e8\u27e8hb, le_rfl, hbc\u27e9, (inter_subset_right _ _).trans Icc_subset_Ici_self\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\na b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\nhb : b \u2208 s\nA : IsGreatest (s \u2229 Icc a b) b\nB : IsLeast (s \u2229 Icc b c) b\n\u22a2 eVariationOn f (s \u2229 Icc a b) + eVariationOn f (s \u2229 Icc b c) = eVariationOn f (s \u2229 Icc a c)\n[PROOFSTEP]\nrw [\u2190 eVariationOn.union f A B, \u2190 inter_union_distrib_left, Icc_union_Icc_eq_Icc hab hbc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\ns : Set \u03b1\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\n\u03c6st : MapsTo \u03c6 t s\n\u22a2 eVariationOn (f \u2218 \u03c6) t \u2264 eVariationOn f s\n[PROOFSTEP]\nrefine' iSup_le _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\ns : Set \u03b1\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\n\u03c6st : MapsTo \u03c6 t s\n\u22a2 \u2200 (i : \u2115 \u00d7 { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t }),\n \u2211 i_1 in Finset.range i.fst, edist ((f \u2218 \u03c6) (\u2191i.snd (i_1 + 1))) ((f \u2218 \u03c6) (\u2191i.snd i_1)) \u2264 eVariationOn f s\n[PROOFSTEP]\nrintro \u27e8n, u, hu, ut\u27e9\n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\ns : Set \u03b1\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\n\u03c6st : MapsTo \u03c6 t s\nn : \u2115\nu : \u2115 \u2192 \u03b2\nhu : Monotone u\nut : \u2200 (i : \u2115), u i \u2208 t\n\u22a2 \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).fst,\n edist ((f \u2218 \u03c6) (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).snd (i + 1)))\n ((f \u2218 \u03c6) (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).snd i)) \u2264\n eVariationOn f s\n[PROOFSTEP]\nrw [\u2190 Finset.sum_range_reflect]\n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\ns : Set \u03b1\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\n\u03c6st : MapsTo \u03c6 t s\nn : \u2115\nu : \u2115 \u2192 \u03b2\nhu : Monotone u\nut : \u2200 (i : \u2115), u i \u2208 t\n\u22a2 \u2211 j in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).fst,\n edist\n ((f \u2218 \u03c6)\n (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).snd\n ((n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).fst - 1 - j + 1)))\n ((f \u2218 \u03c6)\n (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).snd\n ((n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).fst - 1 - j))) \u2264\n eVariationOn f s\n[PROOFSTEP]\nrefine'\n (Finset.sum_congr rfl fun x hx => _).trans_le <|\n le_iSup_of_le\n \u27e8n, fun i => \u03c6 (u <| n - i), fun x y xy => h\u03c6 (ut _) (ut _) (hu <| Nat.sub_le_sub_left n xy), fun i => \u03c6st (ut _)\u27e9\n le_rfl\n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\ns : Set \u03b1\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\n\u03c6st : MapsTo \u03c6 t s\nn : \u2115\nu : \u2115 \u2192 \u03b2\nhu : Monotone u\nut : \u2200 (i : \u2115), u i \u2208 t\nx : \u2115\nhx : x \u2208 Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).fst\n\u22a2 edist\n ((f \u2218 \u03c6)\n (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).snd\n ((n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).fst - 1 - x + 1)))\n ((f \u2218 \u03c6)\n (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).snd\n ((n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).fst - 1 - x))) =\n edist\n (f\n (\u2191(n,\n { val := fun i => \u03c6 (u (n - i)),\n property := (_ : (Monotone fun i => \u03c6 (u (n - i))) \u2227 \u2200 (i : \u2115), (fun i => \u03c6 (u (n - i))) i \u2208 s) }).snd\n (x + 1)))\n (f\n (\u2191(n,\n { val := fun i => \u03c6 (u (n - i)),\n property := (_ : (Monotone fun i => \u03c6 (u (n - i))) \u2227 \u2200 (i : \u2115), (fun i => \u03c6 (u (n - i))) i \u2208 s) }).snd\n x))\n[PROOFSTEP]\ndsimp only [Subtype.coe_mk]\n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\ns : Set \u03b1\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\n\u03c6st : MapsTo \u03c6 t s\nn : \u2115\nu : \u2115 \u2192 \u03b2\nhu : Monotone u\nut : \u2200 (i : \u2115), u i \u2208 t\nx : \u2115\nhx : x \u2208 Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).fst\n\u22a2 edist ((f \u2218 \u03c6) (u (n - 1 - x + 1))) ((f \u2218 \u03c6) (u (n - 1 - x))) = edist (f (\u03c6 (u (n - (x + 1))))) (f (\u03c6 (u (n - x))))\n[PROOFSTEP]\nrw [edist_comm, Nat.sub_sub, add_comm, Nat.sub_succ, Nat.add_one, Nat.succ_pred_eq_of_pos]\n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\ns : Set \u03b1\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\n\u03c6st : MapsTo \u03c6 t s\nn : \u2115\nu : \u2115 \u2192 \u03b2\nhu : Monotone u\nut : \u2200 (i : \u2115), u i \u2208 t\nx : \u2115\nhx : x \u2208 Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).fst\n\u22a2 edist ((f \u2218 \u03c6) (u (Nat.pred (n - x)))) ((f \u2218 \u03c6) (u (n - x))) =\n edist (f (\u03c6 (u (Nat.pred (n - x))))) (f (\u03c6 (u (n - x))))\ncase mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\ns : Set \u03b1\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\n\u03c6st : MapsTo \u03c6 t s\nn : \u2115\nu : \u2115 \u2192 \u03b2\nhu : Monotone u\nut : \u2200 (i : \u2115), u i \u2208 t\nx : \u2115\nhx : x \u2208 Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).fst\n\u22a2 0 < n - x\n[PROOFSTEP]\nsimp only [Function.comp_apply]\n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\ns : Set \u03b1\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\n\u03c6st : MapsTo \u03c6 t s\nn : \u2115\nu : \u2115 \u2192 \u03b2\nhu : Monotone u\nut : \u2200 (i : \u2115), u i \u2208 t\nx : \u2115\nhx : x \u2208 Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 t) }).fst\n\u22a2 0 < n - x\n[PROOFSTEP]\nsimpa only [tsub_pos_iff_lt, Finset.mem_range] using hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\n\u22a2 eVariationOn (f \u2218 \u03c6) t = eVariationOn f (\u03c6 '' t)\n[PROOFSTEP]\napply le_antisymm (comp_le_of_monotoneOn f \u03c6 h\u03c6 (mapsTo_image \u03c6 t))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\n\u22a2 eVariationOn f (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b2\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nh\u271d : IsEmpty \u03b2\n\u22a2 eVariationOn f (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nconvert zero_le (_ : \u211d\u22650\u221e)\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nh\u271d : IsEmpty \u03b2\n\u22a2 eVariationOn f (\u03c6 '' t) = 0\n[PROOFSTEP]\nexact eVariationOn.subsingleton f <| (subsingleton_of_subsingleton.image _).anti (surjOn_image \u03c6 t)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nh\u271d : Nonempty \u03b2\n\u22a2 eVariationOn f (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nlet \u03c8 := \u03c6.invFunOn t\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nh\u271d : Nonempty \u03b2\n\u03c8 : \u03b1 \u2192 \u03b2 := Function.invFunOn \u03c6 t\n\u22a2 eVariationOn f (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nhave \u03c8\u03c6s : EqOn (\u03c6 \u2218 \u03c8) id (\u03c6 '' t) := (surjOn_image \u03c6 t).rightInvOn_invFunOn\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nh\u271d : Nonempty \u03b2\n\u03c8 : \u03b1 \u2192 \u03b2 := Function.invFunOn \u03c6 t\n\u03c8\u03c6s : EqOn (\u03c6 \u2218 \u03c8) id (\u03c6 '' t)\n\u22a2 eVariationOn f (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nhave \u03c8ts : MapsTo \u03c8 (\u03c6 '' t) t := (surjOn_image \u03c6 t).mapsTo_invFunOn\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nh\u271d : Nonempty \u03b2\n\u03c8 : \u03b1 \u2192 \u03b2 := Function.invFunOn \u03c6 t\n\u03c8\u03c6s : EqOn (\u03c6 \u2218 \u03c8) id (\u03c6 '' t)\n\u03c8ts : MapsTo \u03c8 (\u03c6 '' t) t\n\u22a2 eVariationOn f (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nhave h\u03c8 : MonotoneOn \u03c8 (\u03c6 '' t) := Function.monotoneOn_of_rightInvOn_of_mapsTo h\u03c6 \u03c8\u03c6s \u03c8ts\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nh\u271d : Nonempty \u03b2\n\u03c8 : \u03b1 \u2192 \u03b2 := Function.invFunOn \u03c6 t\n\u03c8\u03c6s : EqOn (\u03c6 \u2218 \u03c8) id (\u03c6 '' t)\n\u03c8ts : MapsTo \u03c8 (\u03c6 '' t) t\nh\u03c8 : MonotoneOn \u03c8 (\u03c6 '' t)\n\u22a2 eVariationOn f (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nchange eVariationOn (f \u2218 id) (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nh\u271d : Nonempty \u03b2\n\u03c8 : \u03b1 \u2192 \u03b2 := Function.invFunOn \u03c6 t\n\u03c8\u03c6s : EqOn (\u03c6 \u2218 \u03c8) id (\u03c6 '' t)\n\u03c8ts : MapsTo \u03c8 (\u03c6 '' t) t\nh\u03c8 : MonotoneOn \u03c8 (\u03c6 '' t)\n\u22a2 eVariationOn (f \u2218 id) (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nrw [\u2190 eq_of_eqOn (\u03c8\u03c6s.comp_left : EqOn (f \u2218 \u03c6 \u2218 \u03c8) (f \u2218 id) (\u03c6 '' t))]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nh\u271d : Nonempty \u03b2\n\u03c8 : \u03b1 \u2192 \u03b2 := Function.invFunOn \u03c6 t\n\u03c8\u03c6s : EqOn (\u03c6 \u2218 \u03c8) id (\u03c6 '' t)\n\u03c8ts : MapsTo \u03c8 (\u03c6 '' t) t\nh\u03c8 : MonotoneOn \u03c8 (\u03c6 '' t)\n\u22a2 eVariationOn (f \u2218 \u03c6 \u2218 \u03c8) (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nexact comp_le_of_monotoneOn _ \u03c8 h\u03c8 \u03c8ts\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\n\u22a2 eVariationOn (f \u2218 \u03c6) (t \u2229 Icc x y) = eVariationOn f (\u03c6 '' t \u2229 Icc (\u03c6 x) (\u03c6 y))\n[PROOFSTEP]\nrcases le_total x y with (h | h)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : x \u2264 y\n\u22a2 eVariationOn (f \u2218 \u03c6) (t \u2229 Icc x y) = eVariationOn f (\u03c6 '' t \u2229 Icc (\u03c6 x) (\u03c6 y))\n[PROOFSTEP]\nconvert comp_eq_of_monotoneOn f \u03c6 (h\u03c6.mono (Set.inter_subset_left t (Icc x y)))\n[GOAL]\ncase h.e'_3.h.e'_6\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : x \u2264 y\n\u22a2 \u03c6 '' t \u2229 Icc (\u03c6 x) (\u03c6 y) = \u03c6 '' (t \u2229 Icc x y)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase h.e'_3.h.e'_6.a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : x \u2264 y\n\u22a2 \u03c6 '' t \u2229 Icc (\u03c6 x) (\u03c6 y) \u2264 \u03c6 '' (t \u2229 Icc x y)\n[PROOFSTEP]\nrintro _ \u27e8\u27e8u, us, rfl\u27e9, v\u03c6x, v\u03c6y\u27e9\n[GOAL]\ncase h.e'_3.h.e'_6.a.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : x \u2264 y\nu : \u03b2\nus : u \u2208 t\nv\u03c6x : \u03c6 x \u2264 \u03c6 u\nv\u03c6y : \u03c6 u \u2264 \u03c6 y\n\u22a2 \u03c6 u \u2208 \u03c6 '' (t \u2229 Icc x y)\n[PROOFSTEP]\nrcases le_total x u with (xu | ux)\n[GOAL]\ncase h.e'_3.h.e'_6.a.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : x \u2264 y\nu : \u03b2\nus : u \u2208 t\nv\u03c6x : \u03c6 x \u2264 \u03c6 u\nv\u03c6y : \u03c6 u \u2264 \u03c6 y\nxu : x \u2264 u\n\u22a2 \u03c6 u \u2208 \u03c6 '' (t \u2229 Icc x y)\n[PROOFSTEP]\nrcases le_total u y with (uy | yu)\n[GOAL]\ncase h.e'_3.h.e'_6.a.intro.intro.intro.intro.inl.inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : x \u2264 y\nu : \u03b2\nus : u \u2208 t\nv\u03c6x : \u03c6 x \u2264 \u03c6 u\nv\u03c6y : \u03c6 u \u2264 \u03c6 y\nxu : x \u2264 u\nuy : u \u2264 y\n\u22a2 \u03c6 u \u2208 \u03c6 '' (t \u2229 Icc x y)\n[PROOFSTEP]\nexact \u27e8u, \u27e8us, \u27e8xu, uy\u27e9\u27e9, rfl\u27e9\n[GOAL]\ncase h.e'_3.h.e'_6.a.intro.intro.intro.intro.inl.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : x \u2264 y\nu : \u03b2\nus : u \u2208 t\nv\u03c6x : \u03c6 x \u2264 \u03c6 u\nv\u03c6y : \u03c6 u \u2264 \u03c6 y\nxu : x \u2264 u\nyu : y \u2264 u\n\u22a2 \u03c6 u \u2208 \u03c6 '' (t \u2229 Icc x y)\n[PROOFSTEP]\nrw [le_antisymm v\u03c6y (h\u03c6 hy us yu)]\n[GOAL]\ncase h.e'_3.h.e'_6.a.intro.intro.intro.intro.inl.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : x \u2264 y\nu : \u03b2\nus : u \u2208 t\nv\u03c6x : \u03c6 x \u2264 \u03c6 u\nv\u03c6y : \u03c6 u \u2264 \u03c6 y\nxu : x \u2264 u\nyu : y \u2264 u\n\u22a2 \u03c6 y \u2208 \u03c6 '' (t \u2229 Icc x y)\n[PROOFSTEP]\nexact \u27e8y, \u27e8hy, \u27e8h, le_rfl\u27e9\u27e9, rfl\u27e9\n[GOAL]\ncase h.e'_3.h.e'_6.a.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : x \u2264 y\nu : \u03b2\nus : u \u2208 t\nv\u03c6x : \u03c6 x \u2264 \u03c6 u\nv\u03c6y : \u03c6 u \u2264 \u03c6 y\nux : u \u2264 x\n\u22a2 \u03c6 u \u2208 \u03c6 '' (t \u2229 Icc x y)\n[PROOFSTEP]\nrw [\u2190 le_antisymm v\u03c6x (h\u03c6 us hx ux)]\n[GOAL]\ncase h.e'_3.h.e'_6.a.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : x \u2264 y\nu : \u03b2\nus : u \u2208 t\nv\u03c6x : \u03c6 x \u2264 \u03c6 u\nv\u03c6y : \u03c6 u \u2264 \u03c6 y\nux : u \u2264 x\n\u22a2 \u03c6 x \u2208 \u03c6 '' (t \u2229 Icc x y)\n[PROOFSTEP]\nexact \u27e8x, \u27e8hx, \u27e8le_rfl, h\u27e9\u27e9, rfl\u27e9\n[GOAL]\ncase h.e'_3.h.e'_6.a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : x \u2264 y\n\u22a2 \u03c6 '' (t \u2229 Icc x y) \u2264 \u03c6 '' t \u2229 Icc (\u03c6 x) (\u03c6 y)\n[PROOFSTEP]\nrintro _ \u27e8u, \u27e8\u27e8hu, xu, uy\u27e9, rfl\u27e9\u27e9\n[GOAL]\ncase h.e'_3.h.e'_6.a.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : x \u2264 y\nu : \u03b2\nhu : u \u2208 t\nxu : x \u2264 u\nuy : u \u2264 y\n\u22a2 \u03c6 u \u2208 \u03c6 '' t \u2229 Icc (\u03c6 x) (\u03c6 y)\n[PROOFSTEP]\nrefine' \u27e8\u27e8u, hu, rfl\u27e9, \u27e8h\u03c6 hx hu xu, h\u03c6 hu hy uy\u27e9\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : y \u2264 x\n\u22a2 eVariationOn (f \u2218 \u03c6) (t \u2229 Icc x y) = eVariationOn f (\u03c6 '' t \u2229 Icc (\u03c6 x) (\u03c6 y))\n[PROOFSTEP]\nrw [eVariationOn.subsingleton, eVariationOn.subsingleton]\n[GOAL]\ncase inr.hs\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : y \u2264 x\n\u22a2 Set.Subsingleton (\u03c6 '' t \u2229 Icc (\u03c6 x) (\u03c6 y))\ncase inr.hs\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : y \u2264 x\n\u22a2 Set.Subsingleton (t \u2229 Icc x y)\n[PROOFSTEP]\nexacts [(Set.subsingleton_Icc_of_ge (h\u03c6 hy hx h)).anti (Set.inter_subset_right _ _),\n (Set.subsingleton_Icc_of_ge h).anti (Set.inter_subset_right _ _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\n\u22a2 eVariationOn (f \u2218 \u03c6) t = eVariationOn f (\u03c6 '' t)\n[PROOFSTEP]\napply le_antisymm (comp_le_of_antitoneOn f \u03c6 h\u03c6 (mapsTo_image \u03c6 t))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\n\u22a2 eVariationOn f (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b2\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\nh\u271d : IsEmpty \u03b2\n\u22a2 eVariationOn f (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nconvert zero_le (_ : \u211d\u22650\u221e)\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\nh\u271d : IsEmpty \u03b2\n\u22a2 eVariationOn f (\u03c6 '' t) = 0\n[PROOFSTEP]\nexact eVariationOn.subsingleton f <| (subsingleton_of_subsingleton.image _).anti (surjOn_image \u03c6 t)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\nh\u271d : Nonempty \u03b2\n\u22a2 eVariationOn f (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nlet \u03c8 := \u03c6.invFunOn t\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\nh\u271d : Nonempty \u03b2\n\u03c8 : \u03b1 \u2192 \u03b2 := Function.invFunOn \u03c6 t\n\u22a2 eVariationOn f (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nhave \u03c8\u03c6s : EqOn (\u03c6 \u2218 \u03c8) id (\u03c6 '' t) := (surjOn_image \u03c6 t).rightInvOn_invFunOn\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\nh\u271d : Nonempty \u03b2\n\u03c8 : \u03b1 \u2192 \u03b2 := Function.invFunOn \u03c6 t\n\u03c8\u03c6s : EqOn (\u03c6 \u2218 \u03c8) id (\u03c6 '' t)\n\u22a2 eVariationOn f (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nhave \u03c8ts := (surjOn_image \u03c6 t).mapsTo_invFunOn\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\nh\u271d : Nonempty \u03b2\n\u03c8 : \u03b1 \u2192 \u03b2 := Function.invFunOn \u03c6 t\n\u03c8\u03c6s : EqOn (\u03c6 \u2218 \u03c8) id (\u03c6 '' t)\n\u03c8ts : MapsTo (Function.invFunOn \u03c6 t) (\u03c6 '' t) t\n\u22a2 eVariationOn f (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nhave h\u03c8 : AntitoneOn \u03c8 (\u03c6 '' t) := Function.antitoneOn_of_rightInvOn_of_mapsTo h\u03c6 \u03c8\u03c6s \u03c8ts\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\nh\u271d : Nonempty \u03b2\n\u03c8 : \u03b1 \u2192 \u03b2 := Function.invFunOn \u03c6 t\n\u03c8\u03c6s : EqOn (\u03c6 \u2218 \u03c8) id (\u03c6 '' t)\n\u03c8ts : MapsTo (Function.invFunOn \u03c6 t) (\u03c6 '' t) t\nh\u03c8 : AntitoneOn \u03c8 (\u03c6 '' t)\n\u22a2 eVariationOn f (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nchange eVariationOn (f \u2218 id) (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\nh\u271d : Nonempty \u03b2\n\u03c8 : \u03b1 \u2192 \u03b2 := Function.invFunOn \u03c6 t\n\u03c8\u03c6s : EqOn (\u03c6 \u2218 \u03c8) id (\u03c6 '' t)\n\u03c8ts : MapsTo (Function.invFunOn \u03c6 t) (\u03c6 '' t) t\nh\u03c8 : AntitoneOn \u03c8 (\u03c6 '' t)\n\u22a2 eVariationOn (f \u2218 id) (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nrw [\u2190 eq_of_eqOn (\u03c8\u03c6s.comp_left : EqOn (f \u2218 \u03c6 \u2218 \u03c8) (f \u2218 id) (\u03c6 '' t))]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : AntitoneOn \u03c6 t\nh\u271d : Nonempty \u03b2\n\u03c8 : \u03b1 \u2192 \u03b2 := Function.invFunOn \u03c6 t\n\u03c8\u03c6s : EqOn (\u03c6 \u2218 \u03c8) id (\u03c6 '' t)\n\u03c8ts : MapsTo (Function.invFunOn \u03c6 t) (\u03c6 '' t) t\nh\u03c8 : AntitoneOn \u03c8 (\u03c6 '' t)\n\u22a2 eVariationOn (f \u2218 \u03c6 \u2218 \u03c8) (\u03c6 '' t) \u2264 eVariationOn (f \u2218 \u03c6) t\n[PROOFSTEP]\nexact comp_le_of_antitoneOn _ \u03c8 h\u03c8 \u03c8ts\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\ns : Set \u03b1\n\u22a2 eVariationOn (f \u2218 \u2191ofDual) (\u2191ofDual \u207b\u00b9' s) = eVariationOn f s\n[PROOFSTEP]\nconvert comp_eq_of_antitoneOn f ofDual fun _ _ _ _ => id\n[GOAL]\ncase h.e'_3.h.e'_6\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\ns : Set \u03b1\n\u22a2 s = \u2191ofDual '' (\u2191ofDual \u207b\u00b9' s)\n[PROOFSTEP]\nsimp only [Equiv.image_preimage]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : MonotoneOn f s\na b : \u03b1\nas : a \u2208 s\nbs : b \u2208 s\n\u22a2 eVariationOn f (s \u2229 Icc a b) \u2264 ENNReal.ofReal (f b - f a)\n[PROOFSTEP]\napply iSup_le _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : MonotoneOn f s\na b : \u03b1\nas : a \u2208 s\nbs : b \u2208 s\n\u22a2 \u2200 (i : \u2115 \u00d7 { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u2229 Icc a b }),\n \u2211 i_1 in Finset.range i.fst, edist (f (\u2191i.snd (i_1 + 1))) (f (\u2191i.snd i_1)) \u2264 ENNReal.ofReal (f b - f a)\n[PROOFSTEP]\nrintro \u27e8n, \u27e8u, hu, us\u27e9\u27e9\n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : MonotoneOn f s\na b : \u03b1\nas : a \u2208 s\nbs : b \u2208 s\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s \u2229 Icc a b\n\u22a2 \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u2229 Icc a b) }).fst,\n edist (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u2229 Icc a b) }).snd (i + 1)))\n (f (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s \u2229 Icc a b) }).snd i)) \u2264\n ENNReal.ofReal (f b - f a)\n[PROOFSTEP]\ncalc\n (\u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i))) =\n \u2211 i in Finset.range n, ENNReal.ofReal (f (u (i + 1)) - f (u i)) :=\n by\n refine Finset.sum_congr rfl fun i hi => ?_\n simp only [Finset.mem_range] at hi \n rw [edist_dist, Real.dist_eq, abs_of_nonneg]\n exact sub_nonneg_of_le (hf (us i).1 (us (i + 1)).1 (hu (Nat.le_succ _)))\n _ = ENNReal.ofReal (\u2211 i in Finset.range n, (f (u (i + 1)) - f (u i))) :=\n by\n rw [ENNReal.ofReal_sum_of_nonneg]\n intro i _\n exact sub_nonneg_of_le (hf (us i).1 (us (i + 1)).1 (hu (Nat.le_succ _)))\n _ = ENNReal.ofReal (f (u n) - f (u 0)) := by rw [Finset.sum_range_sub fun i => f (u i)]\n _ \u2264 ENNReal.ofReal (f b - f a) := by\n apply ENNReal.ofReal_le_ofReal\n exact sub_le_sub (hf (us n).1 bs (us n).2.2) (hf as (us 0).1 (us 0).2.1)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : MonotoneOn f s\na b : \u03b1\nas : a \u2208 s\nbs : b \u2208 s\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s \u2229 Icc a b\n\u22a2 \u2211 i in Finset.range n, edist (f (u (i + 1))) (f (u i)) =\n \u2211 i in Finset.range n, ENNReal.ofReal (f (u (i + 1)) - f (u i))\n[PROOFSTEP]\nrefine Finset.sum_congr rfl fun i hi => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : MonotoneOn f s\na b : \u03b1\nas : a \u2208 s\nbs : b \u2208 s\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s \u2229 Icc a b\ni : \u2115\nhi : i \u2208 Finset.range n\n\u22a2 edist (f (u (i + 1))) (f (u i)) = ENNReal.ofReal (f (u (i + 1)) - f (u i))\n[PROOFSTEP]\nsimp only [Finset.mem_range] at hi \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : MonotoneOn f s\na b : \u03b1\nas : a \u2208 s\nbs : b \u2208 s\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s \u2229 Icc a b\ni : \u2115\nhi : i < n\n\u22a2 edist (f (u (i + 1))) (f (u i)) = ENNReal.ofReal (f (u (i + 1)) - f (u i))\n[PROOFSTEP]\nrw [edist_dist, Real.dist_eq, abs_of_nonneg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : MonotoneOn f s\na b : \u03b1\nas : a \u2208 s\nbs : b \u2208 s\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s \u2229 Icc a b\ni : \u2115\nhi : i < n\n\u22a2 0 \u2264 f (u (i + 1)) - f (u i)\n[PROOFSTEP]\nexact sub_nonneg_of_le (hf (us i).1 (us (i + 1)).1 (hu (Nat.le_succ _)))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : MonotoneOn f s\na b : \u03b1\nas : a \u2208 s\nbs : b \u2208 s\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s \u2229 Icc a b\n\u22a2 \u2211 i in Finset.range n, ENNReal.ofReal (f (u (i + 1)) - f (u i)) =\n ENNReal.ofReal (\u2211 i in Finset.range n, (f (u (i + 1)) - f (u i)))\n[PROOFSTEP]\nrw [ENNReal.ofReal_sum_of_nonneg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : MonotoneOn f s\na b : \u03b1\nas : a \u2208 s\nbs : b \u2208 s\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s \u2229 Icc a b\n\u22a2 \u2200 (i : \u2115), i \u2208 Finset.range n \u2192 0 \u2264 f (u (i + 1)) - f (u i)\n[PROOFSTEP]\nintro i _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : MonotoneOn f s\na b : \u03b1\nas : a \u2208 s\nbs : b \u2208 s\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s \u2229 Icc a b\ni : \u2115\na\u271d : i \u2208 Finset.range n\n\u22a2 0 \u2264 f (u (i + 1)) - f (u i)\n[PROOFSTEP]\nexact sub_nonneg_of_le (hf (us i).1 (us (i + 1)).1 (hu (Nat.le_succ _)))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : MonotoneOn f s\na b : \u03b1\nas : a \u2208 s\nbs : b \u2208 s\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s \u2229 Icc a b\n\u22a2 ENNReal.ofReal (\u2211 i in Finset.range n, (f (u (i + 1)) - f (u i))) = ENNReal.ofReal (f (u n) - f (u 0))\n[PROOFSTEP]\nrw [Finset.sum_range_sub fun i => f (u i)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : MonotoneOn f s\na b : \u03b1\nas : a \u2208 s\nbs : b \u2208 s\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s \u2229 Icc a b\n\u22a2 ENNReal.ofReal (f (u n) - f (u 0)) \u2264 ENNReal.ofReal (f b - f a)\n[PROOFSTEP]\napply ENNReal.ofReal_le_ofReal\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : MonotoneOn f s\na b : \u03b1\nas : a \u2208 s\nbs : b \u2208 s\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s \u2229 Icc a b\n\u22a2 f (u n) - f (u 0) \u2264 f b - f a\n[PROOFSTEP]\nexact sub_le_sub (hf (us n).1 bs (us n).2.2) (hf as (us 0).1 (us 0).2.1)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\na : \u03b1\n\u22a2 variationOnFromTo f s a a = 0\n[PROOFSTEP]\ndsimp only [variationOnFromTo]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\na : \u03b1\n\u22a2 (if a \u2264 a then ENNReal.toReal (eVariationOn f (s \u2229 Icc a a)) else -ENNReal.toReal (eVariationOn f (s \u2229 Icc a a))) = 0\n[PROOFSTEP]\nrw [if_pos le_rfl, Icc_self, eVariationOn.subsingleton, ENNReal.zero_toReal]\n[GOAL]\ncase hs\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\na : \u03b1\n\u22a2 Set.Subsingleton (s \u2229 {a})\n[PROOFSTEP]\nexact fun x hx y hy => hx.2.trans hy.2.symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 0 \u2264 variationOnFromTo f s a b\n[PROOFSTEP]\nsimp only [variationOnFromTo, if_pos h, ENNReal.toReal_nonneg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\na b : \u03b1\n\u22a2 variationOnFromTo f s a b = -variationOnFromTo f s b a\n[PROOFSTEP]\nrcases lt_trichotomy a b with (ab | rfl | ba)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\na b : \u03b1\nab : a < b\n\u22a2 variationOnFromTo f s a b = -variationOnFromTo f s b a\n[PROOFSTEP]\nsimp only [variationOnFromTo, if_pos ab.le, if_neg ab.not_le, neg_neg]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\na : \u03b1\n\u22a2 variationOnFromTo f s a a = -variationOnFromTo f s a a\n[PROOFSTEP]\nsimp only [variationOnFromTo.self, neg_zero]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\na b : \u03b1\nba : b < a\n\u22a2 variationOnFromTo f s a b = -variationOnFromTo f s b a\n[PROOFSTEP]\nsimp only [variationOnFromTo, if_pos ba.le, if_neg ba.not_le, neg_neg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\na b : \u03b1\nh : b \u2264 a\n\u22a2 variationOnFromTo f s a b \u2264 0\n[PROOFSTEP]\nrw [variationOnFromTo.eq_neg_swap]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\na b : \u03b1\nh : b \u2264 a\n\u22a2 -variationOnFromTo f s b a \u2264 0\n[PROOFSTEP]\nexact neg_nonpos_of_nonneg (variationOnFromTo.nonneg_of_le f s h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\na b : \u03b1\nh : b \u2264 a\n\u22a2 variationOnFromTo f s a b = -ENNReal.toReal (eVariationOn f (s \u2229 Icc b a))\n[PROOFSTEP]\nrw [variationOnFromTo.eq_neg_swap, neg_inj, variationOnFromTo.eq_of_le f s h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b c : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nhc : c \u2208 s\n\u22a2 variationOnFromTo f s a b + variationOnFromTo f s b c = variationOnFromTo f s a c\n[PROOFSTEP]\nsymm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b c : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nhc : c \u2208 s\n\u22a2 variationOnFromTo f s a c = variationOnFromTo f s a b + variationOnFromTo f s b c\n[PROOFSTEP]\nrefine' additive_of_isTotal ((\u00b7 : \u03b1) \u2264 \u00b7) (variationOnFromTo f s) (\u00b7 \u2208 s) _ _ ha hb hc\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b c : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nhc : c \u2208 s\n\u22a2 \u2200 {a b : \u03b1}, (fun x => x \u2208 s) a \u2192 (fun x => x \u2208 s) b \u2192 variationOnFromTo f s a b + variationOnFromTo f s b a = 0\n[PROOFSTEP]\nrintro x y _xs _ys\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b c : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nhc : c \u2208 s\nx y : \u03b1\n_xs : x \u2208 s\n_ys : y \u2208 s\n\u22a2 variationOnFromTo f s x y + variationOnFromTo f s y x = 0\n[PROOFSTEP]\nsimp only [variationOnFromTo.eq_neg_swap f s y x, Subtype.coe_mk, add_right_neg, forall_true_left]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b c : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nhc : c \u2208 s\n\u22a2 \u2200 {a b c : \u03b1},\n (fun x x_1 => x \u2264 x_1) a b \u2192\n (fun x x_1 => x \u2264 x_1) b c \u2192\n (fun x => x \u2208 s) a \u2192\n (fun x => x \u2208 s) b \u2192\n (fun x => x \u2208 s) c \u2192 variationOnFromTo f s a c = variationOnFromTo f s a b + variationOnFromTo f s b c\n[PROOFSTEP]\nrintro x y z xy yz xs ys zs\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b c : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nhc : c \u2208 s\nx y z : \u03b1\nxy : x \u2264 y\nyz : y \u2264 z\nxs : x \u2208 s\nys : y \u2208 s\nzs : z \u2208 s\n\u22a2 variationOnFromTo f s x z = variationOnFromTo f s x y + variationOnFromTo f s y z\n[PROOFSTEP]\nrw [variationOnFromTo.eq_of_le f s xy, variationOnFromTo.eq_of_le f s yz, variationOnFromTo.eq_of_le f s (xy.trans yz),\n \u2190 ENNReal.toReal_add (hf x y xs ys) (hf y z ys zs), eVariationOn.Icc_add_Icc f xy yz ys]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nh : variationOnFromTo f s a b = 0\n\u22a2 edist (f a) (f b) = 0\n[PROOFSTEP]\nwlog h' : a \u2264 b\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nh : variationOnFromTo f s a b = 0\nthis :\n \u2200 {\u03b1 : Type u_1} [inst : LinearOrder \u03b1] {E : Type u_2} [inst_1 : PseudoEMetricSpace E],\n (\u03b1 \u2192 E) \u2192\n Set \u03b1 \u2192\n \u2200 {f : \u03b1 \u2192 E} {s : Set \u03b1},\n LocallyBoundedVariationOn f s \u2192\n \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 variationOnFromTo f s a b = 0 \u2192 a \u2264 b \u2192 edist (f a) (f b) = 0\nh' : \u00aca \u2264 b\n\u22a2 edist (f a) (f b) = 0\n[PROOFSTEP]\nrw [edist_comm]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nh : variationOnFromTo f s a b = 0\nthis :\n \u2200 {\u03b1 : Type u_1} [inst : LinearOrder \u03b1] {E : Type u_2} [inst_1 : PseudoEMetricSpace E],\n (\u03b1 \u2192 E) \u2192\n Set \u03b1 \u2192\n \u2200 {f : \u03b1 \u2192 E} {s : Set \u03b1},\n LocallyBoundedVariationOn f s \u2192\n \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 variationOnFromTo f s a b = 0 \u2192 a \u2264 b \u2192 edist (f a) (f b) = 0\nh' : \u00aca \u2264 b\n\u22a2 edist (f b) (f a) = 0\n[PROOFSTEP]\napply this f s hf hb ha _ (le_of_not_le h')\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nh : variationOnFromTo f s a b = 0\nthis :\n \u2200 {\u03b1 : Type u_1} [inst : LinearOrder \u03b1] {E : Type u_2} [inst_1 : PseudoEMetricSpace E],\n (\u03b1 \u2192 E) \u2192\n Set \u03b1 \u2192\n \u2200 {f : \u03b1 \u2192 E} {s : Set \u03b1},\n LocallyBoundedVariationOn f s \u2192\n \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 variationOnFromTo f s a b = 0 \u2192 a \u2264 b \u2192 edist (f a) (f b) = 0\nh' : \u00aca \u2264 b\n\u22a2 variationOnFromTo f s b a = 0\n[PROOFSTEP]\nrw [variationOnFromTo.eq_neg_swap, h, neg_zero]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : LinearOrder \u03b1\u271d\nE\u271d : Type u_2\ninst\u271d\u00b2 : PseudoEMetricSpace E\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nh : variationOnFromTo f s a b = 0\nh' : a \u2264 b\n\u22a2 edist (f a) (f b) = 0\n[PROOFSTEP]\napply le_antisymm _ (zero_le _)\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : LinearOrder \u03b1\u271d\nE\u271d : Type u_2\ninst\u271d\u00b2 : PseudoEMetricSpace E\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nh : variationOnFromTo f s a b = 0\nh' : a \u2264 b\n\u22a2 edist (f a) (f b) \u2264 0\n[PROOFSTEP]\nrw [\u2190 ENNReal.ofReal_zero, \u2190 h, variationOnFromTo.eq_of_le f s h', ENNReal.ofReal_toReal (hf a b ha hb)]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : LinearOrder \u03b1\u271d\nE\u271d : Type u_2\ninst\u271d\u00b2 : PseudoEMetricSpace E\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nh : variationOnFromTo f s a b = 0\nh' : a \u2264 b\n\u22a2 edist (f a) (f b) \u2264 eVariationOn f (s \u2229 Icc a b)\n[PROOFSTEP]\napply eVariationOn.edist_le\n[GOAL]\ncase hx\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : LinearOrder \u03b1\u271d\nE\u271d : Type u_2\ninst\u271d\u00b2 : PseudoEMetricSpace E\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nh : variationOnFromTo f s a b = 0\nh' : a \u2264 b\n\u22a2 a \u2208 s \u2229 Icc a b\ncase hy\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : LinearOrder \u03b1\u271d\nE\u271d : Type u_2\ninst\u271d\u00b2 : PseudoEMetricSpace E\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nh : variationOnFromTo f s a b = 0\nh' : a \u2264 b\n\u22a2 b \u2208 s \u2229 Icc a b\n[PROOFSTEP]\nexacts [\u27e8ha, \u27e8le_rfl, h'\u27e9\u27e9, \u27e8hb, \u27e8h', le_rfl\u27e9\u27e9]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b c : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nhc : c \u2208 s\n\u22a2 variationOnFromTo f s a b = variationOnFromTo f s a c \u2194 variationOnFromTo f s b c = 0\n[PROOFSTEP]\nsimp only [\u2190 variationOnFromTo.add hf ha hb hc, self_eq_add_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nab : a \u2264 b\n\u22a2 variationOnFromTo f s a b = 0 \u2194 \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2229 Icc a b \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2229 Icc a b \u2192 edist (f x) (f y) = 0\n[PROOFSTEP]\nrw [variationOnFromTo.eq_of_le _ _ ab, ENNReal.toReal_eq_zero_iff, or_iff_left (hf a b ha hb), eVariationOn.eq_zero_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nba : b \u2264 a\n\u22a2 variationOnFromTo f s a b = 0 \u2194 \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2229 Icc b a \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2229 Icc b a \u2192 edist (f x) (f y) = 0\n[PROOFSTEP]\nrw [variationOnFromTo.eq_of_ge _ _ ba, neg_eq_zero, ENNReal.toReal_eq_zero_iff, or_iff_left (hf b a hb ha),\n eVariationOn.eq_zero_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 variationOnFromTo f s a b = 0 \u2194 \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2229 uIcc a b \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2229 uIcc a b \u2192 edist (f x) (f y) = 0\n[PROOFSTEP]\nrcases le_total a b with (ab | ba)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nab : a \u2264 b\n\u22a2 variationOnFromTo f s a b = 0 \u2194 \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2229 uIcc a b \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2229 uIcc a b \u2192 edist (f x) (f y) = 0\n[PROOFSTEP]\nrw [uIcc_of_le ab]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nab : a \u2264 b\n\u22a2 variationOnFromTo f s a b = 0 \u2194 \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2229 Icc a b \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2229 Icc a b \u2192 edist (f x) (f y) = 0\n[PROOFSTEP]\nexact variationOnFromTo.eq_zero_iff_of_le hf ha hb ab\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nba : b \u2264 a\n\u22a2 variationOnFromTo f s a b = 0 \u2194 \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2229 uIcc a b \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2229 uIcc a b \u2192 edist (f x) (f y) = 0\n[PROOFSTEP]\nrw [uIcc_of_ge ba]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nba : b \u2264 a\n\u22a2 variationOnFromTo f s a b = 0 \u2194 \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2229 Icc b a \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2229 Icc b a \u2192 edist (f x) (f y) = 0\n[PROOFSTEP]\nexact variationOnFromTo.eq_zero_iff_of_ge hf ha hb ba\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\n\u22a2 MonotoneOn (variationOnFromTo f s a) s\n[PROOFSTEP]\nrintro b bs c cs bc\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nc : \u03b1\ncs : c \u2208 s\nbc : b \u2264 c\n\u22a2 variationOnFromTo f s a b \u2264 variationOnFromTo f s a c\n[PROOFSTEP]\nrw [\u2190 variationOnFromTo.add hf as bs cs]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nc : \u03b1\ncs : c \u2208 s\nbc : b \u2264 c\n\u22a2 variationOnFromTo f s a b \u2264 variationOnFromTo f s a b + variationOnFromTo f s b c\n[PROOFSTEP]\nexact le_add_of_nonneg_right (variationOnFromTo.nonneg_of_le f s bc)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\nb : \u03b1\nbs : b \u2208 s\n\u22a2 AntitoneOn (fun a => variationOnFromTo f s a b) s\n[PROOFSTEP]\nrintro a as c cs ac\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\nb : \u03b1\nbs : b \u2208 s\na : \u03b1\nas : a \u2208 s\nc : \u03b1\ncs : c \u2208 s\nac : a \u2264 c\n\u22a2 (fun a => variationOnFromTo f s a b) c \u2264 (fun a => variationOnFromTo f s a b) a\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\nb : \u03b1\nbs : b \u2208 s\na : \u03b1\nas : a \u2208 s\nc : \u03b1\ncs : c \u2208 s\nac : a \u2264 c\n\u22a2 variationOnFromTo f s c b \u2264 variationOnFromTo f s a b\n[PROOFSTEP]\nrw [\u2190 variationOnFromTo.add hf as cs bs]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\nb : \u03b1\nbs : b \u2208 s\na : \u03b1\nas : a \u2208 s\nc : \u03b1\ncs : c \u2208 s\nac : a \u2264 c\n\u22a2 variationOnFromTo f s c b \u2264 variationOnFromTo f s a c + variationOnFromTo f s c b\n[PROOFSTEP]\nexact le_add_of_nonneg_left (variationOnFromTo.nonneg_of_le f s ac)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\n\u22a2 MonotoneOn (variationOnFromTo f s a - f) s\n[PROOFSTEP]\nrintro b bs c cs bc\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nc : \u03b1\ncs : c \u2208 s\nbc : b \u2264 c\n\u22a2 (variationOnFromTo f s a - f) b \u2264 (variationOnFromTo f s a - f) c\n[PROOFSTEP]\nrw [Pi.sub_apply, Pi.sub_apply, le_sub_iff_add_le, add_comm_sub, \u2190 le_sub_iff_add_le']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nc : \u03b1\ncs : c \u2208 s\nbc : b \u2264 c\n\u22a2 f c - f b \u2264 variationOnFromTo f s a c - variationOnFromTo f s a b\n[PROOFSTEP]\ncalc\n f c - f b \u2264 |f c - f b| := le_abs_self _\n _ = dist (f b) (f c) := by rw [dist_comm, Real.dist_eq]\n _ \u2264 variationOnFromTo f s b c := by\n rw [variationOnFromTo.eq_of_le f s bc, dist_edist]\n apply ENNReal.toReal_mono (hf b c bs cs)\n apply eVariationOn.edist_le f\n exacts [\u27e8bs, le_rfl, bc\u27e9, \u27e8cs, bc, le_rfl\u27e9]\n _ = variationOnFromTo f s a c - variationOnFromTo f s a b := by\n rw [\u2190 variationOnFromTo.add hf as bs cs, add_sub_cancel']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nc : \u03b1\ncs : c \u2208 s\nbc : b \u2264 c\n\u22a2 |f c - f b| = dist (f b) (f c)\n[PROOFSTEP]\nrw [dist_comm, Real.dist_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nc : \u03b1\ncs : c \u2208 s\nbc : b \u2264 c\n\u22a2 dist (f b) (f c) \u2264 variationOnFromTo f s b c\n[PROOFSTEP]\nrw [variationOnFromTo.eq_of_le f s bc, dist_edist]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nc : \u03b1\ncs : c \u2208 s\nbc : b \u2264 c\n\u22a2 ENNReal.toReal (edist (f b) (f c)) \u2264 ENNReal.toReal (eVariationOn f (s \u2229 Icc b c))\n[PROOFSTEP]\napply ENNReal.toReal_mono (hf b c bs cs)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nc : \u03b1\ncs : c \u2208 s\nbc : b \u2264 c\n\u22a2 edist (f b) (f c) \u2264 eVariationOn f (s \u2229 Icc b c)\n[PROOFSTEP]\napply eVariationOn.edist_le f\n[GOAL]\ncase hx\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nc : \u03b1\ncs : c \u2208 s\nbc : b \u2264 c\n\u22a2 b \u2208 s \u2229 Icc b c\ncase hy\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nc : \u03b1\ncs : c \u2208 s\nbc : b \u2264 c\n\u22a2 c \u2208 s \u2229 Icc b c\n[PROOFSTEP]\nexacts [\u27e8bs, le_rfl, bc\u27e9, \u27e8cs, bc, le_rfl\u27e9]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nc : \u03b1\ncs : c \u2208 s\nbc : b \u2264 c\n\u22a2 variationOnFromTo f s b c = variationOnFromTo f s a c - variationOnFromTo f s a b\n[PROOFSTEP]\nrw [\u2190 variationOnFromTo.add hf as bs cs, add_sub_cancel']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns : Set \u03b1\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\n\u22a2 variationOnFromTo (f \u2218 \u03c6) t x y = variationOnFromTo f (\u03c6 '' t) (\u03c6 x) (\u03c6 y)\n[PROOFSTEP]\nrcases le_total x y with (h | h)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns : Set \u03b1\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : x \u2264 y\n\u22a2 variationOnFromTo (f \u2218 \u03c6) t x y = variationOnFromTo f (\u03c6 '' t) (\u03c6 x) (\u03c6 y)\n[PROOFSTEP]\nrw [variationOnFromTo.eq_of_le _ _ h, variationOnFromTo.eq_of_le _ _ (h\u03c6 hx hy h),\n eVariationOn.comp_inter_Icc_eq_of_monotoneOn f \u03c6 h\u03c6 hx hy]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns : Set \u03b1\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 E\nt : Set \u03b2\n\u03c6 : \u03b2 \u2192 \u03b1\nh\u03c6 : MonotoneOn \u03c6 t\nx y : \u03b2\nhx : x \u2208 t\nhy : y \u2208 t\nh : y \u2264 x\n\u22a2 variationOnFromTo (f \u2218 \u03c6) t x y = variationOnFromTo f (\u03c6 '' t) (\u03c6 x) (\u03c6 y)\n[PROOFSTEP]\nrw [variationOnFromTo.eq_of_ge _ _ h, variationOnFromTo.eq_of_ge _ _ (h\u03c6 hy hx h),\n eVariationOn.comp_inter_Icc_eq_of_monotoneOn f \u03c6 h\u03c6 hy hx]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nh : LocallyBoundedVariationOn f s\n\u22a2 \u2203 p q, MonotoneOn p s \u2227 MonotoneOn q s \u2227 f = p - q\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | \u27e8c, cs\u27e9)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\nh : LocallyBoundedVariationOn f \u2205\n\u22a2 \u2203 p q, MonotoneOn p \u2205 \u2227 MonotoneOn q \u2205 \u2227 f = p - q\n[PROOFSTEP]\nexact \u27e8f, 0, subsingleton_empty.monotoneOn _, subsingleton_empty.monotoneOn _, (sub_zero f).symm\u27e9\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nh : LocallyBoundedVariationOn f s\nc : \u03b1\ncs : c \u2208 s\n\u22a2 \u2203 p q, MonotoneOn p s \u2227 MonotoneOn q s \u2227 f = p - q\n[PROOFSTEP]\nexact \u27e8_, _, variationOnFromTo.monotoneOn h cs, variationOnFromTo.sub_self_monotoneOn h cs, (sub_sub_cancel _ _).symm\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\nF : Type u_3\ninst\u271d : PseudoEMetricSpace F\nf : E \u2192 F\nC : \u211d\u22650\nt : Set E\nh : LipschitzOnWith C f t\ng : \u03b1 \u2192 E\ns : Set \u03b1\nhg : MapsTo g s t\n\u22a2 eVariationOn (f \u2218 g) s \u2264 \u2191C * eVariationOn g s\n[PROOFSTEP]\napply iSup_le _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\nF : Type u_3\ninst\u271d : PseudoEMetricSpace F\nf : E \u2192 F\nC : \u211d\u22650\nt : Set E\nh : LipschitzOnWith C f t\ng : \u03b1 \u2192 E\ns : Set \u03b1\nhg : MapsTo g s t\n\u22a2 \u2200 (i : \u2115 \u00d7 { u // Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s }),\n \u2211 i_1 in Finset.range i.fst, edist ((f \u2218 g) (\u2191i.snd (i_1 + 1))) ((f \u2218 g) (\u2191i.snd i_1)) \u2264 \u2191C * eVariationOn g s\n[PROOFSTEP]\nrintro \u27e8n, \u27e8u, hu, us\u27e9\u27e9\n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\nF : Type u_3\ninst\u271d : PseudoEMetricSpace F\nf : E \u2192 F\nC : \u211d\u22650\nt : Set E\nh : LipschitzOnWith C f t\ng : \u03b1 \u2192 E\ns : Set \u03b1\nhg : MapsTo g s t\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\n\u22a2 \u2211 i in Finset.range (n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).fst,\n edist ((f \u2218 g) (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd (i + 1)))\n ((f \u2218 g) (\u2191(n, { val := u, property := (_ : Monotone u \u2227 \u2200 (i : \u2115), u i \u2208 s) }).snd i)) \u2264\n \u2191C * eVariationOn g s\n[PROOFSTEP]\ncalc\n (\u2211 i in Finset.range n, edist (f (g (u (i + 1)))) (f (g (u i)))) \u2264\n \u2211 i in Finset.range n, C * edist (g (u (i + 1))) (g (u i)) :=\n Finset.sum_le_sum fun i _ => h (hg (us _)) (hg (us _))\n _ = C * \u2211 i in Finset.range n, edist (g (u (i + 1))) (g (u i)) := by rw [Finset.mul_sum]\n _ \u2264 C * eVariationOn g s := mul_le_mul_left' (eVariationOn.sum_le _ _ hu us) _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b9 : PseudoEMetricSpace E\nF : Type u_3\ninst\u271d : PseudoEMetricSpace F\nf : E \u2192 F\nC : \u211d\u22650\nt : Set E\nh : LipschitzOnWith C f t\ng : \u03b1 \u2192 E\ns : Set \u03b1\nhg : MapsTo g s t\nn : \u2115\nu : \u2115 \u2192 \u03b1\nhu : Monotone u\nus : \u2200 (i : \u2115), u i \u2208 s\n\u22a2 \u2211 i in Finset.range n, \u2191C * edist (g (u (i + 1))) (g (u i)) =\n \u2191C * \u2211 i in Finset.range n, edist (g (u (i + 1))) (g (u i))\n[PROOFSTEP]\nrw [Finset.mul_sum]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 \u211d\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nobtain \u27e8p, q, hp, hq, fpq\u27e9 : \u2203 p q, MonotoneOn p s \u2227 MonotoneOn q s \u2227 f = p - q := h.exists_monotoneOn_sub_monotoneOn\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 \u211d\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\np q : \u211d \u2192 \u211d\nhp : MonotoneOn p s\nhq : MonotoneOn q s\nfpq : f = p - q\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nsubst f\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\ns : Set \u211d\np q : \u211d \u2192 \u211d\nhp : MonotoneOn p s\nhq : MonotoneOn q s\nh : LocallyBoundedVariationOn (p - q) s\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d (p - q) s x\n[PROOFSTEP]\nfilter_upwards [hp.ae_differentiableWithinAt_of_mem, hq.ae_differentiableWithinAt_of_mem] with x hxp hxq xs\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\ns : Set \u211d\np q : \u211d \u2192 \u211d\nhp : MonotoneOn p s\nhq : MonotoneOn q s\nh : LocallyBoundedVariationOn (p - q) s\nx : \u211d\nhxp : x \u2208 s \u2192 DifferentiableWithinAt \u211d p s x\nhxq : x \u2208 s \u2192 DifferentiableWithinAt \u211d q s x\nxs : x \u2208 s\n\u22a2 DifferentiableWithinAt \u211d (p - q) s x\n[PROOFSTEP]\nexact (hxp xs).sub (hxq xs)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u2074 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\nf : \u211d \u2192 \u03b9 \u2192 \u211d\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nhave A : \u2200 i : \u03b9, LipschitzWith 1 fun x : \u03b9 \u2192 \u211d => x i := fun i => LipschitzWith.eval i\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u2074 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\nf : \u211d \u2192 \u03b9 \u2192 \u211d\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\nA : \u2200 (i : \u03b9), LipschitzWith 1 fun x => x i\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nhave : \u2200 i : \u03b9, \u2200\u1d50 x, x \u2208 s \u2192 DifferentiableWithinAt \u211d (fun x : \u211d => f x i) s x := fun i \u21a6\n by\n apply ae_differentiableWithinAt_of_mem_real\n exact LipschitzWith.comp_locallyBoundedVariationOn (A i) h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u2074 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\nf : \u211d \u2192 \u03b9 \u2192 \u211d\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\nA : \u2200 (i : \u03b9), LipschitzWith 1 fun x => x i\ni : \u03b9\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d (fun x => f x i) s x\n[PROOFSTEP]\napply ae_differentiableWithinAt_of_mem_real\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2075 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u2074 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\nf : \u211d \u2192 \u03b9 \u2192 \u211d\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\nA : \u2200 (i : \u03b9), LipschitzWith 1 fun x => x i\ni : \u03b9\n\u22a2 LocallyBoundedVariationOn (fun x => f x i) s\n[PROOFSTEP]\nexact LipschitzWith.comp_locallyBoundedVariationOn (A i) h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u2074 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\nf : \u211d \u2192 \u03b9 \u2192 \u211d\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\nA : \u2200 (i : \u03b9), LipschitzWith 1 fun x => x i\nthis : \u2200 (i : \u03b9), \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d (fun x => f x i) s x\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nfilter_upwards [ae_all_iff.2 this] with x hx xs\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2075 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u2074 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\nf : \u211d \u2192 \u03b9 \u2192 \u211d\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\nA : \u2200 (i : \u03b9), LipschitzWith 1 fun x => x i\nthis : \u2200 (i : \u03b9), \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d (fun x => f x i) s x\nx : \u211d\nhx : \u2200 (i : \u03b9), x \u2208 s \u2192 DifferentiableWithinAt \u211d (fun x => f x i) s x\nxs : x \u2208 s\n\u22a2 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nexact differentiableWithinAt_pi.2 fun i => hx i xs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 V\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nlet A := (Basis.ofVectorSpace \u211d V).equivFun.toContinuousLinearEquiv\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 V\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\nA : V \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d V) \u2192 \u211d :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d V))\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nsuffices H : \u2200\u1d50 x, x \u2208 s \u2192 DifferentiableWithinAt \u211d (A \u2218 f) s x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 V\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\nA : V \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d V) \u2192 \u211d :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d V))\nH : \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d (\u2191A \u2218 f) s x\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nfilter_upwards [H] with x hx xs\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 V\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\nA : V \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d V) \u2192 \u211d :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d V))\nH : \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d (\u2191A \u2218 f) s x\nx : \u211d\nhx :\n x \u2208 s \u2192\n DifferentiableWithinAt \u211d (\u2191(LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d V))) \u2218 f) s x\nxs : x \u2208 s\n\u22a2 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nhave : f = (A.symm \u2218 A) \u2218 f := by simp only [ContinuousLinearEquiv.symm_comp_self, Function.comp.left_id]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 V\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\nA : V \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d V) \u2192 \u211d :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d V))\nH : \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d (\u2191A \u2218 f) s x\nx : \u211d\nhx :\n x \u2208 s \u2192\n DifferentiableWithinAt \u211d (\u2191(LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d V))) \u2218 f) s x\nxs : x \u2208 s\n\u22a2 f = (\u2191(ContinuousLinearEquiv.symm A) \u2218 \u2191A) \u2218 f\n[PROOFSTEP]\nsimp only [ContinuousLinearEquiv.symm_comp_self, Function.comp.left_id]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 V\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\nA : V \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d V) \u2192 \u211d :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d V))\nH : \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d (\u2191A \u2218 f) s x\nx : \u211d\nhx :\n x \u2208 s \u2192\n DifferentiableWithinAt \u211d (\u2191(LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d V))) \u2218 f) s x\nxs : x \u2208 s\nthis : f = (\u2191(ContinuousLinearEquiv.symm A) \u2218 \u2191A) \u2218 f\n\u22a2 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 V\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\nA : V \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d V) \u2192 \u211d :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d V))\nH : \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d (\u2191A \u2218 f) s x\nx : \u211d\nhx :\n x \u2208 s \u2192\n DifferentiableWithinAt \u211d (\u2191(LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d V))) \u2218 f) s x\nxs : x \u2208 s\nthis : f = (\u2191(ContinuousLinearEquiv.symm A) \u2218 \u2191A) \u2218 f\n\u22a2 DifferentiableWithinAt \u211d ((\u2191(ContinuousLinearEquiv.symm A) \u2218 \u2191A) \u2218 f) s x\n[PROOFSTEP]\nexact A.symm.differentiableAt.comp_differentiableWithinAt x (hx xs)\n[GOAL]\ncase H\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 V\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\nA : V \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d V) \u2192 \u211d :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d V))\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d (\u2191A \u2218 f) s x\n[PROOFSTEP]\napply ae_differentiableWithinAt_of_mem_pi\n[GOAL]\ncase H.h\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 V\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\nA : V \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d V) \u2192 \u211d :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d V))\n\u22a2 LocallyBoundedVariationOn (\u2191A \u2218 f) s\n[PROOFSTEP]\nexact A.lipschitz.comp_locallyBoundedVariationOn h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 V\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\nhs : MeasurableSet s\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume s, DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nrw [ae_restrict_iff' hs]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 V\ns : Set \u211d\nh : LocallyBoundedVariationOn f s\nhs : MeasurableSet s\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nexact h.ae_differentiableWithinAt_of_mem\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 V\nh : LocallyBoundedVariationOn f univ\n\u22a2 \u2200\u1d50 (x : \u211d), DifferentiableAt \u211d f x\n[PROOFSTEP]\nfilter_upwards [h.ae_differentiableWithinAt_of_mem] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 V\nh : LocallyBoundedVariationOn f univ\nx : \u211d\nhx : x \u2208 univ \u2192 DifferentiableWithinAt \u211d f univ x\n\u22a2 DifferentiableAt \u211d f x\n[PROOFSTEP]\nrw [differentiableWithinAt_univ] at hx \n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d\u00b3 : PseudoEMetricSpace E\nV : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nf : \u211d \u2192 V\nh : LocallyBoundedVariationOn f univ\nx : \u211d\nhx : x \u2208 univ \u2192 DifferentiableAt \u211d f x\n\u22a2 DifferentiableAt \u211d f x\n[PROOFSTEP]\nexact hx (mem_univ _)\n", "meta": {"mathlib_filename": "Mathlib.Analysis.BoundedVariation", "llama_tokens": 119388, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952052, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.42690164157965366}} {"text": "[GOAL]\n\u22a2 IsPrincipalIdealRing { x // x \u2208 \ud835\udcde \u211a }\n[PROOFSTEP]\nconvert\n IsPrincipalIdealRing.of_surjective (Rat.ringOfIntegersEquiv.symm : \u2124 \u2192+* ringOfIntegers \u211a)\n Rat.ringOfIntegersEquiv.symm.surjective\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.NumberField.ClassNumber", "llama_tokens": 98, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7490872243177518, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.42686934091794226}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n\u22a2 IsRadical I \u2194 IsReduced (R \u29f8 I)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 @Ideal.mk_ker R _ I]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n| IsRadical I\n[PROOFSTEP]\nrw [\u2190 @Ideal.mk_ker R _ I]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n| IsRadical I\n[PROOFSTEP]\nrw [\u2190 @Ideal.mk_ker R _ I]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n| IsRadical I\n[PROOFSTEP]\nrw [\u2190 @Ideal.mk_ker R _ I]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n\u22a2 IsRadical (RingHom.ker (Quotient.mk I)) \u2194 IsReduced (R \u29f8 I)\n[PROOFSTEP]\nexact RingHom.ker_isRadical_iff_reduced_of_surjective (@Ideal.Quotient.mk_surjective R _ I)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI : IsNilpotent I\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\n\u22a2 P I\n[PROOFSTEP]\nobtain \u27e8n, hI : I ^ n = \u22a5\u27e9 := hI\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn : \u2115\nhI : I ^ n = \u22a5\n\u22a2 P I\n[PROOFSTEP]\ninduction' n using Nat.strong_induction_on with n H generalizing S\n[GOAL]\ncase intro.h\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn\u271d : \u2115\nhI\u271d : I\u271d ^ n\u271d = \u22a5\nn : \u2115\nH : \u2200 (m : \u2115), m < n \u2192 \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI : I ^ n = \u22a5\n\u22a2 P I\n[PROOFSTEP]\nby_cases hI' : I = \u22a5\n[GOAL]\ncase pos\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn\u271d : \u2115\nhI\u271d : I\u271d ^ n\u271d = \u22a5\nn : \u2115\nH : \u2200 (m : \u2115), m < n \u2192 \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI : I ^ n = \u22a5\nhI' : I = \u22a5\n\u22a2 P I\n[PROOFSTEP]\nsubst hI'\n[GOAL]\ncase pos\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn\u271d : \u2115\nhI\u271d : I ^ n\u271d = \u22a5\nn : \u2115\nH : \u2200 (m : \u2115), m < n \u2192 \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nhI : \u22a5 ^ n = \u22a5\n\u22a2 P \u22a5\n[PROOFSTEP]\napply h\u2081\n[GOAL]\ncase pos.a\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn\u271d : \u2115\nhI\u271d : I ^ n\u271d = \u22a5\nn : \u2115\nH : \u2200 (m : \u2115), m < n \u2192 \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nhI : \u22a5 ^ n = \u22a5\n\u22a2 \u22a5 ^ 2 = \u22a5\n[PROOFSTEP]\nrw [\u2190 Ideal.zero_eq_bot, zero_pow]\n[GOAL]\ncase pos.a\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn\u271d : \u2115\nhI\u271d : I ^ n\u271d = \u22a5\nn : \u2115\nH : \u2200 (m : \u2115), m < n \u2192 \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nhI : \u22a5 ^ n = \u22a5\n\u22a2 0 < 2\n[PROOFSTEP]\nexact zero_lt_two\n[GOAL]\ncase neg\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn\u271d : \u2115\nhI\u271d : I\u271d ^ n\u271d = \u22a5\nn : \u2115\nH : \u2200 (m : \u2115), m < n \u2192 \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI : I ^ n = \u22a5\nhI' : \u00acI = \u22a5\n\u22a2 P I\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase neg.zero\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn : \u2115\nhI\u271d : I\u271d ^ n = \u22a5\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI' : \u00acI = \u22a5\nH : \u2200 (m : \u2115), m < Nat.zero \u2192 \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nhI : I ^ Nat.zero = \u22a5\n\u22a2 P I\n[PROOFSTEP]\nrw [pow_zero, Ideal.one_eq_top] at hI \n[GOAL]\ncase neg.zero\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn : \u2115\nhI\u271d : I\u271d ^ n = \u22a5\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI' : \u00acI = \u22a5\nH : \u2200 (m : \u2115), m < Nat.zero \u2192 \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nhI : \u22a4 = \u22a5\n\u22a2 P I\n[PROOFSTEP]\nhaveI := subsingleton_of_bot_eq_top hI.symm\n[GOAL]\ncase neg.zero\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn : \u2115\nhI\u271d : I\u271d ^ n = \u22a5\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI' : \u00acI = \u22a5\nH : \u2200 (m : \u2115), m < Nat.zero \u2192 \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nhI : \u22a4 = \u22a5\nthis : Subsingleton (Ideal S)\n\u22a2 P I\n[PROOFSTEP]\nexact (hI' (Subsingleton.elim _ _)).elim\n[GOAL]\ncase neg.succ\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn\u271d : \u2115\nhI\u271d : I\u271d ^ n\u271d = \u22a5\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI' : \u00acI = \u22a5\nn : \u2115\nH :\n \u2200 (m : \u2115), m < Nat.succ n \u2192 \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nhI : I ^ Nat.succ n = \u22a5\n\u22a2 P I\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase neg.succ.zero\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn : \u2115\nhI\u271d : I\u271d ^ n = \u22a5\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI' : \u00acI = \u22a5\nH :\n \u2200 (m : \u2115),\n m < Nat.succ Nat.zero \u2192 \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nhI : I ^ Nat.succ Nat.zero = \u22a5\n\u22a2 P I\n[PROOFSTEP]\nrw [pow_one] at hI \n[GOAL]\ncase neg.succ.zero\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn : \u2115\nhI\u271d : I\u271d ^ n = \u22a5\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI' : \u00acI = \u22a5\nH :\n \u2200 (m : \u2115),\n m < Nat.succ Nat.zero \u2192 \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nhI : I = \u22a5\n\u22a2 P I\n[PROOFSTEP]\nexact (hI' hI).elim\n[GOAL]\ncase neg.succ.succ\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn\u271d : \u2115\nhI\u271d : I\u271d ^ n\u271d = \u22a5\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI' : \u00acI = \u22a5\nn : \u2115\nH :\n \u2200 (m : \u2115),\n m < Nat.succ (Nat.succ n) \u2192\n \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nhI : I ^ Nat.succ (Nat.succ n) = \u22a5\n\u22a2 P I\n[PROOFSTEP]\napply h\u2082 (I ^ 2) _ (Ideal.pow_le_self two_ne_zero)\n[GOAL]\ncase neg.succ.succ.a\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn\u271d : \u2115\nhI\u271d : I\u271d ^ n\u271d = \u22a5\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI' : \u00acI = \u22a5\nn : \u2115\nH :\n \u2200 (m : \u2115),\n m < Nat.succ (Nat.succ n) \u2192\n \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nhI : I ^ Nat.succ (Nat.succ n) = \u22a5\n\u22a2 P (I ^ 2)\n[PROOFSTEP]\napply H n.succ _ (I ^ 2)\n[GOAL]\ncase neg.succ.succ.a\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn\u271d : \u2115\nhI\u271d : I\u271d ^ n\u271d = \u22a5\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI' : \u00acI = \u22a5\nn : \u2115\nH :\n \u2200 (m : \u2115),\n m < Nat.succ (Nat.succ n) \u2192\n \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nhI : I ^ Nat.succ (Nat.succ n) = \u22a5\n\u22a2 (I ^ 2) ^ Nat.succ n = \u22a5\n[PROOFSTEP]\nrw [\u2190 pow_mul, eq_bot_iff, \u2190 hI, Nat.succ_eq_add_one, Nat.succ_eq_add_one]\n[GOAL]\ncase neg.succ.succ.a\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn\u271d : \u2115\nhI\u271d : I\u271d ^ n\u271d = \u22a5\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI' : \u00acI = \u22a5\nn : \u2115\nH :\n \u2200 (m : \u2115),\n m < Nat.succ (Nat.succ n) \u2192\n \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nhI : I ^ Nat.succ (Nat.succ n) = \u22a5\n\u22a2 I ^ (2 * (n + 1)) \u2264 I ^ (n + 1 + 1)\n[PROOFSTEP]\napply Ideal.pow_le_pow (by linarith)\n[GOAL]\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn\u271d : \u2115\nhI\u271d : I\u271d ^ n\u271d = \u22a5\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI' : \u00acI = \u22a5\nn : \u2115\nH :\n \u2200 (m : \u2115),\n m < Nat.succ (Nat.succ n) \u2192\n \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nhI : I ^ Nat.succ (Nat.succ n) = \u22a5\n\u22a2 n + 1 + 1 \u2264 2 * (n + 1)\n[PROOFSTEP]\nlinarith\n[GOAL]\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn\u271d : \u2115\nhI\u271d : I\u271d ^ n\u271d = \u22a5\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI' : \u00acI = \u22a5\nn : \u2115\nH :\n \u2200 (m : \u2115),\n m < Nat.succ (Nat.succ n) \u2192\n \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nhI : I ^ Nat.succ (Nat.succ n) = \u22a5\n\u22a2 Nat.succ n < Nat.succ (Nat.succ n)\n[PROOFSTEP]\nexact n.succ.lt_succ_self\n[GOAL]\ncase neg.succ.succ.a\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn\u271d : \u2115\nhI\u271d : I\u271d ^ n\u271d = \u22a5\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI' : \u00acI = \u22a5\nn : \u2115\nH :\n \u2200 (m : \u2115),\n m < Nat.succ (Nat.succ n) \u2192\n \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nhI : I ^ Nat.succ (Nat.succ n) = \u22a5\n\u22a2 P (map (Quotient.mk (I ^ 2)) I)\n[PROOFSTEP]\napply h\u2081\n[GOAL]\ncase neg.succ.succ.a.a\nR : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nI\u271d : Ideal S\u271d\nP : \u2983S : Type u_2\u2984 \u2192 [inst : CommRing S] \u2192 Ideal S \u2192 Prop\nh\u2081 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I : Ideal S), I ^ 2 = \u22a5 \u2192 P I\nh\u2082 : \u2200 \u2983S : Type u_2\u2984 [inst : CommRing S] (I J : Ideal S), I \u2264 J \u2192 P I \u2192 P (map (Quotient.mk I) J) \u2192 P J\nn\u271d : \u2115\nhI\u271d : I\u271d ^ n\u271d = \u22a5\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nI : Ideal S\nhI' : \u00acI = \u22a5\nn : \u2115\nH :\n \u2200 (m : \u2115),\n m < Nat.succ (Nat.succ n) \u2192\n \u2200 {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = \u22a5 \u2192 P I\nhI : I ^ Nat.succ (Nat.succ n) = \u22a5\n\u22a2 map (Quotient.mk (I ^ 2)) I ^ 2 = \u22a5\n[PROOFSTEP]\nrw [\u2190 Ideal.map_pow, Ideal.map_quotient_self]\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring R\u271d\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R\u271d S\nI\u271d : Ideal S\nR : Type u_3\ninst\u271d : CommRing R\nI : Ideal R\nhI : IsNilpotent I\nx : R\n\u22a2 IsUnit (\u2191(Ideal.Quotient.mk I) x) \u2194 IsUnit x\n[PROOFSTEP]\nrefine' \u27e8_, fun h => h.map <| Ideal.Quotient.mk I\u27e9\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring R\u271d\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R\u271d S\nI\u271d : Ideal S\nR : Type u_3\ninst\u271d : CommRing R\nI : Ideal R\nhI : IsNilpotent I\nx : R\n\u22a2 IsUnit (\u2191(Ideal.Quotient.mk I) x) \u2192 IsUnit x\n[PROOFSTEP]\nrevert x\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring R\u271d\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R\u271d S\nI\u271d : Ideal S\nR : Type u_3\ninst\u271d : CommRing R\nI : Ideal R\nhI : IsNilpotent I\n\u22a2 \u2200 {x : R}, IsUnit (\u2191(Ideal.Quotient.mk I) x) \u2192 IsUnit x\n[PROOFSTEP]\napply Ideal.IsNilpotent.induction_on (R := R) (S := R) I hI\n[GOAL]\ncase h\u2081\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring R\u271d\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R\u271d S\nI\u271d : Ideal S\nR : Type u_3\ninst\u271d : CommRing R\nI : Ideal R\nhI : IsNilpotent I\n\u22a2 \u2200 \u2983S : Type u_3\u2984 [inst : CommRing S] (I : Ideal S),\n I ^ 2 = \u22a5 \u2192 \u2200 {x : S}, IsUnit (\u2191(Ideal.Quotient.mk I) x) \u2192 IsUnit x\n[PROOFSTEP]\nclear hI I\n[GOAL]\ncase h\u2082\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring R\u271d\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R\u271d S\nI\u271d : Ideal S\nR : Type u_3\ninst\u271d : CommRing R\nI : Ideal R\nhI : IsNilpotent I\n\u22a2 \u2200 \u2983S : Type u_3\u2984 [inst : CommRing S] (I J : Ideal S),\n I \u2264 J \u2192\n (\u2200 {x : S}, IsUnit (\u2191(Ideal.Quotient.mk I) x) \u2192 IsUnit x) \u2192\n (\u2200 {x : S \u29f8 I}, IsUnit (\u2191(Ideal.Quotient.mk (Ideal.map (Ideal.Quotient.mk I) J)) x) \u2192 IsUnit x) \u2192\n \u2200 {x : S}, IsUnit (\u2191(Ideal.Quotient.mk J) x) \u2192 IsUnit x\n[PROOFSTEP]\nclear hI I\n[GOAL]\ncase h\u2081\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring R\u271d\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R\u271d S\nI : Ideal S\nR : Type u_3\ninst\u271d : CommRing R\n\u22a2 \u2200 \u2983S : Type u_3\u2984 [inst : CommRing S] (I : Ideal S),\n I ^ 2 = \u22a5 \u2192 \u2200 {x : S}, IsUnit (\u2191(Ideal.Quotient.mk I) x) \u2192 IsUnit x\ncase h\u2082\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring R\u271d\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R\u271d S\nI : Ideal S\nR : Type u_3\ninst\u271d : CommRing R\n\u22a2 \u2200 \u2983S : Type u_3\u2984 [inst : CommRing S] (I J : Ideal S),\n I \u2264 J \u2192\n (\u2200 {x : S}, IsUnit (\u2191(Ideal.Quotient.mk I) x) \u2192 IsUnit x) \u2192\n (\u2200 {x : S \u29f8 I}, IsUnit (\u2191(Ideal.Quotient.mk (Ideal.map (Ideal.Quotient.mk I) J)) x) \u2192 IsUnit x) \u2192\n \u2200 {x : S}, IsUnit (\u2191(Ideal.Quotient.mk J) x) \u2192 IsUnit x\n[PROOFSTEP]\nswap\n[GOAL]\ncase h\u2082\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring R\u271d\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R\u271d S\nI : Ideal S\nR : Type u_3\ninst\u271d : CommRing R\n\u22a2 \u2200 \u2983S : Type u_3\u2984 [inst : CommRing S] (I J : Ideal S),\n I \u2264 J \u2192\n (\u2200 {x : S}, IsUnit (\u2191(Ideal.Quotient.mk I) x) \u2192 IsUnit x) \u2192\n (\u2200 {x : S \u29f8 I}, IsUnit (\u2191(Ideal.Quotient.mk (Ideal.map (Ideal.Quotient.mk I) J)) x) \u2192 IsUnit x) \u2192\n \u2200 {x : S}, IsUnit (\u2191(Ideal.Quotient.mk J) x) \u2192 IsUnit x\n[PROOFSTEP]\nintrov e h\u2081 h\u2082 h\u2083\n[GOAL]\ncase h\u2082\nR\u271d : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\u271d\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R\u271d S\u271d\nI\u271d : Ideal S\u271d\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nS : Type u_3\ninst\u271d : CommRing S\nI J : Ideal S\ne : I \u2264 J\nh\u2081 : \u2200 {x : S}, IsUnit (\u2191(Ideal.Quotient.mk I) x) \u2192 IsUnit x\nh\u2082 : \u2200 {x : S \u29f8 I}, IsUnit (\u2191(Ideal.Quotient.mk (Ideal.map (Ideal.Quotient.mk I) J)) x) \u2192 IsUnit x\nx : S\nh\u2083 : IsUnit (\u2191(Ideal.Quotient.mk J) x)\n\u22a2 IsUnit x\n[PROOFSTEP]\napply h\u2081\n[GOAL]\ncase h\u2082.a\nR\u271d : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\u271d\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R\u271d S\u271d\nI\u271d : Ideal S\u271d\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nS : Type u_3\ninst\u271d : CommRing S\nI J : Ideal S\ne : I \u2264 J\nh\u2081 : \u2200 {x : S}, IsUnit (\u2191(Ideal.Quotient.mk I) x) \u2192 IsUnit x\nh\u2082 : \u2200 {x : S \u29f8 I}, IsUnit (\u2191(Ideal.Quotient.mk (Ideal.map (Ideal.Quotient.mk I) J)) x) \u2192 IsUnit x\nx : S\nh\u2083 : IsUnit (\u2191(Ideal.Quotient.mk J) x)\n\u22a2 IsUnit (\u2191(Ideal.Quotient.mk I) x)\n[PROOFSTEP]\napply h\u2082\n[GOAL]\ncase h\u2082.a.a\nR\u271d : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\u271d\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R\u271d S\u271d\nI\u271d : Ideal S\u271d\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nS : Type u_3\ninst\u271d : CommRing S\nI J : Ideal S\ne : I \u2264 J\nh\u2081 : \u2200 {x : S}, IsUnit (\u2191(Ideal.Quotient.mk I) x) \u2192 IsUnit x\nh\u2082 : \u2200 {x : S \u29f8 I}, IsUnit (\u2191(Ideal.Quotient.mk (Ideal.map (Ideal.Quotient.mk I) J)) x) \u2192 IsUnit x\nx : S\nh\u2083 : IsUnit (\u2191(Ideal.Quotient.mk J) x)\n\u22a2 IsUnit (\u2191(Ideal.Quotient.mk (Ideal.map (Ideal.Quotient.mk I) J)) (\u2191(Ideal.Quotient.mk I) x))\n[PROOFSTEP]\nexact h\u2083.map ((DoubleQuot.quotQuotEquivQuotSup I J).trans (Ideal.quotEquivOfEq (sup_eq_right.mpr e))).symm.toRingHom\n[GOAL]\ncase h\u2081\nR\u271d : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring R\u271d\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R\u271d S\nI : Ideal S\nR : Type u_3\ninst\u271d : CommRing R\n\u22a2 \u2200 \u2983S : Type u_3\u2984 [inst : CommRing S] (I : Ideal S),\n I ^ 2 = \u22a5 \u2192 \u2200 {x : S}, IsUnit (\u2191(Ideal.Quotient.mk I) x) \u2192 IsUnit x\n[PROOFSTEP]\nintrov e H\n[GOAL]\ncase h\u2081\nR\u271d : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\u271d\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R\u271d S\u271d\nI\u271d : Ideal S\u271d\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nS : Type u_3\ninst\u271d : CommRing S\nI : Ideal S\ne : I ^ 2 = \u22a5\nx : S\nH : IsUnit (\u2191(Ideal.Quotient.mk I) x)\n\u22a2 IsUnit x\n[PROOFSTEP]\nskip\n[GOAL]\ncase h\u2081\nR\u271d : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\u271d\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R\u271d S\u271d\nI\u271d : Ideal S\u271d\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nS : Type u_3\ninst\u271d : CommRing S\nI : Ideal S\ne : I ^ 2 = \u22a5\nx : S\nH : IsUnit (\u2191(Ideal.Quotient.mk I) x)\n\u22a2 IsUnit x\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := Ideal.Quotient.mk_surjective (\u2191H.unit\u207b\u00b9 : S \u29f8 I)\n[GOAL]\ncase h\u2081.intro\nR\u271d : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\u271d\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R\u271d S\u271d\nI\u271d : Ideal S\u271d\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nS : Type u_3\ninst\u271d : CommRing S\nI : Ideal S\ne : I ^ 2 = \u22a5\nx : S\nH : IsUnit (\u2191(Ideal.Quotient.mk I) x)\ny : S\nhy : \u2191(Ideal.Quotient.mk I) y = \u2191(IsUnit.unit H)\u207b\u00b9\n\u22a2 IsUnit x\n[PROOFSTEP]\nhave : Ideal.Quotient.mk I (x * y) = Ideal.Quotient.mk I 1 := by rw [map_one, _root_.map_mul, hy, IsUnit.mul_val_inv]\n[GOAL]\nR\u271d : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\u271d\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R\u271d S\u271d\nI\u271d : Ideal S\u271d\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nS : Type u_3\ninst\u271d : CommRing S\nI : Ideal S\ne : I ^ 2 = \u22a5\nx : S\nH : IsUnit (\u2191(Ideal.Quotient.mk I) x)\ny : S\nhy : \u2191(Ideal.Quotient.mk I) y = \u2191(IsUnit.unit H)\u207b\u00b9\n\u22a2 \u2191(Ideal.Quotient.mk I) (x * y) = \u2191(Ideal.Quotient.mk I) 1\n[PROOFSTEP]\nrw [map_one, _root_.map_mul, hy, IsUnit.mul_val_inv]\n[GOAL]\ncase h\u2081.intro\nR\u271d : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\u271d\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R\u271d S\u271d\nI\u271d : Ideal S\u271d\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nS : Type u_3\ninst\u271d : CommRing S\nI : Ideal S\ne : I ^ 2 = \u22a5\nx : S\nH : IsUnit (\u2191(Ideal.Quotient.mk I) x)\ny : S\nhy : \u2191(Ideal.Quotient.mk I) y = \u2191(IsUnit.unit H)\u207b\u00b9\nthis : \u2191(Ideal.Quotient.mk I) (x * y) = \u2191(Ideal.Quotient.mk I) 1\n\u22a2 IsUnit x\n[PROOFSTEP]\nrw [Ideal.Quotient.eq] at this \n[GOAL]\ncase h\u2081.intro\nR\u271d : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\u271d\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R\u271d S\u271d\nI\u271d : Ideal S\u271d\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nS : Type u_3\ninst\u271d : CommRing S\nI : Ideal S\ne : I ^ 2 = \u22a5\nx : S\nH : IsUnit (\u2191(Ideal.Quotient.mk I) x)\ny : S\nhy : \u2191(Ideal.Quotient.mk I) y = \u2191(IsUnit.unit H)\u207b\u00b9\nthis : x * y - 1 \u2208 I\n\u22a2 IsUnit x\n[PROOFSTEP]\nhave : (x * y - 1) ^ 2 = 0 := by\n rw [\u2190 Ideal.mem_bot, \u2190 e]\n exact Ideal.pow_mem_pow this _\n[GOAL]\nR\u271d : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\u271d\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R\u271d S\u271d\nI\u271d : Ideal S\u271d\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nS : Type u_3\ninst\u271d : CommRing S\nI : Ideal S\ne : I ^ 2 = \u22a5\nx : S\nH : IsUnit (\u2191(Ideal.Quotient.mk I) x)\ny : S\nhy : \u2191(Ideal.Quotient.mk I) y = \u2191(IsUnit.unit H)\u207b\u00b9\nthis : x * y - 1 \u2208 I\n\u22a2 (x * y - 1) ^ 2 = 0\n[PROOFSTEP]\nrw [\u2190 Ideal.mem_bot, \u2190 e]\n[GOAL]\nR\u271d : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\u271d\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R\u271d S\u271d\nI\u271d : Ideal S\u271d\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nS : Type u_3\ninst\u271d : CommRing S\nI : Ideal S\ne : I ^ 2 = \u22a5\nx : S\nH : IsUnit (\u2191(Ideal.Quotient.mk I) x)\ny : S\nhy : \u2191(Ideal.Quotient.mk I) y = \u2191(IsUnit.unit H)\u207b\u00b9\nthis : x * y - 1 \u2208 I\n\u22a2 (x * y - 1) ^ 2 \u2208 I ^ 2\n[PROOFSTEP]\nexact Ideal.pow_mem_pow this _\n[GOAL]\ncase h\u2081.intro\nR\u271d : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\u271d\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R\u271d S\u271d\nI\u271d : Ideal S\u271d\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nS : Type u_3\ninst\u271d : CommRing S\nI : Ideal S\ne : I ^ 2 = \u22a5\nx : S\nH : IsUnit (\u2191(Ideal.Quotient.mk I) x)\ny : S\nhy : \u2191(Ideal.Quotient.mk I) y = \u2191(IsUnit.unit H)\u207b\u00b9\nthis\u271d : x * y - 1 \u2208 I\nthis : (x * y - 1) ^ 2 = 0\n\u22a2 IsUnit x\n[PROOFSTEP]\nhave : x * (y * (2 - x * y)) = 1 := by\n rw [eq_comm, \u2190 sub_eq_zero, \u2190 this]\n ring\n[GOAL]\nR\u271d : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\u271d\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R\u271d S\u271d\nI\u271d : Ideal S\u271d\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nS : Type u_3\ninst\u271d : CommRing S\nI : Ideal S\ne : I ^ 2 = \u22a5\nx : S\nH : IsUnit (\u2191(Ideal.Quotient.mk I) x)\ny : S\nhy : \u2191(Ideal.Quotient.mk I) y = \u2191(IsUnit.unit H)\u207b\u00b9\nthis\u271d : x * y - 1 \u2208 I\nthis : (x * y - 1) ^ 2 = 0\n\u22a2 x * (y * (2 - x * y)) = 1\n[PROOFSTEP]\nrw [eq_comm, \u2190 sub_eq_zero, \u2190 this]\n[GOAL]\nR\u271d : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\u271d\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R\u271d S\u271d\nI\u271d : Ideal S\u271d\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nS : Type u_3\ninst\u271d : CommRing S\nI : Ideal S\ne : I ^ 2 = \u22a5\nx : S\nH : IsUnit (\u2191(Ideal.Quotient.mk I) x)\ny : S\nhy : \u2191(Ideal.Quotient.mk I) y = \u2191(IsUnit.unit H)\u207b\u00b9\nthis\u271d : x * y - 1 \u2208 I\nthis : (x * y - 1) ^ 2 = 0\n\u22a2 1 - x * (y * (2 - x * y)) = (x * y - 1) ^ 2\n[PROOFSTEP]\nring\n[GOAL]\ncase h\u2081.intro\nR\u271d : Type u_1\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring R\u271d\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : Algebra R\u271d S\u271d\nI\u271d : Ideal S\u271d\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nS : Type u_3\ninst\u271d : CommRing S\nI : Ideal S\ne : I ^ 2 = \u22a5\nx : S\nH : IsUnit (\u2191(Ideal.Quotient.mk I) x)\ny : S\nhy : \u2191(Ideal.Quotient.mk I) y = \u2191(IsUnit.unit H)\u207b\u00b9\nthis\u271d\u00b9 : x * y - 1 \u2208 I\nthis\u271d : (x * y - 1) ^ 2 = 0\nthis : x * (y * (2 - x * y)) = 1\n\u22a2 IsUnit x\n[PROOFSTEP]\nexact isUnit_of_mul_eq_one _ _ this\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.QuotientNilpotent", "llama_tokens": 14477, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872019117031, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.42686932814979606}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\nh : \u00acisSome (\u2191e (some x)) = true\n\u22a2 isSome (\u2191e none) = true\n[PROOFSTEP]\nrw [\u2190 Option.ne_none_iff_isSome]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\nh : \u00acisSome (\u2191e (some x)) = true\n\u22a2 \u2191e none \u2260 none\n[PROOFSTEP]\nintro hn\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\nh : \u00acisSome (\u2191e (some x)) = true\nhn : \u2191e none = none\n\u22a2 False\n[PROOFSTEP]\nrw [Option.not_isSome_iff_eq_none, \u2190 hn] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\nh : \u2191e (some x) = \u2191e none\nhn : \u2191e none = none\n\u22a2 False\n[PROOFSTEP]\nexact\n Option.some_ne_none _\n (e.injective h)\n -- Porting note: private\n -- #align equiv.remove_none_aux Equiv.removeNone_aux\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\nh : \u2203 x', \u2191e (some x) = some x'\n\u22a2 some (removeNone_aux e x) = \u2191e (some x)\n[PROOFSTEP]\nsimp [removeNone_aux, Option.isSome_iff_exists.mpr h]\n -- Porting note: private\n -- #align equiv.remove_none_aux_some Equiv.removeNone_aux_some\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\nh : \u2191e (some x) = none\n\u22a2 some (removeNone_aux e x) = \u2191e none\n[PROOFSTEP]\nsimp [removeNone_aux, Option.not_isSome_iff_eq_none.mpr h]\n -- Porting note: private\n -- #align equiv.remove_none_aux_none Equiv.removeNone_aux_none\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\n\u22a2 some (removeNone_aux e.symm (removeNone_aux e x)) = some x\n[PROOFSTEP]\ncases h1 : e.symm (some (removeNone_aux e x))\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\nh1 : \u2191e.symm (some (removeNone_aux e x)) = none\n\u22a2 some (removeNone_aux e.symm (removeNone_aux e x)) = some x\n[PROOFSTEP]\ncases h2 : e (some x)\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx val\u271d : \u03b1\nh1 : \u2191e.symm (some (removeNone_aux e x)) = some val\u271d\n\u22a2 some (removeNone_aux e.symm (removeNone_aux e x)) = some x\n[PROOFSTEP]\ncases h2 : e (some x)\n[GOAL]\ncase none.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\nh1 : \u2191e.symm (some (removeNone_aux e x)) = none\nh2 : \u2191e (some x) = none\n\u22a2 some (removeNone_aux e.symm (removeNone_aux e x)) = some x\n[PROOFSTEP]\nrw [removeNone_aux_none _ h1]\n[GOAL]\ncase none.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\nh1 : \u2191e.symm (some (removeNone_aux e x)) = none\nh2 : \u2191e (some x) = none\n\u22a2 \u2191e.symm none = some x\n[PROOFSTEP]\nexact (e.eq_symm_apply.mpr h2).symm\n[GOAL]\ncase none.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\nh1 : \u2191e.symm (some (removeNone_aux e x)) = none\nval\u271d : \u03b2\nh2 : \u2191e (some x) = some val\u271d\n\u22a2 some (removeNone_aux e.symm (removeNone_aux e x)) = some x\n[PROOFSTEP]\nrw [removeNone_aux_some _ \u27e8_, h2\u27e9] at h1 \n[GOAL]\ncase none.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\nh1 : \u2191e.symm (\u2191e (some x)) = none\nval\u271d : \u03b2\nh2 : \u2191e (some x) = some val\u271d\n\u22a2 some (removeNone_aux e.symm (removeNone_aux e x)) = some x\n[PROOFSTEP]\nsimp at h1 \n[GOAL]\ncase some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx val\u271d : \u03b1\nh1 : \u2191e.symm (some (removeNone_aux e x)) = some val\u271d\nh2 : \u2191e (some x) = none\n\u22a2 some (removeNone_aux e.symm (removeNone_aux e x)) = some x\n[PROOFSTEP]\nrw [removeNone_aux_none _ h2] at h1 \n[GOAL]\ncase some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx val\u271d : \u03b1\nh1 : \u2191e.symm (\u2191e none) = some val\u271d\nh2 : \u2191e (some x) = none\n\u22a2 some (removeNone_aux e.symm (removeNone_aux e x)) = some x\n[PROOFSTEP]\nsimp at h1 \n[GOAL]\ncase some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx val\u271d\u00b9 : \u03b1\nh1 : \u2191e.symm (some (removeNone_aux e x)) = some val\u271d\u00b9\nval\u271d : \u03b2\nh2 : \u2191e (some x) = some val\u271d\n\u22a2 some (removeNone_aux e.symm (removeNone_aux e x)) = some x\n[PROOFSTEP]\nrw [removeNone_aux_some _ \u27e8_, h1\u27e9]\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx val\u271d\u00b9 : \u03b1\nh1 : \u2191e.symm (some (removeNone_aux e x)) = some val\u271d\u00b9\nval\u271d : \u03b2\nh2 : \u2191e (some x) = some val\u271d\n\u22a2 \u2191e.symm (some (removeNone_aux e x)) = some x\n[PROOFSTEP]\nrw [removeNone_aux_some _ \u27e8_, h2\u27e9]\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx val\u271d\u00b9 : \u03b1\nh1 : \u2191e.symm (some (removeNone_aux e x)) = some val\u271d\u00b9\nval\u271d : \u03b2\nh2 : \u2191e (some x) = some val\u271d\n\u22a2 \u2191e.symm (\u2191e (some x)) = some x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nh : \u2191e.symm none = none\n\u22a2 \u2191e none = none\n[PROOFSTEP]\nsimpa using (congr_arg e h).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nh : \u2191e none = none\n\u22a2 \u2191e.symm none = none\n[PROOFSTEP]\nsimpa using (congr_arg e.symm h).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\n\u22a2 some (\u2191(removeNone e) x) = \u2191e none \u2194 \u2191e.symm none = some x\n[PROOFSTEP]\ncases' h : e (some x) with a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\nh : \u2191e (some x) = none\n\u22a2 some (\u2191(removeNone e) x) = \u2191e none \u2194 \u2191e.symm none = some x\n[PROOFSTEP]\nrw [removeNone_none _ h]\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\nh : \u2191e (some x) = none\n\u22a2 \u2191e none = \u2191e none \u2194 \u2191e.symm none = some x\n[PROOFSTEP]\nsimpa using (congr_arg e.symm h).symm\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\na : \u03b2\nh : \u2191e (some x) = some a\n\u22a2 some (\u2191(removeNone e) x) = \u2191e none \u2194 \u2191e.symm none = some x\n[PROOFSTEP]\nrw [removeNone_some _ \u27e8a, h\u27e9]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\na : \u03b2\nh : \u2191e (some x) = some a\n\u22a2 \u2191e (some x) = \u2191e none \u2194 \u2191e.symm none = some x\n[PROOFSTEP]\nhave h1 := congr_arg e.symm h\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\na : \u03b2\nh : \u2191e (some x) = some a\nh1 : \u2191e.symm (\u2191e (some x)) = \u2191e.symm (some a)\n\u22a2 \u2191e (some x) = \u2191e none \u2194 \u2191e.symm none = some x\n[PROOFSTEP]\nrw [symm_apply_apply] at h1 \n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\na : \u03b2\nh : \u2191e (some x) = some a\nh1 : some x = \u2191e.symm (some a)\n\u22a2 \u2191e (some x) = \u2191e none \u2194 \u2191e.symm none = some x\n[PROOFSTEP]\nsimp only [false_iff_iff, apply_eq_iff_eq]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Option \u03b1 \u2243 Option \u03b2\nx : \u03b1\na : \u03b2\nh : \u2191e (some x) = some a\nh1 : some x = \u2191e.symm (some a)\n\u22a2 \u00ac\u2191e.symm none = some x\n[PROOFSTEP]\nsimp [h1, apply_eq_iff_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne\u271d : Option \u03b1 \u2243 Option \u03b2\ne : \u03b1 \u2243 \u03b2\nx : \u03b1\n\u22a2 \u2191(optionCongr e) (some x) = some (\u2191e x)\n[PROOFSTEP]\nsimp [EquivFunctor.map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : { e // \u2191e none = x }\na : \u03b1\n\u22a2 (fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true))\n ((fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) }) a) =\n a\n[PROOFSTEP]\nrw [\u2190 some_inj, some_get]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : { e // \u2191e none = x }\na : \u03b1\n\u22a2 \u2191(\u2191e).symm \u2191((fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) }) a) = some a\n[PROOFSTEP]\nexact symm_apply_apply (e : Option \u03b1 \u2243 \u03b2) a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : { e // \u2191e none = x }\nb : { y // y \u2260 x }\n\u22a2 (fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) })\n ((fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true)) b) =\n b\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : { e // \u2191e none = x }\nb : { y // y \u2260 x }\n\u22a2 \u2191((fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) })\n ((fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true)) b)) =\n \u2191b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : \u03b1 \u2243 { y // y \u2260 x }\na : Option \u03b1\n\u22a2 (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a\n[PROOFSTEP]\ncases a with\n| none => simp\n| some a =>\n simp only [casesOn'_some, Function.comp_apply, Subtype.coe_eta, symm_apply_apply, dite_eq_ite]\n exact if_neg (e a).property\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : \u03b1 \u2243 { y // y \u2260 x }\na : Option \u03b1\n\u22a2 (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a\n[PROOFSTEP]\ncases a with\n| none => simp\n| some a =>\n simp only [casesOn'_some, Function.comp_apply, Subtype.coe_eta, symm_apply_apply, dite_eq_ite]\n exact if_neg (e a).property\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : \u03b1 \u2243 { y // y \u2260 x }\n\u22a2 (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) none) =\n none\n[PROOFSTEP]\n\n| none => simp\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : \u03b1 \u2243 { y // y \u2260 x }\n\u22a2 (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) none) =\n none\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : \u03b1 \u2243 { y // y \u2260 x }\na : \u03b1\n\u22a2 (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) (some a)) =\n some a\n[PROOFSTEP]\n\n| some a =>\n simp only [casesOn'_some, Function.comp_apply, Subtype.coe_eta, symm_apply_apply, dite_eq_ite]\n exact if_neg (e a).property\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : \u03b1 \u2243 { y // y \u2260 x }\na : \u03b1\n\u22a2 (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) (some a)) =\n some a\n[PROOFSTEP]\nsimp only [casesOn'_some, Function.comp_apply, Subtype.coe_eta, symm_apply_apply, dite_eq_ite]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : \u03b1 \u2243 { y // y \u2260 x }\na : \u03b1\n\u22a2 (if \u2191(\u2191e a) = x then none else some a) = some a\n[PROOFSTEP]\nexact if_neg (e a).property\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : \u03b1 \u2243 { y // y \u2260 x }\nb : \u03b2\n\u22a2 (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h })) b) =\n b\n[PROOFSTEP]\nby_cases h : b = x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : \u03b1 \u2243 { y // y \u2260 x }\nb : \u03b2\nh : b = x\n\u22a2 (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h })) b) =\n b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : \u03b1 \u2243 { y // y \u2260 x }\nb : \u03b2\nh : \u00acb = x\n\u22a2 (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h })) b) =\n b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : { e // \u2191e none = x }\n\u22a2 (fun e =>\n {\n val :=\n { toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h })) b) =\n b) },\n property :=\n (_ :\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h })) b) =\n b) }\n none =\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h })) b) =\n b) }\n none) })\n ((fun e =>\n { toFun := fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) },\n invFun := fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true),\n left_inv :=\n (_ :\n \u2200 (a : \u03b1),\n (fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true))\n ((fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) }) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : { y // y \u2260 x }),\n (fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) })\n ((fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true)) b) =\n b) })\n e) =\n e\n[PROOFSTEP]\next a\n[GOAL]\ncase a.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : { e // \u2191e none = x }\na : Option \u03b1\n\u22a2 \u2191\u2191((fun e =>\n {\n val :=\n { toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h })) b) =\n b) },\n property :=\n (_ :\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none =\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none) })\n ((fun e =>\n { toFun := fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) },\n invFun := fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true),\n left_inv :=\n (_ :\n \u2200 (a : \u03b1),\n (fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true))\n ((fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) }) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : { y // y \u2260 x }),\n (fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) })\n ((fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true)) b) =\n b) })\n e))\n a =\n \u2191\u2191e a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase a.H.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : { e // \u2191e none = x }\n\u22a2 \u2191\u2191((fun e =>\n {\n val :=\n { toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h })) b) =\n b) },\n property :=\n (_ :\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none =\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none) })\n ((fun e =>\n { toFun := fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) },\n invFun := fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true),\n left_inv :=\n (_ :\n \u2200 (a : \u03b1),\n (fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true))\n ((fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) }) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : { y // y \u2260 x }),\n (fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) })\n ((fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true)) b) =\n b) })\n e))\n none =\n \u2191\u2191e none\n[PROOFSTEP]\nsimpa using e.property.symm\n[GOAL]\ncase a.H.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : { e // \u2191e none = x }\nval\u271d : \u03b1\n\u22a2 \u2191\u2191((fun e =>\n {\n val :=\n { toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h })) b) =\n b) },\n property :=\n (_ :\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none =\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none) })\n ((fun e =>\n { toFun := fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) },\n invFun := fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true),\n left_inv :=\n (_ :\n \u2200 (a : \u03b1),\n (fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true))\n ((fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) }) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : { y // y \u2260 x }),\n (fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) })\n ((fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true)) b) =\n b) })\n e))\n (some val\u271d) =\n \u2191\u2191e (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : \u03b1 \u2243 { y // y \u2260 x }\n\u22a2 (fun e =>\n { toFun := fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) },\n invFun := fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true),\n left_inv :=\n (_ :\n \u2200 (a : \u03b1),\n (fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true))\n ((fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) }) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : { y // y \u2260 x }),\n (fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) })\n ((fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true)) b) =\n b) })\n ((fun e =>\n {\n val :=\n { toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h })) b) =\n b) },\n property :=\n (_ :\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none =\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none) })\n e) =\n e\n[PROOFSTEP]\next a\n[GOAL]\ncase H.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : \u03b1 \u2243 { y // y \u2260 x }\na : \u03b1\n\u22a2 \u2191(\u2191((fun e =>\n { toFun := fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) },\n invFun := fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true),\n left_inv :=\n (_ :\n \u2200 (a : \u03b1),\n (fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true))\n ((fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) }) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : { y // y \u2260 x }),\n (fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) })\n ((fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true)) b) =\n b) })\n ((fun e =>\n {\n val :=\n { toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h })) b) =\n b) },\n property :=\n (_ :\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b =>\n if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none =\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b =>\n if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none) })\n e))\n a) =\n \u2191(\u2191e a)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : { e // \u2191e none = x }\nb : { y // y \u2260 x }\n\u22a2 some (\u2191(\u2191(optionSubtype x) e).symm b) = \u2191(\u2191e).symm \u2191b\n[PROOFSTEP]\ndsimp only [optionSubtype]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : { e // \u2191e none = x }\nb : { y // y \u2260 x }\n\u22a2 some\n (\u2191(\u2191{\n toFun := fun e =>\n { toFun := fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) },\n invFun := fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true),\n left_inv :=\n (_ :\n \u2200 (a : \u03b1),\n (fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true))\n ((fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) }) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : { y // y \u2260 x }),\n (fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) })\n ((fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true)) b) =\n b) },\n invFun := fun e =>\n {\n val :=\n { toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b => if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) },\n property :=\n (_ :\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b =>\n if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b =>\n if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b =>\n if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none =\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b =>\n if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b =>\n if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b =>\n if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none) },\n left_inv :=\n (_ :\n \u2200 (e : { e // \u2191e none = x }),\n (fun e =>\n {\n val :=\n { toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b =>\n if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b =>\n if h : b = x then none else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b =>\n if h : b = x then none\n else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) },\n property :=\n (_ :\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b =>\n if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b =>\n if h : b = x then none\n else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b =>\n if h : b = x then none\n else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none =\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b =>\n if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b =>\n if h : b = x then none\n else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b =>\n if h : b = x then none\n else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none) })\n ((fun e =>\n { toFun := fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) },\n invFun := fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true),\n left_inv :=\n (_ :\n \u2200 (a : \u03b1),\n (fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true))\n ((fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) }) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : { y // y \u2260 x }),\n (fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) })\n ((fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true))\n b) =\n b) })\n e) =\n e),\n right_inv :=\n (_ :\n \u2200 (e : \u03b1 \u2243 { y // y \u2260 x }),\n (fun e =>\n { toFun := fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) },\n invFun := fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true),\n left_inv :=\n (_ :\n \u2200 (a : \u03b1),\n (fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true))\n ((fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) }) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : { y // y \u2260 x }),\n (fun a => { val := \u2191\u2191e (some a), property := (_ : \u2191\u2191e (some a) \u2260 x) })\n ((fun b => Option.get (\u2191(\u2191e).symm \u2191b) (_ : isSome (\u2191(\u2191e).symm \u2191b) = true))\n b) =\n b) })\n ((fun e =>\n {\n val :=\n { toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b =>\n if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b =>\n if h : b = x then none\n else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b =>\n if h : b = x then none\n else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) },\n property :=\n (_ :\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b =>\n if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b =>\n if h : b = x then none\n else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b =>\n if h : b = x then none\n else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none =\n \u2191{ toFun := fun a => casesOn' a x (Subtype.val \u2218 \u2191e),\n invFun := fun b =>\n if h : b = x then none else some (\u2191e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n \u2200 (a : Option \u03b1),\n (fun b =>\n if h : b = x then none\n else some (\u2191e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val \u2218 \u2191e)) a) =\n a),\n right_inv :=\n (_ :\n \u2200 (b : \u03b2),\n (fun a => casesOn' a x (Subtype.val \u2218 \u2191e))\n ((fun b =>\n if h : b = x then none\n else some (\u2191e.symm { val := b, property := h }))\n b) =\n b) }\n none) })\n e) =\n e) }\n e).symm\n b) =\n \u2191(\u2191e).symm \u2191b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : \u03b1 \u2243 { y // y \u2260 x }\nb : { y // y \u2260 x }\n\u22a2 \u2191(\u2191(\u2191(optionSubtype x).symm e)).symm \u2191b = some (\u2191e.symm b)\n[PROOFSTEP]\nsimp only [optionSubtype, coe_fn_symm_mk, Subtype.coe_mk, Subtype.coe_eta, dite_eq_ite, ite_eq_right_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nx : \u03b2\ne : \u03b1 \u2243 { y // y \u2260 x }\nb : { y // y \u2260 x }\n\u22a2 \u2191b = x \u2192 False\n[PROOFSTEP]\nexact fun h => False.elim (b.property h)\n", "meta": {"mathlib_filename": "Mathlib.Logic.Equiv.Option", "llama_tokens": 16395, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702880639791, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4268682338574247}} {"text": "[GOAL]\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : SMul M \u03b1\ninst\u271d\u00b2 : SMul N \u03b1\na\u271d : M\nb\u271d : \u03b1\nx\u271d : Option \u03b1\ninst\u271d\u00b9 : SMul M N\ninst\u271d : IsScalarTower M N \u03b1\na : M\nb : N\nx : Option \u03b1\n\u22a2 (a \u2022 b) \u2022 x = a \u2022 b \u2022 x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase none\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : SMul M \u03b1\ninst\u271d\u00b2 : SMul N \u03b1\na\u271d : M\nb\u271d : \u03b1\nx : Option \u03b1\ninst\u271d\u00b9 : SMul M N\ninst\u271d : IsScalarTower M N \u03b1\na : M\nb : N\n\u22a2 (a \u2022 b) \u2022 none = a \u2022 b \u2022 none\ncase some\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : SMul M \u03b1\ninst\u271d\u00b2 : SMul N \u03b1\na\u271d : M\nb\u271d : \u03b1\nx : Option \u03b1\ninst\u271d\u00b9 : SMul M N\ninst\u271d : IsScalarTower M N \u03b1\na : M\nb : N\nval\u271d : \u03b1\n\u22a2 (a \u2022 b) \u2022 some val\u271d = a \u2022 b \u2022 some val\u271d\n[PROOFSTEP]\nexacts [rfl, congr_arg some (smul_assoc _ _ _)]\n[GOAL]\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : SMul M \u03b1\ninst\u271d\u00b2 : SMul N \u03b1\na\u271d : M\nb : \u03b1\nx\u271d : Option \u03b1\ninst\u271d\u00b9 : SMul M\u1d50\u1d52\u1d56 \u03b1\ninst\u271d : IsCentralScalar M \u03b1\na : M\nx : Option \u03b1\n\u22a2 MulOpposite.op a \u2022 x = a \u2022 x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase none\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : SMul M \u03b1\ninst\u271d\u00b2 : SMul N \u03b1\na\u271d : M\nb : \u03b1\nx : Option \u03b1\ninst\u271d\u00b9 : SMul M\u1d50\u1d52\u1d56 \u03b1\ninst\u271d : IsCentralScalar M \u03b1\na : M\n\u22a2 MulOpposite.op a \u2022 none = a \u2022 none\ncase some\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : SMul M \u03b1\ninst\u271d\u00b2 : SMul N \u03b1\na\u271d : M\nb : \u03b1\nx : Option \u03b1\ninst\u271d\u00b9 : SMul M\u1d50\u1d52\u1d56 \u03b1\ninst\u271d : IsCentralScalar M \u03b1\na : M\nval\u271d : \u03b1\n\u22a2 MulOpposite.op a \u2022 some val\u271d = a \u2022 some val\u271d\n[PROOFSTEP]\nexacts [rfl, congr_arg some (op_smul_eq_smul _ _)]\n[GOAL]\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : SMul N \u03b1\na : M\nb\u271d : \u03b1\nx : Option \u03b1\ninst\u271d : FaithfulSMul M \u03b1\nm\u2081\u271d m\u2082\u271d : M\nh : \u2200 (a : Option \u03b1), m\u2081\u271d \u2022 a = m\u2082\u271d \u2022 a\nb : \u03b1\n\u22a2 m\u2081\u271d \u2022 b = m\u2082\u271d \u2022 b\n[PROOFSTEP]\ninjection h (some b)\n[GOAL]\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\nb : Option \u03b1\n\u22a2 1 \u2022 b = b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\n\u22a2 1 \u2022 none = none\ncase some\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\nval\u271d : \u03b1\n\u22a2 1 \u2022 some val\u271d = some val\u271d\n[PROOFSTEP]\nexacts [rfl, congr_arg some (one_smul _ _)]\n[GOAL]\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\na\u2081 a\u2082 : M\nb : Option \u03b1\n\u22a2 (a\u2081 * a\u2082) \u2022 b = a\u2081 \u2022 a\u2082 \u2022 b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\na\u2081 a\u2082 : M\n\u22a2 (a\u2081 * a\u2082) \u2022 none = a\u2081 \u2022 a\u2082 \u2022 none\ncase some\nM : Type u_1\nN : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\na\u2081 a\u2082 : M\nval\u271d : \u03b1\n\u22a2 (a\u2081 * a\u2082) \u2022 some val\u271d = a\u2081 \u2022 a\u2082 \u2022 some val\u271d\n[PROOFSTEP]\nexacts [rfl, congr_arg some (mul_smul _ _ _)]\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.GroupAction.Option", "llama_tokens": 1475, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195152660687, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.4267459431264709}} {"text": "[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhfg : Exact f g\nhgh : Exact g h\nhf'g' : Exact f' g'\nh\u03b1 : Epi \u03b1\nh\u03b2 : Mono \u03b2\nh\u03b4 : Mono \u03b4\nc : Pseudoelement C\nhc : pseudoApply \u03b3 c = 0\n\u22a2 pseudoApply \u03b4 (pseudoApply h c) = pseudoApply h' (pseudoApply \u03b3 c)\n[PROOFSTEP]\nrw [\u2190 Pseudoelement.comp_apply, \u2190 comm\u2083, Pseudoelement.comp_apply]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhfg : Exact f g\nhgh : Exact g h\nhf'g' : Exact f' g'\nh\u03b1 : Epi \u03b1\nh\u03b2 : Mono \u03b2\nh\u03b4 : Mono \u03b4\nc : Pseudoelement C\nhc : pseudoApply \u03b3 c = 0\n\u22a2 pseudoApply h' (pseudoApply \u03b3 c) = pseudoApply h' 0\n[PROOFSTEP]\nrw [hc]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhfg : Exact f g\nhgh : Exact g h\nhf'g' : Exact f' g'\nh\u03b1 : Epi \u03b1\nh\u03b2 : Mono \u03b2\nh\u03b4 : Mono \u03b4\nc : Pseudoelement C\nhc : pseudoApply \u03b3 c = 0\nthis : pseudoApply h c = 0\nb : Pseudoelement B\nhb : pseudoApply g b = c\n\u22a2 pseudoApply g' (pseudoApply \u03b2 b) = pseudoApply \u03b3 (pseudoApply g b)\n[PROOFSTEP]\nrw [\u2190 Pseudoelement.comp_apply, comm\u2082, Pseudoelement.comp_apply]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhfg : Exact f g\nhgh : Exact g h\nhf'g' : Exact f' g'\nh\u03b1 : Epi \u03b1\nh\u03b2 : Mono \u03b2\nh\u03b4 : Mono \u03b4\nc : Pseudoelement C\nhc : pseudoApply \u03b3 c = 0\nthis : pseudoApply h c = 0\nb : Pseudoelement B\nhb : pseudoApply g b = c\n\u22a2 pseudoApply \u03b3 (pseudoApply g b) = pseudoApply \u03b3 c\n[PROOFSTEP]\nrw [hb]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhfg : Exact f g\nhgh : Exact g h\nhf'g' : Exact f' g'\nh\u03b1 : Epi \u03b1\nh\u03b2 : Mono \u03b2\nh\u03b4 : Mono \u03b4\nc : Pseudoelement C\nhc : pseudoApply \u03b3 c = 0\nthis\u271d : pseudoApply h c = 0\nb : Pseudoelement B\nhb : pseudoApply g b = c\nthis : pseudoApply g' (pseudoApply \u03b2 b) = 0\na' : Pseudoelement A'\nha' : pseudoApply f' a' = pseudoApply \u03b2 b\na : Pseudoelement A\nha : pseudoApply \u03b1 a = a'\n\u22a2 pseudoApply \u03b2 (pseudoApply f a) = pseudoApply f' (pseudoApply \u03b1 a)\n[PROOFSTEP]\nrw [\u2190 Pseudoelement.comp_apply, \u2190 comm\u2081, Pseudoelement.comp_apply]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhfg : Exact f g\nhgh : Exact g h\nhf'g' : Exact f' g'\nh\u03b1 : Epi \u03b1\nh\u03b2 : Mono \u03b2\nh\u03b4 : Mono \u03b4\nc : Pseudoelement C\nhc : pseudoApply \u03b3 c = 0\nthis\u271d : pseudoApply h c = 0\nb : Pseudoelement B\nhb : pseudoApply g b = c\nthis : pseudoApply g' (pseudoApply \u03b2 b) = 0\na' : Pseudoelement A'\nha' : pseudoApply f' a' = pseudoApply \u03b2 b\na : Pseudoelement A\nha : pseudoApply \u03b1 a = a'\n\u22a2 pseudoApply f' (pseudoApply \u03b1 a) = pseudoApply f' a'\n[PROOFSTEP]\nrw [ha]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhfg : Exact f g\nhgh : Exact g h\nhf'g' : Exact f' g'\nh\u03b1 : Epi \u03b1\nh\u03b2 : Mono \u03b2\nh\u03b4 : Mono \u03b4\nc : Pseudoelement C\nhc : pseudoApply \u03b3 c = 0\nthis\u271d\u00b9 : pseudoApply h c = 0\nb : Pseudoelement B\nhb : pseudoApply g b = c\nthis\u271d : pseudoApply g' (pseudoApply \u03b2 b) = 0\na' : Pseudoelement A'\nha' : pseudoApply f' a' = pseudoApply \u03b2 b\na : Pseudoelement A\nha : pseudoApply \u03b1 a = a'\nthis : pseudoApply f a = b\n\u22a2 pseudoApply g b = pseudoApply g (pseudoApply f a)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\n\u22a2 r = 0\n[PROOFSTEP]\nhave hf'r : f' \u226b r = 0 :=\n Limits.zero_of_epi_comp \u03b1 <|\n calc\n \u03b1 \u226b f' \u226b r = f \u226b \u03b2 \u226b r := by rw [reassoc_of% comm\u2081]\n _ = f \u226b 0 := by rw [h\u03b2r]\n _ = 0 := HasZeroMorphisms.comp_zero _ _\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\n\u22a2 \u03b1 \u226b f' \u226b r = f \u226b \u03b2 \u226b r\n[PROOFSTEP]\nrw [reassoc_of% comm\u2081]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\n\u22a2 f \u226b \u03b2 \u226b r = f \u226b 0\n[PROOFSTEP]\nrw [h\u03b2r]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\n\u22a2 r = 0\n[PROOFSTEP]\nlet y : R \u27f6 pushout r g' := pushout.inl\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\n\u22a2 r = 0\n[PROOFSTEP]\nlet z : C' \u27f6 pushout r g' := pushout.inr\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\n\u22a2 r = 0\n[PROOFSTEP]\nhave : Mono (cokernel.desc f' g' hf'g'.w) := mono_cokernel_desc_of_exact _ _ hf'g'\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\n\u22a2 r = 0\n[PROOFSTEP]\nhave : Mono y :=\n mono_inl_of_factor_thru_epi_mono_factorization r g' (cokernel.\u03c0 f') (cokernel.desc f' g' hf'g'.w) (by simp)\n (cokernel.desc f' r hf'r) (by simp) _ (colimit.isColimit _)\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\n\u22a2 cokernel.\u03c0 f' \u226b cokernel.desc f' g' (_ : f' \u226b g' = 0) = g'\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\n\u22a2 cokernel.\u03c0 f' \u226b cokernel.desc f' r hf'r = r\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis : Mono y\n\u22a2 r = 0\n[PROOFSTEP]\nhave hz : g \u226b \u03b3 \u226b z = 0 :=\n calc\n g \u226b \u03b3 \u226b z = \u03b2 \u226b g' \u226b z := by rw [\u2190 reassoc_of% comm\u2082]\n _ = \u03b2 \u226b r \u226b y := by rw [\u2190 pushout.condition]\n _ = 0 \u226b y := by rw [reassoc_of% h\u03b2r]\n _ = 0 := HasZeroMorphisms.zero_comp _ _\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis : Mono y\n\u22a2 g \u226b \u03b3 \u226b z = \u03b2 \u226b g' \u226b z\n[PROOFSTEP]\nrw [\u2190 reassoc_of% comm\u2082]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis : Mono y\n\u22a2 \u03b2 \u226b g' \u226b z = \u03b2 \u226b r \u226b y\n[PROOFSTEP]\nrw [\u2190 pushout.condition]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis : Mono y\n\u22a2 \u03b2 \u226b r \u226b y = 0 \u226b y\n[PROOFSTEP]\nrw [reassoc_of% h\u03b2r]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis : Mono y\nhz : g \u226b \u03b3 \u226b z = 0\n\u22a2 r = 0\n[PROOFSTEP]\nlet v : pushout r g' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inl\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis : Mono y\nhz : g \u226b \u03b3 \u226b z = 0\nv : pushout r g' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inl\n\u22a2 r = 0\n[PROOFSTEP]\nlet w : D' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inr\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis : Mono y\nhz : g \u226b \u03b3 \u226b z = 0\nv : pushout r g' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inl\nw : D' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inr\n\u22a2 r = 0\n[PROOFSTEP]\nhave : Mono (cokernel.desc g h hgh.w) := mono_cokernel_desc_of_exact _ _ hgh\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d\u00b9 : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis\u271d : Mono y\nhz : g \u226b \u03b3 \u226b z = 0\nv : pushout r g' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inl\nw : D' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inr\nthis : Mono (cokernel.desc g h (_ : g \u226b h = 0))\n\u22a2 r = 0\n[PROOFSTEP]\nhave : Mono v :=\n mono_inl_of_factor_thru_epi_mono_factorization _ _ (cokernel.\u03c0 g) (cokernel.desc g h hgh.w \u226b \u03b4) (by simp)\n (cokernel.desc _ _ hz) (by simp) _ (colimit.isColimit _)\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d\u00b9 : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis\u271d : Mono y\nhz : g \u226b \u03b3 \u226b z = 0\nv : pushout r g' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inl\nw : D' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inr\nthis : Mono (cokernel.desc g h (_ : g \u226b h = 0))\n\u22a2 cokernel.\u03c0 g \u226b cokernel.desc g h (_ : g \u226b h = 0) \u226b \u03b4 = h \u226b \u03b4\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d\u00b9 : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis\u271d : Mono y\nhz : g \u226b \u03b3 \u226b z = 0\nv : pushout r g' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inl\nw : D' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inr\nthis : Mono (cokernel.desc g h (_ : g \u226b h = 0))\n\u22a2 cokernel.\u03c0 g \u226b cokernel.desc g (\u03b3 \u226b z) hz = \u03b3 \u226b z\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d\u00b2 : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis\u271d\u00b9 : Mono y\nhz : g \u226b \u03b3 \u226b z = 0\nv : pushout r g' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inl\nw : D' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inr\nthis\u271d : Mono (cokernel.desc g h (_ : g \u226b h = 0))\nthis : Mono v\n\u22a2 r = 0\n[PROOFSTEP]\nhave hzv : z \u226b v = h' \u226b w :=\n (cancel_epi \u03b3).1 <|\n calc\n \u03b3 \u226b z \u226b v = h \u226b \u03b4 \u226b w := by rw [\u2190 Category.assoc, pushout.condition, Category.assoc]\n _ = \u03b3 \u226b h' \u226b w := by rw [reassoc_of% comm\u2083]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d\u00b2 : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis\u271d\u00b9 : Mono y\nhz : g \u226b \u03b3 \u226b z = 0\nv : pushout r g' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inl\nw : D' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inr\nthis\u271d : Mono (cokernel.desc g h (_ : g \u226b h = 0))\nthis : Mono v\n\u22a2 \u03b3 \u226b z \u226b v = h \u226b \u03b4 \u226b w\n[PROOFSTEP]\nrw [\u2190 Category.assoc, pushout.condition, Category.assoc]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d\u00b2 : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis\u271d\u00b9 : Mono y\nhz : g \u226b \u03b3 \u226b z = 0\nv : pushout r g' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inl\nw : D' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inr\nthis\u271d : Mono (cokernel.desc g h (_ : g \u226b h = 0))\nthis : Mono v\n\u22a2 h \u226b \u03b4 \u226b w = \u03b3 \u226b h' \u226b w\n[PROOFSTEP]\nrw [reassoc_of% comm\u2083]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d\u00b2 : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis\u271d\u00b9 : Mono y\nhz : g \u226b \u03b3 \u226b z = 0\nv : pushout r g' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inl\nw : D' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inr\nthis\u271d : Mono (cokernel.desc g h (_ : g \u226b h = 0))\nthis : Mono v\nhzv : z \u226b v = h' \u226b w\n\u22a2 r = 0\n[PROOFSTEP]\nsuffices (r \u226b y) \u226b v = 0 from zero_of_comp_mono _ (zero_of_comp_mono _ this)\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d\u00b2 : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis\u271d\u00b9 : Mono y\nhz : g \u226b \u03b3 \u226b z = 0\nv : pushout r g' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inl\nw : D' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inr\nthis\u271d : Mono (cokernel.desc g h (_ : g \u226b h = 0))\nthis : Mono v\nhzv : z \u226b v = h' \u226b w\n\u22a2 (r \u226b y) \u226b v = 0\n[PROOFSTEP]\ncalc\n (r \u226b y) \u226b v = g' \u226b z \u226b v := by rw [pushout.condition, Category.assoc]\n _ = g' \u226b h' \u226b w := by rw [hzv]\n _ = 0 \u226b w := (hg'h'.w_assoc _)\n _ = 0 := HasZeroMorphisms.zero_comp _ _\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d\u00b2 : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis\u271d\u00b9 : Mono y\nhz : g \u226b \u03b3 \u226b z = 0\nv : pushout r g' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inl\nw : D' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inr\nthis\u271d : Mono (cokernel.desc g h (_ : g \u226b h = 0))\nthis : Mono v\nhzv : z \u226b v = h' \u226b w\n\u22a2 (r \u226b y) \u226b v = g' \u226b z \u226b v\n[PROOFSTEP]\nrw [pushout.condition, Category.assoc]\n[GOAL]\nV : Type u\ninst\u271d\u00b9 : Category.{v, u} V\ninst\u271d : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nhgh : Exact g h\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nh\u03b1 : Epi \u03b1\nh\u03b3 : Epi \u03b3\nh\u03b4 : Mono \u03b4\nR : V\nr : B' \u27f6 R\nh\u03b2r : \u03b2 \u226b r = 0\nhf'r : f' \u226b r = 0\ny : R \u27f6 pushout r g' := pushout.inl\nz : C' \u27f6 pushout r g' := pushout.inr\nthis\u271d\u00b2 : Mono (cokernel.desc f' g' (_ : f' \u226b g' = 0))\nthis\u271d\u00b9 : Mono y\nhz : g \u226b \u03b3 \u226b z = 0\nv : pushout r g' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inl\nw : D' \u27f6 pushout (\u03b3 \u226b z) (h \u226b \u03b4) := pushout.inr\nthis\u271d : Mono (cokernel.desc g h (_ : g \u226b h = 0))\nthis : Mono v\nhzv : z \u226b v = h' \u226b w\n\u22a2 g' \u226b z \u226b v = g' \u226b h' \u226b w\n[PROOFSTEP]\nrw [hzv]\n[GOAL]\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nE E' : V\ni : D \u27f6 E\ni' : D' \u27f6 E'\n\u03b5 : E \u27f6 E'\ncomm\u2084 : \u03b4 \u226b i' = i \u226b \u03b5\nhfg : Exact f g\nhgh : Exact g h\nhhi : Exact h i\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nhh'i' : Exact h' i'\ninst\u271d\u00b3 : Epi \u03b1\ninst\u271d\u00b2 : IsIso \u03b2\ninst\u271d\u00b9 : IsIso \u03b4\ninst\u271d : Mono \u03b5\n\u22a2 Mono \u03b3\n[PROOFSTEP]\napply mono_of_epi_of_mono_of_mono comm\u2081 comm\u2082 comm\u2083 hfg hgh hf'g'\n[GOAL]\ncase h\u03b1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nE E' : V\ni : D \u27f6 E\ni' : D' \u27f6 E'\n\u03b5 : E \u27f6 E'\ncomm\u2084 : \u03b4 \u226b i' = i \u226b \u03b5\nhfg : Exact f g\nhgh : Exact g h\nhhi : Exact h i\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nhh'i' : Exact h' i'\ninst\u271d\u00b3 : Epi \u03b1\ninst\u271d\u00b2 : IsIso \u03b2\ninst\u271d\u00b9 : IsIso \u03b4\ninst\u271d : Mono \u03b5\n\u22a2 Epi \u03b1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase h\u03b2\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nE E' : V\ni : D \u27f6 E\ni' : D' \u27f6 E'\n\u03b5 : E \u27f6 E'\ncomm\u2084 : \u03b4 \u226b i' = i \u226b \u03b5\nhfg : Exact f g\nhgh : Exact g h\nhhi : Exact h i\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nhh'i' : Exact h' i'\ninst\u271d\u00b3 : Epi \u03b1\ninst\u271d\u00b2 : IsIso \u03b2\ninst\u271d\u00b9 : IsIso \u03b4\ninst\u271d : Mono \u03b5\n\u22a2 Mono \u03b2\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase h\u03b4\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nE E' : V\ni : D \u27f6 E\ni' : D' \u27f6 E'\n\u03b5 : E \u27f6 E'\ncomm\u2084 : \u03b4 \u226b i' = i \u226b \u03b5\nhfg : Exact f g\nhgh : Exact g h\nhhi : Exact h i\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nhh'i' : Exact h' i'\ninst\u271d\u00b3 : Epi \u03b1\ninst\u271d\u00b2 : IsIso \u03b2\ninst\u271d\u00b9 : IsIso \u03b4\ninst\u271d : Mono \u03b5\n\u22a2 Mono \u03b4\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nE E' : V\ni : D \u27f6 E\ni' : D' \u27f6 E'\n\u03b5 : E \u27f6 E'\ncomm\u2084 : \u03b4 \u226b i' = i \u226b \u03b5\nhfg : Exact f g\nhgh : Exact g h\nhhi : Exact h i\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nhh'i' : Exact h' i'\ninst\u271d\u00b3 : Epi \u03b1\ninst\u271d\u00b2 : IsIso \u03b2\ninst\u271d\u00b9 : IsIso \u03b4\ninst\u271d : Mono \u03b5\nthis : Mono \u03b3\n\u22a2 Epi \u03b3\n[PROOFSTEP]\napply epi_of_epi_of_epi_of_mono comm\u2082 comm\u2083 comm\u2084 hhi hg'h' hh'i'\n[GOAL]\ncase h\u03b1\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nE E' : V\ni : D \u27f6 E\ni' : D' \u27f6 E'\n\u03b5 : E \u27f6 E'\ncomm\u2084 : \u03b4 \u226b i' = i \u226b \u03b5\nhfg : Exact f g\nhgh : Exact g h\nhhi : Exact h i\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nhh'i' : Exact h' i'\ninst\u271d\u00b3 : Epi \u03b1\ninst\u271d\u00b2 : IsIso \u03b2\ninst\u271d\u00b9 : IsIso \u03b4\ninst\u271d : Mono \u03b5\nthis : Mono \u03b3\n\u22a2 Epi \u03b2\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase h\u03b3\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nE E' : V\ni : D \u27f6 E\ni' : D' \u27f6 E'\n\u03b5 : E \u27f6 E'\ncomm\u2084 : \u03b4 \u226b i' = i \u226b \u03b5\nhfg : Exact f g\nhgh : Exact g h\nhhi : Exact h i\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nhh'i' : Exact h' i'\ninst\u271d\u00b3 : Epi \u03b1\ninst\u271d\u00b2 : IsIso \u03b2\ninst\u271d\u00b9 : IsIso \u03b4\ninst\u271d : Mono \u03b5\nthis : Mono \u03b3\n\u22a2 Epi \u03b4\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase h\u03b4\nV : Type u\ninst\u271d\u2075 : Category.{v, u} V\ninst\u271d\u2074 : Abelian V\nA B C D A' B' C' D' : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\nh' : C' \u27f6 D'\n\u03b1 : A \u27f6 A'\n\u03b2 : B \u27f6 B'\n\u03b3 : C \u27f6 C'\n\u03b4 : D \u27f6 D'\ncomm\u2081 : \u03b1 \u226b f' = f \u226b \u03b2\ncomm\u2082 : \u03b2 \u226b g' = g \u226b \u03b3\ncomm\u2083 : \u03b3 \u226b h' = h \u226b \u03b4\nE E' : V\ni : D \u27f6 E\ni' : D' \u27f6 E'\n\u03b5 : E \u27f6 E'\ncomm\u2084 : \u03b4 \u226b i' = i \u226b \u03b5\nhfg : Exact f g\nhgh : Exact g h\nhhi : Exact h i\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nhh'i' : Exact h' i'\ninst\u271d\u00b3 : Epi \u03b1\ninst\u271d\u00b2 : IsIso \u03b2\ninst\u271d\u00b9 : IsIso \u03b4\ninst\u271d : Mono \u03b5\nthis : Mono \u03b3\n\u22a2 Mono \u03b5\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four", "llama_tokens": 16667, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833737577158, "lm_q2_score": 0.5506073655352403, "lm_q1q2_score": 0.42649131081213426}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nM : Type u_4\nF : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EB\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b2 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b9 : TopologicalSpace B\ninst\u271d\u00b9\u2070 : ChartedSpace HB B\ninst\u271d\u2079 : SmoothManifoldWithCorners IB B\nEB' : Type u_9\ninst\u271d\u2078 : NormedAddCommGroup EB'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c EB'\nHB' : Type u_10\ninst\u271d\u2076 : TopologicalSpace HB'\nIB' : ModelWithCorners \ud835\udd5c EB' HB'\ninst\u271d\u2075 : TopologicalSpace B'\ninst\u271d\u2074 : ChartedSpace HB' B'\ninst\u271d\u00b3 : SmoothManifoldWithCorners IB' B'\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\nf : SmoothMap IB' IB B' B\n\u22a2 \u2200 (e e' : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e] [inst_1 : MemTrivializationAtlas e'],\n SmoothOn IB' \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' b)) (e.baseSet \u2229 e'.baseSet)\n[PROOFSTEP]\nrintro _ _ \u27e8e, he, rfl\u27e9 \u27e8e', he', rfl\u27e9\n[GOAL]\ncase mk.intro.intro.mk.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nM : Type u_4\nF : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EB\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b2 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b9 : TopologicalSpace B\ninst\u271d\u00b9\u2070 : ChartedSpace HB B\ninst\u271d\u2079 : SmoothManifoldWithCorners IB B\nEB' : Type u_9\ninst\u271d\u2078 : NormedAddCommGroup EB'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c EB'\nHB' : Type u_10\ninst\u271d\u2076 : TopologicalSpace HB'\nIB' : ModelWithCorners \ud835\udd5c EB' HB'\ninst\u271d\u2075 : TopologicalSpace B'\ninst\u271d\u2074 : ChartedSpace HB' B'\ninst\u271d\u00b3 : SmoothManifoldWithCorners IB' B'\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\nf : SmoothMap IB' IB B' B\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\n\u22a2 SmoothOn IB' \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F)\n (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c (Trivialization.pullback e f) (Trivialization.pullback e' f) b))\n ((Trivialization.pullback e f).baseSet \u2229 (Trivialization.pullback e' f).baseSet)\n[PROOFSTEP]\nskip\n[GOAL]\ncase mk.intro.intro.mk.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nM : Type u_4\nF : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EB\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b2 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b9 : TopologicalSpace B\ninst\u271d\u00b9\u2070 : ChartedSpace HB B\ninst\u271d\u2079 : SmoothManifoldWithCorners IB B\nEB' : Type u_9\ninst\u271d\u2078 : NormedAddCommGroup EB'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c EB'\nHB' : Type u_10\ninst\u271d\u2076 : TopologicalSpace HB'\nIB' : ModelWithCorners \ud835\udd5c EB' HB'\ninst\u271d\u2075 : TopologicalSpace B'\ninst\u271d\u2074 : ChartedSpace HB' B'\ninst\u271d\u00b3 : SmoothManifoldWithCorners IB' B'\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\nf : SmoothMap IB' IB B' B\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\n\u22a2 SmoothOn IB' \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F)\n (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c (Trivialization.pullback e f) (Trivialization.pullback e' f) b))\n ((Trivialization.pullback e f).baseSet \u2229 (Trivialization.pullback e' f).baseSet)\n[PROOFSTEP]\nrefine' ((smoothOn_coordChangeL _ e e').comp f.smooth.smoothOn fun b hb => hb).congr _\n[GOAL]\ncase mk.intro.intro.mk.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nM : Type u_4\nF : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EB\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b2 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b9 : TopologicalSpace B\ninst\u271d\u00b9\u2070 : ChartedSpace HB B\ninst\u271d\u2079 : SmoothManifoldWithCorners IB B\nEB' : Type u_9\ninst\u271d\u2078 : NormedAddCommGroup EB'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c EB'\nHB' : Type u_10\ninst\u271d\u2076 : TopologicalSpace HB'\nIB' : ModelWithCorners \ud835\udd5c EB' HB'\ninst\u271d\u2075 : TopologicalSpace B'\ninst\u271d\u2074 : ChartedSpace HB' B'\ninst\u271d\u00b3 : SmoothManifoldWithCorners IB' B'\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\nf : SmoothMap IB' IB B' B\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\n\u22a2 \u2200 (y : B'),\n y \u2208 \u2191f \u207b\u00b9' (e.baseSet \u2229 e'.baseSet) \u2192\n \u2191(Trivialization.coordChangeL \ud835\udd5c (Trivialization.pullback e f) (Trivialization.pullback e' f) y) =\n ((fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' b)) \u2218 \u2191f) y\n[PROOFSTEP]\nrintro b (hb : f b \u2208 e.baseSet \u2229 e'.baseSet)\n[GOAL]\ncase mk.intro.intro.mk.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nM : Type u_4\nF : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EB\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b2 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b9 : TopologicalSpace B\ninst\u271d\u00b9\u2070 : ChartedSpace HB B\ninst\u271d\u2079 : SmoothManifoldWithCorners IB B\nEB' : Type u_9\ninst\u271d\u2078 : NormedAddCommGroup EB'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c EB'\nHB' : Type u_10\ninst\u271d\u2076 : TopologicalSpace HB'\nIB' : ModelWithCorners \ud835\udd5c EB' HB'\ninst\u271d\u2075 : TopologicalSpace B'\ninst\u271d\u2074 : ChartedSpace HB' B'\ninst\u271d\u00b3 : SmoothManifoldWithCorners IB' B'\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\nf : SmoothMap IB' IB B' B\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\nb : B'\nhb : \u2191f b \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 \u2191(Trivialization.coordChangeL \ud835\udd5c (Trivialization.pullback e f) (Trivialization.pullback e' f) b) =\n ((fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' b)) \u2218 \u2191f) b\n[PROOFSTEP]\next v\n[GOAL]\ncase mk.intro.intro.mk.intro.intro.h\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nM : Type u_4\nF : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EB\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b2 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b9 : TopologicalSpace B\ninst\u271d\u00b9\u2070 : ChartedSpace HB B\ninst\u271d\u2079 : SmoothManifoldWithCorners IB B\nEB' : Type u_9\ninst\u271d\u2078 : NormedAddCommGroup EB'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c EB'\nHB' : Type u_10\ninst\u271d\u2076 : TopologicalSpace HB'\nIB' : ModelWithCorners \ud835\udd5c EB' HB'\ninst\u271d\u2075 : TopologicalSpace B'\ninst\u271d\u2074 : ChartedSpace HB' B'\ninst\u271d\u00b3 : SmoothManifoldWithCorners IB' B'\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\nf : SmoothMap IB' IB B' B\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\nb : B'\nhb : \u2191f b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 \u2191\u2191(Trivialization.coordChangeL \ud835\udd5c (Trivialization.pullback e f) (Trivialization.pullback e' f) b) v =\n \u2191(((fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' b)) \u2218 \u2191f) b) v\n[PROOFSTEP]\nshow ((e.pullback f).coordChangeL \ud835\udd5c (e'.pullback f) b) v = (e.coordChangeL \ud835\udd5c e' (f b)) v\n[GOAL]\ncase mk.intro.intro.mk.intro.intro.h\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nM : Type u_4\nF : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EB\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b2 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b9 : TopologicalSpace B\ninst\u271d\u00b9\u2070 : ChartedSpace HB B\ninst\u271d\u2079 : SmoothManifoldWithCorners IB B\nEB' : Type u_9\ninst\u271d\u2078 : NormedAddCommGroup EB'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c EB'\nHB' : Type u_10\ninst\u271d\u2076 : TopologicalSpace HB'\nIB' : ModelWithCorners \ud835\udd5c EB' HB'\ninst\u271d\u2075 : TopologicalSpace B'\ninst\u271d\u2074 : ChartedSpace HB' B'\ninst\u271d\u00b3 : SmoothManifoldWithCorners IB' B'\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\nf : SmoothMap IB' IB B' B\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\nb : B'\nhb : \u2191f b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 \u2191(Trivialization.coordChangeL \ud835\udd5c (Trivialization.pullback e f) (Trivialization.pullback e' f) b) v =\n \u2191(Trivialization.coordChangeL \ud835\udd5c e e' (\u2191f b)) v\n[PROOFSTEP]\nrw [e.coordChangeL_apply e' hb, (e.pullback f).coordChangeL_apply' _]\n[GOAL]\ncase mk.intro.intro.mk.intro.intro.h\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nM : Type u_4\nF : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EB\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b2 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b9 : TopologicalSpace B\ninst\u271d\u00b9\u2070 : ChartedSpace HB B\ninst\u271d\u2079 : SmoothManifoldWithCorners IB B\nEB' : Type u_9\ninst\u271d\u2078 : NormedAddCommGroup EB'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c EB'\nHB' : Type u_10\ninst\u271d\u2076 : TopologicalSpace HB'\nIB' : ModelWithCorners \ud835\udd5c EB' HB'\ninst\u271d\u2075 : TopologicalSpace B'\ninst\u271d\u2074 : ChartedSpace HB' B'\ninst\u271d\u00b3 : SmoothManifoldWithCorners IB' B'\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\nf : SmoothMap IB' IB B' B\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\nb : B'\nhb : \u2191f b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 (\u2191(Trivialization.pullback e' f)\n (\u2191(LocalHomeomorph.symm (Trivialization.pullback e f).toLocalHomeomorph) (b, v))).snd =\n (\u2191e' { proj := \u2191f b, snd := Trivialization.symm e (\u2191f b) v }).snd\ncase mk.intro.intro.mk.intro.intro.h.hb\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nM : Type u_4\nF : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EB\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b2 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b9 : TopologicalSpace B\ninst\u271d\u00b9\u2070 : ChartedSpace HB B\ninst\u271d\u2079 : SmoothManifoldWithCorners IB B\nEB' : Type u_9\ninst\u271d\u2078 : NormedAddCommGroup EB'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c EB'\nHB' : Type u_10\ninst\u271d\u2076 : TopologicalSpace HB'\nIB' : ModelWithCorners \ud835\udd5c EB' HB'\ninst\u271d\u2075 : TopologicalSpace B'\ninst\u271d\u2074 : ChartedSpace HB' B'\ninst\u271d\u00b3 : SmoothManifoldWithCorners IB' B'\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\nf : SmoothMap IB' IB B' B\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\nb : B'\nhb : \u2191f b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 b \u2208 (Trivialization.pullback e f).baseSet \u2229 (Trivialization.pullback e' f).baseSet\n[PROOFSTEP]\nexacts [rfl, hb]\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Manifold.VectorBundle.Pullback", "llama_tokens": 5760, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929002541068, "lm_q2_score": 0.546738151984614, "lm_q1q2_score": 0.4264518768460497}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b2\nh\u2081 : abv (a\u2081 - b\u2081) < \u03b5 / 2\nh\u2082 : abv (a\u2082 - b\u2082) < \u03b5 / 2\n\u22a2 abv (a\u2081 + a\u2082 - (b\u2081 + b\u2082)) < \u03b5\n[PROOFSTEP]\nsimpa [add_halves, sub_eq_add_neg, add_comm, add_left_comm, add_assoc] using\n lt_of_le_of_lt (abv_add abv _ _) (add_lt_add h\u2081 h\u2082)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K\u2081 K\u2082 : \u03b1\n\u03b50 : 0 < \u03b5\n\u22a2 \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 : \u03b2},\n abv a\u2081 < K\u2081 \u2192 abv b\u2082 < K\u2082 \u2192 abv (a\u2081 - b\u2081) < \u03b4 \u2192 abv (a\u2082 - b\u2082) < \u03b4 \u2192 abv (a\u2081 * a\u2082 - b\u2081 * b\u2082) < \u03b5\n[PROOFSTEP]\nhave K0 : (0 : \u03b1) < max 1 (max K\u2081 K\u2082) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K\u2081 K\u2082 : \u03b1\n\u03b50 : 0 < \u03b5\nK0 : 0 < max 1 (max K\u2081 K\u2082)\n\u22a2 \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 : \u03b2},\n abv a\u2081 < K\u2081 \u2192 abv b\u2082 < K\u2082 \u2192 abv (a\u2081 - b\u2081) < \u03b4 \u2192 abv (a\u2082 - b\u2082) < \u03b4 \u2192 abv (a\u2081 * a\u2082 - b\u2081 * b\u2082) < \u03b5\n[PROOFSTEP]\nhave \u03b5K := div_pos (half_pos \u03b50) K0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K\u2081 K\u2082 : \u03b1\n\u03b50 : 0 < \u03b5\nK0 : 0 < max 1 (max K\u2081 K\u2082)\n\u03b5K : 0 < \u03b5 / 2 / max 1 (max K\u2081 K\u2082)\n\u22a2 \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 : \u03b2},\n abv a\u2081 < K\u2081 \u2192 abv b\u2082 < K\u2082 \u2192 abv (a\u2081 - b\u2081) < \u03b4 \u2192 abv (a\u2082 - b\u2082) < \u03b4 \u2192 abv (a\u2081 * a\u2082 - b\u2081 * b\u2082) < \u03b5\n[PROOFSTEP]\nrefine' \u27e8_, \u03b5K, fun {a\u2081 a\u2082 b\u2081 b\u2082} ha\u2081 hb\u2082 h\u2081 h\u2082 => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K\u2081 K\u2082 : \u03b1\n\u03b50 : 0 < \u03b5\nK0 : 0 < max 1 (max K\u2081 K\u2082)\n\u03b5K : 0 < \u03b5 / 2 / max 1 (max K\u2081 K\u2082)\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b2\nha\u2081 : abv a\u2081 < K\u2081\nhb\u2082 : abv b\u2082 < K\u2082\nh\u2081 : abv (a\u2081 - b\u2081) < \u03b5 / 2 / max 1 (max K\u2081 K\u2082)\nh\u2082 : abv (a\u2082 - b\u2082) < \u03b5 / 2 / max 1 (max K\u2081 K\u2082)\n\u22a2 abv (a\u2081 * a\u2082 - b\u2081 * b\u2082) < \u03b5\n[PROOFSTEP]\nreplace ha\u2081 := lt_of_lt_of_le ha\u2081 (le_trans (le_max_left _ K\u2082) (le_max_right 1 _))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K\u2081 K\u2082 : \u03b1\n\u03b50 : 0 < \u03b5\nK0 : 0 < max 1 (max K\u2081 K\u2082)\n\u03b5K : 0 < \u03b5 / 2 / max 1 (max K\u2081 K\u2082)\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b2\nhb\u2082 : abv b\u2082 < K\u2082\nh\u2081 : abv (a\u2081 - b\u2081) < \u03b5 / 2 / max 1 (max K\u2081 K\u2082)\nh\u2082 : abv (a\u2082 - b\u2082) < \u03b5 / 2 / max 1 (max K\u2081 K\u2082)\nha\u2081 : abv a\u2081 < max 1 (max K\u2081 K\u2082)\n\u22a2 abv (a\u2081 * a\u2082 - b\u2081 * b\u2082) < \u03b5\n[PROOFSTEP]\nreplace hb\u2082 := lt_of_lt_of_le hb\u2082 (le_trans (le_max_right K\u2081 _) (le_max_right 1 _))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K\u2081 K\u2082 : \u03b1\n\u03b50 : 0 < \u03b5\nK0 : 0 < max 1 (max K\u2081 K\u2082)\n\u03b5K : 0 < \u03b5 / 2 / max 1 (max K\u2081 K\u2082)\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b2\nh\u2081 : abv (a\u2081 - b\u2081) < \u03b5 / 2 / max 1 (max K\u2081 K\u2082)\nh\u2082 : abv (a\u2082 - b\u2082) < \u03b5 / 2 / max 1 (max K\u2081 K\u2082)\nha\u2081 : abv a\u2081 < max 1 (max K\u2081 K\u2082)\nhb\u2082 : abv b\u2082 < max 1 (max K\u2081 K\u2082)\n\u22a2 abv (a\u2081 * a\u2082 - b\u2081 * b\u2082) < \u03b5\n[PROOFSTEP]\nset M := max 1 (max K\u2081 K\u2082)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K\u2081 K\u2082 : \u03b1\n\u03b50 : 0 < \u03b5\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b2\nM : \u03b1 := max 1 (max K\u2081 K\u2082)\nK0 : 0 < M\n\u03b5K : 0 < \u03b5 / 2 / M\nh\u2081 : abv (a\u2081 - b\u2081) < \u03b5 / 2 / M\nh\u2082 : abv (a\u2082 - b\u2082) < \u03b5 / 2 / M\nha\u2081 : abv a\u2081 < M\nhb\u2082 : abv b\u2082 < M\n\u22a2 abv (a\u2081 * a\u2082 - b\u2081 * b\u2082) < \u03b5\n[PROOFSTEP]\nhave : abv (a\u2081 - b\u2081) * abv b\u2082 + abv (a\u2082 - b\u2082) * abv a\u2081 < \u03b5 / 2 / M * M + \u03b5 / 2 / M * M\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K\u2081 K\u2082 : \u03b1\n\u03b50 : 0 < \u03b5\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b2\nM : \u03b1 := max 1 (max K\u2081 K\u2082)\nK0 : 0 < M\n\u03b5K : 0 < \u03b5 / 2 / M\nh\u2081 : abv (a\u2081 - b\u2081) < \u03b5 / 2 / M\nh\u2082 : abv (a\u2082 - b\u2082) < \u03b5 / 2 / M\nha\u2081 : abv a\u2081 < M\nhb\u2082 : abv b\u2082 < M\n\u22a2 abv (a\u2081 - b\u2081) * abv b\u2082 + abv (a\u2082 - b\u2082) * abv a\u2081 < \u03b5 / 2 / M * M + \u03b5 / 2 / M * M\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K\u2081 K\u2082 : \u03b1\n\u03b50 : 0 < \u03b5\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b2\nM : \u03b1 := max 1 (max K\u2081 K\u2082)\nK0 : 0 < M\n\u03b5K : 0 < \u03b5 / 2 / M\nh\u2081 : abv (a\u2081 - b\u2081) < \u03b5 / 2 / M\nh\u2082 : abv (a\u2082 - b\u2082) < \u03b5 / 2 / M\nha\u2081 : abv a\u2081 < M\nhb\u2082 : abv b\u2082 < M\nthis : abv (a\u2081 - b\u2081) * abv b\u2082 + abv (a\u2082 - b\u2082) * abv a\u2081 < \u03b5 / 2 / M * M + \u03b5 / 2 / M * M\n\u22a2 abv (a\u2081 * a\u2082 - b\u2081 * b\u2082) < \u03b5\n[PROOFSTEP]\nrw [\u2190 abv_mul abv, mul_comm, div_mul_cancel _ (ne_of_gt K0), \u2190 abv_mul abv, add_halves] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K\u2081 K\u2082 : \u03b1\n\u03b50 : 0 < \u03b5\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b2\nM : \u03b1 := max 1 (max K\u2081 K\u2082)\nK0 : 0 < M\n\u03b5K : 0 < \u03b5 / 2 / M\nh\u2081 : abv (a\u2081 - b\u2081) < \u03b5 / 2 / M\nh\u2082 : abv (a\u2082 - b\u2082) < \u03b5 / 2 / M\nha\u2081 : abv a\u2081 < M\nhb\u2082 : abv b\u2082 < M\nthis : abv ((a\u2081 - b\u2081) * b\u2082) + abv (a\u2081 * (a\u2082 - b\u2082)) < \u03b5\n\u22a2 abv (a\u2081 * a\u2082 - b\u2081 * b\u2082) < \u03b5\n[PROOFSTEP]\nsimpa [sub_eq_add_neg, mul_add, add_mul, add_left_comm] using lt_of_le_of_lt (abv_add abv _ _) this\n[GOAL]\n\u03b1 : Type u_2\n\u03b2\u271d : Type ?u.21207\ninst\u271d\u2074 : LinearOrderedField \u03b1\ninst\u271d\u00b3 : Ring \u03b2\u271d\nabv\u271d : \u03b2\u271d \u2192 \u03b1\ninst\u271d\u00b2 : IsAbsoluteValue abv\u271d\n\u03b2 : Type u_1\ninst\u271d\u00b9 : DivisionRing \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K : \u03b1\n\u03b50 : 0 < \u03b5\nK0 : 0 < K\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 {a b : \u03b2}, K \u2264 abv a \u2192 K \u2264 abv b \u2192 abv (a - b) < \u03b4 \u2192 abv (a\u207b\u00b9 - b\u207b\u00b9) < \u03b5\n[PROOFSTEP]\nrefine' \u27e8K * \u03b5 * K, mul_pos (mul_pos K0 \u03b50) K0, fun {a b} ha hb h => _\u27e9\n[GOAL]\n\u03b1 : Type u_2\n\u03b2\u271d : Type ?u.21207\ninst\u271d\u2074 : LinearOrderedField \u03b1\ninst\u271d\u00b3 : Ring \u03b2\u271d\nabv\u271d : \u03b2\u271d \u2192 \u03b1\ninst\u271d\u00b2 : IsAbsoluteValue abv\u271d\n\u03b2 : Type u_1\ninst\u271d\u00b9 : DivisionRing \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K : \u03b1\n\u03b50 : 0 < \u03b5\nK0 : 0 < K\na b : \u03b2\nha : K \u2264 abv a\nhb : K \u2264 abv b\nh : abv (a - b) < K * \u03b5 * K\n\u22a2 abv (a\u207b\u00b9 - b\u207b\u00b9) < \u03b5\n[PROOFSTEP]\nhave a0 := K0.trans_le ha\n[GOAL]\n\u03b1 : Type u_2\n\u03b2\u271d : Type ?u.21207\ninst\u271d\u2074 : LinearOrderedField \u03b1\ninst\u271d\u00b3 : Ring \u03b2\u271d\nabv\u271d : \u03b2\u271d \u2192 \u03b1\ninst\u271d\u00b2 : IsAbsoluteValue abv\u271d\n\u03b2 : Type u_1\ninst\u271d\u00b9 : DivisionRing \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K : \u03b1\n\u03b50 : 0 < \u03b5\nK0 : 0 < K\na b : \u03b2\nha : K \u2264 abv a\nhb : K \u2264 abv b\nh : abv (a - b) < K * \u03b5 * K\na0 : 0 < abv a\n\u22a2 abv (a\u207b\u00b9 - b\u207b\u00b9) < \u03b5\n[PROOFSTEP]\nhave b0 := K0.trans_le hb\n[GOAL]\n\u03b1 : Type u_2\n\u03b2\u271d : Type ?u.21207\ninst\u271d\u2074 : LinearOrderedField \u03b1\ninst\u271d\u00b3 : Ring \u03b2\u271d\nabv\u271d : \u03b2\u271d \u2192 \u03b1\ninst\u271d\u00b2 : IsAbsoluteValue abv\u271d\n\u03b2 : Type u_1\ninst\u271d\u00b9 : DivisionRing \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K : \u03b1\n\u03b50 : 0 < \u03b5\nK0 : 0 < K\na b : \u03b2\nha : K \u2264 abv a\nhb : K \u2264 abv b\nh : abv (a - b) < K * \u03b5 * K\na0 : 0 < abv a\nb0 : 0 < abv b\n\u22a2 abv (a\u207b\u00b9 - b\u207b\u00b9) < \u03b5\n[PROOFSTEP]\nrw [inv_sub_inv' ((abv_pos abv).1 a0) ((abv_pos abv).1 b0), abv_mul abv, abv_mul abv, abv_inv abv, abv_inv abv,\n abv_sub abv]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2\u271d : Type ?u.21207\ninst\u271d\u2074 : LinearOrderedField \u03b1\ninst\u271d\u00b3 : Ring \u03b2\u271d\nabv\u271d : \u03b2\u271d \u2192 \u03b1\ninst\u271d\u00b2 : IsAbsoluteValue abv\u271d\n\u03b2 : Type u_1\ninst\u271d\u00b9 : DivisionRing \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K : \u03b1\n\u03b50 : 0 < \u03b5\nK0 : 0 < K\na b : \u03b2\nha : K \u2264 abv a\nhb : K \u2264 abv b\nh : abv (a - b) < K * \u03b5 * K\na0 : 0 < abv a\nb0 : 0 < abv b\n\u22a2 (abv a)\u207b\u00b9 * abv (a - b) * (abv b)\u207b\u00b9 < \u03b5\n[PROOFSTEP]\nrefine' lt_of_mul_lt_mul_left (lt_of_mul_lt_mul_right _ b0.le) a0.le\n[GOAL]\n\u03b1 : Type u_2\n\u03b2\u271d : Type ?u.21207\ninst\u271d\u2074 : LinearOrderedField \u03b1\ninst\u271d\u00b3 : Ring \u03b2\u271d\nabv\u271d : \u03b2\u271d \u2192 \u03b1\ninst\u271d\u00b2 : IsAbsoluteValue abv\u271d\n\u03b2 : Type u_1\ninst\u271d\u00b9 : DivisionRing \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K : \u03b1\n\u03b50 : 0 < \u03b5\nK0 : 0 < K\na b : \u03b2\nha : K \u2264 abv a\nhb : K \u2264 abv b\nh : abv (a - b) < K * \u03b5 * K\na0 : 0 < abv a\nb0 : 0 < abv b\n\u22a2 abv a * ((abv a)\u207b\u00b9 * abv (a - b) * (abv b)\u207b\u00b9) * abv b < abv a * \u03b5 * abv b\n[PROOFSTEP]\nrw [mul_assoc, inv_mul_cancel_right\u2080 b0.ne', \u2190 mul_assoc, mul_inv_cancel a0.ne', one_mul]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2\u271d : Type ?u.21207\ninst\u271d\u2074 : LinearOrderedField \u03b1\ninst\u271d\u00b3 : Ring \u03b2\u271d\nabv\u271d : \u03b2\u271d \u2192 \u03b1\ninst\u271d\u00b2 : IsAbsoluteValue abv\u271d\n\u03b2 : Type u_1\ninst\u271d\u00b9 : DivisionRing \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K : \u03b1\n\u03b50 : 0 < \u03b5\nK0 : 0 < K\na b : \u03b2\nha : K \u2264 abv a\nhb : K \u2264 abv b\nh : abv (a - b) < K * \u03b5 * K\na0 : 0 < abv a\nb0 : 0 < abv b\n\u22a2 abv (a - b) < abv a * \u03b5 * abv b\n[PROOFSTEP]\nrefine' h.trans_le _\n[GOAL]\n\u03b1 : Type u_2\n\u03b2\u271d : Type ?u.21207\ninst\u271d\u2074 : LinearOrderedField \u03b1\ninst\u271d\u00b3 : Ring \u03b2\u271d\nabv\u271d : \u03b2\u271d \u2192 \u03b1\ninst\u271d\u00b2 : IsAbsoluteValue abv\u271d\n\u03b2 : Type u_1\ninst\u271d\u00b9 : DivisionRing \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 K : \u03b1\n\u03b50 : 0 < \u03b5\nK0 : 0 < K\na b : \u03b2\nha : K \u2264 abv a\nhb : K \u2264 abv b\nh : abv (a - b) < K * \u03b5 * K\na0 : 0 < abv a\nb0 : 0 < abv b\n\u22a2 K * \u03b5 * K \u2264 abv a * \u03b5 * abv b\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : \u2115 \u2192 \u03b2\nhf : IsCauSeq abv f\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2200 (k : \u2115), k \u2265 i \u2192 abv (f j - f k) < \u03b5\n[PROOFSTEP]\nrefine' (hf _ (half_pos \u03b50)).imp fun i hi j ij k ik => _\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : \u2115 \u2192 \u03b2\nhf : IsCauSeq abv f\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 abv (f j - f i) < \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nk : \u2115\nik : k \u2265 i\n\u22a2 abv (f j - f k) < \u03b5\n[PROOFSTEP]\nrw [\u2190 add_halves \u03b5]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : \u2115 \u2192 \u03b2\nhf : IsCauSeq abv f\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 abv (f j - f i) < \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nk : \u2115\nik : k \u2265 i\n\u22a2 abv (f j - f k) < \u03b5 / 2 + \u03b5 / 2\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (abv_sub_le abv _ _ _) (add_lt_add (hi _ ij) _)\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : \u2115 \u2192 \u03b2\nhf : IsCauSeq abv f\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 abv (f j - f i) < \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nk : \u2115\nik : k \u2265 i\n\u22a2 abv (f i - f k) < \u03b5 / 2\n[PROOFSTEP]\nrw [abv_sub abv]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : \u2115 \u2192 \u03b2\nhf : IsCauSeq abv f\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 abv (f j - f i) < \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nk : \u2115\nik : k \u2265 i\n\u22a2 abv (f k - f i) < \u03b5 / 2\n[PROOFSTEP]\nexact hi _ ik\n[GOAL]\n\u03b1 : Type ?u.36935\n\u03b2 : Type ?u.36938\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\nf : CauSeq \u03b2 abv\ng : \u2115 \u2192 \u03b2\ne : \u2200 (i : \u2115), \u2191f i = g i\n\u03b5 : \u03b1\n\u22a2 \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abv (g j - g i) < \u03b5\n[PROOFSTEP]\nrw [show g = f from (funext e).symm]\n[GOAL]\n\u03b1 : Type ?u.36935\n\u03b2 : Type ?u.36938\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\nf : CauSeq \u03b2 abv\ng : \u2115 \u2192 \u03b2\ne : \u2200 (i : \u2115), \u2191f i = g i\n\u03b5 : \u03b1\n\u22a2 \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j - \u2191f i) < \u03b5\n[PROOFSTEP]\nexact f.cauchy\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\n\u22a2 \u2203 r, \u2200 (i : \u2115), abv (\u2191f i) < r\n[PROOFSTEP]\ncases' f.cauchy zero_lt_one with i h\n[GOAL]\ncase intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j - \u2191f i) < 1\n\u22a2 \u2203 r, \u2200 (i : \u2115), abv (\u2191f i) < r\n[PROOFSTEP]\nset R : \u2115 \u2192 \u03b1 := @Nat.rec (fun _ => \u03b1) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR\n[GOAL]\ncase intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j - \u2191f i) < 1\nR : \u2115 \u2192 \u03b1 := Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nhR : R = Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\n\u22a2 \u2203 r, \u2200 (i : \u2115), abv (\u2191f i) < r\n[PROOFSTEP]\nhave : \u2200 i, \u2200 j \u2264 i, abv (f j) \u2264 R i := by\n refine' Nat.rec (by simp [hR]) _\n rintro i hi j (rfl | hj)\n \u00b7 simp\n exact (hi j hj).trans (le_max_left _ _)\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j - \u2191f i) < 1\nR : \u2115 \u2192 \u03b1 := Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nhR : R = Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\n\u22a2 \u2200 (i j : \u2115), j \u2264 i \u2192 abv (\u2191f j) \u2264 R i\n[PROOFSTEP]\nrefine' Nat.rec (by simp [hR]) _\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j - \u2191f i) < 1\nR : \u2115 \u2192 \u03b1 := Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nhR : R = Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\n\u22a2 \u2200 (j : \u2115), j \u2264 Nat.zero \u2192 abv (\u2191f j) \u2264 R Nat.zero\n[PROOFSTEP]\nsimp [hR]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j - \u2191f i) < 1\nR : \u2115 \u2192 \u03b1 := Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nhR : R = Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\n\u22a2 \u2200 (n : \u2115), (\u2200 (j : \u2115), j \u2264 n \u2192 abv (\u2191f j) \u2264 R n) \u2192 \u2200 (j : \u2115), j \u2264 Nat.succ n \u2192 abv (\u2191f j) \u2264 R (Nat.succ n)\n[PROOFSTEP]\nrintro i hi j (rfl | hj)\n[GOAL]\ncase refl\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\ni\u271d : \u2115\nh : \u2200 (j : \u2115), j \u2265 i\u271d \u2192 abv (\u2191f j - \u2191f i\u271d) < 1\nR : \u2115 \u2192 \u03b1 := Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nhR : R = Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2264 i \u2192 abv (\u2191f j) \u2264 R i\n\u22a2 abv (\u2191f (Nat.succ i)) \u2264 R (Nat.succ i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase step\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\ni\u271d : \u2115\nh : \u2200 (j : \u2115), j \u2265 i\u271d \u2192 abv (\u2191f j - \u2191f i\u271d) < 1\nR : \u2115 \u2192 \u03b1 := Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nhR : R = Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2264 i \u2192 abv (\u2191f j) \u2264 R i\nj : \u2115\nhj : Nat.le j i\n\u22a2 abv (\u2191f j) \u2264 R (Nat.succ i)\n[PROOFSTEP]\nexact (hi j hj).trans (le_max_left _ _)\n[GOAL]\ncase intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j - \u2191f i) < 1\nR : \u2115 \u2192 \u03b1 := Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nhR : R = Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nthis : \u2200 (i j : \u2115), j \u2264 i \u2192 abv (\u2191f j) \u2264 R i\n\u22a2 \u2203 r, \u2200 (i : \u2115), abv (\u2191f i) < r\n[PROOFSTEP]\nrefine' \u27e8R i + 1, fun j => _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j - \u2191f i) < 1\nR : \u2115 \u2192 \u03b1 := Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nhR : R = Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nthis : \u2200 (i j : \u2115), j \u2264 i \u2192 abv (\u2191f j) \u2264 R i\nj : \u2115\n\u22a2 abv (\u2191f j) < R i + 1\n[PROOFSTEP]\ncases' lt_or_le j i with ij ij\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j - \u2191f i) < 1\nR : \u2115 \u2192 \u03b1 := Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nhR : R = Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nthis : \u2200 (i j : \u2115), j \u2264 i \u2192 abv (\u2191f j) \u2264 R i\nj : \u2115\nij : j < i\n\u22a2 abv (\u2191f j) < R i + 1\n[PROOFSTEP]\nexact lt_of_le_of_lt (this i _ (le_of_lt ij)) (lt_add_one _)\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j - \u2191f i) < 1\nR : \u2115 \u2192 \u03b1 := Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nhR : R = Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nthis : \u2200 (i j : \u2115), j \u2264 i \u2192 abv (\u2191f j) \u2264 R i\nj : \u2115\nij : i \u2264 j\n\u22a2 abv (\u2191f j) < R i + 1\n[PROOFSTEP]\nhave := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add_of_le_of_lt (this i _ le_rfl) (h _ ij))\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j - \u2191f i) < 1\nR : \u2115 \u2192 \u03b1 := Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nhR : R = Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nthis\u271d : \u2200 (i j : \u2115), j \u2264 i \u2192 abv (\u2191f j) \u2264 R i\nj : \u2115\nij : i \u2264 j\nthis : abv (\u2191f i + (\u2191f j - \u2191f i)) < R i + 1\n\u22a2 abv (\u2191f j) < R i + 1\n[PROOFSTEP]\nrw [add_sub, add_comm] at this \n[GOAL]\ncase intro.inr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j - \u2191f i) < 1\nR : \u2115 \u2192 \u03b1 := Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nhR : R = Nat.rec (abv (\u2191f 0)) fun i c => max c (abv (\u2191f (Nat.succ i)))\nthis\u271d : \u2200 (i j : \u2115), j \u2264 i \u2192 abv (\u2191f j) \u2264 R i\nj : \u2115\nij : i \u2264 j\nthis : abv (\u2191f j + \u2191f i - \u2191f i) < R i + 1\n\u22a2 abv (\u2191f j) < R i + 1\n[PROOFSTEP]\nsimpa using this\n[GOAL]\n\u03b1 : Type ?u.58870\n\u03b2 : Type ?u.58873\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nx : \u03b2\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nj : \u2115\nx\u271d : j \u2265 0\n\u22a2 abv ((fun x_1 => x) j - (fun x_1 => x) 0) < \u03b5\n[PROOFSTEP]\nsimpa [abv_zero] using \u03b50\n[GOAL]\n\u03b1 : Type ?u.84925\n\u03b2 : Type ?u.84928\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\ni : \u2115\n\u22a2 \u2191(const (-1) * f) i = (fun x => -\u2191f x) i\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.88209\n\u03b2 : Type ?u.88212\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\ni : \u2115\n\u22a2 \u2191(f + -g) i = (fun x => \u2191f x - \u2191g x) i\n[PROOFSTEP]\nsimp [sub_eq_add_neg]\n[GOAL]\n\u03b1 : Type ?u.110288\n\u03b2 : Type ?u.110291\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u22a2 \u2200 (x : CauSeq \u03b2 abv) (n : \u2115), \u2191(n \u2022 x) = n \u2022 \u2191x\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type ?u.110288\n\u03b2 : Type ?u.110291\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nx\u271d : CauSeq \u03b2 abv\nn\u271d : \u2115\n\u22a2 \u2191(n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type ?u.110288\n\u03b2 : Type ?u.110291\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u22a2 \u2200 (x : CauSeq \u03b2 abv) (n : \u2124), \u2191(n \u2022 x) = n \u2022 \u2191x\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type ?u.110288\n\u03b2 : Type ?u.110291\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nx\u271d : CauSeq \u03b2 abv\nn\u271d : \u2124\n\u22a2 \u2191(n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type ?u.110288\n\u03b2 : Type ?u.110291\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u22a2 \u2200 (n : \u2115), \u2191\u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type ?u.110288\n\u03b2 : Type ?u.110291\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nn\u271d : \u2115\n\u22a2 \u2191\u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type ?u.110288\n\u03b2 : Type ?u.110291\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u22a2 \u2200 (n : \u2124), \u2191\u2191n = \u2191n\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type ?u.110288\n\u03b2 : Type ?u.110291\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nn\u271d : \u2124\n\u22a2 \u2191\u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type ?u.112041\n\u03b2 : Type ?u.112044\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nn : \u2115\n\u22a2 \u2200 (i : \u2115), \u2191(npowRec n f) i = \u2191f i ^ n\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u03b1 : Type ?u.112041\n\u03b2 : Type ?u.112044\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\n\u22a2 \u2200 (i : \u2115), \u2191(npowRec Nat.zero f) i = \u2191f i ^ Nat.zero\n[PROOFSTEP]\nsimp [*, npowRec, pow_succ]\n[GOAL]\ncase succ\n\u03b1 : Type ?u.112041\n\u03b2 : Type ?u.112044\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nn\u271d : \u2115\nn_ih\u271d : \u2200 (i : \u2115), \u2191(npowRec n\u271d f) i = \u2191f i ^ n\u271d\n\u22a2 \u2200 (i : \u2115), \u2191(npowRec (Nat.succ n\u271d) f) i = \u2191f i ^ Nat.succ n\u271d\n[PROOFSTEP]\nsimp [*, npowRec, pow_succ]\n[GOAL]\n\u03b1 : Type ?u.120684\n\u03b2\u271d : Type ?u.120687\ninst\u271d\u2074 : LinearOrderedField \u03b1\ninst\u271d\u00b3 : Ring \u03b2\u271d\nabv\u271d : \u03b2\u271d \u2192 \u03b1\ninst\u271d\u00b2 : IsAbsoluteValue abv\u271d\n\u03b2 : Type u_1\ninst\u271d\u00b9 : CommRing \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nsrc\u271d : Ring (CauSeq \u03b2 abv) := ring\na b : CauSeq \u03b2 abv\nn : \u2115\n\u22a2 \u2191(a * b) n = \u2191(b * a) n\n[PROOFSTEP]\nsimp [mul_left_comm, mul_comm]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : LimZero f\nhg : LimZero g\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j) < \u03b5 / 2 \u2227 abv (\u2191g j) < \u03b5 / 2\nj : \u2115\nij : j \u2265 i\n\u22a2 abv (\u2191(f + g) j) < \u03b5\n[PROOFSTEP]\nlet \u27e8H\u2081, H\u2082\u27e9 := H _ ij\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : LimZero f\nhg : LimZero g\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j) < \u03b5 / 2 \u2227 abv (\u2191g j) < \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nH\u2081 : abv (\u2191f j) < \u03b5 / 2\nH\u2082 : abv (\u2191g j) < \u03b5 / 2\n\u22a2 abv (\u2191(f + g) j) < \u03b5\n[PROOFSTEP]\nsimpa [add_halves \u03b5] using lt_of_le_of_lt (abv_add abv _ _) (add_lt_add H\u2081 H\u2082)\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhg : LimZero g\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nF : \u03b1\nF0 : F > 0\nhF : \u2200 (i : \u2115), abv (\u2191f i) < F\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191g j) < \u03b5 / F\nj : \u2115\nij : j \u2265 i\n\u22a2 abv (\u2191(f * g) j) < \u03b5\n[PROOFSTEP]\nhave := mul_lt_mul' (le_of_lt <| hF j) (H _ ij) (abv_nonneg abv _) F0\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhg : LimZero g\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nF : \u03b1\nF0 : F > 0\nhF : \u2200 (i : \u2115), abv (\u2191f i) < F\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191g j) < \u03b5 / F\nj : \u2115\nij : j \u2265 i\nthis : abv (\u2191f j) * abv (\u2191g j) < F * (\u03b5 / F)\n\u22a2 abv (\u2191(f * g) j) < \u03b5\n[PROOFSTEP]\nrwa [mul_comm F, div_mul_cancel _ (ne_of_gt F0), \u2190 abv_mul] at this \n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhg : LimZero f\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nG : \u03b1\nG0 : G > 0\nhG : \u2200 (i : \u2115), abv (\u2191g i) < G\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j) < \u03b5 / G\nj : \u2115\nij : j \u2265 i\n\u22a2 abv (\u2191(f * g) j) < \u03b5\n[PROOFSTEP]\nhave := mul_lt_mul'' (H _ ij) (hG j) (abv_nonneg abv _) (abv_nonneg abv _)\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhg : LimZero f\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nG : \u03b1\nG0 : G > 0\nhG : \u2200 (i : \u2115), abv (\u2191g i) < G\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j) < \u03b5 / G\nj : \u2115\nij : j \u2265 i\nthis : abv (\u2191f j) * abv (\u2191g j) < \u03b5 / G * G\n\u22a2 abv (\u2191(f * g) j) < \u03b5\n[PROOFSTEP]\nrwa [div_mul_cancel _ (ne_of_gt G0), \u2190 abv_mul] at this \n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : LimZero f\n\u22a2 LimZero (-f)\n[PROOFSTEP]\nrw [\u2190 neg_one_mul f]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : LimZero f\n\u22a2 LimZero (-1 * f)\n[PROOFSTEP]\nexact mul_limZero_right _ hf\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : LimZero f\nhg : LimZero g\n\u22a2 LimZero (f - g)\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using add_limZero hf (neg_limZero hg)\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhfg : LimZero (f - g)\n\u22a2 LimZero (g - f)\n[PROOFSTEP]\nsimpa using neg_limZero hfg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nj : \u2115\nx\u271d : j \u2265 0\n\u22a2 abv (\u21910 j) < \u03b5\n[PROOFSTEP]\nsimpa [abv_zero abv] using \u03b50\n[GOAL]\n\u03b1 : Type ?u.134602\n\u03b2 : Type ?u.134605\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\n\u22a2 LimZero (f - f)\n[PROOFSTEP]\nsimp [zero_limZero]\n[GOAL]\n\u03b1 : Type ?u.134602\n\u03b2 : Type ?u.134605\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nx\u271d y\u271d : CauSeq \u03b2 abv\nf : LimZero (x\u271d - y\u271d)\n\u03b5 : \u03b1\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191(y\u271d - x\u271d) j) < \u03b5\n[PROOFSTEP]\nsimpa using neg_limZero f \u03b5 h\u03b5\n[GOAL]\n\u03b1 : Type ?u.134602\n\u03b2 : Type ?u.134605\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nx\u271d y\u271d z\u271d : CauSeq \u03b2 abv\nfg : LimZero (x\u271d - y\u271d)\ngh : LimZero (y\u271d - z\u271d)\n\u22a2 LimZero (x\u271d - z\u271d)\n[PROOFSTEP]\nsimpa using add_limZero fg gh\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf1 f2 g1 g2 : CauSeq \u03b2 abv\nhf : f1 \u2248 f2\nhg : g1 \u2248 g2\n\u22a2 f1 + g1 \u2248 f2 + g2\n[PROOFSTEP]\nsimpa only [\u2190 add_sub_add_comm] using add_limZero hf hg\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : f \u2248 g\n\u22a2 -f \u2248 -g\n[PROOFSTEP]\nsimpa only [neg_sub'] using neg_limZero hf\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf1 f2 g1 g2 : CauSeq \u03b2 abv\nhf : f1 \u2248 f2\nhg : g1 \u2248 g2\n\u22a2 f1 - g1 \u2248 f2 - g2\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using add_equiv_add hf (neg_equiv_neg hg)\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nh : f \u2248 g\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191(f - g) j) < \u03b5 / 2 \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 abv (\u2191f k - \u2191f j) < \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nk : \u2115\njk : k \u2265 j\n\u22a2 abv (\u2191f k - \u2191g j) < \u03b5\n[PROOFSTEP]\nlet \u27e8h\u2081, h\u2082\u27e9 := H _ ij\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nh : f \u2248 g\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191(f - g) j) < \u03b5 / 2 \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 abv (\u2191f k - \u2191f j) < \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nk : \u2115\njk : k \u2265 j\nh\u2081 : abv (\u2191(f - g) j) < \u03b5 / 2\nh\u2082 : \u2200 (k : \u2115), k \u2265 j \u2192 abv (\u2191f k - \u2191f j) < \u03b5 / 2\n\u22a2 abv (\u2191f k - \u2191g j) < \u03b5\n[PROOFSTEP]\nhave := lt_of_le_of_lt (abv_add abv (f j - g j) _) (add_lt_add h\u2081 (h\u2082 _ jk))\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nh : f \u2248 g\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191(f - g) j) < \u03b5 / 2 \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 abv (\u2191f k - \u2191f j) < \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nk : \u2115\njk : k \u2265 j\nh\u2081 : abv (\u2191(f - g) j) < \u03b5 / 2\nh\u2082 : \u2200 (k : \u2115), k \u2265 j \u2192 abv (\u2191f k - \u2191f j) < \u03b5 / 2\nthis : abv (\u2191f j - \u2191g j + (\u2191f k - \u2191f j)) < \u03b5 / 2 + \u03b5 / 2\n\u22a2 abv (\u2191f k - \u2191g j) < \u03b5\n[PROOFSTEP]\nrwa [sub_add_sub_cancel', add_halves] at this \n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nh : f \u2248 g\nl : LimZero f\n\u22a2 LimZero g\n[PROOFSTEP]\nsimpa using add_limZero (Setoid.symm h) l\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nh : f \u2248 g\nl : LimZero g\n\u22a2 LimZero f\n[PROOFSTEP]\nsimpa using add_limZero h l\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acLimZero f\n\u22a2 \u2203 K, K > 0 \u2227 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 abv (\u2191f j)\n[PROOFSTEP]\nhaveI := Classical.propDecidable\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acLimZero f\nthis : (a : Prop) \u2192 Decidable a\n\u22a2 \u2203 K, K > 0 \u2227 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 abv (\u2191f j)\n[PROOFSTEP]\nby_contra nk\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acLimZero f\nthis : (a : Prop) \u2192 Decidable a\nnk : \u00ac\u2203 K, K > 0 \u2227 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 abv (\u2191f j)\n\u22a2 False\n[PROOFSTEP]\nrefine' hf fun \u03b5 \u03b50 => _\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acLimZero f\nthis : (a : Prop) \u2192 Decidable a\nnk : \u00ac\u2203 K, K > 0 \u2227 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 abv (\u2191f j)\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j) < \u03b5\n[PROOFSTEP]\nsimp [not_forall] at nk \n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acLimZero f\nthis : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, x_1 \u2264 x_2 \u2227 abv (\u2191f x_2) < x\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j) < \u03b5\n[PROOFSTEP]\ncases' f.cauchy\u2083 (half_pos \u03b50) with i hi\n[GOAL]\ncase intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acLimZero f\nthis : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, x_1 \u2264 x_2 \u2227 abv (\u2191f x_2) < x\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 \u2200 (k : \u2115), k \u2265 j \u2192 abv (\u2191f k - \u2191f j) < \u03b5 / 2\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j) < \u03b5\n[PROOFSTEP]\nrcases nk _ (half_pos \u03b50) i with \u27e8j, ij, hj\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acLimZero f\nthis : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, x_1 \u2264 x_2 \u2227 abv (\u2191f x_2) < x\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 \u2200 (k : \u2115), k \u2265 j \u2192 abv (\u2191f k - \u2191f j) < \u03b5 / 2\nj : \u2115\nij : i \u2264 j\nhj : abv (\u2191f j) < \u03b5 / 2\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abv (\u2191f j) < \u03b5\n[PROOFSTEP]\nrefine' \u27e8j, fun k jk => _\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acLimZero f\nthis : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, x_1 \u2264 x_2 \u2227 abv (\u2191f x_2) < x\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 \u2200 (k : \u2115), k \u2265 j \u2192 abv (\u2191f k - \u2191f j) < \u03b5 / 2\nj : \u2115\nij : i \u2264 j\nhj : abv (\u2191f j) < \u03b5 / 2\nk : \u2115\njk : k \u2265 j\n\u22a2 abv (\u2191f k) < \u03b5\n[PROOFSTEP]\nhave := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add (hi j ij k jk) hj)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acLimZero f\nthis\u271d : (a : Prop) \u2192 Decidable a\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nnk : \u2200 (x : \u03b1), 0 < x \u2192 \u2200 (x_1 : \u2115), \u2203 x_2, x_1 \u2264 x_2 \u2227 abv (\u2191f x_2) < x\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 \u2200 (k : \u2115), k \u2265 j \u2192 abv (\u2191f k - \u2191f j) < \u03b5 / 2\nj : \u2115\nij : i \u2264 j\nhj : abv (\u2191f j) < \u03b5 / 2\nk : \u2115\njk : k \u2265 j\nthis : abv (\u2191f k - \u2191f j + \u2191f j) < \u03b5 / 2 + \u03b5 / 2\n\u22a2 abv (\u2191f k) < \u03b5\n[PROOFSTEP]\nrwa [sub_add_cancel, add_halves] at this \n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : \u2115 \u2192 \u03b2\ng : CauSeq \u03b2 abv\nh : \u2200 (\u03b5 : \u03b1), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abv (f j - \u2191g j) < \u03b5\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 abv (f j - \u2191g j) < \u03b5 / 2 / 2 \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 abv (\u2191g k - \u2191g j) < \u03b5 / 2\nj : \u2115\nij : j \u2265 i\n\u22a2 abv (f j - f i) < \u03b5\n[PROOFSTEP]\ncases' hi _ le_rfl with h\u2081 h\u2082\n[GOAL]\ncase intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : \u2115 \u2192 \u03b2\ng : CauSeq \u03b2 abv\nh : \u2200 (\u03b5 : \u03b1), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abv (f j - \u2191g j) < \u03b5\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 abv (f j - \u2191g j) < \u03b5 / 2 / 2 \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 abv (\u2191g k - \u2191g j) < \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nh\u2081 : abv (f i - \u2191g i) < \u03b5 / 2 / 2\nh\u2082 : \u2200 (k : \u2115), k \u2265 i \u2192 abv (\u2191g k - \u2191g i) < \u03b5 / 2\n\u22a2 abv (f j - f i) < \u03b5\n[PROOFSTEP]\nrw [abv_sub abv] at h\u2081 \n[GOAL]\ncase intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : \u2115 \u2192 \u03b2\ng : CauSeq \u03b2 abv\nh : \u2200 (\u03b5 : \u03b1), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abv (f j - \u2191g j) < \u03b5\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 abv (f j - \u2191g j) < \u03b5 / 2 / 2 \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 abv (\u2191g k - \u2191g j) < \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nh\u2081 : abv (\u2191g i - f i) < \u03b5 / 2 / 2\nh\u2082 : \u2200 (k : \u2115), k \u2265 i \u2192 abv (\u2191g k - \u2191g i) < \u03b5 / 2\n\u22a2 abv (f j - f i) < \u03b5\n[PROOFSTEP]\nhave := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add (hi _ ij).1 h\u2081)\n[GOAL]\ncase intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : \u2115 \u2192 \u03b2\ng : CauSeq \u03b2 abv\nh : \u2200 (\u03b5 : \u03b1), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abv (f j - \u2191g j) < \u03b5\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 abv (f j - \u2191g j) < \u03b5 / 2 / 2 \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 abv (\u2191g k - \u2191g j) < \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nh\u2081 : abv (\u2191g i - f i) < \u03b5 / 2 / 2\nh\u2082 : \u2200 (k : \u2115), k \u2265 i \u2192 abv (\u2191g k - \u2191g i) < \u03b5 / 2\nthis : abv (f j - \u2191g j + (\u2191g i - f i)) < \u03b5 / 2 / 2 + \u03b5 / 2 / 2\n\u22a2 abv (f j - f i) < \u03b5\n[PROOFSTEP]\nhave := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add this (h\u2082 _ ij))\n[GOAL]\ncase intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : \u2115 \u2192 \u03b2\ng : CauSeq \u03b2 abv\nh : \u2200 (\u03b5 : \u03b1), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abv (f j - \u2191g j) < \u03b5\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 abv (f j - \u2191g j) < \u03b5 / 2 / 2 \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 abv (\u2191g k - \u2191g j) < \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nh\u2081 : abv (\u2191g i - f i) < \u03b5 / 2 / 2\nh\u2082 : \u2200 (k : \u2115), k \u2265 i \u2192 abv (\u2191g k - \u2191g i) < \u03b5 / 2\nthis\u271d : abv (f j - \u2191g j + (\u2191g i - f i)) < \u03b5 / 2 / 2 + \u03b5 / 2 / 2\nthis : abv (f j - \u2191g j + (\u2191g i - f i) + (\u2191g j - \u2191g i)) < \u03b5 / 2 / 2 + \u03b5 / 2 / 2 + \u03b5 / 2\n\u22a2 abv (f j - f i) < \u03b5\n[PROOFSTEP]\nrwa [add_halves, add_halves, add_right_comm, sub_add_sub_cancel, sub_add_sub_cancel] at this \n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\n\u22a2 \u00acLimZero f\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nh : LimZero f\n\u22a2 False\n[PROOFSTEP]\nhave : LimZero (f - 0) := by simp [h]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nh : LimZero f\n\u22a2 LimZero (f - 0)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nh : LimZero f\nthis : LimZero (f - 0)\n\u22a2 False\n[PROOFSTEP]\nexact hf this\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\ng f : CauSeq \u03b2 abv\nhf : f \u2248 0\nthis : LimZero (f - 0)\n\u22a2 LimZero f\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\ng f : CauSeq \u03b2 abv\nhf : f \u2248 0\nthis\u271d : LimZero (f - 0)\nthis : LimZero (g * f)\n\u22a2 LimZero (g * f - 0)\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\ng f : CauSeq \u03b2 abv\nhf : f \u2248 0\nthis : LimZero (f - 0)\n\u22a2 LimZero f\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\ng f : CauSeq \u03b2 abv\nhf : f \u2248 0\nthis\u271d : LimZero (f - 0)\nthis : LimZero (f * g)\n\u22a2 LimZero (f * g - 0)\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis : LimZero (f * g - 0)\n\u22a2 False\n[PROOFSTEP]\nhave hlz : LimZero (f * g) := by simpa\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis : LimZero (f * g - 0)\n\u22a2 LimZero (f * g)\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis : LimZero (f * g - 0)\nhlz : LimZero (f * g)\n\u22a2 False\n[PROOFSTEP]\nhave hf' : \u00acLimZero f := by simpa using show \u00acLimZero (f - 0) from hf\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis : LimZero (f * g - 0)\nhlz : LimZero (f * g)\n\u22a2 \u00acLimZero f\n[PROOFSTEP]\nsimpa using show \u00acLimZero (f - 0) from hf\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis : LimZero (f * g - 0)\nhlz : LimZero (f * g)\nhf' : \u00acLimZero f\n\u22a2 False\n[PROOFSTEP]\nhave hg' : \u00acLimZero g := by simpa using show \u00acLimZero (g - 0) from hg\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis : LimZero (f * g - 0)\nhlz : LimZero (f * g)\nhf' : \u00acLimZero f\n\u22a2 \u00acLimZero g\n[PROOFSTEP]\nsimpa using show \u00acLimZero (g - 0) from hg\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis : LimZero (f * g - 0)\nhlz : LimZero (f * g)\nhf' : \u00acLimZero f\nhg' : \u00acLimZero g\n\u22a2 False\n[PROOFSTEP]\nrcases abv_pos_of_not_limZero hf' with \u27e8a1, ha1, N1, hN1\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis : LimZero (f * g - 0)\nhlz : LimZero (f * g)\nhf' : \u00acLimZero f\nhg' : \u00acLimZero g\na1 : \u03b1\nha1 : a1 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 a1 \u2264 abv (\u2191f j)\n\u22a2 False\n[PROOFSTEP]\nrcases abv_pos_of_not_limZero hg' with \u27e8a2, ha2, N2, hN2\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis : LimZero (f * g - 0)\nhlz : LimZero (f * g)\nhf' : \u00acLimZero f\nhg' : \u00acLimZero g\na1 : \u03b1\nha1 : a1 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 a1 \u2264 abv (\u2191f j)\na2 : \u03b1\nha2 : a2 > 0\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 a2 \u2264 abv (\u2191g j)\n\u22a2 False\n[PROOFSTEP]\nhave : 0 < a1 * a2 := mul_pos ha1 ha2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis\u271d : LimZero (f * g - 0)\nhlz : LimZero (f * g)\nhf' : \u00acLimZero f\nhg' : \u00acLimZero g\na1 : \u03b1\nha1 : a1 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 a1 \u2264 abv (\u2191f j)\na2 : \u03b1\nha2 : a2 > 0\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 a2 \u2264 abv (\u2191g j)\nthis : 0 < a1 * a2\n\u22a2 False\n[PROOFSTEP]\ncases' hlz _ this with N hN\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis\u271d : LimZero (f * g - 0)\nhlz : LimZero (f * g)\nhf' : \u00acLimZero f\nhg' : \u00acLimZero g\na1 : \u03b1\nha1 : a1 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 a1 \u2264 abv (\u2191f j)\na2 : \u03b1\nha2 : a2 > 0\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 a2 \u2264 abv (\u2191g j)\nthis : 0 < a1 * a2\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 abv (\u2191(f * g) j) < a1 * a2\n\u22a2 False\n[PROOFSTEP]\nlet i := max N (max N1 N2)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis\u271d : LimZero (f * g - 0)\nhlz : LimZero (f * g)\nhf' : \u00acLimZero f\nhg' : \u00acLimZero g\na1 : \u03b1\nha1 : a1 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 a1 \u2264 abv (\u2191f j)\na2 : \u03b1\nha2 : a2 > 0\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 a2 \u2264 abv (\u2191g j)\nthis : 0 < a1 * a2\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 abv (\u2191(f * g) j) < a1 * a2\ni : \u2115 := max N (max N1 N2)\n\u22a2 False\n[PROOFSTEP]\nhave hN' := hN i (le_max_left _ _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis\u271d : LimZero (f * g - 0)\nhlz : LimZero (f * g)\nhf' : \u00acLimZero f\nhg' : \u00acLimZero g\na1 : \u03b1\nha1 : a1 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 a1 \u2264 abv (\u2191f j)\na2 : \u03b1\nha2 : a2 > 0\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 a2 \u2264 abv (\u2191g j)\nthis : 0 < a1 * a2\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 abv (\u2191(f * g) j) < a1 * a2\ni : \u2115 := max N (max N1 N2)\nhN' : abv (\u2191(f * g) i) < a1 * a2\n\u22a2 False\n[PROOFSTEP]\nhave hN1' := hN1 i (le_trans (le_max_left _ _) (le_max_right _ _))\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis\u271d : LimZero (f * g - 0)\nhlz : LimZero (f * g)\nhf' : \u00acLimZero f\nhg' : \u00acLimZero g\na1 : \u03b1\nha1 : a1 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 a1 \u2264 abv (\u2191f j)\na2 : \u03b1\nha2 : a2 > 0\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 a2 \u2264 abv (\u2191g j)\nthis : 0 < a1 * a2\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 abv (\u2191(f * g) j) < a1 * a2\ni : \u2115 := max N (max N1 N2)\nhN' : abv (\u2191(f * g) i) < a1 * a2\nhN1' : a1 \u2264 abv (\u2191f i)\n\u22a2 False\n[PROOFSTEP]\nhave hN1' := hN2 i (le_trans (le_max_right _ _) (le_max_right _ _))\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis\u271d : LimZero (f * g - 0)\nhlz : LimZero (f * g)\nhf' : \u00acLimZero f\nhg' : \u00acLimZero g\na1 : \u03b1\nha1 : a1 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 a1 \u2264 abv (\u2191f j)\na2 : \u03b1\nha2 : a2 > 0\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 a2 \u2264 abv (\u2191g j)\nthis : 0 < a1 * a2\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 abv (\u2191(f * g) j) < a1 * a2\ni : \u2115 := max N (max N1 N2)\nhN' : abv (\u2191(f * g) i) < a1 * a2\nhN1'\u271d : a1 \u2264 abv (\u2191f i)\nhN1' : a2 \u2264 abv (\u2191g i)\n\u22a2 False\n[PROOFSTEP]\napply not_le_of_lt hN'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis\u271d : LimZero (f * g - 0)\nhlz : LimZero (f * g)\nhf' : \u00acLimZero f\nhg' : \u00acLimZero g\na1 : \u03b1\nha1 : a1 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 a1 \u2264 abv (\u2191f j)\na2 : \u03b1\nha2 : a2 > 0\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 a2 \u2264 abv (\u2191g j)\nthis : 0 < a1 * a2\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 abv (\u2191(f * g) j) < a1 * a2\ni : \u2115 := max N (max N1 N2)\nhN' : abv (\u2191(f * g) i) < a1 * a2\nhN1'\u271d : a1 \u2264 abv (\u2191f i)\nhN1' : a2 \u2264 abv (\u2191g i)\n\u22a2 a1 * a2 \u2264 abv (\u2191(f * g) i)\n[PROOFSTEP]\nchange _ \u2264 abv (_ * _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis\u271d : LimZero (f * g - 0)\nhlz : LimZero (f * g)\nhf' : \u00acLimZero f\nhg' : \u00acLimZero g\na1 : \u03b1\nha1 : a1 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 a1 \u2264 abv (\u2191f j)\na2 : \u03b1\nha2 : a2 > 0\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 a2 \u2264 abv (\u2191g j)\nthis : 0 < a1 * a2\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 abv (\u2191(f * g) j) < a1 * a2\ni : \u2115 := max N (max N1 N2)\nhN' : abv (\u2191(f * g) i) < a1 * a2\nhN1'\u271d : a1 \u2264 abv (\u2191f i)\nhN1' : a2 \u2264 abv (\u2191g i)\n\u22a2 a1 * a2 \u2264 abv (\u2191f i * \u2191g i)\n[PROOFSTEP]\nrw [abv_mul abv]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nthis\u271d : LimZero (f * g - 0)\nhlz : LimZero (f * g)\nhf' : \u00acLimZero f\nhg' : \u00acLimZero g\na1 : \u03b1\nha1 : a1 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 a1 \u2264 abv (\u2191f j)\na2 : \u03b1\nha2 : a2 > 0\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 a2 \u2264 abv (\u2191g j)\nthis : 0 < a1 * a2\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 abv (\u2191(f * g) j) < a1 * a2\ni : \u2115 := max N (max N1 N2)\nhN' : abv (\u2191(f * g) i) < a1 * a2\nhN1'\u271d : a1 \u2264 abv (\u2191f i)\nhN1' : a2 \u2264 abv (\u2191g i)\n\u22a2 a1 * a2 \u2264 abv (\u2191f i) * abv (\u2191g i)\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nx y : \u03b2\n\u22a2 LimZero (const x - const y) \u2194 x = y\n[PROOFSTEP]\nrw [\u2190 const_sub, const_limZero, sub_eq_zero]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf1 f2 g1 g2 : CauSeq \u03b2 abv\nhf : f1 \u2248 f2\nhg : g1 \u2248 g2\n\u22a2 f1 * g1 \u2248 f2 * g2\n[PROOFSTEP]\nchange LimZero (f1 * g1 - f2 * g2)\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf1 f2 g1 g2 : CauSeq \u03b2 abv\nhf : f1 \u2248 f2\nhg : g1 \u2248 g2\n\u22a2 LimZero (f1 * g1 - f2 * g2)\n[PROOFSTEP]\nconvert add_limZero (mul_limZero_left g1 hf) (mul_limZero_right f2 hg) using 1\n[GOAL]\ncase h.e'_6\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf1 f2 g1 g2 : CauSeq \u03b2 abv\nhf : f1 \u2248 f2\nhg : g1 \u2248 g2\n\u22a2 f1 * g1 - f2 * g2 = (f1 - f2) * g1 + f2 * (g1 - g2)\n[PROOFSTEP]\nrw [mul_sub, sub_mul]\n -- Porting note: doesn't work with `rw`, but did in Lean 3\n[GOAL]\ncase h.e'_6\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf1 f2 g1 g2 : CauSeq \u03b2 abv\nhf : f1 \u2248 f2\nhg : g1 \u2248 g2\n\u22a2 f1 * g1 - f2 * g2 = f1 * g1 - f2 * g1 + (f2 * g1 - f2 * g2)\n[PROOFSTEP]\nexact (sub_add_sub_cancel (f1 * g1) (f2 * g1) (f2 * g2)).symm\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_2\ninst\u271d\u2074 : LinearOrderedField \u03b1\ninst\u271d\u00b3 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d\u00b2 : IsAbsoluteValue abv\nG : Type u_1\ninst\u271d\u00b9 : SMul G \u03b2\ninst\u271d : IsScalarTower G \u03b2 \u03b2\nf1 f2 : CauSeq \u03b2 abv\nc : G\nhf : f1 \u2248 f2\n\u22a2 c \u2022 f1 \u2248 c \u2022 f2\n[PROOFSTEP]\nsimpa [const_smul, smul_one_mul _ _] using mul_equiv_mul (const_equiv.mpr <| Eq.refl <| c \u2022 (1 : \u03b2)) hf\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf1 f2 : CauSeq \u03b2 abv\nhf : f1 \u2248 f2\nn : \u2115\n\u22a2 f1 ^ n \u2248 f2 ^ n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf1 f2 : CauSeq \u03b2 abv\nhf : f1 \u2248 f2\n\u22a2 f1 ^ Nat.zero \u2248 f2 ^ Nat.zero\n[PROOFSTEP]\nsimp only [Nat.zero_eq, pow_zero, Setoid.refl]\n[GOAL]\ncase succ\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf1 f2 : CauSeq \u03b2 abv\nhf : f1 \u2248 f2\nn : \u2115\nih : f1 ^ n \u2248 f2 ^ n\n\u22a2 f1 ^ Nat.succ n \u2248 f2 ^ Nat.succ n\n[PROOFSTEP]\nsimpa only [pow_succ] using mul_equiv_mul hf ih\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : Ring \u03b2\ninst\u271d\u00b9 : IsDomain \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nh : const abv 1 \u2248 const abv 0\nthis : \u2200 (\u03b5 : \u03b1), \u03b5 > 0 \u2192 \u2203 i, \u2200 (k : \u2115), i \u2264 k \u2192 abv (1 - 0) < \u03b5\nh2 : 0 < abv 1\ni : \u2115\nhi : \u2200 (k : \u2115), i \u2264 k \u2192 abv (1 - 0) < abv 1\n\u22a2 abv 1 < abv 1\n[PROOFSTEP]\nsimpa using hi _ le_rfl\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : DivisionRing \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acLimZero f\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nK : \u03b1\nK0 : K > 0\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 abv (\u2191f j)\nj : \u2115\nij : j \u2265 i\n\u22a2 abv (\u2191(inv f hf * f - 1) j) < \u03b5\n[PROOFSTEP]\nsimpa [(abv_pos abv).1 (lt_of_lt_of_le K0 (H _ ij)), abv_zero abv] using \u03b50\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : DivisionRing \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf : CauSeq \u03b2 abv\nhf : \u00acLimZero f\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nK : \u03b1\nK0 : K > 0\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 abv (\u2191f j)\nj : \u2115\nij : j \u2265 i\n\u22a2 abv (\u2191(f * inv f hf - 1) j) < \u03b5\n[PROOFSTEP]\nsimpa [(abv_pos abv).1 (lt_of_lt_of_le K0 (H _ ij)), abv_zero abv] using \u03b50\n[GOAL]\n\u03b1 : Type ?u.200476\n\u03b2 : Type ?u.200479\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : DivisionRing \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nx : \u03b2\nhx : x \u2260 0\n\u22a2 \u00acLimZero (const abv x)\n[PROOFSTEP]\nrwa [const_limZero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g : CauSeq \u03b1 abs\nF : \u03b1\nF0 : F > 0\nhF : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 F \u2264 \u2191f j\nH : LimZero g\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 F \u2264 \u2191f j \u2227 |\u2191g j| < F / 2\nj : \u2115\nij : j \u2265 i\n\u22a2 F / 2 \u2264 \u2191(f + g) j\n[PROOFSTEP]\ncases' h j ij with h\u2081 h\u2082\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g : CauSeq \u03b1 abs\nF : \u03b1\nF0 : F > 0\nhF : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 F \u2264 \u2191f j\nH : LimZero g\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 F \u2264 \u2191f j \u2227 |\u2191g j| < F / 2\nj : \u2115\nij : j \u2265 i\nh\u2081 : F \u2264 \u2191f j\nh\u2082 : |\u2191g j| < F / 2\n\u22a2 F / 2 \u2264 \u2191(f + g) j\n[PROOFSTEP]\nhave := add_le_add h\u2081 (le_of_lt (abs_lt.1 h\u2082).1)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g : CauSeq \u03b1 abs\nF : \u03b1\nF0 : F > 0\nhF : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 F \u2264 \u2191f j\nH : LimZero g\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 F \u2264 \u2191f j \u2227 |\u2191g j| < F / 2\nj : \u2115\nij : j \u2265 i\nh\u2081 : F \u2264 \u2191f j\nh\u2082 : |\u2191g j| < F / 2\nthis : F + -(F / 2) \u2264 \u2191f j + \u2191g j\n\u22a2 F / 2 \u2264 \u2191(f + g) j\n[PROOFSTEP]\nrwa [\u2190 sub_eq_add_neg, sub_self_div_two] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\n\u22a2 Pos f \u2228 LimZero f \u2228 Pos (-f)\n[PROOFSTEP]\ncases' Classical.em (LimZero f) with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh : LimZero f\n\u22a2 Pos f \u2228 LimZero f \u2228 Pos (-f)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh : \u00acLimZero f\n\u22a2 Pos f \u2228 LimZero f \u2228 Pos (-f)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh : \u00acLimZero f\n\u22a2 Pos f \u2228 Pos (-f)\n[PROOFSTEP]\nrcases abv_pos_of_not_limZero h with \u27e8K, K0, hK\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\n\u22a2 Pos f \u2228 Pos (-f)\n[PROOFSTEP]\nrcases exists_forall_ge_and hK (f.cauchy\u2083 K0) with \u27e8i, hi\u27e9\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j| \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K\n\u22a2 Pos f \u2228 Pos (-f)\n[PROOFSTEP]\nrefine' (le_total 0 (f i)).imp _ _\n[GOAL]\ncase inr.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j| \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K\n\u22a2 0 \u2264 \u2191f i \u2192 Pos f\n[PROOFSTEP]\nrefine' fun h => \u27e8K, K0, i, fun j ij => _\u27e9\n[GOAL]\ncase inr.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j| \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K\n\u22a2 \u2191f i \u2264 0 \u2192 Pos (-f)\n[PROOFSTEP]\nrefine' fun h => \u27e8K, K0, i, fun j ij => _\u27e9\n[GOAL]\ncase inr.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh\u271d : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j| \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K\nh : 0 \u2264 \u2191f i\nj : \u2115\nij : j \u2265 i\n\u22a2 K \u2264 \u2191f j\n[PROOFSTEP]\nhave := (hi _ ij).1\n[GOAL]\ncase inr.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh\u271d : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j| \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K\nh : \u2191f i \u2264 0\nj : \u2115\nij : j \u2265 i\n\u22a2 K \u2264 \u2191(-f) j\n[PROOFSTEP]\nhave := (hi _ ij).1\n[GOAL]\ncase inr.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh\u271d : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j| \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K\nh : 0 \u2264 \u2191f i\nj : \u2115\nij : j \u2265 i\nthis : K \u2264 |\u2191f j|\n\u22a2 K \u2264 \u2191f j\n[PROOFSTEP]\ncases' hi _ le_rfl with h\u2081 h\u2082\n[GOAL]\ncase inr.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh\u271d : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j| \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K\nh : \u2191f i \u2264 0\nj : \u2115\nij : j \u2265 i\nthis : K \u2264 |\u2191f j|\n\u22a2 K \u2264 \u2191(-f) j\n[PROOFSTEP]\ncases' hi _ le_rfl with h\u2081 h\u2082\n[GOAL]\ncase inr.intro.intro.intro.refine'_1.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh\u271d : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j| \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K\nh : 0 \u2264 \u2191f i\nj : \u2115\nij : j \u2265 i\nthis : K \u2264 |\u2191f j|\nh\u2081 : K \u2264 |\u2191f i|\nh\u2082 : \u2200 (k : \u2115), k \u2265 i \u2192 |\u2191f k - \u2191f i| < K\n\u22a2 K \u2264 \u2191f j\n[PROOFSTEP]\nrwa [abs_of_nonneg] at this \n[GOAL]\ncase inr.intro.intro.intro.refine'_1.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh\u271d : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j| \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K\nh : 0 \u2264 \u2191f i\nj : \u2115\nij : j \u2265 i\nthis : K \u2264 |\u2191f j|\nh\u2081 : K \u2264 |\u2191f i|\nh\u2082 : \u2200 (k : \u2115), k \u2265 i \u2192 |\u2191f k - \u2191f i| < K\n\u22a2 0 \u2264 \u2191f j\n[PROOFSTEP]\nrw [abs_of_nonneg h] at h\u2081 \n[GOAL]\ncase inr.intro.intro.intro.refine'_1.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh\u271d : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j| \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K\nh : 0 \u2264 \u2191f i\nj : \u2115\nij : j \u2265 i\nthis : K \u2264 |\u2191f j|\nh\u2081 : K \u2264 \u2191f i\nh\u2082 : \u2200 (k : \u2115), k \u2265 i \u2192 |\u2191f k - \u2191f i| < K\n\u22a2 0 \u2264 \u2191f j\n[PROOFSTEP]\nexact (le_add_iff_nonneg_right _).1 (le_trans h\u2081 <| neg_le_sub_iff_le_add'.1 <| le_of_lt (abs_lt.1 <| h\u2082 _ ij).1)\n[GOAL]\ncase inr.intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh\u271d : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j| \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K\nh : \u2191f i \u2264 0\nj : \u2115\nij : j \u2265 i\nthis : K \u2264 |\u2191f j|\nh\u2081 : K \u2264 |\u2191f i|\nh\u2082 : \u2200 (k : \u2115), k \u2265 i \u2192 |\u2191f k - \u2191f i| < K\n\u22a2 K \u2264 \u2191(-f) j\n[PROOFSTEP]\nrwa [abs_of_nonpos] at this \n[GOAL]\ncase inr.intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh\u271d : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j| \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K\nh : \u2191f i \u2264 0\nj : \u2115\nij : j \u2265 i\nthis : K \u2264 |\u2191f j|\nh\u2081 : K \u2264 |\u2191f i|\nh\u2082 : \u2200 (k : \u2115), k \u2265 i \u2192 |\u2191f k - \u2191f i| < K\n\u22a2 \u2191f j \u2264 0\n[PROOFSTEP]\nrw [abs_of_nonpos h] at h\u2081 \n[GOAL]\ncase inr.intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh\u271d : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j| \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K\nh : \u2191f i \u2264 0\nj : \u2115\nij : j \u2265 i\nthis : K \u2264 |\u2191f j|\nh\u2081 : K \u2264 -\u2191f i\nh\u2082 : \u2200 (k : \u2115), k \u2265 i \u2192 |\u2191f k - \u2191f i| < K\n\u22a2 \u2191f j \u2264 0\n[PROOFSTEP]\nrw [\u2190 sub_le_sub_iff_right, zero_sub]\n[GOAL]\ncase inr.intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh\u271d : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j| \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K\nh : \u2191f i \u2264 0\nj : \u2115\nij : j \u2265 i\nthis : K \u2264 |\u2191f j|\nh\u2081 : K \u2264 -\u2191f i\nh\u2082 : \u2200 (k : \u2115), k \u2265 i \u2192 |\u2191f k - \u2191f i| < K\n\u22a2 \u2191f j - ?inr.intro.intro.intro.refine'_2.intro \u2264 -?inr.intro.intro.intro.refine'_2.intro\ncase inr.intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nh\u271d : \u00acLimZero f\nK : \u03b1\nK0 : K > 0\nhK : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j|\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 K \u2264 |\u2191f j| \u2227 \u2200 (k : \u2115), k \u2265 j \u2192 |\u2191f k - \u2191f j| < K\nh : \u2191f i \u2264 0\nj : \u2115\nij : j \u2265 i\nthis : K \u2264 |\u2191f j|\nh\u2081 : K \u2264 -\u2191f i\nh\u2082 : \u2200 (k : \u2115), k \u2265 i \u2192 |\u2191f k - \u2191f i| < K\n\u22a2 \u03b1\n[PROOFSTEP]\nexact le_trans (le_of_lt (abs_lt.1 <| h\u2082 _ ij).2) h\u2081\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g h : CauSeq \u03b1 abs\nfg : f < g\ngh : g \u2248 h\n\u22a2 Pos (h - f)\n[PROOFSTEP]\nconvert pos_add_limZero fg (neg_limZero gh) using 1\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g h : CauSeq \u03b1 abs\nfg : f < g\ngh : g \u2248 h\n\u22a2 h - f = g - f + -(g - h)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g h : CauSeq \u03b1 abs\nfg : f \u2248 g\ngh : g < h\n\u22a2 f < h\n[PROOFSTEP]\nhave := pos_add_limZero gh (neg_limZero fg)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g h : CauSeq \u03b1 abs\nfg : f \u2248 g\ngh : g < h\nthis : Pos (h - g + -(f - g))\n\u22a2 f < h\n[PROOFSTEP]\nrwa [\u2190 sub_eq_add_neg, sub_sub_sub_cancel_right] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g h : CauSeq \u03b1 abs\nfg : f < g\ngh : g < h\n\u22a2 Pos (h - f)\n[PROOFSTEP]\nconvert add_pos fg gh using 1\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g h : CauSeq \u03b1 abs\nfg : f < g\ngh : g < h\n\u22a2 h - f = g - f + (h - g)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nx\u271d : f < f\nh : f < f := x\u271d\n\u22a2 LimZero (f - f)\n[PROOFSTEP]\nsimp [zero_limZero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g : CauSeq \u03b1 abs\nh\u271d : LimZero (g - f) \u2228 Pos (-(g - f))\nh : Pos (-(g - f))\n\u22a2 g < f\n[PROOFSTEP]\nrwa [neg_sub] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nx y : \u03b1\n\u22a2 Pos (const y - const x) \u2194 x < y\n[PROOFSTEP]\nrw [\u2190 const_sub, const_pos, sub_pos]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nx y : \u03b1\n\u22a2 const x \u2264 const y \u2194 x \u2264 y\n[PROOFSTEP]\nrw [le_iff_lt_or_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nx y : \u03b1\n\u22a2 const x \u2264 const y \u2194 x < y \u2228 x = y\n[PROOFSTEP]\nexact or_congr const_lt const_equiv\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nK : \u03b1\nH : \u2200 (i : \u2115), |\u2191f i| < K\ni : \u2115\nx\u271d : i \u2265 0\n\u22a2 1 \u2264 \u2191(const (K + 1) - f) i\n[PROOFSTEP]\nrw [sub_apply, const_apply, le_sub_iff_add_le', add_le_add_iff_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\nK : \u03b1\nH : \u2200 (i : \u2115), |\u2191f i| < K\ni : \u2115\nx\u271d : i \u2265 0\n\u22a2 \u2191f i \u2264 K\n[PROOFSTEP]\nexact le_of_lt (abs_lt.1 (H _)).2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf : CauSeq \u03b1 abs\na : \u03b1\nh : -f < const a\n\u22a2 Pos (f - const (-a))\n[PROOFSTEP]\nrwa [const_neg, sub_neg_eq_add, add_comm, \u2190 sub_neg_eq_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g : CauSeq \u03b1 abs\nhf : LimZero f\nhg : LimZero g\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 |\u2191f j| < \u03b5 \u2227 |\u2191g j| < \u03b5\nj : \u2115\nij : j \u2265 i\n\u22a2 |\u2191(f \u2294 g) j| < \u03b5\n[PROOFSTEP]\nlet \u27e8H\u2081, H\u2082\u27e9 := H _ ij\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g : CauSeq \u03b1 abs\nhf : LimZero f\nhg : LimZero g\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 |\u2191f j| < \u03b5 \u2227 |\u2191g j| < \u03b5\nj : \u2115\nij : j \u2265 i\nH\u2081 : |\u2191f j| < \u03b5\nH\u2082 : |\u2191g j| < \u03b5\n\u22a2 |\u2191(f \u2294 g) j| < \u03b5\n[PROOFSTEP]\nrw [abs_lt] at H\u2081 H\u2082 \u22a2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g : CauSeq \u03b1 abs\nhf : LimZero f\nhg : LimZero g\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 |\u2191f j| < \u03b5 \u2227 |\u2191g j| < \u03b5\nj : \u2115\nij : j \u2265 i\nH\u2081 : -\u03b5 < \u2191f j \u2227 \u2191f j < \u03b5\nH\u2082 : -\u03b5 < \u2191g j \u2227 \u2191g j < \u03b5\n\u22a2 -\u03b5 < \u2191(f \u2294 g) j \u2227 \u2191(f \u2294 g) j < \u03b5\n[PROOFSTEP]\nexact \u27e8lt_sup_iff.mpr (Or.inl H\u2081.1), sup_lt_iff.mpr \u27e8H\u2081.2, H\u2082.2\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g : CauSeq \u03b1 abs\nhf : LimZero f\nhg : LimZero g\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 |\u2191f j| < \u03b5 \u2227 |\u2191g j| < \u03b5\nj : \u2115\nij : j \u2265 i\n\u22a2 |\u2191(f \u2293 g) j| < \u03b5\n[PROOFSTEP]\nlet \u27e8H\u2081, H\u2082\u27e9 := H _ ij\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g : CauSeq \u03b1 abs\nhf : LimZero f\nhg : LimZero g\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 |\u2191f j| < \u03b5 \u2227 |\u2191g j| < \u03b5\nj : \u2115\nij : j \u2265 i\nH\u2081 : |\u2191f j| < \u03b5\nH\u2082 : |\u2191g j| < \u03b5\n\u22a2 |\u2191(f \u2293 g) j| < \u03b5\n[PROOFSTEP]\nrw [abs_lt] at H\u2081 H\u2082 \u22a2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nf g : CauSeq \u03b1 abs\nhf : LimZero f\nhg : LimZero g\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 |\u2191f j| < \u03b5 \u2227 |\u2191g j| < \u03b5\nj : \u2115\nij : j \u2265 i\nH\u2081 : -\u03b5 < \u2191f j \u2227 \u2191f j < \u03b5\nH\u2082 : -\u03b5 < \u2191g j \u2227 \u2191g j < \u03b5\n\u22a2 -\u03b5 < \u2191(f \u2293 g) j \u2227 \u2191(f \u2293 g) j < \u03b5\n[PROOFSTEP]\nexact \u27e8lt_inf_iff.mpr \u27e8H\u2081.1, H\u2082.1\u27e9, inf_lt_iff.mpr (Or.inl H\u2081.2)\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u2081 b\u2081 a\u2082 b\u2082 : CauSeq \u03b1 abs\nha : a\u2081 \u2248 a\u2082\nhb : b\u2081 \u2248 b\u2082\n\u22a2 a\u2081 \u2294 b\u2081 \u2248 a\u2082 \u2294 b\u2082\n[PROOFSTEP]\nintro \u03b5 \u03b50\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u2081 b\u2081 a\u2082 b\u2082 : CauSeq \u03b1 abs\nha : a\u2081 \u2248 a\u2082\nhb : b\u2081 \u2248 b\u2082\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 |\u2191(a\u2081 \u2294 b\u2081 - a\u2082 \u2294 b\u2082) j| < \u03b5\n[PROOFSTEP]\nobtain \u27e8ai, hai\u27e9 := ha \u03b5 \u03b50\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u2081 b\u2081 a\u2082 b\u2082 : CauSeq \u03b1 abs\nha : a\u2081 \u2248 a\u2082\nhb : b\u2081 \u2248 b\u2082\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nai : \u2115\nhai : \u2200 (j : \u2115), j \u2265 ai \u2192 |\u2191(a\u2081 - a\u2082) j| < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 |\u2191(a\u2081 \u2294 b\u2081 - a\u2082 \u2294 b\u2082) j| < \u03b5\n[PROOFSTEP]\nobtain \u27e8bi, hbi\u27e9 := hb \u03b5 \u03b50\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u2081 b\u2081 a\u2082 b\u2082 : CauSeq \u03b1 abs\nha : a\u2081 \u2248 a\u2082\nhb : b\u2081 \u2248 b\u2082\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nai : \u2115\nhai : \u2200 (j : \u2115), j \u2265 ai \u2192 |\u2191(a\u2081 - a\u2082) j| < \u03b5\nbi : \u2115\nhbi : \u2200 (j : \u2115), j \u2265 bi \u2192 |\u2191(b\u2081 - b\u2082) j| < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 |\u2191(a\u2081 \u2294 b\u2081 - a\u2082 \u2294 b\u2082) j| < \u03b5\n[PROOFSTEP]\nexact\n \u27e8ai \u2294 bi, fun i hi =>\n (abs_max_sub_max_le_max (a\u2081 i) (b\u2081 i) (a\u2082 i) (b\u2082 i)).trans_lt\n (max_lt (hai i (sup_le_iff.mp hi).1) (hbi i (sup_le_iff.mp hi).2))\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u2081 b\u2081 a\u2082 b\u2082 : CauSeq \u03b1 abs\nha : a\u2081 \u2248 a\u2082\nhb : b\u2081 \u2248 b\u2082\n\u22a2 a\u2081 \u2293 b\u2081 \u2248 a\u2082 \u2293 b\u2082\n[PROOFSTEP]\nintro \u03b5 \u03b50\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u2081 b\u2081 a\u2082 b\u2082 : CauSeq \u03b1 abs\nha : a\u2081 \u2248 a\u2082\nhb : b\u2081 \u2248 b\u2082\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 |\u2191(a\u2081 \u2293 b\u2081 - a\u2082 \u2293 b\u2082) j| < \u03b5\n[PROOFSTEP]\nobtain \u27e8ai, hai\u27e9 := ha \u03b5 \u03b50\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u2081 b\u2081 a\u2082 b\u2082 : CauSeq \u03b1 abs\nha : a\u2081 \u2248 a\u2082\nhb : b\u2081 \u2248 b\u2082\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nai : \u2115\nhai : \u2200 (j : \u2115), j \u2265 ai \u2192 |\u2191(a\u2081 - a\u2082) j| < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 |\u2191(a\u2081 \u2293 b\u2081 - a\u2082 \u2293 b\u2082) j| < \u03b5\n[PROOFSTEP]\nobtain \u27e8bi, hbi\u27e9 := hb \u03b5 \u03b50\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u2081 b\u2081 a\u2082 b\u2082 : CauSeq \u03b1 abs\nha : a\u2081 \u2248 a\u2082\nhb : b\u2081 \u2248 b\u2082\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nai : \u2115\nhai : \u2200 (j : \u2115), j \u2265 ai \u2192 |\u2191(a\u2081 - a\u2082) j| < \u03b5\nbi : \u2115\nhbi : \u2200 (j : \u2115), j \u2265 bi \u2192 |\u2191(b\u2081 - b\u2082) j| < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 |\u2191(a\u2081 \u2293 b\u2081 - a\u2082 \u2293 b\u2082) j| < \u03b5\n[PROOFSTEP]\nexact\n \u27e8ai \u2294 bi, fun i hi =>\n (abs_min_sub_min_le_max (a\u2081 i) (b\u2081 i) (a\u2082 i) (b\u2082 i)).trans_lt\n (max_lt (hai i (sup_le_iff.mp hi).1) (hbi i (sup_le_iff.mp hi).2))\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nha : a < c\nhb : b < c\n\u22a2 a \u2294 b < c\n[PROOFSTEP]\nobtain \u27e8\u27e8\u03b5a, \u03b5a0, ia, ha\u27e9, \u27e8\u03b5b, \u03b5b0, ib, hb\u27e9\u27e9 := ha, hb\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\n\u03b5a : \u03b1\n\u03b5a0 : \u03b5a > 0\nia : \u2115\nha : \u2200 (j : \u2115), j \u2265 ia \u2192 \u03b5a \u2264 \u2191(c - a) j\n\u03b5b : \u03b1\n\u03b5b0 : \u03b5b > 0\nib : \u2115\nhb : \u2200 (j : \u2115), j \u2265 ib \u2192 \u03b5b \u2264 \u2191(c - b) j\n\u22a2 a \u2294 b < c\n[PROOFSTEP]\nrefine' \u27e8\u03b5a \u2293 \u03b5b, lt_inf_iff.mpr \u27e8\u03b5a0, \u03b5b0\u27e9, ia \u2294 ib, fun i hi => _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\n\u03b5a : \u03b1\n\u03b5a0 : \u03b5a > 0\nia : \u2115\nha : \u2200 (j : \u2115), j \u2265 ia \u2192 \u03b5a \u2264 \u2191(c - a) j\n\u03b5b : \u03b1\n\u03b5b0 : \u03b5b > 0\nib : \u2115\nhb : \u2200 (j : \u2115), j \u2265 ib \u2192 \u03b5b \u2264 \u2191(c - b) j\ni : \u2115\nhi : i \u2265 ia \u2294 ib\n\u22a2 \u03b5a \u2293 \u03b5b \u2264 \u2191(c - a \u2294 b) i\n[PROOFSTEP]\nhave := min_le_min (ha _ (sup_le_iff.mp hi).1) (hb _ (sup_le_iff.mp hi).2)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\n\u03b5a : \u03b1\n\u03b5a0 : \u03b5a > 0\nia : \u2115\nha : \u2200 (j : \u2115), j \u2265 ia \u2192 \u03b5a \u2264 \u2191(c - a) j\n\u03b5b : \u03b1\n\u03b5b0 : \u03b5b > 0\nib : \u2115\nhb : \u2200 (j : \u2115), j \u2265 ib \u2192 \u03b5b \u2264 \u2191(c - b) j\ni : \u2115\nhi : i \u2265 ia \u2294 ib\nthis : min \u03b5a \u03b5b \u2264 min (\u2191(c - a) i) (\u2191(c - b) i)\n\u22a2 \u03b5a \u2293 \u03b5b \u2264 \u2191(c - a \u2294 b) i\n[PROOFSTEP]\nexact this.trans_eq (min_sub_sub_left _ _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nhb : a < b\nhc : a < c\n\u22a2 a < b \u2293 c\n[PROOFSTEP]\nobtain \u27e8\u27e8\u03b5b, \u03b5b0, ib, hb\u27e9, \u27e8\u03b5c, \u03b5c0, ic, hc\u27e9\u27e9 := hb, hc\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\n\u03b5b : \u03b1\n\u03b5b0 : \u03b5b > 0\nib : \u2115\nhb : \u2200 (j : \u2115), j \u2265 ib \u2192 \u03b5b \u2264 \u2191(b - a) j\n\u03b5c : \u03b1\n\u03b5c0 : \u03b5c > 0\nic : \u2115\nhc : \u2200 (j : \u2115), j \u2265 ic \u2192 \u03b5c \u2264 \u2191(c - a) j\n\u22a2 a < b \u2293 c\n[PROOFSTEP]\nrefine' \u27e8\u03b5b \u2293 \u03b5c, lt_inf_iff.mpr \u27e8\u03b5b0, \u03b5c0\u27e9, ib \u2294 ic, fun i hi => _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\n\u03b5b : \u03b1\n\u03b5b0 : \u03b5b > 0\nib : \u2115\nhb : \u2200 (j : \u2115), j \u2265 ib \u2192 \u03b5b \u2264 \u2191(b - a) j\n\u03b5c : \u03b1\n\u03b5c0 : \u03b5c > 0\nic : \u2115\nhc : \u2200 (j : \u2115), j \u2265 ic \u2192 \u03b5c \u2264 \u2191(c - a) j\ni : \u2115\nhi : i \u2265 ib \u2294 ic\n\u22a2 \u03b5b \u2293 \u03b5c \u2264 \u2191(b \u2293 c - a) i\n[PROOFSTEP]\nhave := min_le_min (hb _ (sup_le_iff.mp hi).1) (hc _ (sup_le_iff.mp hi).2)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\n\u03b5b : \u03b1\n\u03b5b0 : \u03b5b > 0\nib : \u2115\nhb : \u2200 (j : \u2115), j \u2265 ib \u2192 \u03b5b \u2264 \u2191(b - a) j\n\u03b5c : \u03b1\n\u03b5c0 : \u03b5c > 0\nic : \u2115\nhc : \u2200 (j : \u2115), j \u2265 ic \u2192 \u03b5c \u2264 \u2191(c - a) j\ni : \u2115\nhi : i \u2265 ib \u2294 ic\nthis : min \u03b5b \u03b5c \u2264 min (\u2191(b - a) i) (\u2191(c - a) i)\n\u22a2 \u03b5b \u2293 \u03b5c \u2264 \u2191(b \u2293 c - a) i\n[PROOFSTEP]\nexact this.trans_eq (min_sub_sub_right _ _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\nh : a \u2264 b\n\u22a2 a \u2294 b \u2248 b\n[PROOFSTEP]\nobtain \u27e8\u03b5, \u03b50 : _ < _, i, h\u27e9 | h := h\n[GOAL]\ncase inl.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(b - a) j\n\u22a2 a \u2294 b \u2248 b\n[PROOFSTEP]\nintro _ _\n[GOAL]\ncase inl.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(b - a) j\n\u03b5\u271d : \u03b1\na\u271d : \u03b5\u271d > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 |\u2191(a \u2294 b - b) j| < \u03b5\u271d\n[PROOFSTEP]\nrefine' \u27e8i, fun j hj => _\u27e9\n[GOAL]\ncase inl.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(b - a) j\n\u03b5\u271d : \u03b1\na\u271d : \u03b5\u271d > 0\nj : \u2115\nhj : j \u2265 i\n\u22a2 |\u2191(a \u2294 b - b) j| < \u03b5\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase inl.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(b - a) j\n\u03b5\u271d : \u03b1\na\u271d : \u03b5\u271d > 0\nj : \u2115\nhj : j \u2265 i\n\u22a2 |\u2191a j \u2294 \u2191b j - \u2191b j| < \u03b5\u271d\n[PROOFSTEP]\nerw [\u2190 max_sub_sub_right]\n[GOAL]\ncase inl.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(b - a) j\n\u03b5\u271d : \u03b1\na\u271d : \u03b5\u271d > 0\nj : \u2115\nhj : j \u2265 i\n\u22a2 |max (\u2191a j - \u2191b j) (\u2191b j - \u2191b j)| < \u03b5\u271d\n[PROOFSTEP]\nrwa [sub_self, max_eq_right, abs_zero]\n[GOAL]\ncase inl.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(b - a) j\n\u03b5\u271d : \u03b1\na\u271d : \u03b5\u271d > 0\nj : \u2115\nhj : j \u2265 i\n\u22a2 \u2191a j - \u2191b j \u2264 0\n[PROOFSTEP]\nrw [sub_nonpos, \u2190 sub_nonneg]\n[GOAL]\ncase inl.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(b - a) j\n\u03b5\u271d : \u03b1\na\u271d : \u03b5\u271d > 0\nj : \u2115\nhj : j \u2265 i\n\u22a2 0 \u2264 \u2191b j - \u2191a j\n[PROOFSTEP]\nexact \u03b50.le.trans (h _ hj)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\nh : a \u2248 b\n\u22a2 a \u2294 b \u2248 b\n[PROOFSTEP]\nrefine' Setoid.trans (sup_equiv_sup h (Setoid.refl _)) _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\nh : a \u2248 b\n\u22a2 b \u2294 b \u2248 b\n[PROOFSTEP]\nrw [CauSeq.sup_idem]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\nh : b \u2264 a\n\u22a2 a \u2293 b \u2248 b\n[PROOFSTEP]\nobtain \u27e8\u03b5, \u03b50 : _ < _, i, h\u27e9 | h := h\n[GOAL]\ncase inl.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(a - b) j\n\u22a2 a \u2293 b \u2248 b\n[PROOFSTEP]\nintro _ _\n[GOAL]\ncase inl.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(a - b) j\n\u03b5\u271d : \u03b1\na\u271d : \u03b5\u271d > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 |\u2191(a \u2293 b - b) j| < \u03b5\u271d\n[PROOFSTEP]\nrefine' \u27e8i, fun j hj => _\u27e9\n[GOAL]\ncase inl.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(a - b) j\n\u03b5\u271d : \u03b1\na\u271d : \u03b5\u271d > 0\nj : \u2115\nhj : j \u2265 i\n\u22a2 |\u2191(a \u2293 b - b) j| < \u03b5\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase inl.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(a - b) j\n\u03b5\u271d : \u03b1\na\u271d : \u03b5\u271d > 0\nj : \u2115\nhj : j \u2265 i\n\u22a2 |\u2191a j \u2293 \u2191b j - \u2191b j| < \u03b5\u271d\n[PROOFSTEP]\nerw [\u2190 min_sub_sub_right]\n[GOAL]\ncase inl.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(a - b) j\n\u03b5\u271d : \u03b1\na\u271d : \u03b5\u271d > 0\nj : \u2115\nhj : j \u2265 i\n\u22a2 |min (\u2191a j - \u2191b j) (\u2191b j - \u2191b j)| < \u03b5\u271d\n[PROOFSTEP]\nrwa [sub_self, min_eq_right, abs_zero]\n[GOAL]\ncase inl.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\n\u03b5 : \u03b1\n\u03b50 : 0 < \u03b5\ni : \u2115\nh : \u2200 (j : \u2115), j \u2265 i \u2192 \u03b5 \u2264 \u2191(a - b) j\n\u03b5\u271d : \u03b1\na\u271d : \u03b5\u271d > 0\nj : \u2115\nhj : j \u2265 i\n\u22a2 0 \u2264 \u2191a j - \u2191b j\n[PROOFSTEP]\nexact \u03b50.le.trans (h _ hj)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\nh : b \u2248 a\n\u22a2 a \u2293 b \u2248 b\n[PROOFSTEP]\nrefine' Setoid.trans (inf_equiv_inf (Setoid.symm h) (Setoid.refl _)) _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\nh : b \u2248 a\n\u22a2 b \u2293 b \u2248 b\n[PROOFSTEP]\nrw [CauSeq.inf_idem]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\nh : b \u2264 a\n\u22a2 a \u2294 b \u2248 a\n[PROOFSTEP]\nsimpa only [CauSeq.sup_comm] using CauSeq.sup_eq_right h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b : CauSeq \u03b1 abs\nh : a \u2264 b\n\u22a2 a \u2293 b \u2248 a\n[PROOFSTEP]\nsimpa only [CauSeq.inf_comm] using CauSeq.inf_eq_right h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nha : a \u2264 c\nhb : b \u2264 c\n\u22a2 a \u2294 b \u2264 c\n[PROOFSTEP]\ncases' ha with ha ha\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nhb : b \u2264 c\nha : a < c\n\u22a2 a \u2294 b \u2264 c\n[PROOFSTEP]\ncases' hb with hb hb\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nha : a < c\nhb : b < c\n\u22a2 a \u2294 b \u2264 c\n[PROOFSTEP]\nexact Or.inl (CauSeq.sup_lt ha hb)\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nha : a < c\nhb : b \u2248 c\n\u22a2 a \u2294 b \u2264 c\n[PROOFSTEP]\nreplace ha := le_of_le_of_eq ha.le (Setoid.symm hb)\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nhb : b \u2248 c\nha : a \u2264 b\n\u22a2 a \u2294 b \u2264 c\n[PROOFSTEP]\nrefine' le_of_le_of_eq (Or.inr _) hb\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nhb : b \u2248 c\nha : a \u2264 b\n\u22a2 a \u2294 b \u2248 b\n[PROOFSTEP]\nexact CauSeq.sup_eq_right ha\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nhb : b \u2264 c\nha : a \u2248 c\n\u22a2 a \u2294 b \u2264 c\n[PROOFSTEP]\nreplace hb := le_of_le_of_eq hb (Setoid.symm ha)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nha : a \u2248 c\nhb : b \u2264 a\n\u22a2 a \u2294 b \u2264 c\n[PROOFSTEP]\nrefine' le_of_le_of_eq (Or.inr _) ha\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nha : a \u2248 c\nhb : b \u2264 a\n\u22a2 a \u2294 b \u2248 a\n[PROOFSTEP]\nexact CauSeq.sup_eq_left hb\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nhb : a \u2264 b\nhc : a \u2264 c\n\u22a2 a \u2264 b \u2293 c\n[PROOFSTEP]\ncases' hb with hb hb\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nhc : a \u2264 c\nhb : a < b\n\u22a2 a \u2264 b \u2293 c\n[PROOFSTEP]\ncases' hc with hc hc\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nhb : a < b\nhc : a < c\n\u22a2 a \u2264 b \u2293 c\n[PROOFSTEP]\nexact Or.inl (CauSeq.lt_inf hb hc)\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nhb : a < b\nhc : a \u2248 c\n\u22a2 a \u2264 b \u2293 c\n[PROOFSTEP]\nreplace hb := le_of_eq_of_le (Setoid.symm hc) hb.le\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nhc : a \u2248 c\nhb : c \u2264 b\n\u22a2 a \u2264 b \u2293 c\n[PROOFSTEP]\nrefine' le_of_eq_of_le hc (Or.inr _)\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nhc : a \u2248 c\nhb : c \u2264 b\n\u22a2 c \u2248 b \u2293 c\n[PROOFSTEP]\nexact Setoid.symm (CauSeq.inf_eq_right hb)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nhc : a \u2264 c\nhb : a \u2248 b\n\u22a2 a \u2264 b \u2293 c\n[PROOFSTEP]\nreplace hc := le_of_eq_of_le (Setoid.symm hb) hc\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nhb : a \u2248 b\nhc : b \u2264 c\n\u22a2 a \u2264 b \u2293 c\n[PROOFSTEP]\nrefine' le_of_eq_of_le hb (Or.inr _)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c : CauSeq \u03b1 abs\nhb : a \u2248 b\nhc : b \u2264 c\n\u22a2 b \u2248 b \u2293 c\n[PROOFSTEP]\nexact Setoid.symm (CauSeq.inf_eq_left hc)\n", "meta": {"mathlib_filename": "Mathlib.Data.Real.CauSeq", "llama_tokens": 41787, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707281, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.4264322766404823}} {"text": "[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\nm : M\nn : N\np : Submodule R M\nq : Submodule R N\nhm : m \u2208 p\nhn : n \u2208 q\n\u22a2 \u2191(\u2191f \u2191{ val := m, property := hm }) n = \u2191(\u2191f m) n\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 map\u2082 f (span R s) (span R t) = span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 map\u2082 f (span R s) (span R t) \u2264 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nrw [map\u2082_le]\n[GOAL]\ncase a\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (m : M), m \u2208 span R s \u2192 \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f m) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\napply @span_induction' R M _ _ _ s\n[GOAL]\ncase a.Hs\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (x : M), x \u2208 s \u2192 \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f x) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.H0\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f 0) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.H1\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (x : M),\n x \u2208 span R s \u2192\n \u2200 (y : M),\n y \u2208 span R s \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f x) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f y) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f (x + y)) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.H2\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (a : R) (x : M),\n x \u2208 span R s \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f x) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f (a \u2022 x)) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nintro a ha\n[GOAL]\ncase a.Hs\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\n\u22a2 \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f a) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.H0\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f 0) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.H1\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (x : M),\n x \u2208 span R s \u2192\n \u2200 (y : M),\n y \u2208 span R s \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f x) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f y) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f (x + y)) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.H2\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (a : R) (x : M),\n x \u2208 span R s \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f x) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f (a \u2022 x)) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\napply @span_induction' R N _ _ _ t\n[GOAL]\ncase a.Hs.Hs\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\n\u22a2 \u2200 (x : N), x \u2208 t \u2192 \u2191(\u2191f a) x \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.Hs.H0\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\n\u22a2 \u2191(\u2191f a) 0 \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.Hs.H1\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\n\u22a2 \u2200 (x : N),\n x \u2208 span R t \u2192\n \u2200 (y : N),\n y \u2208 span R t \u2192\n \u2191(\u2191f a) x \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t) \u2192\n \u2191(\u2191f a) y \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t) \u2192\n \u2191(\u2191f a) (x + y) \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.Hs.H2\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\n\u22a2 \u2200 (a_1 : R) (x : N),\n x \u2208 span R t \u2192\n \u2191(\u2191f a) x \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t) \u2192\n \u2191(\u2191f a) (a_1 \u2022 x) \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.H0\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f 0) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.H1\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (x : M),\n x \u2208 span R s \u2192\n \u2200 (y : M),\n y \u2208 span R s \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f x) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f y) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f (x + y)) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.H2\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (a : R) (x : M),\n x \u2208 span R s \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f x) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f (a \u2022 x)) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nintro b hb\n[GOAL]\ncase a.Hs.Hs\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\nb : N\nhb : b \u2208 t\n\u22a2 \u2191(\u2191f a) b \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.Hs.H0\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\n\u22a2 \u2191(\u2191f a) 0 \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.Hs.H1\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\n\u22a2 \u2200 (x : N),\n x \u2208 span R t \u2192\n \u2200 (y : N),\n y \u2208 span R t \u2192\n \u2191(\u2191f a) x \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t) \u2192\n \u2191(\u2191f a) y \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t) \u2192\n \u2191(\u2191f a) (x + y) \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.Hs.H2\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\n\u22a2 \u2200 (a_1 : R) (x : N),\n x \u2208 span R t \u2192\n \u2191(\u2191f a) x \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t) \u2192\n \u2191(\u2191f a) (a_1 \u2022 x) \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.H0\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f 0) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.H1\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (x : M),\n x \u2208 span R s \u2192\n \u2200 (y : M),\n y \u2208 span R s \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f x) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f y) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f (x + y)) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.H2\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (a : R) (x : M),\n x \u2208 span R s \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f x) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f (a \u2022 x)) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nexact subset_span \u27e8_, _, \u2039_\u203a, \u2039_\u203a, rfl\u27e9\n[GOAL]\ncase a.Hs.H0\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\n\u22a2 \u2191(\u2191f a) 0 \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.Hs.H1\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\n\u22a2 \u2200 (x : N),\n x \u2208 span R t \u2192\n \u2200 (y : N),\n y \u2208 span R t \u2192\n \u2191(\u2191f a) x \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t) \u2192\n \u2191(\u2191f a) y \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t) \u2192\n \u2191(\u2191f a) (x + y) \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.Hs.H2\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\n\u22a2 \u2200 (a_1 : R) (x : N),\n x \u2208 span R t \u2192\n \u2191(\u2191f a) x \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t) \u2192\n \u2191(\u2191f a) (a_1 \u2022 x) \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.H0\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f 0) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.H1\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (x : M),\n x \u2208 span R s \u2192\n \u2200 (y : M),\n y \u2208 span R s \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f x) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f y) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f (x + y)) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\ncase a.H2\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (a : R) (x : M),\n x \u2208 span R s \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f x) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f (a \u2022 x)) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nall_goals intros;\n simp only [*, add_mem, smul_mem, zero_mem, _root_.map_zero, map_add, LinearMap.zero_apply, LinearMap.add_apply,\n LinearMap.smul_apply, SMulHomClass.map_smul]\n[GOAL]\ncase a.Hs.H0\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\n\u22a2 \u2191(\u2191f a) 0 \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nintros\n[GOAL]\ncase a.Hs.H0\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\n\u22a2 \u2191(\u2191f a) 0 \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nsimp only [*, add_mem, smul_mem, zero_mem, _root_.map_zero, map_add, LinearMap.zero_apply, LinearMap.add_apply,\n LinearMap.smul_apply, SMulHomClass.map_smul]\n[GOAL]\ncase a.Hs.H1\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\n\u22a2 \u2200 (x : N),\n x \u2208 span R t \u2192\n \u2200 (y : N),\n y \u2208 span R t \u2192\n \u2191(\u2191f a) x \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t) \u2192\n \u2191(\u2191f a) y \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t) \u2192\n \u2191(\u2191f a) (x + y) \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nintros\n[GOAL]\ncase a.Hs.H1\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\nx\u271d : N\nhx\u271d : x\u271d \u2208 span R t\ny\u271d : N\nhy\u271d : y\u271d \u2208 span R t\na\u271d\u00b9 : \u2191(\u2191f a) x\u271d \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\na\u271d : \u2191(\u2191f a) y\u271d \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n\u22a2 \u2191(\u2191f a) (x\u271d + y\u271d) \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nsimp only [*, add_mem, smul_mem, zero_mem, _root_.map_zero, map_add, LinearMap.zero_apply, LinearMap.add_apply,\n LinearMap.smul_apply, SMulHomClass.map_smul]\n[GOAL]\ncase a.Hs.H2\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\n\u22a2 \u2200 (a_1 : R) (x : N),\n x \u2208 span R t \u2192\n \u2191(\u2191f a) x \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t) \u2192\n \u2191(\u2191f a) (a_1 \u2022 x) \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nintros\n[GOAL]\ncase a.Hs.H2\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nha : a \u2208 s\na\u271d\u00b9 : R\nx\u271d : N\nhx\u271d : x\u271d \u2208 span R t\na\u271d : \u2191(\u2191f a) x\u271d \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n\u22a2 \u2191(\u2191f a) (a\u271d\u00b9 \u2022 x\u271d) \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nsimp only [*, add_mem, smul_mem, zero_mem, _root_.map_zero, map_add, LinearMap.zero_apply, LinearMap.add_apply,\n LinearMap.smul_apply, SMulHomClass.map_smul]\n[GOAL]\ncase a.H0\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f 0) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nintros\n[GOAL]\ncase a.H0\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\nn\u271d : N\na\u271d : n\u271d \u2208 span R t\n\u22a2 \u2191(\u2191f 0) n\u271d \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nsimp only [*, add_mem, smul_mem, zero_mem, _root_.map_zero, map_add, LinearMap.zero_apply, LinearMap.add_apply,\n LinearMap.smul_apply, SMulHomClass.map_smul]\n[GOAL]\ncase a.H1\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (x : M),\n x \u2208 span R s \u2192\n \u2200 (y : M),\n y \u2208 span R s \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f x) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f y) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f (x + y)) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nintros\n[GOAL]\ncase a.H1\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\nx\u271d : M\nhx\u271d : x\u271d \u2208 span R s\ny\u271d : M\nhy\u271d : y\u271d \u2208 span R s\na\u271d\u00b2 : \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f x\u271d) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\na\u271d\u00b9 : \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f y\u271d) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\nn\u271d : N\na\u271d : n\u271d \u2208 span R t\n\u22a2 \u2191(\u2191f (x\u271d + y\u271d)) n\u271d \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nsimp only [*, add_mem, smul_mem, zero_mem, _root_.map_zero, map_add, LinearMap.zero_apply, LinearMap.add_apply,\n LinearMap.smul_apply, SMulHomClass.map_smul]\n[GOAL]\ncase a.H2\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 \u2200 (a : R) (x : M),\n x \u2208 span R s \u2192\n (\u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f x) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)) \u2192\n \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f (a \u2022 x)) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nintros\n[GOAL]\ncase a.H2\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na\u271d\u00b2 : R\nx\u271d : M\nhx\u271d : x\u271d \u2208 span R s\na\u271d\u00b9 : \u2200 (n : N), n \u2208 span R t \u2192 \u2191(\u2191f x\u271d) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\nn\u271d : N\na\u271d : n\u271d \u2208 span R t\n\u22a2 \u2191(\u2191f (a\u271d\u00b2 \u2022 x\u271d)) n\u271d \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) s t)\n[PROOFSTEP]\nsimp only [*, add_mem, smul_mem, zero_mem, _root_.map_zero, map_add, LinearMap.zero_apply, LinearMap.add_apply,\n LinearMap.smul_apply, SMulHomClass.map_smul]\n[GOAL]\ncase a\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 span R (image2 (fun m n => \u2191(\u2191f m) n) s t) \u2264 map\u2082 f (span R s) (span R t)\n[PROOFSTEP]\nrw [span_le]\n[GOAL]\ncase a\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\n\u22a2 image2 (fun m n => \u2191(\u2191f m) n) s t \u2286 \u2191(map\u2082 f (span R s) (span R t))\n[PROOFSTEP]\nrintro _ \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase a.intro.intro.intro.intro\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Set M\nt : Set N\na : M\nb : N\nha : a \u2208 s\nhb : b \u2208 t\n\u22a2 (fun m n => \u2191(\u2191f m) n) a b \u2208 \u2191(map\u2082 f (span R s) (span R t))\n[PROOFSTEP]\nexact apply_mem_map\u2082 _ (subset_span ha) (subset_span hb)\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\np : Submodule R M\nm : M\n_hm : m \u2208 p\nn : N\nhn : n \u2208 \u22a5\n\u22a2 \u2191(\u2191f m) n \u2208 \u22a5\n[PROOFSTEP]\nrw [Submodule.mem_bot] at hn \n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\np : Submodule R M\nm : M\n_hm : m \u2208 p\nn : N\nhn : n = 0\n\u22a2 \u2191(\u2191f m) n \u2208 \u22a5\n[PROOFSTEP]\nrw [hn, LinearMap.map_zero]\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\np : Submodule R M\nm : M\n_hm : m \u2208 p\nn : N\nhn : n = 0\n\u22a2 0 \u2208 \u22a5\n[PROOFSTEP]\nsimp only [mem_bot]\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\nq : Submodule R N\nm : M\nhm : m \u2208 \u22a5\nn : N\nhn : n \u2208 q\n\u22a2 \u2191(\u2191f m) n \u2208 \u22a5\n[PROOFSTEP]\nrw [Submodule.mem_bot] at hm \u22a2\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\nq : Submodule R N\nm : M\nhm : m = 0\nn : N\nhn : n \u2208 q\n\u22a2 \u2191(\u2191f m) n = 0\n[PROOFSTEP]\nrw [hm, LinearMap.map_zero\u2082]\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\np : Submodule R M\nq : Submodule R N\n\u22a2 image2 (fun m n => \u2191(\u2191f m) n) \u2191p \u2191q \u2286 \u2191(map\u2082 f p q)\n[PROOFSTEP]\nrintro _ \u27e8i, j, hi, hj, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\np : Submodule R M\nq : Submodule R N\ni : M\nj : N\nhi : i \u2208 \u2191p\nhj : j \u2208 \u2191q\n\u22a2 (fun m n => \u2191(\u2191f m) n) i j \u2208 \u2191(map\u2082 f p q)\n[PROOFSTEP]\nexact apply_mem_map\u2082 _ hi hj\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\np : Submodule R M\nq : Submodule R N\n\u22a2 map\u2082 f p q = span R (image2 (fun m n => \u2191(\u2191f m) n) \u2191p \u2191q)\n[PROOFSTEP]\nrw [\u2190 map\u2082_span_span, span_eq, span_eq]\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\np : Submodule R M\nq : Submodule R N\n\u22a2 map\u2082 (LinearMap.flip f) q p = map\u2082 f p q\n[PROOFSTEP]\nrw [map\u2082_eq_span_image2, map\u2082_eq_span_image2, Set.image2_swap]\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\np : Submodule R M\nq : Submodule R N\n\u22a2 span R (image2 (fun a b => \u2191(\u2191(LinearMap.flip f) b) a) \u2191p \u2191q) = span R (image2 (fun m n => \u2191(\u2191f m) n) \u2191p \u2191q)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : \u03b9 \u2192 Submodule R M\nt : Submodule R N\n\u22a2 map\u2082 f (\u2a06 (i : \u03b9), s i) t = \u2a06 (i : \u03b9), map\u2082 f (s i) t\n[PROOFSTEP]\nsuffices map\u2082 f (\u2a06 i, span R (s i : Set M)) (span R t) = \u2a06 i, map\u2082 f (span R (s i)) (span R t) by\n simpa only [span_eq] using this\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : \u03b9 \u2192 Submodule R M\nt : Submodule R N\nthis : map\u2082 f (\u2a06 (i : \u03b9), span R \u2191(s i)) (span R \u2191t) = \u2a06 (i : \u03b9), map\u2082 f (span R \u2191(s i)) (span R \u2191t)\n\u22a2 map\u2082 f (\u2a06 (i : \u03b9), s i) t = \u2a06 (i : \u03b9), map\u2082 f (s i) t\n[PROOFSTEP]\nsimpa only [span_eq] using this\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : \u03b9 \u2192 Submodule R M\nt : Submodule R N\n\u22a2 map\u2082 f (\u2a06 (i : \u03b9), span R \u2191(s i)) (span R \u2191t) = \u2a06 (i : \u03b9), map\u2082 f (span R \u2191(s i)) (span R \u2191t)\n[PROOFSTEP]\nsimp_rw [map\u2082_span_span, \u2190 span_iUnion, map\u2082_span_span, Set.image2_iUnion_left]\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Submodule R M\nt : \u03b9 \u2192 Submodule R N\n\u22a2 map\u2082 f s (\u2a06 (i : \u03b9), t i) = \u2a06 (i : \u03b9), map\u2082 f s (t i)\n[PROOFSTEP]\nsuffices map\u2082 f (span R s) (\u2a06 i, span R (t i : Set N)) = \u2a06 i, map\u2082 f (span R s) (span R (t i)) by\n simpa only [span_eq] using this\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Submodule R M\nt : \u03b9 \u2192 Submodule R N\nthis : map\u2082 f (span R \u2191s) (\u2a06 (i : \u03b9), span R \u2191(t i)) = \u2a06 (i : \u03b9), map\u2082 f (span R \u2191s) (span R \u2191(t i))\n\u22a2 map\u2082 f s (\u2a06 (i : \u03b9), t i) = \u2a06 (i : \u03b9), map\u2082 f s (t i)\n[PROOFSTEP]\nsimpa only [span_eq] using this\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Submodule R M\nt : \u03b9 \u2192 Submodule R N\n\u22a2 map\u2082 f (span R \u2191s) (\u2a06 (i : \u03b9), span R \u2191(t i)) = \u2a06 (i : \u03b9), map\u2082 f (span R \u2191s) (span R \u2191(t i))\n[PROOFSTEP]\nsimp_rw [map\u2082_span_span, \u2190 span_iUnion, map\u2082_span_span, Set.image2_iUnion_right]\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\nm : M\n\u22a2 map\u2082 f (span R {m}) = map (\u2191f m)\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\nm : M\nx\u271d : Submodule R N\n\u22a2 map\u2082 f (span R {m}) x\u271d = map (\u2191f m) x\u271d\n[PROOFSTEP]\nrw [map\u2082_eq_span_image2]\n[GOAL]\ncase h\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\nm : M\nx\u271d : Submodule R N\n\u22a2 span R (image2 (fun m n => \u2191(\u2191f m) n) \u2191(span R {m}) \u2191x\u271d) = map (\u2191f m) x\u271d\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase h.a\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\nm : M\nx\u271d : Submodule R N\n\u22a2 span R (image2 (fun m n => \u2191(\u2191f m) n) \u2191(span R {m}) \u2191x\u271d) \u2264 map (\u2191f m) x\u271d\n[PROOFSTEP]\nrw [span_le, Set.image2_subset_iff]\n[GOAL]\ncase h.a\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\nm : M\nx\u271d : Submodule R N\n\u22a2 \u2200 (x : M), x \u2208 \u2191(span R {m}) \u2192 \u2200 (y : N), y \u2208 \u2191x\u271d \u2192 \u2191(\u2191f x) y \u2208 \u2191(map (\u2191f m) x\u271d)\n[PROOFSTEP]\nintro x hx y hy\n[GOAL]\ncase h.a\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\nm : M\nx\u271d : Submodule R N\nx : M\nhx : x \u2208 \u2191(span R {m})\ny : N\nhy : y \u2208 \u2191x\u271d\n\u22a2 \u2191(\u2191f x) y \u2208 \u2191(map (\u2191f m) x\u271d)\n[PROOFSTEP]\nobtain \u27e8a, rfl\u27e9 := mem_span_singleton.1 hx\n[GOAL]\ncase h.a.intro\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\nm : M\nx\u271d : Submodule R N\ny : N\nhy : y \u2208 \u2191x\u271d\na : R\nhx : a \u2022 m \u2208 \u2191(span R {m})\n\u22a2 \u2191(\u2191f (a \u2022 m)) y \u2208 \u2191(map (\u2191f m) x\u271d)\n[PROOFSTEP]\nrw [f.map_smul]\n[GOAL]\ncase h.a.intro\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\nm : M\nx\u271d : Submodule R N\ny : N\nhy : y \u2208 \u2191x\u271d\na : R\nhx : a \u2022 m \u2208 \u2191(span R {m})\n\u22a2 \u2191(a \u2022 \u2191f m) y \u2208 \u2191(map (\u2191f m) x\u271d)\n[PROOFSTEP]\nexact smul_mem _ a (mem_map_of_mem hy)\n[GOAL]\ncase h.a\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\nm : M\nx\u271d : Submodule R N\n\u22a2 map (\u2191f m) x\u271d \u2264 span R (image2 (fun m n => \u2191(\u2191f m) n) \u2191(span R {m}) \u2191x\u271d)\n[PROOFSTEP]\nrintro _ \u27e8n, hn, rfl\u27e9\n[GOAL]\ncase h.a.intro.intro\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\nm : M\nx\u271d : Submodule R N\nn : N\nhn : n \u2208 \u2191x\u271d\n\u22a2 \u2191(\u2191f m) n \u2208 span R (image2 (fun m n => \u2191(\u2191f m) n) \u2191(span R {m}) \u2191x\u271d)\n[PROOFSTEP]\nexact subset_span \u27e8m, n, mem_span_singleton_self m, hn, rfl\u27e9\n[GOAL]\n\u03b9 : Sort u\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module R P\nf : M \u2192\u2097[R] N \u2192\u2097[R] P\ns : Submodule R M\nn : N\n\u22a2 map\u2082 f s (span R {n}) = map (\u2191(LinearMap.flip f) n) s\n[PROOFSTEP]\nrw [\u2190 map\u2082_span_singleton_eq_map, map\u2082_flip]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Module.Submodule.Bilinear", "llama_tokens": 18485, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124812, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.42633943965046117}} {"text": "[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\nf : LocallyConstant X Y\ninst\u271d\u00b9 : Monoid Y\ninst\u271d : ContinuousMul Y\n\u22a2 \u21911 = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\nf : LocallyConstant X Y\ninst\u271d\u00b9 : Monoid Y\ninst\u271d : ContinuousMul Y\na\u271d : X\n\u22a2 \u2191\u21911 a\u271d = \u21911 a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\nf : LocallyConstant X Y\ninst\u271d\u00b9 : Monoid Y\ninst\u271d : ContinuousMul Y\nx y : LocallyConstant X Y\n\u22a2 OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\nf : LocallyConstant X Y\ninst\u271d\u00b9 : Monoid Y\ninst\u271d : ContinuousMul Y\nx y : LocallyConstant X Y\na\u271d : X\n\u22a2 \u2191(OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y)) a\u271d =\n \u2191(OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y)\n a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : TopologicalSpace Y\nf : LocallyConstant X Y\nR : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid Y\ninst\u271d\u00b2 : Module R Y\ninst\u271d\u00b9 : ContinuousAdd Y\ninst\u271d : ContinuousConstSMul R Y\nx y : LocallyConstant X Y\n\u22a2 \u2191(x + y) = \u2191x + \u2191y\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : TopologicalSpace Y\nf : LocallyConstant X Y\nR : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid Y\ninst\u271d\u00b2 : Module R Y\ninst\u271d\u00b9 : ContinuousAdd Y\ninst\u271d : ContinuousConstSMul R Y\nx y : LocallyConstant X Y\na\u271d : X\n\u22a2 \u2191\u2191(x + y) a\u271d = \u2191(\u2191x + \u2191y) a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : TopologicalSpace Y\nf : LocallyConstant X Y\nR : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid Y\ninst\u271d\u00b2 : Module R Y\ninst\u271d\u00b9 : ContinuousAdd Y\ninst\u271d : ContinuousConstSMul R Y\nx : R\ny : LocallyConstant X Y\n\u22a2 AddHom.toFun { toFun := toContinuousMap, map_add' := (_ : \u2200 (x y : LocallyConstant X Y), \u2191(x + y) = \u2191x + \u2191y) }\n (x \u2022 y) =\n \u2191(RingHom.id R) x \u2022\n AddHom.toFun { toFun := toContinuousMap, map_add' := (_ : \u2200 (x y : LocallyConstant X Y), \u2191(x + y) = \u2191x + \u2191y) } y\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : TopologicalSpace Y\nf : LocallyConstant X Y\nR : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid Y\ninst\u271d\u00b2 : Module R Y\ninst\u271d\u00b9 : ContinuousAdd Y\ninst\u271d : ContinuousConstSMul R Y\nx : R\ny : LocallyConstant X Y\na\u271d : X\n\u22a2 \u2191(AddHom.toFun { toFun := toContinuousMap, map_add' := (_ : \u2200 (x y : LocallyConstant X Y), \u2191(x + y) = \u2191x + \u2191y) }\n (x \u2022 y))\n a\u271d =\n \u2191(\u2191(RingHom.id R) x \u2022\n AddHom.toFun { toFun := toContinuousMap, map_add' := (_ : \u2200 (x y : LocallyConstant X Y), \u2191(x + y) = \u2191x + \u2191y) }\n y)\n a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\nf : LocallyConstant X Y\nR : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring Y\ninst\u271d\u00b9 : Algebra R Y\ninst\u271d : TopologicalSemiring Y\n\u22a2 \u21911 = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\nf : LocallyConstant X Y\nR : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring Y\ninst\u271d\u00b9 : Algebra R Y\ninst\u271d : TopologicalSemiring Y\na\u271d : X\n\u22a2 \u2191\u21911 a\u271d = \u21911 a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\nf : LocallyConstant X Y\nR : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring Y\ninst\u271d\u00b9 : Algebra R Y\ninst\u271d : TopologicalSemiring Y\nx y : LocallyConstant X Y\n\u22a2 OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\nf : LocallyConstant X Y\nR : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring Y\ninst\u271d\u00b9 : Algebra R Y\ninst\u271d : TopologicalSemiring Y\nx y : LocallyConstant X Y\na\u271d : X\n\u22a2 \u2191(OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y)) a\u271d =\n \u2191(OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y)\n a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\nf : LocallyConstant X Y\nR : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring Y\ninst\u271d\u00b9 : Algebra R Y\ninst\u271d : TopologicalSemiring Y\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n 0 =\n 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\nf : LocallyConstant X Y\nR : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring Y\ninst\u271d\u00b9 : Algebra R Y\ninst\u271d : TopologicalSemiring Y\na\u271d : X\n\u22a2 \u2191(OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n 0)\n a\u271d =\n \u21910 a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\nf : LocallyConstant X Y\nR : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring Y\ninst\u271d\u00b9 : Algebra R Y\ninst\u271d : TopologicalSemiring Y\nx y : LocallyConstant X Y\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n (x + y) =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n x +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n y\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\nf : LocallyConstant X Y\nR : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring Y\ninst\u271d\u00b9 : Algebra R Y\ninst\u271d : TopologicalSemiring Y\nx y : LocallyConstant X Y\na\u271d : X\n\u22a2 \u2191(OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n (x + y))\n a\u271d =\n \u2191(OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n x +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n y)\n a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\nf : LocallyConstant X Y\nR : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring Y\ninst\u271d\u00b9 : Algebra R Y\ninst\u271d : TopologicalSemiring Y\nr : R\n\u22a2 OneHom.toFun\n (\u2191\u2191{\n toMonoidHom :=\n { toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n (x + y) =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n x +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n y) })\n (\u2191(algebraMap R (LocallyConstant X Y)) r) =\n \u2191(algebraMap R C(X, Y)) r\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\nf : LocallyConstant X Y\nR : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring Y\ninst\u271d\u00b9 : Algebra R Y\ninst\u271d : TopologicalSemiring Y\nr : R\nx : X\n\u22a2 \u2191(OneHom.toFun\n (\u2191\u2191{\n toMonoidHom :=\n { toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n 0 =\n 0),\n map_add' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n (x + y) =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n x +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : LocallyConstant X Y),\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } (x * y) =\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } x *\n OneHom.toFun { toFun := toContinuousMap, map_one' := (_ : \u21911 = 1) } y) })\n y) })\n (\u2191(algebraMap R (LocallyConstant X Y)) r))\n x =\n \u2191(\u2191(algebraMap R C(X, Y)) r) x\n[PROOFSTEP]\nsimp [Algebra.smul_def]\n", "meta": {"mathlib_filename": "Mathlib.Topology.ContinuousFunction.LocallyConstant", "llama_tokens": 6510, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267117812622843, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.4262690589816831}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b9 : HasTerminal C\ninst\u271d : PreservesLimit (Functor.empty C) G\nF : Discrete PEmpty \u2964 D\n\u22a2 HasLimit F\n[PROOFSTEP]\nhaveI := HasLimit.mk \u27e8_, isLimitOfHasTerminalOfPreservesLimit G\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b9 : HasTerminal C\ninst\u271d : PreservesLimit (Functor.empty C) G\nF : Discrete PEmpty \u2964 D\nthis : HasLimit (Functor.empty D)\n\u22a2 HasLimit F\n[PROOFSTEP]\napply hasLimitOfIso F.uniqueFromEmpty.symm\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b9 : HasTerminal C\ninst\u271d : HasTerminal D\ni : IsIso (terminalComparison G)\n\u22a2 PreservesLimit (Functor.empty C) G\n[PROOFSTEP]\napply preservesLimitOfPreservesLimitCone terminalIsTerminal\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b9 : HasTerminal C\ninst\u271d : HasTerminal D\ni : IsIso (terminalComparison G)\n\u22a2 IsLimit (G.mapCone (asEmptyCone (\u22a4_ C)))\n[PROOFSTEP]\napply (isLimitMapConeEmptyConeEquiv _ _).symm _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b9 : HasTerminal C\ninst\u271d : HasTerminal D\ni : IsIso (terminalComparison G)\n\u22a2 IsTerminal (G.obj (\u22a4_ C))\n[PROOFSTEP]\nexact @IsLimit.ofPointIso _ _ _ _ _ _ _ (limit.isLimit (Functor.empty.{0} D)) i\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b9 : HasTerminal C\ninst\u271d : HasTerminal D\nf : G.obj (\u22a4_ C) \u27f6 \u22a4_ D\ni : IsIso f\n\u22a2 PreservesLimit (Functor.empty C) G\n[PROOFSTEP]\nrw [Subsingleton.elim f (terminalComparison G)] at i \n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b9 : HasTerminal C\ninst\u271d : HasTerminal D\nf : G.obj (\u22a4_ C) \u27f6 \u22a4_ D\ni : IsIso (terminalComparison G)\n\u22a2 PreservesLimit (Functor.empty C) G\n[PROOFSTEP]\nexact PreservesTerminal.ofIsoComparison G\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b2 : HasTerminal C\ninst\u271d\u00b9 : HasTerminal D\ninst\u271d : PreservesLimit (Functor.empty C) G\n\u22a2 IsIso (terminalComparison G)\n[PROOFSTEP]\nrw [\u2190 PreservesTerminal.iso_hom]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b2 : HasTerminal C\ninst\u271d\u00b9 : HasTerminal D\ninst\u271d : PreservesLimit (Functor.empty C) G\n\u22a2 IsIso (PreservesTerminal.iso G).hom\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b9 : HasInitial C\ninst\u271d : PreservesColimit (Functor.empty C) G\nF : Discrete PEmpty \u2964 D\n\u22a2 HasColimit F\n[PROOFSTEP]\nhaveI := HasColimit.mk \u27e8_, isColimitOfHasInitialOfPreservesColimit G\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b9 : HasInitial C\ninst\u271d : PreservesColimit (Functor.empty C) G\nF : Discrete PEmpty \u2964 D\nthis : HasColimit (Functor.empty D)\n\u22a2 HasColimit F\n[PROOFSTEP]\napply hasColimitOfIso F.uniqueFromEmpty\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b9 : HasInitial C\ninst\u271d : HasInitial D\ni : IsIso (initialComparison G)\n\u22a2 PreservesColimit (Functor.empty C) G\n[PROOFSTEP]\napply preservesColimitOfPreservesColimitCocone initialIsInitial\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b9 : HasInitial C\ninst\u271d : HasInitial D\ni : IsIso (initialComparison G)\n\u22a2 IsColimit (G.mapCocone (asEmptyCocone (\u22a5_ C)))\n[PROOFSTEP]\napply (isColimitMapCoconeEmptyCoconeEquiv _ _).symm _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b9 : HasInitial C\ninst\u271d : HasInitial D\ni : IsIso (initialComparison G)\n\u22a2 IsInitial (G.obj (\u22a5_ C))\n[PROOFSTEP]\nexact @IsColimit.ofPointIso _ _ _ _ _ _ _ (colimit.isColimit (Functor.empty.{0} D)) i\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b9 : HasInitial C\ninst\u271d : HasInitial D\nf : \u22a5_ D \u27f6 G.obj (\u22a5_ C)\ni : IsIso f\n\u22a2 PreservesColimit (Functor.empty C) G\n[PROOFSTEP]\nrw [Subsingleton.elim f (initialComparison G)] at i \n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b9 : HasInitial C\ninst\u271d : HasInitial D\nf : \u22a5_ D \u27f6 G.obj (\u22a5_ C)\ni : IsIso (initialComparison G)\n\u22a2 PreservesColimit (Functor.empty C) G\n[PROOFSTEP]\nexact PreservesInitial.ofIsoComparison G\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b2 : HasInitial C\ninst\u271d\u00b9 : HasInitial D\ninst\u271d : PreservesColimit (Functor.empty C) G\n\u22a2 IsIso (initialComparison G)\n[PROOFSTEP]\nrw [\u2190 PreservesInitial.iso_hom]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX : C\ninst\u271d\u00b2 : HasInitial C\ninst\u271d\u00b9 : HasInitial D\ninst\u271d : PreservesColimit (Functor.empty C) G\n\u22a2 IsIso (PreservesInitial.iso G).inv\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal", "llama_tokens": 2503, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.4260356720467403}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 OrdConnected (range Real.toEReal)\n[PROOFSTEP]\nrw [range_coe_eq_Ioo]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 OrdConnected (Ioo \u22a5 \u22a4)\n[PROOFSTEP]\nexact ordConnected_Ioo\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 IsOpen (range Real.toEReal)\n[PROOFSTEP]\nsimp only [range_coe_eq_Ioo, isOpen_Ioo]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na : EReal\nha : a \u2260 \u22a4\nh'a : a \u2260 \u22a5\n\u22a2 Tendsto toReal (\ud835\udcdd a) (\ud835\udcdd (toReal a))\n[PROOFSTEP]\nlift a to \u211d using \u27e8ha, h'a\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na : \u211d\nha : \u2191a \u2260 \u22a4\nh'a : \u2191a \u2260 \u22a5\n\u22a2 Tendsto toReal (\ud835\udcdd \u2191a) (\ud835\udcdd (toReal \u2191a))\n[PROOFSTEP]\nrw [nhds_coe, tendsto_map'_iff]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na : \u211d\nha : \u2191a \u2260 \u22a4\nh'a : \u2191a \u2260 \u22a5\n\u22a2 Tendsto (toReal \u2218 Real.toEReal) (\ud835\udcdd a) (\ud835\udcdd (toReal \u2191a))\n[PROOFSTEP]\nexact tendsto_id\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 OrdConnected (range ENNReal.toEReal)\n[PROOFSTEP]\nrw [range_coe_ennreal]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 OrdConnected (Ici 0)\n[PROOFSTEP]\nexact ordConnected_Ici\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 IsClosed (range ENNReal.toEReal)\n[PROOFSTEP]\nrw [range_coe_ennreal]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 IsClosed (Ici 0)\n[PROOFSTEP]\nexact isClosed_Ici\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 \u2a05 (l : EReal) (_ : l < \u22a4), \ud835\udcdf (Ioi l) = \u2a05 (a : EReal) (_ : a \u2260 \u22a4), \ud835\udcdf (Ioi a)\n[PROOFSTEP]\nsimp only [lt_top_iff_ne_top]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 HasBasis (\ud835\udcdd \u22a4) (fun x => True) fun x => Ioi \u2191x\n[PROOFSTEP]\nrefine nhds_top_basis.to_hasBasis (fun x hx => ?_) fun _ _ \u21a6 \u27e8_, coe_lt_top _, Subset.rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nx : EReal\nhx : x < \u22a4\n\u22a2 \u2203 i', True \u2227 Ioi \u2191i' \u2286 Ioi x\n[PROOFSTEP]\nrcases exists_rat_btwn_of_lt hx with \u27e8y, hxy, -\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nx : EReal\nhx : x < \u22a4\ny : \u211a\nhxy : x < \u2191\u2191y\n\u22a2 \u2203 i', True \u2227 Ioi \u2191i' \u2286 Ioi x\n[PROOFSTEP]\nexact \u27e8_, trivial, Ioi_subset_Ioi hxy.le\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\ns : Set EReal\n\u22a2 (\u2203 i, True \u2227 Ioi \u2191i \u2286 s) \u2194 \u2203 y, Ioi \u2191y \u2286 s\n[PROOFSTEP]\nsimp only [true_and]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_2\nm : \u03b1 \u2192 EReal\nf : Filter \u03b1\n\u22a2 (\u2200 (i : \u211d), True \u2192 \u2200\u1da0 (x : \u03b1) in f, m x \u2208 Ioi \u2191i) \u2194 \u2200 (x : \u211d), \u2200\u1da0 (a : \u03b1) in f, \u2191x < m a\n[PROOFSTEP]\nsimp only [true_implies, mem_Ioi]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 \u2a05 (l : EReal) (_ : \u22a5 < l), \ud835\udcdf (Iio l) = \u2a05 (a : EReal) (_ : a \u2260 \u22a5), \ud835\udcdf (Iio a)\n[PROOFSTEP]\nsimp only [bot_lt_iff_ne_bot]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 HasBasis (\ud835\udcdd \u22a5) (fun x => True) fun x => Iio \u2191x\n[PROOFSTEP]\nrefine nhds_bot_basis.to_hasBasis (fun x hx => ?_) fun _ _ \u21a6 \u27e8_, bot_lt_coe _, Subset.rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nx : EReal\nhx : \u22a5 < x\n\u22a2 \u2203 i', True \u2227 Iio \u2191i' \u2286 Iio x\n[PROOFSTEP]\nrcases exists_rat_btwn_of_lt hx with \u27e8y, -, hxy\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nx : EReal\nhx : \u22a5 < x\ny : \u211a\nhxy : \u2191\u2191y < x\n\u22a2 \u2203 i', True \u2227 Iio \u2191i' \u2286 Iio x\n[PROOFSTEP]\nexact \u27e8_, trivial, Iio_subset_Iio hxy.le\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\ns : Set EReal\n\u22a2 (\u2203 i, True \u2227 Iio \u2191i \u2286 s) \u2194 \u2203 y, Iio \u2191y \u2286 s\n[PROOFSTEP]\nsimp only [true_and]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d : TopologicalSpace \u03b1\u271d\n\u03b1 : Type u_2\nm : \u03b1 \u2192 EReal\nf : Filter \u03b1\n\u22a2 (\u2200 (i : \u211d), True \u2192 \u2200\u1da0 (x : \u03b1) in f, m x \u2208 Iio \u2191i) \u2194 \u2200 (x : \u211d), \u2200\u1da0 (a : \u03b1) in f, m a < \u2191x\n[PROOFSTEP]\nsimp only [true_implies, mem_Iio]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na b : \u211d\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u2191a, \u2191b)\n[PROOFSTEP]\nsimp only [ContinuousAt, nhds_coe_coe, \u2190 coe_add, tendsto_map'_iff, (\u00b7 \u2218 \u00b7), tendsto_coe, tendsto_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na : \u211d\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u22a4, \u2191a)\n[PROOFSTEP]\nsimp only [ContinuousAt, tendsto_nhds_top_iff_real, top_add_coe]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na : \u211d\n\u22a2 \u2200 (x : \u211d), \u2200\u1da0 (a : EReal \u00d7 EReal) in \ud835\udcdd (\u22a4, \u2191a), \u2191x < a.fst + a.snd\n[PROOFSTEP]\nrefine fun r \u21a6\n ((lt_mem_nhds (coe_lt_top (r - (a - 1)))).prod_nhds (lt_mem_nhds <| EReal.coe_lt_coe_iff.2 <| sub_one_lt _)).mono\n fun _ h \u21a6 ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na r : \u211d\nx\u271d : EReal \u00d7 EReal\nh : \u2191(r - (a - 1)) < x\u271d.fst \u2227 \u2191(a - 1) < x\u271d.snd\n\u22a2 \u2191r < x\u271d.fst + x\u271d.snd\n[PROOFSTEP]\nsimpa only [\u2190 coe_add, sub_add_cancel] using add_lt_add h.1 h.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na : \u211d\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u2191a, \u22a4)\n[PROOFSTEP]\nsimpa only [add_comm, (\u00b7 \u2218 \u00b7), ContinuousAt, Prod.swap] using\n Tendsto.comp (continuousAt_add_top_coe a) (continuous_swap.tendsto ((a : EReal), \u22a4))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u22a4, \u22a4)\n[PROOFSTEP]\nsimp only [ContinuousAt, tendsto_nhds_top_iff_real, top_add_top]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 \u2200 (x : \u211d), \u2200\u1da0 (a : EReal \u00d7 EReal) in \ud835\udcdd (\u22a4, \u22a4), \u2191x < a.fst + a.snd\n[PROOFSTEP]\nrefine fun r \u21a6 ((lt_mem_nhds (coe_lt_top 0)).prod_nhds (lt_mem_nhds <| coe_lt_top r)).mono fun _ h \u21a6 ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nr : \u211d\nx\u271d : EReal \u00d7 EReal\nh : \u21910 < x\u271d.fst \u2227 \u2191r < x\u271d.snd\n\u22a2 \u2191r < x\u271d.fst + x\u271d.snd\n[PROOFSTEP]\nsimpa only [coe_zero, zero_add] using add_lt_add h.1 h.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na : \u211d\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u22a5, \u2191a)\n[PROOFSTEP]\nsimp only [ContinuousAt, tendsto_nhds_bot_iff_real, bot_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na : \u211d\n\u22a2 \u2200 (x : \u211d), \u2200\u1da0 (a : EReal \u00d7 EReal) in \ud835\udcdd (\u22a5, \u2191a), a.fst + a.snd < \u2191x\n[PROOFSTEP]\nrefine fun r \u21a6\n ((gt_mem_nhds (bot_lt_coe (r - (a + 1)))).prod_nhds (gt_mem_nhds <| EReal.coe_lt_coe_iff.2 <| lt_add_one _)).mono\n fun _ h \u21a6 ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na r : \u211d\nx\u271d : EReal \u00d7 EReal\nh : x\u271d.fst < \u2191(r - (a + 1)) \u2227 x\u271d.snd < \u2191(a + 1)\n\u22a2 x\u271d.fst + x\u271d.snd < \u2191r\n[PROOFSTEP]\nsimpa only [\u2190 coe_add, sub_add_cancel] using add_lt_add h.1 h.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na : \u211d\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u2191a, \u22a5)\n[PROOFSTEP]\nsimpa only [add_comm, (\u00b7 \u2218 \u00b7), ContinuousAt, Prod.swap] using\n Tendsto.comp (continuousAt_add_bot_coe a) (continuous_swap.tendsto ((a : EReal), \u22a5))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u22a5, \u22a5)\n[PROOFSTEP]\nsimp only [ContinuousAt, tendsto_nhds_bot_iff_real, bot_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 \u2200 (x : \u211d), \u2200\u1da0 (a : EReal \u00d7 EReal) in \ud835\udcdd (\u22a5, \u22a5), a.fst + a.snd < \u2191x\n[PROOFSTEP]\nrefine fun r \u21a6 ((gt_mem_nhds (bot_lt_coe 0)).prod_nhds (gt_mem_nhds <| bot_lt_coe r)).mono fun _ h \u21a6 ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nr : \u211d\nx\u271d : EReal \u00d7 EReal\nh : x\u271d.fst < \u21910 \u2227 x\u271d.snd < \u2191r\n\u22a2 x\u271d.fst + x\u271d.snd < \u2191r\n[PROOFSTEP]\nsimpa only [coe_zero, zero_add] using add_lt_add h.1 h.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\np : EReal \u00d7 EReal\nh : p.fst \u2260 \u22a4 \u2228 p.snd \u2260 \u22a5\nh' : p.fst \u2260 \u22a5 \u2228 p.snd \u2260 \u22a4\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) p\n[PROOFSTEP]\nrcases p with \u27e8x, y\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nx y : EReal\nh : (x, y).fst \u2260 \u22a4 \u2228 (x, y).snd \u2260 \u22a5\nh' : (x, y).fst \u2260 \u22a5 \u2228 (x, y).snd \u2260 \u22a4\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (x, y)\n[PROOFSTEP]\ninduction x using EReal.rec\n[GOAL]\ncase mk.h_bot\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\ny : EReal\nh : (\u22a5, y).fst \u2260 \u22a4 \u2228 (\u22a5, y).snd \u2260 \u22a5\nh' : (\u22a5, y).fst \u2260 \u22a5 \u2228 (\u22a5, y).snd \u2260 \u22a4\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u22a5, y)\n[PROOFSTEP]\ninduction y using EReal.rec\n[GOAL]\ncase mk.h_real\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\ny : EReal\na\u271d : \u211d\nh : (\u2191a\u271d, y).fst \u2260 \u22a4 \u2228 (\u2191a\u271d, y).snd \u2260 \u22a5\nh' : (\u2191a\u271d, y).fst \u2260 \u22a5 \u2228 (\u2191a\u271d, y).snd \u2260 \u22a4\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u2191a\u271d, y)\n[PROOFSTEP]\ninduction y using EReal.rec\n[GOAL]\ncase mk.h_top\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\ny : EReal\nh : (\u22a4, y).fst \u2260 \u22a4 \u2228 (\u22a4, y).snd \u2260 \u22a5\nh' : (\u22a4, y).fst \u2260 \u22a5 \u2228 (\u22a4, y).snd \u2260 \u22a4\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u22a4, y)\n[PROOFSTEP]\ninduction y using EReal.rec\n[GOAL]\ncase mk.h_bot.h_bot\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nh : (\u22a5, \u22a5).fst \u2260 \u22a4 \u2228 (\u22a5, \u22a5).snd \u2260 \u22a5\nh' : (\u22a5, \u22a5).fst \u2260 \u22a5 \u2228 (\u22a5, \u22a5).snd \u2260 \u22a4\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u22a5, \u22a5)\n[PROOFSTEP]\nexact continuousAt_add_bot_bot\n[GOAL]\ncase mk.h_bot.h_real\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na\u271d : \u211d\nh : (\u22a5, \u2191a\u271d).fst \u2260 \u22a4 \u2228 (\u22a5, \u2191a\u271d).snd \u2260 \u22a5\nh' : (\u22a5, \u2191a\u271d).fst \u2260 \u22a5 \u2228 (\u22a5, \u2191a\u271d).snd \u2260 \u22a4\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u22a5, \u2191a\u271d)\n[PROOFSTEP]\nexact continuousAt_add_bot_coe _\n[GOAL]\ncase mk.h_bot.h_top\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nh : (\u22a5, \u22a4).fst \u2260 \u22a4 \u2228 (\u22a5, \u22a4).snd \u2260 \u22a5\nh' : (\u22a5, \u22a4).fst \u2260 \u22a5 \u2228 (\u22a5, \u22a4).snd \u2260 \u22a4\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u22a5, \u22a4)\n[PROOFSTEP]\nsimp at h' \n[GOAL]\ncase mk.h_real.h_bot\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na\u271d : \u211d\nh : (\u2191a\u271d, \u22a5).fst \u2260 \u22a4 \u2228 (\u2191a\u271d, \u22a5).snd \u2260 \u22a5\nh' : (\u2191a\u271d, \u22a5).fst \u2260 \u22a5 \u2228 (\u2191a\u271d, \u22a5).snd \u2260 \u22a4\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u2191a\u271d, \u22a5)\n[PROOFSTEP]\nexact continuousAt_add_coe_bot _\n[GOAL]\ncase mk.h_real.h_real\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na\u271d\u00b9 a\u271d : \u211d\nh : (\u2191a\u271d\u00b9, \u2191a\u271d).fst \u2260 \u22a4 \u2228 (\u2191a\u271d\u00b9, \u2191a\u271d).snd \u2260 \u22a5\nh' : (\u2191a\u271d\u00b9, \u2191a\u271d).fst \u2260 \u22a5 \u2228 (\u2191a\u271d\u00b9, \u2191a\u271d).snd \u2260 \u22a4\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u2191a\u271d\u00b9, \u2191a\u271d)\n[PROOFSTEP]\nexact continuousAt_add_coe_coe _ _\n[GOAL]\ncase mk.h_real.h_top\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na\u271d : \u211d\nh : (\u2191a\u271d, \u22a4).fst \u2260 \u22a4 \u2228 (\u2191a\u271d, \u22a4).snd \u2260 \u22a5\nh' : (\u2191a\u271d, \u22a4).fst \u2260 \u22a5 \u2228 (\u2191a\u271d, \u22a4).snd \u2260 \u22a4\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u2191a\u271d, \u22a4)\n[PROOFSTEP]\nexact continuousAt_add_coe_top _\n[GOAL]\ncase mk.h_top.h_bot\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nh : (\u22a4, \u22a5).fst \u2260 \u22a4 \u2228 (\u22a4, \u22a5).snd \u2260 \u22a5\nh' : (\u22a4, \u22a5).fst \u2260 \u22a5 \u2228 (\u22a4, \u22a5).snd \u2260 \u22a4\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u22a4, \u22a5)\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase mk.h_top.h_real\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\na\u271d : \u211d\nh : (\u22a4, \u2191a\u271d).fst \u2260 \u22a4 \u2228 (\u22a4, \u2191a\u271d).snd \u2260 \u22a5\nh' : (\u22a4, \u2191a\u271d).fst \u2260 \u22a5 \u2228 (\u22a4, \u2191a\u271d).snd \u2260 \u22a4\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u22a4, \u2191a\u271d)\n[PROOFSTEP]\nexact continuousAt_add_top_coe _\n[GOAL]\ncase mk.h_top.h_top\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nh : (\u22a4, \u22a4).fst \u2260 \u22a4 \u2228 (\u22a4, \u22a4).snd \u2260 \u22a5\nh' : (\u22a4, \u22a4).fst \u2260 \u22a5 \u2228 (\u22a4, \u22a4).snd \u2260 \u22a4\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (\u22a4, \u22a4)\n[PROOFSTEP]\nexact continuousAt_add_top_top\n", "meta": {"mathlib_filename": "Mathlib.Topology.Instances.EReal", "llama_tokens": 5728, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185944046238981, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.42588690253446476}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CircularOrder \u03b1\na b c : \u03b1\n\u22a2 sbtw a b c \u2194 \u00acbtw c b a\n[PROOFSTEP]\nrw [sbtw_iff_btw_not_btw]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CircularOrder \u03b1\na b c : \u03b1\n\u22a2 btw a b c \u2227 \u00acbtw c b a \u2194 \u00acbtw c b a\n[PROOFSTEP]\nexact and_iff_right_of_imp (btw_total _ _ _).resolve_left\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CircularOrder \u03b1\na b : \u03b1\n\u22a2 (cIcc a b)\u1d9c = cIoo b a\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : CircularOrder \u03b1\na b x\u271d : \u03b1\n\u22a2 x\u271d \u2208 (cIcc a b)\u1d9c \u2194 x\u271d \u2208 cIoo b a\n[PROOFSTEP]\nrw [Set.mem_cIoo, sbtw_iff_not_btw, cIcc, mem_compl_iff, mem_setOf]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CircularOrder \u03b1\na b : \u03b1\n\u22a2 (cIoo a b)\u1d9c = cIcc b a\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : CircularOrder \u03b1\na b x\u271d : \u03b1\n\u22a2 x\u271d \u2208 (cIoo a b)\u1d9c \u2194 x\u271d \u2208 cIcc b a\n[PROOFSTEP]\nrw [Set.mem_cIcc, btw_iff_not_sbtw, cIoo, mem_compl_iff, mem_setOf]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\nh : btw a b c\n\u22a2 btw b c a\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\nh : btw a b c\n\u22a2 b \u2264 c \u2227 c \u2264 a \u2228 c \u2264 a \u2227 a \u2264 b \u2228 a \u2264 b \u2227 b \u2264 c\n[PROOFSTEP]\nrwa [\u2190 or_assoc, or_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\n\u22a2 sbtw a b c \u2194 btw a b c \u2227 \u00acbtw c b a\n[PROOFSTEP]\nsimp_rw [lt_iff_le_not_le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\n\u22a2 (a \u2264 b \u2227 \u00acb \u2264 a) \u2227 b \u2264 c \u2227 \u00acc \u2264 b \u2228 (b \u2264 c \u2227 \u00acc \u2264 b) \u2227 c \u2264 a \u2227 \u00aca \u2264 c \u2228 (c \u2264 a \u2227 \u00aca \u2264 c) \u2227 a \u2264 b \u2227 \u00acb \u2264 a \u2194\n (a \u2264 b \u2227 b \u2264 c \u2228 b \u2264 c \u2227 c \u2264 a \u2228 c \u2264 a \u2227 a \u2264 b) \u2227 \u00ac(c \u2264 b \u2227 b \u2264 a \u2228 b \u2264 a \u2227 a \u2264 c \u2228 a \u2264 c \u2227 c \u2264 b)\n[PROOFSTEP]\nhave h1 := le_trans a b c\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\nh1 : a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n\u22a2 (a \u2264 b \u2227 \u00acb \u2264 a) \u2227 b \u2264 c \u2227 \u00acc \u2264 b \u2228 (b \u2264 c \u2227 \u00acc \u2264 b) \u2227 c \u2264 a \u2227 \u00aca \u2264 c \u2228 (c \u2264 a \u2227 \u00aca \u2264 c) \u2227 a \u2264 b \u2227 \u00acb \u2264 a \u2194\n (a \u2264 b \u2227 b \u2264 c \u2228 b \u2264 c \u2227 c \u2264 a \u2228 c \u2264 a \u2227 a \u2264 b) \u2227 \u00ac(c \u2264 b \u2227 b \u2264 a \u2228 b \u2264 a \u2227 a \u2264 c \u2228 a \u2264 c \u2227 c \u2264 b)\n[PROOFSTEP]\nhave h2 := le_trans b c a\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\nh1 : a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nh2 : b \u2264 c \u2192 c \u2264 a \u2192 b \u2264 a\n\u22a2 (a \u2264 b \u2227 \u00acb \u2264 a) \u2227 b \u2264 c \u2227 \u00acc \u2264 b \u2228 (b \u2264 c \u2227 \u00acc \u2264 b) \u2227 c \u2264 a \u2227 \u00aca \u2264 c \u2228 (c \u2264 a \u2227 \u00aca \u2264 c) \u2227 a \u2264 b \u2227 \u00acb \u2264 a \u2194\n (a \u2264 b \u2227 b \u2264 c \u2228 b \u2264 c \u2227 c \u2264 a \u2228 c \u2264 a \u2227 a \u2264 b) \u2227 \u00ac(c \u2264 b \u2227 b \u2264 a \u2228 b \u2264 a \u2227 a \u2264 c \u2228 a \u2264 c \u2227 c \u2264 b)\n[PROOFSTEP]\nhave h3 := le_trans c a b\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\nh1 : a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nh2 : b \u2264 c \u2192 c \u2264 a \u2192 b \u2264 a\nh3 : c \u2264 a \u2192 a \u2264 b \u2192 c \u2264 b\n\u22a2 (a \u2264 b \u2227 \u00acb \u2264 a) \u2227 b \u2264 c \u2227 \u00acc \u2264 b \u2228 (b \u2264 c \u2227 \u00acc \u2264 b) \u2227 c \u2264 a \u2227 \u00aca \u2264 c \u2228 (c \u2264 a \u2227 \u00aca \u2264 c) \u2227 a \u2264 b \u2227 \u00acb \u2264 a \u2194\n (a \u2264 b \u2227 b \u2264 c \u2228 b \u2264 c \u2227 c \u2264 a \u2228 c \u2264 a \u2227 a \u2264 b) \u2227 \u00ac(c \u2264 b \u2227 b \u2264 a \u2228 b \u2264 a \u2227 a \u2264 c \u2228 a \u2264 c \u2227 c \u2264 b)\n[PROOFSTEP]\nrevert h1 h2 h3\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\n\u22a2 (a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c) \u2192\n (b \u2264 c \u2192 c \u2264 a \u2192 b \u2264 a) \u2192\n (c \u2264 a \u2192 a \u2264 b \u2192 c \u2264 b) \u2192\n ((a \u2264 b \u2227 \u00acb \u2264 a) \u2227 b \u2264 c \u2227 \u00acc \u2264 b \u2228 (b \u2264 c \u2227 \u00acc \u2264 b) \u2227 c \u2264 a \u2227 \u00aca \u2264 c \u2228 (c \u2264 a \u2227 \u00aca \u2264 c) \u2227 a \u2264 b \u2227 \u00acb \u2264 a \u2194\n (a \u2264 b \u2227 b \u2264 c \u2228 b \u2264 c \u2227 c \u2264 a \u2228 c \u2264 a \u2227 a \u2264 b) \u2227 \u00ac(c \u2264 b \u2227 b \u2264 a \u2228 b \u2264 a \u2227 a \u2264 c \u2228 a \u2264 c \u2227 c \u2264 b))\n[PROOFSTEP]\ngeneralize (a \u2264 b) = p1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 : Prop\n\u22a2 (p1 \u2192 b \u2264 c \u2192 a \u2264 c) \u2192\n (b \u2264 c \u2192 c \u2264 a \u2192 b \u2264 a) \u2192\n (c \u2264 a \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acb \u2264 a) \u2227 b \u2264 c \u2227 \u00acc \u2264 b \u2228 (b \u2264 c \u2227 \u00acc \u2264 b) \u2227 c \u2264 a \u2227 \u00aca \u2264 c \u2228 (c \u2264 a \u2227 \u00aca \u2264 c) \u2227 p1 \u2227 \u00acb \u2264 a \u2194\n (p1 \u2227 b \u2264 c \u2228 b \u2264 c \u2227 c \u2264 a \u2228 c \u2264 a \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 b \u2264 a \u2228 b \u2264 a \u2227 a \u2264 c \u2228 a \u2264 c \u2227 c \u2264 b))\n[PROOFSTEP]\ngeneralize (b \u2264 a) = p2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 : Prop\n\u22a2 (p1 \u2192 b \u2264 c \u2192 a \u2264 c) \u2192\n (b \u2264 c \u2192 c \u2264 a \u2192 p2) \u2192\n (c \u2264 a \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 b \u2264 c \u2227 \u00acc \u2264 b \u2228 (b \u2264 c \u2227 \u00acc \u2264 b) \u2227 c \u2264 a \u2227 \u00aca \u2264 c \u2228 (c \u2264 a \u2227 \u00aca \u2264 c) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 b \u2264 c \u2228 b \u2264 c \u2227 c \u2264 a \u2228 c \u2264 a \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 a \u2264 c \u2228 a \u2264 c \u2227 c \u2264 b))\n[PROOFSTEP]\ngeneralize (a \u2264 c) = p3\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 : Prop\n\u22a2 (p1 \u2192 b \u2264 c \u2192 p3) \u2192\n (b \u2264 c \u2192 c \u2264 a \u2192 p2) \u2192\n (c \u2264 a \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 b \u2264 c \u2227 \u00acc \u2264 b \u2228 (b \u2264 c \u2227 \u00acc \u2264 b) \u2227 c \u2264 a \u2227 \u00acp3 \u2228 (c \u2264 a \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 b \u2264 c \u2228 b \u2264 c \u2227 c \u2264 a \u2228 c \u2264 a \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\ngeneralize (c \u2264 a) = p4\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 : Prop\n\u22a2 (p1 \u2192 b \u2264 c \u2192 p3) \u2192\n (b \u2264 c \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 b \u2264 c \u2227 \u00acc \u2264 b \u2228 (b \u2264 c \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 b \u2264 c \u2228 b \u2264 c \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\ngeneralize (b \u2264 c) = p5\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh : p1\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p2\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh : p1\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p2\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh : \u00acp1\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p2\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh : \u00acp1\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p2\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d : p1\nh : p2\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p3\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d : p1\nh : p2\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p3\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d : p1\nh : \u00acp2\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p3\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d : p1\nh : \u00acp2\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p3\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d : \u00acp1\nh : p2\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p3\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d : \u00acp1\nh : p2\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p3\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d : \u00acp1\nh : \u00acp2\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p3\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d : \u00acp1\nh : \u00acp2\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p3\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : p1\nh\u271d : p2\nh : p3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : p1\nh\u271d : p2\nh : p3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : p1\nh\u271d : p2\nh : \u00acp3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : p1\nh\u271d : p2\nh : \u00acp3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : p1\nh\u271d : \u00acp2\nh : p3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : p1\nh\u271d : \u00acp2\nh : p3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : p1\nh\u271d : \u00acp2\nh : \u00acp3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : p1\nh\u271d : \u00acp2\nh : \u00acp3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : \u00acp1\nh\u271d : p2\nh : p3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : \u00acp1\nh\u271d : p2\nh : p3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : \u00acp1\nh\u271d : p2\nh : \u00acp3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : \u00acp1\nh\u271d : p2\nh : \u00acp3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : \u00acp1\nh\u271d : \u00acp2\nh : p3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : \u00acp1\nh\u271d : \u00acp2\nh : p3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : \u00acp1\nh\u271d : \u00acp2\nh : \u00acp3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b9 : \u00acp1\nh\u271d : \u00acp2\nh : \u00acp3\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p4\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : p2\nh\u271d : p3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : p2\nh\u271d : p3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : p2\nh\u271d : p3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : p2\nh\u271d : p3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : p2\nh\u271d : \u00acp3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : p2\nh\u271d : \u00acp3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : p2\nh\u271d : \u00acp3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : p2\nh\u271d : \u00acp3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : \u00acp2\nh\u271d : p3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : \u00acp2\nh\u271d : p3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : \u00acp2\nh\u271d : p3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : \u00acp2\nh\u271d : p3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : \u00acp2\nh\u271d : \u00acp3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : \u00acp2\nh\u271d : \u00acp3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : \u00acp2\nh\u271d : \u00acp3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : p1\nh\u271d\u00b9 : \u00acp2\nh\u271d : \u00acp3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : p2\nh\u271d : p3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : p2\nh\u271d : p3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : p2\nh\u271d : p3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : p2\nh\u271d : p3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : p2\nh\u271d : \u00acp3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : p2\nh\u271d : \u00acp3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : p2\nh\u271d : \u00acp3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : p2\nh\u271d : \u00acp3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : \u00acp2\nh\u271d : p3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : \u00acp2\nh\u271d : p3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : \u00acp2\nh\u271d : p3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : \u00acp2\nh\u271d : p3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : \u00acp2\nh\u271d : \u00acp3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : \u00acp2\nh\u271d : \u00acp3\nh : p4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : \u00acp2\nh\u271d : \u00acp3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b2 : \u00acp1\nh\u271d\u00b9 : \u00acp2\nh\u271d : \u00acp3\nh : \u00acp4\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nby_cases p5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : p3\nh\u271d : p4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : p3\nh\u271d : p4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : p3\nh\u271d : \u00acp4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : p3\nh\u271d : \u00acp4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : \u00acp3\nh\u271d : p4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : \u00acp3\nh\u271d : p4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : \u00acp3\nh\u271d : \u00acp4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : \u00acp3\nh\u271d : \u00acp4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : p3\nh\u271d : p4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : p3\nh\u271d : p4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : p3\nh\u271d : \u00acp4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : p3\nh\u271d : \u00acp4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : \u00acp3\nh\u271d : p4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : \u00acp3\nh\u271d : p4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : \u00acp3\nh\u271d : \u00acp4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : p1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : \u00acp3\nh\u271d : \u00acp4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : p3\nh\u271d : p4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : p3\nh\u271d : p4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : p3\nh\u271d : \u00acp4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : p3\nh\u271d : \u00acp4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : \u00acp3\nh\u271d : p4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : \u00acp3\nh\u271d : p4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : \u00acp3\nh\u271d : \u00acp4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : p2\nh\u271d\u00b9 : \u00acp3\nh\u271d : \u00acp4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : p3\nh\u271d : p4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : p3\nh\u271d : p4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : p3\nh\u271d : \u00acp4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : p3\nh\u271d : \u00acp4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : \u00acp3\nh\u271d : p4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : \u00acp3\nh\u271d : p4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : \u00acp3\nh\u271d : \u00acp4\nh : p5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c : \u03b1\np1 p2 p3 p4 p5 : Prop\nh\u271d\u00b3 : \u00acp1\nh\u271d\u00b2 : \u00acp2\nh\u271d\u00b9 : \u00acp3\nh\u271d : \u00acp4\nh : \u00acp5\n\u22a2 (p1 \u2192 p5 \u2192 p3) \u2192\n (p5 \u2192 p4 \u2192 p2) \u2192\n (p4 \u2192 p1 \u2192 c \u2264 b) \u2192\n ((p1 \u2227 \u00acp2) \u2227 p5 \u2227 \u00acc \u2264 b \u2228 (p5 \u2227 \u00acc \u2264 b) \u2227 p4 \u2227 \u00acp3 \u2228 (p4 \u2227 \u00acp3) \u2227 p1 \u2227 \u00acp2 \u2194\n (p1 \u2227 p5 \u2228 p5 \u2227 p4 \u2228 p4 \u2227 p1) \u2227 \u00ac(c \u2264 b \u2227 p2 \u2228 p2 \u2227 p3 \u2228 p3 \u2227 c \u2264 b))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c d : \u03b1\n\u22a2 sbtw a b c \u2192 sbtw b d c \u2192 sbtw a d c\n[PROOFSTEP]\nrintro (\u27e8hab, hbc\u27e9 | \u27e8hbc, hca\u27e9 | \u27e8hca, hab\u27e9) (\u27e8hbd, hdc\u27e9 | \u27e8hdc, hcb\u27e9 | \u27e8hcb, hbd\u27e9)\n[GOAL]\ncase inl.intro.inl.intro\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c d : \u03b1\nhab : a < b\nhbc : b < c\nhbd : b < d\nhdc : d < c\n\u22a2 sbtw a d c\n[PROOFSTEP]\nexact Or.inl \u27e8hab.trans hbd, hdc\u27e9\n[GOAL]\ncase inl.intro.inr.inl.intro\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c d : \u03b1\nhab : a < b\nhbc : b < c\nhdc : d < c\nhcb : c < b\n\u22a2 sbtw a d c\n[PROOFSTEP]\nexact (hbc.not_lt hcb).elim\n[GOAL]\ncase inl.intro.inr.inr.intro\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c d : \u03b1\nhab : a < b\nhbc : b < c\nhcb : c < b\nhbd : b < d\n\u22a2 sbtw a d c\n[PROOFSTEP]\nexact (hbc.not_lt hcb).elim\n[GOAL]\ncase inr.inl.intro.inl.intro\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c d : \u03b1\nhbc : b < c\nhca : c < a\nhbd : b < d\nhdc : d < c\n\u22a2 sbtw a d c\n[PROOFSTEP]\nexact Or.inr (Or.inl \u27e8hdc, hca\u27e9)\n[GOAL]\ncase inr.inl.intro.inr.inl.intro\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c d : \u03b1\nhbc : b < c\nhca : c < a\nhdc : d < c\nhcb : c < b\n\u22a2 sbtw a d c\n[PROOFSTEP]\nexact Or.inr (Or.inl \u27e8hdc, hca\u27e9)\n[GOAL]\ncase inr.inl.intro.inr.inr.intro\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c d : \u03b1\nhbc : b < c\nhca : c < a\nhcb : c < b\nhbd : b < d\n\u22a2 sbtw a d c\n[PROOFSTEP]\nexact (hbc.not_lt hcb).elim\n[GOAL]\ncase inr.inr.intro.inl.intro\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c d : \u03b1\nhca : c < a\nhab : a < b\nhbd : b < d\nhdc : d < c\n\u22a2 sbtw a d c\n[PROOFSTEP]\nexact Or.inr (Or.inl \u27e8hdc, hca\u27e9)\n[GOAL]\ncase inr.inr.intro.inr.inl.intro\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c d : \u03b1\nhca : c < a\nhab : a < b\nhdc : d < c\nhcb : c < b\n\u22a2 sbtw a d c\n[PROOFSTEP]\nexact Or.inr (Or.inl \u27e8hdc, hca\u27e9)\n[GOAL]\ncase inr.inr.intro.inr.inr.intro\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\na b c d : \u03b1\nhca : c < a\nhab : a < b\nhcb : c < b\nhbd : b < d\n\u22a2 sbtw a d c\n[PROOFSTEP]\nexact Or.inr (Or.inr \u27e8hca, hab.trans hbd\u27e9)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nsrc\u271d : CircularPreorder \u03b1 := Preorder.toCircularPreorder \u03b1\na b c : \u03b1\n\u22a2 btw a b c \u2192 btw c b a \u2192 a = b \u2228 b = c \u2228 c = a\n[PROOFSTEP]\nrintro (\u27e8hab, hbc\u27e9 | \u27e8hbc, hca\u27e9 | \u27e8hca, hab\u27e9) (\u27e8hcb, hba\u27e9 | \u27e8hba, hac\u27e9 | \u27e8hac, hcb\u27e9)\n[GOAL]\ncase inl.intro.inl.intro\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nsrc\u271d : CircularPreorder \u03b1 := Preorder.toCircularPreorder \u03b1\na b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\nhcb : c \u2264 b\nhba : b \u2264 a\n\u22a2 a = b \u2228 b = c \u2228 c = a\n[PROOFSTEP]\nexact Or.inl (hab.antisymm hba)\n[GOAL]\ncase inl.intro.inr.inl.intro\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nsrc\u271d : CircularPreorder \u03b1 := Preorder.toCircularPreorder \u03b1\na b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\nhba : b \u2264 a\nhac : a \u2264 c\n\u22a2 a = b \u2228 b = c \u2228 c = a\n[PROOFSTEP]\nexact Or.inl (hab.antisymm hba)\n[GOAL]\ncase inl.intro.inr.inr.intro\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nsrc\u271d : CircularPreorder \u03b1 := Preorder.toCircularPreorder \u03b1\na b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\nhac : a \u2264 c\nhcb : c \u2264 b\n\u22a2 a = b \u2228 b = c \u2228 c = a\n[PROOFSTEP]\nexact Or.inr (Or.inl <| hbc.antisymm hcb)\n[GOAL]\ncase inr.inl.intro.inl.intro\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nsrc\u271d : CircularPreorder \u03b1 := Preorder.toCircularPreorder \u03b1\na b c : \u03b1\nhbc : b \u2264 c\nhca : c \u2264 a\nhcb : c \u2264 b\nhba : b \u2264 a\n\u22a2 a = b \u2228 b = c \u2228 c = a\n[PROOFSTEP]\nexact Or.inr (Or.inl <| hbc.antisymm hcb)\n[GOAL]\ncase inr.inl.intro.inr.inl.intro\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nsrc\u271d : CircularPreorder \u03b1 := Preorder.toCircularPreorder \u03b1\na b c : \u03b1\nhbc : b \u2264 c\nhca : c \u2264 a\nhba : b \u2264 a\nhac : a \u2264 c\n\u22a2 a = b \u2228 b = c \u2228 c = a\n[PROOFSTEP]\nexact Or.inr (Or.inr <| hca.antisymm hac)\n[GOAL]\ncase inr.inl.intro.inr.inr.intro\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nsrc\u271d : CircularPreorder \u03b1 := Preorder.toCircularPreorder \u03b1\na b c : \u03b1\nhbc : b \u2264 c\nhca : c \u2264 a\nhac : a \u2264 c\nhcb : c \u2264 b\n\u22a2 a = b \u2228 b = c \u2228 c = a\n[PROOFSTEP]\nexact Or.inr (Or.inl <| hbc.antisymm hcb)\n[GOAL]\ncase inr.inr.intro.inl.intro\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nsrc\u271d : CircularPreorder \u03b1 := Preorder.toCircularPreorder \u03b1\na b c : \u03b1\nhca : c \u2264 a\nhab : a \u2264 b\nhcb : c \u2264 b\nhba : b \u2264 a\n\u22a2 a = b \u2228 b = c \u2228 c = a\n[PROOFSTEP]\nexact Or.inl (hab.antisymm hba)\n[GOAL]\ncase inr.inr.intro.inr.inl.intro\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nsrc\u271d : CircularPreorder \u03b1 := Preorder.toCircularPreorder \u03b1\na b c : \u03b1\nhca : c \u2264 a\nhab : a \u2264 b\nhba : b \u2264 a\nhac : a \u2264 c\n\u22a2 a = b \u2228 b = c \u2228 c = a\n[PROOFSTEP]\nexact Or.inl (hab.antisymm hba)\n[GOAL]\ncase inr.inr.intro.inr.inr.intro\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nsrc\u271d : CircularPreorder \u03b1 := Preorder.toCircularPreorder \u03b1\na b c : \u03b1\nhca : c \u2264 a\nhab : a \u2264 b\nhac : a \u2264 c\nhcb : c \u2264 b\n\u22a2 a = b \u2228 b = c \u2228 c = a\n[PROOFSTEP]\nexact Or.inr (Or.inr <| hca.antisymm hac)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : CircularPartialOrder \u03b1 := PartialOrder.toCircularPartialOrder \u03b1\na b c : \u03b1\n\u22a2 btw a b c \u2228 btw c b a\n[PROOFSTEP]\ncases' le_total a b with hab hba\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : CircularPartialOrder \u03b1 := PartialOrder.toCircularPartialOrder \u03b1\na b c : \u03b1\nhab : a \u2264 b\n\u22a2 btw a b c \u2228 btw c b a\n[PROOFSTEP]\ncases' le_total b c with hbc hcb\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : CircularPartialOrder \u03b1 := PartialOrder.toCircularPartialOrder \u03b1\na b c : \u03b1\nhba : b \u2264 a\n\u22a2 btw a b c \u2228 btw c b a\n[PROOFSTEP]\ncases' le_total b c with hbc hcb\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : CircularPartialOrder \u03b1 := PartialOrder.toCircularPartialOrder \u03b1\na b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\n\u22a2 btw a b c \u2228 btw c b a\n[PROOFSTEP]\ncases' le_total c a with hca hac\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : CircularPartialOrder \u03b1 := PartialOrder.toCircularPartialOrder \u03b1\na b c : \u03b1\nhab : a \u2264 b\nhcb : c \u2264 b\n\u22a2 btw a b c \u2228 btw c b a\n[PROOFSTEP]\ncases' le_total c a with hca hac\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : CircularPartialOrder \u03b1 := PartialOrder.toCircularPartialOrder \u03b1\na b c : \u03b1\nhba : b \u2264 a\nhbc : b \u2264 c\n\u22a2 btw a b c \u2228 btw c b a\n[PROOFSTEP]\ncases' le_total c a with hca hac\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : CircularPartialOrder \u03b1 := PartialOrder.toCircularPartialOrder \u03b1\na b c : \u03b1\nhba : b \u2264 a\nhcb : c \u2264 b\n\u22a2 btw a b c \u2228 btw c b a\n[PROOFSTEP]\ncases' le_total c a with hca hac\n[GOAL]\ncase inl.inl.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : CircularPartialOrder \u03b1 := PartialOrder.toCircularPartialOrder \u03b1\na b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\nhca : c \u2264 a\n\u22a2 btw a b c \u2228 btw c b a\n[PROOFSTEP]\nexact Or.inl (Or.inl \u27e8hab, hbc\u27e9)\n[GOAL]\ncase inl.inl.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : CircularPartialOrder \u03b1 := PartialOrder.toCircularPartialOrder \u03b1\na b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\nhac : a \u2264 c\n\u22a2 btw a b c \u2228 btw c b a\n[PROOFSTEP]\nexact Or.inl (Or.inl \u27e8hab, hbc\u27e9)\n[GOAL]\ncase inl.inr.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : CircularPartialOrder \u03b1 := PartialOrder.toCircularPartialOrder \u03b1\na b c : \u03b1\nhab : a \u2264 b\nhcb : c \u2264 b\nhca : c \u2264 a\n\u22a2 btw a b c \u2228 btw c b a\n[PROOFSTEP]\nexact Or.inl (Or.inr <| Or.inr \u27e8hca, hab\u27e9)\n[GOAL]\ncase inl.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : CircularPartialOrder \u03b1 := PartialOrder.toCircularPartialOrder \u03b1\na b c : \u03b1\nhab : a \u2264 b\nhcb : c \u2264 b\nhac : a \u2264 c\n\u22a2 btw a b c \u2228 btw c b a\n[PROOFSTEP]\nexact Or.inr (Or.inr <| Or.inr \u27e8hac, hcb\u27e9)\n[GOAL]\ncase inr.inl.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : CircularPartialOrder \u03b1 := PartialOrder.toCircularPartialOrder \u03b1\na b c : \u03b1\nhba : b \u2264 a\nhbc : b \u2264 c\nhca : c \u2264 a\n\u22a2 btw a b c \u2228 btw c b a\n[PROOFSTEP]\nexact Or.inl (Or.inr <| Or.inl \u27e8hbc, hca\u27e9)\n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : CircularPartialOrder \u03b1 := PartialOrder.toCircularPartialOrder \u03b1\na b c : \u03b1\nhba : b \u2264 a\nhbc : b \u2264 c\nhac : a \u2264 c\n\u22a2 btw a b c \u2228 btw c b a\n[PROOFSTEP]\nexact Or.inr (Or.inr <| Or.inl \u27e8hba, hac\u27e9)\n[GOAL]\ncase inr.inr.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : CircularPartialOrder \u03b1 := PartialOrder.toCircularPartialOrder \u03b1\na b c : \u03b1\nhba : b \u2264 a\nhcb : c \u2264 b\nhca : c \u2264 a\n\u22a2 btw a b c \u2228 btw c b a\n[PROOFSTEP]\nexact Or.inr (Or.inl \u27e8hcb, hba\u27e9)\n[GOAL]\ncase inr.inr.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : CircularPartialOrder \u03b1 := PartialOrder.toCircularPartialOrder \u03b1\na b c : \u03b1\nhba : b \u2264 a\nhcb : c \u2264 b\nhac : a \u2264 c\n\u22a2 btw a b c \u2228 btw c b a\n[PROOFSTEP]\nexact Or.inr (Or.inr <| Or.inl \u27e8hba, hac\u27e9)\n", "meta": {"mathlib_filename": "Mathlib.Order.Circular", "llama_tokens": 31533, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723317123102956, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.42586180717343813}} {"text": "[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u22a2 condCount \u2205 = 0\n[PROOFSTEP]\nsimp [condCount]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\ns : Set \u03a9\n\u22a2 \u2191\u2191(condCount s) \u2205 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\ns t : Set \u03a9\nh : \u2191\u2191(condCount s) t \u2260 0\n\u22a2 Set.Finite s\n[PROOFSTEP]\nby_contra hs'\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\ns t : Set \u03a9\nh : \u2191\u2191(condCount s) t \u2260 0\nhs' : \u00acSet.Finite s\n\u22a2 False\n[PROOFSTEP]\nsimp [condCount, cond, Measure.count_apply_infinite hs'] at h \n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : Fintype \u03a9\ns : Set \u03a9\n\u22a2 \u2191\u2191(condCount Set.univ) s = \u2191\u2191Measure.count s / \u2191(Fintype.card \u03a9)\n[PROOFSTEP]\nrw [condCount, cond_apply _ MeasurableSet.univ, \u2190 ENNReal.div_eq_inv_mul, Set.univ_inter]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : Fintype \u03a9\ns : Set \u03a9\n\u22a2 \u2191\u2191Measure.count s / \u2191\u2191Measure.count Set.univ = \u2191\u2191Measure.count s / \u2191(Fintype.card \u03a9)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : Fintype \u03a9\ns : Set \u03a9\n\u22a2 \u2191\u2191Measure.count Set.univ = \u2191(Fintype.card \u03a9)\n[PROOFSTEP]\nrw [\u2190 Finset.coe_univ, Measure.count_apply, Finset.univ.tsum_subtype' fun _ => (1 : ENNReal)]\n[GOAL]\ncase e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : Fintype \u03a9\ns : Set \u03a9\n\u22a2 (Finset.sum Finset.univ fun x => 1) = \u2191(Fintype.card \u03a9)\n[PROOFSTEP]\nsimp [Finset.card_univ]\n[GOAL]\ncase e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : Fintype \u03a9\ns : Set \u03a9\n\u22a2 MeasurableSet \u2191Finset.univ\n[PROOFSTEP]\nexact (@Finset.coe_univ \u03a9 _).symm \u25b8 MeasurableSet.univ\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns : Set \u03a9\nhs : Set.Finite s\nhs' : Set.Nonempty s\n\u22a2 \u2191\u2191(condCount s) Set.univ = 1\n[PROOFSTEP]\nrw [condCount, cond_apply _ hs.measurableSet, Set.inter_univ, ENNReal.inv_mul_cancel]\n[GOAL]\ncase h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns : Set \u03a9\nhs : Set.Finite s\nhs' : Set.Nonempty s\n\u22a2 \u2191\u2191Measure.count s \u2260 0\n[PROOFSTEP]\nexact fun h => hs'.ne_empty <| Measure.empty_of_count_eq_zero h\n[GOAL]\ncase ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns : Set \u03a9\nhs : Set.Finite s\nhs' : Set.Nonempty s\n\u22a2 \u2191\u2191Measure.count s \u2260 \u22a4\n[PROOFSTEP]\nexact (Measure.count_apply_lt_top.2 hs).ne\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03a9\n\u03c9 : \u03a9\nt : Set \u03a9\ninst\u271d : Decidable (\u03c9 \u2208 t)\n\u22a2 \u2191\u2191(condCount {\u03c9}) t = if \u03c9 \u2208 t then 1 else 0\n[PROOFSTEP]\nrw [condCount, cond_apply _ (measurableSet_singleton \u03c9), Measure.count_singleton, inv_one, one_mul]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03a9\n\u03c9 : \u03a9\nt : Set \u03a9\ninst\u271d : Decidable (\u03c9 \u2208 t)\n\u22a2 \u2191\u2191Measure.count ({\u03c9} \u2229 t) = if \u03c9 \u2208 t then 1 else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03a9\n\u03c9 : \u03a9\nt : Set \u03a9\ninst\u271d : Decidable (\u03c9 \u2208 t)\nh\u271d : \u03c9 \u2208 t\n\u22a2 \u2191\u2191Measure.count ({\u03c9} \u2229 t) = 1\n[PROOFSTEP]\nrw [(by simpa : ({ \u03c9 } : Set \u03a9) \u2229 t = { \u03c9 }), Measure.count_singleton]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03a9\n\u03c9 : \u03a9\nt : Set \u03a9\ninst\u271d : Decidable (\u03c9 \u2208 t)\nh\u271d : \u03c9 \u2208 t\n\u22a2 {\u03c9} \u2229 t = {\u03c9}\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03a9\n\u03c9 : \u03a9\nt : Set \u03a9\ninst\u271d : Decidable (\u03c9 \u2208 t)\nh\u271d : \u00ac\u03c9 \u2208 t\n\u22a2 \u2191\u2191Measure.count ({\u03c9} \u2229 t) = 0\n[PROOFSTEP]\nrw [(by simpa : ({ \u03c9 } : Set \u03a9) \u2229 t = \u2205), Measure.count_empty]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03a9\n\u03c9 : \u03a9\nt : Set \u03a9\ninst\u271d : Decidable (\u03c9 \u2208 t)\nh\u271d : \u00ac\u03c9 \u2208 t\n\u22a2 {\u03c9} \u2229 t = \u2205\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount s) (s \u2229 t) = \u2191\u2191(condCount s) t\n[PROOFSTEP]\nrw [condCount, cond_inter_self _ hs.measurableSet]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhs' : Set.Nonempty s\n\u22a2 \u2191\u2191(condCount s) s = 1\n[PROOFSTEP]\nrw [condCount, cond_apply _ hs.measurableSet, Set.inter_self, ENNReal.inv_mul_cancel]\n[GOAL]\ncase h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhs' : Set.Nonempty s\n\u22a2 \u2191\u2191Measure.count s \u2260 0\n[PROOFSTEP]\nexact fun h => hs'.ne_empty <| Measure.empty_of_count_eq_zero h\n[GOAL]\ncase ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhs' : Set.Nonempty s\n\u22a2 \u2191\u2191Measure.count s \u2260 \u22a4\n[PROOFSTEP]\nexact (Measure.count_apply_lt_top.2 hs).ne\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhs' : Set.Nonempty s\nht : s \u2286 t\n\u22a2 \u2191\u2191(condCount s) t = 1\n[PROOFSTEP]\nhaveI := condCount_isProbabilityMeasure hs hs'\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhs' : Set.Nonempty s\nht : s \u2286 t\nthis : IsProbabilityMeasure (condCount s)\n\u22a2 \u2191\u2191(condCount s) t = 1\n[PROOFSTEP]\nrefine' eq_of_le_of_not_lt prob_le_one _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhs' : Set.Nonempty s\nht : s \u2286 t\nthis : IsProbabilityMeasure (condCount s)\n\u22a2 \u00ac\u2191\u2191(condCount s) t < 1\n[PROOFSTEP]\nrw [not_lt, \u2190 condCount_self hs hs']\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhs' : Set.Nonempty s\nht : s \u2286 t\nthis : IsProbabilityMeasure (condCount s)\n\u22a2 \u2191\u2191(condCount s) s \u2264 \u2191\u2191(condCount s) t\n[PROOFSTEP]\nexact measure_mono ht\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : \u2191\u2191(condCount s) t = 1\n\u22a2 s \u2286 t\n[PROOFSTEP]\nhave hsf := finite_of_condCount_ne_zero (by rw [h]; exact one_ne_zero)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : \u2191\u2191(condCount s) t = 1\n\u22a2 \u2191\u2191(condCount ?m.9273) ?m.9274 \u2260 0\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : \u2191\u2191(condCount s) t = 1\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nexact one_ne_zero\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : \u2191\u2191(condCount s) t = 1\nhsf : Set.Finite s\n\u22a2 s \u2286 t\n[PROOFSTEP]\nrw [condCount, cond_apply _ hsf.measurableSet, mul_comm] at h \n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : \u2191\u2191Measure.count (s \u2229 t) * (\u2191\u2191Measure.count s)\u207b\u00b9 = 1\nhsf : Set.Finite s\n\u22a2 s \u2286 t\n[PROOFSTEP]\nreplace h := ENNReal.eq_inv_of_mul_eq_one_left h\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : Set.Finite s\nh : \u2191\u2191Measure.count (s \u2229 t) = (\u2191\u2191Measure.count s)\u207b\u00b9\u207b\u00b9\n\u22a2 s \u2286 t\n[PROOFSTEP]\nrw [inv_inv, Measure.count_apply_finite _ hsf, Measure.count_apply_finite _ (hsf.inter_of_left _), Nat.cast_inj] at h \n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s \u2229 t))) = Finset.card (Set.Finite.toFinset hsf)\n\u22a2 s \u2286 t\n[PROOFSTEP]\nsuffices s \u2229 t = s by exact this \u25b8 fun x hx => hx.2\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s \u2229 t))) = Finset.card (Set.Finite.toFinset hsf)\nthis : s \u2229 t = s\n\u22a2 s \u2286 t\n[PROOFSTEP]\nexact this \u25b8 fun x hx => hx.2\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s \u2229 t))) = Finset.card (Set.Finite.toFinset hsf)\n\u22a2 s \u2229 t = s\n[PROOFSTEP]\nrw [\u2190 @Set.Finite.toFinset_inj _ _ _ (hsf.inter_of_left _) hsf]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : Set.Finite s\nh : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s \u2229 t))) = Finset.card (Set.Finite.toFinset hsf)\n\u22a2 Set.Finite.toFinset (_ : Set.Finite (s \u2229 t)) = Set.Finite.toFinset hsf\n[PROOFSTEP]\nexact Finset.eq_of_subset_of_card_le (Set.Finite.toFinset_mono <| s.inter_subset_left t) h.ge\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount s) t = 0 \u2194 s \u2229 t = \u2205\n[PROOFSTEP]\nsimp [condCount, cond_apply _ hs.measurableSet, Measure.count_apply_eq_top, Set.not_infinite.2 hs,\n Measure.count_apply_finite _ (hs.inter_of_left _)]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount s) (t \u2229 u) = \u2191\u2191(condCount (s \u2229 t)) u * \u2191\u2191(condCount s) t\n[PROOFSTEP]\nby_cases hst : s \u2229 t = \u2205\n[GOAL]\ncase pos\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhst : s \u2229 t = \u2205\n\u22a2 \u2191\u2191(condCount s) (t \u2229 u) = \u2191\u2191(condCount (s \u2229 t)) u * \u2191\u2191(condCount s) t\n[PROOFSTEP]\nrw [hst, condCount_empty_meas, Measure.coe_zero, Pi.zero_apply, zero_mul, condCount_eq_zero_iff hs, \u2190 Set.inter_assoc,\n hst, Set.empty_inter]\n[GOAL]\ncase neg\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhst : \u00acs \u2229 t = \u2205\n\u22a2 \u2191\u2191(condCount s) (t \u2229 u) = \u2191\u2191(condCount (s \u2229 t)) u * \u2191\u2191(condCount s) t\n[PROOFSTEP]\nrw [condCount, condCount, cond_apply _ hs.measurableSet, cond_apply _ hs.measurableSet,\n cond_apply _ (hs.inter_of_left _).measurableSet, mul_comm _ (Measure.count (s \u2229 t)), \u2190 mul_assoc,\n mul_comm _ (Measure.count (s \u2229 t)), \u2190 mul_assoc, ENNReal.mul_inv_cancel, one_mul, mul_comm, Set.inter_assoc]\n[GOAL]\ncase neg.h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhst : \u00acs \u2229 t = \u2205\n\u22a2 \u2191\u2191Measure.count (s \u2229 t) \u2260 0\n[PROOFSTEP]\nrwa [\u2190 Measure.count_eq_zero_iff] at hst \n[GOAL]\ncase neg.ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhst : \u00acs \u2229 t = \u2205\n\u22a2 \u2191\u2191Measure.count (s \u2229 t) \u2260 \u22a4\n[PROOFSTEP]\nexact (Measure.count_apply_lt_top.2 <| hs.inter_of_left _).ne\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount s) (t \u2229 u) = \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u\n[PROOFSTEP]\nrw [\u2190 Set.inter_comm]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount s) (u \u2229 t) = \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u\n[PROOFSTEP]\nexact condCount_inter hs\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhtu : Disjoint t u\n\u22a2 \u2191\u2191(condCount s) (t \u222a u) = \u2191\u2191(condCount s) t + \u2191\u2191(condCount s) u\n[PROOFSTEP]\nrw [condCount, cond_apply _ hs.measurableSet, cond_apply _ hs.measurableSet, cond_apply _ hs.measurableSet,\n Set.inter_union_distrib_left, measure_union, mul_add]\n[GOAL]\ncase hd\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhtu : Disjoint t u\n\u22a2 Disjoint (s \u2229 t) (s \u2229 u)\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nhtu : Disjoint t u\n\u22a2 MeasurableSet (s \u2229 u)\n[PROOFSTEP]\nexacts [htu.mono inf_le_right inf_le_right, (hs.inter_of_left _).measurableSet]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u t : Set \u03a9\nhs : Set.Finite s\nhs' : Set.Nonempty s\n\u22a2 \u2191\u2191(condCount s) t + \u2191\u2191(condCount s) t\u1d9c = 1\n[PROOFSTEP]\nrw [\u2190 condCount_union hs disjoint_compl_right, Set.union_compl_self,\n (condCount_isProbabilityMeasure hs hs').measure_univ]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nht : Set.Finite t\nhst : Disjoint s t\n\u22a2 \u2191\u2191(condCount s) u * \u2191\u2191(condCount (s \u222a t)) s + \u2191\u2191(condCount t) u * \u2191\u2191(condCount (s \u222a t)) t = \u2191\u2191(condCount (s \u222a t)) u\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | hs')\n[GOAL]\ncase inl\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\nt u : Set \u03a9\nht : Set.Finite t\nhs : Set.Finite \u2205\nhst : Disjoint \u2205 t\n\u22a2 \u2191\u2191(condCount \u2205) u * \u2191\u2191(condCount (\u2205 \u222a t)) \u2205 + \u2191\u2191(condCount t) u * \u2191\u2191(condCount (\u2205 \u222a t)) t = \u2191\u2191(condCount (\u2205 \u222a t)) u\n[PROOFSTEP]\nrcases t.eq_empty_or_nonempty with (rfl | ht')\n[GOAL]\ncase inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nht : Set.Finite t\nhst : Disjoint s t\nhs' : Set.Nonempty s\n\u22a2 \u2191\u2191(condCount s) u * \u2191\u2191(condCount (s \u222a t)) s + \u2191\u2191(condCount t) u * \u2191\u2191(condCount (s \u222a t)) t = \u2191\u2191(condCount (s \u222a t)) u\n[PROOFSTEP]\nrcases t.eq_empty_or_nonempty with (rfl | ht')\n[GOAL]\ncase inl.inl\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\nu : Set \u03a9\nhs ht : Set.Finite \u2205\nhst : Disjoint \u2205 \u2205\n\u22a2 \u2191\u2191(condCount \u2205) u * \u2191\u2191(condCount (\u2205 \u222a \u2205)) \u2205 + \u2191\u2191(condCount \u2205) u * \u2191\u2191(condCount (\u2205 \u222a \u2205)) \u2205 = \u2191\u2191(condCount (\u2205 \u222a \u2205)) u\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\nt u : Set \u03a9\nht : Set.Finite t\nhs : Set.Finite \u2205\nhst : Disjoint \u2205 t\nht' : Set.Nonempty t\n\u22a2 \u2191\u2191(condCount \u2205) u * \u2191\u2191(condCount (\u2205 \u222a t)) \u2205 + \u2191\u2191(condCount t) u * \u2191\u2191(condCount (\u2205 \u222a t)) t = \u2191\u2191(condCount (\u2205 \u222a t)) u\n[PROOFSTEP]\nsimp [condCount_self ht ht']\n[GOAL]\ncase inr.inl\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns u : Set \u03a9\nhs : Set.Finite s\nhs' : Set.Nonempty s\nht : Set.Finite \u2205\nhst : Disjoint s \u2205\n\u22a2 \u2191\u2191(condCount s) u * \u2191\u2191(condCount (s \u222a \u2205)) s + \u2191\u2191(condCount \u2205) u * \u2191\u2191(condCount (s \u222a \u2205)) \u2205 = \u2191\u2191(condCount (s \u222a \u2205)) u\n[PROOFSTEP]\nsimp [condCount_self hs hs']\n[GOAL]\ncase inr.inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nht : Set.Finite t\nhst : Disjoint s t\nhs' : Set.Nonempty s\nht' : Set.Nonempty t\n\u22a2 \u2191\u2191(condCount s) u * \u2191\u2191(condCount (s \u222a t)) s + \u2191\u2191(condCount t) u * \u2191\u2191(condCount (s \u222a t)) t = \u2191\u2191(condCount (s \u222a t)) u\n[PROOFSTEP]\nrw [condCount, condCount, condCount, cond_apply _ hs.measurableSet, cond_apply _ ht.measurableSet,\n cond_apply _ (hs.union ht).measurableSet, cond_apply _ (hs.union ht).measurableSet,\n cond_apply _ (hs.union ht).measurableSet]\n[GOAL]\ncase inr.inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nht : Set.Finite t\nhst : Disjoint s t\nhs' : Set.Nonempty s\nht' : Set.Nonempty t\n\u22a2 (\u2191\u2191Measure.count s)\u207b\u00b9 * \u2191\u2191Measure.count (s \u2229 u) * ((\u2191\u2191Measure.count (s \u222a t))\u207b\u00b9 * \u2191\u2191Measure.count ((s \u222a t) \u2229 s)) +\n (\u2191\u2191Measure.count t)\u207b\u00b9 * \u2191\u2191Measure.count (t \u2229 u) * ((\u2191\u2191Measure.count (s \u222a t))\u207b\u00b9 * \u2191\u2191Measure.count ((s \u222a t) \u2229 t)) =\n (\u2191\u2191Measure.count (s \u222a t))\u207b\u00b9 * \u2191\u2191Measure.count ((s \u222a t) \u2229 u)\n[PROOFSTEP]\nconv_lhs =>\n rw [Set.union_inter_cancel_left, Set.union_inter_cancel_right, mul_comm (Measure.count (s \u222a t))\u207b\u00b9,\n mul_comm (Measure.count (s \u222a t))\u207b\u00b9, \u2190 mul_assoc, \u2190 mul_assoc, mul_comm _ (Measure.count s),\n mul_comm _ (Measure.count t), \u2190 mul_assoc, \u2190 mul_assoc]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nht : Set.Finite t\nhst : Disjoint s t\nhs' : Set.Nonempty s\nht' : Set.Nonempty t\n| (\u2191\u2191Measure.count s)\u207b\u00b9 * \u2191\u2191Measure.count (s \u2229 u) * ((\u2191\u2191Measure.count (s \u222a t))\u207b\u00b9 * \u2191\u2191Measure.count ((s \u222a t) \u2229 s)) +\n (\u2191\u2191Measure.count t)\u207b\u00b9 * \u2191\u2191Measure.count (t \u2229 u) * ((\u2191\u2191Measure.count (s \u222a t))\u207b\u00b9 * \u2191\u2191Measure.count ((s \u222a t) \u2229 t))\n[PROOFSTEP]\nrw [Set.union_inter_cancel_left, Set.union_inter_cancel_right, mul_comm (Measure.count (s \u222a t))\u207b\u00b9,\n mul_comm (Measure.count (s \u222a t))\u207b\u00b9, \u2190 mul_assoc, \u2190 mul_assoc, mul_comm _ (Measure.count s),\n mul_comm _ (Measure.count t), \u2190 mul_assoc, \u2190 mul_assoc]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nht : Set.Finite t\nhst : Disjoint s t\nhs' : Set.Nonempty s\nht' : Set.Nonempty t\n| (\u2191\u2191Measure.count s)\u207b\u00b9 * \u2191\u2191Measure.count (s \u2229 u) * ((\u2191\u2191Measure.count (s \u222a t))\u207b\u00b9 * \u2191\u2191Measure.count ((s \u222a t) \u2229 s)) +\n (\u2191\u2191Measure.count t)\u207b\u00b9 * \u2191\u2191Measure.count (t \u2229 u) * ((\u2191\u2191Measure.count (s \u222a t))\u207b\u00b9 * \u2191\u2191Measure.count ((s \u222a t) \u2229 t))\n[PROOFSTEP]\nrw [Set.union_inter_cancel_left, Set.union_inter_cancel_right, mul_comm (Measure.count (s \u222a t))\u207b\u00b9,\n mul_comm (Measure.count (s \u222a t))\u207b\u00b9, \u2190 mul_assoc, \u2190 mul_assoc, mul_comm _ (Measure.count s),\n mul_comm _ (Measure.count t), \u2190 mul_assoc, \u2190 mul_assoc]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nht : Set.Finite t\nhst : Disjoint s t\nhs' : Set.Nonempty s\nht' : Set.Nonempty t\n| (\u2191\u2191Measure.count s)\u207b\u00b9 * \u2191\u2191Measure.count (s \u2229 u) * ((\u2191\u2191Measure.count (s \u222a t))\u207b\u00b9 * \u2191\u2191Measure.count ((s \u222a t) \u2229 s)) +\n (\u2191\u2191Measure.count t)\u207b\u00b9 * \u2191\u2191Measure.count (t \u2229 u) * ((\u2191\u2191Measure.count (s \u222a t))\u207b\u00b9 * \u2191\u2191Measure.count ((s \u222a t) \u2229 t))\n[PROOFSTEP]\nrw [Set.union_inter_cancel_left, Set.union_inter_cancel_right, mul_comm (Measure.count (s \u222a t))\u207b\u00b9,\n mul_comm (Measure.count (s \u222a t))\u207b\u00b9, \u2190 mul_assoc, \u2190 mul_assoc, mul_comm _ (Measure.count s),\n mul_comm _ (Measure.count t), \u2190 mul_assoc, \u2190 mul_assoc]\n[GOAL]\ncase inr.inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nht : Set.Finite t\nhst : Disjoint s t\nhs' : Set.Nonempty s\nht' : Set.Nonempty t\n\u22a2 \u2191\u2191Measure.count s * (\u2191\u2191Measure.count s)\u207b\u00b9 * \u2191\u2191Measure.count (s \u2229 u) * (\u2191\u2191Measure.count (s \u222a t))\u207b\u00b9 +\n \u2191\u2191Measure.count t * (\u2191\u2191Measure.count t)\u207b\u00b9 * \u2191\u2191Measure.count (t \u2229 u) * (\u2191\u2191Measure.count (s \u222a t))\u207b\u00b9 =\n (\u2191\u2191Measure.count (s \u222a t))\u207b\u00b9 * \u2191\u2191Measure.count ((s \u222a t) \u2229 u)\n[PROOFSTEP]\nrw [ENNReal.mul_inv_cancel, ENNReal.mul_inv_cancel, one_mul, one_mul, \u2190 add_mul, \u2190 measure_union,\n Set.union_inter_distrib_right, mul_comm]\n[GOAL]\ncase inr.inr.hd\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nht : Set.Finite t\nhst : Disjoint s t\nhs' : Set.Nonempty s\nht' : Set.Nonempty t\n\u22a2 Disjoint (s \u2229 u) (t \u2229 u)\ncase inr.inr.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nht : Set.Finite t\nhst : Disjoint s t\nhs' : Set.Nonempty s\nht' : Set.Nonempty t\n\u22a2 MeasurableSet (t \u2229 u)\ncase inr.inr.h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nht : Set.Finite t\nhst : Disjoint s t\nhs' : Set.Nonempty s\nht' : Set.Nonempty t\n\u22a2 \u2191\u2191Measure.count t \u2260 0\ncase inr.inr.ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nht : Set.Finite t\nhst : Disjoint s t\nhs' : Set.Nonempty s\nht' : Set.Nonempty t\n\u22a2 \u2191\u2191Measure.count t \u2260 \u22a4\ncase inr.inr.h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nht : Set.Finite t\nhst : Disjoint s t\nhs' : Set.Nonempty s\nht' : Set.Nonempty t\n\u22a2 \u2191\u2191Measure.count s \u2260 0\ncase inr.inr.ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : Set.Finite s\nht : Set.Finite t\nhst : Disjoint s t\nhs' : Set.Nonempty s\nht' : Set.Nonempty t\n\u22a2 \u2191\u2191Measure.count s \u2260 \u22a4\n[PROOFSTEP]\nexacts [hst.mono inf_le_left inf_le_left, (ht.inter_of_left _).measurableSet, Measure.count_ne_zero ht',\n (Measure.count_apply_lt_top.2 ht).ne, Measure.count_ne_zero hs', (Measure.count_apply_lt_top.2 hs).ne]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c = \u2191\u2191(condCount s) t\n[PROOFSTEP]\nhave :\n condCount s t =\n (condCount (s \u2229 u) t * condCount (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u) +\n condCount (s \u2229 u\u1d9c) t * condCount (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u\u1d9c)) :=\n by\n rw [condCount_disjoint_union (hs.inter_of_left _) (hs.inter_of_left _)\n (disjoint_compl_right.mono inf_le_right inf_le_right),\n Set.inter_union_compl]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount s) t =\n \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)\n[PROOFSTEP]\nrw [condCount_disjoint_union (hs.inter_of_left _) (hs.inter_of_left _)\n (disjoint_compl_right.mono inf_le_right inf_le_right),\n Set.inter_union_compl]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\nthis :\n \u2191\u2191(condCount s) t =\n \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)\n\u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c = \u2191\u2191(condCount s) t\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\nthis :\n \u2191\u2191(condCount s) t =\n \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)\n\u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c =\n \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)\n[PROOFSTEP]\nsimp [condCount_inter_self hs]\n", "meta": {"mathlib_filename": "Mathlib.Probability.CondCount", "llama_tokens": 10504, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.63341027059799, "lm_q1q2_score": 0.42586179519144157}} {"text": "[GOAL]\nN : Type u_1\ninst\u271d : DecidableEq N\ni : N\nt\u2080 : \u2191I\nt : { j // j \u2260 i } \u2192 \u2191I\nH : (t\u2080 = 0 \u2228 t\u2080 = 1) \u2228 t \u2208 boundary { j // j \u2260 i }\n\u22a2 \u2191(insertAt i) (t\u2080, t) \u2208 boundary N\n[PROOFSTEP]\nobtain H | \u27e8j, H\u27e9 := H\n[GOAL]\ncase inl\nN : Type u_1\ninst\u271d : DecidableEq N\ni : N\nt\u2080 : \u2191I\nt : { j // j \u2260 i } \u2192 \u2191I\nH : t\u2080 = 0 \u2228 t\u2080 = 1\n\u22a2 \u2191(insertAt i) (t\u2080, t) \u2208 boundary N\n[PROOFSTEP]\nuse i\n[GOAL]\ncase h\nN : Type u_1\ninst\u271d : DecidableEq N\ni : N\nt\u2080 : \u2191I\nt : { j // j \u2260 i } \u2192 \u2191I\nH : t\u2080 = 0 \u2228 t\u2080 = 1\n\u22a2 \u2191(insertAt i) (t\u2080, t) i = 0 \u2228 \u2191(insertAt i) (t\u2080, t) i = 1\n[PROOFSTEP]\nrwa [funSplitAt_symm_apply, dif_pos rfl]\n[GOAL]\ncase inr.intro\nN : Type u_1\ninst\u271d : DecidableEq N\ni : N\nt\u2080 : \u2191I\nt : { j // j \u2260 i } \u2192 \u2191I\nj : { j // j \u2260 i }\nH : t j = 0 \u2228 t j = 1\n\u22a2 \u2191(insertAt i) (t\u2080, t) \u2208 boundary N\n[PROOFSTEP]\nuse j\n[GOAL]\ncase h\nN : Type u_1\ninst\u271d : DecidableEq N\ni : N\nt\u2080 : \u2191I\nt : { j // j \u2260 i } \u2192 \u2191I\nj : { j // j \u2260 i }\nH : t j = 0 \u2228 t j = 1\n\u22a2 \u2191(insertAt i) (t\u2080, t) \u2191j = 0 \u2228 \u2191(insertAt i) (t\u2080, t) \u2191j = 1\n[PROOFSTEP]\nrwa [funSplitAt_symm_apply, dif_neg j.prop, Subtype.coe_eta]\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d : TopologicalSpace X\nx : X\nx\u271d\u00b2 x\u271d\u00b9 : \u2191(\u03a9^ N X x)\nf : (N \u2192 \u2191I) \u2192 X\ncontinuous_toFun\u271d\u00b9 : Continuous f\nproperty\u271d\u00b9 : ContinuousMap.mk f \u2208 \u03a9^ N X x\ng : (N \u2192 \u2191I) \u2192 X\ncontinuous_toFun\u271d : Continuous g\nproperty\u271d : ContinuousMap.mk g \u2208 \u03a9^ N X x\nx\u271d :\n (fun f => \u2191\u2191f) { val := ContinuousMap.mk f, property := property\u271d\u00b9 } =\n (fun f => \u2191\u2191f) { val := ContinuousMap.mk g, property := property\u271d }\n\u22a2 { val := ContinuousMap.mk f, property := property\u271d\u00b9 } = { val := ContinuousMap.mk g, property := property\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d : TopologicalSpace X\nx : X\nf : \u2191(\u03a9^ N X x)\ng : (N \u2192 \u2191I) \u2192 X\nh : g = \u2191f\n\u22a2 ContinuousMap.mk g \u2208 \u03a9^ N X x\n[PROOFSTEP]\nconvert f.2\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d : TopologicalSpace X\nx : X\nf : \u2191(\u03a9^ N X x)\ng : (N \u2192 \u2191I) \u2192 X\nh : g = \u2191f\n\u22a2 copy f g h = f\n[PROOFSTEP]\next x\n[GOAL]\ncase H\nN : Type u_1\nX : Type u_2\ninst\u271d : TopologicalSpace X\nx\u271d : X\nf : \u2191(\u03a9^ N X x\u271d)\ng : (N \u2192 \u2191I) \u2192 X\nh : g = \u2191f\nx : N \u2192 \u2191I\n\u22a2 \u2191(copy f g h) x = \u2191f x\n[PROOFSTEP]\nexact congr_fun h x\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u2191(\u03a9^ N X x)\n\u22a2 ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(ContinuousMap.curry (ContinuousMap.comp (\u2191p) (Homeomorph.toContinuousMap (Cube.insertAt i)))) t,\n property :=\n (_ : \u2200 (y : { j // j \u2260 i } \u2192 \u2191I), y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191\u2191p (\u2191(Cube.insertAt i) (t, y)) = x) })\n 0 =\n const\n[PROOFSTEP]\next t\n[GOAL]\ncase H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u2191(\u03a9^ N X x)\nt : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(ContinuousMap.curry (ContinuousMap.comp (\u2191p) (Homeomorph.toContinuousMap (Cube.insertAt i)))) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191\u2191p (\u2191(Cube.insertAt i) (t, y)) = x) })\n 0)\n t =\n \u2191const t\n[PROOFSTEP]\nrefine' p.property (Cube.insertAt i (0, t)) \u27e8i, Or.inl _\u27e9\n[GOAL]\ncase H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u2191(\u03a9^ N X x)\nt : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(Cube.insertAt i) (0, t) i = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u2191(\u03a9^ N X x)\n\u22a2 ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(ContinuousMap.curry (ContinuousMap.comp (\u2191p) (Homeomorph.toContinuousMap (Cube.insertAt i)))) t,\n property :=\n (_ : \u2200 (y : { j // j \u2260 i } \u2192 \u2191I), y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191\u2191p (\u2191(Cube.insertAt i) (t, y)) = x) })\n 1 =\n const\n[PROOFSTEP]\next t\n[GOAL]\ncase H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u2191(\u03a9^ N X x)\nt : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(ContinuousMap.curry (ContinuousMap.comp (\u2191p) (Homeomorph.toContinuousMap (Cube.insertAt i)))) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191\u2191p (\u2191(Cube.insertAt i) (t, y)) = x) })\n 1)\n t =\n \u2191const t\n[PROOFSTEP]\nrefine' p.property (Cube.insertAt i (1, t)) \u27e8i, Or.inr _\u27e9\n[GOAL]\ncase H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u2191(\u03a9^ N X x)\nt : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(Cube.insertAt i) (1, t) i = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u03a9 (\u2191(\u03a9^ { j // j \u2260 i } X x)) const\n\u22a2 Continuous Subtype.val\n[PROOFSTEP]\ncontinuity\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u03a9 (\u2191(\u03a9^ { j // j \u2260 i } X x)) const\n\u22a2 ContinuousMap.comp (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))\n (Homeomorph.toContinuousMap (Cube.splitAt i)) \u2208\n \u03a9^ N X x\n[PROOFSTEP]\nrintro y \u27e8j, Hj\u27e9\n[GOAL]\ncase intro\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u03a9 (\u2191(\u03a9^ { j // j \u2260 i } X x)) const\ny : N \u2192 \u2191I\nj : N\nHj : y j = 0 \u2228 y j = 1\n\u22a2 \u2191(ContinuousMap.comp (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap))\n (Homeomorph.toContinuousMap (Cube.splitAt i)))\n y =\n x\n[PROOFSTEP]\nsimp only [ContinuousMap.comp_apply, toContinuousMap_apply, funSplitAt_apply, ContinuousMap.uncurry_apply,\n ContinuousMap.coe_mk, Function.uncurry_apply_pair]\n[GOAL]\ncase intro\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u03a9 (\u2191(\u03a9^ { j // j \u2260 i } X x)) const\ny : N \u2192 \u2191I\nj : N\nHj : y j = 0 \u2228 y j = 1\n\u22a2 (\u2191\u2191(\u2191p.toContinuousMap (y i)) fun j => y \u2191j) = x\n[PROOFSTEP]\nobtain rfl | Hne := eq_or_ne j i\n[GOAL]\ncase intro.inl\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ny : N \u2192 \u2191I\nj : N\nHj : y j = 0 \u2228 y j = 1\np : \u03a9 (\u2191(\u03a9^ { j_1 // j_1 \u2260 j } X x)) const\n\u22a2 (\u2191\u2191(\u2191p.toContinuousMap (y j)) fun j_1 => y \u2191j_1) = x\n[PROOFSTEP]\ncases' Hj with Hj Hj\n[GOAL]\ncase intro.inl.inl\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ny : N \u2192 \u2191I\nj : N\np : \u03a9 (\u2191(\u03a9^ { j_1 // j_1 \u2260 j } X x)) const\nHj : y j = 0\n\u22a2 (\u2191\u2191(\u2191p.toContinuousMap (y j)) fun j_1 => y \u2191j_1) = x\n[PROOFSTEP]\nsimp only [Hj, p.coe_toContinuousMap, p.source, p.target]\n[GOAL]\ncase intro.inl.inr\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ny : N \u2192 \u2191I\nj : N\np : \u03a9 (\u2191(\u03a9^ { j_1 // j_1 \u2260 j } X x)) const\nHj : y j = 1\n\u22a2 (\u2191\u2191(\u2191p.toContinuousMap (y j)) fun j_1 => y \u2191j_1) = x\n[PROOFSTEP]\nsimp only [Hj, p.coe_toContinuousMap, p.source, p.target]\n[GOAL]\ncase intro.inl.inl\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ny : N \u2192 \u2191I\nj : N\np : \u03a9 (\u2191(\u03a9^ { j_1 // j_1 \u2260 j } X x)) const\nHj : y j = 0\n\u22a2 (\u2191\u2191const fun j_1 => y \u2191j_1) = x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.inl.inr\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ny : N \u2192 \u2191I\nj : N\np : \u03a9 (\u2191(\u03a9^ { j_1 // j_1 \u2260 j } X x)) const\nHj : y j = 1\n\u22a2 (\u2191\u2191const fun j_1 => y \u2191j_1) = x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.inr\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u03a9 (\u2191(\u03a9^ { j // j \u2260 i } X x)) const\ny : N \u2192 \u2191I\nj : N\nHj : y j = 0 \u2228 y j = 1\nHne : j \u2260 i\n\u22a2 (\u2191\u2191(\u2191p.toContinuousMap (y i)) fun j => y \u2191j) = x\n[PROOFSTEP]\nexact GenLoop.boundary _ _ \u27e8\u27e8j, Hne\u27e9, Hj\u27e9\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u03a9 (\u2191(\u03a9^ { j // j \u2260 i } X x)) const\n\u22a2 toLoop i (fromLoop i p) = p\n[PROOFSTEP]\nsimp_rw [toLoop, fromLoop, ContinuousMap.comp_assoc, toContinuousMap_comp_symm, ContinuousMap.comp_id]\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u03a9 (\u2191(\u03a9^ { j // j \u2260 i } X x)) const\n\u22a2 {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n {\n val :=\n \u2191(ContinuousMap.curry\n (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap)))\n t,\n property :=\n (_ :\n (fun x_1 => x_1 \u2208 \u03a9^ { j // j \u2260 i } X x)\n (\u2191(ContinuousMap.curry\n (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap)))\n t)) },\n source' :=\n (_ :\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n {\n val :=\n \u2191(ContinuousMap.curry\n (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap)))\n t,\n property :=\n (_ :\n (fun x_1 => x_1 \u2208 \u03a9^ { j // j \u2260 i } X x)\n (\u2191(ContinuousMap.curry\n (ContinuousMap.uncurry\n (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap)))\n t)) })\n 0 =\n const),\n target' :=\n (_ :\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n {\n val :=\n \u2191(ContinuousMap.curry\n (ContinuousMap.uncurry (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap)))\n t,\n property :=\n (_ :\n (fun x_1 => x_1 \u2208 \u03a9^ { j // j \u2260 i } X x)\n (\u2191(ContinuousMap.curry\n (ContinuousMap.uncurry\n (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap)))\n t)) })\n 1 =\n const) } =\n p\n[PROOFSTEP]\next\n[GOAL]\ncase a.h.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u03a9 (\u2191(\u03a9^ { j // j \u2260 i } X x)) const\nx\u271d : \u2191I\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(\u2191{\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n {\n val :=\n \u2191(ContinuousMap.curry\n (ContinuousMap.uncurry\n (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap)))\n t,\n property :=\n (_ :\n (fun x_1 => x_1 \u2208 \u03a9^ { j // j \u2260 i } X x)\n (\u2191(ContinuousMap.curry\n (ContinuousMap.uncurry\n (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap)))\n t)) },\n source' :=\n (_ :\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n {\n val :=\n \u2191(ContinuousMap.curry\n (ContinuousMap.uncurry\n (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap)))\n t,\n property :=\n (_ :\n (fun x_1 => x_1 \u2208 \u03a9^ { j // j \u2260 i } X x)\n (\u2191(ContinuousMap.curry\n (ContinuousMap.uncurry\n (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap)))\n t)) })\n 0 =\n const),\n target' :=\n (_ :\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n {\n val :=\n \u2191(ContinuousMap.curry\n (ContinuousMap.uncurry\n (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap)))\n t,\n property :=\n (_ :\n (fun x_1 => x_1 \u2208 \u03a9^ { j // j \u2260 i } X x)\n (\u2191(ContinuousMap.curry\n (ContinuousMap.uncurry\n (ContinuousMap.comp (ContinuousMap.mk Subtype.val) p.toContinuousMap)))\n t)) })\n 1 =\n const) }\n x\u271d)\n y\u271d =\n \u2191(\u2191p x\u271d) y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u2191(\u03a9^ N X x)\n\u22a2 fromLoop i (toLoop i p) = p\n[PROOFSTEP]\next\n[GOAL]\ncase H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\n\u22a2 \u2191(fromLoop i (toLoop i p)) y\u271d = \u2191p y\u271d\n[PROOFSTEP]\nexact congr_arg p (Equiv.apply_symm_apply _ _)\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\n\u22a2 Homotopic p q \u2192 Path.Homotopic (toLoop i p) (toLoop i q)\n[PROOFSTEP]\nrefine' Nonempty.map fun H => \u27e8\u27e8\u27e8fun t => \u27e8homotopyTo i H t, _\u27e9, _\u27e9, _, _\u27e9, _\u27e9\n[GOAL]\ncase refine'_1\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt : \u2191I \u00d7 \u2191I\n\u22a2 \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x\n[PROOFSTEP]\nrintro y \u27e8i, iH\u27e9\n[GOAL]\ncase refine'_1.intro\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni\u271d : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt : \u2191I \u00d7 \u2191I\ny : { j // j \u2260 i\u271d } \u2192 \u2191I\ni : { j // j \u2260 i\u271d }\niH : y i = 0 \u2228 y i = 1\n\u22a2 \u2191(\u2191(homotopyTo i\u271d H) t) y = x\n[PROOFSTEP]\nrw [homotopyTo_apply, H.eq_fst, p.2]\n[GOAL]\ncase refine'_1.intro.a\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni\u271d : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt : \u2191I \u00d7 \u2191I\ny : { j // j \u2260 i\u271d } \u2192 \u2191I\ni : { j // j \u2260 i\u271d }\niH : y i = 0 \u2228 y i = 1\n\u22a2 \u2191(Cube.insertAt i\u271d) (t.snd, y) \u2208 Cube.boundary N\ncase refine'_1.intro.hx\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni\u271d : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt : \u2191I \u00d7 \u2191I\ny : { j // j \u2260 i\u271d } \u2192 \u2191I\ni : { j // j \u2260 i\u271d }\niH : y i = 0 \u2228 y i = 1\n\u22a2 \u2191(Cube.insertAt i\u271d) (t.snd, y) \u2208 Cube.boundary N\n[PROOFSTEP]\nall_goals apply Cube.insertAt_boundary; right; exact \u27e8i, iH\u27e9\n[GOAL]\ncase refine'_1.intro.a\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni\u271d : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt : \u2191I \u00d7 \u2191I\ny : { j // j \u2260 i\u271d } \u2192 \u2191I\ni : { j // j \u2260 i\u271d }\niH : y i = 0 \u2228 y i = 1\n\u22a2 \u2191(Cube.insertAt i\u271d) (t.snd, y) \u2208 Cube.boundary N\n[PROOFSTEP]\napply Cube.insertAt_boundary\n[GOAL]\ncase refine'_1.intro.a.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni\u271d : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt : \u2191I \u00d7 \u2191I\ny : { j // j \u2260 i\u271d } \u2192 \u2191I\ni : { j // j \u2260 i\u271d }\niH : y i = 0 \u2228 y i = 1\n\u22a2 (t.snd = 0 \u2228 t.snd = 1) \u2228 y \u2208 Cube.boundary { j // j \u2260 i\u271d }\n[PROOFSTEP]\nright\n[GOAL]\ncase refine'_1.intro.a.H.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni\u271d : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt : \u2191I \u00d7 \u2191I\ny : { j // j \u2260 i\u271d } \u2192 \u2191I\ni : { j // j \u2260 i\u271d }\niH : y i = 0 \u2228 y i = 1\n\u22a2 y \u2208 Cube.boundary { j // j \u2260 i\u271d }\n[PROOFSTEP]\nexact \u27e8i, iH\u27e9\n[GOAL]\ncase refine'_1.intro.hx\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni\u271d : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt : \u2191I \u00d7 \u2191I\ny : { j // j \u2260 i\u271d } \u2192 \u2191I\ni : { j // j \u2260 i\u271d }\niH : y i = 0 \u2228 y i = 1\n\u22a2 \u2191(Cube.insertAt i\u271d) (t.snd, y) \u2208 Cube.boundary N\n[PROOFSTEP]\napply Cube.insertAt_boundary\n[GOAL]\ncase refine'_1.intro.hx.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni\u271d : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt : \u2191I \u00d7 \u2191I\ny : { j // j \u2260 i\u271d } \u2192 \u2191I\ni : { j // j \u2260 i\u271d }\niH : y i = 0 \u2228 y i = 1\n\u22a2 (t.snd = 0 \u2228 t.snd = 1) \u2228 y \u2208 Cube.boundary { j // j \u2260 i\u271d }\n[PROOFSTEP]\nright\n[GOAL]\ncase refine'_1.intro.hx.H.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni\u271d : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt : \u2191I \u00d7 \u2191I\ny : { j // j \u2260 i\u271d } \u2192 \u2191I\ni : { j // j \u2260 i\u271d }\niH : y i = 0 \u2228 y i = 1\n\u22a2 y \u2208 Cube.boundary { j // j \u2260 i\u271d }\n[PROOFSTEP]\nexact \u27e8i, iH\u27e9\n[GOAL]\ncase refine'_2\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 Continuous fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2200 (y : { j // j \u2260 i } \u2192 \u2191I), y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) }\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_3\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (x_1 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ : \u2200 (y : { j // j \u2260 i } \u2192 \u2191I), y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (0, x_1) =\n \u2191(toLoop i p).toContinuousMap x_1\ncase refine'_4\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (x_1 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ : \u2200 (y : { j // j \u2260 i } \u2192 \u2191I), y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x_1) =\n \u2191(toLoop i q).toContinuousMap x_1\ncase refine'_5\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (t x_1 : \u2191I),\n x_1 \u2208 {0, 1} \u2192\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left := ?refine'_3, map_one_left := ?refine'_4 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i p).toContinuousMap x_1 \u2227\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left := ?refine'_3, map_one_left := ?refine'_4 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i q).toContinuousMap x_1\n[PROOFSTEP]\niterate 2 intro; ext; erw [homotopyTo_apply, toLoop_apply]; swap\n[GOAL]\ncase refine'_3\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (x_1 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ : \u2200 (y : { j // j \u2260 i } \u2192 \u2191I), y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (0, x_1) =\n \u2191(toLoop i p).toContinuousMap x_1\ncase refine'_4\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (x_1 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ : \u2200 (y : { j // j \u2260 i } \u2192 \u2191I), y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x_1) =\n \u2191(toLoop i q).toContinuousMap x_1\ncase refine'_5\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (t x_1 : \u2191I),\n x_1 \u2208 {0, 1} \u2192\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left := ?refine'_3, map_one_left := ?refine'_4 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i p).toContinuousMap x_1 \u2227\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left := ?refine'_3, map_one_left := ?refine'_4 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i q).toContinuousMap x_1\n[PROOFSTEP]\nintro\n[GOAL]\ncase refine'_3\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nx\u271d : \u2191I\n\u22a2 ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ : \u2200 (y : { j // j \u2260 i } \u2192 \u2191I), y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (0, x\u271d) =\n \u2191(toLoop i p).toContinuousMap x\u271d\ncase refine'_4\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (x_1 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ : \u2200 (y : { j // j \u2260 i } \u2192 \u2191I), y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x_1) =\n \u2191(toLoop i q).toContinuousMap x_1\ncase refine'_5\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (t x_1 : \u2191I),\n x_1 \u2208 {0, 1} \u2192\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left := ?refine'_4 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i p).toContinuousMap x_1 \u2227\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left := ?refine'_4 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i q).toContinuousMap x_1\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_3.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nx\u271d : \u2191I\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ : \u2200 (y : { j // j \u2260 i } \u2192 \u2191I), y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (0, x\u271d))\n y\u271d =\n \u2191(\u2191(toLoop i p).toContinuousMap x\u271d) y\u271d\ncase refine'_4\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (x_1 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ : \u2200 (y : { j // j \u2260 i } \u2192 \u2191I), y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x_1) =\n \u2191(toLoop i q).toContinuousMap x_1\ncase refine'_5\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (t x_1 : \u2191I),\n x_1 \u2208 {0, 1} \u2192\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left := ?refine'_4 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i p).toContinuousMap x_1 \u2227\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left := ?refine'_4 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i q).toContinuousMap x_1\n[PROOFSTEP]\nerw [homotopyTo_apply, toLoop_apply]\n[GOAL]\ncase refine'_3.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nx\u271d : \u2191I\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191H ((0, x\u271d).fst, \u2191(Cube.insertAt i) ((0, x\u271d).snd, y\u271d)) = \u2191p (\u2191(Cube.insertAt i) (x\u271d, y\u271d))\ncase refine'_4\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (x_1 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ : \u2200 (y : { j // j \u2260 i } \u2192 \u2191I), y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x_1) =\n \u2191(toLoop i q).toContinuousMap x_1\ncase refine'_5\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (t x_1 : \u2191I),\n x_1 \u2208 {0, 1} \u2192\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left := ?refine'_4 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i p).toContinuousMap x_1 \u2227\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left := ?refine'_4 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i q).toContinuousMap x_1\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine'_4\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (x_1 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ : \u2200 (y : { j // j \u2260 i } \u2192 \u2191I), y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x_1) =\n \u2191(toLoop i q).toContinuousMap x_1\ncase refine'_3.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nx\u271d : \u2191I\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191H ((0, x\u271d).fst, \u2191(Cube.insertAt i) ((0, x\u271d).snd, y\u271d)) = \u2191p (\u2191(Cube.insertAt i) (x\u271d, y\u271d))\ncase refine'_5\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (t x_1 : \u2191I),\n x_1 \u2208 {0, 1} \u2192\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left := ?refine'_4 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i p).toContinuousMap x_1 \u2227\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left := ?refine'_4 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i q).toContinuousMap x_1\n[PROOFSTEP]\nintro\n[GOAL]\ncase refine'_4\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nx\u271d : \u2191I\n\u22a2 ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ : \u2200 (y : { j // j \u2260 i } \u2192 \u2191I), y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x\u271d) =\n \u2191(toLoop i q).toContinuousMap x\u271d\ncase refine'_3.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nx\u271d : \u2191I\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191H ((0, x\u271d).fst, \u2191(Cube.insertAt i) ((0, x\u271d).snd, y\u271d)) = \u2191p (\u2191(Cube.insertAt i) (x\u271d, y\u271d))\ncase refine'_5\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (t x_1 : \u2191I),\n x_1 \u2208 {0, 1} \u2192\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (1, x_3) =\n \u2191(toLoop i q).toContinuousMap x_3) }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i p).toContinuousMap x_1 \u2227\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (1, x_3) =\n \u2191(toLoop i q).toContinuousMap x_3) }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i q).toContinuousMap x_1\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_4.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nx\u271d : \u2191I\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ : \u2200 (y : { j // j \u2260 i } \u2192 \u2191I), y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x\u271d))\n y\u271d =\n \u2191(\u2191(toLoop i q).toContinuousMap x\u271d) y\u271d\ncase refine'_3.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nx\u271d : \u2191I\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191H ((0, x\u271d).fst, \u2191(Cube.insertAt i) ((0, x\u271d).snd, y\u271d)) = \u2191p (\u2191(Cube.insertAt i) (x\u271d, y\u271d))\ncase refine'_5\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (t x_1 : \u2191I),\n x_1 \u2208 {0, 1} \u2192\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (1, x_3) =\n \u2191(toLoop i q).toContinuousMap x_3) }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i p).toContinuousMap x_1 \u2227\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (1, x_3) =\n \u2191(toLoop i q).toContinuousMap x_3) }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i q).toContinuousMap x_1\n[PROOFSTEP]\nerw [homotopyTo_apply, toLoop_apply]\n[GOAL]\ncase refine'_4.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nx\u271d : \u2191I\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191H ((1, x\u271d).fst, \u2191(Cube.insertAt i) ((1, x\u271d).snd, y\u271d)) = \u2191q (\u2191(Cube.insertAt i) (x\u271d, y\u271d))\ncase refine'_3.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nx\u271d : \u2191I\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191H ((0, x\u271d).fst, \u2191(Cube.insertAt i) ((0, x\u271d).snd, y\u271d)) = \u2191p (\u2191(Cube.insertAt i) (x\u271d, y\u271d))\ncase refine'_5\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (t x_1 : \u2191I),\n x_1 \u2208 {0, 1} \u2192\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (1, x_3) =\n \u2191(toLoop i q).toContinuousMap x_3) }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i p).toContinuousMap x_1 \u2227\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property := (_ : \u2191(homotopyTo i H) t \u2208 \u03a9^ { j // j \u2260 i } X x) })\n (1, x_3) =\n \u2191(toLoop i q).toContinuousMap x_3) }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i q).toContinuousMap x_1\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine'_3.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nx\u271d : \u2191I\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191H ((0, x\u271d).fst, \u2191(Cube.insertAt i) ((0, x\u271d).snd, y\u271d)) = \u2191p (\u2191(Cube.insertAt i) (x\u271d, y\u271d))\n[PROOFSTEP]\napply H.apply_zero\n[GOAL]\ncase refine'_4.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nx\u271d : \u2191I\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191H ((1, x\u271d).fst, \u2191(Cube.insertAt i) ((1, x\u271d).snd, y\u271d)) = \u2191q (\u2191(Cube.insertAt i) (x\u271d, y\u271d))\n[PROOFSTEP]\napply H.apply_one\n[GOAL]\ncase refine'_5\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\n\u22a2 \u2200 (t x_1 : \u2191I),\n x_1 \u2208 {0, 1} \u2192\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x_3) =\n \u2191(toLoop i q).toContinuousMap x_3) }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i p).toContinuousMap x_1 \u2227\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (0, x_3) =\n \u2191(toLoop i p).toContinuousMap x_3),\n map_one_left :=\n (_ :\n \u2200 (x_3 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x_3) =\n \u2191(toLoop i q).toContinuousMap x_3) }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191(toLoop i q).toContinuousMap x_1\n[PROOFSTEP]\nintro t y yH\n[GOAL]\ncase refine'_5\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_2 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (0, x_2) =\n \u2191(toLoop i p).toContinuousMap x_2),\n map_one_left :=\n (_ :\n \u2200 (x_2 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x_2) =\n \u2191(toLoop i q).toContinuousMap x_2) }.toContinuousMap\n (t, x_1))\n y =\n \u2191(toLoop i p).toContinuousMap y \u2227\n \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_2 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (0, x_2) =\n \u2191(toLoop i p).toContinuousMap x_2),\n map_one_left :=\n (_ :\n \u2200 (x_2 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x_2) =\n \u2191(toLoop i q).toContinuousMap x_2) }.toContinuousMap\n (t, x_1))\n y =\n \u2191(toLoop i q).toContinuousMap y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase refine'_5.left\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_2 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (0, x_2) =\n \u2191(toLoop i p).toContinuousMap x_2),\n map_one_left :=\n (_ :\n \u2200 (x_2 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x_2) =\n \u2191(toLoop i q).toContinuousMap x_2) }.toContinuousMap\n (t, x_1))\n y =\n \u2191(toLoop i p).toContinuousMap y\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_5.right\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_2 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (0, x_2) =\n \u2191(toLoop i p).toContinuousMap x_2),\n map_one_left :=\n (_ :\n \u2200 (x_2 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x_2) =\n \u2191(toLoop i q).toContinuousMap x_2) }.toContinuousMap\n (t, x_1))\n y =\n \u2191(toLoop i q).toContinuousMap y\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_5.left.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(\u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_2 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (0, x_2) =\n \u2191(toLoop i p).toContinuousMap x_2),\n map_one_left :=\n (_ :\n \u2200 (x_2 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x_2) =\n \u2191(toLoop i q).toContinuousMap x_2) }.toContinuousMap\n (t, x_1))\n y)\n y\u271d =\n \u2191(\u2191(toLoop i p).toContinuousMap y) y\u271d\n[PROOFSTEP]\nerw [homotopyTo_apply]\n[GOAL]\ncase refine'_5.right.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(\u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) },\n map_zero_left :=\n (_ :\n \u2200 (x_2 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (0, x_2) =\n \u2191(toLoop i p).toContinuousMap x_2),\n map_one_left :=\n (_ :\n \u2200 (x_2 : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun t =>\n { val := \u2191(homotopyTo i H) t,\n property :=\n (_ :\n \u2200 (y : { j // j \u2260 i } \u2192 \u2191I),\n y \u2208 Cube.boundary { j // j \u2260 i } \u2192 \u2191(\u2191(homotopyTo i H) t) y = x) })\n (1, x_2) =\n \u2191(toLoop i q).toContinuousMap x_2) }.toContinuousMap\n (t, x_1))\n y)\n y\u271d =\n \u2191(\u2191(toLoop i q).toContinuousMap y) y\u271d\n[PROOFSTEP]\nerw [homotopyTo_apply]\n[GOAL]\ncase refine'_5.left.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191H ((t, y).fst, \u2191(Cube.insertAt i) ((t, y).snd, y\u271d)) = \u2191(\u2191(toLoop i p).toContinuousMap y) y\u271d\ncase refine'_5.right.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191H ((t, y).fst, \u2191(Cube.insertAt i) ((t, y).snd, y\u271d)) = \u2191(\u2191(toLoop i q).toContinuousMap y) y\u271d\n[PROOFSTEP]\napply H.eq_fst\n[GOAL]\ncase refine'_5.left.H.hx\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(Cube.insertAt i) ((t, y).snd, y\u271d) \u2208 Cube.boundary N\ncase refine'_5.right.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191H ((t, y).fst, \u2191(Cube.insertAt i) ((t, y).snd, y\u271d)) = \u2191(\u2191(toLoop i q).toContinuousMap y) y\u271d\n[PROOFSTEP]\non_goal 2 => apply H.eq_snd\n[GOAL]\ncase refine'_5.left.H.hx\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(Cube.insertAt i) ((t, y).snd, y\u271d) \u2208 Cube.boundary N\ncase refine'_5.right.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191H ((t, y).fst, \u2191(Cube.insertAt i) ((t, y).snd, y\u271d)) = \u2191(\u2191(toLoop i q).toContinuousMap y) y\u271d\n[PROOFSTEP]\non_goal 2 => apply H.eq_snd\n[GOAL]\ncase refine'_5.right.H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191H ((t, y).fst, \u2191(Cube.insertAt i) ((t, y).snd, y\u271d)) = \u2191(\u2191(toLoop i q).toContinuousMap y) y\u271d\n[PROOFSTEP]\napply H.eq_snd\n[GOAL]\ncase refine'_5.left.H.hx\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(Cube.insertAt i) ((t, y).snd, y\u271d) \u2208 Cube.boundary N\ncase refine'_5.right.H.hx\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(Cube.insertAt i) ((t, y).snd, y\u271d) \u2208 Cube.boundary N\n[PROOFSTEP]\nall_goals use i; rw [funSplitAt_symm_apply, dif_pos rfl]; exact yH\n[GOAL]\ncase refine'_5.left.H.hx\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(Cube.insertAt i) ((t, y).snd, y\u271d) \u2208 Cube.boundary N\n[PROOFSTEP]\nuse i\n[GOAL]\ncase h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(Cube.insertAt i) ((t, y).snd, y\u271d) i = 0 \u2228 \u2191(Cube.insertAt i) ((t, y).snd, y\u271d) i = 1\n[PROOFSTEP]\nrw [funSplitAt_symm_apply, dif_pos rfl]\n[GOAL]\ncase h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 ((t, y).snd, y\u271d).fst = 0 \u2228 ((t, y).snd, y\u271d).fst = 1\n[PROOFSTEP]\nexact yH\n[GOAL]\ncase refine'_5.right.H.hx\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(Cube.insertAt i) ((t, y).snd, y\u271d) \u2208 Cube.boundary N\n[PROOFSTEP]\nuse i\n[GOAL]\ncase h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 \u2191(Cube.insertAt i) ((t, y).snd, y\u271d) i = 0 \u2228 \u2191(Cube.insertAt i) ((t, y).snd, y\u271d) i = 1\n[PROOFSTEP]\nrw [funSplitAt_symm_apply, dif_pos rfl]\n[GOAL]\ncase h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191p) (\u2191q) (Cube.boundary N)\nt y : \u2191I\nyH : y \u2208 {0, 1}\ny\u271d : { j // j \u2260 i } \u2192 \u2191I\n\u22a2 ((t, y).snd, y\u271d).fst = 0 \u2228 ((t, y).snd, y\u271d).fst = 1\n[PROOFSTEP]\nexact yH\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\n\u22a2 Continuous Subtype.val\n[PROOFSTEP]\ncontinuity\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\n\u22a2 Path.Homotopic (toLoop i p) (toLoop i q) \u2192 Homotopic p q\n[PROOFSTEP]\nrefine' Nonempty.map fun H => \u27e8\u27e8homotopyFrom i H, _, _\u27e9, _\u27e9\n[GOAL]\ncase refine'_1\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\n\u22a2 \u2200 (x_1 : N \u2192 \u2191I), ContinuousMap.toFun (homotopyFrom i H) (0, x_1) = \u2191\u2191p x_1\ncase refine'_2\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\n\u22a2 \u2200 (x_1 : N \u2192 \u2191I), ContinuousMap.toFun (homotopyFrom i H) (1, x_1) = \u2191\u2191q x_1\ncase refine'_3\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\n\u22a2 \u2200 (t : \u2191I) (x_1 : N \u2192 \u2191I),\n x_1 \u2208 Cube.boundary N \u2192\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1,\n map_one_left := ?refine'_2 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191\u2191p x_1 \u2227\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1,\n map_one_left := ?refine'_2 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191\u2191q x_1\n[PROOFSTEP]\npick_goal 3\n[GOAL]\ncase refine'_3\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\n\u22a2 \u2200 (t : \u2191I) (x_1 : N \u2192 \u2191I),\n x_1 \u2208 Cube.boundary N \u2192\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1,\n map_one_left := ?refine'_2 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191\u2191p x_1 \u2227\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1,\n map_one_left := ?refine'_2 }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191\u2191q x_1\n[PROOFSTEP]\nrintro t y \u27e8j, jH\u27e9\n[GOAL]\ncase refine'_3.intro\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nt : \u2191I\ny : N \u2192 \u2191I\nj : N\njH : y j = 0 \u2228 y j = 1\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1,\n map_one_left := ?refine'_2 }.toContinuousMap\n (t, x_1))\n y =\n \u2191\u2191p y \u2227\n \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n { toContinuousMap := homotopyFrom i H, map_zero_left := ?refine'_1,\n map_one_left := ?refine'_2 }.toContinuousMap\n (t, x_1))\n y =\n \u2191\u2191q y\n[PROOFSTEP]\nerw [homotopyFrom_apply]\n[GOAL]\ncase refine'_3.intro\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nt : \u2191I\ny : N \u2192 \u2191I\nj : N\njH : y j = 0 \u2228 y j = 1\n\u22a2 (\u2191\u2191(\u2191H ((t, y).fst, Prod.snd (t, y) i)) fun j => Prod.snd (t, y) \u2191j) = \u2191\u2191p y \u2227\n (\u2191\u2191(\u2191H ((t, y).fst, Prod.snd (t, y) i)) fun j => Prod.snd (t, y) \u2191j) = \u2191\u2191q y\ncase refine'_1\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\n\u22a2 \u2200 (x_1 : N \u2192 \u2191I), ContinuousMap.toFun (homotopyFrom i H) (0, x_1) = \u2191\u2191p x_1\ncase refine'_2\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\n\u22a2 \u2200 (x_1 : N \u2192 \u2191I), ContinuousMap.toFun (homotopyFrom i H) (1, x_1) = \u2191\u2191q x_1\n[PROOFSTEP]\nobtain rfl | h := eq_or_ne j i\n[GOAL]\ncase refine'_3.intro.inl\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\np q : \u2191(\u03a9^ N X x)\nt : \u2191I\ny : N \u2192 \u2191I\nj : N\njH : y j = 0 \u2228 y j = 1\nH : Path.Homotopy (toLoop j p) (toLoop j q)\n\u22a2 (\u2191\u2191(\u2191H ((t, y).fst, Prod.snd (t, y) j)) fun j_1 => Prod.snd (t, y) \u2191j_1) = \u2191\u2191p y \u2227\n (\u2191\u2191(\u2191H ((t, y).fst, Prod.snd (t, y) j)) fun j_1 => Prod.snd (t, y) \u2191j_1) = \u2191\u2191q y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase refine'_3.intro.inl.left\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\np q : \u2191(\u03a9^ N X x)\nt : \u2191I\ny : N \u2192 \u2191I\nj : N\njH : y j = 0 \u2228 y j = 1\nH : Path.Homotopy (toLoop j p) (toLoop j q)\n\u22a2 (\u2191\u2191(\u2191H ((t, y).fst, Prod.snd (t, y) j)) fun j_1 => Prod.snd (t, y) \u2191j_1) = \u2191\u2191p y\n[PROOFSTEP]\nrw [H.eq_fst]\n[GOAL]\ncase refine'_3.intro.inl.left\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\np q : \u2191(\u03a9^ N X x)\nt : \u2191I\ny : N \u2192 \u2191I\nj : N\njH : y j = 0 \u2228 y j = 1\nH : Path.Homotopy (toLoop j p) (toLoop j q)\n\u22a2 (\u2191\u2191(\u2191(toLoop j p).toContinuousMap (Prod.snd (t, y) j)) fun j_1 => Prod.snd (t, y) \u2191j_1) = \u2191\u2191p y\ncase refine'_3.intro.inl.left.hx\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\np q : \u2191(\u03a9^ N X x)\nt : \u2191I\ny : N \u2192 \u2191I\nj : N\njH : y j = 0 \u2228 y j = 1\nH : Path.Homotopy (toLoop j p) (toLoop j q)\n\u22a2 Prod.snd (t, y) j \u2208 {0, 1}\n[PROOFSTEP]\nexacts [congr_arg p ((Cube.splitAt j).left_inv _), jH]\n[GOAL]\ncase refine'_3.intro.inl.right\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\np q : \u2191(\u03a9^ N X x)\nt : \u2191I\ny : N \u2192 \u2191I\nj : N\njH : y j = 0 \u2228 y j = 1\nH : Path.Homotopy (toLoop j p) (toLoop j q)\n\u22a2 (\u2191\u2191(\u2191H ((t, y).fst, Prod.snd (t, y) j)) fun j_1 => Prod.snd (t, y) \u2191j_1) = \u2191\u2191q y\n[PROOFSTEP]\nrw [H.eq_snd]\n[GOAL]\ncase refine'_3.intro.inl.right\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\np q : \u2191(\u03a9^ N X x)\nt : \u2191I\ny : N \u2192 \u2191I\nj : N\njH : y j = 0 \u2228 y j = 1\nH : Path.Homotopy (toLoop j p) (toLoop j q)\n\u22a2 (\u2191\u2191(\u2191(toLoop j q).toContinuousMap (Prod.snd (t, y) j)) fun j_1 => Prod.snd (t, y) \u2191j_1) = \u2191\u2191q y\ncase refine'_3.intro.inl.right.hx\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\np q : \u2191(\u03a9^ N X x)\nt : \u2191I\ny : N \u2192 \u2191I\nj : N\njH : y j = 0 \u2228 y j = 1\nH : Path.Homotopy (toLoop j p) (toLoop j q)\n\u22a2 Prod.snd (t, y) j \u2208 {0, 1}\n[PROOFSTEP]\nexacts [congr_arg q ((Cube.splitAt j).left_inv _), jH]\n[GOAL]\ncase refine'_3.intro.inr\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nt : \u2191I\ny : N \u2192 \u2191I\nj : N\njH : y j = 0 \u2228 y j = 1\nh : j \u2260 i\n\u22a2 (\u2191\u2191(\u2191H ((t, y).fst, Prod.snd (t, y) i)) fun j => Prod.snd (t, y) \u2191j) = \u2191\u2191p y \u2227\n (\u2191\u2191(\u2191H ((t, y).fst, Prod.snd (t, y) i)) fun j => Prod.snd (t, y) \u2191j) = \u2191\u2191q y\n[PROOFSTEP]\nrw [p.2 _ \u27e8j, jH\u27e9, q.2 _ \u27e8j, jH\u27e9]\n[GOAL]\ncase refine'_3.intro.inr\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nt : \u2191I\ny : N \u2192 \u2191I\nj : N\njH : y j = 0 \u2228 y j = 1\nh : j \u2260 i\n\u22a2 (\u2191\u2191(\u2191H ((t, y).fst, Prod.snd (t, y) i)) fun j => Prod.snd (t, y) \u2191j) = x \u2227\n (\u2191\u2191(\u2191H ((t, y).fst, Prod.snd (t, y) i)) fun j => Prod.snd (t, y) \u2191j) = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase refine'_3.intro.inr.left\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nt : \u2191I\ny : N \u2192 \u2191I\nj : N\njH : y j = 0 \u2228 y j = 1\nh : j \u2260 i\n\u22a2 (\u2191\u2191(\u2191H ((t, y).fst, Prod.snd (t, y) i)) fun j => Prod.snd (t, y) \u2191j) = x\n[PROOFSTEP]\napply boundary\n[GOAL]\ncase refine'_3.intro.inr.left.a\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nt : \u2191I\ny : N \u2192 \u2191I\nj : N\njH : y j = 0 \u2228 y j = 1\nh : j \u2260 i\n\u22a2 (fun j => Prod.snd (t, y) \u2191j) \u2208 Cube.boundary { j // \u00acj = i }\n[PROOFSTEP]\nexact\n \u27e8\u27e8j, h\u27e9, jH\u27e9\n /- porting note: the following is indented two spaces more than it should be due to\n strange behavior of `erw` -/\n[GOAL]\ncase refine'_3.intro.inr.right\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nt : \u2191I\ny : N \u2192 \u2191I\nj : N\njH : y j = 0 \u2228 y j = 1\nh : j \u2260 i\n\u22a2 (\u2191\u2191(\u2191H ((t, y).fst, Prod.snd (t, y) i)) fun j => Prod.snd (t, y) \u2191j) = x\n[PROOFSTEP]\napply boundary\n[GOAL]\ncase refine'_3.intro.inr.right.a\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nt : \u2191I\ny : N \u2192 \u2191I\nj : N\njH : y j = 0 \u2228 y j = 1\nh : j \u2260 i\n\u22a2 (fun j => Prod.snd (t, y) \u2191j) \u2208 Cube.boundary { j // \u00acj = i }\n[PROOFSTEP]\nexact\n \u27e8\u27e8j, h\u27e9, jH\u27e9\n /- porting note: the following is indented two spaces more than it should be due to\n strange behavior of `erw` -/\n[GOAL]\ncase refine'_1\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\n\u22a2 \u2200 (x_1 : N \u2192 \u2191I), ContinuousMap.toFun (homotopyFrom i H) (0, x_1) = \u2191\u2191p x_1\ncase refine'_2\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\n\u22a2 \u2200 (x_1 : N \u2192 \u2191I), ContinuousMap.toFun (homotopyFrom i H) (1, x_1) = \u2191\u2191q x_1\n[PROOFSTEP]\nall_goals\n intro\n apply (homotopyFrom_apply _ _ _).trans\n first\n | rw [H.apply_zero]\n | rw [H.apply_one]\n first\n | apply congr_arg p\n | apply congr_arg q\n apply (Cube.splitAt i).left_inv\n[GOAL]\ncase refine'_1\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\n\u22a2 \u2200 (x_1 : N \u2192 \u2191I), ContinuousMap.toFun (homotopyFrom i H) (0, x_1) = \u2191\u2191p x_1\n[PROOFSTEP]\nintro\n[GOAL]\ncase refine'_1\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nx\u271d : N \u2192 \u2191I\n\u22a2 ContinuousMap.toFun (homotopyFrom i H) (0, x\u271d) = \u2191\u2191p x\u271d\n[PROOFSTEP]\napply (homotopyFrom_apply _ _ _).trans\n[GOAL]\ncase refine'_1\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nx\u271d : N \u2192 \u2191I\n\u22a2 (\u2191\u2191(\u2191H ((0, x\u271d).fst, Prod.snd (0, x\u271d) i)) fun j => Prod.snd (0, x\u271d) \u2191j) = \u2191\u2191p x\u271d\n[PROOFSTEP]\nfirst\n| rw [H.apply_zero]\n| rw [H.apply_one]\n[GOAL]\ncase refine'_1\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nx\u271d : N \u2192 \u2191I\n\u22a2 (\u2191\u2191(\u2191H ((0, x\u271d).fst, Prod.snd (0, x\u271d) i)) fun j => Prod.snd (0, x\u271d) \u2191j) = \u2191\u2191p x\u271d\n[PROOFSTEP]\nrw [H.apply_zero]\n[GOAL]\ncase refine'_1\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nx\u271d : N \u2192 \u2191I\n\u22a2 (\u2191\u2191(\u2191(toLoop i p).toContinuousMap (Prod.snd (0, x\u271d) i)) fun j => Prod.snd (0, x\u271d) \u2191j) = \u2191\u2191p x\u271d\n[PROOFSTEP]\nfirst\n| apply congr_arg p\n| apply congr_arg q\n[GOAL]\ncase refine'_1\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nx\u271d : N \u2192 \u2191I\n\u22a2 (\u2191\u2191(\u2191(toLoop i p).toContinuousMap (Prod.snd (0, x\u271d) i)) fun j => Prod.snd (0, x\u271d) \u2191j) = \u2191\u2191p x\u271d\n[PROOFSTEP]\napply congr_arg p\n[GOAL]\ncase refine'_1\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nx\u271d : N \u2192 \u2191I\n\u22a2 \u2191(Homeomorph.toContinuousMap (Cube.insertAt i)) (Prod.snd (0, x\u271d) i, fun j => Prod.snd (0, x\u271d) \u2191j) = x\u271d\n[PROOFSTEP]\napply (Cube.splitAt i).left_inv\n[GOAL]\ncase refine'_2\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\n\u22a2 \u2200 (x_1 : N \u2192 \u2191I), ContinuousMap.toFun (homotopyFrom i H) (1, x_1) = \u2191\u2191q x_1\n[PROOFSTEP]\nintro\n[GOAL]\ncase refine'_2\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nx\u271d : N \u2192 \u2191I\n\u22a2 ContinuousMap.toFun (homotopyFrom i H) (1, x\u271d) = \u2191\u2191q x\u271d\n[PROOFSTEP]\napply (homotopyFrom_apply _ _ _).trans\n[GOAL]\ncase refine'_2\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nx\u271d : N \u2192 \u2191I\n\u22a2 (\u2191\u2191(\u2191H ((1, x\u271d).fst, Prod.snd (1, x\u271d) i)) fun j => Prod.snd (1, x\u271d) \u2191j) = \u2191\u2191q x\u271d\n[PROOFSTEP]\nfirst\n| rw [H.apply_zero]\n| rw [H.apply_one]\n[GOAL]\ncase refine'_2\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nx\u271d : N \u2192 \u2191I\n\u22a2 (\u2191\u2191(\u2191H ((1, x\u271d).fst, Prod.snd (1, x\u271d) i)) fun j => Prod.snd (1, x\u271d) \u2191j) = \u2191\u2191q x\u271d\n[PROOFSTEP]\nrw [H.apply_zero]\n[GOAL]\ncase refine'_2\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nx\u271d : N \u2192 \u2191I\n\u22a2 (\u2191\u2191(\u2191H ((1, x\u271d).fst, Prod.snd (1, x\u271d) i)) fun j => Prod.snd (1, x\u271d) \u2191j) = \u2191\u2191q x\u271d\n[PROOFSTEP]\nrw [H.apply_one]\n[GOAL]\ncase refine'_2\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nx\u271d : N \u2192 \u2191I\n\u22a2 (\u2191\u2191(\u2191(toLoop i q).toContinuousMap (Prod.snd (1, x\u271d) i)) fun j => Prod.snd (1, x\u271d) \u2191j) = \u2191\u2191q x\u271d\n[PROOFSTEP]\nfirst\n| apply congr_arg p\n| apply congr_arg q\n[GOAL]\ncase refine'_2\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nx\u271d : N \u2192 \u2191I\n\u22a2 (\u2191\u2191(\u2191(toLoop i q).toContinuousMap (Prod.snd (1, x\u271d) i)) fun j => Prod.snd (1, x\u271d) \u2191j) = \u2191\u2191q x\u271d\n[PROOFSTEP]\napply congr_arg p\n[GOAL]\ncase refine'_2\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nx\u271d : N \u2192 \u2191I\n\u22a2 (\u2191\u2191(\u2191(toLoop i q).toContinuousMap (Prod.snd (1, x\u271d) i)) fun j => Prod.snd (1, x\u271d) \u2191j) = \u2191\u2191q x\u271d\n[PROOFSTEP]\napply congr_arg q\n[GOAL]\ncase refine'_2\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\np q : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (toLoop i p) (toLoop i q)\nx\u271d : N \u2192 \u2191I\n\u22a2 \u2191(Homeomorph.toContinuousMap (Cube.insertAt i)) (Prod.snd (1, x\u271d) i, fun j => Prod.snd (1, x\u271d) \u2191j) = x\u271d\n[PROOFSTEP]\napply (Cube.splitAt i).left_inv\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf g : \u2191(\u03a9^ N X x)\n\u22a2 (fun t =>\n if \u2191(t i) \u2264 1 / 2 then \u2191f (Function.update t i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(t i))))\n else \u2191g (Function.update t i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(t i) - 1)))) =\n \u2191(fromLoop i (Path.trans (toLoop i f) (toLoop i g)))\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf g : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\n\u22a2 (if \u2191(x\u271d i) \u2264 1 / 2 then \u2191f (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i))))\n else \u2191g (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i) - 1)))) =\n \u2191(fromLoop i (Path.trans (toLoop i f) (toLoop i g))) x\u271d\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf g : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\n\u22a2 \u2191(fromLoop i (Path.trans (toLoop i f) (toLoop i g))) x\u271d =\n if \u2191(x\u271d i) \u2264 1 / 2 then \u2191f (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i))))\n else \u2191g (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i) - 1)))\n[PROOFSTEP]\ndsimp only [Path.trans, fromLoop, Path.coe_mk_mk, Function.comp_apply, mk_apply, ContinuousMap.comp_apply,\n toContinuousMap_apply, funSplitAt_apply, ContinuousMap.uncurry_apply, ContinuousMap.coe_mk,\n Function.uncurry_apply_pair]\n[GOAL]\ncase h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf g : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\n\u22a2 (\u2191\u2191(if \u2191(x\u271d i) \u2264 1 / 2 then Path.extend (toLoop i f) (2 * \u2191(x\u271d i)) else Path.extend (toLoop i g) (2 * \u2191(x\u271d i) - 1))\n fun j => x\u271d \u2191j) =\n if \u2191(x\u271d i) \u2264 1 / 2 then \u2191f (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i))))\n else \u2191g (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i) - 1)))\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf g : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\nh\u271d : \u2191(x\u271d i) \u2264 1 / 2\n\u22a2 (\u2191\u2191(Path.extend (toLoop i f) (2 * \u2191(x\u271d i))) fun j => x\u271d \u2191j) =\n \u2191f (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i))))\ncase neg\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf g : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\nh\u271d : \u00ac\u2191(x\u271d i) \u2264 1 / 2\n\u22a2 (\u2191\u2191(Path.extend (toLoop i g) (2 * \u2191(x\u271d i) - 1)) fun j => x\u271d \u2191j) =\n \u2191g (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i) - 1)))\n[PROOFSTEP]\nchange f _ = _\n[GOAL]\ncase pos\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf g : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\nh\u271d : \u2191(x\u271d i) \u2264 1 / 2\n\u22a2 \u2191f\n (\u2191(Homeomorph.toContinuousMap (Cube.insertAt i))\n (Set.projIcc 0 1 Path.extend.proof_1 (2 * \u2191(x\u271d i)), fun j => x\u271d \u2191j)) =\n \u2191f (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i))))\ncase neg\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf g : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\nh\u271d : \u00ac\u2191(x\u271d i) \u2264 1 / 2\n\u22a2 (\u2191\u2191(Path.extend (toLoop i g) (2 * \u2191(x\u271d i) - 1)) fun j => x\u271d \u2191j) =\n \u2191g (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i) - 1)))\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf g : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\nh\u271d : \u00ac\u2191(x\u271d i) \u2264 1 / 2\n\u22a2 (\u2191\u2191(Path.extend (toLoop i g) (2 * \u2191(x\u271d i) - 1)) fun j => x\u271d \u2191j) =\n \u2191g (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i) - 1)))\ncase pos\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf g : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\nh\u271d : \u2191(x\u271d i) \u2264 1 / 2\n\u22a2 \u2191f\n (\u2191(Homeomorph.toContinuousMap (Cube.insertAt i))\n (Set.projIcc 0 1 Path.extend.proof_1 (2 * \u2191(x\u271d i)), fun j => x\u271d \u2191j)) =\n \u2191f (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i))))\n[PROOFSTEP]\nchange g _ = _\n[GOAL]\ncase neg\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf g : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\nh\u271d : \u00ac\u2191(x\u271d i) \u2264 1 / 2\n\u22a2 \u2191g\n (\u2191(Homeomorph.toContinuousMap (Cube.insertAt i))\n (Set.projIcc 0 1 Path.extend.proof_1 (2 * \u2191(x\u271d i) - 1), fun j => x\u271d \u2191j)) =\n \u2191g (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i) - 1)))\ncase pos\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf g : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\nh\u271d : \u2191(x\u271d i) \u2264 1 / 2\n\u22a2 \u2191f\n (\u2191(Homeomorph.toContinuousMap (Cube.insertAt i))\n (Set.projIcc 0 1 Path.extend.proof_1 (2 * \u2191(x\u271d i)), fun j => x\u271d \u2191j)) =\n \u2191f (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i))))\n[PROOFSTEP]\nall_goals congr 1\n[GOAL]\ncase neg\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf g : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\nh\u271d : \u00ac\u2191(x\u271d i) \u2264 1 / 2\n\u22a2 \u2191g\n (\u2191(Homeomorph.toContinuousMap (Cube.insertAt i))\n (Set.projIcc 0 1 Path.extend.proof_1 (2 * \u2191(x\u271d i) - 1), fun j => x\u271d \u2191j)) =\n \u2191g (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i) - 1)))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase pos\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf g : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\nh\u271d : \u2191(x\u271d i) \u2264 1 / 2\n\u22a2 \u2191f\n (\u2191(Homeomorph.toContinuousMap (Cube.insertAt i))\n (Set.projIcc 0 1 Path.extend.proof_1 (2 * \u2191(x\u271d i)), fun j => x\u271d \u2191j)) =\n \u2191f (Function.update x\u271d i (Set.projIcc 0 1 (_ : 0 \u2264 1) (2 * \u2191(x\u271d i))))\n[PROOFSTEP]\ncongr 1\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf : \u2191(\u03a9^ N X x)\n\u22a2 (fun t => \u2191f fun j => if j = i then \u03c3 (t i) else t j) = \u2191(fromLoop i (Path.symm (toLoop i f)))\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\n\u22a2 (\u2191f fun j => if j = i then \u03c3 (x\u271d i) else x\u271d j) = \u2191(fromLoop i (Path.symm (toLoop i f))) x\u271d\n[PROOFSTEP]\nchange _ = f _\n[GOAL]\ncase h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\n\u22a2 (\u2191f fun j => if j = i then \u03c3 (x\u271d i) else x\u271d j) =\n \u2191f\n (\u2191(Homeomorph.toContinuousMap (Cube.insertAt i))\n (\u03c3 (\u2191(Homeomorph.toContinuousMap (Cube.splitAt i)) x\u271d).fst,\n (\u2191(Homeomorph.toContinuousMap (Cube.splitAt i)) x\u271d).snd))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.h.e_6.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf : \u2191(\u03a9^ N X x)\nx\u271d : N \u2192 \u2191I\n\u22a2 (fun j => if j = i then \u03c3 (x\u271d i) else x\u271d j) =\n \u2191(Homeomorph.toContinuousMap (Cube.insertAt i))\n (\u03c3 (\u2191(Homeomorph.toContinuousMap (Cube.splitAt i)) x\u271d).fst,\n (\u2191(Homeomorph.toContinuousMap (Cube.splitAt i)) x\u271d).snd)\n[PROOFSTEP]\next1\n[GOAL]\ncase h.h.e_6.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\nf : \u2191(\u03a9^ N X x)\nx\u271d\u00b9 : N \u2192 \u2191I\nx\u271d : N\n\u22a2 (if x\u271d = i then \u03c3 (x\u271d\u00b9 i) else x\u271d\u00b9 x\u271d) =\n \u2191(Homeomorph.toContinuousMap (Cube.insertAt i))\n (\u03c3 (\u2191(Homeomorph.toContinuousMap (Cube.splitAt i)) x\u271d\u00b9).fst,\n (\u2191(Homeomorph.toContinuousMap (Cube.splitAt i)) x\u271d\u00b9).snd)\n x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\n\u22a2 transAt i (transAt j a b) (transAt j c d) = transAt j (transAt i a c) (transAt i b d)\n[PROOFSTEP]\next\n[GOAL]\ncase H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\n\u22a2 \u2191(transAt i (transAt j a b) (transAt j c d)) y\u271d = \u2191(transAt j (transAt i a c) (transAt i b d)) y\u271d\n[PROOFSTEP]\nsimp_rw [transAt, coe_copy, Function.update_apply, if_neg h, if_neg h.symm]\n[GOAL]\ncase H\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\n\u22a2 (if \u2191(y\u271d i) \u2264 1 / 2 then\n if \u2191(y\u271d j) \u2264 1 / 2 then\n \u2191a\n (Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i)))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j))))\n else\n \u2191b\n (Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i)))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1)))\n else\n if \u2191(y\u271d j) \u2264 1 / 2 then\n \u2191c\n (Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j))))\n else\n \u2191d\n (Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1)))) =\n if \u2191(y\u271d j) \u2264 1 / 2 then\n if \u2191(y\u271d i) \u2264 1 / 2 then\n \u2191a\n (Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j)))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i))))\n else\n \u2191c\n (Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j)))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1)))\n else\n if \u2191(y\u271d i) \u2264 1 / 2 then\n \u2191b\n (Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i))))\n else\n \u2191d\n (Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1)))\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u2191(y\u271d j) \u2264 1 / 2\n\u22a2 \u2191a\n (Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i)))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j)))) =\n \u2191a\n (Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j)))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i))))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase pos.h.e_6.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u2191(y\u271d j) \u2264 1 / 2\n\u22a2 Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i)))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j))) =\n Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j)))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i)))\n[PROOFSTEP]\next1\n[GOAL]\ncase pos.h.e_6.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u2191(y\u271d j) \u2264 1 / 2\nx\u271d : N\n\u22a2 Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i)))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j))) x\u271d =\n Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j)))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i))) x\u271d\n[PROOFSTEP]\nsimp only [Function.update, eq_rec_constant, dite_eq_ite]\n[GOAL]\ncase pos.h.e_6.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u2191(y\u271d j) \u2264 1 / 2\nx\u271d : N\n\u22a2 (if x\u271d = j then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j))\n else if x\u271d = i then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i)) else y\u271d x\u271d) =\n if x\u271d = i then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i))\n else if x\u271d = j then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j)) else y\u271d x\u271d\n[PROOFSTEP]\napply ite_ite_comm\n[GOAL]\ncase pos.h.e_6.h.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u2191(y\u271d j) \u2264 1 / 2\nx\u271d : N\n\u22a2 x\u271d = j \u2192 \u00acx\u271d = i\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase pos.h.e_6.h.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u2191(y\u271d i) \u2264 1 / 2\nx\u271d : N\nh : i \u2260 x\u271d\nh\u271d : \u2191(y\u271d x\u271d) \u2264 1 / 2\n\u22a2 \u00acx\u271d = i\n[PROOFSTEP]\nexact h.symm\n[GOAL]\ncase neg\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u00ac\u2191(y\u271d j) \u2264 1 / 2\n\u22a2 \u2191b\n (Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i)))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1))) =\n \u2191b\n (Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i))))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase neg.h.e_6.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u00ac\u2191(y\u271d j) \u2264 1 / 2\n\u22a2 Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i)))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1)) =\n Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i)))\n[PROOFSTEP]\next1\n[GOAL]\ncase neg.h.e_6.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u00ac\u2191(y\u271d j) \u2264 1 / 2\nx\u271d : N\n\u22a2 Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i)))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1)) x\u271d =\n Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i))) x\u271d\n[PROOFSTEP]\nsimp only [Function.update, eq_rec_constant, dite_eq_ite]\n[GOAL]\ncase neg.h.e_6.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u00ac\u2191(y\u271d j) \u2264 1 / 2\nx\u271d : N\n\u22a2 (if x\u271d = j then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1)\n else if x\u271d = i then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i)) else y\u271d x\u271d) =\n if x\u271d = i then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i))\n else if x\u271d = j then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1) else y\u271d x\u271d\n[PROOFSTEP]\napply ite_ite_comm\n[GOAL]\ncase neg.h.e_6.h.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u00ac\u2191(y\u271d j) \u2264 1 / 2\nx\u271d : N\n\u22a2 x\u271d = j \u2192 \u00acx\u271d = i\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase neg.h.e_6.h.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u2191(y\u271d i) \u2264 1 / 2\nx\u271d : N\nh : i \u2260 x\u271d\nh\u271d : \u00ac\u2191(y\u271d x\u271d) \u2264 1 / 2\n\u22a2 \u00acx\u271d = i\n[PROOFSTEP]\nexact h.symm\n[GOAL]\ncase pos\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u00ac\u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u2191(y\u271d j) \u2264 1 / 2\n\u22a2 \u2191c\n (Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j)))) =\n \u2191c\n (Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j)))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1)))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase pos.h.e_6.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u00ac\u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u2191(y\u271d j) \u2264 1 / 2\n\u22a2 Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j))) =\n Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j)))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1))\n[PROOFSTEP]\next1\n[GOAL]\ncase pos.h.e_6.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u00ac\u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u2191(y\u271d j) \u2264 1 / 2\nx\u271d : N\n\u22a2 Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j))) x\u271d =\n Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j)))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1)) x\u271d\n[PROOFSTEP]\nsimp only [Function.update, eq_rec_constant, dite_eq_ite]\n[GOAL]\ncase pos.h.e_6.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u00ac\u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u2191(y\u271d j) \u2264 1 / 2\nx\u271d : N\n\u22a2 (if x\u271d = j then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j))\n else if x\u271d = i then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1) else y\u271d x\u271d) =\n if x\u271d = i then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1)\n else if x\u271d = j then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j)) else y\u271d x\u271d\n[PROOFSTEP]\napply ite_ite_comm\n[GOAL]\ncase pos.h.e_6.h.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u00ac\u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u2191(y\u271d j) \u2264 1 / 2\nx\u271d : N\n\u22a2 x\u271d = j \u2192 \u00acx\u271d = i\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase pos.h.e_6.h.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u00ac\u2191(y\u271d i) \u2264 1 / 2\nx\u271d : N\nh : i \u2260 x\u271d\nh\u271d : \u2191(y\u271d x\u271d) \u2264 1 / 2\n\u22a2 \u00acx\u271d = i\n[PROOFSTEP]\nexact h.symm\n[GOAL]\ncase neg\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u00ac\u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u00ac\u2191(y\u271d j) \u2264 1 / 2\n\u22a2 \u2191d\n (Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1))) =\n \u2191d\n (Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1)))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase neg.h.e_6.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u00ac\u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u00ac\u2191(y\u271d j) \u2264 1 / 2\n\u22a2 Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1)) =\n Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1))\n[PROOFSTEP]\next1\n[GOAL]\ncase neg.h.e_6.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u00ac\u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u00ac\u2191(y\u271d j) \u2264 1 / 2\nx\u271d : N\n\u22a2 Function.update (Function.update y\u271d i (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1))) j\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1)) x\u271d =\n Function.update (Function.update y\u271d j (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1))) i\n (Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1)) x\u271d\n[PROOFSTEP]\nsimp only [Function.update, eq_rec_constant, dite_eq_ite]\n[GOAL]\ncase neg.h.e_6.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u00ac\u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u00ac\u2191(y\u271d j) \u2264 1 / 2\nx\u271d : N\n\u22a2 (if x\u271d = j then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1)\n else if x\u271d = i then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1) else y\u271d x\u271d) =\n if x\u271d = i then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d i) - 1)\n else if x\u271d = j then Set.projIcc 0 1 transAt.proof_2 (2 * \u2191(y\u271d j) - 1) else y\u271d x\u271d\n[PROOFSTEP]\napply ite_ite_comm\n[GOAL]\ncase neg.h.e_6.h.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i \u2260 j\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u00ac\u2191(y\u271d i) \u2264 1 / 2\nh\u271d : \u00ac\u2191(y\u271d j) \u2264 1 / 2\nx\u271d : N\n\u22a2 x\u271d = j \u2192 \u00acx\u271d = i\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase neg.h.e_6.h.h.h\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\na b c d : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\nh\u271d\u00b9 : \u00ac\u2191(y\u271d i) \u2264 1 / 2\nx\u271d : N\nh : i \u2260 x\u271d\nh\u271d : \u00ac\u2191(y\u271d x\u271d) \u2264 1 / 2\n\u22a2 \u00acx\u271d = i\n[PROOFSTEP]\nexact h.symm\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\n\u22a2 HomotopyGroup N X x \u2243 FundamentalGroup (\u2191(\u03a9^ { j // j \u2260 i } X x)) const\n[PROOFSTEP]\nrefine' Equiv.trans _ (CategoryTheory.Groupoid.isoEquivHom _ _).symm\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\n\u22a2 HomotopyGroup N X x \u2243 (const \u27f6 const)\n[PROOFSTEP]\napply Quotient.congr (loopHomeo i).toEquiv\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni : N\n\u22a2 \u2200 (a\u2081 a\u2082 : \u2191(\u03a9^ N X x)), Setoid.r a\u2081 a\u2082 \u2194 Setoid.r (\u2191(loopHomeo i).toEquiv a\u2081) (\u2191(loopHomeo i).toEquiv a\u2082)\n[PROOFSTEP]\nexact fun p q => \u27e8homotopicTo i, homotopicFrom i\u27e9\n[GOAL]\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx\u271d : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.533751\nx : X\ninst\u271d : IsEmpty N\nf : \u2191(\u03a9^ N X x)\n\u22a2 (fun y =>\n { val := ContinuousMap.const (N \u2192 \u2191I) y,\n property := (_ : \u2200 (x_1 : N \u2192 \u2191I), x_1 \u2208 Cube.boundary N \u2192 \u2191(ContinuousMap.const (N \u2192 \u2191I) y) x_1 = x) })\n ((fun f => \u2191f 0) f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase H\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx\u271d : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.533751\nx : X\ninst\u271d : IsEmpty N\nf : \u2191(\u03a9^ N X x)\ny\u271d : N \u2192 \u2191I\n\u22a2 \u2191((fun y =>\n { val := ContinuousMap.const (N \u2192 \u2191I) y,\n property := (_ : \u2200 (x_1 : N \u2192 \u2191I), x_1 \u2208 Cube.boundary N \u2192 \u2191(ContinuousMap.const (N \u2192 \u2191I) y) x_1 = x) })\n ((fun f => \u2191f 0) f))\n y\u271d =\n \u2191f y\u271d\n[PROOFSTEP]\nexact congr_arg f (Subsingleton.elim _ _)\n[GOAL]\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx\u271d : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.536140\nx : X\ninst\u271d : IsEmpty N\n\u22a2 \u2200 (a\u2081 a\u2082 : \u2191(\u03a9^ N X x)),\n Setoid.r a\u2081 a\u2082 \u2194 Setoid.r (\u2191(genLoopHomeoOfIsEmpty N x).toEquiv a\u2081) (\u2191(genLoopHomeoOfIsEmpty N x).toEquiv a\u2082)\n[PROOFSTEP]\nintros a\u2081 a\u2082\n[GOAL]\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx\u271d : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.536140\nx : X\ninst\u271d : IsEmpty N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\n\u22a2 Setoid.r a\u2081 a\u2082 \u2194 Setoid.r (\u2191(genLoopHomeoOfIsEmpty N x).toEquiv a\u2081) (\u2191(genLoopHomeoOfIsEmpty N x).toEquiv a\u2082)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx\u271d : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.536140\nx : X\ninst\u271d : IsEmpty N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\n\u22a2 Setoid.r a\u2081 a\u2082 \u2192 Setoid.r (\u2191(genLoopHomeoOfIsEmpty N x).toEquiv a\u2081) (\u2191(genLoopHomeoOfIsEmpty N x).toEquiv a\u2082)\n[PROOFSTEP]\nrintro \u27e8H\u27e9\n[GOAL]\ncase mpr\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx\u271d : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.536140\nx : X\ninst\u271d : IsEmpty N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\n\u22a2 Setoid.r (\u2191(genLoopHomeoOfIsEmpty N x).toEquiv a\u2081) (\u2191(genLoopHomeoOfIsEmpty N x).toEquiv a\u2082) \u2192 Setoid.r a\u2081 a\u2082\n[PROOFSTEP]\nrintro \u27e8H\u27e9\n[GOAL]\ncase mp.intro\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx\u271d : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.536140\nx : X\ninst\u271d : IsEmpty N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191a\u2081) (\u2191a\u2082) (Cube.boundary N)\n\u22a2 Setoid.r (\u2191(genLoopHomeoOfIsEmpty N x).toEquiv a\u2081) (\u2191(genLoopHomeoOfIsEmpty N x).toEquiv a\u2082)\ncase mpr.intro\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx\u271d : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.536140\nx : X\ninst\u271d : IsEmpty N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\nH : Path (\u2191(genLoopHomeoOfIsEmpty N x).toEquiv a\u2081) (\u2191(genLoopHomeoOfIsEmpty N x).toEquiv a\u2082)\n\u22a2 Setoid.r a\u2081 a\u2082\n[PROOFSTEP]\nexacts\n [\u27e8{ toFun := fun t => H \u27e8t, isEmptyElim\u27e9\n source' := (H.apply_zero _).trans (congr_arg a\u2081 <| Subsingleton.elim _ _)\n target' := (H.apply_one _).trans (congr_arg a\u2082 <| Subsingleton.elim _ _) }\u27e9,\n \u27e8{ toFun := fun t0 => H t0.fst\n map_zero_left := fun _ => H.source.trans (congr_arg a\u2081 <| Subsingleton.elim _ _)\n map_one_left := fun _ => H.target.trans (congr_arg a\u2082 <| Subsingleton.elim _ _)\n prop' := fun _ _ \u27e8i, _\u27e9 => isEmptyElim i }\u27e9]\n[GOAL]\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.636077\ninst\u271d : Unique N\np : \u2191(\u03a9^ N X x)\n\u22a2 Continuous fun t => \u2191p fun x => t\n[PROOFSTEP]\ncontinuity\n[GOAL]\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.636077\ninst\u271d : Unique N\np : \u03a9 X x\n\u22a2 Continuous fun c => \u2191p (c default)\n[PROOFSTEP]\ncontinuity\n[GOAL]\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.636077\ninst\u271d : Unique N\np : \u03a9 X x\n\u22a2 (ContinuousMap.mk fun c => \u2191p (c default)) \u2208 \u03a9^ N X x\n[PROOFSTEP]\nrintro y \u27e8i, iH | iH\u27e9\n[GOAL]\ncase intro.inl\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.636077\ninst\u271d : Unique N\np : \u03a9 X x\ny : N \u2192 \u2191I\ni : N\niH : y i = 0\n\u22a2 \u2191(ContinuousMap.mk fun c => \u2191p (c default)) y = x\n[PROOFSTEP]\ncases Unique.eq_default i\n[GOAL]\ncase intro.inr\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.636077\ninst\u271d : Unique N\np : \u03a9 X x\ny : N \u2192 \u2191I\ni : N\niH : y i = 1\n\u22a2 \u2191(ContinuousMap.mk fun c => \u2191p (c default)) y = x\n[PROOFSTEP]\ncases Unique.eq_default i\n[GOAL]\ncase intro.inl.refl\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.636077\ninst\u271d : Unique N\np : \u03a9 X x\ny : N \u2192 \u2191I\niH : y default = 0\n\u22a2 \u2191(ContinuousMap.mk fun c => \u2191p (c default)) y = x\n[PROOFSTEP]\napply (congr_arg p iH).trans\n[GOAL]\ncase intro.inr.refl\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.636077\ninst\u271d : Unique N\np : \u03a9 X x\ny : N \u2192 \u2191I\niH : y default = 1\n\u22a2 \u2191(ContinuousMap.mk fun c => \u2191p (c default)) y = x\n[PROOFSTEP]\napply (congr_arg p iH).trans\n[GOAL]\ncase intro.inl.refl\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.636077\ninst\u271d : Unique N\np : \u03a9 X x\ny : N \u2192 \u2191I\niH : y default = 0\n\u22a2 \u2191p 0 = x\ncase intro.inr.refl\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.636077\ninst\u271d : Unique N\np : \u03a9 X x\ny : N \u2192 \u2191I\niH : y default = 1\n\u22a2 \u2191p 1 = x\n[PROOFSTEP]\nexacts [p.source, p.target]\n[GOAL]\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.636077\ninst\u271d : Unique N\np : \u2191(\u03a9^ N X x)\n\u22a2 (fun p =>\n { val := ContinuousMap.mk fun c => \u2191p (c default),\n property := (_ : \u2200 (y : N \u2192 \u2191I), y \u2208 Cube.boundary N \u2192 \u2191(ContinuousMap.mk fun c => \u2191p (c default)) y = x) })\n ((fun p =>\n { toContinuousMap := ContinuousMap.mk fun t => \u2191p fun x => t, source' := (_ : (\u2191p fun x => 0) = x),\n target' := (_ : (\u2191p fun x => 1) = x) })\n p) =\n p\n[PROOFSTEP]\next y\n[GOAL]\ncase H\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.636077\ninst\u271d : Unique N\np : \u2191(\u03a9^ N X x)\ny : N \u2192 \u2191I\n\u22a2 \u2191((fun p =>\n { val := ContinuousMap.mk fun c => \u2191p (c default),\n property :=\n (_ : \u2200 (y : N \u2192 \u2191I), y \u2208 Cube.boundary N \u2192 \u2191(ContinuousMap.mk fun c => \u2191p (c default)) y = x) })\n ((fun p =>\n { toContinuousMap := ContinuousMap.mk fun t => \u2191p fun x => t, source' := (_ : (\u2191p fun x => 0) = x),\n target' := (_ : (\u2191p fun x => 1) = x) })\n p))\n y =\n \u2191p y\n[PROOFSTEP]\nexact congr_arg p (eq_const_of_unique y).symm\n[GOAL]\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.636077\ninst\u271d : Unique N\np : \u03a9 X x\n\u22a2 (fun p =>\n { toContinuousMap := ContinuousMap.mk fun t => \u2191p fun x => t, source' := (_ : (\u2191p fun x => 0) = x),\n target' := (_ : (\u2191p fun x => 1) = x) })\n ((fun p =>\n { val := ContinuousMap.mk fun c => \u2191p (c default),\n property := (_ : \u2200 (y : N \u2192 \u2191I), y \u2208 Cube.boundary N \u2192 \u2191(ContinuousMap.mk fun c => \u2191p (c default)) y = x) })\n p) =\n p\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.636077\ninst\u271d : Unique N\np : \u03a9 X x\nx\u271d : \u2191I\n\u22a2 \u2191((fun p =>\n { toContinuousMap := ContinuousMap.mk fun t => \u2191p fun x => t, source' := (_ : (\u2191p fun x => 0) = x),\n target' := (_ : (\u2191p fun x => 1) = x) })\n ((fun p =>\n { val := ContinuousMap.mk fun c => \u2191p (c default),\n property :=\n (_ : \u2200 (y : N \u2192 \u2191I), y \u2208 Cube.boundary N \u2192 \u2191(ContinuousMap.mk fun c => \u2191p (c default)) y = x) })\n p))\n x\u271d =\n \u2191p x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\n\u22a2 HomotopyGroup N X x \u2243 FundamentalGroup X x\n[PROOFSTEP]\nrefine' Equiv.trans _ (CategoryTheory.Groupoid.isoEquivHom _ _).symm\n[GOAL]\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\n\u22a2 HomotopyGroup N X x \u2243 (x \u27f6 x)\n[PROOFSTEP]\nrefine' Quotient.congr (genLoopEquivOfUnique N) _\n[GOAL]\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\n\u22a2 \u2200 (a\u2081 a\u2082 : \u2191(\u03a9^ N X x)), Setoid.r a\u2081 a\u2082 \u2194 Setoid.r (\u2191(genLoopEquivOfUnique N) a\u2081) (\u2191(genLoopEquivOfUnique N) a\u2082)\n[PROOFSTEP]\nintros a\u2081 a\u2082\n[GOAL]\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\n\u22a2 Setoid.r a\u2081 a\u2082 \u2194 Setoid.r (\u2191(genLoopEquivOfUnique N) a\u2081) (\u2191(genLoopEquivOfUnique N) a\u2082)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\n\u22a2 Setoid.r a\u2081 a\u2082 \u2192 Setoid.r (\u2191(genLoopEquivOfUnique N) a\u2081) (\u2191(genLoopEquivOfUnique N) a\u2082)\n[PROOFSTEP]\nrintro \u27e8H\u27e9\n[GOAL]\ncase mpr\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\n\u22a2 Setoid.r (\u2191(genLoopEquivOfUnique N) a\u2081) (\u2191(genLoopEquivOfUnique N) a\u2082) \u2192 Setoid.r a\u2081 a\u2082\n[PROOFSTEP]\nrintro \u27e8H\u27e9\n[GOAL]\ncase mp.intro\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\nH : ContinuousMap.HomotopyRel (\u2191a\u2081) (\u2191a\u2082) (Cube.boundary N)\n\u22a2 Setoid.r (\u2191(genLoopEquivOfUnique N) a\u2081) (\u2191(genLoopEquivOfUnique N) a\u2082)\n[PROOFSTEP]\nexact\n \u27e8{ toFun := fun tx => H (tx.fst, fun _ => tx.snd)\n map_zero_left := fun _ => H.apply_zero _\n map_one_left := fun _ => H.apply_one _\n prop' := fun t y iH => H.prop' _ _ \u27e8default, iH\u27e9 }\u27e9\n[GOAL]\ncase mpr.intro\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (\u2191(genLoopEquivOfUnique N) a\u2081) (\u2191(genLoopEquivOfUnique N) a\u2082)\n\u22a2 Setoid.r a\u2081 a\u2082\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u27e8\u27e8fun tx => H (tx.fst, tx.snd default), H.continuous.comp _\u27e9, fun y => _, fun y => _\u27e9, _\u27e9\u27e9\n[GOAL]\ncase mpr.intro.refine'_1\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (\u2191(genLoopEquivOfUnique N) a\u2081) (\u2191(genLoopEquivOfUnique N) a\u2082)\n\u22a2 Continuous fun tx => (tx.fst, Prod.snd tx default)\n[PROOFSTEP]\nexact continuous_fst.prod_mk ((continuous_apply _).comp continuous_snd)\n[GOAL]\ncase mpr.intro.refine'_2\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (\u2191(genLoopEquivOfUnique N) a\u2081) (\u2191(genLoopEquivOfUnique N) a\u2082)\ny : N \u2192 \u2191I\n\u22a2 ContinuousMap.toFun (ContinuousMap.mk fun tx => \u2191H (tx.fst, Prod.snd tx default)) (0, y) = \u2191\u2191a\u2081 y\n[PROOFSTEP]\nexact (H.apply_zero _).trans (congr_arg a\u2081 (eq_const_of_unique y).symm)\n[GOAL]\ncase mpr.intro.refine'_3\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (\u2191(genLoopEquivOfUnique N) a\u2081) (\u2191(genLoopEquivOfUnique N) a\u2082)\ny : N \u2192 \u2191I\n\u22a2 ContinuousMap.toFun (ContinuousMap.mk fun tx => \u2191H (tx.fst, Prod.snd tx default)) (1, y) = \u2191\u2191a\u2082 y\n[PROOFSTEP]\nexact (H.apply_one _).trans (congr_arg a\u2082 (eq_const_of_unique y).symm)\n[GOAL]\ncase mpr.intro.refine'_4\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (\u2191(genLoopEquivOfUnique N) a\u2081) (\u2191(genLoopEquivOfUnique N) a\u2082)\n\u22a2 \u2200 (t : \u2191I) (x_1 : N \u2192 \u2191I),\n x_1 \u2208 Cube.boundary N \u2192\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun tx => \u2191H (tx.fst, Prod.snd tx default),\n map_zero_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (0, Prod.snd (0, y) default) = \u2191\u2191a\u2081 y),\n map_one_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (1, Prod.snd (1, y) default) = \u2191\u2191a\u2082 y) }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191\u2191a\u2081 x_1 \u2227\n \u2191(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun tx => \u2191H (tx.fst, Prod.snd tx default),\n map_zero_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (0, Prod.snd (0, y) default) = \u2191\u2191a\u2081 y),\n map_one_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (1, Prod.snd (1, y) default) = \u2191\u2191a\u2082 y) }.toContinuousMap\n (t, x_2))\n x_1 =\n \u2191\u2191a\u2082 x_1\n[PROOFSTEP]\nrintro t y \u27e8i, iH\u27e9\n[GOAL]\ncase mpr.intro.refine'_4.intro\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (\u2191(genLoopEquivOfUnique N) a\u2081) (\u2191(genLoopEquivOfUnique N) a\u2082)\nt : \u2191I\ny : N \u2192 \u2191I\ni : N\niH : y i = 0 \u2228 y i = 1\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun tx => \u2191H (tx.fst, Prod.snd tx default),\n map_zero_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (0, Prod.snd (0, y) default) = \u2191\u2191a\u2081 y),\n map_one_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (1, Prod.snd (1, y) default) = \u2191\u2191a\u2082 y) }.toContinuousMap\n (t, x_1))\n y =\n \u2191\u2191a\u2081 y \u2227\n \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun tx => \u2191H (tx.fst, Prod.snd tx default),\n map_zero_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (0, Prod.snd (0, y) default) = \u2191\u2191a\u2081 y),\n map_one_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (1, Prod.snd (1, y) default) = \u2191\u2191a\u2082 y) }.toContinuousMap\n (t, x_1))\n y =\n \u2191\u2191a\u2082 y\n[PROOFSTEP]\ncases Unique.eq_default i\n[GOAL]\ncase mpr.intro.refine'_4.intro.refl\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (\u2191(genLoopEquivOfUnique N) a\u2081) (\u2191(genLoopEquivOfUnique N) a\u2082)\nt : \u2191I\ny : N \u2192 \u2191I\niH : y default = 0 \u2228 y default = 1\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun tx => \u2191H (tx.fst, Prod.snd tx default),\n map_zero_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (0, Prod.snd (0, y) default) = \u2191\u2191a\u2081 y),\n map_one_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (1, Prod.snd (1, y) default) = \u2191\u2191a\u2082 y) }.toContinuousMap\n (t, x_1))\n y =\n \u2191\u2191a\u2081 y \u2227\n \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun tx => \u2191H (tx.fst, Prod.snd tx default),\n map_zero_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (0, Prod.snd (0, y) default) = \u2191\u2191a\u2081 y),\n map_one_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (1, Prod.snd (1, y) default) = \u2191\u2191a\u2082 y) }.toContinuousMap\n (t, x_1))\n y =\n \u2191\u2191a\u2082 y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.intro.refine'_4.intro.refl.left\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (\u2191(genLoopEquivOfUnique N) a\u2081) (\u2191(genLoopEquivOfUnique N) a\u2082)\nt : \u2191I\ny : N \u2192 \u2191I\niH : y default = 0 \u2228 y default = 1\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun tx => \u2191H (tx.fst, Prod.snd tx default),\n map_zero_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (0, Prod.snd (0, y) default) = \u2191\u2191a\u2081 y),\n map_one_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (1, Prod.snd (1, y) default) = \u2191\u2191a\u2082 y) }.toContinuousMap\n (t, x_1))\n y =\n \u2191\u2191a\u2081 y\n[PROOFSTEP]\nexact (H.eq_fst _ iH).trans (congr_arg a\u2081 (eq_const_of_unique y).symm)\n[GOAL]\ncase mpr.intro.refine'_4.intro.refl.right\nN\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\u271d\nN : Type ?u.712366\ninst\u271d : Unique N\na\u2081 a\u2082 : \u2191(\u03a9^ N X x)\nH : Path.Homotopy (\u2191(genLoopEquivOfUnique N) a\u2081) (\u2191(genLoopEquivOfUnique N) a\u2082)\nt : \u2191I\ny : N \u2192 \u2191I\niH : y default = 0 \u2228 y default = 1\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun tx => \u2191H (tx.fst, Prod.snd tx default),\n map_zero_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (0, Prod.snd (0, y) default) = \u2191\u2191a\u2081 y),\n map_one_left := (_ : \u2200 (y : N \u2192 \u2191I), \u2191H (1, Prod.snd (1, y) default) = \u2191\u2191a\u2082 y) }.toContinuousMap\n (t, x_1))\n y =\n \u2191\u2191a\u2082 y\n[PROOFSTEP]\nexact (H.eq_snd _ iH).trans (congr_arg a\u2082 (eq_const_of_unique y).symm)\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\n\u22a2 auxGroup i = auxGroup j\n[PROOFSTEP]\nby_cases h : i = j\n[GOAL]\ncase pos\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : i = j\n\u22a2 auxGroup i = auxGroup j\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase neg\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : \u00aci = j\n\u22a2 auxGroup i = auxGroup j\n[PROOFSTEP]\nrefine' Group.ext (EckmannHilton.mul (isUnital_auxGroup i) (isUnital_auxGroup j) _)\n[GOAL]\ncase neg\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : \u00aci = j\n\u22a2 \u2200 (a b c d : HomotopyGroup N X x), Mul.mul (Mul.mul a b) (Mul.mul c d) = Mul.mul (Mul.mul a c) (Mul.mul b d)\n[PROOFSTEP]\nrintro \u27e8a\u27e9 \u27e8b\u27e9 \u27e8c\u27e9 \u27e8d\u27e9\n[GOAL]\ncase neg.mk.mk.mk.mk\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : \u00aci = j\na\u271d : HomotopyGroup N X x\na : \u2191(\u03a9^ N X x)\nb\u271d : HomotopyGroup N X x\nb : \u2191(\u03a9^ N X x)\nc\u271d : HomotopyGroup N X x\nc : \u2191(\u03a9^ N X x)\nd\u271d : HomotopyGroup N X x\nd : \u2191(\u03a9^ N X x)\n\u22a2 Mul.mul (Mul.mul (Quot.mk Setoid.r a) (Quot.mk Setoid.r b)) (Mul.mul (Quot.mk Setoid.r c) (Quot.mk Setoid.r d)) =\n Mul.mul (Mul.mul (Quot.mk Setoid.r a) (Quot.mk Setoid.r c)) (Mul.mul (Quot.mk Setoid.r b) (Quot.mk Setoid.r d))\n[PROOFSTEP]\nchange Quotient.mk' _ = _\n[GOAL]\ncase neg.mk.mk.mk.mk\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : \u00aci = j\na\u271d : HomotopyGroup N X x\na : \u2191(\u03a9^ N X x)\nb\u271d : HomotopyGroup N X x\nb : \u2191(\u03a9^ N X x)\nc\u271d : HomotopyGroup N X x\nc : \u2191(\u03a9^ N X x)\nd\u271d : HomotopyGroup N X x\nd : \u2191(\u03a9^ N X x)\n\u22a2 Quotient.mk'\n (\u2191(loopHomeo i).symm\n (Path.trans\n (\u2191(loopHomeo i).toEquiv\n (\u2191(loopHomeo j).symm (Path.trans (\u2191(loopHomeo j).toEquiv d) (\u2191(loopHomeo j).toEquiv c))))\n (\u2191(loopHomeo i).toEquiv\n (\u2191(loopHomeo j).symm (Path.trans (\u2191(loopHomeo j).toEquiv b) (\u2191(loopHomeo j).toEquiv a)))))) =\n Mul.mul (Mul.mul (Quot.mk Setoid.r a) (Quot.mk Setoid.r c)) (Mul.mul (Quot.mk Setoid.r b) (Quot.mk Setoid.r d))\n[PROOFSTEP]\napply congr_arg Quotient.mk'\n[GOAL]\ncase neg.mk.mk.mk.mk\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nh : \u00aci = j\na\u271d : HomotopyGroup N X x\na : \u2191(\u03a9^ N X x)\nb\u271d : HomotopyGroup N X x\nb : \u2191(\u03a9^ N X x)\nc\u271d : HomotopyGroup N X x\nc : \u2191(\u03a9^ N X x)\nd\u271d : HomotopyGroup N X x\nd : \u2191(\u03a9^ N X x)\n\u22a2 \u2191(loopHomeo i).symm\n (Path.trans\n (\u2191(loopHomeo i).toEquiv\n (\u2191(loopHomeo j).symm (Path.trans (\u2191(loopHomeo j).toEquiv d) (\u2191(loopHomeo j).toEquiv c))))\n (\u2191(loopHomeo i).toEquiv\n (\u2191(loopHomeo j).symm (Path.trans (\u2191(loopHomeo j).toEquiv b) (\u2191(loopHomeo j).toEquiv a))))) =\n \u2191(loopHomeo j).symm\n (Path.trans\n (\u2191(loopHomeo j).toEquiv\n (\u2191(loopHomeo i).symm (Path.trans (\u2191(loopHomeo i).toEquiv d) (\u2191(loopHomeo i).toEquiv b))))\n (\u2191(loopHomeo j).toEquiv\n (\u2191(loopHomeo i).symm (Path.trans (\u2191(loopHomeo i).toEquiv c) (\u2191(loopHomeo i).toEquiv a)))))\n[PROOFSTEP]\nsimp only [fromLoop_trans_toLoop, transAt_distrib h, coe_toEquiv, loopHomeo_apply, coe_symm_toEquiv,\n loopHomeo_symm_apply]\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nf g : \u2191(\u03a9^ N X x)\n\u22a2 Quotient.mk (Homotopic.setoid N x) (transAt i f g) = Quotient.mk (Homotopic.setoid N x) (transAt j f g)\n[PROOFSTEP]\nsimp_rw [\u2190 fromLoop_trans_toLoop]\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nf g : \u2191(\u03a9^ N X x)\n\u22a2 Quotient.mk (Homotopic.setoid N x) (fromLoop i (Path.trans (toLoop i f) (toLoop i g))) =\n Quotient.mk (Homotopic.setoid N x) (fromLoop j (Path.trans (toLoop j f) (toLoop j g)))\n[PROOFSTEP]\nlet m := fun (G) (_ : Group G) => ((\u00b7 * \u00b7) : G \u2192 G \u2192 G)\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nf g : \u2191(\u03a9^ N X x)\nm : (G : Type ?u.799968) \u2192 Group G \u2192 G \u2192 G \u2192 G := fun G x x_1 x_2 => x_1 * x_2\n\u22a2 Quotient.mk (Homotopic.setoid N x) (fromLoop i (Path.trans (toLoop i f) (toLoop i g))) =\n Quotient.mk (Homotopic.setoid N x) (fromLoop j (Path.trans (toLoop j f) (toLoop j g)))\n[PROOFSTEP]\nexact congr_fun\u2082 (congr_arg (m <| HomotopyGroup N X x) <| auxGroup_indep i j) \u27e6g\u27e7 \u27e6f\u27e7\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nf : \u2191(\u03a9^ N X x)\n\u22a2 Quotient.mk (Homotopic.setoid N x) (symmAt i f) = Quotient.mk (Homotopic.setoid N x) (symmAt j f)\n[PROOFSTEP]\nsimp_rw [\u2190 fromLoop_symm_toLoop]\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nf : \u2191(\u03a9^ N X x)\n\u22a2 Quotient.mk (Homotopic.setoid N x) (fromLoop i (Path.symm (toLoop i f))) =\n Quotient.mk (Homotopic.setoid N x) (fromLoop j (Path.symm (toLoop j f)))\n[PROOFSTEP]\nlet inv := fun (G) (_ : Group G) => ((\u00b7\u207b\u00b9) : G \u2192 G)\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\nx : X\ninst\u271d : DecidableEq N\ni j : N\nf : \u2191(\u03a9^ N X x)\ninv : (G : Type ?u.802068) \u2192 Group G \u2192 G \u2192 G := fun G x x_1 => x_1\u207b\u00b9\n\u22a2 Quotient.mk (Homotopic.setoid N x) (fromLoop i (Path.symm (toLoop i f))) =\n Quotient.mk (Homotopic.setoid N x) (fromLoop j (Path.symm (toLoop j f)))\n[PROOFSTEP]\nexact congr_fun (congr_arg (inv <| HomotopyGroup N X x) <| auxGroup_indep i j) \u27e6f\u27e7\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Nonempty N\ni : N\np q : \u2191(\u03a9^ N X x)\n\u22a2 (fun x_1 x_2 => x_1 * x_2) (Quotient.mk (Homotopic.setoid N x) p) (Quotient.mk (Homotopic.setoid N x) q) =\n Quotient.mk (Homotopic.setoid N x) (transAt i q p)\n[PROOFSTEP]\nrw [transAt_indep _ q, \u2190 fromLoop_trans_toLoop]\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Nonempty N\ni : N\np q : \u2191(\u03a9^ N X x)\n\u22a2 (fun x_1 x_2 => x_1 * x_2) (Quotient.mk (Homotopic.setoid N x) p) (Quotient.mk (Homotopic.setoid N x) q) =\n Quotient.mk (Homotopic.setoid N x) (fromLoop ?m.837545 (Path.trans (toLoop ?m.837545 q) (toLoop ?m.837545 p)))\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Nonempty N\ni : N\np q : \u2191(\u03a9^ N X x)\n\u22a2 N\n[PROOFSTEP]\napply Quotient.sound\n[GOAL]\ncase a\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Nonempty N\ni : N\np q : \u2191(\u03a9^ N X x)\n\u22a2 \u2191(loopHomeo (Classical.arbitrary N)).symm\n (Path.trans (\u2191(loopHomeo (Classical.arbitrary N)).toEquiv q) (\u2191(loopHomeo (Classical.arbitrary N)).toEquiv p)) \u2248\n fromLoop ?m.837545 (Path.trans (toLoop ?m.837545 q) (toLoop ?m.837545 p))\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Nonempty N\ni : N\np q : \u2191(\u03a9^ N X x)\n\u22a2 N\n[PROOFSTEP]\nrfl\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Nonempty N\ni : N\np : \u2191(\u03a9^ N X x)\n\u22a2 (Quotient.mk (Homotopic.setoid N x) p)\u207b\u00b9 = Quotient.mk (Homotopic.setoid N x) (symmAt i p)\n[PROOFSTEP]\nrw [symmAt_indep _ p, \u2190 fromLoop_symm_toLoop]\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Nonempty N\ni : N\np : \u2191(\u03a9^ N X x)\n\u22a2 (Quotient.mk (Homotopic.setoid N x) p)\u207b\u00b9 =\n Quotient.mk (Homotopic.setoid N x) (fromLoop ?m.838706 (Path.symm (toLoop ?m.838706 p)))\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Nonempty N\ni : N\np : \u2191(\u03a9^ N X x)\n\u22a2 N\n[PROOFSTEP]\napply Quotient.sound\n[GOAL]\ncase a\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Nonempty N\ni : N\np : \u2191(\u03a9^ N X x)\n\u22a2 \u2191(loopHomeo (Classical.arbitrary N)).symm (Path.symm (\u2191(loopHomeo (Classical.arbitrary N)).toEquiv p)) \u2248\n fromLoop ?m.838706 (Path.symm (toLoop ?m.838706 p))\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Nonempty N\ni : N\np : \u2191(\u03a9^ N X x)\n\u22a2 N\n[PROOFSTEP]\nrfl\n[GOAL]\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Nontrivial N\nh : \u2203 y, y \u2260 Classical.arbitrary N := exists_ne (Classical.arbitrary N)\n\u22a2 \u2200 (a b c d : HomotopyGroup N X x), Mul.mul (a * b) (c * d) = Mul.mul a c * Mul.mul b d\n[PROOFSTEP]\nrintro \u27e8a\u27e9 \u27e8b\u27e9 \u27e8c\u27e9 \u27e8d\u27e9\n[GOAL]\ncase mk.mk.mk.mk\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Nontrivial N\nh : \u2203 y, y \u2260 Classical.arbitrary N := exists_ne (Classical.arbitrary N)\na\u271d : HomotopyGroup N X x\na : \u2191(\u03a9^ N X x)\nb\u271d : HomotopyGroup N X x\nb : \u2191(\u03a9^ N X x)\nc\u271d : HomotopyGroup N X x\nc : \u2191(\u03a9^ N X x)\nd\u271d : HomotopyGroup N X x\nd : \u2191(\u03a9^ N X x)\n\u22a2 Mul.mul (Quot.mk Setoid.r a * Quot.mk Setoid.r b) (Quot.mk Setoid.r c * Quot.mk Setoid.r d) =\n Mul.mul (Quot.mk Setoid.r a) (Quot.mk Setoid.r c) * Mul.mul (Quot.mk Setoid.r b) (Quot.mk Setoid.r d)\n[PROOFSTEP]\napply congr_arg Quotient.mk'\n[GOAL]\ncase mk.mk.mk.mk\nN : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\nx : X\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Nontrivial N\nh : \u2203 y, y \u2260 Classical.arbitrary N := exists_ne (Classical.arbitrary N)\na\u271d : HomotopyGroup N X x\na : \u2191(\u03a9^ N X x)\nb\u271d : HomotopyGroup N X x\nb : \u2191(\u03a9^ N X x)\nc\u271d : HomotopyGroup N X x\nc : \u2191(\u03a9^ N X x)\nd\u271d : HomotopyGroup N X x\nd : \u2191(\u03a9^ N X x)\n\u22a2 \u2191(loopHomeo (Classical.choose h)).symm\n (Path.trans\n (\u2191(loopHomeo (Classical.choose h)).toEquiv\n (\u2191(loopHomeo (Classical.arbitrary N)).symm\n (Path.trans (\u2191(loopHomeo (Classical.arbitrary N)).toEquiv d)\n (\u2191(loopHomeo (Classical.arbitrary N)).toEquiv c))))\n (\u2191(loopHomeo (Classical.choose h)).toEquiv\n (\u2191(loopHomeo (Classical.arbitrary N)).symm\n (Path.trans (\u2191(loopHomeo (Classical.arbitrary N)).toEquiv b)\n (\u2191(loopHomeo (Classical.arbitrary N)).toEquiv a))))) =\n \u2191(loopHomeo (Classical.arbitrary N)).symm\n (Path.trans\n (\u2191(loopHomeo (Classical.arbitrary N)).toEquiv\n (\u2191(loopHomeo (Classical.choose h)).symm\n (Path.trans (\u2191(loopHomeo (Classical.choose h)).toEquiv d) (\u2191(loopHomeo (Classical.choose h)).toEquiv b))))\n (\u2191(loopHomeo (Classical.arbitrary N)).toEquiv\n (\u2191(loopHomeo (Classical.choose h)).symm\n (Path.trans (\u2191(loopHomeo (Classical.choose h)).toEquiv c) (\u2191(loopHomeo (Classical.choose h)).toEquiv a)))))\n[PROOFSTEP]\nsimp only [fromLoop_trans_toLoop, transAt_distrib <| Classical.choose_spec h, coe_toEquiv, loopHomeo_apply,\n coe_symm_toEquiv, loopHomeo_symm_apply]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Homotopy.HomotopyGroup", "llama_tokens": 61049, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.737158174177441, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.42570542899630753}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonAssocSemiring \u03b1\nm n : \u2115\n\u22a2 \u2191(m * n) = \u2191m * \u2191n\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonAssocSemiring \u03b1\nm : \u2115\n\u22a2 \u2191(m * zero) = \u2191m * \u2191zero\n[PROOFSTEP]\nsimp [mul_succ, mul_add, *]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonAssocSemiring \u03b1\nm n\u271d : \u2115\nn_ih\u271d : \u2191(m * n\u271d) = \u2191m * \u2191n\u271d\n\u22a2 \u2191(m * succ n\u271d) = \u2191m * \u2191(succ n\u271d)\n[PROOFSTEP]\nsimp [mul_succ, mul_add, *]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonAssocSemiring \u03b1\nn : \u2115\nx : \u03b1\n\u22a2 Commute (\u2191n) x\n[PROOFSTEP]\ninduction n with\n| zero => rw [Nat.cast_zero]; exact Commute.zero_left x\n| succ n ihn => rw [Nat.cast_succ]; exact ihn.add_left (Commute.one_left x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonAssocSemiring \u03b1\nn : \u2115\nx : \u03b1\n\u22a2 Commute (\u2191n) x\n[PROOFSTEP]\ninduction n with\n| zero => rw [Nat.cast_zero]; exact Commute.zero_left x\n| succ n ihn => rw [Nat.cast_succ]; exact ihn.add_left (Commute.one_left x)\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonAssocSemiring \u03b1\nx : \u03b1\n\u22a2 Commute (\u2191zero) x\n[PROOFSTEP]\n\n| zero => rw [Nat.cast_zero]; exact Commute.zero_left x\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonAssocSemiring \u03b1\nx : \u03b1\n\u22a2 Commute (\u2191zero) x\n[PROOFSTEP]\nrw [Nat.cast_zero]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonAssocSemiring \u03b1\nx : \u03b1\n\u22a2 Commute 0 x\n[PROOFSTEP]\nexact Commute.zero_left x\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonAssocSemiring \u03b1\nx : \u03b1\nn : \u2115\nihn : Commute (\u2191n) x\n\u22a2 Commute (\u2191(succ n)) x\n[PROOFSTEP]\n\n| succ n ihn => rw [Nat.cast_succ]; exact ihn.add_left (Commute.one_left x)\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonAssocSemiring \u03b1\nx : \u03b1\nn : \u2115\nihn : Commute (\u2191n) x\n\u22a2 Commute (\u2191(succ n)) x\n[PROOFSTEP]\nrw [Nat.cast_succ]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonAssocSemiring \u03b1\nx : \u03b1\nn : \u2115\nihn : Commute (\u2191n) x\n\u22a2 Commute (\u2191n + 1) x\n[PROOFSTEP]\nexact ihn.add_left (Commute.one_left x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : AddCommMonoidWithOne \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ZeroLEOneClass \u03b1\nn : \u2115\n\u22a2 \u2191n \u2264 \u2191(n + 1)\n[PROOFSTEP]\nrw [Nat.cast_succ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : AddCommMonoidWithOne \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ZeroLEOneClass \u03b1\nn : \u2115\n\u22a2 \u2191n \u2264 \u2191n + 1\n[PROOFSTEP]\nexact le_add_of_nonneg_right zero_le_one\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : AddCommMonoidWithOne \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : NeZero 1\nn : \u2115\n\u22a2 0 < \u2191n \u2194 0 < n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : AddCommMonoidWithOne \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : NeZero 1\n\u22a2 0 < \u2191zero \u2194 0 < zero\n[PROOFSTEP]\nsimp [cast_add_one_pos]\n -- without this more specific version Lean often chokes\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : AddCommMonoidWithOne \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : NeZero 1\nn\u271d : \u2115\n\u22a2 0 < \u2191(succ n\u271d) \u2194 0 < succ n\u271d\n[PROOFSTEP]\nsimp [cast_add_one_pos]\n -- without this more specific version Lean often chokes\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : AddCommMonoidWithOne \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CharZero \u03b1\nm n : \u2115\n\u22a2 1 < \u2191n \u2194 1 < n\n[PROOFSTEP]\nrw [\u2190 cast_one, cast_lt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : AddCommMonoidWithOne \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CharZero \u03b1\nm n : \u2115\n\u22a2 1 \u2264 \u2191n \u2194 1 \u2264 n\n[PROOFSTEP]\nrw [\u2190 cast_one, cast_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : AddCommMonoidWithOne \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CharZero \u03b1\nm n : \u2115\n\u22a2 \u2191n < 1 \u2194 n = 0\n[PROOFSTEP]\nrw [\u2190 cast_one, cast_lt, lt_succ_iff, \u2190 bot_eq_zero, le_bot_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : AddCommMonoidWithOne \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CharZero \u03b1\nm n : \u2115\n\u22a2 \u2191n \u2264 1 \u2194 n \u2264 1\n[PROOFSTEP]\nrw [\u2190 cast_one, cast_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : CanonicallyOrderedCommSemiring \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nm n : \u2115\n\u22a2 \u2191(m - n) = \u2191m - \u2191n\n[PROOFSTEP]\ncases' le_total m n with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : CanonicallyOrderedCommSemiring \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nm n : \u2115\nh : m \u2264 n\n\u22a2 \u2191(m - n) = \u2191m - \u2191n\n[PROOFSTEP]\nrw [tsub_eq_zero_of_le h, cast_zero, tsub_eq_zero_of_le]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : CanonicallyOrderedCommSemiring \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nm n : \u2115\nh : m \u2264 n\n\u22a2 \u2191m \u2264 \u2191n\n[PROOFSTEP]\nexact mono_cast h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : CanonicallyOrderedCommSemiring \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nm n : \u2115\nh : n \u2264 m\n\u22a2 \u2191(m - n) = \u2191m - \u2191n\n[PROOFSTEP]\nrcases le_iff_exists_add'.mp h with \u27e8m, rfl\u27e9\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : CanonicallyOrderedCommSemiring \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nn m : \u2115\nh : n \u2264 m + n\n\u22a2 \u2191(m + n - n) = \u2191(m + n) - \u2191n\n[PROOFSTEP]\nrw [add_tsub_cancel_right, cast_add, add_tsub_cancel_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : AddMonoidWithOne \u03b1\ninst\u271d : CharZero \u03b1\n\u22a2 1 \u2260 0\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nA : Type u_3\nB : Type u_4\nF : Type u_5\ninst\u271d\u00b2 : AddMonoidWithOne B\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : AddMonoidHomClass F \u2115 A\nf g : F\nh : \u2191f 1 = \u2191g 1\n\u22a2 \u2200 (x : \u2115), \u2191f x = \u2191g x\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nA : Type u_3\nB : Type u_4\nF : Type u_5\ninst\u271d\u00b2 : AddMonoidWithOne B\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : AddMonoidHomClass F \u2115 A\nf g : F\nh : \u2191f 1 = \u2191g 1\nn : \u2115\n\u22a2 \u2191f n = \u2191g n\n[PROOFSTEP]\ninduction n with\n| zero => simp_rw [Nat.zero_eq, map_zero f, map_zero g]\n| succ n ihn => simp [Nat.succ_eq_add_one, h, ihn]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nA : Type u_3\nB : Type u_4\nF : Type u_5\ninst\u271d\u00b2 : AddMonoidWithOne B\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : AddMonoidHomClass F \u2115 A\nf g : F\nh : \u2191f 1 = \u2191g 1\nn : \u2115\n\u22a2 \u2191f n = \u2191g n\n[PROOFSTEP]\ninduction n with\n| zero => simp_rw [Nat.zero_eq, map_zero f, map_zero g]\n| succ n ihn => simp [Nat.succ_eq_add_one, h, ihn]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nA : Type u_3\nB : Type u_4\nF : Type u_5\ninst\u271d\u00b2 : AddMonoidWithOne B\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : AddMonoidHomClass F \u2115 A\nf g : F\nh : \u2191f 1 = \u2191g 1\n\u22a2 \u2191f Nat.zero = \u2191g Nat.zero\n[PROOFSTEP]\n\n| zero => simp_rw [Nat.zero_eq, map_zero f, map_zero g]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nA : Type u_3\nB : Type u_4\nF : Type u_5\ninst\u271d\u00b2 : AddMonoidWithOne B\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : AddMonoidHomClass F \u2115 A\nf g : F\nh : \u2191f 1 = \u2191g 1\n\u22a2 \u2191f Nat.zero = \u2191g Nat.zero\n[PROOFSTEP]\nsimp_rw [Nat.zero_eq, map_zero f, map_zero g]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nA : Type u_3\nB : Type u_4\nF : Type u_5\ninst\u271d\u00b2 : AddMonoidWithOne B\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : AddMonoidHomClass F \u2115 A\nf g : F\nh : \u2191f 1 = \u2191g 1\nn : \u2115\nihn : \u2191f n = \u2191g n\n\u22a2 \u2191f (Nat.succ n) = \u2191g (Nat.succ n)\n[PROOFSTEP]\n\n| succ n ihn => simp [Nat.succ_eq_add_one, h, ihn]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nA : Type u_3\nB : Type u_4\nF : Type u_5\ninst\u271d\u00b2 : AddMonoidWithOne B\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : AddMonoidHomClass F \u2115 A\nf g : F\nh : \u2191f 1 = \u2191g 1\nn : \u2115\nihn : \u2191f n = \u2191g n\n\u22a2 \u2191f (Nat.succ n) = \u2191g (Nat.succ n)\n[PROOFSTEP]\nsimp [Nat.succ_eq_add_one, h, ihn]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nA : Type u_3\nB : Type u_4\nF : Type u_5\ninst\u271d\u00b2 : AddMonoidWithOne B\ninst\u271d\u00b9 : AddMonoidWithOne A\ninst\u271d : AddMonoidHomClass F \u2115 A\nf : F\nh1 : \u2191f 1 = 1\n\u22a2 \u2191f 0 = \u21910\n[PROOFSTEP]\nsimp [map_zero f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nA : Type u_3\nB : Type u_4\nF : Type u_5\ninst\u271d\u00b2 : AddMonoidWithOne B\ninst\u271d\u00b9 : AddMonoidWithOne A\ninst\u271d : AddMonoidHomClass F \u2115 A\nf : F\nh1 : \u2191f 1 = 1\nn : \u2115\n\u22a2 \u2191f (n + 1) = \u2191(n + 1)\n[PROOFSTEP]\nrw [map_add, h1, eq_natCast' f h1 n, Nat.cast_add_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nA\u271d : Type u_3\nB : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : AddMonoidWithOne B\ninst\u271d\u00b2 : AddMonoidWithOne A\u271d\nA : Type u_6\ninst\u271d\u00b9 : AddMonoidWithOne A\ninst\u271d : AddMonoidHomClass F A B\nf : F\nh : \u2191f 1 = 1\n\u22a2 \u2191f \u21910 = \u21910\n[PROOFSTEP]\nsimp [map_zero f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nA\u271d : Type u_3\nB : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : AddMonoidWithOne B\ninst\u271d\u00b2 : AddMonoidWithOne A\u271d\nA : Type u_6\ninst\u271d\u00b9 : AddMonoidWithOne A\ninst\u271d : AddMonoidHomClass F A B\nf : F\nh : \u2191f 1 = 1\nn : \u2115\n\u22a2 \u2191f \u2191(n + 1) = \u2191(n + 1)\n[PROOFSTEP]\nrw [Nat.cast_add, map_add, Nat.cast_add, map_natCast' f h n, Nat.cast_one, h, Nat.cast_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nA : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MulZeroOneClass A\ninst\u271d : MonoidWithZeroHomClass F \u2115 A\nf g : F\nh_pos : \u2200 {n : \u2115}, 0 < n \u2192 \u2191f n = \u2191g n\n\u22a2 f = g\n[PROOFSTEP]\napply FunLike.ext\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nA : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MulZeroOneClass A\ninst\u271d : MonoidWithZeroHomClass F \u2115 A\nf g : F\nh_pos : \u2200 {n : \u2115}, 0 < n \u2192 \u2191f n = \u2191g n\n\u22a2 \u2200 (x : \u2115), \u2191f x = \u2191g x\n[PROOFSTEP]\nrintro (_ | n)\n[GOAL]\ncase h.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nA : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MulZeroOneClass A\ninst\u271d : MonoidWithZeroHomClass F \u2115 A\nf g : F\nh_pos : \u2200 {n : \u2115}, 0 < n \u2192 \u2191f n = \u2191g n\n\u22a2 \u2191f Nat.zero = \u2191g Nat.zero\n[PROOFSTEP]\nsimp [map_zero f, map_zero g]\n[GOAL]\ncase h.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nA : Type u_3\nF : Type u_4\ninst\u271d\u00b9 : MulZeroOneClass A\ninst\u271d : MonoidWithZeroHomClass F \u2115 A\nf g : F\nh_pos : \u2200 {n : \u2115}, 0 < n \u2192 \u2191f n = \u2191g n\nn : \u2115\n\u22a2 \u2191f (Nat.succ n) = \u2191g (Nat.succ n)\n[PROOFSTEP]\nexact h_pos n.succ_pos\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b2 : NonAssocSemiring R\ninst\u271d\u00b9 : NonAssocSemiring S\ninst\u271d : RingHomClass F \u2115 R\nf g : F\n\u22a2 \u2191f 1 = \u2191g 1\n[PROOFSTEP]\nsimp only [map_one f, map_one g]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b2 : NonAssocSemiring R\ninst\u271d\u00b9 : NonAssocSemiring S\nn : \u2115\nh\u271d : NeZero \u2191n\ninst\u271d : RingHomClass F R S\nf : F\nhf : Function.Injective \u2191f\nh : \u2191n = 0\n\u22a2 \u2191f \u2191n = \u2191f 0\n[PROOFSTEP]\nsimpa only [map_natCast, map_zero f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR\u271d : Type u_3\nS\u271d : Type u_4\nF\u271d : Type u_5\ninst\u271d\u2074 : NonAssocSemiring R\u271d\ninst\u271d\u00b3 : NonAssocSemiring S\u271d\nR : Type u_6\nS : Type u_7\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Semiring S\nF : Type u_8\ninst\u271d : RingHomClass F R S\nf : F\nn : \u2115\nhn : NeZero \u2191n\n\u22a2 NeZero (\u2191f \u2191n)\n[PROOFSTEP]\nsimp only [map_natCast, hn]\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Cast.Basic", "llama_tokens": 5872, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850933, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.4255450305905669}} {"text": "[GOAL]\nG H : AddCommGroupCat\nf : G \u27f6 H\ns : Fork f 0\n\u22a2 \u2191s.pt \u2192+ \u2191G\n[PROOFSTEP]\nexact Fork.\u03b9 s\n[GOAL]\nG H : AddCommGroupCat\nf : G \u27f6 H\ns : Fork f 0\nc : \u2191s.pt\n\u22a2 \u2191f (\u2191(Fork.\u03b9 s) c) = 0\n[PROOFSTEP]\nexact FunLike.congr_fun s.condition c\n[GOAL]\nG H : AddCommGroupCat\nf : G \u27f6 H\nx\u271d : Fork f 0\n\u22a2 (fun s => codRestrict (Fork.\u03b9 s) (ker f) (_ : \u2200 (c : \u2191s.pt), \u2191(Fork.\u03b9 s) c \u2208 ker f)) x\u271d \u226b Fork.\u03b9 (kernelCone f) =\n Fork.\u03b9 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nG H : AddCommGroupCat\nf : G \u27f6 H\nx\u271d\u00b9 : Fork f 0\nx\u271d : x\u271d\u00b9.pt \u27f6 (kernelCone f).pt\nh : x\u271d \u226b Fork.\u03b9 (kernelCone f) = Fork.\u03b9 x\u271d\u00b9\nx : \u2191x\u271d\u00b9.pt\n\u22a2 \u2191(\u2191x\u271d x) = \u2191(\u2191((fun s => codRestrict (Fork.\u03b9 s) (ker f) (_ : \u2200 (c : \u2191s.pt), \u2191(Fork.\u03b9 s) c \u2208 ker f)) x\u271d\u00b9) x)\n[PROOFSTEP]\nexact FunLike.congr_fun h x\n[GOAL]\nG H : AddCommGroupCat\nf : G \u27f6 H\nx\u271d\u00b9 : Cofork f 0\nx\u271d : (cokernelCocone f).pt \u27f6 x\u271d\u00b9.pt\nh : Cofork.\u03c0 (cokernelCocone f) \u226b x\u271d = Cofork.\u03c0 x\u271d\u00b9\nthis : Epi (Cofork.\u03c0 (cokernelCocone f))\n\u22a2 ?m.55829 x\u271d\u00b9 x\u271d h this \u226b x\u271d =\n ?m.55829 x\u271d\u00b9 x\u271d h this \u226b (fun s => lift (range f) (Cofork.\u03c0 s) (_ : range f \u2264 ker (Cofork.\u03c0 s))) x\u271d\u00b9\n[PROOFSTEP]\nsimpa only [parallelPair_obj_one] using h\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.GroupCat.Kernels", "llama_tokens": 653, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583376458152, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.42549638180041555}} {"text": "[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nb\u271d b : Basis \u03b9 R M\nN : Submodule R M\n\u03d5 : M \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : M \u2192\u2097[R] R), \u00acmap \u03d5 N < map \u03c8 N\ninst\u271d : IsPrincipal (map \u03d5 N)\nhgen : generator (map \u03d5 N) = 0\n\u22a2 N = \u22a5\n[PROOFSTEP]\nrw [Submodule.eq_bot_iff]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nb\u271d b : Basis \u03b9 R M\nN : Submodule R M\n\u03d5 : M \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : M \u2192\u2097[R] R), \u00acmap \u03d5 N < map \u03c8 N\ninst\u271d : IsPrincipal (map \u03d5 N)\nhgen : generator (map \u03d5 N) = 0\n\u22a2 \u2200 (x : M), x \u2208 N \u2192 x = 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nb\u271d b : Basis \u03b9 R M\nN : Submodule R M\n\u03d5 : M \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : M \u2192\u2097[R] R), \u00acmap \u03d5 N < map \u03c8 N\ninst\u271d : IsPrincipal (map \u03d5 N)\nhgen : generator (map \u03d5 N) = 0\nx : M\nhx : x \u2208 N\n\u22a2 x = 0\n[PROOFSTEP]\nrefine' b.ext_elem fun i \u21a6 _\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nb\u271d b : Basis \u03b9 R M\nN : Submodule R M\n\u03d5 : M \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : M \u2192\u2097[R] R), \u00acmap \u03d5 N < map \u03c8 N\ninst\u271d : IsPrincipal (map \u03d5 N)\nhgen : generator (map \u03d5 N) = 0\nx : M\nhx : x \u2208 N\ni : \u03b9\n\u22a2 \u2191(\u2191b.repr x) i = \u2191(\u2191b.repr 0) i\n[PROOFSTEP]\nrw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at h\u03d5 \n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nb\u271d b : Basis \u03b9 R M\nN : Submodule R M\n\u03d5 : M \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : M \u2192\u2097[R] R), \u00ac\u22a5 < map \u03c8 N\ninst\u271d : IsPrincipal (map \u03d5 N)\nhgen : generator (map \u03d5 N) = 0\nx : M\nhx : x \u2208 N\ni : \u03b9\n\u22a2 \u2191(\u2191b.repr x) i = \u2191(\u2191b.repr 0) i\n[PROOFSTEP]\nrw [LinearEquiv.map_zero, Finsupp.zero_apply]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nb\u271d b : Basis \u03b9 R M\nN : Submodule R M\n\u03d5 : M \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : M \u2192\u2097[R] R), \u00ac\u22a5 < map \u03c8 N\ninst\u271d : IsPrincipal (map \u03d5 N)\nhgen : generator (map \u03d5 N) = 0\nx : M\nhx : x \u2208 N\ni : \u03b9\n\u22a2 \u2191(\u2191b.repr x) i = 0\n[PROOFSTEP]\nexact (Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| h\u03d5 (Finsupp.lapply i \u2218\u2097 \u2191b.repr)) _ \u27e8x, hx, rfl\u27e9\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nb\u271d : Basis \u03b9 R M\nN O : Submodule R M\nb : Basis \u03b9 R { x // x \u2208 O }\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\nhgen : generator (LinearMap.submoduleImage \u03d5 N) = 0\n\u22a2 N = \u22a5\n[PROOFSTEP]\nrw [Submodule.eq_bot_iff]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nb\u271d : Basis \u03b9 R M\nN O : Submodule R M\nb : Basis \u03b9 R { x // x \u2208 O }\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\nhgen : generator (LinearMap.submoduleImage \u03d5 N) = 0\n\u22a2 \u2200 (x : M), x \u2208 N \u2192 x = 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nb\u271d : Basis \u03b9 R M\nN O : Submodule R M\nb : Basis \u03b9 R { x // x \u2208 O }\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\nhgen : generator (LinearMap.submoduleImage \u03d5 N) = 0\nx : M\nhx : x \u2208 N\n\u22a2 x = 0\n[PROOFSTEP]\nrefine (mk_eq_zero _ _).mp (show (\u27e8x, hNO hx\u27e9 : O) = 0 from b.ext_elem fun i \u21a6 ?_)\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nb\u271d : Basis \u03b9 R M\nN O : Submodule R M\nb : Basis \u03b9 R { x // x \u2208 O }\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\nhgen : generator (LinearMap.submoduleImage \u03d5 N) = 0\nx : M\nhx : x \u2208 N\ni : \u03b9\n\u22a2 \u2191(\u2191b.repr { val := x, property := (_ : x \u2208 O) }) i = \u2191(\u2191b.repr 0) i\n[PROOFSTEP]\nrw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at h\u03d5 \n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nb\u271d : Basis \u03b9 R M\nN O : Submodule R M\nb : Basis \u03b9 R { x // x \u2208 O }\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00ac\u22a5 < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\nhgen : generator (LinearMap.submoduleImage \u03d5 N) = 0\nx : M\nhx : x \u2208 N\ni : \u03b9\n\u22a2 \u2191(\u2191b.repr { val := x, property := (_ : x \u2208 O) }) i = \u2191(\u2191b.repr 0) i\n[PROOFSTEP]\nrw [LinearEquiv.map_zero, Finsupp.zero_apply]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nb\u271d : Basis \u03b9 R M\nN O : Submodule R M\nb : Basis \u03b9 R { x // x \u2208 O }\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00ac\u22a5 < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\nhgen : generator (LinearMap.submoduleImage \u03d5 N) = 0\nx : M\nhx : x \u2208 N\ni : \u03b9\n\u22a2 \u2191(\u2191b.repr { val := x, property := (_ : x \u2208 O) }) i = 0\n[PROOFSTEP]\nrefine (Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| h\u03d5 (Finsupp.lapply i \u2218\u2097 \u2191b.repr)) _ ?_\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nb\u271d : Basis \u03b9 R M\nN O : Submodule R M\nb : Basis \u03b9 R { x // x \u2208 O }\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00ac\u22a5 < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\nhgen : generator (LinearMap.submoduleImage \u03d5 N) = 0\nx : M\nhx : x \u2208 N\ni : \u03b9\n\u22a2 \u2191(\u2191b.repr { val := x, property := (_ : x \u2208 O) }) i \u2208\n LinearMap.submoduleImage (LinearMap.comp (Finsupp.lapply i) \u2191b.repr) N\n[PROOFSTEP]\nexact (LinearMap.mem_submoduleImage_of_le hNO).mpr \u27e8x, hx, rfl\u27e9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nI : Ideal R\ninst\u271d : IsPrincipal I\nx : R\nhx : x \u2208 I\n\u22a2 x \u2223 generator I \u2194 I = Ideal.span {x}\n[PROOFSTEP]\nconv_rhs => rw [\u2190 span_singleton_generator I]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nI : Ideal R\ninst\u271d : IsPrincipal I\nx : R\nhx : x \u2208 I\n| I = Ideal.span {x}\n[PROOFSTEP]\nrw [\u2190 span_singleton_generator I]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nI : Ideal R\ninst\u271d : IsPrincipal I\nx : R\nhx : x \u2208 I\n| I = Ideal.span {x}\n[PROOFSTEP]\nrw [\u2190 span_singleton_generator I]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nI : Ideal R\ninst\u271d : IsPrincipal I\nx : R\nhx : x \u2208 I\n| I = Ideal.span {x}\n[PROOFSTEP]\nrw [\u2190 span_singleton_generator I]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nI : Ideal R\ninst\u271d : IsPrincipal I\nx : R\nhx : x \u2208 I\n\u22a2 x \u2223 generator I \u2194 span R {generator I} = Ideal.span {x}\n[PROOFSTEP]\nrw [Ideal.submodule_span_eq, Ideal.span_singleton_eq_span_singleton, \u2190 dvd_dvd_iff_associated, \u2190 mem_iff_generator_dvd]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nI : Ideal R\ninst\u271d : IsPrincipal I\nx : R\nhx : x \u2208 I\n\u22a2 x \u2223 generator I \u2194 x \u2208 I \u2227 x \u2223 generator I\n[PROOFSTEP]\nexact \u27e8fun h \u21a6 \u27e8hx, h\u27e9, fun h \u21a6 h.2\u27e9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n[PROOFSTEP]\nlet a : R := generator (\u03d5.submoduleImage N)\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n[PROOFSTEP]\nlet d : R := IsPrincipal.generator (Submodule.span R {a, \u03c8 \u27e8y, hNO yN\u27e9})\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n[PROOFSTEP]\nhave d_dvd_left : d \u2223 a := (mem_iff_generator_dvd _).mp (subset_span (mem_insert _ _))\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n[PROOFSTEP]\nhave d_dvd_right : d \u2223 \u03c8 \u27e8y, hNO yN\u27e9 :=\n (mem_iff_generator_dvd _).mp (subset_span (mem_insert_of_mem _ (mem_singleton _)))\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n[PROOFSTEP]\nrefine' dvd_trans _ d_dvd_right\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2223 d\n[PROOFSTEP]\nrw [dvd_generator_iff, Ideal.span, \u2190 span_singleton_generator (Submodule.span R {a, \u03c8 \u27e8y, hNO yN\u27e9})]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u22a2 span R {generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})} =\n span R {generator (LinearMap.submoduleImage \u03d5 N)}\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2208 span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }}\n[PROOFSTEP]\nobtain \u27e8r\u2081, r\u2082, d_eq\u27e9 : \u2203 r\u2081 r\u2082 : R, d = r\u2081 * a + r\u2082 * \u03c8 \u27e8y, hNO yN\u27e9 :=\n by\n obtain \u27e8r\u2081, r\u2082', hr\u2082', hr\u2081\u27e9 := mem_span_insert.mp (IsPrincipal.generator_mem (Submodule.span R {a, \u03c8 \u27e8y, hNO yN\u27e9}))\n obtain \u27e8r\u2082, rfl\u27e9 := mem_span_singleton.mp hr\u2082'\n exact \u27e8r\u2081, r\u2082, hr\u2081\u27e9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u22a2 \u2203 r\u2081 r\u2082, d = r\u2081 * a + r\u2082 * \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n[PROOFSTEP]\nobtain \u27e8r\u2081, r\u2082', hr\u2082', hr\u2081\u27e9 := mem_span_insert.mp (IsPrincipal.generator_mem (Submodule.span R {a, \u03c8 \u27e8y, hNO yN\u27e9}))\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\nr\u2081 r\u2082' : R\nhr\u2082' : r\u2082' \u2208 span R {\u2191\u03c8 { val := y, property := (_ : y \u2208 O) }}\nhr\u2081 : generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }}) = r\u2081 \u2022 a + r\u2082'\n\u22a2 \u2203 r\u2081 r\u2082, d = r\u2081 * a + r\u2082 * \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n[PROOFSTEP]\nobtain \u27e8r\u2082, rfl\u27e9 := mem_span_singleton.mp hr\u2082'\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\nr\u2081 r\u2082 : R\nhr\u2082' : r\u2082 \u2022 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) } \u2208 span R {\u2191\u03c8 { val := y, property := (_ : y \u2208 O) }}\nhr\u2081 :\n generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }}) =\n r\u2081 \u2022 a + r\u2082 \u2022 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u22a2 \u2203 r\u2081 r\u2082, d = r\u2081 * a + r\u2082 * \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n[PROOFSTEP]\nexact \u27e8r\u2081, r\u2082, hr\u2081\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\nr\u2081 r\u2082 : R\nd_eq : d = r\u2081 * a + r\u2082 * \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u22a2 span R {generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})} =\n span R {generator (LinearMap.submoduleImage \u03d5 N)}\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2208 span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }}\n[PROOFSTEP]\nlet \u03c8' : O \u2192\u2097[R] R := r\u2081 \u2022 \u03d5 + r\u2082 \u2022 \u03c8\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\nr\u2081 r\u2082 : R\nd_eq : d = r\u2081 * a + r\u2082 * \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u03c8' : { x // x \u2208 O } \u2192\u2097[R] R := r\u2081 \u2022 \u03d5 + r\u2082 \u2022 \u03c8\n\u22a2 span R {generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})} =\n span R {generator (LinearMap.submoduleImage \u03d5 N)}\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2208 span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }}\n[PROOFSTEP]\nhave : span R { d } \u2264 \u03c8'.submoduleImage N :=\n by\n rw [span_le, singleton_subset_iff, SetLike.mem_coe, LinearMap.mem_submoduleImage_of_le hNO]\n refine' \u27e8y, yN, _\u27e9\n change r\u2081 * \u03d5 \u27e8y, hNO yN\u27e9 + r\u2082 * \u03c8 \u27e8y, hNO yN\u27e9 = d\n rw [d_eq, \u03d5y_eq]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\nr\u2081 r\u2082 : R\nd_eq : d = r\u2081 * a + r\u2082 * \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u03c8' : { x // x \u2208 O } \u2192\u2097[R] R := r\u2081 \u2022 \u03d5 + r\u2082 \u2022 \u03c8\n\u22a2 span R {d} \u2264 LinearMap.submoduleImage \u03c8' N\n[PROOFSTEP]\nrw [span_le, singleton_subset_iff, SetLike.mem_coe, LinearMap.mem_submoduleImage_of_le hNO]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\nr\u2081 r\u2082 : R\nd_eq : d = r\u2081 * a + r\u2082 * \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u03c8' : { x // x \u2208 O } \u2192\u2097[R] R := r\u2081 \u2022 \u03d5 + r\u2082 \u2022 \u03c8\n\u22a2 \u2203 y yN, \u2191\u03c8' { val := y, property := (_ : y \u2208 O) } = d\n[PROOFSTEP]\nrefine' \u27e8y, yN, _\u27e9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\nr\u2081 r\u2082 : R\nd_eq : d = r\u2081 * a + r\u2082 * \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u03c8' : { x // x \u2208 O } \u2192\u2097[R] R := r\u2081 \u2022 \u03d5 + r\u2082 \u2022 \u03c8\n\u22a2 \u2191\u03c8' { val := y, property := (_ : y \u2208 O) } = d\n[PROOFSTEP]\nchange r\u2081 * \u03d5 \u27e8y, hNO yN\u27e9 + r\u2082 * \u03c8 \u27e8y, hNO yN\u27e9 = d\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\nr\u2081 r\u2082 : R\nd_eq : d = r\u2081 * a + r\u2082 * \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u03c8' : { x // x \u2208 O } \u2192\u2097[R] R := r\u2081 \u2022 \u03d5 + r\u2082 \u2022 \u03c8\n\u22a2 r\u2081 * \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } + r\u2082 * \u2191\u03c8 { val := y, property := (_ : y \u2208 O) } = d\n[PROOFSTEP]\nrw [d_eq, \u03d5y_eq]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\nr\u2081 r\u2082 : R\nd_eq : d = r\u2081 * a + r\u2082 * \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u03c8' : { x // x \u2208 O } \u2192\u2097[R] R := r\u2081 \u2022 \u03d5 + r\u2082 \u2022 \u03c8\nthis : span R {d} \u2264 LinearMap.submoduleImage \u03c8' N\n\u22a2 span R {generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})} =\n span R {generator (LinearMap.submoduleImage \u03d5 N)}\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2208 span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }}\n[PROOFSTEP]\nrefine' le_antisymm (this.trans (le_of_eq _)) (Ideal.span_singleton_le_span_singleton.mpr d_dvd_left)\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\nr\u2081 r\u2082 : R\nd_eq : d = r\u2081 * a + r\u2082 * \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u03c8' : { x // x \u2208 O } \u2192\u2097[R] R := r\u2081 \u2022 \u03d5 + r\u2082 \u2022 \u03c8\nthis : span R {d} \u2264 LinearMap.submoduleImage \u03c8' N\n\u22a2 LinearMap.submoduleImage \u03c8' N = span R {generator (LinearMap.submoduleImage \u03d5 N)}\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2208 span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }}\n[PROOFSTEP]\nrw [span_singleton_generator]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\nr\u2081 r\u2082 : R\nd_eq : d = r\u2081 * a + r\u2082 * \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u03c8' : { x // x \u2208 O } \u2192\u2097[R] R := r\u2081 \u2022 \u03d5 + r\u2082 \u2022 \u03c8\nthis : span R {d} \u2264 LinearMap.submoduleImage \u03c8' N\n\u22a2 LinearMap.submoduleImage \u03c8' N = LinearMap.submoduleImage \u03d5 N\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2208 span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }}\n[PROOFSTEP]\napply (le_trans _ this).eq_of_not_gt (h\u03d5 \u03c8')\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\nr\u2081 r\u2082 : R\nd_eq : d = r\u2081 * a + r\u2082 * \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u03c8' : { x // x \u2208 O } \u2192\u2097[R] R := r\u2081 \u2022 \u03d5 + r\u2082 \u2022 \u03c8\nthis : span R {d} \u2264 LinearMap.submoduleImage \u03c8' N\n\u22a2 LinearMap.submoduleImage \u03d5 N \u2264 span R {d}\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2208 span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }}\n[PROOFSTEP]\nrw [\u2190 span_singleton_generator (\u03d5.submoduleImage N)]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\nr\u2081 r\u2082 : R\nd_eq : d = r\u2081 * a + r\u2082 * \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u03c8' : { x // x \u2208 O } \u2192\u2097[R] R := r\u2081 \u2022 \u03d5 + r\u2082 \u2022 \u03c8\nthis : span R {d} \u2264 LinearMap.submoduleImage \u03c8' N\n\u22a2 span R {generator (LinearMap.submoduleImage \u03d5 N)} \u2264 span R {d}\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2208 span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }}\n[PROOFSTEP]\nexact Ideal.span_singleton_le_span_singleton.mpr d_dvd_left\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : \u03b9 \u2192 M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\nh\u03d5 : \u2200 (\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\ny : M\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = generator (LinearMap.submoduleImage \u03d5 N)\n\u03c8 : { x // x \u2208 O } \u2192\u2097[R] R\na : R := generator (LinearMap.submoduleImage \u03d5 N)\nd : R := generator (span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }})\nd_dvd_left : d \u2223 a\nd_dvd_right : d \u2223 \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2208 span R {a, \u2191\u03c8 { val := y, property := (_ : y \u2208 O) }}\n[PROOFSTEP]\nexact subset_span (mem_insert _ _)\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave : \u2203 \u03d5 : M \u2192\u2097[R] R, \u2200 \u03c8 : M \u2192\u2097[R] R, \u00ac\u03d5.submoduleImage N < \u03c8.submoduleImage N :=\n by\n obtain \u27e8P, P_eq, P_max\u27e9 :=\n set_has_maximal_iff_noetherian.mpr (inferInstance : IsNoetherian R R) _\n (show (Set.range fun \u03c8 : M \u2192\u2097[R] R \u21a6 \u03c8.submoduleImage N).Nonempty from \u27e8_, Set.mem_range.mpr \u27e80, rfl\u27e9\u27e9)\n obtain \u27e8\u03d5, rfl\u27e9 := Set.mem_range.mp P_eq\n exact \u27e8\u03d5, fun \u03c8 h\u03c8 \u21a6 P_max _ \u27e8_, rfl\u27e9 h\u03c8\u27e9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\n\u22a2 \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n[PROOFSTEP]\nobtain \u27e8P, P_eq, P_max\u27e9 :=\n set_has_maximal_iff_noetherian.mpr (inferInstance : IsNoetherian R R) _\n (show (Set.range fun \u03c8 : M \u2192\u2097[R] R \u21a6 \u03c8.submoduleImage N).Nonempty from \u27e8_, Set.mem_range.mpr \u27e80, rfl\u27e9\u27e9)\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nP : Submodule R R\nP_eq : P \u2208 range fun \u03c8 => LinearMap.submoduleImage \u03c8 N\nP_max : \u2200 (I : Submodule R R), (I \u2208 range fun \u03c8 => LinearMap.submoduleImage \u03c8 N) \u2192 \u00acP < I\n\u22a2 \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n[PROOFSTEP]\nobtain \u27e8\u03d5, rfl\u27e9 := Set.mem_range.mp P_eq\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R\nP_eq : LinearMap.submoduleImage \u03d5 N \u2208 range fun \u03c8 => LinearMap.submoduleImage \u03c8 N\nP_max : \u2200 (I : Submodule R R), (I \u2208 range fun \u03c8 => LinearMap.submoduleImage \u03c8 N) \u2192 \u00acLinearMap.submoduleImage \u03d5 N < I\n\u22a2 \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n[PROOFSTEP]\nexact \u27e8\u03d5, fun \u03c8 h\u03c8 \u21a6 P_max _ \u27e8_, rfl\u27e9 h\u03c8\u27e9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nlet \u03d5 := this.choose\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave \u03d5_max := this.choose_spec\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nlet a := generator (\u03d5.submoduleImage N)\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave a_mem : a \u2208 \u03d5.submoduleImage N :=\n generator_mem\n _\n -- If `a` is zero, then the submodule is trivial. So let's assume `a \u2260 0`, `N \u2260 \u22a5`.\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nby_cases a_zero : a = 0\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : a = 0\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave := eq_bot_of_generator_maximal_submoduleImage_eq_zero b'M N_le_M \u03d5_max a_zero\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis\u271d : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\u271d\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this\u271d) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : a = 0\nthis : N = \u22a5\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\ncontradiction\n -- We claim that `\u03d5\u207b\u00b9 a = y` can be taken as basis element of `N`.\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nobtain \u27e8y, yN, \u03d5y_eq\u27e9 := (LinearMap.mem_submoduleImage_of_le N_le_M).mp a_mem\n[GOAL]\ncase neg.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave _\u03d5y_ne_zero : \u03d5 \u27e8y, N_le_M yN\u27e9 \u2260 0 := fun h \u21a6\n a_zero\n (\u03d5y_eq.symm.trans h)\n -- Write `y` as `a \u2022 y'` for some `y'`.\n[GOAL]\ncase neg.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave hdvd : \u2200 i, a \u2223 b'M.coord i \u27e8y, N_le_M yN\u27e9 := fun i \u21a6\n generator_maximal_submoduleImage_dvd N_le_M \u03d5_max y yN \u03d5y_eq (b'M.coord i)\n[GOAL]\ncase neg.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nhdvd : \u2200 (i : \u03b9), a \u2223 \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) }\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nchoose c hc using hdvd\n[GOAL]\ncase neg.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nlet y' : O := \u2211 i, c i \u2022 b'M i\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave y'M : y' \u2208 M := M.sum_mem fun i _ \u21a6 M.smul_mem (c i) (b'M i).2\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave mk_y' : (\u27e8y', y'M\u27e9 : M) = \u2211 i, c i \u2022 b'M i :=\n Subtype.ext\n (show y' = M.subtype _ by\n simp only [LinearMap.map_sum, LinearMap.map_smul]\n rfl)\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\n\u22a2 y' = \u2191(Submodule.subtype M) (\u2211 i : \u03b9, c i \u2022 \u2191b'M i)\n[PROOFSTEP]\nsimp only [LinearMap.map_sum, LinearMap.map_smul]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\n\u22a2 \u2211 x : \u03b9, \u2191(c x \u2022 \u2191b'M x) = \u2211 x : \u03b9, c x \u2022 \u2191(Submodule.subtype M) (\u2191b'M x)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave a_smul_y' : a \u2022 y' = y :=\n by\n refine Subtype.mk_eq_mk.mp (show (a \u2022 \u27e8y', y'M\u27e9 : M) = \u27e8y, N_le_M yN\u27e9 from ?_)\n rw [\u2190 b'M.sum_repr \u27e8y, N_le_M yN\u27e9, mk_y', Finset.smul_sum]\n refine' Finset.sum_congr rfl fun i _ \u21a6 _\n rw [\u2190 mul_smul, \u2190 hc]\n rfl\n -- We found a `y` and an `a`!\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\n\u22a2 a \u2022 y' = y\n[PROOFSTEP]\nrefine Subtype.mk_eq_mk.mp (show (a \u2022 \u27e8y', y'M\u27e9 : M) = \u27e8y, N_le_M yN\u27e9 from ?_)\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\n\u22a2 a \u2022 { val := y', property := y'M } = { val := y, property := (_ : y \u2208 M) }\n[PROOFSTEP]\nrw [\u2190 b'M.sum_repr \u27e8y, N_le_M yN\u27e9, mk_y', Finset.smul_sum]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\n\u22a2 \u2211 x : \u03b9, a \u2022 c x \u2022 \u2191b'M x = \u2211 i : \u03b9, \u2191(\u2191b'M.repr { val := y, property := (_ : y \u2208 M) }) i \u2022 \u2191b'M i\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun i _ \u21a6 _\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 a \u2022 c i \u2022 \u2191b'M i = \u2191(\u2191b'M.repr { val := y, property := (_ : y \u2208 M) }) i \u2022 \u2191b'M i\n[PROOFSTEP]\nrw [\u2190 mul_smul, \u2190 hc]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } \u2022 \u2191b'M i =\n \u2191(\u2191b'M.repr { val := y, property := (_ : y \u2208 M) }) i \u2022 \u2191b'M i\n[PROOFSTEP]\nrfl\n -- We found a `y` and an `a`!\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u22a2 \u2203 y,\n y \u2208 M \u2227\n \u2203 a x M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nrefine' \u27e8y', y'M, a, a_smul_y'.symm \u25b8 yN, _\u27e9\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u22a2 \u2203 M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave \u03d5y'_eq : \u03d5 \u27e8y', y'M\u27e9 = 1 :=\n mul_left_cancel\u2080 a_zero\n (calc\n a \u2022 \u03d5 \u27e8y', y'M\u27e9 = \u03d5 \u27e8a \u2022 y', _\u27e9 := (\u03d5.map_smul a \u27e8y', y'M\u27e9).symm\n _ = \u03d5 \u27e8y, N_le_M yN\u27e9 := by simp only [a_smul_y']\n _ = a := \u03d5y_eq\n _ = a * 1 := (mul_one a).symm)\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u22a2 \u2191\u03d5 { val := a \u2022 y', property := (_ : a \u2022 \u2191{ val := y', property := y'M } \u2208 M) } =\n \u2191\u03d5 { val := y, property := (_ : y \u2208 M) }\n[PROOFSTEP]\nsimp only [a_smul_y']\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u22a2 \u2203 M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave \u03d5y'_ne_zero : \u03d5 \u27e8y', y'M\u27e9 \u2260 0 := by simpa only [\u03d5y'_eq] using one_ne_zero\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u22a2 \u2191\u03d5 { val := y', property := y'M } \u2260 0\n[PROOFSTEP]\nsimpa only [\u03d5y'_eq] using one_ne_zero\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\n\u22a2 \u2203 M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nlet M' : Submodule R O := \u03d5.ker.map M.subtype\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\n\u22a2 \u2203 M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nlet N' : Submodule R O := (\u03d5.comp (ofLe N_le_M)).ker.map N.subtype\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\n\u22a2 \u2203 M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave M'_le_M : M' \u2264 M := M.map_subtype_le (LinearMap.ker \u03d5)\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\n\u22a2 \u2203 M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave N'_le_M' : N' \u2264 M' := by\n intro x hx\n simp only [mem_map, LinearMap.mem_ker] at hx \u22a2\n obtain \u27e8\u27e8x, xN\u27e9, hx, rfl\u27e9 := hx\n exact \u27e8\u27e8x, N_le_M xN\u27e9, hx, rfl\u27e9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\n\u22a2 N' \u2264 M'\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nx : O\nhx : x \u2208 N'\n\u22a2 x \u2208 M'\n[PROOFSTEP]\nsimp only [mem_map, LinearMap.mem_ker] at hx \u22a2\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nx : O\nhx : \u2203 y, \u2191(LinearMap.comp (Exists.choose this) (ofLe N_le_M)) y = 0 \u2227 \u2191(Submodule.subtype N) y = x\n\u22a2 \u2203 y, \u2191(Exists.choose this) y = 0 \u2227 \u2191(Submodule.subtype M) y = x\n[PROOFSTEP]\nobtain \u27e8\u27e8x, xN\u27e9, hx, rfl\u27e9 := hx\n[GOAL]\ncase intro.mk.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nx : O\nxN : x \u2208 N\nhx : \u2191(LinearMap.comp (Exists.choose this) (ofLe N_le_M)) { val := x, property := xN } = 0\n\u22a2 \u2203 y, \u2191(Exists.choose this) y = 0 \u2227 \u2191(Submodule.subtype M) y = \u2191(Submodule.subtype N) { val := x, property := xN }\n[PROOFSTEP]\nexact \u27e8\u27e8x, N_le_M xN\u27e9, hx, rfl\u27e9\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\n\u22a2 \u2203 M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave N'_le_N : N' \u2264 N :=\n N.map_subtype_le\n (LinearMap.ker (\u03d5.comp (ofLe N_le_M)))\n -- So fill in those results as well.\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\n\u22a2 \u2203 M',\n M' \u2264 M \u2227\n \u2203 N',\n N' \u2264 N \u2227\n \u2203 _N'_le_M' _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nrefine'\n \u27e8M', M'_le_M, N', N'_le_N, N'_le_M', _\u27e9\n -- Note that `y'` is orthogonal to `M'`.\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\n\u22a2 \u2203 _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave y'_ortho_M' : \u2200 (c : R), \u2200 z \u2208 M', c \u2022 y' + z = 0 \u2192 c = 0 :=\n by\n intro c x xM' hc\n obtain \u27e8\u27e8x, xM\u27e9, hx', rfl\u27e9 := Submodule.mem_map.mp xM'\n rw [LinearMap.mem_ker] at hx' \n have hc' : (c \u2022 \u27e8y', y'M\u27e9 + \u27e8x, xM\u27e9 : M) = 0 := by exact @Subtype.coe_injective O (\u00b7 \u2208 M) _ _ hc\n simpa only [LinearMap.map_add, LinearMap.map_zero, LinearMap.map_smul, smul_eq_mul, add_zero, mul_eq_zero,\n \u03d5y'_ne_zero, hx', or_false_iff] using congr_arg \u03d5 hc'\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\n\u22a2 \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\n[PROOFSTEP]\nintro c x xM' hc\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc\u271d : \u03b9 \u2192 R\nhc\u271d : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c\u271d i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c\u271d i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c\u271d i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\nc : R\nx : O\nxM' : x \u2208 M'\nhc : c \u2022 y' + x = 0\n\u22a2 c = 0\n[PROOFSTEP]\nobtain \u27e8\u27e8x, xM\u27e9, hx', rfl\u27e9 := Submodule.mem_map.mp xM'\n[GOAL]\ncase intro.mk.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc\u271d : \u03b9 \u2192 R\nhc\u271d : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c\u271d i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c\u271d i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c\u271d i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\nc : R\nx : O\nxM : x \u2208 M\nhx' : { val := x, property := xM } \u2208 LinearMap.ker \u03d5\nxM' : \u2191(Submodule.subtype M) { val := x, property := xM } \u2208 M'\nhc : c \u2022 y' + \u2191(Submodule.subtype M) { val := x, property := xM } = 0\n\u22a2 c = 0\n[PROOFSTEP]\nrw [LinearMap.mem_ker] at hx' \n[GOAL]\ncase intro.mk.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc\u271d : \u03b9 \u2192 R\nhc\u271d : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c\u271d i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c\u271d i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c\u271d i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\nc : R\nx : O\nxM : x \u2208 M\nhx' : \u2191\u03d5 { val := x, property := xM } = 0\nxM' : \u2191(Submodule.subtype M) { val := x, property := xM } \u2208 M'\nhc : c \u2022 y' + \u2191(Submodule.subtype M) { val := x, property := xM } = 0\n\u22a2 c = 0\n[PROOFSTEP]\nhave hc' : (c \u2022 \u27e8y', y'M\u27e9 + \u27e8x, xM\u27e9 : M) = 0 := by exact @Subtype.coe_injective O (\u00b7 \u2208 M) _ _ hc\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc\u271d : \u03b9 \u2192 R\nhc\u271d : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c\u271d i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c\u271d i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c\u271d i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\nc : R\nx : O\nxM : x \u2208 M\nhx' : \u2191\u03d5 { val := x, property := xM } = 0\nxM' : \u2191(Submodule.subtype M) { val := x, property := xM } \u2208 M'\nhc : c \u2022 y' + \u2191(Submodule.subtype M) { val := x, property := xM } = 0\n\u22a2 c \u2022 { val := y', property := y'M } + { val := x, property := xM } = 0\n[PROOFSTEP]\nexact @Subtype.coe_injective O (\u00b7 \u2208 M) _ _ hc\n[GOAL]\ncase intro.mk.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc\u271d : \u03b9 \u2192 R\nhc\u271d : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c\u271d i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c\u271d i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c\u271d i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\nc : R\nx : O\nxM : x \u2208 M\nhx' : \u2191\u03d5 { val := x, property := xM } = 0\nxM' : \u2191(Submodule.subtype M) { val := x, property := xM } \u2208 M'\nhc : c \u2022 y' + \u2191(Submodule.subtype M) { val := x, property := xM } = 0\nhc' : c \u2022 { val := y', property := y'M } + { val := x, property := xM } = 0\n\u22a2 c = 0\n[PROOFSTEP]\nsimpa only [LinearMap.map_add, LinearMap.map_zero, LinearMap.map_smul, smul_eq_mul, add_zero, mul_eq_zero, \u03d5y'_ne_zero,\n hx', or_false_iff] using congr_arg \u03d5 hc'\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\n\u22a2 \u2203 _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nhave ay'_ortho_N' : \u2200 (c : R), \u2200 z \u2208 N', c \u2022 a \u2022 y' + z = 0 \u2192 c = 0 :=\n by\n intro c z zN' hc\n refine' (mul_eq_zero.mp (y'_ortho_M' (a * c) z (N'_le_M' zN') _)).resolve_left a_zero\n rw [mul_comm, mul_smul, hc]\n -- So we can extend a basis for `N'` with `y`\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\n\u22a2 \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\n[PROOFSTEP]\nintro c z zN' hc\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc\u271d : \u03b9 \u2192 R\nhc\u271d : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c\u271d i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c\u271d i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c\u271d i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nc : R\nz : O\nzN' : z \u2208 N'\nhc : c \u2022 a \u2022 y' + z = 0\n\u22a2 c = 0\n[PROOFSTEP]\nrefine' (mul_eq_zero.mp (y'_ortho_M' (a * c) z (N'_le_M' zN') _)).resolve_left a_zero\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc\u271d : \u03b9 \u2192 R\nhc\u271d : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c\u271d i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c\u271d i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c\u271d i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nc : R\nz : O\nzN' : z \u2208 N'\nhc : c \u2022 a \u2022 y' + z = 0\n\u22a2 (a * c) \u2022 y' + z = 0\n[PROOFSTEP]\nrw [mul_comm, mul_smul, hc]\n -- So we can extend a basis for `N'` with `y`\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\n\u22a2 \u2203 _y_ortho_M' _ay_ortho_N',\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nrefine' \u27e8y'_ortho_M', ay'_ortho_N', fun n' bN' \u21a6 \u27e8_, _\u27e9\u27e9\n[GOAL]\ncase neg.intro.intro.intro.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\n\u22a2 Basis (Fin (n' + 1)) R { x // x \u2208 N }\n[PROOFSTEP]\nrefine' Basis.mkFinConsOfLE y yN bN' N'_le_N _ _\n[GOAL]\ncase neg.intro.intro.intro.refine'_1.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\n\u22a2 \u2200 (c : R) (x : O), x \u2208 N' \u2192 c \u2022 y + x = 0 \u2192 c = 0\n[PROOFSTEP]\nintro c z zN' hc\n[GOAL]\ncase neg.intro.intro.intro.refine'_1.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc\u271d : \u03b9 \u2192 R\nhc\u271d : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c\u271d i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c\u271d i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c\u271d i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nc : R\nz : O\nzN' : z \u2208 N'\nhc : c \u2022 y + z = 0\n\u22a2 c = 0\n[PROOFSTEP]\nrefine' ay'_ortho_N' c z zN' _\n[GOAL]\ncase neg.intro.intro.intro.refine'_1.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc\u271d : \u03b9 \u2192 R\nhc\u271d : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c\u271d i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c\u271d i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c\u271d i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nc : R\nz : O\nzN' : z \u2208 N'\nhc : c \u2022 y + z = 0\n\u22a2 c \u2022 a \u2022 y' + z = 0\n[PROOFSTEP]\nrwa [\u2190 a_smul_y'] at hc \n[GOAL]\ncase neg.intro.intro.intro.refine'_1.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\n\u22a2 \u2200 (z : O), z \u2208 N \u2192 \u2203 c, z + c \u2022 y \u2208 N'\n[PROOFSTEP]\nintro z zN\n[GOAL]\ncase neg.intro.intro.intro.refine'_1.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nz : O\nzN : z \u2208 N\n\u22a2 \u2203 c, z + c \u2022 y \u2208 N'\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 : _ \u2223 \u03d5 \u27e8z, N_le_M zN\u27e9 := generator_submoduleImage_dvd_of_mem N_le_M \u03d5 zN\n[GOAL]\ncase neg.intro.intro.intro.refine'_1.refine'_2.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nz : O\nzN : z \u2208 N\nb : R\nhb : \u2191\u03d5 { val := z, property := (_ : z \u2208 M) } = generator (LinearMap.submoduleImage \u03d5 N) * b\n\u22a2 \u2203 c, z + c \u2022 y \u2208 N'\n[PROOFSTEP]\nrefine' \u27e8-b, Submodule.mem_map.mpr \u27e8\u27e8_, N.sub_mem zN (N.smul_mem b yN)\u27e9, _, _\u27e9\u27e9\n[GOAL]\ncase neg.intro.intro.intro.refine'_1.refine'_2.intro.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nz : O\nzN : z \u2208 N\nb : R\nhb : \u2191\u03d5 { val := z, property := (_ : z \u2208 M) } = generator (LinearMap.submoduleImage \u03d5 N) * b\n\u22a2 { val := z - b \u2022 y, property := (_ : z - b \u2022 y \u2208 N) } \u2208 LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M))\n[PROOFSTEP]\nrefine' LinearMap.mem_ker.mpr (show \u03d5 (\u27e8z, N_le_M zN\u27e9 - b \u2022 \u27e8y, N_le_M yN\u27e9) = 0 from _)\n[GOAL]\ncase neg.intro.intro.intro.refine'_1.refine'_2.intro.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nz : O\nzN : z \u2208 N\nb : R\nhb : \u2191\u03d5 { val := z, property := (_ : z \u2208 M) } = generator (LinearMap.submoduleImage \u03d5 N) * b\n\u22a2 \u2191\u03d5 ({ val := z, property := (_ : z \u2208 M) } - b \u2022 { val := y, property := (_ : y \u2208 M) }) = 0\n[PROOFSTEP]\nrw [LinearMap.map_sub, LinearMap.map_smul, hb, \u03d5y_eq, smul_eq_mul, mul_comm, sub_self]\n[GOAL]\ncase neg.intro.intro.intro.refine'_1.refine'_2.intro.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nz : O\nzN : z \u2208 N\nb : R\nhb : \u2191\u03d5 { val := z, property := (_ : z \u2208 M) } = generator (LinearMap.submoduleImage \u03d5 N) * b\n\u22a2 \u2191(Submodule.subtype N) { val := z - b \u2022 y, property := (_ : z - b \u2022 y \u2208 N) } = z + -b \u2022 y\n[PROOFSTEP]\nsimp only [sub_eq_add_neg, neg_smul, coeSubtype]\n -- And extend a basis for `M'` with `y'`\n[GOAL]\ncase neg.intro.intro.intro.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\n\u22a2 \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as',\n \u2200 (i : Fin (n' + 1)),\n \u2191(\u2191(Basis.mkFinConsOfLE y yN bN' N'_le_N (_ : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 y + z = 0 \u2192 c = 0)\n (_ : \u2200 (z : O), z \u2208 N \u2192 \u2203 c, z + c \u2022 y \u2208 N'))\n i) =\n as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nintro m' hn'm' bM'\n[GOAL]\ncase neg.intro.intro.intro.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\n\u22a2 \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as',\n \u2200 (i : Fin (n' + 1)),\n \u2191(\u2191(Basis.mkFinConsOfLE y yN bN' N'_le_N (_ : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 y + z = 0 \u2192 c = 0)\n (_ : \u2200 (z : O), z \u2208 N \u2192 \u2203 c, z + c \u2022 y \u2208 N'))\n i) =\n as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n[PROOFSTEP]\nrefine' \u27e8Nat.succ_le_succ hn'm', _, _\u27e9\n[GOAL]\ncase neg.intro.intro.intro.refine'_2.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\n\u22a2 Basis (Fin (m' + 1)) R { x // x \u2208 M }\n[PROOFSTEP]\nrefine' Basis.mkFinConsOfLE y' y'M bM' M'_le_M y'_ortho_M' _\n[GOAL]\ncase neg.intro.intro.intro.refine'_2.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\n\u22a2 \u2200 (z : O), z \u2208 M \u2192 \u2203 c, z + c \u2022 y' \u2208 M'\n[PROOFSTEP]\nintro z zM\n[GOAL]\ncase neg.intro.intro.intro.refine'_2.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nz : O\nzM : z \u2208 M\n\u22a2 \u2203 c, z + c \u2022 y' \u2208 M'\n[PROOFSTEP]\nrefine' \u27e8-\u03d5 \u27e8z, zM\u27e9, \u27e8\u27e8z, zM\u27e9 - \u03d5 \u27e8z, zM\u27e9 \u2022 \u27e8y', y'M\u27e9, LinearMap.mem_ker.mpr _, _\u27e9\u27e9\n[GOAL]\ncase neg.intro.intro.intro.refine'_2.refine'_1.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nz : O\nzM : z \u2208 M\n\u22a2 \u2191\u03d5 ({ val := z, property := zM } - \u2191\u03d5 { val := z, property := zM } \u2022 { val := y', property := y'M }) = 0\n[PROOFSTEP]\nrw [LinearMap.map_sub, LinearMap.map_smul, \u03d5y'_eq, smul_eq_mul, mul_one, sub_self]\n[GOAL]\ncase neg.intro.intro.intro.refine'_2.refine'_1.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nz : O\nzM : z \u2208 M\n\u22a2 \u2191(Submodule.subtype M)\n ({ val := z, property := zM } - \u2191\u03d5 { val := z, property := zM } \u2022 { val := y', property := y'M }) =\n z + -\u2191\u03d5 { val := z, property := zM } \u2022 y'\n[PROOFSTEP]\nrw [LinearMap.map_sub, LinearMap.map_smul, sub_eq_add_neg, neg_smul]\n[GOAL]\ncase neg.intro.intro.intro.refine'_2.refine'_1.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nz : O\nzM : z \u2208 M\n\u22a2 \u2191(Submodule.subtype M) { val := z, property := zM } +\n -(\u2191\u03d5 { val := z, property := zM } \u2022 \u2191(Submodule.subtype M) { val := y', property := y'M }) =\n z + -(\u2191\u03d5 { val := z, property := zM } \u2022 y')\n[PROOFSTEP]\nrfl\n -- It remains to show the extended bases are compatible with each other.\n[GOAL]\ncase neg.intro.intro.intro.refine'_2.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\n\u22a2 \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as',\n \u2200 (i : Fin (n' + 1)),\n \u2191(\u2191(Basis.mkFinConsOfLE y yN bN' N'_le_N (_ : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 y + z = 0 \u2192 c = 0)\n (_ : \u2200 (z : O), z \u2208 N \u2192 \u2203 c, z + c \u2022 y \u2208 N'))\n i) =\n as' i \u2022\n \u2191(\u2191(Basis.mkFinConsOfLE y' y'M bM' M'_le_M y'_ortho_M' (_ : \u2200 (z : O), z \u2208 M \u2192 \u2203 c, z + c \u2022 y' \u2208 M'))\n (Fin.castLE (_ : Nat.succ n' \u2264 Nat.succ m') i))\n[PROOFSTEP]\nintro as h\n[GOAL]\ncase neg.intro.intro.intro.refine'_2.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nas : Fin n' \u2192 R\nh : \u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))\n\u22a2 \u2203 as',\n \u2200 (i : Fin (n' + 1)),\n \u2191(\u2191(Basis.mkFinConsOfLE y yN bN' N'_le_N (_ : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 y + z = 0 \u2192 c = 0)\n (_ : \u2200 (z : O), z \u2208 N \u2192 \u2203 c, z + c \u2022 y \u2208 N'))\n i) =\n as' i \u2022\n \u2191(\u2191(Basis.mkFinConsOfLE y' y'M bM' M'_le_M y'_ortho_M' (_ : \u2200 (z : O), z \u2208 M \u2192 \u2203 c, z + c \u2022 y' \u2208 M'))\n (Fin.castLE (_ : Nat.succ n' \u2264 Nat.succ m') i))\n[PROOFSTEP]\nrefine' \u27e8Fin.cons a as, _\u27e9\n[GOAL]\ncase neg.intro.intro.intro.refine'_2.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nas : Fin n' \u2192 R\nh : \u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))\n\u22a2 \u2200 (i : Fin (n' + 1)),\n \u2191(\u2191(Basis.mkFinConsOfLE y yN bN' N'_le_N (_ : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 y + z = 0 \u2192 c = 0)\n (_ : \u2200 (z : O), z \u2208 N \u2192 \u2203 c, z + c \u2022 y \u2208 N'))\n i) =\n Fin.cons a as i \u2022\n \u2191(\u2191(Basis.mkFinConsOfLE y' y'M bM' M'_le_M y'_ortho_M' (_ : \u2200 (z : O), z \u2208 M \u2192 \u2203 c, z + c \u2022 y' \u2208 M'))\n (Fin.castLE (_ : Nat.succ n' \u2264 Nat.succ m') i))\n[PROOFSTEP]\nintro i\n[GOAL]\ncase neg.intro.intro.intro.refine'_2.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nas : Fin n' \u2192 R\nh : \u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))\ni : Fin (n' + 1)\n\u22a2 \u2191(\u2191(Basis.mkFinConsOfLE y yN bN' N'_le_N (_ : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 y + z = 0 \u2192 c = 0)\n (_ : \u2200 (z : O), z \u2208 N \u2192 \u2203 c, z + c \u2022 y \u2208 N'))\n i) =\n Fin.cons a as i \u2022\n \u2191(\u2191(Basis.mkFinConsOfLE y' y'M bM' M'_le_M y'_ortho_M' (_ : \u2200 (z : O), z \u2208 M \u2192 \u2203 c, z + c \u2022 y' \u2208 M'))\n (Fin.castLE (_ : Nat.succ n' \u2264 Nat.succ m') i))\n[PROOFSTEP]\nrw [Basis.coe_mkFinConsOfLE, Basis.coe_mkFinConsOfLE]\n[GOAL]\ncase neg.intro.intro.intro.refine'_2.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nas : Fin n' \u2192 R\nh : \u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))\ni : Fin (n' + 1)\n\u22a2 \u2191(Fin.cons { val := y, property := yN } (\u2191(ofLe N'_le_N) \u2218 \u2191bN') i) =\n Fin.cons a as i \u2022\n \u2191(Fin.cons { val := y', property := y'M } (\u2191(ofLe M'_le_M) \u2218 \u2191bM') (Fin.castLE (_ : Nat.succ n' \u2264 Nat.succ m') i))\n[PROOFSTEP]\nrefine' Fin.cases _ (fun i \u21a6 _) i\n[GOAL]\ncase neg.intro.intro.intro.refine'_2.refine'_2.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nas : Fin n' \u2192 R\nh : \u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))\ni : Fin (n' + 1)\n\u22a2 \u2191(Fin.cons { val := y, property := yN } (\u2191(ofLe N'_le_N) \u2218 \u2191bN') 0) =\n Fin.cons a as 0 \u2022\n \u2191(Fin.cons { val := y', property := y'M } (\u2191(ofLe M'_le_M) \u2218 \u2191bM') (Fin.castLE (_ : Nat.succ n' \u2264 Nat.succ m') 0))\n[PROOFSTEP]\nsimp only [Fin.cons_zero, Fin.castLE_zero]\n[GOAL]\ncase neg.intro.intro.intro.refine'_2.refine'_2.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nas : Fin n' \u2192 R\nh : \u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))\ni : Fin (n' + 1)\n\u22a2 y = generator (LinearMap.submoduleImage (Exists.choose this) N) \u2022 \u2211 x : \u03b9, \u2191(c x \u2022 \u2191b'M x)\n[PROOFSTEP]\nexact a_smul_y'.symm\n[GOAL]\ncase neg.intro.intro.intro.refine'_2.refine'_2.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nas : Fin n' \u2192 R\nh : \u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))\ni\u271d : Fin (n' + 1)\ni : Fin n'\n\u22a2 \u2191(Fin.cons { val := y, property := yN } (\u2191(ofLe N'_le_N) \u2218 \u2191bN') (Fin.succ i)) =\n Fin.cons a as (Fin.succ i) \u2022\n \u2191(Fin.cons { val := y', property := y'M } (\u2191(ofLe M'_le_M) \u2218 \u2191bM')\n (Fin.castLE (_ : Nat.succ n' \u2264 Nat.succ m') (Fin.succ i)))\n[PROOFSTEP]\nrw [Fin.castLE_succ]\n[GOAL]\ncase neg.intro.intro.intro.refine'_2.refine'_2.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM\u271d : Type u_3\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nb : \u03b9 \u2192 M\u271d\ninst\u271d\u00b2 : Finite \u03b9\nO : Type u_4\ninst\u271d\u00b9 : AddCommGroup O\ninst\u271d : Module R O\nM N : Submodule R O\nb'M : Basis \u03b9 R { x // x \u2208 M }\nN_bot : N \u2260 \u22a5\nN_le_M : N \u2264 M\nthis : \u2203 \u03d5, \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage \u03d5 N < LinearMap.submoduleImage \u03c8 N\n\u03d5 : { x // x \u2208 M } \u2192\u2097[R] R := Exists.choose this\n\u03d5_max : \u2200 (\u03c8 : { x // x \u2208 M } \u2192\u2097[R] R), \u00acLinearMap.submoduleImage (Exists.choose this) N < LinearMap.submoduleImage \u03c8 N\na : R := generator (LinearMap.submoduleImage \u03d5 N)\na_mem : a \u2208 LinearMap.submoduleImage \u03d5 N\na_zero : \u00aca = 0\ny : O\nyN : y \u2208 N\n\u03d5y_eq : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } = a\n_\u03d5y_ne_zero : \u2191\u03d5 { val := y, property := (_ : y \u2208 M) } \u2260 0\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(Basis.coord b'M i) { val := y, property := (_ : y \u2208 M) } = a * c i\nval\u271d : Fintype \u03b9\ny' : O := \u2211 i : \u03b9, \u2191(c i \u2022 \u2191b'M i)\ny'M : y' \u2208 M\nmk_y' : { val := y', property := y'M } = \u2211 i : \u03b9, c i \u2022 \u2191b'M i\na_smul_y' : a \u2022 y' = y\n\u03d5y'_eq : \u2191\u03d5 { val := y', property := y'M } = 1\n\u03d5y'_ne_zero : \u2191\u03d5 { val := y', property := y'M } \u2260 0\nM' : Submodule R O := map (Submodule.subtype M) (LinearMap.ker \u03d5)\nN' : Submodule R O := map (Submodule.subtype N) (LinearMap.ker (LinearMap.comp \u03d5 (ofLe N_le_M)))\nM'_le_M : M' \u2264 M\nN'_le_M' : N' \u2264 M'\nN'_le_N : N' \u2264 N\ny'_ortho_M' : \u2200 (c : R) (z : O), z \u2208 M' \u2192 c \u2022 y' + z = 0 \u2192 c = 0\nay'_ortho_N' : \u2200 (c : R) (z : O), z \u2208 N' \u2192 c \u2022 a \u2022 y' + z = 0 \u2192 c = 0\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nm' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nas : Fin n' \u2192 R\nh : \u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))\ni\u271d : Fin (n' + 1)\ni : Fin n'\n\u22a2 \u2191(Fin.cons { val := y, property := yN } (\u2191(ofLe N'_le_N) \u2218 \u2191bN') (Fin.succ i)) =\n Fin.cons a as (Fin.succ i) \u2022\n \u2191(Fin.cons { val := y', property := y'M } (\u2191(ofLe M'_le_M) \u2218 \u2191bM') (Fin.succ (Fin.castLE (_ : n' \u2264 m') i)))\n[PROOFSTEP]\nsimp only [Fin.cons_succ, Function.comp_apply, coe_ofLe, map_coe, coeSubtype, h i]\n[GOAL]\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN : Submodule R M\n\u22a2 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N })\n[PROOFSTEP]\nhaveI := Classical.decEq M\n[GOAL]\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN : Submodule R M\nthis : DecidableEq M\n\u22a2 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N })\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN : Submodule R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N })\n[PROOFSTEP]\ninduction' N using inductionOnRank with N ih\n[GOAL]\ncase intro.b\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN : Submodule R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\n\u22a2 Basis ?m.114794 R M\n[PROOFSTEP]\nexact b\n[GOAL]\ncase intro.ih\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\nN : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 N \u2192\n \u2200 (x : M),\n x \u2208 N \u2192 (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N' })\n\u22a2 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N })\ncase intro.inst\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN : Submodule R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\nlet b' := (b.reindex (Fintype.equivFin \u03b9)).map (LinearEquiv.ofTop _ rfl).symm\n[GOAL]\ncase intro.ih\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\nN : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 N \u2192\n \u2200 (x : M),\n x \u2208 N \u2192 (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N' })\nb' : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 \u22a4 } :=\n Basis.map (Basis.reindex b (Fintype.equivFin \u03b9)) (LinearEquiv.symm (LinearEquiv.ofTop \u22a4 (_ : \u22a4 = \u22a4)))\n\u22a2 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N })\ncase intro.inst\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN : Submodule R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\nby_cases N_bot : N = \u22a5\n[GOAL]\ncase pos\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\nN : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 N \u2192\n \u2200 (x : M),\n x \u2208 N \u2192 (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N' })\nb' : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 \u22a4 } :=\n Basis.map (Basis.reindex b (Fintype.equivFin \u03b9)) (LinearEquiv.symm (LinearEquiv.ofTop \u22a4 (_ : \u22a4 = \u22a4)))\nN_bot : N = \u22a5\n\u22a2 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N })\n[PROOFSTEP]\nsubst N_bot\n[GOAL]\ncase pos\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\nb' : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 \u22a4 } :=\n Basis.map (Basis.reindex b (Fintype.equivFin \u03b9)) (LinearEquiv.symm (LinearEquiv.ofTop \u22a4 (_ : \u22a4 = \u22a4)))\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 \u22a5 \u2192\n \u2200 (x : M),\n x \u2208 \u22a5 \u2192 (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N' })\n\u22a2 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 \u22a5 })\n[PROOFSTEP]\nexact \u27e80, \u27e8Basis.empty _\u27e9\u27e9\n[GOAL]\ncase neg\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\nN : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 N \u2192\n \u2200 (x : M),\n x \u2208 N \u2192 (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N' })\nb' : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 \u22a4 } :=\n Basis.map (Basis.reindex b (Fintype.equivFin \u03b9)) (LinearEquiv.symm (LinearEquiv.ofTop \u22a4 (_ : \u22a4 = \u22a4)))\nN_bot : \u00acN = \u22a5\n\u22a2 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N })\ncase intro.inst\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN : Submodule R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\nobtain \u27e8y, -, a, hay, M', -, N', N'_le_N, -, -, ay_ortho, h'\u27e9 := Submodule.basis_of_pid_aux \u22a4 N b' N_bot le_top\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\nN : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 N \u2192\n \u2200 (x : M),\n x \u2208 N \u2192 (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N' })\nb' : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 \u22a4 } :=\n Basis.map (Basis.reindex b (Fintype.equivFin \u03b9)) (LinearEquiv.symm (LinearEquiv.ofTop \u22a4 (_ : \u22a4 = \u22a4)))\nN_bot : \u00acN = \u22a5\ny : M\na : R\nhay : a \u2022 y \u2208 N\nM' N' : Submodule R M\nN'_le_N : N' \u2264 N\nay_ortho : \u2200 (c : R) (z : M), z \u2208 N' \u2192 c \u2022 a \u2022 y + z = 0 \u2192 c = 0\nh' :\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n\u22a2 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N })\ncase intro.inst\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN : Submodule R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\nobtain \u27e8n', \u27e8bN'\u27e9\u27e9 := ih N' N'_le_N _ hay ay_ortho\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\nN : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 N \u2192\n \u2200 (x : M),\n x \u2208 N \u2192 (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N' })\nb' : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 \u22a4 } :=\n Basis.map (Basis.reindex b (Fintype.equivFin \u03b9)) (LinearEquiv.symm (LinearEquiv.ofTop \u22a4 (_ : \u22a4 = \u22a4)))\nN_bot : \u00acN = \u22a5\ny : M\na : R\nhay : a \u2022 y \u2208 N\nM' N' : Submodule R M\nN'_le_N : N' \u2264 N\nay_ortho : \u2200 (c : R) (z : M), z \u2208 N' \u2192 c \u2022 a \u2022 y + z = 0 \u2192 c = 0\nh' :\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\n\u22a2 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N })\ncase intro.inst\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN : Submodule R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\nobtain \u27e8bN, _hbN\u27e9 := h' n' bN'\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\nN : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 N \u2192\n \u2200 (x : M),\n x \u2208 N \u2192 (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N' })\nb' : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 \u22a4 } :=\n Basis.map (Basis.reindex b (Fintype.equivFin \u03b9)) (LinearEquiv.symm (LinearEquiv.ofTop \u22a4 (_ : \u22a4 = \u22a4)))\nN_bot : \u00acN = \u22a5\ny : M\na : R\nhay : a \u2022 y \u2208 N\nM' N' : Submodule R M\nN'_le_N : N' \u2264 N\nay_ortho : \u2200 (c : R) (z : M), z \u2208 N' \u2192 c \u2022 a \u2022 y + z = 0 \u2192 c = 0\nh' :\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\nn' : \u2115\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nbN : Basis (Fin (n' + 1)) R { x // x \u2208 N }\n_hbN :\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n\u22a2 \u2203 n, Nonempty (Basis (Fin n) R { x // x \u2208 N })\ncase intro.inst\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN : Submodule R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\nexact \u27e8n' + 1, \u27e8bN\u27e9\u27e9\n[GOAL]\ncase intro.inst\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN : Submodule R M\nthis : DecidableEq M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\n\u22a2 basisOfPid b \u22a5 = { fst := 0, snd := Basis.empty { x // x \u2208 \u22a5 } }\n[PROOFSTEP]\nobtain \u27e8n, b'\u27e9 := Submodule.basisOfPid b \u22a5\n[GOAL]\ncase mk\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nn : \u2115\nb' : Basis (Fin n) R { x // x \u2208 \u22a5 }\n\u22a2 { fst := n, snd := b' } = { fst := 0, snd := Basis.empty { x // x \u2208 \u22a5 } }\n[PROOFSTEP]\nlet e : Fin n \u2243 Fin 0 := b'.indexEquiv (Basis.empty _ : Basis (Fin 0) R (\u22a5 : Submodule R M))\n[GOAL]\ncase mk\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nn : \u2115\nb' : Basis (Fin n) R { x // x \u2208 \u22a5 }\ne : Fin n \u2243 Fin 0 := Basis.indexEquiv b' (Basis.empty { x // x \u2208 \u22a5 })\n\u22a2 { fst := n, snd := b' } = { fst := 0, snd := Basis.empty { x // x \u2208 \u22a5 } }\n[PROOFSTEP]\nobtain rfl : n = 0 := by simpa using Fintype.card_eq.mpr \u27e8e\u27e9\n[GOAL]\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nn : \u2115\nb' : Basis (Fin n) R { x // x \u2208 \u22a5 }\ne : Fin n \u2243 Fin 0 := Basis.indexEquiv b' (Basis.empty { x // x \u2208 \u22a5 })\n\u22a2 n = 0\n[PROOFSTEP]\nsimpa using Fintype.card_eq.mpr \u27e8e\u27e9\n[GOAL]\ncase mk\n\u03b9\u271d : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9\u271d \u2192 M\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nb' : Basis (Fin 0) R { x // x \u2208 \u22a5 }\ne : Fin 0 \u2243 Fin 0 := Basis.indexEquiv b' (Basis.empty { x // x \u2208 \u22a5 })\n\u22a2 { fst := 0, snd := b' } = { fst := 0, snd := Basis.empty { x // x \u2208 \u22a5 } }\n[PROOFSTEP]\nexact Sigma.eq rfl (Basis.eq_of_apply_eq <| finZeroElim)\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nclassical\n -- We define `N` as the submodule spanned by a maximal linear independent subfamily of `s`\nhave := exists_maximal_independent R s\nlet I : Set \u03b9 := this.choose\nobtain\n \u27e8indepI : LinearIndependent R (s \u2218 (fun x => x) : I \u2192 M), hI :\n \u2200 (i) (_ : i \u2209 I), \u2203 a : R, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\u27e9 :=\n this.choose_spec\nlet N :=\n span R\n (range <| (s \u2218 (fun x => x) : I \u2192 M))\n -- same as `span R (s '' I)` but more convenient\nlet _sI : I \u2192 N := fun i \u21a6\n \u27e8s i.1, subset_span (mem_range_self i)\u27e9\n -- `s` restricted to `I` is a basis of `N`\nlet sI_basis : Basis I R N := Basis.span indepI\nhave exists_a : \u2200 i : \u03b9, \u2203 a : R, a \u2260 0 \u2227 a \u2022 s i \u2208 N :=\n by\n intro i\n by_cases hi : i \u2208 I\n \u00b7 use 1, zero_ne_one.symm\n rw [one_smul]\n exact subset_span (mem_range_self (\u27e8i, hi\u27e9 : I))\n \u00b7 simpa [image_eq_range s I] using hI i hi\nchoose a ha ha' using exists_a\nlet A := \u220f i, a i\nhave hA : A \u2260 0 := by\n rw [Finset.prod_ne_zero_iff]\n simpa using ha\nlet \u03c6 : M \u2192\u2097[R] M := LinearMap.lsmul R M A\nhave : LinearMap.ker \u03c6 = \u22a5 := @LinearMap.ker_lsmul R M _ _ _ _ _ hA\nlet \u03c8 := LinearEquiv.ofInjective \u03c6 (LinearMap.ker_eq_bot.mp this)\nhave : LinearMap.range \u03c6 \u2264 N := by\n -- as announced, `A \u2022 M \u2286 N`\n suffices \u2200 i, \u03c6 (s i) \u2208 N by\n rw [LinearMap.range_eq_map, \u2190 hs, map_span_le]\n rintro _ \u27e8i, rfl\u27e9\n apply this\n intro i\n calc\n (\u220f j, a j) \u2022 s i = (\u220f j in { i }\u1d9c, a j) \u2022 a i \u2022 s i := by rw [Fintype.prod_eq_prod_compl_mul i, mul_smul]\n _ \u2208 N :=\n N.smul_mem _\n (ha' i)\n -- Since a submodule of a free `R`-module is free, we get that `A \u2022 M` is free\nobtain \u27e8n, b : Basis (Fin n) R (LinearMap.range \u03c6)\u27e9 := Submodule.basisOfPidOfLE this sI_basis\nexact \u27e8n, b.map \u03c8.symm\u27e9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nhave := exists_maximal_independent R s\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nlet I : Set \u03b9 := this.choose\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nobtain\n \u27e8indepI : LinearIndependent R (s \u2218 (fun x => x) : I \u2192 M), hI :\n \u2200 (i) (_ : i \u2209 I), \u2203 a : R, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\u27e9 :=\n this.choose_spec\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nlet N :=\n span R\n (range <| (s \u2218 (fun x => x) : I \u2192 M))\n -- same as `span R (s '' I)` but more convenient\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nlet _sI : I \u2192 N := fun i \u21a6\n \u27e8s i.1, subset_span (mem_range_self i)\u27e9\n -- `s` restricted to `I` is a basis of `N`\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nlet sI_basis : Basis I R N := Basis.span indepI\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nhave exists_a : \u2200 i : \u03b9, \u2203 a : R, a \u2260 0 \u2227 a \u2022 s i \u2208 N :=\n by\n intro i\n by_cases hi : i \u2208 I\n \u00b7 use 1, zero_ne_one.symm\n rw [one_smul]\n exact subset_span (mem_range_self (\u27e8i, hi\u27e9 : I))\n \u00b7 simpa [image_eq_range s I] using hI i hi\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\n\u22a2 \u2200 (i : \u03b9), \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 N\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\ni : \u03b9\n\u22a2 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 N\n[PROOFSTEP]\nby_cases hi : i \u2208 I\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\ni : \u03b9\nhi : i \u2208 I\n\u22a2 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 N\n[PROOFSTEP]\nuse 1, zero_ne_one.symm\n[GOAL]\ncase right\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\ni : \u03b9\nhi : i \u2208 I\n\u22a2 1 \u2022 s i \u2208 N\n[PROOFSTEP]\nrw [one_smul]\n[GOAL]\ncase right\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\ni : \u03b9\nhi : i \u2208 I\n\u22a2 s i \u2208 N\n[PROOFSTEP]\nexact subset_span (mem_range_self (\u27e8i, hi\u27e9 : I))\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\ni : \u03b9\nhi : \u00aci \u2208 I\n\u22a2 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 N\n[PROOFSTEP]\nsimpa [image_eq_range s I] using hI i hi\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\nexists_a : \u2200 (i : \u03b9), \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 N\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nchoose a ha ha' using exists_a\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nlet A := \u220f i, a i\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nhave hA : A \u2260 0 := by\n rw [Finset.prod_ne_zero_iff]\n simpa using ha\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\n\u22a2 A \u2260 0\n[PROOFSTEP]\nrw [Finset.prod_ne_zero_iff]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\n\u22a2 \u2200 (a_1 : \u03b9), a_1 \u2208 Finset.univ \u2192 a a_1 \u2260 0\n[PROOFSTEP]\nsimpa using ha\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\nhA : A \u2260 0\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nlet \u03c6 : M \u2192\u2097[R] M := LinearMap.lsmul R M A\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\nhA : A \u2260 0\n\u03c6 : M \u2192\u2097[R] M := \u2191(LinearMap.lsmul R M) A\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nhave : LinearMap.ker \u03c6 = \u22a5 := @LinearMap.ker_lsmul R M _ _ _ _ _ hA\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis\u271d : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\u271d\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\nhA : A \u2260 0\n\u03c6 : M \u2192\u2097[R] M := \u2191(LinearMap.lsmul R M) A\nthis : LinearMap.ker \u03c6 = \u22a5\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nlet \u03c8 := LinearEquiv.ofInjective \u03c6 (LinearMap.ker_eq_bot.mp this)\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis\u271d : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\u271d\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\nhA : A \u2260 0\n\u03c6 : M \u2192\u2097[R] M := \u2191(LinearMap.lsmul R M) A\nthis : LinearMap.ker \u03c6 = \u22a5\n\u03c8 : M \u2243\u2097[R] { x // x \u2208 LinearMap.range \u03c6 } := LinearEquiv.ofInjective \u03c6 (_ : Function.Injective \u2191\u03c6)\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nhave : LinearMap.range \u03c6 \u2264 N := by\n -- as announced, `A \u2022 M \u2286 N`\n suffices \u2200 i, \u03c6 (s i) \u2208 N by\n rw [LinearMap.range_eq_map, \u2190 hs, map_span_le]\n rintro _ \u27e8i, rfl\u27e9\n apply this\n intro i\n calc\n (\u220f j, a j) \u2022 s i = (\u220f j in { i }\u1d9c, a j) \u2022 a i \u2022 s i := by rw [Fintype.prod_eq_prod_compl_mul i, mul_smul]\n _ \u2208 N :=\n N.smul_mem _\n (ha' i)\n -- Since a submodule of a free `R`-module is free, we get that `A \u2022 M` is free\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis\u271d : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\u271d\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\nhA : A \u2260 0\n\u03c6 : M \u2192\u2097[R] M := \u2191(LinearMap.lsmul R M) A\nthis : LinearMap.ker \u03c6 = \u22a5\n\u03c8 : M \u2243\u2097[R] { x // x \u2208 LinearMap.range \u03c6 } := LinearEquiv.ofInjective \u03c6 (_ : Function.Injective \u2191\u03c6)\n\u22a2 LinearMap.range \u03c6 \u2264 N\n[PROOFSTEP]\nsuffices \u2200 i, \u03c6 (s i) \u2208 N by\n rw [LinearMap.range_eq_map, \u2190 hs, map_span_le]\n rintro _ \u27e8i, rfl\u27e9\n apply this\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis\u271d\u00b9 : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\u271d\u00b9\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\nhA : A \u2260 0\n\u03c6 : M \u2192\u2097[R] M := \u2191(LinearMap.lsmul R M) A\nthis\u271d : LinearMap.ker \u03c6 = \u22a5\n\u03c8 : M \u2243\u2097[R] { x // x \u2208 LinearMap.range \u03c6 } := LinearEquiv.ofInjective \u03c6 (_ : Function.Injective \u2191\u03c6)\nthis : \u2200 (i : \u03b9), \u2191\u03c6 (s i) \u2208 N\n\u22a2 LinearMap.range \u03c6 \u2264 N\n[PROOFSTEP]\nrw [LinearMap.range_eq_map, \u2190 hs, map_span_le]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis\u271d\u00b9 : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\u271d\u00b9\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\nhA : A \u2260 0\n\u03c6 : M \u2192\u2097[R] M := \u2191(LinearMap.lsmul R M) A\nthis\u271d : LinearMap.ker \u03c6 = \u22a5\n\u03c8 : M \u2243\u2097[R] { x // x \u2208 LinearMap.range \u03c6 } := LinearEquiv.ofInjective \u03c6 (_ : Function.Injective \u2191\u03c6)\nthis : \u2200 (i : \u03b9), \u2191\u03c6 (s i) \u2208 N\n\u22a2 \u2200 (m : M), m \u2208 range s \u2192 \u2191\u03c6 m \u2208 N\n[PROOFSTEP]\nrintro _ \u27e8i, rfl\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis\u271d\u00b9 : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\u271d\u00b9\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\nhA : A \u2260 0\n\u03c6 : M \u2192\u2097[R] M := \u2191(LinearMap.lsmul R M) A\nthis\u271d : LinearMap.ker \u03c6 = \u22a5\n\u03c8 : M \u2243\u2097[R] { x // x \u2208 LinearMap.range \u03c6 } := LinearEquiv.ofInjective \u03c6 (_ : Function.Injective \u2191\u03c6)\nthis : \u2200 (i : \u03b9), \u2191\u03c6 (s i) \u2208 N\ni : \u03b9\n\u22a2 \u2191\u03c6 (s i) \u2208 N\n[PROOFSTEP]\napply this\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis\u271d : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\u271d\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\nhA : A \u2260 0\n\u03c6 : M \u2192\u2097[R] M := \u2191(LinearMap.lsmul R M) A\nthis : LinearMap.ker \u03c6 = \u22a5\n\u03c8 : M \u2243\u2097[R] { x // x \u2208 LinearMap.range \u03c6 } := LinearEquiv.ofInjective \u03c6 (_ : Function.Injective \u2191\u03c6)\n\u22a2 \u2200 (i : \u03b9), \u2191\u03c6 (s i) \u2208 N\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis\u271d : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\u271d\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\nhA : A \u2260 0\n\u03c6 : M \u2192\u2097[R] M := \u2191(LinearMap.lsmul R M) A\nthis : LinearMap.ker \u03c6 = \u22a5\n\u03c8 : M \u2243\u2097[R] { x // x \u2208 LinearMap.range \u03c6 } := LinearEquiv.ofInjective \u03c6 (_ : Function.Injective \u2191\u03c6)\ni : \u03b9\n\u22a2 \u2191\u03c6 (s i) \u2208 N\n[PROOFSTEP]\ncalc\n (\u220f j, a j) \u2022 s i = (\u220f j in { i }\u1d9c, a j) \u2022 a i \u2022 s i := by rw [Fintype.prod_eq_prod_compl_mul i, mul_smul]\n _ \u2208 N :=\n N.smul_mem _\n (ha' i)\n -- Since a submodule of a free `R`-module is free, we get that `A \u2022 M` is free\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis\u271d : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\u271d\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\nhA : A \u2260 0\n\u03c6 : M \u2192\u2097[R] M := \u2191(LinearMap.lsmul R M) A\nthis : LinearMap.ker \u03c6 = \u22a5\n\u03c8 : M \u2243\u2097[R] { x // x \u2208 LinearMap.range \u03c6 } := LinearEquiv.ofInjective \u03c6 (_ : Function.Injective \u2191\u03c6)\ni : \u03b9\n\u22a2 (\u220f j : \u03b9, a j) \u2022 s i = (\u220f j in {i}\u1d9c, a j) \u2022 a i \u2022 s i\n[PROOFSTEP]\nrw [Fintype.prod_eq_prod_compl_mul i, mul_smul]\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis\u271d\u00b9 : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\u271d\u00b9\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\nhA : A \u2260 0\n\u03c6 : M \u2192\u2097[R] M := \u2191(LinearMap.lsmul R M) A\nthis\u271d : LinearMap.ker \u03c6 = \u22a5\n\u03c8 : M \u2243\u2097[R] { x // x \u2208 LinearMap.range \u03c6 } := LinearEquiv.ofInjective \u03c6 (_ : Function.Injective \u2191\u03c6)\nthis : LinearMap.range \u03c6 \u2264 N\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nobtain \u27e8n, b : Basis (Fin n) R (LinearMap.range \u03c6)\u27e9 := Submodule.basisOfPidOfLE this sI_basis\n[GOAL]\ncase intro.mk\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9 : Fintype \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nthis\u271d\u00b9 : \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nI : Set \u03b9 := Exists.choose this\u271d\u00b9\nindepI : LinearIndependent R (s \u2218 fun x => \u2191x)\nhI : \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\nN : Submodule R M := span R (range (s \u2218 fun x => \u2191x))\n_sI : \u2191I \u2192 { x // x \u2208 N } := fun i => { val := s \u2191i, property := (_ : s \u2191i \u2208 \u2191(span R (range (s \u2218 fun x => \u2191x)))) }\nsI_basis : Basis (\u2191I) R { x // x \u2208 N } := Basis.span indepI\na : \u03b9 \u2192 R\nha : \u2200 (i : \u03b9), a i \u2260 0\nha' : \u2200 (i : \u03b9), a i \u2022 s i \u2208 N\nA : R := \u220f i : \u03b9, a i\nhA : A \u2260 0\n\u03c6 : M \u2192\u2097[R] M := \u2191(LinearMap.lsmul R M) A\nthis\u271d : LinearMap.ker \u03c6 = \u22a5\n\u03c8 : M \u2243\u2097[R] { x // x \u2208 LinearMap.range \u03c6 } := LinearEquiv.ofInjective \u03c6 (_ : Function.Injective \u2191\u03c6)\nthis : LinearMap.range \u03c6 \u2264 N\nn : \u2115\nb : Basis (Fin n) R { x // x \u2208 LinearMap.range \u03c6 }\n\u22a2 (n : \u2115) \u00d7 Basis (Fin n) R M\n[PROOFSTEP]\nexact \u27e8n, b.map \u03c8.symm\u27e9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : _root_.Finite \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\n\u22a2 Free R M\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : _root_.Finite \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nval\u271d : Fintype \u03b9\n\u22a2 Free R M\n[PROOFSTEP]\nobtain \u27e8n, b\u27e9 : \u03a3 n, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree hs\n[GOAL]\ncase intro.mk\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9 : _root_.Finite \u03b9\ns : \u03b9 \u2192 M\nhs : span R (range s) = \u22a4\ninst\u271d : NoZeroSMulDivisors R M\nval\u271d : Fintype \u03b9\nn : \u2115\nb : Basis (Fin n) R M\n\u22a2 Free R M\n[PROOFSTEP]\nexact Module.Free.of_basis b\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb : \u03b9 \u2192 M\ninst\u271d\u00b9 : Finite R M\ninst\u271d : NoZeroSMulDivisors R M\n\u22a2 Free R M\n[PROOFSTEP]\nobtain \u27e8n, b\u27e9 : \u03a3 n, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree'\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9 : Finite R M\ninst\u271d : NoZeroSMulDivisors R M\nn : \u2115\nb : Basis (Fin n) R M\n\u22a2 Free R M\n[PROOFSTEP]\nexact Module.Free.of_basis b\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN O : Submodule R M\nN_le_O : N \u2264 O\n\u22a2 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN O : Submodule R M\nN_le_O : N \u2264 O\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\n[PROOFSTEP]\nrevert N\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nO : Submodule R M\nval\u271d : Fintype \u03b9\n\u22a2 \u2200 (N : Submodule R M), N \u2264 O \u2192 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\n[PROOFSTEP]\ninduction' O using inductionOnRank with M0 ih\n[GOAL]\ncase intro.b\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nO : Submodule R M\nval\u271d : Fintype \u03b9\n\u22a2 Basis ?m.235193 R M\n[PROOFSTEP]\nexact b\n[GOAL]\ncase intro.ih\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\nM0 : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 M0 \u2192\n \u2200 (x : M),\n x \u2208 M0 \u2192\n (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192\n \u2200 (N : Submodule R M), N \u2264 N' \u2192 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\n\u22a2 \u2200 (N : Submodule R M), N \u2264 M0 \u2192 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\ncase intro.inst\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nO : Submodule R M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\nintro N N_le_M0\n[GOAL]\ncase intro.ih\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\nM0 : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 M0 \u2192\n \u2200 (x : M),\n x \u2208 M0 \u2192\n (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192\n \u2200 (N : Submodule R M), N \u2264 N' \u2192 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\nN : Submodule R M\nN_le_M0 : N \u2264 M0\n\u22a2 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\ncase intro.inst\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nO : Submodule R M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\nobtain \u27e8m, b'M\u27e9 := M0.basisOfPid b\n[GOAL]\ncase intro.ih.mk\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\nM0 : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 M0 \u2192\n \u2200 (x : M),\n x \u2208 M0 \u2192\n (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192\n \u2200 (N : Submodule R M), N \u2264 N' \u2192 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\nN : Submodule R M\nN_le_M0 : N \u2264 M0\nm : \u2115\nb'M : Basis (Fin m) R { x // x \u2208 M0 }\n\u22a2 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\ncase intro.inst\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nO : Submodule R M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\nby_cases N_bot : N = \u22a5\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\nM0 : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 M0 \u2192\n \u2200 (x : M),\n x \u2208 M0 \u2192\n (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192\n \u2200 (N : Submodule R M), N \u2264 N' \u2192 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\nN : Submodule R M\nN_le_M0 : N \u2264 M0\nm : \u2115\nb'M : Basis (Fin m) R { x // x \u2208 M0 }\nN_bot : N = \u22a5\n\u22a2 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\n[PROOFSTEP]\nsubst N_bot\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\nM0 : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 M0 \u2192\n \u2200 (x : M),\n x \u2208 M0 \u2192\n (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192\n \u2200 (N : Submodule R M), N \u2264 N' \u2192 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\nm : \u2115\nb'M : Basis (Fin m) R { x // x \u2208 M0 }\nN_le_M0 : \u22a5 \u2264 M0\n\u22a2 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\n[PROOFSTEP]\nexact \u27e80, m, Nat.zero_le _, b'M, Basis.empty _, finZeroElim, finZeroElim\u27e9\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\nM0 : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 M0 \u2192\n \u2200 (x : M),\n x \u2208 M0 \u2192\n (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192\n \u2200 (N : Submodule R M), N \u2264 N' \u2192 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\nN : Submodule R M\nN_le_M0 : N \u2264 M0\nm : \u2115\nb'M : Basis (Fin m) R { x // x \u2208 M0 }\nN_bot : \u00acN = \u22a5\n\u22a2 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\ncase intro.inst\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nO : Submodule R M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\nobtain \u27e8y, hy, a, _, M', M'_le_M, N', _, N'_le_M', y_ortho, _, h\u27e9 := Submodule.basis_of_pid_aux M0 N b'M N_bot N_le_M0\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\nM0 : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 M0 \u2192\n \u2200 (x : M),\n x \u2208 M0 \u2192\n (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192\n \u2200 (N : Submodule R M), N \u2264 N' \u2192 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\nN : Submodule R M\nN_le_M0 : N \u2264 M0\nm : \u2115\nb'M : Basis (Fin m) R { x // x \u2208 M0 }\nN_bot : \u00acN = \u22a5\ny : M\nhy : y \u2208 M0\na : R\nw\u271d\u00b9 : a \u2022 y \u2208 N\nM' : Submodule R M\nM'_le_M : M' \u2264 M0\nN' : Submodule R M\nleft\u271d : N' \u2264 N\nN'_le_M' : N' \u2264 M'\ny_ortho : \u2200 (c : R) (z : M), z \u2208 M' \u2192 c \u2022 y + z = 0 \u2192 c = 0\nw\u271d : \u2200 (c : R) (z : M), z \u2208 N' \u2192 c \u2022 a \u2022 y + z = 0 \u2192 c = 0\nh :\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n\u22a2 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\ncase intro.inst\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nO : Submodule R M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\nobtain \u27e8n', m', hn'm', bM', bN', as', has'\u27e9 := ih M' M'_le_M y hy y_ortho N' N'_le_M'\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\nM0 : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 M0 \u2192\n \u2200 (x : M),\n x \u2208 M0 \u2192\n (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192\n \u2200 (N : Submodule R M), N \u2264 N' \u2192 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\nN : Submodule R M\nN_le_M0 : N \u2264 M0\nm : \u2115\nb'M : Basis (Fin m) R { x // x \u2208 M0 }\nN_bot : \u00acN = \u22a5\ny : M\nhy : y \u2208 M0\na : R\nw\u271d\u00b9 : a \u2022 y \u2208 N\nM' : Submodule R M\nM'_le_M : M' \u2264 M0\nN' : Submodule R M\nleft\u271d : N' \u2264 N\nN'_le_M' : N' \u2264 M'\ny_ortho : \u2200 (c : R) (z : M), z \u2208 M' \u2192 c \u2022 y + z = 0 \u2192 c = 0\nw\u271d : \u2200 (c : R) (z : M), z \u2208 N' \u2192 c \u2022 a \u2022 y + z = 0 \u2192 c = 0\nh :\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\nn' m' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nas' : Fin n' \u2192 R\nhas' : \u2200 (i : Fin n'), \u2191(\u2191bN' i) = \u2191(as' i \u2022 \u2191bM' (Fin.castLE hn'm' i))\n\u22a2 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\ncase intro.inst\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nO : Submodule R M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\nobtain \u27e8bN, h'\u27e9 := h n' bN'\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\nM0 : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 M0 \u2192\n \u2200 (x : M),\n x \u2208 M0 \u2192\n (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192\n \u2200 (N : Submodule R M), N \u2264 N' \u2192 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\nN : Submodule R M\nN_le_M0 : N \u2264 M0\nm : \u2115\nb'M : Basis (Fin m) R { x // x \u2208 M0 }\nN_bot : \u00acN = \u22a5\ny : M\nhy : y \u2208 M0\na : R\nw\u271d\u00b9 : a \u2022 y \u2208 N\nM' : Submodule R M\nM'_le_M : M' \u2264 M0\nN' : Submodule R M\nleft\u271d : N' \u2264 N\nN'_le_M' : N' \u2264 M'\ny_ortho : \u2200 (c : R) (z : M), z \u2208 M' \u2192 c \u2022 y + z = 0 \u2192 c = 0\nw\u271d : \u2200 (c : R) (z : M), z \u2208 N' \u2192 c \u2022 a \u2022 y + z = 0 \u2192 c = 0\nh :\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\nn' m' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nas' : Fin n' \u2192 R\nhas' : \u2200 (i : Fin n'), \u2191(\u2191bN' i) = \u2191(as' i \u2022 \u2191bM' (Fin.castLE hn'm' i))\nbN : Basis (Fin (n' + 1)) R { x // x \u2208 N }\nh' :\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\n\u22a2 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\ncase intro.inst\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nO : Submodule R M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\nobtain \u27e8hmn, bM, h''\u27e9 := h' m' hn'm' bM'\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\nM0 : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 M0 \u2192\n \u2200 (x : M),\n x \u2208 M0 \u2192\n (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192\n \u2200 (N : Submodule R M), N \u2264 N' \u2192 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\nN : Submodule R M\nN_le_M0 : N \u2264 M0\nm : \u2115\nb'M : Basis (Fin m) R { x // x \u2208 M0 }\nN_bot : \u00acN = \u22a5\ny : M\nhy : y \u2208 M0\na : R\nw\u271d\u00b9 : a \u2022 y \u2208 N\nM' : Submodule R M\nM'_le_M : M' \u2264 M0\nN' : Submodule R M\nleft\u271d : N' \u2264 N\nN'_le_M' : N' \u2264 M'\ny_ortho : \u2200 (c : R) (z : M), z \u2208 M' \u2192 c \u2022 y + z = 0 \u2192 c = 0\nw\u271d : \u2200 (c : R) (z : M), z \u2208 N' \u2192 c \u2022 a \u2022 y + z = 0 \u2192 c = 0\nh :\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\nn' m' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nas' : Fin n' \u2192 R\nhas' : \u2200 (i : Fin n'), \u2191(\u2191bN' i) = \u2191(as' i \u2022 \u2191bM' (Fin.castLE hn'm' i))\nbN : Basis (Fin (n' + 1)) R { x // x \u2208 N }\nh' :\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\nhmn : n' + 1 \u2264 m' + 1\nbM : Basis (Fin (m' + 1)) R { x // x \u2208 M0 }\nh'' :\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hmn i))\n\u22a2 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\ncase intro.inst\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nO : Submodule R M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\nobtain \u27e8as, has\u27e9 := h'' as' has'\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\nM0 : Submodule R M\nih :\n \u2200 (N' : Submodule R M),\n N' \u2264 M0 \u2192\n \u2200 (x : M),\n x \u2208 M0 \u2192\n (\u2200 (c : R) (y : M), y \u2208 N' \u2192 c \u2022 x + y = 0 \u2192 c = 0) \u2192\n \u2200 (N : Submodule R M), N \u2264 N' \u2192 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\nN : Submodule R M\nN_le_M0 : N \u2264 M0\nm : \u2115\nb'M : Basis (Fin m) R { x // x \u2208 M0 }\nN_bot : \u00acN = \u22a5\ny : M\nhy : y \u2208 M0\na : R\nw\u271d\u00b9 : a \u2022 y \u2208 N\nM' : Submodule R M\nM'_le_M : M' \u2264 M0\nN' : Submodule R M\nleft\u271d : N' \u2264 N\nN'_le_M' : N' \u2264 M'\ny_ortho : \u2200 (c : R) (z : M), z \u2208 M' \u2192 c \u2022 y + z = 0 \u2192 c = 0\nw\u271d : \u2200 (c : R) (z : M), z \u2208 N' \u2192 c \u2022 a \u2022 y + z = 0 \u2192 c = 0\nh :\n \u2200 (n' : \u2115) (bN' : Basis (Fin n') R { x // x \u2208 N' }),\n \u2203 bN,\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\nn' m' : \u2115\nhn'm' : n' \u2264 m'\nbM' : Basis (Fin m') R { x // x \u2208 M' }\nbN' : Basis (Fin n') R { x // x \u2208 N' }\nas' : Fin n' \u2192 R\nhas' : \u2200 (i : Fin n'), \u2191(\u2191bN' i) = \u2191(as' i \u2022 \u2191bM' (Fin.castLE hn'm' i))\nbN : Basis (Fin (n' + 1)) R { x // x \u2208 N }\nh' :\n \u2200 (m' : \u2115) (hn'm' : n' \u2264 m') (bM' : Basis (Fin m') R { x // x \u2208 M' }),\n \u2203 hnm bM,\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hnm i))\nhmn : n' + 1 \u2264 m' + 1\nbM : Basis (Fin (m' + 1)) R { x // x \u2208 M0 }\nh'' :\n \u2200 (as : Fin n' \u2192 R),\n (\u2200 (i : Fin n'), \u2191(\u2191bN' i) = as i \u2022 \u2191(\u2191bM' (Fin.castLE hn'm' i))) \u2192\n \u2203 as', \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as' i \u2022 \u2191(\u2191bM (Fin.castLE hmn i))\nas : Fin (n' + 1) \u2192 R\nhas : \u2200 (i : Fin (n' + 1)), \u2191(\u2191bN i) = as i \u2022 \u2191(\u2191bM (Fin.castLE hmn i))\n\u22a2 \u2203 n o hno bO bN a, \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\ncase intro.inst\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nO : Submodule R M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\nexact\n \u27e8_, _, hmn, bM, bN, as, has\u27e9\n -- Porting note: Lean generates a goal Fintype \u03b9 for some reason\n[GOAL]\ncase intro.inst\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nO : Submodule R M\nval\u271d : Fintype \u03b9\n\u22a2 Fintype \u03b9\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN O : Submodule R M\nN_le_O : N \u2264 O\n\u22a2 (o : \u2115) \u00d7 (n : \u2115) \u00d7 Basis.SmithNormalForm (comap (Submodule.subtype O) N) (Fin o) n\n[PROOFSTEP]\nchoose n o hno bO bN a snf using N.exists_smith_normal_form_of_le b O N_le_O\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN O : Submodule R M\nN_le_O : N \u2264 O\nn o : \u2115\nhno : n \u2264 o\nbO : Basis (Fin o) R { x // x \u2208 O }\nbN : Basis (Fin n) R { x // x \u2208 N }\na : Fin n \u2192 R\nsnf : \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\n\u22a2 (o : \u2115) \u00d7 (n : \u2115) \u00d7 Basis.SmithNormalForm (comap (Submodule.subtype O) N) (Fin o) n\n[PROOFSTEP]\nrefine' \u27e8o, n, bO, bN.map (comapSubtypeEquivOfLe N_le_O).symm, (Fin.castLEEmb hno).toEmbedding, a, fun i \u21a6 _\u27e9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN O : Submodule R M\nN_le_O : N \u2264 O\nn o : \u2115\nhno : n \u2264 o\nbO : Basis (Fin o) R { x // x \u2208 O }\nbN : Basis (Fin n) R { x // x \u2208 N }\na : Fin n \u2192 R\nsnf : \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\ni : Fin n\n\u22a2 \u2191(\u2191(Basis.map bN (LinearEquiv.symm (comapSubtypeEquivOfLe N_le_O))) i) =\n a i \u2022 \u2191bO (\u2191(Fin.castLEEmb hno).toEmbedding i)\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN O : Submodule R M\nN_le_O : N \u2264 O\nn o : \u2115\nhno : n \u2264 o\nbO : Basis (Fin o) R { x // x \u2208 O }\nbN : Basis (Fin n) R { x // x \u2208 N }\na : Fin n \u2192 R\nsnf : \u2200 (i : Fin n), \u2191(\u2191bN i) = \u2191(a i \u2022 \u2191bO (Fin.castLE hno i))\ni : Fin n\n\u22a2 \u2191\u2191(\u2191(Basis.map bN (LinearEquiv.symm (comapSubtypeEquivOfLe N_le_O))) i) =\n \u2191(a i \u2022 \u2191bO (\u2191(Fin.castLEEmb hno).toEmbedding i))\n[PROOFSTEP]\nsimp only [snf, Basis.map_apply, Submodule.comapSubtypeEquivOfLe_symm_apply, Submodule.coe_smul_of_tower,\n RelEmbedding.coe_toEmbedding, Fin.castLEEmb_apply]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb\u271d : \u03b9 \u2192 M\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R M\nN : Submodule R M\nm n : \u2115\nbM : Basis (Fin m) R { x // x \u2208 \u22a4 }\nbN : Basis (Fin n) R { x // x \u2208 comap (Submodule.subtype \u22a4) N }\nf : Fin n \u21aa Fin m\na : Fin n \u2192 R\nsnf : \u2200 (i : Fin n), \u2191(\u2191bN i) = a i \u2022 \u2191bM (\u2191f i)\nbM' : Basis (Fin m) R M := Basis.map bM (LinearEquiv.ofTop \u22a4 (_ : \u22a4 = \u22a4))\ne : Fin m \u2243 \u03b9 := Basis.indexEquiv bM' b\ni : Fin n\n\u22a2 \u2191(\u2191(Basis.map bN (comapSubtypeEquivOfLe (_ : N \u2264 \u22a4))) i) =\n a i \u2022 \u2191(Basis.reindex bM' e) (\u2191(Function.Embedding.trans f (Equiv.toEmbedding e)) i)\n[PROOFSTEP]\nsimp only [snf, Basis.map_apply, LinearEquiv.ofTop_apply, Submodule.coe_smul_of_tower,\n Submodule.comapSubtypeEquivOfLe_apply_coe, Basis.reindex_apply, Equiv.toEmbedding_apply,\n Function.Embedding.trans_apply, Equiv.symm_apply_apply]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_4\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\nn : \u2115\nbS : Basis \u03b9 R S\nbI : Basis (Fin n) R { x // x \u2208 restrictScalars R I }\nf : Fin n \u21aa \u03b9\na : Fin n \u2192 R\nsnf : \u2200 (i : Fin n), \u2191(\u2191bI i) = a i \u2022 \u2191bS (\u2191f i)\neq : Fintype.card (Fin n) = Fintype.card \u03b9\n\u22a2 Fintype.card (Fin n) = Fintype.card (Fin (Fintype.card \u03b9))\n[PROOFSTEP]\nrw [eq, Fintype.card_fin]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_4\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\nn : \u2115\nbS : Basis \u03b9 R S\nbI : Basis (Fin n) R { x // x \u2208 restrictScalars R I }\nf : Fin n \u21aa \u03b9\na : Fin n \u2192 R\nsnf : \u2200 (i : Fin n), \u2191(\u2191bI i) = a i \u2022 \u2191bS (\u2191f i)\neq : Fintype.card (Fin n) = Fintype.card \u03b9\ne : Fin n \u2243 Fin (Fintype.card \u03b9) :=\n Fintype.equivOfCardEq (_ : Fintype.card (Fin n) = Fintype.card (Fin (Fintype.card \u03b9)))\ni : Fin (Fintype.card \u03b9)\n\u22a2 \u2191(\u2191(Basis.reindex bI e) i) = (a \u2218 \u2191e.symm) i \u2022 \u2191bS (\u2191(Function.Embedding.trans (Equiv.toEmbedding e.symm) f) i)\n[PROOFSTEP]\nsimp only [snf, Basis.coe_reindex, Function.Embedding.trans_apply, Equiv.toEmbedding_apply, (\u00b7 \u2218 \u00b7)]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_4\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\n\u22a2 \u2203 b' a ab', \u2200 (i : \u03b9), \u2191(\u2191ab' i) = a i \u2022 \u2191b' i\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_4\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 b' a ab', \u2200 (i : \u03b9), \u2191(\u2191ab' i) = a i \u2022 \u2191b' i\n[PROOFSTEP]\nlet \u27e8bS, bI, f, a, snf\u27e9 := I.smithNormalForm b hI\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_4\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\nval\u271d : Fintype \u03b9\nbS : Basis \u03b9 R S\nbI : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 restrictScalars R I }\nf : Fin (Fintype.card \u03b9) \u21aa \u03b9\na : Fin (Fintype.card \u03b9) \u2192 R\nsnf : \u2200 (i : Fin (Fintype.card \u03b9)), \u2191(\u2191bI i) = a i \u2022 \u2191bS (\u2191f i)\n\u22a2 \u2203 b' a ab', \u2200 (i : \u03b9), \u2191(\u2191ab' i) = a i \u2022 \u2191b' i\n[PROOFSTEP]\nlet e : Fin (Fintype.card \u03b9) \u2243 \u03b9 :=\n Equiv.ofBijective f ((Fintype.bijective_iff_injective_and_card f).mpr \u27e8f.injective, Fintype.card_fin _\u27e9)\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_4\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\nval\u271d : Fintype \u03b9\nbS : Basis \u03b9 R S\nbI : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 restrictScalars R I }\nf : Fin (Fintype.card \u03b9) \u21aa \u03b9\na : Fin (Fintype.card \u03b9) \u2192 R\nsnf : \u2200 (i : Fin (Fintype.card \u03b9)), \u2191(\u2191bI i) = a i \u2022 \u2191bS (\u2191f i)\ne : Fin (Fintype.card \u03b9) \u2243 \u03b9 := Equiv.ofBijective \u2191f (_ : Function.Bijective \u2191f)\n\u22a2 \u2203 b' a ab', \u2200 (i : \u03b9), \u2191(\u2191ab' i) = a i \u2022 \u2191b' i\n[PROOFSTEP]\nhave fe : \u2200 i, f (e.symm i) = i := e.apply_symm_apply\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_4\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\nval\u271d : Fintype \u03b9\nbS : Basis \u03b9 R S\nbI : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 restrictScalars R I }\nf : Fin (Fintype.card \u03b9) \u21aa \u03b9\na : Fin (Fintype.card \u03b9) \u2192 R\nsnf : \u2200 (i : Fin (Fintype.card \u03b9)), \u2191(\u2191bI i) = a i \u2022 \u2191bS (\u2191f i)\ne : Fin (Fintype.card \u03b9) \u2243 \u03b9 := Equiv.ofBijective \u2191f (_ : Function.Bijective \u2191f)\nfe : \u2200 (i : \u03b9), \u2191f (\u2191e.symm i) = i\n\u22a2 \u2203 b' a ab', \u2200 (i : \u03b9), \u2191(\u2191ab' i) = a i \u2022 \u2191b' i\n[PROOFSTEP]\nexact\n \u27e8bS, a \u2218 e.symm, (bI.reindex e).map ((restrictScalarsEquiv R S _ _).restrictScalars R), fun i \u21a6 by\n simp only [snf, fe, Basis.map_apply, LinearEquiv.restrictScalars_apply R, Submodule.restrictScalarsEquiv_apply,\n Basis.coe_reindex, (\u00b7 \u2218 \u00b7)]\u27e9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_4\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\nval\u271d : Fintype \u03b9\nbS : Basis \u03b9 R S\nbI : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 restrictScalars R I }\nf : Fin (Fintype.card \u03b9) \u21aa \u03b9\na : Fin (Fintype.card \u03b9) \u2192 R\nsnf : \u2200 (i : Fin (Fintype.card \u03b9)), \u2191(\u2191bI i) = a i \u2022 \u2191bS (\u2191f i)\ne : Fin (Fintype.card \u03b9) \u2243 \u03b9 := Equiv.ofBijective \u2191f (_ : Function.Bijective \u2191f)\nfe : \u2200 (i : \u03b9), \u2191f (\u2191e.symm i) = i\ni : \u03b9\n\u22a2 \u2191(\u2191(Basis.map (Basis.reindex bI e) (LinearEquiv.restrictScalars R (restrictScalarsEquiv R S S I))) i) =\n (a \u2218 \u2191e.symm) i \u2022 \u2191bS i\n[PROOFSTEP]\nsimp only [snf, fe, Basis.map_apply, LinearEquiv.restrictScalars_apply R, Submodule.restrictScalarsEquiv_apply,\n Basis.coe_reindex, (\u00b7 \u2218 \u00b7)]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_4\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\ni : \u03b9\n\u22a2 smithCoeffs b I hI i \u2260 0\n[PROOFSTEP]\nintro hi\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_4\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\ni : \u03b9\nhi : smithCoeffs b I hI i = 0\n\u22a2 False\n[PROOFSTEP]\napply Basis.ne_zero (Ideal.selfBasis b I hI) i\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_4\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\ni : \u03b9\nhi : smithCoeffs b I hI i = 0\n\u22a2 \u2191(selfBasis b I hI) i = 0\n[PROOFSTEP]\nrefine' Subtype.coe_injective _\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type u_3\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_4\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\ni : \u03b9\nhi : smithCoeffs b I hI i = 0\n\u22a2 (fun a => \u2191a) (\u2191(selfBasis b I hI) i) = (fun a => \u2191a) 0\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nhinj : Function.Injective \u2191(algebraMap R S)\nv : \u03b9 \u2192 M\nli : LinearIndependent S v\n\u22a2 Function.Injective fun r => r \u2022 1\n[PROOFSTEP]\nrwa [Algebra.algebraMap_eq_smul_one'] at hinj \n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.FreeModule.PID", "llama_tokens": 126757, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.6261241842048092, "lm_q1q2_score": 0.42525020895047644}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Type u_4\nn : Type u_5\nR : Type u_6\nA B C : Matrix m n \u03b1\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : MulZeroOneClass \u03b1\nM : Matrix n n \u03b1\n\u22a2 M \u2299 1 = diagonal fun i => M i i\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Type u_4\nn : Type u_5\nR : Type u_6\nA B C : Matrix m n \u03b1\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : MulZeroOneClass \u03b1\nM : Matrix n n \u03b1\ni j : n\n\u22a2 (M \u2299 1) i j = diagonal (fun i => M i i) i j\n[PROOFSTEP]\nby_cases h : i = j\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Type u_4\nn : Type u_5\nR : Type u_6\nA B C : Matrix m n \u03b1\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : MulZeroOneClass \u03b1\nM : Matrix n n \u03b1\ni j : n\nh : i = j\n\u22a2 (M \u2299 1) i j = diagonal (fun i => M i i) i j\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Type u_4\nn : Type u_5\nR : Type u_6\nA B C : Matrix m n \u03b1\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : MulZeroOneClass \u03b1\nM : Matrix n n \u03b1\ni j : n\nh : \u00aci = j\n\u22a2 (M \u2299 1) i j = diagonal (fun i => M i i) i j\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Type u_4\nn : Type u_5\nR : Type u_6\nA B C : Matrix m n \u03b1\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : MulZeroOneClass \u03b1\nM : Matrix n n \u03b1\n\u22a2 1 \u2299 M = diagonal fun i => M i i\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Type u_4\nn : Type u_5\nR : Type u_6\nA B C : Matrix m n \u03b1\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : MulZeroOneClass \u03b1\nM : Matrix n n \u03b1\ni j : n\n\u22a2 (1 \u2299 M) i j = diagonal (fun i => M i i) i j\n[PROOFSTEP]\nby_cases h : i = j\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Type u_4\nn : Type u_5\nR : Type u_6\nA B C : Matrix m n \u03b1\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : MulZeroOneClass \u03b1\nM : Matrix n n \u03b1\ni j : n\nh : i = j\n\u22a2 (1 \u2299 M) i j = diagonal (fun i => M i i) i j\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Type u_4\nn : Type u_5\nR : Type u_6\nA B C : Matrix m n \u03b1\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : MulZeroOneClass \u03b1\nM : Matrix n n \u03b1\ni j : n\nh : \u00aci = j\n\u22a2 (1 \u2299 M) i j = diagonal (fun i => M i i) i j\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Type u_4\nn : Type u_5\nR : Type u_6\nA B C : Matrix m n \u03b1\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Semiring \u03b1\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nv : m \u2192 \u03b1\nw : n \u2192 \u03b1\n\u22a2 vecMul v (A \u2299 B) \u2b1d\u1d65 w = trace (diagonal v * A * (B * diagonal w)\u1d40)\n[PROOFSTEP]\nrw [\u2190 sum_hadamard_eq, Finset.sum_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : Type u_4\nn : Type u_5\nR : Type u_6\nA B C : Matrix m n \u03b1\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Semiring \u03b1\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nv : m \u2192 \u03b1\nw : n \u2192 \u03b1\n\u22a2 vecMul v (A \u2299 B) \u2b1d\u1d65 w = \u2211 y : n, \u2211 x : m, ((diagonal v * A) \u2299 (B * diagonal w)) x y\n[PROOFSTEP]\nsimp [dotProduct, vecMul, Finset.sum_mul, mul_assoc]\n", "meta": {"mathlib_filename": "Mathlib.Data.Matrix.Hadamard", "llama_tokens": 1466, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4250267293941805}} {"text": "[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nA : Matrix m (n\u2081 \u2295 n\u2082) R\n\u22a2 fromColumns (toColumns\u2081 A) (toColumns\u2082 A) = A\n[PROOFSTEP]\next i (j | j)\n[GOAL]\ncase a.h.inl\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nA : Matrix m (n\u2081 \u2295 n\u2082) R\ni : m\nj : n\u2081\n\u22a2 fromColumns (toColumns\u2081 A) (toColumns\u2082 A) i (Sum.inl j) = A i (Sum.inl j)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.h.inr\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nA : Matrix m (n\u2081 \u2295 n\u2082) R\ni : m\nj : n\u2082\n\u22a2 fromColumns (toColumns\u2081 A) (toColumns\u2082 A) i (Sum.inr j) = A i (Sum.inr j)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nA : Matrix (m\u2081 \u2295 m\u2082) n R\n\u22a2 fromRows (toRows\u2081 A) (toRows\u2082 A) = A\n[PROOFSTEP]\next (i | i) j\n[GOAL]\ncase a.inl.h\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nA : Matrix (m\u2081 \u2295 m\u2082) n R\ni : m\u2081\nj : n\n\u22a2 fromRows (toRows\u2081 A) (toRows\u2082 A) (Sum.inl i) j = A (Sum.inl i) j\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.inr.h\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nA : Matrix (m\u2081 \u2295 m\u2082) n R\ni : m\u2082\nj : n\n\u22a2 fromRows (toRows\u2081 A) (toRows\u2082 A) (Sum.inr i) j = A (Sum.inr i) j\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\n\u22a2 Function.Injective2 fromRows\n[PROOFSTEP]\nintros x1 x2 y1 y2\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nx1 x2 : Matrix m\u2081 n R\ny1 y2 : Matrix m\u2082 n R\n\u22a2 fromRows x1 y1 = fromRows x2 y2 \u2192 x1 = x2 \u2227 y1 = y2\n[PROOFSTEP]\nsimp only [Function.funext_iff, \u2190 Matrix.ext_iff]\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nx1 x2 : Matrix m\u2081 n R\ny1 y2 : Matrix m\u2082 n R\n\u22a2 (\u2200 (i : m\u2081 \u2295 m\u2082) (j : n), fromRows x1 y1 i j = fromRows x2 y2 i j) \u2192\n (\u2200 (i : m\u2081) (j : n), x1 i j = x2 i j) \u2227 \u2200 (i : m\u2082) (j : n), y1 i j = y2 i j\n[PROOFSTEP]\naesop\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\n\u22a2 Function.Injective2 fromColumns\n[PROOFSTEP]\nintros x1 x2 y1 y2\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nx1 x2 : Matrix m n\u2081 R\ny1 y2 : Matrix m n\u2082 R\n\u22a2 fromColumns x1 y1 = fromColumns x2 y2 \u2192 x1 = x2 \u2227 y1 = y2\n[PROOFSTEP]\nsimp only [Function.funext_iff, \u2190 Matrix.ext_iff]\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nx1 x2 : Matrix m n\u2081 R\ny1 y2 : Matrix m n\u2082 R\n\u22a2 (\u2200 (i : m) (j : n\u2081 \u2295 n\u2082), fromColumns x1 y1 i j = fromColumns x2 y2 i j) \u2192\n (\u2200 (i : m) (j : n\u2081), x1 i j = x2 i j) \u2227 \u2200 (i : m) (j : n\u2082), y1 i j = y2 i j\n[PROOFSTEP]\naesop\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nA\u2081 : Matrix m n\u2081 R\nA\u2082 : Matrix m n\u2082 R\n\u22a2 (fromColumns A\u2081 A\u2082)\u1d40 = fromRows A\u2081\u1d40 A\u2082\u1d40\n[PROOFSTEP]\next (i | i) j\n[GOAL]\ncase a.inl.h\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nA\u2081 : Matrix m n\u2081 R\nA\u2082 : Matrix m n\u2082 R\ni : n\u2081\nj : m\n\u22a2 (fromColumns A\u2081 A\u2082)\u1d40 (Sum.inl i) j = fromRows A\u2081\u1d40 A\u2082\u1d40 (Sum.inl i) j\n[PROOFSTEP]\nsimp\n /- A row partioned matrix when transposed gives a column partioned matrix with rows of the initial\n matrix tranposed to become columns. -/\n[GOAL]\ncase a.inr.h\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nA\u2081 : Matrix m n\u2081 R\nA\u2082 : Matrix m n\u2082 R\ni : n\u2082\nj : m\n\u22a2 (fromColumns A\u2081 A\u2082)\u1d40 (Sum.inr i) j = fromRows A\u2081\u1d40 A\u2082\u1d40 (Sum.inr i) j\n[PROOFSTEP]\nsimp\n /- A row partioned matrix when transposed gives a column partioned matrix with rows of the initial\n matrix tranposed to become columns. -/\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nA\u2081 : Matrix m\u2081 n R\nA\u2082 : Matrix m\u2082 n R\n\u22a2 (fromRows A\u2081 A\u2082)\u1d40 = fromColumns A\u2081\u1d40 A\u2082\u1d40\n[PROOFSTEP]\next i (j | j)\n[GOAL]\ncase a.h.inl\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nA\u2081 : Matrix m\u2081 n R\nA\u2082 : Matrix m\u2082 n R\ni : n\nj : m\u2081\n\u22a2 (fromRows A\u2081 A\u2082)\u1d40 i (Sum.inl j) = fromColumns A\u2081\u1d40 A\u2082\u1d40 i (Sum.inl j)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.h.inr\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b9 : Fintype m\ninst\u271d\u00b9\u2070 : Fintype m\u2081\ninst\u271d\u2079 : Fintype m\u2082\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype n\u2081\ninst\u271d\u2076 : Fintype n\u2082\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq m\u2081\ninst\u271d\u00b3 : DecidableEq m\u2082\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq n\u2081\ninst\u271d : DecidableEq n\u2082\nA\u2081 : Matrix m\u2081 n R\nA\u2082 : Matrix m\u2082 n R\ni : n\nj : m\u2082\n\u22a2 (fromRows A\u2081 A\u2082)\u1d40 i (Sum.inr j) = fromColumns A\u2081\u1d40 A\u2082\u1d40 i (Sum.inr j)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix m\u2081 n R\nA\u2082 : Matrix m\u2082 n R\nB : Matrix n m R\n\u22a2 fromRows A\u2081 A\u2082 * B = fromRows (A\u2081 * B) (A\u2082 * B)\n[PROOFSTEP]\next (_ | _) _\n[GOAL]\ncase a.inl.h\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix m\u2081 n R\nA\u2082 : Matrix m\u2082 n R\nB : Matrix n m R\nval\u271d : m\u2081\nx\u271d : m\n\u22a2 (fromRows A\u2081 A\u2082 * B) (Sum.inl val\u271d) x\u271d = fromRows (A\u2081 * B) (A\u2082 * B) (Sum.inl val\u271d) x\u271d\n[PROOFSTEP]\nsimp [mul_apply]\n[GOAL]\ncase a.inr.h\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix m\u2081 n R\nA\u2082 : Matrix m\u2082 n R\nB : Matrix n m R\nval\u271d : m\u2082\nx\u271d : m\n\u22a2 (fromRows A\u2081 A\u2082 * B) (Sum.inr val\u271d) x\u271d = fromRows (A\u2081 * B) (A\u2082 * B) (Sum.inr val\u271d) x\u271d\n[PROOFSTEP]\nsimp [mul_apply]\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA : Matrix m n R\nB\u2081 : Matrix n n\u2081 R\nB\u2082 : Matrix n n\u2082 R\n\u22a2 A * fromColumns B\u2081 B\u2082 = fromColumns (A * B\u2081) (A * B\u2082)\n[PROOFSTEP]\next _ (_ | _)\n[GOAL]\ncase a.h.inl\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA : Matrix m n R\nB\u2081 : Matrix n n\u2081 R\nB\u2082 : Matrix n n\u2082 R\ni\u271d : m\nval\u271d : n\u2081\n\u22a2 (A * fromColumns B\u2081 B\u2082) i\u271d (Sum.inl val\u271d) = fromColumns (A * B\u2081) (A * B\u2082) i\u271d (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp [mul_apply]\n[GOAL]\ncase a.h.inr\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA : Matrix m n R\nB\u2081 : Matrix n n\u2081 R\nB\u2082 : Matrix n n\u2082 R\ni\u271d : m\nval\u271d : n\u2082\n\u22a2 (A * fromColumns B\u2081 B\u2082) i\u271d (Sum.inr val\u271d) = fromColumns (A * B\u2081) (A * B\u2082) i\u271d (Sum.inr val\u271d)\n[PROOFSTEP]\nsimp [mul_apply]\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\n\u22a2 fromRows 0 0 = 0\n[PROOFSTEP]\next (_ | _) _\n[GOAL]\ncase a.inl.h\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nval\u271d : m\u2081\nx\u271d : n\n\u22a2 fromRows 0 0 (Sum.inl val\u271d) x\u271d = OfNat.ofNat 0 (Sum.inl val\u271d) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.inr.h\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nval\u271d : m\u2082\nx\u271d : n\n\u22a2 fromRows 0 0 (Sum.inr val\u271d) x\u271d = OfNat.ofNat 0 (Sum.inr val\u271d) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\n\u22a2 fromColumns 0 0 = 0\n[PROOFSTEP]\next _ (_ | _)\n[GOAL]\ncase a.h.inl\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\ni\u271d : m\nval\u271d : n\u2081\n\u22a2 fromColumns 0 0 i\u271d (Sum.inl val\u271d) = OfNat.ofNat 0 i\u271d (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.h.inr\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\ni\u271d : m\nval\u271d : n\u2082\n\u22a2 fromColumns 0 0 i\u271d (Sum.inr val\u271d) = OfNat.ofNat 0 i\u271d (Sum.inr val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\n\u22a2 fromColumns (fromRows B\u2081\u2081 B\u2082\u2081) (fromRows B\u2081\u2082 B\u2082\u2082) = fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082\n[PROOFSTEP]\next (_ | _) (_ | _)\n[GOAL]\ncase a.inl.h.inl\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\nval\u271d\u00b9 : m\u2081\nval\u271d : n\u2081\n\u22a2 fromColumns (fromRows B\u2081\u2081 B\u2082\u2081) (fromRows B\u2081\u2082 B\u2082\u2082) (Sum.inl val\u271d\u00b9) (Sum.inl val\u271d) =\n fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082 (Sum.inl val\u271d\u00b9) (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.inl.h.inr\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\nval\u271d\u00b9 : m\u2081\nval\u271d : n\u2082\n\u22a2 fromColumns (fromRows B\u2081\u2081 B\u2082\u2081) (fromRows B\u2081\u2082 B\u2082\u2082) (Sum.inl val\u271d\u00b9) (Sum.inr val\u271d) =\n fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082 (Sum.inl val\u271d\u00b9) (Sum.inr val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.inr.h.inl\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\nval\u271d\u00b9 : m\u2082\nval\u271d : n\u2081\n\u22a2 fromColumns (fromRows B\u2081\u2081 B\u2082\u2081) (fromRows B\u2081\u2082 B\u2082\u2082) (Sum.inr val\u271d\u00b9) (Sum.inl val\u271d) =\n fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082 (Sum.inr val\u271d\u00b9) (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.inr.h.inr\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\nval\u271d\u00b9 : m\u2082\nval\u271d : n\u2082\n\u22a2 fromColumns (fromRows B\u2081\u2081 B\u2082\u2081) (fromRows B\u2081\u2082 B\u2082\u2082) (Sum.inr val\u271d\u00b9) (Sum.inr val\u271d) =\n fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082 (Sum.inr val\u271d\u00b9) (Sum.inr val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\n\u22a2 fromRows (fromColumns B\u2081\u2081 B\u2081\u2082) (fromColumns B\u2082\u2081 B\u2082\u2082) = fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082\n[PROOFSTEP]\next (_ | _) (_ | _)\n[GOAL]\ncase a.inl.h.inl\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\nval\u271d\u00b9 : m\u2081\nval\u271d : n\u2081\n\u22a2 fromRows (fromColumns B\u2081\u2081 B\u2081\u2082) (fromColumns B\u2082\u2081 B\u2082\u2082) (Sum.inl val\u271d\u00b9) (Sum.inl val\u271d) =\n fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082 (Sum.inl val\u271d\u00b9) (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.inl.h.inr\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\nval\u271d\u00b9 : m\u2081\nval\u271d : n\u2082\n\u22a2 fromRows (fromColumns B\u2081\u2081 B\u2081\u2082) (fromColumns B\u2082\u2081 B\u2082\u2082) (Sum.inl val\u271d\u00b9) (Sum.inr val\u271d) =\n fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082 (Sum.inl val\u271d\u00b9) (Sum.inr val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.inr.h.inl\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\nval\u271d\u00b9 : m\u2082\nval\u271d : n\u2081\n\u22a2 fromRows (fromColumns B\u2081\u2081 B\u2081\u2082) (fromColumns B\u2082\u2081 B\u2082\u2082) (Sum.inr val\u271d\u00b9) (Sum.inl val\u271d) =\n fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082 (Sum.inr val\u271d\u00b9) (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.inr.h.inr\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\nval\u271d\u00b9 : m\u2082\nval\u271d : n\u2082\n\u22a2 fromRows (fromColumns B\u2081\u2081 B\u2081\u2082) (fromColumns B\u2082\u2081 B\u2082\u2082) (Sum.inr val\u271d\u00b9) (Sum.inr val\u271d) =\n fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082 (Sum.inr val\u271d\u00b9) (Sum.inr val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix m\u2081 n R\nA\u2082 : Matrix m\u2082 n R\nB\u2081 : Matrix n n\u2081 R\nB\u2082 : Matrix n n\u2082 R\n\u22a2 fromRows A\u2081 A\u2082 * fromColumns B\u2081 B\u2082 = fromBlocks (A\u2081 * B\u2081) (A\u2081 * B\u2082) (A\u2082 * B\u2081) (A\u2082 * B\u2082)\n[PROOFSTEP]\next (_ | _) (_ | _)\n[GOAL]\ncase a.inl.h.inl\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix m\u2081 n R\nA\u2082 : Matrix m\u2082 n R\nB\u2081 : Matrix n n\u2081 R\nB\u2082 : Matrix n n\u2082 R\nval\u271d\u00b9 : m\u2081\nval\u271d : n\u2081\n\u22a2 (fromRows A\u2081 A\u2082 * fromColumns B\u2081 B\u2082) (Sum.inl val\u271d\u00b9) (Sum.inl val\u271d) =\n fromBlocks (A\u2081 * B\u2081) (A\u2081 * B\u2082) (A\u2082 * B\u2081) (A\u2082 * B\u2082) (Sum.inl val\u271d\u00b9) (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.inl.h.inr\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix m\u2081 n R\nA\u2082 : Matrix m\u2082 n R\nB\u2081 : Matrix n n\u2081 R\nB\u2082 : Matrix n n\u2082 R\nval\u271d\u00b9 : m\u2081\nval\u271d : n\u2082\n\u22a2 (fromRows A\u2081 A\u2082 * fromColumns B\u2081 B\u2082) (Sum.inl val\u271d\u00b9) (Sum.inr val\u271d) =\n fromBlocks (A\u2081 * B\u2081) (A\u2081 * B\u2082) (A\u2082 * B\u2081) (A\u2082 * B\u2082) (Sum.inl val\u271d\u00b9) (Sum.inr val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.inr.h.inl\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix m\u2081 n R\nA\u2082 : Matrix m\u2082 n R\nB\u2081 : Matrix n n\u2081 R\nB\u2082 : Matrix n n\u2082 R\nval\u271d\u00b9 : m\u2082\nval\u271d : n\u2081\n\u22a2 (fromRows A\u2081 A\u2082 * fromColumns B\u2081 B\u2082) (Sum.inr val\u271d\u00b9) (Sum.inl val\u271d) =\n fromBlocks (A\u2081 * B\u2081) (A\u2081 * B\u2082) (A\u2082 * B\u2081) (A\u2082 * B\u2082) (Sum.inr val\u271d\u00b9) (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.inr.h.inr\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix m\u2081 n R\nA\u2082 : Matrix m\u2082 n R\nB\u2081 : Matrix n n\u2081 R\nB\u2082 : Matrix n n\u2082 R\nval\u271d\u00b9 : m\u2082\nval\u271d : n\u2082\n\u22a2 (fromRows A\u2081 A\u2082 * fromColumns B\u2081 B\u2082) (Sum.inr val\u271d\u00b9) (Sum.inr val\u271d) =\n fromBlocks (A\u2081 * B\u2081) (A\u2081 * B\u2082) (A\u2082 * B\u2081) (A\u2082 * B\u2082) (Sum.inr val\u271d\u00b9) (Sum.inr val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix m n\u2081 R\nA\u2082 : Matrix m n\u2082 R\nB\u2081 : Matrix n\u2081 n R\nB\u2082 : Matrix n\u2082 n R\n\u22a2 fromColumns A\u2081 A\u2082 * fromRows B\u2081 B\u2082 = A\u2081 * B\u2081 + A\u2082 * B\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix m n\u2081 R\nA\u2082 : Matrix m n\u2082 R\nB\u2081 : Matrix n\u2081 n R\nB\u2082 : Matrix n\u2082 n R\ni\u271d : m\nx\u271d : n\n\u22a2 (fromColumns A\u2081 A\u2082 * fromRows B\u2081 B\u2082) i\u271d x\u271d = (A\u2081 * B\u2081 + A\u2082 * B\u2082) i\u271d x\u271d\n[PROOFSTEP]\nsimp [mul_apply]\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix m m\u2081 R\nA\u2082 : Matrix m m\u2082 R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\n\u22a2 fromColumns A\u2081 A\u2082 * fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082 = fromColumns (A\u2081 * B\u2081\u2081 + A\u2082 * B\u2082\u2081) (A\u2081 * B\u2081\u2082 + A\u2082 * B\u2082\u2082)\n[PROOFSTEP]\next _ (_ | _)\n[GOAL]\ncase a.h.inl\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix m m\u2081 R\nA\u2082 : Matrix m m\u2082 R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\ni\u271d : m\nval\u271d : n\u2081\n\u22a2 (fromColumns A\u2081 A\u2082 * fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082) i\u271d (Sum.inl val\u271d) =\n fromColumns (A\u2081 * B\u2081\u2081 + A\u2082 * B\u2082\u2081) (A\u2081 * B\u2081\u2082 + A\u2082 * B\u2082\u2082) i\u271d (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp [mul_apply]\n[GOAL]\ncase a.h.inr\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix m m\u2081 R\nA\u2082 : Matrix m m\u2082 R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\ni\u271d : m\nval\u271d : n\u2082\n\u22a2 (fromColumns A\u2081 A\u2082 * fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082) i\u271d (Sum.inr val\u271d) =\n fromColumns (A\u2081 * B\u2081\u2081 + A\u2082 * B\u2082\u2081) (A\u2081 * B\u2081\u2082 + A\u2082 * B\u2082\u2082) i\u271d (Sum.inr val\u271d)\n[PROOFSTEP]\nsimp [mul_apply]\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix n\u2081 n R\nA\u2082 : Matrix n\u2082 n R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\n\u22a2 fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082 * fromRows A\u2081 A\u2082 = fromRows (B\u2081\u2081 * A\u2081 + B\u2081\u2082 * A\u2082) (B\u2082\u2081 * A\u2081 + B\u2082\u2082 * A\u2082)\n[PROOFSTEP]\next (_ | _) _\n[GOAL]\ncase a.inl.h\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix n\u2081 n R\nA\u2082 : Matrix n\u2082 n R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\nval\u271d : m\u2081\nx\u271d : n\n\u22a2 (fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082 * fromRows A\u2081 A\u2082) (Sum.inl val\u271d) x\u271d =\n fromRows (B\u2081\u2081 * A\u2081 + B\u2081\u2082 * A\u2082) (B\u2082\u2081 * A\u2081 + B\u2082\u2082 * A\u2082) (Sum.inl val\u271d) x\u271d\n[PROOFSTEP]\nsimp [mul_apply]\n[GOAL]\ncase a.inr.h\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Semiring R\nA\u2081 : Matrix n\u2081 n R\nA\u2082 : Matrix n\u2082 n R\nB\u2081\u2081 : Matrix m\u2081 n\u2081 R\nB\u2081\u2082 : Matrix m\u2081 n\u2082 R\nB\u2082\u2081 : Matrix m\u2082 n\u2081 R\nB\u2082\u2082 : Matrix m\u2082 n\u2082 R\nval\u271d : m\u2082\nx\u271d : n\n\u22a2 (fromBlocks B\u2081\u2081 B\u2081\u2082 B\u2082\u2081 B\u2082\u2082 * fromRows A\u2081 A\u2082) (Sum.inr val\u271d) x\u271d =\n fromRows (B\u2081\u2081 * A\u2081 + B\u2081\u2082 * A\u2082) (B\u2082\u2081 * A\u2081 + B\u2082\u2082 * A\u2082) (Sum.inr val\u271d) x\u271d\n[PROOFSTEP]\nsimp [mul_apply]\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : CommRing R\ne : n \u2243 n\u2081 \u2295 n\u2082\nA\u2081 : Matrix n n\u2081 R\nA\u2082 : Matrix n n\u2082 R\nB\u2081 : Matrix n\u2081 n R\nB\u2082 : Matrix n\u2082 n R\n\u22a2 fromColumns A\u2081 A\u2082 * fromRows B\u2081 B\u2082 = 1 \u2194 fromRows B\u2081 B\u2082 * fromColumns A\u2081 A\u2082 = 1\n[PROOFSTEP]\ncalc\n fromColumns A\u2081 A\u2082 * fromRows B\u2081 B\u2082 = 1\n _ \u2194 submatrix (fromColumns A\u2081 A\u2082) id e * submatrix (fromRows B\u2081 B\u2082) e id = 1 := by simp\n _ \u2194 submatrix (fromRows B\u2081 B\u2082) e id * submatrix (fromColumns A\u2081 A\u2082) id e = 1 := mul_eq_one_comm\n _ \u2194 reindex e.symm e.symm (fromRows B\u2081 B\u2082 * fromColumns A\u2081 A\u2082) = reindex e.symm e.symm 1 := by\n simp only [reindex_apply, Equiv.symm_symm, submatrix_one_equiv, submatrix_mul (he\u2082 := Function.bijective_id)]\n _ \u2194 fromRows B\u2081 B\u2082 * fromColumns A\u2081 A\u2082 = 1 := (reindex _ _).injective.eq_iff\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : CommRing R\ne : n \u2243 n\u2081 \u2295 n\u2082\nA\u2081 : Matrix n n\u2081 R\nA\u2082 : Matrix n n\u2082 R\nB\u2081 : Matrix n\u2081 n R\nB\u2082 : Matrix n\u2082 n R\n\u22a2 fromColumns A\u2081 A\u2082 * fromRows B\u2081 B\u2082 = 1 \u2194 submatrix (fromColumns A\u2081 A\u2082) id \u2191e * submatrix (fromRows B\u2081 B\u2082) (\u2191e) id = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : CommRing R\ne : n \u2243 n\u2081 \u2295 n\u2082\nA\u2081 : Matrix n n\u2081 R\nA\u2082 : Matrix n n\u2082 R\nB\u2081 : Matrix n\u2081 n R\nB\u2082 : Matrix n\u2082 n R\n\u22a2 submatrix (fromRows B\u2081 B\u2082) (\u2191e) id * submatrix (fromColumns A\u2081 A\u2082) id \u2191e = 1 \u2194\n \u2191(reindex e.symm e.symm) (fromRows B\u2081 B\u2082 * fromColumns A\u2081 A\u2082) = \u2191(reindex e.symm e.symm) 1\n[PROOFSTEP]\nsimp only [reindex_apply, Equiv.symm_symm, submatrix_one_equiv, submatrix_mul (he\u2082 := Function.bijective_id)]\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Star R\nA\u2081 : Matrix m n\u2081 R\nA\u2082 : Matrix m n\u2082 R\n\u22a2 (fromColumns A\u2081 A\u2082)\u1d34 = fromRows A\u2081\u1d34 A\u2082\u1d34\n[PROOFSTEP]\next (_ | _) _\n[GOAL]\ncase a.inl.h\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Star R\nA\u2081 : Matrix m n\u2081 R\nA\u2082 : Matrix m n\u2082 R\nval\u271d : n\u2081\nx\u271d : m\n\u22a2 (fromColumns A\u2081 A\u2082)\u1d34 (Sum.inl val\u271d) x\u271d = fromRows A\u2081\u1d34 A\u2082\u1d34 (Sum.inl val\u271d) x\u271d\n[PROOFSTEP]\nsimp\n /- A row partioned matrix in a Star ring when conjugate transposed gives a column partitioned matrix\n with the rows of the initial matrix conjugate transposed to become columns. -/\n[GOAL]\ncase a.inr.h\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Star R\nA\u2081 : Matrix m n\u2081 R\nA\u2082 : Matrix m n\u2082 R\nval\u271d : n\u2082\nx\u271d : m\n\u22a2 (fromColumns A\u2081 A\u2082)\u1d34 (Sum.inr val\u271d) x\u271d = fromRows A\u2081\u1d34 A\u2082\u1d34 (Sum.inr val\u271d) x\u271d\n[PROOFSTEP]\nsimp\n /- A row partioned matrix in a Star ring when conjugate transposed gives a column partitioned matrix\n with the rows of the initial matrix conjugate transposed to become columns. -/\n[GOAL]\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Star R\nA\u2081 : Matrix m\u2081 n R\nA\u2082 : Matrix m\u2082 n R\n\u22a2 (fromRows A\u2081 A\u2082)\u1d34 = fromColumns A\u2081\u1d34 A\u2082\u1d34\n[PROOFSTEP]\next _ (_ | _)\n[GOAL]\ncase a.h.inl\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Star R\nA\u2081 : Matrix m\u2081 n R\nA\u2082 : Matrix m\u2082 n R\ni\u271d : n\nval\u271d : m\u2081\n\u22a2 (fromRows A\u2081 A\u2082)\u1d34 i\u271d (Sum.inl val\u271d) = fromColumns A\u2081\u1d34 A\u2082\u1d34 i\u271d (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.h.inr\nR : Type u_1\nm : Type u_2\nm\u2081 : Type u_3\nm\u2082 : Type u_4\nn : Type u_5\nn\u2081 : Type u_6\nn\u2082 : Type u_7\ninst\u271d\u00b9\u00b2 : Fintype m\ninst\u271d\u00b9\u00b9 : Fintype m\u2081\ninst\u271d\u00b9\u2070 : Fintype m\u2082\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype n\u2081\ninst\u271d\u2077 : Fintype n\u2082\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq m\u2081\ninst\u271d\u2074 : DecidableEq m\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq n\u2081\ninst\u271d\u00b9 : DecidableEq n\u2082\ninst\u271d : Star R\nA\u2081 : Matrix m\u2081 n R\nA\u2082 : Matrix m\u2082 n R\ni\u271d : n\nval\u271d : m\u2082\n\u22a2 (fromRows A\u2081 A\u2082)\u1d34 i\u271d (Sum.inr val\u271d) = fromColumns A\u2081\u1d34 A\u2082\u1d34 i\u271d (Sum.inr val\u271d)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Matrix.ColumnRowPartitioned", "llama_tokens": 20206, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585786300049, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.42496661119966056}} {"text": "[GOAL]\ns : Set \u2115\n\u22a2 sInf s = 0 \u2194 0 \u2208 s \u2228 s = \u2205\n[PROOFSTEP]\ncases eq_empty_or_nonempty s with\n| inl h =>\n subst h\n simp only [or_true_iff, eq_self_iff_true, iff_true_iff, iInf, InfSet.sInf, mem_empty_iff_false, exists_false, dif_neg,\n not_false_iff]\n| inr h => simp only [h.ne_empty, or_false_iff, Nat.sInf_def, h, Nat.find_eq_zero]\n[GOAL]\ns : Set \u2115\nx\u271d : s = \u2205 \u2228 Set.Nonempty s\n\u22a2 sInf s = 0 \u2194 0 \u2208 s \u2228 s = \u2205\n[PROOFSTEP]\ncases eq_empty_or_nonempty s with\n| inl h =>\n subst h\n simp only [or_true_iff, eq_self_iff_true, iff_true_iff, iInf, InfSet.sInf, mem_empty_iff_false, exists_false, dif_neg,\n not_false_iff]\n| inr h => simp only [h.ne_empty, or_false_iff, Nat.sInf_def, h, Nat.find_eq_zero]\n[GOAL]\ncase inl\ns : Set \u2115\nh : s = \u2205\n\u22a2 sInf s = 0 \u2194 0 \u2208 s \u2228 s = \u2205\n[PROOFSTEP]\n\n| inl h =>\n subst h\n simp only [or_true_iff, eq_self_iff_true, iff_true_iff, iInf, InfSet.sInf, mem_empty_iff_false, exists_false, dif_neg,\n not_false_iff]\n[GOAL]\ncase inl\ns : Set \u2115\nh : s = \u2205\n\u22a2 sInf s = 0 \u2194 0 \u2208 s \u2228 s = \u2205\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase inl\n\u22a2 sInf \u2205 = 0 \u2194 0 \u2208 \u2205 \u2228 \u2205 = \u2205\n[PROOFSTEP]\nsimp only [or_true_iff, eq_self_iff_true, iff_true_iff, iInf, InfSet.sInf, mem_empty_iff_false, exists_false, dif_neg,\n not_false_iff]\n[GOAL]\ncase inr\ns : Set \u2115\nh : Set.Nonempty s\n\u22a2 sInf s = 0 \u2194 0 \u2208 s \u2228 s = \u2205\n[PROOFSTEP]\n\n| inr h => simp only [h.ne_empty, or_false_iff, Nat.sInf_def, h, Nat.find_eq_zero]\n[GOAL]\ncase inr\ns : Set \u2115\nh : Set.Nonempty s\n\u22a2 sInf s = 0 \u2194 0 \u2208 s \u2228 s = \u2205\n[PROOFSTEP]\nsimp only [h.ne_empty, or_false_iff, Nat.sInf_def, h, Nat.find_eq_zero]\n[GOAL]\n\u22a2 sInf \u2205 = 0\n[PROOFSTEP]\nrw [sInf_eq_zero]\n[GOAL]\n\u22a2 0 \u2208 \u2205 \u2228 \u2205 = \u2205\n[PROOFSTEP]\nright\n[GOAL]\ncase h\n\u22a2 \u2205 = \u2205\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Sort u_1\ninst\u271d : IsEmpty \u03b9\nf : \u03b9 \u2192 \u2115\n\u22a2 iInf f = 0\n[PROOFSTEP]\nrw [iInf_of_empty', sInf_empty]\n[GOAL]\ns : Set \u2115\nh : Set.Nonempty s\n\u22a2 sInf s \u2208 s\n[PROOFSTEP]\nrw [Nat.sInf_def h]\n[GOAL]\ns : Set \u2115\nh : Set.Nonempty s\n\u22a2 Nat.find h \u2208 s\n[PROOFSTEP]\nexact Nat.find_spec h\n[GOAL]\ns : Set \u2115\nm : \u2115\nhm : m < sInf s\n\u22a2 \u00acm \u2208 s\n[PROOFSTEP]\ncases eq_empty_or_nonempty s with\n| inl h => subst h; apply not_mem_empty\n| inr h => rw [Nat.sInf_def h] at hm ; exact Nat.find_min h hm\n[GOAL]\ns : Set \u2115\nm : \u2115\nhm : m < sInf s\nx\u271d : s = \u2205 \u2228 Set.Nonempty s\n\u22a2 \u00acm \u2208 s\n[PROOFSTEP]\ncases eq_empty_or_nonempty s with\n| inl h => subst h; apply not_mem_empty\n| inr h => rw [Nat.sInf_def h] at hm ; exact Nat.find_min h hm\n[GOAL]\ncase inl\ns : Set \u2115\nm : \u2115\nhm : m < sInf s\nh : s = \u2205\n\u22a2 \u00acm \u2208 s\n[PROOFSTEP]\n\n| inl h => subst h; apply not_mem_empty\n[GOAL]\ncase inl\ns : Set \u2115\nm : \u2115\nhm : m < sInf s\nh : s = \u2205\n\u22a2 \u00acm \u2208 s\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase inl\nm : \u2115\nhm : m < sInf \u2205\n\u22a2 \u00acm \u2208 \u2205\n[PROOFSTEP]\napply not_mem_empty\n[GOAL]\ncase inr\ns : Set \u2115\nm : \u2115\nhm : m < sInf s\nh : Set.Nonempty s\n\u22a2 \u00acm \u2208 s\n[PROOFSTEP]\n\n| inr h => rw [Nat.sInf_def h] at hm ; exact Nat.find_min h hm\n[GOAL]\ncase inr\ns : Set \u2115\nm : \u2115\nhm : m < sInf s\nh : Set.Nonempty s\n\u22a2 \u00acm \u2208 s\n[PROOFSTEP]\nrw [Nat.sInf_def h] at hm \n[GOAL]\ncase inr\ns : Set \u2115\nm : \u2115\nh : Set.Nonempty s\nhm : m < Nat.find h\n\u22a2 \u00acm \u2208 s\n[PROOFSTEP]\nexact Nat.find_min h hm\n[GOAL]\ns : Set \u2115\nm : \u2115\nhm : m \u2208 s\n\u22a2 sInf s \u2264 m\n[PROOFSTEP]\nrw [Nat.sInf_def \u27e8m, hm\u27e9]\n[GOAL]\ns : Set \u2115\nm : \u2115\nhm : m \u2208 s\n\u22a2 Nat.find (_ : \u2203 x, x \u2208 s) \u2264 m\n[PROOFSTEP]\nexact Nat.find_min' \u27e8m, hm\u27e9 hm\n[GOAL]\ns : Set \u2115\nh : 0 < sInf s\n\u22a2 Set.Nonempty s\n[PROOFSTEP]\nby_contra contra\n[GOAL]\ns : Set \u2115\nh : 0 < sInf s\ncontra : \u00acSet.Nonempty s\n\u22a2 False\n[PROOFSTEP]\nrw [Set.not_nonempty_iff_eq_empty] at contra \n[GOAL]\ns : Set \u2115\nh : 0 < sInf s\ncontra : s = \u2205\n\u22a2 False\n[PROOFSTEP]\nhave h' : sInf s \u2260 0 := ne_of_gt h\n[GOAL]\ns : Set \u2115\nh : 0 < sInf s\ncontra : s = \u2205\nh' : sInf s \u2260 0\n\u22a2 False\n[PROOFSTEP]\napply h'\n[GOAL]\ns : Set \u2115\nh : 0 < sInf s\ncontra : s = \u2205\nh' : sInf s \u2260 0\n\u22a2 sInf s = 0\n[PROOFSTEP]\nrw [Nat.sInf_eq_zero]\n[GOAL]\ns : Set \u2115\nh : 0 < sInf s\ncontra : s = \u2205\nh' : sInf s \u2260 0\n\u22a2 0 \u2208 s \u2228 s = \u2205\n[PROOFSTEP]\nright\n[GOAL]\ncase h\ns : Set \u2115\nh : 0 < sInf s\ncontra : s = \u2205\nh' : sInf s \u2260 0\n\u22a2 s = \u2205\n[PROOFSTEP]\nassumption\n[GOAL]\ns : Set \u2115\nhs : \u2200 (k\u2081 k\u2082 : \u2115), k\u2081 \u2264 k\u2082 \u2192 k\u2081 \u2208 s \u2192 k\u2082 \u2208 s\nk : \u2115\n\u22a2 sInf s = k + 1 \u2194 k + 1 \u2208 s \u2227 \u00ack \u2208 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\ns : Set \u2115\nhs : \u2200 (k\u2081 k\u2082 : \u2115), k\u2081 \u2264 k\u2082 \u2192 k\u2081 \u2208 s \u2192 k\u2082 \u2208 s\nk : \u2115\n\u22a2 sInf s = k + 1 \u2192 k + 1 \u2208 s \u2227 \u00ack \u2208 s\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\ns : Set \u2115\nhs : \u2200 (k\u2081 k\u2082 : \u2115), k\u2081 \u2264 k\u2082 \u2192 k\u2081 \u2208 s \u2192 k\u2082 \u2208 s\nk : \u2115\nH : sInf s = k + 1\n\u22a2 k + 1 \u2208 s \u2227 \u00ack \u2208 s\n[PROOFSTEP]\nrw [eq_Ici_of_nonempty_of_upward_closed (nonempty_of_sInf_eq_succ _) hs, H, mem_Ici, mem_Ici]\n[GOAL]\ncase mp\ns : Set \u2115\nhs : \u2200 (k\u2081 k\u2082 : \u2115), k\u2081 \u2264 k\u2082 \u2192 k\u2081 \u2208 s \u2192 k\u2082 \u2208 s\nk : \u2115\nH : sInf s = k + 1\n\u22a2 k + 1 \u2264 k + 1 \u2227 \u00ack + 1 \u2264 k\ns : Set \u2115\nhs : \u2200 (k\u2081 k\u2082 : \u2115), k\u2081 \u2264 k\u2082 \u2192 k\u2081 \u2208 s \u2192 k\u2082 \u2208 s\nk : \u2115\nH : sInf s = k + 1\n\u22a2 \u2115\ns : Set \u2115 hs : \u2200 (k\u2081 k\u2082 : \u2115), k\u2081 \u2264 k\u2082 \u2192 k\u2081 \u2208 s \u2192 k\u2082 \u2208 s k : \u2115 H : sInf s = k + 1 \u22a2 sInf s = ?m.6254 + 1\n[PROOFSTEP]\nexact \u27e8le_rfl, k.not_succ_le_self\u27e9\n[GOAL]\ns : Set \u2115\nhs : \u2200 (k\u2081 k\u2082 : \u2115), k\u2081 \u2264 k\u2082 \u2192 k\u2081 \u2208 s \u2192 k\u2082 \u2208 s\nk : \u2115\nH : sInf s = k + 1\n\u22a2 \u2115\ns : Set \u2115 hs : \u2200 (k\u2081 k\u2082 : \u2115), k\u2081 \u2264 k\u2082 \u2192 k\u2081 \u2208 s \u2192 k\u2082 \u2208 s k : \u2115 H : sInf s = k + 1 \u22a2 sInf s = ?m.6254 + 1\n[PROOFSTEP]\nexact k\n[GOAL]\ns : Set \u2115\nhs : \u2200 (k\u2081 k\u2082 : \u2115), k\u2081 \u2264 k\u2082 \u2192 k\u2081 \u2208 s \u2192 k\u2082 \u2208 s\nk : \u2115\nH : sInf s = k + 1\n\u22a2 sInf s = k + 1\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mpr\ns : Set \u2115\nhs : \u2200 (k\u2081 k\u2082 : \u2115), k\u2081 \u2264 k\u2082 \u2192 k\u2081 \u2208 s \u2192 k\u2082 \u2208 s\nk : \u2115\n\u22a2 k + 1 \u2208 s \u2227 \u00ack \u2208 s \u2192 sInf s = k + 1\n[PROOFSTEP]\nrintro \u27e8H, H'\u27e9\n[GOAL]\ncase mpr.intro\ns : Set \u2115\nhs : \u2200 (k\u2081 k\u2082 : \u2115), k\u2081 \u2264 k\u2082 \u2192 k\u2081 \u2208 s \u2192 k\u2082 \u2208 s\nk : \u2115\nH : k + 1 \u2208 s\nH' : \u00ack \u2208 s\n\u22a2 sInf s = k + 1\n[PROOFSTEP]\nrw [sInf_def (\u27e8_, H\u27e9 : s.Nonempty), find_eq_iff]\n[GOAL]\ncase mpr.intro\ns : Set \u2115\nhs : \u2200 (k\u2081 k\u2082 : \u2115), k\u2081 \u2264 k\u2082 \u2192 k\u2081 \u2208 s \u2192 k\u2082 \u2208 s\nk : \u2115\nH : k + 1 \u2208 s\nH' : \u00ack \u2208 s\n\u22a2 k + 1 \u2208 s \u2227 \u2200 (n : \u2115), n < k + 1 \u2192 \u00acn \u2208 s\n[PROOFSTEP]\nexact \u27e8H, fun n hnk hns \u21a6 H' <| hs n k (lt_succ_iff.mp hnk) hns\u27e9\n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\ns : Set \u2115\na : \u2115\nhb : BddAbove s\nha : a \u2208 s\n\u22a2 a \u2264 sSup s\n[PROOFSTEP]\nrw [sSup_def hb]\n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\ns : Set \u2115\na : \u2115\nhb : BddAbove s\nha : a \u2208 s\n\u22a2 a \u2264 Nat.find hb\n[PROOFSTEP]\nrevert a ha\n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\ns : Set \u2115\nhb : BddAbove s\n\u22a2 \u2200 (a : \u2115), a \u2208 s \u2192 a \u2264 Nat.find hb\n[PROOFSTEP]\nexact @Nat.find_spec _ _ hb\n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\ns : Set \u2115\na : \u2115\nx\u271d : Set.Nonempty s\nha : a \u2208 upperBounds s\n\u22a2 sSup s \u2264 a\n[PROOFSTEP]\nrw [sSup_def \u27e8a, ha\u27e9]\n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\ns : Set \u2115\na : \u2115\nx\u271d : Set.Nonempty s\nha : a \u2208 upperBounds s\n\u22a2 Nat.find (_ : \u2203 n, \u2200 (a : \u2115), a \u2208 s \u2192 a \u2264 n) \u2264 a\n[PROOFSTEP]\nexact Nat.find_min' _ ha\n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\ns : Set \u2115\na : \u2115\nx\u271d : BddBelow s\nha : a \u2208 s\n\u22a2 sInf s \u2264 a\n[PROOFSTEP]\nrw [sInf_def \u27e8a, ha\u27e9]\n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\ns : Set \u2115\na : \u2115\nx\u271d : BddBelow s\nha : a \u2208 s\n\u22a2 Nat.find (_ : \u2203 x, x \u2208 s) \u2264 a\n[PROOFSTEP]\nexact Nat.find_min' _ ha\n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\ns : Set \u2115\na : \u2115\nhs : Set.Nonempty s\nhb : a \u2208 lowerBounds s\n\u22a2 a \u2264 sInf s\n[PROOFSTEP]\nrw [sInf_def hs]\n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\ns : Set \u2115\na : \u2115\nhs : Set.Nonempty s\nhb : a \u2208 lowerBounds s\n\u22a2 a \u2264 Nat.find hs\n[PROOFSTEP]\nexact hb (@Nat.find_spec (fun n \u21a6 n \u2208 s) _ _)\n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\n\u22a2 \u2200 (s : Set \u2115), \u00acBddAbove s \u2192 sSup s = sSup univ\n[PROOFSTEP]\nintro s hs\n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\ns : Set \u2115\nhs : \u00acBddAbove s\n\u22a2 sSup s = sSup univ\n[PROOFSTEP]\nsimp only [mem_univ, forall_true_left, sSup]\n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\ns : Set \u2115\nhs : \u00acBddAbove s\n\u22a2 (if h : \u2203 n, \u2200 (a : \u2115), a \u2208 s \u2192 a \u2264 n then Nat.find h else 0) =\n if h : \u2203 n, \u2200 (a : \u2115), a \u2264 n then Nat.find (_ : \u2203 n, (fun n => \u2200 (a : \u2115), a \u2264 n) n) else 0\n[PROOFSTEP]\nrw [dif_neg, dif_neg]\n[GOAL]\ncase hnc\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\ns : Set \u2115\nhs : \u00acBddAbove s\n\u22a2 \u00ac\u2203 n, \u2200 (a : \u2115), a \u2264 n\n[PROOFSTEP]\nsimp only [not_exists, not_forall, not_le]\n[GOAL]\ncase hnc\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\ns : Set \u2115\nhs : \u00acBddAbove s\n\u22a2 \u2200 (x : \u2115), \u2203 x_1, x < x_1\n[PROOFSTEP]\nexact fun n \u21a6 \u27e8n + 1, lt.base n\u27e9\n[GOAL]\ncase hnc\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\ns : Set \u2115\nhs : \u00acBddAbove s\n\u22a2 \u00ac\u2203 n, \u2200 (a : \u2115), a \u2208 s \u2192 a \u2264 n\n[PROOFSTEP]\nexact hs\n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\ns : Set \u2115\nhs : \u00acBddBelow s\n\u22a2 sInf s = sInf univ\n[PROOFSTEP]\nsimp at hs \n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\n\u22a2 sSup \u2205 = \u22a5\n[PROOFSTEP]\nsimp only [sSup_def, Set.mem_empty_iff_false, forall_const, forall_prop_of_false, not_false_iff, exists_const]\n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\n\u22a2 Nat.find (_ : \u2203 n, (fun n => True) n) = \u22a5\n[PROOFSTEP]\napply bot_unique (Nat.find_min' _ _)\n[GOAL]\nsrc\u271d\u00b2 : OrderBot \u2115 := inferInstance\nsrc\u271d\u00b9 : Lattice \u2115 := LinearOrder.toLattice\nsrc\u271d : LinearOrder \u2115 := inferInstance\n\u22a2 True\n[PROOFSTEP]\ntrivial\n[GOAL]\nn : \u2115\np : \u2115 \u2192 Prop\nhn : n \u2264 sInf {m | p m}\n\u22a2 sInf {m | p (m + n)} + n = sInf {m | p m}\n[PROOFSTEP]\nobtain h | \u27e8m, hm\u27e9 := {m | p (m + n)}.eq_empty_or_nonempty\n[GOAL]\ncase inl\nn : \u2115\np : \u2115 \u2192 Prop\nhn : n \u2264 sInf {m | p m}\nh : {m | p (m + n)} = \u2205\n\u22a2 sInf {m | p (m + n)} + n = sInf {m | p m}\n[PROOFSTEP]\nrw [h, Nat.sInf_empty, zero_add]\n[GOAL]\ncase inl\nn : \u2115\np : \u2115 \u2192 Prop\nhn : n \u2264 sInf {m | p m}\nh : {m | p (m + n)} = \u2205\n\u22a2 n = sInf {m | p m}\n[PROOFSTEP]\nobtain hnp | hnp := hn.eq_or_lt\n[GOAL]\ncase inl.inl\nn : \u2115\np : \u2115 \u2192 Prop\nhn : n \u2264 sInf {m | p m}\nh : {m | p (m + n)} = \u2205\nhnp : n = sInf {m | p m}\n\u22a2 n = sInf {m | p m}\n[PROOFSTEP]\nexact hnp\n[GOAL]\ncase inl.inr\nn : \u2115\np : \u2115 \u2192 Prop\nhn : n \u2264 sInf {m | p m}\nh : {m | p (m + n)} = \u2205\nhnp : n < sInf {m | p m}\n\u22a2 n = sInf {m | p m}\n[PROOFSTEP]\nsuffices hp : p (sInf {m | p m} - n + n)\n[GOAL]\ncase inl.inr\nn : \u2115\np : \u2115 \u2192 Prop\nhn : n \u2264 sInf {m | p m}\nh : {m | p (m + n)} = \u2205\nhnp : n < sInf {m | p m}\nhp : p (sInf {m | p m} - n + n)\n\u22a2 n = sInf {m | p m}\n[PROOFSTEP]\nexact (h.subset hp).elim\n[GOAL]\ncase hp\nn : \u2115\np : \u2115 \u2192 Prop\nhn : n \u2264 sInf {m | p m}\nh : {m | p (m + n)} = \u2205\nhnp : n < sInf {m | p m}\n\u22a2 p (sInf {m | p m} - n + n)\n[PROOFSTEP]\nrw [tsub_add_cancel_of_le hn]\n[GOAL]\ncase hp\nn : \u2115\np : \u2115 \u2192 Prop\nhn : n \u2264 sInf {m | p m}\nh : {m | p (m + n)} = \u2205\nhnp : n < sInf {m | p m}\n\u22a2 p (sInf {m | p m})\n[PROOFSTEP]\nexact csInf_mem (nonempty_of_pos_sInf <| n.zero_le.trans_lt hnp)\n[GOAL]\ncase inr.intro\nn : \u2115\np : \u2115 \u2192 Prop\nhn : n \u2264 sInf {m | p m}\nm : \u2115\nhm : m \u2208 {m | p (m + n)}\n\u22a2 sInf {m | p (m + n)} + n = sInf {m | p m}\n[PROOFSTEP]\nhave hp : \u2203 n, n \u2208 {m | p m} := \u27e8_, hm\u27e9\n[GOAL]\ncase inr.intro\nn : \u2115\np : \u2115 \u2192 Prop\nhn : n \u2264 sInf {m | p m}\nm : \u2115\nhm : m \u2208 {m | p (m + n)}\nhp : \u2203 n, n \u2208 {m | p m}\n\u22a2 sInf {m | p (m + n)} + n = sInf {m | p m}\n[PROOFSTEP]\nrw [Nat.sInf_def \u27e8m, hm\u27e9, Nat.sInf_def hp]\n[GOAL]\ncase inr.intro\nn : \u2115\np : \u2115 \u2192 Prop\nhn : n \u2264 sInf {m | p m}\nm : \u2115\nhm : m \u2208 {m | p (m + n)}\nhp : \u2203 n, n \u2208 {m | p m}\n\u22a2 Nat.find (_ : \u2203 x, x \u2208 {m | p (m + n)}) + n = Nat.find hp\n[PROOFSTEP]\nrw [Nat.sInf_def hp] at hn \n[GOAL]\ncase inr.intro\nn : \u2115\np : \u2115 \u2192 Prop\nm : \u2115\nhm : m \u2208 {m | p (m + n)}\nhp : \u2203 n, n \u2208 {m | p m}\nhn : n \u2264 Nat.find hp\n\u22a2 Nat.find (_ : \u2203 x, x \u2208 {m | p (m + n)}) + n = Nat.find hp\n[PROOFSTEP]\nexact find_add hn\n[GOAL]\nn : \u2115\np : \u2115 \u2192 Prop\nh : 0 < sInf {m | p m}\n\u22a2 sInf {m | p m} + n = sInf {m | p (m - n)}\n[PROOFSTEP]\nsuffices h\u2081 : n \u2264 sInf {m | p (m - n)}\n[GOAL]\nn : \u2115\np : \u2115 \u2192 Prop\nh : 0 < sInf {m | p m}\nh\u2081 : n \u2264 sInf {m | p (m - n)}\n\u22a2 sInf {m | p m} + n = sInf {m | p (m - n)}\ncase h\u2081 n : \u2115 p : \u2115 \u2192 Prop h : 0 < sInf {m | p m} \u22a2 n \u2264 sInf {m | p (m - n)}\n[PROOFSTEP]\nconvert sInf_add h\u2081\n[GOAL]\ncase h.e'_2.h.e'_5.h.e'_3.h.e'_2.h.h.e'_1\nn : \u2115\np : \u2115 \u2192 Prop\nh : 0 < sInf {m | p m}\nh\u2081 : n \u2264 sInf {m | p (m - n)}\nx\u271d : \u2115\n\u22a2 x\u271d = x\u271d + n - n\n[PROOFSTEP]\nsimp_rw [add_tsub_cancel_right]\n[GOAL]\ncase h\u2081\nn : \u2115\np : \u2115 \u2192 Prop\nh : 0 < sInf {m | p m}\n\u22a2 n \u2264 sInf {m | p (m - n)}\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := nonempty_of_pos_sInf h\n[GOAL]\ncase h\u2081.intro\nn : \u2115\np : \u2115 \u2192 Prop\nh : 0 < sInf {m | p m}\nm : \u2115\nhm : m \u2208 {m | p m}\n\u22a2 n \u2264 sInf {m | p (m - n)}\n[PROOFSTEP]\nrefine'\n le_csInf \u27e8m + n, _\u27e9 fun b hb \u21a6\n le_of_not_lt fun hbn \u21a6 ne_of_mem_of_not_mem _ (not_mem_of_lt_sInf h) (tsub_eq_zero_of_le hbn.le)\n[GOAL]\ncase h\u2081.intro.refine'_1\nn : \u2115\np : \u2115 \u2192 Prop\nh : 0 < sInf {m | p m}\nm : \u2115\nhm : m \u2208 {m | p m}\n\u22a2 m + n \u2208 {m | p (m - n)}\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2081.intro.refine'_1\nn : \u2115\np : \u2115 \u2192 Prop\nh : 0 < sInf {m | p m}\nm : \u2115\nhm : m \u2208 {m | p m}\n\u22a2 p (m + n - n)\n[PROOFSTEP]\nrwa [add_tsub_cancel_right]\n[GOAL]\ncase h\u2081.intro.refine'_2\nn : \u2115\np : \u2115 \u2192 Prop\nh : 0 < sInf {m | p m}\nm : \u2115\nhm : m \u2208 {m | p m}\nb : \u2115\nhb : b \u2208 {m | p (m - n)}\nhbn : b < n\n\u22a2 b - n \u2208 {m | p m}\n[PROOFSTEP]\nexact hb\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CompleteLattice \u03b1\nu : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 \u2a06 (k : \u2115) (_ : k < n + 1), u k = (\u2a06 (k : \u2115) (_ : k < n), u k) \u2294 u n\n[PROOFSTEP]\nsimp [Nat.lt_succ_iff_lt_or_eq, iSup_or, iSup_sup_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CompleteLattice \u03b1\nu : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 \u2a06 (k : \u2115) (_ : k < n + 1), u k = u 0 \u2294 \u2a06 (k : \u2115) (_ : k < n), u (k + 1)\n[PROOFSTEP]\nrw [\u2190 sup_iSup_nat_succ]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CompleteLattice \u03b1\nu : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 (\u2a06 (_ : 0 < n + 1), u 0) \u2294 \u2a06 (i : \u2115) (_ : i + 1 < n + 1), u (i + 1) = u 0 \u2294 \u2a06 (k : \u2115) (_ : k < n), u (k + 1)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Lattice", "llama_tokens": 7852, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.4247988825165614}} {"text": "[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nA : Type u_2\ninst\u271d : AddGroup A\nH K : Subgroup G\n\u22a2 \u2200 (x : { x // x \u2208 \u22a5 }), x \u2208 {1}\n[PROOFSTEP]\nrintro \u27e8x, \u27e8hx\u27e9\u27e9\n[GOAL]\ncase mk.refl\nG : Type u_1\ninst\u271d\u00b9 : Group G\nA : Type u_2\ninst\u271d : AddGroup A\nH K : Subgroup G\n\u22a2 { val := 1, property := (_ : 1 = 1) } \u2208 {1}\n[PROOFSTEP]\nexact Finset.mem_singleton_self _\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\nA : Type u_2\ninst\u271d\u00b9 : AddGroup A\nH K : Subgroup G\ninst\u271d : Fintype G\n\u22a2 Fintype.card { x // x \u2208 \u22a4 } = Fintype.card G\n[PROOFSTEP]\nrw [Fintype.card_eq]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\nA : Type u_2\ninst\u271d\u00b9 : AddGroup A\nH K : Subgroup G\ninst\u271d : Fintype G\n\u22a2 Nonempty ({ x // x \u2208 \u22a4 } \u2243 G)\n[PROOFSTEP]\nexact Nonempty.intro Subgroup.topEquiv.toEquiv\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\nH K : Subgroup G\ninst\u271d\u00b9 : Fintype { x // x \u2208 H }\ninst\u271d : Fintype G\nh : Fintype.card { x // x \u2208 H } = Fintype.card G\n\u22a2 H = \u22a4\n[PROOFSTEP]\nletI : Fintype (H : Set G) := \u2039Fintype H\u203a\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\nH K : Subgroup G\ninst\u271d\u00b9 : Fintype { x // x \u2208 H }\ninst\u271d : Fintype G\nh : Fintype.card { x // x \u2208 H } = Fintype.card G\nthis : Fintype \u2191\u2191H := inst\u271d\u00b9\n\u22a2 H = \u22a4\n[PROOFSTEP]\nrw [SetLike.ext'_iff, coe_top, \u2190 Finset.coe_univ, \u2190 (H : Set G).coe_toFinset, Finset.coe_inj, \u2190\n Finset.card_eq_iff_eq_univ, \u2190 h, Set.toFinset_card]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\nH K : Subgroup G\ninst\u271d\u00b9 : Fintype { x // x \u2208 H }\ninst\u271d : Fintype G\nh : Fintype.card { x // x \u2208 H } = Fintype.card G\nthis : Fintype \u2191\u2191H := inst\u271d\u00b9\n\u22a2 Fintype.card \u2191\u2191H = Fintype.card { x // x \u2208 H }\n[PROOFSTEP]\ncongr\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\nH K : Subgroup G\ninst\u271d\u00b9 : Fintype { x // x \u2208 H }\ninst\u271d : Fintype G\nh : H = \u22a4\n\u22a2 Fintype.card { x // x \u2208 H } = Fintype.card G\n[PROOFSTEP]\nsimpa only [h] using card_top\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\nA : Type u_2\ninst\u271d\u00b9 : AddGroup A\nH K : Subgroup G\ninst\u271d : Fintype { x // x \u2208 H }\nh : Fintype.card { x // x \u2208 H } \u2264 1\nx : G\nhx : x \u2208 H\n\u22a2 x = 1\n[PROOFSTEP]\nsimpa [Subtype.ext_iff] using Fintype.card_le_one_iff.1 h \u27e8x, hx\u27e9 1\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\nA : Type u_2\ninst\u271d\u00b9 : AddGroup A\nH K : Subgroup G\ninst\u271d : Fintype { x // x \u2208 H }\nh : H = \u22a5\n\u22a2 Fintype.card { x // x \u2208 H } \u2264 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x i = 1\nh2 : \u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x i) \u2208 H\n\u22a2 x \u2208 H\n[PROOFSTEP]\ninduction' I using Finset.induction_on with i I hnmem ih generalizing x\n[GOAL]\ncase empty\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i : \u03b7), \u00aci \u2208 \u2205 \u2192 x i = 1\nh2 : \u2200 (i : \u03b7), i \u2208 \u2205 \u2192 Pi.mulSingle i (x i) \u2208 H\n\u22a2 x \u2208 H\n[PROOFSTEP]\nconvert one_mem H\n[GOAL]\ncase h.e'_4\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i : \u03b7), \u00aci \u2208 \u2205 \u2192 x i = 1\nh2 : \u2200 (i : \u03b7), i \u2208 \u2205 \u2192 Pi.mulSingle i (x i) \u2208 H\n\u22a2 x = 1\n[PROOFSTEP]\next i\n[GOAL]\ncase h.e'_4.h\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i : \u03b7), \u00aci \u2208 \u2205 \u2192 x i = 1\nh2 : \u2200 (i : \u03b7), i \u2208 \u2205 \u2192 Pi.mulSingle i (x i) \u2208 H\ni : \u03b7\n\u22a2 x i = OfNat.ofNat 1 i\n[PROOFSTEP]\nexact h1 i (Finset.not_mem_empty i)\n[GOAL]\ncase insert\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nih : \u2200 (x : (i : \u03b7) \u2192 f i), (\u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x i = 1) \u2192 (\u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x i) \u2208 H) \u2192 x \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\n\u22a2 x \u2208 H\n[PROOFSTEP]\nhave : x = Function.update x i 1 * Pi.mulSingle i (x i) :=\n by\n ext j\n by_cases heq : j = i\n \u00b7 subst heq\n simp\n \u00b7 simp [heq]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nih : \u2200 (x : (i : \u03b7) \u2192 f i), (\u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x i = 1) \u2192 (\u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x i) \u2208 H) \u2192 x \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\n\u22a2 x = Function.update x i 1 * Pi.mulSingle i (x i)\n[PROOFSTEP]\next j\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nih : \u2200 (x : (i : \u03b7) \u2192 f i), (\u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x i = 1) \u2192 (\u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x i) \u2208 H) \u2192 x \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\nj : \u03b7\n\u22a2 x j = (Function.update x i 1 * Pi.mulSingle i (x i)) j\n[PROOFSTEP]\nby_cases heq : j = i\n[GOAL]\ncase pos\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nih : \u2200 (x : (i : \u03b7) \u2192 f i), (\u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x i = 1) \u2192 (\u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x i) \u2208 H) \u2192 x \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\nj : \u03b7\nheq : j = i\n\u22a2 x j = (Function.update x i 1 * Pi.mulSingle i (x i)) j\n[PROOFSTEP]\nsubst heq\n[GOAL]\ncase pos\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\nI : Finset \u03b7\nih : \u2200 (x : (i : \u03b7) \u2192 f i), (\u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x i = 1) \u2192 (\u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x i) \u2208 H) \u2192 x \u2208 H\nx : (i : \u03b7) \u2192 f i\nj : \u03b7\nhnmem : \u00acj \u2208 I\nh1 : \u2200 (i : \u03b7), \u00aci \u2208 insert j I \u2192 x i = 1\nh2 : \u2200 (i : \u03b7), i \u2208 insert j I \u2192 Pi.mulSingle i (x i) \u2208 H\n\u22a2 x j = (Function.update x j 1 * Pi.mulSingle j (x j)) j\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nih : \u2200 (x : (i : \u03b7) \u2192 f i), (\u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x i = 1) \u2192 (\u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x i) \u2208 H) \u2192 x \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\nj : \u03b7\nheq : \u00acj = i\n\u22a2 x j = (Function.update x i 1 * Pi.mulSingle i (x i)) j\n[PROOFSTEP]\nsimp [heq]\n[GOAL]\ncase insert\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nih : \u2200 (x : (i : \u03b7) \u2192 f i), (\u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x i = 1) \u2192 (\u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x i) \u2208 H) \u2192 x \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\nthis : x = Function.update x i 1 * Pi.mulSingle i (x i)\n\u22a2 x \u2208 H\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase insert\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nih : \u2200 (x : (i : \u03b7) \u2192 f i), (\u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x i = 1) \u2192 (\u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x i) \u2208 H) \u2192 x \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\nthis : x = Function.update x i 1 * Pi.mulSingle i (x i)\n\u22a2 Function.update x i 1 * Pi.mulSingle i (x i) \u2208 H\n[PROOFSTEP]\nclear this\n[GOAL]\ncase insert\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nih : \u2200 (x : (i : \u03b7) \u2192 f i), (\u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x i = 1) \u2192 (\u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x i) \u2208 H) \u2192 x \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\n\u22a2 Function.update x i 1 * Pi.mulSingle i (x i) \u2208 H\n[PROOFSTEP]\napply mul_mem\n[GOAL]\ncase insert.a\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nih : \u2200 (x : (i : \u03b7) \u2192 f i), (\u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x i = 1) \u2192 (\u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x i) \u2208 H) \u2192 x \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\n\u22a2 Function.update x i 1 \u2208 H\n[PROOFSTEP]\napply ih\n[GOAL]\ncase insert.a.h1\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nih : \u2200 (x : (i : \u03b7) \u2192 f i), (\u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x i = 1) \u2192 (\u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x i) \u2208 H) \u2192 x \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\n\u22a2 \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 I \u2192 Function.update x i 1 i_1 = 1\n[PROOFSTEP]\nclear ih\n[GOAL]\ncase insert.a.h2\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nih : \u2200 (x : (i : \u03b7) \u2192 f i), (\u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x i = 1) \u2192 (\u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x i) \u2208 H) \u2192 x \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\n\u22a2 \u2200 (i_1 : \u03b7), i_1 \u2208 I \u2192 Pi.mulSingle i_1 (Function.update x i 1 i_1) \u2208 H\n[PROOFSTEP]\nclear ih\n[GOAL]\ncase insert.a.h1\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\n\u22a2 \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 I \u2192 Function.update x i 1 i_1 = 1\n[PROOFSTEP]\nintro j hj\n[GOAL]\ncase insert.a.h1\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\nj : \u03b7\nhj : \u00acj \u2208 I\n\u22a2 Function.update x i 1 j = 1\n[PROOFSTEP]\nby_cases heq : j = i\n[GOAL]\ncase pos\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\nj : \u03b7\nhj : \u00acj \u2208 I\nheq : j = i\n\u22a2 Function.update x i 1 j = 1\n[PROOFSTEP]\nsubst heq\n[GOAL]\ncase pos\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\nI : Finset \u03b7\nx : (i : \u03b7) \u2192 f i\nj : \u03b7\nhj hnmem : \u00acj \u2208 I\nh1 : \u2200 (i : \u03b7), \u00aci \u2208 insert j I \u2192 x i = 1\nh2 : \u2200 (i : \u03b7), i \u2208 insert j I \u2192 Pi.mulSingle i (x i) \u2208 H\n\u22a2 Function.update x j 1 j = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\nj : \u03b7\nhj : \u00acj \u2208 I\nheq : \u00acj = i\n\u22a2 Function.update x i 1 j = 1\n[PROOFSTEP]\nsimp [heq]\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\nj : \u03b7\nhj : \u00acj \u2208 I\nheq : \u00acj = i\n\u22a2 x j = 1\n[PROOFSTEP]\napply h1 j\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\nj : \u03b7\nhj : \u00acj \u2208 I\nheq : \u00acj = i\n\u22a2 \u00acj \u2208 insert i I\n[PROOFSTEP]\nsimpa [heq] using hj\n[GOAL]\ncase insert.a.h2\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\n\u22a2 \u2200 (i_1 : \u03b7), i_1 \u2208 I \u2192 Pi.mulSingle i_1 (Function.update x i 1 i_1) \u2208 H\n[PROOFSTEP]\nintro j hj\n[GOAL]\ncase insert.a.h2\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\nj : \u03b7\nhj : j \u2208 I\n\u22a2 Pi.mulSingle j (Function.update x i 1 j) \u2208 H\n[PROOFSTEP]\nhave : j \u2260 i := by\n rintro rfl\n contradiction\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\nj : \u03b7\nhj : j \u2208 I\n\u22a2 j \u2260 i\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\nI : Finset \u03b7\nx : (i : \u03b7) \u2192 f i\nj : \u03b7\nhj : j \u2208 I\nhnmem : \u00acj \u2208 I\nh1 : \u2200 (i : \u03b7), \u00aci \u2208 insert j I \u2192 x i = 1\nh2 : \u2200 (i : \u03b7), i \u2208 insert j I \u2192 Pi.mulSingle i (x i) \u2208 H\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase insert.a.h2\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\nj : \u03b7\nhj : j \u2208 I\nthis : j \u2260 i\n\u22a2 Pi.mulSingle j (Function.update x i 1 j) \u2208 H\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase insert.a.h2\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\nj : \u03b7\nhj : j \u2208 I\nthis : j \u2260 i\n\u22a2 Pi.mulSingle j (x j) \u2208 H\n[PROOFSTEP]\nexact h2 _ (Finset.mem_insert_of_mem hj)\n[GOAL]\ncase insert.a\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nih : \u2200 (x : (i : \u03b7) \u2192 f i), (\u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x i = 1) \u2192 (\u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x i) \u2208 H) \u2192 x \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\n\u22a2 Pi.mulSingle i (x i) \u2208 H\n[PROOFSTEP]\napply h2\n[GOAL]\ncase insert.a.a\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA : Type u_2\ninst\u271d\u00b2 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d : DecidableEq \u03b7\nI\u271d : Finset \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx\u271d : (i : \u03b7) \u2192 f i\nh1\u271d : \u2200 (i : \u03b7), \u00aci \u2208 I\u271d \u2192 x\u271d i = 1\nh2\u271d : \u2200 (i : \u03b7), i \u2208 I\u271d \u2192 Pi.mulSingle i (x\u271d i) \u2208 H\ni : \u03b7\nI : Finset \u03b7\nhnmem : \u00aci \u2208 I\nih : \u2200 (x : (i : \u03b7) \u2192 f i), (\u2200 (i : \u03b7), \u00aci \u2208 I \u2192 x i = 1) \u2192 (\u2200 (i : \u03b7), i \u2208 I \u2192 Pi.mulSingle i (x i) \u2208 H) \u2192 x \u2208 H\nx : (i : \u03b7) \u2192 f i\nh1 : \u2200 (i_1 : \u03b7), \u00aci_1 \u2208 insert i I \u2192 x i_1 = 1\nh2 : \u2200 (i_1 : \u03b7), i_1 \u2208 insert i I \u2192 Pi.mulSingle i_1 (x i_1) \u2208 H\n\u22a2 i \u2208 insert i I\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u_1\ninst\u271d\u2074 : Group G\nA : Type u_2\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : Finite \u03b7\ninst\u271d : DecidableEq \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx : (i : \u03b7) \u2192 f i\nh : \u2200 (i : \u03b7), Pi.mulSingle i (x i) \u2208 H\n\u22a2 x \u2208 H\n[PROOFSTEP]\ncases nonempty_fintype \u03b7\n[GOAL]\ncase intro\nG : Type u_1\ninst\u271d\u2074 : Group G\nA : Type u_2\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : Finite \u03b7\ninst\u271d : DecidableEq \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx : (i : \u03b7) \u2192 f i\nh : \u2200 (i : \u03b7), Pi.mulSingle i (x i) \u2208 H\nval\u271d : Fintype \u03b7\n\u22a2 x \u2208 H\n[PROOFSTEP]\nexact pi_mem_of_mulSingle_mem_aux Finset.univ x (by simp) fun i _ => h i\n[GOAL]\nG : Type u_1\ninst\u271d\u2074 : Group G\nA : Type u_2\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : Finite \u03b7\ninst\u271d : DecidableEq \u03b7\nH : Subgroup ((i : \u03b7) \u2192 f i)\nx : (i : \u03b7) \u2192 f i\nh : \u2200 (i : \u03b7), Pi.mulSingle i (x i) \u2208 H\nval\u271d : Fintype \u03b7\n\u22a2 \u2200 (i : \u03b7), \u00aci \u2208 Finset.univ \u2192 x i = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u_1\ninst\u271d\u2074 : Group G\nA : Type u_2\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : DecidableEq \u03b7\ninst\u271d : Finite \u03b7\nH : (i : \u03b7) \u2192 Subgroup (f i)\nJ : Subgroup ((i : \u03b7) \u2192 f i)\n\u22a2 pi univ H \u2264 J \u2194 \u2200 (i : \u03b7), map (MonoidHom.single f i) (H i) \u2264 J\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nG : Type u_1\ninst\u271d\u2074 : Group G\nA : Type u_2\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : DecidableEq \u03b7\ninst\u271d : Finite \u03b7\nH : (i : \u03b7) \u2192 Subgroup (f i)\nJ : Subgroup ((i : \u03b7) \u2192 f i)\n\u22a2 pi univ H \u2264 J \u2192 \u2200 (i : \u03b7), map (MonoidHom.single f i) (H i) \u2264 J\n[PROOFSTEP]\nrintro h i _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase mp.intro.intro\nG : Type u_1\ninst\u271d\u2074 : Group G\nA : Type u_2\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : DecidableEq \u03b7\ninst\u271d : Finite \u03b7\nH : (i : \u03b7) \u2192 Subgroup (f i)\nJ : Subgroup ((i : \u03b7) \u2192 f i)\nh : pi univ H \u2264 J\ni : \u03b7\nx : f i\nhx : x \u2208 \u2191(H i)\n\u22a2 \u2191(MonoidHom.single f i) x \u2208 J\n[PROOFSTEP]\napply h\n[GOAL]\ncase mp.intro.intro.a\nG : Type u_1\ninst\u271d\u2074 : Group G\nA : Type u_2\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : DecidableEq \u03b7\ninst\u271d : Finite \u03b7\nH : (i : \u03b7) \u2192 Subgroup (f i)\nJ : Subgroup ((i : \u03b7) \u2192 f i)\nh : pi univ H \u2264 J\ni : \u03b7\nx : f i\nhx : x \u2208 \u2191(H i)\n\u22a2 \u2191(MonoidHom.single f i) x \u2208 pi univ H\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\ncase mpr\nG : Type u_1\ninst\u271d\u2074 : Group G\nA : Type u_2\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_3\nf : \u03b7 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : DecidableEq \u03b7\ninst\u271d : Finite \u03b7\nH : (i : \u03b7) \u2192 Subgroup (f i)\nJ : Subgroup ((i : \u03b7) \u2192 f i)\n\u22a2 (\u2200 (i : \u03b7), map (MonoidHom.single f i) (H i) \u2264 J) \u2192 pi univ H \u2264 J\n[PROOFSTEP]\nexact fun h x hx => pi_mem_of_mulSingle_mem x fun i => h i (mem_map_of_mem _ (hx i trivial))\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\nA : Type u_2\ninst\u271d\u00b9 : AddGroup A\nS : Set G\ninst\u271d : Finite \u2191S\nx : G\nh : \u2200 (n : G), n \u2208 S \u2192 x * n * x\u207b\u00b9 \u2208 S\n\u22a2 x \u2208 setNormalizer S\n[PROOFSTEP]\nhaveI := Classical.propDecidable\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\nA : Type u_2\ninst\u271d\u00b9 : AddGroup A\nS : Set G\ninst\u271d : Finite \u2191S\nx : G\nh : \u2200 (n : G), n \u2208 S \u2192 x * n * x\u207b\u00b9 \u2208 S\nthis : (a : Prop) \u2192 Decidable a\n\u22a2 x \u2208 setNormalizer S\n[PROOFSTEP]\ncases nonempty_fintype S\n[GOAL]\ncase intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\nA : Type u_2\ninst\u271d\u00b9 : AddGroup A\nS : Set G\ninst\u271d : Finite \u2191S\nx : G\nh : \u2200 (n : G), n \u2208 S \u2192 x * n * x\u207b\u00b9 \u2208 S\nthis : (a : Prop) \u2192 Decidable a\nval\u271d : Fintype \u2191S\n\u22a2 x \u2208 setNormalizer S\n[PROOFSTEP]\nhaveI := Set.fintypeImage S fun n => x * n * x\u207b\u00b9\n[GOAL]\ncase intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\nA : Type u_2\ninst\u271d\u00b9 : AddGroup A\nS : Set G\ninst\u271d : Finite \u2191S\nx : G\nh : \u2200 (n : G), n \u2208 S \u2192 x * n * x\u207b\u00b9 \u2208 S\nthis\u271d : (a : Prop) \u2192 Decidable a\nval\u271d : Fintype \u2191S\nthis : Fintype \u2191((fun n => x * n * x\u207b\u00b9) '' S)\n\u22a2 x \u2208 setNormalizer S\n[PROOFSTEP]\nexact fun n =>\n \u27e8h n, fun h\u2081 =>\n have heq : (fun n => x * n * x\u207b\u00b9) '' S = S :=\n Set.eq_of_subset_of_card_le (fun n \u27e8y, hy\u27e9 => hy.2 \u25b8 h y hy.1)\n (by rw [Set.card_image_of_injective S conj_injective])\n have : x * n * x\u207b\u00b9 \u2208 (fun n => x * n * x\u207b\u00b9) '' S := heq.symm \u25b8 h\u2081\n let \u27e8y, hy\u27e9 := this\n conj_injective hy.2 \u25b8 hy.1\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\nA : Type u_2\ninst\u271d\u00b9 : AddGroup A\nS : Set G\ninst\u271d : Finite \u2191S\nx : G\nh : \u2200 (n : G), n \u2208 S \u2192 x * n * x\u207b\u00b9 \u2208 S\nthis\u271d : (a : Prop) \u2192 Decidable a\nval\u271d : Fintype \u2191S\nthis : Fintype \u2191((fun n => x * n * x\u207b\u00b9) '' S)\nn : G\nh\u2081 : x * n * x\u207b\u00b9 \u2208 S\n\u22a2 Fintype.card \u2191S \u2264 Fintype.card \u2191((fun n => x * n * x\u207b\u00b9) '' S)\n[PROOFSTEP]\nrw [Set.card_image_of_injective S conj_injective]\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Subgroup.Finite", "llama_tokens": 15117, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.4247988825165614}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF G F' G' : C \u2964 D\n\u03b1 : F \u27f6 G\n\u03b2 : F' \u27f6 G'\nX Y : C\nf : X \u27f6 Y\n\u22a2 (tensorObj F F').map f \u226b (fun X => NatTrans.app \u03b1 X \u2297 NatTrans.app \u03b2 X) Y =\n (fun X => NatTrans.app \u03b1 X \u2297 NatTrans.app \u03b2 X) X \u226b (tensorObj G G').map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF G F' G' : C \u2964 D\n\u03b1 : F \u27f6 G\n\u03b2 : F' \u27f6 G'\nX Y : C\nf : X \u27f6 Y\n\u22a2 (F.map f \u2297 F'.map f) \u226b (NatTrans.app \u03b1 Y \u2297 NatTrans.app \u03b2 Y) =\n (NatTrans.app \u03b1 X \u2297 NatTrans.app \u03b2 X) \u226b (G.map f \u2297 G'.map f)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u03b1.naturality, \u03b2.naturality, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF\u271d G F' G' : C \u2964 D\n\u03b1 : F\u271d \u27f6 G\n\u03b2\u271d : F' \u27f6 G'\nF : C \u2964 D\n\u03b2 : F' \u27f6 G'\nX Y : C\nf : X \u27f6 Y\n\u22a2 (tensorObj F F').map f \u226b (fun X => F.obj X \u25c1 NatTrans.app \u03b2 X) Y =\n (fun X => F.obj X \u25c1 NatTrans.app \u03b2 X) X \u226b (tensorObj F G').map f\n[PROOFSTEP]\nsimp only [\u2190 id_tensorHom]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF\u271d G F' G' : C \u2964 D\n\u03b1 : F\u271d \u27f6 G\n\u03b2\u271d : F' \u27f6 G'\nF : C \u2964 D\n\u03b2 : F' \u27f6 G'\nX Y : C\nf : X \u27f6 Y\n\u22a2 (tensorObj F F').map f \u226b (\ud835\udfd9 (F.obj Y) \u2297 NatTrans.app \u03b2 Y) = (\ud835\udfd9 (F.obj X) \u2297 NatTrans.app \u03b2 X) \u226b (tensorObj F G').map f\n[PROOFSTEP]\napply (tensorHom (\ud835\udfd9 F) \u03b2).naturality\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF G F'\u271d G' : C \u2964 D\n\u03b1 : F \u27f6 G\n\u03b2 : F'\u271d \u27f6 G'\nF' : C \u2964 D\nX Y : C\nf : X \u27f6 Y\n\u22a2 (tensorObj F F').map f \u226b (fun X => NatTrans.app \u03b1 X \u25b7 F'.obj X) Y =\n (fun X => NatTrans.app \u03b1 X \u25b7 F'.obj X) X \u226b (tensorObj G F').map f\n[PROOFSTEP]\nsimp only [\u2190 tensorHom_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF G F'\u271d G' : C \u2964 D\n\u03b1 : F \u27f6 G\n\u03b2 : F'\u271d \u27f6 G'\nF' : C \u2964 D\nX Y : C\nf : X \u27f6 Y\n\u22a2 (tensorObj F F').map f \u226b (NatTrans.app \u03b1 Y \u2297 \ud835\udfd9 (F'.obj Y)) =\n (NatTrans.app \u03b1 X \u2297 \ud835\udfd9 (F'.obj X)) \u226b (tensorObj G F').map f\n[PROOFSTEP]\napply (tensorHom \u03b1 (\ud835\udfd9 F')).naturality\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\n\u22a2 \u2200 {X\u2081 Y\u2081 X\u2082 Y\u2082 : C \u2964 D} (f : X\u2081 \u27f6 Y\u2081) (g : X\u2082 \u27f6 Y\u2082),\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} \u03b1 \u03b2 => FunctorCategory.tensorHom \u03b1 \u03b2) f g =\n (fun {X\u2081 X\u2082} \u03b1 F => FunctorCategory.whiskerRight \u03b1 F) f X\u2082 \u226b\n (fun F x x_1 \u03b1 => FunctorCategory.whiskerLeft F \u03b1) Y\u2081 X\u2082 Y\u2082 g\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : C \u2964 D\nf\u271d : X\u2081\u271d \u27f6 Y\u2081\u271d\ng\u271d : X\u2082\u271d \u27f6 Y\u2082\u271d\n\u22a2 (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} \u03b1 \u03b2 => FunctorCategory.tensorHom \u03b1 \u03b2) f\u271d g\u271d =\n (fun {X\u2081 X\u2082} \u03b1 F => FunctorCategory.whiskerRight \u03b1 F) f\u271d X\u2082\u271d \u226b\n (fun F x x_1 \u03b1 => FunctorCategory.whiskerLeft F \u03b1) Y\u2081\u271d X\u2082\u271d Y\u2082\u271d g\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : C \u2964 D\nf\u271d : X\u2081\u271d \u27f6 Y\u2081\u271d\ng\u271d : X\u2082\u271d \u27f6 Y\u2082\u271d\nx\u271d : C\n\u22a2 NatTrans.app ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} \u03b1 \u03b2 => FunctorCategory.tensorHom \u03b1 \u03b2) f\u271d g\u271d) x\u271d =\n NatTrans.app\n ((fun {X\u2081 X\u2082} \u03b1 F => FunctorCategory.whiskerRight \u03b1 F) f\u271d X\u2082\u271d \u226b\n (fun F x x_1 \u03b1 => FunctorCategory.whiskerLeft F \u03b1) Y\u2081\u271d X\u2082\u271d Y\u2082\u271d g\u271d)\n x\u271d\n[PROOFSTEP]\nsimp [tensorHom_def]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF G H K : C \u2964 D\n\u22a2 (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} \u03b1 \u03b2 => FunctorCategory.tensorHom \u03b1 \u03b2)\n ((fun F G H => NatIso.ofComponents fun X => \u03b1_ (F.obj X) (G.obj X) (H.obj X)) F G H).hom (\ud835\udfd9 K) \u226b\n ((fun F G H => NatIso.ofComponents fun X => \u03b1_ (F.obj X) (G.obj X) (H.obj X)) F\n ((fun F G => FunctorCategory.tensorObj F G) G H) K).hom \u226b\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} \u03b1 \u03b2 => FunctorCategory.tensorHom \u03b1 \u03b2) (\ud835\udfd9 F)\n ((fun F G H => NatIso.ofComponents fun X => \u03b1_ (F.obj X) (G.obj X) (H.obj X)) G H K).hom =\n ((fun F G H => NatIso.ofComponents fun X => \u03b1_ (F.obj X) (G.obj X) (H.obj X))\n ((fun F G => FunctorCategory.tensorObj F G) F G) H K).hom \u226b\n ((fun F G H => NatIso.ofComponents fun X => \u03b1_ (F.obj X) (G.obj X) (H.obj X)) F G\n ((fun F G => FunctorCategory.tensorObj F G) H K)).hom\n[PROOFSTEP]\next X\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF G H K : C \u2964 D\nX : C\n\u22a2 NatTrans.app\n ((fun {X\u2081 Y\u2081 X\u2082 Y\u2082} \u03b1 \u03b2 => FunctorCategory.tensorHom \u03b1 \u03b2)\n ((fun F G H => NatIso.ofComponents fun X => \u03b1_ (F.obj X) (G.obj X) (H.obj X)) F G H).hom (\ud835\udfd9 K) \u226b\n ((fun F G H => NatIso.ofComponents fun X => \u03b1_ (F.obj X) (G.obj X) (H.obj X)) F\n ((fun F G => FunctorCategory.tensorObj F G) G H) K).hom \u226b\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} \u03b1 \u03b2 => FunctorCategory.tensorHom \u03b1 \u03b2) (\ud835\udfd9 F)\n ((fun F G H => NatIso.ofComponents fun X => \u03b1_ (F.obj X) (G.obj X) (H.obj X)) G H K).hom)\n X =\n NatTrans.app\n (((fun F G H => NatIso.ofComponents fun X => \u03b1_ (F.obj X) (G.obj X) (H.obj X))\n ((fun F G => FunctorCategory.tensorObj F G) F G) H K).hom \u226b\n ((fun F G H => NatIso.ofComponents fun X => \u03b1_ (F.obj X) (G.obj X) (H.obj X)) F G\n ((fun F G => FunctorCategory.tensorObj F G) H K)).hom)\n X\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF G H K : C \u2964 D\nX : C\n\u22a2 ((\u03b1_ (F.obj X) (G.obj X) (H.obj X)).hom \u2297 \ud835\udfd9 (K.obj X)) \u226b\n (\u03b1_ (F.obj X) (G.obj X \u2297 H.obj X) (K.obj X)).hom \u226b (\ud835\udfd9 (F.obj X) \u2297 (\u03b1_ (G.obj X) (H.obj X) (K.obj X)).hom) =\n (\u03b1_ (F.obj X \u2297 G.obj X) (H.obj X) (K.obj X)).hom \u226b (\u03b1_ (F.obj X) (G.obj X) (H.obj X \u2297 K.obj X)).hom\n[PROOFSTEP]\nrw [pentagon]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : BraidedCategory D\nF G H : C \u2964 D\n\u22a2 (\u03b1_ F G H).hom \u226b ((fun F G => NatIso.ofComponents fun X => \u03b2_ (F.obj X) (G.obj X)) F (G \u2297 H)).hom \u226b (\u03b1_ G H F).hom =\n (((fun F G => NatIso.ofComponents fun X => \u03b2_ (F.obj X) (G.obj X)) F G).hom \u2297 \ud835\udfd9 H) \u226b\n (\u03b1_ G F H).hom \u226b (\ud835\udfd9 G \u2297 ((fun F G => NatIso.ofComponents fun X => \u03b2_ (F.obj X) (G.obj X)) F H).hom)\n[PROOFSTEP]\next X\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : BraidedCategory D\nF G H : C \u2964 D\nX : C\n\u22a2 NatTrans.app\n ((\u03b1_ F G H).hom \u226b\n ((fun F G => NatIso.ofComponents fun X => \u03b2_ (F.obj X) (G.obj X)) F (G \u2297 H)).hom \u226b (\u03b1_ G H F).hom)\n X =\n NatTrans.app\n ((((fun F G => NatIso.ofComponents fun X => \u03b2_ (F.obj X) (G.obj X)) F G).hom \u2297 \ud835\udfd9 H) \u226b\n (\u03b1_ G F H).hom \u226b (\ud835\udfd9 G \u2297 ((fun F G => NatIso.ofComponents fun X => \u03b2_ (F.obj X) (G.obj X)) F H).hom))\n X\n[PROOFSTEP]\napply hexagon_forward\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : BraidedCategory D\nF G H : C \u2964 D\n\u22a2 (\u03b1_ F G H).inv \u226b ((fun F G => NatIso.ofComponents fun X => \u03b2_ (F.obj X) (G.obj X)) (F \u2297 G) H).hom \u226b (\u03b1_ H F G).inv =\n (\ud835\udfd9 F \u2297 ((fun F G => NatIso.ofComponents fun X => \u03b2_ (F.obj X) (G.obj X)) G H).hom) \u226b\n (\u03b1_ F H G).inv \u226b (((fun F G => NatIso.ofComponents fun X => \u03b2_ (F.obj X) (G.obj X)) F H).hom \u2297 \ud835\udfd9 G)\n[PROOFSTEP]\next X\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : BraidedCategory D\nF G H : C \u2964 D\nX : C\n\u22a2 NatTrans.app\n ((\u03b1_ F G H).inv \u226b\n ((fun F G => NatIso.ofComponents fun X => \u03b2_ (F.obj X) (G.obj X)) (F \u2297 G) H).hom \u226b (\u03b1_ H F G).inv)\n X =\n NatTrans.app\n ((\ud835\udfd9 F \u2297 ((fun F G => NatIso.ofComponents fun X => \u03b2_ (F.obj X) (G.obj X)) G H).hom) \u226b\n (\u03b1_ F H G).inv \u226b (((fun F G => NatIso.ofComponents fun X => \u03b2_ (F.obj X) (G.obj X)) F H).hom \u2297 \ud835\udfd9 G))\n X\n[PROOFSTEP]\napply hexagon_reverse\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : SymmetricCategory D\nF G : C \u2964 D\n\u22a2 (\u03b2_ F G).hom \u226b (\u03b2_ G F).hom = \ud835\udfd9 (F \u2297 G)\n[PROOFSTEP]\next X\n[GOAL]\ncase w.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : SymmetricCategory D\nF G : C \u2964 D\nX : C\n\u22a2 NatTrans.app ((\u03b2_ F G).hom \u226b (\u03b2_ G F).hom) X = NatTrans.app (\ud835\udfd9 (F \u2297 G)) X\n[PROOFSTEP]\napply symmetry\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monoidal.FunctorCategory", "llama_tokens": 4426, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506418255928, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.4246917865897567}} {"text": "[GOAL]\np x t : \u211d\n\u22a2 \u2016\u2191(t * x)\u2016 = |t| * \u2016\u2191x\u2016\n[PROOFSTEP]\nhave aux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b) := fun {a b c} h =>\n by\n simp only [mem_zmultiples_iff] at h \u22a2\n obtain \u27e8n, rfl\u27e9 := h\n exact \u27e8n, (mul_smul_comm n c b).symm\u27e9\n[GOAL]\np x t a b c : \u211d\nh : a \u2208 zmultiples b\n\u22a2 c * a \u2208 zmultiples (c * b)\n[PROOFSTEP]\nsimp only [mem_zmultiples_iff] at h \u22a2\n[GOAL]\np x t a b c : \u211d\nh : \u2203 k, k \u2022 b = a\n\u22a2 \u2203 k, k \u2022 (c * b) = c * a\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := h\n[GOAL]\ncase intro\np x t b c : \u211d\nn : \u2124\n\u22a2 \u2203 k, k \u2022 (c * b) = c * n \u2022 b\n[PROOFSTEP]\nexact \u27e8n, (mul_smul_comm n c b).symm\u27e9\n[GOAL]\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\n\u22a2 \u2016\u2191(t * x)\u2016 = |t| * \u2016\u2191x\u2016\n[PROOFSTEP]\nrcases eq_or_ne t 0 with (rfl | ht)\n[GOAL]\ncase inl\np x : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\n\u22a2 \u2016\u2191(0 * x)\u2016 = |0| * \u2016\u2191x\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\n\u22a2 \u2016\u2191(t * x)\u2016 = |t| * \u2016\u2191x\u2016\n[PROOFSTEP]\nhave ht' : |t| \u2260 0 := (not_congr abs_eq_zero).mpr ht\n[GOAL]\ncase inr\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\n\u22a2 \u2016\u2191(t * x)\u2016 = |t| * \u2016\u2191x\u2016\n[PROOFSTEP]\nsimp only [quotient_norm_eq, Real.norm_eq_abs]\n[GOAL]\ncase inr\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\n\u22a2 sInf ((fun a => |a|) '' {m | \u2191m = \u2191(t * x)}) = |t| * sInf ((fun a => |a|) '' {m | \u2191m = \u2191x})\n[PROOFSTEP]\nconv_rhs => rw [\u2190 smul_eq_mul, \u2190 Real.sInf_smul_of_nonneg (abs_nonneg t)]\n[GOAL]\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\n| |t| * sInf ((fun a => |a|) '' {m | \u2191m = \u2191x})\n[PROOFSTEP]\nrw [\u2190 smul_eq_mul, \u2190 Real.sInf_smul_of_nonneg (abs_nonneg t)]\n[GOAL]\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\n| |t| * sInf ((fun a => |a|) '' {m | \u2191m = \u2191x})\n[PROOFSTEP]\nrw [\u2190 smul_eq_mul, \u2190 Real.sInf_smul_of_nonneg (abs_nonneg t)]\n[GOAL]\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\n| |t| * sInf ((fun a => |a|) '' {m | \u2191m = \u2191x})\n[PROOFSTEP]\nrw [\u2190 smul_eq_mul, \u2190 Real.sInf_smul_of_nonneg (abs_nonneg t)]\n[GOAL]\ncase inr\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\n\u22a2 sInf ((fun a => |a|) '' {m | \u2191m = \u2191(t * x)}) = sInf (|t| \u2022 (fun a => |a|) '' {m | \u2191m = \u2191x})\n[PROOFSTEP]\nsimp only [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_iff_sub_mem]\n[GOAL]\ncase inr\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\n\u22a2 sInf ((fun a => |a|) '' {m | m - t * x \u2208 zmultiples (t * p)}) =\n sInf (|t| \u2022 (fun a => |a|) '' {m | m - x \u2208 zmultiples p})\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase inr.e_a\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\n\u22a2 (fun a => |a|) '' {m | m - t * x \u2208 zmultiples (t * p)} = |t| \u2022 (fun a => |a|) '' {m | m - x \u2208 zmultiples p}\n[PROOFSTEP]\next z\n[GOAL]\ncase inr.e_a.h\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\nz : \u211d\n\u22a2 z \u2208 (fun a => |a|) '' {m | m - t * x \u2208 zmultiples (t * p)} \u2194 z \u2208 |t| \u2022 (fun a => |a|) '' {m | m - x \u2208 zmultiples p}\n[PROOFSTEP]\nrw [mem_smul_set_iff_inv_smul_mem\u2080 ht']\n[GOAL]\ncase inr.e_a.h\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\nz : \u211d\n\u22a2 z \u2208 (fun a => |a|) '' {m | m - t * x \u2208 zmultiples (t * p)} \u2194 |t|\u207b\u00b9 \u2022 z \u2208 (fun a => |a|) '' {m | m - x \u2208 zmultiples p}\n[PROOFSTEP]\nshow (\u2203 y, y - t * x \u2208 zmultiples (t * p) \u2227 |y| = z) \u2194 \u2203 w, w - x \u2208 zmultiples p \u2227 |w| = |t|\u207b\u00b9 * z\n[GOAL]\ncase inr.e_a.h\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\nz : \u211d\n\u22a2 (\u2203 y, y - t * x \u2208 zmultiples (t * p) \u2227 |y| = z) \u2194 \u2203 w, w - x \u2208 zmultiples p \u2227 |w| = |t|\u207b\u00b9 * z\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase inr.e_a.h.mp\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\nz : \u211d\n\u22a2 (\u2203 y, y - t * x \u2208 zmultiples (t * p) \u2227 |y| = z) \u2192 \u2203 w, w - x \u2208 zmultiples p \u2227 |w| = |t|\u207b\u00b9 * z\n[PROOFSTEP]\nrintro \u27e8y, hy, rfl\u27e9\n[GOAL]\ncase inr.e_a.h.mp.intro.intro\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\ny : \u211d\nhy : y - t * x \u2208 zmultiples (t * p)\n\u22a2 \u2203 w, w - x \u2208 zmultiples p \u2227 |w| = |t|\u207b\u00b9 * |y|\n[PROOFSTEP]\nrefine' \u27e8t\u207b\u00b9 * y, _, by rw [abs_mul, abs_inv]\u27e9\n[GOAL]\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\ny : \u211d\nhy : y - t * x \u2208 zmultiples (t * p)\n\u22a2 |t\u207b\u00b9 * y| = |t|\u207b\u00b9 * |y|\n[PROOFSTEP]\nrw [abs_mul, abs_inv]\n[GOAL]\ncase inr.e_a.h.mp.intro.intro\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\ny : \u211d\nhy : y - t * x \u2208 zmultiples (t * p)\n\u22a2 t\u207b\u00b9 * y - x \u2208 zmultiples p\n[PROOFSTEP]\nrw [\u2190 inv_mul_cancel_left\u2080 ht x, \u2190 inv_mul_cancel_left\u2080 ht p, \u2190 mul_sub]\n[GOAL]\ncase inr.e_a.h.mp.intro.intro\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\ny : \u211d\nhy : y - t * x \u2208 zmultiples (t * p)\n\u22a2 t\u207b\u00b9 * (y - t * x) \u2208 zmultiples (t\u207b\u00b9 * (t * p))\n[PROOFSTEP]\nexact aux hy\n[GOAL]\ncase inr.e_a.h.mpr\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\nz : \u211d\n\u22a2 (\u2203 w, w - x \u2208 zmultiples p \u2227 |w| = |t|\u207b\u00b9 * z) \u2192 \u2203 y, y - t * x \u2208 zmultiples (t * p) \u2227 |y| = z\n[PROOFSTEP]\nrintro \u27e8w, hw, hw'\u27e9\n[GOAL]\ncase inr.e_a.h.mpr.intro.intro\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\nz w : \u211d\nhw : w - x \u2208 zmultiples p\nhw' : |w| = |t|\u207b\u00b9 * z\n\u22a2 \u2203 y, y - t * x \u2208 zmultiples (t * p) \u2227 |y| = z\n[PROOFSTEP]\nrefine' \u27e8t * w, _, by rw [\u2190 (eq_inv_mul_iff_mul_eq\u2080 ht').mp hw', abs_mul]\u27e9\n[GOAL]\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\nz w : \u211d\nhw : w - x \u2208 zmultiples p\nhw' : |w| = |t|\u207b\u00b9 * z\n\u22a2 |t * w| = z\n[PROOFSTEP]\nrw [\u2190 (eq_inv_mul_iff_mul_eq\u2080 ht').mp hw', abs_mul]\n[GOAL]\ncase inr.e_a.h.mpr.intro.intro\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\nz w : \u211d\nhw : w - x \u2208 zmultiples p\nhw' : |w| = |t|\u207b\u00b9 * z\n\u22a2 t * w - t * x \u2208 zmultiples (t * p)\n[PROOFSTEP]\nrw [\u2190 mul_sub]\n[GOAL]\ncase inr.e_a.h.mpr.intro.intro\np x t : \u211d\naux : \u2200 {a b c : \u211d}, a \u2208 zmultiples b \u2192 c * a \u2208 zmultiples (c * b)\nht : t \u2260 0\nht' : |t| \u2260 0\nz w : \u211d\nhw : w - x \u2208 zmultiples p\nhw' : |w| = |t|\u207b\u00b9 * z\n\u22a2 t * (w - x) \u2208 zmultiples (t * p)\n[PROOFSTEP]\nexact aux hw\n[GOAL]\np x : \u211d\n\u22a2 \u2016\u2191x\u2016 = \u2016\u2191x\u2016\n[PROOFSTEP]\nsuffices \u2016(\u2191(-1 * x) : AddCircle (-1 * p))\u2016 = \u2016(x : AddCircle p)\u2016\n by\n rw [\u2190 this, neg_one_mul]\n simp\n[GOAL]\np x : \u211d\nthis : \u2016\u2191(-1 * x)\u2016 = \u2016\u2191x\u2016\n\u22a2 \u2016\u2191x\u2016 = \u2016\u2191x\u2016\n[PROOFSTEP]\nrw [\u2190 this, neg_one_mul]\n[GOAL]\np x : \u211d\nthis : \u2016\u2191(-1 * x)\u2016 = \u2016\u2191x\u2016\n\u22a2 \u2016\u2191x\u2016 = \u2016\u2191(-1 * x)\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\np x : \u211d\n\u22a2 \u2016\u2191(-1 * x)\u2016 = \u2016\u2191x\u2016\n[PROOFSTEP]\nsimp only [norm_coe_mul, abs_neg, abs_one, one_mul]\n[GOAL]\np x : \u211d\n\u22a2 \u2016\u2191x\u2016 = |x|\n[PROOFSTEP]\nsuffices {y : \u211d | (y : AddCircle (0 : \u211d)) = (x : AddCircle (0 : \u211d))} = { x } by\n rw [quotient_norm_eq, this, image_singleton, Real.norm_eq_abs, csInf_singleton]\n[GOAL]\np x : \u211d\nthis : {y | \u2191y = \u2191x} = {x}\n\u22a2 \u2016\u2191x\u2016 = |x|\n[PROOFSTEP]\nrw [quotient_norm_eq, this, image_singleton, Real.norm_eq_abs, csInf_singleton]\n[GOAL]\np x : \u211d\n\u22a2 {y | \u2191y = \u2191x} = {x}\n[PROOFSTEP]\next y\n[GOAL]\ncase h\np x y : \u211d\n\u22a2 y \u2208 {y | \u2191y = \u2191x} \u2194 y \u2208 {x}\n[PROOFSTEP]\nsimp [QuotientAddGroup.eq_iff_sub_mem, mem_zmultiples_iff, sub_eq_zero]\n[GOAL]\np x : \u211d\n\u22a2 \u2016\u2191x\u2016 = |x - \u2191(round (p\u207b\u00b9 * x)) * p|\n[PROOFSTEP]\nsuffices \u2200 x : \u211d, \u2016(x : AddCircle (1 : \u211d))\u2016 = |x - round x|\n by\n rcases eq_or_ne p 0 with (rfl | hp)\n \u00b7 simp\n intros\n have hx := norm_coe_mul p x p\u207b\u00b9\n rw [abs_inv, eq_inv_mul_iff_mul_eq\u2080 ((not_congr abs_eq_zero).mpr hp)] at hx \n rw [\u2190 hx, inv_mul_cancel hp, this, \u2190 abs_mul, mul_sub, mul_inv_cancel_left\u2080 hp, mul_comm p]\n[GOAL]\np x : \u211d\nthis : \u2200 (x : \u211d), \u2016\u2191x\u2016 = |x - \u2191(round x)|\n\u22a2 \u2016\u2191x\u2016 = |x - \u2191(round (p\u207b\u00b9 * x)) * p|\n[PROOFSTEP]\nrcases eq_or_ne p 0 with (rfl | hp)\n[GOAL]\ncase inl\nx : \u211d\nthis : \u2200 (x : \u211d), \u2016\u2191x\u2016 = |x - \u2191(round x)|\n\u22a2 \u2016\u2191x\u2016 = |x - \u2191(round (0\u207b\u00b9 * x)) * 0|\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\np x : \u211d\nthis : \u2200 (x : \u211d), \u2016\u2191x\u2016 = |x - \u2191(round x)|\nhp : p \u2260 0\n\u22a2 \u2016\u2191x\u2016 = |x - \u2191(round (p\u207b\u00b9 * x)) * p|\n[PROOFSTEP]\nintros\n[GOAL]\ncase inr\np x : \u211d\nthis : \u2200 (x : \u211d), \u2016\u2191x\u2016 = |x - \u2191(round x)|\nhp : p \u2260 0\n\u22a2 \u2016\u2191x\u2016 = |x - \u2191(round (p\u207b\u00b9 * x)) * p|\n[PROOFSTEP]\nhave hx := norm_coe_mul p x p\u207b\u00b9\n[GOAL]\ncase inr\np x : \u211d\nthis : \u2200 (x : \u211d), \u2016\u2191x\u2016 = |x - \u2191(round x)|\nhp : p \u2260 0\nhx : \u2016\u2191(p\u207b\u00b9 * x)\u2016 = |p\u207b\u00b9| * \u2016\u2191x\u2016\n\u22a2 \u2016\u2191x\u2016 = |x - \u2191(round (p\u207b\u00b9 * x)) * p|\n[PROOFSTEP]\nrw [abs_inv, eq_inv_mul_iff_mul_eq\u2080 ((not_congr abs_eq_zero).mpr hp)] at hx \n[GOAL]\ncase inr\np x : \u211d\nthis : \u2200 (x : \u211d), \u2016\u2191x\u2016 = |x - \u2191(round x)|\nhp : p \u2260 0\nhx : |p| * \u2016\u2191(p\u207b\u00b9 * x)\u2016 = \u2016\u2191x\u2016\n\u22a2 \u2016\u2191x\u2016 = |x - \u2191(round (p\u207b\u00b9 * x)) * p|\n[PROOFSTEP]\nrw [\u2190 hx, inv_mul_cancel hp, this, \u2190 abs_mul, mul_sub, mul_inv_cancel_left\u2080 hp, mul_comm p]\n[GOAL]\np x : \u211d\n\u22a2 \u2200 (x : \u211d), \u2016\u2191x\u2016 = |x - \u2191(round x)|\n[PROOFSTEP]\nclear! x p\n[GOAL]\n\u22a2 \u2200 (x : \u211d), \u2016\u2191x\u2016 = |x - \u2191(round x)|\n[PROOFSTEP]\nintros x\n[GOAL]\nx : \u211d\n\u22a2 \u2016\u2191x\u2016 = |x - \u2191(round x)|\n[PROOFSTEP]\nrw [quotient_norm_eq, abs_sub_round_eq_min]\n[GOAL]\nx : \u211d\n\u22a2 sInf (norm '' {m | \u2191m = \u2191x}) = min (fract x) (1 - fract x)\n[PROOFSTEP]\nhave h\u2081 : BddBelow (abs '' {m : \u211d | (m : AddCircle (1 : \u211d)) = x}) := \u27e80, by simp [mem_lowerBounds]\u27e9\n[GOAL]\nx : \u211d\n\u22a2 0 \u2208 lowerBounds (abs '' {m | \u2191m = \u2191x})\n[PROOFSTEP]\nsimp [mem_lowerBounds]\n[GOAL]\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\n\u22a2 sInf (norm '' {m | \u2191m = \u2191x}) = min (fract x) (1 - fract x)\n[PROOFSTEP]\nhave h\u2082 : (abs '' {m : \u211d | (m : AddCircle (1 : \u211d)) = x}).Nonempty := \u27e8|x|, \u27e8x, rfl, rfl\u27e9\u27e9\n[GOAL]\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\n\u22a2 sInf (norm '' {m | \u2191m = \u2191x}) = min (fract x) (1 - fract x)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\n\u22a2 sInf (norm '' {m | \u2191m = \u2191x}) \u2264 min (fract x) (1 - fract x)\n[PROOFSTEP]\nsimp_rw [Real.norm_eq_abs, csInf_le_iff h\u2081 h\u2082, le_min_iff]\n[GOAL]\ncase a\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\n\u22a2 \u2200 (b : \u211d), b \u2208 lowerBounds (abs '' {m | \u2191m = \u2191x}) \u2192 b \u2264 fract x \u2227 b \u2264 1 - fract x\n[PROOFSTEP]\nintro b h\n[GOAL]\ncase a\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\nb : \u211d\nh : b \u2208 lowerBounds (abs '' {m | \u2191m = \u2191x})\n\u22a2 b \u2264 fract x \u2227 b \u2264 1 - fract x\n[PROOFSTEP]\nrefine'\n \u27e8mem_lowerBounds.1 h _ \u27e8fract x, _, abs_fract\u27e9,\n mem_lowerBounds.1 h _ \u27e8fract x - 1, _, by rw [abs_sub_comm, abs_one_sub_fract]\u27e9\u27e9\n[GOAL]\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\nb : \u211d\nh : b \u2208 lowerBounds (abs '' {m | \u2191m = \u2191x})\n\u22a2 |fract x - 1| = 1 - fract x\n[PROOFSTEP]\nrw [abs_sub_comm, abs_one_sub_fract]\n[GOAL]\ncase a.refine'_1\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\nb : \u211d\nh : b \u2208 lowerBounds (abs '' {m | \u2191m = \u2191x})\n\u22a2 fract x \u2208 {m | \u2191m = \u2191x}\n[PROOFSTEP]\nsimp only [mem_setOf, fract, sub_eq_self, QuotientAddGroup.mk_sub, QuotientAddGroup.eq_zero_iff,\n int_cast_mem_zmultiples_one]\n[GOAL]\ncase a.refine'_2\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\nb : \u211d\nh : b \u2208 lowerBounds (abs '' {m | \u2191m = \u2191x})\n\u22a2 fract x - 1 \u2208 {m | \u2191m = \u2191x}\n[PROOFSTEP]\nsimp only [mem_setOf, fract, sub_eq_self, QuotientAddGroup.mk_sub, QuotientAddGroup.eq_zero_iff,\n int_cast_mem_zmultiples_one, sub_sub, (by norm_cast : (\u230ax\u230b : \u211d) + 1 = (\u2191(\u230ax\u230b + 1) : \u211d))]\n[GOAL]\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\nb : \u211d\nh : b \u2208 lowerBounds (abs '' {m | \u2191m = \u2191x})\n\u22a2 \u2191\u230ax\u230b + 1 = \u2191(\u230ax\u230b + 1)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase a\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\n\u22a2 min (fract x) (1 - fract x) \u2264 sInf (norm '' {m | \u2191m = \u2191x})\n[PROOFSTEP]\nsimp only [QuotientAddGroup.mk'_apply, Real.norm_eq_abs, le_csInf_iff h\u2081 h\u2082]\n[GOAL]\ncase a\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\n\u22a2 \u2200 (b : \u211d), b \u2208 abs '' {m | \u2191m = \u2191x} \u2192 min (fract x) (1 - fract x) \u2264 b\n[PROOFSTEP]\nrintro b' \u27e8b, hb, rfl\u27e9\n[GOAL]\ncase a.intro.intro\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\nb : \u211d\nhb : b \u2208 {m | \u2191m = \u2191x}\n\u22a2 min (fract x) (1 - fract x) \u2264 |b|\n[PROOFSTEP]\nsimp only [mem_setOf, QuotientAddGroup.eq_iff_sub_mem, mem_zmultiples_iff, smul_one_eq_coe] at hb \n[GOAL]\ncase a.intro.intro\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\nb : \u211d\nhb : \u2203 k, \u2191k = b - x\n\u22a2 min (fract x) (1 - fract x) \u2264 |b|\n[PROOFSTEP]\nobtain \u27e8z, hz\u27e9 := hb\n[GOAL]\ncase a.intro.intro.intro\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\nb : \u211d\nz : \u2124\nhz : \u2191z = b - x\n\u22a2 min (fract x) (1 - fract x) \u2264 |b|\n[PROOFSTEP]\nrw [(by rw [hz]; abel : x = b - z), fract_sub_int, \u2190 abs_sub_round_eq_min]\n[GOAL]\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\nb : \u211d\nz : \u2124\nhz : \u2191z = b - x\n\u22a2 x = b - \u2191z\n[PROOFSTEP]\nrw [hz]\n[GOAL]\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\nb : \u211d\nz : \u2124\nhz : \u2191z = b - x\n\u22a2 x = b - (b - x)\n[PROOFSTEP]\nabel\n[GOAL]\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\nb : \u211d\nz : \u2124\nhz : \u2191z = b - x\n\u22a2 x = b - (b - x)\n[PROOFSTEP]\nabel\n[GOAL]\ncase a.intro.intro.intro\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\nb : \u211d\nz : \u2124\nhz : \u2191z = b - x\n\u22a2 |b - \u2191(round b)| \u2264 |b|\n[PROOFSTEP]\nconvert round_le b 0\n[GOAL]\ncase h.e'_4.h.e'_3\nx : \u211d\nh\u2081 : BddBelow (abs '' {m | \u2191m = \u2191x})\nh\u2082 : Set.Nonempty (abs '' {m | \u2191m = \u2191x})\nb : \u211d\nz : \u2124\nhz : \u2191z = b - x\n\u22a2 b = b - \u21910\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u211d\nhp : 0 < p\nx : \u211d\n\u22a2 \u2016\u2191x\u2016 = p * |p\u207b\u00b9 * x - \u2191(round (p\u207b\u00b9 * x))|\n[PROOFSTEP]\nconv_rhs =>\n congr\n rw [\u2190 abs_eq_self.mpr hp.le]\n[GOAL]\np : \u211d\nhp : 0 < p\nx : \u211d\n| p * |p\u207b\u00b9 * x - \u2191(round (p\u207b\u00b9 * x))|\n[PROOFSTEP]\n congr\n rw [\u2190 abs_eq_self.mpr hp.le]\n[GOAL]\np : \u211d\nhp : 0 < p\nx : \u211d\n| p * |p\u207b\u00b9 * x - \u2191(round (p\u207b\u00b9 * x))|\n[PROOFSTEP]\n congr\n rw [\u2190 abs_eq_self.mpr hp.le]\n[GOAL]\np : \u211d\nhp : 0 < p\nx : \u211d\n| p * |p\u207b\u00b9 * x - \u2191(round (p\u207b\u00b9 * x))|\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\np : \u211d\nhp : 0 < p\nx : \u211d\n| p\ncase a p : \u211d hp : 0 < p x : \u211d | |p\u207b\u00b9 * x - \u2191(round (p\u207b\u00b9 * x))|\n[PROOFSTEP]\nrw [\u2190 abs_eq_self.mpr hp.le]\n[GOAL]\np : \u211d\nhp : 0 < p\nx : \u211d\n\u22a2 \u2016\u2191x\u2016 = |p| * |p\u207b\u00b9 * x - \u2191(round (p\u207b\u00b9 * x))|\n[PROOFSTEP]\nrw [\u2190 abs_mul, mul_sub, mul_inv_cancel_left\u2080 hp.ne.symm, norm_eq, mul_comm p]\n[GOAL]\np : \u211d\nx : AddCircle p\nhp : p \u2260 0\n\u22a2 \u2016x\u2016 \u2264 |p| / 2\n[PROOFSTEP]\nobtain \u27e8x\u27e9 := x\n[GOAL]\ncase mk\np : \u211d\nx\u271d : AddCircle p\nhp : p \u2260 0\nx : \u211d\n\u22a2 \u2016Quot.mk Setoid.r x\u2016 \u2264 |p| / 2\n[PROOFSTEP]\nchange \u2016(x : AddCircle p)\u2016 \u2264 |p| / 2\n[GOAL]\ncase mk\np : \u211d\nx\u271d : AddCircle p\nhp : p \u2260 0\nx : \u211d\n\u22a2 \u2016\u2191x\u2016 \u2264 |p| / 2\n[PROOFSTEP]\nrw [norm_eq, \u2190 mul_le_mul_left (abs_pos.mpr (inv_ne_zero hp)), \u2190 abs_mul, mul_sub, mul_left_comm, \u2190 mul_div_assoc, \u2190\n abs_mul, inv_mul_cancel hp, mul_one, abs_one]\n[GOAL]\ncase mk\np : \u211d\nx\u271d : AddCircle p\nhp : p \u2260 0\nx : \u211d\n\u22a2 |p\u207b\u00b9 * x - \u2191(round (p\u207b\u00b9 * x))| \u2264 1 / 2\n[PROOFSTEP]\nexact abs_sub_round (p\u207b\u00b9 * x)\n[GOAL]\np : \u211d\n\u22a2 \u2016\u2191(p / 2)\u2016 = |p| / 2\n[PROOFSTEP]\nrcases eq_or_ne p 0 with (rfl | hp)\n[GOAL]\ncase inl\n\u22a2 \u2016\u2191(0 / 2)\u2016 = |0| / 2\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\np : \u211d\nhp : p \u2260 0\n\u22a2 \u2016\u2191(p / 2)\u2016 = |p| / 2\n[PROOFSTEP]\nrw [norm_eq, \u2190 mul_div_assoc, inv_mul_cancel hp, one_div, round_two_inv, Int.cast_one, one_mul,\n (by linarith : p / 2 - p = -(p / 2)), abs_neg, abs_div, abs_two]\n[GOAL]\np : \u211d\nhp : p \u2260 0\n\u22a2 p / 2 - p = -(p / 2)\n[PROOFSTEP]\nlinarith\n[GOAL]\np x : \u211d\nhp : p \u2260 0\n\u22a2 \u2016\u2191x\u2016 = |x| \u2194 |x| \u2264 |p| / 2\n[PROOFSTEP]\nrefine' \u27e8fun hx => hx \u25b8 norm_le_half_period p hp, fun hx => _\u27e9\n[GOAL]\np x : \u211d\nhp : p \u2260 0\nhx : |x| \u2264 |p| / 2\n\u22a2 \u2016\u2191x\u2016 = |x|\n[PROOFSTEP]\nsuffices \u2200 p : \u211d, 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016(x : AddCircle p)\u2016 = |x| by\n -- Porting note: replaced `lt_trichotomy` which had trouble substituting `p = 0`.\n rcases hp.symm.lt_or_lt with (hp | hp)\n \u00b7 rw [abs_eq_self.mpr hp.le] at hx \n exact this p hp hx\n \u00b7 rw [\u2190 norm_neg_period]\n rw [abs_eq_neg_self.mpr hp.le] at hx \n exact this (-p) (neg_pos.mpr hp) hx\n[GOAL]\np x : \u211d\nhp : p \u2260 0\nhx : |x| \u2264 |p| / 2\nthis : \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\n\u22a2 \u2016\u2191x\u2016 = |x|\n[PROOFSTEP]\nrcases hp.symm.lt_or_lt with (hp | hp)\n[GOAL]\ncase inl\np x : \u211d\nhp\u271d : p \u2260 0\nhx : |x| \u2264 |p| / 2\nthis : \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\nhp : 0 < p\n\u22a2 \u2016\u2191x\u2016 = |x|\n[PROOFSTEP]\nrw [abs_eq_self.mpr hp.le] at hx \n[GOAL]\ncase inl\np x : \u211d\nhp\u271d : p \u2260 0\nhx : |x| \u2264 p / 2\nthis : \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\nhp : 0 < p\n\u22a2 \u2016\u2191x\u2016 = |x|\n[PROOFSTEP]\nexact this p hp hx\n[GOAL]\ncase inr\np x : \u211d\nhp\u271d : p \u2260 0\nhx : |x| \u2264 |p| / 2\nthis : \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\nhp : p < 0\n\u22a2 \u2016\u2191x\u2016 = |x|\n[PROOFSTEP]\nrw [\u2190 norm_neg_period]\n[GOAL]\ncase inr\np x : \u211d\nhp\u271d : p \u2260 0\nhx : |x| \u2264 |p| / 2\nthis : \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\nhp : p < 0\n\u22a2 \u2016\u2191x\u2016 = |x|\n[PROOFSTEP]\nrw [abs_eq_neg_self.mpr hp.le] at hx \n[GOAL]\ncase inr\np x : \u211d\nhp\u271d : p \u2260 0\nhx : |x| \u2264 -p / 2\nthis : \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\nhp : p < 0\n\u22a2 \u2016\u2191x\u2016 = |x|\n[PROOFSTEP]\nexact this (-p) (neg_pos.mpr hp) hx\n[GOAL]\np x : \u211d\nhp : p \u2260 0\nhx : |x| \u2264 |p| / 2\n\u22a2 \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\n[PROOFSTEP]\nclear hx\n[GOAL]\np x : \u211d\nhp : p \u2260 0\n\u22a2 \u2200 (p : \u211d), 0 < p \u2192 |x| \u2264 p / 2 \u2192 \u2016\u2191x\u2016 = |x|\n[PROOFSTEP]\nintro p hp hx\n[GOAL]\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\n\u22a2 \u2016\u2191x\u2016 = |x|\n[PROOFSTEP]\nrcases eq_or_ne x (p / (2 : \u211d)) with (rfl | hx')\n[GOAL]\ncase inl\np\u271d : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |p / 2| \u2264 p / 2\n\u22a2 \u2016\u2191(p / 2)\u2016 = |p / 2|\n[PROOFSTEP]\nsimp [abs_div, abs_two]\n[GOAL]\ncase inr\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\n\u22a2 \u2016\u2191x\u2016 = |x|\n[PROOFSTEP]\nsuffices round (p\u207b\u00b9 * x) = 0 by simp [norm_eq, this]\n[GOAL]\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\nthis : round (p\u207b\u00b9 * x) = 0\n\u22a2 \u2016\u2191x\u2016 = |x|\n[PROOFSTEP]\nsimp [norm_eq, this]\n[GOAL]\ncase inr\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\n\u22a2 round (p\u207b\u00b9 * x) = 0\n[PROOFSTEP]\nrw [round_eq_zero_iff]\n[GOAL]\ncase inr\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\n\u22a2 p\u207b\u00b9 * x \u2208 Ico (-(1 / 2)) (1 / 2)\n[PROOFSTEP]\nobtain \u27e8hx\u2081, hx\u2082\u27e9 := abs_le.mp hx\n[GOAL]\ncase inr.intro\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\nhx\u2081 : -(p / 2) \u2264 x\nhx\u2082 : x \u2264 p / 2\n\u22a2 p\u207b\u00b9 * x \u2208 Ico (-(1 / 2)) (1 / 2)\n[PROOFSTEP]\nreplace hx\u2082 := Ne.lt_of_le hx' hx\u2082\n[GOAL]\ncase inr.intro\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\nhx\u2081 : -(p / 2) \u2264 x\nhx\u2082 : x < p / 2\n\u22a2 p\u207b\u00b9 * x \u2208 Ico (-(1 / 2)) (1 / 2)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase inr.intro.left\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\nhx\u2081 : -(p / 2) \u2264 x\nhx\u2082 : x < p / 2\n\u22a2 -(1 / 2) \u2264 p\u207b\u00b9 * x\n[PROOFSTEP]\nrwa [\u2190 mul_le_mul_left hp, \u2190 mul_assoc, mul_inv_cancel hp.ne.symm, one_mul, mul_neg, \u2190 mul_div_assoc, mul_one]\n[GOAL]\ncase inr.intro.right\np\u271d x : \u211d\nhp\u271d : p\u271d \u2260 0\np : \u211d\nhp : 0 < p\nhx : |x| \u2264 p / 2\nhx' : x \u2260 p / 2\nhx\u2081 : -(p / 2) \u2264 x\nhx\u2082 : x < p / 2\n\u22a2 p\u207b\u00b9 * x < 1 / 2\n[PROOFSTEP]\nrwa [\u2190 mul_lt_mul_left hp, \u2190 mul_assoc, mul_inv_cancel hp.ne.symm, one_mul, \u2190 mul_div_assoc, mul_one]\n[GOAL]\np : \u211d\nhp : p \u2260 0\nx\u271d : AddCircle p\n\u03b5 : \u211d\nh\u03b5 : |p| / 2 \u2264 \u03b5\nx : AddCircle p\n\u22a2 x \u2208 closedBall x\u271d \u03b5\n[PROOFSTEP]\nsimpa only [mem_closedBall, dist_eq_norm] using (norm_le_half_period p hp).trans h\u03b5\n[GOAL]\np x \u03b5 : \u211d\n\u22a2 QuotientAddGroup.mk \u207b\u00b9' closedBall (\u2191x) \u03b5 = closedBall x \u03b5\n[PROOFSTEP]\next y\n[GOAL]\ncase h\np x \u03b5 y : \u211d\n\u22a2 y \u2208 QuotientAddGroup.mk \u207b\u00b9' closedBall (\u2191x) \u03b5 \u2194 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nsimp only [Set.mem_preimage, dist_eq_norm, AddCircle.norm_eq_of_zero, iff_self, \u2190 QuotientAddGroup.mk_sub,\n Metric.mem_closedBall, Real.norm_eq_abs]\n[GOAL]\np x \u03b5 : \u211d\n\u22a2 QuotientAddGroup.mk \u207b\u00b9' closedBall (\u2191x) \u03b5 = \u22c3 (z : \u2124), closedBall (x + z \u2022 p) \u03b5\n[PROOFSTEP]\nrcases eq_or_ne p 0 with (rfl | hp)\n[GOAL]\ncase inl\nx \u03b5 : \u211d\n\u22a2 QuotientAddGroup.mk \u207b\u00b9' closedBall (\u2191x) \u03b5 = \u22c3 (z : \u2124), closedBall (x + z \u2022 0) \u03b5\n[PROOFSTEP]\nsimp [iUnion_const]\n[GOAL]\ncase inr\np x \u03b5 : \u211d\nhp : p \u2260 0\n\u22a2 QuotientAddGroup.mk \u207b\u00b9' closedBall (\u2191x) \u03b5 = \u22c3 (z : \u2124), closedBall (x + z \u2022 p) \u03b5\n[PROOFSTEP]\next y\n[GOAL]\ncase inr.h\np x \u03b5 : \u211d\nhp : p \u2260 0\ny : \u211d\n\u22a2 y \u2208 QuotientAddGroup.mk \u207b\u00b9' closedBall (\u2191x) \u03b5 \u2194 y \u2208 \u22c3 (z : \u2124), closedBall (x + z \u2022 p) \u03b5\n[PROOFSTEP]\nsimp only [dist_eq_norm, mem_preimage, mem_closedBall, zsmul_eq_mul, mem_iUnion, Real.norm_eq_abs, \u2190\n QuotientAddGroup.mk_sub, norm_eq, \u2190 sub_sub]\n[GOAL]\ncase inr.h\np x \u03b5 : \u211d\nhp : p \u2260 0\ny : \u211d\n\u22a2 |y - x - \u2191(round (p\u207b\u00b9 * (y - x))) * p| \u2264 \u03b5 \u2194 \u2203 i, |y - x - \u2191i * p| \u2264 \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8round (p\u207b\u00b9 * (y - x)), h\u27e9, _\u27e9\n[GOAL]\ncase inr.h\np x \u03b5 : \u211d\nhp : p \u2260 0\ny : \u211d\n\u22a2 (\u2203 i, |y - x - \u2191i * p| \u2264 \u03b5) \u2192 |y - x - \u2191(round (p\u207b\u00b9 * (y - x))) * p| \u2264 \u03b5\n[PROOFSTEP]\nrintro \u27e8n, hn\u27e9\n[GOAL]\ncase inr.h.intro\np x \u03b5 : \u211d\nhp : p \u2260 0\ny : \u211d\nn : \u2124\nhn : |y - x - \u2191n * p| \u2264 \u03b5\n\u22a2 |y - x - \u2191(round (p\u207b\u00b9 * (y - x))) * p| \u2264 \u03b5\n[PROOFSTEP]\nrw [\u2190 mul_le_mul_left (abs_pos.mpr <| inv_ne_zero hp), \u2190 abs_mul, mul_sub, mul_comm _ p, inv_mul_cancel_left\u2080 hp] at hn\n \u22a2\n[GOAL]\ncase inr.h.intro\np x \u03b5 : \u211d\nhp : p \u2260 0\ny : \u211d\nn : \u2124\nhn : |p\u207b\u00b9 * (y - x) - \u2191n| \u2264 |p\u207b\u00b9| * \u03b5\n\u22a2 |p\u207b\u00b9 * (y - x) - \u2191(round (p\u207b\u00b9 * (y - x)))| \u2264 |p\u207b\u00b9| * \u03b5\n[PROOFSTEP]\nexact (round_le (p\u207b\u00b9 * (y - x)) n).trans hn\n[GOAL]\np x \u03b5 : \u211d\ns : Set \u211d\nhs : s \u2286 closedBall x (|p| / 2)\n\u22a2 QuotientAddGroup.mk \u207b\u00b9' closedBall (\u2191x) \u03b5 \u2229 s = if \u03b5 < |p| / 2 then closedBall x \u03b5 \u2229 s else s\n[PROOFSTEP]\ncases' le_or_lt (|p| / 2) \u03b5 with h\u03b5 h\u03b5\n[GOAL]\ncase inl\np x \u03b5 : \u211d\ns : Set \u211d\nhs : s \u2286 closedBall x (|p| / 2)\nh\u03b5 : |p| / 2 \u2264 \u03b5\n\u22a2 QuotientAddGroup.mk \u207b\u00b9' closedBall (\u2191x) \u03b5 \u2229 s = if \u03b5 < |p| / 2 then closedBall x \u03b5 \u2229 s else s\n[PROOFSTEP]\nrcases eq_or_ne p 0 with (rfl | hp)\n[GOAL]\ncase inl.inl\nx \u03b5 : \u211d\ns : Set \u211d\nhs : s \u2286 closedBall x (|0| / 2)\nh\u03b5 : |0| / 2 \u2264 \u03b5\n\u22a2 QuotientAddGroup.mk \u207b\u00b9' closedBall (\u2191x) \u03b5 \u2229 s = if \u03b5 < |0| / 2 then closedBall x \u03b5 \u2229 s else s\n[PROOFSTEP]\nsimp only [abs_zero, zero_div] at h\u03b5 \n[GOAL]\ncase inl.inl\nx \u03b5 : \u211d\ns : Set \u211d\nhs : s \u2286 closedBall x (|0| / 2)\nh\u03b5 : 0 \u2264 \u03b5\n\u22a2 QuotientAddGroup.mk \u207b\u00b9' closedBall (\u2191x) \u03b5 \u2229 s = if \u03b5 < |0| / 2 then closedBall x \u03b5 \u2229 s else s\n[PROOFSTEP]\nsimp only [not_lt.mpr h\u03b5, coe_real_preimage_closedBall_period_zero, abs_zero, zero_div, if_false,\n inter_eq_right_iff_subset]\n[GOAL]\ncase inl.inl\nx \u03b5 : \u211d\ns : Set \u211d\nhs : s \u2286 closedBall x (|0| / 2)\nh\u03b5 : 0 \u2264 \u03b5\n\u22a2 s \u2286 closedBall x \u03b5\n[PROOFSTEP]\nexact\n hs.trans\n (closedBall_subset_closedBall <| by simp [h\u03b5])\n -- Porting note: was\n -- simp [closedBall_eq_univ_of_half_period_le p hp (\u2191x) h\u03b5, not_lt.mpr h\u03b5]\n[GOAL]\nx \u03b5 : \u211d\ns : Set \u211d\nhs : s \u2286 closedBall x (|0| / 2)\nh\u03b5 : 0 \u2264 \u03b5\n\u22a2 |0| / 2 \u2264 \u03b5\n[PROOFSTEP]\nsimp [h\u03b5]\n[GOAL]\ncase inl.inr\np x \u03b5 : \u211d\ns : Set \u211d\nhs : s \u2286 closedBall x (|p| / 2)\nh\u03b5 : |p| / 2 \u2264 \u03b5\nhp : p \u2260 0\n\u22a2 QuotientAddGroup.mk \u207b\u00b9' closedBall (\u2191x) \u03b5 \u2229 s = if \u03b5 < |p| / 2 then closedBall x \u03b5 \u2229 s else s\n[PROOFSTEP]\nsimp only [not_lt.mpr h\u03b5, ite_false, inter_eq_right_iff_subset]\n[GOAL]\ncase inl.inr\np x \u03b5 : \u211d\ns : Set \u211d\nhs : s \u2286 closedBall x (|p| / 2)\nh\u03b5 : |p| / 2 \u2264 \u03b5\nhp : p \u2260 0\n\u22a2 s \u2286 QuotientAddGroup.mk \u207b\u00b9' closedBall (\u2191x) \u03b5\n[PROOFSTEP]\nrw [closedBall_eq_univ_of_half_period_le p hp (\u2191x : \u211d \u29f8 zmultiples p) h\u03b5, preimage_univ]\n[GOAL]\ncase inl.inr\np x \u03b5 : \u211d\ns : Set \u211d\nhs : s \u2286 closedBall x (|p| / 2)\nh\u03b5 : |p| / 2 \u2264 \u03b5\nhp : p \u2260 0\n\u22a2 s \u2286 univ\n[PROOFSTEP]\napply subset_univ\n[GOAL]\ncase inr\np x \u03b5 : \u211d\ns : Set \u211d\nhs : s \u2286 closedBall x (|p| / 2)\nh\u03b5 : \u03b5 < |p| / 2\n\u22a2 QuotientAddGroup.mk \u207b\u00b9' closedBall (\u2191x) \u03b5 \u2229 s = if \u03b5 < |p| / 2 then closedBall x \u03b5 \u2229 s else s\n[PROOFSTEP]\nsuffices \u2200 z : \u2124, closedBall (x + z \u2022 p) \u03b5 \u2229 s = if z = 0 then closedBall x \u03b5 \u2229 s else \u2205 by\n simp [-zsmul_eq_mul, \u2190 QuotientAddGroup.mk_zero, coe_real_preimage_closedBall_eq_iUnion, iUnion_inter, iUnion_ite,\n this, h\u03b5]\n[GOAL]\np x \u03b5 : \u211d\ns : Set \u211d\nhs : s \u2286 closedBall x (|p| / 2)\nh\u03b5 : \u03b5 < |p| / 2\nthis : \u2200 (z : \u2124), closedBall (x + z \u2022 p) \u03b5 \u2229 s = if z = 0 then closedBall x \u03b5 \u2229 s else \u2205\n\u22a2 QuotientAddGroup.mk \u207b\u00b9' closedBall (\u2191x) \u03b5 \u2229 s = if \u03b5 < |p| / 2 then closedBall x \u03b5 \u2229 s else s\n[PROOFSTEP]\nsimp [-zsmul_eq_mul, \u2190 QuotientAddGroup.mk_zero, coe_real_preimage_closedBall_eq_iUnion, iUnion_inter, iUnion_ite, this,\n h\u03b5]\n[GOAL]\ncase inr\np x \u03b5 : \u211d\ns : Set \u211d\nhs : s \u2286 closedBall x (|p| / 2)\nh\u03b5 : \u03b5 < |p| / 2\n\u22a2 \u2200 (z : \u2124), closedBall (x + z \u2022 p) \u03b5 \u2229 s = if z = 0 then closedBall x \u03b5 \u2229 s else \u2205\n[PROOFSTEP]\nintro z\n[GOAL]\ncase inr\np x \u03b5 : \u211d\ns : Set \u211d\nhs : s \u2286 closedBall x (|p| / 2)\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\n\u22a2 closedBall (x + z \u2022 p) \u03b5 \u2229 s = if z = 0 then closedBall x \u03b5 \u2229 s else \u2205\n[PROOFSTEP]\nsimp only [Real.closedBall_eq_Icc, zero_sub, zero_add] at hs \u22a2\n[GOAL]\ncase inr\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\n\u22a2 Icc (x + z \u2022 p - \u03b5) (x + z \u2022 p + \u03b5) \u2229 s = if z = 0 then Icc (x - \u03b5) (x + \u03b5) \u2229 s else \u2205\n[PROOFSTEP]\nrcases eq_or_ne z 0 with (rfl | hz)\n[GOAL]\ncase inr.inl\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\n\u22a2 Icc (x + 0 \u2022 p - \u03b5) (x + 0 \u2022 p + \u03b5) \u2229 s = if 0 = 0 then Icc (x - \u03b5) (x + \u03b5) \u2229 s else \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\n\u22a2 Icc (x + z \u2022 p - \u03b5) (x + z \u2022 p + \u03b5) \u2229 s = if z = 0 then Icc (x - \u03b5) (x + \u03b5) \u2229 s else \u2205\n[PROOFSTEP]\nsimp only [hz, zsmul_eq_mul, if_false, eq_empty_iff_forall_not_mem]\n[GOAL]\ncase inr.inr\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\n\u22a2 \u2200 (x_1 : \u211d), \u00acx_1 \u2208 Icc (x + \u2191z * p - \u03b5) (x + \u2191z * p + \u03b5) \u2229 s\n[PROOFSTEP]\nrintro y \u27e8\u27e8hy\u2081, hy\u2082\u27e9, hy\u2080\u27e9\n[GOAL]\ncase inr.inr.intro.intro\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8hy\u2083, hy\u2084\u27e9 := hs hy\u2080\n[GOAL]\ncase inr.inr.intro.intro.intro\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\n\u22a2 False\n[PROOFSTEP]\nrcases lt_trichotomy 0 p with (hp | (rfl : 0 = p) | hp)\n[GOAL]\ncase inr.inr.intro.intro.intro.inl\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : 0 < p\n\u22a2 False\n[PROOFSTEP]\ncases' Int.cast_le_neg_one_or_one_le_cast_of_ne_zero \u211d hz with hz' hz'\n[GOAL]\ncase inr.inr.intro.intro.intro.inl.inl\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : 0 < p\nhz' : \u2191z \u2264 -1\n\u22a2 False\n[PROOFSTEP]\nhave : \u2191z * p \u2264 -p\n[GOAL]\ncase this\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : 0 < p\nhz' : \u2191z \u2264 -1\n\u22a2 \u2191z * p \u2264 -p\ncase inr.inr.intro.intro.intro.inl.inl\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : 0 < p\nhz' : \u2191z \u2264 -1\nthis : \u2191z * p \u2264 -p\n\u22a2 False\n[PROOFSTEP]\nnlinarith\n[GOAL]\ncase inr.inr.intro.intro.intro.inl.inl\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : 0 < p\nhz' : \u2191z \u2264 -1\nthis : \u2191z * p \u2264 -p\n\u22a2 False\n[PROOFSTEP]\nlinarith [abs_eq_self.mpr hp.le]\n[GOAL]\ncase inr.inr.intro.intro.intro.inl.inr\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : 0 < p\nhz' : 1 \u2264 \u2191z\n\u22a2 False\n[PROOFSTEP]\nhave : p \u2264 \u2191z * p\n[GOAL]\ncase this\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : 0 < p\nhz' : 1 \u2264 \u2191z\n\u22a2 p \u2264 \u2191z * p\ncase inr.inr.intro.intro.intro.inl.inr\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : 0 < p\nhz' : 1 \u2264 \u2191z\nthis : p \u2264 \u2191z * p\n\u22a2 False\n[PROOFSTEP]\nnlinarith\n[GOAL]\ncase inr.inr.intro.intro.intro.inl.inr\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : 0 < p\nhz' : 1 \u2264 \u2191z\nthis : p \u2264 \u2191z * p\n\u22a2 False\n[PROOFSTEP]\nlinarith [abs_eq_self.mpr hp.le]\n[GOAL]\ncase inr.inr.intro.intro.intro.inr.inl\nx \u03b5 : \u211d\ns : Set \u211d\nz : \u2124\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nh\u03b5 : \u03b5 < |0| / 2\nhs : s \u2286 Icc (x - |0| / 2) (x + |0| / 2)\nhy\u2081 : x + \u2191z * 0 - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * 0 + \u03b5\nhy\u2083 : x - |0| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |0| / 2\n\u22a2 False\n[PROOFSTEP]\nsimp only [mul_zero, add_zero, abs_zero, zero_div] at hy\u2081 hy\u2082 h\u03b5 \n[GOAL]\ncase inr.inr.intro.intro.intro.inr.inl\nx \u03b5 : \u211d\ns : Set \u211d\nz : \u2124\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhs : s \u2286 Icc (x - |0| / 2) (x + |0| / 2)\nhy\u2083 : x - |0| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |0| / 2\nhy\u2081 : x - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u03b5\nh\u03b5 : \u03b5 < 0\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase inr.inr.intro.intro.intro.inr.inr\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : p < 0\n\u22a2 False\n[PROOFSTEP]\ncases' Int.cast_le_neg_one_or_one_le_cast_of_ne_zero \u211d hz with hz' hz'\n[GOAL]\ncase inr.inr.intro.intro.intro.inr.inr.inl\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : p < 0\nhz' : \u2191z \u2264 -1\n\u22a2 False\n[PROOFSTEP]\nhave : -p \u2264 \u2191z * p\n[GOAL]\ncase this\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : p < 0\nhz' : \u2191z \u2264 -1\n\u22a2 -p \u2264 \u2191z * p\ncase inr.inr.intro.intro.intro.inr.inr.inl\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : p < 0\nhz' : \u2191z \u2264 -1\nthis : -p \u2264 \u2191z * p\n\u22a2 False\n[PROOFSTEP]\nnlinarith\n[GOAL]\ncase inr.inr.intro.intro.intro.inr.inr.inl\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : p < 0\nhz' : \u2191z \u2264 -1\nthis : -p \u2264 \u2191z * p\n\u22a2 False\n[PROOFSTEP]\nlinarith [abs_eq_neg_self.mpr hp.le]\n[GOAL]\ncase inr.inr.intro.intro.intro.inr.inr.inr\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : p < 0\nhz' : 1 \u2264 \u2191z\n\u22a2 False\n[PROOFSTEP]\nhave : \u2191z * p \u2264 p\n[GOAL]\ncase this\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : p < 0\nhz' : 1 \u2264 \u2191z\n\u22a2 \u2191z * p \u2264 p\ncase inr.inr.intro.intro.intro.inr.inr.inr\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : p < 0\nhz' : 1 \u2264 \u2191z\nthis : \u2191z * p \u2264 p\n\u22a2 False\n[PROOFSTEP]\nnlinarith\n[GOAL]\ncase inr.inr.intro.intro.intro.inr.inr.inr\np x \u03b5 : \u211d\ns : Set \u211d\nh\u03b5 : \u03b5 < |p| / 2\nz : \u2124\nhs : s \u2286 Icc (x - |p| / 2) (x + |p| / 2)\nhz : z \u2260 0\ny : \u211d\nhy\u2080 : y \u2208 s\nhy\u2081 : x + \u2191z * p - \u03b5 \u2264 y\nhy\u2082 : y \u2264 x + \u2191z * p + \u03b5\nhy\u2083 : x - |p| / 2 \u2264 y\nhy\u2084 : y \u2264 x + |p| / 2\nhp : p < 0\nhz' : 1 \u2264 \u2191z\nthis : \u2191z * p \u2264 p\n\u22a2 False\n[PROOFSTEP]\nlinarith [abs_eq_neg_self.mpr hp.le]\n[GOAL]\np : \u211d\nhp : Fact (0 < p)\nm n : \u2115\n\u22a2 \u2016\u2191(\u2191m / \u2191n * p)\u2016 = p * (\u2191(min (m % n) (n - m % n)) / \u2191n)\n[PROOFSTEP]\nhave : p\u207b\u00b9 * (\u2191m / \u2191n * p) = \u2191m / \u2191n := by rw [mul_comm _ p, inv_mul_cancel_left\u2080 hp.out.ne.symm]\n[GOAL]\np : \u211d\nhp : Fact (0 < p)\nm n : \u2115\n\u22a2 p\u207b\u00b9 * (\u2191m / \u2191n * p) = \u2191m / \u2191n\n[PROOFSTEP]\nrw [mul_comm _ p, inv_mul_cancel_left\u2080 hp.out.ne.symm]\n[GOAL]\np : \u211d\nhp : Fact (0 < p)\nm n : \u2115\nthis : p\u207b\u00b9 * (\u2191m / \u2191n * p) = \u2191m / \u2191n\n\u22a2 \u2016\u2191(\u2191m / \u2191n * p)\u2016 = p * (\u2191(min (m % n) (n - m % n)) / \u2191n)\n[PROOFSTEP]\nrw [norm_eq' p hp.out, this, abs_sub_round_div_natCast_eq]\n[GOAL]\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\n\u22a2 \u2203 k, \u2016u\u2016 = p * (\u2191k / \u2191(addOrderOf u))\n[PROOFSTEP]\nlet n := addOrderOf u\n[GOAL]\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nn : \u2115 := addOrderOf u\n\u22a2 \u2203 k, \u2016u\u2016 = p * (\u2191k / \u2191(addOrderOf u))\n[PROOFSTEP]\nchange \u2203 k : \u2115, \u2016u\u2016 = p * (k / n)\n[GOAL]\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nn : \u2115 := addOrderOf u\n\u22a2 \u2203 k, \u2016u\u2016 = p * (\u2191k / \u2191n)\n[PROOFSTEP]\nobtain \u27e8m, -, -, hm\u27e9 := exists_gcd_eq_one_of_isOfFinAddOrder hu\n[GOAL]\ncase intro.intro.intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nn : \u2115 := addOrderOf u\nm : \u2115\nhm : \u2191(\u2191m / \u2191(addOrderOf u) * p) = u\n\u22a2 \u2203 k, \u2016u\u2016 = p * (\u2191k / \u2191n)\n[PROOFSTEP]\nrefine' \u27e8min (m % n) (n - m % n), _\u27e9\n[GOAL]\ncase intro.intro.intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nn : \u2115 := addOrderOf u\nm : \u2115\nhm : \u2191(\u2191m / \u2191(addOrderOf u) * p) = u\n\u22a2 \u2016u\u2016 = p * (\u2191(min (m % n) (n - m % n)) / \u2191n)\n[PROOFSTEP]\nrw [\u2190 hm, norm_div_nat_cast]\n[GOAL]\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nhu' : u \u2260 0\n\u22a2 p \u2264 addOrderOf u \u2022 \u2016u\u2016\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := exists_norm_eq_of_isOfFinAddOrder hu\n[GOAL]\ncase intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\n\u22a2 p \u2264 addOrderOf u \u2022 \u2016u\u2016\n[PROOFSTEP]\nreplace hu : (addOrderOf u : \u211d) \u2260 0\n[GOAL]\ncase hu\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\n\u22a2 \u2191(addOrderOf u) \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase hu\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\n\u22a2 \u00acaddOrderOf u = 0\n[PROOFSTEP]\nexact (addOrderOf_pos_iff.mpr hu).ne'\n[GOAL]\ncase intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\nhu : \u2191(addOrderOf u) \u2260 0\n\u22a2 p \u2264 addOrderOf u \u2022 \u2016u\u2016\n[PROOFSTEP]\nconv_lhs => rw [\u2190 mul_one p]\n[GOAL]\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\nhu : \u2191(addOrderOf u) \u2260 0\n| p\n[PROOFSTEP]\nrw [\u2190 mul_one p]\n[GOAL]\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\nhu : \u2191(addOrderOf u) \u2260 0\n| p\n[PROOFSTEP]\nrw [\u2190 mul_one p]\n[GOAL]\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\nhu : \u2191(addOrderOf u) \u2260 0\n| p\n[PROOFSTEP]\nrw [\u2190 mul_one p]\n[GOAL]\ncase intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\nhu : \u2191(addOrderOf u) \u2260 0\n\u22a2 p * 1 \u2264 addOrderOf u \u2022 \u2016u\u2016\n[PROOFSTEP]\nrw [hn, nsmul_eq_mul, \u2190 mul_assoc, mul_comm _ p, mul_assoc, mul_div_cancel' _ hu, mul_le_mul_left hp.out,\n Nat.one_le_cast, Nat.one_le_iff_ne_zero]\n[GOAL]\ncase intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\nhu : \u2191(addOrderOf u) \u2260 0\n\u22a2 n \u2260 0\n[PROOFSTEP]\ncontrapose! hu'\n[GOAL]\ncase intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\nhu : \u2191(addOrderOf u) \u2260 0\nhu' : n = 0\n\u22a2 u = 0\n[PROOFSTEP]\nsimpa only [hu', Nat.cast_zero, zero_div, mul_zero, norm_eq_zero] using hn\n[GOAL]\nx : \u211d\n\u22a2 \u2016\u2191x\u2016 = |x - \u2191(round x)|\n[PROOFSTEP]\nsimp [AddCircle.norm_eq]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Normed.Group.AddCircle", "llama_tokens": 21704, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4246220492517202}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\nR : Type u_2\ninst\u271d : CommRing R\nabv : AbsoluteValue R \ud835\udd5c\n\u22a2 \u2200 (x : R), (fun x y => \u2191abv (y - x)) x x = 0\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Topology.UniformSpace.AbsoluteValue", "llama_tokens": 89, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7401743505760728, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.4246220426756771}} {"text": "[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : UniformSpace F\ninst\u271d : UniformAddGroup F\n\ud835\udd16 : Set (Set E)\n\u22a2 strongTopology \u03c3 F \ud835\udd16 = UniformSpace.toTopologicalSpace\n[PROOFSTEP]\nrw [strongTopology, UniformAddGroup.toUniformSpace_eq]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : UniformSpace F\ninst\u271d : UniformAddGroup F\n\ud835\udd16 : Set (Set E)\n\u22a2 TopologicalSpace.induced FunLike.coe (UniformOnFun.topologicalSpace E F \ud835\udd16) = UniformSpace.toTopologicalSpace\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : UniformSpace F\ninst\u271d : UniformAddGroup F\n\ud835\udd16 : Set (Set E)\n\u22a2 UniformAddGroup (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nletI : UniformSpace (E \u2192SL[\u03c3] F) := strongUniformity \u03c3 F \ud835\udd16\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : UniformSpace F\ninst\u271d : UniformAddGroup F\n\ud835\udd16 : Set (Set E)\nthis : UniformSpace (E \u2192SL[\u03c3] F) := strongUniformity \u03c3 F \ud835\udd16\n\u22a2 UniformAddGroup (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nrw [strongUniformity, UniformSpace.replaceTopology_eq]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : UniformSpace F\ninst\u271d : UniformAddGroup F\n\ud835\udd16 : Set (Set E)\nthis : UniformSpace (E \u2192SL[\u03c3] F) := strongUniformity \u03c3 F \ud835\udd16\n\u22a2 UniformAddGroup (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nlet \u03c6 : (E \u2192SL[\u03c3] F) \u2192+ E \u2192\u1d64[\ud835\udd16] F := \u27e8\u27e8(FunLike.coe : (E \u2192SL[\u03c3] F) \u2192 E \u2192\u1d64[\ud835\udd16] F), rfl\u27e9, fun _ _ => rfl\u27e9\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : UniformSpace F\ninst\u271d : UniformAddGroup F\n\ud835\udd16 : Set (Set E)\nthis : UniformSpace (E \u2192SL[\u03c3] F) := strongUniformity \u03c3 F \ud835\udd16\n\u03c6 : (E \u2192SL[\u03c3] F) \u2192+ E \u2192\u1d64[\ud835\udd16] F :=\n { toZeroHom := { toFun := FunLike.coe, map_zero' := (_ : \u21910 = \u21910) },\n map_add' :=\n (_ :\n \u2200 (x x_1 : E \u2192SL[\u03c3] F),\n ZeroHom.toFun { toFun := FunLike.coe, map_zero' := (_ : \u21910 = \u21910) } (x + x_1) =\n ZeroHom.toFun { toFun := FunLike.coe, map_zero' := (_ : \u21910 = \u21910) } (x + x_1)) }\n\u22a2 UniformAddGroup (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nexact uniformAddGroup_comap \u03c6\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : TopologicalAddGroup F\n\ud835\udd16 : Set (Set E)\n\u22a2 TopologicalAddGroup (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nletI : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : TopologicalAddGroup F\n\ud835\udd16 : Set (Set E)\nthis : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n\u22a2 TopologicalAddGroup (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nhaveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : TopologicalAddGroup F\n\ud835\udd16 : Set (Set E)\nthis\u271d : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis : UniformAddGroup F\n\u22a2 TopologicalAddGroup (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nletI : UniformSpace (E \u2192SL[\u03c3] F) := strongUniformity \u03c3 F \ud835\udd16\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : TopologicalAddGroup F\n\ud835\udd16 : Set (Set E)\nthis\u271d\u00b9 : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis\u271d : UniformAddGroup F\nthis : UniformSpace (E \u2192SL[\u03c3] F) := strongUniformity \u03c3 F \ud835\udd16\n\u22a2 TopologicalAddGroup (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nhaveI : UniformAddGroup (E \u2192SL[\u03c3] F) := strongUniformity.uniformAddGroup \u03c3 F \ud835\udd16\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : TopologicalAddGroup F\n\ud835\udd16 : Set (Set E)\nthis\u271d\u00b2 : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis\u271d\u00b9 : UniformAddGroup F\nthis\u271d : UniformSpace (E \u2192SL[\u03c3] F) := strongUniformity \u03c3 F \ud835\udd16\nthis : UniformAddGroup (E \u2192SL[\u03c3] F)\n\u22a2 TopologicalAddGroup (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2070 : AddCommGroup E'\ninst\u271d\u2079 : Module \u211d E'\ninst\u271d\u2078 : AddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \u211d F'\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : TopologicalSpace E'\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : T2Space F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ\n\u22a2 T2Space (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nletI : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2070 : AddCommGroup E'\ninst\u271d\u2079 : Module \u211d E'\ninst\u271d\u2078 : AddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \u211d F'\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : TopologicalSpace E'\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : T2Space F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ\nthis : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n\u22a2 T2Space (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nhaveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2070 : AddCommGroup E'\ninst\u271d\u2079 : Module \u211d E'\ninst\u271d\u2078 : AddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \u211d F'\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : TopologicalSpace E'\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : T2Space F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ\nthis\u271d : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis : UniformAddGroup F\n\u22a2 T2Space (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nletI : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2070 : AddCommGroup E'\ninst\u271d\u2079 : Module \u211d E'\ninst\u271d\u2078 : AddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \u211d F'\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : TopologicalSpace E'\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : T2Space F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ\nthis\u271d\u00b9 : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis\u271d : UniformAddGroup F\nthis : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\n\u22a2 T2Space (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nhaveI : T2Space (E \u2192\u1d64[\ud835\udd16] F) := UniformOnFun.t2Space_of_covering h\ud835\udd16\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2070 : AddCommGroup E'\ninst\u271d\u2079 : Module \u211d E'\ninst\u271d\u2078 : AddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \u211d F'\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : TopologicalSpace E'\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : T2Space F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ\nthis\u271d\u00b2 : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis\u271d\u00b9 : UniformAddGroup F\nthis\u271d : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\nthis : T2Space (E \u2192\u1d64[\ud835\udd16] F)\n\u22a2 T2Space (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nexact (strongTopology.embedding_coeFn \u03c3 F \ud835\udd16).t2Space\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2076 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2075 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u2074 : AddCommGroup E\ninst\u271d\u00b9\u00b3 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u00b2 : AddCommGroup E'\ninst\u271d\u00b9\u00b9 : Module \u211d E'\ninst\u271d\u00b9\u2070 : AddCommGroup F\ninst\u271d\u2079 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \u211d F'\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace E'\ninst\u271d\u2074 : RingHomSurjective \u03c3\ninst\u271d\u00b3 : RingHomIsometric \u03c3\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16\u2081 : Set.Nonempty \ud835\udd16\nh\ud835\udd16\u2082 : DirectedOn (fun x x_1 => x \u2286 x_1) \ud835\udd16\nh\ud835\udd16\u2083 : \u2200 (S : Set E), S \u2208 \ud835\udd16 \u2192 Bornology.IsVonNBounded \ud835\udd5c\u2081 S\n\u22a2 ContinuousSMul \ud835\udd5c\u2082 (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nletI : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2076 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2075 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u2074 : AddCommGroup E\ninst\u271d\u00b9\u00b3 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u00b2 : AddCommGroup E'\ninst\u271d\u00b9\u00b9 : Module \u211d E'\ninst\u271d\u00b9\u2070 : AddCommGroup F\ninst\u271d\u2079 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \u211d F'\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace E'\ninst\u271d\u2074 : RingHomSurjective \u03c3\ninst\u271d\u00b3 : RingHomIsometric \u03c3\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16\u2081 : Set.Nonempty \ud835\udd16\nh\ud835\udd16\u2082 : DirectedOn (fun x x_1 => x \u2286 x_1) \ud835\udd16\nh\ud835\udd16\u2083 : \u2200 (S : Set E), S \u2208 \ud835\udd16 \u2192 Bornology.IsVonNBounded \ud835\udd5c\u2081 S\nthis : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n\u22a2 ContinuousSMul \ud835\udd5c\u2082 (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nhaveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2076 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2075 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u2074 : AddCommGroup E\ninst\u271d\u00b9\u00b3 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u00b2 : AddCommGroup E'\ninst\u271d\u00b9\u00b9 : Module \u211d E'\ninst\u271d\u00b9\u2070 : AddCommGroup F\ninst\u271d\u2079 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \u211d F'\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace E'\ninst\u271d\u2074 : RingHomSurjective \u03c3\ninst\u271d\u00b3 : RingHomIsometric \u03c3\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16\u2081 : Set.Nonempty \ud835\udd16\nh\ud835\udd16\u2082 : DirectedOn (fun x x_1 => x \u2286 x_1) \ud835\udd16\nh\ud835\udd16\u2083 : \u2200 (S : Set E), S \u2208 \ud835\udd16 \u2192 Bornology.IsVonNBounded \ud835\udd5c\u2081 S\nthis\u271d : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis : UniformAddGroup F\n\u22a2 ContinuousSMul \ud835\udd5c\u2082 (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nletI : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2076 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2075 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u2074 : AddCommGroup E\ninst\u271d\u00b9\u00b3 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u00b2 : AddCommGroup E'\ninst\u271d\u00b9\u00b9 : Module \u211d E'\ninst\u271d\u00b9\u2070 : AddCommGroup F\ninst\u271d\u2079 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \u211d F'\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace E'\ninst\u271d\u2074 : RingHomSurjective \u03c3\ninst\u271d\u00b3 : RingHomIsometric \u03c3\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16\u2081 : Set.Nonempty \ud835\udd16\nh\ud835\udd16\u2082 : DirectedOn (fun x x_1 => x \u2286 x_1) \ud835\udd16\nh\ud835\udd16\u2083 : \u2200 (S : Set E), S \u2208 \ud835\udd16 \u2192 Bornology.IsVonNBounded \ud835\udd5c\u2081 S\nthis\u271d\u00b9 : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis\u271d : UniformAddGroup F\nthis : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\n\u22a2 ContinuousSMul \ud835\udd5c\u2082 (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nlet \u03c6 : (E \u2192SL[\u03c3] F) \u2192\u2097[\ud835\udd5c\u2082] E \u2192\u1d64[\ud835\udd16] F := \u27e8\u27e8(FunLike.coe : (E \u2192SL[\u03c3] F) \u2192 E \u2192 F), fun _ _ => rfl\u27e9, fun _ _ => rfl\u27e9\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2076 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2075 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u2074 : AddCommGroup E\ninst\u271d\u00b9\u00b3 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u00b2 : AddCommGroup E'\ninst\u271d\u00b9\u00b9 : Module \u211d E'\ninst\u271d\u00b9\u2070 : AddCommGroup F\ninst\u271d\u2079 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \u211d F'\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace E'\ninst\u271d\u2074 : RingHomSurjective \u03c3\ninst\u271d\u00b3 : RingHomIsometric \u03c3\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16\u2081 : Set.Nonempty \ud835\udd16\nh\ud835\udd16\u2082 : DirectedOn (fun x x_1 => x \u2286 x_1) \ud835\udd16\nh\ud835\udd16\u2083 : \u2200 (S : Set E), S \u2208 \ud835\udd16 \u2192 Bornology.IsVonNBounded \ud835\udd5c\u2081 S\nthis\u271d\u00b9 : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis\u271d : UniformAddGroup F\nthis : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\n\u03c6 : (E \u2192SL[\u03c3] F) \u2192\u2097[\ud835\udd5c\u2082] E \u2192\u1d64[\ud835\udd16] F :=\n { toAddHom := { toFun := FunLike.coe, map_add' := (_ : \u2200 (x x_1 : E \u2192SL[\u03c3] F), \u2191(x + x_1) = \u2191(x + x_1)) },\n map_smul' :=\n (_ :\n \u2200 (x : \ud835\udd5c\u2082) (x_1 : E \u2192SL[\u03c3] F),\n AddHom.toFun { toFun := FunLike.coe, map_add' := (_ : \u2200 (x x_2 : E \u2192SL[\u03c3] F), \u2191(x + x_2) = \u2191(x + x_2)) }\n (x \u2022 x_1) =\n AddHom.toFun { toFun := FunLike.coe, map_add' := (_ : \u2200 (x x_2 : E \u2192SL[\u03c3] F), \u2191(x + x_2) = \u2191(x + x_2)) }\n (x \u2022 x_1)) }\n\u22a2 ContinuousSMul \ud835\udd5c\u2082 (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nexact\n UniformOnFun.continuousSMul_induced_of_image_bounded \ud835\udd5c\u2082 E F (E \u2192SL[\u03c3] F) h\ud835\udd16\u2081 h\ud835\udd16\u2082 \u03c6 \u27e8rfl\u27e9 fun u s hs =>\n (h\ud835\udd16\u2083 s hs).image u\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : TopologicalAddGroup F\n\u03b9 : Type u_7\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16\u2081 : Set.Nonempty \ud835\udd16\nh\ud835\udd16\u2082 : DirectedOn (fun x x_1 => x \u2286 x_1) \ud835\udd16\np : \u03b9 \u2192 Prop\nb : \u03b9 \u2192 Set F\nh : Filter.HasBasis (\ud835\udcdd 0) p b\n\u22a2 Filter.HasBasis (\ud835\udcdd 0) (fun Si => Si.fst \u2208 \ud835\udd16 \u2227 p Si.snd) fun Si => {f | \u2200 (x : E), x \u2208 Si.fst \u2192 \u2191f x \u2208 b Si.snd}\n[PROOFSTEP]\nletI : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : TopologicalAddGroup F\n\u03b9 : Type u_7\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16\u2081 : Set.Nonempty \ud835\udd16\nh\ud835\udd16\u2082 : DirectedOn (fun x x_1 => x \u2286 x_1) \ud835\udd16\np : \u03b9 \u2192 Prop\nb : \u03b9 \u2192 Set F\nh : Filter.HasBasis (\ud835\udcdd 0) p b\nthis : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n\u22a2 Filter.HasBasis (\ud835\udcdd 0) (fun Si => Si.fst \u2208 \ud835\udd16 \u2227 p Si.snd) fun Si => {f | \u2200 (x : E), x \u2208 Si.fst \u2192 \u2191f x \u2208 b Si.snd}\n[PROOFSTEP]\nhaveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : TopologicalAddGroup F\n\u03b9 : Type u_7\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16\u2081 : Set.Nonempty \ud835\udd16\nh\ud835\udd16\u2082 : DirectedOn (fun x x_1 => x \u2286 x_1) \ud835\udd16\np : \u03b9 \u2192 Prop\nb : \u03b9 \u2192 Set F\nh : Filter.HasBasis (\ud835\udcdd 0) p b\nthis\u271d : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis : UniformAddGroup F\n\u22a2 Filter.HasBasis (\ud835\udcdd 0) (fun Si => Si.fst \u2208 \ud835\udd16 \u2227 p Si.snd) fun Si => {f | \u2200 (x : E), x \u2208 Si.fst \u2192 \u2191f x \u2208 b Si.snd}\n[PROOFSTEP]\nletI : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : TopologicalAddGroup F\n\u03b9 : Type u_7\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16\u2081 : Set.Nonempty \ud835\udd16\nh\ud835\udd16\u2082 : DirectedOn (fun x x_1 => x \u2286 x_1) \ud835\udd16\np : \u03b9 \u2192 Prop\nb : \u03b9 \u2192 Set F\nh : Filter.HasBasis (\ud835\udcdd 0) p b\nthis\u271d\u00b9 : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis\u271d : UniformAddGroup F\nthis : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\n\u22a2 Filter.HasBasis (\ud835\udcdd 0) (fun Si => Si.fst \u2208 \ud835\udd16 \u2227 p Si.snd) fun Si => {f | \u2200 (x : E), x \u2208 Si.fst \u2192 \u2191f x \u2208 b Si.snd}\n[PROOFSTEP]\nrw [(strongTopology.embedding_coeFn \u03c3 F \ud835\udd16).toInducing.nhds_eq_comap]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b2 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2079 : AddCommGroup E'\ninst\u271d\u2078 : Module \u211d E'\ninst\u271d\u2077 : AddCommGroup F\ninst\u271d\u2076 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \u211d F'\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace E'\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : TopologicalAddGroup F\n\u03b9 : Type u_7\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16\u2081 : Set.Nonempty \ud835\udd16\nh\ud835\udd16\u2082 : DirectedOn (fun x x_1 => x \u2286 x_1) \ud835\udd16\np : \u03b9 \u2192 Prop\nb : \u03b9 \u2192 Set F\nh : Filter.HasBasis (\ud835\udcdd 0) p b\nthis\u271d\u00b9 : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis\u271d : UniformAddGroup F\nthis : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\n\u22a2 Filter.HasBasis (Filter.comap (\u2191(UniformOnFun.ofFun \ud835\udd16) \u2218 FunLike.coe) (\ud835\udcdd ((\u2191(UniformOnFun.ofFun \ud835\udd16) \u2218 FunLike.coe) 0)))\n (fun Si => Si.fst \u2208 \ud835\udd16 \u2227 p Si.snd) fun Si => {f | \u2200 (x : E), x \u2208 Si.fst \u2192 \u2191f x \u2208 b Si.snd}\n[PROOFSTEP]\nexact (UniformOnFun.hasBasis_nhds_zero_of_basis \ud835\udd16 h\ud835\udd16\u2081 h\ud835\udd16\u2082 h).comap FunLike.coe\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\ninst\u271d\u00b2\u2070 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2079 : NormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2078 : NormedField \ud835\udd5c\u2083\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\n\u03c4 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c1 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2077 : RingHomCompTriple \u03c3 \u03c4 \u03c1\nE : Type u_4\nE' : Type u_5\nF : Type u_6\nF' : Type u_7\nG : Type u_8\ninst\u271d\u00b9\u2076 : AddCommGroup E\ninst\u271d\u00b9\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2074 : AddCommGroup E'\ninst\u271d\u00b9\u00b3 : Module \u211d E'\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u2070 : AddCommGroup F'\ninst\u271d\u2079 : Module \u211d F'\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c\u2083 G\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalAddGroup G\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2083 G\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\nL : E \u2192SL[\u03c3] F\n\u22a2 Continuous\n {\n toAddHom :=\n { toFun := fun f => comp f L,\n map_add' := (_ : \u2200 (f g : F \u2192SL[\u03c4] G), comp (f + g) L = comp f L + comp g L) },\n map_smul' := (_ : \u2200 (a : \ud835\udd5c\u2083) (f : F \u2192SL[\u03c4] G), comp (a \u2022 f) L = a \u2022 comp f L) }.toAddHom.toFun\n[PROOFSTEP]\nletI : UniformSpace G := TopologicalAddGroup.toUniformSpace G\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\ninst\u271d\u00b2\u2070 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2079 : NormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2078 : NormedField \ud835\udd5c\u2083\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\n\u03c4 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c1 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2077 : RingHomCompTriple \u03c3 \u03c4 \u03c1\nE : Type u_4\nE' : Type u_5\nF : Type u_6\nF' : Type u_7\nG : Type u_8\ninst\u271d\u00b9\u2076 : AddCommGroup E\ninst\u271d\u00b9\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2074 : AddCommGroup E'\ninst\u271d\u00b9\u00b3 : Module \u211d E'\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u2070 : AddCommGroup F'\ninst\u271d\u2079 : Module \u211d F'\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c\u2083 G\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalAddGroup G\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2083 G\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\nL : E \u2192SL[\u03c3] F\nthis : UniformSpace G := TopologicalAddGroup.toUniformSpace G\n\u22a2 Continuous\n {\n toAddHom :=\n { toFun := fun f => comp f L,\n map_add' := (_ : \u2200 (f g : F \u2192SL[\u03c4] G), comp (f + g) L = comp f L + comp g L) },\n map_smul' := (_ : \u2200 (a : \ud835\udd5c\u2083) (f : F \u2192SL[\u03c4] G), comp (a \u2022 f) L = a \u2022 comp f L) }.toAddHom.toFun\n[PROOFSTEP]\nhaveI : UniformAddGroup G := comm_topologicalAddGroup_is_uniform\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\ninst\u271d\u00b2\u2070 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2079 : NormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2078 : NormedField \ud835\udd5c\u2083\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\n\u03c4 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c1 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2077 : RingHomCompTriple \u03c3 \u03c4 \u03c1\nE : Type u_4\nE' : Type u_5\nF : Type u_6\nF' : Type u_7\nG : Type u_8\ninst\u271d\u00b9\u2076 : AddCommGroup E\ninst\u271d\u00b9\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2074 : AddCommGroup E'\ninst\u271d\u00b9\u00b3 : Module \u211d E'\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u2070 : AddCommGroup F'\ninst\u271d\u2079 : Module \u211d F'\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c\u2083 G\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalAddGroup G\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2083 G\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\nL : E \u2192SL[\u03c3] F\nthis\u271d : UniformSpace G := TopologicalAddGroup.toUniformSpace G\nthis : UniformAddGroup G\n\u22a2 Continuous\n {\n toAddHom :=\n { toFun := fun f => comp f L,\n map_add' := (_ : \u2200 (f g : F \u2192SL[\u03c4] G), comp (f + g) L = comp f L + comp g L) },\n map_smul' := (_ : \u2200 (a : \ud835\udd5c\u2083) (f : F \u2192SL[\u03c4] G), comp (a \u2022 f) L = a \u2022 comp f L) }.toAddHom.toFun\n[PROOFSTEP]\nrw [(strongTopology.embedding_coeFn _ _ _).continuous_iff]\n -- Porting note: without this, the following doesn't work\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\ninst\u271d\u00b2\u2070 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2079 : NormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2078 : NormedField \ud835\udd5c\u2083\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\n\u03c4 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c1 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2077 : RingHomCompTriple \u03c3 \u03c4 \u03c1\nE : Type u_4\nE' : Type u_5\nF : Type u_6\nF' : Type u_7\nG : Type u_8\ninst\u271d\u00b9\u2076 : AddCommGroup E\ninst\u271d\u00b9\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2074 : AddCommGroup E'\ninst\u271d\u00b9\u00b3 : Module \u211d E'\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u2070 : AddCommGroup F'\ninst\u271d\u2079 : Module \u211d F'\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c\u2083 G\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalAddGroup G\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2083 G\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\nL : E \u2192SL[\u03c3] F\nthis\u271d : UniformSpace G := TopologicalAddGroup.toUniformSpace G\nthis : UniformAddGroup G\n\u22a2 Continuous\n ((\u2191(UniformOnFun.ofFun {S | Bornology.IsVonNBounded \ud835\udd5c\u2081 S}) \u2218 FunLike.coe) \u2218\n {\n toAddHom :=\n { toFun := fun f => comp f L,\n map_add' := (_ : \u2200 (f g : F \u2192SL[\u03c4] G), comp (f + g) L = comp f L + comp g L) },\n map_smul' := (_ : \u2200 (a : \ud835\udd5c\u2083) (f : F \u2192SL[\u03c4] G), comp (a \u2022 f) L = a \u2022 comp f L) }.toAddHom.toFun)\n[PROOFSTEP]\nchange Continuous ((\u03bb f \u21a6 UniformOnFun.ofFun _ (f \u2218 L)) \u2218 FunLike.coe)\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\ninst\u271d\u00b2\u2070 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2079 : NormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2078 : NormedField \ud835\udd5c\u2083\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\n\u03c4 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c1 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2077 : RingHomCompTriple \u03c3 \u03c4 \u03c1\nE : Type u_4\nE' : Type u_5\nF : Type u_6\nF' : Type u_7\nG : Type u_8\ninst\u271d\u00b9\u2076 : AddCommGroup E\ninst\u271d\u00b9\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2074 : AddCommGroup E'\ninst\u271d\u00b9\u00b3 : Module \u211d E'\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u2070 : AddCommGroup F'\ninst\u271d\u2079 : Module \u211d F'\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c\u2083 G\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalAddGroup G\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2083 G\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\nL : E \u2192SL[\u03c3] F\nthis\u271d : UniformSpace G := TopologicalAddGroup.toUniformSpace G\nthis : UniformAddGroup G\n\u22a2 Continuous ((fun f => \u2191(UniformOnFun.ofFun {S | Bornology.IsVonNBounded \ud835\udd5c\u2081 S}) (f \u2218 \u2191L)) \u2218 FunLike.coe)\n[PROOFSTEP]\nexact\n (UniformOnFun.precomp_uniformContinuous fun S hS => hS.image L).continuous.comp\n (strongTopology.embedding_coeFn _ _ _).continuous\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\ninst\u271d\u00b2\u2070 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2079 : NormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2078 : NormedField \ud835\udd5c\u2083\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\n\u03c4 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c1 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2077 : RingHomCompTriple \u03c3 \u03c4 \u03c1\nE : Type u_4\nE' : Type u_5\nF : Type u_6\nF' : Type u_7\nG : Type u_8\ninst\u271d\u00b9\u2076 : AddCommGroup E\ninst\u271d\u00b9\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2074 : AddCommGroup E'\ninst\u271d\u00b9\u00b3 : Module \u211d E'\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u2070 : AddCommGroup F'\ninst\u271d\u2079 : Module \u211d F'\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c\u2083 G\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalAddGroup F\ninst\u271d\u00b2 : TopologicalAddGroup G\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c\u2083 G\ninst\u271d : ContinuousConstSMul \ud835\udd5c\u2082 F\nL : F \u2192SL[\u03c4] G\n\u22a2 Continuous\n {\n toAddHom :=\n { toFun := fun f => comp L f,\n map_add' := (_ : \u2200 (f\u2081 f\u2082 : E \u2192SL[\u03c3] F), comp L (f\u2081 + f\u2082) = comp L f\u2081 + comp L f\u2082) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c\u2082) (f : E \u2192SL[\u03c3] F), comp L (c \u2022 f) = \u2191\u03c4 c \u2022 comp L f) }.toAddHom.toFun\n[PROOFSTEP]\nletI : UniformSpace G := TopologicalAddGroup.toUniformSpace G\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\ninst\u271d\u00b2\u2070 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2079 : NormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2078 : NormedField \ud835\udd5c\u2083\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\n\u03c4 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c1 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2077 : RingHomCompTriple \u03c3 \u03c4 \u03c1\nE : Type u_4\nE' : Type u_5\nF : Type u_6\nF' : Type u_7\nG : Type u_8\ninst\u271d\u00b9\u2076 : AddCommGroup E\ninst\u271d\u00b9\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2074 : AddCommGroup E'\ninst\u271d\u00b9\u00b3 : Module \u211d E'\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u2070 : AddCommGroup F'\ninst\u271d\u2079 : Module \u211d F'\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c\u2083 G\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalAddGroup F\ninst\u271d\u00b2 : TopologicalAddGroup G\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c\u2083 G\ninst\u271d : ContinuousConstSMul \ud835\udd5c\u2082 F\nL : F \u2192SL[\u03c4] G\nthis : UniformSpace G := TopologicalAddGroup.toUniformSpace G\n\u22a2 Continuous\n {\n toAddHom :=\n { toFun := fun f => comp L f,\n map_add' := (_ : \u2200 (f\u2081 f\u2082 : E \u2192SL[\u03c3] F), comp L (f\u2081 + f\u2082) = comp L f\u2081 + comp L f\u2082) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c\u2082) (f : E \u2192SL[\u03c3] F), comp L (c \u2022 f) = \u2191\u03c4 c \u2022 comp L f) }.toAddHom.toFun\n[PROOFSTEP]\nhaveI : UniformAddGroup G := comm_topologicalAddGroup_is_uniform\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\ninst\u271d\u00b2\u2070 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2079 : NormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2078 : NormedField \ud835\udd5c\u2083\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\n\u03c4 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c1 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2077 : RingHomCompTriple \u03c3 \u03c4 \u03c1\nE : Type u_4\nE' : Type u_5\nF : Type u_6\nF' : Type u_7\nG : Type u_8\ninst\u271d\u00b9\u2076 : AddCommGroup E\ninst\u271d\u00b9\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2074 : AddCommGroup E'\ninst\u271d\u00b9\u00b3 : Module \u211d E'\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u2070 : AddCommGroup F'\ninst\u271d\u2079 : Module \u211d F'\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c\u2083 G\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalAddGroup F\ninst\u271d\u00b2 : TopologicalAddGroup G\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c\u2083 G\ninst\u271d : ContinuousConstSMul \ud835\udd5c\u2082 F\nL : F \u2192SL[\u03c4] G\nthis\u271d : UniformSpace G := TopologicalAddGroup.toUniformSpace G\nthis : UniformAddGroup G\n\u22a2 Continuous\n {\n toAddHom :=\n { toFun := fun f => comp L f,\n map_add' := (_ : \u2200 (f\u2081 f\u2082 : E \u2192SL[\u03c3] F), comp L (f\u2081 + f\u2082) = comp L f\u2081 + comp L f\u2082) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c\u2082) (f : E \u2192SL[\u03c3] F), comp L (c \u2022 f) = \u2191\u03c4 c \u2022 comp L f) }.toAddHom.toFun\n[PROOFSTEP]\nletI : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\ninst\u271d\u00b2\u2070 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2079 : NormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2078 : NormedField \ud835\udd5c\u2083\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\n\u03c4 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c1 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2077 : RingHomCompTriple \u03c3 \u03c4 \u03c1\nE : Type u_4\nE' : Type u_5\nF : Type u_6\nF' : Type u_7\nG : Type u_8\ninst\u271d\u00b9\u2076 : AddCommGroup E\ninst\u271d\u00b9\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2074 : AddCommGroup E'\ninst\u271d\u00b9\u00b3 : Module \u211d E'\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u2070 : AddCommGroup F'\ninst\u271d\u2079 : Module \u211d F'\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c\u2083 G\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalAddGroup F\ninst\u271d\u00b2 : TopologicalAddGroup G\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c\u2083 G\ninst\u271d : ContinuousConstSMul \ud835\udd5c\u2082 F\nL : F \u2192SL[\u03c4] G\nthis\u271d\u00b9 : UniformSpace G := TopologicalAddGroup.toUniformSpace G\nthis\u271d : UniformAddGroup G\nthis : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n\u22a2 Continuous\n {\n toAddHom :=\n { toFun := fun f => comp L f,\n map_add' := (_ : \u2200 (f\u2081 f\u2082 : E \u2192SL[\u03c3] F), comp L (f\u2081 + f\u2082) = comp L f\u2081 + comp L f\u2082) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c\u2082) (f : E \u2192SL[\u03c3] F), comp L (c \u2022 f) = \u2191\u03c4 c \u2022 comp L f) }.toAddHom.toFun\n[PROOFSTEP]\nhaveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\ninst\u271d\u00b2\u2070 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2079 : NormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2078 : NormedField \ud835\udd5c\u2083\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\n\u03c4 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c1 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2077 : RingHomCompTriple \u03c3 \u03c4 \u03c1\nE : Type u_4\nE' : Type u_5\nF : Type u_6\nF' : Type u_7\nG : Type u_8\ninst\u271d\u00b9\u2076 : AddCommGroup E\ninst\u271d\u00b9\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2074 : AddCommGroup E'\ninst\u271d\u00b9\u00b3 : Module \u211d E'\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u2070 : AddCommGroup F'\ninst\u271d\u2079 : Module \u211d F'\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c\u2083 G\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalAddGroup F\ninst\u271d\u00b2 : TopologicalAddGroup G\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c\u2083 G\ninst\u271d : ContinuousConstSMul \ud835\udd5c\u2082 F\nL : F \u2192SL[\u03c4] G\nthis\u271d\u00b2 : UniformSpace G := TopologicalAddGroup.toUniformSpace G\nthis\u271d\u00b9 : UniformAddGroup G\nthis\u271d : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis : UniformAddGroup F\n\u22a2 Continuous\n {\n toAddHom :=\n { toFun := fun f => comp L f,\n map_add' := (_ : \u2200 (f\u2081 f\u2082 : E \u2192SL[\u03c3] F), comp L (f\u2081 + f\u2082) = comp L f\u2081 + comp L f\u2082) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c\u2082) (f : E \u2192SL[\u03c3] F), comp L (c \u2022 f) = \u2191\u03c4 c \u2022 comp L f) }.toAddHom.toFun\n[PROOFSTEP]\nrw [(strongTopology.embedding_coeFn _ _ _).continuous_iff]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\ninst\u271d\u00b2\u2070 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2079 : NormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2078 : NormedField \ud835\udd5c\u2083\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\n\u03c4 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c1 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b9\u2077 : RingHomCompTriple \u03c3 \u03c4 \u03c1\nE : Type u_4\nE' : Type u_5\nF : Type u_6\nF' : Type u_7\nG : Type u_8\ninst\u271d\u00b9\u2076 : AddCommGroup E\ninst\u271d\u00b9\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2074 : AddCommGroup E'\ninst\u271d\u00b9\u00b3 : Module \u211d E'\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u2070 : AddCommGroup F'\ninst\u271d\u2079 : Module \u211d F'\ninst\u271d\u2078 : AddCommGroup G\ninst\u271d\u2077 : Module \ud835\udd5c\u2083 G\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalAddGroup F\ninst\u271d\u00b2 : TopologicalAddGroup G\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c\u2083 G\ninst\u271d : ContinuousConstSMul \ud835\udd5c\u2082 F\nL : F \u2192SL[\u03c4] G\nthis\u271d\u00b2 : UniformSpace G := TopologicalAddGroup.toUniformSpace G\nthis\u271d\u00b9 : UniformAddGroup G\nthis\u271d : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis : UniformAddGroup F\n\u22a2 Continuous\n ((\u2191(UniformOnFun.ofFun {S | Bornology.IsVonNBounded \ud835\udd5c\u2081 S}) \u2218 FunLike.coe) \u2218\n {\n toAddHom :=\n { toFun := fun f => comp L f,\n map_add' := (_ : \u2200 (f\u2081 f\u2082 : E \u2192SL[\u03c3] F), comp L (f\u2081 + f\u2082) = comp L f\u2081 + comp L f\u2082) },\n map_smul' := (_ : \u2200 (c : \ud835\udd5c\u2082) (f : E \u2192SL[\u03c3] F), comp L (c \u2022 f) = \u2191\u03c4 c \u2022 comp L f) }.toAddHom.toFun)\n[PROOFSTEP]\nexact\n (UniformOnFun.postcomp_uniformContinuous L.uniformContinuous).continuous.comp\n (strongTopology.embedding_coeFn _ _ _).continuous\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\n\ud835\udd5c\u2084 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\ninst\u271d\u00b3\u00b9 : AddCommGroup E\ninst\u271d\u00b3\u2070 : AddCommGroup F\ninst\u271d\u00b2\u2079 : AddCommGroup G\ninst\u271d\u00b2\u2078 : AddCommGroup H\ninst\u271d\u00b2\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b2\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b2\u2074 : NontriviallyNormedField \ud835\udd5c\u2084\ninst\u271d\u00b2\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2\u00b2 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b2\u00b9 : Module \ud835\udd5c\u2083 G\ninst\u271d\u00b2\u2070 : Module \ud835\udd5c\u2084 H\ninst\u271d\u00b9\u2079 : TopologicalSpace E\ninst\u271d\u00b9\u2078 : TopologicalSpace F\ninst\u271d\u00b9\u2077 : TopologicalSpace G\ninst\u271d\u00b9\u2076 : TopologicalSpace H\ninst\u271d\u00b9\u2075 : TopologicalAddGroup G\ninst\u271d\u00b9\u2074 : TopologicalAddGroup H\ninst\u271d\u00b9\u00b3 : ContinuousConstSMul \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b2 : ContinuousConstSMul \ud835\udd5c\u2084 H\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\n\u03c3\u2083\u2084 : \ud835\udd5c\u2083 \u2192+* \ud835\udd5c\u2084\n\u03c3\u2084\u2083 : \ud835\udd5c\u2084 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2082\u2084 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2084\n\u03c3\u2081\u2084 : \ud835\udd5c \u2192+* \ud835\udd5c\u2084\ninst\u271d\u00b9\u00b9 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9\u2070 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u2079 : RingHomInvPair \u03c3\u2083\u2084 \u03c3\u2084\u2083\ninst\u271d\u2078 : RingHomInvPair \u03c3\u2084\u2083 \u03c3\u2083\u2084\ninst\u271d\u2077 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2084 \u03c3\u2082\u2084\ninst\u271d\u2076 : RingHomCompTriple \u03c3\u2082\u2084 \u03c3\u2084\u2083 \u03c3\u2082\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d : RingHomIsometric \u03c3\u2082\u2081\ne\u2081\u2082 : E \u2243SL[\u03c3\u2081\u2082] F\ne\u2084\u2083 : H \u2243SL[\u03c3\u2084\u2083] G\nsrc\u271d : (E \u2192SL[\u03c3\u2081\u2084] H) \u2243 (F \u2192SL[\u03c3\u2082\u2083] G) := arrowCongrEquiv e\u2081\u2082 e\u2084\u2083\nf g : E \u2192SL[\u03c3\u2081\u2084] H\n\u22a2 (fun L => comp (\u2191e\u2084\u2083) (comp L \u2191(ContinuousLinearEquiv.symm e\u2081\u2082))) (f + g) =\n (fun L => comp (\u2191e\u2084\u2083) (comp L \u2191(ContinuousLinearEquiv.symm e\u2081\u2082))) f +\n (fun L => comp (\u2191e\u2084\u2083) (comp L \u2191(ContinuousLinearEquiv.symm e\u2081\u2082))) g\n[PROOFSTEP]\nsimp only [add_comp, comp_add]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\n\ud835\udd5c\u2084 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\ninst\u271d\u00b3\u00b9 : AddCommGroup E\ninst\u271d\u00b3\u2070 : AddCommGroup F\ninst\u271d\u00b2\u2079 : AddCommGroup G\ninst\u271d\u00b2\u2078 : AddCommGroup H\ninst\u271d\u00b2\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b2\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b2\u2074 : NontriviallyNormedField \ud835\udd5c\u2084\ninst\u271d\u00b2\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2\u00b2 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b2\u00b9 : Module \ud835\udd5c\u2083 G\ninst\u271d\u00b2\u2070 : Module \ud835\udd5c\u2084 H\ninst\u271d\u00b9\u2079 : TopologicalSpace E\ninst\u271d\u00b9\u2078 : TopologicalSpace F\ninst\u271d\u00b9\u2077 : TopologicalSpace G\ninst\u271d\u00b9\u2076 : TopologicalSpace H\ninst\u271d\u00b9\u2075 : TopologicalAddGroup G\ninst\u271d\u00b9\u2074 : TopologicalAddGroup H\ninst\u271d\u00b9\u00b3 : ContinuousConstSMul \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b2 : ContinuousConstSMul \ud835\udd5c\u2084 H\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\n\u03c3\u2083\u2084 : \ud835\udd5c\u2083 \u2192+* \ud835\udd5c\u2084\n\u03c3\u2084\u2083 : \ud835\udd5c\u2084 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2082\u2084 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2084\n\u03c3\u2081\u2084 : \ud835\udd5c \u2192+* \ud835\udd5c\u2084\ninst\u271d\u00b9\u00b9 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9\u2070 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u2079 : RingHomInvPair \u03c3\u2083\u2084 \u03c3\u2084\u2083\ninst\u271d\u2078 : RingHomInvPair \u03c3\u2084\u2083 \u03c3\u2083\u2084\ninst\u271d\u2077 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2084 \u03c3\u2082\u2084\ninst\u271d\u2076 : RingHomCompTriple \u03c3\u2082\u2084 \u03c3\u2084\u2083 \u03c3\u2082\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d : RingHomIsometric \u03c3\u2082\u2081\ne\u2081\u2082 : E \u2243SL[\u03c3\u2081\u2082] F\ne\u2084\u2083 : H \u2243SL[\u03c3\u2084\u2083] G\nsrc\u271d : (E \u2192SL[\u03c3\u2081\u2084] H) \u2243 (F \u2192SL[\u03c3\u2082\u2083] G) := arrowCongrEquiv e\u2081\u2082 e\u2084\u2083\nt : \ud835\udd5c\u2084\nf : E \u2192SL[\u03c3\u2081\u2084] H\n\u22a2 AddHom.toFun\n { toFun := fun L => comp (\u2191e\u2084\u2083) (comp L \u2191(ContinuousLinearEquiv.symm e\u2081\u2082)),\n map_add' :=\n (_ :\n \u2200 (f g : E \u2192SL[\u03c3\u2081\u2084] H),\n comp (\u2191e\u2084\u2083) (comp (f + g) \u2191(ContinuousLinearEquiv.symm e\u2081\u2082)) =\n comp (\u2191e\u2084\u2083) (comp f \u2191(ContinuousLinearEquiv.symm e\u2081\u2082)) +\n comp (\u2191e\u2084\u2083) (comp g \u2191(ContinuousLinearEquiv.symm e\u2081\u2082))) }\n (t \u2022 f) =\n \u2191\u03c3\u2084\u2083 t \u2022\n AddHom.toFun\n { toFun := fun L => comp (\u2191e\u2084\u2083) (comp L \u2191(ContinuousLinearEquiv.symm e\u2081\u2082)),\n map_add' :=\n (_ :\n \u2200 (f g : E \u2192SL[\u03c3\u2081\u2084] H),\n comp (\u2191e\u2084\u2083) (comp (f + g) \u2191(ContinuousLinearEquiv.symm e\u2081\u2082)) =\n comp (\u2191e\u2084\u2083) (comp f \u2191(ContinuousLinearEquiv.symm e\u2081\u2082)) +\n comp (\u2191e\u2084\u2083) (comp g \u2191(ContinuousLinearEquiv.symm e\u2081\u2082))) }\n f\n[PROOFSTEP]\nsimp only [smul_comp, comp_smul\u209b\u2097]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Module.StrongTopology", "llama_tokens": 18902, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743505760728, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.42462204267567705}} {"text": "[GOAL]\n\u22a2 Continuous sin\n[PROOFSTEP]\nchange Continuous fun z => (exp (-z * I) - exp (z * I)) * I / 2\n[GOAL]\n\u22a2 Continuous fun z => (exp (-z * I) - exp (z * I)) * I / 2\n[PROOFSTEP]\ncontinuity\n[GOAL]\n\u22a2 Continuous cos\n[PROOFSTEP]\nchange Continuous fun z => (exp (z * I) + exp (-z * I)) / 2\n[GOAL]\n\u22a2 Continuous fun z => (exp (z * I) + exp (-z * I)) / 2\n[PROOFSTEP]\ncontinuity\n[GOAL]\n\u22a2 Continuous sinh\n[PROOFSTEP]\nchange Continuous fun z => (exp z - exp (-z)) / 2\n[GOAL]\n\u22a2 Continuous fun z => (exp z - exp (-z)) / 2\n[PROOFSTEP]\ncontinuity\n[GOAL]\n\u22a2 Continuous cosh\n[PROOFSTEP]\nchange Continuous fun z => (exp z + exp (-z)) / 2\n[GOAL]\n\u22a2 Continuous fun z => (exp z + exp (-z)) / 2\n[PROOFSTEP]\ncontinuity\n[GOAL]\n\u22a2 1 \u2264 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 cos (\u03c0 / 2) = 0\n[PROOFSTEP]\nrw [Real.pi, mul_div_cancel_left _ (two_ne_zero' \u211d)]\n[GOAL]\n\u22a2 cos (choose exists_cos_eq_zero) = 0\n[PROOFSTEP]\nexact (Classical.choose_spec exists_cos_eq_zero).2\n[GOAL]\n\u22a2 1 \u2264 \u03c0 / 2\n[PROOFSTEP]\nrw [Real.pi, mul_div_cancel_left _ (two_ne_zero' \u211d)]\n[GOAL]\n\u22a2 1 \u2264 choose exists_cos_eq_zero\n[PROOFSTEP]\nexact (Classical.choose_spec exists_cos_eq_zero).1.1\n[GOAL]\n\u22a2 \u03c0 / 2 \u2264 2\n[PROOFSTEP]\nrw [Real.pi, mul_div_cancel_left _ (two_ne_zero' \u211d)]\n[GOAL]\n\u22a2 choose exists_cos_eq_zero \u2264 2\n[PROOFSTEP]\nexact (Classical.choose_spec exists_cos_eq_zero).1.2\n[GOAL]\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 2 / 2 \u2264 \u03c0 / 2\n[PROOFSTEP]\nrw [div_self (two_ne_zero' \u211d)]\n[GOAL]\n\u22a2 1 \u2264 \u03c0 / 2\n[PROOFSTEP]\nexact one_le_pi_div_two\n[GOAL]\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 2 = 4 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 0 < 2 * \u03c0\n[PROOFSTEP]\nlinarith [pi_pos]\n[GOAL]\n\u22a2 0 < pi\n[PROOFSTEP]\nexact_mod_cast Real.pi_pos\n[GOAL]\n\u22a2 sin \u03c0 = 0\n[PROOFSTEP]\nrw [\u2190 mul_div_cancel_left \u03c0 (two_ne_zero' \u211d), two_mul, add_div, sin_add, cos_pi_div_two]\n[GOAL]\n\u22a2 sin (\u03c0 / 2) * 0 + 0 * sin (\u03c0 / 2) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 cos \u03c0 = -1\n[PROOFSTEP]\nrw [\u2190 mul_div_cancel_left \u03c0 (two_ne_zero' \u211d), mul_div_assoc, cos_two_mul, cos_pi_div_two]\n[GOAL]\n\u22a2 2 * 0 ^ 2 - 1 = -1\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 sin (2 * \u03c0) = 0\n[PROOFSTEP]\nsimp [two_mul, sin_add]\n[GOAL]\n\u22a2 cos (2 * \u03c0) = 1\n[PROOFSTEP]\nsimp [two_mul, cos_add]\n[GOAL]\n\u22a2 Function.Antiperiodic sin \u03c0\n[PROOFSTEP]\nsimp [sin_add]\n[GOAL]\n\u22a2 Function.Antiperiodic cos \u03c0\n[PROOFSTEP]\nsimp [cos_add]\n[GOAL]\nn : \u2115\n\u22a2 cos (\u2191n * (2 * \u03c0) + \u03c0) = -1\n[PROOFSTEP]\nsimpa only [cos_zero] using (cos_periodic.nat_mul n).add_antiperiod_eq cos_antiperiodic\n[GOAL]\nn : \u2124\n\u22a2 cos (\u2191n * (2 * \u03c0) + \u03c0) = -1\n[PROOFSTEP]\nsimpa only [cos_zero] using (cos_periodic.int_mul n).add_antiperiod_eq cos_antiperiodic\n[GOAL]\nn : \u2115\n\u22a2 cos (\u2191n * (2 * \u03c0) - \u03c0) = -1\n[PROOFSTEP]\nsimpa only [cos_zero] using (cos_periodic.nat_mul n).sub_antiperiod_eq cos_antiperiodic\n[GOAL]\nn : \u2124\n\u22a2 cos (\u2191n * (2 * \u03c0) - \u03c0) = -1\n[PROOFSTEP]\nsimpa only [cos_zero] using (cos_periodic.int_mul n).sub_antiperiod_eq cos_antiperiodic\n[GOAL]\nx : \u211d\nh0x : 0 < x\nhxp : x < \u03c0\nhx2 : \u00acx \u2264 2\n\u22a2 2 + 2 = 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\nhx : x \u2208 Icc 0 \u03c0\n\u22a2 0 \u2264 sin x\n[PROOFSTEP]\nrw [\u2190 closure_Ioo pi_ne_zero.symm] at hx \n[GOAL]\nx : \u211d\nhx : x \u2208 closure (Ioo 0 \u03c0)\n\u22a2 0 \u2264 sin x\n[PROOFSTEP]\nexact closure_lt_subset_le continuous_const continuous_sin (closure_mono (fun y => sin_pos_of_mem_Ioo) hx)\n[GOAL]\n\u22a2 sin (\u03c0 / 2) = 1 \u2228 sin (\u03c0 / 2) = -1\n[PROOFSTEP]\nsimpa [sq, mul_self_eq_one_iff] using sin_sq_add_cos_sq (\u03c0 / 2)\n[GOAL]\nthis : sin (\u03c0 / 2) = 1 \u2228 sin (\u03c0 / 2) = -1\nh : sin (\u03c0 / 2) = -1\n\u22a2 \u00ac0 < -1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\n\u22a2 sin (x + \u03c0 / 2) = cos x\n[PROOFSTEP]\nsimp [sin_add]\n[GOAL]\nx : \u211d\n\u22a2 sin (x - \u03c0 / 2) = -cos x\n[PROOFSTEP]\nsimp [sub_eq_add_neg, sin_add]\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 2 - x) = cos x\n[PROOFSTEP]\nsimp [sub_eq_add_neg, sin_add]\n[GOAL]\nx : \u211d\n\u22a2 cos (x + \u03c0 / 2) = -sin x\n[PROOFSTEP]\nsimp [cos_add]\n[GOAL]\nx : \u211d\n\u22a2 cos (x - \u03c0 / 2) = sin x\n[PROOFSTEP]\nsimp [sub_eq_add_neg, cos_add]\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 2 - x) = sin x\n[PROOFSTEP]\nrw [\u2190 cos_neg, neg_sub, cos_sub_pi_div_two]\n[GOAL]\nx : \u211d\nhx : x \u2208 Ioo (-(\u03c0 / 2)) (\u03c0 / 2)\n\u22a2 0 < x + \u03c0 / 2\n[PROOFSTEP]\nlinarith [hx.1]\n[GOAL]\nx : \u211d\nhx : x \u2208 Ioo (-(\u03c0 / 2)) (\u03c0 / 2)\n\u22a2 x + \u03c0 / 2 < \u03c0\n[PROOFSTEP]\nlinarith [hx.2]\n[GOAL]\nx : \u211d\nhx : x \u2208 Icc (-(\u03c0 / 2)) (\u03c0 / 2)\n\u22a2 0 \u2264 x + \u03c0 / 2\n[PROOFSTEP]\nlinarith [hx.1]\n[GOAL]\nx : \u211d\nhx : x \u2208 Icc (-(\u03c0 / 2)) (\u03c0 / 2)\n\u22a2 x + \u03c0 / 2 \u2264 \u03c0\n[PROOFSTEP]\nlinarith [hx.2]\n[GOAL]\nx : \u211d\nhx\u2081 : \u03c0 / 2 < x\nhx\u2082 : x < \u03c0 + \u03c0 / 2\n\u22a2 -(\u03c0 / 2) < \u03c0 - x\n[PROOFSTEP]\nlinarith\n[GOAL]\nx : \u211d\nhx\u2081 : \u03c0 / 2 < x\nhx\u2082 : x < \u03c0 + \u03c0 / 2\n\u22a2 \u03c0 - x < \u03c0 / 2\n[PROOFSTEP]\nlinarith\n[GOAL]\nx : \u211d\nhx\u2081 : \u03c0 / 2 \u2264 x\nhx\u2082 : x \u2264 \u03c0 + \u03c0 / 2\n\u22a2 -(\u03c0 / 2) \u2264 \u03c0 - x\n[PROOFSTEP]\nlinarith\n[GOAL]\nx : \u211d\nhx\u2081 : \u03c0 / 2 \u2264 x\nhx\u2082 : x \u2264 \u03c0 + \u03c0 / 2\n\u22a2 \u03c0 - x \u2264 \u03c0 / 2\n[PROOFSTEP]\nlinarith\n[GOAL]\nx : \u211d\nhl : 0 \u2264 x\nhu : x \u2264 \u03c0\n\u22a2 sin x = sqrt (1 - cos x ^ 2)\n[PROOFSTEP]\nrw [\u2190 abs_sin_eq_sqrt_one_sub_cos_sq, abs_of_nonneg (sin_nonneg_of_nonneg_of_le_pi hl hu)]\n[GOAL]\nx : \u211d\nhl : -(\u03c0 / 2) \u2264 x\nhu : x \u2264 \u03c0 / 2\n\u22a2 cos x = sqrt (1 - sin x ^ 2)\n[PROOFSTEP]\nrw [\u2190 abs_cos_eq_sqrt_one_sub_sin_sq, abs_of_nonneg (cos_nonneg_of_mem_Icc \u27e8hl, hu\u27e9)]\n[GOAL]\nx : \u211d\nhx\u2081 : -\u03c0 < x\nhx\u2082 : x < \u03c0\nh : x = 0\n\u22a2 sin x = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nx : \u211d\nh : sin x = 0\nh\u2083 : x - \u2191\u230ax / \u03c0\u230b * \u03c0 > 0\n\u22a2 0 = sin (x - \u2191\u230ax / \u03c0\u230b * \u03c0)\n[PROOFSTEP]\nsimp [sub_eq_add_neg, sin_add, h, sin_int_mul_pi]\n[GOAL]\nx : \u211d\n\u22a2 sin x \u2260 0 \u2194 \u2200 (n : \u2124), \u2191n * \u03c0 \u2260 x\n[PROOFSTEP]\nrw [\u2190 not_exists, not_iff_not, sin_eq_zero_iff]\n[GOAL]\nx : \u211d\n\u22a2 sin x = 0 \u2194 cos x = 1 \u2228 cos x = -1\n[PROOFSTEP]\nrw [\u2190 mul_self_eq_one_iff, \u2190 sin_sq_add_cos_sq x, sq, sq, \u2190 sub_eq_iff_eq_add, sub_self]\n[GOAL]\nx : \u211d\n\u22a2 sin x = 0 \u2194 0 = sin x * sin x\n[PROOFSTEP]\nexact \u27e8fun h => by rw [h, mul_zero], eq_zero_of_mul_self_eq_zero \u2218 Eq.symm\u27e9\n[GOAL]\nx : \u211d\nh : sin x = 0\n\u22a2 0 = sin x * sin x\n[PROOFSTEP]\nrw [h, mul_zero]\n[GOAL]\nx : \u211d\nh : cos x = 1\nn : \u2124\nhn : \u2191n * \u03c0 = x\nhn0 : n % 2 = 0\n\u22a2 \u2191(n / 2) * (2 * \u03c0) = x\n[PROOFSTEP]\nrwa [\u2190 mul_assoc, \u2190 @Int.cast_two \u211d, \u2190 Int.cast_mul, Int.ediv_mul_cancel ((Int.dvd_iff_emod_eq_zero _ _).2 hn0)]\n[GOAL]\nx : \u211d\nh : cos x = 1\nn : \u2124\nhn : \u2191n * \u03c0 = x\nhn1 : n % 2 = 1\n\u22a2 \u2191(n / 2) * (2 * \u03c0) = x\n[PROOFSTEP]\nrw [\u2190 Int.emod_add_ediv n 2, hn1, Int.cast_add, Int.cast_one, add_mul, one_mul, add_comm, mul_comm (2 : \u2124),\n Int.cast_mul, mul_assoc, Int.cast_two] at hn \n[GOAL]\nx : \u211d\nh : cos x = 1\nn : \u2124\nhn : \u2191(n / 2) * (2 * \u03c0) + \u03c0 = x\nhn1 : n % 2 = 1\n\u22a2 \u2191(n / 2) * (2 * \u03c0) = x\n[PROOFSTEP]\nrw [\u2190 hn, cos_int_mul_two_pi_add_pi] at h \n[GOAL]\nx : \u211d\nn : \u2124\nh : -1 = 1\nhn : \u2191(n / 2) * (2 * \u03c0) + \u03c0 = x\nhn1 : n % 2 = 1\n\u22a2 \u2191(n / 2) * (2 * \u03c0) = x\n[PROOFSTEP]\nexact absurd h (by norm_num)\n[GOAL]\nx : \u211d\nn : \u2124\nh : -1 = 1\nhn : \u2191(n / 2) * (2 * \u03c0) + \u03c0 = x\nhn1 : n % 2 = 1\n\u22a2 \u00ac-1 = 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\nhx\u2081 : -(2 * \u03c0) < x\nhx\u2082 : x < 2 * \u03c0\nh : cos x = 1\n\u22a2 x = 0\n[PROOFSTEP]\nrcases(cos_eq_one_iff _).1 h with \u27e8n, rfl\u27e9\n[GOAL]\ncase intro\nn : \u2124\nhx\u2081 : -(2 * \u03c0) < \u2191n * (2 * \u03c0)\nhx\u2082 : \u2191n * (2 * \u03c0) < 2 * \u03c0\nh : cos (\u2191n * (2 * \u03c0)) = 1\n\u22a2 \u2191n * (2 * \u03c0) = 0\n[PROOFSTEP]\nrw [mul_lt_iff_lt_one_left two_pi_pos] at hx\u2082 \n[GOAL]\ncase intro\nn : \u2124\nhx\u2081 : -(2 * \u03c0) < \u2191n * (2 * \u03c0)\nhx\u2082 : \u2191n < 1\nh : cos (\u2191n * (2 * \u03c0)) = 1\n\u22a2 \u2191n * (2 * \u03c0) = 0\n[PROOFSTEP]\nrw [neg_lt, neg_mul_eq_neg_mul, mul_lt_iff_lt_one_left two_pi_pos] at hx\u2081 \n[GOAL]\ncase intro\nn : \u2124\nhx\u2081 : -\u2191n < 1\nhx\u2082 : \u2191n < 1\nh : cos (\u2191n * (2 * \u03c0)) = 1\n\u22a2 \u2191n * (2 * \u03c0) = 0\n[PROOFSTEP]\nnorm_cast at hx\u2081 hx\u2082 \n[GOAL]\ncase intro\nn : \u2124\nh : cos (\u2191n * (2 * \u03c0)) = 1\nhx\u2081 : -n < 1\nhx\u2082 : n < 1\n\u22a2 \u2191n * (2 * \u03c0) = 0\n[PROOFSTEP]\nobtain rfl : n = 0 := le_antisymm (by linarith) (by linarith)\n[GOAL]\nn : \u2124\nh : cos (\u2191n * (2 * \u03c0)) = 1\nhx\u2081 : -n < 1\nhx\u2082 : n < 1\n\u22a2 n \u2264 0\n[PROOFSTEP]\nlinarith\n[GOAL]\nn : \u2124\nh : cos (\u2191n * (2 * \u03c0)) = 1\nhx\u2081 : -n < 1\nhx\u2082 : n < 1\n\u22a2 0 \u2264 n\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase intro\nh : cos (\u21910 * (2 * \u03c0)) = 1\nhx\u2081 : -0 < 1\nhx\u2082 : 0 < 1\n\u22a2 \u21910 * (2 * \u03c0) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\nhx\u2081 : -(2 * \u03c0) < x\nhx\u2082 : x < 2 * \u03c0\nh : x = 0\n\u22a2 cos x = 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0 / 2\nhxy : x < y\n\u22a2 cos y < cos x\n[PROOFSTEP]\nrw [\u2190 sub_lt_zero, cos_sub_cos]\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0 / 2\nhxy : x < y\n\u22a2 -2 * sin ((y + x) / 2) * sin ((y - x) / 2) < 0\n[PROOFSTEP]\nhave : 0 < sin ((y + x) / 2) := by refine' sin_pos_of_pos_of_lt_pi _ _ <;> linarith\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0 / 2\nhxy : x < y\n\u22a2 0 < sin ((y + x) / 2)\n[PROOFSTEP]\nrefine' sin_pos_of_pos_of_lt_pi _ _\n[GOAL]\ncase refine'_1\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0 / 2\nhxy : x < y\n\u22a2 0 < (y + x) / 2\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine'_2\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0 / 2\nhxy : x < y\n\u22a2 (y + x) / 2 < \u03c0\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0 / 2\nhxy : x < y\nthis : 0 < sin ((y + x) / 2)\n\u22a2 -2 * sin ((y + x) / 2) * sin ((y - x) / 2) < 0\n[PROOFSTEP]\nhave : 0 < sin ((y - x) / 2) := by refine' sin_pos_of_pos_of_lt_pi _ _ <;> linarith\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0 / 2\nhxy : x < y\nthis : 0 < sin ((y + x) / 2)\n\u22a2 0 < sin ((y - x) / 2)\n[PROOFSTEP]\nrefine' sin_pos_of_pos_of_lt_pi _ _\n[GOAL]\ncase refine'_1\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0 / 2\nhxy : x < y\nthis : 0 < sin ((y + x) / 2)\n\u22a2 0 < (y - x) / 2\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine'_2\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0 / 2\nhxy : x < y\nthis : 0 < sin ((y + x) / 2)\n\u22a2 (y - x) / 2 < \u03c0\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0 / 2\nhxy : x < y\nthis\u271d : 0 < sin ((y + x) / 2)\nthis : 0 < sin ((y - x) / 2)\n\u22a2 -2 * sin ((y + x) / 2) * sin ((y - x) / 2) < 0\n[PROOFSTEP]\nnlinarith\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0\nhxy : x < y\nhx\u271d : x \u2264 \u03c0 / 2\nhy : \u03c0 / 2 \u2264 y\nhx : x < \u03c0 / 2\n\u22a2 y \u2264 \u03c0 + \u03c0 / 2\n[PROOFSTEP]\nlinarith [pi_pos]\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0\nhxy : x < y\nhx\u271d : x \u2264 \u03c0 / 2\nhy : \u03c0 / 2 \u2264 y\nhx : x < \u03c0 / 2\n\u22a2 -(\u03c0 / 2) < x\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0\nhxy : x < y\nhx\u271d : x \u2264 \u03c0 / 2\nhy : \u03c0 / 2 \u2264 y\nhx : x = \u03c0 / 2\n\u22a2 \u03c0 / 2 < y\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0\nhxy : x < y\nhx\u271d : x \u2264 \u03c0 / 2\nhy : \u03c0 / 2 \u2264 y\nhx : x = \u03c0 / 2\n\u22a2 y < \u03c0 + \u03c0 / 2\n[PROOFSTEP]\nlinarith [pi_pos]\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0\nhxy : x < y\nhx\u271d : x \u2264 \u03c0 / 2\nhy : \u03c0 / 2 \u2264 y\nhx : x = \u03c0 / 2\n\u22a2 0 = cos x\n[PROOFSTEP]\nrw [hx, cos_pi_div_two]\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0\nhxy : x < y\nhx : \u03c0 / 2 \u2264 x\nhy : y \u2264 \u03c0 / 2\n\u22a2 cos y < cos x\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0\nhxy : x < y\nhx : \u03c0 / 2 \u2264 x\nhy : \u03c0 / 2 \u2264 y\n\u22a2 -cos x < -cos y\n[PROOFSTEP]\nrw [\u2190 cos_pi_sub, \u2190 cos_pi_sub]\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0\nhxy : x < y\nhx : \u03c0 / 2 \u2264 x\nhy : \u03c0 / 2 \u2264 y\n\u22a2 cos (\u03c0 - x) < cos (\u03c0 - y)\n[PROOFSTEP]\napply cos_lt_cos_of_nonneg_of_le_pi_div_two\n[GOAL]\ncase hx\u2081\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0\nhxy : x < y\nhx : \u03c0 / 2 \u2264 x\nhy : \u03c0 / 2 \u2264 y\n\u22a2 0 \u2264 \u03c0 - y\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hy\u2082\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0\nhxy : x < y\nhx : \u03c0 / 2 \u2264 x\nhy : \u03c0 / 2 \u2264 y\n\u22a2 \u03c0 - x \u2264 \u03c0 / 2\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hxy\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y \u2264 \u03c0\nhxy : x < y\nhx : \u03c0 / 2 \u2264 x\nhy : \u03c0 / 2 \u2264 y\n\u22a2 \u03c0 - y < \u03c0 - x\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : \u211d\nhx\u2081 : -(\u03c0 / 2) \u2264 x\nhy\u2082 : y \u2264 \u03c0 / 2\nhxy : x < y\n\u22a2 sin x < sin y\n[PROOFSTEP]\nrw [\u2190 cos_sub_pi_div_two, \u2190 cos_sub_pi_div_two, \u2190 cos_neg (x - _), \u2190 cos_neg (y - _)]\n[GOAL]\nx y : \u211d\nhx\u2081 : -(\u03c0 / 2) \u2264 x\nhy\u2082 : y \u2264 \u03c0 / 2\nhxy : x < y\n\u22a2 cos (-(x - \u03c0 / 2)) < cos (-(y - \u03c0 / 2))\n[PROOFSTEP]\napply cos_lt_cos_of_nonneg_of_le_pi\n[GOAL]\ncase hx\u2081\nx y : \u211d\nhx\u2081 : -(\u03c0 / 2) \u2264 x\nhy\u2082 : y \u2264 \u03c0 / 2\nhxy : x < y\n\u22a2 0 \u2264 -(y - \u03c0 / 2)\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hy\u2082\nx y : \u211d\nhx\u2081 : -(\u03c0 / 2) \u2264 x\nhy\u2082 : y \u2264 \u03c0 / 2\nhxy : x < y\n\u22a2 -(x - \u03c0 / 2) \u2264 \u03c0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hxy\nx y : \u211d\nhx\u2081 : -(\u03c0 / 2) \u2264 x\nhy\u2082 : y \u2264 \u03c0 / 2\nhxy : x < y\n\u22a2 -(y - \u03c0 / 2) < -(x - \u03c0 / 2)\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u22a2 SurjOn sin (Icc (-(\u03c0 / 2)) (\u03c0 / 2)) (Icc (-1) 1)\n[PROOFSTEP]\nsimpa only [sin_neg, sin_pi_div_two] using\n intermediate_value_Icc (neg_le_self pi_div_two_pos.le) continuous_sin.continuousOn\n[GOAL]\n\u22a2 SurjOn cos (Icc 0 \u03c0) (Icc (-1) 1)\n[PROOFSTEP]\nsimpa only [cos_zero, cos_pi] using intermediate_value_Icc' pi_pos.le continuous_cos.continuousOn\n[GOAL]\n\u22a2 Set.Infinite (range cos)\n[PROOFSTEP]\nrw [Real.range_cos]\n[GOAL]\n\u22a2 Set.Infinite (Icc (-1) 1)\n[PROOFSTEP]\nexact Icc_infinite (by norm_num)\n[GOAL]\n\u22a2 -1 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 Set.Infinite (range sin)\n[PROOFSTEP]\nrw [Real.range_sin]\n[GOAL]\n\u22a2 Set.Infinite (Icc (-1) 1)\n[PROOFSTEP]\nexact Icc_infinite (by norm_num)\n[GOAL]\n\u22a2 -1 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\n\u22a2 sqrtTwoAddSeries x 0 = x\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\n\u22a2 sqrtTwoAddSeries 0 1 = sqrt 2\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\n\u22a2 sqrtTwoAddSeries 0 2 = sqrt (2 + sqrt 2)\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\n\u22a2 sqrtTwoAddSeries 0 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 sqrtTwoAddSeries 0 (n + 1) < 2\n[PROOFSTEP]\nrefine' lt_of_lt_of_le _ (sqrt_sq zero_lt_two.le).le\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 sqrtTwoAddSeries 0 (n + 1) < sqrt (2 ^ 2)\n[PROOFSTEP]\nrw [sqrtTwoAddSeries, sqrt_lt_sqrt_iff, \u2190 lt_sub_iff_add_lt']\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 sqrtTwoAddSeries 0 n < 2 ^ 2 - 2\n[PROOFSTEP]\nrefine' (sqrtTwoAddSeries_lt_two n).trans_le _\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 2 \u2264 2 ^ 2 - 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 0 \u2264 2 + sqrtTwoAddSeries 0 n\n[PROOFSTEP]\nexact add_nonneg zero_le_two (sqrtTwoAddSeries_zero_nonneg n)\n[GOAL]\nx\u271d x : \u211d\nn : \u2115\n\u22a2 sqrtTwoAddSeries x (n + 1 + 1) = sqrtTwoAddSeries (sqrt (2 + x)) (n + 1)\n[PROOFSTEP]\nrw [sqrtTwoAddSeries, sqrtTwoAddSeries_succ _ _, sqrtTwoAddSeries]\n[GOAL]\nx\u271d x y : \u211d\nh : x \u2264 y\nn : \u2115\n\u22a2 sqrtTwoAddSeries x (n + 1) \u2264 sqrtTwoAddSeries y (n + 1)\n[PROOFSTEP]\nrw [sqrtTwoAddSeries, sqrtTwoAddSeries]\n[GOAL]\nx\u271d x y : \u211d\nh : x \u2264 y\nn : \u2115\n\u22a2 sqrt (2 + sqrtTwoAddSeries x n) \u2264 sqrt (2 + sqrtTwoAddSeries y n)\n[PROOFSTEP]\nexact sqrt_le_sqrt (add_le_add_left (sqrtTwoAddSeries_monotone_left h _) _)\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 2 ^ (0 + 1)) = sqrtTwoAddSeries 0 0 / 2\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 cos (\u03c0 / 2 ^ (n + 1 + 1)) = sqrtTwoAddSeries 0 (n + 1) / 2\n[PROOFSTEP]\nhave : (2 : \u211d) \u2260 0 := two_ne_zero\n[GOAL]\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 cos (\u03c0 / 2 ^ (n + 1 + 1)) = sqrtTwoAddSeries 0 (n + 1) / 2\n[PROOFSTEP]\nsymm\n[GOAL]\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 sqrtTwoAddSeries 0 (n + 1) / 2 = cos (\u03c0 / 2 ^ (n + 1 + 1))\n[PROOFSTEP]\nrw [div_eq_iff_mul_eq this]\n[GOAL]\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 cos (\u03c0 / 2 ^ (n + 1 + 1)) * 2 = sqrtTwoAddSeries 0 (n + 1)\n[PROOFSTEP]\nsymm\n[GOAL]\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 sqrtTwoAddSeries 0 (n + 1) = cos (\u03c0 / 2 ^ (n + 1 + 1)) * 2\n[PROOFSTEP]\nrw [sqrtTwoAddSeries, sqrt_eq_iff_sq_eq, mul_pow, cos_sq, \u2190 mul_div_assoc, Nat.add_succ, pow_succ,\n mul_div_mul_left _ _ this, cos_pi_over_two_pow _, add_mul]\n[GOAL]\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 1 / 2 * 2 ^ 2 + sqrtTwoAddSeries 0 n / 2 / 2 * 2 ^ 2 = 2 + sqrtTwoAddSeries 0 n\ncase hx\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 0 \u2264 2 + sqrtTwoAddSeries 0 n\ncase hy x : \u211d n : \u2115 this : 2 \u2260 0 \u22a2 0 \u2264 cos (\u03c0 / 2 ^ (n + 1 + 1)) * 2\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 1 / 2 * 2 ^ 2 = 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase e_a\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 sqrtTwoAddSeries 0 n / 2 / 2 * 2 ^ 2 = sqrtTwoAddSeries 0 n\n[PROOFSTEP]\nrw [mul_comm, sq, mul_assoc, \u2190 mul_div_assoc, mul_div_cancel_left, \u2190 mul_div_assoc, mul_div_cancel_left]\n[GOAL]\ncase e_a.ha\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 2 \u2260 0\n[PROOFSTEP]\ntry exact this\n[GOAL]\ncase e_a.ha\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nexact this\n[GOAL]\ncase e_a.ha\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 2 \u2260 0\n[PROOFSTEP]\ntry exact this\n[GOAL]\ncase e_a.ha\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nexact this\n[GOAL]\ncase hx\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 0 \u2264 2 + sqrtTwoAddSeries 0 n\ncase hy x : \u211d n : \u2115 this : 2 \u2260 0 \u22a2 0 \u2264 cos (\u03c0 / 2 ^ (n + 1 + 1)) * 2\n[PROOFSTEP]\napply add_nonneg\n[GOAL]\ncase hx.ha\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 0 \u2264 2\ncase hx.hb\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 0 \u2264 sqrtTwoAddSeries 0 n\ncase hy x : \u211d n : \u2115 this : 2 \u2260 0 \u22a2 0 \u2264 cos (\u03c0 / 2 ^ (n + 1 + 1)) * 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase hx.hb\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 0 \u2264 sqrtTwoAddSeries 0 n\ncase hy x : \u211d n : \u2115 this : 2 \u2260 0 \u22a2 0 \u2264 cos (\u03c0 / 2 ^ (n + 1 + 1)) * 2\n[PROOFSTEP]\napply sqrtTwoAddSeries_zero_nonneg\n[GOAL]\ncase hy\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 0 \u2264 cos (\u03c0 / 2 ^ (n + 1 + 1)) * 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase hy\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 0 \u2264 cos (\u03c0 / 2 ^ (n + 1 + 1))\n[PROOFSTEP]\napply le_of_lt\n[GOAL]\ncase hy.a\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 0 < cos (\u03c0 / 2 ^ (n + 1 + 1))\n[PROOFSTEP]\napply cos_pos_of_mem_Ioo \u27e8_, _\u27e9\n[GOAL]\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 -(\u03c0 / 2) < \u03c0 / 2 ^ (n + 1 + 1)\n[PROOFSTEP]\ntrans (0 : \u211d)\n[GOAL]\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 -(\u03c0 / 2) < 0\nx : \u211d n : \u2115 this : 2 \u2260 0 \u22a2 0 < \u03c0 / 2 ^ (n + 1 + 1)\n[PROOFSTEP]\nrw [neg_lt_zero]\n[GOAL]\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 0 < \u03c0 / 2\nx : \u211d n : \u2115 this : 2 \u2260 0 \u22a2 0 < \u03c0 / 2 ^ (n + 1 + 1)\n[PROOFSTEP]\napply pi_div_two_pos\n[GOAL]\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 0 < \u03c0 / 2 ^ (n + 1 + 1)\n[PROOFSTEP]\napply div_pos pi_pos\n[GOAL]\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 0 < 2 ^ (n + 1 + 1)\n[PROOFSTEP]\napply pow_pos\n[GOAL]\ncase H\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 \u03c0 / 2 ^ (n + 1 + 1) < \u03c0 / 2\n[PROOFSTEP]\napply div_lt_div' (le_refl \u03c0) _ pi_pos _\n[GOAL]\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 2 < 2 ^ (n + 1 + 1)\nx : \u211d n : \u2115 this : 2 \u2260 0 \u22a2 0 < 2\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (le_of_eq (pow_one _).symm) _\n[GOAL]\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 2 ^ 1 < 2 ^ (n + 1 + 1)\nx : \u211d n : \u2115 this : 2 \u2260 0 \u22a2 0 < 2\n[PROOFSTEP]\napply pow_lt_pow\n[GOAL]\ncase h\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 1 < 2\ncase h2\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 1 < n + 1 + 1\nx : \u211d n : \u2115 this : 2 \u2260 0 \u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h2\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 1 < n + 1 + 1\nx : \u211d n : \u2115 this : 2 \u2260 0 \u22a2 0 < 2\n[PROOFSTEP]\napply Nat.succ_lt_succ\n[GOAL]\ncase h2.a\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 0 < n + 1\nx : \u211d n : \u2115 this : 2 \u2260 0 \u22a2 0 < 2\n[PROOFSTEP]\napply Nat.succ_pos\n[GOAL]\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 0 < 2\n[PROOFSTEP]\nall_goals norm_num\n[GOAL]\nx : \u211d\nn : \u2115\nthis : 2 \u2260 0\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 sin (\u03c0 / 2 ^ (n + 1)) ^ 2 = 1 - (sqrtTwoAddSeries 0 n / 2) ^ 2\n[PROOFSTEP]\nrw [sin_sq, cos_pi_over_two_pow]\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 sin (\u03c0 / 2 ^ (n + 2)) ^ 2 = 1 / 2 - sqrtTwoAddSeries 0 n / 4\n[PROOFSTEP]\nrw [sin_sq_pi_over_two_pow, sqrtTwoAddSeries, div_pow, sq_sqrt, add_div, \u2190 sub_sub]\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 1 - 2 / 2 ^ 2 - sqrtTwoAddSeries 0 n / 2 ^ 2 = 1 / 2 - sqrtTwoAddSeries 0 n / 4\nx : \u211d n : \u2115 \u22a2 0 \u2264 2 + sqrtTwoAddSeries 0 n\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nx : \u211d\nn : \u2115\n\u22a2 1 - 2 / 2 ^ 2 = 1 / 2\ncase e_a.e_a\nx : \u211d\nn : \u2115\n\u22a2 2 ^ 2 = 4\nx : \u211d n : \u2115 \u22a2 0 \u2264 2 + sqrtTwoAddSeries 0 n\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase e_a.e_a\nx : \u211d\nn : \u2115\n\u22a2 2 ^ 2 = 4\nx : \u211d n : \u2115 \u22a2 0 \u2264 2 + sqrtTwoAddSeries 0 n\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 0 \u2264 2 + sqrtTwoAddSeries 0 n\n[PROOFSTEP]\napply add_nonneg\n[GOAL]\ncase ha\nx : \u211d\nn : \u2115\n\u22a2 0 \u2264 2\ncase hb x : \u211d n : \u2115 \u22a2 0 \u2264 sqrtTwoAddSeries 0 n\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase hb\nx : \u211d\nn : \u2115\n\u22a2 0 \u2264 sqrtTwoAddSeries 0 n\n[PROOFSTEP]\napply sqrtTwoAddSeries_zero_nonneg\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 sin (\u03c0 / 2 ^ (n + 2)) = sqrt (2 - sqrtTwoAddSeries 0 n) / 2\n[PROOFSTEP]\nsymm\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 sqrt (2 - sqrtTwoAddSeries 0 n) / 2 = sin (\u03c0 / 2 ^ (n + 2))\n[PROOFSTEP]\nrw [div_eq_iff_mul_eq]\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 sin (\u03c0 / 2 ^ (n + 2)) * 2 = sqrt (2 - sqrtTwoAddSeries 0 n)\nx : \u211d n : \u2115 \u22a2 2 \u2260 0\n[PROOFSTEP]\nsymm\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 sqrt (2 - sqrtTwoAddSeries 0 n) = sin (\u03c0 / 2 ^ (n + 2)) * 2\nx : \u211d n : \u2115 \u22a2 2 \u2260 0\n[PROOFSTEP]\nrw [sqrt_eq_iff_sq_eq, mul_pow, sin_sq_pi_over_two_pow_succ, sub_mul]\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 1 / 2 * 2 ^ 2 - sqrtTwoAddSeries 0 n / 4 * 2 ^ 2 = 2 - sqrtTwoAddSeries 0 n\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nx : \u211d\nn : \u2115\n\u22a2 1 / 2 * 2 ^ 2 = 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase e_a\nx : \u211d\nn : \u2115\n\u22a2 sqrtTwoAddSeries 0 n / 4 * 2 ^ 2 = sqrtTwoAddSeries 0 n\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase hx\nx : \u211d\nn : \u2115\n\u22a2 0 \u2264 2 - sqrtTwoAddSeries 0 n\n[PROOFSTEP]\nrw [sub_nonneg]\n[GOAL]\ncase hx\nx : \u211d\nn : \u2115\n\u22a2 sqrtTwoAddSeries 0 n \u2264 2\n[PROOFSTEP]\napply le_of_lt\n[GOAL]\ncase hx.a\nx : \u211d\nn : \u2115\n\u22a2 sqrtTwoAddSeries 0 n < 2\n[PROOFSTEP]\napply sqrtTwoAddSeries_lt_two\n[GOAL]\ncase hy\nx : \u211d\nn : \u2115\n\u22a2 0 \u2264 sin (\u03c0 / 2 ^ (n + 2)) * 2\nx : \u211d n : \u2115 \u22a2 2 \u2260 0\n[PROOFSTEP]\napply le_of_lt\n[GOAL]\ncase hy.a\nx : \u211d\nn : \u2115\n\u22a2 0 < sin (\u03c0 / 2 ^ (n + 2)) * 2\nx : \u211d n : \u2115 \u22a2 2 \u2260 0\n[PROOFSTEP]\napply mul_pos\n[GOAL]\ncase hy.a.ha\nx : \u211d\nn : \u2115\n\u22a2 0 < sin (\u03c0 / 2 ^ (n + 2))\ncase hy.a.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2\nx : \u211d n : \u2115 \u22a2 2 \u2260 0\n[PROOFSTEP]\napply sin_pos_of_pos_of_lt_pi\n[GOAL]\ncase hy.a.ha.h0x\nx : \u211d\nn : \u2115\n\u22a2 0 < \u03c0 / 2 ^ (n + 2)\n[PROOFSTEP]\napply div_pos pi_pos\n[GOAL]\ncase hy.a.ha.h0x\nx : \u211d\nn : \u2115\n\u22a2 0 < 2 ^ (n + 2)\n[PROOFSTEP]\napply pow_pos\n[GOAL]\ncase hy.a.ha.h0x.H\nx : \u211d\nn : \u2115\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase hy.a.ha.hxp\nx : \u211d\nn : \u2115\n\u22a2 \u03c0 / 2 ^ (n + 2) < \u03c0\ncase hy.a.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2\nx : \u211d n : \u2115 \u22a2 2 \u2260 0\n[PROOFSTEP]\nrefine' lt_of_lt_of_le _ (le_of_eq (div_one _))\n[GOAL]\ncase hy.a.ha.hxp\nx : \u211d\nn : \u2115\n\u22a2 \u03c0 / 2 ^ (n + 2) < \u03c0 / 1\ncase hy.a.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2\nx : \u211d n : \u2115 \u22a2 2 \u2260 0\n[PROOFSTEP]\nrw [div_lt_div_left]\n[GOAL]\ncase hy.a.ha.hxp\nx : \u211d\nn : \u2115\n\u22a2 1 < 2 ^ (n + 2)\ncase hy.a.ha.hxp.ha\nx : \u211d\nn : \u2115\n\u22a2 0 < \u03c0\ncase hy.a.ha.hxp.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2 ^ (n + 2)\ncase hy.a.ha.hxp.hc\nx : \u211d\nn : \u2115\n\u22a2 0 < 1\ncase hy.a.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2\nx : \u211d n : \u2115 \u22a2 2 \u2260 0\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (le_of_eq (pow_zero 2).symm) _\n[GOAL]\ncase hy.a.ha.hxp\nx : \u211d\nn : \u2115\n\u22a2 2 ^ 0 < 2 ^ (n + 2)\ncase hy.a.ha.hxp.ha\nx : \u211d\nn : \u2115\n\u22a2 0 < \u03c0\ncase hy.a.ha.hxp.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2 ^ (n + 2)\ncase hy.a.ha.hxp.hc\nx : \u211d\nn : \u2115\n\u22a2 0 < 1\ncase hy.a.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2\nx : \u211d n : \u2115 \u22a2 2 \u2260 0\n[PROOFSTEP]\napply pow_lt_pow\n[GOAL]\ncase hy.a.ha.hxp.h\nx : \u211d\nn : \u2115\n\u22a2 1 < 2\ncase hy.a.ha.hxp.h2\nx : \u211d\nn : \u2115\n\u22a2 0 < n + 2\ncase hy.a.ha.hxp.ha\nx : \u211d\nn : \u2115\n\u22a2 0 < \u03c0\ncase hy.a.ha.hxp.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2 ^ (n + 2)\ncase hy.a.ha.hxp.hc\nx : \u211d\nn : \u2115\n\u22a2 0 < 1\ncase hy.a.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2\nx : \u211d n : \u2115 \u22a2 2 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase hy.a.ha.hxp.h2\nx : \u211d\nn : \u2115\n\u22a2 0 < n + 2\ncase hy.a.ha.hxp.ha\nx : \u211d\nn : \u2115\n\u22a2 0 < \u03c0\ncase hy.a.ha.hxp.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2 ^ (n + 2)\ncase hy.a.ha.hxp.hc\nx : \u211d\nn : \u2115\n\u22a2 0 < 1\ncase hy.a.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2\nx : \u211d n : \u2115 \u22a2 2 \u2260 0\n[PROOFSTEP]\napply Nat.succ_pos\n[GOAL]\ncase hy.a.ha.hxp.ha\nx : \u211d\nn : \u2115\n\u22a2 0 < \u03c0\ncase hy.a.ha.hxp.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2 ^ (n + 2)\ncase hy.a.ha.hxp.hc\nx : \u211d\nn : \u2115\n\u22a2 0 < 1\ncase hy.a.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2\nx : \u211d n : \u2115 \u22a2 2 \u2260 0\n[PROOFSTEP]\napply pi_pos\n[GOAL]\ncase hy.a.ha.hxp.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2 ^ (n + 2)\ncase hy.a.ha.hxp.hc\nx : \u211d\nn : \u2115\n\u22a2 0 < 1\ncase hy.a.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2\nx : \u211d n : \u2115 \u22a2 2 \u2260 0\n[PROOFSTEP]\napply pow_pos\n[GOAL]\ncase hy.a.ha.hxp.hb.H\nx : \u211d\nn : \u2115\n\u22a2 0 < 2\ncase hy.a.ha.hxp.hc\nx : \u211d\nn : \u2115\n\u22a2 0 < 1\ncase hy.a.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2\nx : \u211d n : \u2115 \u22a2 2 \u2260 0\n[PROOFSTEP]\nall_goals norm_num\n[GOAL]\ncase hy.a.ha.hxp.hb.H\nx : \u211d\nn : \u2115\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase hy.a.ha.hxp.hc\nx : \u211d\nn : \u2115\n\u22a2 0 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase hy.a.hb\nx : \u211d\nn : \u2115\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\nn : \u2115\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 4) = sqrt 2 / 2\n[PROOFSTEP]\ntrans cos (\u03c0 / 2 ^ 2)\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 4) = cos (\u03c0 / 2 ^ 2)\nx : \u211d \u22a2 cos (\u03c0 / 2 ^ 2) = sqrt 2 / 2\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_x.e_a\nx : \u211d\n\u22a2 4 = 2 ^ 2\nx : \u211d \u22a2 cos (\u03c0 / 2 ^ 2) = sqrt 2 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 2 ^ 2) = sqrt 2 / 2\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 4) = sqrt 2 / 2\n[PROOFSTEP]\ntrans sin (\u03c0 / 2 ^ 2)\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 4) = sin (\u03c0 / 2 ^ 2)\nx : \u211d \u22a2 sin (\u03c0 / 2 ^ 2) = sqrt 2 / 2\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_x.e_a\nx : \u211d\n\u22a2 4 = 2 ^ 2\nx : \u211d \u22a2 sin (\u03c0 / 2 ^ 2) = sqrt 2 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 2 ^ 2) = sqrt 2 / 2\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 8) = sqrt (2 + sqrt 2) / 2\n[PROOFSTEP]\ntrans cos (\u03c0 / 2 ^ 3)\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 8) = cos (\u03c0 / 2 ^ 3)\nx : \u211d \u22a2 cos (\u03c0 / 2 ^ 3) = sqrt (2 + sqrt 2) / 2\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_x.e_a\nx : \u211d\n\u22a2 8 = 2 ^ 3\nx : \u211d \u22a2 cos (\u03c0 / 2 ^ 3) = sqrt (2 + sqrt 2) / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 2 ^ 3) = sqrt (2 + sqrt 2) / 2\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 8) = sqrt (2 - sqrt 2) / 2\n[PROOFSTEP]\ntrans sin (\u03c0 / 2 ^ 3)\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 8) = sin (\u03c0 / 2 ^ 3)\nx : \u211d \u22a2 sin (\u03c0 / 2 ^ 3) = sqrt (2 - sqrt 2) / 2\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_x.e_a\nx : \u211d\n\u22a2 8 = 2 ^ 3\nx : \u211d \u22a2 sin (\u03c0 / 2 ^ 3) = sqrt (2 - sqrt 2) / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 2 ^ 3) = sqrt (2 - sqrt 2) / 2\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 16) = sqrt (2 + sqrt (2 + sqrt 2)) / 2\n[PROOFSTEP]\ntrans cos (\u03c0 / 2 ^ 4)\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 16) = cos (\u03c0 / 2 ^ 4)\nx : \u211d \u22a2 cos (\u03c0 / 2 ^ 4) = sqrt (2 + sqrt (2 + sqrt 2)) / 2\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_x.e_a\nx : \u211d\n\u22a2 16 = 2 ^ 4\nx : \u211d \u22a2 cos (\u03c0 / 2 ^ 4) = sqrt (2 + sqrt (2 + sqrt 2)) / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 2 ^ 4) = sqrt (2 + sqrt (2 + sqrt 2)) / 2\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 16) = sqrt (2 - sqrt (2 + sqrt 2)) / 2\n[PROOFSTEP]\ntrans sin (\u03c0 / 2 ^ 4)\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 16) = sin (\u03c0 / 2 ^ 4)\nx : \u211d \u22a2 sin (\u03c0 / 2 ^ 4) = sqrt (2 - sqrt (2 + sqrt 2)) / 2\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_x.e_a\nx : \u211d\n\u22a2 16 = 2 ^ 4\nx : \u211d \u22a2 sin (\u03c0 / 2 ^ 4) = sqrt (2 - sqrt (2 + sqrt 2)) / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 2 ^ 4) = sqrt (2 - sqrt (2 + sqrt 2)) / 2\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 32) = sqrt (2 + sqrt (2 + sqrt (2 + sqrt 2))) / 2\n[PROOFSTEP]\ntrans cos (\u03c0 / 2 ^ 5)\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 32) = cos (\u03c0 / 2 ^ 5)\nx : \u211d \u22a2 cos (\u03c0 / 2 ^ 5) = sqrt (2 + sqrt (2 + sqrt (2 + sqrt 2))) / 2\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_x.e_a\nx : \u211d\n\u22a2 32 = 2 ^ 5\nx : \u211d \u22a2 cos (\u03c0 / 2 ^ 5) = sqrt (2 + sqrt (2 + sqrt (2 + sqrt 2))) / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 2 ^ 5) = sqrt (2 + sqrt (2 + sqrt (2 + sqrt 2))) / 2\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 32) = sqrt (2 - sqrt (2 + sqrt (2 + sqrt 2))) / 2\n[PROOFSTEP]\ntrans sin (\u03c0 / 2 ^ 5)\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 32) = sin (\u03c0 / 2 ^ 5)\nx : \u211d \u22a2 sin (\u03c0 / 2 ^ 5) = sqrt (2 - sqrt (2 + sqrt (2 + sqrt 2))) / 2\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_x.e_a\nx : \u211d\n\u22a2 32 = 2 ^ 5\nx : \u211d \u22a2 sin (\u03c0 / 2 ^ 5) = sqrt (2 - sqrt (2 + sqrt (2 + sqrt 2))) / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 2 ^ 5) = sqrt (2 - sqrt (2 + sqrt (2 + sqrt 2))) / 2\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 3) = 1 / 2\n[PROOFSTEP]\nhave h\u2081 : (2 * cos (\u03c0 / 3) - 1) ^ 2 * (2 * cos (\u03c0 / 3) + 2) = 0 :=\n by\n have : cos (3 * (\u03c0 / 3)) = cos \u03c0 := by\n congr 1\n ring\n linarith [cos_pi, cos_three_mul (\u03c0 / 3)]\n[GOAL]\nx : \u211d\n\u22a2 (2 * cos (\u03c0 / 3) - 1) ^ 2 * (2 * cos (\u03c0 / 3) + 2) = 0\n[PROOFSTEP]\nhave : cos (3 * (\u03c0 / 3)) = cos \u03c0 := by\n congr 1\n ring\n[GOAL]\nx : \u211d\n\u22a2 cos (3 * (\u03c0 / 3)) = cos \u03c0\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_x\nx : \u211d\n\u22a2 3 * (\u03c0 / 3) = \u03c0\n[PROOFSTEP]\nring\n[GOAL]\nx : \u211d\nthis : cos (3 * (\u03c0 / 3)) = cos \u03c0\n\u22a2 (2 * cos (\u03c0 / 3) - 1) ^ 2 * (2 * cos (\u03c0 / 3) + 2) = 0\n[PROOFSTEP]\nlinarith [cos_pi, cos_three_mul (\u03c0 / 3)]\n[GOAL]\nx : \u211d\nh\u2081 : (2 * cos (\u03c0 / 3) - 1) ^ 2 * (2 * cos (\u03c0 / 3) + 2) = 0\n\u22a2 cos (\u03c0 / 3) = 1 / 2\n[PROOFSTEP]\ncases' mul_eq_zero.mp h\u2081 with h h\n[GOAL]\ncase inl\nx : \u211d\nh\u2081 : (2 * cos (\u03c0 / 3) - 1) ^ 2 * (2 * cos (\u03c0 / 3) + 2) = 0\nh : (2 * cos (\u03c0 / 3) - 1) ^ 2 = 0\n\u22a2 cos (\u03c0 / 3) = 1 / 2\n[PROOFSTEP]\nlinarith [pow_eq_zero h]\n[GOAL]\ncase inr\nx : \u211d\nh\u2081 : (2 * cos (\u03c0 / 3) - 1) ^ 2 * (2 * cos (\u03c0 / 3) + 2) = 0\nh : 2 * cos (\u03c0 / 3) + 2 = 0\n\u22a2 cos (\u03c0 / 3) = 1 / 2\n[PROOFSTEP]\nhave : cos \u03c0 < cos (\u03c0 / 3) := by refine' cos_lt_cos_of_nonneg_of_le_pi _ rfl.ge _ <;> linarith [pi_pos]\n[GOAL]\nx : \u211d\nh\u2081 : (2 * cos (\u03c0 / 3) - 1) ^ 2 * (2 * cos (\u03c0 / 3) + 2) = 0\nh : 2 * cos (\u03c0 / 3) + 2 = 0\n\u22a2 cos \u03c0 < cos (\u03c0 / 3)\n[PROOFSTEP]\nrefine' cos_lt_cos_of_nonneg_of_le_pi _ rfl.ge _\n[GOAL]\ncase refine'_1\nx : \u211d\nh\u2081 : (2 * cos (\u03c0 / 3) - 1) ^ 2 * (2 * cos (\u03c0 / 3) + 2) = 0\nh : 2 * cos (\u03c0 / 3) + 2 = 0\n\u22a2 0 \u2264 \u03c0 / 3\n[PROOFSTEP]\nlinarith [pi_pos]\n[GOAL]\ncase refine'_2\nx : \u211d\nh\u2081 : (2 * cos (\u03c0 / 3) - 1) ^ 2 * (2 * cos (\u03c0 / 3) + 2) = 0\nh : 2 * cos (\u03c0 / 3) + 2 = 0\n\u22a2 \u03c0 / 3 < \u03c0\n[PROOFSTEP]\nlinarith [pi_pos]\n[GOAL]\ncase inr\nx : \u211d\nh\u2081 : (2 * cos (\u03c0 / 3) - 1) ^ 2 * (2 * cos (\u03c0 / 3) + 2) = 0\nh : 2 * cos (\u03c0 / 3) + 2 = 0\nthis : cos \u03c0 < cos (\u03c0 / 3)\n\u22a2 cos (\u03c0 / 3) = 1 / 2\n[PROOFSTEP]\nlinarith [cos_pi]\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 6) ^ 2 = 3 / 4\n[PROOFSTEP]\nhave h1 : cos (\u03c0 / 6) ^ 2 = 1 / 2 + 1 / 2 / 2 :=\n by\n convert cos_sq (\u03c0 / 6) using 3\n have h2 : 2 * (\u03c0 / 6) = \u03c0 / 3 := by linarith\n rw [h2, cos_pi_div_three]\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 6) ^ 2 = 1 / 2 + 1 / 2 / 2\n[PROOFSTEP]\nconvert cos_sq (\u03c0 / 6) using 3\n[GOAL]\ncase h.e'_3.h.e'_6.h.e'_5\nx : \u211d\n\u22a2 1 / 2 = cos (2 * (\u03c0 / 6))\n[PROOFSTEP]\nhave h2 : 2 * (\u03c0 / 6) = \u03c0 / 3 := by linarith\n[GOAL]\nx : \u211d\n\u22a2 2 * (\u03c0 / 6) = \u03c0 / 3\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h.e'_3.h.e'_6.h.e'_5\nx : \u211d\nh2 : 2 * (\u03c0 / 6) = \u03c0 / 3\n\u22a2 1 / 2 = cos (2 * (\u03c0 / 6))\n[PROOFSTEP]\nrw [h2, cos_pi_div_three]\n[GOAL]\nx : \u211d\nh1 : cos (\u03c0 / 6) ^ 2 = 1 / 2 + 1 / 2 / 2\n\u22a2 cos (\u03c0 / 6) ^ 2 = 3 / 4\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero] at h1 \u22a2\n[GOAL]\nx : \u211d\nh1\u271d : cos (\u03c0 / 6) ^ 2 = 1 / 2 + 1 / 2 / 2\nh1 : cos (\u03c0 / 6) ^ 2 - (1 / 2 + 1 / 2 / 2) = 0\n\u22a2 cos (\u03c0 / 6) ^ 2 - 3 / 4 = 0\n[PROOFSTEP]\nconvert h1 using 1\n[GOAL]\ncase h.e'_2\nx : \u211d\nh1\u271d : cos (\u03c0 / 6) ^ 2 = 1 / 2 + 1 / 2 / 2\nh1 : cos (\u03c0 / 6) ^ 2 - (1 / 2 + 1 / 2 / 2) = 0\n\u22a2 cos (\u03c0 / 6) ^ 2 - 3 / 4 = cos (\u03c0 / 6) ^ 2 - (1 / 2 + 1 / 2 / 2)\n[PROOFSTEP]\nring\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 6) = sqrt 3 / 2\n[PROOFSTEP]\nsuffices sqrt 3 = cos (\u03c0 / 6) * 2 by\n field_simp [(by norm_num : 0 \u2260 2)]\n exact this.symm\n[GOAL]\nx : \u211d\nthis : sqrt 3 = cos (\u03c0 / 6) * 2\n\u22a2 cos (\u03c0 / 6) = sqrt 3 / 2\n[PROOFSTEP]\nfield_simp [(by norm_num : 0 \u2260 2)]\n[GOAL]\nx : \u211d\nthis : sqrt 3 = cos (\u03c0 / 6) * 2\n\u22a2 0 \u2260 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\nthis : sqrt 3 = cos (\u03c0 / 6) * 2\n\u22a2 cos (\u03c0 / 6) * 2 = sqrt 3\n[PROOFSTEP]\nexact this.symm\n[GOAL]\nx : \u211d\n\u22a2 sqrt 3 = cos (\u03c0 / 6) * 2\n[PROOFSTEP]\nrw [sqrt_eq_iff_sq_eq]\n[GOAL]\nx : \u211d\n\u22a2 (cos (\u03c0 / 6) * 2) ^ 2 = 3\n[PROOFSTEP]\nhave h1 := (mul_right_inj' (by norm_num : (4 : \u211d) \u2260 0)).mpr sq_cos_pi_div_six\n[GOAL]\nx : \u211d\n\u22a2 4 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u211d\nh1 : 4 * cos (\u03c0 / 6) ^ 2 = 4 * (3 / 4)\n\u22a2 (cos (\u03c0 / 6) * 2) ^ 2 = 3\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero] at h1 \u22a2\n[GOAL]\nx : \u211d\nh1\u271d : 4 * cos (\u03c0 / 6) ^ 2 = 4 * (3 / 4)\nh1 : 4 * cos (\u03c0 / 6) ^ 2 - 4 * (3 / 4) = 0\n\u22a2 (cos (\u03c0 / 6) * 2) ^ 2 - 3 = 0\n[PROOFSTEP]\nconvert h1 using 1\n[GOAL]\ncase h.e'_2\nx : \u211d\nh1\u271d : 4 * cos (\u03c0 / 6) ^ 2 = 4 * (3 / 4)\nh1 : 4 * cos (\u03c0 / 6) ^ 2 - 4 * (3 / 4) = 0\n\u22a2 (cos (\u03c0 / 6) * 2) ^ 2 - 3 = 4 * cos (\u03c0 / 6) ^ 2 - 4 * (3 / 4)\n[PROOFSTEP]\nring\n[GOAL]\ncase hx\nx : \u211d\n\u22a2 0 \u2264 3\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase hy\nx : \u211d\n\u22a2 0 \u2264 cos (\u03c0 / 6) * 2\n[PROOFSTEP]\nhave : 0 < cos (\u03c0 / 6) := by apply cos_pos_of_mem_Ioo; constructor <;> linarith [pi_pos]\n[GOAL]\nx : \u211d\n\u22a2 0 < cos (\u03c0 / 6)\n[PROOFSTEP]\napply cos_pos_of_mem_Ioo\n[GOAL]\ncase hx\nx : \u211d\n\u22a2 \u03c0 / 6 \u2208 Ioo (-(\u03c0 / 2)) (\u03c0 / 2)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase hx.left\nx : \u211d\n\u22a2 -(\u03c0 / 2) < \u03c0 / 6\n[PROOFSTEP]\nlinarith [pi_pos]\n[GOAL]\ncase hx.right\nx : \u211d\n\u22a2 \u03c0 / 6 < \u03c0 / 2\n[PROOFSTEP]\nlinarith [pi_pos]\n[GOAL]\ncase hy\nx : \u211d\nthis : 0 < cos (\u03c0 / 6)\n\u22a2 0 \u2264 cos (\u03c0 / 6) * 2\n[PROOFSTEP]\nlinarith\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 6) = 1 / 2\n[PROOFSTEP]\nrw [\u2190 cos_pi_div_two_sub, \u2190 cos_pi_div_three]\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 2 - \u03c0 / 6) = cos (\u03c0 / 3)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_x\nx : \u211d\n\u22a2 \u03c0 / 2 - \u03c0 / 6 = \u03c0 / 3\n[PROOFSTEP]\nring\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 3) ^ 2 = 3 / 4\n[PROOFSTEP]\nrw [\u2190 cos_pi_div_two_sub, \u2190 sq_cos_pi_div_six]\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 2 - \u03c0 / 3) ^ 2 = cos (\u03c0 / 6) ^ 2\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_x\nx : \u211d\n\u22a2 \u03c0 / 2 - \u03c0 / 3 = \u03c0 / 6\n[PROOFSTEP]\nring\n[GOAL]\nx : \u211d\n\u22a2 sin (\u03c0 / 3) = sqrt 3 / 2\n[PROOFSTEP]\nrw [\u2190 cos_pi_div_two_sub, \u2190 cos_pi_div_six]\n[GOAL]\nx : \u211d\n\u22a2 cos (\u03c0 / 2 - \u03c0 / 3) = cos (\u03c0 / 6)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_x\nx : \u211d\n\u22a2 \u03c0 / 2 - \u03c0 / 3 = \u03c0 / 6\n[PROOFSTEP]\nring\n[GOAL]\n\u22a2 tan (\u03c0 / 4) = 1\n[PROOFSTEP]\nrw [tan_eq_sin_div_cos, cos_pi_div_four, sin_pi_div_four]\n[GOAL]\n\u22a2 sqrt 2 / 2 / (sqrt 2 / 2) = 1\n[PROOFSTEP]\nhave h : sqrt 2 / 2 > 0 := by cancel_denoms\n[GOAL]\n\u22a2 sqrt 2 / 2 > 0\n[PROOFSTEP]\ncancel_denoms\n[GOAL]\n\u22a2 sqrt 2 / 2 > 0\n[PROOFSTEP]\ncancel_denoms\n[GOAL]\nh : sqrt 2 / 2 > 0\n\u22a2 sqrt 2 / 2 / (sqrt 2 / 2) = 1\n[PROOFSTEP]\nexact div_self (ne_of_gt h)\n[GOAL]\n\u22a2 tan (\u03c0 / 2) = 0\n[PROOFSTEP]\nsimp [tan_eq_sin_div_cos]\n[GOAL]\nx : \u211d\nh0x : 0 < x\nhxp : x < \u03c0 / 2\n\u22a2 0 < tan x\n[PROOFSTEP]\nrw [tan_eq_sin_div_cos]\n[GOAL]\nx : \u211d\nh0x : 0 < x\nhxp : x < \u03c0 / 2\n\u22a2 0 < sin x / cos x\n[PROOFSTEP]\nexact div_pos (sin_pos_of_pos_of_lt_pi h0x (by linarith)) (cos_pos_of_mem_Ioo \u27e8by linarith, hxp\u27e9)\n[GOAL]\nx : \u211d\nh0x : 0 < x\nhxp : x < \u03c0 / 2\n\u22a2 x < \u03c0\n[PROOFSTEP]\nlinarith\n[GOAL]\nx : \u211d\nh0x : 0 < x\nhxp : x < \u03c0 / 2\n\u22a2 -(\u03c0 / 2) < x\n[PROOFSTEP]\nlinarith\n[GOAL]\nx : \u211d\nh0x : 0 \u2264 x\nhxp\u271d : x \u2264 \u03c0 / 2\nh\u271d : 0 < x\nhxp : x = \u03c0 / 2\n\u22a2 0 \u2264 tan x\n[PROOFSTEP]\nsimp [hxp, tan_eq_sin_div_cos]\n[GOAL]\nx : \u211d\nh0x : 0 \u2264 x\nhxp : x \u2264 \u03c0 / 2\nhx0 : 0 = x\nx\u271d : x < \u03c0 / 2 \u2228 x = \u03c0 / 2\n\u22a2 0 \u2264 tan x\n[PROOFSTEP]\nsimp [hx0.symm]\n[GOAL]\nx : \u211d\nhx0 : x < 0\nhpx : -(\u03c0 / 2) < x\n\u22a2 0 < -x\n[PROOFSTEP]\nlinarith\n[GOAL]\nx : \u211d\nhx0 : x < 0\nhpx : -(\u03c0 / 2) < x\n\u22a2 -x < \u03c0 / 2\n[PROOFSTEP]\nlinarith [pi_pos]\n[GOAL]\nx : \u211d\nhx0 : x \u2264 0\nhpx : -(\u03c0 / 2) \u2264 x\n\u22a2 0 \u2264 -x\n[PROOFSTEP]\nlinarith\n[GOAL]\nx : \u211d\nhx0 : x \u2264 0\nhpx : -(\u03c0 / 2) \u2264 x\n\u22a2 -x \u2264 \u03c0 / 2\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y < \u03c0 / 2\nhxy : x < y\n\u22a2 tan x < tan y\n[PROOFSTEP]\nrw [tan_eq_sin_div_cos, tan_eq_sin_div_cos]\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y < \u03c0 / 2\nhxy : x < y\n\u22a2 sin x / cos x < sin y / cos y\n[PROOFSTEP]\nexact\n div_lt_div (sin_lt_sin_of_lt_of_le_pi_div_two (by linarith) (le_of_lt hy\u2082) hxy)\n (cos_le_cos_of_nonneg_of_le_pi hx\u2081 (by linarith) (le_of_lt hxy))\n (sin_nonneg_of_nonneg_of_le_pi (by linarith) (by linarith)) (cos_pos_of_mem_Ioo \u27e8by linarith, hy\u2082\u27e9)\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y < \u03c0 / 2\nhxy : x < y\n\u22a2 -(\u03c0 / 2) \u2264 x\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y < \u03c0 / 2\nhxy : x < y\n\u22a2 y \u2264 \u03c0\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y < \u03c0 / 2\nhxy : x < y\n\u22a2 0 \u2264 y\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y < \u03c0 / 2\nhxy : x < y\n\u22a2 y \u2264 \u03c0\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : \u211d\nhx\u2081 : 0 \u2264 x\nhy\u2082 : y < \u03c0 / 2\nhxy : x < y\n\u22a2 -(\u03c0 / 2) < y\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : \u211d\nhx\u2081 : -(\u03c0 / 2) < x\nhy\u2082 : y < \u03c0 / 2\nhxy : x < y\nhx0 : x \u2264 0\nhy0 : y \u2264 0\n\u22a2 -tan y < -tan x\n[PROOFSTEP]\nrw [\u2190 tan_neg, \u2190 tan_neg]\n[GOAL]\nx y : \u211d\nhx\u2081 : -(\u03c0 / 2) < x\nhy\u2082 : y < \u03c0 / 2\nhxy : x < y\nhx0 : x \u2264 0\nhy0 : y \u2264 0\n\u22a2 tan (-y) < tan (-x)\n[PROOFSTEP]\nexact tan_lt_tan_of_nonneg_of_lt_pi_div_two (neg_nonneg.2 hy0) (neg_lt.2 hx\u2081) (neg_lt_neg hxy)\n[GOAL]\nx y : \u211d\nhx\u2081 : -(\u03c0 / 2) < x\nhy\u2082 : y < \u03c0 / 2\nhxy : x < y\nhx0 : x \u2264 0\nhy0\u271d : 0 \u2264 y\nhy0 : 0 = y\n\u22a2 tan x < tan y\n[PROOFSTEP]\nrw [\u2190 hy0, tan_zero]\n[GOAL]\nx y : \u211d\nhx\u2081 : -(\u03c0 / 2) < x\nhy\u2082 : y < \u03c0 / 2\nhxy : x < y\nhx0 : x \u2264 0\nhy0\u271d : 0 \u2264 y\nhy0 : 0 = y\n\u22a2 tan x < 0\n[PROOFSTEP]\nexact tan_neg_of_neg_of_pi_div_two_lt (hy0.symm \u25b8 hxy) hx\u2081\n[GOAL]\nx y : \u211d\nhx\u2081 : -(\u03c0 / 2) < x\nhy\u2082 : y < \u03c0 / 2\nhxy : x < y\nhx0 : 0 \u2264 x\nhy0 : y \u2264 0\n\u22a2 tan x < tan y\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u22a2 Function.Periodic tan \u03c0\n[PROOFSTEP]\nsimpa only [Function.Periodic, tan_eq_sin_div_cos] using sin_antiperiodic.div cos_antiperiodic\n[GOAL]\n\u22a2 tan \u03c0 = 0\n[PROOFSTEP]\nrw [tan_periodic.eq, tan_zero]\n[GOAL]\nx : \u211d\n\u22a2 tan (\u03c0 / 2 - x) = (tan x)\u207b\u00b9\n[PROOFSTEP]\nrw [tan_eq_sin_div_cos, tan_eq_sin_div_cos, inv_div, sin_pi_div_two_sub, cos_pi_div_two_sub]\n[GOAL]\n\u22a2 Tendsto sin (\ud835\udcdd[Iio (\u03c0 / 2)] (\u03c0 / 2)) (\ud835\udcdd 1)\n[PROOFSTEP]\nconvert continuous_sin.continuousWithinAt.tendsto\n[GOAL]\ncase h.e'_5.h.e'_3\n\u22a2 1 = sin (\u03c0 / 2)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 Tendsto cos (\ud835\udcdd[Iio (\u03c0 / 2)] (\u03c0 / 2)) (\ud835\udcdd[Ioi 0] 0)\n[PROOFSTEP]\napply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within\n[GOAL]\ncase h1\n\u22a2 Tendsto cos (\ud835\udcdd[Iio (\u03c0 / 2)] (\u03c0 / 2)) (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert continuous_cos.continuousWithinAt.tendsto\n[GOAL]\ncase h.e'_5.h.e'_3\n\u22a2 0 = cos (\u03c0 / 2)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio (\u03c0 / 2)] (\u03c0 / 2), cos x \u2208 Ioi 0\n[PROOFSTEP]\nfilter_upwards [Ioo_mem_nhdsWithin_Iio (right_mem_Ioc.mpr (neg_lt_self pi_div_two_pos))] with x hx using\n cos_pos_of_mem_Ioo hx\n[GOAL]\n\u22a2 Tendsto tan (\ud835\udcdd[Iio (\u03c0 / 2)] (\u03c0 / 2)) atTop\n[PROOFSTEP]\nconvert tendsto_cos_pi_div_two.inv_tendsto_zero.atTop_mul zero_lt_one tendsto_sin_pi_div_two using 1\n[GOAL]\ncase h.e'_3\n\u22a2 tan = fun x => cos\u207b\u00b9 x * sin x\n[PROOFSTEP]\nsimp only [Pi.inv_apply, \u2190 div_eq_inv_mul, \u2190 tan_eq_sin_div_cos]\n[GOAL]\n\u22a2 Tendsto sin (\ud835\udcdd[Ioi (-(\u03c0 / 2))] (-(\u03c0 / 2))) (\ud835\udcdd (-1))\n[PROOFSTEP]\nconvert continuous_sin.continuousWithinAt.tendsto using 2\n[GOAL]\ncase h.e'_5.h.e'_3\n\u22a2 -1 = sin (-(\u03c0 / 2))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 Tendsto cos (\ud835\udcdd[Ioi (-(\u03c0 / 2))] (-(\u03c0 / 2))) (\ud835\udcdd[Ioi 0] 0)\n[PROOFSTEP]\napply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within\n[GOAL]\ncase h1\n\u22a2 Tendsto cos (\ud835\udcdd[Ioi (-(\u03c0 / 2))] (-(\u03c0 / 2))) (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert continuous_cos.continuousWithinAt.tendsto\n[GOAL]\ncase h.e'_5.h.e'_3\n\u22a2 0 = cos (-(\u03c0 / 2))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi (-(\u03c0 / 2))] (-(\u03c0 / 2)), cos x \u2208 Ioi 0\n[PROOFSTEP]\nfilter_upwards [Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.mpr (neg_lt_self pi_div_two_pos))] with x hx using\n cos_pos_of_mem_Ioo hx\n[GOAL]\n\u22a2 Tendsto tan (\ud835\udcdd[Ioi (-(\u03c0 / 2))] (-(\u03c0 / 2))) atBot\n[PROOFSTEP]\nconvert tendsto_cos_neg_pi_div_two.inv_tendsto_zero.atTop_mul_neg (by norm_num) tendsto_sin_neg_pi_div_two using 1\n[GOAL]\n\u22a2 -1 < 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_3\n\u22a2 tan = fun x => cos\u207b\u00b9 x * sin x\n[PROOFSTEP]\nsimp only [Pi.inv_apply, \u2190 div_eq_inv_mul, \u2190 tan_eq_sin_div_cos]\n[GOAL]\nz : \u2102\n\u22a2 sin z = 0 \u2194 cos z = 1 \u2228 cos z = -1\n[PROOFSTEP]\nrw [\u2190 mul_self_eq_one_iff, \u2190 sin_sq_add_cos_sq, sq, sq, \u2190 sub_eq_iff_eq_add, sub_self]\n[GOAL]\nz : \u2102\n\u22a2 sin z = 0 \u2194 0 = sin z * sin z\n[PROOFSTEP]\nexact \u27e8fun h => by rw [h, mul_zero], eq_zero_of_mul_self_eq_zero \u2218 Eq.symm\u27e9\n[GOAL]\nz : \u2102\nh : sin z = 0\n\u22a2 0 = sin z * sin z\n[PROOFSTEP]\nrw [h, mul_zero]\n[GOAL]\n\u22a2 cos (\u2191\u03c0 / 2) = \u2191(Real.cos (\u03c0 / 2))\n[PROOFSTEP]\nrw [ofReal_cos]\n[GOAL]\n\u22a2 cos (\u2191\u03c0 / 2) = cos \u2191(\u03c0 / 2)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2191(Real.cos (\u03c0 / 2)) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 sin (\u2191\u03c0 / 2) = \u2191(Real.sin (\u03c0 / 2))\n[PROOFSTEP]\nrw [ofReal_sin]\n[GOAL]\n\u22a2 sin (\u2191\u03c0 / 2) = sin \u2191(\u03c0 / 2)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2191(Real.sin (\u03c0 / 2)) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 sin \u2191\u03c0 = 0\n[PROOFSTEP]\nrw [\u2190 ofReal_sin, Real.sin_pi]\n[GOAL]\n\u22a2 \u21910 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 cos \u2191\u03c0 = -1\n[PROOFSTEP]\nrw [\u2190 ofReal_cos, Real.cos_pi]\n[GOAL]\n\u22a2 \u2191(-1) = -1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 sin (2 * \u2191\u03c0) = 0\n[PROOFSTEP]\nsimp [two_mul, sin_add]\n[GOAL]\n\u22a2 cos (2 * \u2191\u03c0) = 1\n[PROOFSTEP]\nsimp [two_mul, cos_add]\n[GOAL]\n\u22a2 Function.Antiperiodic sin \u2191\u03c0\n[PROOFSTEP]\nsimp [sin_add]\n[GOAL]\n\u22a2 Function.Antiperiodic cos \u2191\u03c0\n[PROOFSTEP]\nsimp [cos_add]\n[GOAL]\nn : \u2115\n\u22a2 cos (\u2191n * (2 * \u2191\u03c0) + \u2191\u03c0) = -1\n[PROOFSTEP]\nsimpa only [cos_zero] using (cos_periodic.nat_mul n).add_antiperiod_eq cos_antiperiodic\n[GOAL]\nn : \u2124\n\u22a2 cos (\u2191n * (2 * \u2191\u03c0) + \u2191\u03c0) = -1\n[PROOFSTEP]\nsimpa only [cos_zero] using (cos_periodic.int_mul n).add_antiperiod_eq cos_antiperiodic\n[GOAL]\nn : \u2115\n\u22a2 cos (\u2191n * (2 * \u2191\u03c0) - \u2191\u03c0) = -1\n[PROOFSTEP]\nsimpa only [cos_zero] using (cos_periodic.nat_mul n).sub_antiperiod_eq cos_antiperiodic\n[GOAL]\nn : \u2124\n\u22a2 cos (\u2191n * (2 * \u2191\u03c0) - \u2191\u03c0) = -1\n[PROOFSTEP]\nsimpa only [cos_zero] using (cos_periodic.int_mul n).sub_antiperiod_eq cos_antiperiodic\n[GOAL]\nx : \u2102\n\u22a2 sin (x + \u2191\u03c0 / 2) = cos x\n[PROOFSTEP]\nsimp [sin_add]\n[GOAL]\nx : \u2102\n\u22a2 sin (x - \u2191\u03c0 / 2) = -cos x\n[PROOFSTEP]\nsimp [sub_eq_add_neg, sin_add]\n[GOAL]\nx : \u2102\n\u22a2 sin (\u2191\u03c0 / 2 - x) = cos x\n[PROOFSTEP]\nsimp [sub_eq_add_neg, sin_add]\n[GOAL]\nx : \u2102\n\u22a2 cos (x + \u2191\u03c0 / 2) = -sin x\n[PROOFSTEP]\nsimp [cos_add]\n[GOAL]\nx : \u2102\n\u22a2 cos (x - \u2191\u03c0 / 2) = sin x\n[PROOFSTEP]\nsimp [sub_eq_add_neg, cos_add]\n[GOAL]\nx : \u2102\n\u22a2 cos (\u2191\u03c0 / 2 - x) = sin x\n[PROOFSTEP]\nrw [\u2190 cos_neg, neg_sub, cos_sub_pi_div_two]\n[GOAL]\n\u22a2 Function.Periodic tan \u2191\u03c0\n[PROOFSTEP]\nsimpa only [tan_eq_sin_div_cos] using sin_antiperiodic.div cos_antiperiodic\n[GOAL]\nx : \u2102\n\u22a2 tan (\u2191\u03c0 / 2 - x) = (tan x)\u207b\u00b9\n[PROOFSTEP]\nrw [tan_eq_sin_div_cos, tan_eq_sin_div_cos, inv_div, sin_pi_div_two_sub, cos_pi_div_two_sub]\n[GOAL]\n\u22a2 Function.Antiperiodic exp (\u2191\u03c0 * I)\n[PROOFSTEP]\nsimp [exp_add, exp_mul_I]\n[GOAL]\n\u22a2 Function.Antiperiodic (fun x => exp (x * I)) \u2191\u03c0\n[PROOFSTEP]\nsimpa only [mul_inv_cancel_right\u2080 I_ne_zero] using exp_antiperiodic.mul_const I_ne_zero\n[GOAL]\na b : \u211d\nha : a \u2264 0\nz : \u2102\nhz : |z.im| \u2264 b\nhb : b \u2264 \u03c0 / 2\n\u22a2 \u2191abs (exp (\u2191a * (exp z + exp (-z)))) \u2264 Real.exp (a * Real.cos b * Real.exp |z.re|)\n[PROOFSTEP]\nsimp only [abs_exp, Real.exp_le_exp, ofReal_mul_re, add_re, exp_re, neg_im, Real.cos_neg, \u2190 add_mul, mul_assoc,\n mul_comm (Real.cos b), neg_re, \u2190 Real.cos_abs z.im]\n[GOAL]\na b : \u211d\nha : a \u2264 0\nz : \u2102\nhz : |z.im| \u2264 b\nhb : b \u2264 \u03c0 / 2\n\u22a2 a * ((Real.exp z.re + Real.exp (-z.re)) * Real.cos |z.im|) \u2264 a * (Real.exp |z.re| * Real.cos b)\n[PROOFSTEP]\nhave : Real.exp |z.re| \u2264 Real.exp z.re + Real.exp (-z.re) :=\n apply_abs_le_add_of_nonneg (fun x => (Real.exp_pos x).le) z.re\n[GOAL]\na b : \u211d\nha : a \u2264 0\nz : \u2102\nhz : |z.im| \u2264 b\nhb : b \u2264 \u03c0 / 2\nthis : Real.exp |z.re| \u2264 Real.exp z.re + Real.exp (-z.re)\n\u22a2 a * ((Real.exp z.re + Real.exp (-z.re)) * Real.cos |z.im|) \u2264 a * (Real.exp |z.re| * Real.cos b)\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonpos_left (mul_le_mul this _ _ ((Real.exp_pos _).le.trans this)) ha\n[GOAL]\ncase refine'_1\na b : \u211d\nha : a \u2264 0\nz : \u2102\nhz : |z.im| \u2264 b\nhb : b \u2264 \u03c0 / 2\nthis : Real.exp |z.re| \u2264 Real.exp z.re + Real.exp (-z.re)\n\u22a2 Real.cos b \u2264 Real.cos |z.im|\n[PROOFSTEP]\nexact Real.cos_le_cos_of_nonneg_of_le_pi (_root_.abs_nonneg _) (hb.trans <| half_le_self <| Real.pi_pos.le) hz\n[GOAL]\ncase refine'_2\na b : \u211d\nha : a \u2264 0\nz : \u2102\nhz : |z.im| \u2264 b\nhb : b \u2264 \u03c0 / 2\nthis : Real.exp |z.re| \u2264 Real.exp z.re + Real.exp (-z.re)\n\u22a2 0 \u2264 Real.cos b\n[PROOFSTEP]\nrefine' Real.cos_nonneg_of_mem_Icc \u27e8_, hb\u27e9\n[GOAL]\ncase refine'_2\na b : \u211d\nha : a \u2264 0\nz : \u2102\nhz : |z.im| \u2264 b\nhb : b \u2264 \u03c0 / 2\nthis : Real.exp |z.re| \u2264 Real.exp z.re + Real.exp (-z.re)\n\u22a2 -(\u03c0 / 2) \u2264 b\n[PROOFSTEP]\nexact (neg_nonpos.2 <| Real.pi_div_two_pos.le).trans ((_root_.abs_nonneg _).trans hz)\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "llama_tokens": 24732, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494421679929, "lm_q2_score": 0.6187804407739559, "lm_q1q2_score": 0.4244520981733598}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : TopologicalSpace B\nb : B\n\u22a2 Trivialization.coordChangeL \ud835\udd5c (trivialization B F) (trivialization B F) b = ContinuousLinearEquiv.refl \ud835\udd5c F\n[PROOFSTEP]\next v\n[GOAL]\ncase h.h\n\ud835\udd5c : Type u_1\nB : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : TopologicalSpace B\nb : B\nv : F\n\u22a2 \u2191(Trivialization.coordChangeL \ud835\udd5c (trivialization B F) (trivialization B F) b) v = \u2191(ContinuousLinearEquiv.refl \ud835\udd5c F) v\n[PROOFSTEP]\nrw [Trivialization.coordChangeL_apply']\n[GOAL]\ncase h.h\n\ud835\udd5c : Type u_1\nB : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : TopologicalSpace B\nb : B\nv : F\n\u22a2 (\u2191(trivialization B F) (\u2191(LocalHomeomorph.symm (trivialization B F).toLocalHomeomorph) (b, v))).snd =\n \u2191(ContinuousLinearEquiv.refl \ud835\udd5c F) v\ncase h.h.hb\n\ud835\udd5c : Type u_1\nB : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : TopologicalSpace B\nb : B\nv : F\n\u22a2 b \u2208 (trivialization B F).baseSet \u2229 (trivialization B F).baseSet\n[PROOFSTEP]\nexacts [rfl, \u27e8mem_univ _, mem_univ _\u27e9]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : TopologicalSpace B\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\n\u22a2 Trivialization.IsLinear \ud835\udd5c e\n[PROOFSTEP]\nrw [eq_trivialization B F e]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : TopologicalSpace B\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\n\u22a2 Trivialization.IsLinear \ud835\udd5c (trivialization B F)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : TopologicalSpace B\ne e' : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\nhe' : MemTrivializationAtlas e'\n\u22a2 ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' b)) (e.baseSet \u2229 e'.baseSet)\n[PROOFSTEP]\nobtain rfl := eq_trivialization B F e\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : TopologicalSpace B\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\nhe : MemTrivializationAtlas (trivialization B F)\n\u22a2 ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c (trivialization B F) e' b))\n ((trivialization B F).baseSet \u2229 e'.baseSet)\n[PROOFSTEP]\nobtain rfl := eq_trivialization B F e'\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : TopologicalSpace B\nhe he' : MemTrivializationAtlas (trivialization B F)\n\u22a2 ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c (trivialization B F) (trivialization B F) b))\n ((trivialization B F).baseSet \u2229 (trivialization B F).baseSet)\n[PROOFSTEP]\nsimp only [trivialization.coordChangeL]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : TopologicalSpace B\nhe he' : MemTrivializationAtlas (trivialization B F)\n\u22a2 ContinuousOn (fun b => \u2191(ContinuousLinearEquiv.refl \ud835\udd5c F))\n ((trivialization B F).baseSet \u2229 (trivialization B F).baseSet)\n[PROOFSTEP]\nexact continuous_const.continuousOn\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\u2082\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u2078 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c e\u2082'\nb : B\nhb : b \u2208 (prod e\u2081 e\u2082).baseSet \u2229 (prod e\u2081' e\u2082').baseSet\n\u22a2 \u2191(coordChangeL \ud835\udd5c (prod e\u2081 e\u2082) (prod e\u2081' e\u2082') b) =\n ContinuousLinearMap.prodMap \u2191(coordChangeL \ud835\udd5c e\u2081 e\u2081' b) \u2191(coordChangeL \ud835\udd5c e\u2082 e\u2082' b)\n[PROOFSTEP]\nrw [ContinuousLinearMap.ext_iff, ContinuousLinearMap.coe_prodMap']\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\u2082\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u2078 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c e\u2082'\nb : B\nhb : b \u2208 (prod e\u2081 e\u2082).baseSet \u2229 (prod e\u2081' e\u2082').baseSet\n\u22a2 \u2200 (x : F\u2081 \u00d7 F\u2082),\n \u2191\u2191(coordChangeL \ud835\udd5c (prod e\u2081 e\u2082) (prod e\u2081' e\u2082') b) x =\n Prod.map (\u2191\u2191(coordChangeL \ud835\udd5c e\u2081 e\u2081' b)) (\u2191\u2191(coordChangeL \ud835\udd5c e\u2082 e\u2082' b)) x\n[PROOFSTEP]\nrintro \u27e8v\u2081, v\u2082\u27e9\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\u2082\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u2078 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c e\u2082'\nb : B\nhb : b \u2208 (prod e\u2081 e\u2082).baseSet \u2229 (prod e\u2081' e\u2082').baseSet\nv\u2081 : F\u2081\nv\u2082 : F\u2082\n\u22a2 \u2191\u2191(coordChangeL \ud835\udd5c (prod e\u2081 e\u2082) (prod e\u2081' e\u2082') b) (v\u2081, v\u2082) =\n Prod.map \u2191\u2191(coordChangeL \ud835\udd5c e\u2081 e\u2081' b) \u2191\u2191(coordChangeL \ud835\udd5c e\u2082 e\u2082' b) (v\u2081, v\u2082)\n[PROOFSTEP]\nshow (e\u2081.prod e\u2082).coordChangeL \ud835\udd5c (e\u2081'.prod e\u2082') b (v\u2081, v\u2082) = (e\u2081.coordChangeL \ud835\udd5c e\u2081' b v\u2081, e\u2082.coordChangeL \ud835\udd5c e\u2082' b v\u2082)\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\u2082\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u2078 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c e\u2082'\nb : B\nhb : b \u2208 (prod e\u2081 e\u2082).baseSet \u2229 (prod e\u2081' e\u2082').baseSet\nv\u2081 : F\u2081\nv\u2082 : F\u2082\n\u22a2 \u2191(coordChangeL \ud835\udd5c (prod e\u2081 e\u2082) (prod e\u2081' e\u2082') b) (v\u2081, v\u2082) =\n (\u2191(coordChangeL \ud835\udd5c e\u2081 e\u2081' b) v\u2081, \u2191(coordChangeL \ud835\udd5c e\u2082 e\u2082' b) v\u2082)\n[PROOFSTEP]\nrw [e\u2081.coordChangeL_apply e\u2081', e\u2082.coordChangeL_apply e\u2082', (e\u2081.prod e\u2082).coordChangeL_apply']\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\u2082\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u2078 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c e\u2082'\nb : B\nhb : b \u2208 (prod e\u2081 e\u2082).baseSet \u2229 (prod e\u2081' e\u2082').baseSet\nv\u2081 : F\u2081\nv\u2082 : F\u2082\n\u22a2 (\u2191(prod e\u2081' e\u2082') (\u2191(LocalHomeomorph.symm (prod e\u2081 e\u2082).toLocalHomeomorph) (b, v\u2081, v\u2082))).snd =\n ((\u2191e\u2081' { proj := b, snd := Trivialization.symm e\u2081 b v\u2081 }).snd,\n (\u2191e\u2082' { proj := b, snd := Trivialization.symm e\u2082 b v\u2082 }).snd)\ncase mk.hb\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\u2082\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u2078 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c e\u2082'\nb : B\nhb : b \u2208 (prod e\u2081 e\u2082).baseSet \u2229 (prod e\u2081' e\u2082').baseSet\nv\u2081 : F\u2081\nv\u2082 : F\u2082\n\u22a2 b \u2208 (prod e\u2081 e\u2082).baseSet \u2229 (prod e\u2081' e\u2082').baseSet\ncase mk.hb\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\u2082\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u2078 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c e\u2082'\nb : B\nhb : b \u2208 (prod e\u2081 e\u2082).baseSet \u2229 (prod e\u2081' e\u2082').baseSet\nv\u2081 : F\u2081\nv\u2082 : F\u2082\n\u22a2 b \u2208 e\u2082.baseSet \u2229 e\u2082'.baseSet\ncase mk.hb\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\u2082\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u2078 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c e\u2082'\nb : B\nhb : b \u2208 (prod e\u2081 e\u2082).baseSet \u2229 (prod e\u2081' e\u2082').baseSet\nv\u2081 : F\u2081\nv\u2082 : F\u2082\n\u22a2 b \u2208 e\u2081.baseSet \u2229 e\u2081'.baseSet\n[PROOFSTEP]\nexacts [rfl, hb, \u27e8hb.1.2, hb.2.2\u27e9, \u27e8hb.1.1, hb.2.1\u27e9]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\n\u22a2 \u2200 (e : Trivialization (F\u2081 \u00d7 F\u2082) TotalSpace.proj) [inst : MemTrivializationAtlas e], Trivialization.IsLinear \ud835\udd5c e\n[PROOFSTEP]\nrintro _ \u27e8e\u2081, e\u2082, he\u2081, he\u2082, rfl\u27e9\n[GOAL]\ncase mk.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\n\u22a2 Trivialization.IsLinear \ud835\udd5c (Trivialization.prod e\u2081 e\u2082)\n[PROOFSTEP]\nskip\n[GOAL]\ncase mk.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\n\u22a2 Trivialization.IsLinear \ud835\udd5c (Trivialization.prod e\u2081 e\u2082)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\n\u22a2 \u2200 (e e' : Trivialization (F\u2081 \u00d7 F\u2082) TotalSpace.proj) [inst : MemTrivializationAtlas e]\n [inst_1 : MemTrivializationAtlas e'], ContinuousOn (fun b => \u2191(coordChangeL \ud835\udd5c e e' b)) (e.baseSet \u2229 e'.baseSet)\n[PROOFSTEP]\nrintro _ _ \u27e8e\u2081, e\u2082, he\u2081, he\u2082, rfl\u27e9 \u27e8e\u2081', e\u2082', he\u2081', he\u2082', rfl\u27e9\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 ContinuousOn (fun b => \u2191(coordChangeL \ud835\udd5c (Trivialization.prod e\u2081 e\u2082) (Trivialization.prod e\u2081' e\u2082') b))\n ((Trivialization.prod e\u2081 e\u2082).baseSet \u2229 (Trivialization.prod e\u2081' e\u2082').baseSet)\n[PROOFSTEP]\nskip\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 ContinuousOn (fun b => \u2191(coordChangeL \ud835\udd5c (Trivialization.prod e\u2081 e\u2082) (Trivialization.prod e\u2081' e\u2082') b))\n ((Trivialization.prod e\u2081 e\u2082).baseSet \u2229 (Trivialization.prod e\u2081' e\u2082').baseSet)\n[PROOFSTEP]\nrefine' (((continuousOn_coordChange \ud835\udd5c e\u2081 e\u2081').mono _).prod_mapL \ud835\udd5c ((continuousOn_coordChange \ud835\udd5c e\u2082 e\u2082').mono _)).congr _\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 (Trivialization.prod e\u2081 e\u2082).baseSet \u2229 (Trivialization.prod e\u2081' e\u2082').baseSet \u2286 e\u2081.baseSet \u2229 e\u2081'.baseSet\n[PROOFSTEP]\ndsimp only [baseSet_prod, mfld_simps]\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 (Trivialization.prod e\u2081 e\u2082).baseSet \u2229 (Trivialization.prod e\u2081' e\u2082').baseSet \u2286 e\u2082.baseSet \u2229 e\u2082'.baseSet\n[PROOFSTEP]\ndsimp only [baseSet_prod, mfld_simps]\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 EqOn (fun b => \u2191(coordChangeL \ud835\udd5c (Trivialization.prod e\u2081 e\u2082) (Trivialization.prod e\u2081' e\u2082') b))\n (fun x => ContinuousLinearMap.prodMap \u2191(coordChangeL \ud835\udd5c e\u2081 e\u2081' x) \u2191(coordChangeL \ud835\udd5c e\u2082 e\u2082' x))\n ((Trivialization.prod e\u2081 e\u2082).baseSet \u2229 (Trivialization.prod e\u2081' e\u2082').baseSet)\n[PROOFSTEP]\ndsimp only [baseSet_prod, mfld_simps]\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet) \u2286 e\u2081.baseSet \u2229 e\u2081'.baseSet\n[PROOFSTEP]\nmfld_set_tac\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet) \u2286 e\u2082.baseSet \u2229 e\u2082'.baseSet\n[PROOFSTEP]\nmfld_set_tac\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 EqOn (fun b => \u2191(coordChangeL \ud835\udd5c (Trivialization.prod e\u2081 e\u2082) (Trivialization.prod e\u2081' e\u2082') b))\n (fun x => ContinuousLinearMap.prodMap \u2191(coordChangeL \ud835\udd5c e\u2081 e\u2081' x) \u2191(coordChangeL \ud835\udd5c e\u2082 e\u2082' x))\n (e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet))\n[PROOFSTEP]\nrintro b hb\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\n\u22a2 (fun b => \u2191(coordChangeL \ud835\udd5c (Trivialization.prod e\u2081 e\u2082) (Trivialization.prod e\u2081' e\u2082') b)) b =\n (fun x => ContinuousLinearMap.prodMap \u2191(coordChangeL \ud835\udd5c e\u2081 e\u2081' x) \u2191(coordChangeL \ud835\udd5c e\u2082 e\u2082' x)) b\n[PROOFSTEP]\nrw [ContinuousLinearMap.ext_iff]\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\n\u22a2 \u2200 (x : F\u2081 \u00d7 F\u2082),\n \u2191((fun b => \u2191(coordChangeL \ud835\udd5c (Trivialization.prod e\u2081 e\u2082) (Trivialization.prod e\u2081' e\u2082') b)) b) x =\n \u2191((fun x => ContinuousLinearMap.prodMap \u2191(coordChangeL \ud835\udd5c e\u2081 e\u2081' x) \u2191(coordChangeL \ud835\udd5c e\u2082 e\u2082' x)) b) x\n[PROOFSTEP]\nrintro \u27e8v\u2081, v\u2082\u27e9\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3.mk\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\nv\u2081 : F\u2081\nv\u2082 : F\u2082\n\u22a2 \u2191((fun b => \u2191(coordChangeL \ud835\udd5c (Trivialization.prod e\u2081 e\u2082) (Trivialization.prod e\u2081' e\u2082') b)) b) (v\u2081, v\u2082) =\n \u2191((fun x => ContinuousLinearMap.prodMap \u2191(coordChangeL \ud835\udd5c e\u2081 e\u2081' x) \u2191(coordChangeL \ud835\udd5c e\u2082 e\u2082' x)) b) (v\u2081, v\u2082)\n[PROOFSTEP]\nshow (e\u2081.prod e\u2082).coordChangeL \ud835\udd5c (e\u2081'.prod e\u2082') b (v\u2081, v\u2082) = (e\u2081.coordChangeL \ud835\udd5c e\u2081' b v\u2081, e\u2082.coordChangeL \ud835\udd5c e\u2082' b v\u2082)\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3.mk\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\nv\u2081 : F\u2081\nv\u2082 : F\u2082\n\u22a2 \u2191(coordChangeL \ud835\udd5c (Trivialization.prod e\u2081 e\u2082) (Trivialization.prod e\u2081' e\u2082') b) (v\u2081, v\u2082) =\n (\u2191(coordChangeL \ud835\udd5c e\u2081 e\u2081' b) v\u2081, \u2191(coordChangeL \ud835\udd5c e\u2082 e\u2082' b) v\u2082)\n[PROOFSTEP]\nrw [e\u2081.coordChangeL_apply e\u2081', e\u2082.coordChangeL_apply e\u2082', (e\u2081.prod e\u2082).coordChangeL_apply']\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3.mk\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\nv\u2081 : F\u2081\nv\u2082 : F\u2082\n\u22a2 (\u2191(Trivialization.prod e\u2081' e\u2082')\n (\u2191(LocalHomeomorph.symm (Trivialization.prod e\u2081 e\u2082).toLocalHomeomorph) (b, v\u2081, v\u2082))).snd =\n ((\u2191e\u2081' { proj := b, snd := Trivialization.symm e\u2081 b v\u2081 }).snd,\n (\u2191e\u2082' { proj := b, snd := Trivialization.symm e\u2082 b v\u2082 }).snd)\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3.mk.hb\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\nv\u2081 : F\u2081\nv\u2082 : F\u2082\n\u22a2 b \u2208 (Trivialization.prod e\u2081 e\u2082).baseSet \u2229 (Trivialization.prod e\u2081' e\u2082').baseSet\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3.mk.hb\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\nv\u2081 : F\u2081\nv\u2082 : F\u2082\n\u22a2 b \u2208 e\u2082.baseSet \u2229 e\u2082'.baseSet\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3.mk.hb\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d : VectorBundle \ud835\udd5c F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\nhe\u2081' : MemTrivializationAtlas e\u2081'\nhe\u2082' : MemTrivializationAtlas e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\nv\u2081 : F\u2081\nv\u2082 : F\u2082\n\u22a2 b \u2208 e\u2081.baseSet \u2229 e\u2081'.baseSet\n[PROOFSTEP]\nexacts [rfl, hb, \u27e8hb.1.2, hb.2.2\u27e9, \u27e8hb.1.1, hb.2.1\u27e9]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c e\u2081\ninst\u271d : Trivialization.IsLinear \ud835\udd5c e\u2082\nx : B\nhx : x \u2208 (prod e\u2081 e\u2082).baseSet\n\u22a2 continuousLinearEquivAt \ud835\udd5c (prod e\u2081 e\u2082) x hx =\n ContinuousLinearEquiv.prod (continuousLinearEquivAt \ud835\udd5c e\u2081 x (_ : x \u2208 e\u2081.baseSet))\n (continuousLinearEquivAt \ud835\udd5c e\u2082 x (_ : x \u2208 e\u2082.baseSet))\n[PROOFSTEP]\next v : 2\n[GOAL]\ncase h.h\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c e\u2081\ninst\u271d : Trivialization.IsLinear \ud835\udd5c e\u2082\nx : B\nhx : x \u2208 (prod e\u2081 e\u2082).baseSet\nv : E\u2081 x \u00d7 E\u2082 x\n\u22a2 \u2191(continuousLinearEquivAt \ud835\udd5c (prod e\u2081 e\u2082) x hx) v =\n \u2191(ContinuousLinearEquiv.prod (continuousLinearEquivAt \ud835\udd5c e\u2081 x (_ : x \u2208 e\u2081.baseSet))\n (continuousLinearEquivAt \ud835\udd5c e\u2082 x (_ : x \u2208 e\u2082.baseSet)))\n v\n[PROOFSTEP]\nobtain \u27e8v\u2081, v\u2082\u27e9 := v\n[GOAL]\ncase h.h.mk\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c e\u2081\ninst\u271d : Trivialization.IsLinear \ud835\udd5c e\u2082\nx : B\nhx : x \u2208 (prod e\u2081 e\u2082).baseSet\nv\u2081 : E\u2081 x\nv\u2082 : E\u2082 x\n\u22a2 \u2191(continuousLinearEquivAt \ud835\udd5c (prod e\u2081 e\u2082) x hx) (v\u2081, v\u2082) =\n \u2191(ContinuousLinearEquiv.prod (continuousLinearEquivAt \ud835\udd5c e\u2081 x (_ : x \u2208 e\u2081.baseSet))\n (continuousLinearEquivAt \ud835\udd5c e\u2082 x (_ : x \u2208 e\u2082.baseSet)))\n (v\u2081, v\u2082)\n[PROOFSTEP]\nrw [(e\u2081.prod e\u2082).continuousLinearEquivAt_apply \ud835\udd5c, Trivialization.prod]\n[GOAL]\ncase h.h.mk\n\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : TopologicalSpace B\nF\u2081 : Type u_3\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2076 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2081 E\u2081\ninst\u271d\u00b2 : FiberBundle F\u2082 E\u2082\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c e\u2081\ninst\u271d : Trivialization.IsLinear \ud835\udd5c e\u2082\nx : B\nhx : x \u2208 (prod e\u2081 e\u2082).baseSet\nv\u2081 : E\u2081 x\nv\u2082 : E\u2082 x\n\u22a2 (fun y =>\n (\u2191{\n toLocalHomeomorph :=\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227 (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192 x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192 Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227\n (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192 x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) }.source),\n open_target := (_ : IsOpen ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)),\n continuous_toFun :=\n (_ : ContinuousOn (Prod.toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))),\n continuous_invFun := (_ : ContinuousOn (Prod.invFun' e\u2081 e\u2082) ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)) },\n baseSet := e\u2081.baseSet \u2229 e\u2082.baseSet, open_baseSet := (_ : IsOpen (e\u2081.baseSet \u2229 e\u2082.baseSet)),\n source_eq :=\n (_ :\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227\n (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192 x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227\n (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192 x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) }.source),\n open_target := (_ : IsOpen ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)),\n continuous_toFun :=\n (_ : ContinuousOn (Prod.toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))),\n continuous_invFun :=\n (_ :\n ContinuousOn (Prod.invFun' e\u2081 e\u2082)\n ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)) }.toLocalEquiv.source =\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227\n (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192 x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227\n (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192 x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) }.source),\n open_target := (_ : IsOpen ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)),\n continuous_toFun :=\n (_ : ContinuousOn (Prod.toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))),\n continuous_invFun :=\n (_ :\n ContinuousOn (Prod.invFun' e\u2081 e\u2082)\n ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)) }.toLocalEquiv.source),\n target_eq :=\n (_ :\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227\n (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192 x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227\n (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192 x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) }.source),\n open_target := (_ : IsOpen ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)),\n continuous_toFun :=\n (_ : ContinuousOn (Prod.toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))),\n continuous_invFun :=\n (_ :\n ContinuousOn (Prod.invFun' e\u2081 e\u2082)\n ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)) }.toLocalEquiv.target =\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227\n (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192 x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227\n (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192 x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) }.source),\n open_target := (_ : IsOpen ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)),\n continuous_toFun :=\n (_ : ContinuousOn (Prod.toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))),\n continuous_invFun :=\n (_ :\n ContinuousOn (Prod.invFun' e\u2081 e\u2082)\n ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)) }.toLocalEquiv.target),\n proj_toFun :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227\n (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192 x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227\n (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) }.source),\n open_target := (_ : IsOpen ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)),\n continuous_toFun :=\n (_ :\n ContinuousOn (Prod.toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))),\n continuous_invFun :=\n (_ :\n ContinuousOn (Prod.invFun' e\u2081 e\u2082)\n ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)) }.toLocalEquiv.source \u2192\n (\u2191{\n toLocalEquiv :=\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227\n (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192 x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227\n (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) }.source),\n open_target := (_ : IsOpen ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)),\n continuous_toFun :=\n (_ :\n ContinuousOn (Prod.toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))),\n continuous_invFun :=\n (_ : ContinuousOn (Prod.invFun' e\u2081 e\u2082) ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)) }\n x).fst =\n (\u2191{\n toLocalEquiv :=\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227\n (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192 x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e\u2081 e\u2082, invFun := Prod.invFun' e\u2081 e\u2082,\n source := TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet),\n target := (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ,\n map_source' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n (Prod.toFun' e\u2081 e\u2082 x).fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2227\n (Prod.toFun' e\u2081 e\u2082 x).snd \u2208 univ),\n map_target' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n x.fst \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet),\n left_inv' :=\n (_ :\n \u2200 (x : TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x),\n x \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet) \u2192\n Prod.invFun' e\u2081 e\u2082 (Prod.toFun' e\u2081 e\u2082 x) = x),\n right_inv' :=\n (_ :\n \u2200 (x : B \u00d7 F\u2081 \u00d7 F\u2082),\n x \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ \u2192\n Prod.toFun' e\u2081 e\u2082 (Prod.invFun' e\u2081 e\u2082 x) = x) }.source),\n open_target := (_ : IsOpen ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)),\n continuous_toFun :=\n (_ :\n ContinuousOn (Prod.toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))),\n continuous_invFun :=\n (_ : ContinuousOn (Prod.invFun' e\u2081 e\u2082) ((e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ)) }\n x).fst) }\n { proj := x, snd := y }).snd)\n (v\u2081, v\u2082) =\n \u2191(ContinuousLinearEquiv.prod (continuousLinearEquivAt \ud835\udd5c e\u2081 x (_ : x \u2208 e\u2081.baseSet))\n (continuousLinearEquivAt \ud835\udd5c e\u2082 x (_ : x \u2208 e\u2082.baseSet)))\n (v\u2081, v\u2082)\n[PROOFSTEP]\nexact (congr_arg Prod.snd (prod_apply \ud835\udd5c hx.1 hx.2 v\u2081 v\u2082) : _)\n[GOAL]\nR : Type u_1\n\ud835\udd5c : Type u_2\nB : Type u_3\nF : Type u_4\nE : B \u2192 Type u_5\nB' : Type u_6\nf\u271d : B' \u2192 B\ninst\u271d\u00b9\u00b9 : TopologicalSpace B'\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E x)\nK : Type u_7\ninst\u271d\u00b3 : ContinuousMapClass K B' B\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ninst\u271d : VectorBundle \ud835\udd5c F E\nf : K\n\u22a2 \u2200 (e : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e], Trivialization.IsLinear \ud835\udd5c e\n[PROOFSTEP]\nrintro _ \u27e8e, he, rfl\u27e9\n[GOAL]\ncase mk.intro.intro\nR : Type u_1\n\ud835\udd5c : Type u_2\nB : Type u_3\nF : Type u_4\nE : B \u2192 Type u_5\nB' : Type u_6\nf\u271d : B' \u2192 B\ninst\u271d\u00b9\u00b9 : TopologicalSpace B'\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E x)\nK : Type u_7\ninst\u271d\u00b3 : ContinuousMapClass K B' B\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ninst\u271d : VectorBundle \ud835\udd5c F E\nf : K\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\n\u22a2 Trivialization.IsLinear \ud835\udd5c (Trivialization.pullback e f)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\n\ud835\udd5c : Type u_2\nB : Type u_3\nF : Type u_4\nE : B \u2192 Type u_5\nB' : Type u_6\nf\u271d : B' \u2192 B\ninst\u271d\u00b9\u00b9 : TopologicalSpace B'\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E x)\nK : Type u_7\ninst\u271d\u00b3 : ContinuousMapClass K B' B\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ninst\u271d : VectorBundle \ud835\udd5c F E\nf : K\n\u22a2 \u2200 (e e' : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e] [inst_1 : MemTrivializationAtlas e'],\n ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' b)) (e.baseSet \u2229 e'.baseSet)\n[PROOFSTEP]\nrintro _ _ \u27e8e, he, rfl\u27e9 \u27e8e', he', rfl\u27e9\n[GOAL]\ncase mk.intro.intro.mk.intro.intro\nR : Type u_1\n\ud835\udd5c : Type u_2\nB : Type u_3\nF : Type u_4\nE : B \u2192 Type u_5\nB' : Type u_6\nf\u271d : B' \u2192 B\ninst\u271d\u00b9\u00b9 : TopologicalSpace B'\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E x)\nK : Type u_7\ninst\u271d\u00b3 : ContinuousMapClass K B' B\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ninst\u271d : VectorBundle \ud835\udd5c F E\nf : K\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\n\u22a2 ContinuousOn\n (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c (Trivialization.pullback e f) (Trivialization.pullback e' f) b))\n ((Trivialization.pullback e f).baseSet \u2229 (Trivialization.pullback e' f).baseSet)\n[PROOFSTEP]\nrefine' ((continuousOn_coordChange \ud835\udd5c e e').comp (map_continuous f).continuousOn fun b hb => hb).congr _\n[GOAL]\ncase mk.intro.intro.mk.intro.intro\nR : Type u_1\n\ud835\udd5c : Type u_2\nB : Type u_3\nF : Type u_4\nE : B \u2192 Type u_5\nB' : Type u_6\nf\u271d : B' \u2192 B\ninst\u271d\u00b9\u00b9 : TopologicalSpace B'\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E x)\nK : Type u_7\ninst\u271d\u00b3 : ContinuousMapClass K B' B\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ninst\u271d : VectorBundle \ud835\udd5c F E\nf : K\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\n\u22a2 EqOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c (Trivialization.pullback e f) (Trivialization.pullback e' f) b))\n ((fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' b)) \u2218 \u2191f) fun b => (e.baseSet \u2229 e'.baseSet) (\u2191f b)\n[PROOFSTEP]\nrintro b (hb : f b \u2208 e.baseSet \u2229 e'.baseSet)\n[GOAL]\ncase mk.intro.intro.mk.intro.intro\nR : Type u_1\n\ud835\udd5c : Type u_2\nB : Type u_3\nF : Type u_4\nE : B \u2192 Type u_5\nB' : Type u_6\nf\u271d : B' \u2192 B\ninst\u271d\u00b9\u00b9 : TopologicalSpace B'\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E x)\nK : Type u_7\ninst\u271d\u00b3 : ContinuousMapClass K B' B\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ninst\u271d : VectorBundle \ud835\udd5c F E\nf : K\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\nb : B'\nhb : \u2191f b \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c (Trivialization.pullback e f) (Trivialization.pullback e' f) b)) b =\n ((fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' b)) \u2218 \u2191f) b\n[PROOFSTEP]\next v\n[GOAL]\ncase mk.intro.intro.mk.intro.intro.h\nR : Type u_1\n\ud835\udd5c : Type u_2\nB : Type u_3\nF : Type u_4\nE : B \u2192 Type u_5\nB' : Type u_6\nf\u271d : B' \u2192 B\ninst\u271d\u00b9\u00b9 : TopologicalSpace B'\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E x)\nK : Type u_7\ninst\u271d\u00b3 : ContinuousMapClass K B' B\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ninst\u271d : VectorBundle \ud835\udd5c F E\nf : K\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\nb : B'\nhb : \u2191f b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 \u2191((fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c (Trivialization.pullback e f) (Trivialization.pullback e' f) b)) b) v =\n \u2191(((fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' b)) \u2218 \u2191f) b) v\n[PROOFSTEP]\nshow ((e.pullback f).coordChangeL \ud835\udd5c (e'.pullback f) b) v = (e.coordChangeL \ud835\udd5c e' (f b)) v\n[GOAL]\ncase mk.intro.intro.mk.intro.intro.h\nR : Type u_1\n\ud835\udd5c : Type u_2\nB : Type u_3\nF : Type u_4\nE : B \u2192 Type u_5\nB' : Type u_6\nf\u271d : B' \u2192 B\ninst\u271d\u00b9\u00b9 : TopologicalSpace B'\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E x)\nK : Type u_7\ninst\u271d\u00b3 : ContinuousMapClass K B' B\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ninst\u271d : VectorBundle \ud835\udd5c F E\nf : K\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\nb : B'\nhb : \u2191f b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 \u2191(Trivialization.coordChangeL \ud835\udd5c (Trivialization.pullback e f) (Trivialization.pullback e' f) b) v =\n \u2191(Trivialization.coordChangeL \ud835\udd5c e e' (\u2191f b)) v\n[PROOFSTEP]\nrw [e.coordChangeL_apply e' hb, (e.pullback f).coordChangeL_apply' _]\n[GOAL]\ncase mk.intro.intro.mk.intro.intro.h\nR : Type u_1\n\ud835\udd5c : Type u_2\nB : Type u_3\nF : Type u_4\nE : B \u2192 Type u_5\nB' : Type u_6\nf\u271d : B' \u2192 B\ninst\u271d\u00b9\u00b9 : TopologicalSpace B'\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E x)\nK : Type u_7\ninst\u271d\u00b3 : ContinuousMapClass K B' B\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ninst\u271d : VectorBundle \ud835\udd5c F E\nf : K\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\nb : B'\nhb : \u2191f b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 (\u2191(Trivialization.pullback e' f)\n (\u2191(LocalHomeomorph.symm (Trivialization.pullback e f).toLocalHomeomorph) (b, v))).snd =\n (\u2191e' { proj := \u2191f b, snd := Trivialization.symm e (\u2191f b) v }).snd\ncase mk.intro.intro.mk.intro.intro.h.hb\nR : Type u_1\n\ud835\udd5c : Type u_2\nB : Type u_3\nF : Type u_4\nE : B \u2192 Type u_5\nB' : Type u_6\nf\u271d : B' \u2192 B\ninst\u271d\u00b9\u00b9 : TopologicalSpace B'\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module \ud835\udd5c (E x)\nK : Type u_7\ninst\u271d\u00b3 : ContinuousMapClass K B' B\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ninst\u271d : VectorBundle \ud835\udd5c F E\nf : K\ne : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\nb : B'\nhb : \u2191f b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 b \u2208 (Trivialization.pullback e f).baseSet \u2229 (Trivialization.pullback e' f).baseSet\n[PROOFSTEP]\nexacts [rfl, hb]\n", "meta": {"mathlib_filename": "Mathlib.Topology.VectorBundle.Constructions", "llama_tokens": 32904, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952052, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.424280265810289}} {"text": "[GOAL]\na\u2080 a\u2081 a\u2082 : Type u\ni : \u2115\n\u22a2 lift #(Sequence\u2082 a\u2080 a\u2081 a\u2082 i) = #(Sequence\u2082 (ULift a\u2080) (ULift a\u2081) (ULift a\u2082) i)\n[PROOFSTEP]\nrcases i with (_ | _ | _ | i)\n[GOAL]\ncase zero\na\u2080 a\u2081 a\u2082 : Type u\n\u22a2 lift #(Sequence\u2082 a\u2080 a\u2081 a\u2082 Nat.zero) = #(Sequence\u2082 (ULift a\u2080) (ULift a\u2081) (ULift a\u2082) Nat.zero)\n[PROOFSTEP]\nsimp only [Sequence\u2082, mk_uLift, mk_fintype, Fintype.card_of_isEmpty, Nat.cast_zero, lift_zero]\n[GOAL]\ncase succ.zero\na\u2080 a\u2081 a\u2082 : Type u\n\u22a2 lift #(Sequence\u2082 a\u2080 a\u2081 a\u2082 (Nat.succ Nat.zero)) = #(Sequence\u2082 (ULift a\u2080) (ULift a\u2081) (ULift a\u2082) (Nat.succ Nat.zero))\n[PROOFSTEP]\nsimp only [Sequence\u2082, mk_uLift, mk_fintype, Fintype.card_of_isEmpty, Nat.cast_zero, lift_zero]\n[GOAL]\ncase succ.succ.zero\na\u2080 a\u2081 a\u2082 : Type u\n\u22a2 lift #(Sequence\u2082 a\u2080 a\u2081 a\u2082 (Nat.succ (Nat.succ Nat.zero))) =\n #(Sequence\u2082 (ULift a\u2080) (ULift a\u2081) (ULift a\u2082) (Nat.succ (Nat.succ Nat.zero)))\n[PROOFSTEP]\nsimp only [Sequence\u2082, mk_uLift, mk_fintype, Fintype.card_of_isEmpty, Nat.cast_zero, lift_zero]\n[GOAL]\ncase succ.succ.succ\na\u2080 a\u2081 a\u2082 : Type u\ni : \u2115\n\u22a2 lift #(Sequence\u2082 a\u2080 a\u2081 a\u2082 (Nat.succ (Nat.succ (Nat.succ i)))) =\n #(Sequence\u2082 (ULift a\u2080) (ULift a\u2081) (ULift a\u2082) (Nat.succ (Nat.succ (Nat.succ i))))\n[PROOFSTEP]\nsimp only [Sequence\u2082, mk_uLift, mk_fintype, Fintype.card_of_isEmpty, Nat.cast_zero, lift_zero]\n[GOAL]\na\u2080 a\u2081 a\u2082 : Type u\n\u22a2 (sum fun i => #(Sequence\u2082 a\u2080 a\u2081 a\u2082 i)) = #a\u2080 + #a\u2081 + #a\u2082\n[PROOFSTEP]\nrw [sum_nat_eq_add_sum_succ, sum_nat_eq_add_sum_succ, sum_nat_eq_add_sum_succ]\n[GOAL]\na\u2080 a\u2081 a\u2082 : Type u\n\u22a2 #(Sequence\u2082 a\u2080 a\u2081 a\u2082 0) +\n (#(Sequence\u2082 a\u2080 a\u2081 a\u2082 (0 + 1)) +\n (#(Sequence\u2082 a\u2080 a\u2081 a\u2082 (0 + 1 + 1)) + sum fun i => #(Sequence\u2082 a\u2080 a\u2081 a\u2082 (i + 1 + 1 + 1)))) =\n #a\u2080 + #a\u2081 + #a\u2082\n[PROOFSTEP]\nsimp [add_assoc, Sequence\u2082]\n[GOAL]\nL : Language\nL' : Language\n\u22a2 card L = (sum fun l => lift #(Functions L l)) + sum fun l => lift #(Relations L l)\n[PROOFSTEP]\nsimp [card, Symbols]\n[GOAL]\nL : Language\nL' : Language\n\u22a2 card Language.empty = 0\n[PROOFSTEP]\nsimp [card_eq_card_functions_add_card_relations]\n[GOAL]\nL : Language\nL' : Language\n\u22a2 IsEmpty (Symbols Language.empty)\n[PROOFSTEP]\nsimp only [Language.Symbols, isEmpty_sum, isEmpty_sigma]\n[GOAL]\nL : Language\nL' : Language\n\u22a2 (\u2200 (a : \u2115), IsEmpty (Functions Language.empty a)) \u2227 \u2200 (a : \u2115), IsEmpty (Relations Language.empty a)\n[PROOFSTEP]\nexact \u27e8fun _ => inferInstance, fun _ => inferInstance\u27e9\n[GOAL]\nL : Language\nL' : Language\ni : \u2115\n\u22a2 #(Functions (Language.sum L L') i) = lift #(Functions L i) + lift #(Functions L' i)\n[PROOFSTEP]\nsimp [Language.sum]\n[GOAL]\nL : Language\nL' : Language\ni : \u2115\n\u22a2 #(Relations (Language.sum L L') i) = lift #(Relations L i) + lift #(Relations L' i)\n[PROOFSTEP]\nsimp [Language.sum]\n[GOAL]\nL : Language\nL' : Language\n\u22a2 card (Language.sum L L') = lift (card L) + lift (card L')\n[PROOFSTEP]\nsimp only [card_eq_card_functions_add_card_relations, card_functions_sum, card_relations_sum, sum_add_distrib',\n lift_add, lift_sum, lift_lift]\n[GOAL]\nL : Language\nL' : Language\n\u22a2 ((sum fun i => lift #(Functions L i)) + sum fun i => lift #(Functions L' i)) +\n ((sum fun i => lift #(Relations L i)) + sum fun i => lift #(Relations L' i)) =\n ((sum fun i => lift #(Functions L i)) + sum fun i => lift #(Relations L i)) +\n ((sum fun i => lift #(Functions L' i)) + sum fun i => lift #(Relations L' i))\n[PROOFSTEP]\nsimp only [add_assoc, add_comm (Cardinal.sum fun i => (#(L'.Functions i)).lift)]\n[GOAL]\nL : Language\nL' : Language\nc f\u2081 f\u2082 : Type u\nr\u2081 r\u2082 : Type v\n\u22a2 card (Language.mk\u2082 c f\u2081 f\u2082 r\u2081 r\u2082) = lift #c + lift #f\u2081 + lift #f\u2082 + lift #r\u2081 + lift #r\u2082\n[PROOFSTEP]\nsimp [card_eq_card_functions_add_card_relations, add_assoc]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nf g : M \u2192[L] N\nh : f.toFun = g.toFun\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\ng : M \u2192[L] N\ntoFun\u271d : M \u2192 N\nmap_fun'\u271d : \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M), toFun\u271d (funMap f x) = funMap f (toFun\u271d \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r x \u2192 RelMap r (toFun\u271d \u2218 x)\nh : (mk toFun\u271d).toFun = g.toFun\n\u22a2 mk toFun\u271d = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\ntoFun\u271d\u00b9 : M \u2192 N\nmap_fun'\u271d\u00b9 : \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M), toFun\u271d\u00b9 (funMap f x) = funMap f (toFun\u271d\u00b9 \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r x \u2192 RelMap r (toFun\u271d\u00b9 \u2218 x)\ntoFun\u271d : M \u2192 N\nmap_fun'\u271d : \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M), toFun\u271d (funMap f x) = funMap f (toFun\u271d \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r x \u2192 RelMap r (toFun\u271d \u2218 x)\nh : (mk toFun\u271d\u00b9).toFun = (mk toFun\u271d).toFun\n\u22a2 mk toFun\u271d\u00b9 = mk toFun\u271d\n[PROOFSTEP]\ncases h\n[GOAL]\ncase mk.mk.refl\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\ntoFun\u271d : M \u2192 N\nmap_fun'\u271d\u00b9 : \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M), toFun\u271d (funMap f x) = funMap f (toFun\u271d \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r x \u2192 RelMap r (toFun\u271d \u2218 x)\nmap_fun'\u271d : \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M), toFun\u271d (funMap f x) = funMap f (toFun\u271d \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r x \u2192 RelMap r (toFun\u271d \u2218 x)\n\u22a2 mk toFun\u271d = mk toFun\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nhnp : N \u2192[L] P\nhmn : M \u2192[L] N\nn\u271d : \u2115\nx\u271d\u00b9 : Functions L n\u271d\nx\u271d : Fin n\u271d \u2192 M\n\u22a2 (\u2191hnp \u2218 \u2191hmn) (funMap x\u271d\u00b9 x\u271d) = funMap x\u271d\u00b9 ((\u2191hnp \u2218 \u2191hmn) \u2218 x\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nhnp : N \u2192[L] P\nhmn : M \u2192[L] N\nn\u271d : \u2115\nx\u271d\u00b9 : Functions L n\u271d\nx\u271d : Fin n\u271d \u2192 M\n\u22a2 funMap x\u271d\u00b9 (\u2191hnp \u2218 \u2191hmn \u2218 x\u271d) = funMap x\u271d\u00b9 ((\u2191hnp \u2218 \u2191hmn) \u2218 x\u271d)\n[PROOFSTEP]\nrfl\n -- Porting note: should be done by autoparam?\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nf g : M \u21aa[L] N\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\ng : M \u21aa[L] N\ntoEmbedding\u271d : M \u21aa N\nmap_fun'\u271d :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Function.Embedding.toFun toEmbedding\u271d (funMap f x) = funMap f (toEmbedding\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d.toFun \u2218 x) \u2194 RelMap r x\nh : (fun f => f.toFun) (mk toEmbedding\u271d) = (fun f => f.toFun) g\n\u22a2 mk toEmbedding\u271d = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\ntoEmbedding\u271d\u00b9 : M \u21aa N\nmap_fun'\u271d\u00b9 :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Function.Embedding.toFun toEmbedding\u271d\u00b9 (funMap f x) = funMap f (toEmbedding\u271d\u00b9.toFun \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d\u00b9.toFun \u2218 x) \u2194 RelMap r x\ntoEmbedding\u271d : M \u21aa N\nmap_fun'\u271d :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Function.Embedding.toFun toEmbedding\u271d (funMap f x) = funMap f (toEmbedding\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d.toFun \u2218 x) \u2194 RelMap r x\nh : (fun f => f.toFun) (mk toEmbedding\u271d\u00b9) = (fun f => f.toFun) (mk toEmbedding\u271d)\n\u22a2 mk toEmbedding\u271d\u00b9 = mk toEmbedding\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toEmbedding\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\ntoEmbedding\u271d\u00b9 : M \u21aa N\nmap_fun'\u271d\u00b9 :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Function.Embedding.toFun toEmbedding\u271d\u00b9 (funMap f x) = funMap f (toEmbedding\u271d\u00b9.toFun \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d\u00b9.toFun \u2218 x) \u2194 RelMap r x\ntoEmbedding\u271d : M \u21aa N\nmap_fun'\u271d :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Function.Embedding.toFun toEmbedding\u271d (funMap f x) = funMap f (toEmbedding\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d.toFun \u2218 x) \u2194 RelMap r x\nh : (fun f => f.toFun) (mk toEmbedding\u271d\u00b9) = (fun f => f.toFun) (mk toEmbedding\u271d)\n\u22a2 toEmbedding\u271d\u00b9 = toEmbedding\u271d\n[PROOFSTEP]\next x\n[GOAL]\ncase mk.mk.e_toEmbedding.h\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\ntoEmbedding\u271d\u00b9 : M \u21aa N\nmap_fun'\u271d\u00b9 :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Function.Embedding.toFun toEmbedding\u271d\u00b9 (funMap f x) = funMap f (toEmbedding\u271d\u00b9.toFun \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d\u00b9.toFun \u2218 x) \u2194 RelMap r x\ntoEmbedding\u271d : M \u21aa N\nmap_fun'\u271d :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Function.Embedding.toFun toEmbedding\u271d (funMap f x) = funMap f (toEmbedding\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d.toFun \u2218 x) \u2194 RelMap r x\nh : (fun f => f.toFun) (mk toEmbedding\u271d\u00b9) = (fun f => f.toFun) (mk toEmbedding\u271d)\nx : M\n\u22a2 \u2191toEmbedding\u271d\u00b9 x = \u2191toEmbedding\u271d x\n[PROOFSTEP]\nexact Function.funext_iff.1 h x\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nf g : M \u21aa[L] N\nh : \u2191f = \u2191g\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\ng : M \u21aa[L] N\ntoEmbedding\u271d : M \u21aa N\nmap_fun'\u271d :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Function.Embedding.toFun toEmbedding\u271d (funMap f x) = funMap f (toEmbedding\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d.toFun \u2218 x) \u2194 RelMap r x\nh : \u2191(mk toEmbedding\u271d) = \u2191g\n\u22a2 mk toEmbedding\u271d = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\ntoEmbedding\u271d\u00b9 : M \u21aa N\nmap_fun'\u271d\u00b9 :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Function.Embedding.toFun toEmbedding\u271d\u00b9 (funMap f x) = funMap f (toEmbedding\u271d\u00b9.toFun \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d\u00b9.toFun \u2218 x) \u2194 RelMap r x\ntoEmbedding\u271d : M \u21aa N\nmap_fun'\u271d :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Function.Embedding.toFun toEmbedding\u271d (funMap f x) = funMap f (toEmbedding\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d.toFun \u2218 x) \u2194 RelMap r x\nh : \u2191(mk toEmbedding\u271d\u00b9) = \u2191(mk toEmbedding\u271d)\n\u22a2 mk toEmbedding\u271d\u00b9 = mk toEmbedding\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toEmbedding\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\ntoEmbedding\u271d\u00b9 : M \u21aa N\nmap_fun'\u271d\u00b9 :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Function.Embedding.toFun toEmbedding\u271d\u00b9 (funMap f x) = funMap f (toEmbedding\u271d\u00b9.toFun \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d\u00b9.toFun \u2218 x) \u2194 RelMap r x\ntoEmbedding\u271d : M \u21aa N\nmap_fun'\u271d :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Function.Embedding.toFun toEmbedding\u271d (funMap f x) = funMap f (toEmbedding\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d.toFun \u2218 x) \u2194 RelMap r x\nh : \u2191(mk toEmbedding\u271d\u00b9) = \u2191(mk toEmbedding\u271d)\n\u22a2 toEmbedding\u271d\u00b9 = toEmbedding\u271d\n[PROOFSTEP]\next x\n[GOAL]\ncase mk.mk.e_toEmbedding.h\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\ntoEmbedding\u271d\u00b9 : M \u21aa N\nmap_fun'\u271d\u00b9 :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Function.Embedding.toFun toEmbedding\u271d\u00b9 (funMap f x) = funMap f (toEmbedding\u271d\u00b9.toFun \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d\u00b9.toFun \u2218 x) \u2194 RelMap r x\ntoEmbedding\u271d : M \u21aa N\nmap_fun'\u271d :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Function.Embedding.toFun toEmbedding\u271d (funMap f x) = funMap f (toEmbedding\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEmbedding\u271d.toFun \u2218 x) \u2194 RelMap r x\nh : \u2191(mk toEmbedding\u271d\u00b9) = \u2191(mk toEmbedding\u271d)\nx : M\n\u22a2 \u2191toEmbedding\u271d\u00b9 x = \u2191toEmbedding\u271d x\n[PROOFSTEP]\nexact Function.funext_iff.1 h x\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\nP : Type u_1\ninst\u271d\u00b2 : Structure L P\nQ : Type u_2\ninst\u271d\u00b9 : Structure L Q\ninst\u271d : IsAlgebraic L\nf : M \u2192[L] N\nhf : Function.Injective \u2191f\n\u22a2 toHom (ofInjective hf) = f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\nP : Type u_1\ninst\u271d\u00b2 : Structure L P\nQ : Type u_2\ninst\u271d\u00b9 : Structure L Q\ninst\u271d : IsAlgebraic L\nf : M \u2192[L] N\nhf : Function.Injective \u2191f\nx\u271d : M\n\u22a2 \u2191(toHom (ofInjective hf)) x\u271d = \u2191f x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nhnp : N \u21aa[L] P\nhmn : M \u21aa[L] N\n\u22a2 \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Function.Embedding.toFun { toFun := \u2191hnp \u2218 \u2191hmn, inj' := (_ : Function.Injective (\u2191hnp \u2218 \u2191hmn)) } (funMap f x) =\n funMap f ({ toFun := \u2191hnp \u2218 \u2191hmn, inj' := (_ : Function.Injective (\u2191hnp \u2218 \u2191hmn)) }.toFun \u2218 x)\n[PROOFSTEP]\nintros\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nhnp : N \u21aa[L] P\nhmn : M \u21aa[L] N\nn\u271d : \u2115\nf\u271d : Functions L n\u271d\nx\u271d : Fin n\u271d \u2192 M\n\u22a2 Function.Embedding.toFun { toFun := \u2191hnp \u2218 \u2191hmn, inj' := (_ : Function.Injective (\u2191hnp \u2218 \u2191hmn)) } (funMap f\u271d x\u271d) =\n funMap f\u271d ({ toFun := \u2191hnp \u2218 \u2191hmn, inj' := (_ : Function.Injective (\u2191hnp \u2218 \u2191hmn)) }.toFun \u2218 x\u271d)\n[PROOFSTEP]\nsimp only [Function.comp_apply, map_fun]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nhnp : N \u21aa[L] P\nhmn : M \u21aa[L] N\nn\u271d : \u2115\nf\u271d : Functions L n\u271d\nx\u271d : Fin n\u271d \u2192 M\n\u22a2 funMap f\u271d (\u2191hnp \u2218 \u2191hmn \u2218 x\u271d) = funMap f\u271d ((\u2191hnp \u2218 \u2191hmn) \u2218 x\u271d)\n[PROOFSTEP]\ntrivial\n -- Porting note: should be done by autoparam?\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nhnp : N \u21aa[L] P\nhmn : M \u21aa[L] N\n\u22a2 \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M),\n RelMap r ({ toFun := \u2191hnp \u2218 \u2191hmn, inj' := (_ : Function.Injective (\u2191hnp \u2218 \u2191hmn)) }.toFun \u2218 x) \u2194 RelMap r x\n[PROOFSTEP]\nintros\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nhnp : N \u21aa[L] P\nhmn : M \u21aa[L] N\nn\u271d : \u2115\nr\u271d : Relations L n\u271d\nx\u271d : Fin n\u271d \u2192 M\n\u22a2 RelMap r\u271d ({ toFun := \u2191hnp \u2218 \u2191hmn, inj' := (_ : Function.Injective (\u2191hnp \u2218 \u2191hmn)) }.toFun \u2218 x\u271d) \u2194 RelMap r\u271d x\u271d\n[PROOFSTEP]\nrw [Function.comp.assoc, map_rel, map_rel]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nhnp : N \u21aa[L] P\nhmn : M \u21aa[L] N\n\u22a2 toHom (comp hnp hmn) = Hom.comp (toHom hnp) (toHom hmn)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nhnp : N \u21aa[L] P\nhmn : M \u21aa[L] N\nx\u271d : M\n\u22a2 \u2191(toHom (comp hnp hmn)) x\u271d = \u2191(Hom.comp (toHom hnp) (toHom hmn)) x\u271d\n[PROOFSTEP]\nsimp only [coe_toHom, comp_apply, Hom.comp_apply]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nf g : M \u2243[L] N\nh\u2081 : (fun f => f.toFun) f = (fun f => f.toFun) g\nh\u2082 : (fun f => f.invFun) f = (fun f => f.invFun) g\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\ng : M \u2243[L] N\ntoEquiv\u271d : M \u2243 N\nmap_fun'\u271d :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M), Equiv.toFun toEquiv\u271d (funMap f x) = funMap f (toEquiv\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEquiv\u271d.toFun \u2218 x) \u2194 RelMap r x\nh\u2081 : (fun f => f.toFun) (mk toEquiv\u271d) = (fun f => f.toFun) g\nh\u2082 : (fun f => f.invFun) (mk toEquiv\u271d) = (fun f => f.invFun) g\n\u22a2 mk toEquiv\u271d = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\ntoEquiv\u271d\u00b9 : M \u2243 N\nmap_fun'\u271d\u00b9 :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M), Equiv.toFun toEquiv\u271d\u00b9 (funMap f x) = funMap f (toEquiv\u271d\u00b9.toFun \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEquiv\u271d\u00b9.toFun \u2218 x) \u2194 RelMap r x\ntoEquiv\u271d : M \u2243 N\nmap_fun'\u271d :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M), Equiv.toFun toEquiv\u271d (funMap f x) = funMap f (toEquiv\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEquiv\u271d.toFun \u2218 x) \u2194 RelMap r x\nh\u2081 : (fun f => f.toFun) (mk toEquiv\u271d\u00b9) = (fun f => f.toFun) (mk toEquiv\u271d)\nh\u2082 : (fun f => f.invFun) (mk toEquiv\u271d\u00b9) = (fun f => f.invFun) (mk toEquiv\u271d)\n\u22a2 mk toEquiv\u271d\u00b9 = mk toEquiv\u271d\n[PROOFSTEP]\nsimp only [mk.injEq]\n[GOAL]\ncase mk.mk\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\ntoEquiv\u271d\u00b9 : M \u2243 N\nmap_fun'\u271d\u00b9 :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M), Equiv.toFun toEquiv\u271d\u00b9 (funMap f x) = funMap f (toEquiv\u271d\u00b9.toFun \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEquiv\u271d\u00b9.toFun \u2218 x) \u2194 RelMap r x\ntoEquiv\u271d : M \u2243 N\nmap_fun'\u271d :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M), Equiv.toFun toEquiv\u271d (funMap f x) = funMap f (toEquiv\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEquiv\u271d.toFun \u2218 x) \u2194 RelMap r x\nh\u2081 : (fun f => f.toFun) (mk toEquiv\u271d\u00b9) = (fun f => f.toFun) (mk toEquiv\u271d)\nh\u2082 : (fun f => f.invFun) (mk toEquiv\u271d\u00b9) = (fun f => f.invFun) (mk toEquiv\u271d)\n\u22a2 toEquiv\u271d\u00b9 = toEquiv\u271d\n[PROOFSTEP]\next x\n[GOAL]\ncase mk.mk.H\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\ntoEquiv\u271d\u00b9 : M \u2243 N\nmap_fun'\u271d\u00b9 :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M), Equiv.toFun toEquiv\u271d\u00b9 (funMap f x) = funMap f (toEquiv\u271d\u00b9.toFun \u2218 x)\nmap_rel'\u271d\u00b9 : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEquiv\u271d\u00b9.toFun \u2218 x) \u2194 RelMap r x\ntoEquiv\u271d : M \u2243 N\nmap_fun'\u271d :\n \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M), Equiv.toFun toEquiv\u271d (funMap f x) = funMap f (toEquiv\u271d.toFun \u2218 x)\nmap_rel'\u271d : \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M), RelMap r (toEquiv\u271d.toFun \u2218 x) \u2194 RelMap r x\nh\u2081 : (fun f => f.toFun) (mk toEquiv\u271d\u00b9) = (fun f => f.toFun) (mk toEquiv\u271d)\nh\u2082 : (fun f => f.invFun) (mk toEquiv\u271d\u00b9) = (fun f => f.invFun) (mk toEquiv\u271d)\nx : M\n\u22a2 \u2191toEquiv\u271d\u00b9 x = \u2191toEquiv\u271d x\n[PROOFSTEP]\nexact Function.funext_iff.1 h\u2081 x\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nf : M \u2243[L] N\nsrc\u271d : N \u2243 M := f.symm\nn : \u2115\nf' : Functions L n\nx : Fin n \u2192 N\n\u22a2 Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (funMap f' x) =\n funMap f'\n ({ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }.toFun \u2218\n x)\n[PROOFSTEP]\nsimp only [Equiv.toFun_as_coe]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nf : M \u2243[L] N\nsrc\u271d : N \u2243 M := f.symm\nn : \u2115\nf' : Functions L n\nx : Fin n \u2192 N\n\u22a2 \u2191f.symm (funMap f' x) = funMap f' (\u2191f.symm \u2218 x)\n[PROOFSTEP]\nrw [Equiv.symm_apply_eq]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nf : M \u2243[L] N\nsrc\u271d : N \u2243 M := f.symm\nn : \u2115\nf' : Functions L n\nx : Fin n \u2192 N\n\u22a2 funMap f' x = \u2191f.toEquiv (funMap f' (\u2191f.symm \u2218 x))\n[PROOFSTEP]\nrefine' Eq.trans _ (f.map_fun' f' (f.toEquiv.symm \u2218 x)).symm\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nf : M \u2243[L] N\nsrc\u271d : N \u2243 M := f.symm\nn : \u2115\nf' : Functions L n\nx : Fin n \u2192 N\n\u22a2 funMap f' x = funMap f' (f.toFun \u2218 \u2191f.symm \u2218 x)\n[PROOFSTEP]\nrw [\u2190 Function.comp.assoc, Equiv.toFun_as_coe, Equiv.self_comp_symm, Function.comp.left_id]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nf : M \u2243[L] N\nsrc\u271d : N \u2243 M := f.symm\nn : \u2115\nr : Relations L n\nx : Fin n \u2192 N\n\u22a2 RelMap r\n ({ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }.toFun \u2218\n x) \u2194\n RelMap r x\n[PROOFSTEP]\nsimp only [Equiv.toFun_as_coe]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nf : M \u2243[L] N\nsrc\u271d : N \u2243 M := f.symm\nn : \u2115\nr : Relations L n\nx : Fin n \u2192 N\n\u22a2 RelMap r (\u2191f.symm \u2218 x) \u2194 RelMap r x\n[PROOFSTEP]\nrefine' (f.map_rel' r (f.toEquiv.symm \u2218 x)).symm.trans _\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nf : M \u2243[L] N\nsrc\u271d : N \u2243 M := f.symm\nn : \u2115\nr : Relations L n\nx : Fin n \u2192 N\n\u22a2 RelMap r (f.toFun \u2218 \u2191f.symm \u2218 x) \u2194 RelMap r x\n[PROOFSTEP]\nrw [\u2190 Function.comp.assoc, Equiv.toFun_as_coe, Equiv.self_comp_symm, Function.comp.left_id]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nx : M\n\u22a2 \u2191(refl L M) x = x\n[PROOFSTEP]\nsimp [refl]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nx : M\n\u22a2 \u2191(mk (_root_.Equiv.refl M)) x = x\n[PROOFSTEP]\nrfl\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nhnp : N \u2243[L] P\nhmn : M \u2243[L] N\nsrc\u271d : M \u2243 P := hmn.trans hnp.toEquiv\n\u22a2 \u2200 {n : \u2115} (f : Functions L n) (x : Fin n \u2192 M),\n Equiv.toFun\n { toFun := \u2191hnp \u2218 \u2191hmn, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (funMap f x) =\n funMap f\n ({ toFun := \u2191hnp \u2218 \u2191hmn, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }.toFun \u2218\n x)\n[PROOFSTEP]\nintros\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nhnp : N \u2243[L] P\nhmn : M \u2243[L] N\nsrc\u271d : M \u2243 P := hmn.trans hnp.toEquiv\nn\u271d : \u2115\nf\u271d : Functions L n\u271d\nx\u271d : Fin n\u271d \u2192 M\n\u22a2 Equiv.toFun\n { toFun := \u2191hnp \u2218 \u2191hmn, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (funMap f\u271d x\u271d) =\n funMap f\u271d\n ({ toFun := \u2191hnp \u2218 \u2191hmn, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }.toFun \u2218\n x\u271d)\n[PROOFSTEP]\nsimp only [Function.comp_apply, map_fun]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nhnp : N \u2243[L] P\nhmn : M \u2243[L] N\nsrc\u271d : M \u2243 P := hmn.trans hnp.toEquiv\nn\u271d : \u2115\nf\u271d : Functions L n\u271d\nx\u271d : Fin n\u271d \u2192 M\n\u22a2 funMap f\u271d (\u2191hnp \u2218 \u2191hmn \u2218 x\u271d) = funMap f\u271d ((\u2191hnp \u2218 \u2191hmn) \u2218 x\u271d)\n[PROOFSTEP]\ntrivial\n -- Porting note: should be done by autoparam?\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nhnp : N \u2243[L] P\nhmn : M \u2243[L] N\nsrc\u271d : M \u2243 P := hmn.trans hnp.toEquiv\n\u22a2 \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 M),\n RelMap r\n ({ toFun := \u2191hnp \u2218 \u2191hmn, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }.toFun \u2218\n x) \u2194\n RelMap r x\n[PROOFSTEP]\nintros\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\nhnp : N \u2243[L] P\nhmn : M \u2243[L] N\nsrc\u271d : M \u2243 P := hmn.trans hnp.toEquiv\nn\u271d : \u2115\nr\u271d : Relations L n\u271d\nx\u271d : Fin n\u271d \u2192 M\n\u22a2 RelMap r\u271d\n ({ toFun := \u2191hnp \u2218 \u2191hmn, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }.toFun \u2218\n x\u271d) \u2194\n RelMap r\u271d x\u271d\n[PROOFSTEP]\nrw [Function.comp.assoc, map_rel, map_rel]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\na : Structure Language.empty M\n\u22a2 a = default\n[PROOFSTEP]\next _ f\n[GOAL]\ncase funMap.h.h.h\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\na : Structure Language.empty M\nx\u271d\u00b9 : \u2115\nf : Functions Language.empty x\u271d\u00b9\nx\u271d : Fin x\u271d\u00b9 \u2192 M\n\u22a2 funMap f x\u271d = funMap f x\u271d\n[PROOFSTEP]\nexact Empty.elim f\n[GOAL]\ncase RelMap.h.h.h.a\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\nP : Type u_1\ninst\u271d\u00b9 : Structure L P\nQ : Type u_2\ninst\u271d : Structure L Q\na : Structure Language.empty M\nx\u271d\u00b9 : \u2115\nf : Relations Language.empty x\u271d\u00b9\nx\u271d : Fin x\u271d\u00b9 \u2192 M\n\u22a2 RelMap f x\u271d \u2194 RelMap f x\u271d\n[PROOFSTEP]\nexact Empty.elim f\n[GOAL]\nL : Language\nM : Type u_1\nN : Type u_2\ninst\u271d : Structure L M\ne : M \u2243 N\n\u22a2 M \u2243[L] N\n[PROOFSTEP]\nletI : L.Structure N := inducedStructure e\n[GOAL]\nL : Language\nM : Type u_1\nN : Type u_2\ninst\u271d : Structure L M\ne : M \u2243 N\nthis : Structure L N := inducedStructure e\n\u22a2 M \u2243[L] N\n[PROOFSTEP]\nexact\n { e with\n map_fun' := @fun n f x => by simp [\u2190 Function.comp.assoc e.symm e x]\n map_rel' := @fun n r x => by simp [\u2190 Function.comp.assoc e.symm e x] }\n[GOAL]\nL : Language\nM : Type u_1\nN : Type u_2\ninst\u271d : Structure L M\ne : M \u2243 N\nthis : Structure L N := inducedStructure e\nn : \u2115\nf : Functions L n\nx : Fin n \u2192 M\n\u22a2 toFun\n { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n (funMap f x) =\n funMap f\n ({ toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }.toFun \u2218\n x)\n[PROOFSTEP]\nsimp [\u2190 Function.comp.assoc e.symm e x]\n[GOAL]\nL : Language\nM : Type u_1\nN : Type u_2\ninst\u271d : Structure L M\ne : M \u2243 N\nthis : Structure L N := inducedStructure e\nn : \u2115\nr : Relations L n\nx : Fin n \u2192 M\n\u22a2 RelMap r\n ({ toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n right_inv := (_ : Function.RightInverse e.invFun e.toFun) }.toFun \u2218\n x) \u2194\n RelMap r x\n[PROOFSTEP]\nsimp [\u2190 Function.comp.assoc e.symm e x]\n[GOAL]\nL : Language\nM : Type u_1\nN : Type u_2\ninst\u271d : Structure L M\ne : M \u2243 N\n\u22a2 (a : N) \u2192 (fun x => M) a\n[PROOFSTEP]\nletI : L.Structure N := inducedStructure e\n[GOAL]\nL : Language\nM : Type u_1\nN : Type u_2\ninst\u271d : Structure L M\ne : M \u2243 N\nthis : Structure L N := inducedStructure e\n\u22a2 (a : N) \u2192 (fun x => M) a\n[PROOFSTEP]\nexact FunLike.coe (@inducedStructureEquiv L M N _ e).symm\n", "meta": {"mathlib_filename": "Mathlib.ModelTheory.Basic", "llama_tokens": 14990, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300449389326, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.4242802582875494}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\n\u22a2 Equivalence fun rs rs' => \u2203 u v, rs'.fst * \u2191u = rs.fst * v \u2227 \u2191rs'.snd * \u2191u = \u2191rs.snd * v\n[PROOFSTEP]\nrefine \u27e8fun _ => \u27e81, 1, by simp\u27e9, ?_, ?_\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nx\u271d : R \u00d7 { x // x \u2208 S }\n\u22a2 x\u271d.fst * \u21911 = x\u271d.fst * 1 \u2227 \u2191x\u271d.snd * \u21911 = \u2191x\u271d.snd * 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_1\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\n\u22a2 \u2200 {x y : R \u00d7 { x // x \u2208 S }},\n (\u2203 u v, y.fst * \u2191u = x.fst * v \u2227 \u2191y.snd * \u2191u = \u2191x.snd * v) \u2192\n \u2203 u v, x.fst * \u2191u = y.fst * v \u2227 \u2191x.snd * \u2191u = \u2191y.snd * v\n[PROOFSTEP]\nrintro \u27e8r, s\u27e9 \u27e8r', s'\u27e9 \u27e8u, v, hru, hsu\u27e9\n[GOAL]\ncase refine_1.mk.mk.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nr' : R\ns' u : { x // x \u2208 S }\nv : R\nhru : (r', s').fst * \u2191u = (r, s).fst * v\nhsu : \u2191(r', s').snd * \u2191u = \u2191(r, s).snd * v\n\u22a2 \u2203 u v, (r, s).fst * \u2191u = (r', s').fst * v \u2227 \u2191(r, s).snd * \u2191u = \u2191(r', s').snd * v\n[PROOFSTEP]\ndsimp only at *\n[GOAL]\ncase refine_1.mk.mk.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nr' : R\ns' u : { x // x \u2208 S }\nv : R\nhru : r' * \u2191u = r * v\nhsu : \u2191s' * \u2191u = \u2191s * v\n\u22a2 \u2203 u v, r * \u2191u = r' * v \u2227 \u2191s * \u2191u = \u2191s' * v\n[PROOFSTEP]\nrcases oreCondition (s : R) s' with \u27e8r\u2082, s\u2082, h\u2081\u27e9\n[GOAL]\ncase refine_1.mk.mk.intro.intro.intro.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nr' : R\ns' u : { x // x \u2208 S }\nv : R\nhru : r' * \u2191u = r * v\nhsu : \u2191s' * \u2191u = \u2191s * v\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nh\u2081 : \u2191s * \u2191s\u2082 = \u2191s' * r\u2082\n\u22a2 \u2203 u v, r * \u2191u = r' * v \u2227 \u2191s * \u2191u = \u2191s' * v\n[PROOFSTEP]\nrcases oreCondition r\u2082 u with \u27e8r\u2083, s\u2083, h\u2082\u27e9\n[GOAL]\ncase refine_1.mk.mk.intro.intro.intro.mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nr' : R\ns' u : { x // x \u2208 S }\nv : R\nhru : r' * \u2191u = r * v\nhsu : \u2191s' * \u2191u = \u2191s * v\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nh\u2081 : \u2191s * \u2191s\u2082 = \u2191s' * r\u2082\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2082 : r\u2082 * \u2191s\u2083 = \u2191u * r\u2083\n\u22a2 \u2203 u v, r * \u2191u = r' * v \u2227 \u2191s * \u2191u = \u2191s' * v\n[PROOFSTEP]\nhave : (s : R) * ((v : R) * r\u2083) = (s : R) * (s\u2082 * s\u2083) := by\n -- porting note: the proof used `assoc_rw`rw [\u2190 mul_assoc _ (s\u2082 : R), h\u2081, mul_assoc, h\u2082, \u2190 mul_assoc, \u2190 hsu,\n mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nr' : R\ns' u : { x // x \u2208 S }\nv : R\nhru : r' * \u2191u = r * v\nhsu : \u2191s' * \u2191u = \u2191s * v\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nh\u2081 : \u2191s * \u2191s\u2082 = \u2191s' * r\u2082\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2082 : r\u2082 * \u2191s\u2083 = \u2191u * r\u2083\n\u22a2 \u2191s * (v * r\u2083) = \u2191s * (\u2191s\u2082 * \u2191s\u2083)\n[PROOFSTEP]\nrw [\u2190 mul_assoc _ (s\u2082 : R), h\u2081, mul_assoc, h\u2082, \u2190 mul_assoc, \u2190 hsu, mul_assoc]\n[GOAL]\ncase refine_1.mk.mk.intro.intro.intro.mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nr' : R\ns' u : { x // x \u2208 S }\nv : R\nhru : r' * \u2191u = r * v\nhsu : \u2191s' * \u2191u = \u2191s * v\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nh\u2081 : \u2191s * \u2191s\u2082 = \u2191s' * r\u2082\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2082 : r\u2082 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s * (v * r\u2083) = \u2191s * (\u2191s\u2082 * \u2191s\u2083)\n\u22a2 \u2203 u v, r * \u2191u = r' * v \u2227 \u2191s * \u2191u = \u2191s' * v\n[PROOFSTEP]\nrcases ore_left_cancel (v * r\u2083) (s\u2082 * s\u2083) s this with \u27e8w, hw\u27e9\n[GOAL]\ncase refine_1.mk.mk.intro.intro.intro.mk.mk.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nr' : R\ns' u : { x // x \u2208 S }\nv : R\nhru : r' * \u2191u = r * v\nhsu : \u2191s' * \u2191u = \u2191s * v\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nh\u2081 : \u2191s * \u2191s\u2082 = \u2191s' * r\u2082\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2082 : r\u2082 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s * (v * r\u2083) = \u2191s * (\u2191s\u2082 * \u2191s\u2083)\nw : { x // x \u2208 S }\nhw : v * r\u2083 * \u2191w = \u2191s\u2082 * \u2191s\u2083 * \u2191w\n\u22a2 \u2203 u v, r * \u2191u = r' * v \u2227 \u2191s * \u2191u = \u2191s' * v\n[PROOFSTEP]\nrefine \u27e8s\u2082 * s\u2083 * w, u * r\u2083 * w, ?_, ?_\u27e9\n[GOAL]\ncase refine_1.mk.mk.intro.intro.intro.mk.mk.mk.mk.intro.refine_1\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nr' : R\ns' u : { x // x \u2208 S }\nv : R\nhru : r' * \u2191u = r * v\nhsu : \u2191s' * \u2191u = \u2191s * v\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nh\u2081 : \u2191s * \u2191s\u2082 = \u2191s' * r\u2082\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2082 : r\u2082 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s * (v * r\u2083) = \u2191s * (\u2191s\u2082 * \u2191s\u2083)\nw : { x // x \u2208 S }\nhw : v * r\u2083 * \u2191w = \u2191s\u2082 * \u2191s\u2083 * \u2191w\n\u22a2 r * \u2191(s\u2082 * s\u2083 * w) = r' * (\u2191u * r\u2083 * \u2191w)\n[PROOFSTEP]\nsimp only [Submonoid.coe_mul, \u2190 hw]\n[GOAL]\ncase refine_1.mk.mk.intro.intro.intro.mk.mk.mk.mk.intro.refine_2\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nr' : R\ns' u : { x // x \u2208 S }\nv : R\nhru : r' * \u2191u = r * v\nhsu : \u2191s' * \u2191u = \u2191s * v\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nh\u2081 : \u2191s * \u2191s\u2082 = \u2191s' * r\u2082\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2082 : r\u2082 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s * (v * r\u2083) = \u2191s * (\u2191s\u2082 * \u2191s\u2083)\nw : { x // x \u2208 S }\nhw : v * r\u2083 * \u2191w = \u2191s\u2082 * \u2191s\u2083 * \u2191w\n\u22a2 \u2191s * \u2191(s\u2082 * s\u2083 * w) = \u2191s' * (\u2191u * r\u2083 * \u2191w)\n[PROOFSTEP]\nsimp only [Submonoid.coe_mul, \u2190 hw]\n[GOAL]\ncase refine_1.mk.mk.intro.intro.intro.mk.mk.mk.mk.intro.refine_1\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nr' : R\ns' u : { x // x \u2208 S }\nv : R\nhru : r' * \u2191u = r * v\nhsu : \u2191s' * \u2191u = \u2191s * v\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nh\u2081 : \u2191s * \u2191s\u2082 = \u2191s' * r\u2082\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2082 : r\u2082 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s * (v * r\u2083) = \u2191s * (\u2191s\u2082 * \u2191s\u2083)\nw : { x // x \u2208 S }\nhw : v * r\u2083 * \u2191w = \u2191s\u2082 * \u2191s\u2083 * \u2191w\n\u22a2 r * (v * r\u2083 * \u2191w) = r' * (\u2191u * r\u2083 * \u2191w)\n[PROOFSTEP]\nsimp only [\u2190 mul_assoc, hru]\n[GOAL]\ncase refine_1.mk.mk.intro.intro.intro.mk.mk.mk.mk.intro.refine_2\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nr' : R\ns' u : { x // x \u2208 S }\nv : R\nhru : r' * \u2191u = r * v\nhsu : \u2191s' * \u2191u = \u2191s * v\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nh\u2081 : \u2191s * \u2191s\u2082 = \u2191s' * r\u2082\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2082 : r\u2082 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s * (v * r\u2083) = \u2191s * (\u2191s\u2082 * \u2191s\u2083)\nw : { x // x \u2208 S }\nhw : v * r\u2083 * \u2191w = \u2191s\u2082 * \u2191s\u2083 * \u2191w\n\u22a2 \u2191s * (v * r\u2083 * \u2191w) = \u2191s' * (\u2191u * r\u2083 * \u2191w)\n[PROOFSTEP]\nsimp only [\u2190 mul_assoc, hsu]\n[GOAL]\ncase refine_2\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\n\u22a2 \u2200 {x y z : R \u00d7 { x // x \u2208 S }},\n (\u2203 u v, y.fst * \u2191u = x.fst * v \u2227 \u2191y.snd * \u2191u = \u2191x.snd * v) \u2192\n (\u2203 u v, z.fst * \u2191u = y.fst * v \u2227 \u2191z.snd * \u2191u = \u2191y.snd * v) \u2192\n \u2203 u v, z.fst * \u2191u = x.fst * v \u2227 \u2191z.snd * \u2191u = \u2191x.snd * v\n[PROOFSTEP]\nrintro \u27e8r\u2081, s\u2081\u27e9 \u27e8r\u2082, s\u2082\u27e9 \u27e8r\u2083, s\u2083\u27e9 \u27e8u, v, hur\u2081, hs\u2081u\u27e9 \u27e8u', v', hur\u2082, hs\u2082u\u27e9\n[GOAL]\ncase refine_2.mk.mk.mk.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 u : { x // x \u2208 S }\nv : R\nhur\u2081 : (r\u2082, s\u2082).fst * \u2191u = (r\u2081, s\u2081).fst * v\nhs\u2081u : \u2191(r\u2082, s\u2082).snd * \u2191u = \u2191(r\u2081, s\u2081).snd * v\nu' : { x // x \u2208 S }\nv' : R\nhur\u2082 : (r\u2083, s\u2083).fst * \u2191u' = (r\u2082, s\u2082).fst * v'\nhs\u2082u : \u2191(r\u2083, s\u2083).snd * \u2191u' = \u2191(r\u2082, s\u2082).snd * v'\n\u22a2 \u2203 u v, (r\u2083, s\u2083).fst * \u2191u = (r\u2081, s\u2081).fst * v \u2227 \u2191(r\u2083, s\u2083).snd * \u2191u = \u2191(r\u2081, s\u2081).snd * v\n[PROOFSTEP]\nrcases oreCondition v' u with \u27e8r', s', h\u27e9\n[GOAL]\ncase refine_2.mk.mk.mk.intro.intro.intro.intro.intro.intro.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 u : { x // x \u2208 S }\nv : R\nhur\u2081 : (r\u2082, s\u2082).fst * \u2191u = (r\u2081, s\u2081).fst * v\nhs\u2081u : \u2191(r\u2082, s\u2082).snd * \u2191u = \u2191(r\u2081, s\u2081).snd * v\nu' : { x // x \u2208 S }\nv' : R\nhur\u2082 : (r\u2083, s\u2083).fst * \u2191u' = (r\u2082, s\u2082).fst * v'\nhs\u2082u : \u2191(r\u2083, s\u2083).snd * \u2191u' = \u2191(r\u2082, s\u2082).snd * v'\nr' : R\ns' : { x // x \u2208 S }\nh : v' * \u2191s' = \u2191u * r'\n\u22a2 \u2203 u v, (r\u2083, s\u2083).fst * \u2191u = (r\u2081, s\u2081).fst * v \u2227 \u2191(r\u2083, s\u2083).snd * \u2191u = \u2191(r\u2081, s\u2081).snd * v\n[PROOFSTEP]\ndsimp only at *\n[GOAL]\ncase refine_2.mk.mk.mk.intro.intro.intro.intro.intro.intro.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 u : { x // x \u2208 S }\nv : R\nhur\u2081 : r\u2082 * \u2191u = r\u2081 * v\nhs\u2081u : \u2191s\u2082 * \u2191u = \u2191s\u2081 * v\nu' : { x // x \u2208 S }\nv' : R\nhur\u2082 : r\u2083 * \u2191u' = r\u2082 * v'\nhs\u2082u : \u2191s\u2083 * \u2191u' = \u2191s\u2082 * v'\nr' : R\ns' : { x // x \u2208 S }\nh : v' * \u2191s' = \u2191u * r'\n\u22a2 \u2203 u v, r\u2083 * \u2191u = r\u2081 * v \u2227 \u2191s\u2083 * \u2191u = \u2191s\u2081 * v\n[PROOFSTEP]\nrefine \u27e8u' * s', v * r', ?_, ?_\u27e9\n[GOAL]\ncase refine_2.mk.mk.mk.intro.intro.intro.intro.intro.intro.mk.mk.refine_1\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 u : { x // x \u2208 S }\nv : R\nhur\u2081 : r\u2082 * \u2191u = r\u2081 * v\nhs\u2081u : \u2191s\u2082 * \u2191u = \u2191s\u2081 * v\nu' : { x // x \u2208 S }\nv' : R\nhur\u2082 : r\u2083 * \u2191u' = r\u2082 * v'\nhs\u2082u : \u2191s\u2083 * \u2191u' = \u2191s\u2082 * v'\nr' : R\ns' : { x // x \u2208 S }\nh : v' * \u2191s' = \u2191u * r'\n\u22a2 r\u2083 * \u2191(u' * s') = r\u2081 * (v * r')\n[PROOFSTEP]\nsimp only [Submonoid.coe_mul, \u2190 mul_assoc]\n[GOAL]\ncase refine_2.mk.mk.mk.intro.intro.intro.intro.intro.intro.mk.mk.refine_2\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 u : { x // x \u2208 S }\nv : R\nhur\u2081 : r\u2082 * \u2191u = r\u2081 * v\nhs\u2081u : \u2191s\u2082 * \u2191u = \u2191s\u2081 * v\nu' : { x // x \u2208 S }\nv' : R\nhur\u2082 : r\u2083 * \u2191u' = r\u2082 * v'\nhs\u2082u : \u2191s\u2083 * \u2191u' = \u2191s\u2082 * v'\nr' : R\ns' : { x // x \u2208 S }\nh : v' * \u2191s' = \u2191u * r'\n\u22a2 \u2191s\u2083 * \u2191(u' * s') = \u2191s\u2081 * (v * r')\n[PROOFSTEP]\nsimp only [Submonoid.coe_mul, \u2190 mul_assoc]\n[GOAL]\ncase refine_2.mk.mk.mk.intro.intro.intro.intro.intro.intro.mk.mk.refine_1\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 u : { x // x \u2208 S }\nv : R\nhur\u2081 : r\u2082 * \u2191u = r\u2081 * v\nhs\u2081u : \u2191s\u2082 * \u2191u = \u2191s\u2081 * v\nu' : { x // x \u2208 S }\nv' : R\nhur\u2082 : r\u2083 * \u2191u' = r\u2082 * v'\nhs\u2082u : \u2191s\u2083 * \u2191u' = \u2191s\u2082 * v'\nr' : R\ns' : { x // x \u2208 S }\nh : v' * \u2191s' = \u2191u * r'\n\u22a2 r\u2083 * \u2191u' * \u2191s' = r\u2081 * v * r'\n[PROOFSTEP]\nrw [hur\u2082, mul_assoc, h, \u2190 mul_assoc, hur\u2081]\n[GOAL]\ncase refine_2.mk.mk.mk.intro.intro.intro.intro.intro.intro.mk.mk.refine_2\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 u : { x // x \u2208 S }\nv : R\nhur\u2081 : r\u2082 * \u2191u = r\u2081 * v\nhs\u2081u : \u2191s\u2082 * \u2191u = \u2191s\u2081 * v\nu' : { x // x \u2208 S }\nv' : R\nhur\u2082 : r\u2083 * \u2191u' = r\u2082 * v'\nhs\u2082u : \u2191s\u2083 * \u2191u' = \u2191s\u2082 * v'\nr' : R\ns' : { x // x \u2208 S }\nh : v' * \u2191s' = \u2191u * r'\n\u22a2 \u2191s\u2083 * \u2191u' * \u2191s' = \u2191s\u2081 * v * r'\n[PROOFSTEP]\nrw [hs\u2082u, mul_assoc, h, \u2190 mul_assoc, hs\u2081u]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\n\u03b2 : OreLocalization R S \u2192 Prop\nc : \u2200 (r : R) (s : { x // x \u2208 S }), \u03b2 (r /\u2092 s)\n\u22a2 \u2200 (q : OreLocalization R S), \u03b2 q\n[PROOFSTEP]\napply Quotient.ind\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\n\u03b2 : OreLocalization R S \u2192 Prop\nc : \u2200 (r : R) (s : { x // x \u2208 S }), \u03b2 (r /\u2092 s)\n\u22a2 \u2200 (a : R \u00d7 { x // x \u2208 S }), \u03b2 (Quotient.mk (oreEqv R S) a)\n[PROOFSTEP]\nrintro \u27e8r, s\u27e9\n[GOAL]\ncase a.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\n\u03b2 : OreLocalization R S \u2192 Prop\nc : \u2200 (r : R) (s : { x // x \u2208 S }), \u03b2 (r /\u2092 s)\nr : R\ns : { x // x \u2208 S }\n\u22a2 \u03b2 (Quotient.mk (oreEqv R S) (r, s))\n[PROOFSTEP]\nexact c r s\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nt : R\nhst : \u2191s * t \u2208 S\n\u22a2 r /\u2092 s = r * t /\u2092 { val := \u2191s * t, property := hst }\n[PROOFSTEP]\napply Quotient.sound\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nt : R\nhst : \u2191s * t \u2208 S\n\u22a2 (r, s) \u2248 (r * t, { val := \u2191s * t, property := hst })\n[PROOFSTEP]\nrefine' \u27e8s, t * s, _, _\u27e9\n[GOAL]\ncase a.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nt : R\nhst : \u2191s * t \u2208 S\n\u22a2 (r * t, { val := \u2191s * t, property := hst }).fst * \u2191s = (r, s).fst * (t * \u2191s)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nt : R\nhst : \u2191s * t \u2208 S\n\u22a2 \u2191(r * t, { val := \u2191s * t, property := hst }).snd * \u2191s = \u2191(r, s).snd * (t * \u2191s)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nt : R\nhst : \u2191s * t \u2208 S\n\u22a2 r * t * \u2191s = r * (t * \u2191s)\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\ncase a.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\nt : R\nhst : \u2191s * t \u2208 S\n\u22a2 \u2191s * t * \u2191s = \u2191s * (t * \u2191s)\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns s' : { x // x \u2208 S }\n\u22a2 \u2191s * \u2191s' \u2208 S\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns s' : { x // x \u2208 S }\n\u22a2 \u2191(s * s') \u2208 S\n[PROOFSTEP]\napply SetLike.coe_mem\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr r' r\u2081 r\u2082 : R\ns t : { x // x \u2208 S }\nh : r * \u2191t = r' * \u2191t\n\u22a2 r\u2081 * r * r\u2082 /\u2092 s = r\u2081 * r' * r\u2082 /\u2092 s\n[PROOFSTEP]\nrcases oreCondition r\u2082 t with\n \u27e8r\u2082', t', hr\u2082\u27e9\n -- porting note: todo: use `assoc_rw`?\n[GOAL]\ncase mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr r' r\u2081 r\u2082 : R\ns t : { x // x \u2208 S }\nh : r * \u2191t = r' * \u2191t\nr\u2082' : R\nt' : { x // x \u2208 S }\nhr\u2082 : r\u2082 * \u2191t' = \u2191t * r\u2082'\n\u22a2 r\u2081 * r * r\u2082 /\u2092 s = r\u2081 * r' * r\u2082 /\u2092 s\n[PROOFSTEP]\ncalc\n r\u2081 * r * r\u2082 /\u2092 s = r\u2081 * r * r\u2082 * t' /\u2092 (s * t') := OreLocalization.expand (r\u2081 * r * r\u2082) s t' _\n _ = r\u2081 * r * (r\u2082 * t') /\u2092 (s * t') := by simp [\u2190 mul_assoc]\n _ = r\u2081 * r * (t * r\u2082') /\u2092 (s * t') := by rw [hr\u2082]\n _ = r\u2081 * (r * t) * r\u2082' /\u2092 (s * t') := by simp [\u2190 mul_assoc]\n _ = r\u2081 * (r' * t) * r\u2082' /\u2092 (s * t') := by rw [h]\n _ = r\u2081 * r' * (t * r\u2082') /\u2092 (s * t') := by simp [\u2190 mul_assoc]\n _ = r\u2081 * r' * (r\u2082 * t') /\u2092 (s * t') := by rw [hr\u2082]\n _ = r\u2081 * r' * r\u2082 * t' /\u2092 (s * t') := by simp [\u2190 mul_assoc]\n _ = r\u2081 * r' * r\u2082 /\u2092 s := (OreLocalization.expand _ _ _ _).symm\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr r' r\u2081 r\u2082 : R\ns t : { x // x \u2208 S }\nh : r * \u2191t = r' * \u2191t\nr\u2082' : R\nt' : { x // x \u2208 S }\nhr\u2082 : r\u2082 * \u2191t' = \u2191t * r\u2082'\n\u22a2 r\u2081 * r * r\u2082 * \u2191t' /\u2092 (s * t') = r\u2081 * r * (r\u2082 * \u2191t') /\u2092 (s * t')\n[PROOFSTEP]\nsimp [\u2190 mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr r' r\u2081 r\u2082 : R\ns t : { x // x \u2208 S }\nh : r * \u2191t = r' * \u2191t\nr\u2082' : R\nt' : { x // x \u2208 S }\nhr\u2082 : r\u2082 * \u2191t' = \u2191t * r\u2082'\n\u22a2 r\u2081 * r * (r\u2082 * \u2191t') /\u2092 (s * t') = r\u2081 * r * (\u2191t * r\u2082') /\u2092 (s * t')\n[PROOFSTEP]\nrw [hr\u2082]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr r' r\u2081 r\u2082 : R\ns t : { x // x \u2208 S }\nh : r * \u2191t = r' * \u2191t\nr\u2082' : R\nt' : { x // x \u2208 S }\nhr\u2082 : r\u2082 * \u2191t' = \u2191t * r\u2082'\n\u22a2 r\u2081 * r * (\u2191t * r\u2082') /\u2092 (s * t') = r\u2081 * (r * \u2191t) * r\u2082' /\u2092 (s * t')\n[PROOFSTEP]\nsimp [\u2190 mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr r' r\u2081 r\u2082 : R\ns t : { x // x \u2208 S }\nh : r * \u2191t = r' * \u2191t\nr\u2082' : R\nt' : { x // x \u2208 S }\nhr\u2082 : r\u2082 * \u2191t' = \u2191t * r\u2082'\n\u22a2 r\u2081 * (r * \u2191t) * r\u2082' /\u2092 (s * t') = r\u2081 * (r' * \u2191t) * r\u2082' /\u2092 (s * t')\n[PROOFSTEP]\nrw [h]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr r' r\u2081 r\u2082 : R\ns t : { x // x \u2208 S }\nh : r * \u2191t = r' * \u2191t\nr\u2082' : R\nt' : { x // x \u2208 S }\nhr\u2082 : r\u2082 * \u2191t' = \u2191t * r\u2082'\n\u22a2 r\u2081 * (r' * \u2191t) * r\u2082' /\u2092 (s * t') = r\u2081 * r' * (\u2191t * r\u2082') /\u2092 (s * t')\n[PROOFSTEP]\nsimp [\u2190 mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr r' r\u2081 r\u2082 : R\ns t : { x // x \u2208 S }\nh : r * \u2191t = r' * \u2191t\nr\u2082' : R\nt' : { x // x \u2208 S }\nhr\u2082 : r\u2082 * \u2191t' = \u2191t * r\u2082'\n\u22a2 r\u2081 * r' * (\u2191t * r\u2082') /\u2092 (s * t') = r\u2081 * r' * (r\u2082 * \u2191t') /\u2092 (s * t')\n[PROOFSTEP]\nrw [hr\u2082]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr r' r\u2081 r\u2082 : R\ns t : { x // x \u2208 S }\nh : r * \u2191t = r' * \u2191t\nr\u2082' : R\nt' : { x // x \u2208 S }\nhr\u2082 : r\u2082 * \u2191t' = \u2191t * r\u2082'\n\u22a2 r\u2081 * r' * (r\u2082 * \u2191t') /\u2092 (s * t') = r\u2081 * r' * r\u2082 * \u2191t' /\u2092 (s * t')\n[PROOFSTEP]\nsimp [\u2190 mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 C\nhP : \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S), P r s = P (r * t) { val := \u2191s * t, property := ht }\nx\u271d\u00b2 x\u271d\u00b9 : R \u00d7 { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nx\u271d : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\nu : { x // x \u2208 S }\nv : R\nhr\u2082 : (r\u2082, s\u2082).fst * \u2191u = (r\u2081, s\u2081).fst * v\nhs\u2082 : \u2191(r\u2082, s\u2082).snd * \u2191u = \u2191(r\u2081, s\u2081).snd * v\n\u22a2 (fun p => P p.fst p.snd) (r\u2081, s\u2081) = (fun p => P p.fst p.snd) (r\u2082, s\u2082)\n[PROOFSTEP]\ndsimp at *\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 C\nhP : \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S), P r s = P (r * t) { val := \u2191s * t, property := ht }\nx\u271d\u00b2 x\u271d\u00b9 : R \u00d7 { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nx\u271d : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\nu : { x // x \u2208 S }\nv : R\nhr\u2082 : r\u2082 * \u2191u = r\u2081 * v\nhs\u2082 : \u2191s\u2082 * \u2191u = \u2191s\u2081 * v\n\u22a2 P r\u2081 s\u2081 = P r\u2082 s\u2082\n[PROOFSTEP]\nhave s\u2081vS : (s\u2081 : R) * v \u2208 S := by\n rw [\u2190 hs\u2082, \u2190 S.coe_mul]\n exact SetLike.coe_mem (s\u2082 * u)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 C\nhP : \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S), P r s = P (r * t) { val := \u2191s * t, property := ht }\nx\u271d\u00b2 x\u271d\u00b9 : R \u00d7 { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nx\u271d : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\nu : { x // x \u2208 S }\nv : R\nhr\u2082 : r\u2082 * \u2191u = r\u2081 * v\nhs\u2082 : \u2191s\u2082 * \u2191u = \u2191s\u2081 * v\n\u22a2 \u2191s\u2081 * v \u2208 S\n[PROOFSTEP]\nrw [\u2190 hs\u2082, \u2190 S.coe_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 C\nhP : \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S), P r s = P (r * t) { val := \u2191s * t, property := ht }\nx\u271d\u00b2 x\u271d\u00b9 : R \u00d7 { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nx\u271d : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\nu : { x // x \u2208 S }\nv : R\nhr\u2082 : r\u2082 * \u2191u = r\u2081 * v\nhs\u2082 : \u2191s\u2082 * \u2191u = \u2191s\u2081 * v\n\u22a2 \u2191(s\u2082 * u) \u2208 S\n[PROOFSTEP]\nexact SetLike.coe_mem (s\u2082 * u)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 C\nhP : \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S), P r s = P (r * t) { val := \u2191s * t, property := ht }\nx\u271d\u00b2 x\u271d\u00b9 : R \u00d7 { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nx\u271d : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\nu : { x // x \u2208 S }\nv : R\nhr\u2082 : r\u2082 * \u2191u = r\u2081 * v\nhs\u2082 : \u2191s\u2082 * \u2191u = \u2191s\u2081 * v\ns\u2081vS : \u2191s\u2081 * v \u2208 S\n\u22a2 P r\u2081 s\u2081 = P r\u2082 s\u2082\n[PROOFSTEP]\nreplace hs\u2082 : s\u2082 * u = \u27e8(s\u2081 : R) * v, s\u2081vS\u27e9\n[GOAL]\ncase hs\u2082\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 C\nhP : \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S), P r s = P (r * t) { val := \u2191s * t, property := ht }\nx\u271d\u00b2 x\u271d\u00b9 : R \u00d7 { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nx\u271d : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\nu : { x // x \u2208 S }\nv : R\nhr\u2082 : r\u2082 * \u2191u = r\u2081 * v\nhs\u2082 : \u2191s\u2082 * \u2191u = \u2191s\u2081 * v\ns\u2081vS : \u2191s\u2081 * v \u2208 S\n\u22a2 s\u2082 * u = { val := \u2191s\u2081 * v, property := s\u2081vS }\n[PROOFSTEP]\next\n[GOAL]\ncase hs\u2082.a\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 C\nhP : \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S), P r s = P (r * t) { val := \u2191s * t, property := ht }\nx\u271d\u00b2 x\u271d\u00b9 : R \u00d7 { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nx\u271d : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\nu : { x // x \u2208 S }\nv : R\nhr\u2082 : r\u2082 * \u2191u = r\u2081 * v\nhs\u2082 : \u2191s\u2082 * \u2191u = \u2191s\u2081 * v\ns\u2081vS : \u2191s\u2081 * v \u2208 S\n\u22a2 \u2191(s\u2082 * u) = \u2191{ val := \u2191s\u2081 * v, property := s\u2081vS }\n[PROOFSTEP]\nsimp [hs\u2082]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 C\nhP : \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S), P r s = P (r * t) { val := \u2191s * t, property := ht }\nx\u271d\u00b2 x\u271d\u00b9 : R \u00d7 { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nx\u271d : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\nu : { x // x \u2208 S }\nv : R\nhr\u2082 : r\u2082 * \u2191u = r\u2081 * v\ns\u2081vS : \u2191s\u2081 * v \u2208 S\nhs\u2082 : s\u2082 * u = { val := \u2191s\u2081 * v, property := s\u2081vS }\n\u22a2 P r\u2081 s\u2081 = P r\u2082 s\u2082\n[PROOFSTEP]\nrw [hP r\u2081 v s\u2081 s\u2081vS, hP r\u2082 u s\u2082 (by norm_cast; rwa [hs\u2082]), hr\u2082]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 C\nhP : \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S), P r s = P (r * t) { val := \u2191s * t, property := ht }\nx\u271d\u00b2 x\u271d\u00b9 : R \u00d7 { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nx\u271d : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\nu : { x // x \u2208 S }\nv : R\nhr\u2082 : r\u2082 * \u2191u = r\u2081 * v\ns\u2081vS : \u2191s\u2081 * v \u2208 S\nhs\u2082 : s\u2082 * u = { val := \u2191s\u2081 * v, property := s\u2081vS }\n\u22a2 \u2191s\u2082 * \u2191u \u2208 S\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 C\nhP : \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S), P r s = P (r * t) { val := \u2191s * t, property := ht }\nx\u271d\u00b2 x\u271d\u00b9 : R \u00d7 { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nx\u271d : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\nu : { x // x \u2208 S }\nv : R\nhr\u2082 : r\u2082 * \u2191u = r\u2081 * v\ns\u2081vS : \u2191s\u2081 * v \u2208 S\nhs\u2082 : s\u2082 * u = { val := \u2191s\u2081 * v, property := s\u2081vS }\n\u22a2 \u2191(s\u2082 * u) \u2208 S\n[PROOFSTEP]\nrwa [hs\u2082]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 C\nhP : \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S), P r s = P (r * t) { val := \u2191s * t, property := ht }\nx\u271d\u00b2 x\u271d\u00b9 : R \u00d7 { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nx\u271d : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\nu : { x // x \u2208 S }\nv : R\nhr\u2082 : r\u2082 * \u2191u = r\u2081 * v\ns\u2081vS : \u2191s\u2081 * v \u2208 S\nhs\u2082 : s\u2082 * u = { val := \u2191s\u2081 * v, property := s\u2081vS }\n\u22a2 P (r\u2081 * v) { val := \u2191s\u2081 * v, property := s\u2081vS } = P (r\u2081 * v) { val := \u2191s\u2082 * \u2191u, property := (_ : \u2191s\u2082 * \u2191u \u2208 S) }\n[PROOFSTEP]\nsimp only [\u2190 hs\u2082]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 C\nhP : \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S), P r s = P (r * t) { val := \u2191s * t, property := ht }\nx\u271d\u00b2 x\u271d\u00b9 : R \u00d7 { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nx\u271d : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\nu : { x // x \u2208 S }\nv : R\nhr\u2082 : r\u2082 * \u2191u = r\u2081 * v\ns\u2081vS : \u2191s\u2081 * v \u2208 S\nhs\u2082 : s\u2082 * u = { val := \u2191s\u2081 * v, property := s\u2081vS }\n\u22a2 P (r\u2081 * v) (s\u2082 * u) = P (r\u2081 * v) { val := \u2191s\u2082 * \u2191u, property := (_ : \u2191s\u2082 * \u2191u \u2208 S) }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 R \u2192 { x // x \u2208 S } \u2192 C\nhP :\n \u2200 (r\u2081 t\u2081 : R) (s\u2081 : { x // x \u2208 S }) (ht\u2081 : \u2191s\u2081 * t\u2081 \u2208 S) (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 } (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 t\u2082 : R\ns\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S\n\u22a2 P r\u2081 s\u2081 r\u2082 s\u2082 = P r\u2081 s\u2081 (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }\n[PROOFSTEP]\nhave := hP r\u2081 1 s\u2081 (by simp) r\u2082 t\u2082 s\u2082 ht\u2082\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 R \u2192 { x // x \u2208 S } \u2192 C\nhP :\n \u2200 (r\u2081 t\u2081 : R) (s\u2081 : { x // x \u2208 S }) (ht\u2081 : \u2191s\u2081 * t\u2081 \u2208 S) (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 } (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 t\u2082 : R\ns\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S\n\u22a2 \u2191s\u2081 * 1 \u2208 S\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 R \u2192 { x // x \u2208 S } \u2192 C\nhP :\n \u2200 (r\u2081 t\u2081 : R) (s\u2081 : { x // x \u2208 S }) (ht\u2081 : \u2191s\u2081 * t\u2081 \u2208 S) (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 } (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 t\u2082 : R\ns\u2082 : { x // x \u2208 S }\nht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S\nthis :\n P r\u2081 s\u2081 r\u2082 s\u2082 =\n P (r\u2081 * 1) { val := \u2191s\u2081 * 1, property := (_ : \u2191s\u2081 * 1 \u2208 S) } (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }\n\u22a2 P r\u2081 s\u2081 r\u2082 s\u2082 = P r\u2081 s\u2081 (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }\n[PROOFSTEP]\nsimp [this]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 R \u2192 { x // x \u2208 S } \u2192 C\nhP :\n \u2200 (r\u2081 t\u2081 : R) (s\u2081 : { x // x \u2208 S }) (ht\u2081 : \u2191s\u2081 * t\u2081 \u2208 S) (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 } (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }\nr\u2081 t\u2081 : R\ns\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191s\u2081 * t\u2081 \u2208 S\n\u22a2 (fun r\u2081 s\u2081 =>\n liftExpand (P r\u2081 s\u2081)\n (_ :\n \u2200 (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P r\u2081 s\u2081 (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }))\n r\u2081 s\u2081 =\n (fun r\u2081 s\u2081 =>\n liftExpand (P r\u2081 s\u2081)\n (_ :\n \u2200 (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P r\u2081 s\u2081 (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }))\n (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 }\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 R \u2192 { x // x \u2208 S } \u2192 C\nhP :\n \u2200 (r\u2081 t\u2081 : R) (s\u2081 : { x // x \u2208 S }) (ht\u2081 : \u2191s\u2081 * t\u2081 \u2208 S) (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 } (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }\nr\u2081 t\u2081 : R\ns\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191s\u2081 * t\u2081 \u2208 S\nx : OreLocalization R S\n\u22a2 (fun r\u2081 s\u2081 =>\n liftExpand (P r\u2081 s\u2081)\n (_ :\n \u2200 (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P r\u2081 s\u2081 (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }))\n r\u2081 s\u2081 x =\n (fun r\u2081 s\u2081 =>\n liftExpand (P r\u2081 s\u2081)\n (_ :\n \u2200 (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P r\u2081 s\u2081 (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }))\n (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 } x\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r\u2082 s\u2082\n[GOAL]\ncase h.c\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 R \u2192 { x // x \u2208 S } \u2192 C\nhP :\n \u2200 (r\u2081 t\u2081 : R) (s\u2081 : { x // x \u2208 S }) (ht\u2081 : \u2191s\u2081 * t\u2081 \u2208 S) (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 } (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }\nr\u2081 t\u2081 : R\ns\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191s\u2081 * t\u2081 \u2208 S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\n\u22a2 (fun r\u2081 s\u2081 =>\n liftExpand (P r\u2081 s\u2081)\n (_ :\n \u2200 (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P r\u2081 s\u2081 (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }))\n r\u2081 s\u2081 (r\u2082 /\u2092 s\u2082) =\n (fun r\u2081 s\u2081 =>\n liftExpand (P r\u2081 s\u2081)\n (_ :\n \u2200 (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P r\u2081 s\u2081 (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }))\n (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 } (r\u2082 /\u2092 s\u2082)\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h.c\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 R \u2192 { x // x \u2208 S } \u2192 C\nhP :\n \u2200 (r\u2081 t\u2081 : R) (s\u2081 : { x // x \u2208 S }) (ht\u2081 : \u2191s\u2081 * t\u2081 \u2208 S) (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 } (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }\nr\u2081 t\u2081 : R\ns\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191s\u2081 * t\u2081 \u2208 S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\n\u22a2 liftExpand (P r\u2081 s\u2081)\n (_ :\n \u2200 (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P r\u2081 s\u2081 (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 })\n (r\u2082 /\u2092 s\u2082) =\n liftExpand (P (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 })\n (_ :\n \u2200 (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 } r\u2082 s\u2082 =\n P (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 } (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 })\n (r\u2082 /\u2092 s\u2082)\n[PROOFSTEP]\nrw [liftExpand_of, liftExpand_of, hP r\u2081 t\u2081 s\u2081 ht\u2081 r\u2082 1 s\u2082 (by simp)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 R \u2192 { x // x \u2208 S } \u2192 C\nhP :\n \u2200 (r\u2081 t\u2081 : R) (s\u2081 : { x // x \u2208 S }) (ht\u2081 : \u2191s\u2081 * t\u2081 \u2208 S) (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 } (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }\nr\u2081 t\u2081 : R\ns\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191s\u2081 * t\u2081 \u2208 S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\n\u22a2 \u2191s\u2082 * 1 \u2208 S\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.c\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nC : Sort u_2\nP : R \u2192 { x // x \u2208 S } \u2192 R \u2192 { x // x \u2208 S } \u2192 C\nhP :\n \u2200 (r\u2081 t\u2081 : R) (s\u2081 : { x // x \u2208 S }) (ht\u2081 : \u2191s\u2081 * t\u2081 \u2208 S) (r\u2082 t\u2082 : R) (s\u2082 : { x // x \u2208 S }) (ht\u2082 : \u2191s\u2082 * t\u2082 \u2208 S),\n P r\u2081 s\u2081 r\u2082 s\u2082 = P (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 } (r\u2082 * t\u2082) { val := \u2191s\u2082 * t\u2082, property := ht\u2082 }\nr\u2081 t\u2081 : R\ns\u2081 : { x // x \u2208 S }\nht\u2081 : \u2191s\u2081 * t\u2081 \u2208 S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\n\u22a2 P (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 } (r\u2082 * 1) { val := \u2191s\u2082 * 1, property := (_ : \u2191s\u2082 * 1 \u2208 S) } =\n P (r\u2081 * t\u2081) { val := \u2191s\u2081 * t\u2081, property := ht\u2081 } r\u2082 s\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\n\u22a2 OreLocalization.mul' r\u2081 s\u2081 r\u2082 s\u2082 = r\u2081 * v /\u2092 (s\u2082 * u)\n[PROOFSTEP]\nsimp only [mul']\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\n\u22a2 r\u2081 * oreNum r\u2082 s\u2081 /\u2092 (s\u2082 * oreDenom r\u2082 s\u2081) = r\u2081 * v /\u2092 (s\u2082 * u)\n[PROOFSTEP]\nhave h\u2080 := ore_eq r\u2082 s\u2081\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nh\u2080 : r\u2082 * \u2191(oreDenom r\u2082 s\u2081) = \u2191s\u2081 * oreNum r\u2082 s\u2081\n\u22a2 r\u2081 * oreNum r\u2082 s\u2081 /\u2092 (s\u2082 * oreDenom r\u2082 s\u2081) = r\u2081 * v /\u2092 (s\u2082 * u)\n[PROOFSTEP]\nset v\u2080 := oreNum r\u2082 s\u2081\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191(oreDenom r\u2082 s\u2081) = \u2191s\u2081 * v\u2080\n\u22a2 r\u2081 * v\u2080 /\u2092 (s\u2082 * oreDenom r\u2082 s\u2081) = r\u2081 * v /\u2092 (s\u2082 * u)\n[PROOFSTEP]\nset u\u2080 := oreDenom r\u2082 s\u2081\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\n\u22a2 r\u2081 * v\u2080 /\u2092 (s\u2082 * u\u2080) = r\u2081 * v /\u2092 (s\u2082 * u)\n[PROOFSTEP]\nrcases oreCondition (u\u2080 : R) u with \u27e8r\u2083, s\u2083, h\u2083\u27e9\n[GOAL]\ncase mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191u\u2080 * \u2191s\u2083 = \u2191u * r\u2083\n\u22a2 r\u2081 * v\u2080 /\u2092 (s\u2082 * u\u2080) = r\u2081 * v /\u2092 (s\u2082 * u)\n[PROOFSTEP]\nhave :=\n calc\n (s\u2081 : R) * (v * r\u2083) = r\u2082 * u * r\u2083 := by rw [\u2190 mul_assoc, \u2190 huv]\n _ = r\u2082 * u\u2080 * s\u2083 := by rw [mul_assoc, mul_assoc, h\u2083]\n _ = s\u2081 * (v\u2080 * s\u2083) := by rw [\u2190 mul_assoc, h\u2080]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191u\u2080 * \u2191s\u2083 = \u2191u * r\u2083\n\u22a2 \u2191s\u2081 * (v * r\u2083) = r\u2082 * \u2191u * r\u2083\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 huv]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191u\u2080 * \u2191s\u2083 = \u2191u * r\u2083\n\u22a2 r\u2082 * \u2191u * r\u2083 = r\u2082 * \u2191u\u2080 * \u2191s\u2083\n[PROOFSTEP]\nrw [mul_assoc, mul_assoc, h\u2083]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191u\u2080 * \u2191s\u2083 = \u2191u * r\u2083\n\u22a2 r\u2082 * \u2191u\u2080 * \u2191s\u2083 = \u2191s\u2081 * (v\u2080 * \u2191s\u2083)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, h\u2080]\n[GOAL]\ncase mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191u\u2080 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s\u2081 * (v * r\u2083) = \u2191s\u2081 * (v\u2080 * \u2191s\u2083)\n\u22a2 r\u2081 * v\u2080 /\u2092 (s\u2082 * u\u2080) = r\u2081 * v /\u2092 (s\u2082 * u)\n[PROOFSTEP]\nrcases ore_left_cancel _ _ _ this with \u27e8s\u2084, hs\u2084\u27e9\n[GOAL]\ncase mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191u\u2080 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s\u2081 * (v * r\u2083) = \u2191s\u2081 * (v\u2080 * \u2191s\u2083)\ns\u2084 : { x // x \u2208 S }\nhs\u2084 : v * r\u2083 * \u2191s\u2084 = v\u2080 * \u2191s\u2083 * \u2191s\u2084\n\u22a2 r\u2081 * v\u2080 /\u2092 (s\u2082 * u\u2080) = r\u2081 * v /\u2092 (s\u2082 * u)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191u\u2080 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s\u2081 * (v * r\u2083) = \u2191s\u2081 * (v\u2080 * \u2191s\u2083)\ns\u2084 : { x // x \u2208 S }\nhs\u2084 : v * r\u2083 * \u2191s\u2084 = v\u2080 * \u2191s\u2083 * \u2191s\u2084\n\u22a2 r\u2081 * v /\u2092 (s\u2082 * u) = r\u2081 * v\u2080 /\u2092 (s\u2082 * u\u2080)\n[PROOFSTEP]\nrw [oreDiv_eq_iff]\n[GOAL]\ncase mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191u\u2080 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s\u2081 * (v * r\u2083) = \u2191s\u2081 * (v\u2080 * \u2191s\u2083)\ns\u2084 : { x // x \u2208 S }\nhs\u2084 : v * r\u2083 * \u2191s\u2084 = v\u2080 * \u2191s\u2083 * \u2191s\u2084\n\u22a2 \u2203 u_1 v_1, r\u2081 * v\u2080 * \u2191u_1 = r\u2081 * v * v_1 \u2227 \u2191(s\u2082 * u\u2080) * \u2191u_1 = \u2191(s\u2082 * u) * v_1\n[PROOFSTEP]\nuse s\u2083 * s\u2084\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191u\u2080 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s\u2081 * (v * r\u2083) = \u2191s\u2081 * (v\u2080 * \u2191s\u2083)\ns\u2084 : { x // x \u2208 S }\nhs\u2084 : v * r\u2083 * \u2191s\u2084 = v\u2080 * \u2191s\u2083 * \u2191s\u2084\n\u22a2 \u2203 v_1, r\u2081 * v\u2080 * \u2191(s\u2083 * s\u2084) = r\u2081 * v * v_1 \u2227 \u2191(s\u2082 * u\u2080) * \u2191(s\u2083 * s\u2084) = \u2191(s\u2082 * u) * v_1\n[PROOFSTEP]\nuse r\u2083 * s\u2084\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191u\u2080 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s\u2081 * (v * r\u2083) = \u2191s\u2081 * (v\u2080 * \u2191s\u2083)\ns\u2084 : { x // x \u2208 S }\nhs\u2084 : v * r\u2083 * \u2191s\u2084 = v\u2080 * \u2191s\u2083 * \u2191s\u2084\n\u22a2 r\u2081 * v\u2080 * \u2191(s\u2083 * s\u2084) = r\u2081 * v * (r\u2083 * \u2191s\u2084) \u2227 \u2191(s\u2082 * u\u2080) * \u2191(s\u2083 * s\u2084) = \u2191(s\u2082 * u) * (r\u2083 * \u2191s\u2084)\n[PROOFSTEP]\nsimp only [Submonoid.coe_mul]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191u\u2080 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s\u2081 * (v * r\u2083) = \u2191s\u2081 * (v\u2080 * \u2191s\u2083)\ns\u2084 : { x // x \u2208 S }\nhs\u2084 : v * r\u2083 * \u2191s\u2084 = v\u2080 * \u2191s\u2083 * \u2191s\u2084\n\u22a2 r\u2081 * oreNum r\u2082 s\u2081 * (\u2191s\u2083 * \u2191s\u2084) = r\u2081 * v * (r\u2083 * \u2191s\u2084) \u2227 \u2191s\u2082 * \u2191(oreDenom r\u2082 s\u2081) * (\u2191s\u2083 * \u2191s\u2084) = \u2191s\u2082 * \u2191u * (r\u2083 * \u2191s\u2084)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191u\u2080 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s\u2081 * (v * r\u2083) = \u2191s\u2081 * (v\u2080 * \u2191s\u2083)\ns\u2084 : { x // x \u2208 S }\nhs\u2084 : v * r\u2083 * \u2191s\u2084 = v\u2080 * \u2191s\u2083 * \u2191s\u2084\n\u22a2 r\u2081 * oreNum r\u2082 s\u2081 * (\u2191s\u2083 * \u2191s\u2084) = r\u2081 * v * (r\u2083 * \u2191s\u2084)\n[PROOFSTEP]\nrw [mul_assoc (b := v\u2080), \u2190 mul_assoc (a := v\u2080), \u2190 hs\u2084]\n[GOAL]\ncase h.left\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191u\u2080 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s\u2081 * (v * r\u2083) = \u2191s\u2081 * (v\u2080 * \u2191s\u2083)\ns\u2084 : { x // x \u2208 S }\nhs\u2084 : v * r\u2083 * \u2191s\u2084 = v\u2080 * \u2191s\u2083 * \u2191s\u2084\n\u22a2 r\u2081 * (v * r\u2083 * \u2191s\u2084) = r\u2081 * v * (r\u2083 * \u2191s\u2084)\n[PROOFSTEP]\nsimp only [mul_assoc]\n[GOAL]\ncase h.right\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191u\u2080 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s\u2081 * (v * r\u2083) = \u2191s\u2081 * (v\u2080 * \u2191s\u2083)\ns\u2084 : { x // x \u2208 S }\nhs\u2084 : v * r\u2083 * \u2191s\u2084 = v\u2080 * \u2191s\u2083 * \u2191s\u2084\n\u22a2 \u2191s\u2082 * \u2191(oreDenom r\u2082 s\u2081) * (\u2191s\u2083 * \u2191s\u2084) = \u2191s\u2082 * \u2191u * (r\u2083 * \u2191s\u2084)\n[PROOFSTEP]\nrw [mul_assoc (b := (u\u2080 : R)), \u2190 mul_assoc (a := (u\u2080 : R)), h\u2083]\n[GOAL]\ncase h.right\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 u : { x // x \u2208 S }\nv : R\nhuv : r\u2082 * \u2191u = \u2191s\u2081 * v\nv\u2080 : R := oreNum r\u2082 s\u2081\nu\u2080 : { x // x \u2208 S } := oreDenom r\u2082 s\u2081\nh\u2080 : r\u2082 * \u2191u\u2080 = \u2191s\u2081 * v\u2080\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191u\u2080 * \u2191s\u2083 = \u2191u * r\u2083\nthis : \u2191s\u2081 * (v * r\u2083) = \u2191s\u2081 * (v\u2080 * \u2191s\u2083)\ns\u2084 : { x // x \u2208 S }\nhs\u2084 : v * r\u2083 * \u2191s\u2084 = v\u2080 * \u2191s\u2083 * \u2191s\u2084\n\u22a2 \u2191s\u2082 * (\u2191u * r\u2083 * \u2191s\u2084) = \u2191s\u2082 * \u2191u * (r\u2083 * \u2191s\u2084)\n[PROOFSTEP]\nsimp only [mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 =\n OreLocalization.mul' (r\u2082 * p) { val := \u2191s\u2082 * p, property := hp } (r\u2081 * r) { val := \u2191s\u2081 * r, property := hr }\n[PROOFSTEP]\nhave h\u2081 := ore_eq r\u2081 s\u2082\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nh\u2081 : r\u2081 * \u2191(oreDenom r\u2081 s\u2082) = \u2191s\u2082 * oreNum r\u2081 s\u2082\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 =\n OreLocalization.mul' (r\u2082 * p) { val := \u2191s\u2082 * p, property := hp } (r\u2081 * r) { val := \u2191s\u2081 * r, property := hr }\n[PROOFSTEP]\nset r\u2081' := oreNum r\u2081 s\u2082\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191(oreDenom r\u2081 s\u2082) = \u2191s\u2082 * r\u2081'\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 =\n OreLocalization.mul' (r\u2082 * p) { val := \u2191s\u2082 * p, property := hp } (r\u2081 * r) { val := \u2191s\u2081 * r, property := hr }\n[PROOFSTEP]\nset s\u2082' := oreDenom r\u2081 s\u2082\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 =\n OreLocalization.mul' (r\u2082 * p) { val := \u2191s\u2082 * p, property := hp } (r\u2081 * r) { val := \u2191s\u2081 * r, property := hr }\n[PROOFSTEP]\nrcases oreCondition (\u2191s\u2082 * r\u2081') \u27e8s\u2082 * p, hp\u27e9 with \u27e8p', s_star, h\u2082\u27e9\n[GOAL]\ncase mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * r\u2081' * \u2191s_star = \u2191{ val := \u2191s\u2082 * p, property := hp } * p'\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 =\n OreLocalization.mul' (r\u2082 * p) { val := \u2191s\u2082 * p, property := hp } (r\u2081 * r) { val := \u2191s\u2081 * r, property := hr }\n[PROOFSTEP]\ndsimp at h\u2082 \n[GOAL]\ncase mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 =\n OreLocalization.mul' (r\u2082 * p) { val := \u2191s\u2082 * p, property := hp } (r\u2081 * r) { val := \u2191s\u2081 * r, property := hr }\n[PROOFSTEP]\nrcases oreCondition r (s\u2082' * s_star) with \u27e8p_flat, s_flat, h\u2083\u27e9\n[GOAL]\ncase mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(s\u2082' * s_star) * p_flat\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 =\n OreLocalization.mul' (r\u2082 * p) { val := \u2191s\u2082 * p, property := hp } (r\u2081 * r) { val := \u2191s\u2081 * r, property := hr }\n[PROOFSTEP]\nsimp only [S.coe_mul] at h\u2083 \n[GOAL]\ncase mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 =\n OreLocalization.mul' (r\u2082 * p) { val := \u2191s\u2082 * p, property := hp } (r\u2081 * r) { val := \u2191s\u2081 * r, property := hr }\n[PROOFSTEP]\nhave : r\u2081 * r * s_flat = s\u2082 * p * (p' * p_flat) :=\n by\n rw [\u2190 mul_assoc, \u2190 h\u2082, \u2190 h\u2081, mul_assoc, h\u2083]\n simp only [mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\n\u22a2 r\u2081 * r * \u2191s_flat = \u2191s\u2082 * p * (p' * p_flat)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 h\u2082, \u2190 h\u2081, mul_assoc, h\u2083]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\n\u22a2 r\u2081 * (\u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat) = r\u2081 * \u2191s\u2082' * \u2191s_star * p_flat\n[PROOFSTEP]\nsimp only [mul_assoc]\n[GOAL]\ncase mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\nthis : r\u2081 * r * \u2191s_flat = \u2191s\u2082 * p * (p' * p_flat)\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 =\n OreLocalization.mul' (r\u2082 * p) { val := \u2191s\u2082 * p, property := hp } (r\u2081 * r) { val := \u2191s\u2081 * r, property := hr }\n[PROOFSTEP]\nrw [mul'_char (r\u2082 * p) (r\u2081 * r) \u27e8\u2191s\u2082 * p, hp\u27e9 \u27e8\u2191s\u2081 * r, hr\u27e9 _ _ this]\n[GOAL]\ncase mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\nthis : r\u2081 * r * \u2191s_flat = \u2191s\u2082 * p * (p' * p_flat)\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 = r\u2082 * p * (p' * p_flat) /\u2092 ({ val := \u2191s\u2081 * r, property := hr } * s_flat)\n[PROOFSTEP]\nclear this\n[GOAL]\ncase mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 = r\u2082 * p * (p' * p_flat) /\u2092 ({ val := \u2191s\u2081 * r, property := hr } * s_flat)\n[PROOFSTEP]\nhave hsssp : \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat \u2208 S :=\n by\n rw [mul_assoc, mul_assoc, \u2190 mul_assoc (s\u2082' : R), \u2190 h\u2083, \u2190 mul_assoc]\n exact S.mul_mem hr (SetLike.coe_mem s_flat)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\n\u22a2 \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat \u2208 S\n[PROOFSTEP]\nrw [mul_assoc, mul_assoc, \u2190 mul_assoc (s\u2082' : R), \u2190 h\u2083, \u2190 mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\n\u22a2 \u2191s\u2081 * r * \u2191s_flat \u2208 S\n[PROOFSTEP]\nexact S.mul_mem hr (SetLike.coe_mem s_flat)\n[GOAL]\ncase mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\nhsssp : \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat \u2208 S\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 = r\u2082 * p * (p' * p_flat) /\u2092 ({ val := \u2191s\u2081 * r, property := hr } * s_flat)\n[PROOFSTEP]\nhave : (\u27e8\u2191s\u2081 * r, hr\u27e9 : S) * s_flat = \u27e8s\u2081 * s\u2082' * s_star * p_flat, hsssp\u27e9 :=\n by\n ext\n simp only [Submonoid.coe_mul]\n rw [mul_assoc, h\u2083, \u2190 mul_assoc, \u2190 mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\nhsssp : \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat \u2208 S\n\u22a2 { val := \u2191s\u2081 * r, property := hr } * s_flat = { val := \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat, property := hsssp }\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\nhsssp : \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat \u2208 S\n\u22a2 \u2191({ val := \u2191s\u2081 * r, property := hr } * s_flat) = \u2191{ val := \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat, property := hsssp }\n[PROOFSTEP]\nsimp only [Submonoid.coe_mul]\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\nhsssp : \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat \u2208 S\n\u22a2 \u2191s\u2081 * r * \u2191s_flat = \u2191s\u2081 * \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\n[PROOFSTEP]\nrw [mul_assoc, h\u2083, \u2190 mul_assoc, \u2190 mul_assoc]\n[GOAL]\ncase mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\nhsssp : \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat \u2208 S\nthis : { val := \u2191s\u2081 * r, property := hr } * s_flat = { val := \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat, property := hsssp }\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 = r\u2082 * p * (p' * p_flat) /\u2092 ({ val := \u2191s\u2081 * r, property := hr } * s_flat)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\nhsssp : \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat \u2208 S\nthis : { val := \u2191s\u2081 * r, property := hr } * s_flat = { val := \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat, property := hsssp }\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 =\n r\u2082 * p * (p' * p_flat) /\u2092 { val := \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat, property := hsssp }\n[PROOFSTEP]\nclear this\n[GOAL]\ncase mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\nhsssp : \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat \u2208 S\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 =\n r\u2082 * p * (p' * p_flat) /\u2092 { val := \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat, property := hsssp }\n[PROOFSTEP]\nrcases ore_left_cancel (p * p') (r\u2081' * (s_star : R)) s\u2082 (by simp [\u2190 mul_assoc, h\u2082]) with \u27e8s\u2082'', h\u2082''\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\nhsssp : \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat \u2208 S\n\u22a2 \u2191s\u2082 * (p * p') = \u2191s\u2082 * (r\u2081' * \u2191s_star)\n[PROOFSTEP]\nsimp [\u2190 mul_assoc, h\u2082]\n[GOAL]\ncase mk.mk.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\nhsssp : \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat \u2208 S\ns\u2082'' : { x // x \u2208 S }\nh\u2082'' : p * p' * \u2191s\u2082'' = r\u2081' * \u2191s_star * \u2191s\u2082''\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 =\n r\u2082 * p * (p' * p_flat) /\u2092 { val := \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat, property := hsssp }\n[PROOFSTEP]\nrw [\u2190 mul_assoc, mul_assoc r\u2082, OreLocalization.eq_of_num_factor_eq h\u2082'']\n[GOAL]\ncase mk.mk.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\nhsssp : \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat \u2208 S\ns\u2082'' : { x // x \u2208 S }\nh\u2082'' : p * p' * \u2191s\u2082'' = r\u2081' * \u2191s_star * \u2191s\u2082''\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 =\n r\u2082 * (r\u2081' * \u2191s_star) * p_flat /\u2092 { val := \u2191s\u2081 * \u2191s\u2082' * \u2191s_star * p_flat, property := hsssp }\n[PROOFSTEP]\nnorm_cast at hsssp \u22a2\n[GOAL]\ncase mk.mk.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\nhsssp : \u2191(s\u2081 * s\u2082' * s_star) * p_flat \u2208 S\ns\u2082'' : { x // x \u2208 S }\nh\u2082'' : p * p' * \u2191s\u2082'' = r\u2081' * \u2191s_star * \u2191s\u2082''\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 =\n r\u2082 * (r\u2081' * \u2191s_star) * p_flat /\u2092\n { val := \u2191(s\u2081 * s\u2082' * s_star) * p_flat, property := (_ : (fun x => x \u2208 S) (\u2191(s\u2081 * s\u2082' * s_star) * p_flat)) }\n[PROOFSTEP]\nrw [\u2190 OreLocalization.expand _ _ _ hsssp, \u2190 mul_assoc]\n[GOAL]\ncase mk.mk.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 p : R\ns\u2082 : { x // x \u2208 S }\nhp : \u2191s\u2082 * p \u2208 S\nr\u2081 r : R\ns\u2081 : { x // x \u2208 S }\nhr : \u2191s\u2081 * r \u2208 S\nr\u2081' : R := oreNum r\u2081 s\u2082\ns\u2082' : { x // x \u2208 S } := oreDenom r\u2081 s\u2082\nh\u2081 : r\u2081 * \u2191s\u2082' = \u2191s\u2082 * r\u2081'\np' : R\ns_star : { x // x \u2208 S }\nh\u2082 : \u2191s\u2082 * oreNum r\u2081 s\u2082 * \u2191s_star = \u2191s\u2082 * p * p'\np_flat : R\ns_flat : { x // x \u2208 S }\nh\u2083 : r * \u2191s_flat = \u2191(oreDenom r\u2081 s\u2082) * \u2191s_star * p_flat\nhsssp : \u2191(s\u2081 * s\u2082' * s_star) * p_flat \u2208 S\ns\u2082'' : { x // x \u2208 S }\nh\u2082'' : p * p' * \u2191s\u2082'' = r\u2081' * \u2191s_star * \u2191s\u2082''\n\u22a2 OreLocalization.mul' r\u2082 s\u2082 r\u2081 s\u2081 = r\u2082 * r\u2081' * \u2191s_star /\u2092 (s\u2081 * s\u2082' * s_star)\n[PROOFSTEP]\napply OreLocalization.expand\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\nhr : r \u2208 S\n\u22a2 r /\u2092 { val := r, property := hr } = 1\n[PROOFSTEP]\nrw [OreLocalization.one_def, oreDiv_eq_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\nhr : r \u2208 S\n\u22a2 \u2203 u v, 1 * \u2191u = r * v \u2227 \u21911 * \u2191u = \u2191{ val := r, property := hr } * v\n[PROOFSTEP]\nexact \u27e8\u27e8r, hr\u27e9, 1, by simp, by simp\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\nhr : r \u2208 S\n\u22a2 1 * \u2191{ val := r, property := hr } = r * 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\nhr : r \u2208 S\n\u22a2 \u21911 * \u2191{ val := r, property := hr } = \u2191{ val := r, property := hr } * 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nx : OreLocalization R S\n\u22a2 1 * x = x\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r s\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 1 * (r /\u2092 s) = r /\u2092 s\n[PROOFSTEP]\nsimp [OreLocalization.one_def, oreDiv_mul_char (1 : R) r (1 : S) s r 1 (by simp)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 r * \u21911 = \u21911 * r\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nx : OreLocalization R S\n\u22a2 x * 1 = x\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r s\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 r /\u2092 s * 1 = r /\u2092 s\n[PROOFSTEP]\nsimp [OreLocalization.one_def, oreDiv_mul_char r 1 s 1 1 s (by simp)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 1 * \u2191s = \u2191s * 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nx y z : OreLocalization R S\n\u22a2 x * y * z = x * (y * z)\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r\u2081 s\u2081\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\ny z : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\n\u22a2 r\u2081 /\u2092 s\u2081 * y * z = r\u2081 /\u2092 s\u2081 * (y * z)\n[PROOFSTEP]\ninduction' y using OreLocalization.ind with r\u2082 s\u2082\n[GOAL]\ncase c.c\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nz : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\n\u22a2 r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082) * z = r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082 * z)\n[PROOFSTEP]\ninduction' z using OreLocalization.ind with r\u2083 s\u2083\n[GOAL]\ncase c.c.c\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\n\u22a2 r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083))\n[PROOFSTEP]\nrcases oreDivMulChar' r\u2081 r\u2082 s\u2081 s\u2082 with \u27e8ra, sa, ha, ha'\u27e9\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\nha' : r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082) = r\u2081 * ra /\u2092 (s\u2082 * sa)\n\u22a2 r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083))\n[PROOFSTEP]\nrw [ha']\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\nha' : r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082) = r\u2081 * ra /\u2092 (s\u2082 * sa)\n\u22a2 r\u2081 * ra /\u2092 (s\u2082 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083))\n[PROOFSTEP]\nclear ha'\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n\u22a2 r\u2081 * ra /\u2092 (s\u2082 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083))\n[PROOFSTEP]\nrcases oreDivMulChar' r\u2082 r\u2083 s\u2082 s\u2083 with \u27e8rb, sb, hb, hb'\u27e9\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2083 * \u2191sb = \u2191s\u2082 * rb\nhb' : r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083) = r\u2082 * rb /\u2092 (s\u2083 * sb)\n\u22a2 r\u2081 * ra /\u2092 (s\u2082 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083))\n[PROOFSTEP]\nrw [hb']\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2083 * \u2191sb = \u2191s\u2082 * rb\nhb' : r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083) = r\u2082 * rb /\u2092 (s\u2083 * sb)\n\u22a2 r\u2081 * ra /\u2092 (s\u2082 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2082 * rb /\u2092 (s\u2083 * sb))\n[PROOFSTEP]\nclear hb'\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2083 * \u2191sb = \u2191s\u2082 * rb\n\u22a2 r\u2081 * ra /\u2092 (s\u2082 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2082 * rb /\u2092 (s\u2083 * sb))\n[PROOFSTEP]\nrcases oreCondition rb sa with \u27e8rc, sc, hc\u27e9\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2083 * \u2191sb = \u2191s\u2082 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : rb * \u2191sc = \u2191sa * rc\n\u22a2 r\u2081 * ra /\u2092 (s\u2082 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2082 * rb /\u2092 (s\u2083 * sb))\n[PROOFSTEP]\nrw [oreDiv_mul_char (r\u2081 * ra) r\u2083 (s\u2082 * sa) s\u2083 rc (sb * sc)\n (by\n simp only [Submonoid.coe_mul]\n rw [\u2190 mul_assoc, hb, mul_assoc, hc, \u2190 mul_assoc])]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2083 * \u2191sb = \u2191s\u2082 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : rb * \u2191sc = \u2191sa * rc\n\u22a2 r\u2083 * \u2191(sb * sc) = \u2191(s\u2082 * sa) * rc\n[PROOFSTEP]\nsimp only [Submonoid.coe_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2083 * \u2191sb = \u2191s\u2082 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : rb * \u2191sc = \u2191sa * rc\n\u22a2 r\u2083 * (\u2191sb * \u2191sc) = \u2191s\u2082 * \u2191sa * rc\n[PROOFSTEP]\nrw [\u2190 mul_assoc, hb, mul_assoc, hc, \u2190 mul_assoc]\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2083 * \u2191sb = \u2191s\u2082 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : rb * \u2191sc = \u2191sa * rc\n\u22a2 r\u2081 * ra * rc /\u2092 (s\u2083 * (sb * sc)) = r\u2081 /\u2092 s\u2081 * (r\u2082 * rb /\u2092 (s\u2083 * sb))\n[PROOFSTEP]\nrw [mul_assoc, \u2190 mul_assoc s\u2083]\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2083 * \u2191sb = \u2191s\u2082 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : rb * \u2191sc = \u2191sa * rc\n\u22a2 r\u2081 * (ra * rc) /\u2092 (s\u2083 * sb * sc) = r\u2081 /\u2092 s\u2081 * (r\u2082 * rb /\u2092 (s\u2083 * sb))\n[PROOFSTEP]\nsymm\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2083 * \u2191sb = \u2191s\u2082 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : rb * \u2191sc = \u2191sa * rc\n\u22a2 r\u2081 /\u2092 s\u2081 * (r\u2082 * rb /\u2092 (s\u2083 * sb)) = r\u2081 * (ra * rc) /\u2092 (s\u2083 * sb * sc)\n[PROOFSTEP]\napply oreDiv_mul_char\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.huv\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2083 * \u2191sb = \u2191s\u2082 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : rb * \u2191sc = \u2191sa * rc\n\u22a2 r\u2082 * rb * \u2191sc = \u2191s\u2081 * (ra * rc)\n[PROOFSTEP]\nrw [mul_assoc, hc, \u2190 mul_assoc (b := ra), \u2190 ha, mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\ns s' : { x // x \u2208 S }\n\u22a2 \u2191s /\u2092 s' * (\u2191s' /\u2092 s) = 1\n[PROOFSTEP]\nsimp [oreDiv_mul_char (s : R) s' s' s 1 1 (by simp)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\ns s' : { x // x \u2208 S }\n\u22a2 \u2191s' * \u21911 = \u2191s' * 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns t : { x // x \u2208 S }\n\u22a2 r /\u2092 s * (1 /\u2092 t) = r /\u2092 (t * s)\n[PROOFSTEP]\nsimp [oreDiv_mul_char r 1 s t 1 s (by simp)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns t : { x // x \u2208 S }\n\u22a2 1 * \u2191s = \u2191s * 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns t : { x // x \u2208 S }\n\u22a2 r /\u2092 s * (\u2191s /\u2092 t) = r /\u2092 t\n[PROOFSTEP]\nsimp [oreDiv_mul_char r s s t 1 1 (by simp)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns t : { x // x \u2208 S }\n\u22a2 \u2191s * \u21911 = \u2191s * 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns t : { x // x \u2208 S }\n\u22a2 r\u2081 /\u2092 s * (\u2191s * r\u2082 /\u2092 t) = r\u2081 * r\u2082 /\u2092 t\n[PROOFSTEP]\nsimp [oreDiv_mul_char r\u2081 (s * r\u2082) s t r\u2082 1 (by simp)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns t : { x // x \u2208 S }\n\u22a2 \u2191s * r\u2082 * \u21911 = \u2191s * r\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\np r : R\ns : { x // x \u2208 S }\n\u22a2 r /\u2092 1 * (p /\u2092 s) = r * p /\u2092 s\n[PROOFSTEP]\nsimp [oreDiv_mul_char r p 1 s p 1 (by simp)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\np r : R\ns : { x // x \u2208 S }\n\u22a2 p * \u21911 = \u21911 * p\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nx : OreLocalization R S\nr t : R\ns : { x // x \u2208 S }\nht : \u2191s * t \u2208 S\n\u22a2 (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s = (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht }\n[PROOFSTEP]\nsimp only []\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nx : OreLocalization R S\nr t : R\ns : { x // x \u2208 S }\nht : \u2191s * t \u2208 S\n\u22a2 \u2191f r * \u2191(\u2191fS s)\u207b\u00b9 = \u2191f (r * t) * \u2191(\u2191fS { val := \u2191s * t, property := ht })\u207b\u00b9\n[PROOFSTEP]\nhave : (fS \u27e8s * t, ht\u27e9 : T) = fS s * f t := by simp only [\u2190 hf, MonoidHom.map_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nx : OreLocalization R S\nr t : R\ns : { x // x \u2208 S }\nht : \u2191s * t \u2208 S\n\u22a2 \u2191(\u2191fS { val := \u2191s * t, property := ht }) = \u2191(\u2191fS s) * \u2191f t\n[PROOFSTEP]\nsimp only [\u2190 hf, MonoidHom.map_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nx : OreLocalization R S\nr t : R\ns : { x // x \u2208 S }\nht : \u2191s * t \u2208 S\nthis : \u2191(\u2191fS { val := \u2191s * t, property := ht }) = \u2191(\u2191fS s) * \u2191f t\n\u22a2 \u2191f r * \u2191(\u2191fS s)\u207b\u00b9 = \u2191f (r * t) * \u2191(\u2191fS { val := \u2191s * t, property := ht })\u207b\u00b9\n[PROOFSTEP]\nconv_rhs =>\n rw [MonoidHom.map_mul, \u2190 mul_one (f r), \u2190 Units.val_one, \u2190 mul_left_inv (fS s)]\n rw [Units.val_mul, \u2190 mul_assoc, mul_assoc _ (fS s : T), \u2190 this, mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nx : OreLocalization R S\nr t : R\ns : { x // x \u2208 S }\nht : \u2191s * t \u2208 S\nthis : \u2191(\u2191fS { val := \u2191s * t, property := ht }) = \u2191(\u2191fS s) * \u2191f t\n| \u2191f (r * t) * \u2191(\u2191fS { val := \u2191s * t, property := ht })\u207b\u00b9\n[PROOFSTEP]\n rw [MonoidHom.map_mul, \u2190 mul_one (f r), \u2190 Units.val_one, \u2190 mul_left_inv (fS s)]\n rw [Units.val_mul, \u2190 mul_assoc, mul_assoc _ (fS s : T), \u2190 this, mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nx : OreLocalization R S\nr t : R\ns : { x // x \u2208 S }\nht : \u2191s * t \u2208 S\nthis : \u2191(\u2191fS { val := \u2191s * t, property := ht }) = \u2191(\u2191fS s) * \u2191f t\n| \u2191f (r * t) * \u2191(\u2191fS { val := \u2191s * t, property := ht })\u207b\u00b9\n[PROOFSTEP]\n rw [MonoidHom.map_mul, \u2190 mul_one (f r), \u2190 Units.val_one, \u2190 mul_left_inv (fS s)]\n rw [Units.val_mul, \u2190 mul_assoc, mul_assoc _ (fS s : T), \u2190 this, mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nx : OreLocalization R S\nr t : R\ns : { x // x \u2208 S }\nht : \u2191s * t \u2208 S\nthis : \u2191(\u2191fS { val := \u2191s * t, property := ht }) = \u2191(\u2191fS s) * \u2191f t\n| \u2191f (r * t) * \u2191(\u2191fS { val := \u2191s * t, property := ht })\u207b\u00b9\n[PROOFSTEP]\nrw [MonoidHom.map_mul, \u2190 mul_one (f r), \u2190 Units.val_one, \u2190 mul_left_inv (fS s)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nx : OreLocalization R S\nr t : R\ns : { x // x \u2208 S }\nht : \u2191s * t \u2208 S\nthis : \u2191(\u2191fS { val := \u2191s * t, property := ht }) = \u2191(\u2191fS s) * \u2191f t\n| \u2191f r * \u2191((\u2191fS s)\u207b\u00b9 * \u2191fS s) * \u2191f t * \u2191(\u2191fS { val := \u2191s * t, property := ht })\u207b\u00b9\n[PROOFSTEP]\nrw [Units.val_mul, \u2190 mul_assoc, mul_assoc _ (fS s : T), \u2190 this, mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nx : OreLocalization R S\nr t : R\ns : { x // x \u2208 S }\nht : \u2191s * t \u2208 S\nthis : \u2191(\u2191fS { val := \u2191s * t, property := ht }) = \u2191(\u2191fS s) * \u2191f t\n\u22a2 \u2191f r * \u2191(\u2191fS s)\u207b\u00b9 =\n \u2191f r * \u2191(\u2191fS s)\u207b\u00b9 * (\u2191(\u2191fS { val := \u2191s * t, property := ht }) * \u2191(\u2191fS { val := \u2191s * t, property := ht })\u207b\u00b9)\n[PROOFSTEP]\nsimp only [mul_one, Units.mul_inv]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\n\u22a2 (fun x =>\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n x)\n 1 =\n 1\n[PROOFSTEP]\nsimp only []\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\n\u22a2 liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s = (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n 1 =\n 1\n[PROOFSTEP]\nrw [OreLocalization.one_def, liftExpand_of]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\n\u22a2 \u2191f 1 * \u2191(\u2191fS 1)\u207b\u00b9 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nx y : OreLocalization R S\n\u22a2 OneHom.toFun\n {\n toFun := fun x =>\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n x,\n map_one' :=\n (_ :\n (fun x =>\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n x)\n 1 =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n x,\n map_one' :=\n (_ :\n (fun x =>\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n x)\n 1 =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n x,\n map_one' :=\n (_ :\n (fun x =>\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n x)\n 1 =\n 1) }\n y\n[PROOFSTEP]\nsimp only []\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nx y : OreLocalization R S\n\u22a2 liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s = (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n (x * y) =\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n x *\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n y\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r\u2081 s\u2081\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\ny : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\n\u22a2 liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s = (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n (r\u2081 /\u2092 s\u2081 * y) =\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n (r\u2081 /\u2092 s\u2081) *\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n y\n[PROOFSTEP]\ninduction' y using OreLocalization.ind with r\u2082 s\u2082\n[GOAL]\ncase c.c\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\n\u22a2 liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s = (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n (r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082)) =\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n (r\u2081 /\u2092 s\u2081) *\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n (r\u2082 /\u2092 s\u2082)\n[PROOFSTEP]\nrcases oreDivMulChar' r\u2081 r\u2082 s\u2081 s\u2082 with \u27e8ra, sa, ha, ha'\u27e9\n[GOAL]\ncase c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\nha' : r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082) = r\u2081 * ra /\u2092 (s\u2082 * sa)\n\u22a2 liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s = (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n (r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082)) =\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n (r\u2081 /\u2092 s\u2081) *\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n (r\u2082 /\u2092 s\u2082)\n[PROOFSTEP]\nrw [ha']\n[GOAL]\ncase c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\nha' : r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082) = r\u2081 * ra /\u2092 (s\u2082 * sa)\n\u22a2 liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s = (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n (r\u2081 * ra /\u2092 (s\u2082 * sa)) =\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n (r\u2081 /\u2092 s\u2081) *\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n (r\u2082 /\u2092 s\u2082)\n[PROOFSTEP]\nclear ha'\n[GOAL]\ncase c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n\u22a2 liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s = (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n (r\u2081 * ra /\u2092 (s\u2082 * sa)) =\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n (r\u2081 /\u2092 s\u2081) *\n liftExpand (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9)\n (_ :\n \u2200 (r t : R) (s : { x // x \u2208 S }) (ht : \u2191s * t \u2208 S),\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) r s =\n (fun r s => \u2191f r * \u2191(\u2191fS s)\u207b\u00b9) (r * t) { val := \u2191s * t, property := ht })\n (r\u2082 /\u2092 s\u2082)\n[PROOFSTEP]\nrw [liftExpand_of, liftExpand_of, liftExpand_of]\n[GOAL]\ncase c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n\u22a2 \u2191f (r\u2081 * ra) * \u2191(\u2191fS (s\u2082 * sa))\u207b\u00b9 = \u2191f r\u2081 * \u2191(\u2191fS s\u2081)\u207b\u00b9 * (\u2191f r\u2082 * \u2191(\u2191fS s\u2082)\u207b\u00b9)\n[PROOFSTEP]\nconv_rhs =>\n congr\n \u00b7skip\n congr\n rw [\u2190 mul_one (f r\u2082), \u2190 (fS sa).mul_inv, \u2190 mul_assoc, \u2190 hf, \u2190 f.map_mul, ha, f.map_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n| \u2191f r\u2081 * \u2191(\u2191fS s\u2081)\u207b\u00b9 * (\u2191f r\u2082 * \u2191(\u2191fS s\u2082)\u207b\u00b9)\n[PROOFSTEP]\n congr\n \u00b7skip\n congr\n rw [\u2190 mul_one (f r\u2082), \u2190 (fS sa).mul_inv, \u2190 mul_assoc, \u2190 hf, \u2190 f.map_mul, ha, f.map_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n| \u2191f r\u2081 * \u2191(\u2191fS s\u2081)\u207b\u00b9 * (\u2191f r\u2082 * \u2191(\u2191fS s\u2082)\u207b\u00b9)\n[PROOFSTEP]\n congr\n \u00b7skip\n congr\n rw [\u2190 mul_one (f r\u2082), \u2190 (fS sa).mul_inv, \u2190 mul_assoc, \u2190 hf, \u2190 f.map_mul, ha, f.map_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n| \u2191f r\u2081 * \u2191(\u2191fS s\u2081)\u207b\u00b9 * (\u2191f r\u2082 * \u2191(\u2191fS s\u2082)\u207b\u00b9)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n| \u2191f r\u2081 * \u2191(\u2191fS s\u2081)\u207b\u00b9\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n| \u2191f r\u2082 * \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n| \u2191f r\u2081 * \u2191(\u2191fS s\u2081)\u207b\u00b9\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n| \u2191f r\u2081 * \u2191(\u2191fS s\u2081)\u207b\u00b9\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n| \u2191f r\u2081 * \u2191(\u2191fS s\u2081)\u207b\u00b9\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n| \u2191f r\u2082 * \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.a\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n| \u2191f r\u2082\ncase a.a\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n| \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 mul_one (f r\u2082), \u2190 (fS sa).mul_inv, \u2190 mul_assoc, \u2190 hf, \u2190 f.map_mul, ha, f.map_mul]\n[GOAL]\ncase c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n\u22a2 \u2191f (r\u2081 * ra) * \u2191(\u2191fS (s\u2082 * sa))\u207b\u00b9 = \u2191f r\u2081 * \u2191(\u2191fS s\u2081)\u207b\u00b9 * (\u2191f \u2191s\u2081 * \u2191f ra * \u2191(\u2191fS sa)\u207b\u00b9 * \u2191(\u2191fS s\u2082)\u207b\u00b9)\n[PROOFSTEP]\nrw [mul_assoc, mul_assoc, mul_assoc, \u2190 mul_assoc _ (f s\u2081), hf s\u2081, (fS s\u2081).inv_mul, one_mul, f.map_mul, mul_assoc,\n fS.map_mul, \u2190 Units.val_mul]\n[GOAL]\ncase c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : r\u2082 * \u2191sa = \u2191s\u2081 * ra\n\u22a2 \u2191f r\u2081 * (\u2191f ra * \u2191(\u2191fS s\u2082 * \u2191fS sa)\u207b\u00b9) = \u2191f r\u2081 * (\u2191f ra * \u2191((\u2191fS sa)\u207b\u00b9 * (\u2191fS s\u2082)\u207b\u00b9))\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr : R\n\u22a2 \u2191(universalMulHom f fS hf) (\u2191numeratorHom r) = \u2191f r\n[PROOFSTEP]\nsimp [numeratorHom_apply, universalMulHom_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\n\u03c6 : OreLocalization R S \u2192* T\nhuniv : \u2200 (r : R), \u2191\u03c6 (\u2191numeratorHom r) = \u2191f r\n\u22a2 \u03c6 = universalMulHom f fS hf\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\n\u03c6 : OreLocalization R S \u2192* T\nhuniv : \u2200 (r : R), \u2191\u03c6 (\u2191numeratorHom r) = \u2191f r\nx : OreLocalization R S\n\u22a2 \u2191\u03c6 x = \u2191(universalMulHom f fS hf) x\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r s\n[GOAL]\ncase h.c\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Monoid T\nf : R \u2192* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\n\u03c6 : OreLocalization R S \u2192* T\nhuniv : \u2200 (r : R), \u2191\u03c6 (\u2191numeratorHom r) = \u2191f r\nr : R\ns : { x // x \u2208 S }\n\u22a2 \u2191\u03c6 (r /\u2092 s) = \u2191(universalMulHom f fS hf) (r /\u2092 s)\n[PROOFSTEP]\nrw [universalMulHom_apply, \u2190 huniv r, numeratorHom_apply, \u2190 mul_one (\u03c6 (r /\u2092 s)), \u2190 Units.val_one, \u2190\n mul_right_inv (fS s), Units.val_mul, \u2190 mul_assoc, \u2190 hf, \u2190 huniv, \u2190 \u03c6.map_mul, numeratorHom_apply,\n OreLocalization.mul_cancel]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\n\u22a2 r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082) = r\u2081 * r\u2082 /\u2092 (s\u2081 * s\u2082)\n[PROOFSTEP]\nrw [oreDiv_mul_char r\u2081 r\u2082 s\u2081 s\u2082 r\u2082 s\u2081 (by simp [mul_comm]), mul_comm s\u2082]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\n\u22a2 r\u2082 * \u2191s\u2081 = \u2191s\u2081 * r\u2082\n[PROOFSTEP]\nsimp [mul_comm]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nsrc\u271d : Monoid (OreLocalization R S) := instMonoidOreLocalization\nx y : OreLocalization R S\n\u22a2 x * y = y * x\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r\u2081 s\u2081\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nsrc\u271d : Monoid (OreLocalization R S) := instMonoidOreLocalization\ny : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\n\u22a2 r\u2081 /\u2092 s\u2081 * y = y * (r\u2081 /\u2092 s\u2081)\n[PROOFSTEP]\ninduction' y using OreLocalization.ind with r\u2082 s\u2082\n[GOAL]\ncase c.c\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nsrc\u271d : Monoid (OreLocalization R S) := instMonoidOreLocalization\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\n\u22a2 r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082) = r\u2082 /\u2092 s\u2082 * (r\u2081 /\u2092 s\u2081)\n[PROOFSTEP]\nrw [oreDiv_mul_oreDiv_comm, oreDiv_mul_oreDiv_comm, mul_comm r\u2081, mul_comm s\u2081]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\n\u22a2 OneHom.toFun { toFun := \u2191numeratorHom, map_one' := (_ : \u2191numeratorHom 1 = \u2191numeratorHom 1) } (r\u2081 * r\u2082) =\n OneHom.toFun { toFun := \u2191numeratorHom, map_one' := (_ : \u2191numeratorHom 1 = \u2191numeratorHom 1) } r\u2081 *\n OneHom.toFun { toFun := \u2191numeratorHom, map_one' := (_ : \u2191numeratorHom 1 = \u2191numeratorHom 1) } r\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nz : OreLocalization R S\n\u22a2 \u2203 x,\n z *\n OneHom.toFun\n \u2191{ toOneHom := { toFun := \u2191numeratorHom, map_one' := (_ : \u2191numeratorHom 1 = \u2191numeratorHom 1) },\n map_mul' := (_ : \u2200 (r\u2081 r\u2082 : R), \u2191numeratorHom (r\u2081 * r\u2082) = \u2191numeratorHom r\u2081 * \u2191numeratorHom r\u2082) }\n \u2191x.snd =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191numeratorHom, map_one' := (_ : \u2191numeratorHom 1 = \u2191numeratorHom 1) },\n map_mul' := (_ : \u2200 (r\u2081 r\u2082 : R), \u2191numeratorHom (r\u2081 * r\u2082) = \u2191numeratorHom r\u2081 * \u2191numeratorHom r\u2082) })\n x.fst\n[PROOFSTEP]\ninduction' z using OreLocalization.ind with r s\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 \u2203 x,\n r /\u2092 s *\n OneHom.toFun\n \u2191{ toOneHom := { toFun := \u2191numeratorHom, map_one' := (_ : \u2191numeratorHom 1 = \u2191numeratorHom 1) },\n map_mul' := (_ : \u2200 (r\u2081 r\u2082 : R), \u2191numeratorHom (r\u2081 * r\u2082) = \u2191numeratorHom r\u2081 * \u2191numeratorHom r\u2082) }\n \u2191x.snd =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191numeratorHom, map_one' := (_ : \u2191numeratorHom 1 = \u2191numeratorHom 1) },\n map_mul' := (_ : \u2200 (r\u2081 r\u2082 : R), \u2191numeratorHom (r\u2081 * r\u2082) = \u2191numeratorHom r\u2081 * \u2191numeratorHom r\u2082) })\n x.fst\n[PROOFSTEP]\nuse(r, s)\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 r /\u2092 s *\n OneHom.toFun\n \u2191{ toOneHom := { toFun := \u2191numeratorHom, map_one' := (_ : \u2191numeratorHom 1 = \u2191numeratorHom 1) },\n map_mul' := (_ : \u2200 (r\u2081 r\u2082 : R), \u2191numeratorHom (r\u2081 * r\u2082) = \u2191numeratorHom r\u2081 * \u2191numeratorHom r\u2082) }\n \u2191(r, s).snd =\n OneHom.toFun\n \u2191{ toOneHom := { toFun := \u2191numeratorHom, map_one' := (_ : \u2191numeratorHom 1 = \u2191numeratorHom 1) },\n map_mul' := (_ : \u2200 (r\u2081 r\u2082 : R), \u2191numeratorHom (r\u2081 * r\u2082) = \u2191numeratorHom r\u2081 * \u2191numeratorHom r\u2082) }\n (r, s).fst\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 r /\u2092 s * \u2191numeratorHom \u2191s = \u2191numeratorHom r\n[PROOFSTEP]\nrw [numeratorHom_apply, numeratorHom_apply]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 r /\u2092 s * (\u2191s /\u2092 1) = r /\u2092 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191numeratorHom, map_one' := (_ : \u2191numeratorHom 1 = \u2191numeratorHom 1) },\n map_mul' := (_ : \u2200 (r\u2081 r\u2082 : R), \u2191numeratorHom (r\u2081 * r\u2082) = \u2191numeratorHom r\u2081 * \u2191numeratorHom r\u2082) })\n r\u2081 =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191numeratorHom, map_one' := (_ : \u2191numeratorHom 1 = \u2191numeratorHom 1) },\n map_mul' := (_ : \u2200 (r\u2081 r\u2082 : R), \u2191numeratorHom (r\u2081 * r\u2082) = \u2191numeratorHom r\u2081 * \u2191numeratorHom r\u2082) })\n r\u2082 \u2194\n \u2203 c, \u2191c * r\u2081 = \u2191c * r\u2082\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\n\u22a2 \u2191numeratorHom r\u2081 = \u2191numeratorHom r\u2082 \u2194 \u2203 c, \u2191c * r\u2081 = \u2191c * r\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\n\u22a2 \u2191numeratorHom r\u2081 = \u2191numeratorHom r\u2082 \u2192 \u2203 c, \u2191c * r\u2081 = \u2191c * r\u2082\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\nh : \u2191numeratorHom r\u2081 = \u2191numeratorHom r\u2082\n\u22a2 \u2203 c, \u2191c * r\u2081 = \u2191c * r\u2082\n[PROOFSTEP]\nrw [numeratorHom_apply, numeratorHom_apply, oreDiv_eq_iff] at h \n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\nh : \u2203 u v, r\u2082 * \u2191u = r\u2081 * v \u2227 \u21911 * \u2191u = \u21911 * v\n\u22a2 \u2203 c, \u2191c * r\u2081 = \u2191c * r\u2082\n[PROOFSTEP]\nrcases h with \u27e8u, v, h\u2081, h\u2082\u27e9\n[GOAL]\ncase mp.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\nu : { x // x \u2208 S }\nv : R\nh\u2081 : r\u2082 * \u2191u = r\u2081 * v\nh\u2082 : \u21911 * \u2191u = \u21911 * v\n\u22a2 \u2203 c, \u2191c * r\u2081 = \u2191c * r\u2082\n[PROOFSTEP]\ndsimp at h\u2082 \n[GOAL]\ncase mp.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\nu : { x // x \u2208 S }\nv : R\nh\u2081 : r\u2082 * \u2191u = r\u2081 * v\nh\u2082 : 1 * \u2191u = 1 * v\n\u22a2 \u2203 c, \u2191c * r\u2081 = \u2191c * r\u2082\n[PROOFSTEP]\nrw [one_mul, one_mul] at h\u2082 \n[GOAL]\ncase mp.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\nu : { x // x \u2208 S }\nv : R\nh\u2081 : r\u2082 * \u2191u = r\u2081 * v\nh\u2082 : \u2191u = v\n\u22a2 \u2203 c, \u2191c * r\u2081 = \u2191c * r\u2082\n[PROOFSTEP]\nsubst h\u2082\n[GOAL]\ncase mp.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\nu : { x // x \u2208 S }\nh\u2081 : r\u2082 * \u2191u = r\u2081 * \u2191u\n\u22a2 \u2203 c, \u2191c * r\u2081 = \u2191c * r\u2082\n[PROOFSTEP]\nuse u\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\nu : { x // x \u2208 S }\nh\u2081 : r\u2082 * \u2191u = r\u2081 * \u2191u\n\u22a2 \u2191u * r\u2081 = \u2191u * r\u2082\n[PROOFSTEP]\nsimpa only [mul_comm] using h\u2081.symm\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\n\u22a2 (\u2203 c, \u2191c * r\u2081 = \u2191c * r\u2082) \u2192 \u2191numeratorHom r\u2081 = \u2191numeratorHom r\u2082\n[PROOFSTEP]\nrintro \u27e8s, h\u27e9\n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns : { x // x \u2208 S }\nh : \u2191s * r\u2081 = \u2191s * r\u2082\n\u22a2 \u2191numeratorHom r\u2081 = \u2191numeratorHom r\u2082\n[PROOFSTEP]\nrw [numeratorHom_apply, numeratorHom_apply, oreDiv_eq_iff]\n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns : { x // x \u2208 S }\nh : \u2191s * r\u2081 = \u2191s * r\u2082\n\u22a2 \u2203 u v, r\u2082 * \u2191u = r\u2081 * v \u2227 \u21911 * \u2191u = \u21911 * v\n[PROOFSTEP]\nrefine' \u27e8s, s, _, _\u27e9\n[GOAL]\ncase mpr.intro.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns : { x // x \u2208 S }\nh : \u2191s * r\u2081 = \u2191s * r\u2082\n\u22a2 r\u2082 * \u2191s = r\u2081 * \u2191s\n[PROOFSTEP]\nsimpa [mul_comm] using h.symm\n[GOAL]\ncase mpr.intro.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : CommMonoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns : { x // x \u2208 S }\nh : \u2191s * r\u2081 = \u2191s * r\u2082\n\u22a2 \u21911 * \u2191s = \u21911 * \u2191s\n[PROOFSTEP]\nsimp [one_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\n\u22a2 OreLocalization.add'' r\u2081 s\u2081 r\u2082 s\u2082 = (r\u2081 * \u2191sb + r\u2082 * rb) /\u2092 (s\u2081 * sb)\n[PROOFSTEP]\nsimp only [add'']\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\n\u22a2 (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * oreNum (\u2191s\u2081) s\u2082) /\u2092 (s\u2081 * oreDenom (\u2191s\u2081) s\u2082) = (r\u2081 * \u2191sb + r\u2082 * rb) /\u2092 (s\u2081 * sb)\n[PROOFSTEP]\nhave ha := ore_eq (s\u2081 : R) s\u2082\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\n\u22a2 (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * oreNum (\u2191s\u2081) s\u2082) /\u2092 (s\u2081 * oreDenom (\u2191s\u2081) s\u2082) = (r\u2081 * \u2191sb + r\u2082 * rb) /\u2092 (s\u2081 * sb)\n[PROOFSTEP]\nset! ra := oreNum (s\u2081 : R) s\u2082 with h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nh : ra = oreNum (\u2191s\u2081) s\u2082\n\u22a2 (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * oreNum (\u2191s\u2081) s\u2082) /\u2092 (s\u2081 * oreDenom (\u2191s\u2081) s\u2082) = (r\u2081 * \u2191sb + r\u2082 * rb) /\u2092 (s\u2081 * sb)\n[PROOFSTEP]\nrw [\u2190 h] at *\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nh : ra = ra\n\u22a2 (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * ra) /\u2092 (s\u2081 * oreDenom (\u2191s\u2081) s\u2082) = (r\u2081 * \u2191sb + r\u2082 * rb) /\u2092 (s\u2081 * sb)\n[PROOFSTEP]\nclear h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\n\u22a2 (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * ra) /\u2092 (s\u2081 * oreDenom (\u2191s\u2081) s\u2082) = (r\u2081 * \u2191sb + r\u2082 * rb) /\u2092 (s\u2081 * sb)\n[PROOFSTEP]\nset! sa := oreDenom (s\u2081 : R) s\u2082 with h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nh : sa = oreDenom (\u2191s\u2081) s\u2082\n\u22a2 (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * ra) /\u2092 (s\u2081 * oreDenom (\u2191s\u2081) s\u2082) = (r\u2081 * \u2191sb + r\u2082 * rb) /\u2092 (s\u2081 * sb)\n[PROOFSTEP]\nrw [\u2190 h] at *\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nh : sa = sa\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) = (r\u2081 * \u2191sb + r\u2082 * rb) /\u2092 (s\u2081 * sb)\n[PROOFSTEP]\nclear h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) = (r\u2081 * \u2191sb + r\u2082 * rb) /\u2092 (s\u2081 * sb)\n[PROOFSTEP]\nrcases oreCondition (sa : R) sb with\n \u27e8rc, sc, hc\u27e9\n -- s*, r*\n[GOAL]\ncase mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191sa * \u2191sc = \u2191sb * rc\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) = (r\u2081 * \u2191sb + r\u2082 * rb) /\u2092 (s\u2081 * sb)\n[PROOFSTEP]\nhave : (s\u2082 : R) * (rb * rc) = s\u2082 * (ra * sc) := by rw [\u2190 mul_assoc, \u2190 hb, mul_assoc, \u2190 hc, \u2190 mul_assoc, \u2190 mul_assoc, ha]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191sa * \u2191sc = \u2191sb * rc\n\u22a2 \u2191s\u2082 * (rb * rc) = \u2191s\u2082 * (ra * \u2191sc)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 hb, mul_assoc, \u2190 hc, \u2190 mul_assoc, \u2190 mul_assoc, ha]\n[GOAL]\ncase mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191sa * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2082 * (rb * rc) = \u2191s\u2082 * (ra * \u2191sc)\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) = (r\u2081 * \u2191sb + r\u2082 * rb) /\u2092 (s\u2081 * sb)\n[PROOFSTEP]\nrcases ore_left_cancel _ _ s\u2082 this with\n \u27e8sd, hd\u27e9\n -- s#\n[GOAL]\ncase mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191sa * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2082 * (rb * rc) = \u2191s\u2082 * (ra * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = ra * \u2191sc * \u2191sd\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) = (r\u2081 * \u2191sb + r\u2082 * rb) /\u2092 (s\u2081 * sb)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191sa * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2082 * (rb * rc) = \u2191s\u2082 * (ra * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = ra * \u2191sc * \u2191sd\n\u22a2 (r\u2081 * \u2191sb + r\u2082 * rb) /\u2092 (s\u2081 * sb) = (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa)\n[PROOFSTEP]\nrw [oreDiv_eq_iff]\n[GOAL]\ncase mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191sa * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2082 * (rb * rc) = \u2191s\u2082 * (ra * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = ra * \u2191sc * \u2191sd\n\u22a2 \u2203 u v, (r\u2081 * \u2191sa + r\u2082 * ra) * \u2191u = (r\u2081 * \u2191sb + r\u2082 * rb) * v \u2227 \u2191(s\u2081 * sa) * \u2191u = \u2191(s\u2081 * sb) * v\n[PROOFSTEP]\nuse sc * sd\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191sa * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2082 * (rb * rc) = \u2191s\u2082 * (ra * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = ra * \u2191sc * \u2191sd\n\u22a2 \u2203 v, (r\u2081 * \u2191sa + r\u2082 * ra) * \u2191(sc * sd) = (r\u2081 * \u2191sb + r\u2082 * rb) * v \u2227 \u2191(s\u2081 * sa) * \u2191(sc * sd) = \u2191(s\u2081 * sb) * v\n[PROOFSTEP]\nuse rc * sd\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191sa * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2082 * (rb * rc) = \u2191s\u2082 * (ra * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = ra * \u2191sc * \u2191sd\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) * \u2191(sc * sd) = (r\u2081 * \u2191sb + r\u2082 * rb) * (rc * \u2191sd) \u2227\n \u2191(s\u2081 * sa) * \u2191(sc * sd) = \u2191(s\u2081 * sb) * (rc * \u2191sd)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191sa * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2082 * (rb * rc) = \u2191s\u2082 * (ra * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = ra * \u2191sc * \u2191sd\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) * \u2191(sc * sd) = (r\u2081 * \u2191sb + r\u2082 * rb) * (rc * \u2191sd)\n[PROOFSTEP]\nsimp only [Submonoid.coe_mul]\n[GOAL]\ncase h.right\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191sa * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2082 * (rb * rc) = \u2191s\u2082 * (ra * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = ra * \u2191sc * \u2191sd\n\u22a2 \u2191(s\u2081 * sa) * \u2191(sc * sd) = \u2191(s\u2081 * sb) * (rc * \u2191sd)\n[PROOFSTEP]\nsimp only [Submonoid.coe_mul]\n[GOAL]\ncase h.left\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191sa * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2082 * (rb * rc) = \u2191s\u2082 * (ra * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = ra * \u2191sc * \u2191sd\n\u22a2 (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * oreNum (\u2191s\u2081) s\u2082) * (\u2191sc * \u2191sd) = (r\u2081 * \u2191sb + r\u2082 * rb) * (rc * \u2191sd)\n[PROOFSTEP]\nnoncomm_ring\n[GOAL]\ncase h.left\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191sa * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2082 * (rb * rc) = \u2191s\u2082 * (ra * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = ra * \u2191sc * \u2191sd\n\u22a2 r\u2081 * (\u2191(oreDenom (\u2191s\u2081) s\u2082) * (\u2191sc * \u2191sd)) + r\u2082 * (oreNum (\u2191s\u2081) s\u2082 * (\u2191sc * \u2191sd)) =\n r\u2081 * (\u2191sb * (rc * \u2191sd)) + r\u2082 * (rb * (rc * \u2191sd))\n[PROOFSTEP]\nrw [\u2190 mul_assoc (a := rb), hd, \u2190 mul_assoc (a := (sa : R)), hc]\n[GOAL]\ncase h.left\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191sa * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2082 * (rb * rc) = \u2191s\u2082 * (ra * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = ra * \u2191sc * \u2191sd\n\u22a2 r\u2081 * (\u2191sb * rc * \u2191sd) + r\u2082 * (oreNum (\u2191s\u2081) s\u2082 * (\u2191sc * \u2191sd)) = r\u2081 * (\u2191sb * (rc * \u2191sd)) + r\u2082 * (ra * \u2191sc * \u2191sd)\n[PROOFSTEP]\nnoncomm_ring\n[GOAL]\ncase h.right\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191sa * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2082 * (rb * rc) = \u2191s\u2082 * (ra * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = ra * \u2191sc * \u2191sd\n\u22a2 \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) * (\u2191sc * \u2191sd) = \u2191s\u2081 * \u2191sb * (rc * \u2191sd)\n[PROOFSTEP]\nrw [mul_assoc (a := (s\u2081 : R)), \u2190 mul_assoc (a := (sa : R)), hc]\n[GOAL]\ncase h.right\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2081 * \u2191sb = \u2191s\u2082 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191sa * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2082 * (rb * rc) = \u2191s\u2082 * (ra * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = ra * \u2191sc * \u2191sd\n\u22a2 \u2191s\u2081 * (\u2191sb * rc * \u2191sd) = \u2191s\u2081 * \u2191sb * (rc * \u2191sd)\n[PROOFSTEP]\nnoncomm_ring\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\n\u22a2 \u2200 (a b : R \u00d7 { x // x \u2208 S }),\n a \u2248 b \u2192 OreLocalization.add'' a.fst a.snd r\u2082 s\u2082 = OreLocalization.add'' b.fst b.snd r\u2082 s\u2082\n[PROOFSTEP]\nrintro \u27e8r\u2081', s\u2081'\u27e9 \u27e8r\u2081, s\u2081\u27e9\n \u27e8sb, rb, hb, hb'\u27e9\n -- s*, r*\n[GOAL]\ncase mk.mk.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : (r\u2081, s\u2081).fst * \u2191sb = (r\u2081', s\u2081').fst * rb\nhb' : \u2191(r\u2081, s\u2081).snd * \u2191sb = \u2191(r\u2081', s\u2081').snd * rb\n\u22a2 OreLocalization.add'' (r\u2081', s\u2081').fst (r\u2081', s\u2081').snd r\u2082 s\u2082 = OreLocalization.add'' (r\u2081, s\u2081).fst (r\u2081, s\u2081).snd r\u2082 s\u2082\n[PROOFSTEP]\nrcases oreCondition (s\u2081' : R) s\u2082 with\n \u27e8rc, sc, hc\u27e9\n --s~~, r~~\n[GOAL]\ncase mk.mk.intro.intro.intro.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : (r\u2081, s\u2081).fst * \u2191sb = (r\u2081', s\u2081').fst * rb\nhb' : \u2191(r\u2081, s\u2081).snd * \u2191sb = \u2191(r\u2081', s\u2081').snd * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191s\u2081' * \u2191sc = \u2191s\u2082 * rc\n\u22a2 OreLocalization.add'' (r\u2081', s\u2081').fst (r\u2081', s\u2081').snd r\u2082 s\u2082 = OreLocalization.add'' (r\u2081, s\u2081).fst (r\u2081, s\u2081).snd r\u2082 s\u2082\n[PROOFSTEP]\nrcases oreCondition rb sc with\n \u27e8rd, sd, hd\u27e9\n -- s#, r#\n[GOAL]\ncase mk.mk.intro.intro.intro.mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : (r\u2081, s\u2081).fst * \u2191sb = (r\u2081', s\u2081').fst * rb\nhb' : \u2191(r\u2081, s\u2081).snd * \u2191sb = \u2191(r\u2081', s\u2081').snd * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191s\u2081' * \u2191sc = \u2191s\u2082 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : rb * \u2191sd = \u2191sc * rd\n\u22a2 OreLocalization.add'' (r\u2081', s\u2081').fst (r\u2081', s\u2081').snd r\u2082 s\u2082 = OreLocalization.add'' (r\u2081, s\u2081).fst (r\u2081, s\u2081).snd r\u2082 s\u2082\n[PROOFSTEP]\ndsimp at *\n[GOAL]\ncase mk.mk.intro.intro.intro.mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : r\u2081 * \u2191sb = r\u2081' * rb\nhb' : \u2191s\u2081 * \u2191sb = \u2191s\u2081' * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191s\u2081' * \u2191sc = \u2191s\u2082 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : rb * \u2191sd = \u2191sc * rd\n\u22a2 OreLocalization.add'' r\u2081' s\u2081' r\u2082 s\u2082 = OreLocalization.add'' r\u2081 s\u2081 r\u2082 s\u2082\n[PROOFSTEP]\nrw [add''_char _ _ _ _ rc sc hc]\n[GOAL]\ncase mk.mk.intro.intro.intro.mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : r\u2081 * \u2191sb = r\u2081' * rb\nhb' : \u2191s\u2081 * \u2191sb = \u2191s\u2081' * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191s\u2081' * \u2191sc = \u2191s\u2082 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : rb * \u2191sd = \u2191sc * rd\n\u22a2 (r\u2081' * \u2191sc + r\u2082 * rc) /\u2092 (s\u2081' * sc) = OreLocalization.add'' r\u2081 s\u2081 r\u2082 s\u2082\n[PROOFSTEP]\nhave : \u2191s\u2081 * \u2191(sb * sd) = \u2191s\u2082 * (rc * rd) := by\n simp only [Submonoid.coe_mul]\n rw [\u2190 mul_assoc, hb', mul_assoc, hd, \u2190 mul_assoc, hc, mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : r\u2081 * \u2191sb = r\u2081' * rb\nhb' : \u2191s\u2081 * \u2191sb = \u2191s\u2081' * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191s\u2081' * \u2191sc = \u2191s\u2082 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : rb * \u2191sd = \u2191sc * rd\n\u22a2 \u2191s\u2081 * \u2191(sb * sd) = \u2191s\u2082 * (rc * rd)\n[PROOFSTEP]\nsimp only [Submonoid.coe_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : r\u2081 * \u2191sb = r\u2081' * rb\nhb' : \u2191s\u2081 * \u2191sb = \u2191s\u2081' * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191s\u2081' * \u2191sc = \u2191s\u2082 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : rb * \u2191sd = \u2191sc * rd\n\u22a2 \u2191s\u2081 * (\u2191sb * \u2191sd) = \u2191s\u2082 * (rc * rd)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, hb', mul_assoc, hd, \u2190 mul_assoc, hc, mul_assoc]\n[GOAL]\ncase mk.mk.intro.intro.intro.mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : r\u2081 * \u2191sb = r\u2081' * rb\nhb' : \u2191s\u2081 * \u2191sb = \u2191s\u2081' * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191s\u2081' * \u2191sc = \u2191s\u2082 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : rb * \u2191sd = \u2191sc * rd\nthis : \u2191s\u2081 * \u2191(sb * sd) = \u2191s\u2082 * (rc * rd)\n\u22a2 (r\u2081' * \u2191sc + r\u2082 * rc) /\u2092 (s\u2081' * sc) = OreLocalization.add'' r\u2081 s\u2081 r\u2082 s\u2082\n[PROOFSTEP]\nrw [add''_char _ _ _ _ (rc * rd : R) (sb * sd : S) this]\n[GOAL]\ncase mk.mk.intro.intro.intro.mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : r\u2081 * \u2191sb = r\u2081' * rb\nhb' : \u2191s\u2081 * \u2191sb = \u2191s\u2081' * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191s\u2081' * \u2191sc = \u2191s\u2082 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : rb * \u2191sd = \u2191sc * rd\nthis : \u2191s\u2081 * \u2191(sb * sd) = \u2191s\u2082 * (rc * rd)\n\u22a2 (r\u2081' * \u2191sc + r\u2082 * rc) /\u2092 (s\u2081' * sc) = (r\u2081 * \u2191(sb * sd) + r\u2082 * (rc * rd)) /\u2092 (s\u2081 * (sb * sd))\n[PROOFSTEP]\nsimp only [Submonoid.coe_mul]\n[GOAL]\ncase mk.mk.intro.intro.intro.mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : r\u2081 * \u2191sb = r\u2081' * rb\nhb' : \u2191s\u2081 * \u2191sb = \u2191s\u2081' * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191s\u2081' * \u2191sc = \u2191s\u2082 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : rb * \u2191sd = \u2191sc * rd\nthis : \u2191s\u2081 * \u2191(sb * sd) = \u2191s\u2082 * (rc * rd)\n\u22a2 (r\u2081' * \u2191sc + r\u2082 * rc) /\u2092 (s\u2081' * sc) = (r\u2081 * (\u2191sb * \u2191sd) + r\u2082 * (rc * rd)) /\u2092 (s\u2081 * (sb * sd))\n[PROOFSTEP]\nrw [\u2190 mul_assoc (a := r\u2081) (b := (sb : R)), hb, mul_assoc (a := r\u2081') (b := (rb : R)), hd, \u2190 mul_assoc, \u2190 mul_assoc, \u2190\n add_mul, oreDiv_eq_iff]\n[GOAL]\ncase mk.mk.intro.intro.intro.mk.mk.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : r\u2081 * \u2191sb = r\u2081' * rb\nhb' : \u2191s\u2081 * \u2191sb = \u2191s\u2081' * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191s\u2081' * \u2191sc = \u2191s\u2082 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : rb * \u2191sd = \u2191sc * rd\nthis : \u2191s\u2081 * \u2191(sb * sd) = \u2191s\u2082 * (rc * rd)\n\u22a2 \u2203 u v, (r\u2081' * \u2191sc + r\u2082 * rc) * rd * \u2191u = (r\u2081' * \u2191sc + r\u2082 * rc) * v \u2227 \u2191(s\u2081 * (sb * sd)) * \u2191u = \u2191(s\u2081' * sc) * v\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : r\u2081 * \u2191sb = r\u2081' * rb\nhb' : \u2191s\u2081 * \u2191sb = \u2191s\u2081' * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191s\u2081' * \u2191sc = \u2191s\u2082 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : rb * \u2191sd = \u2191sc * rd\nthis : \u2191s\u2081 * \u2191(sb * sd) = \u2191s\u2082 * (rc * rd)\n\u22a2 \u2203 v, (r\u2081' * \u2191sc + r\u2082 * rc) * rd * \u21911 = (r\u2081' * \u2191sc + r\u2082 * rc) * v \u2227 \u2191(s\u2081 * (sb * sd)) * \u21911 = \u2191(s\u2081' * sc) * v\n[PROOFSTEP]\nuse rd\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : r\u2081 * \u2191sb = r\u2081' * rb\nhb' : \u2191s\u2081 * \u2191sb = \u2191s\u2081' * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191s\u2081' * \u2191sc = \u2191s\u2082 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : rb * \u2191sd = \u2191sc * rd\nthis : \u2191s\u2081 * \u2191(sb * sd) = \u2191s\u2082 * (rc * rd)\n\u22a2 (r\u2081' * \u2191sc + r\u2082 * rc) * rd * \u21911 = (r\u2081' * \u2191sc + r\u2082 * rc) * rd \u2227 \u2191(s\u2081 * (sb * sd)) * \u21911 = \u2191(s\u2081' * sc) * rd\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : r\u2081 * \u2191sb = r\u2081' * rb\nhb' : \u2191s\u2081 * \u2191sb = \u2191s\u2081' * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191s\u2081' * \u2191sc = \u2191s\u2082 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : rb * \u2191sd = \u2191sc * rd\nthis : \u2191s\u2081 * \u2191(sb * sd) = \u2191s\u2082 * (rc * rd)\n\u22a2 (r\u2081' * \u2191sc + r\u2082 * rc) * rd * \u21911 = (r\u2081' * \u2191sc + r\u2082 * rc) * rd\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.right\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : r\u2081 * \u2191sb = r\u2081' * rb\nhb' : \u2191s\u2081 * \u2191sb = \u2191s\u2081' * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191s\u2081' * \u2191sc = \u2191s\u2082 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : rb * \u2191sd = \u2191sc * rd\nthis : \u2191s\u2081 * \u2191(sb * sd) = \u2191s\u2082 * (rc * rd)\n\u22a2 \u2191(s\u2081 * (sb * sd)) * \u21911 = \u2191(s\u2081' * sc) * rd\n[PROOFSTEP]\nsimp only [mul_one, Submonoid.coe_one, Submonoid.coe_mul] at this \u22a2\n[GOAL]\ncase h.right\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2081' : R\ns\u2081' : { x // x \u2208 S }\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nsb : { x // x \u2208 S }\nrb : R\nhb : r\u2081 * \u2191sb = r\u2081' * rb\nhb' : \u2191s\u2081 * \u2191sb = \u2191s\u2081' * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191s\u2081' * \u2191sc = \u2191s\u2082 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : rb * \u2191sd = \u2191sc * rd\nthis : \u2191s\u2081 * (\u2191sb * \u2191sd) = \u2191s\u2082 * (rc * rd)\n\u22a2 \u2191s\u2081 * (\u2191sb * \u2191sd) = \u2191s\u2081' * \u2191sc * rd\n[PROOFSTEP]\nrw [hc, this, mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\n\u22a2 OreLocalization.add' r\u2081 s\u2081 (r\u2082 /\u2092 s\u2082) = OreLocalization.add' r\u2082 s\u2082 (r\u2081 /\u2092 s\u2081)\n[PROOFSTEP]\nsimp only [add', oreDiv, add'', Quotient.mk', Quotient.lift_mk]\n -- Porting note: `Quotient.eq` should be used in `rw` instead of `simp`\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\n\u22a2 Quotient.mk (oreEqv R S) (r\u2082 * \u2191(oreDenom (\u2191s\u2082) s\u2081) + r\u2081 * oreNum (\u2191s\u2082) s\u2081, s\u2082 * oreDenom (\u2191s\u2082) s\u2081) =\n Quotient.mk (oreEqv R S) (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * oreNum (\u2191s\u2081) s\u2082, s\u2081 * oreDenom (\u2191s\u2081) s\u2082)\n[PROOFSTEP]\nrw [Quotient.eq]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\n\u22a2 (r\u2082 * \u2191(oreDenom (\u2191s\u2082) s\u2081) + r\u2081 * oreNum (\u2191s\u2082) s\u2081, s\u2082 * oreDenom (\u2191s\u2082) s\u2081) \u2248\n (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * oreNum (\u2191s\u2081) s\u2082, s\u2081 * oreDenom (\u2191s\u2081) s\u2082)\n[PROOFSTEP]\nhave hb := ore_eq (\u2191s\u2082) s\u2081\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191(oreDenom (\u2191s\u2082) s\u2081) = \u2191s\u2081 * oreNum (\u2191s\u2082) s\u2081\n\u22a2 (r\u2082 * \u2191(oreDenom (\u2191s\u2082) s\u2081) + r\u2081 * oreNum (\u2191s\u2082) s\u2081, s\u2082 * oreDenom (\u2191s\u2082) s\u2081) \u2248\n (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * oreNum (\u2191s\u2081) s\u2082, s\u2081 * oreDenom (\u2191s\u2081) s\u2082)\n[PROOFSTEP]\nset rb := oreNum (\u2191s\u2082) s\u2081\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191(oreDenom (\u2191s\u2082) s\u2081) = \u2191s\u2081 * rb\n\u22a2 (r\u2082 * \u2191(oreDenom (\u2191s\u2082) s\u2081) + r\u2081 * rb, s\u2082 * oreDenom (\u2191s\u2082) s\u2081) \u2248\n (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * oreNum (\u2191s\u2081) s\u2082, s\u2081 * oreDenom (\u2191s\u2081) s\u2082)\n[PROOFSTEP]\nset sb := oreDenom (\u2191s\u2082) s\u2081\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nsb : { x // x \u2208 S } := oreDenom (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2081 * rb\n\u22a2 (r\u2082 * \u2191sb + r\u2081 * rb, s\u2082 * sb) \u2248 (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * oreNum (\u2191s\u2081) s\u2082, s\u2081 * oreDenom (\u2191s\u2081) s\u2082)\n[PROOFSTEP]\nhave ha := ore_eq (\u2191s\u2081) s\u2082\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nsb : { x // x \u2208 S } := oreDenom (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2081 * rb\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * oreNum (\u2191s\u2081) s\u2082\n\u22a2 (r\u2082 * \u2191sb + r\u2081 * rb, s\u2082 * sb) \u2248 (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * oreNum (\u2191s\u2081) s\u2082, s\u2081 * oreDenom (\u2191s\u2081) s\u2082)\n[PROOFSTEP]\nset ra := oreNum (\u2191s\u2081) s\u2082\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nsb : { x // x \u2208 S } := oreDenom (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2081 * rb\nra : R := oreNum (\u2191s\u2081) s\u2082\nha : \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) = \u2191s\u2082 * ra\n\u22a2 (r\u2082 * \u2191sb + r\u2081 * rb, s\u2082 * sb) \u2248 (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * ra, s\u2081 * oreDenom (\u2191s\u2081) s\u2082)\n[PROOFSTEP]\nset sa := oreDenom (\u2191s\u2081) s\u2082\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nsb : { x // x \u2208 S } := oreDenom (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2081 * rb\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\n\u22a2 (r\u2082 * \u2191sb + r\u2081 * rb, s\u2082 * sb) \u2248 (r\u2081 * \u2191sa + r\u2082 * ra, s\u2081 * sa)\n[PROOFSTEP]\nrcases oreCondition ra sb with\n \u27e8rc, sc, hc\u27e9\n -- r#, s#\n[GOAL]\ncase mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nsb : { x // x \u2208 S } := oreDenom (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2081 * rb\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrc : R\nsc : { x // x \u2208 S }\nhc : ra * \u2191sc = \u2191sb * rc\n\u22a2 (r\u2082 * \u2191sb + r\u2081 * rb, s\u2082 * sb) \u2248 (r\u2081 * \u2191sa + r\u2082 * ra, s\u2081 * sa)\n[PROOFSTEP]\nhave : (s\u2081 : R) * (rb * rc) = s\u2081 * (sa * sc) := by rw [\u2190 mul_assoc, \u2190 hb, mul_assoc, \u2190 hc, \u2190 mul_assoc, \u2190 ha, mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nsb : { x // x \u2208 S } := oreDenom (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2081 * rb\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrc : R\nsc : { x // x \u2208 S }\nhc : ra * \u2191sc = \u2191sb * rc\n\u22a2 \u2191s\u2081 * (rb * rc) = \u2191s\u2081 * (\u2191sa * \u2191sc)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 hb, mul_assoc, \u2190 hc, \u2190 mul_assoc, \u2190 ha, mul_assoc]\n[GOAL]\ncase mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nsb : { x // x \u2208 S } := oreDenom (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2081 * rb\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrc : R\nsc : { x // x \u2208 S }\nhc : ra * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2081 * (rb * rc) = \u2191s\u2081 * (\u2191sa * \u2191sc)\n\u22a2 (r\u2082 * \u2191sb + r\u2081 * rb, s\u2082 * sb) \u2248 (r\u2081 * \u2191sa + r\u2082 * ra, s\u2081 * sa)\n[PROOFSTEP]\nrcases ore_left_cancel _ _ s\u2081 this with\n \u27e8sd, hd\u27e9\n -- s+\n[GOAL]\ncase mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nsb : { x // x \u2208 S } := oreDenom (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2081 * rb\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrc : R\nsc : { x // x \u2208 S }\nhc : ra * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2081 * (rb * rc) = \u2191s\u2081 * (\u2191sa * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = \u2191sa * \u2191sc * \u2191sd\n\u22a2 (r\u2082 * \u2191sb + r\u2081 * rb, s\u2082 * sb) \u2248 (r\u2081 * \u2191sa + r\u2082 * ra, s\u2081 * sa)\n[PROOFSTEP]\nuse sc * sd\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nsb : { x // x \u2208 S } := oreDenom (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2081 * rb\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrc : R\nsc : { x // x \u2208 S }\nhc : ra * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2081 * (rb * rc) = \u2191s\u2081 * (\u2191sa * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = \u2191sa * \u2191sc * \u2191sd\n\u22a2 \u2203 v,\n (r\u2081 * \u2191sa + r\u2082 * ra, s\u2081 * sa).fst * \u2191(sc * sd) = (r\u2082 * \u2191sb + r\u2081 * rb, s\u2082 * sb).fst * v \u2227\n \u2191(r\u2081 * \u2191sa + r\u2082 * ra, s\u2081 * sa).snd * \u2191(sc * sd) = \u2191(r\u2082 * \u2191sb + r\u2081 * rb, s\u2082 * sb).snd * v\n[PROOFSTEP]\nuse rc * sd\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nsb : { x // x \u2208 S } := oreDenom (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2081 * rb\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrc : R\nsc : { x // x \u2208 S }\nhc : ra * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2081 * (rb * rc) = \u2191s\u2081 * (\u2191sa * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = \u2191sa * \u2191sc * \u2191sd\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra, s\u2081 * sa).fst * \u2191(sc * sd) = (r\u2082 * \u2191sb + r\u2081 * rb, s\u2082 * sb).fst * (rc * \u2191sd) \u2227\n \u2191(r\u2081 * \u2191sa + r\u2082 * ra, s\u2081 * sa).snd * \u2191(sc * sd) = \u2191(r\u2082 * \u2191sb + r\u2081 * rb, s\u2082 * sb).snd * (rc * \u2191sd)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nsb : { x // x \u2208 S } := oreDenom (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2081 * rb\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrc : R\nsc : { x // x \u2208 S }\nhc : ra * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2081 * (rb * rc) = \u2191s\u2081 * (\u2191sa * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = \u2191sa * \u2191sc * \u2191sd\n\u22a2 (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * oreNum (\u2191s\u2081) s\u2082) * (\u2191sc * \u2191sd) =\n (r\u2082 * \u2191(oreDenom (\u2191s\u2082) s\u2081) + r\u2081 * oreNum (\u2191s\u2082) s\u2081) * (rc * \u2191sd) \u2227\n \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) * (\u2191sc * \u2191sd) = \u2191s\u2082 * \u2191(oreDenom (\u2191s\u2082) s\u2081) * (rc * \u2191sd)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nsb : { x // x \u2208 S } := oreDenom (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2081 * rb\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrc : R\nsc : { x // x \u2208 S }\nhc : ra * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2081 * (rb * rc) = \u2191s\u2081 * (\u2191sa * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = \u2191sa * \u2191sc * \u2191sd\n\u22a2 (r\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) + r\u2082 * oreNum (\u2191s\u2081) s\u2082) * (\u2191sc * \u2191sd) =\n (r\u2082 * \u2191(oreDenom (\u2191s\u2082) s\u2081) + r\u2081 * oreNum (\u2191s\u2082) s\u2081) * (rc * \u2191sd)\n[PROOFSTEP]\nrw [add_mul, add_mul, add_comm, mul_assoc (a := r\u2081) (b := (sa : R)), \u2190 mul_assoc (a := (sa : R)), \u2190 hd,\n mul_assoc (a := r\u2082) (b := ra), \u2190 mul_assoc (a := ra) (b := (sc : R)), hc]\n[GOAL]\ncase h.left\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nsb : { x // x \u2208 S } := oreDenom (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2081 * rb\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrc : R\nsc : { x // x \u2208 S }\nhc : ra * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2081 * (rb * rc) = \u2191s\u2081 * (\u2191sa * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = \u2191sa * \u2191sc * \u2191sd\n\u22a2 r\u2082 * (\u2191sb * rc * \u2191sd) + r\u2081 * (rb * rc * \u2191sd) =\n r\u2082 * \u2191(oreDenom (\u2191s\u2082) s\u2081) * (rc * \u2191sd) + r\u2081 * oreNum (\u2191s\u2082) s\u2081 * (rc * \u2191sd)\n[PROOFSTEP]\nsimp only [mul_assoc]\n[GOAL]\ncase h.right\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nsb : { x // x \u2208 S } := oreDenom (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2081 * rb\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrc : R\nsc : { x // x \u2208 S }\nhc : ra * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2081 * (rb * rc) = \u2191s\u2081 * (\u2191sa * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = \u2191sa * \u2191sc * \u2191sd\n\u22a2 \u2191s\u2081 * \u2191(oreDenom (\u2191s\u2081) s\u2082) * (\u2191sc * \u2191sd) = \u2191s\u2082 * \u2191(oreDenom (\u2191s\u2082) s\u2081) * (rc * \u2191sd)\n[PROOFSTEP]\nrw [mul_assoc, \u2190 mul_assoc (sa : R), \u2190 hd, hb]\n[GOAL]\ncase h.right\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 r\u2082 : R\ns\u2081 s\u2082 : { x // x \u2208 S }\nrb : R := oreNum (\u2191s\u2082) s\u2081\nsb : { x // x \u2208 S } := oreDenom (\u2191s\u2082) s\u2081\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2081 * rb\nra : R := oreNum (\u2191s\u2081) s\u2082\nsa : { x // x \u2208 S } := oreDenom (\u2191s\u2081) s\u2082\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrc : R\nsc : { x // x \u2208 S }\nhc : ra * \u2191sc = \u2191sb * rc\nthis : \u2191s\u2081 * (rb * rc) = \u2191s\u2081 * (\u2191sa * \u2191sc)\nsd : { x // x \u2208 S }\nhd : rb * rc * \u2191sd = \u2191sa * \u2191sc * \u2191sd\n\u22a2 \u2191s\u2081 * (rb * rc * \u2191sd) = \u2191s\u2081 * rb * (rc * \u2191sd)\n[PROOFSTEP]\nsimp only [mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nx : OreLocalization R S\n\u22a2 \u2200 (a b : R \u00d7 { x // x \u2208 S }),\n a \u2248 b \u2192 (fun rs => OreLocalization.add' rs.fst rs.snd x) a = (fun rs => OreLocalization.add' rs.fst rs.snd x) b\n[PROOFSTEP]\nrintro \u27e8r\u2081, s\u2081\u27e9 \u27e8r\u2082, s\u2082\u27e9 hyz\n[GOAL]\ncase mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nx : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nhyz : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\n\u22a2 (fun rs => OreLocalization.add' rs.fst rs.snd x) (r\u2081, s\u2081) = (fun rs => OreLocalization.add' rs.fst rs.snd x) (r\u2082, s\u2082)\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r\u2083 s\u2083\n[GOAL]\ncase mk.mk.c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nhyz : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\n\u22a2 (fun rs => OreLocalization.add' rs.fst rs.snd (r\u2083 /\u2092 s\u2083)) (r\u2081, s\u2081) =\n (fun rs => OreLocalization.add' rs.fst rs.snd (r\u2083 /\u2092 s\u2083)) (r\u2082, s\u2082)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.mk.c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nhyz : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\n\u22a2 OreLocalization.add' r\u2081 s\u2081 (r\u2083 /\u2092 s\u2083) = OreLocalization.add' r\u2082 s\u2082 (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrw [add'_comm, add'_comm r\u2082]\n -- Porting note: `Quotient.mk'` required\n[GOAL]\ncase mk.mk.c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nhyz : (r\u2081, s\u2081) \u2248 (r\u2082, s\u2082)\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\n\u22a2 OreLocalization.add' r\u2083 s\u2083 (r\u2081 /\u2092 s\u2081) = OreLocalization.add' r\u2083 s\u2083 (r\u2082 /\u2092 s\u2082)\n[PROOFSTEP]\nsimp [(\u00b7 /\u2092 \u00b7), Quotient.mk', Quotient.sound hyz]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr r' : R\ns : { x // x \u2208 S }\n\u22a2 r /\u2092 s + r' /\u2092 s = (r + r') /\u2092 s\n[PROOFSTEP]\nsimp [oreDiv_add_char s s 1 1 (by simp)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr r' : R\ns : { x // x \u2208 S }\n\u22a2 \u2191s * \u21911 = \u2191s * 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nx y z : OreLocalization R S\n\u22a2 x + y + z = x + (y + z)\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r\u2081 s\u2081\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\ny z : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\n\u22a2 r\u2081 /\u2092 s\u2081 + y + z = r\u2081 /\u2092 s\u2081 + (y + z)\n[PROOFSTEP]\ninduction' y using OreLocalization.ind with r\u2082 s\u2082\n[GOAL]\ncase c.c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nz : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\n\u22a2 r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 + z = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + z)\n[PROOFSTEP]\ninduction' z using OreLocalization.ind with r\u2083 s\u2083\n[GOAL]\ncase c.c.c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\n\u22a2 r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrcases oreDivAddChar' r\u2081 r\u2082 s\u2081 s\u2082 with \u27e8ra, sa, ha, ha'\u27e9\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nha' : r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 = (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa)\n\u22a2 r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrw [ha']\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nha' : r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 = (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa)\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nclear ha'\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrcases oreDivAddChar' r\u2082 r\u2083 s\u2082 s\u2083 with \u27e8rb, sb, hb, hb'\u27e9\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nhb' : r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083 = (r\u2082 * \u2191sb + r\u2083 * rb) /\u2092 (s\u2082 * sb)\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrw [hb']\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nhb' : r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083 = (r\u2082 * \u2191sb + r\u2083 * rb) /\u2092 (s\u2082 * sb)\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 * \u2191sb + r\u2083 * rb) /\u2092 (s\u2082 * sb)\n[PROOFSTEP]\nclear hb'\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 * \u2191sb + r\u2083 * rb) /\u2092 (s\u2082 * sb)\n[PROOFSTEP]\nrcases oreDivAddChar' (r\u2081 * sa + r\u2082 * ra) r\u2083 (s\u2081 * sa) s\u2083 with \u27e8rc, sc, hc, q\u27e9\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa) * \u2191sc = \u2191s\u2083 * rc\nq : (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) + r\u2083 /\u2092 s\u2083 = ((r\u2081 * \u2191sa + r\u2082 * ra) * \u2191sc + r\u2083 * rc) /\u2092 (s\u2081 * sa * sc)\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) + r\u2083 /\u2092 s\u2083 = r\u2081 /\u2092 s\u2081 + (r\u2082 * \u2191sb + r\u2083 * rb) /\u2092 (s\u2082 * sb)\n[PROOFSTEP]\nrw [q]\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa) * \u2191sc = \u2191s\u2083 * rc\nq : (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) + r\u2083 /\u2092 s\u2083 = ((r\u2081 * \u2191sa + r\u2082 * ra) * \u2191sc + r\u2083 * rc) /\u2092 (s\u2081 * sa * sc)\n\u22a2 ((r\u2081 * \u2191sa + r\u2082 * ra) * \u2191sc + r\u2083 * rc) /\u2092 (s\u2081 * sa * sc) = r\u2081 /\u2092 s\u2081 + (r\u2082 * \u2191sb + r\u2083 * rb) /\u2092 (s\u2082 * sb)\n[PROOFSTEP]\nclear q\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa) * \u2191sc = \u2191s\u2083 * rc\n\u22a2 ((r\u2081 * \u2191sa + r\u2082 * ra) * \u2191sc + r\u2083 * rc) /\u2092 (s\u2081 * sa * sc) = r\u2081 /\u2092 s\u2081 + (r\u2082 * \u2191sb + r\u2083 * rb) /\u2092 (s\u2082 * sb)\n[PROOFSTEP]\nrcases oreDivAddChar' r\u2081 (r\u2082 * sb + r\u2083 * rb) s\u2081 (s\u2082 * sb) with \u27e8rd, sd, hd, q\u27e9\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa) * \u2191sc = \u2191s\u2083 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : \u2191s\u2081 * \u2191sd = \u2191(s\u2082 * sb) * rd\nq : r\u2081 /\u2092 s\u2081 + (r\u2082 * \u2191sb + r\u2083 * rb) /\u2092 (s\u2082 * sb) = (r\u2081 * \u2191sd + (r\u2082 * \u2191sb + r\u2083 * rb) * rd) /\u2092 (s\u2081 * sd)\n\u22a2 ((r\u2081 * \u2191sa + r\u2082 * ra) * \u2191sc + r\u2083 * rc) /\u2092 (s\u2081 * sa * sc) = r\u2081 /\u2092 s\u2081 + (r\u2082 * \u2191sb + r\u2083 * rb) /\u2092 (s\u2082 * sb)\n[PROOFSTEP]\nrw [q]\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa) * \u2191sc = \u2191s\u2083 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : \u2191s\u2081 * \u2191sd = \u2191(s\u2082 * sb) * rd\nq : r\u2081 /\u2092 s\u2081 + (r\u2082 * \u2191sb + r\u2083 * rb) /\u2092 (s\u2082 * sb) = (r\u2081 * \u2191sd + (r\u2082 * \u2191sb + r\u2083 * rb) * rd) /\u2092 (s\u2081 * sd)\n\u22a2 ((r\u2081 * \u2191sa + r\u2082 * ra) * \u2191sc + r\u2083 * rc) /\u2092 (s\u2081 * sa * sc) = (r\u2081 * \u2191sd + (r\u2082 * \u2191sb + r\u2083 * rb) * rd) /\u2092 (s\u2081 * sd)\n[PROOFSTEP]\nclear q\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa) * \u2191sc = \u2191s\u2083 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : \u2191s\u2081 * \u2191sd = \u2191(s\u2082 * sb) * rd\n\u22a2 ((r\u2081 * \u2191sa + r\u2082 * ra) * \u2191sc + r\u2083 * rc) /\u2092 (s\u2081 * sa * sc) = (r\u2081 * \u2191sd + (r\u2082 * \u2191sb + r\u2083 * rb) * rd) /\u2092 (s\u2081 * sd)\n[PROOFSTEP]\nsimp only [right_distrib, mul_assoc, add_assoc]\n -- Porting note: `simp` required because `repeat' rw` behaves differently\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa) * \u2191sc = \u2191s\u2083 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : \u2191s\u2081 * \u2191sd = \u2191(s\u2082 * sb) * rd\n\u22a2 (r\u2081 * (\u2191sa * \u2191sc) + (r\u2082 * (ra * \u2191sc) + r\u2083 * rc)) /\u2092 (s\u2081 * (sa * sc)) =\n (r\u2081 * \u2191sd + (r\u2082 * (\u2191sb * rd) + r\u2083 * (rb * rd))) /\u2092 (s\u2081 * sd)\n[PROOFSTEP]\nsimp only [\u2190 add_oreDiv]\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa) * \u2191sc = \u2191s\u2083 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : \u2191s\u2081 * \u2191sd = \u2191(s\u2082 * sb) * rd\n\u22a2 r\u2081 * (\u2191sa * \u2191sc) /\u2092 (s\u2081 * (sa * sc)) + (r\u2082 * (ra * \u2191sc) /\u2092 (s\u2081 * (sa * sc)) + r\u2083 * rc /\u2092 (s\u2081 * (sa * sc))) =\n r\u2081 * \u2191sd /\u2092 (s\u2081 * sd) + (r\u2082 * (\u2191sb * rd) /\u2092 (s\u2081 * sd) + r\u2083 * (rb * rd) /\u2092 (s\u2081 * sd))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa) * \u2191sc = \u2191s\u2083 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : \u2191s\u2081 * \u2191sd = \u2191(s\u2082 * sb) * rd\n\u22a2 r\u2081 * (\u2191sa * \u2191sc) /\u2092 (s\u2081 * (sa * sc)) = r\u2081 * \u2191sd /\u2092 (s\u2081 * sd)\n[PROOFSTEP]\nrw [\u2190 OreLocalization.expand', \u2190 mul_assoc, \u2190 mul_assoc, \u2190 OreLocalization.expand', \u2190 OreLocalization.expand']\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa) * \u2191sc = \u2191s\u2083 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : \u2191s\u2081 * \u2191sd = \u2191(s\u2082 * sb) * rd\n\u22a2 r\u2082 * (ra * \u2191sc) /\u2092 (s\u2081 * (sa * sc)) + r\u2083 * rc /\u2092 (s\u2081 * (sa * sc)) =\n r\u2082 * (\u2191sb * rd) /\u2092 (s\u2081 * sd) + r\u2083 * (rb * rd) /\u2092 (s\u2081 * sd)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa) * \u2191sc = \u2191s\u2083 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : \u2191s\u2081 * \u2191sd = \u2191(s\u2082 * sb) * rd\n\u22a2 r\u2082 * (ra * \u2191sc) /\u2092 (s\u2081 * (sa * sc)) = r\u2082 * (\u2191sb * rd) /\u2092 (s\u2081 * sd)\n[PROOFSTEP]\nsimp_rw [\u2190 Submonoid.coe_mul] at ha hd \n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa) * \u2191sc = \u2191s\u2083 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : \u2191(s\u2081 * sd) = \u2191(s\u2082 * sb) * rd\n\u22a2 r\u2082 * (ra * \u2191sc) /\u2092 (s\u2081 * (sa * sc)) = r\u2082 * (\u2191sb * rd) /\u2092 (s\u2081 * sd)\n[PROOFSTEP]\nrw [Subtype.coe_eq_of_eq_mk hd, \u2190 mul_assoc, \u2190 mul_assoc, \u2190 mul_assoc, \u2190 OreLocalization.expand, \u2190\n OreLocalization.expand', Subtype.coe_eq_of_eq_mk ha, \u2190 OreLocalization.expand]\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa) * \u2191sc = \u2191s\u2083 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : \u2191(s\u2081 * sd) = \u2191(s\u2082 * sb) * rd\n\u22a2 r\u2082 /\u2092 s\u2082 = r\u2082 * \u2191sb /\u2092 (s\u2082 * sb)\n[PROOFSTEP]\napply OreLocalization.expand'\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa) * \u2191sc = \u2191s\u2083 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : \u2191s\u2081 * \u2191sd = \u2191(s\u2082 * sb) * rd\n\u22a2 r\u2083 * rc /\u2092 (s\u2081 * (sa * sc)) = r\u2083 * (rb * rd) /\u2092 (s\u2081 * sd)\n[PROOFSTEP]\nrcases oreCondition (sd : R) (sa * sc) with \u27e8re, _, _\u27e9\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191s\u2082 * \u2191sb = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa) * \u2191sc = \u2191s\u2083 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : \u2191s\u2081 * \u2191sd = \u2191(s\u2082 * sb) * rd\nre : R\nfst\u271d : { x // x \u2208 S }\nsnd\u271d : \u2191sd * \u2191fst\u271d = \u2191(sa * sc) * re\n\u22a2 r\u2083 * rc /\u2092 (s\u2081 * (sa * sc)) = r\u2083 * (rb * rd) /\u2092 (s\u2081 * sd)\n[PROOFSTEP]\nsimp_rw [\u2190 Submonoid.coe_mul] at hb hc hd \n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191(s\u2082 * sb) = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa * sc) = \u2191s\u2083 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : \u2191(s\u2081 * sd) = \u2191(s\u2082 * sb) * rd\nre : R\nfst\u271d : { x // x \u2208 S }\nsnd\u271d : \u2191sd * \u2191fst\u271d = \u2191(sa * sc) * re\n\u22a2 r\u2083 * rc /\u2092 (s\u2081 * (sa * sc)) = r\u2083 * (rb * rd) /\u2092 (s\u2081 * sd)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, Subtype.coe_eq_of_eq_mk hc]\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191(s\u2082 * sb) = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa * sc) = \u2191s\u2083 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : \u2191(s\u2081 * sd) = \u2191(s\u2082 * sb) * rd\nre : R\nfst\u271d : { x // x \u2208 S }\nsnd\u271d : \u2191sd * \u2191fst\u271d = \u2191(sa * sc) * re\n\u22a2 r\u2083 * rc /\u2092 { val := \u2191s\u2083 * rc, property := (_ : \u2191s\u2083 * rc \u2208 \u2191S) } = r\u2083 * (rb * rd) /\u2092 (s\u2081 * sd)\n[PROOFSTEP]\nrw [\u2190 OreLocalization.expand, Subtype.coe_eq_of_eq_mk hd, \u2190 mul_assoc, \u2190 OreLocalization.expand,\n Subtype.coe_eq_of_eq_mk hb]\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a.mk.mk\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : \u2191(s\u2082 * sb) = \u2191s\u2083 * rb\nrc : R\nsc : { x // x \u2208 S }\nhc : \u2191(s\u2081 * sa * sc) = \u2191s\u2083 * rc\nrd : R\nsd : { x // x \u2208 S }\nhd : \u2191(s\u2081 * sd) = \u2191(s\u2082 * sb) * rd\nre : R\nfst\u271d : { x // x \u2208 S }\nsnd\u271d : \u2191sd * \u2191fst\u271d = \u2191(sa * sc) * re\n\u22a2 r\u2083 /\u2092 s\u2083 = r\u2083 * rb /\u2092 { val := \u2191s\u2083 * rb, property := (_ : \u2191s\u2083 * rb \u2208 \u2191S) }\n[PROOFSTEP]\napply OreLocalization.expand\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\ns : { x // x \u2208 S }\n\u22a2 0 /\u2092 s = 0\n[PROOFSTEP]\nrw [OreLocalization.zero_def, oreDiv_eq_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\ns : { x // x \u2208 S }\n\u22a2 \u2203 u v, 0 * \u2191u = 0 * v \u2227 \u21911 * \u2191u = \u2191s * v\n[PROOFSTEP]\nexact \u27e8s, 1, by simp\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\ns : { x // x \u2208 S }\n\u22a2 0 * \u2191s = 0 * 1 \u2227 \u21911 * \u2191s = \u2191s * 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nx : OreLocalization R S\n\u22a2 0 + x = x\n[PROOFSTEP]\ninduction x using OreLocalization.ind\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u271d : R\ns\u271d : { x // x \u2208 S }\n\u22a2 0 + r\u271d /\u2092 s\u271d = r\u271d /\u2092 s\u271d\n[PROOFSTEP]\nrw [\u2190 zero_div_eq_zero, add_oreDiv]\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u271d : R\ns\u271d : { x // x \u2208 S }\n\u22a2 (0 + r\u271d) /\u2092 s\u271d = r\u271d /\u2092 s\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nx y : OreLocalization R S\n\u22a2 x + y = y + x\n[PROOFSTEP]\ninduction x using OreLocalization.ind\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\ny : OreLocalization R S\nr\u271d : R\ns\u271d : { x // x \u2208 S }\n\u22a2 r\u271d /\u2092 s\u271d + y = y + r\u271d /\u2092 s\u271d\n[PROOFSTEP]\ninduction y using OreLocalization.ind\n[GOAL]\ncase c.c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u271d\u00b9 : R\ns\u271d\u00b9 : { x // x \u2208 S }\nr\u271d : R\ns\u271d : { x // x \u2208 S }\n\u22a2 r\u271d\u00b9 /\u2092 s\u271d\u00b9 + r\u271d /\u2092 s\u271d = r\u271d /\u2092 s\u271d + r\u271d\u00b9 /\u2092 s\u271d\u00b9\n[PROOFSTEP]\nchange add' _ _ (_ /\u2092 _) = _\n[GOAL]\ncase c.c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u271d\u00b9 : R\ns\u271d\u00b9 : { x // x \u2208 S }\nr\u271d : R\ns\u271d : { x // x \u2208 S }\n\u22a2 OreLocalization.add' (r\u271d, s\u271d).fst (r\u271d, s\u271d).snd (r\u271d\u00b9 /\u2092 s\u271d\u00b9) = r\u271d /\u2092 s\u271d + r\u271d\u00b9 /\u2092 s\u271d\u00b9\n[PROOFSTEP]\napply add'_comm\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nsrc\u271d : Add (OreLocalization R S) := instAddOreLocalization\nx : OreLocalization R S\n\u22a2 x + 0 = x\n[PROOFSTEP]\nrw [OreLocalization.add_comm, OreLocalization.zero_add]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nx : OreLocalization R S\n\u22a2 0 * x = 0\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r s\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 0 * (r /\u2092 s) = 0\n[PROOFSTEP]\nrw [OreLocalization.zero_def, oreDiv_mul_char 0 r 1 s r 1 (by simp)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 r * \u21911 = \u21911 * r\n[PROOFSTEP]\nsimp\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 0 * r /\u2092 (s * 1) = 0 /\u2092 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nx : OreLocalization R S\n\u22a2 x * 0 = 0\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r s\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 r /\u2092 s * 0 = 0\n[PROOFSTEP]\nrw [OreLocalization.zero_def, oreDiv_mul_char r 0 s 1 0 1 (by simp)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 0 * \u21911 = \u2191s * 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 r * 0 /\u2092 (1 * 1) = 0 /\u2092 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nx y z : OreLocalization R S\n\u22a2 x * (y + z) = x * y + x * z\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r\u2081 s\u2081\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\ny z : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\n\u22a2 r\u2081 /\u2092 s\u2081 * (y + z) = r\u2081 /\u2092 s\u2081 * y + r\u2081 /\u2092 s\u2081 * z\n[PROOFSTEP]\ninduction' y using OreLocalization.ind with r\u2082 s\u2082\n[GOAL]\ncase c.c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nz : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\n\u22a2 r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082 + z) = r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082) + r\u2081 /\u2092 s\u2081 * z\n[PROOFSTEP]\ninduction' z using OreLocalization.ind with r\u2083 s\u2083\n[GOAL]\ncase c.c.c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\n\u22a2 r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082) + r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrcases oreDivAddChar' r\u2082 r\u2083 s\u2082 s\u2083 with \u27e8ra, sa, ha, q\u27e9\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nq : r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083 = (r\u2082 * \u2191sa + r\u2083 * ra) /\u2092 (s\u2082 * sa)\n\u22a2 r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082) + r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrw [q]\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nq : r\u2082 /\u2092 s\u2082 + r\u2083 /\u2092 s\u2083 = (r\u2082 * \u2191sa + r\u2083 * ra) /\u2092 (s\u2082 * sa)\n\u22a2 r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) /\u2092 (s\u2082 * sa)) = r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082) + r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nclear q\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\n\u22a2 r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) /\u2092 (s\u2082 * sa)) = r\u2081 /\u2092 s\u2081 * (r\u2082 /\u2092 s\u2082) + r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrw [OreLocalization.expand' r\u2082 s\u2082 sa]\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\n\u22a2 r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) /\u2092 (s\u2082 * sa)) = r\u2081 /\u2092 s\u2081 * (r\u2082 * \u2191sa /\u2092 (s\u2082 * sa)) + r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrcases oreDivMulChar' r\u2081 (r\u2082 * sa) s\u2081 (s\u2082 * sa) with \u27e8rb, sb, hb, q\u27e9\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nq : r\u2081 /\u2092 s\u2081 * (r\u2082 * \u2191sa /\u2092 (s\u2082 * sa)) = r\u2081 * rb /\u2092 (s\u2082 * sa * sb)\n\u22a2 r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) /\u2092 (s\u2082 * sa)) = r\u2081 /\u2092 s\u2081 * (r\u2082 * \u2191sa /\u2092 (s\u2082 * sa)) + r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrw [q]\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nq : r\u2081 /\u2092 s\u2081 * (r\u2082 * \u2191sa /\u2092 (s\u2082 * sa)) = r\u2081 * rb /\u2092 (s\u2082 * sa * sb)\n\u22a2 r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) /\u2092 (s\u2082 * sa)) = r\u2081 * rb /\u2092 (s\u2082 * sa * sb) + r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nclear q\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\n\u22a2 r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) /\u2092 (s\u2082 * sa)) = r\u2081 * rb /\u2092 (s\u2082 * sa * sb) + r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nhave hs\u2083rasb : \u2191s\u2083 * (ra * sb) \u2208 S := by\n rw [\u2190 mul_assoc, \u2190 ha]\n norm_cast\n apply SetLike.coe_mem\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\n\u22a2 \u2191s\u2083 * (ra * \u2191sb) \u2208 S\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 ha]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\n\u22a2 \u2191s\u2082 * \u2191sa * \u2191sb \u2208 S\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\n\u22a2 \u2191(s\u2082 * sa * sb) \u2208 S\n[PROOFSTEP]\napply SetLike.coe_mem\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\n\u22a2 r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) /\u2092 (s\u2082 * sa)) = r\u2081 * rb /\u2092 (s\u2082 * sa * sb) + r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrw [OreLocalization.expand _ _ _ hs\u2083rasb]\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\n\u22a2 r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) /\u2092 (s\u2082 * sa)) =\n r\u2081 * rb /\u2092 (s\u2082 * sa * sb) + r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 { val := \u2191s\u2083 * (ra * \u2191sb), property := hs\u2083rasb })\n[PROOFSTEP]\nhave ha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * sb) := by simp [ha, \u2190 mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\n\u22a2 \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\n[PROOFSTEP]\nsimp [ha, \u2190 mul_assoc]\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\n\u22a2 r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) /\u2092 (s\u2082 * sa)) =\n r\u2081 * rb /\u2092 (s\u2082 * sa * sb) + r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 { val := \u2191s\u2083 * (ra * \u2191sb), property := hs\u2083rasb })\n[PROOFSTEP]\nrw [\u2190 Subtype.coe_eq_of_eq_mk ha']\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\n\u22a2 r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) /\u2092 (s\u2082 * sa)) =\n r\u2081 * rb /\u2092 (s\u2082 * sa * sb) + r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb))\n[PROOFSTEP]\nrcases oreDivMulChar' r\u2081 (r\u2083 * (ra * sb)) s\u2081 (s\u2082 * sa * sb) with \u27e8rc, sc, hc, hc'\u27e9\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n\u22a2 r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) /\u2092 (s\u2082 * sa)) =\n r\u2081 * rb /\u2092 (s\u2082 * sa * sb) + r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb))\n[PROOFSTEP]\nrw [hc']\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n\u22a2 r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) /\u2092 (s\u2082 * sa)) = r\u2081 * rb /\u2092 (s\u2082 * sa * sb) + r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n[PROOFSTEP]\nrw [oreDiv_add_char (s\u2082 * sa * sb) (s\u2082 * sa * sb * sc) 1 sc (by simp)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n\u22a2 \u2191(s\u2082 * sa * sb) * \u2191sc = \u2191(s\u2082 * sa * sb * sc) * 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n\u22a2 r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) /\u2092 (s\u2082 * sa)) = (r\u2081 * rb * \u2191sc + r\u2081 * rc * 1) /\u2092 (s\u2082 * sa * sb * sc)\n[PROOFSTEP]\nrw [OreLocalization.expand' (r\u2082 * \u2191sa + r\u2083 * ra) (s\u2082 * sa) (sb * sc)]\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n\u22a2 r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) * \u2191(sb * sc) /\u2092 (s\u2082 * sa * (sb * sc))) =\n (r\u2081 * rb * \u2191sc + r\u2081 * rc * 1) /\u2092 (s\u2082 * sa * sb * sc)\n[PROOFSTEP]\nconv_lhs =>\n congr\n \u00b7skip\n congr\n rw [add_mul, S.coe_mul, \u2190 mul_assoc, hb, \u2190 mul_assoc, mul_assoc r\u2083, hc, mul_assoc, \u2190 mul_add]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n| r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) * \u2191(sb * sc) /\u2092 (s\u2082 * sa * (sb * sc)))\n[PROOFSTEP]\n congr\n \u00b7skip\n congr\n rw [add_mul, S.coe_mul, \u2190 mul_assoc, hb, \u2190 mul_assoc, mul_assoc r\u2083, hc, mul_assoc, \u2190 mul_add]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n| r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) * \u2191(sb * sc) /\u2092 (s\u2082 * sa * (sb * sc)))\n[PROOFSTEP]\n congr\n \u00b7skip\n congr\n rw [add_mul, S.coe_mul, \u2190 mul_assoc, hb, \u2190 mul_assoc, mul_assoc r\u2083, hc, mul_assoc, \u2190 mul_add]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n| r\u2081 /\u2092 s\u2081 * ((r\u2082 * \u2191sa + r\u2083 * ra) * \u2191(sb * sc) /\u2092 (s\u2082 * sa * (sb * sc)))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n| r\u2081 /\u2092 s\u2081\ncase a\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n| (r\u2082 * \u2191sa + r\u2083 * ra) * \u2191(sb * sc) /\u2092 (s\u2082 * sa * (sb * sc))\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n| r\u2081 /\u2092 s\u2081\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n| r\u2081 /\u2092 s\u2081\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n| r\u2081 /\u2092 s\u2081\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n| (r\u2082 * \u2191sa + r\u2083 * ra) * \u2191(sb * sc) /\u2092 (s\u2082 * sa * (sb * sc))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.r\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n| (r\u2082 * \u2191sa + r\u2083 * ra) * \u2191(sb * sc)\ncase a.s\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n| s\u2082 * sa * (sb * sc)\n[PROOFSTEP]\nrw [add_mul, S.coe_mul, \u2190 mul_assoc, hb, \u2190 mul_assoc, mul_assoc r\u2083, hc, mul_assoc, \u2190 mul_add]\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n\u22a2 r\u2081 /\u2092 s\u2081 * (\u2191s\u2081 * (rb * \u2191sc + rc) /\u2092 (s\u2082 * sa * (sb * sc))) = (r\u2081 * rb * \u2191sc + r\u2081 * rc * 1) /\u2092 (s\u2082 * sa * sb * sc)\n[PROOFSTEP]\nrw [OreLocalization.mul_cancel']\n[GOAL]\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2082 * \u2191sa = \u2191s\u2083 * ra\nrb : R\nsb : { x // x \u2208 S }\nhb : r\u2082 * \u2191sa * \u2191sb = \u2191s\u2081 * rb\nhs\u2083rasb : \u2191s\u2083 * (ra * \u2191sb) \u2208 S\nha' : \u2191(s\u2082 * sa * sb) = \u2191s\u2083 * (ra * \u2191sb)\nrc : R\nsc : { x // x \u2208 S }\nhc : r\u2083 * (ra * \u2191sb) * \u2191sc = \u2191s\u2081 * rc\nhc' : r\u2081 /\u2092 s\u2081 * (r\u2083 * (ra * \u2191sb) /\u2092 (s\u2082 * sa * sb)) = r\u2081 * rc /\u2092 (s\u2082 * sa * sb * sc)\n\u22a2 r\u2081 * (rb * \u2191sc + rc) /\u2092 (s\u2082 * sa * (sb * sc)) = (r\u2081 * rb * \u2191sc + r\u2081 * rc * 1) /\u2092 (s\u2082 * sa * sb * sc)\n[PROOFSTEP]\nsimp only [mul_one, Submonoid.coe_mul, mul_add, \u2190 mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nx y z : OreLocalization R S\n\u22a2 (x + y) * z = x * z + y * z\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r\u2081 s\u2081\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\ny z : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\n\u22a2 (r\u2081 /\u2092 s\u2081 + y) * z = r\u2081 /\u2092 s\u2081 * z + y * z\n[PROOFSTEP]\ninduction' y using OreLocalization.ind with r\u2082 s\u2082\n[GOAL]\ncase c.c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nz : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\n\u22a2 (r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082) * z = r\u2081 /\u2092 s\u2081 * z + r\u2082 /\u2092 s\u2082 * z\n[PROOFSTEP]\ninduction' z using OreLocalization.ind with r\u2083 s\u2083\n[GOAL]\ncase c.c.c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\n\u22a2 (r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrcases oreDivAddChar' r\u2081 r\u2082 s\u2081 s\u2082 with \u27e8ra, sa, ha, ha'\u27e9\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nha' : r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 = (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa)\n\u22a2 (r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrw [ha']\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nha' : r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 = (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa)\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nclear ha'\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nnorm_cast at ha \n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrw [OreLocalization.expand' r\u2081 s\u2081 sa]\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 * \u2191sa /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrw [OreLocalization.expand r\u2082 s\u2082 ra (by rw [\u2190 ha]; apply SetLike.coe_mem)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 \u2191s\u2082 * ra \u2208 S\n[PROOFSTEP]\nrw [\u2190 ha]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 \u2191(s\u2081 * sa) \u2208 S\n[PROOFSTEP]\napply SetLike.coe_mem\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) =\n r\u2081 * \u2191sa /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) + r\u2082 * ra /\u2092 { val := \u2191s\u2082 * ra, property := (_ : \u2191s\u2082 * ra \u2208 S) } * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrw [\u2190 Subtype.coe_eq_of_eq_mk ha]\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) =\n r\u2081 * \u2191sa /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) + r\u2082 * ra /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrepeat rw [oreDiv_mul_oreDiv]\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) =\n r\u2081 * \u2191sa /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) + r\u2082 * ra /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrw [oreDiv_mul_oreDiv]\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) * oreNum r\u2083 (s\u2081 * sa) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa)) =\n r\u2081 * \u2191sa /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) + r\u2082 * ra /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrw [oreDiv_mul_oreDiv]\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) * oreNum r\u2083 (s\u2081 * sa) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa)) =\n r\u2081 * \u2191sa * oreNum r\u2083 (s\u2081 * sa) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa)) + r\u2082 * ra /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083)\n[PROOFSTEP]\nrw [oreDiv_mul_oreDiv]\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) * oreNum r\u2083 (s\u2081 * sa) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa)) =\n r\u2081 * \u2191sa * oreNum r\u2083 (s\u2081 * sa) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa)) +\n r\u2082 * ra * oreNum r\u2083 (s\u2081 * sa) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa))\n[PROOFSTEP]\nrw [oreDiv_mul_oreDiv]\n[GOAL]\ncase c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) * oreNum r\u2083 (s\u2081 * sa) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa)) =\n r\u2081 * \u2191sa * oreNum r\u2083 (s\u2081 * sa) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa)) +\n r\u2082 * ra * oreNum r\u2083 (s\u2081 * sa) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa))\n[PROOFSTEP]\nsimp only [add_mul, add_oreDiv]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := \u2191src\u271d,\n map_mul' :=\n (_ :\n \u2200 (x y : OreLocalization R S),\n OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n 0 =\n 0\n[PROOFSTEP]\nchange (universalMulHom f.toMonoidHom fS hf : R[S\u207b\u00b9] \u2192 T) 0 = 0\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\n\u22a2 \u2191(universalMulHom (\u2191f) fS hf) 0 = 0\n[PROOFSTEP]\nrw [OreLocalization.zero_def, universalMulHom_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\n\u22a2 \u2191\u2191f 0 * \u2191(\u2191fS 1)\u207b\u00b9 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nx y : OreLocalization R S\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := \u2191src\u271d,\n map_mul' :=\n (_ :\n \u2200 (x y : OreLocalization R S),\n OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n (x + y) =\n OneHom.toFun\n (\u2191{ toOneHom := \u2191src\u271d,\n map_mul' :=\n (_ :\n \u2200 (x y : OreLocalization R S),\n OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n x +\n OneHom.toFun\n (\u2191{ toOneHom := \u2191src\u271d,\n map_mul' :=\n (_ :\n \u2200 (x y : OreLocalization R S),\n OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n y\n[PROOFSTEP]\nchange\n (universalMulHom f.toMonoidHom fS hf : R[S\u207b\u00b9] \u2192 T) (x + y) =\n (universalMulHom f.toMonoidHom fS hf : R[S\u207b\u00b9] \u2192 T) x + (universalMulHom f.toMonoidHom fS hf : R[S\u207b\u00b9] \u2192 T) y\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nx y : OreLocalization R S\n\u22a2 \u2191(universalMulHom (\u2191f) fS hf) (x + y) = \u2191(universalMulHom (\u2191f) fS hf) x + \u2191(universalMulHom (\u2191f) fS hf) y\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r\u2081 s\u2081\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\ny : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\n\u22a2 \u2191(universalMulHom (\u2191f) fS hf) (r\u2081 /\u2092 s\u2081 + y) =\n \u2191(universalMulHom (\u2191f) fS hf) (r\u2081 /\u2092 s\u2081) + \u2191(universalMulHom (\u2191f) fS hf) y\n[PROOFSTEP]\ninduction' y using OreLocalization.ind with r\u2082 s\u2082\n[GOAL]\ncase c.c\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\n\u22a2 \u2191(universalMulHom (\u2191f) fS hf) (r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082) =\n \u2191(universalMulHom (\u2191f) fS hf) (r\u2081 /\u2092 s\u2081) + \u2191(universalMulHom (\u2191f) fS hf) (r\u2082 /\u2092 s\u2082)\n[PROOFSTEP]\nrcases oreDivAddChar' r\u2081 r\u2082 s\u2081 s\u2082 with \u27e8r\u2083, s\u2083, h\u2083, h\u2083'\u27e9\n[GOAL]\ncase c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191s\u2081 * \u2191s\u2083 = \u2191s\u2082 * r\u2083\nh\u2083' : r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 = (r\u2081 * \u2191s\u2083 + r\u2082 * r\u2083) /\u2092 (s\u2081 * s\u2083)\n\u22a2 \u2191(universalMulHom (\u2191f) fS hf) (r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082) =\n \u2191(universalMulHom (\u2191f) fS hf) (r\u2081 /\u2092 s\u2081) + \u2191(universalMulHom (\u2191f) fS hf) (r\u2082 /\u2092 s\u2082)\n[PROOFSTEP]\nrw [h\u2083']\n[GOAL]\ncase c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191s\u2081 * \u2191s\u2083 = \u2191s\u2082 * r\u2083\nh\u2083' : r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 = (r\u2081 * \u2191s\u2083 + r\u2082 * r\u2083) /\u2092 (s\u2081 * s\u2083)\n\u22a2 \u2191(universalMulHom (\u2191f) fS hf) ((r\u2081 * \u2191s\u2083 + r\u2082 * r\u2083) /\u2092 (s\u2081 * s\u2083)) =\n \u2191(universalMulHom (\u2191f) fS hf) (r\u2081 /\u2092 s\u2081) + \u2191(universalMulHom (\u2191f) fS hf) (r\u2082 /\u2092 s\u2082)\n[PROOFSTEP]\nclear h\u2083'\n[GOAL]\ncase c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191s\u2081 * \u2191s\u2083 = \u2191s\u2082 * r\u2083\n\u22a2 \u2191(universalMulHom (\u2191f) fS hf) ((r\u2081 * \u2191s\u2083 + r\u2082 * r\u2083) /\u2092 (s\u2081 * s\u2083)) =\n \u2191(universalMulHom (\u2191f) fS hf) (r\u2081 /\u2092 s\u2081) + \u2191(universalMulHom (\u2191f) fS hf) (r\u2082 /\u2092 s\u2082)\n[PROOFSTEP]\nsimp only [universalMulHom_apply, RingHom.toMonoidHom_eq_coe, MonoidHom.coe_coe]\n[GOAL]\ncase c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191s\u2081 * \u2191s\u2083 = \u2191s\u2082 * r\u2083\n\u22a2 \u2191f (r\u2081 * \u2191s\u2083 + r\u2082 * r\u2083) * \u2191(\u2191fS (s\u2081 * s\u2083))\u207b\u00b9 = \u2191f r\u2081 * \u2191(\u2191fS s\u2081)\u207b\u00b9 + \u2191f r\u2082 * \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\nsimp only [mul_inv_rev, MonoidHom.map_mul, RingHom.map_add, RingHom.map_mul, Units.val_mul]\n[GOAL]\ncase c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191s\u2081 * \u2191s\u2083 = \u2191s\u2082 * r\u2083\n\u22a2 (\u2191f r\u2081 * \u2191f \u2191s\u2083 + \u2191f r\u2082 * \u2191f r\u2083) * (\u2191(\u2191fS s\u2083)\u207b\u00b9 * \u2191(\u2191fS s\u2081)\u207b\u00b9) = \u2191f r\u2081 * \u2191(\u2191fS s\u2081)\u207b\u00b9 + \u2191f r\u2082 * \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\nrw [add_mul, \u2190 mul_assoc, mul_assoc (f r\u2081), hf, \u2190 Units.val_mul]\n[GOAL]\ncase c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191s\u2081 * \u2191s\u2083 = \u2191s\u2082 * r\u2083\n\u22a2 \u2191f r\u2081 * \u2191(\u2191fS s\u2083 * (\u2191fS s\u2083)\u207b\u00b9) * \u2191(\u2191fS s\u2081)\u207b\u00b9 + \u2191f r\u2082 * \u2191f r\u2083 * (\u2191(\u2191fS s\u2083)\u207b\u00b9 * \u2191(\u2191fS s\u2081)\u207b\u00b9) =\n \u2191f r\u2081 * \u2191(\u2191fS s\u2081)\u207b\u00b9 + \u2191f r\u2082 * \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\nsimp only [mul_one, mul_right_inv, Units.val_one]\n[GOAL]\ncase c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191s\u2081 * \u2191s\u2083 = \u2191s\u2082 * r\u2083\n\u22a2 \u2191f r\u2081 * \u2191(\u2191fS s\u2081)\u207b\u00b9 + \u2191f r\u2082 * \u2191f r\u2083 * (\u2191(\u2191fS s\u2083)\u207b\u00b9 * \u2191(\u2191fS s\u2081)\u207b\u00b9) = \u2191f r\u2081 * \u2191(\u2191fS s\u2081)\u207b\u00b9 + \u2191f r\u2082 * \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase c.c.mk.mk.intro.e_a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191s\u2081 * \u2191s\u2083 = \u2191s\u2082 * r\u2083\n\u22a2 \u2191f r\u2082 * \u2191f r\u2083 * (\u2191(\u2191fS s\u2083)\u207b\u00b9 * \u2191(\u2191fS s\u2081)\u207b\u00b9) = \u2191f r\u2082 * \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\ncase c.c.mk.mk.intro.e_a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191s\u2081 * \u2191s\u2083 = \u2191s\u2082 * r\u2083\n\u22a2 \u2191f r\u2082 * (\u2191f r\u2083 * (\u2191(\u2191fS s\u2083)\u207b\u00b9 * \u2191(\u2191fS s\u2081)\u207b\u00b9)) = \u2191f r\u2082 * \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase c.c.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191s\u2081 * \u2191s\u2083 = \u2191s\u2082 * r\u2083\n\u22a2 \u2191f r\u2083 * (\u2191(\u2191fS s\u2083)\u207b\u00b9 * \u2191(\u2191fS s\u2081)\u207b\u00b9) = \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\nnorm_cast at h\u2083 \n[GOAL]\ncase c.c.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\n\u22a2 \u2191f r\u2083 * (\u2191(\u2191fS s\u2083)\u207b\u00b9 * \u2191(\u2191fS s\u2081)\u207b\u00b9) = \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\nhave h\u2083' := Subtype.coe_eq_of_eq_mk h\u2083\n[GOAL]\ncase c.c.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\n\u22a2 \u2191f r\u2083 * (\u2191(\u2191fS s\u2083)\u207b\u00b9 * \u2191(\u2191fS s\u2081)\u207b\u00b9) = \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 Units.val_mul, \u2190 mul_inv_rev, \u2190 fS.map_mul, h\u2083']\n[GOAL]\ncase c.c.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\n\u22a2 \u2191f r\u2083 * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9 = \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\nhave hs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S := by\n rw [\u2190 h\u2083]\n exact SetLike.coe_mem (s\u2081 * s\u2083)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\n\u22a2 \u2191s\u2082 * r\u2083 \u2208 S\n[PROOFSTEP]\nrw [\u2190 h\u2083]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\n\u22a2 \u2191(s\u2081 * s\u2083) \u2208 S\n[PROOFSTEP]\nexact SetLike.coe_mem (s\u2081 * s\u2083)\n[GOAL]\ncase c.c.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n\u22a2 \u2191f r\u2083 * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9 = \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\napply (Units.inv_mul_cancel_left (fS s\u2082) _).symm.trans\n[GOAL]\ncase c.c.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n\u22a2 \u2191(\u2191fS s\u2082)\u207b\u00b9 * (\u2191(\u2191fS s\u2082) * (\u2191f r\u2083 * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9)) = \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\nconv_lhs =>\n congr\n \u00b7skip\n rw [\u2190 Units.mul_inv_cancel_left (fS \u27e8s\u2082 * r\u2083, hs\u2082r\u2083\u27e9) (fS s\u2082), mul_assoc, mul_assoc]\n congr\n \u00b7skip\n rw [\u2190 hf, \u2190 mul_assoc (f s\u2082), \u2190 f.map_mul]\n conv =>\n congr\n \u00b7skip\n congr\n rw [\u2190 h\u2083]\n rw [hf, \u2190 mul_assoc, \u2190 h\u2083', Units.inv_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS s\u2082)\u207b\u00b9 * (\u2191(\u2191fS s\u2082) * (\u2191f r\u2083 * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9))\n[PROOFSTEP]\n congr\n \u00b7skip\n rw [\u2190 Units.mul_inv_cancel_left (fS \u27e8s\u2082 * r\u2083, hs\u2082r\u2083\u27e9) (fS s\u2082), mul_assoc, mul_assoc]\n congr\n \u00b7skip\n rw [\u2190 hf, \u2190 mul_assoc (f s\u2082), \u2190 f.map_mul]\n conv =>\n congr\n \u00b7skip\n congr\n rw [\u2190 h\u2083]\n rw [hf, \u2190 mul_assoc, \u2190 h\u2083', Units.inv_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS s\u2082)\u207b\u00b9 * (\u2191(\u2191fS s\u2082) * (\u2191f r\u2083 * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9))\n[PROOFSTEP]\n congr\n \u00b7skip\n rw [\u2190 Units.mul_inv_cancel_left (fS \u27e8s\u2082 * r\u2083, hs\u2082r\u2083\u27e9) (fS s\u2082), mul_assoc, mul_assoc]\n congr\n \u00b7skip\n rw [\u2190 hf, \u2190 mul_assoc (f s\u2082), \u2190 f.map_mul]\n conv =>\n congr\n \u00b7skip\n congr\n rw [\u2190 h\u2083]\n rw [hf, \u2190 mul_assoc, \u2190 h\u2083', Units.inv_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS s\u2082)\u207b\u00b9 * (\u2191(\u2191fS s\u2082) * (\u2191f r\u2083 * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS s\u2082)\u207b\u00b9\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS s\u2082) * (\u2191f r\u2083 * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9)\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS s\u2082) * (\u2191f r\u2083 * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9)\n[PROOFSTEP]\nrw [\u2190 Units.mul_inv_cancel_left (fS \u27e8s\u2082 * r\u2083, hs\u2082r\u2083\u27e9) (fS s\u2082), mul_assoc, mul_assoc]\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 }) *\n (\u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 })\u207b\u00b9 *\n (\u2191(\u2191fS s\u2082) * (\u2191f r\u2083 * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9)))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 })\ncase a.a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 })\u207b\u00b9 *\n (\u2191(\u2191fS s\u2082) * (\u2191f r\u2083 * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9))\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a.a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 })\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 })\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 })\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 })\u207b\u00b9 *\n (\u2191(\u2191fS s\u2082) * (\u2191f r\u2083 * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9))\n[PROOFSTEP]\nrw [\u2190 hf, \u2190 mul_assoc (f s\u2082), \u2190 f.map_mul]\n[GOAL]\ncase a.a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 })\u207b\u00b9 *\n (\u2191f (\u2191s\u2082 * r\u2083) * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9)\n[PROOFSTEP]\nconv =>\n congr\n \u00b7skip\n congr\n rw [\u2190 h\u2083]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 })\u207b\u00b9 *\n (\u2191f (\u2191s\u2082 * r\u2083) * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9)\n[PROOFSTEP]\n congr\n \u00b7skip\n congr\n rw [\u2190 h\u2083]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 })\u207b\u00b9 *\n (\u2191f (\u2191s\u2082 * r\u2083) * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 })\u207b\u00b9\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191f (\u2191s\u2082 * r\u2083) * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 })\u207b\u00b9\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 })\u207b\u00b9\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 })\u207b\u00b9\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191f (\u2191s\u2082 * r\u2083) * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191f (\u2191s\u2082 * r\u2083)\ncase a.a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 h\u2083]\n[GOAL]\ncase a.a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n| \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 })\u207b\u00b9 *\n (\u2191f \u2191(s\u2081 * s\u2083) * \u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) })\u207b\u00b9)\n[PROOFSTEP]\nrw [hf, \u2190 mul_assoc, \u2190 h\u2083', Units.inv_mul]\n[GOAL]\ncase c.c.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nsrc\u271d : OreLocalization R S \u2192* T := universalMulHom (\u2191f) fS hf\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nh\u2083 : \u2191(s\u2081 * s\u2083) = \u2191s\u2082 * r\u2083\nh\u2083' : s\u2081 * s\u2083 = { val := \u2191s\u2082 * r\u2083, property := (_ : \u2191s\u2082 * r\u2083 \u2208 \u2191S) }\nhs\u2082r\u2083 : \u2191s\u2082 * r\u2083 \u2208 S\n\u22a2 \u2191(\u2191fS s\u2082)\u207b\u00b9 * (\u2191(\u2191fS { val := \u2191s\u2082 * r\u2083, property := hs\u2082r\u2083 }) * (1 * \u2191(\u2191fS (s\u2081 * s\u2083))\u207b\u00b9)) = \u2191(\u2191fS s\u2082)\u207b\u00b9\n[PROOFSTEP]\nrw [one_mul, \u2190 h\u2083', Units.mul_inv, mul_one]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\nT : Type u_2\ninst\u271d : Semiring T\nf : R \u2192+* T\nfS : { x // x \u2208 S } \u2192* T\u02e3\nhf : \u2200 (s : { x // x \u2208 S }), \u2191f \u2191s = \u2191(\u2191fS s)\nr : R\n\u22a2 \u2191(universalHom f fS hf) (\u2191numeratorHom r) = \u2191f r\n[PROOFSTEP]\nsimp [numeratorHom_apply, universalHom_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nS : Submonoid R\ninst\u271d : OreSet S\nr t : R\ns : { x // x \u2208 S }\nht : \u2191s * t \u2208 S\n\u22a2 (fun r s => -r /\u2092 s) r s = (fun r s => -r /\u2092 s) (r * t) { val := \u2191s * t, property := ht }\n[PROOFSTEP]\nsimp only []\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nS : Submonoid R\ninst\u271d : OreSet S\nr t : R\ns : { x // x \u2208 S }\nht : \u2191s * t \u2208 S\n\u22a2 -r /\u2092 s = -(r * t) /\u2092 { val := \u2191s * t, property := ht }\n[PROOFSTEP]\nrw [neg_mul_eq_neg_mul, \u2190 OreLocalization.expand]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nS : Submonoid R\ninst\u271d : OreSet S\nx : OreLocalization R S\n\u22a2 -x + x = 0\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r s\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 -(r /\u2092 s) + r /\u2092 s = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nS : Submonoid R\ninst\u271d : OreSet S\nhS : S \u2264 R\u2070\nr\u2081 r\u2082 : R\nh : \u2191numeratorHom r\u2081 = \u2191numeratorHom r\u2082\n\u22a2 r\u2081 = r\u2082\n[PROOFSTEP]\nrw [numeratorHom_apply, numeratorHom_apply, oreDiv_eq_iff] at h \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nS : Submonoid R\ninst\u271d : OreSet S\nhS : S \u2264 R\u2070\nr\u2081 r\u2082 : R\nh : \u2203 u v, r\u2082 * \u2191u = r\u2081 * v \u2227 \u21911 * \u2191u = \u21911 * v\n\u22a2 r\u2081 = r\u2082\n[PROOFSTEP]\nrcases h with \u27e8u, v, h\u2081, h\u2082\u27e9\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nS : Submonoid R\ninst\u271d : OreSet S\nhS : S \u2264 R\u2070\nr\u2081 r\u2082 : R\nu : { x // x \u2208 S }\nv : R\nh\u2081 : r\u2082 * \u2191u = r\u2081 * v\nh\u2082 : \u21911 * \u2191u = \u21911 * v\n\u22a2 r\u2081 = r\u2082\n[PROOFSTEP]\nsimp only [S.coe_one, one_mul] at h\u2082 \n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nS : Submonoid R\ninst\u271d : OreSet S\nhS : S \u2264 R\u2070\nr\u2081 r\u2082 : R\nu : { x // x \u2208 S }\nv : R\nh\u2081 : r\u2082 * \u2191u = r\u2081 * v\nh\u2082 : \u2191u = v\n\u22a2 r\u2081 = r\u2082\n[PROOFSTEP]\nrwa [\u2190 h\u2082, mul_cancel_right_mem_nonZeroDivisors (hS (SetLike.coe_mem u)), eq_comm] at h\u2081 \n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\ninst\u271d : Nontrivial R\nhS : S \u2264 R\u2070\nh : 0 = 1\n\u22a2 False\n[PROOFSTEP]\nrw [OreLocalization.one_def, OreLocalization.zero_def] at h \n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Ring R\nS : Submonoid R\ninst\u271d\u00b9 : OreSet S\ninst\u271d : Nontrivial R\nhS : S \u2264 R\u2070\nh : 0 /\u2092 1 = 1 /\u2092 1\n\u22a2 False\n[PROOFSTEP]\napply nonZeroDivisors.coe_ne_zero 1 (numeratorHom_inj hS h).symm\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\n\u22a2 \u2200 (r t : R) (s : { x // x \u2208 R\u2070 }) (ht : \u2191s * t \u2208 R\u2070),\n (fun r s => if hr : r = 0 then 0 else \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) }) r s =\n (fun r s => if hr : r = 0 then 0 else \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) }) (r * t)\n { val := \u2191s * t, property := ht }\n[PROOFSTEP]\nintro r t s hst\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nr t : R\ns : { x // x \u2208 R\u2070 }\nhst : \u2191s * t \u2208 R\u2070\n\u22a2 (fun r s => if hr : r = 0 then 0 else \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) }) r s =\n (fun r s => if hr : r = 0 then 0 else \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) }) (r * t)\n { val := \u2191s * t, property := hst }\n[PROOFSTEP]\nby_cases hr : r = 0\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nr t : R\ns : { x // x \u2208 R\u2070 }\nhst : \u2191s * t \u2208 R\u2070\nhr : r = 0\n\u22a2 (fun r s => if hr : r = 0 then 0 else \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) }) r s =\n (fun r s => if hr : r = 0 then 0 else \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) }) (r * t)\n { val := \u2191s * t, property := hst }\n[PROOFSTEP]\nsimp [hr]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nr t : R\ns : { x // x \u2208 R\u2070 }\nhst : \u2191s * t \u2208 R\u2070\nhr : \u00acr = 0\n\u22a2 (fun r s => if hr : r = 0 then 0 else \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) }) r s =\n (fun r s => if hr : r = 0 then 0 else \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) }) (r * t)\n { val := \u2191s * t, property := hst }\n[PROOFSTEP]\nby_cases ht : t = 0\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nr t : R\ns : { x // x \u2208 R\u2070 }\nhst : \u2191s * t \u2208 R\u2070\nhr : \u00acr = 0\nht : t = 0\n\u22a2 (fun r s => if hr : r = 0 then 0 else \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) }) r s =\n (fun r s => if hr : r = 0 then 0 else \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) }) (r * t)\n { val := \u2191s * t, property := hst }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nr t : R\ns : { x // x \u2208 R\u2070 }\nhst : \u2191s * t \u2208 R\u2070\nhr : \u00acr = 0\nht : t = 0\n\u22a2 False\n[PROOFSTEP]\napply nonZeroDivisors.coe_ne_zero \u27e8_, hst\u27e9\n[GOAL]\ncase pos.h\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nr t : R\ns : { x // x \u2208 R\u2070 }\nhst : \u2191s * t \u2208 R\u2070\nhr : \u00acr = 0\nht : t = 0\n\u22a2 \u2191{ val := \u2191s * t, property := hst } = 0\n[PROOFSTEP]\nsimp [ht, mul_zero]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nr t : R\ns : { x // x \u2208 R\u2070 }\nhst : \u2191s * t \u2208 R\u2070\nhr : \u00acr = 0\nht : \u00act = 0\n\u22a2 (fun r s => if hr : r = 0 then 0 else \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) }) r s =\n (fun r s => if hr : r = 0 then 0 else \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) }) (r * t)\n { val := \u2191s * t, property := hst }\n[PROOFSTEP]\nsimp only [hr, ht, dif_neg, not_false_iff, or_self_iff, mul_eq_zero]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nr t : R\ns : { x // x \u2208 R\u2070 }\nhst : \u2191s * t \u2208 R\u2070\nhr : \u00acr = 0\nht : \u00act = 0\n\u22a2 \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) } =\n \u2191s * t /\u2092 { val := r * t, property := (_ : \u2200 (x : R), x * (r * t) = 0 \u2192 x = 0) }\n[PROOFSTEP]\napply OreLocalization.expand\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nx : OreLocalization R R\u2070\nh : x \u2260 0\n\u22a2 x * x\u207b\u00b9 = 1\n[PROOFSTEP]\ninduction' x using OreLocalization.ind with r s\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nx : OreLocalization R R\u2070\nh\u271d : x \u2260 0\nr : R\ns : { x // x \u2208 R\u2070 }\nh : r /\u2092 s \u2260 0\n\u22a2 r /\u2092 s * (r /\u2092 s)\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [OreLocalization.inv_def, OreLocalization.one_def]\n[GOAL]\ncase c\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nx : OreLocalization R R\u2070\nh\u271d : x \u2260 0\nr : R\ns : { x // x \u2208 R\u2070 }\nh : r /\u2092 s \u2260 0\n\u22a2 (r /\u2092 s * if hr : r = 0 then 0 else \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) }) = 1 /\u2092 1\n[PROOFSTEP]\nby_cases hr : r = 0\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nx : OreLocalization R R\u2070\nh\u271d : x \u2260 0\nr : R\ns : { x // x \u2208 R\u2070 }\nh : r /\u2092 s \u2260 0\nhr : r = 0\n\u22a2 (r /\u2092 s * if hr : r = 0 then 0 else \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) }) = 1 /\u2092 1\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nx : OreLocalization R R\u2070\nh\u271d : x \u2260 0\nr : R\ns : { x // x \u2208 R\u2070 }\nh : r /\u2092 s \u2260 0\nhr : r = 0\n\u22a2 False\n[PROOFSTEP]\napply h\n[GOAL]\ncase pos.h\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nx : OreLocalization R R\u2070\nh\u271d : x \u2260 0\nr : R\ns : { x // x \u2208 R\u2070 }\nh : r /\u2092 s \u2260 0\nhr : r = 0\n\u22a2 r /\u2092 s = 0\n[PROOFSTEP]\nsimp [hr]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nx : OreLocalization R R\u2070\nh\u271d : x \u2260 0\nr : R\ns : { x // x \u2208 R\u2070 }\nh : r /\u2092 s \u2260 0\nhr : \u00acr = 0\n\u22a2 (r /\u2092 s * if hr : r = 0 then 0 else \u2191s /\u2092 { val := r, property := (_ : \u2200 (x : R), x * r = 0 \u2192 x = 0) }) = 1 /\u2092 1\n[PROOFSTEP]\nsimp [hr]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\nx : OreLocalization R R\u2070\nh\u271d : x \u2260 0\nr : R\ns : { x // x \u2208 R\u2070 }\nh : r /\u2092 s \u2260 0\nhr : \u00acr = 0\n\u22a2 1 = 1 /\u2092 1\n[PROOFSTEP]\napply OreLocalization.div_eq_one'\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\nrw [OreLocalization.zero_def, OreLocalization.inv_def]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : OreSet R\u2070\ninst\u271d : NoZeroDivisors R\n\u22a2 (if hr : 0 = 0 then 0 else \u21911 /\u2092 { val := 0, property := (_ : \u2200 (x : R), x * 0 = 0 \u2192 x = 0) }) = 0 /\u2092 1\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.OreLocalization.Basic", "llama_tokens": 122969, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702880639791, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.42412244219386286}} {"text": "[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM : Type u_6\nN : Type u_7\nP : Type u_8\nQ : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u00b9 : One M\ninst\u271d : One N\nf : OneHom M N\ng : N \u2192 M\nh\u2081 : Function.LeftInverse g \u2191f\n\u22a2 g 1 = 1\n[PROOFSTEP]\nrw [\u2190 f.map_one, h\u2081]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM : Type u_6\nN : Type u_7\nP : Type u_8\nQ : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u00b9 : Mul M\ninst\u271d : Mul N\nf : M \u2192\u2099* N\ng : N \u2192 M\nh\u2081 : Function.LeftInverse g \u2191f\nh\u2082 : Function.RightInverse g \u2191f\nx y : N\n\u22a2 g (x * y) = g (\u2191f (g x) * \u2191f (g y))\n[PROOFSTEP]\nrw [h\u2082 x, h\u2082 y]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM : Type u_6\nN : Type u_7\nP : Type u_8\nQ : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u00b9 : Mul M\ninst\u271d : Mul N\nf : M \u2192\u2099* N\ng : N \u2192 M\nh\u2081 : Function.LeftInverse g \u2191f\nh\u2082 : Function.RightInverse g \u2191f\nx y : N\n\u22a2 g (\u2191f (g x) * \u2191f (g y)) = g (\u2191f (g x * g y))\n[PROOFSTEP]\nrw [f.map_mul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM : Type u_6\nN : Type u_7\nP : Type u_8\nQ : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulEquivClass F M N\nsrc\u271d : MulHomClass F M N := instMulHomClass F\ne : F\n\u22a2 \u2191e 1 * \u2191e (EquivLike.inv e 1) = \u2191e (EquivLike.inv e 1)\n[PROOFSTEP]\nrw [\u2190 map_mul, one_mul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM : Type u_6\nN : Type u_7\nP : Type u_8\nQ : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u00b2 : MulZeroClass \u03b1\ninst\u271d\u00b9 : MulZeroClass \u03b2\ninst\u271d : MulEquivClass F \u03b1 \u03b2\ne : F\n\u22a2 \u2191e 0 = \u2191e 0 * \u2191e (EquivLike.inv e 0)\n[PROOFSTEP]\nrw [\u2190 map_mul, zero_mul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM : Type u_6\nN : Type u_7\nP : Type u_8\nQ : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u00b2 : MulZeroClass \u03b1\ninst\u271d\u00b9 : MulZeroClass \u03b2\ninst\u271d : MulEquivClass F \u03b1 \u03b2\ne : F\n\u22a2 \u2191e 0 * \u2191e (EquivLike.inv e 0) = 0\n[PROOFSTEP]\nsimp\n -- See note [lower instance priority]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM : Type u_6\nN : Type u_7\nP : Type u_8\nQ : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u00b9 : Mul M\ninst\u271d : Mul N\nf g : M \u2243* N\nh\u2081 : (fun f => f.toFun) f = (fun f => f.toFun) g\nh\u2082 : (fun f => f.invFun) f = (fun f => f.invFun) g\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM : Type u_6\nN : Type u_7\nP : Type u_8\nQ : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u00b9 : Mul M\ninst\u271d : Mul N\ng : M \u2243* N\ntoEquiv\u271d : M \u2243 N\nmap_mul'\u271d : \u2200 (x y : M), Equiv.toFun toEquiv\u271d (x * y) = Equiv.toFun toEquiv\u271d x * Equiv.toFun toEquiv\u271d y\nh\u2081 : (fun f => f.toFun) { toEquiv := toEquiv\u271d, map_mul' := map_mul'\u271d } = (fun f => f.toFun) g\nh\u2082 : (fun f => f.invFun) { toEquiv := toEquiv\u271d, map_mul' := map_mul'\u271d } = (fun f => f.invFun) g\n\u22a2 { toEquiv := toEquiv\u271d, map_mul' := map_mul'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM : Type u_6\nN : Type u_7\nP : Type u_8\nQ : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u00b9 : Mul M\ninst\u271d : Mul N\ntoEquiv\u271d\u00b9 : M \u2243 N\nmap_mul'\u271d\u00b9 : \u2200 (x y : M), Equiv.toFun toEquiv\u271d\u00b9 (x * y) = Equiv.toFun toEquiv\u271d\u00b9 x * Equiv.toFun toEquiv\u271d\u00b9 y\ntoEquiv\u271d : M \u2243 N\nmap_mul'\u271d : \u2200 (x y : M), Equiv.toFun toEquiv\u271d (x * y) = Equiv.toFun toEquiv\u271d x * Equiv.toFun toEquiv\u271d y\nh\u2081 :\n (fun f => f.toFun) { toEquiv := toEquiv\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } =\n (fun f => f.toFun) { toEquiv := toEquiv\u271d, map_mul' := map_mul'\u271d }\nh\u2082 :\n (fun f => f.invFun) { toEquiv := toEquiv\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } =\n (fun f => f.invFun) { toEquiv := toEquiv\u271d, map_mul' := map_mul'\u271d }\n\u22a2 { toEquiv := toEquiv\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } = { toEquiv := toEquiv\u271d, map_mul' := map_mul'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toEquiv\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM : Type u_6\nN : Type u_7\nP : Type u_8\nQ : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u00b9 : Mul M\ninst\u271d : Mul N\ntoEquiv\u271d\u00b9 : M \u2243 N\nmap_mul'\u271d\u00b9 : \u2200 (x y : M), Equiv.toFun toEquiv\u271d\u00b9 (x * y) = Equiv.toFun toEquiv\u271d\u00b9 x * Equiv.toFun toEquiv\u271d\u00b9 y\ntoEquiv\u271d : M \u2243 N\nmap_mul'\u271d : \u2200 (x y : M), Equiv.toFun toEquiv\u271d (x * y) = Equiv.toFun toEquiv\u271d x * Equiv.toFun toEquiv\u271d y\nh\u2081 :\n (fun f => f.toFun) { toEquiv := toEquiv\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } =\n (fun f => f.toFun) { toEquiv := toEquiv\u271d, map_mul' := map_mul'\u271d }\nh\u2082 :\n (fun f => f.invFun) { toEquiv := toEquiv\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 } =\n (fun f => f.invFun) { toEquiv := toEquiv\u271d, map_mul' := map_mul'\u271d }\n\u22a2 toEquiv\u271d\u00b9 = toEquiv\u271d\n[PROOFSTEP]\napply Equiv.coe_fn_injective h\u2081\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM : Type u_6\nN : Type u_7\nP : Type u_8\nQ : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u00b3 : Mul M\ninst\u271d\u00b2 : Mul N\ninst\u271d\u00b9 : Mul P\ninst\u271d : Mul Q\nh1 : M \u2243* N\nh2 : N \u2243* P\nsrc\u271d : M \u2243 P := h1.trans h2.toEquiv\nx y : M\n\u22a2 \u2191h2 (\u2191h1 (x * y)) = \u2191h2 (\u2191h1 x) * \u2191h2 (\u2191h1 y)\n[PROOFSTEP]\nrw [h1.map_mul, h2.map_mul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM\u271d : Type u_6\nN\u271d : Type u_7\nP\u271d : Type u_8\nQ\u271d : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u2075 : Mul M\u271d\ninst\u271d\u2074 : Mul N\u271d\ninst\u271d\u00b3 : Mul P\u271d\ninst\u271d\u00b2 : Mul Q\u271d\nM : Type u_12\nN : Type u_13\nP : Type u_14\nQ : Type u_15\ninst\u271d\u00b9 : Mul P\ninst\u271d : Mul Q\nf : M \u2243 N\ng : P \u2243* Q\nh : M \u2192 P\n\u22a2 (fun k m => \u2191(symm g) (k (\u2191f m))) ((fun h n => \u2191g (h (\u2191f.symm n))) h) = h\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM\u271d : Type u_6\nN\u271d : Type u_7\nP\u271d : Type u_8\nQ\u271d : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u2075 : Mul M\u271d\ninst\u271d\u2074 : Mul N\u271d\ninst\u271d\u00b3 : Mul P\u271d\ninst\u271d\u00b2 : Mul Q\u271d\nM : Type u_12\nN : Type u_13\nP : Type u_14\nQ : Type u_15\ninst\u271d\u00b9 : Mul P\ninst\u271d : Mul Q\nf : M \u2243 N\ng : P \u2243* Q\nh : M \u2192 P\nx\u271d : M\n\u22a2 (fun k m => \u2191(symm g) (k (\u2191f m))) ((fun h n => \u2191g (h (\u2191f.symm n))) h) x\u271d = h x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM\u271d : Type u_6\nN\u271d : Type u_7\nP\u271d : Type u_8\nQ\u271d : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u2075 : Mul M\u271d\ninst\u271d\u2074 : Mul N\u271d\ninst\u271d\u00b3 : Mul P\u271d\ninst\u271d\u00b2 : Mul Q\u271d\nM : Type u_12\nN : Type u_13\nP : Type u_14\nQ : Type u_15\ninst\u271d\u00b9 : Mul P\ninst\u271d : Mul Q\nf : M \u2243 N\ng : P \u2243* Q\nk : N \u2192 Q\n\u22a2 (fun h n => \u2191g (h (\u2191f.symm n))) ((fun k m => \u2191(symm g) (k (\u2191f m))) k) = k\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM\u271d : Type u_6\nN\u271d : Type u_7\nP\u271d : Type u_8\nQ\u271d : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u2075 : Mul M\u271d\ninst\u271d\u2074 : Mul N\u271d\ninst\u271d\u00b3 : Mul P\u271d\ninst\u271d\u00b2 : Mul Q\u271d\nM : Type u_12\nN : Type u_13\nP : Type u_14\nQ : Type u_15\ninst\u271d\u00b9 : Mul P\ninst\u271d : Mul Q\nf : M \u2243 N\ng : P \u2243* Q\nk : N \u2192 Q\nx\u271d : N\n\u22a2 (fun h n => \u2191g (h (\u2191f.symm n))) ((fun k m => \u2191(symm g) (k (\u2191f m))) k) x\u271d = k x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM\u271d : Type u_6\nN\u271d : Type u_7\nP\u271d : Type u_8\nQ\u271d : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u2075 : Mul M\u271d\ninst\u271d\u2074 : Mul N\u271d\ninst\u271d\u00b3 : Mul P\u271d\ninst\u271d\u00b2 : Mul Q\u271d\nM : Type u_12\nN : Type u_13\nP : Type u_14\nQ : Type u_15\ninst\u271d\u00b9 : Mul P\ninst\u271d : Mul Q\nf : M \u2243 N\ng : P \u2243* Q\nh k : M \u2192 P\n\u22a2 Equiv.toFun\n { toFun := fun h n => \u2191g (h (\u2191f.symm n)), invFun := fun k m => \u2191(symm g) (k (\u2191f m)),\n left_inv := (_ : \u2200 (h : M \u2192 P), (fun k m => \u2191(symm g) (k (\u2191f m))) ((fun h n => \u2191g (h (\u2191f.symm n))) h) = h),\n right_inv := (_ : \u2200 (k : N \u2192 Q), (fun h n => \u2191g (h (\u2191f.symm n))) ((fun k m => \u2191(symm g) (k (\u2191f m))) k) = k) }\n (h * k) =\n Equiv.toFun\n { toFun := fun h n => \u2191g (h (\u2191f.symm n)), invFun := fun k m => \u2191(symm g) (k (\u2191f m)),\n left_inv := (_ : \u2200 (h : M \u2192 P), (fun k m => \u2191(symm g) (k (\u2191f m))) ((fun h n => \u2191g (h (\u2191f.symm n))) h) = h),\n right_inv := (_ : \u2200 (k : N \u2192 Q), (fun h n => \u2191g (h (\u2191f.symm n))) ((fun k m => \u2191(symm g) (k (\u2191f m))) k) = k) }\n h *\n Equiv.toFun\n { toFun := fun h n => \u2191g (h (\u2191f.symm n)), invFun := fun k m => \u2191(symm g) (k (\u2191f m)),\n left_inv := (_ : \u2200 (h : M \u2192 P), (fun k m => \u2191(symm g) (k (\u2191f m))) ((fun h n => \u2191g (h (\u2191f.symm n))) h) = h),\n right_inv := (_ : \u2200 (k : N \u2192 Q), (fun h n => \u2191g (h (\u2191f.symm n))) ((fun k m => \u2191(symm g) (k (\u2191f m))) k) = k) }\n k\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM\u271d : Type u_6\nN\u271d : Type u_7\nP\u271d : Type u_8\nQ\u271d : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u2075 : Mul M\u271d\ninst\u271d\u2074 : Mul N\u271d\ninst\u271d\u00b3 : Mul P\u271d\ninst\u271d\u00b2 : Mul Q\u271d\nM : Type u_12\nN : Type u_13\nP : Type u_14\nQ : Type u_15\ninst\u271d\u00b9 : Mul P\ninst\u271d : Mul Q\nf : M \u2243 N\ng : P \u2243* Q\nh k : M \u2192 P\nx\u271d : N\n\u22a2 Equiv.toFun\n { toFun := fun h n => \u2191g (h (\u2191f.symm n)), invFun := fun k m => \u2191(symm g) (k (\u2191f m)),\n left_inv := (_ : \u2200 (h : M \u2192 P), (fun k m => \u2191(symm g) (k (\u2191f m))) ((fun h n => \u2191g (h (\u2191f.symm n))) h) = h),\n right_inv := (_ : \u2200 (k : N \u2192 Q), (fun h n => \u2191g (h (\u2191f.symm n))) ((fun k m => \u2191(symm g) (k (\u2191f m))) k) = k) }\n (h * k) x\u271d =\n (Equiv.toFun\n { toFun := fun h n => \u2191g (h (\u2191f.symm n)), invFun := fun k m => \u2191(symm g) (k (\u2191f m)),\n left_inv := (_ : \u2200 (h : M \u2192 P), (fun k m => \u2191(symm g) (k (\u2191f m))) ((fun h n => \u2191g (h (\u2191f.symm n))) h) = h),\n right_inv :=\n (_ : \u2200 (k : N \u2192 Q), (fun h n => \u2191g (h (\u2191f.symm n))) ((fun k m => \u2191(symm g) (k (\u2191f m))) k) = k) }\n h *\n Equiv.toFun\n { toFun := fun h n => \u2191g (h (\u2191f.symm n)), invFun := fun k m => \u2191(symm g) (k (\u2191f m)),\n left_inv := (_ : \u2200 (h : M \u2192 P), (fun k m => \u2191(symm g) (k (\u2191f m))) ((fun h n => \u2191g (h (\u2191f.symm n))) h) = h),\n right_inv :=\n (_ : \u2200 (k : N \u2192 Q), (fun h n => \u2191g (h (\u2191f.symm n))) ((fun k m => \u2191(symm g) (k (\u2191f m))) k) = k) }\n k)\n x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM\u271d : Type u_6\nN\u271d : Type u_7\nP\u271d : Type u_8\nQ\u271d : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u2077 : Mul M\u271d\ninst\u271d\u2076 : Mul N\u271d\ninst\u271d\u2075 : Mul P\u271d\ninst\u271d\u2074 : Mul Q\u271d\nM : Type ?u.66798\nN : Type ?u.66801\nP : Type ?u.66804\nQ : Type ?u.66807\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : CommMonoid P\ninst\u271d : CommMonoid Q\nf : M \u2243* N\ng : P \u2243* Q\nh : M \u2192* P\n\u22a2 (fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f)))\n ((fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f)))) h) =\n h\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM\u271d : Type u_6\nN\u271d : Type u_7\nP\u271d : Type u_8\nQ\u271d : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u2077 : Mul M\u271d\ninst\u271d\u2076 : Mul N\u271d\ninst\u271d\u2075 : Mul P\u271d\ninst\u271d\u2074 : Mul Q\u271d\nM : Type ?u.66798\nN : Type ?u.66801\nP : Type ?u.66804\nQ : Type ?u.66807\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : CommMonoid P\ninst\u271d : CommMonoid Q\nf : M \u2243* N\ng : P \u2243* Q\nh : M \u2192* P\nx\u271d : M\n\u22a2 \u2191((fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f)))\n ((fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f)))) h))\n x\u271d =\n \u2191h x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM\u271d : Type u_6\nN\u271d : Type u_7\nP\u271d : Type u_8\nQ\u271d : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u2077 : Mul M\u271d\ninst\u271d\u2076 : Mul N\u271d\ninst\u271d\u2075 : Mul P\u271d\ninst\u271d\u2074 : Mul Q\u271d\nM : Type ?u.66798\nN : Type ?u.66801\nP : Type ?u.66804\nQ : Type ?u.66807\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : CommMonoid P\ninst\u271d : CommMonoid Q\nf : M \u2243* N\ng : P \u2243* Q\nk : N \u2192* Q\n\u22a2 (fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f))))\n ((fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f))) k) =\n k\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM\u271d : Type u_6\nN\u271d : Type u_7\nP\u271d : Type u_8\nQ\u271d : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u2077 : Mul M\u271d\ninst\u271d\u2076 : Mul N\u271d\ninst\u271d\u2075 : Mul P\u271d\ninst\u271d\u2074 : Mul Q\u271d\nM : Type ?u.66798\nN : Type ?u.66801\nP : Type ?u.66804\nQ : Type ?u.66807\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : CommMonoid P\ninst\u271d : CommMonoid Q\nf : M \u2243* N\ng : P \u2243* Q\nk : N \u2192* Q\nx\u271d : N\n\u22a2 \u2191((fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f))))\n ((fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f))) k))\n x\u271d =\n \u2191k x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM\u271d : Type u_6\nN\u271d : Type u_7\nP\u271d : Type u_8\nQ\u271d : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u2077 : Mul M\u271d\ninst\u271d\u2076 : Mul N\u271d\ninst\u271d\u2075 : Mul P\u271d\ninst\u271d\u2074 : Mul Q\u271d\nM : Type ?u.66798\nN : Type ?u.66801\nP : Type ?u.66804\nQ : Type ?u.66807\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : CommMonoid P\ninst\u271d : CommMonoid Q\nf : M \u2243* N\ng : P \u2243* Q\nh k : M \u2192* P\n\u22a2 Equiv.toFun\n { toFun := fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f))),\n invFun := fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f)),\n left_inv :=\n (_ :\n \u2200 (h : M \u2192* P),\n (fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f)))\n ((fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f)))) h) =\n h),\n right_inv :=\n (_ :\n \u2200 (k : N \u2192* Q),\n (fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f))))\n ((fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f))) k) =\n k) }\n (h * k) =\n Equiv.toFun\n { toFun := fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f))),\n invFun := fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f)),\n left_inv :=\n (_ :\n \u2200 (h : M \u2192* P),\n (fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f)))\n ((fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f)))) h) =\n h),\n right_inv :=\n (_ :\n \u2200 (k : N \u2192* Q),\n (fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f))))\n ((fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f))) k) =\n k) }\n h *\n Equiv.toFun\n { toFun := fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f))),\n invFun := fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f)),\n left_inv :=\n (_ :\n \u2200 (h : M \u2192* P),\n (fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f)))\n ((fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f)))) h) =\n h),\n right_inv :=\n (_ :\n \u2200 (k : N \u2192* Q),\n (fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f))))\n ((fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f))) k) =\n k) }\n k\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nA : Type u_4\nB : Type u_5\nM\u271d : Type u_6\nN\u271d : Type u_7\nP\u271d : Type u_8\nQ\u271d : Type u_9\nG : Type u_10\nH : Type u_11\ninst\u271d\u2077 : Mul M\u271d\ninst\u271d\u2076 : Mul N\u271d\ninst\u271d\u2075 : Mul P\u271d\ninst\u271d\u2074 : Mul Q\u271d\nM : Type ?u.66798\nN : Type ?u.66801\nP : Type ?u.66804\nQ : Type ?u.66807\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : CommMonoid P\ninst\u271d : CommMonoid Q\nf : M \u2243* N\ng : P \u2243* Q\nh k : M \u2192* P\nx\u271d : N\n\u22a2 \u2191(Equiv.toFun\n { toFun := fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f))),\n invFun := fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f)),\n left_inv :=\n (_ :\n \u2200 (h : M \u2192* P),\n (fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f)))\n ((fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f)))) h) =\n h),\n right_inv :=\n (_ :\n \u2200 (k : N \u2192* Q),\n (fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f))))\n ((fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f))) k) =\n k) }\n (h * k))\n x\u271d =\n \u2191(Equiv.toFun\n { toFun := fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f))),\n invFun := fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f)),\n left_inv :=\n (_ :\n \u2200 (h : M \u2192* P),\n (fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f)))\n ((fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f)))) h) =\n h),\n right_inv :=\n (_ :\n \u2200 (k : N \u2192* Q),\n (fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f))))\n ((fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f))) k) =\n k) }\n h *\n Equiv.toFun\n { toFun := fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f))),\n invFun := fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f)),\n left_inv :=\n (_ :\n \u2200 (h : M \u2192* P),\n (fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f)))\n ((fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f)))) h) =\n h),\n right_inv :=\n (_ :\n \u2200 (k : N \u2192* Q),\n (fun h => MonoidHom.comp (toMonoidHom g) (MonoidHom.comp h (toMonoidHom (symm f))))\n ((fun k => MonoidHom.comp (toMonoidHom (symm g)) (MonoidHom.comp k (toMonoidHom f))) k) =\n k) }\n k)\n x\u271d\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Hom.Equiv.Basic", "llama_tokens": 9216, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154240079185319, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.4240079142985898}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\n\u22a2 pullbackFst F = pullback.snd\n[PROOFSTEP]\nconvert\n (eq_whisker pullback.condition Limits.prod.fst : (_ : constructEqualizer F \u27f6 F.obj WalkingParallelPair.zero) = _)\n[GOAL]\ncase h.e'_2\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\n\u22a2 pullbackFst F =\n (pullback.fst \u226b prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.left)) \u226b prod.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\n\u22a2 pullback.snd =\n (pullback.snd \u226b prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.right)) \u226b prod.fst\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\n\u22a2 pullbackFst F \u226b F.map WalkingParallelPairHom.left = pullbackFst F \u226b F.map WalkingParallelPairHom.right\n[PROOFSTEP]\nconv_rhs => rw [pullbackFst_eq_pullback_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\n| pullbackFst F \u226b F.map WalkingParallelPairHom.right\n[PROOFSTEP]\nrw [pullbackFst_eq_pullback_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\n| pullbackFst F \u226b F.map WalkingParallelPairHom.right\n[PROOFSTEP]\nrw [pullbackFst_eq_pullback_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\n| pullbackFst F \u226b F.map WalkingParallelPairHom.right\n[PROOFSTEP]\nrw [pullbackFst_eq_pullback_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\n\u22a2 pullbackFst F \u226b F.map WalkingParallelPairHom.left = pullback.snd \u226b F.map WalkingParallelPairHom.right\n[PROOFSTEP]\nconvert\n (eq_whisker pullback.condition Limits.prod.snd : (_ : constructEqualizer F \u27f6 F.obj WalkingParallelPair.one) = _) using\n 1\n[GOAL]\ncase h.e'_2\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\n\u22a2 pullbackFst F \u226b F.map WalkingParallelPairHom.left =\n (pullback.fst \u226b prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.left)) \u226b prod.snd\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\n\u22a2 pullback.snd \u226b F.map WalkingParallelPairHom.right =\n (pullback.snd \u226b prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.right)) \u226b prod.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\n\u22a2 (s : Cone F) \u2192 s.pt \u27f6 (equalizerCone F).pt\n[PROOFSTEP]\nintro c\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\nc : Cone F\n\u22a2 c.pt \u27f6 (equalizerCone F).pt\n[PROOFSTEP]\napply pullback.lift (c.\u03c0.app _) (c.\u03c0.app _)\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\nc : Cone F\n\u22a2 NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.left) =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.right)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\nc : Cone F\n\u22a2 (NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.left)) \u226b\n prod.fst =\n (NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.right)) \u226b\n prod.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\nc : Cone F\n\u22a2 (NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.left)) \u226b\n prod.snd =\n (NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.right)) \u226b\n prod.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\n\u22a2 \u2200 (s : Cone F) (j : WalkingParallelPair),\n pullback.lift (NatTrans.app s.\u03c0 WalkingParallelPair.zero) (NatTrans.app s.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app s.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.left) =\n NatTrans.app s.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app (equalizerCone F).\u03c0 j =\n NatTrans.app s.\u03c0 j\n[PROOFSTEP]\nrintro c (_ | _)\n[GOAL]\ncase zero\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\nc : Cone F\n\u22a2 pullback.lift (NatTrans.app c.\u03c0 WalkingParallelPair.zero) (NatTrans.app c.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.left) =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app (equalizerCone F).\u03c0 WalkingParallelPair.zero =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase one\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\nc : Cone F\n\u22a2 pullback.lift (NatTrans.app c.\u03c0 WalkingParallelPair.zero) (NatTrans.app c.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.left) =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app (equalizerCone F).\u03c0 WalkingParallelPair.one =\n NatTrans.app c.\u03c0 WalkingParallelPair.one\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\n\u22a2 \u2200 (s : Cone F) (m : s.pt \u27f6 (equalizerCone F).pt),\n (\u2200 (j : WalkingParallelPair), m \u226b NatTrans.app (equalizerCone F).\u03c0 j = NatTrans.app s.\u03c0 j) \u2192\n m =\n pullback.lift (NatTrans.app s.\u03c0 WalkingParallelPair.zero) (NatTrans.app s.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app s.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.left) =\n NatTrans.app s.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.right))\n[PROOFSTEP]\nintro c _ J\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\nc : Cone F\nm\u271d : c.pt \u27f6 (equalizerCone F).pt\nJ : \u2200 (j : WalkingParallelPair), m\u271d \u226b NatTrans.app (equalizerCone F).\u03c0 j = NatTrans.app c.\u03c0 j\n\u22a2 m\u271d =\n pullback.lift (NatTrans.app c.\u03c0 WalkingParallelPair.zero) (NatTrans.app c.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.left) =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.right))\n[PROOFSTEP]\nhave J0 := J WalkingParallelPair.zero\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\nc : Cone F\nm\u271d : c.pt \u27f6 (equalizerCone F).pt\nJ : \u2200 (j : WalkingParallelPair), m\u271d \u226b NatTrans.app (equalizerCone F).\u03c0 j = NatTrans.app c.\u03c0 j\nJ0 : m\u271d \u226b NatTrans.app (equalizerCone F).\u03c0 WalkingParallelPair.zero = NatTrans.app c.\u03c0 WalkingParallelPair.zero\n\u22a2 m\u271d =\n pullback.lift (NatTrans.app c.\u03c0 WalkingParallelPair.zero) (NatTrans.app c.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.left) =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.right))\n[PROOFSTEP]\nsimp at J0 \n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\nc : Cone F\nm\u271d : c.pt \u27f6 (equalizerCone F).pt\nJ : \u2200 (j : WalkingParallelPair), m\u271d \u226b NatTrans.app (equalizerCone F).\u03c0 j = NatTrans.app c.\u03c0 j\nJ0 : m\u271d \u226b pullbackFst F = NatTrans.app c.\u03c0 WalkingParallelPair.zero\n\u22a2 m\u271d =\n pullback.lift (NatTrans.app c.\u03c0 WalkingParallelPair.zero) (NatTrans.app c.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.left) =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.right))\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\nc : Cone F\nm\u271d : c.pt \u27f6 (equalizerCone F).pt\nJ : \u2200 (j : WalkingParallelPair), m\u271d \u226b NatTrans.app (equalizerCone F).\u03c0 j = NatTrans.app c.\u03c0 j\nJ0 : m\u271d \u226b pullbackFst F = NatTrans.app c.\u03c0 WalkingParallelPair.zero\n\u22a2 m\u271d \u226b pullback.fst =\n pullback.lift (NatTrans.app c.\u03c0 WalkingParallelPair.zero) (NatTrans.app c.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.left) =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.right)) \u226b\n pullback.fst\n[PROOFSTEP]\nrwa [limit.lift_\u03c0]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryProducts C\ninst\u271d : HasPullbacks C\nF : WalkingParallelPair \u2964 C\nc : Cone F\nm\u271d : c.pt \u27f6 (equalizerCone F).pt\nJ : \u2200 (j : WalkingParallelPair), m\u271d \u226b NatTrans.app (equalizerCone F).\u03c0 j = NatTrans.app c.\u03c0 j\nJ0 : m\u271d \u226b pullbackFst F = NatTrans.app c.\u03c0 WalkingParallelPair.zero\n\u22a2 m\u271d \u226b pullback.snd =\n pullback.lift (NatTrans.app c.\u03c0 WalkingParallelPair.zero) (NatTrans.app c.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.left) =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n prod.lift (\ud835\udfd9 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.right)) \u226b\n pullback.snd\n[PROOFSTEP]\nerw [limit.lift_\u03c0, \u2190 J0, pullbackFst_eq_pullback_snd]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\nc : Cone (K \u22d9 G)\n\u22a2 c.pt \u27f6 (G.mapCone (equalizerCone K)).pt\n[PROOFSTEP]\nrefine' pullback.lift ?_ ?_ ?_ \u226b (PreservesPullback.iso _ _ _).inv\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\nc : Cone (K \u22d9 G)\n\u22a2 c.pt \u27f6 G.obj (K.obj WalkingParallelPair.zero)\n[PROOFSTEP]\nexact c.\u03c0.app WalkingParallelPair.zero\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\nc : Cone (K \u22d9 G)\n\u22a2 c.pt \u27f6 G.obj (K.obj WalkingParallelPair.zero)\n[PROOFSTEP]\nexact c.\u03c0.app WalkingParallelPair.zero\n[GOAL]\ncase refine'_3\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\nc : Cone (K \u22d9 G)\n\u22a2 NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left)) =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))\n[PROOFSTEP]\napply (mapIsLimitOfPreservesOfIsLimit G _ _ (prodIsProd _ _)).hom_ext\n[GOAL]\ncase refine'_3\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\nc : Cone (K \u22d9 G)\n\u22a2 \u2200 (j : Discrete WalkingPair),\n (NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left))) \u226b\n NatTrans.app (BinaryFan.mk (G.map prod.fst) (G.map prod.snd)).\u03c0 j =\n (NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))) \u226b\n NatTrans.app (BinaryFan.mk (G.map prod.fst) (G.map prod.snd)).\u03c0 j\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase refine'_3.mk.left\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\nc : Cone (K \u22d9 G)\n\u22a2 (NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left))) \u226b\n NatTrans.app (BinaryFan.mk (G.map prod.fst) (G.map prod.snd)).\u03c0 { as := WalkingPair.left } =\n (NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))) \u226b\n NatTrans.app (BinaryFan.mk (G.map prod.fst) (G.map prod.snd)).\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\nsimp only [Category.assoc, \u2190 G.map_comp, prod.lift_fst, BinaryFan.\u03c0_app_left, BinaryFan.mk_fst]\n[GOAL]\ncase refine'_3.mk.right\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\nc : Cone (K \u22d9 G)\n\u22a2 (NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left))) \u226b\n NatTrans.app (BinaryFan.mk (G.map prod.fst) (G.map prod.snd)).\u03c0 { as := WalkingPair.right } =\n (NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))) \u226b\n NatTrans.app (BinaryFan.mk (G.map prod.fst) (G.map prod.snd)).\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\nsimp only [BinaryFan.\u03c0_app_right, BinaryFan.mk_snd, Category.assoc, \u2190 G.map_comp, prod.lift_snd]\n[GOAL]\ncase refine'_3.mk.right\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\nc : Cone (K \u22d9 G)\n\u22a2 NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b G.map (K.map WalkingParallelPairHom.left) =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b G.map (K.map WalkingParallelPairHom.right)\n[PROOFSTEP]\nexact (c.\u03c0.naturality WalkingParallelPairHom.left).symm.trans (c.\u03c0.naturality WalkingParallelPairHom.right)\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\nc : Cone (K \u22d9 G)\nj : WalkingParallelPair\n\u22a2 (fun c =>\n pullback.lift (NatTrans.app c.\u03c0 WalkingParallelPair.zero) (NatTrans.app c.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left)) =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))) \u226b\n (PreservesPullback.iso G\n (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left))\n (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))).inv)\n c \u226b\n NatTrans.app (G.mapCone (equalizerCone K)).\u03c0 j =\n NatTrans.app c.\u03c0 j\n[PROOFSTEP]\nrcases j with (_ | _)\n[GOAL]\ncase zero\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\nc : Cone (K \u22d9 G)\n\u22a2 (fun c =>\n pullback.lift (NatTrans.app c.\u03c0 WalkingParallelPair.zero) (NatTrans.app c.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left)) =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))) \u226b\n (PreservesPullback.iso G\n (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left))\n (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))).inv)\n c \u226b\n NatTrans.app (G.mapCone (equalizerCone K)).\u03c0 WalkingParallelPair.zero =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero\n[PROOFSTEP]\nsimp only [Category.comp_id, PreservesPullback.iso_inv_fst, Cone.ofFork_\u03c0, G.map_comp,\n PreservesPullback.iso_inv_fst_assoc, Functor.mapCone_\u03c0_app, eqToHom_refl, Category.assoc, Fork.of\u03b9_\u03c0_app,\n pullback.lift_fst, pullback.lift_fst_assoc]\n[GOAL]\ncase one\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\nc : Cone (K \u22d9 G)\n\u22a2 (fun c =>\n pullback.lift (NatTrans.app c.\u03c0 WalkingParallelPair.zero) (NatTrans.app c.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left)) =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))) \u226b\n (PreservesPullback.iso G\n (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left))\n (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))).inv)\n c \u226b\n NatTrans.app (G.mapCone (equalizerCone K)).\u03c0 WalkingParallelPair.one =\n NatTrans.app c.\u03c0 WalkingParallelPair.one\n[PROOFSTEP]\nsimp only [Category.comp_id, PreservesPullback.iso_inv_fst, Cone.ofFork_\u03c0, G.map_comp,\n PreservesPullback.iso_inv_fst_assoc, Functor.mapCone_\u03c0_app, eqToHom_refl, Category.assoc, Fork.of\u03b9_\u03c0_app,\n pullback.lift_fst, pullback.lift_fst_assoc]\n[GOAL]\ncase one\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\nc : Cone (K \u22d9 G)\n\u22a2 NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b G.map (K.map WalkingParallelPairHom.left) =\n NatTrans.app c.\u03c0 WalkingParallelPair.one\n[PROOFSTEP]\nexact (c.\u03c0.naturality WalkingParallelPairHom.left).symm.trans (Category.id_comp _)\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\ns : Cone (K \u22d9 G)\nm : s.pt \u27f6 (G.mapCone (equalizerCone K)).pt\nh : \u2200 (j : WalkingParallelPair), m \u226b NatTrans.app (G.mapCone (equalizerCone K)).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m =\n (fun c =>\n pullback.lift (NatTrans.app c.\u03c0 WalkingParallelPair.zero) (NatTrans.app c.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left)) =\n NatTrans.app c.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))) \u226b\n (PreservesPullback.iso G (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left))\n (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))).inv)\n s\n[PROOFSTEP]\nrw [Iso.eq_comp_inv]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\ns : Cone (K \u22d9 G)\nm : s.pt \u27f6 (G.mapCone (equalizerCone K)).pt\nh : \u2200 (j : WalkingParallelPair), m \u226b NatTrans.app (G.mapCone (equalizerCone K)).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m \u226b\n (PreservesPullback.iso G (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left))\n (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))).hom =\n pullback.lift (NatTrans.app s.\u03c0 WalkingParallelPair.zero) (NatTrans.app s.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app s.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left)) =\n NatTrans.app s.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right)))\n[PROOFSTEP]\nhave := h WalkingParallelPair.zero\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\ns : Cone (K \u22d9 G)\nm : s.pt \u27f6 (G.mapCone (equalizerCone K)).pt\nh : \u2200 (j : WalkingParallelPair), m \u226b NatTrans.app (G.mapCone (equalizerCone K)).\u03c0 j = NatTrans.app s.\u03c0 j\nthis :\n m \u226b NatTrans.app (G.mapCone (equalizerCone K)).\u03c0 WalkingParallelPair.zero = NatTrans.app s.\u03c0 WalkingParallelPair.zero\n\u22a2 m \u226b\n (PreservesPullback.iso G (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left))\n (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))).hom =\n pullback.lift (NatTrans.app s.\u03c0 WalkingParallelPair.zero) (NatTrans.app s.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app s.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left)) =\n NatTrans.app s.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right)))\n[PROOFSTEP]\ndsimp [equalizerCone] at this \n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\ns : Cone (K \u22d9 G)\nm : s.pt \u27f6 (G.mapCone (equalizerCone K)).pt\nh : \u2200 (j : WalkingParallelPair), m \u226b NatTrans.app (G.mapCone (equalizerCone K)).\u03c0 j = NatTrans.app s.\u03c0 j\nthis : m \u226b G.map (pullbackFst K \u226b \ud835\udfd9 (K.obj WalkingParallelPair.zero)) = NatTrans.app s.\u03c0 WalkingParallelPair.zero\n\u22a2 m \u226b\n (PreservesPullback.iso G (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left))\n (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))).hom =\n pullback.lift (NatTrans.app s.\u03c0 WalkingParallelPair.zero) (NatTrans.app s.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app s.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left)) =\n NatTrans.app s.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right)))\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\ns : Cone (K \u22d9 G)\nm : s.pt \u27f6 (G.mapCone (equalizerCone K)).pt\nh : \u2200 (j : WalkingParallelPair), m \u226b NatTrans.app (G.mapCone (equalizerCone K)).\u03c0 j = NatTrans.app s.\u03c0 j\nthis : m \u226b G.map (pullbackFst K \u226b \ud835\udfd9 (K.obj WalkingParallelPair.zero)) = NatTrans.app s.\u03c0 WalkingParallelPair.zero\n\u22a2 (m \u226b\n (PreservesPullback.iso G (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left))\n (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))).hom) \u226b\n pullback.fst =\n pullback.lift (NatTrans.app s.\u03c0 WalkingParallelPair.zero) (NatTrans.app s.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app s.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left)) =\n NatTrans.app s.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))) \u226b\n pullback.fst\n[PROOFSTEP]\nsimp only [PreservesPullback.iso_hom_snd, Category.assoc, PreservesPullback.iso_hom_fst, pullback.lift_fst,\n pullback.lift_snd, Category.comp_id, \u2190 pullbackFst_eq_pullback_snd, \u2190 this]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryProducts C\ninst\u271d\u00b2 : HasPullbacks C\ninst\u271d\u00b9 : PreservesLimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesLimitsOfShape WalkingCospan G\nK : WalkingParallelPair \u2964 C\ns : Cone (K \u22d9 G)\nm : s.pt \u27f6 (G.mapCone (equalizerCone K)).pt\nh : \u2200 (j : WalkingParallelPair), m \u226b NatTrans.app (G.mapCone (equalizerCone K)).\u03c0 j = NatTrans.app s.\u03c0 j\nthis : m \u226b G.map (pullbackFst K \u226b \ud835\udfd9 (K.obj WalkingParallelPair.zero)) = NatTrans.app s.\u03c0 WalkingParallelPair.zero\n\u22a2 (m \u226b\n (PreservesPullback.iso G (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left))\n (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))).hom) \u226b\n pullback.snd =\n pullback.lift (NatTrans.app s.\u03c0 WalkingParallelPair.zero) (NatTrans.app s.\u03c0 WalkingParallelPair.zero)\n (_ :\n NatTrans.app s.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left)) =\n NatTrans.app s.\u03c0 WalkingParallelPair.zero \u226b\n G.map (prod.lift (\ud835\udfd9 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))) \u226b\n pullback.snd\n[PROOFSTEP]\nsimp only [PreservesPullback.iso_hom_snd, Category.assoc, PreservesPullback.iso_hom_fst, pullback.lift_fst,\n pullback.lift_snd, Category.comp_id, \u2190 pullbackFst_eq_pullback_snd, \u2190 this]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\n\u22a2 pushoutInl F = pushout.inr\n[PROOFSTEP]\nconvert (whisker_eq Limits.coprod.inl pushout.condition : (_ : F.obj _ \u27f6 constructCoequalizer _) = _)\n[GOAL]\ncase h.e'_2\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\n\u22a2 pushoutInl F =\n coprod.inl \u226b coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left) \u226b pushout.inl\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\n\u22a2 pushout.inr =\n coprod.inl \u226b coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right) \u226b pushout.inr\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\n\u22a2 F.map WalkingParallelPairHom.left \u226b pushoutInl F = F.map WalkingParallelPairHom.right \u226b pushoutInl F\n[PROOFSTEP]\nconv_rhs => rw [pushoutInl_eq_pushout_inr]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\n| F.map WalkingParallelPairHom.right \u226b pushoutInl F\n[PROOFSTEP]\nrw [pushoutInl_eq_pushout_inr]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\n| F.map WalkingParallelPairHom.right \u226b pushoutInl F\n[PROOFSTEP]\nrw [pushoutInl_eq_pushout_inr]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\n| F.map WalkingParallelPairHom.right \u226b pushoutInl F\n[PROOFSTEP]\nrw [pushoutInl_eq_pushout_inr]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\n\u22a2 F.map WalkingParallelPairHom.left \u226b pushoutInl F = F.map WalkingParallelPairHom.right \u226b pushout.inr\n[PROOFSTEP]\nconvert (whisker_eq Limits.coprod.inr pushout.condition : (_ : F.obj _ \u27f6 constructCoequalizer _) = _) using 1\n[GOAL]\ncase h.e'_2\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\n\u22a2 F.map WalkingParallelPairHom.left \u226b pushoutInl F =\n coprod.inr \u226b coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left) \u226b pushout.inl\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\n\u22a2 F.map WalkingParallelPairHom.right \u226b pushout.inr =\n coprod.inr \u226b coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right) \u226b pushout.inr\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\n\u22a2 (s : Cocone F) \u2192 (coequalizerCocone F).pt \u27f6 s.pt\n[PROOFSTEP]\nintro c\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\nc : Cocone F\n\u22a2 (coequalizerCocone F).pt \u27f6 c.pt\n[PROOFSTEP]\napply pushout.desc (c.\u03b9.app _) (c.\u03b9.app _)\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\nc : Cocone F\n\u22a2 coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\nc : Cocone F\n\u22a2 coprod.inl \u226b\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n coprod.inl \u226b\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\nc : Cocone F\n\u22a2 coprod.inr \u226b\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n coprod.inr \u226b\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\n\u22a2 \u2200 (s : Cocone F) (j : WalkingParallelPair),\n NatTrans.app (coequalizerCocone F).\u03b9 j \u226b\n pushout.desc (NatTrans.app s.\u03b9 WalkingParallelPair.one) (NatTrans.app s.\u03b9 WalkingParallelPair.one)\n (_ :\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left) \u226b\n NatTrans.app s.\u03b9 WalkingParallelPair.one =\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right) \u226b\n NatTrans.app s.\u03b9 WalkingParallelPair.one) =\n NatTrans.app s.\u03b9 j\n[PROOFSTEP]\nrintro c (_ | _)\n[GOAL]\ncase zero\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\nc : Cocone F\n\u22a2 NatTrans.app (coequalizerCocone F).\u03b9 WalkingParallelPair.zero \u226b\n pushout.desc (NatTrans.app c.\u03b9 WalkingParallelPair.one) (NatTrans.app c.\u03b9 WalkingParallelPair.one)\n (_ :\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one) =\n NatTrans.app c.\u03b9 WalkingParallelPair.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase one\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\nc : Cocone F\n\u22a2 NatTrans.app (coequalizerCocone F).\u03b9 WalkingParallelPair.one \u226b\n pushout.desc (NatTrans.app c.\u03b9 WalkingParallelPair.one) (NatTrans.app c.\u03b9 WalkingParallelPair.one)\n (_ :\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one) =\n NatTrans.app c.\u03b9 WalkingParallelPair.one\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\n\u22a2 \u2200 (s : Cocone F) (m : (coequalizerCocone F).pt \u27f6 s.pt),\n (\u2200 (j : WalkingParallelPair), NatTrans.app (coequalizerCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j) \u2192\n m =\n pushout.desc (NatTrans.app s.\u03b9 WalkingParallelPair.one) (NatTrans.app s.\u03b9 WalkingParallelPair.one)\n (_ :\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left) \u226b\n NatTrans.app s.\u03b9 WalkingParallelPair.one =\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right) \u226b\n NatTrans.app s.\u03b9 WalkingParallelPair.one)\n[PROOFSTEP]\nintro c m J\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\nc : Cocone F\nm : (coequalizerCocone F).pt \u27f6 c.pt\nJ : \u2200 (j : WalkingParallelPair), NatTrans.app (coequalizerCocone F).\u03b9 j \u226b m = NatTrans.app c.\u03b9 j\n\u22a2 m =\n pushout.desc (NatTrans.app c.\u03b9 WalkingParallelPair.one) (NatTrans.app c.\u03b9 WalkingParallelPair.one)\n (_ :\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one)\n[PROOFSTEP]\nhave J1 : pushoutInl F \u226b m = c.\u03b9.app WalkingParallelPair.one := by simpa using J WalkingParallelPair.one\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\nc : Cocone F\nm : (coequalizerCocone F).pt \u27f6 c.pt\nJ : \u2200 (j : WalkingParallelPair), NatTrans.app (coequalizerCocone F).\u03b9 j \u226b m = NatTrans.app c.\u03b9 j\n\u22a2 pushoutInl F \u226b m = NatTrans.app c.\u03b9 WalkingParallelPair.one\n[PROOFSTEP]\nsimpa using J WalkingParallelPair.one\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\nc : Cocone F\nm : (coequalizerCocone F).pt \u27f6 c.pt\nJ : \u2200 (j : WalkingParallelPair), NatTrans.app (coequalizerCocone F).\u03b9 j \u226b m = NatTrans.app c.\u03b9 j\nJ1 : pushoutInl F \u226b m = NatTrans.app c.\u03b9 WalkingParallelPair.one\n\u22a2 m =\n pushout.desc (NatTrans.app c.\u03b9 WalkingParallelPair.one) (NatTrans.app c.\u03b9 WalkingParallelPair.one)\n (_ :\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one)\n[PROOFSTEP]\napply pushout.hom_ext\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\nc : Cocone F\nm : (coequalizerCocone F).pt \u27f6 c.pt\nJ : \u2200 (j : WalkingParallelPair), NatTrans.app (coequalizerCocone F).\u03b9 j \u226b m = NatTrans.app c.\u03b9 j\nJ1 : pushoutInl F \u226b m = NatTrans.app c.\u03b9 WalkingParallelPair.one\n\u22a2 pushout.inl \u226b m =\n pushout.inl \u226b\n pushout.desc (NatTrans.app c.\u03b9 WalkingParallelPair.one) (NatTrans.app c.\u03b9 WalkingParallelPair.one)\n (_ :\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one)\n[PROOFSTEP]\nrw [colimit.\u03b9_desc]\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\nc : Cocone F\nm : (coequalizerCocone F).pt \u27f6 c.pt\nJ : \u2200 (j : WalkingParallelPair), NatTrans.app (coequalizerCocone F).\u03b9 j \u226b m = NatTrans.app c.\u03b9 j\nJ1 : pushoutInl F \u226b m = NatTrans.app c.\u03b9 WalkingParallelPair.one\n\u22a2 pushout.inl \u226b m =\n NatTrans.app\n (PushoutCocone.mk (NatTrans.app c.\u03b9 WalkingParallelPair.one) (NatTrans.app c.\u03b9 WalkingParallelPair.one)\n (_ :\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one)).\u03b9\n WalkingSpan.left\n[PROOFSTEP]\nexact J1\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\nc : Cocone F\nm : (coequalizerCocone F).pt \u27f6 c.pt\nJ : \u2200 (j : WalkingParallelPair), NatTrans.app (coequalizerCocone F).\u03b9 j \u226b m = NatTrans.app c.\u03b9 j\nJ1 : pushoutInl F \u226b m = NatTrans.app c.\u03b9 WalkingParallelPair.one\n\u22a2 pushout.inr \u226b m =\n pushout.inr \u226b\n pushout.desc (NatTrans.app c.\u03b9 WalkingParallelPair.one) (NatTrans.app c.\u03b9 WalkingParallelPair.one)\n (_ :\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one)\n[PROOFSTEP]\nrw [colimit.\u03b9_desc, \u2190 pushoutInl_eq_pushout_inr]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b9 : HasBinaryCoproducts C\ninst\u271d : HasPushouts C\nF : WalkingParallelPair \u2964 C\nc : Cocone F\nm : (coequalizerCocone F).pt \u27f6 c.pt\nJ : \u2200 (j : WalkingParallelPair), NatTrans.app (coequalizerCocone F).\u03b9 j \u226b m = NatTrans.app c.\u03b9 j\nJ1 : pushoutInl F \u226b m = NatTrans.app c.\u03b9 WalkingParallelPair.one\n\u22a2 pushoutInl F \u226b m =\n NatTrans.app\n (PushoutCocone.mk (NatTrans.app c.\u03b9 WalkingParallelPair.one) (NatTrans.app c.\u03b9 WalkingParallelPair.one)\n (_ :\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n coprod.desc (\ud835\udfd9 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one)).\u03b9\n WalkingSpan.right\n[PROOFSTEP]\nexact J1\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\nc : Cocone (K \u22d9 G)\n\u22a2 (G.mapCocone (coequalizerCocone K)).pt \u27f6 c.pt\n[PROOFSTEP]\nrefine' (PreservesPushout.iso _ _ _).inv \u226b pushout.desc _ _ _\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\nc : Cocone (K \u22d9 G)\n\u22a2 G.obj (K.obj WalkingParallelPair.one) \u27f6 c.pt\n[PROOFSTEP]\nexact c.\u03b9.app WalkingParallelPair.one\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\nc : Cocone (K \u22d9 G)\n\u22a2 G.obj (K.obj WalkingParallelPair.one) \u27f6 c.pt\n[PROOFSTEP]\nexact c.\u03b9.app WalkingParallelPair.one\n[GOAL]\ncase refine'_3\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\nc : Cocone (K \u22d9 G)\n\u22a2 G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one\n[PROOFSTEP]\napply (mapIsColimitOfPreservesOfIsColimit G _ _ (coprodIsCoprod _ _)).hom_ext\n[GOAL]\ncase refine'_3\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\nc : Cocone (K \u22d9 G)\n\u22a2 \u2200 (j : Discrete WalkingPair),\n NatTrans.app (BinaryCofan.mk (G.map coprod.inl) (G.map coprod.inr)).\u03b9 j \u226b\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n NatTrans.app (BinaryCofan.mk (G.map coprod.inl) (G.map coprod.inr)).\u03b9 j \u226b\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase refine'_3.mk.left\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\nc : Cocone (K \u22d9 G)\n\u22a2 NatTrans.app (BinaryCofan.mk (G.map coprod.inl) (G.map coprod.inr)).\u03b9 { as := WalkingPair.left } \u226b\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n NatTrans.app (BinaryCofan.mk (G.map coprod.inl) (G.map coprod.inr)).\u03b9 { as := WalkingPair.left } \u226b\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one\n[PROOFSTEP]\nsimp only [BinaryCofan.\u03b9_app_left, BinaryCofan.mk_inl, Category.assoc, \u2190 G.map_comp_assoc, coprod.inl_desc]\n[GOAL]\ncase refine'_3.mk.right\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\nc : Cocone (K \u22d9 G)\n\u22a2 NatTrans.app (BinaryCofan.mk (G.map coprod.inl) (G.map coprod.inr)).\u03b9 { as := WalkingPair.right } \u226b\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n NatTrans.app (BinaryCofan.mk (G.map coprod.inl) (G.map coprod.inr)).\u03b9 { as := WalkingPair.right } \u226b\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one\n[PROOFSTEP]\nsimp only [BinaryCofan.\u03b9_app_right, BinaryCofan.mk_inr, Category.assoc, \u2190 G.map_comp_assoc, coprod.inr_desc]\n[GOAL]\ncase refine'_3.mk.right\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\nc : Cocone (K \u22d9 G)\n\u22a2 G.map (K.map WalkingParallelPairHom.left) \u226b NatTrans.app c.\u03b9 WalkingParallelPair.one =\n G.map (K.map WalkingParallelPairHom.right) \u226b NatTrans.app c.\u03b9 WalkingParallelPair.one\n[PROOFSTEP]\nexact (c.\u03b9.naturality WalkingParallelPairHom.left).trans (c.\u03b9.naturality WalkingParallelPairHom.right).symm\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\nc : Cocone (K \u22d9 G)\nj : WalkingParallelPair\n\u22a2 NatTrans.app (G.mapCocone (coequalizerCocone K)).\u03b9 j \u226b\n (fun c =>\n (PreservesPushout.iso G (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left))\n (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right))).inv \u226b\n pushout.desc (NatTrans.app c.\u03b9 WalkingParallelPair.one) (NatTrans.app c.\u03b9 WalkingParallelPair.one)\n (_ :\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one))\n c =\n NatTrans.app c.\u03b9 j\n[PROOFSTEP]\nrcases j with (_ | _)\n[GOAL]\ncase zero\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\nc : Cocone (K \u22d9 G)\n\u22a2 NatTrans.app (G.mapCocone (coequalizerCocone K)).\u03b9 WalkingParallelPair.zero \u226b\n (fun c =>\n (PreservesPushout.iso G (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left))\n (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right))).inv \u226b\n pushout.desc (NatTrans.app c.\u03b9 WalkingParallelPair.one) (NatTrans.app c.\u03b9 WalkingParallelPair.one)\n (_ :\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one))\n c =\n NatTrans.app c.\u03b9 WalkingParallelPair.zero\n[PROOFSTEP]\nsimp only [Functor.mapCocone_\u03b9_app, Cocone.ofCofork_\u03b9, Category.id_comp, eqToHom_refl, Category.assoc, Functor.map_comp,\n Cofork.of\u03c0_\u03b9_app, pushout.inl_desc, PreservesPushout.inl_iso_inv_assoc]\n[GOAL]\ncase one\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\nc : Cocone (K \u22d9 G)\n\u22a2 NatTrans.app (G.mapCocone (coequalizerCocone K)).\u03b9 WalkingParallelPair.one \u226b\n (fun c =>\n (PreservesPushout.iso G (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left))\n (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right))).inv \u226b\n pushout.desc (NatTrans.app c.\u03b9 WalkingParallelPair.one) (NatTrans.app c.\u03b9 WalkingParallelPair.one)\n (_ :\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one))\n c =\n NatTrans.app c.\u03b9 WalkingParallelPair.one\n[PROOFSTEP]\nsimp only [Functor.mapCocone_\u03b9_app, Cocone.ofCofork_\u03b9, Category.id_comp, eqToHom_refl, Category.assoc, Functor.map_comp,\n Cofork.of\u03c0_\u03b9_app, pushout.inl_desc, PreservesPushout.inl_iso_inv_assoc]\n[GOAL]\ncase zero\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\nc : Cocone (K \u22d9 G)\n\u22a2 G.map (K.map WalkingParallelPairHom.left) \u226b NatTrans.app c.\u03b9 WalkingParallelPair.one =\n NatTrans.app c.\u03b9 WalkingParallelPair.zero\n[PROOFSTEP]\nexact (c.\u03b9.naturality WalkingParallelPairHom.left).trans (Category.comp_id _)\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\ns : Cocone (K \u22d9 G)\nm : (G.mapCocone (coequalizerCocone K)).pt \u27f6 s.pt\nh : \u2200 (j : WalkingParallelPair), NatTrans.app (G.mapCocone (coequalizerCocone K)).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n\u22a2 m =\n (fun c =>\n (PreservesPushout.iso G (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left))\n (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right))).inv \u226b\n pushout.desc (NatTrans.app c.\u03b9 WalkingParallelPair.one) (NatTrans.app c.\u03b9 WalkingParallelPair.one)\n (_ :\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one =\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app c.\u03b9 WalkingParallelPair.one))\n s\n[PROOFSTEP]\nrw [Iso.eq_inv_comp]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\ns : Cocone (K \u22d9 G)\nm : (G.mapCocone (coequalizerCocone K)).pt \u27f6 s.pt\nh : \u2200 (j : WalkingParallelPair), NatTrans.app (G.mapCocone (coequalizerCocone K)).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n\u22a2 (PreservesPushout.iso G (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left))\n (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right))).hom \u226b\n m =\n pushout.desc (NatTrans.app s.\u03b9 WalkingParallelPair.one) (NatTrans.app s.\u03b9 WalkingParallelPair.one)\n (_ :\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left)) \u226b\n NatTrans.app s.\u03b9 WalkingParallelPair.one =\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app s.\u03b9 WalkingParallelPair.one)\n[PROOFSTEP]\nhave := h WalkingParallelPair.one\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\ns : Cocone (K \u22d9 G)\nm : (G.mapCocone (coequalizerCocone K)).pt \u27f6 s.pt\nh : \u2200 (j : WalkingParallelPair), NatTrans.app (G.mapCocone (coequalizerCocone K)).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nthis :\n NatTrans.app (G.mapCocone (coequalizerCocone K)).\u03b9 WalkingParallelPair.one \u226b m =\n NatTrans.app s.\u03b9 WalkingParallelPair.one\n\u22a2 (PreservesPushout.iso G (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left))\n (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right))).hom \u226b\n m =\n pushout.desc (NatTrans.app s.\u03b9 WalkingParallelPair.one) (NatTrans.app s.\u03b9 WalkingParallelPair.one)\n (_ :\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left)) \u226b\n NatTrans.app s.\u03b9 WalkingParallelPair.one =\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app s.\u03b9 WalkingParallelPair.one)\n[PROOFSTEP]\ndsimp [coequalizerCocone] at this \n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\ns : Cocone (K \u22d9 G)\nm : (G.mapCocone (coequalizerCocone K)).pt \u27f6 s.pt\nh : \u2200 (j : WalkingParallelPair), NatTrans.app (G.mapCocone (coequalizerCocone K)).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nthis : G.map (\ud835\udfd9 (K.obj WalkingParallelPair.one) \u226b pushoutInl K) \u226b m = NatTrans.app s.\u03b9 WalkingParallelPair.one\n\u22a2 (PreservesPushout.iso G (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left))\n (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right))).hom \u226b\n m =\n pushout.desc (NatTrans.app s.\u03b9 WalkingParallelPair.one) (NatTrans.app s.\u03b9 WalkingParallelPair.one)\n (_ :\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left)) \u226b\n NatTrans.app s.\u03b9 WalkingParallelPair.one =\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app s.\u03b9 WalkingParallelPair.one)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\ns : Cocone (K \u22d9 G)\nm : (G.mapCocone (coequalizerCocone K)).pt \u27f6 s.pt\nh : \u2200 (j : WalkingParallelPair), NatTrans.app (G.mapCocone (coequalizerCocone K)).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nthis : G.map (\ud835\udfd9 (K.obj WalkingParallelPair.one) \u226b pushoutInl K) \u226b m = NatTrans.app s.\u03b9 WalkingParallelPair.one\n\u22a2 pushout.inl \u226b\n (PreservesPushout.iso G (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left))\n (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right))).hom \u226b\n m =\n pushout.inl \u226b\n pushout.desc (NatTrans.app s.\u03b9 WalkingParallelPair.one) (NatTrans.app s.\u03b9 WalkingParallelPair.one)\n (_ :\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left)) \u226b\n NatTrans.app s.\u03b9 WalkingParallelPair.one =\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app s.\u03b9 WalkingParallelPair.one)\n[PROOFSTEP]\nsimp only [PreservesPushout.inl_iso_hom_assoc, Category.id_comp, pushout.inl_desc, pushout.inr_desc,\n PreservesPushout.inr_iso_hom_assoc, \u2190 pushoutInl_eq_pushout_inr, \u2190 this]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\nG : C \u2964 D\ninst\u271d\u00b3 : HasBinaryCoproducts C\ninst\u271d\u00b2 : HasPushouts C\ninst\u271d\u00b9 : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst\u271d : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair \u2964 C\ns : Cocone (K \u22d9 G)\nm : (G.mapCocone (coequalizerCocone K)).pt \u27f6 s.pt\nh : \u2200 (j : WalkingParallelPair), NatTrans.app (G.mapCocone (coequalizerCocone K)).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nthis : G.map (\ud835\udfd9 (K.obj WalkingParallelPair.one) \u226b pushoutInl K) \u226b m = NatTrans.app s.\u03b9 WalkingParallelPair.one\n\u22a2 pushout.inr \u226b\n (PreservesPushout.iso G (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left))\n (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right))).hom \u226b\n m =\n pushout.inr \u226b\n pushout.desc (NatTrans.app s.\u03b9 WalkingParallelPair.one) (NatTrans.app s.\u03b9 WalkingParallelPair.one)\n (_ :\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left)) \u226b\n NatTrans.app s.\u03b9 WalkingParallelPair.one =\n G.map (coprod.desc (\ud835\udfd9 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right)) \u226b\n NatTrans.app s.\u03b9 WalkingParallelPair.one)\n[PROOFSTEP]\nsimp only [PreservesPushout.inl_iso_hom_assoc, Category.id_comp, pushout.inl_desc, pushout.inr_desc,\n PreservesPushout.inr_iso_hom_assoc, \u2190 pushoutInl_eq_pushout_inr, \u2190 this]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Constructions.Equalizers", "llama_tokens": 26575, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154240079185319, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.4240079142985898}} {"text": "[GOAL]\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\n\u22a2 #\u2191t \u2264 a\n[PROOFSTEP]\nrcases lt_or_le a \u2135\u2080 with ha | ha\n[GOAL]\ncase inl\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : a < \u2135\u2080\n\u22a2 #\u2191t \u2264 a\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 : \u2203 (n : \u2115), a = n := lt_aleph0.1 ha\n[GOAL]\ncase inl.intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\n\u22a2 #\u2191t \u2264 \u2191n\n[PROOFSTEP]\napply mk_le_iff_forall_finset_subset_card_le.2 (fun s hs \u21a6 ?_)\n[GOAL]\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\n\u22a2 Finset.card s \u2264 n\n[PROOFSTEP]\nhave A : \u2200 x \u2208 s, \u2200\u1da0 i in l, x \u2208 f i := fun x hx \u21a6 ht x (hs hx)\n[GOAL]\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\n\u22a2 Finset.card s \u2264 n\n[PROOFSTEP]\nhave B : \u2200\u1da0 i in l, \u2200 x \u2208 s, x \u2208 f i := (s.eventually_all).2 A\n[GOAL]\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\n\u22a2 Finset.card s \u2264 n\n[PROOFSTEP]\nrcases B.exists with \u27e8i, hi\u27e9\n[GOAL]\ncase intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\n\u22a2 Finset.card s \u2264 n\n[PROOFSTEP]\nhave : \u2200 i, Fintype (f i) := fun i \u21a6 (lt_aleph0_iff_fintype.1 ((h'f i).trans_lt ha)).some\n[GOAL]\ncase intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\n\u22a2 Finset.card s \u2264 n\n[PROOFSTEP]\nlet u : Finset \u03b1 := (f i).toFinset\n[GOAL]\ncase intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := toFinset (f i)\n\u22a2 Finset.card s \u2264 n\n[PROOFSTEP]\nhave I1 : s.card \u2264 u.card :=\n by\n have : s \u2286 u := fun x hx \u21a6 by simpa only [Set.mem_toFinset] using hi x hx\n exact Finset.card_le_of_subset this\n[GOAL]\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := toFinset (f i)\n\u22a2 Finset.card s \u2264 Finset.card u\n[PROOFSTEP]\nhave : s \u2286 u := fun x hx \u21a6 by simpa only [Set.mem_toFinset] using hi x hx\n[GOAL]\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := toFinset (f i)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 u\n[PROOFSTEP]\nsimpa only [Set.mem_toFinset] using hi x hx\n[GOAL]\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\nthis\u271d : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := toFinset (f i)\nthis : s \u2286 u\n\u22a2 Finset.card s \u2264 Finset.card u\n[PROOFSTEP]\nexact Finset.card_le_of_subset this\n[GOAL]\ncase intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := toFinset (f i)\nI1 : Finset.card s \u2264 Finset.card u\n\u22a2 Finset.card s \u2264 n\n[PROOFSTEP]\nhave I2 : (u.card : Cardinal) \u2264 n := by convert h'f i; simp only [Set.toFinset_card, mk_fintype]\n[GOAL]\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := toFinset (f i)\nI1 : Finset.card s \u2264 Finset.card u\n\u22a2 \u2191(Finset.card u) \u2264 \u2191n\n[PROOFSTEP]\nconvert h'f i\n[GOAL]\ncase h.e'_3\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := toFinset (f i)\nI1 : Finset.card s \u2264 Finset.card u\n\u22a2 \u2191(Finset.card u) = #\u2191(f i)\n[PROOFSTEP]\nsimp only [Set.toFinset_card, mk_fintype]\n[GOAL]\ncase intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := toFinset (f i)\nI1 : Finset.card s \u2264 Finset.card u\nI2 : \u2191(Finset.card u) \u2264 \u2191n\n\u22a2 Finset.card s \u2264 n\n[PROOFSTEP]\nexact\n I1.trans\n (Nat.cast_le.1 I2)\n -- case `a` infinite:\n[GOAL]\ncase inr\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\n\u22a2 #\u2191t \u2264 a\n[PROOFSTEP]\nhave : t \u2286 \u22c3 i, f i := by\n intro x hx\n obtain \u27e8i, hi\u27e9 : \u2203 i, x \u2208 f i := (ht x hx).exists\n exact mem_iUnion_of_mem i hi\n[GOAL]\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\n\u22a2 t \u2286 \u22c3 (i : \u03b9), f i\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 \u22c3 (i : \u03b9), f i\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 : \u2203 i, x \u2208 f i := (ht x hx).exists\n[GOAL]\ncase intro\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\nx : \u03b1\nhx : x \u2208 t\ni : \u03b9\nhi : x \u2208 f i\n\u22a2 x \u2208 \u22c3 (i : \u03b9), f i\n[PROOFSTEP]\nexact mem_iUnion_of_mem i hi\n[GOAL]\ncase inr\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\nthis : t \u2286 \u22c3 (i : \u03b9), f i\n\u22a2 #\u2191t \u2264 a\n[PROOFSTEP]\ncalc\n #t \u2264 #(\u22c3 i, f i) := mk_le_mk_of_subset this\n _ \u2264 sum (fun i \u21a6 #(f i)) := mk_iUnion_le_sum_mk\n _ \u2264 sum (fun _ \u21a6 a) := (sum_le_sum _ _ h'f)\n _ = #\u03b9 * a := by simp\n _ \u2264 \u2135\u2080 * a := (mul_le_mul_right' mk_le_aleph0 a)\n _ = a := aleph0_mul_eq ha\n[GOAL]\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\nthis : t \u2286 \u22c3 (i : \u03b9), f i\n\u22a2 (sum fun x => a) = #\u03b9 * a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\n\u22a2 #\u2191t \u2264 a\n[PROOFSTEP]\nlet g : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (ULift.down \u207b\u00b9' \u00b7) \u2218 f \u2218 ULift.down\n[GOAL]\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift \u03b9 \u2192 Set (ULift \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\n\u22a2 #\u2191t \u2264 a\n[PROOFSTEP]\nsuffices #(ULift.down.{v} \u207b\u00b9' t) \u2264 Cardinal.lift.{v, u} a by simpa\n[GOAL]\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift \u03b9 \u2192 Set (ULift \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nthis : #\u2191(ULift.down \u207b\u00b9' t) \u2264 lift a\n\u22a2 #\u2191t \u2264 a\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift \u03b9 \u2192 Set (ULift \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\n\u22a2 #\u2191(ULift.down \u207b\u00b9' t) \u2264 lift a\n[PROOFSTEP]\nlet l' : Filter (ULift.{u} \u03b9) := Filter.map ULift.up l\n[GOAL]\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift \u03b9 \u2192 Set (ULift \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift \u03b9) := Filter.map ULift.up l\n\u22a2 #\u2191(ULift.down \u207b\u00b9' t) \u2264 lift a\n[PROOFSTEP]\nhave : NeBot l' := map_neBot\n[GOAL]\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift \u03b9 \u2192 Set (ULift \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift \u03b9) := Filter.map ULift.up l\nthis : NeBot l'\n\u22a2 #\u2191(ULift.down \u207b\u00b9' t) \u2264 lift a\n[PROOFSTEP]\napply mk_subtype_le_of_countable_eventually_mem_aux (\u03b9 := ULift.{u} \u03b9) (l := l') (f := g)\n[GOAL]\ncase ht\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift \u03b9 \u2192 Set (ULift \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift \u03b9) := Filter.map ULift.up l\nthis : NeBot l'\n\u22a2 \u2200 (x : ULift \u03b1), x \u2208 ULift.down \u207b\u00b9' t \u2192 \u2200\u1da0 (i : ULift \u03b9) in l', x \u2208 g i\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase ht\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift \u03b9 \u2192 Set (ULift \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift \u03b9) := Filter.map ULift.up l\nthis : NeBot l'\nx : ULift \u03b1\nhx : x \u2208 ULift.down \u207b\u00b9' t\n\u22a2 \u2200\u1da0 (i : ULift \u03b9) in l', x \u2208 g i\n[PROOFSTEP]\nsimpa only [Function.comp_apply, mem_preimage, eventually_map] using ht _ hx\n[GOAL]\ncase h'f\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift \u03b9 \u2192 Set (ULift \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift \u03b9) := Filter.map ULift.up l\nthis : NeBot l'\n\u22a2 \u2200 (i : ULift \u03b9), #\u2191(g i) \u2264 lift a\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h'f\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nt : Set \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift \u03b9 \u2192 Set (ULift \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift \u03b9) := Filter.map ULift.up l\nthis : NeBot l'\ni : ULift \u03b9\n\u22a2 #\u2191(g i) \u2264 lift a\n[PROOFSTEP]\nsimpa using h'f i.down\n[GOAL]\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\n\u22a2 #\u03b1 \u2264 a\n[PROOFSTEP]\nrw [\u2190 mk_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\n\u22a2 #\u2191Set.univ \u2264 a\n[PROOFSTEP]\nexact mk_subtype_le_of_countable_eventually_mem (l := l) (fun x _ \u21a6 ht x) h'f\n[GOAL]\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) = a\n\u22a2 #\u03b1 = a\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) = a\n\u22a2 #\u03b1 \u2264 a\n[PROOFSTEP]\napply mk_le_of_countable_eventually_mem ht (fun i \u21a6 (h'f i).le)\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) = a\n\u22a2 a \u2264 #\u03b1\n[PROOFSTEP]\nobtain \u27e8i\u27e9 : Nonempty \u03b9 := nonempty_of_neBot l\n[GOAL]\ncase a.intro\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) = a\ni : \u03b9\n\u22a2 a \u2264 #\u03b1\n[PROOFSTEP]\nrw [\u2190 (h'f i)]\n[GOAL]\ncase a.intro\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : NeBot l\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) = a\ni : \u03b9\n\u22a2 #\u2191(f i) \u2264 #\u03b1\n[PROOFSTEP]\nexact mk_set_le (f i)\n", "meta": {"mathlib_filename": "Mathlib.SetTheory.Cardinal.CountableCover", "llama_tokens": 8950, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147276, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.42399724710132797}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\ny x d : \u03b1\nh : x \u2260 y\n\u22a2 nextOr (y :: xs) x d = nextOr xs x d\n[PROOFSTEP]\ncases' xs with z zs\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny x d : \u03b1\nh : x \u2260 y\n\u22a2 nextOr [y] x d = nextOr [] x d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny x d : \u03b1\nh : x \u2260 y\nz : \u03b1\nzs : List \u03b1\n\u22a2 nextOr (y :: z :: zs) x d = nextOr (z :: zs) x d\n[PROOFSTEP]\nexact if_neg h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d d' : \u03b1\nx_mem : x \u2208 xs\nx_ne : x \u2260 getLast xs (_ : xs \u2260 [])\n\u22a2 nextOr xs x d = nextOr xs x d'\n[PROOFSTEP]\ninduction' xs with y ys IH\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d d' : \u03b1\nx_mem\u271d : x \u2208 xs\nx_ne\u271d : x \u2260 getLast xs (_ : xs \u2260 [])\nx_mem : x \u2208 []\nx_ne : x \u2260 getLast [] (_ : [] \u2260 [])\n\u22a2 nextOr [] x d = nextOr [] x d'\n[PROOFSTEP]\ncases x_mem\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d d' : \u03b1\nx_mem\u271d : x \u2208 xs\nx_ne\u271d : x \u2260 getLast xs (_ : xs \u2260 [])\ny : \u03b1\nys : List \u03b1\nIH : \u2200 (x_mem : x \u2208 ys), x \u2260 getLast ys (_ : ys \u2260 []) \u2192 nextOr ys x d = nextOr ys x d'\nx_mem : x \u2208 y :: ys\nx_ne : x \u2260 getLast (y :: ys) (_ : y :: ys \u2260 [])\n\u22a2 nextOr (y :: ys) x d = nextOr (y :: ys) x d'\n[PROOFSTEP]\ncases' ys with z zs\n[GOAL]\ncase cons.nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d d' : \u03b1\nx_mem\u271d : x \u2208 xs\nx_ne\u271d : x \u2260 getLast xs (_ : xs \u2260 [])\ny : \u03b1\nIH : \u2200 (x_mem : x \u2208 []), x \u2260 getLast [] (_ : [] \u2260 []) \u2192 nextOr [] x d = nextOr [] x d'\nx_mem : x \u2208 [y]\nx_ne : x \u2260 getLast [y] (_ : [y] \u2260 [])\n\u22a2 nextOr [y] x d = nextOr [y] x d'\n[PROOFSTEP]\nsimp at x_mem x_ne \n[GOAL]\ncase cons.nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d d' : \u03b1\nx_mem\u271d : x \u2208 xs\nx_ne\u271d : x \u2260 getLast xs (_ : xs \u2260 [])\ny : \u03b1\nIH : \u2200 (x_mem : x \u2208 []), x \u2260 getLast [] (_ : [] \u2260 []) \u2192 nextOr [] x d = nextOr [] x d'\nx_ne : \u00acx = y\nx_mem : x = y\n\u22a2 nextOr [y] x d = nextOr [y] x d'\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase cons.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d d' : \u03b1\nx_mem\u271d : x \u2208 xs\nx_ne\u271d : x \u2260 getLast xs (_ : xs \u2260 [])\ny z : \u03b1\nzs : List \u03b1\nIH : \u2200 (x_mem : x \u2208 z :: zs), x \u2260 getLast (z :: zs) (_ : z :: zs \u2260 []) \u2192 nextOr (z :: zs) x d = nextOr (z :: zs) x d'\nx_mem : x \u2208 y :: z :: zs\nx_ne : x \u2260 getLast (y :: z :: zs) (_ : y :: z :: zs \u2260 [])\n\u22a2 nextOr (y :: z :: zs) x d = nextOr (y :: z :: zs) x d'\n[PROOFSTEP]\nby_cases h : x = y\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d d' : \u03b1\nx_mem\u271d : x \u2208 xs\nx_ne\u271d : x \u2260 getLast xs (_ : xs \u2260 [])\ny z : \u03b1\nzs : List \u03b1\nIH : \u2200 (x_mem : x \u2208 z :: zs), x \u2260 getLast (z :: zs) (_ : z :: zs \u2260 []) \u2192 nextOr (z :: zs) x d = nextOr (z :: zs) x d'\nx_mem : x \u2208 y :: z :: zs\nx_ne : x \u2260 getLast (y :: z :: zs) (_ : y :: z :: zs \u2260 [])\nh : x = y\n\u22a2 nextOr (y :: z :: zs) x d = nextOr (y :: z :: zs) x d'\n[PROOFSTEP]\nrw [h, nextOr_self_cons_cons, nextOr_self_cons_cons]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d d' : \u03b1\nx_mem\u271d : x \u2208 xs\nx_ne\u271d : x \u2260 getLast xs (_ : xs \u2260 [])\ny z : \u03b1\nzs : List \u03b1\nIH : \u2200 (x_mem : x \u2208 z :: zs), x \u2260 getLast (z :: zs) (_ : z :: zs \u2260 []) \u2192 nextOr (z :: zs) x d = nextOr (z :: zs) x d'\nx_mem : x \u2208 y :: z :: zs\nx_ne : x \u2260 getLast (y :: z :: zs) (_ : y :: z :: zs \u2260 [])\nh : \u00acx = y\n\u22a2 nextOr (y :: z :: zs) x d = nextOr (y :: z :: zs) x d'\n[PROOFSTEP]\nrw [nextOr, nextOr, IH]\n[GOAL]\ncase neg.x_mem\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d d' : \u03b1\nx_mem\u271d : x \u2208 xs\nx_ne\u271d : x \u2260 getLast xs (_ : xs \u2260 [])\ny z : \u03b1\nzs : List \u03b1\nIH : \u2200 (x_mem : x \u2208 z :: zs), x \u2260 getLast (z :: zs) (_ : z :: zs \u2260 []) \u2192 nextOr (z :: zs) x d = nextOr (z :: zs) x d'\nx_mem : x \u2208 y :: z :: zs\nx_ne : x \u2260 getLast (y :: z :: zs) (_ : y :: z :: zs \u2260 [])\nh : \u00acx = y\n\u22a2 x \u2208 z :: zs\n[PROOFSTEP]\nsimpa [h] using x_mem\n[GOAL]\ncase neg.x_ne\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d d' : \u03b1\nx_mem\u271d : x \u2208 xs\nx_ne\u271d : x \u2260 getLast xs (_ : xs \u2260 [])\ny z : \u03b1\nzs : List \u03b1\nIH : \u2200 (x_mem : x \u2208 z :: zs), x \u2260 getLast (z :: zs) (_ : z :: zs \u2260 []) \u2192 nextOr (z :: zs) x d = nextOr (z :: zs) x d'\nx_mem : x \u2208 y :: z :: zs\nx_ne : x \u2260 getLast (y :: z :: zs) (_ : y :: z :: zs \u2260 [])\nh : \u00acx = y\n\u22a2 x \u2260 getLast (z :: zs) (_ : z :: zs \u2260 [])\n[PROOFSTEP]\nsimpa using x_ne\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh : nextOr xs x d \u2260 d\n\u22a2 x \u2208 xs\n[PROOFSTEP]\ninduction' xs with y ys IH\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\nh : nextOr [] x d \u2260 d\n\u22a2 x \u2208 []\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny : \u03b1\nys : List \u03b1\nIH : nextOr ys x d \u2260 d \u2192 x \u2208 ys\nh : nextOr (y :: ys) x d \u2260 d\n\u22a2 x \u2208 y :: ys\n[PROOFSTEP]\ncases' ys with z zs\n[GOAL]\ncase cons.nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny : \u03b1\nIH : nextOr [] x d \u2260 d \u2192 x \u2208 []\nh : nextOr [y] x d \u2260 d\n\u22a2 x \u2208 [y]\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase cons.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny z : \u03b1\nzs : List \u03b1\nIH : nextOr (z :: zs) x d \u2260 d \u2192 x \u2208 z :: zs\nh : nextOr (y :: z :: zs) x d \u2260 d\n\u22a2 x \u2208 y :: z :: zs\n[PROOFSTEP]\nby_cases hx : x = y\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny z : \u03b1\nzs : List \u03b1\nIH : nextOr (z :: zs) x d \u2260 d \u2192 x \u2208 z :: zs\nh : nextOr (y :: z :: zs) x d \u2260 d\nhx : x = y\n\u22a2 x \u2208 y :: z :: zs\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny z : \u03b1\nzs : List \u03b1\nIH : nextOr (z :: zs) x d \u2260 d \u2192 x \u2208 z :: zs\nh : nextOr (y :: z :: zs) x d \u2260 d\nhx : \u00acx = y\n\u22a2 x \u2208 y :: z :: zs\n[PROOFSTEP]\nrw [nextOr_cons_of_ne _ _ _ _ hx] at h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny z : \u03b1\nzs : List \u03b1\nIH : nextOr (z :: zs) x d \u2260 d \u2192 x \u2208 z :: zs\nh : nextOr (z :: zs) x d \u2260 d\nhx : \u00acx = y\n\u22a2 x \u2208 y :: z :: zs\n[PROOFSTEP]\nsimpa [hx] using IH h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh : \u00acx \u2208 xs\n\u22a2 nextOr (xs ++ [x]) x d = d\n[PROOFSTEP]\ninduction' xs with z zs IH\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : \u00acx \u2208 xs\nh : \u00acx \u2208 []\n\u22a2 nextOr ([] ++ [x]) x d = d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : \u00acx \u2208 xs\nz : \u03b1\nzs : List \u03b1\nIH : \u00acx \u2208 zs \u2192 nextOr (zs ++ [x]) x d = d\nh : \u00acx \u2208 z :: zs\n\u22a2 nextOr (z :: zs ++ [x]) x d = d\n[PROOFSTEP]\nobtain \u27e8hz, hzs\u27e9 := not_or.mp (mt mem_cons.2 h)\n[GOAL]\ncase cons.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : \u00acx \u2208 xs\nz : \u03b1\nzs : List \u03b1\nIH : \u00acx \u2208 zs \u2192 nextOr (zs ++ [x]) x d = d\nh : \u00acx \u2208 z :: zs\nhz : \u00acx = z\nhzs : \u00acx \u2208 zs\n\u22a2 nextOr (z :: zs ++ [x]) x d = d\n[PROOFSTEP]\nrw [cons_append, nextOr_cons_of_ne _ _ _ _ hz, IH hzs]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nhd : d \u2208 xs\n\u22a2 nextOr xs x d \u2208 xs\n[PROOFSTEP]\nrevert hd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\n\u22a2 d \u2208 xs \u2192 nextOr xs x d \u2208 xs\n[PROOFSTEP]\nsuffices \u2200 (xs' : List \u03b1) (_ : \u2200 x \u2208 xs, x \u2208 xs') (_ : d \u2208 xs'), nextOr xs x d \u2208 xs' by exact this xs fun _ => id\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nthis : \u2200 (xs' : List \u03b1), (\u2200 (x : \u03b1), x \u2208 xs \u2192 x \u2208 xs') \u2192 d \u2208 xs' \u2192 nextOr xs x d \u2208 xs'\n\u22a2 d \u2208 xs \u2192 nextOr xs x d \u2208 xs\n[PROOFSTEP]\nexact this xs fun _ => id\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\n\u22a2 \u2200 (xs' : List \u03b1), (\u2200 (x : \u03b1), x \u2208 xs \u2192 x \u2208 xs') \u2192 d \u2208 xs' \u2192 nextOr xs x d \u2208 xs'\n[PROOFSTEP]\nintro xs' hxs' hd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nxs' : List \u03b1\nhxs' : \u2200 (x : \u03b1), x \u2208 xs \u2192 x \u2208 xs'\nhd : d \u2208 xs'\n\u22a2 nextOr xs x d \u2208 xs'\n[PROOFSTEP]\ninduction' xs with y ys ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nxs' : List \u03b1\nhxs'\u271d : \u2200 (x : \u03b1), x \u2208 xs \u2192 x \u2208 xs'\nhd : d \u2208 xs'\nhxs' : \u2200 (x : \u03b1), x \u2208 [] \u2192 x \u2208 xs'\n\u22a2 nextOr [] x d \u2208 xs'\n[PROOFSTEP]\nexact hd\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nxs' : List \u03b1\nhxs'\u271d : \u2200 (x : \u03b1), x \u2208 xs \u2192 x \u2208 xs'\nhd : d \u2208 xs'\ny : \u03b1\nys : List \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 ys \u2192 x \u2208 xs') \u2192 nextOr ys x d \u2208 xs'\nhxs' : \u2200 (x : \u03b1), x \u2208 y :: ys \u2192 x \u2208 xs'\n\u22a2 nextOr (y :: ys) x d \u2208 xs'\n[PROOFSTEP]\ncases' ys with z zs\n[GOAL]\ncase cons.nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nxs' : List \u03b1\nhxs'\u271d : \u2200 (x : \u03b1), x \u2208 xs \u2192 x \u2208 xs'\nhd : d \u2208 xs'\ny : \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 [] \u2192 x \u2208 xs') \u2192 nextOr [] x d \u2208 xs'\nhxs' : \u2200 (x : \u03b1), x \u2208 [y] \u2192 x \u2208 xs'\n\u22a2 nextOr [y] x d \u2208 xs'\n[PROOFSTEP]\nexact hd\n[GOAL]\ncase cons.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nxs' : List \u03b1\nhxs'\u271d : \u2200 (x : \u03b1), x \u2208 xs \u2192 x \u2208 xs'\nhd : d \u2208 xs'\ny z : \u03b1\nzs : List \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 z :: zs \u2192 x \u2208 xs') \u2192 nextOr (z :: zs) x d \u2208 xs'\nhxs' : \u2200 (x : \u03b1), x \u2208 y :: z :: zs \u2192 x \u2208 xs'\n\u22a2 nextOr (y :: z :: zs) x d \u2208 xs'\n[PROOFSTEP]\nrw [nextOr]\n[GOAL]\ncase cons.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nxs' : List \u03b1\nhxs'\u271d : \u2200 (x : \u03b1), x \u2208 xs \u2192 x \u2208 xs'\nhd : d \u2208 xs'\ny z : \u03b1\nzs : List \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 z :: zs \u2192 x \u2208 xs') \u2192 nextOr (z :: zs) x d \u2208 xs'\nhxs' : \u2200 (x : \u03b1), x \u2208 y :: z :: zs \u2192 x \u2208 xs'\n\u22a2 (if x = y then z else nextOr (z :: zs) x d) \u2208 xs'\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nxs' : List \u03b1\nhxs'\u271d : \u2200 (x : \u03b1), x \u2208 xs \u2192 x \u2208 xs'\nhd : d \u2208 xs'\ny z : \u03b1\nzs : List \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 z :: zs \u2192 x \u2208 xs') \u2192 nextOr (z :: zs) x d \u2208 xs'\nhxs' : \u2200 (x : \u03b1), x \u2208 y :: z :: zs \u2192 x \u2208 xs'\nh : x = y\n\u22a2 z \u2208 xs'\n[PROOFSTEP]\nexact hxs' _ (mem_cons_of_mem _ (mem_cons_self _ _))\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nxs' : List \u03b1\nhxs'\u271d : \u2200 (x : \u03b1), x \u2208 xs \u2192 x \u2208 xs'\nhd : d \u2208 xs'\ny z : \u03b1\nzs : List \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 z :: zs \u2192 x \u2208 xs') \u2192 nextOr (z :: zs) x d \u2208 xs'\nhxs' : \u2200 (x : \u03b1), x \u2208 y :: z :: zs \u2192 x \u2208 xs'\nh : \u00acx = y\n\u22a2 nextOr (z :: zs) x d \u2208 xs'\n[PROOFSTEP]\nexact ih fun _ h => hxs' _ (mem_cons_of_mem _ h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx\u271d : \u03b1\nh : x\u271d \u2208 []\n\u22a2 \u03b1\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny z : \u03b1\nxs : List \u03b1\nx : \u03b1\nh : x \u2208 y :: z :: xs\nhx : \u00acx = y\n\u22a2 x \u2208 z :: xs\n[PROOFSTEP]\nsimpa [hx] using h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx y z : \u03b1\nh : x \u2208 y :: z :: l\nhx : x = y\n\u22a2 next (y :: z :: l) x h = z\n[PROOFSTEP]\nrw [next, nextOr, if_pos hx]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh\u271d : x \u2208 l\ny : \u03b1\nh : x \u2208 y :: l\nhy : x \u2260 y\nhx : x \u2260 getLast (y :: l) (_ : y :: l \u2260 [])\n\u22a2 x \u2208 l\n[PROOFSTEP]\nsimpa [hy] using h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh\u271d : x \u2208 l\ny : \u03b1\nh : x \u2208 y :: l\nhy : x \u2260 y\nhx : x \u2260 getLast (y :: l) (_ : y :: l \u2260 [])\n\u22a2 next (y :: l) x h = next l x (_ : x \u2208 l)\n[PROOFSTEP]\nrw [next, next, nextOr_cons_of_ne _ _ _ _ hy, nextOr_eq_nextOr_of_mem_of_ne]\n[GOAL]\ncase x_mem\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh\u271d : x \u2208 l\ny : \u03b1\nh : x \u2208 y :: l\nhy : x \u2260 y\nhx : x \u2260 getLast (y :: l) (_ : y :: l \u2260 [])\n\u22a2 x \u2208 l\n[PROOFSTEP]\nrwa [getLast_cons] at hx \n[GOAL]\ncase x_mem\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh\u271d : x \u2208 l\ny : \u03b1\nh : x \u2208 y :: l\nhy : x \u2260 y\nhx : x \u2260 getLast (y :: l) (_ : y :: l \u2260 [])\n\u22a2 l \u2260 []\n[PROOFSTEP]\nexact ne_nil_of_mem (by assumption)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh\u271d : x \u2208 l\ny : \u03b1\nh : x \u2208 y :: l\nhy : x \u2260 y\nhx : x \u2260 getLast (y :: l) (_ : y :: l \u2260 [])\n\u22a2 ?m.65756 \u2208 l\n[PROOFSTEP]\nassumption\n[GOAL]\ncase x_ne\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh\u271d : x \u2208 l\ny : \u03b1\nh : x \u2208 y :: l\nhy : x \u2260 y\nhx : x \u2260 getLast (y :: l) (_ : y :: l \u2260 [])\n\u22a2 x \u2260 getLast l (_ : l \u2260 [])\n[PROOFSTEP]\nrwa [getLast_cons] at hx \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx y : \u03b1\nhy : x \u2260 y\nhx : \u00acx \u2208 l\nh : optParam (x \u2208 y :: l ++ [x]) (_ : x \u2208 y :: l ++ [x])\n\u22a2 next (y :: l ++ [x]) x h = y\n[PROOFSTEP]\nrw [next, nextOr_concat]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx y : \u03b1\nhy : x \u2260 y\nhx : \u00acx \u2208 l\nh : optParam (x \u2208 y :: l ++ [x]) (_ : x \u2208 y :: l ++ [x])\n\u22a2 get (y :: l ++ [x]) { val := 0, isLt := (_ : 0 < length (y :: l ++ [x])) } = y\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx y : \u03b1\nhy : x \u2260 y\nhx : \u00acx \u2208 l\nh : optParam (x \u2208 y :: l ++ [x]) (_ : x \u2208 y :: l ++ [x])\n\u22a2 \u00acx \u2208 y :: l\n[PROOFSTEP]\nsimp [hy, hx]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh\u271d : x \u2208 l\ny : \u03b1\nh : x \u2208 y :: l\nhy : x \u2260 y\nhx : x = getLast (y :: l) (_ : y :: l \u2260 [])\nhl : Nodup l\n\u22a2 next (y :: l) x h = y\n[PROOFSTEP]\nrw [next, get, \u2190 dropLast_append_getLast (cons_ne_nil y l), hx, nextOr_concat]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh\u271d : x \u2208 l\ny : \u03b1\nh : x \u2208 y :: l\nhy : x \u2260 y\nhx : x = getLast (y :: l) (_ : y :: l \u2260 [])\nhl : Nodup l\n\u22a2 \u00acgetLast (y :: l) (_ : y :: l \u2260 []) \u2208 dropLast (y :: l)\n[PROOFSTEP]\nsubst hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\ny : \u03b1\nhl : Nodup l\nh\u271d : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 l\nh : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 y :: l\nhy : getLast (y :: l) (_ : y :: l \u2260 []) \u2260 y\n\u22a2 \u00acgetLast (y :: l) (_ : y :: l \u2260 []) \u2208 dropLast (y :: l)\n[PROOFSTEP]\nintro H\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\ny : \u03b1\nhl : Nodup l\nh\u271d : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 l\nh : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 y :: l\nhy : getLast (y :: l) (_ : y :: l \u2260 []) \u2260 y\nH : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 dropLast (y :: l)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8\u27e8_ | k, hk\u27e9, hk'\u27e9 := get_of_mem H\n[GOAL]\ncase h.intro.mk.zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\ny : \u03b1\nhl : Nodup l\nh\u271d : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 l\nh : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 y :: l\nhy : getLast (y :: l) (_ : y :: l \u2260 []) \u2260 y\nH : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 dropLast (y :: l)\nhk : Nat.zero < length (dropLast (y :: l))\nhk' : get (dropLast (y :: l)) { val := Nat.zero, isLt := hk } = getLast (y :: l) (_ : y :: l \u2260 [])\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 Option.some_inj] at hk' \n[GOAL]\ncase h.intro.mk.zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\ny : \u03b1\nhl : Nodup l\nh\u271d : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 l\nh : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 y :: l\nhy : getLast (y :: l) (_ : y :: l \u2260 []) \u2260 y\nH : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 dropLast (y :: l)\nhk : Nat.zero < length (dropLast (y :: l))\nhk' : some (get (dropLast (y :: l)) { val := Nat.zero, isLt := hk }) = some (getLast (y :: l) (_ : y :: l \u2260 []))\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 get?_eq_get, dropLast_eq_take, get?_take, get?_zero, head?_cons, Option.some_inj] at hk' \n[GOAL]\ncase h.intro.mk.zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\ny : \u03b1\nhl : Nodup l\nh\u271d : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 l\nh : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 y :: l\nhy : getLast (y :: l) (_ : y :: l \u2260 []) \u2260 y\nH : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 dropLast (y :: l)\nhk : Nat.zero < length (dropLast (y :: l))\nhk' : y = getLast (y :: l) (_ : y :: l \u2260 [])\n\u22a2 False\ncase h.intro.mk.zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\ny : \u03b1\nhl : Nodup l\nh\u271d : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 l\nh : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 y :: l\nhy : getLast (y :: l) (_ : y :: l \u2260 []) \u2260 y\nH : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 dropLast (y :: l)\nhk : Nat.zero < length (dropLast (y :: l))\nhk' : get? (take (Nat.pred (length (y :: l))) (y :: l)) Nat.zero = some (getLast (y :: l) (_ : y :: l \u2260 []))\n\u22a2 Nat.zero < Nat.pred (length (y :: l))\n[PROOFSTEP]\nexact hy (Eq.symm hk')\n[GOAL]\ncase h.intro.mk.zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\ny : \u03b1\nhl : Nodup l\nh\u271d : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 l\nh : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 y :: l\nhy : getLast (y :: l) (_ : y :: l \u2260 []) \u2260 y\nH : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 dropLast (y :: l)\nhk : Nat.zero < length (dropLast (y :: l))\nhk' : get? (take (Nat.pred (length (y :: l))) (y :: l)) Nat.zero = some (getLast (y :: l) (_ : y :: l \u2260 []))\n\u22a2 Nat.zero < Nat.pred (length (y :: l))\n[PROOFSTEP]\nrw [Nat.zero_eq, length_cons, Nat.pred_succ]\n[GOAL]\ncase h.intro.mk.zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\ny : \u03b1\nhl : Nodup l\nh\u271d : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 l\nh : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 y :: l\nhy : getLast (y :: l) (_ : y :: l \u2260 []) \u2260 y\nH : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 dropLast (y :: l)\nhk : Nat.zero < length (dropLast (y :: l))\nhk' : get? (take (Nat.pred (length (y :: l))) (y :: l)) Nat.zero = some (getLast (y :: l) (_ : y :: l \u2260 []))\n\u22a2 0 < length l\n[PROOFSTEP]\nexact length_pos_of_mem (by assumption)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\ny : \u03b1\nhl : Nodup l\nh\u271d : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 l\nh : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 y :: l\nhy : getLast (y :: l) (_ : y :: l \u2260 []) \u2260 y\nH : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 dropLast (y :: l)\nhk : Nat.zero < length (dropLast (y :: l))\nhk' : get? (take (Nat.pred (length (y :: l))) (y :: l)) Nat.zero = some (getLast (y :: l) (_ : y :: l \u2260 []))\n\u22a2 ?m.71859 \u2208 l\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h.intro.mk.succ\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\ny : \u03b1\nhl : Nodup l\nh\u271d : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 l\nh : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 y :: l\nhy : getLast (y :: l) (_ : y :: l \u2260 []) \u2260 y\nH : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 dropLast (y :: l)\nk : \u2115\nhk : Nat.succ k < length (dropLast (y :: l))\nhk' : get (dropLast (y :: l)) { val := Nat.succ k, isLt := hk } = getLast (y :: l) (_ : y :: l \u2260 [])\n\u22a2 False\n[PROOFSTEP]\nsuffices k.succ = l.length by simp [this] at hk \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\ny : \u03b1\nhl : Nodup l\nh\u271d : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 l\nh : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 y :: l\nhy : getLast (y :: l) (_ : y :: l \u2260 []) \u2260 y\nH : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 dropLast (y :: l)\nk : \u2115\nhk : Nat.succ k < length (dropLast (y :: l))\nhk' : get (dropLast (y :: l)) { val := Nat.succ k, isLt := hk } = getLast (y :: l) (_ : y :: l \u2260 [])\nthis : Nat.succ k = length l\n\u22a2 False\n[PROOFSTEP]\nsimp [this] at hk \n[GOAL]\ncase h.intro.mk.succ\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\ny : \u03b1\nhl : Nodup l\nh\u271d : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 l\nh : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 y :: l\nhy : getLast (y :: l) (_ : y :: l \u2260 []) \u2260 y\nH : getLast (y :: l) (_ : y :: l \u2260 []) \u2208 dropLast (y :: l)\nk : \u2115\nhk : Nat.succ k < length (dropLast (y :: l))\nhk' : get (dropLast (y :: l)) { val := Nat.succ k, isLt := hk } = getLast (y :: l) (_ : y :: l \u2260 [])\n\u22a2 Nat.succ k = length l\n[PROOFSTEP]\ncases' l with hd tl\n[GOAL]\ncase h.intro.mk.succ.nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny : \u03b1\nk : \u2115\nhl : Nodup []\nh\u271d : getLast [y] (_ : [y] \u2260 []) \u2208 []\nh : getLast [y] (_ : [y] \u2260 []) \u2208 [y]\nhy : getLast [y] (_ : [y] \u2260 []) \u2260 y\nH : getLast [y] (_ : [y] \u2260 []) \u2208 dropLast [y]\nhk : Nat.succ k < length (dropLast [y])\nhk' : get (dropLast [y]) { val := Nat.succ k, isLt := hk } = getLast [y] (_ : [y] \u2260 [])\n\u22a2 Nat.succ k = length []\n[PROOFSTEP]\nsimp at hk \n[GOAL]\ncase h.intro.mk.succ.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny : \u03b1\nk : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : Nodup (hd :: tl)\nh\u271d : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 hd :: tl\nh : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 y :: hd :: tl\nhy : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2260 y\nH : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 dropLast (y :: hd :: tl)\nhk : Nat.succ k < length (dropLast (y :: hd :: tl))\nhk' :\n get (dropLast (y :: hd :: tl)) { val := Nat.succ k, isLt := hk } = getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 [])\n\u22a2 Nat.succ k = length (hd :: tl)\n[PROOFSTEP]\nrw [nodup_iff_injective_get] at hl \n[GOAL]\ncase h.intro.mk.succ.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny : \u03b1\nk : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : Function.Injective (get (hd :: tl))\nh\u271d : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 hd :: tl\nh : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 y :: hd :: tl\nhy : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2260 y\nH : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 dropLast (y :: hd :: tl)\nhk : Nat.succ k < length (dropLast (y :: hd :: tl))\nhk' :\n get (dropLast (y :: hd :: tl)) { val := Nat.succ k, isLt := hk } = getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 [])\n\u22a2 Nat.succ k = length (hd :: tl)\n[PROOFSTEP]\nrw [length, Nat.succ_inj']\n[GOAL]\ncase h.intro.mk.succ.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny : \u03b1\nk : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : Function.Injective (get (hd :: tl))\nh\u271d : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 hd :: tl\nh : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 y :: hd :: tl\nhy : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2260 y\nH : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 dropLast (y :: hd :: tl)\nhk : Nat.succ k < length (dropLast (y :: hd :: tl))\nhk' :\n get (dropLast (y :: hd :: tl)) { val := Nat.succ k, isLt := hk } = getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 [])\n\u22a2 k = length tl\n[PROOFSTEP]\nrefine' Fin.veq_of_eq (@hl \u27e8k, Nat.lt_of_succ_lt <| by simpa using hk\u27e9 \u27e8tl.length, by simp\u27e9 _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny : \u03b1\nk : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : Function.Injective (get (hd :: tl))\nh\u271d : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 hd :: tl\nh : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 y :: hd :: tl\nhy : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2260 y\nH : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 dropLast (y :: hd :: tl)\nhk : Nat.succ k < length (dropLast (y :: hd :: tl))\nhk' :\n get (dropLast (y :: hd :: tl)) { val := Nat.succ k, isLt := hk } = getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 [])\n\u22a2 Nat.succ k < length (hd :: tl)\n[PROOFSTEP]\nsimpa using hk\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny : \u03b1\nk : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : Function.Injective (get (hd :: tl))\nh\u271d : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 hd :: tl\nh : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 y :: hd :: tl\nhy : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2260 y\nH : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 dropLast (y :: hd :: tl)\nhk : Nat.succ k < length (dropLast (y :: hd :: tl))\nhk' :\n get (dropLast (y :: hd :: tl)) { val := Nat.succ k, isLt := hk } = getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 [])\n\u22a2 length tl < length (hd :: tl)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.intro.mk.succ.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny : \u03b1\nk : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : Function.Injective (get (hd :: tl))\nh\u271d : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 hd :: tl\nh : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 y :: hd :: tl\nhy : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2260 y\nH : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 dropLast (y :: hd :: tl)\nhk : Nat.succ k < length (dropLast (y :: hd :: tl))\nhk' :\n get (dropLast (y :: hd :: tl)) { val := Nat.succ k, isLt := hk } = getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 [])\n\u22a2 get (hd :: tl) { val := k, isLt := (_ : k < length (hd :: tl)) } =\n get (hd :: tl) { val := length tl, isLt := (_ : length tl < Nat.succ (length tl)) }\n[PROOFSTEP]\nrw [\u2190 Option.some_inj] at hk' \n[GOAL]\ncase h.intro.mk.succ.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny : \u03b1\nk : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : Function.Injective (get (hd :: tl))\nh\u271d : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 hd :: tl\nh : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 y :: hd :: tl\nhy : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2260 y\nH : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 dropLast (y :: hd :: tl)\nhk : Nat.succ k < length (dropLast (y :: hd :: tl))\nhk' :\n some (get (dropLast (y :: hd :: tl)) { val := Nat.succ k, isLt := hk }) =\n some (getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []))\n\u22a2 get (hd :: tl) { val := k, isLt := (_ : k < length (hd :: tl)) } =\n get (hd :: tl) { val := length tl, isLt := (_ : length tl < Nat.succ (length tl)) }\n[PROOFSTEP]\nrw [\u2190 get?_eq_get, dropLast_eq_take, get?_take, get?, get?_eq_get, Option.some_inj] at hk' \n[GOAL]\ncase h.intro.mk.succ.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny : \u03b1\nk : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : Function.Injective (get (hd :: tl))\nh\u271d : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 hd :: tl\nh : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 y :: hd :: tl\nhy : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2260 y\nH : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 dropLast (y :: hd :: tl)\nhk : Nat.succ k < length (dropLast (y :: hd :: tl))\nhk'\u271d : get? (hd :: tl) k = some (getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []))\nhk' : get (hd :: tl) { val := k, isLt := ?h.intro.mk.succ.cons } = getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 [])\n\u22a2 get (hd :: tl) { val := k, isLt := (_ : k < length (hd :: tl)) } =\n get (hd :: tl) { val := length tl, isLt := (_ : length tl < Nat.succ (length tl)) }\ncase h.intro.mk.succ.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny : \u03b1\nk : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : Function.Injective (get (hd :: tl))\nh\u271d : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 hd :: tl\nh : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 y :: hd :: tl\nhy : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2260 y\nH : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 dropLast (y :: hd :: tl)\nhk : Nat.succ k < length (dropLast (y :: hd :: tl))\nhk' : get? (hd :: tl) k = some (getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []))\n\u22a2 k < length (hd :: tl)\ncase h.intro.mk.succ.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny : \u03b1\nk : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : Function.Injective (get (hd :: tl))\nh\u271d : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 hd :: tl\nh : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 y :: hd :: tl\nhy : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2260 y\nH : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 dropLast (y :: hd :: tl)\nhk : Nat.succ k < length (dropLast (y :: hd :: tl))\nhk' : get? (hd :: tl) k = some (getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []))\n\u22a2 k < length (hd :: tl)\ncase h.intro.mk.succ.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny : \u03b1\nk : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : Function.Injective (get (hd :: tl))\nh\u271d : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 hd :: tl\nh : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 y :: hd :: tl\nhy : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2260 y\nH : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 dropLast (y :: hd :: tl)\nhk : Nat.succ k < length (dropLast (y :: hd :: tl))\nhk' :\n get? (take (Nat.pred (length (y :: hd :: tl))) (y :: hd :: tl)) (Nat.succ k) =\n some (getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []))\n\u22a2 Nat.succ k < Nat.pred (length (y :: hd :: tl))\n[PROOFSTEP]\nrw [hk']\n[GOAL]\ncase h.intro.mk.succ.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny : \u03b1\nk : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : Function.Injective (get (hd :: tl))\nh\u271d : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 hd :: tl\nh : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 y :: hd :: tl\nhy : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2260 y\nH : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 dropLast (y :: hd :: tl)\nhk : Nat.succ k < length (dropLast (y :: hd :: tl))\nhk'\u271d : get? (hd :: tl) k = some (getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []))\nhk' :\n get (hd :: tl) { val := k, isLt := (_ : k < length (hd :: tl)) } = getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 [])\n\u22a2 getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) =\n get (hd :: tl) { val := length tl, isLt := (_ : length tl < Nat.succ (length tl)) }\ncase h.intro.mk.succ.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny : \u03b1\nk : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : Function.Injective (get (hd :: tl))\nh\u271d : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 hd :: tl\nh : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 y :: hd :: tl\nhy : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2260 y\nH : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 dropLast (y :: hd :: tl)\nhk : Nat.succ k < length (dropLast (y :: hd :: tl))\nhk' :\n get? (take (Nat.pred (length (y :: hd :: tl))) (y :: hd :: tl)) (Nat.succ k) =\n some (getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []))\n\u22a2 Nat.succ k < Nat.pred (length (y :: hd :: tl))\n[PROOFSTEP]\nsimp [getLast_eq_get]\n[GOAL]\ncase h.intro.mk.succ.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ny : \u03b1\nk : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : Function.Injective (get (hd :: tl))\nh\u271d : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 hd :: tl\nh : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 y :: hd :: tl\nhy : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2260 y\nH : getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []) \u2208 dropLast (y :: hd :: tl)\nhk : Nat.succ k < length (dropLast (y :: hd :: tl))\nhk' :\n get? (take (Nat.pred (length (y :: hd :: tl))) (y :: hd :: tl)) (Nat.succ k) =\n some (getLast (y :: hd :: tl) (_ : y :: hd :: tl \u2260 []))\n\u22a2 Nat.succ k < Nat.pred (length (y :: hd :: tl))\n[PROOFSTEP]\nsimpa using hk\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx y : \u03b1\nhxy : x \u2208 y :: l\nhx : x = y\n\u22a2 prev (y :: l) x hxy = getLast (y :: l) (_ : y :: l \u2260 [])\n[PROOFSTEP]\ncases l\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nhx : x = y\nhxy : x \u2208 [y]\n\u22a2 prev [y] x hxy = getLast [y] (_ : [y] \u2260 [])\n[PROOFSTEP]\nsimp [prev, hx]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nhx : x = y\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nhxy : x \u2208 y :: head\u271d :: tail\u271d\n\u22a2 prev (y :: head\u271d :: tail\u271d) x hxy = getLast (y :: head\u271d :: tail\u271d) (_ : y :: head\u271d :: tail\u271d \u2260 [])\n[PROOFSTEP]\nsimp [prev, hx]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx y z : \u03b1\nh : x \u2208 y :: z :: l\nhx : x = y\n\u22a2 prev (y :: z :: l) x h = getLast (z :: l) (_ : z :: l \u2260 [])\n[PROOFSTEP]\nrw [prev, dif_pos hx]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx y z : \u03b1\nh : x \u2208 y :: z :: l\nhy : x \u2260 y\nhz : x = z\n\u22a2 prev (y :: z :: l) x h = y\n[PROOFSTEP]\ncases l\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhy : x \u2260 y\nhz : x = z\nh : x \u2208 [y, z]\n\u22a2 prev [y, z] x h = y\n[PROOFSTEP]\nsimp [prev, hy, hz]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhy : x \u2260 y\nhz : x = z\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh : x \u2208 y :: z :: head\u271d :: tail\u271d\n\u22a2 prev (y :: z :: head\u271d :: tail\u271d) x h = y\n[PROOFSTEP]\nrw [prev, dif_neg hy, if_pos hz]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx y z : \u03b1\nh : x \u2208 y :: z :: l\nhy : x \u2260 y\nhz : x \u2260 z\n\u22a2 x \u2208 z :: l\n[PROOFSTEP]\nsimpa [hy] using h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx y z : \u03b1\nh : x \u2208 y :: z :: l\nhy : x \u2260 y\nhz : x \u2260 z\n\u22a2 prev (y :: z :: l) x h = prev (z :: l) x (_ : x \u2208 z :: l)\n[PROOFSTEP]\ncases l\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhy : x \u2260 y\nhz : x \u2260 z\nh : x \u2208 [y, z]\n\u22a2 prev [y, z] x h = prev [z] x (_ : x \u2208 [z])\n[PROOFSTEP]\nsimp [hy, hz] at h \n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhy : x \u2260 y\nhz : x \u2260 z\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh : x \u2208 y :: z :: head\u271d :: tail\u271d\n\u22a2 prev (y :: z :: head\u271d :: tail\u271d) x h = prev (z :: head\u271d :: tail\u271d) x (_ : x \u2208 z :: head\u271d :: tail\u271d)\n[PROOFSTEP]\nrw [prev, dif_neg hy, if_neg hz]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh : x \u2208 l\n\u22a2 prev l x h \u2208 l\n[PROOFSTEP]\ncases' l with hd tl\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nh : x \u2208 []\n\u22a2 prev [] x h \u2208 []\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd : \u03b1\ntl : List \u03b1\nh : x \u2208 hd :: tl\n\u22a2 prev (hd :: tl) x h \u2208 hd :: tl\n[PROOFSTEP]\ninduction' tl with hd' tl hl generalizing hd\n[GOAL]\ncase cons.nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\nhd : \u03b1\nh : x \u2208 [hd]\n\u22a2 prev [hd] x h \u2208 [hd]\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\n\u22a2 prev (hd :: hd' :: tl) x h \u2208 hd :: hd' :: tl\n[PROOFSTEP]\nby_cases hx : x = hd\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : x = hd\n\u22a2 prev (hd :: hd' :: tl) x h \u2208 hd :: hd' :: tl\n[PROOFSTEP]\nsimp only [hx, prev_cons_cons_eq]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : x = hd\n\u22a2 getLast (hd' :: tl) (_ : hd' :: tl \u2260 []) \u2208 hd :: hd' :: tl\n[PROOFSTEP]\nexact mem_cons_of_mem _ (getLast_mem _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : \u00acx = hd\n\u22a2 prev (hd :: hd' :: tl) x h \u2208 hd :: hd' :: tl\n[PROOFSTEP]\nrw [prev, dif_neg hx]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : \u00acx = hd\n\u22a2 (if x = hd' then hd else prev (hd' :: tl) x (_ : x \u2208 hd' :: tl)) \u2208 hd :: hd' :: tl\n[PROOFSTEP]\nsplit_ifs with hm\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : \u00acx = hd\nhm : x = hd'\n\u22a2 hd \u2208 hd :: hd' :: tl\n[PROOFSTEP]\nexact mem_cons_self _ _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : \u00acx = hd\nhm : \u00acx = hd'\n\u22a2 prev (hd' :: tl) x (_ : x \u2208 hd' :: tl) \u2208 hd :: hd' :: tl\n[PROOFSTEP]\nexact mem_cons_of_mem _ (hl _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nx\u271d : Nodup []\ni : Fin (length [])\n\u22a2 next [] (get [] i) (_ : get [] { val := \u2191i, isLt := (_ : \u2191i < length []) } \u2208 []) =\n get [] { val := (\u2191i + 1) % length [], isLt := (_ : (\u2191i + 1) % length [] < length []) }\n[PROOFSTEP]\nsimpa using i.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx head\u271d : \u03b1\nx\u271d\u00b9 : Nodup [head\u271d]\nx\u271d : Fin (length [head\u271d])\n\u22a2 next [head\u271d] (get [head\u271d] x\u271d) (_ : get [head\u271d] { val := \u2191x\u271d, isLt := (_ : \u2191x\u271d < length [head\u271d]) } \u2208 [head\u271d]) =\n get [head\u271d] { val := (\u2191x\u271d + 1) % length [head\u271d], isLt := (_ : (\u2191x\u271d + 1) % length [head\u271d] < length [head\u271d]) }\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\n_h : Nodup (x :: y :: l)\nh0 : 0 < length (x :: y :: l)\n\u22a2 next (x :: y :: l) (get (x :: y :: l) { val := 0, isLt := h0 })\n (_ :\n get (x :: y :: l)\n { val := \u2191{ val := 0, isLt := h0 }, isLt := (_ : \u2191{ val := 0, isLt := h0 } < length (x :: y :: l)) } \u2208\n x :: y :: l) =\n get (x :: y :: l)\n { val := (\u2191{ val := 0, isLt := h0 } + 1) % length (x :: y :: l),\n isLt := (_ : (\u2191{ val := 0, isLt := h0 } + 1) % length (x :: y :: l) < length (x :: y :: l)) }\n[PROOFSTEP]\nhave h\u2081 : get (x :: y :: l) { val := 0, isLt := h0 } = x := by simp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\n_h : Nodup (x :: y :: l)\nh0 : 0 < length (x :: y :: l)\n\u22a2 get (x :: y :: l) { val := 0, isLt := h0 } = x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\n_h : Nodup (x :: y :: l)\nh0 : 0 < length (x :: y :: l)\nh\u2081 : get (x :: y :: l) { val := 0, isLt := h0 } = x\n\u22a2 next (x :: y :: l) (get (x :: y :: l) { val := 0, isLt := h0 })\n (_ :\n get (x :: y :: l)\n { val := \u2191{ val := 0, isLt := h0 }, isLt := (_ : \u2191{ val := 0, isLt := h0 } < length (x :: y :: l)) } \u2208\n x :: y :: l) =\n get (x :: y :: l)\n { val := (\u2191{ val := 0, isLt := h0 } + 1) % length (x :: y :: l),\n isLt := (_ : (\u2191{ val := 0, isLt := h0 } + 1) % length (x :: y :: l) < length (x :: y :: l)) }\n[PROOFSTEP]\nrw [next_cons_cons_eq' _ _ _ _ _ h\u2081]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\n_h : Nodup (x :: y :: l)\nh0 : 0 < length (x :: y :: l)\nh\u2081 : get (x :: y :: l) { val := 0, isLt := h0 } = x\n\u22a2 y =\n get (x :: y :: l)\n { val := (\u2191{ val := 0, isLt := h0 } + 1) % length (x :: y :: l),\n isLt := (_ : (\u2191{ val := 0, isLt := h0 } + 1) % length (x :: y :: l) < length (x :: y :: l)) }\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\n\u22a2 next (x :: y :: l) (get (x :: y :: l) { val := i + 1, isLt := hi })\n (_ :\n get (x :: y :: l)\n { val := \u2191{ val := i + 1, isLt := hi },\n isLt := (_ : \u2191{ val := i + 1, isLt := hi } < length (x :: y :: l)) } \u2208\n x :: y :: l) =\n get (x :: y :: l)\n { val := (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l),\n isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) }\n[PROOFSTEP]\nhave hx' : (x :: y :: l).get \u27e8i + 1, hi\u27e9 \u2260 x := by\n intro H\n suffices (i + 1 : \u2115) = 0 by simpa\n rw [nodup_iff_injective_get] at hn \n refine' Fin.veq_of_eq (@hn \u27e8i + 1, hi\u27e9 \u27e80, by simp\u27e9 _)\n simpa using H\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\n\u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\n[PROOFSTEP]\nintro H\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nH : get (x :: y :: l) { val := i + 1, isLt := hi } = x\n\u22a2 False\n[PROOFSTEP]\nsuffices (i + 1 : \u2115) = 0 by simpa\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nH : get (x :: y :: l) { val := i + 1, isLt := hi } = x\nthis : i + 1 = 0\n\u22a2 False\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nH : get (x :: y :: l) { val := i + 1, isLt := hi } = x\n\u22a2 i + 1 = 0\n[PROOFSTEP]\nrw [nodup_iff_injective_get] at hn \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Function.Injective (get (x :: y :: l))\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nH : get (x :: y :: l) { val := i + 1, isLt := hi } = x\n\u22a2 i + 1 = 0\n[PROOFSTEP]\nrefine' Fin.veq_of_eq (@hn \u27e8i + 1, hi\u27e9 \u27e80, by simp\u27e9 _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Function.Injective (get (x :: y :: l))\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nH : get (x :: y :: l) { val := i + 1, isLt := hi } = x\n\u22a2 0 < length (x :: y :: l)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Function.Injective (get (x :: y :: l))\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nH : get (x :: y :: l) { val := i + 1, isLt := hi } = x\n\u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } =\n get (x :: y :: l) { val := 0, isLt := (_ : 0 < Nat.succ (length l + 1)) }\n[PROOFSTEP]\nsimpa using H\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\n\u22a2 next (x :: y :: l) (get (x :: y :: l) { val := i + 1, isLt := hi })\n (_ :\n get (x :: y :: l)\n { val := \u2191{ val := i + 1, isLt := hi },\n isLt := (_ : \u2191{ val := i + 1, isLt := hi } < length (x :: y :: l)) } \u2208\n x :: y :: l) =\n get (x :: y :: l)\n { val := (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l),\n isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) }\n[PROOFSTEP]\nhave hi' : i \u2264 l.length := Nat.le_of_lt_succ (Nat.succ_lt_succ_iff.1 hi)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi' : i \u2264 length l\n\u22a2 next (x :: y :: l) (get (x :: y :: l) { val := i + 1, isLt := hi })\n (_ :\n get (x :: y :: l)\n { val := \u2191{ val := i + 1, isLt := hi },\n isLt := (_ : \u2191{ val := i + 1, isLt := hi } < length (x :: y :: l)) } \u2208\n x :: y :: l) =\n get (x :: y :: l)\n { val := (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l),\n isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) }\n[PROOFSTEP]\nrcases hi'.eq_or_lt with (hi' | hi')\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i = length l\n\u22a2 next (x :: y :: l) (get (x :: y :: l) { val := i + 1, isLt := hi })\n (_ :\n get (x :: y :: l)\n { val := \u2191{ val := i + 1, isLt := hi },\n isLt := (_ : \u2191{ val := i + 1, isLt := hi } < length (x :: y :: l)) } \u2208\n x :: y :: l) =\n get (x :: y :: l)\n { val := (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l),\n isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) }\n[PROOFSTEP]\nsubst hi'\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\nhi : length l + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x\nhi' : length l \u2264 length l\n\u22a2 next (x :: y :: l) (get (x :: y :: l) { val := length l + 1, isLt := hi })\n (_ :\n get (x :: y :: l)\n { val := \u2191{ val := length l + 1, isLt := hi },\n isLt := (_ : \u2191{ val := length l + 1, isLt := hi } < length (x :: y :: l)) } \u2208\n x :: y :: l) =\n get (x :: y :: l)\n { val := (\u2191{ val := length l + 1, isLt := hi } + 1) % length (x :: y :: l),\n isLt := (_ : (\u2191{ val := length l + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) }\n[PROOFSTEP]\nrw [next_getLast_cons]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\nhi : length l + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x\nhi' : length l \u2264 length l\n\u22a2 x =\n get (x :: y :: l)\n { val := (\u2191{ val := length l + 1, isLt := hi } + 1) % length (x :: y :: l),\n isLt := (_ : (\u2191{ val := length l + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) }\n[PROOFSTEP]\nsimp [hi', get]\n[GOAL]\ncase inl.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\nhi : length l + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x\nhi' : length l \u2264 length l\n\u22a2 get (x :: y :: l) { val := length l + 1, isLt := hi } \u2208 y :: l\n[PROOFSTEP]\nrw [get_cons_succ]\n[GOAL]\ncase inl.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\nhi : length l + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x\nhi' : length l \u2264 length l\n\u22a2 get (y :: l) { val := length l, isLt := (_ : length l < length (y :: l)) } \u2208 y :: l\n[PROOFSTEP]\nexact get_mem _ _ _\n[GOAL]\ncase inl.hy\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\nhi : length l + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x\nhi' : length l \u2264 length l\n\u22a2 get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x\n[PROOFSTEP]\nexact hx'\n[GOAL]\ncase inl.hx\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\nhi : length l + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x\nhi' : length l \u2264 length l\n\u22a2 get (x :: y :: l) { val := length l + 1, isLt := hi } = getLast (x :: y :: l) (_ : x :: y :: l \u2260 [])\n[PROOFSTEP]\nsimp [getLast_eq_get]\n[GOAL]\ncase inl.hl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\nhi : length l + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x\nhi' : length l \u2264 length l\n\u22a2 Nodup (y :: l)\n[PROOFSTEP]\nexact hn.of_cons\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 next (x :: y :: l) (get (x :: y :: l) { val := i + 1, isLt := hi })\n (_ :\n get (x :: y :: l)\n { val := \u2191{ val := i + 1, isLt := hi },\n isLt := (_ : \u2191{ val := i + 1, isLt := hi } < length (x :: y :: l)) } \u2208\n x :: y :: l) =\n get (x :: y :: l)\n { val := (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l),\n isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) }\n[PROOFSTEP]\nrw [next_ne_head_ne_getLast _ _ _ _ _ hx']\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 next (y :: l) (get (x :: y :: l) { val := i + 1, isLt := hi })\n (_ : get (x :: y :: l) { val := i + 1, isLt := hi } \u2208 y :: l) =\n get (x :: y :: l)\n { val := (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l),\n isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) }\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 getLast (x :: y :: l) (_ : x :: y :: l \u2260 [])\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2208 y :: l\n[PROOFSTEP]\nsimp only [get_cons_succ]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 next (y :: l) (get (y :: l) { val := i, isLt := (_ : i < length (y :: l)) })\n (_ : get (y :: l) { val := i, isLt := (_ : i < length (y :: l)) } \u2208 y :: l) =\n get (x :: y :: l)\n { val := (i + 1 + 1) % length (x :: y :: l),\n isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) }\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 getLast (x :: y :: l) (_ : x :: y :: l \u2260 [])\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2208 y :: l\n[PROOFSTEP]\nrw [next_get (y :: l), \u2190 get_cons_succ (a := x)]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 get (x :: y :: l)\n { val := (\u2191{ val := i, isLt := (_ : i < length (y :: l)) } + 1) % length (y :: l) + 1, isLt := ?m.114701 } =\n get (x :: y :: l)\n { val := (i + 1 + 1) % length (x :: y :: l),\n isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) }\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 (\u2191{ val := i, isLt := (_ : i < length (y :: l)) } + 1) % length (y :: l) + 1 < length (x :: y :: l)\ncase inr._h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 Nodup (y :: l)\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 getLast (x :: y :: l) (_ : x :: y :: l \u2260 [])\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2208 y :: l\n[PROOFSTEP]\ncongr\n[GOAL]\ncase inr.e_a.e_val\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 (\u2191{ val := i, isLt := (_ : i < length (y :: l)) } + 1) % length (y :: l) + 1 = (i + 1 + 1) % length (x :: y :: l)\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 (\u2191{ val := i, isLt := (_ : i < length (y :: l)) } + 1) % length (y :: l) + 1 < length (x :: y :: l)\ncase inr._h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 Nodup (y :: l)\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 getLast (x :: y :: l) (_ : x :: y :: l \u2260 [])\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2208 y :: l\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase inr.e_a.e_val\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 (i + 1) % Nat.succ (length l) + 1 = (i + 1 + 1) % Nat.succ (Nat.succ (length l))\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 (\u2191{ val := i, isLt := (_ : i < length (y :: l)) } + 1) % length (y :: l) + 1 < length (x :: y :: l)\ncase inr._h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 Nodup (y :: l)\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 getLast (x :: y :: l) (_ : x :: y :: l \u2260 [])\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2208 y :: l\n[PROOFSTEP]\nrw [Nat.mod_eq_of_lt (Nat.succ_lt_succ_iff.2 hi'),\n Nat.mod_eq_of_lt (Nat.succ_lt_succ_iff.2 (Nat.succ_lt_succ_iff.2 hi'))]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 (\u2191{ val := i, isLt := (_ : i < length (y :: l)) } + 1) % length (y :: l) + 1 < length (x :: y :: l)\n[PROOFSTEP]\nsimp [Nat.mod_eq_of_lt (Nat.succ_lt_succ_iff.2 hi'), Nat.succ_eq_add_one, hi']\n[GOAL]\ncase inr._h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 Nodup (y :: l)\n[PROOFSTEP]\nexact hn.of_cons\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 getLast (x :: y :: l) (_ : x :: y :: l \u2260 [])\n[PROOFSTEP]\nrw [getLast_eq_get]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260\n get (x :: y :: l) { val := length (x :: y :: l) - 1, isLt := (_ : length (x :: y :: l) - 1 < length (x :: y :: l)) }\n[PROOFSTEP]\nintro h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\nh :\n get (x :: y :: l) { val := i + 1, isLt := hi } =\n get (x :: y :: l) { val := length (x :: y :: l) - 1, isLt := (_ : length (x :: y :: l) - 1 < length (x :: y :: l)) }\n\u22a2 False\n[PROOFSTEP]\nhave := nodup_iff_injective_get.1 hn h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\nh :\n get (x :: y :: l) { val := i + 1, isLt := hi } =\n get (x :: y :: l) { val := length (x :: y :: l) - 1, isLt := (_ : length (x :: y :: l) - 1 < length (x :: y :: l)) }\nthis :\n { val := i + 1, isLt := hi } =\n { val := length (x :: y :: l) - 1, isLt := (_ : length (x :: y :: l) - 1 < length (x :: y :: l)) }\n\u22a2 False\n[PROOFSTEP]\nsimp at this \n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\nh :\n get (x :: y :: l) { val := i + 1, isLt := hi } =\n get (x :: y :: l) { val := length (x :: y :: l) - 1, isLt := (_ : length (x :: y :: l) - 1 < length (x :: y :: l)) }\nthis : i = length l\n\u22a2 False\n[PROOFSTEP]\nsimp [this] at hi' \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2208 y :: l\n[PROOFSTEP]\nrw [get_cons_succ]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d x y : \u03b1\nl : List \u03b1\nhn : Nodup (x :: y :: l)\ni : \u2115\nhi : i + 1 < length (x :: y :: l)\nhx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x\nhi'\u271d : i \u2264 length l\nhi' : i < length l\n\u22a2 get (y :: l) { val := i, isLt := (_ : i < length (y :: l)) } \u2208 y :: l\n[PROOFSTEP]\nexact get_mem _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nn : \u2115\nhn : n < length l\n\u22a2 prev l (nthLe l n hn) (_ : nthLe l n hn \u2208 l) =\n nthLe l ((n + (length l - 1)) % length l) (_ : (n + (length l - 1)) % length l < length l)\n[PROOFSTEP]\ncases' l with x l\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nn : \u2115\nh : Nodup []\nhn : n < length []\n\u22a2 prev [] (nthLe [] n hn) (_ : nthLe [] n hn \u2208 []) =\n nthLe [] ((n + (length [] - 1)) % length []) (_ : (n + (length [] - 1)) % length [] < length [])\n[PROOFSTEP]\nsimp at hn \n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nn : \u2115\nx : \u03b1\nl : List \u03b1\nh : Nodup (x :: l)\nhn : n < length (x :: l)\n\u22a2 prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\n[PROOFSTEP]\ninduction' l with y l hl generalizing n x\n[GOAL]\ncase cons.nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl : List \u03b1\nh\u271d : Nodup (x\u271d :: l)\nhn\u271d : n\u271d < length (x\u271d :: l)\nn : \u2115\nx : \u03b1\nh : Nodup [x]\nhn : n < length [x]\n\u22a2 prev [x] (nthLe [x] n hn) (_ : nthLe [x] n hn \u2208 [x]) =\n nthLe [x] ((n + (length [x] - 1)) % length [x]) (_ : (n + (length [x] - 1)) % length [x] < length [x])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nn : \u2115\nx : \u03b1\nh : Nodup (x :: y :: l)\nhn : n < length (x :: y :: l)\n\u22a2 prev (x :: y :: l) (nthLe (x :: y :: l) n hn) (_ : nthLe (x :: y :: l) n hn \u2208 x :: y :: l) =\n nthLe (x :: y :: l) ((n + (length (x :: y :: l) - 1)) % length (x :: y :: l))\n (_ : (n + (length (x :: y :: l) - 1)) % length (x :: y :: l) < length (x :: y :: l))\n[PROOFSTEP]\nrcases n with (_ | _ | n)\n[GOAL]\ncase cons.cons.zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nhn : Nat.zero < length (x :: y :: l)\n\u22a2 prev (x :: y :: l) (nthLe (x :: y :: l) Nat.zero hn) (_ : nthLe (x :: y :: l) Nat.zero hn \u2208 x :: y :: l) =\n nthLe (x :: y :: l) ((Nat.zero + (length (x :: y :: l) - 1)) % length (x :: y :: l))\n (_ : (Nat.zero + (length (x :: y :: l) - 1)) % length (x :: y :: l) < length (x :: y :: l))\n[PROOFSTEP]\nsimp [Nat.add_succ_sub_one, add_zero, List.prev_cons_cons_eq, Nat.zero_eq, List.length, List.nthLe,\n Nat.succ_add_sub_one, zero_add, getLast_eq_get, Nat.mod_eq_of_lt (Nat.succ_lt_succ l.length.lt_succ_self)]\n[GOAL]\ncase cons.cons.succ.zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nhn : Nat.succ Nat.zero < length (x :: y :: l)\n\u22a2 prev (x :: y :: l) (nthLe (x :: y :: l) (Nat.succ Nat.zero) hn)\n (_ : nthLe (x :: y :: l) (Nat.succ Nat.zero) hn \u2208 x :: y :: l) =\n nthLe (x :: y :: l) ((Nat.succ Nat.zero + (length (x :: y :: l) - 1)) % length (x :: y :: l))\n (_ : (Nat.succ Nat.zero + (length (x :: y :: l) - 1)) % length (x :: y :: l) < length (x :: y :: l))\n[PROOFSTEP]\nsimp only [mem_cons, nodup_cons] at h \n[GOAL]\ncase cons.cons.succ.zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nhn : Nat.succ Nat.zero < length (x :: y :: l)\nh : \u00ac(x = y \u2228 x \u2208 l) \u2227 \u00acy \u2208 l \u2227 Nodup l\n\u22a2 prev (x :: y :: l) (nthLe (x :: y :: l) (Nat.succ Nat.zero) hn)\n (_ : nthLe (x :: y :: l) (Nat.succ Nat.zero) hn \u2208 x :: y :: l) =\n nthLe (x :: y :: l) ((Nat.succ Nat.zero + (length (x :: y :: l) - 1)) % length (x :: y :: l))\n (_ : (Nat.succ Nat.zero + (length (x :: y :: l) - 1)) % length (x :: y :: l) < length (x :: y :: l))\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase cons.cons.succ.zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nhn : Nat.succ Nat.zero < length (x :: y :: l)\nh : (x \u2260 y \u2227 \u00acx \u2208 l) \u2227 \u00acy \u2208 l \u2227 Nodup l\n\u22a2 prev (x :: y :: l) (nthLe (x :: y :: l) (Nat.succ Nat.zero) hn)\n (_ : nthLe (x :: y :: l) (Nat.succ Nat.zero) hn \u2208 x :: y :: l) =\n nthLe (x :: y :: l) ((Nat.succ Nat.zero + (length (x :: y :: l) - 1)) % length (x :: y :: l))\n (_ : (Nat.succ Nat.zero + (length (x :: y :: l) - 1)) % length (x :: y :: l) < length (x :: y :: l))\n[PROOFSTEP]\nsimp only [List.prev_cons_cons_of_ne _ _ _ _ h.left.left.symm, Nat.zero_eq, List.length, List.nthLe, add_comm,\n eq_self_iff_true, Nat.succ_add_sub_one, Nat.mod_self, zero_add, List.get]\n[GOAL]\ncase cons.cons.succ.succ\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\n\u22a2 prev (x :: y :: l) (nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn)\n (_ : nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn \u2208 x :: y :: l) =\n nthLe (x :: y :: l) ((Nat.succ (Nat.succ n) + (length (x :: y :: l) - 1)) % length (x :: y :: l))\n (_ : (Nat.succ (Nat.succ n) + (length (x :: y :: l) - 1)) % length (x :: y :: l) < length (x :: y :: l))\n[PROOFSTEP]\nrw [prev_ne_cons_cons]\n[GOAL]\ncase cons.cons.succ.succ\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\n\u22a2 prev (y :: l) (nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn)\n (_ : nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn \u2208 y :: l) =\n nthLe (x :: y :: l) ((Nat.succ (Nat.succ n) + (length (x :: y :: l) - 1)) % length (x :: y :: l))\n (_ : (Nat.succ (Nat.succ n) + (length (x :: y :: l) - 1)) % length (x :: y :: l) < length (x :: y :: l))\n[PROOFSTEP]\nconvert hl n.succ y h.of_cons (Nat.le_of_succ_le_succ hn) using 1\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\n\u22a2 nthLe (x :: y :: l) ((Nat.succ (Nat.succ n) + (length (x :: y :: l) - 1)) % length (x :: y :: l))\n (_ : (Nat.succ (Nat.succ n) + (length (x :: y :: l) - 1)) % length (x :: y :: l) < length (x :: y :: l)) =\n nthLe (y :: l) ((Nat.succ n + (length (y :: l) - 1)) % length (y :: l))\n (_ : (Nat.succ n + (length (y :: l) - 1)) % length (y :: l) < length (y :: l))\n[PROOFSTEP]\nhave : \u2200 k hk, (y :: l).nthLe k hk = (x :: y :: l).nthLe (k + 1) (Nat.succ_lt_succ hk) :=\n by\n intros\n simp [List.nthLe]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\n\u22a2 \u2200 (k : \u2115) (hk : k < length (y :: l)),\n nthLe (y :: l) k hk = nthLe (x :: y :: l) (k + 1) (_ : Nat.succ k < Nat.succ (length (y :: l)))\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\nk\u271d : \u2115\nhk\u271d : k\u271d < length (y :: l)\n\u22a2 nthLe (y :: l) k\u271d hk\u271d = nthLe (x :: y :: l) (k\u271d + 1) (_ : Nat.succ k\u271d < Nat.succ (length (y :: l)))\n[PROOFSTEP]\nsimp [List.nthLe]\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\nthis :\n \u2200 (k : \u2115) (hk : k < length (y :: l)),\n nthLe (y :: l) k hk = nthLe (x :: y :: l) (k + 1) (_ : Nat.succ k < Nat.succ (length (y :: l)))\n\u22a2 nthLe (x :: y :: l) ((Nat.succ (Nat.succ n) + (length (x :: y :: l) - 1)) % length (x :: y :: l))\n (_ : (Nat.succ (Nat.succ n) + (length (x :: y :: l) - 1)) % length (x :: y :: l) < length (x :: y :: l)) =\n nthLe (y :: l) ((Nat.succ n + (length (y :: l) - 1)) % length (y :: l))\n (_ : (Nat.succ n + (length (y :: l) - 1)) % length (y :: l) < length (y :: l))\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\nthis :\n \u2200 (k : \u2115) (hk : k < length (y :: l)),\n nthLe (y :: l) k hk = nthLe (x :: y :: l) (k + 1) (_ : Nat.succ k < Nat.succ (length (y :: l)))\n\u22a2 nthLe (x :: y :: l) ((Nat.succ (Nat.succ n) + (length (x :: y :: l) - 1)) % length (x :: y :: l))\n (_ : (Nat.succ (Nat.succ n) + (length (x :: y :: l) - 1)) % length (x :: y :: l) < length (x :: y :: l)) =\n nthLe (x :: y :: l) ((Nat.succ n + (length (y :: l) - 1)) % length (y :: l) + 1)\n (_ : Nat.succ ((Nat.succ n + (length (y :: l) - 1)) % length (y :: l)) < Nat.succ (length (y :: l)))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e'_3.e_n\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\nthis :\n \u2200 (k : \u2115) (hk : k < length (y :: l)),\n nthLe (y :: l) k hk = nthLe (x :: y :: l) (k + 1) (_ : Nat.succ k < Nat.succ (length (y :: l)))\n\u22a2 (Nat.succ (Nat.succ n) + (length (x :: y :: l) - 1)) % length (x :: y :: l) =\n (Nat.succ n + (length (y :: l) - 1)) % length (y :: l) + 1\n[PROOFSTEP]\nsimp only [Nat.add_succ_sub_one, add_zero, length]\n[GOAL]\ncase h.e'_3.e_n\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\nthis :\n \u2200 (k : \u2115) (hk : k < length (y :: l)),\n nthLe (y :: l) k hk = nthLe (x :: y :: l) (k + 1) (_ : Nat.succ k < Nat.succ (length (y :: l)))\n\u22a2 (Nat.succ (Nat.succ n) + (length l + 1)) % (length l + 1 + 1) = (Nat.succ n + length l) % (length l + 1) + 1\n[PROOFSTEP]\nsimp only [length, Nat.succ_lt_succ_iff] at hn \n[GOAL]\ncase h.e'_3.e_n\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nthis :\n \u2200 (k : \u2115) (hk : k < length (y :: l)),\n nthLe (y :: l) k hk = nthLe (x :: y :: l) (k + 1) (_ : Nat.succ k < Nat.succ (length (y :: l)))\nhn : n < length l\n\u22a2 (Nat.succ (Nat.succ n) + (length l + 1)) % (length l + 1 + 1) = (Nat.succ n + length l) % (length l + 1) + 1\n[PROOFSTEP]\nset k := l.length\n[GOAL]\ncase h.e'_3.e_n\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nthis :\n \u2200 (k : \u2115) (hk : k < length (y :: l)),\n nthLe (y :: l) k hk = nthLe (x :: y :: l) (k + 1) (_ : Nat.succ k < Nat.succ (length (y :: l)))\nk : \u2115 := length l\nhn : n < k\n\u22a2 (Nat.succ (Nat.succ n) + (k + 1)) % (k + 1 + 1) = (Nat.succ n + k) % (k + 1) + 1\n[PROOFSTEP]\nrw [Nat.succ_add, \u2190 Nat.add_succ, Nat.add_mod_right, Nat.succ_add, \u2190 Nat.add_succ _ k, Nat.add_mod_right,\n Nat.mod_eq_of_lt, Nat.mod_eq_of_lt]\n[GOAL]\ncase h.e'_3.e_n\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nthis :\n \u2200 (k : \u2115) (hk : k < length (y :: l)),\n nthLe (y :: l) k hk = nthLe (x :: y :: l) (k + 1) (_ : Nat.succ k < Nat.succ (length (y :: l)))\nk : \u2115 := length l\nhn : n < k\n\u22a2 n < Nat.succ k\n[PROOFSTEP]\nexact Nat.lt_succ_of_lt hn\n[GOAL]\ncase h.e'_3.e_n\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nthis :\n \u2200 (k : \u2115) (hk : k < length (y :: l)),\n nthLe (y :: l) k hk = nthLe (x :: y :: l) (k + 1) (_ : Nat.succ k < Nat.succ (length (y :: l)))\nk : \u2115 := length l\nhn : n < k\n\u22a2 n + 1 < Nat.succ (k + 1)\n[PROOFSTEP]\nexact Nat.succ_lt_succ (Nat.lt_succ_of_lt hn)\n[GOAL]\ncase cons.cons.succ.succ.hy\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\n\u22a2 nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn \u2260 x\n[PROOFSTEP]\nintro H\n[GOAL]\ncase cons.cons.succ.succ.hy\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\nH : nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn = x\n\u22a2 False\n[PROOFSTEP]\nsuffices n.succ.succ = 0 by simpa\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\nH : nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn = x\nthis : Nat.succ (Nat.succ n) = 0\n\u22a2 False\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase cons.cons.succ.succ.hy\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\nH : nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn = x\n\u22a2 Nat.succ (Nat.succ n) = 0\n[PROOFSTEP]\nrw [nodup_iff_nthLe_inj] at h \n[GOAL]\ncase cons.cons.succ.succ.hy\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (x :: y :: l)) (h\u2082 : j < length (x :: y :: l)),\n nthLe (x :: y :: l) i h\u2081 = nthLe (x :: y :: l) j h\u2082 \u2192 i = j\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\nH : nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn = x\n\u22a2 Nat.succ (Nat.succ n) = 0\n[PROOFSTEP]\nrefine' h _ _ hn Nat.succ_pos' _\n[GOAL]\ncase cons.cons.succ.succ.hy\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (x :: y :: l)) (h\u2082 : j < length (x :: y :: l)),\n nthLe (x :: y :: l) i h\u2081 = nthLe (x :: y :: l) j h\u2082 \u2192 i = j\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\nH : nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn = x\n\u22a2 nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn = nthLe (x :: y :: l) 0 (_ : 0 < Nat.succ (length (y :: l)))\n[PROOFSTEP]\nsimpa using H\n[GOAL]\ncase cons.cons.succ.succ.hz\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\n\u22a2 nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn \u2260 y\n[PROOFSTEP]\nintro H\n[GOAL]\ncase cons.cons.succ.succ.hz\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\nH : nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn = y\n\u22a2 False\n[PROOFSTEP]\nsuffices n.succ.succ = 1 by simpa\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\nH : nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn = y\nthis : Nat.succ (Nat.succ n) = 1\n\u22a2 False\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase cons.cons.succ.succ.hz\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh : Nodup (x :: y :: l)\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\nH : nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn = y\n\u22a2 Nat.succ (Nat.succ n) = 1\n[PROOFSTEP]\nrw [nodup_iff_nthLe_inj] at h \n[GOAL]\ncase cons.cons.succ.succ.hz\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (x :: y :: l)) (h\u2082 : j < length (x :: y :: l)),\n nthLe (x :: y :: l) i h\u2081 = nthLe (x :: y :: l) j h\u2082 \u2192 i = j\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\nH : nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn = y\n\u22a2 Nat.succ (Nat.succ n) = 1\n[PROOFSTEP]\nrefine' h _ _ hn (Nat.succ_lt_succ Nat.succ_pos') _\n[GOAL]\ncase cons.cons.succ.succ.hz\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d\u00b9 : List \u03b1\nx\u271d\u00b9 : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nl\u271d : List \u03b1\nh\u271d : Nodup (x\u271d :: l\u271d)\nhn\u271d : n\u271d < length (x\u271d :: l\u271d)\ny : \u03b1\nl : List \u03b1\nhl :\n \u2200 (n : \u2115) (x : \u03b1),\n Nodup (x :: l) \u2192\n \u2200 (hn : n < length (x :: l)),\n prev (x :: l) (nthLe (x :: l) n hn) (_ : nthLe (x :: l) n hn \u2208 x :: l) =\n nthLe (x :: l) ((n + (length (x :: l) - 1)) % length (x :: l))\n (_ : (n + (length (x :: l) - 1)) % length (x :: l) < length (x :: l))\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (x :: y :: l)) (h\u2082 : j < length (x :: y :: l)),\n nthLe (x :: y :: l) i h\u2081 = nthLe (x :: y :: l) j h\u2082 \u2192 i = j\nn : \u2115\nhn : Nat.succ (Nat.succ n) < length (x :: y :: l)\nH : nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn = y\n\u22a2 nthLe (x :: y :: l) (Nat.succ (Nat.succ n)) hn = nthLe (x :: y :: l) 1 (_ : Nat.succ 0 < Nat.succ (length (y :: l)))\n[PROOFSTEP]\nsimpa using H\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh : Nodup l\n\u22a2 pmap (next l) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l) = rotate l 1\n[PROOFSTEP]\napply List.ext_nthLe\n[GOAL]\ncase hl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh : Nodup l\n\u22a2 length (pmap (next l) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l)) = length (rotate l 1)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh : Nodup l\n\u22a2 \u2200 (n : \u2115) (h\u2081 : n < length (pmap (next l) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l))) (h\u2082 : n < length (rotate l 1)),\n nthLe (pmap (next l) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l)) n h\u2081 = nthLe (rotate l 1) n h\u2082\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh : Nodup l\nn\u271d : \u2115\nh\u2081\u271d : n\u271d < length (pmap (next l) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l))\nh\u2082\u271d : n\u271d < length (rotate l 1)\n\u22a2 nthLe (pmap (next l) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l)) n\u271d h\u2081\u271d = nthLe (rotate l 1) n\u271d h\u2082\u271d\n[PROOFSTEP]\nrw [nthLe_pmap, nthLe_rotate, next_nthLe _ h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh : Nodup l\n\u22a2 pmap (prev l) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l) = rotate l (length l - 1)\n[PROOFSTEP]\napply List.ext_nthLe\n[GOAL]\ncase hl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh : Nodup l\n\u22a2 length (pmap (prev l) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l)) = length (rotate l (length l - 1))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh : Nodup l\n\u22a2 \u2200 (n : \u2115) (h\u2081 : n < length (pmap (prev l) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l)))\n (h\u2082 : n < length (rotate l (length l - 1))),\n nthLe (pmap (prev l) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l)) n h\u2081 = nthLe (rotate l (length l - 1)) n h\u2082\n[PROOFSTEP]\nintro n hn hn'\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh : Nodup l\nn : \u2115\nhn : n < length (pmap (prev l) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l))\nhn' : n < length (rotate l (length l - 1))\n\u22a2 nthLe (pmap (prev l) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l)) n hn = nthLe (rotate l (length l - 1)) n hn'\n[PROOFSTEP]\nrw [nthLe_rotate, nthLe_pmap, prev_nthLe _ h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl : List \u03b1\nh : Nodup l\nx : \u03b1\nhx : x \u2208 l\n\u22a2 prev l (next l x hx) (_ : next l x hx \u2208 l) = x\n[PROOFSTEP]\nobtain \u27e8n, hn, rfl\u27e9 := nthLe_of_mem hx\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nn : \u2115\nhn : n < length l\nhx : nthLe l n hn \u2208 l\n\u22a2 prev l (next l (nthLe l n hn) hx) (_ : next l (nthLe l n hn) hx \u2208 l) = nthLe l n hn\n[PROOFSTEP]\nsimp only [next_nthLe, prev_nthLe, h, Nat.mod_add_mod]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nn : \u2115\nhn : n < length l\nhx : nthLe l n hn \u2208 l\n\u22a2 nthLe l ((n + 1 + (length l - 1)) % length l) (_ : (n + 1 + (length l - 1)) % length l < length l) = nthLe l n hn\n[PROOFSTEP]\ncases' l with hd tl\n[GOAL]\ncase intro.intro.nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nn : \u2115\nh : Nodup []\nhn : n < length []\nhx : nthLe [] n hn \u2208 []\n\u22a2 nthLe [] ((n + 1 + (length [] - 1)) % length []) (_ : (n + 1 + (length [] - 1)) % length [] < length []) =\n nthLe [] n hn\n[PROOFSTEP]\nsimp at hx \n[GOAL]\ncase intro.intro.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nn : \u2115\nhd : \u03b1\ntl : List \u03b1\nh : Nodup (hd :: tl)\nhn : n < length (hd :: tl)\nhx : nthLe (hd :: tl) n hn \u2208 hd :: tl\n\u22a2 nthLe (hd :: tl) ((n + 1 + (length (hd :: tl) - 1)) % length (hd :: tl))\n (_ : (n + 1 + (length (hd :: tl) - 1)) % length (hd :: tl) < length (hd :: tl)) =\n nthLe (hd :: tl) n hn\n[PROOFSTEP]\nhave : (n + 1 + length tl) % (length tl + 1) = n :=\n by\n rw [length_cons] at hn \n rw [add_assoc, add_comm 1, Nat.add_mod_right, Nat.mod_eq_of_lt hn]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nn : \u2115\nhd : \u03b1\ntl : List \u03b1\nh : Nodup (hd :: tl)\nhn : n < length (hd :: tl)\nhx : nthLe (hd :: tl) n hn \u2208 hd :: tl\n\u22a2 (n + 1 + length tl) % (length tl + 1) = n\n[PROOFSTEP]\nrw [length_cons] at hn \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nn : \u2115\nhd : \u03b1\ntl : List \u03b1\nh : Nodup (hd :: tl)\nhn\u271d : n < length (hd :: tl)\nhn : n < Nat.succ (length tl)\nhx : nthLe (hd :: tl) n hn\u271d \u2208 hd :: tl\n\u22a2 (n + 1 + length tl) % (length tl + 1) = n\n[PROOFSTEP]\nrw [add_assoc, add_comm 1, Nat.add_mod_right, Nat.mod_eq_of_lt hn]\n[GOAL]\ncase intro.intro.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nn : \u2115\nhd : \u03b1\ntl : List \u03b1\nh : Nodup (hd :: tl)\nhn : n < length (hd :: tl)\nhx : nthLe (hd :: tl) n hn \u2208 hd :: tl\nthis : (n + 1 + length tl) % (length tl + 1) = n\n\u22a2 nthLe (hd :: tl) ((n + 1 + (length (hd :: tl) - 1)) % length (hd :: tl))\n (_ : (n + 1 + (length (hd :: tl) - 1)) % length (hd :: tl) < length (hd :: tl)) =\n nthLe (hd :: tl) n hn\n[PROOFSTEP]\nsimp only [length_cons, Nat.succ_sub_succ_eq_sub, tsub_zero, Nat.succ_eq_add_one, this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl : List \u03b1\nh : Nodup l\nx : \u03b1\nhx : x \u2208 l\n\u22a2 next l (prev l x hx) (_ : prev l x hx \u2208 l) = x\n[PROOFSTEP]\nobtain \u27e8n, hn, rfl\u27e9 := nthLe_of_mem hx\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nn : \u2115\nhn : n < length l\nhx : nthLe l n hn \u2208 l\n\u22a2 next l (prev l (nthLe l n hn) hx) (_ : prev l (nthLe l n hn) hx \u2208 l) = nthLe l n hn\n[PROOFSTEP]\nsimp only [next_nthLe, prev_nthLe, h, Nat.mod_add_mod]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nn : \u2115\nhn : n < length l\nhx : nthLe l n hn \u2208 l\n\u22a2 nthLe l ((n + (length l - 1) + 1) % length l) (_ : (n + (length l - 1) + 1) % length l < length l) = nthLe l n hn\n[PROOFSTEP]\ncases' l with hd tl\n[GOAL]\ncase intro.intro.nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nn : \u2115\nh : Nodup []\nhn : n < length []\nhx : nthLe [] n hn \u2208 []\n\u22a2 nthLe [] ((n + (length [] - 1) + 1) % length []) (_ : (n + (length [] - 1) + 1) % length [] < length []) =\n nthLe [] n hn\n[PROOFSTEP]\nsimp at hx \n[GOAL]\ncase intro.intro.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nn : \u2115\nhd : \u03b1\ntl : List \u03b1\nh : Nodup (hd :: tl)\nhn : n < length (hd :: tl)\nhx : nthLe (hd :: tl) n hn \u2208 hd :: tl\n\u22a2 nthLe (hd :: tl) ((n + (length (hd :: tl) - 1) + 1) % length (hd :: tl))\n (_ : (n + (length (hd :: tl) - 1) + 1) % length (hd :: tl) < length (hd :: tl)) =\n nthLe (hd :: tl) n hn\n[PROOFSTEP]\nhave : (n + length tl + 1) % (length tl + 1) = n :=\n by\n rw [length_cons] at hn \n rw [add_assoc, Nat.add_mod_right, Nat.mod_eq_of_lt hn]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nn : \u2115\nhd : \u03b1\ntl : List \u03b1\nh : Nodup (hd :: tl)\nhn : n < length (hd :: tl)\nhx : nthLe (hd :: tl) n hn \u2208 hd :: tl\n\u22a2 (n + length tl + 1) % (length tl + 1) = n\n[PROOFSTEP]\nrw [length_cons] at hn \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nn : \u2115\nhd : \u03b1\ntl : List \u03b1\nh : Nodup (hd :: tl)\nhn\u271d : n < length (hd :: tl)\nhn : n < Nat.succ (length tl)\nhx : nthLe (hd :: tl) n hn\u271d \u2208 hd :: tl\n\u22a2 (n + length tl + 1) % (length tl + 1) = n\n[PROOFSTEP]\nrw [add_assoc, Nat.add_mod_right, Nat.mod_eq_of_lt hn]\n[GOAL]\ncase intro.intro.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nn : \u2115\nhd : \u03b1\ntl : List \u03b1\nh : Nodup (hd :: tl)\nhn : n < length (hd :: tl)\nhx : nthLe (hd :: tl) n hn \u2208 hd :: tl\nthis : (n + length tl + 1) % (length tl + 1) = n\n\u22a2 nthLe (hd :: tl) ((n + (length (hd :: tl) - 1) + 1) % length (hd :: tl))\n (_ : (n + (length (hd :: tl) - 1) + 1) % length (hd :: tl) < length (hd :: tl)) =\n nthLe (hd :: tl) n hn\n[PROOFSTEP]\nsimp [this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl : List \u03b1\nh : Nodup l\nx : \u03b1\nhx : x \u2208 l\n\u22a2 prev (reverse l) x (_ : x \u2208 reverse l) = next l x hx\n[PROOFSTEP]\nobtain \u27e8k, hk, rfl\u27e9 := nthLe_of_mem hx\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\n\u22a2 prev (reverse l) (nthLe l k hk) (_ : nthLe l k hk \u2208 reverse l) = next l (nthLe l k hk) hx\n[PROOFSTEP]\nhave lpos : 0 < l.length := k.zero_le.trans_lt hk\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\nlpos : 0 < length l\n\u22a2 prev (reverse l) (nthLe l k hk) (_ : nthLe l k hk \u2208 reverse l) = next l (nthLe l k hk) hx\n[PROOFSTEP]\nhave key : l.length - 1 - k < l.length := (Nat.sub_le _ _).trans_lt (tsub_lt_self lpos Nat.succ_pos')\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n\u22a2 prev (reverse l) (nthLe l k hk) (_ : nthLe l k hk \u2208 reverse l) = next l (nthLe l k hk) hx\n[PROOFSTEP]\nrw [\u2190 nthLe_pmap l.next (fun _ h => h) (by simpa using hk)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n\u22a2 k < length (pmap (next l) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l))\n[PROOFSTEP]\nsimpa using hk\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n\u22a2 prev (reverse l) (nthLe l k hk) (_ : nthLe l k hk \u2208 reverse l) =\n nthLe (pmap (next l) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l)) k\n (_ : k < length (pmap (next l) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l)))\n[PROOFSTEP]\nsimp_rw [\u2190 nthLe_reverse l k (key.trans_le (by simp)), pmap_next_eq_rotate_one _ h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n\u22a2 length l \u2264 length (reverse l)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n\u22a2 prev (reverse l) (nthLe (reverse l) (length l - 1 - k) (_ : length l - 1 - k < length (reverse l)))\n (_ : nthLe (reverse l) (length l - 1 - k) (_ : length l - 1 - k < length (reverse l)) \u2208 reverse l) =\n nthLe (rotate l 1) k (_ : k < length (rotate l 1))\n[PROOFSTEP]\nrw [\u2190 nthLe_pmap l.reverse.prev fun _ h => h]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n\u22a2 nthLe (pmap (prev (reverse l)) (reverse l) (_ : \u2200 (x : \u03b1), x \u2208 reverse l \u2192 x \u2208 reverse l)) (length l - 1 - k)\n ?intro.intro =\n nthLe (rotate l 1) k (_ : k < length (rotate l 1))\n[PROOFSTEP]\nsimp_rw [pmap_prev_eq_rotate_length_sub_one _ (nodup_reverse.mpr h), rotate_reverse, length_reverse,\n Nat.mod_eq_of_lt (tsub_lt_self lpos Nat.succ_pos'), tsub_tsub_cancel_of_le (Nat.succ_le_of_lt lpos)]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n\u22a2 nthLe (reverse (rotate l (Nat.succ 0))) (length l - 1 - k)\n (_ : length l - 1 - k < length (reverse (rotate l (Nat.succ 0)))) =\n nthLe (rotate l 1) k (_ : k < length (rotate l 1))\n[PROOFSTEP]\nrw [\u2190 nthLe_reverse]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n\u22a2 nthLe (reverse (reverse (rotate l (Nat.succ 0)))) (length (reverse (rotate l (Nat.succ 0))) - 1 - (length l - 1 - k))\n ?intro.intro.h1 =\n nthLe (rotate l 1) k (_ : k < length (rotate l 1))\n[PROOFSTEP]\nsimp [tsub_tsub_cancel_of_le (Nat.le_pred_of_lt hk)]\n[GOAL]\ncase intro.intro.h1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n\u22a2 length (reverse (rotate l (Nat.succ 0))) - 1 - (length l - 1 - k) < length (reverse (reverse (rotate l (Nat.succ 0))))\n[PROOFSTEP]\nsimpa using (Nat.sub_le _ _).trans_lt (tsub_lt_self lpos Nat.succ_pos')\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n\u22a2 length l - 1 - k < length (pmap (prev (reverse l)) (reverse l) (_ : \u2200 (x : \u03b1), x \u2208 reverse l \u2192 x \u2208 reverse l))\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl : List \u03b1\nh : Nodup l\nx : \u03b1\nhx : x \u2208 l\n\u22a2 next (reverse l) x (_ : x \u2208 reverse l) = prev l x hx\n[PROOFSTEP]\nconvert (prev_reverse_eq_next l.reverse (nodup_reverse.mpr h) x (mem_reverse.mpr hx)).symm\n[GOAL]\ncase h.e'_3.h.e'_3\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl : List \u03b1\nh : Nodup l\nx : \u03b1\nhx : x \u2208 l\n\u22a2 l = reverse (reverse l)\n[PROOFSTEP]\nexact (reverse_reverse l).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl l' : List \u03b1\nh : l ~r l'\nhn : Nodup l\nx : \u03b1\nhx : x \u2208 l\n\u22a2 next l x hx = next l' x (_ : x \u2208 l')\n[PROOFSTEP]\nobtain \u27e8k, hk, rfl\u27e9 := nthLe_of_mem hx\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl l' : List \u03b1\nh : l ~r l'\nhn : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\n\u22a2 next l (nthLe l k hk) hx = next l' (nthLe l k hk) (_ : nthLe l k hk \u2208 l')\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := id h\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nhn : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\nn : \u2115\nh : l ~r rotate l n\n\u22a2 next l (nthLe l k hk) hx = next (rotate l n) (nthLe l k hk) (_ : nthLe l k hk \u2208 rotate l n)\n[PROOFSTEP]\nrw [next_nthLe _ hn]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nhn : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\nn : \u2115\nh : l ~r rotate l n\n\u22a2 nthLe l ((k + 1) % length l) (_ : (k + 1) % length l < length l) =\n next (rotate l n) (nthLe l k hk) (_ : nthLe l k hk \u2208 rotate l n)\n[PROOFSTEP]\nsimp_rw [\u2190 nthLe_rotate' _ n k]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nhn : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\nn : \u2115\nh : l ~r rotate l n\n\u22a2 nthLe l ((k + 1) % length l) (_ : (k + 1) % length l < length l) =\n next (rotate l n)\n (nthLe (rotate l n) ((length l - n % length l + k) % length l)\n (_ : (length l - n % length l + k) % length l < length (rotate l n)))\n (_ :\n nthLe (rotate l n) ((length l - n % length l + k) % length l)\n (_ : (length l - n % length l + k) % length l < length (rotate l n)) \u2208\n rotate l n)\n[PROOFSTEP]\nrw [next_nthLe _ (h.nodup_iff.mp hn), \u2190 nthLe_rotate' _ n]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\nhn : Nodup l\nk : \u2115\nhk : k < length l\nhx : nthLe l k hk \u2208 l\nn : \u2115\nh : l ~r rotate l n\n\u22a2 nthLe (rotate l n) ((length l - n % length l + (k + 1) % length l) % length l)\n (_ : (length l - n % length l + (k + 1) % length l) % length l < length (rotate l n)) =\n nthLe (rotate l n) (((length l - n % length l + k) % length l + 1) % length (rotate l n))\n (_ : ((length l - n % length l + k) % length l + 1) % length (rotate l n) < length (rotate l n))\n[PROOFSTEP]\nsimp [add_assoc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl l' : List \u03b1\nh : l ~r l'\nhn : Nodup l\nx : \u03b1\nhx : x \u2208 l\n\u22a2 prev l x hx = prev l' x (_ : x \u2208 l')\n[PROOFSTEP]\nrw [\u2190 next_reverse_eq_prev _ hn, \u2190 next_reverse_eq_prev _ (h.nodup_iff.mp hn)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl l' : List \u03b1\nh : l ~r l'\nhn : Nodup l\nx : \u03b1\nhx : x \u2208 l\n\u22a2 next (reverse l) x (_ : x \u2208 reverse l) = next (reverse l') x (_ : x \u2208 reverse l')\n[PROOFSTEP]\nexact isRotated_next_eq h.reverse (nodup_reverse.mpr hn) _\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\na : \u03b1\n\u22a2 rotate (a :: l) 1 = l ++ [a]\n[PROOFSTEP]\nrw [rotate_cons_succ, rotate_zero]\n[GOAL]\n\u03b1 : Type u_1\nC : Cycle \u03b1 \u2192 Prop\ns : Cycle \u03b1\nH0 : C nil\nHI : \u2200 (a : \u03b1) (l : List \u03b1), C \u2191l \u2192 C \u2191(a :: l)\nl : List \u03b1\n\u22a2 C (Quotient.mk'' l)\n[PROOFSTEP]\nrefine List.recOn l ?_ ?_\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\nC : Cycle \u03b1 \u2192 Prop\ns : Cycle \u03b1\nH0 : C nil\nHI : \u2200 (a : \u03b1) (l : List \u03b1), C \u2191l \u2192 C \u2191(a :: l)\nl : List \u03b1\n\u22a2 C (Quotient.mk'' [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\nC : Cycle \u03b1 \u2192 Prop\ns : Cycle \u03b1\nH0 : C nil\nHI : \u2200 (a : \u03b1) (l : List \u03b1), C \u2191l \u2192 C \u2191(a :: l)\nl : List \u03b1\n\u22a2 \u2200 (head : \u03b1) (tail : List \u03b1), C (Quotient.mk'' tail) \u2192 C (Quotient.mk'' (head :: tail))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\nC : Cycle \u03b1 \u2192 Prop\ns : Cycle \u03b1\nH0 : C nil\nHI : \u2200 (a : \u03b1) (l : List \u03b1), C \u2191l \u2192 C \u2191(a :: l)\nl : List \u03b1\n\u22a2 C nil\ncase refine_2\n\u03b1 : Type u_1\nC : Cycle \u03b1 \u2192 Prop\ns : Cycle \u03b1\nH0 : C nil\nHI : \u2200 (a : \u03b1) (l : List \u03b1), C \u2191l \u2192 C \u2191(a :: l)\nl : List \u03b1\n\u22a2 \u2200 (head : \u03b1) (tail : List \u03b1), C \u2191tail \u2192 C \u2191(head :: tail)\n[PROOFSTEP]\nassumption'\n[GOAL]\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx\u271d : List \u03b1\n\u22a2 reverse (reverse (Quot.mk Setoid.r x\u271d)) = Quot.mk Setoid.r x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ns : Cycle \u03b1\n\u22a2 Subsingleton (reverse s) \u2194 Subsingleton s\n[PROOFSTEP]\nsimp [length_subsingleton_iff]\n[GOAL]\n\u03b1 : Type u_1\ns : Cycle \u03b1\nh : Subsingleton s\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 x = y\n[PROOFSTEP]\ninduction' s using Quot.inductionOn with l\n[GOAL]\ncase h\n\u03b1 : Type u_1\ns : Cycle \u03b1\nh\u271d : Subsingleton s\nl : List \u03b1\nh : Subsingleton (Quot.mk Setoid.r l)\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 Quot.mk Setoid.r l \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 Quot.mk Setoid.r l \u2192 x = y\n[PROOFSTEP]\nsimp only [length_subsingleton_iff, length_coe, mk_eq_coe, le_iff_lt_or_eq, Nat.lt_add_one_iff, length_eq_zero,\n length_eq_one, Nat.not_lt_zero, false_or_iff] at h \n[GOAL]\ncase h\n\u03b1 : Type u_1\ns : Cycle \u03b1\nh\u271d : Subsingleton s\nl : List \u03b1\nh : l = [] \u2228 \u2203 a, l = [a]\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 Quot.mk Setoid.r l \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 Quot.mk Setoid.r l \u2192 x = y\n[PROOFSTEP]\nrcases h with (rfl | \u27e8z, rfl\u27e9)\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\ns : Cycle \u03b1\nh : Subsingleton s\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 Quot.mk Setoid.r [] \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 Quot.mk Setoid.r [] \u2192 x = y\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.inr.intro\n\u03b1 : Type u_1\ns : Cycle \u03b1\nh : Subsingleton s\nz : \u03b1\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 Quot.mk Setoid.r [z] \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 Quot.mk Setoid.r [z] \u2192 x = y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nhl : Nodup l\n\u22a2 Nontrivial \u2191l \u2194 2 \u2264 List.length l\n[PROOFSTEP]\nrw [Nontrivial]\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nhl : Nodup l\n\u22a2 (\u2203 x y _h, x \u2208 \u2191l \u2227 y \u2208 \u2191l) \u2194 2 \u2264 List.length l\n[PROOFSTEP]\nrcases l with (_ | \u27e8hd, _ | \u27e8hd', tl\u27e9\u27e9)\n[GOAL]\ncase nil\n\u03b1 : Type u_1\nhl : Nodup []\n\u22a2 (\u2203 x y _h, x \u2208 \u2191[] \u2227 y \u2208 \u2191[]) \u2194 2 \u2264 List.length []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.nil\n\u03b1 : Type u_1\nhd : \u03b1\nhl : Nodup [hd]\n\u22a2 (\u2203 x y _h, x \u2208 \u2191[hd] \u2227 y \u2208 \u2191[hd]) \u2194 2 \u2264 List.length [hd]\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.cons\n\u03b1 : Type u_1\nhd hd' : \u03b1\ntl : List \u03b1\nhl : Nodup (hd :: hd' :: tl)\n\u22a2 (\u2203 x y _h, x \u2208 \u2191(hd :: hd' :: tl) \u2227 y \u2208 \u2191(hd :: hd' :: tl)) \u2194 2 \u2264 List.length (hd :: hd' :: tl)\n[PROOFSTEP]\nsimp only [mem_cons, exists_prop, mem_coe_iff, List.length, Ne.def, Nat.succ_le_succ_iff, zero_le, iff_true_iff]\n[GOAL]\ncase cons.cons\n\u03b1 : Type u_1\nhd hd' : \u03b1\ntl : List \u03b1\nhl : Nodup (hd :: hd' :: tl)\n\u22a2 \u2203 x y, \u00acx = y \u2227 (x = hd \u2228 x = hd' \u2228 x \u2208 tl) \u2227 (y = hd \u2228 y = hd' \u2228 y \u2208 tl)\n[PROOFSTEP]\nrefine' \u27e8hd, hd', _, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\nhd hd' : \u03b1\ntl : List \u03b1\nhl : Nodup (hd :: hd' :: tl)\n\u22a2 (hd = hd \u2228 hd = hd' \u2228 hd \u2208 tl) \u2227 (hd' = hd \u2228 hd' = hd' \u2228 hd' \u2208 tl)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.cons\n\u03b1 : Type u_1\nhd hd' : \u03b1\ntl : List \u03b1\nhl : Nodup (hd :: hd' :: tl)\n\u22a2 \u00achd = hd'\n[PROOFSTEP]\nsimp only [not_or, mem_cons, nodup_cons] at hl \n[GOAL]\ncase cons.cons\n\u03b1 : Type u_1\nhd hd' : \u03b1\ntl : List \u03b1\nhl : (\u00achd = hd' \u2227 \u00achd \u2208 tl) \u2227 \u00achd' \u2208 tl \u2227 Nodup tl\n\u22a2 \u00achd = hd'\n[PROOFSTEP]\nexact hl.left.left\n[GOAL]\n\u03b1 : Type u_1\ns : Cycle \u03b1\n\u22a2 Nontrivial (reverse s) \u2194 Nontrivial s\n[PROOFSTEP]\nsimp [Nontrivial]\n[GOAL]\n\u03b1 : Type u_1\ns : Cycle \u03b1\nh : Nontrivial s\n\u22a2 2 \u2264 length s\n[PROOFSTEP]\nobtain \u27e8x, y, hxy, hx, hy\u27e9 := h\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 2 \u2264 length s\n[PROOFSTEP]\ninduction' s using Quot.inductionOn with l\n[GOAL]\ncase intro.intro.intro.intro.h\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx\u271d : x \u2208 s\nhy\u271d : y \u2208 s\nl : List \u03b1\nhx : x \u2208 Quot.mk Setoid.r l\nhy : y \u2208 Quot.mk Setoid.r l\n\u22a2 2 \u2264 length (Quot.mk Setoid.r l)\n[PROOFSTEP]\nrcases l with (_ | \u27e8hd, _ | \u27e8hd', tl\u27e9\u27e9)\n[GOAL]\ncase intro.intro.intro.intro.h.nil\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx\u271d : x \u2208 s\nhy\u271d : y \u2208 s\nhx : x \u2208 Quot.mk Setoid.r []\nhy : y \u2208 Quot.mk Setoid.r []\n\u22a2 2 \u2264 length (Quot.mk Setoid.r [])\n[PROOFSTEP]\nsimp at hx \n[GOAL]\ncase intro.intro.intro.intro.h.cons.nil\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx\u271d : x \u2208 s\nhy\u271d : y \u2208 s\nhd : \u03b1\nhx : x \u2208 Quot.mk Setoid.r [hd]\nhy : y \u2208 Quot.mk Setoid.r [hd]\n\u22a2 2 \u2264 length (Quot.mk Setoid.r [hd])\n[PROOFSTEP]\nsimp only [mem_coe_iff, mk_eq_coe, mem_singleton] at hx hy \n[GOAL]\ncase intro.intro.intro.intro.h.cons.nil\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx\u271d : x \u2208 s\nhy\u271d : y \u2208 s\nhd : \u03b1\nhx : x = hd\nhy : y = hd\n\u22a2 2 \u2264 length (Quot.mk Setoid.r [hd])\n[PROOFSTEP]\nsimp [hx, hy] at hxy \n[GOAL]\ncase intro.intro.intro.intro.h.cons.cons\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx\u271d : x \u2208 s\nhy\u271d : y \u2208 s\nhd hd' : \u03b1\ntl : List \u03b1\nhx : x \u2208 Quot.mk Setoid.r (hd :: hd' :: tl)\nhy : y \u2208 Quot.mk Setoid.r (hd :: hd' :: tl)\n\u22a2 2 \u2264 length (Quot.mk Setoid.r (hd :: hd' :: tl))\n[PROOFSTEP]\nsimp [Nat.succ_le_succ_iff]\n[GOAL]\n\u03b1 : Type u_1\ns : Cycle \u03b1\nh : Subsingleton s\n\u22a2 Nodup s\n[PROOFSTEP]\ninduction' s using Quot.inductionOn with l\n[GOAL]\ncase h\n\u03b1 : Type u_1\ns : Cycle \u03b1\nh\u271d : Subsingleton s\nl : List \u03b1\nh : Subsingleton (Quot.mk Setoid.r l)\n\u22a2 Nodup (Quot.mk Setoid.r l)\n[PROOFSTEP]\ncases' l with hd tl\n[GOAL]\ncase h.nil\n\u03b1 : Type u_1\ns : Cycle \u03b1\nh\u271d : Subsingleton s\nh : Subsingleton (Quot.mk Setoid.r [])\n\u22a2 Nodup (Quot.mk Setoid.r [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.cons\n\u03b1 : Type u_1\ns : Cycle \u03b1\nh\u271d : Subsingleton s\nhd : \u03b1\ntl : List \u03b1\nh : Subsingleton (Quot.mk Setoid.r (hd :: tl))\n\u22a2 Nodup (Quot.mk Setoid.r (hd :: tl))\n[PROOFSTEP]\nhave : tl = [] := by simpa [Subsingleton, length_eq_zero, Nat.succ_le_succ_iff] using h\n[GOAL]\n\u03b1 : Type u_1\ns : Cycle \u03b1\nh\u271d : Subsingleton s\nhd : \u03b1\ntl : List \u03b1\nh : Subsingleton (Quot.mk Setoid.r (hd :: tl))\n\u22a2 tl = []\n[PROOFSTEP]\nsimpa [Subsingleton, length_eq_zero, Nat.succ_le_succ_iff] using h\n[GOAL]\ncase h.cons\n\u03b1 : Type u_1\ns : Cycle \u03b1\nh\u271d : Subsingleton s\nhd : \u03b1\ntl : List \u03b1\nh : Subsingleton (Quot.mk Setoid.r (hd :: tl))\nthis : tl = []\n\u22a2 Nodup (Quot.mk Setoid.r (hd :: tl))\n[PROOFSTEP]\nsimp [this]\n[GOAL]\n\u03b1 : Type u_1\ns : Cycle \u03b1\nh : Nodup s\n\u22a2 Nontrivial s \u2194 \u00acSubsingleton s\n[PROOFSTEP]\nrw [length_subsingleton_iff]\n[GOAL]\n\u03b1 : Type u_1\ns : Cycle \u03b1\nh : Nodup s\n\u22a2 Nontrivial s \u2194 \u00aclength s \u2264 1\n[PROOFSTEP]\ninduction s using Quotient.inductionOn'\n[GOAL]\ncase h\n\u03b1 : Type u_1\na\u271d : List \u03b1\nh : Nodup (Quotient.mk'' a\u271d)\n\u22a2 Nontrivial (Quotient.mk'' a\u271d) \u2194 \u00aclength (Quotient.mk'' a\u271d) \u2264 1\n[PROOFSTEP]\nsimp only [mk''_eq_coe, nodup_coe_iff] at h \n[GOAL]\ncase h\n\u03b1 : Type u_1\na\u271d : List \u03b1\nh : List.Nodup a\u271d\n\u22a2 Nontrivial (Quotient.mk'' a\u271d) \u2194 \u00aclength (Quotient.mk'' a\u271d) \u2264 1\n[PROOFSTEP]\nsimp [h, Nat.succ_le_iff]\n[GOAL]\n\u03b1 : Type u_1\ns : Cycle \u03b1\n\u22a2 \u2200 (a : List \u03b1), \u2191Multiset.card (toMultiset (Quotient.mk'' a)) = length (Quotient.mk'' a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ns : Cycle \u03b1\n\u22a2 \u2200 (a : List \u03b1), toMultiset (Quotient.mk'' a) = 0 \u2194 Quotient.mk'' a = nil\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\ns : Cycle \u03b1\n\u22a2 \u2200 (a : List \u03b1), map f (Quotient.mk'' a) = nil \u2194 Quotient.mk'' a = nil\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Cycle \u03b1\n\u22a2 \u2200 (a : List \u03b1), b \u2208 map f (Quotient.mk'' a) \u2194 \u2203 a_1, a_1 \u2208 Quotient.mk'' a \u2227 f a_1 = b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ns : Cycle \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : Setoid.r l\u2081 l\u2082\n\u22a2 (fun l => \u2191(cyclicPermutations l)) l\u2081 = (fun l => \u2191(cyclicPermutations l)) l\u2082\n[PROOFSTEP]\nsimpa using h.cyclicPermutations.perm\n[GOAL]\n\u03b1 : Type u_1\ns : Cycle \u03b1\nl\u271d l : List \u03b1\n\u22a2 l\u271d \u2208 lists (Quotient.mk'' l) \u2194 \u2191l\u271d = Quotient.mk'' l\n[PROOFSTEP]\nrw [lists, Quotient.liftOn'_mk'']\n[GOAL]\n\u03b1 : Type u_1\ns : Cycle \u03b1\nl\u271d l : List \u03b1\n\u22a2 l\u271d \u2208 \u2191(cyclicPermutations l) \u2194 \u2191l\u271d = Quotient.mk'' l\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 lists nil = \u2191[[]]\n[PROOFSTEP]\nrw [nil, lists_coe, cyclicPermutations_nil]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u22a2 \u00acNontrivial \u2191[]\n[PROOFSTEP]\nsimp [Nontrivial]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\n\u22a2 \u00acNontrivial \u2191[x]\n[PROOFSTEP]\nsimp [Nontrivial]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nl : List \u03b1\nh : x = y\n\u22a2 Nontrivial \u2191(x :: y :: l) \u2194 Nontrivial \u2191(x :: l)\n[PROOFSTEP]\nsimp [h, Nontrivial]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nl : List \u03b1\nh : \u00acx = y\n\u22a2 x \u2208 \u2191(x :: y :: l)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nl : List \u03b1\nh : \u00acx = y\n\u22a2 y \u2208 \u2191(x :: y :: l)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nl : { l // List.Nodup l }\n\u22a2 Nodup \u2191\u2191l\n[PROOFSTEP]\nsimpa using l.prop\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nx\u271d : { s // Nodup s }\ns : Cycle \u03b1\nhs : Nodup s\n\u22a2 \u2203 a, (fun l => { val := \u2191\u2191l, property := (_ : Nodup \u2191\u2191l) }) a = { val := s, property := hs }\n[PROOFSTEP]\ninduction' s using Quotient.inductionOn' with s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nx\u271d : { s // Nodup s }\ns\u271d : Cycle \u03b1\nhs\u271d : Nodup s\u271d\ns : List \u03b1\nhs : Nodup (Quotient.mk'' s)\n\u22a2 \u2203 a, (fun l => { val := \u2191\u2191l, property := (_ : Nodup \u2191\u2191l) }) a = { val := Quotient.mk'' s, property := hs }\n[PROOFSTEP]\nexact \u27e8\u27e8s, hs\u27e9, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nx\u271d : { s // Nodup s }\ns\u271d : Cycle \u03b1\nhs\u271d : Nodup s\u271d\ns : List \u03b1\nhs : Nodup (Quotient.mk'' s)\n\u22a2 (fun l => { val := \u2191\u2191l, property := (_ : Nodup \u2191\u2191l) }) { val := s, property := hs } =\n { val := Quotient.mk'' s, property := hs }\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u22a2 \u2200 (x : Cycle \u03b1),\n x \u2208 Finset.filter Nontrivial (Finset.map (Function.Embedding.subtype fun s => Nodup s) Finset.univ) \u2194\n Nodup x \u2227 Nontrivial x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\n\u22a2 \u2200 (a : List \u03b1), toFinset (Quotient.mk'' a) = \u2205 \u2194 Quotient.mk'' a = nil\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : Setoid.r l\u2081 l\u2082\nh\u2081 : Nodup (Quot.mk Setoid.r l\u2081)\nh\u2082 : Nodup (Quot.mk Setoid.r l\u2082)\n_he : HEq h\u2081 h\u2082\nx y : \u03b1\nhxy : HEq x y\n\u22a2 x \u2208 Quot.mk Setoid.r l\u2081 \u2194 y \u2208 Quot.mk Setoid.r l\u2082\n[PROOFSTEP]\nrw [eq_of_heq hxy]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : Setoid.r l\u2081 l\u2082\nh\u2081 : Nodup (Quot.mk Setoid.r l\u2081)\nh\u2082 : Nodup (Quot.mk Setoid.r l\u2082)\n_he : HEq h\u2081 h\u2082\nx y : \u03b1\nhxy : HEq x y\n\u22a2 y \u2208 Quot.mk Setoid.r l\u2081 \u2194 y \u2208 Quot.mk Setoid.r l\u2082\n[PROOFSTEP]\nsimpa [eq_of_heq hxy] using h.mem_iff\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : Setoid.r l\u2081 l\u2082\nh\u2081 : Nodup (Quot.mk Setoid.r l\u2081)\nh\u2082 : Nodup (Quot.mk Setoid.r l\u2082)\n_he : HEq h\u2081 h\u2082\nx y : \u03b1\nhxy : HEq x y\nhm : x \u2208 Quot.mk Setoid.r l\u2081\nhm' : y \u2208 Quot.mk Setoid.r l\u2082\nhe' : HEq hm hm'\n\u22a2 (fun l _hn x hx => List.next l x hx) l\u2081 h\u2081 x hm = (fun l _hn x hx => List.next l x hx) l\u2082 h\u2082 y hm'\n[PROOFSTEP]\nrw [heq_iff_eq] at hxy \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : Setoid.r l\u2081 l\u2082\nh\u2081 : Nodup (Quot.mk Setoid.r l\u2081)\nh\u2082 : Nodup (Quot.mk Setoid.r l\u2082)\n_he : HEq h\u2081 h\u2082\nx y : \u03b1\nhxy : x = y\nhm : x \u2208 Quot.mk Setoid.r l\u2081\nhm' : y \u2208 Quot.mk Setoid.r l\u2082\nhe' : HEq hm hm'\n\u22a2 (fun l _hn x hx => List.next l x hx) l\u2081 h\u2081 x hm = (fun l _hn x hx => List.next l x hx) l\u2082 h\u2082 y hm'\n[PROOFSTEP]\nsubst x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : Setoid.r l\u2081 l\u2082\nh\u2081 : Nodup (Quot.mk Setoid.r l\u2081)\nh\u2082 : Nodup (Quot.mk Setoid.r l\u2082)\n_he : HEq h\u2081 h\u2082\ny : \u03b1\nhm' : y \u2208 Quot.mk Setoid.r l\u2082\nhm : y \u2208 Quot.mk Setoid.r l\u2081\nhe' : HEq hm hm'\n\u22a2 (fun l _hn x hx => List.next l x hx) l\u2081 h\u2081 y hm = (fun l _hn x hx => List.next l x hx) l\u2082 h\u2082 y hm'\n[PROOFSTEP]\nsimpa using isRotated_next_eq h h\u2081 _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : Setoid.r l\u2081 l\u2082\nh\u2081 : Nodup (Quot.mk Setoid.r l\u2081)\nh\u2082 : Nodup (Quot.mk Setoid.r l\u2082)\n_he : HEq h\u2081 h\u2082\nx y : \u03b1\nhxy : HEq x y\n\u22a2 x \u2208 Quot.mk Setoid.r l\u2081 \u2194 y \u2208 Quot.mk Setoid.r l\u2082\n[PROOFSTEP]\nrw [eq_of_heq hxy]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : Setoid.r l\u2081 l\u2082\nh\u2081 : Nodup (Quot.mk Setoid.r l\u2081)\nh\u2082 : Nodup (Quot.mk Setoid.r l\u2082)\n_he : HEq h\u2081 h\u2082\nx y : \u03b1\nhxy : HEq x y\n\u22a2 y \u2208 Quot.mk Setoid.r l\u2081 \u2194 y \u2208 Quot.mk Setoid.r l\u2082\n[PROOFSTEP]\nsimpa [eq_of_heq hxy] using h.mem_iff\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : Setoid.r l\u2081 l\u2082\nh\u2081 : Nodup (Quot.mk Setoid.r l\u2081)\nh\u2082 : Nodup (Quot.mk Setoid.r l\u2082)\n_he : HEq h\u2081 h\u2082\nx y : \u03b1\nhxy : HEq x y\nhm : x \u2208 Quot.mk Setoid.r l\u2081\nhm' : y \u2208 Quot.mk Setoid.r l\u2082\nhe' : HEq hm hm'\n\u22a2 (fun l _hn x hx => List.prev l x hx) l\u2081 h\u2081 x hm = (fun l _hn x hx => List.prev l x hx) l\u2082 h\u2082 y hm'\n[PROOFSTEP]\nrw [heq_iff_eq] at hxy \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : Setoid.r l\u2081 l\u2082\nh\u2081 : Nodup (Quot.mk Setoid.r l\u2081)\nh\u2082 : Nodup (Quot.mk Setoid.r l\u2082)\n_he : HEq h\u2081 h\u2082\nx y : \u03b1\nhxy : x = y\nhm : x \u2208 Quot.mk Setoid.r l\u2081\nhm' : y \u2208 Quot.mk Setoid.r l\u2082\nhe' : HEq hm hm'\n\u22a2 (fun l _hn x hx => List.prev l x hx) l\u2081 h\u2081 x hm = (fun l _hn x hx => List.prev l x hx) l\u2082 h\u2082 y hm'\n[PROOFSTEP]\nsubst x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : Setoid.r l\u2081 l\u2082\nh\u2081 : Nodup (Quot.mk Setoid.r l\u2081)\nh\u2082 : Nodup (Quot.mk Setoid.r l\u2082)\n_he : HEq h\u2081 h\u2082\ny : \u03b1\nhm' : y \u2208 Quot.mk Setoid.r l\u2082\nhm : y \u2208 Quot.mk Setoid.r l\u2081\nhe' : HEq hm hm'\n\u22a2 (fun l _hn x hx => List.prev l x hx) l\u2081 h\u2081 y hm = (fun l _hn x hx => List.prev l x hx) l\u2082 h\u2082 y hm'\n[PROOFSTEP]\nsimpa using isRotated_prev_eq h h\u2081 _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\nhs : Nodup s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 next (reverse s) (_ : Nodup (reverse s)) x (_ : x \u2208 reverse s) = prev s hs x hx\n[PROOFSTEP]\nsimp [\u2190 prev_reverse_eq_next]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\nhs : Nodup (reverse s)\nx : \u03b1\nhx : x \u2208 reverse s\n\u22a2 next (reverse s) hs x hx = prev s (_ : Nodup s) x (_ : x \u2208 s)\n[PROOFSTEP]\nsimp [\u2190 prev_reverse_eq_next]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\nhs : Nodup s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 next s hs x hx \u2208 s\n[PROOFSTEP]\ninduction s using Quot.inductionOn\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\na\u271d : List \u03b1\nhs : Nodup (Quot.mk Setoid.r a\u271d)\nhx : x \u2208 Quot.mk Setoid.r a\u271d\n\u22a2 next (Quot.mk Setoid.r a\u271d) hs x hx \u2208 Quot.mk Setoid.r a\u271d\n[PROOFSTEP]\napply next_mem\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\na\u271d : List \u03b1\nhs : Nodup (Quot.mk Setoid.r a\u271d)\nhx : x \u2208 Quot.mk Setoid.r a\u271d\n\u22a2 x \u2208 a\u271d\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\nhs : Nodup s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 prev s hs x hx \u2208 s\n[PROOFSTEP]\nrw [\u2190 next_reverse_eq_prev, \u2190 mem_reverse_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns : Cycle \u03b1\nhs : Nodup s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 next (reverse s) (_ : Nodup (reverse s)) x (_ : x \u2208 reverse s) \u2208 reverse s\n[PROOFSTEP]\napply next_mem\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\na b : List \u03b1\nhab : Setoid.r a b\n\u22a2 (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n a \u2194\n (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n b\n[PROOFSTEP]\ncases' a with a l\n[GOAL]\ncase nil\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\nb : List \u03b1\nhab : Setoid.r [] b\n\u22a2 (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n [] \u2194\n (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n b\n[PROOFSTEP]\ncases' b with b m\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\nb : List \u03b1\na : \u03b1\nl : List \u03b1\nhab : Setoid.r (a :: l) b\n\u22a2 (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n (a :: l) \u2194\n (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n b\n[PROOFSTEP]\ncases' b with b m\n[GOAL]\ncase nil.nil\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\nhab : Setoid.r [] []\n\u22a2 (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n [] \u2194\n (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase nil.cons\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\nb : \u03b1\nm : List \u03b1\nhab : Setoid.r [] (b :: m)\n\u22a2 (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n [] \u2194\n (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n (b :: m)\n[PROOFSTEP]\nhave := isRotated_nil_iff'.1 hab\n[GOAL]\ncase nil.cons\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\nb : \u03b1\nm : List \u03b1\nhab : Setoid.r [] (b :: m)\nthis : [] = b :: m\n\u22a2 (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n [] \u2194\n (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n (b :: m)\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase cons.nil\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\na : \u03b1\nl : List \u03b1\nhab : Setoid.r (a :: l) []\n\u22a2 (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n (a :: l) \u2194\n (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n []\n[PROOFSTEP]\nhave := isRotated_nil_iff.1 hab\n[GOAL]\ncase cons.nil\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\na : \u03b1\nl : List \u03b1\nhab : Setoid.r (a :: l) []\nthis : a :: l = []\n\u22a2 (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n (a :: l) \u2194\n (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n []\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase cons.cons\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\na : \u03b1\nl : List \u03b1\nb : \u03b1\nm : List \u03b1\nhab : Setoid.r (a :: l) (b :: m)\n\u22a2 (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n (a :: l) \u2194\n (fun l =>\n match l with\n | [] => True\n | a :: m => List.Chain r a (m ++ [a]))\n (b :: m)\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase cons.cons\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\na : \u03b1\nl : List \u03b1\nb : \u03b1\nm : List \u03b1\nhab : Setoid.r (a :: l) (b :: m)\n\u22a2 List.Chain r a (l ++ [a]) \u2194 List.Chain r b (m ++ [b])\n[PROOFSTEP]\ncases' hab with n hn\n[GOAL]\ncase cons.cons.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\na : \u03b1\nl : List \u03b1\nb : \u03b1\nm : List \u03b1\nn : \u2115\nhn : rotate (a :: l) n = b :: m\n\u22a2 List.Chain r a (l ++ [a]) \u2194 List.Chain r b (m ++ [b])\n[PROOFSTEP]\ninduction' n with d hd generalizing a b l m\n[GOAL]\ncase cons.cons.intro.zero\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\na\u271d : \u03b1\nl\u271d : List \u03b1\nb\u271d : \u03b1\nm\u271d : List \u03b1\nn : \u2115\nhn\u271d : rotate (a\u271d :: l\u271d) n = b\u271d :: m\u271d\na : \u03b1\nl : List \u03b1\nb : \u03b1\nm : List \u03b1\nhn : rotate (a :: l) Nat.zero = b :: m\n\u22a2 List.Chain r a (l ++ [a]) \u2194 List.Chain r b (m ++ [b])\n[PROOFSTEP]\nsimp only [Nat.zero_eq, rotate_zero, cons.injEq] at hn \n[GOAL]\ncase cons.cons.intro.zero\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\na\u271d : \u03b1\nl\u271d : List \u03b1\nb\u271d : \u03b1\nm\u271d : List \u03b1\nn : \u2115\nhn\u271d : rotate (a\u271d :: l\u271d) n = b\u271d :: m\u271d\na : \u03b1\nl : List \u03b1\nb : \u03b1\nm : List \u03b1\nhn : a = b \u2227 l = m\n\u22a2 List.Chain r a (l ++ [a]) \u2194 List.Chain r b (m ++ [b])\n[PROOFSTEP]\nrw [hn.1, hn.2]\n[GOAL]\ncase cons.cons.intro.succ\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\na\u271d : \u03b1\nl\u271d : List \u03b1\nb\u271d : \u03b1\nm\u271d : List \u03b1\nn : \u2115\nhn\u271d : rotate (a\u271d :: l\u271d) n = b\u271d :: m\u271d\nd : \u2115\nhd :\n \u2200 (a : \u03b1) (l : List \u03b1) (b : \u03b1) (m : List \u03b1),\n rotate (a :: l) d = b :: m \u2192 (List.Chain r a (l ++ [a]) \u2194 List.Chain r b (m ++ [b]))\na : \u03b1\nl : List \u03b1\nb : \u03b1\nm : List \u03b1\nhn : rotate (a :: l) (Nat.succ d) = b :: m\n\u22a2 List.Chain r a (l ++ [a]) \u2194 List.Chain r b (m ++ [b])\n[PROOFSTEP]\ncases' l with c s\n[GOAL]\ncase cons.cons.intro.succ.nil\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\na\u271d : \u03b1\nl : List \u03b1\nb\u271d : \u03b1\nm\u271d : List \u03b1\nn : \u2115\nhn\u271d : rotate (a\u271d :: l) n = b\u271d :: m\u271d\nd : \u2115\nhd :\n \u2200 (a : \u03b1) (l : List \u03b1) (b : \u03b1) (m : List \u03b1),\n rotate (a :: l) d = b :: m \u2192 (List.Chain r a (l ++ [a]) \u2194 List.Chain r b (m ++ [b]))\na b : \u03b1\nm : List \u03b1\nhn : rotate [a] (Nat.succ d) = b :: m\n\u22a2 List.Chain r a ([] ++ [a]) \u2194 List.Chain r b (m ++ [b])\n[PROOFSTEP]\nsimp only [rotate_cons_succ, nil_append, rotate_singleton, cons.injEq] at hn \n[GOAL]\ncase cons.cons.intro.succ.nil\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Cycle \u03b1\na\u271d : \u03b1\nl : List \u03b1\nb\u271d : \u03b1\nm\u271d : List \u03b1\nn : \u2115\nhn\u271d : rotate (a\u271d :: l) n = b\u271d :: m\u271d\nd : \u2115\nhd :\n \u2200 (a : \u03b1) (l : List \u03b1) (b : \u03b1) (m : List \u03b1),\n rotate (a :: l) d = b :: m \u2192 (List.Chain r a (l ++ [a]) \u2194 List.Chain r b (m ++ [b]))\na b : \u03b1\nm : List \u03b1\nhn : a = b \u2227 [] = m\n\u22a2 List.Chain r a ([] ++ [a]) \u2194 List.Chain r b (m ++ [b])\n[PROOFSTEP]\nrw [hn.1, hn.2]\n[GOAL]\ncase cons.cons.intro.succ.cons\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc\u271d : Cycle \u03b1\na\u271d : \u03b1\nl : List \u03b1\nb\u271d : \u03b1\nm\u271d : List \u03b1\nn : \u2115\nhn\u271d : rotate (a\u271d :: l) n = b\u271d :: m\u271d\nd : \u2115\nhd :\n \u2200 (a : \u03b1) (l : List \u03b1) (b : \u03b1) (m : List \u03b1),\n rotate (a :: l) d = b :: m \u2192 (List.Chain r a (l ++ [a]) \u2194 List.Chain r b (m ++ [b]))\na b : \u03b1\nm : List \u03b1\nc : \u03b1\ns : List \u03b1\nhn : rotate (a :: c :: s) (Nat.succ d) = b :: m\n\u22a2 List.Chain r a (c :: s ++ [a]) \u2194 List.Chain r b (m ++ [b])\n[PROOFSTEP]\nrw [Nat.succ_eq_one_add, \u2190 rotate_rotate, rotate_cons_succ, rotate_zero, cons_append] at hn \n[GOAL]\ncase cons.cons.intro.succ.cons\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc\u271d : Cycle \u03b1\na\u271d : \u03b1\nl : List \u03b1\nb\u271d : \u03b1\nm\u271d : List \u03b1\nn : \u2115\nhn\u271d : rotate (a\u271d :: l) n = b\u271d :: m\u271d\nd : \u2115\nhd :\n \u2200 (a : \u03b1) (l : List \u03b1) (b : \u03b1) (m : List \u03b1),\n rotate (a :: l) d = b :: m \u2192 (List.Chain r a (l ++ [a]) \u2194 List.Chain r b (m ++ [b]))\na b : \u03b1\nm : List \u03b1\nc : \u03b1\ns : List \u03b1\nhn : rotate (c :: (s ++ [a])) d = b :: m\n\u22a2 List.Chain r a (c :: s ++ [a]) \u2194 List.Chain r b (m ++ [b])\n[PROOFSTEP]\nrw [\u2190 hd c _ _ _ hn]\n[GOAL]\ncase cons.cons.intro.succ.cons\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc\u271d : Cycle \u03b1\na\u271d : \u03b1\nl : List \u03b1\nb\u271d : \u03b1\nm\u271d : List \u03b1\nn : \u2115\nhn\u271d : rotate (a\u271d :: l) n = b\u271d :: m\u271d\nd : \u2115\nhd :\n \u2200 (a : \u03b1) (l : List \u03b1) (b : \u03b1) (m : List \u03b1),\n rotate (a :: l) d = b :: m \u2192 (List.Chain r a (l ++ [a]) \u2194 List.Chain r b (m ++ [b]))\na b : \u03b1\nm : List \u03b1\nc : \u03b1\ns : List \u03b1\nhn : rotate (c :: (s ++ [a])) d = b :: m\n\u22a2 List.Chain r a (c :: s ++ [a]) \u2194 List.Chain r c (s ++ [a] ++ [c])\n[PROOFSTEP]\nsimp [and_comm]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 Chain r Cycle.nil\n[PROOFSTEP]\ntrivial\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\n\u22a2 Chain r \u2191[a] \u2194 r a a\n[PROOFSTEP]\nrw [chain_coe_cons, nil_append, List.chain_singleton]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nl : List \u03b1\n\u22a2 \u2200 (l : List \u03b1) (a : \u03b1),\n (\u2200 (hl : l \u2260 []), Chain r \u2191l \u2194 List.Chain r (getLast l hl) l) \u2192\n \u2200 (hl : l ++ [a] \u2260 []), Chain r \u2191(l ++ [a]) \u2194 List.Chain r (getLast (l ++ [a]) hl) (l ++ [a])\n[PROOFSTEP]\nintro m a _H _\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nl m : List \u03b1\na : \u03b1\n_H : \u2200 (hl : m \u2260 []), Chain r \u2191m \u2194 List.Chain r (getLast m hl) m\nhl\u271d : m ++ [a] \u2260 []\n\u22a2 Chain r \u2191(m ++ [a]) \u2194 List.Chain r (getLast (m ++ [a]) hl\u271d) (m ++ [a])\n[PROOFSTEP]\nrw [\u2190 coe_cons_eq_coe_append, chain_coe_cons, getLast_append_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 \u03b1\ns : Cycle \u03b2\nl : List \u03b2\n\u22a2 Chain r (map f (Quotient.mk'' l)) \u2194 Chain (fun a b => r (f a) (f b)) (Quotient.mk'' l)\n[PROOFSTEP]\ncases' l with a l\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 \u03b1\ns : Cycle \u03b2\n\u22a2 Chain r (map f (Quotient.mk'' [])) \u2194 Chain (fun a b => r (f a) (f b)) (Quotient.mk'' [])\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 \u03b1\ns : Cycle \u03b2\na : \u03b2\nl : List \u03b2\n\u22a2 Chain r (map f (Quotient.mk'' (a :: l))) \u2194 Chain (fun a b => r (f a) (f b)) (Quotient.mk'' (a :: l))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 \u03b1\ns : Cycle \u03b2\na : \u03b2\nl : List \u03b2\n\u22a2 Chain r (map f (Quotient.mk'' (a :: l))) \u2194 Chain (fun a b => r (f a) (f b)) (Quotient.mk'' (a :: l))\n[PROOFSTEP]\ndsimp only [Chain, \u2190 mk''_eq_coe, Quotient.liftOn'_mk'', Cycle.map, Quotient.map', Quot.map, Quotient.mk'',\n Quotient.liftOn', Quotient.liftOn, Quot.liftOn_mk, List.map]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 \u03b1\ns : Cycle \u03b2\na : \u03b2\nl : List \u03b2\n\u22a2 List.Chain r (f a) (List.map f l ++ [f a]) \u2194 List.Chain (fun a b => r (f a) (f b)) a (l ++ [a])\n[PROOFSTEP]\nrw [\u2190 concat_eq_append, \u2190 List.map_concat, List.chain_map f]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b2 \u2192 \u03b1\ns : Cycle \u03b2\na : \u03b2\nl : List \u03b2\n\u22a2 List.Chain (fun a b => r (f a) (f b)) a (concat l a) \u2194 List.Chain (fun a b => r (f a) (f b)) a (l ++ [a])\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nr : \u2115 \u2192 \u2115 \u2192 Prop\nn : \u2115\n\u22a2 Chain r \u2191(range (Nat.succ n)) \u2194 r n 0 \u2227 \u2200 (m : \u2115), m < n \u2192 r m (Nat.succ m)\n[PROOFSTEP]\nrw [range_succ, \u2190 coe_cons_eq_coe_append, chain_coe_cons, \u2190 range_succ, chain_range_succ]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\nr\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\nH : \u2200 (a b : \u03b1), r\u2081 a b \u2192 r\u2082 a b\np : Chain r\u2081 s\n\u22a2 Chain r\u2082 s\n[PROOFSTEP]\ninduction s using Cycle.induction_on\n[GOAL]\ncase H0\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\nr\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\nH : \u2200 (a b : \u03b1), r\u2081 a b \u2192 r\u2082 a b\np : Chain r\u2081 Cycle.nil\n\u22a2 Chain r\u2082 Cycle.nil\n[PROOFSTEP]\ntriv\n[GOAL]\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\nr\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\nH : \u2200 (a b : \u03b1), r\u2081 a b \u2192 r\u2082 a b\na\u271d\u00b9 : \u03b1\nl\u271d : List \u03b1\na\u271d : Chain r\u2081 \u2191l\u271d \u2192 Chain r\u2082 \u2191l\u271d\np : Chain r\u2081 \u2191(a\u271d\u00b9 :: l\u271d)\n\u22a2 Chain r\u2082 \u2191(a\u271d\u00b9 :: l\u271d)\n[PROOFSTEP]\nrw [chain_coe_cons] at p \u22a2\n[GOAL]\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\nr\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\nH : \u2200 (a b : \u03b1), r\u2081 a b \u2192 r\u2082 a b\na\u271d\u00b9 : \u03b1\nl\u271d : List \u03b1\na\u271d : Chain r\u2081 \u2191l\u271d \u2192 Chain r\u2082 \u2191l\u271d\np : List.Chain r\u2081 a\u271d\u00b9 (l\u271d ++ [a\u271d\u00b9])\n\u22a2 List.Chain r\u2082 a\u271d\u00b9 (l\u271d ++ [a\u271d\u00b9])\n[PROOFSTEP]\nexact p.imp H\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\n\u22a2 (\u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 r a b) \u2192 Chain r s\n[PROOFSTEP]\ninduction' s using Cycle.induction_on with a l _\n[GOAL]\ncase H0\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\n\u22a2 (\u2200 (a : \u03b1), a \u2208 nil \u2192 \u2200 (b : \u03b1), b \u2208 nil \u2192 r a b) \u2192 Chain r nil\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\n\u22a2 (\u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b) \u2192 Chain r \u2191(a :: l)\n[PROOFSTEP]\nexact fun _ => Cycle.Chain.nil r\n[GOAL]\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\n\u22a2 (\u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b) \u2192 Chain r \u2191(a :: l)\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\n\u22a2 Chain r \u2191(a :: l)\n[PROOFSTEP]\nhave Ha : a \u2208 (a :: l : Cycle \u03b1) := by simp\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\n\u22a2 a \u2208 \u2191(a :: l)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\nHa : a \u2208 \u2191(a :: l)\n\u22a2 Chain r \u2191(a :: l)\n[PROOFSTEP]\nhave Hl : \u2200 {b} (_hb : b \u2208 l), b \u2208 (a :: l : Cycle \u03b1) := @fun b hb => by simp [hb]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\nHa : a \u2208 \u2191(a :: l)\nb : \u03b1\nhb : b \u2208 l\n\u22a2 b \u2208 \u2191(a :: l)\n[PROOFSTEP]\nsimp [hb]\n[GOAL]\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\nHa : a \u2208 \u2191(a :: l)\nHl : \u2200 {b : \u03b1}, b \u2208 l \u2192 b \u2208 \u2191(a :: l)\n\u22a2 Chain r \u2191(a :: l)\n[PROOFSTEP]\nrw [Cycle.chain_coe_cons]\n[GOAL]\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\nHa : a \u2208 \u2191(a :: l)\nHl : \u2200 {b : \u03b1}, b \u2208 l \u2192 b \u2208 \u2191(a :: l)\n\u22a2 List.Chain r a (l ++ [a])\n[PROOFSTEP]\napply Pairwise.chain\n[GOAL]\ncase HI.p\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\nHa : a \u2208 \u2191(a :: l)\nHl : \u2200 {b : \u03b1}, b \u2208 l \u2192 b \u2208 \u2191(a :: l)\n\u22a2 List.Pairwise r (a :: (l ++ [a]))\n[PROOFSTEP]\nrw [pairwise_cons]\n[GOAL]\ncase HI.p\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\nHa : a \u2208 \u2191(a :: l)\nHl : \u2200 {b : \u03b1}, b \u2208 l \u2192 b \u2208 \u2191(a :: l)\n\u22a2 (\u2200 (a' : \u03b1), a' \u2208 l ++ [a] \u2192 r a a') \u2227 List.Pairwise r (l ++ [a])\n[PROOFSTEP]\nrefine'\n \u27e8fun b hb => _,\n pairwise_append.2\n \u27e8pairwise_of_forall_mem_list fun b hb c hc => hs b (Hl hb) c (Hl hc), pairwise_singleton r a, fun b hb c hc => _\u27e9\u27e9\n[GOAL]\ncase HI.p.refine'_1\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\nHa : a \u2208 \u2191(a :: l)\nHl : \u2200 {b : \u03b1}, b \u2208 l \u2192 b \u2208 \u2191(a :: l)\nb : \u03b1\nhb : b \u2208 l ++ [a]\n\u22a2 r a b\n[PROOFSTEP]\nrw [mem_append] at hb \n[GOAL]\ncase HI.p.refine'_1\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\nHa : a \u2208 \u2191(a :: l)\nHl : \u2200 {b : \u03b1}, b \u2208 l \u2192 b \u2208 \u2191(a :: l)\nb : \u03b1\nhb : b \u2208 l \u2228 b \u2208 [a]\n\u22a2 r a b\n[PROOFSTEP]\ncases' hb with hb hb\n[GOAL]\ncase HI.p.refine'_1.inl\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\nHa : a \u2208 \u2191(a :: l)\nHl : \u2200 {b : \u03b1}, b \u2208 l \u2192 b \u2208 \u2191(a :: l)\nb : \u03b1\nhb : b \u2208 l\n\u22a2 r a b\n[PROOFSTEP]\nexact hs a Ha b (Hl hb)\n[GOAL]\ncase HI.p.refine'_1.inr\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\nHa : a \u2208 \u2191(a :: l)\nHl : \u2200 {b : \u03b1}, b \u2208 l \u2192 b \u2208 \u2191(a :: l)\nb : \u03b1\nhb : b \u2208 [a]\n\u22a2 r a b\n[PROOFSTEP]\nrw [mem_singleton] at hb \n[GOAL]\ncase HI.p.refine'_1.inr\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\nHa : a \u2208 \u2191(a :: l)\nHl : \u2200 {b : \u03b1}, b \u2208 l \u2192 b \u2208 \u2191(a :: l)\nb : \u03b1\nhb : b = a\n\u22a2 r a b\n[PROOFSTEP]\nrw [hb]\n[GOAL]\ncase HI.p.refine'_1.inr\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\nHa : a \u2208 \u2191(a :: l)\nHl : \u2200 {b : \u03b1}, b \u2208 l \u2192 b \u2208 \u2191(a :: l)\nb : \u03b1\nhb : b = a\n\u22a2 r a a\n[PROOFSTEP]\nexact hs a Ha a Ha\n[GOAL]\ncase HI.p.refine'_2\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\nHa : a \u2208 \u2191(a :: l)\nHl : \u2200 {b : \u03b1}, b \u2208 l \u2192 b \u2208 \u2191(a :: l)\nb : \u03b1\nhb : b \u2208 l\nc : \u03b1\nhc : c \u2208 [a]\n\u22a2 r b c\n[PROOFSTEP]\nrw [mem_singleton] at hc \n[GOAL]\ncase HI.p.refine'_2\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\nHa : a \u2208 \u2191(a :: l)\nHl : \u2200 {b : \u03b1}, b \u2208 l \u2192 b \u2208 \u2191(a :: l)\nb : \u03b1\nhb : b \u2208 l\nc : \u03b1\nhc : c = a\n\u22a2 r b c\n[PROOFSTEP]\nrw [hc]\n[GOAL]\ncase HI.p.refine'_2\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\na : \u03b1\nl : List \u03b1\na\u271d : (\u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b) \u2192 Chain r \u2191l\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_1 b\nHa : a \u2208 \u2191(a :: l)\nHl : \u2200 {b : \u03b1}, b \u2208 l \u2192 b \u2208 \u2191(a :: l)\nb : \u03b1\nhb : b \u2208 l\nc : \u03b1\nhc : c = a\n\u22a2 r b a\n[PROOFSTEP]\nexact hs b (Hl hb) a Ha\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d : IsTrans \u03b1 r\n\u22a2 Chain r s \u2192 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 r a b\n[PROOFSTEP]\ninduction' s using Cycle.induction_on with a l _\n[GOAL]\ncase H0\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d : IsTrans \u03b1 r\n\u22a2 Chain r nil \u2192 \u2200 (a : \u03b1), a \u2208 nil \u2192 \u2200 (b : \u03b1), b \u2208 nil \u2192 r a b\n[PROOFSTEP]\nexact fun _ b hb => (not_mem_nil _ hb).elim\n[GOAL]\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b\n\u22a2 Chain r \u2191(a :: l) \u2192 \u2200 (a_2 : \u03b1), a_2 \u2208 \u2191(a :: l) \u2192 \u2200 (b : \u03b1), b \u2208 \u2191(a :: l) \u2192 r a_2 b\n[PROOFSTEP]\nintro hs b hb c hc\n[GOAL]\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b\nhs : Chain r \u2191(a :: l)\nb : \u03b1\nhb : b \u2208 \u2191(a :: l)\nc : \u03b1\nhc : c \u2208 \u2191(a :: l)\n\u22a2 r b c\n[PROOFSTEP]\nrw [Cycle.chain_coe_cons, List.chain_iff_pairwise] at hs \n[GOAL]\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b\nhs : List.Pairwise r (a :: (l ++ [a]))\nb : \u03b1\nhb : b \u2208 \u2191(a :: l)\nc : \u03b1\nhc : c \u2208 \u2191(a :: l)\n\u22a2 r b c\n[PROOFSTEP]\nsimp only [pairwise_append, pairwise_cons, mem_append, mem_singleton, List.not_mem_nil, IsEmpty.forall_iff,\n imp_true_iff, Pairwise.nil, forall_eq, true_and_iff] at hs \n[GOAL]\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b\nb : \u03b1\nhb : b \u2208 \u2191(a :: l)\nc : \u03b1\nhc : c \u2208 \u2191(a :: l)\nhs : (\u2200 (a' : \u03b1), a' \u2208 l \u2228 a' = a \u2192 r a a') \u2227 List.Pairwise r l \u2227 \u2200 (a_1 : \u03b1), a_1 \u2208 l \u2192 r a_1 a\n\u22a2 r b c\n[PROOFSTEP]\nsimp only [mem_coe_iff, mem_cons] at hb hc \n[GOAL]\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b\nb c : \u03b1\nhs : (\u2200 (a' : \u03b1), a' \u2208 l \u2228 a' = a \u2192 r a a') \u2227 List.Pairwise r l \u2227 \u2200 (a_1 : \u03b1), a_1 \u2208 l \u2192 r a_1 a\nhb : b = a \u2228 b \u2208 l\nhc : c = a \u2228 c \u2208 l\n\u22a2 r b c\n[PROOFSTEP]\nrcases hb with (rfl | hb)\n[GOAL]\ncase HI.inl\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d : IsTrans \u03b1 r\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b\nb c : \u03b1\nhs : (\u2200 (a' : \u03b1), a' \u2208 l \u2228 a' = b \u2192 r b a') \u2227 List.Pairwise r l \u2227 \u2200 (a : \u03b1), a \u2208 l \u2192 r a b\nhc : c = b \u2228 c \u2208 l\n\u22a2 r b c\n[PROOFSTEP]\nrcases hc with (rfl | hc)\n[GOAL]\ncase HI.inr\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b\nb c : \u03b1\nhs : (\u2200 (a' : \u03b1), a' \u2208 l \u2228 a' = a \u2192 r a a') \u2227 List.Pairwise r l \u2227 \u2200 (a_1 : \u03b1), a_1 \u2208 l \u2192 r a_1 a\nhc : c = a \u2228 c \u2208 l\nhb : b \u2208 l\n\u22a2 r b c\n[PROOFSTEP]\nrcases hc with (rfl | hc)\n[GOAL]\ncase HI.inl.inl\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d : IsTrans \u03b1 r\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b\nc : \u03b1\nhs : (\u2200 (a' : \u03b1), a' \u2208 l \u2228 a' = c \u2192 r c a') \u2227 List.Pairwise r l \u2227 \u2200 (a : \u03b1), a \u2208 l \u2192 r a c\n\u22a2 r c c\n[PROOFSTEP]\nexact hs.1 c (Or.inr rfl)\n[GOAL]\ncase HI.inl.inr\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d : IsTrans \u03b1 r\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b\nb c : \u03b1\nhs : (\u2200 (a' : \u03b1), a' \u2208 l \u2228 a' = b \u2192 r b a') \u2227 List.Pairwise r l \u2227 \u2200 (a : \u03b1), a \u2208 l \u2192 r a b\nhc : c \u2208 l\n\u22a2 r b c\n[PROOFSTEP]\nexact hs.1 c (Or.inl hc)\n[GOAL]\ncase HI.inr.inl\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d : IsTrans \u03b1 r\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b\nb c : \u03b1\nhb : b \u2208 l\nhs : (\u2200 (a' : \u03b1), a' \u2208 l \u2228 a' = c \u2192 r c a') \u2227 List.Pairwise r l \u2227 \u2200 (a : \u03b1), a \u2208 l \u2192 r a c\n\u22a2 r b c\n[PROOFSTEP]\nexact hs.2.2 b hb\n[GOAL]\ncase HI.inr.inr\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d : IsTrans \u03b1 r\na : \u03b1\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2200 (a : \u03b1), a \u2208 \u2191l \u2192 \u2200 (b : \u03b1), b \u2208 \u2191l \u2192 r a b\nb c : \u03b1\nhs : (\u2200 (a' : \u03b1), a' \u2208 l \u2228 a' = a \u2192 r a a') \u2227 List.Pairwise r l \u2227 \u2200 (a_1 : \u03b1), a_1 \u2208 l \u2192 r a_1 a\nhb : b \u2208 l\nhc : c \u2208 l\n\u22a2 r b c\n[PROOFSTEP]\nexact _root_.trans (hs.2.2 b hb) (hs.1 c (Or.inl hc))\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsIrrefl \u03b1 r\nh : Chain r s\n\u22a2 s = Cycle.nil\n[PROOFSTEP]\ninduction' s using Cycle.induction_on with a l _ h\n[GOAL]\ncase H0\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsIrrefl \u03b1 r\nh\u271d : Chain r s\nh : Chain r Cycle.nil\n\u22a2 Cycle.nil = Cycle.nil\n[PROOFSTEP]\nrfl\n[GOAL]\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsIrrefl \u03b1 r\nh\u271d : Chain r s\na : \u03b1\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2191l = Cycle.nil\nh : Chain r \u2191(a :: l)\n\u22a2 \u2191(a :: l) = Cycle.nil\n[PROOFSTEP]\nhave ha := mem_cons_self a l\n[GOAL]\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsIrrefl \u03b1 r\nh\u271d : Chain r s\na : \u03b1\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2191l = Cycle.nil\nh : Chain r \u2191(a :: l)\nha : a \u2208 a :: l\n\u22a2 \u2191(a :: l) = Cycle.nil\n[PROOFSTEP]\nexact (irrefl_of r a <| chain_iff_pairwise.1 h a ha a ha).elim\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsAntisymm \u03b1 r\nhs : Chain r s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 a = b\n[PROOFSTEP]\nrw [chain_iff_pairwise] at hs \n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsAntisymm \u03b1 r\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 r a b\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 a = b\n[PROOFSTEP]\nexact antisymm (hs a ha b hb) (hs b hb a ha)\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Cycle", "llama_tokens": 71583, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056295505783, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.42398352147541735}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b x y : \u03b1\nhx : x \u2208 I\nh : y * x = 1\nz : \u03b1\nx\u271d : z \u2208 \u22a4\n\u22a2 z = z * (y * x)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\n\u22a2 I = \u22a4 \u2192 1 \u2208 I\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\na b : \u03b1\n\u22a2 1 \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\nh : 1 \u2208 I\n\u22a2 1 * 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b x y : \u03b1\nhy : IsUnit y\n\u22a2 y * x \u2208 I \u2194 x \u2208 I\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => I.mul_mem_left y h\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b x y : \u03b1\nhy : IsUnit y\nh : y * x \u2208 I\n\u22a2 x \u2208 I\n[PROOFSTEP]\nobtain \u27e8y', hy'\u27e9 := hy.exists_left_inv\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b x y : \u03b1\nhy : IsUnit y\nh : y * x \u2208 I\ny' : \u03b1\nhy' : y' * y = 1\n\u22a2 x \u2208 I\n[PROOFSTEP]\nhave := I.mul_mem_left y' h\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b x y : \u03b1\nhy : IsUnit y\nh : y * x \u2208 I\ny' : \u03b1\nhy' : y' * y = 1\nthis : y' * (y * x) \u2208 I\n\u22a2 x \u2208 I\n[PROOFSTEP]\nrwa [\u2190 mul_assoc, hy', one_mul] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na\u271d b a : \u03b1\nh2 : IsUnit a\nx : \u03b1\n\u22a2 span {a * x} = span {x}\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na\u271d b a : \u03b1\nh2 : IsUnit a\nx : \u03b1\n\u22a2 span {a * x} \u2264 span {x}\n[PROOFSTEP]\nrw [span_singleton_le_iff_mem, mem_span_singleton']\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na\u271d b a : \u03b1\nh2 : IsUnit a\nx : \u03b1\n\u22a2 span {x} \u2264 span {a * x}\n[PROOFSTEP]\nrw [span_singleton_le_iff_mem, mem_span_singleton']\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na\u271d b a : \u03b1\nh2 : IsUnit a\nx : \u03b1\n\u22a2 \u2203 a_1, a_1 * x = a * x\ncase a \u03b1 : Type u \u03b2 : Type v inst\u271d : Semiring \u03b1 I : Ideal \u03b1 a\u271d b a : \u03b1 h2 : IsUnit a x : \u03b1 \u22a2 \u2203 a_1, a_1 * (a * x) = x\n[PROOFSTEP]\nexacts [\u27e8a, rfl\u27e9, \u27e8_, h2.unit.inv_mul_cancel_left x\u27e9]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\n\u22a2 span 0 = \u22a5\n[PROOFSTEP]\nrw [\u2190 Set.singleton_zero, span_singleton_eq_bot]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\n\u22a2 span 1 = \u22a4\n[PROOFSTEP]\nrw [\u2190 Set.singleton_one, span_singleton_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\ns : Set \u03b1\n\u22a2 span s = \u22a4 \u2194 \u2203 s', \u2191s' \u2286 s \u2227 span \u2191s' = \u22a4\n[PROOFSTEP]\nsimp_rw [eq_top_iff_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\ns : Set \u03b1\n\u22a2 1 \u2208 span s \u2194 \u2203 s', \u2191s' \u2286 s \u2227 1 \u2208 span \u2191s'\n[PROOFSTEP]\nexact \u27e8Submodule.mem_span_finite_of_mem_span, fun \u27e8s', h\u2081, h\u2082\u27e9 => span_mono h\u2081 h\u2082\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nS : Type u_1\ninst\u271d : CommSemiring S\nx y : S\nI : Ideal S\n\u22a2 x \u2208 span {y} \u2294 I \u2194 \u2203 a b, b \u2208 I \u2227 a * y + b = x\n[PROOFSTEP]\nrw [Submodule.mem_sup]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nS : Type u_1\ninst\u271d : CommSemiring S\nx y : S\nI : Ideal S\n\u22a2 (\u2203 y_1, y_1 \u2208 span {y} \u2227 \u2203 z, z \u2208 I \u2227 y_1 + z = x) \u2194 \u2203 a b, b \u2208 I \u2227 a * y + b = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nS : Type u_1\ninst\u271d : CommSemiring S\nx y : S\nI : Ideal S\n\u22a2 (\u2203 y_1, y_1 \u2208 span {y} \u2227 \u2203 z, z \u2208 I \u2227 y_1 + z = x) \u2192 \u2203 a b, b \u2208 I \u2227 a * y + b = x\n[PROOFSTEP]\nrintro \u27e8ya, hya, b, hb, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b\u271d : \u03b1\nS : Type u_1\ninst\u271d : CommSemiring S\ny : S\nI : Ideal S\nya : S\nhya : ya \u2208 span {y}\nb : S\nhb : b \u2208 I\n\u22a2 \u2203 a b_1, b_1 \u2208 I \u2227 a * y + b_1 = ya + b\n[PROOFSTEP]\nobtain \u27e8a, rfl\u27e9 := mem_span_singleton'.mp hya\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI\u271d : Ideal \u03b1\na\u271d b\u271d : \u03b1\nS : Type u_1\ninst\u271d : CommSemiring S\ny : S\nI : Ideal S\nb : S\nhb : b \u2208 I\na : S\nhya : a * y \u2208 span {y}\n\u22a2 \u2203 a_1 b_1, b_1 \u2208 I \u2227 a_1 * y + b_1 = a * y + b\n[PROOFSTEP]\nexact \u27e8a, b, hb, rfl\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nS : Type u_1\ninst\u271d : CommSemiring S\nx y : S\nI : Ideal S\n\u22a2 (\u2203 a b, b \u2208 I \u2227 a * y + b = x) \u2192 \u2203 y_1, y_1 \u2208 span {y} \u2227 \u2203 z, z \u2208 I \u2227 y_1 + z = x\n[PROOFSTEP]\nrintro \u27e8a, b, hb, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI\u271d : Ideal \u03b1\na\u271d b\u271d : \u03b1\nS : Type u_1\ninst\u271d : CommSemiring S\ny : S\nI : Ideal S\na b : S\nhb : b \u2208 I\n\u22a2 \u2203 y_1, y_1 \u2208 span {y} \u2227 \u2203 z, z \u2208 I \u2227 y_1 + z = a * y + b\n[PROOFSTEP]\nexact \u27e8a * y, Ideal.mem_span_singleton'.mpr \u27e8a, rfl\u27e9, b, hb, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI : Ideal \u03b1\nhI : IsPrime I\nr : \u03b1\nn : \u2115\nH : r ^ n \u2208 I\n\u22a2 r \u2208 I\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI : Ideal \u03b1\nhI : IsPrime I\nr : \u03b1\nn : \u2115\nH\u271d : r ^ n \u2208 I\nH : r ^ Nat.zero \u2208 I\n\u22a2 r \u2208 I\n[PROOFSTEP]\nrw [pow_zero] at H \n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI : Ideal \u03b1\nhI : IsPrime I\nr : \u03b1\nn : \u2115\nH\u271d : r ^ n \u2208 I\nH : 1 \u2208 I\n\u22a2 r \u2208 I\n[PROOFSTEP]\nexact (mt (eq_top_iff_one _).2 hI.1).elim H\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI : Ideal \u03b1\nhI : IsPrime I\nr : \u03b1\nn\u271d : \u2115\nH\u271d : r ^ n\u271d \u2208 I\nn : \u2115\nih : r ^ n \u2208 I \u2192 r \u2208 I\nH : r ^ Nat.succ n \u2208 I\n\u22a2 r \u2208 I\n[PROOFSTEP]\nrw [pow_succ] at H \n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI : Ideal \u03b1\nhI : IsPrime I\nr : \u03b1\nn\u271d : \u2115\nH\u271d : r ^ n\u271d \u2208 I\nn : \u2115\nih : r ^ n \u2208 I \u2192 r \u2208 I\nH : r * r ^ n \u2208 I\n\u22a2 r \u2208 I\n[PROOFSTEP]\nexact Or.casesOn (hI.mem_or_mem H) id ih\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI : Ideal \u03b1\n\u22a2 \u00acIsPrime I \u2194 I = \u22a4 \u2228 \u2203 x _hx y _hy, x * y \u2208 I\n[PROOFSTEP]\nsimp_rw [Ideal.isPrime_iff, not_and_or, Ne.def, Classical.not_not, not_forall, not_or]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI : Ideal \u03b1\n\u22a2 (I = \u22a4 \u2228 \u2203 x x_1 h, \u00acx \u2208 I \u2227 \u00acx_1 \u2208 I) \u2194 I = \u22a4 \u2228 \u2203 x _hx y _hy, x * y \u2208 I\n[PROOFSTEP]\nexact or_congr Iff.rfl \u27e8fun \u27e8x, y, hxy, hx, hy\u27e9 => \u27e8x, hx, y, hy, hxy\u27e9, fun \u27e8x, hx, y, hy, hxy\u27e9 => \u27e8x, y, hxy, hx, hy\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nh : \u22a5 = \u22a4\n\u22a2 1 = 0\n[PROOFSTEP]\nrwa [Ideal.eq_top_iff_one, Submodule.mem_bot] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx\u271d y\u271d : R\nh : x\u271d * y\u271d \u2208 \u22a5\n\u22a2 x\u271d * y\u271d = 0\n[PROOFSTEP]\nsimpa only [Submodule.mem_bot] using h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI J : Ideal \u03b1\n\u22a2 I < J \u2192 J = \u22a4 \u2194 \u2200 (x : \u03b1), I \u2264 J \u2192 \u00acx \u2208 I \u2192 x \u2208 J \u2192 1 \u2208 J\n[PROOFSTEP]\nrw [lt_iff_le_not_le]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI J : Ideal \u03b1\n\u22a2 I \u2264 J \u2227 \u00acJ \u2264 I \u2192 J = \u22a4 \u2194 \u2200 (x : \u03b1), I \u2264 J \u2192 \u00acx \u2208 I \u2192 x \u2208 J \u2192 1 \u2208 J\n[PROOFSTEP]\nexact\n \u27e8fun H x h hx\u2081 hx\u2082 => J.eq_top_iff_one.1 <| H \u27e8h, not_subset.2 \u27e8_, hx\u2082, hx\u2081\u27e9\u27e9, fun H \u27e8h\u2081, h\u2082\u27e9 =>\n let \u27e8x, xJ, xI\u27e9 := not_subset.1 h\u2082\n J.eq_top_iff_one.2 <| H x h\u2081 xI xJ\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\n\u22a2 IsCoatomic (Ideal \u03b1)\n[PROOFSTEP]\napply CompleteLattice.coatomic_of_top_compact\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\n\u22a2 CompleteLattice.IsCompactElement \u22a4\n[PROOFSTEP]\nrw [\u2190 span_singleton_one]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\n\u22a2 CompleteLattice.IsCompactElement (span {1})\n[PROOFSTEP]\nexact Submodule.singleton_span_isCompactElement 1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\nM M' : Ideal \u03b1\nhM : IsMaximal M\nhM' : IsMaximal M'\nhne : M \u2260 M'\n\u22a2 M \u2294 M' = \u22a4\n[PROOFSTEP]\ncontrapose! hne with h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\nM M' : Ideal \u03b1\nhM : IsMaximal M\nhM' : IsMaximal M'\nh : M \u2294 M' \u2260 \u22a4\n\u22a2 M = M'\n[PROOFSTEP]\nexact hM.eq_of_le hM'.ne_top (le_sup_left.trans_eq (hM'.eq_of_le h le_sup_right).symm)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\ninst\u271d : Nontrivial \u03b1\n\u22a2 Nontrivial (Ideal \u03b1)\n[PROOFSTEP]\nrcases@exists_maximal \u03b1 _ _ with \u27e8M, hM, _\u27e9\n[GOAL]\ncase intro.mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\ninst\u271d : Nontrivial \u03b1\nM : Ideal \u03b1\nhM : M \u2260 \u22a4\nright\u271d : \u2200 (b : Ideal \u03b1), M < b \u2192 b = \u22a4\n\u22a2 Nontrivial (Ideal \u03b1)\n[PROOFSTEP]\nexact nontrivial_of_ne M \u22a4 hM\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\nR : Type u\ninst\u271d : Semiring R\nP : Ideal R\nhmax : \u2200 (m : Ideal R), P < m \u2192 \u00acIsMaximal m\nJ : Ideal R\nhPJ : P < J\n\u22a2 J = \u22a4\n[PROOFSTEP]\nby_contra hnonmax\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\nR : Type u\ninst\u271d : Semiring R\nP : Ideal R\nhmax : \u2200 (m : Ideal R), P < m \u2192 \u00acIsMaximal m\nJ : Ideal R\nhPJ : P < J\nhnonmax : \u00acJ = \u22a4\n\u22a2 False\n[PROOFSTEP]\nrcases exists_le_maximal J hnonmax with \u27e8M, hM1, hM2\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\nR : Type u\ninst\u271d : Semiring R\nP : Ideal R\nhmax : \u2200 (m : Ideal R), P < m \u2192 \u00acIsMaximal m\nJ : Ideal R\nhPJ : P < J\nhnonmax : \u00acJ = \u22a4\nM : Ideal R\nhM1 : IsMaximal M\nhM2 : J \u2264 M\n\u22a2 False\n[PROOFSTEP]\nexact hmax M (lt_of_lt_of_le hPJ hM2) hM1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b x y : \u03b1\n\u22a2 span {x, y} = span {y, x}\n[PROOFSTEP]\nsimp only [span_insert, sup_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\nR : Type u\ninst\u271d : CommRing R\nx y z : R\n\u22a2 span {x + y * z, y} = span {x, y}\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\nR : Type u\ninst\u271d : CommRing R\nx y z x\u271d : R\n\u22a2 x\u271d \u2208 span {x + y * z, y} \u2194 x\u271d \u2208 span {x, y}\n[PROOFSTEP]\nrw [mem_span_pair, mem_span_pair]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\nR : Type u\ninst\u271d : CommRing R\nx y z x\u271d : R\n\u22a2 (\u2203 a b, a * (x + y * z) + b * y = x\u271d) \u2194 \u2203 a b, a * x + b * y = x\u271d\n[PROOFSTEP]\nexact\n \u27e8fun \u27e8a, b, h\u27e9 =>\n \u27e8a, b + a * z, by\n rw [\u2190 h]\n ring1\u27e9,\n fun \u27e8a, b, h\u27e9 =>\n \u27e8a, b - a * z, by\n rw [\u2190 h]\n ring1\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI : Ideal \u03b1\na\u271d b\u271d : \u03b1\nR : Type u\ninst\u271d : CommRing R\nx y z x\u271d\u00b9 : R\nx\u271d : \u2203 a b, a * (x + y * z) + b * y = x\u271d\u00b9\na b : R\nh : a * (x + y * z) + b * y = x\u271d\u00b9\n\u22a2 a * x + (b + a * z) * y = x\u271d\u00b9\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI : Ideal \u03b1\na\u271d b\u271d : \u03b1\nR : Type u\ninst\u271d : CommRing R\nx y z x\u271d\u00b9 : R\nx\u271d : \u2203 a b, a * (x + y * z) + b * y = x\u271d\u00b9\na b : R\nh : a * (x + y * z) + b * y = x\u271d\u00b9\n\u22a2 a * x + (b + a * z) * y = a * (x + y * z) + b * y\n[PROOFSTEP]\nring1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI : Ideal \u03b1\na\u271d b\u271d : \u03b1\nR : Type u\ninst\u271d : CommRing R\nx y z x\u271d\u00b9 : R\nx\u271d : \u2203 a b, a * x + b * y = x\u271d\u00b9\na b : R\nh : a * x + b * y = x\u271d\u00b9\n\u22a2 a * (x + y * z) + (b - a * z) * y = x\u271d\u00b9\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI : Ideal \u03b1\na\u271d b\u271d : \u03b1\nR : Type u\ninst\u271d : CommRing R\nx y z x\u271d\u00b9 : R\nx\u271d : \u2203 a b, a * x + b * y = x\u271d\u00b9\na b : R\nh : a * x + b * y = x\u271d\u00b9\n\u22a2 a * (x + y * z) + (b - a * z) * y = a * x + b * y\n[PROOFSTEP]\nring1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\nR : Type u\ninst\u271d : CommRing R\nx y z : R\n\u22a2 span {x, y + x * z} = span {x, y}\n[PROOFSTEP]\nrw [span_pair_comm, span_pair_add_mul_left, span_pair_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI : Ideal \u03b1\nhI : IsMaximal I\nx : \u03b1\nhx : \u00acx \u2208 I\n\u22a2 \u2203 y i, i \u2208 I \u2227 y * x + i = 1\n[PROOFSTEP]\ncases' isMaximal_iff.1 hI with H\u2081 H\u2082\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI : Ideal \u03b1\nhI : IsMaximal I\nx : \u03b1\nhx : \u00acx \u2208 I\nH\u2081 : \u00ac1 \u2208 I\nH\u2082 : \u2200 (J : Ideal \u03b1) (x : \u03b1), I \u2264 J \u2192 \u00acx \u2208 I \u2192 x \u2208 J \u2192 1 \u2208 J\n\u22a2 \u2203 y i, i \u2208 I \u2227 y * x + i = 1\n[PROOFSTEP]\nrcases mem_span_insert.1\n (H\u2082 (span (insert x I)) x (Set.Subset.trans (subset_insert _ _) subset_span) hx (subset_span (mem_insert _ _))) with\n \u27e8y, z, hz, hy\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI : Ideal \u03b1\nhI : IsMaximal I\nx : \u03b1\nhx : \u00acx \u2208 I\nH\u2081 : \u00ac1 \u2208 I\nH\u2082 : \u2200 (J : Ideal \u03b1) (x : \u03b1), I \u2264 J \u2192 \u00acx \u2208 I \u2192 x \u2208 J \u2192 1 \u2208 J\ny z : \u03b1\nhz : z \u2208 span \u2191I\nhy : 1 = y * x + z\n\u22a2 \u2203 y i, i \u2208 I \u2227 y * x + i = 1\n[PROOFSTEP]\nrefine' \u27e8y, z, _, hy.symm\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\nI : Ideal \u03b1\nhI : IsMaximal I\nx : \u03b1\nhx : \u00acx \u2208 I\nH\u2081 : \u00ac1 \u2208 I\nH\u2082 : \u2200 (J : Ideal \u03b1) (x : \u03b1), I \u2264 J \u2192 \u00acx \u2208 I \u2192 x \u2208 J \u2192 1 \u2208 J\ny z : \u03b1\nhz : z \u2208 span \u2191I\nhy : 1 = y * x + z\n\u22a2 z \u2208 I\n[PROOFSTEP]\nrwa [\u2190 span_eq I]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : x\u271d * y\u271d \u2208 sInf s\nhx : \u00acx\u271d \u2208 sInf s\n\u22a2 y\u271d \u2208 sInf s\n[PROOFSTEP]\nrw [Ideal.mem_sInf] at hx e \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nhx : \u00ac\u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d \u2208 I\n\u22a2 \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 y\u271d \u2208 I\n[PROOFSTEP]\npush_neg at hx \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nhx : Exists fun \u2983I\u2984 => I \u2208 s \u2227 \u00acx\u271d \u2208 I\n\u22a2 \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 y\u271d \u2208 I\n[PROOFSTEP]\nobtain \u27e8I, hI, hI'\u27e9 := hx\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nI : Ideal \u03b1\nhI : I \u2208 s\nhI' : \u00acx\u271d \u2208 I\n\u22a2 \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 y\u271d \u2208 I\n[PROOFSTEP]\nintro J hJ\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nI : Ideal \u03b1\nhI : I \u2208 s\nhI' : \u00acx\u271d \u2208 I\nJ : Ideal \u03b1\nhJ : J \u2208 s\n\u22a2 y\u271d \u2208 J\n[PROOFSTEP]\ncases' hs'.total hI hJ with h h\n[GOAL]\ncase intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nI : Ideal \u03b1\nhI : I \u2208 s\nhI' : \u00acx\u271d \u2208 I\nJ : Ideal \u03b1\nhJ : J \u2208 s\nh : I \u2264 J\n\u22a2 y\u271d \u2208 J\n[PROOFSTEP]\nexact h (((H I hI).mem_or_mem (e hI)).resolve_left hI')\n[GOAL]\ncase intro.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nI : Ideal \u03b1\nhI : I \u2208 s\nhI' : \u00acx\u271d \u2208 I\nJ : Ideal \u03b1\nhJ : J \u2208 s\nh : J \u2264 I\n\u22a2 y\u271d \u2208 J\n[PROOFSTEP]\nexact ((H J hJ).mem_or_mem (e hJ)).resolve_left fun x => hI' <| h x\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\nx y x\u271d : \u03b1\n\u22a2 x\u271d * y = x \u2194 x = y * x\u271d\n[PROOFSTEP]\nrw [eq_comm, mul_comm]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\na b : \u03b1\u271d\ninst\u271d\u00b2 : CommSemiring \u03b1\u271d\nI : Ideal \u03b1\u271d\n\u03b1 : Type u\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : IsDomain \u03b1\nx y : \u03b1\n\u22a2 span {x} = span {y} \u2194 Associated x y\n[PROOFSTEP]\nrw [\u2190 dvd_dvd_iff_associated, le_antisymm_iff, and_comm]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\na b : \u03b1\u271d\ninst\u271d\u00b2 : CommSemiring \u03b1\u271d\nI : Ideal \u03b1\u271d\n\u03b1 : Type u\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : IsDomain \u03b1\nx y : \u03b1\n\u22a2 span {y} \u2264 span {x} \u2227 span {x} \u2264 span {y} \u2194 x \u2223 y \u2227 y \u2223 x\n[PROOFSTEP]\napply and_congr\n[GOAL]\ncase h\u2081\n\u03b1\u271d : Type u\n\u03b2 : Type v\na b : \u03b1\u271d\ninst\u271d\u00b2 : CommSemiring \u03b1\u271d\nI : Ideal \u03b1\u271d\n\u03b1 : Type u\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : IsDomain \u03b1\nx y : \u03b1\n\u22a2 span {y} \u2264 span {x} \u2194 x \u2223 y\n[PROOFSTEP]\nrw [span_singleton_le_span_singleton]\n[GOAL]\ncase h\u2082\n\u03b1\u271d : Type u\n\u03b2 : Type v\na b : \u03b1\u271d\ninst\u271d\u00b2 : CommSemiring \u03b1\u271d\nI : Ideal \u03b1\u271d\n\u03b1 : Type u\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : IsDomain \u03b1\nx y : \u03b1\n\u22a2 span {x} \u2264 span {y} \u2194 y \u2223 x\n[PROOFSTEP]\nrw [span_singleton_le_span_singleton]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\na : \u03b1\nh2 : IsUnit a\nx : \u03b1\n\u22a2 span {x * a} = span {x}\n[PROOFSTEP]\nrw [mul_comm, span_singleton_mul_left_unit h2]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\nx : \u03b1\n\u22a2 span {x} = \u22a4 \u2194 IsUnit x\n[PROOFSTEP]\nrw [isUnit_iff_dvd_one, \u2190 span_singleton_le_span_singleton, span_singleton_one, eq_top_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\np : \u03b1\nhp : p \u2260 0\n\u22a2 IsPrime (span {p}) \u2194 Prime p\n[PROOFSTEP]\nsimp [isPrime_iff, Prime, span_singleton_eq_top, hp, mem_span_singleton]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\n\u22a2 y \u2208 I\n[PROOFSTEP]\nlet J : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\n\u22a2 y \u2208 I\n[PROOFSTEP]\nhave IJ : I \u2264 J := Set.Subset.trans (subset_insert _ _) subset_span\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\n\u22a2 y \u2208 I\n[PROOFSTEP]\nhave xJ : x \u2208 J := Ideal.subset_span (Set.mem_insert x I)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\n\u22a2 y \u2208 I\n[PROOFSTEP]\ncases' isMaximal_iff.1 H with _ oJ\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : \u2200 (J : Ideal \u03b1) (x : \u03b1), I \u2264 J \u2192 \u00acx \u2208 I \u2192 x \u2208 J \u2192 1 \u2208 J\n\u22a2 y \u2208 I\n[PROOFSTEP]\nspecialize oJ J x IJ hx xJ\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : 1 \u2208 J\n\u22a2 y \u2208 I\n[PROOFSTEP]\nrcases Submodule.mem_span_insert.mp oJ with \u27e8a, b, h, oe\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b\u271d : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : 1 \u2208 J\na b : \u03b1\nh : b \u2208 Submodule.span \u03b1 \u2191I\noe : 1 = a \u2022 x + b\n\u22a2 y \u2208 I\n[PROOFSTEP]\nobtain F : y * 1 = y * (a \u2022 x + b) := congr_arg (fun g : \u03b1 => y * g) oe\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b\u271d : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : 1 \u2208 J\na b : \u03b1\nh : b \u2208 Submodule.span \u03b1 \u2191I\noe : 1 = a \u2022 x + b\nF : y * 1 = y * (a \u2022 x + b)\n\u22a2 y \u2208 I\n[PROOFSTEP]\nrw [\u2190 mul_one y, F, mul_add, mul_comm, smul_eq_mul, mul_assoc]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b\u271d : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : 1 \u2208 J\na b : \u03b1\nh : b \u2208 Submodule.span \u03b1 \u2191I\noe : 1 = a \u2022 x + b\nF : y * 1 = y * (a \u2022 x + b)\n\u22a2 a * (x * y) + y * b \u2208 I\n[PROOFSTEP]\nrefine' Submodule.add_mem I (I.mul_mem_left a hxy) (Submodule.smul_mem I y _)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b\u271d : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : 1 \u2208 J\na b : \u03b1\nh : b \u2208 Submodule.span \u03b1 \u2191I\noe : 1 = a \u2022 x + b\nF : y * 1 = y * (a \u2022 x + b)\n\u22a2 b \u2208 I\n[PROOFSTEP]\nrwa [Submodule.span_eq] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d\u00b2 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d\u00b9 : CommRing \u03b2\ninst\u271d : IsDomain \u03b2\nx y : \u03b2\n\u22a2 span {x} < span {y} \u2194 DvdNotUnit y x\n[PROOFSTEP]\nrw [lt_iff_le_not_le, span_singleton_le_span_singleton, span_singleton_le_span_singleton, dvd_and_not_dvd_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d\u00b2 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d\u00b9 : CommRing \u03b2\ninst\u271d : IsDomain \u03b2\nb\u2081 b\u2082 : \u03b2\nh\u2081 : b\u2081 \u2260 0\nh\u2082 : \u00acIsUnit b\u2082\nh : span {b\u2081} \u2264 span {b\u2081 * b\u2082}\n\u22a2 b\u2081 * b\u2082 \u2223 b\u2081 * 1\n[PROOFSTEP]\nrwa [mul_one, \u2190 Ideal.span_singleton_le_span_singleton]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\nha : a \u2208 I\nn : \u2115\nhn : 0 < n\n\u22a2 \u00ac0 < Nat.zero\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nhI : IsPrime I\nx y : \u03b1\n\u22a2 x \u2208 I \u2228 y \u2208 I \u2192 x * y \u2208 I\n[PROOFSTEP]\nrintro (h | h)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nhI : IsPrime I\nx y : \u03b1\nh : x \u2208 I\n\u22a2 x * y \u2208 I\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nhI : IsPrime I\nx y : \u03b1\nh : y \u2208 I\n\u22a2 x * y \u2208 I\n[PROOFSTEP]\nexacts [I.mul_mem_right y h, I.mul_mem_left x h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\nn : \u2115\n\u22a2 Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\n[PROOFSTEP]\ninduction' s using Multiset.induction_on with a s hs\n[GOAL]\ncase empty\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u22a2 Multiset.sum 0 ^ (\u2191Multiset.card 0 * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) 0))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\n\u22a2 Multiset.sum (a ::\u2098 s) ^ (\u2191Multiset.card (a ::\u2098 s) * n + 1) \u2208\n span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) (a ::\u2098 s)))\n[PROOFSTEP]\nsimp only [Finset.coe_insert, Multiset.map_cons, Multiset.toFinset_cons, Multiset.sum_cons, Multiset.card_cons, add_pow]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\n\u22a2 \u2211 m in Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1),\n a ^ m * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - m) * \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) m) \u2208\n span (insert (a ^ (n + 1)) \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)))\n[PROOFSTEP]\nrefine' Submodule.sum_mem _ _\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\n\u22a2 \u2200 (c : \u2115),\n c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1) \u2192\n a ^ c * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) * \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) \u2208\n span (insert (a ^ (n + 1)) \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)))\n[PROOFSTEP]\nintro c _hc\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : \u2115\n_hc : c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1)\n\u22a2 a ^ c * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) * \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) \u2208\n span (insert (a ^ (n + 1)) \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)))\n[PROOFSTEP]\nrw [mem_span_insert]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : \u2115\n_hc : c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1)\n\u22a2 \u2203 a_1 z,\n z \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)) \u2227\n a ^ c * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) * \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) =\n a_1 * a ^ (n + 1) + z\n[PROOFSTEP]\nby_cases h : n + 1 \u2264 c\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : \u2115\n_hc : c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1)\nh : n + 1 \u2264 c\n\u22a2 \u2203 a_1 z,\n z \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)) \u2227\n a ^ c * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) * \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) =\n a_1 * a ^ (n + 1) + z\n[PROOFSTEP]\nrefine'\n \u27e8a ^ (c - (n + 1)) * s.sum ^ ((Multiset.card s + 1) * n + 1 - c) * ((Multiset.card s + 1) * n + 1).choose c, 0,\n Submodule.zero_mem _, _\u27e9\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : \u2115\n_hc : c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1)\nh : n + 1 \u2264 c\n\u22a2 a ^ c * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) * \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) =\n a ^ (c - (n + 1)) * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) *\n \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) *\n a ^ (n + 1) +\n 0\n[PROOFSTEP]\nrw [mul_comm _ (a ^ (n + 1))]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : \u2115\n_hc : c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1)\nh : n + 1 \u2264 c\n\u22a2 a ^ c * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) * \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) =\n a ^ (n + 1) *\n (a ^ (c - (n + 1)) * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) *\n \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c)) +\n 0\n[PROOFSTEP]\nsimp_rw [\u2190 mul_assoc]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : \u2115\n_hc : c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1)\nh : n + 1 \u2264 c\n\u22a2 a ^ c * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) * \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) =\n a ^ (n + 1) * a ^ (c - (n + 1)) * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) *\n \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) +\n 0\n[PROOFSTEP]\nrw [\u2190 pow_add, add_zero, add_tsub_cancel_of_le h]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : \u2115\n_hc : c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1)\nh : \u00acn + 1 \u2264 c\n\u22a2 \u2203 a_1 z,\n z \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)) \u2227\n a ^ c * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) * \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) =\n a_1 * a ^ (n + 1) + z\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : \u2115\n_hc : c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1)\nh : \u00acn + 1 \u2264 c\n\u22a2 \u2203 z,\n z \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)) \u2227\n a ^ c * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) * \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) =\n 0 * a ^ (n + 1) + z\n[PROOFSTEP]\nsimp_rw [zero_mul, zero_add]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : \u2115\n_hc : c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1)\nh : \u00acn + 1 \u2264 c\n\u22a2 \u2203 z,\n z \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s)) \u2227\n a ^ c * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) * \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) =\n z\n[PROOFSTEP]\nrefine' \u27e8_, _, rfl\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : \u2115\n_hc : c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1)\nh : \u00acn + 1 \u2264 c\n\u22a2 a ^ c * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) * \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) \u2208\n span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\n[PROOFSTEP]\nreplace h : c \u2264 n := Nat.lt_succ_iff.mp (not_le.mp h)\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : \u2115\n_hc : c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1)\nh : c \u2264 n\n\u22a2 a ^ c * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) * \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) \u2208\n span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\n[PROOFSTEP]\nhave : (Multiset.card s + 1) * n + 1 - c = Multiset.card s * n + 1 + (n - c) := by\n rw [add_mul, one_mul, add_assoc, add_comm n 1, \u2190 add_assoc, add_tsub_assoc_of_le h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : \u2115\n_hc : c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1)\nh : c \u2264 n\n\u22a2 (\u2191Multiset.card s + 1) * n + 1 - c = \u2191Multiset.card s * n + 1 + (n - c)\n[PROOFSTEP]\nrw [add_mul, one_mul, add_assoc, add_comm n 1, \u2190 add_assoc, add_tsub_assoc_of_le h]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : \u2115\n_hc : c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1)\nh : c \u2264 n\nthis : (\u2191Multiset.card s + 1) * n + 1 - c = \u2191Multiset.card s * n + 1 + (n - c)\n\u22a2 a ^ c * Multiset.sum s ^ ((\u2191Multiset.card s + 1) * n + 1 - c) * \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) \u2208\n span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\n[PROOFSTEP]\nrw [this, pow_add]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : \u2115\n_hc : c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1)\nh : c \u2264 n\nthis : (\u2191Multiset.card s + 1) * n + 1 - c = \u2191Multiset.card s * n + 1 + (n - c)\n\u22a2 a ^ c * (Multiset.sum s ^ (\u2191Multiset.card s * n + 1) * Multiset.sum s ^ (n - c)) *\n \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) \u2208\n span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\n[PROOFSTEP]\nsimp_rw [mul_assoc, mul_comm (s.sum ^ (Multiset.card s * n + 1)), \u2190 mul_assoc]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b : \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b1\nI : Ideal \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nhs : Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208 span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\nc : \u2115\n_hc : c \u2208 Finset.range ((\u2191Multiset.card s + 1) * n + 1 + 1)\nh : c \u2264 n\nthis : (\u2191Multiset.card s + 1) * n + 1 - c = \u2191Multiset.card s * n + 1 + (n - c)\n\u22a2 a ^ c * Multiset.sum s ^ (n - c) * \u2191(Nat.choose ((\u2191Multiset.card s + 1) * n + 1) c) *\n Multiset.sum s ^ (\u2191Multiset.card s * n + 1) \u2208\n span \u2191(Multiset.toFinset (Multiset.map (fun x => x ^ (n + 1)) s))\n[PROOFSTEP]\nexact mul_mem_left _ _ hs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\n\u03b9 : Type u_1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nn : \u2115\n\u22a2 (\u2211 i in s, f i) ^ (Finset.card s * n + 1) \u2208 span ((fun i => f i ^ (n + 1)) '' \u2191s)\n[PROOFSTEP]\nclassical simpa only [Multiset.card_map, Multiset.map_map, comp_apply, Multiset.toFinset_map, Finset.coe_image,\n Finset.val_toFinset] using pow_multiset_sum_mem_span_pow (s.1.map f) n\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\n\u03b9 : Type u_1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nn : \u2115\n\u22a2 (\u2211 i in s, f i) ^ (Finset.card s * n + 1) \u2208 span ((fun i => f i ^ (n + 1)) '' \u2191s)\n[PROOFSTEP]\nsimpa only [Multiset.card_map, Multiset.map_map, comp_apply, Multiset.toFinset_map, Finset.coe_image,\n Finset.val_toFinset] using pow_multiset_sum_mem_span_pow (s.1.map f) n\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nhs : span s = \u22a4\nn : \u2115\n\u22a2 span ((fun x => x ^ n) '' s) = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nhs : span s = \u22a4\nn : \u2115\n\u22a2 1 \u2208 span ((fun x => x ^ n) '' s)\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nhs : span s = \u22a4\n\u22a2 1 \u2208 span ((fun x => x ^ Nat.zero) '' s)\n[PROOFSTEP]\nobtain rfl | \u27e8x, hx\u27e9 := eq_empty_or_nonempty s\n[GOAL]\ncase zero.inl\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\nhs : span \u2205 = \u22a4\n\u22a2 1 \u2208 span ((fun x => x ^ Nat.zero) '' \u2205)\n[PROOFSTEP]\nrw [Set.image_empty, hs]\n[GOAL]\ncase zero.inl\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\nhs : span \u2205 = \u22a4\n\u22a2 1 \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase zero.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nhs : span s = \u22a4\nx : \u03b1\nhx : x \u2208 s\n\u22a2 1 \u2208 span ((fun x => x ^ Nat.zero) '' s)\n[PROOFSTEP]\nexact subset_span \u27e8_, hx, pow_zero _\u27e9\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nhs : span s = \u22a4\nn : \u2115\n\u22a2 1 \u2208 span ((fun x => x ^ Nat.succ n) '' s)\n[PROOFSTEP]\nrw [eq_top_iff_one, span, Finsupp.mem_span_iff_total] at hs \n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nhs : \u2203 l, \u2191(Finsupp.total (\u2191s) \u03b1 \u03b1 Subtype.val) l = 1\nn : \u2115\n\u22a2 1 \u2208 span ((fun x => x ^ Nat.succ n) '' s)\n[PROOFSTEP]\nrcases hs with \u27e8f, hf\u27e9\n[GOAL]\ncase succ.intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nn : \u2115\nf : \u2191s \u2192\u2080 \u03b1\nhf : \u2191(Finsupp.total (\u2191s) \u03b1 \u03b1 Subtype.val) f = 1\n\u22a2 1 \u2208 span ((fun x => x ^ Nat.succ n) '' s)\n[PROOFSTEP]\nhave hf : (f.support.sum fun a => f a * a) = 1 := hf\n[GOAL]\ncase succ.intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nn : \u2115\nf : \u2191s \u2192\u2080 \u03b1\nhf\u271d : \u2191(Finsupp.total (\u2191s) \u03b1 \u03b1 Subtype.val) f = 1\nhf : \u2211 a in f.support, \u2191f a * \u2191a = 1\n\u22a2 1 \u2208 span ((fun x => x ^ Nat.succ n) '' s)\n[PROOFSTEP]\nhave := sum_pow_mem_span_pow f.support (fun a => f a * a) n\n[GOAL]\ncase succ.intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nn : \u2115\nf : \u2191s \u2192\u2080 \u03b1\nhf\u271d : \u2191(Finsupp.total (\u2191s) \u03b1 \u03b1 Subtype.val) f = 1\nhf : \u2211 a in f.support, \u2191f a * \u2191a = 1\nthis :\n (\u2211 i in f.support, \u2191f i * \u2191i) ^ (Finset.card f.support * n + 1) \u2208\n span ((fun i => (\u2191f i * \u2191i) ^ (n + 1)) '' \u2191f.support)\n\u22a2 1 \u2208 span ((fun x => x ^ Nat.succ n) '' s)\n[PROOFSTEP]\nrw [hf, one_pow] at this \n[GOAL]\ncase succ.intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nn : \u2115\nf : \u2191s \u2192\u2080 \u03b1\nhf\u271d : \u2191(Finsupp.total (\u2191s) \u03b1 \u03b1 Subtype.val) f = 1\nhf : \u2211 a in f.support, \u2191f a * \u2191a = 1\nthis : 1 \u2208 span ((fun i => (\u2191f i * \u2191i) ^ (n + 1)) '' \u2191f.support)\n\u22a2 1 \u2208 span ((fun x => x ^ Nat.succ n) '' s)\n[PROOFSTEP]\nrefine' span_le.mpr _ this\n[GOAL]\ncase succ.intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nn : \u2115\nf : \u2191s \u2192\u2080 \u03b1\nhf\u271d : \u2191(Finsupp.total (\u2191s) \u03b1 \u03b1 Subtype.val) f = 1\nhf : \u2211 a in f.support, \u2191f a * \u2191a = 1\nthis : 1 \u2208 span ((fun i => (\u2191f i * \u2191i) ^ (n + 1)) '' \u2191f.support)\n\u22a2 (fun i => (\u2191f i * \u2191i) ^ (n + 1)) '' \u2191f.support \u2286 \u2191(span ((fun x => x ^ Nat.succ n) '' s))\n[PROOFSTEP]\nrintro _ hx\n[GOAL]\ncase succ.intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nn : \u2115\nf : \u2191s \u2192\u2080 \u03b1\nhf\u271d : \u2191(Finsupp.total (\u2191s) \u03b1 \u03b1 Subtype.val) f = 1\nhf : \u2211 a in f.support, \u2191f a * \u2191a = 1\nthis : 1 \u2208 span ((fun i => (\u2191f i * \u2191i) ^ (n + 1)) '' \u2191f.support)\na\u271d : \u03b1\nhx : a\u271d \u2208 (fun i => (\u2191f i * \u2191i) ^ (n + 1)) '' \u2191f.support\n\u22a2 a\u271d \u2208 \u2191(span ((fun x => x ^ Nat.succ n) '' s))\n[PROOFSTEP]\nsimp_rw [Set.mem_image] at hx \n[GOAL]\ncase succ.intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nn : \u2115\nf : \u2191s \u2192\u2080 \u03b1\nhf\u271d : \u2191(Finsupp.total (\u2191s) \u03b1 \u03b1 Subtype.val) f = 1\nhf : \u2211 a in f.support, \u2191f a * \u2191a = 1\nthis : 1 \u2208 span ((fun i => (\u2191f i * \u2191i) ^ (n + 1)) '' \u2191f.support)\na\u271d : \u03b1\nhx : \u2203 x, x \u2208 \u2191f.support \u2227 (\u2191f x * \u2191x) ^ (n + 1) = a\u271d\n\u22a2 a\u271d \u2208 \u2191(span ((fun x => x ^ Nat.succ n) '' s))\n[PROOFSTEP]\nrcases hx with \u27e8x, _, rfl\u27e9\n[GOAL]\ncase succ.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nn : \u2115\nf : \u2191s \u2192\u2080 \u03b1\nhf\u271d : \u2191(Finsupp.total (\u2191s) \u03b1 \u03b1 Subtype.val) f = 1\nhf : \u2211 a in f.support, \u2191f a * \u2191a = 1\nthis : 1 \u2208 span ((fun i => (\u2191f i * \u2191i) ^ (n + 1)) '' \u2191f.support)\nx : \u2191s\nleft\u271d : x \u2208 \u2191f.support\n\u22a2 (\u2191f x * \u2191x) ^ (n + 1) \u2208 \u2191(span ((fun x => x ^ Nat.succ n) '' s))\n[PROOFSTEP]\nhave : span ({(x : \u03b1) ^ (n + 1)} : Set \u03b1) \u2264 span ((fun x : \u03b1 => x ^ (n + 1)) '' s) :=\n by\n rw [span_le, Set.singleton_subset_iff]\n exact subset_span \u27e8x, x.prop, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nn : \u2115\nf : \u2191s \u2192\u2080 \u03b1\nhf\u271d : \u2191(Finsupp.total (\u2191s) \u03b1 \u03b1 Subtype.val) f = 1\nhf : \u2211 a in f.support, \u2191f a * \u2191a = 1\nthis : 1 \u2208 span ((fun i => (\u2191f i * \u2191i) ^ (n + 1)) '' \u2191f.support)\nx : \u2191s\nleft\u271d : x \u2208 \u2191f.support\n\u22a2 span {\u2191x ^ (n + 1)} \u2264 span ((fun x => x ^ (n + 1)) '' s)\n[PROOFSTEP]\nrw [span_le, Set.singleton_subset_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nn : \u2115\nf : \u2191s \u2192\u2080 \u03b1\nhf\u271d : \u2191(Finsupp.total (\u2191s) \u03b1 \u03b1 Subtype.val) f = 1\nhf : \u2211 a in f.support, \u2191f a * \u2191a = 1\nthis : 1 \u2208 span ((fun i => (\u2191f i * \u2191i) ^ (n + 1)) '' \u2191f.support)\nx : \u2191s\nleft\u271d : x \u2208 \u2191f.support\n\u22a2 \u2191x ^ (n + 1) \u2208 \u2191(span ((fun x => x ^ (n + 1)) '' s))\n[PROOFSTEP]\nexact subset_span \u27e8x, x.prop, rfl\u27e9\n[GOAL]\ncase succ.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nn : \u2115\nf : \u2191s \u2192\u2080 \u03b1\nhf\u271d : \u2191(Finsupp.total (\u2191s) \u03b1 \u03b1 Subtype.val) f = 1\nhf : \u2211 a in f.support, \u2191f a * \u2191a = 1\nthis\u271d : 1 \u2208 span ((fun i => (\u2191f i * \u2191i) ^ (n + 1)) '' \u2191f.support)\nx : \u2191s\nleft\u271d : x \u2208 \u2191f.support\nthis : span {\u2191x ^ (n + 1)} \u2264 span ((fun x => x ^ (n + 1)) '' s)\n\u22a2 (\u2191f x * \u2191x) ^ (n + 1) \u2208 \u2191(span ((fun x => x ^ Nat.succ n) '' s))\n[PROOFSTEP]\nrefine' this _\n[GOAL]\ncase succ.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nn : \u2115\nf : \u2191s \u2192\u2080 \u03b1\nhf\u271d : \u2191(Finsupp.total (\u2191s) \u03b1 \u03b1 Subtype.val) f = 1\nhf : \u2211 a in f.support, \u2191f a * \u2191a = 1\nthis\u271d : 1 \u2208 span ((fun i => (\u2191f i * \u2191i) ^ (n + 1)) '' \u2191f.support)\nx : \u2191s\nleft\u271d : x \u2208 \u2191f.support\nthis : span {\u2191x ^ (n + 1)} \u2264 span ((fun x => x ^ (n + 1)) '' s)\n\u22a2 (\u2191f x * \u2191x) ^ (n + 1) \u2208 span {\u2191x ^ (n + 1)}\n[PROOFSTEP]\nrw [mul_pow, mem_span_singleton]\n[GOAL]\ncase succ.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI : Ideal \u03b1\ns : Set \u03b1\nn : \u2115\nf : \u2191s \u2192\u2080 \u03b1\nhf\u271d : \u2191(Finsupp.total (\u2191s) \u03b1 \u03b1 Subtype.val) f = 1\nhf : \u2211 a in f.support, \u2191f a * \u2191a = 1\nthis\u271d : 1 \u2208 span ((fun i => (\u2191f i * \u2191i) ^ (n + 1)) '' \u2191f.support)\nx : \u2191s\nleft\u271d : x \u2208 \u2191f.support\nthis : span {\u2191x ^ (n + 1)} \u2264 span ((fun x => x ^ (n + 1)) '' s)\n\u22a2 \u2191x ^ (n + 1) \u2223 \u2191f x ^ (n + 1) * \u2191x ^ (n + 1)\n[PROOFSTEP]\nexact \u27e8f x ^ (n + 1), mul_comm _ _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Ring \u03b1\nI : Ideal \u03b1\na b x : \u03b1\n\u22a2 span {-x} = span {x}\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Ring \u03b1\nI : Ideal \u03b1\na b x x\u271d : \u03b1\n\u22a2 x\u271d \u2208 span {-x} \u2194 x\u271d \u2208 span {x}\n[PROOFSTEP]\nsimp only [mem_span_singleton']\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Ring \u03b1\nI : Ideal \u03b1\na b x x\u271d : \u03b1\n\u22a2 (\u2203 a, a * -x = x\u271d) \u2194 \u2203 a, a * x = x\u271d\n[PROOFSTEP]\nexact \u27e8fun \u27e8y, h\u27e9 => \u27e8-y, h \u25b8 neg_mul_comm y x\u27e9, fun \u27e8y, h\u27e9 => \u27e8-y, h \u25b8 neg_mul_neg y x\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nK : Type u\ninst\u271d : DivisionSemiring K\nI : Ideal K\n\u22a2 I = \u22a5 \u2228 I = \u22a4\n[PROOFSTEP]\nrw [or_iff_not_imp_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nK : Type u\ninst\u271d : DivisionSemiring K\nI : Ideal K\n\u22a2 \u00acI = \u22a4 \u2192 I = \u22a5\n[PROOFSTEP]\nchange _ \u2260 _ \u2192 _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nK : Type u\ninst\u271d : DivisionSemiring K\nI : Ideal K\n\u22a2 I \u2260 \u22a4 \u2192 I = \u22a5\n[PROOFSTEP]\nrw [Ideal.ne_top_iff_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nK : Type u\ninst\u271d : DivisionSemiring K\nI : Ideal K\n\u22a2 \u00ac1 \u2208 I \u2192 I = \u22a5\n[PROOFSTEP]\nintro h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nK : Type u\ninst\u271d : DivisionSemiring K\nI : Ideal K\nh1 : \u00ac1 \u2208 I\n\u22a2 I = \u22a5\n[PROOFSTEP]\nrw [eq_bot_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nK : Type u\ninst\u271d : DivisionSemiring K\nI : Ideal K\nh1 : \u00ac1 \u2208 I\n\u22a2 I \u2264 \u22a5\n[PROOFSTEP]\nintro r hr\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nK : Type u\ninst\u271d : DivisionSemiring K\nI : Ideal K\nh1 : \u00ac1 \u2208 I\nr : K\nhr : r \u2208 I\n\u22a2 r \u2208 \u22a5\n[PROOFSTEP]\nby_cases H : r = 0\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\nK : Type u\ninst\u271d : DivisionSemiring K\nI : Ideal K\nh1 : \u00ac1 \u2208 I\nr : K\nhr : r \u2208 I\nH : r = 0\n\u22a2 r \u2208 \u22a5\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\nK : Type u\ninst\u271d : DivisionSemiring K\nI : Ideal K\nh1 : \u00ac1 \u2208 I\nr : K\nhr : r \u2208 I\nH : \u00acr = 0\n\u22a2 r \u2208 \u22a5\n[PROOFSTEP]\nsimpa [H, h1] using I.mul_mem_left r\u207b\u00b9 hr\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nK : Type u\ninst\u271d : DivisionSemiring K\nI : Ideal K\nh : \u22a5 = \u22a4\n\u22a2 \u00ac1 \u2208 \u22a4\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nK : Type u\ninst\u271d : DivisionSemiring K\nI : Ideal K\nh : \u22a5 = \u22a4\n\u22a2 \u00ac1 \u2208 \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\na b c d : R\nh1 : a - b \u2208 I\nh2 : c - d \u2208 I\n\u22a2 a * c - b * d \u2208 I\n[PROOFSTEP]\nrw [show a * c - b * d = (a - b) * c + b * (c - d) by rw [sub_mul, mul_sub]; abel]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\na b c d : R\nh1 : a - b \u2208 I\nh2 : c - d \u2208 I\n\u22a2 a * c - b * d = (a - b) * c + b * (c - d)\n[PROOFSTEP]\nrw [sub_mul, mul_sub]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\na b c d : R\nh1 : a - b \u2208 I\nh2 : c - d \u2208 I\n\u22a2 a * c - b * d = a * c - b * c + (b * c - b * d)\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\na b c d : R\nh1 : a - b \u2208 I\nh2 : c - d \u2208 I\n\u22a2 a * c - b * d = a * c - b * c + (b * c - b * d)\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\na b c d : R\nh1 : a - b \u2208 I\nh2 : c - d \u2208 I\n\u22a2 (a - b) * c + b * (c - d) \u2208 I\n[PROOFSTEP]\nexact I.add_mem (I.mul_mem_right _ h1) (I.mul_mem_left _ h2)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nhf : \u00acIsField R\n\u22a2 \u2203 x _hx, \u00acIsUnit x\n[PROOFSTEP]\nhave : \u00ac_ := fun h => hf \u27e8exists_pair_ne R, mul_comm, h\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nhf : \u00acIsField R\nthis : \u00ac\u2200 {a : R}, a \u2260 0 \u2192 \u2203 b, a * b = 1\n\u22a2 \u2203 x _hx, \u00acIsUnit x\n[PROOFSTEP]\nsimp_rw [isUnit_iff_exists_inv]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nhf : \u00acIsField R\nthis : \u00ac\u2200 {a : R}, a \u2260 0 \u2192 \u2203 b, a * b = 1\n\u22a2 \u2203 x h, \u00ac\u2203 b, x * b = 1\n[PROOFSTEP]\npush_neg at this \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nhf : \u00acIsField R\nthis : Exists fun {a} => a \u2260 0 \u2227 \u2200 (b : R), a * b \u2260 1\n\u22a2 \u2203 x h, \u2200 (b : R), x * b \u2260 1\n[PROOFSTEP]\nobtain \u27e8x, hx, not_unit\u27e9 := this\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nhf : \u00acIsField R\nx : R\nhx : x \u2260 0\nnot_unit : \u2200 (b : R), x * b \u2260 1\n\u22a2 \u2203 x h, \u2200 (b : R), x * b \u2260 1\n[PROOFSTEP]\nexact \u27e8x, hx, not_unit\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\n\u22a2 \u00acIsField R \u2194 \u2203 I, \u22a5 < I \u2227 I < \u22a4\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\n\u22a2 \u00acIsField R \u2192 \u2203 I, \u22a5 < I \u2227 I < \u22a4\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nh : \u00acIsField R\n\u22a2 \u2203 I, \u22a5 < I \u2227 I < \u22a4\n[PROOFSTEP]\nobtain \u27e8x, nz, nu\u27e9 := exists_not_isUnit_of_not_isField h\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nh : \u00acIsField R\nx : R\nnz : x \u2260 0\nnu : \u00acIsUnit x\n\u22a2 \u2203 I, \u22a5 < I \u2227 I < \u22a4\n[PROOFSTEP]\nuse Ideal.span { x }\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nh : \u00acIsField R\nx : R\nnz : x \u2260 0\nnu : \u00acIsUnit x\n\u22a2 \u22a5 < Ideal.span {x} \u2227 Ideal.span {x} < \u22a4\n[PROOFSTEP]\nrw [bot_lt_iff_ne_bot, lt_top_iff_ne_top]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nh : \u00acIsField R\nx : R\nnz : x \u2260 0\nnu : \u00acIsUnit x\n\u22a2 Ideal.span {x} \u2260 \u22a5 \u2227 Ideal.span {x} \u2260 \u22a4\n[PROOFSTEP]\nexact \u27e8mt Ideal.span_singleton_eq_bot.mp nz, mt Ideal.span_singleton_eq_top.mp nu\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\n\u22a2 (\u2203 I, \u22a5 < I \u2227 I < \u22a4) \u2192 \u00acIsField R\n[PROOFSTEP]\nrintro \u27e8I, bot_lt, lt_top\u27e9 hf\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nI : Ideal R\nbot_lt : \u22a5 < I\nlt_top : I < \u22a4\nhf : IsField R\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8x, mem, ne_zero\u27e9 := SetLike.exists_of_lt bot_lt\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nI : Ideal R\nbot_lt : \u22a5 < I\nlt_top : I < \u22a4\nhf : IsField R\nx : R\nmem : x \u2208 I\nne_zero : \u00acx \u2208 \u22a5\n\u22a2 False\n[PROOFSTEP]\nrw [Submodule.mem_bot] at ne_zero \n[GOAL]\ncase mpr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nI : Ideal R\nbot_lt : \u22a5 < I\nlt_top : I < \u22a4\nhf : IsField R\nx : R\nmem : x \u2208 I\nne_zero : \u00acx = 0\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := hf.mul_inv_cancel ne_zero\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nI : Ideal R\nbot_lt : \u22a5 < I\nlt_top : I < \u22a4\nhf : IsField R\nx : R\nmem : x \u2208 I\nne_zero : \u00acx = 0\ny : R\nhy : x * y = 1\n\u22a2 False\n[PROOFSTEP]\nrw [lt_top_iff_ne_top, Ne.def, Ideal.eq_top_iff_one, \u2190 hy] at lt_top \n[GOAL]\ncase mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nI : Ideal R\nbot_lt : \u22a5 < I\nhf : IsField R\nx : R\nmem : x \u2208 I\nne_zero : \u00acx = 0\ny : R\nlt_top : \u00acx * y \u2208 I\nhy : x * y = 1\n\u22a2 False\n[PROOFSTEP]\nexact lt_top (I.mul_mem_right _ mem)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\n\u22a2 IsField R \u2194 IsSimpleOrder (Ideal R)\n[PROOFSTEP]\ncases subsingleton_or_nontrivial R\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\nh\u271d : Subsingleton R\n\u22a2 IsField R \u2194 IsSimpleOrder (Ideal R)\n[PROOFSTEP]\nexact \u27e8fun h => (not_isField_of_subsingleton _ h).elim, fun h => (false_of_nontrivial_of_subsingleton <| Ideal R).elim\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\nh\u271d : Nontrivial R\n\u22a2 IsField R \u2194 IsSimpleOrder (Ideal R)\n[PROOFSTEP]\nrw [\u2190 not_iff_not, Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_top, \u2190 not_iff_not]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\nh\u271d : Nontrivial R\n\u22a2 (\u00ac\u2203 I, \u22a5 < I \u2227 I < \u22a4) \u2194 \u00ac\u00acIsSimpleOrder (Ideal R)\n[PROOFSTEP]\npush_neg\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\nh\u271d : Nontrivial R\n\u22a2 (\u2200 (I : Ideal R), \u22a5 < I \u2192 \u00acI < \u22a4) \u2194 IsSimpleOrder (Ideal R)\n[PROOFSTEP]\nsimp_rw [lt_top_iff_ne_top, bot_lt_iff_ne_bot, \u2190 or_iff_not_imp_left, not_ne_iff]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\nh\u271d : Nontrivial R\n\u22a2 (\u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4) \u2194 IsSimpleOrder (Ideal R)\n[PROOFSTEP]\nexact \u27e8fun h => \u27e8h\u27e9, fun h => h.2\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nM : Ideal R\nmax : Ideal.IsMaximal M\nnot_field : \u00acIsField R\n\u22a2 M \u2260 \u22a5\n[PROOFSTEP]\nrintro h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nM : Ideal R\nmax : Ideal.IsMaximal M\nnot_field : \u00acIsField R\nh : M = \u22a5\n\u22a2 False\n[PROOFSTEP]\nrw [h] at max \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nM : Ideal R\nmax : Ideal.IsMaximal \u22a5\nnot_field : \u00acIsField R\nh : M = \u22a5\n\u22a2 False\n[PROOFSTEP]\nrcases max with \u27e8\u27e8_h1, h2\u27e9\u27e9\n[GOAL]\ncase mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nM : Ideal R\nnot_field : \u00acIsField R\nh : M = \u22a5\n_h1 : \u22a5 \u2260 \u22a4\nh2 : \u2200 (b : Ideal R), \u22a5 < b \u2192 b = \u22a4\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8I, hIbot, hItop\u27e9 := not_isField_iff_exists_ideal_bot_lt_and_lt_top.mp not_field\n[GOAL]\ncase mk.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\nM : Ideal R\nnot_field : \u00acIsField R\nh : M = \u22a5\n_h1 : \u22a5 \u2260 \u22a4\nh2 : \u2200 (b : Ideal R), \u22a5 < b \u2192 b = \u22a4\nI : Ideal R\nhIbot : \u22a5 < I\nhItop : I < \u22a4\n\u22a2 False\n[PROOFSTEP]\nexact ne_of_lt hItop (h2 I hIbot)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : \u00acIsField R\n\u22a2 \u22a5 < M\n[PROOFSTEP]\nrcases Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_top.1 non_field with \u27e8I, Ibot, Itop\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : \u00acIsField R\nI : Ideal R\nIbot : \u22a5 < I\nItop : I < \u22a4\n\u22a2 \u22a5 < M\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.left\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : \u00acIsField R\nI : Ideal R\nIbot : \u22a5 < I\nItop : I < \u22a4\n\u22a2 \u2191\u22a5 \u2286 \u2191M\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.right\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : \u00acIsField R\nI : Ideal R\nIbot : \u22a5 < I\nItop : I < \u22a4\n\u22a2 \u00ac\u2191M \u2286 \u2191\u22a5\n[PROOFSTEP]\nintro mle\n[GOAL]\ncase intro.intro.right\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : \u00acIsField R\nI : Ideal R\nIbot : \u22a5 < I\nItop : I < \u22a4\nmle : \u2191M \u2286 \u2191\u22a5\n\u22a2 False\n[PROOFSTEP]\napply lt_irrefl (\u22a4 : Ideal R)\n[GOAL]\ncase intro.intro.right\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : \u00acIsField R\nI : Ideal R\nIbot : \u22a5 < I\nItop : I < \u22a4\nmle : \u2191M \u2286 \u2191\u22a5\n\u22a2 \u22a4 < \u22a4\n[PROOFSTEP]\nhave : M = \u22a5 := eq_bot_iff.mpr mle\n[GOAL]\ncase intro.intro.right\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : \u00acIsField R\nI : Ideal R\nIbot : \u22a5 < I\nItop : I < \u22a4\nmle : \u2191M \u2286 \u2191\u22a5\nthis : M = \u22a5\n\u22a2 \u22a4 < \u22a4\n[PROOFSTEP]\nrw [\u2190 this] at Ibot \n[GOAL]\ncase intro.intro.right\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : \u00acIsField R\nI : Ideal R\nIbot : M < I\nItop : I < \u22a4\nmle : \u2191M \u2286 \u2191\u22a5\nthis : M = \u22a5\n\u22a2 \u22a4 < \u22a4\n[PROOFSTEP]\nrwa [hm.1.2 I Ibot] at Itop \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nh : a \u2208 nonunits \u03b1\n\u22a2 \u2203 I, Ideal.IsMaximal I \u2227 a \u2208 I\n[PROOFSTEP]\nhave : Ideal.span ({ a } : Set \u03b1) \u2260 \u22a4 := by\n intro H\n rw [Ideal.span_singleton_eq_top] at H \n contradiction\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nh : a \u2208 nonunits \u03b1\n\u22a2 Ideal.span {a} \u2260 \u22a4\n[PROOFSTEP]\nintro H\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nh : a \u2208 nonunits \u03b1\nH : Ideal.span {a} = \u22a4\n\u22a2 False\n[PROOFSTEP]\nrw [Ideal.span_singleton_eq_top] at H \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nh : a \u2208 nonunits \u03b1\nH : IsUnit a\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nh : a \u2208 nonunits \u03b1\nthis : Ideal.span {a} \u2260 \u22a4\n\u22a2 \u2203 I, Ideal.IsMaximal I \u2227 a \u2208 I\n[PROOFSTEP]\nrcases Ideal.exists_le_maximal _ this with \u27e8I, Imax, H\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nh : a \u2208 nonunits \u03b1\nthis : Ideal.span {a} \u2260 \u22a4\nI : Ideal \u03b1\nImax : Ideal.IsMaximal I\nH : Ideal.span {a} \u2264 I\n\u22a2 \u2203 I, Ideal.IsMaximal I \u2227 a \u2208 I\n[PROOFSTEP]\nuse I, Imax\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nh : a \u2208 nonunits \u03b1\nthis : Ideal.span {a} \u2260 \u22a4\nI : Ideal \u03b1\nImax : Ideal.IsMaximal I\nH : Ideal.span {a} \u2264 I\n\u22a2 a \u2208 I\n[PROOFSTEP]\napply H\n[GOAL]\ncase right.a\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nh : a \u2208 nonunits \u03b1\nthis : Ideal.span {a} \u2260 \u22a4\nI : Ideal \u03b1\nImax : Ideal.IsMaximal I\nH : Ideal.span {a} \u2264 I\n\u22a2 a \u2208 Ideal.span {a}\n[PROOFSTEP]\napply Ideal.subset_span\n[GOAL]\ncase right.a.a\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nh : a \u2208 nonunits \u03b1\nthis : Ideal.span {a} \u2260 \u22a4\nI : Ideal \u03b1\nImax : Ideal.IsMaximal I\nH : Ideal.span {a} \u2264 I\n\u22a2 a \u2208 {a}\n[PROOFSTEP]\nexact Set.mem_singleton a\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Ideal.Basic", "llama_tokens": 28635, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195152660688, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.4239506175586095}} {"text": "[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\n\u22a2 snorm f p \u03bc = snorm' f (ENNReal.toReal p) \u03bc\n[PROOFSTEP]\nsimp [snorm, hp_ne_zero, hp_ne_top]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\n\u22a2 snorm f p \u03bc = (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nrw [snorm_eq_snorm' hp_ne_zero hp_ne_top, snorm']\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\n\u22a2 snorm f 1 \u03bc = \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [snorm_eq_lintegral_rpow_nnnorm one_ne_zero ENNReal.coe_ne_top, ENNReal.one_toReal, one_div_one,\n ENNReal.rpow_one]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\n\u22a2 snorm f \u22a4 \u03bc = snormEssSup f \u03bc\n[PROOFSTEP]\nsimp [snorm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq0_lt : 0 < q\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc = snorm' f q \u03bc ^ q\n[PROOFSTEP]\nrw [snorm', \u2190 ENNReal.rpow_mul, one_div, inv_mul_cancel, ENNReal.rpow_one]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq0_lt : 0 < q\n\u22a2 q \u2260 0\n[PROOFSTEP]\nexact (ne_of_lt hq0_lt).symm\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq0_lt : 0 < q\nhfq : snorm' f q \u03bc < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc < \u22a4\n[PROOFSTEP]\nrw [lintegral_rpow_nnnorm_eq_rpow_snorm' hq0_lt]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq0_lt : 0 < q\nhfq : snorm' f q \u03bc < \u22a4\n\u22a2 snorm' (fun a => f a) q \u03bc ^ q < \u22a4\n[PROOFSTEP]\nexact ENNReal.rpow_lt_top_of_nonneg (le_of_lt hq0_lt) (ne_of_lt hfq)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhfp : snorm f p \u03bc < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc < \u22a4\n[PROOFSTEP]\napply lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top\n[GOAL]\ncase hq0_lt\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhfp : snorm f p \u03bc < \u22a4\n\u22a2 0 < ENNReal.toReal p\n[PROOFSTEP]\nexact ENNReal.toReal_pos hp_ne_zero hp_ne_top\n[GOAL]\ncase hfq\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhfp : snorm f p \u03bc < \u22a4\n\u22a2 snorm' (fun a => f a) (ENNReal.toReal p) \u03bc < \u22a4\n[PROOFSTEP]\nsimpa [snorm_eq_snorm' hp_ne_zero hp_ne_top] using hfp\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc < \u22a4 \u2192 snorm f p \u03bc < \u22a4\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nh : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc < \u22a4\n\u22a2 snorm f p \u03bc < \u22a4\n[PROOFSTEP]\nhave hp' := ENNReal.toReal_pos hp_ne_zero hp_ne_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nh : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc < \u22a4\nhp' : 0 < ENNReal.toReal p\n\u22a2 snorm f p \u03bc < \u22a4\n[PROOFSTEP]\nhave : 0 < 1 / p.toReal := div_pos zero_lt_one hp'\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nh : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc < \u22a4\nhp' : 0 < ENNReal.toReal p\nthis : 0 < 1 / ENNReal.toReal p\n\u22a2 snorm f p \u03bc < \u22a4\n[PROOFSTEP]\nsimpa [snorm_eq_lintegral_rpow_nnnorm hp_ne_zero hp_ne_top] using\n ENNReal.rpow_lt_top_of_nonneg (le_of_lt this) (ne_of_lt h)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\n\u22a2 snorm' f 0 \u03bc = 1\n[PROOFSTEP]\nrw [snorm', _root_.div_zero, ENNReal.rpow_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\n\u22a2 snorm f 0 \u03bc = 0\n[PROOFSTEP]\nsimp [snorm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\n\u22a2 Mem\u2112p f 0 \u2194 AEStronglyMeasurable f \u03bc\n[PROOFSTEP]\nsimp [Mem\u2112p, snorm_exponent_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp0_lt : 0 < q\n\u22a2 snorm' 0 q \u03bc = 0\n[PROOFSTEP]\nsimp [snorm', hp0_lt]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq0_ne : q \u2260 0\nh\u03bc : \u03bc \u2260 0\n\u22a2 snorm' 0 q \u03bc = 0\n[PROOFSTEP]\ncases' le_or_lt 0 q with hq0 hq_neg\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq0_ne : q \u2260 0\nh\u03bc : \u03bc \u2260 0\nhq0 : 0 \u2264 q\n\u22a2 snorm' 0 q \u03bc = 0\n[PROOFSTEP]\nexact snorm'_zero (lt_of_le_of_ne hq0 hq0_ne.symm)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq0_ne : q \u2260 0\nh\u03bc : \u03bc \u2260 0\nhq_neg : q < 0\n\u22a2 snorm' 0 q \u03bc = 0\n[PROOFSTEP]\nsimp [snorm', ENNReal.rpow_eq_zero_iff, h\u03bc, hq_neg]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u22a2 snormEssSup 0 \u03bc = 0\n[PROOFSTEP]\nsimp_rw [snormEssSup, Pi.zero_apply, nnnorm_zero, ENNReal.coe_zero, \u2190 ENNReal.bot_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u22a2 essSup (fun x => \u22a5) \u03bc = \u22a5\n[PROOFSTEP]\nexact essSup_const_bot\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u22a2 snorm 0 p \u03bc = 0\n[PROOFSTEP]\nby_cases h0 : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh0 : p = 0\n\u22a2 snorm 0 p \u03bc = 0\n[PROOFSTEP]\nsimp [h0]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh0 : \u00acp = 0\n\u22a2 snorm 0 p \u03bc = 0\n[PROOFSTEP]\nby_cases h_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 snorm 0 p \u03bc = 0\n[PROOFSTEP]\nsimp only [h_top, snorm_exponent_top, snormEssSup_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm 0 p \u03bc = 0\n[PROOFSTEP]\nrw [\u2190 Ne.def] at h0 \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh0 : p \u2260 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm 0 p \u03bc = 0\n[PROOFSTEP]\nsimp [snorm_eq_snorm' h0 h_top, ENNReal.toReal_pos h0 h_top]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u22a2 snorm (fun x => 0) p \u03bc = 0\n[PROOFSTEP]\nconvert snorm_zero (F := F)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u22a2 snorm 0 p \u03bc < \u22a4\n[PROOFSTEP]\nrw [snorm_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u22a2 0 < \u22a4\n[PROOFSTEP]\nexact ENNReal.coe_lt_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\n\u22a2 snorm' f q 0 = 0\n[PROOFSTEP]\nsimp [snorm', hq_pos]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 F\n\u22a2 snorm' f 0 0 = 1\n[PROOFSTEP]\nsimp [snorm']\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 F\nhq_neg : q < 0\n\u22a2 snorm' f q 0 = \u22a4\n[PROOFSTEP]\nsimp [snorm', hq_neg]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 F\n\u22a2 snormEssSup f 0 = 0\n[PROOFSTEP]\nsimp [snormEssSup]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 F\n\u22a2 snorm f p 0 = 0\n[PROOFSTEP]\nby_cases h0 : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 F\nh0 : p = 0\n\u22a2 snorm f p 0 = 0\n[PROOFSTEP]\nsimp [h0]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 F\nh0 : \u00acp = 0\n\u22a2 snorm f p 0 = 0\n[PROOFSTEP]\nby_cases h_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 F\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 snorm f p 0 = 0\n[PROOFSTEP]\nsimp [h_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 F\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm f p 0 = 0\n[PROOFSTEP]\nrw [\u2190 Ne.def] at h0 \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 F\nh0 : p \u2260 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm f p 0 = 0\n[PROOFSTEP]\nsimp [snorm_eq_snorm' h0 h_top, snorm', ENNReal.toReal_pos h0 h_top]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n\u22a2 snorm' (fun x => c) q \u03bc = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / q)\n[PROOFSTEP]\nrw [snorm', lintegral_const, ENNReal.mul_rpow_of_nonneg _ _ (by simp [hq_pos.le] : 0 \u2264 1 / q)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n\u22a2 0 \u2264 1 / q\n[PROOFSTEP]\nsimp [hq_pos.le]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n\u22a2 (\u2191\u2016c\u2016\u208a ^ q) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / q) = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / q)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n\u22a2 (\u2191\u2016c\u2016\u208a ^ q) ^ (1 / q) = \u2191\u2016c\u2016\u208a\n[PROOFSTEP]\nrw [\u2190 ENNReal.rpow_mul]\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n\u22a2 \u2191\u2016c\u2016\u208a ^ (q * (1 / q)) = \u2191\u2016c\u2016\u208a\n[PROOFSTEP]\nsuffices hq_cancel : q * (1 / q) = 1\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\nhq_cancel : q * (1 / q) = 1\n\u22a2 \u2191\u2016c\u2016\u208a ^ (q * (1 / q)) = \u2191\u2016c\u2016\u208a\n[PROOFSTEP]\nrw [hq_cancel, ENNReal.rpow_one]\n[GOAL]\ncase hq_cancel\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n\u22a2 q * (1 / q) = 1\n[PROOFSTEP]\nrw [one_div, mul_inv_cancel (ne_of_lt hq_pos).symm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : IsFiniteMeasure \u03bc\nc : F\nhc_ne_zero : c \u2260 0\nhq_ne_zero : q \u2260 0\n\u22a2 snorm' (fun x => c) q \u03bc = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / q)\n[PROOFSTEP]\nrw [snorm', lintegral_const, ENNReal.mul_rpow_of_ne_top _ (measure_ne_top \u03bc Set.univ)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : IsFiniteMeasure \u03bc\nc : F\nhc_ne_zero : c \u2260 0\nhq_ne_zero : q \u2260 0\n\u22a2 (\u2191\u2016c\u2016\u208a ^ q) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / q) = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / q)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : IsFiniteMeasure \u03bc\nc : F\nhc_ne_zero : c \u2260 0\nhq_ne_zero : q \u2260 0\n\u22a2 (\u2191\u2016c\u2016\u208a ^ q) ^ (1 / q) = \u2191\u2016c\u2016\u208a\n[PROOFSTEP]\nrw [\u2190 ENNReal.rpow_mul]\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : IsFiniteMeasure \u03bc\nc : F\nhc_ne_zero : c \u2260 0\nhq_ne_zero : q \u2260 0\n\u22a2 \u2191\u2016c\u2016\u208a ^ (q * (1 / q)) = \u2191\u2016c\u2016\u208a\n[PROOFSTEP]\nsuffices hp_cancel : q * (1 / q) = 1\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : IsFiniteMeasure \u03bc\nc : F\nhc_ne_zero : c \u2260 0\nhq_ne_zero : q \u2260 0\nhp_cancel : q * (1 / q) = 1\n\u22a2 \u2191\u2016c\u2016\u208a ^ (q * (1 / q)) = \u2191\u2016c\u2016\u208a\n[PROOFSTEP]\nrw [hp_cancel, ENNReal.rpow_one]\n[GOAL]\ncase hp_cancel\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : IsFiniteMeasure \u03bc\nc : F\nhc_ne_zero : c \u2260 0\nhq_ne_zero : q \u2260 0\n\u22a2 q * (1 / q) = 1\n[PROOFSTEP]\nrw [one_div, mul_inv_cancel hq_ne_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : IsFiniteMeasure \u03bc\nc : F\nhc_ne_zero : c \u2260 0\nhq_ne_zero : q \u2260 0\n\u22a2 \u2191\u2016c\u2016\u208a ^ q \u2260 \u22a4\n[PROOFSTEP]\nrw [Ne.def, ENNReal.rpow_eq_top_iff, not_or, not_and_or, not_and_or]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : IsFiniteMeasure \u03bc\nc : F\nhc_ne_zero : c \u2260 0\nhq_ne_zero : q \u2260 0\n\u22a2 (\u00ac\u2191\u2016c\u2016\u208a = 0 \u2228 \u00acq < 0) \u2227 (\u00ac\u2191\u2016c\u2016\u208a = \u22a4 \u2228 \u00ac0 < q)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : IsFiniteMeasure \u03bc\nc : F\nhc_ne_zero : c \u2260 0\nhq_ne_zero : q \u2260 0\n\u22a2 \u00ac\u2191\u2016c\u2016\u208a = 0 \u2228 \u00acq < 0\n[PROOFSTEP]\nleft\n[GOAL]\ncase left.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : IsFiniteMeasure \u03bc\nc : F\nhc_ne_zero : c \u2260 0\nhq_ne_zero : q \u2260 0\n\u22a2 \u00ac\u2191\u2016c\u2016\u208a = 0\n[PROOFSTEP]\nrwa [ENNReal.coe_eq_zero, nnnorm_eq_zero]\n[GOAL]\ncase right\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : IsFiniteMeasure \u03bc\nc : F\nhc_ne_zero : c \u2260 0\nhq_ne_zero : q \u2260 0\n\u22a2 \u00ac\u2191\u2016c\u2016\u208a = \u22a4 \u2228 \u00ac0 < q\n[PROOFSTEP]\nexact Or.inl ENNReal.coe_ne_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : F\nh\u03bc : \u03bc \u2260 0\n\u22a2 snormEssSup (fun x => c) \u03bc = \u2191\u2016c\u2016\u208a\n[PROOFSTEP]\nrw [snormEssSup, essSup_const _ h\u03bc]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\ninst\u271d : IsProbabilityMeasure \u03bc\n\u22a2 snorm' (fun x => c) q \u03bc = \u2191\u2016c\u2016\u208a\n[PROOFSTEP]\nsimp [snorm'_const c hq_pos, measure_univ]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : F\nh0 : p \u2260 0\nh\u03bc : \u03bc \u2260 0\n\u22a2 snorm (fun x => c) p \u03bc = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nby_cases h_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : F\nh0 : p \u2260 0\nh\u03bc : \u03bc \u2260 0\nh_top : p = \u22a4\n\u22a2 snorm (fun x => c) p \u03bc = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nsimp [h_top, snormEssSup_const c h\u03bc]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : F\nh0 : p \u2260 0\nh\u03bc : \u03bc \u2260 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm (fun x => c) p \u03bc = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nsimp [snorm_eq_snorm' h0 h_top, snorm'_const, ENNReal.toReal_pos h0 h_top]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : F\nh0 : p \u2260 0\nh_top : p \u2260 \u22a4\n\u22a2 snorm (fun x => c) p \u03bc = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nsimp [snorm_eq_snorm' h0 h_top, snorm'_const, ENNReal.toReal_pos h0 h_top]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4\n[PROOFSTEP]\nhave hp : 0 < p.toReal := ENNReal.toReal_pos hp_ne_zero hp_ne_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4\n[PROOFSTEP]\nby_cases h\u03bc : \u03bc = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u03bc = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4\n[PROOFSTEP]\nsimp only [h\u03bc, Measure.coe_zero, Pi.zero_apply, or_true_iff, WithTop.zero_lt_top, snorm_measure_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4\n[PROOFSTEP]\nby_cases hc : c = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : c = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4\n[PROOFSTEP]\nsimp only [hc, true_or_iff, eq_self_iff_true, WithTop.zero_lt_top, snorm_zero']\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4\n[PROOFSTEP]\nrw [snorm_const' c hp_ne_zero hp_ne_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4\n[PROOFSTEP]\nby_cases h\u03bc_top : \u03bc Set.univ = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\nh\u03bc_top : \u2191\u2191\u03bc Set.univ = \u22a4\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4\n[PROOFSTEP]\nsimp [hc, h\u03bc_top, hp]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\nh\u03bc_top : \u00ac\u2191\u2191\u03bc Set.univ = \u22a4\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4\n[PROOFSTEP]\nrw [ENNReal.mul_lt_top_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\nh\u03bc_top : \u00ac\u2191\u2191\u03bc Set.univ = \u22a4\n\u22a2 \u2191\u2016c\u2016\u208a < \u22a4 \u2227 \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 \u2228 \u2191\u2016c\u2016\u208a = 0 \u2228 \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) = 0 \u2194\n c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4\n[PROOFSTEP]\nsimp only [true_and_iff, one_div, ENNReal.rpow_eq_zero_iff, h\u03bc, false_or_iff, or_false_iff, ENNReal.coe_lt_top,\n nnnorm_eq_zero, ENNReal.coe_eq_zero, MeasureTheory.Measure.measure_univ_eq_zero, hp, inv_lt_zero, hc, and_false_iff,\n false_and_iff, _root_.inv_pos, or_self_iff, h\u03bc_top, Ne.lt_top h\u03bc_top, iff_true_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : F\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhp : 0 < ENNReal.toReal p\nh\u03bc : \u00ac\u03bc = 0\nhc : \u00acc = 0\nh\u03bc_top : \u00ac\u2191\u2191\u03bc Set.univ = \u22a4\n\u22a2 \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9 < \u22a4\n[PROOFSTEP]\nexact ENNReal.rpow_lt_top_of_nonneg (inv_nonneg.mpr hp.le) h\u03bc_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nc : E\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 Mem\u2112p (fun x => c) p\n[PROOFSTEP]\nrefine' \u27e8aestronglyMeasurable_const, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nc : E\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4\n[PROOFSTEP]\nby_cases h0 : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nc : E\ninst\u271d : IsFiniteMeasure \u03bc\nh0 : p = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4\n[PROOFSTEP]\nsimp [h0]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nc : E\ninst\u271d : IsFiniteMeasure \u03bc\nh0 : \u00acp = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4\n[PROOFSTEP]\nby_cases h\u03bc : \u03bc = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nc : E\ninst\u271d : IsFiniteMeasure \u03bc\nh0 : \u00acp = 0\nh\u03bc : \u03bc = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4\n[PROOFSTEP]\nsimp [h\u03bc]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nc : E\ninst\u271d : IsFiniteMeasure \u03bc\nh0 : \u00acp = 0\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 snorm (fun x => c) p \u03bc < \u22a4\n[PROOFSTEP]\nrw [snorm_const c h0 h\u03bc]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nc : E\ninst\u271d : IsFiniteMeasure \u03bc\nh0 : \u00acp = 0\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4\n[PROOFSTEP]\nrefine' ENNReal.mul_lt_top ENNReal.coe_ne_top _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nc : E\ninst\u271d : IsFiniteMeasure \u03bc\nh0 : \u00acp = 0\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) \u2260 \u22a4\n[PROOFSTEP]\nrefine' (ENNReal.rpow_lt_top_of_nonneg _ (measure_ne_top \u03bc Set.univ)).ne\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nc : E\ninst\u271d : IsFiniteMeasure \u03bc\nh0 : \u00acp = 0\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 0 \u2264 1 / ENNReal.toReal p\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\n\u22a2 Mem\u2112p (fun x => c) \u22a4\n[PROOFSTEP]\nrefine' \u27e8aestronglyMeasurable_const, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\n\u22a2 snorm (fun x => c) \u22a4 \u03bc < \u22a4\n[PROOFSTEP]\nby_cases h : \u03bc = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nh : \u03bc = 0\n\u22a2 snorm (fun x => c) \u22a4 \u03bc < \u22a4\n[PROOFSTEP]\nsimp only [h, snorm_measure_zero, WithTop.zero_lt_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nh : \u00ac\u03bc = 0\n\u22a2 snorm (fun x => c) \u22a4 \u03bc < \u22a4\n[PROOFSTEP]\nrw [snorm_const _ ENNReal.top_ne_zero h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : E\nh : \u00ac\u03bc = 0\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal \u22a4) < \u22a4\n[PROOFSTEP]\nsimp only [ENNReal.top_toReal, _root_.div_zero, ENNReal.rpow_zero, mul_one, ENNReal.coe_lt_top]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\n\u22a2 Mem\u2112p (fun x => c) p \u2194 c = 0 \u2228 \u2191\u2191\u03bc Set.univ < \u22a4\n[PROOFSTEP]\nrw [\u2190 snorm_const_lt_top_iff hp_ne_zero hp_ne_top]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nc : E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\n\u22a2 Mem\u2112p (fun x => c) p \u2194 snorm (fun x => c) p \u03bc < \u22a4\n[PROOFSTEP]\nexact \u27e8fun h => h.2, fun h => \u27e8aestronglyMeasurable_const, h\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nhq : 0 \u2264 q\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\n\u22a2 snorm' f q \u03bc \u2264 snorm' g q \u03bc\n[PROOFSTEP]\nrw [snorm']\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nhq : 0 \u2264 q\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) \u2264 snorm' g q \u03bc\n[PROOFSTEP]\nrefine' ENNReal.rpow_le_rpow _ (one_div_nonneg.2 hq)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nhq : 0 \u2264 q\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ q \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_mono_ae (h.mono fun x hx => _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nhq : 0 \u2264 q\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\nx : \u03b1\nhx : \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\n\u22a2 \u2191\u2016f x\u2016\u208a ^ q \u2264 \u2191\u2016g x\u2016\u208a ^ q\n[PROOFSTEP]\nexact ENNReal.rpow_le_rpow (ENNReal.coe_le_coe.2 hx) hq\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 F\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a = \u2016g x\u2016\u208a\n\u22a2 snorm' f q \u03bc = snorm' g q \u03bc\n[PROOFSTEP]\nhave : (fun x => (\u2016f x\u2016\u208a : \u211d\u22650\u221e) ^ q) =\u1d50[\u03bc] fun x => (\u2016g x\u2016\u208a : \u211d\u22650\u221e) ^ q := hfg.mono fun x hx => by simp_rw [hx]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 F\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a = \u2016g x\u2016\u208a\nx : \u03b1\nhx : \u2016f x\u2016\u208a = \u2016g x\u2016\u208a\n\u22a2 (fun x => \u2191\u2016f x\u2016\u208a ^ q) x = (fun x => \u2191\u2016g x\u2016\u208a ^ q) x\n[PROOFSTEP]\nsimp_rw [hx]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 F\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a = \u2016g x\u2016\u208a\nthis : (fun x => \u2191\u2016f x\u2016\u208a ^ q) =\u1d50[\u03bc] fun x => \u2191\u2016g x\u2016\u208a ^ q\n\u22a2 snorm' f q \u03bc = snorm' g q \u03bc\n[PROOFSTEP]\nsimp only [snorm', lintegral_congr_ae this]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\n\u22a2 snorm f p \u03bc \u2264 snorm g p \u03bc\n[PROOFSTEP]\nsimp only [snorm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\n\u22a2 (if p = 0 then 0 else if p = \u22a4 then snormEssSup f \u03bc else snorm' f (ENNReal.toReal p) \u03bc) \u2264\n if p = 0 then 0 else if p = \u22a4 then snormEssSup g \u03bc else snorm' g (ENNReal.toReal p) \u03bc\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\nh\u271d : p = 0\n\u22a2 0 \u2264 0\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\nh\u271d\u00b9 : \u00acp = 0\nh\u271d : p = \u22a4\n\u22a2 snormEssSup f \u03bc \u2264 snormEssSup g \u03bc\n[PROOFSTEP]\nexact essSup_mono_ae (h.mono fun x hx => ENNReal.coe_le_coe.mpr hx)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\nh\u271d\u00b9 : \u00acp = 0\nh\u271d : \u00acp = \u22a4\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc \u2264 snorm' g (ENNReal.toReal p) \u03bc\n[PROOFSTEP]\nexact snorm'_mono_nnnorm_ae ENNReal.toReal_nonneg h\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nC : \u211d\u22650\nhfC : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 C\n\u22a2 snorm f p \u03bc \u2264 C \u2022 \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\nrcases eq_zero_or_neZero \u03bc with rfl | h\u03bc\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nC : \u211d\u22650\nhfC : \u2200\u1d50 (x : \u03b1) \u22020, \u2016f x\u2016\u208a \u2264 C\n\u22a2 snorm f p 0 \u2264 C \u2022 \u2191\u21910 Set.univ ^ (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nC : \u211d\u22650\nhfC : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 C\nh\u03bc : NeZero \u03bc\n\u22a2 snorm f p \u03bc \u2264 C \u2022 \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\nby_cases hp : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nC : \u211d\u22650\nhfC : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 C\nh\u03bc : NeZero \u03bc\nhp : p = 0\n\u22a2 snorm f p \u03bc \u2264 C \u2022 \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nC : \u211d\u22650\nhfC : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 C\nh\u03bc : NeZero \u03bc\nhp : \u00acp = 0\n\u22a2 snorm f p \u03bc \u2264 C \u2022 \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\nhave : \u2200\u1d50 x \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016(C : \u211d)\u2016\u208a := hfC.mono fun x hx => hx.trans_eq C.nnnorm_eq.symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nC : \u211d\u22650\nhfC : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 C\nh\u03bc : NeZero \u03bc\nhp : \u00acp = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016\u2191C\u2016\u208a\n\u22a2 snorm f p \u03bc \u2264 C \u2022 \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\nrefine' (snorm_mono_ae this).trans_eq _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nC : \u211d\u22650\nhfC : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 C\nh\u03bc : NeZero \u03bc\nhp : \u00acp = 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016\u2191C\u2016\u208a\n\u22a2 snorm (fun x => \u2191C) p \u03bc = C \u2022 \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\nrw [snorm_const _ hp (NeZero.ne \u03bc), C.nnnorm_eq, one_div, ENNReal.smul_def, smul_eq_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nC : \u211d\nhfC : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 C\n\u22a2 snorm f p \u03bc \u2264 \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9 * ENNReal.ofReal C\n[PROOFSTEP]\nrw [\u2190 mul_comm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nC : \u211d\nhfC : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 C\n\u22a2 snorm f p \u03bc \u2264 ENNReal.ofReal C * \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\nexact snorm_le_of_ae_nnnorm_bound (hfC.mono fun x hx => hx.trans C.le_coe_toNNReal)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\n\u22a2 snorm' (fun a => \u2016f a\u2016) q \u03bc = snorm' f q \u03bc\n[PROOFSTEP]\nsimp [snorm']\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\np q : \u211d\nhq_pos : 0 < q\n\u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm' f (p * q) \u03bc ^ q\n[PROOFSTEP]\nsimp_rw [snorm']\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\np q : \u211d\nhq_pos : 0 < q\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016\u2016f a\u2016 ^ q\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) = ((\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ (p * q) \u2202\u03bc) ^ (1 / (p * q))) ^ q\n[PROOFSTEP]\nrw [\u2190 ENNReal.rpow_mul, \u2190 one_div_mul_one_div]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\np q : \u211d\nhq_pos : 0 < q\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016\u2016f a\u2016 ^ q\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) = (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ (p * q) \u2202\u03bc) ^ (1 / p * (1 / q) * q)\n[PROOFSTEP]\nsimp_rw [one_div]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\np q : \u211d\nhq_pos : 0 < q\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016\u2016f a\u2016 ^ q\u2016\u208a ^ p \u2202\u03bc) ^ p\u207b\u00b9 = (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ (p * q) \u2202\u03bc) ^ (p\u207b\u00b9 * q\u207b\u00b9 * q)\n[PROOFSTEP]\nrw [mul_assoc, inv_mul_cancel hq_pos.ne.symm, mul_one]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\np q : \u211d\nhq_pos : 0 < q\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016\u2016f a\u2016 ^ q\u2016\u208a ^ p \u2202\u03bc) ^ p\u207b\u00b9 = (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ (p * q) \u2202\u03bc) ^ p\u207b\u00b9\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_f\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\np q : \u211d\nhq_pos : 0 < q\n\u22a2 (fun a => \u2191\u2016\u2016f a\u2016 ^ q\u2016\u208a ^ p) = fun a => \u2191\u2016f a\u2016\u208a ^ (p * q)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase e_a.e_f.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\np q : \u211d\nhq_pos : 0 < q\nx : \u03b1\n\u22a2 \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a ^ p = \u2191\u2016f x\u2016\u208a ^ (p * q)\n[PROOFSTEP]\nsimp_rw [\u2190 ofReal_norm_eq_coe_nnnorm]\n[GOAL]\ncase e_a.e_f.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\np q : \u211d\nhq_pos : 0 < q\nx : \u03b1\n\u22a2 ENNReal.ofReal \u2016\u2016f x\u2016 ^ q\u2016 ^ p = ENNReal.ofReal \u2016f x\u2016 ^ (p * q)\n[PROOFSTEP]\nrw [Real.norm_eq_abs, abs_eq_self.mpr (Real.rpow_nonneg_of_nonneg (norm_nonneg _) _), mul_comm, \u2190\n ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hq_pos.le, ENNReal.rpow_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\n\u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q\n[PROOFSTEP]\nby_cases h0 : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : p = 0\n\u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q\n[PROOFSTEP]\nsimp [h0, ENNReal.zero_rpow_of_pos hq_pos]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\n\u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q\n[PROOFSTEP]\nby_cases hp_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q\n[PROOFSTEP]\nsimp only [hp_top, snorm_exponent_top, ENNReal.top_mul', hq_pos.not_le, ENNReal.ofReal_eq_zero, if_false,\n snorm_exponent_top, snormEssSup]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q\n[PROOFSTEP]\nhave h_rpow : essSup (fun x : \u03b1 => (\u2016\u2016f x\u2016 ^ q\u2016\u208a : \u211d\u22650\u221e)) \u03bc = essSup (fun x : \u03b1 => (\u2016f x\u2016\u208a : \u211d\u22650\u221e) ^ q) \u03bc :=\n by\n congr\n ext1 x\n conv_rhs => rw [\u2190 nnnorm_norm]\n rw [ENNReal.coe_rpow_of_nonneg _ hq_pos.le, ENNReal.coe_eq_coe]\n ext\n push_cast\n rw [Real.norm_rpow_of_nonneg (norm_nonneg _)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) = fun x => \u2191\u2016f x\u2016\u208a ^ q\n[PROOFSTEP]\next1 x\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n\u22a2 \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2191\u2016f x\u2016\u208a ^ q\n[PROOFSTEP]\nconv_rhs => rw [\u2190 nnnorm_norm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n| \u2191\u2016f x\u2016\u208a ^ q\n[PROOFSTEP]\nrw [\u2190 nnnorm_norm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n| \u2191\u2016f x\u2016\u208a ^ q\n[PROOFSTEP]\nrw [\u2190 nnnorm_norm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n| \u2191\u2016f x\u2016\u208a ^ q\n[PROOFSTEP]\nrw [\u2190 nnnorm_norm]\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n\u22a2 \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2191\u2016\u2016f x\u2016\u2016\u208a ^ q\n[PROOFSTEP]\nrw [ENNReal.coe_rpow_of_nonneg _ hq_pos.le, ENNReal.coe_eq_coe]\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n\u22a2 \u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2016\u2016f x\u2016\u2016\u208a ^ q\n[PROOFSTEP]\next\n[GOAL]\ncase e_f.h.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n\u22a2 \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2191(\u2016\u2016f x\u2016\u2016\u208a ^ q)\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase e_f.h.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nx : \u03b1\n\u22a2 \u2016\u2016f x\u2016 ^ q\u2016 = \u2016\u2016f x\u2016\u2016 ^ q\n[PROOFSTEP]\nrw [Real.norm_rpow_of_nonneg (norm_nonneg _)]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nh_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\n\u22a2 essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q\n[PROOFSTEP]\nrw [h_rpow]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nh_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q\n[PROOFSTEP]\nhave h_rpow_mono := ENNReal.strictMono_rpow_of_pos hq_pos\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nh_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\nh_rpow_mono : StrictMono fun x => x ^ q\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q\n[PROOFSTEP]\nhave h_rpow_surj := (ENNReal.rpow_left_bijective hq_pos.ne.symm).2\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nh_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\nh_rpow_mono : StrictMono fun x => x ^ q\nh_rpow_surj : Function.Surjective fun y => y ^ q\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q\n[PROOFSTEP]\nlet iso := h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : p = \u22a4\nh_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc\nh_rpow_mono : StrictMono fun x => x ^ q\nh_rpow_surj : Function.Surjective fun y => y ^ q\niso : \u211d\u22650\u221e \u2243o \u211d\u22650\u221e := StrictMono.orderIsoOfSurjective (fun x => x ^ q) h_rpow_mono h_rpow_surj\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q\n[PROOFSTEP]\nexact (iso.essSup_apply (fun x => (\u2016f x\u2016\u208a : \u211d\u22650\u221e)) \u03bc).symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q\n[PROOFSTEP]\nrw [snorm_eq_snorm' h0 hp_top, snorm_eq_snorm' _ _]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal (p * ENNReal.ofReal q)) \u03bc ^ q\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 p * ENNReal.ofReal q \u2260 0\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 p * ENNReal.ofReal q \u2260 \u22a4\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 p * ENNReal.ofReal q \u2260 0\n[PROOFSTEP]\nrefine' mul_ne_zero h0 _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 ENNReal.ofReal q \u2260 0\n[PROOFSTEP]\nrwa [Ne.def, ENNReal.ofReal_eq_zero, not_le]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal (p * ENNReal.ofReal q)) \u03bc ^ q\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 p * ENNReal.ofReal q \u2260 \u22a4\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 p * ENNReal.ofReal q \u2260 \u22a4\n[PROOFSTEP]\nexact ENNReal.mul_ne_top hp_top ENNReal.ofReal_ne_top\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal (p * ENNReal.ofReal q)) \u03bc ^ q\n[PROOFSTEP]\nrw [ENNReal.toReal_mul, ENNReal.toReal_ofReal hq_pos.le]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\nh0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal p * q) \u03bc ^ q\n[PROOFSTEP]\nexact snorm'_norm_rpow f p.toReal q hq_pos\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhfg : f =\u1d50[\u03bc] g\n\u22a2 Mem\u2112p f p \u2194 Mem\u2112p g p\n[PROOFSTEP]\nsimp only [Mem\u2112p, snorm_congr_ae hfg, aestronglyMeasurable_congr hfg]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nC : \u211d\nhfC : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 C\n\u22a2 snorm f \u22a4 \u03bc < \u22a4\n[PROOFSTEP]\nrw [snorm_exponent_top]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nC : \u211d\nhfC : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 C\n\u22a2 snormEssSup f \u03bc < \u22a4\n[PROOFSTEP]\nexact snormEssSup_lt_top_of_ae_bound hfC\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhq : 0 \u2264 q\n\u22a2 snorm' f q \u03bd \u2264 snorm' f q \u03bc\n[PROOFSTEP]\nsimp_rw [snorm']\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhq : 0 \u2264 q\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bd) ^ (1 / q) \u2264 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nsuffices h_integral_mono : (\u222b\u207b a, (\u2016f a\u2016\u208a : \u211d\u22650\u221e) ^ q \u2202\u03bd) \u2264 \u222b\u207b a, (\u2016f a\u2016\u208a : \u211d\u22650\u221e) ^ q \u2202\u03bc from\n ENNReal.rpow_le_rpow h_integral_mono (by simp [hq])\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhq : 0 \u2264 q\nh_integral_mono : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bd \u2264 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc\n\u22a2 0 \u2264 1 / q\n[PROOFSTEP]\nsimp [hq]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhq : 0 \u2264 q\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bd \u2264 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc\n[PROOFSTEP]\nexact lintegral_mono' h\u03bc\u03bd le_rfl\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u226a \u03bc\n\u22a2 snormEssSup f \u03bd \u2264 snormEssSup f \u03bc\n[PROOFSTEP]\nsimp_rw [snormEssSup]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u226a \u03bc\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bd \u2264 essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc\n[PROOFSTEP]\nexact essSup_mono_measure h\u03bc\u03bd\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\n\u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc\n[PROOFSTEP]\nby_cases hp0 : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhp0 : p = 0\n\u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc\n[PROOFSTEP]\nsimp [hp0]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhp0 : \u00acp = 0\n\u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc\n[PROOFSTEP]\nby_cases hp_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhp0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc\n[PROOFSTEP]\nsimp [hp_top, snormEssSup_mono_measure f (Measure.absolutelyContinuous_of_le h\u03bc\u03bd)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc\n[PROOFSTEP]\nsimp_rw [snorm_eq_snorm' hp0 hp_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh\u03bc\u03bd : \u03bd \u2264 \u03bc\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' f (ENNReal.toReal p) \u03bd \u2264 snorm' f (ENNReal.toReal p) \u03bc\n[PROOFSTEP]\nexact snorm'_mono_measure f h\u03bc\u03bd ENNReal.toReal_nonneg\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\nhp : 0 \u2264 p\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 snorm' f p (c \u2022 \u03bc) = c ^ (1 / p) * snorm' f p \u03bc\n[PROOFSTEP]\nrw [snorm', lintegral_smul_measure, ENNReal.mul_rpow_of_nonneg, snorm']\n[GOAL]\ncase hz\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\nhp : 0 \u2264 p\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 0 \u2264 1 / p\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\n\u22a2 snormEssSup f (c \u2022 \u03bc) = snormEssSup f \u03bc\n[PROOFSTEP]\nsimp_rw [snormEssSup]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\n\u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a) (c \u2022 \u03bc) = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc\n[PROOFSTEP]\nexact essSup_smul_measure hc\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f p \u03bc\n[PROOFSTEP]\nsimp_rw [snorm_eq_snorm' hp_ne_zero hp_ne_top]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 snorm' f (ENNReal.toReal p) (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm' f (ENNReal.toReal p) \u03bc\n[PROOFSTEP]\nrw [snorm'_smul_measure ENNReal.toReal_nonneg]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 c ^ (1 / ENNReal.toReal p) * snorm' f (ENNReal.toReal p) \u03bc =\n c ^ ENNReal.toReal (1 / p) \u2022 snorm' f (ENNReal.toReal p) \u03bc\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 1 / ENNReal.toReal p = ENNReal.toReal (1 / p)\n[PROOFSTEP]\nsimp_rw [one_div]\n[GOAL]\ncase e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 (ENNReal.toReal p)\u207b\u00b9 = ENNReal.toReal p\u207b\u00b9\n[PROOFSTEP]\nrw [ENNReal.toReal_inv]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f p \u03bc\n[PROOFSTEP]\nby_cases hp0 : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\nhp0 : p = 0\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f p \u03bc\n[PROOFSTEP]\nsimp [hp0]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\nhp0 : \u00acp = 0\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f p \u03bc\n[PROOFSTEP]\nby_cases hp_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\nhp0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f p \u03bc\n[PROOFSTEP]\nsimp [hp_top, snormEssSup_smul_measure hc]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\nhc : c \u2260 0\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f p \u03bc\n[PROOFSTEP]\nexact snorm_smul_measure_of_ne_zero_of_ne_top hp0 hp_top c\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f p \u03bc\n[PROOFSTEP]\nby_cases hp0 : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\nhp0 : p = 0\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f p \u03bc\n[PROOFSTEP]\nsimp [hp0]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\nhp0 : \u00acp = 0\n\u22a2 snorm f p (c \u2022 \u03bc) = c ^ ENNReal.toReal (1 / p) \u2022 snorm f p \u03bc\n[PROOFSTEP]\nexact snorm_smul_measure_of_ne_zero_of_ne_top hp0 hp_ne_top c\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 snorm f 1 (c \u2022 \u03bc) = c * snorm f 1 \u03bc\n[PROOFSTEP]\nrw [@snorm_smul_measure_of_ne_top _ _ _ \u03bc _ 1 (@ENNReal.coe_ne_top 1) f c]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nc : \u211d\u22650\u221e\n\u22a2 c ^ ENNReal.toReal (1 / 1) \u2022 snorm f 1 \u03bc = c * snorm f 1 \u03bc\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u22a2 Mem\u2112p f p\n[PROOFSTEP]\nrefine' \u27e8hf.1.mono' (Measure.absolutelyContinuous_of_le_smul h\u03bc'_le), _\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u22a2 snorm f p \u03bc' < \u22a4\n[PROOFSTEP]\nrefine' (snorm_mono_measure f h\u03bc'_le).trans_lt _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u22a2 snorm f p (c \u2022 \u03bc) < \u22a4\n[PROOFSTEP]\nby_cases hc0 : c = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhc0 : c = 0\n\u22a2 snorm f p (c \u2022 \u03bc) < \u22a4\n[PROOFSTEP]\nsimp [hc0]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhc0 : \u00acc = 0\n\u22a2 snorm f p (c \u2022 \u03bc) < \u22a4\n[PROOFSTEP]\nrw [snorm_smul_measure_of_ne_zero hc0, smul_eq_mul]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhc0 : \u00acc = 0\n\u22a2 c ^ ENNReal.toReal (1 / p) * snorm f p \u03bc < \u22a4\n[PROOFSTEP]\nrefine' ENNReal.mul_lt_top _ hf.2.ne\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03bc' : Measure \u03b1\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhc0 : \u00acc = 0\n\u22a2 c ^ ENNReal.toReal (1 / p) \u2260 \u22a4\n[PROOFSTEP]\nsimp [hc, hc0]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc\u271d \u03bd\u271d : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\n\u03bc \u03bd : Measure \u03b1\n\u22a2 snorm f 1 (\u03bc + \u03bd) = snorm f 1 \u03bc + snorm f 1 \u03bd\n[PROOFSTEP]\nsimp_rw [snorm_one_eq_lintegral_nnnorm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc\u271d \u03bd\u271d : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\n\u03bc \u03bd : Measure \u03b1\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202(\u03bc + \u03bd) = \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202\u03bd\n[PROOFSTEP]\nrw [lintegral_add_measure _ \u03bc \u03bd]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nh : Mem\u2112p f p\nx : \u03b1\n\u22a2 \u2016\u2016f x\u2016\u2016 \u2264 \u2016f x\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nh : Mem\u2112p (fun x => \u2016f x\u2016) p\n\u22a2 snorm f p \u03bc < \u22a4\n[PROOFSTEP]\nrw [\u2190 snorm_norm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nh : Mem\u2112p (fun x => \u2016f x\u2016) p\n\u22a2 snorm (fun x => \u2016f x\u2016) p \u03bc < \u22a4\n[PROOFSTEP]\nexact h.2\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nhq0_lt : 0 < q\nhf_zero : f =\u1d50[\u03bc] 0\n\u22a2 snorm' f q \u03bc = 0\n[PROOFSTEP]\nrw [snorm'_congr_ae hf_zero, snorm'_zero hq0_lt]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq0_ne : q \u2260 0\nh\u03bc : \u03bc \u2260 0\nf : \u03b1 \u2192 F\nhf_zero : f =\u1d50[\u03bc] 0\n\u22a2 snorm' f q \u03bc = 0\n[PROOFSTEP]\nrw [snorm'_congr_ae hf_zero, snorm'_zero' hq0_ne h\u03bc]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : snorm' f q \u03bc = 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nrw [snorm', ENNReal.rpow_eq_zero_iff] at h \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc = 0 \u2227 0 < 1 / q \u2228 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc = \u22a4 \u2227 1 / q < 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\ncases h with\n| inl h =>\n rw [lintegral_eq_zero_iff' (hf.ennnorm.pow_const q)] at h \n refine' h.left.mono fun x hx => _\n rw [Pi.zero_apply, ENNReal.rpow_eq_zero_iff] at hx \n cases hx with\n | inl hx =>\n cases' hx with hx _\n rwa [\u2190 ENNReal.coe_zero, ENNReal.coe_eq_coe, nnnorm_eq_zero] at hx \n | inr hx => exact absurd hx.left ENNReal.coe_ne_top\n| inr h =>\n exfalso\n rw [one_div, inv_lt_zero] at h \n exact hq0.not_lt h.right\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc = 0 \u2227 0 < 1 / q \u2228 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc = \u22a4 \u2227 1 / q < 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\ncases h with\n| inl h =>\n rw [lintegral_eq_zero_iff' (hf.ennnorm.pow_const q)] at h \n refine' h.left.mono fun x hx => _\n rw [Pi.zero_apply, ENNReal.rpow_eq_zero_iff] at hx \n cases hx with\n | inl hx =>\n cases' hx with hx _\n rwa [\u2190 ENNReal.coe_zero, ENNReal.coe_eq_coe, nnnorm_eq_zero] at hx \n | inr hx => exact absurd hx.left ENNReal.coe_ne_top\n| inr h =>\n exfalso\n rw [one_div, inv_lt_zero] at h \n exact hq0.not_lt h.right\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc = 0 \u2227 0 < 1 / q\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\n\n| inl h =>\n rw [lintegral_eq_zero_iff' (hf.ennnorm.pow_const q)] at h \n refine' h.left.mono fun x hx => _\n rw [Pi.zero_apply, ENNReal.rpow_eq_zero_iff] at hx \n cases hx with\n | inl hx =>\n cases' hx with hx _\n rwa [\u2190 ENNReal.coe_zero, ENNReal.coe_eq_coe, nnnorm_eq_zero] at hx \n | inr hx => exact absurd hx.left ENNReal.coe_ne_top\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc = 0 \u2227 0 < 1 / q\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nrw [lintegral_eq_zero_iff' (hf.ennnorm.pow_const q)] at h \n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : (fun x => \u2191\u2016f x\u2016\u208a ^ q) =\u1d50[\u03bc] 0 \u2227 0 < 1 / q\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nrefine' h.left.mono fun x hx => _\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : (fun x => \u2191\u2016f x\u2016\u208a ^ q) =\u1d50[\u03bc] 0 \u2227 0 < 1 / q\nx : \u03b1\nhx : (fun x => \u2191\u2016f x\u2016\u208a ^ q) x = OfNat.ofNat 0 x\n\u22a2 f x = OfNat.ofNat 0 x\n[PROOFSTEP]\nrw [Pi.zero_apply, ENNReal.rpow_eq_zero_iff] at hx \n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : (fun x => \u2191\u2016f x\u2016\u208a ^ q) =\u1d50[\u03bc] 0 \u2227 0 < 1 / q\nx : \u03b1\nhx : \u2191\u2016f x\u2016\u208a = 0 \u2227 0 < q \u2228 \u2191\u2016f x\u2016\u208a = \u22a4 \u2227 q < 0\n\u22a2 f x = OfNat.ofNat 0 x\n[PROOFSTEP]\ncases hx with\n| inl hx =>\n cases' hx with hx _\n rwa [\u2190 ENNReal.coe_zero, ENNReal.coe_eq_coe, nnnorm_eq_zero] at hx \n| inr hx => exact absurd hx.left ENNReal.coe_ne_top\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : (fun x => \u2191\u2016f x\u2016\u208a ^ q) =\u1d50[\u03bc] 0 \u2227 0 < 1 / q\nx : \u03b1\nhx : \u2191\u2016f x\u2016\u208a = 0 \u2227 0 < q \u2228 \u2191\u2016f x\u2016\u208a = \u22a4 \u2227 q < 0\n\u22a2 f x = OfNat.ofNat 0 x\n[PROOFSTEP]\ncases hx with\n| inl hx =>\n cases' hx with hx _\n rwa [\u2190 ENNReal.coe_zero, ENNReal.coe_eq_coe, nnnorm_eq_zero] at hx \n| inr hx => exact absurd hx.left ENNReal.coe_ne_top\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : (fun x => \u2191\u2016f x\u2016\u208a ^ q) =\u1d50[\u03bc] 0 \u2227 0 < 1 / q\nx : \u03b1\nhx : \u2191\u2016f x\u2016\u208a = 0 \u2227 0 < q\n\u22a2 f x = OfNat.ofNat 0 x\n[PROOFSTEP]\n\n| inl hx =>\n cases' hx with hx _\n rwa [\u2190 ENNReal.coe_zero, ENNReal.coe_eq_coe, nnnorm_eq_zero] at hx \n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : (fun x => \u2191\u2016f x\u2016\u208a ^ q) =\u1d50[\u03bc] 0 \u2227 0 < 1 / q\nx : \u03b1\nhx : \u2191\u2016f x\u2016\u208a = 0 \u2227 0 < q\n\u22a2 f x = OfNat.ofNat 0 x\n[PROOFSTEP]\ncases' hx with hx _\n[GOAL]\ncase inl.inl.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : (fun x => \u2191\u2016f x\u2016\u208a ^ q) =\u1d50[\u03bc] 0 \u2227 0 < 1 / q\nx : \u03b1\nhx : \u2191\u2016f x\u2016\u208a = 0\nright\u271d : 0 < q\n\u22a2 f x = OfNat.ofNat 0 x\n[PROOFSTEP]\nrwa [\u2190 ENNReal.coe_zero, ENNReal.coe_eq_coe, nnnorm_eq_zero] at hx \n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : (fun x => \u2191\u2016f x\u2016\u208a ^ q) =\u1d50[\u03bc] 0 \u2227 0 < 1 / q\nx : \u03b1\nhx : \u2191\u2016f x\u2016\u208a = \u22a4 \u2227 q < 0\n\u22a2 f x = OfNat.ofNat 0 x\n[PROOFSTEP]\n\n| inr hx => exact absurd hx.left ENNReal.coe_ne_top\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : (fun x => \u2191\u2016f x\u2016\u208a ^ q) =\u1d50[\u03bc] 0 \u2227 0 < 1 / q\nx : \u03b1\nhx : \u2191\u2016f x\u2016\u208a = \u22a4 \u2227 q < 0\n\u22a2 f x = OfNat.ofNat 0 x\n[PROOFSTEP]\nexact absurd hx.left ENNReal.coe_ne_top\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc = \u22a4 \u2227 1 / q < 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\n\n| inr h =>\n exfalso\n rw [one_div, inv_lt_zero] at h \n exact hq0.not_lt h.right\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc = \u22a4 \u2227 1 / q < 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase inr.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc = \u22a4 \u2227 1 / q < 0\n\u22a2 False\n[PROOFSTEP]\nrw [one_div, inv_lt_zero] at h \n[GOAL]\ncase inr.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhq0 : 0 \u2264 q\nhf : AEStronglyMeasurable f \u03bc\nh : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc = \u22a4 \u2227 q < 0\n\u22a2 False\n[PROOFSTEP]\nexact hq0.not_lt h.right\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\n\u22a2 snormEssSup f \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nsimp [EventuallyEq, snormEssSup]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nh0 : p \u2260 0\n\u22a2 snorm f p \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nby_cases h_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nh0 : p \u2260 0\nh_top : p = \u22a4\n\u22a2 snorm f p \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nrw [h_top, snorm_exponent_top, snormEssSup_eq_zero_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nh0 : p \u2260 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm f p \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nrw [snorm_eq_snorm' h0 h_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nh0 : p \u2260 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nexact snorm'_eq_zero_iff (ENNReal.toReal_pos h0 h_top) hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhq1 : 1 \u2264 q\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016(f + g) a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) \u2264\n (\u222b\u207b (a : \u03b1), ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nrefine' ENNReal.rpow_le_rpow _ (by simp [le_trans zero_le_one hq1] : 0 \u2264 1 / q)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhq1 : 1 \u2264 q\n\u22a2 0 \u2264 1 / q\n[PROOFSTEP]\nsimp [le_trans zero_le_one hq1]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhq1 : 1 \u2264 q\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016(f + g) a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ^ q \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_mono fun a => ENNReal.rpow_le_rpow _ (le_trans zero_le_one hq1)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhq1 : 1 \u2264 q\na : \u03b1\n\u22a2 \u2191\u2016(f + g) a\u2016\u208a \u2264 ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a\n[PROOFSTEP]\nsimp only [Pi.add_apply, \u2190 ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0 : 0 \u2264 q\nhq1 : q \u2264 1\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016(f + g) a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) \u2264\n (\u222b\u207b (a : \u03b1), ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ^ q \u2202\u03bc) ^ (1 / q)\n[PROOFSTEP]\nrefine' ENNReal.rpow_le_rpow _ (by simp [hq0] : 0 \u2264 1 / q)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0 : 0 \u2264 q\nhq1 : q \u2264 1\n\u22a2 0 \u2264 1 / q\n[PROOFSTEP]\nsimp [hq0]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0 : 0 \u2264 q\nhq1 : q \u2264 1\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016(f + g) a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ^ q \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_mono fun a => ENNReal.rpow_le_rpow _ hq0\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0 : 0 \u2264 q\nhq1 : q \u2264 1\na : \u03b1\n\u22a2 \u2191\u2016(f + g) a\u2016\u208a \u2264 ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a\n[PROOFSTEP]\nsimp only [Pi.add_apply, \u2190 ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 F\n\u22a2 snormEssSup (f + g) \u03bc \u2264 snormEssSup f \u03bc + snormEssSup g \u03bc\n[PROOFSTEP]\nrefine' le_trans (essSup_mono_ae (eventually_of_forall fun x => _)) (ENNReal.essSup_add_le _ _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 F\nx : \u03b1\n\u22a2 (fun x => \u2191\u2016(f + g) x\u2016\u208a) x \u2264 ((fun x => \u2191\u2016f x\u2016\u208a) + fun x => \u2191\u2016g x\u2016\u208a) x\n[PROOFSTEP]\nsimp_rw [Pi.add_apply, \u2190 ENNReal.coe_add, ENNReal.coe_le_coe]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 F\nx : \u03b1\n\u22a2 \u2016f x + g x\u2016\u208a \u2264 \u2016f x\u2016\u208a + \u2016g x\u2016\u208a\n[PROOFSTEP]\nexact nnnorm_add_le _ _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc\n[PROOFSTEP]\nby_cases hp0 : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : p = 0\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc\n[PROOFSTEP]\nsimp [hp0]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc\n[PROOFSTEP]\nby_cases hp_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc\n[PROOFSTEP]\nsimp [hp_top, snormEssSup_add_le]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc\n[PROOFSTEP]\nhave hp1_real : 1 \u2264 p.toReal := by rwa [\u2190 ENNReal.one_toReal, ENNReal.toReal_le_toReal ENNReal.one_ne_top hp_top]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 1 \u2264 ENNReal.toReal p\n[PROOFSTEP]\nrwa [\u2190 ENNReal.one_toReal, ENNReal.toReal_le_toReal ENNReal.one_ne_top hp_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\nhp1_real : 1 \u2264 ENNReal.toReal p\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc\n[PROOFSTEP]\nrepeat rw [snorm_eq_snorm' hp0 hp_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\nhp1_real : 1 \u2264 ENNReal.toReal p\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc\n[PROOFSTEP]\nrw [snorm_eq_snorm' hp0 hp_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\nhp1_real : 1 \u2264 ENNReal.toReal p\n\u22a2 snorm' (f + g) (ENNReal.toReal p) \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc\n[PROOFSTEP]\nrw [snorm_eq_snorm' hp0 hp_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\nhp1_real : 1 \u2264 ENNReal.toReal p\n\u22a2 snorm' (f + g) (ENNReal.toReal p) \u03bc \u2264 snorm' f (ENNReal.toReal p) \u03bc + snorm g p \u03bc\n[PROOFSTEP]\nrw [snorm_eq_snorm' hp0 hp_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\nhp1_real : 1 \u2264 ENNReal.toReal p\n\u22a2 snorm' (f + g) (ENNReal.toReal p) \u03bc \u2264 snorm' f (ENNReal.toReal p) \u03bc + snorm' g (ENNReal.toReal p) \u03bc\n[PROOFSTEP]\nrw [snorm_eq_snorm' hp0 hp_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp1 : 1 \u2264 p\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\nhp1_real : 1 \u2264 ENNReal.toReal p\n\u22a2 snorm' (f + g) (ENNReal.toReal p) \u03bc \u2264 snorm' f (ENNReal.toReal p) \u03bc + snorm' g (ENNReal.toReal p) \u03bc\n[PROOFSTEP]\nexact snorm'_add_le hf hg hp1_real\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\n\u22a2 LpAddConst p = 1\n[PROOFSTEP]\nrw [LpAddConst, if_neg]\n[GOAL]\ncase hnc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\n\u22a2 \u00acp \u2208 Set.Ioo 0 1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase hnc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nh : p \u2208 Set.Ioo 0 1\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ (h.2.trans_le hp)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u22a2 LpAddConst 0 = 1\n[PROOFSTEP]\nrw [LpAddConst, if_neg]\n[GOAL]\ncase hnc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u22a2 \u00ac0 \u2208 Set.Ioo 0 1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase hnc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nh : 0 \u2208 Set.Ioo 0 1\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ h.1\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\n\u22a2 LpAddConst p < \u22a4\n[PROOFSTEP]\nrw [LpAddConst]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\n\u22a2 (if p \u2208 Set.Ioo 0 1 then 2 ^ (1 / ENNReal.toReal p - 1) else 1) < \u22a4\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nh : p \u2208 Set.Ioo 0 1\n\u22a2 2 ^ (1 / ENNReal.toReal p - 1) < \u22a4\n[PROOFSTEP]\napply ENNReal.rpow_lt_top_of_nonneg _ ENNReal.two_ne_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nh : p \u2208 Set.Ioo 0 1\n\u22a2 0 \u2264 1 / ENNReal.toReal p - 1\n[PROOFSTEP]\nsimp only [one_div, sub_nonneg]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nh : p \u2208 Set.Ioo 0 1\n\u22a2 1 \u2264 (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\napply one_le_inv (ENNReal.toReal_pos h.1.ne' (h.2.trans ENNReal.one_lt_top).ne)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nh : p \u2208 Set.Ioo 0 1\n\u22a2 ENNReal.toReal p \u2264 1\n[PROOFSTEP]\nsimpa using ENNReal.toReal_mono ENNReal.one_ne_top h.2.le\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nh : \u00acp \u2208 Set.Ioo 0 1\n\u22a2 1 < \u22a4\n[PROOFSTEP]\nexact ENNReal.one_lt_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\n\u22a2 snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc)\n[PROOFSTEP]\nrcases eq_or_ne p 0 with (rfl | hp)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\n\u22a2 snorm (f + g) 0 \u03bc \u2264 LpAddConst 0 * (snorm f 0 \u03bc + snorm g 0 \u03bc)\n[PROOFSTEP]\nsimp only [snorm_exponent_zero, add_zero, mul_zero, le_zero_iff]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\n\u22a2 snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc)\n[PROOFSTEP]\nrcases lt_or_le p 1 with (h'p | h'p)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : p < 1\n\u22a2 snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc)\n[PROOFSTEP]\nsimp only [snorm_eq_snorm' hp (h'p.trans ENNReal.one_lt_top).ne]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : p < 1\n\u22a2 snorm' (f + g) (ENNReal.toReal p) \u03bc \u2264 LpAddConst p * (snorm' f (ENNReal.toReal p) \u03bc + snorm' g (ENNReal.toReal p) \u03bc)\n[PROOFSTEP]\nconvert snorm'_add_le_of_le_one hf ENNReal.toReal_nonneg _\n[GOAL]\ncase h.e'_4.h.e'_5\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : p < 1\n\u22a2 LpAddConst p = 2 ^ (1 / ENNReal.toReal p - 1)\n[PROOFSTEP]\nhave : p \u2208 Set.Ioo (0 : \u211d\u22650\u221e) 1 := \u27e8hp.bot_lt, h'p\u27e9\n[GOAL]\ncase h.e'_4.h.e'_5\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : p < 1\nthis : p \u2208 Set.Ioo 0 1\n\u22a2 LpAddConst p = 2 ^ (1 / ENNReal.toReal p - 1)\n[PROOFSTEP]\nsimp only [LpAddConst, if_pos this]\n[GOAL]\ncase inr.inl.convert_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : p < 1\n\u22a2 ENNReal.toReal p \u2264 1\n[PROOFSTEP]\nsimpa using ENNReal.toReal_mono ENNReal.one_ne_top h'p.le\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : 1 \u2264 p\n\u22a2 snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc)\n[PROOFSTEP]\nsimp [LpAddConst_of_one_le h'p]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nhp : p \u2260 0\nh'p : 1 \u2264 p\n\u22a2 snorm (f + g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc\n[PROOFSTEP]\nexact snorm_add_le hf hg h'p\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np \u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 \u2260 0\n\u22a2 \u2203 \u03b7,\n 0 < \u03b7 \u2227\n \u2200 (f g : \u03b1 \u2192 E),\n AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\n[PROOFSTEP]\nhave : Tendsto (fun \u03b7 : \u211d\u22650\u221e => LpAddConst p * (\u03b7 + \u03b7)) (\ud835\udcdd[>] 0) (\ud835\udcdd (LpAddConst p * (0 + 0))) :=\n (ENNReal.Tendsto.const_mul (tendsto_id.add tendsto_id) (Or.inr (LpAddConst_lt_top p).ne)).mono_left nhdsWithin_le_nhds\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np \u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 \u2260 0\nthis : Tendsto (fun \u03b7 => LpAddConst p * (\u03b7 + \u03b7)) (\ud835\udcdd[Set.Ioi 0] 0) (\ud835\udcdd (LpAddConst p * (0 + 0)))\n\u22a2 \u2203 \u03b7,\n 0 < \u03b7 \u2227\n \u2200 (f g : \u03b1 \u2192 E),\n AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\n[PROOFSTEP]\nsimp only [add_zero, mul_zero] at this \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np \u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 \u2260 0\nthis : Tendsto (fun \u03b7 => LpAddConst p * (\u03b7 + \u03b7)) (\ud835\udcdd[Set.Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2203 \u03b7,\n 0 < \u03b7 \u2227\n \u2200 (f g : \u03b1 \u2192 E),\n AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\n[PROOFSTEP]\nrcases(((tendsto_order.1 this).2 \u03b4 h\u03b4.bot_lt).and self_mem_nhdsWithin).exists with \u27e8\u03b7, h\u03b7, \u03b7pos\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np \u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 \u2260 0\nthis : Tendsto (fun \u03b7 => LpAddConst p * (\u03b7 + \u03b7)) (\ud835\udcdd[Set.Ioi 0] 0) (\ud835\udcdd 0)\n\u03b7 : \u211d\u22650\u221e\nh\u03b7 : LpAddConst p * (\u03b7 + \u03b7) < \u03b4\n\u03b7pos : 0 < \u03b7\n\u22a2 \u2203 \u03b7,\n 0 < \u03b7 \u2227\n \u2200 (f g : \u03b1 \u2192 E),\n AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\n[PROOFSTEP]\nrefine' \u27e8\u03b7, \u03b7pos, fun f g hf hg Hf Hg => _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np \u03b4 : \u211d\u22650\u221e\nh\u03b4 : \u03b4 \u2260 0\nthis : Tendsto (fun \u03b7 => LpAddConst p * (\u03b7 + \u03b7)) (\ud835\udcdd[Set.Ioi 0] 0) (\ud835\udcdd 0)\n\u03b7 : \u211d\u22650\u221e\nh\u03b7 : LpAddConst p * (\u03b7 + \u03b7) < \u03b4\n\u03b7pos : 0 < \u03b7\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nHf : snorm f p \u03bc \u2264 \u03b7\nHg : snorm g p \u03bc \u2264 \u03b7\n\u22a2 snorm (f + g) p \u03bc < \u03b4\n[PROOFSTEP]\ncalc\n snorm (f + g) p \u03bc \u2264 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc) := snorm_add_le' hf hg p\n _ \u2264 LpAddConst p * (\u03b7 + \u03b7) := (mul_le_mul_of_nonneg_left (add_le_add Hf Hg) bot_le)\n _ < \u03b4 := h\u03b7\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\n\u22a2 snorm (f - g) p \u03bc = snorm (f + -g) p \u03bc\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n -- We cannot use snorm_add_le on f and (-g) because we don't have `AEMeasurable (-g) \u03bc`, since\n -- we don't suppose `[BorelSpace E]`.\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\n\u22a2 snorm (fun x => \u2016f x + -g x\u2016) p \u03bc \u2264 snorm (fun x => \u2016f x\u2016 + \u2016-g x\u2016) p \u03bc\n[PROOFSTEP]\nrefine' snorm_mono_real fun x => _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nx : \u03b1\n\u22a2 \u2016\u2016f x + -g x\u2016\u2016 \u2264 \u2016f x\u2016 + \u2016-g x\u2016\n[PROOFSTEP]\nrw [norm_norm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\nx : \u03b1\n\u22a2 \u2016f x + -g x\u2016 \u2264 \u2016f x\u2016 + \u2016-g x\u2016\n[PROOFSTEP]\nexact norm_add_le _ _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\n\u22a2 snorm (fun x => \u2016f x\u2016 + \u2016-g x\u2016) p \u03bc = snorm (fun x => \u2016f x\u2016 + \u2016g x\u2016) p \u03bc\n[PROOFSTEP]\nsimp_rw [norm_neg]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\np : \u211d\u22650\u221e\n\u22a2 LpAddConst p * (snorm (fun x => \u2016f x\u2016) p \u03bc + snorm (fun x => \u2016g x\u2016) p \u03bc) = LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc)\n[PROOFSTEP]\nrw [\u2190 snorm_norm f, \u2190 snorm_norm g]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhg : AEStronglyMeasurable g \u03bc\nhp : 1 \u2264 p\n\u22a2 snorm (f - g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc\n[PROOFSTEP]\nsimpa [LpAddConst_of_one_le hp] using snorm_sub_le' hf hg p\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\n\u22a2 LpAddConst p * (snorm f p \u03bc + snorm g p \u03bc) < \u22a4\n[PROOFSTEP]\napply ENNReal.mul_lt_top (LpAddConst_lt_top p).ne\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\n\u22a2 snorm f p \u03bc + snorm g p \u03bc \u2260 \u22a4\n[PROOFSTEP]\nexact (ENNReal.add_lt_top.2 \u27e8hf.2, hg.2\u27e9).ne\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n[PROOFSTEP]\nby_cases hp_zero : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : p = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n[PROOFSTEP]\nsimp only [hp_zero, snorm_exponent_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n[PROOFSTEP]\nby_cases hp_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n[PROOFSTEP]\nsimp_rw [hp_top, snorm_exponent_top]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 snormEssSup g (Measure.map f \u03bc) = snormEssSup (g \u2218 f) \u03bc\n[PROOFSTEP]\nexact snormEssSup_map_measure hg hf\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n[PROOFSTEP]\nsimp_rw [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (x : \u03b2), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202Measure.map f \u03bc) ^ (1 / ENNReal.toReal p) =\n (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nrw [lintegral_map' (hg.ennnorm.pow_const p.toReal) hf]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\nhp_zero : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016g (f a)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 E\nhg : AEStronglyMeasurable g (Measure.map f \u03bc)\nhf : AEMeasurable f\n\u22a2 Mem\u2112p g p \u2194 Mem\u2112p (g \u2218 f) p\n[PROOFSTEP]\nsimp [Mem\u2112p, snorm_map_measure hg hf, hg.comp_aemeasurable hf, hg]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd\u271d : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\n\u03bd : Measure \u03b2\ng : \u03b2 \u2192\u2098[\u03bd] E\nhf : MeasurePreserving f\n\u22a2 snorm (\u2191(compMeasurePreserving g f hf)) p \u03bc = snorm (\u2191g) p \u03bd\n[PROOFSTEP]\nrw [snorm_congr_ae (g.coeFn_compMeasurePreserving _)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd\u271d : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\n\u03bd : Measure \u03b2\ng : \u03b2 \u2192\u2098[\u03bd] E\nhf : MeasurePreserving f\n\u22a2 snorm (\u2191g \u2218 f) p \u03bc = snorm (\u2191g) p \u03bd\n[PROOFSTEP]\nexact snorm_comp_measurePreserving g.aestronglyMeasurable hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n[PROOFSTEP]\nby_cases hp_zero : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : p = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n[PROOFSTEP]\nsimp only [hp_zero, snorm_exponent_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n[PROOFSTEP]\nby_cases hp : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : p = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n[PROOFSTEP]\nsimp_rw [hp, snorm_exponent_top]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : p = \u22a4\n\u22a2 snormEssSup g (Measure.map f \u03bc) = snormEssSup (g \u2218 f) \u03bc\n[PROOFSTEP]\nexact hf.essSup_map_measure\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : \u00acp = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n[PROOFSTEP]\nsimp_rw [snorm_eq_lintegral_rpow_nnnorm hp_zero hp]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (x : \u03b2), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202Measure.map f \u03bc) ^ (1 / ENNReal.toReal p) =\n (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nrw [hf.lintegral_map]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016g (f a)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b2 : Type u_5\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\n\u22a2 Mem\u2112p g p \u2194 Mem\u2112p (g \u2218 f) p\n[PROOFSTEP]\nsimp_rw [Mem\u2112p, hf.aestronglyMeasurable_map_iff, hf.snorm_map_measure]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 snorm' f q (Measure.trim \u03bd hm) = snorm' f q \u03bd\n[PROOFSTEP]\nsimp_rw [snorm']\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202Measure.trim \u03bd hm) ^ (1 / q) = (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bd) ^ (1 / q)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202Measure.trim \u03bd hm = \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bd\n[PROOFSTEP]\nrefine' lintegral_trim hm _\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 Measurable fun a => \u2191\u2016f a\u2016\u208a ^ q\n[PROOFSTEP]\nrefine' @Measurable.pow_const _ _ _ _ _ _ _ m _ (@Measurable.coe_nnreal_ennreal _ m _ _) q\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 Measurable fun a => \u2016f a\u2016\u208a\n[PROOFSTEP]\napply @StronglyMeasurable.measurable\n[GOAL]\ncase e_a.hf\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 StronglyMeasurable fun a => \u2016f a\u2016\u208a\n[PROOFSTEP]\nexact @StronglyMeasurable.nnnorm \u03b1 m _ _ _ hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 limsup f (Measure.ae (Measure.trim \u03bd hm)) = limsup f (Measure.ae \u03bd)\n[PROOFSTEP]\nsimp_rw [limsup_eq]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 sInf {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} = sInf {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}\n[PROOFSTEP]\nsuffices h_set_eq : {a : \u211d\u22650\u221e | \u2200\u1d50 n \u2202\u03bd.trim hm, f n \u2264 a} = {a : \u211d\u22650\u221e | \u2200\u1d50 n \u2202\u03bd, f n \u2264 a}\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh_set_eq : {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} = {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}\n\u22a2 sInf {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} = sInf {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}\n[PROOFSTEP]\nrw [h_set_eq]\n[GOAL]\ncase h_set_eq\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} = {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}\n[PROOFSTEP]\next1 a\n[GOAL]\ncase h_set_eq.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\n\u22a2 a \u2208 {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} \u2194 a \u2208 {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}\n[PROOFSTEP]\nsuffices h_meas_eq : \u03bd {x | \u00acf x \u2264 a} = \u03bd.trim hm {x | \u00acf x \u2264 a}\n[GOAL]\ncase h_set_eq.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\nh_meas_eq : \u2191\u2191\u03bd {x | \u00acf x \u2264 a} = \u2191\u2191(Measure.trim \u03bd hm) {x | \u00acf x \u2264 a}\n\u22a2 a \u2208 {a | \u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a} \u2194 a \u2208 {a | \u2200\u1d50 (n : \u03b1) \u2202\u03bd, f n \u2264 a}\n[PROOFSTEP]\nsimp_rw [Set.mem_setOf_eq, ae_iff, h_meas_eq]\n[GOAL]\ncase h_set_eq.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\nh_meas_eq : \u2191\u2191\u03bd {x | \u00acf x \u2264 a} = \u2191\u2191(Measure.trim \u03bd hm) {x | \u00acf x \u2264 a}\n\u22a2 (\u2200\u1d50 (n : \u03b1) \u2202Measure.trim \u03bd hm, f n \u2264 a) \u2194 \u2191\u2191(Measure.trim \u03bd hm) {a_1 | \u00acf a_1 \u2264 a} = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_meas_eq\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\n\u22a2 \u2191\u2191\u03bd {x | \u00acf x \u2264 a} = \u2191\u2191(Measure.trim \u03bd hm) {x | \u00acf x \u2264 a}\n[PROOFSTEP]\nrefine' (trim_measurableSet_eq hm _).symm\n[GOAL]\ncase h_meas_eq\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\n\u22a2 MeasurableSet {x | \u00acf x \u2264 a}\n[PROOFSTEP]\nrefine' @MeasurableSet.compl _ _ m (@measurableSet_le \u211d\u22650\u221e _ _ _ _ m _ _ _ _ _ hf _)\n[GOAL]\ncase h_meas_eq\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na : \u211d\u22650\u221e\n\u22a2 Measurable fun x => a\n[PROOFSTEP]\nexact @measurable_const _ _ _ m _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 essSup f (Measure.trim \u03bd hm) = essSup f \u03bd\n[PROOFSTEP]\nsimp_rw [essSup]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 limsup f (Measure.ae (Measure.trim \u03bd hm)) = limsup f (Measure.ae \u03bd)\n[PROOFSTEP]\nexact limsup_trim hm hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd\n[PROOFSTEP]\nby_cases h0 : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\nh0 : p = 0\n\u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd\n[PROOFSTEP]\nsimp [h0]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\nh0 : \u00acp = 0\n\u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd\n[PROOFSTEP]\nby_cases h_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd\n[PROOFSTEP]\nsimpa only [h_top, snorm_exponent_top] using snormEssSup_trim hm hf\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : StronglyMeasurable f\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd\n[PROOFSTEP]\nsimpa only [snorm_eq_snorm' h0 h_top] using snorm'_trim hm hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f (Measure.trim \u03bd hm)\n\u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd\n[PROOFSTEP]\nrw [snorm_congr_ae hf.ae_eq_mk, snorm_congr_ae (ae_eq_of_ae_eq_trim hf.ae_eq_mk)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f (Measure.trim \u03bd hm)\n\u22a2 snorm (AEStronglyMeasurable.mk f hf) p (Measure.trim \u03bd hm) = snorm (AEStronglyMeasurable.mk f hf) p \u03bd\n[PROOFSTEP]\nexact snorm_trim hm hf.stronglyMeasurable_mk\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\n\u22a2 snorm' (-f) q \u03bc = snorm' f q \u03bc\n[PROOFSTEP]\nsimp [snorm']\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\n\u22a2 snorm (-f) p \u03bc = snorm f p \u03bc\n[PROOFSTEP]\nby_cases h0 : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh0 : p = 0\n\u22a2 snorm (-f) p \u03bc = snorm f p \u03bc\n[PROOFSTEP]\nsimp [h0]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh0 : \u00acp = 0\n\u22a2 snorm (-f) p \u03bc = snorm f p \u03bc\n[PROOFSTEP]\nby_cases h_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 snorm (-f) p \u03bc = snorm f p \u03bc\n[PROOFSTEP]\nsimp [h_top, snormEssSup]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm (-f) p \u03bc = snorm f p \u03bc\n[PROOFSTEP]\nsimp [snorm_eq_snorm' h0 h_top]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u22a2 snorm (-f) p \u03bc < \u22a4\n[PROOFSTEP]\nsimp [hf.right]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n[PROOFSTEP]\nhave hq0_lt : 0 < q := lt_of_lt_of_le hp0_lt hpq\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n[PROOFSTEP]\nby_cases hpq_eq : p = q\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : p = q\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n[PROOFSTEP]\nrw [hpq_eq, sub_self, ENNReal.rpow_zero, mul_one]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n[PROOFSTEP]\nhave hpq : p < q := lt_of_le_of_ne hpq hpq_eq\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n[PROOFSTEP]\nlet g := fun _ : \u03b1 => (1 : \u211d\u22650\u221e)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n[PROOFSTEP]\nhave h_rw : (\u222b\u207b a, (\u2016f a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) = \u222b\u207b a, ((\u2016f a\u2016\u208a : \u211d\u22650\u221e) * g a) ^ p \u2202\u03bc := lintegral_congr fun a => by simp\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\na : \u03b1\n\u22a2 \u2191\u2016f a\u2016\u208a ^ p = (\u2191\u2016f a\u2016\u208a * g a) ^ p\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n[PROOFSTEP]\nrepeat' rw [snorm']\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n[PROOFSTEP]\nrw [snorm']\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n[PROOFSTEP]\nrw [snorm']\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n[PROOFSTEP]\nrw [snorm']\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n[PROOFSTEP]\nrw [h_rw]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n[PROOFSTEP]\nlet r := p * q / (q - p)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n[PROOFSTEP]\nhave hpqr : 1 / p = 1 / q + 1 / r :=\n by\n field_simp [(ne_of_lt hp0_lt).symm, (ne_of_lt hq0_lt).symm]\n ring\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\n\u22a2 1 / p = 1 / q + 1 / r\n[PROOFSTEP]\nfield_simp [(ne_of_lt hp0_lt).symm, (ne_of_lt hq0_lt).symm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\n\u22a2 p * q = q * p\n[PROOFSTEP]\nring\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n[PROOFSTEP]\ncalc\n (\u222b\u207b a : \u03b1, (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n (\u222b\u207b a : \u03b1, \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b a : \u03b1, g a ^ r \u2202\u03bc) ^ (1 / r) :=\n ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr hp0_lt hpq hpqr \u03bc hf.ennnorm aemeasurable_const\n _ = (\u222b\u207b a : \u03b1, \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u03bc Set.univ ^ (1 / p - 1 / q) := by rw [hpqr]; simp\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r) =\n (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n[PROOFSTEP]\nrw [hpqr]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\nhp0_lt : 0 < p\nhpq\u271d : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhq0_lt : 0 < q\nhpq_eq : \u00acp = q\nhpq : p < q\ng : \u03b1 \u2192 \u211d\u22650\u221e := fun x => 1\nh_rw : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc = \u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * g a) ^ p \u2202\u03bc\nr : \u211d := p * q / (q - p)\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), g a ^ r \u2202\u03bc) ^ (1 / r) =\n (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * \u2191\u2191\u03bc Set.univ ^ (1 / q + 1 / r - 1 / q)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq_pos : 0 < q\nf : \u03b1 \u2192 F\n\u22a2 snorm' f q \u03bc \u2264 snormEssSup f \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / q)\n[PROOFSTEP]\nhave h_le : (\u222b\u207b a : \u03b1, (\u2016f a\u2016\u208a : \u211d\u22650\u221e) ^ q \u2202\u03bc) \u2264 \u222b\u207b _ : \u03b1, snormEssSup f \u03bc ^ q \u2202\u03bc :=\n by\n refine' lintegral_mono_ae _\n have h_nnnorm_le_snorm_ess_sup := coe_nnnorm_ae_le_snormEssSup f \u03bc\n refine' h_nnnorm_le_snorm_ess_sup.mono fun x hx => ENNReal.rpow_le_rpow hx (le_of_lt hq_pos)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq_pos : 0 < q\nf : \u03b1 \u2192 F\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), snormEssSup f \u03bc ^ q \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_mono_ae _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq_pos : 0 < q\nf : \u03b1 \u2192 F\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2016f a\u2016\u208a ^ q \u2264 snormEssSup f \u03bc ^ q\n[PROOFSTEP]\nhave h_nnnorm_le_snorm_ess_sup := coe_nnnorm_ae_le_snormEssSup f \u03bc\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq_pos : 0 < q\nf : \u03b1 \u2192 F\nh_nnnorm_le_snorm_ess_sup : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2016f x\u2016\u208a \u2264 snormEssSup f \u03bc\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2016f a\u2016\u208a ^ q \u2264 snormEssSup f \u03bc ^ q\n[PROOFSTEP]\nrefine' h_nnnorm_le_snorm_ess_sup.mono fun x hx => ENNReal.rpow_le_rpow hx (le_of_lt hq_pos)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq_pos : 0 < q\nf : \u03b1 \u2192 F\nh_le : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), snormEssSup f \u03bc ^ q \u2202\u03bc\n\u22a2 snorm' f q \u03bc \u2264 snormEssSup f \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / q)\n[PROOFSTEP]\nrw [snorm', \u2190 ENNReal.rpow_one (snormEssSup f \u03bc)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq_pos : 0 < q\nf : \u03b1 \u2192 F\nh_le : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), snormEssSup f \u03bc ^ q \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) \u2264 snormEssSup f \u03bc ^ 1 * \u2191\u2191\u03bc Set.univ ^ (1 / q)\n[PROOFSTEP]\nnth_rw 2 [\u2190 mul_inv_cancel (ne_of_lt hq_pos).symm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq_pos : 0 < q\nf : \u03b1 \u2192 F\nh_le : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), snormEssSup f \u03bc ^ q \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) \u2264 snormEssSup f \u03bc ^ (q * q\u207b\u00b9) * \u2191\u2191\u03bc Set.univ ^ (1 / q)\n[PROOFSTEP]\nrw [ENNReal.rpow_mul, one_div, \u2190 ENNReal.mul_rpow_of_nonneg _ _ (by simp [hq_pos.le] : 0 \u2264 q\u207b\u00b9)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq_pos : 0 < q\nf : \u03b1 \u2192 F\nh_le : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), snormEssSup f \u03bc ^ q \u2202\u03bc\n\u22a2 0 \u2264 q\u207b\u00b9\n[PROOFSTEP]\nsimp [hq_pos.le]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq_pos : 0 < q\nf : \u03b1 \u2192 F\nh_le : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), snormEssSup f \u03bc ^ q \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ q\u207b\u00b9 \u2264 (snormEssSup f \u03bc ^ q * \u2191\u2191\u03bc Set.univ) ^ q\u207b\u00b9\n[PROOFSTEP]\nrefine' ENNReal.rpow_le_rpow _ (by simp [hq_pos.le])\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq_pos : 0 < q\nf : \u03b1 \u2192 F\nh_le : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), snormEssSup f \u03bc ^ q \u2202\u03bc\n\u22a2 0 \u2264 q\u207b\u00b9\n[PROOFSTEP]\nsimp [hq_pos.le]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhq_pos : 0 < q\nf : \u03b1 \u2192 F\nh_le : \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), snormEssSup f \u03bc ^ q \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc \u2264 snormEssSup f \u03bc ^ q * \u2191\u2191\u03bc Set.univ\n[PROOFSTEP]\nrwa [lintegral_const] at h_le \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 snorm f p \u03bc \u2264 snorm f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q)\n[PROOFSTEP]\nby_cases hp0 : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p = 0\n\u22a2 snorm f p \u03bc \u2264 snorm f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q)\n[PROOFSTEP]\nsimp [hp0, zero_le]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : \u00acp = 0\n\u22a2 snorm f p \u03bc \u2264 snorm f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q)\n[PROOFSTEP]\nrw [\u2190 Ne.def] at hp0 \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\n\u22a2 snorm f p \u03bc \u2264 snorm f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q)\n[PROOFSTEP]\nhave hp0_lt : 0 < p := lt_of_le_of_ne (zero_le _) hp0.symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\n\u22a2 snorm f p \u03bc \u2264 snorm f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q)\n[PROOFSTEP]\nhave hq0_lt : 0 < q := lt_of_lt_of_le hp0_lt hpq\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\n\u22a2 snorm f p \u03bc \u2264 snorm f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q)\n[PROOFSTEP]\nby_cases hq_top : q = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\nhq_top : q = \u22a4\n\u22a2 snorm f p \u03bc \u2264 snorm f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q)\n[PROOFSTEP]\nsimp only [hq_top, _root_.div_zero, one_div, ENNReal.top_toReal, sub_zero, snorm_exponent_top, GroupWithZero.inv_zero]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\nhq_top : q = \u22a4\n\u22a2 snorm f p \u03bc \u2264 snormEssSup f \u03bc * \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\nby_cases hp_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\nhq_top : q = \u22a4\nhp_top : p = \u22a4\n\u22a2 snorm f p \u03bc \u2264 snormEssSup f \u03bc * \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\nsimp only [hp_top, ENNReal.rpow_zero, mul_one, ENNReal.top_toReal, sub_zero, GroupWithZero.inv_zero, snorm_exponent_top]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\nhq_top : q = \u22a4\nhp_top : p = \u22a4\n\u22a2 snormEssSup f \u03bc \u2264 snormEssSup f \u03bc\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\nhq_top : q = \u22a4\nhp_top : \u00acp = \u22a4\n\u22a2 snorm f p \u03bc \u2264 snormEssSup f \u03bc * \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\nrw [snorm_eq_snorm' hp0 hp_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\nhq_top : q = \u22a4\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc \u2264 snormEssSup f \u03bc * \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\nhave hp_pos : 0 < p.toReal := ENNReal.toReal_pos hp0_lt.ne' hp_top\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\nhq_top : q = \u22a4\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc \u2264 snormEssSup f \u03bc * \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\nrefine' (snorm'_le_snormEssSup_mul_rpow_measure_univ hp_pos).trans (le_of_eq _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\nhq_top : q = \u22a4\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 snormEssSup f \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) = snormEssSup f \u03bc * \u2191\u2191\u03bc Set.univ ^ (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\ncongr\n[GOAL]\ncase neg.e_a.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\nhq_top : q = \u22a4\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 1 / ENNReal.toReal p = (ENNReal.toReal p)\u207b\u00b9\n[PROOFSTEP]\nexact one_div _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\nhq_top : \u00acq = \u22a4\n\u22a2 snorm f p \u03bc \u2264 snorm f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q)\n[PROOFSTEP]\nhave hp_lt_top : p < \u221e := hpq.trans_lt (lt_top_iff_ne_top.mpr hq_top)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\nhq_top : \u00acq = \u22a4\nhp_lt_top : p < \u22a4\n\u22a2 snorm f p \u03bc \u2264 snorm f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q)\n[PROOFSTEP]\nhave hp_pos : 0 < p.toReal := ENNReal.toReal_pos hp0_lt.ne' hp_lt_top.ne\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\nhq_top : \u00acq = \u22a4\nhp_lt_top : p < \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 snorm f p \u03bc \u2264 snorm f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q)\n[PROOFSTEP]\nrw [snorm_eq_snorm' hp0_lt.ne.symm hp_lt_top.ne, snorm_eq_snorm' hq0_lt.ne.symm hq_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\nhq_top : \u00acq = \u22a4\nhp_lt_top : p < \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc \u2264\n snorm' f (ENNReal.toReal q) \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q)\n[PROOFSTEP]\nhave hpq_real : p.toReal \u2264 q.toReal := by rwa [ENNReal.toReal_le_toReal hp_lt_top.ne hq_top]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\nhq_top : \u00acq = \u22a4\nhp_lt_top : p < \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 ENNReal.toReal p \u2264 ENNReal.toReal q\n[PROOFSTEP]\nrwa [ENNReal.toReal_le_toReal hp_lt_top.ne hq_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhp0 : p \u2260 0\nhp0_lt : 0 < p\nhq0_lt : 0 < q\nhq_top : \u00acq = \u22a4\nhp_lt_top : p < \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhpq_real : ENNReal.toReal p \u2264 ENNReal.toReal q\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc \u2264\n snorm' f (ENNReal.toReal q) \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q)\n[PROOFSTEP]\nexact snorm'_le_snorm'_mul_rpow_measure_univ hp_pos hpq_real hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\np q : \u211d\nhp0_lt : 0 < p\nhpq : p \u2264 q\n\u03bc : Measure \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc\n[PROOFSTEP]\nhave h_le_\u03bc := snorm'_le_snorm'_mul_rpow_measure_univ hp0_lt hpq hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm\u271d m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\np q : \u211d\nhp0_lt : 0 < p\nhpq : p \u2264 q\n\u03bc : Measure \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nh_le_\u03bc : snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q)\n\u22a2 snorm' f p \u03bc \u2264 snorm' f q \u03bc\n[PROOFSTEP]\nrwa [measure_univ, ENNReal.one_rpow, mul_one] at h_le_\u03bc \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nhq_pos : 0 < q\nf : \u03b1 \u2192 F\ninst\u271d : IsProbabilityMeasure \u03bc\n\u22a2 snormEssSup f \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / q) = snormEssSup f \u03bc\n[PROOFSTEP]\nsimp [measure_univ]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\ninst\u271d : IsProbabilityMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\n\u22a2 snorm f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q) = snorm f q \u03bc\n[PROOFSTEP]\nsimp [measure_univ]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm' f q \u03bc < \u22a4\nhp_nonneg : 0 \u2264 p\nhpq : p \u2264 q\n\u22a2 snorm' f p \u03bc < \u22a4\n[PROOFSTEP]\ncases' le_or_lt p 0 with hp_nonpos hp_pos\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm' f q \u03bc < \u22a4\nhp_nonneg : 0 \u2264 p\nhpq : p \u2264 q\nhp_nonpos : p \u2264 0\n\u22a2 snorm' f p \u03bc < \u22a4\n[PROOFSTEP]\nrw [le_antisymm hp_nonpos hp_nonneg]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm' f q \u03bc < \u22a4\nhp_nonneg : 0 \u2264 p\nhpq : p \u2264 q\nhp_nonpos : p \u2264 0\n\u22a2 snorm' f 0 \u03bc < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm' f q \u03bc < \u22a4\nhp_nonneg : 0 \u2264 p\nhpq : p \u2264 q\nhp_pos : 0 < p\n\u22a2 snorm' f p \u03bc < \u22a4\n[PROOFSTEP]\nhave hq_pos : 0 < q := lt_of_lt_of_le hp_pos hpq\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm' f q \u03bc < \u22a4\nhp_nonneg : 0 \u2264 p\nhpq : p \u2264 q\nhp_pos : 0 < p\nhq_pos : 0 < q\n\u22a2 snorm' f p \u03bc < \u22a4\n[PROOFSTEP]\ncalc\n snorm' f p \u03bc \u2264 snorm' f q \u03bc * \u03bc Set.univ ^ (1 / p - 1 / q) := snorm'_le_snorm'_mul_rpow_measure_univ hp_pos hpq hf\n _ < \u221e := by\n rw [ENNReal.mul_lt_top_iff]\n refine' Or.inl \u27e8hfq_lt_top, ENNReal.rpow_lt_top_of_nonneg _ (measure_ne_top \u03bc Set.univ)\u27e9\n rwa [le_sub_comm, sub_zero, one_div, one_div, inv_le_inv hq_pos hp_pos]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm' f q \u03bc < \u22a4\nhp_nonneg : 0 \u2264 p\nhpq : p \u2264 q\nhp_pos : 0 < p\nhq_pos : 0 < q\n\u22a2 snorm' f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) < \u22a4\n[PROOFSTEP]\nrw [ENNReal.mul_lt_top_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm' f q \u03bc < \u22a4\nhp_nonneg : 0 \u2264 p\nhpq : p \u2264 q\nhp_pos : 0 < p\nhq_pos : 0 < q\n\u22a2 snorm' f q \u03bc < \u22a4 \u2227 \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) < \u22a4 \u2228 snorm' f q \u03bc = 0 \u2228 \u2191\u2191\u03bc Set.univ ^ (1 / p - 1 / q) = 0\n[PROOFSTEP]\nrefine' Or.inl \u27e8hfq_lt_top, ENNReal.rpow_lt_top_of_nonneg _ (measure_ne_top \u03bc Set.univ)\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm' f q \u03bc < \u22a4\nhp_nonneg : 0 \u2264 p\nhpq : p \u2264 q\nhp_pos : 0 < p\nhq_pos : 0 < q\n\u22a2 0 \u2264 1 / p - 1 / q\n[PROOFSTEP]\nrwa [le_sub_comm, sub_zero, one_div, one_div, inv_le_inv hq_pos hp_pos]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u22a2 (\u03b5 * \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p}) ^ (1 / ENNReal.toReal p) \u2264 snorm f p \u03bc\n[PROOFSTEP]\nrw [snorm_eq_lintegral_rpow_nnnorm hp_ne_zero hp_ne_top]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u22a2 (\u03b5 * \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p}) ^ (1 / ENNReal.toReal p) \u2264\n (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nexact\n ENNReal.rpow_le_rpow (mul_meas_ge_le_lintegral\u2080 (hf.ennnorm.pow_const _) \u03b5) (one_div_nonneg.2 ENNReal.toReal_nonneg)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u22a2 \u03b5 * \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p} \u2264 snorm f p \u03bc ^ ENNReal.toReal p\n[PROOFSTEP]\nhave : 1 / p.toReal * p.toReal = 1 := by\n refine' one_div_mul_cancel _\n rw [Ne, ENNReal.toReal_eq_zero_iff]\n exact not_or_of_not hp_ne_zero hp_ne_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u22a2 1 / ENNReal.toReal p * ENNReal.toReal p = 1\n[PROOFSTEP]\nrefine' one_div_mul_cancel _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u22a2 ENNReal.toReal p \u2260 0\n[PROOFSTEP]\nrw [Ne, ENNReal.toReal_eq_zero_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u22a2 \u00ac(p = 0 \u2228 p = \u22a4)\n[PROOFSTEP]\nexact not_or_of_not hp_ne_zero hp_ne_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nthis : 1 / ENNReal.toReal p * ENNReal.toReal p = 1\n\u22a2 \u03b5 * \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p} \u2264 snorm f p \u03bc ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [\u2190 ENNReal.rpow_one (\u03b5 * \u03bc {x | \u03b5 \u2264 (\u2016f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal}), \u2190 this, ENNReal.rpow_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nthis : 1 / ENNReal.toReal p * ENNReal.toReal p = 1\n\u22a2 ((\u03b5 * \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p}) ^ (1 / ENNReal.toReal p)) ^ ENNReal.toReal p \u2264\n snorm f p \u03bc ^ ENNReal.toReal p\n[PROOFSTEP]\nexact ENNReal.rpow_le_rpow (pow_mul_meas_ge_le_snorm \u03bc hp_ne_zero hp_ne_top hf \u03b5) ENNReal.toReal_nonneg\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u22a2 \u03b5 ^ ENNReal.toReal p * \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 snorm f p \u03bc ^ ENNReal.toReal p\n[PROOFSTEP]\nconvert mul_meas_ge_le_pow_snorm \u03bc hp_ne_zero hp_ne_top hf (\u03b5 ^ p.toReal) using 4\n[GOAL]\ncase h.e'_3.h.e'_6.h.e'_3.h.e'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\n\u22a2 (fun x => \u03b5 \u2264 \u2191\u2016f x\u2016\u208a) = fun x => \u03b5 ^ ENNReal.toReal p \u2264 \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_3.h.e'_6.h.e'_3.h.e'_2.h.a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nx : \u03b1\n\u22a2 \u03b5 \u2264 \u2191\u2016f x\u2016\u208a \u2194 \u03b5 ^ ENNReal.toReal p \u2264 \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [ENNReal.rpow_le_rpow_iff (ENNReal.toReal_pos hp_ne_zero hp_ne_top)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p\n[PROOFSTEP]\nby_cases \u03b5 = \u221e\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p\n[PROOFSTEP]\nby_cases \u03b5 = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u03b5 = \u22a4\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u00ac\u03b5 = \u22a4\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p\n[PROOFSTEP]\nhave h\u03b5pow : \u03b5 ^ p.toReal \u2260 0 := (ENNReal.rpow_pos (pos_iff_ne_zero.2 h\u03b5) h).ne.symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u00ac\u03b5 = \u22a4\nh\u03b5pow : \u03b5 ^ ENNReal.toReal p \u2260 0\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p\n[PROOFSTEP]\nhave h\u03b5pow' : \u03b5 ^ p.toReal \u2260 \u221e := ENNReal.rpow_ne_top_of_nonneg ENNReal.toReal_nonneg h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u00ac\u03b5 = \u22a4\nh\u03b5pow : \u03b5 ^ ENNReal.toReal p \u2260 0\nh\u03b5pow' : \u03b5 ^ ENNReal.toReal p \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [ENNReal.inv_rpow, \u2190 ENNReal.mul_le_mul_left h\u03b5pow h\u03b5pow', \u2190 mul_assoc, ENNReal.mul_inv_cancel h\u03b5pow h\u03b5pow', one_mul]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nhf : AEStronglyMeasurable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh : \u00ac\u03b5 = \u22a4\nh\u03b5pow : \u03b5 ^ ENNReal.toReal p \u2260 0\nh\u03b5pow' : \u03b5 ^ ENNReal.toReal p \u2260 \u22a4\n\u22a2 \u03b5 ^ ENNReal.toReal p * \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 snorm f p \u03bc ^ ENNReal.toReal p\n[PROOFSTEP]\nexact mul_meas_ge_le_pow_snorm' \u03bc hp_ne_zero hp_ne_top hf \u03b5\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhfq : Mem\u2112p f q\nhpq : p \u2264 q\n\u22a2 Mem\u2112p f p\n[PROOFSTEP]\ncases' hfq with hfq_m hfq_lt_top\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\n\u22a2 Mem\u2112p f p\n[PROOFSTEP]\nby_cases hp0 : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p = 0\n\u22a2 Mem\u2112p f p\n[PROOFSTEP]\nrwa [hp0, mem\u2112p_zero_iff_aestronglyMeasurable]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : \u00acp = 0\n\u22a2 Mem\u2112p f p\n[PROOFSTEP]\nrw [\u2190 Ne.def] at hp0 \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\n\u22a2 Mem\u2112p f p\n[PROOFSTEP]\nrefine' \u27e8hfq_m, _\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\n\u22a2 snorm f p \u03bc < \u22a4\n[PROOFSTEP]\nby_cases hp_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : p = \u22a4\n\u22a2 snorm f p \u03bc < \u22a4\n[PROOFSTEP]\nhave hq_top : q = \u221e := by rwa [hp_top, top_le_iff] at hpq \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : p = \u22a4\n\u22a2 q = \u22a4\n[PROOFSTEP]\nrwa [hp_top, top_le_iff] at hpq \n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : p = \u22a4\nhq_top : q = \u22a4\n\u22a2 snorm f p \u03bc < \u22a4\n[PROOFSTEP]\nrw [hp_top]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : p = \u22a4\nhq_top : q = \u22a4\n\u22a2 snorm f \u22a4 \u03bc < \u22a4\n[PROOFSTEP]\nrwa [hq_top] at hfq_lt_top \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm f p \u03bc < \u22a4\n[PROOFSTEP]\nhave hp_pos : 0 < p.toReal := ENNReal.toReal_pos hp0 hp_top\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 snorm f p \u03bc < \u22a4\n[PROOFSTEP]\nby_cases hq_top : q = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 snorm f p \u03bc < \u22a4\n[PROOFSTEP]\nrw [snorm_eq_snorm' hp0 hp_top]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4\n[PROOFSTEP]\nrw [hq_top, snorm_exponent_top] at hfq_lt_top \n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snormEssSup f \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (snorm'_le_snormEssSup_mul_rpow_measure_univ hp_pos) _\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snormEssSup f \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 snormEssSup f \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4\n[PROOFSTEP]\nrefine' ENNReal.mul_lt_top hfq_lt_top.ne _\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snormEssSup f \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) \u2260 \u22a4\n[PROOFSTEP]\nexact (ENNReal.rpow_lt_top_of_nonneg (by simp [hp_pos.le]) (measure_ne_top \u03bc Set.univ)).ne\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snormEssSup f \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : q = \u22a4\n\u22a2 0 \u2264 1 / ENNReal.toReal p\n[PROOFSTEP]\nsimp [hp_pos.le]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\n\u22a2 snorm f p \u03bc < \u22a4\n[PROOFSTEP]\nhave hq0 : q \u2260 0 := by\n by_contra hq_eq_zero\n have hp_eq_zero : p = 0 := le_antisymm (by rwa [hq_eq_zero] at hpq ) (zero_le _)\n rw [hp_eq_zero, ENNReal.zero_toReal] at hp_pos \n exact (lt_irrefl _) hp_pos\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\n\u22a2 q \u2260 0\n[PROOFSTEP]\nby_contra hq_eq_zero\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq_eq_zero : q = 0\n\u22a2 False\n[PROOFSTEP]\nhave hp_eq_zero : p = 0 := le_antisymm (by rwa [hq_eq_zero] at hpq ) (zero_le _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq_eq_zero : q = 0\n\u22a2 p \u2264 0\n[PROOFSTEP]\nrwa [hq_eq_zero] at hpq \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq_eq_zero : q = 0\nhp_eq_zero : p = 0\n\u22a2 False\n[PROOFSTEP]\nrw [hp_eq_zero, ENNReal.zero_toReal] at hp_pos \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < 0\nhq_top : \u00acq = \u22a4\nhq_eq_zero : q = 0\nhp_eq_zero : p = 0\n\u22a2 False\n[PROOFSTEP]\nexact (lt_irrefl _) hp_pos\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq0 : q \u2260 0\n\u22a2 snorm f p \u03bc < \u22a4\n[PROOFSTEP]\nhave hpq_real : p.toReal \u2264 q.toReal := by rwa [ENNReal.toReal_le_toReal hp_top hq_top]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq0 : q \u2260 0\n\u22a2 ENNReal.toReal p \u2264 ENNReal.toReal q\n[PROOFSTEP]\nrwa [ENNReal.toReal_le_toReal hp_top hq_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq0 : q \u2260 0\nhpq_real : ENNReal.toReal p \u2264 ENNReal.toReal q\n\u22a2 snorm f p \u03bc < \u22a4\n[PROOFSTEP]\nrw [snorm_eq_snorm' hp0 hp_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm f q \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq0 : q \u2260 0\nhpq_real : ENNReal.toReal p \u2264 ENNReal.toReal q\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4\n[PROOFSTEP]\nrw [snorm_eq_snorm' hq0 hq_top] at hfq_lt_top \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\np q : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhpq : p \u2264 q\nhfq_m : AEStronglyMeasurable f \u03bc\nhfq_lt_top : snorm' f (ENNReal.toReal q) \u03bc < \u22a4\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < ENNReal.toReal p\nhq_top : \u00acq = \u22a4\nhq0 : q \u2260 0\nhpq_real : ENNReal.toReal p \u2264 ENNReal.toReal q\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4\n[PROOFSTEP]\nexact snorm'_lt_top_of_snorm'_lt_top_of_exponent_le hfq_m hfq_lt_top (le_of_lt hp_pos) hpq_real\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\n\u22a2 Mem\u2112p (f - g) p\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf g : \u03b1 \u2192 E\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\n\u22a2 Mem\u2112p (f + -g) p\n[PROOFSTEP]\nexact hf.add hg.neg\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p\n\u22a2 Mem\u2112p (fun a => \u2211 i in s, f i a) p\n[PROOFSTEP]\nhaveI : DecidableEq \u03b9 := Classical.decEq _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p\nthis : DecidableEq \u03b9\n\u22a2 Mem\u2112p (fun a => \u2211 i in s, f i a) p\n[PROOFSTEP]\nrevert hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nthis : DecidableEq \u03b9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) p\n[PROOFSTEP]\nrefine' Finset.induction_on s _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nthis : DecidableEq \u03b9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in \u2205, f i a) p\n[PROOFSTEP]\nsimp only [zero_mem_\u2112p', Finset.sum_empty, imp_true_iff]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nthis : DecidableEq \u03b9\n\u22a2 \u2200 \u2983a : \u03b9\u2984 {s : Finset \u03b9},\n \u00aca \u2208 s \u2192\n ((\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) p) \u2192\n (\u2200 (i : \u03b9), i \u2208 insert a s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a_3 => \u2211 i in insert a s, f i a_3) p\n[PROOFSTEP]\nintro i s his ih hf\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b9 : Type u_5\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nthis : DecidableEq \u03b9\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) p\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Mem\u2112p (f i_1) p\n\u22a2 Mem\u2112p (fun a => \u2211 i in insert i s, f i a) p\n[PROOFSTEP]\nsimp only [his, Finset.sum_insert, not_false_iff]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b9 : Type u_5\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nthis : DecidableEq \u03b9\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p) \u2192 Mem\u2112p (fun a => \u2211 i in s, f i a) p\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Mem\u2112p (f i_1) p\n\u22a2 Mem\u2112p (fun a => f i a + \u2211 i in s, f i a) p\n[PROOFSTEP]\nexact (hf i (s.mem_insert_self i)).add (ih fun j hj => hf j (Finset.mem_insert_of_mem hj))\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p\n\u22a2 Mem\u2112p (\u2211 i in s, f i) p\n[PROOFSTEP]\nconvert mem\u2112p_finset_sum s hf using 1\n[GOAL]\ncase h.e'_5\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p\n\u22a2 \u2211 i in s, f i = fun a => \u2211 i in s, f i a\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\n\u03b9 : Type u_5\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 E\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (f i) p\nx : \u03b1\n\u22a2 Finset.sum s (fun i => f i) x = \u2211 i in s, f i x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 c * \u2016g x\u2016\u208a\np : \u211d\nhp : 0 < p\n\u22a2 snorm' f p \u03bc \u2264 c \u2022 snorm' g p \u03bc\n[PROOFSTEP]\nsimp_rw [snorm']\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 c * \u2016g x\u2016\u208a\np : \u211d\nhp : 0 < p\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) \u2264 c \u2022 (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p)\n[PROOFSTEP]\nrw [\u2190 ENNReal.rpow_le_rpow_iff hp, ENNReal.smul_def, smul_eq_mul, ENNReal.mul_rpow_of_nonneg _ _ hp.le]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 c * \u2016g x\u2016\u208a\np : \u211d\nhp : 0 < p\n\u22a2 ((\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p)) ^ p \u2264 \u2191c ^ p * ((\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p)) ^ p\n[PROOFSTEP]\nsimp_rw [\u2190 ENNReal.rpow_mul, one_div, inv_mul_cancel hp.ne.symm, ENNReal.rpow_one, ENNReal.coe_rpow_of_nonneg _ hp.le, \u2190\n lintegral_const_mul' _ _ ENNReal.coe_ne_top, \u2190 ENNReal.coe_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 c * \u2016g x\u2016\u208a\np : \u211d\nhp : 0 < p\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a ^ p) \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191(c ^ p * \u2016g a\u2016\u208a ^ p) \u2202\u03bc\n[PROOFSTEP]\napply lintegral_mono_ae\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 c * \u2016g x\u2016\u208a\np : \u211d\nhp : 0 < p\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191(\u2016f a\u2016\u208a ^ p) \u2264 \u2191(c ^ p * \u2016g a\u2016\u208a ^ p)\n[PROOFSTEP]\nsimp_rw [ENNReal.coe_le_coe, \u2190 NNReal.mul_rpow, NNReal.rpow_le_rpow_iff hp]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 c * \u2016g x\u2016\u208a\np : \u211d\nhp : 0 < p\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016f a\u2016\u208a \u2264 c * \u2016g a\u2016\u208a\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 c * \u2016g x\u2016\u208a\n\u22a2 essSup (fun x => \u2191(c * \u2016g x\u2016\u208a)) \u03bc = essSup (fun x => \u2191c * \u2191\u2016g x\u2016\u208a) \u03bc\n[PROOFSTEP]\nsimp_rw [ENNReal.coe_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 c * \u2016g x\u2016\u208a\np : \u211d\u22650\u221e\n\u22a2 snorm f p \u03bc \u2264 c \u2022 snorm g p \u03bc\n[PROOFSTEP]\nby_cases h0 : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 c * \u2016g x\u2016\u208a\np : \u211d\u22650\u221e\nh0 : p = 0\n\u22a2 snorm f p \u03bc \u2264 c \u2022 snorm g p \u03bc\n[PROOFSTEP]\nsimp [h0]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 c * \u2016g x\u2016\u208a\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\n\u22a2 snorm f p \u03bc \u2264 c \u2022 snorm g p \u03bc\n[PROOFSTEP]\nby_cases h_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 c * \u2016g x\u2016\u208a\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 snorm f p \u03bc \u2264 c \u2022 snorm g p \u03bc\n[PROOFSTEP]\nrw [h_top]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 c * \u2016g x\u2016\u208a\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : p = \u22a4\n\u22a2 snorm f \u22a4 \u03bc \u2264 c \u2022 snorm g \u22a4 \u03bc\n[PROOFSTEP]\nexact snormEssSup_le_nnreal_smul_snormEssSup_of_ae_le_mul h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 c * \u2016g x\u2016\u208a\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm f p \u03bc \u2264 c \u2022 snorm g p \u03bc\n[PROOFSTEP]\nsimp_rw [snorm_eq_snorm' h0 h_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\u22650\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 c * \u2016g x\u2016\u208a\np : \u211d\u22650\u221e\nh0 : \u00acp = 0\nh_top : \u00acp = \u22a4\n\u22a2 snorm' f (ENNReal.toReal p) \u03bc \u2264 c \u2022 snorm' g (ENNReal.toReal p) \u03bc\n[PROOFSTEP]\nexact snorm'_le_nnreal_smul_snorm'_of_ae_le_mul h (ENNReal.toReal_pos h0 h_top)\n[GOAL]\n\u03b1\u271d : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\u271d\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\u271d\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\u03b1 : Type u_5\ninst\u271d : LinearOrderedSemiring \u03b1\na b c : \u03b1\nha : 0 \u2264 a\nhb : b < 0\nhc : 0 \u2264 c\n\u22a2 a \u2264 b * c \u2194 a = 0 \u2227 c = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1\u271d : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\u271d\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\u271d\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\u03b1 : Type u_5\ninst\u271d : LinearOrderedSemiring \u03b1\na b c : \u03b1\nha : 0 \u2264 a\nhb : b < 0\nhc : 0 \u2264 c\n\u22a2 a \u2264 b * c \u2192 a = 0 \u2227 c = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1\u271d : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\u271d\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\u271d\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\u03b1 : Type u_5\ninst\u271d : LinearOrderedSemiring \u03b1\na b c : \u03b1\nha : 0 \u2264 a\nhb : b < 0\nhc : 0 \u2264 c\nh : a \u2264 b * c\n\u22a2 a = 0 \u2227 c = 0\n[PROOFSTEP]\nexact\n \u27e8(h.trans (mul_nonpos_of_nonpos_of_nonneg hb.le hc)).antisymm ha,\n (nonpos_of_mul_nonneg_right (ha.trans h) hb).antisymm hc\u27e9\n[GOAL]\ncase mpr\n\u03b1\u271d : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\u271d\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\u271d\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\u03b1 : Type u_5\ninst\u271d : LinearOrderedSemiring \u03b1\na b c : \u03b1\nha : 0 \u2264 a\nhb : b < 0\nhc : 0 \u2264 c\n\u22a2 a = 0 \u2227 c = 0 \u2192 a \u2264 b * c\n[PROOFSTEP]\nrintro \u27e8rfl, rfl\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1\u271d : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\u271d\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\u271d\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\u03b1 : Type u_5\ninst\u271d : LinearOrderedSemiring \u03b1\nb : \u03b1\nhb : b < 0\nha hc : 0 \u2264 0\n\u22a2 0 \u2264 b * 0\n[PROOFSTEP]\nrw [mul_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nhc : c < 0\np : \u211d\u22650\u221e\n\u22a2 snorm f p \u03bc = 0 \u2227 snorm g p \u03bc = 0\n[PROOFSTEP]\nsimp_rw [le_mul_iff_eq_zero_of_nonneg_of_neg_of_nonneg (norm_nonneg _) hc (norm_nonneg _), norm_eq_zero,\n eventually_and] at h \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\nhc : c < 0\np : \u211d\u22650\u221e\nh : (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x = 0) \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x = 0\n\u22a2 snorm f p \u03bc = 0 \u2227 snorm g p \u03bc = 0\n[PROOFSTEP]\nchange f =\u1d50[\u03bc] 0 \u2227 g =\u1d50[\u03bc] 0 at h \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nc : \u211d\nhc : c < 0\np : \u211d\u22650\u221e\nh : f =\u1d50[\u03bc] 0 \u2227 g =\u1d50[\u03bc] 0\n\u22a2 snorm f p \u03bc = 0 \u2227 snorm g p \u03bc = 0\n[PROOFSTEP]\nsimp [snorm_congr_ae h.1, snorm_congr_ae h.2]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 snorm' (fun x => b (f x) (g x)) p \u03bc \u2264 snorm' f q \u03bc * snorm' g r \u03bc\n[PROOFSTEP]\nrw [snorm']\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016b (f a) (g a)\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) \u2264 snorm' f q \u03bc * snorm' g r \u03bc\n[PROOFSTEP]\ncalc\n (\u222b\u207b a : \u03b1, \u2191\u2016b (f a) (g a)\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b a : \u03b1, \u2191(\u2016f a\u2016\u208a * \u2016g a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) :=\n (ENNReal.rpow_le_rpow_iff <| one_div_pos.mpr hp0_lt).mpr <|\n lintegral_mono_ae <| h.mono fun a ha => (ENNReal.rpow_le_rpow_iff hp0_lt).mpr <| ENNReal.coe_le_coe.mpr <| ha\n _ \u2264 _ := ?_\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a * \u2016g a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 snorm' f q \u03bc * snorm' g r \u03bc\n[PROOFSTEP]\nsimp_rw [snorm', ENNReal.coe_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * \u2191\u2016g a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ r \u2202\u03bc) ^ (1 / r)\n[PROOFSTEP]\nexact ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr hp0_lt hpq hpqr \u03bc hf.ennnorm hg.ennnorm\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc\n[PROOFSTEP]\nby_cases hp_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : p = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc\n[PROOFSTEP]\nsimp_rw [hp_top, snorm_exponent_top]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : p = \u22a4\n\u22a2 snormEssSup (fun x => b (f x) (g x)) \u03bc \u2264 snormEssSup f \u03bc * snormEssSup g \u03bc\n[PROOFSTEP]\nrefine' le_trans (essSup_mono_ae <| h.mono fun a ha => _) (ENNReal.essSup_mul_le _ _)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : p = \u22a4\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 (fun x => \u2191\u2016(fun x => b (f x) (g x)) x\u2016\u208a) a \u2264 ((fun x => \u2191\u2016f x\u2016\u208a) * fun x => \u2191\u2016g x\u2016\u208a) a\n[PROOFSTEP]\nsimp_rw [Pi.mul_apply, \u2190 ENNReal.coe_mul, ENNReal.coe_le_coe]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : p = \u22a4\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n[PROOFSTEP]\nexact ha\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc\n[PROOFSTEP]\nby_cases hp_zero : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : p = 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc\n[PROOFSTEP]\nsimp only [hp_zero, snorm_exponent_zero, mul_zero, le_zero_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc\n[PROOFSTEP]\nsimp_rw [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_top, snorm_exponent_top, snormEssSup]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016b (f x) (g x)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) \u2264\n essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\ncalc\n (\u222b\u207b x, (\u2016b (f x) (g x)\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal \u2202\u03bc) ^ (1 / p.toReal) \u2264\n (\u222b\u207b x, (\u2016f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal * (\u2016g x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal \u2202\u03bc) ^ (1 / p.toReal) :=\n by\n refine' ENNReal.rpow_le_rpow _ (one_div_nonneg.mpr ENNReal.toReal_nonneg)\n refine' lintegral_mono_ae (h.mono fun a ha => _)\n rw [\u2190 ENNReal.mul_rpow_of_nonneg _ _ ENNReal.toReal_nonneg]\n refine' ENNReal.rpow_le_rpow _ ENNReal.toReal_nonneg\n rw [\u2190 ENNReal.coe_mul, ENNReal.coe_le_coe]\n exact ha\n _ \u2264 (\u222b\u207b x, essSup (fun x => (\u2016f x\u2016\u208a : \u211d\u22650\u221e)) \u03bc ^ p.toReal * (\u2016g x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal \u2202\u03bc) ^ (1 / p.toReal) :=\n by\n refine' ENNReal.rpow_le_rpow _ _\n swap;\n \u00b7 rw [one_div_nonneg]\n exact ENNReal.toReal_nonneg\n refine' lintegral_mono_ae _\n filter_upwards [@ENNReal.ae_le_essSup _ _ \u03bc fun x => (\u2016f x\u2016\u208a : \u211d\u22650\u221e)] with x hx\n exact mul_le_mul_right' (ENNReal.rpow_le_rpow hx ENNReal.toReal_nonneg) _\n _ = essSup (fun x => (\u2016f x\u2016\u208a : \u211d\u22650\u221e)) \u03bc * (\u222b\u207b x, (\u2016g x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal \u2202\u03bc) ^ (1 / p.toReal) :=\n by\n rw [lintegral_const_mul'']\n swap; \u00b7 exact hg.nnnorm.aemeasurable.coe_nnreal_ennreal.pow aemeasurable_const\n rw [ENNReal.mul_rpow_of_nonneg]\n swap;\n \u00b7 rw [one_div_nonneg]\n exact ENNReal.toReal_nonneg\n rw [\u2190 ENNReal.rpow_mul, one_div, mul_inv_cancel, ENNReal.rpow_one]\n rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]\n exact \u27e8hp_zero, hp_top\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016b (f x) (g x)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) \u2264\n (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nrefine' ENNReal.rpow_le_rpow _ (one_div_nonneg.mpr ENNReal.toReal_nonneg)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016b (f x) (g x)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2264\n \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_mono_ae (h.mono fun a ha => _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 \u2191\u2016b (f a) (g a)\u2016\u208a ^ ENNReal.toReal p \u2264 \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [\u2190 ENNReal.mul_rpow_of_nonneg _ _ ENNReal.toReal_nonneg]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 \u2191\u2016b (f a) (g a)\u2016\u208a ^ ENNReal.toReal p \u2264 (\u2191\u2016f a\u2016\u208a * \u2191\u2016g a\u2016\u208a) ^ ENNReal.toReal p\n[PROOFSTEP]\nrefine' ENNReal.rpow_le_rpow _ ENNReal.toReal_nonneg\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 \u2191\u2016b (f a) (g a)\u2016\u208a \u2264 \u2191\u2016f a\u2016\u208a * \u2191\u2016g a\u2016\u208a\n[PROOFSTEP]\nrw [\u2190 ENNReal.coe_mul, ENNReal.coe_le_coe]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\na : \u03b1\nha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n\u22a2 \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a\n[PROOFSTEP]\nexact ha\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) \u2264\n (\u222b\u207b (x : \u03b1), essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^\n (1 / ENNReal.toReal p)\n[PROOFSTEP]\nrefine' ENNReal.rpow_le_rpow _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2264\n \u222b\u207b (x : \u03b1), essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 1 / ENNReal.toReal p\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 1 / ENNReal.toReal p\n[PROOFSTEP]\nrw [one_div_nonneg]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 ENNReal.toReal p\n[PROOFSTEP]\nexact ENNReal.toReal_nonneg\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2264\n \u222b\u207b (x : \u03b1), essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_mono_ae _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc,\n \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2264\n essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nfilter_upwards [@ENNReal.ae_le_essSup _ _ \u03bc fun x => (\u2016f x\u2016\u208a : \u211d\u22650\u221e)] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\nx : \u03b1\nhx : \u2191\u2016f x\u2016\u208a \u2264 essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc\n\u22a2 \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2264\n essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nexact mul_le_mul_right' (ENNReal.rpow_le_rpow hx ENNReal.toReal_nonneg) _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (\u222b\u207b (x : \u03b1), essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^\n (1 / ENNReal.toReal p) =\n essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nrw [lintegral_const_mul'']\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^\n (1 / ENNReal.toReal p) =\n essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 AEMeasurable fun x => \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nswap\n[GOAL]\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 AEMeasurable fun x => \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nexact hg.nnnorm.aemeasurable.coe_nnreal_ennreal.pow aemeasurable_const\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^\n (1 / ENNReal.toReal p) =\n essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nrw [ENNReal.mul_rpow_of_nonneg]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) *\n (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\ncase hz\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 1 / ENNReal.toReal p\n[PROOFSTEP]\nswap\n[GOAL]\ncase hz\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 1 / ENNReal.toReal p\n[PROOFSTEP]\nrw [one_div_nonneg]\n[GOAL]\ncase hz\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 0 \u2264 ENNReal.toReal p\n[PROOFSTEP]\nexact ENNReal.toReal_nonneg\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) *\n (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nrw [\u2190 ENNReal.rpow_mul, one_div, mul_inv_cancel, ENNReal.rpow_one]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 ENNReal.toReal p \u2260 0\n[PROOFSTEP]\nrw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp_top : \u00acp = \u22a4\nhp_zero : \u00acp = 0\n\u22a2 \u00acp = 0 \u2227 \u00acp = \u22a4\n[PROOFSTEP]\nexact \u27e8hp_zero, hp_top\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f p \u03bc * snorm g \u22a4 \u03bc\n[PROOFSTEP]\nrw [\u2190 snorm_norm f, \u2190 snorm_norm g]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm (fun x => \u2016f x\u2016) p \u03bc * snorm (fun x => \u2016g x\u2016) \u22a4 \u03bc\n[PROOFSTEP]\nrefine' (snorm_mono_ae_real h).trans _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\n\u22a2 snorm (fun x => (fun a => \u2191a) (\u2016f x\u2016\u208a * \u2016g x\u2016\u208a)) p \u03bc \u2264 snorm (fun x => \u2016f x\u2016) p \u03bc * snorm (fun x => \u2016g x\u2016) \u22a4 \u03bc\n[PROOFSTEP]\nsimp_rw [mul_comm \u2016f _\u2016\u208a, val_eq_coe, NNReal.coe_mul, coe_nnnorm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\n\u22a2 snorm (fun x => \u2016g x\u2016 * \u2016f x\u2016) p \u03bc \u2264 snorm (fun x => \u2016f x\u2016) p \u03bc * snorm (fun x => \u2016g x\u2016) \u22a4 \u03bc\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\n\u22a2 snorm (fun x => \u2016g x\u2016 * \u2016f x\u2016) p \u03bc \u2264 snorm (fun x => \u2016g x\u2016) \u22a4 \u03bc * snorm (fun x => \u2016f x\u2016) p \u03bc\n[PROOFSTEP]\nrefine' snorm_le_snorm_top_mul_snorm p (fun x => \u2016g x\u2016) hf.norm _ (h.mono fun x _ => _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nx : \u03b1\nx\u271d : \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\n\u22a2 \u2016(fun x => \u2016g x\u2016) x * \u2016f x\u2016\u2016\u208a \u2264 \u2016(fun x => \u2016g x\u2016) x\u2016\u208a * \u2016\u2016f x\u2016\u2016\u208a\n[PROOFSTEP]\nsimp_rw [nnnorm_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nx : \u03b1\nx\u271d : \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\n\u22a2 \u2016\u2016g x\u2016\u2016\u208a * \u2016\u2016f x\u2016\u2016\u208a \u2264 \u2016\u2016g x\u2016\u2016\u208a * \u2016\u2016f x\u2016\u2016\u208a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc\n[PROOFSTEP]\nby_cases hp_zero : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : p = 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc\n[PROOFSTEP]\nsimp [hp_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc\n[PROOFSTEP]\nhave hq_ne_zero : q \u2260 0 := by\n intro hq_zero\n simp only [hq_zero, hp_zero, one_div, ENNReal.inv_zero, top_add, ENNReal.inv_eq_top] at hpqr \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\n\u22a2 q \u2260 0\n[PROOFSTEP]\nintro hq_zero\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_zero : q = 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [hq_zero, hp_zero, one_div, ENNReal.inv_zero, top_add, ENNReal.inv_eq_top] at hpqr \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc\n[PROOFSTEP]\nhave hr_ne_zero : r \u2260 0 := by\n intro hr_zero\n simp only [hr_zero, hp_zero, one_div, ENNReal.inv_zero, add_top, ENNReal.inv_eq_top] at hpqr \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\n\u22a2 r \u2260 0\n[PROOFSTEP]\nintro hr_zero\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_zero : r = 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [hr_zero, hp_zero, one_div, ENNReal.inv_zero, add_top, ENNReal.inv_eq_top] at hpqr \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc\n[PROOFSTEP]\nby_cases hq_top : q = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : q = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc\n[PROOFSTEP]\nhave hpr : p = r := by simpa only [hq_top, one_div, ENNReal.inv_top, zero_add, inv_inj] using hpqr\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : q = \u22a4\n\u22a2 p = r\n[PROOFSTEP]\nsimpa only [hq_top, one_div, ENNReal.inv_top, zero_add, inv_inj] using hpqr\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : q = \u22a4\nhpr : p = r\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc\n[PROOFSTEP]\nrw [\u2190 hpr, hq_top]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : q = \u22a4\nhpr : p = r\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc\n[PROOFSTEP]\nexact snorm_le_snorm_top_mul_snorm p f hg b h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc\n[PROOFSTEP]\nby_cases hr_top : r = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : r = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc\n[PROOFSTEP]\nhave hpq : p = q := by simpa only [hr_top, one_div, ENNReal.inv_top, add_zero, inv_inj] using hpqr\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : r = \u22a4\n\u22a2 p = q\n[PROOFSTEP]\nsimpa only [hr_top, one_div, ENNReal.inv_top, add_zero, inv_inj] using hpqr\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : r = \u22a4\nhpq : p = q\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc\n[PROOFSTEP]\nrw [\u2190 hpq, hr_top]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : r = \u22a4\nhpq : p = q\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f p \u03bc * snorm g \u22a4 \u03bc\n[PROOFSTEP]\nexact snorm_le_snorm_mul_snorm_top p hf g b h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc\n[PROOFSTEP]\nhave hpq : p < q := by\n suffices 1 / q < 1 / p by rwa [one_div, one_div, ENNReal.inv_lt_inv] at this \n rw [hpqr]\n refine' ENNReal.lt_add_right _ _\n \u00b7 simp only [hq_ne_zero, one_div, Ne.def, ENNReal.inv_eq_top, not_false_iff]\n \u00b7 simp only [hr_top, one_div, Ne.def, ENNReal.inv_eq_zero, not_false_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 p < q\n[PROOFSTEP]\nsuffices 1 / q < 1 / p by rwa [one_div, one_div, ENNReal.inv_lt_inv] at this \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nthis : 1 / q < 1 / p\n\u22a2 p < q\n[PROOFSTEP]\nrwa [one_div, one_div, ENNReal.inv_lt_inv] at this \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 1 / q < 1 / p\n[PROOFSTEP]\nrw [hpqr]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 1 / q < 1 / q + 1 / r\n[PROOFSTEP]\nrefine' ENNReal.lt_add_right _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 1 / q \u2260 \u22a4\n[PROOFSTEP]\nsimp only [hq_ne_zero, one_div, Ne.def, ENNReal.inv_eq_top, not_false_iff]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\n\u22a2 1 / r \u2260 0\n[PROOFSTEP]\nsimp only [hr_top, one_div, Ne.def, ENNReal.inv_eq_zero, not_false_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc\n[PROOFSTEP]\nrw [snorm_eq_snorm' hp_zero (hpq.trans_le le_top).ne, snorm_eq_snorm' hq_ne_zero hq_top,\n snorm_eq_snorm' hr_ne_zero hr_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 snorm' (fun x => b (f x) (g x)) (ENNReal.toReal p) \u03bc \u2264 snorm' f (ENNReal.toReal q) \u03bc * snorm' g (ENNReal.toReal r) \u03bc\n[PROOFSTEP]\nrefine' snorm'_le_snorm'_mul_snorm' hf hg _ h _ _ _\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 0 < ENNReal.toReal p\n[PROOFSTEP]\nexact ENNReal.toReal_pos hp_zero (hpq.trans_le le_top).ne\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 ENNReal.toReal p < ENNReal.toReal q\n[PROOFSTEP]\nexact ENNReal.toReal_strict_mono hq_top hpq\n[GOAL]\ncase neg.refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 1 / ENNReal.toReal p = 1 / ENNReal.toReal q + 1 / ENNReal.toReal r\n[PROOFSTEP]\nrw [\u2190 ENNReal.one_toReal, \u2190 ENNReal.toReal_div, \u2190 ENNReal.toReal_div, \u2190 ENNReal.toReal_div, hpqr, ENNReal.toReal_add]\n[GOAL]\ncase neg.refine'_3.ha\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 1 / q \u2260 \u22a4\n[PROOFSTEP]\nsimp only [hq_ne_zero, one_div, Ne.def, ENNReal.inv_eq_top, not_false_iff]\n[GOAL]\ncase neg.refine'_3.hb\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhpqr : 1 / p = 1 / q + 1 / r\nhp_zero : \u00acp = 0\nhq_ne_zero : q \u2260 0\nhr_ne_zero : r \u2260 0\nhq_top : \u00acq = \u22a4\nhr_top : \u00acr = \u22a4\nhpq : p < q\n\u22a2 1 / r \u2260 \u22a4\n[PROOFSTEP]\nsimp only [hr_ne_zero, one_div, Ne.def, ENNReal.inv_eq_top, not_false_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : MulActionWithZero \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nh\u03c6 : Mem\u2112p \u03c6 \u22a4\n\u22a2 Mem\u2112p (\u03c6 \u2022 f) p\n[PROOFSTEP]\napply hf.smul h\u03c6\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : MulActionWithZero \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nh\u03c6 : Mem\u2112p \u03c6 \u22a4\n\u22a2 1 / p = 1 / \u22a4 + 1 / p\n[PROOFSTEP]\nsimp only [ENNReal.div_top, zero_add]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : MulActionWithZero \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f \u22a4\nh\u03c6 : Mem\u2112p \u03c6 p\n\u22a2 Mem\u2112p (\u03c6 \u2022 f) p\n[PROOFSTEP]\napply hf.smul h\u03c6\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : MulActionWithZero \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f \u22a4\nh\u03c6 : Mem\u2112p \u03c6 p\n\u22a2 1 / p = 1 / p + 1 / \u22a4\n[PROOFSTEP]\nsimp only [ENNReal.div_top, add_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nf : \u03b1 \u2192 F\nc : \ud835\udd5c\nhq_pos : 0 < q\n\u22a2 snorm' (c \u2022 f) q \u03bc = \u2016c\u2016\u208a \u2022 snorm' f q \u03bc\n[PROOFSTEP]\nobtain rfl | hc := eq_or_ne c 0\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\n\u22a2 snorm' (0 \u2022 f) q \u03bc = \u20160\u2016\u208a \u2022 snorm' f q \u03bc\n[PROOFSTEP]\nsimp [snorm', hq_pos]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nf : \u03b1 \u2192 F\nc : \ud835\udd5c\nhq_pos : 0 < q\nhc : c \u2260 0\n\u22a2 snorm' (c \u2022 f) q \u03bc = \u2016c\u2016\u208a \u2022 snorm' f q \u03bc\n[PROOFSTEP]\nrefine' le_antisymm (snorm'_const_smul_le _ _ hq_pos) _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nf : \u03b1 \u2192 F\nc : \ud835\udd5c\nhq_pos : 0 < q\nhc : c \u2260 0\n\u22a2 \u2016c\u2016\u208a \u2022 snorm' f q \u03bc \u2264 snorm' (c \u2022 f) q \u03bc\n[PROOFSTEP]\nhave : snorm' _ q \u03bc \u2264 _ := snorm'_const_smul_le c\u207b\u00b9 (c \u2022 f) hq_pos\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nf : \u03b1 \u2192 F\nc : \ud835\udd5c\nhq_pos : 0 < q\nhc : c \u2260 0\nthis : snorm' (c\u207b\u00b9 \u2022 c \u2022 f) q \u03bc \u2264 \u2016c\u207b\u00b9\u2016\u208a \u2022 snorm' (c \u2022 f) q \u03bc\n\u22a2 \u2016c\u2016\u208a \u2022 snorm' f q \u03bc \u2264 snorm' (c \u2022 f) q \u03bc\n[PROOFSTEP]\nrwa [inv_smul_smul\u2080 hc, nnnorm_inv, ENNReal.le_inv_smul_iff (nnnorm_ne_zero_iff.mpr hc)] at this \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nc : \ud835\udd5c\nf : \u03b1 \u2192 F\n\u22a2 snormEssSup (c \u2022 f) \u03bc = \u2191\u2016c\u2016\u208a * snormEssSup f \u03bc\n[PROOFSTEP]\nsimp_rw [snormEssSup, Pi.smul_apply, nnnorm_smul, ENNReal.coe_mul, ENNReal.essSup_const_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nc : \ud835\udd5c\nf : \u03b1 \u2192 F\n\u22a2 snorm (c \u2022 f) p \u03bc = \u2191\u2016c\u2016\u208a * snorm f p \u03bc\n[PROOFSTEP]\nobtain rfl | hc := eq_or_ne c 0\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nf : \u03b1 \u2192 F\n\u22a2 snorm (0 \u2022 f) p \u03bc = \u2191\u20160\u2016\u208a * snorm f p \u03bc\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nc : \ud835\udd5c\nf : \u03b1 \u2192 F\nhc : c \u2260 0\n\u22a2 snorm (c \u2022 f) p \u03bc = \u2191\u2016c\u2016\u208a * snorm f p \u03bc\n[PROOFSTEP]\nrefine' le_antisymm (snorm_const_smul_le _ _) _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nc : \ud835\udd5c\nf : \u03b1 \u2192 F\nhc : c \u2260 0\n\u22a2 \u2191\u2016c\u2016\u208a * snorm f p \u03bc \u2264 snorm (c \u2022 f) p \u03bc\n[PROOFSTEP]\nhave : snorm _ p \u03bc \u2264 _ := snorm_const_smul_le c\u207b\u00b9 (c \u2022 f)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nc : \ud835\udd5c\nf : \u03b1 \u2192 F\nhc : c \u2260 0\nthis : snorm (c\u207b\u00b9 \u2022 c \u2022 f) p \u03bc \u2264 \u2016c\u207b\u00b9\u2016\u208a \u2022 snorm (c \u2022 f) p \u03bc\n\u22a2 \u2191\u2016c\u2016\u208a * snorm f p \u03bc \u2264 snorm (c \u2022 f) p \u03bc\n[PROOFSTEP]\nrwa [inv_smul_smul\u2080 hc, nnnorm_inv, ENNReal.le_inv_smul_iff (nnnorm_ne_zero_iff.mpr hc)] at this \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nC : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 C \u2264 \u2016Set.indicator s f x\u2016\u208a\n\u22a2 C \u2022 \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) \u2264 snorm (Set.indicator s f) p \u03bc\n[PROOFSTEP]\nrw [ENNReal.smul_def, smul_eq_mul, snorm_eq_lintegral_rpow_nnnorm hp hp',\n ENNReal.le_rpow_one_div_iff (ENNReal.toReal_pos hp hp'), ENNReal.mul_rpow_of_nonneg _ _ ENNReal.toReal_nonneg, \u2190\n ENNReal.rpow_mul, one_div_mul_cancel (ENNReal.toReal_pos hp hp').ne.symm, ENNReal.rpow_one, \u2190 set_lintegral_const, \u2190\n lintegral_indicator _ hs]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nC : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 C \u2264 \u2016Set.indicator s f x\u2016\u208a\n\u22a2 \u222b\u207b (a : \u03b1), Set.indicator s (fun x => \u2191C ^ ENNReal.toReal p) a \u2202\u03bc \u2264\n \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator s f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_mono_ae _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nC : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 C \u2264 \u2016Set.indicator s f x\u2016\u208a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Set.indicator s (fun x => \u2191C ^ ENNReal.toReal p) a \u2264 \u2191\u2016Set.indicator s f a\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nfilter_upwards [hf] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nC : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 C \u2264 \u2016Set.indicator s f x\u2016\u208a\nx : \u03b1\nhx : x \u2208 s \u2192 C \u2264 \u2016Set.indicator s f x\u2016\u208a\n\u22a2 Set.indicator s (fun x => \u2191C ^ ENNReal.toReal p) x \u2264 \u2191\u2016Set.indicator s f x\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [nnnorm_indicator_eq_indicator_nnnorm]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nC : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 C \u2264 \u2016Set.indicator s f x\u2016\u208a\nx : \u03b1\nhx : x \u2208 s \u2192 C \u2264 \u2016Set.indicator s f x\u2016\u208a\n\u22a2 Set.indicator s (fun x => \u2191C ^ ENNReal.toReal p) x \u2264 \u2191(Set.indicator s (fun a => \u2016f a\u2016\u208a) x) ^ ENNReal.toReal p\n[PROOFSTEP]\nby_cases hxs : x \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nC : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 C \u2264 \u2016Set.indicator s f x\u2016\u208a\nx : \u03b1\nhx : x \u2208 s \u2192 C \u2264 \u2016Set.indicator s f x\u2016\u208a\nhxs : x \u2208 s\n\u22a2 Set.indicator s (fun x => \u2191C ^ ENNReal.toReal p) x \u2264 \u2191(Set.indicator s (fun a => \u2016f a\u2016\u208a) x) ^ ENNReal.toReal p\n[PROOFSTEP]\nsimp only [Set.indicator_of_mem hxs] at hx \u22a2\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nC : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 C \u2264 \u2016Set.indicator s f x\u2016\u208a\nx : \u03b1\nhxs : x \u2208 s\nhx : x \u2208 s \u2192 C \u2264 \u2016f x\u2016\u208a\n\u22a2 \u2191C ^ ENNReal.toReal p \u2264 \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nexact ENNReal.rpow_le_rpow (ENNReal.coe_le_coe.2 (hx hxs)) ENNReal.toReal_nonneg\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nf : \u03b1 \u2192 F\nC : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 C \u2264 \u2016Set.indicator s f x\u2016\u208a\nx : \u03b1\nhx : x \u2208 s \u2192 C \u2264 \u2016Set.indicator s f x\u2016\u208a\nhxs : \u00acx \u2208 s\n\u22a2 Set.indicator s (fun x => \u2191C ^ ENNReal.toReal p) x \u2264 \u2191(Set.indicator s (fun a => \u2016f a\u2016\u208a) x) ^ ENNReal.toReal p\n[PROOFSTEP]\nsimp [Set.indicator_of_not_mem hxs]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\n\u22a2 Mem\u2112p (fun x => \u2191IsROrC.re (f x)) p\n[PROOFSTEP]\nhave : \u2200 x, \u2016IsROrC.re (f x)\u2016 \u2264 1 * \u2016f x\u2016 := by\n intro x\n rw [one_mul]\n exact IsROrC.norm_re_le_norm (f x)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\n\u22a2 \u2200 (x : \u03b1), \u2016\u2191IsROrC.re (f x)\u2016 \u2264 1 * \u2016f x\u2016\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nx : \u03b1\n\u22a2 \u2016\u2191IsROrC.re (f x)\u2016 \u2264 1 * \u2016f x\u2016\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nx : \u03b1\n\u22a2 \u2016\u2191IsROrC.re (f x)\u2016 \u2264 \u2016f x\u2016\n[PROOFSTEP]\nexact IsROrC.norm_re_le_norm (f x)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nthis : \u2200 (x : \u03b1), \u2016\u2191IsROrC.re (f x)\u2016 \u2264 1 * \u2016f x\u2016\n\u22a2 Mem\u2112p (fun x => \u2191IsROrC.re (f x)) p\n[PROOFSTEP]\nrefine' hf.of_le_mul _ (eventually_of_forall this)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nthis : \u2200 (x : \u03b1), \u2016\u2191IsROrC.re (f x)\u2016 \u2264 1 * \u2016f x\u2016\n\u22a2 AEStronglyMeasurable (fun x => \u2191IsROrC.re (f x)) \u03bc\n[PROOFSTEP]\nexact IsROrC.continuous_re.comp_aestronglyMeasurable hf.1\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\n\u22a2 Mem\u2112p (fun x => \u2191IsROrC.im (f x)) p\n[PROOFSTEP]\nhave : \u2200 x, \u2016IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016 := by\n intro x\n rw [one_mul]\n exact IsROrC.norm_im_le_norm (f x)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\n\u22a2 \u2200 (x : \u03b1), \u2016\u2191IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nx : \u03b1\n\u22a2 \u2016\u2191IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nx : \u03b1\n\u22a2 \u2016\u2191IsROrC.im (f x)\u2016 \u2264 \u2016f x\u2016\n[PROOFSTEP]\nexact IsROrC.norm_im_le_norm (f x)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nthis : \u2200 (x : \u03b1), \u2016\u2191IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016\n\u22a2 Mem\u2112p (fun x => \u2191IsROrC.im (f x)) p\n[PROOFSTEP]\nrefine' hf.of_le_mul _ (eventually_of_forall this)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\n\ud835\udd5c : Type u_5\ninst\u271d : IsROrC \ud835\udd5c\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2112p f p\nthis : \u2200 (x : \u03b1), \u2016\u2191IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016\n\u22a2 AEStronglyMeasurable (fun x => \u2191IsROrC.im (f x)) \u03bc\n[PROOFSTEP]\nexact IsROrC.continuous_im.comp_aestronglyMeasurable hf.1\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\n[PROOFSTEP]\nby_cases hp0 : p.toReal = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp0 : ENNReal.toReal p = 0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\n[PROOFSTEP]\nsimp only [hp0, ENNReal.rpow_zero]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp0 : ENNReal.toReal p = 0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => 1) atTop < \u22a4\n[PROOFSTEP]\nrefine' eventually_of_forall fun x => _\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp0 : ENNReal.toReal p = 0\nx : \u03b1\n\u22a2 liminf (fun n => 1) atTop < \u22a4\n[PROOFSTEP]\nrw [liminf_const (1 : \u211d\u22650\u221e)]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp0 : ENNReal.toReal p = 0\nx : \u03b1\n\u22a2 1 < \u22a4\n[PROOFSTEP]\nexact ENNReal.one_lt_top\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp0 : \u00acENNReal.toReal p = 0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\n[PROOFSTEP]\nhave hp : p \u2260 0 := fun h => by simp [h] at hp0 \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp0 : \u00acENNReal.toReal p = 0\nh : p = 0\n\u22a2 False\n[PROOFSTEP]\nsimp [h] at hp0 \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp0 : \u00acENNReal.toReal p = 0\nhp : p \u2260 0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\n[PROOFSTEP]\nhave hp' : p \u2260 \u221e := fun h => by simp [h] at hp0 \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp0 : \u00acENNReal.toReal p = 0\nhp : p \u2260 0\nh : p = \u22a4\n\u22a2 False\n[PROOFSTEP]\nsimp [h] at hp0 \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp0 : \u00acENNReal.toReal p = 0\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\n[PROOFSTEP]\nrefine'\n ae_lt_top (measurable_liminf fun n => (hfmeas n).nnnorm.coe_nnreal_ennreal.pow_const p.toReal)\n (lt_of_le_of_lt (lintegral_liminf_le fun n => (hfmeas n).nnnorm.coe_nnreal_ennreal.pow_const p.toReal)\n (lt_of_le_of_lt _\n (ENNReal.rpow_lt_top_of_nonneg ENNReal.toReal_nonneg ENNReal.coe_ne_top : (R : \u211d\u22650\u221e) ^ p.toReal < \u221e))).ne\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp0 : \u00acENNReal.toReal p = 0\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\n\u22a2 liminf (fun n => \u222b\u207b (a : \u03b1), \u2191\u2016f n a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop \u2264 \u2191R ^ ENNReal.toReal p\n[PROOFSTEP]\nsimp_rw [snorm_eq_lintegral_rpow_nnnorm hp hp'] at hbdd \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhp0 : \u00acENNReal.toReal p = 0\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhbdd : \u2200 (n : \u2115), (\u222b\u207b (x : \u03b1), \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) \u2264 \u2191R\n\u22a2 liminf (fun n => \u222b\u207b (a : \u03b1), \u2191\u2016f n a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop \u2264 \u2191R ^ ENNReal.toReal p\n[PROOFSTEP]\nsimp_rw [liminf_eq, eventually_atTop]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhp0 : \u00acENNReal.toReal p = 0\nhp : p \u2260 0\nhp' : p \u2260 \u22a4\nhbdd : \u2200 (n : \u2115), (\u222b\u207b (x : \u03b1), \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) \u2264 \u2191R\n\u22a2 sSup {a | \u2203 a_1, \u2200 (b : \u2115), b \u2265 a_1 \u2192 a \u2264 \u222b\u207b (a : \u03b1), \u2191\u2016f b a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc} \u2264 \u2191R ^ ENNReal.toReal p\n[PROOFSTEP]\nexact\n sSup_le fun b \u27e8a, ha\u27e9 => (ha a le_rfl).trans ((ENNReal.rpow_one_div_le_iff (ENNReal.toReal_pos hp hp')).1 (hbdd _))\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4\n[PROOFSTEP]\nby_cases hp' : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : p = \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4\n[PROOFSTEP]\nsubst hp'\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhp : \u22a4 \u2260 0\nhbdd : \u2200 (n : \u2115), snorm (f n) \u22a4 \u03bc \u2264 \u2191R\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4\n[PROOFSTEP]\nsimp_rw [snorm_exponent_top] at hbdd \n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhp : \u22a4 \u2260 0\nhbdd : \u2200 (n : \u2115), snormEssSup (f n) \u03bc \u2264 \u2191R\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4\n[PROOFSTEP]\nhave : \u2200 n, \u2200\u1d50 x \u2202\u03bc, (\u2016f n x\u2016\u208a : \u211d\u22650\u221e) < R + 1 := fun n =>\n ae_lt_of_essSup_lt (lt_of_le_of_lt (hbdd n) <| ENNReal.lt_add_right ENNReal.coe_ne_top one_ne_zero)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhp : \u22a4 \u2260 0\nhbdd : \u2200 (n : \u2115), snormEssSup (f n) \u03bc \u2264 \u2191R\nthis : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2016f n x\u2016\u208a < \u2191R + 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4\n[PROOFSTEP]\nrw [\u2190 ae_all_iff] at this \n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhp : \u22a4 \u2260 0\nhbdd : \u2200 (n : \u2115), snormEssSup (f n) \u03bc \u2264 \u2191R\nthis : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2016f i a\u2016\u208a < \u2191R + 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4\n[PROOFSTEP]\nfilter_upwards [this] with x hx using\n lt_of_le_of_lt (liminf_le_of_frequently_le' <| frequently_of_forall fun n => (hx n).le)\n (ENNReal.add_lt_top.2 \u27e8ENNReal.coe_lt_top, ENNReal.one_lt_top\u27e9)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4\n[PROOFSTEP]\nfilter_upwards [ae_bdd_liminf_atTop_rpow_of_snorm_bdd hfmeas hbdd] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\n\u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4\n[PROOFSTEP]\nhave hppos : 0 < p.toReal := ENNReal.toReal_pos hp hp'\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\nhppos : 0 < ENNReal.toReal p\n\u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4\n[PROOFSTEP]\nhave : liminf (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) atTop = liminf (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) atTop ^ p.toReal :=\n by\n change\n liminf (fun n => ENNReal.orderIsoRpow p.toReal hppos (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) atTop =\n ENNReal.orderIsoRpow p.toReal hppos (liminf (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) atTop)\n refine' (OrderIso.liminf_apply (ENNReal.orderIsoRpow p.toReal _) _ _ _ _).symm <;> isBoundedDefault\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\nhppos : 0 < ENNReal.toReal p\n\u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop = liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p\n[PROOFSTEP]\nchange\n liminf (fun n => ENNReal.orderIsoRpow p.toReal hppos (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) atTop =\n ENNReal.orderIsoRpow p.toReal hppos (liminf (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) atTop)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\nhppos : 0 < ENNReal.toReal p\n\u22a2 liminf (fun n => \u2191(ENNReal.orderIsoRpow (ENNReal.toReal p) hppos) \u2191\u2016f n x\u2016\u208a) atTop =\n \u2191(ENNReal.orderIsoRpow (ENNReal.toReal p) hppos) (liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop)\n[PROOFSTEP]\nrefine' (OrderIso.liminf_apply (ENNReal.orderIsoRpow p.toReal _) _ _ _ _).symm\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\nhppos : 0 < ENNReal.toReal p\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => \u2191\u2016f n x\u2016\u208a\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\nhppos : 0 < ENNReal.toReal p\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => \u2191\u2016f n x\u2016\u208a\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\nhppos : 0 < ENNReal.toReal p\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun x_1 => \u2191(ENNReal.orderIsoRpow (ENNReal.toReal p) hppos) \u2191\u2016f x_1 x\u2016\u208a\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\nhppos : 0 < ENNReal.toReal p\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) atTop fun x_1 => \u2191(ENNReal.orderIsoRpow (ENNReal.toReal p) hppos) \u2191\u2016f x_1 x\u2016\u208a\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4\nhppos : 0 < ENNReal.toReal p\nthis : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop = liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p\n\u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4\n[PROOFSTEP]\nrw [this] at hx \n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p < \u22a4\nhppos : 0 < ENNReal.toReal p\nthis : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop = liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p\n\u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4\n[PROOFSTEP]\nrw [\u2190 ENNReal.rpow_one (liminf (fun n => \u2016f n x\u2016\u208a) atTop), \u2190 mul_inv_cancel hppos.ne.symm, ENNReal.rpow_mul]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : OpensMeasurableSpace E\nR : \u211d\u22650\np : \u211d\u22650\u221e\nhp : p \u2260 0\nf : \u2115 \u2192 \u03b1 \u2192 E\nhfmeas : \u2200 (n : \u2115), Measurable (f n)\nhbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R\nhp' : \u00acp = \u22a4\nx : \u03b1\nhx : liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p < \u22a4\nhppos : 0 < ENNReal.toReal p\nthis : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop = liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p\n\u22a2 (liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p) ^ (ENNReal.toReal p)\u207b\u00b9 < \u22a4\n[PROOFSTEP]\nexact ENNReal.rpow_lt_top_of_nonneg (inv_nonneg.2 hppos.le) hx.ne\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\nX : Type u_5\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : OpensMeasurableSpace X\nf : X \u2192 E\nhf : Continuous f\nh'f : HasCompactSupport f\n\u03bc : Measure X\n\u22a2 Mem\u2112p f \u22a4\n[PROOFSTEP]\nborelize E\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\nX : Type u_5\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : OpensMeasurableSpace X\nf : X \u2192 E\nhf : Continuous f\nh'f : HasCompactSupport f\n\u03bc : Measure X\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 Mem\u2112p f \u22a4\n[PROOFSTEP]\nrcases hf.bounded_above_of_compact_support h'f with \u27e8C, hC\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\nX : Type u_5\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : OpensMeasurableSpace X\nf : X \u2192 E\nhf : Continuous f\nh'f : HasCompactSupport f\n\u03bc : Measure X\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nC : \u211d\nhC : \u2200 (x : X), \u2016f x\u2016 \u2264 C\n\u22a2 Mem\u2112p f \u22a4\n[PROOFSTEP]\napply mem\u2112p_top_of_bound ?_ C (Filter.eventually_of_forall hC)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedAddCommGroup G\nX : Type u_5\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : OpensMeasurableSpace X\nf : X \u2192 E\nhf : Continuous f\nh'f : HasCompactSupport f\n\u03bc : Measure X\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nC : \u211d\nhC : \u2200 (x : X), \u2016f x\u2016 \u2264 C\n\u22a2 AEStronglyMeasurable (fun x => f x) \u03bc\n[PROOFSTEP]\nexact (hf.stronglyMeasurable_of_hasCompactSupport h'f).aestronglyMeasurable\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.LpSeminorm", "llama_tokens": 155716, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195152660688, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.42395061755860947}} {"text": "[GOAL]\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\n\u22a2 Tendsto (fun y => (f (y + c * (y - x) ^ 2) - d) / (y - x)) l (\ud835\udcdd a)\n[PROOFSTEP]\nhave L : Tendsto (fun y => (y + c * (y - x) ^ 2 - x) / (y - x)) l (\ud835\udcdd 1) :=\n by\n have : Tendsto (fun y => 1 + c * (y - x)) l (\ud835\udcdd (1 + c * (x - x))) :=\n by\n apply Tendsto.mono_left _ (hl.trans nhdsWithin_le_nhds)\n exact ((tendsto_id.sub_const x).const_mul c).const_add 1\n simp only [_root_.sub_self, add_zero, mul_zero] at this \n apply Tendsto.congr' (Eventually.filter_mono hl _) this\n filter_upwards [self_mem_nhdsWithin] with y hy\n field_simp [sub_ne_zero.2 hy]\n ring\n[GOAL]\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\n\u22a2 Tendsto (fun y => (y + c * (y - x) ^ 2 - x) / (y - x)) l (\ud835\udcdd 1)\n[PROOFSTEP]\nhave : Tendsto (fun y => 1 + c * (y - x)) l (\ud835\udcdd (1 + c * (x - x))) :=\n by\n apply Tendsto.mono_left _ (hl.trans nhdsWithin_le_nhds)\n exact ((tendsto_id.sub_const x).const_mul c).const_add 1\n[GOAL]\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\n\u22a2 Tendsto (fun y => 1 + c * (y - x)) l (\ud835\udcdd (1 + c * (x - x)))\n[PROOFSTEP]\napply Tendsto.mono_left _ (hl.trans nhdsWithin_le_nhds)\n[GOAL]\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\n\u22a2 Tendsto (fun y => 1 + c * (y - x)) (\ud835\udcdd x) (\ud835\udcdd (1 + c * (x - x)))\n[PROOFSTEP]\nexact ((tendsto_id.sub_const x).const_mul c).const_add 1\n[GOAL]\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\nthis : Tendsto (fun y => 1 + c * (y - x)) l (\ud835\udcdd (1 + c * (x - x)))\n\u22a2 Tendsto (fun y => (y + c * (y - x) ^ 2 - x) / (y - x)) l (\ud835\udcdd 1)\n[PROOFSTEP]\nsimp only [_root_.sub_self, add_zero, mul_zero] at this \n[GOAL]\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\nthis : Tendsto (fun y => 1 + c * (y - x)) l (\ud835\udcdd 1)\n\u22a2 Tendsto (fun y => (y + c * (y - x) ^ 2 - x) / (y - x)) l (\ud835\udcdd 1)\n[PROOFSTEP]\napply Tendsto.congr' (Eventually.filter_mono hl _) this\n[GOAL]\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\nthis : Tendsto (fun y => 1 + c * (y - x)) l (\ud835\udcdd 1)\n\u22a2 \u2200\u1da0 (x_1 : \u211d) in \ud835\udcdd[{x}\u1d9c] x, 1 + c * (x_1 - x) = (x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin] with y hy\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\nthis : Tendsto (fun y => 1 + c * (y - x)) l (\ud835\udcdd 1)\ny : \u211d\nhy : y \u2208 {x}\u1d9c\n\u22a2 1 + c * (y - x) = (y + c * (y - x) ^ 2 - x) / (y - x)\n[PROOFSTEP]\nfield_simp [sub_ne_zero.2 hy]\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\nthis : Tendsto (fun y => 1 + c * (y - x)) l (\ud835\udcdd 1)\ny : \u211d\nhy : y \u2208 {x}\u1d9c\n\u22a2 (1 + c * (y - x)) * (y - x) = y + c * (y - x) ^ 2 - x\n[PROOFSTEP]\nring\n[GOAL]\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\nL : Tendsto (fun y => (y + c * (y - x) ^ 2 - x) / (y - x)) l (\ud835\udcdd 1)\n\u22a2 Tendsto (fun y => (f (y + c * (y - x) ^ 2) - d) / (y - x)) l (\ud835\udcdd a)\n[PROOFSTEP]\nhave Z := (hf.comp h').mul L\n[GOAL]\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\nL : Tendsto (fun y => (y + c * (y - x) ^ 2 - x) / (y - x)) l (\ud835\udcdd 1)\nZ :\n Tendsto\n (fun x_1 =>\n ((fun y => (f y - d) / (y - x)) \u2218 fun y => y + c * (y - x) ^ 2) x_1 * ((x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x)))\n l (\ud835\udcdd (a * 1))\n\u22a2 Tendsto (fun y => (f (y + c * (y - x) ^ 2) - d) / (y - x)) l (\ud835\udcdd a)\n[PROOFSTEP]\nrw [mul_one] at Z \n[GOAL]\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\nL : Tendsto (fun y => (y + c * (y - x) ^ 2 - x) / (y - x)) l (\ud835\udcdd 1)\nZ :\n Tendsto\n (fun x_1 =>\n ((fun y => (f y - d) / (y - x)) \u2218 fun y => y + c * (y - x) ^ 2) x_1 * ((x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x)))\n l (\ud835\udcdd a)\n\u22a2 Tendsto (fun y => (f (y + c * (y - x) ^ 2) - d) / (y - x)) l (\ud835\udcdd a)\n[PROOFSTEP]\napply Tendsto.congr' _ Z\n[GOAL]\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\nL : Tendsto (fun y => (y + c * (y - x) ^ 2 - x) / (y - x)) l (\ud835\udcdd 1)\nZ :\n Tendsto\n (fun x_1 =>\n ((fun y => (f y - d) / (y - x)) \u2218 fun y => y + c * (y - x) ^ 2) x_1 * ((x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x)))\n l (\ud835\udcdd a)\n\u22a2 (fun x_1 =>\n ((fun y => (f y - d) / (y - x)) \u2218 fun y => y + c * (y - x) ^ 2) x_1 *\n ((x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x))) =\u1da0[l]\n fun y => (f (y + c * (y - x) ^ 2) - d) / (y - x)\n[PROOFSTEP]\nhave : \u2200\u1da0 y in l, y + c * (y - x) ^ 2 \u2260 x := by apply Tendsto.mono_right h' hl self_mem_nhdsWithin\n[GOAL]\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\nL : Tendsto (fun y => (y + c * (y - x) ^ 2 - x) / (y - x)) l (\ud835\udcdd 1)\nZ :\n Tendsto\n (fun x_1 =>\n ((fun y => (f y - d) / (y - x)) \u2218 fun y => y + c * (y - x) ^ 2) x_1 * ((x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x)))\n l (\ud835\udcdd a)\n\u22a2 \u2200\u1da0 (y : \u211d) in l, y + c * (y - x) ^ 2 \u2260 x\n[PROOFSTEP]\napply Tendsto.mono_right h' hl self_mem_nhdsWithin\n[GOAL]\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\nL : Tendsto (fun y => (y + c * (y - x) ^ 2 - x) / (y - x)) l (\ud835\udcdd 1)\nZ :\n Tendsto\n (fun x_1 =>\n ((fun y => (f y - d) / (y - x)) \u2218 fun y => y + c * (y - x) ^ 2) x_1 * ((x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x)))\n l (\ud835\udcdd a)\nthis : \u2200\u1da0 (y : \u211d) in l, y + c * (y - x) ^ 2 \u2260 x\n\u22a2 (fun x_1 =>\n ((fun y => (f y - d) / (y - x)) \u2218 fun y => y + c * (y - x) ^ 2) x_1 *\n ((x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x))) =\u1da0[l]\n fun y => (f (y + c * (y - x) ^ 2) - d) / (y - x)\n[PROOFSTEP]\nfilter_upwards [this] with y hy\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nx a c d : \u211d\nl : Filter \u211d\nhl : l \u2264 \ud835\udcdd[{x}\u1d9c] x\nhf : Tendsto (fun y => (f y - d) / (y - x)) l (\ud835\udcdd a)\nh' : Tendsto (fun y => y + c * (y - x) ^ 2) l l\nL : Tendsto (fun y => (y + c * (y - x) ^ 2 - x) / (y - x)) l (\ud835\udcdd 1)\nZ :\n Tendsto\n (fun x_1 =>\n ((fun y => (f y - d) / (y - x)) \u2218 fun y => y + c * (y - x) ^ 2) x_1 * ((x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x)))\n l (\ud835\udcdd a)\nthis : \u2200\u1da0 (y : \u211d) in l, y + c * (y - x) ^ 2 \u2260 x\ny : \u211d\nhy : y + c * (y - x) ^ 2 \u2260 x\n\u22a2 ((fun y => (f y - d) / (y - x)) \u2218 fun y => y + c * (y - x) ^ 2) y * ((y + c * (y - x) ^ 2 - x) / (y - x)) =\n (f (y + c * (y - x) ^ 2) - d) / (y - x)\n[PROOFSTEP]\nfield_simp [sub_ne_zero.2 hy]\n[GOAL]\nf : StieltjesFunction\n\u22a2 \u2200\u1d50 (x : \u211d), HasDerivAt (\u2191f) (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)) x\n[PROOFSTEP]\nfilter_upwards [VitaliFamily.ae_tendsto_rnDeriv (vitaliFamily (volume : Measure \u211d) 1) f.measure,\n rnDeriv_lt_top f.measure volume, f.countable_leftLim_ne.ae_not_mem volume] with x hx h'x h''x\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\n\u22a2 HasDerivAt (\u2191f) (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)) x\n[PROOFSTEP]\nhave L1 : Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[>] x) (\ud835\udcdd (rnDeriv f.measure volume x).toReal) :=\n by\n apply Tendsto.congr' _ ((ENNReal.tendsto_toReal h'x.ne).comp (hx.comp (Real.tendsto_Icc_vitaliFamily_right x)))\n filter_upwards [self_mem_nhdsWithin]\n rintro y (hxy : x < y)\n simp only [comp_apply, StieltjesFunction.measure_Icc, Real.volume_Icc, Classical.not_not.1 h''x]\n rw [\u2190 ENNReal.ofReal_div_of_pos (sub_pos.2 hxy), ENNReal.toReal_ofReal]\n exact div_nonneg (sub_nonneg.2 (f.mono hxy.le)) (sub_pos.2 hxy).le\n[GOAL]\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\n\u22a2 Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n[PROOFSTEP]\napply Tendsto.congr' _ ((ENNReal.tendsto_toReal h'x.ne).comp (hx.comp (Real.tendsto_Icc_vitaliFamily_right x)))\n[GOAL]\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\n\u22a2 (ENNReal.toReal \u2218 (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) \u2218 fun y => Icc x y) =\u1da0[\ud835\udcdd[Ioi x] x]\n fun y => (\u2191f y - \u2191f x) / (y - x)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\n\u22a2 \u2200 (a : \u211d),\n a \u2208 Ioi x \u2192\n (ENNReal.toReal \u2218 (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) \u2218 fun y => Icc x y) a =\n (\u2191f a - \u2191f x) / (a - x)\n[PROOFSTEP]\nrintro y (hxy : x < y)\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\ny : \u211d\nhxy : x < y\n\u22a2 (ENNReal.toReal \u2218 (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) \u2218 fun y => Icc x y) y =\n (\u2191f y - \u2191f x) / (y - x)\n[PROOFSTEP]\nsimp only [comp_apply, StieltjesFunction.measure_Icc, Real.volume_Icc, Classical.not_not.1 h''x]\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\ny : \u211d\nhxy : x < y\n\u22a2 ENNReal.toReal (ENNReal.ofReal (\u2191f y - \u2191f x) / ENNReal.ofReal (y - x)) = (\u2191f y - \u2191f x) / (y - x)\n[PROOFSTEP]\nrw [\u2190 ENNReal.ofReal_div_of_pos (sub_pos.2 hxy), ENNReal.toReal_ofReal]\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\ny : \u211d\nhxy : x < y\n\u22a2 0 \u2264 (\u2191f y - \u2191f x) / (y - x)\n[PROOFSTEP]\nexact div_nonneg (sub_nonneg.2 (f.mono hxy.le)) (sub_pos.2 hxy).le\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 HasDerivAt (\u2191f) (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)) x\n[PROOFSTEP]\nhave L2 : Tendsto (fun y => (leftLim f y - f x) / (y - x)) (\ud835\udcdd[<] x) (\ud835\udcdd (rnDeriv f.measure volume x).toReal) :=\n by\n apply Tendsto.congr' _ ((ENNReal.tendsto_toReal h'x.ne).comp (hx.comp (Real.tendsto_Icc_vitaliFamily_left x)))\n filter_upwards [self_mem_nhdsWithin]\n rintro y (hxy : y < x)\n simp only [comp_apply, StieltjesFunction.measure_Icc, Real.volume_Icc]\n rw [\u2190 ENNReal.ofReal_div_of_pos (sub_pos.2 hxy), ENNReal.toReal_ofReal, \u2190 neg_neg (y - x), div_neg, neg_div', neg_sub,\n neg_sub]\n exact div_nonneg (sub_nonneg.2 (f.mono.leftLim_le hxy.le)) (sub_pos.2 hxy).le\n[GOAL]\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n[PROOFSTEP]\napply Tendsto.congr' _ ((ENNReal.tendsto_toReal h'x.ne).comp (hx.comp (Real.tendsto_Icc_vitaliFamily_left x)))\n[GOAL]\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 (ENNReal.toReal \u2218 (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) \u2218 fun y => Icc y x) =\u1da0[\ud835\udcdd[Iio x] x]\n fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 \u2200 (a : \u211d),\n a \u2208 Iio x \u2192\n (ENNReal.toReal \u2218 (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) \u2218 fun y => Icc y x) a =\n (leftLim (\u2191f) a - \u2191f x) / (a - x)\n[PROOFSTEP]\nrintro y (hxy : y < x)\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : \u211d\nhxy : y < x\n\u22a2 (ENNReal.toReal \u2218 (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) \u2218 fun y => Icc y x) y =\n (leftLim (\u2191f) y - \u2191f x) / (y - x)\n[PROOFSTEP]\nsimp only [comp_apply, StieltjesFunction.measure_Icc, Real.volume_Icc]\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : \u211d\nhxy : y < x\n\u22a2 ENNReal.toReal (ENNReal.ofReal (\u2191f x - leftLim (\u2191f) y) / ENNReal.ofReal (x - y)) = (leftLim (\u2191f) y - \u2191f x) / (y - x)\n[PROOFSTEP]\nrw [\u2190 ENNReal.ofReal_div_of_pos (sub_pos.2 hxy), ENNReal.toReal_ofReal, \u2190 neg_neg (y - x), div_neg, neg_div', neg_sub,\n neg_sub]\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : \u211d\nhxy : y < x\n\u22a2 0 \u2264 (\u2191f x - leftLim (\u2191f) y) / (x - y)\n[PROOFSTEP]\nexact div_nonneg (sub_nonneg.2 (f.mono.leftLim_le hxy.le)) (sub_pos.2 hxy).le\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 HasDerivAt (\u2191f) (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)) x\n[PROOFSTEP]\nhave L3 :\n Tendsto (fun y => (leftLim f (y + 1 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[<] x)\n (\ud835\udcdd (rnDeriv f.measure volume x).toReal) :=\n by\n apply tendsto_apply_add_mul_sq_div_sub (nhds_left'_le_nhds_ne x) L2\n apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within\n \u00b7 apply Tendsto.mono_left _ nhdsWithin_le_nhds\n have : Tendsto (fun y : \u211d => y + \u21911 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd (x + \u21911 * (x - x) ^ 2)) :=\n tendsto_id.add (((tendsto_id.sub_const x).pow 2).const_mul \u21911)\n simpa using this\n \u00b7 have : Ioo (x - 1) x \u2208 \ud835\udcdd[<] x := by apply Ioo_mem_nhdsWithin_Iio; exact \u27e8by linarith, le_refl _\u27e9\n filter_upwards [this]\n rintro y \u27e8hy : x - 1 < y, h'y : y < x\u27e9\n rw [mem_Iio]\n norm_num;\n nlinarith\n -- Deduce the correct limit on the left, by sandwiching.\n[GOAL]\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n[PROOFSTEP]\napply tendsto_apply_add_mul_sq_div_sub (nhds_left'_le_nhds_ne x) L2\n[GOAL]\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 Tendsto (fun y => y + \u21911 * (y - x) ^ 2) (\ud835\udcdd[Iio x] x) (\ud835\udcdd[Iio x] x)\n[PROOFSTEP]\napply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within\n[GOAL]\ncase h1\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 Tendsto (fun y => y + \u21911 * (y - x) ^ 2) (\ud835\udcdd[Iio x] x) (\ud835\udcdd x)\n[PROOFSTEP]\napply Tendsto.mono_left _ nhdsWithin_le_nhds\n[GOAL]\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 Tendsto (fun y => y + \u21911 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd x)\n[PROOFSTEP]\nhave : Tendsto (fun y : \u211d => y + \u21911 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd (x + \u21911 * (x - x) ^ 2)) :=\n tendsto_id.add (((tendsto_id.sub_const x).pow 2).const_mul \u21911)\n[GOAL]\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Tendsto (fun y => y + 1 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd (x + 1 * (x - x) ^ 2))\n\u22a2 Tendsto (fun y => y + \u21911 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd x)\n[PROOFSTEP]\nsimpa using this\n[GOAL]\ncase h2\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 \u2200\u1da0 (x_1 : \u211d) in \ud835\udcdd[Iio x] x, x_1 + \u21911 * (x_1 - x) ^ 2 \u2208 Iio x\n[PROOFSTEP]\nhave : Ioo (x - 1) x \u2208 \ud835\udcdd[<] x := by apply Ioo_mem_nhdsWithin_Iio; exact \u27e8by linarith, le_refl _\u27e9\n[GOAL]\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 Ioo (x - 1) x \u2208 \ud835\udcdd[Iio x] x\n[PROOFSTEP]\napply Ioo_mem_nhdsWithin_Iio\n[GOAL]\ncase H\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 x \u2208 Ioc (x - 1) x\n[PROOFSTEP]\nexact \u27e8by linarith, le_refl _\u27e9\n[GOAL]\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 x - 1 < x\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h2\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Ioo (x - 1) x \u2208 \ud835\udcdd[Iio x] x\n\u22a2 \u2200\u1da0 (x_1 : \u211d) in \ud835\udcdd[Iio x] x, x_1 + \u21911 * (x_1 - x) ^ 2 \u2208 Iio x\n[PROOFSTEP]\nfilter_upwards [this]\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Ioo (x - 1) x \u2208 \ud835\udcdd[Iio x] x\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioo (x - 1) x \u2192 a + \u21911 * (a - x) ^ 2 \u2208 Iio x\n[PROOFSTEP]\nrintro y \u27e8hy : x - 1 < y, h'y : y < x\u27e9\n[GOAL]\ncase h.intro\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Ioo (x - 1) x \u2208 \ud835\udcdd[Iio x] x\ny : \u211d\nhy : x - 1 < y\nh'y : y < x\n\u22a2 y + \u21911 * (y - x) ^ 2 \u2208 Iio x\n[PROOFSTEP]\nrw [mem_Iio]\n[GOAL]\ncase h.intro\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Ioo (x - 1) x \u2208 \ud835\udcdd[Iio x] x\ny : \u211d\nhy : x - 1 < y\nh'y : y < x\n\u22a2 y + \u21911 * (y - x) ^ 2 < x\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.intro\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Ioo (x - 1) x \u2208 \ud835\udcdd[Iio x] x\ny : \u211d\nhy : x - 1 < y\nh'y : y < x\n\u22a2 y + (y - x) ^ 2 < x\n[PROOFSTEP]\nnlinarith\n -- Deduce the correct limit on the left, by sandwiching.\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 HasDerivAt (\u2191f) (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)) x\n[PROOFSTEP]\nhave L4 : Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[<] x) (\ud835\udcdd (rnDeriv f.measure volume x).toReal) :=\n by\n apply tendsto_of_tendsto_of_tendsto_of_le_of_le' L3 L2\n \u00b7 filter_upwards [self_mem_nhdsWithin]\n rintro y (hy : y < x)\n refine' div_le_div_of_nonpos_of_le (by linarith) ((sub_le_sub_iff_right _).2 _)\n apply f.mono.le_leftLim\n have : \u21910 < (x - y) ^ 2 := sq_pos_of_pos (sub_pos.2 hy)\n norm_num; linarith\n \u00b7 filter_upwards [self_mem_nhdsWithin]\n rintro y (hy : y < x)\n refine' div_le_div_of_nonpos_of_le (by linarith) _\n simpa only [sub_le_sub_iff_right] using\n f.mono.leftLim_le\n (le_refl y)\n -- prove the result by splitting into left and right limits.\n[GOAL]\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n[PROOFSTEP]\napply tendsto_of_tendsto_of_tendsto_of_le_of_le' L3 L2\n[GOAL]\ncase hgf\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Iio x] x, (leftLim (\u2191f) (b + \u21911 * (b - x) ^ 2) - \u2191f x) / (b - x) \u2264 (\u2191f b - \u2191f x) / (b - x)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 \u2200 (a : \u211d), a \u2208 Iio x \u2192 (leftLim (\u2191f) (a + \u21911 * (a - x) ^ 2) - \u2191f x) / (a - x) \u2264 (\u2191f a - \u2191f x) / (a - x)\n[PROOFSTEP]\nrintro y (hy : y < x)\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : \u211d\nhy : y < x\n\u22a2 (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x) \u2264 (\u2191f y - \u2191f x) / (y - x)\n[PROOFSTEP]\nrefine' div_le_div_of_nonpos_of_le (by linarith) ((sub_le_sub_iff_right _).2 _)\n[GOAL]\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : \u211d\nhy : y < x\n\u22a2 y - x \u2264 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : \u211d\nhy : y < x\n\u22a2 \u2191f y \u2264 leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2)\n[PROOFSTEP]\napply f.mono.le_leftLim\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : \u211d\nhy : y < x\n\u22a2 y < y + \u21911 * (y - x) ^ 2\n[PROOFSTEP]\nhave : \u21910 < (x - y) ^ 2 := sq_pos_of_pos (sub_pos.2 hy)\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : \u211d\nhy : y < x\nthis : 0 < (x - y) ^ 2\n\u22a2 y < y + \u21911 * (y - x) ^ 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : \u211d\nhy : y < x\nthis : 0 < (x - y) ^ 2\n\u22a2 0 < (y - x) ^ 2\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hfh\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Iio x] x, (\u2191f b - \u2191f x) / (b - x) \u2264 (leftLim (\u2191f) b - \u2191f x) / (b - x)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 \u2200 (a : \u211d), a \u2208 Iio x \u2192 (\u2191f a - \u2191f x) / (a - x) \u2264 (leftLim (\u2191f) a - \u2191f x) / (a - x)\n[PROOFSTEP]\nrintro y (hy : y < x)\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : \u211d\nhy : y < x\n\u22a2 (\u2191f y - \u2191f x) / (y - x) \u2264 (leftLim (\u2191f) y - \u2191f x) / (y - x)\n[PROOFSTEP]\nrefine' div_le_div_of_nonpos_of_le (by linarith) _\n[GOAL]\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : \u211d\nhy : y < x\n\u22a2 y - x \u2264 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : \u211d\nhy : y < x\n\u22a2 leftLim (\u2191f) y - \u2191f x \u2264 \u2191f y - \u2191f x\n[PROOFSTEP]\nsimpa only [sub_le_sub_iff_right] using\n f.mono.leftLim_le\n (le_refl y)\n -- prove the result by splitting into left and right limits.\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL4 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 HasDerivAt (\u2191f) (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)) x\n[PROOFSTEP]\nrw [hasDerivAt_iff_tendsto_slope, slope_fun_def_field, \u2190 nhds_left'_sup_nhds_right', tendsto_sup]\n[GOAL]\ncase h\nf : StieltjesFunction\nx : \u211d\nhx :\n Tendsto (fun a => \u2191\u2191(StieltjesFunction.measure f) a / \u2191\u2191volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (\ud835\udcdd (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < \u22a4\nh''x : \u00acleftLim (\u2191f) x \u2260 \u2191f x\nL1 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (\u2191f) y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (\u2191f) (y + \u21911 * (y - x) ^ 2) - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL4 :\n Tendsto (fun y => (\u2191f y - \u2191f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n\u22a2 Tendsto (fun b => (\u2191f b - \u2191f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x))) \u2227\n Tendsto (fun b => (\u2191f b - \u2191f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n[PROOFSTEP]\nexact \u27e8L4, L1\u27e9\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\n\u22a2 \u2200\u1d50 (x : \u211d), HasDerivAt f (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)) x\n[PROOFSTEP]\nfilter_upwards [hf.stieltjesFunction.ae_hasDerivAt, hf.countable_not_continuousAt.ae_not_mem volume] with x hx h'x\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n HasDerivAt (\u2191(stieltjesFunction hf))\n (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)) x\nh'x : \u00ac\u00acContinuousAt f x\n\u22a2 HasDerivAt f (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)) x\n[PROOFSTEP]\nhave A : hf.stieltjesFunction x = f x :=\n by\n rw [Classical.not_not, hf.continuousAt_iff_leftLim_eq_rightLim] at h'x \n apply le_antisymm _ (hf.le_rightLim (le_refl _))\n rw [\u2190 h'x]\n exact hf.leftLim_le (le_refl _)\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n HasDerivAt (\u2191(stieltjesFunction hf))\n (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)) x\nh'x : \u00ac\u00acContinuousAt f x\n\u22a2 \u2191(stieltjesFunction hf) x = f x\n[PROOFSTEP]\nrw [Classical.not_not, hf.continuousAt_iff_leftLim_eq_rightLim] at h'x \n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n HasDerivAt (\u2191(stieltjesFunction hf))\n (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)) x\nh'x : leftLim f x = rightLim f x\n\u22a2 \u2191(stieltjesFunction hf) x = f x\n[PROOFSTEP]\napply le_antisymm _ (hf.le_rightLim (le_refl _))\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n HasDerivAt (\u2191(stieltjesFunction hf))\n (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)) x\nh'x : leftLim f x = rightLim f x\n\u22a2 rightLim f x \u2264 f x\n[PROOFSTEP]\nrw [\u2190 h'x]\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n HasDerivAt (\u2191(stieltjesFunction hf))\n (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)) x\nh'x : leftLim f x = rightLim f x\n\u22a2 leftLim f x \u2264 f x\n[PROOFSTEP]\nexact hf.leftLim_le (le_refl _)\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n HasDerivAt (\u2191(stieltjesFunction hf))\n (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)) x\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\n\u22a2 HasDerivAt f (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)) x\n[PROOFSTEP]\nrw [hasDerivAt_iff_tendsto_slope, (nhds_left'_sup_nhds_right' x).symm, tendsto_sup, slope_fun_def_field, A] at hx \n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\n\u22a2 HasDerivAt f (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)) x\n[PROOFSTEP]\nhave L1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[>] x) (\ud835\udcdd (rnDeriv hf.stieltjesFunction.measure volume x).toReal) := by\n -- limit of a helper function, with a small shift compared to `g`\n have :\n Tendsto (fun y => (hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[>] x)\n (\ud835\udcdd (rnDeriv hf.stieltjesFunction.measure volume x).toReal) :=\n by\n apply tendsto_apply_add_mul_sq_div_sub (nhds_right'_le_nhds_ne x) hx.2\n apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within\n \u00b7 apply Tendsto.mono_left _ nhdsWithin_le_nhds\n have : Tendsto (fun y : \u211d => y + -\u21911 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd (x + -\u21911 * (x - x) ^ 2)) :=\n tendsto_id.add (((tendsto_id.sub_const x).pow 2).const_mul (-1))\n simpa using this\n \u00b7 have : Ioo x (x + 1) \u2208 \ud835\udcdd[>] x := by apply Ioo_mem_nhdsWithin_Ioi; exact \u27e8le_refl _, by linarith\u27e9\n filter_upwards [this]\n rintro y \u27e8hy : x < y, h'y : y < x + 1\u27e9\n rw [mem_Ioi]\n norm_num;\n nlinarith\n -- apply the sandwiching argument, with the helper function and `g`\n apply tendsto_of_tendsto_of_tendsto_of_le_of_le' this hx.2\n \u00b7 filter_upwards [self_mem_nhdsWithin]\n rintro y (hy : x < y)\n have : \u21910 < (y - x) ^ 2 := sq_pos_of_pos (sub_pos.2 hy)\n apply div_le_div_of_le_of_nonneg _ (sub_pos.2 hy).le\n exact (sub_le_sub_iff_right _).2 (hf.rightLim_le (by norm_num; linarith))\n \u00b7 filter_upwards [self_mem_nhdsWithin]\n rintro y (hy : x < y)\n apply div_le_div_of_le_of_nonneg _ (sub_pos.2 hy).le\n exact\n (sub_le_sub_iff_right _).2\n (hf.le_rightLim (le_refl y))\n -- prove differentiability on the left, by sandwiching with values of `g`\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\n\u22a2 Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n[PROOFSTEP]\nhave :\n Tendsto (fun y => (hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[>] x)\n (\ud835\udcdd (rnDeriv hf.stieltjesFunction.measure volume x).toReal) :=\n by\n apply tendsto_apply_add_mul_sq_div_sub (nhds_right'_le_nhds_ne x) hx.2\n apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within\n \u00b7 apply Tendsto.mono_left _ nhdsWithin_le_nhds\n have : Tendsto (fun y : \u211d => y + -\u21911 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd (x + -\u21911 * (x - x) ^ 2)) :=\n tendsto_id.add (((tendsto_id.sub_const x).pow 2).const_mul (-1))\n simpa using this\n \u00b7 have : Ioo x (x + 1) \u2208 \ud835\udcdd[>] x := by apply Ioo_mem_nhdsWithin_Ioi; exact \u27e8le_refl _, by linarith\u27e9\n filter_upwards [this]\n rintro y \u27e8hy : x < y, h'y : y < x + 1\u27e9\n rw [mem_Ioi]\n norm_num;\n nlinarith\n -- apply the sandwiching argument, with the helper function and `g`\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\n\u22a2 Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n[PROOFSTEP]\napply tendsto_apply_add_mul_sq_div_sub (nhds_right'_le_nhds_ne x) hx.2\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\n\u22a2 Tendsto (fun y => y + -\u21911 * (y - x) ^ 2) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd[Ioi x] x)\n[PROOFSTEP]\napply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within\n[GOAL]\ncase h1\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\n\u22a2 Tendsto (fun y => y + -\u21911 * (y - x) ^ 2) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd x)\n[PROOFSTEP]\napply Tendsto.mono_left _ nhdsWithin_le_nhds\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\n\u22a2 Tendsto (fun y => y + -\u21911 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd x)\n[PROOFSTEP]\nhave : Tendsto (fun y : \u211d => y + -\u21911 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd (x + -\u21911 * (x - x) ^ 2)) :=\n tendsto_id.add (((tendsto_id.sub_const x).pow 2).const_mul (-1))\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis : Tendsto (fun y => y + -1 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd (x + -1 * (x - x) ^ 2))\n\u22a2 Tendsto (fun y => y + -\u21911 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd x)\n[PROOFSTEP]\nsimpa using this\n[GOAL]\ncase h2\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\n\u22a2 \u2200\u1da0 (x_1 : \u211d) in \ud835\udcdd[Ioi x] x, x_1 + -\u21911 * (x_1 - x) ^ 2 \u2208 Ioi x\n[PROOFSTEP]\nhave : Ioo x (x + 1) \u2208 \ud835\udcdd[>] x := by apply Ioo_mem_nhdsWithin_Ioi; exact \u27e8le_refl _, by linarith\u27e9\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\n\u22a2 Ioo x (x + 1) \u2208 \ud835\udcdd[Ioi x] x\n[PROOFSTEP]\napply Ioo_mem_nhdsWithin_Ioi\n[GOAL]\ncase H\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\n\u22a2 x \u2208 Ico x (x + 1)\n[PROOFSTEP]\nexact \u27e8le_refl _, by linarith\u27e9\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\n\u22a2 x < x + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h2\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis : Ioo x (x + 1) \u2208 \ud835\udcdd[Ioi x] x\n\u22a2 \u2200\u1da0 (x_1 : \u211d) in \ud835\udcdd[Ioi x] x, x_1 + -\u21911 * (x_1 - x) ^ 2 \u2208 Ioi x\n[PROOFSTEP]\nfilter_upwards [this]\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis : Ioo x (x + 1) \u2208 \ud835\udcdd[Ioi x] x\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioo x (x + 1) \u2192 a + -\u21911 * (a - x) ^ 2 \u2208 Ioi x\n[PROOFSTEP]\nrintro y \u27e8hy : x < y, h'y : y < x + 1\u27e9\n[GOAL]\ncase h.intro\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis : Ioo x (x + 1) \u2208 \ud835\udcdd[Ioi x] x\ny : \u211d\nhy : x < y\nh'y : y < x + 1\n\u22a2 y + -\u21911 * (y - x) ^ 2 \u2208 Ioi x\n[PROOFSTEP]\nrw [mem_Ioi]\n[GOAL]\ncase h.intro\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis : Ioo x (x + 1) \u2208 \ud835\udcdd[Ioi x] x\ny : \u211d\nhy : x < y\nh'y : y < x + 1\n\u22a2 x < y + -\u21911 * (y - x) ^ 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.intro\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis : Ioo x (x + 1) \u2208 \ud835\udcdd[Ioi x] x\ny : \u211d\nhy : x < y\nh'y : y < x + 1\n\u22a2 x + (y - x) ^ 2 < y\n[PROOFSTEP]\nnlinarith\n -- apply the sandwiching argument, with the helper function and `g`\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n[PROOFSTEP]\napply tendsto_of_tendsto_of_tendsto_of_le_of_le' this hx.2\n[GOAL]\ncase hgf\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi x] x, (\u2191(stieltjesFunction hf) (b + -\u21911 * (b - x) ^ 2) - f x) / (b - x) \u2264 (f b - f x) / (b - x)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi x \u2192 (\u2191(stieltjesFunction hf) (a + -\u21911 * (a - x) ^ 2) - f x) / (a - x) \u2264 (f a - f x) / (a - x)\n[PROOFSTEP]\nrintro y (hy : x < y)\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\ny : \u211d\nhy : x < y\n\u22a2 (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x) \u2264 (f y - f x) / (y - x)\n[PROOFSTEP]\nhave : \u21910 < (y - x) ^ 2 := sq_pos_of_pos (sub_pos.2 hy)\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis\u271d :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\ny : \u211d\nhy : x < y\nthis : 0 < (y - x) ^ 2\n\u22a2 (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x) \u2264 (f y - f x) / (y - x)\n[PROOFSTEP]\napply div_le_div_of_le_of_nonneg _ (sub_pos.2 hy).le\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis\u271d :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\ny : \u211d\nhy : x < y\nthis : 0 < (y - x) ^ 2\n\u22a2 \u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x \u2264 f y - f x\n[PROOFSTEP]\nexact (sub_le_sub_iff_right _).2 (hf.rightLim_le (by norm_num; linarith))\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis\u271d :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\ny : \u211d\nhy : x < y\nthis : 0 < (y - x) ^ 2\n\u22a2 y + -\u21911 * (y - x) ^ 2 < y\n[PROOFSTEP]\nnorm_num\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis\u271d :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\ny : \u211d\nhy : x < y\nthis : 0 < (y - x) ^ 2\n\u22a2 0 < (y - x) ^ 2\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hfh\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi x] x, (f b - f x) / (b - x) \u2264 (\u2191(stieltjesFunction hf) b - f x) / (b - x)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi x \u2192 (f a - f x) / (a - x) \u2264 (\u2191(stieltjesFunction hf) a - f x) / (a - x)\n[PROOFSTEP]\nrintro y (hy : x < y)\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\ny : \u211d\nhy : x < y\n\u22a2 (f y - f x) / (y - x) \u2264 (\u2191(stieltjesFunction hf) y - f x) / (y - x)\n[PROOFSTEP]\napply div_le_div_of_le_of_nonneg _ (sub_pos.2 hy).le\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\ny : \u211d\nhy : x < y\n\u22a2 f y - f x \u2264 \u2191(stieltjesFunction hf) y - f x\n[PROOFSTEP]\nexact\n (sub_le_sub_iff_right _).2\n (hf.le_rightLim (le_refl y))\n -- prove differentiability on the left, by sandwiching with values of `g`\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 HasDerivAt f (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)) x\n[PROOFSTEP]\nhave L2 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[<] x) (\ud835\udcdd (rnDeriv hf.stieltjesFunction.measure volume x).toReal) := by\n -- limit of a helper function, with a small shift compared to `g`\n have :\n Tendsto (fun y => (hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[<] x)\n (\ud835\udcdd (rnDeriv hf.stieltjesFunction.measure volume x).toReal) :=\n by\n apply tendsto_apply_add_mul_sq_div_sub (nhds_left'_le_nhds_ne x) hx.1\n apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within\n \u00b7 apply Tendsto.mono_left _ nhdsWithin_le_nhds\n have : Tendsto (fun y : \u211d => y + -\u21911 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd (x + -\u21911 * (x - x) ^ 2)) :=\n tendsto_id.add (((tendsto_id.sub_const x).pow 2).const_mul (-1))\n simpa using this\n \u00b7 have : Ioo (x - 1) x \u2208 \ud835\udcdd[<] x := by apply Ioo_mem_nhdsWithin_Iio; exact \u27e8by linarith, le_refl _\u27e9\n filter_upwards [this]\n rintro y \u27e8hy : x - 1 < y, h'y : y < x\u27e9\n rw [mem_Iio]\n norm_num;\n nlinarith\n -- apply the sandwiching argument, with `g` and the helper function\n apply tendsto_of_tendsto_of_tendsto_of_le_of_le' hx.1 this\n \u00b7 filter_upwards [self_mem_nhdsWithin]\n rintro y (hy : y < x)\n apply div_le_div_of_nonpos_of_le (sub_neg.2 hy).le\n exact (sub_le_sub_iff_right _).2 (hf.le_rightLim (le_refl _))\n \u00b7 filter_upwards [self_mem_nhdsWithin]\n rintro y (hy : y < x)\n have : \u21910 < (y - x) ^ 2 := sq_pos_of_neg (sub_neg.2 hy)\n apply div_le_div_of_nonpos_of_le (sub_neg.2 hy).le\n exact\n (sub_le_sub_iff_right _).2\n (hf.rightLim_le (by norm_num; linarith))\n -- conclude global differentiability\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n[PROOFSTEP]\nhave :\n Tendsto (fun y => (hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[<] x)\n (\ud835\udcdd (rnDeriv hf.stieltjesFunction.measure volume x).toReal) :=\n by\n apply tendsto_apply_add_mul_sq_div_sub (nhds_left'_le_nhds_ne x) hx.1\n apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within\n \u00b7 apply Tendsto.mono_left _ nhdsWithin_le_nhds\n have : Tendsto (fun y : \u211d => y + -\u21911 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd (x + -\u21911 * (x - x) ^ 2)) :=\n tendsto_id.add (((tendsto_id.sub_const x).pow 2).const_mul (-1))\n simpa using this\n \u00b7 have : Ioo (x - 1) x \u2208 \ud835\udcdd[<] x := by apply Ioo_mem_nhdsWithin_Iio; exact \u27e8by linarith, le_refl _\u27e9\n filter_upwards [this]\n rintro y \u27e8hy : x - 1 < y, h'y : y < x\u27e9\n rw [mem_Iio]\n norm_num;\n nlinarith\n -- apply the sandwiching argument, with `g` and the helper function\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n[PROOFSTEP]\napply tendsto_apply_add_mul_sq_div_sub (nhds_left'_le_nhds_ne x) hx.1\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 Tendsto (fun y => y + -\u21911 * (y - x) ^ 2) (\ud835\udcdd[Iio x] x) (\ud835\udcdd[Iio x] x)\n[PROOFSTEP]\napply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within\n[GOAL]\ncase h1\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 Tendsto (fun y => y + -\u21911 * (y - x) ^ 2) (\ud835\udcdd[Iio x] x) (\ud835\udcdd x)\n[PROOFSTEP]\napply Tendsto.mono_left _ nhdsWithin_le_nhds\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 Tendsto (fun y => y + -\u21911 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd x)\n[PROOFSTEP]\nhave : Tendsto (fun y : \u211d => y + -\u21911 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd (x + -\u21911 * (x - x) ^ 2)) :=\n tendsto_id.add (((tendsto_id.sub_const x).pow 2).const_mul (-1))\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis : Tendsto (fun y => y + -1 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd (x + -1 * (x - x) ^ 2))\n\u22a2 Tendsto (fun y => y + -\u21911 * (y - x) ^ 2) (\ud835\udcdd x) (\ud835\udcdd x)\n[PROOFSTEP]\nsimpa using this\n[GOAL]\ncase h2\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 \u2200\u1da0 (x_1 : \u211d) in \ud835\udcdd[Iio x] x, x_1 + -\u21911 * (x_1 - x) ^ 2 \u2208 Iio x\n[PROOFSTEP]\nhave : Ioo (x - 1) x \u2208 \ud835\udcdd[<] x := by apply Ioo_mem_nhdsWithin_Iio; exact \u27e8by linarith, le_refl _\u27e9\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 Ioo (x - 1) x \u2208 \ud835\udcdd[Iio x] x\n[PROOFSTEP]\napply Ioo_mem_nhdsWithin_Iio\n[GOAL]\ncase H\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 x \u2208 Ioc (x - 1) x\n[PROOFSTEP]\nexact \u27e8by linarith, le_refl _\u27e9\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 x - 1 < x\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h2\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis : Ioo (x - 1) x \u2208 \ud835\udcdd[Iio x] x\n\u22a2 \u2200\u1da0 (x_1 : \u211d) in \ud835\udcdd[Iio x] x, x_1 + -\u21911 * (x_1 - x) ^ 2 \u2208 Iio x\n[PROOFSTEP]\nfilter_upwards [this]\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis : Ioo (x - 1) x \u2208 \ud835\udcdd[Iio x] x\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioo (x - 1) x \u2192 a + -\u21911 * (a - x) ^ 2 \u2208 Iio x\n[PROOFSTEP]\nrintro y \u27e8hy : x - 1 < y, h'y : y < x\u27e9\n[GOAL]\ncase h.intro\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis : Ioo (x - 1) x \u2208 \ud835\udcdd[Iio x] x\ny : \u211d\nhy : x - 1 < y\nh'y : y < x\n\u22a2 y + -\u21911 * (y - x) ^ 2 \u2208 Iio x\n[PROOFSTEP]\nrw [mem_Iio]\n[GOAL]\ncase h.intro\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis : Ioo (x - 1) x \u2208 \ud835\udcdd[Iio x] x\ny : \u211d\nhy : x - 1 < y\nh'y : y < x\n\u22a2 y + -\u21911 * (y - x) ^ 2 < x\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.intro\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis : Ioo (x - 1) x \u2208 \ud835\udcdd[Iio x] x\ny : \u211d\nhy : x - 1 < y\nh'y : y < x\n\u22a2 y < x + (y - x) ^ 2\n[PROOFSTEP]\nnlinarith\n -- apply the sandwiching argument, with `g` and the helper function\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n[PROOFSTEP]\napply tendsto_of_tendsto_of_tendsto_of_le_of_le' hx.1 this\n[GOAL]\ncase hgf\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Iio x] x, (\u2191(stieltjesFunction hf) b - f x) / (b - x) \u2264 (f b - f x) / (b - x)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 \u2200 (a : \u211d), a \u2208 Iio x \u2192 (\u2191(stieltjesFunction hf) a - f x) / (a - x) \u2264 (f a - f x) / (a - x)\n[PROOFSTEP]\nrintro y (hy : y < x)\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\ny : \u211d\nhy : y < x\n\u22a2 (\u2191(stieltjesFunction hf) y - f x) / (y - x) \u2264 (f y - f x) / (y - x)\n[PROOFSTEP]\napply div_le_div_of_nonpos_of_le (sub_neg.2 hy).le\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\ny : \u211d\nhy : y < x\n\u22a2 f y - f x \u2264 \u2191(stieltjesFunction hf) y - f x\n[PROOFSTEP]\nexact (sub_le_sub_iff_right _).2 (hf.le_rightLim (le_refl _))\n[GOAL]\ncase hfh\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Iio x] x, (f b - f x) / (b - x) \u2264 (\u2191(stieltjesFunction hf) (b + -\u21911 * (b - x) ^ 2) - f x) / (b - x)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 \u2200 (a : \u211d), a \u2208 Iio x \u2192 (f a - f x) / (a - x) \u2264 (\u2191(stieltjesFunction hf) (a + -\u21911 * (a - x) ^ 2) - f x) / (a - x)\n[PROOFSTEP]\nrintro y (hy : y < x)\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\ny : \u211d\nhy : y < x\n\u22a2 (f y - f x) / (y - x) \u2264 (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)\n[PROOFSTEP]\nhave : \u21910 < (y - x) ^ 2 := sq_pos_of_neg (sub_neg.2 hy)\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis\u271d :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\ny : \u211d\nhy : y < x\nthis : 0 < (y - x) ^ 2\n\u22a2 (f y - f x) / (y - x) \u2264 (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)\n[PROOFSTEP]\napply div_le_div_of_nonpos_of_le (sub_neg.2 hy).le\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis\u271d :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\ny : \u211d\nhy : y < x\nthis : 0 < (y - x) ^ 2\n\u22a2 \u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x \u2264 f y - f x\n[PROOFSTEP]\nexact\n (sub_le_sub_iff_right _).2\n (hf.rightLim_le (by norm_num; linarith))\n -- conclude global differentiability\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis\u271d :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\ny : \u211d\nhy : y < x\nthis : 0 < (y - x) ^ 2\n\u22a2 y + -\u21911 * (y - x) ^ 2 < y\n[PROOFSTEP]\nnorm_num\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nthis\u271d :\n Tendsto (fun y => (\u2191(stieltjesFunction hf) (y + -\u21911 * (y - x) ^ 2) - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\ny : \u211d\nhy : y < x\nthis : 0 < (y - x) ^ 2\n\u22a2 0 < (y - x) ^ 2\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nL2 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 HasDerivAt f (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)) x\n[PROOFSTEP]\nrw [hasDerivAt_iff_tendsto_slope, slope_fun_def_field, (nhds_left'_sup_nhds_right' x).symm, tendsto_sup]\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nhf : Monotone f\nx : \u211d\nhx :\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (\u2191(stieltjesFunction hf) b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nh'x : \u00ac\u00acContinuousAt f x\nA : \u2191(stieltjesFunction hf) x = f x\nL1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\nL2 :\n Tendsto (fun y => (f y - f x) / (y - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n\u22a2 Tendsto (fun b => (f b - f x) / (b - x)) (\ud835\udcdd[Iio x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x))) \u2227\n Tendsto (fun b => (f b - f x) / (b - x)) (\ud835\udcdd[Ioi x] x)\n (\ud835\udcdd (ENNReal.toReal (rnDeriv (StieltjesFunction.measure (stieltjesFunction hf)) volume x)))\n[PROOFSTEP]\nexact \u27e8L2, L1\u27e9\n[GOAL]\nf : \u211d \u2192 \u211d\nhf : Monotone f\n\u22a2 \u2200\u1d50 (x : \u211d), DifferentiableAt \u211d f x\n[PROOFSTEP]\nfilter_upwards [hf.ae_hasDerivAt] with x hx using hx.differentiableAt\n[GOAL]\nf : \u211d \u2192 \u211d\ns : Set \u211d\nhf : MonotoneOn f s\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\napply ae_of_mem_of_ae_of_mem_inter_Ioo\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\ns : Set \u211d\nhf : MonotoneOn f s\n\u22a2 \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d), x \u2208 s \u2229 Ioo a b \u2192 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nintro a b as bs _\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\ns : Set \u211d\nhf : MonotoneOn f s\na b : \u211d\nas : a \u2208 s\nbs : b \u2208 s\na\u271d : a < b\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2229 Ioo a b \u2192 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nobtain \u27e8g, hg, gf\u27e9 : \u2203 g : \u211d \u2192 \u211d, Monotone g \u2227 EqOn f g (s \u2229 Icc a b) :=\n (hf.mono (inter_subset_left s (Icc a b))).exists_monotone_extension\n (hf.map_bddBelow (inter_subset_left _ _) \u27e8a, fun x hx => hx.2.1, as\u27e9)\n (hf.map_bddAbove (inter_subset_left _ _) \u27e8b, fun x hx => hx.2.2, bs\u27e9)\n[GOAL]\ncase h.intro.intro\nf : \u211d \u2192 \u211d\ns : Set \u211d\nhf : MonotoneOn f s\na b : \u211d\nas : a \u2208 s\nbs : b \u2208 s\na\u271d : a < b\ng : \u211d \u2192 \u211d\nhg : Monotone g\ngf : EqOn f g (s \u2229 Icc a b)\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2229 Ioo a b \u2192 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nfilter_upwards [hg.ae_differentiableAt] with x hx\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\ns : Set \u211d\nhf : MonotoneOn f s\na b : \u211d\nas : a \u2208 s\nbs : b \u2208 s\na\u271d : a < b\ng : \u211d \u2192 \u211d\nhg : Monotone g\ngf : EqOn f g (s \u2229 Icc a b)\nx : \u211d\nhx : DifferentiableAt \u211d g x\n\u22a2 x \u2208 s \u2229 Ioo a b \u2192 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nintro h'x\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\ns : Set \u211d\nhf : MonotoneOn f s\na b : \u211d\nas : a \u2208 s\nbs : b \u2208 s\na\u271d : a < b\ng : \u211d \u2192 \u211d\nhg : Monotone g\ngf : EqOn f g (s \u2229 Icc a b)\nx : \u211d\nhx : DifferentiableAt \u211d g x\nh'x : x \u2208 s \u2229 Ioo a b\n\u22a2 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\napply hx.differentiableWithinAt.congr_of_eventuallyEq _ (gf \u27e8h'x.1, h'x.2.1.le, h'x.2.2.le\u27e9)\n[GOAL]\nf : \u211d \u2192 \u211d\ns : Set \u211d\nhf : MonotoneOn f s\na b : \u211d\nas : a \u2208 s\nbs : b \u2208 s\na\u271d : a < b\ng : \u211d \u2192 \u211d\nhg : Monotone g\ngf : EqOn f g (s \u2229 Icc a b)\nx : \u211d\nhx : DifferentiableAt \u211d g x\nh'x : x \u2208 s \u2229 Ioo a b\n\u22a2 f =\u1da0[\ud835\udcdd[s] x] g\n[PROOFSTEP]\nhave : Ioo a b \u2208 \ud835\udcdd[s] x := nhdsWithin_le_nhds (Ioo_mem_nhds h'x.2.1 h'x.2.2)\n[GOAL]\nf : \u211d \u2192 \u211d\ns : Set \u211d\nhf : MonotoneOn f s\na b : \u211d\nas : a \u2208 s\nbs : b \u2208 s\na\u271d : a < b\ng : \u211d \u2192 \u211d\nhg : Monotone g\ngf : EqOn f g (s \u2229 Icc a b)\nx : \u211d\nhx : DifferentiableAt \u211d g x\nh'x : x \u2208 s \u2229 Ioo a b\nthis : Ioo a b \u2208 \ud835\udcdd[s] x\n\u22a2 f =\u1da0[\ud835\udcdd[s] x] g\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin, this] with y hy h'y\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\ns : Set \u211d\nhf : MonotoneOn f s\na b : \u211d\nas : a \u2208 s\nbs : b \u2208 s\na\u271d : a < b\ng : \u211d \u2192 \u211d\nhg : Monotone g\ngf : EqOn f g (s \u2229 Icc a b)\nx : \u211d\nhx : DifferentiableAt \u211d g x\nh'x : x \u2208 s \u2229 Ioo a b\nthis : Ioo a b \u2208 \ud835\udcdd[s] x\ny : \u211d\nhy : y \u2208 s\nh'y : y \u2208 Ioo a b\n\u22a2 f y = g y\n[PROOFSTEP]\nexact gf \u27e8hy, h'y.1.le, h'y.2.le\u27e9\n[GOAL]\nf : \u211d \u2192 \u211d\ns : Set \u211d\nhf : MonotoneOn f s\nhs : MeasurableSet s\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume s, DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nrw [ae_restrict_iff' hs]\n[GOAL]\nf : \u211d \u2192 \u211d\ns : Set \u211d\nhf : MonotoneOn f s\nhs : MeasurableSet s\n\u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x\n[PROOFSTEP]\nexact hf.ae_differentiableWithinAt_of_mem\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.Monotone", "llama_tokens": 48256, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891392358014, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.4239370946811427}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\nf : List \u03b1\n\u22a2 f \u2208 sections L \u2194 Forall\u2082 (fun x x_1 => x \u2208 x_1) f L\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\nf : List \u03b1\nh : f \u2208 sections L\n\u22a2 Forall\u2082 (fun x x_1 => x \u2208 x_1) f L\n[PROOFSTEP]\ninduction L generalizing f\n[GOAL]\ncase refine'_1.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : List \u03b1\nh : f \u2208 sections []\n\u22a2 Forall\u2082 (fun x x_1 => x \u2208 x_1) f []\n[PROOFSTEP]\ncases mem_singleton.1 h\n[GOAL]\ncase refine'_1.nil.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nh : [] \u2208 sections []\n\u22a2 Forall\u2082 (fun x x_1 => x \u2208 x_1) [] []\n[PROOFSTEP]\nexact Forall\u2082.nil\n[GOAL]\ncase refine'_1.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhead\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\ntail_ih\u271d : \u2200 {f : List \u03b1}, f \u2208 sections tail\u271d \u2192 Forall\u2082 (fun x x_1 => x \u2208 x_1) f tail\u271d\nf : List \u03b1\nh : f \u2208 sections (head\u271d :: tail\u271d)\n\u22a2 Forall\u2082 (fun x x_1 => x \u2208 x_1) f (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp only [sections, bind_eq_bind, mem_bind, mem_map] at h \n[GOAL]\ncase refine'_1.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhead\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\ntail_ih\u271d : \u2200 {f : List \u03b1}, f \u2208 sections tail\u271d \u2192 Forall\u2082 (fun x x_1 => x \u2208 x_1) f tail\u271d\nf : List \u03b1\nh : \u2203 a, a \u2208 sections tail\u271d \u2227 \u2203 a_1, a_1 \u2208 head\u271d \u2227 a_1 :: a = f\n\u22a2 Forall\u2082 (fun x x_1 => x \u2208 x_1) f (head\u271d :: tail\u271d)\n[PROOFSTEP]\nrcases h with \u27e8_, _, _, _, rfl\u27e9\n[GOAL]\ncase refine'_1.cons.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhead\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\ntail_ih\u271d : \u2200 {f : List \u03b1}, f \u2208 sections tail\u271d \u2192 Forall\u2082 (fun x x_1 => x \u2208 x_1) f tail\u271d\nw\u271d\u00b9 : List \u03b1\nleft\u271d\u00b9 : w\u271d\u00b9 \u2208 sections tail\u271d\nw\u271d : \u03b1\nleft\u271d : w\u271d \u2208 head\u271d\n\u22a2 Forall\u2082 (fun x x_1 => x \u2208 x_1) (w\u271d :: w\u271d\u00b9) (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp only [*, forall\u2082_cons, true_and_iff]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\nf : List \u03b1\nh : Forall\u2082 (fun x x_1 => x \u2208 x_1) f L\n\u22a2 f \u2208 sections L\n[PROOFSTEP]\ninduction' h with a l f L al fL fs\n[GOAL]\ncase refine'_2.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\nf : List \u03b1\n\u22a2 [] \u2208 sections []\n[PROOFSTEP]\nsimp only [sections, mem_singleton]\n[GOAL]\ncase refine'_2.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL\u271d : List (List \u03b1)\nf\u271d : List \u03b1\na : \u03b1\nl f : List \u03b1\nL : List (List \u03b1)\nal : a \u2208 l\nfL : Forall\u2082 (fun x x_1 => x \u2208 x_1) f L\nfs : f \u2208 sections L\n\u22a2 a :: f \u2208 sections (l :: L)\n[PROOFSTEP]\nsimp only [sections, bind_eq_bind, mem_bind, mem_map]\n[GOAL]\ncase refine'_2.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL\u271d : List (List \u03b1)\nf\u271d : List \u03b1\na : \u03b1\nl f : List \u03b1\nL : List (List \u03b1)\nal : a \u2208 l\nfL : Forall\u2082 (fun x x_1 => x \u2208 x_1) f L\nfs : f \u2208 sections L\n\u22a2 \u2203 a_1, a_1 \u2208 sections L \u2227 \u2203 a_2, a_2 \u2208 l \u2227 a_2 :: a_1 = a :: f\n[PROOFSTEP]\nexact \u27e8f, fs, a, al, rfl\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Sections", "llama_tokens": 1361, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.4238870168941186}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\n\u22a2 LipschitzWith K f \u2194 \u2200 (x y : \u03b1), dist (f x) (f y) \u2264 \u2191K * dist x y\n[PROOFSTEP]\nsimp only [LipschitzWith, edist_nndist, dist_nndist]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 (x y : \u03b1), \u2191(nndist (f x) (f y)) \u2264 \u2191K * \u2191(nndist x y)) \u2194 \u2200 (x y : \u03b1), \u2191(nndist (f x) (f y)) \u2264 \u2191K * \u2191(nndist x y)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nK : \u211d\u22650\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 LipschitzOnWith K f s \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (f x) (f y) \u2264 \u2191K * dist x y\n[PROOFSTEP]\nsimp only [LipschitzOnWith, edist_nndist, dist_nndist]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nK : \u211d\u22650\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 \u2191(nndist (f x) (f y)) \u2264 \u2191K * \u2191(nndist x y)) \u2194\n \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 \u2191(nndist (f x) (f y)) \u2264 \u2191K * \u2191(nndist x y)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\n\u22a2 LipschitzOnWith K f univ \u2194 LipschitzWith K f\n[PROOFSTEP]\nsimp [LipschitzOnWith, LipschitzWith]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 LipschitzOnWith K f s \u2194 LipschitzWith K (restrict s f)\n[PROOFSTEP]\nsimp only [LipschitzOnWith, LipschitzWith, SetCoe.forall', restrict, Subtype.edist_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx\u271d y\u271d : \u03b1\nr : \u211d\u22650\u221e\nh : LipschitzWith K f\nx y : \u03b1\n\u22a2 (\u2191K)\u207b\u00b9 * edist (f x) (f y) \u2264 edist x y\n[PROOFSTEP]\nrw [mul_comm, \u2190 div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx\u271d y\u271d : \u03b1\nr : \u211d\u22650\u221e\nh : LipschitzWith K f\nx y : \u03b1\n\u22a2 edist (f x) (f y) / \u2191K \u2264 edist x y\n[PROOFSTEP]\nexact ENNReal.div_le_of_le_mul' (h x y)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx\u271d y\u271d : \u03b1\nr : \u211d\u22650\u221e\nh : \u2200 (x y : \u03b1), edist (f x) (f y) \u2264 edist x y\nx y : \u03b1\n\u22a2 edist (f x) (f y) \u2264 \u21911 * edist x y\n[PROOFSTEP]\nsimp only [ENNReal.coe_one, one_mul, h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nhf : LipschitzWith K f\ns : Set \u03b1\n\u22a2 diam (f '' s) \u2264 \u2191K * diam s\n[PROOFSTEP]\napply EMetric.diam_le\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nhf : LipschitzWith K f\ns : Set \u03b1\n\u22a2 \u2200 (x : \u03b2), x \u2208 f '' s \u2192 \u2200 (y : \u03b2), y \u2208 f '' s \u2192 edist x y \u2264 \u2191K * diam s\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9 _ \u27e8y, hy, rfl\u27e9\n[GOAL]\ncase h.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx\u271d y\u271d : \u03b1\nr : \u211d\u22650\u221e\nhf : LipschitzWith K f\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\n\u22a2 edist (f x) (f y) \u2264 \u2191K * diam s\n[PROOFSTEP]\nexact hf.edist_le_mul_of_le (EMetric.edist_le_diam_of_mem hx hy)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx\u271d y\u271d : \u03b1\nr : \u211d\u22650\u221e\nb : \u03b2\nx y : \u03b1\n\u22a2 edist ((fun x => b) x) ((fun x => b) y) \u2264 \u21910 * edist x y\n[PROOFSTEP]\nsimp only [edist_self, zero_le]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx\u271d y\u271d : \u03b1\nr : \u211d\u22650\u221e\nb : \u03b2\nK : \u211d\u22650\nx y : \u03b1\n\u22a2 edist ((fun x => b) x) ((fun x => b) y) \u2264 \u2191K * edist x y\n[PROOFSTEP]\nsimp only [edist_self, zero_le]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u2074 : PseudoEMetricSpace \u03b1\u271d\ninst\u271d\u00b3 : PseudoEMetricSpace \u03b2\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1\u271d \u2192 \u03b2\nx y : \u03b1\u271d\nr : \u211d\u22650\u221e\n\u03b1 : \u03b9 \u2192 Type u\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PseudoEMetricSpace (\u03b1 i)\ninst\u271d : Fintype \u03b9\ni : \u03b9\nf g : (x : \u03b9) \u2192 \u03b1 x\n\u22a2 edist (eval i f) (eval i g) \u2264 edist f g\n[PROOFSTEP]\nconvert edist_le_pi_edist f g i\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx\u271d y\u271d : \u03b1\nr : \u211d\u22650\u221e\nKf Kg : \u211d\u22650\nf : \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b2\nhf : LipschitzWith Kf f\nhg : LipschitzWith Kg g\nx y : \u03b1\n\u22a2 \u2191Kf * (\u2191Kg * edist x y) = \u2191(Kf * Kg) * edist x y\n[PROOFSTEP]\nrw [\u2190 mul_assoc, ENNReal.coe_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nKf : \u211d\u22650\nhf : LipschitzWith Kf f\ng : \u03b1 \u2192 \u03b3\nKg : \u211d\u22650\nhg : LipschitzWith Kg g\n\u22a2 LipschitzWith (max Kf Kg) fun x => (f x, g x)\n[PROOFSTEP]\nintro x y\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx\u271d y\u271d : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nKf : \u211d\u22650\nhf : LipschitzWith Kf f\ng : \u03b1 \u2192 \u03b3\nKg : \u211d\u22650\nhg : LipschitzWith Kg g\nx y : \u03b1\n\u22a2 edist ((fun x => (f x, g x)) x) ((fun x => (f x, g x)) y) \u2264 \u2191(max Kf Kg) * edist x y\n[PROOFSTEP]\nrw [ENNReal.coe_mono.map_max, Prod.edist_eq, ENNReal.max_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx\u271d y\u271d : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nKf : \u211d\u22650\nhf : LipschitzWith Kf f\ng : \u03b1 \u2192 \u03b3\nKg : \u211d\u22650\nhg : LipschitzWith Kg g\nx y : \u03b1\n\u22a2 max (edist ((fun x => (f x, g x)) x).fst ((fun x => (f x, g x)) y).fst)\n (edist ((fun x => (f x, g x)) x).snd ((fun x => (f x, g x)) y).snd) \u2264\n max (\u2191Kf * edist x y) (\u2191Kg * edist x y)\n[PROOFSTEP]\nexact max_le_max (hf x y) (hg x y)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\na : \u03b1\n\u22a2 LipschitzWith 1 (Prod.mk a)\n[PROOFSTEP]\nsimpa only [max_eq_right zero_le_one] using (LipschitzWith.const a).prod LipschitzWith.id\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nb : \u03b2\n\u22a2 LipschitzWith 1 fun a => (a, b)\n[PROOFSTEP]\nsimpa only [max_eq_left zero_le_one] using LipschitzWith.id.prod (LipschitzWith.const b)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nK\u03b1 K\u03b2 : \u211d\u22650\nh\u03b1 : \u2200 (b : \u03b2), LipschitzWith K\u03b1 fun a => f a b\nh\u03b2 : \u2200 (a : \u03b1), LipschitzWith K\u03b2 (f a)\n\u22a2 LipschitzWith (K\u03b1 + K\u03b2) (uncurry f)\n[PROOFSTEP]\nrintro \u27e8a\u2081, b\u2081\u27e9 \u27e8a\u2082, b\u2082\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nK\u03b1 K\u03b2 : \u211d\u22650\nh\u03b1 : \u2200 (b : \u03b2), LipschitzWith K\u03b1 fun a => f a b\nh\u03b2 : \u2200 (a : \u03b1), LipschitzWith K\u03b2 (f a)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\n\u22a2 edist (uncurry f (a\u2081, b\u2081)) (uncurry f (a\u2082, b\u2082)) \u2264 \u2191(K\u03b1 + K\u03b2) * edist (a\u2081, b\u2081) (a\u2082, b\u2082)\n[PROOFSTEP]\nsimp only [Function.uncurry, ENNReal.coe_add, add_mul]\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nK\u03b1 K\u03b2 : \u211d\u22650\nh\u03b1 : \u2200 (b : \u03b2), LipschitzWith K\u03b1 fun a => f a b\nh\u03b2 : \u2200 (a : \u03b1), LipschitzWith K\u03b2 (f a)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\n\u22a2 edist (f a\u2081 b\u2081) (f a\u2082 b\u2082) \u2264 \u2191K\u03b1 * edist (a\u2081, b\u2081) (a\u2082, b\u2082) + \u2191K\u03b2 * edist (a\u2081, b\u2081) (a\u2082, b\u2082)\n[PROOFSTEP]\napply le_trans (edist_triangle _ (f a\u2082 b\u2081) _)\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nK\u03b1 K\u03b2 : \u211d\u22650\nh\u03b1 : \u2200 (b : \u03b2), LipschitzWith K\u03b1 fun a => f a b\nh\u03b2 : \u2200 (a : \u03b1), LipschitzWith K\u03b2 (f a)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\n\u22a2 edist (f a\u2081 b\u2081) (f a\u2082 b\u2081) + edist (f a\u2082 b\u2081) (f a\u2082 b\u2082) \u2264 \u2191K\u03b1 * edist (a\u2081, b\u2081) (a\u2082, b\u2082) + \u2191K\u03b2 * edist (a\u2081, b\u2081) (a\u2082, b\u2082)\n[PROOFSTEP]\nexact\n add_le_add (le_trans (h\u03b1 _ _ _) <| ENNReal.mul_left_mono <| le_max_left _ _)\n (le_trans (h\u03b2 _ _ _) <| ENNReal.mul_left_mono <| le_max_right _ _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b1\nhf : LipschitzWith K f\n\u22a2 LipschitzWith (K ^ 0) f^[0]\n[PROOFSTEP]\nsimpa only [pow_zero] using LipschitzWith.id\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b1\nhf : LipschitzWith K f\nn : \u2115\n\u22a2 LipschitzWith (K ^ (n + 1)) f^[n + 1]\n[PROOFSTEP]\nrw [pow_succ']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b1\nhf : LipschitzWith K f\nn : \u2115\n\u22a2 LipschitzWith (K ^ n * K) f^[n + 1]\n[PROOFSTEP]\nexact (LipschitzWith.iterate hf n).comp hf\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx\u271d y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b1\nhf : LipschitzWith K f\nx : \u03b1\nn : \u2115\n\u22a2 edist (f^[n] x) (f^[n + 1] x) \u2264 edist x (f x) * \u2191K ^ n\n[PROOFSTEP]\nrw [iterate_succ, mul_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx\u271d y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b1\nhf : LipschitzWith K f\nx : \u03b1\nn : \u2115\n\u22a2 edist (f^[n] x) ((f^[n] \u2218 f) x) \u2264 \u2191K ^ n * edist x (f x)\n[PROOFSTEP]\nsimpa only [ENNReal.coe_pow] using (hf.iterate n) x (f x)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b9 \u2192 Function.End \u03b1\nK : \u03b9 \u2192 \u211d\u22650\nh : \u2200 (i : \u03b9), LipschitzWith (K i) (f i)\n\u22a2 LipschitzWith (List.prod (List.map K [])) (List.prod (List.map f []))\n[PROOFSTEP]\nsimpa using LipschitzWith.id\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b9 \u2192 Function.End \u03b1\nK : \u03b9 \u2192 \u211d\u22650\nh : \u2200 (i : \u03b9), LipschitzWith (K i) (f i)\ni : \u03b9\nl : List \u03b9\n\u22a2 LipschitzWith (List.prod (List.map K (i :: l))) (List.prod (List.map f (i :: l)))\n[PROOFSTEP]\nsimp only [List.map_cons, List.prod_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b9 \u2192 Function.End \u03b1\nK : \u03b9 \u2192 \u211d\u22650\nh : \u2200 (i : \u03b9), LipschitzWith (K i) (f i)\ni : \u03b9\nl : List \u03b9\n\u22a2 LipschitzWith (K i * List.prod (List.map K l)) (f i * List.prod (List.map f l))\n[PROOFSTEP]\nexact (h i).mul (LipschitzWith.list_prod f K h l)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : Function.End \u03b1\nK : \u211d\u22650\nh : LipschitzWith K f\n\u22a2 LipschitzWith (K ^ 0) (f ^ 0)\n[PROOFSTEP]\nsimpa only [pow_zero] using LipschitzWith.id\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : Function.End \u03b1\nK : \u211d\u22650\nh : LipschitzWith K f\nn : \u2115\n\u22a2 LipschitzWith (K ^ (n + 1)) (f ^ (n + 1))\n[PROOFSTEP]\nrw [pow_succ, pow_succ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : Function.End \u03b1\nK : \u211d\u22650\nh : LipschitzWith K f\nn : \u2115\n\u22a2 LipschitzWith (K * K ^ n) (f * f ^ n)\n[PROOFSTEP]\nexact h.mul (LipschitzWith.pow h n)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx\u271d y\u271d : \u03b1\nr K : \u211d\nh : \u2200 (x y : \u03b1), dist (f x) (f y) \u2264 K * dist x y\nx y : \u03b1\n\u22a2 K * dist x y \u2264 \u2191(Real.toNNReal K) * dist x y\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx\u271d y\u271d : \u03b1\nr K : \u211d\nh : \u2200 (x y : \u03b1), dist (f x) (f y) \u2264 K * dist x y\nx y : \u03b1\n\u22a2 K \u2264 \u2191(Real.toNNReal K)\n[PROOFSTEP]\napply Real.le_coe_toNNReal\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\nh : \u2200 (x y : \u03b1), dist (f x) (f y) \u2264 dist x y\n\u22a2 \u2200 (x y : \u03b1), dist (f x) (f y) \u2264 \u21911 * dist x y\n[PROOFSTEP]\nsimpa only [NNReal.coe_one, one_mul] using h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\nf : \u03b1 \u2192 \u211d\nK : \u211d\u22650\nh : \u2200 (x y : \u03b1), f x \u2264 f y + \u2191K * dist x y\n\u22a2 LipschitzWith K f\n[PROOFSTEP]\nsimpa only [Real.toNNReal_coe] using LipschitzWith.of_le_add_mul' K h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\nf : \u03b1 \u2192 \u211d\nh : \u2200 (x y : \u03b1), f x \u2264 f y + dist x y\n\u22a2 \u2200 (x y : \u03b1), f x \u2264 f y + \u21911 * dist x y\n[PROOFSTEP]\nsimpa only [NNReal.coe_one, one_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\nhf : LipschitzWith K f\nhr : dist x y \u2264 r\n\u22a2 \u2191K * dist x y \u2264 \u2191K * r\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\n\u22a2 LipschitzWith 2 (uncurry dist)\n[PROOFSTEP]\nrw [\u2190 one_add_one_eq_two]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\n\u22a2 LipschitzWith (1 + 1) (uncurry dist)\n[PROOFSTEP]\nexact LipschitzWith.uncurry LipschitzWith.dist_left LipschitzWith.dist_right\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx\u271d y : \u03b1\nr : \u211d\nf : \u03b1 \u2192 \u03b1\nhf : LipschitzWith K f\nx : \u03b1\nn : \u2115\n\u22a2 dist (f^[n] x) (f^[n + 1] x) \u2264 dist x (f x) * \u2191K ^ n\n[PROOFSTEP]\nrw [iterate_succ, mul_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx\u271d y : \u03b1\nr : \u211d\nf : \u03b1 \u2192 \u03b1\nhf : LipschitzWith K f\nx : \u03b1\nn : \u2115\n\u22a2 dist (f^[n] x) ((f^[n] \u2218 f) x) \u2264 \u2191K ^ n * dist x (f x)\n[PROOFSTEP]\nsimpa only [NNReal.coe_pow] using (hf.iterate n).dist_le_mul x (f x)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoEMetricSpace \u03b1\nf g : \u03b1 \u2192 \u211d\nKf Kg : \u211d\u22650\nhf : LipschitzWith Kf f\nhg : LipschitzWith Kg g\n\u22a2 LipschitzWith (max Kf Kg) fun x => max (f x) (g x)\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), one_mul] using lipschitzWith_max.comp (hf.prod hg)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoEMetricSpace \u03b1\nf g : \u03b1 \u2192 \u211d\nKf Kg : \u211d\u22650\nhf : LipschitzWith Kf f\nhg : LipschitzWith Kg g\n\u22a2 LipschitzWith (max Kf Kg) fun x => min (f x) (g x)\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), one_mul] using lipschitzWith_min.comp (hf.prod hg)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoEMetricSpace \u03b1\nf g : \u03b1 \u2192 \u211d\nKf Kg : \u211d\u22650\nhf : LipschitzWith Kf f\na : \u211d\n\u22a2 LipschitzWith Kf fun x => max (f x) a\n[PROOFSTEP]\nsimpa only [max_eq_left (zero_le Kf)] using hf.max (LipschitzWith.const a)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoEMetricSpace \u03b1\nf g : \u03b1 \u2192 \u211d\nKf Kg : \u211d\u22650\nhf : LipschitzWith Kf f\na : \u211d\n\u22a2 LipschitzWith Kf fun x => max a (f x)\n[PROOFSTEP]\nsimpa only [max_comm] using hf.max_const a\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoEMetricSpace \u03b1\nf g : \u03b1 \u2192 \u211d\nKf Kg : \u211d\u22650\nhf : LipschitzWith Kf f\na : \u211d\n\u22a2 LipschitzWith Kf fun x => min (f x) a\n[PROOFSTEP]\nsimpa only [max_eq_left (zero_le Kf)] using hf.min (LipschitzWith.const a)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoEMetricSpace \u03b1\nf g : \u03b1 \u2192 \u211d\nKf Kg : \u211d\u22650\nhf : LipschitzWith Kf f\na : \u211d\n\u22a2 LipschitzWith Kf fun x => min a (f x)\n[PROOFSTEP]\nsimpa only [min_comm] using hf.min_const a\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : Bounded (s \u00d7\u02e2 t)\nht : Set.Nonempty t\n\u22a2 Bounded s\n[PROOFSTEP]\nsimpa only [fst_image_prod s ht] using (@LipschitzWith.prod_fst \u03b1 \u03b2 _ _).bounded_image h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : Bounded (s \u00d7\u02e2 t)\nhs : Set.Nonempty s\n\u22a2 Bounded t\n[PROOFSTEP]\nsimpa only [snd_image_prod hs t] using (@LipschitzWith.prod_snd \u03b1 \u03b2 _ _).bounded_image h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 Bounded (s \u00d7\u02e2 t) \u2194 s = \u2205 \u2228 t = \u2205 \u2228 Bounded s \u2227 Bounded t\n[PROOFSTEP]\nsimp only [bounded_iff_isBounded, Bornology.isBounded_prod]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK\u271d : \u211d\u22650\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nK : \u211d\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (f x) (f y) \u2264 K * dist x y\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\n\u22a2 K * dist x y \u2264 \u2191(Real.toNNReal K) * dist x y\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK\u271d : \u211d\u22650\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nK : \u211d\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (f x) (f y) \u2264 K * dist x y\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\n\u22a2 K \u2264 \u2191(Real.toNNReal K)\n[PROOFSTEP]\napply Real.le_coe_toNNReal\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK : \u211d\u22650\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (f x) (f y) \u2264 dist x y\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (f x) (f y) \u2264 \u21911 * dist x y\n[PROOFSTEP]\nsimpa only [NNReal.coe_one, one_mul] using h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK\u271d : \u211d\u22650\ns : Set \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u211d\nK : \u211d\u22650\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 f x \u2264 f y + \u2191K * dist x y\n\u22a2 LipschitzOnWith K f s\n[PROOFSTEP]\nsimpa only [Real.toNNReal_coe] using LipschitzOnWith.of_le_add_mul' K h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK : \u211d\u22650\ns : Set \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u211d\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 f x \u2264 f y + dist x y\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 f x \u2264 f y + \u21911 * dist x y\n[PROOFSTEP]\nsimpa only [NNReal.coe_one, one_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\n\u22a2 ContinuousOn f (s \u00d7\u02e2 t)\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9 \u27e8hx : x \u2208 s, hy : y \u2208 t\u27e9\n[GOAL]\ncase mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u22a2 ContinuousWithinAt f (s \u00d7\u02e2 t) (x, y)\n[PROOFSTEP]\nrefine' EMetric.nhds_basis_closed_eball.tendsto_right_iff.2 fun \u03b5 (\u03b50 : 0 < \u03b5) => _\n[GOAL]\ncase mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x_1 : \u03b1 \u00d7 \u03b2) in \ud835\udcdd[s \u00d7\u02e2 t] (x, y), f x_1 \u2208 EMetric.closedBall (f (x, y)) \u03b5\n[PROOFSTEP]\nreplace \u03b50 : 0 < \u03b5 / 2 := ENNReal.half_pos \u03b50.ne'\n[GOAL]\ncase mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5 / 2\n\u22a2 \u2200\u1da0 (x_1 : \u03b1 \u00d7 \u03b2) in \ud835\udcdd[s \u00d7\u02e2 t] (x, y), f x_1 \u2208 EMetric.closedBall (f (x, y)) \u03b5\n[PROOFSTEP]\nobtain \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9 : \u2203 \u03b4 : \u211d\u22650, 0 < \u03b4 \u2227 (\u03b4 : \u211d\u22650\u221e) * \u2191(3 * K) < \u03b5 / 2 :=\n ENNReal.exists_nnreal_pos_mul_lt ENNReal.coe_ne_top \u03b50.ne'\n[GOAL]\ncase mk.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5 / 2\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191\u03b4 * \u2191(3 * K) < \u03b5 / 2\n\u22a2 \u2200\u1da0 (x_1 : \u03b1 \u00d7 \u03b2) in \ud835\udcdd[s \u00d7\u02e2 t] (x, y), f x_1 \u2208 EMetric.closedBall (f (x, y)) \u03b5\n[PROOFSTEP]\nrw [\u2190 ENNReal.coe_pos] at \u03b4pos \n[GOAL]\ncase mk.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5 / 2\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 : \u2191\u03b4 * \u2191(3 * K) < \u03b5 / 2\n\u22a2 \u2200\u1da0 (x_1 : \u03b1 \u00d7 \u03b2) in \ud835\udcdd[s \u00d7\u02e2 t] (x, y), f x_1 \u2208 EMetric.closedBall (f (x, y)) \u03b5\n[PROOFSTEP]\nrcases EMetric.mem_closure_iff.1 (hss' hx) \u03b4 \u03b4pos with \u27e8x', hx', hxx'\u27e9\n[GOAL]\ncase mk.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5 / 2\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 : \u2191\u03b4 * \u2191(3 * K) < \u03b5 / 2\nx' : \u03b1\nhx' : x' \u2208 s'\nhxx' : edist x x' < \u2191\u03b4\n\u22a2 \u2200\u1da0 (x_1 : \u03b1 \u00d7 \u03b2) in \ud835\udcdd[s \u00d7\u02e2 t] (x, y), f x_1 \u2208 EMetric.closedBall (f (x, y)) \u03b5\n[PROOFSTEP]\nhave A : s \u2229 EMetric.ball x \u03b4 \u2208 \ud835\udcdd[s] x := inter_mem_nhdsWithin _ (EMetric.ball_mem_nhds _ \u03b4pos)\n[GOAL]\ncase mk.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5 / 2\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 : \u2191\u03b4 * \u2191(3 * K) < \u03b5 / 2\nx' : \u03b1\nhx' : x' \u2208 s'\nhxx' : edist x x' < \u2191\u03b4\nA : s \u2229 EMetric.ball x \u2191\u03b4 \u2208 \ud835\udcdd[s] x\n\u22a2 \u2200\u1da0 (x_1 : \u03b1 \u00d7 \u03b2) in \ud835\udcdd[s \u00d7\u02e2 t] (x, y), f x_1 \u2208 EMetric.closedBall (f (x, y)) \u03b5\n[PROOFSTEP]\nhave B : t \u2229 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2} \u2208 \ud835\udcdd[t] y :=\n inter_mem self_mem_nhdsWithin (ha x' hx' y hy (EMetric.closedBall_mem_nhds (f (x', y)) \u03b50))\n[GOAL]\ncase mk.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5 / 2\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 : \u2191\u03b4 * \u2191(3 * K) < \u03b5 / 2\nx' : \u03b1\nhx' : x' \u2208 s'\nhxx' : edist x x' < \u2191\u03b4\nA : s \u2229 EMetric.ball x \u2191\u03b4 \u2208 \ud835\udcdd[s] x\nB : t \u2229 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2} \u2208 \ud835\udcdd[t] y\n\u22a2 \u2200\u1da0 (x_1 : \u03b1 \u00d7 \u03b2) in \ud835\udcdd[s \u00d7\u02e2 t] (x, y), f x_1 \u2208 EMetric.closedBall (f (x, y)) \u03b5\n[PROOFSTEP]\nfilter_upwards [nhdsWithin_prod A B] with \u27e8a, b\u27e9 \u27e8\u27e8has, hax\u27e9, \u27e8hbt, hby\u27e9\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5 / 2\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 : \u2191\u03b4 * \u2191(3 * K) < \u03b5 / 2\nx' : \u03b1\nhx' : x' \u2208 s'\nhxx' : edist x x' < \u2191\u03b4\nA : s \u2229 EMetric.ball x \u2191\u03b4 \u2208 \ud835\udcdd[s] x\nB : t \u2229 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2} \u2208 \ud835\udcdd[t] y\na : \u03b1\nb : \u03b2\nhas : (a, b).fst \u2208 s\nhax : (a, b).fst \u2208 EMetric.ball x \u2191\u03b4\nhbt : (a, b).snd \u2208 t\nhby : (a, b).snd \u2208 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2}\n\u22a2 f (a, b) \u2208 EMetric.closedBall (f (x, y)) \u03b5\n[PROOFSTEP]\ncalc\n edist (f (a, b)) (f (x, y)) \u2264\n edist (f (a, b)) (f (x', b)) + edist (f (x', b)) (f (x', y)) + edist (f (x', y)) (f (x, y)) :=\n edist_triangle4 _ _ _ _\n _ \u2264 K * (\u03b4 + \u03b4) + \u03b5 / 2 + K * \u03b4 := by\n gcongr\n \u00b7 refine (hb b hbt).edist_le_mul_of_le has (hs' hx') ?_\n refine (edist_triangle _ _ _).trans (add_le_add (le_of_lt hax) hxx'.le)\n \u00b7 exact hby\n \u00b7 exact (hb y hy).edist_le_mul_of_le (hs' hx') hx ((edist_comm _ _).trans_le hxx'.le)\n _ = \u03b4 * \u2191(3 * K) + \u03b5 / 2 := by push_cast ; ring\n _ \u2264 \u03b5 / 2 + \u03b5 / 2 := by gcongr\n _ = \u03b5 := ENNReal.add_halves _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5 / 2\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 : \u2191\u03b4 * \u2191(3 * K) < \u03b5 / 2\nx' : \u03b1\nhx' : x' \u2208 s'\nhxx' : edist x x' < \u2191\u03b4\nA : s \u2229 EMetric.ball x \u2191\u03b4 \u2208 \ud835\udcdd[s] x\nB : t \u2229 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2} \u2208 \ud835\udcdd[t] y\na : \u03b1\nb : \u03b2\nhas : (a, b).fst \u2208 s\nhax : (a, b).fst \u2208 EMetric.ball x \u2191\u03b4\nhbt : (a, b).snd \u2208 t\nhby : (a, b).snd \u2208 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2}\n\u22a2 edist (f (a, b)) (f (x', b)) + edist (f (x', b)) (f (x', y)) + edist (f (x', y)) (f (x, y)) \u2264\n \u2191K * (\u2191\u03b4 + \u2191\u03b4) + \u03b5 / 2 + \u2191K * \u2191\u03b4\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081.h\u2081\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5 / 2\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 : \u2191\u03b4 * \u2191(3 * K) < \u03b5 / 2\nx' : \u03b1\nhx' : x' \u2208 s'\nhxx' : edist x x' < \u2191\u03b4\nA : s \u2229 EMetric.ball x \u2191\u03b4 \u2208 \ud835\udcdd[s] x\nB : t \u2229 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2} \u2208 \ud835\udcdd[t] y\na : \u03b1\nb : \u03b2\nhas : (a, b).fst \u2208 s\nhax : (a, b).fst \u2208 EMetric.ball x \u2191\u03b4\nhbt : (a, b).snd \u2208 t\nhby : (a, b).snd \u2208 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2}\n\u22a2 edist (f (a, b)) (f (x', b)) \u2264 \u2191K * (\u2191\u03b4 + \u2191\u03b4)\n[PROOFSTEP]\nrefine (hb b hbt).edist_le_mul_of_le has (hs' hx') ?_\n[GOAL]\ncase h\u2081.h\u2081\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5 / 2\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 : \u2191\u03b4 * \u2191(3 * K) < \u03b5 / 2\nx' : \u03b1\nhx' : x' \u2208 s'\nhxx' : edist x x' < \u2191\u03b4\nA : s \u2229 EMetric.ball x \u2191\u03b4 \u2208 \ud835\udcdd[s] x\nB : t \u2229 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2} \u2208 \ud835\udcdd[t] y\na : \u03b1\nb : \u03b2\nhas : (a, b).fst \u2208 s\nhax : (a, b).fst \u2208 EMetric.ball x \u2191\u03b4\nhbt : (a, b).snd \u2208 t\nhby : (a, b).snd \u2208 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2}\n\u22a2 edist a x' \u2264 \u2191\u03b4 + \u2191\u03b4\n[PROOFSTEP]\nrefine (edist_triangle _ _ _).trans (add_le_add (le_of_lt hax) hxx'.le)\n[GOAL]\ncase h\u2081.h\u2082\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5 / 2\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 : \u2191\u03b4 * \u2191(3 * K) < \u03b5 / 2\nx' : \u03b1\nhx' : x' \u2208 s'\nhxx' : edist x x' < \u2191\u03b4\nA : s \u2229 EMetric.ball x \u2191\u03b4 \u2208 \ud835\udcdd[s] x\nB : t \u2229 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2} \u2208 \ud835\udcdd[t] y\na : \u03b1\nb : \u03b2\nhas : (a, b).fst \u2208 s\nhax : (a, b).fst \u2208 EMetric.ball x \u2191\u03b4\nhbt : (a, b).snd \u2208 t\nhby : (a, b).snd \u2208 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2}\n\u22a2 edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2\n[PROOFSTEP]\nexact hby\n[GOAL]\ncase h\u2082\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5 / 2\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 : \u2191\u03b4 * \u2191(3 * K) < \u03b5 / 2\nx' : \u03b1\nhx' : x' \u2208 s'\nhxx' : edist x x' < \u2191\u03b4\nA : s \u2229 EMetric.ball x \u2191\u03b4 \u2208 \ud835\udcdd[s] x\nB : t \u2229 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2} \u2208 \ud835\udcdd[t] y\na : \u03b1\nb : \u03b2\nhas : (a, b).fst \u2208 s\nhax : (a, b).fst \u2208 EMetric.ball x \u2191\u03b4\nhbt : (a, b).snd \u2208 t\nhby : (a, b).snd \u2208 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2}\n\u22a2 edist (f (x', y)) (f (x, y)) \u2264 \u2191K * \u2191\u03b4\n[PROOFSTEP]\nexact (hb y hy).edist_le_mul_of_le (hs' hx') hx ((edist_comm _ _).trans_le hxx'.le)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5 / 2\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 : \u2191\u03b4 * \u2191(3 * K) < \u03b5 / 2\nx' : \u03b1\nhx' : x' \u2208 s'\nhxx' : edist x x' < \u2191\u03b4\nA : s \u2229 EMetric.ball x \u2191\u03b4 \u2208 \ud835\udcdd[s] x\nB : t \u2229 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2} \u2208 \ud835\udcdd[t] y\na : \u03b1\nb : \u03b2\nhas : (a, b).fst \u2208 s\nhax : (a, b).fst \u2208 EMetric.ball x \u2191\u03b4\nhbt : (a, b).snd \u2208 t\nhby : (a, b).snd \u2208 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2}\n\u22a2 \u2191K * (\u2191\u03b4 + \u2191\u03b4) + \u03b5 / 2 + \u2191K * \u2191\u03b4 = \u2191\u03b4 * \u2191(3 * K) + \u03b5 / 2\n[PROOFSTEP]\npush_cast\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5 / 2\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 : \u2191\u03b4 * \u2191(3 * K) < \u03b5 / 2\nx' : \u03b1\nhx' : x' \u2208 s'\nhxx' : edist x x' < \u2191\u03b4\nA : s \u2229 EMetric.ball x \u2191\u03b4 \u2208 \ud835\udcdd[s] x\nB : t \u2229 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2} \u2208 \ud835\udcdd[t] y\na : \u03b1\nb : \u03b2\nhas : (a, b).fst \u2208 s\nhax : (a, b).fst \u2208 EMetric.ball x \u2191\u03b4\nhbt : (a, b).snd \u2208 t\nhby : (a, b).snd \u2208 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2}\n\u22a2 \u2191K * (\u2191\u03b4 + \u2191\u03b4) + \u03b5 / 2 + \u2191K * \u2191\u03b4 = \u2191\u03b4 * (3 * \u2191K) + \u03b5 / 2\n[PROOFSTEP]\nring\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\ns s' : Set \u03b1\nt : Set \u03b2\nhs' : s' \u2286 s\nhss' : s \u2286 closure s'\nK : \u211d\u22650\nha : \u2200 (a : \u03b1), a \u2208 s' \u2192 ContinuousOn (fun y => f (a, y)) t\nhb : \u2200 (b : \u03b2), b \u2208 t \u2192 LipschitzOnWith K (fun x => f (x, b)) s\nx : \u03b1\ny : \u03b2\nhx : x \u2208 s\nhy : y \u2208 t\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5 / 2\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u2191\u03b4\nh\u03b4 : \u2191\u03b4 * \u2191(3 * K) < \u03b5 / 2\nx' : \u03b1\nhx' : x' \u2208 s'\nhxx' : edist x x' < \u2191\u03b4\nA : s \u2229 EMetric.ball x \u2191\u03b4 \u2208 \ud835\udcdd[s] x\nB : t \u2229 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2} \u2208 \ud835\udcdd[t] y\na : \u03b1\nb : \u03b2\nhas : (a, b).fst \u2208 s\nhax : (a, b).fst \u2208 EMetric.ball x \u2191\u03b4\nhbt : (a, b).snd \u2208 t\nhby : (a, b).snd \u2208 {b | edist (f (x', b)) (f (x', y)) \u2264 \u03b5 / 2}\n\u22a2 \u2191\u03b4 * \u2191(3 * K) + \u03b5 / 2 \u2264 \u03b5 / 2 + \u03b5 / 2\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\nK : \u211d\u22650\ns : Set \u03b1\nhs : Dense s\nha : \u2200 (a : \u03b1), a \u2208 s \u2192 Continuous fun y => f (a, y)\nhb : \u2200 (b : \u03b2), LipschitzWith K fun x => f (x, b)\n\u22a2 Continuous f\n[PROOFSTEP]\nsimp only [continuous_iff_continuousOn_univ, \u2190 univ_prod_univ, \u2190 lipschitz_on_univ] at *\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\nK : \u211d\u22650\ns : Set \u03b1\nhs : Dense s\nha : \u2200 (a : \u03b1), a \u2208 s \u2192 ContinuousOn (fun y => f (a, y)) univ\nhb : \u2200 (b : \u03b2), LipschitzOnWith K (fun x => f (x, b)) univ\n\u22a2 ContinuousOn f (univ \u00d7\u02e2 univ)\n[PROOFSTEP]\nexact\n continuousOn_prod_of_subset_closure_continuousOn_lipschitzOnWith f (subset_univ _) hs.closure_eq.ge K ha fun b _ =>\n hb b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\nr : \u211d\nhr : 0 < r\nK : \u211d\nh : \u2200 (y : \u03b1), dist y x < r \u2192 dist (f y) (f x) \u2264 K * dist y x\n\u22a2 ContinuousAt f x\n[PROOFSTEP]\nrefine\n tendsto_iff_dist_tendsto_zero.2\n (squeeze_zero' (eventually_of_forall fun _ => dist_nonneg) (mem_of_superset (ball_mem_nhds _ hr) h) ?_)\n -- Then show that `K * dist y x` tends to zero as `y \u2192 x`\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\nr : \u211d\nhr : 0 < r\nK : \u211d\nh : \u2200 (y : \u03b1), dist y x < r \u2192 dist (f y) (f x) \u2264 K * dist y x\n\u22a2 Tendsto (fun a => K * dist a x) (\ud835\udcdd x) (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine (continuous_const.mul (continuous_id.dist continuous_const)).tendsto' _ _ ?_\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\nr : \u211d\nhr : 0 < r\nK : \u211d\nh : \u2200 (y : \u03b1), dist y x < r \u2192 dist (f y) (f x) \u2264 K * dist y x\n\u22a2 K * dist (id x) x = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn f g s\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | hs)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\nK : \u211d\u22650\nhf : LipschitzOnWith K f \u2205\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn f g \u2205\n[PROOFSTEP]\nexact \u27e8fun _ => 0, (LipschitzWith.const _).weaken (zero_le _), eqOn_empty _ _\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn f g s\n[PROOFSTEP]\nhave : Nonempty s := by simp only [hs, nonempty_coe_sort]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\n\u22a2 Nonempty \u2191s\n[PROOFSTEP]\nsimp only [hs, nonempty_coe_sort]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\nthis : Nonempty \u2191s\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn f g s\n[PROOFSTEP]\nlet g := fun y : \u03b1 => iInf fun x : s => f x + K * dist y x\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn f g s\n[PROOFSTEP]\nhave B : \u2200 y : \u03b1, BddBelow (range fun x : s => f x + K * dist y x) := fun y =>\n by\n rcases hs with \u27e8z, hz\u27e9\n refine' \u27e8f z - K * dist y z, _\u27e9\n rintro w \u27e8t, rfl\u27e9\n dsimp\n rw [sub_le_iff_le_add, add_assoc, \u2190 mul_add, add_comm (dist y t)]\n calc\n f z \u2264 f t + K * dist z t := hf.le_add_mul hz t.2\n _ \u2264 f t + K * (dist y z + dist y t) := by gcongr; apply dist_triangle_left\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\ny : \u03b1\n\u22a2 BddBelow (range fun x => f \u2191x + \u2191K * dist y \u2191x)\n[PROOFSTEP]\nrcases hs with \u27e8z, hz\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\ny z : \u03b1\nhz : z \u2208 s\n\u22a2 BddBelow (range fun x => f \u2191x + \u2191K * dist y \u2191x)\n[PROOFSTEP]\nrefine' \u27e8f z - K * dist y z, _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\ny z : \u03b1\nhz : z \u2208 s\n\u22a2 f z - \u2191K * dist y z \u2208 lowerBounds (range fun x => f \u2191x + \u2191K * dist y \u2191x)\n[PROOFSTEP]\nrintro w \u27e8t, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\ny z : \u03b1\nhz : z \u2208 s\nt : \u2191s\n\u22a2 f z - \u2191K * dist y z \u2264 (fun x => f \u2191x + \u2191K * dist y \u2191x) t\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\ny z : \u03b1\nhz : z \u2208 s\nt : \u2191s\n\u22a2 f z - \u2191K * dist y z \u2264 f \u2191t + \u2191K * dist y \u2191t\n[PROOFSTEP]\nrw [sub_le_iff_le_add, add_assoc, \u2190 mul_add, add_comm (dist y t)]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\ny z : \u03b1\nhz : z \u2208 s\nt : \u2191s\n\u22a2 f z \u2264 f \u2191t + \u2191K * (dist y z + dist y \u2191t)\n[PROOFSTEP]\ncalc\n f z \u2264 f t + K * dist z t := hf.le_add_mul hz t.2\n _ \u2264 f t + K * (dist y z + dist y t) := by gcongr; apply dist_triangle_left\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\ny z : \u03b1\nhz : z \u2208 s\nt : \u2191s\n\u22a2 f \u2191t + \u2191K * dist z \u2191t \u2264 f \u2191t + \u2191K * (dist y z + dist y \u2191t)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase bc.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\ny z : \u03b1\nhz : z \u2208 s\nt : \u2191s\n\u22a2 dist z \u2191t \u2264 dist y z + dist y \u2191t\n[PROOFSTEP]\napply dist_triangle_left\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\nB : \u2200 (y : \u03b1), BddBelow (range fun x => f \u2191x + \u2191K * dist y \u2191x)\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn f g s\n[PROOFSTEP]\nhave E : EqOn f g s := fun x hx =>\n by\n refine' le_antisymm (le_ciInf fun y => hf.le_add_mul hx y.2) _\n simpa only [add_zero, Subtype.coe_mk, mul_zero, dist_self] using ciInf_le (B x) \u27e8x, hx\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\nB : \u2200 (y : \u03b1), BddBelow (range fun x => f \u2191x + \u2191K * dist y \u2191x)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 f x = g x\n[PROOFSTEP]\nrefine' le_antisymm (le_ciInf fun y => hf.le_add_mul hx y.2) _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\nB : \u2200 (y : \u03b1), BddBelow (range fun x => f \u2191x + \u2191K * dist y \u2191x)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 g x \u2264 f x\n[PROOFSTEP]\nsimpa only [add_zero, Subtype.coe_mk, mul_zero, dist_self] using ciInf_le (B x) \u27e8x, hx\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\nB : \u2200 (y : \u03b1), BddBelow (range fun x => f \u2191x + \u2191K * dist y \u2191x)\nE : EqOn f g s\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn f g s\n[PROOFSTEP]\nrefine' \u27e8g, LipschitzWith.of_le_add_mul K fun x y => _, E\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\nB : \u2200 (y : \u03b1), BddBelow (range fun x => f \u2191x + \u2191K * dist y \u2191x)\nE : EqOn f g s\nx y : \u03b1\n\u22a2 g x \u2264 g y + \u2191K * dist x y\n[PROOFSTEP]\nrw [\u2190 sub_le_iff_le_add]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\nB : \u2200 (y : \u03b1), BddBelow (range fun x => f \u2191x + \u2191K * dist y \u2191x)\nE : EqOn f g s\nx y : \u03b1\n\u22a2 g x - \u2191K * dist x y \u2264 g y\n[PROOFSTEP]\nrefine' le_ciInf fun z => _\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\nB : \u2200 (y : \u03b1), BddBelow (range fun x => f \u2191x + \u2191K * dist y \u2191x)\nE : EqOn f g s\nx y : \u03b1\nz : \u2191s\n\u22a2 g x - \u2191K * dist x y \u2264 f \u2191z + \u2191K * dist y \u2191z\n[PROOFSTEP]\nrw [sub_le_iff_le_add]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\nB : \u2200 (y : \u03b1), BddBelow (range fun x => f \u2191x + \u2191K * dist y \u2191x)\nE : EqOn f g s\nx y : \u03b1\nz : \u2191s\n\u22a2 g x \u2264 f \u2191z + \u2191K * dist y \u2191z + \u2191K * dist x y\n[PROOFSTEP]\ncalc\n g x \u2264 f z + K * dist x z := ciInf_le (B x) _\n _ \u2264 f z + K * dist y z + K * dist x y :=\n by\n rw [add_assoc, \u2190 mul_add, add_comm (dist y z)]\n gcongr\n apply dist_triangle\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\nB : \u2200 (y : \u03b1), BddBelow (range fun x => f \u2191x + \u2191K * dist y \u2191x)\nE : EqOn f g s\nx y : \u03b1\nz : \u2191s\n\u22a2 f \u2191z + \u2191K * dist x \u2191z \u2264 f \u2191z + \u2191K * dist y \u2191z + \u2191K * dist x y\n[PROOFSTEP]\nrw [add_assoc, \u2190 mul_add, add_comm (dist y z)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\nB : \u2200 (y : \u03b1), BddBelow (range fun x => f \u2191x + \u2191K * dist y \u2191x)\nE : EqOn f g s\nx y : \u03b1\nz : \u2191s\n\u22a2 f \u2191z + \u2191K * dist x \u2191z \u2264 f \u2191z + \u2191K * (dist x y + dist y \u2191z)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase bc.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nhs : Set.Nonempty s\nthis : Nonempty \u2191s\ng : \u03b1 \u2192 \u211d := fun y => \u2a05 (x : \u2191s), f \u2191x + \u2191K * dist y \u2191x\nB : \u2200 (y : \u03b1), BddBelow (range fun x => f \u2191x + \u2191K * dist y \u2191x)\nE : EqOn f g s\nx y : \u03b1\nz : \u2191s\n\u22a2 dist x \u2191z \u2264 dist x y + dist y \u2191z\n[PROOFSTEP]\napply dist_triangle\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn f g s\n[PROOFSTEP]\nhave : \u2200 i, \u2203 g : \u03b1 \u2192 \u211d, LipschitzWith K g \u2227 EqOn (fun x => f x i) g s := fun i =>\n by\n have : LipschitzOnWith K (fun x : \u03b1 => f x i) s :=\n LipschitzOnWith.of_dist_le_mul fun x hx y hy => (dist_le_pi_dist _ _ i).trans (hf.dist_le_mul x hx y hy)\n exact this.extend_real\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ni : \u03b9\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn (fun x => f x i) g s\n[PROOFSTEP]\nhave : LipschitzOnWith K (fun x : \u03b1 => f x i) s :=\n LipschitzOnWith.of_dist_le_mul fun x hx y hy => (dist_le_pi_dist _ _ i).trans (hf.dist_le_mul x hx y hy)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ni : \u03b9\nthis : LipschitzOnWith K (fun x => f x i) s\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn (fun x => f x i) g s\n[PROOFSTEP]\nexact this.extend_real\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\nthis : \u2200 (i : \u03b9), \u2203 g, LipschitzWith K g \u2227 EqOn (fun x => f x i) g s\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn f g s\n[PROOFSTEP]\nchoose g hg using this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhg : \u2200 (i : \u03b9), LipschitzWith K (g i) \u2227 EqOn (fun x => f x i) (g i) s\n\u22a2 \u2203 g, LipschitzWith K g \u2227 EqOn f g s\n[PROOFSTEP]\nrefine' \u27e8fun x i => g i x, LipschitzWith.of_dist_le_mul fun x y => _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhg : \u2200 (i : \u03b9), LipschitzWith K (g i) \u2227 EqOn (fun x => f x i) (g i) s\nx y : \u03b1\n\u22a2 (dist (fun i => g i x) fun i => g i y) \u2264 \u2191K * dist x y\n[PROOFSTEP]\nexact (dist_pi_le_iff (mul_nonneg K.2 dist_nonneg)).2 fun i => (hg i).1.dist_le_mul x y\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhg : \u2200 (i : \u03b9), LipschitzWith K (g i) \u2227 EqOn (fun x => f x i) (g i) s\n\u22a2 EqOn f (fun x i => g i x) s\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhg : \u2200 (i : \u03b9), LipschitzWith K (g i) \u2227 EqOn (fun x => f x i) (g i) s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 f x = (fun x i => g i x) x\n[PROOFSTEP]\next1 i\n[GOAL]\ncase refine'_2.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : Fintype \u03b9\nf : \u03b1 \u2192 \u03b9 \u2192 \u211d\ns : Set \u03b1\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhg : \u2200 (i : \u03b9), LipschitzWith K (g i) \u2227 EqOn (fun x => f x i) (g i) s\nx : \u03b1\nhx : x \u2208 s\ni : \u03b9\n\u22a2 f x i = (fun x i => g i x) x i\n[PROOFSTEP]\nexact (hg i).2 hx\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.Lipschitz", "llama_tokens": 26536, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850402140659, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.423603965276478}} {"text": "[GOAL]\n\u22a2 ConcreteCategory \u03c9CPO\n[PROOFSTEP]\nunfold \u03c9CPO\n[GOAL]\n\u22a2 ConcreteCategory (Bundled OmegaCompletePartialOrder)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nJ : Type v\nf : J \u2192 \u03c9CPO\ns : Cone (Discrete.functor f)\nm : s.pt \u27f6 (product f).pt\nw : \u2200 (j : Discrete J), m \u226b NatTrans.app (product f).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m =\n (fun s =>\n {\n toOrderHom :=\n { toFun := fun t j => OrderHom.toFun (NatTrans.app s.\u03c0 { as := j }).toOrderHom t,\n monotone' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n x \u2264 y \u2192 \u2200 (j : J), \u2191(NatTrans.app s.\u03c0 { as := j }) x \u2264 \u2191(NatTrans.app s.\u03c0 { as := j }) y) },\n cont :=\n (_ :\n \u2200 (x : Chain \u2191s.pt),\n \u2191{ toFun := fun t j => OrderHom.toFun (NatTrans.app s.\u03c0 { as := j }).toOrderHom t,\n monotone' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n x \u2264 y \u2192\n \u2200 (j : J), \u2191(NatTrans.app s.\u03c0 { as := j }) x \u2264 \u2191(NatTrans.app s.\u03c0 { as := j }) y) }\n (\u03c9Sup x) =\n \u03c9Sup\n (Chain.map x\n { toFun := fun t j => OrderHom.toFun (NatTrans.app s.\u03c0 { as := j }).toOrderHom t,\n monotone' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n x \u2264 y \u2192\n \u2200 (j : J), \u2191(NatTrans.app s.\u03c0 { as := j }) x \u2264 \u2191(NatTrans.app s.\u03c0 { as := j }) y) })) })\n s\n[PROOFSTEP]\next t\n[GOAL]\ncase w\nJ : Type v\nf : J \u2192 \u03c9CPO\ns : Cone (Discrete.functor f)\nm : s.pt \u27f6 (product f).pt\nw : \u2200 (j : Discrete J), m \u226b NatTrans.app (product f).\u03c0 j = NatTrans.app s.\u03c0 j\nt : (forget \u03c9CPO).obj s.pt\n\u22a2 \u2191m t =\n \u2191((fun s =>\n {\n toOrderHom :=\n { toFun := fun t j => OrderHom.toFun (NatTrans.app s.\u03c0 { as := j }).toOrderHom t,\n monotone' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n x \u2264 y \u2192 \u2200 (j : J), \u2191(NatTrans.app s.\u03c0 { as := j }) x \u2264 \u2191(NatTrans.app s.\u03c0 { as := j }) y) },\n cont :=\n (_ :\n \u2200 (x : Chain \u2191s.pt),\n \u2191{ toFun := fun t j => OrderHom.toFun (NatTrans.app s.\u03c0 { as := j }).toOrderHom t,\n monotone' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n x \u2264 y \u2192\n \u2200 (j : J), \u2191(NatTrans.app s.\u03c0 { as := j }) x \u2264 \u2191(NatTrans.app s.\u03c0 { as := j }) y) }\n (\u03c9Sup x) =\n \u03c9Sup\n (Chain.map x\n { toFun := fun t j => OrderHom.toFun (NatTrans.app s.\u03c0 { as := j }).toOrderHom t,\n monotone' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n x \u2264 y \u2192\n \u2200 (j : J),\n \u2191(NatTrans.app s.\u03c0 { as := j }) x \u2264 \u2191(NatTrans.app s.\u03c0 { as := j }) y) })) })\n s)\n t\n[PROOFSTEP]\nfunext j\n[GOAL]\ncase w.h\nJ : Type v\nf : J \u2192 \u03c9CPO\ns : Cone (Discrete.functor f)\nm : s.pt \u27f6 (product f).pt\nw : \u2200 (j : Discrete J), m \u226b NatTrans.app (product f).\u03c0 j = NatTrans.app s.\u03c0 j\nt : (forget \u03c9CPO).obj s.pt\nj : J\n\u22a2 \u2191m t j =\n \u2191((fun s =>\n {\n toOrderHom :=\n { toFun := fun t j => OrderHom.toFun (NatTrans.app s.\u03c0 { as := j }).toOrderHom t,\n monotone' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n x \u2264 y \u2192 \u2200 (j : J), \u2191(NatTrans.app s.\u03c0 { as := j }) x \u2264 \u2191(NatTrans.app s.\u03c0 { as := j }) y) },\n cont :=\n (_ :\n \u2200 (x : Chain \u2191s.pt),\n \u2191{ toFun := fun t j => OrderHom.toFun (NatTrans.app s.\u03c0 { as := j }).toOrderHom t,\n monotone' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n x \u2264 y \u2192\n \u2200 (j : J), \u2191(NatTrans.app s.\u03c0 { as := j }) x \u2264 \u2191(NatTrans.app s.\u03c0 { as := j }) y) }\n (\u03c9Sup x) =\n \u03c9Sup\n (Chain.map x\n { toFun := fun t j => OrderHom.toFun (NatTrans.app s.\u03c0 { as := j }).toOrderHom t,\n monotone' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n x \u2264 y \u2192\n \u2200 (j : J),\n \u2191(NatTrans.app s.\u03c0 { as := j }) x \u2264 \u2191(NatTrans.app s.\u03c0 { as := j }) y) })) })\n s)\n t j\n[PROOFSTEP]\nchange m.toFun t j = (s.\u03c0.app \u27e8j\u27e9).toFun t\n[GOAL]\ncase w.h\nJ : Type v\nf : J \u2192 \u03c9CPO\ns : Cone (Discrete.functor f)\nm : s.pt \u27f6 (product f).pt\nw : \u2200 (j : Discrete J), m \u226b NatTrans.app (product f).\u03c0 j = NatTrans.app s.\u03c0 j\nt : (forget \u03c9CPO).obj s.pt\nj : J\n\u22a2 OrderHom.toFun m.toOrderHom t j = OrderHom.toFun (NatTrans.app s.\u03c0 { as := j }).toOrderHom t\n[PROOFSTEP]\nrw [\u2190 w \u27e8j\u27e9]\n[GOAL]\ncase w.h\nJ : Type v\nf : J \u2192 \u03c9CPO\ns : Cone (Discrete.functor f)\nm : s.pt \u27f6 (product f).pt\nw : \u2200 (j : Discrete J), m \u226b NatTrans.app (product f).\u03c0 j = NatTrans.app s.\u03c0 j\nt : (forget \u03c9CPO).obj s.pt\nj : J\n\u22a2 OrderHom.toFun m.toOrderHom t j = OrderHom.toFun (m \u226b NatTrans.app (product f).\u03c0 { as := j }).toOrderHom t\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf g : \u03b1 \u2192\ud835\udc84 \u03b2\nc : Chain \u03b1\nhc : \u2200 (i : \u03b1), i \u2208 c \u2192 \u2191f i = \u2191g i\n\u22a2 \u2191f (\u03c9Sup c) = \u2191g (\u03c9Sup c)\n[PROOFSTEP]\nrw [f.continuous, g.continuous]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf g : \u03b1 \u2192\ud835\udc84 \u03b2\nc : Chain \u03b1\nhc : \u2200 (i : \u03b1), i \u2208 c \u2192 \u2191f i = \u2191g i\n\u22a2 \u03c9Sup (Chain.map c \u2191f) = \u03c9Sup (Chain.map c \u2191g)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf g : \u03b1 \u2192\ud835\udc84 \u03b2\nc : Chain \u03b1\nhc : \u2200 (i : \u03b1), i \u2208 c \u2192 \u2191f i = \u2191g i\n\u22a2 Chain.map c \u2191f = Chain.map c \u2191g\n[PROOFSTEP]\napply OrderHom.ext\n[GOAL]\ncase e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf g : \u03b1 \u2192\ud835\udc84 \u03b2\nc : Chain \u03b1\nhc : \u2200 (i : \u03b1), i \u2208 c \u2192 \u2191f i = \u2191g i\n\u22a2 \u2191(Chain.map c \u2191f) = \u2191(Chain.map c \u2191g)\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase e_a.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf g : \u03b1 \u2192\ud835\udc84 \u03b2\nc : Chain \u03b1\nhc : \u2200 (i : \u03b1), i \u2208 c \u2192 \u2191f i = \u2191g i\nx : \u2115\n\u22a2 \u2191(Chain.map c \u2191f) x = \u2191(Chain.map c \u2191g) x\n[PROOFSTEP]\napply hc _ \u27e8_, rfl\u27e9\n[GOAL]\nX Y : \u03c9CPO\nf g : X \u27f6 Y\ns : Fork f g\nx : \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)\n\u22a2 OrderHom.toFun f.toOrderHom (OrderHom.toFun (Fork.\u03b9 s).toOrderHom x) =\n OrderHom.toFun g.toOrderHom (OrderHom.toFun (Fork.\u03b9 s).toOrderHom x)\n[PROOFSTEP]\napply ContinuousHom.congr_fun s.condition\n[GOAL]\nX Y : \u03c9CPO\nf g : X \u27f6 Y\ns : Fork f g\n\u22a2 {\n toOrderHom :=\n {\n toFun := fun x =>\n { val := OrderHom.toFun (Fork.\u03b9 s).toOrderHom x,\n property := (_ : \u2191(Fork.\u03b9 s \u226b f) x = \u2191(Fork.\u03b9 s \u226b g) x) },\n monotone' :=\n (_ :\n \u2200 (x y : \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n x \u2264 y \u2192 \u2191(Fork.\u03b9 s) x \u2264 \u2191(Fork.\u03b9 s) y) },\n cont :=\n (_ :\n \u2200 (x : Chain \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n \u2191{\n toFun := fun x =>\n { val := OrderHom.toFun (Fork.\u03b9 s).toOrderHom x,\n property := (_ : \u2191(Fork.\u03b9 s \u226b f) x = \u2191(Fork.\u03b9 s \u226b g) x) },\n monotone' :=\n (_ :\n \u2200 (x y : \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n x \u2264 y \u2192 \u2191(Fork.\u03b9 s) x \u2264 \u2191(Fork.\u03b9 s) y) }\n (\u03c9Sup x) =\n \u03c9Sup\n (Chain.map x\n {\n toFun := fun x =>\n { val := OrderHom.toFun (Fork.\u03b9 s).toOrderHom x,\n property := (_ : \u2191(Fork.\u03b9 s \u226b f) x = \u2191(Fork.\u03b9 s \u226b g) x) },\n monotone' :=\n (_ :\n \u2200 (x y : \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n x \u2264 y \u2192 \u2191(Fork.\u03b9 s) x \u2264 \u2191(Fork.\u03b9 s) y) })) } \u226b\n Fork.\u03b9 (equalizer f g) =\n Fork.\u03b9 s\n[PROOFSTEP]\next\n[GOAL]\ncase w\nX Y : \u03c9CPO\nf g : X \u27f6 Y\ns : Fork f g\nx\u271d : (forget \u03c9CPO).obj (((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)\n\u22a2 \u2191({\n toOrderHom :=\n {\n toFun := fun x =>\n { val := OrderHom.toFun (Fork.\u03b9 s).toOrderHom x,\n property := (_ : \u2191(Fork.\u03b9 s \u226b f) x = \u2191(Fork.\u03b9 s \u226b g) x) },\n monotone' :=\n (_ :\n \u2200 (x y : \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n x \u2264 y \u2192 \u2191(Fork.\u03b9 s) x \u2264 \u2191(Fork.\u03b9 s) y) },\n cont :=\n (_ :\n \u2200 (x : Chain \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n \u2191{\n toFun := fun x =>\n { val := OrderHom.toFun (Fork.\u03b9 s).toOrderHom x,\n property := (_ : \u2191(Fork.\u03b9 s \u226b f) x = \u2191(Fork.\u03b9 s \u226b g) x) },\n monotone' :=\n (_ :\n \u2200 (x y : \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n x \u2264 y \u2192 \u2191(Fork.\u03b9 s) x \u2264 \u2191(Fork.\u03b9 s) y) }\n (\u03c9Sup x) =\n \u03c9Sup\n (Chain.map x\n {\n toFun := fun x =>\n { val := OrderHom.toFun (Fork.\u03b9 s).toOrderHom x,\n property := (_ : \u2191(Fork.\u03b9 s \u226b f) x = \u2191(Fork.\u03b9 s \u226b g) x) },\n monotone' :=\n (_ :\n \u2200 (x y : \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n x \u2264 y \u2192 \u2191(Fork.\u03b9 s) x \u2264 \u2191(Fork.\u03b9 s) y) })) } \u226b\n Fork.\u03b9 (equalizer f g))\n x\u271d =\n \u2191(Fork.\u03b9 s) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nX Y : \u03c9CPO\nf g : X \u27f6 Y\ns : Fork f g\nm\u271d :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero \u27f6\n ((Functor.const WalkingParallelPair).obj (equalizer f g).pt).obj WalkingParallelPair.zero\nhm : m\u271d \u226b Fork.\u03b9 (equalizer f g) = Fork.\u03b9 s\n\u22a2 m\u271d =\n {\n toOrderHom :=\n {\n toFun := fun x =>\n { val := OrderHom.toFun (Fork.\u03b9 s).toOrderHom x, property := (_ : \u2191(Fork.\u03b9 s \u226b f) x = \u2191(Fork.\u03b9 s \u226b g) x) },\n monotone' :=\n (_ :\n \u2200 (x y : \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n x \u2264 y \u2192 \u2191(Fork.\u03b9 s) x \u2264 \u2191(Fork.\u03b9 s) y) },\n cont :=\n (_ :\n \u2200 (x : Chain \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n \u2191{\n toFun := fun x =>\n { val := OrderHom.toFun (Fork.\u03b9 s).toOrderHom x,\n property := (_ : \u2191(Fork.\u03b9 s \u226b f) x = \u2191(Fork.\u03b9 s \u226b g) x) },\n monotone' :=\n (_ :\n \u2200 (x y : \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n x \u2264 y \u2192 \u2191(Fork.\u03b9 s) x \u2264 \u2191(Fork.\u03b9 s) y) }\n (\u03c9Sup x) =\n \u03c9Sup\n (Chain.map x\n {\n toFun := fun x =>\n { val := OrderHom.toFun (Fork.\u03b9 s).toOrderHom x,\n property := (_ : \u2191(Fork.\u03b9 s \u226b f) x = \u2191(Fork.\u03b9 s \u226b g) x) },\n monotone' :=\n (_ :\n \u2200 (x y : \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n x \u2264 y \u2192 \u2191(Fork.\u03b9 s) x \u2264 \u2191(Fork.\u03b9 s) y) })) }\n[PROOFSTEP]\napply\n ContinuousHom.ext _ _ fun x =>\n Subtype.ext\n ?_\n -- Porting note: Originally `ext`\n[GOAL]\nX Y : \u03c9CPO\nf g : X \u27f6 Y\ns : Fork f g\nm\u271d :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero \u27f6\n ((Functor.const WalkingParallelPair).obj (equalizer f g).pt).obj WalkingParallelPair.zero\nhm : m\u271d \u226b Fork.\u03b9 (equalizer f g) = Fork.\u03b9 s\nx : \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)\n\u22a2 \u2191(\u2191m\u271d x) =\n \u2191(\u2191{\n toOrderHom :=\n {\n toFun := fun x =>\n { val := OrderHom.toFun (Fork.\u03b9 s).toOrderHom x,\n property := (_ : \u2191(Fork.\u03b9 s \u226b f) x = \u2191(Fork.\u03b9 s \u226b g) x) },\n monotone' :=\n (_ :\n \u2200 (x y : \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n x \u2264 y \u2192 \u2191(Fork.\u03b9 s) x \u2264 \u2191(Fork.\u03b9 s) y) },\n cont :=\n (_ :\n \u2200 (x : Chain \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n \u2191{\n toFun := fun x =>\n { val := OrderHom.toFun (Fork.\u03b9 s).toOrderHom x,\n property := (_ : \u2191(Fork.\u03b9 s \u226b f) x = \u2191(Fork.\u03b9 s \u226b g) x) },\n monotone' :=\n (_ :\n \u2200 (x y : \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n x \u2264 y \u2192 \u2191(Fork.\u03b9 s) x \u2264 \u2191(Fork.\u03b9 s) y) }\n (\u03c9Sup x) =\n \u03c9Sup\n (Chain.map x\n {\n toFun := fun x =>\n { val := OrderHom.toFun (Fork.\u03b9 s).toOrderHom x,\n property := (_ : \u2191(Fork.\u03b9 s \u226b f) x = \u2191(Fork.\u03b9 s \u226b g) x) },\n monotone' :=\n (_ :\n \u2200 (x y : \u2191(((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero)),\n x \u2264 y \u2192 \u2191(Fork.\u03b9 s) x \u2264 \u2191(Fork.\u03b9 s) y) })) }\n x)\n[PROOFSTEP]\napply ContinuousHom.congr_fun hm\n", "meta": {"mathlib_filename": "Mathlib.Order.Category.OmegaCompletePartialOrder", "llama_tokens": 5444, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737869342623, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.42352680899577644}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b2 : Type u_3\ns : Set E\nf g : E \u2192 \u03b2\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : SMul \ud835\udd5c E\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nhf : ConvexOn \ud835\udd5c s f\nhfg : EqOn f g s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 g (a \u2022 x + b \u2022 y) \u2264 a \u2022 g x + b \u2022 g y\n[PROOFSTEP]\nsimpa only [\u2190 hfg hx, \u2190 hfg hy, \u2190 hfg (hf.1 hx hy ha hb hab)] using hf.2 hx hy ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b2 : Type u_3\ns : Set E\nf g : E \u2192 \u03b2\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : SMul \ud835\udd5c E\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nhf : ConcaveOn \ud835\udd5c s f\nhfg : EqOn f g s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 g x + b \u2022 g y \u2264 g (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nsimpa only [\u2190 hfg hx, \u2190 hfg hy, \u2190 hfg (hf.1 hx hy ha hb hab)] using hf.2 hx hy ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b2 : Type u_3\ns : Set E\nf g : E \u2192 \u03b2\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : SMul \ud835\udd5c E\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nhf : StrictConvexOn \ud835\udd5c s f\nhfg : EqOn f g s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 g (a \u2022 x + b \u2022 y) < a \u2022 g x + b \u2022 g y\n[PROOFSTEP]\nsimpa only [\u2190 hfg hx, \u2190 hfg hy, \u2190 hfg (hf.1 hx hy ha.le hb.le hab)] using hf.2 hx hy hxy ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b2 : Type u_3\ns : Set E\nf g : E \u2192 \u03b2\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : SMul \ud835\udd5c E\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : SMul \ud835\udd5c \u03b2\nhf : StrictConcaveOn \ud835\udd5c s f\nhfg : EqOn f g s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhxy : x \u2260 y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 g x + b \u2022 g y < g (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nsimpa only [\u2190 hfg hx, \u2190 hfg hy, \u2190 hfg (hf.1 hx hy ha.le hb.le hab)] using hf.2 hx hy hxy ha hb hab\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b\n[PROOFSTEP]\nlet f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => exp (-c * x) * x ^ (c * (u - 1))\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\n\u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b\n[PROOFSTEP]\nhave e : IsConjugateExponent (1 / a) (1 / b) := Real.isConjugateExponent_one_div ha hb hab\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\n\u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b\n[PROOFSTEP]\nhave hab' : b = 1 - a := by linarith\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\n\u22a2 b = 1 - a\n[PROOFSTEP]\nlinarith\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\n\u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b\n[PROOFSTEP]\nhave hst : 0 < a * s + b * t :=\n add_pos (mul_pos ha hs)\n (mul_pos hb ht)\n -- some properties of f:\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\n\u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b\n[PROOFSTEP]\nhave posf : \u2200 c u x : \u211d, x \u2208 Ioi (0 : \u211d) \u2192 0 \u2264 f c u x := fun c u x hx =>\n mul_nonneg (exp_pos _).le (rpow_pos_of_pos hx _).le\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\n\u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b\n[PROOFSTEP]\nhave posf' : \u2200 c u : \u211d, \u2200\u1d50 x : \u211d \u2202volume.restrict (Ioi 0), 0 \u2264 f c u x := fun c u =>\n (ae_restrict_iff' measurableSet_Ioi).mpr (ae_of_all _ (posf c u))\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\n\u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b\n[PROOFSTEP]\nhave fpow : \u2200 {c x : \u211d} (_ : 0 < c) (u : \u211d) (_ : 0 < x), exp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) :=\n by\n intro c x hc u hx\n dsimp only\n rw [mul_rpow (exp_pos _).le ((rpow_nonneg_of_nonneg hx.le) _), \u2190 exp_mul, \u2190 rpow_mul hx.le]\n congr 2 <;>\n \u00b7 field_simp [hc.ne'];\n ring\n -- show `f c u` is in `\u2112p` for `p = 1/c`:\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\n\u22a2 \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\n[PROOFSTEP]\nintro c x hc u hx\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nc x : \u211d\nhc : 0 < c\nu : \u211d\nhx : 0 < x\n\u22a2 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\n[PROOFSTEP]\ndsimp only\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nc x : \u211d\nhc : 0 < c\nu : \u211d\nhx : 0 < x\n\u22a2 rexp (-x) * x ^ (u - 1) = (rexp (-c * x) * x ^ (c * (u - 1))) ^ (1 / c)\n[PROOFSTEP]\nrw [mul_rpow (exp_pos _).le ((rpow_nonneg_of_nonneg hx.le) _), \u2190 exp_mul, \u2190 rpow_mul hx.le]\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nc x : \u211d\nhc : 0 < c\nu : \u211d\nhx : 0 < x\n\u22a2 rexp (-x) * x ^ (u - 1) = rexp (-c * x * (1 / c)) * x ^ (c * (u - 1) * (1 / c))\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_x\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nc x : \u211d\nhc : 0 < c\nu : \u211d\nhx : 0 < x\n\u22a2 -x = -c * x * (1 / c)\n[PROOFSTEP]\nfield_simp [hc.ne']\n[GOAL]\ncase e_a.e_x\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nc x : \u211d\nhc : 0 < c\nu : \u211d\nhx : 0 < x\n\u22a2 x * c = c * x\n[PROOFSTEP]\nring\n -- show `f c u` is in `\u2112p` for `p = 1/c`:\n[GOAL]\ncase e_a.e_a\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nc x : \u211d\nhc : 0 < c\nu : \u211d\nhx : 0 < x\n\u22a2 u - 1 = c * (u - 1) * (1 / c)\n[PROOFSTEP]\nfield_simp [hc.ne']\n[GOAL]\ncase e_a.e_a\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nc x : \u211d\nhc : 0 < c\nu : \u211d\nhx : 0 < x\n\u22a2 (u - 1) * c = c * (u - 1)\n[PROOFSTEP]\nring\n -- show `f c u` is in `\u2112p` for `p = 1/c`:\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\n\u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b\n[PROOFSTEP]\nhave f_mem_Lp :\n \u2200 {c u : \u211d} (hc : 0 < c) (hu : 0 < u), Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) (volume.restrict (Ioi 0)) :=\n by\n intro c u hc hu\n have A : ENNReal.ofReal (1 / c) \u2260 0 := by rwa [Ne.def, ENNReal.ofReal_eq_zero, not_le, one_div_pos]\n have B : ENNReal.ofReal (1 / c) \u2260 \u221e := ENNReal.ofReal_ne_top\n rw [\u2190 mem\u2112p_norm_rpow_iff _ A B, ENNReal.toReal_ofReal (one_div_nonneg.mpr hc.le), ENNReal.div_self A B,\n mem\u2112p_one_iff_integrable]\n \u00b7 apply Integrable.congr (GammaIntegral_convergent hu)\n refine' eventuallyEq_of_mem (self_mem_ae_restrict measurableSet_Ioi) fun x hx => _\n dsimp only\n rw [fpow hc u hx]\n congr 1\n exact (norm_of_nonneg (posf _ _ x hx)).symm\n \u00b7 refine' ContinuousOn.aestronglyMeasurable _ measurableSet_Ioi\n refine' (Continuous.continuousOn _).mul (ContinuousAt.continuousOn fun x hx => _)\n \u00b7 exact continuous_exp.comp (continuous_const.mul continuous_id')\n \u00b7\n exact\n continuousAt_rpow_const _ _\n (Or.inl (mem_Ioi.mp hx).ne')\n -- now apply H\u00f6lder:\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\n\u22a2 \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\n[PROOFSTEP]\nintro c u hc hu\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nc u : \u211d\nhc : 0 < c\nhu : 0 < u\n\u22a2 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\n[PROOFSTEP]\nhave A : ENNReal.ofReal (1 / c) \u2260 0 := by rwa [Ne.def, ENNReal.ofReal_eq_zero, not_le, one_div_pos]\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nc u : \u211d\nhc : 0 < c\nhu : 0 < u\n\u22a2 ENNReal.ofReal (1 / c) \u2260 0\n[PROOFSTEP]\nrwa [Ne.def, ENNReal.ofReal_eq_zero, not_le, one_div_pos]\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nc u : \u211d\nhc : 0 < c\nhu : 0 < u\nA : ENNReal.ofReal (1 / c) \u2260 0\n\u22a2 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\n[PROOFSTEP]\nhave B : ENNReal.ofReal (1 / c) \u2260 \u221e := ENNReal.ofReal_ne_top\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nc u : \u211d\nhc : 0 < c\nhu : 0 < u\nA : ENNReal.ofReal (1 / c) \u2260 0\nB : ENNReal.ofReal (1 / c) \u2260 \u22a4\n\u22a2 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\n[PROOFSTEP]\nrw [\u2190 mem\u2112p_norm_rpow_iff _ A B, ENNReal.toReal_ofReal (one_div_nonneg.mpr hc.le), ENNReal.div_self A B,\n mem\u2112p_one_iff_integrable]\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nc u : \u211d\nhc : 0 < c\nhu : 0 < u\nA : ENNReal.ofReal (1 / c) \u2260 0\nB : ENNReal.ofReal (1 / c) \u2260 \u22a4\n\u22a2 Integrable fun x => \u2016f c u x\u2016 ^ (1 / c)\n[PROOFSTEP]\napply Integrable.congr (GammaIntegral_convergent hu)\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nc u : \u211d\nhc : 0 < c\nhu : 0 < u\nA : ENNReal.ofReal (1 / c) \u2260 0\nB : ENNReal.ofReal (1 / c) \u2260 \u22a4\n\u22a2 (fun x => rexp (-x) * x ^ (u - 1)) =\u1d50[Measure.restrict volume (Ioi 0)] fun x => \u2016f c u x\u2016 ^ (1 / c)\n[PROOFSTEP]\nrefine' eventuallyEq_of_mem (self_mem_ae_restrict measurableSet_Ioi) fun x hx => _\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nc u : \u211d\nhc : 0 < c\nhu : 0 < u\nA : ENNReal.ofReal (1 / c) \u2260 0\nB : ENNReal.ofReal (1 / c) \u2260 \u22a4\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 rexp (-x) * x ^ (u - 1) = \u2016f c u x\u2016 ^ (1 / c)\n[PROOFSTEP]\ndsimp only\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nc u : \u211d\nhc : 0 < c\nhu : 0 < u\nA : ENNReal.ofReal (1 / c) \u2260 0\nB : ENNReal.ofReal (1 / c) \u2260 \u22a4\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 rexp (-x) * x ^ (u - 1) = \u2016rexp (-c * x) * x ^ (c * (u - 1))\u2016 ^ (1 / c)\n[PROOFSTEP]\nrw [fpow hc u hx]\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nc u : \u211d\nhc : 0 < c\nhu : 0 < u\nA : ENNReal.ofReal (1 / c) \u2260 0\nB : ENNReal.ofReal (1 / c) \u2260 \u22a4\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 f c u x ^ (1 / c) = \u2016rexp (-c * x) * x ^ (c * (u - 1))\u2016 ^ (1 / c)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nc u : \u211d\nhc : 0 < c\nhu : 0 < u\nA : ENNReal.ofReal (1 / c) \u2260 0\nB : ENNReal.ofReal (1 / c) \u2260 \u22a4\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 f c u x = \u2016rexp (-c * x) * x ^ (c * (u - 1))\u2016\n[PROOFSTEP]\nexact (norm_of_nonneg (posf _ _ x hx)).symm\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nc u : \u211d\nhc : 0 < c\nhu : 0 < u\nA : ENNReal.ofReal (1 / c) \u2260 0\nB : ENNReal.ofReal (1 / c) \u2260 \u22a4\n\u22a2 AEStronglyMeasurable (f c u) (Measure.restrict volume (Ioi 0))\n[PROOFSTEP]\nrefine' ContinuousOn.aestronglyMeasurable _ measurableSet_Ioi\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nc u : \u211d\nhc : 0 < c\nhu : 0 < u\nA : ENNReal.ofReal (1 / c) \u2260 0\nB : ENNReal.ofReal (1 / c) \u2260 \u22a4\n\u22a2 ContinuousOn (f c u) (Ioi 0)\n[PROOFSTEP]\nrefine' (Continuous.continuousOn _).mul (ContinuousAt.continuousOn fun x hx => _)\n[GOAL]\ncase refine'_1\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nc u : \u211d\nhc : 0 < c\nhu : 0 < u\nA : ENNReal.ofReal (1 / c) \u2260 0\nB : ENNReal.ofReal (1 / c) \u2260 \u22a4\n\u22a2 Continuous fun x => rexp (-c * x)\n[PROOFSTEP]\nexact continuous_exp.comp (continuous_const.mul continuous_id')\n[GOAL]\ncase refine'_2\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nc u : \u211d\nhc : 0 < c\nhu : 0 < u\nA : ENNReal.ofReal (1 / c) \u2260 0\nB : ENNReal.ofReal (1 / c) \u2260 \u22a4\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 ContinuousAt (fun x => x ^ (c * (u - 1))) x\n[PROOFSTEP]\nexact\n continuousAt_rpow_const _ _\n (Or.inl (mem_Ioi.mp hx).ne')\n -- now apply H\u00f6lder:\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\n\u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b\n[PROOFSTEP]\nrw [Gamma_eq_integral hs, Gamma_eq_integral ht, Gamma_eq_integral hst]\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\n\u22a2 \u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (a * s + b * t - 1) \u2264\n (\u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (s - 1)) ^ a * (\u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (t - 1)) ^ b\n[PROOFSTEP]\nconvert\n MeasureTheory.integral_mul_le_Lp_mul_Lq_of_nonneg e (posf' a s) (posf' b t) (f_mem_Lp ha hs) (f_mem_Lp hb ht) using 1\n[GOAL]\ncase h.e'_3\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\n\u22a2 \u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (a * s + b * t - 1) = \u222b (a_1 : \u211d) in Ioi 0, f a s a_1 * f b t a_1\n[PROOFSTEP]\nrefine' set_integral_congr measurableSet_Ioi fun x hx => _\n[GOAL]\ncase h.e'_3\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 rexp (-x) * x ^ (a * s + b * t - 1) = f a s x * f b t x\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h.e'_3\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 rexp (-x) * x ^ (a * s + b * t - 1) = rexp (-a * x) * x ^ (a * (s - 1)) * (rexp (-b * x) * x ^ (b * (t - 1)))\n[PROOFSTEP]\nhave A : exp (-x) = exp (-a * x) * exp (-b * x) := by rw [\u2190 exp_add, \u2190 add_mul, \u2190 neg_add, hab, neg_one_mul]\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 rexp (-x) = rexp (-a * x) * rexp (-b * x)\n[PROOFSTEP]\nrw [\u2190 exp_add, \u2190 add_mul, \u2190 neg_add, hab, neg_one_mul]\n[GOAL]\ncase h.e'_3\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\nx : \u211d\nhx : x \u2208 Ioi 0\nA : rexp (-x) = rexp (-a * x) * rexp (-b * x)\n\u22a2 rexp (-x) * x ^ (a * s + b * t - 1) = rexp (-a * x) * x ^ (a * (s - 1)) * (rexp (-b * x) * x ^ (b * (t - 1)))\n[PROOFSTEP]\nhave B : x ^ (a * s + b * t - 1) = x ^ (a * (s - 1)) * x ^ (b * (t - 1)) := by rw [\u2190 rpow_add hx, hab']; congr 1; ring\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\nx : \u211d\nhx : x \u2208 Ioi 0\nA : rexp (-x) = rexp (-a * x) * rexp (-b * x)\n\u22a2 x ^ (a * s + b * t - 1) = x ^ (a * (s - 1)) * x ^ (b * (t - 1))\n[PROOFSTEP]\nrw [\u2190 rpow_add hx, hab']\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\nx : \u211d\nhx : x \u2208 Ioi 0\nA : rexp (-x) = rexp (-a * x) * rexp (-b * x)\n\u22a2 x ^ (a * s + (1 - a) * t - 1) = x ^ (a * (s - 1) + (1 - a) * (t - 1))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\nx : \u211d\nhx : x \u2208 Ioi 0\nA : rexp (-x) = rexp (-a * x) * rexp (-b * x)\n\u22a2 a * s + (1 - a) * t - 1 = a * (s - 1) + (1 - a) * (t - 1)\n[PROOFSTEP]\nring\n[GOAL]\ncase h.e'_3\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\nx : \u211d\nhx : x \u2208 Ioi 0\nA : rexp (-x) = rexp (-a * x) * rexp (-b * x)\nB : x ^ (a * s + b * t - 1) = x ^ (a * (s - 1)) * x ^ (b * (t - 1))\n\u22a2 rexp (-x) * x ^ (a * s + b * t - 1) = rexp (-a * x) * x ^ (a * (s - 1)) * (rexp (-b * x) * x ^ (b * (t - 1)))\n[PROOFSTEP]\nrw [A, B]\n[GOAL]\ncase h.e'_3\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\nx : \u211d\nhx : x \u2208 Ioi 0\nA : rexp (-x) = rexp (-a * x) * rexp (-b * x)\nB : x ^ (a * s + b * t - 1) = x ^ (a * (s - 1)) * x ^ (b * (t - 1))\n\u22a2 rexp (-a * x) * rexp (-b * x) * (x ^ (a * (s - 1)) * x ^ (b * (t - 1))) =\n rexp (-a * x) * x ^ (a * (s - 1)) * (rexp (-b * x) * x ^ (b * (t - 1)))\n[PROOFSTEP]\nring\n[GOAL]\ncase h.e'_4\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\n\u22a2 (\u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (s - 1)) ^ a * (\u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (t - 1)) ^ b =\n (\u222b (a_1 : \u211d) in Ioi 0, f a s a_1 ^ (1 / a)) ^ (1 / (1 / a)) *\n (\u222b (a : \u211d) in Ioi 0, f b t a ^ (1 / b)) ^ (1 / (1 / b))\n[PROOFSTEP]\nrw [one_div_one_div, one_div_one_div]\n[GOAL]\ncase h.e'_4\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\n\u22a2 (\u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (s - 1)) ^ a * (\u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (t - 1)) ^ b =\n (\u222b (a_1 : \u211d) in Ioi 0, f a s a_1 ^ (1 / a)) ^ a * (\u222b (a : \u211d) in Ioi 0, f b t a ^ (1 / b)) ^ b\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase h.e'_4.e_a.e_a\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\n\u22a2 \u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (s - 1) = \u222b (a_1 : \u211d) in Ioi 0, f a s a_1 ^ (1 / a)\n[PROOFSTEP]\nexact set_integral_congr measurableSet_Ioi fun x hx => fpow (by assumption) _ hx\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 0 < a\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h.e'_4.e_a.e_a\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\n\u22a2 \u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (t - 1) = \u222b (a : \u211d) in Ioi 0, f b t a ^ (1 / b)\n[PROOFSTEP]\nexact set_integral_congr measurableSet_Ioi fun x hx => fpow (by assumption) _ hx\n[GOAL]\ns t a b : \u211d\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1))\ne : IsConjugateExponent (1 / a) (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x\nposf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x\nfpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)\nf_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c))\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 0 < b\n[PROOFSTEP]\nassumption\n[GOAL]\n\u22a2 ConvexOn \u211d (Ioi 0) (log \u2218 Gamma)\n[PROOFSTEP]\nrefine' convexOn_iff_forall_pos.mpr \u27e8convex_Ioi _, fun x hx y hy a b ha hb hab => _\u27e9\n[GOAL]\nx : \u211d\nhx : x \u2208 Ioi 0\ny : \u211d\nhy : y \u2208 Ioi 0\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 (log \u2218 Gamma) (a \u2022 x + b \u2022 y) \u2264 a \u2022 (log \u2218 Gamma) x + b \u2022 (log \u2218 Gamma) y\n[PROOFSTEP]\nhave : b = 1 - a := by linarith\n[GOAL]\nx : \u211d\nhx : x \u2208 Ioi 0\ny : \u211d\nhy : y \u2208 Ioi 0\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 b = 1 - a\n[PROOFSTEP]\nlinarith\n[GOAL]\nx : \u211d\nhx : x \u2208 Ioi 0\ny : \u211d\nhy : y \u2208 Ioi 0\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nthis : b = 1 - a\n\u22a2 (log \u2218 Gamma) (a \u2022 x + b \u2022 y) \u2264 a \u2022 (log \u2218 Gamma) x + b \u2022 (log \u2218 Gamma) y\n[PROOFSTEP]\nsubst this\n[GOAL]\nx : \u211d\nhx : x \u2208 Ioi 0\ny : \u211d\nhy : y \u2208 Ioi 0\na : \u211d\nha : 0 < a\nhb : 0 < 1 - a\nhab : a + (1 - a) = 1\n\u22a2 (log \u2218 Gamma) (a \u2022 x + (1 - a) \u2022 y) \u2264 a \u2022 (log \u2218 Gamma) x + (1 - a) \u2022 (log \u2218 Gamma) y\n[PROOFSTEP]\nsimp_rw [Function.comp_apply, smul_eq_mul]\n[GOAL]\nx : \u211d\nhx : x \u2208 Ioi 0\ny : \u211d\nhy : y \u2208 Ioi 0\na : \u211d\nha : 0 < a\nhb : 0 < 1 - a\nhab : a + (1 - a) = 1\n\u22a2 log (Gamma (a * x + (1 - a) * y)) \u2264 a * log (Gamma x) + (1 - a) * log (Gamma y)\n[PROOFSTEP]\nrw [\u2190 log_rpow (Gamma_pos_of_pos hy), \u2190 log_rpow (Gamma_pos_of_pos hx), \u2190\n log_mul (rpow_pos_of_pos (Gamma_pos_of_pos hx) _).ne' (rpow_pos_of_pos (Gamma_pos_of_pos hy) _).ne',\n log_le_log (Gamma_pos_of_pos (add_pos (mul_pos ha hx) (mul_pos hb hy)))\n (mul_pos (rpow_pos_of_pos (Gamma_pos_of_pos hx) _) (rpow_pos_of_pos (Gamma_pos_of_pos hy) _))]\n[GOAL]\nx : \u211d\nhx : x \u2208 Ioi 0\ny : \u211d\nhy : y \u2208 Ioi 0\na : \u211d\nha : 0 < a\nhb : 0 < 1 - a\nhab : a + (1 - a) = 1\n\u22a2 Gamma (a * x + (1 - a) * y) \u2264 Gamma x ^ a * Gamma y ^ (1 - a)\n[PROOFSTEP]\nexact Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gamma hx hy ha hb hab\n[GOAL]\n\u22a2 ConvexOn \u211d (Ioi 0) Gamma\n[PROOFSTEP]\nrefine'\n ((convexOn_exp.subset (subset_univ _) _).comp convexOn_log_Gamma (exp_monotone.monotoneOn _)).congr fun x hx =>\n exp_log (Gamma_pos_of_pos hx)\n[GOAL]\n\u22a2 Convex \u211d (log \u2218 Gamma '' Ioi 0)\n[PROOFSTEP]\nrw [convex_iff_isPreconnected]\n[GOAL]\n\u22a2 IsPreconnected (log \u2218 Gamma '' Ioi 0)\n[PROOFSTEP]\nrefine' isPreconnected_Ioi.image _ fun x hx => ContinuousAt.continuousWithinAt _\n[GOAL]\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 ContinuousAt (log \u2218 Gamma) x\n[PROOFSTEP]\nrefine' (differentiableAt_Gamma fun m => _).continuousAt.log (Gamma_pos_of_pos hx).ne'\n[GOAL]\nx : \u211d\nhx : x \u2208 Ioi 0\nm : \u2115\n\u22a2 x \u2260 -\u2191m\n[PROOFSTEP]\nexact (neg_lt_iff_pos_add.mpr (add_pos_of_pos_of_nonneg (mem_Ioi.mp hx) (Nat.cast_nonneg m))).ne'\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\n\u22a2 f \u2191n = f 1 + log \u2191(n - 1)!\n[PROOFSTEP]\nrefine' Nat.le_induction (by simp) (fun m hm IH => _) n (Nat.one_le_iff_ne_zero.2 hn)\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\n\u22a2 f \u21911 = f 1 + log \u2191(1 - 1)!\n[PROOFSTEP]\nsimp\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nm : \u2115\nhm : 1 \u2264 m\nIH : f \u2191m = f 1 + log \u2191(m - 1)!\n\u22a2 f \u2191(m + 1) = f 1 + log \u2191(m + 1 - 1)!\n[PROOFSTEP]\nhave A : 0 < (m : \u211d) := Nat.cast_pos.2 hm\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nm : \u2115\nhm : 1 \u2264 m\nIH : f \u2191m = f 1 + log \u2191(m - 1)!\nA : 0 < \u2191m\n\u22a2 f \u2191(m + 1) = f 1 + log \u2191(m + 1 - 1)!\n[PROOFSTEP]\nsimp only [hf_feq A, Nat.cast_add, Nat.cast_one, Nat.add_succ_sub_one, add_zero]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nm : \u2115\nhm : 1 \u2264 m\nIH : f \u2191m = f 1 + log \u2191(m - 1)!\nA : 0 < \u2191m\n\u22a2 f \u2191m + log \u2191m = f 1 + log \u2191m !\n[PROOFSTEP]\nrw [IH, add_assoc, \u2190 log_mul (Nat.cast_ne_zero.mpr (Nat.factorial_ne_zero _)) A.ne', \u2190 Nat.cast_mul]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nm : \u2115\nhm : 1 \u2264 m\nIH : f \u2191m = f 1 + log \u2191(m - 1)!\nA : 0 < \u2191m\n\u22a2 f 1 + log \u2191((m - 1)! * m) = f 1 + log \u2191m !\n[PROOFSTEP]\nconv_rhs => rw [\u2190 Nat.succ_pred_eq_of_pos hm, Nat.factorial_succ, mul_comm]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nm : \u2115\nhm : 1 \u2264 m\nIH : f \u2191m = f 1 + log \u2191(m - 1)!\nA : 0 < \u2191m\n| f 1 + log \u2191m !\n[PROOFSTEP]\nrw [\u2190 Nat.succ_pred_eq_of_pos hm, Nat.factorial_succ, mul_comm]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nm : \u2115\nhm : 1 \u2264 m\nIH : f \u2191m = f 1 + log \u2191(m - 1)!\nA : 0 < \u2191m\n| f 1 + log \u2191m !\n[PROOFSTEP]\nrw [\u2190 Nat.succ_pred_eq_of_pos hm, Nat.factorial_succ, mul_comm]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nm : \u2115\nhm : 1 \u2264 m\nIH : f \u2191m = f 1 + log \u2191(m - 1)!\nA : 0 < \u2191m\n| f 1 + log \u2191m !\n[PROOFSTEP]\nrw [\u2190 Nat.succ_pred_eq_of_pos hm, Nat.factorial_succ, mul_comm]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nm : \u2115\nhm : 1 \u2264 m\nIH : f \u2191m = f 1 + log \u2191(m - 1)!\nA : 0 < \u2191m\n\u22a2 f 1 + log \u2191((m - 1)! * m) = f 1 + log \u2191((Nat.pred m)! * (Nat.pred m + 1))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_x.e_a.e_a\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nm : \u2115\nhm : 1 \u2264 m\nIH : f \u2191m = f 1 + log \u2191(m - 1)!\nA : 0 < \u2191m\n\u22a2 m = Nat.pred m + 1\n[PROOFSTEP]\nexact (Nat.succ_pred_eq_of_pos hm).symm\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nn : \u2115\n\u22a2 f (x + \u2191n) = f x + \u2211 m in Finset.range n, log (x + \u2191m)\n[PROOFSTEP]\ninduction' n with n hn\n[GOAL]\ncase zero\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\n\u22a2 f (x + \u2191Nat.zero) = f x + \u2211 m in Finset.range Nat.zero, log (x + \u2191m)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nn : \u2115\nhn : f (x + \u2191n) = f x + \u2211 m in Finset.range n, log (x + \u2191m)\n\u22a2 f (x + \u2191(Nat.succ n)) = f x + \u2211 m in Finset.range (Nat.succ n), log (x + \u2191m)\n[PROOFSTEP]\nhave : x + n.succ = x + n + 1 := by push_cast ; ring\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nn : \u2115\nhn : f (x + \u2191n) = f x + \u2211 m in Finset.range n, log (x + \u2191m)\n\u22a2 x + \u2191(Nat.succ n) = x + \u2191n + 1\n[PROOFSTEP]\npush_cast\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nn : \u2115\nhn : f (x + \u2191n) = f x + \u2211 m in Finset.range n, log (x + \u2191m)\n\u22a2 x + (\u2191n + 1) = x + \u2191n + 1\n[PROOFSTEP]\nring\n[GOAL]\ncase succ\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nn : \u2115\nhn : f (x + \u2191n) = f x + \u2211 m in Finset.range n, log (x + \u2191m)\nthis : x + \u2191(Nat.succ n) = x + \u2191n + 1\n\u22a2 f (x + \u2191(Nat.succ n)) = f x + \u2211 m in Finset.range (Nat.succ n), log (x + \u2191m)\n[PROOFSTEP]\nrw [this, hf_feq, hn]\n[GOAL]\ncase succ\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nn : \u2115\nhn : f (x + \u2191n) = f x + \u2211 m in Finset.range n, log (x + \u2191m)\nthis : x + \u2191(Nat.succ n) = x + \u2191n + 1\n\u22a2 f x + \u2211 m in Finset.range n, log (x + \u2191m) + log (x + \u2191n) = f x + \u2211 m in Finset.range (Nat.succ n), log (x + \u2191m)\ncase succ\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nn : \u2115\nhn : f (x + \u2191n) = f x + \u2211 m in Finset.range n, log (x + \u2191m)\nthis : x + \u2191(Nat.succ n) = x + \u2191n + 1\n\u22a2 0 < x + \u2191n\n[PROOFSTEP]\nrw [Finset.range_succ, Finset.sum_insert Finset.not_mem_range_self]\n[GOAL]\ncase succ\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nn : \u2115\nhn : f (x + \u2191n) = f x + \u2211 m in Finset.range n, log (x + \u2191m)\nthis : x + \u2191(Nat.succ n) = x + \u2191n + 1\n\u22a2 f x + \u2211 m in Finset.range n, log (x + \u2191m) + log (x + \u2191n) =\n f x + (log (x + \u2191n) + \u2211 x_1 in Finset.range n, log (x + \u2191x_1))\ncase succ\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nn : \u2115\nhn : f (x + \u2191n) = f x + \u2211 m in Finset.range n, log (x + \u2191m)\nthis : x + \u2191(Nat.succ n) = x + \u2191n + 1\n\u22a2 0 < x + \u2191n\n[PROOFSTEP]\nabel\n[GOAL]\ncase succ\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nn : \u2115\nhn : f (x + \u2191n) = f x + \u2211 m in Finset.range n, log (x + \u2191m)\nthis : x + \u2191(Nat.succ n) = x + \u2191n + 1\n\u22a2 f x + \u2211 m in Finset.range n, log (x + \u2191m) + log (x + \u2191n) =\n f x + (log (x + \u2191n) + \u2211 x_1 in Finset.range n, log (x + \u2191x_1))\ncase succ\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nn : \u2115\nhn : f (x + \u2191n) = f x + \u2211 m in Finset.range n, log (x + \u2191m)\nthis : x + \u2191(Nat.succ n) = x + \u2191n + 1\n\u22a2 0 < x + \u2191n\n[PROOFSTEP]\nabel\n[GOAL]\ncase succ\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nn : \u2115\nhn : f (x + \u2191n) = f x + \u2211 m in Finset.range n, log (x + \u2191m)\nthis : x + \u2191(Nat.succ n) = x + \u2191n + 1\n\u22a2 0 < x + \u2191n\n[PROOFSTEP]\nlinarith [(Nat.cast_nonneg n : 0 \u2264 (n : \u211d))]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 f (\u2191n + x) \u2264 f \u2191n + x * log \u2191n\n[PROOFSTEP]\nhave hn' : 0 < (n : \u211d) := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn)\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nhx : 0 < x\nhx' : x \u2264 1\nhn' : 0 < \u2191n\n\u22a2 f (\u2191n + x) \u2264 f \u2191n + x * log \u2191n\n[PROOFSTEP]\nhave : f n + x * log n = (1 - x) * f n + x * f (n + 1) := by rw [hf_feq hn']; ring\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nhx : 0 < x\nhx' : x \u2264 1\nhn' : 0 < \u2191n\n\u22a2 f \u2191n + x * log \u2191n = (1 - x) * f \u2191n + x * f (\u2191n + 1)\n[PROOFSTEP]\nrw [hf_feq hn']\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nhx : 0 < x\nhx' : x \u2264 1\nhn' : 0 < \u2191n\n\u22a2 f \u2191n + x * log \u2191n = (1 - x) * f \u2191n + x * (f \u2191n + log \u2191n)\n[PROOFSTEP]\nring\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nhx : 0 < x\nhx' : x \u2264 1\nhn' : 0 < \u2191n\nthis : f \u2191n + x * log \u2191n = (1 - x) * f \u2191n + x * f (\u2191n + 1)\n\u22a2 f (\u2191n + x) \u2264 f \u2191n + x * log \u2191n\n[PROOFSTEP]\nrw [this, (by ring : (n : \u211d) + x = (1 - x) * n + x * (n + 1))]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nhx : 0 < x\nhx' : x \u2264 1\nhn' : 0 < \u2191n\nthis : f \u2191n + x * log \u2191n = (1 - x) * f \u2191n + x * f (\u2191n + 1)\n\u22a2 \u2191n + x = (1 - x) * \u2191n + x * (\u2191n + 1)\n[PROOFSTEP]\nring\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nhx : 0 < x\nhx' : x \u2264 1\nhn' : 0 < \u2191n\nthis : f \u2191n + x * log \u2191n = (1 - x) * f \u2191n + x * f (\u2191n + 1)\n\u22a2 f ((1 - x) * \u2191n + x * (\u2191n + 1)) \u2264 (1 - x) * f \u2191n + x * f (\u2191n + 1)\n[PROOFSTEP]\nsimpa only [smul_eq_mul] using\n hf_conv.2 hn' (by linarith : 0 < (n + 1 : \u211d)) (by linarith : 0 \u2264 1 - x) hx.le (by linarith)\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nhx : 0 < x\nhx' : x \u2264 1\nhn' : 0 < \u2191n\nthis : f \u2191n + x * log \u2191n = (1 - x) * f \u2191n + x * f (\u2191n + 1)\n\u22a2 0 < \u2191n + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nhx : 0 < x\nhx' : x \u2264 1\nhn' : 0 < \u2191n\nthis : f \u2191n + x * log \u2191n = (1 - x) * f \u2191n + x * f (\u2191n + 1)\n\u22a2 0 \u2264 1 - x\n[PROOFSTEP]\nlinarith\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : n \u2260 0\nhx : 0 < x\nhx' : x \u2264 1\nhn' : 0 < \u2191n\nthis : f \u2191n + x * log \u2191n = (1 - x) * f \u2191n + x * f (\u2191n + 1)\n\u22a2 1 - x + x = 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : 2 \u2264 n\nhx : 0 < x\n\u22a2 f \u2191n + x * log (\u2191n - 1) \u2264 f (\u2191n + x)\n[PROOFSTEP]\nhave npos : 0 < (n : \u211d) - 1 := by rw [\u2190 Nat.cast_one, sub_pos, Nat.cast_lt]; linarith\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : 2 \u2264 n\nhx : 0 < x\n\u22a2 0 < \u2191n - 1\n[PROOFSTEP]\nrw [\u2190 Nat.cast_one, sub_pos, Nat.cast_lt]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : 2 \u2264 n\nhx : 0 < x\n\u22a2 1 < n\n[PROOFSTEP]\nlinarith\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : 2 \u2264 n\nhx : 0 < x\nnpos : 0 < \u2191n - 1\n\u22a2 f \u2191n + x * log (\u2191n - 1) \u2264 f (\u2191n + x)\n[PROOFSTEP]\nhave c :=\n (convexOn_iff_slope_mono_adjacent.mp <| hf_conv).2 npos (by linarith : 0 < (n : \u211d) + x)\n (by linarith : (n : \u211d) - 1 < (n : \u211d)) (by linarith)\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : 2 \u2264 n\nhx : 0 < x\nnpos : 0 < \u2191n - 1\n\u22a2 0 < \u2191n + x\n[PROOFSTEP]\nlinarith\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : 2 \u2264 n\nhx : 0 < x\nnpos : 0 < \u2191n - 1\n\u22a2 \u2191n - 1 < \u2191n\n[PROOFSTEP]\nlinarith\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : 2 \u2264 n\nhx : 0 < x\nnpos : 0 < \u2191n - 1\n\u22a2 \u2191n < \u2191n + x\n[PROOFSTEP]\nlinarith\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : 2 \u2264 n\nhx : 0 < x\nnpos : 0 < \u2191n - 1\nc : (f \u2191n - f (\u2191n - 1)) / (\u2191n - (\u2191n - 1)) \u2264 (f (\u2191n + x) - f \u2191n) / (\u2191n + x - \u2191n)\n\u22a2 f \u2191n + x * log (\u2191n - 1) \u2264 f (\u2191n + x)\n[PROOFSTEP]\nrw [add_sub_cancel', sub_sub_cancel, div_one] at c \n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : 2 \u2264 n\nhx : 0 < x\nnpos : 0 < \u2191n - 1\nc : f \u2191n - f (\u2191n - 1) \u2264 (f (\u2191n + x) - f \u2191n) / x\n\u22a2 f \u2191n + x * log (\u2191n - 1) \u2264 f (\u2191n + x)\n[PROOFSTEP]\nhave : f (\u2191n - 1) = f n - log (\u2191n - 1) := by\n -- Porting note: was\n -- nth_rw_rhs 1 [(by ring : (n : \u211d) = \u2191n - 1 + 1)]\n -- rw [hf_feq npos, add_sub_cancel]rw [eq_sub_iff_add_eq, \u2190 hf_feq npos, sub_add_cancel]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : 2 \u2264 n\nhx : 0 < x\nnpos : 0 < \u2191n - 1\nc : f \u2191n - f (\u2191n - 1) \u2264 (f (\u2191n + x) - f \u2191n) / x\n\u22a2 f (\u2191n - 1) = f \u2191n - log (\u2191n - 1)\n[PROOFSTEP]\nrw [eq_sub_iff_add_eq, \u2190 hf_feq npos, sub_add_cancel]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhn : 2 \u2264 n\nhx : 0 < x\nnpos : 0 < \u2191n - 1\nc : f \u2191n - f (\u2191n - 1) \u2264 (f (\u2191n + x) - f \u2191n) / x\nthis : f (\u2191n - 1) = f \u2191n - log (\u2191n - 1)\n\u22a2 f \u2191n + x * log (\u2191n - 1) \u2264 f (\u2191n + x)\n[PROOFSTEP]\nrwa [this, le_div_iff hx, sub_sub_cancel, le_sub_iff_add_le, mul_comm _ x, add_comm] at c \n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\n\u22a2 logGammaSeq (x + 1) n = logGammaSeq x (n + 1) + log x - (x + 1) * (log (\u2191n + 1) - log \u2191n)\n[PROOFSTEP]\ndsimp only [Nat.factorial_succ, logGammaSeq]\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\n\u22a2 (x + 1) * log \u2191n + log \u2191n ! - \u2211 m in Finset.range (n + 1), log (x + 1 + \u2191m) =\n x * log \u2191(n + 1) + log \u2191((n + 1) * n !) - \u2211 m in Finset.range (n + 1 + 1), log (x + \u2191m) + log x -\n (x + 1) * (log (\u2191n + 1) - log \u2191n)\n[PROOFSTEP]\nconv_rhs => rw [Finset.sum_range_succ', Nat.cast_zero, add_zero]\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\n| x * log \u2191(n + 1) + log \u2191((n + 1) * n !) - \u2211 m in Finset.range (n + 1 + 1), log (x + \u2191m) + log x -\n (x + 1) * (log (\u2191n + 1) - log \u2191n)\n[PROOFSTEP]\nrw [Finset.sum_range_succ', Nat.cast_zero, add_zero]\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\n| x * log \u2191(n + 1) + log \u2191((n + 1) * n !) - \u2211 m in Finset.range (n + 1 + 1), log (x + \u2191m) + log x -\n (x + 1) * (log (\u2191n + 1) - log \u2191n)\n[PROOFSTEP]\nrw [Finset.sum_range_succ', Nat.cast_zero, add_zero]\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\n| x * log \u2191(n + 1) + log \u2191((n + 1) * n !) - \u2211 m in Finset.range (n + 1 + 1), log (x + \u2191m) + log x -\n (x + 1) * (log (\u2191n + 1) - log \u2191n)\n[PROOFSTEP]\nrw [Finset.sum_range_succ', Nat.cast_zero, add_zero]\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\n\u22a2 (x + 1) * log \u2191n + log \u2191n ! - \u2211 m in Finset.range (n + 1), log (x + 1 + \u2191m) =\n x * log \u2191(n + 1) + log \u2191((n + 1) * n !) - (\u2211 k in Finset.range (n + 1), log (x + \u2191(k + 1)) + log x) + log x -\n (x + 1) * (log (\u2191n + 1) - log \u2191n)\n[PROOFSTEP]\nrw [Nat.cast_mul, log_mul]\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\n\u22a2 (x + 1) * log \u2191n + log \u2191n ! - \u2211 m in Finset.range (n + 1), log (x + 1 + \u2191m) =\n x * log \u2191(n + 1) + (log \u2191(n + 1) + log \u2191n !) - (\u2211 k in Finset.range (n + 1), log (x + \u2191(k + 1)) + log x) + log x -\n (x + 1) * (log (\u2191n + 1) - log \u2191n)\ncase hx\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\n\u22a2 \u2191(n + 1) \u2260 0\ncase hy f : \u211d \u2192 \u211d x\u271d : \u211d n\u271d : \u2115 x : \u211d n : \u2115 \u22a2 \u2191n ! \u2260 0\n[PROOFSTEP]\nrotate_left\n[GOAL]\ncase hx\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\n\u22a2 \u2191(n + 1) \u2260 0\n[PROOFSTEP]\nrw [Nat.cast_ne_zero]\n[GOAL]\ncase hx\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\n\u22a2 n + 1 \u2260 0\n[PROOFSTEP]\nexact Nat.succ_ne_zero n\n[GOAL]\ncase hy\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\n\u22a2 \u2191n ! \u2260 0\n[PROOFSTEP]\nrw [Nat.cast_ne_zero]\n[GOAL]\ncase hy\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\n\u22a2 n ! \u2260 0\n[PROOFSTEP]\nexact Nat.factorial_ne_zero n\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\n\u22a2 (x + 1) * log \u2191n + log \u2191n ! - \u2211 m in Finset.range (n + 1), log (x + 1 + \u2191m) =\n x * log \u2191(n + 1) + (log \u2191(n + 1) + log \u2191n !) - (\u2211 k in Finset.range (n + 1), log (x + \u2191(k + 1)) + log x) + log x -\n (x + 1) * (log (\u2191n + 1) - log \u2191n)\n[PROOFSTEP]\nhave : \u2211 m : \u2115 in Finset.range (n + 1), log (x + 1 + \u2191m) = \u2211 k : \u2115 in Finset.range (n + 1), log (x + \u2191(k + 1)) :=\n by\n refine' Finset.sum_congr (by rfl) fun m _ => _\n congr 1\n push_cast\n abel\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\n\u22a2 \u2211 m in Finset.range (n + 1), log (x + 1 + \u2191m) = \u2211 k in Finset.range (n + 1), log (x + \u2191(k + 1))\n[PROOFSTEP]\nrefine' Finset.sum_congr (by rfl) fun m _ => _\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\n\u22a2 Finset.range (n + 1) = Finset.range (n + 1)\n[PROOFSTEP]\nrfl\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d\u00b9 : \u211d\nn\u271d : \u2115\nx : \u211d\nn m : \u2115\nx\u271d : m \u2208 Finset.range (n + 1)\n\u22a2 log (x + 1 + \u2191m) = log (x + \u2191(m + 1))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_x\nf : \u211d \u2192 \u211d\nx\u271d\u00b9 : \u211d\nn\u271d : \u2115\nx : \u211d\nn m : \u2115\nx\u271d : m \u2208 Finset.range (n + 1)\n\u22a2 x + 1 + \u2191m = x + \u2191(m + 1)\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase e_x\nf : \u211d \u2192 \u211d\nx\u271d\u00b9 : \u211d\nn\u271d : \u2115\nx : \u211d\nn m : \u2115\nx\u271d : m \u2208 Finset.range (n + 1)\n\u22a2 x + 1 + \u2191m = x + (\u2191m + 1)\n[PROOFSTEP]\nabel\n[GOAL]\ncase e_x\nf : \u211d \u2192 \u211d\nx\u271d\u00b9 : \u211d\nn\u271d : \u2115\nx : \u211d\nn m : \u2115\nx\u271d : m \u2208 Finset.range (n + 1)\n\u22a2 x + 1 + \u2191m = x + (\u2191m + 1)\n[PROOFSTEP]\nabel\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\nthis : \u2211 m in Finset.range (n + 1), log (x + 1 + \u2191m) = \u2211 k in Finset.range (n + 1), log (x + \u2191(k + 1))\n\u22a2 (x + 1) * log \u2191n + log \u2191n ! - \u2211 m in Finset.range (n + 1), log (x + 1 + \u2191m) =\n x * log \u2191(n + 1) + (log \u2191(n + 1) + log \u2191n !) - (\u2211 k in Finset.range (n + 1), log (x + \u2191(k + 1)) + log x) + log x -\n (x + 1) * (log (\u2191n + 1) - log \u2191n)\n[PROOFSTEP]\nrw [\u2190 this, Nat.cast_add_one n]\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nx : \u211d\nn : \u2115\nthis : \u2211 m in Finset.range (n + 1), log (x + 1 + \u2191m) = \u2211 k in Finset.range (n + 1), log (x + \u2191(k + 1))\n\u22a2 (x + 1) * log \u2191n + log \u2191n ! - \u2211 m in Finset.range (n + 1), log (x + 1 + \u2191m) =\n x * log (\u2191n + 1) + (log (\u2191n + 1) + log \u2191n !) - (\u2211 m in Finset.range (n + 1), log (x + 1 + \u2191m) + log x) + log x -\n (x + 1) * (log (\u2191n + 1) - log \u2191n)\n[PROOFSTEP]\nring\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\nn : \u2115\n\u22a2 f x \u2264 f 1 + x * log (\u2191n + 1) - x * log \u2191n + logGammaSeq x n\n[PROOFSTEP]\nrw [logGammaSeq, \u2190 add_sub_assoc, le_sub_iff_add_le, \u2190 f_add_nat_eq (@hf_feq) hx, add_comm x]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\nn : \u2115\n\u22a2 f (\u2191(n + 1) + x) \u2264 f 1 + x * log (\u2191n + 1) - x * log \u2191n + (x * log \u2191n + log \u2191n !)\n[PROOFSTEP]\nrefine' (f_add_nat_le hf_conv (@hf_feq) (Nat.add_one_ne_zero n) hx hx').trans (le_of_eq _)\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\nn : \u2115\n\u22a2 f \u2191(n + 1) + x * log \u2191(n + 1) = f 1 + x * log (\u2191n + 1) - x * log \u2191n + (x * log \u2191n + log \u2191n !)\n[PROOFSTEP]\nrw [f_nat_eq @hf_feq (by linarith : n + 1 \u2260 0), Nat.add_sub_cancel, Nat.cast_add_one]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\nn : \u2115\n\u22a2 n + 1 \u2260 0\n[PROOFSTEP]\nlinarith\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\nn : \u2115\n\u22a2 f 1 + log \u2191n ! + x * log (\u2191n + 1) = f 1 + x * log (\u2191n + 1) - x * log \u2191n + (x * log \u2191n + log \u2191n !)\n[PROOFSTEP]\nring\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhn : n \u2260 0\n\u22a2 f 1 + logGammaSeq x n \u2264 f x\n[PROOFSTEP]\ndsimp [logGammaSeq]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhn : n \u2260 0\n\u22a2 f 1 + (x * log \u2191n + log \u2191n ! - \u2211 m in Finset.range (n + 1), log (x + \u2191m)) \u2264 f x\n[PROOFSTEP]\nrw [\u2190 add_sub_assoc, sub_le_iff_le_add, \u2190 f_add_nat_eq (@hf_feq) hx, add_comm x _]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhn : n \u2260 0\n\u22a2 f 1 + (x * log \u2191n + log \u2191n !) \u2264 f (\u2191(n + 1) + x)\n[PROOFSTEP]\nrefine' le_trans (le_of_eq _) (f_add_nat_ge hf_conv @hf_feq _ hx)\n[GOAL]\ncase refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhn : n \u2260 0\n\u22a2 f 1 + (x * log \u2191n + log \u2191n !) = f \u2191(n + 1) + x * log (\u2191(n + 1) - 1)\n[PROOFSTEP]\nrw [f_nat_eq @hf_feq, Nat.add_sub_cancel, Nat.cast_add_one, add_sub_cancel]\n[GOAL]\ncase refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhn : n \u2260 0\n\u22a2 f 1 + (x * log \u2191n + log \u2191n !) = f 1 + log \u2191n ! + x * log \u2191n\n[PROOFSTEP]\nring\n[GOAL]\ncase refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhn : n \u2260 0\n\u22a2 n + 1 \u2260 0\n[PROOFSTEP]\nexact Nat.succ_ne_zero _\n[GOAL]\ncase refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhn : n \u2260 0\n\u22a2 2 \u2264 n + 1\n[PROOFSTEP]\napply Nat.succ_le_succ\n[GOAL]\ncase refine'_2.a\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhn : n \u2260 0\n\u22a2 1 \u2264 n\n[PROOFSTEP]\nlinarith [Nat.pos_of_ne_zero hn]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nrefine'\n tendsto_of_tendsto_of_tendsto_of_le_of_le' _ tendsto_const_nhds _\n _\n -- Porting note: `show` no longer reorders goals\n[GOAL]\ncase refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 \u2115 \u2192 \u211d\ncase refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 Tendsto ?refine'_1 atTop (\ud835\udcdd (f x - f 1))\ncase refine'_3\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 \u2200\u1da0 (b : \u2115) in atTop, ?refine'_1 b \u2264 logGammaSeq x b\ncase refine'_4\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 \u2200\u1da0 (b : \u2115) in atTop, logGammaSeq x b \u2264 f x - f 1\n[PROOFSTEP]\npick_goal 4\n[GOAL]\ncase refine'_4\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 \u2200\u1da0 (b : \u2115) in atTop, logGammaSeq x b \u2264 f x - f 1\ncase refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 \u2115 \u2192 \u211d\ncase refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 Tendsto ?refine'_1 atTop (\ud835\udcdd (f x - f 1))\ncase refine'_3\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 \u2200\u1da0 (b : \u2115) in atTop, ?refine'_1 b \u2264 logGammaSeq x b\n[PROOFSTEP]\nshow \u2200\u1da0 n : \u2115 in atTop, logGammaSeq x n \u2264 f x - f 1\n[GOAL]\ncase refine'_4\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, logGammaSeq x n \u2264 f x - f 1\n[PROOFSTEP]\nrefine' Eventually.mp (eventually_ne_atTop 0) (eventually_of_forall fun n hn => _)\n[GOAL]\ncase refine'_4\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\nn : \u2115\nhn : n \u2260 0\n\u22a2 logGammaSeq x n \u2264 f x - f 1\n[PROOFSTEP]\nexact\n le_sub_iff_add_le'.mpr\n (ge_logGammaSeq hf_conv (@hf_feq) hx hn)\n -- Porting note: `show` no longer reorders goals\n[GOAL]\ncase refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 \u2115 \u2192 \u211d\ncase refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 Tendsto ?refine'_1 atTop (\ud835\udcdd (f x - f 1))\ncase refine'_3\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 \u2200\u1da0 (b : \u2115) in atTop, ?refine'_1 b \u2264 logGammaSeq x b\n[PROOFSTEP]\npick_goal 3\n[GOAL]\ncase refine'_3\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 \u2200\u1da0 (b : \u2115) in atTop, ?refine'_1 b \u2264 logGammaSeq x b\ncase refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 \u2115 \u2192 \u211d\ncase refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 Tendsto ?refine'_1 atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nshow \u2200\u1da0 n : \u2115 in atTop, f x - f 1 - x * (log (n + 1) - log n) \u2264 logGammaSeq x n\n[GOAL]\ncase refine'_3\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, f x - f 1 - x * (log (\u2191n + 1) - log \u2191n) \u2264 logGammaSeq x n\n[PROOFSTEP]\nrefine' eventually_of_forall fun n => _\n[GOAL]\ncase refine'_3\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\nn : \u2115\n\u22a2 f x - f 1 - x * (log (\u2191n + 1) - log \u2191n) \u2264 logGammaSeq x n\n[PROOFSTEP]\nrw [sub_le_iff_le_add', sub_le_iff_le_add']\n[GOAL]\ncase refine'_3\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\nn : \u2115\n\u22a2 f x \u2264 f 1 + (x * (log (\u2191n + 1) - log \u2191n) + logGammaSeq x n)\n[PROOFSTEP]\nconvert le_logGammaSeq hf_conv (@hf_feq) hx hx' n using 1\n[GOAL]\ncase h.e'_4\nf : \u211d \u2192 \u211d\nx : \u211d\nn\u271d : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\nn : \u2115\n\u22a2 f 1 + (x * (log (\u2191n + 1) - log \u2191n) + logGammaSeq x n) = f 1 + x * log (\u2191n + 1) - x * log \u2191n + logGammaSeq x n\n[PROOFSTEP]\nring\n[GOAL]\ncase refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 Tendsto (fun b => f x - f 1 - x * (log (\u2191b + 1) - log \u2191b)) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nhave : f x - f 1 = f x - f 1 - x * 0 := by ring\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 f x - f 1 = f x - f 1 - x * 0\n[PROOFSTEP]\nring\n[GOAL]\ncase refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\nthis : f x - f 1 = f x - f 1 - x * 0\n\u22a2 Tendsto (fun b => f x - f 1 - x * (log (\u2191b + 1) - log \u2191b)) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nnth_rw 2 [this]\n[GOAL]\ncase refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nhx' : x \u2264 1\nthis : f x - f 1 = f x - f 1 - x * 0\n\u22a2 Tendsto (fun b => f x - f 1 - x * (log (\u2191b + 1) - log \u2191b)) atTop (\ud835\udcdd (f x - f 1 - x * 0))\n[PROOFSTEP]\nexact Tendsto.sub tendsto_const_nhds (tendsto_log_nat_add_one_sub_log.const_mul _)\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\n\u22a2 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nsuffices \u2200 m : \u2115, \u2191m < x \u2192 x \u2264 m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd <| f x - f 1)\n by\n refine' this \u2308x - 1\u2309\u208a _ _\n \u00b7 rcases lt_or_le x 1 with \u27e8\u27e9\n \u00b7 rwa [Nat.ceil_eq_zero.mpr (by linarith : x - 1 \u2264 0), Nat.cast_zero]\n \u00b7 convert Nat.ceil_lt_add_one (by linarith : 0 \u2264 x - 1)\n abel\n \u00b7 rw [\u2190 sub_le_iff_le_add]; exact Nat.le_ceil _\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nthis : \u2200 (m : \u2115), \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n\u22a2 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nrefine' this \u2308x - 1\u2309\u208a _ _\n[GOAL]\ncase refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nthis : \u2200 (m : \u2115), \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n\u22a2 \u2191\u2308x - 1\u2309\u208a < x\n[PROOFSTEP]\nrcases lt_or_le x 1 with \u27e8\u27e9\n[GOAL]\ncase refine'_1.inl\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nthis : \u2200 (m : \u2115), \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\nh\u271d : x < 1\n\u22a2 \u2191\u2308x - 1\u2309\u208a < x\n[PROOFSTEP]\nrwa [Nat.ceil_eq_zero.mpr (by linarith : x - 1 \u2264 0), Nat.cast_zero]\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nthis : \u2200 (m : \u2115), \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\nh\u271d : x < 1\n\u22a2 x - 1 \u2264 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine'_1.inr\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nthis : \u2200 (m : \u2115), \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\nh\u271d : 1 \u2264 x\n\u22a2 \u2191\u2308x - 1\u2309\u208a < x\n[PROOFSTEP]\nconvert Nat.ceil_lt_add_one (by linarith : 0 \u2264 x - 1)\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nthis : \u2200 (m : \u2115), \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\nh\u271d : 1 \u2264 x\n\u22a2 0 \u2264 x - 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h.e'_4\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nthis : \u2200 (m : \u2115), \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\nh\u271d : 1 \u2264 x\n\u22a2 x = x - 1 + 1\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_4\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nthis : \u2200 (m : \u2115), \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\nh\u271d : 1 \u2264 x\n\u22a2 x = x - 1 + 1\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nthis : \u2200 (m : \u2115), \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n\u22a2 x \u2264 \u2191\u2308x - 1\u2309\u208a + 1\n[PROOFSTEP]\nrw [\u2190 sub_le_iff_le_add]\n[GOAL]\ncase refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nthis : \u2200 (m : \u2115), \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n\u22a2 x - 1 \u2264 \u2191\u2308x - 1\u2309\u208a\n[PROOFSTEP]\nexact Nat.le_ceil _\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\n\u22a2 \u2200 (m : \u2115), \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nintro m\n[GOAL]\nf : \u211d \u2192 \u211d\nx : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx : 0 < x\nm : \u2115\n\u22a2 \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\ninduction' m with m hm generalizing x\n[GOAL]\ncase zero\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nx : \u211d\nhx : 0 < x\n\u22a2 \u2191Nat.zero < x \u2192 x \u2264 \u2191Nat.zero + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nrw [Nat.cast_zero, zero_add]\n[GOAL]\ncase zero\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nx : \u211d\nhx : 0 < x\n\u22a2 0 < x \u2192 x \u2264 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nexact fun _ hx' => tendsto_logGammaSeq_of_le_one hf_conv (@hf_feq) hx hx'\n[GOAL]\ncase succ\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nhm : \u2200 {x : \u211d}, 0 < x \u2192 \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\nx : \u211d\nhx : 0 < x\n\u22a2 \u2191(Nat.succ m) < x \u2192 x \u2264 \u2191(Nat.succ m) + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nintro hy hy'\n[GOAL]\ncase succ\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nhm : \u2200 {x : \u211d}, 0 < x \u2192 \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\nx : \u211d\nhx : 0 < x\nhy : \u2191(Nat.succ m) < x\nhy' : x \u2264 \u2191(Nat.succ m) + 1\n\u22a2 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nrw [Nat.cast_succ, \u2190 sub_le_iff_le_add] at hy' \n[GOAL]\ncase succ\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nhm : \u2200 {x : \u211d}, 0 < x \u2192 \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\nx : \u211d\nhx : 0 < x\nhy : \u2191(Nat.succ m) < x\nhy' : x - 1 \u2264 \u2191m + 1\n\u22a2 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nrw [Nat.cast_succ, \u2190 lt_sub_iff_add_lt] at hy \n[GOAL]\ncase succ\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nhm : \u2200 {x : \u211d}, 0 < x \u2192 \u2191m < x \u2192 x \u2264 \u2191m + 1 \u2192 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\n\u22a2 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nspecialize hm ((Nat.cast_nonneg _).trans_lt hy) hy hy'\n[GOAL]\ncase succ\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nhm : Tendsto (logGammaSeq (x - 1)) atTop (\ud835\udcdd (f (x - 1) - f 1))\n\u22a2 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nhave :\n \u2200\u1da0 n : \u2115 in atTop,\n logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1) :=\n by\n refine' Eventually.mp (eventually_ge_atTop 1) (eventually_of_forall fun n hn => _)\n have := logGammaSeq_add_one (x - 1) (n - 1)\n rw [sub_add_cancel, Nat.sub_add_cancel hn] at this \n rw [this]\n ring\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nhm : Tendsto (logGammaSeq (x - 1)) atTop (\ud835\udcdd (f (x - 1) - f 1))\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop,\n logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\n[PROOFSTEP]\nrefine' Eventually.mp (eventually_ge_atTop 1) (eventually_of_forall fun n hn => _)\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nhm : Tendsto (logGammaSeq (x - 1)) atTop (\ud835\udcdd (f (x - 1) - f 1))\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\n[PROOFSTEP]\nhave := logGammaSeq_add_one (x - 1) (n - 1)\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nhm : Tendsto (logGammaSeq (x - 1)) atTop (\ud835\udcdd (f (x - 1) - f 1))\nn : \u2115\nhn : 1 \u2264 n\nthis :\n logGammaSeq (x - 1 + 1) (n - 1) =\n logGammaSeq (x - 1) (n - 1 + 1) + log (x - 1) - (x - 1 + 1) * (log (\u2191(n - 1) + 1) - log \u2191(n - 1))\n\u22a2 logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\n[PROOFSTEP]\nrw [sub_add_cancel, Nat.sub_add_cancel hn] at this \n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nhm : Tendsto (logGammaSeq (x - 1)) atTop (\ud835\udcdd (f (x - 1) - f 1))\nn : \u2115\nhn : 1 \u2264 n\nthis : logGammaSeq x (n - 1) = logGammaSeq (x - 1) n + log (x - 1) - x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1))\n\u22a2 logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nhm : Tendsto (logGammaSeq (x - 1)) atTop (\ud835\udcdd (f (x - 1) - f 1))\nn : \u2115\nhn : 1 \u2264 n\nthis : logGammaSeq x (n - 1) = logGammaSeq (x - 1) n + log (x - 1) - x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1))\n\u22a2 logGammaSeq (x - 1) n =\n logGammaSeq (x - 1) n + log (x - 1) - x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) +\n x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) -\n log (x - 1)\n[PROOFSTEP]\nring\n[GOAL]\ncase succ\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nhm : Tendsto (logGammaSeq (x - 1)) atTop (\ud835\udcdd (f (x - 1) - f 1))\nthis :\n \u2200\u1da0 (n : \u2115) in atTop,\n logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\n\u22a2 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nreplace hm :=\n ((Tendsto.congr' this hm).add (tendsto_const_nhds : Tendsto (fun _ => log (x - 1)) _ _)).comp\n (tendsto_add_atTop_nat 1)\n[GOAL]\ncase succ\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nthis :\n \u2200\u1da0 (n : \u2115) in atTop,\n logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\nhm :\n Tendsto\n ((fun x_1 => logGammaSeq x (x_1 - 1) + x * (log (\u2191(x_1 - 1) + 1) - log \u2191(x_1 - 1)) - log (x - 1) + log (x - 1)) \u2218\n fun a => a + 1)\n atTop (\ud835\udcdd (f (x - 1) - f 1 + log (x - 1)))\n\u22a2 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nhave :\n ((fun x_1 : \u2115 =>\n (fun n : \u2115 => logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)) x_1 +\n (fun b : \u2115 => log (x - 1)) x_1) \u2218\n fun a : \u2115 => a + 1) =\n fun n => logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n) :=\n by\n ext1 n\n dsimp only [Function.comp_apply]\n rw [sub_add_cancel, Nat.add_sub_cancel]\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nthis :\n \u2200\u1da0 (n : \u2115) in atTop,\n logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\nhm :\n Tendsto\n ((fun x_1 => logGammaSeq x (x_1 - 1) + x * (log (\u2191(x_1 - 1) + 1) - log \u2191(x_1 - 1)) - log (x - 1) + log (x - 1)) \u2218\n fun a => a + 1)\n atTop (\ud835\udcdd (f (x - 1) - f 1 + log (x - 1)))\n\u22a2 ((fun x_1 =>\n (fun n => logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)) x_1 +\n (fun b => log (x - 1)) x_1) \u2218\n fun a => a + 1) =\n fun n => logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nthis :\n \u2200\u1da0 (n : \u2115) in atTop,\n logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\nhm :\n Tendsto\n ((fun x_1 => logGammaSeq x (x_1 - 1) + x * (log (\u2191(x_1 - 1) + 1) - log \u2191(x_1 - 1)) - log (x - 1) + log (x - 1)) \u2218\n fun a => a + 1)\n atTop (\ud835\udcdd (f (x - 1) - f 1 + log (x - 1)))\nn : \u2115\n\u22a2 ((fun x_1 =>\n (fun n => logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)) x_1 +\n (fun b => log (x - 1)) x_1) \u2218\n fun a => a + 1)\n n =\n logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)\n[PROOFSTEP]\ndsimp only [Function.comp_apply]\n[GOAL]\ncase h\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn\u271d : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nthis :\n \u2200\u1da0 (n : \u2115) in atTop,\n logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\nhm :\n Tendsto\n ((fun x_1 => logGammaSeq x (x_1 - 1) + x * (log (\u2191(x_1 - 1) + 1) - log \u2191(x_1 - 1)) - log (x - 1) + log (x - 1)) \u2218\n fun a => a + 1)\n atTop (\ud835\udcdd (f (x - 1) - f 1 + log (x - 1)))\nn : \u2115\n\u22a2 logGammaSeq x (n + 1 - 1) + x * (log (\u2191(n + 1 - 1) + 1) - log \u2191(n + 1 - 1)) - log (x - 1) + log (x - 1) =\n logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)\n[PROOFSTEP]\nrw [sub_add_cancel, Nat.add_sub_cancel]\n[GOAL]\ncase succ\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nthis\u271d :\n \u2200\u1da0 (n : \u2115) in atTop,\n logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\nhm :\n Tendsto\n ((fun x_1 => logGammaSeq x (x_1 - 1) + x * (log (\u2191(x_1 - 1) + 1) - log \u2191(x_1 - 1)) - log (x - 1) + log (x - 1)) \u2218\n fun a => a + 1)\n atTop (\ud835\udcdd (f (x - 1) - f 1 + log (x - 1)))\nthis :\n ((fun x_1 =>\n (fun n => logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)) x_1 +\n (fun b => log (x - 1)) x_1) \u2218\n fun a => a + 1) =\n fun n => logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)\n\u22a2 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nrw [this] at hm \n[GOAL]\ncase succ\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nthis\u271d :\n \u2200\u1da0 (n : \u2115) in atTop,\n logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\nhm : Tendsto (fun n => logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)) atTop (\ud835\udcdd (f (x - 1) - f 1 + log (x - 1)))\nthis :\n ((fun x_1 =>\n (fun n => logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)) x_1 +\n (fun b => log (x - 1)) x_1) \u2218\n fun a => a + 1) =\n fun n => logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)\n\u22a2 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (f x - f 1))\n[PROOFSTEP]\nconvert hm.sub (tendsto_log_nat_add_one_sub_log.const_mul x) using 2\n[GOAL]\ncase h.e'_3.h\nf : \u211d \u2192 \u211d\nx\u271d\u00b9 : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\u00b9\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nthis\u271d :\n \u2200\u1da0 (n : \u2115) in atTop,\n logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\nhm : Tendsto (fun n => logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)) atTop (\ud835\udcdd (f (x - 1) - f 1 + log (x - 1)))\nthis :\n ((fun x_1 =>\n (fun n => logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)) x_1 +\n (fun b => log (x - 1)) x_1) \u2218\n fun a => a + 1) =\n fun n => logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)\nx\u271d : \u2115\n\u22a2 logGammaSeq x x\u271d = logGammaSeq x x\u271d + x * (log (\u2191x\u271d + 1) - log \u2191x\u271d) - x * (log (\u2191x\u271d + 1) - log \u2191x\u271d)\n[PROOFSTEP]\nring\n[GOAL]\ncase h.e'_5.h.e'_3\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nthis\u271d :\n \u2200\u1da0 (n : \u2115) in atTop,\n logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\nhm : Tendsto (fun n => logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)) atTop (\ud835\udcdd (f (x - 1) - f 1 + log (x - 1)))\nthis :\n ((fun x_1 =>\n (fun n => logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)) x_1 +\n (fun b => log (x - 1)) x_1) \u2218\n fun a => a + 1) =\n fun n => logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)\n\u22a2 f x - f 1 = f (x - 1) - f 1 + log (x - 1) - x * 0\n[PROOFSTEP]\nhave := hf_feq ((Nat.cast_nonneg m).trans_lt hy)\n[GOAL]\ncase h.e'_5.h.e'_3\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nthis\u271d\u00b9 :\n \u2200\u1da0 (n : \u2115) in atTop,\n logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\nhm : Tendsto (fun n => logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)) atTop (\ud835\udcdd (f (x - 1) - f 1 + log (x - 1)))\nthis\u271d :\n ((fun x_1 =>\n (fun n => logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)) x_1 +\n (fun b => log (x - 1)) x_1) \u2218\n fun a => a + 1) =\n fun n => logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)\nthis : f (x - 1 + 1) = f (x - 1) + log (x - 1)\n\u22a2 f x - f 1 = f (x - 1) - f 1 + log (x - 1) - x * 0\n[PROOFSTEP]\nrw [sub_add_cancel] at this \n[GOAL]\ncase h.e'_5.h.e'_3\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nthis\u271d\u00b9 :\n \u2200\u1da0 (n : \u2115) in atTop,\n logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\nhm : Tendsto (fun n => logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)) atTop (\ud835\udcdd (f (x - 1) - f 1 + log (x - 1)))\nthis\u271d :\n ((fun x_1 =>\n (fun n => logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)) x_1 +\n (fun b => log (x - 1)) x_1) \u2218\n fun a => a + 1) =\n fun n => logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)\nthis : f x = f (x - 1) + log (x - 1)\n\u22a2 f x - f 1 = f (x - 1) - f 1 + log (x - 1) - x * 0\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase h.e'_5.h.e'_3\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nhf_conv : ConvexOn \u211d (Ioi 0) f\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y\nhx\u271d : 0 < x\u271d\nm : \u2115\nx : \u211d\nhx : 0 < x\nhy : \u2191m < x - 1\nhy' : x - 1 \u2264 \u2191m + 1\nthis\u271d\u00b9 :\n \u2200\u1da0 (n : \u2115) in atTop,\n logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)\nhm : Tendsto (fun n => logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)) atTop (\ud835\udcdd (f (x - 1) - f 1 + log (x - 1)))\nthis\u271d :\n ((fun x_1 =>\n (fun n => logGammaSeq x (n - 1) + x * (log (\u2191(n - 1) + 1) - log \u2191(n - 1)) - log (x - 1)) x_1 +\n (fun b => log (x - 1)) x_1) \u2218\n fun a => a + 1) =\n fun n => logGammaSeq x n + x * (log (\u2191n + 1) - log \u2191n)\nthis : f x = f (x - 1) + log (x - 1)\n\u22a2 f (x - 1) + log (x - 1) - f 1 = f (x - 1) - f 1 + log (x - 1) - x * 0\n[PROOFSTEP]\nring\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nx : \u211d\nhx : 0 < x\n\u22a2 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (log (Gamma x)))\n[PROOFSTEP]\nhave : log (Gamma x) = (log \u2218 Gamma) x - (log \u2218 Gamma) 1 := by\n simp_rw [Function.comp_apply, Gamma_one, log_one, sub_zero]\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nx : \u211d\nhx : 0 < x\n\u22a2 log (Gamma x) = (log \u2218 Gamma) x - (log \u2218 Gamma) 1\n[PROOFSTEP]\nsimp_rw [Function.comp_apply, Gamma_one, log_one, sub_zero]\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nx : \u211d\nhx : 0 < x\nthis : log (Gamma x) = (log \u2218 Gamma) x - (log \u2218 Gamma) 1\n\u22a2 Tendsto (logGammaSeq x) atTop (\ud835\udcdd (log (Gamma x)))\n[PROOFSTEP]\nrw [this]\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nx : \u211d\nhx : 0 < x\nthis : log (Gamma x) = (log \u2218 Gamma) x - (log \u2218 Gamma) 1\n\u22a2 Tendsto (logGammaSeq x) atTop (\ud835\udcdd ((log \u2218 Gamma) x - (log \u2218 Gamma) 1))\n[PROOFSTEP]\nrefine' BohrMollerup.tendsto_logGammaSeq convexOn_log_Gamma (fun {y} hy => _) hx\n[GOAL]\nf : \u211d \u2192 \u211d\nx\u271d : \u211d\nn : \u2115\nx : \u211d\nhx : 0 < x\nthis : log (Gamma x) = (log \u2218 Gamma) x - (log \u2218 Gamma) 1\ny : \u211d\nhy : 0 < y\n\u22a2 (log \u2218 Gamma) (y + 1) = (log \u2218 Gamma) y + log y\n[PROOFSTEP]\nrw [Function.comp_apply, Gamma_add_one hy.ne', log_mul hy.ne' (Gamma_pos_of_pos hy).ne', add_comm, Function.comp_apply]\n[GOAL]\nf : \u211d \u2192 \u211d\nhf_conv : ConvexOn \u211d (Ioi 0) (log \u2218 f)\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = y * f y\nhf_pos : \u2200 {y : \u211d}, 0 < y \u2192 0 < f y\nhf_one : f 1 = 1\n\u22a2 EqOn f Gamma (Ioi 0)\n[PROOFSTEP]\nsuffices : EqOn (log \u2218 f) (log \u2218 Gamma) (Ioi (0 : \u211d))\n[GOAL]\nf : \u211d \u2192 \u211d\nhf_conv : ConvexOn \u211d (Ioi 0) (log \u2218 f)\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = y * f y\nhf_pos : \u2200 {y : \u211d}, 0 < y \u2192 0 < f y\nhf_one : f 1 = 1\nthis : EqOn (log \u2218 f) (log \u2218 Gamma) (Ioi 0)\n\u22a2 EqOn f Gamma (Ioi 0)\ncase this\nf : \u211d \u2192 \u211d\nhf_conv : ConvexOn \u211d (Ioi 0) (log \u2218 f)\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = y * f y\nhf_pos : \u2200 {y : \u211d}, 0 < y \u2192 0 < f y\nhf_one : f 1 = 1\n\u22a2 EqOn (log \u2218 f) (log \u2218 Gamma) (Ioi 0)\n[PROOFSTEP]\nexact fun x hx => log_injOn_pos (hf_pos hx) (Gamma_pos_of_pos hx) (this hx)\n[GOAL]\ncase this\nf : \u211d \u2192 \u211d\nhf_conv : ConvexOn \u211d (Ioi 0) (log \u2218 f)\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = y * f y\nhf_pos : \u2200 {y : \u211d}, 0 < y \u2192 0 < f y\nhf_one : f 1 = 1\n\u22a2 EqOn (log \u2218 f) (log \u2218 Gamma) (Ioi 0)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase this\nf : \u211d \u2192 \u211d\nhf_conv : ConvexOn \u211d (Ioi 0) (log \u2218 f)\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = y * f y\nhf_pos : \u2200 {y : \u211d}, 0 < y \u2192 0 < f y\nhf_one : f 1 = 1\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 (log \u2218 f) x = (log \u2218 Gamma) x\n[PROOFSTEP]\nhave e1 := BohrMollerup.tendsto_logGammaSeq hf_conv ?_ hx\n[GOAL]\ncase this.refine_2\nf : \u211d \u2192 \u211d\nhf_conv : ConvexOn \u211d (Ioi 0) (log \u2218 f)\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = y * f y\nhf_pos : \u2200 {y : \u211d}, 0 < y \u2192 0 < f y\nhf_one : f 1 = 1\nx : \u211d\nhx : x \u2208 Ioi 0\ne1 : Tendsto (BohrMollerup.logGammaSeq x) atTop (\ud835\udcdd ((log \u2218 f) x - (log \u2218 f) 1))\n\u22a2 (log \u2218 f) x = (log \u2218 Gamma) x\n[PROOFSTEP]\nrw [Function.comp_apply (f := log) (g := f) (x := 1), hf_one, log_one, sub_zero] at e1 \n[GOAL]\ncase this.refine_2\nf : \u211d \u2192 \u211d\nhf_conv : ConvexOn \u211d (Ioi 0) (log \u2218 f)\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = y * f y\nhf_pos : \u2200 {y : \u211d}, 0 < y \u2192 0 < f y\nhf_one : f 1 = 1\nx : \u211d\nhx : x \u2208 Ioi 0\ne1 : Tendsto (BohrMollerup.logGammaSeq x) atTop (\ud835\udcdd ((log \u2218 f) x))\n\u22a2 (log \u2218 f) x = (log \u2218 Gamma) x\n[PROOFSTEP]\nexact tendsto_nhds_unique e1 (BohrMollerup.tendsto_log_gamma hx)\n[GOAL]\ncase this.refine_1\nf : \u211d \u2192 \u211d\nhf_conv : ConvexOn \u211d (Ioi 0) (log \u2218 f)\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = y * f y\nhf_pos : \u2200 {y : \u211d}, 0 < y \u2192 0 < f y\nhf_one : f 1 = 1\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 \u2200 {y : \u211d}, 0 < y \u2192 (log \u2218 f) (y + 1) = (log \u2218 f) y + log y\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase this.refine_1\nf : \u211d \u2192 \u211d\nhf_conv : ConvexOn \u211d (Ioi 0) (log \u2218 f)\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = y * f y\nhf_pos : \u2200 {y : \u211d}, 0 < y \u2192 0 < f y\nhf_one : f 1 = 1\nx : \u211d\nhx : x \u2208 Ioi 0\ny : \u211d\nhy : 0 < y\n\u22a2 (log \u2218 f) (y + 1) = (log \u2218 f) y + log y\n[PROOFSTEP]\nrw [Function.comp_apply, Function.comp_apply, hf_feq hy, log_mul hy.ne' (hf_pos hy).ne']\n[GOAL]\ncase this.refine_1\nf : \u211d \u2192 \u211d\nhf_conv : ConvexOn \u211d (Ioi 0) (log \u2218 f)\nhf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = y * f y\nhf_pos : \u2200 {y : \u211d}, 0 < y \u2192 0 < f y\nhf_one : f 1 = 1\nx : \u211d\nhx : x \u2208 Ioi 0\ny : \u211d\nhy : 0 < y\n\u22a2 log y + log (f y) = log (f y) + log y\n[PROOFSTEP]\nring\n[GOAL]\n\u22a2 Gamma 2 = 1\n[PROOFSTEP]\nsimpa [one_add_one_eq_two] using Gamma_nat_eq_factorial 1\n[GOAL]\n\u22a2 Gamma (3 / 2) < 1\n[PROOFSTEP]\nhave A : (0 : \u211d) < 3 / 2 := by norm_num\n[GOAL]\n\u22a2 0 < 3 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nA : 0 < 3 / 2\n\u22a2 Gamma (3 / 2) < 1\n[PROOFSTEP]\nhave :=\n BohrMollerup.f_add_nat_le convexOn_log_Gamma (fun {y} hy => ?_) two_ne_zero one_half_pos\n (by norm_num : 1 / 2 \u2264 (1 : \u211d))\n[GOAL]\nA : 0 < 3 / 2\n\u22a2 1 / 2 \u2264 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine_2\nA : 0 < 3 / 2\nthis : (log \u2218 Gamma) (\u21912 + 1 / 2) \u2264 (log \u2218 Gamma) \u21912 + 1 / 2 * log \u21912\n\u22a2 Gamma (3 / 2) < 1\ncase refine_1 A : 0 < 3 / 2 y : \u211d hy : 0 < y \u22a2 (log \u2218 Gamma) (y + 1) = (log \u2218 Gamma) y + log y\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine_1\nA : 0 < 3 / 2\ny : \u211d\nhy : 0 < y\n\u22a2 (log \u2218 Gamma) (y + 1) = (log \u2218 Gamma) y + log y\n[PROOFSTEP]\nrw [Function.comp_apply, Gamma_add_one hy.ne', log_mul hy.ne' (Gamma_pos_of_pos hy).ne', add_comm, Function.comp_apply]\n[GOAL]\ncase refine_2\nA : 0 < 3 / 2\nthis : (log \u2218 Gamma) (\u21912 + 1 / 2) \u2264 (log \u2218 Gamma) \u21912 + 1 / 2 * log \u21912\n\u22a2 Gamma (3 / 2) < 1\n[PROOFSTEP]\nrw [Function.comp_apply, Function.comp_apply, Nat.cast_two, Gamma_two, log_one, zero_add,\n (by norm_num : (2 : \u211d) + 1 / 2 = 3 / 2 + 1), Gamma_add_one A.ne', log_mul A.ne' (Gamma_pos_of_pos A).ne', \u2190\n le_sub_iff_add_le', log_le_iff_le_exp (Gamma_pos_of_pos A)] at this \n[GOAL]\nA : 0 < 3 / 2\nthis : log (Gamma (2 + 1 / 2)) \u2264 1 / 2 * log 2\n\u22a2 2 + 1 / 2 = 3 / 2 + 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine_2\nA : 0 < 3 / 2\nthis : Gamma (3 / 2) \u2264 rexp (1 / 2 * log 2 - log (3 / 2))\n\u22a2 Gamma (3 / 2) < 1\n[PROOFSTEP]\nrefine' this.trans_lt (exp_lt_one_iff.mpr _)\n[GOAL]\ncase refine_2\nA : 0 < 3 / 2\nthis : Gamma (3 / 2) \u2264 rexp (1 / 2 * log 2 - log (3 / 2))\n\u22a2 1 / 2 * log 2 - log (3 / 2) < 0\n[PROOFSTEP]\nrw [mul_comm, \u2190 mul_div_assoc, div_sub' _ _ (2 : \u211d) two_ne_zero]\n[GOAL]\ncase refine_2\nA : 0 < 3 / 2\nthis : Gamma (3 / 2) \u2264 rexp (1 / 2 * log 2 - log (3 / 2))\n\u22a2 (log 2 * 1 - 2 * log (3 / 2)) / 2 < 0\n[PROOFSTEP]\nrefine' div_neg_of_neg_of_pos _ two_pos\n[GOAL]\ncase refine_2\nA : 0 < 3 / 2\nthis : Gamma (3 / 2) \u2264 rexp (1 / 2 * log 2 - log (3 / 2))\n\u22a2 log 2 * 1 - 2 * log (3 / 2) < 0\n[PROOFSTEP]\nrw [sub_neg, mul_one, \u2190 Nat.cast_two, \u2190 log_pow, \u2190 exp_lt_exp, Nat.cast_two, exp_log two_pos, exp_log]\n[GOAL]\ncase refine_2\nA : 0 < 3 / 2\nthis : Gamma (3 / 2) \u2264 rexp (1 / 2 * log 2 - log (3 / 2))\n\u22a2 2 < (3 / 2) ^ 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine_2\nA : 0 < 3 / 2\nthis : Gamma (3 / 2) \u2264 rexp (1 / 2 * log 2 - log (3 / 2))\n\u22a2 0 < (3 / 2) ^ 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 StrictMonoOn Gamma (Ici 2)\n[PROOFSTEP]\nconvert\n convexOn_Gamma.strict_mono_of_lt (by norm_num : (0 : \u211d) < 3 / 2) (by norm_num : (3 / 2 : \u211d) < 2)\n (Gamma_two.symm \u25b8 Gamma_three_div_two_lt_one)\n[GOAL]\n\u22a2 0 < 3 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 3 / 2 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_6\n\u22a2 Ici 2 = Ioi 0 \u2229 Ici 2\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_6\n\u22a2 Ioi 0 \u2229 Ici 2 = Ici 2\n[PROOFSTEP]\nrw [inter_eq_right_iff_subset]\n[GOAL]\ncase h.e'_6\n\u22a2 Ici 2 \u2286 Ioi 0\n[PROOFSTEP]\nexact fun x hx => two_pos.trans_le <| mem_Ici.mp hx\n[GOAL]\ns : \u211d\nhs : s \u2260 0\n\u22a2 doublingGamma (s + 1) = s * doublingGamma s\n[PROOFSTEP]\nrw [doublingGamma, doublingGamma, (by abel : s + 1 - 1 = s - 1 + 1), add_div, add_assoc, add_halves (1 : \u211d),\n Gamma_add_one (div_ne_zero hs two_ne_zero), rpow_add two_pos, rpow_one]\n[GOAL]\ns : \u211d\nhs : s \u2260 0\n\u22a2 s + 1 - 1 = s - 1 + 1\n[PROOFSTEP]\nabel\n[GOAL]\ns : \u211d\nhs : s \u2260 0\n\u22a2 s + 1 - 1 = s - 1 + 1\n[PROOFSTEP]\nabel\n[GOAL]\ns : \u211d\nhs : s \u2260 0\n\u22a2 Gamma (s / 2 + 1 / 2) * (s / 2 * Gamma (s / 2)) * (2 ^ (s - 1) * 2) / sqrt \u03c0 =\n s * (Gamma (s / 2) * Gamma (s / 2 + 1 / 2) * 2 ^ (s - 1) / sqrt \u03c0)\n[PROOFSTEP]\nring\n[GOAL]\n\u22a2 doublingGamma 1 = 1\n[PROOFSTEP]\nsimp_rw [doublingGamma, Gamma_one_half_eq, add_halves (1 : \u211d), sub_self, Gamma_one, mul_one, rpow_zero, mul_one,\n div_self (sqrt_ne_zero'.mpr pi_pos)]\n[GOAL]\n\u22a2 EqOn (log \u2218 doublingGamma) (fun s => log (Gamma (s / 2)) + log (Gamma (s / 2 + 1 / 2)) + s * log 2 - log (2 * sqrt \u03c0))\n (Ioi 0)\n[PROOFSTEP]\nintro s hs\n[GOAL]\ns : \u211d\nhs : s \u2208 Ioi 0\n\u22a2 (log \u2218 doublingGamma) s =\n (fun s => log (Gamma (s / 2)) + log (Gamma (s / 2 + 1 / 2)) + s * log 2 - log (2 * sqrt \u03c0)) s\n[PROOFSTEP]\nhave h1 : sqrt \u03c0 \u2260 0 := sqrt_ne_zero'.mpr pi_pos\n[GOAL]\ns : \u211d\nhs : s \u2208 Ioi 0\nh1 : sqrt \u03c0 \u2260 0\n\u22a2 (log \u2218 doublingGamma) s =\n (fun s => log (Gamma (s / 2)) + log (Gamma (s / 2 + 1 / 2)) + s * log 2 - log (2 * sqrt \u03c0)) s\n[PROOFSTEP]\nhave h2 : Gamma (s / 2) \u2260 0 := (Gamma_pos_of_pos <| div_pos hs two_pos).ne'\n[GOAL]\ns : \u211d\nhs : s \u2208 Ioi 0\nh1 : sqrt \u03c0 \u2260 0\nh2 : Gamma (s / 2) \u2260 0\n\u22a2 (log \u2218 doublingGamma) s =\n (fun s => log (Gamma (s / 2)) + log (Gamma (s / 2 + 1 / 2)) + s * log 2 - log (2 * sqrt \u03c0)) s\n[PROOFSTEP]\nhave h3 : Gamma (s / 2 + 1 / 2) \u2260 0 := (Gamma_pos_of_pos <| add_pos (div_pos hs two_pos) one_half_pos).ne'\n[GOAL]\ns : \u211d\nhs : s \u2208 Ioi 0\nh1 : sqrt \u03c0 \u2260 0\nh2 : Gamma (s / 2) \u2260 0\nh3 : Gamma (s / 2 + 1 / 2) \u2260 0\n\u22a2 (log \u2218 doublingGamma) s =\n (fun s => log (Gamma (s / 2)) + log (Gamma (s / 2 + 1 / 2)) + s * log 2 - log (2 * sqrt \u03c0)) s\n[PROOFSTEP]\nhave h4 : (2 : \u211d) ^ (s - 1) \u2260 0 := (rpow_pos_of_pos two_pos _).ne'\n[GOAL]\ns : \u211d\nhs : s \u2208 Ioi 0\nh1 : sqrt \u03c0 \u2260 0\nh2 : Gamma (s / 2) \u2260 0\nh3 : Gamma (s / 2 + 1 / 2) \u2260 0\nh4 : 2 ^ (s - 1) \u2260 0\n\u22a2 (log \u2218 doublingGamma) s =\n (fun s => log (Gamma (s / 2)) + log (Gamma (s / 2 + 1 / 2)) + s * log 2 - log (2 * sqrt \u03c0)) s\n[PROOFSTEP]\nrw [Function.comp_apply, doublingGamma, log_div (mul_ne_zero (mul_ne_zero h2 h3) h4) h1, log_mul (mul_ne_zero h2 h3) h4,\n log_mul h2 h3, log_rpow two_pos, log_mul two_ne_zero h1]\n[GOAL]\ns : \u211d\nhs : s \u2208 Ioi 0\nh1 : sqrt \u03c0 \u2260 0\nh2 : Gamma (s / 2) \u2260 0\nh3 : Gamma (s / 2 + 1 / 2) \u2260 0\nh4 : 2 ^ (s - 1) \u2260 0\n\u22a2 log (Gamma (s / 2)) + log (Gamma (s / 2 + 1 / 2)) + (s - 1) * log 2 - log (sqrt \u03c0) =\n (fun s => log (Gamma (s / 2)) + log (Gamma (s / 2 + 1 / 2)) + s * log 2 - (log 2 + log (sqrt \u03c0))) s\n[PROOFSTEP]\nring_nf\n[GOAL]\n\u22a2 ConvexOn \u211d (Ioi 0) (log \u2218 doublingGamma)\n[PROOFSTEP]\nrefine' (((ConvexOn.add _ _).add _).add_const _).congr log_doublingGamma_eq.symm\n[GOAL]\ncase refine'_1\n\u22a2 ConvexOn \u211d (Ioi 0) fun s => log (Gamma (s / 2))\n[PROOFSTEP]\nconvert convexOn_log_Gamma.comp_affineMap (DistribMulAction.toLinearMap \u211d \u211d (1 / 2 : \u211d)).toAffineMap using 1\n[GOAL]\ncase h.e'_9\n\u22a2 Ioi 0 = \u2191(LinearMap.toAffineMap (DistribMulAction.toLinearMap \u211d \u211d (1 / 2))) \u207b\u00b9' Ioi 0\n[PROOFSTEP]\nsimpa only [zero_div] using (preimage_const_mul_Ioi (0 : \u211d) one_half_pos).symm\n[GOAL]\ncase h.e'_10\n\u22a2 (fun s => log (Gamma (s / 2))) = (log \u2218 Gamma) \u2218 \u2191(LinearMap.toAffineMap (DistribMulAction.toLinearMap \u211d \u211d (1 / 2)))\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h.e'_10.h\nx : \u211d\n\u22a2 log (Gamma (x / 2)) = ((log \u2218 Gamma) \u2218 \u2191(LinearMap.toAffineMap (DistribMulAction.toLinearMap \u211d \u211d (1 / 2)))) x\n[PROOFSTEP]\nsimp only [LinearMap.coe_toAffineMap, Function.comp_apply, DistribMulAction.toLinearMap_apply]\n[GOAL]\ncase h.e'_10.h\nx : \u211d\n\u22a2 log (Gamma (x / 2)) = log (Gamma ((1 / 2) \u2022 x))\n[PROOFSTEP]\nrw [smul_eq_mul, mul_comm, mul_one_div]\n[GOAL]\ncase refine'_2\n\u22a2 ConvexOn \u211d (Ioi 0) fun s => log (Gamma (s / 2 + 1 / 2))\n[PROOFSTEP]\nrefine' ConvexOn.subset _ (Ioi_subset_Ioi <| neg_one_lt_zero.le) (convex_Ioi _)\n[GOAL]\ncase refine'_2\n\u22a2 ConvexOn \u211d (Ioi (-1)) fun s => log (Gamma (s / 2 + 1 / 2))\n[PROOFSTEP]\nconvert\n convexOn_log_Gamma.comp_affineMap\n ((DistribMulAction.toLinearMap \u211d \u211d (1 / 2 : \u211d)).toAffineMap + AffineMap.const \u211d \u211d (1 / 2 : \u211d)) using\n 1\n[GOAL]\ncase h.e'_9\n\u22a2 Ioi (-1) = \u2191(LinearMap.toAffineMap (DistribMulAction.toLinearMap \u211d \u211d (1 / 2)) + AffineMap.const \u211d \u211d (1 / 2)) \u207b\u00b9' Ioi 0\n[PROOFSTEP]\nchange Ioi (-1 : \u211d) = ((fun x : \u211d => x + 1 / 2) \u2218 fun x : \u211d => (1 / 2 : \u211d) * x) \u207b\u00b9' Ioi 0\n[GOAL]\ncase h.e'_9\n\u22a2 Ioi (-1) = ((fun x => x + 1 / 2) \u2218 fun x => 1 / 2 * x) \u207b\u00b9' Ioi 0\n[PROOFSTEP]\nrw [preimage_comp, preimage_add_const_Ioi, zero_sub, preimage_const_mul_Ioi (_ : \u211d) one_half_pos, neg_div,\n div_self (@one_half_pos \u211d _).ne']\n[GOAL]\ncase h.e'_10\n\u22a2 (fun s => log (Gamma (s / 2 + 1 / 2))) =\n (log \u2218 Gamma) \u2218 \u2191(LinearMap.toAffineMap (DistribMulAction.toLinearMap \u211d \u211d (1 / 2)) + AffineMap.const \u211d \u211d (1 / 2))\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h.e'_10.h\nx : \u211d\n\u22a2 log (Gamma (x / 2 + 1 / 2)) =\n ((log \u2218 Gamma) \u2218 \u2191(LinearMap.toAffineMap (DistribMulAction.toLinearMap \u211d \u211d (1 / 2)) + AffineMap.const \u211d \u211d (1 / 2)))\n x\n[PROOFSTEP]\nchange log (Gamma (x / 2 + 1 / 2)) = log (Gamma ((1 / 2 : \u211d) \u2022 x + 1 / 2))\n[GOAL]\ncase h.e'_10.h\nx : \u211d\n\u22a2 log (Gamma (x / 2 + 1 / 2)) = log (Gamma ((1 / 2) \u2022 x + 1 / 2))\n[PROOFSTEP]\nrw [smul_eq_mul, mul_comm, mul_one_div]\n[GOAL]\ncase refine'_3\n\u22a2 ConvexOn \u211d (Ioi 0) fun s => s * log 2\n[PROOFSTEP]\nsimpa only [mul_comm _ (log _)] using (convexOn_id (convex_Ioi (0 : \u211d))).smul (log_pos one_lt_two).le\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 doublingGamma s = Gamma s\n[PROOFSTEP]\nrefine'\n eq_Gamma_of_log_convex doublingGamma_log_convex_Ioi (fun {y} hy => doublingGamma_add_one y hy.ne') (fun {y} hy => _)\n doublingGamma_one hs\n[GOAL]\ns : \u211d\nhs : 0 < s\ny : \u211d\nhy : 0 < y\n\u22a2 0 < doublingGamma y\n[PROOFSTEP]\napply_rules [mul_pos, Gamma_pos_of_pos, add_pos, inv_pos_of_pos, rpow_pos_of_pos, two_pos, one_pos,\n sqrt_pos_of_pos pi_pos]\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt \u03c0\n[PROOFSTEP]\nrw [\u2190 doublingGamma_eq_Gamma (mul_pos two_pos hs), doublingGamma, mul_div_cancel_left _ (two_ne_zero' \u211d),\n (by abel : 1 - 2 * s = -(2 * s - 1)), rpow_neg zero_le_two]\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 1 - 2 * s = -(2 * s - 1)\n[PROOFSTEP]\nabel\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 1 - 2 * s = -(2 * s - 1)\n[PROOFSTEP]\nabel\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 Gamma s * Gamma (s + 1 / 2) = Gamma s * Gamma (s + 1 / 2) * 2 ^ (2 * s - 1) / sqrt \u03c0 * (2 ^ (2 * s - 1))\u207b\u00b9 * sqrt \u03c0\n[PROOFSTEP]\nfield_simp [(sqrt_pos_of_pos pi_pos).ne', (rpow_pos_of_pos two_pos (2 * s - 1)).ne']\n[GOAL]\ns : \u211d\nhs : 0 < s\n\u22a2 Gamma s * Gamma ((s * 2 + 1) / 2) * (sqrt \u03c0 * 2 ^ (2 * s - 1)) =\n Gamma s * Gamma ((s * 2 + 1) / 2) * 2 ^ (2 * s - 1) * sqrt \u03c0\n[PROOFSTEP]\nring\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "llama_tokens": 55891, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124811, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4234962601540015}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\n\u03b9 : Sort u_3\ninst\u271d : CompleteLattice \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192o \u03b2\n\u22a2 \u2191(\u2a05 (i : \u03b9), f i) = \u2a05 (i : \u03b9), \u2191(f i)\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\n\u03b9 : Sort u_3\ninst\u271d : CompleteLattice \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192o \u03b2\nx : \u03b1\n\u22a2 \u2191(\u2a05 (i : \u03b9), f i) x = iInf (fun i => \u2191(f i)) x\n[PROOFSTEP]\nsimp [iInf_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\n\u03b9 : Sort u_3\ninst\u271d : CompleteLattice \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192o \u03b2\n\u22a2 \u2191(\u2a06 (i : \u03b9), f i) = \u2a06 (i : \u03b9), \u2191(f i)\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\n\u03b9 : Sort u_3\ninst\u271d : CompleteLattice \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192o \u03b2\nx : \u03b1\n\u22a2 \u2191(\u2a06 (i : \u03b9), f i) x = iSup (fun i => \u2191(f i)) x\n[PROOFSTEP]\nsimp [iSup_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : CompleteLattice \u03b2\nsrc\u271d\u00b2 : Lattice (\u03b1 \u2192o \u03b2) := lattice\nsrc\u271d\u00b9 : OrderTop (\u03b1 \u2192o \u03b2) := orderTop\nsrc\u271d : OrderBot (\u03b1 \u2192o \u03b2) := orderBot\ns : Set (\u03b1 \u2192o \u03b2)\nf : \u03b1 \u2192o \u03b2\nhf : f \u2208 s\nx : \u03b1\n\u22a2 toFun f x \u2264 \u2a06 (_ : f \u2208 s), \u2191f x\n[PROOFSTEP]\napply le_iSup _ hf\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\n\u22a2 (\u2200 (n\u2081 n\u2082 : \u2115) (a\u2081 a\u2082 : \u03b1), (\u2191f)^[n\u2081 + n\u2082] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[n\u2081] a\u2081 \u2294 (\u2191f)^[n\u2082] a\u2082) \u2194\n \u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\n\u22a2 (\u2200 (n\u2081 n\u2082 : \u2115) (a\u2081 a\u2082 : \u03b1), (\u2191f)^[n\u2081 + n\u2082] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[n\u2081] a\u2081 \u2294 (\u2191f)^[n\u2082] a\u2082) \u2192\n \u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\n\u22a2 (\u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082) \u2192\n \u2200 (n\u2081 n\u2082 : \u2115) (a\u2081 a\u2082 : \u03b1), (\u2191f)^[n\u2081 + n\u2082] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[n\u2081] a\u2081 \u2294 (\u2191f)^[n\u2082] a\u2082\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\nh : \u2200 (n\u2081 n\u2082 : \u2115) (a\u2081 a\u2082 : \u03b1), (\u2191f)^[n\u2081 + n\u2082] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[n\u2081] a\u2081 \u2294 (\u2191f)^[n\u2082] a\u2082\n\u22a2 \u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082\n[PROOFSTEP]\nexact h 1 0\n[GOAL]\ncase mpr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\nh : \u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082\n\u22a2 \u2200 (n\u2081 n\u2082 : \u2115) (a\u2081 a\u2082 : \u03b1), (\u2191f)^[n\u2081 + n\u2082] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[n\u2081] a\u2081 \u2294 (\u2191f)^[n\u2082] a\u2082\n[PROOFSTEP]\nintro n\u2081 n\u2082 a\u2081 a\u2082\n[GOAL]\ncase mpr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\nh : \u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082\nn\u2081 n\u2082 : \u2115\na\u2081 a\u2082 : \u03b1\n\u22a2 (\u2191f)^[n\u2081 + n\u2082] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[n\u2081] a\u2081 \u2294 (\u2191f)^[n\u2082] a\u2082\n[PROOFSTEP]\nhave h' : \u2200 n a\u2081 a\u2082, f^[n] (a\u2081 \u2294 a\u2082) \u2264 f^[n] a\u2081 \u2294 a\u2082 := by\n intro n\n induction' n with n ih <;> intro a\u2081 a\u2082\n \u00b7 rfl\n \u00b7\n calc\n f^[n + 1] (a\u2081 \u2294 a\u2082) = f^[n] (f (a\u2081 \u2294 a\u2082)) := Function.iterate_succ_apply f n _\n _ \u2264 f^[n] (f a\u2081 \u2294 a\u2082) := (f.mono.iterate n (h a\u2081 a\u2082))\n _ \u2264 f^[n] (f a\u2081) \u2294 a\u2082 := (ih _ _)\n _ = f^[n + 1] a\u2081 \u2294 a\u2082 := by rw [\u2190 Function.iterate_succ_apply]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\nh : \u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082\nn\u2081 n\u2082 : \u2115\na\u2081 a\u2082 : \u03b1\n\u22a2 \u2200 (n : \u2115) (a\u2081 a\u2082 : \u03b1), (\u2191f)^[n] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[n] a\u2081 \u2294 a\u2082\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\nh : \u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082\nn\u2081 n\u2082 : \u2115\na\u2081 a\u2082 : \u03b1\nn : \u2115\n\u22a2 \u2200 (a\u2081 a\u2082 : \u03b1), (\u2191f)^[n] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[n] a\u2081 \u2294 a\u2082\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\nh : \u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082\nn\u2081 n\u2082 : \u2115\na\u2081 a\u2082 : \u03b1\n\u22a2 \u2200 (a\u2081 a\u2082 : \u03b1), (\u2191f)^[Nat.zero] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[Nat.zero] a\u2081 \u2294 a\u2082\n[PROOFSTEP]\nintro a\u2081 a\u2082\n[GOAL]\ncase succ\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\nh : \u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082\nn\u2081 n\u2082 : \u2115\na\u2081 a\u2082 : \u03b1\nn : \u2115\nih : \u2200 (a\u2081 a\u2082 : \u03b1), (\u2191f)^[n] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[n] a\u2081 \u2294 a\u2082\n\u22a2 \u2200 (a\u2081 a\u2082 : \u03b1), (\u2191f)^[Nat.succ n] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[Nat.succ n] a\u2081 \u2294 a\u2082\n[PROOFSTEP]\nintro a\u2081 a\u2082\n[GOAL]\ncase zero\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\nh : \u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082\nn\u2081 n\u2082 : \u2115\na\u2081\u271d a\u2082\u271d a\u2081 a\u2082 : \u03b1\n\u22a2 (\u2191f)^[Nat.zero] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[Nat.zero] a\u2081 \u2294 a\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\nh : \u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082\nn\u2081 n\u2082 : \u2115\na\u2081\u271d a\u2082\u271d : \u03b1\nn : \u2115\nih : \u2200 (a\u2081 a\u2082 : \u03b1), (\u2191f)^[n] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[n] a\u2081 \u2294 a\u2082\na\u2081 a\u2082 : \u03b1\n\u22a2 (\u2191f)^[Nat.succ n] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[Nat.succ n] a\u2081 \u2294 a\u2082\n[PROOFSTEP]\ncalc\n f^[n + 1] (a\u2081 \u2294 a\u2082) = f^[n] (f (a\u2081 \u2294 a\u2082)) := Function.iterate_succ_apply f n _\n _ \u2264 f^[n] (f a\u2081 \u2294 a\u2082) := (f.mono.iterate n (h a\u2081 a\u2082))\n _ \u2264 f^[n] (f a\u2081) \u2294 a\u2082 := (ih _ _)\n _ = f^[n + 1] a\u2081 \u2294 a\u2082 := by rw [\u2190 Function.iterate_succ_apply]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\nh : \u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082\nn\u2081 n\u2082 : \u2115\na\u2081\u271d a\u2082\u271d : \u03b1\nn : \u2115\nih : \u2200 (a\u2081 a\u2082 : \u03b1), (\u2191f)^[n] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[n] a\u2081 \u2294 a\u2082\na\u2081 a\u2082 : \u03b1\n\u22a2 (\u2191f)^[n] (\u2191f a\u2081) \u2294 a\u2082 = (\u2191f)^[n + 1] a\u2081 \u2294 a\u2082\n[PROOFSTEP]\nrw [\u2190 Function.iterate_succ_apply]\n[GOAL]\ncase mpr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\nh : \u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082\nn\u2081 n\u2082 : \u2115\na\u2081 a\u2082 : \u03b1\nh' : \u2200 (n : \u2115) (a\u2081 a\u2082 : \u03b1), (\u2191f)^[n] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[n] a\u2081 \u2294 a\u2082\n\u22a2 (\u2191f)^[n\u2081 + n\u2082] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[n\u2081] a\u2081 \u2294 (\u2191f)^[n\u2082] a\u2082\n[PROOFSTEP]\ncalc\n f^[n\u2081 + n\u2082] (a\u2081 \u2294 a\u2082) = f^[n\u2081] (f^[n\u2082] (a\u2081 \u2294 a\u2082)) := Function.iterate_add_apply f n\u2081 n\u2082 _\n _ = f^[n\u2081] (f^[n\u2082] (a\u2082 \u2294 a\u2081)) := by rw [sup_comm]\n _ \u2264 f^[n\u2081] (f^[n\u2082] a\u2082 \u2294 a\u2081) := (f.mono.iterate n\u2081 (h' n\u2082 _ _))\n _ = f^[n\u2081] (a\u2081 \u2294 f^[n\u2082] a\u2082) := by rw [sup_comm]\n _ \u2264 f^[n\u2081] a\u2081 \u2294 f^[n\u2082] a\u2082 := h' n\u2081 a\u2081 _\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\nh : \u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082\nn\u2081 n\u2082 : \u2115\na\u2081 a\u2082 : \u03b1\nh' : \u2200 (n : \u2115) (a\u2081 a\u2082 : \u03b1), (\u2191f)^[n] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[n] a\u2081 \u2294 a\u2082\n\u22a2 (\u2191f)^[n\u2081] ((\u2191f)^[n\u2082] (a\u2081 \u2294 a\u2082)) = (\u2191f)^[n\u2081] ((\u2191f)^[n\u2082] (a\u2082 \u2294 a\u2081))\n[PROOFSTEP]\nrw [sup_comm]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : SemilatticeSup \u03b1\nf : \u03b1 \u2192o \u03b1\nh : \u2200 (a\u2081 a\u2082 : \u03b1), \u2191f (a\u2081 \u2294 a\u2082) \u2264 \u2191f a\u2081 \u2294 a\u2082\nn\u2081 n\u2082 : \u2115\na\u2081 a\u2082 : \u03b1\nh' : \u2200 (n : \u2115) (a\u2081 a\u2082 : \u03b1), (\u2191f)^[n] (a\u2081 \u2294 a\u2082) \u2264 (\u2191f)^[n] a\u2081 \u2294 a\u2082\n\u22a2 (\u2191f)^[n\u2081] ((\u2191f)^[n\u2082] a\u2082 \u2294 a\u2081) = (\u2191f)^[n\u2081] (a\u2081 \u2294 (\u2191f)^[n\u2082] a\u2082)\n[PROOFSTEP]\nrw [sup_comm]\n", "meta": {"mathlib_filename": "Mathlib.Order.Hom.Order", "llama_tokens": 4424, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802264851919, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.42346123739025754}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommMonoid \u03b1\nf : Bool \u2192 \u03b1\n\u22a2 \u220f b : Bool, f b = f true * f false\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : CommMonoid \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 (\u220f i : \u03b9, if i \u2208 s then f i else 1) = \u220f i in s, f i\n[PROOFSTEP]\nrw [\u2190 prod_filter, filter_mem_eq_inter, univ_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : CommMonoid M\nf : \u03b1 \u2192 M\na b : \u03b1\nh\u2081 : a \u2260 b\nh\u2082 : \u2200 (x : \u03b1), x \u2260 a \u2227 x \u2260 b \u2192 f x = 1\n\u22a2 \u220f x : \u03b1, f x = f a * f b\n[PROOFSTEP]\napply Finset.prod_eq_mul a b h\u2081 fun x _ hx => h\u2082 x hx\n[GOAL]\ncase ha\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : CommMonoid M\nf : \u03b1 \u2192 M\na b : \u03b1\nh\u2081 : a \u2260 b\nh\u2082 : \u2200 (x : \u03b1), x \u2260 a \u2227 x \u2260 b \u2192 f x = 1\n\u22a2 \u00aca \u2208 Finset.univ \u2192 f a = 1\n[PROOFSTEP]\nexact fun hc => (hc (Finset.mem_univ _)).elim\n[GOAL]\ncase hb\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : CommMonoid M\nf : \u03b1 \u2192 M\na b : \u03b1\nh\u2081 : a \u2260 b\nh\u2082 : \u2200 (x : \u03b1), x \u2260 a \u2227 x \u2260 b \u2192 f x = 1\n\u22a2 \u00acb \u2208 Finset.univ \u2192 f b = 1\n[PROOFSTEP]\nexact fun hc => (hc (Finset.mem_univ _)).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b4 : \u03b1 \u2192 Type u_4\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\n\u22a2 Finset.card (piFinset t) = \u220f a : \u03b1, Finset.card (t a)\n[PROOFSTEP]\nsimp [Fintype.piFinset, card_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u03b2\n\u22a2 card (\u03b1 \u2192 \u03b2) = card \u03b2 ^ card \u03b1\n[PROOFSTEP]\nrw [Fintype.card_pi, Finset.prod_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u03b2\n\u22a2 card \u03b2 ^ Finset.card univ = card \u03b2 ^ card \u03b1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Fintype \u03b1\nn : \u2115\n\u22a2 Fintype.card (Vector \u03b1 n) = Fintype.card \u03b1 ^ n\n[PROOFSTEP]\nrw [Fintype.ofEquiv_card]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Fintype \u03b1\nn : \u2115\n\u22a2 Fintype.card (Fin n \u2192 \u03b1) = Fintype.card \u03b1 ^ n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : CommMonoid \u03b2\nf : { a // a \u2208 univ } \u2192 \u03b2\nx : { x // x \u2208 univ }\n\u22a2 f x =\n f\n { val := \u2191(Equiv.subtypeUnivEquiv (_ : \u2200 (x : \u03b1), x \u2208 univ)) x,\n property := (_ : \u2191(Equiv.subtypeUnivEquiv (_ : \u2200 (x : \u03b1), x \u2208 univ)) x \u2208 univ) }\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : CommMonoid \u03b2\n\u03b4 : \u03b1 \u2192 Type u_4\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : ((a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a) \u2192 \u03b2\n\u22a2 \u220f x in pi univ t, f x = \u220f x in Fintype.piFinset t, f fun a x_1 => x a\n[PROOFSTEP]\nrefine\n prod_bij (fun x _ a => x a (mem_univ _)) ?_ (by simp)\n (by simp (config := { contextual := true }) [Function.funext_iff]) fun x hx =>\n \u27e8fun a _ => x a, by simp_all\u27e9\n -- Porting note: old proof was `by simp`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : CommMonoid \u03b2\n\u03b4 : \u03b1 \u2192 Type u_4\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : ((a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a) \u2192 \u03b2\n\u22a2 \u2200 (a : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a) (ha : a \u2208 pi univ t),\n f a = f fun a_1 x => (fun x x_1 a => x a (_ : a \u2208 univ)) a ha a_1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : CommMonoid \u03b2\n\u03b4 : \u03b1 \u2192 Type u_4\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : ((a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a) \u2192 \u03b2\n\u22a2 \u2200 (a\u2081 a\u2082 : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a) (ha\u2081 : a\u2081 \u2208 pi univ t) (ha\u2082 : a\u2082 \u2208 pi univ t),\n (fun x x_1 a => x a (_ : a \u2208 univ)) a\u2081 ha\u2081 = (fun x x_1 a => x a (_ : a \u2208 univ)) a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\n[PROOFSTEP]\nsimp (config := { contextual := true }) [Function.funext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : CommMonoid \u03b2\n\u03b4 : \u03b1 \u2192 Type u_4\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : ((a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a) \u2192 \u03b2\nx : (a : \u03b1) \u2192 \u03b4 a\nhx : x \u2208 Fintype.piFinset t\n\u22a2 \u2203 ha, x = (fun x x_1 a => x a (_ : a \u2208 univ)) (fun a x_1 => x a) ha\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : CommMonoid \u03b2\n\u03b4 : \u03b1 \u2192 Type u_4\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : ((a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a) \u2192 \u03b2\n\u22a2 \u2200 (a : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a) (ha : a \u2208 pi univ t), (fun x x_1 a => x a (_ : a \u2208 univ)) a ha \u2208 Fintype.piFinset t\n[PROOFSTEP]\nintro a ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : CommMonoid \u03b2\n\u03b4 : \u03b1 \u2192 Type u_4\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : ((a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a) \u2192 \u03b2\na : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a\nha : a \u2208 pi univ t\n\u22a2 (fun x x_1 a => x a (_ : a \u2208 univ)) a ha \u2208 Fintype.piFinset t\n[PROOFSTEP]\nsimp only [Fintype.piFinset, mem_map, mem_pi, Function.Embedding.coeFn_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : CommMonoid \u03b2\n\u03b4 : \u03b1 \u2192 Type u_4\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : ((a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a) \u2192 \u03b2\na : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a\nha : a \u2208 pi univ t\n\u22a2 \u2203 a_1, (\u2200 (a : \u03b1) (h : a \u2208 univ), a_1 a h \u2208 t a) \u2227 (fun a => a_1 a (_ : a \u2208 univ)) = fun a_2 => a a_2 (_ : a_2 \u2208 univ)\n[PROOFSTEP]\nexact \u27e8a, by simpa using ha, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : CommMonoid \u03b2\n\u03b4 : \u03b1 \u2192 Type u_4\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : ((a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a) \u2192 \u03b2\na : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a\nha : a \u2208 pi univ t\n\u22a2 \u2200 (a_1 : \u03b1) (h : a_1 \u2208 univ), a a_1 h \u2208 t a_1\n[PROOFSTEP]\nsimpa using ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : CommMonoid \u03b2\n\u03b4 : \u03b1 \u2192 Type u_4\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : ((a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a) \u2192 \u03b2\na : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a\nha : a \u2208 pi univ t\n\u22a2 (fun a_1 => a a_1 (_ : a_1 \u2208 univ)) = fun a_1 => a a_1 (_ : a_1 \u2208 univ)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\n\u22a2 \u220f a : \u03b1, \u2211 b in t a, f a b = \u2211 p in Fintype.piFinset t, \u220f x : \u03b1, f x (p x)\n[PROOFSTEP]\nsimp only [Finset.prod_attach_univ, prod_sum, Finset.sum_univ_pi]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommMonoid \u03b1\nf : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 \u2200 (x : \u2115), x < n \u2194 x \u2208 range n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommMonoid \u03b1\nf : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 \u2200 (i : Fin n), f \u2191i = f \u2191(\u2191(equivSubtype.trans (Equiv.subtypeEquivRight (_ : \u2200 (a : \u2115), a < n \u2194 a \u2208 range n))) i)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommMonoid \u03b1\nf : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 \u220f i : { x // x \u2208 range n }, f \u2191i = \u220f i in range n, f i\n[PROOFSTEP]\nrw [\u2190 attach_eq_univ, prod_attach]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommMonoid \u03b2\nn : \u2115\nc : Fin n \u2192 \u03b2\n\u22a2 \u220f i : Fin n, c i = \u220f i in range n, if h : i < n then c { val := i, isLt := h } else 1\n[PROOFSTEP]\nrw [\u2190 Fin.prod_univ_eq_prod_range, Finset.prod_congr rfl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommMonoid \u03b2\nn : \u2115\nc : Fin n \u2192 \u03b2\n\u22a2 \u2200 (x : Fin n), x \u2208 univ \u2192 c x = if h : \u2191x < n then c { val := \u2191x, isLt := h } else 1\n[PROOFSTEP]\nrintro \u27e8i, hi\u27e9 _\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommMonoid \u03b2\nn : \u2115\nc : Fin n \u2192 \u03b2\ni : \u2115\nhi : i < n\na\u271d : { val := i, isLt := hi } \u2208 univ\n\u22a2 c { val := i, isLt := hi } =\n if h : \u2191{ val := i, isLt := hi } < n then c { val := \u2191{ val := i, isLt := hi }, isLt := h } else 1\n[PROOFSTEP]\nsimp only [hi, dif_pos]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : Fintype \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2192 M\n\u22a2 \u220f a in Set.toFinset {x | p x}, f a = \u220f a : Subtype p, f \u2191a\n[PROOFSTEP]\nrw [\u2190 Finset.prod_subtype]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : Fintype \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2192 M\n\u22a2 \u2200 (x : \u03b1), x \u2208 Set.toFinset {x | p x} \u2194 p x\n[PROOFSTEP]\nsimp_rw [Set.mem_toFinset]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : Fintype \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2192 M\n\u22a2 \u2200 (x : \u03b1), x \u2208 {x | p x} \u2194 p x\n[PROOFSTEP]\nintro\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : Fintype \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2192 M\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 {x | p x} \u2194 p x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : CommMonoid \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\n\u22a2 \u220f b : \u03b2, \u220f a : { a // f a = b }, g \u2191a = \u220f a : \u03b1, g a\n[PROOFSTEP]\nrw [\u2190 (Equiv.sigmaFiberEquiv f).prod_comp, \u2190 univ_sigma_univ, prod_sigma]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : CommMonoid \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\n\u22a2 \u220f b : \u03b2, \u220f a : { a // f a = b }, g \u2191a =\n \u220f a : \u03b2, \u220f s : { x // f x = a }, g (\u2191(Equiv.sigmaFiberEquiv f) { fst := a, snd := s })\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Fintype \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : CommMonoid \u03b2\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\ng : (a : \u03b1) \u2192 \u00acp a \u2192 \u03b2\n\u22a2 \u220f a : \u03b1, dite (p a) (f a) (g a) = (\u220f a : { a // p a }, f \u2191a (_ : p \u2191a)) * \u220f a : { a // \u00acp a }, g \u2191a (_ : \u00acp \u2191a)\n[PROOFSTEP]\nsimp only [prod_dite, attach_eq_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Fintype \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : CommMonoid \u03b2\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\ng : (a : \u03b1) \u2192 \u00acp a \u2192 \u03b2\n\u22a2 (\u220f x : { x // x \u2208 filter (fun x => p x) univ }, f \u2191x (_ : p \u2191x)) *\n \u220f x : { x // x \u2208 filter (fun x => \u00acp x) univ }, g \u2191x (_ : \u00acp \u2191x) =\n (\u220f a : { a // p a }, f \u2191a (_ : p \u2191a)) * \u220f a : { a // \u00acp a }, g \u2191a (_ : \u00acp \u2191a)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Fintype \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : CommMonoid \u03b2\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\ng : (a : \u03b1) \u2192 \u00acp a \u2192 \u03b2\n\u22a2 \u220f x : { x // x \u2208 filter (fun x => p x) univ }, f \u2191x (_ : p \u2191x) = \u220f a : { a // p a }, f \u2191a (_ : p \u2191a)\n[PROOFSTEP]\nexact (Equiv.subtypeEquivRight $ by simp).prod_comp fun x : { x // p x } => f x x.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Fintype \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : CommMonoid \u03b2\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\ng : (a : \u03b1) \u2192 \u00acp a \u2192 \u03b2\n\u22a2 \u2200 (x : \u03b1), x \u2208 filter (fun x => p x) univ \u2194 p x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Fintype \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : CommMonoid \u03b2\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\ng : (a : \u03b1) \u2192 \u00acp a \u2192 \u03b2\n\u22a2 \u220f x : { x // x \u2208 filter (fun x => \u00acp x) univ }, g \u2191x (_ : \u00acp \u2191x) = \u220f a : { a // \u00acp a }, g \u2191a (_ : \u00acp \u2191a)\n[PROOFSTEP]\nexact (Equiv.subtypeEquivRight $ by simp).prod_comp fun x : { x // \u00acp x } => g x x.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Fintype \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : CommMonoid \u03b2\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\ng : (a : \u03b1) \u2192 \u00acp a \u2192 \u03b2\n\u22a2 \u2200 (x : \u03b1), x \u2208 filter (fun x => \u00acp x) univ \u2194 \u00acp x\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Fintype.BigOperators", "llama_tokens": 6000, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.42288251272526256}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : PosNum\n\u22a2 \u2191\u2191n = \u2191n\n[PROOFSTEP]\nrw [\u2190 to_nat_to_int, Int.cast_ofNat, cast_to_nat]\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 \u2191p + 1 + \u2191p + 1 = \u2191p + \u2191p + 1 + 1\n[PROOFSTEP]\nsimp [add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 1 + n = succ n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase one\n\u03b1 : Type u_1\n\u22a2 1 + one = succ one\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit1\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 1 + bit1 a\u271d = succ (bit1 a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit0\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 1 + bit0 a\u271d = succ (bit0 a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 n + 1 = succ n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase one\n\u03b1 : Type u_1\n\u22a2 one + 1 = succ one\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit1\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 bit1 a\u271d + 1 = succ (bit1 a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit0\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 bit0 a\u271d + 1 = succ (bit0 a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nb : PosNum\n\u22a2 \u2191(1 + b) = \u21911 + \u2191b\n[PROOFSTEP]\nrw [one_add b, succ_to_nat, add_comm]\n[GOAL]\n\u03b1 : Type u_1\nb : PosNum\n\u22a2 1 + \u2191b = \u21911 + \u2191b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\na : PosNum\n\u22a2 \u2191(a + 1) = \u2191a + \u21911\n[PROOFSTEP]\nrw [add_one a, succ_to_nat]\n[GOAL]\n\u03b1 : Type u_1\na : PosNum\n\u22a2 \u2191a + 1 = \u2191a + \u21911\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191a + \u2191b + (\u2191a + \u2191b) + 1 = \u2191a + \u2191a + (\u2191b + \u2191b + 1)\n[PROOFSTEP]\nsimp [add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191a + \u2191b + (\u2191a + \u2191b) + 1 = \u2191a + \u2191a + 1 + (\u2191b + \u2191b)\n[PROOFSTEP]\nsimp [add_comm, add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191(succ (a + b)) + \u2191(succ (a + b)) = \u2191a + \u2191a + 1 + (\u2191b + \u2191b + 1)\n[PROOFSTEP]\nrw [succ_to_nat, add_to_nat a b]\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191a + \u2191b + 1 + (\u2191a + \u2191b + 1) = \u2191a + \u2191a + 1 + (\u2191b + \u2191b + 1)\n[PROOFSTEP]\nsimp [add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\nb : PosNum\n\u22a2 1 + succ b = succ (1 + b)\n[PROOFSTEP]\nsimp [one_add]\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 bit0 (succ (_root_.bit0 p)) = bit0 (bit1 p)\n[PROOFSTEP]\nrw [bit0_of_bit0 p]\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 bit0 (succ (bit0 p)) = bit0 (bit1 p)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 _root_.bit0 n + 1 = bit1 n\n[PROOFSTEP]\nrw [add_one, bit0_of_bit0]\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 succ (bit0 n) = bit1 n\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nm p : PosNum\n\u22a2 \u2191(m * p) + \u2191(m * p) = \u2191m * (\u2191p + \u2191p)\n[PROOFSTEP]\nrw [mul_to_nat m p, left_distrib]\n[GOAL]\n\u03b1 : Type u_1\nm p : PosNum\n\u22a2 \u2191(m * p) + \u2191(m * p) + \u2191m = \u2191m * (\u2191p + \u2191p) + \u2191m\n[PROOFSTEP]\nrw [mul_to_nat m p, left_distrib]\n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\n\u22a2 \u2191m < \u2191n \u2192 \u2191m + \u2191m + 1 + 1 \u2264 \u2191n + \u2191n\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\nh : \u2191m < \u2191n\n\u22a2 \u2191m + \u2191m + 1 + 1 \u2264 \u2191n + \u2191n\n[PROOFSTEP]\nrw [Nat.add_right_comm m m 1, add_assoc]\n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\nh : \u2191m < \u2191n\n\u22a2 \u2191m + 1 + (\u2191m + 1) \u2264 \u2191n + \u2191n\n[PROOFSTEP]\nexact add_le_add h h\n[GOAL]\n\u03b1 : Type u_1\nm : PosNum\n\u22a2 \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\n[PROOFSTEP]\ninduction' m with m IH m IH\n[GOAL]\ncase one\n\u03b1 : Type u_1\n\u22a2 \u2200 (n : PosNum), Ordering.swap (cmp one n) = cmp n one\n[PROOFSTEP]\nintro n\n[GOAL]\ncase bit1\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\n\u22a2 \u2200 (n : PosNum), Ordering.swap (cmp (bit1 m) n) = cmp n (bit1 m)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase bit0\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\n\u22a2 \u2200 (n : PosNum), Ordering.swap (cmp (bit0 m) n) = cmp n (bit0 m)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase one\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 Ordering.swap (cmp one n) = cmp n one\n[PROOFSTEP]\ncases' n with n n\n[GOAL]\ncase bit1\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (cmp (bit1 m) n) = cmp n (bit1 m)\n[PROOFSTEP]\ncases' n with n n\n[GOAL]\ncase bit0\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (cmp (bit0 m) n) = cmp n (bit0 m)\n[PROOFSTEP]\ncases' n with n n\n[GOAL]\ncase one.one\n\u03b1 : Type u_1\n\u22a2 Ordering.swap (cmp one one) = cmp one one\n[PROOFSTEP]\nunfold cmp\n[GOAL]\ncase one.bit1\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 Ordering.swap (cmp one (bit1 n)) = cmp (bit1 n) one\n[PROOFSTEP]\nunfold cmp\n[GOAL]\ncase one.bit0\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 Ordering.swap (cmp one (bit0 n)) = cmp (bit0 n) one\n[PROOFSTEP]\nunfold cmp\n[GOAL]\ncase bit1.one\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\n\u22a2 Ordering.swap (cmp (bit1 m) one) = cmp one (bit1 m)\n[PROOFSTEP]\nunfold cmp\n[GOAL]\ncase bit1.bit1\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (cmp (bit1 m) (bit1 n)) = cmp (bit1 n) (bit1 m)\n[PROOFSTEP]\nunfold cmp\n[GOAL]\ncase bit1.bit0\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (cmp (bit1 m) (bit0 n)) = cmp (bit0 n) (bit1 m)\n[PROOFSTEP]\nunfold cmp\n[GOAL]\ncase bit0.one\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\n\u22a2 Ordering.swap (cmp (bit0 m) one) = cmp one (bit0 m)\n[PROOFSTEP]\nunfold cmp\n[GOAL]\ncase bit0.bit1\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (cmp (bit0 m) (bit1 n)) = cmp (bit1 n) (bit0 m)\n[PROOFSTEP]\nunfold cmp\n[GOAL]\ncase bit0.bit0\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (cmp (bit0 m) (bit0 n)) = cmp (bit0 n) (bit0 m)\n[PROOFSTEP]\nunfold cmp\n[GOAL]\ncase one.one\n\u03b1 : Type u_1\n\u22a2 Ordering.swap Ordering.eq = Ordering.eq\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase one.one\n\u03b1 : Type u_1\n\u22a2 Ordering.swap Ordering.eq = Ordering.eq\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase one.one\n\u03b1 : Type u_1\n\u22a2 Ordering.swap Ordering.eq = Ordering.eq\n[PROOFSTEP]\nrfl\n[GOAL]\ncase one.bit1\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 Ordering.swap Ordering.lt = Ordering.gt\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase one.bit1\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 Ordering.swap Ordering.lt = Ordering.gt\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase one.bit1\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 Ordering.swap Ordering.lt = Ordering.gt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase one.bit0\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 Ordering.swap Ordering.lt = Ordering.gt\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase one.bit0\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 Ordering.swap Ordering.lt = Ordering.gt\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase one.bit0\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 Ordering.swap Ordering.lt = Ordering.gt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit1.one\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\n\u22a2 Ordering.swap Ordering.gt = Ordering.lt\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase bit1.one\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\n\u22a2 Ordering.swap Ordering.gt = Ordering.lt\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase bit1.one\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\n\u22a2 Ordering.swap Ordering.gt = Ordering.lt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit1.bit1\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (cmp m n) = cmp n m\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase bit1.bit1\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (cmp m n) = cmp n m\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase bit1.bit1\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (cmp m n) = cmp n m\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit1.bit0\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn (cmp m n) Ordering.lt Ordering.gt Ordering.gt) =\n Ordering.casesOn (cmp n m) Ordering.lt Ordering.lt Ordering.gt\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase bit1.bit0\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn (cmp m n) Ordering.lt Ordering.gt Ordering.gt) =\n Ordering.casesOn (cmp n m) Ordering.lt Ordering.lt Ordering.gt\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase bit1.bit0\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn (cmp m n) Ordering.lt Ordering.gt Ordering.gt) =\n Ordering.casesOn (cmp n m) Ordering.lt Ordering.lt Ordering.gt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit0.one\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\n\u22a2 Ordering.swap Ordering.gt = Ordering.lt\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase bit0.one\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\n\u22a2 Ordering.swap Ordering.gt = Ordering.lt\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase bit0.one\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\n\u22a2 Ordering.swap Ordering.gt = Ordering.lt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit0.bit1\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn (cmp m n) Ordering.lt Ordering.lt Ordering.gt) =\n Ordering.casesOn (cmp n m) Ordering.lt Ordering.gt Ordering.gt\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase bit0.bit1\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn (cmp m n) Ordering.lt Ordering.lt Ordering.gt) =\n Ordering.casesOn (cmp n m) Ordering.lt Ordering.gt Ordering.gt\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase bit0.bit1\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn (cmp m n) Ordering.lt Ordering.lt Ordering.gt) =\n Ordering.casesOn (cmp n m) Ordering.lt Ordering.gt Ordering.gt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit0.bit0\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (cmp m n) = cmp n m\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase bit0.bit0\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (cmp m n) = cmp n m\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase bit0.bit0\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (cmp m n) = cmp n m\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit1.bit1\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (cmp m n) = cmp n m\n[PROOFSTEP]\nrw [\u2190 IH]\n[GOAL]\ncase bit1.bit0\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn (cmp m n) Ordering.lt Ordering.gt Ordering.gt) =\n Ordering.casesOn (cmp n m) Ordering.lt Ordering.lt Ordering.gt\n[PROOFSTEP]\nrw [\u2190 IH]\n[GOAL]\ncase bit0.bit1\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn (cmp m n) Ordering.lt Ordering.lt Ordering.gt) =\n Ordering.casesOn (cmp n m) Ordering.lt Ordering.gt Ordering.gt\n[PROOFSTEP]\nrw [\u2190 IH]\n[GOAL]\ncase bit0.bit0\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (cmp m n) = cmp n m\n[PROOFSTEP]\nrw [\u2190 IH]\n[GOAL]\ncase bit1.bit0\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn (cmp m n) Ordering.lt Ordering.gt Ordering.gt) =\n Ordering.casesOn (Ordering.swap (cmp m n)) Ordering.lt Ordering.lt Ordering.gt\n[PROOFSTEP]\ncases cmp m n\n[GOAL]\ncase bit0.bit1\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn (cmp m n) Ordering.lt Ordering.lt Ordering.gt) =\n Ordering.casesOn (Ordering.swap (cmp m n)) Ordering.lt Ordering.gt Ordering.gt\n[PROOFSTEP]\ncases cmp m n\n[GOAL]\ncase bit1.bit0.lt\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn Ordering.lt Ordering.lt Ordering.gt Ordering.gt) =\n Ordering.casesOn (Ordering.swap Ordering.lt) Ordering.lt Ordering.lt Ordering.gt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit1.bit0.eq\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn Ordering.eq Ordering.lt Ordering.gt Ordering.gt) =\n Ordering.casesOn (Ordering.swap Ordering.eq) Ordering.lt Ordering.lt Ordering.gt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit1.bit0.gt\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn Ordering.gt Ordering.lt Ordering.gt Ordering.gt) =\n Ordering.casesOn (Ordering.swap Ordering.gt) Ordering.lt Ordering.lt Ordering.gt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit0.bit1.lt\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn Ordering.lt Ordering.lt Ordering.lt Ordering.gt) =\n Ordering.casesOn (Ordering.swap Ordering.lt) Ordering.lt Ordering.gt Ordering.gt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit0.bit1.eq\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn Ordering.eq Ordering.lt Ordering.lt Ordering.gt) =\n Ordering.casesOn (Ordering.swap Ordering.eq) Ordering.lt Ordering.gt Ordering.gt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit0.bit1.gt\n\u03b1 : Type u_1\nm : PosNum\nIH : \u2200 (n : PosNum), Ordering.swap (cmp m n) = cmp n m\nn : PosNum\n\u22a2 Ordering.swap (Ordering.casesOn Ordering.gt Ordering.lt Ordering.lt Ordering.gt) =\n Ordering.casesOn (Ordering.swap Ordering.gt) Ordering.lt Ordering.gt Ordering.gt\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn (cmp (bit0 a) (bit0 b)) (\u2191(bit0 a) < \u2191(bit0 b)) (bit0 a = bit0 b) (\u2191(bit0 b) < \u2191(bit0 a))\n[PROOFSTEP]\nhave := cmp_to_nat a b\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\nthis : Ordering.casesOn (cmp a b) (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a)\n\u22a2 Ordering.casesOn (cmp (bit0 a) (bit0 b)) (\u2191(bit0 a) < \u2191(bit0 b)) (bit0 a = bit0 b) (\u2191(bit0 b) < \u2191(bit0 a))\n[PROOFSTEP]\nrevert this\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn (cmp a b) (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.casesOn (cmp (bit0 a) (bit0 b)) (\u2191(bit0 a) < \u2191(bit0 b)) (bit0 a = bit0 b) (\u2191(bit0 b) < \u2191(bit0 a))\n[PROOFSTEP]\ncases cmp a b\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn Ordering.lt (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.casesOn Ordering.lt (\u2191(bit0 a) < \u2191(bit0 b)) (bit0 a = bit0 b) (\u2191(bit0 b) < \u2191(bit0 a))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn Ordering.eq (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.casesOn Ordering.eq (\u2191(bit0 a) < \u2191(bit0 b)) (bit0 a = bit0 b) (\u2191(bit0 b) < \u2191(bit0 a))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn Ordering.gt (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.casesOn Ordering.gt (\u2191(bit0 a) < \u2191(bit0 b)) (bit0 a = bit0 b) (\u2191(bit0 b) < \u2191(bit0 a))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191a < \u2191b \u2192 _root_.bit0 \u2191a < _root_.bit0 \u2191b\n[PROOFSTEP]\nintro this\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 a = b \u2192 bit0 a = bit0 b\n[PROOFSTEP]\nintro this\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191b < \u2191a \u2192 _root_.bit0 \u2191b < _root_.bit0 \u2191a\n[PROOFSTEP]\nintro this\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\nthis : \u2191a < \u2191b\n\u22a2 _root_.bit0 \u2191a < _root_.bit0 \u2191b\n[PROOFSTEP]\nexact add_lt_add this this\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\nthis : a = b\n\u22a2 bit0 a = bit0 b\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\nthis : \u2191b < \u2191a\n\u22a2 _root_.bit0 \u2191b < _root_.bit0 \u2191a\n[PROOFSTEP]\nexact add_lt_add this this\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn (cmp (bit0 a) (bit1 b)) (\u2191(bit0 a) < \u2191(bit1 b)) (bit0 a = bit1 b) (\u2191(bit1 b) < \u2191(bit0 a))\n[PROOFSTEP]\ndsimp [cmp]\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (_root_.bit0 \u2191a < _root_.bit1 \u2191b) (bit0 a = bit1 b) (_root_.bit1 \u2191b < _root_.bit0 \u2191a)\n (Ordering.rec Ordering.lt Ordering.lt Ordering.gt (cmp a b))\n[PROOFSTEP]\nhave := cmp_to_nat a b\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\nthis : Ordering.casesOn (cmp a b) (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a)\n\u22a2 Ordering.rec (_root_.bit0 \u2191a < _root_.bit1 \u2191b) (bit0 a = bit1 b) (_root_.bit1 \u2191b < _root_.bit0 \u2191a)\n (Ordering.rec Ordering.lt Ordering.lt Ordering.gt (cmp a b))\n[PROOFSTEP]\nrevert this\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn (cmp a b) (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.rec (_root_.bit0 \u2191a < _root_.bit1 \u2191b) (bit0 a = bit1 b) (_root_.bit1 \u2191b < _root_.bit0 \u2191a)\n (Ordering.rec Ordering.lt Ordering.lt Ordering.gt (cmp a b))\n[PROOFSTEP]\ncases cmp a b\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn Ordering.lt (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.rec (_root_.bit0 \u2191a < _root_.bit1 \u2191b) (bit0 a = bit1 b) (_root_.bit1 \u2191b < _root_.bit0 \u2191a)\n (Ordering.rec Ordering.lt Ordering.lt Ordering.gt Ordering.lt)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn Ordering.eq (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.rec (_root_.bit0 \u2191a < _root_.bit1 \u2191b) (bit0 a = bit1 b) (_root_.bit1 \u2191b < _root_.bit0 \u2191a)\n (Ordering.rec Ordering.lt Ordering.lt Ordering.gt Ordering.eq)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn Ordering.gt (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.rec (_root_.bit0 \u2191a < _root_.bit1 \u2191b) (bit0 a = bit1 b) (_root_.bit1 \u2191b < _root_.bit0 \u2191a)\n (Ordering.rec Ordering.lt Ordering.lt Ordering.gt Ordering.gt)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191a < \u2191b \u2192 _root_.bit0 \u2191a < _root_.bit1 \u2191b\n[PROOFSTEP]\nintro this\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 a = b \u2192 _root_.bit0 \u2191a < _root_.bit1 \u2191b\n[PROOFSTEP]\nintro this\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191b < \u2191a \u2192 _root_.bit1 \u2191b < _root_.bit0 \u2191a\n[PROOFSTEP]\nintro this\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\nthis : \u2191a < \u2191b\n\u22a2 _root_.bit0 \u2191a < _root_.bit1 \u2191b\n[PROOFSTEP]\nexact Nat.le_succ_of_le (add_lt_add this this)\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\nthis : a = b\n\u22a2 _root_.bit0 \u2191a < _root_.bit1 \u2191b\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\nthis : a = b\n\u22a2 _root_.bit0 \u2191b < _root_.bit1 \u2191b\n[PROOFSTEP]\napply Nat.lt_succ_self\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\nthis : \u2191b < \u2191a\n\u22a2 _root_.bit1 \u2191b < _root_.bit0 \u2191a\n[PROOFSTEP]\nexact cmp_to_nat_lemma this\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn (cmp (bit1 a) (bit0 b)) (\u2191(bit1 a) < \u2191(bit0 b)) (bit1 a = bit0 b) (\u2191(bit0 b) < \u2191(bit1 a))\n[PROOFSTEP]\ndsimp [cmp]\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (_root_.bit1 \u2191a < _root_.bit0 \u2191b) (bit1 a = bit0 b) (_root_.bit0 \u2191b < _root_.bit1 \u2191a)\n (Ordering.rec Ordering.lt Ordering.gt Ordering.gt (cmp a b))\n[PROOFSTEP]\nhave := cmp_to_nat a b\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\nthis : Ordering.casesOn (cmp a b) (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a)\n\u22a2 Ordering.rec (_root_.bit1 \u2191a < _root_.bit0 \u2191b) (bit1 a = bit0 b) (_root_.bit0 \u2191b < _root_.bit1 \u2191a)\n (Ordering.rec Ordering.lt Ordering.gt Ordering.gt (cmp a b))\n[PROOFSTEP]\nrevert this\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn (cmp a b) (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.rec (_root_.bit1 \u2191a < _root_.bit0 \u2191b) (bit1 a = bit0 b) (_root_.bit0 \u2191b < _root_.bit1 \u2191a)\n (Ordering.rec Ordering.lt Ordering.gt Ordering.gt (cmp a b))\n[PROOFSTEP]\ncases cmp a b\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn Ordering.lt (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.rec (_root_.bit1 \u2191a < _root_.bit0 \u2191b) (bit1 a = bit0 b) (_root_.bit0 \u2191b < _root_.bit1 \u2191a)\n (Ordering.rec Ordering.lt Ordering.gt Ordering.gt Ordering.lt)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn Ordering.eq (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.rec (_root_.bit1 \u2191a < _root_.bit0 \u2191b) (bit1 a = bit0 b) (_root_.bit0 \u2191b < _root_.bit1 \u2191a)\n (Ordering.rec Ordering.lt Ordering.gt Ordering.gt Ordering.eq)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn Ordering.gt (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.rec (_root_.bit1 \u2191a < _root_.bit0 \u2191b) (bit1 a = bit0 b) (_root_.bit0 \u2191b < _root_.bit1 \u2191a)\n (Ordering.rec Ordering.lt Ordering.gt Ordering.gt Ordering.gt)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191a < \u2191b \u2192 _root_.bit1 \u2191a < _root_.bit0 \u2191b\n[PROOFSTEP]\nintro this\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 a = b \u2192 _root_.bit0 \u2191b < _root_.bit1 \u2191a\n[PROOFSTEP]\nintro this\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191b < \u2191a \u2192 _root_.bit0 \u2191b < _root_.bit1 \u2191a\n[PROOFSTEP]\nintro this\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\nthis : \u2191a < \u2191b\n\u22a2 _root_.bit1 \u2191a < _root_.bit0 \u2191b\n[PROOFSTEP]\nexact cmp_to_nat_lemma this\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\nthis : a = b\n\u22a2 _root_.bit0 \u2191b < _root_.bit1 \u2191a\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\nthis : a = b\n\u22a2 _root_.bit0 \u2191b < _root_.bit1 \u2191b\n[PROOFSTEP]\napply Nat.lt_succ_self\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\nthis : \u2191b < \u2191a\n\u22a2 _root_.bit0 \u2191b < _root_.bit1 \u2191a\n[PROOFSTEP]\nexact Nat.le_succ_of_le (add_lt_add this this)\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn (cmp (bit1 a) (bit1 b)) (\u2191(bit1 a) < \u2191(bit1 b)) (bit1 a = bit1 b) (\u2191(bit1 b) < \u2191(bit1 a))\n[PROOFSTEP]\nhave := cmp_to_nat a b\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\nthis : Ordering.casesOn (cmp a b) (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a)\n\u22a2 Ordering.casesOn (cmp (bit1 a) (bit1 b)) (\u2191(bit1 a) < \u2191(bit1 b)) (bit1 a = bit1 b) (\u2191(bit1 b) < \u2191(bit1 a))\n[PROOFSTEP]\nrevert this\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn (cmp a b) (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.casesOn (cmp (bit1 a) (bit1 b)) (\u2191(bit1 a) < \u2191(bit1 b)) (bit1 a = bit1 b) (\u2191(bit1 b) < \u2191(bit1 a))\n[PROOFSTEP]\ncases cmp a b\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn Ordering.lt (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.casesOn Ordering.lt (\u2191(bit1 a) < \u2191(bit1 b)) (bit1 a = bit1 b) (\u2191(bit1 b) < \u2191(bit1 a))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn Ordering.eq (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.casesOn Ordering.eq (\u2191(bit1 a) < \u2191(bit1 b)) (bit1 a = bit1 b) (\u2191(bit1 b) < \u2191(bit1 a))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn Ordering.gt (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.casesOn Ordering.gt (\u2191(bit1 a) < \u2191(bit1 b)) (bit1 a = bit1 b) (\u2191(bit1 b) < \u2191(bit1 a))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191a < \u2191b \u2192 _root_.bit1 \u2191a < _root_.bit1 \u2191b\n[PROOFSTEP]\nintro this\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 a = b \u2192 bit1 a = bit1 b\n[PROOFSTEP]\nintro this\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191b < \u2191a \u2192 _root_.bit1 \u2191b < _root_.bit1 \u2191a\n[PROOFSTEP]\nintro this\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\nthis : \u2191a < \u2191b\n\u22a2 _root_.bit1 \u2191a < _root_.bit1 \u2191b\n[PROOFSTEP]\nexact Nat.succ_lt_succ (add_lt_add this this)\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\nthis : a = b\n\u22a2 bit1 a = bit1 b\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\nthis : \u2191b < \u2191a\n\u22a2 _root_.bit1 \u2191b < _root_.bit1 \u2191a\n[PROOFSTEP]\nexact Nat.succ_lt_succ (add_lt_add this this)\n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\nh : Ordering.casesOn Ordering.lt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 \u2191m < \u2191n \u2194 Ordering.lt = Ordering.lt\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\nh : \u2191m < \u2191n\n\u22a2 \u2191m < \u2191n \u2194 Ordering.lt = Ordering.lt\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\nh : Ordering.casesOn Ordering.eq (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 \u2191m < \u2191n \u2194 Ordering.eq = Ordering.lt\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\nh : m = n\n\u22a2 \u2191m < \u2191n \u2194 Ordering.eq = Ordering.lt\n[PROOFSTEP]\nsimp [h, lt_irrefl]\n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\nh : Ordering.casesOn Ordering.gt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 \u2191m < \u2191n \u2194 Ordering.gt = Ordering.lt\n[PROOFSTEP]\nsimp [not_lt_of_gt h]\n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\n\u22a2 \u2191m \u2264 \u2191n \u2194 m \u2264 n\n[PROOFSTEP]\nrw [\u2190 not_lt]\n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\n\u22a2 \u00ac\u2191n < \u2191m \u2194 m \u2264 n\n[PROOFSTEP]\nexact not_congr lt_to_nat\n[GOAL]\n\u03b1 : Type u_1\nn : Num\n\u22a2 n + 0 = n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u22a2 zero + 0 = zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 pos a\u271d + 0 = pos a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : Num\n\u22a2 0 + n = n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u22a2 0 + zero = zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 0 + pos a\u271d = pos a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 pos p + 1 = succ (pos p)\n[PROOFSTEP]\ncases p\n[GOAL]\ncase one\n\u03b1 : Type u_1\n\u22a2 pos one + 1 = succ (pos one)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit1\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 pos (PosNum.bit1 a\u271d) + 1 = succ (pos (PosNum.bit1 a\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit0\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 pos (PosNum.bit0 a\u271d) + 1 = succ (pos (PosNum.bit0 a\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : Num\n\u22a2 0 + succ n = succ (0 + n)\n[PROOFSTEP]\nsimp [zero_add]\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 pos (p + 1) = succ (pos p + 0)\n[PROOFSTEP]\nrw [PosNum.add_one, add_zero]\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 pos (PosNum.succ p) = succ (pos p)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 ofNat' 0 = 0\n[PROOFSTEP]\nsimp [Num.ofNat']\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 ofNat' 1 = 1\n[PROOFSTEP]\nerw [ofNat'_bit true 0, cond, ofNat'_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 (match (motive := Bool \u2192 Num \u2192 Num) true with\n | true => Num.bit1\n | false => Num.bit0)\n 0 =\n 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 ofNat' (0 + 1) = ofNat' 0 + 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nb : Bool\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 ofNat' (Nat.bit b n + 1) = ofNat' (Nat.bit b n) + 1\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 ofNat' (Nat.bit false n + 1) = ofNat' (Nat.bit false n) + 1\n[PROOFSTEP]\nerw [ofNat'_bit true n, ofNat'_bit]\n[GOAL]\ncase false\n\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 cond true Num.bit1 Num.bit0 (ofNat' n) = cond false Num.bit1 Num.bit0 (ofNat' n) + 1\n[PROOFSTEP]\nsimp only [\u2190 bit1_of_bit1, \u2190 bit0_of_bit0, cond, _root_.bit1]\n -- Porting note: `cc` was not ported yet so `exact Nat.add_left_comm n 1 1` is used.\n[GOAL]\ncase true\n\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 ofNat' (Nat.bit true n + 1) = ofNat' (Nat.bit true n) + 1\n[PROOFSTEP]\nerw [show n.bit true + 1 = (n + 1).bit false by simp [Nat.bit, _root_.bit1, _root_.bit0]; exact Nat.add_left_comm n 1 1,\n ofNat'_bit, ofNat'_bit, ih]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 Nat.bit true n + 1 = Nat.bit false (n + 1)\n[PROOFSTEP]\nsimp [Nat.bit, _root_.bit1, _root_.bit0]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 n + (1 + 1) = 1 + (n + 1)\n[PROOFSTEP]\nexact Nat.add_left_comm n 1 1\n[GOAL]\ncase true\n\u03b1 : Type u_1\nn : \u2115\nih : ofNat' (n + 1) = ofNat' n + 1\n\u22a2 cond false Num.bit1 Num.bit0 (ofNat' n + 1) = cond true Num.bit1 Num.bit0 (ofNat' n) + 1\n[PROOFSTEP]\nsimp only [cond, add_one, bit1_succ]\n[GOAL]\n\u03b1 : Type u_1\nm n : \u2115\n\u22a2 ofNat' (m + n) = ofNat' m + ofNat' n\n[PROOFSTEP]\nhave : \u2200 {n}, ofNat' n.succ = ofNat' n + 1 := ofNat'_succ\n[GOAL]\n\u03b1 : Type u_1\nm n : \u2115\nthis : \u2200 {n : \u2115}, ofNat' (Nat.succ n) = ofNat' n + 1\n\u22a2 ofNat' (m + n) = ofNat' m + ofNat' n\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nm : \u2115\nthis : \u2200 {n : \u2115}, ofNat' (Nat.succ n) = ofNat' n + 1\n\u22a2 ofNat' (m + Nat.zero) = ofNat' m + ofNat' Nat.zero\n[PROOFSTEP]\nsimp [Nat.add_zero, this, add_zero, Nat.add_succ, add_one, add_succ, *]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nm : \u2115\nthis : \u2200 {n : \u2115}, ofNat' (Nat.succ n) = ofNat' n + 1\nn\u271d : \u2115\nn_ih\u271d : ofNat' (m + n\u271d) = ofNat' m + ofNat' n\u271d\n\u22a2 ofNat' (m + Nat.succ n\u271d) = ofNat' m + ofNat' (Nat.succ n\u271d)\n[PROOFSTEP]\nsimp [Nat.add_zero, this, add_zero, Nat.add_succ, add_one, add_succ, *]\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn (cmp (pos a) (pos b)) (\u2191(pos a) < \u2191(pos b)) (pos a = pos b) (\u2191(pos b) < \u2191(pos a))\n[PROOFSTEP]\nhave := PosNum.cmp_to_nat a b\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\nthis : Ordering.casesOn (PosNum.cmp a b) (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a)\n\u22a2 Ordering.casesOn (cmp (pos a) (pos b)) (\u2191(pos a) < \u2191(pos b)) (pos a = pos b) (\u2191(pos b) < \u2191(pos a))\n[PROOFSTEP]\nrevert this\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn (PosNum.cmp a b) (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.casesOn (cmp (pos a) (pos b)) (\u2191(pos a) < \u2191(pos b)) (pos a = pos b) (\u2191(pos b) < \u2191(pos a))\n[PROOFSTEP]\ndsimp [cmp]\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) (PosNum.cmp a b) \u2192\n Ordering.rec (\u2191a < \u2191b) (pos a = pos b) (\u2191b < \u2191a) (PosNum.cmp a b)\n[PROOFSTEP]\ncases PosNum.cmp a b\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) Ordering.lt \u2192 Ordering.rec (\u2191a < \u2191b) (pos a = pos b) (\u2191b < \u2191a) Ordering.lt\ncase eq\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) Ordering.eq \u2192 Ordering.rec (\u2191a < \u2191b) (pos a = pos b) (\u2191b < \u2191a) Ordering.eq\ncase gt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) Ordering.gt \u2192 Ordering.rec (\u2191a < \u2191b) (pos a = pos b) (\u2191b < \u2191a) Ordering.gt\n[PROOFSTEP]\nexacts [id, congr_arg pos, id]\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\nh : Ordering.casesOn Ordering.lt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 \u2191m < \u2191n \u2194 Ordering.lt = Ordering.lt\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u_1\nm n : Num\nh : \u2191m < \u2191n\n\u22a2 \u2191m < \u2191n \u2194 Ordering.lt = Ordering.lt\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\nh : Ordering.casesOn Ordering.eq (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 \u2191m < \u2191n \u2194 Ordering.eq = Ordering.lt\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u_1\nm n : Num\nh : m = n\n\u22a2 \u2191m < \u2191n \u2194 Ordering.eq = Ordering.lt\n[PROOFSTEP]\nsimp [h, lt_irrefl]\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\nh : Ordering.casesOn Ordering.gt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 \u2191m < \u2191n \u2194 Ordering.gt = Ordering.lt\n[PROOFSTEP]\nsimp [not_lt_of_gt h]\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\n\u22a2 \u2191m \u2264 \u2191n \u2194 m \u2264 n\n[PROOFSTEP]\nrw [\u2190 not_lt]\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\n\u22a2 \u00ac\u2191n < \u2191m \u2194 m \u2264 n\n[PROOFSTEP]\nexact not_congr lt_to_nat\n[GOAL]\n\u22a2 Num.ofNat' \u21911 = Num.pos 1\n[PROOFSTEP]\nerw [@Num.ofNat'_bit true 0, Num.ofNat'_zero]\n[GOAL]\n\u22a2 cond true Num.bit1 Num.bit0 0 = Num.pos 1\n[PROOFSTEP]\nrfl\n[GOAL]\np : PosNum\n\u22a2 Num.ofNat' \u2191(bit0 p) = Num.pos (bit0 p)\n[PROOFSTEP]\nerw [@Num.ofNat'_bit false, of_to_nat' p]\n[GOAL]\np : PosNum\n\u22a2 cond false Num.bit1 Num.bit0 (Num.pos p) = Num.pos (bit0 p)\n[PROOFSTEP]\nrfl\n[GOAL]\np : PosNum\n\u22a2 Num.ofNat' \u2191(bit1 p) = Num.pos (bit1 p)\n[PROOFSTEP]\nerw [@Num.ofNat'_bit true, of_to_nat' p]\n[GOAL]\np : PosNum\n\u22a2 cond true Num.bit1 Num.bit0 (Num.pos p) = Num.pos (bit1 p)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u2200 (a b c : Num), a + b + c = a + (b + c)\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u22a2 CommSemiring Num\n[PROOFSTEP]\nrefine'\n { Num.addMonoid, Num.addMonoidWithOne with\n mul := (\u00b7 * \u00b7)\n one := 1\n add := (\u00b7 + \u00b7)\n zero := 0\n npow := @npowRec Num \u27e81\u27e9 \u27e8(\u00b7 * \u00b7)\u27e9, .. }\n[GOAL]\ncase refine'_1\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b : Num), a + b = b + a\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_1\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b : Num), a + b = b + a\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_1\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b : Num), a + b = b + a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\na\u271d b\u271d : Num\n\u22a2 a\u271d + b\u271d = b\u271d + a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b c : Num), a * (b + c) = a * b + a * c\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_2\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b c : Num), a * (b + c) = a * b + a * c\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_2\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b c : Num), a * (b + c) = a * b + a * c\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\na\u271d b\u271d c\u271d : Num\n\u22a2 a\u271d * (b\u271d + c\u271d) = a\u271d * b\u271d + a\u271d * c\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_3\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b c : Num), (a + b) * c = a * c + b * c\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_3\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b c : Num), (a + b) * c = a * c + b * c\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_3\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b c : Num), (a + b) * c = a * c + b * c\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\na\u271d b\u271d c\u271d : Num\n\u22a2 (a\u271d + b\u271d) * c\u271d = a\u271d * c\u271d + b\u271d * c\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_4\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), 0 * a = 0\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_4\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), 0 * a = 0\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_4\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), 0 * a = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_4\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\na\u271d : Num\n\u22a2 0 * a\u271d = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_5\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), a * 0 = 0\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_5\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), a * 0 = 0\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_5\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), a * 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_5\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\na\u271d : Num\n\u22a2 a\u271d * 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_6\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b c : Num), a * b * c = a * (b * c)\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_6\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b c : Num), a * b * c = a * (b * c)\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_6\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b c : Num), a * b * c = a * (b * c)\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_6\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\na\u271d b\u271d c\u271d : Num\n\u22a2 a\u271d * b\u271d * c\u271d = a\u271d * (b\u271d * c\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_7\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), 1 * a = a\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_7\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), 1 * a = a\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_7\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), 1 * a = a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_7\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\na\u271d : Num\n\u22a2 1 * a\u271d = a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_8\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), a * 1 = a\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_8\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), a * 1 = a\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_8\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), a * 1 = a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_8\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\na\u271d : Num\n\u22a2 a\u271d * 1 = a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_9\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (x : Num), npowRec 0 x = 1\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_9\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (x : Num), npowRec 0 x = 1\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_9\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (x : Num), npowRec 0 x = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_9\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\nx\u271d : Num\n\u22a2 npowRec 0 x\u271d = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_10\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (n : \u2115) (x : Num), npowRec (n + 1) x = x * npowRec n x\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_10\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (n : \u2115) (x : Num), npowRec (n + 1) x = x * npowRec n x\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_10\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (n : \u2115) (x : Num), npowRec (n + 1) x = x * npowRec n x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_10\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\nn\u271d : \u2115\nx\u271d : Num\n\u22a2 npowRec (n\u271d + 1) x\u271d = x\u271d * npowRec n\u271d x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_11\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b : Num), a * b = b * a\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_11\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b : Num), a * b = b * a\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_11\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b : Num), a * b = b * a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_11\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\na\u271d b\u271d : Num\n\u22a2 a\u271d * b\u271d = b\u271d * a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b : Num), a + b = b + a\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_2\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b c : Num), a * (b + c) = a * b + a * c\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_3\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b c : Num), (a + b) * c = a * c + b * c\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_4\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), 0 * a = 0\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_5\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), a * 0 = 0\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_6\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b c : Num), a * b * c = a * (b * c)\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_7\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), 1 * a = a\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_8\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a : Num), a * 1 = a\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_11\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\n\u22a2 \u2200 (a b : Num), a * b = b * a\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_1\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\na\u271d b\u271d : Num\n\u22a2 \u2191a\u271d + \u2191b\u271d = \u2191b\u271d + \u2191a\u271d\n[PROOFSTEP]\nsimp [add_comm, mul_add, add_mul, mul_assoc, mul_comm, mul_left_comm]\n[GOAL]\ncase refine'_2\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\na\u271d b\u271d c\u271d : Num\n\u22a2 \u2191a\u271d * (\u2191b\u271d + \u2191c\u271d) = \u2191a\u271d * \u2191b\u271d + \u2191a\u271d * \u2191c\u271d\n[PROOFSTEP]\nsimp [add_comm, mul_add, add_mul, mul_assoc, mul_comm, mul_left_comm]\n[GOAL]\ncase refine'_3\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\na\u271d b\u271d c\u271d : Num\n\u22a2 (\u2191a\u271d + \u2191b\u271d) * \u2191c\u271d = \u2191a\u271d * \u2191c\u271d + \u2191b\u271d * \u2191c\u271d\n[PROOFSTEP]\nsimp [add_comm, mul_add, add_mul, mul_assoc, mul_comm, mul_left_comm]\n[GOAL]\ncase refine'_6\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\na\u271d b\u271d c\u271d : Num\n\u22a2 \u2191a\u271d * \u2191b\u271d * \u2191c\u271d = \u2191a\u271d * (\u2191b\u271d * \u2191c\u271d)\n[PROOFSTEP]\nsimp [add_comm, mul_add, add_mul, mul_assoc, mul_comm, mul_left_comm]\n[GOAL]\ncase refine'_11\nsrc\u271d\u00b9 : AddMonoid Num := addMonoid\nsrc\u271d : AddMonoidWithOne Num := addMonoidWithOne\na\u271d b\u271d : Num\n\u22a2 \u2191a\u271d * \u2191b\u271d = \u2191b\u271d * \u2191a\u271d\n[PROOFSTEP]\nsimp [add_comm, mul_add, add_mul, mul_assoc, mul_comm, mul_left_comm]\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\n\u22a2 \u2200 (a : Num), a \u2264 a\n[PROOFSTEP]\ntransfer\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\n\u22a2 \u2200 (a b c : Num), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n[PROOFSTEP]\nintro a b c\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\na b c : Num\n\u22a2 a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\na b c : Num\n\u22a2 \u2191a \u2264 \u2191b \u2192 \u2191b \u2264 \u2191c \u2192 \u2191a \u2264 \u2191c\n[PROOFSTEP]\napply le_trans\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\n\u22a2 \u2200 (a b : Num), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n[PROOFSTEP]\nintro a b\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\na b : Num\n\u22a2 a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\na b : Num\n\u22a2 \u2191a < \u2191b \u2194 \u2191a \u2264 \u2191b \u2227 \u00ac\u2191b \u2264 \u2191a\n[PROOFSTEP]\napply lt_iff_le_not_le\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\n\u22a2 \u2200 (a b : Num), a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\nintro a b\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\na b : Num\n\u22a2 a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\na b : Num\n\u22a2 \u2191a \u2264 \u2191b \u2192 \u2191b \u2264 \u2191a \u2192 \u2191a = \u2191b\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\n\u22a2 \u2200 (a b : Num), a \u2264 b \u2192 \u2200 (c : Num), c + a \u2264 c + b\n[PROOFSTEP]\nintro a b h c\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\na b : Num\nh : a \u2264 b\nc : Num\n\u22a2 c + a \u2264 c + b\n[PROOFSTEP]\nrevert h\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\na b c : Num\n\u22a2 a \u2264 b \u2192 c + a \u2264 c + b\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\na b c : Num\n\u22a2 \u2191a \u2264 \u2191b \u2192 \u2191c + \u2191a \u2264 \u2191c + \u2191b\n[PROOFSTEP]\nexact fun h => add_le_add_left h c\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\n\u22a2 \u2200 (a b c : Num), a + b \u2264 a + c \u2192 b \u2264 c\n[PROOFSTEP]\nintro a b c\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\na b c : Num\n\u22a2 a + b \u2264 a + c \u2192 b \u2264 c\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\nsrc\u271d : CommSemiring Num := commSemiring\na b c : Num\n\u22a2 \u2191a + \u2191b \u2264 \u2191a + \u2191c \u2192 \u2191b \u2264 \u2191c\n[PROOFSTEP]\napply le_of_add_le_add_left\n[GOAL]\nsrc\u271d\u00b9 : CommSemiring Num := commSemiring\nsrc\u271d : OrderedCancelAddCommMonoid Num := orderedCancelAddCommMonoid\n\u22a2 0 \u2260 1\n[PROOFSTEP]\ndecide\n[GOAL]\nsrc\u271d\u00b9 : CommSemiring Num := commSemiring\nsrc\u271d : OrderedCancelAddCommMonoid Num := orderedCancelAddCommMonoid\n\u22a2 0 \u2264 1\n[PROOFSTEP]\ndecide\n[GOAL]\nsrc\u271d\u00b9 : CommSemiring Num := commSemiring\nsrc\u271d : OrderedCancelAddCommMonoid Num := orderedCancelAddCommMonoid\n\u22a2 \u2200 (a b c : Num), a < b \u2192 0 < c \u2192 c * a < c * b\n[PROOFSTEP]\nintro a b c\n[GOAL]\nsrc\u271d\u00b9 : CommSemiring Num := commSemiring\nsrc\u271d : OrderedCancelAddCommMonoid Num := orderedCancelAddCommMonoid\na b c : Num\n\u22a2 a < b \u2192 0 < c \u2192 c * a < c * b\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\nsrc\u271d\u00b9 : CommSemiring Num := commSemiring\nsrc\u271d : OrderedCancelAddCommMonoid Num := orderedCancelAddCommMonoid\na b c : Num\n\u22a2 \u2191a < \u2191b \u2192 0 < \u2191c \u2192 \u2191c * \u2191a < \u2191c * \u2191b\n[PROOFSTEP]\napply mul_lt_mul_of_pos_left\n[GOAL]\nsrc\u271d\u00b9 : CommSemiring Num := commSemiring\nsrc\u271d : OrderedCancelAddCommMonoid Num := orderedCancelAddCommMonoid\n\u22a2 \u2200 (a b c : Num), a < b \u2192 0 < c \u2192 a * c < b * c\n[PROOFSTEP]\nintro a b c\n[GOAL]\nsrc\u271d\u00b9 : CommSemiring Num := commSemiring\nsrc\u271d : OrderedCancelAddCommMonoid Num := orderedCancelAddCommMonoid\na b c : Num\n\u22a2 a < b \u2192 0 < c \u2192 a * c < b * c\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\nsrc\u271d\u00b9 : CommSemiring Num := commSemiring\nsrc\u271d : OrderedCancelAddCommMonoid Num := orderedCancelAddCommMonoid\na b c : Num\n\u22a2 \u2191a < \u2191b \u2192 0 < \u2191c \u2192 \u2191a * \u2191c < \u2191b * \u2191c\n[PROOFSTEP]\napply mul_lt_mul_of_pos_right\n[GOAL]\nsrc\u271d\u00b9 : CommSemiring Num := commSemiring\nsrc\u271d : OrderedCancelAddCommMonoid Num := orderedCancelAddCommMonoid\n\u22a2 \u2200 (a b : Num), a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\nintro a b\n[GOAL]\nsrc\u271d\u00b9 : CommSemiring Num := commSemiring\nsrc\u271d : OrderedCancelAddCommMonoid Num := orderedCancelAddCommMonoid\na b : Num\n\u22a2 a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\nsrc\u271d\u00b9 : CommSemiring Num := commSemiring\nsrc\u271d : OrderedCancelAddCommMonoid Num := orderedCancelAddCommMonoid\na b : Num\n\u22a2 \u2191a \u2264 \u2191b \u2228 \u2191b \u2264 \u2191a\n[PROOFSTEP]\napply le_total\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : Num\n\u22a2 \u2191\u2191n = \u2191n\n[PROOFSTEP]\nrw [\u2190 to_nat_to_int, Int.cast_ofNat, cast_to_nat]\n[GOAL]\n\u22a2 \u2191\u21910 = 0\n[PROOFSTEP]\nrw [Nat.cast_zero, cast_zero]\n[GOAL]\nn : \u2115\n\u22a2 \u2191\u2191(n + 1) = n + 1\n[PROOFSTEP]\nrw [Nat.cast_succ, add_one, succ_to_nat, to_of_nat n]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddMonoidWithOne \u03b1\nn : \u2115\n\u22a2 \u2191\u2191n = \u2191n\n[PROOFSTEP]\nrw [\u2190 cast_to_nat, to_of_nat]\n[GOAL]\nm n : Num\nx\u271d : \u2191m \u2223 \u2191n\nk : \u2115\ne : \u2191n = \u2191m * k\n\u22a2 n = m * \u2191k\n[PROOFSTEP]\nrw [\u2190 of_to_nat n, e]\n[GOAL]\nm n : Num\nx\u271d : \u2191m \u2223 \u2191n\nk : \u2115\ne : \u2191n = \u2191m * k\n\u22a2 \u2191(\u2191m * k) = m * \u2191k\n[PROOFSTEP]\nsimp\n[GOAL]\nm n : Num\nx\u271d : m \u2223 n\nk : Num\ne : n = m * k\n\u22a2 \u2191n = \u2191m * \u2191k\n[PROOFSTEP]\nsimp [e, mul_to_nat]\n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\nh : \u2191m = \u2191n\n\u22a2 pos m = pos n\n[PROOFSTEP]\nrw [\u2190 PosNum.of_to_nat, \u2190 PosNum.of_to_nat, h]\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 Nat.succ \u2191(pred' n) = \u2191n\n[PROOFSTEP]\nrw [pred'_to_nat n, Nat.succ_pred_eq_of_pos (to_nat_pos n)]\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\nthis : Nat.succ \u2191(pred' n) = \u2191n\nh : \u21911 = \u2191n\n\u22a2 \u2191(Num.casesOn 0 1 bit1) = Nat.pred (_root_.bit0 \u2191n)\n[PROOFSTEP]\nrw [\u2190 to_nat_inj.1 h]\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\nthis : Nat.succ \u2191(pred' n) = \u2191n\nh : \u21911 = \u2191n\n\u22a2 \u2191(Num.casesOn 0 1 bit1) = Nat.pred (_root_.bit0 \u21911)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\nthis : Nat.succ \u2191(pred' n) = \u2191n\np : PosNum\nh : Nat.succ \u2191p = \u2191n\n\u22a2 \u2191(Num.casesOn (pos p) 1 bit1) = Nat.pred (_root_.bit0 \u2191n)\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\nthis : Nat.succ \u2191(pred' n) = \u2191n\np : PosNum\nh : Nat.succ \u2191p = \u2191n\n\u22a2 \u2191(Num.casesOn (pos p) 1 bit1) = Nat.pred (_root_.bit0 (Nat.succ \u2191p))\n[PROOFSTEP]\nexact (Nat.succ_add p p).symm\n[GOAL]\n\u03b1 : Type u_1\nn : Num\n\u22a2 \u2191(pred' (succ' n)) = \u2191n\n[PROOFSTEP]\nrw [pred'_to_nat, succ'_to_nat, Nat.add_one, Nat.pred_succ]\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 \u2191(succ' (pred' n)) = \u2191n\n[PROOFSTEP]\nrw [succ'_to_nat, pred'_to_nat, Nat.add_one, Nat.succ_pred_eq_of_pos (to_nat_pos _)]\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 \u2191(size (bit0 n)) = Nat.size \u2191(bit0 n)\n[PROOFSTEP]\nrw [size, succ_to_nat, size_to_nat n, cast_bit0, Nat.size_bit0 <| ne_of_gt <| to_nat_pos n]\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 \u2191(size (bit1 n)) = Nat.size \u2191(bit1 n)\n[PROOFSTEP]\nrw [size, succ_to_nat, size_to_nat n, cast_bit1, Nat.size_bit1]\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 \u2191(size (bit0 n)) = natSize (bit0 n)\n[PROOFSTEP]\nrw [size, succ_to_nat, natSize, size_eq_natSize n]\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 \u2191(size (bit1 n)) = natSize (bit1 n)\n[PROOFSTEP]\nrw [size, succ_to_nat, natSize, size_eq_natSize n]\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 natSize n = Nat.size \u2191n\n[PROOFSTEP]\nrw [\u2190 size_eq_natSize, size_to_nat]\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 0 < natSize n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase one\n\u03b1 : Type u_1\n\u22a2 0 < natSize one\n[PROOFSTEP]\napply Nat.succ_pos\n[GOAL]\ncase bit1\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 0 < natSize (bit1 a\u271d)\n[PROOFSTEP]\napply Nat.succ_pos\n[GOAL]\ncase bit0\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 0 < natSize (bit0 a\u271d)\n[PROOFSTEP]\napply Nat.succ_pos\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 AddCommSemigroup PosNum\n[PROOFSTEP]\nrefine' { add := (\u00b7 + \u00b7) .. }\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b c : PosNum), a + b + c = a + (b + c)\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b c : PosNum), a + b + c = a + (b + c)\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : PosNum), a + b = b + a\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : PosNum), a + b = b + a\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 CommMonoid PosNum\n[PROOFSTEP]\nrefine'\n { mul := (\u00b7 * \u00b7)\n one := (1 : PosNum)\n npow := @npowRec PosNum \u27e81\u27e9 \u27e8(\u00b7 * \u00b7)\u27e9, .. }\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b c : PosNum), a * b * c = a * (b * c)\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b c : PosNum), a * b * c = a * (b * c)\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b c : PosNum), a * b * c = a * (b * c)\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d : PosNum\n\u22a2 a\u271d * b\u271d * c\u271d = a\u271d * (b\u271d * c\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : PosNum), 1 * a = a\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : PosNum), 1 * a = a\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : PosNum), 1 * a = a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 1 * a\u271d = a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : PosNum), a * 1 = a\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : PosNum), a * 1 = a\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : PosNum), a * 1 = a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 a\u271d * 1 = a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u22a2 \u2200 (x : PosNum), npowRec 0 x = 1\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u22a2 \u2200 (x : PosNum), npowRec 0 x = 1\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u22a2 \u2200 (x : PosNum), npowRec 0 x = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\nx\u271d : PosNum\n\u22a2 npowRec 0 x\u271d = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_5\n\u03b1 : Type u_1\n\u22a2 \u2200 (n : \u2115) (x : PosNum), npowRec (n + 1) x = x * npowRec n x\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_5\n\u03b1 : Type u_1\n\u22a2 \u2200 (n : \u2115) (x : PosNum), npowRec (n + 1) x = x * npowRec n x\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_5\n\u03b1 : Type u_1\n\u22a2 \u2200 (n : \u2115) (x : PosNum), npowRec (n + 1) x = x * npowRec n x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_5\n\u03b1 : Type u_1\nn\u271d : \u2115\nx\u271d : PosNum\n\u22a2 npowRec (n\u271d + 1) x\u271d = x\u271d * npowRec n\u271d x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_6\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : PosNum), a * b = b * a\n[PROOFSTEP]\ntry {intros; rfl\n}\n[GOAL]\ncase refine'_6\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : PosNum), a * b = b * a\n[PROOFSTEP]\n{intros; rfl\n}\n[GOAL]\ncase refine'_6\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : PosNum), a * b = b * a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_6\n\u03b1 : Type u_1\na\u271d b\u271d : PosNum\n\u22a2 a\u271d * b\u271d = b\u271d * a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b c : PosNum), a * b * c = a * (b * c)\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b c : PosNum), a * b * c = a * (b * c)\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : PosNum), 1 * a = a\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : PosNum), 1 * a = a\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_6\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : PosNum), a * b = b * a\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_6\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : PosNum), a * b = b * a\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 Distrib PosNum\n[PROOFSTEP]\nrefine'\n { add := (\u00b7 + \u00b7)\n mul := (\u00b7 * \u00b7) .. }\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b c : PosNum), a * (b + c) = a * b + a * c\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b c : PosNum), a * (b + c) = a * b + a * c\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d : PosNum\n\u22a2 \u2191a\u271d * (\u2191b\u271d + \u2191c\u271d) = \u2191a\u271d * \u2191b\u271d + \u2191a\u271d * \u2191c\u271d\n[PROOFSTEP]\nsimp [mul_add, mul_comm]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b c : PosNum), (a + b) * c = a * c + b * c\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b c : PosNum), (a + b) * c = a * c + b * c\n[PROOFSTEP]\ntransfer\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\na\u271d b\u271d c\u271d : PosNum\n\u22a2 \u2191c\u271d * (\u2191a\u271d + \u2191b\u271d) = \u2191a\u271d * \u2191c\u271d + \u2191b\u271d * \u2191c\u271d\n[PROOFSTEP]\nsimp [mul_add, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : PosNum), a \u2264 a\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : PosNum), a \u2264 a\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b c : PosNum), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n[PROOFSTEP]\nintro a b c\n[GOAL]\n\u03b1 : Type u_1\na b c : PosNum\n\u22a2 a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\n\u03b1 : Type u_1\na b c : PosNum\n\u22a2 a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\n\u03b1 : Type u_1\na b c : PosNum\n\u22a2 \u2191a \u2264 \u2191b \u2192 \u2191b \u2264 \u2191c \u2192 \u2191a \u2264 \u2191c\n[PROOFSTEP]\napply le_trans\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : PosNum), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n[PROOFSTEP]\nintro a b\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191a < \u2191b \u2194 \u2191a \u2264 \u2191b \u2227 \u00ac\u2191b \u2264 \u2191a\n[PROOFSTEP]\napply lt_iff_le_not_le\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : PosNum), a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\nintro a b\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191a \u2264 \u2191b \u2192 \u2191b \u2264 \u2191a \u2192 \u2191a = \u2191b\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : PosNum), a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\nintro a b\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191a \u2264 \u2191b \u2228 \u2191b \u2264 \u2191a\n[PROOFSTEP]\napply le_total\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 DecidableRel fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 DecidableEq PosNum\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 DecidableRel fun x x_1 => x < x_1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 \u2191n = pos n\n[PROOFSTEP]\nrw [\u2190 cast_to_nat, \u2190 of_to_nat n]\n[GOAL]\n\u03b1 : Type u_1\nb : Bool\nn : PosNum\n\u22a2 \u2191(bit b n) = Nat.bit b \u2191n\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 \u2191(bit false n) = Nat.bit false \u2191n\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 \u2191(bit true n) = Nat.bit true \u2191n\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddMonoidWithOne \u03b1\nm n : PosNum\n\u22a2 \u2191(m + n) = \u2191m + \u2191n\n[PROOFSTEP]\nrw [\u2190 cast_to_nat, add_to_nat, Nat.cast_add, cast_to_nat, cast_to_nat]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddMonoidWithOne \u03b1\nn : PosNum\n\u22a2 \u2191(succ n) = \u2191n + 1\n[PROOFSTEP]\nrw [\u2190 add_one, cast_add, cast_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddMonoidWithOne \u03b1\ninst\u271d : CharZero \u03b1\nm n : PosNum\n\u22a2 \u2191m = \u2191n \u2194 m = n\n[PROOFSTEP]\nrw [\u2190 cast_to_nat m, \u2190 cast_to_nat n, Nat.cast_inj, to_nat_inj]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\nn : PosNum\n\u22a2 1 \u2264 \u2191n\n[PROOFSTEP]\nrw [\u2190 cast_to_nat, \u2190 Nat.cast_one, Nat.cast_le (\u03b1 := \u03b1)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\nn : PosNum\n\u22a2 1 \u2264 \u2191n\n[PROOFSTEP]\napply to_nat_pos\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nm n : PosNum\n\u22a2 \u2191(m * n) = \u2191m * \u2191n\n[PROOFSTEP]\nrw [\u2190 cast_to_nat, mul_to_nat, Nat.cast_mul, cast_to_nat, cast_to_nat]\n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\n\u22a2 cmp m n = Ordering.eq \u2194 m = n\n[PROOFSTEP]\nhave := cmp_to_nat m n\n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\nthis : Ordering.casesOn (cmp m n) (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 cmp m n = Ordering.eq \u2194 m = n\n[PROOFSTEP]\nrevert this\n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\n\u22a2 Ordering.casesOn (cmp m n) (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m) \u2192 (cmp m n = Ordering.eq \u2194 m = n)\n[PROOFSTEP]\ncases cmp m n\n[GOAL]\ncase lt\n\u03b1 : Type u_1\nm n : PosNum\n\u22a2 Ordering.casesOn Ordering.lt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m) \u2192 (Ordering.lt = Ordering.eq \u2194 m = n)\n[PROOFSTEP]\nintro this\n[GOAL]\ncase eq\n\u03b1 : Type u_1\nm n : PosNum\n\u22a2 Ordering.casesOn Ordering.eq (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m) \u2192 (Ordering.eq = Ordering.eq \u2194 m = n)\n[PROOFSTEP]\nintro this\n[GOAL]\ncase gt\n\u03b1 : Type u_1\nm n : PosNum\n\u22a2 Ordering.casesOn Ordering.gt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m) \u2192 (Ordering.gt = Ordering.eq \u2194 m = n)\n[PROOFSTEP]\nintro this\n[GOAL]\ncase lt\n\u03b1 : Type u_1\nm n : PosNum\nthis : Ordering.casesOn Ordering.lt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 Ordering.lt = Ordering.eq \u2194 m = n\n[PROOFSTEP]\nsimp at this \u22a2\n[GOAL]\ncase eq\n\u03b1 : Type u_1\nm n : PosNum\nthis : Ordering.casesOn Ordering.eq (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 Ordering.eq = Ordering.eq \u2194 m = n\n[PROOFSTEP]\nsimp at this \u22a2\n[GOAL]\ncase gt\n\u03b1 : Type u_1\nm n : PosNum\nthis : Ordering.casesOn Ordering.gt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 Ordering.gt = Ordering.eq \u2194 m = n\n[PROOFSTEP]\nsimp at this \u22a2\n[GOAL]\ncase lt\n\u03b1 : Type u_1\nm n : PosNum\nthis : \u2191m < \u2191n\n\u22a2 \u00acm = n\n[PROOFSTEP]\ntry {exact this\n}\n[GOAL]\ncase lt\n\u03b1 : Type u_1\nm n : PosNum\nthis : \u2191m < \u2191n\n\u22a2 \u00acm = n\n[PROOFSTEP]\n{exact this\n}\n[GOAL]\ncase lt\n\u03b1 : Type u_1\nm n : PosNum\nthis : \u2191m < \u2191n\n\u22a2 \u00acm = n\n[PROOFSTEP]\nexact this\n[GOAL]\ncase eq\n\u03b1 : Type u_1\nm n : PosNum\nthis : m = n\n\u22a2 m = n\n[PROOFSTEP]\ntry {exact this\n}\n[GOAL]\ncase eq\n\u03b1 : Type u_1\nm n : PosNum\nthis : m = n\n\u22a2 m = n\n[PROOFSTEP]\n{exact this\n}\n[GOAL]\ncase eq\n\u03b1 : Type u_1\nm n : PosNum\nthis : m = n\n\u22a2 m = n\n[PROOFSTEP]\nexact this\n[GOAL]\ncase gt\n\u03b1 : Type u_1\nm n : PosNum\nthis : \u2191n < \u2191m\n\u22a2 \u00acm = n\n[PROOFSTEP]\ntry {exact this\n}\n[GOAL]\ncase gt\n\u03b1 : Type u_1\nm n : PosNum\nthis : \u2191n < \u2191m\n\u22a2 \u00acm = n\n[PROOFSTEP]\n{exact this\n}\n[GOAL]\ncase gt\n\u03b1 : Type u_1\nm n : PosNum\nthis : \u2191n < \u2191m\n\u22a2 \u00acm = n\n[PROOFSTEP]\nexact this\n[GOAL]\ncase lt\n\u03b1 : Type u_1\nm n : PosNum\nthis : \u2191m < \u2191n\n\u22a2 \u00acm = n\n[PROOFSTEP]\nsimp [show m \u2260 n from fun e => by rw [e] at this ; exact lt_irrefl _ this]\n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\nthis : \u2191m < \u2191n\ne : m = n\n\u22a2 False\n[PROOFSTEP]\nrw [e] at this \n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\nthis : \u2191n < \u2191n\ne : m = n\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ this\n[GOAL]\ncase gt\n\u03b1 : Type u_1\nm n : PosNum\nthis : \u2191n < \u2191m\n\u22a2 \u00acm = n\n[PROOFSTEP]\nsimp [show m \u2260 n from fun e => by rw [e] at this ; exact lt_irrefl _ this]\n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\nthis : \u2191n < \u2191m\ne : m = n\n\u22a2 False\n[PROOFSTEP]\nrw [e] at this \n[GOAL]\n\u03b1 : Type u_1\nm n : PosNum\nthis : \u2191n < \u2191n\ne : m = n\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\nm n : PosNum\n\u22a2 \u2191m < \u2191n \u2194 m < n\n[PROOFSTEP]\nrw [\u2190 cast_to_nat m, \u2190 cast_to_nat n, Nat.cast_lt (\u03b1 := \u03b1), lt_to_nat]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\nm n : PosNum\n\u22a2 \u2191m \u2264 \u2191n \u2194 m \u2264 n\n[PROOFSTEP]\nrw [\u2190 not_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\nm n : PosNum\n\u22a2 \u00ac\u2191n < \u2191m \u2194 m \u2264 n\n[PROOFSTEP]\nexact not_congr cast_lt\n[GOAL]\n\u03b1 : Type u_1\nb : Bool\nn : Num\n\u22a2 \u2191(bit b n) = Nat.bit b \u2191n\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nn : Num\n\u22a2 \u2191(bit false n) = Nat.bit false \u2191n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase true\n\u03b1 : Type u_1\nn : Num\n\u22a2 \u2191(bit true n) = Nat.bit true \u2191n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase false.zero\n\u03b1 : Type u_1\n\u22a2 \u2191(bit false zero) = Nat.bit false \u2191zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase false.pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 \u2191(bit false (pos a\u271d)) = Nat.bit false \u2191(pos a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true.zero\n\u03b1 : Type u_1\n\u22a2 \u2191(bit true zero) = Nat.bit true \u2191zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true.pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 \u2191(bit true (pos a\u271d)) = Nat.bit true \u2191(pos a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddMonoidWithOne \u03b1\nn : Num\n\u22a2 \u2191(succ' n) = \u2191n + 1\n[PROOFSTEP]\nrw [\u2190 PosNum.cast_to_nat, succ'_to_nat, Nat.cast_add_one, cast_to_nat]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nm n : Num\n\u22a2 \u2191(m + n) = \u2191m + \u2191n\n[PROOFSTEP]\nrw [\u2190 cast_to_nat, add_to_nat, Nat.cast_add, cast_to_nat, cast_to_nat]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nn : Num\n\u22a2 \u2191(Num.bit0 n) = _root_.bit0 \u2191n\n[PROOFSTEP]\nrw [\u2190 bit0_of_bit0, _root_.bit0, cast_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nn : Num\n\u22a2 \u2191n + \u2191n = _root_.bit0 \u2191n\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nn : Num\n\u22a2 \u2191(Num.bit1 n) = _root_.bit1 \u2191n\n[PROOFSTEP]\nrw [\u2190 bit1_of_bit1, _root_.bit1, bit0_of_bit0, cast_add, cast_bit0]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nn : Num\n\u22a2 _root_.bit0 \u2191n + \u21911 = _root_.bit1 \u2191n\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : Num\n\u22a2 natSize n = Nat.size \u2191n\n[PROOFSTEP]\nrw [\u2190 size_eq_natSize, size_to_nat]\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 ofNat' 0 = \u21910\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nb : Bool\nn : \u2115\nIH : ofNat' n = \u2191n\n\u22a2 ofNat' (Nat.bit b n) = \u2191(Nat.bit b n)\n[PROOFSTEP]\nrw [ofNat'] at IH \u22a2\n[GOAL]\n\u03b1 : Type u_1\nb : Bool\nn : \u2115\nIH : Nat.binaryRec 0 (fun b x => bif b then Num.bit1 else Num.bit0) n = \u2191n\n\u22a2 Nat.binaryRec 0 (fun b x => bif b then Num.bit1 else Num.bit0) (Nat.bit b n) = \u2191(Nat.bit b n)\n[PROOFSTEP]\nrw [Nat.binaryRec_eq, IH]\n -- Porting note: `Nat.cast_bit0` & `Nat.cast_bit1` are not `simp` theorems anymore.\n[GOAL]\n\u03b1 : Type u_1\nb : Bool\nn : \u2115\nIH : Nat.binaryRec 0 (fun b x => bif b then Num.bit1 else Num.bit0) n = \u2191n\n\u22a2 cond b Num.bit1 Num.bit0 \u2191n = \u2191(Nat.bit b n)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nn : \u2115\nIH : Nat.binaryRec 0 (fun b x => bif b then Num.bit1 else Num.bit0) n = \u2191n\n\u22a2 cond false Num.bit1 Num.bit0 \u2191n = \u2191(Nat.bit false n)\n[PROOFSTEP]\nsimp [Nat.bit, bit0_of_bit0, bit1_of_bit1, Nat.cast_bit0, Nat.cast_bit1]\n[GOAL]\ncase true\n\u03b1 : Type u_1\nn : \u2115\nIH : Nat.binaryRec 0 (fun b x => bif b then Num.bit1 else Num.bit0) n = \u2191n\n\u22a2 cond true Num.bit1 Num.bit0 \u2191n = \u2191(Nat.bit true n)\n[PROOFSTEP]\nsimp [Nat.bit, bit0_of_bit0, bit1_of_bit1, Nat.cast_bit0, Nat.cast_bit1]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nb : Bool\nn : \u2115\nIH : Nat.binaryRec 0 (fun b x => bif b then Num.bit1 else Num.bit0) n = \u2191n\n\u22a2 cond false Num.bit1 Num.bit0 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : Num\n\u22a2 -toZNum n = toZNumNeg n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u22a2 -toZNum zero = toZNumNeg zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 -toZNum (pos a\u271d) = toZNumNeg (pos a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : Num\n\u22a2 -toZNumNeg n = toZNum n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u22a2 -toZNumNeg zero = toZNum zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 -toZNumNeg (pos a\u271d) = toZNum (pos a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\nh : toZNum m = toZNum n\n\u22a2 m = n\n[PROOFSTEP]\ncases m\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nn : Num\nh : toZNum zero = toZNum n\n\u22a2 zero = n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nn : Num\na\u271d : PosNum\nh : toZNum (pos a\u271d) = toZNum n\n\u22a2 pos a\u271d = n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero.zero\n\u03b1 : Type u_1\nh : toZNum zero = toZNum zero\n\u22a2 zero = zero\n[PROOFSTEP]\ncases h\n[GOAL]\ncase zero.pos\n\u03b1 : Type u_1\na\u271d : PosNum\nh : toZNum zero = toZNum (pos a\u271d)\n\u22a2 zero = pos a\u271d\n[PROOFSTEP]\ncases h\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\na\u271d : PosNum\nh : toZNum (pos a\u271d) = toZNum zero\n\u22a2 pos a\u271d = zero\n[PROOFSTEP]\ncases h\n[GOAL]\ncase pos.pos\n\u03b1 : Type u_1\na\u271d\u00b9 a\u271d : PosNum\nh : toZNum (pos a\u271d\u00b9) = toZNum (pos a\u271d)\n\u22a2 pos a\u271d\u00b9 = pos a\u271d\n[PROOFSTEP]\ncases h\n[GOAL]\ncase zero.zero.refl\n\u03b1 : Type u_1\n\u22a2 zero = zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.refl\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 pos a\u271d = pos a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\n\u22a2 toZNum (m + n) = toZNum m + toZNum n\n[PROOFSTEP]\ncases m\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nn : Num\n\u22a2 toZNum (zero + n) = toZNum zero + toZNum n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nn : Num\na\u271d : PosNum\n\u22a2 toZNum (pos a\u271d + n) = toZNum (pos a\u271d) + toZNum n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero.zero\n\u03b1 : Type u_1\n\u22a2 toZNum (zero + zero) = toZNum zero + toZNum zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase zero.pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 toZNum (zero + pos a\u271d) = toZNum zero + toZNum (pos a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 toZNum (pos a\u271d + zero) = toZNum (pos a\u271d) + toZNum zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos\n\u03b1 : Type u_1\na\u271d\u00b9 a\u271d : PosNum\n\u22a2 toZNum (pos a\u271d\u00b9 + pos a\u271d) = toZNum (pos a\u271d\u00b9) + toZNum (pos a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nn : PosNum\nh : 1 < n\n\u22a2 \u2191(pred n) = Nat.pred \u2191n\n[PROOFSTEP]\nunfold pred\n[GOAL]\nn : PosNum\nh : 1 < n\n\u22a2 \u2191(Num.casesOn (pred' n) 1 id) = Nat.pred \u2191n\n[PROOFSTEP]\ncases e : pred' n\n[GOAL]\ncase zero\nn : PosNum\nh : 1 < n\ne : pred' n = zero\n\u22a2 \u2191(Num.casesOn zero 1 id) = Nat.pred \u2191n\n[PROOFSTEP]\nhave : (1 : \u2115) \u2264 Nat.pred n := Nat.pred_le_pred ((@cast_lt \u2115 _ _ _).2 h)\n[GOAL]\ncase zero\nn : PosNum\nh : 1 < n\ne : pred' n = zero\nthis : 1 \u2264 Nat.pred \u2191n\n\u22a2 \u2191(Num.casesOn zero 1 id) = Nat.pred \u2191n\n[PROOFSTEP]\nrw [\u2190 pred'_to_nat, e] at this \n[GOAL]\ncase zero\nn : PosNum\nh : 1 < n\ne : pred' n = zero\nthis : 1 \u2264 \u2191zero\n\u22a2 \u2191(Num.casesOn zero 1 id) = Nat.pred \u2191n\n[PROOFSTEP]\nexact absurd this (by decide)\n[GOAL]\nn : PosNum\nh : 1 < n\ne : pred' n = zero\nthis : 1 \u2264 \u2191zero\n\u22a2 \u00ac1 \u2264 \u2191zero\n[PROOFSTEP]\ndecide\n[GOAL]\ncase pos\nn : PosNum\nh : 1 < n\na\u271d : PosNum\ne : pred' n = pos a\u271d\n\u22a2 \u2191(Num.casesOn (pos a\u271d) 1 id) = Nat.pred \u2191n\n[PROOFSTEP]\nrw [\u2190 pred'_to_nat, e]\n[GOAL]\ncase pos\nn : PosNum\nh : 1 < n\na\u271d : PosNum\ne : pred' n = pos a\u271d\n\u22a2 \u2191(Num.casesOn (pos a\u271d) 1 id) = \u2191(pos a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\na : PosNum\n\u22a2 sub' a 1 = toZNum (pred' a)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase one\n\u22a2 sub' one 1 = toZNum (pred' one)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit1\na\u271d : PosNum\n\u22a2 sub' (bit1 a\u271d) 1 = toZNum (pred' (bit1 a\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit0\na\u271d : PosNum\n\u22a2 sub' (bit0 a\u271d) 1 = toZNum (pred' (bit0 a\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\na : PosNum\n\u22a2 sub' 1 a = toZNumNeg (pred' a)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase one\n\u22a2 sub' 1 one = toZNumNeg (pred' one)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit1\na\u271d : PosNum\n\u22a2 sub' 1 (bit1 a\u271d) = toZNumNeg (pred' (bit1 a\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit0\na\u271d : PosNum\n\u22a2 sub' 1 (bit0 a\u271d) = toZNumNeg (pred' (bit0 a\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\nm n : PosNum\n\u22a2 cmp n m = Ordering.lt \u2194 cmp m n = Ordering.gt\n[PROOFSTEP]\nrw [\u2190 cmp_swap]\n[GOAL]\nm n : PosNum\n\u22a2 Ordering.swap (cmp m n) = Ordering.lt \u2194 cmp m n = Ordering.gt\n[PROOFSTEP]\ncases cmp m n\n[GOAL]\ncase lt\nm n : PosNum\n\u22a2 Ordering.swap Ordering.lt = Ordering.lt \u2194 Ordering.lt = Ordering.gt\n[PROOFSTEP]\nexact by decide\n[GOAL]\nm n : PosNum\n\u22a2 Ordering.swap Ordering.lt = Ordering.lt \u2194 Ordering.lt = Ordering.gt\n[PROOFSTEP]\ndecide\n[GOAL]\ncase eq\nm n : PosNum\n\u22a2 Ordering.swap Ordering.eq = Ordering.lt \u2194 Ordering.eq = Ordering.gt\n[PROOFSTEP]\nexact by decide\n[GOAL]\nm n : PosNum\n\u22a2 Ordering.swap Ordering.eq = Ordering.lt \u2194 Ordering.eq = Ordering.gt\n[PROOFSTEP]\ndecide\n[GOAL]\ncase gt\nm n : PosNum\n\u22a2 Ordering.swap Ordering.gt = Ordering.lt \u2194 Ordering.gt = Ordering.gt\n[PROOFSTEP]\nexact by decide\n[GOAL]\nm n : PosNum\n\u22a2 Ordering.swap Ordering.gt = Ordering.lt \u2194 Ordering.gt = Ordering.gt\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 \u2191(pred (pos p)) = Nat.pred \u2191(pos p)\n[PROOFSTEP]\nrw [pred, PosNum.pred'_to_nat]\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 Nat.pred \u2191p = Nat.pred \u2191(pos p)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 castNum <$> ppred (pos p) = Nat.ppred \u2191(pos p)\n[PROOFSTEP]\nrw [ppred, Option.map_some, Nat.ppred_eq_some.2]\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 Nat.succ \u2191(pred' p) = \u2191(pos p)\n[PROOFSTEP]\nrw [PosNum.pred'_to_nat, Nat.succ_pred_eq_of_pos (PosNum.to_nat_pos _)]\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 \u2191p = \u2191(pos p)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\n\u22a2 Ordering.swap (cmp m n) = cmp n m\n[PROOFSTEP]\ncases m\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nn : Num\n\u22a2 Ordering.swap (cmp zero n) = cmp n zero\n[PROOFSTEP]\ncases n\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nn : Num\na\u271d : PosNum\n\u22a2 Ordering.swap (cmp (pos a\u271d) n) = cmp n (pos a\u271d)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero.zero\n\u03b1 : Type u_1\n\u22a2 Ordering.swap (cmp zero zero) = cmp zero zero\n[PROOFSTEP]\ntry {unfold cmp\n}\n[GOAL]\ncase zero.zero\n\u03b1 : Type u_1\n\u22a2 Ordering.swap (cmp zero zero) = cmp zero zero\n[PROOFSTEP]\n{unfold cmp\n}\n[GOAL]\ncase zero.zero\n\u03b1 : Type u_1\n\u22a2 Ordering.swap (cmp zero zero) = cmp zero zero\n[PROOFSTEP]\nunfold cmp\n[GOAL]\ncase zero.pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 Ordering.swap (cmp zero (pos a\u271d)) = cmp (pos a\u271d) zero\n[PROOFSTEP]\ntry {unfold cmp\n}\n[GOAL]\ncase zero.pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 Ordering.swap (cmp zero (pos a\u271d)) = cmp (pos a\u271d) zero\n[PROOFSTEP]\n{unfold cmp\n}\n[GOAL]\ncase zero.pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 Ordering.swap (cmp zero (pos a\u271d)) = cmp (pos a\u271d) zero\n[PROOFSTEP]\nunfold cmp\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 Ordering.swap (cmp (pos a\u271d) zero) = cmp zero (pos a\u271d)\n[PROOFSTEP]\ntry {unfold cmp\n}\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 Ordering.swap (cmp (pos a\u271d) zero) = cmp zero (pos a\u271d)\n[PROOFSTEP]\n{unfold cmp\n}\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 Ordering.swap (cmp (pos a\u271d) zero) = cmp zero (pos a\u271d)\n[PROOFSTEP]\nunfold cmp\n[GOAL]\ncase pos.pos\n\u03b1 : Type u_1\na\u271d\u00b9 a\u271d : PosNum\n\u22a2 Ordering.swap (cmp (pos a\u271d\u00b9) (pos a\u271d)) = cmp (pos a\u271d) (pos a\u271d\u00b9)\n[PROOFSTEP]\ntry {unfold cmp\n}\n[GOAL]\ncase pos.pos\n\u03b1 : Type u_1\na\u271d\u00b9 a\u271d : PosNum\n\u22a2 Ordering.swap (cmp (pos a\u271d\u00b9) (pos a\u271d)) = cmp (pos a\u271d) (pos a\u271d\u00b9)\n[PROOFSTEP]\n{unfold cmp\n}\n[GOAL]\ncase pos.pos\n\u03b1 : Type u_1\na\u271d\u00b9 a\u271d : PosNum\n\u22a2 Ordering.swap (cmp (pos a\u271d\u00b9) (pos a\u271d)) = cmp (pos a\u271d) (pos a\u271d\u00b9)\n[PROOFSTEP]\nunfold cmp\n[GOAL]\ncase zero.zero\n\u03b1 : Type u_1\n\u22a2 Ordering.swap (cmp zero zero) = cmp zero zero\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase zero.zero\n\u03b1 : Type u_1\n\u22a2 Ordering.swap (cmp zero zero) = cmp zero zero\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase zero.zero\n\u03b1 : Type u_1\n\u22a2 Ordering.swap (cmp zero zero) = cmp zero zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase zero.pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 Ordering.swap (cmp zero (pos a\u271d)) = cmp (pos a\u271d) zero\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase zero.pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 Ordering.swap (cmp zero (pos a\u271d)) = cmp (pos a\u271d) zero\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase zero.pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 Ordering.swap (cmp zero (pos a\u271d)) = cmp (pos a\u271d) zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 Ordering.swap (cmp (pos a\u271d) zero) = cmp zero (pos a\u271d)\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 Ordering.swap (cmp (pos a\u271d) zero) = cmp zero (pos a\u271d)\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 Ordering.swap (cmp (pos a\u271d) zero) = cmp zero (pos a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos\n\u03b1 : Type u_1\na\u271d\u00b9 a\u271d : PosNum\n\u22a2 Ordering.swap (cmp (pos a\u271d\u00b9) (pos a\u271d)) = cmp (pos a\u271d) (pos a\u271d\u00b9)\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase pos.pos\n\u03b1 : Type u_1\na\u271d\u00b9 a\u271d : PosNum\n\u22a2 Ordering.swap (cmp (pos a\u271d\u00b9) (pos a\u271d)) = cmp (pos a\u271d) (pos a\u271d\u00b9)\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase pos.pos\n\u03b1 : Type u_1\na\u271d\u00b9 a\u271d : PosNum\n\u22a2 Ordering.swap (cmp (pos a\u271d\u00b9) (pos a\u271d)) = cmp (pos a\u271d) (pos a\u271d\u00b9)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos\n\u03b1 : Type u_1\na\u271d\u00b9 a\u271d : PosNum\n\u22a2 Ordering.swap (cmp (pos a\u271d\u00b9) (pos a\u271d)) = cmp (pos a\u271d) (pos a\u271d\u00b9)\n[PROOFSTEP]\napply PosNum.cmp_swap\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\n\u22a2 cmp m n = Ordering.eq \u2194 m = n\n[PROOFSTEP]\nhave := cmp_to_nat m n\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\nthis : Ordering.casesOn (cmp m n) (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 cmp m n = Ordering.eq \u2194 m = n\n[PROOFSTEP]\nrevert this\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\n\u22a2 Ordering.casesOn (cmp m n) (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m) \u2192 (cmp m n = Ordering.eq \u2194 m = n)\n[PROOFSTEP]\ncases cmp m n\n[GOAL]\ncase lt\n\u03b1 : Type u_1\nm n : Num\n\u22a2 Ordering.casesOn Ordering.lt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m) \u2192 (Ordering.lt = Ordering.eq \u2194 m = n)\n[PROOFSTEP]\nintro this\n[GOAL]\ncase eq\n\u03b1 : Type u_1\nm n : Num\n\u22a2 Ordering.casesOn Ordering.eq (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m) \u2192 (Ordering.eq = Ordering.eq \u2194 m = n)\n[PROOFSTEP]\nintro this\n[GOAL]\ncase gt\n\u03b1 : Type u_1\nm n : Num\n\u22a2 Ordering.casesOn Ordering.gt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m) \u2192 (Ordering.gt = Ordering.eq \u2194 m = n)\n[PROOFSTEP]\nintro this\n[GOAL]\ncase lt\n\u03b1 : Type u_1\nm n : Num\nthis : Ordering.casesOn Ordering.lt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 Ordering.lt = Ordering.eq \u2194 m = n\n[PROOFSTEP]\nsimp at this \u22a2\n[GOAL]\ncase eq\n\u03b1 : Type u_1\nm n : Num\nthis : Ordering.casesOn Ordering.eq (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 Ordering.eq = Ordering.eq \u2194 m = n\n[PROOFSTEP]\nsimp at this \u22a2\n[GOAL]\ncase gt\n\u03b1 : Type u_1\nm n : Num\nthis : Ordering.casesOn Ordering.gt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 Ordering.gt = Ordering.eq \u2194 m = n\n[PROOFSTEP]\nsimp at this \u22a2\n[GOAL]\ncase lt\n\u03b1 : Type u_1\nm n : Num\nthis : \u2191m < \u2191n\n\u22a2 \u00acm = n\n[PROOFSTEP]\ntry {exact this\n}\n[GOAL]\ncase lt\n\u03b1 : Type u_1\nm n : Num\nthis : \u2191m < \u2191n\n\u22a2 \u00acm = n\n[PROOFSTEP]\n{exact this\n}\n[GOAL]\ncase lt\n\u03b1 : Type u_1\nm n : Num\nthis : \u2191m < \u2191n\n\u22a2 \u00acm = n\n[PROOFSTEP]\nexact this\n[GOAL]\ncase eq\n\u03b1 : Type u_1\nm n : Num\nthis : m = n\n\u22a2 m = n\n[PROOFSTEP]\ntry {exact this\n}\n[GOAL]\ncase eq\n\u03b1 : Type u_1\nm n : Num\nthis : m = n\n\u22a2 m = n\n[PROOFSTEP]\n{exact this\n}\n[GOAL]\ncase eq\n\u03b1 : Type u_1\nm n : Num\nthis : m = n\n\u22a2 m = n\n[PROOFSTEP]\nexact this\n[GOAL]\ncase gt\n\u03b1 : Type u_1\nm n : Num\nthis : \u2191n < \u2191m\n\u22a2 \u00acm = n\n[PROOFSTEP]\ntry {exact this\n}\n[GOAL]\ncase gt\n\u03b1 : Type u_1\nm n : Num\nthis : \u2191n < \u2191m\n\u22a2 \u00acm = n\n[PROOFSTEP]\n{exact this\n}\n[GOAL]\ncase gt\n\u03b1 : Type u_1\nm n : Num\nthis : \u2191n < \u2191m\n\u22a2 \u00acm = n\n[PROOFSTEP]\nexact this\n[GOAL]\ncase lt\n\u03b1 : Type u_1\nm n : Num\nthis : \u2191m < \u2191n\n\u22a2 \u00acm = n\n[PROOFSTEP]\nsimp [show m \u2260 n from fun e => by rw [e] at this ; exact lt_irrefl _ this]\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\nthis : \u2191m < \u2191n\ne : m = n\n\u22a2 False\n[PROOFSTEP]\nrw [e] at this \n[GOAL]\n\u03b1 : Type u_1\nm n : Num\nthis : \u2191n < \u2191n\ne : m = n\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ this\n[GOAL]\ncase gt\n\u03b1 : Type u_1\nm n : Num\nthis : \u2191n < \u2191m\n\u22a2 \u00acm = n\n[PROOFSTEP]\nsimp [show m \u2260 n from fun e => by rw [e] at this ; exact lt_irrefl _ this]\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\nthis : \u2191n < \u2191m\ne : m = n\n\u22a2 False\n[PROOFSTEP]\nrw [e] at this \n[GOAL]\n\u03b1 : Type u_1\nm n : Num\nthis : \u2191n < \u2191n\ne : m = n\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\nm n : Num\n\u22a2 \u2191m < \u2191n \u2194 m < n\n[PROOFSTEP]\nrw [\u2190 cast_to_nat m, \u2190 cast_to_nat n, Nat.cast_lt (\u03b1 := \u03b1), lt_to_nat]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\nm n : Num\n\u22a2 \u2191m \u2264 \u2191n \u2194 m \u2264 n\n[PROOFSTEP]\nrw [\u2190 not_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\nm n : Num\n\u22a2 \u00ac\u2191n < \u2191m \u2194 m \u2264 n\n[PROOFSTEP]\nexact not_congr cast_lt\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemiring \u03b1\nm n : Num\n\u22a2 \u2191m = \u2191n \u2194 m = n\n[PROOFSTEP]\nrw [\u2190 cast_to_nat m, \u2190 cast_to_nat n, Nat.cast_inj, to_nat_inj]\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\n\u22a2 cmp n m = Ordering.lt \u2194 cmp m n = Ordering.gt\n[PROOFSTEP]\nrw [\u2190 cmp_swap]\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\n\u22a2 Ordering.swap (cmp m n) = Ordering.lt \u2194 cmp m n = Ordering.gt\n[PROOFSTEP]\ncases cmp m n\n[GOAL]\ncase lt\n\u03b1 : Type u_1\nm n : Num\n\u22a2 Ordering.swap Ordering.lt = Ordering.lt \u2194 Ordering.lt = Ordering.gt\n[PROOFSTEP]\nexact by decide\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\n\u22a2 Ordering.swap Ordering.lt = Ordering.lt \u2194 Ordering.lt = Ordering.gt\n[PROOFSTEP]\ndecide\n[GOAL]\ncase eq\n\u03b1 : Type u_1\nm n : Num\n\u22a2 Ordering.swap Ordering.eq = Ordering.lt \u2194 Ordering.eq = Ordering.gt\n[PROOFSTEP]\nexact by decide\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\n\u22a2 Ordering.swap Ordering.eq = Ordering.lt \u2194 Ordering.eq = Ordering.gt\n[PROOFSTEP]\ndecide\n[GOAL]\ncase gt\n\u03b1 : Type u_1\nm n : Num\n\u22a2 Ordering.swap Ordering.gt = Ordering.lt \u2194 Ordering.gt = Ordering.gt\n[PROOFSTEP]\nexact by decide\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\n\u22a2 Ordering.swap Ordering.gt = Ordering.lt \u2194 Ordering.gt = Ordering.gt\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\n\u22a2 \u2200 (m n : Num), \u2191(f m n) = Nat.bitwise' g \u2191m \u2191n\n[PROOFSTEP]\nintros m n\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm n : Num\n\u22a2 \u2191(f m n) = Nat.bitwise' g \u2191m \u2191n\n[PROOFSTEP]\ncases' m with m\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : Num\n\u22a2 \u2191(f zero n) = Nat.bitwise' g \u2191zero \u2191n\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : Num\nm : PosNum\n\u22a2 \u2191(f (pos m) n) = Nat.bitwise' g \u2191(pos m) \u2191n\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero.zero\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\n\u22a2 \u2191(f zero zero) = Nat.bitwise' g \u2191zero \u2191zero\n[PROOFSTEP]\ntry simp only [show zero = 0 from rfl, show ((0 : Num) : \u2115) = 0 from rfl]\n[GOAL]\ncase zero.zero\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\n\u22a2 \u2191(f zero zero) = Nat.bitwise' g \u2191zero \u2191zero\n[PROOFSTEP]\nsimp only [show zero = 0 from rfl, show ((0 : Num) : \u2115) = 0 from rfl]\n[GOAL]\ncase zero.pos\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\n\u22a2 \u2191(f zero (pos n)) = Nat.bitwise' g \u2191zero \u2191(pos n)\n[PROOFSTEP]\ntry simp only [show zero = 0 from rfl, show ((0 : Num) : \u2115) = 0 from rfl]\n[GOAL]\ncase zero.pos\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\n\u22a2 \u2191(f zero (pos n)) = Nat.bitwise' g \u2191zero \u2191(pos n)\n[PROOFSTEP]\nsimp only [show zero = 0 from rfl, show ((0 : Num) : \u2115) = 0 from rfl]\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm : PosNum\n\u22a2 \u2191(f (pos m) zero) = Nat.bitwise' g \u2191(pos m) \u2191zero\n[PROOFSTEP]\ntry simp only [show zero = 0 from rfl, show ((0 : Num) : \u2115) = 0 from rfl]\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm : PosNum\n\u22a2 \u2191(f (pos m) zero) = Nat.bitwise' g \u2191(pos m) \u2191zero\n[PROOFSTEP]\nsimp only [show zero = 0 from rfl, show ((0 : Num) : \u2115) = 0 from rfl]\n[GOAL]\ncase pos.pos\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm n : PosNum\n\u22a2 \u2191(f (pos m) (pos n)) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n[PROOFSTEP]\ntry simp only [show zero = 0 from rfl, show ((0 : Num) : \u2115) = 0 from rfl]\n[GOAL]\ncase pos.pos\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm n : PosNum\n\u22a2 \u2191(f (pos m) (pos n)) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n[PROOFSTEP]\nsimp only [show zero = 0 from rfl, show ((0 : Num) : \u2115) = 0 from rfl]\n[GOAL]\ncase zero.zero\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\n\u22a2 \u2191(f 0 0) = Nat.bitwise' g 0 0\n[PROOFSTEP]\nrw [f00, Nat.bitwise'_zero]\n[GOAL]\ncase zero.zero\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\n\u22a2 \u21910 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase zero.pos\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\n\u22a2 \u2191(f 0 (pos n)) = Nat.bitwise' g 0 \u2191(pos n)\n[PROOFSTEP]\nunfold Nat.bitwise'\n[GOAL]\ncase zero.pos\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\n\u22a2 \u2191(f 0 (pos n)) =\n Nat.binaryRec (fun n => bif g false true then n else 0)\n (fun a m Ia => Nat.binaryRec (bif g true false then Nat.bit a m else 0) fun b n x => Nat.bit (g a b) (Ia n)) 0\n \u2191(pos n)\n[PROOFSTEP]\nrw [f0n, Nat.binaryRec_zero]\n[GOAL]\ncase zero.pos\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\n\u22a2 \u2191(bif g false true then pos n else 0) = bif g false true then \u2191(pos n) else 0\n[PROOFSTEP]\ncases g false true\n[GOAL]\ncase zero.pos.false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\n\u22a2 \u2191(bif false then pos n else 0) = bif false then \u2191(pos n) else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase zero.pos.true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\n\u22a2 \u2191(bif true then pos n else 0) = bif true then \u2191(pos n) else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm : PosNum\n\u22a2 \u2191(f (pos m) 0) = Nat.bitwise' g (\u2191(pos m)) 0\n[PROOFSTEP]\nunfold Nat.bitwise'\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm : PosNum\n\u22a2 \u2191(f (pos m) 0) =\n Nat.binaryRec (fun n => bif g false true then n else 0)\n (fun a m Ia => Nat.binaryRec (bif g true false then Nat.bit a m else 0) fun b n x => Nat.bit (g a b) (Ia n))\n (\u2191(pos m)) 0\n[PROOFSTEP]\ngeneralize h : (pos m : \u2115) = m'\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm : PosNum\nm' : \u2115\nh : \u2191(pos m) = m'\n\u22a2 \u2191(f (pos m) 0) =\n Nat.binaryRec (fun n => bif g false true then n else 0)\n (fun a m Ia => Nat.binaryRec (bif g true false then Nat.bit a m else 0) fun b n x => Nat.bit (g a b) (Ia n)) m' 0\n[PROOFSTEP]\nrevert h\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm : PosNum\nm' : \u2115\n\u22a2 \u2191(pos m) = m' \u2192\n \u2191(f (pos m) 0) =\n Nat.binaryRec (fun n => bif g false true then n else 0)\n (fun a m Ia => Nat.binaryRec (bif g true false then Nat.bit a m else 0) fun b n x => Nat.bit (g a b) (Ia n)) m'\n 0\n[PROOFSTEP]\napply Nat.bitCasesOn m' _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm : PosNum\nm' : \u2115\n\u22a2 \u2200 (b : Bool) (n : \u2115),\n \u2191(pos m) = Nat.bit b n \u2192\n \u2191(f (pos m) 0) =\n Nat.binaryRec (fun n => bif g false true then n else 0)\n (fun a m Ia => Nat.binaryRec (bif g true false then Nat.bit a m else 0) fun b n x => Nat.bit (g a b) (Ia n))\n (Nat.bit b n) 0\n[PROOFSTEP]\nintro b m' h\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm : PosNum\nm'\u271d : \u2115\nb : Bool\nm' : \u2115\nh : \u2191(pos m) = Nat.bit b m'\n\u22a2 \u2191(f (pos m) 0) =\n Nat.binaryRec (fun n => bif g false true then n else 0)\n (fun a m Ia => Nat.binaryRec (bif g true false then Nat.bit a m else 0) fun b n x => Nat.bit (g a b) (Ia n))\n (Nat.bit b m') 0\n[PROOFSTEP]\nrw [fn0, Nat.binaryRec_eq, Nat.binaryRec_zero, \u2190 h]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm : PosNum\nm'\u271d : \u2115\nb : Bool\nm' : \u2115\nh : \u2191(pos m) = Nat.bit b m'\n\u22a2 \u2191(bif g true false then pos m else 0) = bif g true false then \u2191(pos m) else 0\n[PROOFSTEP]\ncases g true false\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm : PosNum\nm'\u271d : \u2115\nb : Bool\nm' : \u2115\nh : \u2191(pos m) = Nat.bit b m'\n\u22a2 \u2191(bif false then pos m else 0) = bif false then \u2191(pos m) else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm : PosNum\nm'\u271d : \u2115\nb : Bool\nm' : \u2115\nh : \u2191(pos m) = Nat.bit b m'\n\u22a2 \u2191(bif true then pos m else 0) = bif true then \u2191(pos m) else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm : PosNum\nm'\u271d : \u2115\nb : Bool\nm' : \u2115\nh : \u2191(pos m) = Nat.bit b m'\n\u22a2 (Nat.binaryRec (bif g true false then Nat.bit false 0 else 0) fun b n x =>\n Nat.bit (g false b) (bif g false true then n else 0)) =\n fun n => bif g false true then n else 0\n[PROOFSTEP]\napply Nat.bitwise'_bit_aux gff\n[GOAL]\ncase pos.pos\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm n : PosNum\n\u22a2 \u2191(f (pos m) (pos n)) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n[PROOFSTEP]\nrw [fnn]\n[GOAL]\ncase pos.pos\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm n : PosNum\n\u22a2 \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n[PROOFSTEP]\nhave : \u2200 (b) (n : PosNum), (cond b (\u2191n) 0 : \u2115) = \u2191(cond b (pos n) 0 : Num) := by intros b _; cases b <;> rfl\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm n : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm n : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 (bif b then \u2191n\u271d else 0) = \u2191(bif b then pos n\u271d else 0)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm n n\u271d : PosNum\n\u22a2 (bif false then \u2191n\u271d else 0) = \u2191(bif false then pos n\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm n n\u271d : PosNum\n\u22a2 (bif true then \u2191n\u271d else 0) = \u2191(bif true then pos n\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nm n : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n[PROOFSTEP]\ninduction' m with m IH m IH generalizing n\n[GOAL]\ncase pos.pos.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p one n) = Nat.bitwise' g \u2191(pos one) \u2191(pos n)\n[PROOFSTEP]\ncases' n with n n\n[GOAL]\ncase pos.pos.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit1 m) n) = Nat.bitwise' g \u2191(pos (PosNum.bit1 m)) \u2191(pos n)\n[PROOFSTEP]\ncases' n with n n\n[GOAL]\ncase pos.pos.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit0 m) n) = Nat.bitwise' g \u2191(pos (PosNum.bit0 m)) \u2191(pos n)\n[PROOFSTEP]\ncases' n with n n\n[GOAL]\ncase pos.pos.one.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 \u2191(p one one) = Nat.bitwise' g \u2191(pos one) \u2191(pos one)\ncase pos.pos.one.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p one (PosNum.bit1 n)) = Nat.bitwise' g \u2191(pos one) \u2191(pos (PosNum.bit1 n))\ncase pos.pos.one.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p one (PosNum.bit0 n)) = Nat.bitwise' g \u2191(pos one) \u2191(pos (PosNum.bit0 n))\ncase pos.pos.bit1.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit1 m) one) = Nat.bitwise' g \u2191(pos (PosNum.bit1 m)) \u2191(pos one)\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit1 m) (PosNum.bit1 n)) = Nat.bitwise' g \u2191(pos (PosNum.bit1 m)) \u2191(pos (PosNum.bit1 n))\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit1 m) (PosNum.bit0 n)) = Nat.bitwise' g \u2191(pos (PosNum.bit1 m)) \u2191(pos (PosNum.bit0 n))\ncase pos.pos.bit0.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit0 m) one) = Nat.bitwise' g \u2191(pos (PosNum.bit0 m)) \u2191(pos one)\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit0 m) (PosNum.bit1 n)) = Nat.bitwise' g \u2191(pos (PosNum.bit0 m)) \u2191(pos (PosNum.bit1 n))\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit0 m) (PosNum.bit0 n)) = Nat.bitwise' g \u2191(pos (PosNum.bit0 m)) \u2191(pos (PosNum.bit0 n))\n[PROOFSTEP]\nany_goals\n simp only [show one = 1 from rfl, show pos 1 = 1 from rfl, show PosNum.bit0 = PosNum.bit false from rfl,\n show PosNum.bit1 = PosNum.bit true from rfl, show ((1 : Num) : \u2115) = Nat.bit true 0 from rfl]\n[GOAL]\ncase pos.pos.one.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 \u2191(p one one) = Nat.bitwise' g \u2191(pos one) \u2191(pos one)\n[PROOFSTEP]\nsimp only [show one = 1 from rfl, show pos 1 = 1 from rfl, show PosNum.bit0 = PosNum.bit false from rfl,\n show PosNum.bit1 = PosNum.bit true from rfl, show ((1 : Num) : \u2115) = Nat.bit true 0 from rfl]\n[GOAL]\ncase pos.pos.one.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p one (PosNum.bit1 n)) = Nat.bitwise' g \u2191(pos one) \u2191(pos (PosNum.bit1 n))\n[PROOFSTEP]\nsimp only [show one = 1 from rfl, show pos 1 = 1 from rfl, show PosNum.bit0 = PosNum.bit false from rfl,\n show PosNum.bit1 = PosNum.bit true from rfl, show ((1 : Num) : \u2115) = Nat.bit true 0 from rfl]\n[GOAL]\ncase pos.pos.one.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p one (PosNum.bit0 n)) = Nat.bitwise' g \u2191(pos one) \u2191(pos (PosNum.bit0 n))\n[PROOFSTEP]\nsimp only [show one = 1 from rfl, show pos 1 = 1 from rfl, show PosNum.bit0 = PosNum.bit false from rfl,\n show PosNum.bit1 = PosNum.bit true from rfl, show ((1 : Num) : \u2115) = Nat.bit true 0 from rfl]\n[GOAL]\ncase pos.pos.bit1.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit1 m) one) = Nat.bitwise' g \u2191(pos (PosNum.bit1 m)) \u2191(pos one)\n[PROOFSTEP]\nsimp only [show one = 1 from rfl, show pos 1 = 1 from rfl, show PosNum.bit0 = PosNum.bit false from rfl,\n show PosNum.bit1 = PosNum.bit true from rfl, show ((1 : Num) : \u2115) = Nat.bit true 0 from rfl]\n[GOAL]\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit1 m) (PosNum.bit1 n)) = Nat.bitwise' g \u2191(pos (PosNum.bit1 m)) \u2191(pos (PosNum.bit1 n))\n[PROOFSTEP]\nsimp only [show one = 1 from rfl, show pos 1 = 1 from rfl, show PosNum.bit0 = PosNum.bit false from rfl,\n show PosNum.bit1 = PosNum.bit true from rfl, show ((1 : Num) : \u2115) = Nat.bit true 0 from rfl]\n[GOAL]\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit1 m) (PosNum.bit0 n)) = Nat.bitwise' g \u2191(pos (PosNum.bit1 m)) \u2191(pos (PosNum.bit0 n))\n[PROOFSTEP]\nsimp only [show one = 1 from rfl, show pos 1 = 1 from rfl, show PosNum.bit0 = PosNum.bit false from rfl,\n show PosNum.bit1 = PosNum.bit true from rfl, show ((1 : Num) : \u2115) = Nat.bit true 0 from rfl]\n[GOAL]\ncase pos.pos.bit0.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit0 m) one) = Nat.bitwise' g \u2191(pos (PosNum.bit0 m)) \u2191(pos one)\n[PROOFSTEP]\nsimp only [show one = 1 from rfl, show pos 1 = 1 from rfl, show PosNum.bit0 = PosNum.bit false from rfl,\n show PosNum.bit1 = PosNum.bit true from rfl, show ((1 : Num) : \u2115) = Nat.bit true 0 from rfl]\n[GOAL]\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit0 m) (PosNum.bit1 n)) = Nat.bitwise' g \u2191(pos (PosNum.bit0 m)) \u2191(pos (PosNum.bit1 n))\n[PROOFSTEP]\nsimp only [show one = 1 from rfl, show pos 1 = 1 from rfl, show PosNum.bit0 = PosNum.bit false from rfl,\n show PosNum.bit1 = PosNum.bit true from rfl, show ((1 : Num) : \u2115) = Nat.bit true 0 from rfl]\n[GOAL]\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit0 m) (PosNum.bit0 n)) = Nat.bitwise' g \u2191(pos (PosNum.bit0 m)) \u2191(pos (PosNum.bit0 n))\n[PROOFSTEP]\nsimp only [show one = 1 from rfl, show pos 1 = 1 from rfl, show PosNum.bit0 = PosNum.bit false from rfl,\n show PosNum.bit1 = PosNum.bit true from rfl, show ((1 : Num) : \u2115) = Nat.bit true 0 from rfl]\n[GOAL]\ncase pos.pos.one.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 \u2191(p 1 1) = Nat.bitwise' g (Nat.bit true 0) (Nat.bit true 0)\ncase pos.pos.one.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit true n)) = Nat.bitwise' g (Nat.bit true 0) \u2191(pos (PosNum.bit true n))\ncase pos.pos.one.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit false n)) = Nat.bitwise' g (Nat.bit true 0) \u2191(pos (PosNum.bit false n))\ncase pos.pos.bit1.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit true m) 1) = Nat.bitwise' g (\u2191(pos (PosNum.bit true m))) (Nat.bit true 0)\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bitwise' g \u2191(pos (PosNum.bit true m)) \u2191(pos (PosNum.bit true n))\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bitwise' g \u2191(pos (PosNum.bit true m)) \u2191(pos (PosNum.bit false n))\ncase pos.pos.bit0.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit false m) 1) = Nat.bitwise' g (\u2191(pos (PosNum.bit false m))) (Nat.bit true 0)\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bitwise' g \u2191(pos (PosNum.bit false m)) \u2191(pos (PosNum.bit true n))\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) =\n Nat.bitwise' g \u2191(pos (PosNum.bit false m)) \u2191(pos (PosNum.bit false n))\n[PROOFSTEP]\nall_goals\n repeat' rw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n rw [Nat.bitwise'_bit]\n[GOAL]\ncase pos.pos.one.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 \u2191(p 1 1) = Nat.bitwise' g (Nat.bit true 0) (Nat.bit true 0)\n[PROOFSTEP]\nrepeat' rw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\ncase pos.pos.one.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 \u2191(p 1 1) = Nat.bitwise' g (Nat.bit true 0) (Nat.bit true 0)\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.one.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 \u2191(p 1 1) = Nat.bitwise' g (Nat.bit true 0) (Nat.bit true 0)\n[PROOFSTEP]\nrw [Nat.bitwise'_bit]\n[GOAL]\ncase pos.pos.one.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit true n)) = Nat.bitwise' g (Nat.bit true 0) \u2191(pos (PosNum.bit true n))\n[PROOFSTEP]\nrepeat' rw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\ncase pos.pos.one.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit true n)) = Nat.bitwise' g (Nat.bit true 0) \u2191(pos (PosNum.bit true n))\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.one.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit true n)) = Nat.bitwise' g (Nat.bit true 0) (Nat.bit true \u2191n)\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.one.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit true n)) = Nat.bitwise' g (Nat.bit true 0) (Nat.bit true \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_bit]\n[GOAL]\ncase pos.pos.one.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit false n)) = Nat.bitwise' g (Nat.bit true 0) \u2191(pos (PosNum.bit false n))\n[PROOFSTEP]\nrepeat' rw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\ncase pos.pos.one.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit false n)) = Nat.bitwise' g (Nat.bit true 0) \u2191(pos (PosNum.bit false n))\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.one.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit false n)) = Nat.bitwise' g (Nat.bit true 0) (Nat.bit false \u2191n)\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.one.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit false n)) = Nat.bitwise' g (Nat.bit true 0) (Nat.bit false \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_bit]\n[GOAL]\ncase pos.pos.bit1.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit true m) 1) = Nat.bitwise' g (\u2191(pos (PosNum.bit true m))) (Nat.bit true 0)\n[PROOFSTEP]\nrepeat' rw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\ncase pos.pos.bit1.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit true m) 1) = Nat.bitwise' g (\u2191(pos (PosNum.bit true m))) (Nat.bit true 0)\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit1.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit true m) 1) = Nat.bitwise' g (Nat.bit true \u2191m) (Nat.bit true 0)\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit1.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit true m) 1) = Nat.bitwise' g (Nat.bit true \u2191m) (Nat.bit true 0)\n[PROOFSTEP]\nrw [Nat.bitwise'_bit]\n[GOAL]\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bitwise' g \u2191(pos (PosNum.bit true m)) \u2191(pos (PosNum.bit true n))\n[PROOFSTEP]\nrepeat' rw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bitwise' g \u2191(pos (PosNum.bit true m)) \u2191(pos (PosNum.bit true n))\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bitwise' g (Nat.bit true \u2191m) \u2191(pos (PosNum.bit true n))\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bitwise' g (Nat.bit true \u2191m) (Nat.bit true \u2191n)\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bitwise' g (Nat.bit true \u2191m) (Nat.bit true \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_bit]\n[GOAL]\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bitwise' g \u2191(pos (PosNum.bit true m)) \u2191(pos (PosNum.bit false n))\n[PROOFSTEP]\nrepeat' rw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bitwise' g \u2191(pos (PosNum.bit true m)) \u2191(pos (PosNum.bit false n))\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bitwise' g (Nat.bit true \u2191m) \u2191(pos (PosNum.bit false n))\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bitwise' g (Nat.bit true \u2191m) (Nat.bit false \u2191n)\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bitwise' g (Nat.bit true \u2191m) (Nat.bit false \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_bit]\n[GOAL]\ncase pos.pos.bit0.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit false m) 1) = Nat.bitwise' g (\u2191(pos (PosNum.bit false m))) (Nat.bit true 0)\n[PROOFSTEP]\nrepeat' rw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\ncase pos.pos.bit0.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit false m) 1) = Nat.bitwise' g (\u2191(pos (PosNum.bit false m))) (Nat.bit true 0)\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit0.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit false m) 1) = Nat.bitwise' g (Nat.bit false \u2191m) (Nat.bit true 0)\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit0.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit false m) 1) = Nat.bitwise' g (Nat.bit false \u2191m) (Nat.bit true 0)\n[PROOFSTEP]\nrw [Nat.bitwise'_bit]\n[GOAL]\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bitwise' g \u2191(pos (PosNum.bit false m)) \u2191(pos (PosNum.bit true n))\n[PROOFSTEP]\nrepeat' rw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bitwise' g \u2191(pos (PosNum.bit false m)) \u2191(pos (PosNum.bit true n))\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bitwise' g (Nat.bit false \u2191m) \u2191(pos (PosNum.bit true n))\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bitwise' g (Nat.bit false \u2191m) (Nat.bit true \u2191n)\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bitwise' g (Nat.bit false \u2191m) (Nat.bit true \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_bit]\n[GOAL]\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) =\n Nat.bitwise' g \u2191(pos (PosNum.bit false m)) \u2191(pos (PosNum.bit false n))\n[PROOFSTEP]\nrepeat' rw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) =\n Nat.bitwise' g \u2191(pos (PosNum.bit false m)) \u2191(pos (PosNum.bit false n))\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) = Nat.bitwise' g (Nat.bit false \u2191m) \u2191(pos (PosNum.bit false n))\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) = Nat.bitwise' g (Nat.bit false \u2191m) (Nat.bit false \u2191n)\n[PROOFSTEP]\nrw [show \u2200 b n, (pos (PosNum.bit b n) : \u2115) = Nat.bit b \u2191n by intros b _; cases b <;> rfl]\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2200 (b : Bool) (n : PosNum), \u2191(pos (PosNum.bit b n)) = Nat.bit b \u2191n\n[PROOFSTEP]\nintros b _\n[GOAL]\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\nb : Bool\nn\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit b n\u271d)) = Nat.bit b \u2191n\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit false n\u271d)) = Nat.bit false \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d\u00b9 : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn n\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.bit true n\u271d)) = Nat.bit true \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) = Nat.bitwise' g (Nat.bit false \u2191m) (Nat.bit false \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_bit]\n[GOAL]\ncase pos.pos.one.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 \u2191(p 1 1) = Nat.bit (g true true) (Nat.bitwise' g 0 0)\ncase pos.pos.one.one.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 g false false = false\ncase pos.pos.one.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g 0 \u2191n)\ncase pos.pos.one.bit1.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 g false false = false\ncase pos.pos.one.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g 0 \u2191n)\ncase pos.pos.one.bit0.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 g false false = false\ncase pos.pos.bit1.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit true m) 1) = Nat.bit (g true true) (Nat.bitwise' g (\u2191m) 0)\ncase pos.pos.bit1.one.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 g false false = false\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit1.bit1.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 g false false = false\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit1.bit0.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 g false false = false\ncase pos.pos.bit0.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit false m) 1) = Nat.bit (g false true) (Nat.bitwise' g (\u2191m) 0)\ncase pos.pos.bit0.one.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 g false false = false\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bit (g false true) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit0.bit1.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 g false false = false\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) = Nat.bit (g false false) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit0.bit0.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 g false false = false\n[PROOFSTEP]\nany_goals assumption\n[GOAL]\ncase pos.pos.one.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 \u2191(p 1 1) = Nat.bit (g true true) (Nat.bitwise' g 0 0)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.one.one.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 g false false = false\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.one.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g 0 \u2191n)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.one.bit1.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 g false false = false\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.one.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g 0 \u2191n)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.one.bit0.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 g false false = false\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.bit1.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit true m) 1) = Nat.bit (g true true) (Nat.bitwise' g (\u2191m) 0)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.bit1.one.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 g false false = false\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.bit1.bit1.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 g false false = false\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.bit1.bit0.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 g false false = false\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.bit0.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit false m) 1) = Nat.bit (g false true) (Nat.bitwise' g (\u2191m) 0)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.bit0.one.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 g false false = false\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bit (g false true) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.bit0.bit1.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 g false false = false\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) = Nat.bit (g false false) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.bit0.bit0.h\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 g false false = false\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.pos.one.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 \u2191(p 1 1) = Nat.bit (g true true) (Nat.bitwise' g 0 0)\ncase pos.pos.one.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g 0 \u2191n)\ncase pos.pos.one.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g 0 \u2191n)\ncase pos.pos.bit1.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit true m) 1) = Nat.bit (g true true) (Nat.bitwise' g (\u2191m) 0)\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit0.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit false m) 1) = Nat.bit (g false true) (Nat.bitwise' g (\u2191m) 0)\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bit (g false true) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) = Nat.bit (g false false) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nany_goals rw [Nat.bitwise'_zero, p11]; cases g true true <;> rfl\n[GOAL]\ncase pos.pos.one.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 \u2191(p 1 1) = Nat.bit (g true true) (Nat.bitwise' g 0 0)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero, p11]\n[GOAL]\ncase pos.pos.one.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 \u2191(bif g true true then 1 else 0) = Nat.bit (g true true) 0\n[PROOFSTEP]\ncases g true true\n[GOAL]\ncase pos.pos.one.one.false\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 \u2191(bif false then 1 else 0) = Nat.bit false 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.one.one.true\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\n\u22a2 \u2191(bif true then 1 else 0) = Nat.bit true 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.pos.one.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g 0 \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero, p11]\n[GOAL]\ncase pos.pos.one.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g 0 \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero, p11]\n[GOAL]\ncase pos.pos.bit1.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit true m) 1) = Nat.bit (g true true) (Nat.bitwise' g (\u2191m) 0)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero, p11]\n[GOAL]\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero, p11]\n[GOAL]\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero, p11]\n[GOAL]\ncase pos.pos.bit0.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit false m) 1) = Nat.bit (g false true) (Nat.bitwise' g (\u2191m) 0)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero, p11]\n[GOAL]\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bit (g false true) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero, p11]\n[GOAL]\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) = Nat.bit (g false false) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero, p11]\n[GOAL]\ncase pos.pos.one.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g 0 \u2191n)\ncase pos.pos.one.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g 0 \u2191n)\ncase pos.pos.bit1.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit true m) 1) = Nat.bit (g true true) (Nat.bitwise' g (\u2191m) 0)\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit0.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit false m) 1) = Nat.bit (g false true) (Nat.bitwise' g (\u2191m) 0)\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bit (g false true) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) = Nat.bit (g false false) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nany_goals rw [Nat.bitwise'_zero_left, this, \u2190 bit_to_nat, p1b]\n[GOAL]\ncase pos.pos.one.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g 0 \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero_left, this, \u2190 bit_to_nat, p1b]\n[GOAL]\ncase pos.pos.one.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nn : PosNum\n\u22a2 \u2191(p 1 (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g 0 \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero_left, this, \u2190 bit_to_nat, p1b]\n[GOAL]\ncase pos.pos.bit1.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit true m) 1) = Nat.bit (g true true) (Nat.bitwise' g (\u2191m) 0)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero_left, this, \u2190 bit_to_nat, p1b]\n[GOAL]\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero_left, this, \u2190 bit_to_nat, p1b]\n[GOAL]\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero_left, this, \u2190 bit_to_nat, p1b]\n[GOAL]\ncase pos.pos.bit0.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit false m) 1) = Nat.bit (g false true) (Nat.bitwise' g (\u2191m) 0)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero_left, this, \u2190 bit_to_nat, p1b]\n[GOAL]\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bit (g false true) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero_left, this, \u2190 bit_to_nat, p1b]\n[GOAL]\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) = Nat.bit (g false false) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero_left, this, \u2190 bit_to_nat, p1b]\n[GOAL]\ncase pos.pos.bit1.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit true m) 1) = Nat.bit (g true true) (Nat.bitwise' g (\u2191m) 0)\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit0.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit false m) 1) = Nat.bit (g false true) (Nat.bitwise' g (\u2191m) 0)\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bit (g false true) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) = Nat.bit (g false false) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nany_goals rw [Nat.bitwise'_zero_right _ gff, this, \u2190 bit_to_nat, pb1]\n[GOAL]\ncase pos.pos.bit1.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit true m) 1) = Nat.bit (g true true) (Nat.bitwise' g (\u2191m) 0)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero_right _ gff, this, \u2190 bit_to_nat, pb1]\n[GOAL]\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero_right _ gff, this, \u2190 bit_to_nat, pb1]\n[GOAL]\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero_right _ gff, this, \u2190 bit_to_nat, pb1]\n[GOAL]\ncase pos.pos.bit0.one\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\n\u22a2 \u2191(p (PosNum.bit false m) 1) = Nat.bit (g false true) (Nat.bitwise' g (\u2191m) 0)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero_right _ gff, this, \u2190 bit_to_nat, pb1]\n[GOAL]\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bit (g false true) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero_right _ gff, this, \u2190 bit_to_nat, pb1]\n[GOAL]\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) = Nat.bit (g false false) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [Nat.bitwise'_zero_right _ gff, this, \u2190 bit_to_nat, pb1]\n[GOAL]\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bit (g false true) (Nat.bitwise' g \u2191m \u2191n)\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) = Nat.bit (g false false) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nall_goals\n rw [\u2190 show \u2200 n : PosNum, \u2191(p m n) = Nat.bitwise' g \u2191m \u2191n from IH]\n rw [\u2190 bit_to_nat, pbb]\n[GOAL]\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bit (g true true) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [\u2190 show \u2200 n : PosNum, \u2191(p m n) = Nat.bitwise' g \u2191m \u2191n from IH]\n[GOAL]\ncase pos.pos.bit1.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit true n)) = Nat.bit (g true true) \u2191(p m n)\n[PROOFSTEP]\nrw [\u2190 bit_to_nat, pbb]\n[GOAL]\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bit (g true false) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [\u2190 show \u2200 n : PosNum, \u2191(p m n) = Nat.bitwise' g \u2191m \u2191n from IH]\n[GOAL]\ncase pos.pos.bit1.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit true m) (PosNum.bit false n)) = Nat.bit (g true false) \u2191(p m n)\n[PROOFSTEP]\nrw [\u2190 bit_to_nat, pbb]\n[GOAL]\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bit (g false true) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [\u2190 show \u2200 n : PosNum, \u2191(p m n) = Nat.bitwise' g \u2191m \u2191n from IH]\n[GOAL]\ncase pos.pos.bit0.bit1\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit true n)) = Nat.bit (g false true) \u2191(p m n)\n[PROOFSTEP]\nrw [\u2190 bit_to_nat, pbb]\n[GOAL]\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) = Nat.bit (g false false) (Nat.bitwise' g \u2191m \u2191n)\n[PROOFSTEP]\nrw [\u2190 show \u2200 n : PosNum, \u2191(p m n) = Nat.bitwise' g \u2191m \u2191n from IH]\n[GOAL]\ncase pos.pos.bit0.bit0\n\u03b1 : Type u_1\nf : Num \u2192 Num \u2192 Num\ng : Bool \u2192 Bool \u2192 Bool\np : PosNum \u2192 PosNum \u2192 Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : \u2200 (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : \u2200 (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : \u2200 (m n : PosNum), f (pos m) (pos n) = p m n\np11 : p 1 1 = bif g true true then 1 else 0\np1b : \u2200 (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)\npb1 : \u2200 (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)\npbb : \u2200 (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)\nn\u271d : PosNum\nthis : \u2200 (b : Bool) (n : PosNum), (bif b then \u2191n else 0) = \u2191(bif b then pos n else 0)\nm : PosNum\nIH : \u2200 (n : PosNum), \u2191(p m n) = Nat.bitwise' g \u2191(pos m) \u2191(pos n)\nn : PosNum\n\u22a2 \u2191(p (PosNum.bit false m) (PosNum.bit false n)) = Nat.bit (g false false) \u2191(p m n)\n[PROOFSTEP]\nrw [\u2190 bit_to_nat, pbb]\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (m n : Num), \u2191(lor m n) = Nat.lor' \u2191m \u2191n\n[PROOFSTEP]\napply bitwise'_to_nat fun x y => pos (PosNum.lor x y)\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 (false || false) = false\n[PROOFSTEP]\nintros\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 lor 0 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\n\u22a2 \u2200 (n : PosNum), lor 0 (pos n) = bif false || true then pos n else 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\n\u22a2 \u2200 (n : PosNum), lor (pos n) 0 = bif true || false then pos n else 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\n\u22a2 \u2200 (m n : PosNum), lor (pos m) (pos n) = pos (PosNum.lor m n)\n[PROOFSTEP]\nintros\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 pos (PosNum.lor 1 1) = bif true || true then 1 else 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase p1b\n\u03b1 : Type u_1\n\u22a2 \u2200 (b : Bool) (n : PosNum), pos (PosNum.lor 1 (PosNum.bit b n)) = bit (true || b) (bif false || true then pos n else 0)\n[PROOFSTEP]\nintros\n[GOAL]\ncase pb1\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : Bool) (m : PosNum), pos (PosNum.lor (PosNum.bit a m) 1) = bit (a || true) (bif true || false then pos m else 0)\n[PROOFSTEP]\nintros\n[GOAL]\ncase pbb\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : Bool) (m n : PosNum),\n pos (PosNum.lor (PosNum.bit a m) (PosNum.bit b n)) = bit (a || b) (pos (PosNum.lor m n))\n[PROOFSTEP]\nintros\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 (false || false) = false\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 (false || false) = false\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 lor 0 0 = 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 lor 0 0 = 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 lor 0 (pos n\u271d) = bif false || true then pos n\u271d else 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 lor 0 (pos n\u271d) = bif false || true then pos n\u271d else 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 lor (pos n\u271d) 0 = bif true || false then pos n\u271d else 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 lor (pos n\u271d) 0 = bif true || false then pos n\u271d else 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 lor (pos m\u271d) (pos n\u271d) = pos (PosNum.lor m\u271d n\u271d)\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 lor (pos m\u271d) (pos n\u271d) = pos (PosNum.lor m\u271d n\u271d)\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 pos (PosNum.lor 1 1) = bif true || true then 1 else 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 pos (PosNum.lor 1 1) = bif true || true then 1 else 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase p1b\n\u03b1 : Type u_1\nb\u271d : Bool\nn\u271d : PosNum\n\u22a2 pos (PosNum.lor 1 (PosNum.bit b\u271d n\u271d)) = bit (true || b\u271d) (bif false || true then pos n\u271d else 0)\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase p1b\n\u03b1 : Type u_1\nb\u271d : Bool\nn\u271d : PosNum\n\u22a2 pos (PosNum.lor 1 (PosNum.bit b\u271d n\u271d)) = bit (true || b\u271d) (bif false || true then pos n\u271d else 0)\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase pb1\n\u03b1 : Type u_1\na\u271d : Bool\nm\u271d : PosNum\n\u22a2 pos (PosNum.lor (PosNum.bit a\u271d m\u271d) 1) = bit (a\u271d || true) (bif true || false then pos m\u271d else 0)\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase pb1\n\u03b1 : Type u_1\na\u271d : Bool\nm\u271d : PosNum\n\u22a2 pos (PosNum.lor (PosNum.bit a\u271d m\u271d) 1) = bit (a\u271d || true) (bif true || false then pos m\u271d else 0)\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase pbb\n\u03b1 : Type u_1\na\u271d b\u271d : Bool\nm\u271d n\u271d : PosNum\n\u22a2 pos (PosNum.lor (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d)) = bit (a\u271d || b\u271d) (pos (PosNum.lor m\u271d n\u271d))\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase pbb\n\u03b1 : Type u_1\na\u271d b\u271d : Bool\nm\u271d n\u271d : PosNum\n\u22a2 pos (PosNum.lor (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d)) = bit (a\u271d || b\u271d) (pos (PosNum.lor m\u271d n\u271d))\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 (false || false) = false\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 lor 0 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 lor 0 (pos n\u271d) = bif false || true then pos n\u271d else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 lor (pos n\u271d) 0 = bif true || false then pos n\u271d else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 lor (pos m\u271d) (pos n\u271d) = pos (PosNum.lor m\u271d n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 pos (PosNum.lor 1 1) = bif true || true then 1 else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase p1b.false\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 pos (PosNum.lor 1 (PosNum.bit false n\u271d)) = bit (true || false) (bif false || true then pos n\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase p1b.true\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 pos (PosNum.lor 1 (PosNum.bit true n\u271d)) = bit (true || true) (bif false || true then pos n\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pb1.false\n\u03b1 : Type u_1\nm\u271d : PosNum\n\u22a2 pos (PosNum.lor (PosNum.bit false m\u271d) 1) = bit (false || true) (bif true || false then pos m\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pb1.true\n\u03b1 : Type u_1\nm\u271d : PosNum\n\u22a2 pos (PosNum.lor (PosNum.bit true m\u271d) 1) = bit (true || true) (bif true || false then pos m\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.false.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 pos (PosNum.lor (PosNum.bit false m\u271d) (PosNum.bit false n\u271d)) = bit (false || false) (pos (PosNum.lor m\u271d n\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.false.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 pos (PosNum.lor (PosNum.bit false m\u271d) (PosNum.bit true n\u271d)) = bit (false || true) (pos (PosNum.lor m\u271d n\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.true.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 pos (PosNum.lor (PosNum.bit true m\u271d) (PosNum.bit false n\u271d)) = bit (true || false) (pos (PosNum.lor m\u271d n\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.true.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 pos (PosNum.lor (PosNum.bit true m\u271d) (PosNum.bit true n\u271d)) = bit (true || true) (pos (PosNum.lor m\u271d n\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (m n : Num), \u2191(land m n) = Nat.land' \u2191m \u2191n\n[PROOFSTEP]\napply bitwise'_to_nat PosNum.land\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 (false && false) = false\n[PROOFSTEP]\nintros\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 land 0 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\n\u22a2 \u2200 (n : PosNum), land 0 (pos n) = bif false && true then pos n else 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\n\u22a2 \u2200 (n : PosNum), land (pos n) 0 = bif true && false then pos n else 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\n\u22a2 \u2200 (m n : PosNum), land (pos m) (pos n) = PosNum.land m n\n[PROOFSTEP]\nintros\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 PosNum.land 1 1 = bif true && true then 1 else 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase p1b\n\u03b1 : Type u_1\n\u22a2 \u2200 (b : Bool) (n : PosNum), PosNum.land 1 (PosNum.bit b n) = bit (true && b) (bif false && true then pos n else 0)\n[PROOFSTEP]\nintros\n[GOAL]\ncase pb1\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : Bool) (m : PosNum), PosNum.land (PosNum.bit a m) 1 = bit (a && true) (bif true && false then pos m else 0)\n[PROOFSTEP]\nintros\n[GOAL]\ncase pbb\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : Bool) (m n : PosNum), PosNum.land (PosNum.bit a m) (PosNum.bit b n) = bit (a && b) (PosNum.land m n)\n[PROOFSTEP]\nintros\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 (false && false) = false\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 (false && false) = false\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 land 0 0 = 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 land 0 0 = 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 land 0 (pos n\u271d) = bif false && true then pos n\u271d else 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 land 0 (pos n\u271d) = bif false && true then pos n\u271d else 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 land (pos n\u271d) 0 = bif true && false then pos n\u271d else 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 land (pos n\u271d) 0 = bif true && false then pos n\u271d else 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 land (pos m\u271d) (pos n\u271d) = PosNum.land m\u271d n\u271d\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 land (pos m\u271d) (pos n\u271d) = PosNum.land m\u271d n\u271d\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 PosNum.land 1 1 = bif true && true then 1 else 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 PosNum.land 1 1 = bif true && true then 1 else 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase p1b\n\u03b1 : Type u_1\nb\u271d : Bool\nn\u271d : PosNum\n\u22a2 PosNum.land 1 (PosNum.bit b\u271d n\u271d) = bit (true && b\u271d) (bif false && true then pos n\u271d else 0)\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase p1b\n\u03b1 : Type u_1\nb\u271d : Bool\nn\u271d : PosNum\n\u22a2 PosNum.land 1 (PosNum.bit b\u271d n\u271d) = bit (true && b\u271d) (bif false && true then pos n\u271d else 0)\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase pb1\n\u03b1 : Type u_1\na\u271d : Bool\nm\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit a\u271d m\u271d) 1 = bit (a\u271d && true) (bif true && false then pos m\u271d else 0)\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase pb1\n\u03b1 : Type u_1\na\u271d : Bool\nm\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit a\u271d m\u271d) 1 = bit (a\u271d && true) (bif true && false then pos m\u271d else 0)\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase pbb\n\u03b1 : Type u_1\na\u271d b\u271d : Bool\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (a\u271d && b\u271d) (PosNum.land m\u271d n\u271d)\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase pbb\n\u03b1 : Type u_1\na\u271d b\u271d : Bool\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (a\u271d && b\u271d) (PosNum.land m\u271d n\u271d)\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 (false && false) = false\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 land 0 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 land 0 (pos n\u271d) = bif false && true then pos n\u271d else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 land (pos n\u271d) 0 = bif true && false then pos n\u271d else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 land (pos m\u271d) (pos n\u271d) = PosNum.land m\u271d n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 PosNum.land 1 1 = bif true && true then 1 else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase p1b.false\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 PosNum.land 1 (PosNum.bit false n\u271d) = bit (true && false) (bif false && true then pos n\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase p1b.true\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 PosNum.land 1 (PosNum.bit true n\u271d) = bit (true && true) (bif false && true then pos n\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pb1.false\n\u03b1 : Type u_1\nm\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit false m\u271d) 1 = bit (false && true) (bif true && false then pos m\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pb1.true\n\u03b1 : Type u_1\nm\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit true m\u271d) 1 = bit (true && true) (bif true && false then pos m\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.false.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit false m\u271d) (PosNum.bit false n\u271d) = bit (false && false) (PosNum.land m\u271d n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.false.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit false m\u271d) (PosNum.bit true n\u271d) = bit (false && true) (PosNum.land m\u271d n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.true.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit true m\u271d) (PosNum.bit false n\u271d) = bit (true && false) (PosNum.land m\u271d n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.true.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.land (PosNum.bit true m\u271d) (PosNum.bit true n\u271d) = bit (true && true) (PosNum.land m\u271d n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (m n : Num), \u2191(ldiff m n) = Nat.ldiff' \u2191m \u2191n\n[PROOFSTEP]\napply bitwise'_to_nat PosNum.ldiff\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 (false && !false) = false\n[PROOFSTEP]\nintros\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 ldiff 0 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\n\u22a2 \u2200 (n : PosNum), ldiff 0 (pos n) = bif false && !true then pos n else 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\n\u22a2 \u2200 (n : PosNum), ldiff (pos n) 0 = bif true && !false then pos n else 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\n\u22a2 \u2200 (m n : PosNum), ldiff (pos m) (pos n) = PosNum.ldiff m n\n[PROOFSTEP]\nintros\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 PosNum.ldiff 1 1 = bif true && !true then 1 else 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase p1b\n\u03b1 : Type u_1\n\u22a2 \u2200 (b : Bool) (n : PosNum), PosNum.ldiff 1 (PosNum.bit b n) = bit (true && !b) (bif false && !true then pos n else 0)\n[PROOFSTEP]\nintros\n[GOAL]\ncase pb1\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : Bool) (m : PosNum), PosNum.ldiff (PosNum.bit a m) 1 = bit (a && !true) (bif true && !false then pos m else 0)\n[PROOFSTEP]\nintros\n[GOAL]\ncase pbb\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : Bool) (m n : PosNum), PosNum.ldiff (PosNum.bit a m) (PosNum.bit b n) = bit (a && !b) (PosNum.ldiff m n)\n[PROOFSTEP]\nintros\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 (false && !false) = false\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 (false && !false) = false\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 ldiff 0 0 = 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 ldiff 0 0 = 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 ldiff 0 (pos n\u271d) = bif false && !true then pos n\u271d else 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 ldiff 0 (pos n\u271d) = bif false && !true then pos n\u271d else 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 ldiff (pos n\u271d) 0 = bif true && !false then pos n\u271d else 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 ldiff (pos n\u271d) 0 = bif true && !false then pos n\u271d else 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 ldiff (pos m\u271d) (pos n\u271d) = PosNum.ldiff m\u271d n\u271d\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 ldiff (pos m\u271d) (pos n\u271d) = PosNum.ldiff m\u271d n\u271d\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 PosNum.ldiff 1 1 = bif true && !true then 1 else 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 PosNum.ldiff 1 1 = bif true && !true then 1 else 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase p1b\n\u03b1 : Type u_1\nb\u271d : Bool\nn\u271d : PosNum\n\u22a2 PosNum.ldiff 1 (PosNum.bit b\u271d n\u271d) = bit (true && !b\u271d) (bif false && !true then pos n\u271d else 0)\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase p1b\n\u03b1 : Type u_1\nb\u271d : Bool\nn\u271d : PosNum\n\u22a2 PosNum.ldiff 1 (PosNum.bit b\u271d n\u271d) = bit (true && !b\u271d) (bif false && !true then pos n\u271d else 0)\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase pb1\n\u03b1 : Type u_1\na\u271d : Bool\nm\u271d : PosNum\n\u22a2 PosNum.ldiff (PosNum.bit a\u271d m\u271d) 1 = bit (a\u271d && !true) (bif true && !false then pos m\u271d else 0)\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase pb1\n\u03b1 : Type u_1\na\u271d : Bool\nm\u271d : PosNum\n\u22a2 PosNum.ldiff (PosNum.bit a\u271d m\u271d) 1 = bit (a\u271d && !true) (bif true && !false then pos m\u271d else 0)\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase pbb\n\u03b1 : Type u_1\na\u271d b\u271d : Bool\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.ldiff (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (a\u271d && !b\u271d) (PosNum.ldiff m\u271d n\u271d)\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase pbb\n\u03b1 : Type u_1\na\u271d b\u271d : Bool\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.ldiff (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (a\u271d && !b\u271d) (PosNum.ldiff m\u271d n\u271d)\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 (false && !false) = false\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 ldiff 0 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 ldiff 0 (pos n\u271d) = bif false && !true then pos n\u271d else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 ldiff (pos n\u271d) 0 = bif true && !false then pos n\u271d else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 ldiff (pos m\u271d) (pos n\u271d) = PosNum.ldiff m\u271d n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 PosNum.ldiff 1 1 = bif true && !true then 1 else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase p1b.false\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 PosNum.ldiff 1 (PosNum.bit false n\u271d) = bit (true && !false) (bif false && !true then pos n\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase p1b.true\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 PosNum.ldiff 1 (PosNum.bit true n\u271d) = bit (true && !true) (bif false && !true then pos n\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pb1.false\n\u03b1 : Type u_1\nm\u271d : PosNum\n\u22a2 PosNum.ldiff (PosNum.bit false m\u271d) 1 = bit (false && !true) (bif true && !false then pos m\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pb1.true\n\u03b1 : Type u_1\nm\u271d : PosNum\n\u22a2 PosNum.ldiff (PosNum.bit true m\u271d) 1 = bit (true && !true) (bif true && !false then pos m\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.false.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.ldiff (PosNum.bit false m\u271d) (PosNum.bit false n\u271d) = bit (false && !false) (PosNum.ldiff m\u271d n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.false.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.ldiff (PosNum.bit false m\u271d) (PosNum.bit true n\u271d) = bit (false && !true) (PosNum.ldiff m\u271d n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.true.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.ldiff (PosNum.bit true m\u271d) (PosNum.bit false n\u271d) = bit (true && !false) (PosNum.ldiff m\u271d n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.true.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.ldiff (PosNum.bit true m\u271d) (PosNum.bit true n\u271d) = bit (true && !true) (PosNum.ldiff m\u271d n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (m n : Num), \u2191(lxor m n) = Nat.lxor' \u2191m \u2191n\n[PROOFSTEP]\napply bitwise'_to_nat PosNum.lxor\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 xor false false = false\n[PROOFSTEP]\nintros\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 lxor 0 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\n\u22a2 \u2200 (n : PosNum), lxor 0 (pos n) = bif xor false true then pos n else 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\n\u22a2 \u2200 (n : PosNum), lxor (pos n) 0 = bif xor true false then pos n else 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\n\u22a2 \u2200 (m n : PosNum), lxor (pos m) (pos n) = PosNum.lxor m n\n[PROOFSTEP]\nintros\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 PosNum.lxor 1 1 = bif xor true true then 1 else 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase p1b\n\u03b1 : Type u_1\n\u22a2 \u2200 (b : Bool) (n : PosNum), PosNum.lxor 1 (PosNum.bit b n) = bit (xor true b) (bif xor false true then pos n else 0)\n[PROOFSTEP]\nintros\n[GOAL]\ncase pb1\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : Bool) (m : PosNum), PosNum.lxor (PosNum.bit a m) 1 = bit (xor a true) (bif xor true false then pos m else 0)\n[PROOFSTEP]\nintros\n[GOAL]\ncase pbb\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : Bool) (m n : PosNum), PosNum.lxor (PosNum.bit a m) (PosNum.bit b n) = bit (xor a b) (PosNum.lxor m n)\n[PROOFSTEP]\nintros\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 xor false false = false\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 xor false false = false\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 lxor 0 0 = 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 lxor 0 0 = 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 lxor 0 (pos n\u271d) = bif xor false true then pos n\u271d else 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 lxor 0 (pos n\u271d) = bif xor false true then pos n\u271d else 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 lxor (pos n\u271d) 0 = bif xor true false then pos n\u271d else 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 lxor (pos n\u271d) 0 = bif xor true false then pos n\u271d else 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 lxor (pos m\u271d) (pos n\u271d) = PosNum.lxor m\u271d n\u271d\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 lxor (pos m\u271d) (pos n\u271d) = PosNum.lxor m\u271d n\u271d\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 PosNum.lxor 1 1 = bif xor true true then 1 else 0\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 PosNum.lxor 1 1 = bif xor true true then 1 else 0\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase p1b\n\u03b1 : Type u_1\nb\u271d : Bool\nn\u271d : PosNum\n\u22a2 PosNum.lxor 1 (PosNum.bit b\u271d n\u271d) = bit (xor true b\u271d) (bif xor false true then pos n\u271d else 0)\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase p1b\n\u03b1 : Type u_1\nb\u271d : Bool\nn\u271d : PosNum\n\u22a2 PosNum.lxor 1 (PosNum.bit b\u271d n\u271d) = bit (xor true b\u271d) (bif xor false true then pos n\u271d else 0)\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase pb1\n\u03b1 : Type u_1\na\u271d : Bool\nm\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit a\u271d m\u271d) 1 = bit (xor a\u271d true) (bif xor true false then pos m\u271d else 0)\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase pb1\n\u03b1 : Type u_1\na\u271d : Bool\nm\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit a\u271d m\u271d) 1 = bit (xor a\u271d true) (bif xor true false then pos m\u271d else 0)\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase pbb\n\u03b1 : Type u_1\na\u271d b\u271d : Bool\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (xor a\u271d b\u271d) (PosNum.lxor m\u271d n\u271d)\n[PROOFSTEP]\ntry cases_type* Bool\n[GOAL]\ncase pbb\n\u03b1 : Type u_1\na\u271d b\u271d : Bool\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (xor a\u271d b\u271d) (PosNum.lxor m\u271d n\u271d)\n[PROOFSTEP]\ncases_type* Bool\n[GOAL]\ncase gff\n\u03b1 : Type u_1\n\u22a2 xor false false = false\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f00\n\u03b1 : Type u_1\n\u22a2 lxor 0 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f0n\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 lxor 0 (pos n\u271d) = bif xor false true then pos n\u271d else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase fn0\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 lxor (pos n\u271d) 0 = bif xor true false then pos n\u271d else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase fnn\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 lxor (pos m\u271d) (pos n\u271d) = PosNum.lxor m\u271d n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase p11\n\u03b1 : Type u_1\n\u22a2 PosNum.lxor 1 1 = bif xor true true then 1 else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase p1b.false\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 PosNum.lxor 1 (PosNum.bit false n\u271d) = bit (xor true false) (bif xor false true then pos n\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase p1b.true\n\u03b1 : Type u_1\nn\u271d : PosNum\n\u22a2 PosNum.lxor 1 (PosNum.bit true n\u271d) = bit (xor true true) (bif xor false true then pos n\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pb1.false\n\u03b1 : Type u_1\nm\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit false m\u271d) 1 = bit (xor false true) (bif xor true false then pos m\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pb1.true\n\u03b1 : Type u_1\nm\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit true m\u271d) 1 = bit (xor true true) (bif xor true false then pos m\u271d else 0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.false.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit false m\u271d) (PosNum.bit false n\u271d) = bit (xor false false) (PosNum.lxor m\u271d n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.false.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit false m\u271d) (PosNum.bit true n\u271d) = bit (xor false true) (PosNum.lxor m\u271d n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.true.false\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit true m\u271d) (PosNum.bit false n\u271d) = bit (xor true false) (PosNum.lxor m\u271d n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pbb.true.true\n\u03b1 : Type u_1\nm\u271d n\u271d : PosNum\n\u22a2 PosNum.lxor (PosNum.bit true m\u271d) (PosNum.bit true n\u271d) = bit (xor true true) (PosNum.lxor m\u271d n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nm : Num\nn : \u2115\n\u22a2 \u2191(shiftl m n) = Nat.shiftl (\u2191m) n\n[PROOFSTEP]\ncases m\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 \u2191(shiftl zero n) = Nat.shiftl (\u2191zero) n\n[PROOFSTEP]\ndsimp only [shiftl]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nn : \u2115\na\u271d : PosNum\n\u22a2 \u2191(shiftl (pos a\u271d) n) = Nat.shiftl (\u2191(pos a\u271d)) n\n[PROOFSTEP]\ndsimp only [shiftl]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 \u21910 = Nat.shiftl (\u2191zero) n\n[PROOFSTEP]\nsymm\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 Nat.shiftl (\u2191zero) n = \u21910\n[PROOFSTEP]\napply Nat.zero_shiftl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nn : \u2115\na\u271d : PosNum\n\u22a2 \u2191(pos (PosNum.shiftl a\u271d n)) = Nat.shiftl (\u2191(pos a\u271d)) n\n[PROOFSTEP]\nsimp only [cast_pos]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nn : \u2115\na\u271d : PosNum\n\u22a2 \u2191(PosNum.shiftl a\u271d n) = Nat.shiftl (\u2191a\u271d) n\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 \u2191(PosNum.shiftl a\u271d Nat.zero) = Nat.shiftl (\u2191a\u271d) Nat.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.succ\n\u03b1 : Type u_1\na\u271d : PosNum\nn : \u2115\nIH : \u2191(PosNum.shiftl a\u271d n) = Nat.shiftl (\u2191a\u271d) n\n\u22a2 \u2191(PosNum.shiftl a\u271d (Nat.succ n)) = Nat.shiftl (\u2191a\u271d) (Nat.succ n)\n[PROOFSTEP]\nsimp [PosNum.shiftl_succ_eq_bit0_shiftl, Nat.shiftl_succ, IH]\n[GOAL]\n\u03b1 : Type u_1\nm : Num\nn : \u2115\n\u22a2 \u2191(shiftr m n) = Nat.shiftr (\u2191m) n\n[PROOFSTEP]\ncases' m with m\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 \u2191(shiftr zero n) = Nat.shiftr (\u2191zero) n\n[PROOFSTEP]\ndsimp only [shiftr]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nn : \u2115\nm : PosNum\n\u22a2 \u2191(shiftr (pos m) n) = Nat.shiftr (\u2191(pos m)) n\n[PROOFSTEP]\ndsimp only [shiftr]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 \u21910 = Nat.shiftr (\u2191zero) n\n[PROOFSTEP]\nsymm\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 Nat.shiftr (\u2191zero) n = \u21910\n[PROOFSTEP]\napply Nat.zero_shiftr\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nn : \u2115\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\n[PROOFSTEP]\ninduction' n with n IH generalizing m\n[GOAL]\ncase pos.zero\n\u03b1 : Type u_1\nm\u271d m : PosNum\n\u22a2 \u2191(PosNum.shiftr m Nat.zero) = Nat.shiftr (\u2191(pos m)) Nat.zero\n[PROOFSTEP]\ncases m\n[GOAL]\ncase pos.zero.one\n\u03b1 : Type u_1\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr one Nat.zero) = Nat.shiftr (\u2191(pos one)) Nat.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.zero.bit1\n\u03b1 : Type u_1\nm a\u271d : PosNum\n\u22a2 \u2191(PosNum.shiftr (PosNum.bit1 a\u271d) Nat.zero) = Nat.shiftr (\u2191(pos (PosNum.bit1 a\u271d))) Nat.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.zero.bit0\n\u03b1 : Type u_1\nm a\u271d : PosNum\n\u22a2 \u2191(PosNum.shiftr (PosNum.bit0 a\u271d) Nat.zero) = Nat.shiftr (\u2191(pos (PosNum.bit0 a\u271d))) Nat.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.succ\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr m (Nat.succ n)) = Nat.shiftr (\u2191(pos m)) (Nat.succ n)\n[PROOFSTEP]\ncases' m with m m\n[GOAL]\ncase pos.succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\n\u22a2 \u2191(PosNum.shiftr one (Nat.succ n)) = Nat.shiftr (\u2191(pos one)) (Nat.succ n)\n[PROOFSTEP]\ndsimp only [PosNum.shiftr]\n[GOAL]\ncase pos.succ.bit1\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr (PosNum.bit1 m) (Nat.succ n)) = Nat.shiftr (\u2191(pos (PosNum.bit1 m))) (Nat.succ n)\n[PROOFSTEP]\ndsimp only [PosNum.shiftr]\n[GOAL]\ncase pos.succ.bit0\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr (PosNum.bit0 m) (Nat.succ n)) = Nat.shiftr (\u2191(pos (PosNum.bit0 m))) (Nat.succ n)\n[PROOFSTEP]\ndsimp only [PosNum.shiftr]\n[GOAL]\ncase pos.succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\n\u22a2 \u21910 = Nat.shiftr (\u2191(pos one)) (Nat.succ n)\n[PROOFSTEP]\nrw [Nat.shiftr_eq_div_pow]\n[GOAL]\ncase pos.succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\n\u22a2 \u21910 = \u2191(pos one) / 2 ^ Nat.succ n\n[PROOFSTEP]\nsymm\n[GOAL]\ncase pos.succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\n\u22a2 \u2191(pos one) / 2 ^ Nat.succ n = \u21910\n[PROOFSTEP]\napply Nat.div_eq_of_lt\n[GOAL]\ncase pos.succ.one.h\u2080\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\n\u22a2 \u2191(pos one) < 2 ^ Nat.succ n\n[PROOFSTEP]\nexact @Nat.pow_lt_pow_of_lt_right 2 (by decide) 0 (n + 1) (Nat.succ_pos _)\n[GOAL]\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\n\u22a2 1 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\ncase pos.succ.bit1\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos (PosNum.bit1 m))) (Nat.succ n)\n[PROOFSTEP]\ntrans\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr m n) = ?m.648288\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 ?m.648288 = Nat.shiftr (\u2191(pos (PosNum.bit1 m))) (Nat.succ n)\n\u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n m : PosNum \u22a2 \u2115\n[PROOFSTEP]\napply IH\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 Nat.shiftr (\u2191(pos m)) n = Nat.shiftr (\u2191(pos (PosNum.bit1 m))) (Nat.succ n)\n[PROOFSTEP]\nchange Nat.shiftr m n = Nat.shiftr (_root_.bit1 m) (n + 1)\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 Nat.shiftr (\u2191m) n = Nat.shiftr (_root_.bit1 \u2191m) (n + 1)\n[PROOFSTEP]\nrw [add_comm n 1, Nat.shiftr_add]\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 Nat.shiftr (\u2191m) n = Nat.shiftr (Nat.shiftr (_root_.bit1 \u2191m) 1) n\n[PROOFSTEP]\napply congr_arg fun x => Nat.shiftr x n\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191m = Nat.shiftr (_root_.bit1 \u2191m) 1\n[PROOFSTEP]\nrepeat unfold Nat.shiftr\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191m = Nat.shiftr (_root_.bit1 \u2191m) 1\n[PROOFSTEP]\nunfold Nat.shiftr\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191m = Nat.div2 (Nat.shiftr (_root_.bit1 \u2191m) 0)\n[PROOFSTEP]\nunfold Nat.shiftr\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191m = Nat.div2 (_root_.bit1 \u2191m)\n[PROOFSTEP]\nunfold Nat.shiftr\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191m = Nat.div2 (_root_.bit1 \u2191m)\n[PROOFSTEP]\nchange (m : \u2115) = Nat.div2 (Nat.bit true m)\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191m = Nat.div2 (Nat.bit true \u2191m)\n[PROOFSTEP]\nrw [Nat.div2_bit]\n[GOAL]\ncase pos.succ.bit0\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos (PosNum.bit0 m))) (Nat.succ n)\n[PROOFSTEP]\ntrans\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191(PosNum.shiftr m n) = ?m.652080\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 ?m.652080 = Nat.shiftr (\u2191(pos (PosNum.bit0 m))) (Nat.succ n)\n\u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n m : PosNum \u22a2 \u2115\n[PROOFSTEP]\napply IH\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 Nat.shiftr (\u2191(pos m)) n = Nat.shiftr (\u2191(pos (PosNum.bit0 m))) (Nat.succ n)\n[PROOFSTEP]\nchange Nat.shiftr m n = Nat.shiftr (_root_.bit0 m) (n + 1)\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 Nat.shiftr (\u2191m) n = Nat.shiftr (_root_.bit0 \u2191m) (n + 1)\n[PROOFSTEP]\nrw [add_comm n 1, Nat.shiftr_add]\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 Nat.shiftr (\u2191m) n = Nat.shiftr (Nat.shiftr (_root_.bit0 \u2191m) 1) n\n[PROOFSTEP]\napply congr_arg fun x => Nat.shiftr x n\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191m = Nat.shiftr (_root_.bit0 \u2191m) 1\n[PROOFSTEP]\nrepeat unfold Nat.shiftr\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191m = Nat.shiftr (_root_.bit0 \u2191m) 1\n[PROOFSTEP]\nunfold Nat.shiftr\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191m = Nat.div2 (Nat.shiftr (_root_.bit0 \u2191m) 0)\n[PROOFSTEP]\nunfold Nat.shiftr\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191m = Nat.div2 (_root_.bit0 \u2191m)\n[PROOFSTEP]\nunfold Nat.shiftr\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191m = Nat.div2 (_root_.bit0 \u2191m)\n[PROOFSTEP]\nchange (m : \u2115) = Nat.div2 (Nat.bit false m)\n[GOAL]\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), \u2191(PosNum.shiftr m n) = Nat.shiftr (\u2191(pos m)) n\nm : PosNum\n\u22a2 \u2191m = Nat.div2 (Nat.bit false \u2191m)\n[PROOFSTEP]\nrw [Nat.div2_bit]\n[GOAL]\n\u03b1 : Type u_1\nm : Num\nn : \u2115\n\u22a2 testBit m n = Nat.testBit (\u2191m) n\n[PROOFSTEP]\ncases m\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 testBit zero n = Nat.testBit (\u2191zero) n\n[PROOFSTEP]\ndsimp only [testBit, Nat.testBit]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nn : \u2115\na\u271d : PosNum\n\u22a2 testBit (pos a\u271d) n = Nat.testBit (\u2191(pos a\u271d)) n\n[PROOFSTEP]\ndsimp only [testBit, Nat.testBit]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 false = Nat.bodd (Nat.shiftr (\u2191zero) n)\ncase pos \u03b1 : Type u_1 n : \u2115 a\u271d : PosNum \u22a2 PosNum.testBit a\u271d n = Nat.bodd (Nat.shiftr (\u2191(pos a\u271d)) n)\n[PROOFSTEP]\ncase zero =>\n change false = Nat.bodd (Nat.shiftr 0 n)\n rw [Nat.zero_shiftr]\n rfl\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 false = Nat.bodd (Nat.shiftr (\u2191zero) n)\n[PROOFSTEP]\ncase zero =>\n change false = Nat.bodd (Nat.shiftr 0 n)\n rw [Nat.zero_shiftr]\n rfl\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 false = Nat.bodd (Nat.shiftr (\u2191zero) n)\n[PROOFSTEP]\nchange false = Nat.bodd (Nat.shiftr 0 n)\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 false = Nat.bodd (Nat.shiftr 0 n)\n[PROOFSTEP]\nrw [Nat.zero_shiftr]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 false = Nat.bodd 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nn : \u2115\na\u271d : PosNum\n\u22a2 PosNum.testBit a\u271d n = Nat.bodd (Nat.shiftr (\u2191(pos a\u271d)) n)\n[PROOFSTEP]\ncase pos m =>\n induction' n with n IH generalizing m <;> cases' m with m m <;> dsimp only [PosNum.testBit]\n \u00b7 rfl\n \u00b7 exact (Nat.bodd_bit _ _).symm\n \u00b7 exact (Nat.bodd_bit _ _).symm\n \u00b7 change false = Nat.bodd (Nat.shiftr 1 (n + 1))\n rw [add_comm, Nat.shiftr_add]\n change false = Nat.bodd (Nat.shiftr 0 n)\n rw [Nat.zero_shiftr]; rfl\n \u00b7 change PosNum.testBit m n = Nat.bodd (Nat.shiftr (Nat.bit true m) (n + 1))\n rw [add_comm, Nat.shiftr_add]\n dsimp only [Nat.shiftr]\n rw [Nat.div2_bit]\n apply IH\n \u00b7 change PosNum.testBit m n = Nat.bodd (Nat.shiftr (Nat.bit false m) (n + 1))\n rw [add_comm, Nat.shiftr_add]\n dsimp only [Nat.shiftr]\n rw [Nat.div2_bit]\n apply IH\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nm : PosNum\n\u22a2 PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\n[PROOFSTEP]\ncase pos m =>\n induction' n with n IH generalizing m <;> cases' m with m m <;> dsimp only [PosNum.testBit]\n \u00b7 rfl\n \u00b7 exact (Nat.bodd_bit _ _).symm\n \u00b7 exact (Nat.bodd_bit _ _).symm\n \u00b7 change false = Nat.bodd (Nat.shiftr 1 (n + 1))\n rw [add_comm, Nat.shiftr_add]\n change false = Nat.bodd (Nat.shiftr 0 n)\n rw [Nat.zero_shiftr]; rfl\n \u00b7 change PosNum.testBit m n = Nat.bodd (Nat.shiftr (Nat.bit true m) (n + 1))\n rw [add_comm, Nat.shiftr_add]\n dsimp only [Nat.shiftr]\n rw [Nat.div2_bit]\n apply IH\n \u00b7 change PosNum.testBit m n = Nat.bodd (Nat.shiftr (Nat.bit false m) (n + 1))\n rw [add_comm, Nat.shiftr_add]\n dsimp only [Nat.shiftr]\n rw [Nat.div2_bit]\n apply IH\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nm : PosNum\n\u22a2 PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\n[PROOFSTEP]\ninduction' n with n IH generalizing m\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nm\u271d m : PosNum\n\u22a2 PosNum.testBit m Nat.zero = Nat.bodd (Nat.shiftr (\u2191(pos m)) Nat.zero)\n[PROOFSTEP]\ncases' m with m m\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\nm : PosNum\n\u22a2 PosNum.testBit m (Nat.succ n) = Nat.bodd (Nat.shiftr (\u2191(pos m)) (Nat.succ n))\n[PROOFSTEP]\ncases' m with m m\n[GOAL]\ncase zero.one\n\u03b1 : Type u_1\nm : PosNum\n\u22a2 PosNum.testBit one Nat.zero = Nat.bodd (Nat.shiftr (\u2191(pos one)) Nat.zero)\n[PROOFSTEP]\ndsimp only [PosNum.testBit]\n[GOAL]\ncase zero.bit1\n\u03b1 : Type u_1\nm\u271d m : PosNum\n\u22a2 PosNum.testBit (PosNum.bit1 m) Nat.zero = Nat.bodd (Nat.shiftr (\u2191(pos (PosNum.bit1 m))) Nat.zero)\n[PROOFSTEP]\ndsimp only [PosNum.testBit]\n[GOAL]\ncase zero.bit0\n\u03b1 : Type u_1\nm\u271d m : PosNum\n\u22a2 PosNum.testBit (PosNum.bit0 m) Nat.zero = Nat.bodd (Nat.shiftr (\u2191(pos (PosNum.bit0 m))) Nat.zero)\n[PROOFSTEP]\ndsimp only [PosNum.testBit]\n[GOAL]\ncase succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\n\u22a2 PosNum.testBit one (Nat.succ n) = Nat.bodd (Nat.shiftr (\u2191(pos one)) (Nat.succ n))\n[PROOFSTEP]\ndsimp only [PosNum.testBit]\n[GOAL]\ncase succ.bit1\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\nm : PosNum\n\u22a2 PosNum.testBit (PosNum.bit1 m) (Nat.succ n) = Nat.bodd (Nat.shiftr (\u2191(pos (PosNum.bit1 m))) (Nat.succ n))\n[PROOFSTEP]\ndsimp only [PosNum.testBit]\n[GOAL]\ncase succ.bit0\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\nm : PosNum\n\u22a2 PosNum.testBit (PosNum.bit0 m) (Nat.succ n) = Nat.bodd (Nat.shiftr (\u2191(pos (PosNum.bit0 m))) (Nat.succ n))\n[PROOFSTEP]\ndsimp only [PosNum.testBit]\n[GOAL]\ncase zero.one\n\u03b1 : Type u_1\nm : PosNum\n\u22a2 true = Nat.bodd (Nat.shiftr (\u2191(pos one)) Nat.zero)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase zero.bit1\n\u03b1 : Type u_1\nm\u271d m : PosNum\n\u22a2 true = Nat.bodd (Nat.shiftr (\u2191(pos (PosNum.bit1 m))) Nat.zero)\n[PROOFSTEP]\nexact (Nat.bodd_bit _ _).symm\n[GOAL]\ncase zero.bit0\n\u03b1 : Type u_1\nm\u271d m : PosNum\n\u22a2 false = Nat.bodd (Nat.shiftr (\u2191(pos (PosNum.bit0 m))) Nat.zero)\n[PROOFSTEP]\nexact (Nat.bodd_bit _ _).symm\n[GOAL]\ncase succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\n\u22a2 false = Nat.bodd (Nat.shiftr (\u2191(pos one)) (Nat.succ n))\n[PROOFSTEP]\nchange false = Nat.bodd (Nat.shiftr 1 (n + 1))\n[GOAL]\ncase succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\n\u22a2 false = Nat.bodd (Nat.shiftr 1 (n + 1))\n[PROOFSTEP]\nrw [add_comm, Nat.shiftr_add]\n[GOAL]\ncase succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\n\u22a2 false = Nat.bodd (Nat.shiftr (Nat.shiftr 1 1) n)\n[PROOFSTEP]\nchange false = Nat.bodd (Nat.shiftr 0 n)\n[GOAL]\ncase succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\n\u22a2 false = Nat.bodd (Nat.shiftr 0 n)\n[PROOFSTEP]\nrw [Nat.zero_shiftr]\n[GOAL]\ncase succ.one\n\u03b1 : Type u_1\nm : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\n\u22a2 false = Nat.bodd 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.bit1\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\nm : PosNum\n\u22a2 PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos (PosNum.bit1 m))) (Nat.succ n))\n[PROOFSTEP]\nchange PosNum.testBit m n = Nat.bodd (Nat.shiftr (Nat.bit true m) (n + 1))\n[GOAL]\ncase succ.bit1\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\nm : PosNum\n\u22a2 PosNum.testBit m n = Nat.bodd (Nat.shiftr (Nat.bit true \u2191m) (n + 1))\n[PROOFSTEP]\nrw [add_comm, Nat.shiftr_add]\n[GOAL]\ncase succ.bit1\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\nm : PosNum\n\u22a2 PosNum.testBit m n = Nat.bodd (Nat.shiftr (Nat.shiftr (Nat.bit true \u2191m) 1) n)\n[PROOFSTEP]\ndsimp only [Nat.shiftr]\n[GOAL]\ncase succ.bit1\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\nm : PosNum\n\u22a2 PosNum.testBit m n = Nat.bodd (Nat.shiftr (Nat.div2 (Nat.bit true \u2191m)) n)\n[PROOFSTEP]\nrw [Nat.div2_bit]\n[GOAL]\ncase succ.bit1\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\nm : PosNum\n\u22a2 PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191m) n)\n[PROOFSTEP]\napply IH\n[GOAL]\ncase succ.bit0\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\nm : PosNum\n\u22a2 PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos (PosNum.bit0 m))) (Nat.succ n))\n[PROOFSTEP]\nchange PosNum.testBit m n = Nat.bodd (Nat.shiftr (Nat.bit false m) (n + 1))\n[GOAL]\ncase succ.bit0\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\nm : PosNum\n\u22a2 PosNum.testBit m n = Nat.bodd (Nat.shiftr (Nat.bit false \u2191m) (n + 1))\n[PROOFSTEP]\nrw [add_comm, Nat.shiftr_add]\n[GOAL]\ncase succ.bit0\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\nm : PosNum\n\u22a2 PosNum.testBit m n = Nat.bodd (Nat.shiftr (Nat.shiftr (Nat.bit false \u2191m) 1) n)\n[PROOFSTEP]\ndsimp only [Nat.shiftr]\n[GOAL]\ncase succ.bit0\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\nm : PosNum\n\u22a2 PosNum.testBit m n = Nat.bodd (Nat.shiftr (Nat.div2 (Nat.bit false \u2191m)) n)\n[PROOFSTEP]\nrw [Nat.div2_bit]\n[GOAL]\ncase succ.bit0\n\u03b1 : Type u_1\nm\u271d : PosNum\nn : \u2115\nIH : \u2200 (m : PosNum), PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191(pos m)) n)\nm : PosNum\n\u22a2 PosNum.testBit m n = Nat.bodd (Nat.shiftr (\u2191m) n)\n[PROOFSTEP]\napply IH\n[GOAL]\n\u03b1 : Type u_1\nn : ZNum\n\u22a2 - -n = n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u22a2 - -zero = zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 - -pos a\u271d = pos a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 - -neg a\u271d = neg a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : ZNum\n\u22a2 -ZNum.bit1 n = ZNum.bitm1 (-n)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u22a2 -ZNum.bit1 zero = ZNum.bitm1 (-zero)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 -ZNum.bit1 (pos a\u271d) = ZNum.bitm1 (-pos a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 -ZNum.bit1 (neg a\u271d) = ZNum.bitm1 (-neg a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : ZNum\n\u22a2 -ZNum.bitm1 n = ZNum.bit1 (-n)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u22a2 -ZNum.bitm1 zero = ZNum.bit1 (-zero)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 -ZNum.bitm1 (pos a\u271d) = ZNum.bit1 (-pos a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 -ZNum.bitm1 (neg a\u271d) = ZNum.bit1 (-neg a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : ZNum\n\u22a2 -succ n = pred (-n)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u22a2 -succ zero = pred (-zero)\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u22a2 -succ zero = pred (-zero)\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u22a2 -succ zero = pred (-zero)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 -succ (pos a\u271d) = pred (-pos a\u271d)\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 -succ (pos a\u271d) = pred (-pos a\u271d)\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 -succ (pos a\u271d) = pred (-pos a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 -succ (neg a\u271d) = pred (-neg a\u271d)\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 -succ (neg a\u271d) = pred (-neg a\u271d)\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 -succ (neg a\u271d) = pred (-neg a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 -succ (neg a\u271d) = pred (-neg a\u271d)\n[PROOFSTEP]\nrw [succ, Num.zneg_toZNumNeg]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 Num.toZNum (pred' a\u271d) = pred (-neg a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : ZNum\n\u22a2 -pred n = succ (-n)\n[PROOFSTEP]\nrw [\u2190 zneg_zneg (succ (-n)), zneg_succ, zneg_zneg]\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 natAbs \u2191\u2191p = natAbs (-\u2191p)\n[PROOFSTEP]\nrw [p.to_nat_to_int, Int.natAbs_neg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\n\u22a2 \u2191\u21910 = \u21910\n[PROOFSTEP]\nrw [cast_zero, cast_zero, Int.cast_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np : PosNum\n\u22a2 \u2191\u2191(pos p) = \u2191(pos p)\n[PROOFSTEP]\nrw [cast_pos, cast_pos, PosNum.cast_to_int]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np : PosNum\n\u22a2 \u2191\u2191(neg p) = \u2191(neg p)\n[PROOFSTEP]\nrw [cast_neg, cast_neg, Int.cast_neg, PosNum.cast_to_int]\n[GOAL]\n\u03b1 : Type u_1\na : PosNum\n\u22a2 sub' 1 (_root_.bit0 a) = ZNum.bit1 (neg a)\n[PROOFSTEP]\nrw [PosNum.one_sub', a.bit0_of_bit0]\n[GOAL]\n\u03b1 : Type u_1\na : PosNum\n\u22a2 Num.toZNumNeg (pred' (PosNum.bit0 a)) = ZNum.bit1 (neg a)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np : PosNum\n\u22a2 \u2191(ZNum.bit0 (pos p)) = _root_.bit0 \u2191(pos p)\n[PROOFSTEP]\nrw [ZNum.bit0, cast_pos, cast_pos]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np : PosNum\n\u22a2 \u2191(PosNum.bit0 p) = _root_.bit0 \u2191p\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np : PosNum\n\u22a2 \u2191(ZNum.bit0 (neg p)) = _root_.bit0 \u2191(neg p)\n[PROOFSTEP]\nrw [ZNum.bit0, cast_neg, cast_neg, PosNum.cast_bit0, _root_.bit0, _root_.bit0, neg_add_rev]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\n\u22a2 \u2191(ZNum.bit1 0) = _root_.bit1 \u21910\n[PROOFSTEP]\nsimp [ZNum.bit1, _root_.bit1, _root_.bit0]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np : PosNum\n\u22a2 \u2191(ZNum.bit1 (pos p)) = _root_.bit1 \u2191(pos p)\n[PROOFSTEP]\nrw [ZNum.bit1, cast_pos, cast_pos]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np : PosNum\n\u22a2 \u2191(PosNum.bit1 p) = _root_.bit1 \u2191p\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np : PosNum\n\u22a2 \u2191(ZNum.bit1 (neg p)) = _root_.bit1 \u2191(neg p)\n[PROOFSTEP]\nrw [ZNum.bit1, cast_neg, cast_neg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np : PosNum\n\u22a2 -\u2191(Num.casesOn (pred' p) 1 PosNum.bit1) = _root_.bit1 (-\u2191p)\n[PROOFSTEP]\ncases' e : pred' p with a\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np : PosNum\ne : pred' p = Num.zero\n\u22a2 -\u2191(Num.casesOn Num.zero 1 PosNum.bit1) = _root_.bit1 (-\u2191p)\n[PROOFSTEP]\nhave ep : p = _ := (succ'_pred' p).symm.trans (congr_arg Num.succ' e)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np a : PosNum\ne : pred' p = Num.pos a\n\u22a2 -\u2191(Num.casesOn (Num.pos a) 1 PosNum.bit1) = _root_.bit1 (-\u2191p)\n[PROOFSTEP]\nhave ep : p = _ := (succ'_pred' p).symm.trans (congr_arg Num.succ' e)\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np : PosNum\ne : pred' p = Num.zero\nep : p = Num.succ' Num.zero\n\u22a2 -\u2191(Num.casesOn Num.zero 1 PosNum.bit1) = _root_.bit1 (-\u2191p)\n[PROOFSTEP]\nconv at ep => change p = 1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np : PosNum\ne : pred' p = Num.zero\nep : p = Num.succ' Num.zero\n| p = Num.succ' Num.zero\n[PROOFSTEP]\nchange p = 1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np : PosNum\ne : pred' p = Num.zero\nep : p = Num.succ' Num.zero\n| p = Num.succ' Num.zero\n[PROOFSTEP]\nchange p = 1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np : PosNum\ne : pred' p = Num.zero\nep : p = Num.succ' Num.zero\n| p = Num.succ' Num.zero\n[PROOFSTEP]\nchange p = 1\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np : PosNum\ne : pred' p = Num.zero\nep : p = 1\n\u22a2 -\u2191(Num.casesOn Num.zero 1 PosNum.bit1) = _root_.bit1 (-\u2191p)\n[PROOFSTEP]\nsubst p\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\ne : pred' 1 = Num.zero\n\u22a2 -\u2191(Num.casesOn Num.zero 1 PosNum.bit1) = _root_.bit1 (-\u21911)\n[PROOFSTEP]\nsimp [_root_.bit1, _root_.bit0]\n -- Porting note: `rw [Num.succ']` yields a `match` pattern.\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np a : PosNum\ne : pred' p = Num.pos a\nep : p = Num.succ' (Num.pos a)\n\u22a2 -\u2191(Num.casesOn (Num.pos a) 1 PosNum.bit1) = _root_.bit1 (-\u2191p)\n[PROOFSTEP]\ndsimp only [Num.succ'] at ep \n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\np a : PosNum\ne : pred' p = Num.pos a\nep : p = PosNum.succ a\n\u22a2 -\u2191(Num.casesOn (Num.pos a) 1 PosNum.bit1) = _root_.bit1 (-\u2191p)\n[PROOFSTEP]\nsubst p\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na : PosNum\ne : pred' (PosNum.succ a) = Num.pos a\n\u22a2 -\u2191(Num.casesOn (Num.pos a) 1 PosNum.bit1) = _root_.bit1 (-\u2191(PosNum.succ a))\n[PROOFSTEP]\nhave : (\u2191(-\u2191a : \u2124) : \u03b1) = -1 + \u2191(-\u2191a + 1 : \u2124) := by simp [add_comm (-\u2191a : \u2124) 1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na : PosNum\ne : pred' (PosNum.succ a) = Num.pos a\n\u22a2 \u2191(-\u2191a) = -1 + \u2191(-\u2191a + 1)\n[PROOFSTEP]\nsimp [add_comm (-\u2191a : \u2124) 1]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na : PosNum\ne : pred' (PosNum.succ a) = Num.pos a\nthis : \u2191(-\u2191a) = -1 + \u2191(-\u2191a + 1)\n\u22a2 -\u2191(Num.casesOn (Num.pos a) 1 PosNum.bit1) = _root_.bit1 (-\u2191(PosNum.succ a))\n[PROOFSTEP]\nsimpa [_root_.bit1, _root_.bit0] using this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n\u22a2 \u2191(ZNum.bitm1 n) = _root_.bit0 \u2191n - 1\n[PROOFSTEP]\nconv =>\n lhs\n rw [\u2190 zneg_zneg n]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n| \u2191(ZNum.bitm1 n) = _root_.bit0 \u2191n - 1\n[PROOFSTEP]\n lhs\n rw [\u2190 zneg_zneg n]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n| \u2191(ZNum.bitm1 n) = _root_.bit0 \u2191n - 1\n[PROOFSTEP]\n lhs\n rw [\u2190 zneg_zneg n]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n| \u2191(ZNum.bitm1 n) = _root_.bit0 \u2191n - 1\n[PROOFSTEP]\nlhs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n| \u2191(ZNum.bitm1 n)\n[PROOFSTEP]\nrw [\u2190 zneg_zneg n]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n\u22a2 \u2191(ZNum.bitm1 (- -n)) = _root_.bit0 \u2191n - 1\n[PROOFSTEP]\nrw [\u2190 zneg_bit1, cast_zneg, cast_bit1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n\u22a2 -_root_.bit1 \u2191(-n) = _root_.bit0 \u2191n - 1\n[PROOFSTEP]\nhave : ((-1 + n + n : \u2124) : \u03b1) = (n + n + -1 : \u2124) := by simp [add_comm, add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n\u22a2 \u2191(-1 + \u2191n + \u2191n) = \u2191(\u2191n + \u2191n + -1)\n[PROOFSTEP]\nsimp [add_comm, add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\nthis : \u2191(-1 + \u2191n + \u2191n) = \u2191(\u2191n + \u2191n + -1)\n\u22a2 -_root_.bit1 \u2191(-n) = _root_.bit0 \u2191n - 1\n[PROOFSTEP]\nsimpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this\n[GOAL]\n\u03b1 : Type u_1\nn : ZNum\n\u22a2 n + 0 = n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u22a2 zero + 0 = zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 pos a\u271d + 0 = pos a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 neg a\u271d + 0 = neg a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : ZNum\n\u22a2 0 + n = n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u22a2 0 + zero = zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 0 + pos a\u271d = pos a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 0 + neg a\u271d = neg a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 neg p + 1 = succ (neg p)\n[PROOFSTEP]\ncases p\n[GOAL]\ncase one\n\u03b1 : Type u_1\n\u22a2 neg one + 1 = succ (neg one)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit1\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 neg (PosNum.bit1 a\u271d) + 1 = succ (neg (PosNum.bit1 a\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bit0\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 neg (PosNum.bit0 a\u271d) + 1 = succ (neg (PosNum.bit0 a\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na : PosNum\n\u22a2 \u2191(sub' a 1) = \u2191a - \u21911\n[PROOFSTEP]\nrw [sub'_one, Num.cast_toZNum, \u2190 Num.cast_to_nat, pred'_to_nat, \u2190 Nat.sub_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na : PosNum\n\u22a2 \u2191(\u2191a - 1) = \u2191a - \u21911\n[PROOFSTEP]\nsimp [PosNum.cast_pos]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nb : PosNum\n\u22a2 \u2191(sub' 1 b) = \u21911 - \u2191b\n[PROOFSTEP]\nrw [one_sub', Num.cast_toZNumNeg, \u2190 neg_sub, neg_inj, \u2190 Num.cast_to_nat, pred'_to_nat, \u2190 Nat.sub_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nb : PosNum\n\u22a2 \u2191(\u2191b - 1) = \u2191b - \u21911\n[PROOFSTEP]\nsimp [PosNum.cast_pos]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(sub' (bit0 a) (bit0 b)) = \u2191(bit0 a) - \u2191(bit0 b)\n[PROOFSTEP]\nrw [sub', ZNum.cast_bit0, cast_sub' a b]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit0 a) - \u2191(bit0 b)\n[PROOFSTEP]\nhave : ((a + -b + (a + -b) : \u2124) : \u03b1) = a + a + (-b + -b) := by simp [add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(\u2191a + -\u2191b + (\u2191a + -\u2191b)) = \u2191a + \u2191a + (-\u2191b + -\u2191b)\n[PROOFSTEP]\nsimp [add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\nthis : \u2191(\u2191a + -\u2191b + (\u2191a + -\u2191b)) = \u2191a + \u2191a + (-\u2191b + -\u2191b)\n\u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit0 a) - \u2191(bit0 b)\n[PROOFSTEP]\nsimpa [_root_.bit0, sub_eq_add_neg] using this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(sub' (bit0 a) (bit1 b)) = \u2191(bit0 a) - \u2191(bit1 b)\n[PROOFSTEP]\nrw [sub', ZNum.cast_bitm1, cast_sub' a b]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 _root_.bit0 (\u2191a - \u2191b) - 1 = \u2191(bit0 a) - \u2191(bit1 b)\n[PROOFSTEP]\nhave : ((-b + (a + (-b + -1)) : \u2124) : \u03b1) = (a + -1 + (-b + -b) : \u2124) := by simp [add_comm, add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(-\u2191b + (\u2191a + (-\u2191b + -1))) = \u2191(\u2191a + -1 + (-\u2191b + -\u2191b))\n[PROOFSTEP]\nsimp [add_comm, add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\nthis : \u2191(-\u2191b + (\u2191a + (-\u2191b + -1))) = \u2191(\u2191a + -1 + (-\u2191b + -\u2191b))\n\u22a2 _root_.bit0 (\u2191a - \u2191b) - 1 = \u2191(bit0 a) - \u2191(bit1 b)\n[PROOFSTEP]\nsimpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(sub' (bit1 a) (bit0 b)) = \u2191(bit1 a) - \u2191(bit0 b)\n[PROOFSTEP]\nrw [sub', ZNum.cast_bit1, cast_sub' a b]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 _root_.bit1 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit0 b)\n[PROOFSTEP]\nhave : ((-b + (a + (-b + 1)) : \u2124) : \u03b1) = (a + 1 + (-b + -b) : \u2124) := by simp [add_comm, add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(-\u2191b + (\u2191a + (-\u2191b + 1))) = \u2191(\u2191a + 1 + (-\u2191b + -\u2191b))\n[PROOFSTEP]\nsimp [add_comm, add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\nthis : \u2191(-\u2191b + (\u2191a + (-\u2191b + 1))) = \u2191(\u2191a + 1 + (-\u2191b + -\u2191b))\n\u22a2 _root_.bit1 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit0 b)\n[PROOFSTEP]\nsimpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(sub' (bit1 a) (bit1 b)) = \u2191(bit1 a) - \u2191(bit1 b)\n[PROOFSTEP]\nrw [sub', ZNum.cast_bit0, cast_sub' a b]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit1 b)\n[PROOFSTEP]\nhave : ((-b + (a + -b) : \u2124) : \u03b1) = a + (-b + -b) := by simp [add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(-\u2191b + (\u2191a + -\u2191b)) = \u2191a + (-\u2191b + -\u2191b)\n[PROOFSTEP]\nsimp [add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\nthis : \u2191(-\u2191b + (\u2191a + -\u2191b)) = \u2191a + (-\u2191b + -\u2191b)\n\u22a2 _root_.bit0 (\u2191a - \u2191b) = \u2191(bit1 a) - \u2191(bit1 b)\n[PROOFSTEP]\nsimpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 \u2191n = \u2191(pred' n) + 1\n[PROOFSTEP]\nrw [\u2190 Num.succ'_to_nat, n.succ'_pred']\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 \u2191n = \u2191\u2191(pred' n) + 1\n[PROOFSTEP]\nrw [\u2190 n.to_nat_to_int, to_nat_eq_succ_pred]\n[GOAL]\n\u03b1 : Type u_1\nn : PosNum\n\u22a2 \u2191(\u2191(pred' n) + 1) = \u2191\u2191(pred' n) + 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 toZNumNeg (PosNum.pred' (succ' (pos p))) = ZNum.neg p\n[PROOFSTEP]\nrw [PosNum.pred'_succ']\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 toZNumNeg (pos p) = ZNum.neg p\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 ZNum.succ (ZNum.pred (ZNum.pos p)) = ZNum.pos p\n[PROOFSTEP]\nrw [ZNum.pred, \u2190 toZNum_succ, Num.succ, PosNum.succ'_pred', toZNum]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 ZNum.ofInt' (\u2191n + 1) = ZNum.ofInt' \u2191n + 1\n[PROOFSTEP]\nchange ZNum.ofInt' (n + 1 : \u2115) = ZNum.ofInt' (n : \u2115) + 1\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 ZNum.ofInt' \u2191(n + 1) = ZNum.ofInt' \u2191n + 1\n[PROOFSTEP]\ndsimp only [ZNum.ofInt', ZNum.ofInt']\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 toZNum (ofNat' (n + 1)) = toZNum (ofNat' n) + 1\n[PROOFSTEP]\nrw [Num.ofNat'_succ, Num.add_one, toZNum_succ, ZNum.add_one]\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 ZNum.ofInt' (-[0+1] + 1) = ZNum.ofInt' -[0+1] + 1\n[PROOFSTEP]\nchange ZNum.ofInt' 0 = ZNum.ofInt' (-[0+1]) + 1\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 ZNum.ofInt' 0 = ZNum.ofInt' -[0+1] + 1\n[PROOFSTEP]\ndsimp only [ZNum.ofInt', ZNum.ofInt']\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 toZNum (ofNat' 0) = toZNumNeg (ofNat' (0 + 1)) + 1\n[PROOFSTEP]\nrw [ofNat'_succ, ofNat'_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 toZNum 0 = toZNumNeg (0 + 1) + 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 ZNum.ofInt' (-[n + 1+1] + 1) = ZNum.ofInt' -[n + 1+1] + 1\n[PROOFSTEP]\nchange ZNum.ofInt' -[n+1] = ZNum.ofInt' -[(n + 1)+1] + 1\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 ZNum.ofInt' -[n+1] = ZNum.ofInt' -[n + 1+1] + 1\n[PROOFSTEP]\ndsimp only [ZNum.ofInt', ZNum.ofInt']\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 toZNumNeg (ofNat' (n + 1)) = toZNumNeg (ofNat' (n + 1 + 1)) + 1\n[PROOFSTEP]\nrw [@Num.ofNat'_succ (n + 1), Num.add_one, toZNumNeg_succ, @ofNat'_succ n, Num.add_one, ZNum.add_one, pred_succ]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 toZNum \u2191(n + 1) = ZNum.ofInt' \u2191(n + 1)\n[PROOFSTEP]\nrw [Nat.cast_succ, Num.add_one, toZNum_succ, ofInt'_toZNum n, Nat.cast_succ, succ_ofInt', ZNum.add_one]\n[GOAL]\n\u03b1 : Type u_1\nm : Num\np : PosNum\n\u22a2 pos p = m \u2194 ZNum.pos p = toZNum m\n[PROOFSTEP]\ncases m\n[GOAL]\ncase zero\n\u03b1 : Type u_1\np : PosNum\n\u22a2 pos p = zero \u2194 ZNum.pos p = toZNum zero\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase pos\n\u03b1 : Type u_1\np a\u271d : PosNum\n\u22a2 pos p = pos a\u271d \u2194 ZNum.pos p = toZNum (pos a\u271d)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase zero.mp\n\u03b1 : Type u_1\np : PosNum\n\u22a2 pos p = zero \u2192 ZNum.pos p = toZNum zero\n[PROOFSTEP]\nintro h\n[GOAL]\ncase zero.mpr\n\u03b1 : Type u_1\np : PosNum\n\u22a2 ZNum.pos p = toZNum zero \u2192 pos p = zero\n[PROOFSTEP]\nintro h\n[GOAL]\ncase pos.mp\n\u03b1 : Type u_1\np a\u271d : PosNum\n\u22a2 pos p = pos a\u271d \u2192 ZNum.pos p = toZNum (pos a\u271d)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase pos.mpr\n\u03b1 : Type u_1\np a\u271d : PosNum\n\u22a2 ZNum.pos p = toZNum (pos a\u271d) \u2192 pos p = pos a\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase zero.mp\n\u03b1 : Type u_1\np : PosNum\nh : pos p = zero\n\u22a2 ZNum.pos p = toZNum zero\n[PROOFSTEP]\ntry cases h <;> rfl\n[GOAL]\ncase zero.mp\n\u03b1 : Type u_1\np : PosNum\nh : pos p = zero\n\u22a2 ZNum.pos p = toZNum zero\n[PROOFSTEP]\ncases h\n[GOAL]\ncase zero.mpr\n\u03b1 : Type u_1\np : PosNum\nh : ZNum.pos p = toZNum zero\n\u22a2 pos p = zero\n[PROOFSTEP]\ntry cases h <;> rfl\n[GOAL]\ncase zero.mpr\n\u03b1 : Type u_1\np : PosNum\nh : ZNum.pos p = toZNum zero\n\u22a2 pos p = zero\n[PROOFSTEP]\ncases h\n[GOAL]\ncase pos.mp\n\u03b1 : Type u_1\np a\u271d : PosNum\nh : pos p = pos a\u271d\n\u22a2 ZNum.pos p = toZNum (pos a\u271d)\n[PROOFSTEP]\ntry cases h <;> rfl\n[GOAL]\ncase pos.mp\n\u03b1 : Type u_1\np a\u271d : PosNum\nh : pos p = pos a\u271d\n\u22a2 ZNum.pos p = toZNum (pos a\u271d)\n[PROOFSTEP]\ncases h\n[GOAL]\ncase pos.mp.refl\n\u03b1 : Type u_1\np : PosNum\n\u22a2 ZNum.pos p = toZNum (pos p)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.mpr\n\u03b1 : Type u_1\np a\u271d : PosNum\nh : ZNum.pos p = toZNum (pos a\u271d)\n\u22a2 pos p = pos a\u271d\n[PROOFSTEP]\ntry cases h <;> rfl\n[GOAL]\ncase pos.mpr\n\u03b1 : Type u_1\np a\u271d : PosNum\nh : ZNum.pos p = toZNum (pos a\u271d)\n\u22a2 pos p = pos a\u271d\n[PROOFSTEP]\ncases h\n[GOAL]\ncase pos.mpr.refl\n\u03b1 : Type u_1\np : PosNum\n\u22a2 pos p = pos p\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nm : Num\np : PosNum\nh : ZNum.neg p = toZNum m\n\u22a2 m \u2208 ofZNum' (ZNum.neg p)\n[PROOFSTEP]\ncases m\n[GOAL]\ncase zero\n\u03b1 : Type u_1\np : PosNum\nh : ZNum.neg p = toZNum zero\n\u22a2 zero \u2208 ofZNum' (ZNum.neg p)\n[PROOFSTEP]\ncases h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\np a\u271d : PosNum\nh : ZNum.neg p = toZNum (pos a\u271d)\n\u22a2 pos a\u271d \u2208 ofZNum' (ZNum.neg p)\n[PROOFSTEP]\ncases h\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 castNum <$> ofZNum' (ZNum.pos p) = toNat' \u2191p\n[PROOFSTEP]\nrw [\u2190 PosNum.to_nat_to_int p]\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 castNum <$> ofZNum' (ZNum.pos p) = toNat' \u2191\u2191p\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 \u2191(\u2191(PosNum.pred' p) + 1) = \u2191p\n[PROOFSTEP]\nrw [\u2190 succ'_to_nat]\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 \u2191\u2191(succ' (PosNum.pred' p)) = \u2191p\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 \u2191(ofZNum (ZNum.pos p)) = toNat \u2191p\n[PROOFSTEP]\nrw [\u2190 PosNum.to_nat_to_int p]\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 \u2191(ofZNum (ZNum.pos p)) = toNat \u2191\u2191p\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 \u2191(\u2191(PosNum.pred' p) + 1) = \u2191p\n[PROOFSTEP]\nrw [\u2190 succ'_to_nat]\n[GOAL]\n\u03b1 : Type u_1\np : PosNum\n\u22a2 \u2191\u2191(succ' (PosNum.pred' p)) = \u2191p\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n\u22a2 \u2191(ofZNum n) = \u2191(toNat \u2191n)\n[PROOFSTEP]\nrw [\u2190 cast_to_nat, ofZNum_toNat]\n[GOAL]\n\u03b1 : Type u_1\nm n : Num\n\u22a2 \u2191(ofZNum (sub' m n)) = \u2191m - \u2191n\n[PROOFSTEP]\nrw [ofZNum_toNat, cast_sub', \u2190 to_nat_to_int, \u2190 to_nat_to_int, Int.toNat_sub]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na : ZNum\n\u22a2 \u2191(0 + a) = \u21910 + \u2191a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\n\u22a2 \u2191(0 + zero) = \u21910 + \u2191zero\n[PROOFSTEP]\nexact (_root_.zero_add _).symm\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na\u271d : PosNum\n\u22a2 \u2191(0 + pos a\u271d) = \u21910 + \u2191(pos a\u271d)\n[PROOFSTEP]\nexact (_root_.zero_add _).symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na\u271d : PosNum\n\u22a2 \u2191(0 + neg a\u271d) = \u21910 + \u2191(neg a\u271d)\n[PROOFSTEP]\nexact (_root_.zero_add _).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nb : ZNum\n\u22a2 \u2191(b + 0) = \u2191b + \u21910\n[PROOFSTEP]\ncases b\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\n\u22a2 \u2191(zero + 0) = \u2191zero + \u21910\n[PROOFSTEP]\nexact (_root_.add_zero _).symm\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na\u271d : PosNum\n\u22a2 \u2191(pos a\u271d + 0) = \u2191(pos a\u271d) + \u21910\n[PROOFSTEP]\nexact (_root_.add_zero _).symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na\u271d : PosNum\n\u22a2 \u2191(neg a\u271d + 0) = \u2191(neg a\u271d) + \u21910\n[PROOFSTEP]\nexact (_root_.add_zero _).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191(pos a + neg b) = \u2191(pos a) + \u2191(neg b)\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using PosNum.cast_sub' (\u03b1 := \u03b1) _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191b + -\u2191a = -\u2191a + \u2191b\n[PROOFSTEP]\nrw [\u2190 PosNum.cast_to_int a, \u2190 PosNum.cast_to_int b, \u2190 Int.cast_neg, \u2190 Int.cast_add (-a)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 \u2191\u2191b + \u2191(-\u2191a) = \u2191(-\u2191a + \u2191b)\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\na b : PosNum\n\u22a2 -\u2191(a + b) = -\u2191a + -\u2191b\n[PROOFSTEP]\nrw [PosNum.cast_add, neg_eq_iff_eq_neg, neg_add_rev, neg_neg, neg_neg, \u2190 PosNum.cast_to_int a, \u2190 PosNum.cast_to_int b, \u2190\n Int.cast_add, \u2190 Int.cast_add, add_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : ZNum\n\u22a2 \u2191(succ n) = \u2191n + 1\n[PROOFSTEP]\nrw [\u2190 add_one, cast_add, cast_one]\n[GOAL]\n\u03b1 : Type u_1\na : ZNum\n\u22a2 \u2191(0 * a) = \u21910 * \u2191a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u22a2 \u2191(0 * zero) = \u21910 * \u2191zero\n[PROOFSTEP]\nexact (zero_mul _).symm\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 \u2191(0 * pos a\u271d) = \u21910 * \u2191(pos a\u271d)\n[PROOFSTEP]\nexact (zero_mul _).symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 \u2191(0 * neg a\u271d) = \u21910 * \u2191(neg a\u271d)\n[PROOFSTEP]\nexact (zero_mul _).symm\n[GOAL]\n\u03b1 : Type u_1\nb : ZNum\n\u22a2 \u2191(b * 0) = \u2191b * \u21910\n[PROOFSTEP]\ncases b\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u22a2 \u2191(zero * 0) = \u2191zero * \u21910\n[PROOFSTEP]\nexact (mul_zero _).symm\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 \u2191(pos a\u271d * 0) = \u2191(pos a\u271d) * \u21910\n[PROOFSTEP]\nexact (mul_zero _).symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na\u271d : PosNum\n\u22a2 \u2191(neg a\u271d * 0) = \u2191(neg a\u271d) * \u21910\n[PROOFSTEP]\nexact (mul_zero _).symm\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 -\u2191(a * b) = \u2191a * -\u2191b\n[PROOFSTEP]\nrw [PosNum.cast_mul, neg_mul_eq_mul_neg]\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 -\u2191(a * b) = -\u2191a * \u2191b\n[PROOFSTEP]\nrw [PosNum.cast_mul, neg_mul_eq_neg_mul]\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191(a * b) = -\u2191a * -\u2191b\n[PROOFSTEP]\nrw [PosNum.cast_mul, neg_mul_neg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nm n : ZNum\n\u22a2 \u2191(m * n) = \u2191m * \u2191n\n[PROOFSTEP]\nrw [\u2190 cast_to_int, mul_to_int, Int.cast_mul, cast_to_int, cast_to_int]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 ofInt' \u2191(n + 1) = -ofInt' -[n+1]\n[PROOFSTEP]\nsimp only [ofInt', Num.zneg_toZNumNeg]\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 Num.toZNum (Num.ofNat' 0) = -Num.toZNum (Num.ofNat' 0)\n[PROOFSTEP]\nrw [Num.ofNat'_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 Num.toZNum 0 = -Num.toZNum 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 Num.toZNumNeg (Num.ofNat' (Nat.add n 0 + 1)) = -Num.toZNum (Num.ofNat' (n + 1))\n[PROOFSTEP]\nrw [Num.zneg_toZNum]\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 ofInt' \u21910 = 0\n[PROOFSTEP]\ndsimp [ofInt', cast_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 Num.toZNum (Num.ofNat' 0) = 0\n[PROOFSTEP]\nerw [Num.ofNat'_zero, Num.toZNum]\n[GOAL]\n\u03b1 : Type u_1\na : PosNum\n\u22a2 ofInt' \u2191(pos a) = pos a\n[PROOFSTEP]\nrw [cast_pos, \u2190 PosNum.cast_to_nat, \u2190 Num.ofInt'_toZNum, PosNum.of_to_nat]\n[GOAL]\n\u03b1 : Type u_1\na : PosNum\n\u22a2 Num.toZNum (Num.pos a) = pos a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\na : PosNum\n\u22a2 ofInt' \u2191(neg a) = neg a\n[PROOFSTEP]\nrw [cast_neg, ofInt'_neg, \u2190 PosNum.cast_to_nat, \u2190 Num.ofInt'_toZNum, PosNum.of_to_nat]\n[GOAL]\n\u03b1 : Type u_1\na : PosNum\n\u22a2 -Num.toZNum (Num.pos a) = neg a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn (cmp (pos a) (pos b)) (\u2191(pos a) < \u2191(pos b)) (pos a = pos b) (\u2191(pos b) < \u2191(pos a))\n[PROOFSTEP]\nhave := PosNum.cmp_to_nat a b\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\nthis : Ordering.casesOn (PosNum.cmp a b) (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a)\n\u22a2 Ordering.casesOn (cmp (pos a) (pos b)) (\u2191(pos a) < \u2191(pos b)) (pos a = pos b) (\u2191(pos b) < \u2191(pos a))\n[PROOFSTEP]\nrevert this\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn (PosNum.cmp a b) (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) \u2192\n Ordering.casesOn (cmp (pos a) (pos b)) (\u2191(pos a) < \u2191(pos b)) (pos a = pos b) (\u2191(pos b) < \u2191(pos a))\n[PROOFSTEP]\ndsimp [cmp]\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) (PosNum.cmp a b) \u2192\n Ordering.rec (\u2191a < \u2191b) (pos a = pos b) (\u2191b < \u2191a) (PosNum.cmp a b)\n[PROOFSTEP]\ncases PosNum.cmp a b <;> dsimp <;> [simp; exact congr_arg pos; simp [GT.gt]]\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) (PosNum.cmp a b) \u2192\n Ordering.rec (\u2191a < \u2191b) (pos a = pos b) (\u2191b < \u2191a) (PosNum.cmp a b)\n[PROOFSTEP]\ncases PosNum.cmp a b\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) Ordering.lt \u2192 Ordering.rec (\u2191a < \u2191b) (pos a = pos b) (\u2191b < \u2191a) Ordering.lt\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) Ordering.eq \u2192 Ordering.rec (\u2191a < \u2191b) (pos a = pos b) (\u2191b < \u2191a) Ordering.eq\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (\u2191a < \u2191b) (a = b) (\u2191b < \u2191a) Ordering.gt \u2192 Ordering.rec (\u2191a < \u2191b) (pos a = pos b) (\u2191b < \u2191a) Ordering.gt\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191a < \u2191b \u2192 \u2191a < \u2191b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 a = b \u2192 pos a = pos b\n[PROOFSTEP]\nexact congr_arg pos\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191b < \u2191a \u2192 \u2191b < \u2191a\n[PROOFSTEP]\nsimp [GT.gt]\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn (cmp (neg a) (neg b)) (\u2191(neg a) < \u2191(neg b)) (neg a = neg b) (\u2191(neg b) < \u2191(neg a))\n[PROOFSTEP]\nhave := PosNum.cmp_to_nat b a\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\nthis : Ordering.casesOn (PosNum.cmp b a) (\u2191b < \u2191a) (b = a) (\u2191a < \u2191b)\n\u22a2 Ordering.casesOn (cmp (neg a) (neg b)) (\u2191(neg a) < \u2191(neg b)) (neg a = neg b) (\u2191(neg b) < \u2191(neg a))\n[PROOFSTEP]\nrevert this\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.casesOn (PosNum.cmp b a) (\u2191b < \u2191a) (b = a) (\u2191a < \u2191b) \u2192\n Ordering.casesOn (cmp (neg a) (neg b)) (\u2191(neg a) < \u2191(neg b)) (neg a = neg b) (\u2191(neg b) < \u2191(neg a))\n[PROOFSTEP]\ndsimp [cmp]\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (\u2191b < \u2191a) (b = a) (\u2191a < \u2191b) (PosNum.cmp b a) \u2192\n Ordering.rec (-\u2191a < -\u2191b) (neg a = neg b) (-\u2191b < -\u2191a) (PosNum.cmp b a)\n[PROOFSTEP]\ncases PosNum.cmp b a <;> dsimp <;> [simp; simp (config := { contextual := true }); simp [GT.gt]]\n[GOAL]\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (\u2191b < \u2191a) (b = a) (\u2191a < \u2191b) (PosNum.cmp b a) \u2192\n Ordering.rec (-\u2191a < -\u2191b) (neg a = neg b) (-\u2191b < -\u2191a) (PosNum.cmp b a)\n[PROOFSTEP]\ncases PosNum.cmp b a\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (\u2191b < \u2191a) (b = a) (\u2191a < \u2191b) Ordering.lt \u2192\n Ordering.rec (-\u2191a < -\u2191b) (neg a = neg b) (-\u2191b < -\u2191a) Ordering.lt\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (\u2191b < \u2191a) (b = a) (\u2191a < \u2191b) Ordering.eq \u2192\n Ordering.rec (-\u2191a < -\u2191b) (neg a = neg b) (-\u2191b < -\u2191a) Ordering.eq\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 Ordering.rec (\u2191b < \u2191a) (b = a) (\u2191a < \u2191b) Ordering.gt \u2192\n Ordering.rec (-\u2191a < -\u2191b) (neg a = neg b) (-\u2191b < -\u2191a) Ordering.gt\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase lt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191b < \u2191a \u2192 -\u2191a < -\u2191b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase eq\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 b = a \u2192 neg a = neg b\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\ncase gt\n\u03b1 : Type u_1\na b : PosNum\n\u22a2 \u2191a < \u2191b \u2192 -\u2191b < -\u2191a\n[PROOFSTEP]\nsimp [GT.gt]\n[GOAL]\n\u03b1 : Type u_1\nm n : ZNum\nh : Ordering.casesOn Ordering.lt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 \u2191m < \u2191n \u2194 Ordering.lt = Ordering.lt\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u_1\nm n : ZNum\nh : \u2191m < \u2191n\n\u22a2 \u2191m < \u2191n \u2194 Ordering.lt = Ordering.lt\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\nm n : ZNum\nh : Ordering.casesOn Ordering.eq (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 \u2191m < \u2191n \u2194 Ordering.eq = Ordering.lt\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u_1\nm n : ZNum\nh : m = n\n\u22a2 \u2191m < \u2191n \u2194 Ordering.eq = Ordering.lt\n[PROOFSTEP]\nsimp [h, lt_irrefl]\n[GOAL]\n\u03b1 : Type u_1\nm n : ZNum\nh : Ordering.casesOn Ordering.gt (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m)\n\u22a2 \u2191m < \u2191n \u2194 Ordering.gt = Ordering.lt\n[PROOFSTEP]\nsimp [not_lt_of_gt h]\n[GOAL]\n\u03b1 : Type u_1\nm n : ZNum\n\u22a2 \u2191m \u2264 \u2191n \u2194 m \u2264 n\n[PROOFSTEP]\nrw [\u2190 not_lt]\n[GOAL]\n\u03b1 : Type u_1\nm n : ZNum\n\u22a2 \u00ac\u2191n < \u2191m \u2194 m \u2264 n\n[PROOFSTEP]\nexact not_congr lt_to_int\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\nm n : ZNum\n\u22a2 \u2191m < \u2191n \u2194 m < n\n[PROOFSTEP]\nrw [\u2190 cast_to_int m, \u2190 cast_to_int n, Int.cast_lt, lt_to_int]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\nm n : ZNum\n\u22a2 \u2191m \u2264 \u2191n \u2194 m \u2264 n\n[PROOFSTEP]\nrw [\u2190 not_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\nm n : ZNum\n\u22a2 \u00ac\u2191n < \u2191m \u2194 m \u2264 n\n[PROOFSTEP]\nexact not_congr cast_lt\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedRing \u03b1\nm n : ZNum\n\u22a2 \u2191m = \u2191n \u2194 m = n\n[PROOFSTEP]\nrw [\u2190 cast_to_int m, \u2190 cast_to_int n, Int.cast_inj (\u03b1 := \u03b1), to_int_inj]\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : ZNum), a \u2264 a\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b c : ZNum), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n[PROOFSTEP]\nintro a b c\n[GOAL]\n\u03b1 : Type u_1\na b c : ZNum\n\u22a2 a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\n\u03b1 : Type u_1\na b c : ZNum\n\u22a2 \u2191a \u2264 \u2191b \u2192 \u2191b \u2264 \u2191c \u2192 \u2191a \u2264 \u2191c\n[PROOFSTEP]\napply le_trans\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : ZNum), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n[PROOFSTEP]\nintro a b\n[GOAL]\n\u03b1 : Type u_1\na b : ZNum\n\u22a2 a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\n\u03b1 : Type u_1\na b : ZNum\n\u22a2 \u2191a < \u2191b \u2194 \u2191a \u2264 \u2191b \u2227 \u00ac\u2191b \u2264 \u2191a\n[PROOFSTEP]\napply lt_iff_le_not_le\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : ZNum), a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\nintro a b\n[GOAL]\n\u03b1 : Type u_1\na b : ZNum\n\u22a2 a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\n\u03b1 : Type u_1\na b : ZNum\n\u22a2 \u2191a \u2264 \u2191b \u2192 \u2191b \u2264 \u2191a \u2192 \u2191a = \u2191b\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : ZNum), a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\nintro a b\n[GOAL]\n\u03b1 : Type u_1\na b : ZNum\n\u22a2 a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\n\u03b1 : Type u_1\na b : ZNum\n\u22a2 \u2191a \u2264 \u2191b \u2228 \u2191b \u2264 \u2191a\n[PROOFSTEP]\napply le_total\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b c : ZNum), a + b + c = a + (b + c)\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a : ZNum), -a + a = 0\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : ZNum), a + b = b + a\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d : AddCommGroup ZNum := addCommGroup\n\u22a2 Num.toZNum (Num.ofNat' 0) = 0\n[PROOFSTEP]\nrw [Num.ofNat'_zero]\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d : AddCommGroup ZNum := addCommGroup\n\u22a2 Num.toZNum 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d : AddCommGroup ZNum := addCommGroup\nn : \u2115\n\u22a2 Num.toZNum (Num.ofNat' (n + 1)) = Num.toZNum (Num.ofNat' n) + 1\n[PROOFSTEP]\nrw [Num.ofNat'_succ, Num.add_one, Num.toZNum_succ, ZNum.add_one]\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : ZNum), a * b = b * a\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 (a b : ZNum), a \u2264 b \u2192 \u2200 (c : ZNum), c + a \u2264 c + b\n[PROOFSTEP]\nintro a b h c\n[GOAL]\n\u03b1 : Type u_1\na b : ZNum\nh : a \u2264 b\nc : ZNum\n\u22a2 c + a \u2264 c + b\n[PROOFSTEP]\nrevert h\n[GOAL]\n\u03b1 : Type u_1\na b c : ZNum\n\u22a2 a \u2264 b \u2192 c + a \u2264 c + b\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\n\u03b1 : Type u_1\na b c : ZNum\n\u22a2 \u2191a \u2264 \u2191b \u2192 \u2191c + \u2191a \u2264 \u2191c + \u2191b\n[PROOFSTEP]\nexact fun h => _root_.add_le_add_left h c\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d\u00b2 : LinearOrder ZNum := linearOrder\nsrc\u271d\u00b9 : AddCommGroup ZNum := addCommGroup\nsrc\u271d : AddMonoidWithOne ZNum := addMonoidWithOne\n\u22a2 \u2200 (a b c : ZNum), a * (b + c) = a * b + a * c\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d\u00b2 : LinearOrder ZNum := linearOrder\nsrc\u271d\u00b9 : AddCommGroup ZNum := addCommGroup\nsrc\u271d : AddMonoidWithOne ZNum := addMonoidWithOne\na\u271d b\u271d c\u271d : ZNum\n\u22a2 \u2191a\u271d * (\u2191b\u271d + \u2191c\u271d) = \u2191a\u271d * \u2191b\u271d + \u2191a\u271d * \u2191c\u271d\n[PROOFSTEP]\nsimp [mul_add]\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d\u00b2 : LinearOrder ZNum := linearOrder\nsrc\u271d\u00b9 : AddCommGroup ZNum := addCommGroup\nsrc\u271d : AddMonoidWithOne ZNum := addMonoidWithOne\n\u22a2 \u2200 (a b c : ZNum), (a + b) * c = a * c + b * c\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d\u00b2 : LinearOrder ZNum := linearOrder\nsrc\u271d\u00b9 : AddCommGroup ZNum := addCommGroup\nsrc\u271d : AddMonoidWithOne ZNum := addMonoidWithOne\na\u271d b\u271d c\u271d : ZNum\n\u22a2 \u2191c\u271d * (\u2191a\u271d + \u2191b\u271d) = \u2191a\u271d * \u2191c\u271d + \u2191b\u271d * \u2191c\u271d\n[PROOFSTEP]\nsimp [mul_add, _root_.mul_comm]\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d\u00b2 : LinearOrder ZNum := linearOrder\nsrc\u271d\u00b9 : AddCommGroup ZNum := addCommGroup\nsrc\u271d : AddMonoidWithOne ZNum := addMonoidWithOne\n\u22a2 \u2200 (a : ZNum), 0 * a = 0\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d\u00b2 : LinearOrder ZNum := linearOrder\nsrc\u271d\u00b9 : AddCommGroup ZNum := addCommGroup\nsrc\u271d : AddMonoidWithOne ZNum := addMonoidWithOne\n\u22a2 \u2200 (a : ZNum), a * 0 = 0\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d\u00b2 : LinearOrder ZNum := linearOrder\nsrc\u271d\u00b9 : AddCommGroup ZNum := addCommGroup\nsrc\u271d : AddMonoidWithOne ZNum := addMonoidWithOne\n\u22a2 \u2200 (a b c : ZNum), a * b * c = a * (b * c)\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d\u00b2 : LinearOrder ZNum := linearOrder\nsrc\u271d\u00b9 : AddCommGroup ZNum := addCommGroup\nsrc\u271d : AddMonoidWithOne ZNum := addMonoidWithOne\n\u22a2 \u2200 (a : ZNum), 1 * a = a\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d\u00b2 : LinearOrder ZNum := linearOrder\nsrc\u271d\u00b9 : AddCommGroup ZNum := addCommGroup\nsrc\u271d : AddMonoidWithOne ZNum := addMonoidWithOne\n\u22a2 \u2200 (a : ZNum), a * 1 = a\n[PROOFSTEP]\ntransfer\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d\u00b2 : LinearOrder ZNum := linearOrder\nsrc\u271d\u00b9 : AddCommGroup ZNum := addCommGroup\nsrc\u271d : AddMonoidWithOne ZNum := addMonoidWithOne\n\u22a2 0 \u2260 1\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d\u00b2 : LinearOrder ZNum := linearOrder\nsrc\u271d\u00b9 : AddCommGroup ZNum := addCommGroup\nsrc\u271d : AddMonoidWithOne ZNum := addMonoidWithOne\n\u22a2 0 \u2264 1\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d\u00b2 : LinearOrder ZNum := linearOrder\nsrc\u271d\u00b9 : AddCommGroup ZNum := addCommGroup\nsrc\u271d : AddMonoidWithOne ZNum := addMonoidWithOne\na b : ZNum\n\u22a2 0 < a \u2192 0 < b \u2192 0 < a * b\n[PROOFSTEP]\ntransfer_rw\n[GOAL]\n\u03b1 : Type u_1\nsrc\u271d\u00b2 : LinearOrder ZNum := linearOrder\nsrc\u271d\u00b9 : AddCommGroup ZNum := addCommGroup\nsrc\u271d : AddMonoidWithOne ZNum := addMonoidWithOne\na b : ZNum\n\u22a2 0 < \u2191a \u2192 0 < \u2191b \u2192 0 < \u2191a * \u2191b\n[PROOFSTEP]\napply mul_pos\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nm n : ZNum\n\u22a2 \u2191(m - n) = \u2191m - \u2191n\n[PROOFSTEP]\nsimp [sub_eq_neg_add]\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2191(-0) = -\u21910\n[PROOFSTEP]\nrw [Int.cast_neg, Int.cast_zero]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 ofInt' -[n+1] = \u2191-[n+1]\n[PROOFSTEP]\nshow Num.toZNumNeg (n + 1 : \u2115) = -(n + 1 : \u2115)\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 Num.toZNumNeg \u2191(n + 1) = -\u2191(n + 1)\n[PROOFSTEP]\nrw [\u2190 neg_inj, neg_neg, Nat.cast_succ, Num.add_one, Num.zneg_toZNumNeg, Num.toZNum_succ, Nat.cast_succ, ZNum.add_one]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 succ (Num.toZNum \u2191n) = succ \u2191n\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : ZNum\n\u22a2 \u2191\u2191n = n\n[PROOFSTEP]\nrw [\u2190 ofInt'_eq, of_to_int']\n[GOAL]\n\u03b1 : Type u_1\nn : \u2124\n\u22a2 \u2191\u21910 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nn : \u2124\n\u22a2 \u2200 (k : \u2124), 0 \u2264 k \u2192 \u2191\u2191k = k \u2192 \u2191\u2191(k + 1) = k + 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nn : \u2124\n\u22a2 \u2200 (k : \u2124), k \u2264 0 \u2192 \u2191\u2191k = k \u2192 \u2191\u2191(k - 1) = k - 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 Num.toZNumNeg \u2191n = -\u2191n\n[PROOFSTEP]\nrw [\u2190 of_nat_toZNum, Num.zneg_toZNum]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : \u2124\n\u22a2 \u2191\u2191n = \u2191n\n[PROOFSTEP]\nrw [\u2190 cast_to_int, to_of_int]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : AddGroupWithOne \u03b1\nn : \u2115\n\u22a2 \u2191\u2191n = \u2191n\n[PROOFSTEP]\nrw [\u2190 Int.cast_ofNat, of_int_cast, Int.cast_ofNat]\n[GOAL]\n\u03b1 : Type u_1\nm n : ZNum\nx\u271d : \u2191m \u2223 \u2191n\nk : \u2124\ne : \u2191n = \u2191m * k\n\u22a2 n = m * \u2191k\n[PROOFSTEP]\nrw [\u2190 of_to_int n, e]\n[GOAL]\n\u03b1 : Type u_1\nm n : ZNum\nx\u271d : \u2191m \u2223 \u2191n\nk : \u2124\ne : \u2191n = \u2191m * k\n\u22a2 \u2191(\u2191m * k) = m * \u2191k\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nm n : ZNum\nx\u271d : m \u2223 n\nk : ZNum\ne : n = m * k\n\u22a2 \u2191n = \u2191m * \u2191k\n[PROOFSTEP]\nsimp [e]\n[GOAL]\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\n\u22a2 \u2191(divModAux d q r).snd + \u2191d * \u2191(divModAux d q r).fst = \u2191n \u2227 \u2191(divModAux d q r).snd < \u2191d\n[PROOFSTEP]\nunfold divModAux\n[GOAL]\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\n\u22a2 \u2191(match Num.ofZNum' (Num.sub' r (Num.pos d)) with\n | some r' => (Num.bit1 q, r')\n | none => (Num.bit0 q, r)).snd +\n \u2191d *\n \u2191(match Num.ofZNum' (Num.sub' r (Num.pos d)) with\n | some r' => (Num.bit1 q, r')\n | none => (Num.bit0 q, r)).fst =\n \u2191n \u2227\n \u2191(match Num.ofZNum' (Num.sub' r (Num.pos d)) with\n | some r' => (Num.bit1 q, r')\n | none => (Num.bit0 q, r)).snd <\n \u2191d\n[PROOFSTEP]\nhave : \u2200 {r\u2082}, Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082 \u2194 (r : \u2115) = r\u2082 + d :=\n by\n intro r\u2082\n apply Num.mem_ofZNum'.trans\n rw [\u2190 ZNum.to_int_inj, Num.cast_toZNum, Num.cast_sub', sub_eq_iff_eq_add, \u2190 Int.coe_nat_inj']\n simp\n[GOAL]\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\n\u22a2 \u2200 {r\u2082 : Num}, Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082 \u2194 \u2191r = \u2191r\u2082 + \u2191d\n[PROOFSTEP]\nintro r\u2082\n[GOAL]\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\nr\u2082 : Num\n\u22a2 Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082 \u2194 \u2191r = \u2191r\u2082 + \u2191d\n[PROOFSTEP]\napply Num.mem_ofZNum'.trans\n[GOAL]\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\nr\u2082 : Num\n\u22a2 Num.sub' r (Num.pos d) = Num.toZNum r\u2082 \u2194 \u2191r = \u2191r\u2082 + \u2191d\n[PROOFSTEP]\nrw [\u2190 ZNum.to_int_inj, Num.cast_toZNum, Num.cast_sub', sub_eq_iff_eq_add, \u2190 Int.coe_nat_inj']\n[GOAL]\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\nr\u2082 : Num\n\u22a2 \u2191r = \u2191r\u2082 + \u2191(Num.pos d) \u2194 \u2191\u2191r = \u2191(\u2191r\u2082 + \u2191d)\n[PROOFSTEP]\nsimp\n[GOAL]\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\nthis : \u2200 {r\u2082 : Num}, Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082 \u2194 \u2191r = \u2191r\u2082 + \u2191d\n\u22a2 \u2191(match Num.ofZNum' (Num.sub' r (Num.pos d)) with\n | some r' => (Num.bit1 q, r')\n | none => (Num.bit0 q, r)).snd +\n \u2191d *\n \u2191(match Num.ofZNum' (Num.sub' r (Num.pos d)) with\n | some r' => (Num.bit1 q, r')\n | none => (Num.bit0 q, r)).fst =\n \u2191n \u2227\n \u2191(match Num.ofZNum' (Num.sub' r (Num.pos d)) with\n | some r' => (Num.bit1 q, r')\n | none => (Num.bit0 q, r)).snd <\n \u2191d\n[PROOFSTEP]\ncases' e : Num.ofZNum' (Num.sub' r (Num.pos d)) with r\u2082\n[GOAL]\ncase none\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\nthis : \u2200 {r\u2082 : Num}, Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082 \u2194 \u2191r = \u2191r\u2082 + \u2191d\ne : Num.ofZNum' (Num.sub' r (Num.pos d)) = none\n\u22a2 \u2191(match none with\n | some r' => (Num.bit1 q, r')\n | none => (Num.bit0 q, r)).snd +\n \u2191d *\n \u2191(match none with\n | some r' => (Num.bit1 q, r')\n | none => (Num.bit0 q, r)).fst =\n \u2191n \u2227\n \u2191(match none with\n | some r' => (Num.bit1 q, r')\n | none => (Num.bit0 q, r)).snd <\n \u2191d\n[PROOFSTEP]\nsimp [divModAux]\n[GOAL]\ncase some\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\nthis : \u2200 {r\u2082 : Num}, Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082 \u2194 \u2191r = \u2191r\u2082 + \u2191d\nr\u2082 : Num\ne : Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082\n\u22a2 \u2191(match some r\u2082 with\n | some r' => (Num.bit1 q, r')\n | none => (Num.bit0 q, r)).snd +\n \u2191d *\n \u2191(match some r\u2082 with\n | some r' => (Num.bit1 q, r')\n | none => (Num.bit0 q, r)).fst =\n \u2191n \u2227\n \u2191(match some r\u2082 with\n | some r' => (Num.bit1 q, r')\n | none => (Num.bit0 q, r)).snd <\n \u2191d\n[PROOFSTEP]\nsimp [divModAux]\n[GOAL]\ncase none\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\nthis : \u2200 {r\u2082 : Num}, Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082 \u2194 \u2191r = \u2191r\u2082 + \u2191d\ne : Num.ofZNum' (Num.sub' r (Num.pos d)) = none\n\u22a2 \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n \u2227 \u2191r < \u2191d\n[PROOFSTEP]\nrefine' \u27e8h\u2081, lt_of_not_ge fun h => _\u27e9\n[GOAL]\ncase none\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\nthis : \u2200 {r\u2082 : Num}, Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082 \u2194 \u2191r = \u2191r\u2082 + \u2191d\ne : Num.ofZNum' (Num.sub' r (Num.pos d)) = none\nh : \u2191r \u2265 \u2191d\n\u22a2 False\n[PROOFSTEP]\ncases' Nat.le.dest h with r\u2082 e'\n[GOAL]\ncase none.intro\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\nthis : \u2200 {r\u2082 : Num}, Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082 \u2194 \u2191r = \u2191r\u2082 + \u2191d\ne : Num.ofZNum' (Num.sub' r (Num.pos d)) = none\nh : \u2191r \u2265 \u2191d\nr\u2082 : \u2115\ne' : \u2191d + r\u2082 = \u2191r\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 Num.to_of_nat r\u2082, add_comm] at e' \n[GOAL]\ncase none.intro\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\nthis : \u2200 {r\u2082 : Num}, Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082 \u2194 \u2191r = \u2191r\u2082 + \u2191d\ne : Num.ofZNum' (Num.sub' r (Num.pos d)) = none\nh : \u2191r \u2265 \u2191d\nr\u2082 : \u2115\ne' : \u2191\u2191r\u2082 + \u2191d = \u2191r\n\u22a2 False\n[PROOFSTEP]\ncases e.symm.trans (this.2 e'.symm)\n[GOAL]\ncase some\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\nthis : \u2200 {r\u2082 : Num}, Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082 \u2194 \u2191r = \u2191r\u2082 + \u2191d\nr\u2082 : Num\ne : Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082\n\u22a2 \u2191r\u2082 + \u2191d * _root_.bit1 \u2191q = \u2191n \u2227 \u2191r\u2082 < \u2191d\n[PROOFSTEP]\nhave := this.1 e\n[GOAL]\ncase some\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\nthis\u271d : \u2200 {r\u2082 : Num}, Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082 \u2194 \u2191r = \u2191r\u2082 + \u2191d\nr\u2082 : Num\ne : Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082\nthis : \u2191r = \u2191r\u2082 + \u2191d\n\u22a2 \u2191r\u2082 + \u2191d * _root_.bit1 \u2191q = \u2191n \u2227 \u2191r\u2082 < \u2191d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase some.left\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\nthis\u271d : \u2200 {r\u2082 : Num}, Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082 \u2194 \u2191r = \u2191r\u2082 + \u2191d\nr\u2082 : Num\ne : Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082\nthis : \u2191r = \u2191r\u2082 + \u2191d\n\u22a2 \u2191r\u2082 + \u2191d * _root_.bit1 \u2191q = \u2191n\n[PROOFSTEP]\nrwa [_root_.bit1, add_comm _ 1, mul_add, mul_one, \u2190 add_assoc, \u2190 this]\n[GOAL]\ncase some.right\nn d : PosNum\nq r : Num\nh\u2081 : \u2191r + \u2191d * _root_.bit0 \u2191q = \u2191n\nh\u2082 : \u2191r < 2 * \u2191d\nthis\u271d : \u2200 {r\u2082 : Num}, Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082 \u2194 \u2191r = \u2191r\u2082 + \u2191d\nr\u2082 : Num\ne : Num.ofZNum' (Num.sub' r (Num.pos d)) = some r\u2082\nthis : \u2191r = \u2191r\u2082 + \u2191d\n\u22a2 \u2191r\u2082 < \u2191d\n[PROOFSTEP]\nrwa [this, two_mul, add_lt_add_iff_right] at h\u2082 \n[GOAL]\nd n : PosNum\n\u22a2 \u2191n / \u2191d = \u2191(divMod d n).fst \u2227 \u2191n % \u2191d = \u2191(divMod d n).snd\n[PROOFSTEP]\nrw [Nat.div_mod_unique (PosNum.cast_pos _)]\n[GOAL]\nd n : PosNum\n\u22a2 \u2191(divMod d n).snd + \u2191d * \u2191(divMod d n).fst = \u2191n \u2227 \u2191(divMod d n).snd < \u2191d\n[PROOFSTEP]\ninduction' n with n IH n IH\n[GOAL]\ncase one\nd : PosNum\n\u22a2 \u2191(divMod d one).snd + \u2191d * \u2191(divMod d one).fst = \u2191one \u2227 \u2191(divMod d one).snd < \u2191d\n[PROOFSTEP]\nexact divMod_to_nat_aux (by simp) (Nat.mul_le_mul_left 2 (PosNum.cast_pos d : (0 : \u2115) < d))\n[GOAL]\nd : PosNum\n\u22a2 \u21911 + \u2191d * _root_.bit0 \u21910 = \u2191one\n[PROOFSTEP]\nsimp\n[GOAL]\ncase bit1\nd n : PosNum\nIH : \u2191(divMod d n).snd + \u2191d * \u2191(divMod d n).fst = \u2191n \u2227 \u2191(divMod d n).snd < \u2191d\n\u22a2 \u2191(divMod d (bit1 n)).snd + \u2191d * \u2191(divMod d (bit1 n)).fst = \u2191(bit1 n) \u2227 \u2191(divMod d (bit1 n)).snd < \u2191d\n[PROOFSTEP]\nunfold divMod\n[GOAL]\ncase bit1\nd n : PosNum\nIH : \u2191(divMod d n).snd + \u2191d * \u2191(divMod d n).fst = \u2191n \u2227 \u2191(divMod d n).snd < \u2191d\n\u22a2 \u2191(match divMod d n with\n | (q, r\u2081) => divModAux d q (Num.bit1 r\u2081)).snd +\n \u2191d *\n \u2191(match divMod d n with\n | (q, r\u2081) => divModAux d q (Num.bit1 r\u2081)).fst =\n \u2191(bit1 n) \u2227\n \u2191(match divMod d n with\n | (q, r\u2081) => divModAux d q (Num.bit1 r\u2081)).snd <\n \u2191d\n[PROOFSTEP]\nrevert IH\n[GOAL]\ncase bit1\nd n : PosNum\n\u22a2 \u2191(divMod d n).snd + \u2191d * \u2191(divMod d n).fst = \u2191n \u2227 \u2191(divMod d n).snd < \u2191d \u2192\n \u2191(match divMod d n with\n | (q, r\u2081) => divModAux d q (Num.bit1 r\u2081)).snd +\n \u2191d *\n \u2191(match divMod d n with\n | (q, r\u2081) => divModAux d q (Num.bit1 r\u2081)).fst =\n \u2191(bit1 n) \u2227\n \u2191(match divMod d n with\n | (q, r\u2081) => divModAux d q (Num.bit1 r\u2081)).snd <\n \u2191d\n[PROOFSTEP]\ncases' divMod d n with q r\n[GOAL]\ncase bit1.mk\nd n : PosNum\nq r : Num\n\u22a2 \u2191(q, r).snd + \u2191d * \u2191(q, r).fst = \u2191n \u2227 \u2191(q, r).snd < \u2191d \u2192\n \u2191(match (q, r) with\n | (q, r\u2081) => divModAux d q (Num.bit1 r\u2081)).snd +\n \u2191d *\n \u2191(match (q, r) with\n | (q, r\u2081) => divModAux d q (Num.bit1 r\u2081)).fst =\n \u2191(bit1 n) \u2227\n \u2191(match (q, r) with\n | (q, r\u2081) => divModAux d q (Num.bit1 r\u2081)).snd <\n \u2191d\n[PROOFSTEP]\nintro IH\n[GOAL]\ncase bit1.mk\nd n : PosNum\nq r : Num\nIH : \u2191(q, r).snd + \u2191d * \u2191(q, r).fst = \u2191n \u2227 \u2191(q, r).snd < \u2191d\n\u22a2 \u2191(match (q, r) with\n | (q, r\u2081) => divModAux d q (Num.bit1 r\u2081)).snd +\n \u2191d *\n \u2191(match (q, r) with\n | (q, r\u2081) => divModAux d q (Num.bit1 r\u2081)).fst =\n \u2191(bit1 n) \u2227\n \u2191(match (q, r) with\n | (q, r\u2081) => divModAux d q (Num.bit1 r\u2081)).snd <\n \u2191d\n[PROOFSTEP]\nsimp only [divMod] at IH \u22a2\n[GOAL]\ncase bit1.mk\nd n : PosNum\nq r : Num\nIH : \u2191r + \u2191d * \u2191q = \u2191n \u2227 \u2191r < \u2191d\n\u22a2 \u2191(divModAux d q (Num.bit1 r)).snd + \u2191d * \u2191(divModAux d q (Num.bit1 r)).fst = \u2191(bit1 n) \u2227\n \u2191(divModAux d q (Num.bit1 r)).snd < \u2191d\n[PROOFSTEP]\napply divMod_to_nat_aux\n[GOAL]\ncase bit1.mk.h\u2081\nd n : PosNum\nq r : Num\nIH : \u2191r + \u2191d * \u2191q = \u2191n \u2227 \u2191r < \u2191d\n\u22a2 \u2191(Num.bit1 r) + \u2191d * _root_.bit0 \u2191q = \u2191(bit1 n)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase bit1.mk.h\u2082\nd n : PosNum\nq r : Num\nIH : \u2191r + \u2191d * \u2191q = \u2191n \u2227 \u2191r < \u2191d\n\u22a2 \u2191(Num.bit1 r) < 2 * \u2191d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase bit1.mk.h\u2081\nd n : PosNum\nq r : Num\nIH : \u2191r + \u2191d * \u2191q = \u2191n \u2227 \u2191r < \u2191d\n\u22a2 _root_.bit1 \u2191r + \u2191d * _root_.bit0 \u2191q = _root_.bit1 \u2191n\n[PROOFSTEP]\nrw [_root_.bit1, _root_.bit1, add_right_comm, bit0_eq_two_mul (n : \u2115), \u2190 IH.1, mul_add, \u2190 bit0_eq_two_mul,\n mul_left_comm, \u2190 bit0_eq_two_mul]\n[GOAL]\ncase bit1.mk.h\u2082\nd n : PosNum\nq r : Num\nIH : \u2191r + \u2191d * \u2191q = \u2191n \u2227 \u2191r < \u2191d\n\u22a2 _root_.bit1 \u2191r < 2 * \u2191d\n[PROOFSTEP]\nrw [\u2190 bit0_eq_two_mul]\n[GOAL]\ncase bit1.mk.h\u2082\nd n : PosNum\nq r : Num\nIH : \u2191r + \u2191d * \u2191q = \u2191n \u2227 \u2191r < \u2191d\n\u22a2 _root_.bit1 \u2191r < _root_.bit0 \u2191d\n[PROOFSTEP]\nexact Nat.bit1_lt_bit0 IH.2\n[GOAL]\ncase bit0\nd n : PosNum\nIH : \u2191(divMod d n).snd + \u2191d * \u2191(divMod d n).fst = \u2191n \u2227 \u2191(divMod d n).snd < \u2191d\n\u22a2 \u2191(divMod d (bit0 n)).snd + \u2191d * \u2191(divMod d (bit0 n)).fst = \u2191(bit0 n) \u2227 \u2191(divMod d (bit0 n)).snd < \u2191d\n[PROOFSTEP]\nunfold divMod\n[GOAL]\ncase bit0\nd n : PosNum\nIH : \u2191(divMod d n).snd + \u2191d * \u2191(divMod d n).fst = \u2191n \u2227 \u2191(divMod d n).snd < \u2191d\n\u22a2 \u2191(match divMod d n with\n | (q, r\u2081) => divModAux d q (Num.bit0 r\u2081)).snd +\n \u2191d *\n \u2191(match divMod d n with\n | (q, r\u2081) => divModAux d q (Num.bit0 r\u2081)).fst =\n \u2191(bit0 n) \u2227\n \u2191(match divMod d n with\n | (q, r\u2081) => divModAux d q (Num.bit0 r\u2081)).snd <\n \u2191d\n[PROOFSTEP]\nrevert IH\n[GOAL]\ncase bit0\nd n : PosNum\n\u22a2 \u2191(divMod d n).snd + \u2191d * \u2191(divMod d n).fst = \u2191n \u2227 \u2191(divMod d n).snd < \u2191d \u2192\n \u2191(match divMod d n with\n | (q, r\u2081) => divModAux d q (Num.bit0 r\u2081)).snd +\n \u2191d *\n \u2191(match divMod d n with\n | (q, r\u2081) => divModAux d q (Num.bit0 r\u2081)).fst =\n \u2191(bit0 n) \u2227\n \u2191(match divMod d n with\n | (q, r\u2081) => divModAux d q (Num.bit0 r\u2081)).snd <\n \u2191d\n[PROOFSTEP]\ncases' divMod d n with q r\n[GOAL]\ncase bit0.mk\nd n : PosNum\nq r : Num\n\u22a2 \u2191(q, r).snd + \u2191d * \u2191(q, r).fst = \u2191n \u2227 \u2191(q, r).snd < \u2191d \u2192\n \u2191(match (q, r) with\n | (q, r\u2081) => divModAux d q (Num.bit0 r\u2081)).snd +\n \u2191d *\n \u2191(match (q, r) with\n | (q, r\u2081) => divModAux d q (Num.bit0 r\u2081)).fst =\n \u2191(bit0 n) \u2227\n \u2191(match (q, r) with\n | (q, r\u2081) => divModAux d q (Num.bit0 r\u2081)).snd <\n \u2191d\n[PROOFSTEP]\nintro IH\n[GOAL]\ncase bit0.mk\nd n : PosNum\nq r : Num\nIH : \u2191(q, r).snd + \u2191d * \u2191(q, r).fst = \u2191n \u2227 \u2191(q, r).snd < \u2191d\n\u22a2 \u2191(match (q, r) with\n | (q, r\u2081) => divModAux d q (Num.bit0 r\u2081)).snd +\n \u2191d *\n \u2191(match (q, r) with\n | (q, r\u2081) => divModAux d q (Num.bit0 r\u2081)).fst =\n \u2191(bit0 n) \u2227\n \u2191(match (q, r) with\n | (q, r\u2081) => divModAux d q (Num.bit0 r\u2081)).snd <\n \u2191d\n[PROOFSTEP]\nsimp only [divMod] at IH \u22a2\n[GOAL]\ncase bit0.mk\nd n : PosNum\nq r : Num\nIH : \u2191r + \u2191d * \u2191q = \u2191n \u2227 \u2191r < \u2191d\n\u22a2 \u2191(divModAux d q (Num.bit0 r)).snd + \u2191d * \u2191(divModAux d q (Num.bit0 r)).fst = \u2191(bit0 n) \u2227\n \u2191(divModAux d q (Num.bit0 r)).snd < \u2191d\n[PROOFSTEP]\napply divMod_to_nat_aux\n[GOAL]\ncase bit0.mk.h\u2081\nd n : PosNum\nq r : Num\nIH : \u2191r + \u2191d * \u2191q = \u2191n \u2227 \u2191r < \u2191d\n\u22a2 \u2191(Num.bit0 r) + \u2191d * _root_.bit0 \u2191q = \u2191(bit0 n)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase bit0.mk.h\u2082\nd n : PosNum\nq r : Num\nIH : \u2191r + \u2191d * \u2191q = \u2191n \u2227 \u2191r < \u2191d\n\u22a2 \u2191(Num.bit0 r) < 2 * \u2191d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase bit0.mk.h\u2081\nd n : PosNum\nq r : Num\nIH : \u2191r + \u2191d * \u2191q = \u2191n \u2227 \u2191r < \u2191d\n\u22a2 _root_.bit0 \u2191r + \u2191d * _root_.bit0 \u2191q = _root_.bit0 \u2191n\n[PROOFSTEP]\nrw [bit0_eq_two_mul (n : \u2115), \u2190 IH.1, mul_add, \u2190 bit0_eq_two_mul, mul_left_comm, \u2190 bit0_eq_two_mul]\n[GOAL]\ncase bit0.mk.h\u2082\nd n : PosNum\nq r : Num\nIH : \u2191r + \u2191d * \u2191q = \u2191n \u2227 \u2191r < \u2191d\n\u22a2 _root_.bit0 \u2191r < 2 * \u2191d\n[PROOFSTEP]\nrw [\u2190 bit0_eq_two_mul]\n[GOAL]\ncase bit0.mk.h\u2082\nd n : PosNum\nq r : Num\nIH : \u2191r + \u2191d * \u2191q = \u2191n \u2227 \u2191r < \u2191d\n\u22a2 _root_.bit0 \u2191r < _root_.bit0 \u2191d\n[PROOFSTEP]\nexact Nat.bit0_lt IH.2\n[GOAL]\nn : Num\n\u22a2 div n 0 = 0\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u22a2 div zero 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\na\u271d : PosNum\n\u22a2 div (pos a\u271d) 0 = 0\n[PROOFSTEP]\nsimp [Num.div]\n[GOAL]\n\u22a2 \u2191(0 / 0) = \u21910 / \u21910\n[PROOFSTEP]\nsimp\n[GOAL]\nn : Num\n\u22a2 mod n 0 = n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u22a2 mod zero 0 = zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\na\u271d : PosNum\n\u22a2 mod (pos a\u271d) 0 = pos a\u271d\n[PROOFSTEP]\nsimp [Num.mod]\n[GOAL]\n\u22a2 \u2191(0 % 0) = \u21910 % \u21910\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 \u2191(gcdAux (Nat.succ n) (pos a) b) = Nat.gcd \u2191(pos a) \u2191b\n[PROOFSTEP]\nsimp [gcdAux]\n[GOAL]\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 \u2191(gcdAux n (b % pos a) (pos a)) = Nat.gcd \u2191a \u2191b\n[PROOFSTEP]\nrw [Nat.gcd_rec, gcd_to_nat_aux, mod_to_nat]\n[GOAL]\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 Nat.gcd (\u2191b % \u2191(pos a)) \u2191(pos a) = Nat.gcd (\u2191b % \u2191a) \u2191a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 b % pos a \u2264 pos a\n[PROOFSTEP]\nrw [\u2190 le_to_nat, mod_to_nat]\n[GOAL]\ncase a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 \u2191b % \u2191(pos a) \u2264 \u2191(pos a)\n[PROOFSTEP]\nexact le_of_lt (Nat.mod_lt _ (PosNum.cast_pos _))\n[GOAL]\ncase a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : natSize (pos a * b) \u2264 Nat.succ n\n\u22a2 natSize (b % pos a * pos a) \u2264 n\n[PROOFSTEP]\nrw [natSize_to_nat, mul_to_nat, Nat.size_le] at h \u22a2\n[GOAL]\ncase a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * \u2191b < 2 ^ Nat.succ n\n\u22a2 \u2191(b % pos a) * \u2191(pos a) < 2 ^ n\n[PROOFSTEP]\nrw [mod_to_nat, mul_comm]\n[GOAL]\ncase a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * \u2191b < 2 ^ Nat.succ n\n\u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) < 2 ^ n\n[PROOFSTEP]\nrw [pow_succ', \u2190 Nat.mod_add_div b (pos a)] at h \n[GOAL]\ncase a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) < 2 ^ n\n[PROOFSTEP]\nrefine' lt_of_mul_lt_mul_right (lt_of_le_of_lt _ h) (Nat.zero_le 2)\n[GOAL]\ncase a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) * 2 \u2264 \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a)))\n[PROOFSTEP]\nrw [mul_two, mul_add]\n[GOAL]\ncase a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) + \u2191(pos a) * (\u2191b % \u2191(pos a)) \u2264\n \u2191(pos a) * (\u2191b % \u2191(pos a)) + \u2191(pos a) * (\u2191(pos a) * (\u2191b / \u2191(pos a)))\n[PROOFSTEP]\nrefine' add_le_add_left (Nat.mul_le_mul_left _ (le_trans (le_of_lt (Nat.mod_lt _ (PosNum.cast_pos _))) _)) _\n[GOAL]\ncase a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 \u2191a \u2264 \u2191(pos a) * (\u2191b / \u2191(pos a))\n[PROOFSTEP]\nsuffices : 1 \u2264 _\n[GOAL]\ncase a\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\nthis : 1 \u2264 ?m.1208915\n\u22a2 \u2191a \u2264 \u2191(pos a) * (\u2191b / \u2191(pos a))\ncase this\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 1 \u2264 ?m.1208915\n[PROOFSTEP]\nsimpa using Nat.mul_le_mul_left (pos a) this\n[GOAL]\ncase this\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 1 \u2264 \u2191b / \u2191a\n[PROOFSTEP]\nrw [Nat.le_div_iff_mul_le a.cast_pos, one_mul]\n[GOAL]\ncase this\nn : \u2115\na : PosNum\nb : Num\nab : pos a \u2264 b\nh : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2\n\u22a2 \u2191a \u2264 \u2191b\n[PROOFSTEP]\nexact le_to_nat.2 ab\n[GOAL]\n\u22a2 \u2200 (a b : Num), \u2191(gcd a b) = Nat.gcd \u2191a \u2191b\n[PROOFSTEP]\nhave : \u2200 a b : Num, (a * b).natSize \u2264 a.natSize + b.natSize :=\n by\n intros\n simp [natSize_to_nat]\n rw [Nat.size_le, pow_add]\n exact mul_lt_mul'' (Nat.lt_size_self _) (Nat.lt_size_self _) (Nat.zero_le _) (Nat.zero_le _)\n[GOAL]\n\u22a2 \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\n[PROOFSTEP]\nintros\n[GOAL]\na\u271d b\u271d : Num\n\u22a2 natSize (a\u271d * b\u271d) \u2264 natSize a\u271d + natSize b\u271d\n[PROOFSTEP]\nsimp [natSize_to_nat]\n[GOAL]\na\u271d b\u271d : Num\n\u22a2 Nat.size (\u2191a\u271d * \u2191b\u271d) \u2264 Nat.size \u2191a\u271d + Nat.size \u2191b\u271d\n[PROOFSTEP]\nrw [Nat.size_le, pow_add]\n[GOAL]\na\u271d b\u271d : Num\n\u22a2 \u2191a\u271d * \u2191b\u271d < 2 ^ Nat.size \u2191a\u271d * 2 ^ Nat.size \u2191b\u271d\n[PROOFSTEP]\nexact mul_lt_mul'' (Nat.lt_size_self _) (Nat.lt_size_self _) (Nat.zero_le _) (Nat.zero_le _)\n[GOAL]\nthis : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\n\u22a2 \u2200 (a b : Num), \u2191(gcd a b) = Nat.gcd \u2191a \u2191b\n[PROOFSTEP]\nintros\n[GOAL]\nthis : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\na\u271d b\u271d : Num\n\u22a2 \u2191(gcd a\u271d b\u271d) = Nat.gcd \u2191a\u271d \u2191b\u271d\n[PROOFSTEP]\nunfold gcd\n[GOAL]\nthis : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\na\u271d b\u271d : Num\n\u22a2 \u2191(if a\u271d \u2264 b\u271d then gcdAux (natSize a\u271d + natSize b\u271d) a\u271d b\u271d else gcdAux (natSize b\u271d + natSize a\u271d) b\u271d a\u271d) =\n Nat.gcd \u2191a\u271d \u2191b\u271d\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nthis : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\na\u271d b\u271d : Num\nh : a\u271d \u2264 b\u271d\n\u22a2 \u2191(gcdAux (natSize a\u271d + natSize b\u271d) a\u271d b\u271d) = Nat.gcd \u2191a\u271d \u2191b\u271d\n[PROOFSTEP]\nexact gcd_to_nat_aux h (this _ _)\n[GOAL]\ncase neg\nthis : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\na\u271d b\u271d : Num\nh : \u00aca\u271d \u2264 b\u271d\n\u22a2 \u2191(gcdAux (natSize b\u271d + natSize a\u271d) b\u271d a\u271d) = Nat.gcd \u2191a\u271d \u2191b\u271d\n[PROOFSTEP]\nrw [Nat.gcd_comm]\n[GOAL]\ncase neg\nthis : \u2200 (a b : Num), natSize (a * b) \u2264 natSize a + natSize b\na\u271d b\u271d : Num\nh : \u00aca\u271d \u2264 b\u271d\n\u22a2 \u2191(gcdAux (natSize b\u271d + natSize a\u271d) b\u271d a\u271d) = Nat.gcd \u2191b\u271d \u2191a\u271d\n[PROOFSTEP]\nexact gcd_to_nat_aux (le_of_not_le h) (this _ _)\n[GOAL]\nm n : Num\n\u22a2 m \u2223 n \u2194 n % m = 0\n[PROOFSTEP]\nrw [\u2190 dvd_to_nat, Nat.dvd_iff_mod_eq_zero, \u2190 to_nat_inj, mod_to_nat]\n[GOAL]\nm n : Num\n\u22a2 \u2191n % \u2191m = 0 \u2194 \u2191n % \u2191m = \u21910\n[PROOFSTEP]\nrfl\n[GOAL]\nn : ZNum\n\u22a2 div n 0 = 0\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u22a2 div zero 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\na\u271d : PosNum\n\u22a2 div (pos a\u271d) 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\na\u271d : PosNum\n\u22a2 div (neg a\u271d) 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u2191(0 / 0) = \u21910 / \u21910\n[PROOFSTEP]\nsimp [Int.ediv_zero]\n[GOAL]\nn d : PosNum\n\u22a2 \u2191(PosNum.div' n d) = \u2191(pos n) / \u2191(pos d)\n[PROOFSTEP]\nrw [\u2190 Num.to_nat_to_int]\n[GOAL]\nn d : PosNum\n\u22a2 \u2191\u2191(PosNum.div' n d) = \u2191(pos n) / \u2191(pos d)\n[PROOFSTEP]\nsimp\n[GOAL]\nn d : PosNum\n\u22a2 -\u2191(PosNum.div' n d) = \u2191(pos n) / \u2191(neg d)\n[PROOFSTEP]\nrw [\u2190 Num.to_nat_to_int]\n[GOAL]\nn d : PosNum\n\u22a2 -\u2191\u2191(PosNum.div' n d) = \u2191(pos n) / \u2191(neg d)\n[PROOFSTEP]\nsimp\n[GOAL]\nn d : PosNum\n\u22a2 -\u2191(Num.succ' (PosNum.pred' n / Num.pos d)) = -\u2191n / \u2191d\n[PROOFSTEP]\nrw [n.to_int_eq_succ_pred, d.to_int_eq_succ_pred, \u2190 PosNum.to_nat_to_int, Num.succ'_to_nat, Num.div_to_nat]\n[GOAL]\nn d : PosNum\n\u22a2 -\u2191(\u2191(PosNum.pred' n) / \u2191(Num.pos d) + 1) = -(\u2191\u2191(PosNum.pred' n) + 1) / (\u2191\u2191(PosNum.pred' d) + 1)\n[PROOFSTEP]\nchange -[n.pred' / \u2191d+1] = -[n.pred' / (d.pred' + 1)+1]\n[GOAL]\nn d : PosNum\n\u22a2 -[\u2191(PosNum.pred' n) / \u2191d+1] = -[\u2191(PosNum.pred' n) / (\u2191(PosNum.pred' d) + 1)+1]\n[PROOFSTEP]\nrw [d.to_nat_eq_succ_pred]\n[GOAL]\nn d : PosNum\n\u22a2 \u2191(Num.succ' (PosNum.pred' n / Num.pos d)) = -\u2191n / -\u2191d\n[PROOFSTEP]\nrw [n.to_int_eq_succ_pred, d.to_int_eq_succ_pred, \u2190 PosNum.to_nat_to_int, Num.succ'_to_nat, Num.div_to_nat]\n[GOAL]\nn d : PosNum\n\u22a2 \u2191(\u2191(PosNum.pred' n) / \u2191(Num.pos d) + 1) = -(\u2191\u2191(PosNum.pred' n) + 1) / -(\u2191\u2191(PosNum.pred' d) + 1)\n[PROOFSTEP]\nchange (Nat.succ (_ / d) : \u2124) = Nat.succ (n.pred' / (d.pred' + 1))\n[GOAL]\nn d : PosNum\n\u22a2 \u2191(Nat.succ (\u2191(PosNum.pred' n) / \u2191d)) = \u2191(Nat.succ (\u2191(PosNum.pred' n) / (\u2191(PosNum.pred' d) + 1)))\n[PROOFSTEP]\nrw [d.to_nat_eq_succ_pred]\n[GOAL]\nn : PosNum\nd : ZNum\n\u22a2 \u2191(Num.pos n % abs d) = \u2191(pos n) % \u2191d\n[PROOFSTEP]\nrw [\u2190 Num.to_nat_to_int, cast_pos, Num.mod_to_nat, \u2190 PosNum.to_nat_to_int, abs_to_nat]\n[GOAL]\nn : PosNum\nd : ZNum\n\u22a2 \u2191(\u2191(Num.pos n) % natAbs \u2191d) = \u2191\u2191n % \u2191d\n[PROOFSTEP]\nrfl\n[GOAL]\nn : PosNum\nd : ZNum\n\u22a2 \u2191(abs d) - \u2191(Num.succ (PosNum.pred' n % abs d)) = \u2191(neg n) % \u2191d\n[PROOFSTEP]\nrw [\u2190 Num.to_nat_to_int, cast_neg, \u2190 Num.to_nat_to_int, Num.succ_to_nat, Num.mod_to_nat, abs_to_nat, \u2190\n Int.subNatNat_eq_coe, n.to_int_eq_succ_pred]\n[GOAL]\nn : PosNum\nd : ZNum\n\u22a2 subNatNat (natAbs \u2191d) (\u2191(PosNum.pred' n) % natAbs \u2191d + 1) = -(\u2191\u2191(PosNum.pred' n) + 1) % \u2191d\n[PROOFSTEP]\nrfl\n[GOAL]\na b : ZNum\n\u22a2 Nat.gcd \u2191(abs a) \u2191(abs b) = Int.gcd \u2191a \u2191b\n[PROOFSTEP]\nsimp\n[GOAL]\na b : ZNum\n\u22a2 Nat.gcd (natAbs \u2191a) (natAbs \u2191b) = Int.gcd \u2191a \u2191b\n[PROOFSTEP]\nrfl\n[GOAL]\nm n : ZNum\n\u22a2 m \u2223 n \u2194 n % m = 0\n[PROOFSTEP]\nrw [\u2190 dvd_to_int, Int.dvd_iff_emod_eq_zero, \u2190 to_int_inj, mod_to_int]\n[GOAL]\nm n : ZNum\n\u22a2 \u2191n % \u2191m = 0 \u2194 \u2191n % \u2191m = \u21910\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Data.Num.Lemmas", "llama_tokens": 220338, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191214879992, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.422882505390868}} {"text": "[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : EquicontinuousAt (FunLike.coe \u2218 F) 1\n\u22a2 Equicontinuous (FunLike.coe \u2218 F)\n[PROOFSTEP]\nrw [equicontinuous_iff_continuous]\n[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : EquicontinuousAt (FunLike.coe \u2218 F) 1\n\u22a2 Continuous (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F))\n[PROOFSTEP]\nrw [equicontinuousAt_iff_continuousAt] at hf \n[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\n\u22a2 Continuous (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F))\n[PROOFSTEP]\nlet \u03c6 : G \u2192* (\u03b9 \u2192\u1d64 M) :=\n { toFun := swap ((\u2191) \u2218 F)\n map_one' := by dsimp [UniformFun]; ext; exact map_one _\n map_mul' := fun a b => by dsimp [UniformFun]; ext; exact map_mul _ _ _ }\n[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\n\u22a2 swap (FunLike.coe \u2218 F) 1 = 1\n[PROOFSTEP]\ndsimp [UniformFun]\n[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\n\u22a2 swap (FunLike.coe \u2218 F) 1 = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\nx\u271d : \u03b9\n\u22a2 swap (FunLike.coe \u2218 F) 1 x\u271d = OfNat.ofNat 1 x\u271d\n[PROOFSTEP]\nexact map_one _\n[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\na b : G\n\u22a2 OneHom.toFun { toFun := swap (FunLike.coe \u2218 F), map_one' := (_ : swap (FunLike.coe \u2218 F) 1 = 1) } (a * b) =\n OneHom.toFun { toFun := swap (FunLike.coe \u2218 F), map_one' := (_ : swap (FunLike.coe \u2218 F) 1 = 1) } a *\n OneHom.toFun { toFun := swap (FunLike.coe \u2218 F), map_one' := (_ : swap (FunLike.coe \u2218 F) 1 = 1) } b\n[PROOFSTEP]\ndsimp [UniformFun]\n[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\na b : G\n\u22a2 swap (FunLike.coe \u2218 F) (a * b) = swap (FunLike.coe \u2218 F) a * swap (FunLike.coe \u2218 F) b\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\na b : G\nx\u271d : \u03b9\n\u22a2 swap (FunLike.coe \u2218 F) (a * b) x\u271d = (swap (FunLike.coe \u2218 F) a * swap (FunLike.coe \u2218 F) b) x\u271d\n[PROOFSTEP]\nexact map_mul _ _ _\n[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\n\u03c6 : G \u2192* \u03b9 \u2192\u1d64 M :=\n { toOneHom := { toFun := swap (FunLike.coe \u2218 F), map_one' := (_ : swap (FunLike.coe \u2218 F) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (a b : G),\n OneHom.toFun { toFun := swap (FunLike.coe \u2218 F), map_one' := (_ : swap (FunLike.coe \u2218 F) 1 = 1) } (a * b) =\n OneHom.toFun { toFun := swap (FunLike.coe \u2218 F), map_one' := (_ : swap (FunLike.coe \u2218 F) 1 = 1) } a *\n OneHom.toFun { toFun := swap (FunLike.coe \u2218 F), map_one' := (_ : swap (FunLike.coe \u2218 F) 1 = 1) } b) }\n\u22a2 Continuous (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F))\n[PROOFSTEP]\nexact continuous_of_continuousAt_one \u03c6 hf\n[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : UniformSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : UniformGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : EquicontinuousAt (FunLike.coe \u2218 F) 1\n\u22a2 UniformEquicontinuous (FunLike.coe \u2218 F)\n[PROOFSTEP]\nrw [uniformEquicontinuous_iff_uniformContinuous]\n[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : UniformSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : UniformGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : EquicontinuousAt (FunLike.coe \u2218 F) 1\n\u22a2 UniformContinuous (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F))\n[PROOFSTEP]\nrw [equicontinuousAt_iff_continuousAt] at hf \n[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : UniformSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : UniformGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\n\u22a2 UniformContinuous (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F))\n[PROOFSTEP]\nlet \u03c6 : G \u2192* (\u03b9 \u2192\u1d64 M) :=\n { toFun := swap ((\u2191) \u2218 F)\n map_one' := by dsimp [UniformFun]; ext; exact map_one _\n map_mul' := fun a b => by dsimp [UniformFun]; ext; exact map_mul _ _ _ }\n[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : UniformSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : UniformGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\n\u22a2 swap (FunLike.coe \u2218 F) 1 = 1\n[PROOFSTEP]\ndsimp [UniformFun]\n[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : UniformSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : UniformGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\n\u22a2 swap (FunLike.coe \u2218 F) 1 = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : UniformSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : UniformGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\nx\u271d : \u03b9\n\u22a2 swap (FunLike.coe \u2218 F) 1 x\u271d = OfNat.ofNat 1 x\u271d\n[PROOFSTEP]\nexact map_one _\n[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : UniformSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : UniformGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\na b : G\n\u22a2 OneHom.toFun { toFun := swap (FunLike.coe \u2218 F), map_one' := (_ : swap (FunLike.coe \u2218 F) 1 = 1) } (a * b) =\n OneHom.toFun { toFun := swap (FunLike.coe \u2218 F), map_one' := (_ : swap (FunLike.coe \u2218 F) 1 = 1) } a *\n OneHom.toFun { toFun := swap (FunLike.coe \u2218 F), map_one' := (_ : swap (FunLike.coe \u2218 F) 1 = 1) } b\n[PROOFSTEP]\ndsimp [UniformFun]\n[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : UniformSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : UniformGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\na b : G\n\u22a2 swap (FunLike.coe \u2218 F) (a * b) = swap (FunLike.coe \u2218 F) a * swap (FunLike.coe \u2218 F) b\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : UniformSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : UniformGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\na b : G\nx\u271d : \u03b9\n\u22a2 swap (FunLike.coe \u2218 F) (a * b) x\u271d = (swap (FunLike.coe \u2218 F) a * swap (FunLike.coe \u2218 F) b) x\u271d\n[PROOFSTEP]\nexact map_mul _ _ _\n[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst\u271d\u2076 : UniformSpace G\ninst\u271d\u2075 : UniformSpace M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group M\ninst\u271d\u00b2 : UniformGroup G\ninst\u271d\u00b9 : UniformGroup M\ninst\u271d : MonoidHomClass hom G M\nF : \u03b9 \u2192 hom\nhf : ContinuousAt (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F)) 1\n\u03c6 : G \u2192* \u03b9 \u2192\u1d64 M :=\n { toOneHom := { toFun := swap (FunLike.coe \u2218 F), map_one' := (_ : swap (FunLike.coe \u2218 F) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (a b : G),\n OneHom.toFun { toFun := swap (FunLike.coe \u2218 F), map_one' := (_ : swap (FunLike.coe \u2218 F) 1 = 1) } (a * b) =\n OneHom.toFun { toFun := swap (FunLike.coe \u2218 F), map_one' := (_ : swap (FunLike.coe \u2218 F) 1 = 1) } a *\n OneHom.toFun { toFun := swap (FunLike.coe \u2218 F), map_one' := (_ : swap (FunLike.coe \u2218 F) 1 = 1) } b) }\n\u22a2 UniformContinuous (\u2191UniformFun.ofFun \u2218 swap (FunLike.coe \u2218 F))\n[PROOFSTEP]\nexact uniformContinuous_of_continuousAt_one \u03c6 hf\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Equicontinuity", "llama_tokens": 4738, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125848754471, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.4227638035873631}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nhs : IsCompact s\nf : Filter \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 s\u1d9c \u2208 \ud835\udcdd a \u2293 f\n\u22a2 s\u1d9c \u2208 f\n[PROOFSTEP]\ncontrapose! hf\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nhs : IsCompact s\nf : Filter \u03b1\nhf : \u00acs\u1d9c \u2208 f\n\u22a2 \u2203 a, a \u2208 s \u2227 \u00acs\u1d9c \u2208 \ud835\udcdd a \u2293 f\n[PROOFSTEP]\nsimp only [not_mem_iff_inf_principal_compl, compl_compl, inf_assoc] at hf \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nhs : IsCompact s\nf : Filter \u03b1\nhf : NeBot (f \u2293 \ud835\udcdf s)\n\u22a2 \u2203 a, a \u2208 s \u2227 NeBot (\ud835\udcdd a \u2293 (f \u2293 \ud835\udcdf s))\n[PROOFSTEP]\nexact @hs _ hf inf_le_right\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nhs : IsCompact s\nf : Filter \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] a \u2227 t\u1d9c \u2208 f\n\u22a2 s\u1d9c \u2208 f\n[PROOFSTEP]\nrefine' hs.compl_mem_sets fun a ha => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nhs : IsCompact s\nf : Filter \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] a \u2227 t\u1d9c \u2208 f\na : \u03b1\nha : a \u2208 s\n\u22a2 s\u1d9c \u2208 \ud835\udcdd a \u2293 f\n[PROOFSTEP]\nrcases hf a ha with \u27e8t, ht, hst\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nhs : IsCompact s\nf : Filter \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] a \u2227 t\u1d9c \u2208 f\na : \u03b1\nha : a \u2208 s\nt : Set \u03b1\nht : t \u2208 \ud835\udcdd[s] a\nhst : t\u1d9c \u2208 f\n\u22a2 s\u1d9c \u2208 \ud835\udcdd a \u2293 f\n[PROOFSTEP]\nreplace ht := mem_inf_principal.1 ht\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nhs : IsCompact s\nf : Filter \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] a \u2227 t\u1d9c \u2208 f\na : \u03b1\nha : a \u2208 s\nt : Set \u03b1\nhst : t\u1d9c \u2208 f\nht : {x | x \u2208 s \u2192 x \u2208 t} \u2208 \ud835\udcdd a\n\u22a2 s\u1d9c \u2208 \ud835\udcdd a \u2293 f\n[PROOFSTEP]\napply mem_inf_of_inter ht hst\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nhs : IsCompact s\nf : Filter \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] a \u2227 t\u1d9c \u2208 f\na : \u03b1\nha : a \u2208 s\nt : Set \u03b1\nhst : t\u1d9c \u2208 f\nht : {x | x \u2208 s \u2192 x \u2208 t} \u2208 \ud835\udcdd a\n\u22a2 {x | x \u2208 s \u2192 x \u2208 t} \u2229 t\u1d9c \u2286 s\u1d9c\n[PROOFSTEP]\nrintro x \u27e8h\u2081, h\u2082\u27e9 hs\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nhs\u271d : IsCompact s\nf : Filter \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] a \u2227 t\u1d9c \u2208 f\na : \u03b1\nha : a \u2208 s\nt : Set \u03b1\nhst : t\u1d9c \u2208 f\nht : {x | x \u2208 s \u2192 x \u2208 t} \u2208 \ud835\udcdd a\nx : \u03b1\nh\u2081 : x \u2208 {x | x \u2208 s \u2192 x \u2208 t}\nh\u2082 : x \u2208 t\u1d9c\nhs : x \u2208 s\n\u22a2 False\n[PROOFSTEP]\nexact h\u2082 (h\u2081 hs)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nhs : IsCompact s\np : Set \u03b1 \u2192 Prop\nhe : p \u2205\nhmono : \u2200 \u2983s t : Set \u03b1\u2984, s \u2286 t \u2192 p t \u2192 p s\nhunion : \u2200 \u2983s t : Set \u03b1\u2984, p s \u2192 p t \u2192 p (s \u222a t)\nhnhds : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 p t\n\u22a2 p s\n[PROOFSTEP]\nlet f : Filter \u03b1 :=\n { sets := {t | p t\u1d9c}\n univ_sets := by simpa\n sets_of_superset := fun ht\u2081 ht => hmono (compl_subset_compl.2 ht) ht\u2081\n inter_sets := fun ht\u2081 ht\u2082 => by simp [compl_inter, hunion ht\u2081 ht\u2082] }\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nhs : IsCompact s\np : Set \u03b1 \u2192 Prop\nhe : p \u2205\nhmono : \u2200 \u2983s t : Set \u03b1\u2984, s \u2286 t \u2192 p t \u2192 p s\nhunion : \u2200 \u2983s t : Set \u03b1\u2984, p s \u2192 p t \u2192 p (s \u222a t)\nhnhds : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 p t\n\u22a2 univ \u2208 {t | p t\u1d9c}\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nhs : IsCompact s\np : Set \u03b1 \u2192 Prop\nhe : p \u2205\nhmono : \u2200 \u2983s t : Set \u03b1\u2984, s \u2286 t \u2192 p t \u2192 p s\nhunion : \u2200 \u2983s t : Set \u03b1\u2984, p s \u2192 p t \u2192 p (s \u222a t)\nhnhds : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 p t\nx\u271d y\u271d : Set \u03b1\nht\u2081 : x\u271d \u2208 {t | p t\u1d9c}\nht\u2082 : y\u271d \u2208 {t | p t\u1d9c}\n\u22a2 x\u271d \u2229 y\u271d \u2208 {t | p t\u1d9c}\n[PROOFSTEP]\nsimp [compl_inter, hunion ht\u2081 ht\u2082]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nhs : IsCompact s\np : Set \u03b1 \u2192 Prop\nhe : p \u2205\nhmono : \u2200 \u2983s t : Set \u03b1\u2984, s \u2286 t \u2192 p t \u2192 p s\nhunion : \u2200 \u2983s t : Set \u03b1\u2984, p s \u2192 p t \u2192 p (s \u222a t)\nhnhds : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 p t\nf : Filter \u03b1 :=\n { sets := {t | p t\u1d9c}, univ_sets := (_ : univ \u2208 {t | p t\u1d9c}),\n sets_of_superset := (_ : \u2200 {x y : Set \u03b1}, x \u2208 {t | p t\u1d9c} \u2192 x \u2286 y \u2192 p y\u1d9c),\n inter_sets := (_ : \u2200 {x y : Set \u03b1}, x \u2208 {t | p t\u1d9c} \u2192 y \u2208 {t | p t\u1d9c} \u2192 p (x \u2229 y)\u1d9c) }\n\u22a2 p s\n[PROOFSTEP]\nhave : s\u1d9c \u2208 f := hs.compl_mem_sets_of_nhdsWithin (by simpa using hnhds)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nhs : IsCompact s\np : Set \u03b1 \u2192 Prop\nhe : p \u2205\nhmono : \u2200 \u2983s t : Set \u03b1\u2984, s \u2286 t \u2192 p t \u2192 p s\nhunion : \u2200 \u2983s t : Set \u03b1\u2984, p s \u2192 p t \u2192 p (s \u222a t)\nhnhds : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 p t\nf : Filter \u03b1 :=\n { sets := {t | p t\u1d9c}, univ_sets := (_ : univ \u2208 {t | p t\u1d9c}),\n sets_of_superset := (_ : \u2200 {x y : Set \u03b1}, x \u2208 {t | p t\u1d9c} \u2192 x \u2286 y \u2192 p y\u1d9c),\n inter_sets := (_ : \u2200 {x y : Set \u03b1}, x \u2208 {t | p t\u1d9c} \u2192 y \u2208 {t | p t\u1d9c} \u2192 p (x \u2229 y)\u1d9c) }\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] a \u2227 t\u1d9c \u2208 f\n[PROOFSTEP]\nsimpa using hnhds\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nhs : IsCompact s\np : Set \u03b1 \u2192 Prop\nhe : p \u2205\nhmono : \u2200 \u2983s t : Set \u03b1\u2984, s \u2286 t \u2192 p t \u2192 p s\nhunion : \u2200 \u2983s t : Set \u03b1\u2984, p s \u2192 p t \u2192 p (s \u222a t)\nhnhds : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 p t\nf : Filter \u03b1 :=\n { sets := {t | p t\u1d9c}, univ_sets := (_ : univ \u2208 {t | p t\u1d9c}),\n sets_of_superset := (_ : \u2200 {x y : Set \u03b1}, x \u2208 {t | p t\u1d9c} \u2192 x \u2286 y \u2192 p y\u1d9c),\n inter_sets := (_ : \u2200 {x y : Set \u03b1}, x \u2208 {t | p t\u1d9c} \u2192 y \u2208 {t | p t\u1d9c} \u2192 p (x \u2229 y)\u1d9c) }\nthis : s\u1d9c \u2208 f\n\u22a2 p s\n[PROOFSTEP]\nrwa [\u2190 compl_compl s]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nhs : IsCompact s\nht : IsClosed t\n\u22a2 IsCompact (s \u2229 t)\n[PROOFSTEP]\nintro f hnf hstf\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nhs : IsCompact s\nht : IsClosed t\nf : Filter \u03b1\nhnf : NeBot f\nhstf : f \u2264 \ud835\udcdf (s \u2229 t)\n\u22a2 \u2203 a, a \u2208 s \u2229 t \u2227 ClusterPt a f\n[PROOFSTEP]\nobtain \u27e8a, hsa, ha\u27e9 : \u2203 a \u2208 s, ClusterPt a f := hs (le_trans hstf (le_principal_iff.2 (inter_subset_left _ _)))\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nhs : IsCompact s\nht : IsClosed t\nf : Filter \u03b1\nhnf : NeBot f\nhstf : f \u2264 \ud835\udcdf (s \u2229 t)\na : \u03b1\nhsa : a \u2208 s\nha : ClusterPt a f\n\u22a2 \u2203 a, a \u2208 s \u2229 t \u2227 ClusterPt a f\n[PROOFSTEP]\nhave : a \u2208 t := ht.mem_of_nhdsWithin_neBot <| ha.mono <| le_trans hstf (le_principal_iff.2 (inter_subset_right _ _))\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nhs : IsCompact s\nht : IsClosed t\nf : Filter \u03b1\nhnf : NeBot f\nhstf : f \u2264 \ud835\udcdf (s \u2229 t)\na : \u03b1\nhsa : a \u2208 s\nha : ClusterPt a f\nthis : a \u2208 t\n\u22a2 \u2203 a, a \u2208 s \u2229 t \u2227 ClusterPt a f\n[PROOFSTEP]\nexact \u27e8a, \u27e8hsa, this\u27e9, ha\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhf : ContinuousOn f s\n\u22a2 IsCompact (f '' s)\n[PROOFSTEP]\nintro l lne ls\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhf : ContinuousOn f s\nl : Filter \u03b2\nlne : NeBot l\nls : l \u2264 \ud835\udcdf (f '' s)\n\u22a2 \u2203 a, a \u2208 f '' s \u2227 ClusterPt a l\n[PROOFSTEP]\nhave : NeBot (l.comap f \u2293 \ud835\udcdf s) := comap_inf_principal_neBot_of_image_mem lne (le_principal_iff.1 ls)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhf : ContinuousOn f s\nl : Filter \u03b2\nlne : NeBot l\nls : l \u2264 \ud835\udcdf (f '' s)\nthis : NeBot (comap f l \u2293 \ud835\udcdf s)\n\u22a2 \u2203 a, a \u2208 f '' s \u2227 ClusterPt a l\n[PROOFSTEP]\nobtain \u27e8a, has, ha\u27e9 : \u2203 a \u2208 s, ClusterPt a (l.comap f \u2293 \ud835\udcdf s) := @hs _ this inf_le_right\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhf : ContinuousOn f s\nl : Filter \u03b2\nlne : NeBot l\nls : l \u2264 \ud835\udcdf (f '' s)\nthis : NeBot (comap f l \u2293 \ud835\udcdf s)\na : \u03b1\nhas : a \u2208 s\nha : ClusterPt a (comap f l \u2293 \ud835\udcdf s)\n\u22a2 \u2203 a, a \u2208 f '' s \u2227 ClusterPt a l\n[PROOFSTEP]\nhaveI := ha.neBot\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhf : ContinuousOn f s\nl : Filter \u03b2\nlne : NeBot l\nls : l \u2264 \ud835\udcdf (f '' s)\nthis\u271d : NeBot (comap f l \u2293 \ud835\udcdf s)\na : \u03b1\nhas : a \u2208 s\nha : ClusterPt a (comap f l \u2293 \ud835\udcdf s)\nthis : NeBot (\ud835\udcdd a \u2293 (comap f l \u2293 \ud835\udcdf s))\n\u22a2 \u2203 a, a \u2208 f '' s \u2227 ClusterPt a l\n[PROOFSTEP]\nuse f a, mem_image_of_mem f has\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhf : ContinuousOn f s\nl : Filter \u03b2\nlne : NeBot l\nls : l \u2264 \ud835\udcdf (f '' s)\nthis\u271d : NeBot (comap f l \u2293 \ud835\udcdf s)\na : \u03b1\nhas : a \u2208 s\nha : ClusterPt a (comap f l \u2293 \ud835\udcdf s)\nthis : NeBot (\ud835\udcdd a \u2293 (comap f l \u2293 \ud835\udcdf s))\n\u22a2 ClusterPt (f a) l\n[PROOFSTEP]\nhave : Tendsto f (\ud835\udcdd a \u2293 (comap f l \u2293 \ud835\udcdf s)) (\ud835\udcdd (f a) \u2293 l) :=\n by\n convert (hf a has).inf (@tendsto_comap _ _ f l) using 1\n rw [nhdsWithin]\n ac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhf : ContinuousOn f s\nl : Filter \u03b2\nlne : NeBot l\nls : l \u2264 \ud835\udcdf (f '' s)\nthis\u271d : NeBot (comap f l \u2293 \ud835\udcdf s)\na : \u03b1\nhas : a \u2208 s\nha : ClusterPt a (comap f l \u2293 \ud835\udcdf s)\nthis : NeBot (\ud835\udcdd a \u2293 (comap f l \u2293 \ud835\udcdf s))\n\u22a2 Tendsto f (\ud835\udcdd a \u2293 (comap f l \u2293 \ud835\udcdf s)) (\ud835\udcdd (f a) \u2293 l)\n[PROOFSTEP]\nconvert (hf a has).inf (@tendsto_comap _ _ f l) using 1\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhf : ContinuousOn f s\nl : Filter \u03b2\nlne : NeBot l\nls : l \u2264 \ud835\udcdf (f '' s)\nthis\u271d : NeBot (comap f l \u2293 \ud835\udcdf s)\na : \u03b1\nhas : a \u2208 s\nha : ClusterPt a (comap f l \u2293 \ud835\udcdf s)\nthis : NeBot (\ud835\udcdd a \u2293 (comap f l \u2293 \ud835\udcdf s))\n\u22a2 \ud835\udcdd a \u2293 (comap f l \u2293 \ud835\udcdf s) = \ud835\udcdd[s] a \u2293 comap f l\n[PROOFSTEP]\nrw [nhdsWithin]\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhf : ContinuousOn f s\nl : Filter \u03b2\nlne : NeBot l\nls : l \u2264 \ud835\udcdf (f '' s)\nthis\u271d : NeBot (comap f l \u2293 \ud835\udcdf s)\na : \u03b1\nhas : a \u2208 s\nha : ClusterPt a (comap f l \u2293 \ud835\udcdf s)\nthis : NeBot (\ud835\udcdd a \u2293 (comap f l \u2293 \ud835\udcdf s))\n\u22a2 \ud835\udcdd a \u2293 (comap f l \u2293 \ud835\udcdf s) = \ud835\udcdd a \u2293 \ud835\udcdf s \u2293 comap f l\n[PROOFSTEP]\nac_rfl\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhf : ContinuousOn f s\nl : Filter \u03b2\nlne : NeBot l\nls : l \u2264 \ud835\udcdf (f '' s)\nthis\u271d\u00b9 : NeBot (comap f l \u2293 \ud835\udcdf s)\na : \u03b1\nhas : a \u2208 s\nha : ClusterPt a (comap f l \u2293 \ud835\udcdf s)\nthis\u271d : NeBot (\ud835\udcdd a \u2293 (comap f l \u2293 \ud835\udcdf s))\nthis : Tendsto f (\ud835\udcdd a \u2293 (comap f l \u2293 \ud835\udcdf s)) (\ud835\udcdd (f a) \u2293 l)\n\u22a2 ClusterPt (f a) l\n[PROOFSTEP]\nexact this.neBot\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u22a2 IsCompact s \u2194 \u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 \ud835\udcdf s \u2192 \u2203 a, a \u2208 s \u2227 \u2191f \u2264 \ud835\udcdd a\n[PROOFSTEP]\nrefine' (forall_neBot_le_iff _).trans _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u22a2 Monotone fun f => \u2203 a, a \u2208 s \u2227 ClusterPt a f\n[PROOFSTEP]\nrintro f g hle \u27e8a, has, haf\u27e9\n[GOAL]\ncase refine'_1.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf g : Filter \u03b1\nhle : f \u2264 g\na : \u03b1\nhas : a \u2208 s\nhaf : ClusterPt a f\n\u22a2 \u2203 a, a \u2208 s \u2227 ClusterPt a g\n[PROOFSTEP]\nexact \u27e8a, has, haf.mono hle\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u22a2 (\u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 \ud835\udcdf s \u2192 \u2203 a, a \u2208 s \u2227 ClusterPt a \u2191f) \u2194\n \u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 \ud835\udcdf s \u2192 \u2203 a, a \u2208 s \u2227 \u2191f \u2264 \ud835\udcdd a\n[PROOFSTEP]\nsimp only [Ultrafilter.clusterPt_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 s \u2192 U x \u2208 \ud835\udcdd x\nt : Finset \u2191s\nht : s \u2286 \u22c3 (x : \u2191s) (_ : x \u2208 t), U \u2191x\n\u22a2 s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 Finset.image Subtype.val t), U x\n[PROOFSTEP]\nrwa [Finset.set_biUnion_finset_image]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nl : Filter \u03b1\nhs : IsCompact s\n\u22a2 Disjoint (\ud835\udcdd\u02e2 s) l \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 Disjoint (\ud835\udcdd x) l\n[PROOFSTEP]\nrefine' \u27e8fun h x hx => h.mono_left <| nhds_le_nhdsSet hx, fun H => _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nl : Filter \u03b1\nhs : IsCompact s\nH : \u2200 (x : \u03b1), x \u2208 s \u2192 Disjoint (\ud835\udcdd x) l\n\u22a2 Disjoint (\ud835\udcdd\u02e2 s) l\n[PROOFSTEP]\nchoose! U hxU hUl using fun x hx => (nhds_basis_opens x).disjoint_iff_left.1 (H x hx)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nl : Filter \u03b1\nhs : IsCompact s\nH : \u2200 (x : \u03b1), x \u2208 s \u2192 Disjoint (\ud835\udcdd x) l\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 U x \u2227 IsOpen (U x)\nhUl : \u2200 (x : \u03b1), x \u2208 s \u2192 (U x)\u1d9c \u2208 l\n\u22a2 Disjoint (\ud835\udcdd\u02e2 s) l\n[PROOFSTEP]\nchoose hxU hUo using hxU\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nl : Filter \u03b1\nhs : IsCompact s\nH : \u2200 (x : \u03b1), x \u2208 s \u2192 Disjoint (\ud835\udcdd x) l\nU : \u03b1 \u2192 Set \u03b1\nhUl : \u2200 (x : \u03b1), x \u2208 s \u2192 (U x)\u1d9c \u2208 l\nhxU : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 U x\nhUo : \u2200 (x : \u03b1), x \u2208 s \u2192 IsOpen (U x)\n\u22a2 Disjoint (\ud835\udcdd\u02e2 s) l\n[PROOFSTEP]\nrcases hs.elim_nhds_subcover U fun x hx => (hUo x hx).mem_nhds (hxU x hx) with \u27e8t, hts, hst\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nl : Filter \u03b1\nhs : IsCompact s\nH : \u2200 (x : \u03b1), x \u2208 s \u2192 Disjoint (\ud835\udcdd x) l\nU : \u03b1 \u2192 Set \u03b1\nhUl : \u2200 (x : \u03b1), x \u2208 s \u2192 (U x)\u1d9c \u2208 l\nhxU : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 U x\nhUo : \u2200 (x : \u03b1), x \u2208 s \u2192 IsOpen (U x)\nt : Finset \u03b1\nhts : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 s\nhst : s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), U x\n\u22a2 Disjoint (\ud835\udcdd\u02e2 s) l\n[PROOFSTEP]\nrefine' (hasBasis_nhdsSet _).disjoint_iff_left.2 \u27e8\u22c3 x \u2208 t, U x, \u27e8isOpen_biUnion fun x hx => hUo x (hts x hx), hst\u27e9, _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nl : Filter \u03b1\nhs : IsCompact s\nH : \u2200 (x : \u03b1), x \u2208 s \u2192 Disjoint (\ud835\udcdd x) l\nU : \u03b1 \u2192 Set \u03b1\nhUl : \u2200 (x : \u03b1), x \u2208 s \u2192 (U x)\u1d9c \u2208 l\nhxU : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 U x\nhUo : \u2200 (x : \u03b1), x \u2208 s \u2192 IsOpen (U x)\nt : Finset \u03b1\nhts : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 s\nhst : s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), U x\n\u22a2 (\u22c3 (x : \u03b1) (_ : x \u2208 t), U x)\u1d9c \u2208 l\n[PROOFSTEP]\nrw [compl_iUnion\u2082, biInter_finset_mem]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nl : Filter \u03b1\nhs : IsCompact s\nH : \u2200 (x : \u03b1), x \u2208 s \u2192 Disjoint (\ud835\udcdd x) l\nU : \u03b1 \u2192 Set \u03b1\nhUl : \u2200 (x : \u03b1), x \u2208 s \u2192 (U x)\u1d9c \u2208 l\nhxU : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 U x\nhUo : \u2200 (x : \u03b1), x \u2208 s \u2192 IsOpen (U x)\nt : Finset \u03b1\nhts : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 s\nhst : s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), U x\n\u22a2 \u2200 (i : \u03b1), i \u2208 t \u2192 (U i)\u1d9c \u2208 l\n[PROOFSTEP]\nexact fun x hx => hUl x (hts x hx)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nl : Filter \u03b1\nhs : IsCompact s\n\u22a2 Disjoint l (\ud835\udcdd\u02e2 s) \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 Disjoint l (\ud835\udcdd x)\n[PROOFSTEP]\nsimpa only [disjoint_comm] using hs.disjoint_nhdsSet_left\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nhs : IsCompact s\nZ : \u03b9 \u2192 Set \u03b1\nhZc : \u2200 (i : \u03b9), IsClosed (Z i)\nhsZ : s \u2229 \u22c2 (i : \u03b9), Z i = \u2205\nhdZ : Directed (fun x x_1 => x \u2287 x_1) Z\n\u22a2 s \u2286 \u22c3 (i : \u03b9), (compl \u2218 Z) i\n[PROOFSTEP]\nsimpa only [subset_def, not_forall, eq_empty_iff_forall_not_mem, mem_iUnion, exists_prop, mem_inter_iff, not_and,\n iff_self_iff, mem_iInter, mem_compl_iff] using hsZ\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nhs : IsCompact s\nZ : \u03b9 \u2192 Set \u03b1\nhZc : \u2200 (i : \u03b9), IsClosed (Z i)\nhsZ : s \u2229 \u22c2 (i : \u03b9), Z i = \u2205\nhdZ : Directed (fun x x_1 => x \u2287 x_1) Z\nt : \u03b9\nht : s \u2286 (compl \u2218 Z) t\n\u22a2 s \u2229 Z t = \u2205\n[PROOFSTEP]\nsimpa only [subset_def, not_forall, eq_empty_iff_forall_not_mem, mem_iUnion, exists_prop, mem_inter_iff, not_and,\n iff_self_iff, mem_iInter, mem_compl_iff] using ht\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\n\u03b9 : Type v\nhs : IsCompact s\nZ : \u03b9 \u2192 Set \u03b1\nhZc : \u2200 (i : \u03b9), IsClosed (Z i)\nhsZ : s \u2229 \u22c2 (i : \u03b9), Z i = \u2205\n\u22a2 s \u2229 \u22c2 (i : Finset \u03b9) (i_1 : \u03b9) (_ : i_1 \u2208 i), Z i_1 = \u2205\n[PROOFSTEP]\nrwa [\u2190 iInter_eq_iInter_finset]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\nhf : LocallyFinite f\ns : Set \u03b1\nhs : IsCompact s\n\u22a2 Set.Finite {i | Set.Nonempty (f i \u2229 s)}\n[PROOFSTEP]\nchoose U hxU hUf using hf\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nhUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)}\n\u22a2 Set.Finite {i | Set.Nonempty (f i \u2229 s)}\n[PROOFSTEP]\nrcases hs.elim_nhds_subcover U fun x _ => hxU x with \u27e8t, -, hsU\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nhUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)}\nt : Finset \u03b1\nhsU : s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), U x\n\u22a2 Set.Finite {i | Set.Nonempty (f i \u2229 s)}\n[PROOFSTEP]\nrefine' (t.finite_toSet.biUnion fun x _ => hUf x).subset _\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nhUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)}\nt : Finset \u03b1\nhsU : s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), U x\n\u22a2 {i | Set.Nonempty (f i \u2229 s)} \u2286 \u22c3 (i : \u03b1) (_ : i \u2208 \u2191t), {i_1 | Set.Nonempty (f i_1 \u2229 U i)}\n[PROOFSTEP]\nrintro i \u27e8x, hx\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nhUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)}\nt : Finset \u03b1\nhsU : s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), U x\ni : \u03b9\nx : \u03b1\nhx : x \u2208 f i \u2229 s\n\u22a2 i \u2208 \u22c3 (i : \u03b1) (_ : i \u2208 \u2191t), {i_1 | Set.Nonempty (f i_1 \u2229 U i)}\n[PROOFSTEP]\nrcases mem_iUnion\u2082.1 (hsU hx.2) with \u27e8c, hct, hcx\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nhUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)}\nt : Finset \u03b1\nhsU : s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), U x\ni : \u03b9\nx : \u03b1\nhx : x \u2208 f i \u2229 s\nc : \u03b1\nhct : c \u2208 t\nhcx : x \u2208 U c\n\u22a2 i \u2208 \u22c3 (i : \u03b1) (_ : i \u2208 \u2191t), {i_1 | Set.Nonempty (f i_1 \u2229 U i)}\n[PROOFSTEP]\nexact mem_biUnion hct \u27e8x, hx.1, hcx\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\n\u03b9 : Type v\nhs : IsCompact s\nZ : \u03b9 \u2192 Set \u03b1\nhZc : \u2200 (i : \u03b9), IsClosed (Z i)\nhsZ : \u2200 (t : Finset \u03b9), Set.Nonempty (s \u2229 \u22c2 (i : \u03b9) (_ : i \u2208 t), Z i)\n\u22a2 Set.Nonempty (s \u2229 \u22c2 (i : \u03b9), Z i)\n[PROOFSTEP]\nsimp only [nonempty_iff_ne_empty] at hsZ \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\n\u03b9 : Type v\nhs : IsCompact s\nZ : \u03b9 \u2192 Set \u03b1\nhZc : \u2200 (i : \u03b9), IsClosed (Z i)\nhsZ : \u2200 (t : Finset \u03b9), s \u2229 \u22c2 (i : \u03b9) (_ : i \u2208 t), Z i \u2260 \u2205\n\u22a2 s \u2229 \u22c2 (i : \u03b9), Z i \u2260 \u2205\n[PROOFSTEP]\napply mt (hs.elim_finite_subfamily_closed Z hZc)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\n\u03b9 : Type v\nhs : IsCompact s\nZ : \u03b9 \u2192 Set \u03b1\nhZc : \u2200 (i : \u03b9), IsClosed (Z i)\nhsZ : \u2200 (t : Finset \u03b9), s \u2229 \u22c2 (i : \u03b9) (_ : i \u2208 t), Z i \u2260 \u2205\n\u22a2 \u00ac\u2203 t, s \u2229 \u22c2 (i : \u03b9) (_ : i \u2208 t), Z i = \u2205\n[PROOFSTEP]\npush_neg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\n\u03b9 : Type v\nhs : IsCompact s\nZ : \u03b9 \u2192 Set \u03b1\nhZc : \u2200 (i : \u03b9), IsClosed (Z i)\nhsZ : \u2200 (t : Finset \u03b9), s \u2229 \u22c2 (i : \u03b9) (_ : i \u2208 t), Z i \u2260 \u2205\n\u22a2 \u2200 (t : Finset \u03b9), s \u2229 \u22c2 (i : \u03b9) (_ : i \u2208 t), Z i \u2260 \u2205\n[PROOFSTEP]\nexact hsZ\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nZ : \u03b9 \u2192 Set \u03b1\nhZd : Directed (fun x x_1 => x \u2287 x_1) Z\nhZn : \u2200 (i : \u03b9), Set.Nonempty (Z i)\nhZc : \u2200 (i : \u03b9), IsCompact (Z i)\nhZcl : \u2200 (i : \u03b9), IsClosed (Z i)\n\u22a2 Set.Nonempty (\u22c2 (i : \u03b9), Z i)\n[PROOFSTEP]\nlet i\u2080 := h\u03b9.some\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nZ : \u03b9 \u2192 Set \u03b1\nhZd : Directed (fun x x_1 => x \u2287 x_1) Z\nhZn : \u2200 (i : \u03b9), Set.Nonempty (Z i)\nhZc : \u2200 (i : \u03b9), IsCompact (Z i)\nhZcl : \u2200 (i : \u03b9), IsClosed (Z i)\ni\u2080 : \u03b9 := Nonempty.some h\u03b9\n\u22a2 Set.Nonempty (\u22c2 (i : \u03b9), Z i)\n[PROOFSTEP]\nsuffices (Z i\u2080 \u2229 \u22c2 i, Z i).Nonempty by rwa [inter_eq_right_iff_subset.mpr (iInter_subset _ i\u2080)] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nZ : \u03b9 \u2192 Set \u03b1\nhZd : Directed (fun x x_1 => x \u2287 x_1) Z\nhZn : \u2200 (i : \u03b9), Set.Nonempty (Z i)\nhZc : \u2200 (i : \u03b9), IsCompact (Z i)\nhZcl : \u2200 (i : \u03b9), IsClosed (Z i)\ni\u2080 : \u03b9 := Nonempty.some h\u03b9\nthis : Set.Nonempty (Z i\u2080 \u2229 \u22c2 (i : \u03b9), Z i)\n\u22a2 Set.Nonempty (\u22c2 (i : \u03b9), Z i)\n[PROOFSTEP]\nrwa [inter_eq_right_iff_subset.mpr (iInter_subset _ i\u2080)] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nZ : \u03b9 \u2192 Set \u03b1\nhZd : Directed (fun x x_1 => x \u2287 x_1) Z\nhZn : \u2200 (i : \u03b9), Set.Nonempty (Z i)\nhZc : \u2200 (i : \u03b9), IsCompact (Z i)\nhZcl : \u2200 (i : \u03b9), IsClosed (Z i)\ni\u2080 : \u03b9 := Nonempty.some h\u03b9\n\u22a2 Set.Nonempty (Z i\u2080 \u2229 \u22c2 (i : \u03b9), Z i)\n[PROOFSTEP]\nsimp only [nonempty_iff_ne_empty] at hZn \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nZ : \u03b9 \u2192 Set \u03b1\nhZd : Directed (fun x x_1 => x \u2287 x_1) Z\nhZc : \u2200 (i : \u03b9), IsCompact (Z i)\nhZcl : \u2200 (i : \u03b9), IsClosed (Z i)\ni\u2080 : \u03b9 := Nonempty.some h\u03b9\nhZn : \u2200 (i : \u03b9), Z i \u2260 \u2205\n\u22a2 Z (Nonempty.some h\u03b9) \u2229 \u22c2 (i : \u03b9), Z i \u2260 \u2205\n[PROOFSTEP]\napply mt ((hZc i\u2080).elim_directed_family_closed Z hZcl)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nZ : \u03b9 \u2192 Set \u03b1\nhZd : Directed (fun x x_1 => x \u2287 x_1) Z\nhZc : \u2200 (i : \u03b9), IsCompact (Z i)\nhZcl : \u2200 (i : \u03b9), IsClosed (Z i)\ni\u2080 : \u03b9 := Nonempty.some h\u03b9\nhZn : \u2200 (i : \u03b9), Z i \u2260 \u2205\n\u22a2 \u00ac(Directed (fun x x_1 => x \u2287 x_1) Z \u2192 \u2203 i, Z i\u2080 \u2229 Z i = \u2205)\n[PROOFSTEP]\npush_neg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nZ : \u03b9 \u2192 Set \u03b1\nhZd : Directed (fun x x_1 => x \u2287 x_1) Z\nhZc : \u2200 (i : \u03b9), IsCompact (Z i)\nhZcl : \u2200 (i : \u03b9), IsClosed (Z i)\ni\u2080 : \u03b9 := Nonempty.some h\u03b9\nhZn : \u2200 (i : \u03b9), Z i \u2260 \u2205\n\u22a2 Directed (fun x x_1 => x \u2287 x_1) Z \u2227 \u2200 (i : \u03b9), Z (Nonempty.some h\u03b9) \u2229 Z i \u2260 \u2205\n[PROOFSTEP]\nsimp only [\u2190 nonempty_iff_ne_empty] at hZn \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nZ : \u03b9 \u2192 Set \u03b1\nhZd : Directed (fun x x_1 => x \u2287 x_1) Z\nhZc : \u2200 (i : \u03b9), IsCompact (Z i)\nhZcl : \u2200 (i : \u03b9), IsClosed (Z i)\ni\u2080 : \u03b9 := Nonempty.some h\u03b9\nhZn : \u2200 (i : \u03b9), Set.Nonempty (Z i)\n\u22a2 Directed (fun x x_1 => x \u2287 x_1) Z \u2227 \u2200 (i : \u03b9), Set.Nonempty (Z (Nonempty.some h\u03b9) \u2229 Z i)\n[PROOFSTEP]\nrefine' \u27e8hZd, fun i => _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nZ : \u03b9 \u2192 Set \u03b1\nhZd : Directed (fun x x_1 => x \u2287 x_1) Z\nhZc : \u2200 (i : \u03b9), IsCompact (Z i)\nhZcl : \u2200 (i : \u03b9), IsClosed (Z i)\ni\u2080 : \u03b9 := Nonempty.some h\u03b9\nhZn : \u2200 (i : \u03b9), Set.Nonempty (Z i)\ni : \u03b9\n\u22a2 Set.Nonempty (Z (Nonempty.some h\u03b9) \u2229 Z i)\n[PROOFSTEP]\nrcases hZd i\u2080 i with \u27e8j, hji\u2080, hji\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type v\nh\u03b9 : Nonempty \u03b9\nZ : \u03b9 \u2192 Set \u03b1\nhZd : Directed (fun x x_1 => x \u2287 x_1) Z\nhZc : \u2200 (i : \u03b9), IsCompact (Z i)\nhZcl : \u2200 (i : \u03b9), IsClosed (Z i)\ni\u2080 : \u03b9 := Nonempty.some h\u03b9\nhZn : \u2200 (i : \u03b9), Set.Nonempty (Z i)\ni j : \u03b9\nhji\u2080 : Z i\u2080 \u2287 Z j\nhji : Z i \u2287 Z j\n\u22a2 Set.Nonempty (Z (Nonempty.some h\u03b9) \u2229 Z i)\n[PROOFSTEP]\nexact (hZn j).mono (subset_inter hji\u2080 hji)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nb : Set \u03b9\nc : \u03b9 \u2192 Set \u03b1\nhs : IsCompact s\nhc\u2081 : \u2200 (i : \u03b9), i \u2208 b \u2192 IsOpen (c i)\nhc\u2082 : s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 b), c i\n\u22a2 \u2203 b', b' \u2286 b \u2227 Set.Finite b' \u2227 s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 b'), c i\n[PROOFSTEP]\nsimp only [Subtype.forall', biUnion_eq_iUnion] at hc\u2081 hc\u2082 \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nb : Set \u03b9\nc : \u03b9 \u2192 Set \u03b1\nhs : IsCompact s\nhc\u2081 : \u2200 (x : { a // a \u2208 b }), IsOpen (c \u2191x)\nhc\u2082 : s \u2286 \u22c3 (x : \u2191b), c \u2191x\n\u22a2 \u2203 b', b' \u2286 b \u2227 Set.Finite b' \u2227 s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 b'), c i\n[PROOFSTEP]\nrcases hs.elim_finite_subcover (fun i => c i : b \u2192 Set \u03b1) hc\u2081 hc\u2082 with \u27e8d, hd\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nb : Set \u03b9\nc : \u03b9 \u2192 Set \u03b1\nhs : IsCompact s\nhc\u2081 : \u2200 (x : { a // a \u2208 b }), IsOpen (c \u2191x)\nhc\u2082 : s \u2286 \u22c3 (x : \u2191b), c \u2191x\nd : Finset \u2191b\nhd : s \u2286 \u22c3 (i : \u2191b) (_ : i \u2208 d), c \u2191i\n\u22a2 \u2203 b', b' \u2286 b \u2227 Set.Finite b' \u2227 s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 b'), c i\n[PROOFSTEP]\nrefine' \u27e8Subtype.val '' d.toSet, _, d.finite_toSet.image _, _\u27e9\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nb : Set \u03b9\nc : \u03b9 \u2192 Set \u03b1\nhs : IsCompact s\nhc\u2081 : \u2200 (x : { a // a \u2208 b }), IsOpen (c \u2191x)\nhc\u2082 : s \u2286 \u22c3 (x : \u2191b), c \u2191x\nd : Finset \u2191b\nhd : s \u2286 \u22c3 (i : \u2191b) (_ : i \u2208 d), c \u2191i\n\u22a2 Subtype.val '' \u2191d \u2286 b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nb : Set \u03b9\nc : \u03b9 \u2192 Set \u03b1\nhs : IsCompact s\nhc\u2081 : \u2200 (x : { a // a \u2208 b }), IsOpen (c \u2191x)\nhc\u2082 : s \u2286 \u22c3 (x : \u2191b), c \u2191x\nd : Finset \u2191b\nhd : s \u2286 \u22c3 (i : \u2191b) (_ : i \u2208 d), c \u2191i\n\u22a2 s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 Subtype.val '' \u2191d), c i\n[PROOFSTEP]\nrwa [biUnion_image]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nh : \u2200 {\u03b9 : Type u} (U : \u03b9 \u2192 Set \u03b1), (\u2200 (i : \u03b9), IsOpen (U i)) \u2192 s \u2286 \u22c3 (i : \u03b9), U i \u2192 \u2203 t, s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\nf : Filter \u03b1\nhf : NeBot f\nhfs : f \u2264 \ud835\udcdf s\n\u22a2 \u2203 a, a \u2208 s \u2227 ClusterPt a f\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : Filter \u03b1\nhf : NeBot f\nhfs : f \u2264 \ud835\udcdf s\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 \u00acClusterPt a f\n\u22a2 Exists fun {\u03b9} =>\n \u2203 U, (\u2200 (i : \u03b9), IsOpen (U i)) \u2227 s \u2286 \u22c3 (i : \u03b9), U i \u2227 \u2200 (t : Finset \u03b9), \u00acs \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\n[PROOFSTEP]\nsimp only [ClusterPt, not_neBot, \u2190 disjoint_iff, SetCoe.forall', (nhds_basis_opens _).disjoint_iff_left] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : Filter \u03b1\nhf : NeBot f\nhfs : f \u2264 \ud835\udcdf s\nh : \u2200 (x : \u2191s), \u2203 i, (\u2191x \u2208 i \u2227 IsOpen i) \u2227 i\u1d9c \u2208 f\n\u22a2 Exists fun {\u03b9} =>\n \u2203 U, (\u2200 (i : \u03b9), IsOpen (U i)) \u2227 s \u2286 \u22c3 (i : \u03b9), U i \u2227 \u2200 (t : Finset \u03b9), \u00acs \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\n[PROOFSTEP]\nchoose U hU hUf using h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : Filter \u03b1\nhf : NeBot f\nhfs : f \u2264 \ud835\udcdf s\nU : \u2191s \u2192 Set \u03b1\nhU : \u2200 (x : \u2191s), \u2191x \u2208 U x \u2227 IsOpen (U x)\nhUf : \u2200 (x : \u2191s), (U x)\u1d9c \u2208 f\n\u22a2 Exists fun {\u03b9} =>\n \u2203 U, (\u2200 (i : \u03b9), IsOpen (U i)) \u2227 s \u2286 \u22c3 (i : \u03b9), U i \u2227 \u2200 (t : Finset \u03b9), \u00acs \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\n[PROOFSTEP]\nrefine \u27e8s, U, fun x => (hU x).2, fun x hx => mem_iUnion.2 \u27e8\u27e8x, hx\u27e9, (hU _).1\u27e9, fun t ht => ?_\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nf : Filter \u03b1\nhf : NeBot f\nhfs : f \u2264 \ud835\udcdf s\nU : \u2191s \u2192 Set \u03b1\nhU : \u2200 (x : \u2191s), \u2191x \u2208 U x \u2227 IsOpen (U x)\nhUf : \u2200 (x : \u2191s), (U x)\u1d9c \u2208 f\nt : Finset \u2191s\nht : s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 t), U i\n\u22a2 False\n[PROOFSTEP]\nrefine compl_not_mem (le_principal_iff.1 hfs) ?_\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nf : Filter \u03b1\nhf : NeBot f\nhfs : f \u2264 \ud835\udcdf s\nU : \u2191s \u2192 Set \u03b1\nhU : \u2200 (x : \u2191s), \u2191x \u2208 U x \u2227 IsOpen (U x)\nhUf : \u2200 (x : \u2191s), (U x)\u1d9c \u2208 f\nt : Finset \u2191s\nht : s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 t), U i\n\u22a2 s\u1d9c \u2208 f\n[PROOFSTEP]\nrefine mem_of_superset ((biInter_finset_mem t).2 fun x _ => hUf x) ?_\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nf : Filter \u03b1\nhf : NeBot f\nhfs : f \u2264 \ud835\udcdf s\nU : \u2191s \u2192 Set \u03b1\nhU : \u2200 (x : \u2191s), \u2191x \u2208 U x \u2227 IsOpen (U x)\nhUf : \u2200 (x : \u2191s), (U x)\u1d9c \u2208 f\nt : Finset \u2191s\nht : s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 t), U i\n\u22a2 \u22c2 (i : \u2191s) (_ : i \u2208 t), (U i)\u1d9c \u2286 s\u1d9c\n[PROOFSTEP]\nrw [subset_compl_comm, compl_iInter\u2082]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nf : Filter \u03b1\nhf : NeBot f\nhfs : f \u2264 \ud835\udcdf s\nU : \u2191s \u2192 Set \u03b1\nhU : \u2200 (x : \u2191s), \u2191x \u2208 U x \u2227 IsOpen (U x)\nhUf : \u2200 (x : \u2191s), (U x)\u1d9c \u2208 f\nt : Finset \u2191s\nht : s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 t), U i\n\u22a2 s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 t), (U i)\u1d9c\u1d9c\n[PROOFSTEP]\nsimpa only [compl_compl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nh :\n \u2200 {\u03b9 : Type u} (Z : \u03b9 \u2192 Set \u03b1),\n (\u2200 (i : \u03b9), IsClosed (Z i)) \u2192 s \u2229 \u22c2 (i : \u03b9), Z i = \u2205 \u2192 \u2203 t, s \u2229 \u22c2 (i : \u03b9) (_ : i \u2208 t), Z i = \u2205\n\u03b9\u271d : Type u\nU : \u03b9\u271d \u2192 Set \u03b1\nhUo : \u2200 (i : \u03b9\u271d), IsOpen (U i)\nhsU : s \u2286 \u22c3 (i : \u03b9\u271d), U i\n\u22a2 \u2203 t, s \u2286 \u22c3 (i : \u03b9\u271d) (_ : i \u2208 t), U i\n[PROOFSTEP]\nrw [\u2190 disjoint_compl_right_iff_subset, compl_iUnion, disjoint_iff] at hsU \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nh :\n \u2200 {\u03b9 : Type u} (Z : \u03b9 \u2192 Set \u03b1),\n (\u2200 (i : \u03b9), IsClosed (Z i)) \u2192 s \u2229 \u22c2 (i : \u03b9), Z i = \u2205 \u2192 \u2203 t, s \u2229 \u22c2 (i : \u03b9) (_ : i \u2208 t), Z i = \u2205\n\u03b9\u271d : Type u\nU : \u03b9\u271d \u2192 Set \u03b1\nhUo : \u2200 (i : \u03b9\u271d), IsOpen (U i)\nhsU : s \u2293 \u22c2 (i : \u03b9\u271d), (U i)\u1d9c = \u22a5\n\u22a2 \u2203 t, s \u2286 \u22c3 (i : \u03b9\u271d) (_ : i \u2208 t), U i\n[PROOFSTEP]\nrcases h (fun i => (U i)\u1d9c) (fun i => (hUo _).isClosed_compl) hsU with \u27e8t, ht\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nh :\n \u2200 {\u03b9 : Type u} (Z : \u03b9 \u2192 Set \u03b1),\n (\u2200 (i : \u03b9), IsClosed (Z i)) \u2192 s \u2229 \u22c2 (i : \u03b9), Z i = \u2205 \u2192 \u2203 t, s \u2229 \u22c2 (i : \u03b9) (_ : i \u2208 t), Z i = \u2205\n\u03b9\u271d : Type u\nU : \u03b9\u271d \u2192 Set \u03b1\nhUo : \u2200 (i : \u03b9\u271d), IsOpen (U i)\nhsU : s \u2293 \u22c2 (i : \u03b9\u271d), (U i)\u1d9c = \u22a5\nt : Finset \u03b9\u271d\nht : s \u2229 \u22c2 (i : \u03b9\u271d) (_ : i \u2208 t), (U i)\u1d9c = \u2205\n\u22a2 \u2203 t, s \u2286 \u22c3 (i : \u03b9\u271d) (_ : i \u2208 t), U i\n[PROOFSTEP]\nrefine \u27e8t, ?_\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nh :\n \u2200 {\u03b9 : Type u} (Z : \u03b9 \u2192 Set \u03b1),\n (\u2200 (i : \u03b9), IsClosed (Z i)) \u2192 s \u2229 \u22c2 (i : \u03b9), Z i = \u2205 \u2192 \u2203 t, s \u2229 \u22c2 (i : \u03b9) (_ : i \u2208 t), Z i = \u2205\n\u03b9\u271d : Type u\nU : \u03b9\u271d \u2192 Set \u03b1\nhUo : \u2200 (i : \u03b9\u271d), IsOpen (U i)\nhsU : s \u2293 \u22c2 (i : \u03b9\u271d), (U i)\u1d9c = \u22a5\nt : Finset \u03b9\u271d\nht : s \u2229 \u22c2 (i : \u03b9\u271d) (_ : i \u2208 t), (U i)\u1d9c = \u2205\n\u22a2 s \u2286 \u22c3 (i : \u03b9\u271d) (_ : i \u2208 t), U i\n[PROOFSTEP]\nrwa [\u2190 disjoint_compl_right_iff_subset, compl_iUnion\u2082, disjoint_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nx\u2080 : \u03b1\nK : Set \u03b2\nhK : IsCompact K\nP : \u03b1 \u2192 \u03b2 \u2192 Prop\nhP : \u2200 (y : \u03b2), y \u2208 K \u2192 \u2200\u1da0 (z : \u03b1 \u00d7 \u03b2) in \ud835\udcdd (x\u2080, y), P z.fst z.snd\n\u22a2 \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 K \u2192 P x y\n[PROOFSTEP]\nrefine' hK.induction_on _ _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nx\u2080 : \u03b1\nK : Set \u03b2\nhK : IsCompact K\nP : \u03b1 \u2192 \u03b2 \u2192 Prop\nhP : \u2200 (y : \u03b2), y \u2208 K \u2192 \u2200\u1da0 (z : \u03b1 \u00d7 \u03b2) in \ud835\udcdd (x\u2080, y), P z.fst z.snd\n\u22a2 \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 \u2205 \u2192 P x y\n[PROOFSTEP]\nexact eventually_of_forall fun x y => False.elim\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nx\u2080 : \u03b1\nK : Set \u03b2\nhK : IsCompact K\nP : \u03b1 \u2192 \u03b2 \u2192 Prop\nhP : \u2200 (y : \u03b2), y \u2208 K \u2192 \u2200\u1da0 (z : \u03b1 \u00d7 \u03b2) in \ud835\udcdd (x\u2080, y), P z.fst z.snd\n\u22a2 \u2200 \u2983s t : Set \u03b2\u2984, s \u2286 t \u2192 (\u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 t \u2192 P x y) \u2192 \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 s \u2192 P x y\n[PROOFSTEP]\nintro s t hst ht\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\nx\u2080 : \u03b1\nK : Set \u03b2\nhK : IsCompact K\nP : \u03b1 \u2192 \u03b2 \u2192 Prop\nhP : \u2200 (y : \u03b2), y \u2208 K \u2192 \u2200\u1da0 (z : \u03b1 \u00d7 \u03b2) in \ud835\udcdd (x\u2080, y), P z.fst z.snd\ns t : Set \u03b2\nhst : s \u2286 t\nht : \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 t \u2192 P x y\n\u22a2 \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 s \u2192 P x y\n[PROOFSTEP]\nrefine' ht.mono fun x h y hys => h y <| hst hys\n[GOAL]\ncase refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nx\u2080 : \u03b1\nK : Set \u03b2\nhK : IsCompact K\nP : \u03b1 \u2192 \u03b2 \u2192 Prop\nhP : \u2200 (y : \u03b2), y \u2208 K \u2192 \u2200\u1da0 (z : \u03b1 \u00d7 \u03b2) in \ud835\udcdd (x\u2080, y), P z.fst z.snd\n\u22a2 \u2200 \u2983s t : Set \u03b2\u2984,\n (\u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 s \u2192 P x y) \u2192\n (\u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 t \u2192 P x y) \u2192 \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 s \u222a t \u2192 P x y\n[PROOFSTEP]\nintro s t hs ht\n[GOAL]\ncase refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\nx\u2080 : \u03b1\nK : Set \u03b2\nhK : IsCompact K\nP : \u03b1 \u2192 \u03b2 \u2192 Prop\nhP : \u2200 (y : \u03b2), y \u2208 K \u2192 \u2200\u1da0 (z : \u03b1 \u00d7 \u03b2) in \ud835\udcdd (x\u2080, y), P z.fst z.snd\ns t : Set \u03b2\nhs : \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 s \u2192 P x y\nht : \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 t \u2192 P x y\n\u22a2 \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 s \u222a t \u2192 P x y\n[PROOFSTEP]\nfilter_upwards [hs, ht]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\nx\u2080 : \u03b1\nK : Set \u03b2\nhK : IsCompact K\nP : \u03b1 \u2192 \u03b2 \u2192 Prop\nhP : \u2200 (y : \u03b2), y \u2208 K \u2192 \u2200\u1da0 (z : \u03b1 \u00d7 \u03b2) in \ud835\udcdd (x\u2080, y), P z.fst z.snd\ns t : Set \u03b2\nhs : \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 s \u2192 P x y\nht : \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 t \u2192 P x y\n\u22a2 \u2200 (a : \u03b1), (\u2200 (y : \u03b2), y \u2208 s \u2192 P a y) \u2192 (\u2200 (y : \u03b2), y \u2208 t \u2192 P a y) \u2192 \u2200 (y : \u03b2), y \u2208 s \u222a t \u2192 P a y\n[PROOFSTEP]\nrintro x h1 h2 y (hys | hyt)\n[GOAL]\ncase h.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\nx\u2080 : \u03b1\nK : Set \u03b2\nhK : IsCompact K\nP : \u03b1 \u2192 \u03b2 \u2192 Prop\nhP : \u2200 (y : \u03b2), y \u2208 K \u2192 \u2200\u1da0 (z : \u03b1 \u00d7 \u03b2) in \ud835\udcdd (x\u2080, y), P z.fst z.snd\ns t : Set \u03b2\nhs : \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 s \u2192 P x y\nht : \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 t \u2192 P x y\nx : \u03b1\nh1 : \u2200 (y : \u03b2), y \u2208 s \u2192 P x y\nh2 : \u2200 (y : \u03b2), y \u2208 t \u2192 P x y\ny : \u03b2\nhys : y \u2208 s\n\u22a2 P x y\ncase h.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\nx\u2080 : \u03b1\nK : Set \u03b2\nhK : IsCompact K\nP : \u03b1 \u2192 \u03b2 \u2192 Prop\nhP : \u2200 (y : \u03b2), y \u2208 K \u2192 \u2200\u1da0 (z : \u03b1 \u00d7 \u03b2) in \ud835\udcdd (x\u2080, y), P z.fst z.snd\ns t : Set \u03b2\nhs : \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 s \u2192 P x y\nht : \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 t \u2192 P x y\nx : \u03b1\nh1 : \u2200 (y : \u03b2), y \u2208 s \u2192 P x y\nh2 : \u2200 (y : \u03b2), y \u2208 t \u2192 P x y\ny : \u03b2\nhyt : y \u2208 t\n\u22a2 P x y\n[PROOFSTEP]\nexacts [h1 y hys, h2 y hyt]\n[GOAL]\ncase refine'_4\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nx\u2080 : \u03b1\nK : Set \u03b2\nhK : IsCompact K\nP : \u03b1 \u2192 \u03b2 \u2192 Prop\nhP : \u2200 (y : \u03b2), y \u2208 K \u2192 \u2200\u1da0 (z : \u03b1 \u00d7 \u03b2) in \ud835\udcdd (x\u2080, y), P z.fst z.snd\n\u22a2 \u2200 (x : \u03b2), x \u2208 K \u2192 \u2203 t, t \u2208 \ud835\udcdd[K] x \u2227 \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 t \u2192 P x y\n[PROOFSTEP]\nintro y hyK\n[GOAL]\ncase refine'_4\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nx\u2080 : \u03b1\nK : Set \u03b2\nhK : IsCompact K\nP : \u03b1 \u2192 \u03b2 \u2192 Prop\nhP : \u2200 (y : \u03b2), y \u2208 K \u2192 \u2200\u1da0 (z : \u03b1 \u00d7 \u03b2) in \ud835\udcdd (x\u2080, y), P z.fst z.snd\ny : \u03b2\nhyK : y \u2208 K\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[K] y \u2227 \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 t \u2192 P x y\n[PROOFSTEP]\nspecialize hP y hyK\n[GOAL]\ncase refine'_4\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nx\u2080 : \u03b1\nK : Set \u03b2\nhK : IsCompact K\nP : \u03b1 \u2192 \u03b2 \u2192 Prop\ny : \u03b2\nhyK : y \u2208 K\nhP : \u2200\u1da0 (z : \u03b1 \u00d7 \u03b2) in \ud835\udcdd (x\u2080, y), P z.fst z.snd\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[K] y \u2227 \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 t \u2192 P x y\n[PROOFSTEP]\nrw [nhds_prod_eq, eventually_prod_iff] at hP \n[GOAL]\ncase refine'_4\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nx\u2080 : \u03b1\nK : Set \u03b2\nhK : IsCompact K\nP : \u03b1 \u2192 \u03b2 \u2192 Prop\ny : \u03b2\nhyK : y \u2208 K\nhP :\n \u2203 pa,\n (\u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, pa x) \u2227\n \u2203 pb, (\u2200\u1da0 (y : \u03b2) in \ud835\udcdd y, pb y) \u2227 \u2200 {x : \u03b1}, pa x \u2192 \u2200 {y : \u03b2}, pb y \u2192 P (x, y).fst (x, y).snd\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[K] y \u2227 \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 t \u2192 P x y\n[PROOFSTEP]\nrcases hP with \u27e8p, hp, q, hq, hpq\u27e9\n[GOAL]\ncase refine'_4.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nx\u2080 : \u03b1\nK : Set \u03b2\nhK : IsCompact K\nP : \u03b1 \u2192 \u03b2 \u2192 Prop\ny : \u03b2\nhyK : y \u2208 K\np : \u03b1 \u2192 Prop\nhp : \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, p x\nq : \u03b2 \u2192 Prop\nhq : \u2200\u1da0 (y : \u03b2) in \ud835\udcdd y, q y\nhpq : \u2200 {x : \u03b1}, p x \u2192 \u2200 {y : \u03b2}, q y \u2192 P (x, y).fst (x, y).snd\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[K] y \u2227 \u2200\u1da0 (x : \u03b1) in \ud835\udcdd x\u2080, \u2200 (y : \u03b2), y \u2208 t \u2192 P x y\n[PROOFSTEP]\nexact \u27e8{y | q y}, mem_nhdsWithin_of_mem_nhds hq, eventually_of_mem hp @hpq\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\na : \u03b1\nf : Filter \u03b1\nhf : NeBot f\nhfa : f \u2264 \ud835\udcdf {a}\n\u22a2 \ud835\udcdf {a} \u2264 \ud835\udcdd a\n[PROOFSTEP]\nsimpa only [principal_singleton] using pure_le_nhds a\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ns : Set \u03b9\nf : \u03b9 \u2192 Set \u03b1\nhs : Set.Finite s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 IsCompact (f i)\nl : Ultrafilter \u03b1\nhl : \u2191l \u2264 \ud835\udcdf (\u22c3 (i : \u03b9) (_ : i \u2208 s), f i)\n\u22a2 \u2203 a, a \u2208 \u22c3 (i : \u03b9) (_ : i \u2208 s), f i \u2227 \u2191l \u2264 \ud835\udcdd a\n[PROOFSTEP]\nrw [le_principal_iff, Ultrafilter.mem_coe, Ultrafilter.finite_biUnion_mem_iff hs] at hl \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ns : Set \u03b9\nf : \u03b9 \u2192 Set \u03b1\nhs : Set.Finite s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 IsCompact (f i)\nl : Ultrafilter \u03b1\nhl : \u2203 i, i \u2208 s \u2227 f i \u2208 l\n\u22a2 \u2203 a, a \u2208 \u22c3 (i : \u03b9) (_ : i \u2208 s), f i \u2227 \u2191l \u2264 \ud835\udcdd a\n[PROOFSTEP]\nrcases hl with \u27e8i, his, hi\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ns : Set \u03b9\nf : \u03b9 \u2192 Set \u03b1\nhs : Set.Finite s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 IsCompact (f i)\nl : Ultrafilter \u03b1\ni : \u03b9\nhis : i \u2208 s\nhi : f i \u2208 l\n\u22a2 \u2203 a, a \u2208 \u22c3 (i : \u03b9) (_ : i \u2208 s), f i \u2227 \u2191l \u2264 \ud835\udcdd a\n[PROOFSTEP]\nrcases(hf i his).ultrafilter_le_nhds _ (le_principal_iff.2 hi) with \u27e8x, hxi, hlx\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ns : Set \u03b9\nf : \u03b9 \u2192 Set \u03b1\nhs : Set.Finite s\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 IsCompact (f i)\nl : Ultrafilter \u03b1\ni : \u03b9\nhis : i \u2208 s\nhi : f i \u2208 l\nx : \u03b1\nhxi : x \u2208 f i\nhlx : \u2191l \u2264 \ud835\udcdd x\n\u22a2 \u2203 a, a \u2208 \u22c3 (i : \u03b9) (_ : i \u2208 s), f i \u2227 \u2191l \u2264 \ud835\udcdd a\n[PROOFSTEP]\nexact \u27e8x, mem_iUnion\u2082.2 \u27e8i, his, hxi\u27e9, hlx\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nS : Set (Set \u03b1)\nhf : Set.Finite S\nhc : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsCompact s\n\u22a2 IsCompact (\u22c3\u2080 S)\n[PROOFSTEP]\nrw [sUnion_eq_biUnion]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nS : Set (Set \u03b1)\nhf : Set.Finite S\nhc : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsCompact s\n\u22a2 IsCompact (\u22c3 (i : Set \u03b1) (_ : i \u2208 S), i)\n[PROOFSTEP]\nexact hf.isCompact_biUnion hc\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : DiscreteTopology \u03b1\ns : Set \u03b1\nhs : IsCompact s\n\u22a2 Set.Finite s\n[PROOFSTEP]\nhave : \u2200 x : \u03b1, ({ x } : Set \u03b1) \u2208 \ud835\udcdd x := by simp [nhds_discrete]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : DiscreteTopology \u03b1\ns : Set \u03b1\nhs : IsCompact s\n\u22a2 \u2200 (x : \u03b1), {x} \u2208 \ud835\udcdd x\n[PROOFSTEP]\nsimp [nhds_discrete]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : DiscreteTopology \u03b1\ns : Set \u03b1\nhs : IsCompact s\nthis : \u2200 (x : \u03b1), {x} \u2208 \ud835\udcdd x\n\u22a2 Set.Finite s\n[PROOFSTEP]\nrcases hs.elim_nhds_subcover (fun x => { x }) fun x _ => this x with \u27e8t, _, hst\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : DiscreteTopology \u03b1\ns : Set \u03b1\nhs : IsCompact s\nthis : \u2200 (x : \u03b1), {x} \u2208 \ud835\udcdd x\nt : Finset \u03b1\nleft\u271d : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 s\nhst : s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), {x}\n\u22a2 Set.Finite s\n[PROOFSTEP]\nsimp only [\u2190 t.set_biUnion_coe, biUnion_of_singleton] at hst \n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : DiscreteTopology \u03b1\ns : Set \u03b1\nhs : IsCompact s\nthis : \u2200 (x : \u03b1), {x} \u2208 \ud835\udcdd x\nt : Finset \u03b1\nleft\u271d : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 s\nhst : s \u2286 \u2191t\n\u22a2 Set.Finite s\n[PROOFSTEP]\nexact t.finite_toSet.subset hst\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nhs : IsCompact s\nht : IsCompact t\n\u22a2 IsCompact (s \u222a t)\n[PROOFSTEP]\nrw [union_eq_iUnion]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nhs : IsCompact s\nht : IsCompact t\n\u22a2 IsCompact (\u22c3 (b : Bool), bif b then s else t)\n[PROOFSTEP]\nexact isCompact_iUnion fun b => by cases b <;> assumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nhs : IsCompact s\nht : IsCompact t\nb : Bool\n\u22a2 IsCompact (bif b then s else t)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nhs : IsCompact s\nht : IsCompact t\n\u22a2 IsCompact (bif false then s else t)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase true\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nhs : IsCompact s\nht : IsCompact t\n\u22a2 IsCompact (bif true then s else t)\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\n\u22a2 \u2203 i, V i \u2286 U\n[PROOFSTEP]\nobtain \u27e8W, hsubW, W_op, hWU\u27e9 := exists_open_set_nhds hU\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\nW : Set \u03b1\nhsubW : \u22c2 (i : \u03b9), V i \u2286 W\nW_op : IsOpen W\nhWU : W \u2286 U\n\u22a2 \u2203 i, V i \u2286 U\n[PROOFSTEP]\nsuffices : \u2203 i, V i \u2286 W\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\nW : Set \u03b1\nhsubW : \u22c2 (i : \u03b9), V i \u2286 W\nW_op : IsOpen W\nhWU : W \u2286 U\nthis : \u2203 i, V i \u2286 W\n\u22a2 \u2203 i, V i \u2286 U\n[PROOFSTEP]\nexact this.imp fun i hi => hi.trans hWU\n[GOAL]\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\nW : Set \u03b1\nhsubW : \u22c2 (i : \u03b9), V i \u2286 W\nW_op : IsOpen W\nhWU : W \u2286 U\n\u22a2 \u2203 i, V i \u2286 W\n[PROOFSTEP]\nby_contra' H\n[GOAL]\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\nW : Set \u03b1\nhsubW : \u22c2 (i : \u03b9), V i \u2286 W\nW_op : IsOpen W\nhWU : W \u2286 U\nH : \u2200 (i : \u03b9), \u00acV i \u2286 W\n\u22a2 False\n[PROOFSTEP]\nreplace H : \u2200 i, (V i \u2229 W\u1d9c).Nonempty := fun i => Set.inter_compl_nonempty_iff.mpr (H i)\n[GOAL]\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\nW : Set \u03b1\nhsubW : \u22c2 (i : \u03b9), V i \u2286 W\nW_op : IsOpen W\nhWU : W \u2286 U\nH : \u2200 (i : \u03b9), Set.Nonempty (V i \u2229 W\u1d9c)\n\u22a2 False\n[PROOFSTEP]\nhave : (\u22c2 i, V i \u2229 W\u1d9c).Nonempty :=\n by\n refine'\n IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed _ (fun i j => _) H\n (fun i => (hV_cpct i).inter_right W_op.isClosed_compl) fun i => (hV_closed i).inter W_op.isClosed_compl\n rcases hV i j with \u27e8k, hki, hkj\u27e9\n refine' \u27e8k, \u27e8fun x => _, fun x => _\u27e9\u27e9 <;> simp only [and_imp, mem_inter_iff, mem_compl_iff] <;> tauto\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\nW : Set \u03b1\nhsubW : \u22c2 (i : \u03b9), V i \u2286 W\nW_op : IsOpen W\nhWU : W \u2286 U\nH : \u2200 (i : \u03b9), Set.Nonempty (V i \u2229 W\u1d9c)\n\u22a2 Set.Nonempty (\u22c2 (i : \u03b9), V i \u2229 W\u1d9c)\n[PROOFSTEP]\nrefine'\n IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed _ (fun i j => _) H\n (fun i => (hV_cpct i).inter_right W_op.isClosed_compl) fun i => (hV_closed i).inter W_op.isClosed_compl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\nW : Set \u03b1\nhsubW : \u22c2 (i : \u03b9), V i \u2286 W\nW_op : IsOpen W\nhWU : W \u2286 U\nH : \u2200 (i : \u03b9), Set.Nonempty (V i \u2229 W\u1d9c)\ni j : \u03b9\n\u22a2 \u2203 z, (fun x x_1 => x \u2287 x_1) (V i \u2229 W\u1d9c) (V z \u2229 W\u1d9c) \u2227 (fun x x_1 => x \u2287 x_1) (V j \u2229 W\u1d9c) (V z \u2229 W\u1d9c)\n[PROOFSTEP]\nrcases hV i j with \u27e8k, hki, hkj\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\nW : Set \u03b1\nhsubW : \u22c2 (i : \u03b9), V i \u2286 W\nW_op : IsOpen W\nhWU : W \u2286 U\nH : \u2200 (i : \u03b9), Set.Nonempty (V i \u2229 W\u1d9c)\ni j k : \u03b9\nhki : V i \u2287 V k\nhkj : V j \u2287 V k\n\u22a2 \u2203 z, (fun x x_1 => x \u2287 x_1) (V i \u2229 W\u1d9c) (V z \u2229 W\u1d9c) \u2227 (fun x x_1 => x \u2287 x_1) (V j \u2229 W\u1d9c) (V z \u2229 W\u1d9c)\n[PROOFSTEP]\nrefine' \u27e8k, \u27e8fun x => _, fun x => _\u27e9\u27e9\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\nW : Set \u03b1\nhsubW : \u22c2 (i : \u03b9), V i \u2286 W\nW_op : IsOpen W\nhWU : W \u2286 U\nH : \u2200 (i : \u03b9), Set.Nonempty (V i \u2229 W\u1d9c)\ni j k : \u03b9\nhki : V i \u2287 V k\nhkj : V j \u2287 V k\nx : \u03b1\n\u22a2 x \u2208 V k \u2229 W\u1d9c \u2192 x \u2208 V i \u2229 W\u1d9c\n[PROOFSTEP]\nsimp only [and_imp, mem_inter_iff, mem_compl_iff]\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\nW : Set \u03b1\nhsubW : \u22c2 (i : \u03b9), V i \u2286 W\nW_op : IsOpen W\nhWU : W \u2286 U\nH : \u2200 (i : \u03b9), Set.Nonempty (V i \u2229 W\u1d9c)\ni j k : \u03b9\nhki : V i \u2287 V k\nhkj : V j \u2287 V k\nx : \u03b1\n\u22a2 x \u2208 V k \u2229 W\u1d9c \u2192 x \u2208 V j \u2229 W\u1d9c\n[PROOFSTEP]\nsimp only [and_imp, mem_inter_iff, mem_compl_iff]\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\nW : Set \u03b1\nhsubW : \u22c2 (i : \u03b9), V i \u2286 W\nW_op : IsOpen W\nhWU : W \u2286 U\nH : \u2200 (i : \u03b9), Set.Nonempty (V i \u2229 W\u1d9c)\ni j k : \u03b9\nhki : V i \u2287 V k\nhkj : V j \u2287 V k\nx : \u03b1\n\u22a2 x \u2208 V k \u2192 \u00acx \u2208 W \u2192 x \u2208 V i \u2227 \u00acx \u2208 W\n[PROOFSTEP]\ntauto\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\nW : Set \u03b1\nhsubW : \u22c2 (i : \u03b9), V i \u2286 W\nW_op : IsOpen W\nhWU : W \u2286 U\nH : \u2200 (i : \u03b9), Set.Nonempty (V i \u2229 W\u1d9c)\ni j k : \u03b9\nhki : V i \u2287 V k\nhkj : V j \u2287 V k\nx : \u03b1\n\u22a2 x \u2208 V k \u2192 \u00acx \u2208 W \u2192 x \u2208 V j \u2227 \u00acx \u2208 W\n[PROOFSTEP]\ntauto\n[GOAL]\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\nW : Set \u03b1\nhsubW : \u22c2 (i : \u03b9), V i \u2286 W\nW_op : IsOpen W\nhWU : W \u2286 U\nH : \u2200 (i : \u03b9), Set.Nonempty (V i \u2229 W\u1d9c)\nthis : Set.Nonempty (\u22c2 (i : \u03b9), V i \u2229 W\u1d9c)\n\u22a2 False\n[PROOFSTEP]\nhave : \u00ac\u22c2 i : \u03b9, V i \u2286 W := by simpa [\u2190 iInter_inter, inter_compl_nonempty_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\nW : Set \u03b1\nhsubW : \u22c2 (i : \u03b9), V i \u2286 W\nW_op : IsOpen W\nhWU : W \u2286 U\nH : \u2200 (i : \u03b9), Set.Nonempty (V i \u2229 W\u1d9c)\nthis : Set.Nonempty (\u22c2 (i : \u03b9), V i \u2229 W\u1d9c)\n\u22a2 \u00ac\u22c2 (i : \u03b9), V i \u2286 W\n[PROOFSTEP]\nsimpa [\u2190 iInter_inter, inter_compl_nonempty_iff]\n[GOAL]\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : Nonempty \u03b9\nV : \u03b9 \u2192 Set \u03b1\nhV : Directed (fun x x_1 => x \u2287 x_1) V\nhV_cpct : \u2200 (i : \u03b9), IsCompact (V i)\nhV_closed : \u2200 (i : \u03b9), IsClosed (V i)\nU : Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 \u22c2 (i : \u03b9), V i \u2192 U \u2208 \ud835\udcdd x\nW : Set \u03b1\nhsubW : \u22c2 (i : \u03b9), V i \u2286 W\nW_op : IsOpen W\nhWU : W \u2286 U\nH : \u2200 (i : \u03b9), Set.Nonempty (V i \u2229 W\u1d9c)\nthis\u271d : Set.Nonempty (\u22c2 (i : \u03b9), V i \u2229 W\u1d9c)\nthis : \u00ac\u22c2 (i : \u03b9), V i \u2286 W\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\n\u22a2 IsCompact U \u2227 IsOpen U \u2194 \u2203 s, Set.Finite s \u2227 U = \u22c3 (i : \u03b9) (_ : i \u2208 s), b i\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\n\u22a2 IsCompact U \u2227 IsOpen U \u2192 \u2203 s, Set.Finite s \u2227 U = \u22c3 (i : \u03b9) (_ : i \u2208 s), b i\n[PROOFSTEP]\nrintro \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase mp.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u22a2 \u2203 s, Set.Finite s \u2227 U = \u22c3 (i : \u03b9) (_ : i \u2208 s), b i\n[PROOFSTEP]\nobtain \u27e8\u03b2, f, e, hf\u27e9 := hb.open_eq_iUnion h\u2082\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf : \u03b2 \u2192 Set \u03b1\ne : U = \u22c3 (i : \u03b2), f i\nhf : \u2200 (i : \u03b2), f i \u2208 range b\n\u22a2 \u2203 s, Set.Finite s \u2227 U = \u22c3 (i : \u03b9) (_ : i \u2208 s), b i\n[PROOFSTEP]\nchoose f' hf' using hf\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf : \u03b2 \u2192 Set \u03b1\ne : U = \u22c3 (i : \u03b2), f i\nf' : \u03b2 \u2192 \u03b9\nhf' : \u2200 (i : \u03b2), b (f' i) = f i\n\u22a2 \u2203 s, Set.Finite s \u2227 U = \u22c3 (i : \u03b9) (_ : i \u2208 s), b i\n[PROOFSTEP]\nhave : b \u2218 f' = f := funext hf'\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf : \u03b2 \u2192 Set \u03b1\ne : U = \u22c3 (i : \u03b2), f i\nf' : \u03b2 \u2192 \u03b9\nhf' : \u2200 (i : \u03b2), b (f' i) = f i\nthis : b \u2218 f' = f\n\u22a2 \u2203 s, Set.Finite s \u2227 U = \u22c3 (i : \u03b9) (_ : i \u2208 s), b i\n[PROOFSTEP]\nsubst this\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf' : \u03b2 \u2192 \u03b9\ne : U = \u22c3 (i : \u03b2), (b \u2218 f') i\nhf' : \u2200 (i : \u03b2), b (f' i) = (b \u2218 f') i\n\u22a2 \u2203 s, Set.Finite s \u2227 U = \u22c3 (i : \u03b9) (_ : i \u2208 s), b i\n[PROOFSTEP]\nobtain \u27e8t, ht\u27e9 := h\u2081.elim_finite_subcover (b \u2218 f') (fun i => hb.isOpen (Set.mem_range_self _)) (by rw [e])\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf' : \u03b2 \u2192 \u03b9\ne : U = \u22c3 (i : \u03b2), (b \u2218 f') i\nhf' : \u2200 (i : \u03b2), b (f' i) = (b \u2218 f') i\n\u22a2 U \u2286 \u22c3 (i : \u03b2), (b \u2218 f') i\n[PROOFSTEP]\nrw [e]\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t\u271d : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf' : \u03b2 \u2192 \u03b9\ne : U = \u22c3 (i : \u03b2), (b \u2218 f') i\nhf' : \u2200 (i : \u03b2), b (f' i) = (b \u2218 f') i\nt : Finset \u03b2\nht : U \u2286 \u22c3 (i : \u03b2) (_ : i \u2208 t), (b \u2218 f') i\n\u22a2 \u2203 s, Set.Finite s \u2227 U = \u22c3 (i : \u03b9) (_ : i \u2208 s), b i\n[PROOFSTEP]\nrefine' \u27e8t.image f', Set.Finite.intro inferInstance, le_antisymm _ _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t\u271d : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf' : \u03b2 \u2192 \u03b9\ne : U = \u22c3 (i : \u03b2), (b \u2218 f') i\nhf' : \u2200 (i : \u03b2), b (f' i) = (b \u2218 f') i\nt : Finset \u03b2\nht : U \u2286 \u22c3 (i : \u03b2) (_ : i \u2208 t), (b \u2218 f') i\n\u22a2 U \u2264 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(Finset.image f' t)), b i\n[PROOFSTEP]\nrefine' Set.Subset.trans ht _\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t\u271d : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf' : \u03b2 \u2192 \u03b9\ne : U = \u22c3 (i : \u03b2), (b \u2218 f') i\nhf' : \u2200 (i : \u03b2), b (f' i) = (b \u2218 f') i\nt : Finset \u03b2\nht : U \u2286 \u22c3 (i : \u03b2) (_ : i \u2208 t), (b \u2218 f') i\n\u22a2 \u22c3 (i : \u03b2) (_ : i \u2208 t), (b \u2218 f') i \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(Finset.image f' t)), b i\n[PROOFSTEP]\nsimp only [Set.iUnion_subset_iff]\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t\u271d : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf' : \u03b2 \u2192 \u03b9\ne : U = \u22c3 (i : \u03b2), (b \u2218 f') i\nhf' : \u2200 (i : \u03b2), b (f' i) = (b \u2218 f') i\nt : Finset \u03b2\nht : U \u2286 \u22c3 (i : \u03b2) (_ : i \u2208 t), (b \u2218 f') i\n\u22a2 \u2200 (i : \u03b2), i \u2208 t \u2192 (b \u2218 f') i \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(Finset.image f' t)), b i\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t\u271d : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf' : \u03b2 \u2192 \u03b9\ne : U = \u22c3 (i : \u03b2), (b \u2218 f') i\nhf' : \u2200 (i : \u03b2), b (f' i) = (b \u2218 f') i\nt : Finset \u03b2\nht : U \u2286 \u22c3 (i : \u03b2) (_ : i \u2208 t), (b \u2218 f') i\ni : \u03b2\nhi : i \u2208 t\n\u22a2 (b \u2218 f') i \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(Finset.image f' t)), b i\n[PROOFSTEP]\nerw [\u2190 Set.iUnion_subtype (fun x : \u03b9 => x \u2208 t.image f') fun i => b i.1]\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t\u271d : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf' : \u03b2 \u2192 \u03b9\ne : U = \u22c3 (i : \u03b2), (b \u2218 f') i\nhf' : \u2200 (i : \u03b2), b (f' i) = (b \u2218 f') i\nt : Finset \u03b2\nht : U \u2286 \u22c3 (i : \u03b2) (_ : i \u2208 t), (b \u2218 f') i\ni : \u03b2\nhi : i \u2208 t\n\u22a2 (b \u2218 f') i \u2286 \u22c3 (x : { x // x \u2208 Finset.image f' t }), b \u2191x\n[PROOFSTEP]\nexact Set.subset_iUnion (fun i : t.image f' => b i) \u27e8_, Finset.mem_image_of_mem _ hi\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t\u271d : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf' : \u03b2 \u2192 \u03b9\ne : U = \u22c3 (i : \u03b2), (b \u2218 f') i\nhf' : \u2200 (i : \u03b2), b (f' i) = (b \u2218 f') i\nt : Finset \u03b2\nht : U \u2286 \u22c3 (i : \u03b2) (_ : i \u2208 t), (b \u2218 f') i\n\u22a2 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(Finset.image f' t)), b i \u2264 U\n[PROOFSTEP]\napply Set.iUnion\u2082_subset\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.refine'_2.h\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t\u271d : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf' : \u03b2 \u2192 \u03b9\ne : U = \u22c3 (i : \u03b2), (b \u2218 f') i\nhf' : \u2200 (i : \u03b2), b (f' i) = (b \u2218 f') i\nt : Finset \u03b2\nht : U \u2286 \u22c3 (i : \u03b2) (_ : i \u2208 t), (b \u2218 f') i\n\u22a2 \u2200 (i : \u03b9), i \u2208 \u2191(Finset.image f' t) \u2192 b i \u2286 U\n[PROOFSTEP]\nrintro i hi\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.refine'_2.h\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t\u271d : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf' : \u03b2 \u2192 \u03b9\ne : U = \u22c3 (i : \u03b2), (b \u2218 f') i\nhf' : \u2200 (i : \u03b2), b (f' i) = (b \u2218 f') i\nt : Finset \u03b2\nht : U \u2286 \u22c3 (i : \u03b2) (_ : i \u2208 t), (b \u2218 f') i\ni : \u03b9\nhi : i \u2208 \u2191(Finset.image f' t)\n\u22a2 b i \u2286 U\n[PROOFSTEP]\nobtain \u27e8j, -, rfl\u27e9 := Finset.mem_image.mp hi\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.refine'_2.h.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t\u271d : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf' : \u03b2 \u2192 \u03b9\ne : U = \u22c3 (i : \u03b2), (b \u2218 f') i\nhf' : \u2200 (i : \u03b2), b (f' i) = (b \u2218 f') i\nt : Finset \u03b2\nht : U \u2286 \u22c3 (i : \u03b2) (_ : i \u2208 t), (b \u2218 f') i\nj : \u03b2\nhi : f' j \u2208 \u2191(Finset.image f' t)\n\u22a2 b (f' j) \u2286 U\n[PROOFSTEP]\nrw [e]\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.refine'_2.h.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\u271d\ns t\u271d : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\nh\u2081 : IsCompact U\nh\u2082 : IsOpen U\n\u03b2 : Type u\nf' : \u03b2 \u2192 \u03b9\ne : U = \u22c3 (i : \u03b2), (b \u2218 f') i\nhf' : \u2200 (i : \u03b2), b (f' i) = (b \u2218 f') i\nt : Finset \u03b2\nht : U \u2286 \u22c3 (i : \u03b2) (_ : i \u2208 t), (b \u2218 f') i\nj : \u03b2\nhi : f' j \u2208 \u2191(Finset.image f' t)\n\u22a2 b (f' j) \u2286 \u22c3 (i : \u03b2), (b \u2218 f') i\n[PROOFSTEP]\nexact Set.subset_iUnion (b \u2218 f') j\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\nU : Set \u03b1\n\u22a2 (\u2203 s, Set.Finite s \u2227 U = \u22c3 (i : \u03b9) (_ : i \u2208 s), b i) \u2192 IsCompact U \u2227 IsOpen U\n[PROOFSTEP]\nrintro \u27e8s, hs, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\ns : Set \u03b9\nhs : Set.Finite s\n\u22a2 IsCompact (\u22c3 (i : \u03b9) (_ : i \u2208 s), b i) \u2227 IsOpen (\u22c3 (i : \u03b9) (_ : i \u2208 s), b i)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.intro.intro.left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\ns : Set \u03b9\nhs : Set.Finite s\n\u22a2 IsCompact (\u22c3 (i : \u03b9) (_ : i \u2208 s), b i)\n[PROOFSTEP]\nexact hs.isCompact_biUnion fun i _ => hb' i\n[GOAL]\ncase mpr.intro.intro.right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nb : \u03b9 \u2192 Set \u03b1\nhb : IsTopologicalBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact (b i)\ns : Set \u03b9\nhs : Set.Finite s\n\u22a2 IsOpen (\u22c3 (i : \u03b9) (_ : i \u2208 s), b i)\n[PROOFSTEP]\nexact isOpen_biUnion fun i _ => hb.isOpen (Set.mem_range_self _)\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : DiscreteTopology \u03b1\n\u22a2 cocompact \u03b1 = cofinite\n[PROOFSTEP]\nsimp only [cocompact, hasBasis_cofinite.eq_biInf, isCompact_iff_finite]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nhf : Tendsto f (cocompact \u03b1) (\ud835\udcdd b)\nhfc : Continuous f\n\u22a2 IsCompact (insert b (range f))\n[PROOFSTEP]\nintro l hne hle\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nhf : Tendsto f (cocompact \u03b1) (\ud835\udcdd b)\nhfc : Continuous f\nl : Filter \u03b2\nhne : NeBot l\nhle : l \u2264 \ud835\udcdf (insert b (range f))\n\u22a2 \u2203 a, a \u2208 insert b (range f) \u2227 ClusterPt a l\n[PROOFSTEP]\nby_cases hb : ClusterPt b l\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nhf : Tendsto f (cocompact \u03b1) (\ud835\udcdd b)\nhfc : Continuous f\nl : Filter \u03b2\nhne : NeBot l\nhle : l \u2264 \ud835\udcdf (insert b (range f))\nhb : ClusterPt b l\n\u22a2 \u2203 a, a \u2208 insert b (range f) \u2227 ClusterPt a l\n[PROOFSTEP]\nexact \u27e8b, Or.inl rfl, hb\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nhf : Tendsto f (cocompact \u03b1) (\ud835\udcdd b)\nhfc : Continuous f\nl : Filter \u03b2\nhne : NeBot l\nhle : l \u2264 \ud835\udcdf (insert b (range f))\nhb : \u00acClusterPt b l\n\u22a2 \u2203 a, a \u2208 insert b (range f) \u2227 ClusterPt a l\n[PROOFSTEP]\nsimp only [clusterPt_iff, not_forall, \u2190 not_disjoint_iff_nonempty_inter, not_not] at hb \n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nhf : Tendsto f (cocompact \u03b1) (\ud835\udcdd b)\nhfc : Continuous f\nl : Filter \u03b2\nhne : NeBot l\nhle : l \u2264 \ud835\udcdf (insert b (range f))\nhb : \u2203 x h x_1 h, Disjoint x x_1\n\u22a2 \u2203 a, a \u2208 insert b (range f) \u2227 ClusterPt a l\n[PROOFSTEP]\nrcases hb with \u27e8s, hsb, t, htl, hd\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nhf : Tendsto f (cocompact \u03b1) (\ud835\udcdd b)\nhfc : Continuous f\nl : Filter \u03b2\nhne : NeBot l\nhle : l \u2264 \ud835\udcdf (insert b (range f))\ns : Set \u03b2\nhsb : s \u2208 \ud835\udcdd b\nt : Set \u03b2\nhtl : t \u2208 l\nhd : Disjoint s t\n\u22a2 \u2203 a, a \u2208 insert b (range f) \u2227 ClusterPt a l\n[PROOFSTEP]\nrcases mem_cocompact.1 (hf hsb) with \u27e8K, hKc, hKs\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nhf : Tendsto f (cocompact \u03b1) (\ud835\udcdd b)\nhfc : Continuous f\nl : Filter \u03b2\nhne : NeBot l\nhle : l \u2264 \ud835\udcdf (insert b (range f))\ns : Set \u03b2\nhsb : s \u2208 \ud835\udcdd b\nt : Set \u03b2\nhtl : t \u2208 l\nhd : Disjoint s t\nK : Set \u03b1\nhKc : IsCompact K\nhKs : K\u1d9c \u2286 f \u207b\u00b9' s\n\u22a2 \u2203 a, a \u2208 insert b (range f) \u2227 ClusterPt a l\n[PROOFSTEP]\nhave : f '' K \u2208 l := by\n filter_upwards [htl, le_principal_iff.1 hle] with y hyt hyf\n rcases hyf with (rfl | \u27e8x, rfl\u27e9)\n exacts [(hd.le_bot \u27e8mem_of_mem_nhds hsb, hyt\u27e9).elim,\n mem_image_of_mem _ (not_not.1 fun hxK => hd.le_bot \u27e8hKs hxK, hyt\u27e9)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nhf : Tendsto f (cocompact \u03b1) (\ud835\udcdd b)\nhfc : Continuous f\nl : Filter \u03b2\nhne : NeBot l\nhle : l \u2264 \ud835\udcdf (insert b (range f))\ns : Set \u03b2\nhsb : s \u2208 \ud835\udcdd b\nt : Set \u03b2\nhtl : t \u2208 l\nhd : Disjoint s t\nK : Set \u03b1\nhKc : IsCompact K\nhKs : K\u1d9c \u2286 f \u207b\u00b9' s\n\u22a2 f '' K \u2208 l\n[PROOFSTEP]\nfilter_upwards [htl, le_principal_iff.1 hle] with y hyt hyf\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nhf : Tendsto f (cocompact \u03b1) (\ud835\udcdd b)\nhfc : Continuous f\nl : Filter \u03b2\nhne : NeBot l\nhle : l \u2264 \ud835\udcdf (insert b (range f))\ns : Set \u03b2\nhsb : s \u2208 \ud835\udcdd b\nt : Set \u03b2\nhtl : t \u2208 l\nhd : Disjoint s t\nK : Set \u03b1\nhKc : IsCompact K\nhKs : K\u1d9c \u2286 f \u207b\u00b9' s\ny : \u03b2\nhyt : y \u2208 t\nhyf : y \u2208 insert b (range f)\n\u22a2 y \u2208 f '' K\n[PROOFSTEP]\nrcases hyf with (rfl | \u27e8x, rfl\u27e9)\n[GOAL]\ncase h.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhfc : Continuous f\nl : Filter \u03b2\nhne : NeBot l\ns t : Set \u03b2\nhtl : t \u2208 l\nhd : Disjoint s t\nK : Set \u03b1\nhKc : IsCompact K\nhKs : K\u1d9c \u2286 f \u207b\u00b9' s\ny : \u03b2\nhyt : y \u2208 t\nhf : Tendsto f (cocompact \u03b1) (\ud835\udcdd y)\nhle : l \u2264 \ud835\udcdf (insert y (range f))\nhsb : s \u2208 \ud835\udcdd y\n\u22a2 y \u2208 f '' K\ncase h.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nhf : Tendsto f (cocompact \u03b1) (\ud835\udcdd b)\nhfc : Continuous f\nl : Filter \u03b2\nhne : NeBot l\nhle : l \u2264 \ud835\udcdf (insert b (range f))\ns : Set \u03b2\nhsb : s \u2208 \ud835\udcdd b\nt : Set \u03b2\nhtl : t \u2208 l\nhd : Disjoint s t\nK : Set \u03b1\nhKc : IsCompact K\nhKs : K\u1d9c \u2286 f \u207b\u00b9' s\nx : \u03b1\nhyt : f x \u2208 t\n\u22a2 f x \u2208 f '' K\n[PROOFSTEP]\nexacts [(hd.le_bot \u27e8mem_of_mem_nhds hsb, hyt\u27e9).elim, mem_image_of_mem _ (not_not.1 fun hxK => hd.le_bot \u27e8hKs hxK, hyt\u27e9)]\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nhf : Tendsto f (cocompact \u03b1) (\ud835\udcdd b)\nhfc : Continuous f\nl : Filter \u03b2\nhne : NeBot l\nhle : l \u2264 \ud835\udcdf (insert b (range f))\ns : Set \u03b2\nhsb : s \u2208 \ud835\udcdd b\nt : Set \u03b2\nhtl : t \u2208 l\nhd : Disjoint s t\nK : Set \u03b1\nhKc : IsCompact K\nhKs : K\u1d9c \u2286 f \u207b\u00b9' s\nthis : f '' K \u2208 l\n\u22a2 \u2203 a, a \u2208 insert b (range f) \u2227 ClusterPt a l\n[PROOFSTEP]\nrcases hKc.image hfc (le_principal_iff.2 this) with \u27e8y, hy, hyl\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nhf : Tendsto f (cocompact \u03b1) (\ud835\udcdd b)\nhfc : Continuous f\nl : Filter \u03b2\nhne : NeBot l\nhle : l \u2264 \ud835\udcdf (insert b (range f))\ns : Set \u03b2\nhsb : s \u2208 \ud835\udcdd b\nt : Set \u03b2\nhtl : t \u2208 l\nhd : Disjoint s t\nK : Set \u03b1\nhKc : IsCompact K\nhKs : K\u1d9c \u2286 f \u207b\u00b9' s\nthis : f '' K \u2208 l\ny : \u03b2\nhy : y \u2208 f '' K\nhyl : ClusterPt y l\n\u22a2 \u2203 a, a \u2208 insert b (range f) \u2227 ClusterPt a l\n[PROOFSTEP]\nexact \u27e8y, Or.inr <| image_subset_range _ _ hy, hyl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b9 \u2192 \u03b1\na : \u03b1\nhf : Tendsto f cofinite (\ud835\udcdd a)\n\u22a2 IsCompact (insert a (range f))\n[PROOFSTEP]\nletI : TopologicalSpace \u03b9 := \u22a5\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b9 \u2192 \u03b1\na : \u03b1\nhf : Tendsto f cofinite (\ud835\udcdd a)\nthis : TopologicalSpace \u03b9 := \u22a5\n\u22a2 IsCompact (insert a (range f))\n[PROOFSTEP]\nhaveI h : DiscreteTopology \u03b9 := \u27e8rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b9 \u2192 \u03b1\na : \u03b1\nhf : Tendsto f cofinite (\ud835\udcdd a)\nthis : TopologicalSpace \u03b9 := \u22a5\nh : DiscreteTopology \u03b9\n\u22a2 IsCompact (insert a (range f))\n[PROOFSTEP]\nrw [\u2190 cocompact_eq_cofinite \u03b9] at hf \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b9 \u2192 \u03b1\na : \u03b1\nthis : TopologicalSpace \u03b9 := \u22a5\nhf : Tendsto f (cocompact \u03b9) (\ud835\udcdd a)\nh : DiscreteTopology \u03b9\n\u22a2 IsCompact (insert a (range f))\n[PROOFSTEP]\nexact hf.isCompact_insert_range_of_cocompact continuous_of_discreteTopology\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u22a2 HasBasis (coclosedCompact \u03b1) (fun s => IsClosed s \u2227 IsCompact s) compl\n[PROOFSTEP]\nsimp only [Filter.coclosedCompact, iInf_and']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u22a2 HasBasis (\u2a05 (s : Set \u03b1) (_ : IsClosed s \u2227 IsCompact s), \ud835\udcdf s\u1d9c) (fun s => IsClosed s \u2227 IsCompact s) compl\n[PROOFSTEP]\nrefine' hasBasis_biInf_principal' _ \u27e8\u2205, isClosed_empty, isCompact_empty\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u22a2 \u2200 (i : Set \u03b1),\n IsClosed i \u2227 IsCompact i \u2192\n \u2200 (j : Set \u03b1), IsClosed j \u2227 IsCompact j \u2192 \u2203 k, (IsClosed k \u2227 IsCompact k) \u2227 k\u1d9c \u2286 i\u1d9c \u2227 k\u1d9c \u2286 j\u1d9c\n[PROOFSTEP]\nrintro s \u27e8hs\u2081, hs\u2082\u27e9 t \u27e8ht\u2081, ht\u2082\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nhs\u2081 : IsClosed s\nhs\u2082 : IsCompact s\nt : Set \u03b1\nht\u2081 : IsClosed t\nht\u2082 : IsCompact t\n\u22a2 \u2203 k, (IsClosed k \u2227 IsCompact k) \u2227 k\u1d9c \u2286 s\u1d9c \u2227 k\u1d9c \u2286 t\u1d9c\n[PROOFSTEP]\nexact\n \u27e8s \u222a t,\n \u27e8\u27e8hs\u2081.union ht\u2081, hs\u2082.union ht\u2082\u27e9, compl_subset_compl.2 (subset_union_left _ _),\n compl_subset_compl.2 (subset_union_right _ _)\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u22a2 s \u2208 coclosedCompact \u03b1 \u2194 \u2203 t, IsClosed t \u2227 IsCompact t \u2227 t\u1d9c \u2286 s\n[PROOFSTEP]\nsimp only [hasBasis_coclosedCompact.mem_iff, and_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u22a2 s \u2208 coclosedCompact \u03b1 \u2194 \u2203 t, IsClosed t \u2227 IsCompact t \u2227 s\u1d9c \u2286 t\n[PROOFSTEP]\nsimp only [mem_coclosedCompact, compl_subset_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u22a2 IsBounded s \u2194 \u2203 t, IsCompact t \u2227 s \u2286 t\n[PROOFSTEP]\nchange s\u1d9c \u2208 Filter.cocompact \u03b1 \u2194 _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u22a2 s\u1d9c \u2208 cocompact \u03b1 \u2194 \u2203 t, IsCompact t \u2227 s \u2286 t\n[PROOFSTEP]\nrw [Filter.mem_cocompact]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u22a2 (\u2203 t, IsCompact t \u2227 t\u1d9c \u2286 s\u1d9c) \u2194 \u2203 t, IsCompact t \u2227 s \u2286 t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nH : NhdsContainBoxes s t\nn : Set (\u03b2 \u00d7 \u03b1)\nhn : IsOpen n\nhp : t \u00d7\u02e2 s \u2286 n\n\u22a2 s \u00d7\u02e2 t \u2286 Prod.swap \u207b\u00b9' n\n[PROOFSTEP]\nrwa [\u2190 image_subset_iff, image_swap_prod]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nH : NhdsContainBoxes s t\nn : Set (\u03b2 \u00d7 \u03b1)\nhn : IsOpen n\nhp : t \u00d7\u02e2 s \u2286 n\nu : Set \u03b1\nv : Set \u03b2\nuo : IsOpen u\nvo : IsOpen v\nsu : s \u2286 u\ntv : t \u2286 v\np : u \u00d7\u02e2 v \u2286 Prod.swap \u207b\u00b9' n\n\u22a2 v \u00d7\u02e2 u \u2286 n\n[PROOFSTEP]\nrwa [\u2190 image_subset_iff, image_swap_prod] at p \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nx : \u03b1\ny : \u03b2\nn : Set (\u03b1 \u00d7 \u03b2)\nhn : IsOpen n\nhp : {x} \u00d7\u02e2 {y} \u2286 n\n\u22a2 (x, y) \u2208 {x} \u00d7\u02e2 {y}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nx : \u03b1\ny : \u03b2\nn : Set (\u03b1 \u00d7 \u03b2)\nhn : IsOpen n\nhp : {x} \u00d7\u02e2 {y} \u2286 n\nu : Set \u03b1\nv : Set \u03b2\nuo : IsOpen u\nvo : IsOpen v\nxu : x \u2208 u\nyv : y \u2208 v\nhp' : u \u00d7\u02e2 v \u2286 n\n\u22a2 {x} \u2286 u\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nx : \u03b1\ny : \u03b2\nn : Set (\u03b1 \u00d7 \u03b2)\nhn : IsOpen n\nhp : {x} \u00d7\u02e2 {y} \u2286 n\nu : Set \u03b1\nv : Set \u03b2\nuo : IsOpen u\nvo : IsOpen v\nxu : x \u2208 u\nyv : y \u2208 v\nhp' : u \u00d7\u02e2 v \u2286 n\n\u22a2 {y} \u2286 v\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nhs : IsCompact s\nt : Set \u03b2\nH : \u2200 (x : \u03b1), x \u2208 s \u2192 NhdsContainBoxes {x} t\nn : Set (\u03b1 \u00d7 \u03b2)\nhn : IsOpen n\nhp : s \u00d7\u02e2 t \u2286 n\nx\u271d : \u2191s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 {x} \u2286 s\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nhs : IsCompact s\nt : Set \u03b2\nH : \u2200 (x : \u03b1), x \u2208 s \u2192 NhdsContainBoxes {x} t\nn : Set (\u03b1 \u00d7 \u03b2)\nhn : IsOpen n\nhp : s \u00d7\u02e2 t \u2286 n\nthis : \u2200 (x : \u2191s), \u2203 uv, IsOpen uv.fst \u2227 IsOpen uv.snd \u2227 {\u2191x} \u2286 uv.fst \u2227 t \u2286 uv.snd \u2227 uv.fst \u00d7\u02e2 uv.snd \u2286 n\nuvs : \u2191s \u2192 Set \u03b1 \u00d7 Set \u03b2\nh :\n \u2200 (x : \u2191s),\n IsOpen (uvs x).fst \u2227 IsOpen (uvs x).snd \u2227 {\u2191x} \u2286 (uvs x).fst \u2227 t \u2286 (uvs x).snd \u2227 (uvs x).fst \u00d7\u02e2 (uvs x).snd \u2286 n\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 (uvs { val := x, property := hx }).fst\n[PROOFSTEP]\nsimpa using (h \u27e8x, hx\u27e9).2.2.1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nhs : IsCompact s\nt : Set \u03b2\nH : \u2200 (x : \u03b1), x \u2208 s \u2192 NhdsContainBoxes {x} t\nn : Set (\u03b1 \u00d7 \u03b2)\nhn : IsOpen n\nhp : s \u00d7\u02e2 t \u2286 n\nthis\u271d\u00b2 : \u2200 (x : \u2191s), \u2203 uv, IsOpen uv.fst \u2227 IsOpen uv.snd \u2227 {\u2191x} \u2286 uv.fst \u2227 t \u2286 uv.snd \u2227 uv.fst \u00d7\u02e2 uv.snd \u2286 n\nuvs : \u2191s \u2192 Set \u03b1 \u00d7 Set \u03b2\nh :\n \u2200 (x : \u2191s),\n IsOpen (uvs x).fst \u2227 IsOpen (uvs x).snd \u2227 {\u2191x} \u2286 (uvs x).fst \u2227 t \u2286 (uvs x).snd \u2227 (uvs x).fst \u00d7\u02e2 (uvs x).snd \u2286 n\nus_cover : s \u2286 \u22c3 (i : \u2191s), (uvs i).fst\ns0 : Finset \u2191s\ns0_cover : s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 s0), (uvs i).fst\nu : Set \u03b1 := \u22c3 (i : \u2191s) (_ : i \u2208 s0), (uvs i).fst\nv : Set \u03b2 := \u22c2 (i : \u2191s) (_ : i \u2208 s0), (uvs i).snd\nthis\u271d\u00b9 : IsOpen u\nthis\u271d : IsOpen v\nthis : t \u2286 v\nx\u271d\u00b9 : \u03b1 \u00d7 \u03b2\nx' : \u03b1\ny' : \u03b2\nx\u271d : (x', y') \u2208 u \u00d7\u02e2 v\nhx' : (x', y').fst \u2208 u\nhy' : (x', y').snd \u2208 v\n\u22a2 \u2203 i, i \u2208 s0 \u2227 x' \u2208 (uvs i).fst\n[PROOFSTEP]\nsimpa using hx'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : CompactSpace \u03b1\nf : Filter \u03b1\ninst\u271d : NeBot f\n\u22a2 \u2203 x, ClusterPt x f\n[PROOFSTEP]\nsimpa using isCompact_univ (show f \u2264 \ud835\udcdf univ by simp)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : CompactSpace \u03b1\nf : Filter \u03b1\ninst\u271d : NeBot f\n\u22a2 f \u2264 \ud835\udcdf univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nU : \u03b1 \u2192 Set \u03b1\nhU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\n\u22a2 \u2203 t, \u22c3 (x : \u03b1) (_ : x \u2208 t), U x = \u22a4\n[PROOFSTEP]\nobtain \u27e8t, -, s\u27e9 := IsCompact.elim_nhds_subcover isCompact_univ U fun x _ => hU x\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nU : \u03b1 \u2192 Set \u03b1\nhU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nt : Finset \u03b1\ns : univ \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), U x\n\u22a2 \u2203 t, \u22c3 (x : \u03b1) (_ : x \u2208 t), U x = \u22a4\n[PROOFSTEP]\nexact \u27e8t, top_unique s\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nh :\n \u2200 {\u03b9 : Type u} (Z : \u03b9 \u2192 Set \u03b1), (\u2200 (i : \u03b9), IsClosed (Z i)) \u2192 \u22c2 (i : \u03b9), Z i = \u2205 \u2192 \u2203 t, \u22c2 (i : \u03b9) (_ : i \u2208 t), Z i = \u2205\n\u03b9\u271d : Type u\nZ : \u03b9\u271d \u2192 Set \u03b1\n\u22a2 (\u2200 (i : \u03b9\u271d), IsClosed (Z i)) \u2192 univ \u2229 \u22c2 (i : \u03b9\u271d), Z i = \u2205 \u2192 \u2203 t, univ \u2229 \u22c2 (i : \u03b9\u271d) (_ : i \u2208 t), Z i = \u2205\n[PROOFSTEP]\nsimpa using h Z\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : NoncompactSpace \u03b1\n\u22a2 NeBot (cocompact \u03b1)\n[PROOFSTEP]\nrefine' Filter.hasBasis_cocompact.neBot_iff.2 fun hs => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : NoncompactSpace \u03b1\ni\u271d : Set \u03b1\nhs : IsCompact i\u271d\n\u22a2 Set.Nonempty i\u271d\u1d9c\n[PROOFSTEP]\ncontrapose hs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : NoncompactSpace \u03b1\ni\u271d : Set \u03b1\nhs : \u00acSet.Nonempty i\u271d\u1d9c\n\u22a2 \u00acIsCompact i\u271d\n[PROOFSTEP]\nrw [not_nonempty_iff_eq_empty, compl_empty_iff] at hs \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : NoncompactSpace \u03b1\ni\u271d : Set \u03b1\nhs : i\u271d = univ\n\u22a2 \u00acIsCompact i\u271d\n[PROOFSTEP]\nrw [hs]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : NoncompactSpace \u03b1\ni\u271d : Set \u03b1\nhs : i\u271d = univ\n\u22a2 \u00acIsCompact univ\n[PROOFSTEP]\nexact noncompact_univ \u03b1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u22a2 NeBot (cocompact \u2124)\n[PROOFSTEP]\nsimp only [Filter.cocompact_eq_cofinite, Filter.cofinite_neBot]\n -- Note: We can't make this into an instance because it loops with `Finite.compactSpace`.\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : CompactSpace \u03b1\ninst\u271d : Infinite \u03b1\n\u22a2 \u2203 z, NeBot (\ud835\udcdd[{z}\u1d9c] z)\n[PROOFSTEP]\nby_contra' H\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : CompactSpace \u03b1\ninst\u271d : Infinite \u03b1\nH : \u2200 (z : \u03b1), \u00acNeBot (\ud835\udcdd[{z}\u1d9c] z)\n\u22a2 False\n[PROOFSTEP]\nsimp_rw [not_neBot] at H \n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : CompactSpace \u03b1\ninst\u271d : Infinite \u03b1\nH : \u2200 (z : \u03b1), \ud835\udcdd[{z}\u1d9c] z = \u22a5\n\u22a2 False\n[PROOFSTEP]\nhaveI := discreteTopology_iff_nhds_ne.2 H\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : CompactSpace \u03b1\ninst\u271d : Infinite \u03b1\nH : \u2200 (z : \u03b1), \ud835\udcdd[{z}\u1d9c] z = \u22a5\nthis : DiscreteTopology \u03b1\n\u22a2 False\n[PROOFSTEP]\nexact Infinite.not_finite (finite_of_compact_of_discrete : Finite \u03b1)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : CompactSpace \u03b1\nf : \u03b9 \u2192 Set \u03b1\nhf : LocallyFinite f\n\u22a2 Set.Finite {i | Set.Nonempty (f i)}\n[PROOFSTEP]\nsimpa only [inter_univ] using hf.finite_nonempty_inter_compact isCompact_univ\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : CompactSpace \u03b1\nf : \u03b9 \u2192 Set \u03b1\nhf : LocallyFinite f\nhne : \u2200 (i : \u03b9), Set.Nonempty (f i)\n\u22a2 Set.Finite univ\n[PROOFSTEP]\nsimpa only [hne] using hf.finite_nonempty_of_compact\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\n\u22a2 comap f (cocompact \u03b2) \u2264 cocompact \u03b1\n[PROOFSTEP]\nrw [(Filter.hasBasis_cocompact.comap f).le_basis_iff Filter.hasBasis_cocompact]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\n\u22a2 \u2200 (i' : Set \u03b1), IsCompact i' \u2192 \u2203 i, IsCompact i \u2227 f \u207b\u00b9' i\u1d9c \u2286 i'\u1d9c\n[PROOFSTEP]\nintro t ht\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nt : Set \u03b1\nht : IsCompact t\n\u22a2 \u2203 i, IsCompact i \u2227 f \u207b\u00b9' i\u1d9c \u2286 t\u1d9c\n[PROOFSTEP]\nrefine' \u27e8f '' t, ht.image hf, _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nt : Set \u03b1\nht : IsCompact t\n\u22a2 f \u207b\u00b9' (f '' t)\u1d9c \u2286 t\u1d9c\n[PROOFSTEP]\nsimpa using t.subset_preimage_image f\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\n\u22a2 IsCompact (range f)\n[PROOFSTEP]\nrw [\u2190 image_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\n\u22a2 IsCompact (f '' univ)\n[PROOFSTEP]\nexact isCompact_univ.image hf\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\nY : Type u_4\ninst\u271d : TopologicalSpace Y\ns : Set (X \u00d7 Y)\nhs : IsClosed s\n\u22a2 IsClosed (Prod.snd '' s)\n[PROOFSTEP]\nrw [\u2190 isOpen_compl_iff, isOpen_iff_mem_nhds]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\nY : Type u_4\ninst\u271d : TopologicalSpace Y\ns : Set (X \u00d7 Y)\nhs : IsClosed s\n\u22a2 \u2200 (a : Y), a \u2208 (Prod.snd '' s)\u1d9c \u2192 (Prod.snd '' s)\u1d9c \u2208 \ud835\udcdd a\n[PROOFSTEP]\nintro y hy\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\nY : Type u_4\ninst\u271d : TopologicalSpace Y\ns : Set (X \u00d7 Y)\nhs : IsClosed s\ny : Y\nhy : y \u2208 (Prod.snd '' s)\u1d9c\n\u22a2 (Prod.snd '' s)\u1d9c \u2208 \ud835\udcdd y\n[PROOFSTEP]\nhave : univ \u00d7\u02e2 { y } \u2286 s\u1d9c\n[GOAL]\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\nY : Type u_4\ninst\u271d : TopologicalSpace Y\ns : Set (X \u00d7 Y)\nhs : IsClosed s\ny : Y\nhy : y \u2208 (Prod.snd '' s)\u1d9c\n\u22a2 univ \u00d7\u02e2 {y} \u2286 s\u1d9c\n[PROOFSTEP]\nexact fun (x, y') \u27e8_, rfl\u27e9 hs => hy \u27e8(x, y'), hs, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\nY : Type u_4\ninst\u271d : TopologicalSpace Y\ns : Set (X \u00d7 Y)\nhs : IsClosed s\ny : Y\nhy : y \u2208 (Prod.snd '' s)\u1d9c\nthis : univ \u00d7\u02e2 {y} \u2286 s\u1d9c\n\u22a2 (Prod.snd '' s)\u1d9c \u2208 \ud835\udcdd y\n[PROOFSTEP]\nrcases generalized_tube_lemma isCompact_univ isCompact_singleton hs.isOpen_compl this with \u27e8U, V, -, hVo, hU, hV, hs\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\nY : Type u_4\ninst\u271d : TopologicalSpace Y\ns : Set (X \u00d7 Y)\nhs\u271d : IsClosed s\ny : Y\nhy : y \u2208 (Prod.snd '' s)\u1d9c\nthis : univ \u00d7\u02e2 {y} \u2286 s\u1d9c\nU : Set X\nV : Set Y\nhVo : IsOpen V\nhU : univ \u2286 U\nhV : {y} \u2286 V\nhs : U \u00d7\u02e2 V \u2286 s\u1d9c\n\u22a2 (Prod.snd '' s)\u1d9c \u2208 \ud835\udcdd y\n[PROOFSTEP]\nrefine mem_nhds_iff.2 \u27e8V, ?_, hVo, hV rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\nY : Type u_4\ninst\u271d : TopologicalSpace Y\ns : Set (X \u00d7 Y)\nhs\u271d : IsClosed s\ny : Y\nhy : y \u2208 (Prod.snd '' s)\u1d9c\nthis : univ \u00d7\u02e2 {y} \u2286 s\u1d9c\nU : Set X\nV : Set Y\nhVo : IsOpen V\nhU : univ \u2286 U\nhV : {y} \u2286 V\nhs : U \u00d7\u02e2 V \u2286 s\u1d9c\n\u22a2 V \u2286 (Prod.snd '' s)\u1d9c\n[PROOFSTEP]\nrintro _ hzV \u27e8z, hzs, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\nY : Type u_4\ninst\u271d : TopologicalSpace Y\ns : Set (X \u00d7 Y)\nhs\u271d : IsClosed s\ny : Y\nhy : y \u2208 (Prod.snd '' s)\u1d9c\nthis : univ \u00d7\u02e2 {y} \u2286 s\u1d9c\nU : Set X\nV : Set Y\nhVo : IsOpen V\nhU : univ \u2286 U\nhV : {y} \u2286 V\nhs : U \u00d7\u02e2 V \u2286 s\u1d9c\nz : X \u00d7 Y\nhzs : z \u2208 s\nhzV : z.snd \u2208 V\n\u22a2 False\n[PROOFSTEP]\nexact hs \u27e8hU trivial, hzV\u27e9 hzs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Inducing f\ns : Set \u03b1\n\u22a2 IsCompact (f '' s) \u2194 IsCompact s\n[PROOFSTEP]\nrefine \u27e8fun hs F F_ne_bot F_le => ?_, fun hs => hs.image hf.continuous\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsCompact (f '' s)\nF : Filter \u03b1\nF_ne_bot : NeBot F\nF_le : F \u2264 \ud835\udcdf s\n\u22a2 \u2203 a, a \u2208 s \u2227 ClusterPt a F\n[PROOFSTEP]\nobtain \u27e8_, \u27e8x, x_in : x \u2208 s, rfl\u27e9, hx : ClusterPt (f x) (map f F)\u27e9 := hs ((map_mono F_le).trans_eq map_principal)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsCompact (f '' s)\nF : Filter \u03b1\nF_ne_bot : NeBot F\nF_le : F \u2264 \ud835\udcdf s\nx : \u03b1\nx_in : x \u2208 s\nhx : ClusterPt (f x) (map f F)\n\u22a2 \u2203 a, a \u2208 s \u2227 ClusterPt a F\n[PROOFSTEP]\nexact \u27e8x, x_in, hf.mapClusterPt_iff.1 hx\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nK : Set \u03b2\nhK : IsCompact K\n\u22a2 IsCompact (f \u207b\u00b9' K)\n[PROOFSTEP]\nreplace hK := hK.inter_right hf.closed_range\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nK : Set \u03b2\nhK : IsCompact (K \u2229 range f)\n\u22a2 IsCompact (f \u207b\u00b9' K)\n[PROOFSTEP]\nrwa [\u2190 hf.toInducing.isCompact_iff, image_preimage_eq_inter_range]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\n\u22a2 IsCompact s \u2194 IsCompact univ\n[PROOFSTEP]\nrw [isCompact_iff_isCompact_in_subtype, image_univ, Subtype.range_coe]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nhs : IsCompact s\nhs' : Set.Infinite s\n\u22a2 \u2203 z, z \u2208 s \u2227 NeBot (\ud835\udcdd[{z}\u1d9c] z \u2293 \ud835\udcdf s)\n[PROOFSTEP]\nby_contra' H\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nhs : IsCompact s\nhs' : Set.Infinite s\nH : \u2200 (z : \u03b1), z \u2208 s \u2192 \u00acNeBot (\ud835\udcdd[{z}\u1d9c] z \u2293 \ud835\udcdf s)\n\u22a2 False\n[PROOFSTEP]\nsimp_rw [not_neBot] at H \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nhs : IsCompact s\nhs' : Set.Infinite s\nH : \u2200 (z : \u03b1), z \u2208 s \u2192 \ud835\udcdd[{z}\u1d9c] z \u2293 \ud835\udcdf s = \u22a5\n\u22a2 False\n[PROOFSTEP]\nexact hs' (hs.finite <| discreteTopology_subtype_iff.mpr H)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nh : CompactSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\n\u22a2 IsCompact univ\n[PROOFSTEP]\nrw [\u2190 hf.toInducing.isCompact_iff, image_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nh : CompactSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\n\u22a2 IsCompact (range f)\n[PROOFSTEP]\nexact hf.closed_range.isCompact\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nhs : IsCompact s\nht : IsCompact t\n\u22a2 IsCompact (s \u00d7\u02e2 t)\n[PROOFSTEP]\nrw [isCompact_iff_ultrafilter_le_nhds] at hs ht \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nhs : \u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 \ud835\udcdf s \u2192 \u2203 a, a \u2208 s \u2227 \u2191f \u2264 \ud835\udcdd a\nht : \u2200 (f : Ultrafilter \u03b2), \u2191f \u2264 \ud835\udcdf t \u2192 \u2203 a, a \u2208 t \u2227 \u2191f \u2264 \ud835\udcdd a\n\u22a2 \u2200 (f : Ultrafilter (\u03b1 \u00d7 \u03b2)), \u2191f \u2264 \ud835\udcdf (s \u00d7\u02e2 t) \u2192 \u2203 a, a \u2208 s \u00d7\u02e2 t \u2227 \u2191f \u2264 \ud835\udcdd a\n[PROOFSTEP]\nintro f hfs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nhs : \u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 \ud835\udcdf s \u2192 \u2203 a, a \u2208 s \u2227 \u2191f \u2264 \ud835\udcdd a\nht : \u2200 (f : Ultrafilter \u03b2), \u2191f \u2264 \ud835\udcdf t \u2192 \u2203 a, a \u2208 t \u2227 \u2191f \u2264 \ud835\udcdd a\nf : Ultrafilter (\u03b1 \u00d7 \u03b2)\nhfs : \u2191f \u2264 \ud835\udcdf (s \u00d7\u02e2 t)\n\u22a2 \u2203 a, a \u2208 s \u00d7\u02e2 t \u2227 \u2191f \u2264 \ud835\udcdd a\n[PROOFSTEP]\nrw [le_principal_iff] at hfs \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nhs : \u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 \ud835\udcdf s \u2192 \u2203 a, a \u2208 s \u2227 \u2191f \u2264 \ud835\udcdd a\nht : \u2200 (f : Ultrafilter \u03b2), \u2191f \u2264 \ud835\udcdf t \u2192 \u2203 a, a \u2208 t \u2227 \u2191f \u2264 \ud835\udcdd a\nf : Ultrafilter (\u03b1 \u00d7 \u03b2)\nhfs : s \u00d7\u02e2 t \u2208 \u2191f\n\u22a2 \u2203 a, a \u2208 s \u00d7\u02e2 t \u2227 \u2191f \u2264 \ud835\udcdd a\n[PROOFSTEP]\nobtain \u27e8a : \u03b1, sa : a \u2208 s, ha : map Prod.fst f.1 \u2264 \ud835\udcdd a\u27e9 :=\n hs (f.map Prod.fst) (le_principal_iff.2 <| mem_map.2 <| mem_of_superset hfs fun x => And.left)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nhs : \u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 \ud835\udcdf s \u2192 \u2203 a, a \u2208 s \u2227 \u2191f \u2264 \ud835\udcdd a\nht : \u2200 (f : Ultrafilter \u03b2), \u2191f \u2264 \ud835\udcdf t \u2192 \u2203 a, a \u2208 t \u2227 \u2191f \u2264 \ud835\udcdd a\nf : Ultrafilter (\u03b1 \u00d7 \u03b2)\nhfs : s \u00d7\u02e2 t \u2208 \u2191f\na : \u03b1\nsa : a \u2208 s\nha : map Prod.fst \u2191f \u2264 \ud835\udcdd a\n\u22a2 \u2203 a, a \u2208 s \u00d7\u02e2 t \u2227 \u2191f \u2264 \ud835\udcdd a\n[PROOFSTEP]\nobtain \u27e8b : \u03b2, tb : b \u2208 t, hb : map Prod.snd f.1 \u2264 \ud835\udcdd b\u27e9 :=\n ht (f.map Prod.snd) (le_principal_iff.2 <| mem_map.2 <| mem_of_superset hfs fun x => And.right)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nhs : \u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 \ud835\udcdf s \u2192 \u2203 a, a \u2208 s \u2227 \u2191f \u2264 \ud835\udcdd a\nht : \u2200 (f : Ultrafilter \u03b2), \u2191f \u2264 \ud835\udcdf t \u2192 \u2203 a, a \u2208 t \u2227 \u2191f \u2264 \ud835\udcdd a\nf : Ultrafilter (\u03b1 \u00d7 \u03b2)\nhfs : s \u00d7\u02e2 t \u2208 \u2191f\na : \u03b1\nsa : a \u2208 s\nha : map Prod.fst \u2191f \u2264 \ud835\udcdd a\nb : \u03b2\ntb : b \u2208 t\nhb : map Prod.snd \u2191f \u2264 \ud835\udcdd b\n\u22a2 \u2203 a, a \u2208 s \u00d7\u02e2 t \u2227 \u2191f \u2264 \ud835\udcdd a\n[PROOFSTEP]\nrw [map_le_iff_le_comap] at ha hb \n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nhs : \u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 \ud835\udcdf s \u2192 \u2203 a, a \u2208 s \u2227 \u2191f \u2264 \ud835\udcdd a\nht : \u2200 (f : Ultrafilter \u03b2), \u2191f \u2264 \ud835\udcdf t \u2192 \u2203 a, a \u2208 t \u2227 \u2191f \u2264 \ud835\udcdd a\nf : Ultrafilter (\u03b1 \u00d7 \u03b2)\nhfs : s \u00d7\u02e2 t \u2208 \u2191f\na : \u03b1\nsa : a \u2208 s\nha : \u2191f \u2264 comap Prod.fst (\ud835\udcdd a)\nb : \u03b2\ntb : b \u2208 t\nhb : \u2191f \u2264 comap Prod.snd (\ud835\udcdd b)\n\u22a2 \u2203 a, a \u2208 s \u00d7\u02e2 t \u2227 \u2191f \u2264 \ud835\udcdd a\n[PROOFSTEP]\nrefine' \u27e8\u27e8a, b\u27e9, \u27e8sa, tb\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nhs : \u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 \ud835\udcdf s \u2192 \u2203 a, a \u2208 s \u2227 \u2191f \u2264 \ud835\udcdd a\nht : \u2200 (f : Ultrafilter \u03b2), \u2191f \u2264 \ud835\udcdf t \u2192 \u2203 a, a \u2208 t \u2227 \u2191f \u2264 \ud835\udcdd a\nf : Ultrafilter (\u03b1 \u00d7 \u03b2)\nhfs : s \u00d7\u02e2 t \u2208 \u2191f\na : \u03b1\nsa : a \u2208 s\nha : \u2191f \u2264 comap Prod.fst (\ud835\udcdd a)\nb : \u03b2\ntb : b \u2208 t\nhb : \u2191f \u2264 comap Prod.snd (\ud835\udcdd b)\n\u22a2 \u2191f \u2264 \ud835\udcdd (a, b)\n[PROOFSTEP]\nrw [nhds_prod_eq]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nt : Set \u03b2\nhs : \u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 \ud835\udcdf s \u2192 \u2203 a, a \u2208 s \u2227 \u2191f \u2264 \ud835\udcdd a\nht : \u2200 (f : Ultrafilter \u03b2), \u2191f \u2264 \ud835\udcdf t \u2192 \u2203 a, a \u2208 t \u2227 \u2191f \u2264 \ud835\udcdd a\nf : Ultrafilter (\u03b1 \u00d7 \u03b2)\nhfs : s \u00d7\u02e2 t \u2208 \u2191f\na : \u03b1\nsa : a \u2208 s\nha : \u2191f \u2264 comap Prod.fst (\ud835\udcdd a)\nb : \u03b2\ntb : b \u2208 t\nhb : \u2191f \u2264 comap Prod.snd (\ud835\udcdd b)\n\u22a2 \u2191f \u2264 \ud835\udcdd a \u00d7\u02e2 \ud835\udcdd b\n[PROOFSTEP]\nexact le_inf ha hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : CompactSpace \u03b1\ninst\u271d : CompactSpace \u03b2\n\u22a2 IsCompact univ\n[PROOFSTEP]\nrw [\u2190 univ_prod_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : CompactSpace \u03b1\ninst\u271d : CompactSpace \u03b2\n\u22a2 IsCompact (univ \u00d7\u02e2 univ)\n[PROOFSTEP]\nexact isCompact_univ.prod isCompact_univ\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : CompactSpace \u03b1\ninst\u271d : CompactSpace \u03b2\n\u22a2 IsCompact univ\n[PROOFSTEP]\nrw [\u2190 range_inl_union_range_inr]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : CompactSpace \u03b1\ninst\u271d : CompactSpace \u03b2\n\u22a2 IsCompact (range Sum.inl \u222a range Sum.inr)\n[PROOFSTEP]\nexact (isCompact_range continuous_inl).union (isCompact_range continuous_inr)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\n\u22a2 CompactSpace ((i : \u03b9) \u00d7 \u03c0 i)\n[PROOFSTEP]\nrefine' \u27e8_\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\n\u22a2 IsCompact univ\n[PROOFSTEP]\nrw [Sigma.univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\n\u22a2 IsCompact (\u22c3 (a : \u03b9), range (Sigma.mk a))\n[PROOFSTEP]\nexact isCompact_iUnion fun i => isCompact_range continuous_sigmaMk\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u22a2 Filter.coprod (cocompact \u03b1) (cocompact \u03b2) = cocompact (\u03b1 \u00d7 \u03b2)\n[PROOFSTEP]\next S\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nS : Set (\u03b1 \u00d7 \u03b2)\n\u22a2 S \u2208 Filter.coprod (cocompact \u03b1) (cocompact \u03b2) \u2194 S \u2208 cocompact (\u03b1 \u00d7 \u03b2)\n[PROOFSTEP]\nsimp only [mem_coprod_iff, exists_prop, mem_comap, Filter.mem_cocompact]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nS : Set (\u03b1 \u00d7 \u03b2)\n\u22a2 ((\u2203 t\u2081, (\u2203 t, IsCompact t \u2227 t\u1d9c \u2286 t\u2081) \u2227 Prod.fst \u207b\u00b9' t\u2081 \u2286 S) \u2227\n \u2203 t\u2082, (\u2203 t, IsCompact t \u2227 t\u1d9c \u2286 t\u2082) \u2227 Prod.snd \u207b\u00b9' t\u2082 \u2286 S) \u2194\n \u2203 t, IsCompact t \u2227 t\u1d9c \u2286 S\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nS : Set (\u03b1 \u00d7 \u03b2)\n\u22a2 ((\u2203 t\u2081, (\u2203 t, IsCompact t \u2227 t\u1d9c \u2286 t\u2081) \u2227 Prod.fst \u207b\u00b9' t\u2081 \u2286 S) \u2227\n \u2203 t\u2082, (\u2203 t, IsCompact t \u2227 t\u1d9c \u2286 t\u2082) \u2227 Prod.snd \u207b\u00b9' t\u2082 \u2286 S) \u2192\n \u2203 t, IsCompact t \u2227 t\u1d9c \u2286 S\n[PROOFSTEP]\nrintro \u27e8\u27e8A, \u27e8t, ht, hAt\u27e9, hAS\u27e9, B, \u27e8t', ht', hBt'\u27e9, hBS\u27e9\n[GOAL]\ncase a.mp.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nS : Set (\u03b1 \u00d7 \u03b2)\nA : Set \u03b1\nhAS : Prod.fst \u207b\u00b9' A \u2286 S\nt : Set \u03b1\nht : IsCompact t\nhAt : t\u1d9c \u2286 A\nB : Set \u03b2\nhBS : Prod.snd \u207b\u00b9' B \u2286 S\nt' : Set \u03b2\nht' : IsCompact t'\nhBt' : t'\u1d9c \u2286 B\n\u22a2 \u2203 t, IsCompact t \u2227 t\u1d9c \u2286 S\n[PROOFSTEP]\nrefine' \u27e8t \u00d7\u02e2 t', ht.prod ht', _\u27e9\n[GOAL]\ncase a.mp.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nS : Set (\u03b1 \u00d7 \u03b2)\nA : Set \u03b1\nhAS : Prod.fst \u207b\u00b9' A \u2286 S\nt : Set \u03b1\nht : IsCompact t\nhAt : t\u1d9c \u2286 A\nB : Set \u03b2\nhBS : Prod.snd \u207b\u00b9' B \u2286 S\nt' : Set \u03b2\nht' : IsCompact t'\nhBt' : t'\u1d9c \u2286 B\n\u22a2 (t \u00d7\u02e2 t')\u1d9c \u2286 S\n[PROOFSTEP]\nrefine' Subset.trans _ (union_subset hAS hBS)\n[GOAL]\ncase a.mp.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nS : Set (\u03b1 \u00d7 \u03b2)\nA : Set \u03b1\nhAS : Prod.fst \u207b\u00b9' A \u2286 S\nt : Set \u03b1\nht : IsCompact t\nhAt : t\u1d9c \u2286 A\nB : Set \u03b2\nhBS : Prod.snd \u207b\u00b9' B \u2286 S\nt' : Set \u03b2\nht' : IsCompact t'\nhBt' : t'\u1d9c \u2286 B\n\u22a2 (t \u00d7\u02e2 t')\u1d9c \u2286 Prod.fst \u207b\u00b9' A \u222a Prod.snd \u207b\u00b9' B\n[PROOFSTEP]\nrw [compl_subset_comm] at hAt hBt' \u22a2\n[GOAL]\ncase a.mp.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nS : Set (\u03b1 \u00d7 \u03b2)\nA : Set \u03b1\nhAS : Prod.fst \u207b\u00b9' A \u2286 S\nt : Set \u03b1\nht : IsCompact t\nhAt : A\u1d9c \u2286 t\nB : Set \u03b2\nhBS : Prod.snd \u207b\u00b9' B \u2286 S\nt' : Set \u03b2\nht' : IsCompact t'\nhBt' : B\u1d9c \u2286 t'\n\u22a2 (Prod.fst \u207b\u00b9' A \u222a Prod.snd \u207b\u00b9' B)\u1d9c \u2286 t \u00d7\u02e2 t'\n[PROOFSTEP]\nrefine' Subset.trans (fun x => _) (Set.prod_mono hAt hBt')\n[GOAL]\ncase a.mp.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nS : Set (\u03b1 \u00d7 \u03b2)\nA : Set \u03b1\nhAS : Prod.fst \u207b\u00b9' A \u2286 S\nt : Set \u03b1\nht : IsCompact t\nhAt : A\u1d9c \u2286 t\nB : Set \u03b2\nhBS : Prod.snd \u207b\u00b9' B \u2286 S\nt' : Set \u03b2\nht' : IsCompact t'\nhBt' : B\u1d9c \u2286 t'\nx : \u03b1 \u00d7 \u03b2\n\u22a2 x \u2208 (Prod.fst \u207b\u00b9' A \u222a Prod.snd \u207b\u00b9' B)\u1d9c \u2192 x \u2208 A\u1d9c \u00d7\u02e2 B\u1d9c\n[PROOFSTEP]\nsimp only [compl_union, mem_inter_iff, mem_prod, mem_preimage, mem_compl_iff]\n[GOAL]\ncase a.mp.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nS : Set (\u03b1 \u00d7 \u03b2)\nA : Set \u03b1\nhAS : Prod.fst \u207b\u00b9' A \u2286 S\nt : Set \u03b1\nht : IsCompact t\nhAt : A\u1d9c \u2286 t\nB : Set \u03b2\nhBS : Prod.snd \u207b\u00b9' B \u2286 S\nt' : Set \u03b2\nht' : IsCompact t'\nhBt' : B\u1d9c \u2286 t'\nx : \u03b1 \u00d7 \u03b2\n\u22a2 \u00acx.fst \u2208 A \u2227 \u00acx.snd \u2208 B \u2192 \u00acx.fst \u2208 A \u2227 \u00acx.snd \u2208 B\n[PROOFSTEP]\ntauto\n[GOAL]\ncase a.mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nS : Set (\u03b1 \u00d7 \u03b2)\n\u22a2 (\u2203 t, IsCompact t \u2227 t\u1d9c \u2286 S) \u2192\n (\u2203 t\u2081, (\u2203 t, IsCompact t \u2227 t\u1d9c \u2286 t\u2081) \u2227 Prod.fst \u207b\u00b9' t\u2081 \u2286 S) \u2227\n \u2203 t\u2082, (\u2203 t, IsCompact t \u2227 t\u1d9c \u2286 t\u2082) \u2227 Prod.snd \u207b\u00b9' t\u2082 \u2286 S\n[PROOFSTEP]\nrintro \u27e8t, ht, htS\u27e9\n[GOAL]\ncase a.mpr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nS t : Set (\u03b1 \u00d7 \u03b2)\nht : IsCompact t\nhtS : t\u1d9c \u2286 S\n\u22a2 (\u2203 t\u2081, (\u2203 t, IsCompact t \u2227 t\u1d9c \u2286 t\u2081) \u2227 Prod.fst \u207b\u00b9' t\u2081 \u2286 S) \u2227 \u2203 t\u2082, (\u2203 t, IsCompact t \u2227 t\u1d9c \u2286 t\u2082) \u2227 Prod.snd \u207b\u00b9' t\u2082 \u2286 S\n[PROOFSTEP]\nrefine' \u27e8\u27e8(Prod.fst '' t)\u1d9c, _, _\u27e9, \u27e8(Prod.snd '' t)\u1d9c, _, _\u27e9\u27e9\n[GOAL]\ncase a.mpr.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nS t : Set (\u03b1 \u00d7 \u03b2)\nht : IsCompact t\nhtS : t\u1d9c \u2286 S\n\u22a2 \u2203 t_1, IsCompact t_1 \u2227 t_1\u1d9c \u2286 (Prod.fst '' t)\u1d9c\n[PROOFSTEP]\nexact \u27e8Prod.fst '' t, ht.image continuous_fst, Subset.rfl\u27e9\n[GOAL]\ncase a.mpr.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nS t : Set (\u03b1 \u00d7 \u03b2)\nht : IsCompact t\nhtS : t\u1d9c \u2286 S\n\u22a2 Prod.fst \u207b\u00b9' (Prod.fst '' t)\u1d9c \u2286 S\n[PROOFSTEP]\nrw [preimage_compl]\n[GOAL]\ncase a.mpr.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nS t : Set (\u03b1 \u00d7 \u03b2)\nht : IsCompact t\nhtS : t\u1d9c \u2286 S\n\u22a2 (Prod.fst \u207b\u00b9' (Prod.fst '' t))\u1d9c \u2286 S\n[PROOFSTEP]\nrw [compl_subset_comm] at htS \u22a2\n[GOAL]\ncase a.mpr.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nS t : Set (\u03b1 \u00d7 \u03b2)\nht : IsCompact t\nhtS : S\u1d9c \u2286 t\n\u22a2 S\u1d9c \u2286 Prod.fst \u207b\u00b9' (Prod.fst '' t)\n[PROOFSTEP]\nexact htS.trans (subset_preimage_image Prod.fst _)\n[GOAL]\ncase a.mpr.intro.intro.refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nS t : Set (\u03b1 \u00d7 \u03b2)\nht : IsCompact t\nhtS : t\u1d9c \u2286 S\n\u22a2 \u2203 t_1, IsCompact t_1 \u2227 t_1\u1d9c \u2286 (Prod.snd '' t)\u1d9c\n[PROOFSTEP]\nexact \u27e8Prod.snd '' t, ht.image continuous_snd, Subset.rfl\u27e9\n[GOAL]\ncase a.mpr.intro.intro.refine'_4\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nS t : Set (\u03b1 \u00d7 \u03b2)\nht : IsCompact t\nhtS : t\u1d9c \u2286 S\n\u22a2 Prod.snd \u207b\u00b9' (Prod.snd '' t)\u1d9c \u2286 S\n[PROOFSTEP]\nrw [preimage_compl]\n[GOAL]\ncase a.mpr.intro.intro.refine'_4\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nS t : Set (\u03b1 \u00d7 \u03b2)\nht : IsCompact t\nhtS : t\u1d9c \u2286 S\n\u22a2 (Prod.snd \u207b\u00b9' (Prod.snd '' t))\u1d9c \u2286 S\n[PROOFSTEP]\nrw [compl_subset_comm] at htS \u22a2\n[GOAL]\ncase a.mpr.intro.intro.refine'_4\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nS t : Set (\u03b1 \u00d7 \u03b2)\nht : IsCompact t\nhtS : S\u1d9c \u2286 t\n\u22a2 S\u1d9c \u2286 Prod.snd \u207b\u00b9' (Prod.snd '' t)\n[PROOFSTEP]\nexact htS.trans (subset_preimage_image Prod.snd _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\n\u22a2 NoncompactSpace (\u03b1 \u00d7 \u03b2) \u2194 NoncompactSpace \u03b1 \u2227 Nonempty \u03b2 \u2228 Nonempty \u03b1 \u2227 NoncompactSpace \u03b2\n[PROOFSTEP]\nsimp [\u2190 Filter.cocompact_neBot_iff, \u2190 Filter.coprod_cocompact, Filter.coprod_neBot_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\n\u22a2 (\u2200 (i : \u03b9), IsCompact (s i)) \u2192 IsCompact {x | \u2200 (i : \u03b9), x i \u2208 s i}\n[PROOFSTEP]\nsimp only [isCompact_iff_ultrafilter_le_nhds, nhds_pi, Filter.pi, exists_prop, mem_setOf_eq, le_iInf_iff,\n le_principal_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\n\u22a2 (\u2200 (i : \u03b9) (f : Ultrafilter (\u03c0 i)), s i \u2208 \u2191f \u2192 \u2203 a, a \u2208 s i \u2227 \u2191f \u2264 \ud835\udcdd a) \u2192\n \u2200 (f : Ultrafilter ((i : \u03b9) \u2192 \u03c0 i)),\n {x | \u2200 (i : \u03b9), x i \u2208 s i} \u2208 \u2191f \u2192 \u2203 a, (\u2200 (i : \u03b9), a i \u2208 s i) \u2227 \u2200 (i : \u03b9), \u2191f \u2264 comap (Function.eval i) (\ud835\udcdd (a i))\n[PROOFSTEP]\nintro h f hfs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nh : \u2200 (i : \u03b9) (f : Ultrafilter (\u03c0 i)), s i \u2208 \u2191f \u2192 \u2203 a, a \u2208 s i \u2227 \u2191f \u2264 \ud835\udcdd a\nf : Ultrafilter ((i : \u03b9) \u2192 \u03c0 i)\nhfs : {x | \u2200 (i : \u03b9), x i \u2208 s i} \u2208 \u2191f\n\u22a2 \u2203 a, (\u2200 (i : \u03b9), a i \u2208 s i) \u2227 \u2200 (i : \u03b9), \u2191f \u2264 comap (Function.eval i) (\ud835\udcdd (a i))\n[PROOFSTEP]\nhave : \u2200 i : \u03b9, \u2203 a, a \u2208 s i \u2227 Tendsto (Function.eval i) f (\ud835\udcdd a) :=\n by\n refine fun i => h i (f.map _) (mem_map.2 ?_)\n exact mem_of_superset hfs fun x hx => hx i\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nh : \u2200 (i : \u03b9) (f : Ultrafilter (\u03c0 i)), s i \u2208 \u2191f \u2192 \u2203 a, a \u2208 s i \u2227 \u2191f \u2264 \ud835\udcdd a\nf : Ultrafilter ((i : \u03b9) \u2192 \u03c0 i)\nhfs : {x | \u2200 (i : \u03b9), x i \u2208 s i} \u2208 \u2191f\n\u22a2 \u2200 (i : \u03b9), \u2203 a, a \u2208 s i \u2227 Tendsto (Function.eval i) (\u2191f) (\ud835\udcdd a)\n[PROOFSTEP]\nrefine fun i => h i (f.map _) (mem_map.2 ?_)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nh : \u2200 (i : \u03b9) (f : Ultrafilter (\u03c0 i)), s i \u2208 \u2191f \u2192 \u2203 a, a \u2208 s i \u2227 \u2191f \u2264 \ud835\udcdd a\nf : Ultrafilter ((i : \u03b9) \u2192 \u03c0 i)\nhfs : {x | \u2200 (i : \u03b9), x i \u2208 s i} \u2208 \u2191f\ni : \u03b9\n\u22a2 Function.eval i \u207b\u00b9' s i \u2208 \u2191f\n[PROOFSTEP]\nexact mem_of_superset hfs fun x hx => hx i\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nh : \u2200 (i : \u03b9) (f : Ultrafilter (\u03c0 i)), s i \u2208 \u2191f \u2192 \u2203 a, a \u2208 s i \u2227 \u2191f \u2264 \ud835\udcdd a\nf : Ultrafilter ((i : \u03b9) \u2192 \u03c0 i)\nhfs : {x | \u2200 (i : \u03b9), x i \u2208 s i} \u2208 \u2191f\nthis : \u2200 (i : \u03b9), \u2203 a, a \u2208 s i \u2227 Tendsto (Function.eval i) (\u2191f) (\ud835\udcdd a)\n\u22a2 \u2203 a, (\u2200 (i : \u03b9), a i \u2208 s i) \u2227 \u2200 (i : \u03b9), \u2191f \u2264 comap (Function.eval i) (\ud835\udcdd (a i))\n[PROOFSTEP]\nchoose a ha using this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nh : \u2200 (i : \u03b9) (f : Ultrafilter (\u03c0 i)), s i \u2208 \u2191f \u2192 \u2203 a, a \u2208 s i \u2227 \u2191f \u2264 \ud835\udcdd a\nf : Ultrafilter ((i : \u03b9) \u2192 \u03c0 i)\nhfs : {x | \u2200 (i : \u03b9), x i \u2208 s i} \u2208 \u2191f\na : (i : \u03b9) \u2192 \u03c0 i\nha : \u2200 (i : \u03b9), a i \u2208 s i \u2227 Tendsto (Function.eval i) (\u2191f) (\ud835\udcdd (a i))\n\u22a2 \u2203 a, (\u2200 (i : \u03b9), a i \u2208 s i) \u2227 \u2200 (i : \u03b9), \u2191f \u2264 comap (Function.eval i) (\ud835\udcdd (a i))\n[PROOFSTEP]\nexact \u27e8a, fun i => (ha i).left, fun i => (ha i).right.le_comap\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nh : \u2200 (i : \u03b9), IsCompact (s i)\n\u22a2 IsCompact (Set.pi univ s)\n[PROOFSTEP]\nconvert isCompact_pi_infinite h\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nh : \u2200 (i : \u03b9), IsCompact (s i)\n\u22a2 Set.pi univ s = {x | \u2200 (i : \u03b9), x i \u2208 s i}\n[PROOFSTEP]\nsimp only [\u2190 mem_univ_pi, setOf_mem_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\n\u22a2 IsCompact univ\n[PROOFSTEP]\nrw [\u2190 pi_univ univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\n\u22a2 IsCompact (Set.pi univ fun i => univ)\n[PROOFSTEP]\nexact isCompact_univ_pi fun i => isCompact_univ\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\n\u03b4 : Type u_3\n\u03ba : \u03b4 \u2192 Type u_4\ninst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d)\n\u22a2 (Filter.coprod\u1d62 fun d => cocompact (\u03ba d)) = cocompact ((d : \u03b4) \u2192 \u03ba d)\n[PROOFSTEP]\nrefine' le_antisymm (iSup_le fun i => Filter.comap_cocompact_le (continuous_apply i)) _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\n\u03b4 : Type u_3\n\u03ba : \u03b4 \u2192 Type u_4\ninst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d)\n\u22a2 cocompact ((d : \u03b4) \u2192 \u03ba d) \u2264 Filter.coprod\u1d62 fun d => cocompact (\u03ba d)\n[PROOFSTEP]\nrefine' compl_surjective.forall.2 fun s H => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\n\u03b4 : Type u_3\n\u03ba : \u03b4 \u2192 Type u_4\ninst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d)\ns : Set ((i : \u03b4) \u2192 \u03ba i)\nH : s\u1d9c \u2208 Filter.coprod\u1d62 fun d => cocompact (\u03ba d)\n\u22a2 s\u1d9c \u2208 cocompact ((d : \u03b4) \u2192 \u03ba d)\n[PROOFSTEP]\nsimp only [compl_mem_coprod\u1d62, Filter.mem_cocompact, compl_subset_compl, image_subset_iff] at H \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\n\u03b4 : Type u_3\n\u03ba : \u03b4 \u2192 Type u_4\ninst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d)\ns : Set ((i : \u03b4) \u2192 \u03ba i)\nH : \u2200 (i : \u03b4), \u2203 t, IsCompact t \u2227 s \u2286 Function.eval i \u207b\u00b9' t\n\u22a2 \u2203 t, IsCompact t \u2227 s \u2286 t\n[PROOFSTEP]\nchoose K hKc htK using H\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\n\u03b4 : Type u_3\n\u03ba : \u03b4 \u2192 Type u_4\ninst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d)\ns : Set ((i : \u03b4) \u2192 \u03ba i)\nK : (i : \u03b4) \u2192 Set (\u03ba i)\nhKc : \u2200 (i : \u03b4), IsCompact (K i)\nhtK : \u2200 (i : \u03b4), s \u2286 Function.eval i \u207b\u00b9' K i\n\u22a2 \u2203 t, IsCompact t \u2227 s \u2286 t\n[PROOFSTEP]\nexact \u27e8Set.pi univ K, isCompact_univ_pi hKc, fun f hf i _ => htK i hf\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompactSpace \u03b1\n\u22a2 IsCompact univ\n[PROOFSTEP]\nrw [\u2190 range_quot_mk]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompactSpace \u03b1\n\u22a2 IsCompact (range (mk r))\n[PROOFSTEP]\nexact isCompact_range continuous_quot_mk\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nx : \u03b1\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 \ud835\udcdd x \u2192 \u2203 r, r \u2208 \ud835\udcdd x \u2227 IsCompact r \u2227 r \u2286 t\n[PROOFSTEP]\nsimpa only [and_comm] using LocallyCompactSpace.local_compact_nhds x\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : Finite \u03b9\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\nhn : n \u2208 \ud835\udcdd t\n\u22a2 \u2203 s, s \u2208 \ud835\udcdd t \u2227 s \u2286 n \u2227 IsCompact s\n[PROOFSTEP]\nrw [nhds_pi, Filter.mem_pi] at hn \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : Finite \u03b9\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\nhn : \u2203 I, Set.Finite I \u2227 \u2203 t_1, (\u2200 (i : \u03b9), t_1 i \u2208 \ud835\udcdd (t i)) \u2227 Set.pi I t_1 \u2286 n\n\u22a2 \u2203 s, s \u2208 \ud835\udcdd t \u2227 s \u2286 n \u2227 IsCompact s\n[PROOFSTEP]\nobtain \u27e8s, -, n', hn', hsub\u27e9 := hn\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : Finite \u03b9\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\ns : Set \u03b9\nn' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn' : \u2200 (i : \u03b9), n' i \u2208 \ud835\udcdd (t i)\nhsub : Set.pi s n' \u2286 n\n\u22a2 \u2203 s, s \u2208 \ud835\udcdd t \u2227 s \u2286 n \u2227 IsCompact s\n[PROOFSTEP]\nchoose n'' hn'' hsub' hc using fun i => LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : Finite \u03b9\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\ns : Set \u03b9\nn' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn' : \u2200 (i : \u03b9), n' i \u2208 \ud835\udcdd (t i)\nhsub : Set.pi s n' \u2286 n\nn'' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn'' : \u2200 (i : \u03b9), n'' i \u2208 \ud835\udcdd (t i)\nhsub' : \u2200 (i : \u03b9), n'' i \u2286 n' i\nhc : \u2200 (i : \u03b9), IsCompact (n'' i)\n\u22a2 \u2203 s, s \u2208 \ud835\udcdd t \u2227 s \u2286 n \u2227 IsCompact s\n[PROOFSTEP]\nrefine' \u27e8(Set.univ : Set \u03b9).pi n'', _, subset_trans (fun _ h => _) hsub, isCompact_univ_pi hc\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : Finite \u03b9\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\ns : Set \u03b9\nn' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn' : \u2200 (i : \u03b9), n' i \u2208 \ud835\udcdd (t i)\nhsub : Set.pi s n' \u2286 n\nn'' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn'' : \u2200 (i : \u03b9), n'' i \u2208 \ud835\udcdd (t i)\nhsub' : \u2200 (i : \u03b9), n'' i \u2286 n' i\nhc : \u2200 (i : \u03b9), IsCompact (n'' i)\n\u22a2 Set.pi univ n'' \u2208 \ud835\udcdd t\n[PROOFSTEP]\nexact (set_pi_mem_nhds_iff (@Set.finite_univ \u03b9 _) _).mpr fun i _ => hn'' i\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : Finite \u03b9\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\ns : Set \u03b9\nn' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn' : \u2200 (i : \u03b9), n' i \u2208 \ud835\udcdd (t i)\nhsub : Set.pi s n' \u2286 n\nn'' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn'' : \u2200 (i : \u03b9), n'' i \u2208 \ud835\udcdd (t i)\nhsub' : \u2200 (i : \u03b9), n'' i \u2286 n' i\nhc : \u2200 (i : \u03b9), IsCompact (n'' i)\nx\u271d : (i : \u03b9) \u2192 \u03c0 i\nh : x\u271d \u2208 Set.pi univ n''\n\u22a2 x\u271d \u2208 Set.pi s n'\n[PROOFSTEP]\nexact fun i _ => hsub' i (h i trivial)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\nhn : n \u2208 \ud835\udcdd t\n\u22a2 \u2203 s, s \u2208 \ud835\udcdd t \u2227 s \u2286 n \u2227 IsCompact s\n[PROOFSTEP]\nrw [nhds_pi, Filter.mem_pi] at hn \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\nhn : \u2203 I, Set.Finite I \u2227 \u2203 t_1, (\u2200 (i : \u03b9), t_1 i \u2208 \ud835\udcdd (t i)) \u2227 Set.pi I t_1 \u2286 n\n\u22a2 \u2203 s, s \u2208 \ud835\udcdd t \u2227 s \u2286 n \u2227 IsCompact s\n[PROOFSTEP]\nobtain \u27e8s, hs, n', hn', hsub\u27e9 := hn\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\ns : Set \u03b9\nhs : Set.Finite s\nn' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn' : \u2200 (i : \u03b9), n' i \u2208 \ud835\udcdd (t i)\nhsub : Set.pi s n' \u2286 n\n\u22a2 \u2203 s, s \u2208 \ud835\udcdd t \u2227 s \u2286 n \u2227 IsCompact s\n[PROOFSTEP]\nchoose n'' hn'' hsub' hc using fun i => LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\ns : Set \u03b9\nhs : Set.Finite s\nn' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn' : \u2200 (i : \u03b9), n' i \u2208 \ud835\udcdd (t i)\nhsub : Set.pi s n' \u2286 n\nn'' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn'' : \u2200 (i : \u03b9), n'' i \u2208 \ud835\udcdd (t i)\nhsub' : \u2200 (i : \u03b9), n'' i \u2286 n' i\nhc : \u2200 (i : \u03b9), IsCompact (n'' i)\n\u22a2 \u2203 s, s \u2208 \ud835\udcdd t \u2227 s \u2286 n \u2227 IsCompact s\n[PROOFSTEP]\nrefine' \u27e8s.pi n'', _, subset_trans (fun _ => _) hsub, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\ns : Set \u03b9\nhs : Set.Finite s\nn' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn' : \u2200 (i : \u03b9), n' i \u2208 \ud835\udcdd (t i)\nhsub : Set.pi s n' \u2286 n\nn'' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn'' : \u2200 (i : \u03b9), n'' i \u2208 \ud835\udcdd (t i)\nhsub' : \u2200 (i : \u03b9), n'' i \u2286 n' i\nhc : \u2200 (i : \u03b9), IsCompact (n'' i)\n\u22a2 Set.pi s n'' \u2208 \ud835\udcdd t\n[PROOFSTEP]\nexact (set_pi_mem_nhds_iff hs _).mpr fun i _ => hn'' i\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\ns : Set \u03b9\nhs : Set.Finite s\nn' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn' : \u2200 (i : \u03b9), n' i \u2208 \ud835\udcdd (t i)\nhsub : Set.pi s n' \u2286 n\nn'' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn'' : \u2200 (i : \u03b9), n'' i \u2208 \ud835\udcdd (t i)\nhsub' : \u2200 (i : \u03b9), n'' i \u2286 n' i\nhc : \u2200 (i : \u03b9), IsCompact (n'' i)\nx\u271d : (i : \u03b9) \u2192 \u03c0 i\n\u22a2 x\u271d \u2208 Set.pi s n'' \u2192 x\u271d \u2208 Set.pi s n'\n[PROOFSTEP]\nexact forall\u2082_imp fun i _ hi' => hsub' i hi'\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\ns : Set \u03b9\nhs : Set.Finite s\nn' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn' : \u2200 (i : \u03b9), n' i \u2208 \ud835\udcdd (t i)\nhsub : Set.pi s n' \u2286 n\nn'' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn'' : \u2200 (i : \u03b9), n'' i \u2208 \ud835\udcdd (t i)\nhsub' : \u2200 (i : \u03b9), n'' i \u2286 n' i\nhc : \u2200 (i : \u03b9), IsCompact (n'' i)\n\u22a2 IsCompact (Set.pi s n'')\n[PROOFSTEP]\nrw [\u2190 Set.univ_pi_ite]\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\ns : Set \u03b9\nhs : Set.Finite s\nn' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn' : \u2200 (i : \u03b9), n' i \u2208 \ud835\udcdd (t i)\nhsub : Set.pi s n' \u2286 n\nn'' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn'' : \u2200 (i : \u03b9), n'' i \u2208 \ud835\udcdd (t i)\nhsub' : \u2200 (i : \u03b9), n'' i \u2286 n' i\nhc : \u2200 (i : \u03b9), IsCompact (n'' i)\n\u22a2 IsCompact (Set.pi univ fun i => if i \u2208 s then n'' i else univ)\n[PROOFSTEP]\nrefine' isCompact_univ_pi fun i => _\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\ns : Set \u03b9\nhs : Set.Finite s\nn' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn' : \u2200 (i : \u03b9), n' i \u2208 \ud835\udcdd (t i)\nhsub : Set.pi s n' \u2286 n\nn'' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn'' : \u2200 (i : \u03b9), n'' i \u2208 \ud835\udcdd (t i)\nhsub' : \u2200 (i : \u03b9), n'' i \u2286 n' i\nhc : \u2200 (i : \u03b9), IsCompact (n'' i)\ni : \u03b9\n\u22a2 IsCompact (if i \u2208 s then n'' i else univ)\n[PROOFSTEP]\nby_cases h : i \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\ns : Set \u03b9\nhs : Set.Finite s\nn' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn' : \u2200 (i : \u03b9), n' i \u2208 \ud835\udcdd (t i)\nhsub : Set.pi s n' \u2286 n\nn'' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn'' : \u2200 (i : \u03b9), n'' i \u2208 \ud835\udcdd (t i)\nhsub' : \u2200 (i : \u03b9), n'' i \u2286 n' i\nhc : \u2200 (i : \u03b9), IsCompact (n'' i)\ni : \u03b9\nh : i \u2208 s\n\u22a2 IsCompact (if i \u2208 s then n'' i else univ)\n[PROOFSTEP]\nrw [if_pos h]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\ns : Set \u03b9\nhs : Set.Finite s\nn' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn' : \u2200 (i : \u03b9), n' i \u2208 \ud835\udcdd (t i)\nhsub : Set.pi s n' \u2286 n\nn'' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn'' : \u2200 (i : \u03b9), n'' i \u2208 \ud835\udcdd (t i)\nhsub' : \u2200 (i : \u03b9), n'' i \u2286 n' i\nhc : \u2200 (i : \u03b9), IsCompact (n'' i)\ni : \u03b9\nh : i \u2208 s\n\u22a2 IsCompact (n'' i)\n[PROOFSTEP]\nexact hc i\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\ns : Set \u03b9\nhs : Set.Finite s\nn' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn' : \u2200 (i : \u03b9), n' i \u2208 \ud835\udcdd (t i)\nhsub : Set.pi s n' \u2286 n\nn'' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn'' : \u2200 (i : \u03b9), n'' i \u2208 \ud835\udcdd (t i)\nhsub' : \u2200 (i : \u03b9), n'' i \u2286 n' i\nhc : \u2200 (i : \u03b9), IsCompact (n'' i)\ni : \u03b9\nh : \u00aci \u2208 s\n\u22a2 IsCompact (if i \u2208 s then n'' i else univ)\n[PROOFSTEP]\nrw [if_neg h]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), LocallyCompactSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), CompactSpace (\u03c0 i)\nt : (i : \u03b9) \u2192 \u03c0 i\nn : Set ((i : \u03b9) \u2192 \u03c0 i)\ns : Set \u03b9\nhs : Set.Finite s\nn' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn' : \u2200 (i : \u03b9), n' i \u2208 \ud835\udcdd (t i)\nhsub : Set.pi s n' \u2286 n\nn'' : (i : \u03b9) \u2192 Set (\u03c0 i)\nhn'' : \u2200 (i : \u03b9), n'' i \u2208 \ud835\udcdd (t i)\nhsub' : \u2200 (i : \u03b9), n'' i \u2286 n' i\nhc : \u2200 (i : \u03b9), IsCompact (n'' i)\ni : \u03b9\nh : \u00aci \u2208 s\n\u22a2 IsCompact univ\n[PROOFSTEP]\nexact CompactSpace.isCompact_univ\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nx : \u03b1\nU : Set \u03b1\nhU : IsOpen U\nhx : x \u2208 U\n\u22a2 \u2203 K, IsCompact K \u2227 x \u2208 interior K \u2227 K \u2286 U\n[PROOFSTEP]\nrcases LocallyCompactSpace.local_compact_nhds x U (hU.mem_nhds hx) with \u27e8K, h1K, h2K, h3K\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nx : \u03b1\nU : Set \u03b1\nhU : IsOpen U\nhx : x \u2208 U\nK : Set \u03b1\nh1K : K \u2208 \ud835\udcdd x\nh2K : K \u2286 U\nh3K : IsCompact K\n\u22a2 \u2203 K, IsCompact K \u2227 x \u2208 interior K \u2227 K \u2286 U\n[PROOFSTEP]\nexact \u27e8K, h3K, mem_interior_iff_mem_nhds.2 h1K, h2K\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nh\u03b1 : LocallyCompactSpace \u03b1\nK U : Set \u03b1\nhK : IsCompact K\nhU : IsOpen U\nh_KU : K \u2286 U\n\u22a2 \u2203 L, IsCompact L \u2227 K \u2286 interior L \u2227 L \u2286 U\n[PROOFSTEP]\nchoose V hVc hxV hKV using fun x : K => exists_compact_subset hU (h_KU x.2)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nh\u03b1 : LocallyCompactSpace \u03b1\nK U : Set \u03b1\nhK : IsCompact K\nhU : IsOpen U\nh_KU : K \u2286 U\nV : \u2191K \u2192 Set \u03b1\nhVc : \u2200 (x : \u2191K), IsCompact (V x)\nhxV : \u2200 (x : \u2191K), \u2191x \u2208 interior (V x)\nhKV : \u2200 (x : \u2191K), V x \u2286 U\n\u22a2 \u2203 L, IsCompact L \u2227 K \u2286 interior L \u2227 L \u2286 U\n[PROOFSTEP]\nhave : K \u2286 \u22c3 x, interior (V x) := fun x hx => mem_iUnion.2 \u27e8\u27e8x, hx\u27e9, hxV _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nh\u03b1 : LocallyCompactSpace \u03b1\nK U : Set \u03b1\nhK : IsCompact K\nhU : IsOpen U\nh_KU : K \u2286 U\nV : \u2191K \u2192 Set \u03b1\nhVc : \u2200 (x : \u2191K), IsCompact (V x)\nhxV : \u2200 (x : \u2191K), \u2191x \u2208 interior (V x)\nhKV : \u2200 (x : \u2191K), V x \u2286 U\nthis : K \u2286 \u22c3 (x : \u2191K), interior (V x)\n\u22a2 \u2203 L, IsCompact L \u2227 K \u2286 interior L \u2227 L \u2286 U\n[PROOFSTEP]\nrcases hK.elim_finite_subcover _ (fun x => @isOpen_interior \u03b1 _ (V x)) this with \u27e8t, ht\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nh\u03b1 : LocallyCompactSpace \u03b1\nK U : Set \u03b1\nhK : IsCompact K\nhU : IsOpen U\nh_KU : K \u2286 U\nV : \u2191K \u2192 Set \u03b1\nhVc : \u2200 (x : \u2191K), IsCompact (V x)\nhxV : \u2200 (x : \u2191K), \u2191x \u2208 interior (V x)\nhKV : \u2200 (x : \u2191K), V x \u2286 U\nthis : K \u2286 \u22c3 (x : \u2191K), interior (V x)\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), interior (V i)\n\u22a2 \u2203 L, IsCompact L \u2227 K \u2286 interior L \u2227 L \u2286 U\n[PROOFSTEP]\nrefine' \u27e8_, t.isCompact_biUnion fun x _ => hVc x, fun x hx => _, Set.iUnion\u2082_subset fun i _ => hKV i\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nh\u03b1 : LocallyCompactSpace \u03b1\nK U : Set \u03b1\nhK : IsCompact K\nhU : IsOpen U\nh_KU : K \u2286 U\nV : \u2191K \u2192 Set \u03b1\nhVc : \u2200 (x : \u2191K), IsCompact (V x)\nhxV : \u2200 (x : \u2191K), \u2191x \u2208 interior (V x)\nhKV : \u2200 (x : \u2191K), V x \u2286 U\nthis : K \u2286 \u22c3 (x : \u2191K), interior (V x)\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), interior (V i)\nx : \u03b1\nhx : x \u2208 K\n\u22a2 x \u2208 interior (\u22c3 (i : \u2191K) (_ : i \u2208 t), V i)\n[PROOFSTEP]\nrcases mem_iUnion\u2082.1 (ht hx) with \u27e8y, hyt, hy\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nh\u03b1 : LocallyCompactSpace \u03b1\nK U : Set \u03b1\nhK : IsCompact K\nhU : IsOpen U\nh_KU : K \u2286 U\nV : \u2191K \u2192 Set \u03b1\nhVc : \u2200 (x : \u2191K), IsCompact (V x)\nhxV : \u2200 (x : \u2191K), \u2191x \u2208 interior (V x)\nhKV : \u2200 (x : \u2191K), V x \u2286 U\nthis : K \u2286 \u22c3 (x : \u2191K), interior (V x)\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), interior (V i)\nx : \u03b1\nhx : x \u2208 K\ny : \u2191K\nhyt : y \u2208 t\nhy : x \u2208 interior (V y)\n\u22a2 x \u2208 interior (\u22c3 (i : \u2191K) (_ : i \u2208 t), V i)\n[PROOFSTEP]\nexact interior_mono (subset_iUnion\u2082 y hyt) hy\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\n\u22a2 \u2200 (x : \u03b1), HasBasis (\ud835\udcdd x) (fun s => s \u2208 \ud835\udcdd (f x) \u2227 IsCompact s) fun s => f \u207b\u00b9' s\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nx : \u03b1\n\u22a2 HasBasis (\ud835\udcdd x) (fun s => s \u2208 \ud835\udcdd (f x) \u2227 IsCompact s) fun s => f \u207b\u00b9' s\n[PROOFSTEP]\nrw [hf.toInducing.nhds_eq_comap]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nx : \u03b1\n\u22a2 HasBasis (comap f (\ud835\udcdd (f x))) (fun s => s \u2208 \ud835\udcdd (f x) \u2227 IsCompact s) fun s => f \u207b\u00b9' s\n[PROOFSTEP]\nexact (compact_basis_nhds _).comap _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\n\u22a2 LocallyCompactSpace \u03b1\n[PROOFSTEP]\nhave : \u2200 x : \u03b1, (\ud835\udcdd x).HasBasis (fun s => (s \u2208 \ud835\udcdd (f x) \u2227 IsCompact s) \u2227 s \u2286 range f) fun s => f \u207b\u00b9' s :=\n by\n intro x\n rw [hf.toInducing.nhds_eq_comap]\n exact ((compact_basis_nhds _).restrict_subset <| hf.open_range.mem_nhds <| mem_range_self _).comap _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\n\u22a2 \u2200 (x : \u03b1), HasBasis (\ud835\udcdd x) (fun s => (s \u2208 \ud835\udcdd (f x) \u2227 IsCompact s) \u2227 s \u2286 range f) fun s => f \u207b\u00b9' s\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\nx : \u03b1\n\u22a2 HasBasis (\ud835\udcdd x) (fun s => (s \u2208 \ud835\udcdd (f x) \u2227 IsCompact s) \u2227 s \u2286 range f) fun s => f \u207b\u00b9' s\n[PROOFSTEP]\nrw [hf.toInducing.nhds_eq_comap]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\nx : \u03b1\n\u22a2 HasBasis (comap f (\ud835\udcdd (f x))) (fun s => (s \u2208 \ud835\udcdd (f x) \u2227 IsCompact s) \u2227 s \u2286 range f) fun s => f \u207b\u00b9' s\n[PROOFSTEP]\nexact ((compact_basis_nhds _).restrict_subset <| hf.open_range.mem_nhds <| mem_range_self _).comap _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\nthis : \u2200 (x : \u03b1), HasBasis (\ud835\udcdd x) (fun s => (s \u2208 \ud835\udcdd (f x) \u2227 IsCompact s) \u2227 s \u2286 range f) fun s => f \u207b\u00b9' s\n\u22a2 LocallyCompactSpace \u03b1\n[PROOFSTEP]\nrefine' locallyCompactSpace_of_hasBasis this fun x s hs => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\nthis : \u2200 (x : \u03b1), HasBasis (\ud835\udcdd x) (fun s => (s \u2208 \ud835\udcdd (f x) \u2227 IsCompact s) \u2227 s \u2286 range f) fun s => f \u207b\u00b9' s\nx : \u03b1\ns : Set \u03b2\nhs : (s \u2208 \ud835\udcdd (f x) \u2227 IsCompact s) \u2227 s \u2286 range f\n\u22a2 IsCompact (f \u207b\u00b9' s)\n[PROOFSTEP]\nrw [\u2190 hf.toInducing.isCompact_iff, image_preimage_eq_of_subset hs.2]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\nthis : \u2200 (x : \u03b1), HasBasis (\ud835\udcdd x) (fun s => (s \u2208 \ud835\udcdd (f x) \u2227 IsCompact s) \u2227 s \u2286 range f) fun s => f \u207b\u00b9' s\nx : \u03b1\ns : Set \u03b2\nhs : (s \u2208 \ud835\udcdd (f x) \u2227 IsCompact s) \u2227 s \u2286 range f\n\u22a2 IsCompact s\n[PROOFSTEP]\nexact hs.1.2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nF : Ultrafilter \u03b1\n\u22a2 \u2191F \u2264 \ud835\udcdd (lim F)\n[PROOFSTEP]\nrcases isCompact_univ.ultrafilter_le_nhds F (by simp) with \u27e8x, -, h\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nF : Ultrafilter \u03b1\n\u22a2 \u2191F \u2264 \ud835\udcdf univ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nF : Ultrafilter \u03b1\nx : \u03b1\nh : \u2191F \u2264 \ud835\udcdd x\n\u22a2 \u2191F \u2264 \ud835\udcdd (lim F)\n[PROOFSTEP]\nexact le_nhds_lim \u27e8x, h\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\n\u22a2 \u2203 V, V \u2286 S \u2227 Set.Nonempty V \u2227 IsClosed V \u2227 \u2200 (V' : Set \u03b1), V' \u2286 V \u2192 Set.Nonempty V' \u2192 IsClosed V' \u2192 V' = V\n[PROOFSTEP]\nlet opens := {U : Set \u03b1 | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 U\u1d9c.Nonempty}\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\n\u22a2 \u2203 V, V \u2286 S \u2227 Set.Nonempty V \u2227 IsClosed V \u2227 \u2200 (V' : Set \u03b1), V' \u2286 V \u2192 Set.Nonempty V' \u2192 IsClosed V' \u2192 V' = V\n[PROOFSTEP]\nobtain \u27e8U, \u27e8Uc, Uo, Ucne\u27e9, h\u27e9 :=\n zorn_subset opens fun c hc hz => by\n by_cases hcne : c.Nonempty\n \u00b7 obtain \u27e8U\u2080, hU\u2080\u27e9 := hcne\n haveI : Nonempty { U // U \u2208 c } := \u27e8\u27e8U\u2080, hU\u2080\u27e9\u27e9\n obtain \u27e8U\u2080compl, -, -\u27e9 := hc hU\u2080\n use\u22c3\u2080 c\n refine' \u27e8\u27e8_, _, _\u27e9, fun U hU a ha => \u27e8U, hU, ha\u27e9\u27e9\n \u00b7 exact fun a ha => \u27e8U\u2080, hU\u2080, U\u2080compl ha\u27e9\n \u00b7 exact isOpen_sUnion fun _ h => (hc h).2.1\n \u00b7 convert_to (\u22c2 U : { U // U \u2208 c }, U.1\u1d9c).Nonempty\n \u00b7 ext\n simp only [not_exists, exists_prop, not_and, Set.mem_iInter, Subtype.forall, mem_setOf_eq, mem_compl_iff,\n mem_sUnion]\n apply IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed\n \u00b7 rintro \u27e8U, hU\u27e9 \u27e8U', hU'\u27e9\n obtain \u27e8V, hVc, hVU, hVU'\u27e9 := hz.directedOn U hU U' hU'\n exact \u27e8\u27e8V, hVc\u27e9, Set.compl_subset_compl.mpr hVU, Set.compl_subset_compl.mpr hVU'\u27e9\n \u00b7 exact fun U => (hc U.2).2.2\n \u00b7 exact fun U => (hc U.2).2.1.isClosed_compl.isCompact\n \u00b7 exact fun U => (hc U.2).2.1.isClosed_compl\n \u00b7 use S\u1d9c\n refine' \u27e8\u27e8Set.Subset.refl _, isOpen_compl_iff.mpr hS, _\u27e9, fun U Uc => (hcne \u27e8U, Uc\u27e9).elim\u27e9\n rw [compl_compl]\n exact hne\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub\n[PROOFSTEP]\nby_cases hcne : c.Nonempty\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nhcne : Set.Nonempty c\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub\n[PROOFSTEP]\nobtain \u27e8U\u2080, hU\u2080\u27e9 := hcne\n[GOAL]\ncase pos.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub\n[PROOFSTEP]\nhaveI : Nonempty { U // U \u2208 c } := \u27e8\u27e8U\u2080, hU\u2080\u27e9\u27e9\n[GOAL]\ncase pos.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub\n[PROOFSTEP]\nobtain \u27e8U\u2080compl, -, -\u27e9 := hc hU\u2080\n[GOAL]\ncase pos.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub\n[PROOFSTEP]\nuse\u22c3\u2080 c\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 \u22c3\u2080 c \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 \u22c3\u2080 c\n[PROOFSTEP]\nrefine' \u27e8\u27e8_, _, _\u27e9, fun U hU a ha => \u27e8U, hU, ha\u27e9\u27e9\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 S\u1d9c \u2286 \u22c3\u2080 c\n[PROOFSTEP]\nexact fun a ha => \u27e8U\u2080, hU\u2080, U\u2080compl ha\u27e9\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 IsOpen (\u22c3\u2080 c)\n[PROOFSTEP]\nexact isOpen_sUnion fun _ h => (hc h).2.1\n[GOAL]\ncase h.refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 Set.Nonempty (\u22c3\u2080 c)\u1d9c\n[PROOFSTEP]\nconvert_to (\u22c2 U : { U // U \u2208 c }, U.1\u1d9c).Nonempty\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 (\u22c3\u2080 c)\u1d9c = \u22c2 (U : { U // U \u2208 c }), (\u2191U)\u1d9c\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_2.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 (\u22c3\u2080 c)\u1d9c \u2194 x\u271d \u2208 \u22c2 (U : { U // U \u2208 c }), (\u2191U)\u1d9c\n[PROOFSTEP]\nsimp only [not_exists, exists_prop, not_and, Set.mem_iInter, Subtype.forall, mem_setOf_eq, mem_compl_iff, mem_sUnion]\n[GOAL]\ncase h.refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 Set.Nonempty (\u22c2 (U : { U // U \u2208 c }), (\u2191U)\u1d9c)\n[PROOFSTEP]\napply IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed\n[GOAL]\ncase h.refine'_3.hZd\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 Directed (fun x x_1 => x \u2287 x_1) fun i => (\u2191i)\u1d9c\n[PROOFSTEP]\nrintro \u27e8U, hU\u27e9 \u27e8U', hU'\u27e9\n[GOAL]\ncase h.refine'_3.hZd.mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\nU : Set \u03b1\nhU : U \u2208 c\nU' : Set \u03b1\nhU' : U' \u2208 c\n\u22a2 \u2203 z,\n (fun x x_1 => x \u2287 x_1) ((fun i => (\u2191i)\u1d9c) { val := U, property := hU }) ((fun i => (\u2191i)\u1d9c) z) \u2227\n (fun x x_1 => x \u2287 x_1) ((fun i => (\u2191i)\u1d9c) { val := U', property := hU' }) ((fun i => (\u2191i)\u1d9c) z)\n[PROOFSTEP]\nobtain \u27e8V, hVc, hVU, hVU'\u27e9 := hz.directedOn U hU U' hU'\n[GOAL]\ncase h.refine'_3.hZd.mk.mk.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\nU : Set \u03b1\nhU : U \u2208 c\nU' : Set \u03b1\nhU' : U' \u2208 c\nV : Set \u03b1\nhVc : V \u2208 c\nhVU : U \u2286 V\nhVU' : U' \u2286 V\n\u22a2 \u2203 z,\n (fun x x_1 => x \u2287 x_1) ((fun i => (\u2191i)\u1d9c) { val := U, property := hU }) ((fun i => (\u2191i)\u1d9c) z) \u2227\n (fun x x_1 => x \u2287 x_1) ((fun i => (\u2191i)\u1d9c) { val := U', property := hU' }) ((fun i => (\u2191i)\u1d9c) z)\n[PROOFSTEP]\nexact \u27e8\u27e8V, hVc\u27e9, Set.compl_subset_compl.mpr hVU, Set.compl_subset_compl.mpr hVU'\u27e9\n[GOAL]\ncase h.refine'_3.hZn\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 \u2200 (i : { U // U \u2208 c }), Set.Nonempty (\u2191i)\u1d9c\n[PROOFSTEP]\nexact fun U => (hc U.2).2.2\n[GOAL]\ncase h.refine'_3.hZc\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 \u2200 (i : { U // U \u2208 c }), IsCompact (\u2191i)\u1d9c\n[PROOFSTEP]\nexact fun U => (hc U.2).2.1.isClosed_compl.isCompact\n[GOAL]\ncase h.refine'_3.hZcl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 \u2200 (i : { U // U \u2208 c }), IsClosed (\u2191i)\u1d9c\n[PROOFSTEP]\nexact fun U => (hc U.2).2.1.isClosed_compl\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nhcne : \u00acSet.Nonempty c\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub\n[PROOFSTEP]\nuse S\u1d9c\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nhcne : \u00acSet.Nonempty c\n\u22a2 S\u1d9c \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 S\u1d9c\n[PROOFSTEP]\nrefine' \u27e8\u27e8Set.Subset.refl _, isOpen_compl_iff.mpr hS, _\u27e9, fun U Uc => (hcne \u27e8U, Uc\u27e9).elim\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nhcne : \u00acSet.Nonempty c\n\u22a2 Set.Nonempty S\u1d9c\u1d9c\n[PROOFSTEP]\nrw [compl_compl]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nhcne : \u00acSet.Nonempty c\n\u22a2 Set.Nonempty S\n[PROOFSTEP]\nexact hne\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty U\u1d9c\n\u22a2 \u2203 V, V \u2286 S \u2227 Set.Nonempty V \u2227 IsClosed V \u2227 \u2200 (V' : Set \u03b1), V' \u2286 V \u2192 Set.Nonempty V' \u2192 IsClosed V' \u2192 V' = V\n[PROOFSTEP]\nrefine' \u27e8U\u1d9c, Set.compl_subset_comm.mp Uc, Ucne, Uo.isClosed_compl, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty U\u1d9c\n\u22a2 \u2200 (V' : Set \u03b1), V' \u2286 U\u1d9c \u2192 Set.Nonempty V' \u2192 IsClosed V' \u2192 V' = U\u1d9c\n[PROOFSTEP]\nintro V' V'sub V'ne V'cls\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty U\u1d9c\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\n\u22a2 V' = U\u1d9c\n[PROOFSTEP]\nhave : V'\u1d9c = U := by\n refine' h V'\u1d9c \u27e8_, isOpen_compl_iff.mpr V'cls, _\u27e9 (Set.subset_compl_comm.mp V'sub)\n exact Set.Subset.trans Uc (Set.subset_compl_comm.mp V'sub)\n simp only [compl_compl, V'ne]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty U\u1d9c\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\n\u22a2 V'\u1d9c = U\n[PROOFSTEP]\nrefine' h V'\u1d9c \u27e8_, isOpen_compl_iff.mpr V'cls, _\u27e9 (Set.subset_compl_comm.mp V'sub)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty U\u1d9c\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\n\u22a2 S\u1d9c \u2286 V'\u1d9c\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty U\u1d9c\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\n\u22a2 Set.Nonempty V'\u1d9c\u1d9c\n[PROOFSTEP]\nexact Set.Subset.trans Uc (Set.subset_compl_comm.mp V'sub)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty U\u1d9c\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\n\u22a2 Set.Nonempty V'\u1d9c\u1d9c\n[PROOFSTEP]\nsimp only [compl_compl, V'ne]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty U\u1d9c}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty U\u1d9c\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\nthis : V'\u1d9c = U\n\u22a2 V' = U\u1d9c\n[PROOFSTEP]\nrw [\u2190 this, compl_compl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u22a2 SigmaCompactSpace \u03b1\n[PROOFSTEP]\nchoose K hKc hxK using fun x : \u03b1 => exists_compact_mem_nhds x\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nK : \u03b1 \u2192 Set \u03b1\nhKc : \u2200 (x : \u03b1), IsCompact (K x)\nhxK : \u2200 (x : \u03b1), K x \u2208 \ud835\udcdd x\n\u22a2 SigmaCompactSpace \u03b1\n[PROOFSTEP]\nrcases countable_cover_nhds hxK with \u27e8s, hsc, hsU\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nK : \u03b1 \u2192 Set \u03b1\nhKc : \u2200 (x : \u03b1), IsCompact (K x)\nhxK : \u2200 (x : \u03b1), K x \u2208 \ud835\udcdd x\ns : Set \u03b1\nhsc : Set.Countable s\nhsU : \u22c3 (x : \u03b1) (_ : x \u2208 s), K x = univ\n\u22a2 SigmaCompactSpace \u03b1\n[PROOFSTEP]\nrefine' SigmaCompactSpace.of_countable _ (hsc.image K) (ball_image_iff.2 fun x _ => hKc x) _\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nK : \u03b1 \u2192 Set \u03b1\nhKc : \u2200 (x : \u03b1), IsCompact (K x)\nhxK : \u2200 (x : \u03b1), K x \u2208 \ud835\udcdd x\ns : Set \u03b1\nhsc : Set.Countable s\nhsU : \u22c3 (x : \u03b1) (_ : x \u2208 s), K x = univ\n\u22a2 \u22c3\u2080 (K '' s) = univ\n[PROOFSTEP]\nrwa [sUnion_image]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\n\u22a2 \u22c3 (n : \u2115), compactCovering \u03b1 n = univ\n[PROOFSTEP]\nrw [compactCovering, iUnion_accumulate]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\n\u22a2 \u22c3 (x : \u2115), Exists.choose (_ : \u2203 K, (\u2200 (n : \u2115), IsCompact (K n)) \u2227 \u22c3 (n : \u2115), K n = univ) x = univ\n[PROOFSTEP]\nexact (Classical.choose_spec SigmaCompactSpace.exists_compact_covering).2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : SigmaCompactSpace \u03b1\ninst\u271d : SigmaCompactSpace \u03b2\n\u22a2 \u22c3 (n : \u2115), (fun n => compactCovering \u03b1 n \u00d7\u02e2 compactCovering \u03b2 n) n = univ\n[PROOFSTEP]\nsimp only [iUnion_prod_of_monotone (compactCovering_subset \u03b1) (compactCovering_subset \u03b2), iUnion_compactCovering,\n univ_prod_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\n\u22a2 SigmaCompactSpace ((i : \u03b9) \u2192 \u03c0 i)\n[PROOFSTEP]\nrefine'\n \u27e8\u27e8fun n => Set.pi univ fun i => compactCovering (\u03c0 i) n, fun n =>\n isCompact_univ_pi fun i => isCompact_compactCovering (\u03c0 i) _, _\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\n\u22a2 \u22c3 (n : \u2115), (fun n => Set.pi univ fun i => compactCovering (\u03c0 i) n) n = univ\n[PROOFSTEP]\nrw [iUnion_univ_pi_of_monotone]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\n\u22a2 (Set.pi univ fun i => \u22c3 (j : \u2115), compactCovering (\u03c0 i) j) = univ\n[PROOFSTEP]\nsimp only [iUnion_compactCovering, pi_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\n\u22a2 \u2200 (i : \u03b9), Monotone fun n => compactCovering (\u03c0 i) n\n[PROOFSTEP]\nexact fun i => compactCovering_subset (\u03c0 i)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : SigmaCompactSpace \u03b1\ninst\u271d : SigmaCompactSpace \u03b2\n\u22a2 \u22c3 (n : \u2115), (fun n => Sum.inl '' compactCovering \u03b1 n \u222a Sum.inr '' compactCovering \u03b2 n) n = univ\n[PROOFSTEP]\nsimp only [iUnion_union_distrib, \u2190 image_iUnion, iUnion_compactCovering, image_univ, range_inl_union_range_inr]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\n\u22a2 SigmaCompactSpace ((i : \u03b9) \u00d7 \u03c0 i)\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b9\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\nh\u271d : IsEmpty \u03b9\n\u22a2 SigmaCompactSpace ((i : \u03b9) \u00d7 \u03c0 i)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\nh\u271d : Nonempty \u03b9\n\u22a2 SigmaCompactSpace ((i : \u03b9) \u00d7 \u03c0 i)\n[PROOFSTEP]\nrcases exists_surjective_nat \u03b9 with \u27e8f, hf\u27e9\n[GOAL]\ncase inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\nh\u271d : Nonempty \u03b9\nf : \u2115 \u2192 \u03b9\nhf : Function.Surjective f\n\u22a2 SigmaCompactSpace ((i : \u03b9) \u00d7 \u03c0 i)\n[PROOFSTEP]\nrefine' \u27e8\u27e8fun n => \u22c3 k \u2264 n, Sigma.mk (f k) '' compactCovering (\u03c0 (f k)) n, fun n => _, _\u27e9\u27e9\n[GOAL]\ncase inr.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\nh\u271d : Nonempty \u03b9\nf : \u2115 \u2192 \u03b9\nhf : Function.Surjective f\nn : \u2115\n\u22a2 IsCompact ((fun n => \u22c3 (k : \u2115) (_ : k \u2264 n), Sigma.mk (f k) '' compactCovering (\u03c0 (f k)) n) n)\n[PROOFSTEP]\nrefine' (finite_le_nat _).isCompact_biUnion fun k _ => _\n[GOAL]\ncase inr.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\nh\u271d : Nonempty \u03b9\nf : \u2115 \u2192 \u03b9\nhf : Function.Surjective f\nn k : \u2115\nx\u271d : k \u2208 {i | i \u2264 n}\n\u22a2 IsCompact (Sigma.mk (f k) '' compactCovering (\u03c0 (f k)) n)\n[PROOFSTEP]\nexact (isCompact_compactCovering _ _).image continuous_sigmaMk\n[GOAL]\ncase inr.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\nh\u271d : Nonempty \u03b9\nf : \u2115 \u2192 \u03b9\nhf : Function.Surjective f\n\u22a2 \u22c3 (n : \u2115), (fun n => \u22c3 (k : \u2115) (_ : k \u2264 n), Sigma.mk (f k) '' compactCovering (\u03c0 (f k)) n) n = univ\n[PROOFSTEP]\nsimp only [iUnion_eq_univ_iff, Sigma.forall, mem_iUnion]\n[GOAL]\ncase inr.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\nh\u271d : Nonempty \u03b9\nf : \u2115 \u2192 \u03b9\nhf : Function.Surjective f\n\u22a2 \u2200 (a : \u03b9) (b : \u03c0 a), \u2203 i i_1 i_2, { fst := a, snd := b } \u2208 Sigma.mk (f i_1) '' compactCovering (\u03c0 (f i_1)) i\n[PROOFSTEP]\nrw [hf.forall]\n -- porting note: `simp only` failed to use `hf.forall`\n[GOAL]\ncase inr.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\nh\u271d : Nonempty \u03b9\nf : \u2115 \u2192 \u03b9\nhf : Function.Surjective f\n\u22a2 \u2200 (x : \u2115) (b : \u03c0 (f x)), \u2203 i i_1 i_2, { fst := f x, snd := b } \u2208 Sigma.mk (f i_1) '' compactCovering (\u03c0 (f i_1)) i\n[PROOFSTEP]\nintro k y\n[GOAL]\ncase inr.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\nh\u271d : Nonempty \u03b9\nf : \u2115 \u2192 \u03b9\nhf : Function.Surjective f\nk : \u2115\ny : \u03c0 (f k)\n\u22a2 \u2203 i i_1 i_2, { fst := f k, snd := y } \u2208 Sigma.mk (f i_1) '' compactCovering (\u03c0 (f i_1)) i\n[PROOFSTEP]\nrcases exists_mem_compactCovering y with \u27e8n, hn\u27e9\n[GOAL]\ncase inr.intro.refine'_2.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\nh\u271d : Nonempty \u03b9\nf : \u2115 \u2192 \u03b9\nhf : Function.Surjective f\nk : \u2115\ny : \u03c0 (f k)\nn : \u2115\nhn : y \u2208 compactCovering (\u03c0 (f k)) n\n\u22a2 \u2203 i i_1 i_2, { fst := f k, snd := y } \u2208 Sigma.mk (f i_1) '' compactCovering (\u03c0 (f i_1)) i\n[PROOFSTEP]\nrefine' \u27e8max k n, k, le_max_left _ _, mem_image_of_mem _ _\u27e9\n[GOAL]\ncase inr.intro.refine'_2.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b3 : SigmaCompactSpace \u03b1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), SigmaCompactSpace (\u03c0 i)\nh\u271d : Nonempty \u03b9\nf : \u2115 \u2192 \u03b9\nhf : Function.Surjective f\nk : \u2115\ny : \u03c0 (f k)\nn : \u2115\nhn : y \u2208 compactCovering (\u03c0 (f k)) n\n\u22a2 y \u2208 compactCovering (\u03c0 (f k)) (max k n)\n[PROOFSTEP]\nexact compactCovering_subset _ (le_max_right _ _) hn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\ne : \u03b2 \u2192 \u03b1\nhe : ClosedEmbedding e\n\u22a2 \u22c3 (n : \u2115), (fun n => e \u207b\u00b9' compactCovering \u03b1 n) n = univ\n[PROOFSTEP]\nrw [\u2190 preimage_iUnion, iUnion_compactCovering, preimage_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\nhf : LocallyFinite f\nhne : \u2200 (i : \u03b9), Set.Nonempty (f i)\n\u22a2 Set.Countable univ\n[PROOFSTEP]\nhave := fun n => hf.finite_nonempty_inter_compact (isCompact_compactCovering \u03b1 n)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\nhf : LocallyFinite f\nhne : \u2200 (i : \u03b9), Set.Nonempty (f i)\nthis : \u2200 (n : \u2115), Set.Finite {i | Set.Nonempty (f i \u2229 compactCovering \u03b1 n)}\n\u22a2 Set.Countable univ\n[PROOFSTEP]\nrefine (countable_iUnion fun n => (this n).countable).mono fun i _ => ?_\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\nhf : LocallyFinite f\nhne : \u2200 (i : \u03b9), Set.Nonempty (f i)\nthis : \u2200 (n : \u2115), Set.Finite {i | Set.Nonempty (f i \u2229 compactCovering \u03b1 n)}\ni : \u03b9\nx\u271d : i \u2208 univ\n\u22a2 i \u2208 \u22c3 (i : \u2115), {i_1 | Set.Nonempty (f i_1 \u2229 compactCovering \u03b1 i)}\n[PROOFSTEP]\nrcases hne i with \u27e8x, hx\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\nhf : LocallyFinite f\nhne : \u2200 (i : \u03b9), Set.Nonempty (f i)\nthis : \u2200 (n : \u2115), Set.Finite {i | Set.Nonempty (f i \u2229 compactCovering \u03b1 n)}\ni : \u03b9\nx\u271d : i \u2208 univ\nx : \u03b1\nhx : x \u2208 f i\n\u22a2 i \u2208 \u22c3 (i : \u2115), {i_1 | Set.Nonempty (f i_1 \u2229 compactCovering \u03b1 i)}\n[PROOFSTEP]\nrcases iUnion_eq_univ_iff.1 (iUnion_compactCovering \u03b1) x with \u27e8n, hn\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\nhf : LocallyFinite f\nhne : \u2200 (i : \u03b9), Set.Nonempty (f i)\nthis : \u2200 (n : \u2115), Set.Finite {i | Set.Nonempty (f i \u2229 compactCovering \u03b1 n)}\ni : \u03b9\nx\u271d : i \u2208 univ\nx : \u03b1\nhx : x \u2208 f i\nn : \u2115\nhn : x \u2208 compactCovering \u03b1 n\n\u22a2 i \u2208 \u22c3 (i : \u2115), {i_1 | Set.Nonempty (f i_1 \u2229 compactCovering \u03b1 i)}\n[PROOFSTEP]\nexact mem_iUnion.2 \u27e8n, x, hx, hn\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2208 \ud835\udcdd[s] x\n\u22a2 \u2203 t x, Set.Countable t \u2227 s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), f x\n[PROOFSTEP]\nsimp only [nhdsWithin, mem_inf_principal] at hf \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x\n\u22a2 \u2203 t x, Set.Countable t \u2227 s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), f x\n[PROOFSTEP]\nchoose t ht hsub using fun n =>\n ((isCompact_compactCovering \u03b1 n).inter_right hs).elim_nhds_subcover _ fun x hx => hf x hx.right\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x\nt : \u2115 \u2192 Finset \u03b1\nht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s\nhsub : \u2200 (n : \u2115), compactCovering \u03b1 n \u2229 s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t n), {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x}\n\u22a2 \u2203 t x, Set.Countable t \u2227 s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), f x\n[PROOFSTEP]\nrefine'\n \u27e8\u22c3 n, (t n : Set \u03b1), iUnion_subset fun n x hx => (ht n x hx).2, countable_iUnion fun n => (t n).countable_toSet,\n fun x hx => mem_iUnion\u2082.2 _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x\nt : \u2115 \u2192 Finset \u03b1\nht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s\nhsub : \u2200 (n : \u2115), compactCovering \u03b1 n \u2229 s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t n), {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x}\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2203 i j, x \u2208 f i\n[PROOFSTEP]\nrcases exists_mem_compactCovering x with \u27e8n, hn\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x\nt : \u2115 \u2192 Finset \u03b1\nht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s\nhsub : \u2200 (n : \u2115), compactCovering \u03b1 n \u2229 s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t n), {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x}\nx : \u03b1\nhx : x \u2208 s\nn : \u2115\nhn : x \u2208 compactCovering \u03b1 n\n\u22a2 \u2203 i j, x \u2208 f i\n[PROOFSTEP]\nrcases mem_iUnion\u2082.1 (hsub n \u27e8hn, hx\u27e9) with \u27e8y, hyt : y \u2208 t n, hyf : x \u2208 s \u2192 x \u2208 f y\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x\nt : \u2115 \u2192 Finset \u03b1\nht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s\nhsub : \u2200 (n : \u2115), compactCovering \u03b1 n \u2229 s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t n), {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x}\nx : \u03b1\nhx : x \u2208 s\nn : \u2115\nhn : x \u2208 compactCovering \u03b1 n\ny : \u03b1\nhyt : y \u2208 t n\nhyf : x \u2208 s \u2192 x \u2208 f y\n\u22a2 \u2203 i j, x \u2208 f i\n[PROOFSTEP]\nexact \u27e8y, mem_iUnion.2 \u27e8n, hyt\u27e9, hyf hx\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\nhf : \u2200 (x : \u03b1), f x \u2208 \ud835\udcdd x\n\u22a2 \u2203 s, Set.Countable s \u2227 \u22c3 (x : \u03b1) (_ : x \u2208 s), f x = univ\n[PROOFSTEP]\nsimp only [\u2190 nhdsWithin_univ] at hf \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\nhf : \u2200 (x : \u03b1), f x \u2208 \ud835\udcdd[univ] x\n\u22a2 \u2203 s, Set.Countable s \u2227 \u22c3 (x : \u03b1) (_ : x \u2208 s), f x = univ\n[PROOFSTEP]\nrcases countable_cover_nhdsWithin_of_sigma_compact isClosed_univ fun x _ => hf x with \u27e8s, -, hsc, hsU\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\nhf : \u2200 (x : \u03b1), f x \u2208 \ud835\udcdd[univ] x\ns : Set \u03b1\nhsc : Set.Countable s\nhsU : univ \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 s), f x\n\u22a2 \u2203 s, Set.Countable s \u2227 \u22c3 (x : \u03b1) (_ : x \u2208 s), f x = univ\n[PROOFSTEP]\nexact \u27e8s, hsc, univ_subset_iff.1 hsU\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Set \u03b1\nK : CompactExhaustion \u03b1\nx : \u03b1\n\u22a2 \u00acCompactExhaustion.find (shiftr K) x \u2264 CompactExhaustion.find K x\n[PROOFSTEP]\nsimp only [find_shiftr, not_le, Nat.lt_succ_self]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t : Set \u03b1\nK : CompactExhaustion \u03b1\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : SigmaCompactSpace X\n\u22a2 CompactExhaustion X\n[PROOFSTEP]\napply Classical.choice\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t : Set \u03b1\nK : CompactExhaustion \u03b1\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : SigmaCompactSpace X\n\u22a2 Nonempty (CompactExhaustion X)\n[PROOFSTEP]\nlet K : \u2115 \u2192 { s : Set X // IsCompact s } := fun n =>\n Nat.recOn n \u27e8\u2205, isCompact_empty\u27e9 fun n s =>\n \u27e8(exists_compact_superset s.2).choose \u222a compactCovering X n,\n (exists_compact_superset s.2).choose_spec.1.union (isCompact_compactCovering _ _)\u27e9\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t : Set \u03b1\nK\u271d : CompactExhaustion \u03b1\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : SigmaCompactSpace X\nK : \u2115 \u2192 { s // IsCompact s } :=\n fun n =>\n Nat.recOn n { val := \u2205, property := (_ : IsCompact \u2205) } fun n s =>\n { val := Exists.choose (_ : \u2203 K', IsCompact K' \u2227 \u2191s \u2286 interior K') \u222a compactCovering X n,\n property := (_ : IsCompact (Exists.choose (_ : \u2203 K', IsCompact K' \u2227 \u2191s \u2286 interior K') \u222a compactCovering X n)) }\n\u22a2 Nonempty (CompactExhaustion X)\n[PROOFSTEP]\nrefine' \u27e8\u27e8fun n => (K n).1, fun n => (K n).2, fun n => _, _\u27e9\u27e9\n[GOAL]\ncase a.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t : Set \u03b1\nK\u271d : CompactExhaustion \u03b1\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : SigmaCompactSpace X\nK : \u2115 \u2192 { s // IsCompact s } :=\n fun n =>\n Nat.recOn n { val := \u2205, property := (_ : IsCompact \u2205) } fun n s =>\n { val := Exists.choose (_ : \u2203 K', IsCompact K' \u2227 \u2191s \u2286 interior K') \u222a compactCovering X n,\n property := (_ : IsCompact (Exists.choose (_ : \u2203 K', IsCompact K' \u2227 \u2191s \u2286 interior K') \u222a compactCovering X n)) }\nn : \u2115\n\u22a2 (fun n => \u2191(K n)) n \u2286 interior ((fun n => \u2191(K n)) (n + 1))\n[PROOFSTEP]\nexact Subset.trans (exists_compact_superset (K n).2).choose_spec.2 (interior_mono <| subset_union_left _ _)\n[GOAL]\ncase a.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t : Set \u03b1\nK\u271d : CompactExhaustion \u03b1\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : SigmaCompactSpace X\nK : \u2115 \u2192 { s // IsCompact s } :=\n fun n =>\n Nat.recOn n { val := \u2205, property := (_ : IsCompact \u2205) } fun n s =>\n { val := Exists.choose (_ : \u2203 K', IsCompact K' \u2227 \u2191s \u2286 interior K') \u222a compactCovering X n,\n property := (_ : IsCompact (Exists.choose (_ : \u2203 K', IsCompact K' \u2227 \u2191s \u2286 interior K') \u222a compactCovering X n)) }\n\u22a2 \u22c3 (n : \u2115), (fun n => \u2191(K n)) n = univ\n[PROOFSTEP]\nrefine' univ_subset_iff.1 (iUnion_compactCovering X \u25b8 _)\n[GOAL]\ncase a.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ns t : Set \u03b1\nK\u271d : CompactExhaustion \u03b1\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : SigmaCompactSpace X\nK : \u2115 \u2192 { s // IsCompact s } :=\n fun n =>\n Nat.recOn n { val := \u2205, property := (_ : IsCompact \u2205) } fun n s =>\n { val := Exists.choose (_ : \u2203 K', IsCompact K' \u2227 \u2191s \u2286 interior K') \u222a compactCovering X n,\n property := (_ : IsCompact (Exists.choose (_ : \u2203 K', IsCompact K' \u2227 \u2191s \u2286 interior K') \u222a compactCovering X n)) }\n\u22a2 \u22c3 (n : \u2115), compactCovering X n \u2286 \u22c3 (n : \u2115), (fun n => \u2191(K n)) n\n[PROOFSTEP]\nexact iUnion_mono' fun n => \u27e8n + 1, subset_union_right _ _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\n\u22a2 IsClopen s \u2194 frontier s = \u2205\n[PROOFSTEP]\nrw [IsClopen, \u2190 closure_eq_iff_isClosed, \u2190 interior_eq_iff_isOpen, frontier, diff_eq_empty]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\n\u22a2 interior s = s \u2227 closure s = s \u2194 closure s \u2286 interior s\n[PROOFSTEP]\nrefine' \u27e8fun h => (h.2.trans h.1.symm).subset, fun h => _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : closure s \u2286 interior s\n\u22a2 interior s = s \u2227 closure s = s\n[PROOFSTEP]\nexact \u27e8interior_subset.antisymm (subset_closure.trans h), (h.trans interior_subset).antisymm subset_closure\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t Z a b : Set \u03b1\nh : IsClopen Z\ncover : Z \u2286 a \u222a b\nha : IsOpen a\nhb : IsOpen b\nhab : Disjoint a b\n\u22a2 IsClopen (Z \u2229 a)\n[PROOFSTEP]\nrefine' \u27e8IsOpen.inter h.1 ha, _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t Z a b : Set \u03b1\nh : IsClopen Z\ncover : Z \u2286 a \u222a b\nha : IsOpen a\nhb : IsOpen b\nhab : Disjoint a b\n\u22a2 IsClosed (Z \u2229 a)\n[PROOFSTEP]\nhave : IsClosed (Z \u2229 b\u1d9c) := IsClosed.inter h.2 (isClosed_compl_iff.2 hb)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t Z a b : Set \u03b1\nh : IsClopen Z\ncover : Z \u2286 a \u222a b\nha : IsOpen a\nhb : IsOpen b\nhab : Disjoint a b\nthis : IsClosed (Z \u2229 b\u1d9c)\n\u22a2 IsClosed (Z \u2229 a)\n[PROOFSTEP]\nconvert this using 1\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t Z a b : Set \u03b1\nh : IsClopen Z\ncover : Z \u2286 a \u222a b\nha : IsOpen a\nhb : IsOpen b\nhab : Disjoint a b\nthis : IsClosed (Z \u2229 b\u1d9c)\n\u22a2 Z \u2229 a = Z \u2229 b\u1d9c\n[PROOFSTEP]\nrefine' (inter_subset_inter_right Z hab.subset_compl_right).antisymm _\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t Z a b : Set \u03b1\nh : IsClopen Z\ncover : Z \u2286 a \u222a b\nha : IsOpen a\nhb : IsOpen b\nhab : Disjoint a b\nthis : IsClosed (Z \u2229 b\u1d9c)\n\u22a2 Z \u2229 b\u1d9c \u2286 Z \u2229 a\n[PROOFSTEP]\nrintro x \u27e8hx\u2081, hx\u2082\u27e9\n[GOAL]\ncase h.e'_3.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t Z a b : Set \u03b1\nh : IsClopen Z\ncover : Z \u2286 a \u222a b\nha : IsOpen a\nhb : IsOpen b\nhab : Disjoint a b\nthis : IsClosed (Z \u2229 b\u1d9c)\nx : \u03b1\nhx\u2081 : x \u2208 Z\nhx\u2082 : x \u2208 b\u1d9c\n\u22a2 x \u2208 Z \u2229 a\n[PROOFSTEP]\nexact \u27e8hx\u2081, by simpa [not_mem_of_mem_compl hx\u2082] using cover hx\u2081\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t Z a b : Set \u03b1\nh : IsClopen Z\ncover : Z \u2286 a \u222a b\nha : IsOpen a\nhb : IsOpen b\nhab : Disjoint a b\nthis : IsClosed (Z \u2229 b\u1d9c)\nx : \u03b1\nhx\u2081 : x \u2208 Z\nhx\u2082 : x \u2208 b\u1d9c\n\u22a2 x \u2208 a\n[PROOFSTEP]\nsimpa [not_mem_of_mem_compl hx\u2082] using cover hx\u2081\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nU : Set X\n\u22a2 Continuous (boolIndicator U) \u2194 IsClopen U\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nU : Set X\n\u22a2 Continuous (boolIndicator U) \u2192 IsClopen U\n[PROOFSTEP]\nintro hc\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nU : Set X\nhc : Continuous (boolIndicator U)\n\u22a2 IsClopen U\n[PROOFSTEP]\nrw [\u2190 U.preimage_boolIndicator_true]\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nU : Set X\nhc : Continuous (boolIndicator U)\n\u22a2 IsClopen (boolIndicator U \u207b\u00b9' {true})\n[PROOFSTEP]\nexact \u27e8(isOpen_discrete _).preimage hc, (isClosed_discrete _).preimage hc\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nU : Set X\n\u22a2 IsClopen U \u2192 Continuous (boolIndicator U)\n[PROOFSTEP]\nrefine' fun hU => \u27e8fun s _ => _\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nU : Set X\nhU : IsClopen U\ns : Set Bool\nx\u271d : IsOpen s\n\u22a2 IsOpen (boolIndicator U \u207b\u00b9' s)\n[PROOFSTEP]\nrcases U.preimage_boolIndicator s with (h | h | h | h)\n[GOAL]\ncase mpr.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nU : Set X\nhU : IsClopen U\ns : Set Bool\nx\u271d : IsOpen s\nh : boolIndicator U \u207b\u00b9' s = univ\n\u22a2 IsOpen (boolIndicator U \u207b\u00b9' s)\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr.inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nU : Set X\nhU : IsClopen U\ns : Set Bool\nx\u271d : IsOpen s\nh : boolIndicator U \u207b\u00b9' s = U\n\u22a2 IsOpen (boolIndicator U \u207b\u00b9' s)\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr.inr.inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nU : Set X\nhU : IsClopen U\ns : Set Bool\nx\u271d : IsOpen s\nh : boolIndicator U \u207b\u00b9' s = U\u1d9c\n\u22a2 IsOpen (boolIndicator U \u207b\u00b9' s)\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr.inr.inr.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nU : Set X\nhU : IsClopen U\ns : Set Bool\nx\u271d : IsOpen s\nh : boolIndicator U \u207b\u00b9' s = \u2205\n\u22a2 IsOpen (boolIndicator U \u207b\u00b9' s)\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nU : Set X\nhU : IsClopen U\ns : Set Bool\nx\u271d : IsOpen s\nh : boolIndicator U \u207b\u00b9' s = univ\n\u22a2 IsOpen univ\ncase mpr.inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nU : Set X\nhU : IsClopen U\ns : Set Bool\nx\u271d : IsOpen s\nh : boolIndicator U \u207b\u00b9' s = U\n\u22a2 IsOpen U\ncase mpr.inr.inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nU : Set X\nhU : IsClopen U\ns : Set Bool\nx\u271d : IsOpen s\nh : boolIndicator U \u207b\u00b9' s = U\u1d9c\n\u22a2 IsOpen U\u1d9c\ncase mpr.inr.inr.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nU : Set X\nhU : IsClopen U\ns : Set Bool\nx\u271d : IsOpen s\nh : boolIndicator U \u207b\u00b9' s = \u2205\n\u22a2 IsOpen \u2205\n[PROOFSTEP]\nexacts [isOpen_univ, hU.1, hU.2.isOpen_compl, isOpen_empty]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\ns U : Set X\n\u22a2 ContinuousOn (boolIndicator U) s \u2194 IsClopen (Subtype.val \u207b\u00b9' U)\n[PROOFSTEP]\nrw [continuousOn_iff_continuous_restrict, \u2190 continuous_boolIndicator_iff_clopen]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\ns U : Set X\n\u22a2 Continuous (restrict s (boolIndicator U)) \u2194 Continuous (boolIndicator (Subtype.val \u207b\u00b9' U))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\n\u22a2 Set.Nonempty (closure s \u2229 u) \u2192 Set.Nonempty (closure s \u2229 v) \u2192 Set.Nonempty (closure s \u2229 (u \u2229 v)) \u2194\n Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\niterate 3 rw [closure_inter_open_nonempty_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\n\u22a2 Set.Nonempty (closure s \u2229 u) \u2192 Set.Nonempty (closure s \u2229 v) \u2192 Set.Nonempty (closure s \u2229 (u \u2229 v)) \u2194\n Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrw [closure_inter_open_nonempty_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\n\u22a2 Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (closure s \u2229 v) \u2192 Set.Nonempty (closure s \u2229 (u \u2229 v)) \u2194\n Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\n\u22a2 IsOpen u\n[PROOFSTEP]\nrw [closure_inter_open_nonempty_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\n\u22a2 Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (closure s \u2229 (u \u2229 v)) \u2194\n Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\n\u22a2 IsOpen v\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\n\u22a2 IsOpen u\n[PROOFSTEP]\nrw [closure_inter_open_nonempty_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\n\u22a2 IsOpen (u \u2229 v)\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\n\u22a2 IsOpen v\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\n\u22a2 IsOpen u\n[PROOFSTEP]\nexacts [hu.inter hv, hv, hu]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : s \u2208 irreducibleComponents \u03b1\n\u22a2 IsClosed s\n[PROOFSTEP]\nrw [\u2190 closure_eq_iff_isClosed, eq_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : s \u2208 irreducibleComponents \u03b1\n\u22a2 s = closure s\n[PROOFSTEP]\nexact subset_closure.antisymm (H.2 H.1.closure subset_closure)\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 irreducibleComponents \u03b1 = maximals (fun x x_1 => x \u2264 x_1) {s | IsClosed s \u2227 IsIrreducible s}\n[PROOFSTEP]\next s\n[GOAL]\ncase h\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\n\u22a2 s \u2208 irreducibleComponents \u03b1 \u2194 s \u2208 maximals (fun x x_1 => x \u2264 x_1) {s | IsClosed s \u2227 IsIrreducible s}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\n\u22a2 s \u2208 irreducibleComponents \u03b1 \u2192 s \u2208 maximals (fun x x_1 => x \u2264 x_1) {s | IsClosed s \u2227 IsIrreducible s}\n[PROOFSTEP]\nintro H\n[GOAL]\ncase h.mp\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nH : s \u2208 irreducibleComponents \u03b1\n\u22a2 s \u2208 maximals (fun x x_1 => x \u2264 x_1) {s | IsClosed s \u2227 IsIrreducible s}\n[PROOFSTEP]\nexact \u27e8\u27e8isClosed_of_mem_irreducibleComponents _ H, H.1\u27e9, fun x h e => H.2 h.2 e\u27e9\n[GOAL]\ncase h.mpr\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\n\u22a2 s \u2208 maximals (fun x x_1 => x \u2264 x_1) {s | IsClosed s \u2227 IsIrreducible s} \u2192 s \u2208 irreducibleComponents \u03b1\n[PROOFSTEP]\nintro H\n[GOAL]\ncase h.mpr\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nH : s \u2208 maximals (fun x x_1 => x \u2264 x_1) {s | IsClosed s \u2227 IsIrreducible s}\n\u22a2 s \u2208 irreducibleComponents \u03b1\n[PROOFSTEP]\nrefine' \u27e8H.1.2, fun x h e => _\u27e9\n[GOAL]\ncase h.mpr\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nH : s \u2208 maximals (fun x x_1 => x \u2264 x_1) {s | IsClosed s \u2227 IsIrreducible s}\nx : Set \u03b1\nh : x \u2208 {s | IsIrreducible s}\ne : (fun x x_1 => x \u2264 x_1) s x\n\u22a2 (fun x x_1 => x \u2264 x_1) x s\n[PROOFSTEP]\nhave : closure x \u2264 s := H.2 \u27e8isClosed_closure, h.closure\u27e9 (e.trans subset_closure)\n[GOAL]\ncase h.mpr\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nH : s \u2208 maximals (fun x x_1 => x \u2264 x_1) {s | IsClosed s \u2227 IsIrreducible s}\nx : Set \u03b1\nh : x \u2208 {s | IsIrreducible s}\ne : (fun x x_1 => x \u2264 x_1) s x\nthis : closure x \u2264 s\n\u22a2 (fun x x_1 => x \u2264 x_1) x s\n[PROOFSTEP]\nexact le_trans subset_closure this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : PreirreducibleSpace \u03b1\ns t : Set \u03b1\n\u22a2 IsOpen s \u2192 IsOpen t \u2192 Set.Nonempty s \u2192 Set.Nonempty t \u2192 Set.Nonempty (s \u2229 t)\n[PROOFSTEP]\nsimpa only [univ_inter, univ_subset_iff] using @PreirreducibleSpace.isPreirreducible_univ \u03b1 _ _ s t\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\n\u22a2 IsPreirreducible (f '' s)\n[PROOFSTEP]\nrintro u v hu hv \u27e8_, \u27e8\u27e8x, hx, rfl\u27e9, hxu\u27e9\u27e9 \u27e8_, \u27e8\u27e8y, hy, rfl\u27e9, hyv\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : f x \u2208 u\ny : \u03b1\nhy : y \u2208 s\nhyv : f y \u2208 v\n\u22a2 Set.Nonempty (f '' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrw [\u2190 mem_preimage] at hxu hyv \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\n\u22a2 Set.Nonempty (f '' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrcases continuousOn_iff'.1 hf u hu with \u27e8u', hu', u'_eq\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\n\u22a2 Set.Nonempty (f '' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrcases continuousOn_iff'.1 hf v hv with \u27e8v', hv', v'_eq\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\n\u22a2 Set.Nonempty (f '' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nhave := H u' v' hu' hv'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (s \u2229 u') \u2192 Set.Nonempty (s \u2229 v') \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\n\u22a2 Set.Nonempty (f '' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrw [inter_comm s u', \u2190 u'_eq] at this \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (f \u207b\u00b9' u \u2229 s) \u2192 Set.Nonempty (s \u2229 v') \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\n\u22a2 Set.Nonempty (f '' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrw [inter_comm s v', \u2190 v'_eq] at this \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (f \u207b\u00b9' u \u2229 s) \u2192 Set.Nonempty (f \u207b\u00b9' v \u2229 s) \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\n\u22a2 Set.Nonempty (f '' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrcases this \u27e8x, hxu, hx\u27e9 \u27e8y, hyv, hy\u27e9 with \u27e8z, hzs, hzu', hzv'\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (f \u207b\u00b9' u \u2229 s) \u2192 Set.Nonempty (f \u207b\u00b9' v \u2229 s) \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\nz : \u03b1\nhzs : z \u2208 s\nhzu' : z \u2208 u'\nhzv' : z \u2208 v'\n\u22a2 Set.Nonempty (f '' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrefine' \u27e8f z, mem_image_of_mem f hzs, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (f \u207b\u00b9' u \u2229 s) \u2192 Set.Nonempty (f \u207b\u00b9' v \u2229 s) \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\nz : \u03b1\nhzs : z \u2208 s\nhzu' : z \u2208 u'\nhzv' : z \u2208 v'\n\u22a2 f z \u2208 u\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (f \u207b\u00b9' u \u2229 s) \u2192 Set.Nonempty (f \u207b\u00b9' v \u2229 s) \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\nz : \u03b1\nhzs : z \u2208 s\nhzu' : z \u2208 u'\nhzv' : z \u2208 v'\n\u22a2 f z \u2208 v\n[PROOFSTEP]\nall_goals\n rw [\u2190 mem_preimage]\n apply mem_of_mem_inter_left\n show z \u2208 _ \u2229 s\n simp [*]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (f \u207b\u00b9' u \u2229 s) \u2192 Set.Nonempty (f \u207b\u00b9' v \u2229 s) \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\nz : \u03b1\nhzs : z \u2208 s\nhzu' : z \u2208 u'\nhzv' : z \u2208 v'\n\u22a2 f z \u2208 u\n[PROOFSTEP]\nrw [\u2190 mem_preimage]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (f \u207b\u00b9' u \u2229 s) \u2192 Set.Nonempty (f \u207b\u00b9' v \u2229 s) \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\nz : \u03b1\nhzs : z \u2208 s\nhzu' : z \u2208 u'\nhzv' : z \u2208 v'\n\u22a2 z \u2208 f \u207b\u00b9' u\n[PROOFSTEP]\napply mem_of_mem_inter_left\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (f \u207b\u00b9' u \u2229 s) \u2192 Set.Nonempty (f \u207b\u00b9' v \u2229 s) \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\nz : \u03b1\nhzs : z \u2208 s\nhzu' : z \u2208 u'\nhzv' : z \u2208 v'\n\u22a2 z \u2208 f \u207b\u00b9' u \u2229 ?intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.b\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.b\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (f \u207b\u00b9' u \u2229 s) \u2192 Set.Nonempty (f \u207b\u00b9' v \u2229 s) \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\nz : \u03b1\nhzs : z \u2208 s\nhzu' : z \u2208 u'\nhzv' : z \u2208 v'\n\u22a2 Set \u03b1\n[PROOFSTEP]\nshow z \u2208 _ \u2229 s\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (f \u207b\u00b9' u \u2229 s) \u2192 Set.Nonempty (f \u207b\u00b9' v \u2229 s) \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\nz : \u03b1\nhzs : z \u2208 s\nhzu' : z \u2208 u'\nhzv' : z \u2208 v'\n\u22a2 z \u2208 f \u207b\u00b9' u \u2229 s\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (f \u207b\u00b9' u \u2229 s) \u2192 Set.Nonempty (f \u207b\u00b9' v \u2229 s) \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\nz : \u03b1\nhzs : z \u2208 s\nhzu' : z \u2208 u'\nhzv' : z \u2208 v'\n\u22a2 f z \u2208 v\n[PROOFSTEP]\nrw [\u2190 mem_preimage]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (f \u207b\u00b9' u \u2229 s) \u2192 Set.Nonempty (f \u207b\u00b9' v \u2229 s) \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\nz : \u03b1\nhzs : z \u2208 s\nhzu' : z \u2208 u'\nhzv' : z \u2208 v'\n\u22a2 z \u2208 f \u207b\u00b9' v\n[PROOFSTEP]\napply mem_of_mem_inter_left\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (f \u207b\u00b9' u \u2229 s) \u2192 Set.Nonempty (f \u207b\u00b9' v \u2229 s) \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\nz : \u03b1\nhzs : z \u2208 s\nhzu' : z \u2208 u'\nhzv' : z \u2208 v'\n\u22a2 z \u2208 f \u207b\u00b9' v \u2229 ?intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2.b\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2.b\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (f \u207b\u00b9' u \u2229 s) \u2192 Set.Nonempty (f \u207b\u00b9' v \u2229 s) \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\nz : \u03b1\nhzs : z \u2208 s\nhzu' : z \u2208 u'\nhzv' : z \u2208 v'\n\u22a2 Set \u03b1\n[PROOFSTEP]\nshow z \u2208 _ \u2229 s\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nH : IsPreirreducible s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 s\nhxu : x \u2208 f \u207b\u00b9' u\ny : \u03b1\nhy : y \u2208 s\nhyv : y \u2208 f \u207b\u00b9' v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nthis : Set.Nonempty (f \u207b\u00b9' u \u2229 s) \u2192 Set.Nonempty (f \u207b\u00b9' v \u2229 s) \u2192 Set.Nonempty (s \u2229 (u' \u2229 v'))\nz : \u03b1\nhzs : z \u2208 s\nhzu' : z \u2208 u'\nhzv' : z \u2208 v'\n\u22a2 z \u2208 f \u207b\u00b9' v \u2229 s\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsPreirreducible s\n\u22a2 IsPreirreducible univ\n[PROOFSTEP]\nrintro _ _ \u27e8u, hu, rfl\u27e9 \u27e8v, hv, rfl\u27e9 \u27e8\u27e8x, hxs\u27e9, -, hxu\u27e9 \u27e8\u27e8y, hys\u27e9, -, hyv\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.mk.intro.intro.mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsPreirreducible s\nu : Set \u03b1\nhu : IsOpen u\nv : Set \u03b1\nhv : IsOpen v\nx : \u03b1\nhxs : x \u2208 s\nhxu : { val := x, property := hxs } \u2208 val \u207b\u00b9' u\ny : \u03b1\nhys : y \u2208 s\nhyv : { val := y, property := hys } \u2208 val \u207b\u00b9' v\n\u22a2 Set.Nonempty (univ \u2229 (val \u207b\u00b9' u \u2229 val \u207b\u00b9' v))\n[PROOFSTEP]\nrcases h u v hu hv \u27e8x, hxs, hxu\u27e9 \u27e8y, hys, hyv\u27e9 with \u27e8z, hzs, \u27e8hzu, hzv\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.mk.intro.intro.mk.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsPreirreducible s\nu : Set \u03b1\nhu : IsOpen u\nv : Set \u03b1\nhv : IsOpen v\nx : \u03b1\nhxs : x \u2208 s\nhxu : { val := x, property := hxs } \u2208 val \u207b\u00b9' u\ny : \u03b1\nhys : y \u2208 s\nhyv : { val := y, property := hys } \u2208 val \u207b\u00b9' v\nz : \u03b1\nhzs : z \u2208 s\nhzu : z \u2208 u\nhzv : z \u2208 v\n\u22a2 Set.Nonempty (univ \u2229 (val \u207b\u00b9' u \u2229 val \u207b\u00b9' v))\n[PROOFSTEP]\nexact \u27e8\u27e8z, hzs\u27e9, \u27e8Set.mem_univ _, \u27e8hzu, hzv\u27e9\u27e9\u27e9\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\u271d\n\u03b1 : Type ?u.291097\ninst\u271d : Infinite \u03b1\nu v : Set (CofiniteTopology \u03b1)\n\u22a2 IsOpen u \u2192 IsOpen v \u2192 Set.Nonempty (univ \u2229 u) \u2192 Set.Nonempty (univ \u2229 v) \u2192 Set.Nonempty (univ \u2229 (u \u2229 v))\n[PROOFSTEP]\nhaveI : Infinite (CofiniteTopology \u03b1) := \u2039_\u203a\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\u271d\n\u03b1 : Type ?u.291097\ninst\u271d : Infinite \u03b1\nu v : Set (CofiniteTopology \u03b1)\nthis : Infinite (CofiniteTopology \u03b1)\n\u22a2 IsOpen u \u2192 IsOpen v \u2192 Set.Nonempty (univ \u2229 u) \u2192 Set.Nonempty (univ \u2229 v) \u2192 Set.Nonempty (univ \u2229 (u \u2229 v))\n[PROOFSTEP]\nsimp only [CofiniteTopology.isOpen_iff, univ_inter]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\u271d\n\u03b1 : Type ?u.291097\ninst\u271d : Infinite \u03b1\nu v : Set (CofiniteTopology \u03b1)\nthis : Infinite (CofiniteTopology \u03b1)\n\u22a2 (Set.Nonempty u \u2192 Set.Finite u\u1d9c) \u2192\n (Set.Nonempty v \u2192 Set.Finite v\u1d9c) \u2192 Set.Nonempty u \u2192 Set.Nonempty v \u2192 Set.Nonempty (u \u2229 v)\n[PROOFSTEP]\nintro hu hv hu' hv'\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\u271d\n\u03b1 : Type ?u.291097\ninst\u271d : Infinite \u03b1\nu v : Set (CofiniteTopology \u03b1)\nthis : Infinite (CofiniteTopology \u03b1)\nhu : Set.Nonempty u \u2192 Set.Finite u\u1d9c\nhv : Set.Nonempty v \u2192 Set.Finite v\u1d9c\nhu' : Set.Nonempty u\nhv' : Set.Nonempty v\n\u22a2 Set.Nonempty (u \u2229 v)\n[PROOFSTEP]\nsimpa only [compl_union, compl_compl] using ((hu hu').union (hv hv')).infinite_compl.nonempty\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\n\u22a2 IsIrreducible s \u2194\n \u2200 (U : Finset (Set \u03b1)),\n (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 (\u2200 (u : Set \u03b1), u \u2208 U \u2192 Set.Nonempty (s \u2229 u)) \u2192 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191U)\n[PROOFSTEP]\nrefine \u27e8fun h U hu hU => ?_, fun h => \u27e8?_, ?_\u27e9\u27e9\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsIrreducible s\nU : Finset (Set \u03b1)\nhu : \u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u\nhU : \u2200 (u : Set \u03b1), u \u2208 U \u2192 Set.Nonempty (s \u2229 u)\n\u22a2 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191U)\n[PROOFSTEP]\ninduction U using Finset.induction_on\n[GOAL]\ncase refine_1.empty\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsIrreducible s\nhu : \u2200 (u : Set \u03b1), u \u2208 \u2205 \u2192 IsOpen u\nhU : \u2200 (u : Set \u03b1), u \u2208 \u2205 \u2192 Set.Nonempty (s \u2229 u)\n\u22a2 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191\u2205)\ncase refine_1.insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d\u00b9 t s : Set \u03b1\nh : IsIrreducible s\na\u271d\u00b2 : Set \u03b1\ns\u271d : Finset (Set \u03b1)\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 s\u271d\na\u271d : (\u2200 (u : Set \u03b1), u \u2208 s\u271d \u2192 IsOpen u) \u2192 (\u2200 (u : Set \u03b1), u \u2208 s\u271d \u2192 Set.Nonempty (s \u2229 u)) \u2192 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191s\u271d)\nhu : \u2200 (u : Set \u03b1), u \u2208 insert a\u271d\u00b2 s\u271d \u2192 IsOpen u\nhU : \u2200 (u : Set \u03b1), u \u2208 insert a\u271d\u00b2 s\u271d \u2192 Set.Nonempty (s \u2229 u)\n\u22a2 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191(insert a\u271d\u00b2 s\u271d))\n[PROOFSTEP]\ncase empty => simpa using h.nonempty\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsIrreducible s\nhu : \u2200 (u : Set \u03b1), u \u2208 \u2205 \u2192 IsOpen u\nhU : \u2200 (u : Set \u03b1), u \u2208 \u2205 \u2192 Set.Nonempty (s \u2229 u)\n\u22a2 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191\u2205)\n[PROOFSTEP]\ncase empty => simpa using h.nonempty\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsIrreducible s\nhu : \u2200 (u : Set \u03b1), u \u2208 \u2205 \u2192 IsOpen u\nhU : \u2200 (u : Set \u03b1), u \u2208 \u2205 \u2192 Set.Nonempty (s \u2229 u)\n\u22a2 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191\u2205)\n[PROOFSTEP]\nsimpa using h.nonempty\n[GOAL]\ncase refine_1.insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d\u00b9 t s : Set \u03b1\nh : IsIrreducible s\na\u271d\u00b2 : Set \u03b1\ns\u271d : Finset (Set \u03b1)\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 s\u271d\na\u271d : (\u2200 (u : Set \u03b1), u \u2208 s\u271d \u2192 IsOpen u) \u2192 (\u2200 (u : Set \u03b1), u \u2208 s\u271d \u2192 Set.Nonempty (s \u2229 u)) \u2192 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191s\u271d)\nhu : \u2200 (u : Set \u03b1), u \u2208 insert a\u271d\u00b2 s\u271d \u2192 IsOpen u\nhU : \u2200 (u : Set \u03b1), u \u2208 insert a\u271d\u00b2 s\u271d \u2192 Set.Nonempty (s \u2229 u)\n\u22a2 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191(insert a\u271d\u00b2 s\u271d))\n[PROOFSTEP]\ncase insert u U _ IH =>\n rw [Finset.coe_insert, sInter_insert]\n rw [Finset.forall_mem_insert] at hu hU \n exact h.2 _ _ hu.1 (isOpen_sInter U.finite_toSet hu.2) hU.1 (IH hu.2 hU.2)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsIrreducible s\nu : Set \u03b1\nU : Finset (Set \u03b1)\na\u271d : \u00acu \u2208 U\nIH : (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 (\u2200 (u : Set \u03b1), u \u2208 U \u2192 Set.Nonempty (s \u2229 u)) \u2192 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191U)\nhu : \u2200 (u_1 : Set \u03b1), u_1 \u2208 insert u U \u2192 IsOpen u_1\nhU : \u2200 (u_1 : Set \u03b1), u_1 \u2208 insert u U \u2192 Set.Nonempty (s \u2229 u_1)\n\u22a2 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191(insert u U))\n[PROOFSTEP]\ncase insert u U _ IH =>\n rw [Finset.coe_insert, sInter_insert]\n rw [Finset.forall_mem_insert] at hu hU \n exact h.2 _ _ hu.1 (isOpen_sInter U.finite_toSet hu.2) hU.1 (IH hu.2 hU.2)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsIrreducible s\nu : Set \u03b1\nU : Finset (Set \u03b1)\na\u271d : \u00acu \u2208 U\nIH : (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 (\u2200 (u : Set \u03b1), u \u2208 U \u2192 Set.Nonempty (s \u2229 u)) \u2192 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191U)\nhu : \u2200 (u_1 : Set \u03b1), u_1 \u2208 insert u U \u2192 IsOpen u_1\nhU : \u2200 (u_1 : Set \u03b1), u_1 \u2208 insert u U \u2192 Set.Nonempty (s \u2229 u_1)\n\u22a2 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191(insert u U))\n[PROOFSTEP]\nrw [Finset.coe_insert, sInter_insert]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsIrreducible s\nu : Set \u03b1\nU : Finset (Set \u03b1)\na\u271d : \u00acu \u2208 U\nIH : (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 (\u2200 (u : Set \u03b1), u \u2208 U \u2192 Set.Nonempty (s \u2229 u)) \u2192 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191U)\nhu : \u2200 (u_1 : Set \u03b1), u_1 \u2208 insert u U \u2192 IsOpen u_1\nhU : \u2200 (u_1 : Set \u03b1), u_1 \u2208 insert u U \u2192 Set.Nonempty (s \u2229 u_1)\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 \u22c2\u2080 \u2191U))\n[PROOFSTEP]\nrw [Finset.forall_mem_insert] at hu hU \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsIrreducible s\nu : Set \u03b1\nU : Finset (Set \u03b1)\na\u271d : \u00acu \u2208 U\nIH : (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 (\u2200 (u : Set \u03b1), u \u2208 U \u2192 Set.Nonempty (s \u2229 u)) \u2192 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191U)\nhu : IsOpen u \u2227 \u2200 (x : Set \u03b1), x \u2208 U \u2192 IsOpen x\nhU : Set.Nonempty (s \u2229 u) \u2227 \u2200 (x : Set \u03b1), x \u2208 U \u2192 Set.Nonempty (s \u2229 x)\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 \u22c2\u2080 \u2191U))\n[PROOFSTEP]\nexact h.2 _ _ hu.1 (isOpen_sInter U.finite_toSet hu.2) hU.1 (IH hu.2 hU.2)\n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh :\n \u2200 (U : Finset (Set \u03b1)),\n (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 (\u2200 (u : Set \u03b1), u \u2208 U \u2192 Set.Nonempty (s \u2229 u)) \u2192 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191U)\n\u22a2 Set.Nonempty s\n[PROOFSTEP]\nsimpa using h \u2205\n[GOAL]\ncase refine_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh :\n \u2200 (U : Finset (Set \u03b1)),\n (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 (\u2200 (u : Set \u03b1), u \u2208 U \u2192 Set.Nonempty (s \u2229 u)) \u2192 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191U)\n\u22a2 IsPreirreducible s\n[PROOFSTEP]\nintro u v hu hv hu' hv'\n[GOAL]\ncase refine_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh :\n \u2200 (U : Finset (Set \u03b1)),\n (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 (\u2200 (u : Set \u03b1), u \u2208 U \u2192 Set.Nonempty (s \u2229 u)) \u2192 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191U)\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhu' : Set.Nonempty (s \u2229 u)\nhv' : Set.Nonempty (s \u2229 v)\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nsimpa [*] using h { u, v }\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\n\u22a2 IsPreirreducible s \u2194 \u2200 (z\u2081 z\u2082 : Set \u03b1), IsClosed z\u2081 \u2192 IsClosed z\u2082 \u2192 s \u2286 z\u2081 \u222a z\u2082 \u2192 s \u2286 z\u2081 \u2228 s \u2286 z\u2082\n[PROOFSTEP]\nrefine\n compl_surjective.forall.trans <| forall_congr' fun z\u2081 => compl_surjective.forall.trans <| forall_congr' fun z\u2082 => ?_\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s z\u2081 z\u2082 : Set \u03b1\n\u22a2 IsOpen z\u2081\u1d9c \u2192 IsOpen z\u2082\u1d9c \u2192 Set.Nonempty (s \u2229 z\u2081\u1d9c) \u2192 Set.Nonempty (s \u2229 z\u2082\u1d9c) \u2192 Set.Nonempty (s \u2229 (z\u2081\u1d9c \u2229 z\u2082\u1d9c)) \u2194\n IsClosed z\u2081 \u2192 IsClosed z\u2082 \u2192 s \u2286 z\u2081 \u222a z\u2082 \u2192 s \u2286 z\u2081 \u2228 s \u2286 z\u2082\n[PROOFSTEP]\nsimp only [isOpen_compl_iff, \u2190 compl_union, inter_compl_nonempty_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s z\u2081 z\u2082 : Set \u03b1\n\u22a2 IsClosed z\u2081 \u2192 IsClosed z\u2082 \u2192 \u00acs \u2286 z\u2081 \u2192 \u00acs \u2286 z\u2082 \u2192 \u00acs \u2286 z\u2081 \u222a z\u2082 \u2194\n IsClosed z\u2081 \u2192 IsClosed z\u2082 \u2192 s \u2286 z\u2081 \u222a z\u2082 \u2192 s \u2286 z\u2081 \u2228 s \u2286 z\u2082\n[PROOFSTEP]\nrefine forall\u2082_congr fun _ _ => ?_\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s z\u2081 z\u2082 : Set \u03b1\nx\u271d\u00b9 : IsClosed z\u2081\nx\u271d : IsClosed z\u2082\n\u22a2 \u00acs \u2286 z\u2081 \u2192 \u00acs \u2286 z\u2082 \u2192 \u00acs \u2286 z\u2081 \u222a z\u2082 \u2194 s \u2286 z\u2081 \u222a z\u2082 \u2192 s \u2286 z\u2081 \u2228 s \u2286 z\u2082\n[PROOFSTEP]\nrw [\u2190 and_imp, \u2190 not_or, not_imp_not]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\n\u22a2 IsIrreducible s \u2194 \u2200 (Z : Finset (Set \u03b1)), (\u2200 (z : Set \u03b1), z \u2208 Z \u2192 IsClosed z) \u2192 s \u2286 \u22c3\u2080 \u2191Z \u2192 \u2203 z, z \u2208 Z \u2227 s \u2286 z\n[PROOFSTEP]\nsimp only [isIrreducible_iff_sInter]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\n\u22a2 (\u2200 (U : Finset (Set \u03b1)),\n (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 (\u2200 (u : Set \u03b1), u \u2208 U \u2192 Set.Nonempty (s \u2229 u)) \u2192 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191U)) \u2194\n \u2200 (Z : Finset (Set \u03b1)), (\u2200 (z : Set \u03b1), z \u2208 Z \u2192 IsClosed z) \u2192 s \u2286 \u22c3\u2080 \u2191Z \u2192 \u2203 z, z \u2208 Z \u2227 s \u2286 z\n[PROOFSTEP]\nrefine ((@compl_involutive (Set \u03b1) _).toPerm _).finsetCongr.forall_congr fun {Z} => ?_\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nZ : Finset (Set \u03b1)\n\u22a2 (\u2200 (u : Set \u03b1), u \u2208 Z \u2192 IsOpen u) \u2192 (\u2200 (u : Set \u03b1), u \u2208 Z \u2192 Set.Nonempty (s \u2229 u)) \u2192 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191Z) \u2194\n (\u2200 (z : Set \u03b1),\n z \u2208 \u2191(Equiv.finsetCongr (Function.Involutive.toPerm compl (_ : Function.Involutive compl))) Z \u2192 IsClosed z) \u2192\n s \u2286 \u22c3\u2080 \u2191(\u2191(Equiv.finsetCongr (Function.Involutive.toPerm compl (_ : Function.Involutive compl))) Z) \u2192\n \u2203 z, z \u2208 \u2191(Equiv.finsetCongr (Function.Involutive.toPerm compl (_ : Function.Involutive compl))) Z \u2227 s \u2286 z\n[PROOFSTEP]\nsimp_rw [Equiv.finsetCongr_apply, Finset.forall_mem_map, Finset.mem_map, Finset.coe_map, sUnion_image,\n Equiv.coe_toEmbedding, Function.Involutive.coe_toPerm, isClosed_compl_iff, exists_exists_and_eq_and]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nZ : Finset (Set \u03b1)\n\u22a2 (\u2200 (u : Set \u03b1), u \u2208 Z \u2192 IsOpen u) \u2192 (\u2200 (u : Set \u03b1), u \u2208 Z \u2192 Set.Nonempty (s \u2229 u)) \u2192 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191Z) \u2194\n (\u2200 (u : Set \u03b1), u \u2208 Z \u2192 IsOpen u) \u2192 s \u2286 \u22c3 (x : Set \u03b1) (_ : x \u2208 \u2191Z), x\u1d9c \u2192 \u2203 a, a \u2208 Z \u2227 s \u2286 a\u1d9c\n[PROOFSTEP]\nrefine forall_congr' fun _ => Iff.trans ?_ not_imp_not\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nZ : Finset (Set \u03b1)\nx\u271d : \u2200 (u : Set \u03b1), u \u2208 Z \u2192 IsOpen u\n\u22a2 (\u2200 (u : Set \u03b1), u \u2208 Z \u2192 Set.Nonempty (s \u2229 u)) \u2192 Set.Nonempty (s \u2229 \u22c2\u2080 \u2191Z) \u2194\n (\u00ac\u2203 a, a \u2208 Z \u2227 s \u2286 a\u1d9c) \u2192 \u00acs \u2286 \u22c3 (x : Set \u03b1) (_ : x \u2208 \u2191Z), x\u1d9c\n[PROOFSTEP]\nsimp only [not_exists, not_and, \u2190 compl_iInter\u2082, \u2190 sInter_eq_biInter, subset_compl_iff_disjoint_right,\n not_disjoint_iff_nonempty_inter]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U : Set \u03b1\nhS : IsPreirreducible S\nhU : IsOpen U\nh : Set.Nonempty (S \u2229 U)\n\u22a2 S \u2286 closure (S \u2229 U)\n[PROOFSTEP]\nby_contra h'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U : Set \u03b1\nhS : IsPreirreducible S\nhU : IsOpen U\nh : Set.Nonempty (S \u2229 U)\nh' : \u00acS \u2286 closure (S \u2229 U)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8x, h\u2081, h\u2082, h\u2083\u27e9 := hS _ (closure (S \u2229 U))\u1d9c hU isClosed_closure.isOpen_compl h (inter_compl_nonempty_iff.mpr h')\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U : Set \u03b1\nhS : IsPreirreducible S\nhU : IsOpen U\nh : Set.Nonempty (S \u2229 U)\nh' : \u00acS \u2286 closure (S \u2229 U)\nx : \u03b1\nh\u2081 : x \u2208 S\nh\u2082 : x \u2208 U\nh\u2083 : x \u2208 (closure (S \u2229 U))\u1d9c\n\u22a2 False\n[PROOFSTEP]\nexact h\u2083 (subset_closure \u27e8h\u2081, h\u2082\u27e9)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U Z : Set \u03b1\nhZ : IsPreirreducible Z\nhU : Set.Nonempty U\nhU' : IsOpen U\nh\u2081 : U \u2286 S\nh\u2082 : S \u2286 Z\n\u22a2 IsIrreducible S\n[PROOFSTEP]\nobtain \u27e8z, hz\u27e9 := hU\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U Z : Set \u03b1\nhZ : IsPreirreducible Z\nhU' : IsOpen U\nh\u2081 : U \u2286 S\nh\u2082 : S \u2286 Z\nz : \u03b1\nhz : z \u2208 U\n\u22a2 IsIrreducible S\n[PROOFSTEP]\nreplace hZ : IsIrreducible Z := \u27e8\u27e8z, h\u2082 (h\u2081 hz)\u27e9, hZ\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U Z : Set \u03b1\nhU' : IsOpen U\nh\u2081 : U \u2286 S\nh\u2082 : S \u2286 Z\nz : \u03b1\nhz : z \u2208 U\nhZ : IsIrreducible Z\n\u22a2 IsIrreducible S\n[PROOFSTEP]\nrefine' \u27e8\u27e8z, h\u2081 hz\u27e9, _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U Z : Set \u03b1\nhU' : IsOpen U\nh\u2081 : U \u2286 S\nh\u2082 : S \u2286 Z\nz : \u03b1\nhz : z \u2208 U\nhZ : IsIrreducible Z\n\u22a2 IsPreirreducible S\n[PROOFSTEP]\nrintro u v hu hv \u27e8x, hx, hx'\u27e9 \u27e8y, hy, hy'\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U Z : Set \u03b1\nhU' : IsOpen U\nh\u2081 : U \u2286 S\nh\u2082 : S \u2286 Z\nz : \u03b1\nhz : z \u2208 U\nhZ : IsIrreducible Z\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 S\nhx' : x \u2208 u\ny : \u03b1\nhy : y \u2208 S\nhy' : y \u2208 v\n\u22a2 Set.Nonempty (S \u2229 (u \u2229 v))\n[PROOFSTEP]\nobtain \u27e8a, -, ha'\u27e9 : Set.Nonempty (Z \u2229 \u22c2\u2080 \u2191({ U, u, v } : Finset (Set \u03b1)))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U Z : Set \u03b1\nhU' : IsOpen U\nh\u2081 : U \u2286 S\nh\u2082 : S \u2286 Z\nz : \u03b1\nhz : z \u2208 U\nhZ : IsIrreducible Z\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 S\nhx' : x \u2208 u\ny : \u03b1\nhy : y \u2208 S\nhy' : y \u2208 v\n\u22a2 Set.Nonempty (Z \u2229 \u22c2\u2080 \u2191{U, u, v})\n[PROOFSTEP]\nrefine isIrreducible_iff_sInter.mp hZ { U, u, v } ?_ ?_\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U Z : Set \u03b1\nhU' : IsOpen U\nh\u2081 : U \u2286 S\nh\u2082 : S \u2286 Z\nz : \u03b1\nhz : z \u2208 U\nhZ : IsIrreducible Z\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 S\nhx' : x \u2208 u\ny : \u03b1\nhy : y \u2208 S\nhy' : y \u2208 v\n\u22a2 \u2200 (u_1 : Set \u03b1), u_1 \u2208 {U, u, v} \u2192 IsOpen u_1\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U Z : Set \u03b1\nhU' : IsOpen U\nh\u2081 : U \u2286 S\nh\u2082 : S \u2286 Z\nz : \u03b1\nhz : z \u2208 U\nhZ : IsIrreducible Z\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 S\nhx' : x \u2208 u\ny : \u03b1\nhy : y \u2208 S\nhy' : y \u2208 v\n\u22a2 \u2200 (u_1 : Set \u03b1), u_1 \u2208 {U, u, v} \u2192 Set.Nonempty (Z \u2229 u_1)\n[PROOFSTEP]\nintro U H\n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U\u271d Z : Set \u03b1\nhU' : IsOpen U\u271d\nh\u2081 : U\u271d \u2286 S\nh\u2082 : S \u2286 Z\nz : \u03b1\nhz : z \u2208 U\u271d\nhZ : IsIrreducible Z\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 S\nhx' : x \u2208 u\ny : \u03b1\nhy : y \u2208 S\nhy' : y \u2208 v\nU : Set \u03b1\nH : U \u2208 {U\u271d, u, v}\n\u22a2 Set.Nonempty (Z \u2229 U)\n[PROOFSTEP]\nsimp only [Finset.mem_insert, Finset.mem_singleton] at H \n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U\u271d Z : Set \u03b1\nhU' : IsOpen U\u271d\nh\u2081 : U\u271d \u2286 S\nh\u2082 : S \u2286 Z\nz : \u03b1\nhz : z \u2208 U\u271d\nhZ : IsIrreducible Z\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 S\nhx' : x \u2208 u\ny : \u03b1\nhy : y \u2208 S\nhy' : y \u2208 v\nU : Set \u03b1\nH : U = U\u271d \u2228 U = u \u2228 U = v\n\u22a2 Set.Nonempty (Z \u2229 U)\n[PROOFSTEP]\nrcases H with (rfl | rfl | rfl)\n[GOAL]\ncase refine_2.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S Z : Set \u03b1\nh\u2082 : S \u2286 Z\nz : \u03b1\nhZ : IsIrreducible Z\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 S\nhx' : x \u2208 u\ny : \u03b1\nhy : y \u2208 S\nhy' : y \u2208 v\nU : Set \u03b1\nhU' : IsOpen U\nh\u2081 : U \u2286 S\nhz : z \u2208 U\n\u22a2 Set.Nonempty (Z \u2229 U)\ncase refine_2.inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U\u271d Z : Set \u03b1\nhU' : IsOpen U\u271d\nh\u2081 : U\u271d \u2286 S\nh\u2082 : S \u2286 Z\nz : \u03b1\nhz : z \u2208 U\u271d\nhZ : IsIrreducible Z\nv : Set \u03b1\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 S\ny : \u03b1\nhy : y \u2208 S\nhy' : y \u2208 v\nU : Set \u03b1\nhu : IsOpen U\nhx' : x \u2208 U\n\u22a2 Set.Nonempty (Z \u2229 U)\ncase refine_2.inr.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U\u271d Z : Set \u03b1\nhU' : IsOpen U\u271d\nh\u2081 : U\u271d \u2286 S\nh\u2082 : S \u2286 Z\nz : \u03b1\nhz : z \u2208 U\u271d\nhZ : IsIrreducible Z\nu : Set \u03b1\nhu : IsOpen u\nx : \u03b1\nhx : x \u2208 S\nhx' : x \u2208 u\ny : \u03b1\nhy : y \u2208 S\nU : Set \u03b1\nhv : IsOpen U\nhy' : y \u2208 U\n\u22a2 Set.Nonempty (Z \u2229 U)\n[PROOFSTEP]\nexacts [\u27e8z, h\u2082 (h\u2081 hz), hz\u27e9, \u27e8x, h\u2082 hx, hx'\u27e9, \u27e8y, h\u2082 hy, hy'\u27e9]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U Z : Set \u03b1\nhU' : IsOpen U\nh\u2081 : U \u2286 S\nh\u2082 : S \u2286 Z\nz : \u03b1\nhz : z \u2208 U\nhZ : IsIrreducible Z\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 S\nhx' : x \u2208 u\ny : \u03b1\nhy : y \u2208 S\nhy' : y \u2208 v\na : \u03b1\nha' : a \u2208 \u22c2\u2080 \u2191{U, u, v}\n\u22a2 Set.Nonempty (S \u2229 (u \u2229 v))\n[PROOFSTEP]\nreplace ha' : a \u2208 U \u2227 a \u2208 u \u2227 a \u2208 v := by simpa using ha'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U Z : Set \u03b1\nhU' : IsOpen U\nh\u2081 : U \u2286 S\nh\u2082 : S \u2286 Z\nz : \u03b1\nhz : z \u2208 U\nhZ : IsIrreducible Z\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 S\nhx' : x \u2208 u\ny : \u03b1\nhy : y \u2208 S\nhy' : y \u2208 v\na : \u03b1\nha' : a \u2208 \u22c2\u2080 \u2191{U, u, v}\n\u22a2 a \u2208 U \u2227 a \u2208 u \u2227 a \u2208 v\n[PROOFSTEP]\nsimpa using ha'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t S U Z : Set \u03b1\nhU' : IsOpen U\nh\u2081 : U \u2286 S\nh\u2082 : S \u2286 Z\nz : \u03b1\nhz : z \u2208 U\nhZ : IsIrreducible Z\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nhx : x \u2208 S\nhx' : x \u2208 u\ny : \u03b1\nhy : y \u2208 S\nhy' : y \u2208 v\na : \u03b1\nha' : a \u2208 U \u2227 a \u2208 u \u2227 a \u2208 v\n\u22a2 Set.Nonempty (S \u2229 (u \u2229 v))\n[PROOFSTEP]\nexact \u27e8a, h\u2081 ha'.1, ha'.2\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t Z : Set \u03b1\nhZ : IsPreirreducible Z\nf : \u03b2 \u2192 \u03b1\nhf : OpenEmbedding f\n\u22a2 IsPreirreducible (f \u207b\u00b9' Z)\n[PROOFSTEP]\nrintro U V hU hV \u27e8x, hx, hx'\u27e9 \u27e8y, hy, hy'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t Z : Set \u03b1\nhZ : IsPreirreducible Z\nf : \u03b2 \u2192 \u03b1\nhf : OpenEmbedding f\nU V : Set \u03b2\nhU : IsOpen U\nhV : IsOpen V\nx : \u03b2\nhx : x \u2208 f \u207b\u00b9' Z\nhx' : x \u2208 U\ny : \u03b2\nhy : y \u2208 f \u207b\u00b9' Z\nhy' : y \u2208 V\n\u22a2 Set.Nonempty (f \u207b\u00b9' Z \u2229 (U \u2229 V))\n[PROOFSTEP]\nobtain \u27e8_, h\u2081, \u27e8z, h\u2082, rfl\u27e9, \u27e8z', h\u2083, h\u2084\u27e9\u27e9 :=\n hZ _ _ (hf.isOpenMap _ hU) (hf.isOpenMap _ hV) \u27e8f x, hx, Set.mem_image_of_mem f hx'\u27e9\n \u27e8f y, hy, Set.mem_image_of_mem f hy'\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t Z : Set \u03b1\nhZ : IsPreirreducible Z\nf : \u03b2 \u2192 \u03b1\nhf : OpenEmbedding f\nU V : Set \u03b2\nhU : IsOpen U\nhV : IsOpen V\nx : \u03b2\nhx : x \u2208 f \u207b\u00b9' Z\nhx' : x \u2208 U\ny : \u03b2\nhy : y \u2208 f \u207b\u00b9' Z\nhy' : y \u2208 V\nz : \u03b2\nh\u2082 : z \u2208 U\nh\u2081 : f z \u2208 Z\nz' : \u03b2\nh\u2083 : z' \u2208 V\nh\u2084 : f z' = f z\n\u22a2 Set.Nonempty (f \u207b\u00b9' Z \u2229 (U \u2229 V))\n[PROOFSTEP]\ncases hf.inj h\u2084\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t Z : Set \u03b1\nhZ : IsPreirreducible Z\nf : \u03b2 \u2192 \u03b1\nhf : OpenEmbedding f\nU V : Set \u03b2\nhU : IsOpen U\nhV : IsOpen V\nx : \u03b2\nhx : x \u2208 f \u207b\u00b9' Z\nhx' : x \u2208 U\ny : \u03b2\nhy : y \u2208 f \u207b\u00b9' Z\nhy' : y \u2208 V\nz : \u03b2\nh\u2082 : z \u2208 U\nh\u2081 : f z \u2208 Z\nh\u2083 : z \u2208 V\nh\u2084 : f z = f z\n\u22a2 Set.Nonempty (f \u207b\u00b9' Z \u2229 (U \u2229 V))\n[PROOFSTEP]\nexact \u27e8z, h\u2081, h\u2082, h\u2083\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.SubsetProperties", "llama_tokens": 116943, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.746138993030751, "lm_q2_score": 0.5660185351961016, "lm_q1q2_score": 0.42232849988795995}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc : \ud835\udd5c\nv : E\nhv : v \u2208 (eigenspace T \u03bc)\u15ee\n\u22a2 \u2191T v \u2208 (eigenspace T \u03bc)\u15ee\n[PROOFSTEP]\nintro w hw\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc : \ud835\udd5c\nv : E\nhv : v \u2208 (eigenspace T \u03bc)\u15ee\nw : E\nhw : w \u2208 eigenspace T \u03bc\n\u22a2 inner w (\u2191T v) = 0\n[PROOFSTEP]\nhave : T w = (\u03bc : \ud835\udd5c) \u2022 w := by rwa [mem_eigenspace_iff] at hw \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc : \ud835\udd5c\nv : E\nhv : v \u2208 (eigenspace T \u03bc)\u15ee\nw : E\nhw : w \u2208 eigenspace T \u03bc\n\u22a2 \u2191T w = \u03bc \u2022 w\n[PROOFSTEP]\nrwa [mem_eigenspace_iff] at hw \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc : \ud835\udd5c\nv : E\nhv : v \u2208 (eigenspace T \u03bc)\u15ee\nw : E\nhw : w \u2208 eigenspace T \u03bc\nthis : \u2191T w = \u03bc \u2022 w\n\u22a2 inner w (\u2191T v) = 0\n[PROOFSTEP]\nsimp [\u2190 hT w, this, inner_smul_left, hv w hw]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc : \ud835\udd5c\nh\u03bc : HasEigenvalue T \u03bc\n\u22a2 \u2191(starRingEnd \ud835\udd5c) \u03bc = \u03bc\n[PROOFSTEP]\nobtain \u27e8v, hv\u2081, hv\u2082\u27e9 := h\u03bc.exists_hasEigenvector\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc : \ud835\udd5c\nh\u03bc : HasEigenvalue T \u03bc\nv : E\nhv\u2081 : v \u2208 eigenspace T \u03bc\nhv\u2082 : v \u2260 0\n\u22a2 \u2191(starRingEnd \ud835\udd5c) \u03bc = \u03bc\n[PROOFSTEP]\nrw [mem_eigenspace_iff] at hv\u2081 \n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc : \ud835\udd5c\nh\u03bc : HasEigenvalue T \u03bc\nv : E\nhv\u2081 : \u2191T v = \u03bc \u2022 v\nhv\u2082 : v \u2260 0\n\u22a2 \u2191(starRingEnd \ud835\udd5c) \u03bc = \u03bc\n[PROOFSTEP]\nsimpa [hv\u2082, inner_smul_left, inner_smul_right, hv\u2081] using hT v v\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u22a2 OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T \u03bc }) fun \u03bc => Submodule.subtype\u2097\u1d62 (eigenspace T \u03bc)\n[PROOFSTEP]\nrintro \u03bc \u03bd h\u03bc\u03bd \u27e8v, hv\u27e9 \u27e8w, hw\u27e9\n[GOAL]\ncase mk.mk\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc \u03bd : \ud835\udd5c\nh\u03bc\u03bd : \u03bc \u2260 \u03bd\nv : E\nhv : v \u2208 eigenspace T \u03bc\nw : E\nhw : w \u2208 eigenspace T \u03bd\n\u22a2 inner (\u2191((fun \u03bc => Submodule.subtype\u2097\u1d62 (eigenspace T \u03bc)) \u03bc) { val := v, property := hv })\n (\u2191((fun \u03bc => Submodule.subtype\u2097\u1d62 (eigenspace T \u03bc)) \u03bd) { val := w, property := hw }) =\n 0\n[PROOFSTEP]\nby_cases hv' : v = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc \u03bd : \ud835\udd5c\nh\u03bc\u03bd : \u03bc \u2260 \u03bd\nv : E\nhv : v \u2208 eigenspace T \u03bc\nw : E\nhw : w \u2208 eigenspace T \u03bd\nhv' : v = 0\n\u22a2 inner (\u2191((fun \u03bc => Submodule.subtype\u2097\u1d62 (eigenspace T \u03bc)) \u03bc) { val := v, property := hv })\n (\u2191((fun \u03bc => Submodule.subtype\u2097\u1d62 (eigenspace T \u03bc)) \u03bd) { val := w, property := hw }) =\n 0\n[PROOFSTEP]\nsimp [hv']\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc \u03bd : \ud835\udd5c\nh\u03bc\u03bd : \u03bc \u2260 \u03bd\nv : E\nhv : v \u2208 eigenspace T \u03bc\nw : E\nhw : w \u2208 eigenspace T \u03bd\nhv' : \u00acv = 0\n\u22a2 inner (\u2191((fun \u03bc => Submodule.subtype\u2097\u1d62 (eigenspace T \u03bc)) \u03bc) { val := v, property := hv })\n (\u2191((fun \u03bc => Submodule.subtype\u2097\u1d62 (eigenspace T \u03bc)) \u03bd) { val := w, property := hw }) =\n 0\n[PROOFSTEP]\nhave H := hT.conj_eigenvalue_eq_self (hasEigenvalue_of_hasEigenvector \u27e8hv, hv'\u27e9)\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc \u03bd : \ud835\udd5c\nh\u03bc\u03bd : \u03bc \u2260 \u03bd\nv : E\nhv : v \u2208 eigenspace T \u03bc\nw : E\nhw : w \u2208 eigenspace T \u03bd\nhv' : \u00acv = 0\nH : \u2191(starRingEnd \ud835\udd5c) \u03bc = \u03bc\n\u22a2 inner (\u2191((fun \u03bc => Submodule.subtype\u2097\u1d62 (eigenspace T \u03bc)) \u03bc) { val := v, property := hv })\n (\u2191((fun \u03bc => Submodule.subtype\u2097\u1d62 (eigenspace T \u03bc)) \u03bd) { val := w, property := hw }) =\n 0\n[PROOFSTEP]\nrw [mem_eigenspace_iff] at hv hw \n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc \u03bd : \ud835\udd5c\nh\u03bc\u03bd : \u03bc \u2260 \u03bd\nv : E\nhv\u271d : v \u2208 eigenspace T \u03bc\nhv : \u2191T v = \u03bc \u2022 v\nw : E\nhw\u271d : w \u2208 eigenspace T \u03bd\nhw : \u2191T w = \u03bd \u2022 w\nhv' : \u00acv = 0\nH : \u2191(starRingEnd \ud835\udd5c) \u03bc = \u03bc\n\u22a2 inner (\u2191((fun \u03bc => Submodule.subtype\u2097\u1d62 (eigenspace T \u03bc)) \u03bc) { val := v, property := hv\u271d })\n (\u2191((fun \u03bc => Submodule.subtype\u2097\u1d62 (eigenspace T \u03bc)) \u03bd) { val := w, property := hw\u271d }) =\n 0\n[PROOFSTEP]\nrefine' Or.resolve_left _ h\u03bc\u03bd.symm\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc \u03bd : \ud835\udd5c\nh\u03bc\u03bd : \u03bc \u2260 \u03bd\nv : E\nhv\u271d : v \u2208 eigenspace T \u03bc\nhv : \u2191T v = \u03bc \u2022 v\nw : E\nhw\u271d : w \u2208 eigenspace T \u03bd\nhw : \u2191T w = \u03bd \u2022 w\nhv' : \u00acv = 0\nH : \u2191(starRingEnd \ud835\udd5c) \u03bc = \u03bc\n\u22a2 \u03bd = \u03bc \u2228\n inner (\u2191((fun \u03bc => Submodule.subtype\u2097\u1d62 (eigenspace T \u03bc)) \u03bc) { val := v, property := hv\u271d })\n (\u2191((fun \u03bc => Submodule.subtype\u2097\u1d62 (eigenspace T \u03bc)) \u03bd) { val := w, property := hw\u271d }) =\n 0\n[PROOFSTEP]\nsimpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nv : E\nhv : v \u2208 (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee\n\u22a2 \u2191T v \u2208 (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee\n[PROOFSTEP]\nrw [\u2190 Submodule.iInf_orthogonal] at hv \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nv : E\nhv : v \u2208 \u2a05 (i : \ud835\udd5c), (eigenspace T i)\u15ee\n\u22a2 \u2191T v \u2208 \u2a05 (i : \ud835\udd5c), (eigenspace T i)\u15ee\n[PROOFSTEP]\nexact T.iInf_invariant hT.invariant_orthogonalComplement_eigenspace v hv\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc : \ud835\udd5c\n\u22a2 eigenspace (restrict T (_ : \u2200 \u2983v : E\u2984, v \u2208 (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee \u2192 \u2191T v \u2208 (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee)) \u03bc = \u22a5\n[PROOFSTEP]\nset p : Submodule \ud835\udd5c E := (\u2a06 \u03bc, eigenspace T \u03bc)\u15ee\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc : \ud835\udd5c\np : Submodule \ud835\udd5c E := (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee\n\u22a2 eigenspace (restrict T (_ : \u2200 \u2983v : E\u2984, v \u2208 (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee \u2192 \u2191T v \u2208 (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee)) \u03bc = \u22a5\n[PROOFSTEP]\nrefine' eigenspace_restrict_eq_bot hT.orthogonalComplement_iSup_eigenspaces_invariant _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc : \ud835\udd5c\np : Submodule \ud835\udd5c E := (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee\n\u22a2 Disjoint (eigenspace T \u03bc) (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee\n[PROOFSTEP]\nhave H\u2082 : eigenspace T \u03bc \u27c2 p := (Submodule.isOrtho_orthogonal_right _).mono_left (le_iSup _ _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u03bc : \ud835\udd5c\np : Submodule \ud835\udd5c E := (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee\nH\u2082 : eigenspace T \u03bc \u27c2 p\n\u22a2 Disjoint (eigenspace T \u03bc) (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee\n[PROOFSTEP]\nexact H\u2082.disjoint\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\n\u22a2 (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee = \u22a5\n[PROOFSTEP]\nhave hT' : IsSymmetric _ := hT.restrict_invariant hT.orthogonalComplement_iSup_eigenspaces_invariant\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhT' : IsSymmetric (restrict T (_ : \u2200 \u2983v : E\u2984, v \u2208 (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee \u2192 \u2191T v \u2208 (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee))\n\u22a2 (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee = \u22a5\n[PROOFSTEP]\nhaveI := hT'.subsingleton_of_no_eigenvalue_finiteDimensional hT.orthogonalComplement_iSup_eigenspaces\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhT' : IsSymmetric (restrict T (_ : \u2200 \u2983v : E\u2984, v \u2208 (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee \u2192 \u2191T v \u2208 (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee))\nthis : Subsingleton { x // x \u2208 (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee }\n\u22a2 (\u2a06 (\u03bc : \ud835\udd5c), eigenspace T \u03bc)\u15ee = \u22a5\n[PROOFSTEP]\nexact Submodule.eq_bot_of_subsingleton _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\n\u22a2 (\u2a06 (\u03bc : { \u03bc // eigenspace T \u03bc \u2260 \u22a5 }), eigenspace T \u2191\u03bc)\u15ee = \u22a5\n[PROOFSTEP]\nrw [iSup_ne_bot_subtype, hT.orthogonalComplement_iSup_eigenspaces_eq_bot]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT\u271d : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhT : Fact (IsSymmetric T)\n\u03bc : Eigenvalues T\n\u22a2 CompleteSpace { x // x \u2208 eigenspace T (\u2191T \u03bc) }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nv : E\n\u03bc : Eigenvalues T\n\u22a2 \u2191(diagonalization hT) (\u2191T v) \u03bc = \u2191T \u03bc \u2022 \u2191(diagonalization hT) v \u03bc\n[PROOFSTEP]\nsuffices\n \u2200 w : PiLp 2 fun \u03bc : Eigenvalues T => eigenspace T \u03bc,\n T (hT.diagonalization.symm w) = hT.diagonalization.symm fun \u03bc => (\u03bc : \ud835\udd5c) \u2022 w \u03bc\n by\n simpa only [LinearIsometryEquiv.symm_apply_apply, LinearIsometryEquiv.apply_symm_apply] using\n congr_arg (fun w => hT.diagonalization w \u03bc) (this (hT.diagonalization v))\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nv : E\n\u03bc : Eigenvalues T\nthis :\n \u2200 (w : PiLp 2 fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }),\n \u2191T (\u2191(LinearIsometryEquiv.symm (diagonalization hT)) w) =\n \u2191(LinearIsometryEquiv.symm (diagonalization hT)) fun \u03bc => \u2191T \u03bc \u2022 w \u03bc\n\u22a2 \u2191(diagonalization hT) (\u2191T v) \u03bc = \u2191T \u03bc \u2022 \u2191(diagonalization hT) v \u03bc\n[PROOFSTEP]\nsimpa only [LinearIsometryEquiv.symm_apply_apply, LinearIsometryEquiv.apply_symm_apply] using\n congr_arg (fun w => hT.diagonalization w \u03bc) (this (hT.diagonalization v))\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nv : E\n\u03bc : Eigenvalues T\n\u22a2 \u2200 (w : PiLp 2 fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }),\n \u2191T (\u2191(LinearIsometryEquiv.symm (diagonalization hT)) w) =\n \u2191(LinearIsometryEquiv.symm (diagonalization hT)) fun \u03bc => \u2191T \u03bc \u2022 w \u03bc\n[PROOFSTEP]\nintro w\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nv : E\n\u03bc : Eigenvalues T\nw : PiLp 2 fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }\n\u22a2 \u2191T (\u2191(LinearIsometryEquiv.symm (diagonalization hT)) w) =\n \u2191(LinearIsometryEquiv.symm (diagonalization hT)) fun \u03bc => \u2191T \u03bc \u2022 w \u03bc\n[PROOFSTEP]\nhave hwT : \u2200 \u03bc, T (w \u03bc) = (\u03bc : \ud835\udd5c) \u2022 w \u03bc := fun \u03bc => mem_eigenspace_iff.1 (w \u03bc).2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nv : E\n\u03bc : Eigenvalues T\nw : PiLp 2 fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }\nhwT : \u2200 (\u03bc : Eigenvalues T), \u2191T \u2191(w \u03bc) = \u2191(\u2191T \u03bc \u2022 w \u03bc)\n\u22a2 \u2191T (\u2191(LinearIsometryEquiv.symm (diagonalization hT)) w) =\n \u2191(LinearIsometryEquiv.symm (diagonalization hT)) fun \u03bc => \u2191T \u03bc \u2022 w \u03bc\n[PROOFSTEP]\nsimp only [hwT, diagonalization_symm_apply, map_sum, Submodule.coe_smul_of_tower]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\ni : Fin n\n\u22a2 HasEigenvector T (\u2191(eigenvalues hT hn i)) (\u2191(eigenvectorBasis hT hn) i)\n[PROOFSTEP]\nlet v : E := hT.eigenvectorBasis hn i\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\ni : Fin n\nv : E := \u2191(eigenvectorBasis hT hn) i\n\u22a2 HasEigenvector T (\u2191(eigenvalues hT hn i)) (\u2191(eigenvectorBasis hT hn) i)\n[PROOFSTEP]\nlet \u03bc : \ud835\udd5c := (hT.direct_sum_isInternal.subordinateOrthonormalBasisIndex hn i hT.orthogonalFamily_eigenspaces').val\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\ni : Fin n\nv : E := \u2191(eigenvectorBasis hT hn) i\n\u03bc : \ud835\udd5c :=\n \u2191T\n (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc)) i\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc))))\n\u22a2 HasEigenvector T (\u2191(eigenvalues hT hn i)) (\u2191(eigenvectorBasis hT hn) i)\n[PROOFSTEP]\nsimp_rw [eigenvalues]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\ni : Fin n\nv : E := \u2191(eigenvectorBasis hT hn) i\n\u03bc : \ud835\udd5c :=\n \u2191T\n (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc)) i\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc))))\n\u22a2 HasEigenvector T\n (\u2191(\u2191IsROrC.re\n (\u2191T\n (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn\n (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc)) i\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc)))))))\n (\u2191(eigenvectorBasis hT hn) i)\n[PROOFSTEP]\nchange HasEigenvector T (IsROrC.re \u03bc) v\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\ni : Fin n\nv : E := \u2191(eigenvectorBasis hT hn) i\n\u03bc : \ud835\udd5c :=\n \u2191T\n (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc)) i\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc))))\n\u22a2 HasEigenvector T (\u2191(\u2191IsROrC.re \u03bc)) v\n[PROOFSTEP]\nhave key : HasEigenvector T \u03bc v :=\n by\n have H\u2081 : v \u2208 eigenspace T \u03bc := by\n simp_rw [eigenvectorBasis]\n exact hT.direct_sum_isInternal.subordinateOrthonormalBasis_subordinate hn i hT.orthogonalFamily_eigenspaces'\n have H\u2082 : v \u2260 0 := by simpa using (hT.eigenvectorBasis hn).toBasis.ne_zero i\n exact \u27e8H\u2081, H\u2082\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\ni : Fin n\nv : E := \u2191(eigenvectorBasis hT hn) i\n\u03bc : \ud835\udd5c :=\n \u2191T\n (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc)) i\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc))))\n\u22a2 HasEigenvector T \u03bc v\n[PROOFSTEP]\nhave H\u2081 : v \u2208 eigenspace T \u03bc := by\n simp_rw [eigenvectorBasis]\n exact hT.direct_sum_isInternal.subordinateOrthonormalBasis_subordinate hn i hT.orthogonalFamily_eigenspaces'\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\ni : Fin n\nv : E := \u2191(eigenvectorBasis hT hn) i\n\u03bc : \ud835\udd5c :=\n \u2191T\n (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc)) i\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc))))\n\u22a2 v \u2208 eigenspace T \u03bc\n[PROOFSTEP]\nsimp_rw [eigenvectorBasis]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\ni : Fin n\nv : E := \u2191(eigenvectorBasis hT hn) i\n\u03bc : \ud835\udd5c :=\n \u2191T\n (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc)) i\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc))))\n\u22a2 \u2191(DirectSum.IsInternal.subordinateOrthonormalBasis hn (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc))\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc))))\n i \u2208\n eigenspace T\n (\u2191T\n (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn\n (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc)) i\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc)))))\n[PROOFSTEP]\nexact hT.direct_sum_isInternal.subordinateOrthonormalBasis_subordinate hn i hT.orthogonalFamily_eigenspaces'\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\ni : Fin n\nv : E := \u2191(eigenvectorBasis hT hn) i\n\u03bc : \ud835\udd5c :=\n \u2191T\n (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc)) i\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc))))\nH\u2081 : v \u2208 eigenspace T \u03bc\n\u22a2 HasEigenvector T \u03bc v\n[PROOFSTEP]\nhave H\u2082 : v \u2260 0 := by simpa using (hT.eigenvectorBasis hn).toBasis.ne_zero i\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\ni : Fin n\nv : E := \u2191(eigenvectorBasis hT hn) i\n\u03bc : \ud835\udd5c :=\n \u2191T\n (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc)) i\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc))))\nH\u2081 : v \u2208 eigenspace T \u03bc\n\u22a2 v \u2260 0\n[PROOFSTEP]\nsimpa using (hT.eigenvectorBasis hn).toBasis.ne_zero i\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\ni : Fin n\nv : E := \u2191(eigenvectorBasis hT hn) i\n\u03bc : \ud835\udd5c :=\n \u2191T\n (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc)) i\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc))))\nH\u2081 : v \u2208 eigenspace T \u03bc\nH\u2082 : v \u2260 0\n\u22a2 HasEigenvector T \u03bc v\n[PROOFSTEP]\nexact \u27e8H\u2081, H\u2082\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\ni : Fin n\nv : E := \u2191(eigenvectorBasis hT hn) i\n\u03bc : \ud835\udd5c :=\n \u2191T\n (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc)) i\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc))))\nkey : HasEigenvector T \u03bc v\n\u22a2 HasEigenvector T (\u2191(\u2191IsROrC.re \u03bc)) v\n[PROOFSTEP]\nhave re_\u03bc : \u2191(IsROrC.re \u03bc) = \u03bc := by\n rw [\u2190 IsROrC.conj_eq_iff_re]\n exact hT.conj_eigenvalue_eq_self (hasEigenvalue_of_hasEigenvector key)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\ni : Fin n\nv : E := \u2191(eigenvectorBasis hT hn) i\n\u03bc : \ud835\udd5c :=\n \u2191T\n (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc)) i\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc))))\nkey : HasEigenvector T \u03bc v\n\u22a2 \u2191(\u2191IsROrC.re \u03bc) = \u03bc\n[PROOFSTEP]\nrw [\u2190 IsROrC.conj_eq_iff_re]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\ni : Fin n\nv : E := \u2191(eigenvectorBasis hT hn) i\n\u03bc : \ud835\udd5c :=\n \u2191T\n (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc)) i\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc))))\nkey : HasEigenvector T \u03bc v\n\u22a2 \u2191(starRingEnd \ud835\udd5c) \u03bc = \u03bc\n[PROOFSTEP]\nexact hT.conj_eigenvalue_eq_self (hasEigenvalue_of_hasEigenvector key)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\ni : Fin n\nv : E := \u2191(eigenvectorBasis hT hn) i\n\u03bc : \ud835\udd5c :=\n \u2191T\n (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn (_ : DirectSum.IsInternal fun \u03bc => eigenspace T (\u2191T \u03bc)) i\n (_ :\n OrthogonalFamily \ud835\udd5c (fun \u03bc => { x // x \u2208 eigenspace T (\u2191T \u03bc) }) fun \u03bc =>\n Submodule.subtype\u2097\u1d62 (eigenspace T (\u2191T \u03bc))))\nkey : HasEigenvector T \u03bc v\nre_\u03bc : \u2191(\u2191IsROrC.re \u03bc) = \u03bc\n\u22a2 HasEigenvector T (\u2191(\u2191IsROrC.re \u03bc)) v\n[PROOFSTEP]\nsimpa [re_\u03bc] using key\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\nv : E\ni : Fin n\n\u22a2 \u2191(eigenvectorBasis hT hn).repr (\u2191T v) i = \u2191(eigenvalues hT hn i) * \u2191(eigenvectorBasis hT hn).repr v i\n[PROOFSTEP]\nsuffices\n \u2200 w : EuclideanSpace \ud835\udd5c (Fin n),\n T ((hT.eigenvectorBasis hn).repr.symm w) = (hT.eigenvectorBasis hn).repr.symm fun i => hT.eigenvalues hn i * w i\n by\n simpa [OrthonormalBasis.sum_repr_symm] using\n congr_arg (fun v => (hT.eigenvectorBasis hn).repr v i) (this ((hT.eigenvectorBasis hn).repr v))\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\nv : E\ni : Fin n\nthis :\n \u2200 (w : EuclideanSpace \ud835\udd5c (Fin n)),\n \u2191T (\u2191(LinearIsometryEquiv.symm (eigenvectorBasis hT hn).repr) w) =\n \u2191(LinearIsometryEquiv.symm (eigenvectorBasis hT hn).repr) fun i => \u2191(eigenvalues hT hn i) * w i\n\u22a2 \u2191(eigenvectorBasis hT hn).repr (\u2191T v) i = \u2191(eigenvalues hT hn i) * \u2191(eigenvectorBasis hT hn).repr v i\n[PROOFSTEP]\nsimpa [OrthonormalBasis.sum_repr_symm] using\n congr_arg (fun v => (hT.eigenvectorBasis hn).repr v i) (this ((hT.eigenvectorBasis hn).repr v))\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\nv : E\ni : Fin n\n\u22a2 \u2200 (w : EuclideanSpace \ud835\udd5c (Fin n)),\n \u2191T (\u2191(LinearIsometryEquiv.symm (eigenvectorBasis hT hn).repr) w) =\n \u2191(LinearIsometryEquiv.symm (eigenvectorBasis hT hn).repr) fun i => \u2191(eigenvalues hT hn i) * w i\n[PROOFSTEP]\nintro w\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\nv : E\ni : Fin n\nw : EuclideanSpace \ud835\udd5c (Fin n)\n\u22a2 \u2191T (\u2191(LinearIsometryEquiv.symm (eigenvectorBasis hT hn).repr) w) =\n \u2191(LinearIsometryEquiv.symm (eigenvectorBasis hT hn).repr) fun i => \u2191(eigenvalues hT hn i) * w i\n[PROOFSTEP]\nsimp_rw [\u2190 OrthonormalBasis.sum_repr_symm, LinearMap.map_sum, LinearMap.map_smul, apply_eigenvectorBasis]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\nv : E\ni : Fin n\nw : EuclideanSpace \ud835\udd5c (Fin n)\n\u22a2 \u2211 x : Fin n, w x \u2022 \u2191(eigenvalues hT hn x) \u2022 \u2191(eigenvectorBasis hT hn) x =\n \u2211 x : Fin n, (\u2191(eigenvalues hT hn x) * w x) \u2022 \u2191(eigenvectorBasis hT hn) x\n[PROOFSTEP]\napply Fintype.sum_congr\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\nv : E\ni : Fin n\nw : EuclideanSpace \ud835\udd5c (Fin n)\n\u22a2 \u2200 (a : Fin n),\n w a \u2022 \u2191(eigenvalues hT hn a) \u2022 \u2191(eigenvectorBasis hT hn) a =\n (\u2191(eigenvalues hT hn a) * w a) \u2022 \u2191(eigenvectorBasis hT hn) a\n[PROOFSTEP]\nintro a\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\ninst\u271d : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : FiniteDimensional.finrank \ud835\udd5c E = n\nv : E\ni : Fin n\nw : EuclideanSpace \ud835\udd5c (Fin n)\na : Fin n\n\u22a2 w a \u2022 \u2191(eigenvalues hT hn a) \u2022 \u2191(eigenvectorBasis hT hn) a =\n (\u2191(eigenvalues hT hn a) * w a) \u2022 \u2191(eigenvectorBasis hT hn) a\n[PROOFSTEP]\nrw [smul_smul, mul_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \ud835\udd5c\nv : E\nT : E \u2192\u2097[\ud835\udd5c] E\nh : v \u2208 eigenspace T \u03bc\n\u22a2 inner v (\u2191T v) = \u03bc * \u2191\u2016v\u2016 ^ 2\n[PROOFSTEP]\nsimp only [mem_eigenspace_iff.mp h, inner_smul_right, inner_self_eq_norm_sq_to_K]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 \u2264 \u2191IsROrC.re (inner x (\u2191T x))\n\u22a2 0 \u2264 \u03bc\n[PROOFSTEP]\nobtain \u27e8v, hv\u27e9 := h\u03bc.exists_hasEigenvector\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 \u2264 \u2191IsROrC.re (inner x (\u2191T x))\nv : E\nhv : HasEigenvector T (\u2191\u03bc) v\n\u22a2 0 \u2264 \u03bc\n[PROOFSTEP]\nhave hpos : (0 : \u211d) < \u2016v\u2016 ^ 2 := by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv.2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 \u2264 \u2191IsROrC.re (inner x (\u2191T x))\nv : E\nhv : HasEigenvector T (\u2191\u03bc) v\n\u22a2 0 < \u2016v\u2016 ^ 2\n[PROOFSTEP]\nsimpa only [sq_pos_iff, norm_ne_zero_iff] using hv.2\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 \u2264 \u2191IsROrC.re (inner x (\u2191T x))\nv : E\nhv : HasEigenvector T (\u2191\u03bc) v\nhpos : 0 < \u2016v\u2016 ^ 2\n\u22a2 0 \u2264 \u03bc\n[PROOFSTEP]\nhave : IsROrC.re \u27eav, T v\u27eb = \u03bc * \u2016v\u2016 ^ 2 :=\n by\n have :=\n congr_arg IsROrC.re\n (inner_product_apply_eigenvector hv.1)\n -- porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`\n rw [\u2190 IsROrC.ofReal_pow, \u2190 IsROrC.ofReal_mul] at this \n exact_mod_cast this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 \u2264 \u2191IsROrC.re (inner x (\u2191T x))\nv : E\nhv : HasEigenvector T (\u2191\u03bc) v\nhpos : 0 < \u2016v\u2016 ^ 2\n\u22a2 \u2191IsROrC.re (inner v (\u2191T v)) = \u03bc * \u2016v\u2016 ^ 2\n[PROOFSTEP]\nhave :=\n congr_arg IsROrC.re\n (inner_product_apply_eigenvector hv.1)\n -- porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 \u2264 \u2191IsROrC.re (inner x (\u2191T x))\nv : E\nhv : HasEigenvector T (\u2191\u03bc) v\nhpos : 0 < \u2016v\u2016 ^ 2\nthis : \u2191IsROrC.re (inner v (\u2191T v)) = \u2191IsROrC.re (\u2191\u03bc * \u2191\u2016v\u2016 ^ 2)\n\u22a2 \u2191IsROrC.re (inner v (\u2191T v)) = \u03bc * \u2016v\u2016 ^ 2\n[PROOFSTEP]\nrw [\u2190 IsROrC.ofReal_pow, \u2190 IsROrC.ofReal_mul] at this \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 \u2264 \u2191IsROrC.re (inner x (\u2191T x))\nv : E\nhv : HasEigenvector T (\u2191\u03bc) v\nhpos : 0 < \u2016v\u2016 ^ 2\nthis : \u2191IsROrC.re (inner v (\u2191T v)) = \u2191IsROrC.re \u2191(\u03bc * \u2016v\u2016 ^ 2)\n\u22a2 \u2191IsROrC.re (inner v (\u2191T v)) = \u03bc * \u2016v\u2016 ^ 2\n[PROOFSTEP]\nexact_mod_cast this\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 \u2264 \u2191IsROrC.re (inner x (\u2191T x))\nv : E\nhv : HasEigenvector T (\u2191\u03bc) v\nhpos : 0 < \u2016v\u2016 ^ 2\nthis : \u2191IsROrC.re (inner v (\u2191T v)) = \u03bc * \u2016v\u2016 ^ 2\n\u22a2 0 \u2264 \u03bc\n[PROOFSTEP]\nexact (zero_le_mul_right hpos).mp (this \u25b8 hnn v)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 < \u2191IsROrC.re (inner x (\u2191T x))\n\u22a2 0 < \u03bc\n[PROOFSTEP]\nobtain \u27e8v, hv\u27e9 := h\u03bc.exists_hasEigenvector\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 < \u2191IsROrC.re (inner x (\u2191T x))\nv : E\nhv : HasEigenvector T (\u2191\u03bc) v\n\u22a2 0 < \u03bc\n[PROOFSTEP]\nhave hpos : (0 : \u211d) < \u2016v\u2016 ^ 2 := by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv.2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 < \u2191IsROrC.re (inner x (\u2191T x))\nv : E\nhv : HasEigenvector T (\u2191\u03bc) v\n\u22a2 0 < \u2016v\u2016 ^ 2\n[PROOFSTEP]\nsimpa only [sq_pos_iff, norm_ne_zero_iff] using hv.2\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 < \u2191IsROrC.re (inner x (\u2191T x))\nv : E\nhv : HasEigenvector T (\u2191\u03bc) v\nhpos : 0 < \u2016v\u2016 ^ 2\n\u22a2 0 < \u03bc\n[PROOFSTEP]\nhave : IsROrC.re \u27eav, T v\u27eb = \u03bc * \u2016v\u2016 ^ 2 :=\n by\n have :=\n congr_arg IsROrC.re\n (inner_product_apply_eigenvector hv.1)\n -- porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`\n rw [\u2190 IsROrC.ofReal_pow, \u2190 IsROrC.ofReal_mul] at this \n exact_mod_cast this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 < \u2191IsROrC.re (inner x (\u2191T x))\nv : E\nhv : HasEigenvector T (\u2191\u03bc) v\nhpos : 0 < \u2016v\u2016 ^ 2\n\u22a2 \u2191IsROrC.re (inner v (\u2191T v)) = \u03bc * \u2016v\u2016 ^ 2\n[PROOFSTEP]\nhave :=\n congr_arg IsROrC.re\n (inner_product_apply_eigenvector hv.1)\n -- porting note: why can't `exact_mod_cast` do this? These lemmas are marked `norm_cast`\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 < \u2191IsROrC.re (inner x (\u2191T x))\nv : E\nhv : HasEigenvector T (\u2191\u03bc) v\nhpos : 0 < \u2016v\u2016 ^ 2\nthis : \u2191IsROrC.re (inner v (\u2191T v)) = \u2191IsROrC.re (\u2191\u03bc * \u2191\u2016v\u2016 ^ 2)\n\u22a2 \u2191IsROrC.re (inner v (\u2191T v)) = \u03bc * \u2016v\u2016 ^ 2\n[PROOFSTEP]\nrw [\u2190 IsROrC.ofReal_pow, \u2190 IsROrC.ofReal_mul] at this \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 < \u2191IsROrC.re (inner x (\u2191T x))\nv : E\nhv : HasEigenvector T (\u2191\u03bc) v\nhpos : 0 < \u2016v\u2016 ^ 2\nthis : \u2191IsROrC.re (inner v (\u2191T v)) = \u2191IsROrC.re \u2191(\u03bc * \u2016v\u2016 ^ 2)\n\u22a2 \u2191IsROrC.re (inner v (\u2191T v)) = \u03bc * \u2016v\u2016 ^ 2\n[PROOFSTEP]\nexact_mod_cast this\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ndec_\ud835\udd5c : DecidableEq \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\n\u03bc : \u211d\nT : E \u2192\u2097[\ud835\udd5c] E\nh\u03bc : HasEigenvalue T \u2191\u03bc\nhnn : \u2200 (x : E), 0 < \u2191IsROrC.re (inner x (\u2191T x))\nv : E\nhv : HasEigenvector T (\u2191\u03bc) v\nhpos : 0 < \u2016v\u2016 ^ 2\nthis : \u2191IsROrC.re (inner v (\u2191T v)) = \u03bc * \u2016v\u2016 ^ 2\n\u22a2 0 < \u03bc\n[PROOFSTEP]\nexact (zero_lt_mul_right hpos).mp (this \u25b8 hnn v)\n", "meta": {"mathlib_filename": "Mathlib.Analysis.InnerProductSpace.Spectrum", "llama_tokens": 17358, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389817407016, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.4223284934975826}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsIrrefl \u03b1 r\ninst\u271d : IsIrrefl \u03b2 s\n\u22a2 \u2200 (a : \u03b1 \u2295 \u03b2), \u00acLiftRel r s a a\n[PROOFSTEP]\nrintro _ (\u27e8h\u27e9 | \u27e8h\u27e9)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsIrrefl \u03b1 r\ninst\u271d : IsIrrefl \u03b2 s\na\u271d : \u03b1\nh : r a\u271d a\u271d\n\u22a2 False\n[PROOFSTEP]\nexact irrefl _ h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsIrrefl \u03b1 r\ninst\u271d : IsIrrefl \u03b2 s\nb\u271d : \u03b2\nh : s b\u271d b\u271d\n\u22a2 False\n[PROOFSTEP]\nexact irrefl _ h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsAntisymm \u03b1 r\ninst\u271d : IsAntisymm \u03b2 s\n\u22a2 \u2200 (a b : \u03b1 \u2295 \u03b2), LiftRel r s a b \u2192 LiftRel r s b a \u2192 a = b\n[PROOFSTEP]\nrintro _ _ (\u27e8hab\u27e9 | \u27e8hab\u27e9) (\u27e8hba\u27e9 | \u27e8hba\u27e9)\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsAntisymm \u03b1 r\ninst\u271d : IsAntisymm \u03b2 s\na\u271d c\u271d : \u03b1\nhab : r a\u271d c\u271d\nhba : r c\u271d a\u271d\n\u22a2 inl a\u271d = inl c\u271d\n[PROOFSTEP]\nrw [antisymm hab hba]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsAntisymm \u03b1 r\ninst\u271d : IsAntisymm \u03b2 s\nb\u271d d\u271d : \u03b2\nhab : s b\u271d d\u271d\nhba : s d\u271d b\u271d\n\u22a2 inr b\u271d = inr d\u271d\n[PROOFSTEP]\nrw [antisymm hab hba]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsRefl \u03b1 r\ninst\u271d : IsRefl \u03b2 s\n\u22a2 \u2200 (a : \u03b1 \u2295 \u03b2), Lex r s a a\n[PROOFSTEP]\nrintro (a | a)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsRefl \u03b1 r\ninst\u271d : IsRefl \u03b2 s\na : \u03b1\n\u22a2 Lex r s (inl a) (inl a)\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsRefl \u03b1 r\ninst\u271d : IsRefl \u03b2 s\na : \u03b2\n\u22a2 Lex r s (inr a) (inr a)\n[PROOFSTEP]\nexacts [Lex.inl (refl _), Lex.inr (refl _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsIrrefl \u03b1 r\ninst\u271d : IsIrrefl \u03b2 s\n\u22a2 \u2200 (a : \u03b1 \u2295 \u03b2), \u00acLex r s a a\n[PROOFSTEP]\nrintro _ (\u27e8h\u27e9 | \u27e8h\u27e9)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsIrrefl \u03b1 r\ninst\u271d : IsIrrefl \u03b2 s\na\u2081\u271d : \u03b1\nh : r a\u2081\u271d a\u2081\u271d\n\u22a2 False\n[PROOFSTEP]\nexact irrefl _ h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsIrrefl \u03b1 r\ninst\u271d : IsIrrefl \u03b2 s\nb\u2081\u271d : \u03b2\nh : s b\u2081\u271d b\u2081\u271d\n\u22a2 False\n[PROOFSTEP]\nexact irrefl _ h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsTrans \u03b2 s\n\u22a2 \u2200 (a b c : \u03b1 \u2295 \u03b2), Lex r s a b \u2192 Lex r s b c \u2192 Lex r s a c\n[PROOFSTEP]\nrintro _ _ _ (\u27e8hab\u27e9 | \u27e8hab\u27e9) (\u27e8hbc\u27e9 | \u27e8hbc\u27e9)\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsTrans \u03b2 s\na\u2081\u271d a\u2082\u271d\u00b9 : \u03b1\nhab : r a\u2081\u271d a\u2082\u271d\u00b9\na\u2082\u271d : \u03b1\nhbc : r a\u2082\u271d\u00b9 a\u2082\u271d\n\u22a2 Lex r s (inl a\u2081\u271d) (inl a\u2082\u271d)\ncase inl.sep\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsTrans \u03b2 s\na\u2081\u271d a\u2082\u271d : \u03b1\nhab : r a\u2081\u271d a\u2082\u271d\nb\u271d : \u03b2\n\u22a2 Lex r s (inl a\u2081\u271d) (inr b\u271d)\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsTrans \u03b2 s\nb\u2081\u271d b\u2082\u271d\u00b9 : \u03b2\nhab : s b\u2081\u271d b\u2082\u271d\u00b9\nb\u2082\u271d : \u03b2\nhbc : s b\u2082\u271d\u00b9 b\u2082\u271d\n\u22a2 Lex r s (inr b\u2081\u271d) (inr b\u2082\u271d)\ncase sep.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsTrans \u03b2 s\na\u271d : \u03b1\nb\u271d b\u2082\u271d : \u03b2\nhbc : s b\u271d b\u2082\u271d\n\u22a2 Lex r s (inl a\u271d) (inr b\u2082\u271d)\n[PROOFSTEP]\nexacts [.inl (_root_.trans hab hbc), .sep _ _, .inr (_root_.trans hab hbc), .sep _ _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsAntisymm \u03b1 r\ninst\u271d : IsAntisymm \u03b2 s\n\u22a2 \u2200 (a b : \u03b1 \u2295 \u03b2), Lex r s a b \u2192 Lex r s b a \u2192 a = b\n[PROOFSTEP]\nrintro _ _ (\u27e8hab\u27e9 | \u27e8hab\u27e9) (\u27e8hba\u27e9 | \u27e8hba\u27e9)\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsAntisymm \u03b1 r\ninst\u271d : IsAntisymm \u03b2 s\na\u2081\u271d a\u2082\u271d : \u03b1\nhab : r a\u2081\u271d a\u2082\u271d\nhba : r a\u2082\u271d a\u2081\u271d\n\u22a2 inl a\u2081\u271d = inl a\u2082\u271d\n[PROOFSTEP]\nrw [antisymm hab hba]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsAntisymm \u03b1 r\ninst\u271d : IsAntisymm \u03b2 s\nb\u2081\u271d b\u2082\u271d : \u03b2\nhab : s b\u2081\u271d b\u2082\u271d\nhba : s b\u2082\u271d b\u2081\u271d\n\u22a2 inr b\u2081\u271d = inr b\u2082\u271d\n[PROOFSTEP]\nrw [antisymm hab hba]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (\u03b1 \u2295 \u03b2) := instLESum\nsrc\u271d : LT (\u03b1 \u2295 \u03b2) := instLTSum\na b : \u03b1 \u2295 \u03b2\n\u22a2 a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n[PROOFSTEP]\nrefine' \u27e8fun hab => \u27e8hab.mono (fun _ _ => le_of_lt) fun _ _ => le_of_lt, _\u27e9, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (\u03b1 \u2295 \u03b2) := instLESum\nsrc\u271d : LT (\u03b1 \u2295 \u03b2) := instLTSum\na b : \u03b1 \u2295 \u03b2\nhab : a < b\n\u22a2 \u00acb \u2264 a\n[PROOFSTEP]\nrintro (\u27e8hba\u27e9 | \u27e8hba\u27e9)\n[GOAL]\ncase refine'_1.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (\u03b1 \u2295 \u03b2) := instLESum\nsrc\u271d : LT (\u03b1 \u2295 \u03b2) := instLTSum\na\u271d c\u271d : \u03b1\nhba : a\u271d \u2264 c\u271d\nhab : inl c\u271d < inl a\u271d\n\u22a2 False\n[PROOFSTEP]\nexact hba.not_lt (inl_lt_inl_iff.1 hab)\n[GOAL]\ncase refine'_1.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (\u03b1 \u2295 \u03b2) := instLESum\nsrc\u271d : LT (\u03b1 \u2295 \u03b2) := instLTSum\nb\u271d d\u271d : \u03b2\nhba : b\u271d \u2264 d\u271d\nhab : inr d\u271d < inr b\u271d\n\u22a2 False\n[PROOFSTEP]\nexact hba.not_lt (inr_lt_inr_iff.1 hab)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (\u03b1 \u2295 \u03b2) := instLESum\nsrc\u271d : LT (\u03b1 \u2295 \u03b2) := instLTSum\na b : \u03b1 \u2295 \u03b2\n\u22a2 a \u2264 b \u2227 \u00acb \u2264 a \u2192 a < b\n[PROOFSTEP]\nrintro \u27e8\u27e8hab\u27e9 | \u27e8hab\u27e9, hba\u27e9\n[GOAL]\ncase refine'_2.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (\u03b1 \u2295 \u03b2) := instLESum\nsrc\u271d : LT (\u03b1 \u2295 \u03b2) := instLTSum\na\u271d c\u271d : \u03b1\nhab : a\u271d \u2264 c\u271d\nhba : \u00acinl c\u271d \u2264 inl a\u271d\n\u22a2 inl a\u271d < inl c\u271d\n[PROOFSTEP]\nexact LiftRel.inl (hab.lt_of_not_le fun h => hba <| LiftRel.inl h)\n[GOAL]\ncase refine'_2.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (\u03b1 \u2295 \u03b2) := instLESum\nsrc\u271d : LT (\u03b1 \u2295 \u03b2) := instLTSum\nb\u271d d\u271d : \u03b2\nhab : b\u271d \u2264 d\u271d\nhba : \u00acinr d\u271d \u2264 inr b\u271d\n\u22a2 inr b\u271d < inr d\u271d\n[PROOFSTEP]\nexact LiftRel.inr (hab.lt_of_not_le fun h => hba <| LiftRel.inr h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : NoMinOrder (\u03b1 \u2295 \u03b2)\na : \u03b1\n\u22a2 \u2203 b, b < a\n[PROOFSTEP]\nobtain \u27e8b | b, h\u27e9 := exists_lt (inl a : Sum \u03b1 \u03b2)\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : NoMinOrder (\u03b1 \u2295 \u03b2)\na b : \u03b1\nh : inl b < inl a\n\u22a2 \u2203 b, b < a\n[PROOFSTEP]\nexact \u27e8b, inl_lt_inl_iff.1 h\u27e9\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : NoMinOrder (\u03b1 \u2295 \u03b2)\na : \u03b1\nb : \u03b2\nh : inr b < inl a\n\u22a2 \u2203 b, b < a\n[PROOFSTEP]\nexact (not_inr_lt_inl h).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : NoMinOrder (\u03b1 \u2295 \u03b2)\na : \u03b2\n\u22a2 \u2203 b, b < a\n[PROOFSTEP]\nobtain \u27e8b | b, h\u27e9 := exists_lt (inr a : Sum \u03b1 \u03b2)\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : NoMinOrder (\u03b1 \u2295 \u03b2)\na : \u03b2\nb : \u03b1\nh : inl b < inr a\n\u22a2 \u2203 b, b < a\n[PROOFSTEP]\nexact (not_inl_lt_inr h).elim\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : NoMinOrder (\u03b1 \u2295 \u03b2)\na b : \u03b2\nh : inr b < inr a\n\u22a2 \u2203 b, b < a\n[PROOFSTEP]\nexact \u27e8b, inr_lt_inr_iff.1 h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : NoMaxOrder (\u03b1 \u2295 \u03b2)\na : \u03b1\n\u22a2 \u2203 b, a < b\n[PROOFSTEP]\nobtain \u27e8b | b, h\u27e9 := exists_gt (inl a : Sum \u03b1 \u03b2)\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : NoMaxOrder (\u03b1 \u2295 \u03b2)\na b : \u03b1\nh : inl a < inl b\n\u22a2 \u2203 b, a < b\n[PROOFSTEP]\nexact \u27e8b, inl_lt_inl_iff.1 h\u27e9\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : NoMaxOrder (\u03b1 \u2295 \u03b2)\na : \u03b1\nb : \u03b2\nh : inl a < inr b\n\u22a2 \u2203 b, a < b\n[PROOFSTEP]\nexact (not_inl_lt_inr h).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : NoMaxOrder (\u03b1 \u2295 \u03b2)\na : \u03b2\n\u22a2 \u2203 b, a < b\n[PROOFSTEP]\nobtain \u27e8b | b, h\u27e9 := exists_gt (inr a : Sum \u03b1 \u03b2)\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : NoMaxOrder (\u03b1 \u2295 \u03b2)\na : \u03b2\nb : \u03b1\nh : inr a < inl b\n\u22a2 \u2203 b, a < b\n[PROOFSTEP]\nexact (not_inr_lt_inl h).elim\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : NoMaxOrder (\u03b1 \u2295 \u03b2)\na b : \u03b2\nh : inr a < inr b\n\u22a2 \u2203 b, a < b\n[PROOFSTEP]\nexact \u27e8b, inr_lt_inr_iff.1 h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b1\nh : a < b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b\n[PROOFSTEP]\nobtain \u27e8c | c, ha, hb\u27e9 := @exists_between (Sum \u03b1 \u03b2) _ _ _ _ (inl_lt_inl_iff.2 h)\n[GOAL]\ncase intro.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b1\nh : a < b\nc : \u03b1\nha : inl a < inl c\nhb : inl c < inl b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b\n[PROOFSTEP]\nexact \u27e8c, inl_lt_inl_iff.1 ha, inl_lt_inl_iff.1 hb\u27e9\n[GOAL]\ncase intro.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b1\nh : a < b\nc : \u03b2\nha : inl a < inr c\nhb : inr c < inl b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b\n[PROOFSTEP]\nexact (not_inl_lt_inr ha).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b2\nh : a < b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b\n[PROOFSTEP]\nobtain \u27e8c | c, ha, hb\u27e9 := @exists_between (Sum \u03b1 \u03b2) _ _ _ _ (inr_lt_inr_iff.2 h)\n[GOAL]\ncase intro.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b2\nh : a < b\nc : \u03b1\nha : inr a < inl c\nhb : inl c < inr b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b\n[PROOFSTEP]\nexact (not_inl_lt_inr hb).elim\n[GOAL]\ncase intro.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b2\nh : a < b\nc : \u03b2\nha : inr a < inr c\nhb : inr c < inr b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b\n[PROOFSTEP]\nexact \u27e8c, inr_lt_inr_iff.1 ha, inr_lt_inr_iff.1 hb\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LE\nsrc\u271d : LT (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LT\na b : _root_.Lex (\u03b1 \u2295 \u03b2)\n\u22a2 a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n[PROOFSTEP]\nrefine' \u27e8fun hab => \u27e8hab.mono (fun _ _ => le_of_lt) fun _ _ => le_of_lt, _\u27e9, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LE\nsrc\u271d : LT (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LT\na b : _root_.Lex (\u03b1 \u2295 \u03b2)\nhab : a < b\n\u22a2 \u00acb \u2264 a\n[PROOFSTEP]\nrintro (\u27e8hba\u27e9 | \u27e8hba\u27e9 | \u27e8b, a\u27e9)\n[GOAL]\ncase refine'_1.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LE\nsrc\u271d : LT (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LT\na\u2081\u271d a\u2082\u271d : \u03b1\nhba : a\u2081\u271d \u2264 a\u2082\u271d\nhab : inl a\u2082\u271d < inl a\u2081\u271d\n\u22a2 False\n[PROOFSTEP]\nexact hba.not_lt (inl_lt_inl_iff.1 hab)\n[GOAL]\ncase refine'_1.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LE\nsrc\u271d : LT (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LT\nb\u2081\u271d b\u2082\u271d : \u03b2\nhba : b\u2081\u271d \u2264 b\u2082\u271d\nhab : inr b\u2082\u271d < inr b\u2081\u271d\n\u22a2 False\n[PROOFSTEP]\nexact hba.not_lt (inr_lt_inr_iff.1 hab)\n[GOAL]\ncase refine'_1.sep\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LE\nsrc\u271d : LT (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LT\nb : \u03b1\na : \u03b2\nhab : inr a < inl b\n\u22a2 False\n[PROOFSTEP]\nexact not_inr_lt_inl hab\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LE\nsrc\u271d : LT (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LT\na b : _root_.Lex (\u03b1 \u2295 \u03b2)\n\u22a2 a \u2264 b \u2227 \u00acb \u2264 a \u2192 a < b\n[PROOFSTEP]\nrintro \u27e8\u27e8hab\u27e9 | \u27e8hab\u27e9 | \u27e8a, b\u27e9, hba\u27e9\n[GOAL]\ncase refine'_2.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LE\nsrc\u271d : LT (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LT\na\u2081\u271d a\u2082\u271d : \u03b1\nhab : a\u2081\u271d \u2264 a\u2082\u271d\nhba : \u00acinl a\u2082\u271d \u2264 inl a\u2081\u271d\n\u22a2 inl a\u2081\u271d < inl a\u2082\u271d\n[PROOFSTEP]\nexact Lex.inl (hab.lt_of_not_le fun h => hba <| Lex.inl h)\n[GOAL]\ncase refine'_2.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LE\nsrc\u271d : LT (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LT\nb\u2081\u271d b\u2082\u271d : \u03b2\nhab : b\u2081\u271d \u2264 b\u2082\u271d\nhba : \u00acinr b\u2082\u271d \u2264 inr b\u2081\u271d\n\u22a2 inr b\u2081\u271d < inr b\u2082\u271d\n[PROOFSTEP]\nexact Lex.inr (hab.lt_of_not_le fun h => hba <| Lex.inr h)\n[GOAL]\ncase refine'_2.intro.sep\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LE\nsrc\u271d : LT (_root_.Lex (\u03b1 \u2295 \u03b2)) := Lex.LT\na : \u03b1\nb : \u03b2\nhba : \u00acinr b \u2264 inl a\n\u22a2 inl a < inr b\n[PROOFSTEP]\nexact Lex.sep _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : LE \u03b2\n\u22a2 \u2200 (a : _root_.Lex (\u03b1 \u2295 \u03b2)), \u22a5 \u2264 a\n[PROOFSTEP]\nrintro (a | b)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : LE \u03b2\na : \u03b1\n\u22a2 \u22a5 \u2264 inl a\n[PROOFSTEP]\nexact Lex.inl bot_le\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : LE \u03b2\nb : \u03b2\n\u22a2 \u22a5 \u2264 inr b\n[PROOFSTEP]\nexact Lex.sep _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : OrderTop \u03b2\n\u22a2 \u2200 (a : _root_.Lex (\u03b1 \u2295 \u03b2)), a \u2264 \u22a4\n[PROOFSTEP]\nrintro (a | b)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : OrderTop \u03b2\na : \u03b1\n\u22a2 inl a \u2264 \u22a4\n[PROOFSTEP]\nexact Lex.sep _ _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : OrderTop \u03b2\nb : \u03b2\n\u22a2 inr b \u2264 \u22a4\n[PROOFSTEP]\nexact Lex.inr le_top\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : LE \u03b1\u271d\ninst\u271d\u2074 : LE \u03b2\u271d\ninst\u271d\u00b3 : LE \u03b3\u271d\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : LE \u03b3\nsrc\u271d : (\u03b1 \u2295 \u03b2) \u2295 \u03b3 \u2243 \u03b1 \u2295 \u03b2 \u2295 \u03b3 := Equiv.sumAssoc \u03b1 \u03b2 \u03b3\na b : (\u03b1 \u2295 \u03b2) \u2295 \u03b3\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n a \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n b \u2194\n a \u2264 b\n[PROOFSTEP]\nrcases a with ((_ | _) | _)\n[GOAL]\ncase inl.inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : LE \u03b1\u271d\ninst\u271d\u2074 : LE \u03b2\u271d\ninst\u271d\u00b3 : LE \u03b3\u271d\na : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : LE \u03b3\nsrc\u271d : (\u03b1 \u2295 \u03b2) \u2295 \u03b3 \u2243 \u03b1 \u2295 \u03b2 \u2295 \u03b3 := Equiv.sumAssoc \u03b1 \u03b2 \u03b3\nb : (\u03b1 \u2295 \u03b2) \u2295 \u03b3\nval\u271d : \u03b1\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl (inl val\u271d)) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n b \u2194\n inl (inl val\u271d) \u2264 b\n[PROOFSTEP]\nrcases b with ((_ | _) | _)\n[GOAL]\ncase inl.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : LE \u03b1\u271d\ninst\u271d\u2074 : LE \u03b2\u271d\ninst\u271d\u00b3 : LE \u03b3\u271d\na : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : LE \u03b3\nsrc\u271d : (\u03b1 \u2295 \u03b2) \u2295 \u03b3 \u2243 \u03b1 \u2295 \u03b2 \u2295 \u03b3 := Equiv.sumAssoc \u03b1 \u03b2 \u03b3\nb : (\u03b1 \u2295 \u03b2) \u2295 \u03b3\nval\u271d : \u03b2\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl (inr val\u271d)) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n b \u2194\n inl (inr val\u271d) \u2264 b\n[PROOFSTEP]\nrcases b with ((_ | _) | _)\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : LE \u03b1\u271d\ninst\u271d\u2074 : LE \u03b2\u271d\ninst\u271d\u00b3 : LE \u03b3\u271d\na : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : LE \u03b3\nsrc\u271d : (\u03b1 \u2295 \u03b2) \u2295 \u03b3 \u2243 \u03b1 \u2295 \u03b2 \u2295 \u03b3 := Equiv.sumAssoc \u03b1 \u03b2 \u03b3\nb : (\u03b1 \u2295 \u03b2) \u2295 \u03b3\nval\u271d : \u03b3\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr val\u271d) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n b \u2194\n inr val\u271d \u2264 b\n[PROOFSTEP]\nrcases b with ((_ | _) | _)\n[GOAL]\ncase inl.inl.inl.inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : LE \u03b1\u271d\ninst\u271d\u2074 : LE \u03b2\u271d\ninst\u271d\u00b3 : LE \u03b3\u271d\na : \u03b1\u271d\nb : \u03b2\u271d\nc : \u03b3\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : LE \u03b3\nsrc\u271d : (\u03b1 \u2295 \u03b2) \u2295 \u03b3 \u2243 \u03b1 \u2295 \u03b2 \u2295 \u03b3 := Equiv.sumAssoc \u03b1 \u03b2 \u03b3\nval\u271d\u00b9 val\u271d : \u03b1\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl (inl val\u271d\u00b9)) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl (inl val\u271d)) \u2194\n inl (inl val\u271d\u00b9) \u2264 inl (inl val\u271d)\n[PROOFSTEP]\nsimp [Equiv.sumAssoc]\n[GOAL]\ncase inl.inl.inl.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : LE \u03b1\u271d\ninst\u271d\u2074 : LE \u03b2\u271d\ninst\u271d\u00b3 : LE \u03b3\u271d\na : \u03b1\u271d\nb : \u03b2\u271d\nc : \u03b3\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : LE \u03b3\nsrc\u271d : (\u03b1 \u2295 \u03b2) \u2295 \u03b3 \u2243 \u03b1 \u2295 \u03b2 \u2295 \u03b3 := Equiv.sumAssoc \u03b1 \u03b2 \u03b3\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl (inl val\u271d\u00b9)) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl (inr val\u271d)) \u2194\n inl (inl val\u271d\u00b9) \u2264 inl (inr val\u271d)\n[PROOFSTEP]\nsimp [Equiv.sumAssoc]\n[GOAL]\ncase inl.inl.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : LE \u03b1\u271d\ninst\u271d\u2074 : LE \u03b2\u271d\ninst\u271d\u00b3 : LE \u03b3\u271d\na : \u03b1\u271d\nb : \u03b2\u271d\nc : \u03b3\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : LE \u03b3\nsrc\u271d : (\u03b1 \u2295 \u03b2) \u2295 \u03b3 \u2243 \u03b1 \u2295 \u03b2 \u2295 \u03b3 := Equiv.sumAssoc \u03b1 \u03b2 \u03b3\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b3\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl (inl val\u271d\u00b9)) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr val\u271d) \u2194\n inl (inl val\u271d\u00b9) \u2264 inr val\u271d\n[PROOFSTEP]\nsimp [Equiv.sumAssoc]\n[GOAL]\ncase inl.inr.inl.inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : LE \u03b1\u271d\ninst\u271d\u2074 : LE \u03b2\u271d\ninst\u271d\u00b3 : LE \u03b3\u271d\na : \u03b1\u271d\nb : \u03b2\u271d\nc : \u03b3\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : LE \u03b3\nsrc\u271d : (\u03b1 \u2295 \u03b2) \u2295 \u03b3 \u2243 \u03b1 \u2295 \u03b2 \u2295 \u03b3 := Equiv.sumAssoc \u03b1 \u03b2 \u03b3\nval\u271d\u00b9 : \u03b2\nval\u271d : \u03b1\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl (inr val\u271d\u00b9)) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl (inl val\u271d)) \u2194\n inl (inr val\u271d\u00b9) \u2264 inl (inl val\u271d)\n[PROOFSTEP]\nsimp [Equiv.sumAssoc]\n[GOAL]\ncase inl.inr.inl.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : LE \u03b1\u271d\ninst\u271d\u2074 : LE \u03b2\u271d\ninst\u271d\u00b3 : LE \u03b3\u271d\na : \u03b1\u271d\nb : \u03b2\u271d\nc : \u03b3\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : LE \u03b3\nsrc\u271d : (\u03b1 \u2295 \u03b2) \u2295 \u03b3 \u2243 \u03b1 \u2295 \u03b2 \u2295 \u03b3 := Equiv.sumAssoc \u03b1 \u03b2 \u03b3\nval\u271d\u00b9 val\u271d : \u03b2\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl (inr val\u271d\u00b9)) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl (inr val\u271d)) \u2194\n inl (inr val\u271d\u00b9) \u2264 inl (inr val\u271d)\n[PROOFSTEP]\nsimp [Equiv.sumAssoc]\n[GOAL]\ncase inl.inr.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : LE \u03b1\u271d\ninst\u271d\u2074 : LE \u03b2\u271d\ninst\u271d\u00b3 : LE \u03b3\u271d\na : \u03b1\u271d\nb : \u03b2\u271d\nc : \u03b3\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : LE \u03b3\nsrc\u271d : (\u03b1 \u2295 \u03b2) \u2295 \u03b3 \u2243 \u03b1 \u2295 \u03b2 \u2295 \u03b3 := Equiv.sumAssoc \u03b1 \u03b2 \u03b3\nval\u271d\u00b9 : \u03b2\nval\u271d : \u03b3\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl (inr val\u271d\u00b9)) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr val\u271d) \u2194\n inl (inr val\u271d\u00b9) \u2264 inr val\u271d\n[PROOFSTEP]\nsimp [Equiv.sumAssoc]\n[GOAL]\ncase inr.inl.inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : LE \u03b1\u271d\ninst\u271d\u2074 : LE \u03b2\u271d\ninst\u271d\u00b3 : LE \u03b3\u271d\na : \u03b1\u271d\nb : \u03b2\u271d\nc : \u03b3\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : LE \u03b3\nsrc\u271d : (\u03b1 \u2295 \u03b2) \u2295 \u03b3 \u2243 \u03b1 \u2295 \u03b2 \u2295 \u03b3 := Equiv.sumAssoc \u03b1 \u03b2 \u03b3\nval\u271d\u00b9 : \u03b3\nval\u271d : \u03b1\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr val\u271d\u00b9) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl (inl val\u271d)) \u2194\n inr val\u271d\u00b9 \u2264 inl (inl val\u271d)\n[PROOFSTEP]\nsimp [Equiv.sumAssoc]\n[GOAL]\ncase inr.inl.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : LE \u03b1\u271d\ninst\u271d\u2074 : LE \u03b2\u271d\ninst\u271d\u00b3 : LE \u03b3\u271d\na : \u03b1\u271d\nb : \u03b2\u271d\nc : \u03b3\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : LE \u03b3\nsrc\u271d : (\u03b1 \u2295 \u03b2) \u2295 \u03b3 \u2243 \u03b1 \u2295 \u03b2 \u2295 \u03b3 := Equiv.sumAssoc \u03b1 \u03b2 \u03b3\nval\u271d\u00b9 : \u03b3\nval\u271d : \u03b2\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr val\u271d\u00b9) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl (inr val\u271d)) \u2194\n inr val\u271d\u00b9 \u2264 inl (inr val\u271d)\n[PROOFSTEP]\nsimp [Equiv.sumAssoc]\n[GOAL]\ncase inr.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : LE \u03b1\u271d\ninst\u271d\u2074 : LE \u03b2\u271d\ninst\u271d\u00b3 : LE \u03b3\u271d\na : \u03b1\u271d\nb : \u03b2\u271d\nc : \u03b3\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : LE \u03b2\ninst\u271d : LE \u03b3\nsrc\u271d : (\u03b1 \u2295 \u03b2) \u2295 \u03b3 \u2243 \u03b1 \u2295 \u03b2 \u2295 \u03b3 := Equiv.sumAssoc \u03b1 \u03b2 \u03b3\nval\u271d\u00b9 val\u271d : \u03b3\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr val\u271d\u00b9) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr val\u271d) \u2194\n inr val\u271d\u00b9 \u2264 inr val\u271d\n[PROOFSTEP]\nsimp [Equiv.sumAssoc]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na : \u03b1\u271d\nb : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : (\u03b1 \u2295 \u03b2)\u1d52\u1d48 \u2243 (\u03b1 \u2295 \u03b2)\u1d52\u1d48 := Equiv.refl (\u03b1 \u2295 \u03b2)\u1d52\u1d48\n\u22a2 \u2200 {a b : (\u03b1 \u2295 \u03b2)\u1d52\u1d48},\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n a \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n b \u2194\n a \u2264 b\n[PROOFSTEP]\nrintro (a | a) (b | b)\n[GOAL]\ncase inl.inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : (\u03b1 \u2295 \u03b2)\u1d52\u1d48 \u2243 (\u03b1 \u2295 \u03b2)\u1d52\u1d48 := Equiv.refl (\u03b1 \u2295 \u03b2)\u1d52\u1d48\na b : \u03b1\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl a) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl b) \u2194\n inl a \u2264 inl b\n[PROOFSTEP]\nchange inl (toDual a) \u2264 inl (toDual b) \u2194 toDual (inl a) \u2264 toDual (inl b)\n[GOAL]\ncase inl.inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : (\u03b1 \u2295 \u03b2)\u1d52\u1d48 \u2243 (\u03b1 \u2295 \u03b2)\u1d52\u1d48 := Equiv.refl (\u03b1 \u2295 \u03b2)\u1d52\u1d48\na b : \u03b1\n\u22a2 inl (\u2191toDual a) \u2264 inl (\u2191toDual b) \u2194 \u2191toDual (inl a) \u2264 \u2191toDual (inl b)\n[PROOFSTEP]\nsimp [toDual_le_toDual, inl_le_inl_iff]\n[GOAL]\ncase inl.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : (\u03b1 \u2295 \u03b2)\u1d52\u1d48 \u2243 (\u03b1 \u2295 \u03b2)\u1d52\u1d48 := Equiv.refl (\u03b1 \u2295 \u03b2)\u1d52\u1d48\na : \u03b1\nb : \u03b2\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl a) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr b) \u2194\n inl a \u2264 inr b\n[PROOFSTEP]\nexact iff_of_false (@not_inl_le_inr (OrderDual \u03b2) (OrderDual \u03b1) _ _ _ _) not_inr_le_inl\n[GOAL]\ncase inr.inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : (\u03b1 \u2295 \u03b2)\u1d52\u1d48 \u2243 (\u03b1 \u2295 \u03b2)\u1d52\u1d48 := Equiv.refl (\u03b1 \u2295 \u03b2)\u1d52\u1d48\na : \u03b2\nb : \u03b1\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr a) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl b) \u2194\n inr a \u2264 inl b\n[PROOFSTEP]\nexact iff_of_false (@not_inr_le_inl (OrderDual \u03b1) (OrderDual \u03b2) _ _ _ _) not_inl_le_inr\n[GOAL]\ncase inr.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : (\u03b1 \u2295 \u03b2)\u1d52\u1d48 \u2243 (\u03b1 \u2295 \u03b2)\u1d52\u1d48 := Equiv.refl (\u03b1 \u2295 \u03b2)\u1d52\u1d48\na b : \u03b2\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr a) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr b) \u2194\n inr a \u2264 inr b\n[PROOFSTEP]\nchange inr (toDual a) \u2264 inr (toDual b) \u2194 toDual (inr a) \u2264 toDual (inr b)\n[GOAL]\ncase inr.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : (\u03b1 \u2295 \u03b2)\u1d52\u1d48 \u2243 (\u03b1 \u2295 \u03b2)\u1d52\u1d48 := Equiv.refl (\u03b1 \u2295 \u03b2)\u1d52\u1d48\na b : \u03b2\n\u22a2 inr (\u2191toDual a) \u2264 inr (\u2191toDual b) \u2194 \u2191toDual (inr a) \u2264 \u2191toDual (inr b)\n[PROOFSTEP]\nsimp [toDual_le_toDual, inr_le_inr_iff]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : \u03b1 \u2295 \u03b2 \u2243 \u03b2 \u2295 \u03b1 := Equiv.sumComm \u03b1 \u03b2\na b : (_root_.Lex (\u03b1 \u2295 \u03b2))\u1d52\u1d48\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n a \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n b \u2194\n a \u2264 b\n[PROOFSTEP]\nrcases a with (a | a)\n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : \u03b1 \u2295 \u03b2 \u2243 \u03b2 \u2295 \u03b1 := Equiv.sumComm \u03b1 \u03b2\nb : (_root_.Lex (\u03b1 \u2295 \u03b2))\u1d52\u1d48\na : \u03b1\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl a) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n b \u2194\n inl a \u2264 b\n[PROOFSTEP]\nrcases b with (b | b)\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : \u03b1 \u2295 \u03b2 \u2243 \u03b2 \u2295 \u03b1 := Equiv.sumComm \u03b1 \u03b2\nb : (_root_.Lex (\u03b1 \u2295 \u03b2))\u1d52\u1d48\na : \u03b2\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr a) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n b \u2194\n inr a \u2264 b\n[PROOFSTEP]\nrcases b with (b | b)\n[GOAL]\ncase inl.inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : \u03b1 \u2295 \u03b2 \u2243 \u03b2 \u2295 \u03b1 := Equiv.sumComm \u03b1 \u03b2\na b : \u03b1\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl a) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl b) \u2194\n inl a \u2264 inl b\ncase inl.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : \u03b1 \u2295 \u03b2 \u2243 \u03b2 \u2295 \u03b1 := Equiv.sumComm \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl a) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr b) \u2194\n inl a \u2264 inr b\ncase inr.inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : \u03b1 \u2295 \u03b2 \u2243 \u03b2 \u2295 \u03b1 := Equiv.sumComm \u03b1 \u03b2\na : \u03b2\nb : \u03b1\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr a) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl b) \u2194\n inr a \u2264 inl b\ncase inr.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : \u03b1 \u2295 \u03b2 \u2243 \u03b2 \u2295 \u03b1 := Equiv.sumComm \u03b1 \u03b2\na b : \u03b2\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr a) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr b) \u2194\n inr a \u2264 inr b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : \u03b1 \u2295 \u03b2 \u2243 \u03b2 \u2295 \u03b1 := Equiv.sumComm \u03b1 \u03b2\na b : \u03b1\n\u22a2 inr a \u2264 inr b \u2194 inl a \u2264 inl b\n[PROOFSTEP]\nchange toLex (inr <| toDual a) \u2264 toLex (inr <| toDual b) \u2194 toDual (toLex <| inl a) \u2264 toDual (toLex <| inl b)\n[GOAL]\ncase inl.inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : \u03b1 \u2295 \u03b2 \u2243 \u03b2 \u2295 \u03b1 := Equiv.sumComm \u03b1 \u03b2\na b : \u03b1\n\u22a2 \u2191toLex (inr (\u2191toDual a)) \u2264 \u2191toLex (inr (\u2191toDual b)) \u2194 \u2191toDual (\u2191toLex (inl a)) \u2264 \u2191toDual (\u2191toLex (inl b))\n[PROOFSTEP]\nsimp [toDual_le_toDual, Lex.inl_le_inl_iff, Lex.inr_le_inr_iff]\n[GOAL]\ncase inl.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : \u03b1 \u2295 \u03b2 \u2243 \u03b2 \u2295 \u03b1 := Equiv.sumComm \u03b1 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl a) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr b) \u2194\n inl a \u2264 inr b\n[PROOFSTEP]\nexact iff_of_false (@Lex.not_inr_le_inl (OrderDual \u03b2) (OrderDual \u03b1) _ _ _ _) Lex.not_inr_le_inl\n[GOAL]\ncase inr.inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : \u03b1 \u2295 \u03b2 \u2243 \u03b2 \u2295 \u03b1 := Equiv.sumComm \u03b1 \u03b2\na : \u03b2\nb : \u03b1\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr a) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inl b) \u2194\n inr a \u2264 inl b\n[PROOFSTEP]\nexact iff_of_true (@Lex.inl_le_inr (OrderDual \u03b2) (OrderDual \u03b1) _ _ _ _) (Lex.inl_le_inr _ _)\n[GOAL]\ncase inr.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : \u03b1 \u2295 \u03b2 \u2243 \u03b2 \u2295 \u03b1 := Equiv.sumComm \u03b1 \u03b2\na b : \u03b2\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr a) \u2264\n \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (inr b) \u2194\n inr a \u2264 inr b\n[PROOFSTEP]\nchange toLex (inl <| toDual a) \u2264 toLex (inl <| toDual b) \u2194 toDual (toLex <| inr a) \u2264 toDual (toLex <| inr b)\n[GOAL]\ncase inr.inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : LE \u03b1\u271d\ninst\u271d\u00b3 : LE \u03b2\u271d\ninst\u271d\u00b2 : LE \u03b3\na\u271d : \u03b1\u271d\nb\u271d : \u03b2\u271d\nc : \u03b3\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\nsrc\u271d : \u03b1 \u2295 \u03b2 \u2243 \u03b2 \u2295 \u03b1 := Equiv.sumComm \u03b1 \u03b2\na b : \u03b2\n\u22a2 \u2191toLex (inl (\u2191toDual a)) \u2264 \u2191toLex (inl (\u2191toDual b)) \u2194 \u2191toDual (\u2191toLex (inr a)) \u2264 \u2191toDual (\u2191toLex (inr b))\n[PROOFSTEP]\nsimp [toDual_le_toDual, Lex.inl_le_inl_iff, Lex.inr_le_inr_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\na b : WithBot \u03b1\n\u22a2 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans ((Equiv.sumComm \u03b1 PUnit).trans toLex)) a \u2264\n \u2191((Equiv.optionEquivSumPUnit \u03b1).trans ((Equiv.sumComm \u03b1 PUnit).trans toLex)) b \u2194\n a \u2264 b\n[PROOFSTEP]\nrcases a with (a | _)\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\nb : WithBot \u03b1\n\u22a2 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans ((Equiv.sumComm \u03b1 PUnit).trans toLex)) none \u2264\n \u2191((Equiv.optionEquivSumPUnit \u03b1).trans ((Equiv.sumComm \u03b1 PUnit).trans toLex)) b \u2194\n none \u2264 b\n[PROOFSTEP]\nrcases b with (b | _)\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\nb : WithBot \u03b1\nval\u271d : \u03b1\n\u22a2 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans ((Equiv.sumComm \u03b1 PUnit).trans toLex)) (Option.some val\u271d) \u2264\n \u2191((Equiv.optionEquivSumPUnit \u03b1).trans ((Equiv.sumComm \u03b1 PUnit).trans toLex)) b \u2194\n Option.some val\u271d \u2264 b\n[PROOFSTEP]\nrcases b with (b | _)\n[GOAL]\ncase none.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\n\u22a2 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans ((Equiv.sumComm \u03b1 PUnit).trans toLex)) none \u2264\n \u2191((Equiv.optionEquivSumPUnit \u03b1).trans ((Equiv.sumComm \u03b1 PUnit).trans toLex)) none \u2194\n none \u2264 none\n[PROOFSTEP]\nsimp [swap, Equiv.optionEquivSumPUnit]\n[GOAL]\ncase none.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\nval\u271d : \u03b1\n\u22a2 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans ((Equiv.sumComm \u03b1 PUnit).trans toLex)) none \u2264\n \u2191((Equiv.optionEquivSumPUnit \u03b1).trans ((Equiv.sumComm \u03b1 PUnit).trans toLex)) (Option.some val\u271d) \u2194\n none \u2264 Option.some val\u271d\n[PROOFSTEP]\nsimp [swap, Equiv.optionEquivSumPUnit]\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\nval\u271d : \u03b1\n\u22a2 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans ((Equiv.sumComm \u03b1 PUnit).trans toLex)) (Option.some val\u271d) \u2264\n \u2191((Equiv.optionEquivSumPUnit \u03b1).trans ((Equiv.sumComm \u03b1 PUnit).trans toLex)) none \u2194\n Option.some val\u271d \u2264 none\n[PROOFSTEP]\nsimp [swap, Equiv.optionEquivSumPUnit]\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\n\u22a2 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans ((Equiv.sumComm \u03b1 PUnit).trans toLex)) (Option.some val\u271d\u00b9) \u2264\n \u2191((Equiv.optionEquivSumPUnit \u03b1).trans ((Equiv.sumComm \u03b1 PUnit).trans toLex)) (Option.some val\u271d) \u2194\n Option.some val\u271d\u00b9 \u2264 Option.some val\u271d\n[PROOFSTEP]\nsimp [swap, Equiv.optionEquivSumPUnit]\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\nval\u271d : \u03b1\n\u22a2 \u00acOption.some val\u271d \u2264 none\n[PROOFSTEP]\nexact not_coe_le_bot _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\na b : WithTop \u03b1\n\u22a2 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans toLex) a \u2264 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans toLex) b \u2194 a \u2264 b\n[PROOFSTEP]\nrcases a with (a | _)\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\nb : WithTop \u03b1\n\u22a2 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans toLex) none \u2264 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans toLex) b \u2194 none \u2264 b\n[PROOFSTEP]\nrcases b with (b | _)\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\nb : WithTop \u03b1\nval\u271d : \u03b1\n\u22a2 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans toLex) (Option.some val\u271d) \u2264 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans toLex) b \u2194\n Option.some val\u271d \u2264 b\n[PROOFSTEP]\nrcases b with (b | _)\n[GOAL]\ncase none.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\n\u22a2 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans toLex) none \u2264 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans toLex) none \u2194 none \u2264 none\n[PROOFSTEP]\nsimp [swap, Equiv.optionEquivSumPUnit]\n[GOAL]\ncase none.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\nval\u271d : \u03b1\n\u22a2 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans toLex) none \u2264 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans toLex) (Option.some val\u271d) \u2194\n none \u2264 Option.some val\u271d\n[PROOFSTEP]\nsimp [swap, Equiv.optionEquivSumPUnit]\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\nval\u271d : \u03b1\n\u22a2 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans toLex) (Option.some val\u271d) \u2264 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans toLex) none \u2194\n Option.some val\u271d \u2264 none\n[PROOFSTEP]\nsimp [swap, Equiv.optionEquivSumPUnit]\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\n\u22a2 \u2191((Equiv.optionEquivSumPUnit \u03b1).trans toLex) (Option.some val\u271d\u00b9) \u2264\n \u2191((Equiv.optionEquivSumPUnit \u03b1).trans toLex) (Option.some val\u271d) \u2194\n Option.some val\u271d\u00b9 \u2264 Option.some val\u271d\n[PROOFSTEP]\nsimp [swap, Equiv.optionEquivSumPUnit]\n[GOAL]\ncase none.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : LE \u03b1\nval\u271d : \u03b1\n\u22a2 \u00acnone \u2264 Option.some val\u271d\n[PROOFSTEP]\nexact not_top_le_coe _\n", "meta": {"mathlib_filename": "Mathlib.Data.Sum.Order", "llama_tokens": 24017, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.4223020047509993}} {"text": "[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : MulZeroClass M\u2080\na\u271d b\u271d a b : M\u2080\nh : a \u2260 0 \u2192 b = 0\nha : a = 0\n\u22a2 a * b = 0\n[PROOFSTEP]\nrw [ha, zero_mul]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : MulZeroClass M\u2080\na\u271d b\u271d a b : M\u2080\nh : a \u2260 0 \u2192 b = 0\nha : \u00aca = 0\n\u22a2 a * b = 0\n[PROOFSTEP]\nrw [h ha, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : MulZeroOneClass M\u2080\nh : 0 = 1\na : M\u2080\n\u22a2 a = 0\n[PROOFSTEP]\nrw [\u2190 mul_one a, \u2190 h, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CancelMonoidWithZero M\u2080\na b c : M\u2080\n\u22a2 a * c = b * c \u2194 a = b \u2228 c = 0\n[PROOFSTEP]\nby_cases hc : c = 0 <;> [simp [hc]; simp [mul_left_inj', hc]]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CancelMonoidWithZero M\u2080\na b c : M\u2080\n\u22a2 a * c = b * c \u2194 a = b \u2228 c = 0\n[PROOFSTEP]\nby_cases hc : c = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CancelMonoidWithZero M\u2080\na b c : M\u2080\nhc : c = 0\n\u22a2 a * c = b * c \u2194 a = b \u2228 c = 0\n[PROOFSTEP]\nsimp [hc]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CancelMonoidWithZero M\u2080\na b c : M\u2080\nhc : \u00acc = 0\n\u22a2 a * c = b * c \u2194 a = b \u2228 c = 0\n[PROOFSTEP]\nsimp [mul_left_inj', hc]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CancelMonoidWithZero M\u2080\na b c : M\u2080\n\u22a2 a * b = a * c \u2194 b = c \u2228 a = 0\n[PROOFSTEP]\nby_cases ha : a = 0 <;> [simp [ha]; simp [mul_right_inj', ha]]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CancelMonoidWithZero M\u2080\na b c : M\u2080\n\u22a2 a * b = a * c \u2194 b = c \u2228 a = 0\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CancelMonoidWithZero M\u2080\na b c : M\u2080\nha : a = 0\n\u22a2 a * b = a * c \u2194 b = c \u2228 a = 0\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CancelMonoidWithZero M\u2080\na b c : M\u2080\nha : \u00aca = 0\n\u22a2 a * b = a * c \u2194 b = c \u2228 a = 0\n[PROOFSTEP]\nsimp [mul_right_inj', ha]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CancelMonoidWithZero M\u2080\na b c : M\u2080\n\u22a2 a * b = a \u2194 a * b = a * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CancelMonoidWithZero M\u2080\na b c : M\u2080\n\u22a2 a * b = b \u2194 a * b = 1 * b\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CancelMonoidWithZero M\u2080\na b c : M\u2080\nha : a \u2260 0\n\u22a2 a * b = a \u2194 b = 1\n[PROOFSTEP]\nrw [Iff.comm, \u2190 mul_right_inj' ha, mul_one]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CancelMonoidWithZero M\u2080\na b c : M\u2080\nhb : b \u2260 0\n\u22a2 a * b = b \u2194 a = 1\n[PROOFSTEP]\nrw [Iff.comm, \u2190 mul_left_inj' hb, one_mul]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CancelMonoidWithZero M\u2080\na b c : M\u2080\nha : a \u2260 0\n\u22a2 a = a * b \u2194 b = 1\n[PROOFSTEP]\nrw [eq_comm, mul_eq_left\u2080 ha]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CancelMonoidWithZero M\u2080\na b c : M\u2080\nhb : b \u2260 0\n\u22a2 b = a * b \u2194 a = 1\n[PROOFSTEP]\nrw [eq_comm, mul_eq_right\u2080 hb]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c g h\u271d x : G\u2080\nh : b \u2260 0\na : G\u2080\n\u22a2 a * (b * b\u207b\u00b9) = a\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na b\u271d c g h\u271d x : G\u2080\nh : a \u2260 0\nb : G\u2080\n\u22a2 a * a\u207b\u00b9 * b = b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na b c g h\u271d x : G\u2080\nh : a \u2260 0\na_eq_0 : a\u207b\u00b9 = 0\n\u22a2 False\n[PROOFSTEP]\nhave := mul_inv_cancel h\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na b c g h\u271d x : G\u2080\nh : a \u2260 0\na_eq_0 : a\u207b\u00b9 = 0\nthis : a * a\u207b\u00b9 = 1\n\u22a2 False\n[PROOFSTEP]\nsimp [a_eq_0] at this \n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na b c g h\u271d x : G\u2080\nh : a \u2260 0\n\u22a2 a\u207b\u00b9 * a = a\u207b\u00b9 * a * a\u207b\u00b9 * a\u207b\u00b9\u207b\u00b9\n[PROOFSTEP]\nsimp [inv_ne_zero h]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na b c g h\u271d x : G\u2080\nh : a \u2260 0\n\u22a2 a\u207b\u00b9 * a * a\u207b\u00b9 * a\u207b\u00b9\u207b\u00b9 = a\u207b\u00b9 * a\u207b\u00b9\u207b\u00b9\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na b c g h\u271d x : G\u2080\nh : a \u2260 0\n\u22a2 a\u207b\u00b9 * a\u207b\u00b9\u207b\u00b9 = 1\n[PROOFSTEP]\nsimp [inv_ne_zero h]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na b c g h\u271d x : G\u2080\nh : x \u2260 0\ny y' : G\u2080\nw : (fun y => x * y) y = (fun y => x * y) y'\n\u22a2 y = y'\n[PROOFSTEP]\nsimpa only [\u2190 mul_assoc, inv_mul_cancel h, one_mul] using congr_arg (fun y => x\u207b\u00b9 * y) w\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na b c g h\u271d x : G\u2080\nh : x \u2260 0\ny y' : G\u2080\nw : (fun y => y * x) y = (fun y => y * x) y'\n\u22a2 y = y'\n[PROOFSTEP]\nsimpa only [mul_assoc, mul_inv_cancel _ h, mul_one] using congr_arg (fun y => y * x\u207b\u00b9) w\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c g h\u271d x : G\u2080\nh : b \u2260 0\na : G\u2080\n\u22a2 a * (b\u207b\u00b9 * b) = a\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na b\u271d c g h\u271d x : G\u2080\nh : a \u2260 0\nb : G\u2080\n\u22a2 a\u207b\u00b9 * a * b = b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na b c g h\u271d x : G\u2080\nh : a * b = 1\n\u22a2 a\u207b\u00b9 = b\n[PROOFSTEP]\nrw [\u2190 inv_mul_cancel_left\u2080 (left_ne_zero_of_mul_eq_one h) b, h, mul_one]\n -- See note [lower instance priority]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c g h x : G\u2080\nsrc\u271d : GroupWithZero G\u2080 := inst\u271d\na : G\u2080\n\u22a2 a\u207b\u00b9\u207b\u00b9 = a\n[PROOFSTEP]\nby_cases h : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c g h\u271d x : G\u2080\nsrc\u271d : GroupWithZero G\u2080 := inst\u271d\na : G\u2080\nh : a = 0\n\u22a2 a\u207b\u00b9\u207b\u00b9 = a\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c g h\u271d x : G\u2080\nsrc\u271d : GroupWithZero G\u2080 := inst\u271d\na : G\u2080\nh : \u00aca = 0\n\u22a2 a\u207b\u00b9\u207b\u00b9 = a\n[PROOFSTEP]\nexact left_inv_eq_right_inv (inv_mul_cancel <| inv_ne_zero h) (inv_mul_cancel h)\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b\u271d c g h x : G\u2080\nsrc\u271d : GroupWithZero G\u2080 := inst\u271d\na b : G\u2080\n\u22a2 (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b\u271d c g h x : G\u2080\nsrc\u271d : GroupWithZero G\u2080 := inst\u271d\na b : G\u2080\nha : a = 0\n\u22a2 (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b\u271d c g h x : G\u2080\nsrc\u271d : GroupWithZero G\u2080 := inst\u271d\na b : G\u2080\nha : \u00aca = 0\n\u22a2 (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9\n[PROOFSTEP]\nby_cases hb : b = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b\u271d c g h x : G\u2080\nsrc\u271d : GroupWithZero G\u2080 := inst\u271d\na b : G\u2080\nha : \u00aca = 0\nhb : b = 0\n\u22a2 (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9\n[PROOFSTEP]\nsimp [hb]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b\u271d c g h x : G\u2080\nsrc\u271d : GroupWithZero G\u2080 := inst\u271d\na b : G\u2080\nha : \u00aca = 0\nhb : \u00acb = 0\n\u22a2 (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9\n[PROOFSTEP]\nrefine' inv_eq_of_mul _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b\u271d c g h x : G\u2080\nsrc\u271d : GroupWithZero G\u2080 := inst\u271d\na b : G\u2080\nha : \u00aca = 0\nhb : \u00acb = 0\n\u22a2 a * b * (b\u207b\u00b9 * a\u207b\u00b9) = 1\n[PROOFSTEP]\nsimp [mul_assoc, ha, hb]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na b c g h\u271d x\u271d : G\u2080\nsrc\u271d : GroupWithZero G\u2080 := inst\u271d\nx y z : G\u2080\nhx : x \u2260 0\nh : x * y = x * z\n\u22a2 y = z\n[PROOFSTEP]\nrw [\u2190 inv_mul_cancel_left\u2080 hx y, h, inv_mul_cancel_left\u2080 hx z]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na b c g h\u271d x\u271d : G\u2080\nsrc\u271d : GroupWithZero G\u2080 := inst\u271d\nx y z : G\u2080\nhy : y \u2260 0\nh : x * y = z * y\n\u22a2 x = z\n[PROOFSTEP]\nrw [\u2190 mul_inv_cancel_right\u2080 hy x, h, mul_inv_cancel_right\u2080 hy z]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\n\u22a2 0 / a = 0\n[PROOFSTEP]\nrw [div_eq_mul_inv, zero_mul]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\n\u22a2 a / 0 = 0\n[PROOFSTEP]\nrw [div_eq_mul_inv, inv_zero, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\n\u22a2 a * a * a\u207b\u00b9 = a\n[PROOFSTEP]\nby_cases h : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\nh : a = 0\n\u22a2 a * a * a\u207b\u00b9 = a\n[PROOFSTEP]\nrw [h, inv_zero, mul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\nh : \u00aca = 0\n\u22a2 a * a * a\u207b\u00b9 = a\n[PROOFSTEP]\nrw [mul_assoc, mul_inv_cancel h, mul_one]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\n\u22a2 a * a\u207b\u00b9 * a = a\n[PROOFSTEP]\nby_cases h : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\nh : a = 0\n\u22a2 a * a\u207b\u00b9 * a = a\n[PROOFSTEP]\nrw [h, inv_zero, mul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\nh : \u00aca = 0\n\u22a2 a * a\u207b\u00b9 * a = a\n[PROOFSTEP]\nrw [mul_inv_cancel h, one_mul]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\n\u22a2 a\u207b\u00b9 * a * a = a\n[PROOFSTEP]\nby_cases h : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\nh : a = 0\n\u22a2 a\u207b\u00b9 * a * a = a\n[PROOFSTEP]\nrw [h, inv_zero, mul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\nh : \u00aca = 0\n\u22a2 a\u207b\u00b9 * a * a = a\n[PROOFSTEP]\nrw [inv_mul_cancel h, one_mul]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\n\u22a2 a * a / a = a\n[PROOFSTEP]\nrw [div_eq_mul_inv, mul_self_mul_inv a]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\n\u22a2 a / a * a = a\n[PROOFSTEP]\nrw [div_eq_mul_inv, mul_inv_mul_self a]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\n\u22a2 a / (a * a) = a\u207b\u00b9\u207b\u00b9 * a\u207b\u00b9 * a\u207b\u00b9\n[PROOFSTEP]\nsimp [mul_inv_rev]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\nh : a \u2260 0\n\u22a2 1 / a \u2260 0\n[PROOFSTEP]\nsimpa only [one_div] using inv_ne_zero h\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\n\u22a2 a\u207b\u00b9 = 0 \u2194 a = 0\n[PROOFSTEP]\nrw [inv_eq_iff_eq_inv, inv_zero]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\n\u22a2 a / (a / a) = a\n[PROOFSTEP]\nrw [div_div_eq_mul_div]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\n\u22a2 a * a / a = a\n[PROOFSTEP]\nexact mul_self_div_self a\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\nh : 1 / a \u2260 0\nha : a = 0\n\u22a2 False\n[PROOFSTEP]\nrw [ha, div_zero] at h \n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\nh : 0 \u2260 0\nha : a = 0\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\nh : a \u2260 0\ng : G\u2080\n\u22a2 (fun g => a * g) (a\u207b\u00b9 * g) = g\n[PROOFSTEP]\nsimp [\u2190 mul_assoc, mul_inv_cancel h]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\nh : a \u2260 0\ng : G\u2080\n\u22a2 (fun g => g * a) (g * a\u207b\u00b9) = g\n[PROOFSTEP]\nsimp [mul_assoc, inv_mul_cancel h]\n[GOAL]\n\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : CommGroupWithZero G\u2080\na\u271d b\u271d c\u271d d a b c : G\u2080\n\u22a2 a / c * b = a * b / c\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv, mul_assoc, mul_comm c\u207b\u00b9]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.GroupWithZero.Basic", "llama_tokens": 8357, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.4223020047509993}} {"text": "[GOAL]\nE : Type u_1\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : ContinuousSMul \u211d E\ninst\u271d\u00b9 : LocallyConvexSpace \u211d E\ninst\u271d : T1Space E\nx : E\nhx : x \u2260 0\n\u22a2 \u2203 f, \u2191f x \u2260 0\n[PROOFSTEP]\nrcases geometric_hahn_banach_point_point hx.symm with \u27e8f, hf\u27e9\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : ContinuousSMul \u211d E\ninst\u271d\u00b9 : LocallyConvexSpace \u211d E\ninst\u271d : T1Space E\nx : E\nhx : x \u2260 0\nf : E \u2192L[\u211d] \u211d\nhf : \u2191f 0 < \u2191f x\n\u22a2 \u2203 f, \u2191f x \u2260 0\n[PROOFSTEP]\nsimp only [map_zero] at hf \n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : ContinuousSMul \u211d E\ninst\u271d\u00b9 : LocallyConvexSpace \u211d E\ninst\u271d : T1Space E\nx : E\nhx : x \u2260 0\nf : E \u2192L[\u211d] \u211d\nhf : 0 < \u2191f x\n\u22a2 \u2203 f, \u2191f x \u2260 0\n[PROOFSTEP]\nexact \u27e8f, hf.ne'\u27e9\n[GOAL]\nE : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nhx : x \u2260 0\n\u22a2 \u2203 f, \u2191f x \u2260 0\n[PROOFSTEP]\nrcases exists_dual_vector \ud835\udd5c x hx with \u27e8f, -, hf\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nhx : x \u2260 0\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nhf : \u2191f x = \u2191\u2016x\u2016\n\u22a2 \u2203 f, \u2191f x \u2260 0\n[PROOFSTEP]\nrefine \u27e8f, ?_\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nhx : x \u2260 0\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nhf : \u2191f x = \u2191\u2016x\u2016\n\u22a2 \u2191f x \u2260 0\n[PROOFSTEP]\nsimpa [hf] using hx\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : TopologicalSpace V\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : Module R V\ninst\u271d : SeparatingDual R V\nx y : V\nh : x \u2260 y\n\u22a2 \u2203 f, \u2191f x \u2260 \u2191f y\n[PROOFSTEP]\nrcases exists_ne_zero (R := R) (sub_ne_zero_of_ne h) with \u27e8f, hf\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : TopologicalSpace V\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : Module R V\ninst\u271d : SeparatingDual R V\nx y : V\nh : x \u2260 y\nf : V \u2192L[R] R\nhf : \u2191f (x - y) \u2260 0\n\u22a2 \u2203 f, \u2191f x \u2260 \u2191f y\n[PROOFSTEP]\nexact \u27e8f, by simpa [sub_ne_zero] using hf\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : TopologicalSpace V\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : Module R V\ninst\u271d : SeparatingDual R V\nx y : V\nh : x \u2260 y\nf : V \u2192L[R] R\nhf : \u2191f (x - y) \u2260 0\n\u22a2 \u2191f x \u2260 \u2191f y\n[PROOFSTEP]\nsimpa [sub_ne_zero] using hf\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : T1Space R\n\u22a2 T1Space V\n[PROOFSTEP]\napply t1Space_iff_exists_open.2 (fun x y hxy \u21a6 ?_)\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : T1Space R\nx y : V\nhxy : x \u2260 y\n\u22a2 \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u00acy \u2208 U\n[PROOFSTEP]\nrcases exists_separating_of_ne (R := R) hxy with \u27e8f, hf\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : T1Space R\nx y : V\nhxy : x \u2260 y\nf : V \u2192L[R] R\nhf : \u2191f x \u2260 \u2191f y\n\u22a2 \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u00acy \u2208 U\n[PROOFSTEP]\nexact \u27e8f \u207b\u00b9' {f y}\u1d9c, isOpen_compl_singleton.preimage f.continuous, hf, by simp\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : T1Space R\nx y : V\nhxy : x \u2260 y\nf : V \u2192L[R] R\nhf : \u2191f x \u2260 \u2191f y\n\u22a2 \u00acy \u2208 \u2191f \u207b\u00b9' {\u2191f y}\u1d9c\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : T2Space R\n\u22a2 T2Space V\n[PROOFSTEP]\napply (t2Space_iff _).2 (fun {x} {y} hxy \u21a6 ?_)\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : T2Space R\nx y : V\nhxy : x \u2260 y\n\u22a2 \u2203 u v, IsOpen u \u2227 IsOpen v \u2227 x \u2208 u \u2227 y \u2208 v \u2227 Disjoint u v\n[PROOFSTEP]\nrcases exists_separating_of_ne (R := R) hxy with \u27e8f, hf\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : T2Space R\nx y : V\nhxy : x \u2260 y\nf : V \u2192L[R] R\nhf : \u2191f x \u2260 \u2191f y\n\u22a2 \u2203 u v, IsOpen u \u2227 IsOpen v \u2227 x \u2208 u \u2227 y \u2208 v \u2227 Disjoint u v\n[PROOFSTEP]\nexact separated_by_continuous f.continuous hf\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2077 : Field R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : SeparatingDual R V\nx : V\nhx : x \u2260 0\n\u22a2 \u2203 f, \u2191f x = 1\n[PROOFSTEP]\nrcases exists_ne_zero (R := R) hx with \u27e8f, hf\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\ninst\u271d\u2077 : Field R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : SeparatingDual R V\nx : V\nhx : x \u2260 0\nf : V \u2192L[R] R\nhf : \u2191f x \u2260 0\n\u22a2 \u2203 f, \u2191f x = 1\n[PROOFSTEP]\nexact \u27e8(f x)\u207b\u00b9 \u2022 f, inv_mul_cancel hf\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2077 : Field R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : SeparatingDual R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 \u2203 f, \u2191f x = 1 \u2227 \u2191f y \u2260 0\n[PROOFSTEP]\nobtain \u27e8u, ux\u27e9 : \u2203 u : V \u2192L[R] R, u x = 1 := exists_eq_one hx\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\ninst\u271d\u2077 : Field R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : SeparatingDual R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nu : V \u2192L[R] R\nux : \u2191u x = 1\n\u22a2 \u2203 f, \u2191f x = 1 \u2227 \u2191f y \u2260 0\n[PROOFSTEP]\nrcases ne_or_eq (u y) 0 with uy | uy\n[GOAL]\ncase intro.inl\nR : Type u_1\nV : Type u_2\ninst\u271d\u2077 : Field R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : SeparatingDual R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nu : V \u2192L[R] R\nux : \u2191u x = 1\nuy : \u2191u y \u2260 0\n\u22a2 \u2203 f, \u2191f x = 1 \u2227 \u2191f y \u2260 0\n[PROOFSTEP]\nexact \u27e8u, ux, uy\u27e9\n[GOAL]\ncase intro.inr\nR : Type u_1\nV : Type u_2\ninst\u271d\u2077 : Field R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : SeparatingDual R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nu : V \u2192L[R] R\nux : \u2191u x = 1\nuy : \u2191u y = 0\n\u22a2 \u2203 f, \u2191f x = 1 \u2227 \u2191f y \u2260 0\n[PROOFSTEP]\nobtain \u27e8v, vy\u27e9 : \u2203 v : V \u2192L[R] R, v y = 1 := exists_eq_one hy\n[GOAL]\ncase intro.inr.intro\nR : Type u_1\nV : Type u_2\ninst\u271d\u2077 : Field R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : SeparatingDual R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nu : V \u2192L[R] R\nux : \u2191u x = 1\nuy : \u2191u y = 0\nv : V \u2192L[R] R\nvy : \u2191v y = 1\n\u22a2 \u2203 f, \u2191f x = 1 \u2227 \u2191f y \u2260 0\n[PROOFSTEP]\nrcases ne_or_eq (v x) 0 with vx | vx\n[GOAL]\ncase intro.inr.intro.inl\nR : Type u_1\nV : Type u_2\ninst\u271d\u2077 : Field R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : SeparatingDual R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nu : V \u2192L[R] R\nux : \u2191u x = 1\nuy : \u2191u y = 0\nv : V \u2192L[R] R\nvy : \u2191v y = 1\nvx : \u2191v x \u2260 0\n\u22a2 \u2203 f, \u2191f x = 1 \u2227 \u2191f y \u2260 0\n[PROOFSTEP]\nexact \u27e8(v x)\u207b\u00b9 \u2022 v, inv_mul_cancel vx, show (v x)\u207b\u00b9 * v y \u2260 0 by simp [vx, vy]\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2077 : Field R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : SeparatingDual R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nu : V \u2192L[R] R\nux : \u2191u x = 1\nuy : \u2191u y = 0\nv : V \u2192L[R] R\nvy : \u2191v y = 1\nvx : \u2191v x \u2260 0\n\u22a2 (\u2191v x)\u207b\u00b9 * \u2191v y \u2260 0\n[PROOFSTEP]\nsimp [vx, vy]\n[GOAL]\ncase intro.inr.intro.inr\nR : Type u_1\nV : Type u_2\ninst\u271d\u2077 : Field R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : SeparatingDual R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nu : V \u2192L[R] R\nux : \u2191u x = 1\nuy : \u2191u y = 0\nv : V \u2192L[R] R\nvy : \u2191v y = 1\nvx : \u2191v x = 0\n\u22a2 \u2203 f, \u2191f x = 1 \u2227 \u2191f y \u2260 0\n[PROOFSTEP]\nexact \u27e8u + v, by simp [ux, vx], by simp [uy, vy]\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2077 : Field R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : SeparatingDual R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nu : V \u2192L[R] R\nux : \u2191u x = 1\nuy : \u2191u y = 0\nv : V \u2192L[R] R\nvy : \u2191v y = 1\nvx : \u2191v x = 0\n\u22a2 \u2191(u + v) x = 1\n[PROOFSTEP]\nsimp [ux, vx]\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2077 : Field R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : TopologicalSpace R\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : SeparatingDual R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nu : V \u2192L[R] R\nux : \u2191u x = 1\nuy : \u2191u y = 0\nv : V \u2192L[R] R\nvy : \u2191v y = 1\nvx : \u2191v x = 0\n\u22a2 \u2191(u + v) y \u2260 0\n[PROOFSTEP]\nsimp [uy, vy]\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 \u2203 A, \u2191A x = y\n[PROOFSTEP]\nobtain \u27e8G, Gx, Gy\u27e9 : \u2203 G : V \u2192L[R] R, G x = 1 \u2227 G y \u2260 0 := exists_eq_one_ne_zero_of_ne_zero_pair hx hy\n[GOAL]\ncase intro.intro\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\n\u22a2 \u2203 A, \u2191A x = y\n[PROOFSTEP]\nlet A : V \u2243L[R] V :=\n { toFun := fun z \u21a6 z + G z \u2022 (y - x)\n invFun := fun z \u21a6 z + ((G y)\u207b\u00b9 * G z) \u2022 (x - y)\n map_add' := fun a b \u21a6 by simp [add_smul]; abel\n map_smul' := by simp [smul_smul]\n left_inv := fun z \u21a6\n by\n simp only [id_eq, eq_mpr_eq_cast, RingHom.id_apply, smul_eq_mul, AddHom.toFun_eq_coe, AddHom.coe_mk, map_add,\n map_smul\u209b\u2097, map_sub, Gx, mul_sub, mul_one, add_sub_cancel'_right]\n rw [mul_comm (G z), \u2190 mul_assoc, inv_mul_cancel Gy]\n simp only [smul_sub, one_mul]\n abel\n right_inv := fun z \u21a6\n by\n simp only [map_add, map_smul\u209b\u2097, map_mul, map_inv\u2080, RingHom.id_apply, map_sub, Gx, smul_eq_mul, mul_sub, mul_one]\n rw [mul_comm _ (G y), \u2190 mul_assoc, mul_inv_cancel Gy]\n simp only [smul_sub, one_mul, add_sub_cancel'_right]\n abel\n continuous_toFun := continuous_id.add (G.continuous.smul continuous_const)\n continuous_invFun := continuous_id.add ((continuous_const.mul G.continuous).smul continuous_const) }\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\na b : V\n\u22a2 (fun z => z + \u2191G z \u2022 (y - x)) (a + b) = (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b\n[PROOFSTEP]\nsimp [add_smul]\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\na b : V\n\u22a2 a + b + (\u2191G a \u2022 (y - x) + \u2191G b \u2022 (y - x)) = a + \u2191G a \u2022 (y - x) + (b + \u2191G b \u2022 (y - x))\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\na b : V\n\u22a2 a + b + (\u2191G a \u2022 (y - x) + \u2191G b \u2022 (y - x)) = a + \u2191G a \u2022 (y - x) + (b + \u2191G b \u2022 (y - x))\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\n\u22a2 \u2200 (r : R) (x_1 : V),\n AddHom.toFun\n { toFun := fun z => z + \u2191G z \u2022 (y - x),\n map_add' :=\n (_ :\n \u2200 (a b : V),\n (fun z => z + \u2191G z \u2022 (y - x)) (a + b) =\n (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) }\n (r \u2022 x_1) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun z => z + \u2191G z \u2022 (y - x),\n map_add' :=\n (_ :\n \u2200 (a b : V),\n (fun z => z + \u2191G z \u2022 (y - x)) (a + b) =\n (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) }\n x_1\n[PROOFSTEP]\nsimp [smul_smul]\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\nz : V\n\u22a2 (fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y))\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun z => z + \u2191G z \u2022 (y - x),\n map_add' :=\n (_ :\n \u2200 (a b : V),\n (fun z => z + \u2191G z \u2022 (y - x)) (a + b) =\n (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) },\n map_smul' :=\n (_ : \u2200 (a : R) (a_1 : V), a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) }.toAddHom\n z) =\n z\n[PROOFSTEP]\nsimp only [id_eq, eq_mpr_eq_cast, RingHom.id_apply, smul_eq_mul, AddHom.toFun_eq_coe, AddHom.coe_mk, map_add,\n map_smul\u209b\u2097, map_sub, Gx, mul_sub, mul_one, add_sub_cancel'_right]\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\nz : V\n\u22a2 z + \u2191G z \u2022 (y - x) + ((\u2191G y)\u207b\u00b9 * (\u2191G z * \u2191G y)) \u2022 (x - y) = z\n[PROOFSTEP]\nrw [mul_comm (G z), \u2190 mul_assoc, inv_mul_cancel Gy]\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\nz : V\n\u22a2 z + \u2191G z \u2022 (y - x) + (1 * \u2191G z) \u2022 (x - y) = z\n[PROOFSTEP]\nsimp only [smul_sub, one_mul]\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\nz : V\n\u22a2 z + (\u2191G z \u2022 y - \u2191G z \u2022 x) + (\u2191G z \u2022 x - \u2191G z \u2022 y) = z\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\nz : V\n\u22a2 z + (\u2191G z \u2022 y - \u2191G z \u2022 x) + (\u2191G z \u2022 x - \u2191G z \u2022 y) = z\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\nz : V\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun z => z + \u2191G z \u2022 (y - x),\n map_add' :=\n (_ :\n \u2200 (a b : V),\n (fun z => z + \u2191G z \u2022 (y - x)) (a + b) =\n (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) },\n map_smul' :=\n (_ : \u2200 (a : R) (a_1 : V), a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) }.toAddHom\n ((fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y)) z) =\n z\n[PROOFSTEP]\nsimp only [map_add, map_smul\u209b\u2097, map_mul, map_inv\u2080, RingHom.id_apply, map_sub, Gx, smul_eq_mul, mul_sub, mul_one]\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\nz : V\n\u22a2 z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y) + (\u2191G z + ((\u2191G y)\u207b\u00b9 * \u2191G z - (\u2191G y)\u207b\u00b9 * \u2191G z * \u2191G y)) \u2022 (y - x) = z\n[PROOFSTEP]\nrw [mul_comm _ (G y), \u2190 mul_assoc, mul_inv_cancel Gy]\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\nz : V\n\u22a2 z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y) + (\u2191G z + ((\u2191G y)\u207b\u00b9 * \u2191G z - 1 * \u2191G z)) \u2022 (y - x) = z\n[PROOFSTEP]\nsimp only [smul_sub, one_mul, add_sub_cancel'_right]\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\nz : V\n\u22a2 z + (((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 x - ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 y) + (((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 y - ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 x) = z\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\nz : V\n\u22a2 z + (((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 x - ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 y) + (((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 y - ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 x) = z\n[PROOFSTEP]\nabel\n[GOAL]\ncase intro.intro\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\nA : V \u2243L[R] V :=\n ContinuousLinearEquiv.mk\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := fun z => z + \u2191G z \u2022 (y - x),\n map_add' :=\n (_ :\n \u2200 (a b : V),\n (fun z => z + \u2191G z \u2022 (y - x)) (a + b) =\n (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) },\n map_smul' := (_ : \u2200 (a : R) (a_1 : V), a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) },\n invFun := fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y),\n left_inv :=\n (_ :\n \u2200 (z : V),\n (fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y))\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun z => z + \u2191G z \u2022 (y - x),\n map_add' :=\n (_ :\n \u2200 (a b : V),\n (fun z => z + \u2191G z \u2022 (y - x)) (a + b) =\n (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) },\n map_smul' :=\n (_ :\n \u2200 (a : R) (a_1 : V),\n a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) }.toAddHom\n z) =\n z),\n right_inv :=\n (_ :\n \u2200 (z : V),\n AddHom.toFun\n {\n toAddHom :=\n { toFun := fun z => z + \u2191G z \u2022 (y - x),\n map_add' :=\n (_ :\n \u2200 (a b : V),\n (fun z => z + \u2191G z \u2022 (y - x)) (a + b) =\n (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) },\n map_smul' :=\n (_ :\n \u2200 (a : R) (a_1 : V), a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) }.toAddHom\n ((fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y)) z) =\n z) }\n\u22a2 \u2203 A, \u2191A x = y\n[PROOFSTEP]\nexact \u27e8A, show x + G x \u2022 (y - x) = y by simp [Gx]\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\ninst\u271d\u2078 : Field R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : TopologicalSpace R\ninst\u271d\u2075 : TopologicalSpace V\ninst\u271d\u2074 : TopologicalRing R\ninst\u271d\u00b3 : TopologicalAddGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : SeparatingDual R V\ninst\u271d : ContinuousSMul R V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nG : V \u2192L[R] R\nGx : \u2191G x = 1\nGy : \u2191G y \u2260 0\nA : V \u2243L[R] V :=\n ContinuousLinearEquiv.mk\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := fun z => z + \u2191G z \u2022 (y - x),\n map_add' :=\n (_ :\n \u2200 (a b : V),\n (fun z => z + \u2191G z \u2022 (y - x)) (a + b) =\n (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) },\n map_smul' := (_ : \u2200 (a : R) (a_1 : V), a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) },\n invFun := fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y),\n left_inv :=\n (_ :\n \u2200 (z : V),\n (fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y))\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun z => z + \u2191G z \u2022 (y - x),\n map_add' :=\n (_ :\n \u2200 (a b : V),\n (fun z => z + \u2191G z \u2022 (y - x)) (a + b) =\n (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) },\n map_smul' :=\n (_ :\n \u2200 (a : R) (a_1 : V),\n a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) }.toAddHom\n z) =\n z),\n right_inv :=\n (_ :\n \u2200 (z : V),\n AddHom.toFun\n {\n toAddHom :=\n { toFun := fun z => z + \u2191G z \u2022 (y - x),\n map_add' :=\n (_ :\n \u2200 (a b : V),\n (fun z => z + \u2191G z \u2022 (y - x)) (a + b) =\n (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) },\n map_smul' :=\n (_ :\n \u2200 (a : R) (a_1 : V), a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) }.toAddHom\n ((fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y)) z) =\n z) }\n\u22a2 x + \u2191G x \u2022 (y - x) = y\n[PROOFSTEP]\nsimp [Gx]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "llama_tokens": 12150, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.42229278304306533}} {"text": "[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nN : Type (max u v)\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\n\u22a2 DirectSum.IsInternal fun p => torsionBy R M (IsPrincipal.generator \u2191p ^ Multiset.count (\u2191p) (factors (annihilator \u22a4)))\n[PROOFSTEP]\nconvert isInternal_prime_power_torsion hM\n[GOAL]\ncase h.e'_8.h\nR : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nN : Type (max u v)\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\nx\u271d : { x // x \u2208 Multiset.toFinset (factors (annihilator \u22a4)) }\n\u22a2 torsionBy R M (IsPrincipal.generator \u2191x\u271d ^ Multiset.count (\u2191x\u271d) (factors (annihilator \u22a4))) =\n torsionBySet R M \u2191(\u2191x\u271d ^ Multiset.count (\u2191x\u271d) (factors (annihilator \u22a4)))\n[PROOFSTEP]\next p : 1\n[GOAL]\ncase h.e'_8.h.h\nR : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nN : Type (max u v)\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\nx\u271d : { x // x \u2208 Multiset.toFinset (factors (annihilator \u22a4)) }\np : M\n\u22a2 p \u2208 torsionBy R M (IsPrincipal.generator \u2191x\u271d ^ Multiset.count (\u2191x\u271d) (factors (annihilator \u22a4))) \u2194\n p \u2208 torsionBySet R M \u2191(\u2191x\u271d ^ Multiset.count (\u2191x\u271d) (factors (annihilator \u22a4)))\n[PROOFSTEP]\nrw [\u2190 torsionBySet_span_singleton_eq, Ideal.submodule_span_eq, \u2190 Ideal.span_singleton_pow,\n Ideal.span_singleton_generator]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nN : Type (max u v)\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\n\u22a2 \u2203 \u03b9 x x p x_1 e, DirectSum.IsInternal fun i => torsionBy R M (p i ^ e i)\n[PROOFSTEP]\nrefine' \u27e8_, _, _, _, _, _, Submodule.isInternal_prime_power_torsion_of_pid hM\u27e9\n[GOAL]\ncase refine'_1\nR : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nN : Type (max u v)\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\n\u22a2 Fintype { x // x \u2208 Multiset.toFinset (factors (annihilator \u22a4)) }\ncase refine'_2\nR : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nN : Type (max u v)\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\n\u22a2 \u2200 (i : { x // x \u2208 Multiset.toFinset (factors (annihilator \u22a4)) }), Irreducible (IsPrincipal.generator \u2191i)\n[PROOFSTEP]\nexact Finset.fintypeCoeSort _\n[GOAL]\ncase refine'_2\nR : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nN : Type (max u v)\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\n\u22a2 \u2200 (i : { x // x \u2208 Multiset.toFinset (factors (annihilator \u22a4)) }), Irreducible (IsPrincipal.generator \u2191i)\n[PROOFSTEP]\nrintro \u27e8p, hp\u27e9\n[GOAL]\ncase refine'_2.mk\nR : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nN : Type (max u v)\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\np : Ideal R\nhp : p \u2208 Multiset.toFinset (factors (annihilator \u22a4))\n\u22a2 Irreducible (IsPrincipal.generator \u2191{ val := p, property := hp })\n[PROOFSTEP]\nhave hP := prime_of_factor p (Multiset.mem_toFinset.mp hp)\n[GOAL]\ncase refine'_2.mk\nR : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nN : Type (max u v)\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\np : Ideal R\nhp : p \u2208 Multiset.toFinset (factors (annihilator \u22a4))\nhP : Prime p\n\u22a2 Irreducible (IsPrincipal.generator \u2191{ val := p, property := hp })\n[PROOFSTEP]\nhaveI := Ideal.isPrime_of_prime hP\n[GOAL]\ncase refine'_2.mk\nR : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nN : Type (max u v)\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\np : Ideal R\nhp : p \u2208 Multiset.toFinset (factors (annihilator \u22a4))\nhP : Prime p\nthis : Ideal.IsPrime p\n\u22a2 Irreducible (IsPrincipal.generator \u2191{ val := p, property := hp })\n[PROOFSTEP]\nexact (IsPrincipal.prime_generator_of_isPrime p hP.ne_zero).irreducible\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx : M\n\u22a2 torsionOf R M x = Ideal.span {p ^ pOrder hM x}\n[PROOFSTEP]\ndsimp only [pOrder]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx : M\n\u22a2 torsionOf R M x = Ideal.span {p ^ Nat.find (_ : \u2203 n, p ^ n \u2022 x = 0)}\n[PROOFSTEP]\nrw [\u2190 (torsionOf R M x).span_singleton_generator, Ideal.span_singleton_eq_span_singleton, \u2190\n Associates.mk_eq_mk_iff_associated, Associates.mk_pow]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx : M\n\u22a2 Associates.mk (generator (torsionOf R M x)) = Associates.mk p ^ Nat.find (_ : \u2203 n, p ^ n \u2022 x = 0)\n[PROOFSTEP]\nhave prop :\n (fun n : \u2115 => p ^ n \u2022 x = 0) = fun n : \u2115 => (Associates.mk <| generator <| torsionOf R M x) \u2223 Associates.mk p ^ n :=\n by ext n; rw [\u2190 Associates.mk_pow, Associates.mk_dvd_mk, \u2190 mem_iff_generator_dvd]; rfl\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx : M\n\u22a2 (fun n => p ^ n \u2022 x = 0) = fun n => Associates.mk (generator (torsionOf R M x)) \u2223 Associates.mk p ^ n\n[PROOFSTEP]\next n\n[GOAL]\ncase h.a\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx : M\nn : \u2115\n\u22a2 p ^ n \u2022 x = 0 \u2194 Associates.mk (generator (torsionOf R M x)) \u2223 Associates.mk p ^ n\n[PROOFSTEP]\nrw [\u2190 Associates.mk_pow, Associates.mk_dvd_mk, \u2190 mem_iff_generator_dvd]\n[GOAL]\ncase h.a\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx : M\nn : \u2115\n\u22a2 p ^ n \u2022 x = 0 \u2194 p ^ n \u2208 torsionOf R M x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx : M\nprop : (fun n => p ^ n \u2022 x = 0) = fun n => Associates.mk (generator (torsionOf R M x)) \u2223 Associates.mk p ^ n\n\u22a2 Associates.mk (generator (torsionOf R M x)) = Associates.mk p ^ Nat.find (_ : \u2203 n, p ^ n \u2022 x = 0)\n[PROOFSTEP]\nhave := (isTorsion'_powers_iff p).mp hM x\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx : M\nprop : (fun n => p ^ n \u2022 x = 0) = fun n => Associates.mk (generator (torsionOf R M x)) \u2223 Associates.mk p ^ n\nthis : \u2203 n, p ^ n \u2022 x = 0\n\u22a2 Associates.mk (generator (torsionOf R M x)) = Associates.mk p ^ Nat.find (_ : \u2203 n, p ^ n \u2022 x = 0)\n[PROOFSTEP]\nrw [prop] at this \n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx : M\nprop : (fun n => p ^ n \u2022 x = 0) = fun n => Associates.mk (generator (torsionOf R M x)) \u2223 Associates.mk p ^ n\nthis : \u2203 n, Associates.mk (generator (torsionOf R M x)) \u2223 Associates.mk p ^ n\n\u22a2 Associates.mk (generator (torsionOf R M x)) = Associates.mk p ^ Nat.find (_ : \u2203 n, p ^ n \u2022 x = 0)\n[PROOFSTEP]\nclassical convert Associates.eq_pow_find_of_dvd_irreducible_pow ((Associates.irreducible_mk p).mpr hp) this.choose_spec\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx : M\nprop : (fun n => p ^ n \u2022 x = 0) = fun n => Associates.mk (generator (torsionOf R M x)) \u2223 Associates.mk p ^ n\nthis : \u2203 n, Associates.mk (generator (torsionOf R M x)) \u2223 Associates.mk p ^ n\n\u22a2 Associates.mk (generator (torsionOf R M x)) = Associates.mk p ^ Nat.find (_ : \u2203 n, p ^ n \u2022 x = 0)\n[PROOFSTEP]\nconvert Associates.eq_pow_find_of_dvd_irreducible_pow ((Associates.irreducible_mk p).mpr hp) this.choose_spec\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\n\u22a2 \u2203 a, p ^ k \u2022 x = p ^ k \u2022 a \u2022 y\n[PROOFSTEP]\nletI : MulAction R M := MulActionWithZero.toMulAction\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\n\u22a2 \u2203 a, p ^ k \u2022 x = p ^ k \u2022 a \u2022 y\n[PROOFSTEP]\nby_cases hk : k \u2264 pOrder hM y\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\n\u22a2 \u2203 a, p ^ k \u2022 x = p ^ k \u2022 a \u2022 y\n[PROOFSTEP]\nlet f := ((R \u2219 p ^ (pOrder hM y - k) * p ^ k).quotEquivOfEq _ ?_).trans (quotTorsionOfEquivSpanSingleton R M y)\n[GOAL]\ncase pos.refine_2\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\n\u22a2 \u2203 a, p ^ k \u2022 x = p ^ k \u2022 a \u2022 y\ncase pos.refine_1\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\n\u22a2 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k} = torsionOf R M y\n[PROOFSTEP]\nhave : f.symm \u27e8p ^ k \u2022 x, h\u27e9 \u2208 R \u2219 Ideal.Quotient.mk (R \u2219 p ^ (pOrder hM y - k) * p ^ k) (p ^ k) :=\n by\n rw [\u2190 Quotient.torsionBy_eq_span_singleton, mem_torsionBy_iff, \u2190 f.symm.map_smul]\n convert f.symm.map_zero; ext\n rw [coe_smul_of_tower, coe_mk, coe_zero, smul_smul, \u2190 pow_add, Nat.sub_add_cancel hk, @hM' x]\n \u00b7 exact mem_nonZeroDivisors_of_ne_zero (pow_ne_zero _ hp.ne_zero)\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\n\u22a2 \u2191(LinearEquiv.symm f) { val := p ^ k \u2022 x, property := h } \u2208\n Submodule.span R {\u2191(Ideal.Quotient.mk (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k})) (p ^ k)}\n[PROOFSTEP]\nrw [\u2190 Quotient.torsionBy_eq_span_singleton, mem_torsionBy_iff, \u2190 f.symm.map_smul]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\n\u22a2 \u2191(LinearEquiv.symm f) (p ^ (pOrder hM y - k) \u2022 { val := p ^ k \u2022 x, property := h }) = 0\ncase ha\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\n\u22a2 p ^ (pOrder hM y - k) \u2208 nonZeroDivisors R\n[PROOFSTEP]\nconvert f.symm.map_zero\n[GOAL]\ncase h.e'_2.h.e'_6\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\n\u22a2 p ^ (pOrder hM y - k) \u2022 { val := p ^ k \u2022 x, property := h } = 0\ncase ha\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\n\u22a2 p ^ (pOrder hM y - k) \u2208 nonZeroDivisors R\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_2.h.e'_6.a\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\n\u22a2 \u2191(p ^ (pOrder hM y - k) \u2022 { val := p ^ k \u2022 x, property := h }) = \u21910\ncase ha\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\n\u22a2 p ^ (pOrder hM y - k) \u2208 nonZeroDivisors R\n[PROOFSTEP]\nrw [coe_smul_of_tower, coe_mk, coe_zero, smul_smul, \u2190 pow_add, Nat.sub_add_cancel hk, @hM' x]\n[GOAL]\ncase ha\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\n\u22a2 p ^ (pOrder hM y - k) \u2208 nonZeroDivisors R\n[PROOFSTEP]\nexact mem_nonZeroDivisors_of_ne_zero (pow_ne_zero _ hp.ne_zero)\n[GOAL]\ncase pos.refine_2\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis\u271d : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\nthis :\n \u2191(LinearEquiv.symm f) { val := p ^ k \u2022 x, property := h } \u2208\n Submodule.span R {\u2191(Ideal.Quotient.mk (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k})) (p ^ k)}\n\u22a2 \u2203 a, p ^ k \u2022 x = p ^ k \u2022 a \u2022 y\ncase pos.refine_1\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\n\u22a2 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k} = torsionOf R M y\n[PROOFSTEP]\nrw [Submodule.mem_span_singleton] at this \n[GOAL]\ncase pos.refine_2\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis\u271d : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\nthis :\n \u2203 a,\n a \u2022 \u2191(Ideal.Quotient.mk (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k})) (p ^ k) =\n \u2191(LinearEquiv.symm f) { val := p ^ k \u2022 x, property := h }\n\u22a2 \u2203 a, p ^ k \u2022 x = p ^ k \u2022 a \u2022 y\ncase pos.refine_1\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\n\u22a2 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k} = torsionOf R M y\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := this\n[GOAL]\ncase pos.refine_2.intro\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\na : R\nha :\n a \u2022 \u2191(Ideal.Quotient.mk (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k})) (p ^ k) =\n \u2191(LinearEquiv.symm f) { val := p ^ k \u2022 x, property := h }\n\u22a2 \u2203 a, p ^ k \u2022 x = p ^ k \u2022 a \u2022 y\ncase pos.refine_1\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\n\u22a2 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k} = torsionOf R M y\n[PROOFSTEP]\nuse a\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\na : R\nha :\n a \u2022 \u2191(Ideal.Quotient.mk (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k})) (p ^ k) =\n \u2191(LinearEquiv.symm f) { val := p ^ k \u2022 x, property := h }\n\u22a2 p ^ k \u2022 x = p ^ k \u2022 a \u2022 y\ncase pos.refine_1\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\n\u22a2 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k} = torsionOf R M y\n[PROOFSTEP]\nrw [f.eq_symm_apply, \u2190 Ideal.Quotient.mk_eq_mk, \u2190 Quotient.mk_smul] at ha \n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\na : R\nha : \u2191f (Submodule.Quotient.mk (a \u2022 p ^ k)) = { val := p ^ k \u2022 x, property := h }\n\u22a2 p ^ k \u2022 x = p ^ k \u2022 a \u2022 y\ncase pos.refine_1\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\n\u22a2 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k} = torsionOf R M y\n[PROOFSTEP]\ndsimp only [smul_eq_mul, LinearEquiv.trans_apply, Submodule.quotEquivOfEq_mk,\n quotTorsionOfEquivSpanSingleton_apply_mk] at ha \n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\na : R\nha :\n \u2191(quotTorsionOfEquivSpanSingleton R M y)\n (\u2191(Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (Submodule.Quotient.mk (a * p ^ k))) =\n { val := p ^ k \u2022 x, property := h }\n\u22a2 p ^ k \u2022 x = p ^ k \u2022 a \u2022 y\ncase pos.refine_1\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\n\u22a2 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k} = torsionOf R M y\n[PROOFSTEP]\nrw [smul_smul, mul_comm]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\nf : (R \u29f8 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) \u2243\u2097[R] { x // x \u2208 Submodule.span R {y} } :=\n LinearEquiv.trans\n (Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (quotTorsionOfEquivSpanSingleton R M y)\na : R\nha :\n \u2191(quotTorsionOfEquivSpanSingleton R M y)\n (\u2191(Submodule.quotEquivOfEq (Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}) (torsionOf R M y) ?pos.refine_1\u271d)\n (Submodule.Quotient.mk (a * p ^ k))) =\n { val := p ^ k \u2022 x, property := h }\n\u22a2 p ^ k \u2022 x = (a * p ^ k) \u2022 y\ncase pos.refine_1\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\n\u22a2 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k} = torsionOf R M y\n[PROOFSTEP]\nexact congr_arg ((\u2191) : _ \u2192 M) ha.symm\n[GOAL]\ncase pos.refine_1\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\n\u22a2 Submodule.span R {p ^ (pOrder hM y - k) * p ^ k} = torsionOf R M y\n[PROOFSTEP]\nsymm\n[GOAL]\ncase pos.refine_1\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\n\u22a2 torsionOf R M y = Submodule.span R {p ^ (pOrder hM y - k) * p ^ k}\n[PROOFSTEP]\nconvert Ideal.torsionOf_eq_span_pow_pOrder hp hM y\n[GOAL]\ncase h.e'_3.h.e'_1.h.e'_4\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : k \u2264 pOrder hM y\n\u22a2 p ^ (pOrder hM y - k) * p ^ k = p ^ pOrder hM y\n[PROOFSTEP]\nrw [\u2190 pow_add, Nat.sub_add_cancel hk]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : \u00ack \u2264 pOrder hM y\n\u22a2 \u2203 a, p ^ k \u2022 x = p ^ k \u2022 a \u2022 y\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nx y : M\nk : \u2115\nhM' : IsTorsionBy R M (p ^ pOrder hM y)\nh : p ^ k \u2022 x \u2208 Submodule.span R {y}\nthis : MulAction R M := MulActionWithZero.toMulAction\nhk : \u00ack \u2264 pOrder hM y\n\u22a2 p ^ k \u2022 x = p ^ k \u2022 0 \u2022 y\n[PROOFSTEP]\nrw [zero_smul, smul_zero, \u2190 Nat.sub_add_cancel (le_of_not_le hk), pow_add, mul_smul, hM', smul_zero]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nz : M\nhz : IsTorsionBy R M (p ^ pOrder hM z)\nk : \u2115\nf : R \u29f8 Submodule.span R {p ^ k} \u2192\u2097[R] M \u29f8 Submodule.span R {z}\n\u22a2 \u2203 x, p ^ k \u2022 x = 0 \u2227 Submodule.Quotient.mk x = \u2191f 1\n[PROOFSTEP]\nhave f1 := mk_surjective (R \u2219 z) (f 1)\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nz : M\nhz : IsTorsionBy R M (p ^ pOrder hM z)\nk : \u2115\nf : R \u29f8 Submodule.span R {p ^ k} \u2192\u2097[R] M \u29f8 Submodule.span R {z}\nf1 : \u2203 a, Submodule.Quotient.mk a = \u2191f 1\n\u22a2 \u2203 x, p ^ k \u2022 x = 0 \u2227 Submodule.Quotient.mk x = \u2191f 1\n[PROOFSTEP]\nhave : p ^ k \u2022 f1.choose \u2208 R \u2219 z :=\n by\n rw [\u2190 Quotient.mk_eq_zero, mk_smul, f1.choose_spec, \u2190 f.map_smul]\n convert f.map_zero; change _ \u2022 Submodule.Quotient.mk _ = _\n rw [\u2190 mk_smul, Quotient.mk_eq_zero, Algebra.id.smul_eq_mul, mul_one]\n exact Submodule.mem_span_singleton_self _\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nz : M\nhz : IsTorsionBy R M (p ^ pOrder hM z)\nk : \u2115\nf : R \u29f8 Submodule.span R {p ^ k} \u2192\u2097[R] M \u29f8 Submodule.span R {z}\nf1 : \u2203 a, Submodule.Quotient.mk a = \u2191f 1\n\u22a2 p ^ k \u2022 Exists.choose f1 \u2208 Submodule.span R {z}\n[PROOFSTEP]\nrw [\u2190 Quotient.mk_eq_zero, mk_smul, f1.choose_spec, \u2190 f.map_smul]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nz : M\nhz : IsTorsionBy R M (p ^ pOrder hM z)\nk : \u2115\nf : R \u29f8 Submodule.span R {p ^ k} \u2192\u2097[R] M \u29f8 Submodule.span R {z}\nf1 : \u2203 a, Submodule.Quotient.mk a = \u2191f 1\n\u22a2 \u2191f (p ^ k \u2022 1) = 0\n[PROOFSTEP]\nconvert f.map_zero\n[GOAL]\ncase h.e'_2.h.e'_6\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nz : M\nhz : IsTorsionBy R M (p ^ pOrder hM z)\nk : \u2115\nf : R \u29f8 Submodule.span R {p ^ k} \u2192\u2097[R] M \u29f8 Submodule.span R {z}\nf1 : \u2203 a, Submodule.Quotient.mk a = \u2191f 1\n\u22a2 p ^ k \u2022 1 = 0\n[PROOFSTEP]\nchange _ \u2022 Submodule.Quotient.mk _ = _\n[GOAL]\ncase h.e'_2.h.e'_6\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nz : M\nhz : IsTorsionBy R M (p ^ pOrder hM z)\nk : \u2115\nf : R \u29f8 Submodule.span R {p ^ k} \u2192\u2097[R] M \u29f8 Submodule.span R {z}\nf1 : \u2203 a, Submodule.Quotient.mk a = \u2191f 1\n\u22a2 p ^ k \u2022 Submodule.Quotient.mk 1 = 0\n[PROOFSTEP]\nrw [\u2190 mk_smul, Quotient.mk_eq_zero, Algebra.id.smul_eq_mul, mul_one]\n[GOAL]\ncase h.e'_2.h.e'_6\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nz : M\nhz : IsTorsionBy R M (p ^ pOrder hM z)\nk : \u2115\nf : R \u29f8 Submodule.span R {p ^ k} \u2192\u2097[R] M \u29f8 Submodule.span R {z}\nf1 : \u2203 a, Submodule.Quotient.mk a = \u2191f 1\n\u22a2 p ^ k \u2208 Submodule.span R {p ^ k}\n[PROOFSTEP]\nexact Submodule.mem_span_singleton_self _\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nz : M\nhz : IsTorsionBy R M (p ^ pOrder hM z)\nk : \u2115\nf : R \u29f8 Submodule.span R {p ^ k} \u2192\u2097[R] M \u29f8 Submodule.span R {z}\nf1 : \u2203 a, Submodule.Quotient.mk a = \u2191f 1\nthis : p ^ k \u2022 Exists.choose f1 \u2208 Submodule.span R {z}\n\u22a2 \u2203 x, p ^ k \u2022 x = 0 \u2227 Submodule.Quotient.mk x = \u2191f 1\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := p_pow_smul_lift hp hM hz this\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nz : M\nhz : IsTorsionBy R M (p ^ pOrder hM z)\nk : \u2115\nf : R \u29f8 Submodule.span R {p ^ k} \u2192\u2097[R] M \u29f8 Submodule.span R {z}\nf1 : \u2203 a, Submodule.Quotient.mk a = \u2191f 1\nthis : p ^ k \u2022 Exists.choose f1 \u2208 Submodule.span R {z}\na : R\nha : p ^ k \u2022 Exists.choose f1 = p ^ k \u2022 a \u2022 z\n\u22a2 \u2203 x, p ^ k \u2022 x = 0 \u2227 Submodule.Quotient.mk x = \u2191f 1\n[PROOFSTEP]\nrefine' \u27e8f1.choose - a \u2022 z, by rw [smul_sub, sub_eq_zero, ha], _\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nz : M\nhz : IsTorsionBy R M (p ^ pOrder hM z)\nk : \u2115\nf : R \u29f8 Submodule.span R {p ^ k} \u2192\u2097[R] M \u29f8 Submodule.span R {z}\nf1 : \u2203 a, Submodule.Quotient.mk a = \u2191f 1\nthis : p ^ k \u2022 Exists.choose f1 \u2208 Submodule.span R {z}\na : R\nha : p ^ k \u2022 Exists.choose f1 = p ^ k \u2022 a \u2022 z\n\u22a2 p ^ k \u2022 (Exists.choose f1 - a \u2022 z) = 0\n[PROOFSTEP]\nrw [smul_sub, sub_eq_zero, ha]\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nz : M\nhz : IsTorsionBy R M (p ^ pOrder hM z)\nk : \u2115\nf : R \u29f8 Submodule.span R {p ^ k} \u2192\u2097[R] M \u29f8 Submodule.span R {z}\nf1 : \u2203 a, Submodule.Quotient.mk a = \u2191f 1\nthis : p ^ k \u2022 Exists.choose f1 \u2208 Submodule.span R {z}\na : R\nha : p ^ k \u2022 Exists.choose f1 = p ^ k \u2022 a \u2022 z\n\u22a2 Submodule.Quotient.mk (Exists.choose f1 - a \u2022 z) = \u2191f 1\n[PROOFSTEP]\nrw [mk_sub, mk_smul, (Quotient.mk_eq_zero _).mpr <| Submodule.mem_span_singleton_self _, smul_zero, sub_zero,\n f1.choose_spec]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\nh' : Finite R N\n\u22a2 \u2203 d k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\n[PROOFSTEP]\nobtain \u27e8d, s, hs\u27e9 := @Module.Finite.exists_fin _ _ _ _ _ h'\n[GOAL]\ncase intro.intro\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\nh' : Finite R N\nd : \u2115\ns : Fin d \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\n\u22a2 \u2203 d k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\n[PROOFSTEP]\nuse d\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\nh' : Finite R N\nd : \u2115\ns : Fin d \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\n\u22a2 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\n[PROOFSTEP]\nclear h'\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\nd : \u2115\ns : Fin d \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\n\u22a2 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\n[PROOFSTEP]\ninduction' d with d IH generalizing N\n[GOAL]\ncase h.zero\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd : \u2115\ns\u271d : Fin d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin Nat.zero \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\n\u22a2 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin Nat.zero), R \u29f8 Submodule.span R {p ^ k i})\n[PROOFSTEP]\nsimp only [Nat.zero_eq] at *\n -- Porting note: was `use fun i => finZeroElim i`\n[GOAL]\ncase h.zero\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd : \u2115\ns\u271d : Fin d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin Nat.zero \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\n\u22a2 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin Nat.zero), R \u29f8 Submodule.span R {p ^ k i})\n[PROOFSTEP]\nuse finZeroElim\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd : \u2115\ns\u271d : Fin d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin Nat.zero \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\n\u22a2 Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin Nat.zero), R \u29f8 Submodule.span R {p ^ finZeroElim i})\n[PROOFSTEP]\nrw [Set.range_eq_empty, Submodule.span_empty] at hs \n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd : \u2115\ns\u271d : Fin d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin Nat.zero \u2192 N\nhs : \u22a5 = \u22a4\n\u22a2 Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin Nat.zero), R \u29f8 Submodule.span R {p ^ finZeroElim i})\n[PROOFSTEP]\nhaveI : Unique N := \u27e8\u27e80\u27e9, fun x => by dsimp; rw [\u2190 Submodule.mem_bot R, hs]; exact Submodule.mem_top\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd : \u2115\ns\u271d : Fin d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin Nat.zero \u2192 N\nhs : \u22a5 = \u22a4\nx : N\n\u22a2 x = default\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd : \u2115\ns\u271d : Fin d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin Nat.zero \u2192 N\nhs : \u22a5 = \u22a4\nx : N\n\u22a2 x = 0\n[PROOFSTEP]\nrw [\u2190 Submodule.mem_bot R, hs]\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd : \u2115\ns\u271d : Fin d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin Nat.zero \u2192 N\nhs : \u22a5 = \u22a4\nx : N\n\u22a2 x \u2208 \u22a4\n[PROOFSTEP]\nexact Submodule.mem_top\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd : \u2115\ns\u271d : Fin d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin Nat.zero \u2192 N\nhs : \u22a5 = \u22a4\nthis : Unique N\n\u22a2 Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin Nat.zero), R \u29f8 Submodule.span R {p ^ finZeroElim i})\n[PROOFSTEP]\nhaveI : IsEmpty (Fin Nat.zero) := inferInstanceAs (IsEmpty (Fin 0))\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd : \u2115\ns\u271d : Fin d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin Nat.zero \u2192 N\nhs : \u22a5 = \u22a4\nthis\u271d : Unique N\nthis : IsEmpty (Fin Nat.zero)\n\u22a2 Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin Nat.zero), R \u29f8 Submodule.span R {p ^ finZeroElim i})\n[PROOFSTEP]\nexact \u27e80\u27e9\n[GOAL]\ncase h.succ\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\n\u22a2 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin (Nat.succ d)), R \u29f8 Submodule.span R {p ^ k i})\n[PROOFSTEP]\nhave : \u2200 x : N, Decidable (x = 0) := fun _ => by classical infer_instance\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nx\u271d : N\n\u22a2 Decidable (x\u271d = 0)\n[PROOFSTEP]\nclassical infer_instance\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nx\u271d : N\n\u22a2 Decidable (x\u271d = 0)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase h.succ\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\n\u22a2 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin (Nat.succ d)), R \u29f8 Submodule.span R {p ^ k i})\n[PROOFSTEP]\nobtain \u27e8j, hj\u27e9 := exists_isTorsionBy hN d.succ d.succ_ne_zero s hs\n[GOAL]\ncase h.succ.intro\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\n\u22a2 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin (Nat.succ d)), R \u29f8 Submodule.span R {p ^ k i})\n[PROOFSTEP]\nlet s' : Fin d \u2192 N \u29f8 R \u2219 s j := Submodule.Quotient.mk \u2218 s \u2218 j.succAbove\n[GOAL]\ncase h.succ.intro\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\n\u22a2 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin (Nat.succ d)), R \u29f8 Submodule.span R {p ^ k i})\n[PROOFSTEP]\nhave := IH ?_ s' ?_\n[GOAL]\ncase h.succ.intro.refine_3\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis\u271d : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nthis : \u2203 k, Nonempty ((N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\n\u22a2 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin (Nat.succ d)), R \u29f8 Submodule.span R {p ^ k i})\ncase h.succ.intro.refine_1\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\n\u22a2 IsTorsion' (N \u29f8 Submodule.span R {s j}) { x // x \u2208 Submonoid.powers p }\ncase h.succ.intro.refine_2\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\n\u22a2 Submodule.span R (Set.range s') = \u22a4\n[PROOFSTEP]\nobtain \u27e8k, \u27e8f\u27e9\u27e9 := this\n[GOAL]\ncase h.succ.intro.refine_3.intro.intro\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\n\u22a2 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin (Nat.succ d)), R \u29f8 Submodule.span R {p ^ k i})\ncase h.succ.intro.refine_1\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\n\u22a2 IsTorsion' (N \u29f8 Submodule.span R {s j}) { x // x \u2208 Submonoid.powers p }\ncase h.succ.intro.refine_2\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\n\u22a2 Submodule.span R (Set.range s') = \u22a4\n[PROOFSTEP]\nclear IH\n[GOAL]\ncase h.succ.intro.refine_3.intro.intro\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\n\u22a2 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin (Nat.succ d)), R \u29f8 Submodule.span R {p ^ k i})\n[PROOFSTEP]\nhave : \u2200 i : Fin d, \u2203 x : N, p ^ k i \u2022 x = 0 \u2227 f (Submodule.Quotient.mk x) = DirectSum.lof R _ _ i 1 :=\n by\n intro i\n let fi := f.symm.toLinearMap.comp (DirectSum.lof _ _ _ i)\n obtain \u27e8x, h0, h1\u27e9 := exists_smul_eq_zero_and_mk_eq hp hN hj fi; refine' \u27e8x, h0, _\u27e9; rw [h1]\n simp only [LinearMap.coe_comp, f.symm.coe_toLinearMap, f.apply_symm_apply, Function.comp_apply]\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\n\u22a2 \u2200 (i : Fin d),\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\n[PROOFSTEP]\nintro i\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\ni : Fin d\n\u22a2 \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\n[PROOFSTEP]\nlet fi := f.symm.toLinearMap.comp (DirectSum.lof _ _ _ i)\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\ni : Fin d\nfi : R \u29f8 Submodule.span R {p ^ k i} \u2192\u2097[R] N \u29f8 Submodule.span R {s j} :=\n LinearMap.comp (\u2191(LinearEquiv.symm f)) (DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i)\n\u22a2 \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\n[PROOFSTEP]\nobtain \u27e8x, h0, h1\u27e9 := exists_smul_eq_zero_and_mk_eq hp hN hj fi\n[GOAL]\ncase intro.intro\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\ni : Fin d\nfi : R \u29f8 Submodule.span R {p ^ k i} \u2192\u2097[R] N \u29f8 Submodule.span R {s j} :=\n LinearMap.comp (\u2191(LinearEquiv.symm f)) (DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i)\nx : N\nh0 : p ^ k i \u2022 x = 0\nh1 : Submodule.Quotient.mk x = \u2191fi 1\n\u22a2 \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\n[PROOFSTEP]\nrefine' \u27e8x, h0, _\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\ni : Fin d\nfi : R \u29f8 Submodule.span R {p ^ k i} \u2192\u2097[R] N \u29f8 Submodule.span R {s j} :=\n LinearMap.comp (\u2191(LinearEquiv.symm f)) (DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i)\nx : N\nh0 : p ^ k i \u2022 x = 0\nh1 : Submodule.Quotient.mk x = \u2191fi 1\n\u22a2 \u2191f (Submodule.Quotient.mk x) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\n[PROOFSTEP]\nrw [h1]\n[GOAL]\ncase intro.intro\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\ni : Fin d\nfi : R \u29f8 Submodule.span R {p ^ k i} \u2192\u2097[R] N \u29f8 Submodule.span R {s j} :=\n LinearMap.comp (\u2191(LinearEquiv.symm f)) (DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i)\nx : N\nh0 : p ^ k i \u2022 x = 0\nh1 : Submodule.Quotient.mk x = \u2191fi 1\n\u22a2 \u2191f (\u2191fi 1) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\n[PROOFSTEP]\nsimp only [LinearMap.coe_comp, f.symm.coe_toLinearMap, f.apply_symm_apply, Function.comp_apply]\n[GOAL]\ncase h.succ.intro.refine_3.intro.intro\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis\u271d : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\nthis :\n \u2200 (i : Fin d),\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\n\u22a2 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin (Nat.succ d)), R \u29f8 Submodule.span R {p ^ k i})\n[PROOFSTEP]\nrefine \u27e8?_, \u27e8?_\u27e9\u27e9\n[GOAL]\ncase h.succ.intro.refine_3.intro.intro.refine_1\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis\u271d : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\nthis :\n \u2200 (i : Fin d),\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\n\u22a2 Fin (Nat.succ d) \u2192 \u2115\n[PROOFSTEP]\nexact fun a => (fun i => (Option.rec (pOrder hN (s j)) k i : \u2115)) (finSuccEquiv d a)\n[GOAL]\ncase h.succ.intro.refine_3.intro.intro.refine_2\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis\u271d : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\nthis :\n \u2200 (i : Fin d),\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\n\u22a2 N \u2243\u2097[R]\n \u2a01 (i : Fin (Nat.succ d)),\n R \u29f8 Submodule.span R {p ^ (fun i => Option.rec (pOrder hN (s j)) k i) (\u2191(finSuccEquiv d) i)}\n[PROOFSTEP]\nrefine\n (((@lequivProdOfRightSplitExact _ _ _ _ _ _ _ _ _ _ _ _\n ((f.trans ULift.moduleEquiv.{u, u, v}.symm).toLinearMap.comp <| mkQ _)\n ((DirectSum.toModule _ _ _ fun i =>\n (liftQSpanSingleton (p ^ k i) (LinearMap.toSpanSingleton _ _ _) (this i).choose_spec.left :\n R \u29f8 _ \u2192\u2097[R] _)).comp\n ULift.moduleEquiv.toLinearMap)\n (R \u2219 s j).injective_subtype ?_ ?_).symm.trans\n (((quotTorsionOfEquivSpanSingleton R N (s j)).symm.trans\n (quotEquivOfEq (torsionOf R N (s j)) _ (Ideal.torsionOf_eq_span_pow_pOrder hp hN (s j)))).prod\n (ULift.moduleEquiv))).trans\n (@DirectSum.lequivProdDirectSum R _ _ _ (fun i => R \u29f8 R \u2219 p ^ @Option.rec _ (fun _ => \u2115) (pOrder hN <| s j) k i)\n _ _).symm).trans\n (DirectSum.lequivCongrLeft R (finSuccEquiv d).symm)\n[GOAL]\ncase h.succ.intro.refine_3.intro.intro.refine_2.refine_1\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis\u271d : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\nthis :\n \u2200 (i : Fin d),\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\n\u22a2 LinearMap.range (Submodule.subtype (Submodule.span R {s j})) =\n LinearMap.ker\n (LinearMap.comp (\u2191(LinearEquiv.trans f (LinearEquiv.symm ULift.moduleEquiv))) (mkQ (Submodule.span R {s j})))\n[PROOFSTEP]\nrw [range_subtype, LinearEquiv.ker_comp, ker_mkQ]\n[GOAL]\ncase h.succ.intro.refine_3.intro.intro.refine_2.refine_2\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis\u271d : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\nthis :\n \u2200 (i : Fin d),\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\n\u22a2 LinearMap.comp\n (LinearMap.comp (\u2191(LinearEquiv.trans f (LinearEquiv.symm ULift.moduleEquiv))) (mkQ (Submodule.span R {s j})))\n (LinearMap.comp\n (DirectSum.toModule R (Fin d) N fun i =>\n liftQSpanSingleton (p ^ k i)\n (LinearMap.toSpanSingleton R N\n (Exists.choose\n (_ :\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) =\n \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1)))\n (_ :\n p ^ k i \u2022\n Exists.choose\n (_ :\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) =\n \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1) =\n 0))\n \u2191ULift.moduleEquiv) =\n LinearMap.id\n[PROOFSTEP]\nrw [\u2190 f.comp_coe, LinearMap.comp_assoc, LinearMap.comp_assoc, LinearEquiv.toLinearMap_symm_comp_eq, LinearMap.comp_id, \u2190\n LinearMap.comp_assoc, \u2190 LinearMap.comp_assoc]\n[GOAL]\ncase h.succ.intro.refine_3.intro.intro.refine_2.refine_2\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis\u271d : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\nthis :\n \u2200 (i : Fin d),\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\n\u22a2 LinearMap.comp\n (LinearMap.comp (LinearMap.comp (\u2191f) (mkQ (Submodule.span R {s j})))\n (DirectSum.toModule R (Fin d) N fun i =>\n liftQSpanSingleton (p ^ k i)\n (LinearMap.toSpanSingleton R N\n (Exists.choose\n (_ :\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) =\n \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1)))\n (_ :\n p ^ k i \u2022\n Exists.choose\n (_ :\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) =\n \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1) =\n 0)))\n \u2191ULift.moduleEquiv =\n \u2191ULift.moduleEquiv\n[PROOFSTEP]\nsuffices (f.toLinearMap.comp (R \u2219 s j).mkQ).comp _ = LinearMap.id by rw [this, LinearMap.id_comp]\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis\u271d\u00b9 : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\nthis\u271d :\n \u2200 (i : Fin d),\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\nthis : LinearMap.comp (LinearMap.comp (\u2191f) (mkQ (Submodule.span R {s j}))) ?m.464090 = LinearMap.id\n\u22a2 LinearMap.comp\n (LinearMap.comp (LinearMap.comp (\u2191f) (mkQ (Submodule.span R {s j})))\n (DirectSum.toModule R (Fin d) N fun i =>\n liftQSpanSingleton (p ^ k i)\n (LinearMap.toSpanSingleton R N\n (Exists.choose\n (_ :\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) =\n \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1)))\n (_ :\n p ^ k i \u2022\n Exists.choose\n (_ :\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) =\n \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1) =\n 0)))\n \u2191ULift.moduleEquiv =\n \u2191ULift.moduleEquiv\n[PROOFSTEP]\nrw [this, LinearMap.id_comp]\n[GOAL]\ncase h.succ.intro.refine_3.intro.intro.refine_2.refine_2\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis\u271d : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\nthis :\n \u2200 (i : Fin d),\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\n\u22a2 LinearMap.comp (LinearMap.comp (\u2191f) (mkQ (Submodule.span R {s j})))\n (DirectSum.toModule R (Fin d) N fun i =>\n liftQSpanSingleton (p ^ k i)\n (LinearMap.toSpanSingleton R N\n (Exists.choose\n (_ :\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) =\n \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1)))\n (_ :\n p ^ k i \u2022\n Exists.choose\n (_ :\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) =\n \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1) =\n 0)) =\n LinearMap.id\n[PROOFSTEP]\next i : 3\n[GOAL]\ncase h.succ.intro.refine_3.intro.intro.refine_2.refine_2.H.h.h\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis\u271d : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\nthis :\n \u2200 (i : Fin d),\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\ni : Fin d\n\u22a2 \u2191(LinearMap.comp\n (LinearMap.comp\n (LinearMap.comp (LinearMap.comp (\u2191f) (mkQ (Submodule.span R {s j})))\n (DirectSum.toModule R (Fin d) N fun i =>\n liftQSpanSingleton (p ^ k i)\n (LinearMap.toSpanSingleton R N\n (Exists.choose\n (_ :\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) =\n \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1)))\n (_ :\n p ^ k i \u2022\n Exists.choose\n (_ :\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) =\n \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1) =\n 0)))\n (DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i))\n (mkQ (Submodule.span R {p ^ k i})))\n 1 =\n \u2191(LinearMap.comp (LinearMap.comp LinearMap.id (DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i))\n (mkQ (Submodule.span R {p ^ k i})))\n 1\n[PROOFSTEP]\nsimp only [LinearMap.coe_comp, Function.comp_apply, mkQ_apply]\n[GOAL]\ncase h.succ.intro.refine_3.intro.intro.refine_2.refine_2.H.h.h\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis\u271d : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nk : Fin d \u2192 \u2115\nf : (N \u29f8 Submodule.span R {s j}) \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i}\nthis :\n \u2200 (i : Fin d),\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) = \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1\ni : Fin d\n\u22a2 \u2191\u2191f\n (Submodule.Quotient.mk\n (\u2191(DirectSum.toModule R (Fin d) N fun i =>\n liftQSpanSingleton (p ^ k i)\n (LinearMap.toSpanSingleton R N\n (Exists.choose\n (_ :\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) =\n \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1)))\n (_ :\n p ^ k i \u2022\n Exists.choose\n (_ :\n \u2203 x,\n p ^ k i \u2022 x = 0 \u2227\n \u2191f (Submodule.Quotient.mk x) =\n \u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) 1) =\n 0))\n (\u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) (Submodule.Quotient.mk 1)))) =\n \u2191LinearMap.id (\u2191(DirectSum.lof R (Fin d) (fun i => R \u29f8 Submodule.span R {p ^ k i}) i) (Submodule.Quotient.mk 1))\n[PROOFSTEP]\nrw [LinearEquiv.coe_toLinearMap, LinearMap.id_apply, DirectSum.toModule_lof, liftQSpanSingleton_apply,\n LinearMap.toSpanSingleton_one, Ideal.Quotient.mk_eq_mk, map_one, (this i).choose_spec.right]\n[GOAL]\ncase h.succ.intro.refine_1\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\n\u22a2 IsTorsion' (N \u29f8 Submodule.span R {s j}) { x // x \u2208 Submonoid.powers p }\n[PROOFSTEP]\nexact\n (mk_surjective _).forall.mpr fun x =>\n \u27e8(@hN x).choose, by rw [\u2190 Quotient.mk_smul, (@hN x).choose_spec, Quotient.mk_zero]\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nx : N\n\u22a2 Exists.choose (_ : \u2203 a, a \u2022 x = 0) \u2022 Submodule.Quotient.mk x = 0\n[PROOFSTEP]\nrw [\u2190 Quotient.mk_smul, (@hN x).choose_spec, Quotient.mk_zero]\n[GOAL]\ncase h.succ.intro.refine_2\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\n\u22a2 Submodule.span R (Set.range s') = \u22a4\n[PROOFSTEP]\nhave hs' := congr_arg (Submodule.map <| mkQ <| R \u2219 s j) hs\n[GOAL]\ncase h.succ.intro.refine_2\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nhs' :\n Submodule.map (mkQ (Submodule.span R {s j})) (Submodule.span R (Set.range s)) =\n Submodule.map (mkQ (Submodule.span R {s j})) \u22a4\n\u22a2 Submodule.span R (Set.range s') = \u22a4\n[PROOFSTEP]\nrw [Submodule.map_span, Submodule.map_top, range_mkQ] at hs' \n[GOAL]\ncase h.succ.intro.refine_2\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nhs' : Submodule.span R (\u2191(mkQ (Submodule.span R {s j})) '' Set.range s) = \u22a4\n\u22a2 Submodule.span R (Set.range s') = \u22a4\n[PROOFSTEP]\nsimp only [mkQ_apply] at hs' \n[GOAL]\ncase h.succ.intro.refine_2\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nhs' : Submodule.span R ((fun a => Submodule.Quotient.mk a) '' Set.range s) = \u22a4\n\u22a2 Submodule.span R (Set.range s') = \u22a4\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase h.succ.intro.refine_2\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nhs' : Submodule.span R ((fun a => Submodule.Quotient.mk a) '' Set.range s) = \u22a4\n\u22a2 Submodule.span R (Set.range (Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j)) = \u22a4\n[PROOFSTEP]\nrw [\u2190 Function.comp.assoc, Set.range_comp (_ \u2218 s), Fin.range_succAbove]\n[GOAL]\ncase h.succ.intro.refine_2\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nhs' : Submodule.span R ((fun a => Submodule.Quotient.mk a) '' Set.range s) = \u22a4\n\u22a2 Submodule.span R (Submodule.Quotient.mk \u2218 s '' {j}\u1d9c) = \u22a4\n[PROOFSTEP]\nrw [\u2190 Set.range_comp, \u2190 Set.insert_image_compl_eq_range _ j, Function.comp_apply,\n (Quotient.mk_eq_zero _).mpr (Submodule.mem_span_singleton_self _), span_insert_zero] at hs' \n[GOAL]\ncase h.succ.intro.refine_2\nR : Type u\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nN\u271d : Type (max u v)\ninst\u271d\u00b3 : AddCommGroup N\u271d\ninst\u271d\u00b2 : Module R N\u271d\np : R\nhp : Irreducible p\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\ndec : (x : M) \u2192 Decidable (x = 0)\nhN\u271d : IsTorsion' N\u271d { x // x \u2208 Submonoid.powers p }\nd\u271d : \u2115\ns\u271d : Fin d\u271d \u2192 N\u271d\nhs\u271d : Submodule.span R (Set.range s\u271d) = \u22a4\nd : \u2115\nIH :\n \u2200 {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N],\n IsTorsion' N { x // x \u2208 Submonoid.powers p } \u2192\n \u2200 (s : Fin d \u2192 N),\n Submodule.span R (Set.range s) = \u22a4 \u2192 \u2203 k, Nonempty (N \u2243\u2097[R] \u2a01 (i : Fin d), R \u29f8 Submodule.span R {p ^ k i})\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nhN : IsTorsion' N { x // x \u2208 Submonoid.powers p }\ns : Fin (Nat.succ d) \u2192 N\nhs : Submodule.span R (Set.range s) = \u22a4\nthis : (x : N) \u2192 Decidable (x = 0)\nj : Fin (Nat.succ d)\nhj : IsTorsionBy R N (p ^ pOrder hN (s j))\ns' : Fin d \u2192 N \u29f8 Submodule.span R {s j} := Submodule.Quotient.mk \u2218 s \u2218 Fin.succAbove j\nhs' : Submodule.span R ((fun a => Submodule.Quotient.mk a) \u2218 s '' {j}\u1d9c) = \u22a4\n\u22a2 Submodule.span R (Submodule.Quotient.mk \u2218 s '' {j}\u1d9c) = \u22a4\n[PROOFSTEP]\nexact hs'\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nhN : IsTorsion R N\n\u22a2 \u2203 \u03b9 x p x e, Nonempty (N \u2243\u2097[R] \u2a01 (i : \u03b9), R \u29f8 span R {p i ^ e i})\n[PROOFSTEP]\nobtain \u27e8I, fI, _, p, hp, e, h\u27e9 := Submodule.exists_isInternal_prime_power_torsion_of_pid hN\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nhN : IsTorsion R N\nI : Type u\nfI : Fintype I\nw\u271d : DecidableEq I\np : I \u2192 R\nhp : \u2200 (i : I), Irreducible (p i)\ne : I \u2192 \u2115\nh : DirectSum.IsInternal fun i => torsionBy R N (p i ^ e i)\n\u22a2 \u2203 \u03b9 x p x e, Nonempty (N \u2243\u2097[R] \u2a01 (i : \u03b9), R \u29f8 span R {p i ^ e i})\n[PROOFSTEP]\nhaveI := fI\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nhN : IsTorsion R N\nI : Type u\nfI : Fintype I\nw\u271d : DecidableEq I\np : I \u2192 R\nhp : \u2200 (i : I), Irreducible (p i)\ne : I \u2192 \u2115\nh : DirectSum.IsInternal fun i => torsionBy R N (p i ^ e i)\nthis : Fintype I\n\u22a2 \u2203 \u03b9 x p x e, Nonempty (N \u2243\u2097[R] \u2a01 (i : \u03b9), R \u29f8 span R {p i ^ e i})\n[PROOFSTEP]\nhave : \u2200 i, \u2203 (d : \u2115) (k : Fin d \u2192 \u2115), Nonempty <| torsionBy R N (p i ^ e i) \u2243\u2097[R] \u2a01 j, R \u29f8 R \u2219 p i ^ k j :=\n by\n have := isNoetherian_of_isNoetherianRing_of_finite R N\n haveI := fun i => isNoetherian_submodule' (torsionBy R N <| p i ^ e i)\n exact fun i =>\n torsion_by_prime_power_decomposition.{u, v} (hp i)\n ((isTorsion'_powers_iff <| p i).mpr fun x => \u27e8e i, smul_torsionBy _ _\u27e9)\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nhN : IsTorsion R N\nI : Type u\nfI : Fintype I\nw\u271d : DecidableEq I\np : I \u2192 R\nhp : \u2200 (i : I), Irreducible (p i)\ne : I \u2192 \u2115\nh : DirectSum.IsInternal fun i => torsionBy R N (p i ^ e i)\nthis : Fintype I\n\u22a2 \u2200 (i : I), \u2203 d k, Nonempty ({ x // x \u2208 torsionBy R N (p i ^ e i) } \u2243\u2097[R] \u2a01 (j : Fin d), R \u29f8 span R {p i ^ k j})\n[PROOFSTEP]\nhave := isNoetherian_of_isNoetherianRing_of_finite R N\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nhN : IsTorsion R N\nI : Type u\nfI : Fintype I\nw\u271d : DecidableEq I\np : I \u2192 R\nhp : \u2200 (i : I), Irreducible (p i)\ne : I \u2192 \u2115\nh : DirectSum.IsInternal fun i => torsionBy R N (p i ^ e i)\nthis\u271d : Fintype I\nthis : IsNoetherian R N\n\u22a2 \u2200 (i : I), \u2203 d k, Nonempty ({ x // x \u2208 torsionBy R N (p i ^ e i) } \u2243\u2097[R] \u2a01 (j : Fin d), R \u29f8 span R {p i ^ k j})\n[PROOFSTEP]\nhaveI := fun i => isNoetherian_submodule' (torsionBy R N <| p i ^ e i)\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nhN : IsTorsion R N\nI : Type u\nfI : Fintype I\nw\u271d : DecidableEq I\np : I \u2192 R\nhp : \u2200 (i : I), Irreducible (p i)\ne : I \u2192 \u2115\nh : DirectSum.IsInternal fun i => torsionBy R N (p i ^ e i)\nthis\u271d\u00b9 : Fintype I\nthis\u271d : IsNoetherian R N\nthis : \u2200 (i : I), IsNoetherian R { x // x \u2208 torsionBy R N (p i ^ e i) }\n\u22a2 \u2200 (i : I), \u2203 d k, Nonempty ({ x // x \u2208 torsionBy R N (p i ^ e i) } \u2243\u2097[R] \u2a01 (j : Fin d), R \u29f8 span R {p i ^ k j})\n[PROOFSTEP]\nexact fun i =>\n torsion_by_prime_power_decomposition.{u, v} (hp i)\n ((isTorsion'_powers_iff <| p i).mpr fun x => \u27e8e i, smul_torsionBy _ _\u27e9)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nhN : IsTorsion R N\nI : Type u\nfI : Fintype I\nw\u271d : DecidableEq I\np : I \u2192 R\nhp : \u2200 (i : I), Irreducible (p i)\ne : I \u2192 \u2115\nh : DirectSum.IsInternal fun i => torsionBy R N (p i ^ e i)\nthis\u271d : Fintype I\nthis : \u2200 (i : I), \u2203 d k, Nonempty ({ x // x \u2208 torsionBy R N (p i ^ e i) } \u2243\u2097[R] \u2a01 (j : Fin d), R \u29f8 span R {p i ^ k j})\n\u22a2 \u2203 \u03b9 x p x e, Nonempty (N \u2243\u2097[R] \u2a01 (i : \u03b9), R \u29f8 span R {p i ^ e i})\n[PROOFSTEP]\nclassical\nrefine'\n \u27e8\u03a3 i, Fin (this i).choose, inferInstance, fun \u27e8i, _\u27e9 => p i, fun \u27e8i, _\u27e9 => hp i, fun \u27e8i, j\u27e9 =>\n (this i).choose_spec.choose j,\n \u27e8(LinearEquiv.ofBijective (DirectSum.coeLinearMap _) h).symm.trans <|\n (DFinsupp.mapRange.linearEquiv fun i => (this i).choose_spec.choose_spec.some).trans <|\n (DirectSum.sigmaLcurryEquiv R).symm.trans (DFinsupp.mapRange.linearEquiv fun i => quotEquivOfEq _ _ _)\u27e9\u27e9\ncases' i with i j\nsimp only\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nhN : IsTorsion R N\nI : Type u\nfI : Fintype I\nw\u271d : DecidableEq I\np : I \u2192 R\nhp : \u2200 (i : I), Irreducible (p i)\ne : I \u2192 \u2115\nh : DirectSum.IsInternal fun i => torsionBy R N (p i ^ e i)\nthis\u271d : Fintype I\nthis : \u2200 (i : I), \u2203 d k, Nonempty ({ x // x \u2208 torsionBy R N (p i ^ e i) } \u2243\u2097[R] \u2a01 (j : Fin d), R \u29f8 span R {p i ^ k j})\n\u22a2 \u2203 \u03b9 x p x e, Nonempty (N \u2243\u2097[R] \u2a01 (i : \u03b9), R \u29f8 span R {p i ^ e i})\n[PROOFSTEP]\nrefine'\n \u27e8\u03a3 i, Fin (this i).choose, inferInstance, fun \u27e8i, _\u27e9 => p i, fun \u27e8i, _\u27e9 => hp i, fun \u27e8i, j\u27e9 =>\n (this i).choose_spec.choose j,\n \u27e8(LinearEquiv.ofBijective (DirectSum.coeLinearMap _) h).symm.trans <|\n (DFinsupp.mapRange.linearEquiv fun i => (this i).choose_spec.choose_spec.some).trans <|\n (DirectSum.sigmaLcurryEquiv R).symm.trans (DFinsupp.mapRange.linearEquiv fun i => quotEquivOfEq _ _ _)\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nhN : IsTorsion R N\nI : Type u\nfI : Fintype I\nw\u271d : DecidableEq I\np : I \u2192 R\nhp : \u2200 (i : I), Irreducible (p i)\ne : I \u2192 \u2115\nh : DirectSum.IsInternal fun i => torsionBy R N (p i ^ e i)\nthis\u271d : Fintype I\nthis : \u2200 (i : I), \u2203 d k, Nonempty ({ x // x \u2208 torsionBy R N (p i ^ e i) } \u2243\u2097[R] \u2a01 (j : Fin d), R \u29f8 span R {p i ^ k j})\ni :\n (x : I) \u00d7\n Fin\n (Exists.choose\n (_ : \u2203 d k, Nonempty ({ x_1 // x_1 \u2208 torsionBy R N (p x ^ e x) } \u2243\u2097[R] \u2a01 (j : Fin d), R \u29f8 span R {p x ^ k j})))\n\u22a2 span R\n {p i.fst ^\n Exists.choose\n (_ :\n \u2203 k,\n Nonempty\n ({ x // x \u2208 torsionBy R N (p i.fst ^ e i.fst) } \u2243\u2097[R]\n \u2a01 (j :\n Fin\n (Exists.choose\n (_ :\n \u2203 d k,\n Nonempty\n ({ x // x \u2208 torsionBy R N (p i.fst ^ e i.fst) } \u2243\u2097[R]\n \u2a01 (j : Fin d), R \u29f8 span R {p i.fst ^ k j})))),\n R \u29f8 span R {p i.fst ^ k j}))\n i.snd} =\n span R\n {(fun x =>\n match x with\n | { fst := i, snd := snd } => p i)\n i ^\n (fun x =>\n match x with\n | { fst := i, snd := j } =>\n Exists.choose\n (_ :\n \u2203 k,\n Nonempty\n ({ x // x \u2208 torsionBy R N (p i ^ e i) } \u2243\u2097[R]\n \u2a01 (j :\n Fin\n (Exists.choose\n (_ :\n \u2203 d k,\n Nonempty\n ({ x // x \u2208 torsionBy R N (p i ^ e i) } \u2243\u2097[R]\n \u2a01 (j : Fin d), R \u29f8 span R {p i ^ k j})))),\n R \u29f8 span R {p i ^ k j}))\n j)\n i}\n[PROOFSTEP]\ncases' i with i j\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nhN : IsTorsion R N\nI : Type u\nfI : Fintype I\nw\u271d : DecidableEq I\np : I \u2192 R\nhp : \u2200 (i : I), Irreducible (p i)\ne : I \u2192 \u2115\nh : DirectSum.IsInternal fun i => torsionBy R N (p i ^ e i)\nthis\u271d : Fintype I\nthis : \u2200 (i : I), \u2203 d k, Nonempty ({ x // x \u2208 torsionBy R N (p i ^ e i) } \u2243\u2097[R] \u2a01 (j : Fin d), R \u29f8 span R {p i ^ k j})\ni : I\nj :\n Fin\n (Exists.choose\n (_ : \u2203 d k, Nonempty ({ x // x \u2208 torsionBy R N (p i ^ e i) } \u2243\u2097[R] \u2a01 (j : Fin d), R \u29f8 span R {p i ^ k j})))\n\u22a2 span R\n {p { fst := i, snd := j }.fst ^\n Exists.choose\n (_ :\n \u2203 k,\n Nonempty\n ({ x // x \u2208 torsionBy R N (p { fst := i, snd := j }.fst ^ e { fst := i, snd := j }.fst) } \u2243\u2097[R]\n \u2a01 (j_1 :\n Fin\n (Exists.choose\n (_ :\n \u2203 d k,\n Nonempty\n ({ x //\n x \u2208\n torsionBy R N\n (p { fst := i, snd := j }.fst ^ e { fst := i, snd := j }.fst) } \u2243\u2097[R]\n \u2a01 (j_1 : Fin d), R \u29f8 span R {p { fst := i, snd := j }.fst ^ k j_1})))),\n R \u29f8 span R {p { fst := i, snd := j }.fst ^ k j_1}))\n { fst := i, snd := j }.snd} =\n span R\n {(fun x =>\n match x with\n | { fst := i, snd := snd } => p i)\n { fst := i, snd := j } ^\n (fun x =>\n match x with\n | { fst := i, snd := j } =>\n Exists.choose\n (_ :\n \u2203 k,\n Nonempty\n ({ x // x \u2208 torsionBy R N (p i ^ e i) } \u2243\u2097[R]\n \u2a01 (j :\n Fin\n (Exists.choose\n (_ :\n \u2203 d k,\n Nonempty\n ({ x // x \u2208 torsionBy R N (p i ^ e i) } \u2243\u2097[R]\n \u2a01 (j : Fin d), R \u29f8 span R {p i ^ k j})))),\n R \u29f8 span R {p i ^ k j}))\n j)\n { fst := i, snd := j }}\n[PROOFSTEP]\nsimp only\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\n\u22a2 \u2203 n \u03b9 x p x e, Nonempty (N \u2243\u2097[R] (Fin n \u2192\u2080 R) \u00d7 \u2a01 (i : \u03b9), R \u29f8 span R {p i ^ e i})\n[PROOFSTEP]\nhave := isNoetherian_of_isNoetherianRing_of_finite R N\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nthis : IsNoetherian R N\n\u22a2 \u2203 n \u03b9 x p x e, Nonempty (N \u2243\u2097[R] (Fin n \u2192\u2080 R) \u00d7 \u2a01 (i : \u03b9), R \u29f8 span R {p i ^ e i})\n[PROOFSTEP]\nhaveI := isNoetherian_submodule' (torsion R N)\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nthis\u271d : IsNoetherian R N\nthis : IsNoetherian R { x // x \u2208 torsion R N }\n\u22a2 \u2203 n \u03b9 x p x e, Nonempty (N \u2243\u2097[R] (Fin n \u2192\u2080 R) \u00d7 \u2a01 (i : \u03b9), R \u29f8 span R {p i ^ e i})\n[PROOFSTEP]\nhaveI := Module.Finite.of_surjective _ (torsion R N).mkQ_surjective\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nthis\u271d\u00b9 : IsNoetherian R N\nthis\u271d : IsNoetherian R { x // x \u2208 torsion R N }\nthis : Finite R (N \u29f8 torsion R N)\n\u22a2 \u2203 n \u03b9 x p x e, Nonempty (N \u2243\u2097[R] (Fin n \u2192\u2080 R) \u00d7 \u2a01 (i : \u03b9), R \u29f8 span R {p i ^ e i})\n[PROOFSTEP]\nobtain \u27e8I, fI, p, hp, e, \u27e8h\u27e9\u27e9 := equiv_directSum_of_isTorsion.{u, v} (@torsion_isTorsion R N _ _ _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nthis\u271d\u00b9 : IsNoetherian R N\nthis\u271d : IsNoetherian R { x // x \u2208 torsion R N }\nthis : Finite R (N \u29f8 torsion R N)\nI : Type u\nfI : Fintype I\np : I \u2192 R\nhp : \u2200 (i : I), Irreducible (p i)\ne : I \u2192 \u2115\nh : { x // x \u2208 torsion R N } \u2243\u2097[R] \u2a01 (i : I), R \u29f8 span R {p i ^ e i}\n\u22a2 \u2203 n \u03b9 x p x e, Nonempty (N \u2243\u2097[R] (Fin n \u2192\u2080 R) \u00d7 \u2a01 (i : \u03b9), R \u29f8 span R {p i ^ e i})\n[PROOFSTEP]\nobtain \u27e8n, \u27e8g\u27e9\u27e9 := @Module.basisOfFiniteTypeTorsionFree' R _ _ _ (N \u29f8 torsion R N) _ _ _ _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk.ofRepr\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nthis\u271d\u00b9 : IsNoetherian R N\nthis\u271d : IsNoetherian R { x // x \u2208 torsion R N }\nthis : Finite R (N \u29f8 torsion R N)\nI : Type u\nfI : Fintype I\np : I \u2192 R\nhp : \u2200 (i : I), Irreducible (p i)\ne : I \u2192 \u2115\nh : { x // x \u2208 torsion R N } \u2243\u2097[R] \u2a01 (i : I), R \u29f8 span R {p i ^ e i}\nn : \u2115\ng : (N \u29f8 torsion R N) \u2243\u2097[R] Fin n \u2192\u2080 R\n\u22a2 \u2203 n \u03b9 x p x e, Nonempty (N \u2243\u2097[R] (Fin n \u2192\u2080 R) \u00d7 \u2a01 (i : \u03b9), R \u29f8 span R {p i ^ e i})\n[PROOFSTEP]\nhaveI : Module.Projective R (N \u29f8 torsion R N) := Module.Projective.of_basis \u27e8g\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk.ofRepr\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nthis\u271d\u00b2 : IsNoetherian R N\nthis\u271d\u00b9 : IsNoetherian R { x // x \u2208 torsion R N }\nthis\u271d : Finite R (N \u29f8 torsion R N)\nI : Type u\nfI : Fintype I\np : I \u2192 R\nhp : \u2200 (i : I), Irreducible (p i)\ne : I \u2192 \u2115\nh : { x // x \u2208 torsion R N } \u2243\u2097[R] \u2a01 (i : I), R \u29f8 span R {p i ^ e i}\nn : \u2115\ng : (N \u29f8 torsion R N) \u2243\u2097[R] Fin n \u2192\u2080 R\nthis : Projective R (N \u29f8 torsion R N)\n\u22a2 \u2203 n \u03b9 x p x e, Nonempty (N \u2243\u2097[R] (Fin n \u2192\u2080 R) \u00d7 \u2a01 (i : \u03b9), R \u29f8 span R {p i ^ e i})\n[PROOFSTEP]\nobtain \u27e8f, hf\u27e9 := Module.projective_lifting_property _ LinearMap.id (torsion R N).mkQ_surjective\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk.ofRepr.intro\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nthis\u271d\u00b2 : IsNoetherian R N\nthis\u271d\u00b9 : IsNoetherian R { x // x \u2208 torsion R N }\nthis\u271d : Finite R (N \u29f8 torsion R N)\nI : Type u\nfI : Fintype I\np : I \u2192 R\nhp : \u2200 (i : I), Irreducible (p i)\ne : I \u2192 \u2115\nh : { x // x \u2208 torsion R N } \u2243\u2097[R] \u2a01 (i : I), R \u29f8 span R {p i ^ e i}\nn : \u2115\ng : (N \u29f8 torsion R N) \u2243\u2097[R] Fin n \u2192\u2080 R\nthis : Projective R (N \u29f8 torsion R N)\nf : N \u29f8 torsion R N \u2192\u2097[R] N\nhf : LinearMap.comp (mkQ (torsion R N)) f = LinearMap.id\n\u22a2 \u2203 n \u03b9 x p x e, Nonempty (N \u2243\u2097[R] (Fin n \u2192\u2080 R) \u00d7 \u2a01 (i : \u03b9), R \u29f8 span R {p i ^ e i})\n[PROOFSTEP]\nrefine'\n \u27e8n, I, fI, p, hp, e,\n \u27e8(lequivProdOfRightSplitExact (torsion R N).injective_subtype _ hf).symm.trans <|\n (h.prod g).trans <| LinearEquiv.prodComm.{u, u} R _ (Fin n \u2192\u2080 R)\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk.ofRepr.intro\nR : Type u\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nN : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh' : Finite R N\nthis\u271d\u00b2 : IsNoetherian R N\nthis\u271d\u00b9 : IsNoetherian R { x // x \u2208 torsion R N }\nthis\u271d : Finite R (N \u29f8 torsion R N)\nI : Type u\nfI : Fintype I\np : I \u2192 R\nhp : \u2200 (i : I), Irreducible (p i)\ne : I \u2192 \u2115\nh : { x // x \u2208 torsion R N } \u2243\u2097[R] \u2a01 (i : I), R \u29f8 span R {p i ^ e i}\nn : \u2115\ng : (N \u29f8 torsion R N) \u2243\u2097[R] Fin n \u2192\u2080 R\nthis : Projective R (N \u29f8 torsion R N)\nf : N \u29f8 torsion R N \u2192\u2097[R] N\nhf : LinearMap.comp (mkQ (torsion R N)) f = LinearMap.id\n\u22a2 LinearMap.range (Submodule.subtype (torsion R N)) = LinearMap.ker (mkQ (torsion R N))\n[PROOFSTEP]\nrw [range_subtype, ker_mkQ]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Module.PID", "llama_tokens": 59029, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585786300049, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.42218294083815516}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\ninit : Fin E.order \u2192 \u03b1\nx\u271d : \u2115\nn : \u2115 := x\u271d\nh : \u00acn < E.order\nk : Fin E.order\n\u22a2 n - E.order + \u2191k < n\n[PROOFSTEP]\nrw [add_comm, \u2190 add_tsub_assoc_of_le (not_lt.mp h), tsub_lt_iff_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\ninit : Fin E.order \u2192 \u03b1\nx\u271d : \u2115\nn : \u2115 := x\u271d\nh : \u00acn < E.order\nk : Fin E.order\n\u22a2 \u2191k + n < E.order + n\n[PROOFSTEP]\nexact add_lt_add_right k.is_lt n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\ninit : Fin E.order \u2192 \u03b1\nx\u271d : \u2115\nn : \u2115 := x\u271d\nh : \u00acn < E.order\nk : Fin E.order\n\u22a2 E.order \u2264 \u2191k + n\n[PROOFSTEP]\nconvert add_le_add (zero_le (k : \u2115)) (not_lt.mp h)\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\ninit : Fin E.order \u2192 \u03b1\nx\u271d : \u2115\nn : \u2115 := x\u271d\nh : \u00acn < E.order\nk : Fin E.order\n\u22a2 E.order = 0 + E.order\n[PROOFSTEP]\nsimp only [zero_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\ninit : Fin E.order \u2192 \u03b1\n\u22a2 IsSolution E (mkSol E init)\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\ninit : Fin E.order \u2192 \u03b1\nn : \u2115\n\u22a2 mkSol E init (n + E.order) = \u2211 i : Fin E.order, coeffs E i * mkSol E init (n + \u2191i)\n[PROOFSTEP]\nrw [mkSol]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\ninit : Fin E.order \u2192 \u03b1\nn : \u2115\n\u22a2 (if h : n + E.order < E.order then init { val := n + E.order, isLt := h }\n else\n \u2211 k : Fin E.order,\n let_fun x := (_ : n + E.order - E.order + \u2191k < n + E.order);\n coeffs E k * mkSol E init (n + E.order - E.order + \u2191k)) =\n \u2211 i : Fin E.order, coeffs E i * mkSol E init (n + \u2191i)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\ninit : Fin E.order \u2192 \u03b1\n\u22a2 \u2200 (n : Fin E.order), mkSol E init \u2191n = init n\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\ninit : Fin E.order \u2192 \u03b1\nn : Fin E.order\n\u22a2 mkSol E init \u2191n = init n\n[PROOFSTEP]\nrw [mkSol]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\ninit : Fin E.order \u2192 \u03b1\nn : Fin E.order\n\u22a2 (if h : \u2191n < E.order then init { val := \u2191n, isLt := h }\n else\n \u2211 k : Fin E.order,\n let_fun x := (_ : \u2191n - E.order + \u2191k < \u2191n);\n coeffs E k * mkSol E init (\u2191n - E.order + \u2191k)) =\n init n\n[PROOFSTEP]\nsimp only [n.is_lt, dif_pos, Fin.mk_val, Fin.eta]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\n\u22a2 \u2200 (n : \u2115), u n = mkSol E init n\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\n\u22a2 u n = mkSol E init n\n[PROOFSTEP]\nrw [mkSol]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\n\u22a2 u n =\n if h : n < E.order then init { val := n, isLt := h }\n else\n \u2211 k : Fin E.order,\n let_fun x := (_ : n - E.order + \u2191k < n);\n coeffs E k * mkSol E init (n - E.order + \u2191k)\n[PROOFSTEP]\nsplit_ifs with h'\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : n < E.order\n\u22a2 u n = init { val := n, isLt := h' }\n[PROOFSTEP]\nexact_mod_cast heq \u27e8n, h'\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\n\u22a2 u n =\n \u2211 k : Fin E.order,\n let_fun x := (_ : n - E.order + \u2191k < n);\n coeffs E k * mkSol E init (n - E.order + \u2191k)\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\n\u22a2 u n = \u2211 x : Fin E.order, coeffs E x * mkSol E init (n - E.order + \u2191x)\n[PROOFSTEP]\nrw [\u2190 tsub_add_cancel_of_le (le_of_not_lt h'), h (n - E.order)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\n\u22a2 \u2211 i : Fin E.order, coeffs E i * u (n - E.order + \u2191i) =\n \u2211 x : Fin E.order, coeffs E x * mkSol E init (n - E.order + E.order - E.order + \u2191x)\n[PROOFSTEP]\ncongr with k\n[GOAL]\ncase neg.e_f.h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\n\u22a2 coeffs E k * u (n - E.order + \u2191k) = coeffs E k * mkSol E init (n - E.order + E.order - E.order + \u2191k)\n[PROOFSTEP]\nhave : n - E.order + k < n :=\n by\n rw [add_comm, \u2190 add_tsub_assoc_of_le (not_lt.mp h'), tsub_lt_iff_left]\n \u00b7 exact add_lt_add_right k.is_lt n\n \u00b7 convert add_le_add (zero_le (k : \u2115)) (not_lt.mp h')\n simp only [zero_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\n\u22a2 n - E.order + \u2191k < n\n[PROOFSTEP]\nrw [add_comm, \u2190 add_tsub_assoc_of_le (not_lt.mp h'), tsub_lt_iff_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\n\u22a2 \u2191k + n < E.order + n\n[PROOFSTEP]\nexact add_lt_add_right k.is_lt n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\n\u22a2 E.order \u2264 \u2191k + n\n[PROOFSTEP]\nconvert add_le_add (zero_le (k : \u2115)) (not_lt.mp h')\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\n\u22a2 E.order = 0 + E.order\n[PROOFSTEP]\nsimp only [zero_add]\n[GOAL]\ncase neg.e_f.h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\nthis : n - E.order + \u2191k < n\n\u22a2 coeffs E k * u (n - E.order + \u2191k) = coeffs E k * mkSol E init (n - E.order + E.order - E.order + \u2191k)\n[PROOFSTEP]\nrw [eq_mk_of_is_sol_of_eq_init h heq (n - E.order + k)]\n[GOAL]\ncase neg.e_f.h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\nthis : n - E.order + \u2191k < n\n\u22a2 coeffs E k * mkSol E (fun n => init n) (n - E.order + \u2191k) =\n coeffs E k * mkSol E init (n - E.order + E.order - E.order + \u2191k)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu v : \u2115 \u2192 \u03b1\nhu : u \u2208 {u | IsSolution E u}\nhv : v \u2208 {u | IsSolution E u}\nn : \u2115\n\u22a2 (u + v) (n + E.order) = \u2211 i : Fin E.order, coeffs E i * (u + v) (n + \u2191i)\n[PROOFSTEP]\nsimp [mul_add, sum_add_distrib, hu n, hv n]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nn : \u2115\n\u22a2 OfNat.ofNat 0 (n + E.order) = \u2211 i : Fin E.order, coeffs E i * OfNat.ofNat 0 (n + \u2191i)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\na : \u03b1\nu : \u2115 \u2192 \u03b1\nhu :\n u \u2208\n {\n toAddSubsemigroup :=\n { carrier := {u | IsSolution E u},\n add_mem' :=\n (_ :\n \u2200 {u v : \u2115 \u2192 \u03b1},\n u \u2208 {u | IsSolution E u} \u2192\n v \u2208 {u | IsSolution E u} \u2192\n \u2200 (n : \u2115),\n u (n + E.order) + v (n + E.order) = \u2211 i : Fin E.order, coeffs E i * (u + v) (n + \u2191i)) },\n zero_mem' :=\n (_ : \u2200 (n : \u2115), 0 = \u2211 i : Fin E.order, coeffs E i * OfNat.ofNat 0 (n + \u2191i)) }.toAddSubsemigroup.carrier\nn : \u2115\n\u22a2 (a \u2022 u) (n + E.order) = \u2211 i : Fin E.order, coeffs E i * (a \u2022 u) (n + \u2191i)\n[PROOFSTEP]\nsimp [hu n, mul_sum]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\na : \u03b1\nu : \u2115 \u2192 \u03b1\nhu :\n u \u2208\n {\n toAddSubsemigroup :=\n { carrier := {u | IsSolution E u},\n add_mem' :=\n (_ :\n \u2200 {u v : \u2115 \u2192 \u03b1},\n u \u2208 {u | IsSolution E u} \u2192\n v \u2208 {u | IsSolution E u} \u2192\n \u2200 (n : \u2115),\n u (n + E.order) + v (n + E.order) = \u2211 i : Fin E.order, coeffs E i * (u + v) (n + \u2191i)) },\n zero_mem' :=\n (_ : \u2200 (n : \u2115), 0 = \u2211 i : Fin E.order, coeffs E i * OfNat.ofNat 0 (n + \u2191i)) }.toAddSubsemigroup.carrier\nn : \u2115\n\u22a2 \u2211 x : Fin E.order, a * (coeffs E x * u (n + \u2191x)) = \u2211 x : Fin E.order, coeffs E x * (a * u (n + \u2191x))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\na : \u03b1\nu : \u2115 \u2192 \u03b1\nhu :\n u \u2208\n {\n toAddSubsemigroup :=\n { carrier := {u | IsSolution E u},\n add_mem' :=\n (_ :\n \u2200 {u v : \u2115 \u2192 \u03b1},\n u \u2208 {u | IsSolution E u} \u2192\n v \u2208 {u | IsSolution E u} \u2192\n \u2200 (n : \u2115),\n u (n + E.order) + v (n + E.order) = \u2211 i : Fin E.order, coeffs E i * (u + v) (n + \u2191i)) },\n zero_mem' :=\n (_ : \u2200 (n : \u2115), 0 = \u2211 i : Fin E.order, coeffs E i * OfNat.ofNat 0 (n + \u2191i)) }.toAddSubsemigroup.carrier\nn : \u2115\n\u22a2 (fun x => a * (coeffs E x * u (n + \u2191x))) = fun x => coeffs E x * (a * u (n + \u2191x))\n[PROOFSTEP]\next\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\na : \u03b1\nu : \u2115 \u2192 \u03b1\nhu :\n u \u2208\n {\n toAddSubsemigroup :=\n { carrier := {u | IsSolution E u},\n add_mem' :=\n (_ :\n \u2200 {u v : \u2115 \u2192 \u03b1},\n u \u2208 {u | IsSolution E u} \u2192\n v \u2208 {u | IsSolution E u} \u2192\n \u2200 (n : \u2115),\n u (n + E.order) + v (n + E.order) = \u2211 i : Fin E.order, coeffs E i * (u + v) (n + \u2191i)) },\n zero_mem' :=\n (_ : \u2200 (n : \u2115), 0 = \u2211 i : Fin E.order, coeffs E i * OfNat.ofNat 0 (n + \u2191i)) }.toAddSubsemigroup.carrier\nn : \u2115\nx\u271d : Fin E.order\n\u22a2 a * (coeffs E x\u271d * u (n + \u2191x\u271d)) = coeffs E x\u271d * (a * u (n + \u2191x\u271d))\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu v : { x // x \u2208 solSpace E }\n\u22a2 (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu v : { x // x \u2208 solSpace E }\nx\u271d : Fin E.order\n\u22a2 (fun u x => \u2191u \u2191x) (u + v) x\u271d = ((fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\na : \u03b1\nu : { x // x \u2208 solSpace E }\n\u22a2 AddHom.toFun\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) }\n (a \u2022 u) =\n \u2191(RingHom.id \u03b1) a \u2022\n AddHom.toFun\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) }\n u\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\na : \u03b1\nu : { x // x \u2208 solSpace E }\nx\u271d : Fin E.order\n\u22a2 AddHom.toFun\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) }\n (a \u2022 u) x\u271d =\n (\u2191(RingHom.id \u03b1) a \u2022\n AddHom.toFun\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) }\n u)\n x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : { x // x \u2208 solSpace E }\n\u22a2 (fun u => { val := mkSol E u, property := (_ : IsSolution E (mkSol E u)) })\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) },\n map_smul' :=\n (_ :\n \u2200 (a : \u03b1) (u : { x // x \u2208 solSpace E }),\n AddHom.toFun\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) }\n (a \u2022 u) =\n \u2191(RingHom.id \u03b1) a \u2022\n AddHom.toFun\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) }\n u) }.toAddHom\n u) =\n u\n[PROOFSTEP]\next n\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : { x // x \u2208 solSpace E }\nn : \u2115\n\u22a2 \u2191((fun u => { val := mkSol E u, property := (_ : IsSolution E (mkSol E u)) })\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) },\n map_smul' :=\n (_ :\n \u2200 (a : \u03b1) (u : { x // x \u2208 solSpace E }),\n AddHom.toFun\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) }\n (a \u2022 u) =\n \u2191(RingHom.id \u03b1) a \u2022\n AddHom.toFun\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) }\n u) }.toAddHom\n u))\n n =\n \u2191u n\n[PROOFSTEP]\nsymm\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : { x // x \u2208 solSpace E }\nn : \u2115\n\u22a2 \u2191u n =\n \u2191((fun u => { val := mkSol E u, property := (_ : IsSolution E (mkSol E u)) })\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) },\n map_smul' :=\n (_ :\n \u2200 (a : \u03b1) (u : { x // x \u2208 solSpace E }),\n AddHom.toFun\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) }\n (a \u2022 u) =\n \u2191(RingHom.id \u03b1) a \u2022\n AddHom.toFun\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) }\n u) }.toAddHom\n u))\n n\n[PROOFSTEP]\napply E.eq_mk_of_is_sol_of_eq_init u.2\n[GOAL]\ncase a.h.heq\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : { x // x \u2208 solSpace E }\nn : \u2115\n\u22a2 \u2200 (n : Fin E.order),\n \u2191u \u2191n =\n AddHom.toFun\n {\n toAddHom :=\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) },\n map_smul' :=\n (_ :\n \u2200 (a : \u03b1) (u : { x // x \u2208 solSpace E }),\n AddHom.toFun\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) }\n (a \u2022 u) =\n \u2191(RingHom.id \u03b1) a \u2022\n AddHom.toFun\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) }\n u) }.toAddHom\n u n\n[PROOFSTEP]\nintro k\n[GOAL]\ncase a.h.heq\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : { x // x \u2208 solSpace E }\nn : \u2115\nk : Fin E.order\n\u22a2 \u2191u \u2191k =\n AddHom.toFun\n {\n toAddHom :=\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) },\n map_smul' :=\n (_ :\n \u2200 (a : \u03b1) (u : { x // x \u2208 solSpace E }),\n AddHom.toFun\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) }\n (a \u2022 u) =\n \u2191(RingHom.id \u03b1) a \u2022\n AddHom.toFun\n { toFun := fun u x => \u2191u \u2191x,\n map_add' :=\n (_ :\n \u2200 (u v : { x // x \u2208 solSpace E }),\n (fun u x => \u2191u \u2191x) (u + v) = (fun u x => \u2191u \u2191x) u + (fun u x => \u2191u \u2191x) v) }\n u) }.toAddHom\n u k\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu v : \u2115 \u2192 \u03b1\nhu : IsSolution E u\nhv : IsSolution E v\n\u22a2 u = v \u2194 Set.EqOn u v \u2191(range E.order)\n[PROOFSTEP]\nrefine' Iff.intro (fun h x _ \u21a6 h \u25b8 rfl) _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu v : \u2115 \u2192 \u03b1\nhu : IsSolution E u\nhv : IsSolution E v\n\u22a2 Set.EqOn u v \u2191(range E.order) \u2192 u = v\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu v : \u2115 \u2192 \u03b1\nhu : IsSolution E u\nhv : IsSolution E v\nh : Set.EqOn u v \u2191(range E.order)\n\u22a2 u = v\n[PROOFSTEP]\nset u' : \u21a5E.solSpace := \u27e8u, hu\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu v : \u2115 \u2192 \u03b1\nhu : IsSolution E u\nhv : IsSolution E v\nh : Set.EqOn u v \u2191(range E.order)\nu' : { x // x \u2208 solSpace E } := { val := u, property := hu }\n\u22a2 u = v\n[PROOFSTEP]\nset v' : \u21a5E.solSpace := \u27e8v, hv\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu v : \u2115 \u2192 \u03b1\nhu : IsSolution E u\nhv : IsSolution E v\nh : Set.EqOn u v \u2191(range E.order)\nu' : { x // x \u2208 solSpace E } := { val := u, property := hu }\nv' : { x // x \u2208 solSpace E } := { val := v, property := hv }\n\u22a2 u = v\n[PROOFSTEP]\nchange u'.val = v'.val\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu v : \u2115 \u2192 \u03b1\nhu : IsSolution E u\nhv : IsSolution E v\nh : Set.EqOn u v \u2191(range E.order)\nu' : { x // x \u2208 solSpace E } := { val := u, property := hu }\nv' : { x // x \u2208 solSpace E } := { val := v, property := hv }\n\u22a2 \u2191u' = \u2191v'\n[PROOFSTEP]\nsuffices h' : u' = v'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu v : \u2115 \u2192 \u03b1\nhu : IsSolution E u\nhv : IsSolution E v\nh : Set.EqOn u v \u2191(range E.order)\nu' : { x // x \u2208 solSpace E } := { val := u, property := hu }\nv' : { x // x \u2208 solSpace E } := { val := v, property := hv }\nh' : u' = v'\n\u22a2 \u2191u' = \u2191v'\ncase h'\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu v : \u2115 \u2192 \u03b1\nhu : IsSolution E u\nhv : IsSolution E v\nh : Set.EqOn u v \u2191(range E.order)\nu' : { x // x \u2208 solSpace E } := { val := u, property := hu }\nv' : { x // x \u2208 solSpace E } := { val := v, property := hv }\n\u22a2 u' = v'\n[PROOFSTEP]\nexact h' \u25b8 rfl\n[GOAL]\ncase h'\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu v : \u2115 \u2192 \u03b1\nhu : IsSolution E u\nhv : IsSolution E v\nh : Set.EqOn u v \u2191(range E.order)\nu' : { x // x \u2208 solSpace E } := { val := u, property := hu }\nv' : { x // x \u2208 solSpace E } := { val := v, property := hv }\n\u22a2 u' = v'\n[PROOFSTEP]\nrw [\u2190 E.toInit.toEquiv.apply_eq_iff_eq, LinearEquiv.coe_toEquiv]\n[GOAL]\ncase h'\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu v : \u2115 \u2192 \u03b1\nhu : IsSolution E u\nhv : IsSolution E v\nh : Set.EqOn u v \u2191(range E.order)\nu' : { x // x \u2208 solSpace E } := { val := u, property := hu }\nv' : { x // x \u2208 solSpace E } := { val := v, property := hv }\n\u22a2 \u2191(toInit E) u' = \u2191(toInit E) v'\n[PROOFSTEP]\next x\n[GOAL]\ncase h'.h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu v : \u2115 \u2192 \u03b1\nhu : IsSolution E u\nhv : IsSolution E v\nh : Set.EqOn u v \u2191(range E.order)\nu' : { x // x \u2208 solSpace E } := { val := u, property := hu }\nv' : { x // x \u2208 solSpace E } := { val := v, property := hv }\nx : Fin E.order\n\u22a2 \u2191(toInit E) u' x = \u2191(toInit E) v' x\n[PROOFSTEP]\nexact_mod_cast h (mem_range.mpr x.2)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nx y : Fin E.order \u2192 \u03b1\n\u22a2 (fun X i => if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * X i)\n (x + y) =\n (fun X i => if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * X i)\n x +\n (fun X i => if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * X i)\n y\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nx y : Fin E.order \u2192 \u03b1\ni : Fin E.order\n\u22a2 (fun X i => if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * X i)\n (x + y) i =\n ((fun X i => if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * X i)\n x +\n (fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * X i)\n y)\n i\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nx y : Fin E.order \u2192 \u03b1\ni : Fin E.order\n\u22a2 (if h : \u2191i + 1 < E.order then (x + y) { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * (x + y) i) =\n ((fun i => if h : \u2191i + 1 < E.order then x { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * x i) +\n fun i => if h : \u2191i + 1 < E.order then y { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * y i)\n i\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nx y : Fin E.order \u2192 \u03b1\ni : Fin E.order\nh : \u2191i + 1 < E.order\n\u22a2 (x + y) { val := \u2191i + 1, isLt := h } =\n ((fun i => if h : \u2191i + 1 < E.order then x { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * x i) +\n fun i => if h : \u2191i + 1 < E.order then y { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * y i)\n i\n[PROOFSTEP]\nsimp [h, mul_add, sum_add_distrib]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nx y : Fin E.order \u2192 \u03b1\ni : Fin E.order\nh : \u00ac\u2191i + 1 < E.order\n\u22a2 \u2211 i : Fin E.order, coeffs E i * (x + y) i =\n ((fun i => if h : \u2191i + 1 < E.order then x { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * x i) +\n fun i => if h : \u2191i + 1 < E.order then y { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * y i)\n i\n[PROOFSTEP]\nsimp [h, mul_add, sum_add_distrib]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nx : \u03b1\ny : Fin E.order \u2192 \u03b1\n\u22a2 AddHom.toFun\n {\n toFun := fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * X i,\n map_add' :=\n (_ :\n \u2200 (x y : Fin E.order \u2192 \u03b1),\n (fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h }\n else \u2211 i : Fin E.order, coeffs E i * X i)\n (x + y) =\n (fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h }\n else \u2211 i : Fin E.order, coeffs E i * X i)\n x +\n (fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h }\n else \u2211 i : Fin E.order, coeffs E i * X i)\n y) }\n (x \u2022 y) =\n \u2191(RingHom.id \u03b1) x \u2022\n AddHom.toFun\n {\n toFun := fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * X i,\n map_add' :=\n (_ :\n \u2200 (x y : Fin E.order \u2192 \u03b1),\n (fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h }\n else \u2211 i : Fin E.order, coeffs E i * X i)\n (x + y) =\n (fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h }\n else \u2211 i : Fin E.order, coeffs E i * X i)\n x +\n (fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h }\n else \u2211 i : Fin E.order, coeffs E i * X i)\n y) }\n y\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nx : \u03b1\ny : Fin E.order \u2192 \u03b1\ni : Fin E.order\n\u22a2 AddHom.toFun\n {\n toFun := fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * X i,\n map_add' :=\n (_ :\n \u2200 (x y : Fin E.order \u2192 \u03b1),\n (fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h }\n else \u2211 i : Fin E.order, coeffs E i * X i)\n (x + y) =\n (fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h }\n else \u2211 i : Fin E.order, coeffs E i * X i)\n x +\n (fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h }\n else \u2211 i : Fin E.order, coeffs E i * X i)\n y) }\n (x \u2022 y) i =\n (\u2191(RingHom.id \u03b1) x \u2022\n AddHom.toFun\n {\n toFun := fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * X i,\n map_add' :=\n (_ :\n \u2200 (x y : Fin E.order \u2192 \u03b1),\n (fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h }\n else \u2211 i : Fin E.order, coeffs E i * X i)\n (x + y) =\n (fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h }\n else \u2211 i : Fin E.order, coeffs E i * X i)\n x +\n (fun X i =>\n if h : \u2191i + 1 < E.order then X { val := \u2191i + 1, isLt := h }\n else \u2211 i : Fin E.order, coeffs E i * X i)\n y) }\n y)\n i\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nx : \u03b1\ny : Fin E.order \u2192 \u03b1\ni : Fin E.order\n\u22a2 (if h : \u2191i + 1 < E.order then (x \u2022 y) { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * (x \u2022 y) i) =\n (\u2191(RingHom.id \u03b1) x \u2022 fun i =>\n if h : \u2191i + 1 < E.order then y { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * y i)\n i\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nx : \u03b1\ny : Fin E.order \u2192 \u03b1\ni : Fin E.order\nh : \u2191i + 1 < E.order\n\u22a2 (x \u2022 y) { val := \u2191i + 1, isLt := h } =\n (\u2191(RingHom.id \u03b1) x \u2022 fun i =>\n if h : \u2191i + 1 < E.order then y { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * y i)\n i\n[PROOFSTEP]\nsimp [h, mul_sum]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nx : \u03b1\ny : Fin E.order \u2192 \u03b1\ni : Fin E.order\nh : \u00ac\u2191i + 1 < E.order\n\u22a2 \u2211 i : Fin E.order, coeffs E i * (x \u2022 y) i =\n (\u2191(RingHom.id \u03b1) x \u2022 fun i =>\n if h : \u2191i + 1 < E.order then y { val := \u2191i + 1, isLt := h } else \u2211 i : Fin E.order, coeffs E i * y i)\n i\n[PROOFSTEP]\nsimp [h, mul_sum]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nx : \u03b1\ny : Fin E.order \u2192 \u03b1\ni : Fin E.order\nh : \u00ac\u2191i + 1 < E.order\n\u22a2 \u2211 x_1 : Fin E.order, coeffs E x_1 * (x * y x_1) = \u2211 x_1 : Fin E.order, x * (coeffs E x_1 * y x_1)\n[PROOFSTEP]\nexact sum_congr rfl fun x _ \u21a6 by ac_rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nx\u271d\u00b9 : \u03b1\ny : Fin E.order \u2192 \u03b1\ni : Fin E.order\nh : \u00ac\u2191i + 1 < E.order\nx : Fin E.order\nx\u271d : x \u2208 univ\n\u22a2 coeffs E x * (x\u271d\u00b9 * y x) = x\u271d\u00b9 * (coeffs E x * y x)\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\nE : LinearRecurrence \u03b1\nq : \u03b1\n\u22a2 (IsSolution E fun n => q ^ n) \u2194 IsRoot (charPoly E) q\n[PROOFSTEP]\nrw [charPoly, Polynomial.IsRoot.def, Polynomial.eval]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\nE : LinearRecurrence \u03b1\nq : \u03b1\n\u22a2 (IsSolution E fun n => q ^ n) \u2194\n eval\u2082 (RingHom.id \u03b1) q (\u2191(monomial E.order) 1 - \u2211 i : Fin E.order, \u2191(monomial \u2191i) (coeffs E i)) = 0\n[PROOFSTEP]\nsimp only [Polynomial.eval\u2082_finset_sum, one_mul, RingHom.id_apply, Polynomial.eval\u2082_monomial, Polynomial.eval\u2082_sub]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\nE : LinearRecurrence \u03b1\nq : \u03b1\n\u22a2 (IsSolution E fun n => q ^ n) \u2194 q ^ E.order - \u2211 x : Fin E.order, coeffs E x * q ^ \u2191x = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\nE : LinearRecurrence \u03b1\nq : \u03b1\n\u22a2 (IsSolution E fun n => q ^ n) \u2192 q ^ E.order - \u2211 x : Fin E.order, coeffs E x * q ^ \u2191x = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\nE : LinearRecurrence \u03b1\nq : \u03b1\nh : IsSolution E fun n => q ^ n\n\u22a2 q ^ E.order - \u2211 x : Fin E.order, coeffs E x * q ^ \u2191x = 0\n[PROOFSTEP]\nsimpa [sub_eq_zero] using h 0\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\nE : LinearRecurrence \u03b1\nq : \u03b1\n\u22a2 q ^ E.order - \u2211 x : Fin E.order, coeffs E x * q ^ \u2191x = 0 \u2192 IsSolution E fun n => q ^ n\n[PROOFSTEP]\nintro h n\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\nE : LinearRecurrence \u03b1\nq : \u03b1\nh : q ^ E.order - \u2211 x : Fin E.order, coeffs E x * q ^ \u2191x = 0\nn : \u2115\n\u22a2 (fun n => q ^ n) (n + E.order) = \u2211 i : Fin E.order, coeffs E i * (fun n => q ^ n) (n + \u2191i)\n[PROOFSTEP]\nsimp only [pow_add, sub_eq_zero.mp h, mul_sum]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\nE : LinearRecurrence \u03b1\nq : \u03b1\nh : q ^ E.order - \u2211 x : Fin E.order, coeffs E x * q ^ \u2191x = 0\nn : \u2115\n\u22a2 \u2211 x : Fin E.order, q ^ n * (coeffs E x * q ^ \u2191x) = \u2211 x : Fin E.order, coeffs E x * (q ^ n * q ^ \u2191x)\n[PROOFSTEP]\nexact sum_congr rfl fun _ _ \u21a6 by ring\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\nE : LinearRecurrence \u03b1\nq : \u03b1\nh : q ^ E.order - \u2211 x : Fin E.order, coeffs E x * q ^ \u2191x = 0\nn : \u2115\nx\u271d\u00b9 : Fin E.order\nx\u271d : x\u271d\u00b9 \u2208 univ\n\u22a2 q ^ n * (coeffs E x\u271d\u00b9 * q ^ \u2191x\u271d\u00b9) = coeffs E x\u271d\u00b9 * (q ^ n * q ^ \u2191x\u271d\u00b9)\n[PROOFSTEP]\nring\n", "meta": {"mathlib_filename": "Mathlib.Algebra.LinearRecurrence", "llama_tokens": 14164, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.42211622701075724}} {"text": "[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\nx y : TruncatedWittVector p n R\nh : x = y\ni : Fin n\n\u22a2 coeff i x = coeff i y\n[PROOFSTEP]\nrw [h]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\nx : TruncatedWittVector p n R\n\u22a2 (mk p fun i => coeff i x) = x\n[PROOFSTEP]\next i\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\nx : TruncatedWittVector p n R\ni : Fin n\n\u22a2 coeff i (mk p fun i => coeff i x) = coeff i x\n[PROOFSTEP]\nrw [coeff_mk]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : TruncatedWittVector p n R\ni : Fin n\n\u22a2 WittVector.coeff (out x) \u2191i = coeff i x\n[PROOFSTEP]\nrw [out]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : TruncatedWittVector p n R\ni : Fin n\n\u22a2 WittVector.coeff { coeff := fun i => if h : i < n then coeff { val := i, isLt := h } x else 0 } \u2191i = coeff i x\n[PROOFSTEP]\ndsimp only\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : TruncatedWittVector p n R\ni : Fin n\n\u22a2 (if h : \u2191i < n then coeff { val := \u2191i, isLt := h } x else 0) = coeff i x\n[PROOFSTEP]\nrw [dif_pos i.is_lt, Fin.eta]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 Injective out\n[PROOFSTEP]\nintro x y h\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx y : TruncatedWittVector p n R\nh : out x = out y\n\u22a2 x = y\n[PROOFSTEP]\next i\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx y : TruncatedWittVector p n R\nh : out x = out y\ni : Fin n\n\u22a2 coeff i x = coeff i y\n[PROOFSTEP]\nrw [WittVector.ext_iff] at h \n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx y : TruncatedWittVector p n R\nh : \u2200 (n_1 : \u2115), WittVector.coeff (out x) n_1 = WittVector.coeff (out y) n_1\ni : Fin n\n\u22a2 coeff i x = coeff i y\n[PROOFSTEP]\nsimpa only [coeff_out] using h \u2191i\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\nx : \ud835\udd4e R\ni : Fin n\n\u22a2 TruncatedWittVector.coeff i (truncateFun n x) = coeff x \u2191i\n[PROOFSTEP]\nrw [truncateFun, TruncatedWittVector.coeff_mk]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\n\u22a2 TruncatedWittVector.out (truncateFun n x) = init n x\n[PROOFSTEP]\next i\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\ni : \u2115\n\u22a2 coeff (TruncatedWittVector.out (truncateFun n x)) i = coeff (init n x) i\n[PROOFSTEP]\ndsimp [TruncatedWittVector.out, init, select, coeff_mk]\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\ni : \u2115\n\u22a2 (if h : i < n then TruncatedWittVector.coeff { val := i, isLt := h } (truncateFun n x) else 0) =\n if i < n then coeff x i else 0\n[PROOFSTEP]\nsplit_ifs with hi\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\ni : \u2115\nhi : i < n\n\u22a2 TruncatedWittVector.coeff { val := i, isLt := hi } (truncateFun n x) = coeff x i\ncase neg p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 R : Type u_1 inst\u271d : CommRing R x : \ud835\udd4e R i : \u2115 hi : \u00aci < n \u22a2 0 = 0\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\ni : \u2115\nhi : \u00aci < n\n\u22a2 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\ni : \u2115\nhi : i < n\n\u22a2 TruncatedWittVector.coeff { val := i, isLt := hi } (truncateFun n x) = coeff x i\n[PROOFSTEP]\nrw [coeff_truncateFun, Fin.val_mk]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : TruncatedWittVector p n R\n\u22a2 WittVector.truncateFun n (out x) = x\n[PROOFSTEP]\nsimp only [WittVector.truncateFun, coeff_out, mk_coeff]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\ni : Fin n\n\u22a2 coeff i 0 = 0\n[PROOFSTEP]\nshow coeff i (truncateFun _ 0 : TruncatedWittVector p n R) = 0\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\ni : Fin n\n\u22a2 coeff i (truncateFun n 0) = 0\n[PROOFSTEP]\nrw [coeff_truncateFun, WittVector.zero_coeff]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx y : \ud835\udd4e R\n\u22a2 truncateFun n (x + y) = truncateFun n x + truncateFun n y\n[PROOFSTEP]\nwitt_truncateFun_tac\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx y : \ud835\udd4e R\n\u22a2 truncateFun n (x * y) = truncateFun n x * truncateFun n y\n[PROOFSTEP]\nwitt_truncateFun_tac\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\n\u22a2 truncateFun n (-x) = -truncateFun n x\n[PROOFSTEP]\nwitt_truncateFun_tac\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx y : \ud835\udd4e R\n\u22a2 truncateFun n (x - y) = truncateFun n x - truncateFun n y\n[PROOFSTEP]\nwitt_truncateFun_tac\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nm : \u2115\n\u22a2 truncateFun n (m \u2022 x) = m \u2022 truncateFun n x\n[PROOFSTEP]\nwitt_truncateFun_tac\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nm : \u2124\n\u22a2 truncateFun n (m \u2022 x) = m \u2022 truncateFun n x\n[PROOFSTEP]\nwitt_truncateFun_tac\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nm : \u2115\n\u22a2 truncateFun n (x ^ m) = truncateFun n x ^ m\n[PROOFSTEP]\nwitt_truncateFun_tac\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\n\u22a2 x \u2208 RingHom.ker (truncate n) \u2194 \u2200 (i : \u2115), i < n \u2192 coeff x i = 0\n[PROOFSTEP]\nsimp only [RingHom.mem_ker, truncate, truncateFun, RingHom.coe_mk, TruncatedWittVector.ext_iff,\n TruncatedWittVector.coeff_mk, coeff_zero]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\n\u22a2 (\u2200 (i : Fin n),\n TruncatedWittVector.coeff i\n (\u2191{ toOneHom := { toFun := truncateFun n, map_one' := (_ : truncateFun n 1 = 1) },\n map_mul' := (_ : \u2200 (x y : \ud835\udd4e R), truncateFun n (x * y) = truncateFun n x * truncateFun n y) }\n x) =\n 0) \u2194\n \u2200 (i : \u2115), i < n \u2192 coeff x i = 0\n[PROOFSTEP]\nexact Fin.forall_iff\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nf : \u2115 \u2192 R\n\u22a2 \u2191(truncate n) { coeff := f } = TruncatedWittVector.mk p fun k => f \u2191k\n[PROOFSTEP]\next i\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nf : \u2115 \u2192 R\ni : Fin n\n\u22a2 TruncatedWittVector.coeff i (\u2191(truncate n) { coeff := f }) =\n TruncatedWittVector.coeff i (TruncatedWittVector.mk p fun k => f \u2191k)\n[PROOFSTEP]\nsimp only [coeff_truncate, TruncatedWittVector.coeff_mk]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nm : \u2115\nhm : n \u2264 m\n\u22a2 RingHom.ker (WittVector.truncate m) \u2264 RingHom.ker (WittVector.truncate n)\n[PROOFSTEP]\nintro x\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nm : \u2115\nhm : n \u2264 m\nx : \ud835\udd4e R\n\u22a2 x \u2208 RingHom.ker (WittVector.truncate m) \u2192 x \u2208 RingHom.ker (WittVector.truncate n)\n[PROOFSTEP]\nsimp only [WittVector.mem_ker_truncate]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nm : \u2115\nhm : n \u2264 m\nx : \ud835\udd4e R\n\u22a2 (\u2200 (i : \u2115), i < m \u2192 WittVector.coeff x i = 0) \u2192 \u2200 (i : \u2115), i < n \u2192 WittVector.coeff x i = 0\n[PROOFSTEP]\nintro h i hi\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nm : \u2115\nhm : n \u2264 m\nx : \ud835\udd4e R\nh : \u2200 (i : \u2115), i < m \u2192 WittVector.coeff x i = 0\ni : \u2115\nhi : i < n\n\u22a2 WittVector.coeff x i = 0\n[PROOFSTEP]\nexact h i (lt_of_lt_of_le hi hm)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nn\u2081 n\u2082 n\u2083 : \u2115\nh1 : n\u2081 \u2264 n\u2082\nh2 : n\u2082 \u2264 n\u2083\nx : TruncatedWittVector p n\u2083 R\n\u22a2 \u2191(truncate h1) (\u2191(truncate h2) x) = \u2191(truncate (_ : n\u2081 \u2264 n\u2083)) x\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 := @WittVector.truncate_surjective p _ n\u2083 R _ x\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nn\u2081 n\u2082 n\u2083 : \u2115\nh1 : n\u2081 \u2264 n\u2082\nh2 : n\u2082 \u2264 n\u2083\nx : \ud835\udd4e R\n\u22a2 \u2191(truncate h1) (\u2191(truncate h2) (\u2191(WittVector.truncate n\u2083) x)) =\n \u2191(truncate (_ : n\u2081 \u2264 n\u2083)) (\u2191(WittVector.truncate n\u2083) x)\n[PROOFSTEP]\nsimp only [truncate_wittVector_truncate]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nn\u2081 n\u2082 n\u2083 : \u2115\nh1 : n\u2081 \u2264 n\u2082\nh2 : n\u2082 \u2264 n\u2083\n\u22a2 RingHom.comp (truncate h1) (truncate h2) = truncate (_ : n\u2081 \u2264 n\u2083)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nn\u2081 n\u2082 n\u2083 : \u2115\nh1 : n\u2081 \u2264 n\u2082\nh2 : n\u2082 \u2264 n\u2083\nx : TruncatedWittVector p n\u2083 R\n\u22a2 \u2191(RingHom.comp (truncate h1) (truncate h2)) x = \u2191(truncate (_ : n\u2081 \u2264 n\u2083)) x\n[PROOFSTEP]\nsimp only [truncate_truncate, Function.comp_apply, RingHom.coe_comp]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nm : \u2115\nhm : n \u2264 m\n\u22a2 Surjective \u2191(truncate hm)\n[PROOFSTEP]\nintro x\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nm : \u2115\nhm : n \u2264 m\nx : TruncatedWittVector p n R\n\u22a2 \u2203 a, \u2191(truncate hm) a = x\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 := @WittVector.truncate_surjective p _ _ R _ x\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nm : \u2115\nhm : n \u2264 m\nx : \ud835\udd4e R\n\u22a2 \u2203 a, \u2191(truncate hm) a = \u2191(WittVector.truncate n) x\n[PROOFSTEP]\nexact \u27e8WittVector.truncate _ x, truncate_wittVector_truncate _ _\u27e9\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nm : \u2115\nhm : n \u2264 m\ni : Fin n\nx : TruncatedWittVector p m R\n\u22a2 coeff i (\u2191(truncate hm) x) = coeff (Fin.castLE hm i) x\n[PROOFSTEP]\nobtain \u27e8y, rfl\u27e9 := @WittVector.truncate_surjective p _ _ _ _ x\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nm : \u2115\nhm : n \u2264 m\ni : Fin n\ny : \ud835\udd4e R\n\u22a2 coeff i (\u2191(truncate hm) (\u2191(WittVector.truncate m) y)) = coeff (Fin.castLE hm i) (\u2191(WittVector.truncate m) y)\n[PROOFSTEP]\nsimp only [truncate_wittVector_truncate, WittVector.coeff_truncate, Fin.coe_castLE]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR\u271d : Type u_1\ninst\u271d\u00b9 : CommRing R\u271d\nR : Type u_2\ninst\u271d : Fintype R\n\u22a2 Fintype.card (TruncatedWittVector p n R) = Fintype.card R ^ n\n[PROOFSTEP]\nsimp only [TruncatedWittVector, Fintype.card_fin, Fintype.card_fun]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 \u2a05 (i : \u2115), RingHom.ker (WittVector.truncate i) = \u22a5\n[PROOFSTEP]\nrw [Submodule.eq_bot_iff]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 \u2200 (x : \ud835\udd4e R), x \u2208 \u2a05 (i : \u2115), RingHom.ker (WittVector.truncate i) \u2192 x = 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nhx : x \u2208 \u2a05 (i : \u2115), RingHom.ker (WittVector.truncate i)\n\u22a2 x = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nhx : x \u2208 \u2a05 (i : \u2115), RingHom.ker (WittVector.truncate i)\nn\u271d : \u2115\n\u22a2 WittVector.coeff x n\u271d = WittVector.coeff 0 n\u271d\n[PROOFSTEP]\nsimp only [WittVector.mem_ker_truncate, Ideal.mem_iInf, WittVector.zero_coeff] at hx \u22a2\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nn\u271d : \u2115\nhx : \u2200 (i i_1 : \u2115), i_1 < i \u2192 WittVector.coeff x i_1 = 0\n\u22a2 WittVector.coeff x n\u271d = 0\n[PROOFSTEP]\nexact hx _ _ (Nat.lt_succ_self _)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\ns : S\n\u22a2 \u2191(truncate n) (liftFun f s) = \u2191(f n) s\n[PROOFSTEP]\next i\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\ns : S\ni : Fin n\n\u22a2 TruncatedWittVector.coeff i (\u2191(truncate n) (liftFun f s)) = TruncatedWittVector.coeff i (\u2191(f n) s)\n[PROOFSTEP]\nsimp only [liftFun, TruncatedWittVector.coeff_mk, WittVector.truncate_mk']\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\ns : S\ni : Fin n\n\u22a2 TruncatedWittVector.coeff (Fin.last \u2191i) (\u2191(f (\u2191i + 1)) s) = TruncatedWittVector.coeff i (\u2191(f n) s)\n[PROOFSTEP]\nrw [\u2190 f_compat (i + 1) n i.is_lt, RingHom.comp_apply, TruncatedWittVector.coeff_truncate]\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\ns : S\ni : Fin n\n\u22a2 TruncatedWittVector.coeff (Fin.castLE (_ : \u2191i < n) (Fin.last \u2191i)) (\u2191(f n) s) = TruncatedWittVector.coeff i (\u2191(f n) s)\n[PROOFSTEP]\ncongr 1 with _\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\n\u22a2 S \u2192+* \ud835\udd4e R\n[PROOFSTEP]\nrefine'\n { toFun := liftFun f\n map_zero' := _\n map_one' := _\n map_add' := _\n map_mul' := _ }\n[GOAL]\ncase refine'_1\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\n\u22a2 liftFun f 1 = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\n\u22a2 liftFun f 1 = 1\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase refine'_1\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\n\u22a2 liftFun f 1 = 1\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 Ideal.mem_bot, \u2190 iInf_ker_truncate, Ideal.mem_iInf]\n[GOAL]\ncase refine'_1\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\n\u22a2 \u2200 (i : \u2115), liftFun f 1 - 1 \u2208 RingHom.ker (truncate i)\n[PROOFSTEP]\nsimp [RingHom.mem_ker, f_compat]\n[GOAL]\ncase refine'_2\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\n\u22a2 \u2200 (x y : S),\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } (x * y) =\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } x *\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\nx\u271d y\u271d : S\n\u22a2 OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } (x\u271d * y\u271d) =\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } x\u271d *\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } y\u271d\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase refine'_2\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\nx\u271d y\u271d : S\n\u22a2 liftFun f (x\u271d * y\u271d) = liftFun f x\u271d * liftFun f y\u271d\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 Ideal.mem_bot, \u2190 iInf_ker_truncate, Ideal.mem_iInf]\n[GOAL]\ncase refine'_2\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\nx\u271d y\u271d : S\n\u22a2 \u2200 (i : \u2115), liftFun f (x\u271d * y\u271d) - liftFun f x\u271d * liftFun f y\u271d \u2208 RingHom.ker (truncate i)\n[PROOFSTEP]\nsimp [RingHom.mem_ker, f_compat]\n[GOAL]\ncase refine'_3\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : S),\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } (x * y) =\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } x *\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } y) })\n 0 =\n 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : S),\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } (x * y) =\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } x *\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } y) })\n 0 =\n 0\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase refine'_3\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\n\u22a2 liftFun f 0 = 0\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 Ideal.mem_bot, \u2190 iInf_ker_truncate, Ideal.mem_iInf]\n[GOAL]\ncase refine'_3\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\n\u22a2 \u2200 (i : \u2115), liftFun f 0 - 0 \u2208 RingHom.ker (truncate i)\n[PROOFSTEP]\nsimp [RingHom.mem_ker, f_compat]\n[GOAL]\ncase refine'_4\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\n\u22a2 \u2200 (x y : S),\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : S),\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } (x * y) =\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } x *\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } y) })\n (x + y) =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : S),\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } (x * y) =\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } x *\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } y) })\n x +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : S),\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } (x * y) =\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } x *\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } y) })\n y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_4\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\nx\u271d y\u271d : S\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : S),\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } (x * y) =\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } x *\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } y) })\n (x\u271d + y\u271d) =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : S),\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } (x * y) =\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } x *\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } y) })\n x\u271d +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : S),\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } (x * y) =\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } x *\n OneHom.toFun { toFun := liftFun f, map_one' := (_ : liftFun f 1 = 1) } y) })\n y\u271d\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase refine'_4\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\nx\u271d y\u271d : S\n\u22a2 liftFun f (x\u271d + y\u271d) = liftFun f x\u271d + liftFun f y\u271d\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 Ideal.mem_bot, \u2190 iInf_ker_truncate, Ideal.mem_iInf]\n[GOAL]\ncase refine'_4\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\nx\u271d y\u271d : S\n\u22a2 \u2200 (i : \u2115), liftFun f (x\u271d + y\u271d) - (liftFun f x\u271d + liftFun f y\u271d) \u2208 RingHom.ker (truncate i)\n[PROOFSTEP]\nsimp [RingHom.mem_ker, f_compat]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\n\u22a2 RingHom.comp (truncate n) (lift (fun k\u2082 => f k\u2082) f_compat) = f n\n[PROOFSTEP]\next1\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\nx\u271d : S\n\u22a2 \u2191(RingHom.comp (truncate n) (lift (fun k\u2082 => f k\u2082) f_compat)) x\u271d = \u2191(f n) x\u271d\n[PROOFSTEP]\nrw [RingHom.comp_apply, truncate_lift]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\ng : S \u2192+* \ud835\udd4e R\ng_compat : \u2200 (k : \u2115), RingHom.comp (truncate k) g = f k\n\u22a2 lift (fun k\u2082 => f k\u2082) f_compat = g\n[PROOFSTEP]\next1 x\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\ng : S \u2192+* \ud835\udd4e R\ng_compat : \u2200 (k : \u2115), RingHom.comp (truncate k) g = f k\nx : S\n\u22a2 \u2191(lift (fun k\u2082 => f k\u2082) f_compat) x = \u2191g x\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 Ideal.mem_bot, \u2190 iInf_ker_truncate, Ideal.mem_iInf]\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\ng : S \u2192+* \ud835\udd4e R\ng_compat : \u2200 (k : \u2115), RingHom.comp (truncate k) g = f k\nx : S\n\u22a2 \u2200 (i : \u2115), \u2191(lift (fun k\u2082 => f k\u2082) f_compat) x - \u2191g x \u2208 RingHom.ker (truncate i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\ng : S \u2192+* \ud835\udd4e R\ng_compat : \u2200 (k : \u2115), RingHom.comp (truncate k) g = f k\nx : S\ni : \u2115\n\u22a2 \u2191(lift (fun k\u2082 => f k\u2082) f_compat) x - \u2191g x \u2208 RingHom.ker (truncate i)\n[PROOFSTEP]\nsimp only [RingHom.mem_ker, g_compat, \u2190 RingHom.comp_apply, truncate_comp_lift, RingHom.map_sub, sub_self]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\ng : S \u2192+* \ud835\udd4e R\n\u22a2 \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082),\n RingHom.comp (TruncatedWittVector.truncate hk) ((fun k => RingHom.comp (truncate k) g) k\u2082) =\n (fun k => RingHom.comp (truncate k) g) k\u2081\n[PROOFSTEP]\nintro _ _ h\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\ng : S \u2192+* \ud835\udd4e R\nk\u2081\u271d k\u2082\u271d : \u2115\nh : k\u2081\u271d \u2264 k\u2082\u271d\n\u22a2 RingHom.comp (TruncatedWittVector.truncate h) ((fun k => RingHom.comp (truncate k) g) k\u2082\u271d) =\n (fun k => RingHom.comp (truncate k) g) k\u2081\u271d\n[PROOFSTEP]\nsimp only [\u2190 RingHom.comp_assoc, truncate_comp_wittVector_truncate]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\n\u22a2 Function.LeftInverse\n (fun g =>\n { val := fun k => RingHom.comp (truncate k) g,\n property :=\n (_ :\n \u2200 (k\u2081 k\u2082 : \u2115) (h : k\u2081 \u2264 k\u2082),\n RingHom.comp (RingHom.comp (TruncatedWittVector.truncate h) (truncate k\u2082)) g =\n RingHom.comp (truncate k\u2081) g) })\n fun f => lift \u2191f (_ : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (\u2191f k\u2082) = \u2191f k\u2081)\n[PROOFSTEP]\nrintro \u27e8f, hf\u27e9\n[GOAL]\ncase mk\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : Semiring S\nf\u271d : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nf_compat : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f\u271d k\u2082) = f\u271d k\u2081\nf : (k : \u2115) \u2192 S \u2192+* TruncatedWittVector p k R\nhf : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (f k\u2082) = f k\u2081\n\u22a2 (fun g =>\n { val := fun k => RingHom.comp (truncate k) g,\n property :=\n (_ :\n \u2200 (k\u2081 k\u2082 : \u2115) (h : k\u2081 \u2264 k\u2082),\n RingHom.comp (RingHom.comp (TruncatedWittVector.truncate h) (truncate k\u2082)) g =\n RingHom.comp (truncate k\u2081) g) })\n ((fun f =>\n lift \u2191f (_ : \u2200 (k\u2081 k\u2082 : \u2115) (hk : k\u2081 \u2264 k\u2082), RingHom.comp (TruncatedWittVector.truncate hk) (\u2191f k\u2082) = \u2191f k\u2081))\n { val := f, property := hf }) =\n { val := f, property := hf }\n[PROOFSTEP]\nsimp only [truncate_comp_lift]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.WittVector.Truncated", "llama_tokens": 14223, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.4219191792734037}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf g : X \u27f6 Y\ninst\u271d : HasEqualizer f g\n\u22a2 (equalizerSubobjectIso f g).hom \u226b equalizer.\u03b9 f g = arrow (equalizerSubobject f g)\n[PROOFSTEP]\nsimp [equalizerSubobjectIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf g : X \u27f6 Y\ninst\u271d : HasEqualizer f g\n\u22a2 (equalizerSubobjectIso f g).inv \u226b arrow (equalizerSubobject f g) = equalizer.\u03b9 f g\n[PROOFSTEP]\nsimp [equalizerSubobjectIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf g : X \u27f6 Y\ninst\u271d : HasEqualizer f g\n\u22a2 arrow (equalizerSubobject f g) \u226b f = arrow (equalizerSubobject f g) \u226b g\n[PROOFSTEP]\nrw [\u2190 equalizerSubobject_arrow, Category.assoc, Category.assoc, equalizer.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf g : X \u27f6 Y\ninst\u271d : HasEqualizer f g\nW : C\nh : W \u27f6 X\nw : h \u226b f = h \u226b g\n\u22a2 equalizer.lift h w \u226b MonoOver.arrow (MonoOver.mk' (equalizer.\u03b9 f g)) = h\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf g : X \u27f6 Y\ninst\u271d : HasEqualizer f g\nW : C\nh : W \u27f6 X\nw : Factors (equalizerSubobject f g) h\n\u22a2 h \u226b f = h \u226b g\n[PROOFSTEP]\nrw [\u2190 Subobject.factorThru_arrow _ _ w, Category.assoc, equalizerSubobject_arrow_comp, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\n\u22a2 (kernelSubobjectIso f).hom \u226b kernel.\u03b9 f = arrow (kernelSubobject f)\n[PROOFSTEP]\nsimp [kernelSubobjectIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\n\u22a2 (kernelSubobjectIso f).inv \u226b arrow (kernelSubobject f) = kernel.\u03b9 f\n[PROOFSTEP]\nsimp [kernelSubobjectIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\n\u22a2 arrow (kernelSubobject f) \u226b f = 0\n[PROOFSTEP]\nrw [\u2190 kernelSubobject_arrow]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\n\u22a2 ((kernelSubobjectIso f).hom \u226b kernel.\u03b9 f) \u226b f = 0\n[PROOFSTEP]\nsimp only [Category.assoc, kernel.condition, comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nW : C\nh : W \u27f6 X\nw : h \u226b f = 0\n\u22a2 kernel.lift f h w \u226b MonoOver.arrow (MonoOver.mk' (kernel.\u03b9 f)) = h\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nW : C\nh : W \u27f6 X\nw : Factors (kernelSubobject f) h\n\u22a2 h \u226b f = 0\n[PROOFSTEP]\nrw [\u2190 Subobject.factorThru_arrow _ _ w, Category.assoc, kernelSubobject_arrow_comp, comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nW : C\nh : W \u27f6 X\nw : h \u226b f = 0\n\u22a2 factorThruKernelSubobject f h w \u226b arrow (kernelSubobject f) = h\n[PROOFSTEP]\ndsimp [factorThruKernelSubobject]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nW : C\nh : W \u27f6 X\nw : h \u226b f = 0\n\u22a2 factorThru (kernelSubobject f) h (_ : Factors (kernelSubobject f) h) \u226b arrow (kernelSubobject f) = h\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nW : C\nh : W \u27f6 X\nw : h \u226b f = 0\n\u22a2 (factorThruKernelSubobject f h w \u226b (kernelSubobjectIso f).hom) \u226b kernel.\u03b9 f = kernel.lift f h w \u226b kernel.\u03b9 f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nX' Y' : C\nf' : X' \u27f6 Y'\ninst\u271d : HasKernel f'\nsq : Arrow.mk f \u27f6 Arrow.mk f'\n\u22a2 (arrow (kernelSubobject f) \u226b sq.left) \u226b f' = 0\n[PROOFSTEP]\nsimp [sq.w]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nX' Y' : C\nf' : X' \u27f6 Y'\ninst\u271d : HasKernel f'\nsq : Arrow.mk f \u27f6 Arrow.mk f'\n\u22a2 kernelSubobjectMap sq \u226b arrow (kernelSubobject f') = arrow (kernelSubobject f) \u226b sq.left\n[PROOFSTEP]\nsimp [kernelSubobjectMap]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nX' Y' : C\nf' : X' \u27f6 Y'\ninst\u271d : HasKernel f'\n\u22a2 kernelSubobjectMap (\ud835\udfd9 (Arrow.mk f)) = \ud835\udfd9 (underlying.obj (kernelSubobject f))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\nX' Y' : C\nf' : X' \u27f6 Y'\ninst\u271d\u00b9 : HasKernel f'\nX'' Y'' : C\nf'' : X'' \u27f6 Y''\ninst\u271d : HasKernel f''\nsq : Arrow.mk f \u27f6 Arrow.mk f'\nsq' : Arrow.mk f' \u27f6 Arrow.mk f''\n\u22a2 kernelSubobjectMap (sq \u226b sq') = kernelSubobjectMap sq \u226b kernelSubobjectMap sq'\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nX' Y' : C\nf' : X' \u27f6 Y'\ninst\u271d : HasKernel f'\nsq : Arrow.mk f \u27f6 Arrow.mk f'\n\u22a2 kernel.map f f' sq.left sq.right (_ : (Arrow.mk f).hom \u226b (\ud835\udfed C).map sq.right = (\ud835\udfed C).map sq.left \u226b (Arrow.mk f').hom) \u226b\n (kernelSubobjectIso f').inv =\n (kernelSubobjectIso f).inv \u226b kernelSubobjectMap sq\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nX' Y' : C\nf' : X' \u27f6 Y'\ninst\u271d : HasKernel f'\nsq : Arrow.mk f \u27f6 Arrow.mk f'\n\u22a2 (kernelSubobjectIso f).hom \u226b\n kernel.map f f' sq.left sq.right\n (_ : (Arrow.mk f).hom \u226b (\ud835\udfed C).map sq.right = (\ud835\udfed C).map sq.left \u226b (Arrow.mk f').hom) =\n kernelSubobjectMap sq \u226b (kernelSubobjectIso f').hom\n[PROOFSTEP]\nsimp [\u2190 Iso.comp_inv_eq, kernel_map_comp_kernelSubobjectIso_inv]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nA B : C\n\u22a2 IsIso (kernel.\u03b9 0)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nA : Subobject X\nh : arrow A \u226b f = 0\n\u22a2 kernel.lift f (arrow A) h \u226b kernel.\u03b9 f = arrow A\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nX' : C\nf : X' \u27f6 X\ninst\u271d\u00b9 : IsIso f\ng : X \u27f6 Y\ninst\u271d : HasKernel g\n\u22a2 (kernelSubobjectIsoComp f g).hom \u226b arrow (kernelSubobject g) = arrow (kernelSubobject (f \u226b g)) \u226b f\n[PROOFSTEP]\nsimp [kernelSubobjectIsoComp]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nX' : C\nf : X' \u27f6 X\ninst\u271d\u00b9 : IsIso f\ng : X \u27f6 Y\ninst\u271d : HasKernel g\n\u22a2 (kernelSubobjectIsoComp f g).inv \u226b arrow (kernelSubobject (f \u226b g)) = arrow (kernelSubobject g) \u226b inv f\n[PROOFSTEP]\nsimp [kernelSubobjectIsoComp]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX Y Z\u271d : C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nZ : C\nh : Y \u27f6 Z\ninst\u271d : HasKernel (f \u226b h)\n\u22a2 arrow (kernelSubobject f) \u226b f \u226b h = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX Y Z\u271d : C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nZ : C\nh : Y \u27f6 Z\ninst\u271d : Mono h\n\u22a2 (arrow (kernelSubobject (f \u226b h)) \u226b f) \u226b h = 0 \u226b h\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX Y Z\u271d : C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nZ : C\nh : Y \u27f6 Z\ninst\u271d : Mono h\n\u22a2 IsIso (ofLE (kernelSubobject f) (kernelSubobject (f \u226b h)) (_ : kernelSubobject f \u2264 kernelSubobject (f \u226b h)))\n[PROOFSTEP]\nrw [ofLE_mk_le_mk_of_comm (kernelCompMono f h).inv]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX Y Z\u271d : C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nZ : C\nh : Y \u27f6 Z\ninst\u271d : Mono h\n\u22a2 IsIso ((underlyingIso (kernel.\u03b9 f)).hom \u226b (kernelCompMono f h).inv \u226b (underlyingIso (kernel.\u03b9 (f \u226b h))).inv)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX Y Z\u271d : C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nZ : C\nh : Y \u27f6 Z\ninst\u271d : Mono h\n\u22a2 (kernelCompMono f h).inv \u226b kernel.\u03b9 (f \u226b h) = kernel.\u03b9 f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasCokernels C\nX : C\n\u22a2 \u2200 \u2983A B : C\u2984 (f : A \u27f6 X) (g : B \u27f6 X) [inst : Mono f] [inst_1 : Mono g] (i : A \u2245 B),\n i.hom \u226b g = f \u2192\n (fun A f x => Subobject.mk (cokernel.\u03c0 f).op) A f inst = (fun A f x => Subobject.mk (cokernel.\u03c0 f).op) B g inst_1\n[PROOFSTEP]\nrintro A B f g hf hg i rfl\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasCokernels C\nX A B : C\ng : B \u27f6 X\nhg : Mono g\ni : A \u2245 B\nhf : Mono (i.hom \u226b g)\n\u22a2 (fun A f x => Subobject.mk (cokernel.\u03c0 f).op) A (i.hom \u226b g) hf = (fun A f x => Subobject.mk (cokernel.\u03c0 f).op) B g hg\n[PROOFSTEP]\nrefine' Subobject.mk_eq_mk_of_comm _ _ (Iso.op _) (Quiver.Hom.unop_inj _)\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasCokernels C\nX A B : C\ng : B \u27f6 X\nhg : Mono g\ni : A \u2245 B\nhf : Mono (i.hom \u226b g)\n\u22a2 cokernel g \u2245 cokernel (i.hom \u226b g)\n[PROOFSTEP]\nexact\n (IsColimit.coconePointUniqueUpToIso (colimit.isColimit _) (isCokernelEpiComp (colimit.isColimit _) i.hom rfl)).symm\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasCokernels C\nX A B : C\ng : B \u27f6 X\nhg : Mono g\ni : A \u2245 B\nhf : Mono (i.hom \u226b g)\n\u22a2 ((Iso.op\n (IsColimit.coconePointUniqueUpToIso (colimit.isColimit (parallelPair (i.hom \u226b g) 0))\n (isCokernelEpiComp (colimit.isColimit (parallelPair g 0)) i.hom\n (_ : i.hom \u226b g = i.hom \u226b g))).symm).hom \u226b\n (cokernel.\u03c0 g).op).unop =\n (cokernel.\u03c0 (i.hom \u226b g)).op.unop\n[PROOFSTEP]\nsimp only [Iso.comp_inv_eq, Iso.op_hom, Iso.symm_hom, unop_comp, Quiver.Hom.unop_op,\n colimit.comp_coconePointUniqueUpToIso_hom, Cofork.of\u03c0_\u03b9_app, coequalizer.cofork_\u03c0]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasCokernels C\nX : C\n\u22a2 \u2200 \u2983A B : C\u2984 (f : A \u27f6 X) (g : B \u27f6 X) [inst : Mono f] [inst_1 : Mono g],\n Subobject.mk f \u2264 Subobject.mk g \u2192\n Subobject.lift (fun A f x => Subobject.mk (cokernel.\u03c0 f).op)\n (_ :\n \u2200 \u2983A B : C\u2984 (f : A \u27f6 X) (g : B \u27f6 X) [hf : Mono f] [hg : Mono g] (i : A \u2245 B),\n i.hom \u226b g = f \u2192\n (fun A f x => Subobject.mk (cokernel.\u03c0 f).op) A f hf =\n (fun A f x => Subobject.mk (cokernel.\u03c0 f).op) B g hg)\n (Subobject.mk f) \u2264\n Subobject.lift (fun A f x => Subobject.mk (cokernel.\u03c0 f).op)\n (_ :\n \u2200 \u2983A B : C\u2984 (f : A \u27f6 X) (g : B \u27f6 X) [hf : Mono f] [hg : Mono g] (i : A \u2245 B),\n i.hom \u226b g = f \u2192\n (fun A f x => Subobject.mk (cokernel.\u03c0 f).op) A f hf =\n (fun A f x => Subobject.mk (cokernel.\u03c0 f).op) B g hg)\n (Subobject.mk g)\n[PROOFSTEP]\nintro A B f g hf hg h\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf\u271d : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\u271d\ninst\u271d : HasCokernels C\nX A B : C\nf : A \u27f6 X\ng : B \u27f6 X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f \u2264 Subobject.mk g\n\u22a2 Subobject.lift (fun A f x => Subobject.mk (cokernel.\u03c0 f).op)\n (_ :\n \u2200 \u2983A B : C\u2984 (f : A \u27f6 X) (g : B \u27f6 X) [hf : Mono f] [hg : Mono g] (i : A \u2245 B),\n i.hom \u226b g = f \u2192\n (fun A f x => Subobject.mk (cokernel.\u03c0 f).op) A f hf = (fun A f x => Subobject.mk (cokernel.\u03c0 f).op) B g hg)\n (Subobject.mk f) \u2264\n Subobject.lift (fun A f x => Subobject.mk (cokernel.\u03c0 f).op)\n (_ :\n \u2200 \u2983A B : C\u2984 (f : A \u27f6 X) (g : B \u27f6 X) [hf : Mono f] [hg : Mono g] (i : A \u2245 B),\n i.hom \u226b g = f \u2192\n (fun A f x => Subobject.mk (cokernel.\u03c0 f).op) A f hf = (fun A f x => Subobject.mk (cokernel.\u03c0 f).op) B g hg)\n (Subobject.mk g)\n[PROOFSTEP]\ndsimp only [Subobject.lift_mk]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf\u271d : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\u271d\ninst\u271d : HasCokernels C\nX A B : C\nf : A \u27f6 X\ng : B \u27f6 X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f \u2264 Subobject.mk g\n\u22a2 Subobject.mk (cokernel.\u03c0 f).op \u2264 Subobject.mk (cokernel.\u03c0 g).op\n[PROOFSTEP]\nrefine' Subobject.mk_le_mk_of_comm (cokernel.desc f (cokernel.\u03c0 g) _).op _\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf\u271d : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\u271d\ninst\u271d : HasCokernels C\nX A B : C\nf : A \u27f6 X\ng : B \u27f6 X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f \u2264 Subobject.mk g\n\u22a2 f \u226b cokernel.\u03c0 g = 0\n[PROOFSTEP]\nrw [\u2190 Subobject.ofMkLEMk_comp h, Category.assoc, cokernel.condition, comp_zero]\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf\u271d : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\u271d\ninst\u271d : HasCokernels C\nX A B : C\nf : A \u27f6 X\ng : B \u27f6 X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f \u2264 Subobject.mk g\n\u22a2 (cokernel.desc f (cokernel.\u03c0 g) (_ : f \u226b cokernel.\u03c0 g = 0)).op \u226b (cokernel.\u03c0 f).op = (cokernel.\u03c0 g).op\n[PROOFSTEP]\nexact Quiver.Hom.unop_inj (cokernel.\u03c0_desc _ _ _)\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernels C\nX : C\n\u22a2 \u2200 \u2983A B : C\u1d52\u1d56\u2984 (f : A \u27f6 op X) (g : B \u27f6 op X) [inst : Mono f] [inst_1 : Mono g] (i : A \u2245 B),\n i.hom \u226b g = f \u2192\n (fun A f x => Subobject.mk (kernel.\u03b9 f.unop)) A f inst = (fun A f x => Subobject.mk (kernel.\u03b9 f.unop)) B g inst_1\n[PROOFSTEP]\nrintro A B f g hf hg i rfl\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernels C\nX : C\nA B : C\u1d52\u1d56\ng : B \u27f6 op X\nhg : Mono g\ni : A \u2245 B\nhf : Mono (i.hom \u226b g)\n\u22a2 (fun A f x => Subobject.mk (kernel.\u03b9 f.unop)) A (i.hom \u226b g) hf = (fun A f x => Subobject.mk (kernel.\u03b9 f.unop)) B g hg\n[PROOFSTEP]\nrefine' Subobject.mk_eq_mk_of_comm _ _ _ _\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernels C\nX : C\nA B : C\u1d52\u1d56\ng : B \u27f6 op X\nhg : Mono g\ni : A \u2245 B\nhf : Mono (i.hom \u226b g)\n\u22a2 kernel (i.hom \u226b g).unop \u2245 kernel g.unop\n[PROOFSTEP]\nexact\n IsLimit.conePointUniqueUpToIso (limit.isLimit _)\n (isKernelCompMono (limit.isLimit (parallelPair g.unop 0)) i.unop.hom rfl)\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernels C\nX : C\nA B : C\u1d52\u1d56\ng : B \u27f6 op X\nhg : Mono g\ni : A \u2245 B\nhf : Mono (i.hom \u226b g)\n\u22a2 (IsLimit.conePointUniqueUpToIso (limit.isLimit (parallelPair (g.unop \u226b (Iso.unop i).hom) 0))\n (isKernelCompMono (limit.isLimit (parallelPair g.unop 0)) (Iso.unop i).hom\n (_ : g.unop \u226b (Iso.unop i).hom = g.unop \u226b (Iso.unop i).hom))).hom \u226b\n kernel.\u03b9 g.unop =\n kernel.\u03b9 (i.hom \u226b g).unop\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernels C\nX : C\nA B : C\u1d52\u1d56\ng : B \u27f6 op X\nhg : Mono g\ni : A \u2245 B\nhf : Mono (i.hom \u226b g)\n\u22a2 (IsLimit.conePointUniqueUpToIso (limit.isLimit (parallelPair (g.unop \u226b i.hom.unop) 0))\n (isKernelCompMono (limit.isLimit (parallelPair g.unop 0)) i.hom.unop\n (_ : g.unop \u226b i.hom.unop = g.unop \u226b i.hom.unop))).hom \u226b\n kernel.\u03b9 g.unop =\n kernel.\u03b9 (g.unop \u226b i.hom.unop)\n[PROOFSTEP]\nsimp only [\u2190 Iso.eq_inv_comp, limit.conePointUniqueUpToIso_inv_comp, Fork.of\u03b9_\u03c0_app]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\ninst\u271d : HasKernels C\nX : C\n\u22a2 \u2200 \u2983A B : C\u1d52\u1d56\u2984 (f : A \u27f6 op X) (g : B \u27f6 op X) [inst : Mono f] [inst_1 : Mono g],\n Subobject.mk f \u2264 Subobject.mk g \u2192\n Subobject.lift (fun A f x => Subobject.mk (kernel.\u03b9 f.unop))\n (_ :\n \u2200 \u2983A B : C\u1d52\u1d56\u2984 (f : A \u27f6 op X) (g : B \u27f6 op X) [hf : Mono f] [hg : Mono g] (i : A \u2245 B),\n i.hom \u226b g = f \u2192\n (fun A f x => Subobject.mk (kernel.\u03b9 f.unop)) A f hf =\n (fun A f x => Subobject.mk (kernel.\u03b9 f.unop)) B g hg)\n (Subobject.mk f) \u2264\n Subobject.lift (fun A f x => Subobject.mk (kernel.\u03b9 f.unop))\n (_ :\n \u2200 \u2983A B : C\u1d52\u1d56\u2984 (f : A \u27f6 op X) (g : B \u27f6 op X) [hf : Mono f] [hg : Mono g] (i : A \u2245 B),\n i.hom \u226b g = f \u2192\n (fun A f x => Subobject.mk (kernel.\u03b9 f.unop)) A f hf =\n (fun A f x => Subobject.mk (kernel.\u03b9 f.unop)) B g hg)\n (Subobject.mk g)\n[PROOFSTEP]\nintro A B f g hf hg h\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf\u271d : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\u271d\ninst\u271d : HasKernels C\nX : C\nA B : C\u1d52\u1d56\nf : A \u27f6 op X\ng : B \u27f6 op X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f \u2264 Subobject.mk g\n\u22a2 Subobject.lift (fun A f x => Subobject.mk (kernel.\u03b9 f.unop))\n (_ :\n \u2200 \u2983A B : C\u1d52\u1d56\u2984 (f : A \u27f6 op X) (g : B \u27f6 op X) [hf : Mono f] [hg : Mono g] (i : A \u2245 B),\n i.hom \u226b g = f \u2192\n (fun A f x => Subobject.mk (kernel.\u03b9 f.unop)) A f hf = (fun A f x => Subobject.mk (kernel.\u03b9 f.unop)) B g hg)\n (Subobject.mk f) \u2264\n Subobject.lift (fun A f x => Subobject.mk (kernel.\u03b9 f.unop))\n (_ :\n \u2200 \u2983A B : C\u1d52\u1d56\u2984 (f : A \u27f6 op X) (g : B \u27f6 op X) [hf : Mono f] [hg : Mono g] (i : A \u2245 B),\n i.hom \u226b g = f \u2192\n (fun A f x => Subobject.mk (kernel.\u03b9 f.unop)) A f hf = (fun A f x => Subobject.mk (kernel.\u03b9 f.unop)) B g hg)\n (Subobject.mk g)\n[PROOFSTEP]\ndsimp only [Subobject.lift_mk]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf\u271d : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\u271d\ninst\u271d : HasKernels C\nX : C\nA B : C\u1d52\u1d56\nf : A \u27f6 op X\ng : B \u27f6 op X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f \u2264 Subobject.mk g\n\u22a2 Subobject.mk (kernel.\u03b9 f.unop) \u2264 Subobject.mk (kernel.\u03b9 g.unop)\n[PROOFSTEP]\nrefine' Subobject.mk_le_mk_of_comm (kernel.lift g.unop (kernel.\u03b9 f.unop) _) _\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf\u271d : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\u271d\ninst\u271d : HasKernels C\nX : C\nA B : C\u1d52\u1d56\nf : A \u27f6 op X\ng : B \u27f6 op X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f \u2264 Subobject.mk g\n\u22a2 kernel.\u03b9 f.unop \u226b g.unop = 0\n[PROOFSTEP]\nrw [\u2190 Subobject.ofMkLEMk_comp h, unop_comp, kernel.condition_assoc, zero_comp]\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf\u271d : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\u271d\ninst\u271d : HasKernels C\nX : C\nA B : C\u1d52\u1d56\nf : A \u27f6 op X\ng : B \u27f6 op X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f \u2264 Subobject.mk g\n\u22a2 kernel.lift g.unop (kernel.\u03b9 f.unop) (_ : kernel.\u03b9 f.unop \u226b g.unop = 0) \u226b kernel.\u03b9 g.unop = kernel.\u03b9 f.unop\n[PROOFSTEP]\nexact Quiver.Hom.op_inj (by simp)\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf\u271d : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\u271d\ninst\u271d : HasKernels C\nX : C\nA B : C\u1d52\u1d56\nf : A \u27f6 op X\ng : B \u27f6 op X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f \u2264 Subobject.mk g\n\u22a2 (kernel.lift g.unop (kernel.\u03b9 f.unop) (_ : kernel.\u03b9 f.unop \u226b g.unop = 0) \u226b kernel.\u03b9 g.unop).op = (kernel.\u03b9 f.unop).op\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : HasImage f\n\u22a2 (imageSubobjectIso f).hom \u226b image.\u03b9 f = arrow (imageSubobject f)\n[PROOFSTEP]\nsimp [imageSubobjectIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : HasImage f\n\u22a2 (imageSubobjectIso f).inv \u226b arrow (imageSubobject f) = image.\u03b9 f\n[PROOFSTEP]\nsimp [imageSubobjectIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasImage f\ninst\u271d : HasEqualizers C\n\u22a2 Epi (factorThruImageSubobject f)\n[PROOFSTEP]\ndsimp [factorThruImageSubobject]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasImage f\ninst\u271d : HasEqualizers C\n\u22a2 Epi (factorThruImage f \u226b (imageSubobjectIso f).inv)\n[PROOFSTEP]\napply epi_comp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : HasImage f\n\u22a2 factorThruImageSubobject f \u226b arrow (imageSubobject f) = f\n[PROOFSTEP]\nsimp [factorThruImageSubobject, imageSubobject_arrow]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b3 : HasImage f\u271d\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasImage f\ninst\u271d : Epi (factorThruImageSubobject f)\nh : f \u226b g = 0\n\u22a2 factorThruImageSubobject f \u226b arrow (imageSubobject f) \u226b g = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : HasImage f\nW : C\nk : W \u27f6 X\n\u22a2 (k \u226b factorThruImage f) \u226b MonoOver.arrow (MonoOver.mk' (image.\u03b9 f)) = k \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : HasImage f\nW : C\nk : W \u27f6 X\nh : Factors (imageSubobject f) (k \u226b f)\n\u22a2 factorThru (imageSubobject f) (k \u226b f) h = k \u226b factorThruImageSubobject f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : HasImage f\nW : C\nk : W \u27f6 X\nh : Factors (imageSubobject f) (k \u226b f)\n\u22a2 factorThru (imageSubobject f) (k \u226b f) h \u226b arrow (imageSubobject f) =\n (k \u226b factorThruImageSubobject f) \u226b arrow (imageSubobject f)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : HasImage f\nW W' : C\nk : W \u27f6 W'\nk' : W' \u27f6 X\nh : Factors (imageSubobject f) (k \u226b k' \u226b f)\n\u22a2 factorThru (imageSubobject f) (k \u226b k' \u226b f) h = k \u226b k' \u226b factorThruImageSubobject f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : HasImage f\nW W' : C\nk : W \u27f6 W'\nk' : W' \u27f6 X\nh : Factors (imageSubobject f) (k \u226b k' \u226b f)\n\u22a2 factorThru (imageSubobject f) (k \u226b k' \u226b f) h \u226b arrow (imageSubobject f) =\n (k \u226b k' \u226b factorThruImageSubobject f) \u226b arrow (imageSubobject f)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX Y Z : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasImage f\u271d\nX' : C\nh : X' \u27f6 X\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasImage f\ninst\u271d : HasImage (h \u226b f)\n\u22a2 image.preComp h f \u226b image.\u03b9 f = image.\u03b9 (h \u226b f)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d\u00b2 : HasImage f\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\n\u22a2 arrow (imageSubobject 0) = 0\n[PROOFSTEP]\nrw [\u2190 imageSubobject_arrow]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d\u00b2 : HasImage f\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\n\u22a2 (imageSubobjectIso 0).hom \u226b image.\u03b9 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d\u00b2 : HasImage f\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nA B : C\n\u22a2 (imageSubobjectIso 0 \u226a\u226b imageZero \u226a\u226b botCoeIsoZero.symm).hom \u226b arrow \u22a5 = arrow (imageSubobject 0)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nX Y Z : C\nf\u271d : X \u27f6 Y\ninst\u271d\u2074 : HasImage f\u271d\ninst\u271d\u00b3 : HasEqualizers C\nX' : C\nh : X' \u27f6 X\ninst\u271d\u00b2 : Epi h\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasImage f\ninst\u271d : HasImage (h \u226b f)\n\u22a2 Epi (ofLE (imageSubobject (h \u226b f)) (imageSubobject f) (_ : imageSubobject (h \u226b f) \u2264 imageSubobject f))\n[PROOFSTEP]\nrw [ofLE_mk_le_mk_of_comm (image.preComp h f)]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nX Y Z : C\nf\u271d : X \u27f6 Y\ninst\u271d\u2074 : HasImage f\u271d\ninst\u271d\u00b3 : HasEqualizers C\nX' : C\nh : X' \u27f6 X\ninst\u271d\u00b2 : Epi h\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasImage f\ninst\u271d : HasImage (h \u226b f)\n\u22a2 Epi ((underlyingIso (image.\u03b9 (h \u226b f))).hom \u226b image.preComp h f \u226b (underlyingIso (image.\u03b9 f)).inv)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nX Y Z : C\nf\u271d : X \u27f6 Y\ninst\u271d\u2074 : HasImage f\u271d\ninst\u271d\u00b3 : HasEqualizers C\nX' : C\nh : X' \u27f6 X\ninst\u271d\u00b2 : Epi h\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasImage f\ninst\u271d : HasImage (h \u226b f)\n\u22a2 image.preComp h f \u226b image.\u03b9 f = image.\u03b9 (h \u226b f)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX Y Z : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\u271d\ninst\u271d\u00b2 : HasEqualizers C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasImage f\nY' : C\nh : Y \u27f6 Y'\ninst\u271d : IsIso h\n\u22a2 (imageSubobjectCompIso f h).hom \u226b arrow (imageSubobject f) = arrow (imageSubobject (f \u226b h)) \u226b inv h\n[PROOFSTEP]\nsimp [imageSubobjectCompIso]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX Y Z : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\u271d\ninst\u271d\u00b2 : HasEqualizers C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasImage f\nY' : C\nh : Y \u27f6 Y'\ninst\u271d : IsIso h\n\u22a2 (imageSubobjectCompIso f h).inv \u226b arrow (imageSubobject (f \u226b h)) = arrow (imageSubobject f) \u226b h\n[PROOFSTEP]\nsimp [imageSubobjectCompIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b9 : HasImage f\u271d\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 (imageSubobjectIso f \u226a\u226b imageMonoIsoSource f \u226a\u226b (underlyingIso f).symm).hom \u226b arrow (Subobject.mk f) =\n arrow (imageSubobject f)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX Y Z : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\u271d\ninst\u271d\u00b2 : HasEqualizers C\nX' : C\nh : X' \u27f6 X\ninst\u271d\u00b9 : IsIso h\nf : X \u27f6 Y\ninst\u271d : HasImage f\n\u22a2 inv (image.preComp h f) \u226b image.\u03b9 (h \u226b f) = image.\u03b9 f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y Z : C\nf\u271d : X\u271d \u27f6 Y\ninst\u271d\u00b9 : HasImage f\u271d\nA B : C\nX : Subobject B\nf : A \u27f6 B\ninst\u271d : HasImage f\nh : A \u27f6 underlying.obj X\nw : h \u226b arrow X = f\n\u22a2 ((imageSubobjectIso f).hom \u226b image.lift (MonoFactorisation.mk (underlying.obj X) (arrow X) h)) \u226b arrow X =\n arrow (imageSubobject f)\n[PROOFSTEP]\nrw [assoc, image.lift_fac, imageSubobject_arrow]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nX\u271d Y Z : C\nf\u271d : X\u271d \u27f6 Y\ninst\u271d\u00b2 : HasImage f\u271d\nA B X : C\ng : X \u27f6 B\ninst\u271d\u00b9 : Mono g\nf : A \u27f6 B\ninst\u271d : HasImage f\nh : A \u27f6 X\nw : h \u226b g = f\n\u22a2 (h \u226b (underlyingIso g).inv) \u226b arrow (Subobject.mk g) = f\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b3 : HasImage f\u271d\nW X Y Z : C\nf : W \u27f6 X\ninst\u271d\u00b2 : HasImage f\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasImage g\nsq : Arrow.mk f \u27f6 Arrow.mk g\ninst\u271d : HasImageMap sq\n\u22a2 imageSubobjectMap sq \u226b arrow (imageSubobject g) = arrow (imageSubobject f) \u226b sq.right\n[PROOFSTEP]\nsimp only [imageSubobjectMap, Category.assoc, imageSubobject_arrow']\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b3 : HasImage f\u271d\nW X Y Z : C\nf : W \u27f6 X\ninst\u271d\u00b2 : HasImage f\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasImage g\nsq : Arrow.mk f \u27f6 Arrow.mk g\ninst\u271d : HasImageMap sq\n\u22a2 (imageSubobjectIso f).hom \u226b image.map sq \u226b image.\u03b9 g = arrow (imageSubobject f) \u226b sq.right\n[PROOFSTEP]\nerw [image.map_\u03b9, \u2190 Category.assoc, imageSubobject_arrow]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b3 : HasImage f\u271d\nW X Y Z : C\nf : W \u27f6 X\ninst\u271d\u00b2 : HasImage f\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasImage g\nsq : Arrow.mk f \u27f6 Arrow.mk g\ninst\u271d : HasImageMap sq\n\u22a2 image.map sq \u226b (imageSubobjectIso (Arrow.mk g).hom).inv = (imageSubobjectIso f).inv \u226b imageSubobjectMap sq\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b3 : HasImage f\u271d\nW X Y Z : C\nf : W \u27f6 X\ninst\u271d\u00b2 : HasImage f\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasImage g\nsq : Arrow.mk f \u27f6 Arrow.mk g\ninst\u271d : HasImageMap sq\n\u22a2 (image.map sq \u226b (imageSubobjectIso (Arrow.mk g).hom).inv) \u226b arrow (imageSubobject (Arrow.mk g).hom) =\n ((imageSubobjectIso f).inv \u226b imageSubobjectMap sq) \u226b arrow (imageSubobject (Arrow.mk g).hom)\n[PROOFSTEP]\nsimpa using image.map_\u03b9 sq\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ninst\u271d\u00b3 : HasImage f\u271d\nW X Y Z : C\nf : W \u27f6 X\ninst\u271d\u00b2 : HasImage f\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasImage g\nsq : Arrow.mk f \u27f6 Arrow.mk g\ninst\u271d : HasImageMap sq\n\u22a2 (imageSubobjectIso (Arrow.mk f).hom).hom \u226b image.map sq = imageSubobjectMap sq \u226b (imageSubobjectIso g).hom\n[PROOFSTEP]\nerw [\u2190 Iso.comp_inv_eq, Category.assoc, \u2190 (imageSubobjectIso f).eq_inv_comp, image_map_comp_imageSubobjectIso_inv sq]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Subobject.Limits", "llama_tokens": 13945, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.640635854839898, "lm_q1q2_score": 0.42180585831388284}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\nhx : x \u2264 a\n\u22a2 projIcc a b h x = { val := a, property := (_ : a \u2208 Icc a b) }\n[PROOFSTEP]\nsimp [projIcc, hx, hx.trans h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\nhx : b \u2264 x\n\u22a2 projIcc a b h x = { val := b, property := (_ : b \u2208 Icc a b) }\n[PROOFSTEP]\nsimp [projIcc, hx, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\n\u22a2 projIci a x = { val := a, property := (_ : a \u2264 a) } \u2194 x \u2264 a\n[PROOFSTEP]\nsimp [projIci, Subtype.ext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\n\u22a2 projIic b x = { val := b, property := (_ : b \u2264 b) } \u2194 b \u2264 x\n[PROOFSTEP]\nsimp [projIic, Subtype.ext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d : a \u2264 b\nx : \u03b1\nh : a < b\n\u22a2 projIcc a b (_ : a \u2264 b) x = { val := a, property := (_ : a \u2208 Icc a b) } \u2194 x \u2264 a\n[PROOFSTEP]\nsimp [projIcc, Subtype.ext_iff, h.not_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d : a \u2264 b\nx : \u03b1\nh : a < b\n\u22a2 projIcc a b (_ : a \u2264 b) x = { val := b, property := (_ : b \u2208 Icc a b) } \u2194 b \u2264 x\n[PROOFSTEP]\nsimp [projIcc, Subtype.ext_iff, max_min_distrib_left, h.le, h.not_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\nhx : x \u2208 Ici a\n\u22a2 projIci a x = { val := x, property := hx }\n[PROOFSTEP]\nsimpa [projIci]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\nhx : x \u2208 Iic b\n\u22a2 projIic b x = { val := x, property := hx }\n[PROOFSTEP]\nsimpa [projIic]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\nhx : x \u2208 Icc a b\n\u22a2 projIcc a b h x = { val := x, property := hx }\n[PROOFSTEP]\nsimp [projIcc, hx.1, hx.2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx\u271d : \u03b1\nx : \u2191(Ici a)\n\u22a2 projIci a \u2191x = x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx val\u271d : \u03b1\nproperty\u271d : val\u271d \u2208 Ici a\n\u22a2 projIci a \u2191{ val := val\u271d, property := property\u271d } = { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\napply projIci_of_mem\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx\u271d : \u03b1\nx : \u2191(Iic b)\n\u22a2 projIic b \u2191x = x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx val\u271d : \u03b1\nproperty\u271d : val\u271d \u2208 Iic b\n\u22a2 projIic b \u2191{ val := val\u271d, property := property\u271d } = { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\napply projIic_of_mem\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx\u271d : \u03b1\nx : \u2191(Icc a b)\n\u22a2 projIcc a b h \u2191x = x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx val\u271d : \u03b1\nproperty\u271d : val\u271d \u2208 Icc a b\n\u22a2 projIcc a b h \u2191{ val := val\u271d, property := property\u271d } = { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\napply projIcc_of_mem\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx\u271d x : \u03b1\nhx : x \u2208 Ici a\ny : \u03b1\nhy : y \u2208 Ici a\nhxy : x < y\n\u22a2 projIci a x < projIci a y\n[PROOFSTEP]\nsimpa only [projIci_of_mem, hx, hy]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx\u271d x : \u03b1\nhx : x \u2208 Iic b\ny : \u03b1\nhy : y \u2208 Iic b\nhxy : x < y\n\u22a2 projIic b x < projIic b y\n[PROOFSTEP]\nsimpa only [projIic_of_mem, hx, hy]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx\u271d x : \u03b1\nhx : x \u2208 Icc a b\ny : \u03b1\nhy : y \u2208 Icc a b\nhxy : x < y\n\u22a2 projIcc a b h x < projIcc a b h y\n[PROOFSTEP]\nsimpa only [projIcc_of_mem, hx, hy]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\nf : \u2191(Ici a) \u2192 \u03b2\n\u22a2 range (IciExtend f) = range f\n[PROOFSTEP]\nsimp only [IciExtend, range_comp f, range_projIci, range_id', image_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\nf : \u2191(Iic b) \u2192 \u03b2\n\u22a2 range (IicExtend f) = range f\n[PROOFSTEP]\nsimp only [IicExtend, range_comp f, range_projIic, range_id', image_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\nf : \u2191(Icc a b) \u2192 \u03b2\n\u22a2 range (IccExtend h f) = range f\n[PROOFSTEP]\nsimp only [IccExtend, range_comp f, range_projIcc, image_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx : \u03b1\nf : \u03b1 \u2192 \u03b2\nha : \u2200 (x : \u03b1), x < a \u2192 f x = f a\nhb : \u2200 (x : \u03b1), b < x \u2192 f x = f b\n\u22a2 IccExtend h (f \u2218 Subtype.val) = f\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\nha : \u2200 (x : \u03b1), x < a \u2192 f x = f a\nhb : \u2200 (x : \u03b1), b < x \u2192 f x = f b\nx : \u03b1\n\u22a2 IccExtend h (f \u2218 Subtype.val) x = f x\n[PROOFSTEP]\ncases' lt_or_le x a with hxa hax\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\nha : \u2200 (x : \u03b1), x < a \u2192 f x = f a\nhb : \u2200 (x : \u03b1), b < x \u2192 f x = f b\nx : \u03b1\nhxa : x < a\n\u22a2 IccExtend h (f \u2218 Subtype.val) x = f x\n[PROOFSTEP]\nsimp [IccExtend_of_le_left _ _ hxa.le, ha x hxa]\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\nha : \u2200 (x : \u03b1), x < a \u2192 f x = f a\nhb : \u2200 (x : \u03b1), b < x \u2192 f x = f b\nx : \u03b1\nhax : a \u2264 x\n\u22a2 IccExtend h (f \u2218 Subtype.val) x = f x\n[PROOFSTEP]\ncases' le_or_lt x b with hxb hbx\n[GOAL]\ncase h.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\nha : \u2200 (x : \u03b1), x < a \u2192 f x = f a\nhb : \u2200 (x : \u03b1), b < x \u2192 f x = f b\nx : \u03b1\nhax : a \u2264 x\nhxb : x \u2264 b\n\u22a2 IccExtend h (f \u2218 Subtype.val) x = f x\n[PROOFSTEP]\nlift x to Icc a b using \u27e8hax, hxb\u27e9\n[GOAL]\ncase h.inr.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\nha : \u2200 (x : \u03b1), x < a \u2192 f x = f a\nhb : \u2200 (x : \u03b1), b < x \u2192 f x = f b\nx : { x // x \u2208 Icc a b }\nhax : a \u2264 \u2191x\nhxb : \u2191x \u2264 b\n\u22a2 IccExtend h (f \u2218 Subtype.val) \u2191x = f \u2191x\n[PROOFSTEP]\nrw [IccExtend_val, comp_apply]\n[GOAL]\ncase h.inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nx\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\nha : \u2200 (x : \u03b1), x < a \u2192 f x = f a\nhb : \u2200 (x : \u03b1), b < x \u2192 f x = f b\nx : \u03b1\nhax : a \u2264 x\nhbx : b < x\n\u22a2 IccExtend h (f \u2218 Subtype.val) x = f x\n[PROOFSTEP]\nsimp [IccExtend_of_right_le _ _ hbx.le, hb x hbx]\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Intervals.ProjIcc", "llama_tokens": 3321, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.658417500561683, "lm_q1q2_score": 0.42180584927889736}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nh : IsFormalAdjoint T S\ny : { x // x \u2208 S.domain }\nx\u271d : { x // x \u2208 T.domain }\n\u22a2 inner (\u2191S y) \u2191x\u271d = inner (\u2191y) (\u2191T x\u271d)\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, \u2190 inner_conj_symm (y : F), h]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\na\u271d b\u271d : F\nhx : a\u271d \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)}\nhy : b\u271d \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)}\n\u22a2 a\u271d + b\u271d \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)}\n[PROOFSTEP]\nrw [Set.mem_setOf_eq, LinearMap.map_add] at *\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\na\u271d b\u271d : F\nhx : Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) a\u271d) T.toFun)\nhy : Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) b\u271d) T.toFun)\n\u22a2 Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) a\u271d + \u2191(inner\u209b\u2097 \ud835\udd5c) b\u271d) T.toFun)\n[PROOFSTEP]\nexact hx.add hy\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\n\u22a2 0 \u2208\n { carrier := {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)},\n add_mem' :=\n (_ :\n \u2200 {a b : F},\n a \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)} \u2192\n b \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)} \u2192\n a + b \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)}) }.carrier\n[PROOFSTEP]\nrw [Set.mem_setOf_eq, LinearMap.map_zero, LinearMap.zero_comp]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\n\u22a2 Continuous \u21910\n[PROOFSTEP]\nexact continuous_zero\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\na : \ud835\udd5c\nx : F\nhx :\n x \u2208\n {\n toAddSubsemigroup :=\n { carrier := {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)},\n add_mem' :=\n (_ :\n \u2200 {a b : F},\n a \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)} \u2192\n b \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)} \u2192\n a + b \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)}) },\n zero_mem' :=\n (_ :\n 0 \u2208\n { carrier := {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)},\n add_mem' :=\n (_ :\n \u2200 {a b : F},\n a \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)} \u2192\n b \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)} \u2192\n a + b \u2208\n {y |\n Continuous\n \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)}) }.carrier) }.toAddSubsemigroup.carrier\n\u22a2 a \u2022 x \u2208\n {\n toAddSubsemigroup :=\n { carrier := {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)},\n add_mem' :=\n (_ :\n \u2200 {a b : F},\n a \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)} \u2192\n b \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)} \u2192\n a + b \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)}) },\n zero_mem' :=\n (_ :\n 0 \u2208\n { carrier := {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)},\n add_mem' :=\n (_ :\n \u2200 {a b : F},\n a \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)} \u2192\n b \u2208 {y | Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)} \u2192\n a + b \u2208\n {y |\n Continuous\n \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)}) }.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrw [Set.mem_setOf_eq, LinearMap.map_smul\u209b\u2097] at *\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\na : \ud835\udd5c\nx : F\nhx : Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) x) T.toFun)\n\u22a2 Continuous \u2191(LinearMap.comp (\u2191(starRingEnd \ud835\udd5c) a \u2022 \u2191(inner\u209b\u2097 \ud835\udd5c) x) T.toFun)\n[PROOFSTEP]\nexact hx.const_smul (conj a)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\nx y : { x // x \u2208 adjointDomain T }\nx\u271d : { x // x \u2208 T.domain }\n\u22a2 inner ((fun y => \u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \ud835\udd5c E)) (adjointDomainMkClmExtend hT y)) (x + y))\n \u2191x\u271d =\n inner\n ((fun y => \u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \ud835\udd5c E)) (adjointDomainMkClmExtend hT y)) x +\n (fun y => \u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \ud835\udd5c E)) (adjointDomainMkClmExtend hT y)) y)\n \u2191x\u271d\n[PROOFSTEP]\nsimp only [inner_add_left, Submodule.coe_add, InnerProductSpace.toDual_symm_apply, adjointDomainMkClmExtend_apply]\n -- Porting note(https://github.com/leanprover-community/mathlib4/issues/5026):\n -- mathlib3 was finished here\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\nx y : { x // x \u2208 adjointDomain T }\nx\u271d : { x // x \u2208 T.domain }\n\u22a2 \u2191(adjointDomainMkClmExtend hT (x + y)) \u2191x\u271d =\n \u2191(adjointDomainMkClmExtend hT x) \u2191x\u271d + \u2191(adjointDomainMkClmExtend hT y) \u2191x\u271d\n[PROOFSTEP]\nrw [adjointDomainMkClmExtend_apply, adjointDomainMkClmExtend_apply, adjointDomainMkClmExtend_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\nx y : { x // x \u2208 adjointDomain T }\nx\u271d : { x // x \u2208 T.domain }\n\u22a2 inner (\u2191(x + y)) (\u2191T x\u271d) = inner (\u2191x) (\u2191T x\u271d) + inner (\u2191y) (\u2191T x\u271d)\n[PROOFSTEP]\nsimp only [AddSubmonoid.coe_add, Submodule.coe_toAddSubmonoid, inner_add_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\nx\u271d\u00b2 : \ud835\udd5c\nx\u271d\u00b9 : { x // x \u2208 adjointDomain T }\nx\u271d : { x // x \u2208 T.domain }\n\u22a2 inner\n (AddHom.toFun\n { toFun := fun y => \u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \ud835\udd5c E)) (adjointDomainMkClmExtend hT y),\n map_add' :=\n (_ :\n \u2200 (x y : { x // x \u2208 adjointDomain T }),\n (fun y => \u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \ud835\udd5c E)) (adjointDomainMkClmExtend hT y))\n (x + y) =\n (fun y => \u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \ud835\udd5c E)) (adjointDomainMkClmExtend hT y))\n x +\n (fun y =>\n \u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \ud835\udd5c E)) (adjointDomainMkClmExtend hT y))\n y) }\n (x\u271d\u00b2 \u2022 x\u271d\u00b9))\n \u2191x\u271d =\n inner\n (\u2191(RingHom.id \ud835\udd5c) x\u271d\u00b2 \u2022\n AddHom.toFun\n {\n toFun := fun y =>\n \u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \ud835\udd5c E)) (adjointDomainMkClmExtend hT y),\n map_add' :=\n (_ :\n \u2200 (x y : { x // x \u2208 adjointDomain T }),\n (fun y => \u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \ud835\udd5c E)) (adjointDomainMkClmExtend hT y))\n (x + y) =\n (fun y =>\n \u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \ud835\udd5c E)) (adjointDomainMkClmExtend hT y))\n x +\n (fun y =>\n \u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \ud835\udd5c E)) (adjointDomainMkClmExtend hT y))\n y) }\n x\u271d\u00b9)\n \u2191x\u271d\n[PROOFSTEP]\nsimp only [inner_smul_left, Submodule.coe_smul_of_tower, RingHom.id_apply, InnerProductSpace.toDual_symm_apply,\n adjointDomainMkClmExtend_apply]\n -- Porting note(https://github.com/leanprover-community/mathlib4/issues/5026):\n -- mathlib3 was finished here\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\nx\u271d\u00b2 : \ud835\udd5c\nx\u271d\u00b9 : { x // x \u2208 adjointDomain T }\nx\u271d : { x // x \u2208 T.domain }\n\u22a2 \u2191(adjointDomainMkClmExtend hT (x\u271d\u00b2 \u2022 x\u271d\u00b9)) \u2191x\u271d = \u2191(starRingEnd \ud835\udd5c) x\u271d\u00b2 * \u2191(adjointDomainMkClmExtend hT x\u271d\u00b9) \u2191x\u271d\n[PROOFSTEP]\nrw [adjointDomainMkClmExtend_apply, adjointDomainMkClmExtend_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\nx\u271d\u00b2 : \ud835\udd5c\nx\u271d\u00b9 : { x // x \u2208 adjointDomain T }\nx\u271d : { x // x \u2208 T.domain }\n\u22a2 inner (\u2191(x\u271d\u00b2 \u2022 x\u271d\u00b9)) (\u2191T x\u271d) = \u2191(starRingEnd \ud835\udd5c) x\u271d\u00b2 * inner (\u2191x\u271d\u00b9) (\u2191T x\u271d)\n[PROOFSTEP]\nsimp only [Submodule.coe_smul_of_tower, inner_smul_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\ny : { x // x \u2208 adjointDomain T }\nx : { x // x \u2208 T.domain }\n\u22a2 inner (\u2191(adjointAux hT) y) \u2191x = inner (\u2191y) (\u2191T x)\n[PROOFSTEP]\nsimp only [adjointAux, LinearMap.coe_mk, InnerProductSpace.toDual_symm_apply, adjointDomainMkClmExtend_apply]\n -- Porting note(https://github.com/leanprover-community/mathlib4/issues/5026):\n -- mathlib3 was finished here\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\ny : { x // x \u2208 adjointDomain T }\nx : { x // x \u2208 T.domain }\n\u22a2 inner\n (\u2191{ toFun := fun y => \u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \ud835\udd5c E)) (adjointDomainMkClmExtend hT y),\n map_add' :=\n (_ :\n \u2200 (x y : { x // x \u2208 adjointDomain T }),\n (fun y => \u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \ud835\udd5c E)) (adjointDomainMkClmExtend hT y))\n (x + y) =\n (fun y =>\n \u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \ud835\udd5c E)) (adjointDomainMkClmExtend hT y))\n x +\n (fun y =>\n \u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \ud835\udd5c E)) (adjointDomainMkClmExtend hT y))\n y) }\n y)\n \u2191x =\n inner (\u2191y) (\u2191T x)\n[PROOFSTEP]\nsimp only [AddHom.coe_mk, InnerProductSpace.toDual_symm_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\ny : { x // x \u2208 adjointDomain T }\nx : { x // x \u2208 T.domain }\n\u22a2 \u2191(adjointDomainMkClmExtend hT y) \u2191x = inner (\u2191y) (\u2191T x)\n[PROOFSTEP]\nrw [adjointDomainMkClmExtend_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\ny : F\nh : \u2203 w, \u2200 (x : { x // x \u2208 T.domain }), inner w \u2191x = inner y (\u2191T x)\n\u22a2 y \u2208 T\u2020.domain\n[PROOFSTEP]\ncases' h with w hw\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\ny : F\nw : E\nhw : \u2200 (x : { x // x \u2208 T.domain }), inner w \u2191x = inner y (\u2191T x)\n\u22a2 y \u2208 T\u2020.domain\n[PROOFSTEP]\nrw [T.mem_adjoint_domain_iff]\n -- Porting note: was `by continuity`\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\ny : F\nw : E\nhw : \u2200 (x : { x // x \u2208 T.domain }), inner w \u2191x = inner y (\u2191T x)\n\u22a2 Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)\n[PROOFSTEP]\nhave : Continuous ((innerSL \ud835\udd5c w).comp T.domain.subtypeL) := ContinuousLinearMap.continuous _\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\ny : F\nw : E\nhw : \u2200 (x : { x // x \u2208 T.domain }), inner w \u2191x = inner y (\u2191T x)\nthis : Continuous \u2191(ContinuousLinearMap.comp (\u2191(innerSL \ud835\udd5c) w) (Submodule.subtypeL T.domain))\n\u22a2 Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun)\n[PROOFSTEP]\nconvert this using 1\n[GOAL]\ncase h.e'_5\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\ny : F\nw : E\nhw : \u2200 (x : { x // x \u2208 T.domain }), inner w \u2191x = inner y (\u2191T x)\nthis : Continuous \u2191(ContinuousLinearMap.comp (\u2191(innerSL \ud835\udd5c) w) (Submodule.subtypeL T.domain))\n\u22a2 \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) y) T.toFun) =\n \u2191(ContinuousLinearMap.comp (\u2191(innerSL \ud835\udd5c) w) (Submodule.subtypeL T.domain))\n[PROOFSTEP]\nexact funext fun x => (hw x).symm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT\u271d : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\nhT : \u00acDense \u2191T.domain\ny : { x // x \u2208 T\u2020.domain }\n\u22a2 \u2191T\u2020 y = 0\n[PROOFSTEP]\nchange (if hT : Dense (T.domain : Set E) then adjointAux hT else 0) y = _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT\u271d : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\nhT : \u00acDense \u2191T.domain\ny : { x // x \u2208 T\u2020.domain }\n\u22a2 \u2191(if hT : Dense \u2191T.domain then adjointAux hT else 0) y = 0\n[PROOFSTEP]\nsimp only [hT, not_false_iff, dif_neg, LinearMap.zero_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\ny : { x // x \u2208 T\u2020.domain }\n\u22a2 \u2191T\u2020 y = \u2191(adjointAux hT) y\n[PROOFSTEP]\nchange (if hT : Dense (T.domain : Set E) then adjointAux hT else 0) y = _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\ny : { x // x \u2208 T\u2020.domain }\n\u22a2 \u2191(if hT : Dense \u2191T.domain then adjointAux hT else 0) y = \u2191(adjointAux hT) y\n[PROOFSTEP]\nsimp only [hT, dif_pos, LinearMap.coe_mk]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\nh : IsFormalAdjoint T S\nx\u271d\u00b2 : { x // x \u2208 S.domain }\nx\u271d\u00b9 : { x // x \u2208 T\u2020.domain }\nhxy : \u2191x\u271d\u00b2 = \u2191x\u271d\u00b9\nx\u271d : { x // x \u2208 T.domain }\n\u22a2 inner (\u2191S x\u271d\u00b2) \u2191x\u271d = inner (\u2191x\u271d\u00b9) (\u2191T x\u271d)\n[PROOFSTEP]\nrw [h.symm, hxy]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\np : Submodule \ud835\udd5c E\nhp : Dense \u2191p\n\u22a2 LinearPMap.adjoint (LinearMap.toPMap (\u2191A) p) = LinearMap.toPMap \u2191(\u2191adjoint A) \u22a4\n[PROOFSTEP]\next x y hxy\n[GOAL]\ncase h.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\np : Submodule \ud835\udd5c E\nhp : Dense \u2191p\nx : F\n\u22a2 x \u2208 (LinearPMap.adjoint (LinearMap.toPMap (\u2191A) p)).domain \u2194 x \u2208 (LinearMap.toPMap \u2191(\u2191adjoint A) \u22a4).domain\n[PROOFSTEP]\nsimp only [LinearMap.toPMap_domain, Submodule.mem_top, iff_true_iff, LinearPMap.mem_adjoint_domain_iff,\n LinearMap.coe_comp, inner\u209b\u2097_apply_coe]\n[GOAL]\ncase h.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\np : Submodule \ud835\udd5c E\nhp : Dense \u2191p\nx : F\n\u22a2 Continuous ((fun w => inner x w) \u2218 \u2191(LinearMap.toPMap (\u2191A) p).toFun)\n[PROOFSTEP]\nexact ((innerSL \ud835\udd5c x).comp <| A.comp <| Submodule.subtypeL _).cont\n[GOAL]\ncase h'\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\np : Submodule \ud835\udd5c E\nhp : Dense \u2191p\nx : { x // x \u2208 (LinearPMap.adjoint (LinearMap.toPMap (\u2191A) p)).domain }\ny : { x // x \u2208 (LinearMap.toPMap \u2191(\u2191adjoint A) \u22a4).domain }\nhxy : \u2191x = \u2191y\n\u22a2 \u2191(LinearPMap.adjoint (LinearMap.toPMap (\u2191A) p)) x = \u2191(LinearMap.toPMap \u2191(\u2191adjoint A) \u22a4) y\n[PROOFSTEP]\nrefine' LinearPMap.adjoint_apply_eq _ _ fun v => _\n[GOAL]\ncase h'.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\np : Submodule \ud835\udd5c E\nhp : Dense \u2191p\nx : { x // x \u2208 (LinearPMap.adjoint (LinearMap.toPMap (\u2191A) p)).domain }\ny : { x // x \u2208 (LinearMap.toPMap \u2191(\u2191adjoint A) \u22a4).domain }\nhxy : \u2191x = \u2191y\n\u22a2 Dense \u2191(LinearMap.toPMap (\u2191A) p).domain\n[PROOFSTEP]\nsimpa using hp\n[GOAL]\ncase h'.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\np : Submodule \ud835\udd5c E\nhp : Dense \u2191p\nx : { x // x \u2208 (LinearPMap.adjoint (LinearMap.toPMap (\u2191A) p)).domain }\ny : { x // x \u2208 (LinearMap.toPMap \u2191(\u2191adjoint A) \u22a4).domain }\nhxy : \u2191x = \u2191y\nv : { x // x \u2208 (LinearMap.toPMap (\u2191A) p).domain }\n\u22a2 inner (\u2191(LinearMap.toPMap \u2191(\u2191adjoint A) \u22a4) y) \u2191v = inner (\u2191x) (\u2191(LinearMap.toPMap (\u2191A) p) v)\n[PROOFSTEP]\nsimp only [adjoint_inner_left, hxy, LinearMap.toPMap_apply, coe_coe]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\ninst\u271d : CompleteSpace E\nA : E \u2192\u2097.[\ud835\udd5c] E\nhA : IsSelfAdjoint A\n\u22a2 Dense \u2191A.domain\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\ninst\u271d : CompleteSpace E\nA : E \u2192\u2097.[\ud835\udd5c] E\nhA : IsSelfAdjoint A\nh : \u00acDense \u2191A.domain\n\u22a2 False\n[PROOFSTEP]\nrw [isSelfAdjoint_def] at hA \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\ninst\u271d : CompleteSpace E\nA : E \u2192\u2097.[\ud835\udd5c] E\nhA : A\u2020 = A\nh : \u00acDense \u2191A.domain\n\u22a2 False\n[PROOFSTEP]\nhave h' : A.domain = \u22a4 := by\n rw [\u2190 hA, Submodule.eq_top_iff']\n intro x\n rw [mem_adjoint_domain_iff, \u2190 hA]\n refine (innerSL \ud835\udd5c x).cont.comp ?_\n simp [adjoint, h, continuous_const]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\ninst\u271d : CompleteSpace E\nA : E \u2192\u2097.[\ud835\udd5c] E\nhA : A\u2020 = A\nh : \u00acDense \u2191A.domain\n\u22a2 A.domain = \u22a4\n[PROOFSTEP]\nrw [\u2190 hA, Submodule.eq_top_iff']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\ninst\u271d : CompleteSpace E\nA : E \u2192\u2097.[\ud835\udd5c] E\nhA : A\u2020 = A\nh : \u00acDense \u2191A.domain\n\u22a2 \u2200 (x : E), x \u2208 A\u2020.domain\n[PROOFSTEP]\nintro x\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\ninst\u271d : CompleteSpace E\nA : E \u2192\u2097.[\ud835\udd5c] E\nhA : A\u2020 = A\nh : \u00acDense \u2191A.domain\nx : E\n\u22a2 x \u2208 A\u2020.domain\n[PROOFSTEP]\nrw [mem_adjoint_domain_iff, \u2190 hA]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\ninst\u271d : CompleteSpace E\nA : E \u2192\u2097.[\ud835\udd5c] E\nhA : A\u2020 = A\nh : \u00acDense \u2191A.domain\nx : E\n\u22a2 Continuous \u2191(LinearMap.comp (\u2191(inner\u209b\u2097 \ud835\udd5c) x) A\u2020.toFun)\n[PROOFSTEP]\nrefine (innerSL \ud835\udd5c x).cont.comp ?_\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\ninst\u271d : CompleteSpace E\nA : E \u2192\u2097.[\ud835\udd5c] E\nhA : A\u2020 = A\nh : \u00acDense \u2191A.domain\nx : E\n\u22a2 Continuous fun x => \u2191A\u2020.toFun x\n[PROOFSTEP]\nsimp [adjoint, h, continuous_const]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\ninst\u271d : CompleteSpace E\nA : E \u2192\u2097.[\ud835\udd5c] E\nhA : A\u2020 = A\nh : \u00acDense \u2191A.domain\nh' : A.domain = \u22a4\n\u22a2 False\n[PROOFSTEP]\nsimp [h'] at h \n", "meta": {"mathlib_filename": "Mathlib.Analysis.InnerProductSpace.LinearPMap", "llama_tokens": 11269, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149868676283, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.42148941559653147}} {"text": "[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nx\u271d\u00b9 x\u271d : \u03a0\u2080 (i : \u03b9), \u03b2 i\nf\u2081 : (i : \u03b9) \u2192 \u03b2 i\ns\u2081\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f\u2081 i = 0 }\nf\u2082 : (i : \u03b9) \u2192 \u03b2 i\ns\u2081 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f\u2082 i = 0 }\nh : f\u2081 = f\u2082\n\u22a2 { toFun := f\u2081, support' := s\u2081\u271d } = { toFun := f\u2082, support' := s\u2081 }\n[PROOFSTEP]\nsubst h\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nx\u271d\u00b9 x\u271d : \u03a0\u2080 (i : \u03b9), \u03b2 i\nf\u2081 : (i : \u03b9) \u2192 \u03b2 i\ns\u2081\u271d s\u2081 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f\u2081 i = 0 }\n\u22a2 { toFun := f\u2081, support' := s\u2081\u271d } = { toFun := f\u2081, support' := s\u2081 }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_support'\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nx\u271d\u00b9 x\u271d : \u03a0\u2080 (i : \u03b9), \u03b2 i\nf\u2081 : (i : \u03b9) \u2192 \u03b2 i\ns\u2081\u271d s\u2081 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f\u2081 i = 0 }\n\u22a2 s\u2081\u271d = s\u2081\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\ni : \u03b9\nh : \u2191x i = 0\n\u22a2 (fun i => f i (\u2191x i)) i = 0\n[PROOFSTEP]\nrw [\u2190 hf i, \u2190 h]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nh : optParam (\u2200 (i : \u03b9), id 0 = 0) (_ : \u2200 (i : \u03b9), id 0 = id 0)\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\n\u22a2 mapRange (fun i => id) h g = g\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nh : optParam (\u2200 (i : \u03b9), id 0 = 0) (_ : \u2200 (i : \u03b9), id 0 = id 0)\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ni\u271d : \u03b9\n\u22a2 \u2191(mapRange (fun i => id) h g) i\u271d = \u2191g i\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b2\u2081 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\nhf\u2082 : \u2200 (i : \u03b9), f\u2082 i 0 = 0\nh : \u2200 (i : \u03b9), (f i \u2218 f\u2082 i) 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 mapRange (fun i => f i \u2218 f\u2082 i) h g = mapRange f hf (mapRange f\u2082 hf\u2082 g)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b2\u2081 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\nhf\u2082 : \u2200 (i : \u03b9), f\u2082 i 0 = 0\nh : \u2200 (i : \u03b9), (f i \u2218 f\u2082 i) 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni\u271d : \u03b9\n\u22a2 \u2191(mapRange (fun i => f i \u2218 f\u2082 i) h g) i\u271d = \u2191(mapRange f hf (mapRange f\u2082 hf\u2082 g)) i\u271d\n[PROOFSTEP]\nsimp only [mapRange_apply]\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b2\u2081 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\nhf\u2082 : \u2200 (i : \u03b9), f\u2082 i 0 = 0\nh : \u2200 (i : \u03b9), (f i \u2218 f\u2082 i) 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni\u271d : \u03b9\n\u22a2 (f i\u271d \u2218 f\u2082 i\u271d) (\u2191g i\u271d) = f i\u271d (f\u2082 i\u271d (\u2191g i\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\n\u22a2 mapRange f hf 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ni\u271d : \u03b9\n\u22a2 \u2191(mapRange f hf 0) i\u271d = \u21910 i\u271d\n[PROOFSTEP]\nsimp only [mapRange_apply, coe_zero, Pi.zero_apply, hf]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ny : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\n\u22a2 Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 (fun i => f i (\u2191x i) (\u2191y i)) i = 0 }\n[PROOFSTEP]\nrefine' x.support'.bind fun xs => _\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ny : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\nxs : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\n\u22a2 Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 (fun i => f i (\u2191x i) (\u2191y i)) i = 0 }\n[PROOFSTEP]\nrefine' y.support'.map fun ys => _\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ny : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\nxs : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\nys : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun y i = 0 }\n\u22a2 { s // \u2200 (i : \u03b9), i \u2208 s \u2228 (fun i => f i (\u2191x i) (\u2191y i)) i = 0 }\n[PROOFSTEP]\nrefine' \u27e8xs + ys, fun i => _\u27e9\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ny : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\nxs : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\nys : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun y i = 0 }\ni : \u03b9\n\u22a2 i \u2208 \u2191xs + \u2191ys \u2228 (fun i => f i (\u2191x i) (\u2191y i)) i = 0\n[PROOFSTEP]\nobtain h1 | (h1 : x i = 0) := xs.prop i\n[GOAL]\ncase inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ny : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\nxs : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\nys : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun y i = 0 }\ni : \u03b9\nh1 : i \u2208 \u2191xs\n\u22a2 i \u2208 \u2191xs + \u2191ys \u2228 (fun i => f i (\u2191x i) (\u2191y i)) i = 0\n[PROOFSTEP]\nleft\n[GOAL]\ncase inl.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ny : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\nxs : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\nys : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun y i = 0 }\ni : \u03b9\nh1 : i \u2208 \u2191xs\n\u22a2 i \u2208 \u2191xs + \u2191ys\n[PROOFSTEP]\nrw [Multiset.mem_add]\n[GOAL]\ncase inl.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ny : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\nxs : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\nys : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun y i = 0 }\ni : \u03b9\nh1 : i \u2208 \u2191xs\n\u22a2 i \u2208 \u2191xs \u2228 i \u2208 \u2191ys\n[PROOFSTEP]\nleft\n[GOAL]\ncase inl.h.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ny : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\nxs : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\nys : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun y i = 0 }\ni : \u03b9\nh1 : i \u2208 \u2191xs\n\u22a2 i \u2208 \u2191xs\n[PROOFSTEP]\nexact h1\n[GOAL]\ncase inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ny : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\nxs : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\nys : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun y i = 0 }\ni : \u03b9\nh1 : \u2191x i = 0\n\u22a2 i \u2208 \u2191xs + \u2191ys \u2228 (fun i => f i (\u2191x i) (\u2191y i)) i = 0\n[PROOFSTEP]\nobtain h2 | (h2 : y i = 0) := ys.prop i\n[GOAL]\ncase inr.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ny : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\nxs : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\nys : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun y i = 0 }\ni : \u03b9\nh1 : \u2191x i = 0\nh2 : i \u2208 \u2191ys\n\u22a2 i \u2208 \u2191xs + \u2191ys \u2228 (fun i => f i (\u2191x i) (\u2191y i)) i = 0\n[PROOFSTEP]\nleft\n[GOAL]\ncase inr.inl.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ny : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\nxs : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\nys : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun y i = 0 }\ni : \u03b9\nh1 : \u2191x i = 0\nh2 : i \u2208 \u2191ys\n\u22a2 i \u2208 \u2191xs + \u2191ys\n[PROOFSTEP]\nrw [Multiset.mem_add]\n[GOAL]\ncase inr.inl.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ny : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\nxs : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\nys : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun y i = 0 }\ni : \u03b9\nh1 : \u2191x i = 0\nh2 : i \u2208 \u2191ys\n\u22a2 i \u2208 \u2191xs \u2228 i \u2208 \u2191ys\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.inl.h.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ny : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\nxs : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\nys : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun y i = 0 }\ni : \u03b9\nh1 : \u2191x i = 0\nh2 : i \u2208 \u2191ys\n\u22a2 i \u2208 \u2191ys\n[PROOFSTEP]\nexact h2\n[GOAL]\ncase inr.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ny : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\nxs : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\nys : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun y i = 0 }\ni : \u03b9\nh1 : \u2191x i = 0\nh2 : \u2191y i = 0\n\u22a2 i \u2208 \u2191xs + \u2191ys \u2228 (fun i => f i (\u2191x i) (\u2191y i)) i = 0\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.inr.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ny : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\nxs : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\nys : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun y i = 0 }\ni : \u03b9\nh1 : \u2191x i = 0\nh2 : \u2191y i = 0\n\u22a2 (fun i => f i (\u2191x i) (\u2191y i)) i = 0\n[PROOFSTEP]\nrw [\u2190 hf, \u2190 h1, \u2190 h2]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nx y : \u03a0\u2080 (i : \u03b9), \u03b2 i\ns : Set \u03b9\ninst\u271d : (i : \u03b9) \u2192 Decidable (i \u2208 s)\n\u22a2 \u2191(piecewise x y s) = Set.piecewise s \u2191x \u2191y\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nx y : \u03a0\u2080 (i : \u03b9), \u03b2 i\ns : Set \u03b9\ninst\u271d : (i : \u03b9) \u2192 Decidable (i \u2208 s)\nx\u271d : \u03b9\n\u22a2 \u2191(piecewise x y s) x\u271d = Set.piecewise s (\u2191x) (\u2191y) x\u271d\n[PROOFSTEP]\napply piecewise_apply\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\n\u03b4 : Type u_1\ninst\u271d\u2075 : Monoid \u03b3\ninst\u271d\u2074 : Monoid \u03b4\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DistribMulAction \u03b4 (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), SMulCommClass \u03b3 \u03b4 (\u03b2 i)\nr : \u03b3\ns : \u03b4\nm : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 \u2191(r \u2022 s \u2022 m) i = \u2191(s \u2022 r \u2022 m) i\n[PROOFSTEP]\nsimp only [smul_apply, smul_comm r s (m i)]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\n\u03b4 : Type u_1\ninst\u271d\u2076 : Monoid \u03b3\ninst\u271d\u2075 : Monoid \u03b4\ninst\u271d\u2074 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 DistribMulAction \u03b4 (\u03b2 i)\ninst\u271d\u00b9 : SMul \u03b3 \u03b4\ninst\u271d : \u2200 (i : \u03b9), IsScalarTower \u03b3 \u03b4 (\u03b2 i)\nr : \u03b3\ns : \u03b4\nm : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 \u2191((r \u2022 s) \u2022 m) i = \u2191(r \u2022 s \u2022 m) i\n[PROOFSTEP]\nsimp only [smul_apply, smul_assoc r s (m i)]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u2074 : Monoid \u03b3\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DistribMulAction \u03b3\u1d50\u1d52\u1d56 (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), IsCentralScalar \u03b3 (\u03b2 i)\nr : \u03b3\nm : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 \u2191(MulOpposite.op r \u2022 m) i = \u2191(r \u2022 m) i\n[PROOFSTEP]\nsimp only [smul_apply, op_smul_eq_smul r (m i)]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : Semiring \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 Module \u03b3 (\u03b2 i)\nsrc\u271d : DistribMulAction \u03b3 (\u03a0\u2080 (i : \u03b9), \u03b2 i) := inferInstanceAs (DistribMulAction \u03b3 (\u03a0\u2080 (i : \u03b9), \u03b2 i))\nc x : \u03b3\ny : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 \u2191((c + x) \u2022 y) i = \u2191(c \u2022 y + x \u2022 y) i\n[PROOFSTEP]\nsimp only [add_apply, smul_apply, add_smul]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b2 : Semiring \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 Module \u03b3 (\u03b2 i)\nsrc\u271d : DistribMulAction \u03b3 (\u03a0\u2080 (i : \u03b9), \u03b2 i) := inferInstanceAs (DistribMulAction \u03b3 (\u03a0\u2080 (i : \u03b9), \u03b2 i))\nc : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 \u2191(0 \u2022 c) i = \u21910 i\n[PROOFSTEP]\nsimp only [smul_apply, zero_smul, zero_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\nxs : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun x i = 0 }\ni : \u03b9\nH : \u2191x i = 0\n\u22a2 (fun i => if p i then \u2191x i else 0) i = 0\n[PROOFSTEP]\nsimp only [H, ite_self]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh : p i\n\u22a2 \u2191(filter p f) i = \u2191f i\n[PROOFSTEP]\nsimp only [filter_apply, if_pos h]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh : \u00acp i\n\u22a2 \u2191(filter p f) i = 0\n[PROOFSTEP]\nsimp only [filter_apply, if_neg h]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ni : \u03b9\n\u22a2 \u2191(filter p f + filter (fun i => \u00acp i) f) i = \u2191f i\n[PROOFSTEP]\nsimp only [add_apply, filter_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ni : \u03b9\n\u22a2 ((if p i then \u2191f i else 0) + if \u00acp i then \u2191f i else 0) = \u2191f i\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ni : \u03b9\nh\u271d : p i\n\u22a2 \u2191f i + 0 = \u2191f i\n[PROOFSTEP]\nsimp only [add_zero, zero_add]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ni : \u03b9\nh\u271d : \u00acp i\n\u22a2 0 + \u2191f i = \u2191f i\n[PROOFSTEP]\nsimp only [add_zero, zero_add]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 filter p 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ni\u271d : \u03b9\n\u22a2 \u2191(filter p 0) i\u271d = \u21910 i\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 filter p (f + g) = filter p f + filter p g\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni\u271d : \u03b9\n\u22a2 \u2191(filter p (f + g)) i\u271d = \u2191(filter p f + filter p g) i\u271d\n[PROOFSTEP]\nsimp [ite_add_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b3 : Monoid \u03b3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nr : \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 filter p (r \u2022 f) = r \u2022 filter p f\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u00b3 : Monoid \u03b3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nr : \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni\u271d : \u03b9\n\u22a2 \u2191(filter p (r \u2022 f)) i\u271d = \u2191(r \u2022 filter p f) i\u271d\n[PROOFSTEP]\nsimp [smul_apply, smul_ite]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 Set.Finite {i | \u2191f i \u2260 0}\n[PROOFSTEP]\nclassical!\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 Set.Finite {i | \u2191f i \u2260 0}\n[PROOFSTEP]\nexact\n Trunc.induction_on f.support' fun xs =>\n (Multiset.toFinset xs.1).finite_toSet.subset fun i H => Multiset.mem_toFinset.2 ((xs.prop i).resolve_right H)\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns\u271d : Finset \u03b9\nx : (i : \u2191\u2191s\u271d) \u2192 \u03b2 \u2191i\ni : \u03b9\ns : Finset \u03b9\n\u22a2 Function.Injective (mk s)\n[PROOFSTEP]\nintro x y H\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns\u271d : Finset \u03b9\nx\u271d : (i : \u2191\u2191s\u271d) \u2192 \u03b2 \u2191i\ni : \u03b9\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\nH : mk s x = mk s y\n\u22a2 x = y\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns\u271d : Finset \u03b9\nx\u271d : (i : \u2191\u2191s\u271d) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\nH : mk s x = mk s y\ni : \u2191\u2191s\n\u22a2 x i = y i\n[PROOFSTEP]\nhave h1 : (mk s x : \u2200 i, \u03b2 i) i = (mk s y : \u2200 i, \u03b2 i) i := by rw [H]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns\u271d : Finset \u03b9\nx\u271d : (i : \u2191\u2191s\u271d) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\nH : mk s x = mk s y\ni : \u2191\u2191s\n\u22a2 \u2191(mk s x) \u2191i = \u2191(mk s y) \u2191i\n[PROOFSTEP]\nrw [H]\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns\u271d : Finset \u03b9\nx\u271d : (i : \u2191\u2191s\u271d) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\nH : mk s x = mk s y\ni : \u2191\u2191s\nh1 : \u2191(mk s x) \u2191i = \u2191(mk s y) \u2191i\n\u22a2 x i = y i\n[PROOFSTEP]\nobtain \u27e8i, hi : i \u2208 s\u27e9 := i\n[GOAL]\ncase h.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns\u271d : Finset \u03b9\nx\u271d : (i : \u2191\u2191s\u271d) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\nH : mk s x = mk s y\ni : \u03b9\nhi : i \u2208 s\nh1 : \u2191(mk s x) \u2191{ val := i, property := hi } = \u2191(mk s y) \u2191{ val := i, property := hi }\n\u22a2 x { val := i, property := hi } = y { val := i, property := hi }\n[PROOFSTEP]\ndsimp only [mk_apply, Subtype.coe_mk] at h1 \n[GOAL]\ncase h.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns\u271d : Finset \u03b9\nx\u271d : (i : \u2191\u2191s\u271d) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\nH : mk s x = mk s y\ni : \u03b9\nhi : i \u2208 s\nh1 : (if H : i \u2208 s then x { val := i, property := H } else 0) = if H : i \u2208 s then y { val := i, property := H } else 0\n\u22a2 x { val := i, property := hi } = y { val := i, property := hi }\n[PROOFSTEP]\nsimpa only [dif_pos hi] using h1\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i : \u03b9\nb : \u03b2 i\nj : \u03b9\n\u22a2 j = i \u2192 j \u2208 {i}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i i' : \u03b9\nb : \u03b2 i\n\u22a2 \u2191(single i b) i' = if h : i = i' then Eq.recOn h b else 0\n[PROOFSTEP]\nrw [single_eq_pi_single, Pi.single, Function.update]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i i' : \u03b9\nb : \u03b2 i\n\u22a2 (if h : i' = i then (_ : i = i') \u25b8 b else OfNat.ofNat 0 i') = if h : i = i' then Eq.recOn h b else 0\n[PROOFSTEP]\nsimp [@eq_comm _ i i']\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i : \u03b9\nb : \u03b2 i\n\u22a2 \u2191(single i b) i = b\n[PROOFSTEP]\nsimp only [single_apply, dite_eq_ite, ite_true]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i i' : \u03b9\nb : \u03b2 i\nh : i \u2260 i'\n\u22a2 \u2191(single i b) i' = 0\n[PROOFSTEP]\nsimp only [single_apply, dif_neg h]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nxi : \u03b2 i\nxj : \u03b2 j\n\u22a2 single i xi = single j xj \u2194 i = j \u2227 HEq xi xj \u2228 xi = 0 \u2227 xj = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nxi : \u03b2 i\nxj : \u03b2 j\n\u22a2 single i xi = single j xj \u2192 i = j \u2227 HEq xi xj \u2228 xi = 0 \u2227 xj = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nxi : \u03b2 i\nxj : \u03b2 j\nh : single i xi = single j xj\n\u22a2 i = j \u2227 HEq xi xj \u2228 xi = 0 \u2227 xj = 0\n[PROOFSTEP]\nby_cases hij : i = j\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nxi : \u03b2 i\nxj : \u03b2 j\nh : single i xi = single j xj\nhij : i = j\n\u22a2 i = j \u2227 HEq xi xj \u2228 xi = 0 \u2227 xj = 0\n[PROOFSTEP]\nsubst hij\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i : \u03b9\nxi xj : \u03b2 i\nh : single i xi = single i xj\n\u22a2 i = i \u2227 HEq xi xj \u2228 xi = 0 \u2227 xj = 0\n[PROOFSTEP]\nexact Or.inl \u27e8rfl, heq_of_eq (DFinsupp.single_injective h)\u27e9\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nxi : \u03b2 i\nxj : \u03b2 j\nh : single i xi = single j xj\nhij : \u00aci = j\n\u22a2 i = j \u2227 HEq xi xj \u2228 xi = 0 \u2227 xj = 0\n[PROOFSTEP]\nhave h_coe : \u21d1(DFinsupp.single i xi) = DFinsupp.single j xj := congr_arg (\u21d1) h\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nxi : \u03b2 i\nxj : \u03b2 j\nh : single i xi = single j xj\nhij : \u00aci = j\nh_coe : \u2191(single i xi) = \u2191(single j xj)\n\u22a2 i = j \u2227 HEq xi xj \u2228 xi = 0 \u2227 xj = 0\n[PROOFSTEP]\nhave hci := congr_fun h_coe i\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nxi : \u03b2 i\nxj : \u03b2 j\nh : single i xi = single j xj\nhij : \u00aci = j\nh_coe : \u2191(single i xi) = \u2191(single j xj)\nhci : \u2191(single i xi) i = \u2191(single j xj) i\n\u22a2 i = j \u2227 HEq xi xj \u2228 xi = 0 \u2227 xj = 0\n[PROOFSTEP]\nhave hcj := congr_fun h_coe j\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nxi : \u03b2 i\nxj : \u03b2 j\nh : single i xi = single j xj\nhij : \u00aci = j\nh_coe : \u2191(single i xi) = \u2191(single j xj)\nhci : \u2191(single i xi) i = \u2191(single j xj) i\nhcj : \u2191(single i xi) j = \u2191(single j xj) j\n\u22a2 i = j \u2227 HEq xi xj \u2228 xi = 0 \u2227 xj = 0\n[PROOFSTEP]\nrw [DFinsupp.single_eq_same] at hci hcj \n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nxi : \u03b2 i\nxj : \u03b2 j\nh : single i xi = single j xj\nhij : \u00aci = j\nh_coe : \u2191(single i xi) = \u2191(single j xj)\nhci : xi = \u2191(single j xj) i\nhcj : \u2191(single i xi) j = xj\n\u22a2 i = j \u2227 HEq xi xj \u2228 xi = 0 \u2227 xj = 0\n[PROOFSTEP]\nrw [DFinsupp.single_eq_of_ne (Ne.symm hij)] at hci \n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nxi : \u03b2 i\nxj : \u03b2 j\nh : single i xi = single j xj\nhij : \u00aci = j\nh_coe : \u2191(single i xi) = \u2191(single j xj)\nhci : xi = 0\nhcj : \u2191(single i xi) j = xj\n\u22a2 i = j \u2227 HEq xi xj \u2228 xi = 0 \u2227 xj = 0\n[PROOFSTEP]\nrw [DFinsupp.single_eq_of_ne hij] at hcj \n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nxi : \u03b2 i\nxj : \u03b2 j\nh : single i xi = single j xj\nhij : \u00aci = j\nh_coe : \u2191(single i xi) = \u2191(single j xj)\nhci : xi = 0\nhcj : 0 = xj\n\u22a2 i = j \u2227 HEq xi xj \u2228 xi = 0 \u2227 xj = 0\n[PROOFSTEP]\nexact Or.inr \u27e8hci, hcj.symm\u27e9\n[GOAL]\ncase mpr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nxi : \u03b2 i\nxj : \u03b2 j\n\u22a2 i = j \u2227 HEq xi xj \u2228 xi = 0 \u2227 xj = 0 \u2192 single i xi = single j xj\n[PROOFSTEP]\nrintro (\u27e8rfl, hxi\u27e9 | \u27e8hi, hj\u27e9)\n[GOAL]\ncase mpr.inl.intro\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i : \u03b9\nxi xj : \u03b2 i\nhxi : HEq xi xj\n\u22a2 single i xi = single i xj\n[PROOFSTEP]\nrw [eq_of_heq hxi]\n[GOAL]\ncase mpr.inr.intro\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nxi : \u03b2 i\nxj : \u03b2 j\nhi : xi = 0\nhj : xj = 0\n\u22a2 single i xi = single j xj\n[PROOFSTEP]\nrw [hi, hj, DFinsupp.single_zero, DFinsupp.single_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i : \u03b9\nxi : \u03b2 i\n\u22a2 single i xi = 0 \u2194 xi = 0\n[PROOFSTEP]\nrw [\u2190 single_zero i, single_eq_single_iff]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i : \u03b9\nxi : \u03b2 i\n\u22a2 i = i \u2227 HEq xi 0 \u2228 xi = 0 \u2227 0 = 0 \u2194 xi = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ni : \u03b9\nx : \u03b2 i\n\u22a2 filter p (single i x) = if p i then single i x else 0\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ni : \u03b9\nx : \u03b2 i\nj : \u03b9\n\u22a2 \u2191(filter p (single i x)) j = \u2191(if p i then single i x else 0) j\n[PROOFSTEP]\nhave := apply_ite (fun x : \u03a0\u2080 i, \u03b2 i => x j) (p i) (single i x) 0\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ni : \u03b9\nx : \u03b2 i\nj : \u03b9\nthis : \u2191(if p i then single i x else 0) j = if p i then \u2191(single i x) j else \u21910 j\n\u22a2 \u2191(filter p (single i x)) j = \u2191(if p i then single i x else 0) j\n[PROOFSTEP]\ndsimp at this \n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ni : \u03b9\nx : \u03b2 i\nj : \u03b9\nthis : \u2191(if p i then single i x else 0) j = if p i then \u2191(single i x) j else 0\n\u22a2 \u2191(filter p (single i x)) j = \u2191(if p i then single i x else 0) j\n[PROOFSTEP]\nrw [filter_apply, this]\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ni : \u03b9\nx : \u03b2 i\nj : \u03b9\nthis : \u2191(if p i then single i x else 0) j = if p i then \u2191(single i x) j else 0\n\u22a2 (if p j then \u2191(single i x) j else 0) = if p i then \u2191(single i x) j else 0\n[PROOFSTEP]\nobtain rfl | hij := Decidable.eq_or_ne i j\n[GOAL]\ncase h.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ni : \u03b9\nx : \u03b2 i\nthis : \u2191(if p i then single i x else 0) i = if p i then \u2191(single i x) i else 0\n\u22a2 (if p i then \u2191(single i x) i else 0) = if p i then \u2191(single i x) i else 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ni : \u03b9\nx : \u03b2 i\nj : \u03b9\nthis : \u2191(if p i then single i x else 0) j = if p i then \u2191(single i x) j else 0\nhij : i \u2260 j\n\u22a2 (if p j then \u2191(single i x) j else 0) = if p i then \u2191(single i x) j else 0\n[PROOFSTEP]\nrw [single_eq_of_ne hij, ite_self, ite_self]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ni : \u03b9\nx : \u03b2 i\nh : p i\n\u22a2 filter p (single i x) = single i x\n[PROOFSTEP]\nrw [filter_single, if_pos h]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ni : \u03b9\nx : \u03b2 i\nh : \u00acp i\n\u22a2 filter p (single i x) = 0\n[PROOFSTEP]\nrw [filter_single, if_neg h]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nxi : \u03b2 i\nxj : \u03b2 j\nh : { fst := i, snd := xi } = { fst := j, snd := xj }\n\u22a2 single i xi = single j xj\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i : \u03b9\nxi : \u03b2 i\n\u22a2 single i xi = single i xi\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\ninst\u271d : Fintype \u03b9\ni : \u03b9\nm : \u03b2 i\n\u22a2 \u2191equivFunOnFintype (single i m) = Pi.single i m\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\ninst\u271d : Fintype \u03b9\ni : \u03b9\nm : \u03b2 i\nx : \u03b9\n\u22a2 \u2191equivFunOnFintype (single i m) x = Pi.single i m x\n[PROOFSTEP]\ndsimp [Pi.single, Function.update]\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\ninst\u271d : Fintype \u03b9\ni : \u03b9\nm : \u03b2 i\nx : \u03b9\n\u22a2 \u2191(single i m) x = if h : x = i then (_ : i = x) \u25b8 m else 0\n[PROOFSTEP]\nsimp [DFinsupp.single_eq_pi_single, @eq_comm _ i]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\ninst\u271d : Fintype \u03b9\ni : \u03b9\nm : \u03b2 i\n\u22a2 \u2191equivFunOnFintype.symm (Pi.single i m) = single i m\n[PROOFSTEP]\next i'\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\ninst\u271d : Fintype \u03b9\ni : \u03b9\nm : \u03b2 i\ni' : \u03b9\n\u22a2 \u2191(\u2191equivFunOnFintype.symm (Pi.single i m)) i' = \u2191(single i m) i'\n[PROOFSTEP]\nsimp only [\u2190 single_eq_pi_single, equivFunOnFintype_symm_coe]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 \u2191(erase i f) i = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i i' : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh : i' \u2260 i\n\u22a2 \u2191(erase i f) i' = \u2191f i'\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\ninst\u271d : (i' : \u03b9) \u2192 Decidable (i' \u2208 {i})\n\u22a2 piecewise (single i (\u2191x i)) (erase i x) {i} = x\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\ninst\u271d : (i' : \u03b9) \u2192 Decidable (i' \u2208 {i})\nj : \u03b9\n\u22a2 \u2191(piecewise (single i (\u2191x i)) (erase i x) {i}) j = \u2191x j\n[PROOFSTEP]\nrw [piecewise_apply]\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\ninst\u271d : (i' : \u03b9) \u2192 Decidable (i' \u2208 {i})\nj : \u03b9\n\u22a2 (if j \u2208 {i} then \u2191(single i (\u2191x i)) j else \u2191(erase i x) j) = \u2191x j\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\ninst\u271d : (i' : \u03b9) \u2192 Decidable (i' \u2208 {i})\nj : \u03b9\nh : j \u2208 {i}\n\u22a2 \u2191(single i (\u2191x i)) j = \u2191x j\n[PROOFSTEP]\nrw [(id h : j = i), single_eq_same]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\ninst\u271d : (i' : \u03b9) \u2192 Decidable (i' \u2208 {i})\nj : \u03b9\nh : \u00acj \u2208 {i}\n\u22a2 \u2191(erase i x) j = \u2191x j\n[PROOFSTEP]\nexact erase_ne h\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u271d i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2\u271d \u2191i\ni\u271d : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 erase i f = f - single i (\u2191f i)\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u271d i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2\u271d \u2191i\ni\u271d : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni j : \u03b9\n\u22a2 \u2191(erase i f) j = \u2191(f - single i (\u2191f i)) j\n[PROOFSTEP]\nrcases eq_or_ne i j with (rfl | h)\n[GOAL]\ncase h.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u271d i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2\u271d \u2191i\ni\u271d : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 \u2191(erase i f) i = \u2191(f - single i (\u2191f i)) i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u271d i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2\u271d \u2191i\ni\u271d : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni j : \u03b9\nh : i \u2260 j\n\u22a2 \u2191(erase i f) j = \u2191(f - single i (\u2191f i)) j\n[PROOFSTEP]\nsimp [erase_ne h.symm, single_eq_of_ne h, @eq_comm _ j, h]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 filter (fun x => x \u2260 i) f = erase i f\n[PROOFSTEP]\next1 j\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni j : \u03b9\n\u22a2 \u2191(filter (fun x => x \u2260 i) f) j = \u2191(erase i f) j\n[PROOFSTEP]\nsimp only [DFinsupp.filter_apply, DFinsupp.erase_apply, ite_not]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 filter ((fun x x_1 => x \u2260 x_1) i) f = erase i f\n[PROOFSTEP]\nrw [\u2190 filter_ne_eq_erase f i]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 filter ((fun x x_1 => x \u2260 x_1) i) f = filter (fun x => x \u2260 i) f\n[PROOFSTEP]\ncongr with j\n[GOAL]\ncase e_p.h.a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni j : \u03b9\n\u22a2 (fun x x_1 => x \u2260 x_1) i j \u2194 j \u2260 i\n[PROOFSTEP]\nexact ne_comm\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d j i : \u03b9\nx : \u03b2 i\n\u22a2 erase j (single i x) = if i = j then 0 else single i x\n[PROOFSTEP]\nrw [\u2190 filter_ne_eq_erase, filter_single, ite_not]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i : \u03b9\nx : \u03b2 i\n\u22a2 erase i (single i x) = 0\n[PROOFSTEP]\nrw [erase_single, if_pos rfl]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nx : \u03b2 i\nh : i \u2260 j\n\u22a2 erase j (single i x) = single i x\n[PROOFSTEP]\nrw [erase_single, if_neg h]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns\u271d : Finset \u03b9\nx : (i : \u2191\u2191s\u271d) \u2192 \u03b2 \u2191i\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nb : \u03b2 i\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }\nj : \u03b9\n\u22a2 j \u2208 i ::\u2098 \u2191s \u2228 Function.update (\u2191f) i b j = 0\n[PROOFSTEP]\nrcases eq_or_ne i j with (rfl | hi)\n[GOAL]\ncase inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns\u271d : Finset \u03b9\nx : (i : \u2191\u2191s\u271d) \u2192 \u03b2 \u2191i\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nb : \u03b2 i\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }\n\u22a2 i \u2208 i ::\u2098 \u2191s \u2228 Function.update (\u2191f) i b i = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns\u271d : Finset \u03b9\nx : (i : \u2191\u2191s\u271d) \u2192 \u03b2 \u2191i\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nb : \u03b2 i\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }\nj : \u03b9\nhi : i \u2260 j\n\u22a2 j \u2208 i ::\u2098 \u2191s \u2228 Function.update (\u2191f) i b j = 0\n[PROOFSTEP]\nobtain hj | (hj : f j = 0) := s.prop j\n[GOAL]\ncase inr.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns\u271d : Finset \u03b9\nx : (i : \u2191\u2191s\u271d) \u2192 \u03b2 \u2191i\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nb : \u03b2 i\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }\nj : \u03b9\nhi : i \u2260 j\nhj : j \u2208 \u2191s\n\u22a2 j \u2208 i ::\u2098 \u2191s \u2228 Function.update (\u2191f) i b j = 0\n[PROOFSTEP]\nexact Or.inl (Multiset.mem_cons_of_mem hj)\n[GOAL]\ncase inr.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns\u271d : Finset \u03b9\nx : (i : \u2191\u2191s\u271d) \u2192 \u03b2 \u2191i\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nb : \u03b2 i\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }\nj : \u03b9\nhi : i \u2260 j\nhj : \u2191f j = 0\n\u22a2 j \u2208 i ::\u2098 \u2191s \u2228 Function.update (\u2191f) i b j = 0\n[PROOFSTEP]\nexact Or.inr ((Function.update_noteq hi.symm b _).trans hj)\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nb : \u03b2 i\nj : \u03b9\n\u22a2 update i f (\u2191f i) = f\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nb : \u03b2 i\nj i\u271d : \u03b9\n\u22a2 \u2191(update i f (\u2191f i)) i\u271d = \u2191f i\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nb : \u03b2 i\nj : \u03b9\n\u22a2 update i f 0 = erase i f\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nb : \u03b2 i\nj\u271d j : \u03b9\n\u22a2 \u2191(update i f 0) j = \u2191(erase i f) j\n[PROOFSTEP]\nrcases eq_or_ne i j with (rfl | hi)\n[GOAL]\ncase h.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nb : \u03b2 i\nj : \u03b9\n\u22a2 \u2191(update i f 0) i = \u2191(erase i f) i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nb : \u03b2 i\nj\u271d j : \u03b9\nhi : i \u2260 j\n\u22a2 \u2191(update i f 0) j = \u2191(erase i f) j\n[PROOFSTEP]\nsimp [hi.symm]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u271d i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2\u271d \u2191i\ni\u271d : \u03b9\nf\u271d : \u03a0\u2080 (i : \u03b9), \u03b2\u271d i\nb\u271d : \u03b2\u271d i\u271d\nj : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\n\u22a2 update i f b = single i b + erase i f\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u271d i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2\u271d \u2191i\ni\u271d : \u03b9\nf\u271d : \u03a0\u2080 (i : \u03b9), \u03b2\u271d i\nb\u271d : \u03b2\u271d i\u271d\nj\u271d : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\nj : \u03b9\n\u22a2 \u2191(update i f b) j = \u2191(single i b + erase i f) j\n[PROOFSTEP]\nrcases eq_or_ne i j with (rfl | h)\n[GOAL]\ncase h.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u271d i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2\u271d \u2191i\ni\u271d : \u03b9\nf\u271d : \u03a0\u2080 (i : \u03b9), \u03b2\u271d i\nb\u271d : \u03b2\u271d i\u271d\nj : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\n\u22a2 \u2191(update i f b) i = \u2191(single i b + erase i f) i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u271d i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2\u271d \u2191i\ni\u271d : \u03b9\nf\u271d : \u03a0\u2080 (i : \u03b9), \u03b2\u271d i\nb\u271d : \u03b2\u271d i\u271d\nj\u271d : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\nj : \u03b9\nh : i \u2260 j\n\u22a2 \u2191(update i f b) j = \u2191(single i b + erase i f) j\n[PROOFSTEP]\nsimp [Function.update_noteq h.symm, h, erase_ne, h.symm]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u271d i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2\u271d \u2191i\ni\u271d : \u03b9\nf\u271d : \u03a0\u2080 (i : \u03b9), \u03b2\u271d i\nb\u271d : \u03b2\u271d i\u271d\nj : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\n\u22a2 update i f b = erase i f + single i b\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u271d i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2\u271d \u2191i\ni\u271d : \u03b9\nf\u271d : \u03a0\u2080 (i : \u03b9), \u03b2\u271d i\nb\u271d : \u03b2\u271d i\u271d\nj\u271d : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\nj : \u03b9\n\u22a2 \u2191(update i f b) j = \u2191(erase i f + single i b) j\n[PROOFSTEP]\nrcases eq_or_ne i j with (rfl | h)\n[GOAL]\ncase h.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u271d i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2\u271d \u2191i\ni\u271d : \u03b9\nf\u271d : \u03a0\u2080 (i : \u03b9), \u03b2\u271d i\nb\u271d : \u03b2\u271d i\u271d\nj : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\n\u22a2 \u2191(update i f b) i = \u2191(erase i f + single i b) i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u271d i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2\u271d \u2191i\ni\u271d : \u03b9\nf\u271d : \u03a0\u2080 (i : \u03b9), \u03b2\u271d i\nb\u271d : \u03b2\u271d i\u271d\nj\u271d : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\nj : \u03b9\nh : i \u2260 j\n\u22a2 \u2191(update i f b) j = \u2191(erase i f + single i b) j\n[PROOFSTEP]\nsimp [Function.update_noteq h.symm, h, erase_ne, h.symm]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u271d i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2\u271d \u2191i\ni\u271d : \u03b9\nf\u271d : \u03a0\u2080 (i : \u03b9), \u03b2\u271d i\nb\u271d : \u03b2\u271d i\u271d\nj : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\n\u22a2 update i f b = f - single i (\u2191f i) + single i b\n[PROOFSTEP]\nrw [update_eq_erase_add_single f i b, erase_eq_sub_single f i]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ni : \u03b9\nb\u2081 b\u2082 : \u03b2 i\ni' : \u03b9\n\u22a2 \u2191(single i (b\u2081 + b\u2082)) i' = \u2191(single i b\u2081 + single i b\u2082) i'\n[PROOFSTEP]\nby_cases h : i = i'\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ni : \u03b9\nb\u2081 b\u2082 : \u03b2 i\ni' : \u03b9\nh : i = i'\n\u22a2 \u2191(single i (b\u2081 + b\u2082)) i' = \u2191(single i b\u2081 + single i b\u2082) i'\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ni : \u03b9\nb\u2081 b\u2082 : \u03b2 i\n\u22a2 \u2191(single i (b\u2081 + b\u2082)) i = \u2191(single i b\u2081 + single i b\u2082) i\n[PROOFSTEP]\nsimp only [add_apply, single_eq_same]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ni : \u03b9\nb\u2081 b\u2082 : \u03b2 i\ni' : \u03b9\nh : \u00aci = i'\n\u22a2 \u2191(single i (b\u2081 + b\u2082)) i' = \u2191(single i b\u2081 + single i b\u2082) i'\n[PROOFSTEP]\nsimp only [add_apply, single_eq_of_ne h, zero_add]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ni : \u03b9\nf\u2081 f\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2 i\nx\u271d : \u03b9\n\u22a2 \u2191(erase i (f\u2081 + f\u2082)) x\u271d = \u2191(erase i f\u2081 + erase i f\u2082) x\u271d\n[PROOFSTEP]\nsimp [ite_zero_add]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni' : \u03b9\nh : i = i'\n\u22a2 \u2191(single i (\u2191f i) + erase i f) i' = \u2191f i'\n[PROOFSTEP]\nsubst h\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 \u2191(single i (\u2191f i) + erase i f) i = \u2191f i\n[PROOFSTEP]\nsimp only [add_apply, single_apply, erase_apply, add_zero, dite_eq_ite, if_true]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni' : \u03b9\nh : \u00aci = i'\n\u22a2 \u2191(single i (\u2191f i) + erase i f) i' = \u2191f i'\n[PROOFSTEP]\nsimp only [add_apply, single_apply, erase_apply, dif_neg h, if_neg (Ne.symm h), zero_add]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni' : \u03b9\nh : i = i'\n\u22a2 \u2191(erase i f + single i (\u2191f i)) i' = \u2191f i'\n[PROOFSTEP]\nsubst h\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 \u2191(erase i f + single i (\u2191f i)) i = \u2191f i\n[PROOFSTEP]\nsimp only [add_apply, single_apply, erase_apply, zero_add, dite_eq_ite, if_true]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni' : \u03b9\nh : \u00aci = i'\n\u22a2 \u2191(erase i f + single i (\u2191f i)) i' = \u2191f i'\n[PROOFSTEP]\nsimp only [add_apply, single_apply, erase_apply, dif_neg h, if_neg (Ne.symm h), add_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\n\u22a2 p f\n[PROOFSTEP]\ncases' f with f s\n[GOAL]\ncase mk'\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf : (i : \u03b9) \u2192 \u03b2 i\ns : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\n\u22a2 p { toFun := f, support' := s }\n[PROOFSTEP]\ninduction' s using Trunc.induction_on with s\n[GOAL]\ncase mk'.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf : (i : \u03b9) \u2192 \u03b2 i\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\n\u22a2 p { toFun := f, support' := Trunc.mk s }\n[PROOFSTEP]\ncases' s with s H\n[GOAL]\ncase mk'.h.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf : (i : \u03b9) \u2192 \u03b2 i\ns : Multiset \u03b9\nH : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0\n\u22a2 p { toFun := f, support' := Trunc.mk { val := s, property := H } }\n[PROOFSTEP]\ninduction' s using Multiset.induction_on with i s ih generalizing f\n[GOAL]\ncase mk'.h.mk.empty\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s \u2228 f\u271d i = 0\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i : \u03b9), i \u2208 0 \u2228 f i = 0\n\u22a2 p { toFun := f, support' := Trunc.mk { val := 0, property := H } }\n[PROOFSTEP]\nhave : f = 0 := funext fun i => (H i).resolve_left (Multiset.not_mem_zero _)\n[GOAL]\ncase mk'.h.mk.empty\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s \u2228 f\u271d i = 0\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i : \u03b9), i \u2208 0 \u2228 f i = 0\nthis : f = 0\n\u22a2 p { toFun := f, support' := Trunc.mk { val := 0, property := H } }\n[PROOFSTEP]\nsubst this\n[GOAL]\ncase mk'.h.mk.empty\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf : (i : \u03b9) \u2192 \u03b2 i\ns : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0\nH : \u2200 (i : \u03b9), i \u2208 0 \u2228 OfNat.ofNat 0 i = 0\n\u22a2 p { toFun := 0, support' := Trunc.mk { val := 0, property := H } }\n[PROOFSTEP]\nexact h0\n[GOAL]\ncase mk'.h.mk.cons\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\n\u22a2 p { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\n[PROOFSTEP]\nhave H2 : p (erase i \u27e8f, Trunc.mk \u27e8i ::\u2098 s, H\u27e9\u27e9) :=\n by\n dsimp only [erase, Trunc.map, Trunc.bind, Trunc.liftOn, Trunc.lift_mk, Function.comp, Subtype.coe_mk]\n have H2 : \u2200 j, j \u2208 s \u2228 ite (j = i) 0 (f j) = 0 := by\n intro j\n cases' H j with H2 H2\n \u00b7 cases' Multiset.mem_cons.1 H2 with H3 H3\n \u00b7 right; exact if_pos H3\n \u00b7 left; exact H3\n right\n split_ifs <;> [rfl; exact H2]\n have H3 :\n \u2200 aux,\n (\u27e8fun j : \u03b9 => ite (j = i) 0 (f j), Trunc.mk \u27e8i ::\u2098 s, aux\u27e9\u27e9 : \u03a0\u2080 i, \u03b2 i) =\n \u27e8fun j : \u03b9 => ite (j = i) 0 (f j), Trunc.mk \u27e8s, H2\u27e9\u27e9 :=\n fun _ \u21a6 ext fun _ => rfl\n rw [H3]\n apply ih\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\n\u22a2 p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\n[PROOFSTEP]\ndsimp only [erase, Trunc.map, Trunc.bind, Trunc.liftOn, Trunc.lift_mk, Function.comp, Subtype.coe_mk]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\n\u22a2 p\n { toFun := fun j => if j = i then 0 else f j,\n support' :=\n Trunc.mk\n { val := i ::\u2098 s,\n property :=\n (_ :\n \u2200 (j : \u03b9),\n j \u2208 \u2191{ val := i ::\u2098 s, property := H } \u2228\n (fun j =>\n if j = i then 0\n else toFun { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } j)\n j =\n 0) } }\n[PROOFSTEP]\nhave H2 : \u2200 j, j \u2208 s \u2228 ite (j = i) 0 (f j) = 0 := by\n intro j\n cases' H j with H2 H2\n \u00b7 cases' Multiset.mem_cons.1 H2 with H3 H3\n \u00b7 right; exact if_pos H3\n \u00b7 left; exact H3\n right\n split_ifs <;> [rfl; exact H2]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\n\u22a2 \u2200 (j : \u03b9), j \u2208 s \u2228 (if j = i then 0 else f j) = 0\n[PROOFSTEP]\nintro j\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\n\u22a2 j \u2208 s \u2228 (if j = i then 0 else f j) = 0\n[PROOFSTEP]\ncases' H j with H2 H2\n[GOAL]\ncase inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : j \u2208 i ::\u2098 s\n\u22a2 j \u2208 s \u2228 (if j = i then 0 else f j) = 0\n[PROOFSTEP]\ncases' Multiset.mem_cons.1 H2 with H3 H3\n[GOAL]\ncase inl.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : j \u2208 i ::\u2098 s\nH3 : j = i\n\u22a2 j \u2208 s \u2228 (if j = i then 0 else f j) = 0\n[PROOFSTEP]\nright\n[GOAL]\ncase inl.inl.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : j \u2208 i ::\u2098 s\nH3 : j = i\n\u22a2 (if j = i then 0 else f j) = 0\n[PROOFSTEP]\nexact if_pos H3\n[GOAL]\ncase inl.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : j \u2208 i ::\u2098 s\nH3 : j \u2208 s\n\u22a2 j \u2208 s \u2228 (if j = i then 0 else f j) = 0\n[PROOFSTEP]\nleft\n[GOAL]\ncase inl.inr.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : j \u2208 i ::\u2098 s\nH3 : j \u2208 s\n\u22a2 j \u2208 s\n[PROOFSTEP]\nexact H3\n[GOAL]\ncase inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : f j = 0\n\u22a2 j \u2208 s \u2228 (if j = i then 0 else f j) = 0\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : f j = 0\n\u22a2 (if j = i then 0 else f j) = 0\n[PROOFSTEP]\nsplit_ifs <;> [rfl; exact H2]\n[GOAL]\ncase inr.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : f j = 0\n\u22a2 (if j = i then 0 else f j) = 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : f j = 0\nh\u271d : j = i\n\u22a2 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : f j = 0\nh\u271d : \u00acj = i\n\u22a2 f j = 0\n[PROOFSTEP]\nexact H2\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : \u2200 (j : \u03b9), j \u2208 s \u2228 (if j = i then 0 else f j) = 0\n\u22a2 p\n { toFun := fun j => if j = i then 0 else f j,\n support' :=\n Trunc.mk\n { val := i ::\u2098 s,\n property :=\n (_ :\n \u2200 (j : \u03b9),\n j \u2208 \u2191{ val := i ::\u2098 s, property := H } \u2228\n (fun j =>\n if j = i then 0\n else toFun { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } j)\n j =\n 0) } }\n[PROOFSTEP]\nhave H3 :\n \u2200 aux,\n (\u27e8fun j : \u03b9 => ite (j = i) 0 (f j), Trunc.mk \u27e8i ::\u2098 s, aux\u27e9\u27e9 : \u03a0\u2080 i, \u03b2 i) =\n \u27e8fun j : \u03b9 => ite (j = i) 0 (f j), Trunc.mk \u27e8s, H2\u27e9\u27e9 :=\n fun _ \u21a6 ext fun _ => rfl\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : \u2200 (j : \u03b9), j \u2208 s \u2228 (if j = i then 0 else f j) = 0\nH3 :\n \u2200 (aux : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 (fun j => if j = i then 0 else f j) i_1 = 0),\n { toFun := fun j => if j = i then 0 else f j, support' := Trunc.mk { val := i ::\u2098 s, property := aux } } =\n { toFun := fun j => if j = i then 0 else f j, support' := Trunc.mk { val := s, property := H2 } }\n\u22a2 p\n { toFun := fun j => if j = i then 0 else f j,\n support' :=\n Trunc.mk\n { val := i ::\u2098 s,\n property :=\n (_ :\n \u2200 (j : \u03b9),\n j \u2208 \u2191{ val := i ::\u2098 s, property := H } \u2228\n (fun j =>\n if j = i then 0\n else toFun { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } j)\n j =\n 0) } }\n[PROOFSTEP]\nrw [H3]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : \u2200 (j : \u03b9), j \u2208 s \u2228 (if j = i then 0 else f j) = 0\nH3 :\n \u2200 (aux : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 (fun j => if j = i then 0 else f j) i_1 = 0),\n { toFun := fun j => if j = i then 0 else f j, support' := Trunc.mk { val := i ::\u2098 s, property := aux } } =\n { toFun := fun j => if j = i then 0 else f j, support' := Trunc.mk { val := s, property := H2 } }\n\u22a2 p { toFun := fun j => if j = i then 0 else f j, support' := Trunc.mk { val := s, property := H2 } }\n[PROOFSTEP]\napply ih\n[GOAL]\ncase mk'.h.mk.cons\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\n\u22a2 p { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\n[PROOFSTEP]\nhave H3 : single i _ + _ = (\u27e8f, Trunc.mk \u27e8i ::\u2098 s, H\u27e9\u27e9 : \u03a0\u2080 i, \u03b2 i) := single_add_erase _ _\n[GOAL]\ncase mk'.h.mk.cons\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\n\u22a2 p { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\n[PROOFSTEP]\nrw [\u2190 H3]\n[GOAL]\ncase mk'.h.mk.cons\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\n\u22a2 p\n (single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\n[PROOFSTEP]\nchange p (single i (f i) + _)\n[GOAL]\ncase mk'.h.mk.cons\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\n\u22a2 p (single i (f i) + erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\n[PROOFSTEP]\ncases' Classical.em (f i = 0) with h h\n[GOAL]\ncase mk'.h.mk.cons.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\nh : f i = 0\n\u22a2 p (single i (f i) + erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\n[PROOFSTEP]\nrw [h, single_zero, zero_add]\n[GOAL]\ncase mk'.h.mk.cons.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\nh : f i = 0\n\u22a2 p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\n[PROOFSTEP]\nexact H2\n[GOAL]\ncase mk'.h.mk.cons.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\nh : \u00acf i = 0\n\u22a2 p (single i (f i) + erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\n[PROOFSTEP]\nrefine' ha _ _ _ _ h H2\n[GOAL]\ncase mk'.h.mk.cons.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\nh : \u00acf i = 0\n\u22a2 \u2191(erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }) i = 0\n[PROOFSTEP]\nrw [erase_same]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nf\u271d : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single i b)\ni : \u03b9\nb : \u03b2 i\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh1 : \u2191f i = 0\nh2 : b \u2260 0\nh3 : p f\n\u22a2 f + single i b = single i b + f\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nf\u271d : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single i b)\ni : \u03b9\nb : \u03b2 i\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh1 : \u2191f i = 0\nh2 : b \u2260 0\nh3 : p f\nj : \u03b9\n\u22a2 \u2191(f + single i b) j = \u2191(single i b + f) j\n[PROOFSTEP]\nby_cases H : i = j\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nf\u271d : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single i b)\ni : \u03b9\nb : \u03b2 i\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh1 : \u2191f i = 0\nh2 : b \u2260 0\nh3 : p f\nj : \u03b9\nH : i = j\n\u22a2 \u2191(f + single i b) j = \u2191(single i b + f) j\n[PROOFSTEP]\nsubst H\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nf\u271d : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single i b)\ni : \u03b9\nb : \u03b2 i\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh1 : \u2191f i = 0\nh2 : b \u2260 0\nh3 : p f\n\u22a2 \u2191(f + single i b) i = \u2191(single i b + f) i\n[PROOFSTEP]\nsimp [h1]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nf\u271d : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single i b)\ni : \u03b9\nb : \u03b2 i\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh1 : \u2191f i = 0\nh2 : b \u2260 0\nh3 : p f\nj : \u03b9\nH : \u00aci = j\n\u22a2 \u2191(f + single i b) j = \u2191(single i b + f) j\n[PROOFSTEP]\nsimp [H]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\nx\u271d : x \u2208 \u22a4\n\u22a2 x \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), Set.range (single i))\n[PROOFSTEP]\napply DFinsupp.induction x\n[GOAL]\ncase h0\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\nx\u271d : x \u2208 \u22a4\n\u22a2 0 \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), Set.range (single i))\ncase ha\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\nx\u271d : x \u2208 \u22a4\n\u22a2 \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i),\n \u2191f i = 0 \u2192\n b \u2260 0 \u2192\n f \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), Set.range (single i)) \u2192\n single i b + f \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), Set.range (single i))\n[PROOFSTEP]\nexact AddSubmonoid.zero_mem _\n[GOAL]\ncase ha\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\nx\u271d : x \u2208 \u22a4\n\u22a2 \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i),\n \u2191f i = 0 \u2192\n b \u2260 0 \u2192\n f \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), Set.range (single i)) \u2192\n single i b + f \u2208 AddSubmonoid.closure (\u22c3 (i : \u03b9), Set.range (single i))\n[PROOFSTEP]\nexact fun a b f _ _ hf =>\n AddSubmonoid.add_mem _ (AddSubmonoid.subset_closure <| Set.mem_iUnion.2 \u27e8a, Set.mem_range_self _\u27e9) hf\n[GOAL]\n\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\n\u03b3 : Type w\ninst\u271d : AddZeroClass \u03b3\nf g : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nH : \u2200 (i : \u03b9) (y : \u03b2 i), \u2191f (single i y) = \u2191g (single i y)\n\u22a2 f = g\n[PROOFSTEP]\nrefine' AddMonoidHom.eq_of_eqOn_denseM add_closure_iUnion_range_single fun f hf => _\n[GOAL]\n\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\n\u03b3 : Type w\ninst\u271d : AddZeroClass \u03b3\nf\u271d g : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nH : \u2200 (i : \u03b9) (y : \u03b2 i), \u2191f\u271d (single i y) = \u2191g (single i y)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nhf : f \u2208 \u22c3 (i : \u03b9), Set.range (single i)\n\u22a2 \u2191f\u271d f = \u2191g f\n[PROOFSTEP]\nsimp only [Set.mem_iUnion, Set.mem_range] at hf \n[GOAL]\n\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\n\u03b3 : Type w\ninst\u271d : AddZeroClass \u03b3\nf\u271d g : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nH : \u2200 (i : \u03b9) (y : \u03b2 i), \u2191f\u271d (single i y) = \u2191g (single i y)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nhf : \u2203 i y, single i y = f\n\u22a2 \u2191f\u271d f = \u2191g f\n[PROOFSTEP]\nrcases hf with \u27e8x, y, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\n\u03b3 : Type w\ninst\u271d : AddZeroClass \u03b3\nf g : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nH : \u2200 (i : \u03b9) (y : \u03b2 i), \u2191f (single i y) = \u2191g (single i y)\nx : \u03b9\ny : \u03b2 x\n\u22a2 \u2191f (single x y) = \u2191g (single x y)\n[PROOFSTEP]\napply H\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\n\u22a2 \u2191(mk s (x + y)) i = \u2191(mk s x + mk s y) i\n[PROOFSTEP]\nsimp only [add_apply, mk_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\n\u22a2 (if H : i \u2208 s then (x + y) { val := i, property := H } else 0) =\n (if H : i \u2208 s then x { val := i, property := H } else 0) + if H : i \u2208 s then y { val := i, property := H } else 0\n[PROOFSTEP]\nsplit_ifs <;> [rfl; rw [zero_add]]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\n\u22a2 (if H : i \u2208 s then (x + y) { val := i, property := H } else 0) =\n (if H : i \u2208 s then x { val := i, property := H } else 0) + if H : i \u2208 s then y { val := i, property := H } else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\nh\u271d : i \u2208 s\n\u22a2 (x + y) { val := i, property := h\u271d } = x { val := i, property := h\u271d } + y { val := i, property := h\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\nh\u271d : \u00aci \u2208 s\n\u22a2 0 = 0 + 0\n[PROOFSTEP]\nrw [zero_add]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\ni : \u03b9\n\u22a2 \u2191(mk s 0) i = \u21910 i\n[PROOFSTEP]\nsimp only [mk_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\ni : \u03b9\n\u22a2 (if H : i \u2208 s then OfNat.ofNat 0 { val := i, property := H } else 0) = \u21910 i\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\ni : \u03b9\nh\u271d : i \u2208 s\n\u22a2 OfNat.ofNat 0 { val := i, property := h\u271d } = \u21910 i\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\ni : \u03b9\nh\u271d : \u00aci \u2208 s\n\u22a2 0 = \u21910 i\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\n\u22a2 \u2191(mk s (-x)) i = \u2191(-mk s x) i\n[PROOFSTEP]\nsimp only [neg_apply, mk_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\n\u22a2 (if H : i \u2208 s then (-x) { val := i, property := H } else 0) = -if H : i \u2208 s then x { val := i, property := H } else 0\n[PROOFSTEP]\nsplit_ifs <;> [rfl; rw [neg_zero]]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\n\u22a2 (if H : i \u2208 s then (-x) { val := i, property := H } else 0) = -if H : i \u2208 s then x { val := i, property := H } else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\nh\u271d : i \u2208 s\n\u22a2 (-x) { val := i, property := h\u271d } = -x { val := i, property := h\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\ns : Finset \u03b9\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\nh\u271d : \u00aci \u2208 s\n\u22a2 0 = -0\n[PROOFSTEP]\nrw [neg_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\n\u22a2 \u2191(mk s (x - y)) i = \u2191(mk s x - mk s y) i\n[PROOFSTEP]\nsimp only [sub_apply, mk_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\n\u22a2 (if H : i \u2208 s then (x - y) { val := i, property := H } else 0) =\n (if H : i \u2208 s then x { val := i, property := H } else 0) - if H : i \u2208 s then y { val := i, property := H } else 0\n[PROOFSTEP]\nsplit_ifs <;> [rfl; rw [sub_zero]]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\n\u22a2 (if H : i \u2208 s then (x - y) { val := i, property := H } else 0) =\n (if H : i \u2208 s then x { val := i, property := H } else 0) - if H : i \u2208 s then y { val := i, property := H } else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\nh\u271d : i \u2208 s\n\u22a2 (x - y) { val := i, property := h\u271d } = x { val := i, property := h\u271d } - y { val := i, property := h\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\ns : Finset \u03b9\nx y : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\nh\u271d : \u00aci \u2208 s\n\u22a2 0 = 0 - 0\n[PROOFSTEP]\nrw [sub_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\ns : Finset \u03b9\nc : \u03b3\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\n\u22a2 \u2191(mk s (c \u2022 x)) i = \u2191(c \u2022 mk s x) i\n[PROOFSTEP]\nsimp only [smul_apply, mk_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\ns : Finset \u03b9\nc : \u03b3\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\n\u22a2 (if H : i \u2208 s then (c \u2022 x) { val := i, property := H } else 0) =\n c \u2022 if H : i \u2208 s then x { val := i, property := H } else 0\n[PROOFSTEP]\nsplit_ifs <;> [rfl; rw [smul_zero]]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\ns : Finset \u03b9\nc : \u03b3\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\n\u22a2 (if H : i \u2208 s then (c \u2022 x) { val := i, property := H } else 0) =\n c \u2022 if H : i \u2208 s then x { val := i, property := H } else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\ns : Finset \u03b9\nc : \u03b3\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\nh\u271d : i \u2208 s\n\u22a2 (c \u2022 x) { val := i, property := h\u271d } = c \u2022 x { val := i, property := h\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\ns : Finset \u03b9\nc : \u03b3\nx : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni : \u03b9\nh\u271d : \u00aci \u2208 s\n\u22a2 0 = c \u2022 0\n[PROOFSTEP]\nrw [smul_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\ni\u271d : \u03b9\nc : \u03b3\nx : \u03b2 i\u271d\ni : \u03b9\n\u22a2 \u2191(single i\u271d (c \u2022 x)) i = \u2191(c \u2022 single i\u271d x) i\n[PROOFSTEP]\nsimp only [smul_apply, single_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\ni\u271d : \u03b9\nc : \u03b3\nx : \u03b2 i\u271d\ni : \u03b9\n\u22a2 (if h : i\u271d = i then (_ : i\u271d = i) \u25b8 (c \u2022 x) else 0) = c \u2022 if h : i\u271d = i then (_ : i\u271d = i) \u25b8 x else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\ni\u271d : \u03b9\nc : \u03b3\nx : \u03b2 i\u271d\ni : \u03b9\nh : i\u271d = i\n\u22a2 (_ : i\u271d = i) \u25b8 (c \u2022 x) = c \u2022 (_ : i\u271d = i) \u25b8 x\n[PROOFSTEP]\ncases h\n[GOAL]\ncase pos.refl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\ni : \u03b9\nc : \u03b3\nx : \u03b2 i\n\u22a2 (_ : i = i) \u25b8 (c \u2022 x) = c \u2022 (_ : i = i) \u25b8 x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\ni\u271d : \u03b9\nc : \u03b3\nx : \u03b2 i\u271d\ni : \u03b9\nh : \u00aci\u271d = i\n\u22a2 0 = c \u2022 0\n[PROOFSTEP]\nrw [smul_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }),\n (fun xs => Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset \u2191xs)) a =\n (fun xs => Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset \u2191xs)) b\n[PROOFSTEP]\nrintro \u27e8sx, hx\u27e9 \u27e8sy, hy\u27e9\n[GOAL]\ncase mk.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\n\u22a2 (fun xs => Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset \u2191xs)) { val := sx, property := hx } =\n (fun xs => Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset \u2191xs)) { val := sy, property := hy }\n[PROOFSTEP]\ndsimp only [Subtype.coe_mk, toFun_eq_coe] at *\n[GOAL]\ncase mk.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\n\u22a2 Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sx) = Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sy)\n[PROOFSTEP]\next i\n[GOAL]\ncase mk.mk.a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\ni : \u03b9\n\u22a2 i \u2208 Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sx) \u2194\n i \u2208 Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sy)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.mk.a.mp\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\ni : \u03b9\n\u22a2 i \u2208 Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sx) \u2192\n i \u2208 Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sy)\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mk.mk.a.mp\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\ni : \u03b9\nH : i \u2208 Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sx)\n\u22a2 i \u2208 Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sy)\n[PROOFSTEP]\nrcases Finset.mem_filter.1 H with \u27e8_, h\u27e9\n[GOAL]\ncase mk.mk.a.mp.intro\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\ni : \u03b9\nH : i \u2208 Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sx)\nleft\u271d : i \u2208 Multiset.toFinset sx\nh : \u2191f i \u2260 0\n\u22a2 i \u2208 Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sy)\n[PROOFSTEP]\nexact Finset.mem_filter.2 \u27e8Multiset.mem_toFinset.2 <| (hy i).resolve_right h, h\u27e9\n[GOAL]\ncase mk.mk.a.mpr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\ni : \u03b9\n\u22a2 i \u2208 Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sy) \u2192\n i \u2208 Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sx)\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mk.mk.a.mpr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\ni : \u03b9\nH : i \u2208 Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sy)\n\u22a2 i \u2208 Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sx)\n[PROOFSTEP]\nrcases Finset.mem_filter.1 H with \u27e8_, h\u27e9\n[GOAL]\ncase mk.mk.a.mpr.intro\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\ni : \u03b9\nH : i \u2208 Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sy)\nleft\u271d : i \u2208 Multiset.toFinset sy\nh : \u2191f i \u2260 0\n\u22a2 i \u2208 Finset.filter (fun i => \u2191f i \u2260 0) (Multiset.toFinset sx)\n[PROOFSTEP]\nexact Finset.mem_filter.2 \u27e8Multiset.mem_toFinset.2 <| (hx i).resolve_right h, h\u27e9\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2 i\ns : Multiset \u03b9\nh : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0\ni : \u03b9\nH : i \u2208 support { toFun := f, support' := Trunc.mk { val := s, property := h } }\n\u22a2 i \u2208 Multiset.toFinset (Multiset.toFinset s).val\n[PROOFSTEP]\nsimpa using (Finset.mem_filter.1 H).1\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 i \u2208 support f \u2194 \u2191f i \u2260 0\n[PROOFSTEP]\ncases' f with f s\n[GOAL]\ncase mk'\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b2 i\ns : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\n\u22a2 i \u2208 support { toFun := f, support' := s } \u2194 \u2191{ toFun := f, support' := s } i \u2260 0\n[PROOFSTEP]\ninduction' s using Trunc.induction_on with s\n[GOAL]\ncase mk'.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b2 i\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\n\u22a2 i \u2208 support { toFun := f, support' := Trunc.mk s } \u2194 \u2191{ toFun := f, support' := Trunc.mk s } i \u2260 0\n[PROOFSTEP]\ndsimp only [support, Trunc.lift_mk]\n[GOAL]\ncase mk'.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b2 i\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\n\u22a2 i \u2208 Finset.filter (fun i => \u2191{ toFun := f, support' := Trunc.mk s } i \u2260 0) (Multiset.toFinset \u2191s) \u2194\n \u2191{ toFun := f, support' := Trunc.mk s } i \u2260 0\n[PROOFSTEP]\nrw [Finset.mem_filter, Multiset.mem_toFinset, coe_mk']\n[GOAL]\ncase mk'.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b2 i\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\n\u22a2 i \u2208 \u2191s \u2227 f i \u2260 0 \u2194 f i \u2260 0\n[PROOFSTEP]\nexact and_iff_right_of_imp (s.prop i).resolve_right\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 f = mk (support f) fun i => \u2191f \u2191i\n[PROOFSTEP]\nchange f = mk f.support fun i => f i.1\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 f = mk (support f) fun i => \u2191f \u2191i\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 \u2191f i = \u2191(mk (support f) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nby_cases h : f i \u2260 0 <;> [skip; rw [not_not] at h ]\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 \u2191f i = \u2191(mk (support f) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nby_cases h : f i \u2260 0\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh : \u2191f i \u2260 0\n\u22a2 \u2191f i = \u2191(mk (support f) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nskip\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh : \u00ac\u2191f i \u2260 0\n\u22a2 \u2191f i = \u2191(mk (support f) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nrw [not_not] at h \n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh : \u2191f i \u2260 0\n\u22a2 \u2191f i = \u2191(mk (support f) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh : \u2191f i = 0\n\u22a2 \u2191f i = \u2191(mk (support f) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ns : Finset \u03b9\nf : (i : { x // x \u2208 s }) \u2192 { x // x \u2260 0 }\ni : \u03b9\n\u22a2 i \u2208 support (mk s fun i => \u2191(f i)) \u2194 i \u2208 s\n[PROOFSTEP]\ncalc\n i \u2208 support (mk s fun i \u21a6 (f i).1) \u2194 \u2203 h : i \u2208 s, (f \u27e8i, h\u27e9).1 \u2260 0 := by simp\n _ \u2194 \u2203 _ : i \u2208 s, True := (exists_congr fun h \u21a6 (iff_true _).mpr (f _).2)\n _ \u2194 i \u2208 s := by simp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ns : Finset \u03b9\nf : (i : { x // x \u2208 s }) \u2192 { x // x \u2260 0 }\ni : \u03b9\n\u22a2 i \u2208 support (mk s fun i => \u2191(f i)) \u2194 \u2203 h, \u2191(f { val := i, property := h }) \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ns : Finset \u03b9\nf : (i : { x // x \u2208 s }) \u2192 { x // x \u2260 0 }\ni : \u03b9\n\u22a2 (\u2203 x, True) \u2194 i \u2208 s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ns : Finset \u03b9\n\u22a2 Function.LeftInverse (fun f => { val := mk s fun i => \u2191(f i), property := (_ : support (mk s fun i => \u2191(f i)) = s) })\n fun x =>\n match (motive := { f // support f = s } \u2192 (i : { x // x \u2208 s }) \u2192 { x // x \u2260 0 }) x with\n | { val := f, property := hf } => fun x =>\n match x with\n | { val := i, property := hi } => { val := \u2191f i, property := (_ : \u2191f i \u2260 0) }\n[PROOFSTEP]\nrintro \u27e8f, rfl\u27e9\n[GOAL]\ncase mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 (fun f_1 =>\n { val := mk (support f) fun i => \u2191(f_1 i),\n property := (_ : support (mk (support f) fun i => \u2191(f_1 i)) = support f) })\n ((fun x =>\n match (motive := { f_1 // support f_1 = support f } \u2192 (i : { x // x \u2208 support f }) \u2192 { x // x \u2260 0 }) x with\n | { val := f_1, property := hf } => fun x =>\n match x with\n | { val := i, property := hi } => { val := \u2191f_1 i, property := (_ : \u2191f_1 i \u2260 0) })\n { val := f, property := (_ : support f = support f) }) =\n { val := f, property := (_ : support f = support f) }\n[PROOFSTEP]\next i\n[GOAL]\ncase mk.a.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 \u2191\u2191((fun f_1 =>\n { val := mk (support f) fun i => \u2191(f_1 i),\n property := (_ : support (mk (support f) fun i => \u2191(f_1 i)) = support f) })\n ((fun x =>\n match (motive := { f_1 // support f_1 = support f } \u2192 (i : { x // x \u2208 support f }) \u2192 { x // x \u2260 0 })\n x with\n | { val := f_1, property := hf } => fun x =>\n match x with\n | { val := i, property := hi } => { val := \u2191f_1 i, property := (_ : \u2191f_1 i \u2260 0) })\n { val := f, property := (_ : support f = support f) }))\n i =\n \u2191\u2191{ val := f, property := (_ : support f = support f) } i\n[PROOFSTEP]\nsimpa using Eq.symm\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ns : Finset \u03b9\nf : (i : { x // x \u2208 s }) \u2192 { x // x \u2260 0 }\n\u22a2 (fun x =>\n match (motive := { f // support f = s } \u2192 (i : { x // x \u2208 s }) \u2192 { x // x \u2260 0 }) x with\n | { val := f, property := hf } => fun x =>\n match x with\n | { val := i, property := hi } => { val := \u2191f i, property := (_ : \u2191f i \u2260 0) })\n ((fun f => { val := mk s fun i => \u2191(f i), property := (_ : support (mk s fun i => \u2191(f i)) = s) }) f) =\n f\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ns : Finset \u03b9\nf : (i : { x // x \u2208 s }) \u2192 { x // x \u2260 0 }\nx\u271d : { x // x \u2208 s }\n\u22a2 (fun x =>\n match (motive := { f // support f = s } \u2192 (i : { x // x \u2208 s }) \u2192 { x // x \u2260 0 }) x with\n | { val := f, property := hf } => fun x =>\n match x with\n | { val := i, property := hi } => { val := \u2191f i, property := (_ : \u2191f i \u2260 0) })\n ((fun f => { val := mk s fun i => \u2191(f i), property := (_ : support (mk s fun i => \u2191(f i)) = s) }) f) x\u271d =\n f x\u271d\n[PROOFSTEP]\nsimp [Subtype.eta]\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ns : Finset \u03b9\nf : (i : { x // x \u2208 s }) \u2192 { x // x \u2260 0 }\nx\u271d : { x // x \u2208 s }\n\u22a2 f { val := \u2191x\u271d, property := (_ : \u2191x\u271d \u2208 s) } = f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nH : support f = \u2205\n\u22a2 \u2200 (i : \u03b9), \u2191f i = \u21910 i\n[PROOFSTEP]\nsimpa [Finset.ext_iff] using H\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 f = 0 \u2192 support f = \u2205\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ns : Set \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 \u2191(support f) \u2286 s \u2194 \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 \u2191f i = 0\n[PROOFSTEP]\nsimp [Set.subset_def]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ns : Set \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 (\u2200 (x : \u03b9), \u00ac\u2191f x = 0 \u2192 x \u2208 s) \u2194 \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 \u2191f i = 0\n[PROOFSTEP]\nexact forall_congr' fun i => not_imp_comm\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nb : \u03b2 i\nhb : b \u2260 0\n\u22a2 support (single i b) = {i}\n[PROOFSTEP]\next j\n[GOAL]\ncase a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nb : \u03b2 i\nhb : b \u2260 0\nj : \u03b9\n\u22a2 j \u2208 support (single i b) \u2194 j \u2208 {i}\n[PROOFSTEP]\nby_cases h : i = j\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nb : \u03b2 i\nhb : b \u2260 0\nj : \u03b9\nh : i = j\n\u22a2 j \u2208 support (single i b) \u2194 j \u2208 {i}\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nb : \u03b2 i\nhb : b \u2260 0\n\u22a2 i \u2208 support (single i b) \u2194 i \u2208 {i}\n[PROOFSTEP]\nsimp [hb]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nb : \u03b2 i\nhb : b \u2260 0\nj : \u03b9\nh : \u00aci = j\n\u22a2 j \u2208 support (single i b) \u2194 j \u2208 {i}\n[PROOFSTEP]\nsimp [Ne.symm h, h]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\n\u22a2 mapRange f hf g = mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ni : \u03b9\n\u22a2 \u2191(mapRange f hf g) i = \u2191(mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)) i\n[PROOFSTEP]\nby_cases h : g i \u2260 0\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ni : \u03b9\nh : \u2191g i \u2260 0\n\u22a2 \u2191(mapRange f hf g) i = \u2191(mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)) i\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ni : \u03b9\nh : \u00ac\u2191g i \u2260 0\n\u22a2 \u2191(mapRange f hf g) i = \u2191(mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)) i\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ni : \u03b9\nh : \u00ac\u2191g i = 0\n\u22a2 \u2191(mapRange f hf g) i = \u2191(mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)) i\n[PROOFSTEP]\nsimp [h, hf]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ni : \u03b9\nh : \u2191g i = 0\n\u22a2 \u2191(mapRange f hf g) i = \u2191(mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)) i\n[PROOFSTEP]\nsimp [h, hf]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ni : \u03b9\nb : \u03b2\u2081 i\ni' : \u03b9\n\u22a2 \u2191(mapRange f hf (single i b)) i' = \u2191(single i (f i b)) i'\n[PROOFSTEP]\nby_cases h : i = i'\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ni : \u03b9\nb : \u03b2\u2081 i\ni' : \u03b9\nh : i = i'\n\u22a2 \u2191(mapRange f hf (single i b)) i' = \u2191(single i (f i b)) i'\n[PROOFSTEP]\nsubst i'\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ni : \u03b9\nb : \u03b2\u2081 i\n\u22a2 \u2191(mapRange f hf (single i b)) i = \u2191(single i (f i b)) i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ni : \u03b9\nb : \u03b2\u2081 i\ni' : \u03b9\nh : \u00aci = i'\n\u22a2 \u2191(mapRange f hf (single i b)) i' = \u2191(single i (f i b)) i'\n[PROOFSTEP]\nsimp [h, hf]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2075 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\n\u22a2 support (mapRange f hf g) \u2286 support g\n[PROOFSTEP]\nsimp [mapRange_def]\n[GOAL]\n\u03b9\u271d : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9\u271d \u2192 Type v\n\u03b2\u2081\u271d : \u03b9\u271d \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9\u271d \u2192 Type v\u2082\ndec\u271d : DecidableEq \u03b9\u271d\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u271d i)\ninst\u271d\u2079 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2078 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2081\u271d i)\ninst\u271d\u2077 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2082\u271d i)\ninst\u271d\u2076 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2081\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2075 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2082\u271d i) \u2192 Decidable (x \u2260 0)\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\ng\u2081 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ng\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\n\u22a2 zipWith f hf g\u2081 g\u2082 = mk (support g\u2081 \u222a support g\u2082) fun i => f (\u2191i) (\u2191g\u2081 \u2191i) (\u2191g\u2082 \u2191i)\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9\u271d : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9\u271d \u2192 Type v\n\u03b2\u2081\u271d : \u03b9\u271d \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9\u271d \u2192 Type v\u2082\ndec\u271d : DecidableEq \u03b9\u271d\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u271d i)\ninst\u271d\u2079 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2078 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2081\u271d i)\ninst\u271d\u2077 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2082\u271d i)\ninst\u271d\u2076 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2081\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2075 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2082\u271d i) \u2192 Decidable (x \u2260 0)\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\ng\u2081 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ng\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\ni : \u03b9\n\u22a2 \u2191(zipWith f hf g\u2081 g\u2082) i = \u2191(mk (support g\u2081 \u222a support g\u2082) fun i => f (\u2191i) (\u2191g\u2081 \u2191i) (\u2191g\u2082 \u2191i)) i\n[PROOFSTEP]\nby_cases h1 : g\u2081 i \u2260 0\n[GOAL]\ncase pos\n\u03b9\u271d : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9\u271d \u2192 Type v\n\u03b2\u2081\u271d : \u03b9\u271d \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9\u271d \u2192 Type v\u2082\ndec\u271d : DecidableEq \u03b9\u271d\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u271d i)\ninst\u271d\u2079 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2078 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2081\u271d i)\ninst\u271d\u2077 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2082\u271d i)\ninst\u271d\u2076 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2081\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2075 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2082\u271d i) \u2192 Decidable (x \u2260 0)\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\ng\u2081 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ng\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\ni : \u03b9\nh1 : \u2191g\u2081 i \u2260 0\n\u22a2 \u2191(zipWith f hf g\u2081 g\u2082) i = \u2191(mk (support g\u2081 \u222a support g\u2082) fun i => f (\u2191i) (\u2191g\u2081 \u2191i) (\u2191g\u2082 \u2191i)) i\n[PROOFSTEP]\nby_cases h2 : g\u2082 i \u2260 0\n[GOAL]\ncase neg\n\u03b9\u271d : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9\u271d \u2192 Type v\n\u03b2\u2081\u271d : \u03b9\u271d \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9\u271d \u2192 Type v\u2082\ndec\u271d : DecidableEq \u03b9\u271d\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u271d i)\ninst\u271d\u2079 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2078 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2081\u271d i)\ninst\u271d\u2077 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2082\u271d i)\ninst\u271d\u2076 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2081\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2075 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2082\u271d i) \u2192 Decidable (x \u2260 0)\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\ng\u2081 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ng\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\ni : \u03b9\nh1 : \u00ac\u2191g\u2081 i \u2260 0\n\u22a2 \u2191(zipWith f hf g\u2081 g\u2082) i = \u2191(mk (support g\u2081 \u222a support g\u2082) fun i => f (\u2191i) (\u2191g\u2081 \u2191i) (\u2191g\u2082 \u2191i)) i\n[PROOFSTEP]\nby_cases h2 : g\u2082 i \u2260 0\n[GOAL]\ncase pos\n\u03b9\u271d : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9\u271d \u2192 Type v\n\u03b2\u2081\u271d : \u03b9\u271d \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9\u271d \u2192 Type v\u2082\ndec\u271d : DecidableEq \u03b9\u271d\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u271d i)\ninst\u271d\u2079 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2078 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2081\u271d i)\ninst\u271d\u2077 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2082\u271d i)\ninst\u271d\u2076 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2081\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2075 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2082\u271d i) \u2192 Decidable (x \u2260 0)\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\ng\u2081 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ng\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\ni : \u03b9\nh1 : \u2191g\u2081 i \u2260 0\nh2 : \u2191g\u2082 i \u2260 0\n\u22a2 \u2191(zipWith f hf g\u2081 g\u2082) i = \u2191(mk (support g\u2081 \u222a support g\u2082) fun i => f (\u2191i) (\u2191g\u2081 \u2191i) (\u2191g\u2082 \u2191i)) i\n[PROOFSTEP]\nsimp only [not_not, Ne.def] at h1 h2 \n[GOAL]\ncase neg\n\u03b9\u271d : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9\u271d \u2192 Type v\n\u03b2\u2081\u271d : \u03b9\u271d \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9\u271d \u2192 Type v\u2082\ndec\u271d : DecidableEq \u03b9\u271d\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u271d i)\ninst\u271d\u2079 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2078 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2081\u271d i)\ninst\u271d\u2077 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2082\u271d i)\ninst\u271d\u2076 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2081\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2075 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2082\u271d i) \u2192 Decidable (x \u2260 0)\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\ng\u2081 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ng\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\ni : \u03b9\nh1 : \u2191g\u2081 i \u2260 0\nh2 : \u00ac\u2191g\u2082 i \u2260 0\n\u22a2 \u2191(zipWith f hf g\u2081 g\u2082) i = \u2191(mk (support g\u2081 \u222a support g\u2082) fun i => f (\u2191i) (\u2191g\u2081 \u2191i) (\u2191g\u2082 \u2191i)) i\n[PROOFSTEP]\nsimp only [not_not, Ne.def] at h1 h2 \n[GOAL]\ncase pos\n\u03b9\u271d : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9\u271d \u2192 Type v\n\u03b2\u2081\u271d : \u03b9\u271d \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9\u271d \u2192 Type v\u2082\ndec\u271d : DecidableEq \u03b9\u271d\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u271d i)\ninst\u271d\u2079 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2078 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2081\u271d i)\ninst\u271d\u2077 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2082\u271d i)\ninst\u271d\u2076 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2081\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2075 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2082\u271d i) \u2192 Decidable (x \u2260 0)\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\ng\u2081 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ng\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\ni : \u03b9\nh1 : \u00ac\u2191g\u2081 i \u2260 0\nh2 : \u2191g\u2082 i \u2260 0\n\u22a2 \u2191(zipWith f hf g\u2081 g\u2082) i = \u2191(mk (support g\u2081 \u222a support g\u2082) fun i => f (\u2191i) (\u2191g\u2081 \u2191i) (\u2191g\u2082 \u2191i)) i\n[PROOFSTEP]\nsimp only [not_not, Ne.def] at h1 h2 \n[GOAL]\ncase neg\n\u03b9\u271d : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9\u271d \u2192 Type v\n\u03b2\u2081\u271d : \u03b9\u271d \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9\u271d \u2192 Type v\u2082\ndec\u271d : DecidableEq \u03b9\u271d\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u271d i)\ninst\u271d\u2079 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2078 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2081\u271d i)\ninst\u271d\u2077 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2082\u271d i)\ninst\u271d\u2076 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2081\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2075 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2082\u271d i) \u2192 Decidable (x \u2260 0)\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\ng\u2081 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ng\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\ni : \u03b9\nh1 : \u00ac\u2191g\u2081 i \u2260 0\nh2 : \u00ac\u2191g\u2082 i \u2260 0\n\u22a2 \u2191(zipWith f hf g\u2081 g\u2082) i = \u2191(mk (support g\u2081 \u222a support g\u2082) fun i => f (\u2191i) (\u2191g\u2081 \u2191i) (\u2191g\u2082 \u2191i)) i\n[PROOFSTEP]\nsimp only [not_not, Ne.def] at h1 h2 \n[GOAL]\ncase pos\n\u03b9\u271d : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9\u271d \u2192 Type v\n\u03b2\u2081\u271d : \u03b9\u271d \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9\u271d \u2192 Type v\u2082\ndec\u271d : DecidableEq \u03b9\u271d\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u271d i)\ninst\u271d\u2079 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2078 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2081\u271d i)\ninst\u271d\u2077 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2082\u271d i)\ninst\u271d\u2076 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2081\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2075 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2082\u271d i) \u2192 Decidable (x \u2260 0)\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\ng\u2081 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ng\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\ni : \u03b9\nh1 : \u00ac\u2191g\u2081 i = 0\nh2 : \u00ac\u2191g\u2082 i = 0\n\u22a2 \u2191(zipWith f hf g\u2081 g\u2082) i = \u2191(mk (support g\u2081 \u222a support g\u2082) fun i => f (\u2191i) (\u2191g\u2081 \u2191i) (\u2191g\u2082 \u2191i)) i\n[PROOFSTEP]\nsimp [h1, h2, hf]\n[GOAL]\ncase neg\n\u03b9\u271d : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9\u271d \u2192 Type v\n\u03b2\u2081\u271d : \u03b9\u271d \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9\u271d \u2192 Type v\u2082\ndec\u271d : DecidableEq \u03b9\u271d\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u271d i)\ninst\u271d\u2079 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2078 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2081\u271d i)\ninst\u271d\u2077 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2082\u271d i)\ninst\u271d\u2076 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2081\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2075 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2082\u271d i) \u2192 Decidable (x \u2260 0)\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\ng\u2081 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ng\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\ni : \u03b9\nh1 : \u00ac\u2191g\u2081 i = 0\nh2 : \u2191g\u2082 i = 0\n\u22a2 \u2191(zipWith f hf g\u2081 g\u2082) i = \u2191(mk (support g\u2081 \u222a support g\u2082) fun i => f (\u2191i) (\u2191g\u2081 \u2191i) (\u2191g\u2082 \u2191i)) i\n[PROOFSTEP]\nsimp [h1, h2, hf]\n[GOAL]\ncase pos\n\u03b9\u271d : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9\u271d \u2192 Type v\n\u03b2\u2081\u271d : \u03b9\u271d \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9\u271d \u2192 Type v\u2082\ndec\u271d : DecidableEq \u03b9\u271d\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u271d i)\ninst\u271d\u2079 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2078 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2081\u271d i)\ninst\u271d\u2077 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2082\u271d i)\ninst\u271d\u2076 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2081\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2075 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2082\u271d i) \u2192 Decidable (x \u2260 0)\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\ng\u2081 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ng\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\ni : \u03b9\nh2 : \u00ac\u2191g\u2082 i = 0\nh1 : \u2191g\u2081 i = 0\n\u22a2 \u2191(zipWith f hf g\u2081 g\u2082) i = \u2191(mk (support g\u2081 \u222a support g\u2082) fun i => f (\u2191i) (\u2191g\u2081 \u2191i) (\u2191g\u2082 \u2191i)) i\n[PROOFSTEP]\nsimp [h1, h2, hf]\n[GOAL]\ncase neg\n\u03b9\u271d : Type u\n\u03b3 : Type w\n\u03b2\u271d : \u03b9\u271d \u2192 Type v\n\u03b2\u2081\u271d : \u03b9\u271d \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9\u271d \u2192 Type v\u2082\ndec\u271d : DecidableEq \u03b9\u271d\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u271d i)\ninst\u271d\u2079 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2078 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2081\u271d i)\ninst\u271d\u2077 : (i : \u03b9\u271d) \u2192 Zero (\u03b2\u2082\u271d i)\ninst\u271d\u2076 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2081\u271d i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u2075 : (i : \u03b9\u271d) \u2192 (x : \u03b2\u2082\u271d i) \u2192 Decidable (x \u2260 0)\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\ng\u2081 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ng\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\ni : \u03b9\nh1 : \u2191g\u2081 i = 0\nh2 : \u2191g\u2082 i = 0\n\u22a2 \u2191(zipWith f hf g\u2081 g\u2082) i = \u2191(mk (support g\u2081 \u222a support g\u2082) fun i => f (\u2191i) (\u2191g\u2081 \u2191i) (\u2191g\u2082 \u2191i)) i\n[PROOFSTEP]\nsimp [h1, h2, hf]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u2075 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\ng\u2081 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ng\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\n\u22a2 support (zipWith f hf g\u2081 g\u2082) \u2286 support g\u2081 \u222a support g\u2082\n[PROOFSTEP]\nsimp [zipWith_def]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 erase i f = mk (Finset.erase (support f) i) fun j => \u2191f \u2191j\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\n\u22a2 \u2191(erase i f) j = \u2191(mk (Finset.erase (support f) i) fun j => \u2191f \u2191j) j\n[PROOFSTEP]\nby_cases h1 : j = i\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : j = i\n\u22a2 \u2191(erase i f) j = \u2191(mk (Finset.erase (support f) i) fun j => \u2191f \u2191j) j\n[PROOFSTEP]\nby_cases h2 : f j \u2260 0\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : \u00acj = i\n\u22a2 \u2191(erase i f) j = \u2191(mk (Finset.erase (support f) i) fun j => \u2191f \u2191j) j\n[PROOFSTEP]\nby_cases h2 : f j \u2260 0\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : j = i\nh2 : \u2191f j \u2260 0\n\u22a2 \u2191(erase i f) j = \u2191(mk (Finset.erase (support f) i) fun j => \u2191f \u2191j) j\n[PROOFSTEP]\nsimp at h2 \n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : j = i\nh2 : \u00ac\u2191f j \u2260 0\n\u22a2 \u2191(erase i f) j = \u2191(mk (Finset.erase (support f) i) fun j => \u2191f \u2191j) j\n[PROOFSTEP]\nsimp at h2 \n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : \u00acj = i\nh2 : \u2191f j \u2260 0\n\u22a2 \u2191(erase i f) j = \u2191(mk (Finset.erase (support f) i) fun j => \u2191f \u2191j) j\n[PROOFSTEP]\nsimp at h2 \n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : \u00acj = i\nh2 : \u00ac\u2191f j \u2260 0\n\u22a2 \u2191(erase i f) j = \u2191(mk (Finset.erase (support f) i) fun j => \u2191f \u2191j) j\n[PROOFSTEP]\nsimp at h2 \n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : j = i\nh2 : \u00ac\u2191f j = 0\n\u22a2 \u2191(erase i f) j = \u2191(mk (Finset.erase (support f) i) fun j => \u2191f \u2191j) j\n[PROOFSTEP]\nsimp [h1, h2]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : j = i\nh2 : \u2191f j = 0\n\u22a2 \u2191(erase i f) j = \u2191(mk (Finset.erase (support f) i) fun j => \u2191f \u2191j) j\n[PROOFSTEP]\nsimp [h1, h2]\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : \u00acj = i\nh2 : \u00ac\u2191f j = 0\n\u22a2 \u2191(erase i f) j = \u2191(mk (Finset.erase (support f) i) fun j => \u2191f \u2191j) j\n[PROOFSTEP]\nsimp [h1, h2]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : \u00acj = i\nh2 : \u2191f j = 0\n\u22a2 \u2191(erase i f) j = \u2191(mk (Finset.erase (support f) i) fun j => \u2191f \u2191j) j\n[PROOFSTEP]\nsimp [h1, h2]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 support (erase i f) = Finset.erase (support f) i\n[PROOFSTEP]\next j\n[GOAL]\ncase a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\n\u22a2 j \u2208 support (erase i f) \u2194 j \u2208 Finset.erase (support f) i\n[PROOFSTEP]\nby_cases h1 : j = i\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : j = i\n\u22a2 j \u2208 support (erase i f) \u2194 j \u2208 Finset.erase (support f) i\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : \u00acj = i\n\u22a2 j \u2208 support (erase i f) \u2194 j \u2208 Finset.erase (support f) i\n[PROOFSTEP]\nsimp [h1]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : \u00acj = i\n\u22a2 j \u2208 support (erase i f) \u2194 j \u2208 Finset.erase (support f) i\n[PROOFSTEP]\nby_cases h2 : f j \u2260 0\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : \u00acj = i\nh2 : \u2191f j \u2260 0\n\u22a2 j \u2208 support (erase i f) \u2194 j \u2208 Finset.erase (support f) i\n[PROOFSTEP]\nsimp at h2 \n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : \u00acj = i\nh2 : \u00ac\u2191f j \u2260 0\n\u22a2 j \u2208 support (erase i f) \u2194 j \u2208 Finset.erase (support f) i\n[PROOFSTEP]\nsimp at h2 \n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : \u00acj = i\nh2 : \u00ac\u2191f j = 0\n\u22a2 j \u2208 support (erase i f) \u2194 j \u2208 Finset.erase (support f) i\n[PROOFSTEP]\nsimp [h1, h2]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nj : \u03b9\nh1 : \u00acj = i\nh2 : \u2191f j = 0\n\u22a2 j \u2208 support (erase i f) \u2194 j \u2208 Finset.erase (support f) i\n[PROOFSTEP]\nsimp [h1, h2]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\nh : b \u2260 0\n\u22a2 support (update i f b) = insert i (support f)\n[PROOFSTEP]\next j\n[GOAL]\ncase a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\nh : b \u2260 0\nj : \u03b9\n\u22a2 j \u2208 support (update i f b) \u2194 j \u2208 insert i (support f)\n[PROOFSTEP]\nrcases eq_or_ne i j with (rfl | hi)\n[GOAL]\ncase a.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\nh : b \u2260 0\n\u22a2 i \u2208 support (update i f b) \u2194 i \u2208 insert i (support f)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase a.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\nh : b \u2260 0\nj : \u03b9\nhi : i \u2260 j\n\u22a2 j \u2208 support (update i f b) \u2194 j \u2208 insert i (support f)\n[PROOFSTEP]\nsimp [hi.symm]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\ninst\u271d : Decidable (b = 0)\n\u22a2 support (update i f b) = if b = 0 then support (erase i f) else insert i (support f)\n[PROOFSTEP]\next j\n[GOAL]\ncase a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\ninst\u271d : Decidable (b = 0)\nj : \u03b9\n\u22a2 j \u2208 support (update i f b) \u2194 j \u2208 if b = 0 then support (erase i f) else insert i (support f)\n[PROOFSTEP]\nsplit_ifs with hb\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\ninst\u271d : Decidable (b = 0)\nj : \u03b9\nhb : b = 0\n\u22a2 j \u2208 support (update i f b) \u2194 j \u2208 support (erase i f)\n[PROOFSTEP]\nsubst hb\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni j : \u03b9\ninst\u271d : Decidable (0 = 0)\n\u22a2 j \u2208 support (update i f 0) \u2194 j \u2208 support (erase i f)\n[PROOFSTEP]\nsimp [update_eq_erase, support_erase]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nb : \u03b2 i\ninst\u271d : Decidable (b = 0)\nj : \u03b9\nhb : \u00acb = 0\n\u22a2 j \u2208 support (update i f b) \u2194 j \u2208 insert i (support f)\n[PROOFSTEP]\nrw [support_update_ne_zero f _ hb]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 filter p f = mk (Finset.filter p (support f)) fun i => \u2191f \u2191i\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 \u2191(filter p f) i = \u2191(mk (Finset.filter p (support f)) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nby_cases h1 : p i\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh1 : p i\n\u22a2 \u2191(filter p f) i = \u2191(mk (Finset.filter p (support f)) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nby_cases h2 : f i \u2260 0\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh1 : \u00acp i\n\u22a2 \u2191(filter p f) i = \u2191(mk (Finset.filter p (support f)) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nby_cases h2 : f i \u2260 0\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh1 : p i\nh2 : \u2191f i \u2260 0\n\u22a2 \u2191(filter p f) i = \u2191(mk (Finset.filter p (support f)) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nsimp at h2 \n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh1 : p i\nh2 : \u00ac\u2191f i \u2260 0\n\u22a2 \u2191(filter p f) i = \u2191(mk (Finset.filter p (support f)) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nsimp at h2 \n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh1 : \u00acp i\nh2 : \u2191f i \u2260 0\n\u22a2 \u2191(filter p f) i = \u2191(mk (Finset.filter p (support f)) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nsimp at h2 \n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh1 : \u00acp i\nh2 : \u00ac\u2191f i \u2260 0\n\u22a2 \u2191(filter p f) i = \u2191(mk (Finset.filter p (support f)) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nsimp at h2 \n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh1 : p i\nh2 : \u00ac\u2191f i = 0\n\u22a2 \u2191(filter p f) i = \u2191(mk (Finset.filter p (support f)) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nsimp [h1, h2]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh1 : p i\nh2 : \u2191f i = 0\n\u22a2 \u2191(filter p f) i = \u2191(mk (Finset.filter p (support f)) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nsimp [h1, h2]\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh1 : \u00acp i\nh2 : \u00ac\u2191f i = 0\n\u22a2 \u2191(filter p f) i = \u2191(mk (Finset.filter p (support f)) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nsimp [h1, h2]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh1 : \u00acp i\nh2 : \u2191f i = 0\n\u22a2 \u2191(filter p f) i = \u2191(mk (Finset.filter p (support f)) fun i => \u2191f \u2191i) i\n[PROOFSTEP]\nsimp [h1, h2]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 support (filter p f) = Finset.filter p (support f)\n[PROOFSTEP]\next i\n[GOAL]\ncase a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 i \u2208 support (filter p f) \u2194 i \u2208 Finset.filter p (support f)\n[PROOFSTEP]\nby_cases h : p i\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh : p i\n\u22a2 i \u2208 support (filter p f) \u2194 i \u2208 Finset.filter p (support f)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\nh : \u00acp i\n\u22a2 i \u2208 support (filter p f) \u2194 i \u2208 Finset.filter p (support f)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 subtypeDomain p f = mk (Finset.subtype p (support f)) fun i => \u2191f \u2191\u2191i\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : Subtype p\n\u22a2 \u2191(subtypeDomain p f) i = \u2191(mk (Finset.subtype p (support f)) fun i => \u2191f \u2191\u2191i) i\n[PROOFSTEP]\nby_cases h2 : f i \u2260 0\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : Subtype p\nh2 : \u2191f \u2191i \u2260 0\n\u22a2 \u2191(subtypeDomain p f) i = \u2191(mk (Finset.subtype p (support f)) fun i => \u2191f \u2191\u2191i) i\n[PROOFSTEP]\ntry simp at h2 ; dsimp; simp [h2]\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : Subtype p\nh2 : \u2191f \u2191i \u2260 0\n\u22a2 \u2191(subtypeDomain p f) i = \u2191(mk (Finset.subtype p (support f)) fun i => \u2191f \u2191\u2191i) i\n[PROOFSTEP]\nsimp at h2 \n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : Subtype p\nh2 : \u00ac\u2191f \u2191i = 0\n\u22a2 \u2191(subtypeDomain p f) i = \u2191(mk (Finset.subtype p (support f)) fun i => \u2191f \u2191\u2191i) i\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : Subtype p\nh2 : \u00ac\u2191f \u2191i = 0\n\u22a2 \u2191f \u2191i = if i \u2208 Finset.subtype p (support f) then \u2191f \u2191i else 0\n[PROOFSTEP]\nsimp [h2]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : Subtype p\nh2 : \u00ac\u2191f \u2191i \u2260 0\n\u22a2 \u2191(subtypeDomain p f) i = \u2191(mk (Finset.subtype p (support f)) fun i => \u2191f \u2191\u2191i) i\n[PROOFSTEP]\ntry simp at h2 ; dsimp; simp [h2]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : Subtype p\nh2 : \u00ac\u2191f \u2191i \u2260 0\n\u22a2 \u2191(subtypeDomain p f) i = \u2191(mk (Finset.subtype p (support f)) fun i => \u2191f \u2191\u2191i) i\n[PROOFSTEP]\nsimp at h2 \n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : Subtype p\nh2 : \u2191f \u2191i = 0\n\u22a2 \u2191(subtypeDomain p f) i = \u2191(mk (Finset.subtype p (support f)) fun i => \u2191f \u2191\u2191i) i\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : Subtype p\nh2 : \u2191f \u2191i = 0\n\u22a2 \u2191f \u2191i = if i \u2208 Finset.subtype p (support f) then \u2191f \u2191i else 0\n[PROOFSTEP]\nsimp [h2]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 support (subtypeDomain p f) = Finset.subtype p (support f)\n[PROOFSTEP]\next i\n[GOAL]\ncase a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : Subtype p\n\u22a2 i \u2208 support (subtypeDomain p f) \u2194 i \u2208 Finset.subtype p (support f)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 support (-f) = support f\n[PROOFSTEP]\next i\n[GOAL]\ncase a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 i \u2208 support (-f) \u2194 i \u2208 support f\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b2 i)\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\nx\u271d : support f = support g \u2227 \u2200 (i : \u03b9), i \u2208 support f \u2192 \u2191f i = \u2191g i\nh\u2081 : support f = support g\nh\u2082 : \u2200 (i : \u03b9), i \u2208 support f \u2192 \u2191f i = \u2191g i\ni : \u03b9\nh : \u00aci \u2208 support f\n\u22a2 \u2191f i = \u2191g i\n[PROOFSTEP]\nhave hf : f i = 0 := by rwa [mem_support_iff, not_not] at h \n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b2 i)\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\nx\u271d : support f = support g \u2227 \u2200 (i : \u03b9), i \u2208 support f \u2192 \u2191f i = \u2191g i\nh\u2081 : support f = support g\nh\u2082 : \u2200 (i : \u03b9), i \u2208 support f \u2192 \u2191f i = \u2191g i\ni : \u03b9\nh : \u00aci \u2208 support f\n\u22a2 \u2191f i = 0\n[PROOFSTEP]\nrwa [mem_support_iff, not_not] at h \n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b2 i)\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\nx\u271d : support f = support g \u2227 \u2200 (i : \u03b9), i \u2208 support f \u2192 \u2191f i = \u2191g i\nh\u2081 : support f = support g\nh\u2082 : \u2200 (i : \u03b9), i \u2208 support f \u2192 \u2191f i = \u2191g i\ni : \u03b9\nh : \u00aci \u2208 support f\nhf : \u2191f i = 0\n\u22a2 \u2191f i = \u2191g i\n[PROOFSTEP]\nhave hg : g i = 0 := by rwa [h\u2081, mem_support_iff, not_not] at h \n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b2 i)\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\nx\u271d : support f = support g \u2227 \u2200 (i : \u03b9), i \u2208 support f \u2192 \u2191f i = \u2191g i\nh\u2081 : support f = support g\nh\u2082 : \u2200 (i : \u03b9), i \u2208 support f \u2192 \u2191f i = \u2191g i\ni : \u03b9\nh : \u00aci \u2208 support f\nhf : \u2191f i = 0\n\u22a2 \u2191g i = 0\n[PROOFSTEP]\nrwa [h\u2081, mem_support_iff, not_not] at h \n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b2 i)\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\nx\u271d : support f = support g \u2227 \u2200 (i : \u03b9), i \u2208 support f \u2192 \u2191f i = \u2191g i\nh\u2081 : support f = support g\nh\u2082 : \u2200 (i : \u03b9), i \u2208 support f \u2192 \u2191f i = \u2191g i\ni : \u03b9\nh : \u00aci \u2208 support f\nhf : \u2191f i = 0\nhg : \u2191g i = 0\n\u22a2 \u2191f i = \u2191g i\n[PROOFSTEP]\nrw [hf, hg]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b2 i)\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 f = g \u2192 support f = support g \u2227 \u2200 (i : \u03b9), i \u2208 support f \u2192 \u2191f i = \u2191g i\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b2 i)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 support f = support f \u2227 \u2200 (i : \u03b9), i \u2208 support f \u2192 \u2191f i = \u2191f i\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nhh : Function.Injective h\n\u22a2 comapDomain h hh 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nhh : Function.Injective h\ni\u271d : \u03ba\n\u22a2 \u2191(comapDomain h hh 0) i\u271d = \u21910 i\u271d\n[PROOFSTEP]\nrw [zero_apply, comapDomain_apply, zero_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nhh : Function.Injective h\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 comapDomain h hh (f + g) = comapDomain h hh f + comapDomain h hh g\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nhh : Function.Injective h\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni\u271d : \u03ba\n\u22a2 \u2191(comapDomain h hh (f + g)) i\u271d = \u2191(comapDomain h hh f + comapDomain h hh g) i\u271d\n[PROOFSTEP]\nrw [add_apply, comapDomain_apply, comapDomain_apply, comapDomain_apply, add_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nhh : Function.Injective h\nr : \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 comapDomain h hh (r \u2022 f) = r \u2022 comapDomain h hh f\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nhh : Function.Injective h\nr : \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni\u271d : \u03ba\n\u22a2 \u2191(comapDomain h hh (r \u2022 f)) i\u271d = \u2191(r \u2022 comapDomain h hh f) i\u271d\n[PROOFSTEP]\nrw [smul_apply, comapDomain_apply, smul_apply, comapDomain_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nhh : Function.Injective h\nk : \u03ba\nx : \u03b2 (h k)\n\u22a2 comapDomain h hh (single (h k) x) = single k x\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nhh : Function.Injective h\nk : \u03ba\nx : \u03b2 (h k)\ni : \u03ba\n\u22a2 \u2191(comapDomain h hh (single (h k) x)) i = \u2191(single k x) i\n[PROOFSTEP]\nrw [comapDomain_apply]\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nhh : Function.Injective h\nk : \u03ba\nx : \u03b2 (h k)\ni : \u03ba\n\u22a2 \u2191(single (h k) x) (h i) = \u2191(single k x) i\n[PROOFSTEP]\nobtain rfl | hik := Decidable.eq_or_ne i k\n[GOAL]\ncase h.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nhh : Function.Injective h\ni : \u03ba\nx : \u03b2 (h i)\n\u22a2 \u2191(single (h i) x) (h i) = \u2191(single i x) i\n[PROOFSTEP]\nrw [single_eq_same, single_eq_same]\n[GOAL]\ncase h.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nhh : Function.Injective h\nk : \u03ba\nx : \u03b2 (h k)\ni : \u03ba\nhik : i \u2260 k\n\u22a2 \u2191(single (h k) x) (h i) = \u2191(single k x) i\n[PROOFSTEP]\nrw [single_eq_of_ne hik.symm, single_eq_of_ne (hh.ne hik.symm)]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\n\u22a2 comapDomain' h hh' 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\ni\u271d : \u03ba\n\u22a2 \u2191(comapDomain' h hh' 0) i\u271d = \u21910 i\u271d\n[PROOFSTEP]\nrw [zero_apply, comapDomain'_apply, zero_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 comapDomain' h hh' (f + g) = comapDomain' h hh' f + comapDomain' h hh' g\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni\u271d : \u03ba\n\u22a2 \u2191(comapDomain' h hh' (f + g)) i\u271d = \u2191(comapDomain' h hh' f + comapDomain' h hh' g) i\u271d\n[PROOFSTEP]\nrw [add_apply, comapDomain'_apply, comapDomain'_apply, comapDomain'_apply, add_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\nr : \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 comapDomain' h hh' (r \u2022 f) = r \u2022 comapDomain' h hh' f\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\nr : \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni\u271d : \u03ba\n\u22a2 \u2191(comapDomain' h hh' (r \u2022 f)) i\u271d = \u2191(r \u2022 comapDomain' h hh' f) i\u271d\n[PROOFSTEP]\nrw [smul_apply, comapDomain'_apply, smul_apply, comapDomain'_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\nk : \u03ba\nx : \u03b2 (h k)\n\u22a2 comapDomain' h hh' (single (h k) x) = single k x\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\nk : \u03ba\nx : \u03b2 (h k)\ni : \u03ba\n\u22a2 \u2191(comapDomain' h hh' (single (h k) x)) i = \u2191(single k x) i\n[PROOFSTEP]\nrw [comapDomain'_apply]\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\nk : \u03ba\nx : \u03b2 (h k)\ni : \u03ba\n\u22a2 \u2191(single (h k) x) (h i) = \u2191(single k x) i\n[PROOFSTEP]\nobtain rfl | hik := Decidable.eq_or_ne i k\n[GOAL]\ncase h.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\ni : \u03ba\nx : \u03b2 (h i)\n\u22a2 \u2191(single (h i) x) (h i) = \u2191(single i x) i\n[PROOFSTEP]\nrw [single_eq_same, single_eq_same]\n[GOAL]\ncase h.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\nk : \u03ba\nx : \u03b2 (h k)\ni : \u03ba\nhik : i \u2260 k\n\u22a2 \u2191(single (h k) x) (h i) = \u2191(single k x) i\n[PROOFSTEP]\nrw [single_eq_of_ne hik.symm, single_eq_of_ne (hh'.injective.ne hik.symm)]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03b9 \u2243 \u03ba\nf : \u03a0\u2080 (k : \u03ba), \u03b2 (\u2191h.symm k)\ni : \u03b9\n\u22a2 HEq 0 0\n[PROOFSTEP]\nrw [Equiv.symm_apply_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03b9 \u2243 \u03ba\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 (fun f =>\n mapRange (fun i => \u2191(Equiv.cast (_ : \u03b2 (\u2191h.symm (\u2191h i)) = \u03b2 i)))\n (_ : \u2200 (i : \u03b9), \u2191(Equiv.cast (_ : \u03b2 (\u2191h.symm (\u2191h i)) = \u03b2 i)) 0 = 0)\n (comapDomain' \u2191h (_ : Function.LeftInverse h.invFun h.toFun) f))\n (comapDomain' \u2191h.symm (_ : Function.RightInverse h.invFun h.toFun) f) =\n f\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03b9 \u2243 \u03ba\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\n\u22a2 \u2191((fun f =>\n mapRange (fun i => \u2191(Equiv.cast (_ : \u03b2 (\u2191h.symm (\u2191h i)) = \u03b2 i)))\n (_ : \u2200 (i : \u03b9), \u2191(Equiv.cast (_ : \u03b2 (\u2191h.symm (\u2191h i)) = \u03b2 i)) 0 = 0)\n (comapDomain' \u2191h (_ : Function.LeftInverse h.invFun h.toFun) f))\n (comapDomain' \u2191h.symm (_ : Function.RightInverse h.invFun h.toFun) f))\n i =\n \u2191f i\n[PROOFSTEP]\nrw [mapRange_apply, comapDomain'_apply, comapDomain'_apply, Equiv.cast_eq_iff_heq, h.symm_apply_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03b9 \u2243 \u03ba\nf : \u03a0\u2080 (k : \u03ba), \u03b2 (\u2191h.symm k)\n\u22a2 comapDomain' \u2191h.symm (_ : Function.RightInverse h.invFun h.toFun)\n ((fun f =>\n mapRange (fun i => \u2191(Equiv.cast (_ : \u03b2 (\u2191h.symm (\u2191h i)) = \u03b2 i)))\n (_ : \u2200 (i : \u03b9), \u2191(Equiv.cast (_ : \u03b2 (\u2191h.symm (\u2191h i)) = \u03b2 i)) 0 = 0)\n (comapDomain' \u2191h (_ : Function.LeftInverse h.invFun h.toFun) f))\n f) =\n f\n[PROOFSTEP]\next k\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03b9 \u2243 \u03ba\nf : \u03a0\u2080 (k : \u03ba), \u03b2 (\u2191h.symm k)\nk : \u03ba\n\u22a2 \u2191(comapDomain' \u2191h.symm (_ : Function.RightInverse h.invFun h.toFun)\n ((fun f =>\n mapRange (fun i => \u2191(Equiv.cast (_ : \u03b2 (\u2191h.symm (\u2191h i)) = \u03b2 i)))\n (_ : \u2200 (i : \u03b9), \u2191(Equiv.cast (_ : \u03b2 (\u2191h.symm (\u2191h i)) = \u03b2 i)) 0 = 0)\n (comapDomain' \u2191h (_ : Function.LeftInverse h.invFun h.toFun) f))\n f))\n k =\n \u2191f k\n[PROOFSTEP]\nrw [comapDomain'_apply, mapRange_apply, comapDomain'_apply, Equiv.cast_eq_iff_heq, h.apply_symm_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 AddZeroClass (\u03b4 i j)\nf g : \u03a0\u2080 (i : (x : \u03b9) \u00d7 \u03b1 x), \u03b4 i.fst i.snd\n\u22a2 sigmaCurry (f + g) = sigmaCurry f + sigmaCurry g\n[PROOFSTEP]\next (i j)\n[GOAL]\ncase h.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 AddZeroClass (\u03b4 i j)\nf g : \u03a0\u2080 (i : (x : \u03b9) \u00d7 \u03b1 x), \u03b4 i.fst i.snd\ni : \u03b9\nj : \u03b1 i\n\u22a2 \u2191(\u2191(sigmaCurry (f + g)) i) j = \u2191(\u2191(sigmaCurry f + sigmaCurry g) i) j\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 AddMonoid (\u03b4 i j)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 DistribMulAction \u03b3 (\u03b4 i j)\nr : \u03b3\nf : \u03a0\u2080 (i : (x : \u03b9) \u00d7 \u03b1 x), \u03b4 i.fst i.snd\n\u22a2 sigmaCurry (r \u2022 f) = r \u2022 sigmaCurry f\n[PROOFSTEP]\next (i j)\n[GOAL]\ncase h.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 AddMonoid (\u03b4 i j)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 DistribMulAction \u03b3 (\u03b4 i j)\nr : \u03b3\nf : \u03a0\u2080 (i : (x : \u03b9) \u00d7 \u03b1 x), \u03b4 i.fst i.snd\ni : \u03b9\nj : \u03b1 i\n\u22a2 \u2191(\u2191(sigmaCurry (r \u2022 f)) i) j = \u2191(\u2191(r \u2022 sigmaCurry f) i) j\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\nij : (i : \u03b9) \u00d7 \u03b1 i\nx : \u03b4 ij.fst ij.snd\n\u22a2 sigmaCurry (single ij x) = single ij.fst (single ij.snd x)\n[PROOFSTEP]\nobtain \u27e8i, j\u27e9 := ij\n[GOAL]\ncase mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 { fst := i, snd := j }.fst { fst := i, snd := j }.snd\n\u22a2 sigmaCurry (single { fst := i, snd := j } x) = single { fst := i, snd := j }.fst (single { fst := i, snd := j }.snd x)\n[PROOFSTEP]\next i' j'\n[GOAL]\ncase mk.h.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 { fst := i, snd := j }.fst { fst := i, snd := j }.snd\ni' : \u03b9\nj' : \u03b1 i'\n\u22a2 \u2191(\u2191(sigmaCurry (single { fst := i, snd := j } x)) i') j' =\n \u2191(\u2191(single { fst := i, snd := j }.fst (single { fst := i, snd := j }.snd x)) i') j'\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase mk.h.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 { fst := i, snd := j }.fst { fst := i, snd := j }.snd\ni' : \u03b9\nj' : \u03b1 i'\n\u22a2 \u2191(\u2191(sigmaCurry (single { fst := i, snd := j } x)) i') j' = \u2191(\u2191(single i (single j x)) i') j'\n[PROOFSTEP]\nrw [sigmaCurry_apply]\n[GOAL]\ncase mk.h.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 { fst := i, snd := j }.fst { fst := i, snd := j }.snd\ni' : \u03b9\nj' : \u03b1 i'\n\u22a2 \u2191(single { fst := i, snd := j } x) { fst := i', snd := j' } = \u2191(\u2191(single i (single j x)) i') j'\n[PROOFSTEP]\nobtain rfl | hi := eq_or_ne i i'\n[GOAL]\ncase mk.h.h.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 { fst := i, snd := j }.fst { fst := i, snd := j }.snd\nj' : \u03b1 i\n\u22a2 \u2191(single { fst := i, snd := j } x) { fst := i, snd := j' } = \u2191(\u2191(single i (single j x)) i) j'\n[PROOFSTEP]\nrw [single_eq_same]\n[GOAL]\ncase mk.h.h.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 { fst := i, snd := j }.fst { fst := i, snd := j }.snd\nj' : \u03b1 i\n\u22a2 \u2191(single { fst := i, snd := j } x) { fst := i, snd := j' } = \u2191(single j x) j'\n[PROOFSTEP]\nobtain rfl | hj := eq_or_ne j j'\n[GOAL]\ncase mk.h.h.inl.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 { fst := i, snd := j }.fst { fst := i, snd := j }.snd\n\u22a2 \u2191(single { fst := i, snd := j } x) { fst := i, snd := j } = \u2191(single j x) j\n[PROOFSTEP]\nrw [single_eq_same, single_eq_same]\n[GOAL]\ncase mk.h.h.inl.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 { fst := i, snd := j }.fst { fst := i, snd := j }.snd\nj' : \u03b1 i\nhj : j \u2260 j'\n\u22a2 \u2191(single { fst := i, snd := j } x) { fst := i, snd := j' } = \u2191(single j x) j'\n[PROOFSTEP]\nrw [single_eq_of_ne, single_eq_of_ne hj]\n[GOAL]\ncase mk.h.h.inl.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 { fst := i, snd := j }.fst { fst := i, snd := j }.snd\nj' : \u03b1 i\nhj : j \u2260 j'\n\u22a2 { fst := i, snd := j } \u2260 { fst := i, snd := j' }\n[PROOFSTEP]\nsimpa using hj\n[GOAL]\ncase mk.h.h.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 { fst := i, snd := j }.fst { fst := i, snd := j }.snd\ni' : \u03b9\nj' : \u03b1 i'\nhi : i \u2260 i'\n\u22a2 \u2191(single { fst := i, snd := j } x) { fst := i', snd := j' } = \u2191(\u2191(single i (single j x)) i') j'\n[PROOFSTEP]\nrw [single_eq_of_ne, single_eq_of_ne hi, zero_apply]\n[GOAL]\ncase mk.h.h.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 { fst := i, snd := j }.fst { fst := i, snd := j }.snd\ni' : \u03b9\nj' : \u03b1 i'\nhi : i \u2260 i'\n\u22a2 { fst := i, snd := j } \u2260 { fst := i', snd := j' }\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9) (j : \u03b1 i), \u03b4 i j\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }\ni : (i : \u03b9) \u00d7 \u03b1 i\n\u22a2 (i \u2208\n Multiset.bind \u2191s fun i =>\n (map { toFun := Sigma.mk i, inj' := (_ : Function.Injective (Sigma.mk i)) } (support (\u2191f i))).val) \u2228\n (fun i => \u2191(\u2191f i.fst) i.snd) i = 0\n[PROOFSTEP]\nsimp_rw [Multiset.mem_bind, map_val, Multiset.mem_map, Function.Embedding.coeFn_mk, \u2190 Finset.mem_def, mem_support_toFun]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9) (j : \u03b1 i), \u03b4 i j\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }\ni : (i : \u03b9) \u00d7 \u03b1 i\n\u22a2 (\u2203 a, a \u2208 \u2191s \u2227 \u2203 a_1, \u2191(\u2191f a) a_1 \u2260 0 \u2227 { fst := a, snd := a_1 } = i) \u2228 \u2191(\u2191f i.fst) i.snd = 0\n[PROOFSTEP]\nobtain hi | (hi : f i.1 = 0) := s.prop i.1\n[GOAL]\ncase inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9) (j : \u03b1 i), \u03b4 i j\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }\ni : (i : \u03b9) \u00d7 \u03b1 i\nhi : i.fst \u2208 \u2191s\n\u22a2 (\u2203 a, a \u2208 \u2191s \u2227 \u2203 a_1, \u2191(\u2191f a) a_1 \u2260 0 \u2227 { fst := a, snd := a_1 } = i) \u2228 \u2191(\u2191f i.fst) i.snd = 0\n[PROOFSTEP]\nby_cases hi' : f i.1 i.2 = 0\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9) (j : \u03b1 i), \u03b4 i j\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }\ni : (i : \u03b9) \u00d7 \u03b1 i\nhi : i.fst \u2208 \u2191s\nhi' : \u2191(\u2191f i.fst) i.snd = 0\n\u22a2 (\u2203 a, a \u2208 \u2191s \u2227 \u2203 a_1, \u2191(\u2191f a) a_1 \u2260 0 \u2227 { fst := a, snd := a_1 } = i) \u2228 \u2191(\u2191f i.fst) i.snd = 0\n[PROOFSTEP]\nexact Or.inr hi'\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9) (j : \u03b1 i), \u03b4 i j\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }\ni : (i : \u03b9) \u00d7 \u03b1 i\nhi : i.fst \u2208 \u2191s\nhi' : \u00ac\u2191(\u2191f i.fst) i.snd = 0\n\u22a2 (\u2203 a, a \u2208 \u2191s \u2227 \u2203 a_1, \u2191(\u2191f a) a_1 \u2260 0 \u2227 { fst := a, snd := a_1 } = i) \u2228 \u2191(\u2191f i.fst) i.snd = 0\n[PROOFSTEP]\nexact Or.inl \u27e8_, hi, i.2, hi', Sigma.eta _\u27e9\n[GOAL]\ncase inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9) (j : \u03b1 i), \u03b4 i j\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }\ni : (i : \u03b9) \u00d7 \u03b1 i\nhi : \u2191f i.fst = 0\n\u22a2 (\u2203 a, a \u2208 \u2191s \u2227 \u2203 a_1, \u2191(\u2191f a) a_1 \u2260 0 \u2227 { fst := a, snd := a_1 } = i) \u2228 \u2191(\u2191f i.fst) i.snd = 0\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9) (j : \u03b1 i), \u03b4 i j\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }\ni : (i : \u03b9) \u00d7 \u03b1 i\nhi : \u2191f i.fst = 0\n\u22a2 \u2191(\u2191f i.fst) i.snd = 0\n[PROOFSTEP]\nrw [hi, zero_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 i j\n\u22a2 \u03b4 { fst := i, snd := j }.fst { fst := i, snd := j }.snd\n[PROOFSTEP]\nexact x\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 i j\n\u22a2 sigmaUncurry (single i (single j x)) = single { fst := i, snd := j } x\n[PROOFSTEP]\next \u27e8i', j'\u27e9\n[GOAL]\ncase h.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 i j\ni' : \u03b9\nj' : \u03b1 i'\n\u22a2 \u2191(sigmaUncurry (single i (single j x))) { fst := i', snd := j' } =\n \u2191(single { fst := i, snd := j } x) { fst := i', snd := j' }\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 i j\ni' : \u03b9\nj' : \u03b1 i'\n\u22a2 \u2191(sigmaUncurry (single i (single j x))) { fst := i', snd := j' } =\n \u2191(single { fst := i, snd := j } x) { fst := i', snd := j' }\n[PROOFSTEP]\nrw [sigmaUncurry_apply]\n[GOAL]\ncase h.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 i j\ni' : \u03b9\nj' : \u03b1 i'\n\u22a2 \u2191(\u2191(single i (single j x)) i') j' = \u2191(single { fst := i, snd := j } x) { fst := i', snd := j' }\n[PROOFSTEP]\nobtain rfl | hi := eq_or_ne i i'\n[GOAL]\ncase h.mk.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 i j\nj' : \u03b1 i\n\u22a2 \u2191(\u2191(single i (single j x)) i) j' = \u2191(single { fst := i, snd := j } x) { fst := i, snd := j' }\n[PROOFSTEP]\nrw [single_eq_same]\n[GOAL]\ncase h.mk.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 i j\nj' : \u03b1 i\n\u22a2 \u2191(single j x) j' = \u2191(single { fst := i, snd := j } x) { fst := i, snd := j' }\n[PROOFSTEP]\nobtain rfl | hj := eq_or_ne j j'\n[GOAL]\ncase h.mk.inl.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 i j\n\u22a2 \u2191(single j x) j = \u2191(single { fst := i, snd := j } x) { fst := i, snd := j }\n[PROOFSTEP]\nrw [single_eq_same, single_eq_same]\n[GOAL]\ncase h.mk.inl.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 i j\nj' : \u03b1 i\nhj : j \u2260 j'\n\u22a2 \u2191(single j x) j' = \u2191(single { fst := i, snd := j } x) { fst := i, snd := j' }\n[PROOFSTEP]\nrw [single_eq_of_ne hj, single_eq_of_ne]\n[GOAL]\ncase h.mk.inl.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 i j\nj' : \u03b1 i\nhj : j \u2260 j'\n\u22a2 { fst := i, snd := j } \u2260 { fst := i, snd := j' }\n[PROOFSTEP]\nsimpa using hj\n[GOAL]\ncase h.mk.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 i j\ni' : \u03b9\nj' : \u03b1 i'\nhi : i \u2260 i'\n\u22a2 \u2191(\u2191(single i (single j x)) i') j' = \u2191(single { fst := i, snd := j } x) { fst := i', snd := j' }\n[PROOFSTEP]\nrw [single_eq_of_ne hi, single_eq_of_ne, zero_apply]\n[GOAL]\ncase h.mk.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\ni : \u03b9\nj : \u03b1 i\nx : \u03b4 i j\ni' : \u03b9\nj' : \u03b1 i'\nhi : i \u2260 i'\n\u22a2 { fst := i, snd := j } \u2260 { fst := i', snd := j' }\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : (i : \u03b9) \u00d7 \u03b1 i), \u03b4 i.fst i.snd\n\u22a2 sigmaUncurry (sigmaCurry f) = f\n[PROOFSTEP]\next \u27e8i, j\u27e9\n[GOAL]\ncase h.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : (i : \u03b9) \u00d7 \u03b1 i), \u03b4 i.fst i.snd\ni : \u03b9\nj : \u03b1 i\n\u22a2 \u2191(sigmaUncurry (sigmaCurry f)) { fst := i, snd := j } = \u2191f { fst := i, snd := j }\n[PROOFSTEP]\nrw [sigmaUncurry_apply, sigmaCurry_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9) (j : \u03b1 i), \u03b4 i j\n\u22a2 sigmaCurry (sigmaUncurry f) = f\n[PROOFSTEP]\next i j\n[GOAL]\ncase h.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b4 : (i : \u03b9) \u2192 \u03b1 i \u2192 Type v\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 Zero (\u03b4 i j)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (j : \u03b1 i) \u2192 (x : \u03b4 i j) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9) (j : \u03b1 i), \u03b4 i j\ni : \u03b9\nj : \u03b1 i\n\u22a2 \u2191(\u2191(sigmaCurry (sigmaUncurry f)) i) j = \u2191(\u2191f i) j\n[PROOFSTEP]\nrw [sigmaCurry_apply, sigmaUncurry_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\ni : \u03b9\nx : \u03b1 (some i)\n\u22a2 extendWith 0 (single i x) = single (some i) x\n[PROOFSTEP]\next (_ | j)\n[GOAL]\ncase h.none\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\ni : \u03b9\nx : \u03b1 (some i)\n\u22a2 \u2191(extendWith 0 (single i x)) none = \u2191(single (some i) x) none\n[PROOFSTEP]\nrw [extendWith_none, single_eq_of_ne (Option.some_ne_none _)]\n[GOAL]\ncase h.some\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\ni : \u03b9\nx : \u03b1 (some i)\nj : \u03b9\n\u22a2 \u2191(extendWith 0 (single i x)) (some j) = \u2191(single (some i) x) (some j)\n[PROOFSTEP]\nrw [extendWith_some]\n[GOAL]\ncase h.some\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\ni : \u03b9\nx : \u03b1 (some i)\nj : \u03b9\n\u22a2 \u2191(single i x) j = \u2191(single (some i) x) (some j)\n[PROOFSTEP]\nobtain rfl | hij := Decidable.eq_or_ne i j\n[GOAL]\ncase h.some.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\ni : \u03b9\nx : \u03b1 (some i)\n\u22a2 \u2191(single i x) i = \u2191(single (some i) x) (some i)\n[PROOFSTEP]\nrw [single_eq_same, single_eq_same]\n[GOAL]\ncase h.some.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\ni : \u03b9\nx : \u03b1 (some i)\nj : \u03b9\nhij : i \u2260 j\n\u22a2 \u2191(single i x) j = \u2191(single (some i) x) (some j)\n[PROOFSTEP]\nrw [single_eq_of_ne hij, single_eq_of_ne ((Option.some_injective _).ne hij)]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\nx : \u03b1 none\n\u22a2 extendWith x 0 = single none x\n[PROOFSTEP]\next (_ | j)\n[GOAL]\ncase h.none\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\nx : \u03b1 none\n\u22a2 \u2191(extendWith x 0) none = \u2191(single none x) none\n[PROOFSTEP]\nrw [extendWith_none, single_eq_same]\n[GOAL]\ncase h.some\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\nx : \u03b1 none\nj : \u03b9\n\u22a2 \u2191(extendWith x 0) (some j) = \u2191(single none x) (some j)\n[PROOFSTEP]\nrw [extendWith_some, single_eq_of_ne (Option.some_ne_none _).symm, zero_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\nf : \u03a0\u2080 (i : Option \u03b9), \u03b1 i\n\u22a2 (fun f => extendWith f.fst f.snd) ((fun f => (\u2191f none, comapDomain some (_ : Function.Injective some) f)) f) = f\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\nf : \u03a0\u2080 (i : Option \u03b9), \u03b1 i\ni : Option \u03b9\n\u22a2 \u2191((fun f => extendWith f.fst f.snd) ((fun f => (\u2191f none, comapDomain some (_ : Function.Injective some) f)) f)) i =\n \u2191f i\n[PROOFSTEP]\ncases' i with i\n[GOAL]\ncase h.none\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\nf : \u03a0\u2080 (i : Option \u03b9), \u03b1 i\n\u22a2 \u2191((fun f => extendWith f.fst f.snd) ((fun f => (\u2191f none, comapDomain some (_ : Function.Injective some) f)) f)) none =\n \u2191f none\n[PROOFSTEP]\nrw [extendWith_none]\n[GOAL]\ncase h.some\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\nf : \u03a0\u2080 (i : Option \u03b9), \u03b1 i\ni : \u03b9\n\u22a2 \u2191((fun f => extendWith f.fst f.snd) ((fun f => (\u2191f none, comapDomain some (_ : Function.Injective some) f)) f))\n (some i) =\n \u2191f (some i)\n[PROOFSTEP]\nrw [extendWith_some, comapDomain_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\nx : \u03b1 none \u00d7 \u03a0\u2080 (i : \u03b9), \u03b1 (some i)\n\u22a2 (fun f => (\u2191f none, comapDomain some (_ : Function.Injective some) f)) ((fun f => extendWith f.fst f.snd) x) = x\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\nx : \u03b1 none \u00d7 \u03a0\u2080 (i : \u03b9), \u03b1 (some i)\n\u22a2 (\u2191(extendWith x.fst x.snd) none, comapDomain some (_ : Function.Injective some) (extendWith x.fst x.snd)) = x\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\nx : \u03b1 none \u00d7 \u03a0\u2080 (i : \u03b9), \u03b1 (some i)\n\u22a2 (\u2191(extendWith x.fst x.snd) none, comapDomain some (_ : Function.Injective some) (extendWith x.fst x.snd)).fst = x.fst\n[PROOFSTEP]\nexact extendWith_none x.snd _\n[GOAL]\ncase h\u2082.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\n\u03b1 : Option \u03b9 \u2192 Type v\ninst\u271d : (i : Option \u03b9) \u2192 Zero (\u03b1 i)\nx : \u03b1 none \u00d7 \u03a0\u2080 (i : \u03b9), \u03b1 (some i)\ni\u271d : \u03b9\n\u22a2 \u2191(\u2191(extendWith x.fst x.snd) none, comapDomain some (_ : Function.Injective some) (extendWith x.fst x.snd)).snd i\u271d =\n \u2191x.snd i\u271d\n[PROOFSTEP]\nrw [comapDomain_apply, extendWith_some]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081\u271d : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\nh : (i : \u03b9) \u2192 \u03b2\u2082 i \u2192 \u03b3\nh0 : \u2200 (i : \u03b9), h i 0 = 1\n\u22a2 prod (mapRange f hf g) h = prod g fun i b => h i (f i b)\n[PROOFSTEP]\nrw [mapRange_def]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081\u271d : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\nh : (i : \u03b9) \u2192 \u03b2\u2082 i \u2192 \u03b3\nh0 : \u2200 (i : \u03b9), h i 0 = 1\n\u22a2 prod (mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)) h = prod g fun i b => h i (f i b)\n[PROOFSTEP]\nrefine' (Finset.prod_subset support_mk_subset _).trans _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081\u271d : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\nh : (i : \u03b9) \u2192 \u03b2\u2082 i \u2192 \u03b3\nh0 : \u2200 (i : \u03b9), h i 0 = 1\n\u22a2 \u2200 (x : \u03b9),\n x \u2208 support g \u2192\n \u00acx \u2208 support (mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)) \u2192 h x (\u2191(mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)) x) = 1\n[PROOFSTEP]\nintro i h1 h2\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081\u271d : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\nh : (i : \u03b9) \u2192 \u03b2\u2082 i \u2192 \u03b3\nh0 : \u2200 (i : \u03b9), h i 0 = 1\ni : \u03b9\nh1 : i \u2208 support g\nh2 : \u00aci \u2208 support (mk (support g) fun i => f (\u2191i) (\u2191g \u2191i))\n\u22a2 h i (\u2191(mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)) i) = 1\n[PROOFSTEP]\nsimp only [mem_support_toFun, ne_eq] at h1 \n[GOAL]\ncase refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081\u271d : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\nh : (i : \u03b9) \u2192 \u03b2\u2082 i \u2192 \u03b3\nh0 : \u2200 (i : \u03b9), h i 0 = 1\ni : \u03b9\nh2 : \u00aci \u2208 support (mk (support g) fun i => f (\u2191i) (\u2191g \u2191i))\nh1 : \u00ac\u2191g i = 0\n\u22a2 h i (\u2191(mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)) i) = 1\n[PROOFSTEP]\nsimp only [Finset.coe_sort_coe, mem_support_toFun, mk_apply, ne_eq, h1, not_false_iff, dite_eq_ite, ite_true,\n not_not] at h2 \n[GOAL]\ncase refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081\u271d : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\nh : (i : \u03b9) \u2192 \u03b2\u2082 i \u2192 \u03b3\nh0 : \u2200 (i : \u03b9), h i 0 = 1\ni : \u03b9\nh1 : \u00ac\u2191g i = 0\nh2 : f i (\u2191g i) = 0\n\u22a2 h i (\u2191(mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)) i) = 1\n[PROOFSTEP]\nsimp [h2, h0]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081\u271d : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\nh : (i : \u03b9) \u2192 \u03b2\u2082 i \u2192 \u03b3\nh0 : \u2200 (i : \u03b9), h i 0 = 1\n\u22a2 \u220f x in support g, h x (\u2191(mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)) x) = prod g fun i b => h i (f i b)\n[PROOFSTEP]\nrefine' Finset.prod_congr rfl _\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081\u271d : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\nh : (i : \u03b9) \u2192 \u03b2\u2082 i \u2192 \u03b3\nh0 : \u2200 (i : \u03b9), h i 0 = 1\n\u22a2 \u2200 (x : \u03b9), x \u2208 support g \u2192 h x (\u2191(mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)) x) = (fun i b => h i (f i b)) x (\u2191g x)\n[PROOFSTEP]\nintro i h1\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081\u271d : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\nh : (i : \u03b9) \u2192 \u03b2\u2082 i \u2192 \u03b3\nh0 : \u2200 (i : \u03b9), h i 0 = 1\ni : \u03b9\nh1 : i \u2208 support g\n\u22a2 h i (\u2191(mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)) i) = (fun i b => h i (f i b)) i (\u2191g i)\n[PROOFSTEP]\nsimp only [mem_support_toFun, ne_eq] at h1 \n[GOAL]\ncase refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081\u271d : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ninst\u271d\u2074 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\ng : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\nh : (i : \u03b9) \u2192 \u03b2\u2082 i \u2192 \u03b3\nh0 : \u2200 (i : \u03b9), h i 0 = 1\ni : \u03b9\nh1 : \u00ac\u2191g i = 0\n\u22a2 h i (\u2191(mk (support g) fun i => f (\u2191i) (\u2191g \u2191i)) i) = (fun i b => h i (f i b)) i (\u2191g i)\n[PROOFSTEP]\nsimp [h1]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\ni : \u03b9\nb : \u03b2 i\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_zero : h i 0 = 1\n\u22a2 prod (single i b) h = h i b\n[PROOFSTEP]\nby_cases h : b \u2260 0\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\ni : \u03b9\nb : \u03b2 i\nh\u271d : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_zero : h\u271d i 0 = 1\nh : b \u2260 0\n\u22a2 prod (single i b) h\u271d = h\u271d i b\n[PROOFSTEP]\nsimp [DFinsupp.prod, support_single_ne_zero h]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\ni : \u03b9\nb : \u03b2 i\nh\u271d : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_zero : h\u271d i 0 = 1\nh : \u00acb \u2260 0\n\u22a2 prod (single i b) h\u271d = h\u271d i b\n[PROOFSTEP]\nrw [not_not] at h \n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\ni : \u03b9\nb : \u03b2 i\nh\u271d : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_zero : h\u271d i 0 = 1\nh : b = 0\n\u22a2 prod (single i b) h\u271d = h\u271d i b\n[PROOFSTEP]\nsimp [h, prod_zero_index, h_zero]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\ni : \u03b9\nb : \u03b2 i\nh\u271d : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_zero : h\u271d i 0 = 1\nh : b = 0\n\u22a2 prod 0 h\u271d = 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081\u271d : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b9\u2081 : Type u\u2081\ninst\u271d\u2074 : DecidableEq \u03b9\u2081\n\u03b2\u2081 : \u03b9\u2081 \u2192 Type v\u2081\ninst\u271d\u00b3 : (i\u2081 : \u03b9\u2081) \u2192 Zero (\u03b2\u2081 i\u2081)\ninst\u271d\u00b2 : (i : \u03b9\u2081) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i\u2081 : \u03b9\u2081), \u03b2\u2081 i\u2081\ng : (i\u2081 : \u03b9\u2081) \u2192 \u03b2\u2081 i\u2081 \u2192 \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 support (sum f g) \u2286 Finset.biUnion (support f) fun i => support (g i (\u2191f i))\n[PROOFSTEP]\nhave : \u2200 i\u2081 : \u03b9, (f.sum fun (i : \u03b9\u2081) (b : \u03b2\u2081 i) => (g i b) i\u2081) \u2260 0 \u2192 \u2203 i : \u03b9\u2081, f i \u2260 0 \u2227 \u00ac(g i (f i)) i\u2081 = 0 :=\n fun i\u2081 h =>\n let \u27e8i, hi, Ne\u27e9 := Finset.exists_ne_zero_of_sum_ne_zero h\n \u27e8i, mem_support_iff.1 hi, Ne\u27e9\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081\u271d : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b9\u2081 : Type u\u2081\ninst\u271d\u2074 : DecidableEq \u03b9\u2081\n\u03b2\u2081 : \u03b9\u2081 \u2192 Type v\u2081\ninst\u271d\u00b3 : (i\u2081 : \u03b9\u2081) \u2192 Zero (\u03b2\u2081 i\u2081)\ninst\u271d\u00b2 : (i : \u03b9\u2081) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i\u2081 : \u03b9\u2081), \u03b2\u2081 i\u2081\ng : (i\u2081 : \u03b9\u2081) \u2192 \u03b2\u2081 i\u2081 \u2192 \u03a0\u2080 (i : \u03b9), \u03b2 i\nthis : \u2200 (i\u2081 : \u03b9), (sum f fun i b => \u2191(g i b) i\u2081) \u2260 0 \u2192 \u2203 i, \u2191f i \u2260 0 \u2227 \u00ac\u2191(g i (\u2191f i)) i\u2081 = 0\n\u22a2 support (sum f g) \u2286 Finset.biUnion (support f) fun i => support (g i (\u2191f i))\n[PROOFSTEP]\nsimpa [Finset.subset_iff, mem_support_iff, Finset.mem_biUnion, sum_apply] using this\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_zero : \u2200 (i : \u03b9), h i 0 = 1\nh_add : \u2200 (i : \u03b9) (b\u2081 b\u2082 : \u03b2 i), h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082\n\u22a2 \u2200 (x : \u03b9), x \u2208 support f \u222a support g \u2192 \u00acx \u2208 support f \u2192 h x (\u2191f x) = 1\n[PROOFSTEP]\nsimp (config := { contextual := true }) [mem_support_iff, h_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_zero : \u2200 (i : \u03b9), h i 0 = 1\nh_add : \u2200 (i : \u03b9) (b\u2081 b\u2082 : \u03b2 i), h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082\nf_eq : \u220f i in support f \u222a support g, h i (\u2191f i) = prod f h\n\u22a2 \u2200 (x : \u03b9), x \u2208 support f \u222a support g \u2192 \u00acx \u2208 support g \u2192 h x (\u2191g x) = 1\n[PROOFSTEP]\nsimp (config := { contextual := true }) [mem_support_iff, h_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_zero : \u2200 (i : \u03b9), h i 0 = 1\nh_add : \u2200 (i : \u03b9) (b\u2081 b\u2082 : \u03b2 i), h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082\nf_eq : \u220f i in support f \u222a support g, h i (\u2191f i) = prod f h\ng_eq : \u220f i in support f \u222a support g, h i (\u2191g i) = prod g h\n\u22a2 \u2200 (x : \u03b9), x \u2208 support f \u222a support g \u2192 \u00acx \u2208 support (f + g) \u2192 h x (\u2191(f + g) x) = 1\n[PROOFSTEP]\nsimp (config := { contextual := true }) [mem_support_iff, h_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_zero : \u2200 (i : \u03b9), h i 0 = 1\nh_add : \u2200 (i : \u03b9) (b\u2081 b\u2082 : \u03b2 i), h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082\nf_eq : \u220f i in support f \u222a support g, h i (\u2191f i) = prod f h\ng_eq : \u220f i in support f \u222a support g, h i (\u2191g i) = prod g h\n\u22a2 \u220f i in support f \u222a support g, h i (\u2191(f + g) i) =\n (\u220f i in support f \u222a support g, h i (\u2191f i)) * \u220f i in support f \u222a support g, h i (\u2191g i)\n[PROOFSTEP]\n{simp [h_add, Finset.prod_mul_distrib]\n}\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_zero : \u2200 (i : \u03b9), h i 0 = 1\nh_add : \u2200 (i : \u03b9) (b\u2081 b\u2082 : \u03b2 i), h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082\nf_eq : \u220f i in support f \u222a support g, h i (\u2191f i) = prod f h\ng_eq : \u220f i in support f \u222a support g, h i (\u2191g i) = prod g h\n\u22a2 \u220f i in support f \u222a support g, h i (\u2191(f + g) i) =\n (\u220f i in support f \u222a support g, h i (\u2191f i)) * \u220f i in support f \u222a support g, h i (\u2191g i)\n[PROOFSTEP]\nsimp [h_add, Finset.prod_mul_distrib]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_zero : \u2200 (i : \u03b9), h i 0 = 1\nh_add : \u2200 (i : \u03b9) (b\u2081 b\u2082 : \u03b2 i), h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082\nf_eq : \u220f i in support f \u222a support g, h i (\u2191f i) = prod f h\ng_eq : \u220f i in support f \u222a support g, h i (\u2191g i) = prod g h\n\u22a2 (\u220f i in support f \u222a support g, h i (\u2191f i)) * \u220f i in support f \u222a support g, h i (\u2191g i) = prod f h * prod g h\n[PROOFSTEP]\nrw [f_eq, g_eq]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), \u03b2 i\nf : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nhf : \u2200 (i : \u03b9), f i 0 = 1\n\u22a2 prod v f = \u220f i : \u03b9, f i (\u2191equivFunOnFintype v i)\n[PROOFSTEP]\nsuffices (\u220f i in v.support, f i (v i)) = \u220f i, f i (v i) by simp [DFinsupp.prod, this]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), \u03b2 i\nf : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nhf : \u2200 (i : \u03b9), f i 0 = 1\nthis : \u220f i in support v, f i (\u2191v i) = \u220f i : \u03b9, f i (\u2191v i)\n\u22a2 prod v f = \u220f i : \u03b9, f i (\u2191equivFunOnFintype v i)\n[PROOFSTEP]\nsimp [DFinsupp.prod, this]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), \u03b2 i\nf : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nhf : \u2200 (i : \u03b9), f i 0 = 1\n\u22a2 \u220f i in support v, f i (\u2191v i) = \u220f i : \u03b9, f i (\u2191v i)\n[PROOFSTEP]\napply Finset.prod_subset v.support.subset_univ\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), \u03b2 i\nf : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nhf : \u2200 (i : \u03b9), f i 0 = 1\n\u22a2 \u2200 (x : \u03b9), x \u2208 Finset.univ \u2192 \u00acx \u2208 support v \u2192 f x (\u2191v x) = 1\n[PROOFSTEP]\nintro i _ hi\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), \u03b2 i\nf : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nhf : \u2200 (i : \u03b9), f i 0 = 1\ni : \u03b9\na\u271d : i \u2208 Finset.univ\nhi : \u00aci \u2208 support v\n\u22a2 f i (\u2191v i) = 1\n[PROOFSTEP]\nrw [mem_support_iff, not_not] at hi \n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), \u03b2 i\nf : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nhf : \u2200 (i : \u03b9), f i 0 = 1\ni : \u03b9\na\u271d : i \u2208 Finset.univ\nhi : \u2191v i = 0\n\u22a2 f i (\u2191v i) = 1\n[PROOFSTEP]\nrw [hi, hf]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }),\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) a = (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) b\n[PROOFSTEP]\nrintro \u27e8sx, hx\u27e9 \u27e8sy, hy\u27e9\n[GOAL]\ncase mk.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\n\u22a2 (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) { val := sx, property := hx } =\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) { val := sy, property := hy }\n[PROOFSTEP]\ndsimp only [Subtype.coe_mk, toFun_eq_coe] at *\n[GOAL]\ncase mk.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\n\u22a2 \u2211 i in Multiset.toFinset sx, \u2191(\u03c6 i) (\u2191f i) = \u2211 i in Multiset.toFinset sy, \u2191(\u03c6 i) (\u2191f i)\n[PROOFSTEP]\nhave H1 : sx.toFinset \u2229 sy.toFinset \u2286 sx.toFinset := Finset.inter_subset_left _ _\n[GOAL]\ncase mk.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\nH1 : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sx\n\u22a2 \u2211 i in Multiset.toFinset sx, \u2191(\u03c6 i) (\u2191f i) = \u2211 i in Multiset.toFinset sy, \u2191(\u03c6 i) (\u2191f i)\n[PROOFSTEP]\nhave H2 : sx.toFinset \u2229 sy.toFinset \u2286 sy.toFinset := Finset.inter_subset_right _ _\n[GOAL]\ncase mk.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\nH1 : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sx\nH2 : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sy\n\u22a2 \u2211 i in Multiset.toFinset sx, \u2191(\u03c6 i) (\u2191f i) = \u2211 i in Multiset.toFinset sy, \u2191(\u03c6 i) (\u2191f i)\n[PROOFSTEP]\nrefine' (Finset.sum_subset H1 _).symm.trans ((Finset.sum_congr rfl _).trans (Finset.sum_subset H2 _))\n[GOAL]\ncase mk.mk.refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\nH1 : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sx\nH2 : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sy\n\u22a2 \u2200 (x : \u03b9), x \u2208 Multiset.toFinset sx \u2192 \u00acx \u2208 Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2192 \u2191(\u03c6 x) (\u2191f x) = 0\n[PROOFSTEP]\nintro i H1 H2\n[GOAL]\ncase mk.mk.refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\nH1\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sx\nH2\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sy\ni : \u03b9\nH1 : i \u2208 Multiset.toFinset sx\nH2 : \u00aci \u2208 Multiset.toFinset sx \u2229 Multiset.toFinset sy\n\u22a2 \u2191(\u03c6 i) (\u2191f i) = 0\n[PROOFSTEP]\nrw [Finset.mem_inter] at H2 \n[GOAL]\ncase mk.mk.refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\nH1\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sx\nH2\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sy\ni : \u03b9\nH1 : i \u2208 Multiset.toFinset sx\nH2 : \u00ac(i \u2208 Multiset.toFinset sx \u2227 i \u2208 Multiset.toFinset sy)\n\u22a2 \u2191(\u03c6 i) (\u2191f i) = 0\n[PROOFSTEP]\nsimp only [Multiset.mem_toFinset] at H1 H2 \n[GOAL]\ncase mk.mk.refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\nH1\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sx\nH2\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sy\ni : \u03b9\nH1 : i \u2208 sx\nH2 : \u00ac(i \u2208 sx \u2227 i \u2208 sy)\n\u22a2 \u2191(\u03c6 i) (\u2191f i) = 0\n[PROOFSTEP]\nconvert AddMonoidHom.map_zero (\u03c6 i)\n[GOAL]\ncase h.e'_2.h.e'_6\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\nH1\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sx\nH2\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sy\ni : \u03b9\nH1 : i \u2208 sx\nH2 : \u00ac(i \u2208 sx \u2227 i \u2208 sy)\n\u22a2 \u2191f i = 0\n[PROOFSTEP]\nexact (hy i).resolve_left (mt (And.intro H1) H2)\n[GOAL]\ncase mk.mk.refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\nH1 : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sx\nH2 : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sy\n\u22a2 \u2200 (x : \u03b9), x \u2208 Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2192 \u2191(\u03c6 x) (\u2191f x) = \u2191(\u03c6 x) (\u2191f x)\n[PROOFSTEP]\nintro i _\n[GOAL]\ncase mk.mk.refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\nH1 : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sx\nH2 : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sy\ni : \u03b9\na\u271d : i \u2208 Multiset.toFinset sx \u2229 Multiset.toFinset sy\n\u22a2 \u2191(\u03c6 i) (\u2191f i) = \u2191(\u03c6 i) (\u2191f i)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.mk.refine'_3\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\nH1 : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sx\nH2 : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sy\n\u22a2 \u2200 (x : \u03b9), x \u2208 Multiset.toFinset sy \u2192 \u00acx \u2208 Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2192 \u2191(\u03c6 x) (\u2191f x) = 0\n[PROOFSTEP]\nintro i H1 H2\n[GOAL]\ncase mk.mk.refine'_3\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\nH1\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sx\nH2\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sy\ni : \u03b9\nH1 : i \u2208 Multiset.toFinset sy\nH2 : \u00aci \u2208 Multiset.toFinset sx \u2229 Multiset.toFinset sy\n\u22a2 \u2191(\u03c6 i) (\u2191f i) = 0\n[PROOFSTEP]\nrw [Finset.mem_inter] at H2 \n[GOAL]\ncase mk.mk.refine'_3\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\nH1\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sx\nH2\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sy\ni : \u03b9\nH1 : i \u2208 Multiset.toFinset sy\nH2 : \u00ac(i \u2208 Multiset.toFinset sx \u2227 i \u2208 Multiset.toFinset sy)\n\u22a2 \u2191(\u03c6 i) (\u2191f i) = 0\n[PROOFSTEP]\nsimp only [Multiset.mem_toFinset] at H1 H2 \n[GOAL]\ncase mk.mk.refine'_3\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\nH1\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sx\nH2\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sy\ni : \u03b9\nH1 : i \u2208 sy\nH2 : \u00ac(i \u2208 sx \u2227 i \u2208 sy)\n\u22a2 \u2191(\u03c6 i) (\u2191f i) = 0\n[PROOFSTEP]\nconvert AddMonoidHom.map_zero (\u03c6 i)\n[GOAL]\ncase h.e'_2.h.e'_6\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nsx : Multiset \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 sx \u2228 toFun f i = 0\nsy : Multiset \u03b9\nhy : \u2200 (i : \u03b9), i \u2208 sy \u2228 toFun f i = 0\nH1\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sx\nH2\u271d : Multiset.toFinset sx \u2229 Multiset.toFinset sy \u2286 Multiset.toFinset sy\ni : \u03b9\nH1 : i \u2208 sy\nH2 : \u00ac(i \u2208 sx \u2227 i \u2208 sy)\n\u22a2 \u2191f i = 0\n[PROOFSTEP]\nexact (hx i).resolve_left (mt (fun H3 => And.intro H3 H1) H2)\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\n\u22a2 (fun f =>\n (fun c => Trunc.lift (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) c f.support')\n (_ :\n \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }),\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) a =\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) b))\n 0 =\n 0\n[PROOFSTEP]\nsimp only [toFun_eq_coe, coe_zero, Pi.zero_apply, map_zero, Finset.sum_const_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\n\u22a2 Trunc.lift (fun s => 0) (_ : \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun 0 i = 0 }), (fun s => 0) a = (fun s => 0) b)\n 0.support' =\n 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\n\u22a2 \u2200 (x y : \u03a0\u2080 (i : \u03b9), \u03b2 i),\n ZeroHom.toFun\n {\n toFun := fun f =>\n (fun c => Trunc.lift (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) c f.support')\n (_ :\n \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }),\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) a =\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) b),\n map_zero' :=\n (_ :\n (fun f =>\n (fun c => Trunc.lift (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) c f.support')\n (_ :\n \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }),\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) a =\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) b))\n 0 =\n 0) }\n (x + y) =\n ZeroHom.toFun\n {\n toFun := fun f =>\n (fun c => Trunc.lift (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) c f.support')\n (_ :\n \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }),\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) a =\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) b),\n map_zero' :=\n (_ :\n (fun f =>\n (fun c => Trunc.lift (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) c f.support')\n (_ :\n \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }),\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) a =\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) b))\n 0 =\n 0) }\n x +\n ZeroHom.toFun\n {\n toFun := fun f =>\n (fun c => Trunc.lift (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) c f.support')\n (_ :\n \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }),\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) a =\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) b),\n map_zero' :=\n (_ :\n (fun f =>\n (fun c => Trunc.lift (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) c f.support')\n (_ :\n \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }),\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) a =\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) b))\n 0 =\n 0) }\n y\n[PROOFSTEP]\nrintro \u27e8f, sf, hf\u27e9 \u27e8g, sg, hg\u27e9\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\n\u22a2 ZeroHom.toFun\n {\n toFun := fun f =>\n (fun c => Trunc.lift (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) c f.support')\n (_ :\n \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }),\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) a =\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) b),\n map_zero' :=\n (_ :\n (fun f =>\n (fun c => Trunc.lift (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) c f.support')\n (_ :\n \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }),\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) a =\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) b))\n 0 =\n 0) }\n ({ toFun := f, support' := Quot.mk Setoid.r { val := sf, property := hf } } +\n { toFun := g, support' := Quot.mk Setoid.r { val := sg, property := hg } }) =\n ZeroHom.toFun\n {\n toFun := fun f =>\n (fun c => Trunc.lift (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) c f.support')\n (_ :\n \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }),\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) a =\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) b),\n map_zero' :=\n (_ :\n (fun f =>\n (fun c => Trunc.lift (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) c f.support')\n (_ :\n \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }),\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) a =\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) b))\n 0 =\n 0) }\n { toFun := f, support' := Quot.mk Setoid.r { val := sf, property := hf } } +\n ZeroHom.toFun\n {\n toFun := fun f =>\n (fun c => Trunc.lift (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) c f.support')\n (_ :\n \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }),\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) a =\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) b),\n map_zero' :=\n (_ :\n (fun f =>\n (fun c => Trunc.lift (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) c f.support')\n (_ :\n \u2200 (a b : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 toFun f i = 0 }),\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) a =\n (fun s => \u2211 i in Multiset.toFinset \u2191s, \u2191(\u03c6 i) (\u2191f i)) b))\n 0 =\n 0) }\n { toFun := g, support' := Quot.mk Setoid.r { val := sg, property := hg } }\n[PROOFSTEP]\nchange (\u2211 i in _, _) = (\u2211 i in _, _) + \u2211 i in _, _\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\n\u22a2 \u2211 i in\n Multiset.toFinset\n \u2191((fun ys =>\n { val := \u2191{ val := sf, property := hf } + \u2191ys,\n property :=\n (_ :\n \u2200 (i : \u03b9),\n i \u2208 \u2191{ val := sf, property := hf } + \u2191ys \u2228\n (fun i =>\n \u2191{ toFun := f, support' := Quot.mk Setoid.r { val := sf, property := hf } } i +\n \u2191{ toFun := g, support' := Quot.mk Setoid.r { val := sg, property := hg } } i)\n i =\n 0) })\n { val := sg, property := hg }),\n \u2191(\u03c6 i)\n (\u2191({ toFun := f, support' := Quot.mk Setoid.r { val := sf, property := hf } } +\n { toFun := g, support' := Quot.mk Setoid.r { val := sg, property := hg } })\n i) =\n \u2211 i in Multiset.toFinset \u2191{ val := sf, property := hf },\n \u2191(\u03c6 i) (\u2191{ toFun := f, support' := Quot.mk Setoid.r { val := sf, property := hf } } i) +\n \u2211 i in Multiset.toFinset \u2191{ val := sg, property := hg },\n \u2191(\u03c6 i) (\u2191{ toFun := g, support' := Quot.mk Setoid.r { val := sg, property := hg } } i)\n[PROOFSTEP]\nsimp only [coe_add, coe_mk', Subtype.coe_mk, Pi.add_apply, map_add, Finset.sum_add_distrib]\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\n\u22a2 \u2211 x in Multiset.toFinset (sf + sg), \u2191(\u03c6 x) (f x) + \u2211 x in Multiset.toFinset (sf + sg), \u2191(\u03c6 x) (g x) =\n \u2211 x in Multiset.toFinset sf, \u2191(\u03c6 x) (f x) + \u2211 x in Multiset.toFinset sg, \u2191(\u03c6 x) (g x)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk.e_a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\n\u22a2 \u2211 x in Multiset.toFinset (sf + sg), \u2191(\u03c6 x) (f x) = \u2211 x in Multiset.toFinset sf, \u2191(\u03c6 x) (f x)\n[PROOFSTEP]\nrefine' (Finset.sum_subset _ _).symm\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk.e_a.refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\n\u22a2 Multiset.toFinset sf \u2286 Multiset.toFinset (sf + sg)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk.e_a.refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\ni : \u03b9\n\u22a2 i \u2208 Multiset.toFinset sf \u2192 i \u2208 Multiset.toFinset (sf + sg)\n[PROOFSTEP]\nsimp only [Multiset.mem_toFinset, Multiset.mem_add]\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk.e_a.refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\ni : \u03b9\n\u22a2 i \u2208 sf \u2192 i \u2208 sf \u2228 i \u2208 sg\n[PROOFSTEP]\nexact Or.inl\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk.e_a.refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\n\u22a2 \u2200 (x : \u03b9), x \u2208 Multiset.toFinset (sf + sg) \u2192 \u00acx \u2208 Multiset.toFinset sf \u2192 \u2191(\u03c6 x) (f x) = 0\n[PROOFSTEP]\nintro i _ H2\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk.e_a.refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\ni : \u03b9\na\u271d : i \u2208 Multiset.toFinset (sf + sg)\nH2 : \u00aci \u2208 Multiset.toFinset sf\n\u22a2 \u2191(\u03c6 i) (f i) = 0\n[PROOFSTEP]\nsimp only [Multiset.mem_toFinset, Multiset.mem_add] at H2 \n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk.e_a.refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\ni : \u03b9\na\u271d : i \u2208 Multiset.toFinset (sf + sg)\nH2 : \u00aci \u2208 sf\n\u22a2 \u2191(\u03c6 i) (f i) = 0\n[PROOFSTEP]\nrw [(hf i).resolve_left H2, AddMonoidHom.map_zero]\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk.e_a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\n\u22a2 \u2211 x in Multiset.toFinset (sf + sg), \u2191(\u03c6 x) (g x) = \u2211 x in Multiset.toFinset sg, \u2191(\u03c6 x) (g x)\n[PROOFSTEP]\nrefine' (Finset.sum_subset _ _).symm\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk.e_a.refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\n\u22a2 Multiset.toFinset sg \u2286 Multiset.toFinset (sf + sg)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk.e_a.refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\ni : \u03b9\n\u22a2 i \u2208 Multiset.toFinset sg \u2192 i \u2208 Multiset.toFinset (sf + sg)\n[PROOFSTEP]\nsimp only [Multiset.mem_toFinset, Multiset.mem_add]\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk.e_a.refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\ni : \u03b9\n\u22a2 i \u2208 sg \u2192 i \u2208 sf \u2228 i \u2208 sg\n[PROOFSTEP]\nexact Or.inr\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk.e_a.refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\n\u22a2 \u2200 (x : \u03b9), x \u2208 Multiset.toFinset (sf + sg) \u2192 \u00acx \u2208 Multiset.toFinset sg \u2192 \u2191(\u03c6 x) (g x) = 0\n[PROOFSTEP]\nintro i _ H2\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk.e_a.refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\ni : \u03b9\na\u271d : i \u2208 Multiset.toFinset (sf + sg)\nH2 : \u00aci \u2208 Multiset.toFinset sg\n\u22a2 \u2191(\u03c6 i) (g i) = 0\n[PROOFSTEP]\nsimp only [Multiset.mem_toFinset, Multiset.mem_add] at H2 \n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk.e_a.refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\nsf : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 sf \u2228 f i = 0\ng : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 g i = 0 }\nsg : Multiset \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 sg \u2228 g i = 0\ni : \u03b9\na\u271d : i \u2208 Multiset.toFinset (sf + sg)\nH2 : \u00aci \u2208 sg\n\u22a2 \u2191(\u03c6 i) (g i) = 0\n[PROOFSTEP]\nrw [(hg i).resolve_left H2, AddMonoidHom.map_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\ni : \u03b9\nx : \u03b2 i\n\u22a2 \u2191(sumAddHom \u03c6) (single i x) = \u2191(\u03c6 i) x\n[PROOFSTEP]\ndsimp [sumAddHom, single, Trunc.lift_mk]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\ni : \u03b9\nx : \u03b2 i\n\u22a2 \u2211 i_1 in Multiset.toFinset {i}, \u2191(\u03c6 i_1) (Pi.single i x i_1) = \u2191(\u03c6 i) x\n[PROOFSTEP]\nrw [Multiset.toFinset_singleton, Finset.sum_singleton, Pi.single_eq_same]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 \u2191(sumAddHom \u03c6) f = sum f fun x => \u2191(\u03c6 x)\n[PROOFSTEP]\nrcases f with \u27e8f, s, hf\u27e9\n[GOAL]\ncase mk'.mk.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\ns : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0\n\u22a2 \u2191(sumAddHom \u03c6) { toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } =\n sum { toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } fun x => \u2191(\u03c6 x)\n[PROOFSTEP]\nchange (\u2211 i in _, _) = \u2211 i in Finset.filter _ _, _\n[GOAL]\ncase mk'.mk.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\ns : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0\n\u22a2 \u2211 i in Multiset.toFinset \u2191{ val := s, property := hf },\n \u2191(\u03c6 i) (\u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i) =\n \u2211 i in\n Finset.filter (fun i => \u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i \u2260 0)\n (Multiset.toFinset \u2191{ val := s, property := hf }),\n (fun x => \u2191(\u03c6 x)) i (\u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i)\n[PROOFSTEP]\nrw [Finset.sum_filter, Finset.sum_congr rfl]\n[GOAL]\ncase mk'.mk.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\ns : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0\n\u22a2 \u2200 (x : \u03b9),\n x \u2208 Multiset.toFinset \u2191{ val := s, property := hf } \u2192\n \u2191(\u03c6 x) (\u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } x) =\n if \u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } x \u2260 0 then\n (fun x => \u2191(\u03c6 x)) x (\u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } x)\n else 0\n[PROOFSTEP]\nintro i _\n[GOAL]\ncase mk'.mk.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\ns : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0\ni : \u03b9\na\u271d : i \u2208 Multiset.toFinset \u2191{ val := s, property := hf }\n\u22a2 \u2191(\u03c6 i) (\u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i) =\n if \u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i \u2260 0 then\n (fun x => \u2191(\u03c6 x)) i (\u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i)\n else 0\n[PROOFSTEP]\ndsimp only [coe_mk', Subtype.coe_mk] at *\n[GOAL]\ncase mk'.mk.mk\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\ns : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0\ni : \u03b9\na\u271d : i \u2208 Multiset.toFinset s\n\u22a2 \u2191(\u03c6 i) (f i) = if f i \u2260 0 then \u2191(\u03c6 i) (f i) else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\ns : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0\ni : \u03b9\na\u271d : i \u2208 Multiset.toFinset s\nh : f i \u2260 0\n\u22a2 \u2191(\u03c6 i) (f i) = \u2191(\u03c6 i) (f i)\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\ns : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0\ni : \u03b9\na\u271d : i \u2208 Multiset.toFinset s\nh : \u00acf i \u2260 0\n\u22a2 \u2191(\u03c6 i) (f i) = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\ns : Multiset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0\ni : \u03b9\na\u271d : i \u2208 Multiset.toFinset s\nh : \u00acf i \u2260 0\n\u22a2 \u2191(\u03c6 i) (f i) = 0\n[PROOFSTEP]\nrw [not_not.mp h, AddMonoidHom.map_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b2 : AddCommMonoid \u03b3\nS : Type u_1\ninst\u271d\u00b9 : SetLike S \u03b3\ninst\u271d : AddSubmonoidClass S \u03b3\ns : S\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ng : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nh : \u2200 (c : \u03b9), \u2191f c \u2260 0 \u2192 \u2191(g c) (\u2191f c) \u2208 s\n\u22a2 \u2191(sumAddHom g) f \u2208 s\n[PROOFSTEP]\nclassical\nrw [DFinsupp.sumAddHom_apply]\nexact dfinsupp_sum_mem s f (g \u00b7) h\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b2 : AddCommMonoid \u03b3\nS : Type u_1\ninst\u271d\u00b9 : SetLike S \u03b3\ninst\u271d : AddSubmonoidClass S \u03b3\ns : S\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ng : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nh : \u2200 (c : \u03b9), \u2191f c \u2260 0 \u2192 \u2191(g c) (\u2191f c) \u2208 s\n\u22a2 \u2191(sumAddHom g) f \u2208 s\n[PROOFSTEP]\nrw [DFinsupp.sumAddHom_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b2 : AddCommMonoid \u03b3\nS : Type u_1\ninst\u271d\u00b9 : SetLike S \u03b3\ninst\u271d : AddSubmonoidClass S \u03b3\ns : S\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ng : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nh : \u2200 (c : \u03b9), \u2191f c \u2260 0 \u2192 \u2191(g c) (\u2191f c) \u2208 s\n\u22a2 (sum f fun x => \u2191(g x)) \u2208 s\n[PROOFSTEP]\nexact dfinsupp_sum_mem s f (g \u00b7) h\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\n\u22a2 iSup S = AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i))\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\n\u22a2 iSup S \u2264 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i))\n[PROOFSTEP]\napply iSup_le _\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\n\u22a2 \u2200 (i : \u03b9), S i \u2264 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i))\n[PROOFSTEP]\nintro i y hy\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\ni : \u03b9\ny : \u03b3\nhy : y \u2208 S i\n\u22a2 y \u2208 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i))\n[PROOFSTEP]\nexact \u27e8DFinsupp.single i \u27e8y, hy\u27e9, DFinsupp.sumAddHom_single _ _ _\u27e9\n[GOAL]\ncase a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\n\u22a2 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i)) \u2264 iSup S\n[PROOFSTEP]\nrintro x \u27e8v, rfl\u27e9\n[GOAL]\ncase a.intro\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), { x // x \u2208 S i }\n\u22a2 \u2191(sumAddHom fun i => AddSubmonoid.subtype (S i)) v \u2208 iSup S\n[PROOFSTEP]\nexact dfinsupp_sumAddHom_mem _ v _ fun i _ => (le_iSup S i : S i \u2264 _) (v i).prop\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\n\u22a2 \u2a06 (i : \u03b9) (_ : p i), S i =\n AddMonoidHom.mrange\n (AddMonoidHom.comp (sumAddHom fun i => AddSubmonoid.subtype (S i))\n (filterAddMonoidHom (fun i => { x // x \u2208 S i }) p))\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\n\u22a2 \u2a06 (i : \u03b9) (_ : p i), S i \u2264\n AddMonoidHom.mrange\n (AddMonoidHom.comp (sumAddHom fun i => AddSubmonoid.subtype (S i))\n (filterAddMonoidHom (fun i => { x // x \u2208 S i }) p))\n[PROOFSTEP]\nrefine' iSup\u2082_le fun i hi y hy => \u27e8DFinsupp.single i \u27e8y, hy\u27e9, _\u27e9\n[GOAL]\ncase a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\ni : \u03b9\nhi : p i\ny : \u03b3\nhy : y \u2208 S i\n\u22a2 \u2191(AddMonoidHom.comp (sumAddHom fun i => AddSubmonoid.subtype (S i))\n (filterAddMonoidHom (fun i => { x // x \u2208 S i }) p))\n (single i { val := y, property := hy }) =\n y\n[PROOFSTEP]\nrw [AddMonoidHom.comp_apply, filterAddMonoidHom_apply, filter_single_pos _ _ hi]\n[GOAL]\ncase a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\ni : \u03b9\nhi : p i\ny : \u03b3\nhy : y \u2208 S i\n\u22a2 \u2191(sumAddHom fun i => AddSubmonoid.subtype (S i)) (single i { val := y, property := hy }) = y\n[PROOFSTEP]\nexact sumAddHom_single _ _ _\n[GOAL]\ncase a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\n\u22a2 AddMonoidHom.mrange\n (AddMonoidHom.comp (sumAddHom fun i => AddSubmonoid.subtype (S i))\n (filterAddMonoidHom (fun i => { x // x \u2208 S i }) p)) \u2264\n \u2a06 (i : \u03b9) (_ : p i), S i\n[PROOFSTEP]\nrintro x \u27e8v, rfl\u27e9\n[GOAL]\ncase a.intro\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), { x // x \u2208 S i }\n\u22a2 \u2191(AddMonoidHom.comp (sumAddHom fun i => AddSubmonoid.subtype (S i))\n (filterAddMonoidHom (fun i => { x // x \u2208 S i }) p))\n v \u2208\n \u2a06 (i : \u03b9) (_ : p i), S i\n[PROOFSTEP]\nrefine' dfinsupp_sumAddHom_mem _ _ _ fun i _ => _\n[GOAL]\ncase a.intro\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), { x // x \u2208 S i }\ni : \u03b9\nx\u271d : \u2191(\u2191(filterAddMonoidHom (fun i => { x // x \u2208 S i }) p) v) i \u2260 0\n\u22a2 \u2191((fun i => AddSubmonoid.subtype (S i)) i) (\u2191(\u2191(filterAddMonoidHom (fun i => { x // x \u2208 S i }) p) v) i) \u2208\n \u2a06 (i : \u03b9) (_ : p i), S i\n[PROOFSTEP]\nrefine' AddSubmonoid.mem_iSup_of_mem i _\n[GOAL]\ncase a.intro\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), { x // x \u2208 S i }\ni : \u03b9\nx\u271d : \u2191(\u2191(filterAddMonoidHom (fun i => { x // x \u2208 S i }) p) v) i \u2260 0\n\u22a2 \u2191((fun i => AddSubmonoid.subtype (S i)) i) (\u2191(\u2191(filterAddMonoidHom (fun i => { x // x \u2208 S i }) p) v) i) \u2208\n \u2a06 (_ : p i), S i\n[PROOFSTEP]\nby_cases hp : p i\n[GOAL]\ncase pos\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), { x // x \u2208 S i }\ni : \u03b9\nx\u271d : \u2191(\u2191(filterAddMonoidHom (fun i => { x // x \u2208 S i }) p) v) i \u2260 0\nhp : p i\n\u22a2 \u2191((fun i => AddSubmonoid.subtype (S i)) i) (\u2191(\u2191(filterAddMonoidHom (fun i => { x // x \u2208 S i }) p) v) i) \u2208\n \u2a06 (_ : p i), S i\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\ncase neg\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), { x // x \u2208 S i }\ni : \u03b9\nx\u271d : \u2191(\u2191(filterAddMonoidHom (fun i => { x // x \u2208 S i }) p) v) i \u2260 0\nhp : \u00acp i\n\u22a2 \u2191((fun i => AddSubmonoid.subtype (S i)) i) (\u2191(\u2191(filterAddMonoidHom (fun i => { x // x \u2208 S i }) p) v) i) \u2208\n \u2a06 (_ : p i), S i\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 S i }) \u2192 Decidable (x \u2260 0)\nx : \u03b3\n\u22a2 x \u2208 iSup S \u2194 \u2203 f, (sum f fun i xi => \u2191xi) = x\n[PROOFSTEP]\nrw [AddSubmonoid.mem_iSup_iff_exists_dfinsupp]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 S i }) \u2192 Decidable (x \u2260 0)\nx : \u03b3\n\u22a2 (\u2203 f, \u2191(sumAddHom fun i => AddSubmonoid.subtype (S i)) f = x) \u2194 \u2203 f, (sum f fun i xi => \u2191xi) = x\n[PROOFSTEP]\nsimp_rw [sumAddHom_apply]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 S i }) \u2192 Decidable (x \u2260 0)\nx : \u03b3\n\u22a2 (\u2203 f, (sum f fun x => \u2191(AddSubmonoid.subtype (S x))) = x) \u2194 \u2203 f, (sum f fun i xi => \u2191xi) = x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081\u271d : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b9\u2081 : Type u_4\n\u03b9\u2082 : Type u_5\n\u03b2\u2081 : \u03b9\u2081 \u2192 Type u_1\n\u03b2\u2082 : \u03b9\u2082 \u2192 Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : DecidableEq \u03b9\u2081\ninst\u271d\u00b3 : DecidableEq \u03b9\u2082\ninst\u271d\u00b2 : (i : \u03b9\u2081) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d\u00b9 : (i : \u03b9\u2082) \u2192 AddZeroClass (\u03b2\u2082 i)\ninst\u271d : AddCommMonoid \u03b3\nf\u2081 : \u03a0\u2080 (i : \u03b9\u2081), \u03b2\u2081 i\nf\u2082 : \u03a0\u2080 (i : \u03b9\u2082), \u03b2\u2082 i\nh : (i : \u03b9\u2081) \u2192 (j : \u03b9\u2082) \u2192 \u03b2\u2081 i \u2192+ \u03b2\u2082 j \u2192+ \u03b3\n\u22a2 \u2191(sumAddHom fun i\u2082 => \u2191(sumAddHom fun i\u2081 => h i\u2081 i\u2082) f\u2081) f\u2082 =\n \u2191(sumAddHom fun i\u2081 => \u2191(sumAddHom fun i\u2082 => AddMonoidHom.flip (h i\u2081 i\u2082)) f\u2082) f\u2081\n[PROOFSTEP]\nobtain \u27e8\u27e8f\u2081, s\u2081, h\u2081\u27e9, \u27e8f\u2082, s\u2082, h\u2082\u27e9\u27e9 := f\u2081, f\u2082\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk\n\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081\u271d : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b9\u2081 : Type u_4\n\u03b9\u2082 : Type u_5\n\u03b2\u2081 : \u03b9\u2081 \u2192 Type u_1\n\u03b2\u2082 : \u03b9\u2082 \u2192 Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : DecidableEq \u03b9\u2081\ninst\u271d\u00b3 : DecidableEq \u03b9\u2082\ninst\u271d\u00b2 : (i : \u03b9\u2081) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d\u00b9 : (i : \u03b9\u2082) \u2192 AddZeroClass (\u03b2\u2082 i)\ninst\u271d : AddCommMonoid \u03b3\nh : (i : \u03b9\u2081) \u2192 (j : \u03b9\u2082) \u2192 \u03b2\u2081 i \u2192+ \u03b2\u2082 j \u2192+ \u03b3\nf\u2081 : (i : \u03b9\u2081) \u2192 \u03b2\u2081 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9\u2081), i \u2208 s \u2228 f\u2081 i = 0 }\ns\u2081 : Multiset \u03b9\u2081\nh\u2081 : \u2200 (i : \u03b9\u2081), i \u2208 s\u2081 \u2228 f\u2081 i = 0\nf\u2082 : (i : \u03b9\u2082) \u2192 \u03b2\u2082 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9\u2082), i \u2208 s \u2228 f\u2082 i = 0 }\ns\u2082 : Multiset \u03b9\u2082\nh\u2082 : \u2200 (i : \u03b9\u2082), i \u2208 s\u2082 \u2228 f\u2082 i = 0\n\u22a2 \u2191(sumAddHom fun i\u2082 =>\n \u2191(sumAddHom fun i\u2081 => h i\u2081 i\u2082) { toFun := f\u2081, support' := Quot.mk Setoid.r { val := s\u2081, property := h\u2081 } })\n { toFun := f\u2082, support' := Quot.mk Setoid.r { val := s\u2082, property := h\u2082 } } =\n \u2191(sumAddHom fun i\u2081 =>\n \u2191(sumAddHom fun i\u2082 => AddMonoidHom.flip (h i\u2081 i\u2082))\n { toFun := f\u2082, support' := Quot.mk Setoid.r { val := s\u2082, property := h\u2082 } })\n { toFun := f\u2081, support' := Quot.mk Setoid.r { val := s\u2081, property := h\u2081 } }\n[PROOFSTEP]\nsimp only [sumAddHom, AddMonoidHom.finset_sum_apply, Quotient.liftOn_mk, AddMonoidHom.coe_mk, AddMonoidHom.flip_apply,\n Trunc.lift, toFun_eq_coe, ZeroHom.coe_mk, coe_mk']\n[GOAL]\ncase mk'.mk.mk.mk'.mk.mk\n\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081\u271d : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082\u271d : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b9\u2081 : Type u_4\n\u03b9\u2082 : Type u_5\n\u03b2\u2081 : \u03b9\u2081 \u2192 Type u_1\n\u03b2\u2082 : \u03b9\u2082 \u2192 Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : DecidableEq \u03b9\u2081\ninst\u271d\u00b3 : DecidableEq \u03b9\u2082\ninst\u271d\u00b2 : (i : \u03b9\u2081) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d\u00b9 : (i : \u03b9\u2082) \u2192 AddZeroClass (\u03b2\u2082 i)\ninst\u271d : AddCommMonoid \u03b3\nh : (i : \u03b9\u2081) \u2192 (j : \u03b9\u2082) \u2192 \u03b2\u2081 i \u2192+ \u03b2\u2082 j \u2192+ \u03b3\nf\u2081 : (i : \u03b9\u2081) \u2192 \u03b2\u2081 i\nsupport'\u271d\u00b9 : Trunc { s // \u2200 (i : \u03b9\u2081), i \u2208 s \u2228 f\u2081 i = 0 }\ns\u2081 : Multiset \u03b9\u2081\nh\u2081 : \u2200 (i : \u03b9\u2081), i \u2208 s\u2081 \u2228 f\u2081 i = 0\nf\u2082 : (i : \u03b9\u2082) \u2192 \u03b2\u2082 i\nsupport'\u271d : Trunc { s // \u2200 (i : \u03b9\u2082), i \u2208 s \u2228 f\u2082 i = 0 }\ns\u2082 : Multiset \u03b9\u2082\nh\u2082 : \u2200 (i : \u03b9\u2082), i \u2208 s\u2082 \u2228 f\u2082 i = 0\n\u22a2 \u2211 x in Multiset.toFinset s\u2082, \u2211 x_1 in Multiset.toFinset s\u2081, \u2191(\u2191(h x_1 x) (f\u2081 x_1)) (f\u2082 x) =\n \u2211 x in Multiset.toFinset s\u2081, \u2211 x_1 in Multiset.toFinset s\u2082, \u2191(\u2191(h x x_1) (f\u2081 x)) (f\u2082 x_1)\n[PROOFSTEP]\nexact Finset.sum_comm\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\nx : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\n\u22a2 (fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) (sumAddHom x) = x\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\nx : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nx\u271d\u00b9 : \u03b9\nx\u271d : \u03b2 x\u271d\u00b9\n\u22a2 \u2191((fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) (sumAddHom x) x\u271d\u00b9) x\u271d = \u2191(x x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c8 : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\n\u22a2 sumAddHom ((fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) \u03c8) = \u03c8\n[PROOFSTEP]\next\n[GOAL]\ncase H.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\n\u03c8 : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nx\u271d\u00b9 : \u03b9\nx\u271d : \u03b2 x\u271d\u00b9\n\u22a2 \u2191(AddMonoidHom.comp (sumAddHom ((fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) \u03c8))\n (singleAddHom (fun i => \u03b2 i) x\u271d\u00b9))\n x\u271d =\n \u2191(AddMonoidHom.comp \u03c8 (singleAddHom (fun i => \u03b2 i) x\u271d\u00b9)) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\nF G : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\n\u22a2 Equiv.toFun\n { toFun := sumAddHom, invFun := fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i),\n left_inv :=\n (_ : \u2200 (x : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3), (fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) (sumAddHom x) = x),\n right_inv :=\n (_ : \u2200 (\u03c8 : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3), sumAddHom ((fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) \u03c8) = \u03c8) }\n (F + G) =\n Equiv.toFun\n { toFun := sumAddHom, invFun := fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i),\n left_inv :=\n (_ : \u2200 (x : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3), (fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) (sumAddHom x) = x),\n right_inv :=\n (_ :\n \u2200 (\u03c8 : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3), sumAddHom ((fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) \u03c8) = \u03c8) }\n F +\n Equiv.toFun\n { toFun := sumAddHom, invFun := fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i),\n left_inv :=\n (_ : \u2200 (x : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3), (fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) (sumAddHom x) = x),\n right_inv :=\n (_ :\n \u2200 (\u03c8 : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3), sumAddHom ((fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) \u03c8) = \u03c8) }\n G\n[PROOFSTEP]\next\n[GOAL]\ncase H.h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\nF G : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\nx\u271d\u00b9 : \u03b9\nx\u271d : \u03b2 x\u271d\u00b9\n\u22a2 \u2191(AddMonoidHom.comp\n (Equiv.toFun\n { toFun := sumAddHom, invFun := fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i),\n left_inv :=\n (_ : \u2200 (x : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3), (fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) (sumAddHom x) = x),\n right_inv :=\n (_ :\n \u2200 (\u03c8 : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3),\n sumAddHom ((fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) \u03c8) = \u03c8) }\n (F + G))\n (singleAddHom (fun i => \u03b2 i) x\u271d\u00b9))\n x\u271d =\n \u2191(AddMonoidHom.comp\n (Equiv.toFun\n { toFun := sumAddHom, invFun := fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i),\n left_inv :=\n (_ :\n \u2200 (x : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3), (fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) (sumAddHom x) = x),\n right_inv :=\n (_ :\n \u2200 (\u03c8 : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3),\n sumAddHom ((fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) \u03c8) = \u03c8) }\n F +\n Equiv.toFun\n { toFun := sumAddHom, invFun := fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i),\n left_inv :=\n (_ :\n \u2200 (x : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3), (fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) (sumAddHom x) = x),\n right_inv :=\n (_ :\n \u2200 (\u03c8 : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3),\n sumAddHom ((fun F i => AddMonoidHom.comp F (singleAddHom \u03b2 i)) \u03c8) = \u03c8) }\n G)\n (singleAddHom (fun i => \u03b2 i) x\u271d\u00b9))\n x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\ni : \u03b9\nx : \u03b2 i\n\u22a2 \u2191(\u2191liftAddHom f) (single i x) = \u2191(f i) x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\ni : \u03b9\n\u22a2 AddMonoidHom.comp (\u2191liftAddHom f) (singleAddHom \u03b2 i) = f i\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b4 : Type u_1\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : AddCommMonoid \u03b3\ninst\u271d : AddCommMonoid \u03b4\ng : \u03b3 \u2192+ \u03b4\nf : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\na : \u03b9\n\u22a2 \u2191(AddEquiv.symm liftAddHom) (AddMonoidHom.comp g (\u2191liftAddHom f)) a = AddMonoidHom.comp g (f a)\n[PROOFSTEP]\nrw [liftAddHom_symm_apply, AddMonoidHom.comp_assoc, liftAddHom_comp_single]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : AddCommGroup \u03b3\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_sub : \u2200 (i : \u03b9) (b\u2081 b\u2082 : \u03b2 i), h i (b\u2081 - b\u2082) = h i b\u2081 - h i b\u2082\n\u22a2 sum (f - g) h = sum f h - sum g h\n[PROOFSTEP]\nhave := (liftAddHom (\u03b2 := \u03b2) fun a => AddMonoidHom.ofMapSub (h a) (h_sub a)).map_sub f g\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : AddCommGroup \u03b3\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_sub : \u2200 (i : \u03b9) (b\u2081 b\u2082 : \u03b2 i), h i (b\u2081 - b\u2082) = h i b\u2081 - h i b\u2082\nthis :\n \u2191(\u2191liftAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : \u2200 (b\u2081 b\u2082 : \u03b2 a), h a (b\u2081 - b\u2082) = h a b\u2081 - h a b\u2082)) (f - g) =\n \u2191(\u2191liftAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : \u2200 (b\u2081 b\u2082 : \u03b2 a), h a (b\u2081 - b\u2082) = h a b\u2081 - h a b\u2082)) f -\n \u2191(\u2191liftAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : \u2200 (b\u2081 b\u2082 : \u03b2 a), h a (b\u2081 - b\u2082) = h a b\u2081 - h a b\u2082)) g\n\u22a2 sum (f - g) h = sum f h - sum g h\n[PROOFSTEP]\nrw [liftAddHom_apply, sumAddHom_apply, sumAddHom_apply, sumAddHom_apply] at this \n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddGroup (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : AddCommGroup \u03b3\nf g : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_sub : \u2200 (i : \u03b9) (b\u2081 b\u2082 : \u03b2 i), h i (b\u2081 - b\u2082) = h i b\u2081 - h i b\u2082\nthis\u271d\u00b2 :\n \u2191(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : \u2200 (b\u2081 b\u2082 : \u03b2 a), h a (b\u2081 - b\u2082) = h a b\u2081 - h a b\u2082)) (f - g) =\n \u2191(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : \u2200 (b\u2081 b\u2082 : \u03b2 a), h a (b\u2081 - b\u2082) = h a b\u2081 - h a b\u2082)) f -\n \u2191(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : \u2200 (b\u2081 b\u2082 : \u03b2 a), h a (b\u2081 - b\u2082) = h a b\u2081 - h a b\u2082)) g\nthis\u271d\u00b9 :\n (sum (f - g) fun x => \u2191(AddMonoidHom.ofMapSub (h x) (_ : \u2200 (b\u2081 b\u2082 : \u03b2 x), h x (b\u2081 - b\u2082) = h x b\u2081 - h x b\u2082))) =\n \u2191(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : \u2200 (b\u2081 b\u2082 : \u03b2 a), h a (b\u2081 - b\u2082) = h a b\u2081 - h a b\u2082)) f -\n \u2191(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : \u2200 (b\u2081 b\u2082 : \u03b2 a), h a (b\u2081 - b\u2082) = h a b\u2081 - h a b\u2082)) g\nthis\u271d :\n (sum (f - g) fun x => \u2191(AddMonoidHom.ofMapSub (h x) (_ : \u2200 (b\u2081 b\u2082 : \u03b2 x), h x (b\u2081 - b\u2082) = h x b\u2081 - h x b\u2082))) =\n (sum f fun x => \u2191(AddMonoidHom.ofMapSub (h x) (_ : \u2200 (b\u2081 b\u2082 : \u03b2 x), h x (b\u2081 - b\u2082) = h x b\u2081 - h x b\u2082))) -\n \u2191(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : \u2200 (b\u2081 b\u2082 : \u03b2 a), h a (b\u2081 - b\u2082) = h a b\u2081 - h a b\u2082)) g\nthis :\n (sum (f - g) fun x => \u2191(AddMonoidHom.ofMapSub (h x) (_ : \u2200 (b\u2081 b\u2082 : \u03b2 x), h x (b\u2081 - b\u2082) = h x b\u2081 - h x b\u2082))) =\n (sum f fun x => \u2191(AddMonoidHom.ofMapSub (h x) (_ : \u2200 (b\u2081 b\u2082 : \u03b2 x), h x (b\u2081 - b\u2082) = h x b\u2081 - h x b\u2082))) -\n sum g fun x => \u2191(AddMonoidHom.ofMapSub (h x) (_ : \u2200 (b\u2081 b\u2082 : \u03b2 x), h x (b\u2081 - b\u2082) = h x b\u2081 - h x b\u2082))\n\u22a2 sum (f - g) h = sum f h - sum g h\n[PROOFSTEP]\nexact this\n[GOAL]\n\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b3 : Type w\n\u03b1 : Type x\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\ns : Finset \u03b1\ng : \u03b1 \u2192 \u03a0\u2080 (i : \u03b9), \u03b2 i\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_zero : \u2200 (i : \u03b9), h i 0 = 1\nh_add : \u2200 (i : \u03b9) (b\u2081 b\u2082 : \u03b2 i), h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082\n\u22a2 \u220f i in s, prod (g i) h = prod (\u2211 i in s, g i) h\n[PROOFSTEP]\nclassical exact\n Finset.induction_on s (by simp [prod_zero_index])\n (by simp (config := { contextual := true }) [prod_add_index, h_zero, h_add])\n[GOAL]\n\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b3 : Type w\n\u03b1 : Type x\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\ns : Finset \u03b1\ng : \u03b1 \u2192 \u03a0\u2080 (i : \u03b9), \u03b2 i\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_zero : \u2200 (i : \u03b9), h i 0 = 1\nh_add : \u2200 (i : \u03b9) (b\u2081 b\u2082 : \u03b2 i), h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082\n\u22a2 \u220f i in s, prod (g i) h = prod (\u2211 i in s, g i) h\n[PROOFSTEP]\nexact\n Finset.induction_on s (by simp [prod_zero_index])\n (by simp (config := { contextual := true }) [prod_add_index, h_zero, h_add])\n[GOAL]\n\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b3 : Type w\n\u03b1 : Type x\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\ns : Finset \u03b1\ng : \u03b1 \u2192 \u03a0\u2080 (i : \u03b9), \u03b2 i\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_zero : \u2200 (i : \u03b9), h i 0 = 1\nh_add : \u2200 (i : \u03b9) (b\u2081 b\u2082 : \u03b2 i), h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082\n\u22a2 \u220f i in \u2205, prod (g i) h = prod (\u2211 i in \u2205, g i) h\n[PROOFSTEP]\nsimp [prod_zero_index]\n[GOAL]\n\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03b3 : Type w\n\u03b1 : Type x\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CommMonoid \u03b3\ns : Finset \u03b1\ng : \u03b1 \u2192 \u03a0\u2080 (i : \u03b9), \u03b2 i\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nh_zero : \u2200 (i : \u03b9), h i 0 = 1\nh_add : \u2200 (i : \u03b9) (b\u2081 b\u2082 : \u03b2 i), h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082\n\u22a2 \u2200 \u2983a : \u03b1\u2984 {s : Finset \u03b1},\n \u00aca \u2208 s \u2192\n \u220f i in s, prod (g i) h = prod (\u2211 i in s, g i) h \u2192\n \u220f i in insert a s, prod (g i) h = prod (\u2211 i in insert a s, g i) h\n[PROOFSTEP]\nsimp (config := { contextual := true }) [prod_add_index, h_zero, h_add]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 sum f single = f\n[PROOFSTEP]\nhave := FunLike.congr_fun (liftAddHom_singleAddHom (\u03b2 := \u03b2)) f\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nthis : \u2191(\u2191liftAddHom (singleAddHom \u03b2)) f = \u2191(AddMonoidHom.id (\u03a0\u2080 (i : \u03b9), \u03b2 i)) f\n\u22a2 sum f single = f\n[PROOFSTEP]\nrw [liftAddHom_apply, sumAddHom_apply] at this \n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nthis\u271d : \u2191(sumAddHom (singleAddHom \u03b2)) f = \u2191(AddMonoidHom.id (\u03a0\u2080 (i : \u03b9), \u03b2 i)) f\nthis : (sum f fun x => \u2191(singleAddHom \u03b2 x)) = \u2191(AddMonoidHom.id (\u03a0\u2080 (i : \u03b9), \u03b2 i)) f\n\u22a2 sum f single = f\n[PROOFSTEP]\nexact this\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : CommMonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), \u03b2 i\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nhp : \u2200 (x : \u03b9), x \u2208 support v \u2192 p x\n\u22a2 \u2200 (a : Subtype p) (ha : a \u2208 support (subtypeDomain p v)), (fun p_1 x => \u2191p_1) a ha \u2208 support v\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : CommMonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), \u03b2 i\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nhp : \u2200 (x : \u03b9), x \u2208 support v \u2192 p x\n\u22a2 \u2200 (a : Subtype p) (ha : a \u2208 support (subtypeDomain p v)),\n (fun i b => h (\u2191i) b) a (\u2191(subtypeDomain p v) a) = h ((fun p_1 x => \u2191p_1) a ha) (\u2191v ((fun p_1 x => \u2191p_1) a ha))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : CommMonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), \u03b2 i\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nhp : \u2200 (x : \u03b9), x \u2208 support v \u2192 p x\nx\u271d\u00b9 x\u271d : Subtype p\na\u2080 : \u03b9\nha\u2080 : p a\u2080\na\u2081 : \u03b9\nha\u2081 : p a\u2081\n\u22a2 \u2200 (ha\u2081_1 : { val := a\u2080, property := ha\u2080 } \u2208 support (subtypeDomain p v))\n (ha\u2082 : { val := a\u2081, property := ha\u2081 } \u2208 support (subtypeDomain p v)),\n (fun p_1 x => \u2191p_1) { val := a\u2080, property := ha\u2080 } ha\u2081_1 = (fun p_1 x => \u2191p_1) { val := a\u2081, property := ha\u2081 } ha\u2082 \u2192\n { val := a\u2080, property := ha\u2080 } = { val := a\u2081, property := ha\u2081 }\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b9 : CommMonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), \u03b2 i\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nh : (i : \u03b9) \u2192 \u03b2 i \u2192 \u03b3\nhp : \u2200 (x : \u03b9), x \u2208 support v \u2192 p x\ni : \u03b9\nhi : i \u2208 support v\n\u22a2 { val := i, property := (_ : p i) } \u2208 support (subtypeDomain p v)\n[PROOFSTEP]\nsimpa using hi\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\nhf' : \u2200 (i : \u03b9) (x y : \u03b2\u2081 i), f i (x + y) = f i x + f i y\ng\u2081 g\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\n\u22a2 mapRange f hf (g\u2081 + g\u2082) = mapRange f hf g\u2081 + mapRange f hf g\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i\nhf : \u2200 (i : \u03b9), f i 0 = 0\nhf' : \u2200 (i : \u03b9) (x y : \u03b2\u2081 i), f i (x + y) = f i x + f i y\ng\u2081 g\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ni\u271d : \u03b9\n\u22a2 \u2191(mapRange f hf (g\u2081 + g\u2082)) i\u271d = \u2191(mapRange f hf g\u2081 + mapRange f hf g\u2082) i\u271d\n[PROOFSTEP]\nsimp only [mapRange_apply f, coe_add, Pi.add_apply, hf']\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192+ \u03b2\u2082 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b2\u2081 i\n\u22a2 (addMonoidHom fun i => AddMonoidHom.comp (f i) (f\u2082 i)) = AddMonoidHom.comp (addMonoidHom f) (addMonoidHom f\u2082)\n[PROOFSTEP]\nrefine' AddMonoidHom.ext <| mapRange_comp (fun i x => f i x) (fun i x => f\u2082 i x) _ _ _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192+ \u03b2\u2082 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b2\u2081 i\n\u22a2 \u2200 (i : \u03b9), (fun i x => \u2191(f i) x) i 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192+ \u03b2\u2082 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b2\u2081 i\ni\u271d : \u03b9\n\u22a2 (fun i x => \u2191(f i) x) i\u271d 0 = 0\n[PROOFSTEP]\napply map_zero\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192+ \u03b2\u2082 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b2\u2081 i\n\u22a2 \u2200 (i : \u03b9), (fun i x => \u2191(f\u2082 i) x) i 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192+ \u03b2\u2082 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b2\u2081 i\ni\u271d : \u03b9\n\u22a2 (fun i x => \u2191(f\u2082 i) x) i\u271d 0 = 0\n[PROOFSTEP]\napply map_zero\n[GOAL]\ncase refine'_3\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192+ \u03b2\u2082 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b2\u2081 i\n\u22a2 \u2200 (i : \u03b9), ((fun i x => \u2191(f i) x) i \u2218 (fun i x => \u2191(f\u2082 i) x) i) 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192+ \u03b2\u2082 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b2\u2081 i\ni\u271d : \u03b9\n\u22a2 ((fun i x => \u2191(f i) x) i\u271d \u2218 (fun i x => \u2191(f\u2082 i) x) i\u271d) 0 = 0\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_3\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192+ \u03b2\u2082 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b2\u2081 i\ni\u271d : \u03b9\n\u22a2 \u2191(f i\u271d) (\u2191(f\u2082 i\u271d) 0) = 0\n[PROOFSTEP]\nsimp only [map_zero]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\ne : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\nsrc\u271d : (\u03a0\u2080 (i : \u03b9), \u03b2\u2081 i) \u2192+ \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i := addMonoidHom fun i => AddEquiv.toAddMonoidHom (e i)\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\n\u22a2 mapRange (fun i x => \u2191(AddEquiv.symm (e i)) x) (_ : \u2200 (i : \u03b9), \u2191(AddEquiv.symm (e i)) 0 = 0)\n (mapRange (fun i x => \u2191(e i) x) (_ : \u2200 (i : \u03b9), \u2191(e i) 0 = 0) x) =\n x\n[PROOFSTEP]\nrw [\u2190 mapRange_comp]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\ne : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\nsrc\u271d : (\u03a0\u2080 (i : \u03b9), \u03b2\u2081 i) \u2192+ \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i := addMonoidHom fun i => AddEquiv.toAddMonoidHom (e i)\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\n\u22a2 mapRange (fun i => (fun x => \u2191(AddEquiv.symm (e i)) x) \u2218 fun x => \u2191(e i) x) ?h x = x\n[PROOFSTEP]\nsimp_rw [AddEquiv.symm_comp_self]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\ne : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\nsrc\u271d : (\u03a0\u2080 (i : \u03b9), \u03b2\u2081 i) \u2192+ \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i := addMonoidHom fun i => AddEquiv.toAddMonoidHom (e i)\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\n\u22a2 mapRange (fun i => id) (_ : \u2200 (i : \u03b9), (fun i => id) i 0 = 0) x = x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\ne : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\nsrc\u271d : (\u03a0\u2080 (i : \u03b9), \u03b2\u2081 i) \u2192+ \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i := addMonoidHom fun i => AddEquiv.toAddMonoidHom (e i)\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\n\u22a2 \u2200 (i : \u03b9), ((fun x => \u2191(AddEquiv.symm (e i)) x) \u2218 fun x => \u2191(e i) x) 0 = 0\n[PROOFSTEP]\nsimp_rw [AddEquiv.symm_comp_self]\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\ne : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\nsrc\u271d : (\u03a0\u2080 (i : \u03b9), \u03b2\u2081 i) \u2192+ \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i := addMonoidHom fun i => AddEquiv.toAddMonoidHom (e i)\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\n\u22a2 \u2200 (i : \u03b9), id 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\ne : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\nsrc\u271d : (\u03a0\u2080 (i : \u03b9), \u03b2\u2081 i) \u2192+ \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i := addMonoidHom fun i => AddEquiv.toAddMonoidHom (e i)\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\n\u22a2 mapRange (fun i x => \u2191(e i) x) (_ : \u2200 (i : \u03b9), \u2191(e i) 0 = 0)\n (mapRange (fun i x => \u2191(AddEquiv.symm (e i)) x) (_ : \u2200 (i : \u03b9), \u2191(AddEquiv.symm (e i)) 0 = 0) x) =\n x\n[PROOFSTEP]\nrw [\u2190 mapRange_comp]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\ne : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\nsrc\u271d : (\u03a0\u2080 (i : \u03b9), \u03b2\u2081 i) \u2192+ \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i := addMonoidHom fun i => AddEquiv.toAddMonoidHom (e i)\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\n\u22a2 mapRange (fun i => (fun x => \u2191(e i) x) \u2218 fun x => \u2191(AddEquiv.symm (e i)) x) ?h x = x\n[PROOFSTEP]\nsimp_rw [AddEquiv.self_comp_symm]\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\ne : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\nsrc\u271d : (\u03a0\u2080 (i : \u03b9), \u03b2\u2081 i) \u2192+ \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i := addMonoidHom fun i => AddEquiv.toAddMonoidHom (e i)\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\n\u22a2 mapRange (fun i => id) (_ : \u2200 (i : \u03b9), (fun i => id) i 0 = 0) x = x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\ne : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\nsrc\u271d : (\u03a0\u2080 (i : \u03b9), \u03b2\u2081 i) \u2192+ \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i := addMonoidHom fun i => AddEquiv.toAddMonoidHom (e i)\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\n\u22a2 \u2200 (i : \u03b9), ((fun x => \u2191(e i) x) \u2218 fun x => \u2191(AddEquiv.symm (e i)) x) 0 = 0\n[PROOFSTEP]\nsimp_rw [AddEquiv.self_comp_symm]\n[GOAL]\ncase h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\ne : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\nsrc\u271d : (\u03a0\u2080 (i : \u03b9), \u03b2\u2081 i) \u2192+ \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i := addMonoidHom fun i => AddEquiv.toAddMonoidHom (e i)\nx : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\n\u22a2 \u2200 (i : \u03b9), id 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2 i \u2243+ \u03b2\u2081 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\n\u22a2 (addEquiv fun i => AddEquiv.trans (f i) (f\u2082 i)) = AddEquiv.trans (addEquiv f) (addEquiv f\u2082)\n[PROOFSTEP]\nrefine' AddEquiv.ext <| mapRange_comp (fun i x => f\u2082 i x) (fun i x => f i x) _ _ _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2 i \u2243+ \u03b2\u2081 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\n\u22a2 \u2200 (i : \u03b9), (fun i x => \u2191(f\u2082 i) x) i 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2 i \u2243+ \u03b2\u2081 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\ni\u271d : \u03b9\n\u22a2 (fun i x => \u2191(f\u2082 i) x) i\u271d 0 = 0\n[PROOFSTEP]\napply map_zero\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2 i \u2243+ \u03b2\u2081 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\n\u22a2 \u2200 (i : \u03b9), (fun i x => \u2191(f i) x) i 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2 i \u2243+ \u03b2\u2081 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\ni\u271d : \u03b9\n\u22a2 (fun i x => \u2191(f i) x) i\u271d 0 = 0\n[PROOFSTEP]\napply map_zero\n[GOAL]\ncase refine'_3\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2 i \u2243+ \u03b2\u2081 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\n\u22a2 \u2200 (i : \u03b9), ((fun i x => \u2191(f\u2082 i) x) i \u2218 (fun i x => \u2191(f i) x) i) 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2 i \u2243+ \u03b2\u2081 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\ni\u271d : \u03b9\n\u22a2 ((fun i x => \u2191(f\u2082 i) x) i\u271d \u2218 (fun i x => \u2191(f i) x) i\u271d) 0 = 0\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_3\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2081 i)\ninst\u271d : (i : \u03b9) \u2192 AddZeroClass (\u03b2\u2082 i)\nf : (i : \u03b9) \u2192 \u03b2 i \u2243+ \u03b2\u2081 i\nf\u2082 : (i : \u03b9) \u2192 \u03b2\u2081 i \u2243+ \u03b2\u2082 i\ni\u271d : \u03b9\n\u22a2 \u2191(f\u2082 i\u271d) (\u2191(f i\u271d) 0) = 0\n[PROOFSTEP]\nsimp only [map_zero]\n[GOAL]\n\u03b9\u271d : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9\u271d \u2192 Type v\n\u03b2\u2081 : \u03b9\u271d \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9\u271d \u2192 Type v\u2082\n\u03b9 : Type ?u.1536688\n\u03c0 : \u03b9 \u2192 Type ?u.1536671\ninst\u271d\u00b2 : \u2200 (i : \u03b9), Nontrivial (\u03c0 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03c0 i)\ninst\u271d : Infinite \u03b9\n\u22a2 Infinite (\u03a0\u2080 (i : \u03b9), \u03c0 i)\n[PROOFSTEP]\nletI := Classical.decEq \u03b9\n[GOAL]\n\u03b9\u271d : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9\u271d \u2192 Type v\n\u03b2\u2081 : \u03b9\u271d \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9\u271d \u2192 Type v\u2082\n\u03b9 : Type ?u.1536688\n\u03c0 : \u03b9 \u2192 Type ?u.1536671\ninst\u271d\u00b2 : \u2200 (i : \u03b9), Nontrivial (\u03c0 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03c0 i)\ninst\u271d : Infinite \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 Infinite (\u03a0\u2080 (i : \u03b9), \u03c0 i)\n[PROOFSTEP]\nchoose m hm using fun i => exists_ne (0 : \u03c0 i)\n[GOAL]\n\u03b9\u271d : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9\u271d \u2192 Type v\n\u03b2\u2081 : \u03b9\u271d \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9\u271d \u2192 Type v\u2082\n\u03b9 : Type ?u.1536688\n\u03c0 : \u03b9 \u2192 Type ?u.1536671\ninst\u271d\u00b2 : \u2200 (i : \u03b9), Nontrivial (\u03c0 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03c0 i)\ninst\u271d : Infinite \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\nm : (i : \u03b9) \u2192 \u03c0 i\nhm : \u2200 (i : \u03b9), m i \u2260 0\n\u22a2 Infinite (\u03a0\u2080 (i : \u03b9), \u03c0 i)\n[PROOFSTEP]\nexact Infinite.of_injective _ (DFinsupp.single_left_injective hm)\n", "meta": {"mathlib_filename": "Mathlib.Data.DFinsupp.Basic", "llama_tokens": 151119, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702880639791, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.42136933876782473}} {"text": "[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nb : \u03b2\nf : \u03b1 \u2192 \u03b2\na : Option \u03b1\n\u22a2 Option.elim' b f a = Option.elim a b f\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nb : \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 Option.elim' b f none = Option.elim none b f\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nb : \u03b2\nf : \u03b1 \u2192 \u03b2\nval\u271d : \u03b1\n\u22a2 Option.elim' b f (some val\u271d) = Option.elim (some val\u271d) b f\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\na b : \u03b1\n\u22a2 a \u2208 some b \u2194 b = a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 \u2200 (a : \u03b1), a \u2208 none \u2192 p a\n[PROOFSTEP]\nsimp [false_imp_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : \u2203 a, a \u2208 none \u2227 p a\na : \u03b1\nh : a \u2208 none\nright\u271d : p a\n\u22a2 False\n[PROOFSTEP]\ncases h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : \u03b1\no : Option \u03b1\n\u22a2 a \u2208 toList o \u2194 a \u2208 o\n[PROOFSTEP]\ncases o\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : \u03b1\n\u22a2 a \u2208 toList none \u2194 a \u2208 none\n[PROOFSTEP]\nsimp [toList, eq_comm]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na val\u271d : \u03b1\n\u22a2 a \u2208 toList (some val\u271d) \u2194 a \u2208 some val\u271d\n[PROOFSTEP]\nsimp [toList, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsCommutative \u03b1 f\na b : Option \u03b1\n\u22a2 liftOrGet f a b = liftOrGet f b a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsCommutative \u03b1 f\nb : Option \u03b1\n\u22a2 liftOrGet f none b = liftOrGet f b none\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsCommutative \u03b1 f\nb : Option \u03b1\nval\u271d : \u03b1\n\u22a2 liftOrGet f (some val\u271d) b = liftOrGet f b (some val\u271d)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsCommutative \u03b1 f\n\u22a2 liftOrGet f none none = liftOrGet f none none\n[PROOFSTEP]\nsimp [liftOrGet, IsCommutative.comm]\n[GOAL]\ncase none.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsCommutative \u03b1 f\nval\u271d : \u03b1\n\u22a2 liftOrGet f none (some val\u271d) = liftOrGet f (some val\u271d) none\n[PROOFSTEP]\nsimp [liftOrGet, IsCommutative.comm]\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsCommutative \u03b1 f\nval\u271d : \u03b1\n\u22a2 liftOrGet f (some val\u271d) none = liftOrGet f none (some val\u271d)\n[PROOFSTEP]\nsimp [liftOrGet, IsCommutative.comm]\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsCommutative \u03b1 f\nval\u271d\u00b9 val\u271d : \u03b1\n\u22a2 liftOrGet f (some val\u271d\u00b9) (some val\u271d) = liftOrGet f (some val\u271d) (some val\u271d\u00b9)\n[PROOFSTEP]\nsimp [liftOrGet, IsCommutative.comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsAssociative \u03b1 f\na b c : Option \u03b1\n\u22a2 liftOrGet f (liftOrGet f a b) c = liftOrGet f a (liftOrGet f b c)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsAssociative \u03b1 f\nb c : Option \u03b1\n\u22a2 liftOrGet f (liftOrGet f none b) c = liftOrGet f none (liftOrGet f b c)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsAssociative \u03b1 f\nb c : Option \u03b1\nval\u271d : \u03b1\n\u22a2 liftOrGet f (liftOrGet f (some val\u271d) b) c = liftOrGet f (some val\u271d) (liftOrGet f b c)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsAssociative \u03b1 f\nc : Option \u03b1\n\u22a2 liftOrGet f (liftOrGet f none none) c = liftOrGet f none (liftOrGet f none c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase none.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsAssociative \u03b1 f\nc : Option \u03b1\nval\u271d : \u03b1\n\u22a2 liftOrGet f (liftOrGet f none (some val\u271d)) c = liftOrGet f none (liftOrGet f (some val\u271d) c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsAssociative \u03b1 f\nc : Option \u03b1\nval\u271d : \u03b1\n\u22a2 liftOrGet f (liftOrGet f (some val\u271d) none) c = liftOrGet f (some val\u271d) (liftOrGet f none c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsAssociative \u03b1 f\nc : Option \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\n\u22a2 liftOrGet f (liftOrGet f (some val\u271d\u00b9) (some val\u271d)) c = liftOrGet f (some val\u271d\u00b9) (liftOrGet f (some val\u271d) c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase none.none.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsAssociative \u03b1 f\n\u22a2 liftOrGet f (liftOrGet f none none) none = liftOrGet f none (liftOrGet f none none)\n[PROOFSTEP]\nsimp [liftOrGet, IsAssociative.assoc]\n[GOAL]\ncase none.none.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsAssociative \u03b1 f\nval\u271d : \u03b1\n\u22a2 liftOrGet f (liftOrGet f none none) (some val\u271d) = liftOrGet f none (liftOrGet f none (some val\u271d))\n[PROOFSTEP]\nsimp [liftOrGet, IsAssociative.assoc]\n[GOAL]\ncase none.some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsAssociative \u03b1 f\nval\u271d : \u03b1\n\u22a2 liftOrGet f (liftOrGet f none (some val\u271d)) none = liftOrGet f none (liftOrGet f (some val\u271d) none)\n[PROOFSTEP]\nsimp [liftOrGet, IsAssociative.assoc]\n[GOAL]\ncase none.some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsAssociative \u03b1 f\nval\u271d\u00b9 val\u271d : \u03b1\n\u22a2 liftOrGet f (liftOrGet f none (some val\u271d\u00b9)) (some val\u271d) = liftOrGet f none (liftOrGet f (some val\u271d\u00b9) (some val\u271d))\n[PROOFSTEP]\nsimp [liftOrGet, IsAssociative.assoc]\n[GOAL]\ncase some.none.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsAssociative \u03b1 f\nval\u271d : \u03b1\n\u22a2 liftOrGet f (liftOrGet f (some val\u271d) none) none = liftOrGet f (some val\u271d) (liftOrGet f none none)\n[PROOFSTEP]\nsimp [liftOrGet, IsAssociative.assoc]\n[GOAL]\ncase some.none.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsAssociative \u03b1 f\nval\u271d\u00b9 val\u271d : \u03b1\n\u22a2 liftOrGet f (liftOrGet f (some val\u271d\u00b9) none) (some val\u271d) = liftOrGet f (some val\u271d\u00b9) (liftOrGet f none (some val\u271d))\n[PROOFSTEP]\nsimp [liftOrGet, IsAssociative.assoc]\n[GOAL]\ncase some.some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsAssociative \u03b1 f\nval\u271d\u00b9 val\u271d : \u03b1\n\u22a2 liftOrGet f (liftOrGet f (some val\u271d\u00b9) (some val\u271d)) none = liftOrGet f (some val\u271d\u00b9) (liftOrGet f (some val\u271d) none)\n[PROOFSTEP]\nsimp [liftOrGet, IsAssociative.assoc]\n[GOAL]\ncase some.some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsAssociative \u03b1 f\nval\u271d\u00b2 val\u271d\u00b9 val\u271d : \u03b1\n\u22a2 liftOrGet f (liftOrGet f (some val\u271d\u00b2) (some val\u271d\u00b9)) (some val\u271d) =\n liftOrGet f (some val\u271d\u00b2) (liftOrGet f (some val\u271d\u00b9) (some val\u271d))\n[PROOFSTEP]\nsimp [liftOrGet, IsAssociative.assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsIdempotent \u03b1 f\na : Option \u03b1\n\u22a2 liftOrGet f a a = a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsIdempotent \u03b1 f\n\u22a2 liftOrGet f none none = none\n[PROOFSTEP]\nsimp [liftOrGet, IsIdempotent.idempotent]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d : IsIdempotent \u03b1 f\nval\u271d : \u03b1\n\u22a2 liftOrGet f (some val\u271d) (some val\u271d) = some val\u271d\n[PROOFSTEP]\nsimp [liftOrGet, IsIdempotent.idempotent]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\na : Option \u03b1\n\u22a2 liftOrGet f none a = a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n\u22a2 liftOrGet f none none = none\n[PROOFSTEP]\nsimp [liftOrGet]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nval\u271d : \u03b1\n\u22a2 liftOrGet f none (some val\u271d) = some val\u271d\n[PROOFSTEP]\nsimp [liftOrGet]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\na : Option \u03b1\n\u22a2 liftOrGet f a none = a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n\u22a2 liftOrGet f none none = none\n[PROOFSTEP]\nsimp [liftOrGet]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nval\u271d : \u03b1\n\u22a2 liftOrGet f (some val\u271d) none = some val\u271d\n[PROOFSTEP]\nsimp [liftOrGet]\n", "meta": {"mathlib_filename": "Mathlib.Data.Option.Defs", "llama_tokens": 3607, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.72487026428967, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4213693249477454}} {"text": "[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\n\u22a2 x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) \u2264 4 ^ x\n[PROOFSTEP]\nlet f : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\n\u22a2 x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) \u2264 4 ^ x\n[PROOFSTEP]\nhave hf' : \u2200 x, 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>\n div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\n\u22a2 x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) \u2264 4 ^ x\n[PROOFSTEP]\nhave hf : \u2200 x, 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) :=\n by\n intro x h5\n have h6 := mul_pos (zero_lt_two' \u211d) h5\n have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))\n rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6,\n log_rpow zero_lt_four, \u2190 mul_div_right_comm, \u2190 mul_div, mul_comm x]\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\n\u22a2 \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\n[PROOFSTEP]\nintro x h5\n[GOAL]\nx\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nx : \u211d\nh5 : 0 < x\n\u22a2 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\n[PROOFSTEP]\nhave h6 := mul_pos (zero_lt_two' \u211d) h5\n[GOAL]\nx\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nx : \u211d\nh5 : 0 < x\nh6 : 0 < 2 * x\n\u22a2 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\n[PROOFSTEP]\nhave h7 := rpow_pos_of_pos h6 (sqrt (2 * x))\n[GOAL]\nx\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nx : \u211d\nh5 : 0 < x\nh6 : 0 < 2 * x\nh7 : 0 < (2 * x) ^ sqrt (2 * x)\n\u22a2 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\n[PROOFSTEP]\nrw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', log_rpow h6,\n log_rpow zero_lt_four, \u2190 mul_div_right_comm, \u2190 mul_div, mul_comm x]\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\n\u22a2 x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) \u2264 4 ^ x\n[PROOFSTEP]\nhave h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\n\u22a2 0 < 512\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) \u2264 4 ^ x\n[PROOFSTEP]\nrw [\u2190 div_le_one (rpow_pos_of_pos four_pos x), \u2190 div_div_eq_mul_div, \u2190 rpow_sub four_pos, \u2190 mul_div 2 x,\n mul_div_left_comm, \u2190 mul_one_sub, (by norm_num1 : (1 : \u211d) - 2 / 3 = 1 / 3), mul_one_div, \u2190 log_nonpos_iff (hf' x h5),\n \u2190 hf x h5]\n -- porting note: the proof was rewritten, because it was too slow\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 1 - 2 / 3 = 1 / 3\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 f x \u2264 0\n[PROOFSTEP]\nhave h : ConcaveOn \u211d (Set.Ioi 0.5) f := by\n apply ConcaveOn.sub\n apply ConcaveOn.add\n exact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)\n convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : \u211d) \u2022 LinearMap.id))) using 1\n \u00b7 ext x\n simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul]\n rw [\u2190 mul_lt_mul_left (two_pos)]\n norm_num1\n rfl\n apply ConvexOn.smul\n refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)\n exact convexOn_id (convex_Ioi (0.5 : \u211d))\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConcaveOn \u211d (Set.Ioi 0.5) f\n[PROOFSTEP]\napply ConcaveOn.sub\n[GOAL]\ncase hf\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConcaveOn \u211d (Set.Ioi 0.5) fun x => log x + sqrt (2 * x) * log (2 * x)\ncase hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => log 4 / 3 * x\n[PROOFSTEP]\napply ConcaveOn.add\n[GOAL]\ncase hf.hf\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConcaveOn \u211d (Set.Ioi 0.5) fun x => log x\ncase hf.hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConcaveOn \u211d (Set.Ioi 0.5) fun x => sqrt (2 * x) * log (2 * x)\ncase hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => log 4 / 3 * x\n[PROOFSTEP]\nexact strictConcaveOn_log_Ioi.concaveOn.subset (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 0 \u2264 0.5\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase hf.hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConcaveOn \u211d (Set.Ioi 0.5) fun x => sqrt (2 * x) * log (2 * x)\ncase hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => log 4 / 3 * x\n[PROOFSTEP]\nconvert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap ((2 : \u211d) \u2022 LinearMap.id))) using 1\n[GOAL]\ncase h.e'_9\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 Set.Ioi 0.5 = \u2191(2 \u2022 LinearMap.id) \u207b\u00b9' Set.Ioi 1\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_9.h\nx\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\u271d\nx : \u211d\n\u22a2 x \u2208 Set.Ioi 0.5 \u2194 x \u2208 \u2191(2 \u2022 LinearMap.id) \u207b\u00b9' Set.Ioi 1\n[PROOFSTEP]\nsimp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply, LinearMap.id_coe, id_eq, smul_eq_mul]\n[GOAL]\ncase h.e'_9.h\nx\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\u271d\nx : \u211d\n\u22a2 OfScientific.ofScientific 5 true 1 < x \u2194 1 < 2 * x\n[PROOFSTEP]\nrw [\u2190 mul_lt_mul_left (two_pos)]\n[GOAL]\ncase h.e'_9.h\nx\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\u271d\nx : \u211d\n\u22a2 2 * OfScientific.ofScientific 5 true 1 < 2 * x \u2194 1 < 2 * x\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase h.e'_9.h\nx\u271d : \u211d\nn_large : 512 \u2264 x\u271d\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\u271d\nx : \u211d\n\u22a2 1 < 2 * x \u2194 1 < 2 * x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase hg\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => log 4 / 3 * x\n[PROOFSTEP]\napply ConvexOn.smul\n[GOAL]\ncase hg.hc\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 0 \u2264 log 4 / 3\ncase hg.hf\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => x\n[PROOFSTEP]\nrefine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 1 \u2264 4\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 0 \u2264 3\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase hg.hf\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\n\u22a2 ConvexOn \u211d (Set.Ioi 0.5) fun x => x\n[PROOFSTEP]\nexact convexOn_id (convex_Ioi (0.5 : \u211d))\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 f x \u2264 0\n[PROOFSTEP]\nsuffices \u2203 x1 x2, 0.5 < x1 \u2227 x1 < x2 \u2227 x2 \u2264 x \u2227 0 \u2264 f x1 \u2227 f x2 \u2264 0\n by\n obtain \u27e8x1, x2, h1, h2, h0, h3, h4\u27e9 := this\n exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : \u2203 x1 x2, 0.5 < x1 \u2227 x1 < x2 \u2227 x2 \u2264 x \u2227 0 \u2264 f x1 \u2227 f x2 \u2264 0\n\u22a2 f x \u2264 0\n[PROOFSTEP]\nobtain \u27e8x1, x2, h1, h2, h0, h3, h4\u27e9 := this\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nx1 x2 : \u211d\nh1 : 0.5 < x1\nh2 : x1 < x2\nh0 : x2 \u2264 x\nh3 : 0 \u2264 f x1\nh4 : f x2 \u2264 0\n\u22a2 f x \u2264 0\n[PROOFSTEP]\nexact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 \u2203 x1 x2, 0.5 < x1 \u2227 x1 < x2 \u2227 x2 \u2264 x \u2227 0 \u2264 f x1 \u2227 f x2 \u2264 0\n[PROOFSTEP]\nrefine' \u27e818, 512, by norm_num1, by norm_num1, n_large, _, _\u27e9\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 0.5 < 18\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 18 < 512\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 0 \u2264 f 18\n[PROOFSTEP]\nhave : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 0 < 6\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 6 * 6 = 2 * 18\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 \u2264 f 18\n[PROOFSTEP]\nrw [hf, log_nonneg_iff, this]\n[GOAL]\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 1 \u2264 18 * (2 * 18) ^ 6 / 4 ^ (18 / 3)\n[PROOFSTEP]\nrw [one_le_div]\n[GOAL]\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 4 ^ (18 / 3) \u2264 18 * (2 * 18) ^ 6\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4 ^ (18 / 3)\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 4 ^ 6 \u2264 18 * 36 ^ 6\n[PROOFSTEP]\napply le_trans _ (le_mul_of_one_le_left _ _)\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 4 ^ 6 \u2264 36 ^ 6\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 \u2264 36 ^ 6\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 1 \u2264 18\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 4 ^ 6 \u2264 36 ^ 6\n[PROOFSTEP]\napply Real.rpow_le_rpow\n[GOAL]\ncase h\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 \u2264 4\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase h\u2081\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 4 \u2264 36\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase h\u2082\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 \u2264 6\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 \u2264 36 ^ 6\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4 ^ 6\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18\n[PROOFSTEP]\napply rpow_nonneg_of_nonneg\n[GOAL]\ncase hx\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 \u2264 36\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4 ^ 6\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4 ^ 6\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18\n[PROOFSTEP]\napply rpow_pos_of_pos\n[GOAL]\ncase refine'_1.hx\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 4\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18 * (2 * 18) ^ sqrt (2 * 18) / 4 ^ (18 / 3)\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18\n[PROOFSTEP]\napply hf' 18\n[GOAL]\ncase refine'_1\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_1.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 18) = 6\n\u22a2 0 < 18\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 f 512 \u2264 0\n[PROOFSTEP]\nhave : sqrt (2 * 512) = 32 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 0 < 32\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\n\u22a2 32 * 32 = 2 * 512\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 f 512 \u2264 0\n[PROOFSTEP]\nrw [hf, log_nonpos_iff (hf' _ _), this, div_le_one]\n[GOAL]\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 512 * (2 * 512) ^ 32 \u2264 4 ^ (512 / 3)\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 512\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2.a\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 512\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 512 * 1024 ^ 32 \u2264 4 ^ (512 / 3)\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)\n[PROOFSTEP]\nhave : (512 : \u211d) = 2 ^ (9 : \u2115)\n[GOAL]\ncase this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 512 = 2 ^ \u21919\n[PROOFSTEP]\nrw [rpow_nat_cast 2 9]\n[GOAL]\ncase this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 512 = 2 ^ 9\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d : sqrt (2 * 512) = 32\nthis : 512 = 2 ^ \u21919\n\u22a2 512 * 1024 ^ 32 \u2264 4 ^ (512 / 3)\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)\n[PROOFSTEP]\nconv_lhs => rw [this]\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d : sqrt (2 * 512) = 32\nthis : 512 = 2 ^ \u21919\n| 512 * 1024 ^ 32\n[PROOFSTEP]\nrw [this]\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d : sqrt (2 * 512) = 32\nthis : 512 = 2 ^ \u21919\n| 512 * 1024 ^ 32\n[PROOFSTEP]\nrw [this]\n[GOAL]\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d : sqrt (2 * 512) = 32\nthis : 512 = 2 ^ \u21919\n| 512 * 1024 ^ 32\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d : sqrt (2 * 512) = 32\nthis : 512 = 2 ^ \u21919\n\u22a2 2 ^ \u21919 * 1024 ^ 32 \u2264 4 ^ (512 / 3)\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)\n[PROOFSTEP]\nhave : (1024 : \u211d) = 2 ^ (10 : \u2115)\n[GOAL]\ncase this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d : sqrt (2 * 512) = 32\nthis : 512 = 2 ^ \u21919\n\u22a2 1024 = 2 ^ \u219110\n[PROOFSTEP]\nrw [rpow_nat_cast 2 10]\n[GOAL]\ncase this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d : sqrt (2 * 512) = 32\nthis : 512 = 2 ^ \u21919\n\u22a2 1024 = 2 ^ 10\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b9 : sqrt (2 * 512) = 32\nthis\u271d : 512 = 2 ^ \u21919\nthis : 1024 = 2 ^ \u219110\n\u22a2 2 ^ \u21919 * 1024 ^ 32 \u2264 4 ^ (512 / 3)\n[PROOFSTEP]\nrw [this, \u2190 rpow_mul, \u2190 rpow_add]\n[GOAL]\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b9 : sqrt (2 * 512) = 32\nthis\u271d : 512 = 2 ^ \u21919\nthis : 1024 = 2 ^ \u219110\n\u22a2 2 ^ (\u21919 + \u219110 * 32) \u2264 4 ^ (512 / 3)\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2.hx\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b9 : sqrt (2 * 512) = 32\nthis\u271d : 512 = 2 ^ \u21919\nthis : 1024 = 2 ^ \u219110\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2.hx\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b9 : sqrt (2 * 512) = 32\nthis\u271d : 512 = 2 ^ \u21919\nthis : 1024 = 2 ^ \u219110\n\u22a2 0 \u2264 2\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b9 : sqrt (2 * 512) = 32\nthis\u271d : 512 = 2 ^ \u21919\nthis : 1024 = 2 ^ \u219110\n\u22a2 2 ^ 329 \u2264 4 ^ (512 / 3)\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)\n[PROOFSTEP]\nhave : (4 : \u211d) = 2 ^ (2 : \u2115)\n[GOAL]\ncase this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b9 : sqrt (2 * 512) = 32\nthis\u271d : 512 = 2 ^ \u21919\nthis : 1024 = 2 ^ \u219110\n\u22a2 4 = 2 ^ \u21912\n[PROOFSTEP]\nrw [rpow_nat_cast 2 2]\n[GOAL]\ncase this\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b9 : sqrt (2 * 512) = 32\nthis\u271d : 512 = 2 ^ \u21919\nthis : 1024 = 2 ^ \u219110\n\u22a2 4 = 2 ^ 2\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b2 : sqrt (2 * 512) = 32\nthis\u271d\u00b9 : 512 = 2 ^ \u21919\nthis\u271d : 1024 = 2 ^ \u219110\nthis : 4 = 2 ^ \u21912\n\u22a2 2 ^ 329 \u2264 4 ^ (512 / 3)\n[PROOFSTEP]\nrw [this, \u2190 rpow_mul]\n[GOAL]\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b2 : sqrt (2 * 512) = 32\nthis\u271d\u00b9 : 512 = 2 ^ \u21919\nthis\u271d : 1024 = 2 ^ \u219110\nthis : 4 = 2 ^ \u21912\n\u22a2 2 ^ 329 \u2264 2 ^ (\u21912 * (512 / 3))\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2.hx\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b2 : sqrt (2 * 512) = 32\nthis\u271d\u00b9 : 512 = 2 ^ \u21919\nthis\u271d : 1024 = 2 ^ \u219110\nthis : 4 = 2 ^ \u21912\n\u22a2 0 \u2264 2\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b2 : sqrt (2 * 512) = 32\nthis\u271d\u00b9 : 512 = 2 ^ \u21919\nthis\u271d : 1024 = 2 ^ \u219110\nthis : 4 = 2 ^ \u21912\n\u22a2 2 ^ 329 \u2264 2 ^ (1024 / 3)\n[PROOFSTEP]\napply rpow_le_rpow_of_exponent_le\n[GOAL]\ncase refine'_2.hx\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b2 : sqrt (2 * 512) = 32\nthis\u271d\u00b9 : 512 = 2 ^ \u21919\nthis\u271d : 1024 = 2 ^ \u219110\nthis : 4 = 2 ^ \u21912\n\u22a2 1 \u2264 2\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2.hyz\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis\u271d\u00b2 : sqrt (2 * 512) = 32\nthis\u271d\u00b9 : 512 = 2 ^ \u21919\nthis\u271d : 1024 = 2 ^ \u219110\nthis : 4 = 2 ^ \u21912\n\u22a2 329 \u2264 1024 / 3\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2\nx : \u211d\nn_large : 512 \u2264 x\nf : \u211d \u2192 \u211d := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : \u2200 (x : \u211d), 0 < x \u2192 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)\nhf : \u2200 (x : \u211d), 0 < x \u2192 f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn \u211d (Set.Ioi 0.5) f\nthis : sqrt (2 * 512) = 32\n\u22a2 0 < 4 ^ (512 / 3)\n[PROOFSTEP]\napply rpow_pos_of_pos four_pos\n[GOAL]\nn : \u2115\nn_large : 512 \u2264 n\n\u22a2 n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) \u2264 4 ^ n\n[PROOFSTEP]\nrw [\u2190 @cast_le \u211d]\n[GOAL]\nn : \u2115\nn_large : 512 \u2264 n\n\u22a2 \u2191(n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)) \u2264 \u2191(4 ^ n)\n[PROOFSTEP]\nsimp only [cast_add, cast_one, cast_mul, cast_pow, \u2190 Real.rpow_nat_cast]\n[GOAL]\nn : \u2115\nn_large : 512 \u2264 n\n\u22a2 \u2191n * (\u21912 * \u2191n) ^ \u2191(sqrt (2 * n)) * \u21914 ^ \u2191(2 * n / 3) \u2264 \u21914 ^ \u2191n\n[PROOFSTEP]\nhave n_pos : 0 < n := (by decide : 0 < 512).trans_le n_large\n[GOAL]\nn : \u2115\nn_large : 512 \u2264 n\n\u22a2 0 < 512\n[PROOFSTEP]\ndecide\n[GOAL]\nn : \u2115\nn_large : 512 \u2264 n\nn_pos : 0 < n\n\u22a2 \u2191n * (\u21912 * \u2191n) ^ \u2191(sqrt (2 * n)) * \u21914 ^ \u2191(2 * n / 3) \u2264 \u21914 ^ \u2191n\n[PROOFSTEP]\nhave n2_pos : 1 \u2264 2 * n := mul_pos (by decide) n_pos\n[GOAL]\nn : \u2115\nn_large : 512 \u2264 n\nn_pos : 0 < n\n\u22a2 0 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\nn : \u2115\nn_large : 512 \u2264 n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\n\u22a2 \u2191n * (\u21912 * \u2191n) ^ \u2191(sqrt (2 * n)) * \u21914 ^ \u2191(2 * n / 3) \u2264 \u21914 ^ \u2191n\n[PROOFSTEP]\nrefine' _root_.trans (mul_le_mul _ _ _ _) (Bertrand.real_main_inequality (by exact_mod_cast n_large))\n[GOAL]\nn : \u2115\nn_large : 512 \u2264 n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\n\u22a2 512 \u2264 \u2191n\n[PROOFSTEP]\nexact_mod_cast n_large\n[GOAL]\ncase refine'_1\nn : \u2115\nn_large : 512 \u2264 n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\n\u22a2 \u2191n * (\u21912 * \u2191n) ^ \u2191(sqrt (2 * n)) \u2264 \u2191n * (2 * \u2191n) ^ Real.sqrt (2 * \u2191n)\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)\n[GOAL]\ncase refine'_1\nn : \u2115\nn_large : 512 \u2264 n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\n\u22a2 (\u21912 * \u2191n) ^ \u2191(sqrt (2 * n)) \u2264 (2 * \u2191n) ^ Real.sqrt (2 * \u2191n)\n[PROOFSTEP]\nrefine' Real.rpow_le_rpow_of_exponent_le (by exact_mod_cast n2_pos) _\n[GOAL]\nn : \u2115\nn_large : 512 \u2264 n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\n\u22a2 1 \u2264 \u21912 * \u2191n\n[PROOFSTEP]\nexact_mod_cast n2_pos\n[GOAL]\ncase refine'_1\nn : \u2115\nn_large : 512 \u2264 n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\n\u22a2 \u2191(sqrt (2 * n)) \u2264 Real.sqrt (2 * \u2191n)\n[PROOFSTEP]\nexact_mod_cast Real.nat_sqrt_le_real_sqrt\n[GOAL]\ncase refine'_2\nn : \u2115\nn_large : 512 \u2264 n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\n\u22a2 \u21914 ^ \u2191(2 * n / 3) \u2264 4 ^ (2 * \u2191n / 3)\n[PROOFSTEP]\nexact Real.rpow_le_rpow_of_exponent_le (by norm_num1) (cast_div_le.trans (by norm_cast))\n[GOAL]\nn : \u2115\nn_large : 512 \u2264 n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\n\u22a2 1 \u2264 \u21914\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nn : \u2115\nn_large : 512 \u2264 n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\n\u22a2 \u2191(2 * n) / \u21913 \u2264 2 * \u2191n / 3\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase refine'_3\nn : \u2115\nn_large : 512 \u2264 n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\n\u22a2 0 \u2264 \u21914 ^ \u2191(2 * n / 3)\n[PROOFSTEP]\nexact Real.rpow_nonneg_of_nonneg (by norm_num1) _\n[GOAL]\nn : \u2115\nn_large : 512 \u2264 n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\n\u22a2 0 \u2264 \u21914\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_4\nn : \u2115\nn_large : 512 \u2264 n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\n\u22a2 0 \u2264 \u2191n * (2 * \u2191n) ^ Real.sqrt (2 * \u2191n)\n[PROOFSTEP]\nrefine' mul_nonneg (Nat.cast_nonneg _) _\n[GOAL]\ncase refine'_4\nn : \u2115\nn_large : 512 \u2264 n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\n\u22a2 0 \u2264 (2 * \u2191n) ^ Real.sqrt (2 * \u2191n)\n[PROOFSTEP]\nexact Real.rpow_nonneg_of_nonneg (mul_nonneg zero_le_two (Nat.cast_nonneg _)) _\n[GOAL]\nn : \u2115\nn_large : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\n\u22a2 centralBinom n = \u220f p in Finset.range (2 * n / 3 + 1), p ^ \u2191(Nat.factorization (centralBinom n)) p\n[PROOFSTEP]\nrefine' (Eq.trans _ n.prod_pow_factorization_centralBinom).symm\n[GOAL]\nn : \u2115\nn_large : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\n\u22a2 \u220f p in Finset.range (2 * n / 3 + 1), p ^ \u2191(Nat.factorization (centralBinom n)) p =\n \u220f p in Finset.range (2 * n + 1), p ^ \u2191(Nat.factorization (centralBinom n)) p\n[PROOFSTEP]\napply Finset.prod_subset\n[GOAL]\ncase h\nn : \u2115\nn_large : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\n\u22a2 Finset.range (2 * n / 3 + 1) \u2286 Finset.range (2 * n + 1)\n[PROOFSTEP]\nexact Finset.range_subset.2 (add_le_add_right (Nat.div_le_self _ _) _)\n[GOAL]\ncase hf\nn : \u2115\nn_large : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\n\u22a2 \u2200 (x : \u2115),\n x \u2208 Finset.range (2 * n + 1) \u2192 \u00acx \u2208 Finset.range (2 * n / 3 + 1) \u2192 x ^ \u2191(Nat.factorization (centralBinom n)) x = 1\n[PROOFSTEP]\nintro x hx h2x\n[GOAL]\ncase hf\nn : \u2115\nn_large : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nx : \u2115\nhx : x \u2208 Finset.range (2 * n + 1)\nh2x : \u00acx \u2208 Finset.range (2 * n / 3 + 1)\n\u22a2 x ^ \u2191(Nat.factorization (centralBinom n)) x = 1\n[PROOFSTEP]\nrw [Finset.mem_range, lt_succ_iff] at hx h2x \n[GOAL]\ncase hf\nn : \u2115\nn_large : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nx : \u2115\nhx : x \u2264 2 * n\nh2x : \u00acx \u2264 2 * n / 3\n\u22a2 x ^ \u2191(Nat.factorization (centralBinom n)) x = 1\n[PROOFSTEP]\nrw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x \n[GOAL]\ncase hf\nn : \u2115\nn_large : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nx : \u2115\nhx : x \u2264 2 * n\nh2x : 2 * n < 3 * x\n\u22a2 x ^ \u2191(Nat.factorization (centralBinom n)) x = 1\n[PROOFSTEP]\nreplace no_prime := not_exists.mp no_prime x\n[GOAL]\ncase hf\nn : \u2115\nn_large : 2 < n\nx : \u2115\nhx : x \u2264 2 * n\nh2x : 2 * n < 3 * x\nno_prime : \u00ac(Nat.Prime x \u2227 n < x \u2227 x \u2264 2 * n)\n\u22a2 x ^ \u2191(Nat.factorization (centralBinom n)) x = 1\n[PROOFSTEP]\nrw [\u2190 and_assoc, not_and', not_and_or, not_lt] at no_prime \n[GOAL]\ncase hf\nn : \u2115\nn_large : 2 < n\nx : \u2115\nhx : x \u2264 2 * n\nh2x : 2 * n < 3 * x\nno_prime : x \u2264 2 * n \u2192 \u00acNat.Prime x \u2228 x \u2264 n\n\u22a2 x ^ \u2191(Nat.factorization (centralBinom n)) x = 1\n[PROOFSTEP]\ncases' no_prime hx with h h\n[GOAL]\ncase hf.inl\nn : \u2115\nn_large : 2 < n\nx : \u2115\nhx : x \u2264 2 * n\nh2x : 2 * n < 3 * x\nno_prime : x \u2264 2 * n \u2192 \u00acNat.Prime x \u2228 x \u2264 n\nh : \u00acNat.Prime x\n\u22a2 x ^ \u2191(Nat.factorization (centralBinom n)) x = 1\n[PROOFSTEP]\nrw [factorization_eq_zero_of_non_prime n.centralBinom h, Nat.pow_zero]\n[GOAL]\ncase hf.inr\nn : \u2115\nn_large : 2 < n\nx : \u2115\nhx : x \u2264 2 * n\nh2x : 2 * n < 3 * x\nno_prime : x \u2264 2 * n \u2192 \u00acNat.Prime x \u2228 x \u2264 n\nh : x \u2264 n\n\u22a2 x ^ \u2191(Nat.factorization (centralBinom n)) x = 1\n[PROOFSTEP]\nrw [factorization_centralBinom_of_two_mul_self_lt_three_mul n_large h h2x, Nat.pow_zero]\n[GOAL]\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\n\u22a2 centralBinom n \u2264 (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)\n[PROOFSTEP]\nhave n_pos : 0 < n := (Nat.zero_le _).trans_lt n_big\n[GOAL]\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\n\u22a2 centralBinom n \u2264 (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)\n[PROOFSTEP]\nhave n2_pos : 1 \u2264 2 * n := mul_pos (zero_lt_two' \u2115) n_pos\n[GOAL]\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\n\u22a2 centralBinom n \u2264 (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)\n[PROOFSTEP]\nlet S := (Finset.range (2 * n / 3 + 1)).filter Nat.Prime\n[GOAL]\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\n\u22a2 centralBinom n \u2264 (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)\n[PROOFSTEP]\nlet f x := x ^ n.centralBinom.factorization x\n[GOAL]\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\n\u22a2 centralBinom n \u2264 (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)\n[PROOFSTEP]\nhave : \u220f x : \u2115 in S, f x = \u220f x : \u2115 in Finset.range (2 * n / 3 + 1), f x :=\n by\n refine' Finset.prod_filter_of_ne fun p _ h => _\n contrapose! h; dsimp only\n rw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]\n[GOAL]\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\n\u22a2 \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\n[PROOFSTEP]\nrefine' Finset.prod_filter_of_ne fun p _ h => _\n[GOAL]\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\np : \u2115\nx\u271d : p \u2208 Finset.range (2 * n / 3 + 1)\nh : f p \u2260 1\n\u22a2 Nat.Prime p\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\np : \u2115\nx\u271d : p \u2208 Finset.range (2 * n / 3 + 1)\nh : \u00acNat.Prime p\n\u22a2 (fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x) p = 1\n[PROOFSTEP]\ndsimp only\n[GOAL]\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\np : \u2115\nx\u271d : p \u2208 Finset.range (2 * n / 3 + 1)\nh : \u00acNat.Prime p\n\u22a2 p ^ \u2191(Nat.factorization (centralBinom n)) p = 1\n[PROOFSTEP]\nrw [factorization_eq_zero_of_non_prime n.centralBinom h, _root_.pow_zero]\n[GOAL]\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\n\u22a2 centralBinom n \u2264 (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)\n[PROOFSTEP]\nrw [centralBinom_factorization_small n n_big no_prime, \u2190 this, \u2190\n Finset.prod_filter_mul_prod_filter_not S (\u00b7 \u2264 sqrt (2 * n))]\n[GOAL]\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\n\u22a2 (\u220f x in Finset.filter (fun x => x \u2264 sqrt (2 * n)) S, f x) * \u220f x in Finset.filter (fun x => \u00acx \u2264 sqrt (2 * n)) S, f x \u2264\n (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)\n[PROOFSTEP]\napply mul_le_mul'\n[GOAL]\ncase h\u2081\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\n\u22a2 \u220f x in Finset.filter (fun x => x \u2264 sqrt (2 * n)) S, f x \u2264 (2 * n) ^ sqrt (2 * n)\n[PROOFSTEP]\nrefine' (Finset.prod_le_prod' fun p _ => (_ : f p \u2264 2 * n)).trans _\n[GOAL]\ncase h\u2081.refine'_1\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\np : \u2115\nx\u271d : p \u2208 Finset.filter (fun x => x \u2264 sqrt (2 * n)) S\n\u22a2 f p \u2264 2 * n\n[PROOFSTEP]\nexact pow_factorization_choose_le (mul_pos two_pos n_pos)\n[GOAL]\ncase h\u2081.refine'_2\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\n\u22a2 \u220f i in Finset.filter (fun x => x \u2264 sqrt (2 * n)) S, 2 * n \u2264 (2 * n) ^ sqrt (2 * n)\n[PROOFSTEP]\nhave : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]\n[GOAL]\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\n\u22a2 Finset.card (Finset.Icc 1 (sqrt (2 * n))) = sqrt (2 * n)\n[PROOFSTEP]\nrw [card_Icc, Nat.add_sub_cancel]\n[GOAL]\ncase h\u2081.refine'_2\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis\u271d : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\nthis : Finset.card (Finset.Icc 1 (sqrt (2 * n))) = sqrt (2 * n)\n\u22a2 \u220f i in Finset.filter (fun x => x \u2264 sqrt (2 * n)) S, 2 * n \u2264 (2 * n) ^ sqrt (2 * n)\n[PROOFSTEP]\nrw [Finset.prod_const]\n[GOAL]\ncase h\u2081.refine'_2\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis\u271d : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\nthis : Finset.card (Finset.Icc 1 (sqrt (2 * n))) = sqrt (2 * n)\n\u22a2 (2 * n) ^ Finset.card (Finset.filter (fun x => x \u2264 sqrt (2 * n)) S) \u2264 (2 * n) ^ sqrt (2 * n)\n[PROOFSTEP]\nrefine' pow_le_pow n2_pos ((Finset.card_le_of_subset fun x hx => _).trans this.le)\n[GOAL]\ncase h\u2081.refine'_2\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis\u271d : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\nthis : Finset.card (Finset.Icc 1 (sqrt (2 * n))) = sqrt (2 * n)\nx : \u2115\nhx : x \u2208 Finset.filter (fun x => x \u2264 sqrt (2 * n)) S\n\u22a2 x \u2208 Finset.Icc 1 (sqrt (2 * n))\n[PROOFSTEP]\nobtain \u27e8h1, h2\u27e9 := Finset.mem_filter.1 hx\n[GOAL]\ncase h\u2081.refine'_2.intro\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis\u271d : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\nthis : Finset.card (Finset.Icc 1 (sqrt (2 * n))) = sqrt (2 * n)\nx : \u2115\nhx : x \u2208 Finset.filter (fun x => x \u2264 sqrt (2 * n)) S\nh1 : x \u2208 S\nh2 : x \u2264 sqrt (2 * n)\n\u22a2 x \u2208 Finset.Icc 1 (sqrt (2 * n))\n[PROOFSTEP]\nexact Finset.mem_Icc.mpr \u27e8(Finset.mem_filter.1 h1).2.one_lt.le, h2\u27e9\n[GOAL]\ncase h\u2082\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\n\u22a2 \u220f x in Finset.filter (fun x => \u00acx \u2264 sqrt (2 * n)) S, f x \u2264 4 ^ (2 * n / 3)\n[PROOFSTEP]\nrefine' le_trans _ (primorial_le_4_pow (2 * n / 3))\n[GOAL]\ncase h\u2082\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\n\u22a2 \u220f x in Finset.filter (fun x => \u00acx \u2264 sqrt (2 * n)) S, f x \u2264 primorial (2 * n / 3)\n[PROOFSTEP]\nrefine' (Finset.prod_le_prod' fun p hp => (_ : f p \u2264 p)).trans _\n[GOAL]\ncase h\u2082.refine'_1\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\np : \u2115\nhp : p \u2208 Finset.filter (fun x => \u00acx \u2264 sqrt (2 * n)) S\n\u22a2 f p \u2264 p\n[PROOFSTEP]\nobtain \u27e8h1, h2\u27e9 := Finset.mem_filter.1 hp\n[GOAL]\ncase h\u2082.refine'_1.intro\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\np : \u2115\nhp : p \u2208 Finset.filter (fun x => \u00acx \u2264 sqrt (2 * n)) S\nh1 : p \u2208 S\nh2 : \u00acp \u2264 sqrt (2 * n)\n\u22a2 f p \u2264 p\n[PROOFSTEP]\nrefine' (pow_le_pow (Finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le\n[GOAL]\ncase h\u2082.refine'_1.intro\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\np : \u2115\nhp : p \u2208 Finset.filter (fun x => \u00acx \u2264 sqrt (2 * n)) S\nh1 : p \u2208 S\nh2 : \u00acp \u2264 sqrt (2 * n)\n\u22a2 \u2191(Nat.factorization (centralBinom n)) p \u2264 1\n[PROOFSTEP]\nexact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)\n[GOAL]\ncase h\u2082.refine'_2\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\n\u22a2 \u220f i in Finset.filter (fun x => \u00acx \u2264 sqrt (2 * n)) S, i \u2264 primorial (2 * n / 3)\n[PROOFSTEP]\nrefine' Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) _\n[GOAL]\ncase h\u2082.refine'_2\nn : \u2115\nn_big : 2 < n\nno_prime : \u00ac\u2203 p, Nat.Prime p \u2227 n < p \u2227 p \u2264 2 * n\nn_pos : 0 < n\nn2_pos : 1 \u2264 2 * n\nS : Finset \u2115 := Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1))\nf : \u2115 \u2192 \u2115 := fun x => x ^ \u2191(Nat.factorization (centralBinom n)) x\nthis : \u220f x in S, f x = \u220f x in Finset.range (2 * n / 3 + 1), f x\n\u22a2 \u2200 (i : \u2115),\n i \u2208 Finset.filter Nat.Prime (Finset.range (2 * n / 3 + 1)) \u2192\n \u00aci \u2208 Finset.filter (fun x => \u00acx \u2264 sqrt (2 * n)) S \u2192 1 \u2264 i\n[PROOFSTEP]\nexact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le\n[GOAL]\nn : \u2115\nn_big : 512 \u2264 n\n\u22a2 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\n[PROOFSTEP]\nby_contra no_prime\n[GOAL]\nn : \u2115\nn_big : 512 \u2264 n\nno_prime : \u00ac\u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\n\u22a2 False\n[PROOFSTEP]\nhave H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) \u2264 4 ^ n := bertrand_main_inequality n_big\n[GOAL]\nn : \u2115\nn_big : 512 \u2264 n\nno_prime : \u00ac\u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\nH1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) \u2264 4 ^ n\n\u22a2 False\n[PROOFSTEP]\nhave H2 : 4 ^ n < n * n.centralBinom := Nat.four_pow_lt_mul_centralBinom n (le_trans (by norm_num1) n_big)\n[GOAL]\nn : \u2115\nn_big : 512 \u2264 n\nno_prime : \u00ac\u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\nH1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) \u2264 4 ^ n\n\u22a2 4 \u2264 512\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nn : \u2115\nn_big : 512 \u2264 n\nno_prime : \u00ac\u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\nH1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) \u2264 4 ^ n\nH2 : 4 ^ n < n * centralBinom n\n\u22a2 False\n[PROOFSTEP]\nhave H3 : n.centralBinom \u2264 (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) :=\n centralBinom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime\n[GOAL]\nn : \u2115\nn_big : 512 \u2264 n\nno_prime : \u00ac\u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\nH1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) \u2264 4 ^ n\nH2 : 4 ^ n < n * centralBinom n\n\u22a2 2 < 512\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nn : \u2115\nn_big : 512 \u2264 n\nno_prime : \u00ac\u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\nH1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) \u2264 4 ^ n\nH2 : 4 ^ n < n * centralBinom n\nH3 : centralBinom n \u2264 (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)\n\u22a2 False\n[PROOFSTEP]\nrw [mul_assoc] at H1 \n[GOAL]\nn : \u2115\nn_big : 512 \u2264 n\nno_prime : \u00ac\u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\nH1 : n * ((2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)) \u2264 4 ^ n\nH2 : 4 ^ n < n * centralBinom n\nH3 : centralBinom n \u2264 (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3)\n\u22a2 False\n[PROOFSTEP]\nexact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1)\n[GOAL]\nn q p : \u2115\nprime_p : Prime p\ncovering : p \u2264 2 * q\nH : n < q \u2192 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\nhn : n < p\n\u22a2 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\n[PROOFSTEP]\nby_cases p \u2264 2 * n\n[GOAL]\nn q p : \u2115\nprime_p : Prime p\ncovering : p \u2264 2 * q\nH : n < q \u2192 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\nhn : n < p\n\u22a2 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\n[PROOFSTEP]\nby_cases p \u2264 2 * n\n[GOAL]\ncase pos\nn q p : \u2115\nprime_p : Prime p\ncovering : p \u2264 2 * q\nH : n < q \u2192 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\nhn : n < p\nh : p \u2264 2 * n\n\u22a2 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\n[PROOFSTEP]\nexact \u27e8p, prime_p, hn, h\u27e9\n[GOAL]\ncase neg\nn q p : \u2115\nprime_p : Prime p\ncovering : p \u2264 2 * q\nH : n < q \u2192 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\nhn : n < p\nh : \u00acp \u2264 2 * n\n\u22a2 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\n[PROOFSTEP]\nexact H (lt_of_mul_lt_mul_left' (lt_of_lt_of_le (not_le.1 h) covering))\n[GOAL]\nn : \u2115\nhn0 : n \u2260 0\n\u22a2 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\n[PROOFSTEP]\ncases' lt_or_le 511 n with h h\n[GOAL]\ncase inl\nn : \u2115\nhn0 : n \u2260 0\nh : 511 < n\n\u22a2 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\n[PROOFSTEP]\nexact exists_prime_lt_and_le_two_mul_eventually n h\n[GOAL]\ncase inr\nn : \u2115\nhn0 : n \u2260 0\nh : n \u2264 511\n\u22a2 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\n[PROOFSTEP]\nreplace h : n < 521 := h.trans_lt (by norm_num1)\n[GOAL]\nn : \u2115\nhn0 : n \u2260 0\nh : n \u2264 511\n\u22a2 511 < 521\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase inr\nn : \u2115\nhn0 : n \u2260 0\nh : n < 521\n\u22a2 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\n[PROOFSTEP]\nrevert h\n[GOAL]\ncase inr\nn : \u2115\nhn0 : n \u2260 0\n\u22a2 n < 521 \u2192 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\n[PROOFSTEP]\nopen Lean Elab\n Tactic in\n run_tac\n do\n for i in [317, 163, 83, 43, 23, 13, 7, 5, 3, 2]do\n let i : Term := quote i\n evalTactic <| \u2190 `(tactic| refine' exists_prime_lt_and_le_two_mul_succ $i (by norm_num1) (by norm_num1) _)\n exact fun h2 => \u27e82, prime_two, h2, Nat.mul_le_mul_left 2 (Nat.pos_of_ne_zero hn0)\u27e9\n[GOAL]\ncase inr\nn : \u2115\nhn0 : n \u2260 0\n\u22a2 n < 521 \u2192 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\n[PROOFSTEP]\nrun_tac\n do\n for i in [317, 163, 83, 43, 23, 13, 7, 5, 3, 2]do\n let i : Term := quote i\n evalTactic <| \u2190 `(tactic| refine' exists_prime_lt_and_le_two_mul_succ $i (by norm_num1) (by norm_num1) _)\n[GOAL]\ncase inr\nn : \u2115\nhn0 : n \u2260 0\n\u22a2 n < 2 \u2192 \u2203 p, Prime p \u2227 n < p \u2227 p \u2264 2 * n\n[PROOFSTEP]\nexact fun h2 => \u27e82, prime_two, h2, Nat.mul_le_mul_left 2 (Nat.pos_of_ne_zero hn0)\u27e9\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.Bertrand", "llama_tokens": 37354, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7931059414036511, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.4213053101520366}} {"text": "[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b3 : Groupoid C\ninst\u271d\u00b2 : Category.{?u.41, u_2} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : RightRigidCategory D\nF : C \u2964 D\nX\u271d Y\u271d Z\u271d : C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun X => (F.obj X)\u1601, map := fun {X Y} f => F.map (inv f)\u1601 }.map (f \u226b g) =\n { obj := fun X => (F.obj X)\u1601, map := fun {X Y} f => F.map (inv f)\u1601 }.map f \u226b\n { obj := fun X => (F.obj X)\u1601, map := fun {X Y} f => F.map (inv f)\u1601 }.map g\n[PROOFSTEP]\nsimp [comp_rightAdjointMate]\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b3 : Groupoid C\ninst\u271d\u00b2 : Category.{?u.41, u_2} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : RightRigidCategory D\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\n\u22a2 (\ud835\udfd9_ (C \u2964 D)).map f \u226b\n (fun X => \u03b7_ (F.obj X) ((Functor.mk { obj := fun X => (F.obj X)\u1601, map := fun {X Y} f => F.map (inv f)\u1601 }).obj X))\n Y =\n (fun X => \u03b7_ (F.obj X) ((Functor.mk { obj := fun X => (F.obj X)\u1601, map := fun {X Y} f => F.map (inv f)\u1601 }).obj X))\n X \u226b\n (F \u2297 Functor.mk { obj := fun X => (F.obj X)\u1601, map := fun {X Y} f => F.map (inv f)\u1601 }).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b3 : Groupoid C\ninst\u271d\u00b2 : Category.{?u.41, u_2} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : RightRigidCategory D\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\n\u22a2 \ud835\udfd9 (\ud835\udfd9_ D) \u226b \u03b7_ (F.obj Y) (F.obj Y)\u1601 = \u03b7_ (F.obj X) (F.obj X)\u1601 \u226b (F.map f \u2297 F.map (inv f)\u1601)\n[PROOFSTEP]\nrw [Functor.map_inv, Category.id_comp, \u2190 id_tensor_comp_tensor_id, \u2190 Category.assoc, coevaluation_comp_rightAdjointMate,\n Category.assoc, \u2190 comp_tensor_id, IsIso.inv_hom_id, tensor_id, Category.comp_id]\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b3 : Groupoid C\ninst\u271d\u00b2 : Category.{?u.41, u_2} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : RightRigidCategory D\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\n\u22a2 (Functor.mk { obj := fun X => (F.obj X)\u1601, map := fun {X Y} f => F.map (inv f)\u1601 } \u2297 F).map f \u226b\n (fun X => \u03b5_ (F.obj X) ((Functor.mk { obj := fun X => (F.obj X)\u1601, map := fun {X Y} f => F.map (inv f)\u1601 }).obj X))\n Y =\n (fun X => \u03b5_ (F.obj X) ((Functor.mk { obj := fun X => (F.obj X)\u1601, map := fun {X Y} f => F.map (inv f)\u1601 }).obj X))\n X \u226b\n (\ud835\udfd9_ (C \u2964 D)).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b3 : Groupoid C\ninst\u271d\u00b2 : Category.{?u.41, u_2} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : RightRigidCategory D\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\n\u22a2 (F.map (inv f)\u1601 \u2297 F.map f) \u226b \u03b5_ (F.obj Y) (F.obj Y)\u1601 = \u03b5_ (F.obj X) (F.obj X)\u1601 \u226b \ud835\udfd9 (\ud835\udfd9_ D)\n[PROOFSTEP]\nrw [Category.comp_id, Functor.map_inv, \u2190 id_tensor_comp_tensor_id, Category.assoc, rightAdjointMate_comp_evaluation, \u2190\n Category.assoc, \u2190 id_tensor_comp, IsIso.hom_inv_id, tensor_id, Category.id_comp]\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b3 : Groupoid C\ninst\u271d\u00b2 : Category.{?u.37015, u_2} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : LeftRigidCategory D\nF : C \u2964 D\nX\u271d Y\u271d Z\u271d : C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun X => \u1601(F.obj X), map := fun {X Y} f => \u1601F.map (inv f) }.map (f \u226b g) =\n { obj := fun X => \u1601(F.obj X), map := fun {X Y} f => \u1601F.map (inv f) }.map f \u226b\n { obj := fun X => \u1601(F.obj X), map := fun {X Y} f => \u1601F.map (inv f) }.map g\n[PROOFSTEP]\nsimp [comp_leftAdjointMate]\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b3 : Groupoid C\ninst\u271d\u00b2 : Category.{?u.37015, u_2} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : LeftRigidCategory D\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\n\u22a2 (\ud835\udfd9_ (C \u2964 D)).map f \u226b\n (fun X => \u03b7_ ((Functor.mk { obj := fun X => \u1601(F.obj X), map := fun {X Y} f => \u1601F.map (inv f) }).obj X) (F.obj X))\n Y =\n (fun X => \u03b7_ ((Functor.mk { obj := fun X => \u1601(F.obj X), map := fun {X Y} f => \u1601F.map (inv f) }).obj X) (F.obj X))\n X \u226b\n (Functor.mk { obj := fun X => \u1601(F.obj X), map := fun {X Y} f => \u1601F.map (inv f) } \u2297 F).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b3 : Groupoid C\ninst\u271d\u00b2 : Category.{?u.37015, u_2} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : LeftRigidCategory D\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\n\u22a2 \ud835\udfd9 (\ud835\udfd9_ D) \u226b \u03b7_ (\u1601(F.obj Y)) (F.obj Y) = \u03b7_ (\u1601(F.obj X)) (F.obj X) \u226b ((\u1601F.map (inv f)) \u2297 F.map f)\n[PROOFSTEP]\nrw [Functor.map_inv, Category.id_comp, \u2190 tensor_id_comp_id_tensor, \u2190 Category.assoc, coevaluation_comp_leftAdjointMate,\n Category.assoc, \u2190 id_tensor_comp, IsIso.inv_hom_id, tensor_id, Category.comp_id]\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b3 : Groupoid C\ninst\u271d\u00b2 : Category.{?u.37015, u_2} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : LeftRigidCategory D\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\n\u22a2 (F \u2297 Functor.mk { obj := fun X => \u1601(F.obj X), map := fun {X Y} f => \u1601F.map (inv f) }).map f \u226b\n (fun X => \u03b5_ ((Functor.mk { obj := fun X => \u1601(F.obj X), map := fun {X Y} f => \u1601F.map (inv f) }).obj X) (F.obj X))\n Y =\n (fun X => \u03b5_ ((Functor.mk { obj := fun X => \u1601(F.obj X), map := fun {X Y} f => \u1601F.map (inv f) }).obj X) (F.obj X))\n X \u226b\n (\ud835\udfd9_ (C \u2964 D)).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b3 : Groupoid C\ninst\u271d\u00b2 : Category.{?u.37015, u_2} D\ninst\u271d\u00b9 : MonoidalCategory D\ninst\u271d : LeftRigidCategory D\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\n\u22a2 (F.map f \u2297 \u1601F.map (inv f)) \u226b \u03b5_ (\u1601(F.obj Y)) (F.obj Y) = \u03b5_ (\u1601(F.obj X)) (F.obj X) \u226b \ud835\udfd9 (\ud835\udfd9_ D)\n[PROOFSTEP]\nrw [Category.comp_id, Functor.map_inv, \u2190 tensor_id_comp_id_tensor, Category.assoc, leftAdjointMate_comp_evaluation, \u2190\n Category.assoc, \u2190 comp_tensor_id, IsIso.hom_inv_id, tensor_id, Category.id_comp]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCategory", "llama_tokens": 2790, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195269001831, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4211485210718869}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasColimits C\nX Y Z : TopCat\nF : Presheaf C X\nU V : Opens \u2191X\ni : U \u27f6 V\nx : { x // x \u2208 U }\ninst\u271d : ConcreteCategory C\ns : (forget C).obj (F.obj (op V))\n\u22a2 \u2191(germ F x) (\u2191(F.map i.op) s) = \u2191(germ F ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) s\n[PROOFSTEP]\nrw [\u2190 comp_apply, germ_res]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y\u271d Z : TopCat\nF : Presheaf C X\nx : \u2191X\nY : C\nf\u2081 f\u2082 : stalk F x \u27f6 Y\nih :\n \u2200 (U : Opens \u2191X) (hxU : x \u2208 U), germ F { val := x, property := hxU } \u226b f\u2081 = germ F { val := x, property := hxU } \u226b f\u2082\nU : (OpenNhds x)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F) U \u226b f\u2081 =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F) U \u226b f\u2082\n[PROOFSTEP]\ninduction' U using Opposite.rec with U\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y\u271d Z : TopCat\nF : Presheaf C X\nx : \u2191X\nY : C\nf\u2081 f\u2082 : stalk F x \u27f6 Y\nih :\n \u2200 (U : Opens \u2191X) (hxU : x \u2208 U), germ F { val := x, property := hxU } \u226b f\u2081 = germ F { val := x, property := hxU } \u226b f\u2082\nU : OpenNhds x\n\u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F) { unop := U } \u226b f\u2081 =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F) { unop := U } \u226b f\u2082\n[PROOFSTEP]\ncases' U with U hxU\n[GOAL]\ncase mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y\u271d Z : TopCat\nF : Presheaf C X\nx : \u2191X\nY : C\nf\u2081 f\u2082 : stalk F x \u27f6 Y\nih :\n \u2200 (U : Opens \u2191X) (hxU : x \u2208 U), germ F { val := x, property := hxU } \u226b f\u2081 = germ F { val := x, property := hxU } \u226b f\u2082\nU : Opens \u2191X\nhxU : x \u2208 U\n\u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\n { unop := { obj := U, property := hxU } } \u226b\n f\u2081 =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\n { unop := { obj := U, property := hxU } } \u226b\n f\u2082\n[PROOFSTEP]\nexact ih U hxU\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C X\nx : \u2191X\n\u22a2 stalk (f _* F) (\u2191f x) \u27f6 stalk F x\n[PROOFSTEP]\nrefine' ?_ \u226b colimit.pre _ (OpenNhds.map f x).op\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C X\nx : \u2191X\n\u22a2 stalk (f _* F) (\u2191f x) \u27f6\n colimit\n ((OpenNhds.map f x).op \u22d9 ((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\n[PROOFSTEP]\nexact colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C X\nU : Opens \u2191Y\nx : { x // x \u2208 (Opens.map f).obj U }\n\u22a2 germ (f _* F) { val := \u2191f \u2191x, property := (_ : \u2191x \u2208 (Opens.map f).obj U) } \u226b stalkPushforward C f F \u2191x = germ F x\n[PROOFSTEP]\nrw [stalkPushforward, germ, colimit.\u03b9_map_assoc, colimit.\u03b9_pre, whiskerRight_app]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C X\nU : Opens \u2191Y\nx : { x // x \u2208 (Opens.map f).obj U }\n\u22a2 F.map\n (NatTrans.app (NatTrans.op (OpenNhds.inclusionMapIso f \u2191x).inv)\n (op { obj := U, property := (_ : \u2191{ val := \u2191f \u2191x, property := (_ : \u2191x \u2208 (Opens.map f).obj U) } \u2208 U) })) \u226b\n colimit.\u03b9 (((whiskeringLeft (OpenNhds \u2191x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion \u2191x).op).obj F)\n ((OpenNhds.map f \u2191x).op.obj\n (op { obj := U, property := (_ : \u2191{ val := \u2191f \u2191x, property := (_ : \u2191x \u2208 (Opens.map f).obj U) } \u2208 U) })) =\n germ F x\n[PROOFSTEP]\nerw [CategoryTheory.Functor.map_id, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C X\nU : Opens \u2191Y\nx : { x // x \u2208 (Opens.map f).obj U }\n\u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds \u2191x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion \u2191x).op).obj F)\n ((OpenNhds.map f \u2191x).op.obj\n (op { obj := U, property := (_ : \u2191{ val := \u2191f \u2191x, property := (_ : \u2191x \u2208 (Opens.map f).obj U) } \u2208 U) })) =\n germ F x\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nx : \u2191X\n\u22a2 stalkPushforward C (\ud835\udfd9 X) \u2131 x = (stalkFunctor C x).map (Pushforward.id \u2131).hom\n[PROOFSTEP]\nchange (_ : colimit _ \u27f6 _) = (_ : colimit _ \u27f6 _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nx : \u2191X\n\u22a2 stalkPushforward C (\ud835\udfd9 X) \u2131 x = (stalkFunctor C x).map (Pushforward.id \u2131).hom\n[PROOFSTEP]\next1 j\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nx : \u2191X\nj : (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56\n\u22a2 colimit.\u03b9\n (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n (\ud835\udfd9 X _* \u2131))\n j \u226b\n stalkPushforward C (\ud835\udfd9 X) \u2131 x =\n colimit.\u03b9\n (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n (\ud835\udfd9 X _* \u2131))\n j \u226b\n (stalkFunctor C x).map (Pushforward.id \u2131).hom\n[PROOFSTEP]\ninduction' j with j\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nx : \u2191X\nj : OpenNhds (\u2191(\ud835\udfd9 X) x)\n\u22a2 colimit.\u03b9\n (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n (\ud835\udfd9 X _* \u2131))\n (op j) \u226b\n stalkPushforward C (\ud835\udfd9 X) \u2131 x =\n colimit.\u03b9\n (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n (\ud835\udfd9 X _* \u2131))\n (op j) \u226b\n (stalkFunctor C x).map (Pushforward.id \u2131).hom\n[PROOFSTEP]\nrcases j with \u27e8\u27e8_, _\u27e9, _\u27e9\n[GOAL]\ncase w.h.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nx : \u2191X\ncarrier\u271d : Set \u2191X\nis_open'\u271d : IsOpen carrier\u271d\nproperty\u271d : \u2191(\ud835\udfd9 X) x \u2208 { carrier := carrier\u271d, is_open' := is_open'\u271d }\n\u22a2 colimit.\u03b9\n (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n (\ud835\udfd9 X _* \u2131))\n (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b\n stalkPushforward C (\ud835\udfd9 X) \u2131 x =\n colimit.\u03b9\n (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n (\ud835\udfd9 X _* \u2131))\n (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b\n (stalkFunctor C x).map (Pushforward.id \u2131).hom\n[PROOFSTEP]\nerw [colimit.\u03b9_map_assoc]\n[GOAL]\ncase w.h.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nx : \u2191X\ncarrier\u271d : Set \u2191X\nis_open'\u271d : IsOpen carrier\u271d\nproperty\u271d : \u2191(\ud835\udfd9 X) x \u2208 { carrier := carrier\u271d, is_open' := is_open'\u271d }\n\u22a2 NatTrans.app (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso (\ud835\udfd9 X) x).inv) \u2131)\n (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b\n colimit.\u03b9\n ((OpenNhds.map (\ud835\udfd9 X) x).op \u22d9\n ((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj \u2131)\n (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b\n colimit.pre (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj \u2131)\n (OpenNhds.map (\ud835\udfd9 X) x).op =\n colimit.\u03b9\n (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n (\ud835\udfd9 X _* \u2131))\n (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b\n (stalkFunctor C x).map (Pushforward.id \u2131).hom\n[PROOFSTEP]\nsimp [stalkFunctor, stalkPushforward]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nf : X \u27f6 Y\ng : Y \u27f6 Z\nx : \u2191X\n\u22a2 stalkPushforward C (f \u226b g) \u2131 x = stalkPushforward C g (f _* \u2131) (\u2191f x) \u226b stalkPushforward C f \u2131 x\n[PROOFSTEP]\nchange (_ : colimit _ \u27f6 _) = (_ : colimit _ \u27f6 _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nf : X \u27f6 Y\ng : Y \u27f6 Z\nx : \u2191X\n\u22a2 stalkPushforward C (f \u226b g) \u2131 x = stalkPushforward C g (f _* \u2131) (\u2191f x) \u226b stalkPushforward C f \u2131 x\n[PROOFSTEP]\next U\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nf : X \u27f6 Y\ng : Y \u27f6 Z\nx : \u2191X\nU : (OpenNhds (\u2191(f \u226b g) x))\u1d52\u1d56\n\u22a2 colimit.\u03b9\n (((whiskeringLeft (OpenNhds (\u2191(f \u226b g) x))\u1d52\u1d56 (Opens \u2191Z)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(f \u226b g) x)).op).obj\n ((f \u226b g) _* \u2131))\n U \u226b\n stalkPushforward C (f \u226b g) \u2131 x =\n colimit.\u03b9\n (((whiskeringLeft (OpenNhds (\u2191(f \u226b g) x))\u1d52\u1d56 (Opens \u2191Z)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(f \u226b g) x)).op).obj\n ((f \u226b g) _* \u2131))\n U \u226b\n stalkPushforward C g (f _* \u2131) (\u2191f x) \u226b stalkPushforward C f \u2131 x\n[PROOFSTEP]\nrcases U with \u27e8\u27e8_, _\u27e9, _\u27e9\n[GOAL]\ncase w.mk.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nf : X \u27f6 Y\ng : Y \u27f6 Z\nx : \u2191X\ncarrier\u271d : Set \u2191Z\nis_open'\u271d : IsOpen carrier\u271d\nproperty\u271d : \u2191(f \u226b g) x \u2208 { carrier := carrier\u271d, is_open' := is_open'\u271d }\n\u22a2 colimit.\u03b9\n (((whiskeringLeft (OpenNhds (\u2191(f \u226b g) x))\u1d52\u1d56 (Opens \u2191Z)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(f \u226b g) x)).op).obj\n ((f \u226b g) _* \u2131))\n { unop := { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d } } \u226b\n stalkPushforward C (f \u226b g) \u2131 x =\n colimit.\u03b9\n (((whiskeringLeft (OpenNhds (\u2191(f \u226b g) x))\u1d52\u1d56 (Opens \u2191Z)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(f \u226b g) x)).op).obj\n ((f \u226b g) _* \u2131))\n { unop := { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d } } \u226b\n stalkPushforward C g (f _* \u2131) (\u2191f x) \u226b stalkPushforward C f \u2131 x\n[PROOFSTEP]\nsimp only [colimit.\u03b9_map_assoc, colimit.\u03b9_pre_assoc, whiskerRight_app, Category.assoc]\n[GOAL]\ncase w.mk.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nf : X \u27f6 Y\ng : Y \u27f6 Z\nx : \u2191X\ncarrier\u271d : Set \u2191Z\nis_open'\u271d : IsOpen carrier\u271d\nproperty\u271d : \u2191(f \u226b g) x \u2208 { carrier := carrier\u271d, is_open' := is_open'\u271d }\n\u22a2 colimit.\u03b9\n (((whiskeringLeft (OpenNhds (\u2191(f \u226b g) x))\u1d52\u1d56 (Opens \u2191Z)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(f \u226b g) x)).op).obj\n ((f \u226b g) _* \u2131))\n { unop := { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d } } \u226b\n stalkPushforward C (f \u226b g) \u2131 x =\n colimit.\u03b9\n (((whiskeringLeft (OpenNhds (\u2191(f \u226b g) x))\u1d52\u1d56 (Opens \u2191Z)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(f \u226b g) x)).op).obj\n ((f \u226b g) _* \u2131))\n { unop := { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d } } \u226b\n stalkPushforward C g (f _* \u2131) (\u2191f x) \u226b stalkPushforward C f \u2131 x\n[PROOFSTEP]\nsimp [stalkFunctor, stalkPushforward]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\n\u22a2 IsIso (stalkPushforward C f F x)\n[PROOFSTEP]\nhaveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\n\u22a2 IsIso (stalkPushforward C f F x)\n[PROOFSTEP]\nconvert\n IsIso.of_iso\n ((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op ((OpenNhds.inclusion (f x)).op \u22d9 f _* F) : _).symm \u226a\u226b\n colim.mapIso _)\n[GOAL]\ncase h.e'_5.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\ne_3\u271d : stalk (f _* F) (\u2191f x) = colimit ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)\ne_4\u271d : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\n\u22a2 stalkPushforward C f F x =\n ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op\n ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)).symm \u226a\u226b\n colim.mapIso ?convert_2).hom\ncase convert_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\n\u22a2 (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op \u22d9 (OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F \u2245\n ((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F\n[PROOFSTEP]\nswap\n[GOAL]\ncase convert_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\n\u22a2 (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op \u22d9 (OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F \u2245\n ((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F\n[PROOFSTEP]\nfapply NatIso.ofComponents\n[GOAL]\ncase convert_2.app\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\n\u22a2 (X_1 : (OpenNhds x)\u1d52\u1d56) \u2192\n ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op \u22d9 (OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F).obj X_1 \u2245\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F).obj X_1\n[PROOFSTEP]\nintro U\n[GOAL]\ncase convert_2.app\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\nU : (OpenNhds x)\u1d52\u1d56\n\u22a2 ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op \u22d9 (OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F).obj U \u2245\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F).obj U\n[PROOFSTEP]\nrefine' F.mapIso (eqToIso _)\n[GOAL]\ncase convert_2.app\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\nU : (OpenNhds x)\u1d52\u1d56\n\u22a2 (Opens.map f).op.obj ((OpenNhds.inclusion (\u2191f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj U)) =\n (OpenNhds.inclusion x).op.obj U\n[PROOFSTEP]\ndsimp only [Functor.op]\n[GOAL]\ncase convert_2.app\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\nU : (OpenNhds x)\u1d52\u1d56\n\u22a2 op\n ((Opens.map f).obj\n (op\n ((OpenNhds.inclusion (\u2191f x)).obj\n (op ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).obj U.unop)).unop)).unop) =\n op ((OpenNhds.inclusion x).obj U.unop)\n[PROOFSTEP]\nexact congr_arg op (Opens.ext <| Set.preimage_image_eq (unop U).1.1 hf.inj)\n[GOAL]\ncase convert_2.naturality\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\n\u22a2 autoParam\n (\u2200 {X_1 Y_1 : (OpenNhds x)\u1d52\u1d56} (f_1 : X_1 \u27f6 Y_1),\n ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op \u22d9 (OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F).map f_1 \u226b\n (F.mapIso\n (eqToIso\n (_ :\n (Opens.map f).op.obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj Y_1)) =\n (OpenNhds.inclusion x).op.obj Y_1))).hom =\n (F.mapIso\n (eqToIso\n (_ :\n (Opens.map f).op.obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj X_1)) =\n (OpenNhds.inclusion x).op.obj X_1))).hom \u226b\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F).map f_1)\n _auto\u271d\n[PROOFSTEP]\nintro U V i\n[GOAL]\ncase convert_2.naturality\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\nU V : (OpenNhds x)\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op \u22d9 (OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F).map i \u226b\n (F.mapIso\n (eqToIso\n (_ :\n (Opens.map f).op.obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj V)) =\n (OpenNhds.inclusion x).op.obj V))).hom =\n (F.mapIso\n (eqToIso\n (_ :\n (Opens.map f).op.obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj U)) =\n (OpenNhds.inclusion x).op.obj U))).hom \u226b\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F).map i\n[PROOFSTEP]\nerw [\u2190 F.map_comp, \u2190 F.map_comp]\n[GOAL]\ncase convert_2.naturality\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\nU V : (OpenNhds x)\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 F.map\n ((Opens.map f).op.map\n ((OpenNhds.inclusion (\u2191f x)).op.map ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.map i)) \u226b\n (eqToIso\n (_ :\n (Opens.map f).op.obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj V)) =\n (OpenNhds.inclusion x).op.obj V)).hom) =\n F.map\n ((eqToIso\n (_ :\n (Opens.map f).op.obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj U)) =\n (OpenNhds.inclusion x).op.obj U)).hom \u226b\n (OpenNhds.inclusion x).op.map i)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e'_5.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\ne_3\u271d : stalk (f _* F) (\u2191f x) = colimit ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)\ne_4\u271d : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\n\u22a2 stalkPushforward C f F x =\n ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op\n ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)).symm \u226a\u226b\n colim.mapIso\n (NatIso.ofComponents fun U =>\n F.mapIso\n (eqToIso\n (_ :\n (Opens.map f).op.obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj U)) =\n (OpenNhds.inclusion x).op.obj U)))).hom\n[PROOFSTEP]\nchange (_ : colimit _ \u27f6 _) = (_ : colimit _ \u27f6 _)\n[GOAL]\ncase h.e'_5.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\ne_3\u271d : stalk (f _* F) (\u2191f x) = colimit ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)\ne_4\u271d : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\n\u22a2 stalkPushforward C f F x =\n ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op\n ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)).symm \u226a\u226b\n colim.mapIso\n (NatIso.ofComponents fun U =>\n F.mapIso\n (eqToIso\n (_ :\n (Opens.map f).op.obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj U)) =\n (OpenNhds.inclusion x).op.obj U)))).hom\n[PROOFSTEP]\next U\n[GOAL]\ncase h.e'_5.h.w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\ne_3\u271d : stalk (f _* F) (\u2191f x) = colimit ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)\ne_4\u271d : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\nU : (OpenNhds (\u2191f x))\u1d52\u1d56\n\u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).obj (f _* F)) U \u226b\n stalkPushforward C f F x =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).obj (f _* F))\n U \u226b\n ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op\n ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)).symm \u226a\u226b\n colim.mapIso\n (NatIso.ofComponents fun U =>\n F.mapIso\n (eqToIso\n (_ :\n (Opens.map f).op.obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj U)) =\n (OpenNhds.inclusion x).op.obj U)))).hom\n[PROOFSTEP]\nrw [\u2190 Iso.comp_inv_eq]\n[GOAL]\ncase h.e'_5.h.w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\ne_3\u271d : stalk (f _* F) (\u2191f x) = colimit ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)\ne_4\u271d : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\nU : (OpenNhds (\u2191f x))\u1d52\u1d56\n\u22a2 (colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).obj (f _* F)) U \u226b\n stalkPushforward C f F x) \u226b\n ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op\n ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)).symm \u226a\u226b\n colim.mapIso\n (NatIso.ofComponents fun U =>\n F.mapIso\n (eqToIso\n (_ :\n (Opens.map f).op.obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj U)) =\n (OpenNhds.inclusion x).op.obj U)))).inv =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).obj (f _* F)) U\n[PROOFSTEP]\nerw [colimit.\u03b9_map_assoc]\n[GOAL]\ncase h.e'_5.h.w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\ne_3\u271d : stalk (f _* F) (\u2191f x) = colimit ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)\ne_4\u271d : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\nU : (OpenNhds (\u2191f x))\u1d52\u1d56\n\u22a2 (NatTrans.app (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F) U \u226b\n colimit.\u03b9\n ((OpenNhds.map f x).op \u22d9\n ((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\n U \u226b\n colimit.pre (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\n (OpenNhds.map f x).op) \u226b\n ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op\n ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)).symm \u226a\u226b\n colim.mapIso\n (NatIso.ofComponents fun U =>\n F.mapIso\n (eqToIso\n (_ :\n (Opens.map f).op.obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj U)) =\n (OpenNhds.inclusion x).op.obj U)))).inv =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).obj (f _* F)) U\n[PROOFSTEP]\nrw [colimit.\u03b9_pre, Category.assoc]\n[GOAL]\ncase h.e'_5.h.w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\ne_3\u271d : stalk (f _* F) (\u2191f x) = colimit ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)\ne_4\u271d : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\nU : (OpenNhds (\u2191f x))\u1d52\u1d56\n\u22a2 NatTrans.app (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F) U \u226b\n colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\n ((OpenNhds.map f x).op.obj U) \u226b\n ((Functor.Final.colimitIso (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op\n ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)).symm \u226a\u226b\n colim.mapIso\n (NatIso.ofComponents fun U =>\n F.mapIso\n (eqToIso\n (_ :\n (Opens.map f).op.obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj U)) =\n (OpenNhds.inclusion x).op.obj U)))).inv =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).obj (f _* F)) U\n[PROOFSTEP]\nerw [colimit.\u03b9_map_assoc, colimit.\u03b9_pre, \u2190 F.map_comp_assoc]\n[GOAL]\ncase h.e'_5.h.w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\ne_3\u271d : stalk (f _* F) (\u2191f x) = colimit ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)\ne_4\u271d : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\nU : (OpenNhds (\u2191f x))\u1d52\u1d56\n\u22a2 F.map\n (NatTrans.app (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) U \u226b\n (eqToIso\n (_ :\n (Opens.map f).op.obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj\n ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj ((OpenNhds.map f x).op.obj U))) =\n (OpenNhds.inclusion x).op.obj ((OpenNhds.map f x).op.obj U))).inv) \u226b\n colimit.\u03b9 ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)\n ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj ((OpenNhds.map f x).op.obj U)) =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).obj (f _* F)) U\n[PROOFSTEP]\napply colimit.w ((OpenNhds.inclusion (f x)).op \u22d9 f _* F) _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\ne_3\u271d : stalk (f _* F) (\u2191f x) = colimit ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)\ne_4\u271d : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\nU : (OpenNhds (\u2191f x))\u1d52\u1d56\n\u22a2 U \u27f6 (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).op.obj ((OpenNhds.map f x).op.obj U)\n[PROOFSTEP]\ndsimp only [Functor.op]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\ne_3\u271d : stalk (f _* F) (\u2191f x) = colimit ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)\ne_4\u271d : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\nU : (OpenNhds (\u2191f x))\u1d52\u1d56\n\u22a2 U \u27f6 op ((IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).obj (op ((OpenNhds.map f x).obj U.unop)).unop)\n[PROOFSTEP]\nrefine' ((homOfLE _).op : op (unop U) \u27f6 _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f\nF : Presheaf C X\nx : \u2191X\nthis : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x)\ne_3\u271d : stalk (f _* F) (\u2191f x) = colimit ((OpenNhds.inclusion (\u2191f x)).op \u22d9 f _* F)\ne_4\u271d : stalk F x = colim.obj (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\nU : (OpenNhds (\u2191f x))\u1d52\u1d56\n\u22a2 (IsOpenMap.functorNhds (_ : IsOpenMap \u2191f) x).obj (op ((OpenNhds.map f x).obj U.unop)).unop \u2264 U.unop\n[PROOFSTEP]\nexact Set.image_preimage_subset _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nU : Opens \u2191X\nx : { x // x \u2208 U }\nx\u271d\u00b9 x\u271d : CostructuredArrow (Opens.map f).op (op U)\ni : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 (Lan.diagram (Opens.map f).op F (op U)).map i \u226b\n (fun V => germ F { val := \u2191f \u2191x, property := (_ : \u2191x \u2208 \u2191((Opens.map f).op.obj V.left).unop) }) x\u271d =\n (fun V => germ F { val := \u2191f \u2191x, property := (_ : \u2191x \u2208 \u2191((Opens.map f).op.obj V.left).unop) }) x\u271d\u00b9 \u226b\n ((Functor.const (CostructuredArrow (Opens.map f).op (op U))).obj (stalk F (\u2191f \u2191x))).map i\n[PROOFSTEP]\nerw [Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nU : Opens \u2191X\nx : { x // x \u2208 U }\nx\u271d\u00b9 x\u271d : CostructuredArrow (Opens.map f).op (op U)\ni : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 (Lan.diagram (Opens.map f).op F (op U)).map i \u226b\n (fun V => germ F { val := \u2191f \u2191x, property := (_ : \u2191x \u2208 \u2191((Opens.map f).op.obj V.left).unop) }) x\u271d =\n (fun V => germ F { val := \u2191f \u2191x, property := (_ : \u2191x \u2208 \u2191((Opens.map f).op.obj V.left).unop) }) x\u271d\u00b9\n[PROOFSTEP]\nexact F.germ_res i.left.unop _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\nx\u271d\u00b2 x\u271d\u00b9 : (OpenNhds x)\u1d52\u1d56\nx\u271d : x\u271d\u00b2 \u27f6 x\u271d\u00b9\n\u22a2 ((OpenNhds.inclusion x).op \u22d9 pullbackObj f F).map x\u271d \u226b\n (fun U => germToPullbackStalk C f F U.unop.obj { val := x, property := (_ : x \u2208 U.unop.obj) }) x\u271d\u00b9 =\n (fun U => germToPullbackStalk C f F U.unop.obj { val := x, property := (_ : x \u2208 U.unop.obj) }) x\u271d\u00b2 \u226b\n ((Functor.const (OpenNhds x)\u1d52\u1d56).obj (stalk F (\u2191f x))).map x\u271d\n[PROOFSTEP]\nerw [colimit.pre_desc, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\nx\u271d\u00b2 x\u271d\u00b9 : (OpenNhds x)\u1d52\u1d56\nx\u271d : x\u271d\u00b2 \u27f6 x\u271d\u00b9\n\u22a2 colimit.desc\n (CostructuredArrow.map ((OpenNhds.inclusion x).op.map x\u271d) \u22d9\n Lan.diagram (Opens.map f).op F ((OpenNhds.inclusion x).op.obj x\u271d\u00b9))\n (Cocone.whisker (CostructuredArrow.map ((OpenNhds.inclusion x).op.map x\u271d))\n { pt := stalk F (\u2191f \u2191{ val := x, property := (_ : x \u2208 x\u271d\u00b9.unop.obj) }),\n \u03b9 :=\n NatTrans.mk fun V =>\n germ F\n { val := \u2191f \u2191{ val := x, property := (_ : x \u2208 x\u271d\u00b9.unop.obj) },\n property :=\n (_ : \u2191{ val := x, property := (_ : x \u2208 x\u271d\u00b9.unop.obj) } \u2208 \u2191((Opens.map f).op.obj V.left).unop) } }) =\n (fun U => germToPullbackStalk C f F U.unop.obj { val := x, property := (_ : x \u2208 U.unop.obj) }) x\u271d\u00b2\n[PROOFSTEP]\ncongr\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\n\u22a2 stalkPullbackHom C f F x \u226b stalkPullbackInv C f F x = \ud835\udfd9 (stalk F (\u2191f x))\n[PROOFSTEP]\ndelta stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pullback stalkPushforward germToPullbackStalk germ\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\n\u22a2 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op \u22d9 colim).map\n (NatTrans.app (pushforwardPullbackAdjunction C f).unit F) \u226b\n colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) ((lan (Opens.map f).op).obj F)) \u226b\n colimit.pre\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F))\n (OpenNhds.map f x).op) \u226b\n colimit.desc ((OpenNhds.inclusion x).op \u22d9 pullbackObj f F)\n { pt := stalk F (\u2191f x),\n \u03b9 :=\n NatTrans.mk fun U =>\n colimit.desc (Lan.diagram (Opens.map f).op F (op U.unop.obj))\n { pt := stalk F (\u2191f \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) }),\n \u03b9 :=\n NatTrans.mk fun V =>\n colimit.\u03b9\n ((OpenNhds.inclusion\n \u2191{ val := \u2191f \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) },\n property :=\n (_ :\n \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj V.left).unop) }).op \u22d9\n F)\n (op\n { obj := V.left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) },\n property :=\n (_ :\n \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj V.left).unop) } \u2208\n V.left.unop) }) } } =\n \ud835\udfd9 (stalk F (\u2191f x))\n[PROOFSTEP]\nchange (_ : colimit _ \u27f6 _) = (_ : colimit _ \u27f6 _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\n\u22a2 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op \u22d9 colim).map\n (NatTrans.app (pushforwardPullbackAdjunction C f).unit F) \u226b\n colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) ((lan (Opens.map f).op).obj F)) \u226b\n colimit.pre\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F))\n (OpenNhds.map f x).op) \u226b\n colimit.desc ((OpenNhds.inclusion x).op \u22d9 pullbackObj f F)\n { pt := stalk F (\u2191f x),\n \u03b9 :=\n NatTrans.mk fun U =>\n colimit.desc (Lan.diagram (Opens.map f).op F (op U.unop.obj))\n { pt := stalk F (\u2191f \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) }),\n \u03b9 :=\n NatTrans.mk fun V =>\n colimit.\u03b9\n ((OpenNhds.inclusion\n \u2191{ val := \u2191f \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) },\n property :=\n (_ :\n \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj V.left).unop) }).op \u22d9\n F)\n (op\n { obj := V.left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) },\n property :=\n (_ :\n \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj V.left).unop) } \u2208\n V.left.unop) }) } } =\n \ud835\udfd9 (stalk F (\u2191f x))\n[PROOFSTEP]\next j\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\nj : (OpenNhds (\u2191f x))\u1d52\u1d56\n\u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).obj F) j \u226b\n (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op \u22d9 colim).map\n (NatTrans.app (pushforwardPullbackAdjunction C f).unit F) \u226b\n colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) ((lan (Opens.map f).op).obj F)) \u226b\n colimit.pre\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F))\n (OpenNhds.map f x).op) \u226b\n colimit.desc ((OpenNhds.inclusion x).op \u22d9 pullbackObj f F)\n { pt := stalk F (\u2191f x),\n \u03b9 :=\n NatTrans.mk fun U =>\n colimit.desc (Lan.diagram (Opens.map f).op F (op U.unop.obj))\n { pt := stalk F (\u2191f \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) }),\n \u03b9 :=\n NatTrans.mk fun V =>\n colimit.\u03b9\n ((OpenNhds.inclusion\n \u2191{ val := \u2191f \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) },\n property :=\n (_ :\n \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj V.left).unop) }).op \u22d9\n F)\n (op\n { obj := V.left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) },\n property :=\n (_ :\n \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj V.left).unop) } \u2208\n V.left.unop) }) } } =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).obj F) j \u226b\n \ud835\udfd9 (stalk F (\u2191f x))\n[PROOFSTEP]\ninduction' j with j\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\nj : OpenNhds (\u2191f x)\n\u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).obj F) (op j) \u226b\n (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op \u22d9 colim).map\n (NatTrans.app (pushforwardPullbackAdjunction C f).unit F) \u226b\n colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) ((lan (Opens.map f).op).obj F)) \u226b\n colimit.pre\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F))\n (OpenNhds.map f x).op) \u226b\n colimit.desc ((OpenNhds.inclusion x).op \u22d9 pullbackObj f F)\n { pt := stalk F (\u2191f x),\n \u03b9 :=\n NatTrans.mk fun U =>\n colimit.desc (Lan.diagram (Opens.map f).op F (op U.unop.obj))\n { pt := stalk F (\u2191f \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) }),\n \u03b9 :=\n NatTrans.mk fun V =>\n colimit.\u03b9\n ((OpenNhds.inclusion\n \u2191{ val := \u2191f \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) },\n property :=\n (_ :\n \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj V.left).unop) }).op \u22d9\n F)\n (op\n { obj := V.left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) },\n property :=\n (_ :\n \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj V.left).unop) } \u2208\n V.left.unop) }) } } =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).obj F) (op j) \u226b\n \ud835\udfd9 (stalk F (\u2191f x))\n[PROOFSTEP]\ncases j\n[GOAL]\ncase w.h.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\nobj\u271d : Opens \u2191Y\nproperty\u271d : \u2191f x \u2208 obj\u271d\n\u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).obj F)\n (op { obj := obj\u271d, property := property\u271d }) \u226b\n (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op \u22d9 colim).map\n (NatTrans.app (pushforwardPullbackAdjunction C f).unit F) \u226b\n colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) ((lan (Opens.map f).op).obj F)) \u226b\n colimit.pre\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F))\n (OpenNhds.map f x).op) \u226b\n colimit.desc ((OpenNhds.inclusion x).op \u22d9 pullbackObj f F)\n { pt := stalk F (\u2191f x),\n \u03b9 :=\n NatTrans.mk fun U =>\n colimit.desc (Lan.diagram (Opens.map f).op F (op U.unop.obj))\n { pt := stalk F (\u2191f \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) }),\n \u03b9 :=\n NatTrans.mk fun V =>\n colimit.\u03b9\n ((OpenNhds.inclusion\n \u2191{ val := \u2191f \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) },\n property :=\n (_ :\n \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj V.left).unop) }).op \u22d9\n F)\n (op\n { obj := V.left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) },\n property :=\n (_ :\n \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj V.left).unop) } \u2208\n V.left.unop) }) } } =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).obj F)\n (op { obj := obj\u271d, property := property\u271d }) \u226b\n \ud835\udfd9 (stalk F (\u2191f x))\n[PROOFSTEP]\nsimp only [TopologicalSpace.OpenNhds.inclusionMapIso_inv, whiskerRight_app, whiskerLeft_app, whiskeringLeft_obj_map,\n Functor.comp_map, colimit.\u03b9_map_assoc, NatTrans.op_id, lan_obj_map, pushforwardPullbackAdjunction_unit_app_app,\n Category.assoc, colimit.\u03b9_pre_assoc]\n[GOAL]\ncase w.h.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\nobj\u271d : Opens \u2191Y\nproperty\u271d : \u2191f x \u2208 obj\u271d\n\u22a2 colimit.\u03b9\n (Lan.diagram (Opens.map f).op F\n (op\n ((Opens.map f).obj ((OpenNhds.inclusion (\u2191f x)).op.obj (op { obj := obj\u271d, property := property\u271d })).unop)))\n (CostructuredArrow.mk\n (\ud835\udfd9\n (op\n ((Opens.map f).obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj (op { obj := obj\u271d, property := property\u271d })).unop)))) \u226b\n colimit.pre\n (Lan.diagram (Opens.map f).op F\n ((OpenNhds.map f x \u22d9 OpenNhds.inclusion x).op.obj (op { obj := obj\u271d, property := property\u271d })))\n (CostructuredArrow.map\n (NatTrans.app (\ud835\udfd9 (OpenNhds.map f x \u22d9 OpenNhds.inclusion x).op)\n (op { obj := obj\u271d, property := property\u271d }))) \u226b\n colimit.\u03b9\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F))\n ((OpenNhds.map f x).op.obj (op { obj := obj\u271d, property := property\u271d })) \u226b\n colimit.desc ((OpenNhds.inclusion x).op \u22d9 pullbackObj f F)\n { pt := stalk F (\u2191f x),\n \u03b9 :=\n NatTrans.mk fun U =>\n colimit.desc (Lan.diagram (Opens.map f).op F (op U.unop.obj))\n { pt := stalk F (\u2191f x),\n \u03b9 :=\n NatTrans.mk fun V =>\n colimit.\u03b9 ((OpenNhds.inclusion (\u2191f x)).op \u22d9 F)\n (op\n { obj := V.left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x, property := (_ : x \u2208 U.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj V.left).unop) } \u2208\n V.left.unop) }) } } =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).obj F)\n (op { obj := obj\u271d, property := property\u271d }) \u226b\n \ud835\udfd9 (stalk F (\u2191f x))\n[PROOFSTEP]\nerw [colimit.\u03b9_desc, colimit.pre_desc, colimit.\u03b9_desc, Category.comp_id]\n[GOAL]\ncase w.h.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\nobj\u271d : Opens \u2191Y\nproperty\u271d : \u2191f x \u2208 obj\u271d\n\u22a2 NatTrans.app\n (Cocone.whisker\n (CostructuredArrow.map\n (NatTrans.app (\ud835\udfd9 (OpenNhds.map f x \u22d9 OpenNhds.inclusion x).op) (op { obj := obj\u271d, property := property\u271d })))\n { pt := stalk F (\u2191f x),\n \u03b9 :=\n NatTrans.mk fun V =>\n colimit.\u03b9 ((OpenNhds.inclusion (\u2191f x)).op \u22d9 F)\n (op\n { obj := V.left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ :\n x \u2208\n ((OpenNhds.map f x).op.obj\n (op { obj := obj\u271d, property := property\u271d })).unop.obj) } \u2208\n \u2191((Opens.map f).op.obj V.left).unop) } \u2208\n V.left.unop) }) }).\u03b9\n (CostructuredArrow.mk\n (\ud835\udfd9\n (op\n ((Opens.map f).obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj (op { obj := obj\u271d, property := property\u271d })).unop)))) =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).obj F)\n (op { obj := obj\u271d, property := property\u271d })\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\n\u22a2 stalkPullbackInv C f F x \u226b stalkPullbackHom C f F x = \ud835\udfd9 (stalk (pullbackObj f F) x)\n[PROOFSTEP]\ndelta stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pullback stalkPushforward\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\n\u22a2 colimit.desc ((OpenNhds.inclusion x).op \u22d9 pullbackObj f F)\n { pt := stalk F (\u2191f x),\n \u03b9 :=\n NatTrans.mk fun U => germToPullbackStalk C f F U.unop.obj { val := x, property := (_ : x \u2208 U.unop.obj) } } \u226b\n ((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op \u22d9 colim).map\n (NatTrans.app (pushforwardPullbackAdjunction C f).unit F) \u226b\n colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) ((lan (Opens.map f).op).obj F)) \u226b\n colimit.pre\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F))\n (OpenNhds.map f x).op =\n \ud835\udfd9 (stalk (pullbackObj f F) x)\n[PROOFSTEP]\nchange (_ : colimit _ \u27f6 _) = (_ : colimit _ \u27f6 _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\n\u22a2 colimit.desc ((OpenNhds.inclusion x).op \u22d9 pullbackObj f F)\n { pt := stalk F (\u2191f x),\n \u03b9 :=\n NatTrans.mk fun U => germToPullbackStalk C f F U.unop.obj { val := x, property := (_ : x \u2208 U.unop.obj) } } \u226b\n ((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op \u22d9 colim).map\n (NatTrans.app (pushforwardPullbackAdjunction C f).unit F) \u226b\n colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) ((lan (Opens.map f).op).obj F)) \u226b\n colimit.pre\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F))\n (OpenNhds.map f x).op =\n \ud835\udfd9 (stalk (pullbackObj f F) x)\n[PROOFSTEP]\next \u27e8U_obj, U_property\u27e9\n[GOAL]\ncase w.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\nU_obj : Opens \u2191X\nU_property : x \u2208 U_obj\n\u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj (pullbackObj f F))\n { unop := { obj := U_obj, property := U_property } } \u226b\n colimit.desc ((OpenNhds.inclusion x).op \u22d9 pullbackObj f F)\n { pt := stalk F (\u2191f x),\n \u03b9 :=\n NatTrans.mk fun U =>\n germToPullbackStalk C f F U.unop.obj { val := x, property := (_ : x \u2208 U.unop.obj) } } \u226b\n ((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op \u22d9 colim).map\n (NatTrans.app (pushforwardPullbackAdjunction C f).unit F) \u226b\n colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) ((lan (Opens.map f).op).obj F)) \u226b\n colimit.pre\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F))\n (OpenNhds.map f x).op =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj (pullbackObj f F))\n { unop := { obj := U_obj, property := U_property } } \u226b\n \ud835\udfd9 (stalk (pullbackObj f F) x)\n[PROOFSTEP]\nchange (_ : colimit _ \u27f6 _) = (_ : colimit _ \u27f6 _)\n[GOAL]\ncase w.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\nU_obj : Opens \u2191X\nU_property : x \u2208 U_obj\n\u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj (pullbackObj f F))\n { unop := { obj := U_obj, property := U_property } } \u226b\n colimit.desc ((OpenNhds.inclusion x).op \u22d9 pullbackObj f F)\n { pt := stalk F (\u2191f x),\n \u03b9 :=\n NatTrans.mk fun U =>\n germToPullbackStalk C f F U.unop.obj { val := x, property := (_ : x \u2208 U.unop.obj) } } \u226b\n ((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op \u22d9 colim).map\n (NatTrans.app (pushforwardPullbackAdjunction C f).unit F) \u226b\n colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) ((lan (Opens.map f).op).obj F)) \u226b\n colimit.pre\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F))\n (OpenNhds.map f x).op =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj (pullbackObj f F))\n { unop := { obj := U_obj, property := U_property } } \u226b\n \ud835\udfd9 (stalk (pullbackObj f F) x)\n[PROOFSTEP]\next \u27e8j_left, \u27e8\u27e8\u27e9\u27e9, j_hom\u27e9\n[GOAL]\ncase w.mk.mk.w.mk.mk.unit\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\nU_obj : Opens \u2191X\nU_property : x \u2208 U_obj\nj_left : (Opens \u2191Y)\u1d52\u1d56\nj_hom :\n (Opens.map f).op.obj j_left \u27f6\n (Functor.fromPUnit ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } })).obj\n { as := PUnit.unit }\n\u22a2 colimit.\u03b9\n (Lan.diagram (Opens.map f).op F\n ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } }))\n { left := j_left, right := { as := PUnit.unit }, hom := j_hom } \u226b\n colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj (pullbackObj f F))\n { unop := { obj := U_obj, property := U_property } } \u226b\n colimit.desc ((OpenNhds.inclusion x).op \u22d9 pullbackObj f F)\n { pt := stalk F (\u2191f x),\n \u03b9 :=\n NatTrans.mk fun U =>\n germToPullbackStalk C f F U.unop.obj { val := x, property := (_ : x \u2208 U.unop.obj) } } \u226b\n ((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op \u22d9 colim).map\n (NatTrans.app (pushforwardPullbackAdjunction C f).unit F) \u226b\n colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) ((lan (Opens.map f).op).obj F)) \u226b\n colimit.pre\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F))\n (OpenNhds.map f x).op =\n colimit.\u03b9\n (Lan.diagram (Opens.map f).op F\n ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } }))\n { left := j_left, right := { as := PUnit.unit }, hom := j_hom } \u226b\n colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj (pullbackObj f F))\n { unop := { obj := U_obj, property := U_property } } \u226b\n \ud835\udfd9 (stalk (pullbackObj f F) x)\n[PROOFSTEP]\nerw [colimit.map_desc, colimit.map_desc, colimit.\u03b9_desc_assoc, colimit.\u03b9_desc_assoc, colimit.\u03b9_desc, Category.comp_id]\n[GOAL]\ncase w.mk.mk.w.mk.mk.unit\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\nU_obj : Opens \u2191X\nU_property : x \u2208 U_obj\nj_left : (Opens \u2191Y)\u1d52\u1d56\nj_hom :\n (Opens.map f).op.obj j_left \u27f6\n (Functor.fromPUnit ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } })).obj\n { as := PUnit.unit }\n\u22a2 NatTrans.app\n ((Cocones.precompose\n (((whiskeringLeft (OpenNhds (\u2191f x))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f x)).op).map\n (NatTrans.app (pushforwardPullbackAdjunction C f).unit F))).obj\n ((Cocones.precompose\n (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) ((lan (Opens.map f).op).obj F))).obj\n (Cocone.whisker (OpenNhds.map f x).op\n (colimit.cocone\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F)))))).\u03b9\n (op\n { obj := { left := j_left, right := { as := PUnit.unit }, hom := j_hom }.left.unop,\n property :=\n (_ :\n \u2191{\n val :=\n \u2191f\n \u2191{ val := x,\n property := (_ : x \u2208 { unop := { obj := U_obj, property := U_property } }.unop.obj) },\n property :=\n (_ :\n \u2191{ val := x,\n property := (_ : x \u2208 { unop := { obj := U_obj, property := U_property } }.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj\n { left := j_left, right := { as := PUnit.unit }, hom := j_hom }.left).unop) } \u2208\n { left := j_left, right := { as := PUnit.unit }, hom := j_hom }.left.unop) }) =\n colimit.\u03b9\n (Lan.diagram (Opens.map f).op F\n ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } }))\n { left := j_left, right := { as := PUnit.unit }, hom := j_hom } \u226b\n colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj (pullbackObj f F))\n { unop := { obj := U_obj, property := U_property } }\n[PROOFSTEP]\nsimp only [Cocone.whisker_\u03b9, colimit.cocone_\u03b9, OpenNhds.inclusionMapIso_inv, Cocones.precompose_obj_\u03b9, whiskerRight_app,\n whiskerLeft_app, NatTrans.comp_app, whiskeringLeft_obj_map, NatTrans.op_id, lan_obj_map,\n pushforwardPullbackAdjunction_unit_app_app]\n[GOAL]\ncase w.mk.mk.w.mk.mk.unit\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\nU_obj : Opens \u2191X\nU_property : x \u2208 U_obj\nj_left : (Opens \u2191Y)\u1d52\u1d56\nj_hom :\n (Opens.map f).op.obj j_left \u27f6\n (Functor.fromPUnit ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } })).obj\n { as := PUnit.unit }\n\u22a2 colimit.\u03b9\n (Lan.diagram (Opens.map f).op F\n (op\n ((Opens.map f).obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj\n (op\n { obj := j_left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ : x \u2208 { unop := { obj := U_obj, property := U_property } }.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj\n { left := j_left, right := { as := PUnit.unit },\n hom := j_hom }.left).unop) } \u2208\n j_left.unop) })).unop)))\n (CostructuredArrow.mk\n (\ud835\udfd9\n (op\n ((Opens.map f).obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj\n (op\n { obj := j_left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ :\n x \u2208 { unop := { obj := U_obj, property := U_property } }.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj\n { left := j_left, right := { as := PUnit.unit },\n hom := j_hom }.left).unop) } \u2208\n j_left.unop) })).unop)))) \u226b\n colimit.pre\n (Lan.diagram (Opens.map f).op F\n ((OpenNhds.map f x \u22d9 OpenNhds.inclusion x).op.obj\n (op\n { obj := j_left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ : x \u2208 { unop := { obj := U_obj, property := U_property } }.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj\n { left := j_left, right := { as := PUnit.unit }, hom := j_hom }.left).unop) } \u2208\n j_left.unop) })))\n (CostructuredArrow.map\n (NatTrans.app (\ud835\udfd9 (OpenNhds.map f x \u22d9 OpenNhds.inclusion x).op)\n (op\n { obj := j_left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ : x \u2208 { unop := { obj := U_obj, property := U_property } }.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj\n { left := j_left, right := { as := PUnit.unit }, hom := j_hom }.left).unop) } \u2208\n j_left.unop) }))) \u226b\n colimit.\u03b9\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F))\n ((OpenNhds.map f x).op.obj\n (op\n { obj := j_left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ : x \u2208 { unop := { obj := U_obj, property := U_property } }.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj\n { left := j_left, right := { as := PUnit.unit }, hom := j_hom }.left).unop) } \u2208\n j_left.unop) })) =\n colimit.\u03b9\n (Lan.diagram (Opens.map f).op F\n ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } }))\n { left := j_left, right := { as := PUnit.unit }, hom := j_hom } \u226b\n colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj (pullbackObj f F))\n { unop := { obj := U_obj, property := U_property } }\n[PROOFSTEP]\nerw [\u2190\n colimit.w _\n (@homOfLE (OpenNhds x) _ \u27e8_, U_property\u27e9 \u27e8(Opens.map f).obj (unop j_left), j_hom.unop.le U_property\u27e9\n j_hom.unop.le).op]\n[GOAL]\ncase w.mk.mk.w.mk.mk.unit\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\nU_obj : Opens \u2191X\nU_property : x \u2208 U_obj\nj_left : (Opens \u2191Y)\u1d52\u1d56\nj_hom :\n (Opens.map f).op.obj j_left \u27f6\n (Functor.fromPUnit ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } })).obj\n { as := PUnit.unit }\n\u22a2 colimit.\u03b9\n (Lan.diagram (Opens.map f).op F\n (op\n ((Opens.map f).obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj\n (op\n { obj := j_left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ : x \u2208 { unop := { obj := U_obj, property := U_property } }.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj\n { left := j_left, right := { as := PUnit.unit },\n hom := j_hom }.left).unop) } \u2208\n j_left.unop) })).unop)))\n (CostructuredArrow.mk\n (\ud835\udfd9\n (op\n ((Opens.map f).obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj\n (op\n { obj := j_left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ :\n x \u2208 { unop := { obj := U_obj, property := U_property } }.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj\n { left := j_left, right := { as := PUnit.unit },\n hom := j_hom }.left).unop) } \u2208\n j_left.unop) })).unop)))) \u226b\n colimit.pre\n (Lan.diagram (Opens.map f).op F\n ((OpenNhds.map f x \u22d9 OpenNhds.inclusion x).op.obj\n (op\n { obj := j_left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ : x \u2208 { unop := { obj := U_obj, property := U_property } }.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj\n { left := j_left, right := { as := PUnit.unit }, hom := j_hom }.left).unop) } \u2208\n j_left.unop) })))\n (CostructuredArrow.map\n (NatTrans.app (\ud835\udfd9 (OpenNhds.map f x \u22d9 OpenNhds.inclusion x).op)\n (op\n { obj := j_left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ : x \u2208 { unop := { obj := U_obj, property := U_property } }.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj\n { left := j_left, right := { as := PUnit.unit }, hom := j_hom }.left).unop) } \u2208\n j_left.unop) }))) \u226b\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F)).map\n (homOfLE\n (_ :\n ((Functor.fromPUnit\n ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } })).obj\n { as := PUnit.unit }).unop \u2264\n ((Opens.map f).op.obj j_left).unop)).op \u226b\n colimit.\u03b9\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F))\n (op { obj := U_obj, property := U_property }) =\n colimit.\u03b9\n (Lan.diagram (Opens.map f).op F\n ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } }))\n { left := j_left, right := { as := PUnit.unit }, hom := j_hom } \u226b\n colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj (pullbackObj f F))\n { unop := { obj := U_obj, property := U_property } }\n[PROOFSTEP]\nerw [colimit.\u03b9_pre_assoc (Lan.diagram _ F _) (CostructuredArrow.map _)]\n[GOAL]\ncase w.mk.mk.w.mk.mk.unit\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\nU_obj : Opens \u2191X\nU_property : x \u2208 U_obj\nj_left : (Opens \u2191Y)\u1d52\u1d56\nj_hom :\n (Opens.map f).op.obj j_left \u27f6\n (Functor.fromPUnit ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } })).obj\n { as := PUnit.unit }\n\u22a2 colimit.\u03b9\n (Lan.diagram (Opens.map f).op F\n (op\n ((Opens.map f).obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj\n (op\n { obj := j_left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ : x \u2208 { unop := { obj := U_obj, property := U_property } }.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj\n { left := j_left, right := { as := PUnit.unit },\n hom := j_hom }.left).unop) } \u2208\n j_left.unop) })).unop)))\n ((CostructuredArrow.map\n (NatTrans.app (\ud835\udfd9 (OpenNhds.map f x \u22d9 OpenNhds.inclusion x).op)\n (op\n { obj := j_left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ : x \u2208 { unop := { obj := U_obj, property := U_property } }.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj\n { left := j_left, right := { as := PUnit.unit },\n hom := j_hom }.left).unop) } \u2208\n j_left.unop) }))).obj\n (CostructuredArrow.mk\n (\ud835\udfd9\n (op\n ((Opens.map f).obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj\n (op\n { obj := j_left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ :\n x \u2208 { unop := { obj := U_obj, property := U_property } }.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj\n { left := j_left, right := { as := PUnit.unit },\n hom := j_hom }.left).unop) } \u2208\n j_left.unop) })).unop))))) \u226b\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F)).map\n (homOfLE\n (_ :\n ((Functor.fromPUnit\n ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } })).obj\n { as := PUnit.unit }).unop \u2264\n ((Opens.map f).op.obj j_left).unop)).op \u226b\n colimit.\u03b9\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F))\n (op { obj := U_obj, property := U_property }) =\n colimit.\u03b9\n (Lan.diagram (Opens.map f).op F\n ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } }))\n { left := j_left, right := { as := PUnit.unit }, hom := j_hom } \u226b\n colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj (pullbackObj f F))\n { unop := { obj := U_obj, property := U_property } }\n[PROOFSTEP]\nerw [colimit.\u03b9_pre_assoc (Lan.diagram _ F (op U_obj)) (CostructuredArrow.map _)]\n[GOAL]\ncase w.mk.mk.w.mk.mk.unit\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C Y\nx : \u2191X\nU_obj : Opens \u2191X\nU_property : x \u2208 U_obj\nj_left : (Opens \u2191Y)\u1d52\u1d56\nj_hom :\n (Opens.map f).op.obj j_left \u27f6\n (Functor.fromPUnit ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } })).obj\n { as := PUnit.unit }\n\u22a2 colimit.\u03b9 (Lan.diagram (Opens.map f).op F (op U_obj))\n ((CostructuredArrow.map\n ((OpenNhds.inclusion x).op.map\n (homOfLE\n (_ :\n ((Functor.fromPUnit\n ((OpenNhds.inclusion x).op.obj\n { unop := { obj := U_obj, property := U_property } })).obj\n { as := PUnit.unit }).unop \u2264\n ((Opens.map f).op.obj j_left).unop)).op)).obj\n ((CostructuredArrow.map\n (NatTrans.app (\ud835\udfd9 (OpenNhds.map f x \u22d9 OpenNhds.inclusion x).op)\n (op\n { obj := j_left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ : x \u2208 { unop := { obj := U_obj, property := U_property } }.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj\n { left := j_left, right := { as := PUnit.unit },\n hom := j_hom }.left).unop) } \u2208\n j_left.unop) }))).obj\n (CostructuredArrow.mk\n (\ud835\udfd9\n (op\n ((Opens.map f).obj\n ((OpenNhds.inclusion (\u2191f x)).op.obj\n (op\n { obj := j_left.unop,\n property :=\n (_ :\n \u2191{ val := \u2191f x,\n property :=\n (_ :\n \u2191{ val := x,\n property :=\n (_ :\n x \u2208\n { unop := { obj := U_obj, property := U_property } }.unop.obj) } \u2208\n \u2191((Opens.map f).op.obj\n { left := j_left, right := { as := PUnit.unit },\n hom := j_hom }.left).unop) } \u2208\n j_left.unop) })).unop)))))) \u226b\n colimit.\u03b9\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj\n ((lan (Opens.map f).op).obj F))\n (op { obj := U_obj, property := U_property }) =\n colimit.\u03b9\n (Lan.diagram (Opens.map f).op F\n ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } }))\n { left := j_left, right := { as := PUnit.unit }, hom := j_hom } \u226b\n colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj (pullbackObj f F))\n { unop := { obj := U_obj, property := U_property } }\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nF : Presheaf C X\nx y : \u2191X\nh : x \u2933 y\n\u22a2 stalk F y \u27f6 stalk F x\n[PROOFSTEP]\nrefine' colimit.desc _ \u27e8_, fun U => _, _\u27e9\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nF : Presheaf C X\nx y : \u2191X\nh : x \u2933 y\nU : (OpenNhds y)\u1d52\u1d56\n\u22a2 (((whiskeringLeft (OpenNhds y)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion y).op).obj F).obj U \u27f6\n ((Functor.const (OpenNhds y)\u1d52\u1d56).obj\n (colim.obj (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F))).obj\n U\n[PROOFSTEP]\nexact\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 F)\n (op \u27e8(unop U).1, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)\u27e9)\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nF : Presheaf C X\nx y : \u2191X\nh : x \u2933 y\n\u22a2 \u2200 \u2983X_1 Y : (OpenNhds y)\u1d52\u1d56\u2984 (f : X_1 \u27f6 Y),\n (((whiskeringLeft (OpenNhds y)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion y).op).obj F).map f \u226b\n (fun U =>\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 F)\n (op { obj := U.unop.obj, property := (_ : x \u2208 U.unop.obj.carrier) }))\n Y =\n (fun U =>\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 F)\n (op { obj := U.unop.obj, property := (_ : x \u2208 U.unop.obj.carrier) }))\n X_1 \u226b\n ((Functor.const (OpenNhds y)\u1d52\u1d56).obj\n (colim.obj (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F))).map\n f\n[PROOFSTEP]\nintro U V i\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nF : Presheaf C X\nx y : \u2191X\nh : x \u2933 y\nU V : (OpenNhds y)\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 (((whiskeringLeft (OpenNhds y)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion y).op).obj F).map i \u226b\n (fun U =>\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 F)\n (op { obj := U.unop.obj, property := (_ : x \u2208 U.unop.obj.carrier) }))\n V =\n (fun U =>\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 F)\n (op { obj := U.unop.obj, property := (_ : x \u2208 U.unop.obj.carrier) }))\n U \u226b\n ((Functor.const (OpenNhds y)\u1d52\u1d56).obj\n (colim.obj (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F))).map\n i\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nF : Presheaf C X\nx y : \u2191X\nh : x \u2933 y\nU V : (OpenNhds y)\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 F.map ((OpenNhds.inclusion y).map i.unop).op \u226b\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 F) (op { obj := V.unop.obj, property := (_ : x \u2208 \u2191V.unop.obj) }) =\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 F) (op { obj := U.unop.obj, property := (_ : x \u2208 \u2191U.unop.obj) }) \u226b\n \ud835\udfd9 (colimit ((OpenNhds.inclusion x).op \u22d9 F))\n[PROOFSTEP]\nrw [Category.comp_id]\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nF : Presheaf C X\nx y : \u2191X\nh : x \u2933 y\nU V : (OpenNhds y)\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 F.map ((OpenNhds.inclusion y).map i.unop).op \u226b\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 F) (op { obj := V.unop.obj, property := (_ : x \u2208 \u2191V.unop.obj) }) =\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 F) (op { obj := U.unop.obj, property := (_ : x \u2208 \u2191U.unop.obj) })\n[PROOFSTEP]\nlet U' : OpenNhds x := \u27e8_, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)\u27e9\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nF : Presheaf C X\nx y : \u2191X\nh : x \u2933 y\nU V : (OpenNhds y)\u1d52\u1d56\ni : U \u27f6 V\nU' : OpenNhds x := { obj := U.unop.obj, property := (_ : x \u2208 U.unop.obj.carrier) }\n\u22a2 F.map ((OpenNhds.inclusion y).map i.unop).op \u226b\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 F) (op { obj := V.unop.obj, property := (_ : x \u2208 \u2191V.unop.obj) }) =\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 F) (op { obj := U.unop.obj, property := (_ : x \u2208 \u2191U.unop.obj) })\n[PROOFSTEP]\nlet V' : OpenNhds x := \u27e8_, (specializes_iff_forall_open.mp h _ (unop V).1.2 (unop V).2 : _)\u27e9\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nF : Presheaf C X\nx y : \u2191X\nh : x \u2933 y\nU V : (OpenNhds y)\u1d52\u1d56\ni : U \u27f6 V\nU' : OpenNhds x := { obj := U.unop.obj, property := (_ : x \u2208 U.unop.obj.carrier) }\nV' : OpenNhds x := { obj := V.unop.obj, property := (_ : x \u2208 V.unop.obj.carrier) }\n\u22a2 F.map ((OpenNhds.inclusion y).map i.unop).op \u226b\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 F) (op { obj := V.unop.obj, property := (_ : x \u2208 \u2191V.unop.obj) }) =\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 F) (op { obj := U.unop.obj, property := (_ : x \u2208 \u2191U.unop.obj) })\n[PROOFSTEP]\nexact colimit.w ((OpenNhds.inclusion x).op \u22d9 F) (show V' \u27f6 U' from i.unop).op\n[GOAL]\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : HasColimits C\u271d\nX\u271d Y Z : TopCat\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasColimits C\nX : TopCat\nF : Presheaf C X\nx : \u2191X\n\u22a2 stalkSpecializes F (_ : x \u2933 x) = \ud835\udfd9 (stalk F x)\n[PROOFSTEP]\next\n[GOAL]\ncase ih\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : HasColimits C\u271d\nX\u271d Y Z : TopCat\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasColimits C\nX : TopCat\nF : Presheaf C X\nx : \u2191X\nU\u271d : Opens \u2191X\nhxU\u271d : x \u2208 U\u271d\n\u22a2 germ F { val := x, property := hxU\u271d } \u226b stalkSpecializes F (_ : x \u2933 x) =\n germ F { val := x, property := hxU\u271d } \u226b \ud835\udfd9 (stalk F x)\n[PROOFSTEP]\nsimp\n[GOAL]\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : HasColimits C\u271d\nX\u271d Y Z : TopCat\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasColimits C\nX : TopCat\nF : Presheaf C X\nx y z : \u2191X\nh : x \u2933 y\nh' : y \u2933 z\n\u22a2 stalkSpecializes F h' \u226b stalkSpecializes F h = stalkSpecializes F (_ : x \u2933 z)\n[PROOFSTEP]\next\n[GOAL]\ncase ih\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : HasColimits C\u271d\nX\u271d Y Z : TopCat\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasColimits C\nX : TopCat\nF : Presheaf C X\nx y z : \u2191X\nh : x \u2933 y\nh' : y \u2933 z\nU\u271d : Opens \u2191X\nhxU\u271d : z \u2208 U\u271d\n\u22a2 germ F { val := z, property := hxU\u271d } \u226b stalkSpecializes F h' \u226b stalkSpecializes F h =\n germ F { val := z, property := hxU\u271d } \u226b stalkSpecializes F (_ : x \u2933 z)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nF G : Presheaf C X\nf : F \u27f6 G\nx y : \u2191X\nh : x \u2933 y\n\u22a2 stalkSpecializes F h \u226b (stalkFunctor C x).map f = (stalkFunctor C y).map f \u226b stalkSpecializes G h\n[PROOFSTEP]\nchange (_ : colimit _ \u27f6 _) = (_ : colimit _ \u27f6 _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nF G : Presheaf C X\nf : F \u27f6 G\nx y : \u2191X\nh : x \u2933 y\n\u22a2 stalkSpecializes F h \u226b (stalkFunctor C x).map f = (stalkFunctor C y).map f \u226b stalkSpecializes G h\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nF G : Presheaf C X\nf : F \u27f6 G\nx y : \u2191X\nh : x \u2933 y\nj\u271d : (OpenNhds y)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds y)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion y).op).obj F) j\u271d \u226b\n stalkSpecializes F h \u226b (stalkFunctor C x).map f =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds y)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion y).op).obj F) j\u271d \u226b\n (stalkFunctor C y).map f \u226b stalkSpecializes G h\n[PROOFSTEP]\ndelta stalkFunctor\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nF G : Presheaf C X\nf : F \u27f6 G\nx y : \u2191X\nh : x \u2933 y\nj\u271d : (OpenNhds y)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds y)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion y).op).obj F) j\u271d \u226b\n stalkSpecializes F h \u226b\n ((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op \u22d9 colim).map f =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds y)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion y).op).obj F) j\u271d \u226b\n ((whiskeringLeft (OpenNhds y)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion y).op \u22d9 colim).map f \u226b\n stalkSpecializes G h\n[PROOFSTEP]\nsimpa [stalkSpecializes] using by rfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nF G : Presheaf C X\nf : F \u27f6 G\nx y : \u2191X\nh : x \u2933 y\nj\u271d : (OpenNhds y)\u1d52\u1d56\n\u22a2 NatTrans.app f (op j\u271d.unop.obj) \u226b\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 G) (op { obj := j\u271d.unop.obj, property := (_ : x \u2208 j\u271d.unop.obj.carrier) }) =\n NatTrans.app f (op ((OpenNhds.inclusion y).obj j\u271d.unop)) \u226b\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 G) (op { obj := j\u271d.unop.obj, property := (_ : x \u2208 j\u271d.unop.obj.carrier) })\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C X\nx y : \u2191X\nh : x \u2933 y\n\u22a2 stalkSpecializes (f _* F) (_ : \u2191f x \u2933 \u2191f y) \u226b stalkPushforward C f F x =\n stalkPushforward C f F y \u226b stalkSpecializes F h\n[PROOFSTEP]\nchange (_ : colimit _ \u27f6 _) = (_ : colimit _ \u27f6 _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C X\nx y : \u2191X\nh : x \u2933 y\n\u22a2 stalkSpecializes (f _* F) (_ : \u2191f x \u2933 \u2191f y) \u226b stalkPushforward C f F x =\n stalkPushforward C f F y \u226b stalkSpecializes F h\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C X\nx y : \u2191X\nh : x \u2933 y\nj\u271d : (OpenNhds (\u2191f y))\u1d52\u1d56\n\u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f y))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f y)).op).obj (f _* F)) j\u271d \u226b\n stalkSpecializes (f _* F) (_ : \u2191f x \u2933 \u2191f y) \u226b stalkPushforward C f F x =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f y))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f y)).op).obj (f _* F))\n j\u271d \u226b\n stalkPushforward C f F y \u226b stalkSpecializes F h\n[PROOFSTEP]\ndelta stalkPushforward\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C X\nx y : \u2191X\nh : x \u2933 y\nj\u271d : (OpenNhds (\u2191f y))\u1d52\u1d56\n\u22a2 colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f y))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f y)).op).obj (f _* F)) j\u271d \u226b\n stalkSpecializes (f _* F) (_ : \u2191f x \u2933 \u2191f y) \u226b\n colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F) \u226b\n colimit.pre (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\n (OpenNhds.map f x).op =\n colimit.\u03b9 (((whiskeringLeft (OpenNhds (\u2191f y))\u1d52\u1d56 (Opens \u2191Y)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191f y)).op).obj (f _* F))\n j\u271d \u226b\n (colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f y).inv) F) \u226b\n colimit.pre (((whiskeringLeft (OpenNhds y)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion y).op).obj F)\n (OpenNhds.map f y).op) \u226b\n stalkSpecializes F h\n[PROOFSTEP]\nsimp only [stalkSpecializes, colimit.\u03b9_desc_assoc, colimit.\u03b9_map_assoc, colimit.\u03b9_pre, Category.assoc, colimit.pre_desc,\n colimit.\u03b9_desc]\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\nf : X \u27f6 Y\nF : Presheaf C X\nx y : \u2191X\nh : x \u2933 y\nj\u271d : (OpenNhds (\u2191f y))\u1d52\u1d56\n\u22a2 NatTrans.app (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F)\n (op { obj := j\u271d.unop.obj, property := (_ : \u2191f x \u2208 j\u271d.unop.obj.carrier) }) \u226b\n colimit.\u03b9 (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F)\n ((OpenNhds.map f x).op.obj (op { obj := j\u271d.unop.obj, property := (_ : \u2191f x \u2208 j\u271d.unop.obj.carrier) })) =\n NatTrans.app (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f y).inv) F) j\u271d \u226b\n NatTrans.app\n (Cocone.whisker (OpenNhds.map f y).op\n { pt := colim.obj (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F),\n \u03b9 :=\n NatTrans.mk fun U =>\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 F)\n (op { obj := U.unop.obj, property := (_ : x \u2208 U.unop.obj.carrier) }) }).\u03b9\n j\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : HasColimits C\u271d\nX\u271d Y Z : TopCat\nX : TopCat\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.182834, u_1} C\ninst\u271d : HasColimits C\nF : Presheaf C X\nx y : \u2191X\ne : Inseparable x y\n\u22a2 stalkSpecializes F (_ : nhds y \u2264 nhds x) \u226b stalkSpecializes F (_ : nhds x \u2264 nhds y) = \ud835\udfd9 (stalk F x)\n[PROOFSTEP]\nsimp\n[GOAL]\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : HasColimits C\u271d\nX\u271d Y Z : TopCat\nX : TopCat\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.182834, u_1} C\ninst\u271d : HasColimits C\nF : Presheaf C X\nx y : \u2191X\ne : Inseparable x y\n\u22a2 stalkSpecializes F (_ : nhds x \u2264 nhds y) \u226b stalkSpecializes F (_ : nhds y \u2264 nhds x) = \ud835\udfd9 (stalk F y)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasColimits C\nX Y Z : TopCat\ninst\u271d : ConcreteCategory C\nF : Presheaf C X\nU V : Opens \u2191X\nx : \u2191X\nhxU : x \u2208 U\nhxV : x \u2208 V\nW : Opens \u2191X\nhxW : x \u2208 W\niWU : W \u27f6 U\niWV : W \u27f6 V\nsU : (forget C).obj (F.obj (op U))\nsV : (forget C).obj (F.obj (op V))\nih : \u2191(F.map iWU.op) sU = \u2191(F.map iWV.op) sV\n\u22a2 \u2191(germ F { val := x, property := hxU }) sU = \u2191(germ F { val := x, property := hxV }) sV\n[PROOFSTEP]\nerw [\u2190 F.germ_res iWU \u27e8x, hxW\u27e9, \u2190 F.germ_res iWV \u27e8x, hxW\u27e9, comp_apply, comp_apply, ih]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF : Presheaf C X\nx : \u2191X\nt : (forget C).obj (stalk F x)\n\u22a2 \u2203 U m s, \u2191(germ F { val := x, property := m }) s = t\n[PROOFSTEP]\nobtain \u27e8U, s, e\u27e9 := Types.jointly_surjective.{v, v} _ (isColimitOfPreserves (forget C) (colimit.isColimit _)) t\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF : Presheaf C X\nx : \u2191X\nt : (forget C).obj (stalk F x)\nU : (OpenNhds x)\u1d52\u1d56\ns : (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F \u22d9 forget C).obj U\ne :\n NatTrans.app\n ((forget C).mapCocone\n (colimit.cocone (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F))).\u03b9\n U s =\n t\n\u22a2 \u2203 U m s, \u2191(germ F { val := x, property := m }) s = t\n[PROOFSTEP]\nrevert s e\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF : Presheaf C X\nx : \u2191X\nt : (forget C).obj (stalk F x)\nU : (OpenNhds x)\u1d52\u1d56\n\u22a2 \u2200 (s : (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F \u22d9 forget C).obj U),\n NatTrans.app\n ((forget C).mapCocone\n (colimit.cocone (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F))).\u03b9\n U s =\n t \u2192\n \u2203 U m s, \u2191(germ F { val := x, property := m }) s = t\n[PROOFSTEP]\ninduction U with\n| h U => ?_\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF : Presheaf C X\nx : \u2191X\nt : (forget C).obj (stalk F x)\nU : (OpenNhds x)\u1d52\u1d56\n\u22a2 \u2200 (s : (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F \u22d9 forget C).obj U),\n NatTrans.app\n ((forget C).mapCocone\n (colimit.cocone (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F))).\u03b9\n U s =\n t \u2192\n \u2203 U m s, \u2191(germ F { val := x, property := m }) s = t\n[PROOFSTEP]\ninduction U with\n| h U => ?_\n[GOAL]\ncase intro.intro.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF : Presheaf C X\nx : \u2191X\nt : (forget C).obj (stalk F x)\nU : OpenNhds x\n\u22a2 \u2200 (s : (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F \u22d9 forget C).obj (op U)),\n NatTrans.app\n ((forget C).mapCocone\n (colimit.cocone (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F))).\u03b9\n (op U) s =\n t \u2192\n \u2203 U m s, \u2191(germ F { val := x, property := m }) s = t\n[PROOFSTEP]\ncases' U with V m\n[GOAL]\ncase intro.intro.h.mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF : Presheaf C X\nx : \u2191X\nt : (forget C).obj (stalk F x)\nV : Opens \u2191X\nm : x \u2208 V\n\u22a2 \u2200\n (s :\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F \u22d9 forget C).obj\n (op { obj := V, property := m })),\n NatTrans.app\n ((forget C).mapCocone\n (colimit.cocone (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F))).\u03b9\n (op { obj := V, property := m }) s =\n t \u2192\n \u2203 U m s, \u2191(germ F { val := x, property := m }) s = t\n[PROOFSTEP]\nintro s e\n[GOAL]\ncase intro.intro.h.mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF : Presheaf C X\nx : \u2191X\nt : (forget C).obj (stalk F x)\nV : Opens \u2191X\nm : x \u2208 V\ns :\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F \u22d9 forget C).obj\n (op { obj := V, property := m })\ne :\n NatTrans.app\n ((forget C).mapCocone\n (colimit.cocone (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj F))).\u03b9\n (op { obj := V, property := m }) s =\n t\n\u22a2 \u2203 U m s, \u2191(germ F { val := x, property := m }) s = t\n[PROOFSTEP]\nexact \u27e8V, m, s, e\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF : Presheaf C X\nU V : Opens \u2191X\nx : \u2191X\nmU : x \u2208 U\nmV : x \u2208 V\ns : (forget C).obj (F.obj (op U))\nt : (forget C).obj (F.obj (op V))\nh : \u2191(germ F { val := x, property := mU }) s = \u2191(germ F { val := x, property := mV }) t\n\u22a2 \u2203 W _m iU iV, \u2191(F.map iU.op) s = \u2191(F.map iV.op) t\n[PROOFSTEP]\nobtain \u27e8W, iU, iV, e\u27e9 :=\n (Types.FilteredColimit.isColimit_eq_iff.{v, v} _\n (isColimitOfPreserves _ (colimit.isColimit ((OpenNhds.inclusion x).op \u22d9 F)))).mp\n h\n[GOAL]\ncase intro.intro.intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF : Presheaf C X\nU V : Opens \u2191X\nx : \u2191X\nmU : x \u2208 U\nmV : x \u2208 V\ns : (forget C).obj (F.obj (op U))\nt : (forget C).obj (F.obj (op V))\nh : \u2191(germ F { val := x, property := mU }) s = \u2191(germ F { val := x, property := mV }) t\nW : (OpenNhds x)\u1d52\u1d56\niU : op { obj := U, property := (_ : \u2191{ val := x, property := mU } \u2208 U) } \u27f6 W\niV : op { obj := V, property := (_ : \u2191{ val := x, property := mV } \u2208 V) } \u27f6 W\ne : (((OpenNhds.inclusion x).op \u22d9 F) \u22d9 forget C).map iU s = (((OpenNhds.inclusion x).op \u22d9 F) \u22d9 forget C).map iV t\n\u22a2 \u2203 W _m iU iV, \u2191(F.map iU.op) s = \u2191(F.map iV.op) t\n[PROOFSTEP]\nexact \u27e8(unop W).1, (unop W).2, iU.unop, iV.unop, e\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF G : Presheaf C X\nf : F \u27f6 G\nh : \u2200 (U : Opens \u2191X), Function.Injective \u2191(NatTrans.app f (op U))\nx : \u2191X\ns t : (forget C).obj ((stalkFunctor C x).obj F)\nhst : \u2191((stalkFunctor C x).map f) s = \u2191((stalkFunctor C x).map f) t\n\u22a2 s = t\n[PROOFSTEP]\nrcases germ_exist F x s with \u27e8U\u2081, hxU\u2081, s, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF G : Presheaf C X\nf : F \u27f6 G\nh : \u2200 (U : Opens \u2191X), Function.Injective \u2191(NatTrans.app f (op U))\nx : \u2191X\nt : (forget C).obj ((stalkFunctor C x).obj F)\nU\u2081 : Opens \u2191X\nhxU\u2081 : x \u2208 U\u2081\ns : (forget C).obj (F.obj (op U\u2081))\nhst : \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2081 }) s) = \u2191((stalkFunctor C x).map f) t\n\u22a2 \u2191(germ F { val := x, property := hxU\u2081 }) s = t\n[PROOFSTEP]\nrcases germ_exist F x t with \u27e8U\u2082, hxU\u2082, t, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF G : Presheaf C X\nf : F \u27f6 G\nh : \u2200 (U : Opens \u2191X), Function.Injective \u2191(NatTrans.app f (op U))\nx : \u2191X\nU\u2081 : Opens \u2191X\nhxU\u2081 : x \u2208 U\u2081\ns : (forget C).obj (F.obj (op U\u2081))\nU\u2082 : Opens \u2191X\nhxU\u2082 : x \u2208 U\u2082\nt : (forget C).obj (F.obj (op U\u2082))\nhst :\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2081 }) s) =\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2082 }) t)\n\u22a2 \u2191(germ F { val := x, property := hxU\u2081 }) s = \u2191(germ F { val := x, property := hxU\u2082 }) t\n[PROOFSTEP]\nerw [stalkFunctor_map_germ_apply _ \u27e8x, _\u27e9] at hst \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF G : Presheaf C X\nf : F \u27f6 G\nh : \u2200 (U : Opens \u2191X), Function.Injective \u2191(NatTrans.app f (op U))\nx : \u2191X\nU\u2081 : Opens \u2191X\nhxU\u2081 : x \u2208 U\u2081\ns : (forget C).obj (F.obj (op U\u2081))\nU\u2082 : Opens \u2191X\nhxU\u2082 : x \u2208 U\u2082\nt : (forget C).obj (F.obj (op U\u2082))\nhst\u271d :\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2081 }) s) =\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2082 }) t)\nhst :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2081 }).op \u22d9 G)\n (op { obj := U\u2081, property := (_ : \u2191{ val := x, property := hxU\u2081 } \u2208 U\u2081) }))\n (\u2191(NatTrans.app f (op U\u2081)) s) =\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2082 }) t)\n\u22a2 \u2191(germ F { val := x, property := hxU\u2081 }) s = \u2191(germ F { val := x, property := hxU\u2082 }) t\n[PROOFSTEP]\nerw [stalkFunctor_map_germ_apply _ \u27e8x, _\u27e9] at hst \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF G : Presheaf C X\nf : F \u27f6 G\nh : \u2200 (U : Opens \u2191X), Function.Injective \u2191(NatTrans.app f (op U))\nx : \u2191X\nU\u2081 : Opens \u2191X\nhxU\u2081 : x \u2208 U\u2081\ns : (forget C).obj (F.obj (op U\u2081))\nU\u2082 : Opens \u2191X\nhxU\u2082 : x \u2208 U\u2082\nt : (forget C).obj (F.obj (op U\u2082))\nhst\u271d\u00b9 :\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2081 }) s) =\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2082 }) t)\nhst\u271d :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2081 }).op \u22d9 G)\n (op { obj := U\u2081, property := (_ : \u2191{ val := x, property := hxU\u2081 } \u2208 U\u2081) }))\n (\u2191(NatTrans.app f (op U\u2081)) s) =\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2082 }) t)\nhst :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2081 }).op \u22d9 G)\n (op { obj := U\u2081, property := (_ : \u2191{ val := x, property := hxU\u2081 } \u2208 U\u2081) }))\n (\u2191(NatTrans.app f (op U\u2081)) s) =\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2082 }).op \u22d9 G)\n (op { obj := U\u2082, property := (_ : \u2191{ val := x, property := hxU\u2082 } \u2208 U\u2082) }))\n (\u2191(NatTrans.app f (op U\u2082)) t)\n\u22a2 \u2191(germ F { val := x, property := hxU\u2081 }) s = \u2191(germ F { val := x, property := hxU\u2082 }) t\n[PROOFSTEP]\nobtain \u27e8W, hxW, iWU\u2081, iWU\u2082, heq\u27e9 := G.germ_eq x hxU\u2081 hxU\u2082 _ _ hst\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF G : Presheaf C X\nf : F \u27f6 G\nh : \u2200 (U : Opens \u2191X), Function.Injective \u2191(NatTrans.app f (op U))\nx : \u2191X\nU\u2081 : Opens \u2191X\nhxU\u2081 : x \u2208 U\u2081\ns : (forget C).obj (F.obj (op U\u2081))\nU\u2082 : Opens \u2191X\nhxU\u2082 : x \u2208 U\u2082\nt : (forget C).obj (F.obj (op U\u2082))\nhst\u271d\u00b9 :\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2081 }) s) =\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2082 }) t)\nhst\u271d :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2081 }).op \u22d9 G)\n (op { obj := U\u2081, property := (_ : \u2191{ val := x, property := hxU\u2081 } \u2208 U\u2081) }))\n (\u2191(NatTrans.app f (op U\u2081)) s) =\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2082 }) t)\nhst :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2081 }).op \u22d9 G)\n (op { obj := U\u2081, property := (_ : \u2191{ val := x, property := hxU\u2081 } \u2208 U\u2081) }))\n (\u2191(NatTrans.app f (op U\u2081)) s) =\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2082 }).op \u22d9 G)\n (op { obj := U\u2082, property := (_ : \u2191{ val := x, property := hxU\u2082 } \u2208 U\u2082) }))\n (\u2191(NatTrans.app f (op U\u2082)) t)\nW : Opens \u2191X\nhxW : x \u2208 W\niWU\u2081 : W \u27f6 U\u2081\niWU\u2082 : W \u27f6 U\u2082\nheq : \u2191(G.map iWU\u2081.op) (\u2191(NatTrans.app f (op U\u2081)) s) = \u2191(G.map iWU\u2082.op) (\u2191(NatTrans.app f (op U\u2082)) t)\n\u22a2 \u2191(germ F { val := x, property := hxU\u2081 }) s = \u2191(germ F { val := x, property := hxU\u2082 }) t\n[PROOFSTEP]\nrw [\u2190 comp_apply, \u2190 comp_apply, \u2190 f.naturality, \u2190 f.naturality, comp_apply, comp_apply] at heq \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF G : Presheaf C X\nf : F \u27f6 G\nh : \u2200 (U : Opens \u2191X), Function.Injective \u2191(NatTrans.app f (op U))\nx : \u2191X\nU\u2081 : Opens \u2191X\nhxU\u2081 : x \u2208 U\u2081\ns : (forget C).obj (F.obj (op U\u2081))\nU\u2082 : Opens \u2191X\nhxU\u2082 : x \u2208 U\u2082\nt : (forget C).obj (F.obj (op U\u2082))\nhst\u271d\u00b9 :\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2081 }) s) =\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2082 }) t)\nhst\u271d :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2081 }).op \u22d9 G)\n (op { obj := U\u2081, property := (_ : \u2191{ val := x, property := hxU\u2081 } \u2208 U\u2081) }))\n (\u2191(NatTrans.app f (op U\u2081)) s) =\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2082 }) t)\nhst :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2081 }).op \u22d9 G)\n (op { obj := U\u2081, property := (_ : \u2191{ val := x, property := hxU\u2081 } \u2208 U\u2081) }))\n (\u2191(NatTrans.app f (op U\u2081)) s) =\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2082 }).op \u22d9 G)\n (op { obj := U\u2082, property := (_ : \u2191{ val := x, property := hxU\u2082 } \u2208 U\u2082) }))\n (\u2191(NatTrans.app f (op U\u2082)) t)\nW : Opens \u2191X\nhxW : x \u2208 W\niWU\u2081 : W \u27f6 U\u2081\niWU\u2082 : W \u27f6 U\u2082\nheq : \u2191(NatTrans.app f (op W)) (\u2191(F.map iWU\u2081.op) s) = \u2191(NatTrans.app f (op W)) (\u2191(F.map iWU\u2082.op) t)\n\u22a2 \u2191(germ F { val := x, property := hxU\u2081 }) s = \u2191(germ F { val := x, property := hxU\u2082 }) t\n[PROOFSTEP]\nreplace heq := h W heq\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF G : Presheaf C X\nf : F \u27f6 G\nh : \u2200 (U : Opens \u2191X), Function.Injective \u2191(NatTrans.app f (op U))\nx : \u2191X\nU\u2081 : Opens \u2191X\nhxU\u2081 : x \u2208 U\u2081\ns : (forget C).obj (F.obj (op U\u2081))\nU\u2082 : Opens \u2191X\nhxU\u2082 : x \u2208 U\u2082\nt : (forget C).obj (F.obj (op U\u2082))\nhst\u271d\u00b9 :\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2081 }) s) =\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2082 }) t)\nhst\u271d :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2081 }).op \u22d9 G)\n (op { obj := U\u2081, property := (_ : \u2191{ val := x, property := hxU\u2081 } \u2208 U\u2081) }))\n (\u2191(NatTrans.app f (op U\u2081)) s) =\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2082 }) t)\nhst :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2081 }).op \u22d9 G)\n (op { obj := U\u2081, property := (_ : \u2191{ val := x, property := hxU\u2081 } \u2208 U\u2081) }))\n (\u2191(NatTrans.app f (op U\u2081)) s) =\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2082 }).op \u22d9 G)\n (op { obj := U\u2082, property := (_ : \u2191{ val := x, property := hxU\u2082 } \u2208 U\u2082) }))\n (\u2191(NatTrans.app f (op U\u2082)) t)\nW : Opens \u2191X\nhxW : x \u2208 W\niWU\u2081 : W \u27f6 U\u2081\niWU\u2082 : W \u27f6 U\u2082\nheq : \u2191(F.map iWU\u2081.op) s = \u2191(F.map iWU\u2082.op) t\n\u22a2 \u2191(germ F { val := x, property := hxU\u2081 }) s = \u2191(germ F { val := x, property := hxU\u2082 }) t\n[PROOFSTEP]\nconvert congr_arg (F.germ \u27e8x, hxW\u27e9) heq using 1\n[GOAL]\ncase h.e'_2.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF G : Presheaf C X\nf : F \u27f6 G\nh : \u2200 (U : Opens \u2191X), Function.Injective \u2191(NatTrans.app f (op U))\nx : \u2191X\nU\u2081 : Opens \u2191X\nhxU\u2081 : x \u2208 U\u2081\ns : (forget C).obj (F.obj (op U\u2081))\nU\u2082 : Opens \u2191X\nhxU\u2082 : x \u2208 U\u2082\nt : (forget C).obj (F.obj (op U\u2082))\nhst\u271d\u00b9 :\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2081 }) s) =\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2082 }) t)\nhst\u271d :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2081 }).op \u22d9 G)\n (op { obj := U\u2081, property := (_ : \u2191{ val := x, property := hxU\u2081 } \u2208 U\u2081) }))\n (\u2191(NatTrans.app f (op U\u2081)) s) =\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2082 }) t)\nhst :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2081 }).op \u22d9 G)\n (op { obj := U\u2081, property := (_ : \u2191{ val := x, property := hxU\u2081 } \u2208 U\u2081) }))\n (\u2191(NatTrans.app f (op U\u2081)) s) =\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2082 }).op \u22d9 G)\n (op { obj := U\u2082, property := (_ : \u2191{ val := x, property := hxU\u2082 } \u2208 U\u2082) }))\n (\u2191(NatTrans.app f (op U\u2082)) t)\nW : Opens \u2191X\nhxW : x \u2208 W\niWU\u2081 : W \u27f6 U\u2081\niWU\u2082 : W \u27f6 U\u2082\nheq : \u2191(F.map iWU\u2081.op) s = \u2191(F.map iWU\u2082.op) t\ne_1\u271d : (forget C).obj ((stalkFunctor C x).obj F) = (forget C).obj (stalk F \u2191{ val := x, property := hxW })\n\u22a2 \u2191(germ F { val := x, property := hxU\u2081 }) s = \u2191(germ F { val := x, property := hxW }) (\u2191(F.map iWU\u2081.op) s)\ncase h.e'_3.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u00b9 : ConcreteCategory C\ninst\u271d : PreservesFilteredColimits (forget C)\nF G : Presheaf C X\nf : F \u27f6 G\nh : \u2200 (U : Opens \u2191X), Function.Injective \u2191(NatTrans.app f (op U))\nx : \u2191X\nU\u2081 : Opens \u2191X\nhxU\u2081 : x \u2208 U\u2081\ns : (forget C).obj (F.obj (op U\u2081))\nU\u2082 : Opens \u2191X\nhxU\u2082 : x \u2208 U\u2082\nt : (forget C).obj (F.obj (op U\u2082))\nhst\u271d\u00b9 :\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2081 }) s) =\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2082 }) t)\nhst\u271d :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2081 }).op \u22d9 G)\n (op { obj := U\u2081, property := (_ : \u2191{ val := x, property := hxU\u2081 } \u2208 U\u2081) }))\n (\u2191(NatTrans.app f (op U\u2081)) s) =\n \u2191((stalkFunctor C x).map f) (\u2191(germ F { val := x, property := hxU\u2082 }) t)\nhst :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2081 }).op \u22d9 G)\n (op { obj := U\u2081, property := (_ : \u2191{ val := x, property := hxU\u2081 } \u2208 U\u2081) }))\n (\u2191(NatTrans.app f (op U\u2081)) s) =\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := x, property := hxU\u2082 }).op \u22d9 G)\n (op { obj := U\u2082, property := (_ : \u2191{ val := x, property := hxU\u2082 } \u2208 U\u2082) }))\n (\u2191(NatTrans.app f (op U\u2082)) t)\nW : Opens \u2191X\nhxW : x \u2208 W\niWU\u2081 : W \u27f6 U\u2081\niWU\u2082 : W \u27f6 U\u2082\nheq : \u2191(F.map iWU\u2081.op) s = \u2191(F.map iWU\u2082.op) t\ne_1\u271d : (forget C).obj ((stalkFunctor C x).obj F) = (forget C).obj (stalk F \u2191{ val := x, property := hxW })\n\u22a2 \u2191(germ F { val := x, property := hxU\u2082 }) t = \u2191(germ F { val := x, property := hxW }) (\u2191(F.map iWU\u2082.op) t)\n[PROOFSTEP]\nexacts [(F.germ_res_apply iWU\u2081 \u27e8x, hxW\u27e9 s).symm, (F.germ_res_apply iWU\u2082 \u27e8x, hxW\u27e9 t).symm]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF : Sheaf C X\nU : Opens \u2191X\ns t : (forget C).obj (F.val.obj (op U))\nh : \u2200 (x : { x // x \u2208 U }), \u2191(germ (Sheaf.presheaf F) x) s = \u2191(germ (Sheaf.presheaf F) x) t\n\u22a2 s = t\n[PROOFSTEP]\nchoose V m i\u2081 i\u2082 heq using fun x : U =>\n F.presheaf.germ_eq x.1 x.2 x.2 s t\n (h x)\n -- Since `F` is a sheaf, we can prove the equality locally, if we can show that these\n -- neighborhoods form a cover of `U`.\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF : Sheaf C X\nU : Opens \u2191X\ns t : (forget C).obj (F.val.obj (op U))\nh : \u2200 (x : { x // x \u2208 U }), \u2191(germ (Sheaf.presheaf F) x) s = \u2191(germ (Sheaf.presheaf F) x) t\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nm : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\ni\u2081 i\u2082 : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nheq : \u2200 (x : { x // x \u2208 U }), \u2191((Sheaf.presheaf F).map (i\u2081 x).op) s = \u2191((Sheaf.presheaf F).map (i\u2082 x).op) t\n\u22a2 s = t\n[PROOFSTEP]\napply F.eq_of_locally_eq' V U i\u2081\n[GOAL]\ncase hcover\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF : Sheaf C X\nU : Opens \u2191X\ns t : (forget C).obj (F.val.obj (op U))\nh : \u2200 (x : { x // x \u2208 U }), \u2191(germ (Sheaf.presheaf F) x) s = \u2191(germ (Sheaf.presheaf F) x) t\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nm : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\ni\u2081 i\u2082 : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nheq : \u2200 (x : { x // x \u2208 U }), \u2191((Sheaf.presheaf F).map (i\u2081 x).op) s = \u2191((Sheaf.presheaf F).map (i\u2082 x).op) t\n\u22a2 U \u2264 iSup V\n[PROOFSTEP]\nintro x hxU\n[GOAL]\ncase hcover\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF : Sheaf C X\nU : Opens \u2191X\ns t : (forget C).obj (F.val.obj (op U))\nh : \u2200 (x : { x // x \u2208 U }), \u2191(germ (Sheaf.presheaf F) x) s = \u2191(germ (Sheaf.presheaf F) x) t\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nm : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\ni\u2081 i\u2082 : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nheq : \u2200 (x : { x // x \u2208 U }), \u2191((Sheaf.presheaf F).map (i\u2081 x).op) s = \u2191((Sheaf.presheaf F).map (i\u2082 x).op) t\nx : \u2191X\nhxU : x \u2208 \u2191U\n\u22a2 x \u2208 \u2191(iSup V)\n[PROOFSTEP]\nerw [Opens.mem_iSup]\n[GOAL]\ncase hcover\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF : Sheaf C X\nU : Opens \u2191X\ns t : (forget C).obj (F.val.obj (op U))\nh : \u2200 (x : { x // x \u2208 U }), \u2191(germ (Sheaf.presheaf F) x) s = \u2191(germ (Sheaf.presheaf F) x) t\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nm : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\ni\u2081 i\u2082 : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nheq : \u2200 (x : { x // x \u2208 U }), \u2191((Sheaf.presheaf F).map (i\u2081 x).op) s = \u2191((Sheaf.presheaf F).map (i\u2082 x).op) t\nx : \u2191X\nhxU : x \u2208 \u2191U\n\u22a2 \u2203 i, x \u2208 V i\n[PROOFSTEP]\nexact \u27e8\u27e8x, hxU\u27e9, m \u27e8x, hxU\u27e9\u27e9\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF : Sheaf C X\nU : Opens \u2191X\ns t : (forget C).obj (F.val.obj (op U))\nh : \u2200 (x : { x // x \u2208 U }), \u2191(germ (Sheaf.presheaf F) x) s = \u2191(germ (Sheaf.presheaf F) x) t\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nm : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\ni\u2081 i\u2082 : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nheq : \u2200 (x : { x // x \u2208 U }), \u2191((Sheaf.presheaf F).map (i\u2081 x).op) s = \u2191((Sheaf.presheaf F).map (i\u2082 x).op) t\n\u22a2 \u2200 (i : { x // x \u2208 U }), \u2191(F.val.map (i\u2081 i).op) s = \u2191(F.val.map (i\u2081 i).op) t\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF : Sheaf C X\nU : Opens \u2191X\ns t : (forget C).obj (F.val.obj (op U))\nh : \u2200 (x : { x // x \u2208 U }), \u2191(germ (Sheaf.presheaf F) x) s = \u2191(germ (Sheaf.presheaf F) x) t\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nm : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\ni\u2081 i\u2082 : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nheq : \u2200 (x : { x // x \u2208 U }), \u2191((Sheaf.presheaf F).map (i\u2081 x).op) s = \u2191((Sheaf.presheaf F).map (i\u2082 x).op) t\nx : { x // x \u2208 U }\n\u22a2 \u2191(F.val.map (i\u2081 x).op) s = \u2191(F.val.map (i\u2081 x).op) t\n[PROOFSTEP]\nrw [heq, Subsingleton.elim (i\u2081 x) (i\u2082 x)]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF : Sheaf C X\nG : Presheaf C X\nf : F.val \u27f6 G\nU : Opens \u2191X\nh : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f)\ns t : (forget C).obj (F.val.obj (op U))\nhst : \u2191(NatTrans.app f (op U)) s = \u2191(NatTrans.app f (op U)) t\nx : { x // x \u2208 U }\n\u22a2 \u2191((stalkFunctor C \u2191x).map f) (\u2191(germ (Sheaf.presheaf F) x) s) =\n \u2191((stalkFunctor C \u2191x).map f) (\u2191(germ (Sheaf.presheaf F) x) t)\n[PROOFSTEP]\nerw [stalkFunctor_map_germ_apply, stalkFunctor_map_germ_apply, hst]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\n\u22a2 Function.Surjective \u2191(NatTrans.app f.val (op U))\n[PROOFSTEP]\nintro t\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\n\u22a2 \u2203 a, \u2191(NatTrans.app f.val (op U)) a = t\n[PROOFSTEP]\nchoose V mV iVU sf heq using hsurj t\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\n\u22a2 \u2203 a, \u2191(NatTrans.app f.val (op U)) a = t\n[PROOFSTEP]\nhave V_cover : U \u2264 iSup V := by\n intro x hxU\n erw [Opens.mem_iSup]\n exact \u27e8\u27e8x, hxU\u27e9, mV \u27e8x, hxU\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\n\u22a2 U \u2264 iSup V\n[PROOFSTEP]\nintro x hxU\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nx : \u2191X\nhxU : x \u2208 \u2191U\n\u22a2 x \u2208 \u2191(iSup V)\n[PROOFSTEP]\nerw [Opens.mem_iSup]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nx : \u2191X\nhxU : x \u2208 \u2191U\n\u22a2 \u2203 i, x \u2208 V i\n[PROOFSTEP]\nexact \u27e8\u27e8x, hxU\u27e9, mV \u27e8x, hxU\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nV_cover : U \u2264 iSup V\n\u22a2 \u2203 a, \u2191(NatTrans.app f.val (op U)) a = t\n[PROOFSTEP]\nsuffices IsCompatible F.val V sf by\n -- Since `F` is a sheaf, we can glue all the local preimages together to get a global preimage.\n obtain \u27e8s, s_spec, -\u27e9 := F.existsUnique_gluing' V U iVU V_cover sf this\n \u00b7 use s\n apply G.eq_of_locally_eq' V U iVU V_cover\n intro x\n rw [\u2190 comp_apply, \u2190 f.1.naturality, comp_apply, s_spec, heq]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nV_cover : U \u2264 iSup V\nthis : IsCompatible F.val V sf\n\u22a2 \u2203 a, \u2191(NatTrans.app f.val (op U)) a = t\n[PROOFSTEP]\nobtain \u27e8s, s_spec, -\u27e9 := F.existsUnique_gluing' V U iVU V_cover sf this\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nV_cover : U \u2264 iSup V\nthis : IsCompatible F.val V sf\ns : (forget C).obj (F.val.obj (op U))\ns_spec : \u2200 (i : { x // x \u2208 U }), \u2191(F.val.map (iVU i).op) s = sf i\n\u22a2 \u2203 a, \u2191(NatTrans.app f.val (op U)) a = t\n[PROOFSTEP]\nuse s\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nV_cover : U \u2264 iSup V\nthis : IsCompatible F.val V sf\ns : (forget C).obj (F.val.obj (op U))\ns_spec : \u2200 (i : { x // x \u2208 U }), \u2191(F.val.map (iVU i).op) s = sf i\n\u22a2 \u2191(NatTrans.app f.val (op U)) s = t\n[PROOFSTEP]\napply G.eq_of_locally_eq' V U iVU V_cover\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nV_cover : U \u2264 iSup V\nthis : IsCompatible F.val V sf\ns : (forget C).obj (F.val.obj (op U))\ns_spec : \u2200 (i : { x // x \u2208 U }), \u2191(F.val.map (iVU i).op) s = sf i\n\u22a2 \u2200 (i : { x // x \u2208 U }), \u2191(G.val.map (iVU i).op) (\u2191(NatTrans.app f.val (op U)) s) = \u2191(G.val.map (iVU i).op) t\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nV_cover : U \u2264 iSup V\nthis : IsCompatible F.val V sf\ns : (forget C).obj (F.val.obj (op U))\ns_spec : \u2200 (i : { x // x \u2208 U }), \u2191(F.val.map (iVU i).op) s = sf i\nx : { x // x \u2208 U }\n\u22a2 \u2191(G.val.map (iVU x).op) (\u2191(NatTrans.app f.val (op U)) s) = \u2191(G.val.map (iVU x).op) t\n[PROOFSTEP]\nrw [\u2190 comp_apply, \u2190 f.1.naturality, comp_apply, s_spec, heq]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nV_cover : U \u2264 iSup V\n\u22a2 IsCompatible F.val V sf\n[PROOFSTEP]\nintro x y\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nV_cover : U \u2264 iSup V\nx y : { x // x \u2208 U }\n\u22a2 \u2191(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x) = \u2191(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y)\n[PROOFSTEP]\napply section_ext\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nV_cover : U \u2264 iSup V\nx y : { x // x \u2208 U }\n\u22a2 \u2200 (x_1 : { x_1 // x_1 \u2208 V x \u2293 V y }),\n \u2191(germ (Sheaf.presheaf F) x_1) (\u2191(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x)) =\n \u2191(germ (Sheaf.presheaf F) x_1) (\u2191(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y))\n[PROOFSTEP]\nintro z\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nV_cover : U \u2264 iSup V\nx y : { x // x \u2208 U }\nz : { x_1 // x_1 \u2208 V x \u2293 V y }\n\u22a2 \u2191(germ (Sheaf.presheaf F) z) (\u2191(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x)) =\n \u2191(germ (Sheaf.presheaf F) z) (\u2191(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y))\n[PROOFSTEP]\napply hinj \u27e8z, (iVU x).le ((inf_le_left : V x \u2293 V y \u2264 V x) z.2)\u27e9\n[GOAL]\ncase h.a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nV_cover : U \u2264 iSup V\nx y : { x // x \u2208 U }\nz : { x_1 // x_1 \u2208 V x \u2293 V y }\n\u22a2 \u2191((stalkFunctor C \u2191{ val := \u2191z, property := (_ : \u2191z \u2208 \u2191U) }).map f.val)\n (\u2191(germ (Sheaf.presheaf F) z) (\u2191(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x))) =\n \u2191((stalkFunctor C \u2191{ val := \u2191z, property := (_ : \u2191z \u2208 \u2191U) }).map f.val)\n (\u2191(germ (Sheaf.presheaf F) z) (\u2191(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y)))\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h.a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nV_cover : U \u2264 iSup V\nx y : { x // x \u2208 U }\nz : { x_1 // x_1 \u2208 V x \u2293 V y }\n\u22a2 \u2191((stalkFunctor C \u2191z).map f.val)\n (\u2191(germ (Sheaf.presheaf F) z) (\u2191(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x))) =\n \u2191((stalkFunctor C \u2191z).map f.val)\n (\u2191(germ (Sheaf.presheaf F) z) (\u2191(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y)))\n[PROOFSTEP]\nerw [stalkFunctor_map_germ_apply, stalkFunctor_map_germ_apply]\n[GOAL]\ncase h.a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nV_cover : U \u2264 iSup V\nx y : { x // x \u2208 U }\nz : { x_1 // x_1 \u2208 V x \u2293 V y }\n\u22a2 \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191z).op \u22d9 G.val) (op { obj := V x \u2293 V y, property := (_ : \u2191z \u2208 V x \u2293 V y) }))\n (\u2191(NatTrans.app f.val (op (V x \u2293 V y))) (\u2191(F.val.map (Opens.infLELeft (V x) (V y)).op) (sf x))) =\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191z).op \u22d9 G.val) (op { obj := V x \u2293 V y, property := (_ : \u2191z \u2208 V x \u2293 V y) }))\n (\u2191(NatTrans.app f.val (op (V x \u2293 V y))) (\u2191(F.val.map (Opens.infLERight (V x) (V y)).op) (sf y)))\n[PROOFSTEP]\nsimp_rw [\u2190 comp_apply, f.1.naturality, comp_apply, heq, \u2190 comp_apply, \u2190 G.1.map_comp]\n[GOAL]\ncase h.a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nhinj : \u2200 (x : { x // x \u2208 U }), Function.Injective \u2191((stalkFunctor C \u2191x).map f.val)\nhsurj :\n \u2200 (t : (forget C).obj (G.val.obj (op U))) (x : { x // x \u2208 U }),\n \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\nt : (forget C).obj (G.val.obj (op U))\nV : { x // x \u2208 U } \u2192 Opens \u2191X\nmV : \u2200 (x : { x // x \u2208 U }), \u2191x \u2208 V x\niVU : (x : { x // x \u2208 U }) \u2192 V x \u27f6 U\nsf : (x : { x // x \u2208 U }) \u2192 (forget C).obj (F.val.obj (op (V x)))\nheq : \u2200 (x : { x // x \u2208 U }), \u2191(NatTrans.app f.val (op (V x))) (sf x) = \u2191(G.val.map (iVU x).op) t\nV_cover : U \u2264 iSup V\nx y : { x // x \u2208 U }\nz : { x_1 // x_1 \u2208 V x \u2293 V y }\n\u22a2 \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191z).op \u22d9 G.val) (op { obj := V x \u2293 V y, property := (_ : \u2191z \u2208 V x \u2293 V y) }))\n (\u2191(G.val.map ((iVU x).op \u226b (Opens.infLELeft (V x) (V y)).op)) t) =\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191z).op \u22d9 G.val) (op { obj := V x \u2293 V y, property := (_ : \u2191z \u2208 V x \u2293 V y) }))\n (\u2191(G.val.map ((iVU y).op \u226b (Opens.infLERight (V x) (V y)).op)) t)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nh : \u2200 (x : { x // x \u2208 U }), Function.Bijective \u2191((stalkFunctor C \u2191x).map f.val)\n\u22a2 Function.Surjective \u2191(NatTrans.app f.val (op U))\n[PROOFSTEP]\nrefine'\n app_surjective_of_injective_of_locally_surjective f U (fun x => (h x).1) fun t x =>\n _\n -- Now we need to prove our initial claim: That we can find preimages of `t` locally.\n -- Since `f` is surjective on stalks, we can find a preimage `s\u2080` of the germ of `t` at `x`\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nh : \u2200 (x : { x // x \u2208 U }), Function.Bijective \u2191((stalkFunctor C \u2191x).map f.val)\nt : (forget C).obj (G.val.obj (op U))\nx : { x // x \u2208 U }\n\u22a2 \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\n[PROOFSTEP]\nobtain \u27e8s\u2080, hs\u2080\u27e9 :=\n (h x).2\n (G.presheaf.germ x t)\n -- ... and this preimage must come from some section `s\u2081` defined on some open neighborhood `V\u2081`\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nh : \u2200 (x : { x // x \u2208 U }), Function.Bijective \u2191((stalkFunctor C \u2191x).map f.val)\nt : (forget C).obj (G.val.obj (op U))\nx : { x // x \u2208 U }\ns\u2080 : (forget C).obj ((stalkFunctor C \u2191x).obj F.val)\nhs\u2080 : \u2191((stalkFunctor C \u2191x).map f.val) s\u2080 = \u2191(germ (Sheaf.presheaf G) x) t\n\u22a2 \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\n[PROOFSTEP]\nobtain \u27e8V\u2081, hxV\u2081, s\u2081, hs\u2081\u27e9 := F.presheaf.germ_exist x.1 s\u2080\n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nh : \u2200 (x : { x // x \u2208 U }), Function.Bijective \u2191((stalkFunctor C \u2191x).map f.val)\nt : (forget C).obj (G.val.obj (op U))\nx : { x // x \u2208 U }\ns\u2080 : (forget C).obj ((stalkFunctor C \u2191x).obj F.val)\nhs\u2080 : \u2191((stalkFunctor C \u2191x).map f.val) s\u2080 = \u2191(germ (Sheaf.presheaf G) x) t\nV\u2081 : Opens \u2191X\nhxV\u2081 : \u2191x \u2208 V\u2081\ns\u2081 : (forget C).obj ((Sheaf.presheaf F).obj (op V\u2081))\nhs\u2081 : \u2191(germ (Sheaf.presheaf F) { val := \u2191x, property := hxV\u2081 }) s\u2081 = s\u2080\n\u22a2 \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\n[PROOFSTEP]\nsubst hs\u2081\n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nh : \u2200 (x : { x // x \u2208 U }), Function.Bijective \u2191((stalkFunctor C \u2191x).map f.val)\nt : (forget C).obj (G.val.obj (op U))\nx : { x // x \u2208 U }\nV\u2081 : Opens \u2191X\nhxV\u2081 : \u2191x \u2208 V\u2081\ns\u2081 : (forget C).obj ((Sheaf.presheaf F).obj (op V\u2081))\nhs\u2080 :\n \u2191((stalkFunctor C \u2191x).map f.val) (\u2191(germ (Sheaf.presheaf F) { val := \u2191x, property := hxV\u2081 }) s\u2081) =\n \u2191(germ (Sheaf.presheaf G) x) t\n\u22a2 \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\n[PROOFSTEP]\nrename' hs\u2080 => hs\u2081\n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nh : \u2200 (x : { x // x \u2208 U }), Function.Bijective \u2191((stalkFunctor C \u2191x).map f.val)\nt : (forget C).obj (G.val.obj (op U))\nx : { x // x \u2208 U }\nV\u2081 : Opens \u2191X\nhxV\u2081 : \u2191x \u2208 V\u2081\ns\u2081 : (forget C).obj ((Sheaf.presheaf F).obj (op V\u2081))\nhs\u2081 :\n \u2191((stalkFunctor C \u2191x).map f.val) (\u2191(germ (Sheaf.presheaf F) { val := \u2191x, property := hxV\u2081 }) s\u2081) =\n \u2191(germ (Sheaf.presheaf G) x) t\n\u22a2 \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\n[PROOFSTEP]\nerw [stalkFunctor_map_germ_apply V\u2081 \u27e8x.1, hxV\u2081\u27e9 f.1 s\u2081] at hs\u2081 \n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nh : \u2200 (x : { x // x \u2208 U }), Function.Bijective \u2191((stalkFunctor C \u2191x).map f.val)\nt : (forget C).obj (G.val.obj (op U))\nx : { x // x \u2208 U }\nV\u2081 : Opens \u2191X\nhxV\u2081 : \u2191x \u2208 V\u2081\ns\u2081 : (forget C).obj ((Sheaf.presheaf F).obj (op V\u2081))\nhs\u2081 :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := \u2191x, property := hxV\u2081 }).op \u22d9 G.val)\n (op { obj := V\u2081, property := (_ : \u2191{ val := \u2191x, property := hxV\u2081 } \u2208 V\u2081) }))\n (\u2191(NatTrans.app f.val (op V\u2081)) s\u2081) =\n \u2191(germ (Sheaf.presheaf G) x) t\n\u22a2 \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\n[PROOFSTEP]\nobtain \u27e8V\u2082, hxV\u2082, iV\u2082V\u2081, iV\u2082U, heq\u27e9 := G.presheaf.germ_eq x.1 hxV\u2081 x.2 _ _ hs\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nh : \u2200 (x : { x // x \u2208 U }), Function.Bijective \u2191((stalkFunctor C \u2191x).map f.val)\nt : (forget C).obj (G.val.obj (op U))\nx : { x // x \u2208 U }\nV\u2081 : Opens \u2191X\nhxV\u2081 : \u2191x \u2208 V\u2081\ns\u2081 : (forget C).obj ((Sheaf.presheaf F).obj (op V\u2081))\nhs\u2081 :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := \u2191x, property := hxV\u2081 }).op \u22d9 G.val)\n (op { obj := V\u2081, property := (_ : \u2191{ val := \u2191x, property := hxV\u2081 } \u2208 V\u2081) }))\n (\u2191(NatTrans.app f.val (op V\u2081)) s\u2081) =\n \u2191(germ (Sheaf.presheaf G) x) t\nV\u2082 : Opens \u2191X\nhxV\u2082 : \u2191x \u2208 V\u2082\niV\u2082V\u2081 : V\u2082 \u27f6 V\u2081\niV\u2082U : V\u2082 \u27f6 U\nheq : \u2191((Sheaf.presheaf G).map iV\u2082V\u2081.op) (\u2191(NatTrans.app f.val (op V\u2081)) s\u2081) = \u2191((Sheaf.presheaf G).map iV\u2082U.op) t\n\u22a2 \u2203 V x iVU s, \u2191(NatTrans.app f.val (op V)) s = \u2191(G.val.map iVU.op) t\n[PROOFSTEP]\nuse V\u2082, hxV\u2082, iV\u2082U, F.1.map iV\u2082V\u2081.op s\u2081\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : PreservesLimits (forget C)\ninst\u271d : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\nh : \u2200 (x : { x // x \u2208 U }), Function.Bijective \u2191((stalkFunctor C \u2191x).map f.val)\nt : (forget C).obj (G.val.obj (op U))\nx : { x // x \u2208 U }\nV\u2081 : Opens \u2191X\nhxV\u2081 : \u2191x \u2208 V\u2081\ns\u2081 : (forget C).obj ((Sheaf.presheaf F).obj (op V\u2081))\nhs\u2081 :\n \u2191(colimit.\u03b9 ((OpenNhds.inclusion \u2191{ val := \u2191x, property := hxV\u2081 }).op \u22d9 G.val)\n (op { obj := V\u2081, property := (_ : \u2191{ val := \u2191x, property := hxV\u2081 } \u2208 V\u2081) }))\n (\u2191(NatTrans.app f.val (op V\u2081)) s\u2081) =\n \u2191(germ (Sheaf.presheaf G) x) t\nV\u2082 : Opens \u2191X\nhxV\u2082 : \u2191x \u2208 V\u2082\niV\u2082V\u2081 : V\u2082 \u27f6 V\u2081\niV\u2082U : V\u2082 \u27f6 U\nheq : \u2191((Sheaf.presheaf G).map iV\u2082V\u2081.op) (\u2191(NatTrans.app f.val (op V\u2081)) s\u2081) = \u2191((Sheaf.presheaf G).map iV\u2082U.op) t\n\u22a2 \u2191(NatTrans.app f.val (op V\u2082)) (\u2191(F.val.map iV\u2082V\u2081.op) s\u2081) = \u2191(G.val.map iV\u2082U.op) t\n[PROOFSTEP]\nrw [\u2190 comp_apply, f.1.naturality, comp_apply, heq]\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\ninst\u271d\u2076 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2075 : ConcreteCategory C\ninst\u271d\u2074 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b3 : HasLimits C\ninst\u271d\u00b2 : PreservesLimits (forget C)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\ninst\u271d : \u2200 (x : { x // x \u2208 U }), IsIso ((stalkFunctor C \u2191x).map f.val)\n\u22a2 IsIso (NatTrans.app f.val (op U))\n[PROOFSTEP]\nsuffices IsIso ((forget C).map (f.1.app (op U))) by exact isIso_of_reflects_iso (f.1.app (op U)) (forget C)\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\ninst\u271d\u2076 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2075 : ConcreteCategory C\ninst\u271d\u2074 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b3 : HasLimits C\ninst\u271d\u00b2 : PreservesLimits (forget C)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\ninst\u271d : \u2200 (x : { x // x \u2208 U }), IsIso ((stalkFunctor C \u2191x).map f.val)\nthis : IsIso ((forget C).map (NatTrans.app f.val (op U)))\n\u22a2 IsIso (NatTrans.app f.val (op U))\n[PROOFSTEP]\nexact isIso_of_reflects_iso (f.1.app (op U)) (forget C)\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\ninst\u271d\u2076 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2075 : ConcreteCategory C\ninst\u271d\u2074 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b3 : HasLimits C\ninst\u271d\u00b2 : PreservesLimits (forget C)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\ninst\u271d : \u2200 (x : { x // x \u2208 U }), IsIso ((stalkFunctor C \u2191x).map f.val)\n\u22a2 IsIso ((forget C).map (NatTrans.app f.val (op U)))\n[PROOFSTEP]\nrw [isIso_iff_bijective]\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\ninst\u271d\u2076 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2075 : ConcreteCategory C\ninst\u271d\u2074 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b3 : HasLimits C\ninst\u271d\u00b2 : PreservesLimits (forget C)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\ninst\u271d : \u2200 (x : { x // x \u2208 U }), IsIso ((stalkFunctor C \u2191x).map f.val)\n\u22a2 Function.Bijective ((forget C).map (NatTrans.app f.val (op U)))\n[PROOFSTEP]\napply app_bijective_of_stalkFunctor_map_bijective\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\ninst\u271d\u2076 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2075 : ConcreteCategory C\ninst\u271d\u2074 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b3 : HasLimits C\ninst\u271d\u00b2 : PreservesLimits (forget C)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\ninst\u271d : \u2200 (x : { x // x \u2208 U }), IsIso ((stalkFunctor C \u2191x).map f.val)\n\u22a2 \u2200 (x : { x // x \u2208 U }), Function.Bijective \u2191((stalkFunctor C \u2191x).map f.val)\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\ninst\u271d\u2076 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2075 : ConcreteCategory C\ninst\u271d\u2074 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b3 : HasLimits C\ninst\u271d\u00b2 : PreservesLimits (forget C)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\ninst\u271d : \u2200 (x : { x // x \u2208 U }), IsIso ((stalkFunctor C \u2191x).map f.val)\nx : { x // x \u2208 U }\n\u22a2 Function.Bijective \u2191((stalkFunctor C \u2191x).map f.val)\n[PROOFSTEP]\napply (isIso_iff_bijective _).mp\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\ninst\u271d\u2076 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2075 : ConcreteCategory C\ninst\u271d\u2074 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b3 : HasLimits C\ninst\u271d\u00b2 : PreservesLimits (forget C)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\nU : Opens \u2191X\ninst\u271d : \u2200 (x : { x // x \u2208 U }), IsIso ((stalkFunctor C \u2191x).map f.val)\nx : { x // x \u2208 U }\n\u22a2 IsIso \u2191((stalkFunctor C \u2191x).map f.val)\n[PROOFSTEP]\nexact Functor.map_isIso (forget C) ((stalkFunctor C x.1).map f.1)\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\ninst\u271d\u2076 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2075 : ConcreteCategory C\ninst\u271d\u2074 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b3 : HasLimits C\ninst\u271d\u00b2 : PreservesLimits (forget C)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\ninst\u271d : \u2200 (x : \u2191X), IsIso ((stalkFunctor C x).map f.val)\n\u22a2 IsIso f\n[PROOFSTEP]\nsuffices IsIso ((Sheaf.forget C X).map f) by exact isIso_of_fully_faithful (Sheaf.forget C X) f\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\ninst\u271d\u2076 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2075 : ConcreteCategory C\ninst\u271d\u2074 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b3 : HasLimits C\ninst\u271d\u00b2 : PreservesLimits (forget C)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\ninst\u271d : \u2200 (x : \u2191X), IsIso ((stalkFunctor C x).map f.val)\nthis : IsIso ((Sheaf.forget C X).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nexact isIso_of_fully_faithful (Sheaf.forget C X) f\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\ninst\u271d\u2076 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2075 : ConcreteCategory C\ninst\u271d\u2074 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b3 : HasLimits C\ninst\u271d\u00b2 : PreservesLimits (forget C)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\ninst\u271d : \u2200 (x : \u2191X), IsIso ((stalkFunctor C x).map f.val)\n\u22a2 IsIso ((Sheaf.forget C X).map f)\n[PROOFSTEP]\nsuffices \u2200 U : (Opens X)\u1d52\u1d56, IsIso (f.1.app U) by exact @NatIso.isIso_of_isIso_app _ _ _ _ F.1 G.1 f.1 this\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\ninst\u271d\u2076 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2075 : ConcreteCategory C\ninst\u271d\u2074 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b3 : HasLimits C\ninst\u271d\u00b2 : PreservesLimits (forget C)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\ninst\u271d : \u2200 (x : \u2191X), IsIso ((stalkFunctor C x).map f.val)\nthis : \u2200 (U : (Opens \u2191X)\u1d52\u1d56), IsIso (NatTrans.app f.val U)\n\u22a2 IsIso ((Sheaf.forget C X).map f)\n[PROOFSTEP]\nexact @NatIso.isIso_of_isIso_app _ _ _ _ F.1 G.1 f.1 this\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\ninst\u271d\u2076 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2075 : ConcreteCategory C\ninst\u271d\u2074 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b3 : HasLimits C\ninst\u271d\u00b2 : PreservesLimits (forget C)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\ninst\u271d : \u2200 (x : \u2191X), IsIso ((stalkFunctor C x).map f.val)\n\u22a2 \u2200 (U : (Opens \u2191X)\u1d52\u1d56), IsIso (NatTrans.app f.val U)\n[PROOFSTEP]\nintro U\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\ninst\u271d\u2076 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2075 : ConcreteCategory C\ninst\u271d\u2074 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b3 : HasLimits C\ninst\u271d\u00b2 : PreservesLimits (forget C)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\ninst\u271d : \u2200 (x : \u2191X), IsIso ((stalkFunctor C x).map f.val)\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 IsIso (NatTrans.app f.val U)\n[PROOFSTEP]\ninduction U\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\ninst\u271d\u2076 : HasColimits C\nX Y Z : TopCat\ninst\u271d\u2075 : ConcreteCategory C\ninst\u271d\u2074 : PreservesFilteredColimits (forget C)\ninst\u271d\u00b3 : HasLimits C\ninst\u271d\u00b2 : PreservesLimits (forget C)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F \u27f6 G\ninst\u271d : \u2200 (x : \u2191X), IsIso ((stalkFunctor C x).map f.val)\nX\u271d : Opens \u2191X\n\u22a2 IsIso (NatTrans.app f.val (op X\u271d))\n[PROOFSTEP]\napply app_isIso_of_stalkFunctor_map_iso\n", "meta": {"mathlib_filename": "Mathlib.Topology.Sheaves.Stalks", "llama_tokens": 67644, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.734119526900183, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.42114852107188677}} {"text": "[GOAL]\n\u03b1 : Type u_1\ns : Set \u03b1\nx : \u03b1\n\u22a2 x \u2208 s \u2194 boolIndicator s x = true\n[PROOFSTEP]\nunfold boolIndicator\n[GOAL]\n\u03b1 : Type u_1\ns : Set \u03b1\nx : \u03b1\n\u22a2 x \u2208 s \u2194 (if x \u2208 s then true else false) = true\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ns : Set \u03b1\nx : \u03b1\nh : x \u2208 s\n\u22a2 x \u2208 s \u2194 true = true\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ns : Set \u03b1\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 x \u2208 s \u2194 False\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\ns : Set \u03b1\nx : \u03b1\n\u22a2 \u00acx \u2208 s \u2194 boolIndicator s x = false\n[PROOFSTEP]\nunfold boolIndicator\n[GOAL]\n\u03b1 : Type u_1\ns : Set \u03b1\nx : \u03b1\n\u22a2 \u00acx \u2208 s \u2194 (if x \u2208 s then true else false) = false\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ns : Set \u03b1\nx : \u03b1\nh : x \u2208 s\n\u22a2 \u00acx \u2208 s \u2194 False\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ns : Set \u03b1\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 \u00acx \u2208 s \u2194 false = false\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\n\u22a2 boolIndicator s \u207b\u00b9' t = (if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\nx : \u03b1\n\u22a2 x \u2208 boolIndicator s \u207b\u00b9' t \u2194 x \u2208 (if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205\n[PROOFSTEP]\nsimp only [boolIndicator, mem_preimage]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\nx : \u03b1\n\u22a2 (if x \u2208 s then true else false) \u2208 t \u2194 x \u2208 (if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\nx : \u03b1\nh\u271d\u00b2 : x \u2208 s\nh\u271d\u00b9 : true \u2208 t\nh\u271d : false \u2208 t\n\u22a2 true \u2208 t \u2194 x \u2208 s \u222a s\u1d9c\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\nx : \u03b1\nh\u271d\u00b2 : x \u2208 s\nh\u271d\u00b9 : true \u2208 t\nh\u271d : \u00acfalse \u2208 t\n\u22a2 true \u2208 t \u2194 x \u2208 s \u222a \u2205\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\nx : \u03b1\nh\u271d\u00b2 : x \u2208 s\nh\u271d\u00b9 : \u00actrue \u2208 t\nh\u271d : false \u2208 t\n\u22a2 true \u2208 t \u2194 x \u2208 \u2205 \u222a s\u1d9c\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\nx : \u03b1\nh\u271d\u00b2 : x \u2208 s\nh\u271d\u00b9 : \u00actrue \u2208 t\nh\u271d : \u00acfalse \u2208 t\n\u22a2 true \u2208 t \u2194 x \u2208 \u2205 \u222a \u2205\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\nx : \u03b1\nh\u271d\u00b2 : \u00acx \u2208 s\nh\u271d\u00b9 : true \u2208 t\nh\u271d : false \u2208 t\n\u22a2 false \u2208 t \u2194 x \u2208 s \u222a s\u1d9c\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\nx : \u03b1\nh\u271d\u00b2 : \u00acx \u2208 s\nh\u271d\u00b9 : true \u2208 t\nh\u271d : \u00acfalse \u2208 t\n\u22a2 false \u2208 t \u2194 x \u2208 s \u222a \u2205\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\nx : \u03b1\nh\u271d\u00b2 : \u00acx \u2208 s\nh\u271d\u00b9 : \u00actrue \u2208 t\nh\u271d : false \u2208 t\n\u22a2 false \u2208 t \u2194 x \u2208 \u2205 \u222a s\u1d9c\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\nx : \u03b1\nh\u271d\u00b2 : \u00acx \u2208 s\nh\u271d\u00b9 : \u00actrue \u2208 t\nh\u271d : \u00acfalse \u2208 t\n\u22a2 false \u2208 t \u2194 x \u2208 \u2205 \u222a \u2205\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\n\u22a2 boolIndicator s \u207b\u00b9' t = univ \u2228 boolIndicator s \u207b\u00b9' t = s \u2228 boolIndicator s \u207b\u00b9' t = s\u1d9c \u2228 boolIndicator s \u207b\u00b9' t = \u2205\n[PROOFSTEP]\nsimp only [preimage_boolIndicator_eq_union]\n[GOAL]\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\n\u22a2 ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = univ \u2228\n ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = s \u2228\n ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = s\u1d9c \u2228\n ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = \u2205\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\nh\u271d\u00b9 : true \u2208 t\nh\u271d : false \u2208 t\n\u22a2 s \u222a s\u1d9c = univ \u2228 s \u222a s\u1d9c = s \u2228 s \u222a s\u1d9c = s\u1d9c \u2228 s \u222a s\u1d9c = \u2205\n[PROOFSTEP]\nsimp [s.union_compl_self]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\nh\u271d\u00b9 : true \u2208 t\nh\u271d : \u00acfalse \u2208 t\n\u22a2 s \u222a \u2205 = univ \u2228 s \u222a \u2205 = s \u2228 s \u222a \u2205 = s\u1d9c \u2228 s \u222a \u2205 = \u2205\n[PROOFSTEP]\nsimp [s.union_compl_self]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\nh\u271d\u00b9 : \u00actrue \u2208 t\nh\u271d : false \u2208 t\n\u22a2 \u2205 \u222a s\u1d9c = univ \u2228 \u2205 \u222a s\u1d9c = s \u2228 \u2205 \u222a s\u1d9c = s\u1d9c \u2228 \u2205 \u222a s\u1d9c = \u2205\n[PROOFSTEP]\nsimp [s.union_compl_self]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ns : Set \u03b1\nt : Set Bool\nh\u271d\u00b9 : \u00actrue \u2208 t\nh\u271d : \u00acfalse \u2208 t\n\u22a2 \u2205 \u222a \u2205 = univ \u2228 \u2205 \u222a \u2205 = s \u2228 \u2205 \u222a \u2205 = s\u1d9c \u2228 \u2205 \u222a \u2205 = \u2205\n[PROOFSTEP]\nsimp [s.union_compl_self]\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.BoolIndicator", "llama_tokens": 2151, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124812, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.4206468676394403}} {"text": "[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : AddMonoidWithOne A\n\u22a2 \u2200 (n : \u2115), n \u2022 1 = \u2191n\n[PROOFSTEP]\nlet f : \u2115 \u2192+ A := { toFun := fun n \u21a6 n \u2022 (1 : A), map_zero' := by simp [zero_nsmul], map_add' := by simp [add_nsmul] }\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : AddMonoidWithOne A\n\u22a2 (fun n => n \u2022 1) 0 = 0\n[PROOFSTEP]\nsimp [zero_nsmul]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : AddMonoidWithOne A\n\u22a2 \u2200 (x y : \u2115),\n ZeroHom.toFun { toFun := fun n => n \u2022 1, map_zero' := (_ : 0 \u2022 1 = 0) } (x + y) =\n ZeroHom.toFun { toFun := fun n => n \u2022 1, map_zero' := (_ : 0 \u2022 1 = 0) } x +\n ZeroHom.toFun { toFun := fun n => n \u2022 1, map_zero' := (_ : 0 \u2022 1 = 0) } y\n[PROOFSTEP]\nsimp [add_nsmul]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : AddMonoidWithOne A\nf : \u2115 \u2192+ A :=\n { toZeroHom := { toFun := fun n => n \u2022 1, map_zero' := (_ : 0 \u2022 1 = 0) },\n map_add' := (_ : \u2200 (a a_1 : \u2115), (a + a_1) \u2022 1 = a \u2022 1 + a_1 \u2022 1) }\n\u22a2 \u2200 (n : \u2115), n \u2022 1 = \u2191n\n[PROOFSTEP]\nrefine' eq_natCast' f _\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : AddMonoidWithOne A\nf : \u2115 \u2192+ A :=\n { toZeroHom := { toFun := fun n => n \u2022 1, map_zero' := (_ : 0 \u2022 1 = 0) },\n map_add' := (_ : \u2200 (a a_1 : \u2115), (a + a_1) \u2022 1 = a \u2022 1 + a_1 \u2022 1) }\n\u22a2 \u2191f 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\nm : M\ninst\u271d : Invertible m\nn : \u2115\n\u22a2 \u215fm ^ n * m ^ n = 1\n[PROOFSTEP]\nrw [\u2190 (commute_invOf m).symm.mul_pow, invOf_mul_self, one_pow]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\nm : M\ninst\u271d : Invertible m\nn : \u2115\n\u22a2 m ^ n * \u215fm ^ n = 1\n[PROOFSTEP]\nrw [\u2190 (commute_invOf m).mul_pow, mul_invOf_self, one_pow]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : AddMonoid B\nu : M\u02e3\nx : M\nn : \u2115\nhn : n \u2260 0\nhu : x ^ n = \u2191u\n\u22a2 x * x ^ (n - 1) = \u2191u\n[PROOFSTEP]\nrwa [\u2190 _root_.pow_succ, Nat.sub_add_cancel (Nat.succ_le_of_lt <| Nat.pos_of_ne_zero hn)]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : AddMonoid B\nx : M\nn : \u2115\nhx : x ^ n = 1\nhn : n \u2260 0\n\u22a2 \u2191(ofPowEqOne x n hx hn ^ n) = \u21911\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u2076 : Monoid M\ninst\u271d\u2075 : Monoid N\ninst\u271d\u2074 : AddMonoid A\ninst\u271d\u00b3 : AddMonoid B\ninst\u271d\u00b2 : MulAction M N\ninst\u271d\u00b9 : IsScalarTower M N N\ninst\u271d : SMulCommClass M N N\nk : M\nx : N\np : \u2115\n\u22a2 (k \u2022 x) ^ p = k ^ p \u2022 x ^ p\n[PROOFSTEP]\ninduction' p with p IH\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u2076 : Monoid M\ninst\u271d\u2075 : Monoid N\ninst\u271d\u2074 : AddMonoid A\ninst\u271d\u00b3 : AddMonoid B\ninst\u271d\u00b2 : MulAction M N\ninst\u271d\u00b9 : IsScalarTower M N N\ninst\u271d : SMulCommClass M N N\nk : M\nx : N\n\u22a2 (k \u2022 x) ^ Nat.zero = k ^ Nat.zero \u2022 x ^ Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u2076 : Monoid M\ninst\u271d\u2075 : Monoid N\ninst\u271d\u2074 : AddMonoid A\ninst\u271d\u00b3 : AddMonoid B\ninst\u271d\u00b2 : MulAction M N\ninst\u271d\u00b9 : IsScalarTower M N N\ninst\u271d : SMulCommClass M N N\nk : M\nx : N\np : \u2115\nIH : (k \u2022 x) ^ p = k ^ p \u2022 x ^ p\n\u22a2 (k \u2022 x) ^ Nat.succ p = k ^ Nat.succ p \u2022 x ^ Nat.succ p\n[PROOFSTEP]\nrw [pow_succ', IH, smul_mul_smul, \u2190 pow_succ', \u2190 pow_succ']\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\ninst\u271d : MulDistribMulAction M N\nx : M\nm : N\nn : \u2115\n\u22a2 x \u2022 m ^ n = (x \u2022 m) ^ n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\ninst\u271d : MulDistribMulAction M N\nx : M\nm : N\n\u22a2 x \u2022 m ^ Nat.zero = (x \u2022 m) ^ Nat.zero\n[PROOFSTEP]\nrw [pow_zero, pow_zero]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\ninst\u271d : MulDistribMulAction M N\nx : M\nm : N\n\u22a2 x \u2022 1 = 1\n[PROOFSTEP]\nexact smul_one x\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\ninst\u271d : MulDistribMulAction M N\nx : M\nm : N\nn : \u2115\nih : x \u2022 m ^ n = (x \u2022 m) ^ n\n\u22a2 x \u2022 m ^ Nat.succ n = (x \u2022 m) ^ Nat.succ n\n[PROOFSTEP]\nrw [pow_succ, pow_succ]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\ninst\u271d : MulDistribMulAction M N\nx : M\nm : N\nn : \u2115\nih : x \u2022 m ^ n = (x \u2022 m) ^ n\n\u22a2 x \u2022 (m * m ^ n) = x \u2022 m * (x \u2022 m) ^ n\n[PROOFSTEP]\nexact (smul_mul' x m (m ^ n)).trans (congr_arg _ ih)\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : AddGroupWithOne A\nn : \u2124\n\u22a2 n \u2022 1 = \u2191n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase ofNat\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : AddGroupWithOne A\na\u271d : \u2115\n\u22a2 ofNat a\u271d \u2022 1 = \u2191(ofNat a\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase negSucc\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : AddGroupWithOne A\na\u271d : \u2115\n\u22a2 -[a\u271d+1] \u2022 1 = \u2191-[a\u271d+1]\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nm n : \u2115\n\u22a2 a ^ (\u2191m * \u2191n) = (a ^ \u2191m) ^ \u2191n\n[PROOFSTEP]\nrw [zpow_ofNat, zpow_ofNat, \u2190 pow_mul, \u2190 zpow_ofNat]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nm n : \u2115\n\u22a2 a ^ (\u2191m * \u2191n) = a ^ \u2191(m * n)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nm n : \u2115\n\u22a2 a ^ (\u2191m * -[n+1]) = (a ^ \u2191m) ^ -[n+1]\n[PROOFSTEP]\nrw [zpow_ofNat, zpow_negSucc, \u2190 pow_mul, ofNat_mul_negSucc, zpow_neg, inv_inj, \u2190 zpow_ofNat]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nm n : \u2115\n\u22a2 a ^ (-[m+1] * \u2191n) = (a ^ -[m+1]) ^ \u2191n\n[PROOFSTEP]\nrw [zpow_ofNat, zpow_negSucc, \u2190 inv_pow, \u2190 pow_mul, negSucc_mul_ofNat, zpow_neg, inv_pow, inv_inj, \u2190 zpow_ofNat]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nm n : \u2115\n\u22a2 a ^ (-[m+1] * -[n+1]) = (a ^ -[m+1]) ^ -[n+1]\n[PROOFSTEP]\nrw [zpow_negSucc, zpow_negSucc, negSucc_mul_negSucc, inv_pow, inv_inv, \u2190 pow_mul, \u2190 zpow_ofNat]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nm n : \u2115\n\u22a2 a ^ (\u2191(Nat.succ m) * \u2191(Nat.succ n)) = a ^ \u2191((m + 1) * (n + 1))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nm n : \u2124\n\u22a2 a ^ (m * n) = (a ^ n) ^ m\n[PROOFSTEP]\nrw [mul_comm, zpow_mul]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nn : \u2115\n\u22a2 a ^ bit0 \u2191n = a ^ \u2191n * a ^ \u2191n\n[PROOFSTEP]\nsimp only [zpow_ofNat, \u2190 Int.ofNat_bit0, pow_bit0]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nn : \u2115\n\u22a2 a ^ bit0 -[n+1] = a ^ -[n+1] * a ^ -[n+1]\n[PROOFSTEP]\nsimp [\u2190 mul_inv_rev, \u2190 pow_bit0]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nn : \u2115\n\u22a2 a ^ bit0 -[n+1] = (a ^ bit0 (n + 1))\u207b\u00b9\n[PROOFSTEP]\nrw [negSucc_eq, bit0_neg, zpow_neg]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nn : \u2115\n\u22a2 (a ^ bit0 (\u2191n + 1))\u207b\u00b9 = (a ^ bit0 (n + 1))\u207b\u00b9\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : DivisionMonoid \u03b1\ninst\u271d : HasDistribNeg \u03b1\nx : \u03b1\nn : \u2124\n\u22a2 (-x) ^ bit0 n = x ^ bit0 n\n[PROOFSTEP]\nrw [zpow_bit0', zpow_bit0', neg_mul_neg]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\nn : \u2115\n\u22a2 a ^ (\u2191n + 1) = a ^ \u2191n * a\n[PROOFSTEP]\nsimp only [\u2190 Int.ofNat_succ, zpow_ofNat, pow_succ']\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\n\u22a2 a ^ (-[0+1] + 1) = a ^ -[0+1] * a\n[PROOFSTEP]\nerw [zpow_zero, zpow_negSucc, pow_one, mul_left_inv]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\nn : \u2115\n\u22a2 a ^ (-[n + 1+1] + 1) = a ^ -[n + 1+1] * a\n[PROOFSTEP]\nrw [zpow_negSucc, pow_succ, mul_inv_rev, inv_mul_cancel_right]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\nn : \u2115\n\u22a2 a ^ (-[n + 1+1] + 1) = (a ^ (n + 1))\u207b\u00b9\n[PROOFSTEP]\nrw [Int.negSucc_eq, neg_add, add_assoc, neg_add_self, add_zero]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\nn : \u2115\n\u22a2 a ^ (-\u2191(n + 1)) = (a ^ (n + 1))\u207b\u00b9\n[PROOFSTEP]\nexact zpow_negSucc _ _\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\nn : \u2124\n\u22a2 a ^ (n - 1) * a * a\u207b\u00b9 = a ^ n * a\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 zpow_add_one, sub_add_cancel]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\nm n : \u2124\n\u22a2 a ^ (m + n) = a ^ m * a ^ n\n[PROOFSTEP]\ninduction' n using Int.induction_on with n ihn n ihn\n[GOAL]\ncase hz\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\nm : \u2124\n\u22a2 a ^ (m + 0) = a ^ m * a ^ 0\ncase hp\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\nm : \u2124\nn : \u2115\nihn : a ^ (m + \u2191n) = a ^ m * a ^ \u2191n\n\u22a2 a ^ (m + (\u2191n + 1)) = a ^ m * a ^ (\u2191n + 1)\ncase hn\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\nm : \u2124\nn : \u2115\nihn : a ^ (m + -\u2191n) = a ^ m * a ^ (-\u2191n)\n\u22a2 a ^ (m + (-\u2191n - 1)) = a ^ m * a ^ (-\u2191n - 1)\n[PROOFSTEP]\ncase hz => simp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\nm : \u2124\n\u22a2 a ^ (m + 0) = a ^ m * a ^ 0\n[PROOFSTEP]\ncase hz => simp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\nm : \u2124\n\u22a2 a ^ (m + 0) = a ^ m * a ^ 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hp\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\nm : \u2124\nn : \u2115\nihn : a ^ (m + \u2191n) = a ^ m * a ^ \u2191n\n\u22a2 a ^ (m + (\u2191n + 1)) = a ^ m * a ^ (\u2191n + 1)\n[PROOFSTEP]\nsimp only [\u2190 add_assoc, zpow_add_one, ihn, mul_assoc]\n[GOAL]\ncase hn\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\nm : \u2124\nn : \u2115\nihn : a ^ (m + -\u2191n) = a ^ m * a ^ (-\u2191n)\n\u22a2 a ^ (m + (-\u2191n - 1)) = a ^ m * a ^ (-\u2191n - 1)\n[PROOFSTEP]\nrw [zpow_sub_one, \u2190 mul_assoc, \u2190 ihn, \u2190 zpow_sub_one, add_sub_assoc]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n\u22a2 b * b ^ m = b ^ (m + 1)\n[PROOFSTEP]\nconv_lhs =>\n congr\n rw [\u2190 zpow_one b]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n| b * b ^ m\n[PROOFSTEP]\n congr\n rw [\u2190 zpow_one b]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n| b * b ^ m\n[PROOFSTEP]\n congr\n rw [\u2190 zpow_one b]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n| b * b ^ m\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n| b\ncase a\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n| b ^ m\n[PROOFSTEP]\nrw [\u2190 zpow_one b]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n\u22a2 b ^ 1 * b ^ m = b ^ (m + 1)\n[PROOFSTEP]\nrw [\u2190 zpow_add, add_comm]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n\u22a2 b ^ m * b = b ^ (m + 1)\n[PROOFSTEP]\nconv_lhs =>\n congr\n \u00b7skip\n rw [\u2190 zpow_one b]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n| b ^ m * b\n[PROOFSTEP]\n congr\n \u00b7skip\n rw [\u2190 zpow_one b]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n| b ^ m * b\n[PROOFSTEP]\n congr\n \u00b7skip\n rw [\u2190 zpow_one b]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n| b ^ m * b\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n| b ^ m\ncase a\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n| b\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n| b ^ m\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n| b ^ m\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n| b ^ m\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n| b\n[PROOFSTEP]\nrw [\u2190 zpow_one b]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\nb : G\nm : \u2124\n\u22a2 b ^ m * b ^ 1 = b ^ (m + 1)\n[PROOFSTEP]\nrw [\u2190 zpow_add, add_comm]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\nm n : \u2124\n\u22a2 a ^ (m - n) = a ^ m * (a ^ n)\u207b\u00b9\n[PROOFSTEP]\nrw [sub_eq_add_neg, zpow_add, zpow_neg]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\ni : \u2124\n\u22a2 a ^ (1 + i) = a * a ^ i\n[PROOFSTEP]\nrw [zpow_add, zpow_one]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na : G\nn : \u2124\n\u22a2 a ^ bit1 n = a ^ n * a ^ n * a\n[PROOFSTEP]\nrw [bit1, zpow_add, zpow_bit0, zpow_one]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\ng : G\nP : G \u2192 Prop\nh_one : P 1\nh_mul : \u2200 (a : G), P a \u2192 P (g * a)\nh_inv : \u2200 (a : G), P a \u2192 P (g\u207b\u00b9 * a)\nn : \u2124\n\u22a2 P (g ^ n)\n[PROOFSTEP]\ninduction' n using Int.induction_on with n ih n ih\n[GOAL]\ncase hz\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\ng : G\nP : G \u2192 Prop\nh_one : P 1\nh_mul : \u2200 (a : G), P a \u2192 P (g * a)\nh_inv : \u2200 (a : G), P a \u2192 P (g\u207b\u00b9 * a)\n\u22a2 P (g ^ 0)\n[PROOFSTEP]\nrwa [zpow_zero]\n[GOAL]\ncase hp\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\ng : G\nP : G \u2192 Prop\nh_one : P 1\nh_mul : \u2200 (a : G), P a \u2192 P (g * a)\nh_inv : \u2200 (a : G), P a \u2192 P (g\u207b\u00b9 * a)\nn : \u2115\nih : P (g ^ \u2191n)\n\u22a2 P (g ^ (\u2191n + 1))\n[PROOFSTEP]\nrw [add_comm, zpow_add, zpow_one]\n[GOAL]\ncase hp\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\ng : G\nP : G \u2192 Prop\nh_one : P 1\nh_mul : \u2200 (a : G), P a \u2192 P (g * a)\nh_inv : \u2200 (a : G), P a \u2192 P (g\u207b\u00b9 * a)\nn : \u2115\nih : P (g ^ \u2191n)\n\u22a2 P (g * g ^ \u2191n)\n[PROOFSTEP]\nexact h_mul _ ih\n[GOAL]\ncase hn\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\ng : G\nP : G \u2192 Prop\nh_one : P 1\nh_mul : \u2200 (a : G), P a \u2192 P (g * a)\nh_inv : \u2200 (a : G), P a \u2192 P (g\u207b\u00b9 * a)\nn : \u2115\nih : P (g ^ (-\u2191n))\n\u22a2 P (g ^ (-\u2191n - 1))\n[PROOFSTEP]\nrw [sub_eq_add_neg, add_comm, zpow_add, zpow_neg_one]\n[GOAL]\ncase hn\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\ng : G\nP : G \u2192 Prop\nh_one : P 1\nh_mul : \u2200 (a : G), P a \u2192 P (g * a)\nh_inv : \u2200 (a : G), P a \u2192 P (g\u207b\u00b9 * a)\nn : \u2115\nih : P (g ^ (-\u2191n))\n\u22a2 P (g\u207b\u00b9 * g ^ (-\u2191n))\n[PROOFSTEP]\nexact h_inv _ ih\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\ng : G\nP : G \u2192 Prop\nh_one : P 1\nh_mul : \u2200 (a : G), P a \u2192 P (a * g)\nh_inv : \u2200 (a : G), P a \u2192 P (a * g\u207b\u00b9)\nn : \u2124\n\u22a2 P (g ^ n)\n[PROOFSTEP]\ninduction' n using Int.induction_on with n ih n ih\n[GOAL]\ncase hz\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\ng : G\nP : G \u2192 Prop\nh_one : P 1\nh_mul : \u2200 (a : G), P a \u2192 P (a * g)\nh_inv : \u2200 (a : G), P a \u2192 P (a * g\u207b\u00b9)\n\u22a2 P (g ^ 0)\n[PROOFSTEP]\nrwa [zpow_zero]\n[GOAL]\ncase hp\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\ng : G\nP : G \u2192 Prop\nh_one : P 1\nh_mul : \u2200 (a : G), P a \u2192 P (a * g)\nh_inv : \u2200 (a : G), P a \u2192 P (a * g\u207b\u00b9)\nn : \u2115\nih : P (g ^ \u2191n)\n\u22a2 P (g ^ (\u2191n + 1))\n[PROOFSTEP]\nrw [zpow_add_one]\n[GOAL]\ncase hp\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\ng : G\nP : G \u2192 Prop\nh_one : P 1\nh_mul : \u2200 (a : G), P a \u2192 P (a * g)\nh_inv : \u2200 (a : G), P a \u2192 P (a * g\u207b\u00b9)\nn : \u2115\nih : P (g ^ \u2191n)\n\u22a2 P (g ^ \u2191n * g)\n[PROOFSTEP]\nexact h_mul _ ih\n[GOAL]\ncase hn\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\ng : G\nP : G \u2192 Prop\nh_one : P 1\nh_mul : \u2200 (a : G), P a \u2192 P (a * g)\nh_inv : \u2200 (a : G), P a \u2192 P (a * g\u207b\u00b9)\nn : \u2115\nih : P (g ^ (-\u2191n))\n\u22a2 P (g ^ (-\u2191n - 1))\n[PROOFSTEP]\nrw [zpow_sub_one]\n[GOAL]\ncase hn\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\ng : G\nP : G \u2192 Prop\nh_one : P 1\nh_mul : \u2200 (a : G), P a \u2192 P (a * g)\nh_inv : \u2200 (a : G), P a \u2192 P (a * g\u207b\u00b9)\nn : \u2115\nih : P (g ^ (-\u2191n))\n\u22a2 P (g ^ (-\u2191n) * g\u207b\u00b9)\n[PROOFSTEP]\nexact h_inv _ ih\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedCommGroup \u03b1\nm n : \u2124\na b : \u03b1\nha : 1 < a\nk : \u2124\nhk : 0 < k\n\u22a2 1 < a ^ k\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := Int.eq_ofNat_of_zero_le hk.le\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedCommGroup \u03b1\nm n\u271d : \u2124\na b : \u03b1\nha : 1 < a\nk : \u2124\nhk : 0 < k\nn : \u2115\nhn : k = \u2191n\n\u22a2 1 < a ^ k\n[PROOFSTEP]\nrw [hn, zpow_ofNat]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedCommGroup \u03b1\nm n\u271d : \u2124\na b : \u03b1\nha : 1 < a\nk : \u2124\nhk : 0 < k\nn : \u2115\nhn : k = \u2191n\n\u22a2 1 < a ^ n\n[PROOFSTEP]\nrefine' one_lt_pow' ha (coe_nat_pos.mp _).ne'\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedCommGroup \u03b1\nm n\u271d : \u2124\na b : \u03b1\nha : 1 < a\nk : \u2124\nhk : 0 < k\nn : \u2115\nhn : k = \u2191n\n\u22a2 0 < \u2191n\n[PROOFSTEP]\nrwa [\u2190 hn]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedCommGroup \u03b1\nm\u271d n\u271d : \u2124\na b : \u03b1\nha : 1 < a\nm n : \u2124\nh : m < n\n\u22a2 a ^ m * a ^ (n - m) = a ^ n\n[PROOFSTEP]\nrw [\u2190 zpow_add]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedCommGroup \u03b1\nm\u271d n\u271d : \u2124\na b : \u03b1\nha : 1 < a\nm n : \u2124\nh : m < n\n\u22a2 a ^ (m + (n - m)) = a ^ n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedCommGroup \u03b1\nm\u271d n\u271d : \u2124\na b : \u03b1\nha : 1 \u2264 a\nm n : \u2124\nh : m \u2264 n\n\u22a2 a ^ m * a ^ (n - m) = a ^ n\n[PROOFSTEP]\nrw [\u2190 zpow_add]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedCommGroup \u03b1\nm\u271d n\u271d : \u2124\na b : \u03b1\nha : 1 \u2264 a\nm n : \u2124\nh : m \u2264 n\n\u22a2 a ^ (m + (n - m)) = a ^ n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedCommGroup \u03b1\nm n : \u2124\na\u271d b\u271d : \u03b1\nhn : 0 < n\na b : \u03b1\nhab : a < b\n\u22a2 (fun x => x ^ n) a < (fun x => x ^ n) b\n[PROOFSTEP]\nrw [\u2190 one_lt_div', \u2190 div_zpow]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedCommGroup \u03b1\nm n : \u2124\na\u271d b\u271d : \u03b1\nhn : 0 < n\na b : \u03b1\nhab : a < b\n\u22a2 1 < (b / a) ^ n\n[PROOFSTEP]\nexact one_lt_zpow' (one_lt_div'.2 hab) hn\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedCommGroup \u03b1\nm n : \u2124\na\u271d b\u271d : \u03b1\nhn : 0 \u2264 n\na b : \u03b1\nhab : a \u2264 b\n\u22a2 (fun x => x ^ n) a \u2264 (fun x => x ^ n) b\n[PROOFSTEP]\nrw [\u2190 one_le_div', \u2190 div_zpow]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedCommGroup \u03b1\nm n : \u2124\na\u271d b\u271d : \u03b1\nhn : 0 \u2264 n\na b : \u03b1\nhab : a \u2264 b\n\u22a2 1 \u2264 (b / a) ^ n\n[PROOFSTEP]\nexact one_le_zpow (one_le_div'.2 hab) hn\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedCommGroup \u03b1\nn : \u2124\na b : \u03b1\nhn : n \u2260 0\n\u22a2 Function.Injective fun x => x ^ n\n[PROOFSTEP]\nrcases hn.symm.lt_or_lt with (h | h)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedCommGroup \u03b1\nn : \u2124\na b : \u03b1\nhn : n \u2260 0\nh : 0 < n\n\u22a2 Function.Injective fun x => x ^ n\n[PROOFSTEP]\nexact (zpow_strictMono_left \u03b1 h).injective\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedCommGroup \u03b1\nn : \u2124\na b : \u03b1\nhn : n \u2260 0\nh : n < 0\n\u22a2 Function.Injective fun x => x ^ n\n[PROOFSTEP]\nrefine' fun a b (hab : a ^ n = b ^ n) => (zpow_strictMono_left \u03b1 (neg_pos.mpr h)).injective _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedCommGroup \u03b1\nn : \u2124\na\u271d b\u271d : \u03b1\nhn : n \u2260 0\nh : n < 0\na b : \u03b1\nhab : a ^ n = b ^ n\n\u22a2 a ^ (-n) = b ^ (-n)\n[PROOFSTEP]\nrw [zpow_neg, zpow_neg, hab]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b : \u03b1\nn : \u2115\na : \u03b1\n\u22a2 |n \u2022 a| = n \u2022 |a|\n[PROOFSTEP]\ncases' le_total a 0 with hneg hpos\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b : \u03b1\nn : \u2115\na : \u03b1\nhneg : a \u2264 0\n\u22a2 |n \u2022 a| = n \u2022 |a|\n[PROOFSTEP]\nrw [abs_of_nonpos hneg, \u2190 abs_neg, \u2190 neg_nsmul, abs_of_nonneg]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b : \u03b1\nn : \u2115\na : \u03b1\nhneg : a \u2264 0\n\u22a2 0 \u2264 n \u2022 -a\n[PROOFSTEP]\nexact nsmul_nonneg (neg_nonneg.mpr hneg) n\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b : \u03b1\nn : \u2115\na : \u03b1\nhpos : 0 \u2264 a\n\u22a2 |n \u2022 a| = n \u2022 |a|\n[PROOFSTEP]\nrw [abs_of_nonneg hpos, abs_of_nonneg]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b : \u03b1\nn : \u2115\na : \u03b1\nhpos : 0 \u2264 a\n\u22a2 0 \u2264 n \u2022 a\n[PROOFSTEP]\nexact nsmul_nonneg hpos n\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b : \u03b1\nn : \u2124\na : \u03b1\n\u22a2 |n \u2022 a| = |n| \u2022 |a|\n[PROOFSTEP]\nobtain n0 | n0 := le_total 0 n\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b : \u03b1\nn : \u2124\na : \u03b1\nn0 : 0 \u2264 n\n\u22a2 |n \u2022 a| = |n| \u2022 |a|\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := Int.eq_ofNat_of_zero_le n0\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b a : \u03b1\nn : \u2115\nn0 : 0 \u2264 \u2191n\n\u22a2 |\u2191n \u2022 a| = |\u2191n| \u2022 |a|\n[PROOFSTEP]\nsimp only [abs_nsmul, coe_nat_zsmul, Nat.abs_cast]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b : \u03b1\nn : \u2124\na : \u03b1\nn0 : n \u2264 0\n\u22a2 |n \u2022 a| = |n| \u2022 |a|\n[PROOFSTEP]\nobtain \u27e8m, h\u27e9 := Int.eq_ofNat_of_zero_le (neg_nonneg.2 n0)\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b : \u03b1\nn : \u2124\na : \u03b1\nn0 : n \u2264 0\nm : \u2115\nh : -n = \u2191m\n\u22a2 |n \u2022 a| = |n| \u2022 |a|\n[PROOFSTEP]\nrw [\u2190 abs_neg, \u2190 neg_zsmul, \u2190 abs_neg n, h, coe_nat_zsmul, Nat.abs_cast, coe_nat_zsmul]\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b : \u03b1\nn : \u2124\na : \u03b1\nn0 : n \u2264 0\nm : \u2115\nh : -n = \u2191m\n\u22a2 |m \u2022 a| = m \u2022 |a|\n[PROOFSTEP]\nexact abs_nsmul m _\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\n\u22a2 |a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\n[PROOFSTEP]\nobtain a0 | a0 := le_or_lt 0 a\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : 0 \u2264 a\n\u22a2 |a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\n[PROOFSTEP]\nobtain b0 | b0 := le_or_lt 0 b\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : a < 0\n\u22a2 |a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\n[PROOFSTEP]\nobtain b0 | b0 := le_or_lt 0 b\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : 0 \u2264 a\nb0 : 0 \u2264 b\n\u22a2 |a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\n[PROOFSTEP]\nsimp [a0, b0, abs_of_nonneg, add_nonneg a0 b0]\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : 0 \u2264 a\nb0 : b < 0\n\u22a2 |a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\n[PROOFSTEP]\nexact (lt_irrefl (0 : \u03b1) <| a0.trans_lt <| hle.trans_lt b0).elim\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : a < 0\nb0 : 0 \u2264 b\n\u22a2 |a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\ncase inr.inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : a < 0\nb0 : b < 0\n\u22a2 |a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\n[PROOFSTEP]\nany_goals simp [a0.le, b0.le, abs_of_nonpos, add_nonpos, add_comm]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : a < 0\nb0 : 0 \u2264 b\n\u22a2 |a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\n[PROOFSTEP]\nsimp [a0.le, b0.le, abs_of_nonpos, add_nonpos, add_comm]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : a < 0\nb0 : b < 0\n\u22a2 |a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\n[PROOFSTEP]\nsimp [a0.le, b0.le, abs_of_nonpos, add_nonpos, add_comm]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : a < 0\nb0 : 0 \u2264 b\n\u22a2 |a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\n[PROOFSTEP]\nhave : (|a + b| = -a + b \u2194 b \u2264 0) \u2194 (|a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0) := by\n simp [a0, a0.le, a0.not_le, b0, abs_of_neg, abs_of_nonneg]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : a < 0\nb0 : 0 \u2264 b\n\u22a2 (|a + b| = -a + b \u2194 b \u2264 0) \u2194 (|a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0)\n[PROOFSTEP]\nsimp [a0, a0.le, a0.not_le, b0, abs_of_neg, abs_of_nonneg]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : a < 0\nb0 : 0 \u2264 b\nthis : (|a + b| = -a + b \u2194 b \u2264 0) \u2194 (|a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0)\n\u22a2 |a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\n[PROOFSTEP]\nrefine' this.mp \u27e8fun h => _, fun h => by simp only [le_antisymm h b0, abs_of_neg a0, add_zero]\u27e9\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : a < 0\nb0 : 0 \u2264 b\nthis : (|a + b| = -a + b \u2194 b \u2264 0) \u2194 (|a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0)\nh : b \u2264 0\n\u22a2 |a + b| = -a + b\n[PROOFSTEP]\nsimp only [le_antisymm h b0, abs_of_neg a0, add_zero]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : a < 0\nb0 : 0 \u2264 b\nthis : (|a + b| = -a + b \u2194 b \u2264 0) \u2194 (|a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0)\nh : |a + b| = -a + b\n\u22a2 b \u2264 0\n[PROOFSTEP]\nobtain ab | ab := le_or_lt (a + b) 0\n[GOAL]\ncase inr.inl.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : a < 0\nb0 : 0 \u2264 b\nthis : (|a + b| = -a + b \u2194 b \u2264 0) \u2194 (|a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0)\nh : |a + b| = -a + b\nab : a + b \u2264 0\n\u22a2 b \u2264 0\n[PROOFSTEP]\nrefine' le_of_eq (eq_zero_of_neg_eq _)\n[GOAL]\ncase inr.inl.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : a < 0\nb0 : 0 \u2264 b\nthis : (|a + b| = -a + b \u2194 b \u2264 0) \u2194 (|a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0)\nh : |a + b| = -a + b\nab : a + b \u2264 0\n\u22a2 -b = b\n[PROOFSTEP]\nrwa [abs_of_nonpos ab, neg_add_rev, add_comm, add_right_inj] at h \n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : a < 0\nb0 : 0 \u2264 b\nthis : (|a + b| = -a + b \u2194 b \u2264 0) \u2194 (|a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0)\nh : |a + b| = -a + b\nab : 0 < a + b\n\u22a2 b \u2264 0\n[PROOFSTEP]\nrefine' (lt_irrefl (0 : \u03b1) _).elim\n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : a < 0\nb0 : 0 \u2264 b\nthis : (|a + b| = -a + b \u2194 b \u2264 0) \u2194 (|a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0)\nh : |a + b| = -a + b\nab : 0 < a + b\n\u22a2 0 < 0\n[PROOFSTEP]\nrw [abs_of_pos ab, add_left_inj] at h \n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b : \u03b1\nhle : a \u2264 b\na0 : a < 0\nb0 : 0 \u2264 b\nthis : (|a + b| = -a + b \u2194 b \u2264 0) \u2194 (|a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0)\nh : a = -a\nab : 0 < a + b\n\u22a2 0 < 0\n[PROOFSTEP]\nrwa [eq_zero_of_neg_eq h.symm] at a0 \n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d a b : \u03b1\n\u22a2 |a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\n[PROOFSTEP]\nobtain ab | ab := le_total a b\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d a b : \u03b1\nab : a \u2264 b\n\u22a2 |a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\n[PROOFSTEP]\nexact abs_add_eq_add_abs_le ab\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d a b : \u03b1\nab : b \u2264 a\n\u22a2 |a + b| = |a| + |b| \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\n[PROOFSTEP]\nrw [add_comm a, add_comm (abs _), abs_add_eq_add_abs_le ab, and_comm, @and_comm (b \u2264 0)]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonAssocSemiring R\na : R\nn : \u2115\n\u22a2 n \u2022 a = a * \u2191n\n[PROOFSTEP]\ninduction' n with n ih <;> [rw [zero_nsmul, Nat.cast_zero, mul_zero];\n rw [succ_nsmul', ih, Nat.cast_succ, mul_add, mul_one]]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonAssocSemiring R\na : R\nn : \u2115\n\u22a2 n \u2022 a = a * \u2191n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonAssocSemiring R\na : R\n\u22a2 Nat.zero \u2022 a = a * \u2191Nat.zero\n[PROOFSTEP]\nrw [zero_nsmul, Nat.cast_zero, mul_zero]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonAssocSemiring R\na : R\nn : \u2115\nih : n \u2022 a = a * \u2191n\n\u22a2 Nat.succ n \u2022 a = a * \u2191(Nat.succ n)\n[PROOFSTEP]\nrw [succ_nsmul', ih, Nat.cast_succ, mul_add, mul_one]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonAssocSemiring R\nn : \u2115\na : R\n\u22a2 n \u2022 a = \u2191n * a\n[PROOFSTEP]\nrw [nsmul_eq_mul', (n.cast_commute a).eq]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonUnitalNonAssocSemiring R\nn : \u2115\nx y : R\n\u22a2 n \u2022 x \u2022 y = x \u2022 n \u2022 y\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : R\n\u22a2 Nat.zero \u2022 x \u2022 y = x \u2022 Nat.zero \u2022 y\n[PROOFSTEP]\nsimp [zero_nsmul]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : R\nn : \u2115\nih : n \u2022 x \u2022 y = x \u2022 n \u2022 y\n\u22a2 Nat.succ n \u2022 x \u2022 y = x \u2022 Nat.succ n \u2022 y\n[PROOFSTEP]\nsimp_rw [succ_nsmul, smul_eq_mul, mul_add, \u2190 smul_eq_mul, ih]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonUnitalNonAssocSemiring R\nn : \u2115\nx y : R\n\u22a2 (n \u2022 x) \u2022 y = n \u2022 x \u2022 y\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : R\n\u22a2 (Nat.zero \u2022 x) \u2022 y = Nat.zero \u2022 x \u2022 y\n[PROOFSTEP]\nsimp [zero_nsmul]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : R\nn : \u2115\nih : (n \u2022 x) \u2022 y = n \u2022 x \u2022 y\n\u22a2 (Nat.succ n \u2022 x) \u2022 y = Nat.succ n \u2022 x \u2022 y\n[PROOFSTEP]\nsimp_rw [succ_nsmul, \u2190 ih, smul_eq_mul, add_mul]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Semiring R\nn m : \u2115\n\u22a2 \u2191(n ^ m) = \u2191n ^ m\n[PROOFSTEP]\ninduction' m with m ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Semiring R\nn : \u2115\n\u22a2 \u2191(n ^ zero) = \u2191n ^ zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Semiring R\nn m : \u2115\nih : \u2191(n ^ m) = \u2191n ^ m\n\u22a2 \u2191(n ^ succ m) = \u2191n ^ succ m\n[PROOFSTEP]\nrw [_root_.pow_succ', _root_.pow_succ', Nat.cast_mul, ih]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\nn m : \u2115\n\u22a2 \u2191(n ^ m) = \u2191n ^ m\n[PROOFSTEP]\ninduction' m with m _\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\nn : \u2115\n\u22a2 \u2191(n ^ Nat.zero) = \u2191n ^ Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\nn m : \u2115\nn_ih\u271d : \u2191(n ^ m) = \u2191n ^ m\n\u22a2 \u2191(n ^ Nat.succ m) = \u2191n ^ Nat.succ m\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\nn : \u2124\nk : \u2115\n\u22a2 natAbs (n ^ k) = natAbs n ^ k\n[PROOFSTEP]\ninduction' k with k ih <;> [rfl; rw [pow_succ', Int.natAbs_mul, pow_succ', ih]]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\nn : \u2124\nk : \u2115\n\u22a2 natAbs (n ^ k) = natAbs n ^ k\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\nn : \u2124\n\u22a2 natAbs (n ^ Nat.zero) = natAbs n ^ Nat.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\nn : \u2124\nk : \u2115\nih : natAbs (n ^ k) = natAbs n ^ k\n\u22a2 natAbs (n ^ Nat.succ k) = natAbs n ^ Nat.succ k\n[PROOFSTEP]\nrw [pow_succ', Int.natAbs_mul, pow_succ', ih]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonUnitalNonAssocRing R\nn r : R\n\u22a2 bit0 n * r = 2 \u2022 (n * r)\n[PROOFSTEP]\ndsimp [bit0]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonUnitalNonAssocRing R\nn r : R\n\u22a2 (n + n) * r = 2 \u2022 (n * r)\n[PROOFSTEP]\nrw [add_mul, \u2190 one_add_one_eq_two, add_zsmul, one_zsmul]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonUnitalNonAssocRing R\nn r : R\n\u22a2 r * bit0 n = 2 \u2022 (r * n)\n[PROOFSTEP]\ndsimp [bit0]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonUnitalNonAssocRing R\nn r : R\n\u22a2 r * (n + n) = 2 \u2022 (r * n)\n[PROOFSTEP]\nrw [mul_add, \u2190 one_add_one_eq_two, add_zsmul, one_zsmul]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonAssocRing R\nn r : R\n\u22a2 bit1 n * r = 2 \u2022 (n * r) + r\n[PROOFSTEP]\ndsimp [bit1]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonAssocRing R\nn r : R\n\u22a2 (bit0 n + 1) * r = 2 \u2022 (n * r) + r\n[PROOFSTEP]\nrw [add_mul, bit0_mul, one_mul]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonAssocRing R\nn r : R\n\u22a2 r * bit1 n = 2 \u2022 (r * n) + r\n[PROOFSTEP]\ndsimp [bit1]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonAssocRing R\nn r : R\n\u22a2 r * (bit0 n + 1) = 2 \u2022 (r * n) + r\n[PROOFSTEP]\nrw [mul_add, mul_bit0, mul_one]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : AddCommGroupWithOne \u03b1\nm : \u2124\n\u22a2 \u2200 (n : \u2124), \u2191(0 * n) = 0 \u2022 \u2191n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : AddCommGroupWithOne \u03b1\nm k : \u2124\nx\u271d : 0 \u2264 k\nih : \u2200 (n : \u2124), \u2191(k * n) = k \u2022 \u2191n\nn : \u2124\n\u22a2 \u2191((k + 1) * n) = (k + 1) \u2022 \u2191n\n[PROOFSTEP]\nsimp [add_mul, add_zsmul, ih]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : AddCommGroupWithOne \u03b1\nm k : \u2124\nx\u271d : k \u2264 0\nih : \u2200 (n : \u2124), \u2191(k * n) = k \u2022 \u2191n\nn : \u2124\n\u22a2 \u2191((k - 1) * n) = (k - 1) \u2022 \u2191n\n[PROOFSTEP]\nsimp [sub_mul, sub_zsmul, ih, \u2190 sub_eq_add_neg]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Ring R\na : R\nn : \u2115\n\u22a2 \u2191n \u2022 a = \u2191\u2191n * a\n[PROOFSTEP]\nrw [coe_nat_zsmul, nsmul_eq_mul, Int.cast_ofNat]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Ring R\na : R\nn : \u2115\n\u22a2 -[n+1] \u2022 a = \u2191-[n+1] * a\n[PROOFSTEP]\nsimp [Nat.cast_succ, neg_add_rev, Int.cast_negSucc, add_mul]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Ring R\na : R\nn : \u2124\n\u22a2 n \u2022 a = a * \u2191n\n[PROOFSTEP]\nrw [zsmul_eq_mul, (n.cast_commute a).eq]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonUnitalNonAssocRing R\nn\u271d : \u2124\nx y : R\nn : \u2115\n\u22a2 \u2191n \u2022 x \u2022 y = x \u2022 \u2191n \u2022 y\n[PROOFSTEP]\nsimp_rw [coe_nat_zsmul, smul_comm]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonUnitalNonAssocRing R\nn\u271d : \u2124\nx y : R\nn : \u2115\n\u22a2 -[n+1] \u2022 x \u2022 y = x \u2022 -[n+1] \u2022 y\n[PROOFSTEP]\nsimp_rw [negSucc_zsmul, smul_eq_mul, mul_neg, mul_smul_comm]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonUnitalNonAssocRing R\nn\u271d : \u2124\nx y : R\nn : \u2115\n\u22a2 (\u2191n \u2022 x) \u2022 y = \u2191n \u2022 x \u2022 y\n[PROOFSTEP]\nsimp_rw [coe_nat_zsmul, smul_assoc]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonUnitalNonAssocRing R\nn\u271d : \u2124\nx y : R\nn : \u2115\n\u22a2 (-[n+1] \u2022 x) \u2022 y = -[n+1] \u2022 x \u2022 y\n[PROOFSTEP]\nsimp_rw [negSucc_zsmul, smul_eq_mul, neg_mul, smul_mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\na b : \u2124\n\u22a2 a \u2022 b = a * b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\nn : \u2124\n\u22a2 n \u2022 1 = n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Ring R\nn : \u2124\nm : \u2115\n\u22a2 \u2191(n ^ m) = \u2191n ^ m\n[PROOFSTEP]\ninduction' m with m ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Ring R\nn : \u2124\n\u22a2 \u2191(n ^ Nat.zero) = \u2191n ^ Nat.zero\n[PROOFSTEP]\nrw [pow_zero, pow_zero, Int.cast_one]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Ring R\nn : \u2124\nm : \u2115\nih : \u2191(n ^ m) = \u2191n ^ m\n\u22a2 \u2191(n ^ Nat.succ m) = \u2191n ^ Nat.succ m\n[PROOFSTEP]\nrw [pow_succ, pow_succ, Int.cast_mul, ih]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Ring R\nn : \u2115\n\u22a2 (-1) ^ n = (-1) ^ (n % 2)\n[PROOFSTEP]\nrw [\u2190 Nat.mod_add_div n 2, pow_add, pow_mul]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Ring R\nn : \u2115\n\u22a2 (-1) ^ (n % 2) * ((-1) ^ 2) ^ (n / 2) = (-1) ^ ((n % 2 + 2 * (n / 2)) % 2)\n[PROOFSTEP]\nsimp [sq]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH\u271d : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedSemiring R\na : R\nHsq : 0 \u2264 a * a\nHsq' : 0 \u2264 (1 + a) * (1 + a)\nH : 0 \u2264 2 + a\n\u22a2 1 + \u21910 * a \u2264 (1 + a) ^ 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH\u271d : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedSemiring R\na : R\nHsq : 0 \u2264 a * a\nHsq' : 0 \u2264 (1 + a) * (1 + a)\nH : 0 \u2264 2 + a\n\u22a2 1 + \u21911 * a \u2264 (1 + a) ^ 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH\u271d : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedSemiring R\na : R\nHsq : 0 \u2264 a * a\nHsq' : 0 \u2264 (1 + a) * (1 + a)\nH : 0 \u2264 2 + a\nn : \u2115\nthis : 0 \u2264 \u2191n * (a * a * (2 + a)) + a * a\n\u22a2 1 + \u2191(n + 2) * a + (\u2191n * (a * a * (2 + a)) + a * a) = (1 + a) * (1 + a) * (1 + \u2191n * a)\n[PROOFSTEP]\n{ simp only [Nat.cast_add, add_mul, mul_add, one_mul, mul_one, \u2190 one_add_one_eq_two, Nat.cast_one, add_assoc,\n add_right_inj]\n simp only [\u2190 add_assoc, add_comm _ (\u2191n * a)]\n simp only [add_assoc, (n.cast_commute (_ : R)).left_comm]\n simp only [add_comm, add_left_comm]\n}\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH\u271d : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedSemiring R\na : R\nHsq : 0 \u2264 a * a\nHsq' : 0 \u2264 (1 + a) * (1 + a)\nH : 0 \u2264 2 + a\nn : \u2115\nthis : 0 \u2264 \u2191n * (a * a * (2 + a)) + a * a\n\u22a2 1 + \u2191(n + 2) * a + (\u2191n * (a * a * (2 + a)) + a * a) = (1 + a) * (1 + a) * (1 + \u2191n * a)\n[PROOFSTEP]\nsimp only [Nat.cast_add, add_mul, mul_add, one_mul, mul_one, \u2190 one_add_one_eq_two, Nat.cast_one, add_assoc,\n add_right_inj]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH\u271d : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedSemiring R\na : R\nHsq : 0 \u2264 a * a\nHsq' : 0 \u2264 (1 + a) * (1 + a)\nH : 0 \u2264 2 + a\nn : \u2115\nthis : 0 \u2264 \u2191n * (a * a * (2 + a)) + a * a\n\u22a2 1 + (\u2191n * a + (a + (a + (\u2191n * (a * a) + (\u2191n * (a * a) + (\u2191n * (a * a * a) + a * a)))))) =\n 1 + (a + (a + (a * a + (\u2191n * a + (a * (\u2191n * a) + (a * (\u2191n * a) + a * a * (\u2191n * a)))))))\n[PROOFSTEP]\nsimp only [\u2190 add_assoc, add_comm _ (\u2191n * a)]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH\u271d : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedSemiring R\na : R\nHsq : 0 \u2264 a * a\nHsq' : 0 \u2264 (1 + a) * (1 + a)\nH : 0 \u2264 2 + a\nn : \u2115\nthis : 0 \u2264 \u2191n * (a * a * (2 + a)) + a * a\n\u22a2 \u2191n * a + 1 + a + a + \u2191n * (a * a) + \u2191n * (a * a) + \u2191n * (a * a * a) + a * a =\n \u2191n * a + 1 + a + a + a * a + a * (\u2191n * a) + a * (\u2191n * a) + a * a * (\u2191n * a)\n[PROOFSTEP]\nsimp only [add_assoc, (n.cast_commute (_ : R)).left_comm]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH\u271d : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedSemiring R\na : R\nHsq : 0 \u2264 a * a\nHsq' : 0 \u2264 (1 + a) * (1 + a)\nH : 0 \u2264 2 + a\nn : \u2115\nthis : 0 \u2264 \u2191n * (a * a * (2 + a)) + a * a\n\u22a2 \u2191n * a + (1 + (a + (a + (a * (\u2191n * a) + (a * (\u2191n * a) + (a * a * (\u2191n * a) + a * a)))))) =\n \u2191n * a + (1 + (a + (a + (a * a + (a * (\u2191n * a) + (a * (\u2191n * a) + a * a * (\u2191n * a)))))))\n[PROOFSTEP]\nsimp only [add_comm, add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH\u271d : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedSemiring R\na : R\nHsq : 0 \u2264 a * a\nHsq' : 0 \u2264 (1 + a) * (1 + a)\nH : 0 \u2264 2 + a\nn : \u2115\nthis : 0 \u2264 \u2191n * (a * a * (2 + a)) + a * a\n\u22a2 (1 + a) * (1 + a) * (1 + a) ^ n = (1 + a) ^ (n + 2)\n[PROOFSTEP]\nsimp only [pow_succ, mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedSemiring R\na : R\nh : 0 \u2264 a\nha : a \u2264 1\ni : \u2115\n\u22a2 a ^ (i + 0) \u2264 a ^ i\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedSemiring R\na : R\nh : 0 \u2264 a\nha : a \u2264 1\ni k : \u2115\n\u22a2 a ^ (i + (k + 1)) \u2264 a ^ i\n[PROOFSTEP]\nrw [\u2190 add_assoc, \u2190 one_mul (a ^ i), pow_succ]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedSemiring R\na : R\nh : 0 \u2264 a\nha : a \u2264 1\ni k : \u2115\n\u22a2 a * a ^ (i + k) \u2264 1 * a ^ i\n[PROOFSTEP]\nexact mul_le_mul ha (pow_le_pow_of_le_one_aux h ha _ _) (pow_nonneg h _) zero_le_one\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedSemiring R\na : R\nh : 0 \u2264 a\nha : a \u2264 1\ni j : \u2115\nhij : i \u2264 j\n\u22a2 a ^ j \u2264 a ^ i\n[PROOFSTEP]\nlet \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le hij\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedSemiring R\na : R\nh : 0 \u2264 a\nha : a \u2264 1\ni j : \u2115\nhij : i \u2264 j\nk : \u2115\nhk : j = i + k\n\u22a2 a ^ j \u2264 a ^ i\n[PROOFSTEP]\nrw [hk]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : OrderedSemiring R\na : R\nh : 0 \u2264 a\nha : a \u2264 1\ni j : \u2115\nhij : i \u2264 j\nk : \u2115\nhk : j = i + k\n\u22a2 a ^ (i + k) \u2264 a ^ i\n[PROOFSTEP]\nexact pow_le_pow_of_le_one_aux h ha _ _\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedSemiring R\nC r : R\nh : \u2200 (n : \u2115), 0 \u2264 C * r ^ n\n\u22a2 C = 0 \u2228 0 < C \u2227 0 \u2264 r\n[PROOFSTEP]\nhave : 0 \u2264 C := by simpa only [pow_zero, mul_one] using h 0\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedSemiring R\nC r : R\nh : \u2200 (n : \u2115), 0 \u2264 C * r ^ n\n\u22a2 0 \u2264 C\n[PROOFSTEP]\nsimpa only [pow_zero, mul_one] using h 0\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedSemiring R\nC r : R\nh : \u2200 (n : \u2115), 0 \u2264 C * r ^ n\nthis : 0 \u2264 C\n\u22a2 C = 0 \u2228 0 < C \u2227 0 \u2264 r\n[PROOFSTEP]\nrefine' this.eq_or_lt.elim (fun h => Or.inl h.symm) fun hC => Or.inr \u27e8hC, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedSemiring R\nC r : R\nh : \u2200 (n : \u2115), 0 \u2264 C * r ^ n\nthis : 0 \u2264 C\nhC : 0 < C\n\u22a2 0 \u2264 r\n[PROOFSTEP]\nrefine' nonneg_of_mul_nonneg_right _ hC\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedSemiring R\nC r : R\nh : \u2200 (n : \u2115), 0 \u2264 C * r ^ n\nthis : 0 \u2264 C\nhC : 0 < C\n\u22a2 0 \u2264 C * r\n[PROOFSTEP]\nsimpa only [pow_one] using h 1\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\n\u22a2 a ^ bit1 n \u2264 0 \u2194 a \u2264 0\n[PROOFSTEP]\nsimp only [le_iff_lt_or_eq, pow_bit1_neg_iff]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\n\u22a2 a < 0 \u2228 a ^ bit1 n = 0 \u2194 a < 0 \u2228 a = 0\n[PROOFSTEP]\nrefine' \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\n\u22a2 a < 0 \u2228 a ^ bit1 n = 0 \u2192 a < 0 \u2228 a = 0\n[PROOFSTEP]\nrintro (hpos | hz)\n[GOAL]\ncase refine'_1.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhpos : a < 0\n\u22a2 a < 0 \u2228 a = 0\n[PROOFSTEP]\nleft\n[GOAL]\ncase refine'_1.inl.h\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhpos : a < 0\n\u22a2 a < 0\n[PROOFSTEP]\nexact hpos\n[GOAL]\ncase refine'_1.inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhz : a ^ bit1 n = 0\n\u22a2 a < 0 \u2228 a = 0\n[PROOFSTEP]\nright\n[GOAL]\ncase refine'_1.inr.h\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhz : a ^ bit1 n = 0\n\u22a2 a = 0\n[PROOFSTEP]\nexact (pow_eq_zero_iff'.1 hz).1\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\n\u22a2 a < 0 \u2228 a = 0 \u2192 a < 0 \u2228 a ^ bit1 n = 0\n[PROOFSTEP]\nrintro (hneg | hz)\n[GOAL]\ncase refine'_2.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhneg : a < 0\n\u22a2 a < 0 \u2228 a ^ bit1 n = 0\n[PROOFSTEP]\nleft\n[GOAL]\ncase refine'_2.inl.h\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhneg : a < 0\n\u22a2 a < 0\n[PROOFSTEP]\nexact hneg\n[GOAL]\ncase refine'_2.inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhz : a = 0\n\u22a2 a < 0 \u2228 a ^ bit1 n = 0\n[PROOFSTEP]\nright\n[GOAL]\ncase refine'_2.inr.h\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhz : a = 0\n\u22a2 a ^ bit1 n = 0\n[PROOFSTEP]\nsimp [hz, bit1]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn\u271d n : \u2115\n\u22a2 StrictMono fun a => a ^ bit1 n\n[PROOFSTEP]\nintro a b hab\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na\u271d : R\nn\u271d n : \u2115\na b : R\nhab : a < b\n\u22a2 (fun a => a ^ bit1 n) a < (fun a => a ^ bit1 n) b\n[PROOFSTEP]\ncases' le_total a 0 with ha ha\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na\u271d : R\nn\u271d n : \u2115\na b : R\nhab : a < b\nha : a \u2264 0\n\u22a2 (fun a => a ^ bit1 n) a < (fun a => a ^ bit1 n) b\n[PROOFSTEP]\ncases' le_or_lt b 0 with hb hb\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na\u271d : R\nn\u271d n : \u2115\na b : R\nhab : a < b\nha : a \u2264 0\nhb : b \u2264 0\n\u22a2 (fun a => a ^ bit1 n) a < (fun a => a ^ bit1 n) b\n[PROOFSTEP]\nrw [\u2190 neg_lt_neg_iff, \u2190 neg_pow_bit1, \u2190 neg_pow_bit1]\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na\u271d : R\nn\u271d n : \u2115\na b : R\nhab : a < b\nha : a \u2264 0\nhb : b \u2264 0\n\u22a2 (-b) ^ bit1 n < (-a) ^ bit1 n\n[PROOFSTEP]\nexact pow_lt_pow_of_lt_left (neg_lt_neg hab) (neg_nonneg.2 hb) (bit1_pos (zero_le n))\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na\u271d : R\nn\u271d n : \u2115\na b : R\nhab : a < b\nha : a \u2264 0\nhb : 0 < b\n\u22a2 (fun a => a ^ bit1 n) a < (fun a => a ^ bit1 n) b\n[PROOFSTEP]\nexact (pow_bit1_nonpos_iff.2 ha).trans_lt (pow_bit1_pos_iff.2 hb)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na\u271d : R\nn\u271d n : \u2115\na b : R\nhab : a < b\nha : 0 \u2264 a\n\u22a2 (fun a => a ^ bit1 n) a < (fun a => a ^ bit1 n) b\n[PROOFSTEP]\nexact pow_lt_pow_of_lt_left hab ha (bit1_pos (zero_le n))\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH\u271d : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn\u271d : \u2115\nH : -1 \u2264 a\nn : \u2115\n\u22a2 1 + \u2191n * (a - 1) \u2264 a ^ n\n[PROOFSTEP]\nhave : -2 \u2264 a - 1 := by rwa [\u2190 one_add_one_eq_two, neg_add, \u2190 sub_eq_add_neg, sub_le_sub_iff_right]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH\u271d : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn\u271d : \u2115\nH : -1 \u2264 a\nn : \u2115\n\u22a2 -2 \u2264 a - 1\n[PROOFSTEP]\nrwa [\u2190 one_add_one_eq_two, neg_add, \u2190 sub_eq_add_neg, sub_le_sub_iff_right]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH\u271d : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : LinearOrderedRing R\na : R\nn\u271d : \u2115\nH : -1 \u2264 a\nn : \u2115\nthis : -2 \u2264 a - 1\n\u22a2 1 + \u2191n * (a - 1) \u2264 a ^ n\n[PROOFSTEP]\nsimpa only [add_sub_cancel'_right] using one_add_mul_le_pow this n\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\nx : \u2124\n\u22a2 \u2191(natAbs x) ^ 2 = x ^ 2\n[PROOFSTEP]\nrw [sq, Int.natAbs_mul_self', sq]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\na : \u2124\n\u22a2 \u2191(natAbs a) \u2264 a ^ 2\n[PROOFSTEP]\nrw [\u2190 Int.natAbs_sq a, sq]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\na : \u2124\n\u22a2 \u2191(natAbs a) \u2264 \u2191(natAbs a) * \u2191(natAbs a)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\na : \u2124\n\u22a2 natAbs a \u2264 natAbs a * natAbs a\n[PROOFSTEP]\napply Nat.le_mul_self\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\nx : \u2124\nh : 1 < natAbs x\n\u22a2 Function.Injective ((fun x x_1 => x ^ x_1) x)\n[PROOFSTEP]\nsuffices Function.Injective (natAbs \u2218 ((\u00b7 ^ \u00b7) x : \u2115 \u2192 \u2124)) by exact Function.Injective.of_comp this\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\nx : \u2124\nh : 1 < natAbs x\nthis : Function.Injective (natAbs \u2218 (fun x x_1 => x ^ x_1) x)\n\u22a2 Function.Injective ((fun x x_1 => x ^ x_1) x)\n[PROOFSTEP]\nexact Function.Injective.of_comp this\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\nx : \u2124\nh : 1 < natAbs x\n\u22a2 Function.Injective (natAbs \u2218 (fun x x_1 => x ^ x_1) x)\n[PROOFSTEP]\nconvert Nat.pow_right_injective h using 2\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\nx : \u2124\nh : 1 < natAbs x\nx\u271d : \u2115\n\u22a2 (natAbs \u2218 (fun x x_1 => x ^ x_1) x) x\u271d = natAbs x ^ x\u271d\n[PROOFSTEP]\nrw [Function.comp_apply, natAbs_pow]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Monoid M\nf : Multiplicative \u2115 \u2192* M\nn : Multiplicative \u2115\n\u22a2 \u2191f n = \u2191f (\u2191Multiplicative.ofAdd 1) ^ \u2191Multiplicative.toAdd n\n[PROOFSTEP]\nrw [\u2190 powersHom_symm_apply, \u2190 powersHom_apply, Equiv.apply_symm_apply]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Monoid M\nf g : Multiplicative \u2115 \u2192* M\nh : \u2191f (\u2191Multiplicative.ofAdd 1) = \u2191g (\u2191Multiplicative.ofAdd 1)\nn : Multiplicative \u2115\n\u22a2 \u2191f n = \u2191g n\n[PROOFSTEP]\nrw [f.apply_mnat, g.apply_mnat, h]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group M\nf : Multiplicative \u2124 \u2192* M\nn : Multiplicative \u2124\n\u22a2 \u2191f n = \u2191f (\u2191Multiplicative.ofAdd 1) ^ \u2191Multiplicative.toAdd n\n[PROOFSTEP]\nrw [\u2190 zpowersHom_symm_apply, \u2190 zpowersHom_apply, Equiv.apply_symm_apply]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : AddMonoid M\nf : \u2115 \u2192+ M\nn : \u2115\n\u22a2 \u2191f n = n \u2022 \u2191f 1\n[PROOFSTEP]\nrw [\u2190 multiplesHom_symm_apply, \u2190 multiplesHom_apply, Equiv.apply_symm_apply]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : AddGroup M\nf : \u2124 \u2192+ M\nn : \u2124\n\u22a2 \u2191f n = n \u2022 \u2191f 1\n[PROOFSTEP]\nrw [\u2190 zmultiplesHom_symm_apply, \u2190 zmultiplesHom_apply, Equiv.apply_symm_apply]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : CommMonoid M\nsrc\u271d : M \u2243 (Multiplicative \u2115 \u2192* M) := powersHom M\na b : M\nn : Multiplicative \u2115\n\u22a2 \u2191(Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (a * b))\n n =\n \u2191(Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n a *\n Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n b)\n n\n[PROOFSTEP]\nsimp [mul_pow]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : CommGroup G\nsrc\u271d : G \u2243 (Multiplicative \u2124 \u2192* G) := zpowersHom G\na b : G\nn : Multiplicative \u2124\n\u22a2 \u2191(Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (a * b))\n n =\n \u2191(Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n a *\n Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n b)\n n\n[PROOFSTEP]\nsimp [mul_zpow]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : AddCommMonoid A\nsrc\u271d : A \u2243 (\u2115 \u2192+ A) := multiplesHom A\na b : A\nn : \u2115\n\u22a2 \u2191(Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (a + b))\n n =\n \u2191(Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n a +\n Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n b)\n n\n[PROOFSTEP]\nsimp [nsmul_add]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : AddCommGroup A\nsrc\u271d : A \u2243 (\u2124 \u2192+ A) := zmultiplesHom A\na b : A\nn : \u2124\n\u22a2 \u2191(Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (a + b))\n n =\n \u2191(Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n a +\n Equiv.toFun\n { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n b)\n n\n[PROOFSTEP]\nsimp [zsmul_add]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Group G\ninst\u271d : Ring R\na : M\nx y : M\u02e3\nh : SemiconjBy a \u2191x \u2191y\nn : \u2115\n\u22a2 SemiconjBy a \u2191(x ^ \u2191n) \u2191(y ^ \u2191n)\n[PROOFSTEP]\nsimp only [zpow_ofNat, Units.val_pow_eq_pow_val, h, pow_right]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Group G\ninst\u271d : Ring R\na : M\nx y : M\u02e3\nh : SemiconjBy a \u2191x \u2191y\nn : \u2115\n\u22a2 SemiconjBy a \u2191(x ^ -[n+1]) \u2191(y ^ -[n+1])\n[PROOFSTEP]\nsimp only [zpow_negSucc, Units.val_pow_eq_pow_val, units_inv_right, h, pow_right]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.GroupPower.Lemmas", "llama_tokens": 34434, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124811, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.42064686763944026}} {"text": "[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra R B\ns\u271d t : Set A\ninst\u271d : DecidableEq A\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : { x // x \u2208 s } \u2192 Finset A\n\u22a2 adjoin R \u2191(Finset.biUnion (Finset.attach s) f) = \u2a06 (x : { x // x \u2208 s }), adjoin R \u2191(f x)\n[PROOFSTEP]\nsimp [adjoin_iUnion]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\np : A \u2192 A \u2192 Prop\na b : A\nha : a \u2208 adjoin R s\nhb : b \u2208 adjoin R s\nHs : \u2200 (x : A), x \u2208 s \u2192 \u2200 (y : A), y \u2208 s \u2192 p x y\nHalg : \u2200 (r\u2081 r\u2082 : R), p (\u2191(algebraMap R A) r\u2081) (\u2191(algebraMap R A) r\u2082)\nHalg_left : \u2200 (r : R) (x : A), x \u2208 s \u2192 p (\u2191(algebraMap R A) r) x\nHalg_right : \u2200 (r : R) (x : A), x \u2208 s \u2192 p x (\u2191(algebraMap R A) r)\nHadd_left : \u2200 (x\u2081 x\u2082 y : A), p x\u2081 y \u2192 p x\u2082 y \u2192 p (x\u2081 + x\u2082) y\nHadd_right : \u2200 (x y\u2081 y\u2082 : A), p x y\u2081 \u2192 p x y\u2082 \u2192 p x (y\u2081 + y\u2082)\nHmul_left : \u2200 (x\u2081 x\u2082 y : A), p x\u2081 y \u2192 p x\u2082 y \u2192 p (x\u2081 * x\u2082) y\nHmul_right : \u2200 (x y\u2081 y\u2082 : A), p x y\u2081 \u2192 p x y\u2082 \u2192 p x (y\u2081 * y\u2082)\n\u22a2 p a b\n[PROOFSTEP]\nrefine' adjoin_induction hb _ (fun r => _) (Hadd_right a) (Hmul_right a)\n[GOAL]\ncase refine'_1\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\np : A \u2192 A \u2192 Prop\na b : A\nha : a \u2208 adjoin R s\nhb : b \u2208 adjoin R s\nHs : \u2200 (x : A), x \u2208 s \u2192 \u2200 (y : A), y \u2208 s \u2192 p x y\nHalg : \u2200 (r\u2081 r\u2082 : R), p (\u2191(algebraMap R A) r\u2081) (\u2191(algebraMap R A) r\u2082)\nHalg_left : \u2200 (r : R) (x : A), x \u2208 s \u2192 p (\u2191(algebraMap R A) r) x\nHalg_right : \u2200 (r : R) (x : A), x \u2208 s \u2192 p x (\u2191(algebraMap R A) r)\nHadd_left : \u2200 (x\u2081 x\u2082 y : A), p x\u2081 y \u2192 p x\u2082 y \u2192 p (x\u2081 + x\u2082) y\nHadd_right : \u2200 (x y\u2081 y\u2082 : A), p x y\u2081 \u2192 p x y\u2082 \u2192 p x (y\u2081 + y\u2082)\nHmul_left : \u2200 (x\u2081 x\u2082 y : A), p x\u2081 y \u2192 p x\u2082 y \u2192 p (x\u2081 * x\u2082) y\nHmul_right : \u2200 (x y\u2081 y\u2082 : A), p x y\u2081 \u2192 p x y\u2082 \u2192 p x (y\u2081 * y\u2082)\n\u22a2 \u2200 (x : A), x \u2208 s \u2192 p a x\n[PROOFSTEP]\nexact\n adjoin_induction ha Hs Halg_left (fun x y Hx Hy z hz => Hadd_left x y z (Hx z hz) (Hy z hz)) fun x y Hx Hy z hz =>\n Hmul_left x y z (Hx z hz) (Hy z hz)\n[GOAL]\ncase refine'_2\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\np : A \u2192 A \u2192 Prop\na b : A\nha : a \u2208 adjoin R s\nhb : b \u2208 adjoin R s\nHs : \u2200 (x : A), x \u2208 s \u2192 \u2200 (y : A), y \u2208 s \u2192 p x y\nHalg : \u2200 (r\u2081 r\u2082 : R), p (\u2191(algebraMap R A) r\u2081) (\u2191(algebraMap R A) r\u2082)\nHalg_left : \u2200 (r : R) (x : A), x \u2208 s \u2192 p (\u2191(algebraMap R A) r) x\nHalg_right : \u2200 (r : R) (x : A), x \u2208 s \u2192 p x (\u2191(algebraMap R A) r)\nHadd_left : \u2200 (x\u2081 x\u2082 y : A), p x\u2081 y \u2192 p x\u2082 y \u2192 p (x\u2081 + x\u2082) y\nHadd_right : \u2200 (x y\u2081 y\u2082 : A), p x y\u2081 \u2192 p x y\u2082 \u2192 p x (y\u2081 + y\u2082)\nHmul_left : \u2200 (x\u2081 x\u2082 y : A), p x\u2081 y \u2192 p x\u2082 y \u2192 p (x\u2081 * x\u2082) y\nHmul_right : \u2200 (x y\u2081 y\u2082 : A), p x y\u2081 \u2192 p x y\u2082 \u2192 p x (y\u2081 * y\u2082)\nr : R\n\u22a2 p a (\u2191(algebraMap R A) r)\n[PROOFSTEP]\nexact\n adjoin_induction ha (Halg_right r) (fun r' => Halg r' r) (fun x y => Hadd_left x y ((algebraMap R A) r)) fun x y =>\n Hmul_left x y ((algebraMap R A) r)\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\np : { x // x \u2208 adjoin R s } \u2192 Prop\nHs : \u2200 (x : A) (h : x \u2208 s), p { val := x, property := (_ : x \u2208 \u2191(adjoin R s)) }\nHalg : \u2200 (r : R), p (\u2191(algebraMap R { x // x \u2208 adjoin R s }) r)\nHadd : \u2200 (x y : { x // x \u2208 adjoin R s }), p x \u2192 p y \u2192 p (x + y)\nHmul : \u2200 (x y : { x // x \u2208 adjoin R s }), p x \u2192 p y \u2192 p (x * y)\nx\u271d : { x // x \u2208 adjoin R s }\nx : A\nhx : x \u2208 adjoin R s\n\u22a2 p { val := x, property := hx }\n[PROOFSTEP]\nrefine' Exists.elim _ fun (hx : x \u2208 adjoin R s) (hc : p \u27e8x, hx\u27e9) => hc\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\np : { x // x \u2208 adjoin R s } \u2192 Prop\nHs : \u2200 (x : A) (h : x \u2208 s), p { val := x, property := (_ : x \u2208 \u2191(adjoin R s)) }\nHalg : \u2200 (r : R), p (\u2191(algebraMap R { x // x \u2208 adjoin R s }) r)\nHadd : \u2200 (x y : { x // x \u2208 adjoin R s }), p x \u2192 p y \u2192 p (x + y)\nHmul : \u2200 (x y : { x // x \u2208 adjoin R s }), p x \u2192 p y \u2192 p (x * y)\nx\u271d : { x // x \u2208 adjoin R s }\nx : A\nhx : x \u2208 adjoin R s\n\u22a2 \u2203 x_1, p { val := x, property := x_1 }\n[PROOFSTEP]\nexact\n adjoin_induction hx (fun x hx => \u27e8subset_adjoin hx, Hs x hx\u27e9) (fun r => \u27e8Subalgebra.algebraMap_mem _ r, Halg r\u27e9)\n (fun x y hx hy =>\n Exists.elim hx fun hx' hx => Exists.elim hy fun hy' hy => \u27e8Subalgebra.add_mem _ hx' hy', Hadd _ _ hx hy\u27e9)\n fun x y hx hy =>\n Exists.elim hx fun hx' hx => Exists.elim hy fun hy' hy => \u27e8Subalgebra.mul_mem _ hx' hy', Hmul _ _ hx hy\u27e9\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\n\u22a2 adjoin R (Subtype.val \u207b\u00b9' s) = \u22a4\n[PROOFSTEP]\nrefine'\n eq_top_iff.2\n fun x =>\n -- porting note: Lean could no longer infer the motive\n adjoin_induction' (p := fun y => y \u2208 adjoin R (((\u2191) : adjoin R s \u2192 A) \u207b\u00b9' s)) (fun a ha => _) (fun r => _)\n (fun _ _ => _) (fun _ _ => _) x\n[GOAL]\ncase refine'_1\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nx : { x // x \u2208 \u2191(adjoin R s) }\na : A\nha : a \u2208 s\n\u22a2 (fun y => y \u2208 adjoin R (Subtype.val \u207b\u00b9' s)) { val := a, property := (_ : a \u2208 \u2191(adjoin R s)) }\n[PROOFSTEP]\nexact subset_adjoin ha\n[GOAL]\ncase refine'_2\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nx : { x // x \u2208 \u2191(adjoin R s) }\nr : R\n\u22a2 (fun y => y \u2208 adjoin R (Subtype.val \u207b\u00b9' s)) (\u2191(algebraMap R { x // x \u2208 adjoin R s }) r)\n[PROOFSTEP]\nexact Subalgebra.algebraMap_mem _ r\n[GOAL]\ncase refine'_3\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nx : { x // x \u2208 \u2191(adjoin R s) }\nx\u271d\u00b9 x\u271d : { x // x \u2208 adjoin R s }\n\u22a2 (fun y => y \u2208 adjoin R (Subtype.val \u207b\u00b9' s)) x\u271d\u00b9 \u2192\n (fun y => y \u2208 adjoin R (Subtype.val \u207b\u00b9' s)) x\u271d \u2192 (fun y => y \u2208 adjoin R (Subtype.val \u207b\u00b9' s)) (x\u271d\u00b9 + x\u271d)\n[PROOFSTEP]\nexact Subalgebra.add_mem _\n[GOAL]\ncase refine'_4\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nx : { x // x \u2208 \u2191(adjoin R s) }\nx\u271d\u00b9 x\u271d : { x // x \u2208 adjoin R s }\n\u22a2 (fun y => y \u2208 adjoin R (Subtype.val \u207b\u00b9' s)) x\u271d\u00b9 \u2192\n (fun y => y \u2208 adjoin R (Subtype.val \u207b\u00b9' s)) x\u271d \u2192 (fun y => y \u2208 adjoin R (Subtype.val \u207b\u00b9' s)) (x\u271d\u00b9 * x\u271d)\n[PROOFSTEP]\nexact Subalgebra.mul_mem _\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\n\u22a2 adjoin R \u22a5 = \u22a5\n[PROOFSTEP]\napply GaloisConnection.l_bot\n[GOAL]\ncase gc\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\n\u22a2 GaloisConnection (adjoin R) ?u\ncase u\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\n\u22a2 Subalgebra R A \u2192 Set A\n[PROOFSTEP]\nexact Algebra.gc\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\n\u22a2 \u2191Subalgebra.toSubmodule (adjoin R s) = span R \u2191(Submonoid.closure s)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\n\u22a2 \u2191Subalgebra.toSubmodule (adjoin R s) \u2264 span R \u2191(Submonoid.closure s)\n[PROOFSTEP]\nintro r hr\n[GOAL]\ncase a\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nr : A\nhr : r \u2208 \u2191Subalgebra.toSubmodule (adjoin R s)\n\u22a2 r \u2208 span R \u2191(Submonoid.closure s)\n[PROOFSTEP]\nrcases Subsemiring.mem_closure_iff_exists_list.1 hr with \u27e8L, HL, rfl\u27e9\n[GOAL]\ncase a.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhr : List.sum (List.map List.prod L) \u2208 \u2191Subalgebra.toSubmodule (adjoin R s)\n\u22a2 List.sum (List.map List.prod L) \u2208 span R \u2191(Submonoid.closure s)\n[PROOFSTEP]\nclear hr\n[GOAL]\ncase a.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 List.sum (List.map List.prod L) \u2208 span R \u2191(Submonoid.closure s)\n[PROOFSTEP]\ninduction' L with hd tl ih\n[GOAL]\ncase a.intro.intro.nil\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nHL : \u2200 (t : List A), t \u2208 [] \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 List.sum (List.map List.prod []) \u2208 span R \u2191(Submonoid.closure s)\n[PROOFSTEP]\nexact zero_mem _\n[GOAL]\ncase a.intro.intro.cons\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : List A\ntl : List (List A)\nih :\n (\u2200 (t : List A), t \u2208 tl \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s) \u2192\n List.sum (List.map List.prod tl) \u2208 span R \u2191(Submonoid.closure s)\nHL : \u2200 (t : List A), t \u2208 hd :: tl \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 List.sum (List.map List.prod (hd :: tl)) \u2208 span R \u2191(Submonoid.closure s)\n[PROOFSTEP]\nrw [List.forall_mem_cons] at HL \n[GOAL]\ncase a.intro.intro.cons\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : List A\ntl : List (List A)\nih :\n (\u2200 (t : List A), t \u2208 tl \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s) \u2192\n List.sum (List.map List.prod tl) \u2208 span R \u2191(Submonoid.closure s)\nHL :\n (\u2200 (y : A), y \u2208 hd \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s) \u2227\n \u2200 (x : List A), x \u2208 tl \u2192 \u2200 (y : A), y \u2208 x \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 List.sum (List.map List.prod (hd :: tl)) \u2208 span R \u2191(Submonoid.closure s)\n[PROOFSTEP]\nrw [List.map_cons, List.sum_cons]\n[GOAL]\ncase a.intro.intro.cons\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : List A\ntl : List (List A)\nih :\n (\u2200 (t : List A), t \u2208 tl \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s) \u2192\n List.sum (List.map List.prod tl) \u2208 span R \u2191(Submonoid.closure s)\nHL :\n (\u2200 (y : A), y \u2208 hd \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s) \u2227\n \u2200 (x : List A), x \u2208 tl \u2192 \u2200 (y : A), y \u2208 x \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 List.prod hd + List.sum (List.map List.prod tl) \u2208 span R \u2191(Submonoid.closure s)\n[PROOFSTEP]\nrefine' Submodule.add_mem _ _ (ih HL.2)\n[GOAL]\ncase a.intro.intro.cons\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : List A\ntl : List (List A)\nih :\n (\u2200 (t : List A), t \u2208 tl \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s) \u2192\n List.sum (List.map List.prod tl) \u2208 span R \u2191(Submonoid.closure s)\nHL :\n (\u2200 (y : A), y \u2208 hd \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s) \u2227\n \u2200 (x : List A), x \u2208 tl \u2192 \u2200 (y : A), y \u2208 x \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 List.prod hd \u2208 span R \u2191(Submonoid.closure s)\n[PROOFSTEP]\nreplace HL := HL.1\n[GOAL]\ncase a.intro.intro.cons\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : List A\ntl : List (List A)\nih :\n (\u2200 (t : List A), t \u2208 tl \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s) \u2192\n List.sum (List.map List.prod tl) \u2208 span R \u2191(Submonoid.closure s)\nHL : \u2200 (y : A), y \u2208 hd \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 List.prod hd \u2208 span R \u2191(Submonoid.closure s)\n[PROOFSTEP]\nclear ih tl\n[GOAL]\ncase a.intro.intro.cons\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : List A\nHL : \u2200 (y : A), y \u2208 hd \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 List.prod hd \u2208 span R \u2191(Submonoid.closure s)\n[PROOFSTEP]\nsuffices \u2203 (z r : _) (_hr : r \u2208 Submonoid.closure s), z \u2022 r = List.prod hd\n by\n rcases this with \u27e8z, r, hr, hzr\u27e9\n rw [\u2190 hzr]\n exact smul_mem _ _ (subset_span hr)\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : List A\nHL : \u2200 (y : A), y \u2208 hd \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nthis : \u2203 z r _hr, z \u2022 r = List.prod hd\n\u22a2 List.prod hd \u2208 span R \u2191(Submonoid.closure s)\n[PROOFSTEP]\nrcases this with \u27e8z, r, hr, hzr\u27e9\n[GOAL]\ncase intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : List A\nHL : \u2200 (y : A), y \u2208 hd \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nz : ?m.213266\nr : A\nhr : r \u2208 Submonoid.closure s\nhzr : z \u2022 r = List.prod hd\n\u22a2 List.prod hd \u2208 span R \u2191(Submonoid.closure s)\n[PROOFSTEP]\nrw [\u2190 hzr]\n[GOAL]\ncase intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : List A\nHL : \u2200 (y : A), y \u2208 hd \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nz : ?m.213266\nr : A\nhr : r \u2208 Submonoid.closure s\nhzr : z \u2022 r = List.prod hd\n\u22a2 z \u2022 r \u2208 span R \u2191(Submonoid.closure s)\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : List A\nHL : \u2200 (y : A), y \u2208 hd \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 Type ?u.213363\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : List A\nHL : \u2200 (y : A), y \u2208 hd \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 ?m.213266 \u2192 (r : A) \u2192 r \u2208 Submonoid.closure s \u2192 HSMul ?m.213266 A A\n[PROOFSTEP]\nexact smul_mem _ _ (subset_span hr)\n[GOAL]\ncase a.intro.intro.cons\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : List A\nHL : \u2200 (y : A), y \u2208 hd \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 \u2203 z r _hr, z \u2022 r = List.prod hd\n[PROOFSTEP]\ninduction' hd with hd tl ih\n[GOAL]\ncase a.intro.intro.cons.nil\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d\u00b9 : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : List A\nHL\u271d : \u2200 (y : A), y \u2208 hd \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nHL : \u2200 (y : A), y \u2208 [] \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 \u2203 z r _hr, z \u2022 r = List.prod []\n[PROOFSTEP]\nexact \u27e81, 1, (Submonoid.closure s).one_mem', one_smul _ _\u27e9\n[GOAL]\ncase a.intro.intro.cons.cons\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d\u00b9 : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd\u271d : List A\nHL\u271d : \u2200 (y : A), y \u2208 hd\u271d \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : A\ntl : List A\nih : (\u2200 (y : A), y \u2208 tl \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s) \u2192 \u2203 z r _hr, z \u2022 r = List.prod tl\nHL : \u2200 (y : A), y \u2208 hd :: tl \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 \u2203 z r _hr, z \u2022 r = List.prod (hd :: tl)\n[PROOFSTEP]\nrw [List.forall_mem_cons] at HL \n[GOAL]\ncase a.intro.intro.cons.cons\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d\u00b9 : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd\u271d : List A\nHL\u271d : \u2200 (y : A), y \u2208 hd\u271d \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : A\ntl : List A\nih : (\u2200 (y : A), y \u2208 tl \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s) \u2192 \u2203 z r _hr, z \u2022 r = List.prod tl\nHL : hd \u2208 Set.range \u2191(algebraMap R A) \u222a s \u2227 \u2200 (x : A), x \u2208 tl \u2192 x \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 \u2203 z r _hr, z \u2022 r = List.prod (hd :: tl)\n[PROOFSTEP]\nrcases ih HL.2 with \u27e8z, r, hr, hzr\u27e9\n[GOAL]\ncase a.intro.intro.cons.cons.intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d\u00b9 : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd\u271d : List A\nHL\u271d : \u2200 (y : A), y \u2208 hd\u271d \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : A\ntl : List A\nih : (\u2200 (y : A), y \u2208 tl \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s) \u2192 \u2203 z r _hr, z \u2022 r = List.prod tl\nHL : hd \u2208 Set.range \u2191(algebraMap R A) \u222a s \u2227 \u2200 (x : A), x \u2208 tl \u2192 x \u2208 Set.range \u2191(algebraMap R A) \u222a s\nz : R\nr : A\nhr : r \u2208 Submonoid.closure s\nhzr : z \u2022 r = List.prod tl\n\u22a2 \u2203 z r _hr, z \u2022 r = List.prod (hd :: tl)\n[PROOFSTEP]\nrw [List.prod_cons, \u2190 hzr]\n[GOAL]\ncase a.intro.intro.cons.cons.intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d\u00b9 : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd\u271d : List A\nHL\u271d : \u2200 (y : A), y \u2208 hd\u271d \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : A\ntl : List A\nih : (\u2200 (y : A), y \u2208 tl \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s) \u2192 \u2203 z r _hr, z \u2022 r = List.prod tl\nHL : hd \u2208 Set.range \u2191(algebraMap R A) \u222a s \u2227 \u2200 (x : A), x \u2208 tl \u2192 x \u2208 Set.range \u2191(algebraMap R A) \u222a s\nz : R\nr : A\nhr : r \u2208 Submonoid.closure s\nhzr : z \u2022 r = List.prod tl\n\u22a2 \u2203 z_1 r_1 _hr, z_1 \u2022 r_1 = hd * z \u2022 r\n[PROOFSTEP]\nrcases HL.1 with (\u27e8hd, rfl\u27e9 | hs)\n[GOAL]\ncase a.intro.intro.cons.cons.intro.intro.intro.inl.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d\u00b9 : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd\u271d : List A\nHL\u271d : \u2200 (y : A), y \u2208 hd\u271d \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\ntl : List A\nih : (\u2200 (y : A), y \u2208 tl \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s) \u2192 \u2203 z r _hr, z \u2022 r = List.prod tl\nz : R\nr : A\nhr : r \u2208 Submonoid.closure s\nhzr : z \u2022 r = List.prod tl\nhd : R\nHL : \u2191(algebraMap R A) hd \u2208 Set.range \u2191(algebraMap R A) \u222a s \u2227 \u2200 (x : A), x \u2208 tl \u2192 x \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 \u2203 z_1 r_1 _hr, z_1 \u2022 r_1 = \u2191(algebraMap R A) hd * z \u2022 r\n[PROOFSTEP]\nrefine' \u27e8hd * z, r, hr, _\u27e9\n[GOAL]\ncase a.intro.intro.cons.cons.intro.intro.intro.inl.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d\u00b9 : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd\u271d : List A\nHL\u271d : \u2200 (y : A), y \u2208 hd\u271d \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\ntl : List A\nih : (\u2200 (y : A), y \u2208 tl \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s) \u2192 \u2203 z r _hr, z \u2022 r = List.prod tl\nz : R\nr : A\nhr : r \u2208 Submonoid.closure s\nhzr : z \u2022 r = List.prod tl\nhd : R\nHL : \u2191(algebraMap R A) hd \u2208 Set.range \u2191(algebraMap R A) \u222a s \u2227 \u2200 (x : A), x \u2208 tl \u2192 x \u2208 Set.range \u2191(algebraMap R A) \u222a s\n\u22a2 (hd * z) \u2022 r = \u2191(algebraMap R A) hd * z \u2022 r\n[PROOFSTEP]\nrw [Algebra.smul_def, Algebra.smul_def, (algebraMap _ _).map_mul, _root_.mul_assoc]\n[GOAL]\ncase a.intro.intro.cons.cons.intro.intro.intro.inr\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\nL : List (List A)\nHL\u271d\u00b9 : \u2200 (t : List A), t \u2208 L \u2192 \u2200 (y : A), y \u2208 t \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd\u271d : List A\nHL\u271d : \u2200 (y : A), y \u2208 hd\u271d \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s\nhd : A\ntl : List A\nih : (\u2200 (y : A), y \u2208 tl \u2192 y \u2208 Set.range \u2191(algebraMap R A) \u222a s) \u2192 \u2203 z r _hr, z \u2022 r = List.prod tl\nHL : hd \u2208 Set.range \u2191(algebraMap R A) \u222a s \u2227 \u2200 (x : A), x \u2208 tl \u2192 x \u2208 Set.range \u2191(algebraMap R A) \u222a s\nz : R\nr : A\nhr : r \u2208 Submonoid.closure s\nhzr : z \u2022 r = List.prod tl\nhs : hd \u2208 s\n\u22a2 \u2203 z_1 r_1 _hr, z_1 \u2022 r_1 = hd * z \u2022 r\n[PROOFSTEP]\nexact \u27e8z, hd * r, Submonoid.mul_mem _ (Submonoid.subset_closure hs) hr, (mul_smul_comm _ _ _).symm\u27e9\n[GOAL]\ncase a\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\n\u22a2 span R \u2191(Submonoid.closure s) \u2264 \u2191Subalgebra.toSubmodule (adjoin R s)\n[PROOFSTEP]\nrefine' span_le.2 _\n[GOAL]\ncase a\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\n\u22a2 \u2191(Submonoid.closure s) \u2286 \u2191(\u2191Subalgebra.toSubmodule (adjoin R s))\n[PROOFSTEP]\nchange Submonoid.closure s \u2264 (adjoin R s).toSubsemiring.toSubmonoid\n[GOAL]\ncase a\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns t : Set A\n\u22a2 Submonoid.closure s \u2264 (adjoin R s).toSubsemiring.toSubmonoid\n[PROOFSTEP]\nexact Submonoid.closure_le.2 subset_adjoin\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t\u271d s : Set A\nt : Submodule R A\n\u22a2 \u2191Subalgebra.toSubmodule (adjoin R s) \u2264 t \u2194 \u2191(Submonoid.closure s) \u2286 \u2191t\n[PROOFSTEP]\nrw [adjoin_eq_span, span_le]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t : Set A\nf : A \u2192\u2090[R] B\ns : Set A\n\u22a2 (fun x => x) '' s \u2286 (Subalgebra.comap f (adjoin R (\u2191f '' s))).toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nsimp only [Set.image_id', coe_carrier_toSubmonoid, Subalgebra.coe_toSubsemiring, Subalgebra.coe_comap]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t : Set A\nf : A \u2192\u2090[R] B\ns : Set A\n\u22a2 s \u2286 \u2191f \u207b\u00b9' \u2191(adjoin R ((fun a => \u2191f a) '' s))\n[PROOFSTEP]\nexact fun x hx => subset_adjoin \u27e8x, hx, rfl\u27e9\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nx : A\nf : A \u2192\u2097[R] B\nhf : \u2200 (a\u2081 a\u2082 : A), \u2191f (a\u2081 * a\u2082) = \u2191f a\u2081 * \u2191f a\u2082\nh : x \u2208 adjoin R s\n\u22a2 \u2191f x \u2208 adjoin R (\u2191f '' (s \u222a {1}))\n[PROOFSTEP]\nrefine'\n @adjoin_induction R A _ _ _ _ (fun a => f a \u2208 adjoin R (f '' (s \u222a {1}))) x h\n (fun a ha => subset_adjoin \u27e8a, \u27e8Set.subset_union_left _ _ ha, rfl\u27e9\u27e9) (fun r => _)\n (fun y z hy hz => by simpa [hy, hz] using Subalgebra.add_mem _ hy hz) fun y z hy hz => by\n simpa [hy, hz, hf y z] using Subalgebra.mul_mem _ hy hz\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nx : A\nf : A \u2192\u2097[R] B\nhf : \u2200 (a\u2081 a\u2082 : A), \u2191f (a\u2081 * a\u2082) = \u2191f a\u2081 * \u2191f a\u2082\nh : x \u2208 adjoin R s\ny z : A\nhy : (fun a => \u2191f a \u2208 adjoin R (\u2191f '' (s \u222a {1}))) y\nhz : (fun a => \u2191f a \u2208 adjoin R (\u2191f '' (s \u222a {1}))) z\n\u22a2 (fun a => \u2191f a \u2208 adjoin R (\u2191f '' (s \u222a {1}))) (y + z)\n[PROOFSTEP]\nsimpa [hy, hz] using Subalgebra.add_mem _ hy hz\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nx : A\nf : A \u2192\u2097[R] B\nhf : \u2200 (a\u2081 a\u2082 : A), \u2191f (a\u2081 * a\u2082) = \u2191f a\u2081 * \u2191f a\u2082\nh : x \u2208 adjoin R s\ny z : A\nhy : (fun a => \u2191f a \u2208 adjoin R (\u2191f '' (s \u222a {1}))) y\nhz : (fun a => \u2191f a \u2208 adjoin R (\u2191f '' (s \u222a {1}))) z\n\u22a2 (fun a => \u2191f a \u2208 adjoin R (\u2191f '' (s \u222a {1}))) (y * z)\n[PROOFSTEP]\nsimpa [hy, hz, hf y z] using Subalgebra.mul_mem _ hy hz\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nx : A\nf : A \u2192\u2097[R] B\nhf : \u2200 (a\u2081 a\u2082 : A), \u2191f (a\u2081 * a\u2082) = \u2191f a\u2081 * \u2191f a\u2082\nh : x \u2208 adjoin R s\nr : R\n\u22a2 (fun a => \u2191f a \u2208 adjoin R (\u2191f '' (s \u222a {1}))) (\u2191(algebraMap R A) r)\n[PROOFSTEP]\nhave : f 1 \u2208 adjoin R (f '' (s \u222a {1})) := subset_adjoin \u27e81, \u27e8Set.subset_union_right _ _ <| Set.mem_singleton 1, rfl\u27e9\u27e9\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nx : A\nf : A \u2192\u2097[R] B\nhf : \u2200 (a\u2081 a\u2082 : A), \u2191f (a\u2081 * a\u2082) = \u2191f a\u2081 * \u2191f a\u2082\nh : x \u2208 adjoin R s\nr : R\nthis : \u2191f 1 \u2208 adjoin R (\u2191f '' (s \u222a {1}))\n\u22a2 (fun a => \u2191f a \u2208 adjoin R (\u2191f '' (s \u222a {1}))) (\u2191(algebraMap R A) r)\n[PROOFSTEP]\nconvert Subalgebra.smul_mem (adjoin R (f '' (s \u222a {1}))) this r\n[GOAL]\ncase h.e'_4\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nx : A\nf : A \u2192\u2097[R] B\nhf : \u2200 (a\u2081 a\u2082 : A), \u2191f (a\u2081 * a\u2082) = \u2191f a\u2081 * \u2191f a\u2082\nh : x \u2208 adjoin R s\nr : R\nthis : \u2191f 1 \u2208 adjoin R (\u2191f '' (s \u222a {1}))\n\u22a2 \u2191f (\u2191(algebraMap R A) r) = r \u2022 \u2191f 1\n[PROOFSTEP]\nrw [algebraMap_eq_smul_one]\n[GOAL]\ncase h.e'_4\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nx : A\nf : A \u2192\u2097[R] B\nhf : \u2200 (a\u2081 a\u2082 : A), \u2191f (a\u2081 * a\u2082) = \u2191f a\u2081 * \u2191f a\u2082\nh : x \u2208 adjoin R s\nr : R\nthis : \u2191f 1 \u2208 adjoin R (\u2191f '' (s \u222a {1}))\n\u22a2 \u2191f (r \u2022 1) = r \u2022 \u2191f 1\n[PROOFSTEP]\nexact f.map_smul _ _\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t\u271d s : Set A\nt : Set B\n\u22a2 adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}) \u222a \u2191(LinearMap.inr R A B) '' (t \u222a {1})) =\n Subalgebra.prod (adjoin R s) (adjoin R t)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t\u271d s : Set A\nt : Set B\n\u22a2 adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}) \u222a \u2191(LinearMap.inr R A B) '' (t \u222a {1})) \u2264\n Subalgebra.prod (adjoin R s) (adjoin R t)\n[PROOFSTEP]\nsimp only [adjoin_le_iff, Set.insert_subset_iff, Subalgebra.zero_mem, Subalgebra.one_mem, subset_adjoin,\n -- the rest comes from `squeeze_simp`Set.union_subset_iff, LinearMap.coe_inl, Set.mk_preimage_prod_right,\n Set.image_subset_iff, SetLike.mem_coe, Set.mk_preimage_prod_left, LinearMap.coe_inr, and_self_iff,\n Set.union_singleton, Subalgebra.coe_prod]\n[GOAL]\ncase a\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t\u271d s : Set A\nt : Set B\n\u22a2 Subalgebra.prod (adjoin R s) (adjoin R t) \u2264\n adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}) \u222a \u2191(LinearMap.inr R A B) '' (t \u222a {1}))\n[PROOFSTEP]\nrintro \u27e8a, b\u27e9 \u27e8ha, hb\u27e9\n[GOAL]\ncase a.mk.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t\u271d s : Set A\nt : Set B\na : A\nb : B\nha : (a, b).fst \u2208 \u2191(adjoin R s)\nhb : (a, b).snd \u2208 \u2191(adjoin R t)\n\u22a2 (a, b) \u2208 adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}) \u222a \u2191(LinearMap.inr R A B) '' (t \u222a {1}))\n[PROOFSTEP]\nlet P := adjoin R (LinearMap.inl R A B '' (s \u222a {1}) \u222a LinearMap.inr R A B '' (t \u222a {1}))\n[GOAL]\ncase a.mk.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t\u271d s : Set A\nt : Set B\na : A\nb : B\nha : (a, b).fst \u2208 \u2191(adjoin R s)\nhb : (a, b).snd \u2208 \u2191(adjoin R t)\nP : Subalgebra R (A \u00d7 B) := adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}) \u222a \u2191(LinearMap.inr R A B) '' (t \u222a {1}))\n\u22a2 (a, b) \u2208 adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}) \u222a \u2191(LinearMap.inr R A B) '' (t \u222a {1}))\n[PROOFSTEP]\nhave Ha : (a, (0 : B)) \u2208 adjoin R (LinearMap.inl R A B '' (s \u222a {1})) := mem_adjoin_of_map_mul R LinearMap.inl_map_mul ha\n[GOAL]\ncase a.mk.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t\u271d s : Set A\nt : Set B\na : A\nb : B\nha : (a, b).fst \u2208 \u2191(adjoin R s)\nhb : (a, b).snd \u2208 \u2191(adjoin R t)\nP : Subalgebra R (A \u00d7 B) := adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}) \u222a \u2191(LinearMap.inr R A B) '' (t \u222a {1}))\nHa : (a, 0) \u2208 adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}))\n\u22a2 (a, b) \u2208 adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}) \u222a \u2191(LinearMap.inr R A B) '' (t \u222a {1}))\n[PROOFSTEP]\nhave Hb : ((0 : A), b) \u2208 adjoin R (LinearMap.inr R A B '' (t \u222a {1})) := mem_adjoin_of_map_mul R LinearMap.inr_map_mul hb\n[GOAL]\ncase a.mk.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t\u271d s : Set A\nt : Set B\na : A\nb : B\nha : (a, b).fst \u2208 \u2191(adjoin R s)\nhb : (a, b).snd \u2208 \u2191(adjoin R t)\nP : Subalgebra R (A \u00d7 B) := adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}) \u222a \u2191(LinearMap.inr R A B) '' (t \u222a {1}))\nHa : (a, 0) \u2208 adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}))\nHb : (0, b) \u2208 adjoin R (\u2191(LinearMap.inr R A B) '' (t \u222a {1}))\n\u22a2 (a, b) \u2208 adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}) \u222a \u2191(LinearMap.inr R A B) '' (t \u222a {1}))\n[PROOFSTEP]\nreplace Ha : (a, (0 : B)) \u2208 P := adjoin_mono (Set.subset_union_left _ _) Ha\n[GOAL]\ncase a.mk.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t\u271d s : Set A\nt : Set B\na : A\nb : B\nha : (a, b).fst \u2208 \u2191(adjoin R s)\nhb : (a, b).snd \u2208 \u2191(adjoin R t)\nP : Subalgebra R (A \u00d7 B) := adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}) \u222a \u2191(LinearMap.inr R A B) '' (t \u222a {1}))\nHb : (0, b) \u2208 adjoin R (\u2191(LinearMap.inr R A B) '' (t \u222a {1}))\nHa : (a, 0) \u2208 P\n\u22a2 (a, b) \u2208 adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}) \u222a \u2191(LinearMap.inr R A B) '' (t \u222a {1}))\n[PROOFSTEP]\nreplace Hb : ((0 : A), b) \u2208 P := adjoin_mono (Set.subset_union_right _ _) Hb\n[GOAL]\ncase a.mk.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t\u271d s : Set A\nt : Set B\na : A\nb : B\nha : (a, b).fst \u2208 \u2191(adjoin R s)\nhb : (a, b).snd \u2208 \u2191(adjoin R t)\nP : Subalgebra R (A \u00d7 B) := adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}) \u222a \u2191(LinearMap.inr R A B) '' (t \u222a {1}))\nHa : (a, 0) \u2208 P\nHb : (0, b) \u2208 P\n\u22a2 (a, b) \u2208 adjoin R (\u2191(LinearMap.inl R A B) '' (s \u222a {1}) \u222a \u2191(LinearMap.inr R A B) '' (t \u222a {1}))\n[PROOFSTEP]\nsimpa using Subalgebra.add_mem _ Ha Hb\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nhcomm : \u2200 (a : A), a \u2208 s \u2192 \u2200 (b : A), b \u2208 s \u2192 a * b = b * a\nsrc\u271d : Semiring { x // x \u2208 adjoin R s } := Subalgebra.toSemiring (adjoin R s)\nx y : { x // x \u2208 adjoin R s }\n\u22a2 x * y = y * x\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nhcomm : \u2200 (a : A), a \u2208 s \u2192 \u2200 (b : A), b \u2208 s \u2192 a * b = b * a\nsrc\u271d : Semiring { x // x \u2208 adjoin R s } := Subalgebra.toSemiring (adjoin R s)\nx y : { x // x \u2208 adjoin R s }\n\u22a2 \u2191(x * y) = \u2191(y * x)\n[PROOFSTEP]\nsimp only [Subalgebra.coe_mul]\n[GOAL]\ncase a\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nhcomm : \u2200 (a : A), a \u2208 s \u2192 \u2200 (b : A), b \u2208 s \u2192 a * b = b * a\nsrc\u271d : Semiring { x // x \u2208 adjoin R s } := Subalgebra.toSemiring (adjoin R s)\nx y : { x // x \u2208 adjoin R s }\n\u22a2 \u2191x * \u2191y = \u2191y * \u2191x\n[PROOFSTEP]\nexact\n adjoin_induction\u2082 x.prop y.prop hcomm (fun _ _ => by rw [commutes]) (fun r x _hx => commutes r x)\n (fun r x _hx => (commutes r x).symm) (fun _ _ _ h\u2081 h\u2082 => by simp only [add_mul, mul_add, h\u2081, h\u2082])\n (fun _ _ _ h\u2081 h\u2082 => by simp only [add_mul, mul_add, h\u2081, h\u2082])\n (fun x\u2081 x\u2082 y\u2081 h\u2081 h\u2082 => by rw [mul_assoc, h\u2082, \u2190 mul_assoc y\u2081, \u2190 h\u2081, mul_assoc x\u2081]) fun x\u2081 x\u2082 y\u2081 h\u2081 h\u2082 => by\n rw [mul_assoc x\u2082, \u2190 h\u2082, \u2190 mul_assoc x\u2082, \u2190 h\u2081, \u2190 mul_assoc]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nhcomm : \u2200 (a : A), a \u2208 s \u2192 \u2200 (b : A), b \u2208 s \u2192 a * b = b * a\nsrc\u271d : Semiring { x // x \u2208 adjoin R s } := Subalgebra.toSemiring (adjoin R s)\nx y : { x // x \u2208 adjoin R s }\nx\u271d\u00b9 x\u271d : R\n\u22a2 \u2191(algebraMap R A) x\u271d\u00b9 * \u2191(algebraMap R A) x\u271d = \u2191(algebraMap R A) x\u271d * \u2191(algebraMap R A) x\u271d\u00b9\n[PROOFSTEP]\nrw [commutes]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nhcomm : \u2200 (a : A), a \u2208 s \u2192 \u2200 (b : A), b \u2208 s \u2192 a * b = b * a\nsrc\u271d : Semiring { x // x \u2208 adjoin R s } := Subalgebra.toSemiring (adjoin R s)\nx y : { x // x \u2208 adjoin R s }\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : A\nh\u2081 : x\u271d\u00b2 * x\u271d = x\u271d * x\u271d\u00b2\nh\u2082 : x\u271d\u00b9 * x\u271d = x\u271d * x\u271d\u00b9\n\u22a2 (x\u271d\u00b2 + x\u271d\u00b9) * x\u271d = x\u271d * (x\u271d\u00b2 + x\u271d\u00b9)\n[PROOFSTEP]\nsimp only [add_mul, mul_add, h\u2081, h\u2082]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nhcomm : \u2200 (a : A), a \u2208 s \u2192 \u2200 (b : A), b \u2208 s \u2192 a * b = b * a\nsrc\u271d : Semiring { x // x \u2208 adjoin R s } := Subalgebra.toSemiring (adjoin R s)\nx y : { x // x \u2208 adjoin R s }\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : A\nh\u2081 : x\u271d\u00b2 * x\u271d\u00b9 = x\u271d\u00b9 * x\u271d\u00b2\nh\u2082 : x\u271d\u00b2 * x\u271d = x\u271d * x\u271d\u00b2\n\u22a2 x\u271d\u00b2 * (x\u271d\u00b9 + x\u271d) = (x\u271d\u00b9 + x\u271d) * x\u271d\u00b2\n[PROOFSTEP]\nsimp only [add_mul, mul_add, h\u2081, h\u2082]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nhcomm : \u2200 (a : A), a \u2208 s \u2192 \u2200 (b : A), b \u2208 s \u2192 a * b = b * a\nsrc\u271d : Semiring { x // x \u2208 adjoin R s } := Subalgebra.toSemiring (adjoin R s)\nx y : { x // x \u2208 adjoin R s }\nx\u2081 x\u2082 y\u2081 : A\nh\u2081 : x\u2081 * y\u2081 = y\u2081 * x\u2081\nh\u2082 : x\u2082 * y\u2081 = y\u2081 * x\u2082\n\u22a2 x\u2081 * x\u2082 * y\u2081 = y\u2081 * (x\u2081 * x\u2082)\n[PROOFSTEP]\nrw [mul_assoc, h\u2082, \u2190 mul_assoc y\u2081, \u2190 h\u2081, mul_assoc x\u2081]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\ns\u271d t s : Set A\nhcomm : \u2200 (a : A), a \u2208 s \u2192 \u2200 (b : A), b \u2208 s \u2192 a * b = b * a\nsrc\u271d : Semiring { x // x \u2208 adjoin R s } := Subalgebra.toSemiring (adjoin R s)\nx y : { x // x \u2208 adjoin R s }\nx\u2081 x\u2082 y\u2081 : A\nh\u2081 : x\u2081 * x\u2082 = x\u2082 * x\u2081\nh\u2082 : x\u2081 * y\u2081 = y\u2081 * x\u2081\n\u22a2 x\u2081 * (x\u2082 * y\u2081) = x\u2082 * y\u2081 * x\u2081\n[PROOFSTEP]\nrw [mul_assoc x\u2082, \u2190 h\u2082, \u2190 mul_assoc x\u2082, \u2190 h\u2081, \u2190 mul_assoc]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\n\u22a2 \u2191Subalgebra.toSubmodule (adjoin R (s \u222a t)) =\n \u2191Subalgebra.toSubmodule (adjoin R s) * \u2191Subalgebra.toSubmodule (adjoin R t)\n[PROOFSTEP]\nrw [adjoin_eq_span, adjoin_eq_span, adjoin_eq_span, span_mul_span]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\n\u22a2 span R \u2191(Submonoid.closure (s \u222a t)) = span R (\u2191(Submonoid.closure s) * \u2191(Submonoid.closure t))\n[PROOFSTEP]\ncongr 1 with z\n[GOAL]\ncase e_s.h\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nz : A\n\u22a2 z \u2208 \u2191(Submonoid.closure (s \u222a t)) \u2194 z \u2208 \u2191(Submonoid.closure s) * \u2191(Submonoid.closure t)\n[PROOFSTEP]\nsimp [Submonoid.closure_union, Submonoid.mem_sup, Set.mem_mul]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\ns : Set B\n\u22a2 adjoin A \u2191(adjoin R s) = adjoin A s\n[PROOFSTEP]\napply le_antisymm (adjoin_le _)\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\ns : Set B\n\u22a2 adjoin A s \u2264 adjoin A \u2191(adjoin R s)\n[PROOFSTEP]\nexact adjoin_mono subset_adjoin\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\ns : Set B\n\u22a2 \u2191(adjoin R s) \u2286 \u2191(adjoin A s)\n[PROOFSTEP]\nchange adjoin R s \u2264 (adjoin A s).restrictScalars R\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\ns : Set B\n\u22a2 adjoin R s \u2264 Subalgebra.restrictScalars R (adjoin A s)\n[PROOFSTEP]\nrefine'\n adjoin_le\n _\n -- porting note: unclear why this was broken\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\ns : Set B\n\u22a2 s \u2286 \u2191(Subalgebra.restrictScalars R (adjoin A s))\n[PROOFSTEP]\nhave : (Subalgebra.restrictScalars R (adjoin A s) : Set B) = adjoin A s := rfl\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\ns : Set B\nthis : \u2191(Subalgebra.restrictScalars R (adjoin A s)) = \u2191(adjoin A s)\n\u22a2 s \u2286 \u2191(Subalgebra.restrictScalars R (adjoin A s))\n[PROOFSTEP]\nrw [this]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\ns : Set B\nthis : \u2191(Subalgebra.restrictScalars R (adjoin A s)) = \u2191(adjoin A s)\n\u22a2 s \u2286 \u2191(adjoin A s)\n[PROOFSTEP]\nexact subset_adjoin\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nx : B\nhx : x \u2208 adjoin R s\nhr : \u2191(algebraMap A B) r \u2208 B'\n\u22a2 \u2203 n\u2080, \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 r ^ n \u2022 x \u2208 B'\n[PROOFSTEP]\nreplace hx : x \u2208 Subalgebra.toSubmodule (adjoin R s) := hx\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nx : B\nhr : \u2191(algebraMap A B) r \u2208 B'\nhx : x \u2208 \u2191Subalgebra.toSubmodule (adjoin R s)\n\u22a2 \u2203 n\u2080, \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 r ^ n \u2022 x \u2208 B'\n[PROOFSTEP]\nrw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx \n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nx : B\nhr : \u2191(algebraMap A B) r \u2208 B'\nhx : \u2203 l, \u2191(Finsupp.total (\u2191\u2191(Submonoid.closure s)) B R Subtype.val) l = x\n\u22a2 \u2203 n\u2080, \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 r ^ n \u2022 x \u2208 B'\n[PROOFSTEP]\nrcases hx with \u27e8l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c \u2022 (i : B)) = x\u27e9\n[GOAL]\ncase intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nhr : \u2191(algebraMap A B) r \u2208 B'\nl : \u2191\u2191(Submonoid.closure s) \u2192\u2080 R\n\u22a2 \u2203 n\u2080, \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 (r ^ n \u2022 Finsupp.sum l fun i c => c \u2022 \u2191i) \u2208 B'\n[PROOFSTEP]\nchoose n\u2081 n\u2082 using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop\n[GOAL]\ncase intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nhr : \u2191(algebraMap A B) r \u2208 B'\nl : \u2191\u2191(Submonoid.closure s) \u2192\u2080 R\nn\u2081 : { x // x \u2208 Submonoid.closure s } \u2192 \u2115\nn\u2082 : \u2200 (x : { x // x \u2208 Submonoid.closure s }), r ^ n\u2081 x \u2022 \u2191x \u2208 Submonoid.closure (r \u2022 s)\n\u22a2 \u2203 n\u2080, \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 (r ^ n \u2022 Finsupp.sum l fun i c => c \u2022 \u2191i) \u2208 B'\n[PROOFSTEP]\nuse l.support.sup n\u2081\n[GOAL]\ncase h\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nhr : \u2191(algebraMap A B) r \u2208 B'\nl : \u2191\u2191(Submonoid.closure s) \u2192\u2080 R\nn\u2081 : { x // x \u2208 Submonoid.closure s } \u2192 \u2115\nn\u2082 : \u2200 (x : { x // x \u2208 Submonoid.closure s }), r ^ n\u2081 x \u2022 \u2191x \u2208 Submonoid.closure (r \u2022 s)\n\u22a2 \u2200 (n : \u2115), n \u2265 Finset.sup l.support n\u2081 \u2192 (r ^ n \u2022 Finsupp.sum l fun i c => c \u2022 \u2191i) \u2208 B'\n[PROOFSTEP]\nintro n hn\n[GOAL]\ncase h\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nhr : \u2191(algebraMap A B) r \u2208 B'\nl : \u2191\u2191(Submonoid.closure s) \u2192\u2080 R\nn\u2081 : { x // x \u2208 Submonoid.closure s } \u2192 \u2115\nn\u2082 : \u2200 (x : { x // x \u2208 Submonoid.closure s }), r ^ n\u2081 x \u2022 \u2191x \u2208 Submonoid.closure (r \u2022 s)\nn : \u2115\nhn : n \u2265 Finset.sup l.support n\u2081\n\u22a2 (r ^ n \u2022 Finsupp.sum l fun i c => c \u2022 \u2191i) \u2208 B'\n[PROOFSTEP]\nrw [Finsupp.smul_sum]\n[GOAL]\ncase h\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nhr : \u2191(algebraMap A B) r \u2208 B'\nl : \u2191\u2191(Submonoid.closure s) \u2192\u2080 R\nn\u2081 : { x // x \u2208 Submonoid.closure s } \u2192 \u2115\nn\u2082 : \u2200 (x : { x // x \u2208 Submonoid.closure s }), r ^ n\u2081 x \u2022 \u2191x \u2208 Submonoid.closure (r \u2022 s)\nn : \u2115\nhn : n \u2265 Finset.sup l.support n\u2081\n\u22a2 (Finsupp.sum l fun a b => r ^ n \u2022 b \u2022 \u2191a) \u2208 B'\n[PROOFSTEP]\nrefine' B'.toSubmodule.sum_mem _\n[GOAL]\ncase h\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nhr : \u2191(algebraMap A B) r \u2208 B'\nl : \u2191\u2191(Submonoid.closure s) \u2192\u2080 R\nn\u2081 : { x // x \u2208 Submonoid.closure s } \u2192 \u2115\nn\u2082 : \u2200 (x : { x // x \u2208 Submonoid.closure s }), r ^ n\u2081 x \u2022 \u2191x \u2208 Submonoid.closure (r \u2022 s)\nn : \u2115\nhn : n \u2265 Finset.sup l.support n\u2081\n\u22a2 \u2200 (c : \u2191\u2191(Submonoid.closure s)), c \u2208 l.support \u2192 (fun a b => r ^ n \u2022 b \u2022 \u2191a) c (\u2191l c) \u2208 \u2191Subalgebra.toSubmodule B'\n[PROOFSTEP]\nintro a ha\n[GOAL]\ncase h\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nhr : \u2191(algebraMap A B) r \u2208 B'\nl : \u2191\u2191(Submonoid.closure s) \u2192\u2080 R\nn\u2081 : { x // x \u2208 Submonoid.closure s } \u2192 \u2115\nn\u2082 : \u2200 (x : { x // x \u2208 Submonoid.closure s }), r ^ n\u2081 x \u2022 \u2191x \u2208 Submonoid.closure (r \u2022 s)\nn : \u2115\nhn : n \u2265 Finset.sup l.support n\u2081\na : \u2191\u2191(Submonoid.closure s)\nha : a \u2208 l.support\n\u22a2 (fun a b => r ^ n \u2022 b \u2022 \u2191a) a (\u2191l a) \u2208 \u2191Subalgebra.toSubmodule B'\n[PROOFSTEP]\nhave : n \u2265 n\u2081 a := le_trans (Finset.le_sup ha) hn\n[GOAL]\ncase h\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nhr : \u2191(algebraMap A B) r \u2208 B'\nl : \u2191\u2191(Submonoid.closure s) \u2192\u2080 R\nn\u2081 : { x // x \u2208 Submonoid.closure s } \u2192 \u2115\nn\u2082 : \u2200 (x : { x // x \u2208 Submonoid.closure s }), r ^ n\u2081 x \u2022 \u2191x \u2208 Submonoid.closure (r \u2022 s)\nn : \u2115\nhn : n \u2265 Finset.sup l.support n\u2081\na : \u2191\u2191(Submonoid.closure s)\nha : a \u2208 l.support\nthis : n \u2265 n\u2081 a\n\u22a2 (fun a b => r ^ n \u2022 b \u2022 \u2191a) a (\u2191l a) \u2208 \u2191Subalgebra.toSubmodule B'\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nhr : \u2191(algebraMap A B) r \u2208 B'\nl : \u2191\u2191(Submonoid.closure s) \u2192\u2080 R\nn\u2081 : { x // x \u2208 Submonoid.closure s } \u2192 \u2115\nn\u2082 : \u2200 (x : { x // x \u2208 Submonoid.closure s }), r ^ n\u2081 x \u2022 \u2191x \u2208 Submonoid.closure (r \u2022 s)\nn : \u2115\nhn : n \u2265 Finset.sup l.support n\u2081\na : \u2191\u2191(Submonoid.closure s)\nha : a \u2208 l.support\nthis : n \u2265 n\u2081 a\n\u22a2 r ^ n \u2022 \u2191l a \u2022 \u2191a \u2208 \u2191Subalgebra.toSubmodule B'\n[PROOFSTEP]\nrw [\u2190 tsub_add_cancel_of_le this, pow_add, \u2190 smul_smul, \u2190 IsScalarTower.algebraMap_smul A (l a) (a : B),\n smul_smul (r ^ n\u2081 a), mul_comm, \u2190 smul_smul, smul_def, map_pow, IsScalarTower.algebraMap_smul]\n[GOAL]\ncase h\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nhr : \u2191(algebraMap A B) r \u2208 B'\nl : \u2191\u2191(Submonoid.closure s) \u2192\u2080 R\nn\u2081 : { x // x \u2208 Submonoid.closure s } \u2192 \u2115\nn\u2082 : \u2200 (x : { x // x \u2208 Submonoid.closure s }), r ^ n\u2081 x \u2022 \u2191x \u2208 Submonoid.closure (r \u2022 s)\nn : \u2115\nhn : n \u2265 Finset.sup l.support n\u2081\na : \u2191\u2191(Submonoid.closure s)\nha : a \u2208 l.support\nthis : n \u2265 n\u2081 a\n\u22a2 \u2191(algebraMap A B) r ^ (n - n\u2081 a) * \u2191l a \u2022 r ^ n\u2081 a \u2022 \u2191a \u2208 \u2191Subalgebra.toSubmodule B'\n[PROOFSTEP]\napply Subalgebra.mul_mem _ (Subalgebra.pow_mem _ hr _) _\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nhr : \u2191(algebraMap A B) r \u2208 B'\nl : \u2191\u2191(Submonoid.closure s) \u2192\u2080 R\nn\u2081 : { x // x \u2208 Submonoid.closure s } \u2192 \u2115\nn\u2082 : \u2200 (x : { x // x \u2208 Submonoid.closure s }), r ^ n\u2081 x \u2022 \u2191x \u2208 Submonoid.closure (r \u2022 s)\nn : \u2115\nhn : n \u2265 Finset.sup l.support n\u2081\na : \u2191\u2191(Submonoid.closure s)\nha : a \u2208 l.support\nthis : n \u2265 n\u2081 a\n\u22a2 \u2191l a \u2022 r ^ n\u2081 a \u2022 \u2191a \u2208 B'\n[PROOFSTEP]\nrefine' Subalgebra.smul_mem _ _ _\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nhr : \u2191(algebraMap A B) r \u2208 B'\nl : \u2191\u2191(Submonoid.closure s) \u2192\u2080 R\nn\u2081 : { x // x \u2208 Submonoid.closure s } \u2192 \u2115\nn\u2082 : \u2200 (x : { x // x \u2208 Submonoid.closure s }), r ^ n\u2081 x \u2022 \u2191x \u2208 Submonoid.closure (r \u2022 s)\nn : \u2115\nhn : n \u2265 Finset.sup l.support n\u2081\na : \u2191\u2191(Submonoid.closure s)\nha : a \u2208 l.support\nthis : n \u2265 n\u2081 a\n\u22a2 r ^ n\u2081 a \u2022 \u2191a \u2208 B'\n[PROOFSTEP]\nchange _ \u2208 B'.toSubmonoid\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nhr : \u2191(algebraMap A B) r \u2208 B'\nl : \u2191\u2191(Submonoid.closure s) \u2192\u2080 R\nn\u2081 : { x // x \u2208 Submonoid.closure s } \u2192 \u2115\nn\u2082 : \u2200 (x : { x // x \u2208 Submonoid.closure s }), r ^ n\u2081 x \u2022 \u2191x \u2208 Submonoid.closure (r \u2022 s)\nn : \u2115\nhn : n \u2265 Finset.sup l.support n\u2081\na : \u2191\u2191(Submonoid.closure s)\nha : a \u2208 l.support\nthis : n \u2265 n\u2081 a\n\u22a2 r ^ n\u2081 a \u2022 \u2191a \u2208 B'.toSubmonoid\n[PROOFSTEP]\nrw [\u2190 Submonoid.closure_eq B'.toSubmonoid]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Algebra R A\ns\u271d t : Set A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nr : A\ns : Set B\nB' : Subalgebra R B\nhs : r \u2022 s \u2286 \u2191B'\nhr : \u2191(algebraMap A B) r \u2208 B'\nl : \u2191\u2191(Submonoid.closure s) \u2192\u2080 R\nn\u2081 : { x // x \u2208 Submonoid.closure s } \u2192 \u2115\nn\u2082 : \u2200 (x : { x // x \u2208 Submonoid.closure s }), r ^ n\u2081 x \u2022 \u2191x \u2208 Submonoid.closure (r \u2022 s)\nn : \u2115\nhn : n \u2265 Finset.sup l.support n\u2081\na : \u2191\u2191(Submonoid.closure s)\nha : a \u2208 l.support\nthis : n \u2265 n\u2081 a\n\u22a2 r ^ n\u2081 a \u2022 \u2191a \u2208 Submonoid.closure \u2191B'.toSubmonoid\n[PROOFSTEP]\napply Submonoid.closure_mono hs (n\u2082 a)\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Adjoin.Basic", "llama_tokens": 25597, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833841649233, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.4204928790186402}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsWeakUpperModularLattice \u03b1\na b : \u03b1\n\u22a2 a \u2293 b \u22d6 a \u2192 a \u2293 b \u22d6 b \u2192 b \u22d6 a \u2294 b\n[PROOFSTEP]\nrw [inf_comm, sup_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsWeakUpperModularLattice \u03b1\na b : \u03b1\n\u22a2 b \u2293 a \u22d6 a \u2192 b \u2293 a \u22d6 b \u2192 b \u22d6 b \u2294 a\n[PROOFSTEP]\nexact fun ha hb => covby_sup_of_inf_covby_of_inf_covby_left hb ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsWeakLowerModularLattice \u03b1\na b : \u03b1\n\u22a2 a \u22d6 a \u2294 b \u2192 b \u22d6 a \u2294 b \u2192 a \u2293 b \u22d6 b\n[PROOFSTEP]\nrw [sup_comm, inf_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsWeakLowerModularLattice \u03b1\na b : \u03b1\n\u22a2 a \u22d6 b \u2294 a \u2192 b \u22d6 b \u2294 a \u2192 b \u2293 a \u22d6 b\n[PROOFSTEP]\nexact fun ha hb => inf_covby_of_covby_sup_of_covby_sup_left hb ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsUpperModularLattice \u03b1\na b : \u03b1\n\u22a2 a \u2293 b \u22d6 b \u2192 a \u22d6 a \u2294 b\n[PROOFSTEP]\nrw [sup_comm, inf_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsUpperModularLattice \u03b1\na b : \u03b1\n\u22a2 b \u2293 a \u22d6 b \u2192 a \u22d6 b \u2294 a\n[PROOFSTEP]\nexact covby_sup_of_inf_covby_left\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsLowerModularLattice \u03b1\na b : \u03b1\n\u22a2 b \u22d6 a \u2294 b \u2192 a \u2293 b \u22d6 a\n[PROOFSTEP]\nrw [inf_comm, sup_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsLowerModularLattice \u03b1\na b : \u03b1\n\u22a2 b \u22d6 b \u2294 a \u2192 b \u2293 a \u22d6 a\n[PROOFSTEP]\nexact inf_covby_of_covby_sup_left\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx y z : \u03b1\nh : z \u2264 x\n\u22a2 x \u2293 y \u2294 z = x \u2293 (y \u2294 z)\n[PROOFSTEP]\nrw [inf_comm, sup_comm, \u2190 sup_inf_assoc_of_le y h, inf_comm, sup_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx\u271d y z : \u03b1\u1d52\u1d48\nxz : x\u271d \u2264 z\n\u22a2 (x\u271d \u2294 y) \u2293 z = x\u271d \u2294 y \u2293 z\n[PROOFSTEP]\nrw [inf_comm, sup_comm, eq_comm, inf_comm, sup_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx\u271d y z : \u03b1\u1d52\u1d48\nxz : x\u271d \u2264 z\n\u22a2 z \u2293 y \u2294 x\u271d = z \u2293 (y \u2294 x\u271d)\n[PROOFSTEP]\nexact @sup_inf_assoc_of_le \u03b1 _ _ _ y _ xz\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx y z : \u03b1\nhxy : x \u2264 y\nhinf : y \u2293 z \u2264 x \u2293 z\nhsup : y \u2294 z \u2264 x \u2294 z\nh : y \u2264 x \u2294 z\n\u22a2 z \u2293 y \u2264 z \u2293 x\n[PROOFSTEP]\nrw [inf_comm, @inf_comm _ _ z]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx y z : \u03b1\nhxy : x \u2264 y\nhinf : y \u2293 z \u2264 x \u2293 z\nhsup : y \u2294 z \u2264 x \u2294 z\nh : y \u2264 x \u2294 z\n\u22a2 y \u2293 z \u2264 x \u2293 z\n[PROOFSTEP]\nexact hinf\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\nx y z : \u03b1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : Preorder \u03b3\nh\u2081 : WellFounded fun x x_1 => x < x_1\nh\u2082 : WellFounded fun x x_1 => x < x_1\nK : \u03b1\nf\u2081 : \u03b2 \u2192 \u03b1\nf\u2082 : \u03b1 \u2192 \u03b2\ng\u2081 : \u03b3 \u2192 \u03b1\ng\u2082 : \u03b1 \u2192 \u03b3\ngci : GaloisCoinsertion f\u2081 f\u2082\ngi : GaloisInsertion g\u2082 g\u2081\nhf : \u2200 (a : \u03b1), f\u2081 (f\u2082 a) = a \u2293 K\nhg : \u2200 (a : \u03b1), g\u2081 (g\u2082 a) = a \u2294 K\nA B : \u03b1\nhAB : A < B\n\u22a2 Prod.Lex (fun x x_1 => x < x_1) (fun x x_1 => x < x_1) (f\u2082 A, g\u2082 A) (f\u2082 B, g\u2082 B)\n[PROOFSTEP]\nsimp only [Prod.lex_def, lt_iff_le_not_le, \u2190 gci.l_le_l_iff, \u2190 gi.u_le_u_iff, hf, hg, le_antisymm_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\nx y z : \u03b1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : Preorder \u03b3\nh\u2081 : WellFounded fun x x_1 => x < x_1\nh\u2082 : WellFounded fun x x_1 => x < x_1\nK : \u03b1\nf\u2081 : \u03b2 \u2192 \u03b1\nf\u2082 : \u03b1 \u2192 \u03b2\ng\u2081 : \u03b3 \u2192 \u03b1\ng\u2082 : \u03b1 \u2192 \u03b3\ngci : GaloisCoinsertion f\u2081 f\u2082\ngi : GaloisInsertion g\u2082 g\u2081\nhf : \u2200 (a : \u03b1), f\u2081 (f\u2082 a) = a \u2293 K\nhg : \u2200 (a : \u03b1), g\u2081 (g\u2082 a) = a \u2294 K\nA B : \u03b1\nhAB : A < B\n\u22a2 A \u2293 K \u2264 B \u2293 K \u2227 \u00acB \u2293 K \u2264 A \u2293 K \u2228 (A \u2293 K \u2264 B \u2293 K \u2227 B \u2293 K \u2264 A \u2293 K) \u2227 A \u2294 K \u2264 B \u2294 K \u2227 \u00acB \u2294 K \u2264 A \u2294 K\n[PROOFSTEP]\nsimp only [gci.l_le_l_iff, gi.u_le_u_iff, \u2190 lt_iff_le_not_le, \u2190 le_antisymm_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\nx y z : \u03b1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : Preorder \u03b3\nh\u2081 : WellFounded fun x x_1 => x < x_1\nh\u2082 : WellFounded fun x x_1 => x < x_1\nK : \u03b1\nf\u2081 : \u03b2 \u2192 \u03b1\nf\u2082 : \u03b1 \u2192 \u03b2\ng\u2081 : \u03b3 \u2192 \u03b1\ng\u2082 : \u03b1 \u2192 \u03b3\ngci : GaloisCoinsertion f\u2081 f\u2082\ngi : GaloisInsertion g\u2082 g\u2081\nhf : \u2200 (a : \u03b1), f\u2081 (f\u2082 a) = a \u2293 K\nhg : \u2200 (a : \u03b1), g\u2081 (g\u2082 a) = a \u2294 K\nA B : \u03b1\nhAB : A < B\n\u22a2 A \u2293 K < B \u2293 K \u2228 A \u2293 K = B \u2293 K \u2227 A \u2294 K < B \u2294 K\n[PROOFSTEP]\ncases' lt_or_eq_of_le (inf_le_inf_right K (le_of_lt hAB)) with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\nx y z : \u03b1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : Preorder \u03b3\nh\u2081 : WellFounded fun x x_1 => x < x_1\nh\u2082 : WellFounded fun x x_1 => x < x_1\nK : \u03b1\nf\u2081 : \u03b2 \u2192 \u03b1\nf\u2082 : \u03b1 \u2192 \u03b2\ng\u2081 : \u03b3 \u2192 \u03b1\ng\u2082 : \u03b1 \u2192 \u03b3\ngci : GaloisCoinsertion f\u2081 f\u2082\ngi : GaloisInsertion g\u2082 g\u2081\nhf : \u2200 (a : \u03b1), f\u2081 (f\u2082 a) = a \u2293 K\nhg : \u2200 (a : \u03b1), g\u2081 (g\u2082 a) = a \u2294 K\nA B : \u03b1\nhAB : A < B\nh : A \u2293 K < B \u2293 K\n\u22a2 A \u2293 K < B \u2293 K \u2228 A \u2293 K = B \u2293 K \u2227 A \u2294 K < B \u2294 K\n[PROOFSTEP]\nexact Or.inl h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\nx y z : \u03b1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : Preorder \u03b3\nh\u2081 : WellFounded fun x x_1 => x < x_1\nh\u2082 : WellFounded fun x x_1 => x < x_1\nK : \u03b1\nf\u2081 : \u03b2 \u2192 \u03b1\nf\u2082 : \u03b1 \u2192 \u03b2\ng\u2081 : \u03b3 \u2192 \u03b1\ng\u2082 : \u03b1 \u2192 \u03b3\ngci : GaloisCoinsertion f\u2081 f\u2082\ngi : GaloisInsertion g\u2082 g\u2081\nhf : \u2200 (a : \u03b1), f\u2081 (f\u2082 a) = a \u2293 K\nhg : \u2200 (a : \u03b1), g\u2081 (g\u2082 a) = a \u2294 K\nA B : \u03b1\nhAB : A < B\nh : A \u2293 K = B \u2293 K\n\u22a2 A \u2293 K < B \u2293 K \u2228 A \u2293 K = B \u2293 K \u2227 A \u2294 K < B \u2294 K\n[PROOFSTEP]\nexact Or.inr \u27e8h, sup_lt_sup_of_lt_of_inf_le_inf hAB (le_of_eq h.symm)\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx\u271d y z a b : \u03b1\nx : \u2191(Icc (a \u2293 b) a)\n\u22a2 \u2191((fun x => { val := a \u2293 \u2191x, property := (_ : a \u2293 b \u2264 a \u2293 \u2191x \u2227 a \u2293 \u2191x \u2264 a) })\n ((fun x => { val := \u2191x \u2294 b, property := (_ : b \u2264 \u2191x \u2294 b \u2227 \u2191x \u2294 b \u2264 a \u2294 b) }) x)) =\n \u2191x\n[PROOFSTEP]\nchange a \u2293 (\u2191x \u2294 b) = \u2191x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx\u271d y z a b : \u03b1\nx : \u2191(Icc (a \u2293 b) a)\n\u22a2 a \u2293 (\u2191x \u2294 b) = \u2191x\n[PROOFSTEP]\nrw [sup_comm, \u2190 inf_sup_assoc_of_le _ x.prop.2, sup_eq_right.2 x.prop.1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx\u271d y z a b : \u03b1\nx : \u2191(Icc b (a \u2294 b))\n\u22a2 \u2191((fun x => { val := \u2191x \u2294 b, property := (_ : b \u2264 \u2191x \u2294 b \u2227 \u2191x \u2294 b \u2264 a \u2294 b) })\n ((fun x => { val := a \u2293 \u2191x, property := (_ : a \u2293 b \u2264 a \u2293 \u2191x \u2227 a \u2293 \u2191x \u2264 a) }) x)) =\n \u2191x\n[PROOFSTEP]\nchange a \u2293 \u2191x \u2294 b = \u2191x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx\u271d y z a b : \u03b1\nx : \u2191(Icc b (a \u2294 b))\n\u22a2 a \u2293 \u2191x \u2294 b = \u2191x\n[PROOFSTEP]\nrw [inf_comm, inf_sup_assoc_of_le _ x.prop.1, inf_eq_left.2 x.prop.2]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx\u271d y\u271d z a b : \u03b1\nx y : \u2191(Icc (a \u2293 b) a)\n\u22a2 \u2191{ toFun := fun x => { val := \u2191x \u2294 b, property := (_ : b \u2264 \u2191x \u2294 b \u2227 \u2191x \u2294 b \u2264 a \u2294 b) },\n invFun := fun x => { val := a \u2293 \u2191x, property := (_ : a \u2293 b \u2264 a \u2293 \u2191x \u2227 a \u2293 \u2191x \u2264 a) },\n left_inv :=\n (_ :\n \u2200 (x : \u2191(Icc (a \u2293 b) a)),\n (fun x => { val := a \u2293 \u2191x, property := (_ : a \u2293 b \u2264 a \u2293 \u2191x \u2227 a \u2293 \u2191x \u2264 a) })\n ((fun x => { val := \u2191x \u2294 b, property := (_ : b \u2264 \u2191x \u2294 b \u2227 \u2191x \u2294 b \u2264 a \u2294 b) }) x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : \u2191(Icc b (a \u2294 b))),\n (fun x => { val := \u2191x \u2294 b, property := (_ : b \u2264 \u2191x \u2294 b \u2227 \u2191x \u2294 b \u2264 a \u2294 b) })\n ((fun x => { val := a \u2293 \u2191x, property := (_ : a \u2293 b \u2264 a \u2293 \u2191x \u2227 a \u2293 \u2191x \u2264 a) }) x) =\n x) }\n x \u2264\n \u2191{ toFun := fun x => { val := \u2191x \u2294 b, property := (_ : b \u2264 \u2191x \u2294 b \u2227 \u2191x \u2294 b \u2264 a \u2294 b) },\n invFun := fun x => { val := a \u2293 \u2191x, property := (_ : a \u2293 b \u2264 a \u2293 \u2191x \u2227 a \u2293 \u2191x \u2264 a) },\n left_inv :=\n (_ :\n \u2200 (x : \u2191(Icc (a \u2293 b) a)),\n (fun x => { val := a \u2293 \u2191x, property := (_ : a \u2293 b \u2264 a \u2293 \u2191x \u2227 a \u2293 \u2191x \u2264 a) })\n ((fun x => { val := \u2191x \u2294 b, property := (_ : b \u2264 \u2191x \u2294 b \u2227 \u2191x \u2294 b \u2264 a \u2294 b) }) x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : \u2191(Icc b (a \u2294 b))),\n (fun x => { val := \u2191x \u2294 b, property := (_ : b \u2264 \u2191x \u2294 b \u2227 \u2191x \u2294 b \u2264 a \u2294 b) })\n ((fun x => { val := a \u2293 \u2191x, property := (_ : a \u2293 b \u2264 a \u2293 \u2191x \u2227 a \u2293 \u2191x \u2264 a) }) x) =\n x) }\n y \u2194\n x \u2264 y\n[PROOFSTEP]\nsimp only [Subtype.mk_le_mk, Equiv.coe_fn_mk, and_true_iff, le_sup_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx\u271d y\u271d z a b : \u03b1\nx y : \u2191(Icc (a \u2293 b) a)\n\u22a2 \u2191x \u2294 b \u2264 \u2191y \u2294 b \u2194 x \u2264 y\n[PROOFSTEP]\nrw [\u2190 Subtype.coe_le_coe]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx\u271d y\u271d z a b : \u03b1\nx y : \u2191(Icc (a \u2293 b) a)\n\u22a2 \u2191x \u2294 b \u2264 \u2191y \u2294 b \u2194 \u2191x \u2264 \u2191y\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => sup_le_sup_right h _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx\u271d y\u271d z a b : \u03b1\nx y : \u2191(Icc (a \u2293 b) a)\nh : \u2191x \u2294 b \u2264 \u2191y \u2294 b\n\u22a2 \u2191x \u2264 \u2191y\n[PROOFSTEP]\nrw [\u2190 sup_eq_right.2 x.prop.1, inf_sup_assoc_of_le _ x.prop.2, sup_comm, \u2190 sup_eq_right.2 y.prop.1,\n inf_sup_assoc_of_le _ y.prop.2, @sup_comm _ _ b]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx\u271d y\u271d z a b : \u03b1\nx y : \u2191(Icc (a \u2293 b) a)\nh : \u2191x \u2294 b \u2264 \u2191y \u2294 b\n\u22a2 a \u2293 (\u2191x \u2294 b) \u2264 a \u2293 (\u2191y \u2294 b)\n[PROOFSTEP]\nexact inf_le_inf_left _ h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx y z a b : \u03b1\nc : \u2191(Ioo (a \u2293 b) a)\n\u22a2 \u2191((fun c => { val := a \u2293 \u2191c, property := (_ : a \u2293 b < a \u2293 \u2191c \u2227 a \u2293 \u2191c < a) })\n ((fun c => { val := \u2191c \u2294 b, property := (_ : b < \u2191c \u2294 b \u2227 \u2191c \u2294 b < a \u2294 b) }) c)) =\n \u2191c\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx y z a b : \u03b1\nc : \u2191(Ioo (a \u2293 b) a)\n\u22a2 a \u2293 (\u2191c \u2294 b) = \u2191c\n[PROOFSTEP]\nrw [sup_comm, \u2190 inf_sup_assoc_of_le _ c.prop.2.le, sup_eq_right.2 c.prop.1.le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx y z a b : \u03b1\nc : \u2191(Ioo b (a \u2294 b))\n\u22a2 \u2191((fun c => { val := \u2191c \u2294 b, property := (_ : b < \u2191c \u2294 b \u2227 \u2191c \u2294 b < a \u2294 b) })\n ((fun c => { val := a \u2293 \u2191c, property := (_ : a \u2293 b < a \u2293 \u2191c \u2227 a \u2293 \u2191c < a) }) c)) =\n \u2191c\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx y z a b : \u03b1\nc : \u2191(Ioo b (a \u2294 b))\n\u22a2 a \u2293 \u2191c \u2294 b = \u2191c\n[PROOFSTEP]\nrw [inf_comm, inf_sup_assoc_of_le _ c.prop.1.le, inf_eq_left.2 c.prop.2.le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx y z a b : \u03b1\n\u22a2 a \u22d6 a \u2294 b \u2192 a \u2293 b \u22d6 b\n[PROOFSTEP]\nsimp_rw [covby_iff_Ioo_eq, @sup_comm _ _ a, @inf_comm _ _ a, \u2190 isEmpty_coe_sort, right_lt_sup, inf_lt_left,\n (infIooOrderIsoIooSup b a).symm.toEquiv.isEmpty_congr]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx y z a b : \u03b1\n\u22a2 \u00acb \u2264 a \u2227 IsEmpty \u2191(Ioo (b \u2293 a) b) \u2192 \u00acb \u2264 a \u2227 IsEmpty \u2191(Ioo (b \u2293 a) b)\n[PROOFSTEP]\nexact id\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx y z a b : \u03b1\n\u22a2 a \u2293 b \u22d6 a \u2192 b \u22d6 a \u2294 b\n[PROOFSTEP]\nsimp_rw [covby_iff_Ioo_eq, \u2190 isEmpty_coe_sort, right_lt_sup, inf_lt_left,\n (infIooOrderIsoIooSup a b).toEquiv.isEmpty_congr]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : IsModularLattice \u03b1\nx y z a b : \u03b1\n\u22a2 \u00aca \u2264 b \u2227 IsEmpty \u2191(Ioo b (a \u2294 b)) \u2192 \u00aca \u2264 b \u2227 IsEmpty \u2191(Ioo b (a \u2294 b))\n[PROOFSTEP]\nexact id\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Lattice \u03b1\nh : \u2200 (x y z : \u03b1), x \u2293 z \u2294 y \u2293 z = (x \u2293 z \u2294 y) \u2293 z\nx\u271d y z : \u03b1\nxz : x\u271d \u2264 z\n\u22a2 (x\u271d \u2294 y) \u2293 z \u2264 x\u271d \u2294 y \u2293 z\n[PROOFSTEP]\nrw [\u2190 inf_eq_left.2 xz, h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DistribLattice \u03b1\nx\u271d y z : \u03b1\nxz : x\u271d \u2264 z\n\u22a2 (x\u271d \u2294 y) \u2293 z \u2264 x\u271d \u2294 y \u2293 z\n[PROOFSTEP]\nrw [inf_sup_right, inf_eq_left.2 xz]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : IsModularLattice \u03b1\na b c : \u03b1\nh : Disjoint a b\nhsup : Disjoint (a \u2294 b) c\n\u22a2 Disjoint a (b \u2294 c)\n[PROOFSTEP]\nrw [disjoint_iff_inf_le, \u2190 h.eq_bot, sup_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : IsModularLattice \u03b1\na b c : \u03b1\nh : Disjoint a b\nhsup : Disjoint (a \u2294 b) c\n\u22a2 a \u2293 (c \u2294 b) \u2264 a \u2293 b\n[PROOFSTEP]\napply le_inf inf_le_left\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : IsModularLattice \u03b1\na b c : \u03b1\nh : Disjoint a b\nhsup : Disjoint (a \u2294 b) c\n\u22a2 a \u2293 (c \u2294 b) \u2264 b\n[PROOFSTEP]\napply (inf_le_inf_right (c \u2294 b) le_sup_right).trans\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : IsModularLattice \u03b1\na b c : \u03b1\nh : Disjoint a b\nhsup : Disjoint (a \u2294 b) c\n\u22a2 (?m.28074 \u2294 a) \u2293 (c \u2294 b) \u2264 b\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : IsModularLattice \u03b1\na b c : \u03b1\nh : Disjoint a b\nhsup : Disjoint (a \u2294 b) c\n\u22a2 \u03b1\n[PROOFSTEP]\nrw [sup_comm, IsModularLattice.sup_inf_sup_assoc, hsup.eq_bot, bot_sup_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : IsModularLattice \u03b1\na b c : \u03b1\nh : Disjoint b c\nhsup : Disjoint a (b \u2294 c)\n\u22a2 Disjoint (a \u2294 b) c\n[PROOFSTEP]\nrw [disjoint_comm, sup_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : IsModularLattice \u03b1\na b c : \u03b1\nh : Disjoint b c\nhsup : Disjoint a (b \u2294 c)\n\u22a2 Disjoint c (b \u2294 a)\n[PROOFSTEP]\napply Disjoint.disjoint_sup_right_of_disjoint_sup_left h.symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : IsModularLattice \u03b1\na b c : \u03b1\nh : Disjoint b c\nhsup : Disjoint a (b \u2294 c)\n\u22a2 Disjoint (c \u2294 b) a\n[PROOFSTEP]\nrwa [sup_comm, disjoint_comm] at hsup \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Iic a)\nx : \u03b1\nhx : x \u2208 Iic a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 IsCompl { val := x, property := hx } { val := y \u2293 a, property := (_ : y \u2293 a \u2208 Iic a) }\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Iic a)\nx : \u03b1\nhx : x \u2208 Iic a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 Disjoint { val := x, property := hx } { val := y \u2293 a, property := (_ : y \u2293 a \u2208 Iic a) }\n[PROOFSTEP]\nrw [disjoint_iff_inf_le]\n[GOAL]\ncase disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Iic a)\nx : \u03b1\nhx : x \u2208 Iic a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 { val := x, property := hx } \u2293 { val := y \u2293 a, property := (_ : y \u2293 a \u2208 Iic a) } \u2264 \u22a5\n[PROOFSTEP]\nchange\n x \u2293 (y \u2293 a) \u2264\n \u22a5\n -- improve lattice subtype API\n[GOAL]\ncase disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Iic a)\nx : \u03b1\nhx : x \u2208 Iic a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 x \u2293 (y \u2293 a) \u2264 \u22a5\n[PROOFSTEP]\nrw [\u2190 inf_assoc]\n[GOAL]\ncase disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Iic a)\nx : \u03b1\nhx : x \u2208 Iic a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 x \u2293 y \u2293 a \u2264 \u22a5\n[PROOFSTEP]\nexact le_trans inf_le_left hy.1.le_bot\n[GOAL]\ncase codisjoint\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Iic a)\nx : \u03b1\nhx : x \u2208 Iic a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 Codisjoint { val := x, property := hx } { val := y \u2293 a, property := (_ : y \u2293 a \u2208 Iic a) }\n[PROOFSTEP]\nrw [codisjoint_iff_le_sup]\n[GOAL]\ncase codisjoint\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Iic a)\nx : \u03b1\nhx : x \u2208 Iic a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 \u22a4 \u2264 { val := x, property := hx } \u2294 { val := y \u2293 a, property := (_ : y \u2293 a \u2208 Iic a) }\n[PROOFSTEP]\nchange a \u2264 x \u2294 y \u2293 a\n[GOAL]\ncase codisjoint\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Iic a)\nx : \u03b1\nhx : x \u2208 Iic a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 a \u2264 x \u2294 y \u2293 a\n[PROOFSTEP]\nrw [\u2190 sup_inf_assoc_of_le _ (Set.mem_Iic.1 hx), hy.2.eq_top, top_inf_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Ici a)\nx : \u03b1\nhx : x \u2208 Ici a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 IsCompl { val := x, property := hx } { val := y \u2294 a, property := (_ : y \u2294 a \u2208 Ici a) }\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Ici a)\nx : \u03b1\nhx : x \u2208 Ici a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 Disjoint { val := x, property := hx } { val := y \u2294 a, property := (_ : y \u2294 a \u2208 Ici a) }\n[PROOFSTEP]\nrw [disjoint_iff_inf_le]\n[GOAL]\ncase disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Ici a)\nx : \u03b1\nhx : x \u2208 Ici a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 { val := x, property := hx } \u2293 { val := y \u2294 a, property := (_ : y \u2294 a \u2208 Ici a) } \u2264 \u22a5\n[PROOFSTEP]\nchange x \u2293 (y \u2294 a) \u2264 a\n[GOAL]\ncase disjoint\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Ici a)\nx : \u03b1\nhx : x \u2208 Ici a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 x \u2293 (y \u2294 a) \u2264 a\n[PROOFSTEP]\nrw [\u2190 inf_sup_assoc_of_le _ (Set.mem_Ici.1 hx), hy.1.eq_bot, bot_sup_eq]\n[GOAL]\ncase codisjoint\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Ici a)\nx : \u03b1\nhx : x \u2208 Ici a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 Codisjoint { val := x, property := hx } { val := y \u2294 a, property := (_ : y \u2294 a \u2208 Ici a) }\n[PROOFSTEP]\nrw [codisjoint_iff_le_sup]\n[GOAL]\ncase codisjoint\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Ici a)\nx : \u03b1\nhx : x \u2208 Ici a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 \u22a4 \u2264 { val := x, property := hx } \u2294 { val := y \u2294 a, property := (_ : y \u2294 a \u2208 Ici a) }\n[PROOFSTEP]\nchange\n \u22a4 \u2264\n x \u2294\n (y \u2294 a)\n -- improve lattice subtype API\n[GOAL]\ncase codisjoint\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Ici a)\nx : \u03b1\nhx : x \u2208 Ici a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 \u22a4 \u2264 x \u2294 (y \u2294 a)\n[PROOFSTEP]\nrw [\u2190 sup_assoc]\n[GOAL]\ncase codisjoint\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\na : \u03b1\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : ComplementedLattice \u03b1\nx\u271d : \u2191(Ici a)\nx : \u03b1\nhx : x \u2208 Ici a\ny : \u03b1\nhy : IsCompl x y\n\u22a2 \u22a4 \u2264 x \u2294 y \u2294 a\n[PROOFSTEP]\nexact le_trans hy.2.top_le le_sup_left\n", "meta": {"mathlib_filename": "Mathlib.Order.ModularLattice", "llama_tokens": 10167, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833737577158, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.4204928733689493}} {"text": "[GOAL]\n\u03b1 : Type u_1\nM : Type ?u.8804\nN : Type ?u.8800\ninst\u271d\u00b2 : Mul N\ninst\u271d\u00b9 : SMul M N\ninst\u271d : IsScalarTower M N N\nx : M\ny : N\u1d50\u1d52\u1d56\nz : N\n\u22a2 x \u2022 y \u2022 z = y \u2022 x \u2022 z\n[PROOFSTEP]\ninduction y using MulOpposite.rec'\n[GOAL]\ncase h\n\u03b1 : Type u_1\nM : Type ?u.8804\nN : Type ?u.8800\ninst\u271d\u00b2 : Mul N\ninst\u271d\u00b9 : SMul M N\ninst\u271d : IsScalarTower M N N\nx : M\nz X\u271d : N\n\u22a2 x \u2022 op X\u271d \u2022 z = op X\u271d \u2022 x \u2022 z\n[PROOFSTEP]\nsimp only [smul_mul_assoc, MulOpposite.smul_eq_mul_unop]\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.GroupAction.Opposite", "llama_tokens": 260, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7185943805178139, "lm_q2_score": 0.5851011542032313, "lm_q1q2_score": 0.42045040144492885}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3\u271d : Type u_1\n\u03b4 : Type u_2\n\u03b3 : \u03b1 \u2295 \u03b2 \u2192 Sort u_3\np : ((ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab) \u2192 Prop\nh : \u2200 (fa : (val : \u03b1) \u2192 \u03b3 (inl val)) (fb : (val : \u03b2) \u2192 \u03b3 (inr val)), p fun t => rec fa fb t\nfab : (ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab\n\u22a2 p fab\n[PROOFSTEP]\nhave h1 : fab = Sum.rec (fun a => fab (Sum.inl a)) (fun b => fab (Sum.inr b)) := by ext ab; cases ab <;> rfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3\u271d : Type u_1\n\u03b4 : Type u_2\n\u03b3 : \u03b1 \u2295 \u03b2 \u2192 Sort u_3\np : ((ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab) \u2192 Prop\nh : \u2200 (fa : (val : \u03b1) \u2192 \u03b3 (inl val)) (fb : (val : \u03b2) \u2192 \u03b3 (inr val)), p fun t => rec fa fb t\nfab : (ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab\n\u22a2 fab = fun t => rec (fun a => fab (inl a)) (fun b => fab (inr b)) t\n[PROOFSTEP]\next ab\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3\u271d : Type u_1\n\u03b4 : Type u_2\n\u03b3 : \u03b1 \u2295 \u03b2 \u2192 Sort u_3\np : ((ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab) \u2192 Prop\nh : \u2200 (fa : (val : \u03b1) \u2192 \u03b3 (inl val)) (fb : (val : \u03b2) \u2192 \u03b3 (inr val)), p fun t => rec fa fb t\nfab : (ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab\nab : \u03b1 \u2295 \u03b2\n\u22a2 fab ab = rec (fun a => fab (inl a)) (fun b => fab (inr b)) ab\n[PROOFSTEP]\ncases ab\n[GOAL]\ncase h.inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3\u271d : Type u_1\n\u03b4 : Type u_2\n\u03b3 : \u03b1 \u2295 \u03b2 \u2192 Sort u_3\np : ((ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab) \u2192 Prop\nh : \u2200 (fa : (val : \u03b1) \u2192 \u03b3 (inl val)) (fb : (val : \u03b2) \u2192 \u03b3 (inr val)), p fun t => rec fa fb t\nfab : (ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab\nval\u271d : \u03b1\n\u22a2 fab (inl val\u271d) = rec (fun a => fab (inl a)) (fun b => fab (inr b)) (inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3\u271d : Type u_1\n\u03b4 : Type u_2\n\u03b3 : \u03b1 \u2295 \u03b2 \u2192 Sort u_3\np : ((ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab) \u2192 Prop\nh : \u2200 (fa : (val : \u03b1) \u2192 \u03b3 (inl val)) (fb : (val : \u03b2) \u2192 \u03b3 (inr val)), p fun t => rec fa fb t\nfab : (ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab\nval\u271d : \u03b2\n\u22a2 fab (inr val\u271d) = rec (fun a => fab (inl a)) (fun b => fab (inr b)) (inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3\u271d : Type u_1\n\u03b4 : Type u_2\n\u03b3 : \u03b1 \u2295 \u03b2 \u2192 Sort u_3\np : ((ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab) \u2192 Prop\nh : \u2200 (fa : (val : \u03b1) \u2192 \u03b3 (inl val)) (fb : (val : \u03b2) \u2192 \u03b3 (inr val)), p fun t => rec fa fb t\nfab : (ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab\nh1 : fab = fun t => rec (fun a => fab (inl a)) (fun b => fab (inr b)) t\n\u22a2 p fab\n[PROOFSTEP]\nrw [h1]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3\u271d : Type u_1\n\u03b4 : Type u_2\n\u03b3 : \u03b1 \u2295 \u03b2 \u2192 Sort u_3\np : ((ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab) \u2192 Prop\nh : \u2200 (fa : (val : \u03b1) \u2192 \u03b3 (inl val)) (fb : (val : \u03b2) \u2192 \u03b3 (inr val)), p fun t => rec fa fb t\nfab : (ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab\nh1 : fab = fun t => rec (fun a => fab (inl a)) (fun b => fab (inr b)) t\n\u22a2 p fun t => rec (fun a => fab (inl a)) (fun b => fab (inr b)) t\n[PROOFSTEP]\nexact h _ _\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3\u271d : Type u_1\n\u03b4 : Type u_2\n\u03b3 : \u03b1 \u2295 \u03b2 \u2192 Sort u_3\np : ((ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab) \u2192 Prop\n\u22a2 (\u2203 fab, p fab) \u2194 \u2203 fa fb, p fun t => rec fa fb t\n[PROOFSTEP]\nrw [\u2190 not_forall_not, forall_sum_pi]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3\u271d : Type u_1\n\u03b4 : Type u_2\n\u03b3 : \u03b1 \u2295 \u03b2 \u2192 Sort u_3\np : ((ab : \u03b1 \u2295 \u03b2) \u2192 \u03b3 ab) \u2192 Prop\n\u22a2 (\u00ac\u2200 (fa : (val : \u03b1) \u2192 \u03b3 (inl val)) (fb : (val : \u03b2) \u2192 \u03b3 (inr val)), \u00acp fun t => rec fa fb t) \u2194\n \u2203 fa fb, p fun t => rec fa fb t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx y : \u03b1 \u2295 \u03b2\n\u22a2 getLeft? x = none \u2194 isRight x = true\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\nval\u271d : \u03b1\n\u22a2 getLeft? (inl val\u271d) = none \u2194 isRight (inl val\u271d) = true\n[PROOFSTEP]\nsimp only [getLeft?, isRight, eq_self_iff_true]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\nval\u271d : \u03b2\n\u22a2 getLeft? (inr val\u271d) = none \u2194 isRight (inr val\u271d) = true\n[PROOFSTEP]\nsimp only [getLeft?, isRight, eq_self_iff_true]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx y : \u03b1 \u2295 \u03b2\n\u22a2 getRight? x = none \u2194 isLeft x = true\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\nval\u271d : \u03b1\n\u22a2 getRight? (inl val\u271d) = none \u2194 isLeft (inl val\u271d) = true\n[PROOFSTEP]\nsimp only [getRight?, isLeft, eq_self_iff_true]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\nval\u271d : \u03b2\n\u22a2 getRight? (inr val\u271d) = none \u2194 isLeft (inr val\u271d) = true\n[PROOFSTEP]\nsimp only [getRight?, isLeft, eq_self_iff_true]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx y : \u03b1 \u2295 \u03b2\na : \u03b1\n\u22a2 getLeft? x = some a \u2194 x = inl a\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\na val\u271d : \u03b1\n\u22a2 getLeft? (inl val\u271d) = some a \u2194 inl val\u271d = inl a\n[PROOFSTEP]\nsimp only [getLeft?, Option.some.injEq, inl.injEq]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\na : \u03b1\nval\u271d : \u03b2\n\u22a2 getLeft? (inr val\u271d) = some a \u2194 inr val\u271d = inl a\n[PROOFSTEP]\nsimp only [getLeft?, Option.some.injEq, inl.injEq]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx y : \u03b1 \u2295 \u03b2\nb : \u03b2\n\u22a2 getRight? x = some b \u2194 x = inr b\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\nb : \u03b2\nval\u271d : \u03b1\n\u22a2 getRight? (inl val\u271d) = some b \u2194 inl val\u271d = inr b\n[PROOFSTEP]\nsimp only [getRight?, Option.some.injEq, inr.injEq]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\nb val\u271d : \u03b2\n\u22a2 getRight? (inr val\u271d) = some b \u2194 inr val\u271d = inr b\n[PROOFSTEP]\nsimp only [getRight?, Option.some.injEq, inr.injEq]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx\u271d y x : \u03b1 \u2295 \u03b2\n\u22a2 (!isLeft x) = isRight x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx y : \u03b1 \u2295 \u03b2\nval\u271d : \u03b1\n\u22a2 (!isLeft (inl val\u271d)) = isRight (inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx y : \u03b1 \u2295 \u03b2\nval\u271d : \u03b2\n\u22a2 (!isLeft (inr val\u271d)) = isRight (inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx y : \u03b1 \u2295 \u03b2\n\u22a2 isLeft x = false \u2194 isRight x = true\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\nval\u271d : \u03b1\n\u22a2 isLeft (inl val\u271d) = false \u2194 isRight (inl val\u271d) = true\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\nval\u271d : \u03b2\n\u22a2 isLeft (inr val\u271d) = false \u2194 isRight (inr val\u271d) = true\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx y : \u03b1 \u2295 \u03b2\n\u22a2 \u00acisLeft x = true \u2194 isRight x = true\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx\u271d y x : \u03b1 \u2295 \u03b2\n\u22a2 (!decide (isRight x = isLeft x)) = true\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx y : \u03b1 \u2295 \u03b2\nval\u271d : \u03b1\n\u22a2 (!decide (isRight (inl val\u271d) = isLeft (inl val\u271d))) = true\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx y : \u03b1 \u2295 \u03b2\nval\u271d : \u03b2\n\u22a2 (!decide (isRight (inr val\u271d) = isLeft (inr val\u271d))) = true\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx y : \u03b1 \u2295 \u03b2\n\u22a2 isRight x = false \u2194 isLeft x = true\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\nval\u271d : \u03b1\n\u22a2 isRight (inl val\u271d) = false \u2194 isLeft (inl val\u271d) = true\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\nval\u271d : \u03b2\n\u22a2 isRight (inr val\u271d) = false \u2194 isLeft (inr val\u271d) = true\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx y : \u03b1 \u2295 \u03b2\n\u22a2 \u00acisRight x = true \u2194 isLeft x = true\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx y : \u03b1 \u2295 \u03b2\n\u22a2 isLeft x = true \u2194 \u2203 y, x = inl y\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\nval\u271d : \u03b1\n\u22a2 isLeft (inl val\u271d) = true \u2194 \u2203 y, inl val\u271d = inl y\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\nval\u271d : \u03b2\n\u22a2 isLeft (inr val\u271d) = true \u2194 \u2203 y, inr val\u271d = inl y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx y : \u03b1 \u2295 \u03b2\n\u22a2 isRight x = true \u2194 \u2203 y, x = inr y\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\nval\u271d : \u03b1\n\u22a2 isRight (inl val\u271d) = true \u2194 \u2203 y, inl val\u271d = inr y\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ny : \u03b1 \u2295 \u03b2\nval\u271d : \u03b2\n\u22a2 isRight (inr val\u271d) = true \u2194 \u2203 y, inr val\u271d = inr y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u\n\u03b1' : Type w\n\u03b2\u271d : Type v\n\u03b2' : Type x\n\u03b3\u271d : Type u_1\n\u03b4\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\n\u03b5 : Sort u_7\nf\u2081 : \u03b1 \u2192 \u03b2\nf\u2082 : \u03b2 \u2192 \u03b5\ng\u2081 : \u03b3 \u2192 \u03b4\ng\u2082 : \u03b4 \u2192 \u03b5\nx : \u03b1 \u2295 \u03b3\n\u22a2 Sum.elim f\u2082 g\u2082 (Sum.map f\u2081 g\u2081 x) = Sum.elim (f\u2082 \u2218 f\u2081) (g\u2082 \u2218 g\u2081) x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1\u271d : Type u\n\u03b1' : Type w\n\u03b2\u271d : Type v\n\u03b2' : Type x\n\u03b3\u271d : Type u_1\n\u03b4\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\n\u03b5 : Sort u_7\nf\u2081 : \u03b1 \u2192 \u03b2\nf\u2082 : \u03b2 \u2192 \u03b5\ng\u2081 : \u03b3 \u2192 \u03b4\ng\u2082 : \u03b4 \u2192 \u03b5\nval\u271d : \u03b1\n\u22a2 Sum.elim f\u2082 g\u2082 (Sum.map f\u2081 g\u2081 (inl val\u271d)) = Sum.elim (f\u2082 \u2218 f\u2081) (g\u2082 \u2218 g\u2081) (inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1\u271d : Type u\n\u03b1' : Type w\n\u03b2\u271d : Type v\n\u03b2' : Type x\n\u03b3\u271d : Type u_1\n\u03b4\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\n\u03b5 : Sort u_7\nf\u2081 : \u03b1 \u2192 \u03b2\nf\u2082 : \u03b2 \u2192 \u03b5\ng\u2081 : \u03b3 \u2192 \u03b4\ng\u2082 : \u03b4 \u2192 \u03b5\nval\u271d : \u03b3\n\u22a2 Sum.elim f\u2082 g\u2082 (Sum.map f\u2081 g\u2081 (inr val\u271d)) = Sum.elim (f\u2082 \u2218 f\u2081) (g\u2082 \u2218 g\u2081) (inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nx : \u03b1 \u2295 \u03b3\n\u22a2 isLeft (Sum.map f g x) = isLeft x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nval\u271d : \u03b1\n\u22a2 isLeft (Sum.map f g (inl val\u271d)) = isLeft (inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nval\u271d : \u03b3\n\u22a2 isLeft (Sum.map f g (inr val\u271d)) = isLeft (inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nx : \u03b1 \u2295 \u03b3\n\u22a2 isRight (Sum.map f g x) = isRight x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nval\u271d : \u03b1\n\u22a2 isRight (Sum.map f g (inl val\u271d)) = isRight (inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nval\u271d : \u03b3\n\u22a2 isRight (Sum.map f g (inr val\u271d)) = isRight (inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nx : \u03b1 \u2295 \u03b3\n\u22a2 getLeft? (Sum.map f g x) = Option.map f (getLeft? x)\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nval\u271d : \u03b1\n\u22a2 getLeft? (Sum.map f g (inl val\u271d)) = Option.map f (getLeft? (inl val\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nval\u271d : \u03b3\n\u22a2 getLeft? (Sum.map f g (inr val\u271d)) = Option.map f (getLeft? (inr val\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nx : \u03b1 \u2295 \u03b3\n\u22a2 getRight? (Sum.map f g x) = Option.map g (getRight? x)\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nval\u271d : \u03b1\n\u22a2 getRight? (Sum.map f g (inl val\u271d)) = Option.map g (getRight? (inl val\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nval\u271d : \u03b3\n\u22a2 getRight? (Sum.map f g (inr val\u271d)) = Option.map g (getRight? (inr val\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq (\u03b1 \u2295 \u03b2)\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b1\nx : \u03b3\n\u22a2 x = Sum.elim (update f i x) g (inl i)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq (\u03b1 \u2295 \u03b2)\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b1\nx : \u03b3\n\u22a2 \u2200 (x_1 : \u03b1 \u2295 \u03b2), x_1 \u2260 inl i \u2192 Sum.elim f g x_1 = Sum.elim (update f i x) g x_1\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq (\u03b1 \u2295 \u03b2)\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b2\nx : \u03b3\n\u22a2 x = Sum.elim f (update g i x) (inr i)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq (\u03b1 \u2295 \u03b2)\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b2\nx : \u03b3\n\u22a2 \u2200 (x_1 : \u03b1 \u2295 \u03b2), x_1 \u2260 inr i \u2192 Sum.elim f g x_1 = Sum.elim f (update g i x) x_1\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq (\u03b1 \u2295 \u03b2)\nf : \u03b1 \u2295 \u03b2 \u2192 \u03b3\ni j : \u03b1\nx : \u03b3\n\u22a2 update f (inl i) x (inl j) = update (f \u2218 inl) i x j\n[PROOFSTEP]\nrw [\u2190 update_inl_comp_inl, Function.comp_apply]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq (\u03b1 \u2295 \u03b2)\nf : \u03b1 \u2295 \u03b2 \u2192 \u03b3\ni j : \u03b2\nx : \u03b3\n\u22a2 update f (inr i) x (inr j) = update (f \u2218 inr) i x j\n[PROOFSTEP]\nrw [\u2190 update_inr_comp_inr, Function.comp_apply]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx : \u03b1 \u2295 \u03b2\n\u22a2 swap (swap x) = x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nval\u271d : \u03b1\n\u22a2 swap (swap (inl val\u271d)) = inl val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nval\u271d : \u03b2\n\u22a2 swap (swap (inr val\u271d)) = inr val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx : \u03b1 \u2295 \u03b2\n\u22a2 isLeft (swap x) = isRight x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nval\u271d : \u03b1\n\u22a2 isLeft (swap (inl val\u271d)) = isRight (inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nval\u271d : \u03b2\n\u22a2 isLeft (swap (inr val\u271d)) = isRight (inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx : \u03b1 \u2295 \u03b2\n\u22a2 isRight (swap x) = isLeft x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nval\u271d : \u03b1\n\u22a2 isRight (swap (inl val\u271d)) = isLeft (inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nval\u271d : \u03b2\n\u22a2 isRight (swap (inr val\u271d)) = isLeft (inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx : \u03b1 \u2295 \u03b2\n\u22a2 getLeft? (swap x) = getRight? x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nval\u271d : \u03b1\n\u22a2 getLeft? (swap (inl val\u271d)) = getRight? (inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nval\u271d : \u03b2\n\u22a2 getLeft? (swap (inr val\u271d)) = getRight? (inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nx : \u03b1 \u2295 \u03b2\n\u22a2 getRight? (swap x) = getLeft? x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nval\u271d : \u03b1\n\u22a2 getRight? (swap (inl val\u271d)) = getLeft? (inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nval\u271d : \u03b2\n\u22a2 getRight? (swap (inr val\u271d)) = getLeft? (inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b3 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nb : \u03b2\nc : \u03b3\nd : \u03b4\nx : \u03b1 \u2295 \u03b2\ny : \u03b3 \u2295 \u03b4\nh : LiftRel r s (inl a) (inl c)\n\u22a2 r a c\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b3 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nb : \u03b2\nc : \u03b3\nd : \u03b4\nx : \u03b1 \u2295 \u03b2\ny : \u03b3 \u2295 \u03b4\na\u271d : r a c\n\u22a2 r a c\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b3 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nb : \u03b2\nc : \u03b3\nd : \u03b4\nx : \u03b1 \u2295 \u03b2\ny : \u03b3 \u2295 \u03b4\nh : LiftRel r s (inr b) (inr d)\n\u22a2 s b d\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b3 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nb : \u03b2\nc : \u03b3\nd : \u03b4\nx : \u03b1 \u2295 \u03b2\ny : \u03b3 \u2295 \u03b4\na\u271d : s b d\n\u22a2 s b d\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b3 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nb : \u03b2\nc : \u03b3\nd : \u03b4\nx : \u03b1 \u2295 \u03b2\ny : \u03b3 \u2295 \u03b4\nhr : \u2200 (a : \u03b1) (b : \u03b3), r\u2081 a b \u2192 r\u2082 a b\nhs : \u2200 (a : \u03b2) (b : \u03b4), s\u2081 a b \u2192 s\u2082 a b\nh : LiftRel r\u2081 s\u2081 x y\n\u22a2 LiftRel r\u2082 s\u2082 x y\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b3 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nb : \u03b2\nc : \u03b3\nd : \u03b4\nhr : \u2200 (a : \u03b1) (b : \u03b3), r\u2081 a b \u2192 r\u2082 a b\nhs : \u2200 (a : \u03b2) (b : \u03b4), s\u2081 a b \u2192 s\u2082 a b\na\u271d\u00b9 : \u03b1\nc\u271d : \u03b3\na\u271d : r\u2081 a\u271d\u00b9 c\u271d\n\u22a2 LiftRel r\u2082 s\u2082 (inl a\u271d\u00b9) (inl c\u271d)\n[PROOFSTEP]\nexact LiftRel.inl (hr _ _ \u2039_\u203a)\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b3 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nb : \u03b2\nc : \u03b3\nd : \u03b4\nhr : \u2200 (a : \u03b1) (b : \u03b3), r\u2081 a b \u2192 r\u2082 a b\nhs : \u2200 (a : \u03b2) (b : \u03b4), s\u2081 a b \u2192 s\u2082 a b\nb\u271d : \u03b2\nd\u271d : \u03b4\na\u271d : s\u2081 b\u271d d\u271d\n\u22a2 LiftRel r\u2082 s\u2082 (inr b\u271d) (inr d\u271d)\n[PROOFSTEP]\nexact LiftRel.inr (hs _ _ \u2039_\u203a)\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b3 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nb : \u03b2\nc : \u03b3\nd : \u03b4\nx : \u03b1 \u2295 \u03b2\ny : \u03b3 \u2295 \u03b4\nh : LiftRel r s x y\n\u22a2 LiftRel s r (swap x) (swap y)\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b3 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nb : \u03b2\nc : \u03b3\nd : \u03b4\na\u271d\u00b9 : \u03b1\nc\u271d : \u03b3\na\u271d : r a\u271d\u00b9 c\u271d\n\u22a2 LiftRel s r (swap (inl a\u271d\u00b9)) (swap (inl c\u271d))\n[PROOFSTEP]\nexact LiftRel.inr \u2039_\u203a\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b3 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nb : \u03b2\nc : \u03b3\nd : \u03b4\nb\u271d : \u03b2\nd\u271d : \u03b4\na\u271d : s b\u271d d\u271d\n\u22a2 LiftRel s r (swap (inr b\u271d)) (swap (inr d\u271d))\n[PROOFSTEP]\nexact LiftRel.inl \u2039_\u203a\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b3 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nb : \u03b2\nc : \u03b3\nd : \u03b4\nx : \u03b1 \u2295 \u03b2\ny : \u03b3 \u2295 \u03b4\nh : LiftRel s r (swap x) (swap y)\n\u22a2 LiftRel r s x y\n[PROOFSTEP]\nrw [\u2190 swap_swap x, \u2190 swap_swap y]\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b3 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b4 \u2192 Prop\na : \u03b1\nb : \u03b2\nc : \u03b3\nd : \u03b4\nx : \u03b1 \u2295 \u03b2\ny : \u03b3 \u2295 \u03b4\nh : LiftRel s r (swap x) (swap y)\n\u22a2 LiftRel r s (swap (swap x)) (swap (swap y))\n[PROOFSTEP]\nexact h.swap\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\nh : Lex r s (inl a\u2081) (inl a\u2082)\n\u22a2 r a\u2081 a\u2082\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\nh\u271d : r a\u2081 a\u2082\n\u22a2 r a\u2081 a\u2082\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\nh : Lex r s (inr b\u2081) (inr b\u2082)\n\u22a2 s b\u2081 b\u2082\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\nh\u271d : s b\u2081 b\u2082\n\u22a2 s b\u2081 b\u2082\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\nx y a b : \u03b1 \u2295 \u03b2\nh : LiftRel r s a b\n\u22a2 Lex r s a b\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\na\u271d\u00b9 c\u271d : \u03b1\na\u271d : r a\u271d\u00b9 c\u271d\n\u22a2 Lex r s (inl a\u271d\u00b9) (inl c\u271d)\n[PROOFSTEP]\nexact Lex.inl \u2039_\u203a\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\nb\u271d d\u271d : \u03b2\na\u271d : s b\u271d d\u271d\n\u22a2 Lex r s (inr b\u271d) (inr d\u271d)\n[PROOFSTEP]\nexact Lex.inr \u2039_\u203a\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\nhr : \u2200 (a b : \u03b1), r\u2081 a b \u2192 r\u2082 a b\nhs : \u2200 (a b : \u03b2), s\u2081 a b \u2192 s\u2082 a b\nh : Lex r\u2081 s\u2081 x y\n\u22a2 Lex r\u2082 s\u2082 x y\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nhr : \u2200 (a b : \u03b1), r\u2081 a b \u2192 r\u2082 a b\nhs : \u2200 (a b : \u03b2), s\u2081 a b \u2192 s\u2082 a b\na\u2081\u271d a\u2082\u271d : \u03b1\nh\u271d : r\u2081 a\u2081\u271d a\u2082\u271d\n\u22a2 Lex r\u2082 s\u2082 (inl a\u2081\u271d) (inl a\u2082\u271d)\n[PROOFSTEP]\nexact Lex.inl (hr _ _ \u2039_\u203a)\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nhr : \u2200 (a b : \u03b1), r\u2081 a b \u2192 r\u2082 a b\nhs : \u2200 (a b : \u03b2), s\u2081 a b \u2192 s\u2082 a b\nb\u2081\u271d b\u2082\u271d : \u03b2\nh\u271d : s\u2081 b\u2081\u271d b\u2082\u271d\n\u22a2 Lex r\u2082 s\u2082 (inr b\u2081\u271d) (inr b\u2082\u271d)\n[PROOFSTEP]\nexact Lex.inr (hs _ _ \u2039_\u203a)\n[GOAL]\ncase sep\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nhr : \u2200 (a b : \u03b1), r\u2081 a b \u2192 r\u2082 a b\nhs : \u2200 (a b : \u03b2), s\u2081 a b \u2192 s\u2082 a b\na\u271d : \u03b1\nb\u271d : \u03b2\n\u22a2 Lex r\u2082 s\u2082 (inl a\u271d) (inr b\u271d)\n[PROOFSTEP]\nexact Lex.sep _ _\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na\u271d a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\na : \u03b1\naca : Acc r a\n\u22a2 Acc (Lex r s) (inl a)\n[PROOFSTEP]\ninduction' aca with a _ IH\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na\u271d\u00b9 a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nIH : \u2200 (y : \u03b1), r y a \u2192 Acc (Lex r s) (inl y)\n\u22a2 Acc (Lex r s) (inl a)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.h\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na\u271d\u00b9 a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nIH : \u2200 (y : \u03b1), r y a \u2192 Acc (Lex r s) (inl y)\n\u22a2 \u2200 (y : \u03b1 \u2295 \u03b2), Lex r s y (inl a) \u2192 Acc (Lex r s) y\n[PROOFSTEP]\nintro y h\n[GOAL]\ncase intro.h\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na\u271d\u00b9 a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y\u271d : \u03b1 \u2295 \u03b2\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nIH : \u2200 (y : \u03b1), r y a \u2192 Acc (Lex r s) (inl y)\ny : \u03b1 \u2295 \u03b2\nh : Lex r s y (inl a)\n\u22a2 Acc (Lex r s) y\n[PROOFSTEP]\ncases' h with a' _ h'\n[GOAL]\ncase intro.h.inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na\u271d\u00b9 a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nIH : \u2200 (y : \u03b1), r y a \u2192 Acc (Lex r s) (inl y)\na' : \u03b1\nh' : r a' a\n\u22a2 Acc (Lex r s) (inl a')\n[PROOFSTEP]\nexact IH _ h'\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\naca : \u2200 (a : \u03b1), Acc (Lex r s) (inl a)\nb : \u03b2\nacb : Acc s b\n\u22a2 Acc (Lex r s) (inr b)\n[PROOFSTEP]\ninduction' acb with b _ IH\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na a\u2081 a\u2082 : \u03b1\nb\u271d\u00b9 b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\naca : \u2200 (a : \u03b1), Acc (Lex r s) (inl a)\nb\u271d b : \u03b2\nh\u271d : \u2200 (y : \u03b2), s y b \u2192 Acc s y\nIH : \u2200 (y : \u03b2), s y b \u2192 Acc (Lex r s) (inr y)\n\u22a2 Acc (Lex r s) (inr b)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.h\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na a\u2081 a\u2082 : \u03b1\nb\u271d\u00b9 b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\naca : \u2200 (a : \u03b1), Acc (Lex r s) (inl a)\nb\u271d b : \u03b2\nh\u271d : \u2200 (y : \u03b2), s y b \u2192 Acc s y\nIH : \u2200 (y : \u03b2), s y b \u2192 Acc (Lex r s) (inr y)\n\u22a2 \u2200 (y : \u03b1 \u2295 \u03b2), Lex r s y (inr b) \u2192 Acc (Lex r s) y\n[PROOFSTEP]\nintro y h\n[GOAL]\ncase intro.h\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na a\u2081 a\u2082 : \u03b1\nb\u271d\u00b9 b\u2081 b\u2082 : \u03b2\nx y\u271d : \u03b1 \u2295 \u03b2\naca : \u2200 (a : \u03b1), Acc (Lex r s) (inl a)\nb\u271d b : \u03b2\nh\u271d : \u2200 (y : \u03b2), s y b \u2192 Acc s y\nIH : \u2200 (y : \u03b2), s y b \u2192 Acc (Lex r s) (inr y)\ny : \u03b1 \u2295 \u03b2\nh : Lex r s y (inr b)\n\u22a2 Acc (Lex r s) y\n[PROOFSTEP]\ncases' h with _ _ _ b' _ h' a\n[GOAL]\ncase intro.h.inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na a\u2081 a\u2082 : \u03b1\nb\u271d\u00b9 b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\naca : \u2200 (a : \u03b1), Acc (Lex r s) (inl a)\nb\u271d b : \u03b2\nh\u271d : \u2200 (y : \u03b2), s y b \u2192 Acc s y\nIH : \u2200 (y : \u03b2), s y b \u2192 Acc (Lex r s) (inr y)\nb' : \u03b2\nh' : s b' b\n\u22a2 Acc (Lex r s) (inr b')\n[PROOFSTEP]\nexact IH _ h'\n[GOAL]\ncase intro.h.sep\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\ns s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d\u00b9 b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u2295 \u03b2\naca : \u2200 (a : \u03b1), Acc (Lex r s) (inl a)\nb\u271d b : \u03b2\nh\u271d : \u2200 (y : \u03b2), s y b \u2192 Acc s y\nIH : \u2200 (y : \u03b2), s y b \u2192 Acc (Lex r s) (inr y)\na : \u03b1\n\u22a2 Acc (Lex r s) (inl a)\n[PROOFSTEP]\nexact aca _\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh : Surjective (Sum.map f g)\nc : \u03b3\n\u22a2 \u2203 a, f a = c\n[PROOFSTEP]\nobtain \u27e8a | b, h\u27e9 := h (inl c)\n[GOAL]\ncase intro.inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh\u271d : Surjective (Sum.map f g)\nc : \u03b3\na : \u03b1\nh : Sum.map f g (inl a) = inl c\n\u22a2 \u2203 a, f a = c\n[PROOFSTEP]\nexact \u27e8a, inl_injective h\u27e9\n[GOAL]\ncase intro.inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh\u271d : Surjective (Sum.map f g)\nc : \u03b3\nb : \u03b2\nh : Sum.map f g (inr b) = inl c\n\u22a2 \u2203 a, f a = c\n[PROOFSTEP]\ncases h\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh : Surjective (Sum.map f g)\nd : \u03b4\n\u22a2 \u2203 a, g a = d\n[PROOFSTEP]\nobtain \u27e8a | b, h\u27e9 := h (inr d)\n[GOAL]\ncase intro.inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh\u271d : Surjective (Sum.map f g)\nd : \u03b4\na : \u03b1\nh : Sum.map f g (inl a) = inr d\n\u22a2 \u2203 a, g a = d\n[PROOFSTEP]\ncases h\n[GOAL]\ncase intro.inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh\u271d : Surjective (Sum.map f g)\nd : \u03b4\nb : \u03b2\nh : Sum.map f g (inr b) = inr d\n\u22a2 \u2203 a, g a = d\n[PROOFSTEP]\nexact \u27e8b, inr_injective h\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nc : \u03b3\n\u22a2 Sum.elim (const \u03b1 c) (const \u03b2 c) = const (\u03b1 \u2295 \u03b2) c\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nc : \u03b3\nx : \u03b1 \u2295 \u03b2\n\u22a2 Sum.elim (const \u03b1 c) (const \u03b2 c) x = const (\u03b1 \u2295 \u03b2) c x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase h.inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nc : \u03b3\nval\u271d : \u03b1\n\u22a2 Sum.elim (const \u03b1 c) (const \u03b2 c) (inl val\u271d) = const (\u03b1 \u2295 \u03b2) c (inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nc : \u03b3\nval\u271d : \u03b2\n\u22a2 Sum.elim (const \u03b1 c) (const \u03b2 c) (inr val\u271d) = const (\u03b1 \u2295 \u03b2) c (inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b1\nc : \u03b3\n\u22a2 Sum.elim (update f i c) g = update (Sum.elim f g) (inl i) c\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b1\nc : \u03b3\nx : \u03b1 \u2295 \u03b2\n\u22a2 Sum.elim (update f i c) g x = update (Sum.elim f g) (inl i) c x\n[PROOFSTEP]\nrcases x with x | x\n[GOAL]\ncase h.inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b1\nc : \u03b3\nx : \u03b1\n\u22a2 Sum.elim (update f i c) g (inl x) = update (Sum.elim f g) (inl i) c (inl x)\n[PROOFSTEP]\nby_cases h : x = i\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b1\nc : \u03b3\nx : \u03b1\nh : x = i\n\u22a2 Sum.elim (update f i c) g (inl x) = update (Sum.elim f g) (inl i) c (inl x)\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\nc : \u03b3\nx : \u03b1\n\u22a2 Sum.elim (update f x c) g (inl x) = update (Sum.elim f g) (inl x) c (inl x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b1\nc : \u03b3\nx : \u03b1\nh : \u00acx = i\n\u22a2 Sum.elim (update f i c) g (inl x) = update (Sum.elim f g) (inl i) c (inl x)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase h.inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b1\nc : \u03b3\nx : \u03b2\n\u22a2 Sum.elim (update f i c) g (inr x) = update (Sum.elim f g) (inl i) c (inr x)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b2\nc : \u03b3\n\u22a2 Sum.elim f (update g i c) = update (Sum.elim f g) (inr i) c\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b2\nc : \u03b3\nx : \u03b1 \u2295 \u03b2\n\u22a2 Sum.elim f (update g i c) x = update (Sum.elim f g) (inr i) c x\n[PROOFSTEP]\nrcases x with x | x\n[GOAL]\ncase h.inl\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b2\nc : \u03b3\nx : \u03b1\n\u22a2 Sum.elim f (update g i c) (inl x) = update (Sum.elim f g) (inr i) c (inl x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.inr\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b2\nc : \u03b3\nx : \u03b2\n\u22a2 Sum.elim f (update g i c) (inr x) = update (Sum.elim f g) (inr i) c (inr x)\n[PROOFSTEP]\nby_cases h : x = i\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b2\nc : \u03b3\nx : \u03b2\nh : x = i\n\u22a2 Sum.elim f (update g i c) (inr x) = update (Sum.elim f g) (inr i) c (inr x)\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\nc : \u03b3\nx : \u03b2\n\u22a2 Sum.elim f (update g x c) (inr x) = update (Sum.elim f g) (inr x) c (inr x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b3\ni : \u03b2\nc : \u03b3\nx : \u03b2\nh : \u00acx = i\n\u22a2 Sum.elim f (update g i c) (inr x) = update (Sum.elim f g) (inr i) c (inr x)\n[PROOFSTEP]\nsimp [h]\n", "meta": {"mathlib_filename": "Mathlib.Data.Sum.Basic", "llama_tokens": 17678, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943805178139, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.4204504014449288}} {"text": "[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\n\u22a2 TFAE\n [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]\n[PROOFSTEP]\ntfae_have 1 \u2192 4\n[GOAL]\ncase tfae_1_to_4\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\n\u22a2 IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\u22a2 TFAE\n [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]\n[PROOFSTEP]\nexact fun h y => h { y }\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\u22a2 TFAE\n [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]\n[PROOFSTEP]\ntfae_have 4 \u2192 3\n[GOAL]\ncase tfae_4_to_3\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\u22a2 (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\u22a2 TFAE\n [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]\n[PROOFSTEP]\nexact fun h x => h (f x)\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\u22a2 TFAE\n [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]\n[PROOFSTEP]\ntfae_have 3 \u2192 2\n[GOAL]\ncase tfae_3_to_2\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\u22a2 (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\u22a2 TFAE\n [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]\n[PROOFSTEP]\nexact fun h x => IsOpen.mem_nhds (h x) rfl\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\u22a2 TFAE\n [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]\n[PROOFSTEP]\ntfae_have 2 \u2192 5\n[GOAL]\ncase tfae_2_to_5\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\u22a2 (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\n[PROOFSTEP]\nintro h x\n[GOAL]\ncase tfae_2_to_5\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nh : \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nx : X\n\u22a2 \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\n[PROOFSTEP]\nrcases mem_nhds_iff.1 (h x) with \u27e8U, eq, hU, hx\u27e9\n[GOAL]\ncase tfae_2_to_5.intro.intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nh : \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nx : X\nU : Set X\neq : U \u2286 {x_1 | (fun x' => f x' = f x) x_1}\nhU : IsOpen U\nhx : x \u2208 U\n\u22a2 \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\n[PROOFSTEP]\nexact \u27e8U, hU, hx, eq\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\n\u22a2 TFAE\n [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]\n[PROOFSTEP]\ntfae_have 5 \u2192 1\n[GOAL]\ncase tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\n\u22a2 (\u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x) \u2192 IsLocallyConstant f\n[PROOFSTEP]\nintro h s\n[GOAL]\ncase tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\nh : \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\ns : Set Y\n\u22a2 IsOpen (f \u207b\u00b9' s)\n[PROOFSTEP]\nrefine' isOpen_iff_forall_mem_open.2 fun x hx => _\n[GOAL]\ncase tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\nh : \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\ns : Set Y\nx : X\nhx : x \u2208 f \u207b\u00b9' s\n\u22a2 \u2203 t, t \u2286 f \u207b\u00b9' s \u2227 IsOpen t \u2227 x \u2208 t\n[PROOFSTEP]\nrcases h x with \u27e8U, hU, hxU, eq\u27e9\n[GOAL]\ncase tfae_5_to_1.intro.intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\nh : \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\ns : Set Y\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nU : Set X\nhU : IsOpen U\nhxU : x \u2208 U\neq : \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\n\u22a2 \u2203 t, t \u2286 f \u207b\u00b9' s \u2227 IsOpen t \u2227 x \u2208 t\n[PROOFSTEP]\nexact \u27e8U, fun x' hx' => mem_preimage.2 <| (eq x' hx').symm \u25b8 hx, hU, hxU\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\ntfae_5_to_1 : (\u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x) \u2192 IsLocallyConstant f\n\u22a2 TFAE\n [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]\n[PROOFSTEP]\ntfae_finish\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\nhf : IsLocallyConstant f\ng : Y \u2192 Z\ns : Set Z\n\u22a2 IsOpen (g \u2218 f \u207b\u00b9' s)\n[PROOFSTEP]\nrw [Set.preimage_comp]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\nhf : IsLocallyConstant f\ng : Y \u2192 Z\ns : Set Z\n\u22a2 IsOpen (f \u207b\u00b9' (g \u207b\u00b9' s))\n[PROOFSTEP]\nexact hf _\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ng : Y \u2192 Z\nf : X \u2192 Y\nhg : IsLocallyConstant g\nhf : Continuous f\ns : Set Z\n\u22a2 IsOpen (g \u2218 f \u207b\u00b9' s)\n[PROOFSTEP]\nrw [Set.preimage_comp]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ng : Y \u2192 Z\nf : X \u2192 Y\nhg : IsLocallyConstant g\nhf : Continuous f\ns : Set Z\n\u22a2 IsOpen (f \u207b\u00b9' (g \u207b\u00b9' s))\n[PROOFSTEP]\nexact hf.isOpen_preimage _ (hg _)\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\nhf : IsLocallyConstant f\ns : Set X\nhs : IsPreconnected s\nx y : X\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 f x = f y\n[PROOFSTEP]\nlet U := f \u207b\u00b9' {f y}\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\nhf : IsLocallyConstant f\ns : Set X\nhs : IsPreconnected s\nx y : X\nhx : x \u2208 s\nhy : y \u2208 s\nU : Set X := f \u207b\u00b9' {f y}\n\u22a2 f x = f y\n[PROOFSTEP]\nsuffices : x \u2209 U\u1d9c\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\nhf : IsLocallyConstant f\ns : Set X\nhs : IsPreconnected s\nx y : X\nhx : x \u2208 s\nhy : y \u2208 s\nU : Set X := f \u207b\u00b9' {f y}\nthis : \u00acx \u2208 U\u1d9c\n\u22a2 f x = f y\ncase this\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\nhf : IsLocallyConstant f\ns : Set X\nhs : IsPreconnected s\nx y : X\nhx : x \u2208 s\nhy : y \u2208 s\nU : Set X := f \u207b\u00b9' {f y}\n\u22a2 \u00acx \u2208 U\u1d9c\n[PROOFSTEP]\nexact Classical.not_not.1 this\n[GOAL]\ncase this\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\nhf : IsLocallyConstant f\ns : Set X\nhs : IsPreconnected s\nx y : X\nhx : x \u2208 s\nhy : y \u2208 s\nU : Set X := f \u207b\u00b9' {f y}\n\u22a2 \u00acx \u2208 U\u1d9c\n[PROOFSTEP]\nintro hxV\n[GOAL]\ncase this\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\nhf : IsLocallyConstant f\ns : Set X\nhs : IsPreconnected s\nx y : X\nhx : x \u2208 s\nhy : y \u2208 s\nU : Set X := f \u207b\u00b9' {f y}\nhxV : x \u2208 U\u1d9c\n\u22a2 False\n[PROOFSTEP]\nspecialize hs U U\u1d9c (hf {f y}) (hf {f y}\u1d9c) _ \u27e8y, \u27e8hy, rfl\u27e9\u27e9 \u27e8x, \u27e8hx, hxV\u27e9\u27e9\n[GOAL]\ncase this\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\nhf : IsLocallyConstant f\ns : Set X\nhs : IsPreconnected s\nx y : X\nhx : x \u2208 s\nhy : y \u2208 s\nU : Set X := f \u207b\u00b9' {f y}\nhxV : x \u2208 U\u1d9c\n\u22a2 s \u2286 U \u222a U\u1d9c\n[PROOFSTEP]\nsimp only [union_compl_self, subset_univ]\n[GOAL]\ncase this\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\nhf : IsLocallyConstant f\ns : Set X\nx y : X\nhx : x \u2208 s\nhy : y \u2208 s\nU : Set X := f \u207b\u00b9' {f y}\nhxV : x \u2208 U\u1d9c\nhs : Set.Nonempty (s \u2229 (U \u2229 U\u1d9c))\n\u22a2 False\n[PROOFSTEP]\nsimp only [inter_empty, Set.not_nonempty_empty, inter_compl_self] at hs \n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : PreconnectedSpace X\ninst\u271d : Nonempty Y\nf : X \u2192 Y\nhf : IsLocallyConstant f\n\u22a2 \u2203 y, f = Function.const X y\n[PROOFSTEP]\ncases' isEmpty_or_nonempty X with h h\n[GOAL]\ncase inl\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : PreconnectedSpace X\ninst\u271d : Nonempty Y\nf : X \u2192 Y\nhf : IsLocallyConstant f\nh : IsEmpty X\n\u22a2 \u2203 y, f = Function.const X y\n[PROOFSTEP]\nexact \u27e8Classical.arbitrary Y, funext <| h.elim\u27e9\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : PreconnectedSpace X\ninst\u271d : Nonempty Y\nf : X \u2192 Y\nhf : IsLocallyConstant f\nh : Nonempty X\n\u22a2 \u2203 y, f = Function.const X y\n[PROOFSTEP]\nexact \u27e8f (Classical.arbitrary X), hf.eq_const _\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nf : X \u2192 Y\nhf : IsLocallyConstant f\n\u22a2 Set.Finite (range f)\n[PROOFSTEP]\nletI : TopologicalSpace Y := \u22a5\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nf : X \u2192 Y\nhf : IsLocallyConstant f\nthis : TopologicalSpace Y := \u22a5\n\u22a2 Set.Finite (range f)\n[PROOFSTEP]\nhaveI := discreteTopology_bot Y\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nf : X \u2192 Y\nhf : IsLocallyConstant f\nthis\u271d : TopologicalSpace Y := \u22a5\nthis : DiscreteTopology Y\n\u22a2 Set.Finite (range f)\n[PROOFSTEP]\nexact (isCompact_range hf.continuous).finite_of_discrete\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1\u271d : Type u_4\ninst\u271d : TopologicalSpace X\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nf : X \u2192 \u03b1\ng : \u03b1 \u2192 \u03b2\nh : IsLocallyConstant (g \u2218 f)\ninj : Function.Injective g\ns : Set \u03b1\n\u22a2 IsOpen (f \u207b\u00b9' s)\n[PROOFSTEP]\nrw [\u2190 preimage_image_eq s inj, preimage_preimage]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1\u271d : Type u_4\ninst\u271d : TopologicalSpace X\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nf : X \u2192 \u03b1\ng : \u03b1 \u2192 \u03b2\nh : IsLocallyConstant (g \u2218 f)\ninj : Function.Injective g\ns : Set \u03b1\n\u22a2 IsOpen ((fun x => g (f x)) \u207b\u00b9' (g '' s))\n[PROOFSTEP]\nexact h (g '' s)\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\n\u22a2 Function.Injective LocallyConstant.toFun\n[PROOFSTEP]\nrintro \u27e8_, _\u27e9 \u27e8_, _\u27e9 _\n[GOAL]\ncase mk.mk\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\ntoFun\u271d\u00b9 : X \u2192 Y\nisLocallyConstant\u271d\u00b9 : IsLocallyConstant toFun\u271d\u00b9\ntoFun\u271d : X \u2192 Y\nisLocallyConstant\u271d : IsLocallyConstant toFun\u271d\na\u271d :\n { toFun := toFun\u271d\u00b9, isLocallyConstant := isLocallyConstant\u271d\u00b9 }.toFun =\n { toFun := toFun\u271d, isLocallyConstant := isLocallyConstant\u271d }.toFun\n\u22a2 { toFun := toFun\u271d\u00b9, isLocallyConstant := isLocallyConstant\u271d\u00b9 } =\n { toFun := toFun\u271d, isLocallyConstant := isLocallyConstant\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\nU : Set X\ninst\u271d : (x : X) \u2192 Decidable (x \u2208 U)\nhU : IsClopen U\n\u22a2 IsLocallyConstant fun x => if x \u2208 U then 0 else 1\n[PROOFSTEP]\nrefine IsLocallyConstant.iff_isOpen_fiber.2 <| Fin.forall_fin_two.2 \u27e8?_, ?_\u27e9\n[GOAL]\ncase refine_1\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\nU : Set X\ninst\u271d : (x : X) \u2192 Decidable (x \u2208 U)\nhU : IsClopen U\n\u22a2 IsOpen ((fun x => if x \u2208 U then 0 else 1) \u207b\u00b9' {0})\n[PROOFSTEP]\nconvert hU.1 using 1\n[GOAL]\ncase h.e'_3\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\nU : Set X\ninst\u271d : (x : X) \u2192 Decidable (x \u2208 U)\nhU : IsClopen U\n\u22a2 (fun x => if x \u2208 U then 0 else 1) \u207b\u00b9' {0} = U\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_3.h\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\nU : Set X\ninst\u271d : (x : X) \u2192 Decidable (x \u2208 U)\nhU : IsClopen U\nx\u271d : X\n\u22a2 x\u271d \u2208 (fun x => if x \u2208 U then 0 else 1) \u207b\u00b9' {0} \u2194 x\u271d \u2208 U\n[PROOFSTEP]\nsimp only [mem_singleton_iff, Fin.one_eq_zero_iff, mem_preimage, ite_eq_left_iff, Nat.succ_succ_ne_one]\n[GOAL]\ncase h.e'_3.h\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\nU : Set X\ninst\u271d : (x : X) \u2192 Decidable (x \u2208 U)\nhU : IsClopen U\nx\u271d : X\n\u22a2 \u00acx\u271d \u2208 U \u2192 False \u2194 x\u271d \u2208 U\n[PROOFSTEP]\ntauto\n[GOAL]\ncase refine_2\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\nU : Set X\ninst\u271d : (x : X) \u2192 Decidable (x \u2208 U)\nhU : IsClopen U\n\u22a2 IsOpen ((fun x => if x \u2208 U then 0 else 1) \u207b\u00b9' {1})\n[PROOFSTEP]\nrw [\u2190 isClosed_compl_iff]\n[GOAL]\ncase refine_2\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\nU : Set X\ninst\u271d : (x : X) \u2192 Decidable (x \u2208 U)\nhU : IsClopen U\n\u22a2 IsClosed ((fun x => if x \u2208 U then 0 else 1) \u207b\u00b9' {1})\u1d9c\n[PROOFSTEP]\nconvert hU.2\n[GOAL]\ncase h.e'_3\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\nU : Set X\ninst\u271d : (x : X) \u2192 Decidable (x \u2208 U)\nhU : IsClopen U\n\u22a2 ((fun x => if x \u2208 U then 0 else 1) \u207b\u00b9' {1})\u1d9c = U\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_3.h\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\nU : Set X\ninst\u271d : (x : X) \u2192 Decidable (x \u2208 U)\nhU : IsClopen U\nx\u271d : X\n\u22a2 x\u271d \u2208 ((fun x => if x \u2208 U then 0 else 1) \u207b\u00b9' {1})\u1d9c \u2194 x\u271d \u2208 U\n[PROOFSTEP]\nsimp\n[GOAL]\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\nU : Set X\ninst\u271d : (x : X) \u2192 Decidable (x \u2208 U)\nhU : IsClopen U\n\u22a2 \u2191(ofClopen hU) \u207b\u00b9' {0} = U\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\nU : Set X\ninst\u271d : (x : X) \u2192 Decidable (x \u2208 U)\nhU : IsClopen U\nx\u271d : X\n\u22a2 x\u271d \u2208 \u2191(ofClopen hU) \u207b\u00b9' {0} \u2194 x\u271d \u2208 U\n[PROOFSTEP]\nsimp only [ofClopen, mem_singleton_iff, Fin.one_eq_zero_iff, coe_mk, mem_preimage, ite_eq_left_iff,\n Nat.succ_succ_ne_one]\n[GOAL]\ncase h\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\nU : Set X\ninst\u271d : (x : X) \u2192 Decidable (x \u2208 U)\nhU : IsClopen U\nx\u271d : X\n\u22a2 \u00acx\u271d \u2208 U \u2192 False \u2194 x\u271d \u2208 U\n[PROOFSTEP]\ntauto\n[GOAL]\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\nU : Set X\ninst\u271d : (x : X) \u2192 Decidable (x \u2208 U)\nhU : IsClopen U\n\u22a2 \u2191(ofClopen hU) \u207b\u00b9' {1} = U\u1d9c\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\nU : Set X\ninst\u271d : (x : X) \u2192 Decidable (x \u2208 U)\nhU : IsClopen U\nx\u271d : X\n\u22a2 x\u271d \u2208 \u2191(ofClopen hU) \u207b\u00b9' {1} \u2194 x\u271d \u2208 U\u1d9c\n[PROOFSTEP]\nsimp only [ofClopen, mem_singleton_iff, coe_mk, Fin.zero_eq_one_iff, mem_preimage, ite_eq_right_iff, mem_compl_iff,\n Nat.succ_succ_ne_one]\n[GOAL]\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d : TopologicalSpace X\nf g : LocallyConstant X (Fin 2)\nh : \u2191f \u207b\u00b9' {0} = \u2191g \u207b\u00b9' {0}\n\u22a2 f = g\n[PROOFSTEP]\nsimp only [Set.ext_iff, mem_singleton_iff, mem_preimage] at h \n[GOAL]\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d : TopologicalSpace X\nf g : LocallyConstant X (Fin 2)\nh : \u2200 (x : X), \u2191f x = 0 \u2194 \u2191g x = 0\n\u22a2 f = g\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\u271d\nX : Type u_5\ninst\u271d : TopologicalSpace X\nf g : LocallyConstant X (Fin 2)\nh : \u2200 (x : X), \u2191f x = 0 \u2194 \u2191g x = 0\nx : X\n\u22a2 \u2191f x = \u2191g x\n[PROOFSTEP]\nexact Fin.fin_two_eq_of_eq_zero_iff (h x)\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : PreconnectedSpace X\ninst\u271d : Nonempty Y\nf : LocallyConstant X Y\n\u22a2 \u2203 y, f = const X y\n[PROOFSTEP]\nrcases Classical.em (Nonempty X) with (\u27e8\u27e8x\u27e9\u27e9 | hX)\n[GOAL]\ncase inl.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : PreconnectedSpace X\ninst\u271d : Nonempty Y\nf : LocallyConstant X Y\nx : X\n\u22a2 \u2203 y, f = const X y\n[PROOFSTEP]\nexact \u27e8f x, f.eq_const x\u27e9\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : PreconnectedSpace X\ninst\u271d : Nonempty Y\nf : LocallyConstant X Y\nhX : \u00acNonempty X\n\u22a2 \u2203 y, f = const X y\n[PROOFSTEP]\nexact \u27e8Classical.arbitrary Y, ext fun x => (hX \u27e8x\u27e9).elim\u27e9\n[GOAL]\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1\u271d : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\n\u03b1 : Type u_6\n\u03b2 : Type u_7\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : TopologicalSpace X\nf : \u03b1 \u2192 LocallyConstant X \u03b2\ng : \u03b1 \u2192 \u03b2\n\u22a2 IsOpen ((fun x a => \u2191(f a) x) \u207b\u00b9' {g})\n[PROOFSTEP]\nhave : (fun (x : X) (a : \u03b1) => f a x) \u207b\u00b9' { g } = \u22c2 a : \u03b1, f a \u207b\u00b9' {g a} := by ext; simp [Function.funext_iff]\n[GOAL]\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1\u271d : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\n\u03b1 : Type u_6\n\u03b2 : Type u_7\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : TopologicalSpace X\nf : \u03b1 \u2192 LocallyConstant X \u03b2\ng : \u03b1 \u2192 \u03b2\n\u22a2 (fun x a => \u2191(f a) x) \u207b\u00b9' {g} = \u22c2 (a : \u03b1), \u2191(f a) \u207b\u00b9' {g a}\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1\u271d : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\n\u03b1 : Type u_6\n\u03b2 : Type u_7\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : TopologicalSpace X\nf : \u03b1 \u2192 LocallyConstant X \u03b2\ng : \u03b1 \u2192 \u03b2\nx\u271d : X\n\u22a2 x\u271d \u2208 (fun x a => \u2191(f a) x) \u207b\u00b9' {g} \u2194 x\u271d \u2208 \u22c2 (a : \u03b1), \u2191(f a) \u207b\u00b9' {g a}\n[PROOFSTEP]\nsimp [Function.funext_iff]\n[GOAL]\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1\u271d : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\n\u03b1 : Type u_6\n\u03b2 : Type u_7\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : TopologicalSpace X\nf : \u03b1 \u2192 LocallyConstant X \u03b2\ng : \u03b1 \u2192 \u03b2\nthis : (fun x a => \u2191(f a) x) \u207b\u00b9' {g} = \u22c2 (a : \u03b1), \u2191(f a) \u207b\u00b9' {g a}\n\u22a2 IsOpen ((fun x a => \u2191(f a) x) \u207b\u00b9' {g})\n[PROOFSTEP]\nrw [this]\n[GOAL]\nX\u271d : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1\u271d : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\u271d\nX : Type u_5\n\u03b1 : Type u_6\n\u03b2 : Type u_7\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : TopologicalSpace X\nf : \u03b1 \u2192 LocallyConstant X \u03b2\ng : \u03b1 \u2192 \u03b2\nthis : (fun x a => \u2191(f a) x) \u207b\u00b9' {g} = \u22c2 (a : \u03b1), \u2191(f a) \u207b\u00b9' {g a}\n\u22a2 IsOpen (\u22c2 (a : \u03b1), \u2191(f a) \u207b\u00b9' {g a})\n[PROOFSTEP]\nexact isOpen_iInter fun a => (f a).isLocallyConstant _\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\nhf : \u00acContinuous f\n\u22a2 LocallyConstant Y Z \u2192 LocallyConstant X Z\n[PROOFSTEP]\nby_cases H : Nonempty X\n[GOAL]\ncase pos\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\nhf : \u00acContinuous f\nH : Nonempty X\n\u22a2 LocallyConstant Y Z \u2192 LocallyConstant X Z\n[PROOFSTEP]\nintro g\n[GOAL]\ncase pos\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\nhf : \u00acContinuous f\nH : Nonempty X\ng : LocallyConstant Y Z\n\u22a2 LocallyConstant X Z\n[PROOFSTEP]\nexact const X (g <| f <| Classical.arbitrary X)\n[GOAL]\ncase neg\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\nhf : \u00acContinuous f\nH : \u00acNonempty X\n\u22a2 LocallyConstant Y Z \u2192 LocallyConstant X Z\n[PROOFSTEP]\nrefine fun _ => \u27e8fun x => (H \u27e8x\u27e9).elim, fun s => ?_\u27e9\n[GOAL]\ncase neg\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\nhf : \u00acContinuous f\nH : \u00acNonempty X\nx\u271d : LocallyConstant Y Z\ns : Set Z\n\u22a2 IsOpen ((fun x => False.elim (_ : False)) \u207b\u00b9' s)\n[PROOFSTEP]\nrw [isOpen_iff_nhds]\n[GOAL]\ncase neg\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\nhf : \u00acContinuous f\nH : \u00acNonempty X\nx\u271d : LocallyConstant Y Z\ns : Set Z\n\u22a2 \u2200 (a : X), a \u2208 (fun x => False.elim (_ : False)) \u207b\u00b9' s \u2192 \ud835\udcdd a \u2264 \ud835\udcdf ((fun x => False.elim (_ : False)) \u207b\u00b9' s)\n[PROOFSTEP]\nintro x\n[GOAL]\ncase neg\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\nhf : \u00acContinuous f\nH : \u00acNonempty X\nx\u271d : LocallyConstant Y Z\ns : Set Z\nx : X\n\u22a2 x \u2208 (fun x => False.elim (_ : False)) \u207b\u00b9' s \u2192 \ud835\udcdd x \u2264 \ud835\udcdf ((fun x => False.elim (_ : False)) \u207b\u00b9' s)\n[PROOFSTEP]\nexact (H \u27e8x\u27e9).elim\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\ng : LocallyConstant Y Z\nhf : Continuous f\n\u22a2 \u2191(comap f g) = \u2191g \u2218 f\n[PROOFSTEP]\nrw [comap, dif_pos hf]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\ng : LocallyConstant Y Z\nhf : Continuous f\n\u22a2 \u2191{ toFun := \u2191g \u2218 f, isLocallyConstant := (_ : IsLocallyConstant (g.toFun \u2218 f)) } = \u2191g \u2218 f\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\ng : LocallyConstant Y Z\nhf : Continuous f\nx : X\n\u22a2 \u2191(comap f g) x = \u2191g (f x)\n[PROOFSTEP]\nsimp only [hf, coe_comap, Function.comp_apply]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\n\u22a2 comap id = id\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d\u00b9 : LocallyConstant X Z\nx\u271d : X\n\u22a2 \u2191(comap id x\u271d\u00b9) x\u271d = \u2191(id x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nsimp only [continuous_id, id.def, Function.comp.right_id, coe_comap]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nf : X \u2192 Y\ng : Y \u2192 Z\nhf : Continuous f\nhg : Continuous g\n\u22a2 comap f \u2218 comap g = comap (g \u2218 f)\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nf : X \u2192 Y\ng : Y \u2192 Z\nhf : Continuous f\nhg : Continuous g\nx\u271d\u00b9 : LocallyConstant Z \u03b1\nx\u271d : X\n\u22a2 \u2191((comap f \u2218 comap g) x\u271d\u00b9) x\u271d = \u2191(comap (g \u2218 f) x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nrw [Function.comp_apply]\n[GOAL]\ncase h.h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nf : X \u2192 Y\ng : Y \u2192 Z\nhf : Continuous f\nhg : Continuous g\nx\u271d\u00b9 : LocallyConstant Z \u03b1\nx\u271d : X\n\u22a2 \u2191(comap f (comap g x\u271d\u00b9)) x\u271d = \u2191(comap (g \u2218 f) x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nsimp only [hf, hg, hg.comp hf, coe_comap]\n[GOAL]\ncase h.h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nf : X \u2192 Y\ng : Y \u2192 Z\nhf : Continuous f\nhg : Continuous g\nx\u271d\u00b9 : LocallyConstant Z \u03b1\nx\u271d : X\n\u22a2 ((\u2191x\u271d\u00b9 \u2218 g) \u2218 f) x\u271d = (\u2191x\u271d\u00b9 \u2218 g \u2218 f) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nf : X \u2192 Y\ng : Y \u2192 Z\nhf : Continuous f\nhg : Continuous g\nx : LocallyConstant Z \u03b1\n\u22a2 comap f (comap g x) = comap (g \u2218 f) x\n[PROOFSTEP]\nrw [\u2190 comap_comp f g hf hg]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nf : X \u2192 Y\ng : Y \u2192 Z\nhf : Continuous f\nhg : Continuous g\nx : LocallyConstant Z \u03b1\n\u22a2 comap f (comap g x) = (comap f \u2218 comap g) x\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\ny : Y\nh : \u2200 (x : X), f x = y\n\u22a2 comap f = fun g => const X (\u2191g y)\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\ny : Y\nh : \u2200 (x : X), f x = y\nx\u271d\u00b9 : LocallyConstant Y Z\nx\u271d : X\n\u22a2 \u2191(comap f x\u271d\u00b9) x\u271d = \u2191(const X (\u2191x\u271d\u00b9 y)) x\u271d\n[PROOFSTEP]\nrw [coe_comap]\n[GOAL]\ncase h.h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\ny : Y\nh : \u2200 (x : X), f x = y\nx\u271d\u00b9 : LocallyConstant Y Z\nx\u271d : X\n\u22a2 (\u2191x\u271d\u00b9 \u2218 f) x\u271d = \u2191(const X (\u2191x\u271d\u00b9 y)) x\u271d\n[PROOFSTEP]\nsimp only [Function.comp_apply, h, coe_const, Function.const_apply]\n[GOAL]\ncase h.h.hf\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\ny : Y\nh : \u2200 (x : X), f x = y\nx\u271d\u00b9 : LocallyConstant Y Z\nx\u271d : X\n\u22a2 Continuous f\n[PROOFSTEP]\nrw [show f = fun _ => y by ext; apply h]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\ny : Y\nh : \u2200 (x : X), f x = y\nx\u271d\u00b9 : LocallyConstant Y Z\nx\u271d : X\n\u22a2 f = fun x => y\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\ny : Y\nh : \u2200 (x : X), f x = y\nx\u271d\u00b2 : LocallyConstant Y Z\nx\u271d\u00b9 x\u271d : X\n\u22a2 f x\u271d = y\n[PROOFSTEP]\napply h\n[GOAL]\ncase h.h.hf\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\ny : Y\nh : \u2200 (x : X), f x = y\nx\u271d\u00b9 : LocallyConstant Y Z\nx\u271d : X\n\u22a2 Continuous fun x => y\n[PROOFSTEP]\nexact continuous_const\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\nhf : Continuous f\nhfs : Function.Surjective f\n\u22a2 Function.Injective (comap f)\n[PROOFSTEP]\nintro a b h\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\nhf : Continuous f\nhfs : Function.Surjective f\na b : LocallyConstant Y Z\nh : comap f a = comap f b\n\u22a2 a = b\n[PROOFSTEP]\nrw [LocallyConstant.ext_iff] at h \n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\nhf : Continuous f\nhfs : Function.Surjective f\na b : LocallyConstant Y Z\nh : \u2200 (x : X), \u2191(comap f a) x = \u2191(comap f b) x\n\u22a2 a = b\n[PROOFSTEP]\next y\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\nhf : Continuous f\nhfs : Function.Surjective f\na b : LocallyConstant Y Z\nh : \u2200 (x : X), \u2191(comap f a) x = \u2191(comap f b) x\ny : Y\n\u22a2 \u2191a y = \u2191b y\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := hfs y\n[GOAL]\ncase h.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\nhf : Continuous f\nhfs : Function.Surjective f\na b : LocallyConstant Y Z\nh : \u2200 (x : X), \u2191(comap f a) x = \u2191(comap f b) x\ny : Y\nx : X\nhx : f x = y\n\u22a2 \u2191a y = \u2191b y\n[PROOFSTEP]\nspecialize h x\n[GOAL]\ncase h.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\nhf : Continuous f\nhfs : Function.Surjective f\na b : LocallyConstant Y Z\ny : Y\nx : X\nhx : f x = y\nh : \u2191(comap f a) x = \u2191(comap f b) x\n\u22a2 \u2191a y = \u2191b y\n[PROOFSTEP]\nrwa [coe_comap_apply _ _ hf, coe_comap_apply _ _ hf, hx] at h \n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nR : Type u_5\ninst\u271d : One R\nU : Set X\nf : LocallyConstant X R\nhU : IsClopen U\ns : Set R\n\u22a2 IsOpen (Set.mulIndicator U \u2191f \u207b\u00b9' s)\n[PROOFSTEP]\nrw [mulIndicator_preimage, Set.ite, Set.diff_eq]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nR : Type u_5\ninst\u271d : One R\nU : Set X\nf : LocallyConstant X R\nhU : IsClopen U\ns : Set R\n\u22a2 IsOpen (\u2191f \u207b\u00b9' s \u2229 U \u222a 1 \u207b\u00b9' s \u2229 U\u1d9c)\n[PROOFSTEP]\nexact ((f.2 s).inter hU.isOpen).union ((IsLocallyConstant.const 1 s).inter hU.compl.isOpen)\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ne : X \u2243\u209c Y\n\u22a2 Function.LeftInverse (comap \u2191e) (comap \u2191(Homeomorph.symm e))\n[PROOFSTEP]\nintro\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ne : X \u2243\u209c Y\nx\u271d : LocallyConstant X Z\n\u22a2 comap (\u2191e) (comap (\u2191(Homeomorph.symm e)) x\u271d) = x\u271d\n[PROOFSTEP]\nrw [comap_comap _ _ e.continuous e.symm.continuous]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ne : X \u2243\u209c Y\nx\u271d : LocallyConstant X Z\n\u22a2 comap (\u2191(Homeomorph.symm e) \u2218 \u2191e) x\u271d = x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ne : X \u2243\u209c Y\n\u22a2 Function.RightInverse (comap \u2191e) (comap \u2191(Homeomorph.symm e))\n[PROOFSTEP]\nintro\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ne : X \u2243\u209c Y\nx\u271d : LocallyConstant Y Z\n\u22a2 comap (\u2191(Homeomorph.symm e)) (comap (\u2191e) x\u271d) = x\u271d\n[PROOFSTEP]\nrw [comap_comap _ _ e.symm.continuous e.continuous]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ne : X \u2243\u209c Y\nx\u271d : LocallyConstant Y Z\n\u22a2 comap (\u2191e \u2218 \u2191(Homeomorph.symm e)) x\u271d = x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh : C\u2081 \u222a C\u2082 = univ\nf : LocallyConstant (\u2191C\u2081) Z\ng : LocallyConstant (\u2191C\u2082) Z\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun f { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun g { val := x, property := (_ : x \u2208 C\u2082) }\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\n\u22a2 IsLocallyConstant fun i =>\n if hi : i \u2208 C\u2081 then \u2191f { val := i, property := hi } else \u2191g { val := i, property := (_ : i \u2208 C\u2082) }\n[PROOFSTEP]\nlet dZ : TopologicalSpace Z := \u22a5\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh : C\u2081 \u222a C\u2082 = univ\nf : LocallyConstant (\u2191C\u2081) Z\ng : LocallyConstant (\u2191C\u2082) Z\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun f { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun g { val := x, property := (_ : x \u2208 C\u2082) }\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\n\u22a2 IsLocallyConstant fun i =>\n if hi : i \u2208 C\u2081 then \u2191f { val := i, property := hi } else \u2191g { val := i, property := (_ : i \u2208 C\u2082) }\n[PROOFSTEP]\nhaveI : DiscreteTopology Z := discreteTopology_bot Z\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh : C\u2081 \u222a C\u2082 = univ\nf : LocallyConstant (\u2191C\u2081) Z\ng : LocallyConstant (\u2191C\u2082) Z\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun f { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun g { val := x, property := (_ : x \u2208 C\u2082) }\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\n\u22a2 IsLocallyConstant fun i =>\n if hi : i \u2208 C\u2081 then \u2191f { val := i, property := hi } else \u2191g { val := i, property := (_ : i \u2208 C\u2082) }\n[PROOFSTEP]\nobtain \u27e8f, hf\u27e9 := f\n[GOAL]\ncase mk\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh : C\u2081 \u222a C\u2082 = univ\ng : LocallyConstant (\u2191C\u2082) Z\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf : IsLocallyConstant f\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun g { val := x, property := (_ : x \u2208 C\u2082) }\n\u22a2 IsLocallyConstant fun i =>\n if hi : i \u2208 C\u2081 then \u2191{ toFun := f, isLocallyConstant := hf } { val := i, property := hi }\n else \u2191g { val := i, property := (_ : i \u2208 C\u2082) }\n[PROOFSTEP]\nobtain \u27e8g, hg\u27e9 := g\n[GOAL]\ncase mk.mk\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh : C\u2081 \u222a C\u2082 = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf : IsLocallyConstant f\ng : \u2191C\u2082 \u2192 Z\nhg : IsLocallyConstant g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg } { val := x, property := (_ : x \u2208 C\u2082) }\n\u22a2 IsLocallyConstant fun i =>\n if hi : i \u2208 C\u2081 then \u2191{ toFun := f, isLocallyConstant := hf } { val := i, property := hi }\n else \u2191{ toFun := g, isLocallyConstant := hg } { val := i, property := (_ : i \u2208 C\u2082) }\n[PROOFSTEP]\nrw [IsLocallyConstant.iff_continuous] at hf hg \u22a2\n[GOAL]\ncase mk.mk\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh : C\u2081 \u222a C\u2082 = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\n\u22a2 Continuous fun i =>\n if hi : i \u2208 C\u2081 then \u2191{ toFun := f, isLocallyConstant := hf\u271d } { val := i, property := hi }\n else \u2191{ toFun := g, isLocallyConstant := hg\u271d } { val := i, property := (_ : i \u2208 C\u2082) }\n[PROOFSTEP]\ndsimp only [coe_mk]\n[GOAL]\ncase mk.mk\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh : C\u2081 \u222a C\u2082 = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\n\u22a2 Continuous fun i => if hi : i \u2208 C\u2081 then f { val := i, property := hi } else g { val := i, property := (_ : i \u2208 C\u2082) }\n[PROOFSTEP]\nrw [Set.union_eq_iUnion] at h \n[GOAL]\ncase mk.mk\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh\u271d : C\u2081 \u222a C\u2082 = univ\nh : (\u22c3 (b : Bool), bif b then C\u2081 else C\u2082) = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\n\u22a2 Continuous fun i => if hi : i \u2208 C\u2081 then f { val := i, property := hi } else g { val := i, property := (_ : i \u2208 C\u2082) }\n[PROOFSTEP]\nrefine' (locallyFinite_of_finite _).continuous h (fun i \u21a6 _) (fun i \u21a6 _)\n[GOAL]\ncase mk.mk.refine'_1\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh\u271d : C\u2081 \u222a C\u2082 = univ\nh : (\u22c3 (b : Bool), bif b then C\u2081 else C\u2082) = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\ni : Bool\n\u22a2 IsClosed (bif i then C\u2081 else C\u2082)\n[PROOFSTEP]\ncases i <;> [exact h\u2082; exact h\u2081]\n[GOAL]\ncase mk.mk.refine'_1\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh\u271d : C\u2081 \u222a C\u2082 = univ\nh : (\u22c3 (b : Bool), bif b then C\u2081 else C\u2082) = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\ni : Bool\n\u22a2 IsClosed (bif i then C\u2081 else C\u2082)\n[PROOFSTEP]\ncases i\n[GOAL]\ncase mk.mk.refine'_1.false\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh\u271d : C\u2081 \u222a C\u2082 = univ\nh : (\u22c3 (b : Bool), bif b then C\u2081 else C\u2082) = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\n\u22a2 IsClosed (bif false then C\u2081 else C\u2082)\n[PROOFSTEP]\nexact h\u2082\n[GOAL]\ncase mk.mk.refine'_1.true\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh\u271d : C\u2081 \u222a C\u2082 = univ\nh : (\u22c3 (b : Bool), bif b then C\u2081 else C\u2082) = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\n\u22a2 IsClosed (bif true then C\u2081 else C\u2082)\n[PROOFSTEP]\nexact h\u2081\n[GOAL]\ncase mk.mk.refine'_2\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh\u271d : C\u2081 \u222a C\u2082 = univ\nh : (\u22c3 (b : Bool), bif b then C\u2081 else C\u2082) = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\ni : Bool\n\u22a2 ContinuousOn\n (fun i => if hi : i \u2208 C\u2081 then f { val := i, property := hi } else g { val := i, property := (_ : i \u2208 C\u2082) })\n (bif i then C\u2081 else C\u2082)\n[PROOFSTEP]\ncases i\n[GOAL]\ncase mk.mk.refine'_2.false\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh\u271d : C\u2081 \u222a C\u2082 = univ\nh : (\u22c3 (b : Bool), bif b then C\u2081 else C\u2082) = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\n\u22a2 ContinuousOn\n (fun i => if hi : i \u2208 C\u2081 then f { val := i, property := hi } else g { val := i, property := (_ : i \u2208 C\u2082) })\n (bif false then C\u2081 else C\u2082)\n[PROOFSTEP]\nrw [continuousOn_iff_continuous_restrict]\n[GOAL]\ncase mk.mk.refine'_2.true\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh\u271d : C\u2081 \u222a C\u2082 = univ\nh : (\u22c3 (b : Bool), bif b then C\u2081 else C\u2082) = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\n\u22a2 ContinuousOn\n (fun i => if hi : i \u2208 C\u2081 then f { val := i, property := hi } else g { val := i, property := (_ : i \u2208 C\u2082) })\n (bif true then C\u2081 else C\u2082)\n[PROOFSTEP]\nrw [continuousOn_iff_continuous_restrict]\n[GOAL]\ncase mk.mk.refine'_2.false\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh\u271d : C\u2081 \u222a C\u2082 = univ\nh : (\u22c3 (b : Bool), bif b then C\u2081 else C\u2082) = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\n\u22a2 Continuous\n (restrict (bif false then C\u2081 else C\u2082) fun i =>\n if hi : i \u2208 C\u2081 then f { val := i, property := hi } else g { val := i, property := (_ : i \u2208 C\u2082) })\n[PROOFSTEP]\nconvert hg\n[GOAL]\ncase h.e'_5.h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh\u271d : C\u2081 \u222a C\u2082 = univ\nh : (\u22c3 (b : Bool), bif b then C\u2081 else C\u2082) = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\ne_1\u271d : \u2191(bif false then C\u2081 else C\u2082) = \u2191C\u2082\n\u22a2 (restrict (bif false then C\u2081 else C\u2082) fun i =>\n if hi : i \u2208 C\u2081 then f { val := i, property := hi } else g { val := i, property := (_ : i \u2208 C\u2082) }) =\n g\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_5.h.h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh\u271d : C\u2081 \u222a C\u2082 = univ\nh : (\u22c3 (b : Bool), bif b then C\u2081 else C\u2082) = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\ne_1\u271d : \u2191(bif false then C\u2081 else C\u2082) = \u2191C\u2082\nx : \u2191(bif false then C\u2081 else C\u2082)\n\u22a2 restrict (bif false then C\u2081 else C\u2082)\n (fun i => if hi : i \u2208 C\u2081 then f { val := i, property := hi } else g { val := i, property := (_ : i \u2208 C\u2082) }) x =\n g x\n[PROOFSTEP]\nsimp only [cond_false, restrict_apply, Subtype.coe_eta, dite_eq_right_iff]\n[GOAL]\ncase h.e'_5.h.h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh\u271d : C\u2081 \u222a C\u2082 = univ\nh : (\u22c3 (b : Bool), bif b then C\u2081 else C\u2082) = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\ne_1\u271d : \u2191(bif false then C\u2081 else C\u2082) = \u2191C\u2082\nx : \u2191(bif false then C\u2081 else C\u2082)\n\u22a2 \u2200 (h : \u2191x \u2208 C\u2081), f { val := \u2191x, property := (_ : \u2191x \u2208 C\u2081) } = g x\n[PROOFSTEP]\nexact fun hx \u21a6 hfg x \u27e8hx, x.prop\u27e9\n[GOAL]\ncase mk.mk.refine'_2.true\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh\u271d : C\u2081 \u222a C\u2082 = univ\nh : (\u22c3 (b : Bool), bif b then C\u2081 else C\u2082) = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\n\u22a2 Continuous\n (restrict (bif true then C\u2081 else C\u2082) fun i =>\n if hi : i \u2208 C\u2081 then f { val := i, property := hi } else g { val := i, property := (_ : i \u2208 C\u2082) })\n[PROOFSTEP]\nsimp only [cond_true, restrict_dite, Subtype.coe_eta]\n[GOAL]\ncase mk.mk.refine'_2.true\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace X\nC\u2081 C\u2082 : Set X\nh\u2081 : IsClosed C\u2081\nh\u2082 : IsClosed C\u2082\nh\u271d : C\u2081 \u222a C\u2082 = univ\nh : (\u22c3 (b : Bool), bif b then C\u2081 else C\u2082) = univ\ninst\u271d : (j : X) \u2192 Decidable (j \u2208 C\u2081)\ndZ : TopologicalSpace Z := \u22a5\nthis : DiscreteTopology Z\nf : \u2191C\u2081 \u2192 Z\nhf\u271d : IsLocallyConstant f\nhf : Continuous f\ng : \u2191C\u2082 \u2192 Z\nhg\u271d : IsLocallyConstant g\nhg : Continuous g\nhfg :\n \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082),\n LocallyConstant.toFun { toFun := f, isLocallyConstant := hf\u271d } { val := x, property := (_ : x \u2208 C\u2081) } =\n LocallyConstant.toFun { toFun := g, isLocallyConstant := hg\u271d } { val := x, property := (_ : x \u2208 C\u2082) }\n\u22a2 Continuous fun a => f a\n[PROOFSTEP]\nexact hf\n", "meta": {"mathlib_filename": "Mathlib.Topology.LocallyConstant.Basic", "llama_tokens": 26053, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.4204024477122627}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\np : X = Y\n\u22a2 X \u27f6 Y\n[PROOFSTEP]\nrw [p]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\np : X = Y\n\u22a2 Y \u27f6 Y\n[PROOFSTEP]\nexact \ud835\udfd9 _\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\np : X = Y\nq : Y = Z\n\u22a2 eqToHom p \u226b eqToHom q = eqToHom (_ : X = Z)\n[PROOFSTEP]\ncases p\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Z : C\nq : X = Z\n\u22a2 eqToHom (_ : X = X) \u226b eqToHom q = eqToHom (_ : X = Z)\n[PROOFSTEP]\ncases q\n[GOAL]\ncase refl.refl\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\n\u22a2 eqToHom (_ : X = X) \u226b eqToHom (_ : X = X) = eqToHom (_ : X = X)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Y' : C\np : Y = Y'\nf : X \u27f6 Y\ng : X \u27f6 Y'\nh : f \u226b eqToHom p = g\n\u22a2 f = (f \u226b eqToHom p) \u226b eqToHom (_ : Y' = Y)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Y' : C\np : Y = Y'\nf : X \u27f6 Y\ng : X \u27f6 Y'\nh : f = g \u226b eqToHom (_ : Y' = Y)\n\u22a2 f \u226b eqToHom p = g\n[PROOFSTEP]\nsimp [eq_whisker h (eqToHom p)]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX X' Y : C\np : X = X'\nf : X \u27f6 Y\ng : X' \u27f6 Y\nh : eqToHom p \u226b g = f\n\u22a2 g = eqToHom (_ : X' = X) \u226b eqToHom p \u226b g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX X' Y : C\np : X = X'\nf : X \u27f6 Y\ng : X' \u27f6 Y\nh : g = eqToHom (_ : X' = X) \u226b f\n\u22a2 eqToHom p \u226b eqToHom (_ : X' = X) \u226b f = f\n[PROOFSTEP]\nsimp [whisker_eq _ h]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort ?u.2415\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u27f6 g b\nj j' : \u03b2\nw : j = j'\n\u22a2 g j = g j'\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort ?u.2415\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u27f6 g b\nj j' : \u03b2\nw : j = j'\n\u22a2 f j = f j'\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u27f6 g b\nj j' : \u03b2\nw : j = j'\n\u22a2 z j \u226b eqToHom (_ : g j = g j') = eqToHom (_ : f j = f j') \u226b z j'\n[PROOFSTEP]\ncases w\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u27f6 g b\nj : \u03b2\n\u22a2 z j \u226b eqToHom (_ : g j = g j) = eqToHom (_ : f j = f j) \u226b z j\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort ?u.3466\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u2245 g b\nj j' : \u03b2\nw : j = j'\n\u22a2 g j = g j'\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort ?u.3466\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u2245 g b\nj j' : \u03b2\nw : j = j'\n\u22a2 f j = f j'\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u2245 g b\nj j' : \u03b2\nw : j = j'\n\u22a2 (z j).hom \u226b eqToHom (_ : g j = g j') = eqToHom (_ : f j = f j') \u226b (z j').hom\n[PROOFSTEP]\ncases w\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u2245 g b\nj : \u03b2\n\u22a2 (z j).hom \u226b eqToHom (_ : g j = g j) = eqToHom (_ : f j = f j) \u226b (z j).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort ?u.4552\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u2245 g b\nj j' : \u03b2\nw : j = j'\n\u22a2 f j = f j'\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort ?u.4552\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u2245 g b\nj j' : \u03b2\nw : j = j'\n\u22a2 g j = g j'\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u2245 g b\nj j' : \u03b2\nw : j = j'\n\u22a2 (z j).inv \u226b eqToHom (_ : f j = f j') = eqToHom (_ : g j = g j') \u226b (z j').inv\n[PROOFSTEP]\ncases w\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u2245 g b\nj : \u03b2\n\u22a2 (z j).inv \u226b eqToHom (_ : f j = f j) = eqToHom (_ : g j = g j) \u226b (z j).inv\n[PROOFSTEP]\nsimp\n /- Porting note: simpNF complains about this not reducing but it is clearly used\n in `congrArg_mpr_hom_left`. It has been no-linted. -/\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\np : X = Y\nq : Y \u27f6 Z\n\u22a2 cast (_ : (Y \u27f6 Z) = (X \u27f6 Z)) q = eqToHom p \u226b q\n[PROOFSTEP]\ncases p\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Z : C\nq : X \u27f6 Z\n\u22a2 cast (_ : (X \u27f6 Z) = (X \u27f6 Z)) q = eqToHom (_ : X = X) \u226b q\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\np : X = Y\nq : Y \u27f6 Z\n\u22a2 Eq.mpr (_ : (X \u27f6 Z) = (Y \u27f6 Z)) q = eqToHom p \u226b q\n[PROOFSTEP]\ncases p\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Z : C\nq : X \u27f6 Z\n\u22a2 Eq.mpr (_ : (X \u27f6 Z) = (X \u27f6 Z)) q = eqToHom (_ : X = X) \u226b q\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\np : X \u27f6 Y\nq : Z = Y\n\u22a2 cast (_ : (X \u27f6 Y) = (X \u27f6 Z)) p = p \u226b eqToHom (_ : Y = Z)\n[PROOFSTEP]\ncases q\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\np : X \u27f6 Y\n\u22a2 cast (_ : (X \u27f6 Y) = (X \u27f6 Y)) p = p \u226b eqToHom (_ : Y = Y)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\np : X \u27f6 Y\nq : Z = Y\n\u22a2 Eq.mpr (_ : (X \u27f6 Z) = (X \u27f6 Y)) p = p \u226b eqToHom (_ : Y = Z)\n[PROOFSTEP]\ncases q\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\np : X \u27f6 Y\n\u22a2 Eq.mpr (_ : (X \u27f6 Y) = (X \u27f6 Y)) p = p \u226b eqToHom (_ : Y = Y)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\np : X = Y\n\u22a2 eqToHom p \u226b eqToHom (_ : Y = X) = \ud835\udfd9 X\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\np : X = Y\n\u22a2 eqToHom (_ : Y = X) \u226b eqToHom p = \ud835\udfd9 Y\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\np : X = Y\nq : Y = Z\n\u22a2 eqToIso p \u226a\u226b eqToIso q = eqToIso (_ : X = Z)\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\np : X = Y\nq : Y = Z\n\u22a2 (eqToIso p \u226a\u226b eqToIso q).hom = (eqToIso (_ : X = Z)).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\nh : X = Y\n\u22a2 (eqToHom h).op = eqToHom (_ : op Y = op X)\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\n\u22a2 (eqToHom (_ : X = X)).op = eqToHom (_ : op X = op X)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\u1d52\u1d56\nh : X = Y\n\u22a2 (eqToHom h).unop = eqToHom (_ : Y.unop = X.unop)\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\u1d52\u1d56\n\u22a2 (eqToHom (_ : X = X)).unop = eqToHom (_ : X.unop = X.unop)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\nh : X = Y\n\u22a2 inv (eqToHom h) = eqToHom (_ : Y = X)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nh_obj : \u2200 (X : C), F.obj X = G.obj X\nh_map :\n autoParam\n (\u2200 (X Y : C) (f : X \u27f6 Y), F.map f = eqToHom (_ : F.obj X = G.obj X) \u226b G.map f \u226b eqToHom (_ : G.obj Y = F.obj Y))\n _auto\u271d\n\u22a2 F = G\n[PROOFSTEP]\nmatch F, G with\n| mk F_pre _ _, mk G_pre _ _ =>\n match F_pre, G_pre with\n -- Porting note: did not unfold the Prefunctor unlike Lean3\n | Prefunctor.mk F_obj _,\n Prefunctor.mk G_obj\n _ =>\n obtain rfl : F_obj = G_obj := by\n ext X\n apply h_obj\n congr\n funext X Y f\n simpa using h_map X Y f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nF_pre : C \u2964q D\nmap_id\u271d\u00b9 : \u2200 (X : C), F_pre.map (\ud835\udfd9 X) = \ud835\udfd9 (F_pre.obj X)\nmap_comp\u271d\u00b9 : \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z), F_pre.map (f \u226b g) = F_pre.map f \u226b F_pre.map g\nG_pre : C \u2964q D\nmap_id\u271d : \u2200 (X : C), G_pre.map (\ud835\udfd9 X) = \ud835\udfd9 (G_pre.obj X)\nmap_comp\u271d : \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z), G_pre.map (f \u226b g) = G_pre.map f \u226b G_pre.map g\nh_obj : \u2200 (X : C), (mk F_pre).obj X = (mk G_pre).obj X\nh_map :\n autoParam\n (\u2200 (X Y : C) (f : X \u27f6 Y),\n (mk F_pre).map f =\n eqToHom (_ : (mk F_pre).obj X = (mk G_pre).obj X) \u226b\n (mk G_pre).map f \u226b eqToHom (_ : (mk G_pre).obj Y = (mk F_pre).obj Y))\n _auto\u271d\n\u22a2 mk F_pre = mk G_pre\n[PROOFSTEP]\nmatch F_pre, G_pre with\n -- Porting note: did not unfold the Prefunctor unlike Lean3\n| Prefunctor.mk F_obj _,\n Prefunctor.mk G_obj\n _ =>\n obtain rfl : F_obj = G_obj := by\n ext X\n apply h_obj\n congr\n funext X Y f\n simpa using h_map X Y f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nF_pre G_pre : C \u2964q D\nF_obj : C \u2192 D\nmap\u271d\u00b9 : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (F_obj X \u27f6 F_obj Y)\nG_obj : C \u2192 D\nmap\u271d : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (G_obj X \u27f6 G_obj Y)\nmap_id\u271d\u00b9 : \u2200 (X : C), { obj := F_obj, map := map\u271d\u00b9 }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := F_obj, map := map\u271d\u00b9 }.obj X)\nmap_comp\u271d\u00b9 :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := F_obj, map := map\u271d\u00b9 }.map (f \u226b g) =\n { obj := F_obj, map := map\u271d\u00b9 }.map f \u226b { obj := F_obj, map := map\u271d\u00b9 }.map g\nmap_id\u271d : \u2200 (X : C), { obj := G_obj, map := map\u271d }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := G_obj, map := map\u271d }.obj X)\nmap_comp\u271d :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := G_obj, map := map\u271d }.map (f \u226b g) =\n { obj := G_obj, map := map\u271d }.map f \u226b { obj := G_obj, map := map\u271d }.map g\nh_obj : \u2200 (X : C), (mk { obj := F_obj, map := map\u271d\u00b9 }).obj X = (mk { obj := G_obj, map := map\u271d }).obj X\nh_map :\n autoParam\n (\u2200 (X Y : C) (f : X \u27f6 Y),\n (mk { obj := F_obj, map := map\u271d\u00b9 }).map f =\n eqToHom (_ : (mk { obj := F_obj, map := map\u271d\u00b9 }).obj X = (mk { obj := G_obj, map := map\u271d }).obj X) \u226b\n (mk { obj := G_obj, map := map\u271d }).map f \u226b\n eqToHom (_ : (mk { obj := G_obj, map := map\u271d }).obj Y = (mk { obj := F_obj, map := map\u271d\u00b9 }).obj Y))\n _auto\u271d\n\u22a2 mk { obj := F_obj, map := map\u271d\u00b9 } = mk { obj := G_obj, map := map\u271d }\n[PROOFSTEP]\nobtain rfl : F_obj = G_obj := by\n ext X\n apply h_obj\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nF_pre G_pre : C \u2964q D\nF_obj : C \u2192 D\nmap\u271d\u00b9 : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (F_obj X \u27f6 F_obj Y)\nG_obj : C \u2192 D\nmap\u271d : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (G_obj X \u27f6 G_obj Y)\nmap_id\u271d\u00b9 : \u2200 (X : C), { obj := F_obj, map := map\u271d\u00b9 }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := F_obj, map := map\u271d\u00b9 }.obj X)\nmap_comp\u271d\u00b9 :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := F_obj, map := map\u271d\u00b9 }.map (f \u226b g) =\n { obj := F_obj, map := map\u271d\u00b9 }.map f \u226b { obj := F_obj, map := map\u271d\u00b9 }.map g\nmap_id\u271d : \u2200 (X : C), { obj := G_obj, map := map\u271d }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := G_obj, map := map\u271d }.obj X)\nmap_comp\u271d :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := G_obj, map := map\u271d }.map (f \u226b g) =\n { obj := G_obj, map := map\u271d }.map f \u226b { obj := G_obj, map := map\u271d }.map g\nh_obj : \u2200 (X : C), (mk { obj := F_obj, map := map\u271d\u00b9 }).obj X = (mk { obj := G_obj, map := map\u271d }).obj X\nh_map :\n autoParam\n (\u2200 (X Y : C) (f : X \u27f6 Y),\n (mk { obj := F_obj, map := map\u271d\u00b9 }).map f =\n eqToHom (_ : (mk { obj := F_obj, map := map\u271d\u00b9 }).obj X = (mk { obj := G_obj, map := map\u271d }).obj X) \u226b\n (mk { obj := G_obj, map := map\u271d }).map f \u226b\n eqToHom (_ : (mk { obj := G_obj, map := map\u271d }).obj Y = (mk { obj := F_obj, map := map\u271d\u00b9 }).obj Y))\n _auto\u271d\n\u22a2 F_obj = G_obj\n[PROOFSTEP]\next X\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nF_pre G_pre : C \u2964q D\nF_obj : C \u2192 D\nmap\u271d\u00b9 : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (F_obj X \u27f6 F_obj Y)\nG_obj : C \u2192 D\nmap\u271d : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (G_obj X \u27f6 G_obj Y)\nmap_id\u271d\u00b9 : \u2200 (X : C), { obj := F_obj, map := map\u271d\u00b9 }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := F_obj, map := map\u271d\u00b9 }.obj X)\nmap_comp\u271d\u00b9 :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := F_obj, map := map\u271d\u00b9 }.map (f \u226b g) =\n { obj := F_obj, map := map\u271d\u00b9 }.map f \u226b { obj := F_obj, map := map\u271d\u00b9 }.map g\nmap_id\u271d : \u2200 (X : C), { obj := G_obj, map := map\u271d }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := G_obj, map := map\u271d }.obj X)\nmap_comp\u271d :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := G_obj, map := map\u271d }.map (f \u226b g) =\n { obj := G_obj, map := map\u271d }.map f \u226b { obj := G_obj, map := map\u271d }.map g\nh_obj : \u2200 (X : C), (mk { obj := F_obj, map := map\u271d\u00b9 }).obj X = (mk { obj := G_obj, map := map\u271d }).obj X\nh_map :\n autoParam\n (\u2200 (X Y : C) (f : X \u27f6 Y),\n (mk { obj := F_obj, map := map\u271d\u00b9 }).map f =\n eqToHom (_ : (mk { obj := F_obj, map := map\u271d\u00b9 }).obj X = (mk { obj := G_obj, map := map\u271d }).obj X) \u226b\n (mk { obj := G_obj, map := map\u271d }).map f \u226b\n eqToHom (_ : (mk { obj := G_obj, map := map\u271d }).obj Y = (mk { obj := F_obj, map := map\u271d\u00b9 }).obj Y))\n _auto\u271d\nX : C\n\u22a2 F_obj X = G_obj X\n[PROOFSTEP]\napply h_obj\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nF_pre G_pre : C \u2964q D\nF_obj : C \u2192 D\nmap\u271d\u00b9 : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (F_obj X \u27f6 F_obj Y)\nmap_id\u271d\u00b9 : \u2200 (X : C), { obj := F_obj, map := map\u271d\u00b9 }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := F_obj, map := map\u271d\u00b9 }.obj X)\nmap_comp\u271d\u00b9 :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := F_obj, map := map\u271d\u00b9 }.map (f \u226b g) =\n { obj := F_obj, map := map\u271d\u00b9 }.map f \u226b { obj := F_obj, map := map\u271d\u00b9 }.map g\nmap\u271d : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (F_obj X \u27f6 F_obj Y)\nmap_id\u271d : \u2200 (X : C), { obj := F_obj, map := map\u271d }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := F_obj, map := map\u271d }.obj X)\nmap_comp\u271d :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := F_obj, map := map\u271d }.map (f \u226b g) =\n { obj := F_obj, map := map\u271d }.map f \u226b { obj := F_obj, map := map\u271d }.map g\nh_obj : \u2200 (X : C), (mk { obj := F_obj, map := map\u271d\u00b9 }).obj X = (mk { obj := F_obj, map := map\u271d }).obj X\nh_map :\n autoParam\n (\u2200 (X Y : C) (f : X \u27f6 Y),\n (mk { obj := F_obj, map := map\u271d\u00b9 }).map f =\n eqToHom (_ : (mk { obj := F_obj, map := map\u271d\u00b9 }).obj X = (mk { obj := F_obj, map := map\u271d }).obj X) \u226b\n (mk { obj := F_obj, map := map\u271d }).map f \u226b\n eqToHom (_ : (mk { obj := F_obj, map := map\u271d }).obj Y = (mk { obj := F_obj, map := map\u271d\u00b9 }).obj Y))\n _auto\u271d\n\u22a2 mk { obj := F_obj, map := map\u271d\u00b9 } = mk { obj := F_obj, map := map\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_toPrefunctor.e_map\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nF_pre G_pre : C \u2964q D\nF_obj : C \u2192 D\nmap\u271d\u00b9 : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (F_obj X \u27f6 F_obj Y)\nmap_id\u271d\u00b9 : \u2200 (X : C), { obj := F_obj, map := map\u271d\u00b9 }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := F_obj, map := map\u271d\u00b9 }.obj X)\nmap_comp\u271d\u00b9 :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := F_obj, map := map\u271d\u00b9 }.map (f \u226b g) =\n { obj := F_obj, map := map\u271d\u00b9 }.map f \u226b { obj := F_obj, map := map\u271d\u00b9 }.map g\nmap\u271d : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (F_obj X \u27f6 F_obj Y)\nmap_id\u271d : \u2200 (X : C), { obj := F_obj, map := map\u271d }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := F_obj, map := map\u271d }.obj X)\nmap_comp\u271d :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := F_obj, map := map\u271d }.map (f \u226b g) =\n { obj := F_obj, map := map\u271d }.map f \u226b { obj := F_obj, map := map\u271d }.map g\nh_obj : \u2200 (X : C), (mk { obj := F_obj, map := map\u271d\u00b9 }).obj X = (mk { obj := F_obj, map := map\u271d }).obj X\nh_map :\n autoParam\n (\u2200 (X Y : C) (f : X \u27f6 Y),\n (mk { obj := F_obj, map := map\u271d\u00b9 }).map f =\n eqToHom (_ : (mk { obj := F_obj, map := map\u271d\u00b9 }).obj X = (mk { obj := F_obj, map := map\u271d }).obj X) \u226b\n (mk { obj := F_obj, map := map\u271d }).map f \u226b\n eqToHom (_ : (mk { obj := F_obj, map := map\u271d }).obj Y = (mk { obj := F_obj, map := map\u271d\u00b9 }).obj Y))\n _auto\u271d\n\u22a2 map\u271d\u00b9 = map\u271d\n[PROOFSTEP]\nfunext X Y f\n[GOAL]\ncase e_toPrefunctor.e_map.h.h.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nF_pre G_pre : C \u2964q D\nF_obj : C \u2192 D\nmap\u271d\u00b9 : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (F_obj X \u27f6 F_obj Y)\nmap_id\u271d\u00b9 : \u2200 (X : C), { obj := F_obj, map := map\u271d\u00b9 }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := F_obj, map := map\u271d\u00b9 }.obj X)\nmap_comp\u271d\u00b9 :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := F_obj, map := map\u271d\u00b9 }.map (f \u226b g) =\n { obj := F_obj, map := map\u271d\u00b9 }.map f \u226b { obj := F_obj, map := map\u271d\u00b9 }.map g\nmap\u271d : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (F_obj X \u27f6 F_obj Y)\nmap_id\u271d : \u2200 (X : C), { obj := F_obj, map := map\u271d }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := F_obj, map := map\u271d }.obj X)\nmap_comp\u271d :\n \u2200 {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := F_obj, map := map\u271d }.map (f \u226b g) =\n { obj := F_obj, map := map\u271d }.map f \u226b { obj := F_obj, map := map\u271d }.map g\nh_obj : \u2200 (X : C), (mk { obj := F_obj, map := map\u271d\u00b9 }).obj X = (mk { obj := F_obj, map := map\u271d }).obj X\nh_map :\n autoParam\n (\u2200 (X Y : C) (f : X \u27f6 Y),\n (mk { obj := F_obj, map := map\u271d\u00b9 }).map f =\n eqToHom (_ : (mk { obj := F_obj, map := map\u271d\u00b9 }).obj X = (mk { obj := F_obj, map := map\u271d }).obj X) \u226b\n (mk { obj := F_obj, map := map\u271d }).map f \u226b\n eqToHom (_ : (mk { obj := F_obj, map := map\u271d }).obj Y = (mk { obj := F_obj, map := map\u271d\u00b9 }).obj Y))\n _auto\u271d\nX Y : C\nf : X \u27f6 Y\n\u22a2 map\u271d\u00b9 f = map\u271d f\n[PROOFSTEP]\nsimpa using h_map X Y f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Y Z : C\nf : W \u27f6 X\ng : Y \u27f6 Z\nh : W = Y\nh' : X = Z\n\u22a2 f = eqToHom h \u226b g \u226b eqToHom (_ : Z = X) \u2194 HEq f g\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X Z : C\nf : W \u27f6 X\nh' : X = Z\ng : W \u27f6 Z\n\u22a2 f = eqToHom (_ : W = W) \u226b g \u226b eqToHom (_ : Z = X) \u2194 HEq f g\n[PROOFSTEP]\ncases h'\n[GOAL]\ncase refl.refl\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nW X : C\nf g : W \u27f6 X\n\u22a2 f = eqToHom (_ : W = W) \u226b g \u226b eqToHom (_ : X = X) \u2194 HEq f g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nh : F = G\nX : C\n\u22a2 F.obj X = G.obj X\n[PROOFSTEP]\nrw [h]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nh : F = G\nX Y : C\nf : X \u27f6 Y\n\u22a2 F.map f = eqToHom (_ : F.obj X = G.obj X) \u226b G.map f \u226b eqToHom (_ : G.obj Y = F.obj Y)\n[PROOFSTEP]\nsubst h\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\n\u22a2 F.map f = eqToHom (_ : F.obj X = F.obj X) \u226b F.map f \u226b eqToHom (_ : F.obj Y = F.obj Y)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nX Y : C\ne : X \u2245 Y\nhX : F.obj X = G.obj X\nhY : F.obj Y = G.obj Y\n\u22a2 F.obj X = G.obj X\n[PROOFSTEP]\nrw [hX]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nX Y : C\ne : X \u2245 Y\nhX : F.obj X = G.obj X\nhY : F.obj Y = G.obj Y\n\u22a2 G.obj Y = F.obj Y\n[PROOFSTEP]\nrw [hY]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nX Y : C\ne : X \u2245 Y\nhX : F.obj X = G.obj X\nhY : F.obj Y = G.obj Y\nh\u2082 : F.map e.hom = eqToHom (_ : F.obj X = G.obj X) \u226b G.map e.hom \u226b eqToHom (_ : G.obj Y = F.obj Y)\n\u22a2 F.obj Y = G.obj Y\n[PROOFSTEP]\nrw [hY]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nX Y : C\ne : X \u2245 Y\nhX : F.obj X = G.obj X\nhY : F.obj Y = G.obj Y\nh\u2082 : F.map e.hom = eqToHom (_ : F.obj X = G.obj X) \u226b G.map e.hom \u226b eqToHom (_ : G.obj Y = F.obj Y)\n\u22a2 G.obj X = F.obj X\n[PROOFSTEP]\nrw [hX]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nX Y : C\ne : X \u2245 Y\nhX : F.obj X = G.obj X\nhY : F.obj Y = G.obj Y\nh\u2082 : F.map e.hom = eqToHom (_ : F.obj X = G.obj X) \u226b G.map e.hom \u226b eqToHom (_ : G.obj Y = F.obj Y)\n\u22a2 F.map e.inv = eqToHom (_ : F.obj Y = G.obj Y) \u226b G.map e.inv \u226b eqToHom (_ : G.obj X = F.obj X)\n[PROOFSTEP]\nsimp only [\u2190 IsIso.Iso.inv_hom e, Functor.map_inv, h\u2082, IsIso.inv_comp, inv_eqToHom, Category.assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y : C\nf g : X \u27f6 Y\nh : f = g\n\u22a2 F.map f = F.map g\n[PROOFSTEP]\nrw [h]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nF G : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhx : F.obj X = G.obj X\nhy : F.obj Y = G.obj Y\nhz : F.obj Z = G.obj Z\nhf : HEq (F.map f) (G.map f)\nhg : HEq (F.map g) (G.map g)\n\u22a2 HEq (F.map (f \u226b g)) (G.map (f \u226b g))\n[PROOFSTEP]\nrw [F.map_comp, G.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nF G : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhx : F.obj X = G.obj X\nhy : F.obj Y = G.obj Y\nhz : F.obj Z = G.obj Z\nhf : HEq (F.map f) (G.map f)\nhg : HEq (F.map g) (G.map g)\n\u22a2 HEq (F.map f \u226b F.map g) (G.map f \u226b G.map g)\n[PROOFSTEP]\ncongr\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nF G : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhobj : \u2200 (X : C), F.obj X = G.obj X\nhmap : \u2200 {X Y : C} (f : X \u27f6 Y), HEq (F.map f) (G.map f)\n\u22a2 HEq (F.map (f \u226b g)) (G.map (f \u226b g))\n[PROOFSTEP]\nrw [Functor.hext hobj fun _ _ => hmap]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nF G : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nH : D \u2964 E\nhx : F.obj X = G.obj X\nhy : F.obj Y = G.obj Y\nhmap : HEq (F.map f) (G.map f)\n\u22a2 HEq ((F \u22d9 H).map f) ((G \u22d9 H).map f)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nF G : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nH : D \u2964 E\nhx : F.obj X = G.obj X\nhy : F.obj Y = G.obj Y\nhmap : HEq (F.map f) (G.map f)\n\u22a2 HEq (H.map (F.map f)) (H.map (G.map f))\n[PROOFSTEP]\ncongr\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nF G : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nH : D \u2964 E\nhobj : \u2200 (X : C), F.obj X = G.obj X\nhmap : \u2200 {X Y : C} (f : X \u27f6 Y), HEq (F.map f) (G.map f)\n\u22a2 HEq ((F \u22d9 H).map f) ((G \u22d9 H).map f)\n[PROOFSTEP]\nrw [Functor.hext hobj fun _ _ => hmap]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nF\u271d G\u271d : C \u2964 D\nX\u271d Y\u271d Z : C\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nF G : C \u2964 D\nh : F = G\nX Y : C\nf : X \u27f6 Y\n\u22a2 HEq (F.map f) (G.map f)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y : C\np : X = Y\n\u22a2 F.map (eqToHom p) = eqToHom (_ : F.obj X = F.obj Y)\n[PROOFSTEP]\ncases p\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX : C\n\u22a2 F.map (eqToHom (_ : X = X)) = eqToHom (_ : F.obj X = F.obj X)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y : C\np : X = Y\n\u22a2 F.mapIso (eqToIso p) = eqToIso (_ : F.obj X = F.obj Y)\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y : C\np : X = Y\n\u22a2 (F.mapIso (eqToIso p)).hom = (eqToIso (_ : F.obj X = F.obj Y)).hom\n[PROOFSTEP]\ncases p\n[GOAL]\ncase w.refl\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX : C\n\u22a2 (F.mapIso (eqToIso (_ : X = X))).hom = (eqToIso (_ : F.obj X = F.obj X)).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\nh : F = G\nX : C\n\u22a2 NatTrans.app (eqToHom h) X = eqToHom (_ : F.obj X = G.obj X)\n[PROOFSTEP]\nsubst h\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX : C\n\u22a2 NatTrans.app (eqToHom (_ : F = F)) X = eqToHom (_ : F.obj X = F.obj X)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX Y : C\nh : X = Y\n\u22a2 app \u03b1 X = F.map (eqToHom h) \u226b app \u03b1 Y \u226b G.map (eqToHom (_ : Y = X))\n[PROOFSTEP]\nrw [\u03b1.naturality_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX Y : C\nh : X = Y\n\u22a2 app \u03b1 X = app \u03b1 X \u226b G.map (eqToHom h) \u226b G.map (eqToHom (_ : Y = X))\n[PROOFSTEP]\nsimp [eqToHom_map]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX Y : C\nf : X \u27f6 Y\n\u22a2 f = eqToHom (_ : X = X) \u226b f \u226b eqToHom (_ : Y = Y)\n[PROOFSTEP]\nsimp only [Category.id_comp, eqToHom_refl, Category.comp_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\n\u03b9 : Type u_1\nF G : \u03b9 \u2192 C\n\u03b1 : (i : \u03b9) \u2192 F i \u27f6 G i\ni j : \u03b9\nh : i = j\n\u22a2 \u03b1 i = eqToHom (_ : F i = F j) \u226b \u03b1 j \u226b eqToHom (_ : G j = G i)\n[PROOFSTEP]\nsubst h\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\n\u03b9 : Type u_1\nF G : \u03b9 \u2192 C\n\u03b1 : (i : \u03b9) \u2192 F i \u27f6 G i\ni : \u03b9\n\u22a2 \u03b1 i = eqToHom (_ : F i = F i) \u226b \u03b1 i \u226b eqToHom (_ : G i = G i)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.EqToHom", "llama_tokens": 13180, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754371026368, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.4204024408958882}} {"text": "[GOAL]\nR : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nJ : Ideal R\nhJ : J \u2264 supp v\nq : R \u29f8 J\na b : R\nh : Setoid.r a b\n\u22a2 \u2191v a = \u2191v (b + -(-a + b))\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393\u2080\nv\u271d : Valuation R \u0393\u2080\nJ : Ideal R\nv : Valuation (R \u29f8 J) \u0393\u2080\n\u22a2 J \u2264 supp (comap (Ideal.Quotient.mk J) v)\n[PROOFSTEP]\nrw [comap_supp, \u2190 Ideal.map_le_iff_le_comap]\n[GOAL]\nR : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393\u2080\nv\u271d : Valuation R \u0393\u2080\nJ : Ideal R\nv : Valuation (R \u29f8 J) \u0393\u2080\n\u22a2 Ideal.map (Ideal.Quotient.mk J) J \u2264 supp v\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393\u2080\nv\u271d : Valuation R \u0393\u2080\nJ : Ideal R\nv : Valuation (R \u29f8 J) \u0393\u2080\n\u22a2 \u2200 (r : R \u29f8 J), \u2191(onQuot (comap (Ideal.Quotient.mk J) v) (_ : J \u2264 supp (comap (Ideal.Quotient.mk J) v))) r = \u2191v r\n[PROOFSTEP]\nrintro \u27e8x\u27e9\n[GOAL]\ncase mk\nR : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393\u2080\nv\u271d : Valuation R \u0393\u2080\nJ : Ideal R\nv : Valuation (R \u29f8 J) \u0393\u2080\nr\u271d : R \u29f8 J\nx : R\n\u22a2 \u2191(onQuot (comap (Ideal.Quotient.mk J) v) (_ : J \u2264 supp (comap (Ideal.Quotient.mk J) v))) (Quot.mk Setoid.r x) =\n \u2191v (Quot.mk Setoid.r x)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nJ : Ideal R\nhJ : J \u2264 supp v\n\u22a2 supp (onQuot v hJ) = Ideal.map (Ideal.Quotient.mk J) (supp v)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nR : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nJ : Ideal R\nhJ : J \u2264 supp v\n\u22a2 supp (onQuot v hJ) \u2264 Ideal.map (Ideal.Quotient.mk J) (supp v)\n[PROOFSTEP]\nrintro \u27e8x\u27e9 hx\n[GOAL]\ncase a.mk\nR : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nJ : Ideal R\nhJ : J \u2264 supp v\nx\u271d : R \u29f8 J\nx : R\nhx : Quot.mk Setoid.r x \u2208 supp (onQuot v hJ)\n\u22a2 Quot.mk Setoid.r x \u2208 Ideal.map (Ideal.Quotient.mk J) (supp v)\n[PROOFSTEP]\napply Ideal.subset_span\n[GOAL]\ncase a.mk.a\nR : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nJ : Ideal R\nhJ : J \u2264 supp v\nx\u271d : R \u29f8 J\nx : R\nhx : Quot.mk Setoid.r x \u2208 supp (onQuot v hJ)\n\u22a2 Quot.mk Setoid.r x \u2208 \u2191(Ideal.Quotient.mk J) '' \u2191(supp v)\n[PROOFSTEP]\nexact \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase a\nR : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nJ : Ideal R\nhJ : J \u2264 supp v\n\u22a2 Ideal.map (Ideal.Quotient.mk J) (supp v) \u2264 supp (onQuot v hJ)\n[PROOFSTEP]\nrw [Ideal.map_le_iff_le_comap]\n[GOAL]\ncase a\nR : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nJ : Ideal R\nhJ : J \u2264 supp v\n\u22a2 supp v \u2264 Ideal.comap (Ideal.Quotient.mk J) (supp (onQuot v hJ))\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase a\nR : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nJ : Ideal R\nhJ : J \u2264 supp v\nx : R\nhx : x \u2208 supp v\n\u22a2 x \u2208 Ideal.comap (Ideal.Quotient.mk J) (supp (onQuot v hJ))\n[PROOFSTEP]\nexact hx\n[GOAL]\nR : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\n\u22a2 supp (onQuot v (_ : supp v \u2264 supp v)) = 0\n[PROOFSTEP]\nrw [supp_quot]\n[GOAL]\nR : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LinearOrderedCommMonoidWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\n\u22a2 Ideal.map (Ideal.Quotient.mk (supp v)) (supp v) = 0\n[PROOFSTEP]\nexact Ideal.map_quotient_self _\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Valuation.Quotient", "llama_tokens": 1914, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737963569016, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.42039750874564147}} {"text": "[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\n\u22a2 multinomial \u2205 f = 1\n[PROOFSTEP]\ndsimp [multinomial]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\n\u22a2 1 / 1 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\n\u22a2 multinomial {a} f = 1\n[PROOFSTEP]\nsimp [multinomial, Nat.div_self (factorial_pos (f a))]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : \u00aca \u2208 s\nh\u2081 : f a = 1\n\u22a2 multinomial (insert a s) f = succ (Finset.sum s f) * multinomial s f\n[PROOFSTEP]\nsimp only [multinomial, one_mul, factorial]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : \u00aca \u2208 s\nh\u2081 : f a = 1\n\u22a2 (\u2211 i in insert a s, f i)! / \u220f i in insert a s, (f i)! = succ (Finset.sum s f) * ((\u2211 i in s, f i)! / \u220f i in s, (f i)!)\n[PROOFSTEP]\nrw [Finset.sum_insert h, Finset.prod_insert h, h\u2081, add_comm, \u2190 succ_eq_add_one, factorial_succ]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : \u00aca \u2208 s\nh\u2081 : f a = 1\n\u22a2 (\u2211 x in s, f x + 1) * (\u2211 x in s, f x)! / (1! * \u220f x in s, (f x)!) =\n succ (Finset.sum s f) * ((\u2211 i in s, f i)! / \u220f i in s, (f i)!)\n[PROOFSTEP]\nsimp only [factorial_one, one_mul, Function.comp_apply, factorial, mul_one, \u2190 one_eq_succ_zero]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : \u00aca \u2208 s\nh\u2081 : f a = 1\n\u22a2 (\u2211 x in s, f x + 1) * (\u2211 x in s, f x)! / \u220f x in s, (f x)! =\n succ (Finset.sum s f) * ((\u2211 x in s, f x)! / \u220f x in s, (f x)!)\n[PROOFSTEP]\nrw [Nat.mul_div_assoc _ (prod_factorial_dvd_factorial_sum _ _)]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : \u00aca \u2208 s\n\u22a2 multinomial (insert a s) f = choose (f a + Finset.sum s f) (f a) * multinomial s f\n[PROOFSTEP]\nrw [choose_eq_factorial_div_factorial (le.intro rfl)]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : \u00aca \u2208 s\n\u22a2 multinomial (insert a s) f = (f a + Finset.sum s f)! / ((f a)! * (f a + Finset.sum s f - f a)!) * multinomial s f\n[PROOFSTEP]\nsimp only [multinomial, Nat.add_sub_cancel_left, Finset.sum_insert h, Finset.prod_insert h, Function.comp_apply]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : \u00aca \u2208 s\n\u22a2 (f a + \u2211 i in s, f i)! / ((f a)! * \u220f i in s, (f i)!) =\n (f a + Finset.sum s f)! / ((f a)! * (Finset.sum s f)!) * ((\u2211 i in s, f i)! / \u220f i in s, (f i)!)\n[PROOFSTEP]\nrw [div_mul_div_comm ((f a).factorial_mul_factorial_dvd_factorial_add (s.sum f)) (prod_factorial_dvd_factorial_sum _ _),\n mul_comm (f a)! (s.sum f)!, mul_assoc, mul_comm _ (s.sum f)!, Nat.mul_div_mul_left _ _ (factorial_pos _)]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\nf g : \u03b1 \u2192 \u2115\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 f a = g a\n\u22a2 multinomial s f = multinomial s g\n[PROOFSTEP]\nsimp only [multinomial]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\nf g : \u03b1 \u2192 \u2115\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 f a = g a\n\u22a2 (\u2211 i in s, f i)! / \u220f i in s, (f i)! = (\u2211 i in s, g i)! / \u220f i in s, (g i)!\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\nf g : \u03b1 \u2192 \u2115\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 f a = g a\n\u22a2 (\u2211 i in s, f i)! = (\u2211 i in s, g i)!\n[PROOFSTEP]\nrw [Finset.sum_congr rfl h]\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\nf g : \u03b1 \u2192 \u2115\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 f a = g a\n\u22a2 \u220f i in s, (f i)! = \u220f i in s, (g i)!\n[PROOFSTEP]\nexact Finset.prod_congr rfl fun a ha => by rw [h a ha]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u2115\na\u271d b : \u03b1\nn : \u2115\nf g : \u03b1 \u2192 \u2115\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 f a = g a\na : \u03b1\nha : a \u2208 s\n\u22a2 (f a)! = (g a)!\n[PROOFSTEP]\nrw [h a ha]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : a \u2260 b\n\u22a2 multinomial {a, b} f = (f a + f b)! / ((f a)! * (f b)!)\n[PROOFSTEP]\nsimp [multinomial, Finset.sum_pair h, Finset.prod_pair h]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : a \u2260 b\n\u22a2 multinomial {a, b} f = choose (f a + f b) (f a)\n[PROOFSTEP]\nsimp [binomial_eq _ h, choose_eq_factorial_div_factorial (Nat.le_add_right _ _)]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nhab : a \u2260 b\n\u22a2 (f a)! * (f b)! * multinomial {a, b} f = (f a + f b)!\n[PROOFSTEP]\nsimpa [Finset.sum_pair hab, Finset.prod_pair hab] using multinomial_spec { a, b } f\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : a \u2260 b\nh\u2081 : f a = 1\n\u22a2 multinomial {a, b} f = succ (f b)\n[PROOFSTEP]\nsimp [multinomial_insert_one { b } f (Finset.not_mem_singleton.mpr h) h\u2081]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : a \u2260 b\n\u22a2 multinomial {a, b} (Function.update (Function.update f a (succ (f a))) b (succ (f b))) =\n multinomial {a, b} (Function.update f a (succ (f a))) + multinomial {a, b} (Function.update f b (succ (f b)))\n[PROOFSTEP]\nsimp only [binomial_eq_choose, Function.update_apply, h, Ne.def, ite_true, ite_false]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : a \u2260 b\n\u22a2 choose (succ (f a) + succ (f b)) (succ (f a)) =\n choose (succ (f a) + if b = a then succ (f a) else f b) (succ (f a)) + choose (f a + succ (f b)) (f a)\n[PROOFSTEP]\nrw [if_neg h.symm]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : a \u2260 b\n\u22a2 choose (succ (f a) + succ (f b)) (succ (f a)) =\n choose (succ (f a) + f b) (succ (f a)) + choose (f a + succ (f b)) (f a)\n[PROOFSTEP]\nrw [add_succ, choose_succ_succ, succ_add_eq_succ_add]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : a \u2260 b\n\u22a2 choose (f a + succ (f b)) (f a) + choose (f a + succ (f b)) (succ (f a)) =\n choose (f a + succ (f b)) (succ (f a)) + choose (f a + succ (f b)) (f a)\n[PROOFSTEP]\nring\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : a \u2260 b\n\u22a2 succ (f a + f b) * multinomial {a, b} f = succ (f a) * multinomial {a, b} (Function.update f a (succ (f a)))\n[PROOFSTEP]\nrw [binomial_eq_choose _ h, binomial_eq_choose _ h, mul_comm (f a).succ, Function.update_same,\n Function.update_noteq (ne_comm.mp h)]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : a \u2260 b\n\u22a2 succ (f a + f b) * choose (f a + f b) (f a) = choose (succ (f a) + f b) (succ (f a)) * succ (f a)\n[PROOFSTEP]\nrw [succ_mul_choose_eq (f a + f b) (f a), succ_add (f a) (f b)]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na\u271d b\u271d : \u03b1\nn a b : \u2115\n\u22a2 multinomial Finset.univ ![a, b] = (a + b)! / (a ! * b !)\n[PROOFSTEP]\nrw [multinomial, Fin.sum_univ_two, Fin.prod_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na\u271d b\u271d : \u03b1\nn a b c : \u2115\n\u22a2 multinomial Finset.univ ![a, b, c] = (a + b + c)! / (a ! * b ! * c !)\n[PROOFSTEP]\nrw [multinomial, Fin.sum_univ_three, Fin.prod_univ_three]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na\u271d b\u271d : \u03b1\nn a b c : \u2115\n\u22a2 (Matrix.vecCons a ![b, c] 0 + Matrix.vecCons a ![b, c] 1 + Matrix.vecCons a ![b, c] 2)! /\n ((Matrix.vecCons a ![b, c] 0)! * (Matrix.vecCons a ![b, c] 1)! * (Matrix.vecCons a ![b, c] 2)!) =\n (a + b + c)! / (a ! * b ! * c !)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\na : \u03b1\nf : \u03b1 \u2192\u2080 \u2115\n\u22a2 multinomial f = choose (sum f fun x => id) (\u2191f a) * multinomial (update f a 0)\n[PROOFSTEP]\nsimp only [multinomial_eq]\n[GOAL]\n\u03b1 : Type u_1\na : \u03b1\nf : \u03b1 \u2192\u2080 \u2115\n\u22a2 Nat.multinomial f.support \u2191f =\n choose (sum f fun x => id) (\u2191f a) * Nat.multinomial (update f a 0).support \u2191(update f a 0)\n[PROOFSTEP]\nclassical\nby_cases a \u2208 f.support\n\u00b7 rw [\u2190 Finset.insert_erase h, Nat.multinomial_insert _ f (Finset.not_mem_erase a _), Finset.add_sum_erase _ f h,\n support_update_zero]\n congr 1\n exact Nat.multinomial_congr _ fun _ h => (Function.update_noteq (Finset.mem_erase.1 h).1 0 f).symm\nrw [not_mem_support_iff] at h \nrw [h, Nat.choose_zero_right, one_mul, \u2190 h, update_self]\n[GOAL]\n\u03b1 : Type u_1\na : \u03b1\nf : \u03b1 \u2192\u2080 \u2115\n\u22a2 Nat.multinomial f.support \u2191f =\n choose (sum f fun x => id) (\u2191f a) * Nat.multinomial (update f a 0).support \u2191(update f a 0)\n[PROOFSTEP]\nby_cases a \u2208 f.support\n[GOAL]\n\u03b1 : Type u_1\na : \u03b1\nf : \u03b1 \u2192\u2080 \u2115\n\u22a2 Nat.multinomial f.support \u2191f =\n choose (sum f fun x => id) (\u2191f a) * Nat.multinomial (update f a 0).support \u2191(update f a 0)\n[PROOFSTEP]\nby_cases a \u2208 f.support\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na : \u03b1\nf : \u03b1 \u2192\u2080 \u2115\nh : a \u2208 f.support\n\u22a2 Nat.multinomial f.support \u2191f =\n choose (sum f fun x => id) (\u2191f a) * Nat.multinomial (update f a 0).support \u2191(update f a 0)\n[PROOFSTEP]\nrw [\u2190 Finset.insert_erase h, Nat.multinomial_insert _ f (Finset.not_mem_erase a _), Finset.add_sum_erase _ f h,\n support_update_zero]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na : \u03b1\nf : \u03b1 \u2192\u2080 \u2115\nh : a \u2208 f.support\n\u22a2 choose (\u2211 x in f.support, \u2191f x) (\u2191f a) * Nat.multinomial (Finset.erase f.support a) \u2191f =\n choose (sum f fun x => id) (\u2191f a) * Nat.multinomial (Finset.erase f.support a) \u2191(update f a 0)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase pos.e_a\n\u03b1 : Type u_1\na : \u03b1\nf : \u03b1 \u2192\u2080 \u2115\nh : a \u2208 f.support\n\u22a2 Nat.multinomial (Finset.erase f.support a) \u2191f = Nat.multinomial (Finset.erase f.support a) \u2191(update f a 0)\n[PROOFSTEP]\nexact Nat.multinomial_congr _ fun _ h => (Function.update_noteq (Finset.mem_erase.1 h).1 0 f).symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na : \u03b1\nf : \u03b1 \u2192\u2080 \u2115\nh : \u00aca \u2208 f.support\n\u22a2 Nat.multinomial f.support \u2191f =\n choose (sum f fun x => id) (\u2191f a) * Nat.multinomial (update f a 0).support \u2191(update f a 0)\n[PROOFSTEP]\nrw [not_mem_support_iff] at h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\na : \u03b1\nf : \u03b1 \u2192\u2080 \u2115\nh : \u2191f a = 0\n\u22a2 Nat.multinomial f.support \u2191f =\n choose (sum f fun x => id) (\u2191f a) * Nat.multinomial (update f a 0).support \u2191(update f a 0)\n[PROOFSTEP]\nrw [h, Nat.choose_zero_right, one_mul, \u2190 h, update_self]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : Multiset \u03b1\n\u22a2 multinomial m = Nat.choose (\u2191card m) (count a m) * multinomial (filter ((fun x x_1 => x \u2260 x_1) a) m)\n[PROOFSTEP]\ndsimp only [multinomial]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : Multiset \u03b1\n\u22a2 Finsupp.multinomial (\u2191toFinsupp m) =\n Nat.choose (\u2191card m) (count a m) * Finsupp.multinomial (\u2191toFinsupp (filter (fun x => a \u2260 x) m))\n[PROOFSTEP]\nconvert Finsupp.multinomial_update a _\n[GOAL]\ncase h.e'_3.h.e'_5.h.e'_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : Multiset \u03b1\n\u22a2 \u2191card m = Finsupp.sum (\u2191toFinsupp m) fun x => id\n[PROOFSTEP]\nrw [\u2190 Finsupp.card_toMultiset, m.toFinsupp_toMultiset]\n[GOAL]\ncase h.e'_3.h.e'_6.h.e'_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : Multiset \u03b1\n\u22a2 \u2191toFinsupp (filter (fun x => a \u2260 x) m) = Finsupp.update (\u2191toFinsupp m) a 0\n[PROOFSTEP]\next1 a\n[GOAL]\ncase h.e'_3.h.e'_6.h.e'_2.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na\u271d : \u03b1\nm : Multiset \u03b1\na : \u03b1\n\u22a2 \u2191(\u2191toFinsupp (filter (fun x => a\u271d \u2260 x) m)) a = \u2191(Finsupp.update (\u2191toFinsupp m) a\u271d 0) a\n[PROOFSTEP]\nrw [toFinsupp_apply, count_filter, Finsupp.coe_update]\n[GOAL]\ncase h.e'_3.h.e'_6.h.e'_2.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na\u271d : \u03b1\nm : Multiset \u03b1\na : \u03b1\n\u22a2 (if a\u271d \u2260 a then count a m else 0) = Function.update (\u2191(\u2191toFinsupp m)) a\u271d 0 a\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na\u271d : \u03b1\nm : Multiset \u03b1\na : \u03b1\nh : a\u271d \u2260 a\n\u22a2 count a m = Function.update (\u2191(\u2191toFinsupp m)) a\u271d 0 a\n[PROOFSTEP]\nrw [Function.update_noteq h.symm, toFinsupp_apply]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na\u271d : \u03b1\nm : Multiset \u03b1\na : \u03b1\nh : \u00aca\u271d \u2260 a\n\u22a2 0 = Function.update (\u2191(\u2191toFinsupp m)) a\u271d 0 a\n[PROOFSTEP]\nrw [not_ne_iff.1 h, Function.update_same]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\n\u22a2 \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n[PROOFSTEP]\ninduction' s using Finset.induction with a s ha ih\n[GOAL]\ncase empty\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\nhc : Set.Pairwise \u2191\u2205 fun i j => Commute (x i) (x j)\n\u22a2 \u2200 (n : \u2115),\n Finset.sum \u2205 x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym \u2205 n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n[PROOFSTEP]\nrw [sum_empty]\n[GOAL]\ncase empty\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\nhc : Set.Pairwise \u2191\u2205 fun i j => Commute (x i) (x j)\n\u22a2 \u2200 (n : \u2115),\n 0 ^ n =\n \u2211 k : { x // x \u2208 Finset.sym \u2205 n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n[PROOFSTEP]\nrintro\n (_ | n)\n -- Porting note : Lean cannot infer this instance by itself\n[GOAL]\ncase empty.zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\nhc : Set.Pairwise \u2191\u2205 fun i j => Commute (x i) (x j)\n\u22a2 0 ^ zero =\n \u2211 k : { x // x \u2208 Finset.sym \u2205 zero },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n[PROOFSTEP]\nhaveI : Subsingleton (Sym \u03b1 0) := Unique.instSubsingleton\n[GOAL]\ncase empty.zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\nhc : Set.Pairwise \u2191\u2205 fun i j => Commute (x i) (x j)\nthis : Subsingleton (Sym \u03b1 0)\n\u22a2 0 ^ zero =\n \u2211 k : { x // x \u2208 Finset.sym \u2205 zero },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n[PROOFSTEP]\nrw [_root_.pow_zero, Fintype.sum_subsingleton]\n[GOAL]\ncase empty.zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\nhc : Set.Pairwise \u2191\u2205 fun i j => Commute (x i) (x j)\nthis : Subsingleton (Sym \u03b1 0)\n\u22a2 1 =\n \u2191(Multiset.multinomial \u2191\u2191?empty.zero.a) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191?empty.zero.a)\n (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191?empty.zero.a} Commute)\ncase empty.zero.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\nhc : Set.Pairwise \u2191\u2205 fun i j => Commute (x i) (x j)\nthis : Subsingleton (Sym \u03b1 0)\n\u22a2 { x // x \u2208 Finset.sym \u2205 zero }\n[PROOFSTEP]\nswap\n -- Porting note : Lean cannot infer this instance by itself\n[GOAL]\ncase empty.zero.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\nhc : Set.Pairwise \u2191\u2205 fun i j => Commute (x i) (x j)\nthis : Subsingleton (Sym \u03b1 0)\n\u22a2 { x // x \u2208 Finset.sym \u2205 zero }\n[PROOFSTEP]\nhave : Zero (Sym \u03b1 0) := Sym.instZeroSym\n[GOAL]\ncase empty.zero.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\nhc : Set.Pairwise \u2191\u2205 fun i j => Commute (x i) (x j)\nthis\u271d : Subsingleton (Sym \u03b1 0)\nthis : Zero (Sym \u03b1 0)\n\u22a2 { x // x \u2208 Finset.sym \u2205 zero }\n[PROOFSTEP]\nexact \u27e80, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\nhc : Set.Pairwise \u2191\u2205 fun i j => Commute (x i) (x j)\nthis\u271d : Subsingleton (Sym \u03b1 0)\nthis : Zero (Sym \u03b1 0)\n\u22a2 0 \u2208 Finset.sym \u2205 zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase empty.zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\nhc : Set.Pairwise \u2191\u2205 fun i j => Commute (x i) (x j)\nthis : Subsingleton (Sym \u03b1 0)\n\u22a2 1 =\n \u2191(Multiset.multinomial\n \u2191\u2191(let_fun this_1 := Sym.instZeroSym;\n { val := 0, property := (_ : 0 \u2208 {\u2205}) })) *\n Multiset.noncommProd\n (Multiset.map x\n \u2191\u2191(let_fun this_1 := Sym.instZeroSym;\n { val := 0, property := (_ : 0 \u2208 {\u2205}) }))\n (_ :\n Set.Pairwise\n {b |\n b \u2208\n Multiset.map x\n \u2191\u2191(let_fun this_1 := Sym.instZeroSym;\n { val := 0, property := (_ : 0 \u2208 {\u2205}) })}\n Commute)\n[PROOFSTEP]\nconvert (@one_mul R _ _).symm\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\nhc : Set.Pairwise \u2191\u2205 fun i j => Commute (x i) (x j)\nthis : Subsingleton (Sym \u03b1 0)\n\u22a2 \u2191(Multiset.multinomial\n \u2191\u2191(let_fun this_1 := Sym.instZeroSym;\n { val := 0, property := (_ : 0 \u2208 {\u2205}) })) =\n 1\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\nhc : Set.Pairwise \u2191\u2205 fun i j => Commute (x i) (x j)\nthis : Subsingleton (Sym \u03b1 0)\n\u22a2 \u2191(Multiset.multinomial \u21910) = 1\n[PROOFSTEP]\nconvert @Nat.cast_one R _\n[GOAL]\ncase empty.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\nhc : Set.Pairwise \u2191\u2205 fun i j => Commute (x i) (x j)\nn : \u2115\n\u22a2 0 ^ succ n =\n \u2211 k : { x // x \u2208 Finset.sym \u2205 (succ n) },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n[PROOFSTEP]\nrw [_root_.pow_succ, zero_mul]\n -- Porting note : Lean cannot infer this instance by itself\n[GOAL]\ncase empty.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\nhc : Set.Pairwise \u2191\u2205 fun i j => Commute (x i) (x j)\nn : \u2115\n\u22a2 0 =\n \u2211 k : { x // x \u2208 Finset.sym \u2205 (succ n) },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n[PROOFSTEP]\nhaveI : IsEmpty (Finset.sym (\u2205 : Finset \u03b1) (succ n)) := Finset.instIsEmpty\n[GOAL]\ncase empty.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)\nhc : Set.Pairwise \u2191\u2205 fun i j => Commute (x i) (x j)\nn : \u2115\nthis : IsEmpty { x // x \u2208 Finset.sym \u2205 (succ n) }\n\u22a2 0 =\n \u2211 k : { x // x \u2208 Finset.sym \u2205 (succ n) },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n[PROOFSTEP]\napply (Fintype.sum_empty _).symm\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih :\n \u2200 (hc : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)) (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\n\u22a2 \u2200 (n : \u2115),\n Finset.sum (insert a s) x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym (insert a s) n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih :\n \u2200 (hc : Set.Pairwise \u2191s fun i j => Commute (x i) (x j)) (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\n\u22a2 Finset.sum (insert a s) x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym (insert a s) n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n[PROOFSTEP]\nspecialize ih (hc.mono <| s.subset_insert a)\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n\u22a2 Finset.sum (insert a s) x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym (insert a s) n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n[PROOFSTEP]\nrw [sum_insert ha, (Commute.sum_right s _ _ _).add_pow, sum_range]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n\u22a2 \u2211 i : Fin (n + 1), x a ^ \u2191i * (\u2211 i in s, x i) ^ (n - \u2191i) * \u2191(Nat.choose n \u2191i) =\n \u2211 k : { x // x \u2208 Finset.sym (insert a s) n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n\u22a2 \u2200 (i : \u03b1), i \u2208 s \u2192 Commute (x a) (x i)\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n\u22a2 \u2200 (i : \u03b1), i \u2208 s \u2192 Commute (x a) (x i)\n[PROOFSTEP]\nexact fun _ hb => hc (mem_insert_self a s) (mem_insert_of_mem hb) (ne_of_mem_of_not_mem hb ha).symm\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n\u22a2 \u2211 i : Fin (n + 1), x a ^ \u2191i * (\u2211 i in s, x i) ^ (n - \u2191i) * \u2191(Nat.choose n \u2191i) =\n \u2211 k : { x // x \u2208 Finset.sym (insert a s) n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n[PROOFSTEP]\nsimp_rw [ih, mul_sum, sum_mul, sum_sigma', univ_sigma_univ]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\n\u22a2 \u2211 x_1 : (i : Fin (n + 1)) \u00d7 { x // x \u2208 Finset.sym s (n - \u2191i) },\n x a ^ \u2191x_1.fst *\n (\u2191(Multiset.multinomial \u2191\u2191x_1.snd) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191x_1.snd)\n (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191x_1.snd} Commute)) *\n \u2191(Nat.choose n \u2191x_1.fst) =\n \u2211 x_1 : { x // x \u2208 Finset.sym (insert a s) n },\n \u2191(Multiset.multinomial \u2191\u2191x_1) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191x_1) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191x_1} Commute)\n[PROOFSTEP]\nrefine' (Fintype.sum_equiv (symInsertEquiv ha) _ _ fun m => _).symm\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\nm : { x // x \u2208 Finset.sym (insert a s) n }\n\u22a2 \u2191(Multiset.multinomial \u2191\u2191m) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191m) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191m} Commute) =\n x a ^ \u2191(\u2191(symInsertEquiv ha) m).fst *\n (\u2191(Multiset.multinomial \u2191\u2191(\u2191(symInsertEquiv ha) m).snd) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd)\n (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd} Commute)) *\n \u2191(Nat.choose n \u2191(\u2191(symInsertEquiv ha) m).fst)\n[PROOFSTEP]\nrw [m.1.1.multinomial_filter_ne a]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\nm : { x // x \u2208 Finset.sym (insert a s) n }\n\u22a2 \u2191(Nat.choose (\u2191Multiset.card \u2191\u2191m) (Multiset.count a \u2191\u2191m) *\n Multiset.multinomial (Multiset.filter ((fun x x_1 => x \u2260 x_1) a) \u2191\u2191m)) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191m) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191m} Commute) =\n x a ^ \u2191(\u2191(symInsertEquiv ha) m).fst *\n (\u2191(Multiset.multinomial \u2191\u2191(\u2191(symInsertEquiv ha) m).snd) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd)\n (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd} Commute)) *\n \u2191(Nat.choose n \u2191(\u2191(symInsertEquiv ha) m).fst)\n[PROOFSTEP]\nconv in m.1.1.map _ => rw [\u2190 m.1.1.filter_add_not ((\u00b7 = \u00b7) a), Multiset.map_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\nm : { x // x \u2208 Finset.sym (insert a s) n }\n| Multiset.map x \u2191\u2191m\n[PROOFSTEP]\nrw [\u2190 m.1.1.filter_add_not ((\u00b7 = \u00b7) a), Multiset.map_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\nm : { x // x \u2208 Finset.sym (insert a s) n }\n| Multiset.map x \u2191\u2191m\n[PROOFSTEP]\nrw [\u2190 m.1.1.filter_add_not ((\u00b7 = \u00b7) a), Multiset.map_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\nm : { x // x \u2208 Finset.sym (insert a s) n }\n| Multiset.map x \u2191\u2191m\n[PROOFSTEP]\nrw [\u2190 m.1.1.filter_add_not ((\u00b7 = \u00b7) a), Multiset.map_add]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\nm : { x // x \u2208 Finset.sym (insert a s) n }\n\u22a2 \u2191(Nat.choose (\u2191Multiset.card \u2191\u2191m) (Multiset.count a \u2191\u2191m) *\n Multiset.multinomial (Multiset.filter ((fun x x_1 => x \u2260 x_1) a) \u2191\u2191m)) *\n Multiset.noncommProd\n (Multiset.map x (Multiset.filter ((fun x x_1 => x = x_1) a) \u2191\u2191m) +\n Multiset.map x (Multiset.filter (fun a_1 => \u00ac(fun x x_1 => x = x_1) a a_1) \u2191\u2191m))\n (_ :\n Set.Pairwise\n {x_1 |\n x_1 \u2208\n Multiset.map x (Multiset.filter ((fun x x_2 => x = x_2) a) \u2191\u2191m) +\n Multiset.map x (Multiset.filter (fun a_1 => \u00ac(fun x x_2 => x = x_2) a a_1) \u2191\u2191m)}\n Commute) =\n x a ^ \u2191(\u2191(symInsertEquiv ha) m).fst *\n (\u2191(Multiset.multinomial \u2191\u2191(\u2191(symInsertEquiv ha) m).snd) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd)\n (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd} Commute)) *\n \u2191(Nat.choose n \u2191(\u2191(symInsertEquiv ha) m).fst)\n[PROOFSTEP]\nsimp_rw [Multiset.noncommProd_add, m.1.1.filter_eq, Multiset.map_replicate, m.1.2]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\nm : { x // x \u2208 Finset.sym (insert a s) n }\n\u22a2 \u2191(Nat.choose n (Multiset.count a \u2191\u2191m) * Multiset.multinomial (Multiset.filter (fun x => a \u2260 x) \u2191\u2191m)) *\n (Multiset.noncommProd (Multiset.replicate (Multiset.count a \u2191\u2191m) (x a))\n (_ : Set.Pairwise {x_1 | x_1 \u2208 Multiset.replicate (Multiset.count a \u2191\u2191m) (x a)} Commute) *\n Multiset.noncommProd (Multiset.map x (Multiset.filter (fun a_1 => \u00aca = a_1) \u2191\u2191m))\n (_ : Set.Pairwise {x_1 | x_1 \u2208 Multiset.map x (Multiset.filter (fun a_1 => \u00aca = a_1) \u2191\u2191m)} Commute)) =\n x a ^ \u2191(\u2191(symInsertEquiv ha) m).fst *\n (\u2191(Multiset.multinomial \u2191\u2191(\u2191(symInsertEquiv ha) m).snd) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd)\n (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd} Commute)) *\n \u2191(Nat.choose n \u2191(\u2191(symInsertEquiv ha) m).fst)\n[PROOFSTEP]\nrw [Multiset.noncommProd_eq_pow_card _ _ _ fun _ => Multiset.eq_of_mem_replicate]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\nm : { x // x \u2208 Finset.sym (insert a s) n }\n\u22a2 \u2191(Nat.choose n (Multiset.count a \u2191\u2191m) * Multiset.multinomial (Multiset.filter (fun x => a \u2260 x) \u2191\u2191m)) *\n (x a ^ \u2191Multiset.card (Multiset.replicate (Multiset.count a \u2191\u2191m) (x a)) *\n Multiset.noncommProd (Multiset.map x (Multiset.filter (fun a_1 => \u00aca = a_1) \u2191\u2191m))\n (_ : Set.Pairwise {x_1 | x_1 \u2208 Multiset.map x (Multiset.filter (fun a_1 => \u00aca = a_1) \u2191\u2191m)} Commute)) =\n x a ^ \u2191(\u2191(symInsertEquiv ha) m).fst *\n (\u2191(Multiset.multinomial \u2191\u2191(\u2191(symInsertEquiv ha) m).snd) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd)\n (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd} Commute)) *\n \u2191(Nat.choose n \u2191(\u2191(symInsertEquiv ha) m).fst)\n[PROOFSTEP]\nrw [Multiset.card_replicate, Nat.cast_mul, mul_assoc, Nat.cast_comm]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\nm : { x // x \u2208 Finset.sym (insert a s) n }\n\u22a2 \u2191(Multiset.multinomial (Multiset.filter (fun x => a \u2260 x) \u2191\u2191m)) *\n (x a ^ Multiset.count a \u2191\u2191m *\n Multiset.noncommProd (Multiset.map x (Multiset.filter (fun a_1 => \u00aca = a_1) \u2191\u2191m))\n (_ : Set.Pairwise {x_1 | x_1 \u2208 Multiset.map x (Multiset.filter (fun a_1 => \u00aca = a_1) \u2191\u2191m)} Commute)) *\n \u2191(Nat.choose n (Multiset.count a \u2191\u2191m)) =\n x a ^ \u2191(\u2191(symInsertEquiv ha) m).fst *\n (\u2191(Multiset.multinomial \u2191\u2191(\u2191(symInsertEquiv ha) m).snd) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd)\n (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd} Commute)) *\n \u2191(Nat.choose n \u2191(\u2191(symInsertEquiv ha) m).fst)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase insert.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\nm : { x // x \u2208 Finset.sym (insert a s) n }\n\u22a2 \u2191(Multiset.multinomial (Multiset.filter (fun x => a \u2260 x) \u2191\u2191m)) *\n (x a ^ Multiset.count a \u2191\u2191m *\n Multiset.noncommProd (Multiset.map x (Multiset.filter (fun a_1 => \u00aca = a_1) \u2191\u2191m))\n (_ : Set.Pairwise {x_1 | x_1 \u2208 Multiset.map x (Multiset.filter (fun a_1 => \u00aca = a_1) \u2191\u2191m)} Commute)) =\n x a ^ \u2191(\u2191(symInsertEquiv ha) m).fst *\n (\u2191(Multiset.multinomial \u2191\u2191(\u2191(symInsertEquiv ha) m).snd) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd)\n (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd} Commute))\n[PROOFSTEP]\nsimp_rw [\u2190 mul_assoc, Nat.cast_comm]\n[GOAL]\ncase insert.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nR : Type u_2\ninst\u271d : Semiring R\nx : \u03b1 \u2192 R\nhc\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (x i) (x j)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhc : Set.Pairwise \u2191(insert a s) fun i j => Commute (x i) (x j)\nn : \u2115\nih :\n \u2200 (n : \u2115),\n Finset.sum s x ^ n =\n \u2211 k : { x // x \u2208 Finset.sym s n },\n \u2191(Multiset.multinomial \u2191\u2191k) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191k) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191k} Commute)\nm : { x // x \u2208 Finset.sym (insert a s) n }\n\u22a2 x a ^ Multiset.count a \u2191\u2191m * \u2191(Multiset.multinomial (Multiset.filter (fun x => a \u2260 x) \u2191\u2191m)) *\n Multiset.noncommProd (Multiset.map x (Multiset.filter (fun a_1 => \u00aca = a_1) \u2191\u2191m))\n (_ : Set.Pairwise {x_1 | x_1 \u2208 Multiset.map x (Multiset.filter (fun a_1 => \u00aca = a_1) \u2191\u2191m)} Commute) =\n x a ^ \u2191(\u2191(symInsertEquiv ha) m).fst * \u2191(Multiset.multinomial \u2191\u2191(\u2191(symInsertEquiv ha) m).snd) *\n Multiset.noncommProd (Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd)\n (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191(\u2191(symInsertEquiv ha) m).snd} Commute)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : CommSemiring R\nx : \u03b1 \u2192 R\nn : \u2115\n\u22a2 Finset.sum s x ^ n = \u2211 k in Finset.sym s n, \u2191(Multiset.multinomial \u2191k) * Multiset.prod (Multiset.map x \u2191k)\n[PROOFSTEP]\nconv_rhs => rw [\u2190 sum_coe_sort]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : CommSemiring R\nx : \u03b1 \u2192 R\nn : \u2115\n| \u2211 k in Finset.sym s n, \u2191(Multiset.multinomial \u2191k) * Multiset.prod (Multiset.map x \u2191k)\n[PROOFSTEP]\nrw [\u2190 sum_coe_sort]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : CommSemiring R\nx : \u03b1 \u2192 R\nn : \u2115\n| \u2211 k in Finset.sym s n, \u2191(Multiset.multinomial \u2191k) * Multiset.prod (Multiset.map x \u2191k)\n[PROOFSTEP]\nrw [\u2190 sum_coe_sort]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : CommSemiring R\nx : \u03b1 \u2192 R\nn : \u2115\n| \u2211 k in Finset.sym s n, \u2191(Multiset.multinomial \u2191k) * Multiset.prod (Multiset.map x \u2191k)\n[PROOFSTEP]\nrw [\u2190 sum_coe_sort]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : CommSemiring R\nx : \u03b1 \u2192 R\nn : \u2115\n\u22a2 Finset.sum s x ^ n =\n \u2211 i : { x // x \u2208 Finset.sym s n }, \u2191(Multiset.multinomial \u2191\u2191i) * Multiset.prod (Multiset.map x \u2191\u2191i)\n[PROOFSTEP]\nconvert sum_pow_of_commute s x (fun _ _ _ _ _ => mul_comm _ _) n\n[GOAL]\ncase h.e'_3.a.h.e'_6\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nR : Type u_2\ninst\u271d : CommSemiring R\nx : \u03b1 \u2192 R\nn : \u2115\nx\u271d : { x // x \u2208 Finset.sym s n }\na\u271d : x\u271d \u2208 univ\n\u22a2 Multiset.prod (Multiset.map x \u2191\u2191x\u271d) =\n Multiset.noncommProd (Multiset.map x \u2191\u2191x\u271d) (_ : Set.Pairwise {b | b \u2208 Multiset.map x \u2191\u2191x\u271d} Commute)\n[PROOFSTEP]\nrw [Multiset.noncommProd_eq_prod]\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Choose.Multinomial", "llama_tokens": 18960, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.42026248202888933}} {"text": "[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\n\u22a2 \u2191toSubmodule S * \u2191toSubmodule T \u2264 \u2191toSubmodule (S \u2294 T)\n[PROOFSTEP]\nrw [Submodule.mul_le]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\n\u22a2 \u2200 (m : A), m \u2208 \u2191toSubmodule S \u2192 \u2200 (n : A), n \u2208 \u2191toSubmodule T \u2192 m * n \u2208 \u2191toSubmodule (S \u2294 T)\n[PROOFSTEP]\nintro y hy z hz\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y \u2208 \u2191toSubmodule S\nz : A\nhz : z \u2208 \u2191toSubmodule T\n\u22a2 y * z \u2208 \u2191toSubmodule (S \u2294 T)\n[PROOFSTEP]\nshow y * z \u2208 S \u2294 T\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y \u2208 \u2191toSubmodule S\nz : A\nhz : z \u2208 \u2191toSubmodule T\n\u22a2 y * z \u2208 S \u2294 T\n[PROOFSTEP]\nexact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\n\u22a2 \u2191toSubmodule S * \u2191toSubmodule S = \u2191toSubmodule S\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\n\u22a2 \u2191toSubmodule S * \u2191toSubmodule S \u2264 \u2191toSubmodule S\n[PROOFSTEP]\nrefine' (mul_toSubmodule_le _ _).trans_eq _\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\n\u22a2 \u2191toSubmodule (S \u2294 S) = \u2191toSubmodule S\n[PROOFSTEP]\nrw [sup_idem]\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\n\u22a2 \u2191toSubmodule S \u2264 \u2191toSubmodule S * \u2191toSubmodule S\n[PROOFSTEP]\nintro x hx1\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x \u2208 \u2191toSubmodule S\n\u22a2 x \u2208 \u2191toSubmodule S * \u2191toSubmodule S\n[PROOFSTEP]\nrw [\u2190 mul_one x]\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x \u2208 \u2191toSubmodule S\n\u22a2 x * 1 \u2208 \u2191toSubmodule S * \u2191toSubmodule S\n[PROOFSTEP]\nexact Submodule.mul_mem_mul hx1 (show (1 : A) \u2208 S from one_mem S)\n[GOAL]\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\n\u22a2 \u2191toSubmodule S * \u2191toSubmodule T = \u2191toSubmodule (S \u2294 T)\n[PROOFSTEP]\nrefine' le_antisymm (mul_toSubmodule_le _ _) _\n[GOAL]\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\n\u22a2 \u2191toSubmodule (S \u2294 T) \u2264 \u2191toSubmodule S * \u2191toSubmodule T\n[PROOFSTEP]\nrintro x (hx : x \u2208 Algebra.adjoin R (S \u222a T : Set A))\n[GOAL]\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx : A\nhx : x \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\n\u22a2 x \u2208 \u2191toSubmodule S * \u2191toSubmodule T\n[PROOFSTEP]\nrefine' Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _) fun x y hx hy => _\n[GOAL]\ncase refine'_1\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx\u271d : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx : A\nhx : x \u2208 \u2191S \u222a \u2191T\n\u22a2 x \u2208 \u2191toSubmodule S * \u2191toSubmodule T\n[PROOFSTEP]\ncases' hx with hxS hxT\n[GOAL]\ncase refine'_1.inl\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx : A\nhxS : x \u2208 \u2191S\n\u22a2 x \u2208 \u2191toSubmodule S * \u2191toSubmodule T\n[PROOFSTEP]\nrw [\u2190 mul_one x]\n[GOAL]\ncase refine'_1.inl\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx : A\nhxS : x \u2208 \u2191S\n\u22a2 x * 1 \u2208 \u2191toSubmodule S * \u2191toSubmodule T\n[PROOFSTEP]\nexact Submodule.mul_mem_mul hxS (show (1 : A) \u2208 T from one_mem T)\n[GOAL]\ncase refine'_1.inr\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx : A\nhxT : x \u2208 \u2191T\n\u22a2 x \u2208 \u2191toSubmodule S * \u2191toSubmodule T\n[PROOFSTEP]\nrw [\u2190 one_mul x]\n[GOAL]\ncase refine'_1.inr\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx : A\nhxT : x \u2208 \u2191T\n\u22a2 1 * x \u2208 \u2191toSubmodule S * \u2191toSubmodule T\n[PROOFSTEP]\nexact Submodule.mul_mem_mul (show (1 : A) \u2208 S from one_mem S) hxT\n[GOAL]\ncase refine'_2\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx : A\nhx : x \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nr : R\n\u22a2 \u2191(algebraMap R A) r \u2208 \u2191toSubmodule S * \u2191toSubmodule T\n[PROOFSTEP]\nrw [\u2190 one_mul (algebraMap _ _ _)]\n[GOAL]\ncase refine'_2\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx : A\nhx : x \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nr : R\n\u22a2 1 * \u2191(algebraMap R A) r \u2208 \u2191toSubmodule S * \u2191toSubmodule T\n[PROOFSTEP]\nexact Submodule.mul_mem_mul (show (1 : A) \u2208 S from one_mem S) (algebraMap_mem T _)\n[GOAL]\ncase refine'_3\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx\u271d : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx y : A\nhx : x \u2208 \u2191toSubmodule S * \u2191toSubmodule T\nhy : y \u2208 \u2191toSubmodule S * \u2191toSubmodule T\n\u22a2 x * y \u2208 \u2191toSubmodule S * \u2191toSubmodule T\n[PROOFSTEP]\nhave := Submodule.mul_mem_mul hx hy\n[GOAL]\ncase refine'_3\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx\u271d : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx y : A\nhx : x \u2208 \u2191toSubmodule S * \u2191toSubmodule T\nhy : y \u2208 \u2191toSubmodule S * \u2191toSubmodule T\nthis : x * y \u2208 \u2191toSubmodule S * \u2191toSubmodule T * (\u2191toSubmodule S * \u2191toSubmodule T)\n\u22a2 x * y \u2208 \u2191toSubmodule S * \u2191toSubmodule T\n[PROOFSTEP]\nrwa [mul_assoc, mul_comm _ (Subalgebra.toSubmodule T), \u2190 mul_assoc _ _ (Subalgebra.toSubmodule S), mul_self,\n mul_comm (Subalgebra.toSubmodule T), \u2190 mul_assoc, mul_self] at this \n", "meta": {"mathlib_filename": "Mathlib.Algebra.Algebra.Subalgebra.Pointwise", "llama_tokens": 3875, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.42021195932279093}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\np : CommSq f g h i\n\u22a2 PushoutCocone.inl (PullbackCone.op (cone p)) \u226b (Iso.refl (PullbackCone.op (cone p)).pt).hom =\n PushoutCocone.inl (cocone (_ : CommSq h.op i.op f.op g.op))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\np : CommSq f g h i\n\u22a2 PushoutCocone.inr (PullbackCone.op (cone p)) \u226b (Iso.refl (PullbackCone.op (cone p)).pt).hom =\n PushoutCocone.inr (cocone (_ : CommSq h.op i.op f.op g.op))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\np : CommSq f g h i\n\u22a2 PullbackCone.fst (PushoutCocone.op (cocone p)) =\n (Iso.refl (PushoutCocone.op (cocone p)).pt).hom \u226b PullbackCone.fst (cone (_ : CommSq h.op i.op f.op g.op))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\np : CommSq f g h i\n\u22a2 PullbackCone.snd (PushoutCocone.op (cocone p)) =\n (Iso.refl (PushoutCocone.op (cocone p)).pt).hom \u226b PullbackCone.snd (cone (_ : CommSq h.op i.op f.op g.op))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nW\u271d X\u271d Y\u271d Z\u271d : C\nf\u271d : W\u271d \u27f6 X\u271d\ng\u271d : W\u271d \u27f6 Y\u271d\nh\u271d : X\u271d \u27f6 Z\u271d\ni\u271d : Y\u271d \u27f6 Z\u271d\nW X Y Z : C\u1d52\u1d56\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\np : CommSq f g h i\n\u22a2 PushoutCocone.inl (PullbackCone.unop (cone p)) \u226b (Iso.refl (PullbackCone.unop (cone p)).pt).hom =\n PushoutCocone.inl (cocone (_ : CommSq h.unop i.unop f.unop g.unop))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nW\u271d X\u271d Y\u271d Z\u271d : C\nf\u271d : W\u271d \u27f6 X\u271d\ng\u271d : W\u271d \u27f6 Y\u271d\nh\u271d : X\u271d \u27f6 Z\u271d\ni\u271d : Y\u271d \u27f6 Z\u271d\nW X Y Z : C\u1d52\u1d56\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\np : CommSq f g h i\n\u22a2 PushoutCocone.inr (PullbackCone.unop (cone p)) \u226b (Iso.refl (PullbackCone.unop (cone p)).pt).hom =\n PushoutCocone.inr (cocone (_ : CommSq h.unop i.unop f.unop g.unop))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nW\u271d X\u271d Y\u271d Z\u271d : C\nf\u271d : W\u271d \u27f6 X\u271d\ng\u271d : W\u271d \u27f6 Y\u271d\nh\u271d : X\u271d \u27f6 Z\u271d\ni\u271d : Y\u271d \u27f6 Z\u271d\nW X Y Z : C\u1d52\u1d56\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\np : CommSq f g h i\n\u22a2 PullbackCone.fst (PushoutCocone.unop (cocone p)) =\n (Iso.refl (PushoutCocone.unop (cocone p)).pt).hom \u226b PullbackCone.fst (cone (_ : CommSq h.unop i.unop f.unop g.unop))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nW\u271d X\u271d Y\u271d Z\u271d : C\nf\u271d : W\u271d \u27f6 X\u271d\ng\u271d : W\u271d \u27f6 Y\u271d\nh\u271d : X\u271d \u27f6 Z\u271d\ni\u271d : Y\u271d \u27f6 Z\u271d\nW X Y Z : C\u1d52\u1d56\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\np : CommSq f g h i\n\u22a2 PullbackCone.snd (PushoutCocone.unop (cocone p)) =\n (Iso.refl (PushoutCocone.unop (cocone p)).pt).hom \u226b PullbackCone.snd (cone (_ : CommSq h.unop i.unop f.unop g.unop))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\nc : PullbackCone f g\nh : IsLimit c\n\u22a2 PullbackCone.fst c =\n (Iso.refl c.pt).hom \u226b\n PullbackCone.fst\n (PullbackCone.mk (PullbackCone.fst c) (PullbackCone.snd c)\n (_ : PullbackCone.fst c \u226b f = PullbackCone.snd c \u226b g))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\nc : PullbackCone f g\nh : IsLimit c\n\u22a2 PullbackCone.snd c =\n (Iso.refl c.pt).hom \u226b\n PullbackCone.snd\n (PullbackCone.mk (PullbackCone.fst c) (PullbackCone.snd c)\n (_ : PullbackCone.fst c \u226b f = PullbackCone.snd c \u226b g))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\nc : BinaryFan X Y\nh : IsLimit c\nt : IsTerminal Z\n\u22a2 \u2200 (j : Discrete WalkingPair),\n NatTrans.app c.\u03c0 j =\n (Iso.refl c.pt).hom \u226b\n NatTrans.app (BinaryFan.mk (c.\u03c0.1 { as := WalkingPair.left }) (c.\u03c0.1 { as := WalkingPair.right })).\u03c0 j\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.left\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\nc : BinaryFan X Y\nh : IsLimit c\nt : IsTerminal Z\n\u22a2 NatTrans.app c.\u03c0 { as := WalkingPair.left } =\n (Iso.refl c.pt).hom \u226b\n NatTrans.app (BinaryFan.mk (c.\u03c0.1 { as := WalkingPair.left }) (c.\u03c0.1 { as := WalkingPair.right })).\u03c0\n { as := WalkingPair.left }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.left\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\nc : BinaryFan X Y\nh : IsLimit c\nt : IsTerminal Z\n\u22a2 BinaryFan.fst c = \ud835\udfd9 c.pt \u226b BinaryFan.fst c\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.right\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\nc : BinaryFan X Y\nh : IsLimit c\nt : IsTerminal Z\n\u22a2 NatTrans.app c.\u03c0 { as := WalkingPair.right } =\n (Iso.refl c.pt).hom \u226b\n NatTrans.app (BinaryFan.mk (c.\u03c0.1 { as := WalkingPair.left }) (c.\u03c0.1 { as := WalkingPair.right })).\u03c0\n { as := WalkingPair.right }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.right\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\nc : BinaryFan X Y\nh : IsLimit c\nt : IsTerminal Z\n\u22a2 BinaryFan.snd c = \ud835\udfd9 c.pt \u226b BinaryFan.snd c\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b2 : HasBinaryProduct X Y\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\n\u22a2 IsPullback prod.fst prod.snd 0 0\n[PROOFSTEP]\nconvert @of_is_product _ _ X Y 0 _ (limit.isLimit _) HasZeroObject.zeroIsTerminal\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : IsPullback fst snd f g\ninst\u271d : HasPullback f g\n\u22a2 (isoPullback h).hom \u226b pullback.fst = fst\n[PROOFSTEP]\ndsimp [isoPullback, cone, CommSq.cone]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : IsPullback fst snd f g\ninst\u271d : HasPullback f g\n\u22a2 (limit.isoLimitCone { cone := PullbackCone.mk fst snd (_ : fst \u226b f = snd \u226b g), isLimit := isLimit h }).inv \u226b\n pullback.fst =\n fst\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : IsPullback fst snd f g\ninst\u271d : HasPullback f g\n\u22a2 (isoPullback h).hom \u226b pullback.snd = snd\n[PROOFSTEP]\ndsimp [isoPullback, cone, CommSq.cone]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : IsPullback fst snd f g\ninst\u271d : HasPullback f g\n\u22a2 (limit.isoLimitCone { cone := PullbackCone.mk fst snd (_ : fst \u226b f = snd \u226b g), isLimit := isLimit h }).inv \u226b\n pullback.snd =\n snd\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : IsPullback fst snd f g\ninst\u271d : HasPullback f g\n\u22a2 (isoPullback h).inv \u226b fst = pullback.fst\n[PROOFSTEP]\nsimp [Iso.inv_comp_eq]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : IsPullback fst snd f g\ninst\u271d : HasPullback f g\n\u22a2 (isoPullback h).inv \u226b snd = pullback.snd\n[PROOFSTEP]\nsimp [Iso.inv_comp_eq]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso fst\ninst\u271d : IsIso g\nsq : CommSq fst snd f g\n\u22a2 IsLimit (CommSq.cone sq)\n[PROOFSTEP]\nrefine' PullbackCone.IsLimit.mk _ (fun s => s.fst \u226b inv fst) (by aesop_cat) (fun s => _) (by aesop_cat)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso fst\ninst\u271d : IsIso g\nsq : CommSq fst snd f g\n\u22a2 \u2200 (s : PullbackCone f g), (fun s => PullbackCone.fst s \u226b inv fst) s \u226b fst = PullbackCone.fst s\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso fst\ninst\u271d : IsIso g\nsq : CommSq fst snd f g\n\u22a2 \u2200 (s : PullbackCone f g) (m : s.pt \u27f6 P),\n m \u226b fst = PullbackCone.fst s \u2192 m \u226b snd = PullbackCone.snd s \u2192 m = (fun s => PullbackCone.fst s \u226b inv fst) s\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : IsIso fst\ninst\u271d : IsIso g\nsq : CommSq fst snd f g\ns : PullbackCone f g\n\u22a2 (fun s => PullbackCone.fst s \u226b inv fst) s \u226b snd = PullbackCone.snd s\n[PROOFSTEP]\nsimp only [\u2190 cancel_mono g, Category.assoc, \u2190 sq.w, IsIso.inv_hom_id_assoc, s.condition]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\nc : PushoutCocone f g\nh : IsColimit c\n\u22a2 PushoutCocone.inl c \u226b (Iso.refl c.pt).hom =\n PushoutCocone.inl\n (PushoutCocone.mk (PushoutCocone.inl c) (PushoutCocone.inr c)\n (_ : f \u226b PushoutCocone.inl c = g \u226b PushoutCocone.inr c))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\nc : PushoutCocone f g\nh : IsColimit c\n\u22a2 PushoutCocone.inr c \u226b (Iso.refl c.pt).hom =\n PushoutCocone.inr\n (PushoutCocone.mk (PushoutCocone.inl c) (PushoutCocone.inr c)\n (_ : f \u226b PushoutCocone.inl c = g \u226b PushoutCocone.inr c))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\nc : BinaryCofan X Y\nh : IsColimit c\nt : IsInitial Z\n\u22a2 \u2200 (j : Discrete WalkingPair),\n NatTrans.app c.\u03b9 j \u226b (Iso.refl c.pt).hom =\n NatTrans.app (BinaryCofan.mk (c.\u03b9.1 { as := WalkingPair.left }) (c.\u03b9.1 { as := WalkingPair.right })).\u03b9 j\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.left\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\nc : BinaryCofan X Y\nh : IsColimit c\nt : IsInitial Z\n\u22a2 NatTrans.app c.\u03b9 { as := WalkingPair.left } \u226b (Iso.refl c.pt).hom =\n NatTrans.app (BinaryCofan.mk (c.\u03b9.1 { as := WalkingPair.left }) (c.\u03b9.1 { as := WalkingPair.right })).\u03b9\n { as := WalkingPair.left }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.left\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\nc : BinaryCofan X Y\nh : IsColimit c\nt : IsInitial Z\n\u22a2 BinaryCofan.inl c \u226b \ud835\udfd9 c.pt = BinaryCofan.inl c\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.right\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\nc : BinaryCofan X Y\nh : IsColimit c\nt : IsInitial Z\n\u22a2 NatTrans.app c.\u03b9 { as := WalkingPair.right } \u226b (Iso.refl c.pt).hom =\n NatTrans.app (BinaryCofan.mk (c.\u03b9.1 { as := WalkingPair.left }) (c.\u03b9.1 { as := WalkingPair.right })).\u03b9\n { as := WalkingPair.right }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.right\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\nc : BinaryCofan X Y\nh : IsColimit c\nt : IsInitial Z\n\u22a2 BinaryCofan.inr c \u226b \ud835\udfd9 c.pt = BinaryCofan.inr c\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b2 : HasBinaryCoproduct X Y\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\n\u22a2 IsPushout 0 0 coprod.inl coprod.inr\n[PROOFSTEP]\nconvert @of_is_coproduct _ _ 0 X Y _ (colimit.isColimit _) HasZeroObject.zeroIsInitial\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\nh : IsPushout f g inl inr\ninst\u271d : HasPushout f g\n\u22a2 pushout.inl \u226b (isoPushout h).inv = inl\n[PROOFSTEP]\ndsimp [isoPushout, cocone, CommSq.cocone]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\nh : IsPushout f g inl inr\ninst\u271d : HasPushout f g\n\u22a2 pushout.inl \u226b\n (colimit.isoColimitCocone\n { cocone := PushoutCocone.mk inl inr (_ : f \u226b inl = g \u226b inr), isColimit := isColimit h }).hom =\n inl\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\nh : IsPushout f g inl inr\ninst\u271d : HasPushout f g\n\u22a2 pushout.inr \u226b (isoPushout h).inv = inr\n[PROOFSTEP]\ndsimp [isoPushout, cocone, CommSq.cocone]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\nh : IsPushout f g inl inr\ninst\u271d : HasPushout f g\n\u22a2 pushout.inr \u226b\n (colimit.isoColimitCocone\n { cocone := PushoutCocone.mk inl inr (_ : f \u226b inl = g \u226b inr), isColimit := isColimit h }).hom =\n inr\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\nh : IsPushout f g inl inr\ninst\u271d : HasPushout f g\n\u22a2 inl \u226b (isoPushout h).hom = pushout.inl\n[PROOFSTEP]\nsimp [\u2190 Iso.eq_comp_inv]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\nh : IsPushout f g inl inr\ninst\u271d : HasPushout f g\n\u22a2 inr \u226b (isoPushout h).hom = pushout.inr\n[PROOFSTEP]\nsimp [\u2190 Iso.eq_comp_inv]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X\u271d Y Z : C\nfst : P \u27f6 X\u271d\nsnd : P \u27f6 Y\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\n\u22a2 0 \u226b \ud835\udfd9 X = 0 \u226b 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X\u271d Y Z : C\nfst : P \u27f6 X\u271d\nsnd : P \u27f6 Y\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\ns : Cone (cospan (\ud835\udfd9 X) 0)\n\u22a2 \u2200 (j : WalkingCospan),\n (fun s => 0) s \u226b NatTrans.app (PullbackCone.mk 0 0 (_ : 0 \u226b \ud835\udfd9 X = 0 \u226b 0)).\u03c0 j = NatTrans.app s.\u03c0 j\n[PROOFSTEP]\nsimpa using @PullbackCone.equalizer_ext _ _ _ _ _ _ _ s _ 0 (\ud835\udfd9 _) (by simpa using (PullbackCone.condition s).symm)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X\u271d Y Z : C\nfst : P \u27f6 X\u271d\nsnd : P \u27f6 Y\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\ns : Cone (cospan (\ud835\udfd9 X) 0)\n\u22a2 0 \u226b PullbackCone.fst s = \ud835\udfd9 s.pt \u226b PullbackCone.fst s\n[PROOFSTEP]\nsimpa using (PullbackCone.condition s).symm\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X\u271d Y Z : C\nfst : P \u27f6 X\u271d\nsnd : P \u27f6 Y\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\n\u22a2 CommSq 0 (\ud835\udfd9 X) 0 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X\u271d Y Z : C\nfst : P \u27f6 X\u271d\nsnd : P \u27f6 Y\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\n\u22a2 ((zeroProdIso X).symm \u226a\u226b (pullbackZeroZeroIso 0 X).symm).hom \u226b pullback.fst = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X\u271d Y Z : C\nfst : P \u27f6 X\u271d\nsnd : P \u27f6 Y\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\n\u22a2 ((zeroProdIso X).symm \u226a\u226b (pullbackZeroZeroIso 0 X).symm).hom \u226b pullback.snd = \ud835\udfd9 X\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPullback b.fst b.snd 0 0\n[PROOFSTEP]\nconvert IsPullback.of_is_product' h.isLimit HasZeroObject.zeroIsTerminal\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPullback b.inl 0 b.snd 0\n[PROOFSTEP]\nrefine' of_right _ (by simp) (of_isBilimit h)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 b.inl \u226b b.snd = 0 \u226b 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPullback (b.inl \u226b b.fst) 0 0 (0 \u226b 0)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPullback b.inr 0 b.fst 0\n[PROOFSTEP]\napply flip\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPullback 0 b.inr 0 b.fst\n[PROOFSTEP]\nrefine' of_bot _ (by simp) (of_isBilimit h)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 0 \u226b 0 = b.inr \u226b b.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPullback 0 (b.inr \u226b b.snd) (0 \u226b 0) 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPullback 0 0 b.inl b.inr\n[PROOFSTEP]\nrefine' IsPullback.of_right _ (by simp) (IsPullback.inl_snd' h).flip\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 0 \u226b b.inl = 0 \u226b b.inr\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPullback (0 \u226b 0) 0 0 (b.inr \u226b b.snd)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X\u271d Y P : C\nf : Z \u27f6 X\u271d\ng : Z \u27f6 Y\ninl : X\u271d \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\n\u22a2 0 \u226b 0 = \ud835\udfd9 X \u226b 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X\u271d Y P : C\nf : Z \u27f6 X\u271d\ng : Z \u27f6 Y\ninl : X\u271d \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\ns : Cocone (span 0 (\ud835\udfd9 X))\n\u22a2 \u2200 (j : WalkingSpan),\n NatTrans.app (PushoutCocone.mk 0 0 (_ : 0 \u226b 0 = \ud835\udfd9 X \u226b 0)).\u03b9 j \u226b (fun s => 0) s = NatTrans.app s.\u03b9 j\n[PROOFSTEP]\nhave c := @PushoutCocone.coequalizer_ext _ _ _ _ _ _ _ s _ 0 (\ud835\udfd9 _) (by simp) (by simpa using PushoutCocone.condition s)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X\u271d Y P : C\nf : Z \u27f6 X\u271d\ng : Z \u27f6 Y\ninl : X\u271d \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\ns : Cocone (span 0 (\ud835\udfd9 X))\n\u22a2 PushoutCocone.inl s \u226b 0 = PushoutCocone.inl s \u226b \ud835\udfd9 s.pt\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X\u271d Y P : C\nf : Z \u27f6 X\u271d\ng : Z \u27f6 Y\ninl : X\u271d \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\ns : Cocone (span 0 (\ud835\udfd9 X))\n\u22a2 PushoutCocone.inr s \u226b 0 = PushoutCocone.inr s \u226b \ud835\udfd9 s.pt\n[PROOFSTEP]\nsimpa using PushoutCocone.condition s\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X\u271d Y P : C\nf : Z \u27f6 X\u271d\ng : Z \u27f6 Y\ninl : X\u271d \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\ns : Cocone (span 0 (\ud835\udfd9 X))\nc : \u2200 (j : WalkingSpan), NatTrans.app s.\u03b9 j \u226b 0 = NatTrans.app s.\u03b9 j \u226b \ud835\udfd9 s.pt\n\u22a2 \u2200 (j : WalkingSpan),\n NatTrans.app (PushoutCocone.mk 0 0 (_ : 0 \u226b 0 = \ud835\udfd9 X \u226b 0)).\u03b9 j \u226b (fun s => 0) s = NatTrans.app s.\u03b9 j\n[PROOFSTEP]\ndsimp at c \n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X\u271d Y P : C\nf : Z \u27f6 X\u271d\ng : Z \u27f6 Y\ninl : X\u271d \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\ns : Cocone (span 0 (\ud835\udfd9 X))\nc : \u2200 (j : WalkingSpan), NatTrans.app s.\u03b9 j \u226b 0 = NatTrans.app s.\u03b9 j \u226b \ud835\udfd9 s.pt\n\u22a2 \u2200 (j : WalkingSpan),\n NatTrans.app (PushoutCocone.mk 0 0 (_ : 0 \u226b 0 = \ud835\udfd9 X \u226b 0)).\u03b9 j \u226b (fun s => 0) s = NatTrans.app s.\u03b9 j\n[PROOFSTEP]\nsimpa using c\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X\u271d Y P : C\nf : Z \u27f6 X\u271d\ng : Z \u27f6 Y\ninl : X\u271d \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\n\u22a2 CommSq 0 0 (\ud835\udfd9 X) 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X\u271d Y P : C\nf : Z \u27f6 X\u271d\ng : Z \u27f6 Y\ninl : X\u271d \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\n\u22a2 \ud835\udfd9 X \u226b ((coprodZeroIso X).symm \u226a\u226b (pushoutZeroZeroIso X 0).symm).hom = pushout.inl\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X\u271d Y P : C\nf : Z \u27f6 X\u271d\ng : Z \u27f6 Y\ninl : X\u271d \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\n\u22a2 0 \u226b ((coprodZeroIso X).symm \u226a\u226b (pushoutZeroZeroIso X 0).symm).hom = pushout.inr\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPushout 0 0 b.inl b.inr\n[PROOFSTEP]\nconvert IsPushout.of_is_coproduct' h.isColimit HasZeroObject.zeroIsInitial\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPushout b.inl 0 b.snd 0\n[PROOFSTEP]\napply flip\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPushout 0 b.inl 0 b.snd\n[PROOFSTEP]\nrefine' of_right _ (by simp) (of_isBilimit h)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 0 \u226b 0 = b.inl \u226b b.snd\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPushout (0 \u226b 0) 0 0 (b.inr \u226b b.snd)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPushout b.inr 0 b.fst 0\n[PROOFSTEP]\nrefine' of_bot _ (by simp) (of_isBilimit h)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 b.inr \u226b b.fst = 0 \u226b 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPushout 0 (0 \u226b 0) (b.inl \u226b b.fst) 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPushout b.fst b.snd 0 0\n[PROOFSTEP]\nrefine' IsPushout.of_right _ (by simp) (IsPushout.inl_snd' h)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 b.fst \u226b 0 = b.snd \u226b 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPushout (b.inl \u226b b.fst) 0 0 (0 \u226b 0)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : IsIso f\ninst\u271d : IsIso inr\nsq : CommSq f g inl inr\n\u22a2 IsColimit (CommSq.cocone sq)\n[PROOFSTEP]\nrefine' PushoutCocone.IsColimit.mk _ (fun s => inv inr \u226b s.inr) (fun s => _) (by aesop_cat) (by aesop_cat)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : IsIso f\ninst\u271d : IsIso inr\nsq : CommSq f g inl inr\n\u22a2 \u2200 (s : PushoutCocone f g), inr \u226b (fun s => inv inr \u226b PushoutCocone.inr s) s = PushoutCocone.inr s\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : IsIso f\ninst\u271d : IsIso inr\nsq : CommSq f g inl inr\n\u22a2 \u2200 (s : PushoutCocone f g) (m : P \u27f6 s.pt),\n inl \u226b m = PushoutCocone.inl s \u2192 inr \u226b m = PushoutCocone.inr s \u2192 m = (fun s => inv inr \u226b PushoutCocone.inr s) s\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nZ X Y P : C\nf : Z \u27f6 X\ng : Z \u27f6 Y\ninl : X \u27f6 P\ninr : Y \u27f6 P\ninst\u271d\u00b9 : IsIso f\ninst\u271d : IsIso inr\nsq : CommSq f g inl inr\ns : PushoutCocone f g\n\u22a2 inl \u226b (fun s => inv inr \u226b PushoutCocone.inr s) s = PushoutCocone.inl s\n[PROOFSTEP]\nsimp only [\u2190 cancel_epi f, s.condition, sq.w_assoc, IsIso.hom_inv_id_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPullback f f g g'\n\u22a2 IsLimit (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g'))\n[PROOFSTEP]\nfapply Fork.IsLimit.mk\n[GOAL]\ncase lift\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPullback f f g g'\n\u22a2 (s : Fork g g') \u2192 s.pt \u27f6 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')).pt\n[PROOFSTEP]\nexact fun s => H.isLimit.lift (PullbackCone.mk s.\u03b9 s.\u03b9 s.condition)\n[GOAL]\ncase fac\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPullback f f g g'\n\u22a2 \u2200 (s : Fork g g'),\n IsLimit.lift (isLimit H) (PullbackCone.mk (Fork.\u03b9 s) (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b g = Fork.\u03b9 s \u226b g')) \u226b\n Fork.\u03b9 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')) =\n Fork.\u03b9 s\n[PROOFSTEP]\nexact fun s => H.isLimit.fac _ WalkingCospan.left\n[GOAL]\ncase uniq\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPullback f f g g'\n\u22a2 \u2200 (s : Fork g g') (m : s.pt \u27f6 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')).pt),\n m \u226b Fork.\u03b9 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')) = Fork.\u03b9 s \u2192\n m = IsLimit.lift (isLimit H) (PullbackCone.mk (Fork.\u03b9 s) (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b g = Fork.\u03b9 s \u226b g'))\n[PROOFSTEP]\nintro s m e\n[GOAL]\ncase uniq\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPullback f f g g'\ns : Fork g g'\nm : s.pt \u27f6 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')).pt\ne : m \u226b Fork.\u03b9 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')) = Fork.\u03b9 s\n\u22a2 m = IsLimit.lift (isLimit H) (PullbackCone.mk (Fork.\u03b9 s) (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b g = Fork.\u03b9 s \u226b g'))\n[PROOFSTEP]\napply PullbackCone.IsLimit.hom_ext H.isLimit\n[GOAL]\ncase uniq.h\u2080\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPullback f f g g'\ns : Fork g g'\nm : s.pt \u27f6 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')).pt\ne : m \u226b Fork.\u03b9 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')) = Fork.\u03b9 s\n\u22a2 m \u226b PullbackCone.fst (cone H) =\n IsLimit.lift (isLimit H) (PullbackCone.mk (Fork.\u03b9 s) (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b g = Fork.\u03b9 s \u226b g')) \u226b\n PullbackCone.fst (cone H)\n[PROOFSTEP]\nrefine' e.trans _\n[GOAL]\ncase uniq.h\u2081\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPullback f f g g'\ns : Fork g g'\nm : s.pt \u27f6 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')).pt\ne : m \u226b Fork.\u03b9 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')) = Fork.\u03b9 s\n\u22a2 m \u226b PullbackCone.snd (cone H) =\n IsLimit.lift (isLimit H) (PullbackCone.mk (Fork.\u03b9 s) (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b g = Fork.\u03b9 s \u226b g')) \u226b\n PullbackCone.snd (cone H)\n[PROOFSTEP]\nrefine' e.trans _\n[GOAL]\ncase uniq.h\u2080\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPullback f f g g'\ns : Fork g g'\nm : s.pt \u27f6 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')).pt\ne : m \u226b Fork.\u03b9 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')) = Fork.\u03b9 s\n\u22a2 Fork.\u03b9 s =\n IsLimit.lift (isLimit H) (PullbackCone.mk (Fork.\u03b9 s) (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b g = Fork.\u03b9 s \u226b g')) \u226b\n PullbackCone.fst (cone H)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase uniq.h\u2081\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPullback f f g g'\ns : Fork g g'\nm : s.pt \u27f6 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')).pt\ne : m \u226b Fork.\u03b9 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')) = Fork.\u03b9 s\n\u22a2 Fork.\u03b9 s =\n IsLimit.lift (isLimit H) (PullbackCone.mk (Fork.\u03b9 s) (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b g = Fork.\u03b9 s \u226b g')) \u226b\n PullbackCone.snd (cone H)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase uniq.h\u2080\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPullback f f g g'\ns : Fork g g'\nm : s.pt \u27f6 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')).pt\ne : m \u226b Fork.\u03b9 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')) = Fork.\u03b9 s\n\u22a2 IsLimit.lift (isLimit H) (PullbackCone.mk (Fork.\u03b9 s) (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b g = Fork.\u03b9 s \u226b g')) \u226b\n PullbackCone.fst (cone H) =\n Fork.\u03b9 s\n[PROOFSTEP]\nexact H.isLimit.fac _ _\n[GOAL]\ncase uniq.h\u2081\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPullback f f g g'\ns : Fork g g'\nm : s.pt \u27f6 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')).pt\ne : m \u226b Fork.\u03b9 (Fork.of\u03b9 f (_ : f \u226b g = f \u226b g')) = Fork.\u03b9 s\n\u22a2 IsLimit.lift (isLimit H) (PullbackCone.mk (Fork.\u03b9 s) (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b g = Fork.\u03b9 s \u226b g')) \u226b\n PullbackCone.snd (cone H) =\n Fork.\u03b9 s\n[PROOFSTEP]\nexact H.isLimit.fac _ _\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPushout f f' g g\n\u22a2 IsColimit (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g))\n[PROOFSTEP]\nfapply Cofork.IsColimit.mk\n[GOAL]\ncase desc\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPushout f f' g g\n\u22a2 (s : Cofork f f') \u2192 (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)).pt \u27f6 s.pt\n[PROOFSTEP]\nexact fun s => H.isColimit.desc (PushoutCocone.mk s.\u03c0 s.\u03c0 s.condition)\n[GOAL]\ncase fac\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPushout f f' g g\n\u22a2 \u2200 (s : Cofork f f'),\n Cofork.\u03c0 (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)) \u226b\n IsColimit.desc (isColimit H)\n (PushoutCocone.mk (Cofork.\u03c0 s) (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = f' \u226b Cofork.\u03c0 s)) =\n Cofork.\u03c0 s\n[PROOFSTEP]\nexact fun s => H.isColimit.fac _ WalkingSpan.left\n[GOAL]\ncase uniq\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPushout f f' g g\n\u22a2 \u2200 (s : Cofork f f') (m : (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)).pt \u27f6 s.pt),\n Cofork.\u03c0 (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)) \u226b m = Cofork.\u03c0 s \u2192\n m =\n IsColimit.desc (isColimit H) (PushoutCocone.mk (Cofork.\u03c0 s) (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = f' \u226b Cofork.\u03c0 s))\n[PROOFSTEP]\nintro s m e\n[GOAL]\ncase uniq\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPushout f f' g g\ns : Cofork f f'\nm : (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)).pt \u27f6 s.pt\ne : Cofork.\u03c0 (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)) \u226b m = Cofork.\u03c0 s\n\u22a2 m = IsColimit.desc (isColimit H) (PushoutCocone.mk (Cofork.\u03c0 s) (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = f' \u226b Cofork.\u03c0 s))\n[PROOFSTEP]\napply PushoutCocone.IsColimit.hom_ext H.isColimit\n[GOAL]\ncase uniq.h\u2080\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPushout f f' g g\ns : Cofork f f'\nm : (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)).pt \u27f6 s.pt\ne : Cofork.\u03c0 (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)) \u226b m = Cofork.\u03c0 s\n\u22a2 PushoutCocone.inl (cocone H) \u226b m =\n PushoutCocone.inl (cocone H) \u226b\n IsColimit.desc (isColimit H) (PushoutCocone.mk (Cofork.\u03c0 s) (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = f' \u226b Cofork.\u03c0 s))\n[PROOFSTEP]\nrefine' e.trans _\n[GOAL]\ncase uniq.h\u2081\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPushout f f' g g\ns : Cofork f f'\nm : (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)).pt \u27f6 s.pt\ne : Cofork.\u03c0 (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)) \u226b m = Cofork.\u03c0 s\n\u22a2 PushoutCocone.inr (cocone H) \u226b m =\n PushoutCocone.inr (cocone H) \u226b\n IsColimit.desc (isColimit H) (PushoutCocone.mk (Cofork.\u03c0 s) (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = f' \u226b Cofork.\u03c0 s))\n[PROOFSTEP]\nrefine' e.trans _\n[GOAL]\ncase uniq.h\u2080\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPushout f f' g g\ns : Cofork f f'\nm : (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)).pt \u27f6 s.pt\ne : Cofork.\u03c0 (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)) \u226b m = Cofork.\u03c0 s\n\u22a2 Cofork.\u03c0 s =\n PushoutCocone.inl (cocone H) \u226b\n IsColimit.desc (isColimit H) (PushoutCocone.mk (Cofork.\u03c0 s) (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = f' \u226b Cofork.\u03c0 s))\n[PROOFSTEP]\nsymm\n[GOAL]\ncase uniq.h\u2081\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPushout f f' g g\ns : Cofork f f'\nm : (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)).pt \u27f6 s.pt\ne : Cofork.\u03c0 (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)) \u226b m = Cofork.\u03c0 s\n\u22a2 Cofork.\u03c0 s =\n PushoutCocone.inr (cocone H) \u226b\n IsColimit.desc (isColimit H) (PushoutCocone.mk (Cofork.\u03c0 s) (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = f' \u226b Cofork.\u03c0 s))\n[PROOFSTEP]\nsymm\n[GOAL]\ncase uniq.h\u2080\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPushout f f' g g\ns : Cofork f f'\nm : (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)).pt \u27f6 s.pt\ne : Cofork.\u03c0 (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)) \u226b m = Cofork.\u03c0 s\n\u22a2 PushoutCocone.inl (cocone H) \u226b\n IsColimit.desc (isColimit H) (PushoutCocone.mk (Cofork.\u03c0 s) (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = f' \u226b Cofork.\u03c0 s)) =\n Cofork.\u03c0 s\n[PROOFSTEP]\nexact H.isColimit.fac _ _\n[GOAL]\ncase uniq.h\u2081\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf f' : X \u27f6 Y\ng g' : Y \u27f6 Z\nH : IsPushout f f' g g\ns : Cofork f f'\nm : (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)).pt \u27f6 s.pt\ne : Cofork.\u03c0 (Cofork.of\u03c0 g (_ : f \u226b g = f' \u226b g)) \u226b m = Cofork.\u03c0 s\n\u22a2 PushoutCocone.inr (cocone H) \u226b\n IsColimit.desc (isColimit H) (PushoutCocone.mk (Cofork.\u03c0 s) (Cofork.\u03c0 s) (_ : f \u226b Cofork.\u03c0 s = f' \u226b Cofork.\u03c0 s)) =\n Cofork.\u03c0 s\n[PROOFSTEP]\nexact H.isColimit.fac _ _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d\u00b2 : HasZeroObject C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 BicartesianSq biprod.fst biprod.snd 0 0\n[PROOFSTEP]\nconvert of_is_biproduct\u2081 (BinaryBiproduct.isBilimit X Y)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d\u00b2 : HasZeroObject C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 BicartesianSq 0 0 biprod.inl biprod.inr\n[PROOFSTEP]\nconvert of_is_biproduct\u2082 (BinaryBiproduct.isBilimit X Y)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : PreservesLimit (cospan h i) F\ns : IsPullback f g h i\n\u22a2 IsPullback (F.map f) (F.map g) (F.map h) (F.map i)\n[PROOFSTEP]\nrefine'\n IsPullback.of_isLimit' (F.map_commSq s.toCommSq)\n (IsLimit.equivOfNatIsoOfIso (cospanCompIso F h i) _ _ (WalkingCospan.ext _ _ _) (isLimitOfPreserves F s.isLimit))\n[GOAL]\ncase refine'_1\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : PreservesLimit (cospan h i) F\ns : IsPullback f g h i\n\u22a2 ((Cones.postcompose (cospanCompIso F h i).hom).obj (F.mapCone (IsPullback.cone s))).pt \u2245\n (CommSq.cone (_ : CommSq (F.map f) (F.map g) (F.map h) (F.map i))).pt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : PreservesLimit (cospan h i) F\ns : IsPullback f g h i\n\u22a2 NatTrans.app ((Cones.postcompose (cospanCompIso F h i).hom).obj (F.mapCone (IsPullback.cone s))).\u03c0\n WalkingCospan.left =\n (Iso.refl ((Cones.postcompose (cospanCompIso F h i).hom).obj (F.mapCone (IsPullback.cone s))).pt).hom \u226b\n NatTrans.app (CommSq.cone (_ : CommSq (F.map f) (F.map g) (F.map h) (F.map i))).\u03c0 WalkingCospan.left\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : PreservesLimit (cospan h i) F\ns : IsPullback f g h i\n\u22a2 NatTrans.app ((Cones.postcompose (cospanCompIso F h i).hom).obj (F.mapCone (IsPullback.cone s))).\u03c0\n WalkingCospan.right =\n (Iso.refl ((Cones.postcompose (cospanCompIso F h i).hom).obj (F.mapCone (IsPullback.cone s))).pt).hom \u226b\n NatTrans.app (CommSq.cone (_ : CommSq (F.map f) (F.map g) (F.map h) (F.map i))).\u03c0 WalkingCospan.right\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : PreservesColimit (span f g) F\ns : IsPushout f g h i\n\u22a2 IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n[PROOFSTEP]\nrefine'\n IsPushout.of_isColimit' (F.map_commSq s.toCommSq)\n (IsColimit.equivOfNatIsoOfIso (spanCompIso F f g) _ _ (WalkingSpan.ext _ _ _) (isColimitOfPreserves F s.isColimit))\n[GOAL]\ncase refine'_1\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : PreservesColimit (span f g) F\ns : IsPushout f g h i\n\u22a2 ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (IsPushout.cocone s))).pt \u2245\n (CommSq.cocone (_ : CommSq (F.map f) (F.map g) (F.map h) (F.map i))).pt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : PreservesColimit (span f g) F\ns : IsPushout f g h i\n\u22a2 NatTrans.app ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (IsPushout.cocone s))).\u03b9\n WalkingCospan.left \u226b\n (Iso.refl ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (IsPushout.cocone s))).pt).hom =\n NatTrans.app (CommSq.cocone (_ : CommSq (F.map f) (F.map g) (F.map h) (F.map i))).\u03b9 WalkingCospan.left\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : PreservesColimit (span f g) F\ns : IsPushout f g h i\n\u22a2 NatTrans.app ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (IsPushout.cocone s))).\u03b9\n WalkingCospan.right \u226b\n (Iso.refl ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (IsPushout.cocone s))).pt).hom =\n NatTrans.app (CommSq.cocone (_ : CommSq (F.map f) (F.map g) (F.map h) (F.map i))).\u03b9 WalkingCospan.right\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsLimit (cospan h i) F\ne : f \u226b h = g \u226b i\nH : IsPullback (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 IsPullback f g h i\n[PROOFSTEP]\nrefine' \u27e8\u27e8e\u27e9, \u27e8isLimitOfReflects F <| _\u27e9\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsLimit (cospan h i) F\ne : f \u226b h = g \u226b i\nH : IsPullback (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 IsLimit (F.mapCone (PullbackCone.mk f g (_ : f \u226b h = g \u226b i)))\n[PROOFSTEP]\nrefine' (IsLimit.equivOfNatIsoOfIso (cospanCompIso F h i) _ _ (WalkingCospan.ext _ _ _)).symm H.isLimit\n[GOAL]\ncase refine'_1\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsLimit (cospan h i) F\ne : f \u226b h = g \u226b i\nH : IsPullback (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 ((Cones.postcompose (cospanCompIso F h i).hom).obj (F.mapCone (PullbackCone.mk f g (_ : f \u226b h = g \u226b i)))).pt \u2245\n (cone H).pt\ncase refine'_2\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsLimit (cospan h i) F\ne : f \u226b h = g \u226b i\nH : IsPullback (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 NatTrans.app\n ((Cones.postcompose (cospanCompIso F h i).hom).obj (F.mapCone (PullbackCone.mk f g (_ : f \u226b h = g \u226b i)))).\u03c0\n WalkingCospan.left =\n ?refine'_1.hom \u226b NatTrans.app (cone H).\u03c0 WalkingCospan.left\ncase refine'_3\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsLimit (cospan h i) F\ne : f \u226b h = g \u226b i\nH : IsPullback (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 NatTrans.app\n ((Cones.postcompose (cospanCompIso F h i).hom).obj (F.mapCone (PullbackCone.mk f g (_ : f \u226b h = g \u226b i)))).\u03c0\n WalkingCospan.right =\n ?refine'_1.hom \u226b NatTrans.app (cone H).\u03c0 WalkingCospan.right\n[PROOFSTEP]\nexacts [Iso.refl _, (Category.comp_id _).trans (Category.id_comp _).symm,\n (Category.comp_id _).trans (Category.id_comp _).symm]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d\u00b9 : ReflectsLimit (cospan h i) F\ninst\u271d : Faithful F\nH : IsPullback (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 F.map (f \u226b h) = F.map (g \u226b i)\n[PROOFSTEP]\nsimpa only [F.map_comp] using H.w\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsColimit (span f g) F\ne : f \u226b h = g \u226b i\nH : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 IsPushout f g h i\n[PROOFSTEP]\nrefine' \u27e8\u27e8e\u27e9, \u27e8isColimitOfReflects F <| _\u27e9\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsColimit (span f g) F\ne : f \u226b h = g \u226b i\nH : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 IsColimit (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i)))\n[PROOFSTEP]\nrefine' (IsColimit.equivOfNatIsoOfIso (spanCompIso F f g) _ _ (WalkingSpan.ext _ _ _)).symm H.isColimit\n[GOAL]\ncase refine'_1\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsColimit (span f g) F\ne : f \u226b h = g \u226b i\nH : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i)))).pt \u2245\n (cocone H).pt\ncase refine'_2\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsColimit (span f g) F\ne : f \u226b h = g \u226b i\nH : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 NatTrans.app\n ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i)))).\u03b9\n WalkingCospan.left \u226b\n ?refine'_1.hom =\n NatTrans.app (cocone H).\u03b9 WalkingCospan.left\ncase refine'_3\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsColimit (span f g) F\ne : f \u226b h = g \u226b i\nH : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 NatTrans.app\n ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i)))).\u03b9\n WalkingCospan.right \u226b\n ?refine'_1.hom =\n NatTrans.app (cocone H).\u03b9 WalkingCospan.right\n[PROOFSTEP]\nexacts [Iso.refl _, (Category.comp_id _).trans (Category.id_comp _), (Category.comp_id _).trans (Category.id_comp _)]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d\u00b9 : ReflectsColimit (span f g) F\ninst\u271d : Faithful F\nH : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 F.map (f \u226b h) = F.map (g \u226b i)\n[PROOFSTEP]\nsimpa only [F.map_comp] using H.w\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.CommSq", "llama_tokens": 23410, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.4201745973850446}} {"text": "[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a0d\u207b (_x : \u03b1), 0 \u2202\u03bc = 0\n[PROOFSTEP]\nrw [laverage, lintegral_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g f : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u22020 = 0\n[PROOFSTEP]\nsimp [laverage]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g f : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc = (\u222b\u207b (x : \u03b1), f x \u2202\u03bc) / \u2191\u2191\u03bc univ\n[PROOFSTEP]\nrw [laverage_eq', lintegral_smul_measure, ENNReal.div_eq_inv_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsProbabilityMeasure \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [laverage, measure_univ, inv_one, one_smul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2191\u2191\u03bc univ * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\ncases' eq_or_ne \u03bc 0 with h\u03bc h\u03bc\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u03bc = 0\n\u22a2 \u2191\u2191\u03bc univ * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [h\u03bc, lintegral_zero_measure, laverage_zero_measure, mul_zero]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u03bc \u2260 0\n\u22a2 \u2191\u2191\u03bc univ * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [laverage_eq, ENNReal.mul_div_cancel' (measure_univ_ne_zero.2 h\u03bc) (measure_ne_top _ _)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t : Set \u03b1\nf\u271d g f : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\n\u22a2 \u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc = (\u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc) / \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [laverage_eq, restrict_apply_univ]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t : Set \u03b1\nf\u271d g f : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\n\u22a2 \u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202(\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 Measure.restrict \u03bc s\n[PROOFSTEP]\nsimp only [laverage_eq', restrict_apply_univ]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g\u271d f g : \u03b1 \u2192 \u211d\u22650\u221e\nh : f =\u1da0[ae \u03bc] g\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc = \u2a0d\u207b (x : \u03b1), g x \u2202\u03bc\n[PROOFSTEP]\nsimp only [laverage_eq, lintegral_congr_ae h]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh : s =\u1da0[ae \u03bc] t\n\u22a2 \u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u2a0d\u207b (x : \u03b1) in t, f x \u2202\u03bc\n[PROOFSTEP]\nsimp only [setLaverage_eq, set_lintegral_congr h, measure_congr h]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhs : MeasurableSet s\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = g x\n\u22a2 \u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u2a0d\u207b (x : \u03b1) in s, g x \u2202\u03bc\n[PROOFSTEP]\nsimp only [laverage_eq, set_lintegral_congr_fun hs h]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc < \u22a4\n[PROOFSTEP]\nobtain rfl | h\u03bc := eq_or_ne \u03bc 0\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u222b\u207b (x : \u03b1), f x \u22020 \u2260 \u22a4\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u22020 < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh\u03bc : \u03bc \u2260 0\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc < \u22a4\n[PROOFSTEP]\nrw [laverage_eq]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nh\u03bc : \u03bc \u2260 0\n\u22a2 (\u222b\u207b (x : \u03b1), f x \u2202\u03bc) / \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nexact div_lt_top hf (measure_univ_ne_zero.2 h\u03bc)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n[PROOFSTEP]\nby_cases h\u03bc : IsFiniteMeasure \u03bc\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u00acIsFiniteMeasure \u03bc\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n[PROOFSTEP]\nrw [not_isFiniteMeasure_iff] at h\u03bc \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc univ = \u22a4\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n[PROOFSTEP]\nsimp [laverage_eq, h\u03bc]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n[PROOFSTEP]\nby_cases h\u03bd : IsFiniteMeasure \u03bd\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : IsFiniteMeasure \u03bd\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : \u00acIsFiniteMeasure \u03bd\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : \u00acIsFiniteMeasure \u03bd\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n[PROOFSTEP]\nrw [not_isFiniteMeasure_iff] at h\u03bd \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : \u2191\u2191\u03bd univ = \u22a4\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n[PROOFSTEP]\nsimp [laverage_eq, h\u03bd]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : IsFiniteMeasure \u03bd\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n[PROOFSTEP]\nhaveI := h\u03bc\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : IsFiniteMeasure \u03bd\nthis : IsFiniteMeasure \u03bc\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n[PROOFSTEP]\nhaveI := h\u03bd\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : IsFiniteMeasure \u03bc\nh\u03bd : IsFiniteMeasure \u03bd\nthis\u271d : IsFiniteMeasure \u03bc\nthis : IsFiniteMeasure \u03bd\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd\n[PROOFSTEP]\nsimp only [\u2190 ENNReal.mul_div_right_comm, measure_mul_laverage, \u2190 ENNReal.add_div, \u2190 lintegral_add_measure, \u2190\n Measure.add_apply, \u2190 laverage_eq]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g f : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc s * \u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc\n[PROOFSTEP]\nhave := Fact.mk h.lt_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g f : \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2191\u2191\u03bc s \u2260 \u22a4\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 \u2191\u2191\u03bc s * \u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 measure_mul_laverage, restrict_apply_univ]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\n\u22a2 \u2a0d\u207b (x : \u03b1) in s \u222a t, f x \u2202\u03bc =\n \u2191\u2191\u03bc s / (\u2191\u2191\u03bc s + \u2191\u2191\u03bc t) * \u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc + \u2191\u2191\u03bc t / (\u2191\u2191\u03bc s + \u2191\u2191\u03bc t) * \u2a0d\u207b (x : \u03b1) in t, f x \u2202\u03bc\n[PROOFSTEP]\nrw [restrict_union\u2080 hd ht, laverage_add_measure, restrict_apply_univ, restrict_apply_univ]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u2080 : \u2191\u2191\u03bc s \u2260 0\nht\u2080 : \u2191\u2191\u03bc t \u2260 0\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2a0d\u207b (x : \u03b1) in s \u222a t, f x \u2202\u03bc \u2208 openSegment \u211d\u22650\u221e (\u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc) (\u2a0d\u207b (x : \u03b1) in t, f x \u2202\u03bc)\n[PROOFSTEP]\nrefine'\n \u27e8\u03bc s / (\u03bc s + \u03bc t), \u03bc t / (\u03bc s + \u03bc t), ENNReal.div_pos hs\u2080 <| add_ne_top.2 \u27e8hs\u03bc, ht\u03bc\u27e9,\n ENNReal.div_pos ht\u2080 <| add_ne_top.2 \u27e8hs\u03bc, ht\u03bc\u27e9, _, (laverage_union hd ht).symm\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u2080 : \u2191\u2191\u03bc s \u2260 0\nht\u2080 : \u2191\u2191\u03bc t \u2260 0\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc s / (\u2191\u2191\u03bc s + \u2191\u2191\u03bc t) + \u2191\u2191\u03bc t / (\u2191\u2191\u03bc s + \u2191\u2191\u03bc t) = 1\n[PROOFSTEP]\nrw [\u2190 ENNReal.add_div, ENNReal.div_self (add_eq_zero.not.2 fun h => hs\u2080 h.1) (add_ne_top.2 \u27e8hs\u03bc, ht\u03bc\u27e9)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2a0d\u207b (x : \u03b1) in s \u222a t, f x \u2202\u03bc \u2208 [\u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc-[\u211d\u22650\u221e]\u2a0d\u207b (x : \u03b1) in t, f x \u2202\u03bc]\n[PROOFSTEP]\nby_cases hs\u2080 : \u03bc s = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhs\u2080 : \u2191\u2191\u03bc s = 0\n\u22a2 \u2a0d\u207b (x : \u03b1) in s \u222a t, f x \u2202\u03bc \u2208 [\u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc-[\u211d\u22650\u221e]\u2a0d\u207b (x : \u03b1) in t, f x \u2202\u03bc]\n[PROOFSTEP]\nrw [\u2190 ae_eq_empty] at hs\u2080 \n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhs\u2080 : s =\u1da0[ae \u03bc] \u2205\n\u22a2 \u2a0d\u207b (x : \u03b1) in s \u222a t, f x \u2202\u03bc \u2208 [\u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc-[\u211d\u22650\u221e]\u2a0d\u207b (x : \u03b1) in t, f x \u2202\u03bc]\n[PROOFSTEP]\nrw [restrict_congr_set (hs\u2080.union EventuallyEq.rfl), empty_union]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhs\u2080 : s =\u1da0[ae \u03bc] \u2205\n\u22a2 \u2a0d\u207b (x : \u03b1) in t, f x \u2202\u03bc \u2208 [\u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc-[\u211d\u22650\u221e]\u2a0d\u207b (x : \u03b1) in t, f x \u2202\u03bc]\n[PROOFSTEP]\nexact right_mem_segment _ _ _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhs\u2080 : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2a0d\u207b (x : \u03b1) in s \u222a t, f x \u2202\u03bc \u2208 [\u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc-[\u211d\u22650\u221e]\u2a0d\u207b (x : \u03b1) in t, f x \u2202\u03bc]\n[PROOFSTEP]\nrefine' \u27e8\u03bc s / (\u03bc s + \u03bc t), \u03bc t / (\u03bc s + \u03bc t), zero_le _, zero_le _, _, (laverage_union hd ht).symm\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhs\u2080 : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc s / (\u2191\u2191\u03bc s + \u2191\u2191\u03bc t) + \u2191\u2191\u03bc t / (\u2191\u2191\u03bc s + \u2191\u2191\u03bc t) = 1\n[PROOFSTEP]\nrw [\u2190 ENNReal.add_div, ENNReal.div_self (add_eq_zero.not.2 fun h => hs\u2080 h.1) (add_ne_top.2 \u27e8hs\u03bc, ht\u03bc\u27e9)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhs : NullMeasurableSet s\nhs\u2080 : \u2191\u2191\u03bc s \u2260 0\nhsc\u2080 : \u2191\u2191\u03bc s\u1d9c \u2260 0\n\u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc \u2208 openSegment \u211d\u22650\u221e (\u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc) (\u2a0d\u207b (x : \u03b1) in s\u1d9c, f x \u2202\u03bc)\n[PROOFSTEP]\nsimpa only [union_compl_self, restrict_univ] using\n laverage_union_mem_openSegment aedisjoint_compl_right hs.compl hs\u2080 hsc\u2080 (measure_ne_top _ _) (measure_ne_top _ _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc\u271d \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nh : NeZero \u03bc\nc : \u211d\u22650\u221e\n\u22a2 \u2a0d\u207b (_x : \u03b1), c \u2202\u03bc = c\n[PROOFSTEP]\nsimp only [laverage, lintegral_const, measure_univ, mul_one]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhs\u2080 : \u2191\u2191\u03bc s \u2260 0\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\u22650\u221e\n\u22a2 \u2a0d\u207b (_x : \u03b1) in s, c \u2202\u03bc = c\n[PROOFSTEP]\nsimp only [setLaverage_eq, lintegral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter, div_eq_mul_inv,\n mul_assoc, ENNReal.mul_inv_cancel hs\u2080 hs, mul_one]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc\u271d \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (_x : \u03b1), \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nobtain rfl | h\u03bc := eq_or_ne \u03bc 0\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g f : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure 0\n\u22a2 \u222b\u207b (_x : \u03b1), \u2a0d\u207b (a : \u03b1), f a \u22020 \u22020 = \u222b\u207b (x : \u03b1), f x \u22020\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc\u271d \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u03bc \u2260 0\n\u22a2 \u222b\u207b (_x : \u03b1), \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_const, laverage_eq, ENNReal.div_mul_cancel (measure_univ_ne_zero.2 h\u03bc) (measure_ne_top _ _)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 E\n\u22a2 \u2a0d (x : \u03b1), 0 \u2202\u03bc = 0\n[PROOFSTEP]\nrw [average, integral_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\n\u22a2 \u2a0d (x : \u03b1), f x \u22020 = 0\n[PROOFSTEP]\nrw [average, smul_zero, integral_zero_measure]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\n\u22a2 \u2a0d (x : \u03b1), f x \u2202\u03bc = (ENNReal.toReal (\u2191\u2191\u03bc univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [average_eq', integral_smul_measure, ENNReal.toReal_inv]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsProbabilityMeasure \u03bc\nf : \u03b1 \u2192 E\n\u22a2 \u2a0d (x : \u03b1), f x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [average, measure_univ, inv_one, one_smul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2a0d (x : \u03b1), f x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\ncases' eq_or_ne \u03bc 0 with h\u03bc h\u03bc\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nh\u03bc : \u03bc = 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2a0d (x : \u03b1), f x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [h\u03bc, integral_zero_measure, average_zero_measure, smul_zero]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nh\u03bc : \u03bc \u2260 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2a0d (x : \u03b1), f x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nrw [average_eq, smul_inv_smul\u2080]\n[GOAL]\ncase inr.hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nh\u03bc : \u03bc \u2260 0\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2260 0\n[PROOFSTEP]\nrefine' (ENNReal.toReal_pos _ <| measure_ne_top _ _).ne'\n[GOAL]\ncase inr.hc\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nh\u03bc : \u03bc \u2260 0\n\u22a2 \u2191\u2191\u03bc univ \u2260 0\n[PROOFSTEP]\nrwa [Ne.def, measure_univ_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns : Set \u03b1\n\u22a2 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc = (ENNReal.toReal (\u2191\u2191\u03bc s))\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n[PROOFSTEP]\nrw [average_eq, restrict_apply_univ]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns : Set \u03b1\n\u22a2 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202(\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 Measure.restrict \u03bc s\n[PROOFSTEP]\nsimp only [average_eq', restrict_apply_univ]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\n\u22a2 \u2a0d (x : \u03b1), f x \u2202\u03bc = \u2a0d (x : \u03b1), g x \u2202\u03bc\n[PROOFSTEP]\nsimp only [average_eq, integral_congr_ae h]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 E\nh : s =\u1da0[ae \u03bc] t\n\u22a2 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc = \u2a0d (x : \u03b1) in t, f x \u2202\u03bc\n[PROOFSTEP]\nsimp only [setAverage_eq, set_integral_congr_set_ae h, measure_congr h]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 E\nhs : MeasurableSet s\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = g x\n\u22a2 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc = \u2a0d (x : \u03b1) in s, g x \u2202\u03bc\n[PROOFSTEP]\nsimp only [average_eq, set_integral_congr_ae hs h]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\n\u03bc \u03bd\u271d : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\n\u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nf : \u03b1 \u2192 E\nh\u03bc : Integrable f\nh\u03bd : Integrable f\n\u22a2 \u2a0d (x : \u03b1), f x \u2202(\u03bc + \u03bd) =\n (ENNReal.toReal (\u2191\u2191\u03bc univ) / (ENNReal.toReal (\u2191\u2191\u03bc univ) + ENNReal.toReal (\u2191\u2191\u03bd univ))) \u2022 \u2a0d (x : \u03b1), f x \u2202\u03bc +\n (ENNReal.toReal (\u2191\u2191\u03bd univ) / (ENNReal.toReal (\u2191\u2191\u03bc univ) + ENNReal.toReal (\u2191\u2191\u03bd univ))) \u2022 \u2a0d (x : \u03b1), f x \u2202\u03bd\n[PROOFSTEP]\nsimp only [div_eq_inv_mul, mul_smul, measure_smul_average, \u2190 smul_add, \u2190 integral_add_measure h\u03bc h\u03bd, \u2190\n ENNReal.toReal_add (measure_ne_top \u03bc _) (measure_ne_top \u03bd _)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\n\u03bc \u03bd\u271d : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\n\u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nf : \u03b1 \u2192 E\nh\u03bc : Integrable f\nh\u03bd : Integrable f\n\u22a2 \u2a0d (x : \u03b1), f x \u2202(\u03bc + \u03bd) = (ENNReal.toReal (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd)\n[PROOFSTEP]\nrw [average_eq, Measure.add_apply]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns : Set \u03b1\nh : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc\n[PROOFSTEP]\nhaveI := Fact.mk h.lt_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns : Set \u03b1\nh : \u2191\u2191\u03bc s \u2260 \u22a4\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 measure_smul_average, restrict_apply_univ]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc =\n (ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc +\n (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc\n[PROOFSTEP]\nhaveI := Fact.mk hs\u03bc.lt_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc =\n (ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc +\n (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc\n[PROOFSTEP]\nhaveI := Fact.mk ht\u03bc.lt_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nthis\u271d : Fact (\u2191\u2191\u03bc s < \u22a4)\nthis : Fact (\u2191\u2191\u03bc t < \u22a4)\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc =\n (ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc +\n (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc\n[PROOFSTEP]\nrw [restrict_union\u2080 hd ht, average_add_measure hfs hft, restrict_apply_univ, restrict_apply_univ]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u2080 : \u2191\u2191\u03bc s \u2260 0\nht\u2080 : \u2191\u2191\u03bc t \u2260 0\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc \u2208 openSegment \u211d (\u2a0d (x : \u03b1) in s, f x \u2202\u03bc) (\u2a0d (x : \u03b1) in t, f x \u2202\u03bc)\n[PROOFSTEP]\nreplace hs\u2080 : 0 < (\u03bc s).toReal\n[GOAL]\ncase hs\u2080\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u2080 : \u2191\u2191\u03bc s \u2260 0\nht\u2080 : \u2191\u2191\u03bc t \u2260 0\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\n\u22a2 0 < ENNReal.toReal (\u2191\u2191\u03bc s)\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nht\u2080 : \u2191\u2191\u03bc t \u2260 0\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nhs\u2080 : 0 < ENNReal.toReal (\u2191\u2191\u03bc s)\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc \u2208 openSegment \u211d (\u2a0d (x : \u03b1) in s, f x \u2202\u03bc) (\u2a0d (x : \u03b1) in t, f x \u2202\u03bc)\n[PROOFSTEP]\nexact ENNReal.toReal_pos hs\u2080 hs\u03bc\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nht\u2080 : \u2191\u2191\u03bc t \u2260 0\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nhs\u2080 : 0 < ENNReal.toReal (\u2191\u2191\u03bc s)\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc \u2208 openSegment \u211d (\u2a0d (x : \u03b1) in s, f x \u2202\u03bc) (\u2a0d (x : \u03b1) in t, f x \u2202\u03bc)\n[PROOFSTEP]\nreplace ht\u2080 : 0 < (\u03bc t).toReal\n[GOAL]\ncase ht\u2080\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nht\u2080 : \u2191\u2191\u03bc t \u2260 0\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nhs\u2080 : 0 < ENNReal.toReal (\u2191\u2191\u03bc s)\n\u22a2 0 < ENNReal.toReal (\u2191\u2191\u03bc t)\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nhs\u2080 : 0 < ENNReal.toReal (\u2191\u2191\u03bc s)\nht\u2080 : 0 < ENNReal.toReal (\u2191\u2191\u03bc t)\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc \u2208 openSegment \u211d (\u2a0d (x : \u03b1) in s, f x \u2202\u03bc) (\u2a0d (x : \u03b1) in t, f x \u2202\u03bc)\n[PROOFSTEP]\nexact ENNReal.toReal_pos ht\u2080 ht\u03bc\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nhs\u2080 : 0 < ENNReal.toReal (\u2191\u2191\u03bc s)\nht\u2080 : 0 < ENNReal.toReal (\u2191\u2191\u03bc t)\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc \u2208 openSegment \u211d (\u2a0d (x : \u03b1) in s, f x \u2202\u03bc) (\u2a0d (x : \u03b1) in t, f x \u2202\u03bc)\n[PROOFSTEP]\nrefine' mem_openSegment_iff_div.mpr \u27e8(\u03bc s).toReal, (\u03bc t).toReal, hs\u2080, ht\u2080, (average_union hd ht hs\u03bc ht\u03bc hfs hft).symm\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc \u2208 [\u2a0d (x : \u03b1) in s, f x \u2202\u03bc-[\u211d]\u2a0d (x : \u03b1) in t, f x \u2202\u03bc]\n[PROOFSTEP]\nby_cases hse : \u03bc s = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nhse : \u2191\u2191\u03bc s = 0\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc \u2208 [\u2a0d (x : \u03b1) in s, f x \u2202\u03bc-[\u211d]\u2a0d (x : \u03b1) in t, f x \u2202\u03bc]\n[PROOFSTEP]\nrw [\u2190 ae_eq_empty] at hse \n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nhse : s =\u1da0[ae \u03bc] \u2205\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc \u2208 [\u2a0d (x : \u03b1) in s, f x \u2202\u03bc-[\u211d]\u2a0d (x : \u03b1) in t, f x \u2202\u03bc]\n[PROOFSTEP]\nrw [restrict_congr_set (hse.union EventuallyEq.rfl), empty_union]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nhse : s =\u1da0[ae \u03bc] \u2205\n\u22a2 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc \u2208 [\u2a0d (x : \u03b1) in s, f x \u2202\u03bc-[\u211d]\u2a0d (x : \u03b1) in t, f x \u2202\u03bc]\n[PROOFSTEP]\nexact right_mem_segment _ _ _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nhse : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc \u2208 [\u2a0d (x : \u03b1) in s, f x \u2202\u03bc-[\u211d]\u2a0d (x : \u03b1) in t, f x \u2202\u03bc]\n[PROOFSTEP]\nrefine'\n mem_segment_iff_div.mpr\n \u27e8(\u03bc s).toReal, (\u03bc t).toReal, ENNReal.toReal_nonneg, ENNReal.toReal_nonneg, _,\n (average_union hd ht hs\u03bc ht\u03bc hfs hft).symm\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\nf\u271d g f : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nhse : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 0 < ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t)\n[PROOFSTEP]\ncalc\n 0 < (\u03bc s).toReal := ENNReal.toReal_pos hse hs\u03bc\n _ \u2264 _ := le_add_of_nonneg_right ENNReal.toReal_nonneg\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns\u271d t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhs : NullMeasurableSet s\nhs\u2080 : \u2191\u2191\u03bc s \u2260 0\nhsc\u2080 : \u2191\u2191\u03bc s\u1d9c \u2260 0\nhfi : Integrable f\n\u22a2 \u2a0d (x : \u03b1), f x \u2202\u03bc \u2208 openSegment \u211d (\u2a0d (x : \u03b1) in s, f x \u2202\u03bc) (\u2a0d (x : \u03b1) in s\u1d9c, f x \u2202\u03bc)\n[PROOFSTEP]\nsimpa only [union_compl_self, restrict_univ] using\n average_union_mem_openSegment aedisjoint_compl_right hs.compl hs\u2080 hsc\u2080 (measure_ne_top _ _) (measure_ne_top _ _)\n hfi.integrableOn hfi.integrableOn\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc\u271d \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nh : NeZero \u03bc\nc : E\n\u22a2 \u2a0d (_x : \u03b1), c \u2202\u03bc = c\n[PROOFSTEP]\nrw [average, integral_const, measure_univ, ENNReal.one_toReal, one_smul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc\u271d \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\n\u22a2 \u222b (x : \u03b1), \u2a0d (a : \u03b1), f a \u2202\u03bc \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc\u271d \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\n\u22a2 \u222b (x : \u03b1), f x - \u2a0d (a : \u03b1), f a \u2202\u03bc \u2202\u03bc = 0\n[PROOFSTEP]\nby_cases hf : Integrable f \u03bc\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc\u271d \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 \u222b (x : \u03b1), f x - \u2a0d (a : \u03b1), f a \u2202\u03bc \u2202\u03bc = 0\n[PROOFSTEP]\nrw [integral_sub hf (integrable_const _), integral_average, sub_self]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc\u271d \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b1), f x - \u2a0d (a : \u03b1), f a \u2202\u03bc \u2202\u03bc = 0\n[PROOFSTEP]\nrefine integral_undef fun h => hf ?_\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc\u271d \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\nh : Integrable fun x => f x - \u2a0d (a : \u03b1), f a \u2202\u03bc\n\u22a2 Integrable f\n[PROOFSTEP]\nconvert h.add (integrable_const (\u2a0d a, f a \u2202\u03bc))\n[GOAL]\ncase h.e'_5\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc\u271d \u03bd : Measure \u03b1\ns t : Set \u03b1\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\nh : Integrable fun x => f x - \u2a0d (a : \u03b1), f a \u2202\u03bc\n\u22a2 f = (fun x => f x - \u2a0d (a : \u03b1), f a \u2202\u03bc) + fun x => \u2a0d (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nexact (sub_add_cancel _ _).symm\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 E\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Integrable f\n\u22a2 \u222b (x : \u03b1), \u2a0d (a : \u03b1), f a \u2202\u03bc - f x \u2202\u03bc = 0\n[PROOFSTEP]\nrw [integral_sub (integrable_const _) hf, integral_average, sub_self]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf\u2080 : 0 \u2264\u1da0[ae \u03bc] f\n\u22a2 ENNReal.ofReal (\u2a0d (x : \u03b1), f x \u2202\u03bc) = (\u222b\u207b (x : \u03b1), ENNReal.ofReal (f x) \u2202\u03bc) / \u2191\u2191\u03bc univ\n[PROOFSTEP]\nobtain rfl | h\u03bc := eq_or_ne \u03bc 0\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bd : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf\u2080 : 0 \u2264\u1da0[ae 0] f\n\u22a2 ENNReal.ofReal (\u2a0d (x : \u03b1), f x \u22020) = (\u222b\u207b (x : \u03b1), ENNReal.ofReal (f x) \u22020) / \u2191\u21910 univ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf\u2080 : 0 \u2264\u1da0[ae \u03bc] f\nh\u03bc : \u03bc \u2260 0\n\u22a2 ENNReal.ofReal (\u2a0d (x : \u03b1), f x \u2202\u03bc) = (\u222b\u207b (x : \u03b1), ENNReal.ofReal (f x) \u2202\u03bc) / \u2191\u2191\u03bc univ\n[PROOFSTEP]\nrw [average_eq, smul_eq_mul, \u2190 toReal_inv, ofReal_mul toReal_nonneg,\n ofReal_toReal (inv_ne_top.2 <| measure_univ_ne_zero.2 h\u03bc), ofReal_integral_eq_lintegral_ofReal hf hf\u2080,\n ENNReal.div_eq_inv_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u211d\nhf : IntegrableOn f s\nhf\u2080 : 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] f\n\u22a2 ENNReal.ofReal (\u2a0d (x : \u03b1) in s, f x \u2202\u03bc) = (\u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (f x) \u2202\u03bc) / \u2191\u2191\u03bc s\n[PROOFSTEP]\nsimpa using ofReal_average hf hf\u2080\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 \u22a4\n\u22a2 ENNReal.toReal (\u2a0d\u207b (x : \u03b1), f x \u2202\u03bc) = \u2a0d (x : \u03b1), ENNReal.toReal (f x) \u2202\u03bc\n[PROOFSTEP]\nrw [average_eq, laverage_eq, smul_eq_mul, toReal_div, div_eq_inv_mul, \u2190\n integral_toReal hf (hf'.mono fun _ => lt_top_iff_ne_top.2)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf' : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, f x \u2260 \u22a4\n\u22a2 ENNReal.toReal (\u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc) = \u2a0d (x : \u03b1) in s, ENNReal.toReal (f x) \u2202\u03bc\n[PROOFSTEP]\nsimpa [laverage_eq] using toReal_laverage hf hf'\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\n\u22a2 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc}\n[PROOFSTEP]\nrefine' pos_iff_ne_zero.2 fun H => _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\n\u22a2 False\n[PROOFSTEP]\nreplace H : (\u03bc.restrict s) {x | f x \u2264 \u2a0d a in s, f a \u2202\u03bc} = 0\n[GOAL]\ncase H\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\n\u22a2 \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\n[PROOFSTEP]\nrwa [restrict_apply\u2080, inter_comm]\n[GOAL]\ncase H\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\n\u22a2 NullMeasurableSet {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc}\n[PROOFSTEP]\nexact AEStronglyMeasurable.nullMeasurableSet_le hf.1 aestronglyMeasurable_const\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\n\u22a2 False\n[PROOFSTEP]\nhaveI := Fact.mk h\u03bc\u2081.lt_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 False\n[PROOFSTEP]\nrefine' (integral_sub_average (\u03bc.restrict s) f).not_gt _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 0 < \u222b (x : \u03b1) in s, f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc \u2202\u03bc\n[PROOFSTEP]\nrefine' (set_integral_pos_iff_support_of_nonneg_ae _ _).2 _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc\n[PROOFSTEP]\nrefine' eq_bot_mono (measure_mono fun x hx => _) H\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\nx : \u03b1\nhx : x \u2208 {x | (fun x => OfNat.ofNat 0 x \u2264 (fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc) x) x}\u1d9c\n\u22a2 x \u2208 {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc}\n[PROOFSTEP]\nsimp only [Pi.zero_apply, sub_nonneg, mem_compl_iff, mem_setOf_eq, not_le] at hx \n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\nx : \u03b1\nhx : f x < \u2a0d (a : \u03b1) in s, f a \u2202\u03bc\n\u22a2 x \u2208 {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc}\n[PROOFSTEP]\nexact hx.le\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 IntegrableOn (fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc) s\n[PROOFSTEP]\nexact hf.sub (integrableOn_const.2 <| Or.inr <| lt_top_iff_ne_top.2 h\u03bc\u2081)\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 0 < \u2191\u2191\u03bc ((support fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc) \u2229 s)\n[PROOFSTEP]\nrwa [pos_iff_ne_zero, inter_comm, \u2190 diff_compl, \u2190 diff_inter_self_eq_diff, measure_diff_null]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 \u2191\u2191\u03bc ((support fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc)\u1d9c \u2229 s) = 0\n[PROOFSTEP]\nrefine' eq_bot_mono (measure_mono _) (measure_inter_eq_zero_of_restrict H)\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\nH : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 (support fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc)\u1d9c \u2229 s \u2286 {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} \u2229 s\n[PROOFSTEP]\nexact inter_subset_inter_left _ fun a ha => (sub_eq_zero.1 <| of_not_not ha).le\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : IntegrableOn f s\n\u22a2 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc \u2264 f x}\n[PROOFSTEP]\nsimpa [integral_neg, neg_div] using measure_le_setAverage_pos h\u03bc h\u03bc\u2081 hf.neg\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc : \u03bc \u2260 0\nhf : Integrable f\n\u22a2 0 < \u2191\u2191\u03bc {x | f x \u2264 \u2a0d (a : \u03b1), f a \u2202\u03bc}\n[PROOFSTEP]\nsimpa using measure_le_setAverage_pos (Measure.measure_univ_ne_zero.2 h\u03bc) (measure_ne_top _ _) hf.integrableOn\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc : \u03bc \u2260 0\nhf : Integrable f\n\u22a2 0 < \u2191\u2191\u03bc {x | \u2a0d (a : \u03b1), f a \u2202\u03bc \u2264 f x}\n[PROOFSTEP]\nsimpa using measure_setAverage_le_pos (Measure.measure_univ_ne_zero.2 h\u03bc) (measure_ne_top _ _) hf.integrableOn\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc : \u03bc \u2260 0\nhf : Integrable f\nhN : \u2191\u2191\u03bc N = 0\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 f x \u2264 \u2a0d (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nhave := measure_le_average_pos h\u03bc hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc : \u03bc \u2260 0\nhf : Integrable f\nhN : \u2191\u2191\u03bc N = 0\nthis : 0 < \u2191\u2191\u03bc {x | f x \u2264 \u2a0d (a : \u03b1), f a \u2202\u03bc}\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 f x \u2264 \u2a0d (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 measure_diff_null hN] at this \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc : \u03bc \u2260 0\nhf : Integrable f\nhN : \u2191\u2191\u03bc N = 0\nthis : 0 < \u2191\u2191\u03bc ({x | f x \u2264 \u2a0d (a : \u03b1), f a \u2202\u03bc} \\ N)\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 f x \u2264 \u2a0d (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nobtain \u27e8x, hx, hxN\u27e9 := nonempty_of_measure_ne_zero this.ne'\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc : \u03bc \u2260 0\nhf : Integrable f\nhN : \u2191\u2191\u03bc N = 0\nthis : 0 < \u2191\u2191\u03bc ({x | f x \u2264 \u2a0d (a : \u03b1), f a \u2202\u03bc} \\ N)\nx : \u03b1\nhx : x \u2208 {x | f x \u2264 \u2a0d (a : \u03b1), f a \u2202\u03bc}\nhxN : \u00acx \u2208 N\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 f x \u2264 \u2a0d (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nexact \u27e8x, hxN, hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc : \u03bc \u2260 0\nhf : Integrable f\nhN : \u2191\u2191\u03bc N = 0\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 \u2a0d (a : \u03b1), f a \u2202\u03bc \u2264 f x\n[PROOFSTEP]\nsimpa [integral_neg, neg_div] using exists_not_mem_null_le_average h\u03bc hf.neg hN\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nhf : Integrable f\n\u22a2 0 < \u2191\u2191\u03bc {x | f x \u2264 \u222b (a : \u03b1), f a \u2202\u03bc}\n[PROOFSTEP]\nsimpa only [average_eq_integral] using measure_le_average_pos (IsProbabilityMeasure.ne_zero \u03bc) hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nhf : Integrable f\n\u22a2 0 < \u2191\u2191\u03bc {x | \u222b (a : \u03b1), f a \u2202\u03bc \u2264 f x}\n[PROOFSTEP]\nsimpa only [average_eq_integral] using measure_average_le_pos (IsProbabilityMeasure.ne_zero \u03bc) hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nhf : Integrable f\n\u22a2 \u2203 x, f x \u2264 \u222b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nsimpa only [average_eq_integral] using exists_le_average (IsProbabilityMeasure.ne_zero \u03bc) hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nhf : Integrable f\n\u22a2 \u2203 x, \u222b (a : \u03b1), f a \u2202\u03bc \u2264 f x\n[PROOFSTEP]\nsimpa only [average_eq_integral] using exists_average_le (IsProbabilityMeasure.ne_zero \u03bc) hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nhf : Integrable f\nhN : \u2191\u2191\u03bc N = 0\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 f x \u2264 \u222b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nsimpa only [average_eq_integral] using exists_not_mem_null_le_average (IsProbabilityMeasure.ne_zero \u03bc) hf hN\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsProbabilityMeasure \u03bc\nhf : Integrable f\nhN : \u2191\u2191\u03bc N = 0\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 \u222b (a : \u03b1), f a \u2202\u03bc \u2264 f x\n[PROOFSTEP]\nsimpa only [average_eq_integral] using exists_not_mem_null_average_le (IsProbabilityMeasure.ne_zero \u03bc) hf hN\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : AEMeasurable f\n\u22a2 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc}\n[PROOFSTEP]\nobtain h | h := eq_or_ne (\u222b\u207b a in s, f a \u2202\u03bc) \u221e\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : AEMeasurable f\nh : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u22a4\n\u22a2 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc}\n[PROOFSTEP]\nsimpa [mul_top, h\u03bc\u2081, laverage, h, top_div_of_ne_top h\u03bc\u2081, pos_iff_ne_zero] using h\u03bc\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : AEMeasurable f\nh : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2260 \u22a4\n\u22a2 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc}\n[PROOFSTEP]\nhave := measure_le_setAverage_pos h\u03bc h\u03bc\u2081 (integrable_toReal_of_lintegral_ne_top hf h)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : AEMeasurable f\nh : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2260 \u22a4\nthis : 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 ENNReal.toReal (f x) \u2264 \u2a0d (a : \u03b1) in s, ENNReal.toReal (f a) \u2202\u03bc}\n\u22a2 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc}\n[PROOFSTEP]\nrw [\u2190 setOf_inter_eq_sep, \u2190\n Measure.restrict_apply\u2080 (hf.aestronglyMeasurable.nullMeasurableSet_le aestronglyMeasurable_const)]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : AEMeasurable f\nh : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2260 \u22a4\nthis : 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 ENNReal.toReal (f x) \u2264 \u2a0d (a : \u03b1) in s, ENNReal.toReal (f a) \u2202\u03bc}\n\u22a2 0 < \u2191\u2191(Measure.restrict \u03bc s) {a | f a \u2264 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc}\n[PROOFSTEP]\nrw [\u2190 setOf_inter_eq_sep, \u2190\n Measure.restrict_apply\u2080 (hf.ennreal_toReal.aestronglyMeasurable.nullMeasurableSet_le aestronglyMeasurable_const), \u2190\n measure_diff_null (measure_eq_top_of_lintegral_ne_top hf h)] at this \n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : AEMeasurable f\nh : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2260 \u22a4\nthis :\n 0 < \u2191\u2191(Measure.restrict \u03bc s) ({a | ENNReal.toReal (f a) \u2264 \u2a0d (a : \u03b1) in s, ENNReal.toReal (f a) \u2202\u03bc} \\ {x | f x = \u22a4})\n\u22a2 0 < \u2191\u2191(Measure.restrict \u03bc s) {a | f a \u2264 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc}\n[PROOFSTEP]\nrefine' this.trans_le (measure_mono _)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : AEMeasurable f\nh : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2260 \u22a4\nthis :\n 0 < \u2191\u2191(Measure.restrict \u03bc s) ({a | ENNReal.toReal (f a) \u2264 \u2a0d (a : \u03b1) in s, ENNReal.toReal (f a) \u2202\u03bc} \\ {x | f x = \u22a4})\n\u22a2 {a | ENNReal.toReal (f a) \u2264 \u2a0d (a : \u03b1) in s, ENNReal.toReal (f a) \u2202\u03bc} \\ {x | f x = \u22a4} \u2286\n {a | f a \u2264 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc}\n[PROOFSTEP]\nrintro x \u27e8hfx, hx\u27e9\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : AEMeasurable f\nh : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2260 \u22a4\nthis :\n 0 < \u2191\u2191(Measure.restrict \u03bc s) ({a | ENNReal.toReal (f a) \u2264 \u2a0d (a : \u03b1) in s, ENNReal.toReal (f a) \u2202\u03bc} \\ {x | f x = \u22a4})\nx : \u03b1\nhfx : x \u2208 {a | ENNReal.toReal (f a) \u2264 \u2a0d (a : \u03b1) in s, ENNReal.toReal (f a) \u2202\u03bc}\nhx : \u00acx \u2208 {x | f x = \u22a4}\n\u22a2 x \u2208 {a | f a \u2264 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc}\n[PROOFSTEP]\ndsimp at hfx \n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : AEMeasurable f\nh : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2260 \u22a4\nthis :\n 0 < \u2191\u2191(Measure.restrict \u03bc s) ({a | ENNReal.toReal (f a) \u2264 \u2a0d (a : \u03b1) in s, ENNReal.toReal (f a) \u2202\u03bc} \\ {x | f x = \u22a4})\nx : \u03b1\nhfx : ENNReal.toReal (f x) \u2264 \u2a0d (a : \u03b1) in s, ENNReal.toReal (f a) \u2202\u03bc\nhx : \u00acx \u2208 {x | f x = \u22a4}\n\u22a2 x \u2208 {a | f a \u2264 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc}\n[PROOFSTEP]\nrwa [\u2190 toReal_laverage hf, toReal_le_toReal hx (setLaverage_lt_top h).ne] at hfx \n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : AEMeasurable f\nh : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2260 \u22a4\nthis :\n 0 < \u2191\u2191(Measure.restrict \u03bc s) ({a | ENNReal.toReal (f a) \u2264 \u2a0d (a : \u03b1) in s, ENNReal.toReal (f a) \u2202\u03bc} \\ {x | f x = \u22a4})\nx : \u03b1\nhfx : ENNReal.toReal (f x) \u2264 \u2a0d (a : \u03b1) in s, ENNReal.toReal (f a) \u2202\u03bc\nhx : \u00acx \u2208 {x | f x = \u22a4}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, f x \u2260 \u22a4\n[PROOFSTEP]\nsimp_rw [ae_iff, not_ne_iff]\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\nhf : AEMeasurable f\nh : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2260 \u22a4\nthis :\n 0 < \u2191\u2191(Measure.restrict \u03bc s) ({a | ENNReal.toReal (f a) \u2264 \u2a0d (a : \u03b1) in s, ENNReal.toReal (f a) \u2202\u03bc} \\ {x | f x = \u22a4})\nx : \u03b1\nhfx : ENNReal.toReal (f x) \u2264 \u2a0d (a : \u03b1) in s, ENNReal.toReal (f a) \u2202\u03bc\nhx : \u00acx \u2208 {x | f x = \u22a4}\n\u22a2 \u2191\u2191(Measure.restrict \u03bc s) {a | f a = \u22a4} = 0\n[PROOFSTEP]\nexact measure_eq_top_of_lintegral_ne_top hf h\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nhint : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2260 \u22a4\n\u22a2 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2264 f x}\n[PROOFSTEP]\nobtain h\u03bc\u2081 | h\u03bc\u2081 := eq_or_ne (\u03bc s) \u221e\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nhint : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2260 \u22a4\nh\u03bc\u2081 : \u2191\u2191\u03bc s = \u22a4\n\u22a2 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2264 f x}\n[PROOFSTEP]\nsimp [setLaverage_eq, h\u03bc\u2081]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nhint : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2260 \u22a4\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2264 f x}\n[PROOFSTEP]\nobtain \u27e8g, hg, hgf, hfg\u27e9 := exists_measurable_le_lintegral_eq (\u03bc.restrict s) f\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nhint : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2260 \u22a4\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhg : Measurable g\nhgf : g \u2264 f\nhfg : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\n\u22a2 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2264 f x}\n[PROOFSTEP]\nhave hfg' : \u2a0d\u207b a in s, f a \u2202\u03bc = \u2a0d\u207b a in s, g a \u2202\u03bc := by simp_rw [laverage_eq, hfg]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nhint : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2260 \u22a4\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhg : Measurable g\nhgf : g \u2264 f\nhfg : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\n\u22a2 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [laverage_eq, hfg]\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nhint : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2260 \u22a4\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhg : Measurable g\nhgf : g \u2264 f\nhfg : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\nhfg' : \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc\n\u22a2 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2264 f x}\n[PROOFSTEP]\nrw [hfg] at hint \n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhint : \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2260 \u22a4\nhg : Measurable g\nhgf : g \u2264 f\nhfg : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\nhfg' : \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc\n\u22a2 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2264 f x}\n[PROOFSTEP]\nhave := measure_setAverage_le_pos h\u03bc h\u03bc\u2081 (integrable_toReal_of_lintegral_ne_top hg.aemeasurable hint)\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhint : \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2260 \u22a4\nhg : Measurable g\nhgf : g \u2264 f\nhfg : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\nhfg' : \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc\nthis : 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g x)}\n\u22a2 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc \u2264 f x}\n[PROOFSTEP]\nsimp_rw [\u2190 setOf_inter_eq_sep, \u2190 Measure.restrict_apply\u2080' hs, hfg']\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhint : \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2260 \u22a4\nhg : Measurable g\nhgf : g \u2264 f\nhfg : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\nhfg' : \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc\nthis : 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g x)}\n\u22a2 0 < \u2191\u2191(Measure.restrict \u03bc s) {a | \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2264 f a}\n[PROOFSTEP]\nrw [\u2190 setOf_inter_eq_sep, \u2190 Measure.restrict_apply\u2080' hs, \u2190\n measure_diff_null (measure_eq_top_of_lintegral_ne_top hg.aemeasurable hint)] at this \n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhint : \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2260 \u22a4\nhg : Measurable g\nhgf : g \u2264 f\nhfg : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\nhfg' : \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc\nthis :\n 0 < \u2191\u2191(Measure.restrict \u03bc s) ({a | \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g a)} \\ {x | g x = \u22a4})\n\u22a2 0 < \u2191\u2191(Measure.restrict \u03bc s) {a | \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2264 f a}\n[PROOFSTEP]\nrefine' this.trans_le (measure_mono _)\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhint : \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2260 \u22a4\nhg : Measurable g\nhgf : g \u2264 f\nhfg : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\nhfg' : \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc\nthis :\n 0 < \u2191\u2191(Measure.restrict \u03bc s) ({a | \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g a)} \\ {x | g x = \u22a4})\n\u22a2 {a | \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g a)} \\ {x | g x = \u22a4} \u2286\n {a | \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2264 f a}\n[PROOFSTEP]\nrintro x \u27e8hfx, hx\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhint : \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2260 \u22a4\nhg : Measurable g\nhgf : g \u2264 f\nhfg : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\nhfg' : \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc\nthis :\n 0 < \u2191\u2191(Measure.restrict \u03bc s) ({a | \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g a)} \\ {x | g x = \u22a4})\nx : \u03b1\nhfx : x \u2208 {a | \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g a)}\nhx : \u00acx \u2208 {x | g x = \u22a4}\n\u22a2 x \u2208 {a | \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2264 f a}\n[PROOFSTEP]\ndsimp at hfx \n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhint : \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2260 \u22a4\nhg : Measurable g\nhgf : g \u2264 f\nhfg : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\nhfg' : \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc\nthis :\n 0 < \u2191\u2191(Measure.restrict \u03bc s) ({a | \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g a)} \\ {x | g x = \u22a4})\nx : \u03b1\nhfx : \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g x)\nhx : \u00acx \u2208 {x | g x = \u22a4}\n\u22a2 x \u2208 {a | \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2264 f a}\n[PROOFSTEP]\nrw [\u2190 toReal_laverage hg.aemeasurable, toReal_le_toReal (setLaverage_lt_top hint).ne hx] at hfx \n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhint : \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2260 \u22a4\nhg : Measurable g\nhgf : g \u2264 f\nhfg : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\nhfg' : \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc\nthis :\n 0 < \u2191\u2191(Measure.restrict \u03bc s) ({a | \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g a)} \\ {x | g x = \u22a4})\nx : \u03b1\nhfx : \u2a0d\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2264 g x\nhx : \u00acx \u2208 {x | g x = \u22a4}\n\u22a2 x \u2208 {a | \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2264 f a}\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhint : \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2260 \u22a4\nhg : Measurable g\nhgf : g \u2264 f\nhfg : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\nhfg' : \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc\nthis :\n 0 < \u2191\u2191(Measure.restrict \u03bc s) ({a | \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g a)} \\ {x | g x = \u22a4})\nx : \u03b1\nhfx : \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g x)\nhx : \u00acx \u2208 {x | g x = \u22a4}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, g x \u2260 \u22a4\n[PROOFSTEP]\nexact hfx.trans (hgf _)\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhint : \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2260 \u22a4\nhg : Measurable g\nhgf : g \u2264 f\nhfg : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\nhfg' : \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc\nthis :\n 0 < \u2191\u2191(Measure.restrict \u03bc s) ({a | \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g a)} \\ {x | g x = \u22a4})\nx : \u03b1\nhfx : \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g x)\nhx : \u00acx \u2208 {x | g x = \u22a4}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, g x \u2260 \u22a4\n[PROOFSTEP]\nsimp_rw [ae_iff, not_ne_iff]\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u2191\u2191\u03bc s \u2260 0\nhs : NullMeasurableSet s\nh\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nhint : \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc \u2260 \u22a4\nhg : Measurable g\nhgf : g \u2264 f\nhfg : \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\nhfg' : \u2a0d\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u2a0d\u207b (a : \u03b1) in s, g a \u2202\u03bc\nthis :\n 0 < \u2191\u2191(Measure.restrict \u03bc s) ({a | \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g a)} \\ {x | g x = \u22a4})\nx : \u03b1\nhfx : \u2a0d (a : \u03b1) in s, ENNReal.toReal (g a) \u2202\u03bc \u2264 ENNReal.toReal (g x)\nhx : \u00acx \u2208 {x | g x = \u22a4}\n\u22a2 \u2191\u2191(Measure.restrict \u03bc s) {a | g a = \u22a4} = 0\n[PROOFSTEP]\nexact measure_eq_top_of_lintegral_ne_top hg.aemeasurable hint\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u03bc \u2260 0\nhint : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 \u22a4\n\u22a2 0 < \u2191\u2191\u03bc {x | \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc \u2264 f x}\n[PROOFSTEP]\nsimpa [hint] using @measure_setLaverage_le_pos _ _ _ _ f (measure_univ_ne_zero.2 h\u03bc) nullMeasurableSet_univ\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u03bc \u2260 0\nhint : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 \u22a4\nhN : \u2191\u2191\u03bc N = 0\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc \u2264 f x\n[PROOFSTEP]\nhave := measure_laverage_le_pos h\u03bc hint\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u03bc \u2260 0\nhint : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 \u22a4\nhN : \u2191\u2191\u03bc N = 0\nthis : 0 < \u2191\u2191\u03bc {x | \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc \u2264 f x}\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc \u2264 f x\n[PROOFSTEP]\nrw [\u2190 measure_diff_null hN] at this \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u03bc \u2260 0\nhint : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 \u22a4\nhN : \u2191\u2191\u03bc N = 0\nthis : 0 < \u2191\u2191\u03bc ({x | \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc \u2264 f x} \\ N)\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc \u2264 f x\n[PROOFSTEP]\nobtain \u27e8x, hx, hxN\u27e9 := nonempty_of_measure_ne_zero this.ne'\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh\u03bc : \u03bc \u2260 0\nhint : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 \u22a4\nhN : \u2191\u2191\u03bc N = 0\nthis : 0 < \u2191\u2191\u03bc ({x | \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc \u2264 f x} \\ N)\nx : \u03b1\nhx : x \u2208 {x | \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc \u2264 f x}\nhxN : \u00acx \u2208 N\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc \u2264 f x\n[PROOFSTEP]\nexact \u27e8x, hxN, hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc : \u03bc \u2260 0\nhf : AEMeasurable f\n\u22a2 0 < \u2191\u2191\u03bc {x | f x \u2264 \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc}\n[PROOFSTEP]\nsimpa using measure_le_setLaverage_pos (measure_univ_ne_zero.2 h\u03bc) (measure_ne_top _ _) hf.restrict\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc : \u03bc \u2260 0\nhf : AEMeasurable f\nhN : \u2191\u2191\u03bc N = 0\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 f x \u2264 \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nhave := measure_le_laverage_pos h\u03bc hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc : \u03bc \u2260 0\nhf : AEMeasurable f\nhN : \u2191\u2191\u03bc N = 0\nthis : 0 < \u2191\u2191\u03bc {x | f x \u2264 \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc}\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 f x \u2264 \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 measure_diff_null hN] at this \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc : \u03bc \u2260 0\nhf : AEMeasurable f\nhN : \u2191\u2191\u03bc N = 0\nthis : 0 < \u2191\u2191\u03bc ({x | f x \u2264 \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc} \\ N)\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 f x \u2264 \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nobtain \u27e8x, hx, hxN\u27e9 := nonempty_of_measure_ne_zero this.ne'\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc : \u03bc \u2260 0\nhf : AEMeasurable f\nhN : \u2191\u2191\u03bc N = 0\nthis : 0 < \u2191\u2191\u03bc ({x | f x \u2264 \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc} \\ N)\nx : \u03b1\nhx : x \u2208 {x | f x \u2264 \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc}\nhxN : \u00acx \u2208 N\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 f x \u2264 \u2a0d\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nexact \u27e8x, hxN, hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsProbabilityMeasure \u03bc\nhf : AEMeasurable f\n\u22a2 0 < \u2191\u2191\u03bc {x | f x \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc}\n[PROOFSTEP]\nsimpa only [laverage_eq_lintegral] using measure_le_laverage_pos (IsProbabilityMeasure.ne_zero \u03bc) hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsProbabilityMeasure \u03bc\nhint : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 \u22a4\n\u22a2 0 < \u2191\u2191\u03bc {x | \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2264 f x}\n[PROOFSTEP]\nsimpa only [laverage_eq_lintegral] using measure_laverage_le_pos (IsProbabilityMeasure.ne_zero \u03bc) hint\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsProbabilityMeasure \u03bc\nhf : AEMeasurable f\n\u22a2 \u2203 x, f x \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nsimpa only [laverage_eq_lintegral] using exists_le_laverage (IsProbabilityMeasure.ne_zero \u03bc) hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsProbabilityMeasure \u03bc\nhint : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 x, \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2264 f x\n[PROOFSTEP]\nsimpa only [laverage_eq_lintegral] using exists_laverage_le (IsProbabilityMeasure.ne_zero \u03bc) hint\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsProbabilityMeasure \u03bc\nhf : AEMeasurable f\nhN : \u2191\u2191\u03bc N = 0\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 f x \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc\n[PROOFSTEP]\nsimpa only [laverage_eq_lintegral] using exists_not_mem_null_le_laverage (IsProbabilityMeasure.ne_zero \u03bc) hf hN\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t N : Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : IsProbabilityMeasure \u03bc\nhint : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 \u22a4\nhN : \u2191\u2191\u03bc N = 0\n\u22a2 \u2203 x, \u00acx \u2208 N \u2227 \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2264 f x\n[PROOFSTEP]\nsimpa only [laverage_eq_lintegral] using exists_not_mem_null_laverage_le (IsProbabilityMeasure.ne_zero \u03bc) hint hN\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\n\u22a2 Tendsto (fun i => \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc) l (\ud835\udcdd c)\n[PROOFSTEP]\nhave g_int : \u2200\u1da0 i in l, Integrable (g i) \u03bc :=\n by\n filter_upwards [(tendsto_order.1 hg).1 _ zero_lt_one] with i hi\n contrapose hi\n simp only [integral_undef hi, lt_self_iff_false, not_false_eq_true]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\n[PROOFSTEP]\nfilter_upwards [(tendsto_order.1 hg).1 _ zero_lt_one] with i hi\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ni : \u03b9\nhi : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\n\u22a2 Integrable (g i)\n[PROOFSTEP]\ncontrapose hi\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ni : \u03b9\nhi : \u00acIntegrable (g i)\n\u22a2 \u00ac0 < \u222b (y : \u03b1), g i y \u2202\u03bc\n[PROOFSTEP]\nsimp only [integral_undef hi, lt_self_iff_false, not_false_eq_true]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\n\u22a2 Tendsto (fun i => \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc) l (\ud835\udcdd c)\n[PROOFSTEP]\nhave I : \u2200\u1da0 i in l, \u222b y, g i y \u2022 (f y - c) \u2202\u03bc + (\u222b y, g i y \u2202\u03bc) \u2022 c = \u222b y, g i y \u2022 f y \u2202\u03bc :=\n by\n filter_upwards [f_int, g_int, g_supp, g_bound] with i hif hig hisupp hibound\n rw [\u2190 integral_smul_const, \u2190 integral_add]\n \u00b7 simp only [smul_sub, sub_add_cancel]\n \u00b7 simp_rw [smul_sub]\n apply Integrable.sub _ (hig.smul_const _)\n have A : Function.support (fun y \u21a6 g i y \u2022 f y) \u2286 a i :=\n by\n apply Subset.trans _ hisupp\n exact Function.support_smul_subset_left _ _\n rw [\u2190 integrableOn_iff_integrable_of_support_subset A]\n apply Integrable.smul_of_top_right hif\n exact mem\u2112p_top_of_bound hig.aestronglyMeasurable.restrict (K / (\u03bc (a i)).toReal) (eventually_of_forall hibound)\n \u00b7 exact hig.smul_const _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\n[PROOFSTEP]\nfilter_upwards [f_int, g_int, g_supp, g_bound] with i hif hig hisupp hibound\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\ni : \u03b9\nhif : IntegrableOn f (a i)\nhig : Integrable (g i)\nhisupp : support (g i) \u2286 a i\nhibound : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\n\u22a2 \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 integral_smul_const, \u2190 integral_add]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\ni : \u03b9\nhif : IntegrableOn f (a i)\nhig : Integrable (g i)\nhisupp : support (g i) \u2286 a i\nhibound : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\n\u22a2 \u222b (a : \u03b1), g i a \u2022 (f a - c) + g i a \u2022 c \u2202\u03bc = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\n[PROOFSTEP]\nsimp only [smul_sub, sub_add_cancel]\n[GOAL]\ncase h.hf\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\ni : \u03b9\nhif : IntegrableOn f (a i)\nhig : Integrable (g i)\nhisupp : support (g i) \u2286 a i\nhibound : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\n\u22a2 Integrable fun y => g i y \u2022 (f y - c)\n[PROOFSTEP]\nsimp_rw [smul_sub]\n[GOAL]\ncase h.hf\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\ni : \u03b9\nhif : IntegrableOn f (a i)\nhig : Integrable (g i)\nhisupp : support (g i) \u2286 a i\nhibound : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\n\u22a2 Integrable fun y => g i y \u2022 f y - g i y \u2022 c\n[PROOFSTEP]\napply Integrable.sub _ (hig.smul_const _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\ni : \u03b9\nhif : IntegrableOn f (a i)\nhig : Integrable (g i)\nhisupp : support (g i) \u2286 a i\nhibound : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\n\u22a2 Integrable fun y => g i y \u2022 f y\n[PROOFSTEP]\nhave A : Function.support (fun y \u21a6 g i y \u2022 f y) \u2286 a i :=\n by\n apply Subset.trans _ hisupp\n exact Function.support_smul_subset_left _ _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\ni : \u03b9\nhif : IntegrableOn f (a i)\nhig : Integrable (g i)\nhisupp : support (g i) \u2286 a i\nhibound : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\n\u22a2 (support fun y => g i y \u2022 f y) \u2286 a i\n[PROOFSTEP]\napply Subset.trans _ hisupp\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\ni : \u03b9\nhif : IntegrableOn f (a i)\nhig : Integrable (g i)\nhisupp : support (g i) \u2286 a i\nhibound : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\n\u22a2 (support fun y => g i y \u2022 f y) \u2286 support (g i)\n[PROOFSTEP]\nexact Function.support_smul_subset_left _ _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\ni : \u03b9\nhif : IntegrableOn f (a i)\nhig : Integrable (g i)\nhisupp : support (g i) \u2286 a i\nhibound : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nA : (support fun y => g i y \u2022 f y) \u2286 a i\n\u22a2 Integrable fun y => g i y \u2022 f y\n[PROOFSTEP]\nrw [\u2190 integrableOn_iff_integrable_of_support_subset A]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\ni : \u03b9\nhif : IntegrableOn f (a i)\nhig : Integrable (g i)\nhisupp : support (g i) \u2286 a i\nhibound : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nA : (support fun y => g i y \u2022 f y) \u2286 a i\n\u22a2 IntegrableOn (fun y => g i y \u2022 f y) (a i)\n[PROOFSTEP]\napply Integrable.smul_of_top_right hif\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\ni : \u03b9\nhif : IntegrableOn f (a i)\nhig : Integrable (g i)\nhisupp : support (g i) \u2286 a i\nhibound : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nA : (support fun y => g i y \u2022 f y) \u2286 a i\n\u22a2 Mem\u2112p (fun y => g i y) \u22a4\n[PROOFSTEP]\nexact mem\u2112p_top_of_bound hig.aestronglyMeasurable.restrict (K / (\u03bc (a i)).toReal) (eventually_of_forall hibound)\n[GOAL]\ncase h.hg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\ni : \u03b9\nhif : IntegrableOn f (a i)\nhig : Integrable (g i)\nhisupp : support (g i) \u2286 a i\nhibound : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\n\u22a2 Integrable fun x => g i x \u2022 c\n[PROOFSTEP]\nexact hig.smul_const _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\n\u22a2 Tendsto (fun i => \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc) l (\ud835\udcdd c)\n[PROOFSTEP]\nhave L0 : Tendsto (fun i \u21a6 \u222b y, g i y \u2022 (f y - c) \u2202\u03bc) l (\ud835\udcdd 0) :=\n by\n have := hf.const_mul K\n simp only [mul_zero] at this \n refine' squeeze_zero_norm' _ this\n filter_upwards [g_supp, g_bound, f_int, (tendsto_order.1 hg).1 _ zero_lt_one] with i hi h'i h''i hi_int\n have mu_ai : \u03bc (a i) < \u221e := by\n rw [lt_top_iff_ne_top]\n intro h\n simp only [h, ENNReal.top_toReal, _root_.div_zero, abs_nonpos_iff] at h'i \n have : \u222b (y : \u03b1), g i y \u2202\u03bc = \u222b (y : \u03b1), 0 \u2202\u03bc := by congr; ext y; exact h'i y\n simp [this] at hi_int \n apply (norm_integral_le_integral_norm _).trans\n simp_rw [average_eq, smul_eq_mul, \u2190 integral_mul_left, norm_smul, \u2190 mul_assoc, \u2190 div_eq_mul_inv]\n have : \u2200 x, x \u2209 a i \u2192 \u2016g i x\u2016 * \u2016(f x - c)\u2016 = 0 := by\n intro x hx\n have : g i x = 0 := by rw [\u2190 Function.nmem_support]; exact fun h \u21a6 hx (hi h)\n simp [this]\n rw [\u2190 set_integral_eq_integral_of_forall_compl_eq_zero this (\u03bc := \u03bc)]\n refine' integral_mono_of_nonneg (eventually_of_forall (fun x \u21a6 by positivity)) _ (eventually_of_forall (fun x \u21a6 _))\n \u00b7 apply (Integrable.sub h''i _).norm.const_mul\n change IntegrableOn (fun _ \u21a6 c) (a i) \u03bc\n simp [integrableOn_const, mu_ai]\n \u00b7 dsimp; gcongr; simpa using h'i x\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\n\u22a2 Tendsto (fun i => \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc) l (\ud835\udcdd 0)\n[PROOFSTEP]\nhave := hf.const_mul K\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd (K * 0))\n\u22a2 Tendsto (fun i => \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc) l (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp only [mul_zero] at this \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun i => \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc) l (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' squeeze_zero_norm' _ this\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (n : \u03b9) in l, \u2016\u222b (y : \u03b1), g n y \u2022 (f y - c) \u2202\u03bc\u2016 \u2264 K * \u2a0d (y : \u03b1) in a n, \u2016f y - c\u2016 \u2202\u03bc\n[PROOFSTEP]\nfilter_upwards [g_supp, g_bound, f_int, (tendsto_order.1 hg).1 _ zero_lt_one] with i hi h'i h''i hi_int\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\n\u22a2 \u2016\u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc\u2016 \u2264 K * \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc\n[PROOFSTEP]\nhave mu_ai : \u03bc (a i) < \u221e := by\n rw [lt_top_iff_ne_top]\n intro h\n simp only [h, ENNReal.top_toReal, _root_.div_zero, abs_nonpos_iff] at h'i \n have : \u222b (y : \u03b1), g i y \u2202\u03bc = \u222b (y : \u03b1), 0 \u2202\u03bc := by congr; ext y; exact h'i y\n simp [this] at hi_int \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\n\u22a2 \u2191\u2191\u03bc (a i) < \u22a4\n[PROOFSTEP]\nrw [lt_top_iff_ne_top]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\n\u22a2 \u2191\u2191\u03bc (a i) \u2260 \u22a4\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nh : \u2191\u2191\u03bc (a i) = \u22a4\n\u22a2 False\n[PROOFSTEP]\nsimp only [h, ENNReal.top_toReal, _root_.div_zero, abs_nonpos_iff] at h'i \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nh : \u2191\u2191\u03bc (a i) = \u22a4\nh'i : \u2200 (x : \u03b1), g i x = 0\n\u22a2 False\n[PROOFSTEP]\nhave : \u222b (y : \u03b1), g i y \u2202\u03bc = \u222b (y : \u03b1), 0 \u2202\u03bc := by congr; ext y; exact h'i y\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nh : \u2191\u2191\u03bc (a i) = \u22a4\nh'i : \u2200 (x : \u03b1), g i x = 0\n\u22a2 \u222b (y : \u03b1), g i y \u2202\u03bc = \u222b (y : \u03b1), 0 \u2202\u03bc\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nh : \u2191\u2191\u03bc (a i) = \u22a4\nh'i : \u2200 (x : \u03b1), g i x = 0\n\u22a2 (fun y => g i y) = fun y => 0\n[PROOFSTEP]\next y\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nh : \u2191\u2191\u03bc (a i) = \u22a4\nh'i : \u2200 (x : \u03b1), g i x = 0\ny : \u03b1\n\u22a2 g i y = 0\n[PROOFSTEP]\nexact h'i y\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis\u271d : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nh : \u2191\u2191\u03bc (a i) = \u22a4\nh'i : \u2200 (x : \u03b1), g i x = 0\nthis : \u222b (y : \u03b1), g i y \u2202\u03bc = \u222b (y : \u03b1), 0 \u2202\u03bc\n\u22a2 False\n[PROOFSTEP]\nsimp [this] at hi_int \n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\n\u22a2 \u2016\u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc\u2016 \u2264 K * \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc\n[PROOFSTEP]\napply (norm_integral_le_integral_norm _).trans\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\n\u22a2 \u222b (a : \u03b1), \u2016g i a \u2022 (f a - c)\u2016 \u2202\u03bc \u2264 K * \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [average_eq, smul_eq_mul, \u2190 integral_mul_left, norm_smul, \u2190 mul_assoc, \u2190 div_eq_mul_inv]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\n\u22a2 \u222b (a : \u03b1), \u2016g i a\u2016 * \u2016f a - c\u2016 \u2202\u03bc \u2264\n \u222b (a_1 : \u03b1) in a i, K / ENNReal.toReal (\u2191\u2191(Measure.restrict \u03bc (a i)) univ) * \u2016f a_1 - c\u2016 \u2202\u03bc\n[PROOFSTEP]\nhave : \u2200 x, x \u2209 a i \u2192 \u2016g i x\u2016 * \u2016(f x - c)\u2016 = 0 := by\n intro x hx\n have : g i x = 0 := by rw [\u2190 Function.nmem_support]; exact fun h \u21a6 hx (hi h)\n simp [this]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\n\u22a2 \u2200 (x : \u03b1), \u00acx \u2208 a i \u2192 \u2016g i x\u2016 * \u2016f x - c\u2016 = 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\nx : \u03b1\nhx : \u00acx \u2208 a i\n\u22a2 \u2016g i x\u2016 * \u2016f x - c\u2016 = 0\n[PROOFSTEP]\nhave : g i x = 0 := by rw [\u2190 Function.nmem_support]; exact fun h \u21a6 hx (hi h)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\nx : \u03b1\nhx : \u00acx \u2208 a i\n\u22a2 g i x = 0\n[PROOFSTEP]\nrw [\u2190 Function.nmem_support]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\nx : \u03b1\nhx : \u00acx \u2208 a i\n\u22a2 \u00acx \u2208 support (g i)\n[PROOFSTEP]\nexact fun h \u21a6 hx (hi h)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis\u271d : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\nx : \u03b1\nhx : \u00acx \u2208 a i\nthis : g i x = 0\n\u22a2 \u2016g i x\u2016 * \u2016f x - c\u2016 = 0\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis\u271d : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\nthis : \u2200 (x : \u03b1), \u00acx \u2208 a i \u2192 \u2016g i x\u2016 * \u2016f x - c\u2016 = 0\n\u22a2 \u222b (a : \u03b1), \u2016g i a\u2016 * \u2016f a - c\u2016 \u2202\u03bc \u2264\n \u222b (a_1 : \u03b1) in a i, K / ENNReal.toReal (\u2191\u2191(Measure.restrict \u03bc (a i)) univ) * \u2016f a_1 - c\u2016 \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 set_integral_eq_integral_of_forall_compl_eq_zero this (\u03bc := \u03bc)]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis\u271d : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\nthis : \u2200 (x : \u03b1), \u00acx \u2208 a i \u2192 \u2016g i x\u2016 * \u2016f x - c\u2016 = 0\n\u22a2 \u222b (x : \u03b1) in a i, \u2016g i x\u2016 * \u2016f x - c\u2016 \u2202\u03bc \u2264\n \u222b (a_1 : \u03b1) in a i, K / ENNReal.toReal (\u2191\u2191(Measure.restrict \u03bc (a i)) univ) * \u2016f a_1 - c\u2016 \u2202\u03bc\n[PROOFSTEP]\nrefine' integral_mono_of_nonneg (eventually_of_forall (fun x \u21a6 by positivity)) _ (eventually_of_forall (fun x \u21a6 _))\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis\u271d : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\nthis : \u2200 (x : \u03b1), \u00acx \u2208 a i \u2192 \u2016g i x\u2016 * \u2016f x - c\u2016 = 0\nx : \u03b1\n\u22a2 OfNat.ofNat 0 x \u2264 (fun x => \u2016g i x\u2016 * \u2016f x - c\u2016) x\n[PROOFSTEP]\npositivity\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis\u271d : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\nthis : \u2200 (x : \u03b1), \u00acx \u2208 a i \u2192 \u2016g i x\u2016 * \u2016f x - c\u2016 = 0\n\u22a2 Integrable fun a_1 => K / ENNReal.toReal (\u2191\u2191(Measure.restrict \u03bc (a i)) univ) * \u2016f a_1 - c\u2016\n[PROOFSTEP]\napply (Integrable.sub h''i _).norm.const_mul\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis\u271d : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\nthis : \u2200 (x : \u03b1), \u00acx \u2208 a i \u2192 \u2016g i x\u2016 * \u2016f x - c\u2016 = 0\n\u22a2 Integrable fun x => c\n[PROOFSTEP]\nchange IntegrableOn (fun _ \u21a6 c) (a i) \u03bc\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis\u271d : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\nthis : \u2200 (x : \u03b1), \u00acx \u2208 a i \u2192 \u2016g i x\u2016 * \u2016f x - c\u2016 = 0\n\u22a2 IntegrableOn (fun x => c) (a i)\n[PROOFSTEP]\nsimp [integrableOn_const, mu_ai]\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis\u271d : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\nthis : \u2200 (x : \u03b1), \u00acx \u2208 a i \u2192 \u2016g i x\u2016 * \u2016f x - c\u2016 = 0\nx : \u03b1\n\u22a2 (fun x => \u2016g i x\u2016 * \u2016f x - c\u2016) x \u2264 (fun a_1 => K / ENNReal.toReal (\u2191\u2191(Measure.restrict \u03bc (a i)) univ) * \u2016f a_1 - c\u2016) x\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis\u271d : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\nthis : \u2200 (x : \u03b1), \u00acx \u2208 a i \u2192 \u2016g i x\u2016 * \u2016f x - c\u2016 = 0\nx : \u03b1\n\u22a2 |g i x| * \u2016f x - c\u2016 \u2264 K / ENNReal.toReal (\u2191\u2191(Measure.restrict \u03bc (a i)) univ) * \u2016f x - c\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h.refine'_2.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nthis\u271d : Tendsto (fun k => K * \u2a0d (y : \u03b1) in a k, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\ni : \u03b9\nhi : support (g i) \u2286 a i\nh'i : \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\nh''i : IntegrableOn f (a i)\nhi_int : 0 < \u222b (y : \u03b1), g i y \u2202\u03bc\nmu_ai : \u2191\u2191\u03bc (a i) < \u22a4\nthis : \u2200 (x : \u03b1), \u00acx \u2208 a i \u2192 \u2016g i x\u2016 * \u2016f x - c\u2016 = 0\nx : \u03b1\n\u22a2 |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191(Measure.restrict \u03bc (a i)) univ)\n[PROOFSTEP]\nsimpa using h'i x\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nL0 : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun i => \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc) l (\ud835\udcdd c)\n[PROOFSTEP]\nhave := L0.add (hg.smul_const c)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nL0 : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc) l (\ud835\udcdd 0)\nthis : Tendsto (fun x => \u222b (y : \u03b1), g x y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g x y \u2202\u03bc) \u2022 c) l (\ud835\udcdd (0 + 1 \u2022 c))\n\u22a2 Tendsto (fun i => \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc) l (\ud835\udcdd c)\n[PROOFSTEP]\nsimp only [one_smul, zero_add] at this \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc \u03bd : Measure \u03b1\ns t : Set \u03b1\n\u03b9 : Type u_4\na : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\nf : \u03b1 \u2192 E\nc : E\ng : \u03b9 \u2192 \u03b1 \u2192 \u211d\nK : \u211d\nhf : Tendsto (fun i => \u2a0d (y : \u03b1) in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0)\nf_int : \u2200\u1da0 (i : \u03b9) in l, IntegrableOn f (a i)\nhg : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2202\u03bc) l (\ud835\udcdd 1)\ng_supp : \u2200\u1da0 (i : \u03b9) in l, support (g i) \u2286 a i\ng_bound : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), |g i x| \u2264 K / ENNReal.toReal (\u2191\u2191\u03bc (a i))\ng_int : \u2200\u1da0 (i : \u03b9) in l, Integrable (g i)\nI : \u2200\u1da0 (i : \u03b9) in l, \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g i y \u2202\u03bc) \u2022 c = \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc\nL0 : Tendsto (fun i => \u222b (y : \u03b1), g i y \u2022 (f y - c) \u2202\u03bc) l (\ud835\udcdd 0)\nthis : Tendsto (fun x => \u222b (y : \u03b1), g x y \u2022 (f y - c) \u2202\u03bc + (\u222b (y : \u03b1), g x y \u2202\u03bc) \u2022 c) l (\ud835\udcdd c)\n\u22a2 Tendsto (fun i => \u222b (y : \u03b1), g i y \u2022 f y \u2202\u03bc) l (\ud835\udcdd c)\n[PROOFSTEP]\nexact Tendsto.congr' I this\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Integral.Average", "llama_tokens": 77884, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581741774411, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.42007154006663383}} {"text": "[GOAL]\nu : Level\n\u03b1 :\n let u := u;\n Q(Type u)\nx\u271d : Q(Semiring \u00ab$\u03b1\u00bb)\nn k\u271d : \u2115\nlit k : Q(\u2115)\nthis : \u00ab$lit\u00bb =Q \u00ab$k\u00bb + 2\n\u22a2 Q(\u00ab$\u03b1\u00bb)\n[PROOFSTEP]\nexact q(OfNat.ofNat $lit)\n", "meta": {"mathlib_filename": "Mathlib.Tactic.Linarith.Verification", "llama_tokens": 103, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7905303186696747, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.41993711521347377}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Sort u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\nx : \u2191T\ni : \u03b9\nhx : \u2191x \u2208 S i\n\u22a2 iUnionLift S f hf T hT x = f i { val := \u2191x, property := hx }\n[PROOFSTEP]\ncases' x with x hx\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Sort u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\ni : \u03b9\nx : \u03b1\nhx\u271d : x \u2208 T\nhx : \u2191{ val := x, property := hx\u271d } \u2208 S i\n\u22a2 iUnionLift S f hf T hT { val := x, property := hx\u271d } = f i { val := \u2191{ val := x, property := hx\u271d }, property := hx }\n[PROOFSTEP]\nexact hf _ _ _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Type u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\nt : Set \u03b2\n\u22a2 iUnionLift S f hf T hT \u207b\u00b9' t = inclusion hT \u207b\u00b9' \u22c3 (i : \u03b9), inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) '' (f i \u207b\u00b9' t)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Type u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\nt : Set \u03b2\nx : \u2191T\n\u22a2 x \u2208 iUnionLift S f hf T hT \u207b\u00b9' t \u2194 x \u2208 inclusion hT \u207b\u00b9' \u22c3 (i : \u03b9), inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) '' (f i \u207b\u00b9' t)\n[PROOFSTEP]\nsimp only [mem_preimage, mem_iUnion, mem_image]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Type u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\nt : Set \u03b2\nx : \u2191T\n\u22a2 iUnionLift S f hf T hT x \u2208 t \u2194 \u2203 i x_1, f i x_1 \u2208 t \u2227 inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) x_1 = inclusion hT x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Type u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\nt : Set \u03b2\nx : \u2191T\n\u22a2 iUnionLift S f hf T hT x \u2208 t \u2192 \u2203 i x_1, f i x_1 \u2208 t \u2227 inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) x_1 = inclusion hT x\n[PROOFSTEP]\nrcases mem_iUnion.1 (hT x.prop) with \u27e8i, hi\u27e9\n[GOAL]\ncase h.mp.intro\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Type u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\nt : Set \u03b2\nx : \u2191T\ni : \u03b9\nhi : \u2191x \u2208 S i\n\u22a2 iUnionLift S f hf T hT x \u2208 t \u2192 \u2203 i x_1, f i x_1 \u2208 t \u2227 inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) x_1 = inclusion hT x\n[PROOFSTEP]\nrefine fun h => \u27e8i, \u27e8x, hi\u27e9, ?_, rfl\u27e9\n[GOAL]\ncase h.mp.intro\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Type u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\nt : Set \u03b2\nx : \u2191T\ni : \u03b9\nhi : \u2191x \u2208 S i\nh : iUnionLift S f hf T hT x \u2208 t\n\u22a2 f i { val := \u2191x, property := hi } \u2208 t\n[PROOFSTEP]\nrwa [iUnionLift_of_mem x hi] at h \n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Type u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\nt : Set \u03b2\nx : \u2191T\n\u22a2 (\u2203 i x_1, f i x_1 \u2208 t \u2227 inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) x_1 = inclusion hT x) \u2192 iUnionLift S f hf T hT x \u2208 t\n[PROOFSTEP]\nrintro \u27e8i, \u27e8y, hi\u27e9, h, hxy\u27e9\n[GOAL]\ncase h.mpr.intro.intro.mk.intro\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Type u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\nt : Set \u03b2\nx : \u2191T\ni : \u03b9\ny : \u03b1\nhi : y \u2208 S i\nh : f i { val := y, property := hi } \u2208 t\nhxy : inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) { val := y, property := hi } = inclusion hT x\n\u22a2 iUnionLift S f hf T hT x \u2208 t\n[PROOFSTEP]\nobtain rfl : y = x := congr_arg Subtype.val hxy\n[GOAL]\ncase h.mpr.intro.intro.mk.intro\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Type u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\nt : Set \u03b2\nx : \u2191T\ni : \u03b9\nhi : \u2191x \u2208 S i\nh : f i { val := \u2191x, property := hi } \u2208 t\nhxy : inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191x, property := hi } = inclusion hT x\n\u22a2 iUnionLift S f hf T hT x \u2208 t\n[PROOFSTEP]\nrwa [iUnionLift_of_mem x hi]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Sort u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\nc : \u2191T\nci : (i : \u03b9) \u2192 \u2191(S i)\nhci : \u2200 (i : \u03b9), \u2191(ci i) = \u2191c\nc\u03b2 : \u03b2\nh : \u2200 (i : \u03b9), f i (ci i) = c\u03b2\n\u22a2 iUnionLift S f hf T hT c = c\u03b2\n[PROOFSTEP]\nlet \u27e8i, hi\u27e9 := Set.mem_iUnion.1 (hT c.prop)\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Sort u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\nc : \u2191T\nci : (i : \u03b9) \u2192 \u2191(S i)\nhci : \u2200 (i : \u03b9), \u2191(ci i) = \u2191c\nc\u03b2 : \u03b2\nh : \u2200 (i : \u03b9), f i (ci i) = c\u03b2\ni : \u03b9\nhi : \u2191c \u2208 S i\n\u22a2 iUnionLift S f hf T hT c = c\u03b2\n[PROOFSTEP]\nhave : ci i = \u27e8c, hi\u27e9 := Subtype.ext (hci i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Sort u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\nc : \u2191T\nci : (i : \u03b9) \u2192 \u2191(S i)\nhci : \u2200 (i : \u03b9), \u2191(ci i) = \u2191c\nc\u03b2 : \u03b2\nh : \u2200 (i : \u03b9), f i (ci i) = c\u03b2\ni : \u03b9\nhi : \u2191c \u2208 S i\nthis : ci i = { val := \u2191c, property := hi }\n\u22a2 iUnionLift S f hf T hT c = c\u03b2\n[PROOFSTEP]\nrw [iUnionLift_of_mem _ hi, \u2190 this, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\nu : \u2191T \u2192 \u2191T\nui : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i)\nhui : \u2200 (i : \u03b9) (x : \u2191(S i)), u (inclusion (_ : S i \u2286 T) x) = inclusion (_ : S i \u2286 T) (ui i x)\nu\u03b2 : \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x : \u2191(S i)), f i (ui i x) = u\u03b2 (f i x)\nx : \u2191T\n\u22a2 iUnionLift S f hf T (_ : T \u2264 iUnion S) (u x) = u\u03b2 (iUnionLift S f hf T (_ : T \u2264 iUnion S) x)\n[PROOFSTEP]\nsubst hT'\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nui : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i)\nu\u03b2 : \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x : \u2191(S i)), f i (ui i x) = u\u03b2 (f i x)\nhT : iUnion S \u2286 iUnion S\nu : \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhui : \u2200 (i : \u03b9) (x : \u2191(S i)), u (inclusion (_ : S i \u2286 iUnion S) x) = inclusion (_ : S i \u2286 iUnion S) (ui i x)\nx : \u2191(iUnion S)\n\u22a2 iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (u x) =\n u\u03b2 (iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) x)\n[PROOFSTEP]\ncases' Set.mem_iUnion.1 x.prop with i hi\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nui : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i)\nu\u03b2 : \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x : \u2191(S i)), f i (ui i x) = u\u03b2 (f i x)\nhT : iUnion S \u2286 iUnion S\nu : \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhui : \u2200 (i : \u03b9) (x : \u2191(S i)), u (inclusion (_ : S i \u2286 iUnion S) x) = inclusion (_ : S i \u2286 iUnion S) (ui i x)\nx : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\n\u22a2 iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (u x) =\n u\u03b2 (iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) x)\n[PROOFSTEP]\nrw [iUnionLift_of_mem x hi, \u2190 h i]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nui : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i)\nu\u03b2 : \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x : \u2191(S i)), f i (ui i x) = u\u03b2 (f i x)\nhT : iUnion S \u2286 iUnion S\nu : \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhui : \u2200 (i : \u03b9) (x : \u2191(S i)), u (inclusion (_ : S i \u2286 iUnion S) x) = inclusion (_ : S i \u2286 iUnion S) (ui i x)\nx : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\n\u22a2 iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (u x) = f i (ui i { val := \u2191x, property := hi })\n[PROOFSTEP]\nhave : x = Set.inclusion (Set.subset_iUnion S i) \u27e8x, hi\u27e9 :=\n by\n cases x\n rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nui : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i)\nu\u03b2 : \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x : \u2191(S i)), f i (ui i x) = u\u03b2 (f i x)\nhT : iUnion S \u2286 iUnion S\nu : \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhui : \u2200 (i : \u03b9) (x : \u2191(S i)), u (inclusion (_ : S i \u2286 iUnion S) x) = inclusion (_ : S i \u2286 iUnion S) (ui i x)\nx : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\n\u22a2 x = inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191x, property := hi }\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nui : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i)\nu\u03b2 : \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x : \u2191(S i)), f i (ui i x) = u\u03b2 (f i x)\nhT : iUnion S \u2286 iUnion S\nu : \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhui : \u2200 (i : \u03b9) (x : \u2191(S i)), u (inclusion (_ : S i \u2286 iUnion S) x) = inclusion (_ : S i \u2286 iUnion S) (ui i x)\ni : \u03b9\nval\u271d : \u03b1\nproperty\u271d : val\u271d \u2208 iUnion S\nhi : \u2191{ val := val\u271d, property := property\u271d } \u2208 S i\n\u22a2 { val := val\u271d, property := property\u271d } =\n inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191{ val := val\u271d, property := property\u271d }, property := hi }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nui : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i)\nu\u03b2 : \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x : \u2191(S i)), f i (ui i x) = u\u03b2 (f i x)\nhT : iUnion S \u2286 iUnion S\nu : \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhui : \u2200 (i : \u03b9) (x : \u2191(S i)), u (inclusion (_ : S i \u2286 iUnion S) x) = inclusion (_ : S i \u2286 iUnion S) (ui i x)\nx : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nthis : x = inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191x, property := hi }\n\u22a2 iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (u x) = f i (ui i { val := \u2191x, property := hi })\n[PROOFSTEP]\nconv_lhs => rw [this, hui, iUnionLift_inclusion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nui : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i)\nu\u03b2 : \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x : \u2191(S i)), f i (ui i x) = u\u03b2 (f i x)\nhT : iUnion S \u2286 iUnion S\nu : \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhui : \u2200 (i : \u03b9) (x : \u2191(S i)), u (inclusion (_ : S i \u2286 iUnion S) x) = inclusion (_ : S i \u2286 iUnion S) (ui i x)\nx : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nthis : x = inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191x, property := hi }\n| iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (u x)\n[PROOFSTEP]\nrw [this, hui, iUnionLift_inclusion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nui : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i)\nu\u03b2 : \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x : \u2191(S i)), f i (ui i x) = u\u03b2 (f i x)\nhT : iUnion S \u2286 iUnion S\nu : \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhui : \u2200 (i : \u03b9) (x : \u2191(S i)), u (inclusion (_ : S i \u2286 iUnion S) x) = inclusion (_ : S i \u2286 iUnion S) (ui i x)\nx : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nthis : x = inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191x, property := hi }\n| iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (u x)\n[PROOFSTEP]\nrw [this, hui, iUnionLift_inclusion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nui : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i)\nu\u03b2 : \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x : \u2191(S i)), f i (ui i x) = u\u03b2 (f i x)\nhT : iUnion S \u2286 iUnion S\nu : \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhui : \u2200 (i : \u03b9) (x : \u2191(S i)), u (inclusion (_ : S i \u2286 iUnion S) x) = inclusion (_ : S i \u2286 iUnion S) (ui i x)\nx : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nthis : x = inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191x, property := hi }\n| iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (u x)\n[PROOFSTEP]\nrw [this, hui, iUnionLift_inclusion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set \u03b1\nhT : T \u2286 iUnion S\nhT' : T = iUnion S\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nop : \u2191T \u2192 \u2191T \u2192 \u2191T\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 T) (opi i x y) = op (inclusion (_ : S i \u2286 T) x) (inclusion (_ : S i \u2286 T) y)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nx y : \u2191T\n\u22a2 iUnionLift S f hf T (_ : T \u2264 iUnion S) (op x y) =\n op\u03b2 (iUnionLift S f hf T (_ : T \u2264 iUnion S) x) (iUnionLift S f hf T (_ : T \u2264 iUnion S) y)\n[PROOFSTEP]\nsubst hT'\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nhT : iUnion S \u2286 iUnion S\nop : \u2191(iUnion S) \u2192 \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 iUnion S) (opi i x y) =\n op (inclusion (_ : S i \u2286 iUnion S) x) (inclusion (_ : S i \u2286 iUnion S) y)\nx y : \u2191(iUnion S)\n\u22a2 iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (op x y) =\n op\u03b2 (iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) x)\n (iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) y)\n[PROOFSTEP]\ncases' Set.mem_iUnion.1 x.prop with i hi\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nhT : iUnion S \u2286 iUnion S\nop : \u2191(iUnion S) \u2192 \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 iUnion S) (opi i x y) =\n op (inclusion (_ : S i \u2286 iUnion S) x) (inclusion (_ : S i \u2286 iUnion S) y)\nx y : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\n\u22a2 iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (op x y) =\n op\u03b2 (iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) x)\n (iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) y)\n[PROOFSTEP]\ncases' Set.mem_iUnion.1 y.prop with j hj\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nhT : iUnion S \u2286 iUnion S\nop : \u2191(iUnion S) \u2192 \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 iUnion S) (opi i x y) =\n op (inclusion (_ : S i \u2286 iUnion S) x) (inclusion (_ : S i \u2286 iUnion S) y)\nx y : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nj : \u03b9\nhj : \u2191y \u2208 S j\n\u22a2 iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (op x y) =\n op\u03b2 (iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) x)\n (iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) y)\n[PROOFSTEP]\nrcases dir i j with \u27e8k, hik, hjk\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nhT : iUnion S \u2286 iUnion S\nop : \u2191(iUnion S) \u2192 \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 iUnion S) (opi i x y) =\n op (inclusion (_ : S i \u2286 iUnion S) x) (inclusion (_ : S i \u2286 iUnion S) y)\nx y : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nj : \u03b9\nhj : \u2191y \u2208 S j\nk : \u03b9\nhik : S i \u2264 S k\nhjk : S j \u2264 S k\n\u22a2 iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (op x y) =\n op\u03b2 (iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) x)\n (iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) y)\n[PROOFSTEP]\nrw [iUnionLift_of_mem x (hik hi), iUnionLift_of_mem y (hjk hj), \u2190 h k]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nhT : iUnion S \u2286 iUnion S\nop : \u2191(iUnion S) \u2192 \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 iUnion S) (opi i x y) =\n op (inclusion (_ : S i \u2286 iUnion S) x) (inclusion (_ : S i \u2286 iUnion S) y)\nx y : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nj : \u03b9\nhj : \u2191y \u2208 S j\nk : \u03b9\nhik : S i \u2264 S k\nhjk : S j \u2264 S k\n\u22a2 iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (op x y) =\n f k (opi k { val := \u2191x, property := (_ : \u2191x \u2208 S k) } { val := \u2191y, property := (_ : \u2191y \u2208 S k) })\n[PROOFSTEP]\nhave hx : x = Set.inclusion (Set.subset_iUnion S k) \u27e8x, hik hi\u27e9 :=\n by\n cases x\n rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nhT : iUnion S \u2286 iUnion S\nop : \u2191(iUnion S) \u2192 \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 iUnion S) (opi i x y) =\n op (inclusion (_ : S i \u2286 iUnion S) x) (inclusion (_ : S i \u2286 iUnion S) y)\nx y : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nj : \u03b9\nhj : \u2191y \u2208 S j\nk : \u03b9\nhik : S i \u2264 S k\nhjk : S j \u2264 S k\n\u22a2 x = inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191x, property := (_ : \u2191x \u2208 S k) }\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nhT : iUnion S \u2286 iUnion S\nop : \u2191(iUnion S) \u2192 \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 iUnion S) (opi i x y) =\n op (inclusion (_ : S i \u2286 iUnion S) x) (inclusion (_ : S i \u2286 iUnion S) y)\ny : \u2191(iUnion S)\ni j : \u03b9\nhj : \u2191y \u2208 S j\nk : \u03b9\nhik : S i \u2264 S k\nhjk : S j \u2264 S k\nval\u271d : \u03b1\nproperty\u271d : val\u271d \u2208 iUnion S\nhi : \u2191{ val := val\u271d, property := property\u271d } \u2208 S i\n\u22a2 { val := val\u271d, property := property\u271d } =\n inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i)\n { val := \u2191{ val := val\u271d, property := property\u271d },\n property := (_ : \u2191{ val := val\u271d, property := property\u271d } \u2208 S k) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nhT : iUnion S \u2286 iUnion S\nop : \u2191(iUnion S) \u2192 \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 iUnion S) (opi i x y) =\n op (inclusion (_ : S i \u2286 iUnion S) x) (inclusion (_ : S i \u2286 iUnion S) y)\nx y : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nj : \u03b9\nhj : \u2191y \u2208 S j\nk : \u03b9\nhik : S i \u2264 S k\nhjk : S j \u2264 S k\nhx : x = inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191x, property := (_ : \u2191x \u2208 S k) }\n\u22a2 iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (op x y) =\n f k (opi k { val := \u2191x, property := (_ : \u2191x \u2208 S k) } { val := \u2191y, property := (_ : \u2191y \u2208 S k) })\n[PROOFSTEP]\nhave hy : y = Set.inclusion (Set.subset_iUnion S k) \u27e8y, hjk hj\u27e9 :=\n by\n cases y\n rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nhT : iUnion S \u2286 iUnion S\nop : \u2191(iUnion S) \u2192 \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 iUnion S) (opi i x y) =\n op (inclusion (_ : S i \u2286 iUnion S) x) (inclusion (_ : S i \u2286 iUnion S) y)\nx y : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nj : \u03b9\nhj : \u2191y \u2208 S j\nk : \u03b9\nhik : S i \u2264 S k\nhjk : S j \u2264 S k\nhx : x = inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191x, property := (_ : \u2191x \u2208 S k) }\n\u22a2 y = inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191y, property := (_ : \u2191y \u2208 S k) }\n[PROOFSTEP]\ncases y\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nhT : iUnion S \u2286 iUnion S\nop : \u2191(iUnion S) \u2192 \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 iUnion S) (opi i x y) =\n op (inclusion (_ : S i \u2286 iUnion S) x) (inclusion (_ : S i \u2286 iUnion S) y)\nx : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nj k : \u03b9\nhik : S i \u2264 S k\nhjk : S j \u2264 S k\nhx : x = inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191x, property := (_ : \u2191x \u2208 S k) }\nval\u271d : \u03b1\nproperty\u271d : val\u271d \u2208 iUnion S\nhj : \u2191{ val := val\u271d, property := property\u271d } \u2208 S j\n\u22a2 { val := val\u271d, property := property\u271d } =\n inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i)\n { val := \u2191{ val := val\u271d, property := property\u271d },\n property := (_ : \u2191{ val := val\u271d, property := property\u271d } \u2208 S k) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nhT : iUnion S \u2286 iUnion S\nop : \u2191(iUnion S) \u2192 \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 iUnion S) (opi i x y) =\n op (inclusion (_ : S i \u2286 iUnion S) x) (inclusion (_ : S i \u2286 iUnion S) y)\nx y : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nj : \u03b9\nhj : \u2191y \u2208 S j\nk : \u03b9\nhik : S i \u2264 S k\nhjk : S j \u2264 S k\nhx : x = inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191x, property := (_ : \u2191x \u2208 S k) }\nhy : y = inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191y, property := (_ : \u2191y \u2208 S k) }\n\u22a2 iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (op x y) =\n f k (opi k { val := \u2191x, property := (_ : \u2191x \u2208 S k) } { val := \u2191y, property := (_ : \u2191y \u2208 S k) })\n[PROOFSTEP]\nhave hxy : (Set.inclusion (Set.subset_iUnion S k) (opi k \u27e8x, hik hi\u27e9 \u27e8y, hjk hj\u27e9) : \u03b1) \u2208 S k :=\n (opi k \u27e8x, hik hi\u27e9 \u27e8y, hjk hj\u27e9).prop\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nhT : iUnion S \u2286 iUnion S\nop : \u2191(iUnion S) \u2192 \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 iUnion S) (opi i x y) =\n op (inclusion (_ : S i \u2286 iUnion S) x) (inclusion (_ : S i \u2286 iUnion S) y)\nx y : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nj : \u03b9\nhj : \u2191y \u2208 S j\nk : \u03b9\nhik : S i \u2264 S k\nhjk : S j \u2264 S k\nhx : x = inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191x, property := (_ : \u2191x \u2208 S k) }\nhy : y = inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191y, property := (_ : \u2191y \u2208 S k) }\nhxy :\n \u2191(inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i)\n (opi k { val := \u2191x, property := (_ : \u2191x \u2208 S k) } { val := \u2191y, property := (_ : \u2191y \u2208 S k) })) \u2208\n S k\n\u22a2 iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (op x y) =\n f k (opi k { val := \u2191x, property := (_ : \u2191x \u2208 S k) } { val := \u2191y, property := (_ : \u2191y \u2208 S k) })\n[PROOFSTEP]\nconv_lhs => rw [hx, hy, \u2190 hopi, iUnionLift_of_mem _ hxy]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nhT : iUnion S \u2286 iUnion S\nop : \u2191(iUnion S) \u2192 \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 iUnion S) (opi i x y) =\n op (inclusion (_ : S i \u2286 iUnion S) x) (inclusion (_ : S i \u2286 iUnion S) y)\nx y : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nj : \u03b9\nhj : \u2191y \u2208 S j\nk : \u03b9\nhik : S i \u2264 S k\nhjk : S j \u2264 S k\nhx : x = inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191x, property := (_ : \u2191x \u2208 S k) }\nhy : y = inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191y, property := (_ : \u2191y \u2208 S k) }\nhxy :\n \u2191(inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i)\n (opi k { val := \u2191x, property := (_ : \u2191x \u2208 S k) } { val := \u2191y, property := (_ : \u2191y \u2208 S k) })) \u2208\n S k\n| iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (op x y)\n[PROOFSTEP]\nrw [hx, hy, \u2190 hopi, iUnionLift_of_mem _ hxy]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nhT : iUnion S \u2286 iUnion S\nop : \u2191(iUnion S) \u2192 \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 iUnion S) (opi i x y) =\n op (inclusion (_ : S i \u2286 iUnion S) x) (inclusion (_ : S i \u2286 iUnion S) y)\nx y : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nj : \u03b9\nhj : \u2191y \u2208 S j\nk : \u03b9\nhik : S i \u2264 S k\nhjk : S j \u2264 S k\nhx : x = inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191x, property := (_ : \u2191x \u2208 S k) }\nhy : y = inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191y, property := (_ : \u2191y \u2208 S k) }\nhxy :\n \u2191(inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i)\n (opi k { val := \u2191x, property := (_ : \u2191x \u2208 S k) } { val := \u2191y, property := (_ : \u2191y \u2208 S k) })) \u2208\n S k\n| iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (op x y)\n[PROOFSTEP]\nrw [hx, hy, \u2190 hopi, iUnionLift_of_mem _ hxy]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03b2 : Sort u_3\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\ndir : Directed (fun x x_1 => x \u2264 x_1) S\nopi : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u2191(S i) \u2192 \u2191(S i)\nop\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nh : \u2200 (i : \u03b9) (x y : \u2191(S i)), f i (opi i x y) = op\u03b2 (f i x) (f i y)\nhT : iUnion S \u2286 iUnion S\nop : \u2191(iUnion S) \u2192 \u2191(iUnion S) \u2192 \u2191(iUnion S)\nhopi :\n \u2200 (i : \u03b9) (x y : \u2191(S i)),\n inclusion (_ : S i \u2286 iUnion S) (opi i x y) =\n op (inclusion (_ : S i \u2286 iUnion S) x) (inclusion (_ : S i \u2286 iUnion S) y)\nx y : \u2191(iUnion S)\ni : \u03b9\nhi : \u2191x \u2208 S i\nj : \u03b9\nhj : \u2191y \u2208 S j\nk : \u03b9\nhik : S i \u2264 S k\nhjk : S j \u2264 S k\nhx : x = inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191x, property := (_ : \u2191x \u2208 S k) }\nhy : y = inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i) { val := \u2191y, property := (_ : \u2191y \u2208 S k) }\nhxy :\n \u2191(inclusion (_ : S k \u2286 \u22c3 (i : \u03b9), S i)\n (opi k { val := \u2191x, property := (_ : \u2191x \u2208 S k) } { val := \u2191y, property := (_ : \u2191y \u2208 S k) })) \u2208\n S k\n| iUnionLift S f hf (iUnion S) (_ : iUnion S \u2264 iUnion S) (op x y)\n[PROOFSTEP]\nrw [hx, hy, \u2190 hopi, iUnionLift_of_mem _ hxy]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Type u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nhS : iUnion S = univ\nt : Set \u03b2\n\u22a2 liftCover S f hf hS \u207b\u00b9' t = \u22c3 (i : \u03b9), Subtype.val '' (f i \u207b\u00b9' t)\n[PROOFSTEP]\nchange (iUnionLift S f hf univ hS.symm.subset \u2218 fun a => \u27e8a, mem_univ a\u27e9) \u207b\u00b9' t = _\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Type u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nhS : iUnion S = univ\nt : Set \u03b2\n\u22a2 (iUnionLift S f hf univ (_ : univ \u2286 iUnion S) \u2218 fun a => { val := a, property := (_ : a \u2208 univ) }) \u207b\u00b9' t =\n \u22c3 (i : \u03b9), Subtype.val '' (f i \u207b\u00b9' t)\n[PROOFSTEP]\nrw [preimage_comp, preimage_iUnionLift]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Type u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nhS : iUnion S = univ\nt : Set \u03b2\n\u22a2 (fun a => { val := a, property := (_ : a \u2208 univ) }) \u207b\u00b9'\n (inclusion (_ : univ \u2286 iUnion S) \u207b\u00b9' \u22c3 (i : \u03b9), inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) '' (f i \u207b\u00b9' t)) =\n \u22c3 (i : \u03b9), Subtype.val '' (f i \u207b\u00b9' t)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b9 : Sort u_3\n\u03b2 : Type u_2\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nhS : iUnion S = univ\nt : Set \u03b2\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208\n (fun a => { val := a, property := (_ : a \u2208 univ) }) \u207b\u00b9'\n (inclusion (_ : univ \u2286 iUnion S) \u207b\u00b9' \u22c3 (i : \u03b9), inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) '' (f i \u207b\u00b9' t)) \u2194\n x\u271d \u2208 \u22c3 (i : \u03b9), Subtype.val '' (f i \u207b\u00b9' t)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.UnionLift", "llama_tokens": 17431, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.4198689200097167}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 (List.bind l fun x => [x]) = l\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 (List.bind [] fun x => [x]) = []\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : (List.bind tail\u271d fun x => [x]) = tail\u271d\n\u22a2 (List.bind (head\u271d :: tail\u271d) fun x => [x]) = head\u271d :: tail\u271d\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\n\u22a2 map f l = List.bind l fun x => [f x]\n[PROOFSTEP]\nsimp only [\u2190 map_singleton]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\n\u22a2 map f l = List.bind l fun x => map f [x]\n[PROOFSTEP]\nrw [\u2190 bind_singleton' l, bind_map, bind_singleton']\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\nf : \u03b1 \u2192 List \u03b2\ng : \u03b2 \u2192 List \u03b3\n\u22a2 List.bind (List.bind l f) g = List.bind l fun x => List.bind (f x) g\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase nil\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 List \u03b2\ng : \u03b2 \u2192 List \u03b3\n\u22a2 List.bind (List.bind [] f) g = List.bind [] fun x => List.bind (f x) g\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase cons\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 List \u03b2\ng : \u03b2 \u2192 List \u03b3\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : List.bind (List.bind tail\u271d f) g = List.bind tail\u271d fun x => List.bind (f x) g\n\u22a2 List.bind (List.bind (head\u271d :: tail\u271d) f) g = List.bind (head\u271d :: tail\u271d) fun x => List.bind (f x) g\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 \u00ac\u2203 x, x \u2208 [] \u2227 p x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx : \u03b1\nxs : List \u03b1\nh\u2081 : \u00acp x\nh\u2082 : \u2203 x, x \u2208 xs \u2227 p x\n\u22a2 \u2203 x_1, x_1 \u2208 x :: xs \u2227 p x_1\n[PROOFSTEP]\ncases' h\u2082 with y h\n[GOAL]\ncase intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx : \u03b1\nxs : List \u03b1\nh\u2081 : \u00acp x\ny : \u03b1\nh : y \u2208 xs \u2227 p y\n\u22a2 \u2203 x_1, x_1 \u2208 x :: xs \u2227 p x_1\n[PROOFSTEP]\ncases' h with hm hp\n[GOAL]\ncase intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx : \u03b1\nxs : List \u03b1\nh\u2081 : \u00acp x\ny : \u03b1\nhm : y \u2208 xs\nhp : p y\n\u22a2 \u2203 x_1, x_1 \u2208 x :: xs \u2227 p x_1\n[PROOFSTEP]\nexact \u27e8y, mem_cons_of_mem _ hm, hp\u27e9\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx : \u03b1\nxs : List \u03b1\nh\u2081 : \u00acp x\nh\u2082 : \u00ac\u2203 x, x \u2208 xs \u2227 p x\n\u22a2 \u00ac\u2203 x_1, x_1 \u2208 x :: xs \u2227 p x_1\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx : \u03b1\nxs : List \u03b1\nh\u2081 : \u00acp x\nh\u2082 : \u00ac\u2203 x, x \u2208 xs \u2227 p x\nh : \u2203 x_1, x_1 \u2208 x :: xs \u2227 p x_1\n\u22a2 False\n[PROOFSTEP]\ncases' h with y h\n[GOAL]\ncase intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx : \u03b1\nxs : List \u03b1\nh\u2081 : \u00acp x\nh\u2082 : \u00ac\u2203 x, x \u2208 xs \u2227 p x\ny : \u03b1\nh : y \u2208 x :: xs \u2227 p y\n\u22a2 False\n[PROOFSTEP]\ncases' h with hm hp\n[GOAL]\ncase intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx : \u03b1\nxs : List \u03b1\nh\u2081 : \u00acp x\nh\u2082 : \u00ac\u2203 x, x \u2208 xs \u2227 p x\ny : \u03b1\nhm : y \u2208 x :: xs\nhp : p y\n\u22a2 False\n[PROOFSTEP]\ncases' mem_cons.1 hm with h h\n[GOAL]\ncase intro.intro.inl\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx : \u03b1\nxs : List \u03b1\nh\u2081 : \u00acp x\nh\u2082 : \u00ac\u2203 x, x \u2208 xs \u2227 p x\ny : \u03b1\nhm : y \u2208 x :: xs\nhp : p y\nh : y = x\n\u22a2 False\n[PROOFSTEP]\nrw [h] at hp \n[GOAL]\ncase intro.intro.inl\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx : \u03b1\nxs : List \u03b1\nh\u2081 : \u00acp x\nh\u2082 : \u00ac\u2203 x, x \u2208 xs \u2227 p x\ny : \u03b1\nhm : y \u2208 x :: xs\nhp : p x\nh : y = x\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase intro.intro.inr\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx : \u03b1\nxs : List \u03b1\nh\u2081 : \u00acp x\nh\u2082 : \u00ac\u2203 x, x \u2208 xs \u2227 p x\ny : \u03b1\nhm : y \u2208 x :: xs\nhp : p y\nh : y \u2208 xs\n\u22a2 False\n[PROOFSTEP]\nexact absurd \u27e8y, h, hp\u27e9 h\u2082\n", "meta": {"mathlib_filename": "Mathlib.Init.Data.List.Instances", "llama_tokens": 2076, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.4198428145312696}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\n\u22a2 natDegree (cancelLeads p q) < natDegree q\n[PROOFSTEP]\nby_cases hp : p = 0\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\nhp : p = 0\n\u22a2 natDegree (cancelLeads p q) < natDegree q\n[PROOFSTEP]\nconvert hq\n[GOAL]\ncase h.e'_3\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\nhp : p = 0\n\u22a2 natDegree (cancelLeads p q) = 0\n[PROOFSTEP]\nsimp [hp, cancelLeads]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\nhp : \u00acp = 0\n\u22a2 natDegree (cancelLeads p q) < natDegree q\n[PROOFSTEP]\nrw [cancelLeads, sub_eq_add_neg, tsub_eq_zero_iff_le.mpr h, pow_zero, mul_one]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\nhp : \u00acp = 0\n\u22a2 natDegree (\u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) < natDegree q\n[PROOFSTEP]\nby_cases h0 : C p.leadingCoeff * q + -(C q.leadingCoeff * X ^ (q.natDegree - p.natDegree) * p) = 0\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\nhp : \u00acp = 0\nh0 : \u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n\u22a2 natDegree (\u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) < natDegree q\n[PROOFSTEP]\nexact (le_of_eq (by simp only [h0, natDegree_zero])).trans_lt hq\n[GOAL]\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\nhp : \u00acp = 0\nh0 : \u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n\u22a2 natDegree (\u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = 0\n[PROOFSTEP]\nsimp only [h0, natDegree_zero]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\nhp : \u00acp = 0\nh0 : \u00ac\u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n\u22a2 natDegree (\u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) < natDegree q\n[PROOFSTEP]\napply lt_of_le_of_ne\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\nhp : \u00acp = 0\nh0 : \u00ac\u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n\u22a2 natDegree (\u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) \u2264 natDegree q\n[PROOFSTEP]\nrw [natDegree_add_le_iff_left]\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\nhp : \u00acp = 0\nh0 : \u00ac\u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n\u22a2 natDegree (\u2191C (leadingCoeff p) * q) \u2264 natDegree q\n[PROOFSTEP]\napply natDegree_C_mul_le\n[GOAL]\ncase neg.a.qn\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\nhp : \u00acp = 0\nh0 : \u00ac\u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n\u22a2 natDegree (-(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) \u2264 natDegree q\n[PROOFSTEP]\nrefine (natDegree_neg (C q.leadingCoeff * X ^ (q.natDegree - p.natDegree) * p)).le.trans ?_\n[GOAL]\ncase neg.a.qn\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\nhp : \u00acp = 0\nh0 : \u00ac\u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n\u22a2 natDegree (\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) \u2264 natDegree q\n[PROOFSTEP]\nexact natDegree_mul_le.trans <| Nat.add_le_of_le_sub h <| natDegree_C_mul_X_pow_le _ _\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\nhp : \u00acp = 0\nh0 : \u00ac\u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n\u22a2 natDegree (\u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) \u2260 natDegree q\n[PROOFSTEP]\ncontrapose! h0\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\nhp : \u00acp = 0\nh0 : natDegree (\u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = natDegree q\n\u22a2 \u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0\n[PROOFSTEP]\nrw [\u2190 leadingCoeff_eq_zero, leadingCoeff, h0, mul_assoc, X_pow_mul, \u2190 tsub_add_cancel_of_le h, add_comm _ p.natDegree]\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\nhp : \u00acp = 0\nh0 : natDegree (\u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = natDegree q\n\u22a2 coeff\n (\u2191C (leadingCoeff p) * q +\n -(\u2191C (leadingCoeff q) * (p * X ^ (natDegree p + (natDegree q - natDegree p) - natDegree p))))\n (natDegree p + (natDegree q - natDegree p)) =\n 0\n[PROOFSTEP]\nsimp only [coeff_mul_X_pow, coeff_neg, coeff_C_mul, add_tsub_cancel_left, coeff_add]\n[GOAL]\ncase neg.a\nR : Type u_1\ninst\u271d : Ring R\np q : R[X]\ncomm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p\nh : natDegree p \u2264 natDegree q\nhq : 0 < natDegree q\nhp : \u00acp = 0\nh0 : natDegree (\u2191C (leadingCoeff p) * q + -(\u2191C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = natDegree q\n\u22a2 leadingCoeff p * coeff q (natDegree p + (natDegree q - natDegree p)) + -(leadingCoeff q * coeff p (natDegree p)) = 0\n[PROOFSTEP]\nrw [add_comm p.natDegree, tsub_add_cancel_of_le h, \u2190 leadingCoeff, \u2190 leadingCoeff, comm, add_right_neg]\n", "meta": {"mathlib_filename": "Mathlib.Data.Polynomial.CancelLeads", "llama_tokens": 3308, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956580903722561, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.41956365801392353}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\n\u22a2 Measurable g\n[PROOFSTEP]\nrcases u.exists_seq_tendsto with \u27e8x, hx\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nx : \u2115 \u2192 \u03b9\nhx : Tendsto x atTop u\n\u22a2 Measurable g\n[PROOFSTEP]\nrw [tendsto_pi_nhds] at lim \n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x))\nx : \u2115 \u2192 \u03b9\nhx : Tendsto x atTop u\n\u22a2 Measurable g\n[PROOFSTEP]\nhave : (fun y => liminf (fun n => (f (x n) y : \u211d\u22650\u221e)) atTop) = g :=\n by\n ext1 y\n exact ((lim y).comp hx).liminf_eq\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x))\nx : \u2115 \u2192 \u03b9\nhx : Tendsto x atTop u\n\u22a2 (fun y => liminf (fun n => f (x n) y) atTop) = g\n[PROOFSTEP]\next1 y\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x))\nx : \u2115 \u2192 \u03b9\nhx : Tendsto x atTop u\ny : \u03b1\n\u22a2 liminf (fun n => f (x n) y) atTop = g y\n[PROOFSTEP]\nexact ((lim y).comp hx).liminf_eq\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x))\nx : \u2115 \u2192 \u03b9\nhx : Tendsto x atTop u\nthis : (fun y => liminf (fun n => f (x n) y) atTop) = g\n\u22a2 Measurable g\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x))\nx : \u2115 \u2192 \u03b9\nhx : Tendsto x atTop u\nthis : (fun y => liminf (fun n => f (x n) y) atTop) = g\n\u22a2 Measurable fun y => liminf (fun n => f (x n) y) atTop\n[PROOFSTEP]\nshow Measurable fun y => liminf (fun n => (f (x n) y : \u211d\u22650\u221e)) atTop\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b1 \u2192 \u211d\u22650\u221e\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x))\nx : \u2115 \u2192 \u03b9\nhx : Tendsto x atTop u\nthis : (fun y => liminf (fun n => f (x n) y) atTop) = g\n\u22a2 Measurable fun y => liminf (fun n => f (x n) y) atTop\n[PROOFSTEP]\nexact measurable_liminf fun n => hf (x n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\ng : \u03b1 \u2192 \u211d\u22650\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\n\u22a2 Measurable g\n[PROOFSTEP]\nsimp_rw [\u2190 measurable_coe_nnreal_ennreal_iff] at hf \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\ng : \u03b1 \u2192 \u211d\u22650\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nlim : Tendsto f u (\ud835\udcdd g)\nhf : \u2200 (i : \u03b9), Measurable fun x => \u2191(f i x)\n\u22a2 Measurable fun x => \u2191(g x)\n[PROOFSTEP]\nrefine' measurable_of_tendsto_ennreal' u hf _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\ng : \u03b1 \u2192 \u211d\u22650\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nlim : Tendsto f u (\ud835\udcdd g)\nhf : \u2200 (i : \u03b9), Measurable fun x => \u2191(f i x)\n\u22a2 Tendsto (fun i x => \u2191(f i x)) u (\ud835\udcdd fun x => \u2191(g x))\n[PROOFSTEP]\nrw [tendsto_pi_nhds] at lim \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\ng : \u03b1 \u2192 \u211d\u22650\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nlim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x))\nhf : \u2200 (i : \u03b9), Measurable fun x => \u2191(f i x)\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun i => \u2191(f i x)) u (\ud835\udcdd \u2191(g x))\n[PROOFSTEP]\nexact fun x => (ENNReal.continuous_coe.tendsto (g x)).comp (lim x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\n\u22a2 Measurable g\n[PROOFSTEP]\nletI : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\n\u22a2 Measurable g\n[PROOFSTEP]\napply measurable_of_is_closed'\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\n\u22a2 \u2200 (s : Set \u03b2), IsClosed s \u2192 Set.Nonempty s \u2192 s \u2260 Set.univ \u2192 MeasurableSet (g \u207b\u00b9' s)\n[PROOFSTEP]\nintro s h1s h2s h3s\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\n\u22a2 MeasurableSet (g \u207b\u00b9' s)\n[PROOFSTEP]\nhave : Measurable fun x => infNndist (g x) s :=\n by\n suffices : Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s)\n exact measurable_of_tendsto_nnreal' u (fun i => (hf i).infNndist) this\n rw [tendsto_pi_nhds] at lim \u22a2\n intro x\n exact ((continuous_infNndist_pt s).tendsto (g x)).comp (lim x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\n\u22a2 Measurable fun x => infNndist (g x) s\n[PROOFSTEP]\nsuffices : Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s)\n\u22a2 Measurable fun x => infNndist (g x) s\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\n\u22a2 Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s)\n[PROOFSTEP]\nexact measurable_of_tendsto_nnreal' u (fun i => (hf i).infNndist) this\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\n\u22a2 Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s)\n[PROOFSTEP]\nrw [tendsto_pi_nhds] at lim \u22a2\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x))\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun i => infNndist (f i x) s) u (\ud835\udcdd (infNndist (g x) s))\n[PROOFSTEP]\nintro x\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x))\nthis : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nx : \u03b1\n\u22a2 Tendsto (fun i => infNndist (f i x) s) u (\ud835\udcdd (infNndist (g x) s))\n[PROOFSTEP]\nexact ((continuous_infNndist_pt s).tendsto (g x)).comp (lim x)\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Measurable fun x => infNndist (g x) s\n\u22a2 MeasurableSet (g \u207b\u00b9' s)\n[PROOFSTEP]\nhave h4s : g \u207b\u00b9' s = (fun x => infNndist (g x) s) \u207b\u00b9' {0} :=\n by\n ext x\n simp [h1s, \u2190 h1s.mem_iff_infDist_zero h2s, \u2190 NNReal.coe_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Measurable fun x => infNndist (g x) s\n\u22a2 g \u207b\u00b9' s = (fun x => infNndist (g x) s) \u207b\u00b9' {0}\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Measurable fun x => infNndist (g x) s\nx : \u03b1\n\u22a2 x \u2208 g \u207b\u00b9' s \u2194 x \u2208 (fun x => infNndist (g x) s) \u207b\u00b9' {0}\n[PROOFSTEP]\nsimp [h1s, \u2190 h1s.mem_iff_infDist_zero h2s, \u2190 NNReal.coe_eq_zero]\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Measurable fun x => infNndist (g x) s\nh4s : g \u207b\u00b9' s = (fun x => infNndist (g x) s) \u207b\u00b9' {0}\n\u22a2 MeasurableSet (g \u207b\u00b9' s)\n[PROOFSTEP]\nrw [h4s]\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\ninst\u271d\u00b9 : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (i : \u03b9), Measurable (f i)\nlim : Tendsto f u (\ud835\udcdd g)\nthis\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\ns : Set \u03b2\nh1s : IsClosed s\nh2s : Set.Nonempty s\nh3s : s \u2260 Set.univ\nthis : Measurable fun x => infNndist (g x) s\nh4s : g \u207b\u00b9' s = (fun x => infNndist (g x) s) \u207b\u00b9' {0}\n\u22a2 MeasurableSet ((fun x => infNndist (g x) s) \u207b\u00b9' {0})\n[PROOFSTEP]\nexact this (measurableSet_singleton 0)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\n\u03b9 : Type u_3\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\nhu : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) u (\ud835\udcdd (g x))\n\u22a2 AEMeasurable g\n[PROOFSTEP]\nrcases u.exists_seq_tendsto with \u27e8v, hv\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\n\u03b9 : Type u_3\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\nhu : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) u (\ud835\udcdd (g x))\nv : \u2115 \u2192 \u03b9\nhv : Tendsto v atTop u\n\u22a2 AEMeasurable g\n[PROOFSTEP]\nhave h'f : \u2200 n, AEMeasurable (f (v n)) \u03bc := fun n => hf (v n)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\n\u03b9 : Type u_3\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\nhu : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) u (\ud835\udcdd (g x))\nv : \u2115 \u2192 \u03b9\nhv : Tendsto v atTop u\nh'f : \u2200 (n : \u2115), AEMeasurable (f (v n))\n\u22a2 AEMeasurable g\n[PROOFSTEP]\nset p : \u03b1 \u2192 (\u2115 \u2192 \u03b2) \u2192 Prop := fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x))\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\n\u03b9 : Type u_3\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\nhu : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) u (\ud835\udcdd (g x))\nv : \u2115 \u2192 \u03b9\nhv : Tendsto v atTop u\nh'f : \u2200 (n : \u2115), AEMeasurable (f (v n))\np : \u03b1 \u2192 (\u2115 \u2192 \u03b2) \u2192 Prop := fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x))\n\u22a2 AEMeasurable g\n[PROOFSTEP]\nhave hp : \u2200\u1d50 x \u2202\u03bc, p x fun n => f (v n) x := by filter_upwards [h_tendsto] with x hx using hx.comp hv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\n\u03b9 : Type u_3\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\nhu : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) u (\ud835\udcdd (g x))\nv : \u2115 \u2192 \u03b9\nhv : Tendsto v atTop u\nh'f : \u2200 (n : \u2115), AEMeasurable (f (v n))\np : \u03b1 \u2192 (\u2115 \u2192 \u03b2) \u2192 Prop := fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x))\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f (v n) x\n[PROOFSTEP]\nfilter_upwards [h_tendsto] with x hx using hx.comp hv\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\n\u03b9 : Type u_3\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\nhu : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) u (\ud835\udcdd (g x))\nv : \u2115 \u2192 \u03b9\nhv : Tendsto v atTop u\nh'f : \u2200 (n : \u2115), AEMeasurable (f (v n))\np : \u03b1 \u2192 (\u2115 \u2192 \u03b2) \u2192 Prop := fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x))\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f (v n) x\n\u22a2 AEMeasurable g\n[PROOFSTEP]\nset aeSeqLim := fun x => ite (x \u2208 aeSeqSet h'f p) (g x) (\u27e8f (v 0) x\u27e9 : Nonempty \u03b2).some\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\n\u03b9 : Type u_3\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\nhu : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) u (\ud835\udcdd (g x))\nv : \u2115 \u2192 \u03b9\nhv : Tendsto v atTop u\nh'f : \u2200 (n : \u2115), AEMeasurable (f (v n))\np : \u03b1 \u2192 (\u2115 \u2192 \u03b2) \u2192 Prop := fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x))\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f (v n) x\naeSeqLim : \u03b1 \u2192 \u03b2 := fun x => if x \u2208 aeSeqSet h'f p then g x else Nonempty.some (_ : Nonempty \u03b2)\n\u22a2 AEMeasurable g\n[PROOFSTEP]\nrefine' \u27e8aeSeqLim, measurable_of_tendsto_metrizable' atTop (aeSeq.measurable h'f p) (tendsto_pi_nhds.mpr fun x => _), _\u27e9\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\n\u03b9 : Type u_3\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\nhu : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) u (\ud835\udcdd (g x))\nv : \u2115 \u2192 \u03b9\nhv : Tendsto v atTop u\nh'f : \u2200 (n : \u2115), AEMeasurable (f (v n))\np : \u03b1 \u2192 (\u2115 \u2192 \u03b2) \u2192 Prop := fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x))\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f (v n) x\naeSeqLim : \u03b1 \u2192 \u03b2 := fun x => if x \u2208 aeSeqSet h'f p then g x else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\n\u22a2 Tendsto (fun i => aeSeq h'f p i x) atTop (\ud835\udcdd (aeSeqLim x))\n[PROOFSTEP]\nsimp_rw [aeSeq]\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\n\u03b9 : Type u_3\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\nhu : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) u (\ud835\udcdd (g x))\nv : \u2115 \u2192 \u03b9\nhv : Tendsto v atTop u\nh'f : \u2200 (n : \u2115), AEMeasurable (f (v n))\np : \u03b1 \u2192 (\u2115 \u2192 \u03b2) \u2192 Prop := fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x))\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f (v n) x\naeSeqLim : \u03b1 \u2192 \u03b2 := fun x => if x \u2208 aeSeqSet h'f p then g x else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\n\u22a2 Tendsto\n (fun i =>\n if x \u2208 aeSeqSet h'f fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x)) then\n AEMeasurable.mk (f (v i)) (_ : AEMeasurable (f (v i))) x\n else Nonempty.some (_ : Nonempty \u03b2))\n atTop\n (\ud835\udcdd\n (if x \u2208 aeSeqSet h'f fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x)) then g x\n else Nonempty.some (_ : Nonempty \u03b2)))\n[PROOFSTEP]\nsplit_ifs with hx\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\n\u03b9 : Type u_3\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\nhu : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) u (\ud835\udcdd (g x))\nv : \u2115 \u2192 \u03b9\nhv : Tendsto v atTop u\nh'f : \u2200 (n : \u2115), AEMeasurable (f (v n))\np : \u03b1 \u2192 (\u2115 \u2192 \u03b2) \u2192 Prop := fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x))\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f (v n) x\naeSeqLim : \u03b1 \u2192 \u03b2 := fun x => if x \u2208 aeSeqSet h'f p then g x else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\nhx : x \u2208 aeSeqSet h'f fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x))\n\u22a2 Tendsto (fun i => AEMeasurable.mk (f (v i)) (_ : AEMeasurable (f (v i))) x) atTop (\ud835\udcdd (g x))\n[PROOFSTEP]\nsimp_rw [aeSeq.mk_eq_fun_of_mem_aeSeqSet h'f hx]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\n\u03b9 : Type u_3\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\nhu : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) u (\ud835\udcdd (g x))\nv : \u2115 \u2192 \u03b9\nhv : Tendsto v atTop u\nh'f : \u2200 (n : \u2115), AEMeasurable (f (v n))\np : \u03b1 \u2192 (\u2115 \u2192 \u03b2) \u2192 Prop := fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x))\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f (v n) x\naeSeqLim : \u03b1 \u2192 \u03b2 := fun x => if x \u2208 aeSeqSet h'f p then g x else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\nhx : x \u2208 aeSeqSet h'f fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x))\n\u22a2 Tendsto (fun i => f (v i) x) atTop (\ud835\udcdd (g x))\n[PROOFSTEP]\nexact @aeSeq.fun_prop_of_mem_aeSeqSet _ \u03b1 \u03b2 _ _ _ _ _ h'f x hx\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\n\u03b9 : Type u_3\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\nhu : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) u (\ud835\udcdd (g x))\nv : \u2115 \u2192 \u03b9\nhv : Tendsto v atTop u\nh'f : \u2200 (n : \u2115), AEMeasurable (f (v n))\np : \u03b1 \u2192 (\u2115 \u2192 \u03b2) \u2192 Prop := fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x))\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f (v n) x\naeSeqLim : \u03b1 \u2192 \u03b2 := fun x => if x \u2208 aeSeqSet h'f p then g x else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\nhx : \u00acx \u2208 aeSeqSet h'f fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x))\n\u22a2 Tendsto (fun i => Nonempty.some (_ : Nonempty \u03b2)) atTop (\ud835\udcdd (Nonempty.some (_ : Nonempty \u03b2)))\n[PROOFSTEP]\nexact tendsto_const_nhds\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : PseudoMetrizableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\n\u03b9 : Type u_3\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nu : Filter \u03b9\nhu : NeBot u\ninst\u271d : IsCountablyGenerated u\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) u (\ud835\udcdd (g x))\nv : \u2115 \u2192 \u03b9\nhv : Tendsto v atTop u\nh'f : \u2200 (n : \u2115), AEMeasurable (f (v n))\np : \u03b1 \u2192 (\u2115 \u2192 \u03b2) \u2192 Prop := fun x f' => Tendsto (fun n => f' n) atTop (\ud835\udcdd (g x))\nhp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun n => f (v n) x\naeSeqLim : \u03b1 \u2192 \u03b2 := fun x => if x \u2208 aeSeqSet h'f p then g x else Nonempty.some (_ : Nonempty \u03b2)\n\u22a2 g =\u1d50[\u03bc] aeSeqLim\n[PROOFSTEP]\nexact\n (ite_ae_eq_of_measure_compl_zero g (fun x => (\u27e8f (v 0) x\u27e9 : Nonempty \u03b2).some) (aeSeqSet h'f p)\n (aeSeq.measure_compl_aeSeqSet_eq_zero h'f hp)).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : BorelSpace \u03b2\u271d\n\u03b2 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b1\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f x) (g x) \u2264 \u03b5\n\u22a2 AEMeasurable g\n[PROOFSTEP]\nobtain \u27e8u, -, u_pos, u_lim\u27e9 : \u2203 u : \u2115 \u2192 \u211d, StrictAnti u \u2227 (\u2200 n : \u2115, 0 < u n) \u2227 Tendsto u atTop (\ud835\udcdd 0) :=\n exists_seq_strictAnti_tendsto (0 : \u211d)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : BorelSpace \u03b2\u271d\n\u03b2 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b1\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f x) (g x) \u2264 \u03b5\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\n\u22a2 AEMeasurable g\n[PROOFSTEP]\nchoose f Hf using fun n : \u2115 => hf (u n) (u_pos n)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : BorelSpace \u03b2\u271d\n\u03b2 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b1\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f x) (g x) \u2264 \u03b5\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nHf : \u2200 (n : \u2115), AEMeasurable (f n) \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f n x) (g x) \u2264 u n\n\u22a2 AEMeasurable g\n[PROOFSTEP]\nhave : \u2200\u1d50 x \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) :=\n by\n have : \u2200\u1d50 x \u2202\u03bc, \u2200 n, dist (f n x) (g x) \u2264 u n := ae_all_iff.2 fun n => (Hf n).2\n filter_upwards [this]\n intro x hx\n rw [tendsto_iff_dist_tendsto_zero]\n exact squeeze_zero (fun n => dist_nonneg) hx u_lim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : BorelSpace \u03b2\u271d\n\u03b2 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b1\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f x) (g x) \u2264 \u03b5\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nHf : \u2200 (n : \u2115), AEMeasurable (f n) \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f n x) (g x) \u2264 u n\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n[PROOFSTEP]\nhave : \u2200\u1d50 x \u2202\u03bc, \u2200 n, dist (f n x) (g x) \u2264 u n := ae_all_iff.2 fun n => (Hf n).2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : BorelSpace \u03b2\u271d\n\u03b2 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b1\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f x) (g x) \u2264 \u03b5\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nHf : \u2200 (n : \u2115), AEMeasurable (f n) \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f n x) (g x) \u2264 u n\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), dist (f n x) (g x) \u2264 u n\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n[PROOFSTEP]\nfilter_upwards [this]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : BorelSpace \u03b2\u271d\n\u03b2 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b1\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f x) (g x) \u2264 \u03b5\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nHf : \u2200 (n : \u2115), AEMeasurable (f n) \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f n x) (g x) \u2264 u n\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), dist (f n x) (g x) \u2264 u n\n\u22a2 \u2200 (a : \u03b1), (\u2200 (n : \u2115), dist (f n a) (g a) \u2264 u n) \u2192 Tendsto (fun n => f n a) atTop (\ud835\udcdd (g a))\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : BorelSpace \u03b2\u271d\n\u03b2 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b1\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f x) (g x) \u2264 \u03b5\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nHf : \u2200 (n : \u2115), AEMeasurable (f n) \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f n x) (g x) \u2264 u n\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), dist (f n x) (g x) \u2264 u n\nx : \u03b1\nhx : \u2200 (n : \u2115), dist (f n x) (g x) \u2264 u n\n\u22a2 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n[PROOFSTEP]\nrw [tendsto_iff_dist_tendsto_zero]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : BorelSpace \u03b2\u271d\n\u03b2 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b1\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f x) (g x) \u2264 \u03b5\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nHf : \u2200 (n : \u2115), AEMeasurable (f n) \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f n x) (g x) \u2264 u n\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), dist (f n x) (g x) \u2264 u n\nx : \u03b1\nhx : \u2200 (n : \u2115), dist (f n x) (g x) \u2264 u n\n\u22a2 Tendsto (fun b => dist (f b x) (g x)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact squeeze_zero (fun n => dist_nonneg) hx u_lim\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : BorelSpace \u03b2\u271d\n\u03b2 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b1\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f x) (g x) \u2264 \u03b5\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nHf : \u2200 (n : \u2115), AEMeasurable (f n) \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, dist (f n x) (g x) \u2264 u n\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 AEMeasurable g\n[PROOFSTEP]\nexact aemeasurable_of_tendsto_metrizable_ae' (fun n => (Hf n).1) this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\n[PROOFSTEP]\ninhabit \u03b9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\n[PROOFSTEP]\nrcases eq_or_neBot L with (rfl | hL)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\ninhabited_h : Inhabited \u03b9\ninst\u271d : IsCountablyGenerated \u22a5\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) \u22a5 (\ud835\udcdd l)\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) \u22a5 (\ud835\udcdd (f_lim x))\n[PROOFSTEP]\nexact \u27e8(hf default).mk _, (hf default).measurable_mk, eventually_of_forall fun x => tendsto_bot\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : NeBot L\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\n[PROOFSTEP]\nlet p : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l : \u03b2, Tendsto (fun n => f' n) L (\ud835\udcdd l)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\n[PROOFSTEP]\nhave hp_mem : \u2200 x \u2208 aeSeqSet hf p, p x fun n => f n x := fun x hx => aeSeq.fun_prop_of_mem_aeSeqSet hf hx\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\n[PROOFSTEP]\nhave h_ae_eq : \u2200\u1d50 x \u2202\u03bc, \u2200 n, aeSeq hf p n x = f n x := aeSeq.aeSeq_eq_fun_ae hf h_ae_tendsto\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\n[PROOFSTEP]\nset f_lim : \u03b1 \u2192 \u03b2 := fun x =>\n dite (x \u2208 aeSeqSet hf p) (fun h => (hp_mem x h).choose) fun _ => (\u27e8f default x\u27e9 : Nonempty \u03b2).some\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\n[PROOFSTEP]\nhave hf_lim : \u2200 x, Tendsto (fun n => aeSeq hf p n x) L (\ud835\udcdd (f_lim x)) :=\n by\n intro x\n simp only [aeSeq]\n split_ifs with h\n \u00b7 refine' (hp_mem x h).choose_spec.congr fun n => _\n exact (aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h n).symm\n \u00b7 exact tendsto_const_nhds\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun n => aeSeq hf p n x) L (\ud835\udcdd (f_lim x))\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\n\u22a2 Tendsto (fun n => aeSeq hf p n x) L (\ud835\udcdd (f_lim x))\n[PROOFSTEP]\nsimp only [aeSeq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\n\u22a2 Tendsto\n (fun n =>\n if x \u2208 aeSeqSet hf fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l) then\n AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x\n else Nonempty.some (_ : Nonempty \u03b2))\n L\n (\ud835\udcdd\n (if h : x \u2208 aeSeqSet hf fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l) then\n Exists.choose (_ : p x fun n => f n x)\n else Nonempty.some (_ : Nonempty \u03b2)))\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\nh : x \u2208 aeSeqSet hf fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\n\u22a2 Tendsto (fun n => AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x) L (\ud835\udcdd (Exists.choose (_ : p x fun n => f n x)))\n[PROOFSTEP]\nrefine' (hp_mem x h).choose_spec.congr fun n => _\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\nh : x \u2208 aeSeqSet hf fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nn : \u03b9\n\u22a2 (fun n => f n x) n = AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x\n[PROOFSTEP]\nexact (aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h n).symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\nh : \u00acx \u2208 aeSeqSet hf fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\n\u22a2 Tendsto (fun n => Nonempty.some (_ : Nonempty \u03b2)) L (\ud835\udcdd (Nonempty.some (_ : Nonempty \u03b2)))\n[PROOFSTEP]\nexact tendsto_const_nhds\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nhf_lim : \u2200 (x : \u03b1), Tendsto (fun n => aeSeq hf p n x) L (\ud835\udcdd (f_lim x))\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\n[PROOFSTEP]\nhave h_ae_tendsto_f_lim : \u2200\u1d50 x \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x)) :=\n h_ae_eq.mono fun x hx => (hf_lim x).congr hx\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nhf_lim : \u2200 (x : \u03b1), Tendsto (fun n => aeSeq hf p n x) L (\ud835\udcdd (f_lim x))\nh_ae_tendsto_f_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\n[PROOFSTEP]\nhave h_f_lim_meas : Measurable f_lim :=\n measurable_of_tendsto_metrizable' L (aeSeq.measurable hf p) (tendsto_pi_nhds.mpr fun x => hf_lim x)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nhf_lim : \u2200 (x : \u03b1), Tendsto (fun n => aeSeq hf p n x) L (\ud835\udcdd (f_lim x))\nh_ae_tendsto_f_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\nh_f_lim_meas : Measurable f_lim\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\n[PROOFSTEP]\nexact \u27e8f_lim, h_f_lim_meas, h_ae_tendsto_f_lim\u27e9\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "llama_tokens": 23849, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217431943271999, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4195515495139005}} {"text": "[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np\u271d q p : MvPolynomial \u03c3 R\n\u22a2 degrees (-p) = degrees p\n[PROOFSTEP]\nrw [degrees, support_neg]\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np\u271d q p : MvPolynomial \u03c3 R\n\u22a2 (Finset.sup (support p) fun s => \u2191toMultiset s) = degrees p\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\np\u271d q\u271d : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\np q : MvPolynomial \u03c3 R\n\u22a2 degrees (p - q) \u2264 degrees p \u2294 degrees q\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using le_trans (degrees_add p (-q)) (by rw [degrees_neg])\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\np\u271d q\u271d : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\np q : MvPolynomial \u03c3 R\n\u22a2 degrees p \u2294 degrees (-q) \u2264 degrees p \u2294 degrees q\n[PROOFSTEP]\nrw [degrees_neg]\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q : MvPolynomial \u03c3 R\n\u22a2 vars (-p) = vars p\n[PROOFSTEP]\nsimp [vars, degrees_neg]\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\np q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\n\u22a2 vars (p - q) \u2286 vars p \u222a vars q\n[PROOFSTEP]\nconvert vars_add_subset p (-q) using 2\n[GOAL]\ncase h.e'_3.h.e'_4\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\np q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\n\u22a2 p - q = p + -q\n[PROOFSTEP]\nsimp [sub_eq_add_neg]\n[GOAL]\ncase h.e'_4.h.e'_4\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\np q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\n\u22a2 vars q = vars (-q)\n[PROOFSTEP]\nsimp [sub_eq_add_neg]\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\np q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\nhpq : Disjoint (vars p) (vars q)\n\u22a2 vars (p - q) = vars p \u222a vars q\n[PROOFSTEP]\nrw [\u2190 vars_neg q] at hpq \n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\np q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\nhpq : Disjoint (vars p) (vars (-q))\n\u22a2 vars (p - q) = vars p \u222a vars q\n[PROOFSTEP]\nconvert vars_add_of_disjoint hpq using 2\n[GOAL]\ncase h.e'_2.h.e'_4\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\np q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\nhpq : Disjoint (vars p) (vars (-q))\n\u22a2 p - q = p + -q\n[PROOFSTEP]\nsimp [sub_eq_add_neg]\n[GOAL]\ncase h.e'_3.h.e'_4\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\np q : MvPolynomial \u03c3 R\ninst\u271d : DecidableEq \u03c3\nhpq : Disjoint (vars p) (vars (-q))\n\u22a2 vars q = vars (-q)\n[PROOFSTEP]\nsimp [sub_eq_add_neg]\n[GOAL]\nR\u271d : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\u271d\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\u271d\np q : MvPolynomial \u03c3 R\u271d\ninst\u271d : CommRing S\nf\u271d : R\u271d \u2192+* S\ng : \u03c3 \u2192 S\nR : Type u\nc : \u2124 \u2192+* S\nf : MvPolynomial R \u2124 \u2192+* S\nx : MvPolynomial R \u2124\n\u22a2 eval\u2082 c (\u2191f \u2218 X) x = \u2191f x\n[PROOFSTEP]\napply\n MvPolynomial.induction_on x\n (fun n => by\n rw [hom_C f, eval\u2082_C]\n exact eq_intCast c n)\n (fun p q hp hq => by\n rw [eval\u2082_add, hp, hq]\n exact (f.map_add _ _).symm)\n (fun p n hp => by\n rw [eval\u2082_mul, eval\u2082_X, hp]\n exact (f.map_mul _ _).symm)\n[GOAL]\nR\u271d : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\u271d\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\u271d\np q : MvPolynomial \u03c3 R\u271d\ninst\u271d : CommRing S\nf\u271d : R\u271d \u2192+* S\ng : \u03c3 \u2192 S\nR : Type u\nc : \u2124 \u2192+* S\nf : MvPolynomial R \u2124 \u2192+* S\nx : MvPolynomial R \u2124\nn : \u2124\n\u22a2 eval\u2082 c (\u2191f \u2218 X) (\u2191C n) = \u2191f (\u2191C n)\n[PROOFSTEP]\nrw [hom_C f, eval\u2082_C]\n[GOAL]\nR\u271d : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\u271d\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\u271d\np q : MvPolynomial \u03c3 R\u271d\ninst\u271d : CommRing S\nf\u271d : R\u271d \u2192+* S\ng : \u03c3 \u2192 S\nR : Type u\nc : \u2124 \u2192+* S\nf : MvPolynomial R \u2124 \u2192+* S\nx : MvPolynomial R \u2124\nn : \u2124\n\u22a2 \u2191c n = \u2191n\n[PROOFSTEP]\nexact eq_intCast c n\n[GOAL]\nR\u271d : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\u271d\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\u271d\np\u271d q\u271d : MvPolynomial \u03c3 R\u271d\ninst\u271d : CommRing S\nf\u271d : R\u271d \u2192+* S\ng : \u03c3 \u2192 S\nR : Type u\nc : \u2124 \u2192+* S\nf : MvPolynomial R \u2124 \u2192+* S\nx p q : MvPolynomial R \u2124\nhp : eval\u2082 c (\u2191f \u2218 X) p = \u2191f p\nhq : eval\u2082 c (\u2191f \u2218 X) q = \u2191f q\n\u22a2 eval\u2082 c (\u2191f \u2218 X) (p + q) = \u2191f (p + q)\n[PROOFSTEP]\nrw [eval\u2082_add, hp, hq]\n[GOAL]\nR\u271d : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\u271d\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\u271d\np\u271d q\u271d : MvPolynomial \u03c3 R\u271d\ninst\u271d : CommRing S\nf\u271d : R\u271d \u2192+* S\ng : \u03c3 \u2192 S\nR : Type u\nc : \u2124 \u2192+* S\nf : MvPolynomial R \u2124 \u2192+* S\nx p q : MvPolynomial R \u2124\nhp : eval\u2082 c (\u2191f \u2218 X) p = \u2191f p\nhq : eval\u2082 c (\u2191f \u2218 X) q = \u2191f q\n\u22a2 \u2191f p + \u2191f q = \u2191f (p + q)\n[PROOFSTEP]\nexact (f.map_add _ _).symm\n[GOAL]\nR\u271d : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\u271d\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\u271d\np\u271d q : MvPolynomial \u03c3 R\u271d\ninst\u271d : CommRing S\nf\u271d : R\u271d \u2192+* S\ng : \u03c3 \u2192 S\nR : Type u\nc : \u2124 \u2192+* S\nf : MvPolynomial R \u2124 \u2192+* S\nx p : MvPolynomial R \u2124\nn : R\nhp : eval\u2082 c (\u2191f \u2218 X) p = \u2191f p\n\u22a2 eval\u2082 c (\u2191f \u2218 X) (p * X n) = \u2191f (p * X n)\n[PROOFSTEP]\nrw [eval\u2082_mul, eval\u2082_X, hp]\n[GOAL]\nR\u271d : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\u271d\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\u271d\np\u271d q : MvPolynomial \u03c3 R\u271d\ninst\u271d : CommRing S\nf\u271d : R\u271d \u2192+* S\ng : \u03c3 \u2192 S\nR : Type u\nc : \u2124 \u2192+* S\nf : MvPolynomial R \u2124 \u2192+* S\nx p : MvPolynomial R \u2124\nn : R\nhp : eval\u2082 c (\u2191f \u2218 X) p = \u2191f p\n\u22a2 \u2191f p * (\u2191f \u2218 X) n = \u2191f (p * X n)\n[PROOFSTEP]\nexact (f.map_mul _ _).symm\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommRing R\np q : MvPolynomial \u03c3 R\ninst\u271d : CommRing S\nf\u271d : R \u2192+* S\ng f : \u03c3 \u2192 S\nx : \u03c3\n\u22a2 (fun f => \u2191f \u2218 X) ((fun f => eval\u2082Hom (Int.castRingHom S) f) f) x = f x\n[PROOFSTEP]\nsimp only [coe_eval\u2082Hom, Function.comp_apply, eval\u2082_X]\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\n\u22a2 degreeOf x (f - g) < k\n[PROOFSTEP]\nclassical\nrw [degreeOf_lt_iff h]\nintro m hm\nby_contra' hc\nhave h := support_sub \u03c3 f g hm\nsimp only [mem_support_iff, Ne.def, coeff_sub, sub_eq_zero] at hm \ncases' Finset.mem_union.1 h with cf cg\n\u00b7 exact hm (hf m cf hc)\n\u00b7 exact hm (hg m cg hc)\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\n\u22a2 degreeOf x (f - g) < k\n[PROOFSTEP]\nrw [degreeOf_lt_iff h]\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\n\u22a2 \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support (f - g) \u2192 \u2191m x < k\n[PROOFSTEP]\nintro m hm\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m\u271d : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhm : m \u2208 support (f - g)\n\u22a2 \u2191m x < k\n[PROOFSTEP]\nby_contra' hc\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m\u271d : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhm : m \u2208 support (f - g)\nhc : k \u2264 \u2191m x\n\u22a2 False\n[PROOFSTEP]\nhave h := support_sub \u03c3 f g hm\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m\u271d : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh\u271d : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhm : m \u2208 support (f - g)\nhc : k \u2264 \u2191m x\nh : m \u2208 support f \u222a support g\n\u22a2 False\n[PROOFSTEP]\nsimp only [mem_support_iff, Ne.def, coeff_sub, sub_eq_zero] at hm \n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m\u271d : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh\u271d : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhc : k \u2264 \u2191m x\nh : m \u2208 support f \u222a support g\nhm : \u00accoeff m f = coeff m g\n\u22a2 False\n[PROOFSTEP]\ncases' Finset.mem_union.1 h with cf cg\n[GOAL]\ncase inl\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m\u271d : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh\u271d : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhc : k \u2264 \u2191m x\nh : m \u2208 support f \u222a support g\nhm : \u00accoeff m f = coeff m g\ncf : m \u2208 support f\n\u22a2 False\n[PROOFSTEP]\nexact hm (hf m cf hc)\n[GOAL]\ncase inr\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m\u271d : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q : MvPolynomial \u03c3 R\nx : \u03c3\nf g : MvPolynomial \u03c3 R\nk : \u2115\nh\u271d : 0 < k\nhf : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support f \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nhg : \u2200 (m : \u03c3 \u2192\u2080 \u2115), m \u2208 support g \u2192 k \u2264 \u2191m x \u2192 coeff m f = coeff m g\nm : \u03c3 \u2192\u2080 \u2115\nhc : k \u2264 \u2191m x\nh : m \u2208 support f \u222a support g\nhm : \u00accoeff m f = coeff m g\ncg : m \u2208 support g\n\u22a2 False\n[PROOFSTEP]\nexact hm (hg m cg hc)\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na\u271d a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q a : MvPolynomial \u03c3 R\n\u22a2 totalDegree (-a) = totalDegree a\n[PROOFSTEP]\nsimp only [totalDegree, support_neg]\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na\u271d a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q a b : MvPolynomial \u03c3 R\n\u22a2 totalDegree (a - b) = totalDegree (a + -b)\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\nR : Type u\nS : Type v\n\u03c3 : Type u_1\na\u271d a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommRing R\np q a b : MvPolynomial \u03c3 R\n\u22a2 max (totalDegree a) (totalDegree (-b)) = max (totalDegree a) (totalDegree b)\n[PROOFSTEP]\nrw [totalDegree_neg]\n", "meta": {"mathlib_filename": "Mathlib.Data.MvPolynomial.CommRing", "llama_tokens": 6023, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389930307512, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.41946180827648055}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nP : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 \u0399 a b \u2192 P x) \u2194 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 Ioc a b \u2192 P x) \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 Ioc b a \u2192 P x\n[PROOFSTEP]\nsimp only [uIoc_eq_union, mem_union, or_imp, eventually_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t s : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (s \u222a t) + \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bc s + \u2191\u2191\u03bc t\n[PROOFSTEP]\nrw [\u2190 measure_inter_add_diff (s \u222a t) ht, Set.union_inter_cancel_right, union_diff_right, \u2190 measure_inter_add_diff s ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t s : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc t + \u2191\u2191\u03bc (s \\ t) + \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) + \u2191\u2191\u03bc t\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191\u2191\u03bc (s \u222a t) + \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bc s + \u2191\u2191\u03bc t\n[PROOFSTEP]\nrw [union_comm, inter_comm, measure_union_add_inter t hs, add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (s \u2206 t) = \u2191\u2191\u03bc (s \\ t) + \u2191\u2191\u03bc (t \\ s)\n[PROOFSTEP]\nsimpa only [symmDiff_def, sup_eq_union] using measure_union disjoint_sdiff_sdiff (ht.diff hs)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhs : Set.Countable s\nhd : Set.Pairwise s (AEDisjoint \u03bc on f)\nh : \u2200 (b : \u03b2), b \u2208 s \u2192 NullMeasurableSet (f b)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : \u03b2) (_ : b \u2208 s), f b) = \u2211' (p : \u2191s), \u2191\u2191\u03bc (f \u2191p)\n[PROOFSTEP]\nhaveI := hs.toEncodable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhs : Set.Countable s\nhd : Set.Pairwise s (AEDisjoint \u03bc on f)\nh : \u2200 (b : \u03b2), b \u2208 s \u2192 NullMeasurableSet (f b)\nthis : Encodable \u2191s\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : \u03b2) (_ : b \u2208 s), f b) = \u2211' (p : \u2191s), \u2191\u2191\u03bc (f \u2191p)\n[PROOFSTEP]\nrw [biUnion_eq_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhs : Set.Countable s\nhd : Set.Pairwise s (AEDisjoint \u03bc on f)\nh : \u2200 (b : \u03b2), b \u2208 s \u2192 NullMeasurableSet (f b)\nthis : Encodable \u2191s\n\u22a2 \u2191\u2191\u03bc (\u22c3 (x : \u2191s), f \u2191x) = \u2211' (p : \u2191s), \u2191\u2191\u03bc (f \u2191p)\n[PROOFSTEP]\nexact measure_iUnion\u2080 (hd.on_injective Subtype.coe_injective fun x => x.2) fun x => h x x.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nS : Set (Set \u03b1)\nhs : Set.Countable S\nhd : Set.Pairwise S (AEDisjoint \u03bc)\nh : \u2200 (s : Set \u03b1), s \u2208 S \u2192 NullMeasurableSet s\n\u22a2 \u2191\u2191\u03bc (\u22c3\u2080 S) = \u2211' (s : \u2191S), \u2191\u2191\u03bc \u2191s\n[PROOFSTEP]\nrw [sUnion_eq_biUnion, measure_biUnion\u2080 hs hd h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nS : Set (Set \u03b1)\nhs : Set.Countable S\nhd : Set.Pairwise S Disjoint\nh : \u2200 (s : Set \u03b1), s \u2208 S \u2192 MeasurableSet s\n\u22a2 \u2191\u2191\u03bc (\u22c3\u2080 S) = \u2211' (s : \u2191S), \u2191\u2191\u03bc \u2191s\n[PROOFSTEP]\nrw [sUnion_eq_biUnion, measure_biUnion hs hd h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03b1\nhd : Set.Pairwise (\u2191s) (AEDisjoint \u03bc on f)\nhm : \u2200 (b : \u03b9), b \u2208 s \u2192 NullMeasurableSet (f b)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : \u03b9) (_ : b \u2208 s), f b) = \u2211 p in s, \u2191\u2191\u03bc (f p)\n[PROOFSTEP]\nrw [\u2190 Finset.sum_attach, Finset.attach_eq_univ, \u2190 tsum_fintype]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 Set \u03b1\nhd : Set.Pairwise (\u2191s) (AEDisjoint \u03bc on f)\nhm : \u2200 (b : \u03b9), b \u2208 s \u2192 NullMeasurableSet (f b)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : \u03b9) (_ : b \u2208 s), f b) = \u2211' (b : { x // x \u2208 s }), \u2191\u2191\u03bc (f \u2191b)\n[PROOFSTEP]\nexact measure_biUnion\u2080 s.countable_toSet hd hm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\n\u22a2 \u2211' (i : \u03b9), \u2191\u2191\u03bc (As i) \u2264 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i)\n[PROOFSTEP]\nrcases show Summable fun i => \u03bc (As i) from ENNReal.summable with \u27e8S, hS\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nS : \u211d\u22650\u221e\nhS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S\n\u22a2 \u2211' (i : \u03b9), \u2191\u2191\u03bc (As i) \u2264 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i)\n[PROOFSTEP]\nrw [hS.tsum_eq]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nS : \u211d\u22650\u221e\nhS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S\n\u22a2 S \u2264 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i)\n[PROOFSTEP]\nrefine' tendsto_le_of_eventuallyLE hS tendsto_const_nhds (eventually_of_forall _)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nS : \u211d\u22650\u221e\nhS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S\n\u22a2 \u2200 (x : Finset \u03b9), (fun s => \u2211 b in s, (fun i => \u2191\u2191\u03bc (As i)) b) x \u2264 (fun x => \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i)) x\n[PROOFSTEP]\nintro s\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nS : \u211d\u22650\u221e\nhS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S\ns : Finset \u03b9\n\u22a2 (fun s => \u2211 b in s, (fun i => \u2191\u2191\u03bc (As i)) b) s \u2264 (fun x => \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i)) s\n[PROOFSTEP]\nsimp [\u2190 measure_biUnion_finset (fun _i _hi _j _hj hij => As_disj hij) fun i _ => As_mble i]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nS : \u211d\u22650\u221e\nhS : HasSum (fun i => \u2191\u2191\u03bc (As i)) S\ns : Finset \u03b9\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : \u03b9) (_ : b \u2208 s), As b) \u2264 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i)\n[PROOFSTEP]\nexact measure_mono (iUnion\u2082_subset_iUnion (fun i : \u03b9 => i \u2208 s) fun i : \u03b9 => As i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Set \u03b2\nhs : Set.Countable s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (y : \u03b2), y \u2208 s \u2192 MeasurableSet (f \u207b\u00b9' {y})\n\u22a2 \u2211' (b : \u2191s), \u2191\u2191\u03bc (f \u207b\u00b9' {\u2191b}) = \u2191\u2191\u03bc (f \u207b\u00b9' s)\n[PROOFSTEP]\nrw [\u2190 Set.biUnion_preimage_singleton, measure_biUnion hs (pairwiseDisjoint_fiber f s) hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (y : \u03b2), y \u2208 s \u2192 MeasurableSet (f \u207b\u00b9' {y})\n\u22a2 \u2211 b in s, \u2191\u2191\u03bc (f \u207b\u00b9' {b}) = \u2191\u2191\u03bc (f \u207b\u00b9' \u2191s)\n[PROOFSTEP]\nsimp only [\u2190 measure_biUnion_finset (pairwiseDisjoint_fiber f s) hf, Finset.set_biUnion_preimage_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191\u2191\u03bc s + \u2191\u2191\u03bc (t \\ s) = \u2191\u2191\u03bc (s \u222a t)\n[PROOFSTEP]\nrw [\u2190 measure_union' (@disjoint_sdiff_right _ s t) hs, union_diff_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t s : Set \u03b1\nhm : MeasurableSet t\nh_fin : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (s \\ t) + \u2191\u2191\u03bc t = \u2191\u2191\u03bc (s \u222a t)\n[PROOFSTEP]\nrw [add_comm, measure_add_diff hm, union_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : s\u2082 \u2286 s\u2081\nh\u2082 : MeasurableSet s\u2082\nh_fin : \u2191\u2191\u03bc s\u2082 \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (s\u2081 \\ s\u2082) = \u2191\u2191\u03bc s\u2081 - \u2191\u2191\u03bc s\u2082\n[PROOFSTEP]\nrw [measure_diff' _ h\u2082 h_fin, union_eq_self_of_subset_right h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nhs : MeasurableSet s\nhst : s \u2286 t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh : \u2191\u2191\u03bc t < \u2191\u2191\u03bc s + \u03b5\n\u22a2 \u2191\u2191\u03bc (t \\ s) < \u03b5\n[PROOFSTEP]\nrw [measure_diff hst hs hs']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nhs : MeasurableSet s\nhst : s \u2286 t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh : \u2191\u2191\u03bc t < \u2191\u2191\u03bc s + \u03b5\n\u22a2 \u2191\u2191\u03bc t - \u2191\u2191\u03bc s < \u03b5\n[PROOFSTEP]\nrw [add_comm] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nhs : MeasurableSet s\nhst : s \u2286 t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh : \u2191\u2191\u03bc t < \u03b5 + \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc t - \u2191\u2191\u03bc s < \u03b5\n[PROOFSTEP]\nexact ENNReal.sub_lt_of_lt_add (measure_mono hst) h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nhs : MeasurableSet s\nhst : s \u2286 t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u22a2 \u2191\u2191\u03bc (t \\ s) \u2264 \u03b5 \u2194 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nrw [measure_diff hst hs hs', tsub_le_iff_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081\u271d s\u2082\u271d t s\u2081 s\u2082 s\u2083 : Set \u03b1\nh12 : s\u2081 \u2286 s\u2082\nh23 : s\u2082 \u2286 s\u2083\nh_nulldiff : \u2191\u2191\u03bc (s\u2083 \\ s\u2081) = 0\n\u22a2 \u2191\u2191\u03bc s\u2081 = \u2191\u2191\u03bc s\u2082 \u2227 \u2191\u2191\u03bc s\u2082 = \u2191\u2191\u03bc s\u2083\n[PROOFSTEP]\nhave le12 : \u03bc s\u2081 \u2264 \u03bc s\u2082 := measure_mono h12\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081\u271d s\u2082\u271d t s\u2081 s\u2082 s\u2083 : Set \u03b1\nh12 : s\u2081 \u2286 s\u2082\nh23 : s\u2082 \u2286 s\u2083\nh_nulldiff : \u2191\u2191\u03bc (s\u2083 \\ s\u2081) = 0\nle12 : \u2191\u2191\u03bc s\u2081 \u2264 \u2191\u2191\u03bc s\u2082\n\u22a2 \u2191\u2191\u03bc s\u2081 = \u2191\u2191\u03bc s\u2082 \u2227 \u2191\u2191\u03bc s\u2082 = \u2191\u2191\u03bc s\u2083\n[PROOFSTEP]\nhave le23 : \u03bc s\u2082 \u2264 \u03bc s\u2083 := measure_mono h23\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081\u271d s\u2082\u271d t s\u2081 s\u2082 s\u2083 : Set \u03b1\nh12 : s\u2081 \u2286 s\u2082\nh23 : s\u2082 \u2286 s\u2083\nh_nulldiff : \u2191\u2191\u03bc (s\u2083 \\ s\u2081) = 0\nle12 : \u2191\u2191\u03bc s\u2081 \u2264 \u2191\u2191\u03bc s\u2082\nle23 : \u2191\u2191\u03bc s\u2082 \u2264 \u2191\u2191\u03bc s\u2083\n\u22a2 \u2191\u2191\u03bc s\u2081 = \u2191\u2191\u03bc s\u2082 \u2227 \u2191\u2191\u03bc s\u2082 = \u2191\u2191\u03bc s\u2083\n[PROOFSTEP]\nhave key : \u03bc s\u2083 \u2264 \u03bc s\u2081 :=\n calc\n \u03bc s\u2083 = \u03bc (s\u2083 \\ s\u2081 \u222a s\u2081) := by rw [diff_union_of_subset (h12.trans h23)]\n _ \u2264 \u03bc (s\u2083 \\ s\u2081) + \u03bc s\u2081 := (measure_union_le _ _)\n _ = \u03bc s\u2081 := by simp only [h_nulldiff, zero_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081\u271d s\u2082\u271d t s\u2081 s\u2082 s\u2083 : Set \u03b1\nh12 : s\u2081 \u2286 s\u2082\nh23 : s\u2082 \u2286 s\u2083\nh_nulldiff : \u2191\u2191\u03bc (s\u2083 \\ s\u2081) = 0\nle12 : \u2191\u2191\u03bc s\u2081 \u2264 \u2191\u2191\u03bc s\u2082\nle23 : \u2191\u2191\u03bc s\u2082 \u2264 \u2191\u2191\u03bc s\u2083\n\u22a2 \u2191\u2191\u03bc s\u2083 = \u2191\u2191\u03bc (s\u2083 \\ s\u2081 \u222a s\u2081)\n[PROOFSTEP]\nrw [diff_union_of_subset (h12.trans h23)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081\u271d s\u2082\u271d t s\u2081 s\u2082 s\u2083 : Set \u03b1\nh12 : s\u2081 \u2286 s\u2082\nh23 : s\u2082 \u2286 s\u2083\nh_nulldiff : \u2191\u2191\u03bc (s\u2083 \\ s\u2081) = 0\nle12 : \u2191\u2191\u03bc s\u2081 \u2264 \u2191\u2191\u03bc s\u2082\nle23 : \u2191\u2191\u03bc s\u2082 \u2264 \u2191\u2191\u03bc s\u2083\n\u22a2 \u2191\u2191\u03bc (s\u2083 \\ s\u2081) + \u2191\u2191\u03bc s\u2081 = \u2191\u2191\u03bc s\u2081\n[PROOFSTEP]\nsimp only [h_nulldiff, zero_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081\u271d s\u2082\u271d t s\u2081 s\u2082 s\u2083 : Set \u03b1\nh12 : s\u2081 \u2286 s\u2082\nh23 : s\u2082 \u2286 s\u2083\nh_nulldiff : \u2191\u2191\u03bc (s\u2083 \\ s\u2081) = 0\nle12 : \u2191\u2191\u03bc s\u2081 \u2264 \u2191\u2191\u03bc s\u2082\nle23 : \u2191\u2191\u03bc s\u2082 \u2264 \u2191\u2191\u03bc s\u2083\nkey : \u2191\u2191\u03bc s\u2083 \u2264 \u2191\u2191\u03bc s\u2081\n\u22a2 \u2191\u2191\u03bc s\u2081 = \u2191\u2191\u03bc s\u2082 \u2227 \u2191\u2191\u03bc s\u2082 = \u2191\u2191\u03bc s\u2083\n[PROOFSTEP]\nexact \u27e8le12.antisymm (le23.trans key), le23.antisymm (key.trans le12)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh\u2081 : MeasurableSet s\nh_fin : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc s\u1d9c = \u2191\u2191\u03bc univ - \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [compl_eq_univ_diff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh\u2081 : MeasurableSet s\nh_fin : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (univ \\ s) = \u2191\u2191\u03bc univ - \u2191\u2191\u03bc s\n[PROOFSTEP]\nexact measure_diff (subset_univ s) h\u2081 h_fin\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u22a2 s \u222a t =\u1d50[\u03bc] s \u2194 t \u2264\u1d50[\u03bc] s\n[PROOFSTEP]\nrw [ae_le_set]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u22a2 s \u222a t =\u1d50[\u03bc] s \u2194 \u2191\u2191\u03bc (t \\ s) = 0\n[PROOFSTEP]\nrefine'\n \u27e8fun h => by simpa only [union_diff_left] using (ae_eq_set.mp h).1, fun h =>\n eventuallyLE_antisymm_iff.mpr\n \u27e8by rwa [ae_le_set, union_diff_left], HasSubset.Subset.eventuallyLE <| subset_union_left s t\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : s \u222a t =\u1d50[\u03bc] s\n\u22a2 \u2191\u2191\u03bc (t \\ s) = 0\n[PROOFSTEP]\nsimpa only [union_diff_left] using (ae_eq_set.mp h).1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : \u2191\u2191\u03bc (t \\ s) = 0\n\u22a2 s \u222a t \u2264\u1d50[\u03bc] s\n[PROOFSTEP]\nrwa [ae_le_set, union_diff_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u22a2 s \u222a t =\u1d50[\u03bc] t \u2194 s \u2264\u1d50[\u03bc] t\n[PROOFSTEP]\nrw [union_comm, union_ae_eq_left_iff_ae_subset]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh\u2081 : s \u2264\u1d50[\u03bc] t\nh\u2082 : \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc s\nhsm : MeasurableSet s\nht : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 s =\u1d50[\u03bc] t\n[PROOFSTEP]\nrefine' eventuallyLE_antisymm_iff.mpr \u27e8h\u2081, ae_le_set.mpr _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh\u2081 : s \u2264\u1d50[\u03bc] t\nh\u2082 : \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc s\nhsm : MeasurableSet s\nht : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (t \\ s) = 0\n[PROOFSTEP]\nreplace h\u2082 : \u03bc t = \u03bc s\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh\u2081 : s \u2264\u1d50[\u03bc] t\nh\u2082 : \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc s\nhsm : MeasurableSet s\nht : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc t = \u2191\u2191\u03bc s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh\u2081 : s \u2264\u1d50[\u03bc] t\nhsm : MeasurableSet s\nht : \u2191\u2191\u03bc t \u2260 \u22a4\nh\u2082 : \u2191\u2191\u03bc t = \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (t \\ s) = 0\n[PROOFSTEP]\nexact h\u2082.antisymm (measure_mono_ae h\u2081)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh\u2081 : s \u2264\u1d50[\u03bc] t\nhsm : MeasurableSet s\nht : \u2191\u2191\u03bc t \u2260 \u22a4\nh\u2082 : \u2191\u2191\u03bc t = \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (t \\ s) = 0\n[PROOFSTEP]\nreplace ht : \u03bc s \u2260 \u221e\n[GOAL]\ncase ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh\u2081 : s \u2264\u1d50[\u03bc] t\nhsm : MeasurableSet s\nht : \u2191\u2191\u03bc t \u2260 \u22a4\nh\u2082 : \u2191\u2191\u03bc t = \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh\u2081 : s \u2264\u1d50[\u03bc] t\nhsm : MeasurableSet s\nh\u2082 : \u2191\u2191\u03bc t = \u2191\u2191\u03bc s\nht : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (t \\ s) = 0\n[PROOFSTEP]\nexact h\u2082 \u25b8 ht\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh\u2081 : s \u2264\u1d50[\u03bc] t\nhsm : MeasurableSet s\nh\u2082 : \u2191\u2191\u03bc t = \u2191\u2191\u03bc s\nht : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (t \\ s) = 0\n[PROOFSTEP]\nrw [measure_diff' t hsm ht, measure_congr (union_ae_eq_left_iff_ae_subset.mpr h\u2081), h\u2082, tsub_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b2\ns t : \u03b2 \u2192 Set \u03b1\nhsub : \u2200 (b : \u03b2), s b \u2286 t b\nh_le : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2264 \u2191\u2191\u03bc (s b)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : \u03b2), s b) = \u2191\u2191\u03bc (\u22c3 (b : \u03b2), t b)\n[PROOFSTEP]\nrcases Classical.em (\u2203 b, \u03bc (t b) = \u221e) with (\u27e8b, hb\u27e9 | htop)\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b2\ns t : \u03b2 \u2192 Set \u03b1\nhsub : \u2200 (b : \u03b2), s b \u2286 t b\nh_le : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2264 \u2191\u2191\u03bc (s b)\nb : \u03b2\nhb : \u2191\u2191\u03bc (t b) = \u22a4\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : \u03b2), s b) = \u2191\u2191\u03bc (\u22c3 (b : \u03b2), t b)\n[PROOFSTEP]\ncalc\n \u03bc (\u22c3 b, s b) = \u221e := top_unique (hb \u25b8 (h_le b).trans <| measure_mono <| subset_iUnion _ _)\n _ = \u03bc (\u22c3 b, t b) := Eq.symm <| top_unique <| hb \u25b8 measure_mono (subset_iUnion _ _)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b2\ns t : \u03b2 \u2192 Set \u03b1\nhsub : \u2200 (b : \u03b2), s b \u2286 t b\nh_le : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2264 \u2191\u2191\u03bc (s b)\nhtop : \u00ac\u2203 b, \u2191\u2191\u03bc (t b) = \u22a4\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : \u03b2), s b) = \u2191\u2191\u03bc (\u22c3 (b : \u03b2), t b)\n[PROOFSTEP]\npush_neg at htop \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b2\ns t : \u03b2 \u2192 Set \u03b1\nhsub : \u2200 (b : \u03b2), s b \u2286 t b\nh_le : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2264 \u2191\u2191\u03bc (s b)\nhtop : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : \u03b2), s b) = \u2191\u2191\u03bc (\u22c3 (b : \u03b2), t b)\n[PROOFSTEP]\nrefine' le_antisymm (measure_mono (iUnion_mono hsub)) _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b2\ns t : \u03b2 \u2192 Set \u03b1\nhsub : \u2200 (b : \u03b2), s b \u2286 t b\nh_le : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2264 \u2191\u2191\u03bc (s b)\nhtop : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : \u03b2), t b) \u2264 \u2191\u2191\u03bc (\u22c3 (b : \u03b2), s b)\n[PROOFSTEP]\nset M := toMeasurable \u03bc\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b2\ns t : \u03b2 \u2192 Set \u03b1\nhsub : \u2200 (b : \u03b2), s b \u2286 t b\nh_le : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2264 \u2191\u2191\u03bc (s b)\nhtop : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2260 \u22a4\nM : Set \u03b1 \u2192 Set \u03b1 := toMeasurable \u03bc\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : \u03b2), t b) \u2264 \u2191\u2191\u03bc (\u22c3 (b : \u03b2), s b)\n[PROOFSTEP]\nhave H : \u2200 b, (M (t b) \u2229 M (\u22c3 b, s b) : Set \u03b1) =\u1d50[\u03bc] M (t b) :=\n by\n refine' fun b => ae_eq_of_subset_of_measure_ge (inter_subset_left _ _) _ _ _\n \u00b7\n calc\n \u03bc (M (t b)) = \u03bc (t b) := measure_toMeasurable _\n _ \u2264 \u03bc (s b) := (h_le b)\n _ \u2264 \u03bc (M (t b) \u2229 M (\u22c3 b, s b)) :=\n measure_mono <|\n subset_inter ((hsub b).trans <| subset_toMeasurable _ _)\n ((subset_iUnion _ _).trans <| subset_toMeasurable _ _)\n \u00b7 exact (measurableSet_toMeasurable _ _).inter (measurableSet_toMeasurable _ _)\n \u00b7 rw [measure_toMeasurable]\n exact htop b\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b2\ns t : \u03b2 \u2192 Set \u03b1\nhsub : \u2200 (b : \u03b2), s b \u2286 t b\nh_le : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2264 \u2191\u2191\u03bc (s b)\nhtop : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2260 \u22a4\nM : Set \u03b1 \u2192 Set \u03b1 := toMeasurable \u03bc\n\u22a2 \u2200 (b : \u03b2), M (t b) \u2229 M (\u22c3 (b : \u03b2), s b) =\u1d50[\u03bc] M (t b)\n[PROOFSTEP]\nrefine' fun b => ae_eq_of_subset_of_measure_ge (inter_subset_left _ _) _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b2\ns t : \u03b2 \u2192 Set \u03b1\nhsub : \u2200 (b : \u03b2), s b \u2286 t b\nh_le : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2264 \u2191\u2191\u03bc (s b)\nhtop : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2260 \u22a4\nM : Set \u03b1 \u2192 Set \u03b1 := toMeasurable \u03bc\nb : \u03b2\n\u22a2 \u2191\u2191\u03bc (M (t b)) \u2264 \u2191\u2191\u03bc (M (t b) \u2229 M (\u22c3 (b : \u03b2), s b))\n[PROOFSTEP]\ncalc\n \u03bc (M (t b)) = \u03bc (t b) := measure_toMeasurable _\n _ \u2264 \u03bc (s b) := (h_le b)\n _ \u2264 \u03bc (M (t b) \u2229 M (\u22c3 b, s b)) :=\n measure_mono <|\n subset_inter ((hsub b).trans <| subset_toMeasurable _ _) ((subset_iUnion _ _).trans <| subset_toMeasurable _ _)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b2\ns t : \u03b2 \u2192 Set \u03b1\nhsub : \u2200 (b : \u03b2), s b \u2286 t b\nh_le : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2264 \u2191\u2191\u03bc (s b)\nhtop : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2260 \u22a4\nM : Set \u03b1 \u2192 Set \u03b1 := toMeasurable \u03bc\nb : \u03b2\n\u22a2 MeasurableSet (M (t b) \u2229 M (\u22c3 (b : \u03b2), s b))\n[PROOFSTEP]\nexact (measurableSet_toMeasurable _ _).inter (measurableSet_toMeasurable _ _)\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b2\ns t : \u03b2 \u2192 Set \u03b1\nhsub : \u2200 (b : \u03b2), s b \u2286 t b\nh_le : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2264 \u2191\u2191\u03bc (s b)\nhtop : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2260 \u22a4\nM : Set \u03b1 \u2192 Set \u03b1 := toMeasurable \u03bc\nb : \u03b2\n\u22a2 \u2191\u2191\u03bc (M (t b)) \u2260 \u22a4\n[PROOFSTEP]\nrw [measure_toMeasurable]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b2\ns t : \u03b2 \u2192 Set \u03b1\nhsub : \u2200 (b : \u03b2), s b \u2286 t b\nh_le : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2264 \u2191\u2191\u03bc (s b)\nhtop : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2260 \u22a4\nM : Set \u03b1 \u2192 Set \u03b1 := toMeasurable \u03bc\nb : \u03b2\n\u22a2 \u2191\u2191\u03bc (t b) \u2260 \u22a4\n[PROOFSTEP]\nexact htop b\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b2\ns t : \u03b2 \u2192 Set \u03b1\nhsub : \u2200 (b : \u03b2), s b \u2286 t b\nh_le : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2264 \u2191\u2191\u03bc (s b)\nhtop : \u2200 (b : \u03b2), \u2191\u2191\u03bc (t b) \u2260 \u22a4\nM : Set \u03b1 \u2192 Set \u03b1 := toMeasurable \u03bc\nH : \u2200 (b : \u03b2), M (t b) \u2229 M (\u22c3 (b : \u03b2), s b) =\u1d50[\u03bc] M (t b)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : \u03b2), t b) \u2264 \u2191\u2191\u03bc (\u22c3 (b : \u03b2), s b)\n[PROOFSTEP]\ncalc\n \u03bc (\u22c3 b, t b) \u2264 \u03bc (\u22c3 b, M (t b)) := measure_mono (iUnion_mono fun b => subset_toMeasurable _ _)\n _ = \u03bc (\u22c3 b, M (t b) \u2229 M (\u22c3 b, s b)) := (measure_congr (EventuallyEq.countable_iUnion H).symm)\n _ \u2264 \u03bc (M (\u22c3 b, s b)) := (measure_mono (iUnion_subset fun b => inter_subset_right _ _))\n _ = \u03bc (\u22c3 b, s b) := measure_toMeasurable _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Set \u03b1\nhs : s\u2081 \u2286 s\u2082\nhs\u03bc : \u2191\u2191\u03bc s\u2082 \u2264 \u2191\u2191\u03bc s\u2081\nht : t\u2081 \u2286 t\u2082\nht\u03bc : \u2191\u2191\u03bc t\u2082 \u2264 \u2191\u2191\u03bc t\u2081\n\u22a2 \u2191\u2191\u03bc (s\u2081 \u222a t\u2081) = \u2191\u2191\u03bc (s\u2082 \u222a t\u2082)\n[PROOFSTEP]\nrw [union_eq_iUnion, union_eq_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Set \u03b1\nhs : s\u2081 \u2286 s\u2082\nhs\u03bc : \u2191\u2191\u03bc s\u2082 \u2264 \u2191\u2191\u03bc s\u2081\nht : t\u2081 \u2286 t\u2082\nht\u03bc : \u2191\u2191\u03bc t\u2082 \u2264 \u2191\u2191\u03bc t\u2081\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : Bool), bif b then s\u2081 else t\u2081) = \u2191\u2191\u03bc (\u22c3 (b : Bool), bif b then s\u2082 else t\u2082)\n[PROOFSTEP]\nexact measure_iUnion_congr_of_subset (Bool.forall_bool.2 \u27e8ht, hs\u27e9) (Bool.forall_bool.2 \u27e8ht\u03bc, hs\u03bc\u27e9)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nI : Set \u03b2\nhc : Set.Countable I\ns : \u03b2 \u2192 Set \u03b1\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : \u03b2) (_ : b \u2208 I), toMeasurable \u03bc (s b)) = \u2191\u2191\u03bc (\u22c3 (b : \u03b2) (_ : b \u2208 I), s b)\n[PROOFSTEP]\nhaveI := hc.toEncodable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nI : Set \u03b2\nhc : Set.Countable I\ns : \u03b2 \u2192 Set \u03b1\nthis : Encodable \u2191I\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : \u03b2) (_ : b \u2208 I), toMeasurable \u03bc (s b)) = \u2191\u2191\u03bc (\u22c3 (b : \u03b2) (_ : b \u2208 I), s b)\n[PROOFSTEP]\nsimp only [biUnion_eq_iUnion, measure_iUnion_toMeasurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 MeasurableSet (t i)\nH : PairwiseDisjoint (\u2191s) t\n\u22a2 \u2211 i in s, \u2191\u2191\u03bc (t i) \u2264 \u2191\u2191\u03bc univ\n[PROOFSTEP]\nrw [\u2190 measure_biUnion_finset H h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 MeasurableSet (t i)\nH : PairwiseDisjoint (\u2191s) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (b : \u03b9) (_ : b \u2208 s), t b) \u2264 \u2191\u2191\u03bc univ\n[PROOFSTEP]\nexact measure_mono (subset_univ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : Pairwise (Disjoint on s)\n\u22a2 \u2211' (i : \u03b9), \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc univ\n[PROOFSTEP]\nrw [ENNReal.tsum_eq_iSup_sum]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : Pairwise (Disjoint on s)\n\u22a2 \u2a06 (s_1 : Finset \u03b9), \u2211 a in s_1, \u2191\u2191\u03bc (s a) \u2264 \u2191\u2191\u03bc univ\n[PROOFSTEP]\nexact iSup_le fun s => sum_measure_le_measure_univ (fun i _hi => hs i) fun i _hi j _hj hij => H hij\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : \u2191\u2191\u03bc univ < \u2211' (i : \u03b9), \u2191\u2191\u03bc (s i)\n\u22a2 \u2203 i j _h, Set.Nonempty (s i \u2229 s j)\n[PROOFSTEP]\ncontrapose! H\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j)\n\u22a2 \u2211' (i : \u03b9), \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc univ\n[PROOFSTEP]\napply tsum_measure_le_measure_univ hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j)\n\u22a2 Pairwise (Disjoint on fun i => s i)\n[PROOFSTEP]\nintro i j hij\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j)\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => s i) i j\n[PROOFSTEP]\nrw [Function.onFun, disjoint_iff_inf_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nH : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j)\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 s i \u2293 s j \u2264 \u22a5\n[PROOFSTEP]\nexact fun x hx => H i j hij \u27e8x, hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 MeasurableSet (t i)\nH : \u2191\u2191\u03bc univ < \u2211 i in s, \u2191\u2191\u03bc (t i)\n\u22a2 \u2203 i, i \u2208 s \u2227 \u2203 j, j \u2208 s \u2227 \u2203 _h, Set.Nonempty (t i \u2229 t j)\n[PROOFSTEP]\ncontrapose! H\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 MeasurableSet (t i)\nH : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (j : \u03b9), j \u2208 s \u2192 i \u2260 j \u2192 \u00acSet.Nonempty (t i \u2229 t j)\n\u22a2 \u2211 i in s, \u2191\u2191\u03bc (t i) \u2264 \u2191\u2191\u03bc univ\n[PROOFSTEP]\napply sum_measure_le_measure_univ h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 MeasurableSet (t i)\nH : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (j : \u03b9), j \u2208 s \u2192 i \u2260 j \u2192 \u00acSet.Nonempty (t i \u2229 t j)\n\u22a2 PairwiseDisjoint \u2191s fun i => t i\n[PROOFSTEP]\nintro i hi j hj hij\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 MeasurableSet (t i)\nH : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (j : \u03b9), j \u2208 s \u2192 i \u2260 j \u2192 \u00acSet.Nonempty (t i \u2229 t j)\ni : \u03b9\nhi : i \u2208 \u2191s\nj : \u03b9\nhj : j \u2208 \u2191s\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => t i) i j\n[PROOFSTEP]\nrw [Function.onFun, disjoint_iff_inf_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 MeasurableSet (t i)\nH : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (j : \u03b9), j \u2208 s \u2192 i \u2260 j \u2192 \u00acSet.Nonempty (t i \u2229 t j)\ni : \u03b9\nhi : i \u2208 \u2191s\nj : \u03b9\nhj : j \u2208 \u2191s\nhij : i \u2260 j\n\u22a2 t i \u2293 t j \u2264 \u22a5\n[PROOFSTEP]\nexact fun x hx => H i hi j hj hij \u27e8x, hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t u : Set \u03b1\nht : MeasurableSet t\nh's : s \u2286 u\nh't : t \u2286 u\nh : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u2191\u2191\u03bc t\n\u22a2 Set.Nonempty (s \u2229 t)\n[PROOFSTEP]\nrw [\u2190 Set.not_disjoint_iff_nonempty_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t u : Set \u03b1\nht : MeasurableSet t\nh's : s \u2286 u\nh't : t \u2286 u\nh : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u2191\u2191\u03bc t\n\u22a2 \u00acDisjoint s t\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t u : Set \u03b1\nht : MeasurableSet t\nh's : s \u2286 u\nh't : t \u2286 u\nh : Disjoint s t\n\u22a2 \u2191\u2191\u03bc s + \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc u\n[PROOFSTEP]\ncalc\n \u03bc s + \u03bc t = \u03bc (s \u222a t) := (measure_union h ht).symm\n _ \u2264 \u03bc u := measure_mono (union_subset h's h't)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t u : Set \u03b1\nhs : MeasurableSet s\nh's : s \u2286 u\nh't : t \u2286 u\nh : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u2191\u2191\u03bc t\n\u22a2 Set.Nonempty (s \u2229 t)\n[PROOFSTEP]\nrw [add_comm] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t u : Set \u03b1\nhs : MeasurableSet s\nh's : s \u2286 u\nh't : t \u2286 u\nh : \u2191\u2191\u03bc u < \u2191\u2191\u03bc t + \u2191\u2191\u03bc s\n\u22a2 Set.Nonempty (s \u2229 t)\n[PROOFSTEP]\nrw [inter_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t u : Set \u03b1\nhs : MeasurableSet s\nh's : s \u2286 u\nh't : t \u2286 u\nh : \u2191\u2191\u03bc u < \u2191\u2191\u03bc t + \u2191\u2191\u03bc s\n\u22a2 Set.Nonempty (t \u2229 s)\n[PROOFSTEP]\nexact nonempty_inter_of_measure_lt_add \u03bc hs h't h's h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) s\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), s i) = \u2a06 (i : \u03b9), \u2191\u2191\u03bc (s i)\n[PROOFSTEP]\ncases nonempty_encodable \u03b9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) s\nval\u271d : Encodable \u03b9\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), s i) = \u2a06 (i : \u03b9), \u2191\u2191\u03bc (s i)\n[PROOFSTEP]\ngeneralize ht : Function.extend Encodable.encode s \u22a5 = t\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) s\nval\u271d : Encodable \u03b9\nt : \u2115 \u2192 Set \u03b1\nht : Function.extend Encodable.encode s \u22a5 = t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), s i) = \u2a06 (i : \u03b9), \u2191\u2191\u03bc (s i)\n[PROOFSTEP]\nreplace hd : Directed (\u00b7 \u2286 \u00b7) t := ht \u25b8 hd.extend_bot Encodable.encode_injective\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nval\u271d : Encodable \u03b9\nt : \u2115 \u2192 Set \u03b1\nht : Function.extend Encodable.encode s \u22a5 = t\nhd : Directed (fun x x_1 => x \u2286 x_1) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), s i) = \u2a06 (i : \u03b9), \u2191\u2191\u03bc (s i)\n[PROOFSTEP]\nsuffices \u03bc (\u22c3 n, t n) = \u2a06 n, \u03bc (t n)\n by\n simp only [\u2190 ht, Encodable.encode_injective.apply_extend \u03bc, \u2190 iSup_eq_iUnion,\n iSup_extend_bot Encodable.encode_injective, (\u00b7 \u2218 \u00b7), Pi.bot_apply, bot_eq_empty, measure_empty] at this \n exact this.trans (iSup_extend_bot Encodable.encode_injective _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nval\u271d : Encodable \u03b9\nt : \u2115 \u2192 Set \u03b1\nht : Function.extend Encodable.encode s \u22a5 = t\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nthis : \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) = \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), s i) = \u2a06 (i : \u03b9), \u2191\u2191\u03bc (s i)\n[PROOFSTEP]\nsimp only [\u2190 ht, Encodable.encode_injective.apply_extend \u03bc, \u2190 iSup_eq_iUnion,\n iSup_extend_bot Encodable.encode_injective, (\u00b7 \u2218 \u00b7), Pi.bot_apply, bot_eq_empty, measure_empty] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nval\u271d : Encodable \u03b9\nt : \u2115 \u2192 Set \u03b1\nht : Function.extend Encodable.encode s \u22a5 = t\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nthis : \u2191\u2191\u03bc (\u2a06 (i : \u03b9), s i) = \u2a06 (n : \u2115), Function.extend Encodable.encode (fun x => \u2191\u2191\u03bc (s x)) (fun x => 0) n\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), s i) = \u2a06 (i : \u03b9), \u2191\u2191\u03bc (s i)\n[PROOFSTEP]\nexact this.trans (iSup_extend_bot Encodable.encode_injective _)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nval\u271d : Encodable \u03b9\nt : \u2115 \u2192 Set \u03b1\nht : Function.extend Encodable.encode s \u22a5 = t\nhd : Directed (fun x x_1 => x \u2286 x_1) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) = \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)\n[PROOFSTEP]\nclear! \u03b9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) = \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)\n[PROOFSTEP]\nrefine'\n le_antisymm _\n (iSup_le fun i => measure_mono <| subset_iUnion _ _)\n -- Choose `T n \u2287 t n` of the same measure, put `Td n = disjointed T`\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)\n[PROOFSTEP]\nset T : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)\n[PROOFSTEP]\nset Td : \u2115 \u2192 Set \u03b1 := disjointed T\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)\n[PROOFSTEP]\nhave hm : \u2200 n, MeasurableSet (Td n) := MeasurableSet.disjointed fun n => measurableSet_toMeasurable _ _\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)\n[PROOFSTEP]\ncalc\n \u03bc (\u22c3 n, t n) \u2264 \u03bc (\u22c3 n, T n) := measure_mono (iUnion_mono fun i => subset_toMeasurable _ _)\n _ = \u03bc (\u22c3 n, Td n) := by rw [iUnion_disjointed]\n _ \u2264 \u2211' n, \u03bc (Td n) := (measure_iUnion_le _)\n _ = \u2a06 I : Finset \u2115, \u2211 n in I, \u03bc (Td n) := ENNReal.tsum_eq_iSup_sum\n _ \u2264 \u2a06 n, \u03bc (t n) :=\n iSup_le fun I => by\n rcases hd.finset_le I with \u27e8N, hN\u27e9\n calc\n (\u2211 n in I, \u03bc (Td n)) = \u03bc (\u22c3 n \u2208 I, Td n) :=\n (measure_biUnion_finset ((disjoint_disjointed T).set_pairwise I) fun n _ => hm n).symm\n _ \u2264 \u03bc (\u22c3 n \u2208 I, T n) := (measure_mono (iUnion\u2082_mono fun n _hn => disjointed_subset _ _))\n _ = \u03bc (\u22c3 n \u2208 I, t n) := (measure_biUnion_toMeasurable I.countable_toSet _)\n _ \u2264 \u03bc (t N) := (measure_mono (iUnion\u2082_subset hN))\n _ \u2264 \u2a06 n, \u03bc (t n) := le_iSup (\u03bc \u2218 t) N\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), T n) = \u2191\u2191\u03bc (\u22c3 (n : \u2115), Td n)\n[PROOFSTEP]\nrw [iUnion_disjointed]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\nI : Finset \u2115\n\u22a2 \u2211 n in I, \u2191\u2191\u03bc (Td n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)\n[PROOFSTEP]\nrcases hd.finset_le I with \u27e8N, hN\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\nI : Finset \u2115\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2208 I \u2192 t i \u2286 t N\n\u22a2 \u2211 n in I, \u2191\u2191\u03bc (Td n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)\n[PROOFSTEP]\ncalc\n (\u2211 n in I, \u03bc (Td n)) = \u03bc (\u22c3 n \u2208 I, Td n) :=\n (measure_biUnion_finset ((disjoint_disjointed T).set_pairwise I) fun n _ => hm n).symm\n _ \u2264 \u03bc (\u22c3 n \u2208 I, T n) := (measure_mono (iUnion\u2082_mono fun n _hn => disjointed_subset _ _))\n _ = \u03bc (\u22c3 n \u2208 I, t n) := (measure_biUnion_toMeasurable I.countable_toSet _)\n _ \u2264 \u03bc (t N) := (measure_mono (iUnion\u2082_subset hN))\n _ \u2264 \u2a06 n, \u03bc (t n) := le_iSup (\u03bc \u2218 t) N\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\nht : Set.Countable t\nhd : DirectedOn ((fun x x_1 => x \u2286 x_1) on s) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 t), s i) = \u2a06 (i : \u03b9) (_ : i \u2208 t), \u2191\u2191\u03bc (s i)\n[PROOFSTEP]\nhaveI := ht.toEncodable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\nht : Set.Countable t\nhd : DirectedOn ((fun x x_1 => x \u2286 x_1) on s) t\nthis : Encodable \u2191t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 t), s i) = \u2a06 (i : \u03b9) (_ : i \u2208 t), \u2191\u2191\u03bc (s i)\n[PROOFSTEP]\nrw [biUnion_eq_iUnion, measure_iUnion_eq_iSup hd.directed_val, \u2190 iSup_subtype'']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nhfin : \u2203 i, \u2191\u2191\u03bc (s i) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9), s i) = \u2a05 (i : \u03b9), \u2191\u2191\u03bc (s i)\n[PROOFSTEP]\nrcases hfin with \u27e8k, hk\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9), s i) = \u2a05 (i : \u03b9), \u2191\u2191\u03bc (s i)\n[PROOFSTEP]\nhave : \u2200 (t) (_ : t \u2286 s k), \u03bc t \u2260 \u221e := fun t ht => ne_top_of_le_ne_top hk (measure_mono ht)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\nthis : \u2200 (t : Set \u03b1), t \u2286 s k \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9), s i) = \u2a05 (i : \u03b9), \u2191\u2191\u03bc (s i)\n[PROOFSTEP]\nrw [\u2190 ENNReal.sub_sub_cancel hk (iInf_le _ k), ENNReal.sub_iInf, \u2190\n ENNReal.sub_sub_cancel hk (measure_mono (iInter_subset _ k)), \u2190\n measure_diff (iInter_subset _ k) (MeasurableSet.iInter h) (this _ (iInter_subset _ k)), diff_iInter,\n measure_iUnion_eq_iSup]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\nthis : \u2200 (t : Set \u03b1), t \u2286 s k \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (s k) - \u2a06 (i : \u03b9), \u2191\u2191\u03bc (s k \\ s i) = \u2191\u2191\u03bc (s k) - \u2a06 (i : \u03b9), \u2191\u2191\u03bc (s k) - \u2191\u2191\u03bc (s i)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase intro.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\nthis : \u2200 (t : Set \u03b1), t \u2286 s k \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2a06 (i : \u03b9), \u2191\u2191\u03bc (s k \\ s i) = \u2a06 (i : \u03b9), \u2191\u2191\u03bc (s k) - \u2191\u2191\u03bc (s i)\n[PROOFSTEP]\nrefine' le_antisymm (iSup_mono' fun i => _) (iSup_mono fun i => _)\n[GOAL]\ncase intro.e_a.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\nthis : \u2200 (t : Set \u03b1), t \u2286 s k \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\ni : \u03b9\n\u22a2 \u2203 i', \u2191\u2191\u03bc (s k \\ s i) \u2264 \u2191\u2191\u03bc (s k) - \u2191\u2191\u03bc (s i')\n[PROOFSTEP]\nrcases hd i k with \u27e8j, hji, hjk\u27e9\n[GOAL]\ncase intro.e_a.refine'_1.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\nthis : \u2200 (t : Set \u03b1), t \u2286 s k \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\ni j : \u03b9\nhji : s i \u2287 s j\nhjk : s k \u2287 s j\n\u22a2 \u2203 i', \u2191\u2191\u03bc (s k \\ s i) \u2264 \u2191\u2191\u03bc (s k) - \u2191\u2191\u03bc (s i')\n[PROOFSTEP]\nuse j\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\nthis : \u2200 (t : Set \u03b1), t \u2286 s k \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\ni j : \u03b9\nhji : s i \u2287 s j\nhjk : s k \u2287 s j\n\u22a2 \u2191\u2191\u03bc (s k \\ s i) \u2264 \u2191\u2191\u03bc (s k) - \u2191\u2191\u03bc (s j)\n[PROOFSTEP]\nrw [\u2190 measure_diff hjk (h _) (this _ hjk)]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\nthis : \u2200 (t : Set \u03b1), t \u2286 s k \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\ni j : \u03b9\nhji : s i \u2287 s j\nhjk : s k \u2287 s j\n\u22a2 \u2191\u2191\u03bc (s k \\ s i) \u2264 \u2191\u2191\u03bc (s k \\ s j)\n[PROOFSTEP]\nexact measure_mono (diff_subset_diff_right hji)\n[GOAL]\ncase intro.e_a.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\nthis : \u2200 (t : Set \u03b1), t \u2286 s k \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\ni : \u03b9\n\u22a2 \u2191\u2191\u03bc (s k) - \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc (s k \\ s i)\n[PROOFSTEP]\nrw [tsub_le_iff_right, \u2190 measure_union, Set.union_comm]\n[GOAL]\ncase intro.e_a.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\nthis : \u2200 (t : Set \u03b1), t \u2286 s k \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\ni : \u03b9\n\u22a2 \u2191\u2191\u03bc (s k) \u2264 \u2191\u2191\u03bc (s i \u222a s k \\ s i)\ncase intro.e_a.refine'_2.hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\nthis : \u2200 (t : Set \u03b1), t \u2286 s k \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\ni : \u03b9\n\u22a2 Disjoint (s k \\ s i) (s i)\ncase intro.e_a.refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\nthis : \u2200 (t : Set \u03b1), t \u2286 s k \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\ni : \u03b9\n\u22a2 MeasurableSet (s i)\n[PROOFSTEP]\nexact measure_mono (diff_subset_iff.1 <| Subset.refl _)\n[GOAL]\ncase intro.e_a.refine'_2.hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\nthis : \u2200 (t : Set \u03b1), t \u2286 s k \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\ni : \u03b9\n\u22a2 Disjoint (s k \\ s i) (s i)\ncase intro.e_a.refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\nthis : \u2200 (t : Set \u03b1), t \u2286 s k \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\ni : \u03b9\n\u22a2 MeasurableSet (s i)\n[PROOFSTEP]\napply disjoint_sdiff_left\n[GOAL]\ncase intro.e_a.refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\nthis : \u2200 (t : Set \u03b1), t \u2286 s k \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\ni : \u03b9\n\u22a2 MeasurableSet (s i)\n[PROOFSTEP]\napply h i\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), MeasurableSet (s i)\nhd : Directed (fun x x_1 => x \u2287 x_1) s\nk : \u03b9\nhk : \u2191\u2191\u03bc (s k) \u2260 \u22a4\nthis : \u2200 (t : Set \u03b1), t \u2286 s k \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun i => s k \\ s i\n[PROOFSTEP]\nexact hd.mono_comp _ fun _ _ => diff_subset_diff_right\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhm : Monotone s\n\u22a2 Tendsto (\u2191\u2191\u03bc \u2218 s) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 (n : \u03b9), s n)))\n[PROOFSTEP]\nrw [measure_iUnion_eq_iSup (directed_of_sup hm)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhm : Monotone s\n\u22a2 Tendsto (\u2191\u2191\u03bc \u2218 s) atTop (\ud835\udcdd (\u2a06 (i : \u03b9), \u2191\u2191\u03bc (s i)))\n[PROOFSTEP]\nexact tendsto_atTop_iSup fun n m hnm => measure_mono <| hm hnm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), MeasurableSet (s n)\nhm : Antitone s\nhf : \u2203 i, \u2191\u2191\u03bc (s i) \u2260 \u22a4\n\u22a2 Tendsto (\u2191\u2191\u03bc \u2218 s) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u03b9), s n)))\n[PROOFSTEP]\nrw [measure_iInter_eq_iInf hs (directed_of_sup hm) hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : SemilatticeSup \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), MeasurableSet (s n)\nhm : Antitone s\nhf : \u2203 i, \u2191\u2191\u03bc (s i) \u2260 \u22a4\n\u22a2 Tendsto (\u2191\u2191\u03bc \u2218 s) atTop (\ud835\udcdd (\u2a05 (i : \u03b9), \u2191\u2191\u03bc (s i)))\n[PROOFSTEP]\nexact tendsto_atTop_iInf fun n m hnm => measure_mono <| hm hnm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\n\u22a2 Tendsto (\u2191\u2191\u03bc \u2218 s) (\ud835\udcdd[Ioi a] a) (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)))\n[PROOFSTEP]\nrefine' tendsto_order.2 \u27e8fun l hl => _, fun L hL => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nl : \u211d\u22650\u221e\nhl : l < \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\n\u22a2 \u2200\u1da0 (b : \u03b9) in \ud835\udcdd[Ioi a] a, l < (\u2191\u2191\u03bc \u2218 s) b\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin (s := Ioi a)] with r hr using hl.trans_le (measure_mono (biInter_subset_of_mem hr))\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\n\u22a2 \u2200\u1da0 (b : \u03b9) in \ud835\udcdd[Ioi a] a, (\u2191\u2191\u03bc \u2218 s) b < L\n[PROOFSTEP]\nobtain \u27e8u, u_anti, u_pos, u_lim\u27e9 : \u2203 u : \u2115 \u2192 \u03b9, StrictAnti u \u2227 (\u2200 n : \u2115, a < u n) \u2227 Tendsto u atTop (\ud835\udcdd a) :=\n by\n rcases hf with \u27e8r, ar, _\u27e9\n rcases exists_seq_strictAnti_tendsto' ar with \u27e8w, w_anti, w_mem, w_lim\u27e9\n exact \u27e8w, w_anti, fun n => (w_mem n).1, w_lim\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\n\u22a2 \u2203 u, StrictAnti u \u2227 (\u2200 (n : \u2115), a < u n) \u2227 Tendsto u atTop (\ud835\udcdd a)\n[PROOFSTEP]\nrcases hf with \u27e8r, ar, _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nr : \u03b9\nar : r > a\nright\u271d : \u2191\u2191\u03bc (s r) \u2260 \u22a4\n\u22a2 \u2203 u, StrictAnti u \u2227 (\u2200 (n : \u2115), a < u n) \u2227 Tendsto u atTop (\ud835\udcdd a)\n[PROOFSTEP]\nrcases exists_seq_strictAnti_tendsto' ar with \u27e8w, w_anti, w_mem, w_lim\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nr : \u03b9\nar : r > a\nright\u271d : \u2191\u2191\u03bc (s r) \u2260 \u22a4\nw : \u2115 \u2192 \u03b9\nw_anti : StrictAnti w\nw_mem : \u2200 (n : \u2115), w n \u2208 Ioo a r\nw_lim : Tendsto w atTop (\ud835\udcdd a)\n\u22a2 \u2203 u, StrictAnti u \u2227 (\u2200 (n : \u2115), a < u n) \u2227 Tendsto u atTop (\ud835\udcdd a)\n[PROOFSTEP]\nexact \u27e8w, w_anti, fun n => (w_mem n).1, w_lim\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\n\u22a2 \u2200\u1da0 (b : \u03b9) in \ud835\udcdd[Ioi a] a, (\u2191\u2191\u03bc \u2218 s) b < L\n[PROOFSTEP]\nhave A : Tendsto (\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u03bc (\u22c2 n, s (u n)))) :=\n by\n refine' tendsto_measure_iInter (fun n => hs _ (u_pos n)) _ _\n \u00b7 intro m n hmn\n exact hm _ _ (u_pos n) (u_anti.antitone hmn)\n \u00b7 rcases hf with \u27e8r, rpos, hr\u27e9\n obtain \u27e8n, hn\u27e9 : \u2203 n : \u2115, u n < r := ((tendsto_order.1 u_lim).2 r rpos).exists\n refine' \u27e8n, ne_of_lt (lt_of_le_of_lt _ hr.lt_top)\u27e9\n exact measure_mono (hm _ _ (u_pos n) hn.le)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\n\u22a2 Tendsto (\u2191\u2191\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), s (u n))))\n[PROOFSTEP]\nrefine' tendsto_measure_iInter (fun n => hs _ (u_pos n)) _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\n\u22a2 Antitone (s \u2218 u)\n[PROOFSTEP]\nintro m n hmn\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nm n : \u2115\nhmn : m \u2264 n\n\u22a2 (s \u2218 u) n \u2264 (s \u2218 u) m\n[PROOFSTEP]\nexact hm _ _ (u_pos n) (u_anti.antitone hmn)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\n\u22a2 \u2203 i, \u2191\u2191\u03bc ((s \u2218 u) i) \u2260 \u22a4\n[PROOFSTEP]\nrcases hf with \u27e8r, rpos, hr\u27e9\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nr : \u03b9\nrpos : r > a\nhr : \u2191\u2191\u03bc (s r) \u2260 \u22a4\n\u22a2 \u2203 i, \u2191\u2191\u03bc ((s \u2218 u) i) \u2260 \u22a4\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : \u2203 n : \u2115, u n < r := ((tendsto_order.1 u_lim).2 r rpos).exists\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nr : \u03b9\nrpos : r > a\nhr : \u2191\u2191\u03bc (s r) \u2260 \u22a4\nn : \u2115\nhn : u n < r\n\u22a2 \u2203 i, \u2191\u2191\u03bc ((s \u2218 u) i) \u2260 \u22a4\n[PROOFSTEP]\nrefine' \u27e8n, ne_of_lt (lt_of_le_of_lt _ hr.lt_top)\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nr : \u03b9\nrpos : r > a\nhr : \u2191\u2191\u03bc (s r) \u2260 \u22a4\nn : \u2115\nhn : u n < r\n\u22a2 \u2191\u2191\u03bc ((s \u2218 u) n) \u2264 \u2191\u2191\u03bc (s r)\n[PROOFSTEP]\nexact measure_mono (hm _ _ (u_pos n) hn.le)\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nA : Tendsto (\u2191\u2191\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), s (u n))))\n\u22a2 \u2200\u1da0 (b : \u03b9) in \ud835\udcdd[Ioi a] a, (\u2191\u2191\u03bc \u2218 s) b < L\n[PROOFSTEP]\nhave B : \u22c2 n, s (u n) = \u22c2 r > a, s r := by\n apply Subset.antisymm\n \u00b7 simp only [subset_iInter_iff, gt_iff_lt]\n intro r rpos\n obtain \u27e8n, hn\u27e9 : \u2203 n, u n < r := ((tendsto_order.1 u_lim).2 _ rpos).exists\n exact Subset.trans (iInter_subset _ n) (hm (u n) r (u_pos n) hn.le)\n \u00b7 simp only [subset_iInter_iff, gt_iff_lt]\n intro n\n apply biInter_subset_of_mem\n exact u_pos n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nA : Tendsto (\u2191\u2191\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), s (u n))))\n\u22a2 \u22c2 (n : \u2115), s (u n) = \u22c2 (r : \u03b9) (_ : r > a), s r\n[PROOFSTEP]\napply Subset.antisymm\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nA : Tendsto (\u2191\u2191\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), s (u n))))\n\u22a2 \u22c2 (n : \u2115), s (u n) \u2286 \u22c2 (r : \u03b9) (_ : r > a), s r\n[PROOFSTEP]\nsimp only [subset_iInter_iff, gt_iff_lt]\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nA : Tendsto (\u2191\u2191\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), s (u n))))\n\u22a2 \u2200 (i : \u03b9), a < i \u2192 \u22c2 (n : \u2115), s (u n) \u2286 s i\n[PROOFSTEP]\nintro r rpos\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nA : Tendsto (\u2191\u2191\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), s (u n))))\nr : \u03b9\nrpos : a < r\n\u22a2 \u22c2 (n : \u2115), s (u n) \u2286 s r\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : \u2203 n, u n < r := ((tendsto_order.1 u_lim).2 _ rpos).exists\n[GOAL]\ncase h\u2081.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nA : Tendsto (\u2191\u2191\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), s (u n))))\nr : \u03b9\nrpos : a < r\nn : \u2115\nhn : u n < r\n\u22a2 \u22c2 (n : \u2115), s (u n) \u2286 s r\n[PROOFSTEP]\nexact Subset.trans (iInter_subset _ n) (hm (u n) r (u_pos n) hn.le)\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nA : Tendsto (\u2191\u2191\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), s (u n))))\n\u22a2 \u22c2 (r : \u03b9) (_ : r > a), s r \u2286 \u22c2 (n : \u2115), s (u n)\n[PROOFSTEP]\nsimp only [subset_iInter_iff, gt_iff_lt]\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nA : Tendsto (\u2191\u2191\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), s (u n))))\n\u22a2 \u2200 (i : \u2115), \u22c2 (r : \u03b9) (_ : a < r), s r \u2286 s (u i)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nA : Tendsto (\u2191\u2191\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), s (u n))))\nn : \u2115\n\u22a2 \u22c2 (r : \u03b9) (_ : a < r), s r \u2286 s (u n)\n[PROOFSTEP]\napply biInter_subset_of_mem\n[GOAL]\ncase h\u2082.xs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nA : Tendsto (\u2191\u2191\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), s (u n))))\nn : \u2115\n\u22a2 u n \u2208 fun x => Preorder.toLT.1 a x\n[PROOFSTEP]\nexact u_pos n\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nA : Tendsto (\u2191\u2191\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), s (u n))))\nB : \u22c2 (n : \u2115), s (u n) = \u22c2 (r : \u03b9) (_ : r > a), s r\n\u22a2 \u2200\u1da0 (b : \u03b9) in \ud835\udcdd[Ioi a] a, (\u2191\u2191\u03bc \u2218 s) b < L\n[PROOFSTEP]\nrw [B] at A \n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nA : Tendsto (\u2191\u2191\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)))\nB : \u22c2 (n : \u2115), s (u n) = \u22c2 (r : \u03b9) (_ : r > a), s r\n\u22a2 \u2200\u1da0 (b : \u03b9) in \ud835\udcdd[Ioi a] a, (\u2191\u2191\u03bc \u2218 s) b < L\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : \u2203 n, \u03bc (s (u n)) < L := ((tendsto_order.1 A).2 _ hL).exists\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nA : Tendsto (\u2191\u2191\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)))\nB : \u22c2 (n : \u2115), s (u n) = \u22c2 (r : \u03b9) (_ : r > a), s r\nn : \u2115\nhn : \u2191\u2191\u03bc (s (u n)) < L\n\u22a2 \u2200\u1da0 (b : \u03b9) in \ud835\udcdd[Ioi a] a, (\u2191\u2191\u03bc \u2218 s) b < L\n[PROOFSTEP]\nhave : Ioc a (u n) \u2208 \ud835\udcdd[>] a := Ioc_mem_nhdsWithin_Ioi \u27e8le_rfl, u_pos n\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u2074 : LinearOrder \u03b9\ninst\u271d\u00b3 : TopologicalSpace \u03b9\ninst\u271d\u00b2 : OrderTopology \u03b9\ninst\u271d\u00b9 : DenselyOrdered \u03b9\ninst\u271d : TopologicalSpace.FirstCountableTopology \u03b9\ns : \u03b9 \u2192 Set \u03b1\na : \u03b9\nhs : \u2200 (r : \u03b9), r > a \u2192 MeasurableSet (s r)\nhm : \u2200 (i j : \u03b9), a < i \u2192 i \u2264 j \u2192 s i \u2286 s j\nhf : \u2203 r, r > a \u2227 \u2191\u2191\u03bc (s r) \u2260 \u22a4\nL : \u211d\u22650\u221e\nhL : L > \u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)\nu : \u2115 \u2192 \u03b9\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), a < u n\nu_lim : Tendsto u atTop (\ud835\udcdd a)\nA : Tendsto (\u2191\u2191\u03bc \u2218 s \u2218 u) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (r : \u03b9) (_ : r > a), s r)))\nB : \u22c2 (n : \u2115), s (u n) = \u22c2 (r : \u03b9) (_ : r > a), s r\nn : \u2115\nhn : \u2191\u2191\u03bc (s (u n)) < L\nthis : Ioc a (u n) \u2208 \ud835\udcdd[Ioi a] a\n\u22a2 \u2200\u1da0 (b : \u03b9) in \ud835\udcdd[Ioi a] a, (\u2191\u2191\u03bc \u2218 s) b < L\n[PROOFSTEP]\nfilter_upwards [this] with r hr using lt_of_le_of_lt (measure_mono (hm _ _ hr.1 hr.2)) hn\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (limsup s atTop) = 0\n[PROOFSTEP]\nset t : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (s n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nt : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (s n)\n\u22a2 \u2191\u2191\u03bc (limsup s atTop) = 0\n[PROOFSTEP]\nhave ht : (\u2211' i, \u03bc (t i)) \u2260 \u221e := by simpa only [measure_toMeasurable] using hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nt : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (s n)\n\u22a2 \u2211' (i : \u2115), \u2191\u2191\u03bc (t i) \u2260 \u22a4\n[PROOFSTEP]\nsimpa only [measure_toMeasurable] using hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nt : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (s n)\nht : \u2211' (i : \u2115), \u2191\u2191\u03bc (t i) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (limsup s atTop) = 0\n[PROOFSTEP]\nsuffices \u03bc (limsup t atTop) = 0\n by\n have A : s \u2264 t := fun n =>\n subset_toMeasurable \u03bc\n (s n)\n -- TODO default args fail\n exact measure_mono_null (limsup_le_limsup (eventually_of_forall (Pi.le_def.mp A))) this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nt : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (s n)\nht : \u2211' (i : \u2115), \u2191\u2191\u03bc (t i) \u2260 \u22a4\nthis : \u2191\u2191\u03bc (limsup t atTop) = 0\n\u22a2 \u2191\u2191\u03bc (limsup s atTop) = 0\n[PROOFSTEP]\nhave A : s \u2264 t := fun n =>\n subset_toMeasurable \u03bc\n (s n)\n -- TODO default args fail\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nt : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (s n)\nht : \u2211' (i : \u2115), \u2191\u2191\u03bc (t i) \u2260 \u22a4\nthis : \u2191\u2191\u03bc (limsup t atTop) = 0\nA : s \u2264 t\n\u22a2 \u2191\u2191\u03bc (limsup s atTop) = 0\n[PROOFSTEP]\nexact measure_mono_null (limsup_le_limsup (eventually_of_forall (Pi.le_def.mp A))) this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nt : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (s n)\nht : \u2211' (i : \u2115), \u2191\u2191\u03bc (t i) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (limsup t atTop) = 0\n[PROOFSTEP]\nsimp only [limsup_eq_iInf_iSup_of_nat', Set.iInf_eq_iInter, Set.iSup_eq_iUnion, \u2190 nonpos_iff_eq_zero]\n -- Finally, we estimate `\u03bc (\u22c3 i, t (i + n))` by `\u2211 i', \u03bc (t (i + n))`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nt : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (s n)\nht : \u2211' (i : \u2115), \u2191\u2191\u03bc (t i) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (\u22c2 (n : \u2115), \u22c3 (i : \u2115), toMeasurable \u03bc (s (i + n))) \u2264 0\n[PROOFSTEP]\nrefine'\n le_of_tendsto_of_tendsto'\n (tendsto_measure_iInter (fun i => MeasurableSet.iUnion fun b => measurableSet_toMeasurable _ _) _\n \u27e80, ne_top_of_le_ne_top ht (measure_iUnion_le t)\u27e9)\n (ENNReal.tendsto_sum_nat_add (\u03bc \u2218 t) ht) fun n => measure_iUnion_le _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nt : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (s n)\nht : \u2211' (i : \u2115), \u2191\u2191\u03bc (t i) \u2260 \u22a4\n\u22a2 Antitone fun n => \u22c3 (i : \u2115), toMeasurable \u03bc (s (i + n))\n[PROOFSTEP]\nintro n m hnm x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nt : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (s n)\nht : \u2211' (i : \u2115), \u2191\u2191\u03bc (t i) \u2260 \u22a4\nn m : \u2115\nhnm : n \u2264 m\nx : \u03b1\n\u22a2 x \u2208 (fun n => \u22c3 (i : \u2115), toMeasurable \u03bc (s (i + n))) m \u2192 x \u2208 (fun n => \u22c3 (i : \u2115), toMeasurable \u03bc (s (i + n))) n\n[PROOFSTEP]\nsimp only [Set.mem_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nt : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (s n)\nht : \u2211' (i : \u2115), \u2191\u2191\u03bc (t i) \u2260 \u22a4\nn m : \u2115\nhnm : n \u2264 m\nx : \u03b1\n\u22a2 (\u2203 i, x \u2208 toMeasurable \u03bc (s (i + m))) \u2192 \u2203 i, x \u2208 toMeasurable \u03bc (s (i + n))\n[PROOFSTEP]\nexact fun \u27e8i, hi\u27e9 => \u27e8i + (m - n), by simpa only [add_assoc, tsub_add_cancel_of_le hnm] using hi\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nt : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (s n)\nht : \u2211' (i : \u2115), \u2191\u2191\u03bc (t i) \u2260 \u22a4\nn m : \u2115\nhnm : n \u2264 m\nx : \u03b1\nx\u271d : \u2203 i, x \u2208 toMeasurable \u03bc (s (i + m))\ni : \u2115\nhi : x \u2208 toMeasurable \u03bc (s (i + m))\n\u22a2 x \u2208 toMeasurable \u03bc (s (i + (m - n) + n))\n[PROOFSTEP]\nsimpa only [add_assoc, tsub_add_cancel_of_le hnm] using hi\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (liminf s atTop) = 0\n[PROOFSTEP]\nrw [\u2190 le_zero_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (liminf s atTop) \u2264 0\n[PROOFSTEP]\nhave : liminf s atTop \u2264 limsup s atTop := liminf_le_limsup\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nthis : liminf s atTop \u2264 limsup s atTop\n\u22a2 \u2191\u2191\u03bc (liminf s atTop) \u2264 0\n[PROOFSTEP]\nexact (\u03bc.mono this).trans (by simp [measure_limsup_eq_zero h])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4\nthis : liminf s atTop \u2264 limsup s atTop\n\u22a2 \u2191\u2191\u03bc (limsup s atTop) \u2264 0\n[PROOFSTEP]\nsimp [measure_limsup_eq_zero h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), s n =\u1d50[\u03bc] t\n\u22a2 limsup s atTop =\u1d50[\u03bc] t\n[PROOFSTEP]\nsimp_rw [ae_eq_set] at h \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ s n) = 0\n\u22a2 \u2191\u2191\u03bc (limsup s atTop \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ limsup s atTop) = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ s n) = 0\n\u22a2 \u2191\u2191\u03bc (limsup s atTop \\ t) = 0\n[PROOFSTEP]\nrw [atTop.limsup_sdiff s t]\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ s n) = 0\n\u22a2 \u2191\u2191\u03bc (limsup (fun b => s b \\ t) atTop) = 0\n[PROOFSTEP]\napply measure_limsup_eq_zero\n[GOAL]\ncase left.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ s n) = 0\n\u22a2 \u2211' (i : \u2115), \u2191\u2191\u03bc (s i \\ t) \u2260 \u22a4\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ s n) = 0\n\u22a2 \u2191\u2191\u03bc (t \\ limsup s atTop) = 0\n[PROOFSTEP]\nrw [atTop.sdiff_limsup s t]\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ s n) = 0\n\u22a2 \u2191\u2191\u03bc (liminf (fun b => t \\ s b) atTop) = 0\n[PROOFSTEP]\napply measure_liminf_eq_zero\n[GOAL]\ncase right.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ s n) = 0\n\u22a2 \u2211' (i : \u2115), \u2191\u2191\u03bc (t \\ s i) \u2260 \u22a4\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), s n =\u1d50[\u03bc] t\n\u22a2 liminf s atTop =\u1d50[\u03bc] t\n[PROOFSTEP]\nsimp_rw [ae_eq_set] at h \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ s n) = 0\n\u22a2 \u2191\u2191\u03bc (liminf s atTop \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ liminf s atTop) = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ s n) = 0\n\u22a2 \u2191\u2191\u03bc (liminf s atTop \\ t) = 0\n[PROOFSTEP]\nrw [atTop.liminf_sdiff s t]\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ s n) = 0\n\u22a2 \u2191\u2191\u03bc (liminf (fun b => s b \\ t) atTop) = 0\n[PROOFSTEP]\napply measure_liminf_eq_zero\n[GOAL]\ncase left.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ s n) = 0\n\u22a2 \u2211' (i : \u2115), \u2191\u2191\u03bc (s i \\ t) \u2260 \u22a4\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ s n) = 0\n\u22a2 \u2191\u2191\u03bc (t \\ liminf s atTop) = 0\n[PROOFSTEP]\nrw [atTop.sdiff_liminf s t]\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ s n) = 0\n\u22a2 \u2191\u2191\u03bc (limsup (fun b => t \\ s b) atTop) = 0\n[PROOFSTEP]\napply measure_limsup_eq_zero\n[GOAL]\ncase right.hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nt : Set \u03b1\nh : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n \\ t) = 0 \u2227 \u2191\u2191\u03bc (t \\ s n) = 0\n\u22a2 \u2211' (i : \u2115), \u2191\u2191\u03bc (t \\ s i) \u2260 \u22a4\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nx : \u03b2\nt : Set \u03b2\ns : Set \u03b1\n\u22a2 \u2191\u2191\u03bc (if x \u2208 t then s else \u2205) = indicator t (fun x => \u2191\u2191\u03bc s) x\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nx : \u03b2\nt : Set \u03b2\ns : Set \u03b1\nh : x \u2208 t\n\u22a2 \u2191\u2191\u03bc s = indicator t (fun x => \u2191\u2191\u03bc s) x\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\nx : \u03b2\nt : Set \u03b2\ns : Set \u03b1\nh : \u00acx \u2208 t\n\u22a2 \u2191\u2191\u03bc \u2205 = indicator t (fun x => \u2191\u2191\u03bc s) x\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nms : MeasurableSpace \u03b1\ns\u271d t : Set \u03b1\nm : OuterMeasure \u03b1\nh : ms \u2264 OuterMeasure.caratheodory m\ns : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 \u2191\u2191(OuterMeasure.toMeasure m h) s = \u2191m s\n[PROOFSTEP]\nrefine' le_antisymm _ (le_toMeasure_apply _ _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nms : MeasurableSpace \u03b1\ns\u271d t : Set \u03b1\nm : OuterMeasure \u03b1\nh : ms \u2264 OuterMeasure.caratheodory m\ns : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 \u2191\u2191(OuterMeasure.toMeasure m h) s \u2264 \u2191m s\n[PROOFSTEP]\nrcases hs.exists_measurable_subset_ae_eq with \u27e8t, hts, htm, heq\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nms : MeasurableSpace \u03b1\ns\u271d t\u271d : Set \u03b1\nm : OuterMeasure \u03b1\nh : ms \u2264 OuterMeasure.caratheodory m\ns : Set \u03b1\nhs : NullMeasurableSet s\nt : Set \u03b1\nhts : t \u2286 s\nhtm : MeasurableSet t\nheq : t =\u1d50[OuterMeasure.toMeasure m h] s\n\u22a2 \u2191\u2191(OuterMeasure.toMeasure m h) s \u2264 \u2191m s\n[PROOFSTEP]\ncalc\n m.toMeasure h s = m.toMeasure h t := measure_congr heq.symm\n _ = m t := (toMeasure_apply m h htm)\n _ \u2264 m s := m.mono hts\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t u : Set \u03b1\nhs : MeasurableSet s\nh : \u2191\u2191\u03bc t = \u2191\u2191\u03bc u\nhtu : t \u2286 u\nht_ne_top : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (t \u2229 s) = \u2191\u2191\u03bc (u \u2229 s)\n[PROOFSTEP]\nrw [h] at ht_ne_top \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t u : Set \u03b1\nhs : MeasurableSet s\nh : \u2191\u2191\u03bc t = \u2191\u2191\u03bc u\nhtu : t \u2286 u\nht_ne_top : \u2191\u2191\u03bc u \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (t \u2229 s) = \u2191\u2191\u03bc (u \u2229 s)\n[PROOFSTEP]\nrefine' le_antisymm (measure_mono (inter_subset_inter_left _ htu)) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t u : Set \u03b1\nhs : MeasurableSet s\nh : \u2191\u2191\u03bc t = \u2191\u2191\u03bc u\nhtu : t \u2286 u\nht_ne_top : \u2191\u2191\u03bc u \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (u \u2229 s) \u2264 \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nhave A : \u03bc (u \u2229 s) + \u03bc (u \\ s) \u2264 \u03bc (t \u2229 s) + \u03bc (u \\ s) :=\n calc\n \u03bc (u \u2229 s) + \u03bc (u \\ s) = \u03bc u := measure_inter_add_diff _ hs\n _ = \u03bc t := h.symm\n _ = \u03bc (t \u2229 s) + \u03bc (t \\ s) := (measure_inter_add_diff _ hs).symm\n _ \u2264 \u03bc (t \u2229 s) + \u03bc (u \\ s) := add_le_add le_rfl (measure_mono (diff_subset_diff htu Subset.rfl))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t u : Set \u03b1\nhs : MeasurableSet s\nh : \u2191\u2191\u03bc t = \u2191\u2191\u03bc u\nhtu : t \u2286 u\nht_ne_top : \u2191\u2191\u03bc u \u2260 \u22a4\nA : \u2191\u2191\u03bc (u \u2229 s) + \u2191\u2191\u03bc (u \\ s) \u2264 \u2191\u2191\u03bc (t \u2229 s) + \u2191\u2191\u03bc (u \\ s)\n\u22a2 \u2191\u2191\u03bc (u \u2229 s) \u2264 \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nhave B : \u03bc (u \\ s) \u2260 \u221e := (lt_of_le_of_lt (measure_mono (diff_subset _ _)) ht_ne_top.lt_top).ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t u : Set \u03b1\nhs : MeasurableSet s\nh : \u2191\u2191\u03bc t = \u2191\u2191\u03bc u\nhtu : t \u2286 u\nht_ne_top : \u2191\u2191\u03bc u \u2260 \u22a4\nA : \u2191\u2191\u03bc (u \u2229 s) + \u2191\u2191\u03bc (u \\ s) \u2264 \u2191\u2191\u03bc (t \u2229 s) + \u2191\u2191\u03bc (u \\ s)\nB : \u2191\u2191\u03bc (u \\ s) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (u \u2229 s) \u2264 \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nexact ENNReal.le_of_add_le_add_right B A\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsEmpty \u03b1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\n\u22a2 \u03bc = \u03bd\n[PROOFSTEP]\next1 s _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : IsEmpty \u03b1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\na\u271d : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\n[PROOFSTEP]\nrw [eq_empty_of_isEmpty s]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : IsEmpty \u03b1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\na\u271d : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc \u2205 = \u2191\u2191\u03bd \u2205\n[PROOFSTEP]\nsimp only [measure_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081\u271d \u03bc\u2082\u271d \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), MeasurableSet (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 \u2191\u2191\u03bc\u2081 (\u22c3 (i : \u2115), s i) + \u2191\u2191\u03bc\u2082 (\u22c3 (i : \u2115), s i) = \u2211' (i : \u2115), (\u2191\u2191\u03bc\u2081 (s i) + \u2191\u2191\u03bc\u2082 (s i))\n[PROOFSTEP]\nrw [ENNReal.tsum_add, measure_iUnion hd hs, measure_iUnion hd hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081\u271d \u03bc\u2082\u271d \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u2081 \u03bc\u2082 : Measure \u03b1\n\u22a2 OuterMeasure.trim (\u2191\u03bc\u2081 + \u2191\u03bc\u2082) = \u2191\u03bc\u2081 + \u2191\u03bc\u2082\n[PROOFSTEP]\nrw [OuterMeasure.trim_add, \u03bc\u2081.trimmed, \u03bc\u2082.trimmed]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul R' \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : MeasurableSpace \u03b1\nc : R\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), MeasurableSet (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 \u2191(c \u2022 \u2191\u03bc) (\u22c3 (i : \u2115), s i) = \u2211' (i : \u2115), \u2191(c \u2022 \u2191\u03bc) (s i)\n[PROOFSTEP]\nrw [\u2190 smul_one_smul \u211d\u22650\u221e c (_ : OuterMeasure \u03b1)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul R' \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : MeasurableSpace \u03b1\nc : R\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), MeasurableSet (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 \u2191((c \u2022 1) \u2022 \u2191\u03bc) (\u22c3 (i : \u2115), s i) = \u2211' (i : \u2115), \u2191((c \u2022 1) \u2022 \u2191\u03bc) (s i)\n[PROOFSTEP]\nconv_lhs =>\n change OuterMeasure.measureOf ((c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) \u2022 \u03bc.toOuterMeasure) (\u22c3 i, s i)\n change (c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) * OuterMeasure.measureOf \u03bc.toOuterMeasure (\u22c3 i, s i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul R' \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : MeasurableSpace \u03b1\nc : R\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), MeasurableSet (s i)\nhd : Pairwise (Disjoint on s)\n| \u2191((c \u2022 1) \u2022 \u2191\u03bc) (\u22c3 (i : \u2115), s i)\n[PROOFSTEP]\n change OuterMeasure.measureOf ((c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) \u2022 \u03bc.toOuterMeasure) (\u22c3 i, s i)\n change (c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) * OuterMeasure.measureOf \u03bc.toOuterMeasure (\u22c3 i, s i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul R' \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : MeasurableSpace \u03b1\nc : R\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), MeasurableSet (s i)\nhd : Pairwise (Disjoint on s)\n| \u2191((c \u2022 1) \u2022 \u2191\u03bc) (\u22c3 (i : \u2115), s i)\n[PROOFSTEP]\n change OuterMeasure.measureOf ((c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) \u2022 \u03bc.toOuterMeasure) (\u22c3 i, s i)\n change (c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) * OuterMeasure.measureOf \u03bc.toOuterMeasure (\u22c3 i, s i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul R' \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : MeasurableSpace \u03b1\nc : R\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), MeasurableSet (s i)\nhd : Pairwise (Disjoint on s)\n| \u2191((c \u2022 1) \u2022 \u2191\u03bc) (\u22c3 (i : \u2115), s i)\n[PROOFSTEP]\nchange OuterMeasure.measureOf ((c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) \u2022 \u03bc.toOuterMeasure) (\u22c3 i, s i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul R' \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : MeasurableSpace \u03b1\nc : R\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), MeasurableSet (s i)\nhd : Pairwise (Disjoint on s)\n| \u2191((c \u2022 OfNat.ofNat 1) \u2022 \u2191\u03bc) (\u22c3 (i : \u2115), s i)\n[PROOFSTEP]\nchange (c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) * OuterMeasure.measureOf \u03bc.toOuterMeasure (\u22c3 i, s i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul R' \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : MeasurableSpace \u03b1\nc : R\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), MeasurableSet (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 c \u2022 OfNat.ofNat 1 * \u2191\u2191\u03bc (\u22c3 (i : \u2115), s i) = \u2211' (i : \u2115), \u2191((c \u2022 1) \u2022 \u2191\u03bc) (s i)\n[PROOFSTEP]\nconv_rhs =>\n change \u2211' i, OuterMeasure.measureOf ((c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) \u2022 \u03bc.toOuterMeasure) (s i)\n change \u2211' i, (c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) * OuterMeasure.measureOf (\u03bc.toOuterMeasure) (s i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul R' \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : MeasurableSpace \u03b1\nc : R\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), MeasurableSet (s i)\nhd : Pairwise (Disjoint on s)\n| \u2211' (i : \u2115), \u2191((c \u2022 1) \u2022 \u2191\u03bc) (s i)\n[PROOFSTEP]\n change \u2211' i, OuterMeasure.measureOf ((c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) \u2022 \u03bc.toOuterMeasure) (s i)\n change \u2211' i, (c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) * OuterMeasure.measureOf (\u03bc.toOuterMeasure) (s i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul R' \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : MeasurableSpace \u03b1\nc : R\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), MeasurableSet (s i)\nhd : Pairwise (Disjoint on s)\n| \u2211' (i : \u2115), \u2191((c \u2022 1) \u2022 \u2191\u03bc) (s i)\n[PROOFSTEP]\n change \u2211' i, OuterMeasure.measureOf ((c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) \u2022 \u03bc.toOuterMeasure) (s i)\n change \u2211' i, (c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) * OuterMeasure.measureOf (\u03bc.toOuterMeasure) (s i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul R' \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : MeasurableSpace \u03b1\nc : R\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), MeasurableSet (s i)\nhd : Pairwise (Disjoint on s)\n| \u2211' (i : \u2115), \u2191((c \u2022 1) \u2022 \u2191\u03bc) (s i)\n[PROOFSTEP]\nchange \u2211' i, OuterMeasure.measureOf ((c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) \u2022 \u03bc.toOuterMeasure) (s i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul R' \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : MeasurableSpace \u03b1\nc : R\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), MeasurableSet (s i)\nhd : Pairwise (Disjoint on s)\n| \u2211' (i : \u2115), \u2191((c \u2022 OfNat.ofNat 1) \u2022 \u2191\u03bc) (s i)\n[PROOFSTEP]\nchange \u2211' i, (c \u2022 @OfNat.ofNat _ 1 One.toOfNat1 : \u211d\u22650\u221e) * OuterMeasure.measureOf (\u03bc.toOuterMeasure) (s i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul R' \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : MeasurableSpace \u03b1\nc : R\n\u03bc : Measure \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u2115), MeasurableSet (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 c \u2022 OfNat.ofNat 1 * \u2191\u2191\u03bc (\u22c3 (i : \u2115), s i) = \u2211' (i : \u2115), c \u2022 OfNat.ofNat 1 * \u2191\u2191\u03bc (s i)\n[PROOFSTEP]\nsimp_rw [measure_iUnion hd hs, ENNReal.tsum_mul_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b2 : SMul R' \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R' \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : MeasurableSpace \u03b1\nc : R\n\u03bc : Measure \u03b1\n\u22a2 OuterMeasure.trim (c \u2022 \u2191\u03bc) = c \u2022 \u2191\u03bc\n[PROOFSTEP]\nrw [OuterMeasure.trim_smul, \u03bc.trimmed]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nm : MeasurableSpace \u03b1\nI : Finset \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : Set \u03b1\n\u22a2 \u2191\u2191(\u2211 i in I, \u03bc i) s = \u2211 i in I, \u2191\u2191(\u03bc i) s\n[PROOFSTEP]\nrw [coe_finset_sum, Finset.sum_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n_m : MeasurableSpace \u03b1\nc : \u211d\u22650\n\u03bc : Measure \u03b1\ns : Set \u03b1\n\u22a2 c \u2022 \u2191\u2191\u03bc s = \u2191c * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\np : \u03b1 \u2192 Prop\nc : \u211d\u22650\u221e\nhc : c \u2260 0\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202c \u2022 \u03bc, p x) \u2194 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x\n[PROOFSTEP]\nsimp only [ae_iff, Algebra.id.smul_eq_mul, smul_apply, or_iff_right_iff_imp, mul_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\np : \u03b1 \u2192 Prop\nc : \u211d\u22650\u221e\nhc : c \u2260 0\n\u22a2 c = 0 \u2192 \u2191\u2191\u03bc {a | \u00acp a} = 0\n[PROOFSTEP]\nsimp only [IsEmpty.forall_iff, hc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nh : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\nh' : s \u2286 t\nh'' : \u2191\u2191(\u03bc + \u03bd) s = \u2191\u2191(\u03bc + \u03bd) t\n\u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc t\n[PROOFSTEP]\nrefine' le_antisymm (measure_mono h') _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nh : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\nh' : s \u2286 t\nh'' : \u2191\u2191(\u03bc + \u03bd) s = \u2191\u2191(\u03bc + \u03bd) t\n\u22a2 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc s\n[PROOFSTEP]\nhave : \u03bc t + \u03bd t \u2264 \u03bc s + \u03bd t :=\n calc\n \u03bc t + \u03bd t = \u03bc s + \u03bd s := h''.symm\n _ \u2264 \u03bc s + \u03bd t := add_le_add le_rfl (measure_mono h')\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nh : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\nh' : s \u2286 t\nh'' : \u2191\u2191(\u03bc + \u03bd) s = \u2191\u2191(\u03bc + \u03bd) t\nthis : \u2191\u2191\u03bc t + \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc s + \u2191\u2191\u03bd t\n\u22a2 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc s\n[PROOFSTEP]\napply ENNReal.le_of_add_le_add_right _ this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nh : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\nh' : s \u2286 t\nh'' : \u2191\u2191(\u03bc + \u03bd) s = \u2191\u2191(\u03bc + \u03bd) t\nthis : \u2191\u2191\u03bc t + \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc s + \u2191\u2191\u03bd t\n\u22a2 \u2191\u2191\u03bd t \u2260 \u22a4\n[PROOFSTEP]\nsimp only [not_or, ENNReal.add_eq_top, Pi.add_apply, Ne.def, coe_add] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nh' : s \u2286 t\nh'' : \u2191\u2191(\u03bc + \u03bd) s = \u2191\u2191(\u03bc + \u03bd) t\nthis : \u2191\u2191\u03bc t + \u2191\u2191\u03bd t \u2264 \u2191\u2191\u03bc s + \u2191\u2191\u03bd t\nh : \u00ac\u2191\u2191\u03bc t = \u22a4 \u2227 \u00ac\u2191\u2191\u03bd t = \u22a4\n\u22a2 \u2191\u2191\u03bd t \u2260 \u22a4\n[PROOFSTEP]\nexact h.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nh : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\nh' : s \u2286 t\nh'' : \u2191\u2191(\u03bc + \u03bd) s = \u2191\u2191(\u03bc + \u03bd) t\n\u22a2 \u2191\u2191\u03bd s = \u2191\u2191\u03bd t\n[PROOFSTEP]\nrw [add_comm] at h'' h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nh : \u2191\u2191(\u03bd + \u03bc) t \u2260 \u22a4\nh' : s \u2286 t\nh'' : \u2191\u2191(\u03bd + \u03bc) s = \u2191\u2191(\u03bd + \u03bc) t\n\u22a2 \u2191\u2191\u03bd s = \u2191\u2191\u03bd t\n[PROOFSTEP]\nexact measure_eq_left_of_subset_of_measure_add_eq h h' h''\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (toMeasurable (\u03bc + \u03bd) t \u2229 s) = \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nrefine' (measure_inter_eq_of_measure_eq hs _ (subset_toMeasurable _ _) _).symm\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc t = \u2191\u2191\u03bc (toMeasurable (\u03bc + \u03bd) t)\n[PROOFSTEP]\nrefine' measure_eq_left_of_subset_of_measure_add_eq _ (subset_toMeasurable _ _) (measure_toMeasurable t).symm\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\n\u22a2 \u2191\u2191(\u03bc + \u03bd) (toMeasurable (\u03bc + \u03bd) t) \u2260 \u22a4\n[PROOFSTEP]\nrwa [measure_toMeasurable t]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc t \u2260 \u22a4\n[PROOFSTEP]\nsimp only [not_or, ENNReal.add_eq_top, Pi.add_apply, Ne.def, coe_add] at ht \n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u00ac\u2191\u2191\u03bc t = \u22a4 \u2227 \u00ac\u2191\u2191\u03bd t = \u22a4\n\u22a2 \u2191\u2191\u03bc t \u2260 \u22a4\n[PROOFSTEP]\nexact ht.1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bd (toMeasurable (\u03bc + \u03bd) t \u2229 s) = \u2191\u2191\u03bd (t \u2229 s)\n[PROOFSTEP]\nrw [add_comm] at ht \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u2191\u2191(\u03bd + \u03bc) t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bd (toMeasurable (\u03bd + \u03bc) t \u2229 s) = \u2191\u2191\u03bd (t \u2229 s)\n[PROOFSTEP]\nexact measure_toMeasurable_add_inter_left hs ht\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u22a2 \u2191\u03bc\u2081 \u2264 \u2191\u03bc\u2082 \u2194 \u03bc\u2081 \u2264 \u03bc\u2082\n[PROOFSTEP]\nrw [\u2190 \u03bc\u2082.trimmed, OuterMeasure.le_trim_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u22a2 (\u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc\u2081 s \u2264 \u2191\u2191\u03bc\u2082 s) \u2194 \u03bc\u2081 \u2264 \u03bc\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u22a2 \u00ac\u03bd \u2264 \u03bc \u2194 \u2203 s, MeasurableSet s \u2227 \u2191\u2191\u03bc s < \u2191\u2191\u03bd s\n[PROOFSTEP]\nsimp only [le_iff, not_forall, not_le, exists_prop]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u22a2 \u00ac\u03bd \u2264 \u03bc \u2194 \u2203 s, \u2191\u2191\u03bc s < \u2191\u2191\u03bd s\n[PROOFSTEP]\nsimp only [le_iff', not_forall, not_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nm : Set (Measure \u03b1)\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nrw [OuterMeasure.sInf_eq_boundedBy_sInfGen]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nm : Set (Measure \u03b1)\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nrefine' OuterMeasure.boundedBy_caratheodory fun t => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nm : Set (Measure \u03b1)\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 OuterMeasure.sInfGen (toOuterMeasure '' m) (t \u2229 s) + OuterMeasure.sInfGen (toOuterMeasure '' m) (t \\ s) \u2264\n OuterMeasure.sInfGen (toOuterMeasure '' m) t\n[PROOFSTEP]\nsimp only [OuterMeasure.sInfGen, le_iInf_iff, ball_image_iff, measure_eq_iInf t]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nm : Set (Measure \u03b1)\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2200 (x : Measure \u03b1),\n x \u2208 m \u2192\n \u2200 (i : Set \u03b1),\n t \u2286 i \u2192\n MeasurableSet i \u2192\n (\u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 toOuterMeasure '' m), \u2191\u03bc (t \u2229 s)) +\n \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 toOuterMeasure '' m), \u2191\u03bc (t \\ s) \u2264\n \u2191\u2191x i\n[PROOFSTEP]\nintro \u03bc h\u03bc u htu _hu\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nm : Set (Measure \u03b1)\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2208 m\nu : Set \u03b1\nhtu : t \u2286 u\n_hu : MeasurableSet u\n\u22a2 (\u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 toOuterMeasure '' m), \u2191\u03bc (t \u2229 s)) +\n \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 toOuterMeasure '' m), \u2191\u03bc (t \\ s) \u2264\n \u2191\u2191\u03bc u\n[PROOFSTEP]\nhave hm : \u2200 {s t}, s \u2286 t \u2192 OuterMeasure.sInfGen (toOuterMeasure '' m) s \u2264 \u03bc t :=\n by\n intro s t hst\n rw [OuterMeasure.sInfGen_def]\n refine' iInf_le_of_le \u03bc.toOuterMeasure (iInf_le_of_le (mem_image_of_mem _ h\u03bc) _)\n refine' measure_mono hst\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nm : Set (Measure \u03b1)\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2208 m\nu : Set \u03b1\nhtu : t \u2286 u\n_hu : MeasurableSet u\n\u22a2 \u2200 {s t : Set \u03b1}, s \u2286 t \u2192 OuterMeasure.sInfGen (toOuterMeasure '' m) s \u2264 \u2191\u2191\u03bc t\n[PROOFSTEP]\nintro s t hst\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d\u00b9 s' t\u271d\u00b9 : Set \u03b1\nm : Set (Measure \u03b1)\ns\u271d : Set \u03b1\nhs : MeasurableSet s\u271d\nt\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2208 m\nu : Set \u03b1\nhtu : t\u271d \u2286 u\n_hu : MeasurableSet u\ns t : Set \u03b1\nhst : s \u2286 t\n\u22a2 OuterMeasure.sInfGen (toOuterMeasure '' m) s \u2264 \u2191\u2191\u03bc t\n[PROOFSTEP]\nrw [OuterMeasure.sInfGen_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d\u00b9 s' t\u271d\u00b9 : Set \u03b1\nm : Set (Measure \u03b1)\ns\u271d : Set \u03b1\nhs : MeasurableSet s\u271d\nt\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2208 m\nu : Set \u03b1\nhtu : t\u271d \u2286 u\n_hu : MeasurableSet u\ns t : Set \u03b1\nhst : s \u2286 t\n\u22a2 \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 toOuterMeasure '' m), \u2191\u03bc s \u2264 \u2191\u2191\u03bc t\n[PROOFSTEP]\nrefine' iInf_le_of_le \u03bc.toOuterMeasure (iInf_le_of_le (mem_image_of_mem _ h\u03bc) _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d\u00b9 s' t\u271d\u00b9 : Set \u03b1\nm : Set (Measure \u03b1)\ns\u271d : Set \u03b1\nhs : MeasurableSet s\u271d\nt\u271d : Set \u03b1\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2208 m\nu : Set \u03b1\nhtu : t\u271d \u2286 u\n_hu : MeasurableSet u\ns t : Set \u03b1\nhst : s \u2286 t\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc t\n[PROOFSTEP]\nrefine' measure_mono hst\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nm : Set (Measure \u03b1)\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2208 m\nu : Set \u03b1\nhtu : t \u2286 u\n_hu : MeasurableSet u\nhm : \u2200 {s t : Set \u03b1}, s \u2286 t \u2192 OuterMeasure.sInfGen (toOuterMeasure '' m) s \u2264 \u2191\u2191\u03bc t\n\u22a2 (\u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 toOuterMeasure '' m), \u2191\u03bc (t \u2229 s)) +\n \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 toOuterMeasure '' m), \u2191\u03bc (t \\ s) \u2264\n \u2191\u2191\u03bc u\n[PROOFSTEP]\nrw [\u2190 measure_inter_add_diff u hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nm : Set (Measure \u03b1)\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u03bc : Measure \u03b1\nh\u03bc : \u03bc \u2208 m\nu : Set \u03b1\nhtu : t \u2286 u\n_hu : MeasurableSet u\nhm : \u2200 {s t : Set \u03b1}, s \u2286 t \u2192 OuterMeasure.sInfGen (toOuterMeasure '' m) s \u2264 \u2191\u2191\u03bc t\n\u22a2 (\u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 toOuterMeasure '' m), \u2191\u03bc (t \u2229 s)) +\n \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 toOuterMeasure '' m), \u2191\u03bc (t \\ s) \u2264\n \u2191\u2191\u03bc (u \u2229 s) + \u2191\u2191\u03bc (u \\ s)\n[PROOFSTEP]\nrefine' add_le_add (hm <| inter_subset_inter_left _ htu) (hm <| diff_subset_diff_left htu)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nm : Set (Measure \u03b1)\nh : \u03bc \u2208 m\nthis : sInf (toOuterMeasure '' m) \u2264 \u2191\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(sInf m) s \u2264 \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [sInf_apply hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nm : Set (Measure \u03b1)\nh : \u03bc \u2208 m\nthis : sInf (toOuterMeasure '' m) \u2264 \u2191\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191(sInf (toOuterMeasure '' m)) s \u2264 \u2191\u2191\u03bc s\n[PROOFSTEP]\nexact this s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nm : Set (Measure \u03b1)\nh : \u2200 (\u03bc' : Measure \u03b1), \u03bc' \u2208 m \u2192 \u03bc \u2264 \u03bc'\nthis : \u2191\u03bc \u2264 sInf (toOuterMeasure '' m)\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2191\u2191(sInf m) s\n[PROOFSTEP]\nrw [sInf_apply hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nm : Set (Measure \u03b1)\nh : \u2200 (\u03bc' : Measure \u03b1), \u03bc' \u2208 m \u2192 \u03bc \u2264 \u03bc'\nthis : \u2191\u03bc \u2264 sInf (toOuterMeasure '' m)\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2191(sInf (toOuterMeasure '' m)) s\n[PROOFSTEP]\nexact this s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : Set (Measure \u03b1)\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 PartialOrder (Measure \u03b1)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : Set (Measure \u03b1)\ninst\u271d : MeasurableSpace \u03b1\nsrc\u271d : PartialOrder (Measure \u03b1) := inferInstance\n\u22a2 InfSet (Measure \u03b1)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 inst\u271d \u2264 OuterMeasure.caratheodory \u22a4\n[PROOFSTEP]\nrw [OuterMeasure.top_caratheodory]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 inst\u271d \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u22a4 s \u2264 \u2191\u2191(OuterMeasure.toMeasure \u22a4 (_ : inst\u271d \u2264 OuterMeasure.caratheodory \u22a4)) s\n[PROOFSTEP]\ncases' s.eq_empty_or_nonempty with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nh : s = \u2205\n\u22a2 \u2191\u2191\u22a4 s \u2264 \u2191\u2191(OuterMeasure.toMeasure \u22a4 (_ : inst\u271d \u2264 OuterMeasure.caratheodory \u22a4)) s\n[PROOFSTEP]\nsimp [h, toMeasure_apply \u22a4 _ hs, OuterMeasure.top_apply]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nh : Set.Nonempty s\n\u22a2 \u2191\u2191\u22a4 s \u2264 \u2191\u2191(OuterMeasure.toMeasure \u22a4 (_ : inst\u271d \u2264 OuterMeasure.caratheodory \u22a4)) s\n[PROOFSTEP]\nsimp [h, toMeasure_apply \u22a4 _ hs, OuterMeasure.top_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 \u2191\u22a4 = \u22a4\n[PROOFSTEP]\nrw [\u2190 OuterMeasure.toMeasure_top, toMeasure_toOuterMeasure, OuterMeasure.trim_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081\u271d \u03bc\u2082\u271d \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nm0 : MeasurableSpace \u03b1\nf : OuterMeasure \u03b1 \u2192\u2097[\u211d\u22650\u221e] OuterMeasure \u03b2\nhf : \u2200 (\u03bc : Measure \u03b1), inst\u271d\u00b9 \u2264 OuterMeasure.caratheodory (\u2191f \u2191\u03bc)\n\u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191((fun \u03bc => OuterMeasure.toMeasure (\u2191f \u2191\u03bc) (_ : inst\u271d\u00b9 \u2264 OuterMeasure.caratheodory (\u2191f \u2191\u03bc))) (\u03bc\u2081 + \u03bc\u2082)) s =\n \u2191\u2191((fun \u03bc => OuterMeasure.toMeasure (\u2191f \u2191\u03bc) (_ : inst\u271d\u00b9 \u2264 OuterMeasure.caratheodory (\u2191f \u2191\u03bc))) \u03bc\u2081 +\n (fun \u03bc => OuterMeasure.toMeasure (\u2191f \u2191\u03bc) (_ : inst\u271d\u00b9 \u2264 OuterMeasure.caratheodory (\u2191f \u2191\u03bc))) \u03bc\u2082)\n s\n[PROOFSTEP]\nsimp only [map_add, coe_add, Pi.add_apply, toMeasure_apply, add_toOuterMeasure, OuterMeasure.coe_add, hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nm0 : MeasurableSpace \u03b1\nf : OuterMeasure \u03b1 \u2192\u2097[\u211d\u22650\u221e] OuterMeasure \u03b2\nhf : \u2200 (\u03bc : Measure \u03b1), inst\u271d\u00b9 \u2264 OuterMeasure.caratheodory (\u2191f \u2191\u03bc)\nc : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(AddHom.toFun\n { toFun := fun \u03bc => OuterMeasure.toMeasure (\u2191f \u2191\u03bc) (_ : inst\u271d\u00b9 \u2264 OuterMeasure.caratheodory (\u2191f \u2191\u03bc)),\n map_add' :=\n (_ :\n \u2200 (\u03bc\u2081 \u03bc\u2082 : Measure \u03b1),\n (fun \u03bc => OuterMeasure.toMeasure (\u2191f \u2191\u03bc) (_ : inst\u271d\u00b9 \u2264 OuterMeasure.caratheodory (\u2191f \u2191\u03bc)))\n (\u03bc\u2081 + \u03bc\u2082) =\n (fun \u03bc => OuterMeasure.toMeasure (\u2191f \u2191\u03bc) (_ : inst\u271d\u00b9 \u2264 OuterMeasure.caratheodory (\u2191f \u2191\u03bc))) \u03bc\u2081 +\n (fun \u03bc => OuterMeasure.toMeasure (\u2191f \u2191\u03bc) (_ : inst\u271d\u00b9 \u2264 OuterMeasure.caratheodory (\u2191f \u2191\u03bc))) \u03bc\u2082) }\n (c \u2022 \u03bc))\n s =\n \u2191\u2191(\u2191(RingHom.id \u211d\u22650\u221e) c \u2022\n AddHom.toFun\n { toFun := fun \u03bc => OuterMeasure.toMeasure (\u2191f \u2191\u03bc) (_ : inst\u271d\u00b9 \u2264 OuterMeasure.caratheodory (\u2191f \u2191\u03bc)),\n map_add' :=\n (_ :\n \u2200 (\u03bc\u2081 \u03bc\u2082 : Measure \u03b1),\n (fun \u03bc => OuterMeasure.toMeasure (\u2191f \u2191\u03bc) (_ : inst\u271d\u00b9 \u2264 OuterMeasure.caratheodory (\u2191f \u2191\u03bc)))\n (\u03bc\u2081 + \u03bc\u2082) =\n (fun \u03bc => OuterMeasure.toMeasure (\u2191f \u2191\u03bc) (_ : inst\u271d\u00b9 \u2264 OuterMeasure.caratheodory (\u2191f \u2191\u03bc))) \u03bc\u2081 +\n (fun \u03bc => OuterMeasure.toMeasure (\u2191f \u2191\u03bc) (_ : inst\u271d\u00b9 \u2264 OuterMeasure.caratheodory (\u2191f \u2191\u03bc)))\n \u03bc\u2082) }\n \u03bc)\n s\n[PROOFSTEP]\nsimp only [LinearMap.map_smul\u209b\u2097, coe_smul, Pi.smul_apply, toMeasure_apply, smul_toOuterMeasure (R := \u211d\u22650\u221e),\n OuterMeasure.coe_smul (R := \u211d\u22650\u221e), smul_apply, hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\nh : f =\u1da0[ae \u03bc] g\n\u22a2 \u2191(map\u2097 f) \u03bc = \u2191(map\u2097 g) \u03bc\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nhf : Measurable f\nhg : Measurable g\nh : f =\u1da0[ae \u03bc] g\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(map\u2097 f) \u03bc) s = \u2191\u2191(\u2191(map\u2097 g) \u03bc) s\n[PROOFSTEP]\nsimpa only [map\u2097, hf, hg, hs, dif_pos, liftLinear_apply, OuterMeasure.map_apply] using measure_congr (h.preimage s)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 \u2191(map\u2097 (AEMeasurable.mk f hf)) \u03bc = map f \u03bc\n[PROOFSTEP]\nsimp [map, hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\n\u03bc : Measure \u03b1\n\u22a2 \u2191(map\u2097 f) \u03bc = map f \u03bc\n[PROOFSTEP]\nsimp only [\u2190 map\u2097_mk_apply_of_aemeasurable hf.aemeasurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\n\u03bc : Measure \u03b1\n\u22a2 \u2191(map\u2097 f) \u03bc = \u2191(map\u2097 (AEMeasurable.mk f (_ : AEMeasurable f))) \u03bc\n[PROOFSTEP]\nexact map\u2097_congr hf hf.aemeasurable.measurable_mk hf.aemeasurable.ae_eq_mk\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\n\u22a2 map f (\u03bc + \u03bd) = map f \u03bc + map f \u03bd\n[PROOFSTEP]\nsimp [\u2190 map\u2097_apply_of_measurable hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 map f 0 = 0\n[PROOFSTEP]\nby_cases hf : AEMeasurable f (0 : Measure \u03b1)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 map f 0 = 0\n[PROOFSTEP]\nsimp [map, hf]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u00acAEMeasurable f\n\u22a2 map f 0 = 0\n[PROOFSTEP]\nsimp [map, hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\nhf : \u00acAEMeasurable f\n\u22a2 map f \u03bc = 0\n[PROOFSTEP]\nsimp [map, hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nh : f =\u1da0[ae \u03bc] g\n\u22a2 map f \u03bc = map g \u03bc\n[PROOFSTEP]\nby_cases hf : AEMeasurable f \u03bc\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nh : f =\u1da0[ae \u03bc] g\nhf : AEMeasurable f\n\u22a2 map f \u03bc = map g \u03bc\n[PROOFSTEP]\nhave hg : AEMeasurable g \u03bc := hf.congr h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nh : f =\u1da0[ae \u03bc] g\nhf : AEMeasurable f\nhg : AEMeasurable g\n\u22a2 map f \u03bc = map g \u03bc\n[PROOFSTEP]\nsimp only [\u2190 map\u2097_mk_apply_of_aemeasurable hf, \u2190 map\u2097_mk_apply_of_aemeasurable hg]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nh : f =\u1da0[ae \u03bc] g\nhf : AEMeasurable f\nhg : AEMeasurable g\n\u22a2 \u2191(map\u2097 (AEMeasurable.mk f hf)) \u03bc = \u2191(map\u2097 (AEMeasurable.mk g hg)) \u03bc\n[PROOFSTEP]\nexact map\u2097_congr hf.measurable_mk hg.measurable_mk (hf.ae_eq_mk.symm.trans (h.trans hg.ae_eq_mk))\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nh : f =\u1da0[ae \u03bc] g\nhf : \u00acAEMeasurable f\n\u22a2 map f \u03bc = map g \u03bc\n[PROOFSTEP]\nhave hg : \u00acAEMeasurable g \u03bc := by simpa [\u2190 aemeasurable_congr h] using hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nh : f =\u1da0[ae \u03bc] g\nhf : \u00acAEMeasurable f\n\u22a2 \u00acAEMeasurable g\n[PROOFSTEP]\nsimpa [\u2190 aemeasurable_congr h] using hf\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nh : f =\u1da0[ae \u03bc] g\nhf : \u00acAEMeasurable f\nhg : \u00acAEMeasurable g\n\u22a2 map f \u03bc = map g \u03bc\n[PROOFSTEP]\nsimp [map_of_not_aemeasurable, hf, hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nc : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 map f (c \u2022 \u03bc) = c \u2022 map f \u03bc\n[PROOFSTEP]\nrcases eq_or_ne c 0 with (rfl | hc)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 map f (0 \u2022 \u03bc) = 0 \u2022 map f \u03bc\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nc : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhc : c \u2260 0\n\u22a2 map f (c \u2022 \u03bc) = c \u2022 map f \u03bc\n[PROOFSTEP]\nby_cases hf : AEMeasurable f \u03bc\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nc : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhc : c \u2260 0\nhf : AEMeasurable f\n\u22a2 map f (c \u2022 \u03bc) = c \u2022 map f \u03bc\n[PROOFSTEP]\nhave hfc : AEMeasurable f (c \u2022 \u03bc) := \u27e8hf.mk f, hf.measurable_mk, (ae_smul_measure_iff hc).2 hf.ae_eq_mk\u27e9\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nc : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhc : c \u2260 0\nhf : AEMeasurable f\nhfc : AEMeasurable f\n\u22a2 map f (c \u2022 \u03bc) = c \u2022 map f \u03bc\n[PROOFSTEP]\nsimp only [\u2190 map\u2097_mk_apply_of_aemeasurable hf, \u2190 map\u2097_mk_apply_of_aemeasurable hfc, LinearMap.map_smul\u209b\u2097,\n RingHom.id_apply]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nc : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhc : c \u2260 0\nhf : AEMeasurable f\nhfc : AEMeasurable f\n\u22a2 c \u2022 \u2191(map\u2097 (AEMeasurable.mk f hfc)) \u03bc = c \u2022 \u2191(map\u2097 (AEMeasurable.mk f hf)) \u03bc\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase pos.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nc : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhc : c \u2260 0\nhf : AEMeasurable f\nhfc : AEMeasurable f\n\u22a2 \u2191(map\u2097 (AEMeasurable.mk f hfc)) \u03bc = \u2191(map\u2097 (AEMeasurable.mk f hf)) \u03bc\n[PROOFSTEP]\napply map\u2097_congr hfc.measurable_mk hf.measurable_mk\n[GOAL]\ncase pos.e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nc : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhc : c \u2260 0\nhf : AEMeasurable f\nhfc : AEMeasurable f\n\u22a2 AEMeasurable.mk f hfc =\u1da0[ae \u03bc] AEMeasurable.mk f hf\n[PROOFSTEP]\nexact EventuallyEq.trans ((ae_smul_measure_iff hc).1 hfc.ae_eq_mk.symm) hf.ae_eq_mk\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nc : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhc : c \u2260 0\nhf : \u00acAEMeasurable f\n\u22a2 map f (c \u2022 \u03bc) = c \u2022 map f \u03bc\n[PROOFSTEP]\nhave hfc : \u00acAEMeasurable f (c \u2022 \u03bc) := by\n intro hfc\n exact hf \u27e8hfc.mk f, hfc.measurable_mk, (ae_smul_measure_iff hc).1 hfc.ae_eq_mk\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nc : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhc : c \u2260 0\nhf : \u00acAEMeasurable f\n\u22a2 \u00acAEMeasurable f\n[PROOFSTEP]\nintro hfc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nc : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhc : c \u2260 0\nhf : \u00acAEMeasurable f\nhfc : AEMeasurable f\n\u22a2 False\n[PROOFSTEP]\nexact hf \u27e8hfc.mk f, hfc.measurable_mk, (ae_smul_measure_iff hc).1 hfc.ae_eq_mk\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nc : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhc : c \u2260 0\nhf : \u00acAEMeasurable f\nhfc : \u00acAEMeasurable f\n\u22a2 map f (c \u2022 \u03bc) = c \u2022 map f \u03bc\n[PROOFSTEP]\nsimp [map_of_not_aemeasurable hf, map_of_not_aemeasurable hfc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(map f \u03bc) s = \u2191\u2191\u03bc (f \u207b\u00b9' s)\n[PROOFSTEP]\nsimpa only [map\u2097, hf.measurable_mk, hs, dif_pos, liftLinear_apply, OuterMeasure.map_apply, \u2190\n map\u2097_mk_apply_of_aemeasurable hf] using measure_congr (hf.ae_eq_mk.symm.preimage s)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 \u2191(map f \u03bc) = OuterMeasure.trim (\u2191(OuterMeasure.map f) \u2191\u03bc)\n[PROOFSTEP]\nrw [\u2190 trimmed, OuterMeasure.trim_eq_trim_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 \u2200 (s : Set \u03b2), MeasurableSet s \u2192 \u2191\u2191(map f \u03bc) s = \u2191(\u2191(OuterMeasure.map f) \u2191\u03bc) s\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(map f \u03bc) s = \u2191(\u2191(OuterMeasure.map f) \u2191\u03bc) s\n[PROOFSTEP]\nrw [map_apply_of_aemeasurable hf hs, OuterMeasure.map_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : Measurable g\nhf : Measurable f\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(map g (map f \u03bc)) s = \u2191\u2191(map (g \u2218 f) \u03bc) s\n[PROOFSTEP]\nsimp [hf, hg, hs, hg hs, hg.comp hf, \u2190 preimage_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nh : \u03bc \u2264 \u03bd\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(map f \u03bc) s \u2264 \u2191\u2191(map f \u03bd) s\n[PROOFSTEP]\nsimp [hf.aemeasurable, hs, h _ (hf hs)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Injective f \u2227 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 MeasurableSet (f '' s)\n\u03bc : Measure \u03b2\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191(\u2191(OuterMeasure.comap f) \u2191\u03bc) t = \u2191(\u2191(OuterMeasure.comap f) \u2191\u03bc) (t \u2229 s) + \u2191(\u2191(OuterMeasure.comap f) \u2191\u03bc) (t \\ s)\n[PROOFSTEP]\nsimp only [OuterMeasure.comap_apply, image_inter hf.1, image_diff hf.1]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Injective f \u2227 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 MeasurableSet (f '' s)\n\u03bc : Measure \u03b2\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191\u2191\u03bc (f '' t) = \u2191\u2191\u03bc (f '' t \u2229 f '' s) + \u2191\u2191\u03bc (f '' t \\ f '' s)\n[PROOFSTEP]\napply le_toOuterMeasure_caratheodory\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Injective f \u2227 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 MeasurableSet (f '' s)\n\u03bc : Measure \u03b2\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 MeasurableSet (f '' s)\n[PROOFSTEP]\nexact hf.2 s hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 MeasurableSet (f '' s)\n\u03bc : Measure \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(comap\u2097 f) \u03bc) s = \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nrw [comap\u2097, dif_pos, liftLinear_apply _ hs, OuterMeasure.comap_apply]\n[GOAL]\ncase hc\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 MeasurableSet (f '' s)\n\u03bc : Measure \u03b2\nhs : MeasurableSet s\n\u22a2 Injective f \u2227 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 MeasurableSet (f '' s)\ncase hc\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 MeasurableSet (f '' s)\n\u03bc : Measure \u03b2\nhs : MeasurableSet s\n\u22a2 Injective f \u2227 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 MeasurableSet (f '' s)\n[PROOFSTEP]\nexact \u27e8hfi, hf\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhf : Injective f \u2227 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191(\u2191(OuterMeasure.comap f) \u2191\u03bc) t = \u2191(\u2191(OuterMeasure.comap f) \u2191\u03bc) (t \u2229 s) + \u2191(\u2191(OuterMeasure.comap f) \u2191\u03bc) (t \\ s)\n[PROOFSTEP]\nsimp only [OuterMeasure.comap_apply, image_inter hf.1, image_diff hf.1]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhf : Injective f \u2227 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191\u2191\u03bc (f '' t) = \u2191\u2191\u03bc (f '' t \u2229 f '' s) + \u2191\u2191\u03bc (f '' t \\ f '' s)\n[PROOFSTEP]\nexact (measure_inter_add_diff\u2080 _ (hf.2 s hs)).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\nhs : NullMeasurableSet s\n\u22a2 \u2191\u2191(comap f \u03bc) s = \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nrw [comap, dif_pos (And.intro hfi hf)] at hs \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\nhs : NullMeasurableSet s\n\u22a2 \u2191\u2191(OuterMeasure.toMeasure (\u2191(OuterMeasure.comap f) \u2191\u03bc)\n (_ :\n \u2200 (s : Set \u03b1),\n MeasurableSet s \u2192\n \u2200 (t : Set \u03b1),\n \u2191(\u2191(OuterMeasure.comap f) \u2191\u03bc) t =\n \u2191(\u2191(OuterMeasure.comap f) \u2191\u03bc) (t \u2229 s) + \u2191(\u2191(OuterMeasure.comap f) \u2191\u03bc) (t \\ s)))\n s =\n \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nrw [toMeasure_apply\u2080 _ _ hs, OuterMeasure.comap_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns : Set \u03b1\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u2191(comap f \u03bc) s\n[PROOFSTEP]\nrw [comap, dif_pos (And.intro hfi hf)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns : Set \u03b1\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264\n \u2191\u2191(OuterMeasure.toMeasure (\u2191(OuterMeasure.comap f) \u2191\u03bc)\n (_ :\n \u2200 (s : Set \u03b1),\n MeasurableSet s \u2192\n \u2200 (t : Set \u03b1),\n \u2191(\u2191(OuterMeasure.comap f) \u2191\u03bc) t =\n \u2191(\u2191(OuterMeasure.comap f) \u2191\u03bc) (t \u2229 s) + \u2191(\u2191(OuterMeasure.comap f) \u2191\u03bc) (t \\ s)))\n s\n[PROOFSTEP]\nexact le_toMeasure_apply _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : s =\u1da0[ae (comap f \u03bc)] t\n\u22a2 f '' s =\u1da0[ae \u03bc] f '' t\n[PROOFSTEP]\nrw [EventuallyEq, ae_iff] at hst \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\n\u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0\n[PROOFSTEP]\nhave h_eq_\u03b1 : {a : \u03b1 | \u00acs a = t a} = s \\ t \u222a t \\ s := by\n ext1 x\n simp only [eq_iff_iff, mem_setOf_eq, mem_union, mem_diff]\n tauto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\n\u22a2 {a | \u00acs a = t a} = s \\ t \u222a t \\ s\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nx : \u03b1\n\u22a2 x \u2208 {a | \u00acs a = t a} \u2194 x \u2208 s \\ t \u222a t \\ s\n[PROOFSTEP]\nsimp only [eq_iff_iff, mem_setOf_eq, mem_union, mem_diff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nx : \u03b1\n\u22a2 \u00ac(s x \u2194 t x) \u2194 x \u2208 s \u2227 \u00acx \u2208 t \u2228 x \u2208 t \u2227 \u00acx \u2208 s\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\n\u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0\n[PROOFSTEP]\nhave h_eq_\u03b2 : {a : \u03b2 | \u00ac(f '' s) a = (f '' t) a} = f '' s \\ f '' t \u222a f '' t \\ f '' s :=\n by\n ext1 x\n simp only [eq_iff_iff, mem_setOf_eq, mem_union, mem_diff]\n tauto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\n\u22a2 {a | \u00ac(f '' s) a = (f '' t) a} = f '' s \\ f '' t \u222a f '' t \\ f '' s\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nx : \u03b2\n\u22a2 x \u2208 {a | \u00ac(f '' s) a = (f '' t) a} \u2194 x \u2208 f '' s \\ f '' t \u222a f '' t \\ f '' s\n[PROOFSTEP]\nsimp only [eq_iff_iff, mem_setOf_eq, mem_union, mem_diff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nx : \u03b2\n\u22a2 \u00ac((f '' s) x \u2194 (f '' t) x) \u2194 x \u2208 f '' s \u2227 \u00acx \u2208 f '' t \u2228 x \u2208 f '' t \u2227 \u00acx \u2208 f '' s\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nh_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' s \\ f '' t \u222a f '' t \\ f '' s\n\u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0\n[PROOFSTEP]\nrw [\u2190 Set.image_diff hfi, \u2190 Set.image_diff hfi, \u2190 Set.image_union] at h_eq_\u03b2 \n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nh_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' (s \\ t \u222a t \\ s)\n\u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0\n[PROOFSTEP]\nrw [h_eq_\u03b2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nh_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' (s \\ t \u222a t \\ s)\n\u22a2 \u2191\u2191\u03bc (f '' (s \\ t \u222a t \\ s)) = 0\n[PROOFSTEP]\nrw [h_eq_\u03b1] at hst \n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns t : Set \u03b1\nhst : \u2191\u2191(comap f \u03bc) (s \\ t \u222a t \\ s) = 0\nh_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s\nh_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' (s \\ t \u222a t \\ s)\n\u22a2 \u2191\u2191\u03bc (f '' (s \\ t \u222a t \\ s)) = 0\n[PROOFSTEP]\nexact measure_image_eq_zero_of_comap_eq_zero f \u03bc hfi hf hst\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 NullMeasurableSet (f '' s)\n[PROOFSTEP]\nrefine' \u27e8toMeasurable \u03bc (f '' toMeasurable (\u03bc.comap f) s), measurableSet_toMeasurable _ _, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 f '' s =\u1da0[ae \u03bc] toMeasurable \u03bc (f '' toMeasurable (comap f \u03bc) s)\n[PROOFSTEP]\nrefine' EventuallyEq.trans _ (NullMeasurableSet.toMeasurable_ae_eq _).symm\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 f '' s =\u1da0[ae \u03bc] f '' toMeasurable (comap f \u03bc) s\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 NullMeasurableSet (f '' toMeasurable (comap f \u03bc) s)\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 NullMeasurableSet (f '' toMeasurable (comap f \u03bc) s)\n[PROOFSTEP]\nexact hf _ (measurableSet_toMeasurable _ _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 f '' s =\u1da0[ae \u03bc] f '' toMeasurable (comap f \u03bc) s\n[PROOFSTEP]\nhave h : toMeasurable (comap f \u03bc) s =\u1d50[comap f \u03bc] s :=\n @NullMeasurableSet.toMeasurable_ae_eq _ _ (\u03bc.comap f : Measure \u03b1) s hs\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\nhfi : Injective f\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s)\ns : Set \u03b1\nhs : NullMeasurableSet s\nh : toMeasurable (comap f \u03bc) s =\u1da0[ae (comap f \u03bc)] s\n\u22a2 f '' s =\u1da0[ae \u03bc] f '' toMeasurable (comap f \u03bc) s\n[PROOFSTEP]\nexact ae_eq_image_of_ae_eq_comap f \u03bc hfi hf h.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b2\ns : Set \u03b2\nhf : Injective f\nhf' : Measurable f\nh : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 NullMeasurableSet (f '' t)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(comap f \u03bc) (f \u207b\u00b9' s) = \u2191\u2191\u03bc (s \u2229 range f)\n[PROOFSTEP]\nrw [comap_apply\u2080 _ _ hf h (hf' hs).nullMeasurableSet, image_preimage_eq_inter_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 NullMeasurableSet (Subtype.val '' t)\n[PROOFSTEP]\nrw [Subtype.instMeasurableSpace, comap_eq_generateFrom] at ht \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 NullMeasurableSet (Subtype.val '' t)\n[PROOFSTEP]\nrefine'\n generateFrom_induction (p := fun t : Set s => NullMeasurableSet ((\u2191) '' t) \u03bc)\n {t : Set s | \u2203 s' : Set \u03b1, MeasurableSet s' \u2227 (\u2191) \u207b\u00b9' s' = t} _ _ _ _ ht\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u2191s),\n t \u2208 {t | \u2203 s', MeasurableSet s' \u2227 Subtype.val \u207b\u00b9' s' = t} \u2192 (fun t => NullMeasurableSet (Subtype.val '' t)) t\n[PROOFSTEP]\nrintro t' \u27e8s', hs', rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s'\u271d t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\ns' : Set \u03b1\nhs' : MeasurableSet s'\n\u22a2 NullMeasurableSet (Subtype.val '' (Subtype.val \u207b\u00b9' s'))\n[PROOFSTEP]\nrw [Subtype.image_preimage_coe]\n[GOAL]\ncase refine'_1.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s'\u271d t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\ns' : Set \u03b1\nhs' : MeasurableSet s'\n\u22a2 NullMeasurableSet (s' \u2229 s)\n[PROOFSTEP]\nexact hs'.nullMeasurableSet.inter hs\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 (fun t => NullMeasurableSet (Subtype.val '' t)) \u2205\n[PROOFSTEP]\nsimp only [image_empty, nullMeasurableSet_empty]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2200 (t : Set \u2191s), (fun t => NullMeasurableSet (Subtype.val '' t)) t \u2192 (fun t => NullMeasurableSet (Subtype.val '' t)) t\u1d9c\n[PROOFSTEP]\nintro t'\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nt' : Set \u2191s\n\u22a2 (fun t => NullMeasurableSet (Subtype.val '' t)) t' \u2192 (fun t => NullMeasurableSet (Subtype.val '' t)) t'\u1d9c\n[PROOFSTEP]\nsimp only [\u2190 range_diff_image Subtype.coe_injective, Subtype.range_coe_subtype, setOf_mem_eq]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nt' : Set \u2191s\n\u22a2 NullMeasurableSet (Subtype.val '' t') \u2192 NullMeasurableSet (s \\ (fun a => \u2191a) '' t')\n[PROOFSTEP]\nexact hs.diff\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u2191s),\n (\u2200 (n : \u2115), (fun t => NullMeasurableSet (Subtype.val '' t)) (f n)) \u2192\n (fun t => NullMeasurableSet (Subtype.val '' t)) (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nintro f\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u2191s\n\u22a2 (\u2200 (n : \u2115), (fun t => NullMeasurableSet (Subtype.val '' t)) (f n)) \u2192\n (fun t => NullMeasurableSet (Subtype.val '' t)) (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\ndsimp only []\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u2191s\n\u22a2 (\u2200 (n : \u2115), NullMeasurableSet (Subtype.val '' f n)) \u2192 NullMeasurableSet (Subtype.val '' \u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nrw [image_iUnion]\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : Set \u2191s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nf : \u2115 \u2192 Set \u2191s\n\u22a2 (\u2200 (n : \u2115), NullMeasurableSet (Subtype.val '' f n)) \u2192 NullMeasurableSet (\u22c3 (i : \u2115), Subtype.val '' f i)\n[PROOFSTEP]\nexact NullMeasurableSet.iUnion\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ns : Set \u03b1\ninst\u271d : MeasureSpace \u03b1\np : \u03b1 \u2192 Prop\nhs : NullMeasurableSet s\n\u22a2 \u2191\u2191volume univ = \u2191\u2191volume s\n[PROOFSTEP]\nrw [Subtype.volume_def, comap_apply\u2080 _ _ _ _ MeasurableSet.univ.nullMeasurableSet]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ns : Set \u03b1\ninst\u271d : MeasureSpace \u03b1\np : \u03b1 \u2192 Prop\nhs : NullMeasurableSet s\n\u22a2 \u2191\u2191volume (Subtype.val '' univ) = \u2191\u2191volume s\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ns : Set \u03b1\ninst\u271d : MeasureSpace \u03b1\np : \u03b1 \u2192 Prop\nhs : NullMeasurableSet s\n\u22a2 Subtype.val '' univ = s\n[PROOFSTEP]\nsimp only [image_univ, Subtype.range_coe_subtype, setOf_mem_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ns : Set \u03b1\ninst\u271d : MeasureSpace \u03b1\np : \u03b1 \u2192 Prop\nhs : NullMeasurableSet s\n\u22a2 Injective Subtype.val\n[PROOFSTEP]\nexact Subtype.coe_injective\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ns : Set \u03b1\ninst\u271d : MeasureSpace \u03b1\np : \u03b1 \u2192 Prop\nhs : NullMeasurableSet s\n\u22a2 \u2200 (s_1 : Set \u2191s), MeasurableSet s_1 \u2192 NullMeasurableSet (Subtype.val '' s_1)\n[PROOFSTEP]\nexact fun t => MeasurableSet.nullMeasurableSet_subtype_coe hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s'\u271d t\u271d : Set \u03b1\nm0 : MeasurableSpace \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\ns' : Set \u03b1\nhs' : MeasurableSet s'\nt : Set \u03b1\n\u22a2 \u2191(\u2191(OuterMeasure.restrict s) \u2191\u03bc) t =\n \u2191(\u2191(OuterMeasure.restrict s) \u2191\u03bc) (t \u2229 s') + \u2191(\u2191(OuterMeasure.restrict s) \u2191\u03bc) (t \\ s')\n[PROOFSTEP]\nsuffices \u03bc (s \u2229 t) = \u03bc (s \u2229 t \u2229 s') + \u03bc ((s \u2229 t) \\ s') by\n simpa [\u2190 Set.inter_assoc, Set.inter_comm _ s, \u2190 inter_diff_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s'\u271d t\u271d : Set \u03b1\nm0 : MeasurableSpace \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\ns' : Set \u03b1\nhs' : MeasurableSet s'\nt : Set \u03b1\nthis : \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bc (s \u2229 t \u2229 s') + \u2191\u2191\u03bc ((s \u2229 t) \\ s')\n\u22a2 \u2191(\u2191(OuterMeasure.restrict s) \u2191\u03bc) t =\n \u2191(\u2191(OuterMeasure.restrict s) \u2191\u03bc) (t \u2229 s') + \u2191(\u2191(OuterMeasure.restrict s) \u2191\u03bc) (t \\ s')\n[PROOFSTEP]\nsimpa [\u2190 Set.inter_assoc, Set.inter_comm _ s, \u2190 inter_diff_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s'\u271d t\u271d : Set \u03b1\nm0 : MeasurableSpace \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\ns' : Set \u03b1\nhs' : MeasurableSet s'\nt : Set \u03b1\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bc (s \u2229 t \u2229 s') + \u2191\u2191\u03bc ((s \u2229 t) \\ s')\n[PROOFSTEP]\nexact le_toOuterMeasure_caratheodory _ _ hs' _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nh : MeasurableSet s\n\u22a2 \u2191(restrict \u03bc s) = \u2191(OuterMeasure.restrict s) \u2191\u03bc\n[PROOFSTEP]\nsimp_rw [restrict, restrict\u2097, liftLinear, LinearMap.coe_mk, AddHom.coe_mk, toMeasure_toOuterMeasure,\n OuterMeasure.restrict_trim h, \u03bc.trimmed]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s'\u271d t\u271d : Set \u03b1\nht : NullMeasurableSet t\u271d\ns' : Set \u03b1\nhs' : MeasurableSet s'\nt : Set \u03b1\n\u22a2 \u2191(\u2191(OuterMeasure.restrict s) \u2191\u03bc) t =\n \u2191(\u2191(OuterMeasure.restrict s) \u2191\u03bc) (t \u2229 s') + \u2191(\u2191(OuterMeasure.restrict s) \u2191\u03bc) (t \\ s')\n[PROOFSTEP]\nsuffices \u03bc (s \u2229 t) = \u03bc (s \u2229 t \u2229 s') + \u03bc ((s \u2229 t) \\ s') by\n simpa [\u2190 Set.inter_assoc, Set.inter_comm _ s, \u2190 inter_diff_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s'\u271d t\u271d : Set \u03b1\nht : NullMeasurableSet t\u271d\ns' : Set \u03b1\nhs' : MeasurableSet s'\nt : Set \u03b1\nthis : \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bc (s \u2229 t \u2229 s') + \u2191\u2191\u03bc ((s \u2229 t) \\ s')\n\u22a2 \u2191(\u2191(OuterMeasure.restrict s) \u2191\u03bc) t =\n \u2191(\u2191(OuterMeasure.restrict s) \u2191\u03bc) (t \u2229 s') + \u2191(\u2191(OuterMeasure.restrict s) \u2191\u03bc) (t \\ s')\n[PROOFSTEP]\nsimpa [\u2190 Set.inter_assoc, Set.inter_comm _ s, \u2190 inter_diff_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s'\u271d t\u271d : Set \u03b1\nht : NullMeasurableSet t\u271d\ns' : Set \u03b1\nhs' : MeasurableSet s'\nt : Set \u03b1\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bc (s \u2229 t \u2229 s') + \u2191\u2191\u03bc ((s \u2229 t) \\ s')\n[PROOFSTEP]\nexact le_toOuterMeasure_caratheodory _ _ hs' _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nht : NullMeasurableSet t\n\u22a2 \u2191(\u2191(OuterMeasure.restrict s) \u2191\u03bc) t = \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nsimp only [OuterMeasure.restrict_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(restrict \u03bc s) t = \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nrw [Measure.restrict_toOuterMeasure_eq_toOuterMeasure_restrict hs, OuterMeasure.restrict_apply s t _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 \u2191\u2191(restrict \u03bc s) t = \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nrw [\u2190 restrict_congr_set hs.toMeasurable_ae_eq, restrict_apply' (measurableSet_toMeasurable _ _),\n measure_congr ((ae_eq_refl t).inter hs.toMeasurable_ae_eq)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nh : s \u2286 t\n\u22a2 \u2191\u2191\u03bc (toMeasurable (restrict \u03bc t) s \u2229 t) = \u2191\u2191(restrict \u03bc t) s\n[PROOFSTEP]\nrw [\u2190 restrict_apply (measurableSet_toMeasurable _ _), measure_toMeasurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t s : Set \u03b1\n\u22a2 \u2191\u2191(restrict \u03bc s) univ = \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [restrict_apply MeasurableSet.univ, Set.univ_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : NullMeasurableSet s\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(restrict (restrict \u03bc t) s) u = \u2191\u2191(restrict \u03bc (s \u2229 t)) u\n[PROOFSTEP]\nsimp only [Set.inter_assoc, restrict_apply hu, restrict_apply\u2080 (hu.nullMeasurableSet.inter hs)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nh : s \u2286 t\n\u22a2 restrict (restrict \u03bc t) s = restrict \u03bc s\n[PROOFSTEP]\next1 u hu\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nh : s \u2286 t\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(restrict (restrict \u03bc t) s) u = \u2191\u2191(restrict \u03bc s) u\n[PROOFSTEP]\nrw [restrict_apply hu, restrict_apply hu, restrict_eq_self]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nh : s \u2286 t\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 u \u2229 s \u2286 t\n[PROOFSTEP]\nexact (inter_subset_right _ _).trans h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nht : NullMeasurableSet t\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(restrict (restrict \u03bc t) s) u = \u2191\u2191(restrict \u03bc (s \u2229 t)) u\n[PROOFSTEP]\nsimp only [restrict_apply hu, restrict_apply\u2080' ht, inter_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\n\u22a2 restrict (restrict \u03bc t) s = restrict (restrict \u03bc s) t\n[PROOFSTEP]\nrw [restrict_restrict hs, restrict_restrict' hs, inter_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(restrict \u03bc s) t = 0 \u2194 \u2191\u2191\u03bc (t \u2229 s) = 0\n[PROOFSTEP]\nrw [restrict_apply ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(restrict \u03bc s) t = 0 \u2194 \u2191\u2191\u03bc (t \u2229 s) = 0\n[PROOFSTEP]\nrw [restrict_apply' hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u22a2 restrict \u03bc s = 0 \u2194 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nrw [\u2190 measure_univ_eq_zero, restrict_apply_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t s : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(restrict \u03bc univ) s = \u2191\u2191\u03bc s\n[PROOFSTEP]\nsimp [hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t s : Set \u03b1\nht : NullMeasurableSet t\n\u22a2 restrict \u03bc (s \u2229 t) + restrict \u03bc (s \\ t) = restrict \u03bc s\n[PROOFSTEP]\next1 u hu\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t s : Set \u03b1\nht : NullMeasurableSet t\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(restrict \u03bc (s \u2229 t) + restrict \u03bc (s \\ t)) u = \u2191\u2191(restrict \u03bc s) u\n[PROOFSTEP]\nsimp only [add_apply, restrict_apply hu, \u2190 inter_assoc, diff_eq]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t s : Set \u03b1\nht : NullMeasurableSet t\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03bc (u \u2229 s \u2229 t) + \u2191\u2191\u03bc (u \u2229 s \u2229 t\u1d9c) = \u2191\u2191\u03bc (u \u2229 s)\n[PROOFSTEP]\nexact measure_inter_add_diff\u2080 (u \u2229 s) ht\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t s : Set \u03b1\nht : NullMeasurableSet t\n\u22a2 restrict \u03bc (s \u222a t) + restrict \u03bc (s \u2229 t) = restrict \u03bc s + restrict \u03bc t\n[PROOFSTEP]\nrw [\u2190 restrict_inter_add_diff\u2080 (s \u222a t) ht, union_inter_cancel_right, union_diff_right, \u2190 restrict_inter_add_diff\u2080 s ht,\n add_comm, \u2190 add_assoc, add_right_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 restrict \u03bc (s \u222a t) + restrict \u03bc (s \u2229 t) = restrict \u03bc s + restrict \u03bc t\n[PROOFSTEP]\nsimpa only [union_comm, inter_comm, add_comm] using restrict_union_add_inter t hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nh : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\n\u22a2 restrict \u03bc (s \u222a t) = restrict \u03bc s + restrict \u03bc t\n[PROOFSTEP]\nsimp [\u2190 restrict_union_add_inter\u2080 s ht, restrict_zero_set h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nh : Disjoint s t\nhs : MeasurableSet s\n\u22a2 restrict \u03bc (s \u222a t) = restrict \u03bc s + restrict \u03bc t\n[PROOFSTEP]\nrw [union_comm, restrict_union h.symm hs, add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\n\u22a2 restrict \u03bc s + restrict \u03bc s\u1d9c = \u03bc\n[PROOFSTEP]\nrw [\u2190 restrict_union (@disjoint_compl_right (Set \u03b1) _ _) hs.compl, union_compl_self, restrict_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\n\u22a2 restrict \u03bc s\u1d9c + restrict \u03bc s = \u03bc\n[PROOFSTEP]\nrw [add_comm, restrict_add_restrict_compl hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s'\u271d t s s' : Set \u03b1\n\u22a2 restrict \u03bc (s \u222a s') \u2264 restrict \u03bc s + restrict \u03bc s'\n[PROOFSTEP]\nintro t ht\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s'\u271d t\u271d s s' t : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(restrict \u03bc (s \u222a s')) t \u2264 \u2191\u2191(restrict \u03bc s + restrict \u03bc s') t\n[PROOFSTEP]\nsuffices \u03bc (t \u2229 s \u222a t \u2229 s') \u2264 \u03bc (t \u2229 s) + \u03bc (t \u2229 s') by simpa [ht, inter_union_distrib_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s'\u271d t\u271d s s' t : Set \u03b1\nht : MeasurableSet t\nthis : \u2191\u2191\u03bc (t \u2229 s \u222a t \u2229 s') \u2264 \u2191\u2191\u03bc (t \u2229 s) + \u2191\u2191\u03bc (t \u2229 s')\n\u22a2 \u2191\u2191(restrict \u03bc (s \u222a s')) t \u2264 \u2191\u2191(restrict \u03bc s + restrict \u03bc s') t\n[PROOFSTEP]\nsimpa [ht, inter_union_distrib_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s'\u271d t\u271d s s' t : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (t \u2229 s \u222a t \u2229 s') \u2264 \u2191\u2191\u03bc (t \u2229 s) + \u2191\u2191\u03bc (t \u2229 s')\n[PROOFSTEP]\napply measure_union_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Pairwise (AEDisjoint \u03bc on s)\nhm : \u2200 (i : \u03b9), NullMeasurableSet (s i)\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(restrict \u03bc (\u22c3 (i : \u03b9), s i)) t = \u2211' (i : \u03b9), \u2191\u2191(restrict \u03bc (s i)) t\n[PROOFSTEP]\nsimp only [restrict_apply, ht, inter_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Pairwise (AEDisjoint \u03bc on s)\nhm : \u2200 (i : \u03b9), NullMeasurableSet (s i)\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), t \u2229 s i) = \u2211' (i : \u03b9), \u2191\u2191\u03bc (t \u2229 s i)\n[PROOFSTEP]\nexact\n measure_iUnion\u2080 (hd.mono fun i j h => h.mono (inter_subset_right _ _) (inter_subset_right _ _)) fun i =>\n ht.nullMeasurableSet.inter (hm i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) s\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(restrict \u03bc (\u22c3 (i : \u03b9), s i)) t = \u2a06 (i : \u03b9), \u2191\u2191(restrict \u03bc (s i)) t\n[PROOFSTEP]\nsimp only [restrict_apply ht, inter_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) s\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), t \u2229 s i) = \u2a06 (i : \u03b9), \u2191\u2191\u03bc (t \u2229 s i)\n[PROOFSTEP]\nrw [measure_iUnion_eq_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) s\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun i => t \u2229 s i\n[PROOFSTEP]\nexacts [hd.mono_comp _ fun s\u2081 s\u2082 => inter_subset_inter_right _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\nht : MeasurableSet t\n\u22a2 \u2191\u2191(restrict (map f \u03bc) s) t = \u2191\u2191(map f (restrict \u03bc (f \u207b\u00b9' s))) t\n[PROOFSTEP]\nsimp [*, hf ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nh : \u2191\u2191\u03bc s \u2260 \u22a4\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(restrict \u03bc (toMeasurable \u03bc s)) t = \u2191\u2191(restrict \u03bc s) t\n[PROOFSTEP]\nrw [restrict_apply ht, restrict_apply ht, inter_comm, measure_toMeasurable_inter ht h, inter_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nhs : MeasurableSet s\nH : restrict \u03bc s = restrict \u03bd s\nt : Set \u03b1\nhts : t \u2286 s\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\n[PROOFSTEP]\nrw [\u2190 inter_eq_self_of_subset_left hts, \u2190 restrict_apply ht, H, restrict_apply ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nhs : MeasurableSet s\nH : \u2200 (t : Set \u03b1), t \u2286 s \u2192 MeasurableSet t \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(restrict \u03bc s) t = \u2191\u2191(restrict \u03bd s) t\n[PROOFSTEP]\nrw [restrict_apply ht, restrict_apply ht, H _ (inter_subset_right _ _) (ht.inter hs)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : s \u2286 t\nh : restrict \u03bc t = restrict \u03bd t\n\u22a2 restrict \u03bc s = restrict \u03bd s\n[PROOFSTEP]\nrw [\u2190 restrict_restrict_of_subset hs, h, restrict_restrict_of_subset hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u22a2 restrict \u03bc (s \u222a t) = restrict \u03bd (s \u222a t) \u2194 restrict \u03bc s = restrict \u03bd s \u2227 restrict \u03bc t = restrict \u03bd t\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8restrict_congr_mono (subset_union_left _ _) h, restrict_congr_mono (subset_union_right _ _) h\u27e9, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u22a2 restrict \u03bc s = restrict \u03bd s \u2227 restrict \u03bc t = restrict \u03bd t \u2192 restrict \u03bc (s \u222a t) = restrict \u03bd (s \u222a t)\n[PROOFSTEP]\nrintro \u27e8hs, ht\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\n\u22a2 restrict \u03bc (s \u222a t) = restrict \u03bd (s \u222a t)\n[PROOFSTEP]\next1 u hu\n[GOAL]\ncase intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(restrict \u03bc (s \u222a t)) u = \u2191\u2191(restrict \u03bd (s \u222a t)) u\n[PROOFSTEP]\nsimp only [restrict_apply hu, inter_union_distrib_left]\n[GOAL]\ncase intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03bc (u \u2229 s \u222a u \u2229 t) = \u2191\u2191\u03bd (u \u2229 s \u222a u \u2229 t)\n[PROOFSTEP]\nrcases exists_measurable_superset\u2082 \u03bc \u03bd (u \u2229 s) with \u27e8US, hsub, hm, h\u03bc, h\u03bd\u27e9\n[GOAL]\ncase intro.h.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\nu : Set \u03b1\nhu : MeasurableSet u\nUS : Set \u03b1\nhsub : u \u2229 s \u2286 US\nhm : MeasurableSet US\nh\u03bc : \u2191\u2191\u03bc US = \u2191\u2191\u03bc (u \u2229 s)\nh\u03bd : \u2191\u2191\u03bd US = \u2191\u2191\u03bd (u \u2229 s)\n\u22a2 \u2191\u2191\u03bc (u \u2229 s \u222a u \u2229 t) = \u2191\u2191\u03bd (u \u2229 s \u222a u \u2229 t)\n[PROOFSTEP]\ncalc\n \u03bc (u \u2229 s \u222a u \u2229 t) = \u03bc (US \u222a u \u2229 t) := measure_union_congr_of_subset hsub h\u03bc.le Subset.rfl le_rfl\n _ = \u03bc US + \u03bc ((u \u2229 t) \\ US) := (measure_add_diff hm _).symm\n _ = restrict \u03bc s u + restrict \u03bc t (u \\ US) := by\n simp only [restrict_apply, hu, hu.diff hm, h\u03bc, \u2190 inter_comm t, inter_diff_assoc]\n _ = restrict \u03bd s u + restrict \u03bd t (u \\ US) := by rw [hs, ht]\n _ = \u03bd US + \u03bd ((u \u2229 t) \\ US) := by simp only [restrict_apply, hu, hu.diff hm, h\u03bd, \u2190 inter_comm t, inter_diff_assoc]\n _ = \u03bd (US \u222a u \u2229 t) := (measure_add_diff hm _)\n _ = \u03bd (u \u2229 s \u222a u \u2229 t) := Eq.symm <| measure_union_congr_of_subset hsub h\u03bd.le Subset.rfl le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\nu : Set \u03b1\nhu : MeasurableSet u\nUS : Set \u03b1\nhsub : u \u2229 s \u2286 US\nhm : MeasurableSet US\nh\u03bc : \u2191\u2191\u03bc US = \u2191\u2191\u03bc (u \u2229 s)\nh\u03bd : \u2191\u2191\u03bd US = \u2191\u2191\u03bd (u \u2229 s)\n\u22a2 \u2191\u2191\u03bc US + \u2191\u2191\u03bc ((u \u2229 t) \\ US) = \u2191\u2191(restrict \u03bc s) u + \u2191\u2191(restrict \u03bc t) (u \\ US)\n[PROOFSTEP]\nsimp only [restrict_apply, hu, hu.diff hm, h\u03bc, \u2190 inter_comm t, inter_diff_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\nu : Set \u03b1\nhu : MeasurableSet u\nUS : Set \u03b1\nhsub : u \u2229 s \u2286 US\nhm : MeasurableSet US\nh\u03bc : \u2191\u2191\u03bc US = \u2191\u2191\u03bc (u \u2229 s)\nh\u03bd : \u2191\u2191\u03bd US = \u2191\u2191\u03bd (u \u2229 s)\n\u22a2 \u2191\u2191(restrict \u03bc s) u + \u2191\u2191(restrict \u03bc t) (u \\ US) = \u2191\u2191(restrict \u03bd s) u + \u2191\u2191(restrict \u03bd t) (u \\ US)\n[PROOFSTEP]\nrw [hs, ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : restrict \u03bc s = restrict \u03bd s\nht : restrict \u03bc t = restrict \u03bd t\nu : Set \u03b1\nhu : MeasurableSet u\nUS : Set \u03b1\nhsub : u \u2229 s \u2286 US\nhm : MeasurableSet US\nh\u03bc : \u2191\u2191\u03bc US = \u2191\u2191\u03bc (u \u2229 s)\nh\u03bd : \u2191\u2191\u03bd US = \u2191\u2191\u03bd (u \u2229 s)\n\u22a2 \u2191\u2191(restrict \u03bd s) u + \u2191\u2191(restrict \u03bd t) (u \\ US) = \u2191\u2191\u03bd US + \u2191\u2191\u03bd ((u \u2229 t) \\ US)\n[PROOFSTEP]\nsimp only [restrict_apply, hu, hu.diff hm, h\u03bd, \u2190 inter_comm t, inter_diff_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Set \u03b1\n\u22a2 restrict \u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 s), t i) = restrict \u03bd (\u22c3 (i : \u03b9) (_ : i \u2208 s), t i) \u2194\n \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n[PROOFSTEP]\ninduction' s using Finset.induction_on with i s _ hs\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\n\u22a2 restrict \u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 \u2205), t i) = restrict \u03bd (\u22c3 (i : \u03b9) (_ : i \u2208 \u2205), t i) \u2194\n \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\ni : \u03b9\ns : Finset \u03b9\na\u271d : \u00aci \u2208 s\nhs :\n restrict \u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 s), t i) = restrict \u03bd (\u22c3 (i : \u03b9) (_ : i \u2208 s), t i) \u2194\n \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n\u22a2 restrict \u03bc (\u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 insert i s), t i_1) = restrict \u03bd (\u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 insert i s), t i_1) \u2194\n \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 restrict \u03bc (t i_1) = restrict \u03bd (t i_1)\n[PROOFSTEP]\nsimp only [forall_eq_or_imp, iUnion_iUnion_eq_or_left, Finset.mem_insert]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\ni : \u03b9\ns : Finset \u03b9\na\u271d : \u00aci \u2208 s\nhs :\n restrict \u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 s), t i) = restrict \u03bd (\u22c3 (i : \u03b9) (_ : i \u2208 s), t i) \u2194\n \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n\u22a2 restrict \u03bc (t i \u222a \u22c3 (x : \u03b9) (_ : x \u2208 s), t x) = restrict \u03bd (t i \u222a \u22c3 (x : \u03b9) (_ : x \u2208 s), t x) \u2194\n restrict \u03bc (t i) = restrict \u03bd (t i) \u2227 \u2200 (a : \u03b9), a \u2208 s \u2192 restrict \u03bc (t a) = restrict \u03bd (t a)\n[PROOFSTEP]\nrw [restrict_union_congr, \u2190 hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\n\u22a2 restrict \u03bc (\u22c3 (i : \u03b9), s i) = restrict \u03bd (\u22c3 (i : \u03b9), s i) \u2194 \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i)\n[PROOFSTEP]\nrefine' \u27e8fun h i => restrict_congr_mono (subset_iUnion _ _) h, fun h => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i)\n\u22a2 restrict \u03bc (\u22c3 (i : \u03b9), s i) = restrict \u03bd (\u22c3 (i : \u03b9), s i)\n[PROOFSTEP]\next1 t ht\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i)\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(restrict \u03bc (\u22c3 (i : \u03b9), s i)) t = \u2191\u2191(restrict \u03bd (\u22c3 (i : \u03b9), s i)) t\n[PROOFSTEP]\nhave D : Directed (\u00b7 \u2286 \u00b7) fun t : Finset \u03b9 => \u22c3 i \u2208 t, s i :=\n directed_of_sup fun t\u2081 t\u2082 ht => biUnion_subset_biUnion_left ht\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i)\nt : Set \u03b1\nht : MeasurableSet t\nD : Directed (fun x x_1 => x \u2286 x_1) fun t => \u22c3 (i : \u03b9) (_ : i \u2208 t), s i\n\u22a2 \u2191\u2191(restrict \u03bc (\u22c3 (i : \u03b9), s i)) t = \u2191\u2191(restrict \u03bd (\u22c3 (i : \u03b9), s i)) t\n[PROOFSTEP]\nrw [iUnion_eq_iUnion_finset]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i)\nt : Set \u03b1\nht : MeasurableSet t\nD : Directed (fun x x_1 => x \u2286 x_1) fun t => \u22c3 (i : \u03b9) (_ : i \u2208 t), s i\n\u22a2 \u2191\u2191(restrict \u03bc (\u22c3 (t : Finset \u03b9) (i : \u03b9) (_ : i \u2208 t), s i)) t =\n \u2191\u2191(restrict \u03bd (\u22c3 (t : Finset \u03b9) (i : \u03b9) (_ : i \u2208 t), s i)) t\n[PROOFSTEP]\nsimp only [restrict_iUnion_apply_eq_iSup D ht, restrict_finset_biUnion_congr.2 fun i _ => h i]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : Set \u03b9\nt : \u03b9 \u2192 Set \u03b1\nhc : Set.Countable s\n\u22a2 restrict \u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 s), t i) = restrict \u03bd (\u22c3 (i : \u03b9) (_ : i \u2208 s), t i) \u2194\n \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n[PROOFSTEP]\nhaveI := hc.toEncodable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : Set \u03b9\nt : \u03b9 \u2192 Set \u03b1\nhc : Set.Countable s\nthis : Encodable \u2191s\n\u22a2 restrict \u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 s), t i) = restrict \u03bd (\u22c3 (i : \u03b9) (_ : i \u2208 s), t i) \u2194\n \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n[PROOFSTEP]\nsimp only [biUnion_eq_iUnion, SetCoe.forall', restrict_iUnion_congr]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nS : Set (Set \u03b1)\nhc : Set.Countable S\n\u22a2 restrict \u03bc (\u22c3\u2080 S) = restrict \u03bd (\u22c3\u2080 S) \u2194 \u2200 (s : Set \u03b1), s \u2208 S \u2192 restrict \u03bc s = restrict \u03bd s\n[PROOFSTEP]\nrw [sUnion_eq_biUnion, restrict_biUnion_congr hc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm0 : MeasurableSpace \u03b1\nm : Set (Measure \u03b1)\nhm : Set.Nonempty m\nht : MeasurableSet t\n\u22a2 restrict (sInf m) t = sInf ((fun \u03bc => restrict \u03bc t) '' m)\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nm0 : MeasurableSpace \u03b1\nm : Set (Measure \u03b1)\nhm : Set.Nonempty m\nht : MeasurableSet t\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(restrict (sInf m) t) s = \u2191\u2191(sInf ((fun \u03bc => restrict \u03bc t) '' m)) s\n[PROOFSTEP]\nsimp_rw [sInf_apply hs, restrict_apply hs, sInf_apply (MeasurableSet.inter hs ht), Set.image_image,\n restrict_toOuterMeasure_eq_toOuterMeasure_restrict ht, \u2190 Set.image_image _ toOuterMeasure, \u2190\n OuterMeasure.restrict_sInf_eq_sInf_restrict _ (hm.image _), OuterMeasure.restrict_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : \u2191\u2191\u03bc s \u2260 0\np : \u03b1 \u2192 Prop\nhp : \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc s, p x\n\u22a2 \u2203 x, x \u2208 s \u2227 p x\n[PROOFSTEP]\nrw [\u2190 \u03bc.restrict_apply_self, \u2190 frequently_ae_mem_iff] at hs \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : \u2203\u1d50 (a : \u03b1) \u2202restrict \u03bc s, a \u2208 s\np : \u03b1 \u2192 Prop\nhp : \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc s, p x\n\u22a2 \u2203 x, x \u2208 s \u2227 p x\n[PROOFSTEP]\nexact (hs.and_eventually hp).exists\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u22c3 (i : \u03b9), s i = univ\n\u22a2 \u03bc = \u03bd \u2194 \u2200 (i : \u03b9), restrict \u03bc (s i) = restrict \u03bd (s i)\n[PROOFSTEP]\nrw [\u2190 restrict_iUnion_congr, hs, restrict_univ, restrict_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nS : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhc : Set.Countable S\nhs : \u22c3 (i : \u03b9) (_ : i \u2208 S), s i = univ\n\u22a2 \u03bc = \u03bd \u2194 \u2200 (i : \u03b9), i \u2208 S \u2192 restrict \u03bc (s i) = restrict \u03bd (s i)\n[PROOFSTEP]\nrw [\u2190 restrict_biUnion_congr hc, hs, restrict_univ, restrict_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nS : Set (Set \u03b1)\nhc : Set.Countable S\nhs : \u22c3\u2080 S = univ\n\u22a2 \u22c3 (i : Set \u03b1) (_ : i \u2208 S), i = univ\n[PROOFSTEP]\nrwa [\u2190 sUnion_eq_biUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\n\u22a2 \u03bc = \u03bd\n[PROOFSTEP]\nrefine' ext_of_sUnion_eq_univ hc hU fun t ht => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\n\u22a2 restrict \u03bc t = restrict \u03bd t\n[PROOFSTEP]\next1 u hu\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(restrict \u03bc t) u = \u2191\u2191(restrict \u03bd t) u\n[PROOFSTEP]\nsimp only [restrict_apply hu]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03bc (u \u2229 t) = \u2191\u2191\u03bd (u \u2229 t)\n[PROOFSTEP]\nrefine' induction_on_inter h_gen h_inter _ (ST_eq t ht) _ _ hu\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03bc (\u2205 \u2229 t) = \u2191\u2191\u03bd (\u2205 \u2229 t)\n[PROOFSTEP]\nsimp only [Set.empty_inter, measure_empty]\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2200 (t_1 : Set \u03b1), MeasurableSet t_1 \u2192 \u2191\u2191\u03bc (t_1 \u2229 t) = \u2191\u2191\u03bd (t_1 \u2229 t) \u2192 \u2191\u2191\u03bc (t_1\u1d9c \u2229 t) = \u2191\u2191\u03bd (t_1\u1d9c \u2229 t)\n[PROOFSTEP]\nintro v hv hvt\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nv : Set \u03b1\nhv : MeasurableSet v\nhvt : \u2191\u2191\u03bc (v \u2229 t) = \u2191\u2191\u03bd (v \u2229 t)\n\u22a2 \u2191\u2191\u03bc (v\u1d9c \u2229 t) = \u2191\u2191\u03bd (v\u1d9c \u2229 t)\n[PROOFSTEP]\nhave := T_eq t ht\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nv : Set \u03b1\nhv : MeasurableSet v\nhvt : \u2191\u2191\u03bc (v \u2229 t) = \u2191\u2191\u03bd (v \u2229 t)\nthis : \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\n\u22a2 \u2191\u2191\u03bc (v\u1d9c \u2229 t) = \u2191\u2191\u03bd (v\u1d9c \u2229 t)\n[PROOFSTEP]\nrw [Set.inter_comm] at hvt \u22a2\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nv : Set \u03b1\nhv : MeasurableSet v\nhvt : \u2191\u2191\u03bc (t \u2229 v) = \u2191\u2191\u03bd (t \u2229 v)\nthis : \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\n\u22a2 \u2191\u2191\u03bc (t \u2229 v\u1d9c) = \u2191\u2191\u03bd (t \u2229 v\u1d9c)\n[PROOFSTEP]\nrwa [\u2190 measure_inter_add_diff t hv, \u2190 measure_inter_add_diff t hv, \u2190 hvt, ENNReal.add_right_inj] at this \n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nv : Set \u03b1\nhv : MeasurableSet v\nhvt : \u2191\u2191\u03bc (t \u2229 v) = \u2191\u2191\u03bd (t \u2229 v)\nthis : \u2191\u2191\u03bc (t \u2229 v) + \u2191\u2191\u03bc (t \\ v) = \u2191\u2191\u03bc (t \u2229 v) + \u2191\u2191\u03bd (t \\ v)\n\u22a2 \u2191\u2191\u03bc (t \u2229 v) \u2260 \u22a4\n[PROOFSTEP]\nexact ne_top_of_le_ne_top (htop t ht) (measure_mono <| Set.inter_subset_left _ _)\n[GOAL]\ncase h.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1),\n Pairwise (Disjoint on f) \u2192\n (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n (\u2200 (i : \u2115), \u2191\u2191\u03bc (f i \u2229 t) = \u2191\u2191\u03bd (f i \u2229 t)) \u2192 \u2191\u2191\u03bc ((\u22c3 (i : \u2115), f i) \u2229 t) = \u2191\u2191\u03bd ((\u22c3 (i : \u2115), f i) \u2229 t)\n[PROOFSTEP]\nintro f hfd hfm h_eq\n[GOAL]\ncase h.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nf : \u2115 \u2192 Set \u03b1\nhfd : Pairwise (Disjoint on f)\nhfm : \u2200 (i : \u2115), MeasurableSet (f i)\nh_eq : \u2200 (i : \u2115), \u2191\u2191\u03bc (f i \u2229 t) = \u2191\u2191\u03bd (f i \u2229 t)\n\u22a2 \u2191\u2191\u03bc ((\u22c3 (i : \u2115), f i) \u2229 t) = \u2191\u2191\u03bd ((\u22c3 (i : \u2115), f i) \u2229 t)\n[PROOFSTEP]\nsimp only [\u2190 restrict_apply (hfm _), \u2190 restrict_apply (MeasurableSet.iUnion hfm)] at h_eq \u22a2\n[GOAL]\ncase h.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nhc : Set.Countable T\nh_inter : IsPiSystem S\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\nST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\nT_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nt : Set \u03b1\nht : t \u2208 T\nu : Set \u03b1\nhu : MeasurableSet u\nf : \u2115 \u2192 Set \u03b1\nhfd : Pairwise (Disjoint on f)\nhfm : \u2200 (i : \u2115), MeasurableSet (f i)\nh_eq : \u2200 (i : \u2115), \u2191\u2191(restrict \u03bc t) (f i) = \u2191\u2191(restrict \u03bd t) (f i)\n\u22a2 \u2191\u2191(restrict \u03bc t) (\u22c3 (i : \u2115), f i) = \u2191\u2191(restrict \u03bd t) (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nsimp only [measure_iUnion hfd hfm, h_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nh_inter : IsPiSystem S\nh_sub : T \u2286 S\nhc : Set.Countable T\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\n\u22a2 \u03bc = \u03bd\n[PROOFSTEP]\nrefine' ext_of_generateFrom_of_cover h_gen hc h_inter hU htop _ fun t ht => h_eq t (h_sub ht)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nh_inter : IsPiSystem S\nh_sub : T \u2286 S\nhc : Set.Countable T\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\n[PROOFSTEP]\nintro t ht s hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nh_inter : IsPiSystem S\nh_sub : T \u2286 S\nhc : Set.Countable T\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nt : Set \u03b1\nht : t \u2208 T\ns : Set \u03b1\nhs : s \u2208 S\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\n[PROOFSTEP]\ncases' (s \u2229 t).eq_empty_or_nonempty with H H\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nh_inter : IsPiSystem S\nh_sub : T \u2286 S\nhc : Set.Countable T\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nt : Set \u03b1\nht : t \u2208 T\ns : Set \u03b1\nhs : s \u2208 S\nH : s \u2229 t = \u2205\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\n[PROOFSTEP]\nsimp only [H, measure_empty]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nS T : Set (Set \u03b1)\nh_gen : m0 = generateFrom S\nh_inter : IsPiSystem S\nh_sub : T \u2286 S\nhc : Set.Countable T\nhU : \u22c3\u2080 T = univ\nhtop : \u2200 (s : Set \u03b1), s \u2208 T \u2192 \u2191\u2191\u03bc s \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nt : Set \u03b1\nht : t \u2208 T\ns : Set \u03b1\nhs : s \u2208 S\nH : Set.Nonempty (s \u2229 t)\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t)\n[PROOFSTEP]\nexact h_eq _ (h_inter _ hs _ (h_sub ht) H)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nC : Set (Set \u03b1)\nB : \u2115 \u2192 Set \u03b1\nhA : m0 = generateFrom C\nhC : IsPiSystem C\nh1B : \u22c3 (i : \u2115), B i = univ\nh2B : \u2200 (i : \u2115), B i \u2208 C\nh\u03bcB : \u2200 (i : \u2115), \u2191\u2191\u03bc (B i) \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\n\u22a2 \u03bc = \u03bd\n[PROOFSTEP]\nrefine' ext_of_generateFrom_of_cover_subset hA hC _ (countable_range B) h1B _ h_eq\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nC : Set (Set \u03b1)\nB : \u2115 \u2192 Set \u03b1\nhA : m0 = generateFrom C\nhC : IsPiSystem C\nh1B : \u22c3 (i : \u2115), B i = univ\nh2B : \u2200 (i : \u2115), B i \u2208 C\nh\u03bcB : \u2200 (i : \u2115), \u2191\u2191\u03bc (B i) \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\n\u22a2 range B \u2286 C\n[PROOFSTEP]\nrintro _ \u27e8i, rfl\u27e9\n[GOAL]\ncase refine'_1.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nC : Set (Set \u03b1)\nB : \u2115 \u2192 Set \u03b1\nhA : m0 = generateFrom C\nhC : IsPiSystem C\nh1B : \u22c3 (i : \u2115), B i = univ\nh2B : \u2200 (i : \u2115), B i \u2208 C\nh\u03bcB : \u2200 (i : \u2115), \u2191\u2191\u03bc (B i) \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\ni : \u2115\n\u22a2 B i \u2208 C\n[PROOFSTEP]\napply h2B\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nC : Set (Set \u03b1)\nB : \u2115 \u2192 Set \u03b1\nhA : m0 = generateFrom C\nhC : IsPiSystem C\nh1B : \u22c3 (i : \u2115), B i = univ\nh2B : \u2200 (i : \u2115), B i \u2208 C\nh\u03bcB : \u2200 (i : \u2115), \u2191\u2191\u03bc (B i) \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\n\u22a2 \u2200 (s : Set \u03b1), s \u2208 range B \u2192 \u2191\u2191\u03bc s \u2260 \u22a4\n[PROOFSTEP]\nrintro _ \u27e8i, rfl\u27e9\n[GOAL]\ncase refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nC : Set (Set \u03b1)\nB : \u2115 \u2192 Set \u03b1\nhA : m0 = generateFrom C\nhC : IsPiSystem C\nh1B : \u22c3 (i : \u2115), B i = univ\nh2B : \u2200 (i : \u2115), B i \u2208 C\nh\u03bcB : \u2200 (i : \u2115), \u2191\u2191\u03bc (B i) \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\ni : \u2115\n\u22a2 \u2191\u2191\u03bc (B i) \u2260 \u22a4\n[PROOFSTEP]\napply h\u03bcB\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : \u03b9 \u2192 Measure \u03b1\ni : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u03bc i) s \u2264 \u2191\u2191(sum \u03bc) s\n[PROOFSTEP]\nsimpa only [sum_apply \u03bc hs] using ENNReal.le_tsum i\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : Set \u03b1\n\u22a2 \u2191\u2191(sum \u03bc) s = 0 \u2194 \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) s = 0\n[PROOFSTEP]\nrefine' \u27e8fun h i => nonpos_iff_eq_zero.1 <| h \u25b8 le_iff'.1 (le_sum \u03bc i) _, fun h => nonpos_iff_eq_zero.1 _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : Set \u03b1\nh : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) s = 0\n\u22a2 \u2191\u2191(sum \u03bc) s \u2264 0\n[PROOFSTEP]\nrcases exists_measurable_superset_forall_eq \u03bc s with \u27e8t, hst, htm, ht\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : Set \u03b1\nh : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) s = 0\nt : Set \u03b1\nhst : s \u2286 t\nhtm : MeasurableSet t\nht : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) t = \u2191\u2191(\u03bc i) s\n\u22a2 \u2191\u2191(sum \u03bc) s \u2264 0\n[PROOFSTEP]\ncalc\n sum \u03bc s \u2264 sum \u03bc t := measure_mono hst\n _ = 0 := by simp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : Set \u03b1\nh : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) s = 0\nt : Set \u03b1\nhst : s \u2286 t\nhtm : MeasurableSet t\nht : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) t = \u2191\u2191(\u03bc i) s\n\u22a2 \u2191\u2191(sum \u03bc) t = 0\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(sum \u03bc) s = 0 \u2194 \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) s = 0\n[PROOFSTEP]\nsimp [hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9' : Type u_8\n\u03bc : \u03b9 \u2192 \u03b9' \u2192 Measure \u03b1\n\u22a2 (sum fun n => sum (\u03bc n)) = sum fun m => sum fun n => \u03bc n m\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b9' : Type u_8\n\u03bc : \u03b9 \u2192 \u03b9' \u2192 Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(sum fun n => sum (\u03bc n)) s = \u2191\u2191(sum fun m => sum fun n => \u03bc n m) s\n[PROOFSTEP]\nsimp_rw [sum_apply _ hs]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b9' : Type u_8\n\u03bc : \u03b9 \u2192 \u03b9' \u2192 Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2211' (i : \u03b9) (i_1 : \u03b9'), \u2191\u2191(\u03bc i i_1) s = \u2211' (i : \u03b9') (i_1 : \u03b9), \u2191\u2191(\u03bc i_1 i) s\n[PROOFSTEP]\nrw [ENNReal.tsum_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : Fintype \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 sum \u03bc = \u2211 i : \u03b9, \u03bc i\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Fintype \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(sum \u03bc) s = \u2191\u2191(\u2211 i : \u03b9, \u03bc i) s\n[PROOFSTEP]\nsimp only [sum_apply, finset_sum_apply, hs, tsum_fintype]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ns : Finset \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 (sum fun i => \u03bc \u2191i) = \u2211 i in s, \u03bc i\n[PROOFSTEP]\nrw [sum_fintype, Finset.sum_coe_sort s \u03bc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf : Bool \u2192 Measure \u03b1\n\u22a2 sum f = f true + f false\n[PROOFSTEP]\nrw [sum_fintype, Fintype.sum_bool]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(restrict (sum \u03bc) s) t = \u2191\u2191(sum fun i => restrict (\u03bc i) s) t\n[PROOFSTEP]\nsimp only [sum_apply, restrict_apply, ht, ht.inter hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsEmpty \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 sum \u03bc = 0\n[PROOFSTEP]\nrw [\u2190 measure_univ_eq_zero, sum_apply _ MeasurableSet.univ, tsum_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ns : Set \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 ((sum fun i => \u03bc \u2191i) + sum fun i => \u03bc \u2191i) = sum \u03bc\n[PROOFSTEP]\next1 t ht\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : Set \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191((sum fun i => \u03bc \u2191i) + sum fun i => \u03bc \u2191i) t = \u2191\u2191(sum \u03bc) t\n[PROOFSTEP]\nsimp only [add_apply, sum_apply _ ht]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : Set \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2211' (i : \u2191s), \u2191\u2191(\u03bc \u2191i) t + \u2211' (i : \u2191s\u1d9c), \u2191\u2191(\u03bc \u2191i) t = \u2211' (i : \u03b9), \u2191\u2191(\u03bc i) t\n[PROOFSTEP]\nexact tsum_add_tsum_compl (f := fun i => \u03bc i t) ENNReal.summable ENNReal.summable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc \u03bd : \u2115 \u2192 Measure \u03b1\n\u22a2 sum \u03bc + sum \u03bd = sum fun n => \u03bc n + \u03bd n\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc \u03bd : \u2115 \u2192 Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(sum \u03bc + sum \u03bd) s = \u2191\u2191(sum fun n => \u03bc n + \u03bd n) s\n[PROOFSTEP]\nsimp only [add_apply, sum_apply _ hs, Pi.add_apply, coe_add, tsum_add ENNReal.summable ENNReal.summable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Pairwise (AEDisjoint \u03bc on s)\nhm : \u2200 (i : \u03b9), NullMeasurableSet (s i)\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(restrict \u03bc (\u22c3 (i : \u03b9), s i)) t = \u2191\u2191(sum fun i => restrict \u03bc (s i)) t\n[PROOFSTEP]\nsimp only [sum_apply _ ht, restrict_iUnion_apply_ae hd hm ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\n\u22a2 restrict \u03bc (\u22c3 (i : \u03b9), s i) \u2264 sum fun i => restrict \u03bc (s i)\n[PROOFSTEP]\nintro t ht\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(restrict \u03bc (\u22c3 (i : \u03b9), s i)) t \u2264 \u2191\u2191(sum fun i => restrict \u03bc (s i)) t\n[PROOFSTEP]\nsuffices \u03bc (\u22c3 i, t \u2229 s i) \u2264 \u2211' i, \u03bc (t \u2229 s i) by simpa [ht, inter_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\nht : MeasurableSet t\nthis : \u2191\u2191\u03bc (\u22c3 (i : \u03b9), t \u2229 s i) \u2264 \u2211' (i : \u03b9), \u2191\u2191\u03bc (t \u2229 s i)\n\u22a2 \u2191\u2191(restrict \u03bc (\u22c3 (i : \u03b9), s i)) t \u2264 \u2191\u2191(sum fun i => restrict \u03bc (s i)) t\n[PROOFSTEP]\nsimpa [ht, inter_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), t \u2229 s i) \u2264 \u2211' (i : \u03b9), \u2191\u2191\u03bc (t \u2229 s i)\n[PROOFSTEP]\napply measure_iUnion_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nh : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bd s = 0 \u2192 \u2191\u2191\u03bc s = 0\n\u22a2 \u03bc \u226a \u03bd\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nh : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bd s = 0 \u2192 \u2191\u2191\u03bc s = 0\ns : Set \u03b1\nhs : \u2191\u2191\u03bd s = 0\n\u22a2 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nrcases exists_measurable_superset_of_null hs with \u27e8t, h1t, h2t, h3t\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nh : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bd s = 0 \u2192 \u2191\u2191\u03bc s = 0\ns : Set \u03b1\nhs : \u2191\u2191\u03bd s = 0\nt : Set \u03b1\nh1t : s \u2286 t\nh2t : MeasurableSet t\nh3t : \u2191\u2191\u03bd t = 0\n\u22a2 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nexact measure_mono_null h1t (h h2t h3t)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nh : \u03bc \u226a \u03bd\nf : \u03b1 \u2192 \u03b2\nhf : Measurable f\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(map f \u03bd) s = 0 \u2192 \u2191\u2191(map f \u03bc) s = 0\n[PROOFSTEP]\nsimpa [hf, hs] using @h _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : DistribMulAction R \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\nh : \u03bc \u226a \u03bd\nc : R\ns : Set \u03b1\nh\u03bds : \u2191\u2191\u03bd s = 0\n\u22a2 \u2191\u2191(c \u2022 \u03bc) s = 0\n[PROOFSTEP]\nsimp only [h h\u03bds, smul_eq_mul, smul_apply, smul_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nh : ae \u03bc \u2264 ae \u03bd\ns : Set \u03b1\n\u22a2 \u2191\u2191\u03bd s = 0 \u2192 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nrw [measure_zero_iff_ae_nmem, measure_zero_iff_ae_nmem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nh : ae \u03bc \u2264 ae \u03bd\ns : Set \u03b1\n\u22a2 (\u2200\u1d50 (a : \u03b1) \u2202\u03bd, \u00aca \u2208 s) \u2192 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u00aca \u2208 s\n[PROOFSTEP]\nexact fun hs => h hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : QuasiMeasurePreserving g\nhf : QuasiMeasurePreserving f\n\u22a2 map (g \u2218 f) \u03bca \u226a \u03bcc\n[PROOFSTEP]\nrw [\u2190 map_map hg.1 hf.1]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : QuasiMeasurePreserving g\nhf : QuasiMeasurePreserving f\n\u22a2 map g (map f \u03bca) \u226a \u03bcc\n[PROOFSTEP]\nexact (hf.2.map hg.1).trans hg.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nk : \u2115\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 f^[k] \u207b\u00b9' s =\u1d50[\u03bc] s\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 f^[Nat.zero] \u207b\u00b9' s =\u1d50[\u03bc] s\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\nk : \u2115\nih : f^[k] \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 f^[Nat.succ k] \u207b\u00b9' s =\u1d50[\u03bc] s\n[PROOFSTEP]\nrw [iterate_succ, preimage_comp]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\nk : \u2115\nih : f^[k] \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 f \u207b\u00b9' (f^[k] \u207b\u00b9' s) =\u1d50[\u03bc] s\n[PROOFSTEP]\nexact EventuallyEq.trans (hf.preimage_ae_eq ih) hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nk : \u2124\nhs : \u2191e '' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ k) '' s =\u1d50[\u03bc] s\n[PROOFSTEP]\nrw [Equiv.image_eq_preimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nk : \u2124\nhs : \u2191e '' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ k).symm \u207b\u00b9' s =\u1d50[\u03bc] s\n[PROOFSTEP]\nobtain \u27e8k, rfl | rfl\u27e9 := k.eq_nat_or_neg\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 \u2191(e ^ \u2191k).symm \u207b\u00b9' s =\u1d50[\u03bc] s\n[PROOFSTEP]\nreplace hs : (\u21d1e\u207b\u00b9) \u207b\u00b9' s =\u1d50[\u03bc] s\n[GOAL]\ncase hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 \u2191e\u207b\u00b9 \u207b\u00b9' s =\u1d50[\u03bc] s\n[PROOFSTEP]\nrwa [Equiv.image_eq_preimage] at hs \n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nk : \u2115\nhs : \u2191e\u207b\u00b9 \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ \u2191k).symm \u207b\u00b9' s =\u1d50[\u03bc] s\n[PROOFSTEP]\nreplace he' : (\u21d1e\u207b\u00b9)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s := he'.preimage_iterate_ae_eq k hs\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nk : \u2115\nhs : \u2191e\u207b\u00b9 \u207b\u00b9' s =\u1d50[\u03bc] s\nhe' : (\u2191e\u207b\u00b9)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ \u2191k).symm \u207b\u00b9' s =\u1d50[\u03bc] s\n[PROOFSTEP]\nrwa [Equiv.Perm.iterate_eq_pow e\u207b\u00b9 k, inv_pow e k] at he' \n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 \u2191(e ^ (-\u2191k)).symm \u207b\u00b9' s =\u1d50[\u03bc] s\n[PROOFSTEP]\nrw [zpow_neg, zpow_ofNat]\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 \u2191(e ^ k)\u207b\u00b9.symm \u207b\u00b9' s =\u1d50[\u03bc] s\n[PROOFSTEP]\nreplace hs : e \u207b\u00b9' s =\u1d50[\u03bc] s\n[GOAL]\ncase hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 \u2191e \u207b\u00b9' s =\u1d50[\u03bc] s\n[PROOFSTEP]\nconvert he.preimage_ae_eq hs.symm\n[GOAL]\ncase h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nhs : \u2191e '' s =\u1d50[\u03bc] s\nk : \u2115\n\u22a2 s = \u2191e \u207b\u00b9' (\u2191e '' s)\n[PROOFSTEP]\nrw [Equiv.preimage_image]\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nk : \u2115\nhs : \u2191e \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ k)\u207b\u00b9.symm \u207b\u00b9' s =\u1d50[\u03bc] s\n[PROOFSTEP]\nreplace he : (\u21d1e)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s := he.preimage_iterate_ae_eq k hs\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ne : \u03b1 \u2243 \u03b1\nhe' : QuasiMeasurePreserving \u2191e.symm\nk : \u2115\nhs : \u2191e \u207b\u00b9' s =\u1d50[\u03bc] s\nhe : (\u2191e)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 \u2191(e ^ k)\u207b\u00b9.symm \u207b\u00b9' s =\u1d50[\u03bc] s\n[PROOFSTEP]\nrwa [Equiv.Perm.iterate_eq_pow e k] at he \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 \u2200 (n : \u2115), (preimage f)^[n] s =\u1d50[\u03bc] s\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\nn : \u2115\n\u22a2 (preimage f)^[n] s =\u1d50[\u03bc] s\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 (preimage f)^[Nat.zero] s =\u1d50[\u03bc] s\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\nn : \u2115\nih : (preimage f)^[n] s =\u1d50[\u03bc] s\n\u22a2 (preimage f)^[Nat.succ n] s =\u1d50[\u03bc] s\n[PROOFSTEP]\nsimpa only [iterate_succ', comp_apply] using ae_eq_trans (hf.ae_eq ih) hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 liminf (fun n => (preimage f)^[n] s) atTop =\u1d50[\u03bc] s\n[PROOFSTEP]\nrw [\u2190 ae_eq_set_compl_compl, @Filter.liminf_compl (Set \u03b1)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 limsup (compl \u2218 fun n => (preimage f)^[n] s) atTop =\u1d50[\u03bc] s\u1d9c\n[PROOFSTEP]\nrw [\u2190 ae_eq_set_compl_compl, \u2190 preimage_compl] at hs \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s\u1d9c =\u1d50[\u03bc] s\u1d9c\n\u22a2 limsup (compl \u2218 fun n => (preimage f)^[n] s) atTop =\u1d50[\u03bc] s\u1d9c\n[PROOFSTEP]\nconvert hf.limsup_preimage_iterate_ae_eq hs\n[GOAL]\ncase h.e'_4.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s\u1d9c =\u1d50[\u03bc] s\u1d9c\nx\u271d : \u2115\n\u22a2 (compl \u2218 fun n => (preimage f)^[n] s) x\u271d = (preimage f)^[x\u271d] s\u1d9c\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h.e'_4.h.e'_4.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bca \u03bca' : Measure \u03b1\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b1\nhf : QuasiMeasurePreserving f\nhs : f \u207b\u00b9' s\u1d9c =\u1d50[\u03bc] s\u1d9c\nx\u271d : \u2115\nn : \u03b1\n\u22a2 n \u2208 (compl \u2218 fun n => (preimage f)^[n] s) x\u271d \u2194 n \u2208 (preimage f)^[x\u271d] s\u1d9c\n[PROOFSTEP]\nsimp only [\u2190 Set.preimage_iterate_eq, comp_apply, preimage_compl]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t\u271d : Set \u03b1\u271d\n\u03bca \u03bca' : Measure \u03b1\u271d\n\u03bcb \u03bcb' : Measure \u03b2\n\u03bcc : Measure \u03b3\nf : \u03b1\u271d \u2192 \u03b2\nG : Type u_8\n\u03b1 : Type u_9\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ng : G\nh_qmp : QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g\u207b\u00b9)\nh_ae_eq : s =\u1d50[\u03bc] t\n\u22a2 g \u2022 s =\u1d50[\u03bc] g \u2022 t\n[PROOFSTEP]\nsimpa only [\u2190 preimage_smul_inv] using h_qmp.ae_eq h_ae_eq\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_8\n\u03b1 : Type u_9\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\n\u22a2 Pairwise (AEDisjoint \u03bc on fun g => g \u2022 s)\n[PROOFSTEP]\nintro g\u2081 g\u2082 hg\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_8\n\u03b1 : Type u_9\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\nhg : g\u2081 \u2260 g\u2082\n\u22a2 (AEDisjoint \u03bc on fun g => g \u2022 s) g\u2081 g\u2082\n[PROOFSTEP]\nlet g := g\u2082\u207b\u00b9 * g\u2081\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_8\n\u03b1 : Type u_9\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\nhg : g\u2081 \u2260 g\u2082\ng : G := g\u2082\u207b\u00b9 * g\u2081\n\u22a2 (AEDisjoint \u03bc on fun g => g \u2022 s) g\u2081 g\u2082\n[PROOFSTEP]\nreplace hg : g \u2260 1\n[GOAL]\ncase hg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_8\n\u03b1 : Type u_9\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\nhg : g\u2081 \u2260 g\u2082\ng : G := g\u2082\u207b\u00b9 * g\u2081\n\u22a2 g \u2260 1\n[PROOFSTEP]\nrw [Ne.def, inv_mul_eq_one]\n[GOAL]\ncase hg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_8\n\u03b1 : Type u_9\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\nhg : g\u2081 \u2260 g\u2082\ng : G := g\u2082\u207b\u00b9 * g\u2081\n\u22a2 \u00acg\u2082 = g\u2081\n[PROOFSTEP]\nexact hg.symm\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_8\n\u03b1 : Type u_9\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\ng : G := g\u2082\u207b\u00b9 * g\u2081\nhg : g \u2260 1\n\u22a2 (AEDisjoint \u03bc on fun g => g \u2022 s) g\u2081 g\u2082\n[PROOFSTEP]\nhave : (\u00b7 \u2022 \u00b7) g\u2082\u207b\u00b9 \u207b\u00b9' (g \u2022 s \u2229 s) = g\u2081 \u2022 s \u2229 g\u2082 \u2022 s := by\n rw [preimage_eq_iff_eq_image (MulAction.bijective g\u2082\u207b\u00b9), image_smul, smul_set_inter, smul_smul, smul_smul,\n inv_mul_self, one_smul]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_8\n\u03b1 : Type u_9\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\ng : G := g\u2082\u207b\u00b9 * g\u2081\nhg : g \u2260 1\n\u22a2 (fun x x_1 => x \u2022 x_1) g\u2082\u207b\u00b9 \u207b\u00b9' (g \u2022 s \u2229 s) = g\u2081 \u2022 s \u2229 g\u2082 \u2022 s\n[PROOFSTEP]\nrw [preimage_eq_iff_eq_image (MulAction.bijective g\u2082\u207b\u00b9), image_smul, smul_set_inter, smul_smul, smul_smul, inv_mul_self,\n one_smul]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_8\n\u03b1 : Type u_9\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\ng : G := g\u2082\u207b\u00b9 * g\u2081\nhg : g \u2260 1\nthis : (fun x x_1 => x \u2022 x_1) g\u2082\u207b\u00b9 \u207b\u00b9' (g \u2022 s \u2229 s) = g\u2081 \u2022 s \u2229 g\u2082 \u2022 s\n\u22a2 (AEDisjoint \u03bc on fun g => g \u2022 s) g\u2081 g\u2082\n[PROOFSTEP]\nchange \u03bc (g\u2081 \u2022 s \u2229 g\u2082 \u2022 s) = 0\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_8\n\u03b1 : Type u_9\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\ng : G := g\u2082\u207b\u00b9 * g\u2081\nhg : g \u2260 1\nthis : (fun x x_1 => x \u2022 x_1) g\u2082\u207b\u00b9 \u207b\u00b9' (g \u2022 s \u2229 s) = g\u2081 \u2022 s \u2229 g\u2082 \u2022 s\n\u22a2 \u2191\u2191\u03bc (g\u2081 \u2022 s \u2229 g\u2082 \u2022 s) = 0\n[PROOFSTEP]\nexact this \u25b8 (h_qmp g\u2082\u207b\u00b9).preimage_null (h_ae_disjoint g hg)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u22a2 univ \u2208 {s | \u2191\u2191\u03bc s\u1d9c < \u22a4}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhs : s \u2208 {s | \u2191\u2191\u03bc s\u1d9c < \u22a4}\nht : t \u2208 {s | \u2191\u2191\u03bc s\u1d9c < \u22a4}\n\u22a2 s \u2229 t \u2208 {s | \u2191\u2191\u03bc s\u1d9c < \u22a4}\n[PROOFSTEP]\nsimp only [compl_inter, mem_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhs : s \u2208 {s | \u2191\u2191\u03bc s\u1d9c < \u22a4}\nht : t \u2208 {s | \u2191\u2191\u03bc s\u1d9c < \u22a4}\n\u22a2 \u2191\u2191\u03bc (s\u1d9c \u222a t\u1d9c) < \u22a4\n[PROOFSTEP]\ncalc\n \u03bc (s\u1d9c \u222a t\u1d9c) \u2264 \u03bc s\u1d9c + \u03bc t\u1d9c := measure_union_le _ _\n _ < \u221e := ENNReal.add_lt_top.2 \u27e8hs, ht\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u22a2 s\u1d9c \u2208 cofinite \u03bc \u2194 \u2191\u2191\u03bc s < \u22a4\n[PROOFSTEP]\nrw [mem_cofinite, compl_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u22a2 ae \u03bc = \u22a5 \u2194 \u03bc = 0\n[PROOFSTEP]\nrw [\u2190 empty_mem_iff_bot, mem_ae_iff, compl_empty, measure_univ_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 s \u2208 ae (Measure.map f \u03bc) \u2194 f \u207b\u00b9' s \u2208 ae \u03bc\n[PROOFSTEP]\nsimp only [mem_ae_iff, map_apply_of_aemeasurable hf hs.compl, preimage_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm0 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableSet (range f)\n\u03bc : Measure \u03b1\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202Measure.map f \u03bc, x \u2208 range f\n[PROOFSTEP]\nby_cases h : AEMeasurable f \u03bc\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm0 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableSet (range f)\n\u03bc : Measure \u03b1\nh : AEMeasurable f\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202Measure.map f \u03bc, x \u2208 range f\n[PROOFSTEP]\nchange range f \u2208 (\u03bc.map f).ae\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm0 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableSet (range f)\n\u03bc : Measure \u03b1\nh : AEMeasurable f\n\u22a2 range f \u2208 ae (Measure.map f \u03bc)\n[PROOFSTEP]\nrw [mem_ae_map_iff h hf]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm0 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableSet (range f)\n\u03bc : Measure \u03b1\nh : AEMeasurable f\n\u22a2 f \u207b\u00b9' range f \u2208 ae \u03bc\n[PROOFSTEP]\napply eventually_of_forall\n[GOAL]\ncase pos.hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm0 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableSet (range f)\n\u03bc : Measure \u03b1\nh : AEMeasurable f\n\u22a2 \u2200 (x : \u03b1), f x \u2208 range f\n[PROOFSTEP]\nexact mem_range_self\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm0 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableSet (range f)\n\u03bc : Measure \u03b1\nh : \u00acAEMeasurable f\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202Measure.map f \u03bc, x \u2208 range f\n[PROOFSTEP]\nsimp [map_of_not_aemeasurable h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\n\u22a2 ae (Measure.restrict \u03bc (s \u222a t)) = ae (Measure.restrict \u03bc s) \u2294 ae (Measure.restrict \u03bc t)\n[PROOFSTEP]\nsimp [union_eq_iUnion, iSup_bool_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\nht : Set.Countable t\n\u22a2 ae (Measure.restrict \u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 t), s i)) = \u2a06 (i : \u03b9) (_ : i \u2208 t), ae (Measure.restrict \u03bc (s i))\n[PROOFSTEP]\nhaveI := ht.to_subtype\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\nht : Set.Countable t\nthis : Countable \u2191t\n\u22a2 ae (Measure.restrict \u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 t), s i)) = \u2a06 (i : \u03b9) (_ : i \u2208 t), ae (Measure.restrict \u03bc (s i))\n[PROOFSTEP]\nrw [biUnion_eq_iUnion, ae_restrict_iUnion_eq, \u2190 iSup_subtype'']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (\u22c3 (i : \u03b9), s i), p x) \u2194 \u2200 (i : \u03b9), \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s i), p x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s \u222a t), p x) \u2194\n (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, p x) \u2227 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, p x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\nht : Set.Countable t\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 t), s i), p x) \u2194\n \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s i), p x\n[PROOFSTEP]\nsimp_rw [Filter.Eventually, ae_restrict_biUnion_eq s ht, mem_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : \u03b9 \u2192 Set \u03b1\nt : Finset \u03b9\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 t), s i), p x) \u2194\n \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s i), p x\n[PROOFSTEP]\nsimp_rw [Filter.Eventually, ae_restrict_biUnion_finset_eq s, mem_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf g : \u03b1 \u2192 \u03b4\n\u22a2 f =\u1da0[ae (Measure.restrict \u03bc (\u22c3 (i : \u03b9), s i))] g \u2194 \u2200 (i : \u03b9), f =\u1da0[ae (Measure.restrict \u03bc (s i))] g\n[PROOFSTEP]\nsimp_rw [EventuallyEq, ae_restrict_iUnion_eq, eventually_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\nht : Set.Countable t\nf g : \u03b1 \u2192 \u03b4\n\u22a2 f =\u1da0[ae (Measure.restrict \u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 t), s i))] g \u2194\n \u2200 (i : \u03b9), i \u2208 t \u2192 f =\u1da0[ae (Measure.restrict \u03bc (s i))] g\n[PROOFSTEP]\nsimp_rw [ae_restrict_biUnion_eq s ht, EventuallyEq, eventually_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 ae (Measure.restrict \u03bc (\u0399 a b)) = ae (Measure.restrict \u03bc (Ioc a b)) \u2294 ae (Measure.restrict \u03bc (Ioc b a))\n[PROOFSTEP]\nsimp only [uIoc_eq_union, ae_restrict_union_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nP : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (\u0399 a b), P x) \u2194\n (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (Ioc a b), P x) \u2227 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (Ioc b a), P x\n[PROOFSTEP]\nrw [ae_restrict_uIoc_eq, eventually_sup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\np : \u03b1 \u2192 Prop\nhp : MeasurableSet {x | p x}\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, p x) \u2194 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 p x\n[PROOFSTEP]\nsimp only [ae_iff, \u2190 compl_setOf, Measure.restrict_apply hp.compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\np : \u03b1 \u2192 Prop\nhp : MeasurableSet {x | p x}\n\u22a2 \u2191\u2191\u03bc ({x | p x}\u1d9c \u2229 s) = 0 \u2194 \u2191\u2191\u03bc {x | x \u2208 s \u2192 p x}\u1d9c = 0\n[PROOFSTEP]\nrw [iff_iff_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\np : \u03b1 \u2192 Prop\nhp : MeasurableSet {x | p x}\n\u22a2 (\u2191\u2191\u03bc ({x | p x}\u1d9c \u2229 s) = 0) = (\u2191\u2191\u03bc {x | x \u2208 s \u2192 p x}\u1d9c = 0)\n[PROOFSTEP]\ncongr with x\n[GOAL]\ncase e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\np : \u03b1 \u2192 Prop\nhp : MeasurableSet {x | p x}\nx : \u03b1\n\u22a2 x \u2208 {x | p x}\u1d9c \u2229 s \u2194 x \u2208 {x | x \u2208 s \u2192 p x}\u1d9c\n[PROOFSTEP]\nsimp [and_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t s : Set \u03b1\np : \u03b1 \u2192 Prop\nh : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, p x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 p x\n[PROOFSTEP]\nsimp only [ae_iff] at h \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t s : Set \u03b1\np : \u03b1 \u2192 Prop\nh : \u2191\u2191(Measure.restrict \u03bc s) {a | \u00acp a} = 0\n\u22a2 \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 p a)} = 0\n[PROOFSTEP]\nsimpa [setOf_and, inter_comm] using measure_inter_eq_zero_of_restrict h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\np : \u03b1 \u2192 Prop\nhs : MeasurableSet s\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, p x) \u2194 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 p x\n[PROOFSTEP]\nsimp only [ae_iff, \u2190 compl_setOf, restrict_apply_eq_zero' hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\np : \u03b1 \u2192 Prop\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc ({x | p x}\u1d9c \u2229 s) = 0 \u2194 \u2191\u2191\u03bc {x | x \u2208 s \u2192 p x}\u1d9c = 0\n[PROOFSTEP]\nrw [iff_iff_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\np : \u03b1 \u2192 Prop\nhs : MeasurableSet s\n\u22a2 (\u2191\u2191\u03bc ({x | p x}\u1d9c \u2229 s) = 0) = (\u2191\u2191\u03bc {x | x \u2208 s \u2192 p x}\u1d9c = 0)\n[PROOFSTEP]\ncongr with x\n[GOAL]\ncase e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\np : \u03b1 \u2192 Prop\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 x \u2208 {x | p x}\u1d9c \u2229 s \u2194 x \u2208 {x | x \u2208 s \u2192 p x}\u1d9c\n[PROOFSTEP]\nsimp [and_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nf g : \u03b1 \u2192 \u03b4\ns : Set \u03b1\nhfg : f =\u1da0[ae \u03bc] g\n\u22a2 f =\u1da0[ae (Measure.restrict \u03bc s)] g\n[PROOFSTEP]\nrefine' hfg.filter_mono _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nf g : \u03b1 \u2192 \u03b4\ns : Set \u03b1\nhfg : f =\u1da0[ae \u03bc] g\n\u22a2 ae (Measure.restrict \u03bc s) \u2264 ae \u03bc\n[PROOFSTEP]\nrw [Measure.ae_le_iff_absolutelyContinuous]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nf g : \u03b1 \u2192 \u03b4\ns : Set \u03b1\nhfg : f =\u1da0[ae \u03bc] g\n\u22a2 Measure.restrict \u03bc s \u226a \u03bc\n[PROOFSTEP]\nexact Measure.absolutelyContinuous_of_le Measure.restrict_le_self\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, x \u2208 s\n[PROOFSTEP]\nrcases hs.exists_measurable_subset_ae_eq with \u27e8t, hts, htm, ht_eq\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nhs : NullMeasurableSet s\nt : Set \u03b1\nhts : t \u2286 s\nhtm : MeasurableSet t\nht_eq : t =\u1da0[ae \u03bc] s\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, x \u2208 s\n[PROOFSTEP]\nrw [\u2190 restrict_congr_set ht_eq]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nhs : NullMeasurableSet s\nt : Set \u03b1\nhts : t \u2286 s\nhtm : MeasurableSet t\nht_eq : t =\u1da0[ae \u03bc] s\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, x \u2208 s\n[PROOFSTEP]\nexact (ae_restrict_mem htm).mono hts\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\np : \u03b1 \u2192 Prop\nhs : NullMeasurableSet s\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, p x) \u2194 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 p x\n[PROOFSTEP]\nrefine' \u27e8fun h => ae_imp_of_ae_restrict h, fun h => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\np : \u03b1 \u2192 Prop\nhs : NullMeasurableSet s\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 p x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, p x\n[PROOFSTEP]\nfilter_upwards [ae_restrict_mem\u2080 hs, ae_restrict_of_ae h] with x hx h'x using h'x hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d t : Set \u03b1\np : \u03b1 \u2192 Prop\nht : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, p x\nhtc : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t\u1d9c, p x\n\u22a2 \u2191\u2191\u03bc {x | \u00acp x} = \u2191\u2191\u03bc ({x | \u00acp x} \u2229 t \u222a {x | \u00acp x} \u2229 t\u1d9c)\n[PROOFSTEP]\nrw [\u2190 inter_union_distrib_left, union_compl_self, inter_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d t : Set \u03b1\np : \u03b1 \u2192 Prop\nht : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, p x\nhtc : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t\u1d9c, p x\n\u22a2 \u2191\u2191(Measure.restrict \u03bc t) {x | \u00acp x} + \u2191\u2191(Measure.restrict \u03bc t\u1d9c) {x | \u00acp x} = 0\n[PROOFSTEP]\nrw [ae_iff.1 ht, ae_iff.1 htc, zero_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\u271d\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\n\u03b2 : Type u_8\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : MeasurableSet s\n\u22a2 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s)) \u2194 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u2229 s) = 0\n[PROOFSTEP]\nrw [mem_map, mem_ae_iff, Measure.restrict_apply' hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : DistribMulAction R \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x\nc : R\n\u22a2 \u2191\u2191(c \u2022 \u03bc) {a | \u00acp a} = 0\n[PROOFSTEP]\nrw [smul_apply, ae_iff.1 h, smul_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : AddGroup \u03b2\nf g : \u03b1 \u2192 \u03b2\n\u22a2 f - g =\u1da0[ae \u03bc] 0 \u2194 f =\u1da0[ae \u03bc] g\n[PROOFSTEP]\nrefine' \u27e8fun h => h.mono fun x hx => _, fun h => h.mono fun x hx => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : AddGroup \u03b2\nf g : \u03b1 \u2192 \u03b2\nh : f - g =\u1da0[ae \u03bc] 0\nx : \u03b1\nhx : (f - g) x = OfNat.ofNat 0 x\n\u22a2 f x = g x\n[PROOFSTEP]\nrwa [Pi.sub_apply, Pi.zero_apply, sub_eq_zero] at hx \n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d : AddGroup \u03b2\nf g : \u03b1 \u2192 \u03b2\nh : f =\u1da0[ae \u03bc] g\nx : \u03b1\nhx : f x = g x\n\u22a2 (f - g) x = OfNat.ofNat 0 x\n[PROOFSTEP]\nrwa [Pi.sub_apply, Pi.zero_apply, sub_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\n\u22a2 ae (Measure.restrict \u03bc s) = ae \u03bc \u2293 \ud835\udcdf s\n[PROOFSTEP]\next t\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 t \u2208 ae (Measure.restrict \u03bc s) \u2194 t \u2208 ae \u03bc \u2293 \ud835\udcdf s\n[PROOFSTEP]\nsimp only [mem_inf_principal, mem_ae_iff, restrict_apply_eq_zero' hs, compl_setOf, not_imp, fun a =>\n and_comm (a := a \u2208 s) (b := \u00aca \u2208 t)]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191\u2191\u03bc (t\u1d9c \u2229 s) = 0 \u2194 \u2191\u2191\u03bc {a | \u00aca \u2208 t \u2227 a \u2208 s} = 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t s : Set \u03b1\nhs : MeasurableSet s\n\u22a2 s \u2208 ae (Measure.restrict \u03bc s)\n[PROOFSTEP]\nsimp only [ae_restrict_eq hs, exists_prop, mem_principal, mem_inf_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t s : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2203 t\u2081, t\u2081 \u2208 ae \u03bc \u2227 \u2203 t\u2082, s \u2286 t\u2082 \u2227 s = t\u2081 \u2229 t\u2082\n[PROOFSTEP]\nexact \u27e8_, univ_mem, s, Subset.rfl, (univ_inter s).symm\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns t : \u03b1 \u2192 Prop\nhst : s =\u1da0[ae \u03bc] t\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, p x) \u2192 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, p x\n[PROOFSTEP]\nsimp [Measure.restrict_congr_set hst]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\np : \u2115 \u2192 \u03b1 \u2192 Prop\nhp : \u2211' (i : \u2115), \u2191\u2191\u03bc {x | p i x} \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc {x | \u2203\u1da0 (n : \u2115) in atTop, p n x} = 0\n[PROOFSTEP]\nsimpa only [limsup_eq_iInf_iSup_of_nat, frequently_atTop, \u2190 bex_def, setOf_forall, setOf_exists] using\n measure_limsup_eq_zero hp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhst : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x \u2264 y\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\n\u22a2 \u2a06 (x : \u03b1) (_ : x \u2208 s), \u2191\u2191\u03bc (Iic x) = \u2191\u2191\u03bc univ\n[PROOFSTEP]\nrw [\u2190 measure_biUnion_eq_iSup hsc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhst : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x \u2264 y\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b1) (_ : i \u2208 s), Iic i) = \u2191\u2191\u03bc univ\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhst : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x \u2264 y\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\n\u22a2 \u22c3 (i : \u03b1) (_ : i \u2208 s), Iic i = univ\n[PROOFSTEP]\nsimp only [\u2190 bex_def] at hst \n[GOAL]\ncase e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\nhst : \u2200 (x : \u03b1), \u2203 x_1 x_2, x \u2264 x_1\n\u22a2 \u22c3 (i : \u03b1) (_ : i \u2208 s), Iic i = univ\n[PROOFSTEP]\nexact iUnion\u2082_eq_univ_iff.2 hst\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhst : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 x \u2264 y\nhdir : DirectedOn (fun x x_1 => x \u2264 x_1) s\n\u22a2 DirectedOn ((fun x x_1 => x \u2286 x_1) on fun x => Iic x) s\n[PROOFSTEP]\nexact directedOn_iff_directed.2 (hdir.directed_val.mono_comp _ fun x y => Iic_subset_Iic.2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\na : \u03b1\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Ico a x)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (Ici a)))\n[PROOFSTEP]\nhaveI : Nonempty \u03b1 := \u27e8a\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Ico a x)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (Ici a)))\n[PROOFSTEP]\nhave h_mono : Monotone fun x => \u03bc (Ico a x) := fun i j hij => measure_mono (Ico_subset_Ico_right hij)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Ico a x)\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Ico a x)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (Ici a)))\n[PROOFSTEP]\nconvert tendsto_atTop_iSup h_mono\n[GOAL]\ncase h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Ico a x)\n\u22a2 \u2191\u2191\u03bc (Ici a) = \u2a06 (i : \u03b1), \u2191\u2191\u03bc (Ico a i)\n[PROOFSTEP]\nobtain \u27e8xs, hxs_mono, hxs_tendsto\u27e9 := exists_seq_monotone_tendsto_atTop_atTop \u03b1\n[GOAL]\ncase h.e'_5.h.e'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Ico a x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\n\u22a2 \u2191\u2191\u03bc (Ici a) = \u2a06 (i : \u03b1), \u2191\u2191\u03bc (Ico a i)\n[PROOFSTEP]\nhave h_Ici : Ici a = \u22c3 n, Ico a (xs n) := by\n ext1 x\n simp only [mem_Ici, mem_iUnion, mem_Ico, exists_and_left, iff_self_and]\n intro\n obtain \u27e8y, hxy\u27e9 := NoMaxOrder.exists_gt x\n obtain \u27e8n, hn\u27e9 := tendsto_atTop_atTop.mp hxs_tendsto y\n exact \u27e8n, hxy.trans_le (hn n le_rfl)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Ico a x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\n\u22a2 Ici a = \u22c3 (n : \u2115), Ico a (xs n)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Ico a x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nx : \u03b1\n\u22a2 x \u2208 Ici a \u2194 x \u2208 \u22c3 (n : \u2115), Ico a (xs n)\n[PROOFSTEP]\nsimp only [mem_Ici, mem_iUnion, mem_Ico, exists_and_left, iff_self_and]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Ico a x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nx : \u03b1\n\u22a2 a \u2264 x \u2192 \u2203 x_1, x < xs x_1\n[PROOFSTEP]\nintro\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Ico a x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nx : \u03b1\na\u271d : a \u2264 x\n\u22a2 \u2203 x_1, x < xs x_1\n[PROOFSTEP]\nobtain \u27e8y, hxy\u27e9 := NoMaxOrder.exists_gt x\n[GOAL]\ncase h.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Ico a x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nx : \u03b1\na\u271d : a \u2264 x\ny : \u03b1\nhxy : x < y\n\u22a2 \u2203 x_1, x < xs x_1\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := tendsto_atTop_atTop.mp hxs_tendsto y\n[GOAL]\ncase h.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Ico a x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nx : \u03b1\na\u271d : a \u2264 x\ny : \u03b1\nhxy : x < y\nn : \u2115\nhn : \u2200 (a : \u2115), n \u2264 a \u2192 y \u2264 xs a\n\u22a2 \u2203 x_1, x < xs x_1\n[PROOFSTEP]\nexact \u27e8n, hxy.trans_le (hn n le_rfl)\u27e9\n[GOAL]\ncase h.e'_5.h.e'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Ico a x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nh_Ici : Ici a = \u22c3 (n : \u2115), Ico a (xs n)\n\u22a2 \u2191\u2191\u03bc (Ici a) = \u2a06 (i : \u03b1), \u2191\u2191\u03bc (Ico a i)\n[PROOFSTEP]\nrw [h_Ici, measure_iUnion_eq_iSup, iSup_eq_iSup_subseq_of_monotone h_mono hxs_tendsto]\n[GOAL]\ncase h.e'_5.h.e'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Ico a x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nh_Ici : Ici a = \u22c3 (n : \u2115), Ico a (xs n)\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => Ico a (xs n)\n[PROOFSTEP]\nexact Monotone.directed_le fun i j hij => Ico_subset_Ico_right (hxs_mono hij)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Ioc x a)) atBot (\ud835\udcdd (\u2191\u2191\u03bc (Iic a)))\n[PROOFSTEP]\nhaveI : Nonempty \u03b1 := \u27e8a\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Ioc x a)) atBot (\ud835\udcdd (\u2191\u2191\u03bc (Iic a)))\n[PROOFSTEP]\nhave h_mono : Antitone fun x => \u03bc (Ioc x a) := fun i j hij => measure_mono (Ioc_subset_Ioc_left hij)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Ioc x a)) atBot (\ud835\udcdd (\u2191\u2191\u03bc (Iic a)))\n[PROOFSTEP]\nconvert tendsto_atBot_iSup h_mono\n[GOAL]\ncase h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\n\u22a2 \u2191\u2191\u03bc (Iic a) = \u2a06 (i : \u03b1), \u2191\u2191\u03bc (Ioc i a)\n[PROOFSTEP]\nobtain \u27e8xs, hxs_mono, hxs_tendsto\u27e9 := exists_seq_antitone_tendsto_atTop_atBot \u03b1\n[GOAL]\ncase h.e'_5.h.e'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\n\u22a2 \u2191\u2191\u03bc (Iic a) = \u2a06 (i : \u03b1), \u2191\u2191\u03bc (Ioc i a)\n[PROOFSTEP]\nhave h_Iic : Iic a = \u22c3 n, Ioc (xs n) a := by\n ext1 x\n simp only [mem_Iic, mem_iUnion, mem_Ioc, exists_and_right, iff_and_self]\n intro\n obtain \u27e8y, hxy\u27e9 := NoMinOrder.exists_lt x\n obtain \u27e8n, hn\u27e9 := tendsto_atTop_atBot.mp hxs_tendsto y\n exact \u27e8n, (hn n le_rfl).trans_lt hxy\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\n\u22a2 Iic a = \u22c3 (n : \u2115), Ioc (xs n) a\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nx : \u03b1\n\u22a2 x \u2208 Iic a \u2194 x \u2208 \u22c3 (n : \u2115), Ioc (xs n) a\n[PROOFSTEP]\nsimp only [mem_Iic, mem_iUnion, mem_Ioc, exists_and_right, iff_and_self]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nx : \u03b1\n\u22a2 x \u2264 a \u2192 \u2203 x_1, xs x_1 < x\n[PROOFSTEP]\nintro\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nx : \u03b1\na\u271d : x \u2264 a\n\u22a2 \u2203 x_1, xs x_1 < x\n[PROOFSTEP]\nobtain \u27e8y, hxy\u27e9 := NoMinOrder.exists_lt x\n[GOAL]\ncase h.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nx : \u03b1\na\u271d : x \u2264 a\ny : \u03b1\nhxy : y < x\n\u22a2 \u2203 x_1, xs x_1 < x\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := tendsto_atTop_atBot.mp hxs_tendsto y\n[GOAL]\ncase h.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nx : \u03b1\na\u271d : x \u2264 a\ny : \u03b1\nhxy : y < x\nn : \u2115\nhn : \u2200 (a : \u2115), n \u2264 a \u2192 xs a \u2264 y\n\u22a2 \u2203 x_1, xs x_1 < x\n[PROOFSTEP]\nexact \u27e8n, (hn n le_rfl).trans_lt hxy\u27e9\n[GOAL]\ncase h.e'_5.h.e'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nh_Iic : Iic a = \u22c3 (n : \u2115), Ioc (xs n) a\n\u22a2 \u2191\u2191\u03bc (Iic a) = \u2a06 (i : \u03b1), \u2191\u2191\u03bc (Ioc i a)\n[PROOFSTEP]\nrw [h_Iic, measure_iUnion_eq_iSup, iSup_eq_iSup_subseq_of_antitone h_mono hxs_tendsto]\n[GOAL]\ncase h.e'_5.h.e'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : IsCountablyGenerated atBot\n\u03bc : Measure \u03b1\na : \u03b1\nthis : Nonempty \u03b1\nh_mono : Antitone fun x => \u2191\u2191\u03bc (Ioc x a)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Antitone xs\nhxs_tendsto : Tendsto xs atTop atBot\nh_Iic : Iic a = \u22c3 (n : \u2115), Ioc (xs n) a\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => Ioc (xs n) a\n[PROOFSTEP]\nexact Monotone.directed_le fun i j hij => Ioc_subset_Ioc_left (hxs_mono hij)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Iic x)) atTop (\ud835\udcdd (\u2191\u2191\u03bc univ))\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b1\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\nh\u271d : IsEmpty \u03b1\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Iic x)) atTop (\ud835\udcdd (\u2191\u2191\u03bc univ))\n[PROOFSTEP]\nhave h1 : \u2200 x : \u03b1, Iic x = \u2205 := fun x => Subsingleton.elim _ _\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\nh\u271d : IsEmpty \u03b1\nh1 : \u2200 (x : \u03b1), Iic x = \u2205\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Iic x)) atTop (\ud835\udcdd (\u2191\u2191\u03bc univ))\n[PROOFSTEP]\nhave h2 : (univ : Set \u03b1) = \u2205 := Subsingleton.elim _ _\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\nh\u271d : IsEmpty \u03b1\nh1 : \u2200 (x : \u03b1), Iic x = \u2205\nh2 : univ = \u2205\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Iic x)) atTop (\ud835\udcdd (\u2191\u2191\u03bc univ))\n[PROOFSTEP]\nsimp_rw [h1, h2]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\nh\u271d : IsEmpty \u03b1\nh1 : \u2200 (x : \u03b1), Iic x = \u2205\nh2 : univ = \u2205\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc \u2205) atTop (\ud835\udcdd (\u2191\u2191\u03bc \u2205))\n[PROOFSTEP]\nexact tendsto_const_nhds\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\nh\u271d : Nonempty \u03b1\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Iic x)) atTop (\ud835\udcdd (\u2191\u2191\u03bc univ))\n[PROOFSTEP]\nhave h_mono : Monotone fun x => \u03bc (Iic x) := fun i j hij => measure_mono (Iic_subset_Iic.mpr hij)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\nh\u271d : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x)\n\u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Iic x)) atTop (\ud835\udcdd (\u2191\u2191\u03bc univ))\n[PROOFSTEP]\nconvert tendsto_atTop_iSup h_mono\n[GOAL]\ncase h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\nh\u271d : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x)\n\u22a2 \u2191\u2191\u03bc univ = \u2a06 (i : \u03b1), \u2191\u2191\u03bc (Iic i)\n[PROOFSTEP]\nobtain \u27e8xs, hxs_mono, hxs_tendsto\u27e9 := exists_seq_monotone_tendsto_atTop_atTop \u03b1\n[GOAL]\ncase h.e'_5.h.e'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\nh\u271d : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\n\u22a2 \u2191\u2191\u03bc univ = \u2a06 (i : \u03b1), \u2191\u2191\u03bc (Iic i)\n[PROOFSTEP]\nhave h_univ : (univ : Set \u03b1) = \u22c3 n, Iic (xs n) := by\n ext1 x\n simp only [mem_univ, mem_iUnion, mem_Iic, true_iff_iff]\n obtain \u27e8n, hn\u27e9 := tendsto_atTop_atTop.mp hxs_tendsto x\n exact \u27e8n, hn n le_rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\nh\u271d : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\n\u22a2 univ = \u22c3 (n : \u2115), Iic (xs n)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\nh\u271d : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nx : \u03b1\n\u22a2 x \u2208 univ \u2194 x \u2208 \u22c3 (n : \u2115), Iic (xs n)\n[PROOFSTEP]\nsimp only [mem_univ, mem_iUnion, mem_Iic, true_iff_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\nh\u271d : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nx : \u03b1\n\u22a2 \u2203 i, x \u2264 xs i\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := tendsto_atTop_atTop.mp hxs_tendsto x\n[GOAL]\ncase h.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\nh\u271d : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nx : \u03b1\nn : \u2115\nhn : \u2200 (a : \u2115), n \u2264 a \u2192 x \u2264 xs a\n\u22a2 \u2203 i, x \u2264 xs i\n[PROOFSTEP]\nexact \u27e8n, hn n le_rfl\u27e9\n[GOAL]\ncase h.e'_5.h.e'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\nh\u271d : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nh_univ : univ = \u22c3 (n : \u2115), Iic (xs n)\n\u22a2 \u2191\u2191\u03bc univ = \u2a06 (i : \u03b1), \u2191\u2191\u03bc (Iic i)\n[PROOFSTEP]\nrw [h_univ, measure_iUnion_eq_iSup, iSup_eq_iSup_subseq_of_monotone h_mono hxs_tendsto]\n[GOAL]\ncase h.e'_5.h.e'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : IsCountablyGenerated atTop\n\u03bc : Measure \u03b1\nh\u271d : Nonempty \u03b1\nh_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x)\nxs : \u2115 \u2192 \u03b1\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nh_univ : univ = \u22c3 (n : \u2115), Iic (xs n)\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => Iic (xs n)\n[PROOFSTEP]\nexact Monotone.directed_le fun i j hij => Iic_subset_Iic.mpr (hxs_mono hij)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : PartialOrder \u03b1\na b : \u03b1\nha : \u2191\u2191\u03bc {a} = 0\n\u22a2 Iio a =\u1da0[ae \u03bc] Iic a\n[PROOFSTEP]\nrw [\u2190 Iic_diff_right, diff_ae_eq_self, measure_mono_null (Set.inter_subset_right _ _) ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u22a2 \u00acIsFiniteMeasure \u03bc \u2194 \u2191\u2191\u03bc univ = \u22a4\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => fun h' => h'.measure_univ_lt_top.ne h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nh : \u00acIsFiniteMeasure \u03bc\n\u22a2 \u2191\u2191\u03bc univ = \u22a4\n[PROOFSTEP]\nby_contra h'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nh : \u00acIsFiniteMeasure \u03bc\nh' : \u00ac\u2191\u2191\u03bc univ = \u22a4\n\u22a2 False\n[PROOFSTEP]\nexact h \u27e8lt_top_iff_ne_top.mpr h'\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\nhs : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 \u2191\u2191(Measure.restrict \u03bc s) univ < \u22a4\n[PROOFSTEP]\nsimpa using hs.elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsFiniteMeasure \u03bc\n\u22a2 \u2191\u2191(Measure.restrict \u03bc s) univ < \u22a4\n[PROOFSTEP]\nsimpa using measure_lt_top \u03bc s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhs : MeasurableSet s\nht : MeasurableSet t\n\u03b5 : \u211d\u22650\u221e\nh : \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc t + \u03b5\n\u22a2 \u2191\u2191\u03bc t\u1d9c \u2264 \u2191\u2191\u03bc s\u1d9c + \u03b5\n[PROOFSTEP]\nrw [measure_compl ht (measure_ne_top \u03bc _), measure_compl hs (measure_ne_top \u03bc _), tsub_le_iff_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhs : MeasurableSet s\nht : MeasurableSet t\n\u03b5 : \u211d\u22650\u221e\nh : \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc t + \u03b5\n\u22a2 \u2191\u2191\u03bc univ \u2264 \u2191\u2191\u03bc univ - \u2191\u2191\u03bc s + \u03b5 + \u2191\u2191\u03bc t\n[PROOFSTEP]\ncalc\n \u03bc univ = \u03bc univ - \u03bc s + \u03bc s := (tsub_add_cancel_of_le <| measure_mono s.subset_univ).symm\n _ \u2264 \u03bc univ - \u03bc s + (\u03bc t + \u03b5) := (add_le_add_left h _)\n _ = _ := by rw [add_right_comm, add_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhs : MeasurableSet s\nht : MeasurableSet t\n\u03b5 : \u211d\u22650\u221e\nh : \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc t + \u03b5\n\u22a2 \u2191\u2191\u03bc univ - \u2191\u2191\u03bc s + (\u2191\u2191\u03bc t + \u03b5) = \u2191\u2191\u03bc univ - \u2191\u2191\u03bc s + \u03b5 + \u2191\u2191\u03bc t\n[PROOFSTEP]\nrw [add_right_comm, add_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u22a2 \u2191\u21910 univ < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsEmpty \u03b1\n\u22a2 IsFiniteMeasure \u03bc\n[PROOFSTEP]\nrw [eq_zero_of_isEmpty \u03bc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsEmpty \u03b1\n\u22a2 IsFiniteMeasure 0\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\n\u22a2 \u2191\u2191(\u03bc + \u03bd) univ < \u22a4\n[PROOFSTEP]\nrw [Measure.coe_add, Pi.add_apply, ENNReal.add_lt_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\n\u22a2 \u2191\u2191\u03bc univ < \u22a4 \u2227 \u2191\u2191\u03bd univ < \u22a4\n[PROOFSTEP]\nexact \u27e8measure_lt_top _ _, measure_lt_top _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u22a2 \u2191\u2191((\u2191\u2191\u03bc univ)\u207b\u00b9 \u2022 \u03bc) univ < \u22a4\n[PROOFSTEP]\nrw [smul_apply, smul_eq_mul, \u2190 ENNReal.div_eq_inv_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u22a2 \u2191\u2191\u03bc univ / \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nexact ENNReal.div_self_le_one.trans_lt ENNReal.one_lt_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR\u271d : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nR : Type ?u.578721\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nr : R\n\u22a2 IsFiniteMeasure (r \u2022 \u03bc)\n[PROOFSTEP]\nrw [\u2190 smul_one_smul \u211d\u22650 r \u03bc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR\u271d : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nR : Type ?u.578721\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nr : R\n\u22a2 IsFiniteMeasure ((r \u2022 1) \u2022 \u03bc)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\n\u22a2 IsFiniteMeasure (map f \u03bc)\n[PROOFSTEP]\nby_cases hf : AEMeasurable f \u03bc\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 IsFiniteMeasure (map f \u03bc)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase pos.measure_univ_lt_top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 \u2191\u2191(map f \u03bc) univ < \u22a4\n[PROOFSTEP]\nrw [map_apply_of_aemeasurable hf MeasurableSet.univ]\n[GOAL]\ncase pos.measure_univ_lt_top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 \u2191\u2191\u03bc (f \u207b\u00b9' univ) < \u22a4\n[PROOFSTEP]\nexact measure_lt_top \u03bc _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : \u00acAEMeasurable f\n\u22a2 IsFiniteMeasure (map f \u03bc)\n[PROOFSTEP]\nrw [map_of_not_aemeasurable hf]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : \u00acAEMeasurable f\n\u22a2 IsFiniteMeasure 0\n[PROOFSTEP]\nexact MeasureTheory.isFiniteMeasureZero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 measureUnivNNReal \u03bc = 0 \u2194 \u03bc = 0\n[PROOFSTEP]\nrw [\u2190 MeasureTheory.Measure.measure_univ_eq_zero, \u2190 coe_measureUnivNNReal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 measureUnivNNReal \u03bc = 0 \u2194 \u2191(measureUnivNNReal \u03bc) = 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc : \u03bc \u2260 0\n\u22a2 0 < measureUnivNNReal \u03bc\n[PROOFSTEP]\ncontrapose! h\u03bc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nh\u03bc : measureUnivNNReal \u03bc \u2264 0\n\u22a2 \u03bc = 0\n[PROOFSTEP]\nsimpa [measureUnivNNReal_eq_zero, le_zero_iff] using h\u03bc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nh\u03bc : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u2115), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\n\u22a2 Summable fun x => ENNReal.toReal (\u2191\u2191\u03bc (f x))\n[PROOFSTEP]\napply ENNReal.summable_toReal\n[GOAL]\ncase hsum\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nh\u03bc : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u2115), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\n\u22a2 \u2211' (x : \u2115), \u2191\u2191\u03bc (f x) \u2260 \u22a4\n[PROOFSTEP]\nrw [\u2190 MeasureTheory.measure_iUnion hf\u2082 hf\u2081]\n[GOAL]\ncase hsum\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nh\u03bc : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 Set \u03b1\nhf\u2081 : \u2200 (i : \u2115), MeasurableSet (f i)\nhf\u2082 : Pairwise (Disjoint on f)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u2115), f i) \u2260 \u22a4\n[PROOFSTEP]\nexact ne_of_lt (measure_lt_top _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhs : NullMeasurableSet s\n\u22a2 s =\u1da0[ae \u03bc] univ \u2194 \u2191\u2191\u03bc s = \u2191\u2191\u03bc univ\n[PROOFSTEP]\nrefine' \u27e8measure_congr, fun h => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhs : NullMeasurableSet s\nh : \u2191\u2191\u03bc s = \u2191\u2191\u03bc univ\n\u22a2 s =\u1da0[ae \u03bc] univ\n[PROOFSTEP]\nobtain \u27e8t, -, ht\u2081, ht\u2082\u27e9 := hs.exists_measurable_subset_ae_eq\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nhs : NullMeasurableSet s\nh : \u2191\u2191\u03bc s = \u2191\u2191\u03bc univ\nt : Set \u03b1\nht\u2081 : MeasurableSet t\nht\u2082 : t =\u1da0[ae \u03bc] s\n\u22a2 s =\u1da0[ae \u03bc] univ\n[PROOFSTEP]\nexact\n ht\u2082.symm.trans\n (ae_eq_of_subset_of_measure_ge (subset_univ t) (Eq.le ((measure_congr ht\u2082).trans h).symm) ht\u2081\n (measure_ne_top \u03bc univ))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\np : \u03b1 \u2192 Prop\nhp : NullMeasurableSet {a | p a}\n\u22a2 (\u2200\u1d50 (a : \u03b1) \u2202\u03bc, p a) \u2194 \u2191\u2191\u03bc {a | p a} = \u2191\u2191\u03bc univ\n[PROOFSTEP]\nrw [\u2190 ae_eq_univ_iff_measure_eq hp, eventuallyEq_univ, eventually_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 |ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t)| \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2206 t))\n[PROOFSTEP]\nhave hst : \u03bc (s \\ t) \u2260 \u221e := (measure_lt_top_of_subset (diff_subset s t) hs').ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\n\u22a2 |ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t)| \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2206 t))\n[PROOFSTEP]\nhave hts : \u03bc (t \\ s) \u2260 \u221e := (measure_lt_top_of_subset (diff_subset t s) ht').ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\n\u22a2 |ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t)| \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2206 t))\n[PROOFSTEP]\nsuffices : (\u03bc s).toReal - (\u03bc t).toReal = (\u03bc (s \\ t)).toReal - (\u03bc (t \\ s)).toReal\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\nthis : ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) = ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))\n\u22a2 |ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t)| \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2206 t))\n[PROOFSTEP]\nrw [this, measure_symmDiff_eq hs ht, ENNReal.toReal_add hst hts]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\nthis : ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) = ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))\n\u22a2 |ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))| \u2264\n ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) + ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))\n[PROOFSTEP]\nconvert abs_sub (\u03bc (s \\ t)).toReal (\u03bc (t \\ s)).toReal\n[GOAL]\ncase h.e'_4.h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\nthis : ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) = ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) = |ENNReal.toReal (\u2191\u2191\u03bc (s \\ t))|\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_4.h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\nthis : ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) = ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (t \\ s)) = |ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))|\n[PROOFSTEP]\nsimp\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) = ENNReal.toReal (\u2191\u2191\u03bc (s \\ t)) - ENNReal.toReal (\u2191\u2191\u03bc (t \\ s))\n[PROOFSTEP]\nrw [measure_diff' s ht ht', measure_diff' t hs hs',\n ENNReal.toReal_sub_of_le measure_le_measure_union_right (measure_union_ne_top hs' ht'),\n ENNReal.toReal_sub_of_le measure_le_measure_union_right (measure_union_ne_top ht' hs'), union_comm t s]\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) =\n ENNReal.toReal (\u2191\u2191\u03bc (s \u222a t)) - ENNReal.toReal (\u2191\u2191\u03bc t) - (ENNReal.toReal (\u2191\u2191\u03bc (s \u222a t)) - ENNReal.toReal (\u2191\u2191\u03bc s))\n[PROOFSTEP]\nabel\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\nhs' : \u2191\u2191\u03bc s \u2260 \u22a4\nht' : \u2191\u2191\u03bc t \u2260 \u22a4\nhst : \u2191\u2191\u03bc (s \\ t) \u2260 \u22a4\nhts : \u2191\u2191\u03bc (t \\ s) \u2260 \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc s) - ENNReal.toReal (\u2191\u2191\u03bc t) =\n ENNReal.toReal (\u2191\u2191\u03bc (s \u222a t)) - ENNReal.toReal (\u2191\u2191\u03bc t) - (ENNReal.toReal (\u2191\u2191\u03bc (s \u222a t)) - ENNReal.toReal (\u2191\u2191\u03bc s))\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\n\u22a2 \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nsimp only [measure_univ, ENNReal.one_lt_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\n\u22a2 \u00ac\u2191\u2191\u03bc univ = 0\n[PROOFSTEP]\nsimp [measure_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 \u2191\u2191(Measure.map f \u03bc) univ = 1\n[PROOFSTEP]\nsimp [map_apply_of_aemeasurable, hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s\u1d9c = 1 - \u2191\u2191\u03bc s\n[PROOFSTEP]\nsimpa only [measure_univ] using measure_compl hs (measure_lt_top \u03bc s).ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s\u1d9c = 0 \u2194 \u2191\u2191\u03bc s = 1\n[PROOFSTEP]\nrw [prob_compl_eq_one_sub hs, tsub_eq_zero_iff_le, one_le_prob_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s\u1d9c = 1 \u2194 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nrw [\u2190 prob_compl_eq_zero_iff hs.compl, compl_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : NoAtoms \u03bc\na : \u03b1\n\u22a2 restrict \u03bc {a} = 0\n[PROOFSTEP]\nsimp only [measure_singleton, Measure.restrict_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : NoAtoms \u03bc\ns : Set \u03b1\n\u22a2 NoAtoms (restrict \u03bc s)\n[PROOFSTEP]\nrefine' \u27e8fun x => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : NoAtoms \u03bc\ns : Set \u03b1\nx : \u03b1\n\u22a2 \u2191\u2191(restrict \u03bc s) {x} = 0\n[PROOFSTEP]\nobtain \u27e8t, hxt, ht1, ht2\u27e9 := exists_measurable_superset_of_null (measure_singleton x : \u03bc { x } = 0)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : NoAtoms \u03bc\ns : Set \u03b1\nx : \u03b1\nt : Set \u03b1\nhxt : {x} \u2286 t\nht1 : MeasurableSet t\nht2 : \u2191\u2191\u03bc t = 0\n\u22a2 \u2191\u2191(restrict \u03bc s) {x} = 0\n[PROOFSTEP]\napply measure_mono_null hxt\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : NoAtoms \u03bc\ns : Set \u03b1\nx : \u03b1\nt : Set \u03b1\nhxt : {x} \u2286 t\nht1 : MeasurableSet t\nht2 : \u2191\u2191\u03bc t = 0\n\u22a2 \u2191\u2191(restrict \u03bc s) t = 0\n[PROOFSTEP]\nrw [Measure.restrict_apply ht1]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : NoAtoms \u03bc\ns : Set \u03b1\nx : \u03b1\nt : Set \u03b1\nhxt : {x} \u2286 t\nht1 : MeasurableSet t\nht2 : \u2191\u2191\u03bc t = 0\n\u22a2 \u2191\u2191\u03bc (t \u2229 s) = 0\n[PROOFSTEP]\napply measure_mono_null (inter_subset_left t s) ht2\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\ninst\u271d\u00b9 : NoAtoms \u03bc\u271d\n\u03b1 : Type u_8\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nh : Set.Countable s\n\u03bc : Measure \u03b1\ninst\u271d : NoAtoms \u03bc\n\u22a2 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nrw [\u2190 biUnion_of_singleton s, \u2190 nonpos_iff_eq_zero]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\ninst\u271d\u00b9 : NoAtoms \u03bc\u271d\n\u03b1 : Type u_8\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nh : Set.Countable s\n\u03bc : Measure \u03b1\ninst\u271d : NoAtoms \u03bc\n\u22a2 \u2191\u2191\u03bc (\u22c3 (x : \u03b1) (_ : x \u2208 s), {x}) \u2264 0\n[PROOFSTEP]\nrefine' le_trans (measure_biUnion_le h _) _\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\ninst\u271d\u00b9 : NoAtoms \u03bc\u271d\n\u03b1 : Type u_8\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nh : Set.Countable s\n\u03bc : Measure \u03b1\ninst\u271d : NoAtoms \u03bc\n\u22a2 \u2211' (p : \u2191s), \u2191\u2191\u03bc {\u2191p} \u2264 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\ninst\u271d\u00b9 : NoAtoms \u03bc\u271d\n\u03b1 : Type u_8\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nh : Set.Countable s\n\u03bc : Measure \u03b1\ninst\u271d : NoAtoms \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s\n[PROOFSTEP]\nsimpa only [ae_iff, Classical.not_not] using h.measure_zero \u03bc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b3 : Type u_8\nf g : \u03b1 \u2192 \u03b3\ns : Set \u03b1\nhs_zero : \u2191\u2191\u03bc s = 0\n\u22a2 (fun x => if x \u2208 s then f x else g x) =\u1da0[ae \u03bc] g\n[PROOFSTEP]\nhave h_ss : s\u1d9c \u2286 {a : \u03b1 | ite (a \u2208 s) (f a) (g a) = g a} := fun x hx => by simp [(Set.mem_compl_iff _ _).mp hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b3 : Type u_8\nf g : \u03b1 \u2192 \u03b3\ns : Set \u03b1\nhs_zero : \u2191\u2191\u03bc s = 0\nx : \u03b1\nhx : x \u2208 s\u1d9c\n\u22a2 x \u2208 {a | (if a \u2208 s then f a else g a) = g a}\n[PROOFSTEP]\nsimp [(Set.mem_compl_iff _ _).mp hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b3 : Type u_8\nf g : \u03b1 \u2192 \u03b3\ns : Set \u03b1\nhs_zero : \u2191\u2191\u03bc s = 0\nh_ss : s\u1d9c \u2286 {a | (if a \u2208 s then f a else g a) = g a}\n\u22a2 (fun x => if x \u2208 s then f x else g x) =\u1da0[ae \u03bc] g\n[PROOFSTEP]\nrefine' measure_mono_null _ hs_zero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b3 : Type u_8\nf g : \u03b1 \u2192 \u03b3\ns : Set \u03b1\nhs_zero : \u2191\u2191\u03bc s = 0\nh_ss : s\u1d9c \u2286 {a | (if a \u2208 s then f a else g a) = g a}\n\u22a2 {x | (fun x => (fun x => if x \u2208 s then f x else g x) x = g x) x}\u1d9c \u2286 s\n[PROOFSTEP]\nconv_rhs => rw [\u2190 compl_compl s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b3 : Type u_8\nf g : \u03b1 \u2192 \u03b3\ns : Set \u03b1\nhs_zero : \u2191\u2191\u03bc s = 0\nh_ss : s\u1d9c \u2286 {a | (if a \u2208 s then f a else g a) = g a}\n| s\n[PROOFSTEP]\nrw [\u2190 compl_compl s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b3 : Type u_8\nf g : \u03b1 \u2192 \u03b3\ns : Set \u03b1\nhs_zero : \u2191\u2191\u03bc s = 0\nh_ss : s\u1d9c \u2286 {a | (if a \u2208 s then f a else g a) = g a}\n| s\n[PROOFSTEP]\nrw [\u2190 compl_compl s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b3 : Type u_8\nf g : \u03b1 \u2192 \u03b3\ns : Set \u03b1\nhs_zero : \u2191\u2191\u03bc s = 0\nh_ss : s\u1d9c \u2286 {a | (if a \u2208 s then f a else g a) = g a}\n| s\n[PROOFSTEP]\nrw [\u2190 compl_compl s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b3 : Type u_8\nf g : \u03b1 \u2192 \u03b3\ns : Set \u03b1\nhs_zero : \u2191\u2191\u03bc s = 0\nh_ss : s\u1d9c \u2286 {a | (if a \u2208 s then f a else g a) = g a}\n\u22a2 {x | (fun x => (fun x => if x \u2208 s then f x else g x) x = g x) x}\u1d9c \u2286 s\u1d9c\u1d9c\n[PROOFSTEP]\nrwa [Set.compl_subset_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b3 : Type u_8\nf g : \u03b1 \u2192 \u03b3\ns : Set \u03b1\nhs_zero : \u2191\u2191\u03bc s\u1d9c = 0\n\u22a2 (fun x => if x \u2208 s then f x else g x) =\u1da0[ae \u03bc] f\n[PROOFSTEP]\nchange s \u2208 \u03bc.ae at hs_zero \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b3 : Type u_8\nf g : \u03b1 \u2192 \u03b3\ns : Set \u03b1\nhs_zero : s \u2208 ae \u03bc\n\u22a2 (fun x => if x \u2208 s then f x else g x) =\u1da0[ae \u03bc] f\n[PROOFSTEP]\nfilter_upwards [hs_zero]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b3 : Type u_8\nf g : \u03b1 \u2192 \u03b3\ns : Set \u03b1\nhs_zero : s \u2208 ae \u03bc\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2192 (if a \u2208 s then f a else g a) = f a\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b3 : Type u_8\nf g : \u03b1 \u2192 \u03b3\ns : Set \u03b1\nhs_zero : s \u2208 ae \u03bc\na\u271d\u00b9 : \u03b1\na\u271d : a\u271d\u00b9 \u2208 s\n\u22a2 (if a\u271d\u00b9 \u2208 s then f a\u271d\u00b9 else g a\u271d\u00b9) = f a\u271d\u00b9\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b3 : Type u_8\nf g : \u03b1 \u2192 \u03b3\ns : Set \u03b1\nhs_zero : s \u2208 ae \u03bc\na\u271d\u00b9 : \u03b1\na\u271d : a\u271d\u00b9 \u2208 s\n\u22a2 f a\u271d\u00b9 = f a\u271d\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0\u271d : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u22a2 \u2191\u2191\u03bc \u2205 < \u22a4\n[PROOFSTEP]\nsimp only [measure_empty, WithTop.zero_lt_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh : SigmaFinite \u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc\n ((fun n =>\n toMeasurable \u03bc (FiniteSpanningSetsIn.set (Nonempty.some (_ : Nonempty (FiniteSpanningSetsIn \u03bc univ))) n))\n n) <\n \u22a4\n[PROOFSTEP]\nrw [measure_toMeasurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh : SigmaFinite \u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (FiniteSpanningSetsIn.set (Nonempty.some (_ : Nonempty (FiniteSpanningSetsIn \u03bc univ))) n) < \u22a4\n[PROOFSTEP]\nexact h.out.some.finite n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 \u22c3 (i : \u2115), spanningSets \u03bc i = univ\n[PROOFSTEP]\nsimp_rw [spanningSets, iUnion_accumulate, \u03bc.toFiniteSpanningSetsIn.spanning]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\nn : \u2115\n\u22a2 spanningSetsIndex \u03bc x = n \u2194 x \u2208 disjointed (spanningSets \u03bc) n\n[PROOFSTEP]\nconvert Set.ext_iff.1 (preimage_spanningSetsIndex_singleton \u03bc n) x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(restrict \u03bc (\u22c3 (i : \u2115), spanningSets \u03bc i)) s = \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [iUnion_spanningSets, restrict_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nr : \u211d\u22650\u221e\nhs : MeasurableSet s\nh's : r < \u2191\u2191\u03bc s\n\u22a2 \u2203 t, MeasurableSet t \u2227 t \u2286 s \u2227 r < \u2191\u2191\u03bc t \u2227 \u2191\u2191\u03bc t < \u22a4\n[PROOFSTEP]\nrw [\u2190 iSup_restrict_spanningSets hs, @lt_iSup_iff _ _ _ r fun i : \u2115 => \u03bc.restrict (spanningSets \u03bc i) s] at h's \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nr : \u211d\u22650\u221e\nhs : MeasurableSet s\nh's\u271d : r < \u2191\u2191\u03bc s\nh's : \u2203 i, r < \u2191\u2191(restrict \u03bc (spanningSets \u03bc i)) s\n\u22a2 \u2203 t, MeasurableSet t \u2227 t \u2286 s \u2227 r < \u2191\u2191\u03bc t \u2227 \u2191\u2191\u03bc t < \u22a4\n[PROOFSTEP]\nrcases h's with \u27e8n, hn\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nr : \u211d\u22650\u221e\nhs : MeasurableSet s\nh's : r < \u2191\u2191\u03bc s\nn : \u2115\nhn : r < \u2191\u2191(restrict \u03bc (spanningSets \u03bc n)) s\n\u22a2 \u2203 t, MeasurableSet t \u2227 t \u2286 s \u2227 r < \u2191\u2191\u03bc t \u2227 \u2191\u2191\u03bc t < \u22a4\n[PROOFSTEP]\nsimp only [restrict_apply hs] at hn \n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nr : \u211d\u22650\u221e\nhs : MeasurableSet s\nh's : r < \u2191\u2191\u03bc s\nn : \u2115\nhn : r < \u2191\u2191\u03bc (s \u2229 spanningSets \u03bc n)\n\u22a2 \u2203 t, MeasurableSet t \u2227 t \u2286 s \u2227 r < \u2191\u2191\u03bc t \u2227 \u2191\u2191\u03bc t < \u22a4\n[PROOFSTEP]\nrefine' \u27e8s \u2229 spanningSets \u03bc n, hs.inter (measurable_spanningSets _ _), inter_subset_left _ _, hn, _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nr : \u211d\u22650\u221e\nhs : MeasurableSet s\nh's : r < \u2191\u2191\u03bc s\nn : \u2115\nhn : r < \u2191\u2191\u03bc (s \u2229 spanningSets \u03bc n)\n\u22a2 \u2191\u2191\u03bc (s \u2229 spanningSets \u03bc n) < \u22a4\n[PROOFSTEP]\nexact (measure_mono (inter_subset_right _ _)).trans_lt (measure_spanningSets_lt_top _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\n\u22a2 (\u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 spanningSets \u03bc n) = 0) \u2194 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nnth_rw 2 [show s = \u22c3 n, s \u2229 spanningSets \u03bc n by rw [\u2190 inter_iUnion, iUnion_spanningSets, inter_univ]]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\n\u22a2 s = \u22c3 (n : \u2115), s \u2229 spanningSets \u03bc n\n[PROOFSTEP]\nrw [\u2190 inter_iUnion, iUnion_spanningSets, inter_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\n\u22a2 (\u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 spanningSets \u03bc n) = 0) \u2194 \u2191\u2191\u03bc (\u22c3 (n : \u2115), s \u2229 spanningSets \u03bc n) = 0\n[PROOFSTEP]\nrw [measure_iUnion_null_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\n\u22a2 (\u2203 n, 0 < \u2191\u2191\u03bc (s \u2229 spanningSets \u03bc n)) \u2194 0 < \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [\u2190 not_iff_not]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\n\u22a2 (\u00ac\u2203 n, 0 < \u2191\u2191\u03bc (s \u2229 spanningSets \u03bc n)) \u2194 \u00ac0 < \u2191\u2191\u03bc s\n[PROOFSTEP]\nsimp only [not_exists, not_lt, nonpos_iff_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\n\u22a2 (\u2200 (x : \u2115), \u2191\u2191\u03bc (s \u2229 spanningSets \u03bc x) = 0) \u2194 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nexact forall_measure_inter_spanningSets_eq_zero s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nUnion_As_finite : \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i) \u2260 \u22a4\n\u22a2 Set.Finite {i | \u03b5 \u2264 \u2191\u2191\u03bc (As i)}\n[PROOFSTEP]\nby_contra con\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nUnion_As_finite : \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i) \u2260 \u22a4\ncon : \u00acSet.Finite {i | \u03b5 \u2264 \u2191\u2191\u03bc (As i)}\n\u22a2 False\n[PROOFSTEP]\nhave aux :=\n lt_of_le_of_lt (tsum_meas_le_meas_iUnion_of_disjoint \u03bc As_mble As_disj) (lt_top_iff_ne_top.mpr Union_As_finite)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nUnion_As_finite : \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i) \u2260 \u22a4\ncon : \u00acSet.Finite {i | \u03b5 \u2264 \u2191\u2191\u03bc (As i)}\naux : \u2211' (i : \u03b9), \u2191\u2191\u03bc (As i) < \u22a4\n\u22a2 False\n[PROOFSTEP]\nexact con (ENNReal.finite_const_le_of_tsum_ne_top aux.ne \u03b5_pos.ne.symm)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nUnion_As_finite : \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i) \u2260 \u22a4\n\u22a2 Set.Countable {i | 0 < \u2191\u2191\u03bc (As i)}\n[PROOFSTEP]\nset posmeas := {i : \u03b9 | 0 < \u03bc (As i)} with posmeas_def\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nUnion_As_finite : \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i) \u2260 \u22a4\nposmeas : Set \u03b9 := {i | 0 < \u2191\u2191\u03bc (As i)}\nposmeas_def : posmeas = {i | 0 < \u2191\u2191\u03bc (As i)}\n\u22a2 Set.Countable posmeas\n[PROOFSTEP]\nrcases exists_seq_strictAnti_tendsto' (zero_lt_one : (0 : \u211d\u22650\u221e) < 1) with \u27e8as, _, as_mem, as_lim\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nUnion_As_finite : \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i) \u2260 \u22a4\nposmeas : Set \u03b9 := {i | 0 < \u2191\u2191\u03bc (As i)}\nposmeas_def : posmeas = {i | 0 < \u2191\u2191\u03bc (As i)}\nas : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictAnti as\nas_mem : \u2200 (n : \u2115), as n \u2208 Ioo 0 1\nas_lim : Tendsto as atTop (\ud835\udcdd 0)\n\u22a2 Set.Countable posmeas\n[PROOFSTEP]\nset fairmeas := fun n : \u2115 => {i : \u03b9 | as n \u2264 \u03bc (As i)}\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nUnion_As_finite : \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i) \u2260 \u22a4\nposmeas : Set \u03b9 := {i | 0 < \u2191\u2191\u03bc (As i)}\nposmeas_def : posmeas = {i | 0 < \u2191\u2191\u03bc (As i)}\nas : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictAnti as\nas_mem : \u2200 (n : \u2115), as n \u2208 Ioo 0 1\nas_lim : Tendsto as atTop (\ud835\udcdd 0)\nfairmeas : \u2115 \u2192 Set \u03b9 := fun n => {i | as n \u2264 \u2191\u2191\u03bc (As i)}\n\u22a2 Set.Countable posmeas\n[PROOFSTEP]\nhave countable_union : posmeas = \u22c3 n, fairmeas n :=\n by\n have fairmeas_eq : \u2200 n, fairmeas n = (fun i => \u03bc (As i)) \u207b\u00b9' Ici (as n) := fun n =>\n by\n simp only []\n rfl\n simpa only [fairmeas_eq, posmeas_def, \u2190 preimage_iUnion,\n iUnion_Ici_eq_Ioi_of_lt_of_tendsto (0 : \u211d\u22650\u221e) (fun n => (as_mem n).1) as_lim]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nUnion_As_finite : \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i) \u2260 \u22a4\nposmeas : Set \u03b9 := {i | 0 < \u2191\u2191\u03bc (As i)}\nposmeas_def : posmeas = {i | 0 < \u2191\u2191\u03bc (As i)}\nas : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictAnti as\nas_mem : \u2200 (n : \u2115), as n \u2208 Ioo 0 1\nas_lim : Tendsto as atTop (\ud835\udcdd 0)\nfairmeas : \u2115 \u2192 Set \u03b9 := fun n => {i | as n \u2264 \u2191\u2191\u03bc (As i)}\n\u22a2 posmeas = \u22c3 (n : \u2115), fairmeas n\n[PROOFSTEP]\nhave fairmeas_eq : \u2200 n, fairmeas n = (fun i => \u03bc (As i)) \u207b\u00b9' Ici (as n) := fun n =>\n by\n simp only []\n rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nUnion_As_finite : \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i) \u2260 \u22a4\nposmeas : Set \u03b9 := {i | 0 < \u2191\u2191\u03bc (As i)}\nposmeas_def : posmeas = {i | 0 < \u2191\u2191\u03bc (As i)}\nas : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictAnti as\nas_mem : \u2200 (n : \u2115), as n \u2208 Ioo 0 1\nas_lim : Tendsto as atTop (\ud835\udcdd 0)\nfairmeas : \u2115 \u2192 Set \u03b9 := fun n => {i | as n \u2264 \u2191\u2191\u03bc (As i)}\nn : \u2115\n\u22a2 fairmeas n = (fun i => \u2191\u2191\u03bc (As i)) \u207b\u00b9' Ici (as n)\n[PROOFSTEP]\nsimp only []\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nUnion_As_finite : \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i) \u2260 \u22a4\nposmeas : Set \u03b9 := {i | 0 < \u2191\u2191\u03bc (As i)}\nposmeas_def : posmeas = {i | 0 < \u2191\u2191\u03bc (As i)}\nas : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictAnti as\nas_mem : \u2200 (n : \u2115), as n \u2208 Ioo 0 1\nas_lim : Tendsto as atTop (\ud835\udcdd 0)\nfairmeas : \u2115 \u2192 Set \u03b9 := fun n => {i | as n \u2264 \u2191\u2191\u03bc (As i)}\nn : \u2115\n\u22a2 {i | as n \u2264 \u2191\u2191\u03bc (As i)} = (fun i => \u2191\u2191\u03bc (As i)) \u207b\u00b9' Ici (as n)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nUnion_As_finite : \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i) \u2260 \u22a4\nposmeas : Set \u03b9 := {i | 0 < \u2191\u2191\u03bc (As i)}\nposmeas_def : posmeas = {i | 0 < \u2191\u2191\u03bc (As i)}\nas : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictAnti as\nas_mem : \u2200 (n : \u2115), as n \u2208 Ioo 0 1\nas_lim : Tendsto as atTop (\ud835\udcdd 0)\nfairmeas : \u2115 \u2192 Set \u03b9 := fun n => {i | as n \u2264 \u2191\u2191\u03bc (As i)}\nfairmeas_eq : \u2200 (n : \u2115), fairmeas n = (fun i => \u2191\u2191\u03bc (As i)) \u207b\u00b9' Ici (as n)\n\u22a2 posmeas = \u22c3 (n : \u2115), fairmeas n\n[PROOFSTEP]\nsimpa only [fairmeas_eq, posmeas_def, \u2190 preimage_iUnion,\n iUnion_Ici_eq_Ioi_of_lt_of_tendsto (0 : \u211d\u22650\u221e) (fun n => (as_mem n).1) as_lim]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nUnion_As_finite : \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i) \u2260 \u22a4\nposmeas : Set \u03b9 := {i | 0 < \u2191\u2191\u03bc (As i)}\nposmeas_def : posmeas = {i | 0 < \u2191\u2191\u03bc (As i)}\nas : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictAnti as\nas_mem : \u2200 (n : \u2115), as n \u2208 Ioo 0 1\nas_lim : Tendsto as atTop (\ud835\udcdd 0)\nfairmeas : \u2115 \u2192 Set \u03b9 := fun n => {i | as n \u2264 \u2191\u2191\u03bc (As i)}\ncountable_union : posmeas = \u22c3 (n : \u2115), fairmeas n\n\u22a2 Set.Countable posmeas\n[PROOFSTEP]\nrw [countable_union]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nUnion_As_finite : \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i) \u2260 \u22a4\nposmeas : Set \u03b9 := {i | 0 < \u2191\u2191\u03bc (As i)}\nposmeas_def : posmeas = {i | 0 < \u2191\u2191\u03bc (As i)}\nas : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictAnti as\nas_mem : \u2200 (n : \u2115), as n \u2208 Ioo 0 1\nas_lim : Tendsto as atTop (\ud835\udcdd 0)\nfairmeas : \u2115 \u2192 Set \u03b9 := fun n => {i | as n \u2264 \u2191\u2191\u03bc (As i)}\ncountable_union : posmeas = \u22c3 (n : \u2115), fairmeas n\n\u22a2 Set.Countable (\u22c3 (n : \u2115), fairmeas n)\n[PROOFSTEP]\nrefine' countable_iUnion fun n => Finite.countable _\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nUnion_As_finite : \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i) \u2260 \u22a4\nposmeas : Set \u03b9 := {i | 0 < \u2191\u2191\u03bc (As i)}\nposmeas_def : posmeas = {i | 0 < \u2191\u2191\u03bc (As i)}\nas : \u2115 \u2192 \u211d\u22650\u221e\nleft\u271d : StrictAnti as\nas_mem : \u2200 (n : \u2115), as n \u2208 Ioo 0 1\nas_lim : Tendsto as atTop (\ud835\udcdd 0)\nfairmeas : \u2115 \u2192 Set \u03b9 := fun n => {i | as n \u2264 \u2191\u2191\u03bc (As i)}\ncountable_union : posmeas = \u22c3 (n : \u2115), fairmeas n\nn : \u2115\n\u22a2 Set.Finite (fairmeas n)\n[PROOFSTEP]\nrefine' finite_const_le_meas_of_disjoint_iUnion \u03bc (as_mem n).1 As_mble As_disj Union_As_finite\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\n\u22a2 Set.Countable {i | 0 < \u2191\u2191\u03bc (As i)}\n[PROOFSTEP]\nhave obs : {i : \u03b9 | 0 < \u03bc (As i)} \u2286 \u22c3 n, {i : \u03b9 | 0 < \u03bc (As i \u2229 spanningSets \u03bc n)} :=\n by\n intro i i_in_nonzeroes\n by_contra con\n simp only [mem_iUnion, mem_setOf_eq, not_exists, not_lt, nonpos_iff_eq_zero] at *\n simp [(forall_measure_inter_spanningSets_eq_zero _).mp con] at i_in_nonzeroes \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\n\u22a2 {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 (n : \u2115), {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\n[PROOFSTEP]\nintro i i_in_nonzeroes\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\ni : \u03b9\ni_in_nonzeroes : i \u2208 {i | 0 < \u2191\u2191\u03bc (As i)}\n\u22a2 i \u2208 \u22c3 (n : \u2115), {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\n[PROOFSTEP]\nby_contra con\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\ni : \u03b9\ni_in_nonzeroes : i \u2208 {i | 0 < \u2191\u2191\u03bc (As i)}\ncon : \u00aci \u2208 \u22c3 (n : \u2115), {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\n\u22a2 False\n[PROOFSTEP]\nsimp only [mem_iUnion, mem_setOf_eq, not_exists, not_lt, nonpos_iff_eq_zero] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\ni : \u03b9\ni_in_nonzeroes : 0 < \u2191\u2191\u03bc (As i)\ncon : \u2200 (x : \u2115), \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc x) = 0\n\u22a2 False\n[PROOFSTEP]\nsimp [(forall_measure_inter_spanningSets_eq_zero _).mp con] at i_in_nonzeroes \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 (n : \u2115), {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\n\u22a2 Set.Countable {i | 0 < \u2191\u2191\u03bc (As i)}\n[PROOFSTEP]\napply Countable.mono obs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 (n : \u2115), {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\n\u22a2 Set.Countable (\u22c3 (n : \u2115), {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)})\n[PROOFSTEP]\nrefine' countable_iUnion fun n => countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top \u03bc _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 (n : \u2115), {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\nn : \u2115\n\u22a2 \u2200 (i : \u03b9), MeasurableSet (As i \u2229 spanningSets \u03bc n)\n[PROOFSTEP]\nexact fun i => MeasurableSet.inter (As_mble i) (measurable_spanningSets \u03bc n)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 (n : \u2115), {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\nn : \u2115\n\u22a2 Pairwise (Disjoint on fun i => As i \u2229 spanningSets \u03bc n)\n[PROOFSTEP]\nexact fun i j i_ne_j b hbi hbj => As_disj i_ne_j (hbi.trans (inter_subset_left _ _)) (hbj.trans (inter_subset_left _ _))\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 (n : \u2115), {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), As i \u2229 spanningSets \u03bc n) \u2260 \u22a4\n[PROOFSTEP]\nrefine' (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top \u03bc n)).ne\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type u_8\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nAs : \u03b9 \u2192 Set \u03b1\nAs_mble : \u2200 (i : \u03b9), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < \u2191\u2191\u03bc (As i)} \u2286 \u22c3 (n : \u2115), {i | 0 < \u2191\u2191\u03bc (As i \u2229 spanningSets \u03bc n)}\nn : \u2115\n\u22a2 \u22c3 (i : \u03b9), As i \u2229 spanningSets \u03bc n \u2286 spanningSets \u03bc n\n[PROOFSTEP]\nexact iUnion_subset fun i => inter_subset_right _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc t \u2229 s) = \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nhave A : \u2203 (t' : _) (_ : t' \u2287 t), MeasurableSet t' \u2227 \u2200 u, MeasurableSet u \u2192 \u03bc (t' \u2229 u) = \u03bc (t \u2229 u) :=\n by\n let w n := toMeasurable \u03bc (t \u2229 v n)\n have hw : \u2200 n, \u03bc (w n) < \u221e := by\n intro n\n simp_rw [measure_toMeasurable]\n exact (h'v n).lt_top\n set t' := \u22c3 n, toMeasurable \u03bc (t \u2229 disjointed w n) with ht'\n have tt' : t \u2286 t' :=\n calc\n t \u2286 \u22c3 n, t \u2229 disjointed w n :=\n by\n rw [\u2190 inter_iUnion, iUnion_disjointed, inter_iUnion]\n intro x hx\n rcases mem_iUnion.1 (hv hx) with \u27e8n, hn\u27e9\n refine' mem_iUnion.2 \u27e8n, _\u27e9\n have : x \u2208 t \u2229 v n := \u27e8hx, hn\u27e9\n exact \u27e8hx, subset_toMeasurable \u03bc _ this\u27e9\n _ \u2286 \u22c3 n, toMeasurable \u03bc (t \u2229 disjointed w n) := iUnion_mono fun n => subset_toMeasurable _ _\n refine' \u27e8t', tt', MeasurableSet.iUnion fun n => measurableSet_toMeasurable \u03bc _, fun u hu => _\u27e9\n apply le_antisymm _ (measure_mono (inter_subset_inter tt' Subset.rfl))\n calc\n \u03bc (t' \u2229 u) \u2264 \u2211' n, \u03bc (toMeasurable \u03bc (t \u2229 disjointed w n) \u2229 u) :=\n by\n rw [ht', iUnion_inter]\n exact measure_iUnion_le _\n _ = \u2211' n, \u03bc (t \u2229 disjointed w n \u2229 u) := by\n congr 1\n ext1 n\n apply measure_toMeasurable_inter hu\n apply ne_of_lt\n calc\n \u03bc (t \u2229 disjointed w n) \u2264 \u03bc (t \u2229 w n) := measure_mono (inter_subset_inter_right _ (disjointed_le w n))\n _ \u2264 \u03bc (w n) := (measure_mono (inter_subset_right _ _))\n _ < \u221e := hw n\n _ = \u2211' n, \u03bc.restrict (t \u2229 u) (disjointed w n) := by\n congr 1\n ext1 n\n rw [restrict_apply, inter_comm t _, inter_assoc]\n refine MeasurableSet.disjointed (fun n => ?_) n\n exact measurableSet_toMeasurable _ _\n _ = \u03bc.restrict (t \u2229 u) (\u22c3 n, disjointed w n) := by\n rw [measure_iUnion]\n \u00b7 exact disjoint_disjointed _\n \u00b7 intro i\n refine MeasurableSet.disjointed (fun n => ?_) i\n exact measurableSet_toMeasurable _ _\n _ \u2264 \u03bc.restrict (t \u2229 u) univ := (measure_mono (subset_univ _))\n _ = \u03bc (t \u2229 u) := by\n rw [restrict_apply MeasurableSet.univ, univ_inter]\n -- thanks to the definition of `toMeasurable`, the previous property will also be shared\n -- by `toMeasurable \u03bc t`, which is enough to conclude the proof.\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\n\u22a2 \u2203 t' x, MeasurableSet t' \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t' \u2229 u) = \u2191\u2191\u03bc (t \u2229 u)\n[PROOFSTEP]\nlet w n := toMeasurable \u03bc (t \u2229 v n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\n\u22a2 \u2203 t' x, MeasurableSet t' \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t' \u2229 u) = \u2191\u2191\u03bc (t \u2229 u)\n[PROOFSTEP]\nhave hw : \u2200 n, \u03bc (w n) < \u221e := by\n intro n\n simp_rw [measure_toMeasurable]\n exact (h'v n).lt_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\n\u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (w n) < \u22a4\n[PROOFSTEP]\nsimp_rw [measure_toMeasurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (t \u2229 v n) < \u22a4\n[PROOFSTEP]\nexact (h'v n).lt_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\n\u22a2 \u2203 t' x, MeasurableSet t' \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t' \u2229 u) = \u2191\u2191\u03bc (t \u2229 u)\n[PROOFSTEP]\nset t' := \u22c3 n, toMeasurable \u03bc (t \u2229 disjointed w n) with ht'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\n\u22a2 \u2203 t' x, MeasurableSet t' \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t' \u2229 u) = \u2191\u2191\u03bc (t \u2229 u)\n[PROOFSTEP]\nhave tt' : t \u2286 t' :=\n calc\n t \u2286 \u22c3 n, t \u2229 disjointed w n := by\n rw [\u2190 inter_iUnion, iUnion_disjointed, inter_iUnion]\n intro x hx\n rcases mem_iUnion.1 (hv hx) with \u27e8n, hn\u27e9\n refine' mem_iUnion.2 \u27e8n, _\u27e9\n have : x \u2208 t \u2229 v n := \u27e8hx, hn\u27e9\n exact \u27e8hx, subset_toMeasurable \u03bc _ this\u27e9\n _ \u2286 \u22c3 n, toMeasurable \u03bc (t \u2229 disjointed w n) := iUnion_mono fun n => subset_toMeasurable _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\n\u22a2 t \u2286 \u22c3 (n : \u2115), t \u2229 disjointed w n\n[PROOFSTEP]\nrw [\u2190 inter_iUnion, iUnion_disjointed, inter_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\n\u22a2 t \u2286 \u22c3 (i : \u2115), t \u2229 w i\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 \u22c3 (i : \u2115), t \u2229 w i\n[PROOFSTEP]\nrcases mem_iUnion.1 (hv hx) with \u27e8n, hn\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nx : \u03b1\nhx : x \u2208 t\nn : \u2115\nhn : x \u2208 v n\n\u22a2 x \u2208 \u22c3 (i : \u2115), t \u2229 w i\n[PROOFSTEP]\nrefine' mem_iUnion.2 \u27e8n, _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nx : \u03b1\nhx : x \u2208 t\nn : \u2115\nhn : x \u2208 v n\n\u22a2 x \u2208 t \u2229 w n\n[PROOFSTEP]\nhave : x \u2208 t \u2229 v n := \u27e8hx, hn\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nx : \u03b1\nhx : x \u2208 t\nn : \u2115\nhn : x \u2208 v n\nthis : x \u2208 t \u2229 v n\n\u22a2 x \u2208 t \u2229 w n\n[PROOFSTEP]\nexact \u27e8hx, subset_toMeasurable \u03bc _ this\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\n\u22a2 \u2203 t' x, MeasurableSet t' \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t' \u2229 u) = \u2191\u2191\u03bc (t \u2229 u)\n[PROOFSTEP]\nrefine' \u27e8t', tt', MeasurableSet.iUnion fun n => measurableSet_toMeasurable \u03bc _, fun u hu => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03bc (t' \u2229 u) = \u2191\u2191\u03bc (t \u2229 u)\n[PROOFSTEP]\napply le_antisymm _ (measure_mono (inter_subset_inter tt' Subset.rfl))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03bc (t' \u2229 u) \u2264 \u2191\u2191\u03bc (t \u2229 u)\n[PROOFSTEP]\ncalc\n \u03bc (t' \u2229 u) \u2264 \u2211' n, \u03bc (toMeasurable \u03bc (t \u2229 disjointed w n) \u2229 u) :=\n by\n rw [ht', iUnion_inter]\n exact measure_iUnion_le _\n _ = \u2211' n, \u03bc (t \u2229 disjointed w n \u2229 u) := by\n congr 1\n ext1 n\n apply measure_toMeasurable_inter hu\n apply ne_of_lt\n calc\n \u03bc (t \u2229 disjointed w n) \u2264 \u03bc (t \u2229 w n) := measure_mono (inter_subset_inter_right _ (disjointed_le w n))\n _ \u2264 \u03bc (w n) := (measure_mono (inter_subset_right _ _))\n _ < \u221e := hw n\n _ = \u2211' n, \u03bc.restrict (t \u2229 u) (disjointed w n) := by\n congr 1\n ext1 n\n rw [restrict_apply, inter_comm t _, inter_assoc]\n refine MeasurableSet.disjointed (fun n => ?_) n\n exact measurableSet_toMeasurable _ _\n _ = \u03bc.restrict (t \u2229 u) (\u22c3 n, disjointed w n) := by\n rw [measure_iUnion]\n \u00b7 exact disjoint_disjointed _\n \u00b7 intro i\n refine MeasurableSet.disjointed (fun n => ?_) i\n exact measurableSet_toMeasurable _ _\n _ \u2264 \u03bc.restrict (t \u2229 u) univ := (measure_mono (subset_univ _))\n _ = \u03bc (t \u2229 u) := by\n rw [restrict_apply MeasurableSet.univ, univ_inter]\n -- thanks to the definition of `toMeasurable`, the previous property will also be shared\n -- by `toMeasurable \u03bc t`, which is enough to conclude the proof.\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03bc (t' \u2229 u) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (toMeasurable \u03bc (t \u2229 disjointed w n) \u2229 u)\n[PROOFSTEP]\nrw [ht', iUnion_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u2115), toMeasurable \u03bc (t \u2229 disjointed w i) \u2229 u) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (toMeasurable \u03bc (t \u2229 disjointed w n) \u2229 u)\n[PROOFSTEP]\nexact measure_iUnion_le _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (toMeasurable \u03bc (t \u2229 disjointed w n) \u2229 u) = \u2211' (n : \u2115), \u2191\u2191\u03bc (t \u2229 disjointed w n \u2229 u)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 (fun n => \u2191\u2191\u03bc (toMeasurable \u03bc (t \u2229 disjointed w n) \u2229 u)) = fun n => \u2191\u2191\u03bc (t \u2229 disjointed w n \u2229 u)\n[PROOFSTEP]\next1 n\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc (t \u2229 disjointed w n) \u2229 u) = \u2191\u2191\u03bc (t \u2229 disjointed w n \u2229 u)\n[PROOFSTEP]\napply measure_toMeasurable_inter hu\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (t \u2229 disjointed w n) \u2260 \u22a4\n[PROOFSTEP]\napply ne_of_lt\n[GOAL]\ncase e_f.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (t \u2229 disjointed w n) < \u22a4\n[PROOFSTEP]\ncalc\n \u03bc (t \u2229 disjointed w n) \u2264 \u03bc (t \u2229 w n) := measure_mono (inter_subset_inter_right _ (disjointed_le w n))\n _ \u2264 \u03bc (w n) := (measure_mono (inter_subset_right _ _))\n _ < \u221e := hw n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (t \u2229 disjointed w n \u2229 u) = \u2211' (n : \u2115), \u2191\u2191(restrict \u03bc (t \u2229 u)) (disjointed w n)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_f\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 (fun n => \u2191\u2191\u03bc (t \u2229 disjointed w n \u2229 u)) = fun n => \u2191\u2191(restrict \u03bc (t \u2229 u)) (disjointed w n)\n[PROOFSTEP]\next1 n\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (t \u2229 disjointed w n \u2229 u) = \u2191\u2191(restrict \u03bc (t \u2229 u)) (disjointed w n)\n[PROOFSTEP]\nrw [restrict_apply, inter_comm t _, inter_assoc]\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\nn : \u2115\n\u22a2 MeasurableSet (disjointed w n)\n[PROOFSTEP]\nrefine MeasurableSet.disjointed (fun n => ?_) n\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\nn\u271d n : \u2115\n\u22a2 MeasurableSet (w n)\n[PROOFSTEP]\nexact measurableSet_toMeasurable _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2211' (n : \u2115), \u2191\u2191(restrict \u03bc (t \u2229 u)) (disjointed w n) = \u2191\u2191(restrict \u03bc (t \u2229 u)) (\u22c3 (n : \u2115), disjointed w n)\n[PROOFSTEP]\nrw [measure_iUnion]\n[GOAL]\ncase hn\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 Pairwise (Disjoint on fun n => disjointed w n)\n[PROOFSTEP]\nexact disjoint_disjointed _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2200 (i : \u2115), MeasurableSet (disjointed w i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\ni : \u2115\n\u22a2 MeasurableSet (disjointed w i)\n[PROOFSTEP]\nrefine MeasurableSet.disjointed (fun n => ?_) i\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\ni n : \u2115\n\u22a2 MeasurableSet (w n)\n[PROOFSTEP]\nexact measurableSet_toMeasurable _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nw : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t \u2229 v n)\nhw : \u2200 (n : \u2115), \u2191\u2191\u03bc (w n) < \u22a4\nt' : Set \u03b1 := \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\nht' : t' = \u22c3 (n : \u2115), toMeasurable \u03bc (t \u2229 disjointed w n)\ntt' : t \u2286 t'\nu : Set \u03b1\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(restrict \u03bc (t \u2229 u)) univ = \u2191\u2191\u03bc (t \u2229 u)\n[PROOFSTEP]\nrw [restrict_apply MeasurableSet.univ, univ_inter]\n -- thanks to the definition of `toMeasurable`, the previous property will also be shared\n -- by `toMeasurable \u03bc t`, which is enough to conclude the proof.\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nA : \u2203 t' x, MeasurableSet t' \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t' \u2229 u) = \u2191\u2191\u03bc (t \u2229 u)\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc t \u2229 s) = \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nrw [toMeasurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nA : \u2203 t' x, MeasurableSet t' \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t' \u2229 u) = \u2191\u2191\u03bc (t \u2229 u)\n\u22a2 \u2191\u2191\u03bc\n ((if h : \u2203 t_1 x, MeasurableSet t_1 \u2227 t_1 =\u1da0[ae \u03bc] t then Exists.choose h\n else\n if h' : \u2203 t_1 x, MeasurableSet t_1 \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t_1 \u2229 u) = \u2191\u2191\u03bc (t \u2229 u) then\n Exists.choose h'\n else Exists.choose (_ : \u2203 t_1, t \u2286 t_1 \u2227 MeasurableSet t_1 \u2227 \u2191\u2191\u03bc t_1 = \u2191\u2191\u03bc t)) \u2229\n s) =\n \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nsplit_ifs with ht\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nA : \u2203 t' x, MeasurableSet t' \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t' \u2229 u) = \u2191\u2191\u03bc (t \u2229 u)\nht : \u2203 t_1 x, MeasurableSet t_1 \u2227 t_1 =\u1da0[ae \u03bc] t\n\u22a2 \u2191\u2191\u03bc (Exists.choose (_ : \u2203 t_1 x, MeasurableSet t_1 \u2227 t_1 =\u1da0[ae \u03bc] t) \u2229 s) = \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\napply measure_congr\n[GOAL]\ncase pos.H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nA : \u2203 t' x, MeasurableSet t' \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t' \u2229 u) = \u2191\u2191\u03bc (t \u2229 u)\nht : \u2203 t_1 x, MeasurableSet t_1 \u2227 t_1 =\u1da0[ae \u03bc] t\n\u22a2 Exists.choose (_ : \u2203 t_1 x, MeasurableSet t_1 \u2227 t_1 =\u1da0[ae \u03bc] t) \u2229 s =\u1da0[ae \u03bc] t \u2229 s\n[PROOFSTEP]\nexact ae_eq_set_inter ht.choose_spec.snd.2 (ae_eq_refl _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : t \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (t \u2229 v n) \u2260 \u22a4\nA : \u2203 t' x, MeasurableSet t' \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t' \u2229 u) = \u2191\u2191\u03bc (t \u2229 u)\nht : \u00ac\u2203 t_1 x, MeasurableSet t_1 \u2227 t_1 =\u1da0[ae \u03bc] t\n\u22a2 \u2191\u2191\u03bc (Exists.choose A \u2229 s) = \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nexact A.choose_spec.snd.2 s hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s : Set \u03b1\nv : \u2115 \u2192 Set \u03b1\nhv : s \u2286 \u22c3 (n : \u2115), v n\nh'v : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(restrict \u03bc (toMeasurable \u03bc s)) t = \u2191\u2191(restrict \u03bc s) t\n[PROOFSTEP]\nsimp only [restrict_apply ht, inter_comm t, measure_toMeasurable_inter_of_cover ht hv h'v]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc t \u2229 s) = \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nhave : t \u2286 \u22c3 n, spanningSets \u03bc n := by\n rw [iUnion_spanningSets]\n exact subset_univ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 t \u2286 \u22c3 (n : \u2115), spanningSets \u03bc n\n[PROOFSTEP]\nrw [iUnion_spanningSets]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 t \u2286 univ\n[PROOFSTEP]\nexact subset_univ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nthis : t \u2286 \u22c3 (n : \u2115), spanningSets \u03bc n\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc t \u2229 s) = \u2191\u2191\u03bc (t \u2229 s)\n[PROOFSTEP]\nrefine measure_toMeasurable_inter_of_cover hs this fun n => ne_of_lt ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nthis : t \u2286 \u22c3 (n : \u2115), spanningSets \u03bc n\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (t \u2229 spanningSets \u03bc n) < \u22a4\n[PROOFSTEP]\ncalc\n \u03bc (t \u2229 spanningSets \u03bc n) \u2264 \u03bc (spanningSets \u03bc n) := measure_mono (inter_subset_right _ _)\n _ < \u221e := measure_spanningSets_lt_top \u03bc n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\ns t : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(restrict \u03bc (toMeasurable \u03bc s)) t = \u2191\u2191(restrict \u03bc s) t\n[PROOFSTEP]\nrw [restrict_apply ht, inter_comm t, measure_toMeasurable_inter_of_sigmaFinite ht, restrict_apply ht, inter_comm t]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nS : Set (Set \u03b1)\nhc : Set.Countable S\nh\u03bc : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s < \u22a4\nhU : \u22c3\u2080 S = univ\n\u22a2 SigmaFinite \u03bc\n[PROOFSTEP]\nobtain \u27e8s, h\u03bc, hs\u27e9 : \u2203 s : \u2115 \u2192 Set \u03b1, (\u2200 n, \u03bc (s n) < \u221e) \u2227 \u22c3 n, s n = univ\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nS : Set (Set \u03b1)\nhc : Set.Countable S\nh\u03bc : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s < \u22a4\nhU : \u22c3\u2080 S = univ\n\u22a2 \u2203 s, (\u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4) \u2227 \u22c3 (n : \u2115), s n = univ\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nS : Set (Set \u03b1)\nhc : Set.Countable S\nh\u03bc\u271d : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s < \u22a4\nhU : \u22c3\u2080 S = univ\ns : \u2115 \u2192 Set \u03b1\nh\u03bc : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\nhs : \u22c3 (n : \u2115), s n = univ\n\u22a2 SigmaFinite \u03bc\n[PROOFSTEP]\nexact (@exists_seq_cover_iff_countable _ (fun x => \u03bc x < \u22a4) \u27e8\u2205, by simp\u27e9).2 \u27e8S, hc, h\u03bc, hU\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nS : Set (Set \u03b1)\nhc : Set.Countable S\nh\u03bc : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s < \u22a4\nhU : \u22c3\u2080 S = univ\n\u22a2 (fun x => \u2191\u2191\u03bc x < \u22a4) \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\nS : Set (Set \u03b1)\nhc : Set.Countable S\nh\u03bc\u271d : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc s < \u22a4\nhU : \u22c3\u2080 S = univ\ns : \u2115 \u2192 Set \u03b1\nh\u03bc : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) < \u22a4\nhs : \u22c3 (n : \u2115), s n = univ\n\u22a2 SigmaFinite \u03bc\n[PROOFSTEP]\nexact \u27e8\u27e8\u27e8fun n => s n, fun _ => trivial, h\u03bc, hs\u27e9\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\n\u22a2 SigmaFinite \u03bc \u2194 IsFiniteMeasure \u03bc\n[PROOFSTEP]\nrefine'\n \u27e8fun h => \u27e8_\u27e9, fun h => by\n haveI := h\n infer_instance\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh : IsFiniteMeasure \u03bc\n\u22a2 SigmaFinite \u03bc\n[PROOFSTEP]\nhaveI := h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : IsFiniteMeasure \u03bc\n\u22a2 SigmaFinite \u03bc\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh : SigmaFinite \u03bc\n\u22a2 \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nhaveI : SigmaFinite \u03bc := h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : SigmaFinite \u03bc\n\u22a2 \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nlet s := spanningSets \u03bc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : SigmaFinite \u03bc\ns : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\n\u22a2 \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nhave hs_univ : \u22c3 i, s i = Set.univ := iUnion_spanningSets \u03bc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : SigmaFinite \u03bc\ns : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nhs_univ : \u22c3 (i : \u2115), s i = univ\n\u22a2 \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nhave hs_meas : \u2200 i, MeasurableSet[\u22a5] (s i) := measurable_spanningSets \u03bc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : SigmaFinite \u03bc\ns : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nhs_univ : \u22c3 (i : \u2115), s i = univ\nhs_meas : \u2200 (i : \u2115), MeasurableSet (s i)\n\u22a2 \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nsimp_rw [MeasurableSpace.measurableSet_bot_iff] at hs_meas \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : SigmaFinite \u03bc\ns : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nhs_univ : \u22c3 (i : \u2115), s i = univ\nhs_meas : \u2200 (i : \u2115), spanningSets \u03bc i = \u2205 \u2228 spanningSets \u03bc i = univ\n\u22a2 \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nby_cases h_univ_empty : Set.univ = \u2205\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : SigmaFinite \u03bc\ns : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nhs_univ : \u22c3 (i : \u2115), s i = univ\nhs_meas : \u2200 (i : \u2115), spanningSets \u03bc i = \u2205 \u2228 spanningSets \u03bc i = univ\nh_univ_empty : univ = \u2205\n\u22a2 \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nrw [h_univ_empty, @measure_empty \u03b1 \u22a5]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : SigmaFinite \u03bc\ns : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nhs_univ : \u22c3 (i : \u2115), s i = univ\nhs_meas : \u2200 (i : \u2115), spanningSets \u03bc i = \u2205 \u2228 spanningSets \u03bc i = univ\nh_univ_empty : univ = \u2205\n\u22a2 0 < \u22a4\n[PROOFSTEP]\nexact ENNReal.zero_ne_top.lt_top\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : SigmaFinite \u03bc\ns : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nhs_univ : \u22c3 (i : \u2115), s i = univ\nhs_meas : \u2200 (i : \u2115), spanningSets \u03bc i = \u2205 \u2228 spanningSets \u03bc i = univ\nh_univ_empty : \u00acuniv = \u2205\n\u22a2 \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nobtain \u27e8i, hsi\u27e9 : \u2203 i, s i = Set.univ := by\n by_contra' h_not_univ\n have h_empty : \u2200 i, s i = \u2205 := by simpa [h_not_univ] using hs_meas\n simp only [h_empty, iUnion_empty] at hs_univ \n exact h_univ_empty hs_univ.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : SigmaFinite \u03bc\ns : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nhs_univ : \u22c3 (i : \u2115), s i = univ\nhs_meas : \u2200 (i : \u2115), spanningSets \u03bc i = \u2205 \u2228 spanningSets \u03bc i = univ\nh_univ_empty : \u00acuniv = \u2205\n\u22a2 \u2203 i, s i = univ\n[PROOFSTEP]\nby_contra' h_not_univ\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : SigmaFinite \u03bc\ns : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nhs_univ : \u22c3 (i : \u2115), s i = univ\nhs_meas : \u2200 (i : \u2115), spanningSets \u03bc i = \u2205 \u2228 spanningSets \u03bc i = univ\nh_univ_empty : \u00acuniv = \u2205\nh_not_univ : \u2200 (i : \u2115), spanningSets \u03bc i \u2260 univ\n\u22a2 False\n[PROOFSTEP]\nhave h_empty : \u2200 i, s i = \u2205 := by simpa [h_not_univ] using hs_meas\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : SigmaFinite \u03bc\ns : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nhs_univ : \u22c3 (i : \u2115), s i = univ\nhs_meas : \u2200 (i : \u2115), spanningSets \u03bc i = \u2205 \u2228 spanningSets \u03bc i = univ\nh_univ_empty : \u00acuniv = \u2205\nh_not_univ : \u2200 (i : \u2115), spanningSets \u03bc i \u2260 univ\n\u22a2 \u2200 (i : \u2115), s i = \u2205\n[PROOFSTEP]\nsimpa [h_not_univ] using hs_meas\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : SigmaFinite \u03bc\ns : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nhs_univ : \u22c3 (i : \u2115), s i = univ\nhs_meas : \u2200 (i : \u2115), spanningSets \u03bc i = \u2205 \u2228 spanningSets \u03bc i = univ\nh_univ_empty : \u00acuniv = \u2205\nh_not_univ : \u2200 (i : \u2115), spanningSets \u03bc i \u2260 univ\nh_empty : \u2200 (i : \u2115), s i = \u2205\n\u22a2 False\n[PROOFSTEP]\nsimp only [h_empty, iUnion_empty] at hs_univ \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : SigmaFinite \u03bc\ns : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nhs_meas : \u2200 (i : \u2115), spanningSets \u03bc i = \u2205 \u2228 spanningSets \u03bc i = univ\nh_univ_empty : \u00acuniv = \u2205\nh_not_univ : \u2200 (i : \u2115), spanningSets \u03bc i \u2260 univ\nh_empty : \u2200 (i : \u2115), s i = \u2205\nhs_univ : \u2205 = univ\n\u22a2 False\n[PROOFSTEP]\nexact h_univ_empty hs_univ.symm\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : SigmaFinite \u03bc\ns : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nhs_univ : \u22c3 (i : \u2115), s i = univ\nhs_meas : \u2200 (i : \u2115), spanningSets \u03bc i = \u2205 \u2228 spanningSets \u03bc i = univ\nh_univ_empty : \u00acuniv = \u2205\ni : \u2115\nhsi : s i = univ\n\u22a2 \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nrw [\u2190 hsi]\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\nh this : SigmaFinite \u03bc\ns : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nhs_univ : \u22c3 (i : \u2115), s i = univ\nhs_meas : \u2200 (i : \u2115), spanningSets \u03bc i = \u2205 \u2228 spanningSets \u03bc i = univ\nh_univ_empty : \u00acuniv = \u2205\ni : \u2115\nhsi : s i = univ\n\u22a2 \u2191\u2191\u03bc (s i) < \u22a4\n[PROOFSTEP]\nexact measure_spanningSets_lt_top \u03bc i\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\n\u22a2 SigmaFinite (Measure.restrict \u03bc s)\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u27e8spanningSets \u03bc, fun _ => trivial, fun i => _, iUnion_spanningSets \u03bc\u27e9\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\ni : \u2115\n\u22a2 \u2191\u2191(Measure.restrict \u03bc s) (spanningSets \u03bc i) < \u22a4\n[PROOFSTEP]\nrw [Measure.restrict_apply (measurable_spanningSets \u03bc i)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\ni : \u2115\n\u22a2 \u2191\u2191\u03bc (spanningSets \u03bc i \u2229 s) < \u22a4\n[PROOFSTEP]\nexact (measure_mono <| inter_subset_left _ _).trans_lt (measure_spanningSets_lt_top \u03bc i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type ?u.729029\ninst\u271d\u00b9 : Finite \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\n\u22a2 SigmaFinite (sum \u03bc)\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type ?u.729029\ninst\u271d\u00b9 : Finite \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\nval\u271d : Fintype \u03b9\n\u22a2 SigmaFinite (sum \u03bc)\n[PROOFSTEP]\nhave : \u2200 n, MeasurableSet (\u22c2 i : \u03b9, spanningSets (\u03bc i) n) := fun n =>\n MeasurableSet.iInter fun i => measurable_spanningSets (\u03bc i) n\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type ?u.729029\ninst\u271d\u00b9 : Finite \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\nval\u271d : Fintype \u03b9\nthis : \u2200 (n : \u2115), MeasurableSet (\u22c2 (i : \u03b9), spanningSets (\u03bc i) n)\n\u22a2 SigmaFinite (sum \u03bc)\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u27e8fun n => \u22c2 i, spanningSets (\u03bc i) n, fun _ => trivial, fun n => _, _\u27e9\u27e9\u27e9\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type ?u.729029\ninst\u271d\u00b9 : Finite \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\nval\u271d : Fintype \u03b9\nthis : \u2200 (n : \u2115), MeasurableSet (\u22c2 (i : \u03b9), spanningSets (\u03bc i) n)\nn : \u2115\n\u22a2 \u2191\u2191(sum \u03bc) ((fun n => \u22c2 (i : \u03b9), spanningSets (\u03bc i) n) n) < \u22a4\n[PROOFSTEP]\nrw [sum_apply _ (this n), tsum_fintype, ENNReal.sum_lt_top_iff]\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type ?u.729029\ninst\u271d\u00b9 : Finite \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\nval\u271d : Fintype \u03b9\nthis : \u2200 (n : \u2115), MeasurableSet (\u22c2 (i : \u03b9), spanningSets (\u03bc i) n)\nn : \u2115\n\u22a2 \u2200 (a : \u03b9), a \u2208 Finset.univ \u2192 \u2191\u2191(\u03bc a) (\u22c2 (i : \u03b9), spanningSets (\u03bc i) n) < \u22a4\n[PROOFSTEP]\nrintro i -\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type ?u.729029\ninst\u271d\u00b9 : Finite \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\nval\u271d : Fintype \u03b9\nthis : \u2200 (n : \u2115), MeasurableSet (\u22c2 (i : \u03b9), spanningSets (\u03bc i) n)\nn : \u2115\ni : \u03b9\n\u22a2 \u2191\u2191(\u03bc i) (\u22c2 (i : \u03b9), spanningSets (\u03bc i) n) < \u22a4\n[PROOFSTEP]\nexact (measure_mono <| iInter_subset _ i).trans_lt (measure_spanningSets_lt_top (\u03bc i) n)\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type ?u.729029\ninst\u271d\u00b9 : Finite \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\nval\u271d : Fintype \u03b9\nthis : \u2200 (n : \u2115), MeasurableSet (\u22c2 (i : \u03b9), spanningSets (\u03bc i) n)\n\u22a2 \u22c3 (i : \u2115), (fun n => \u22c2 (i : \u03b9), spanningSets (\u03bc i) n) i = univ\n[PROOFSTEP]\nrw [iUnion_iInter_of_monotone]\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type ?u.729029\ninst\u271d\u00b9 : Finite \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\nval\u271d : Fintype \u03b9\nthis : \u2200 (n : \u2115), MeasurableSet (\u22c2 (i : \u03b9), spanningSets (\u03bc i) n)\n\u22a2 \u22c2 (i : \u03b9), \u22c3 (j : \u2115), spanningSets (\u03bc i) j = univ\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type ?u.729029\ninst\u271d\u00b9 : Finite \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\nval\u271d : Fintype \u03b9\nthis : \u2200 (n : \u2115), MeasurableSet (\u22c2 (i : \u03b9), spanningSets (\u03bc i) n)\n\u22a2 \u2200 (i : \u03b9), Monotone fun i_1 => spanningSets (\u03bc i) i_1\n[PROOFSTEP]\nsimp_rw [iUnion_spanningSets, iInter_univ]\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9\u271d : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9 : Type ?u.729029\ninst\u271d\u00b9 : Finite \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\nval\u271d : Fintype \u03b9\nthis : \u2200 (n : \u2115), MeasurableSet (\u22c2 (i : \u03b9), spanningSets (\u03bc i) n)\n\u22a2 \u2200 (i : \u03b9), Monotone fun i_1 => spanningSets (\u03bc i) i_1\n[PROOFSTEP]\nexact fun i => monotone_spanningSets (\u03bc i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\n\u22a2 SigmaFinite (\u03bc + \u03bd)\n[PROOFSTEP]\nrw [\u2190 sum_cond]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\n\u22a2 SigmaFinite (sum fun b => bif b then \u03bc else \u03bd)\n[PROOFSTEP]\nrefine' @sum.sigmaFinite _ _ _ _ _ (Bool.rec _ _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\n\u22a2 SigmaFinite (bif false then \u03bc else \u03bd)\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\n\u22a2 SigmaFinite (bif true then \u03bc else \u03bd)\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nc : \u211d\u22650\n\u22a2 \u2200 (i : \u2115), \u2191\u2191(c \u2022 \u03bc) (spanningSets \u03bc i) < \u22a4\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nc : \u211d\u22650\ni : \u2115\n\u22a2 \u2191\u2191(c \u2022 \u03bc) (spanningSets \u03bc i) < \u22a4\n[PROOFSTEP]\nsimp only [smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply, nnreal_smul_coe_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nc : \u211d\u22650\ni : \u2115\n\u22a2 \u2191c * \u2191\u2191\u03bc (spanningSets \u03bc i) < \u22a4\n[PROOFSTEP]\nexact ENNReal.mul_lt_top ENNReal.coe_ne_top (measure_spanningSets_lt_top \u03bc i).ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\nh : SigmaFinite (Measure.map f \u03bc)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc ((fun n => f \u207b\u00b9' spanningSets (Measure.map f \u03bc) n) n) < \u22a4\n[PROOFSTEP]\nsimp only [\u2190 map_apply_of_aemeasurable hf, measurable_spanningSets, measure_spanningSets_lt_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\nh : SigmaFinite (Measure.map f \u03bc)\n\u22a2 \u22c3 (i : \u2115), (fun n => f \u207b\u00b9' spanningSets (Measure.map f \u03bc) n) i = univ\n[PROOFSTEP]\nrw [\u2190 preimage_iUnion, iUnion_spanningSets, preimage_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2243\u1d50 \u03b2\nh : SigmaFinite \u03bc\n\u22a2 SigmaFinite (Measure.map (\u2191f) \u03bc)\n[PROOFSTEP]\nrefine' SigmaFinite.of_map _ f.symm.measurable.aemeasurable _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2243\u1d50 \u03b2\nh : SigmaFinite \u03bc\n\u22a2 SigmaFinite (Measure.map (\u2191(MeasurableEquiv.symm f)) (Measure.map (\u2191f) \u03bc))\n[PROOFSTEP]\nrwa [map_map f.symm.measurable f.measurable, f.symm_comp_self, Measure.map_id]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nP : \u03b1 \u2192 Prop\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2191\u2191\u03bd s < \u22a4 \u2192 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, P x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, P x\n[PROOFSTEP]\nhave : \u2200 n, \u2200\u1d50 x \u2202\u03bc, x \u2208 spanningSets (\u03bc + \u03bd) n \u2192 P x :=\n by\n intro n\n have :=\n h (spanningSets (\u03bc + \u03bd) n) (measurable_spanningSets _ _)\n ((self_le_add_right _ _).trans_lt (measure_spanningSets_lt_top (\u03bc + \u03bd) _))\n ((self_le_add_left _ _).trans_lt (measure_spanningSets_lt_top (\u03bc + \u03bd) _))\n exact (ae_restrict_iff' (measurable_spanningSets _ _)).mp this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nP : \u03b1 \u2192 Prop\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2191\u2191\u03bd s < \u22a4 \u2192 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, P x\n\u22a2 \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 spanningSets (\u03bc + \u03bd) n \u2192 P x\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nP : \u03b1 \u2192 Prop\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2191\u2191\u03bd s < \u22a4 \u2192 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, P x\nn : \u2115\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 spanningSets (\u03bc + \u03bd) n \u2192 P x\n[PROOFSTEP]\nhave :=\n h (spanningSets (\u03bc + \u03bd) n) (measurable_spanningSets _ _)\n ((self_le_add_right _ _).trans_lt (measure_spanningSets_lt_top (\u03bc + \u03bd) _))\n ((self_le_add_left _ _).trans_lt (measure_spanningSets_lt_top (\u03bc + \u03bd) _))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nP : \u03b1 \u2192 Prop\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2191\u2191\u03bd s < \u22a4 \u2192 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, P x\nn : \u2115\nthis : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (spanningSets (\u03bc + \u03bd) n), P x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 spanningSets (\u03bc + \u03bd) n \u2192 P x\n[PROOFSTEP]\nexact (ae_restrict_iff' (measurable_spanningSets _ _)).mp this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nP : \u03b1 \u2192 Prop\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2191\u2191\u03bd s < \u22a4 \u2192 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, P x\nthis : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 spanningSets (\u03bc + \u03bd) n \u2192 P x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, P x\n[PROOFSTEP]\nfilter_upwards [ae_all_iff.2 this] with _ hx using hx _ (mem_spanningSetsIndex _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 IsFiniteMeasure (c \u2022 \u03bc)\n[PROOFSTEP]\nlift c to \u211d\u22650 using hc\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nc : \u211d\u22650\n\u22a2 IsFiniteMeasure (\u2191c \u2022 \u03bc)\n[PROOFSTEP]\nexact MeasureTheory.isFiniteMeasureSMulNNReal\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nx : \u03b1\n\u22a2 \u2203 s, x \u2208 s \u2227 IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4\n[PROOFSTEP]\nsimpa only [exists_prop, and_assoc] using (\u03bc.finiteAt_nhds x).exists_mem_basis (nhds_basis_opens x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nc : \u211d\u22650\n\u22a2 IsLocallyFiniteMeasure (c \u2022 \u03bc)\n[PROOFSTEP]\nrefine' \u27e8fun x => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nc : \u211d\u22650\nx : \u03b1\n\u22a2 FiniteAtFilter (c \u2022 \u03bc) (\ud835\udcdd x)\n[PROOFSTEP]\nrcases \u03bc.exists_isOpen_measure_lt_top x with \u27e8o, xo, o_open, \u03bco\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nc : \u211d\u22650\nx : \u03b1\no : Set \u03b1\nxo : x \u2208 o\no_open : IsOpen o\n\u03bco : \u2191\u2191\u03bc o < \u22a4\n\u22a2 FiniteAtFilter (c \u2022 \u03bc) (\ud835\udcdd x)\n[PROOFSTEP]\nrefine' \u27e8o, o_open.mem_nhds xo, _\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nc : \u211d\u22650\nx : \u03b1\no : Set \u03b1\nxo : x \u2208 o\no_open : IsOpen o\n\u03bco : \u2191\u2191\u03bc o < \u22a4\n\u22a2 \u2191\u2191(c \u2022 \u03bc) o < \u22a4\n[PROOFSTEP]\napply ENNReal.mul_lt_top _ \u03bco.ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nc : \u211d\u22650\nx : \u03b1\no : Set \u03b1\nxo : x \u2208 o\no_open : IsOpen o\n\u03bco : \u2191\u2191\u03bc o < \u22a4\n\u22a2 \u2191(RingHom.id \u211d\u22650\u221e) (\u2191ofNNRealHom c) \u2260 \u22a4\n[PROOFSTEP]\nsimp only [RingHom.id_apply, RingHom.toMonoidHom_eq_coe, ENNReal.coe_ne_top, ENNReal.coe_ofNNRealHom, Ne.def,\n not_false_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\n\u22a2 TopologicalSpace.IsTopologicalBasis {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\n[PROOFSTEP]\nrefine' TopologicalSpace.isTopologicalBasis_of_open_of_nhds (fun s hs => hs.1) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\n\u22a2 \u2200 (a : \u03b1) (u : Set \u03b1), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4} \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nintro x s xs hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nx : \u03b1\ns : Set \u03b1\nxs : x \u2208 s\nhs : IsOpen s\n\u22a2 \u2203 v, v \u2208 {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4} \u2227 x \u2208 v \u2227 v \u2286 s\n[PROOFSTEP]\nrcases \u03bc.exists_isOpen_measure_lt_top x with \u27e8v, xv, hv, \u03bcv\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nx : \u03b1\ns : Set \u03b1\nxs : x \u2208 s\nhs : IsOpen s\nv : Set \u03b1\nxv : x \u2208 v\nhv : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v < \u22a4\n\u22a2 \u2203 v, v \u2208 {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4} \u2227 x \u2208 v \u2227 v \u2286 s\n[PROOFSTEP]\nrefine' \u27e8v \u2229 s, \u27e8hv.inter hs, lt_of_le_of_lt _ \u03bcv\u27e9, \u27e8xv, xs\u27e9, inter_subset_right _ _\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nx : \u03b1\ns : Set \u03b1\nxs : x \u2208 s\nhs : IsOpen s\nv : Set \u03b1\nxv : x \u2208 v\nhv : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v < \u22a4\n\u22a2 \u2191\u2191\u03bc (v \u2229 s) \u2264 \u2191\u2191\u03bc v\n[PROOFSTEP]\nexact measure_mono (inter_subset_left _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\n\u22a2 SigmaFinite \u03bc\n[PROOFSTEP]\nchoose s hsx hs\u03bc using \u03bc.finiteAt_nhds\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : \u03b1 \u2192 Set \u03b1\nhsx : \u2200 (x : \u03b1), s x \u2208 \ud835\udcdd x\nhs\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (s x) < \u22a4\n\u22a2 SigmaFinite \u03bc\n[PROOFSTEP]\nrcases TopologicalSpace.countable_cover_nhds hsx with \u27e8t, htc, htU\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : \u03b1 \u2192 Set \u03b1\nhsx : \u2200 (x : \u03b1), s x \u2208 \ud835\udcdd x\nhs\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (s x) < \u22a4\nt : Set \u03b1\nhtc : Set.Countable t\nhtU : \u22c3 (x : \u03b1) (_ : x \u2208 t), s x = univ\n\u22a2 SigmaFinite \u03bc\n[PROOFSTEP]\nrefine' Measure.sigmaFinite_of_countable (htc.image s) (ball_image_iff.2 fun x _ => hs\u03bc x) _\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : \u03b1 \u2192 Set \u03b1\nhsx : \u2200 (x : \u03b1), s x \u2208 \ud835\udcdd x\nhs\u03bc : \u2200 (x : \u03b1), \u2191\u2191\u03bc (s x) < \u22a4\nt : Set \u03b1\nhtc : Set.Countable t\nhtU : \u22c3 (x : \u03b1) (_ : x \u2208 t), s x = univ\n\u22a2 \u22c3\u2080 (s '' t) = univ\n[PROOFSTEP]\nrwa [sUnion_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : IsFiniteMeasureOnCompacts \u03bc\n\u22a2 \u2200 (x : \u03b1), FiniteAtFilter \u03bc (\ud835\udcdd x)\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : IsFiniteMeasureOnCompacts \u03bc\nx : \u03b1\n\u22a2 FiniteAtFilter \u03bc (\ud835\udcdd x)\n[PROOFSTEP]\nrcases exists_compact_mem_nhds x with \u27e8K, K_compact, K_mem\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : IsFiniteMeasureOnCompacts \u03bc\nx : \u03b1\nK : Set \u03b1\nK_compact : IsCompact K\nK_mem : K \u2208 \ud835\udcdd x\n\u22a2 FiniteAtFilter \u03bc (\ud835\udcdd x)\n[PROOFSTEP]\nexact \u27e8K, K_mem, K_compact.measure_lt_top\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : Countable \u03b9\nU : \u03b9 \u2192 Set \u03b1\nhU : \u22c3 (i : \u03b9), U i = univ\nh\u03bc : \u03bc \u2260 0\n\u22a2 \u2203 i, 0 < \u2191\u2191\u03bc (U i)\n[PROOFSTEP]\ncontrapose! h\u03bc with H\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : Countable \u03b9\nU : \u03b9 \u2192 Set \u03b1\nhU : \u22c3 (i : \u03b9), U i = univ\nH : \u2200 (i : \u03b9), \u2191\u2191\u03bc (U i) \u2264 0\n\u22a2 \u03bc = 0\n[PROOFSTEP]\nrw [\u2190 measure_univ_eq_zero, \u2190 hU]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : Countable \u03b9\nU : \u03b9 \u2192 Set \u03b1\nhU : \u22c3 (i : \u03b9), U i = univ\nH : \u2200 (i : \u03b9), \u2191\u2191\u03bc (U i) \u2264 0\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), U i) = 0\n[PROOFSTEP]\nexact measure_iUnion_null fun i => nonpos_iff_eq_zero.1 (H i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : PseudoMetricSpace \u03b4\nf : \u03b1 \u2192 \u03b4\nx : \u03b4\nh\u03bc : \u03bc \u2260 0\n\u22a2 \u22c3 (i : \u2115), f \u207b\u00b9' Metric.ball x \u2191i = univ\n[PROOFSTEP]\nrw [\u2190 preimage_iUnion, Metric.iUnion_ball_nat, preimage_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T1Space \u03b2\ninst\u271d\u00b9 : SecondCountableTopology \u03b2\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (b : \u03b2), \u2203\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 b\n\u22a2 \u2203 a b, a \u2260 b \u2227 (\u2200 (s : Set \u03b2), s \u2208 \ud835\udcdd a \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' s)) \u2227 \u2200 (t : Set \u03b2), t \u2208 \ud835\udcdd b \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' t)\n[PROOFSTEP]\nset m : OuterMeasure \u03b2 := OuterMeasure.map f \u03bc.toOuterMeasure\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T1Space \u03b2\ninst\u271d\u00b9 : SecondCountableTopology \u03b2\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (b : \u03b2), \u2203\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 b\nm : OuterMeasure \u03b2 := \u2191(OuterMeasure.map f) \u2191\u03bc\n\u22a2 \u2203 a b, a \u2260 b \u2227 (\u2200 (s : Set \u03b2), s \u2208 \ud835\udcdd a \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' s)) \u2227 \u2200 (t : Set \u03b2), t \u2208 \ud835\udcdd b \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' t)\n[PROOFSTEP]\nreplace h : \u2200 b : \u03b2, m { b }\u1d9c \u2260 0 := fun b => not_eventually.mpr (h b)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T1Space \u03b2\ninst\u271d\u00b9 : SecondCountableTopology \u03b2\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b2\nm : OuterMeasure \u03b2 := \u2191(OuterMeasure.map f) \u2191\u03bc\nh : \u2200 (b : \u03b2), \u2191m {b}\u1d9c \u2260 0\n\u22a2 \u2203 a b, a \u2260 b \u2227 (\u2200 (s : Set \u03b2), s \u2208 \ud835\udcdd a \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' s)) \u2227 \u2200 (t : Set \u03b2), t \u2208 \ud835\udcdd b \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' t)\n[PROOFSTEP]\ninhabit \u03b2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T1Space \u03b2\ninst\u271d\u00b9 : SecondCountableTopology \u03b2\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b2\nm : OuterMeasure \u03b2 := \u2191(OuterMeasure.map f) \u2191\u03bc\nh : \u2200 (b : \u03b2), \u2191m {b}\u1d9c \u2260 0\ninhabited_h : Inhabited \u03b2\n\u22a2 \u2203 a b, a \u2260 b \u2227 (\u2200 (s : Set \u03b2), s \u2208 \ud835\udcdd a \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' s)) \u2227 \u2200 (t : Set \u03b2), t \u2208 \ud835\udcdd b \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' t)\n[PROOFSTEP]\nhave : m univ \u2260 0 := ne_bot_of_le_ne_bot (h default) (m.mono' <| subset_univ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T1Space \u03b2\ninst\u271d\u00b9 : SecondCountableTopology \u03b2\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b2\nm : OuterMeasure \u03b2 := \u2191(OuterMeasure.map f) \u2191\u03bc\nh : \u2200 (b : \u03b2), \u2191m {b}\u1d9c \u2260 0\ninhabited_h : Inhabited \u03b2\nthis : \u2191m univ \u2260 0\n\u22a2 \u2203 a b, a \u2260 b \u2227 (\u2200 (s : Set \u03b2), s \u2208 \ud835\udcdd a \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' s)) \u2227 \u2200 (t : Set \u03b2), t \u2208 \ud835\udcdd b \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' t)\n[PROOFSTEP]\nrcases m.exists_mem_forall_mem_nhds_within_pos this with \u27e8b, -, hb\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T1Space \u03b2\ninst\u271d\u00b9 : SecondCountableTopology \u03b2\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b2\nm : OuterMeasure \u03b2 := \u2191(OuterMeasure.map f) \u2191\u03bc\nh : \u2200 (b : \u03b2), \u2191m {b}\u1d9c \u2260 0\ninhabited_h : Inhabited \u03b2\nthis : \u2191m univ \u2260 0\nb : \u03b2\nhb : \u2200 (t : Set \u03b2), t \u2208 \ud835\udcdd[univ] b \u2192 0 < \u2191m t\n\u22a2 \u2203 a b, a \u2260 b \u2227 (\u2200 (s : Set \u03b2), s \u2208 \ud835\udcdd a \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' s)) \u2227 \u2200 (t : Set \u03b2), t \u2208 \ud835\udcdd b \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' t)\n[PROOFSTEP]\nsimp only [nhdsWithin_univ] at hb \n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T1Space \u03b2\ninst\u271d\u00b9 : SecondCountableTopology \u03b2\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b2\nm : OuterMeasure \u03b2 := \u2191(OuterMeasure.map f) \u2191\u03bc\nh : \u2200 (b : \u03b2), \u2191m {b}\u1d9c \u2260 0\ninhabited_h : Inhabited \u03b2\nthis : \u2191m univ \u2260 0\nb : \u03b2\nhb : \u2200 (t : Set \u03b2), t \u2208 \ud835\udcdd b \u2192 0 < \u2191(\u2191(OuterMeasure.map f) \u2191\u03bc) t\n\u22a2 \u2203 a b, a \u2260 b \u2227 (\u2200 (s : Set \u03b2), s \u2208 \ud835\udcdd a \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' s)) \u2227 \u2200 (t : Set \u03b2), t \u2208 \ud835\udcdd b \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' t)\n[PROOFSTEP]\nrcases m.exists_mem_forall_mem_nhds_within_pos (h b) with \u27e8a, hab : a \u2260 b, ha\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T1Space \u03b2\ninst\u271d\u00b9 : SecondCountableTopology \u03b2\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b2\nm : OuterMeasure \u03b2 := \u2191(OuterMeasure.map f) \u2191\u03bc\nh : \u2200 (b : \u03b2), \u2191m {b}\u1d9c \u2260 0\ninhabited_h : Inhabited \u03b2\nthis : \u2191m univ \u2260 0\nb : \u03b2\nhb : \u2200 (t : Set \u03b2), t \u2208 \ud835\udcdd b \u2192 0 < \u2191(\u2191(OuterMeasure.map f) \u2191\u03bc) t\na : \u03b2\nhab : a \u2260 b\nha : \u2200 (t : Set \u03b2), t \u2208 \ud835\udcdd[{b}\u1d9c] a \u2192 0 < \u2191m t\n\u22a2 \u2203 a b, a \u2260 b \u2227 (\u2200 (s : Set \u03b2), s \u2208 \ud835\udcdd a \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' s)) \u2227 \u2200 (t : Set \u03b2), t \u2208 \ud835\udcdd b \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' t)\n[PROOFSTEP]\nsimp only [isOpen_compl_singleton.nhdsWithin_eq hab] at ha \n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b2 : Type u_8\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T1Space \u03b2\ninst\u271d\u00b9 : SecondCountableTopology \u03b2\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b2\nm : OuterMeasure \u03b2 := \u2191(OuterMeasure.map f) \u2191\u03bc\nh : \u2200 (b : \u03b2), \u2191m {b}\u1d9c \u2260 0\ninhabited_h : Inhabited \u03b2\nthis : \u2191m univ \u2260 0\nb : \u03b2\nhb : \u2200 (t : Set \u03b2), t \u2208 \ud835\udcdd b \u2192 0 < \u2191(\u2191(OuterMeasure.map f) \u2191\u03bc) t\na : \u03b2\nhab : a \u2260 b\nha : \u2200 (t : Set \u03b2), t \u2208 \ud835\udcdd a \u2192 0 < \u2191(\u2191(OuterMeasure.map f) \u2191\u03bc) t\n\u22a2 \u2203 a b, a \u2260 b \u2227 (\u2200 (s : Set \u03b2), s \u2208 \ud835\udcdd a \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' s)) \u2227 \u2200 (t : Set \u03b2), t \u2208 \ud835\udcdd b \u2192 0 < \u2191\u2191\u03bc (f \u207b\u00b9' t)\n[PROOFSTEP]\nexact \u27e8a, b, hab, ha, hb\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\n\u03b1 : Type u_8\nm\u2080 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nC : Set (Set \u03b1)\nh\u03bc\u03bd : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nm : MeasurableSpace \u03b1\nh : m \u2264 m\u2080\nhA : m = generateFrom C\nhC : IsPiSystem C\nh_univ : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\n[PROOFSTEP]\nhaveI : IsFiniteMeasure \u03bd := by\n constructor\n rw [\u2190 h_univ]\n apply IsFiniteMeasure.measure_univ_lt_top\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\n\u03b1 : Type u_8\nm\u2080 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nC : Set (Set \u03b1)\nh\u03bc\u03bd : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nm : MeasurableSpace \u03b1\nh : m \u2264 m\u2080\nhA : m = generateFrom C\nhC : IsPiSystem C\nh_univ : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 IsFiniteMeasure \u03bd\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase measure_univ_lt_top\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\n\u03b1 : Type u_8\nm\u2080 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nC : Set (Set \u03b1)\nh\u03bc\u03bd : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nm : MeasurableSpace \u03b1\nh : m \u2264 m\u2080\nhA : m = generateFrom C\nhC : IsPiSystem C\nh_univ : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bd univ < \u22a4\n[PROOFSTEP]\nrw [\u2190 h_univ]\n[GOAL]\ncase measure_univ_lt_top\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\n\u03b1 : Type u_8\nm\u2080 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nC : Set (Set \u03b1)\nh\u03bc\u03bd : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nm : MeasurableSpace \u03b1\nh : m \u2264 m\u2080\nhA : m = generateFrom C\nhC : IsPiSystem C\nh_univ : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\napply IsFiniteMeasure.measure_univ_lt_top\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\n\u03b1 : Type u_8\nm\u2080 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nC : Set (Set \u03b1)\nh\u03bc\u03bd : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nm : MeasurableSpace \u03b1\nh : m \u2264 m\u2080\nhA : m = generateFrom C\nhC : IsPiSystem C\nh_univ : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\ns : Set \u03b1\nhs : MeasurableSet s\nthis : IsFiniteMeasure \u03bd\n\u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\n[PROOFSTEP]\nrefine' induction_on_inter hA hC (by simp) h\u03bc\u03bd _ _ hs\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\n\u03b1 : Type u_8\nm\u2080 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nC : Set (Set \u03b1)\nh\u03bc\u03bd : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nm : MeasurableSpace \u03b1\nh : m \u2264 m\u2080\nhA : m = generateFrom C\nhC : IsPiSystem C\nh_univ : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\ns : Set \u03b1\nhs : MeasurableSet s\nthis : IsFiniteMeasure \u03bd\n\u22a2 \u2191\u2191\u03bc \u2205 = \u2191\u2191\u03bd \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\n\u03b1 : Type u_8\nm\u2080 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nC : Set (Set \u03b1)\nh\u03bc\u03bd : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nm : MeasurableSpace \u03b1\nh : m \u2264 m\u2080\nhA : m = generateFrom C\nhC : IsPiSystem C\nh_univ : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\ns : Set \u03b1\nhs : MeasurableSet s\nthis : IsFiniteMeasure \u03bd\n\u22a2 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t \u2192 \u2191\u2191\u03bc t\u1d9c = \u2191\u2191\u03bd t\u1d9c\n[PROOFSTEP]\nintro t h1t h2t\n[GOAL]\ncase refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t\u271d : Set \u03b1\u271d\n\u03b1 : Type u_8\nm\u2080 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nC : Set (Set \u03b1)\nh\u03bc\u03bd : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nm : MeasurableSpace \u03b1\nh : m \u2264 m\u2080\nhA : m = generateFrom C\nhC : IsPiSystem C\nh_univ : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\ns : Set \u03b1\nhs : MeasurableSet s\nthis : IsFiniteMeasure \u03bd\nt : Set \u03b1\nh1t : MeasurableSet t\nh2t : \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\n\u22a2 \u2191\u2191\u03bc t\u1d9c = \u2191\u2191\u03bd t\u1d9c\n[PROOFSTEP]\nhave h1t_ : @MeasurableSet \u03b1 m\u2080 t := h _ h1t\n[GOAL]\ncase refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t\u271d : Set \u03b1\u271d\n\u03b1 : Type u_8\nm\u2080 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nC : Set (Set \u03b1)\nh\u03bc\u03bd : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nm : MeasurableSpace \u03b1\nh : m \u2264 m\u2080\nhA : m = generateFrom C\nhC : IsPiSystem C\nh_univ : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\ns : Set \u03b1\nhs : MeasurableSet s\nthis : IsFiniteMeasure \u03bd\nt : Set \u03b1\nh1t : MeasurableSet t\nh2t : \u2191\u2191\u03bc t = \u2191\u2191\u03bd t\nh1t_ : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc t\u1d9c = \u2191\u2191\u03bd t\u1d9c\n[PROOFSTEP]\nrw [@measure_compl \u03b1 m\u2080 \u03bc t h1t_ (@measure_ne_top \u03b1 m\u2080 \u03bc _ t),\n @measure_compl \u03b1 m\u2080 \u03bd t h1t_ (@measure_ne_top \u03b1 m\u2080 \u03bd _ t), h_univ, h2t]\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\n\u03b1 : Type u_8\nm\u2080 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nC : Set (Set \u03b1)\nh\u03bc\u03bd : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nm : MeasurableSpace \u03b1\nh : m \u2264 m\u2080\nhA : m = generateFrom C\nhC : IsPiSystem C\nh_univ : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\ns : Set \u03b1\nhs : MeasurableSet s\nthis : IsFiniteMeasure \u03bd\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1),\n Pairwise (Disjoint on f) \u2192\n (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n (\u2200 (i : \u2115), \u2191\u2191\u03bc (f i) = \u2191\u2191\u03bd (f i)) \u2192 \u2191\u2191\u03bc (\u22c3 (i : \u2115), f i) = \u2191\u2191\u03bd (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nintro f h1f h2f h3f\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\n\u03b1 : Type u_8\nm\u2080 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nC : Set (Set \u03b1)\nh\u03bc\u03bd : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nm : MeasurableSpace \u03b1\nh : m \u2264 m\u2080\nhA : m = generateFrom C\nhC : IsPiSystem C\nh_univ : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\ns : Set \u03b1\nhs : MeasurableSet s\nthis : IsFiniteMeasure \u03bd\nf : \u2115 \u2192 Set \u03b1\nh1f : Pairwise (Disjoint on f)\nh2f : \u2200 (i : \u2115), MeasurableSet (f i)\nh3f : \u2200 (i : \u2115), \u2191\u2191\u03bc (f i) = \u2191\u2191\u03bd (f i)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u2115), f i) = \u2191\u2191\u03bd (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nhave h2f_ : \u2200 i : \u2115, @MeasurableSet \u03b1 m\u2080 (f i) := fun i => h _ (h2f i)\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\n\u03b1 : Type u_8\nm\u2080 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nC : Set (Set \u03b1)\nh\u03bc\u03bd : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\nm : MeasurableSpace \u03b1\nh : m \u2264 m\u2080\nhA : m = generateFrom C\nhC : IsPiSystem C\nh_univ : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ\ns : Set \u03b1\nhs : MeasurableSet s\nthis : IsFiniteMeasure \u03bd\nf : \u2115 \u2192 Set \u03b1\nh1f : Pairwise (Disjoint on f)\nh2f : \u2200 (i : \u2115), MeasurableSet (f i)\nh3f : \u2200 (i : \u2115), \u2191\u2191\u03bc (f i) = \u2191\u2191\u03bd (f i)\nh2f_ : \u2200 (i : \u2115), MeasurableSet (f i)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u2115), f i) = \u2191\u2191\u03bd (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nsimp [measure_iUnion, h1f, h3f, h2f_]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf g : Filter \u03b1\n\u22a2 FiniteAtFilter \u03bc (f \u2293 ae \u03bc) \u2194 FiniteAtFilter \u03bc f\n[PROOFSTEP]\nrefine' \u27e8_, fun h => h.filter_mono inf_le_left\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf g : Filter \u03b1\n\u22a2 FiniteAtFilter \u03bc (f \u2293 ae \u03bc) \u2192 FiniteAtFilter \u03bc f\n[PROOFSTEP]\nrintro \u27e8s, \u27e8t, ht, u, hu, rfl\u27e9, h\u03bc\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nf g : Filter \u03b1\nt : Set \u03b1\nht : t \u2208 f\nu : Set \u03b1\nhu : u \u2208 ae \u03bc\nh\u03bc : \u2191\u2191\u03bc (t \u2229 u) < \u22a4\n\u22a2 FiniteAtFilter \u03bc f\n[PROOFSTEP]\nsuffices : \u03bc t \u2264 \u03bc (t \u2229 u)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nf g : Filter \u03b1\nt : Set \u03b1\nht : t \u2208 f\nu : Set \u03b1\nhu : u \u2208 ae \u03bc\nh\u03bc : \u2191\u2191\u03bc (t \u2229 u) < \u22a4\nthis : \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (t \u2229 u)\n\u22a2 FiniteAtFilter \u03bc f\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nf g : Filter \u03b1\nt : Set \u03b1\nht : t \u2208 f\nu : Set \u03b1\nhu : u \u2208 ae \u03bc\nh\u03bc : \u2191\u2191\u03bc (t \u2229 u) < \u22a4\n\u22a2 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (t \u2229 u)\n[PROOFSTEP]\nexact \u27e8t, ht, this.trans_lt h\u03bc\u27e9\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nf g : Filter \u03b1\nt : Set \u03b1\nht : t \u2208 f\nu : Set \u03b1\nhu : u \u2208 ae \u03bc\nh\u03bc : \u2191\u2191\u03bc (t \u2229 u) < \u22a4\n\u22a2 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (t \u2229 u)\n[PROOFSTEP]\nexact measure_mono_ae (mem_of_superset hu fun x hu ht => \u27e8ht, hu\u27e9)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s = \u2191\u2191\u03bc (f \u207b\u00b9' s)\n[PROOFSTEP]\nrefine' le_antisymm _ (le_map_apply hf.measurable.aemeasurable s)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s)\n[PROOFSTEP]\nset t := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s)\n[PROOFSTEP]\nhave htm : MeasurableSet t :=\n (hf.measurableSet_image.2 <| measurableSet_toMeasurable _ _).union hf.measurableSet_range.compl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s)\n[PROOFSTEP]\nhave hst : s \u2286 t := by\n rw [subset_union_compl_iff_inter_subset, \u2190 image_preimage_eq_inter_range]\n exact image_subset _ (subset_toMeasurable _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\n\u22a2 s \u2286 t\n[PROOFSTEP]\nrw [subset_union_compl_iff_inter_subset, \u2190 image_preimage_eq_inter_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\n\u22a2 f '' (f \u207b\u00b9' s) \u2286 f '' toMeasurable \u03bc (f \u207b\u00b9' s)\n[PROOFSTEP]\nexact image_subset _ (subset_toMeasurable _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\nhst : s \u2286 t\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s)\n[PROOFSTEP]\nhave hft : f \u207b\u00b9' t = toMeasurable \u03bc (f \u207b\u00b9' s) := by\n rw [preimage_union, preimage_compl, preimage_range, compl_univ, union_empty, hf.injective.preimage_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\nhst : s \u2286 t\n\u22a2 f \u207b\u00b9' t = toMeasurable \u03bc (f \u207b\u00b9' s)\n[PROOFSTEP]\nrw [preimage_union, preimage_compl, preimage_range, compl_univ, union_empty, hf.injective.preimage_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\nhst : s \u2286 t\nhft : f \u207b\u00b9' t = toMeasurable \u03bc (f \u207b\u00b9' s)\n\u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s)\n[PROOFSTEP]\ncalc\n \u03bc.map f s \u2264 \u03bc.map f t := measure_mono hst\n _ = \u03bc (f \u207b\u00b9' s) := by rw [map_apply hf.measurable htm, hft, measure_toMeasurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns : Set \u03b2\nt : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c\nhtm : MeasurableSet t\nhst : s \u2286 t\nhft : f \u207b\u00b9' t = toMeasurable \u03bc (f \u207b\u00b9' s)\n\u22a2 \u2191\u2191(Measure.map f \u03bc) t = \u2191\u2191\u03bc (f \u207b\u00b9' s)\n[PROOFSTEP]\nrw [map_apply hf.measurable htm, hft, measure_toMeasurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b2\n\u22a2 Measure.map f (Measure.comap f \u03bc) = Measure.restrict \u03bc (range f)\n[PROOFSTEP]\next1 t ht\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b2\nt : Set \u03b2\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.map f (Measure.comap f \u03bc)) t = \u2191\u2191(Measure.restrict \u03bc (range f)) t\n[PROOFSTEP]\nrw [hf.map_apply, comap_apply f hf.injective hf.measurableSet_image' _ (hf.measurable ht),\n image_preimage_eq_inter_range, Measure.restrict_apply ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b2\ns : Set \u03b1\n\u22a2 \u2191\u2191(Measure.comap f \u03bc) s = \u2191\u2191(Measure.comap f \u03bc) (f \u207b\u00b9' (f '' s))\n[PROOFSTEP]\nrw [hf.injective.preimage_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b2\ns : Set \u03b1\n\u22a2 \u2191\u2191(Measure.map f (Measure.comap f \u03bc)) (f '' s) = \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nrw [hf.map_comap, restrict_apply' hf.measurableSet_range, inter_eq_self_of_subset_left (image_subset_range _ _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\np : \u03b2 \u2192 Prop\n\u03bc : Measure \u03b1\n\u22a2 (\u2200\u1d50 (x : \u03b2) \u2202Measure.map f \u03bc, p x) \u2194 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p (f x)\n[PROOFSTEP]\nsimp only [ae_iff, hf.map_apply, preimage_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc : Measure \u03b1\ns t : Set \u03b2\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.restrict (Measure.map f \u03bc) s) t = \u2191\u2191(Measure.map f (Measure.restrict \u03bc (f \u207b\u00b9' s))) t\n[PROOFSTEP]\nsimp [hf.map_apply, ht, hf.measurable ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u03bc : Measure \u03b1\n\u22a2 Measure.map Subtype.val (Measure.comap Subtype.val \u03bc) = Measure.restrict \u03bc s\n[PROOFSTEP]\nrw [(MeasurableEmbedding.subtype_coe hs).map_comap, Subtype.range_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, p x) \u2194 \u2200\u1d50 (x : \u2191s) \u2202Measure.comap Subtype.val \u03bc, p \u2191x\n[PROOFSTEP]\nrw [\u2190 map_comap_subtype_coe hs, (MeasurableEmbedding.subtype_coe hs).ae_map_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d : MeasureSpace \u03b1\ns\u271d t s : Set \u03b1\nhs : MeasurableSet s\n\u22a2 Measure.map Subtype.val volume = Measure.restrict volume s\n[PROOFSTEP]\nrw [volume_set_coe_def, (MeasurableEmbedding.subtype_coe hs).map_comap volume, Subtype.range_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d : MeasureSpace \u03b1\ns t : Set \u03b1\nhs : NullMeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191volume (Subtype.val \u207b\u00b9' t) = \u2191\u2191volume (t \u2229 s)\n[PROOFSTEP]\nrw [volume_set_coe_def,\n comap_apply\u2080 _ _ Subtype.coe_injective (fun h => MeasurableSet.nullMeasurableSet_subtype_coe hs)\n (measurable_subtype_coe ht).nullMeasurableSet,\n image_preimage_eq_inter_range, Subtype.range_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\n\u22a2 Measure.map (\u2191(symm e)) (Measure.map (\u2191e) \u03bc) = \u03bc\n[PROOFSTEP]\nsimp [map_map e.symm.measurable e.measurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\n\u22a2 Measure.map (\u2191e) (Measure.map (\u2191(symm e)) \u03bd) = \u03bd\n[PROOFSTEP]\nsimp [map_map e.measurable e.symm.measurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\n\u22a2 Injective (Measure.map \u2191e)\n[PROOFSTEP]\nintro \u03bc\u2081 \u03bc\u2082 h\u03bc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\n\u03bc\u2081 \u03bc\u2082 : Measure \u03b1\nh\u03bc : Measure.map (\u2191e) \u03bc\u2081 = Measure.map (\u2191e) \u03bc\u2082\n\u22a2 \u03bc\u2081 = \u03bc\u2082\n[PROOFSTEP]\napply_fun Measure.map e.symm at h\u03bc \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\n\u03bc\u2081 \u03bc\u2082 : Measure \u03b1\nh\u03bc : Measure.map (\u2191(symm e)) (Measure.map (\u2191e) \u03bc\u2081) = Measure.map (\u2191(symm e)) (Measure.map (\u2191e) \u03bc\u2082)\n\u22a2 \u03bc\u2081 = \u03bc\u2082\n[PROOFSTEP]\nsimpa [map_symm_map e] using h\u03bc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\n\u22a2 Measure.map (\u2191e) \u03bc = \u03bd \u2194 Measure.map (\u2191(symm e)) \u03bd = \u03bc\n[PROOFSTEP]\nrw [\u2190 (map_measurableEquiv_injective e).eq_iff, map_map_symm, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc\u271d : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2243\u1d50 \u03b2\n\u03bc : Measure \u03b1\n\u22a2 Filter.map (\u2191f) (ae \u03bc) = ae (Measure.map (\u2191f) \u03bc)\n[PROOFSTEP]\next s\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc\u271d : Measure \u03b1\n\u03bd : Measure \u03b2\nf : \u03b1 \u2243\u1d50 \u03b2\n\u03bc : Measure \u03b1\ns : Set \u03b2\n\u22a2 s \u2208 Filter.map (\u2191f) (ae \u03bc) \u2194 s \u2208 ae (Measure.map (\u2191f) \u03bc)\n[PROOFSTEP]\nsimp_rw [mem_map, mem_ae_iff, \u2190 preimage_compl, f.map_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc\u271d : Measure \u03b1\n\u03bd : Measure \u03b2\n\u03bc : Measure \u03b1\ne : \u03b1 \u2243\u1d50 \u03b2\n\u22a2 Measure.map (\u2191(symm e)) (Measure.map (\u2191e) \u03bc) \u226a \u03bc\n[PROOFSTEP]\nrw [Measure.map_map, e.symm_comp_self, Measure.map_id]\n[GOAL]\ncase hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc\u271d : Measure \u03b1\n\u03bd : Measure \u03b2\n\u03bc : Measure \u03b1\ne : \u03b1 \u2243\u1d50 \u03b2\n\u22a2 Measurable \u2191(symm e)\n[PROOFSTEP]\nmeasurability\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc\u271d : Measure \u03b1\n\u03bd : Measure \u03b2\n\u03bc : Measure \u03b1\ne : \u03b1 \u2243\u1d50 \u03b2\n\u22a2 Measurable \u2191e\n[PROOFSTEP]\nmeasurability\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 toMeasure 0 (_ : inst\u271d \u2264 OuterMeasure.caratheodory 0) = 0\n[PROOFSTEP]\nrw [\u2190 Measure.measure_univ_eq_zero, toMeasure_apply _ _ MeasurableSet.univ, OuterMeasure.coe_zero, Pi.zero_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u22a2 trim \u03bc (_ : inst\u271d \u2264 inst\u271d) = \u03bc\n[PROOFSTEP]\nsimp [Measure.trim]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\n\u22a2 \u2191(trim \u03bc hm) = OuterMeasure.trim \u2191\u03bc\n[PROOFSTEP]\nrw [Measure.trim, toMeasure_toOuterMeasure (ms := m)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhm : m \u2264 m0\n\u22a2 trim 0 hm = 0\n[PROOFSTEP]\nsimp [Measure.trim, @OuterMeasure.toMeasure_zero _ m]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(trim \u03bc hm) s = \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [Measure.trim, toMeasure_apply (ms := m) _ _ hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhm : m \u2264 m0\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2191\u2191(trim \u03bc hm) s\n[PROOFSTEP]\nsimp_rw [Measure.trim]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhm : m \u2264 m0\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2191\u2191(OuterMeasure.toMeasure \u2191\u03bc (_ : m \u2264 OuterMeasure.caratheodory \u2191\u03bc)) s\n[PROOFSTEP]\nexact @le_toMeasure_apply _ m _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhm : m \u2264 m0\nhs : \u2191\u2191(trim \u03bc hm) s = 0\n\u22a2 \u2191\u2191(trim \u03bc hm) (toMeasurable (trim \u03bc hm) s) = 0\n[PROOFSTEP]\nrwa [@measure_toMeasurable _ m]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nm\u2081 m\u2082 : MeasurableSpace \u03b1\nhm\u2081\u2082 : m\u2081 \u2264 m\u2082\nhm\u2082 : m\u2082 \u2264 m0\n\u22a2 trim (trim \u03bc hm\u2082) hm\u2081\u2082 = trim \u03bc (_ : m\u2081 \u2264 m0)\n[PROOFSTEP]\nrefine @Measure.ext _ m\u2081 _ _ (fun t ht => ?_)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nm\u2081 m\u2082 : MeasurableSpace \u03b1\nhm\u2081\u2082 : m\u2081 \u2264 m\u2082\nhm\u2082 : m\u2082 \u2264 m0\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(trim (trim \u03bc hm\u2082) hm\u2081\u2082) t = \u2191\u2191(trim \u03bc (_ : m\u2081 \u2264 m0)) t\n[PROOFSTEP]\nrw [trim_measurableSet_eq hm\u2081\u2082 ht, trim_measurableSet_eq (hm\u2081\u2082.trans hm\u2082) ht, trim_measurableSet_eq hm\u2082 (hm\u2081\u2082 t ht)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nhm : m \u2264 m0\n\u03bc : Measure \u03b1\nhs : MeasurableSet s\n\u22a2 Measure.restrict (trim \u03bc hm) s = trim (Measure.restrict \u03bc s) hm\n[PROOFSTEP]\nrefine @Measure.ext _ m _ _ (fun t ht => ?_)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nhm : m \u2264 m0\n\u03bc : Measure \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.restrict (trim \u03bc hm) s) t = \u2191\u2191(trim (Measure.restrict \u03bc s) hm) t\n[PROOFSTEP]\nrw [@Measure.restrict_apply \u03b1 m _ _ _ ht, trim_measurableSet_eq hm ht, Measure.restrict_apply (hm t ht),\n trim_measurableSet_eq hm (@MeasurableSet.inter \u03b1 m t s ht hs)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 \u2191\u2191(trim \u03bc hm) univ < \u22a4\n[PROOFSTEP]\nrw [trim_measurableSet_eq hm (@MeasurableSet.univ _ m)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhm : m \u2264 m0\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nexact measure_lt_top _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (trim \u03bc (_ : m\u2082 \u2264 m0))\n\u22a2 SigmaFinite (trim \u03bc hm)\n[PROOFSTEP]\nhave _ := Measure.FiniteSpanningSetsIn (\u03bc.trim (hm\u2082.trans hm)) Set.univ\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\n\u22a2 SigmaFinite (trim \u03bc hm)\n[PROOFSTEP]\nrefine' Measure.FiniteSpanningSetsIn.sigmaFinite _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\n\u22a2 Set (Set \u03b1)\n[PROOFSTEP]\nexact Set.univ\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\n\u22a2 FiniteSpanningSetsIn (trim \u03bc hm) univ\n[PROOFSTEP]\nrefine'\n { set := spanningSets (\u03bc.trim (hm\u2082.trans hm))\n set_mem := fun _ => Set.mem_univ _\n finite := fun i =>\n _\n -- This is the only one left to prove\n spanning := iUnion_spanningSets _ }\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\ni : \u2115\n\u22a2 \u2191\u2191(trim \u03bc hm) (spanningSets (trim \u03bc (_ : m\u2082 \u2264 m0)) i) < \u22a4\n[PROOFSTEP]\ncalc\n (\u03bc.trim hm) (spanningSets (\u03bc.trim (hm\u2082.trans hm)) i) =\n ((\u03bc.trim hm).trim hm\u2082) (spanningSets (\u03bc.trim (hm\u2082.trans hm)) i) :=\n by rw [@trim_measurableSet_eq \u03b1 m\u2082 m (\u03bc.trim hm) _ hm\u2082 (measurable_spanningSets _ _)]\n _ = (\u03bc.trim (hm\u2082.trans hm)) (spanningSets (\u03bc.trim (hm\u2082.trans hm)) i) := by rw [@trim_trim _ _ \u03bc _ _ hm\u2082 hm]\n _ < \u221e := measure_spanningSets_lt_top _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\ni : \u2115\n\u22a2 \u2191\u2191(trim \u03bc hm) (spanningSets (trim \u03bc (_ : m\u2082 \u2264 m0)) i) =\n \u2191\u2191(trim (trim \u03bc hm) hm\u2082) (spanningSets (trim \u03bc (_ : m\u2082 \u2264 m0)) i)\n[PROOFSTEP]\nrw [@trim_measurableSet_eq \u03b1 m\u2082 m (\u03bc.trim hm) _ hm\u2082 (measurable_spanningSets _ _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm\u271d m0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns : Set \u03b1\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhm\u2082 : m\u2082 \u2264 m\ninst\u271d : SigmaFinite (trim \u03bc (_ : m\u2082 \u2264 m0))\nx\u271d : Type u_1\ni : \u2115\n\u22a2 \u2191\u2191(trim (trim \u03bc hm) hm\u2082) (spanningSets (trim \u03bc (_ : m\u2082 \u2264 m0)) i) =\n \u2191\u2191(trim \u03bc (_ : m\u2082 \u2264 m0)) (spanningSets (trim \u03bc (_ : m\u2082 \u2264 m0)) i)\n[PROOFSTEP]\nrw [@trim_trim _ _ \u03bc _ _ hm\u2082 hm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\n\u22a2 SigmaFinite (trim \u03bc (_ : \u22a5 \u2264 m0)) \u2194 IsFiniteMeasure \u03bc\n[PROOFSTEP]\nrw [sigmaFinite_bot_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\n\u22a2 IsFiniteMeasure (trim \u03bc (_ : \u22a5 \u2264 m0)) \u2194 IsFiniteMeasure \u03bc\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8_\u27e9, fun h => \u27e8_\u27e9\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsFiniteMeasure (trim \u03bc (_ : \u22a5 \u2264 m0))\n\u22a2 \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nhave h_univ := h.measure_univ_lt_top\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsFiniteMeasure \u03bc\n\u22a2 \u2191\u2191(trim \u03bc (_ : \u22a5 \u2264 m0)) univ < \u22a4\n[PROOFSTEP]\nhave h_univ := h.measure_univ_lt_top\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsFiniteMeasure (trim \u03bc (_ : \u22a5 \u2264 m0))\nh_univ : \u2191\u2191(trim \u03bc (_ : \u22a5 \u2264 m0)) univ < \u22a4\n\u22a2 \u2191\u2191\u03bc univ < \u22a4\n[PROOFSTEP]\nrwa [trim_measurableSet_eq bot_le MeasurableSet.univ] at h_univ \n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsFiniteMeasure \u03bc\nh_univ : \u2191\u2191\u03bc univ < \u22a4\n\u22a2 \u2191\u2191(trim \u03bc (_ : \u22a5 \u2264 m0)) univ < \u22a4\n[PROOFSTEP]\nrwa [trim_measurableSet_eq bot_le MeasurableSet.univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsCompact s\nh\u03bc : \u2200 (x : \u03b1), x \u2208 s \u2192 FiniteAtFilter \u03bc (\ud835\udcdd x)\n\u22a2 \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u22a4\n[PROOFSTEP]\nrefine' IsCompact.induction_on h _ _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsCompact s\nh\u03bc : \u2200 (x : \u03b1), x \u2208 s \u2192 FiniteAtFilter \u03bc (\ud835\udcdd x)\n\u22a2 \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u22a4\n[PROOFSTEP]\nuse\u2205\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsCompact s\nh\u03bc : \u2200 (x : \u03b1), x \u2208 s \u2192 FiniteAtFilter \u03bc (\ud835\udcdd x)\n\u22a2 \u2203 x, IsOpen \u2205 \u2227 \u2191\u2191\u03bc \u2205 < \u22a4\n[PROOFSTEP]\nsimp [Superset]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsCompact s\nh\u03bc : \u2200 (x : \u03b1), x \u2208 s \u2192 FiniteAtFilter \u03bc (\ud835\udcdd x)\n\u22a2 \u2200 \u2983s t : Set \u03b1\u2984, s \u2286 t \u2192 (\u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u22a4) \u2192 \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u22a4\n[PROOFSTEP]\nrintro s t hst \u27e8U, htU, hUo, hU\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d : Set \u03b1\nh : IsCompact s\u271d\nh\u03bc : \u2200 (x : \u03b1), x \u2208 s\u271d \u2192 FiniteAtFilter \u03bc (\ud835\udcdd x)\ns t : Set \u03b1\nhst : s \u2286 t\nU : Set \u03b1\nhtU : U \u2287 t\nhUo : IsOpen U\nhU : \u2191\u2191\u03bc U < \u22a4\n\u22a2 \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u22a4\n[PROOFSTEP]\nexact \u27e8U, hst.trans htU, hUo, hU\u27e9\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsCompact s\nh\u03bc : \u2200 (x : \u03b1), x \u2208 s \u2192 FiniteAtFilter \u03bc (\ud835\udcdd x)\n\u22a2 \u2200 \u2983s t : Set \u03b1\u2984, (\u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u22a4) \u2192 (\u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u22a4) \u2192 \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u22a4\n[PROOFSTEP]\nrintro s t \u27e8U, hsU, hUo, hU\u27e9 \u27e8V, htV, hVo, hV\u27e9\n[GOAL]\ncase refine'_3.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d : Set \u03b1\nh : IsCompact s\u271d\nh\u03bc : \u2200 (x : \u03b1), x \u2208 s\u271d \u2192 FiniteAtFilter \u03bc (\ud835\udcdd x)\ns t U : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhU : \u2191\u2191\u03bc U < \u22a4\nV : Set \u03b1\nhtV : V \u2287 t\nhVo : IsOpen V\nhV : \u2191\u2191\u03bc V < \u22a4\n\u22a2 \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u22a4\n[PROOFSTEP]\nrefine'\n \u27e8U \u222a V, union_subset_union hsU htV, hUo.union hVo, (measure_union_le _ _).trans_lt <| ENNReal.add_lt_top.2 \u27e8hU, hV\u27e9\u27e9\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsCompact s\nh\u03bc : \u2200 (x : \u03b1), x \u2208 s \u2192 FiniteAtFilter \u03bc (\ud835\udcdd x)\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u22a4\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsCompact s\nh\u03bc : \u2200 (x : \u03b1), x \u2208 s \u2192 FiniteAtFilter \u03bc (\ud835\udcdd x)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u22a4\n[PROOFSTEP]\nrcases(h\u03bc x hx).exists_mem_basis (nhds_basis_opens _) with \u27e8U, \u27e8hx, hUo\u27e9, hU\u27e9\n[GOAL]\ncase refine'_4.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsCompact s\nh\u03bc : \u2200 (x : \u03b1), x \u2208 s \u2192 FiniteAtFilter \u03bc (\ud835\udcdd x)\nx : \u03b1\nhx\u271d : x \u2208 s\nU : Set \u03b1\nhU : \u2191\u2191\u03bc U < \u22a4\nhx : x \u2208 U\nhUo : IsOpen U\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u22a4\n[PROOFSTEP]\nexact \u27e8U, nhdsWithin_le_nhds (hUo.mem_nhds hx), U, Subset.rfl, hUo, hU\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh : IsCompact s\nh\u03bc : \u2200 (x : \u03b1), x \u2208 s \u2192 FiniteAtFilter \u03bc (\ud835\udcdd[s] x)\n\u22a2 \u2191\u2191\u03bc \u2205 < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : IsCompact s\n\u22a2 (\u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] a \u2227 \u2191\u2191\u03bc t = 0) \u2192 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nsimpa only [\u2190 compl_mem_ae_iff] using hs.compl_mem_sets_of_nhdsWithin\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompactSpace \u03b1\n\u22a2 IsFiniteMeasure \u03bc \u2194 IsFiniteMeasureOnCompacts \u03bc\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompactSpace \u03b1\n\u22a2 IsFiniteMeasure \u03bc \u2192 IsFiniteMeasureOnCompacts \u03bc\n[PROOFSTEP]\nintros\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompactSpace \u03b1\n\u22a2 IsFiniteMeasureOnCompacts \u03bc \u2192 IsFiniteMeasure \u03bc\n[PROOFSTEP]\nintros\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompactSpace \u03b1\na\u271d : IsFiniteMeasure \u03bc\n\u22a2 IsFiniteMeasureOnCompacts \u03bc\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompactSpace \u03b1\na\u271d : IsFiniteMeasureOnCompacts \u03bc\n\u22a2 IsFiniteMeasure \u03bc\n[PROOFSTEP]\nexact CompactSpace.isFiniteMeasure\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\n\u22a2 FiniteSpanningSetsIn \u03bc {K | IsOpen K}\n[PROOFSTEP]\nsuffices H : Nonempty (\u03bc.FiniteSpanningSetsIn {K | IsOpen K})\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nH : Nonempty (FiniteSpanningSetsIn \u03bc {K | IsOpen K})\n\u22a2 FiniteSpanningSetsIn \u03bc {K | IsOpen K}\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\n\u22a2 Nonempty (FiniteSpanningSetsIn \u03bc {K | IsOpen K})\n[PROOFSTEP]\nexact H.some\n[GOAL]\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\n\u22a2 Nonempty (FiniteSpanningSetsIn \u03bc {K | IsOpen K})\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b1\n[GOAL]\ncase H.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : IsEmpty \u03b1\n\u22a2 Nonempty (FiniteSpanningSetsIn \u03bc {K | IsOpen K})\n[PROOFSTEP]\nexact\n \u27e8{ set := fun _ => \u2205\n set_mem := fun _ => by simp\n finite := fun _ => by simp\n spanning := by simp }\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : IsEmpty \u03b1\nx\u271d : \u2115\n\u22a2 (fun x => \u2205) x\u271d \u2208 {K | IsOpen K}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : IsEmpty \u03b1\nx\u271d : \u2115\n\u22a2 \u2191\u2191\u03bc ((fun x => \u2205) x\u271d) < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : IsEmpty \u03b1\n\u22a2 \u22c3 (i : \u2115), (fun x => \u2205) i = univ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase H.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\n\u22a2 Nonempty (FiniteSpanningSetsIn \u03bc {K | IsOpen K})\n[PROOFSTEP]\ninhabit \u03b1\n[GOAL]\ncase H.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\n\u22a2 Nonempty (FiniteSpanningSetsIn \u03bc {K | IsOpen K})\n[PROOFSTEP]\nlet S : Set (Set \u03b1) := {s | IsOpen s \u2227 \u03bc s < \u221e}\n[GOAL]\ncase H.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\n\u22a2 Nonempty (FiniteSpanningSetsIn \u03bc {K | IsOpen K})\n[PROOFSTEP]\nobtain \u27e8T, T_count, TS, hT\u27e9 : \u2203 T : Set (Set \u03b1), T.Countable \u2227 T \u2286 S \u2227 \u22c3\u2080 T = \u22c3\u2080 S :=\n isOpen_sUnion_countable S fun s hs => hs.1\n[GOAL]\ncase H.inr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = \u22c3\u2080 S\n\u22a2 Nonempty (FiniteSpanningSetsIn \u03bc {K | IsOpen K})\n[PROOFSTEP]\nrw [\u03bc.isTopologicalBasis_isOpen_lt_top.sUnion_eq] at hT \n[GOAL]\ncase H.inr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\n\u22a2 Nonempty (FiniteSpanningSetsIn \u03bc {K | IsOpen K})\n[PROOFSTEP]\nhave T_ne : T.Nonempty := by\n by_contra h'T\n rw [not_nonempty_iff_eq_empty.1 h'T, sUnion_empty] at hT \n simpa only [\u2190 hT] using mem_univ (default : \u03b1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\n\u22a2 Set.Nonempty T\n[PROOFSTEP]\nby_contra h'T\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nh'T : \u00acSet.Nonempty T\n\u22a2 False\n[PROOFSTEP]\nrw [not_nonempty_iff_eq_empty.1 h'T, sUnion_empty] at hT \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u2205 = univ\nh'T : \u00acSet.Nonempty T\n\u22a2 False\n[PROOFSTEP]\nsimpa only [\u2190 hT] using mem_univ (default : \u03b1)\n[GOAL]\ncase H.inr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\n\u22a2 Nonempty (FiniteSpanningSetsIn \u03bc {K | IsOpen K})\n[PROOFSTEP]\nobtain \u27e8f, hf\u27e9 : \u2203 f : \u2115 \u2192 Set \u03b1, T = range f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\n\u22a2 \u2203 f, T = range f\ncase H.inr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\nf : \u2115 \u2192 Set \u03b1\nhf : T = range f\n\u22a2 Nonempty (FiniteSpanningSetsIn \u03bc {K | IsOpen K})\n[PROOFSTEP]\nexact T_count.exists_eq_range T_ne\n[GOAL]\ncase H.inr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\nf : \u2115 \u2192 Set \u03b1\nhf : T = range f\n\u22a2 Nonempty (FiniteSpanningSetsIn \u03bc {K | IsOpen K})\n[PROOFSTEP]\nhave fS : \u2200 n, f n \u2208 S := by\n intro n\n apply TS\n rw [hf]\n exact mem_range_self n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\nf : \u2115 \u2192 Set \u03b1\nhf : T = range f\n\u22a2 \u2200 (n : \u2115), f n \u2208 S\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\nf : \u2115 \u2192 Set \u03b1\nhf : T = range f\nn : \u2115\n\u22a2 f n \u2208 S\n[PROOFSTEP]\napply TS\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\nf : \u2115 \u2192 Set \u03b1\nhf : T = range f\nn : \u2115\n\u22a2 f n \u2208 T\n[PROOFSTEP]\nrw [hf]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\nf : \u2115 \u2192 Set \u03b1\nhf : T = range f\nn : \u2115\n\u22a2 f n \u2208 range f\n[PROOFSTEP]\nexact mem_range_self n\n[GOAL]\ncase H.inr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\nf : \u2115 \u2192 Set \u03b1\nhf : T = range f\nfS : \u2200 (n : \u2115), f n \u2208 S\n\u22a2 Nonempty (FiniteSpanningSetsIn \u03bc {K | IsOpen K})\n[PROOFSTEP]\nrefine'\n \u27e8{ set := f\n set_mem := fun n => (fS n).1\n finite := fun n => (fS n).2\n spanning := _ }\u27e9\n[GOAL]\ncase H.inr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\nf : \u2115 \u2192 Set \u03b1\nhf : T = range f\nfS : \u2200 (n : \u2115), f n \u2208 S\n\u22a2 \u22c3 (i : \u2115), f i = univ\n[PROOFSTEP]\nrefine eq_univ_of_forall fun x => ?_\n[GOAL]\ncase H.inr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\nf : \u2115 \u2192 Set \u03b1\nhf : T = range f\nfS : \u2200 (n : \u2115), f n \u2208 S\nx : \u03b1\n\u22a2 x \u2208 \u22c3 (i : \u2115), f i\n[PROOFSTEP]\nobtain \u27e8t, tT, xt\u27e9 : \u2203 t : Set \u03b1, t \u2208 range f \u2227 x \u2208 t :=\n by\n have : x \u2208 \u22c3\u2080 T := by simp only [hT, mem_univ]\n simpa only [mem_sUnion, exists_prop, \u2190 hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\nf : \u2115 \u2192 Set \u03b1\nhf : T = range f\nfS : \u2200 (n : \u2115), f n \u2208 S\nx : \u03b1\n\u22a2 \u2203 t, t \u2208 range f \u2227 x \u2208 t\n[PROOFSTEP]\nhave : x \u2208 \u22c3\u2080 T := by simp only [hT, mem_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\nf : \u2115 \u2192 Set \u03b1\nhf : T = range f\nfS : \u2200 (n : \u2115), f n \u2208 S\nx : \u03b1\n\u22a2 x \u2208 \u22c3\u2080 T\n[PROOFSTEP]\nsimp only [hT, mem_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\nf : \u2115 \u2192 Set \u03b1\nhf : T = range f\nfS : \u2200 (n : \u2115), f n \u2208 S\nx : \u03b1\nthis : x \u2208 \u22c3\u2080 T\n\u22a2 \u2203 t, t \u2208 range f \u2227 x \u2208 t\n[PROOFSTEP]\nsimpa only [mem_sUnion, exists_prop, \u2190 hf]\n[GOAL]\ncase H.inr.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\nf : \u2115 \u2192 Set \u03b1\nhf : T = range f\nfS : \u2200 (n : \u2115), f n \u2208 S\nx : \u03b1\nt : Set \u03b1\ntT : t \u2208 range f\nxt : x \u2208 t\n\u22a2 x \u2208 \u22c3 (i : \u2115), f i\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 : \u2203 n : \u2115, f n = t := by simpa only using tT\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\nf : \u2115 \u2192 Set \u03b1\nhf : T = range f\nfS : \u2200 (n : \u2115), f n \u2208 S\nx : \u03b1\nt : Set \u03b1\ntT : t \u2208 range f\nxt : x \u2208 t\n\u22a2 \u2203 n, f n = t\n[PROOFSTEP]\nsimpa only using tT\n[GOAL]\ncase H.inr.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nh\u271d : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\nS : Set (Set \u03b1) := {s | IsOpen s \u2227 \u2191\u2191\u03bc s < \u22a4}\nT : Set (Set \u03b1)\nT_count : Set.Countable T\nTS : T \u2286 S\nhT : \u22c3\u2080 T = univ\nT_ne : Set.Nonempty T\nf : \u2115 \u2192 Set \u03b1\nhf : T = range f\nfS : \u2200 (n : \u2115), f n \u2208 S\nx : \u03b1\nn : \u2115\ntT : f n \u2208 range f\nxt : x \u2208 f n\n\u22a2 x \u2208 \u22c3 (i : \u2115), f i\n[PROOFSTEP]\nexact mem_iUnion_of_mem _ xt\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nhs : MeasurableSet s\n\u22a2 piecewise s f g =\u1da0[ae (Measure.restrict \u03bc s)] f\n[PROOFSTEP]\nrw [ae_restrict_eq hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nhs : MeasurableSet s\n\u22a2 piecewise s f g =\u1da0[ae \u03bc \u2293 \ud835\udcdf s] f\n[PROOFSTEP]\nexact (piecewise_eqOn s f g).eventuallyEq.filter_mono inf_le_right\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nhs : MeasurableSet s\n\u22a2 piecewise s f g =\u1da0[ae (Measure.restrict \u03bc s\u1d9c)] g\n[PROOFSTEP]\nrw [ae_restrict_eq hs.compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nhs : MeasurableSet s\n\u22a2 piecewise s f g =\u1da0[ae \u03bc \u2293 \ud835\udcdf s\u1d9c] g\n[PROOFSTEP]\nexact (piecewise_eqOn_compl s f g).eventuallyEq.filter_mono inf_le_right\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nhst : s =\u1da0[ae \u03bc] t\nx : \u03b1\nhx : x \u2208 s \u2194 x \u2208 t\n\u22a2 piecewise s f g x = piecewise t f g x\n[PROOFSTEP]\nsimp [piecewise, hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nt : Set \u03b2\nht : 0 \u2208 t\nhs : MeasurableSet s\n\u22a2 t \u2208 Filter.map (indicator s f) (ae \u03bc) \u2194 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s))\n[PROOFSTEP]\nsimp_rw [mem_map, mem_ae_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nt : Set \u03b2\nht : 0 \u2208 t\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc (indicator s f \u207b\u00b9' t)\u1d9c = 0 \u2194 \u2191\u2191(Measure.restrict \u03bc s) (f \u207b\u00b9' t)\u1d9c = 0\n[PROOFSTEP]\nrw [Measure.restrict_apply' hs, Set.indicator_preimage, Set.ite]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nt : Set \u03b2\nht : 0 \u2208 t\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc (f \u207b\u00b9' t \u2229 s \u222a 0 \u207b\u00b9' t \\ s)\u1d9c = 0 \u2194 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u2229 s) = 0\n[PROOFSTEP]\nsimp_rw [Set.compl_union, Set.compl_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nt : Set \u03b2\nht : 0 \u2208 t\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc (((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) \u2229 (0 \u207b\u00b9' t \\ s)\u1d9c) = 0 \u2194 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u2229 s) = 0\n[PROOFSTEP]\nchange \u03bc (((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) \u2229 ((fun _ => (0 : \u03b2)) \u207b\u00b9' t \\ s)\u1d9c) = 0 \u2194 \u03bc ((f \u207b\u00b9' t)\u1d9c \u2229 s) = 0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nt : Set \u03b2\nht : 0 \u2208 t\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc (((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) \u2229 ((fun x => 0) \u207b\u00b9' t \\ s)\u1d9c) = 0 \u2194 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u2229 s) = 0\n[PROOFSTEP]\nsimp only [ht, \u2190 Set.compl_eq_univ_diff, compl_compl, Set.compl_union, if_true, Set.preimage_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nt : Set \u03b2\nht : 0 \u2208 t\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc (((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) \u2229 s) = 0 \u2194 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u2229 s) = 0\n[PROOFSTEP]\nsimp_rw [Set.union_inter_distrib_right, Set.compl_inter_self s, Set.union_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nt : Set \u03b2\nht : \u00ac0 \u2208 t\n\u22a2 t \u2208 Filter.map (indicator s f) (ae \u03bc) \u2194 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) = 0\n[PROOFSTEP]\nrw [mem_map, mem_ae_iff, Set.indicator_preimage, Set.ite, Set.compl_union, Set.compl_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nt : Set \u03b2\nht : \u00ac0 \u2208 t\n\u22a2 \u2191\u2191\u03bc (((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) \u2229 (0 \u207b\u00b9' t \\ s)\u1d9c) = 0 \u2194 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) = 0\n[PROOFSTEP]\nchange \u03bc (((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) \u2229 ((fun _ => (0 : \u03b2)) \u207b\u00b9' t \\ s)\u1d9c) = 0 \u2194 \u03bc ((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) = 0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nt : Set \u03b2\nht : \u00ac0 \u2208 t\n\u22a2 \u2191\u2191\u03bc (((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) \u2229 ((fun x => 0) \u207b\u00b9' t \\ s)\u1d9c) = 0 \u2194 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) = 0\n[PROOFSTEP]\nsimp only [ht, if_false, Set.compl_empty, Set.empty_diff, Set.inter_univ, Set.preimage_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\n\u22a2 Filter.map f (ae (Measure.restrict \u03bc s)) \u2264 Filter.map (indicator s f) (ae \u03bc)\n[PROOFSTEP]\nintro t\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\n\u22a2 t \u2208 Filter.map (indicator s f) (ae \u03bc) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s))\n[PROOFSTEP]\nby_cases ht : (0 : \u03b2) \u2208 t\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\nht : 0 \u2208 t\n\u22a2 t \u2208 Filter.map (indicator s f) (ae \u03bc) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s))\n[PROOFSTEP]\nrw [mem_map_indicator_ae_iff_mem_map_restrict_ae_of_zero_mem ht hs]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\nht : 0 \u2208 t\n\u22a2 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s)) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s))\n[PROOFSTEP]\nexact id\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\nht : \u00ac0 \u2208 t\n\u22a2 t \u2208 Filter.map (indicator s f) (ae \u03bc) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s))\n[PROOFSTEP]\nrw [mem_map_indicator_ae_iff_of_zero_nmem ht, mem_map_restrict_ae_iff hs]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nt : Set \u03b2\nht : \u00ac0 \u2208 t\n\u22a2 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) = 0 \u2192 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u2229 s) = 0\n[PROOFSTEP]\nexact fun h => measure_mono_null ((Set.inter_subset_left _ _).trans (Set.subset_union_left _ _)) h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nhf : f =\u1da0[ae (Measure.restrict \u03bc s\u1d9c)] 0\n\u22a2 indicator s f =\u1da0[ae \u03bc] f\n[PROOFSTEP]\nrw [Filter.EventuallyEq, ae_restrict_iff' hs.compl] at hf \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 f x = OfNat.ofNat 0 x\n\u22a2 indicator s f =\u1da0[ae \u03bc] f\n[PROOFSTEP]\nfilter_upwards [hf] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 f x = OfNat.ofNat 0 x\nx : \u03b1\nhx : x \u2208 s\u1d9c \u2192 f x = OfNat.ofNat 0 x\n\u22a2 indicator s f x = f x\n[PROOFSTEP]\nby_cases hxs : x \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 f x = OfNat.ofNat 0 x\nx : \u03b1\nhx : x \u2208 s\u1d9c \u2192 f x = OfNat.ofNat 0 x\nhxs : x \u2208 s\n\u22a2 indicator s f x = f x\n[PROOFSTEP]\nsimp only [hxs, Set.indicator_of_mem]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s\u1d9c \u2192 f x = OfNat.ofNat 0 x\nx : \u03b1\nhx : x \u2208 s\u1d9c \u2192 f x = OfNat.ofNat 0 x\nhxs : \u00acx \u2208 s\n\u22a2 indicator s f x = f x\n[PROOFSTEP]\nsimp only [hx hxs, Pi.zero_apply, Set.indicator_apply_eq_zero, eq_self_iff_true, imp_true_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nhf : f =\u1da0[ae (Measure.restrict \u03bc s)] 0\n\u22a2 indicator s f =\u1da0[ae \u03bc] 0\n[PROOFSTEP]\nrw [Filter.EventuallyEq, ae_restrict_iff' hs] at hf \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = OfNat.ofNat 0 x\n\u22a2 indicator s f =\u1da0[ae \u03bc] 0\n[PROOFSTEP]\nfilter_upwards [hf] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = OfNat.ofNat 0 x\nx : \u03b1\nhx : x \u2208 s \u2192 f x = OfNat.ofNat 0 x\n\u22a2 indicator s f x = OfNat.ofNat 0 x\n[PROOFSTEP]\nby_cases hxs : x \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = OfNat.ofNat 0 x\nx : \u03b1\nhx : x \u2208 s \u2192 f x = OfNat.ofNat 0 x\nhxs : x \u2208 s\n\u22a2 indicator s f x = OfNat.ofNat 0 x\n[PROOFSTEP]\nsimp only [hxs, hx hxs, Set.indicator_of_mem]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = OfNat.ofNat 0 x\nx : \u03b1\nhx : x \u2208 s \u2192 f x = OfNat.ofNat 0 x\nhxs : \u00acx \u2208 s\n\u22a2 indicator s f x = OfNat.ofNat 0 x\n[PROOFSTEP]\nsimp [hx, hxs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\ng : \u03b1 \u2192 \u03b2\nhs : MeasurableSet s\n\u22a2 f =\u1da0[ae (Measure.restrict \u03bc s)] g \u2194 indicator s f =\u1da0[ae \u03bc] indicator s g\n[PROOFSTEP]\nrw [Filter.EventuallyEq, ae_restrict_iff' hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\ng : \u03b1 \u2192 \u03b2\nhs : MeasurableSet s\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = g x) \u2194 indicator s f =\u1da0[ae \u03bc] indicator s g\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\ng : \u03b1 \u2192 \u03b2\nhs : MeasurableSet s\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = g x\n\u22a2 indicator s f =\u1da0[ae \u03bc] indicator s g\n[PROOFSTEP]\nfilter_upwards [h] with x hx\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\ng : \u03b1 \u2192 \u03b2\nhs : MeasurableSet s\nh : indicator s f =\u1da0[ae \u03bc] indicator s g\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = g x\n[PROOFSTEP]\nfilter_upwards [h] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\ng : \u03b1 \u2192 \u03b2\nhs : MeasurableSet s\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = g x\nx : \u03b1\nhx : x \u2208 s \u2192 f x = g x\n\u22a2 indicator s f x = indicator s g x\n[PROOFSTEP]\nby_cases hxs : x \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\ng : \u03b1 \u2192 \u03b2\nhs : MeasurableSet s\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = g x\nx : \u03b1\nhx : x \u2208 s \u2192 f x = g x\nhxs : x \u2208 s\n\u22a2 indicator s f x = indicator s g x\n[PROOFSTEP]\nsimp [hxs, hx hxs]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\ng : \u03b1 \u2192 \u03b2\nhs : MeasurableSet s\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = g x\nx : \u03b1\nhx : x \u2208 s \u2192 f x = g x\nhxs : \u00acx \u2208 s\n\u22a2 indicator s f x = indicator s g x\n[PROOFSTEP]\nsimp [hxs]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\ng : \u03b1 \u2192 \u03b2\nhs : MeasurableSet s\nh : indicator s f =\u1da0[ae \u03bc] indicator s g\nx : \u03b1\nhx : indicator s f x = indicator s g x\n\u22a2 x \u2208 s \u2192 f x = g x\n[PROOFSTEP]\nintro hxs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\ng : \u03b1 \u2192 \u03b2\nhs : MeasurableSet s\nh : indicator s f =\u1da0[ae \u03bc] indicator s g\nx : \u03b1\nhx : indicator s f x = indicator s g x\nhxs : x \u2208 s\n\u22a2 f x = g x\n[PROOFSTEP]\nsimpa [hxs] using hx\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.MeasureSpace", "llama_tokens": 264094, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952052, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.4190122124776902}} {"text": "[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : SecondCountableTopology E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : E), (fun x => inner c (f x)) =\u1d50[\u03bc] 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nlet s := denseSeq E\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : SecondCountableTopology E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : E), (fun x => inner c (f x)) =\u1d50[\u03bc] 0\ns : \u2115 \u2192 E := denseSeq E\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave hs : DenseRange s := denseRange_denseSeq E\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : SecondCountableTopology E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : E), (fun x => inner c (f x)) =\u1d50[\u03bc] 0\ns : \u2115 \u2192 E := denseSeq E\nhs : DenseRange s\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave hf' : \u2200\u1d50 x \u2202\u03bc, \u2200 n : \u2115, inner (s n) (f x) = (0 : \ud835\udd5c) := ae_all_iff.mpr fun n => hf (s n)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : SecondCountableTopology E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : E), (fun x => inner c (f x)) =\u1d50[\u03bc] 0\ns : \u2115 \u2192 E := denseSeq E\nhs : DenseRange s\nhf' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), inner (s n) (f x) = 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nrefine' hf'.mono fun x hx => _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : SecondCountableTopology E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : E), (fun x => inner c (f x)) =\u1d50[\u03bc] 0\ns : \u2115 \u2192 E := denseSeq E\nhs : DenseRange s\nhf' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), inner (s n) (f x) = 0\nx : \u03b1\nhx : \u2200 (n : \u2115), inner (s n) (f x) = 0\n\u22a2 f x = OfNat.ofNat 0 x\n[PROOFSTEP]\nrw [Pi.zero_apply, \u2190 @inner_self_eq_zero \ud835\udd5c]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : SecondCountableTopology E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : E), (fun x => inner c (f x)) =\u1d50[\u03bc] 0\ns : \u2115 \u2192 E := denseSeq E\nhs : DenseRange s\nhf' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), inner (s n) (f x) = 0\nx : \u03b1\nhx : \u2200 (n : \u2115), inner (s n) (f x) = 0\n\u22a2 inner (f x) (f x) = 0\n[PROOFSTEP]\nhave h_closed : IsClosed {c : E | inner c (f x) = (0 : \ud835\udd5c)} :=\n isClosed_eq (continuous_id.inner continuous_const) continuous_const\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : SecondCountableTopology E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : E), (fun x => inner c (f x)) =\u1d50[\u03bc] 0\ns : \u2115 \u2192 E := denseSeq E\nhs : DenseRange s\nhf' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), inner (s n) (f x) = 0\nx : \u03b1\nhx : \u2200 (n : \u2115), inner (s n) (f x) = 0\nh_closed : IsClosed {c | inner c (f x) = 0}\n\u22a2 inner (f x) (f x) = 0\n[PROOFSTEP]\nexact @isClosed_property \u2115 E _ s (fun c => inner c (f x) = (0 : \ud835\udd5c)) hs h_closed (fun n => hx n) _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nht : IsSeparable t\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nrcases ht with \u27e8d, d_count, hd\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhaveI : Encodable d := d_count.toEncodable\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave : \u2200 x : d, \u2203 g : E \u2192L[\ud835\udd5c] \ud835\udd5c, \u2016g\u2016 \u2264 1 \u2227 g x = \u2016(x : E)\u2016 := fun x => exists_dual_vector'' \ud835\udd5c (x : E)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis\u271d : Encodable \u2191d\nthis : \u2200 (x : \u2191d), \u2203 g, \u2016g\u2016 \u2264 1 \u2227 \u2191g \u2191x = \u2191\u2016\u2191x\u2016\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nchoose s hs using this\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave A : \u2200 a : E, a \u2208 t \u2192 (\u2200 x, \u27eaa, s x\u27eb = (0 : \ud835\udd5c)) \u2192 a = 0 :=\n by\n intro a hat ha\n contrapose! ha\n have a_pos : 0 < \u2016a\u2016 := by simp only [ha, norm_pos_iff, Ne.def, not_false_iff]\n have a_mem : a \u2208 closure d := hd hat\n obtain \u27e8x, hx\u27e9 : \u2203 x : d, dist a x < \u2016a\u2016 / 2 :=\n by\n rcases Metric.mem_closure_iff.1 a_mem (\u2016a\u2016 / 2) (half_pos a_pos) with \u27e8x, h'x, hx\u27e9\n exact \u27e8\u27e8x, h'x\u27e9, hx\u27e9\n use x\n have I : \u2016a\u2016 / 2 < \u2016(x : E)\u2016 :=\n by\n have : \u2016a\u2016 \u2264 \u2016(x : E)\u2016 + \u2016a - x\u2016 := norm_le_insert' _ _\n have : \u2016a - x\u2016 < \u2016a\u2016 / 2 := by rwa [dist_eq_norm] at hx \n linarith\n intro h\n apply lt_irrefl \u2016s x x\u2016\n calc\n \u2016s x x\u2016 = \u2016s x (x - a)\u2016 := by simp only [h, sub_zero, ContinuousLinearMap.map_sub]\n _ \u2264 1 * \u2016(x : E) - a\u2016 := (ContinuousLinearMap.le_of_op_norm_le _ (hs x).1 _)\n _ < \u2016a\u2016 / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx \n _ < \u2016(x : E)\u2016 := I\n _ = \u2016s x x\u2016 := by rw [(hs x).2, IsROrC.norm_coe_norm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\n\u22a2 \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0\n[PROOFSTEP]\nintro a hat ha\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : \u2200 (x : \u2191d), \u2191(s x) a = 0\n\u22a2 a = 0\n[PROOFSTEP]\ncontrapose! ha\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\n\u22a2 \u2203 x, \u2191(s x) a \u2260 0\n[PROOFSTEP]\nhave a_pos : 0 < \u2016a\u2016 := by simp only [ha, norm_pos_iff, Ne.def, not_false_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\n\u22a2 0 < \u2016a\u2016\n[PROOFSTEP]\nsimp only [ha, norm_pos_iff, Ne.def, not_false_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\n\u22a2 \u2203 x, \u2191(s x) a \u2260 0\n[PROOFSTEP]\nhave a_mem : a \u2208 closure d := hd hat\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\n\u22a2 \u2203 x, \u2191(s x) a \u2260 0\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 : \u2203 x : d, dist a x < \u2016a\u2016 / 2 :=\n by\n rcases Metric.mem_closure_iff.1 a_mem (\u2016a\u2016 / 2) (half_pos a_pos) with \u27e8x, h'x, hx\u27e9\n exact \u27e8\u27e8x, h'x\u27e9, hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\n\u22a2 \u2203 x, dist a \u2191x < \u2016a\u2016 / 2\n[PROOFSTEP]\nrcases Metric.mem_closure_iff.1 a_mem (\u2016a\u2016 / 2) (half_pos a_pos) with \u27e8x, h'x, hx\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : E\nh'x : x \u2208 d\nhx : dist a x < \u2016a\u2016 / 2\n\u22a2 \u2203 x, dist a \u2191x < \u2016a\u2016 / 2\n[PROOFSTEP]\nexact \u27e8\u27e8x, h'x\u27e9, hx\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\n\u22a2 \u2203 x, \u2191(s x) a \u2260 0\n[PROOFSTEP]\nuse x\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\n\u22a2 \u2191(s x) a \u2260 0\n[PROOFSTEP]\nhave I : \u2016a\u2016 / 2 < \u2016(x : E)\u2016 := by\n have : \u2016a\u2016 \u2264 \u2016(x : E)\u2016 + \u2016a - x\u2016 := norm_le_insert' _ _\n have : \u2016a - x\u2016 < \u2016a\u2016 / 2 := by rwa [dist_eq_norm] at hx \n linarith\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\n\u22a2 \u2016a\u2016 / 2 < \u2016\u2191x\u2016\n[PROOFSTEP]\nhave : \u2016a\u2016 \u2264 \u2016(x : E)\u2016 + \u2016a - x\u2016 := norm_le_insert' _ _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis\u271d : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nthis : \u2016a\u2016 \u2264 \u2016\u2191x\u2016 + \u2016a - \u2191x\u2016\n\u22a2 \u2016a\u2016 / 2 < \u2016\u2191x\u2016\n[PROOFSTEP]\nhave : \u2016a - x\u2016 < \u2016a\u2016 / 2 := by rwa [dist_eq_norm] at hx \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis\u271d : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nthis : \u2016a\u2016 \u2264 \u2016\u2191x\u2016 + \u2016a - \u2191x\u2016\n\u22a2 \u2016a - \u2191x\u2016 < \u2016a\u2016 / 2\n[PROOFSTEP]\nrwa [dist_eq_norm] at hx \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis\u271d\u00b9 : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nthis\u271d : \u2016a\u2016 \u2264 \u2016\u2191x\u2016 + \u2016a - \u2191x\u2016\nthis : \u2016a - \u2191x\u2016 < \u2016a\u2016 / 2\n\u22a2 \u2016a\u2016 / 2 < \u2016\u2191x\u2016\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\n\u22a2 \u2191(s x) a \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\nh : \u2191(s x) a = 0\n\u22a2 False\n[PROOFSTEP]\napply lt_irrefl \u2016s x x\u2016\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\nh : \u2191(s x) a = 0\n\u22a2 \u2016\u2191(s x) \u2191x\u2016 < \u2016\u2191(s x) \u2191x\u2016\n[PROOFSTEP]\ncalc\n \u2016s x x\u2016 = \u2016s x (x - a)\u2016 := by simp only [h, sub_zero, ContinuousLinearMap.map_sub]\n _ \u2264 1 * \u2016(x : E) - a\u2016 := (ContinuousLinearMap.le_of_op_norm_le _ (hs x).1 _)\n _ < \u2016a\u2016 / 2 := by rw [one_mul]; rwa [dist_eq_norm'] at hx \n _ < \u2016(x : E)\u2016 := I\n _ = \u2016s x x\u2016 := by rw [(hs x).2, IsROrC.norm_coe_norm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\nh : \u2191(s x) a = 0\n\u22a2 \u2016\u2191(s x) \u2191x\u2016 = \u2016\u2191(s x) (\u2191x - a)\u2016\n[PROOFSTEP]\nsimp only [h, sub_zero, ContinuousLinearMap.map_sub]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\nh : \u2191(s x) a = 0\n\u22a2 1 * \u2016\u2191x - a\u2016 < \u2016a\u2016 / 2\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\nh : \u2191(s x) a = 0\n\u22a2 \u2016\u2191x - a\u2016 < \u2016a\u2016 / 2\n[PROOFSTEP]\nrwa [dist_eq_norm'] at hx \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\na : E\nhat : a \u2208 t\nha : a \u2260 0\na_pos : 0 < \u2016a\u2016\na_mem : a \u2208 closure d\nx : \u2191d\nhx : dist a \u2191x < \u2016a\u2016 / 2\nI : \u2016a\u2016 / 2 < \u2016\u2191x\u2016\nh : \u2191(s x) a = 0\n\u22a2 \u2016\u2191x\u2016 = \u2016\u2191(s x) \u2191x\u2016\n[PROOFSTEP]\nrw [(hs x).2, IsROrC.norm_coe_norm]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\nA : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave hfs : \u2200 y : d, \u2200\u1d50 x \u2202\u03bc, \u27eaf x, s y\u27eb = (0 : \ud835\udd5c) := fun y => hf (s y)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\nA : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0\nhfs : \u2200 (y : \u2191d), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(s y) (f x) = 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave hf' : \u2200\u1d50 x \u2202\u03bc, \u2200 y : d, \u27eaf x, s y\u27eb = (0 : \ud835\udd5c) := by rwa [ae_all_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\nA : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0\nhfs : \u2200 (y : \u2191d), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(s y) (f x) = 0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (y : \u2191d), \u2191(s y) (f x) = 0\n[PROOFSTEP]\nrwa [ae_all_iff]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\nA : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0\nhfs : \u2200 (y : \u2191d), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(s y) (f x) = 0\nhf' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (y : \u2191d), \u2191(s y) (f x) = 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nfilter_upwards [hf', h't] with x hx h'x\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nt : Set E\nf : \u03b1 \u2192 E\nhf : \u2200 (c : Dual \ud835\udd5c E), (fun x => \u2191c (f x)) =\u1d50[\u03bc] 0\nh't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t\nd : Set E\nd_count : Set.Countable d\nhd : t \u2286 closure d\nthis : Encodable \u2191d\ns : \u2191d \u2192 E \u2192L[\ud835\udd5c] \ud835\udd5c\nhs : \u2200 (x : \u2191d), \u2016s x\u2016 \u2264 1 \u2227 \u2191(s x) \u2191x = \u2191\u2016\u2191x\u2016\nA : \u2200 (a : E), a \u2208 t \u2192 (\u2200 (x : \u2191d), \u2191(s x) a = 0) \u2192 a = 0\nhfs : \u2200 (y : \u2191d), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(s y) (f x) = 0\nhf' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (y : \u2191d), \u2191(s y) (f x) = 0\nx : \u03b1\nhx : \u2200 (y : \u2191d), \u2191(s y) (f x) = 0\nh'x : f x \u2208 t\n\u22a2 f x = OfNat.ofNat 0 x\n[PROOFSTEP]\nexact A (f x) h'x hx\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, c \u2264 f x) \u2194 \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n[PROOFSTEP]\nrw [ae_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 \u2191\u2191\u03bc {a | \u00acc \u2264 f a} = 0 \u2194 \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n[PROOFSTEP]\npush_neg\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0 \u2194 \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0 \u2192 \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n[PROOFSTEP]\nintro h b hb\n[GOAL]\ncase mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nh : \u2191\u2191\u03bc {a | f a < c} = 0\nb : \u03b2\nhb : b < c\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n[PROOFSTEP]\nexact measure_mono_null (fun y hy => (lt_of_le_of_lt hy hb : _)) h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 (\u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0) \u2192 \u2191\u2191\u03bc {a | f a < c} = 0\n[PROOFSTEP]\nintro hc\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0\n[PROOFSTEP]\nby_cases h : \u2200 b, c \u2264 b\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0\n[PROOFSTEP]\nhave : {a : \u03b1 | f a < c} = \u2205 := by\n apply Set.eq_empty_iff_forall_not_mem.2 fun x hx => ?_\n exact (lt_irrefl _ (lt_of_lt_of_le hx (h (f x)))).elim\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\n\u22a2 {a | f a < c} = \u2205\n[PROOFSTEP]\napply Set.eq_empty_iff_forall_not_mem.2 fun x hx => ?_\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\nx : \u03b1\nhx : x \u2208 {a | f a < c}\n\u22a2 False\n[PROOFSTEP]\nexact (lt_irrefl _ (lt_of_lt_of_le hx (h (f x)))).elim\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\nthis : {a | f a < c} = \u2205\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0\n[PROOFSTEP]\nby_cases H : \u00acIsLUB (Set.Iio c) c\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0\n[PROOFSTEP]\nhave : c \u2208 upperBounds (Set.Iio c) := fun y hy => le_of_lt hy\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\nthis : c \u2208 upperBounds (Set.Iio c)\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0\n[PROOFSTEP]\nobtain \u27e8b, b_up, bc\u27e9 : \u2203 b : \u03b2, b \u2208 upperBounds (Set.Iio c) \u2227 b < c := by\n simpa [IsLUB, IsLeast, this, lowerBounds] using H\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\nthis : c \u2208 upperBounds (Set.Iio c)\n\u22a2 \u2203 b, b \u2208 upperBounds (Set.Iio c) \u2227 b < c\n[PROOFSTEP]\nsimpa [IsLUB, IsLeast, this, lowerBounds] using H\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\nthis : c \u2208 upperBounds (Set.Iio c)\nb : \u03b2\nb_up : b \u2208 upperBounds (Set.Iio c)\nbc : b < c\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0\n[PROOFSTEP]\nexact measure_mono_null (fun x hx => b_up hx) (hc b bc)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00ac\u00acIsLUB (Set.Iio c) c\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0\n[PROOFSTEP]\npush_neg at H h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0\n[PROOFSTEP]\nobtain \u27e8u, _, u_lt, u_lim, -\u27e9 :\n \u2203 u : \u2115 \u2192 \u03b2, StrictMono u \u2227 (\u2200 n : \u2115, u n < c) \u2227 Tendsto u atTop (nhds c) \u2227 \u2200 n : \u2115, u n \u2208 Set.Iio c :=\n H.exists_seq_strictMono_tendsto_of_not_mem (lt_irrefl c) h\n[GOAL]\ncase neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0\n[PROOFSTEP]\nhave h_Union : {x | f x < c} = \u22c3 n : \u2115, {x | f x \u2264 u n} :=\n by\n ext1 x\n simp_rw [Set.mem_iUnion, Set.mem_setOf_eq]\n constructor <;> intro h\n \u00b7 obtain \u27e8n, hn\u27e9 := ((tendsto_order.1 u_lim).1 _ h).exists; exact \u27e8n, hn.le\u27e9\n \u00b7 obtain \u27e8n, hn\u27e9 := h; exact hn.trans_lt (u_lt _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\n\u22a2 {x | f x < c} = \u22c3 (n : \u2115), {x | f x \u2264 u n}\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\n\u22a2 x \u2208 {x | f x < c} \u2194 x \u2208 \u22c3 (n : \u2115), {x | f x \u2264 u n}\n[PROOFSTEP]\nsimp_rw [Set.mem_iUnion, Set.mem_setOf_eq]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\n\u22a2 f x < c \u2194 \u2203 i, f x \u2264 u i\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\n\u22a2 f x < c \u2192 \u2203 i, f x \u2264 u i\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\n\u22a2 (\u2203 i, f x \u2264 u i) \u2192 f x < c\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh\u271d : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nh : f x < c\n\u22a2 \u2203 i, f x \u2264 u i\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := ((tendsto_order.1 u_lim).1 _ h).exists\n[GOAL]\ncase h.mp.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh\u271d : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nh : f x < c\nn : \u2115\nhn : f x < u n\n\u22a2 \u2203 i, f x \u2264 u i\n[PROOFSTEP]\nexact \u27e8n, hn.le\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh\u271d : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nh : \u2203 i, f x \u2264 u i\n\u22a2 f x < c\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := h\n[GOAL]\ncase h.mpr.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nn : \u2115\nhn : f x \u2264 u n\n\u22a2 f x < c\n[PROOFSTEP]\nexact hn.trans_lt (u_lt _)\n[GOAL]\ncase neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nh_Union : {x | f x < c} = \u22c3 (n : \u2115), {x | f x \u2264 u n}\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0\n[PROOFSTEP]\nrw [h_Union, measure_iUnion_null_iff]\n[GOAL]\ncase neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nh_Union : {x | f x < c} = \u22c3 (n : \u2115), {x | f x \u2264 u n}\n\u22a2 \u2200 (i : \u2115), \u2191\u2191\u03bc {x | f x \u2264 u i} = 0\n[PROOFSTEP]\nintro n\n[GOAL]\ncase neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nh_Union : {x | f x < c} = \u22c3 (n : \u2115), {x | f x \u2264 u n}\nn : \u2115\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 u n} = 0\n[PROOFSTEP]\nexact hc _ (u_lt n)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 f \u2264\u1d50[\u03bc] g\n[PROOFSTEP]\nhave A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u03bc ({x | g x + \u03b5 \u2264 f x \u2227 g x \u2264 N} \u2229 spanningSets \u03bc p) = 0 :=\n by\n intro \u03b5 N p \u03b5pos\n let s := {x | g x + \u03b5 \u2264 f x \u2227 g x \u2264 N} \u2229 spanningSets \u03bc p\n have s_meas : MeasurableSet s :=\n by\n have A : MeasurableSet {x | g x + \u03b5 \u2264 f x} := measurableSet_le (hg.add measurable_const) hf\n have B : MeasurableSet {x | g x \u2264 N} := measurableSet_le hg measurable_const\n exact (A.inter B).inter (measurable_spanningSets \u03bc p)\n have s_lt_top : \u03bc s < \u221e := (measure_mono (Set.inter_subset_right _ _)).trans_lt (measure_spanningSets_lt_top \u03bc p)\n have A : (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 :=\n calc\n (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s = (\u222b\u207b x in s, g x \u2202\u03bc) + \u222b\u207b _ in s, \u03b5 \u2202\u03bc := by\n simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n _ = \u222b\u207b x in s, g x + \u03b5 \u2202\u03bc := (lintegral_add_right _ measurable_const).symm\n _ \u2264 \u222b\u207b x in s, f x \u2202\u03bc := (set_lintegral_mono (hg.add measurable_const) hf fun x hx => hx.1.1)\n _ \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 := by rw [add_zero]; exact h s s_meas s_lt_top\n have B : (\u222b\u207b x in s, g x \u2202\u03bc) \u2260 \u221e := by\n apply ne_of_lt\n calc\n (\u222b\u207b x in s, g x \u2202\u03bc) \u2264 \u222b\u207b _ in s, N \u2202\u03bc := set_lintegral_mono hg measurable_const fun x hx => hx.1.2\n _ = N * \u03bc s := by simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n _ < \u221e := by\n simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top, ENNReal.mul_eq_top, Ne.def,\n not_false_iff, false_and_iff, or_self_iff]\n have : (\u03b5 : \u211d\u22650\u221e) * \u03bc s \u2264 0 := ENNReal.le_of_add_le_add_left B A\n simpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, \u03b5pos.ne', false_or_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\n[PROOFSTEP]\nintro \u03b5 N p \u03b5pos\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\n[PROOFSTEP]\nlet s := {x | g x + \u03b5 \u2264 f x \u2227 g x \u2264 N} \u2229 spanningSets \u03bc p\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\n[PROOFSTEP]\nhave s_meas : MeasurableSet s :=\n by\n have A : MeasurableSet {x | g x + \u03b5 \u2264 f x} := measurableSet_le (hg.add measurable_const) hf\n have B : MeasurableSet {x | g x \u2264 N} := measurableSet_le hg measurable_const\n exact (A.inter B).inter (measurable_spanningSets \u03bc p)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nhave A : MeasurableSet {x | g x + \u03b5 \u2264 f x} := measurableSet_le (hg.add measurable_const) hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\nA : MeasurableSet {x | g x + \u2191\u03b5 \u2264 f x}\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nhave B : MeasurableSet {x | g x \u2264 N} := measurableSet_le hg measurable_const\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\nA : MeasurableSet {x | g x + \u2191\u03b5 \u2264 f x}\nB : MeasurableSet {x | g x \u2264 \u2191N}\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nexact (A.inter B).inter (measurable_spanningSets \u03bc p)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\n[PROOFSTEP]\nhave s_lt_top : \u03bc s < \u221e := (measure_mono (Set.inter_subset_right _ _)).trans_lt (measure_spanningSets_lt_top \u03bc p)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\n[PROOFSTEP]\nhave A : (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 :=\n calc\n (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s = (\u222b\u207b x in s, g x \u2202\u03bc) + \u222b\u207b _ in s, \u03b5 \u2202\u03bc := by\n simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n _ = \u222b\u207b x in s, g x + \u03b5 \u2202\u03bc := (lintegral_add_right _ measurable_const).symm\n _ \u2264 \u222b\u207b x in s, f x \u2202\u03bc := (set_lintegral_mono (hg.add measurable_const) hf fun x hx => hx.1.1)\n _ \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 := by rw [add_zero]; exact h s s_meas s_lt_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in s, \u2191\u03b5 \u2202\u03bc\n[PROOFSTEP]\nsimp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n[PROOFSTEP]\nexact h s s_meas s_lt_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\n[PROOFSTEP]\nhave B : (\u222b\u207b x in s, g x \u2202\u03bc) \u2260 \u221e := by\n apply ne_of_lt\n calc\n (\u222b\u207b x in s, g x \u2202\u03bc) \u2264 \u222b\u207b _ in s, N \u2202\u03bc := set_lintegral_mono hg measurable_const fun x hx => hx.1.2\n _ = N * \u03bc s := by simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n _ < \u221e := by\n simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top, ENNReal.mul_eq_top, Ne.def,\n not_false_iff, false_and_iff, or_self_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2260 \u22a4\n[PROOFSTEP]\napply ne_of_lt\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc < \u22a4\n[PROOFSTEP]\ncalc\n (\u222b\u207b x in s, g x \u2202\u03bc) \u2264 \u222b\u207b _ in s, N \u2202\u03bc := set_lintegral_mono hg measurable_const fun x hx => hx.1.2\n _ = N * \u03bc s := by simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n _ < \u221e := by\n simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top, ENNReal.mul_eq_top, Ne.def,\n not_false_iff, false_and_iff, or_self_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191N \u2202\u03bc = \u2191N * \u2191\u2191\u03bc s\n[PROOFSTEP]\nsimp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\n\u22a2 \u2191N * \u2191\u2191\u03bc s < \u22a4\n[PROOFSTEP]\nsimp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top, ENNReal.mul_eq_top, Ne.def, not_false_iff,\n false_and_iff, or_self_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\nB : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\n[PROOFSTEP]\nhave : (\u03b5 : \u211d\u22650\u221e) * \u03bc s \u2264 0 := ENNReal.le_of_add_le_add_left B A\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np\u271d : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u03b5 N : \u211d\u22650\np : \u2115\n\u03b5pos : 0 < \u03b5\ns : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p\ns_meas : MeasurableSet s\ns_lt_top : \u2191\u2191\u03bc s < \u22a4\nA : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0\nB : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2260 \u22a4\nthis : \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 0\n\u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\n[PROOFSTEP]\nsimpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, \u03b5pos.ne', false_or_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\n\u22a2 f \u2264\u1d50[\u03bc] g\n[PROOFSTEP]\nobtain \u27e8u, _, u_pos, u_lim\u27e9 : \u2203 u : \u2115 \u2192 \u211d\u22650, StrictAnti u \u2227 (\u2200 n, 0 < u n) \u2227 Tendsto u atTop (nhds 0) :=\n exists_seq_strictAnti_tendsto (0 : \u211d\u22650)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\n\u22a2 f \u2264\u1d50[\u03bc] g\n[PROOFSTEP]\nlet s := fun n : \u2115 => {x | g x + u n \u2264 f x \u2227 g x \u2264 (n : \u211d\u22650)} \u2229 spanningSets \u03bc n\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u22a2 f \u2264\u1d50[\u03bc] g\n[PROOFSTEP]\nhave \u03bcs : \u2200 n, \u03bc (s n) = 0 := fun n => A _ _ _ (u_pos n)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\n\u22a2 f \u2264\u1d50[\u03bc] g\n[PROOFSTEP]\nhave B : {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 n, s n := by\n intro x hx\n simp only [Set.mem_compl_iff, Set.mem_setOf, not_le] at hx \n have L1 : \u2200\u1da0 n in atTop, g x + u n \u2264 f x :=\n by\n have : Tendsto (fun n => g x + u n) atTop (\ud835\udcdd (g x + (0 : \u211d\u22650))) :=\n tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)\n simp only [ENNReal.coe_zero, add_zero] at this \n exact eventually_le_of_tendsto_lt hx this\n have L2 : \u2200\u1da0 n : \u2115 in (atTop : Filter \u2115), g x \u2264 (n : \u211d\u22650) :=\n haveI : Tendsto (fun n : \u2115 => ((n : \u211d\u22650) : \u211d\u22650\u221e)) atTop (\ud835\udcdd \u221e) :=\n by\n simp only [ENNReal.coe_nat]\n exact ENNReal.tendsto_nat_nhds_top\n eventually_ge_of_tendsto_gt (hx.trans_le le_top) this\n apply Set.mem_iUnion.2\n exact ((L1.and L2).and (eventually_mem_spanningSets \u03bc x)).exists\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\n\u22a2 {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 (n : \u2115), s n\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : x \u2208 {x | f x \u2264 g x}\u1d9c\n\u22a2 x \u2208 \u22c3 (n : \u2115), s n\n[PROOFSTEP]\nsimp only [Set.mem_compl_iff, Set.mem_setOf, not_le] at hx \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\n\u22a2 x \u2208 \u22c3 (n : \u2115), s n\n[PROOFSTEP]\nhave L1 : \u2200\u1da0 n in atTop, g x + u n \u2264 f x :=\n by\n have : Tendsto (fun n => g x + u n) atTop (\ud835\udcdd (g x + (0 : \u211d\u22650))) :=\n tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)\n simp only [ENNReal.coe_zero, add_zero] at this \n exact eventually_le_of_tendsto_lt hx this\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\n[PROOFSTEP]\nhave : Tendsto (fun n => g x + u n) atTop (\ud835\udcdd (g x + (0 : \u211d\u22650))) := tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nthis : Tendsto (fun n => g x + \u2191(u n)) atTop (\ud835\udcdd (g x + \u21910))\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\n[PROOFSTEP]\nsimp only [ENNReal.coe_zero, add_zero] at this \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nthis : Tendsto (fun n => g x + \u2191(u n)) atTop (\ud835\udcdd (g x))\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\n[PROOFSTEP]\nexact eventually_le_of_tendsto_lt hx this\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nL1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\n\u22a2 x \u2208 \u22c3 (n : \u2115), s n\n[PROOFSTEP]\nhave L2 : \u2200\u1da0 n : \u2115 in (atTop : Filter \u2115), g x \u2264 (n : \u211d\u22650) :=\n haveI : Tendsto (fun n : \u2115 => ((n : \u211d\u22650) : \u211d\u22650\u221e)) atTop (\ud835\udcdd \u221e) :=\n by\n simp only [ENNReal.coe_nat]\n exact ENNReal.tendsto_nat_nhds_top\n eventually_ge_of_tendsto_gt (hx.trans_le le_top) this\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nL1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\n\u22a2 Tendsto (fun n => \u2191\u2191n) atTop (\ud835\udcdd \u22a4)\n[PROOFSTEP]\nsimp only [ENNReal.coe_nat]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nL1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\n\u22a2 Tendsto (fun n => \u2191n) atTop (\ud835\udcdd \u22a4)\n[PROOFSTEP]\nexact ENNReal.tendsto_nat_nhds_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nL1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\nL2 : \u2200\u1da0 (n : \u2115) in atTop, g x \u2264 \u2191\u2191n\n\u22a2 x \u2208 \u22c3 (n : \u2115), s n\n[PROOFSTEP]\napply Set.mem_iUnion.2\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nx : \u03b1\nhx : g x < f x\nL1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x\nL2 : \u2200\u1da0 (n : \u2115) in atTop, g x \u2264 \u2191\u2191n\n\u22a2 \u2203 i, x \u2208 s i\n[PROOFSTEP]\nexact ((L1.and L2).and (eventually_mem_spanningSets \u03bc x)).exists\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nB : {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 (n : \u2115), s n\n\u22a2 f \u2264\u1d50[\u03bc] g\n[PROOFSTEP]\nrefine' le_antisymm _ bot_le\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nB : {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 (n : \u2115), s n\n\u22a2 \u2191\u2191\u03bc {x | (fun x => f x \u2264 g x) x}\u1d9c \u2264 0\n[PROOFSTEP]\ncalc\n \u03bc {x : \u03b1 | (fun x : \u03b1 => f x \u2264 g x) x}\u1d9c \u2264 \u03bc (\u22c3 n, s n) := measure_mono B\n _ \u2264 \u2211' n, \u03bc (s n) := (measure_iUnion_le _)\n _ = 0 := by simp only [\u03bcs, tsum_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0\nu : \u2115 \u2192 \u211d\u22650\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\ns : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n\n\u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0\nB : {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 (n : \u2115), s n\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = 0\n[PROOFSTEP]\nsimp only [\u03bcs, tsum_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 f =\u1d50[\u03bc] g\n[PROOFSTEP]\nhave A : f \u2264\u1d50[\u03bc] g := ae_le_of_forall_set_lintegral_le_of_sigmaFinite hf hg fun s hs h's => le_of_eq (h s hs h's)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : f \u2264\u1d50[\u03bc] g\n\u22a2 f =\u1d50[\u03bc] g\n[PROOFSTEP]\nhave B : g \u2264\u1d50[\u03bc] f := ae_le_of_forall_set_lintegral_le_of_sigmaFinite hg hf fun s hs h's => ge_of_eq (h s hs h's)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : f \u2264\u1d50[\u03bc] g\nB : g \u2264\u1d50[\u03bc] f\n\u22a2 f =\u1d50[\u03bc] g\n[PROOFSTEP]\nfilter_upwards [A, B] with x using le_antisymm\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] f\n[PROOFSTEP]\nsimp_rw [EventuallyLE, Pi.zero_apply]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 f x\n[PROOFSTEP]\nrw [ae_const_le_iff_forall_lt_measure_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\u22a2 \u2200 (b : \u211d), b < 0 \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n[PROOFSTEP]\nintro b hb_neg\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n[PROOFSTEP]\nlet s := {x | f x \u2264 b}\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n[PROOFSTEP]\nhave hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n[PROOFSTEP]\nhave mus : \u03bc s < \u221e := by\n let c : \u211d\u22650 := \u27e8|b|, abs_nonneg _\u27e9\n have c_pos : (c : \u211d\u22650\u221e) \u2260 0 := by simpa [\u2190 NNReal.coe_eq_zero] using hb_neg.ne\n calc\n \u03bc s \u2264 \u03bc {x | (c : \u211d\u22650\u221e) \u2264 \u2016f x\u2016\u208a} := by\n apply measure_mono\n intro x hx\n simp only [Set.mem_setOf_eq] at hx \n simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs, Subtype.mk_le_mk,\n neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx\n _ \u2264 (\u222b\u207b x, \u2016f x\u2016\u208a \u2202\u03bc) / c := (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)\n _ < \u221e := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s < \u22a4\n[PROOFSTEP]\nlet c : \u211d\u22650 := \u27e8|b|, abs_nonneg _\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\n\u22a2 \u2191\u2191\u03bc s < \u22a4\n[PROOFSTEP]\nhave c_pos : (c : \u211d\u22650\u221e) \u2260 0 := by simpa [\u2190 NNReal.coe_eq_zero] using hb_neg.ne\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\n\u22a2 \u2191c \u2260 0\n[PROOFSTEP]\nsimpa [\u2190 NNReal.coe_eq_zero] using hb_neg.ne\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\nc_pos : \u2191c \u2260 0\n\u22a2 \u2191\u2191\u03bc s < \u22a4\n[PROOFSTEP]\ncalc\n \u03bc s \u2264 \u03bc {x | (c : \u211d\u22650\u221e) \u2264 \u2016f x\u2016\u208a} := by\n apply measure_mono\n intro x hx\n simp only [Set.mem_setOf_eq] at hx \n simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs, Subtype.mk_le_mk,\n neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx\n _ \u2264 (\u222b\u207b x, \u2016f x\u2016\u208a \u2202\u03bc) / c := (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)\n _ < \u221e := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\nc_pos : \u2191c \u2260 0\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc {x | \u2191c \u2264 \u2191\u2016f x\u2016\u208a}\n[PROOFSTEP]\napply measure_mono\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\nc_pos : \u2191c \u2260 0\n\u22a2 s \u2286 {x | \u2191c \u2264 \u2191\u2016f x\u2016\u208a}\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\nc_pos : \u2191c \u2260 0\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 {x | \u2191c \u2264 \u2191\u2016f x\u2016\u208a}\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq] at hx \n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nc : \u211d\u22650 := { val := |b|, property := (_ : 0 \u2264 |b|) }\nc_pos : \u2191c \u2260 0\nx : \u03b1\nhx : f x \u2264 b\n\u22a2 x \u2208 {x | \u2191c \u2264 \u2191\u2016f x\u2016\u208a}\n[PROOFSTEP]\nsimpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs, Subtype.mk_le_mk,\n neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n[PROOFSTEP]\nhave h_int_gt : (\u222b x in s, f x \u2202\u03bc) \u2264 b * (\u03bc s).toReal :=\n by\n have h_const_le : (\u222b x in s, f x \u2202\u03bc) \u2264 \u222b _ in s, b \u2202\u03bc :=\n by\n refine' set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _\n rw [EventuallyLE, ae_restrict_iff hs]\n exact eventually_of_forall fun x hxs => hxs\n rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\n[PROOFSTEP]\nhave h_const_le : (\u222b x in s, f x \u2202\u03bc) \u2264 \u222b _ in s, b \u2202\u03bc :=\n by\n refine' set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _\n rw [EventuallyLE, ae_restrict_iff hs]\n exact eventually_of_forall fun x hxs => hxs\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, b \u2202\u03bc\n[PROOFSTEP]\nrefine' set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\n\u22a2 (fun x => f x) \u2264\u1d50[Measure.restrict \u03bc s] fun x => b\n[PROOFSTEP]\nrw [EventuallyLE, ae_restrict_iff hs]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 b\n[PROOFSTEP]\nexact eventually_of_forall fun x hxs => hxs\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_const_le : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, b \u2202\u03bc\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\n[PROOFSTEP]\nrwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 False\n[PROOFSTEP]\nrefine' (lt_self_iff_false (\u222b x in s, f x \u2202\u03bc)).mp (h_int_gt.trans_lt _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 b * ENNReal.toReal (\u2191\u2191\u03bc s) < \u222b (x : \u03b1) in s, f x \u2202\u03bc\n[PROOFSTEP]\nrefine' (mul_neg_iff.mpr (Or.inr \u27e8hb_neg, _\u27e9)).trans_le _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 0 < ENNReal.toReal (\u2191\u2191\u03bc s)\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n[PROOFSTEP]\nexact hf_zero s hs mus\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 0 < ENNReal.toReal (\u2191\u2191\u03bc s)\n[PROOFSTEP]\nrefine' ENNReal.toReal_nonneg.lt_of_ne fun h_eq => h _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh_eq : 0 = ENNReal.toReal (\u2191\u2191\u03bc s)\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n[PROOFSTEP]\ncases' (ENNReal.toReal_eq_zero_iff _).mp h_eq.symm with h\u03bcs_eq_zero h\u03bcs_eq_top\n[GOAL]\ncase refine'_1.inl\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh_eq : 0 = ENNReal.toReal (\u2191\u2191\u03bc s)\nh\u03bcs_eq_zero : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n[PROOFSTEP]\nexact h\u03bcs_eq_zero\n[GOAL]\ncase refine'_1.inr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhfm : StronglyMeasurable f\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nb : \u211d\nhb_neg : b < 0\ns : Set \u03b1 := {x | f x \u2264 b}\nhs : MeasurableSet s\nmus : \u2191\u2191\u03bc s < \u22a4\nh_int_gt : \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 b * ENNReal.toReal (\u2191\u2191\u03bc s)\nh : \u00ac\u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh_eq : 0 = ENNReal.toReal (\u2191\u2191\u03bc s)\nh\u03bcs_eq_top : \u2191\u2191\u03bc s = \u22a4\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n[PROOFSTEP]\nexact absurd h\u03bcs_eq_top mus.ne\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] f\n[PROOFSTEP]\nrcases hf.1 with \u27e8f', hf'_meas, hf_ae\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\n\u22a2 0 \u2264\u1d50[\u03bc] f\n[PROOFSTEP]\nhave hf'_integrable : Integrable f' \u03bc := Integrable.congr hf hf_ae\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nhf'_integrable : Integrable f'\n\u22a2 0 \u2264\u1d50[\u03bc] f\n[PROOFSTEP]\nhave hf'_zero : \u2200 s, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 0 \u2264 \u222b x in s, f' x \u2202\u03bc :=\n by\n intro s hs h's\n rw [set_integral_congr_ae hs (hf_ae.mono fun x hx _ => hx.symm)]\n exact hf_zero s hs h's\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nhf'_integrable : Integrable f'\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f' x \u2202\u03bc\n[PROOFSTEP]\nintro s hs h's\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nhf'_integrable : Integrable f'\ns : Set \u03b1\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, f' x \u2202\u03bc\n[PROOFSTEP]\nrw [set_integral_congr_ae hs (hf_ae.mono fun x hx _ => hx.symm)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nhf'_integrable : Integrable f'\ns : Set \u03b1\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n[PROOFSTEP]\nexact hf_zero s hs h's\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nf' : \u03b1 \u2192 \u211d\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nhf'_integrable : Integrable f'\nhf'_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f' x \u2202\u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] f\n[PROOFSTEP]\nexact\n (ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable hf'_meas hf'_integrable hf'_zero).trans hf_ae.symm.le\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f g : \u03b1 \u2192 \u211d\nhf : Integrable f\nhg : Integrable g\nhf_le : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 f \u2264\u1d50[\u03bc] g\n[PROOFSTEP]\nrw [\u2190 eventually_sub_nonneg]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f g : \u03b1 \u2192 \u211d\nhf : Integrable f\nhg : Integrable g\nhf_le : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] g - f\n[PROOFSTEP]\nrefine' ae_nonneg_of_forall_set_integral_nonneg (hg.sub hf) fun s hs => _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f g : \u03b1 \u2192 \u211d\nhf : Integrable f\nhg : Integrable g\nhf_le : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, (g - f) x \u2202\u03bc\n[PROOFSTEP]\nrw [integral_sub' hg.integrableOn hf.integrableOn, sub_nonneg]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f g : \u03b1 \u2192 \u211d\nhf : Integrable f\nhg : Integrable g\nhf_le : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (a : \u03b1) in s, f a \u2202\u03bc \u2264 \u222b (a : \u03b1) in s, g a \u2202\u03bc\n[PROOFSTEP]\nexact hf_le s hs\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nt : Set \u03b1\nhf : IntegrableOn f t\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (s \u2229 t) < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc\n\u22a2 0 \u2264\u1d50[Measure.restrict \u03bc t] f\n[PROOFSTEP]\nrefine' ae_nonneg_of_forall_set_integral_nonneg hf fun s hs h's => _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nt : Set \u03b1\nhf : IntegrableOn f t\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (s \u2229 t) < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh's : \u2191\u2191(Measure.restrict \u03bc t) s < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202Measure.restrict \u03bc t\n[PROOFSTEP]\nsimp_rw [Measure.restrict_restrict hs]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nt : Set \u03b1\nhf : IntegrableOn f t\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (s \u2229 t) < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh's : \u2191\u2191(Measure.restrict \u03bc t) s < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc\n[PROOFSTEP]\napply hf_zero s hs\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nt : Set \u03b1\nhf : IntegrableOn f t\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (s \u2229 t) < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh's : \u2191\u2191(Measure.restrict \u03bc t) s < \u22a4\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) < \u22a4\n[PROOFSTEP]\nrwa [Measure.restrict_apply hs] at h's \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] f\n[PROOFSTEP]\napply ae_of_forall_measure_lt_top_ae_restrict\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, OfNat.ofNat 0 x \u2264 f x\n[PROOFSTEP]\nintro t t_meas t_lt_top\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1\nt_meas : MeasurableSet t\nt_lt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, OfNat.ofNat 0 x \u2264 f x\n[PROOFSTEP]\napply ae_nonneg_restrict_of_forall_set_integral_nonneg_inter (hf_int_finite t t_meas t_lt_top)\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1\nt_meas : MeasurableSet t\nt_lt_top : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (s \u2229 t) < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc\n[PROOFSTEP]\nintro s s_meas _\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1\nt_meas : MeasurableSet t\nt_lt_top : \u2191\u2191\u03bc t < \u22a4\ns : Set \u03b1\ns_meas : MeasurableSet s\na\u271d : \u2191\u2191\u03bc (s \u2229 t) < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc\n[PROOFSTEP]\nexact hf_zero _ (s_meas.inter t_meas) (lt_of_le_of_lt (measure_mono (Set.inter_subset_right _ _)) t_lt_top)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf : AEFinStronglyMeasurable f \u03bc\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] f\n[PROOFSTEP]\nlet t := hf.sigmaFiniteSet\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf : AEFinStronglyMeasurable f \u03bc\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\n\u22a2 0 \u2264\u1d50[\u03bc] f\n[PROOFSTEP]\nsuffices 0 \u2264\u1d50[\u03bc.restrict t] f from ae_of_ae_restrict_of_ae_restrict_compl _ this hf.ae_eq_zero_compl.symm.le\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf : AEFinStronglyMeasurable f \u03bc\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\n\u22a2 0 \u2264\u1d50[Measure.restrict \u03bc t] f\n[PROOFSTEP]\nhaveI : SigmaFinite (\u03bc.restrict t) := hf.sigmaFinite_restrict\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf : AEFinStronglyMeasurable f \u03bc\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\n\u22a2 0 \u2264\u1d50[Measure.restrict \u03bc t] f\n[PROOFSTEP]\nrefine' ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite (fun s hs h\u03bcts => _) fun s hs h\u03bcts => _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf : AEFinStronglyMeasurable f \u03bc\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcts : \u2191\u2191(Measure.restrict \u03bc t) s < \u22a4\n\u22a2 IntegrableOn f s\n[PROOFSTEP]\nrw [IntegrableOn, Measure.restrict_restrict hs]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf : AEFinStronglyMeasurable f \u03bc\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcts : \u2191\u2191(Measure.restrict \u03bc t) s < \u22a4\n\u22a2 Integrable f\n[PROOFSTEP]\nrw [Measure.restrict_apply hs] at h\u03bcts \n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf : AEFinStronglyMeasurable f \u03bc\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcts : \u2191\u2191\u03bc (s \u2229 t) < \u22a4\n\u22a2 Integrable f\n[PROOFSTEP]\nexact hf_int_finite (s \u2229 t) (hs.inter hf.measurableSet) h\u03bcts\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf : AEFinStronglyMeasurable f \u03bc\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcts : \u2191\u2191(Measure.restrict \u03bc t) s < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202Measure.restrict \u03bc t\n[PROOFSTEP]\nrw [Measure.restrict_restrict hs]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf : AEFinStronglyMeasurable f \u03bc\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcts : \u2191\u2191(Measure.restrict \u03bc t) s < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc\n[PROOFSTEP]\nrw [Measure.restrict_apply hs] at h\u03bcts \n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf : AEFinStronglyMeasurable f \u03bc\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcts : \u2191\u2191\u03bc (s \u2229 t) < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc\n[PROOFSTEP]\nexact hf_zero (s \u2229 t) (hs.inter hf.measurableSet) h\u03bcts\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 0 \u2264\u1d50[Measure.restrict \u03bc t] f\n[PROOFSTEP]\nrefine'\n ae_nonneg_restrict_of_forall_set_integral_nonneg_inter (hf_int_finite t ht (lt_top_iff_ne_top.mpr h\u03bct)) fun s hs _ =>\n _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nx\u271d : \u2191\u2191\u03bc (s \u2229 t) < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc\n[PROOFSTEP]\nrefine' hf_zero (s \u2229 t) (hs.inter ht) _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 0 \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nx\u271d : \u2191\u2191\u03bc (s \u2229 t) < \u22a4\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) < \u22a4\n[PROOFSTEP]\nexact (measure_mono (Set.inter_subset_right s t)).trans_lt (lt_top_iff_ne_top.mpr h\u03bct)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 f =\u1d50[Measure.restrict \u03bc t] 0\n[PROOFSTEP]\nsuffices h_and : f \u2264\u1d50[\u03bc.restrict t] 0 \u2227 0 \u2264\u1d50[\u03bc.restrict t] f\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nh_and : f \u2264\u1d50[Measure.restrict \u03bc t] 0 \u2227 0 \u2264\u1d50[Measure.restrict \u03bc t] f\n\u22a2 f =\u1d50[Measure.restrict \u03bc t] 0\ncase h_and\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 f \u2264\u1d50[Measure.restrict \u03bc t] 0 \u2227 0 \u2264\u1d50[Measure.restrict \u03bc t] f\n[PROOFSTEP]\nexact h_and.1.mp (h_and.2.mono fun x hx1 hx2 => le_antisymm hx2 hx1)\n[GOAL]\ncase h_and\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 f \u2264\u1d50[Measure.restrict \u03bc t] 0 \u2227 0 \u2264\u1d50[Measure.restrict \u03bc t] f\n[PROOFSTEP]\nrefine'\n \u27e8_,\n ae_nonneg_restrict_of_forall_set_integral_nonneg hf_int_finite (fun s hs h\u03bcs => (hf_zero s hs h\u03bcs).symm.le) ht h\u03bct\u27e9\n[GOAL]\ncase h_and\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 f \u2264\u1d50[Measure.restrict \u03bc t] 0\n[PROOFSTEP]\nsuffices h_neg : 0 \u2264\u1d50[\u03bc.restrict t] -f\n[GOAL]\ncase h_and\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nh_neg : 0 \u2264\u1d50[Measure.restrict \u03bc t] -f\n\u22a2 f \u2264\u1d50[Measure.restrict \u03bc t] 0\n[PROOFSTEP]\nrefine' h_neg.mono fun x hx => _\n[GOAL]\ncase h_and\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nh_neg : 0 \u2264\u1d50[Measure.restrict \u03bc t] -f\nx : \u03b1\nhx : OfNat.ofNat 0 x \u2264 (-f) x\n\u22a2 f x \u2264 OfNat.ofNat 0 x\n[PROOFSTEP]\nrw [Pi.neg_apply] at hx \n[GOAL]\ncase h_and\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nh_neg : 0 \u2264\u1d50[Measure.restrict \u03bc t] -f\nx : \u03b1\nhx : OfNat.ofNat 0 x \u2264 -f x\n\u22a2 f x \u2264 OfNat.ofNat 0 x\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\ncase h_neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 0 \u2264\u1d50[Measure.restrict \u03bc t] -f\n[PROOFSTEP]\nrefine'\n ae_nonneg_restrict_of_forall_set_integral_nonneg (fun s hs h\u03bcs => (hf_int_finite s hs h\u03bcs).neg) (fun s hs h\u03bcs => _) ht\n h\u03bct\n[GOAL]\ncase h_neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, (-f) x \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [Pi.neg_apply]\n[GOAL]\ncase h_neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, -f x \u2202\u03bc\n[PROOFSTEP]\nrw [integral_neg, neg_nonneg]\n[GOAL]\ncase h_neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf\u271d f : \u03b1 \u2192 \u211d\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc \u2264 0\n[PROOFSTEP]\nexact (hf_zero s hs h\u03bcs).le\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 f =\u1d50[Measure.restrict \u03bc t] 0\n[PROOFSTEP]\nrcases(hf_int_finite t ht h\u03bct.lt_top).aestronglyMeasurable.isSeparable_ae_range with \u27e8u, u_sep, hu\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\n\u22a2 f =\u1d50[Measure.restrict \u03bc t] 0\n[PROOFSTEP]\nrefine' ae_eq_zero_of_forall_dual_of_isSeparable \u211d u_sep (fun c => _) hu\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\n\u22a2 (fun x => \u2191c (f x)) =\u1d50[Measure.restrict \u03bc t] 0\n[PROOFSTEP]\nrefine' ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real _ _ ht h\u03bct\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (fun x => \u2191c (f x)) s\n[PROOFSTEP]\nintro s hs h\u03bcs\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (fun x => \u2191c (f x)) s\n[PROOFSTEP]\nexact ContinuousLinearMap.integrable_comp c (hf_int_finite s hs h\u03bcs)\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191c (f x) \u2202\u03bc = 0\n[PROOFSTEP]\nintro s hs h\u03bcs\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191c (f x) \u2202\u03bc = 0\n[PROOFSTEP]\nrw [ContinuousLinearMap.integral_comp_comm c (hf_int_finite s hs h\u03bcs), hf_zero s hs h\u03bcs]\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nu : Set E\nu_sep : IsSeparable u\nhu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u\nc : Dual \u211d E\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2191c 0 = 0\n[PROOFSTEP]\nexact ContinuousLinearMap.map_zero _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 f =\u1d50[Measure.restrict \u03bc t] g\n[PROOFSTEP]\nrw [\u2190 sub_ae_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 f - g =\u1d50[Measure.restrict \u03bc t] 0\n[PROOFSTEP]\nhave hfg' : \u2200 s : Set \u03b1, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 (\u222b x in s, (f - g) x \u2202\u03bc) = 0 :=\n by\n intro s hs h\u03bcs\n rw [integral_sub' (hf_int_finite s hs h\u03bcs) (hg_int_finite s hs h\u03bcs)]\n exact sub_eq_zero.mpr (hfg_zero s hs h\u03bcs)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\n[PROOFSTEP]\nintro s hs h\u03bcs\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\n[PROOFSTEP]\nrw [integral_sub' (hf_int_finite s hs h\u03bcs) (hg_int_finite s hs h\u03bcs)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc - \u222b (a : \u03b1) in s, g a \u2202\u03bc = 0\n[PROOFSTEP]\nexact sub_eq_zero.mpr (hfg_zero s hs h\u03bcs)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\n\u22a2 f - g =\u1d50[Measure.restrict \u03bc t] 0\n[PROOFSTEP]\nhave hfg_int : \u2200 s, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 IntegrableOn (f - g) s \u03bc := fun s hs h\u03bcs =>\n (hf_int_finite s hs h\u03bcs).sub (hg_int_finite s hs h\u03bcs)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\nhfg_int : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (f - g) s\n\u22a2 f - g =\u1d50[Measure.restrict \u03bc t] 0\n[PROOFSTEP]\nexact ae_eq_zero_restrict_of_forall_set_integral_eq_zero hfg_int hfg' ht h\u03bct\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nlet S := spanningSets \u03bc\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nS : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nrw [\u2190 @Measure.restrict_univ _ _ \u03bc, \u2190 iUnion_spanningSets \u03bc, EventuallyEq, ae_iff,\n Measure.restrict_apply' (MeasurableSet.iUnion (measurable_spanningSets \u03bc))]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nS : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\n\u22a2 \u2191\u2191\u03bc ({a | \u00acf a = OfNat.ofNat 0 a} \u2229 \u22c3 (b : \u2115), spanningSets \u03bc b) = 0\n[PROOFSTEP]\nrw [Set.inter_iUnion, measure_iUnion_null_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nS : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\n\u22a2 \u2200 (i : \u2115), \u2191\u2191\u03bc ({a | \u00acf a = OfNat.ofNat 0 a} \u2229 spanningSets \u03bc i) = 0\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nS : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc ({a | \u00acf a = OfNat.ofNat 0 a} \u2229 spanningSets \u03bc n) = 0\n[PROOFSTEP]\nhave h_meas_n : MeasurableSet (S n) := measurable_spanningSets \u03bc n\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nS : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nn : \u2115\nh_meas_n : MeasurableSet (S n)\n\u22a2 \u2191\u2191\u03bc ({a | \u00acf a = OfNat.ofNat 0 a} \u2229 spanningSets \u03bc n) = 0\n[PROOFSTEP]\nhave h\u03bcn : \u03bc (S n) < \u221e := measure_spanningSets_lt_top \u03bc n\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nS : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nn : \u2115\nh_meas_n : MeasurableSet (S n)\nh\u03bcn : \u2191\u2191\u03bc (S n) < \u22a4\n\u22a2 \u2191\u2191\u03bc ({a | \u00acf a = OfNat.ofNat 0 a} \u2229 spanningSets \u03bc n) = 0\n[PROOFSTEP]\nrw [\u2190 Measure.restrict_apply' h_meas_n]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nS : \u2115 \u2192 Set \u03b1 := spanningSets \u03bc\nn : \u2115\nh_meas_n : MeasurableSet (S n)\nh\u03bcn : \u2191\u2191\u03bc (S n) < \u22a4\n\u22a2 \u2191\u2191(Measure.restrict \u03bc (S n)) {a | \u00acf a = OfNat.ofNat 0 a} = 0\n[PROOFSTEP]\nexact ae_eq_zero_restrict_of_forall_set_integral_eq_zero hf_int_finite hf_zero h_meas_n h\u03bcn.ne\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 f =\u1d50[\u03bc] g\n[PROOFSTEP]\nrw [\u2190 sub_ae_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 f - g =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave hfg : \u2200 s : Set \u03b1, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 (\u222b x in s, (f - g) x \u2202\u03bc) = 0 :=\n by\n intro s hs h\u03bcs\n rw [integral_sub' (hf_int_finite s hs h\u03bcs) (hg_int_finite s hs h\u03bcs), sub_eq_zero.mpr (hfg_eq s hs h\u03bcs)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\n[PROOFSTEP]\nintro s hs h\u03bcs\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\n[PROOFSTEP]\nrw [integral_sub' (hf_int_finite s hs h\u03bcs) (hg_int_finite s hs h\u03bcs), sub_eq_zero.mpr (hfg_eq s hs h\u03bcs)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\n\u22a2 f - g =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave hfg_int : \u2200 s, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 IntegrableOn (f - g) s \u03bc := fun s hs h\u03bcs =>\n (hf_int_finite s hs h\u03bcs).sub (hg_int_finite s hs h\u03bcs)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\nhfg_int : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (f - g) s\n\u22a2 f - g =\u1d50[\u03bc] 0\n[PROOFSTEP]\nexact ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite hfg_int hfg\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : AEFinStronglyMeasurable f \u03bc\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nlet t := hf.sigmaFiniteSet\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : AEFinStronglyMeasurable f \u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nsuffices f =\u1d50[\u03bc.restrict t] 0 from ae_of_ae_restrict_of_ae_restrict_compl _ this hf.ae_eq_zero_compl\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : AEFinStronglyMeasurable f \u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\n\u22a2 f =\u1d50[Measure.restrict \u03bc t] 0\n[PROOFSTEP]\nhaveI : SigmaFinite (\u03bc.restrict t) := hf.sigmaFinite_restrict\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : AEFinStronglyMeasurable f \u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\n\u22a2 f =\u1d50[Measure.restrict \u03bc t] 0\n[PROOFSTEP]\nrefine' ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : AEFinStronglyMeasurable f \u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.restrict \u03bc t) s < \u22a4 \u2192 IntegrableOn f s\n[PROOFSTEP]\nintro s hs h\u03bcs\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : AEFinStronglyMeasurable f \u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.restrict \u03bc t) s < \u22a4\n\u22a2 IntegrableOn f s\n[PROOFSTEP]\nrw [IntegrableOn, Measure.restrict_restrict hs]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : AEFinStronglyMeasurable f \u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.restrict \u03bc t) s < \u22a4\n\u22a2 Integrable f\n[PROOFSTEP]\nrw [Measure.restrict_apply hs] at h\u03bcs \n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : AEFinStronglyMeasurable f \u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc (s \u2229 t) < \u22a4\n\u22a2 Integrable f\n[PROOFSTEP]\nexact hf_int_finite _ (hs.inter hf.measurableSet) h\u03bcs\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : AEFinStronglyMeasurable f \u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.restrict \u03bc t) s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202Measure.restrict \u03bc t = 0\n[PROOFSTEP]\nintro s hs h\u03bcs\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : AEFinStronglyMeasurable f \u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.restrict \u03bc t) s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202Measure.restrict \u03bc t = 0\n[PROOFSTEP]\nrw [Measure.restrict_restrict hs]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : AEFinStronglyMeasurable f \u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.restrict \u03bc t) s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc = 0\n[PROOFSTEP]\nrw [Measure.restrict_apply hs] at h\u03bcs \n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : AEFinStronglyMeasurable f \u03bc\nt : Set \u03b1 := sigmaFiniteSet hf\nthis : SigmaFinite (Measure.restrict \u03bc t)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc (s \u2229 t) < \u22a4\n\u22a2 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc = 0\n[PROOFSTEP]\nexact hf_zero _ (hs.inter hf.measurableSet) h\u03bcs\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhf : AEFinStronglyMeasurable f \u03bc\nhg : AEFinStronglyMeasurable g \u03bc\n\u22a2 f =\u1d50[\u03bc] g\n[PROOFSTEP]\nrw [\u2190 sub_ae_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhf : AEFinStronglyMeasurable f \u03bc\nhg : AEFinStronglyMeasurable g \u03bc\n\u22a2 f - g =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave hfg : \u2200 s : Set \u03b1, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 (\u222b x in s, (f - g) x \u2202\u03bc) = 0 :=\n by\n intro s hs h\u03bcs\n rw [integral_sub' (hf_int_finite s hs h\u03bcs) (hg_int_finite s hs h\u03bcs), sub_eq_zero.mpr (hfg_eq s hs h\u03bcs)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhf : AEFinStronglyMeasurable f \u03bc\nhg : AEFinStronglyMeasurable g \u03bc\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\n[PROOFSTEP]\nintro s hs h\u03bcs\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhf : AEFinStronglyMeasurable f \u03bc\nhg : AEFinStronglyMeasurable g \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\n[PROOFSTEP]\nrw [integral_sub' (hf_int_finite s hs h\u03bcs) (hg_int_finite s hs h\u03bcs), sub_eq_zero.mpr (hfg_eq s hs h\u03bcs)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhf : AEFinStronglyMeasurable f \u03bc\nhg : AEFinStronglyMeasurable g \u03bc\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\n\u22a2 f - g =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave hfg_int : \u2200 s, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 IntegrableOn (f - g) s \u03bc := fun s hs h\u03bcs =>\n (hf_int_finite s hs h\u03bcs).sub (hg_int_finite s hs h\u03bcs)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhg_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn g s\nhfg_eq : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhf : AEFinStronglyMeasurable f \u03bc\nhg : AEFinStronglyMeasurable g \u03bc\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\nhfg_int : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (f - g) s\n\u22a2 f - g =\u1d50[\u03bc] 0\n[PROOFSTEP]\nexact (hf.sub hg).ae_eq_zero_of_forall_set_integral_eq_zero hfg_int hfg\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nobtain \u27e8t, ht_meas, htf_zero, ht\u03bc\u27e9 := hf.exists_set_sigmaFinite\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhaveI : SigmaFinite ((\u03bc.restrict t).trim hm) := by rwa [restrict_trim hm \u03bc ht_meas] at ht\u03bc \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\n\u22a2 SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\n[PROOFSTEP]\nrwa [restrict_trim hm \u03bc ht_meas] at ht\u03bc \n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave htf_zero : f =\u1d50[\u03bc.restrict t\u1d9c] 0 :=\n by\n rw [EventuallyEq, ae_restrict_iff' (MeasurableSet.compl (hm _ ht_meas))]\n exact eventually_of_forall htf_zero\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\n\u22a2 f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\n[PROOFSTEP]\nrw [EventuallyEq, ae_restrict_iff' (MeasurableSet.compl (hm _ ht_meas))]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t\u1d9c \u2192 f x = OfNat.ofNat 0 x\n[PROOFSTEP]\nexact eventually_of_forall htf_zero\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave hf_meas_m : StronglyMeasurable[m] f := hf.stronglyMeasurable\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nsuffices : f =\u1d50[\u03bc.restrict t] 0\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis\u271d : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\nthis : f =\u1d50[Measure.restrict \u03bc t] 0\n\u22a2 f =\u1d50[\u03bc] 0\ncase this\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\n\u22a2 f =\u1d50[Measure.restrict \u03bc t] 0\n[PROOFSTEP]\nexact ae_of_ae_restrict_of_ae_restrict_compl _ this htf_zero\n[GOAL]\ncase this\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\n\u22a2 f =\u1d50[Measure.restrict \u03bc t] 0\n[PROOFSTEP]\nrefine' measure_eq_zero_of_trim_eq_zero hm _\n[GOAL]\ncase this\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\n\u22a2 \u2191\u2191(Measure.trim (Measure.restrict \u03bc t) hm) {x | (fun x => f x = OfNat.ofNat 0 x) x}\u1d9c = 0\n[PROOFSTEP]\nrefine' ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite _ _\n[GOAL]\ncase this.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191(Measure.trim (Measure.restrict \u03bc t) hm) s < \u22a4 \u2192 IntegrableOn (fun x => f x) s\n[PROOFSTEP]\nintro s hs h\u03bcs\n[GOAL]\ncase this.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim (Measure.restrict \u03bc t) hm) s < \u22a4\n\u22a2 IntegrableOn (fun x => f x) s\n[PROOFSTEP]\nunfold IntegrableOn\n[GOAL]\ncase this.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim (Measure.restrict \u03bc t) hm) s < \u22a4\n\u22a2 Integrable fun x => f x\n[PROOFSTEP]\nrw [restrict_trim hm (\u03bc.restrict t) hs, Measure.restrict_restrict (hm s hs)]\n[GOAL]\ncase this.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim (Measure.restrict \u03bc t) hm) s < \u22a4\n\u22a2 Integrable fun x => f x\n[PROOFSTEP]\nrw [\u2190 restrict_trim hm \u03bc ht_meas, Measure.restrict_apply hs, trim_measurableSet_eq hm (hs.inter ht_meas)] at h\u03bcs \n[GOAL]\ncase this.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc (s \u2229 t) < \u22a4\n\u22a2 Integrable fun x => f x\n[PROOFSTEP]\nrefine' Integrable.trim hm _ hf_meas_m\n[GOAL]\ncase this.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc (s \u2229 t) < \u22a4\n\u22a2 Integrable fun x => f x\n[PROOFSTEP]\nexact hf_int_finite _ (hs.inter ht_meas) h\u03bcs\n[GOAL]\ncase this.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\n\u22a2 \u2200 (s : Set \u03b1),\n MeasurableSet s \u2192\n \u2191\u2191(Measure.trim (Measure.restrict \u03bc t) hm) s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202Measure.trim (Measure.restrict \u03bc t) hm = 0\n[PROOFSTEP]\nintro s hs h\u03bcs\n[GOAL]\ncase this.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim (Measure.restrict \u03bc t) hm) s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202Measure.trim (Measure.restrict \u03bc t) hm = 0\n[PROOFSTEP]\nrw [restrict_trim hm (\u03bc.restrict t) hs, Measure.restrict_restrict (hm s hs)]\n[GOAL]\ncase this.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim (Measure.restrict \u03bc t) hm) s < \u22a4\n\u22a2 \u222b (x : \u03b1), f x \u2202Measure.trim (Measure.restrict \u03bc (s \u2229 t)) hm = 0\n[PROOFSTEP]\nrw [\u2190 restrict_trim hm \u03bc ht_meas, Measure.restrict_apply hs, trim_measurableSet_eq hm (hs.inter ht_meas)] at h\u03bcs \n[GOAL]\ncase this.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc (s \u2229 t) < \u22a4\n\u22a2 \u222b (x : \u03b1), f x \u2202Measure.trim (Measure.restrict \u03bc (s \u2229 t)) hm = 0\n[PROOFSTEP]\nrw [\u2190 integral_trim hm hf_meas_m]\n[GOAL]\ncase this.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nhm : m \u2264 m0\nf : \u03b1 \u2192 E\nhf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf : FinStronglyMeasurable f (Measure.trim \u03bc hm)\nt : Set \u03b1\nht_meas : MeasurableSet t\nhtf_zero\u271d : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 f x = 0\nht\u03bc : SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) t)\nthis : SigmaFinite (Measure.trim (Measure.restrict \u03bc t) hm)\nhtf_zero : f =\u1d50[Measure.restrict \u03bc t\u1d9c] 0\nhf_meas_m : StronglyMeasurable f\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc (s \u2229 t) < \u22a4\n\u22a2 \u222b (x : \u03b1) in s \u2229 t, f x \u2202\u03bc = 0\n[PROOFSTEP]\nexact hf_zero _ (hs.inter ht_meas) h\u03bcs\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave hf_Lp : Mem\u2112p f 1 \u03bc := mem\u2112p_one_iff_integrable.mpr hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf_Lp : Mem\u2112p f 1\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nlet f_Lp := hf_Lp.toLp f\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave hf_f_Lp : f =\u1d50[\u03bc] f_Lp := (Mem\u2112p.coeFn_toLp hf_Lp).symm\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nrefine' hf_f_Lp.trans _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\n\u22a2 \u2191\u2191f_Lp =\u1d50[\u03bc] 0\n[PROOFSTEP]\nrefine' Lp.ae_eq_zero_of_forall_set_integral_eq_zero f_Lp one_ne_zero ENNReal.coe_ne_top _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f_Lp) s\n[PROOFSTEP]\nexact fun s _ _ => Integrable.integrableOn (L1.integrable_coeFn _)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f_Lp x \u2202\u03bc = 0\n[PROOFSTEP]\nintro s hs h\u03bcs\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191f_Lp x \u2202\u03bc = 0\n[PROOFSTEP]\nrw [integral_congr_ae (ae_restrict_of_ae hf_f_Lp.symm)]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 E\nhf : Integrable f\nhf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0\nhf_Lp : Mem\u2112p f 1\nf_Lp : { x // x \u2208 Lp E 1 } := Mem\u2112p.toLp f hf_Lp\nhf_f_Lp : f =\u1d50[\u03bc] \u2191\u2191f_Lp\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc = 0\n[PROOFSTEP]\nexact hf_zero s hs h\u03bcs\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 f =\u1d50[\u03bc] g\n[PROOFSTEP]\nrw [\u2190 sub_ae_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 f - g =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave hfg' : \u2200 s : Set \u03b1, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 (\u222b x in s, (f - g) x \u2202\u03bc) = 0 :=\n by\n intro s hs h\u03bcs\n rw [integral_sub' hf.integrableOn hg.integrableOn]\n exact sub_eq_zero.mpr (hfg s hs h\u03bcs)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\n[PROOFSTEP]\nintro s hs h\u03bcs\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\n[PROOFSTEP]\nrw [integral_sub' hf.integrableOn hg.integrableOn]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (a : \u03b1) in s, f a \u2202\u03bc - \u222b (a : \u03b1) in s, g a \u2202\u03bc = 0\n[PROOFSTEP]\nexact sub_eq_zero.mpr (hfg s hs h\u03bcs)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in s, g x \u2202\u03bc\nhfg' : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, (f - g) x \u2202\u03bc = 0\n\u22a2 f - g =\u1d50[\u03bc] 0\n[PROOFSTEP]\nexact Integrable.ae_eq_zero_of_forall_set_integral_eq_zero (hf.sub hg) hfg'\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsClosed s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nsuffices : \u2200 s, MeasurableSet s \u2192 \u222b x in s, f x \u2202\u03bc = 0\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsClosed s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nthis : \u2200 (s : Set \u03b2), MeasurableSet s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nexact hf.ae_eq_zero_of_forall_set_integral_eq_zero (fun s hs _ \u21a6 this s hs)\n[GOAL]\ncase this\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsClosed s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\n\u22a2 \u2200 (s : Set \u03b2), MeasurableSet s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\n[PROOFSTEP]\nhave A : \u2200 (t : Set \u03b2), MeasurableSet t \u2192 \u222b (x : \u03b2) in t, f x \u2202\u03bc = 0 \u2192 \u222b (x : \u03b2) in t\u1d9c, f x \u2202\u03bc = 0 :=\n by\n intro t t_meas ht\n have I : \u222b x, f x \u2202\u03bc = 0 := by rw [\u2190 integral_univ]; exact h'f _ isClosed_univ\n simpa [ht, I] using integral_add_compl t_meas hf\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsClosed s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\n\u22a2 \u2200 (t : Set \u03b2), MeasurableSet t \u2192 \u222b (x : \u03b2) in t, f x \u2202\u03bc = 0 \u2192 \u222b (x : \u03b2) in t\u1d9c, f x \u2202\u03bc = 0\n[PROOFSTEP]\nintro t t_meas ht\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsClosed s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nt : Set \u03b2\nt_meas : MeasurableSet t\nht : \u222b (x : \u03b2) in t, f x \u2202\u03bc = 0\n\u22a2 \u222b (x : \u03b2) in t\u1d9c, f x \u2202\u03bc = 0\n[PROOFSTEP]\nhave I : \u222b x, f x \u2202\u03bc = 0 := by rw [\u2190 integral_univ]; exact h'f _ isClosed_univ\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsClosed s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nt : Set \u03b2\nt_meas : MeasurableSet t\nht : \u222b (x : \u03b2) in t, f x \u2202\u03bc = 0\n\u22a2 \u222b (x : \u03b2), f x \u2202\u03bc = 0\n[PROOFSTEP]\nrw [\u2190 integral_univ]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsClosed s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nt : Set \u03b2\nt_meas : MeasurableSet t\nht : \u222b (x : \u03b2) in t, f x \u2202\u03bc = 0\n\u22a2 \u222b (x : \u03b2) in Set.univ, f x \u2202\u03bc = 0\n[PROOFSTEP]\nexact h'f _ isClosed_univ\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsClosed s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nt : Set \u03b2\nt_meas : MeasurableSet t\nht : \u222b (x : \u03b2) in t, f x \u2202\u03bc = 0\nI : \u222b (x : \u03b2), f x \u2202\u03bc = 0\n\u22a2 \u222b (x : \u03b2) in t\u1d9c, f x \u2202\u03bc = 0\n[PROOFSTEP]\nsimpa [ht, I] using integral_add_compl t_meas hf\n[GOAL]\ncase this\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsClosed s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nA : \u2200 (t : Set \u03b2), MeasurableSet t \u2192 \u222b (x : \u03b2) in t, f x \u2202\u03bc = 0 \u2192 \u222b (x : \u03b2) in t\u1d9c, f x \u2202\u03bc = 0\n\u22a2 \u2200 (s : Set \u03b2), MeasurableSet s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\n[PROOFSTEP]\nintro s hs\n[GOAL]\ncase this\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsClosed s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nA : \u2200 (t : Set \u03b2), MeasurableSet t \u2192 \u222b (x : \u03b2) in t, f x \u2202\u03bc = 0 \u2192 \u222b (x : \u03b2) in t\u1d9c, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\n[PROOFSTEP]\nrefine MeasurableSet.induction_on_open (fun U hU \u21a6 ?_) A (fun g g_disj g_meas hg \u21a6 ?_) hs\n[GOAL]\ncase this.refine_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsClosed s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nA : \u2200 (t : Set \u03b2), MeasurableSet t \u2192 \u222b (x : \u03b2) in t, f x \u2202\u03bc = 0 \u2192 \u222b (x : \u03b2) in t\u1d9c, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : MeasurableSet s\nU : Set \u03b2\nhU : IsOpen U\n\u22a2 \u222b (x : \u03b2) in U, f x \u2202\u03bc = 0\n[PROOFSTEP]\nrw [\u2190 compl_compl U]\n[GOAL]\ncase this.refine_1\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsClosed s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nA : \u2200 (t : Set \u03b2), MeasurableSet t \u2192 \u222b (x : \u03b2) in t, f x \u2202\u03bc = 0 \u2192 \u222b (x : \u03b2) in t\u1d9c, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : MeasurableSet s\nU : Set \u03b2\nhU : IsOpen U\n\u22a2 \u222b (x : \u03b2) in U\u1d9c\u1d9c, f x \u2202\u03bc = 0\n[PROOFSTEP]\nexact A _ hU.measurableSet.compl (h'f _ hU.isClosed_compl)\n[GOAL]\ncase this.refine_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsClosed s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nA : \u2200 (t : Set \u03b2), MeasurableSet t \u2192 \u222b (x : \u03b2) in t, f x \u2202\u03bc = 0 \u2192 \u222b (x : \u03b2) in t\u1d9c, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : MeasurableSet s\ng : \u2115 \u2192 Set \u03b2\ng_disj : Pairwise (Disjoint on g)\ng_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nhg : \u2200 (i : \u2115), \u222b (x : \u03b2) in g i, f x \u2202\u03bc = 0\n\u22a2 \u222b (x : \u03b2) in \u22c3 (i : \u2115), g i, f x \u2202\u03bc = 0\n[PROOFSTEP]\nrw [integral_iUnion g_meas g_disj hf.integrableOn]\n[GOAL]\ncase this.refine_2\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsClosed s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nA : \u2200 (t : Set \u03b2), MeasurableSet t \u2192 \u222b (x : \u03b2) in t, f x \u2202\u03bc = 0 \u2192 \u222b (x : \u03b2) in t\u1d9c, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : MeasurableSet s\ng : \u2115 \u2192 Set \u03b2\ng_disj : Pairwise (Disjoint on g)\ng_meas : \u2200 (i : \u2115), MeasurableSet (g i)\nhg : \u2200 (i : \u2115), \u222b (x : \u03b2) in g i, f x \u2202\u03bc = 0\n\u22a2 \u2211' (n : \u2115), \u222b (a : \u03b2) in g n, f a \u2202\u03bc = 0\n[PROOFSTEP]\nsimp [hg]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\napply ae_eq_zero_of_forall_set_integral_isClosed_eq_zero hf (fun s hs \u21a6 ?_)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : IsClosed s\n\u22a2 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\n[PROOFSTEP]\nlet t : \u2115 \u2192 Set \u03b2 := fun n \u21a6 compactCovering \u03b2 n \u2229 s\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : IsClosed s\nt : \u2115 \u2192 Set \u03b2 := fun n => compactCovering \u03b2 n \u2229 s\n\u22a2 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\n[PROOFSTEP]\nsuffices H : Tendsto (fun n \u21a6 \u222b x in t n, f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b x in s, f x \u2202\u03bc))\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : IsClosed s\nt : \u2115 \u2192 Set \u03b2 := fun n => compactCovering \u03b2 n \u2229 s\nH : Tendsto (fun n => \u222b (x : \u03b2) in t n, f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2) in s, f x \u2202\u03bc))\n\u22a2 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\n[PROOFSTEP]\nhave A : \u2200 n, \u222b x in t n, f x \u2202\u03bc = 0 := fun n \u21a6 h'f _ (IsCompact.inter_right (isCompact_compactCovering \u03b2 n) hs)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : IsClosed s\nt : \u2115 \u2192 Set \u03b2 := fun n => compactCovering \u03b2 n \u2229 s\nH : Tendsto (fun n => \u222b (x : \u03b2) in t n, f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2) in s, f x \u2202\u03bc))\nA : \u2200 (n : \u2115), \u222b (x : \u03b2) in t n, f x \u2202\u03bc = 0\n\u22a2 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\n[PROOFSTEP]\nsimp_rw [A, tendsto_const_nhds_iff] at H \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : IsClosed s\nt : \u2115 \u2192 Set \u03b2 := fun n => compactCovering \u03b2 n \u2229 s\nA : \u2200 (n : \u2115), \u222b (x : \u03b2) in t n, f x \u2202\u03bc = 0\nH : 0 = \u222b (x : \u03b2) in s, f x \u2202\u03bc\n\u22a2 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\n[PROOFSTEP]\nexact H.symm\n[GOAL]\ncase H\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : IsClosed s\nt : \u2115 \u2192 Set \u03b2 := fun n => compactCovering \u03b2 n \u2229 s\n\u22a2 Tendsto (fun n => \u222b (x : \u03b2) in t n, f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2) in s, f x \u2202\u03bc))\n[PROOFSTEP]\nhave B : s = \u22c3 n, t n := by rw [\u2190 Set.iUnion_inter, iUnion_compactCovering, Set.univ_inter]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : IsClosed s\nt : \u2115 \u2192 Set \u03b2 := fun n => compactCovering \u03b2 n \u2229 s\n\u22a2 s = \u22c3 (n : \u2115), t n\n[PROOFSTEP]\nrw [\u2190 Set.iUnion_inter, iUnion_compactCovering, Set.univ_inter]\n[GOAL]\ncase H\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : IsClosed s\nt : \u2115 \u2192 Set \u03b2 := fun n => compactCovering \u03b2 n \u2229 s\nB : s = \u22c3 (n : \u2115), t n\n\u22a2 Tendsto (fun n => \u222b (x : \u03b2) in t n, f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2) in s, f x \u2202\u03bc))\n[PROOFSTEP]\nrw [B]\n[GOAL]\ncase H\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : IsClosed s\nt : \u2115 \u2192 Set \u03b2 := fun n => compactCovering \u03b2 n \u2229 s\nB : s = \u22c3 (n : \u2115), t n\n\u22a2 Tendsto (fun n => \u222b (x : \u03b2) in t n, f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b2) in \u22c3 (n : \u2115), t n, f x \u2202\u03bc))\n[PROOFSTEP]\napply tendsto_set_integral_of_monotone\n[GOAL]\ncase H.hsm\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : IsClosed s\nt : \u2115 \u2192 Set \u03b2 := fun n => compactCovering \u03b2 n \u2229 s\nB : s = \u22c3 (n : \u2115), t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (t i)\n[PROOFSTEP]\nintros n\n[GOAL]\ncase H.hsm\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : IsClosed s\nt : \u2115 \u2192 Set \u03b2 := fun n => compactCovering \u03b2 n \u2229 s\nB : s = \u22c3 (n : \u2115), t n\nn : \u2115\n\u22a2 MeasurableSet (t n)\n[PROOFSTEP]\nexact ((isCompact_compactCovering \u03b2 n).inter_right hs).isClosed.measurableSet\n[GOAL]\ncase H.h_mono\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : IsClosed s\nt : \u2115 \u2192 Set \u03b2 := fun n => compactCovering \u03b2 n \u2229 s\nB : s = \u22c3 (n : \u2115), t n\n\u22a2 Monotone fun i => t i\n[PROOFSTEP]\nintros m n hmn\n[GOAL]\ncase H.h_mono\n\u03b1 : Type u_1\nE : Type u_2\nm\u271d m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : IsClosed s\nt : \u2115 \u2192 Set \u03b2 := fun n => compactCovering \u03b2 n \u2229 s\nB : s = \u22c3 (n : \u2115), t n\nm n : \u2115\nhmn : m \u2264 n\n\u22a2 (fun i => t i) m \u2264 (fun i => t i) n\n[PROOFSTEP]\nexact Set.inter_subset_inter_left _ (compactCovering_subset \u03b2 hmn)\n[GOAL]\ncase H.hfi\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : Integrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\ns : Set \u03b2\nhs : IsClosed s\nt : \u2115 \u2192 Set \u03b2 := fun n => compactCovering \u03b2 n \u2229 s\nB : s = \u22c3 (n : \u2115), t n\n\u22a2 IntegrableOn (fun a => f a) (\u22c3 (n : \u2115), t n)\n[PROOFSTEP]\nexact hf.integrableOn\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : LocallyIntegrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\n\u22a2 f =\u1d50[\u03bc] 0\n[PROOFSTEP]\nrw [\u2190 Measure.restrict_univ (\u03bc := \u03bc), \u2190 iUnion_compactCovering]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : LocallyIntegrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\n\u22a2 f =\u1d50[Measure.restrict \u03bc (\u22c3 (n : \u2115), compactCovering \u03b2 n)] 0\n[PROOFSTEP]\napply (ae_restrict_iUnion_iff _ _).2 (fun n \u21a6 ?_)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : LocallyIntegrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nn : \u2115\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202Measure.restrict \u03bc (compactCovering \u03b2 n), f x = OfNat.ofNat 0 x\n[PROOFSTEP]\napply ae_eq_zero_of_forall_set_integral_isCompact_eq_zero\n[GOAL]\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : LocallyIntegrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nn : \u2115\n\u22a2 Integrable fun x => f x\n[PROOFSTEP]\nexact hf.integrableOn_isCompact (isCompact_compactCovering \u03b2 n)\n[GOAL]\ncase h'f\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : LocallyIntegrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nn : \u2115\n\u22a2 \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202Measure.restrict \u03bc (compactCovering \u03b2 n) = 0\n[PROOFSTEP]\nintro s hs\n[GOAL]\ncase h'f\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : LocallyIntegrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nn : \u2115\ns : Set \u03b2\nhs : IsCompact s\n\u22a2 \u222b (x : \u03b2) in s, f x \u2202Measure.restrict \u03bc (compactCovering \u03b2 n) = 0\n[PROOFSTEP]\nrw [Measure.restrict_restrict hs.measurableSet]\n[GOAL]\ncase h'f\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ninst\u271d\u00b9 : SigmaCompactSpace \u03b2\ninst\u271d : T2Space \u03b2\n\u03bc : Measure \u03b2\nf : \u03b2 \u2192 E\nhf : LocallyIntegrable f\nh'f : \u2200 (s : Set \u03b2), IsCompact s \u2192 \u222b (x : \u03b2) in s, f x \u2202\u03bc = 0\nn : \u2115\ns : Set \u03b2\nhs : IsCompact s\n\u22a2 \u222b (x : \u03b2) in s \u2229 compactCovering \u03b2 n, f x \u2202\u03bc = 0\n[PROOFSTEP]\nexact h'f _ (hs.inter (isCompact_compactCovering \u03b2 n))\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 f =\u1d50[\u03bc] g\n[PROOFSTEP]\nrefine'\n ENNReal.eventuallyEq_of_toReal_eventuallyEq (ae_lt_top' hf hfi).ne_of_lt (ae_lt_top' hg hgi).ne_of_lt\n (Integrable.ae_eq_of_forall_set_integral_eq _ _ (integrable_toReal_of_lintegral_ne_top hf hfi)\n (integrable_toReal_of_lintegral_ne_top hg hgi) fun s hs hs' => _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, ENNReal.toReal (f x) \u2202\u03bc = \u222b (x : \u03b1) in s, ENNReal.toReal (g x) \u2202\u03bc\n[PROOFSTEP]\nrw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (f a)) \u2202\u03bc) =\n ENNReal.toReal (\u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (f a)) \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_congr_ae (ofReal_toReal_ae_eq _), lintegral_congr_ae (ofReal_toReal_ae_eq _)]\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc\n[PROOFSTEP]\nexact hfg hs hs'\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, g x < \u22a4\n[PROOFSTEP]\nrefine' ae_lt_top' hg.restrict (ne_of_lt (lt_of_le_of_lt _ hgi.lt_top))\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), g x \u2202\u03bc\n[PROOFSTEP]\nexact @set_lintegral_univ \u03b1 _ \u03bc g \u25b8 lintegral_mono_set (Set.subset_univ _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, f x < \u22a4\n[PROOFSTEP]\nrefine' ae_lt_top' hf.restrict (ne_of_lt (lt_of_le_of_lt _ hfi.lt_top))\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\nexact\n @set_lintegral_univ \u03b1 _ \u03bc f \u25b8\n lintegral_mono_set\n (Set.subset_univ _)\n -- putting the proofs where they are used is extremely slow\n[GOAL]\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 0 \u2264\u1d50[Measure.restrict \u03bc s] fun x => ENNReal.toReal (g x)\ncase hfm\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (g x)) (Measure.restrict \u03bc s)\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 0 \u2264\u1d50[Measure.restrict \u03bc s] fun x => ENNReal.toReal (f x)\ncase hfm\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : \u211d\u22650\u221e\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4\nhgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\nhfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nhs' : \u2191\u2191\u03bc s < \u22a4\n\u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (f x)) (Measure.restrict \u03bc s)\n[PROOFSTEP]\nexacts [ae_of_all _ fun x => ENNReal.toReal_nonneg, hg.ennreal_toReal.restrict.aestronglyMeasurable,\n ae_of_all _ fun x => ENNReal.toReal_nonneg, hf.ennreal_toReal.restrict.aestronglyMeasurable]\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "llama_tokens": 105535, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434873426302, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.4190054896341362}} {"text": "[GOAL]\nk : Type u\ninst\u271d\u2079 : CommRing k\nG : Type u\ninst\u271d\u2078 : Group G\nV : Type v\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module k V\ninst\u271d\u2075 : Module (MonoidAlgebra k G) V\ninst\u271d\u2074 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type w\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module k W\ninst\u271d\u00b9 : Module (MonoidAlgebra k G) W\ninst\u271d : IsScalarTower k (MonoidAlgebra k G) W\n\u03c0 : W \u2192\u2097[k] V\ni : V \u2192\u2097[MonoidAlgebra k G] W\nh : \u2200 (v : V), \u2191\u03c0 (\u2191i v) = v\ng : G\nv : V\n\u22a2 \u2191(conjugate \u03c0 g) (\u2191i v) = v\n[PROOFSTEP]\nrw [conjugate_apply, \u2190 i.map_smul, h, \u2190 mul_smul, single_mul_single, mul_one, mul_left_inv, \u2190 one_def, one_smul]\n[GOAL]\nk : Type u\ninst\u271d\u00b9\u2070 : CommRing k\nG : Type u\ninst\u271d\u2079 : Group G\nV : Type v\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module k V\ninst\u271d\u2076 : Module (MonoidAlgebra k G) V\ninst\u271d\u2075 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type w\ninst\u271d\u2074 : AddCommGroup W\ninst\u271d\u00b3 : Module k W\ninst\u271d\u00b2 : Module (MonoidAlgebra k G) W\ninst\u271d\u00b9 : IsScalarTower k (MonoidAlgebra k G) W\n\u03c0 : W \u2192\u2097[k] V\ni : V \u2192\u2097[MonoidAlgebra k G] W\nh : \u2200 (v : V), \u2191\u03c0 (\u2191i v) = v\ninst\u271d : Fintype G\ng : G\nv : W\n\u22a2 \u2191(sumOfConjugates G \u03c0) (MonoidAlgebra.single g 1 \u2022 v) = MonoidAlgebra.single g 1 \u2022 \u2191(sumOfConjugates G \u03c0) v\n[PROOFSTEP]\nsimp only [sumOfConjugates_apply, Finset.smul_sum, conjugate_apply]\n[GOAL]\nk : Type u\ninst\u271d\u00b9\u2070 : CommRing k\nG : Type u\ninst\u271d\u2079 : Group G\nV : Type v\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module k V\ninst\u271d\u2076 : Module (MonoidAlgebra k G) V\ninst\u271d\u2075 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type w\ninst\u271d\u2074 : AddCommGroup W\ninst\u271d\u00b3 : Module k W\ninst\u271d\u00b2 : Module (MonoidAlgebra k G) W\ninst\u271d\u00b9 : IsScalarTower k (MonoidAlgebra k G) W\n\u03c0 : W \u2192\u2097[k] V\ni : V \u2192\u2097[MonoidAlgebra k G] W\nh : \u2200 (v : V), \u2191\u03c0 (\u2191i v) = v\ninst\u271d : Fintype G\ng : G\nv : W\n\u22a2 \u2211 x : G, MonoidAlgebra.single x\u207b\u00b9 1 \u2022 \u2191\u03c0 (MonoidAlgebra.single x 1 \u2022 MonoidAlgebra.single g 1 \u2022 v) =\n \u2211 x : G, MonoidAlgebra.single g 1 \u2022 MonoidAlgebra.single x\u207b\u00b9 1 \u2022 \u2191\u03c0 (MonoidAlgebra.single x 1 \u2022 v)\n[PROOFSTEP]\nrefine Fintype.sum_bijective (\u00b7 * g) (Group.mulRight_bijective g) _ _ fun i \u21a6 ?_\n[GOAL]\nk : Type u\ninst\u271d\u00b9\u2070 : CommRing k\nG : Type u\ninst\u271d\u2079 : Group G\nV : Type v\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module k V\ninst\u271d\u2076 : Module (MonoidAlgebra k G) V\ninst\u271d\u2075 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type w\ninst\u271d\u2074 : AddCommGroup W\ninst\u271d\u00b3 : Module k W\ninst\u271d\u00b2 : Module (MonoidAlgebra k G) W\ninst\u271d\u00b9 : IsScalarTower k (MonoidAlgebra k G) W\n\u03c0 : W \u2192\u2097[k] V\ni\u271d : V \u2192\u2097[MonoidAlgebra k G] W\nh : \u2200 (v : V), \u2191\u03c0 (\u2191i\u271d v) = v\ninst\u271d : Fintype G\ng : G\nv : W\ni : G\n\u22a2 MonoidAlgebra.single i\u207b\u00b9 1 \u2022 \u2191\u03c0 (MonoidAlgebra.single i 1 \u2022 MonoidAlgebra.single g 1 \u2022 v) =\n MonoidAlgebra.single g 1 \u2022\n MonoidAlgebra.single ((fun x => x * g) i)\u207b\u00b9 1 \u2022 \u2191\u03c0 (MonoidAlgebra.single ((fun x => x * g) i) 1 \u2022 v)\n[PROOFSTEP]\nsimp only [smul_smul, single_mul_single, mul_inv_rev, mul_inv_cancel_left, one_mul]\n[GOAL]\nk : Type u\ninst\u271d\u00b9\u00b9 : CommRing k\nG : Type u\ninst\u271d\u00b9\u2070 : Group G\nV : Type v\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module k V\ninst\u271d\u2077 : Module (MonoidAlgebra k G) V\ninst\u271d\u2076 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type w\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module k W\ninst\u271d\u00b3 : Module (MonoidAlgebra k G) W\ninst\u271d\u00b2 : IsScalarTower k (MonoidAlgebra k G) W\n\u03c0 : W \u2192\u2097[k] V\ni : V \u2192\u2097[MonoidAlgebra k G] W\nh : \u2200 (v : V), \u2191\u03c0 (\u2191i v) = v\ninst\u271d\u00b9 : Fintype G\ninst\u271d : Invertible \u2191(Fintype.card G)\nv : W\n\u22a2 \u2191(equivariantProjection G \u03c0) v = \u215f\u2191(Fintype.card G) \u2022 \u2211 g : G, \u2191(conjugate \u03c0 g) v\n[PROOFSTEP]\nsimp only [equivariantProjection, smul_apply, sumOfConjugatesEquivariant_apply]\n[GOAL]\nk : Type u\ninst\u271d\u00b9\u00b9 : CommRing k\nG : Type u\ninst\u271d\u00b9\u2070 : Group G\nV : Type v\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module k V\ninst\u271d\u2077 : Module (MonoidAlgebra k G) V\ninst\u271d\u2076 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type w\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module k W\ninst\u271d\u00b3 : Module (MonoidAlgebra k G) W\ninst\u271d\u00b2 : IsScalarTower k (MonoidAlgebra k G) W\n\u03c0 : W \u2192\u2097[k] V\ni : V \u2192\u2097[MonoidAlgebra k G] W\nh : \u2200 (v : V), \u2191\u03c0 (\u2191i v) = v\ninst\u271d\u00b9 : Fintype G\ninst\u271d : Invertible \u2191(Fintype.card G)\nv : V\n\u22a2 \u2191(equivariantProjection G \u03c0) (\u2191i v) = v\n[PROOFSTEP]\nrw [equivariantProjection_apply]\n[GOAL]\nk : Type u\ninst\u271d\u00b9\u00b9 : CommRing k\nG : Type u\ninst\u271d\u00b9\u2070 : Group G\nV : Type v\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module k V\ninst\u271d\u2077 : Module (MonoidAlgebra k G) V\ninst\u271d\u2076 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type w\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module k W\ninst\u271d\u00b3 : Module (MonoidAlgebra k G) W\ninst\u271d\u00b2 : IsScalarTower k (MonoidAlgebra k G) W\n\u03c0 : W \u2192\u2097[k] V\ni : V \u2192\u2097[MonoidAlgebra k G] W\nh : \u2200 (v : V), \u2191\u03c0 (\u2191i v) = v\ninst\u271d\u00b9 : Fintype G\ninst\u271d : Invertible \u2191(Fintype.card G)\nv : V\n\u22a2 \u215f\u2191(Fintype.card G) \u2022 \u2211 g : G, \u2191(conjugate \u03c0 g) (\u2191i v) = v\n[PROOFSTEP]\nsimp only [conjugate_i \u03c0 i h]\n[GOAL]\nk : Type u\ninst\u271d\u00b9\u00b9 : CommRing k\nG : Type u\ninst\u271d\u00b9\u2070 : Group G\nV : Type v\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module k V\ninst\u271d\u2077 : Module (MonoidAlgebra k G) V\ninst\u271d\u2076 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type w\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module k W\ninst\u271d\u00b3 : Module (MonoidAlgebra k G) W\ninst\u271d\u00b2 : IsScalarTower k (MonoidAlgebra k G) W\n\u03c0 : W \u2192\u2097[k] V\ni : V \u2192\u2097[MonoidAlgebra k G] W\nh : \u2200 (v : V), \u2191\u03c0 (\u2191i v) = v\ninst\u271d\u00b9 : Fintype G\ninst\u271d : Invertible \u2191(Fintype.card G)\nv : V\n\u22a2 \u215f\u2191(Fintype.card G) \u2022 \u2211 x : G, v = v\n[PROOFSTEP]\nrw [Finset.sum_const, Finset.card_univ, nsmul_eq_smul_cast k, smul_smul, Invertible.invOf_mul_self, one_smul]\n[GOAL]\nk : Type u\ninst\u271d\u00b9\u00b9 : Field k\nG : Type u\ninst\u271d\u00b9\u2070 : Fintype G\ninst\u271d\u2079 : Invertible \u2191(Fintype.card G)\ninst\u271d\u2078 : Group G\nV : Type u\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module k V\ninst\u271d\u2075 : Module (MonoidAlgebra k G) V\ninst\u271d\u2074 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type u\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module k W\ninst\u271d\u00b9 : Module (MonoidAlgebra k G) W\ninst\u271d : IsScalarTower k (MonoidAlgebra k G) W\nf : V \u2192\u2097[MonoidAlgebra k G] W\nhf : LinearMap.ker f = \u22a5\n\u22a2 \u2203 g, LinearMap.comp g f = LinearMap.id\n[PROOFSTEP]\nobtain \u27e8\u03c6, h\u03c6\u27e9 :=\n (f.restrictScalars k).exists_leftInverse_of_injective <| by\n simp only [hf, Submodule.restrictScalars_bot, LinearMap.ker_restrictScalars]\n[GOAL]\nk : Type u\ninst\u271d\u00b9\u00b9 : Field k\nG : Type u\ninst\u271d\u00b9\u2070 : Fintype G\ninst\u271d\u2079 : Invertible \u2191(Fintype.card G)\ninst\u271d\u2078 : Group G\nV : Type u\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module k V\ninst\u271d\u2075 : Module (MonoidAlgebra k G) V\ninst\u271d\u2074 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type u\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module k W\ninst\u271d\u00b9 : Module (MonoidAlgebra k G) W\ninst\u271d : IsScalarTower k (MonoidAlgebra k G) W\nf : V \u2192\u2097[MonoidAlgebra k G] W\nhf : LinearMap.ker f = \u22a5\n\u22a2 LinearMap.ker (\u2191k f) = \u22a5\n[PROOFSTEP]\nsimp only [hf, Submodule.restrictScalars_bot, LinearMap.ker_restrictScalars]\n[GOAL]\ncase intro\nk : Type u\ninst\u271d\u00b9\u00b9 : Field k\nG : Type u\ninst\u271d\u00b9\u2070 : Fintype G\ninst\u271d\u2079 : Invertible \u2191(Fintype.card G)\ninst\u271d\u2078 : Group G\nV : Type u\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module k V\ninst\u271d\u2075 : Module (MonoidAlgebra k G) V\ninst\u271d\u2074 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type u\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module k W\ninst\u271d\u00b9 : Module (MonoidAlgebra k G) W\ninst\u271d : IsScalarTower k (MonoidAlgebra k G) W\nf : V \u2192\u2097[MonoidAlgebra k G] W\nhf : LinearMap.ker f = \u22a5\n\u03c6 : W \u2192\u2097[k] V\nh\u03c6 : LinearMap.comp \u03c6 (\u2191k f) = LinearMap.id\n\u22a2 \u2203 g, LinearMap.comp g f = LinearMap.id\n[PROOFSTEP]\nrefine \u27e8\u03c6.equivariantProjection G, FunLike.ext _ _ ?_\u27e9\n[GOAL]\ncase intro\nk : Type u\ninst\u271d\u00b9\u00b9 : Field k\nG : Type u\ninst\u271d\u00b9\u2070 : Fintype G\ninst\u271d\u2079 : Invertible \u2191(Fintype.card G)\ninst\u271d\u2078 : Group G\nV : Type u\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module k V\ninst\u271d\u2075 : Module (MonoidAlgebra k G) V\ninst\u271d\u2074 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type u\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module k W\ninst\u271d\u00b9 : Module (MonoidAlgebra k G) W\ninst\u271d : IsScalarTower k (MonoidAlgebra k G) W\nf : V \u2192\u2097[MonoidAlgebra k G] W\nhf : LinearMap.ker f = \u22a5\n\u03c6 : W \u2192\u2097[k] V\nh\u03c6 : LinearMap.comp \u03c6 (\u2191k f) = LinearMap.id\n\u22a2 \u2200 (x : V), \u2191(LinearMap.comp (LinearMap.equivariantProjection G \u03c6) f) x = \u2191LinearMap.id x\n[PROOFSTEP]\nexact \u03c6.equivariantProjection_condition G _ <| FunLike.congr_fun h\u03c6\n[GOAL]\nk : Type u\ninst\u271d\u00b9\u00b9 : Field k\nG : Type u\ninst\u271d\u00b9\u2070 : Fintype G\ninst\u271d\u2079 : Invertible \u2191(Fintype.card G)\ninst\u271d\u2078 : Group G\nV : Type u\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module k V\ninst\u271d\u2075 : Module (MonoidAlgebra k G) V\ninst\u271d\u2074 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type u\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module k W\ninst\u271d\u00b9 : Module (MonoidAlgebra k G) W\ninst\u271d : IsScalarTower k (MonoidAlgebra k G) W\np : Submodule (MonoidAlgebra k G) V\n\u22a2 \u2203 q, IsCompl p q\n[PROOFSTEP]\nhave : IsScalarTower k (MonoidAlgebra k G) p := p.isScalarTower'\n[GOAL]\nk : Type u\ninst\u271d\u00b9\u00b9 : Field k\nG : Type u\ninst\u271d\u00b9\u2070 : Fintype G\ninst\u271d\u2079 : Invertible \u2191(Fintype.card G)\ninst\u271d\u2078 : Group G\nV : Type u\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module k V\ninst\u271d\u2075 : Module (MonoidAlgebra k G) V\ninst\u271d\u2074 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type u\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module k W\ninst\u271d\u00b9 : Module (MonoidAlgebra k G) W\ninst\u271d : IsScalarTower k (MonoidAlgebra k G) W\np : Submodule (MonoidAlgebra k G) V\nthis : IsScalarTower k (MonoidAlgebra k G) { x // x \u2208 p }\n\u22a2 \u2203 q, IsCompl p q\n[PROOFSTEP]\nrcases MonoidAlgebra.exists_leftInverse_of_injective p.subtype p.ker_subtype with \u27e8f, hf\u27e9\n[GOAL]\ncase intro\nk : Type u\ninst\u271d\u00b9\u00b9 : Field k\nG : Type u\ninst\u271d\u00b9\u2070 : Fintype G\ninst\u271d\u2079 : Invertible \u2191(Fintype.card G)\ninst\u271d\u2078 : Group G\nV : Type u\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module k V\ninst\u271d\u2075 : Module (MonoidAlgebra k G) V\ninst\u271d\u2074 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type u\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module k W\ninst\u271d\u00b9 : Module (MonoidAlgebra k G) W\ninst\u271d : IsScalarTower k (MonoidAlgebra k G) W\np : Submodule (MonoidAlgebra k G) V\nthis : IsScalarTower k (MonoidAlgebra k G) { x // x \u2208 p }\nf : V \u2192\u2097[MonoidAlgebra k G] { x // x \u2208 p }\nhf : LinearMap.comp f (Submodule.subtype p) = LinearMap.id\n\u22a2 \u2203 q, IsCompl p q\n[PROOFSTEP]\nrefine \u27e8LinearMap.ker f, LinearMap.isCompl_of_proj ?_\u27e9\n[GOAL]\ncase intro\nk : Type u\ninst\u271d\u00b9\u00b9 : Field k\nG : Type u\ninst\u271d\u00b9\u2070 : Fintype G\ninst\u271d\u2079 : Invertible \u2191(Fintype.card G)\ninst\u271d\u2078 : Group G\nV : Type u\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module k V\ninst\u271d\u2075 : Module (MonoidAlgebra k G) V\ninst\u271d\u2074 : IsScalarTower k (MonoidAlgebra k G) V\nW : Type u\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module k W\ninst\u271d\u00b9 : Module (MonoidAlgebra k G) W\ninst\u271d : IsScalarTower k (MonoidAlgebra k G) W\np : Submodule (MonoidAlgebra k G) V\nthis : IsScalarTower k (MonoidAlgebra k G) { x // x \u2208 p }\nf : V \u2192\u2097[MonoidAlgebra k G] { x // x \u2208 p }\nhf : LinearMap.comp f (Submodule.subtype p) = LinearMap.id\n\u22a2 \u2200 (x : { x // x \u2208 p }), \u2191f \u2191x = x\n[PROOFSTEP]\nexact FunLike.congr_fun hf\n", "meta": {"mathlib_filename": "Mathlib.RepresentationTheory.Maschke", "llama_tokens": 5099, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936430859597, "lm_q2_score": 0.546738151984614, "lm_q1q2_score": 0.41896197029837495}} {"text": "[GOAL]\n\u03b1 : Type u\ninst\u271d : AddZeroClass \u03b1\n\u22a2 unop 0 = 0\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\n\u22a2 unop 0 = 0\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d : AddMonoidWithOne \u03b1\nsrc\u271d\u00b2 : AddMonoid \u03b1\u1d50\u1d52\u1d56 := addMonoid \u03b1\nsrc\u271d\u00b9 : One \u03b1\u1d50\u1d52\u1d56 := one \u03b1\nsrc\u271d : NatCast \u03b1\u1d50\u1d52\u1d56 := natCast \u03b1\n\u22a2 op \u21910 = 0\n[PROOFSTEP]\nrw [Nat.cast_zero, op_zero]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : AddMonoidWithOne \u03b1\nsrc\u271d\u00b2 : AddMonoid \u03b1\u1d50\u1d52\u1d56 := addMonoid \u03b1\nsrc\u271d\u00b9 : One \u03b1\u1d50\u1d52\u1d56 := one \u03b1\nsrc\u271d : NatCast \u03b1\u1d50\u1d52\u1d56 := natCast \u03b1\n\u22a2 \u2200 (n : \u2115), op \u2191(n + 1) = op \u2191n + 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\ninst\u271d : SubNegMonoid \u03b1\n\u22a2 unop 0 = 0\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d : AddGroup \u03b1\n\u22a2 unop 0 = 0\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d : AddGroupWithOne \u03b1\nsrc\u271d\u00b9 : AddMonoidWithOne \u03b1\u1d50\u1d52\u1d56 := addMonoidWithOne \u03b1\nsrc\u271d : AddGroup \u03b1\u1d50\u1d52\u1d56 := addGroup \u03b1\nn : \u2115\n\u22a2 op \u2191\u2191n = op \u2191n\n[PROOFSTEP]\nrw [Int.cast_ofNat]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : AddGroupWithOne \u03b1\nsrc\u271d\u00b9 : AddMonoidWithOne \u03b1\u1d50\u1d52\u1d56 := addMonoidWithOne \u03b1\nsrc\u271d : AddGroup \u03b1\u1d50\u1d52\u1d56 := addGroup \u03b1\nn : \u2115\n\u22a2 op \u2191(Int.negSucc n) = op (-unop (op \u2191(n + 1)))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DivInvMonoid \u03b1\nsrc\u271d\u00b9 : Monoid \u03b1\u1d50\u1d52\u1d56 := monoid \u03b1\nsrc\u271d : Inv \u03b1\u1d50\u1d52\u1d56 := inv \u03b1\nn : \u2115\nx : \u03b1\u1d50\u1d52\u1d56\n\u22a2 unop ((fun n x => op (unop x ^ n)) (Int.ofNat (Nat.succ n)) x) =\n unop (x * (fun n x => op (unop x ^ n)) (Int.ofNat n) x)\n[PROOFSTEP]\nsimp only [Int.ofNat_eq_coe]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DivInvMonoid \u03b1\nsrc\u271d\u00b9 : Monoid \u03b1\u1d50\u1d52\u1d56 := monoid \u03b1\nsrc\u271d : Inv \u03b1\u1d50\u1d52\u1d56 := inv \u03b1\nn : \u2115\nx : \u03b1\u1d50\u1d52\u1d56\n\u22a2 unop (op (unop x ^ \u2191(Nat.succ n))) = unop (x * op (unop x ^ \u2191n))\n[PROOFSTEP]\nrw [unop_op, zpow_ofNat, pow_succ', unop_mul, unop_op, zpow_ofNat]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DivInvMonoid \u03b1\nx y : \u03b1\n\u22a2 op (x / y) = (op y)\u207b\u00b9 * op x\n[PROOFSTEP]\nsimp [div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Mul \u03b1\na x y : \u03b1\n\u22a2 SemiconjBy (op a) (op y) (op x) \u2194 SemiconjBy a x y\n[PROOFSTEP]\nsimp only [SemiconjBy, \u2190 op_mul, op_inj, eq_comm]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Mul \u03b1\na x y : \u03b1\u1d50\u1d52\u1d56\n\u22a2 SemiconjBy (unop a) (unop y) (unop x) \u2194 SemiconjBy a x y\n[PROOFSTEP]\nconv_rhs => rw [\u2190 op_unop a, \u2190 op_unop x, \u2190 op_unop y, semiconjBy_op]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Mul \u03b1\na x y : \u03b1\u1d50\u1d52\u1d56\n| SemiconjBy a x y\n[PROOFSTEP]\nrw [\u2190 op_unop a, \u2190 op_unop x, \u2190 op_unop y, semiconjBy_op]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Mul \u03b1\na x y : \u03b1\u1d50\u1d52\u1d56\n| SemiconjBy a x y\n[PROOFSTEP]\nrw [\u2190 op_unop a, \u2190 op_unop x, \u2190 op_unop y, semiconjBy_op]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Mul \u03b1\na x y : \u03b1\u1d50\u1d52\u1d56\n| SemiconjBy a x y\n[PROOFSTEP]\nrw [\u2190 op_unop a, \u2190 op_unop x, \u2190 op_unop y, semiconjBy_op]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : MulOneClass \u03b1\n\u22a2 unop 1 = 1\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\n\u22a2 unop 1 = 1\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d : CommMonoid \u03b1\n\u22a2 unop 1 = 1\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DivInvMonoid \u03b1\n\u22a2 unop 1 = 1\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Group \u03b1\n\u22a2 unop 1 = 1\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d : CommGroup \u03b1\n\u22a2 unop 1 = 1\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d : AddCommMonoidWithOne \u03b1\nsrc\u271d\u00b2 : AddCommMonoid \u03b1\u1d43\u1d52\u1d56 := addCommMonoid \u03b1\nsrc\u271d\u00b9 : One \u03b1\u1d43\u1d52\u1d56 := one\nsrc\u271d : NatCast \u03b1\u1d43\u1d52\u1d56 := natCast \u03b1\n\u22a2 op \u21910 = 0\n[PROOFSTEP]\nrw [Nat.cast_zero, op_zero]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : AddCommMonoidWithOne \u03b1\nsrc\u271d\u00b2 : AddCommMonoid \u03b1\u1d43\u1d52\u1d56 := addCommMonoid \u03b1\nsrc\u271d\u00b9 : One \u03b1\u1d43\u1d52\u1d56 := one\nsrc\u271d : NatCast \u03b1\u1d43\u1d52\u1d56 := natCast \u03b1\n\u22a2 \u2200 (n : \u2115), op \u2191(n + 1) = op \u2191n + 1\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : Mul M\ninst\u271d : Mul N\nf : M \u2192\u2099* N\nhf : \u2200 (x y : M), Commute (\u2191f x) (\u2191f y)\nx y : M\n\u22a2 (op \u2218 \u2191f) (x * y) = (op \u2218 \u2191f) x * (op \u2218 \u2191f) y\n[PROOFSTEP]\nsimp [(hf x y).eq]\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : MulOneClass M\ninst\u271d : MulOneClass N\nf : M \u2192* N\nhf : \u2200 (x y : M), Commute (\u2191f x) (\u2191f y)\nx y : M\n\u22a2 OneHom.toFun { toFun := op \u2218 \u2191f, map_one' := (_ : op (\u2191f 1) = op 1) } (x * y) =\n OneHom.toFun { toFun := op \u2218 \u2191f, map_one' := (_ : op (\u2191f 1) = op 1) } x *\n OneHom.toFun { toFun := op \u2218 \u2191f, map_one' := (_ : op (\u2191f 1) = op 1) } y\n[PROOFSTEP]\nsimp [(hf x y).eq]\n[GOAL]\n\u03b1 : Type u\nM : Type ?u.46725\ninst\u271d : Monoid M\nx : M\u1d50\u1d52\u1d56\u02e3\n\u22a2 \u2191((fun X =>\n MulOpposite.rec'\n (fun u =>\n { val := op \u2191u, inv := op \u2191u\u207b\u00b9, val_inv := (_ : op \u2191u * op \u2191u\u207b\u00b9 = 1),\n inv_val := (_ : op \u2191u\u207b\u00b9 * op \u2191u = 1) })\n X)\n ((fun u =>\n op\n { val := unop \u2191u, inv := unop \u2191u\u207b\u00b9, val_inv := (_ : unop \u2191u * unop \u2191u\u207b\u00b9 = 1),\n inv_val := (_ : unop \u2191u\u207b\u00b9 * unop \u2191u = 1) })\n x)) =\n \u2191x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nM : Type ?u.46725\ninst\u271d : Monoid M\nx : M\u02e3\u1d50\u1d52\u1d56\n\u22a2 \u2191(unop\n ((fun u =>\n op\n { val := unop \u2191u, inv := unop \u2191u\u207b\u00b9, val_inv := (_ : unop \u2191u * unop \u2191u\u207b\u00b9 = 1),\n inv_val := (_ : unop \u2191u\u207b\u00b9 * unop \u2191u = 1) })\n ((fun X =>\n MulOpposite.rec'\n (fun u =>\n { val := op \u2191u, inv := op \u2191u\u207b\u00b9, val_inv := (_ : op \u2191u * op \u2191u\u207b\u00b9 = 1),\n inv_val := (_ : op \u2191u\u207b\u00b9 * op \u2191u = 1) })\n X)\n x))) =\n \u2191(unop x)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u\n\u03b1 : Type ?u.65550\n\u03b2 : Type ?u.65553\ninst\u271d\u00b9 : Mul \u03b1\ninst\u271d : Mul \u03b2\nf : \u03b1\u1d50\u1d52\u1d56 \u2243* \u03b2\u1d50\u1d52\u1d56\nx : \u03b1\n\u22a2 (unop \u2218 \u2191(symm f) \u2218 MulOpposite.op) ((unop \u2218 \u2191f \u2218 MulOpposite.op) x) = x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u\n\u03b1 : Type ?u.65550\n\u03b2 : Type ?u.65553\ninst\u271d\u00b9 : Mul \u03b1\ninst\u271d : Mul \u03b2\nf : \u03b1\u1d50\u1d52\u1d56 \u2243* \u03b2\u1d50\u1d52\u1d56\nx : \u03b2\n\u22a2 (unop \u2218 \u2191f \u2218 MulOpposite.op) ((unop \u2218 \u2191(symm f) \u2218 MulOpposite.op) x) = x\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Group.Opposite", "llama_tokens": 3101, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7772998611746912, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.41895158261602483}} {"text": "[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\ny : L\n\u22a2 \u2203 t z, z \u2208 I \u2227 y = t \u2022 x + z\n[PROOFSTEP]\nhave hy : y \u2208 (\u22a4 : Submodule R L) := Submodule.mem_top\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\ny : L\nhy : y \u2208 \u22a4\n\u22a2 \u2203 t z, z \u2208 I \u2227 y = t \u2022 x + z\n[PROOFSTEP]\nsimp only [\u2190 hxI, Submodule.mem_sup, Submodule.mem_span_singleton] at hy \n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\ny : L\nhy : \u2203 y_1, (\u2203 a, a \u2022 x = y_1) \u2227 \u2203 z, z \u2208 (\u2191R L I).toSubmodule \u2227 y_1 + z = y\n\u22a2 \u2203 t z, z \u2208 I \u2227 y = t \u2022 x + z\n[PROOFSTEP]\nobtain \u27e8-, \u27e8t, rfl\u27e9, z, hz, rfl\u27e9 := hy\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nt : R\nz : L\nhz : z \u2208 (\u2191R L I).toSubmodule\n\u22a2 \u2203 t_1 z_1, z_1 \u2208 I \u2227 t \u2022 x + z = t_1 \u2022 x + z_1\n[PROOFSTEP]\nexact \u27e8t, z, hz, rfl\u27e9\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nN : LieSubmodule R L M\n\u22a2 \u2191\u2045\u22a4, N\u2046 = Submodule.map (\u2191(toEndomorphism R L M) x) \u2191N \u2294 \u2191\u2045I, N\u2046\n[PROOFSTEP]\nsimp only [lieIdeal_oper_eq_linear_span', Submodule.sup_span, mem_top, exists_prop, true_and, Submodule.map_coe,\n toEndomorphism_apply_apply]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nN : LieSubmodule R L M\n\u22a2 Submodule.span R {m | \u2203 x n, n \u2208 N \u2227 \u2045x, n\u2046 = m} =\n Submodule.span R ((fun a => \u2045x, a\u2046) '' \u2191\u2191N \u222a {m | \u2203 x, x \u2208 I \u2227 \u2203 n, n \u2208 N \u2227 \u2045x, n\u2046 = m})\n[PROOFSTEP]\nrefine' le_antisymm (Submodule.span_le.mpr _) (Submodule.span_mono fun z hz => _)\n[GOAL]\ncase refine'_1\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nN : LieSubmodule R L M\n\u22a2 {m | \u2203 x n, n \u2208 N \u2227 \u2045x, n\u2046 = m} \u2286\n \u2191(Submodule.span R ((fun a => \u2045x, a\u2046) '' \u2191\u2191N \u222a {m | \u2203 x, x \u2208 I \u2227 \u2203 n, n \u2208 N \u2227 \u2045x, n\u2046 = m}))\n[PROOFSTEP]\nrintro z \u27e8y, n, hn : n \u2208 N, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nN : LieSubmodule R L M\ny : L\nn : M\nhn : n \u2208 N\n\u22a2 \u2045y, n\u2046 \u2208 \u2191(Submodule.span R ((fun a => \u2045x, a\u2046) '' \u2191\u2191N \u222a {m | \u2203 x, x \u2208 I \u2227 \u2203 n, n \u2208 N \u2227 \u2045x, n\u2046 = m}))\n[PROOFSTEP]\nobtain \u27e8t, z, hz, rfl\u27e9 := exists_smul_add_of_span_sup_eq_top hxI y\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nN : LieSubmodule R L M\nn : M\nhn : n \u2208 N\nt : R\nz : L\nhz : z \u2208 I\n\u22a2 \u2045t \u2022 x + z, n\u2046 \u2208 \u2191(Submodule.span R ((fun a => \u2045x, a\u2046) '' \u2191\u2191N \u222a {m | \u2203 x, x \u2208 I \u2227 \u2203 n, n \u2208 N \u2227 \u2045x, n\u2046 = m}))\n[PROOFSTEP]\nsimp only [SetLike.mem_coe, Submodule.span_union, Submodule.mem_sup]\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nN : LieSubmodule R L M\nn : M\nhn : n \u2208 N\nt : R\nz : L\nhz : z \u2208 I\n\u22a2 \u2203 y,\n y \u2208 Submodule.span R ((fun a => \u2045x, a\u2046) '' \u2191\u2191N) \u2227\n \u2203 z_1, z_1 \u2208 Submodule.span R {m | \u2203 x, x \u2208 I \u2227 \u2203 n, n \u2208 N \u2227 \u2045x, n\u2046 = m} \u2227 y + z_1 = \u2045t \u2022 x + z, n\u2046\n[PROOFSTEP]\nexact\n \u27e8t \u2022 \u2045x, n\u2046, Submodule.subset_span \u27e8t \u2022 n, N.smul_mem' t hn, lie_smul t x n\u27e9, \u2045z, n\u2046,\n Submodule.subset_span \u27e8z, hz, n, hn, rfl\u27e9, by simp\u27e9\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nN : LieSubmodule R L M\nn : M\nhn : n \u2208 N\nt : R\nz : L\nhz : z \u2208 I\n\u22a2 t \u2022 \u2045x, n\u2046 + \u2045z, n\u2046 = \u2045t \u2022 x + z, n\u2046\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nN : LieSubmodule R L M\nz : M\nhz : z \u2208 (fun a => \u2045x, a\u2046) '' \u2191\u2191N \u222a {m | \u2203 x, x \u2208 I \u2227 \u2203 n, n \u2208 N \u2227 \u2045x, n\u2046 = m}\n\u22a2 z \u2208 {m | \u2203 x n, n \u2208 N \u2227 \u2045x, n\u2046 = m}\n[PROOFSTEP]\nrcases hz with (\u27e8m, hm, rfl\u27e9 | \u27e8y, -, m, hm, rfl\u27e9)\n[GOAL]\ncase refine'_2.inl.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nN : LieSubmodule R L M\nm : M\nhm : m \u2208 \u2191\u2191N\n\u22a2 (fun a => \u2045x, a\u2046) m \u2208 {m | \u2203 x n, n \u2208 N \u2227 \u2045x, n\u2046 = m}\ncase refine'_2.inr.intro.intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nN : LieSubmodule R L M\ny : L\nm : M\nhm : m \u2208 N\n\u22a2 \u2045y, m\u2046 \u2208 {m | \u2203 x n, n \u2208 N \u2227 \u2045x, n\u2046 = m}\n[PROOFSTEP]\nexacts [\u27e8x, m, hm, rfl\u27e9, \u27e8y, m, hm, rfl\u27e9]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn i j : \u2115\nhxn : \u2191(toEndomorphism R L M) x ^ n = 0\nhIM : lowerCentralSeries R L M i \u2264 LieIdeal.lcs I M j\n\u22a2 lowerCentralSeries R L M (i + n) \u2264 LieIdeal.lcs I M (j + 1)\n[PROOFSTEP]\nsuffices\n \u2200 l,\n ((\u22a4 : LieIdeal R L).lcs M (i + l) : Submodule R M) \u2264\n (I.lcs M j : Submodule R M).map (toEndomorphism R L M x ^ l) \u2294 (I.lcs M (j + 1) : Submodule R M)\n by simpa only [bot_sup_eq, LieIdeal.incl_coe, Submodule.map_zero, hxn] using this n\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn i j : \u2115\nhxn : \u2191(toEndomorphism R L M) x ^ n = 0\nhIM : lowerCentralSeries R L M i \u2264 LieIdeal.lcs I M j\nthis :\n \u2200 (l : \u2115),\n \u2191(LieIdeal.lcs \u22a4 M (i + l)) \u2264\n Submodule.map (\u2191(toEndomorphism R L M) x ^ l) \u2191(LieIdeal.lcs I M j) \u2294 \u2191(LieIdeal.lcs I M (j + 1))\n\u22a2 lowerCentralSeries R L M (i + n) \u2264 LieIdeal.lcs I M (j + 1)\n[PROOFSTEP]\nsimpa only [bot_sup_eq, LieIdeal.incl_coe, Submodule.map_zero, hxn] using this n\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn i j : \u2115\nhxn : \u2191(toEndomorphism R L M) x ^ n = 0\nhIM : lowerCentralSeries R L M i \u2264 LieIdeal.lcs I M j\n\u22a2 \u2200 (l : \u2115),\n \u2191(LieIdeal.lcs \u22a4 M (i + l)) \u2264\n Submodule.map (\u2191(toEndomorphism R L M) x ^ l) \u2191(LieIdeal.lcs I M j) \u2294 \u2191(LieIdeal.lcs I M (j + 1))\n[PROOFSTEP]\nintro l\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn i j : \u2115\nhxn : \u2191(toEndomorphism R L M) x ^ n = 0\nhIM : lowerCentralSeries R L M i \u2264 LieIdeal.lcs I M j\nl : \u2115\n\u22a2 \u2191(LieIdeal.lcs \u22a4 M (i + l)) \u2264\n Submodule.map (\u2191(toEndomorphism R L M) x ^ l) \u2191(LieIdeal.lcs I M j) \u2294 \u2191(LieIdeal.lcs I M (j + 1))\n[PROOFSTEP]\ninduction' l with l ih\n[GOAL]\ncase zero\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn i j : \u2115\nhxn : \u2191(toEndomorphism R L M) x ^ n = 0\nhIM : lowerCentralSeries R L M i \u2264 LieIdeal.lcs I M j\n\u22a2 \u2191(LieIdeal.lcs \u22a4 M (i + Nat.zero)) \u2264\n Submodule.map (\u2191(toEndomorphism R L M) x ^ Nat.zero) \u2191(LieIdeal.lcs I M j) \u2294 \u2191(LieIdeal.lcs I M (j + 1))\n[PROOFSTEP]\nsimp only [Nat.zero_eq, add_zero, LieIdeal.lcs_succ, pow_zero, LinearMap.one_eq_id, Submodule.map_id]\n[GOAL]\ncase zero\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn i j : \u2115\nhxn : \u2191(toEndomorphism R L M) x ^ n = 0\nhIM : lowerCentralSeries R L M i \u2264 LieIdeal.lcs I M j\n\u22a2 \u2191(LieIdeal.lcs \u22a4 M i) \u2264 \u2191(LieIdeal.lcs I M j) \u2294 \u2191\u2045I, LieIdeal.lcs I M j\u2046\n[PROOFSTEP]\nexact le_sup_of_le_left hIM\n[GOAL]\ncase succ\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn i j : \u2115\nhxn : \u2191(toEndomorphism R L M) x ^ n = 0\nhIM : lowerCentralSeries R L M i \u2264 LieIdeal.lcs I M j\nl : \u2115\nih :\n \u2191(LieIdeal.lcs \u22a4 M (i + l)) \u2264\n Submodule.map (\u2191(toEndomorphism R L M) x ^ l) \u2191(LieIdeal.lcs I M j) \u2294 \u2191(LieIdeal.lcs I M (j + 1))\n\u22a2 \u2191(LieIdeal.lcs \u22a4 M (i + Nat.succ l)) \u2264\n Submodule.map (\u2191(toEndomorphism R L M) x ^ Nat.succ l) \u2191(LieIdeal.lcs I M j) \u2294 \u2191(LieIdeal.lcs I M (j + 1))\n[PROOFSTEP]\nsimp only [LieIdeal.lcs_succ, i.add_succ l, lie_top_eq_of_span_sup_eq_top hxI, sup_le_iff]\n[GOAL]\ncase succ\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn i j : \u2115\nhxn : \u2191(toEndomorphism R L M) x ^ n = 0\nhIM : lowerCentralSeries R L M i \u2264 LieIdeal.lcs I M j\nl : \u2115\nih :\n \u2191(LieIdeal.lcs \u22a4 M (i + l)) \u2264\n Submodule.map (\u2191(toEndomorphism R L M) x ^ l) \u2191(LieIdeal.lcs I M j) \u2294 \u2191(LieIdeal.lcs I M (j + 1))\n\u22a2 Submodule.map (\u2191(toEndomorphism R L M) x) \u2191(LieIdeal.lcs \u22a4 M (i + l)) \u2264\n Submodule.map (\u2191(toEndomorphism R L M) x ^ Nat.succ l) \u2191(LieIdeal.lcs I M j) \u2294 \u2191\u2045I, LieIdeal.lcs I M j\u2046 \u2227\n \u2191\u2045I, LieIdeal.lcs \u22a4 M (i + l)\u2046 \u2264\n Submodule.map (\u2191(toEndomorphism R L M) x ^ Nat.succ l) \u2191(LieIdeal.lcs I M j) \u2294 \u2191\u2045I, LieIdeal.lcs I M j\u2046\n[PROOFSTEP]\nrefine' \u27e8(Submodule.map_mono ih).trans _, le_sup_of_le_right _\u27e9\n[GOAL]\ncase succ.refine'_1\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn i j : \u2115\nhxn : \u2191(toEndomorphism R L M) x ^ n = 0\nhIM : lowerCentralSeries R L M i \u2264 LieIdeal.lcs I M j\nl : \u2115\nih :\n \u2191(LieIdeal.lcs \u22a4 M (i + l)) \u2264\n Submodule.map (\u2191(toEndomorphism R L M) x ^ l) \u2191(LieIdeal.lcs I M j) \u2294 \u2191(LieIdeal.lcs I M (j + 1))\n\u22a2 Submodule.map (\u2191(toEndomorphism R L M) x)\n (Submodule.map (\u2191(toEndomorphism R L M) x ^ l) \u2191(LieIdeal.lcs I M j) \u2294 \u2191(LieIdeal.lcs I M (j + 1))) \u2264\n Submodule.map (\u2191(toEndomorphism R L M) x ^ Nat.succ l) \u2191(LieIdeal.lcs I M j) \u2294 \u2191\u2045I, LieIdeal.lcs I M j\u2046\n[PROOFSTEP]\nrw [Submodule.map_sup, \u2190 Submodule.map_comp, \u2190 LinearMap.mul_eq_comp, \u2190 pow_succ, \u2190 I.lcs_succ]\n[GOAL]\ncase succ.refine'_1\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn i j : \u2115\nhxn : \u2191(toEndomorphism R L M) x ^ n = 0\nhIM : lowerCentralSeries R L M i \u2264 LieIdeal.lcs I M j\nl : \u2115\nih :\n \u2191(LieIdeal.lcs \u22a4 M (i + l)) \u2264\n Submodule.map (\u2191(toEndomorphism R L M) x ^ l) \u2191(LieIdeal.lcs I M j) \u2294 \u2191(LieIdeal.lcs I M (j + 1))\n\u22a2 Submodule.map (\u2191(toEndomorphism R L M) x ^ (l + 1)) \u2191(LieIdeal.lcs I M j) \u2294\n Submodule.map (\u2191(toEndomorphism R L M) x) \u2191(LieIdeal.lcs I M (j + 1)) \u2264\n Submodule.map (\u2191(toEndomorphism R L M) x ^ Nat.succ l) \u2191(LieIdeal.lcs I M j) \u2294 \u2191(LieIdeal.lcs I M (j + 1))\n[PROOFSTEP]\nexact sup_le_sup_left coe_map_toEndomorphism_le _\n[GOAL]\ncase succ.refine'_2\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn i j : \u2115\nhxn : \u2191(toEndomorphism R L M) x ^ n = 0\nhIM : lowerCentralSeries R L M i \u2264 LieIdeal.lcs I M j\nl : \u2115\nih :\n \u2191(LieIdeal.lcs \u22a4 M (i + l)) \u2264\n Submodule.map (\u2191(toEndomorphism R L M) x ^ l) \u2191(LieIdeal.lcs I M j) \u2294 \u2191(LieIdeal.lcs I M (j + 1))\n\u22a2 \u2191\u2045I, LieIdeal.lcs \u22a4 M (i + l)\u2046 \u2264 \u2191\u2045I, LieIdeal.lcs I M j\u2046\n[PROOFSTEP]\nrefine' le_trans (mono_lie_right _ _ I _) (mono_lie_right _ _ I hIM)\n[GOAL]\ncase succ.refine'_2\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn i j : \u2115\nhxn : \u2191(toEndomorphism R L M) x ^ n = 0\nhIM : lowerCentralSeries R L M i \u2264 LieIdeal.lcs I M j\nl : \u2115\nih :\n \u2191(LieIdeal.lcs \u22a4 M (i + l)) \u2264\n Submodule.map (\u2191(toEndomorphism R L M) x ^ l) \u2191(LieIdeal.lcs I M j) \u2294 \u2191(LieIdeal.lcs I M (j + 1))\n\u22a2 LieIdeal.lcs \u22a4 M (i + l) \u2264 lowerCentralSeries R L M i\n[PROOFSTEP]\nexact antitone_lowerCentralSeries R L M le_self_add\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nhnp : _root_.IsNilpotent (\u2191(toEndomorphism R L M) x)\nhIM : LieModule.IsNilpotent R { x // x \u2208 \u2191I } M\n\u22a2 LieModule.IsNilpotent R L M\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := hnp\n[GOAL]\ncase intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nhIM : LieModule.IsNilpotent R { x // x \u2208 \u2191I } M\nn : \u2115\nhn : \u2191(toEndomorphism R L M) x ^ n = 0\n\u22a2 LieModule.IsNilpotent R L M\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := hIM\n[GOAL]\ncase intro.mk.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn : \u2115\nhn : \u2191(toEndomorphism R L M) x ^ n = 0\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 \u2191I } M k = \u22a5\n\u22a2 LieModule.IsNilpotent R L M\n[PROOFSTEP]\nhave hk' : I.lcs M k = \u22a5 := by simp only [\u2190 coe_toSubmodule_eq_iff, I.coe_lcs_eq, hk, bot_coeSubmodule]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn : \u2115\nhn : \u2191(toEndomorphism R L M) x ^ n = 0\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 \u2191I } M k = \u22a5\n\u22a2 LieIdeal.lcs I M k = \u22a5\n[PROOFSTEP]\nsimp only [\u2190 coe_toSubmodule_eq_iff, I.coe_lcs_eq, hk, bot_coeSubmodule]\n[GOAL]\ncase intro.mk.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn : \u2115\nhn : \u2191(toEndomorphism R L M) x ^ n = 0\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 \u2191I } M k = \u22a5\nhk' : LieIdeal.lcs I M k = \u22a5\n\u22a2 LieModule.IsNilpotent R L M\n[PROOFSTEP]\nsuffices \u2200 l, lowerCentralSeries R L M (l * n) \u2264 I.lcs M l\n by\n use k * n\n simpa [hk'] using this k\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn : \u2115\nhn : \u2191(toEndomorphism R L M) x ^ n = 0\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 \u2191I } M k = \u22a5\nhk' : LieIdeal.lcs I M k = \u22a5\nthis : \u2200 (l : \u2115), lowerCentralSeries R L M (l * n) \u2264 LieIdeal.lcs I M l\n\u22a2 LieModule.IsNilpotent R L M\n[PROOFSTEP]\nuse k * n\n[GOAL]\ncase h\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn : \u2115\nhn : \u2191(toEndomorphism R L M) x ^ n = 0\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 \u2191I } M k = \u22a5\nhk' : LieIdeal.lcs I M k = \u22a5\nthis : \u2200 (l : \u2115), lowerCentralSeries R L M (l * n) \u2264 LieIdeal.lcs I M l\n\u22a2 lowerCentralSeries R L M (k * n) = \u22a5\n[PROOFSTEP]\nsimpa [hk'] using this k\n[GOAL]\ncase intro.mk.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn : \u2115\nhn : \u2191(toEndomorphism R L M) x ^ n = 0\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 \u2191I } M k = \u22a5\nhk' : LieIdeal.lcs I M k = \u22a5\n\u22a2 \u2200 (l : \u2115), lowerCentralSeries R L M (l * n) \u2264 LieIdeal.lcs I M l\n[PROOFSTEP]\nintro l\n[GOAL]\ncase intro.mk.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn : \u2115\nhn : \u2191(toEndomorphism R L M) x ^ n = 0\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 \u2191I } M k = \u22a5\nhk' : LieIdeal.lcs I M k = \u22a5\nl : \u2115\n\u22a2 lowerCentralSeries R L M (l * n) \u2264 LieIdeal.lcs I M l\n[PROOFSTEP]\ninduction' l with l ih\n[GOAL]\ncase intro.mk.intro.zero\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn : \u2115\nhn : \u2191(toEndomorphism R L M) x ^ n = 0\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 \u2191I } M k = \u22a5\nhk' : LieIdeal.lcs I M k = \u22a5\n\u22a2 lowerCentralSeries R L M (Nat.zero * n) \u2264 LieIdeal.lcs I M Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.mk.intro.succ\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : Submodule.span R {x} \u2294 (\u2191R L I).toSubmodule = \u22a4\nn : \u2115\nhn : \u2191(toEndomorphism R L M) x ^ n = 0\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 \u2191I } M k = \u22a5\nhk' : LieIdeal.lcs I M k = \u22a5\nl : \u2115\nih : lowerCentralSeries R L M (l * n) \u2264 LieIdeal.lcs I M l\n\u22a2 lowerCentralSeries R L M (Nat.succ l * n) \u2264 LieIdeal.lcs I M (Nat.succ l)\n[PROOFSTEP]\nexact (l.succ_mul n).symm \u25b8 lcs_le_lcs_of_is_nilpotent_span_sup_eq_top hxI hn ih\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : LieRingModule L M\ninst\u271d\u00b9 : LieModule R L M\ninst\u271d : Subsingleton L\n\u22a2 IsEngelian R L\n[PROOFSTEP]\nintro M _i1 _i2 _i3 _i4 _h\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : Subsingleton L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\n_h : \u2200 (x : L), _root_.IsNilpotent (\u2191(toEndomorphism R L M) x)\n\u22a2 LieModule.IsNilpotent R L M\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase h\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : Subsingleton L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\n_h : \u2200 (x : L), _root_.IsNilpotent (\u2191(toEndomorphism R L M) x)\n\u22a2 lowerCentralSeries R L M 1 = \u22a5\n[PROOFSTEP]\nsuffices (\u22a4 : LieIdeal R L) = \u22a5 by simp [this]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : Subsingleton L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\n_h : \u2200 (x : L), _root_.IsNilpotent (\u2191(toEndomorphism R L M) x)\nthis : \u22a4 = \u22a5\n\u22a2 lowerCentralSeries R L M 1 = \u22a5\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase h\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : Subsingleton L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\n_h : \u2200 (x : L), _root_.IsNilpotent (\u2191(toEndomorphism R L M) x)\n\u22a2 \u22a4 = \u22a5\n[PROOFSTEP]\nhaveI := (LieSubmodule.subsingleton_iff R L L).mpr inferInstance\n[GOAL]\ncase h\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : Subsingleton L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\n_h : \u2200 (x : L), _root_.IsNilpotent (\u2191(toEndomorphism R L M) x)\nthis : Subsingleton (LieSubmodule R L L)\n\u22a2 \u22a4 = \u22a5\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nhf : Surjective \u2191f\nh : LieAlgebra.IsEngelian R L\n\u22a2 LieAlgebra.IsEngelian R L\u2082\n[PROOFSTEP]\nintro M _i1 _i2 _i3 _i4 h'\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nhf : Surjective \u2191f\nh : LieAlgebra.IsEngelian R L\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L\u2082 M\n_i4 : LieModule R L\u2082 M\nh' : \u2200 (x : L\u2082), IsNilpotent (\u2191(toEndomorphism R L\u2082 M) x)\n\u22a2 LieModule.IsNilpotent R L\u2082 M\n[PROOFSTEP]\nletI : LieRingModule L M := LieRingModule.compLieHom M f\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nhf : Surjective \u2191f\nh : LieAlgebra.IsEngelian R L\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L\u2082 M\n_i4 : LieModule R L\u2082 M\nh' : \u2200 (x : L\u2082), IsNilpotent (\u2191(toEndomorphism R L\u2082 M) x)\nthis : LieRingModule L M := LieRingModule.compLieHom M f\n\u22a2 LieModule.IsNilpotent R L\u2082 M\n[PROOFSTEP]\nletI : LieModule R L M := compLieHom M f\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nhf : Surjective \u2191f\nh : LieAlgebra.IsEngelian R L\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L\u2082 M\n_i4 : LieModule R L\u2082 M\nh' : \u2200 (x : L\u2082), IsNilpotent (\u2191(toEndomorphism R L\u2082 M) x)\nthis\u271d : LieRingModule L M := LieRingModule.compLieHom M f\nthis : LieModule R L M := compLieHom M f\n\u22a2 LieModule.IsNilpotent R L\u2082 M\n[PROOFSTEP]\nhave hnp : \u2200 x, IsNilpotent (toEndomorphism R L M x) := fun x => h' (f x)\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nhf : Surjective \u2191f\nh : LieAlgebra.IsEngelian R L\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L\u2082 M\n_i4 : LieModule R L\u2082 M\nh' : \u2200 (x : L\u2082), IsNilpotent (\u2191(toEndomorphism R L\u2082 M) x)\nthis\u271d : LieRingModule L M := LieRingModule.compLieHom M f\nthis : LieModule R L M := compLieHom M f\nhnp : \u2200 (x : L), IsNilpotent (\u2191(toEndomorphism R L M) x)\n\u22a2 LieModule.IsNilpotent R L\u2082 M\n[PROOFSTEP]\nhave surj_id : Function.Surjective (LinearMap.id : M \u2192\u2097[R] M) := Function.surjective_id\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nhf : Surjective \u2191f\nh : LieAlgebra.IsEngelian R L\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L\u2082 M\n_i4 : LieModule R L\u2082 M\nh' : \u2200 (x : L\u2082), IsNilpotent (\u2191(toEndomorphism R L\u2082 M) x)\nthis\u271d : LieRingModule L M := LieRingModule.compLieHom M f\nthis : LieModule R L M := compLieHom M f\nhnp : \u2200 (x : L), IsNilpotent (\u2191(toEndomorphism R L M) x)\nsurj_id : Surjective \u2191LinearMap.id\n\u22a2 LieModule.IsNilpotent R L\u2082 M\n[PROOFSTEP]\nhaveI : LieModule.IsNilpotent R L M := h M hnp\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nhf : Surjective \u2191f\nh : LieAlgebra.IsEngelian R L\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L\u2082 M\n_i4 : LieModule R L\u2082 M\nh' : \u2200 (x : L\u2082), IsNilpotent (\u2191(toEndomorphism R L\u2082 M) x)\nthis\u271d\u00b9 : LieRingModule L M := LieRingModule.compLieHom M f\nthis\u271d : LieModule R L M := compLieHom M f\nhnp : \u2200 (x : L), IsNilpotent (\u2191(toEndomorphism R L M) x)\nsurj_id : Surjective \u2191LinearMap.id\nthis : LieModule.IsNilpotent R L M\n\u22a2 LieModule.IsNilpotent R L\u2082 M\n[PROOFSTEP]\napply hf.lieModuleIsNilpotent surj_id\n[GOAL]\ncase hfg\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nhf : Surjective \u2191f\nh : LieAlgebra.IsEngelian R L\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L\u2082 M\n_i4 : LieModule R L\u2082 M\nh' : \u2200 (x : L\u2082), IsNilpotent (\u2191(toEndomorphism R L\u2082 M) x)\nthis\u271d\u00b9 : LieRingModule L M := LieRingModule.compLieHom M f\nthis\u271d : LieModule R L M := compLieHom M f\nhnp : \u2200 (x : L), IsNilpotent (\u2191(toEndomorphism R L M) x)\nsurj_id : Surjective \u2191LinearMap.id\nthis : LieModule.IsNilpotent R L M\n\u22a2 \u2200 (x : L) (m : M), \u2045\u2191f x, \u2191LinearMap.id m\u2046 = \u2191LinearMap.id \u2045x, m\u2046\n[PROOFSTEP]\nintros\n[GOAL]\ncase hfg\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nhf : Surjective \u2191f\nh : LieAlgebra.IsEngelian R L\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L\u2082 M\n_i4 : LieModule R L\u2082 M\nh' : \u2200 (x : L\u2082), IsNilpotent (\u2191(toEndomorphism R L\u2082 M) x)\nthis\u271d\u00b9 : LieRingModule L M := LieRingModule.compLieHom M f\nthis\u271d : LieModule R L M := compLieHom M f\nhnp : \u2200 (x : L), IsNilpotent (\u2191(toEndomorphism R L M) x)\nsurj_id : Surjective \u2191LinearMap.id\nthis : LieModule.IsNilpotent R L M\nx\u271d : L\nm\u271d : M\n\u22a2 \u2045\u2191f x\u271d, \u2191LinearMap.id m\u271d\u2046 = \u2191LinearMap.id \u2045x\u271d, m\u271d\u2046\n[PROOFSTEP]\nsimp only [LinearMap.id_coe, id_eq]\n[GOAL]\ncase hfg\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nhf : Surjective \u2191f\nh : LieAlgebra.IsEngelian R L\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L\u2082 M\n_i4 : LieModule R L\u2082 M\nh' : \u2200 (x : L\u2082), IsNilpotent (\u2191(toEndomorphism R L\u2082 M) x)\nthis\u271d\u00b9 : LieRingModule L M := LieRingModule.compLieHom M f\nthis\u271d : LieModule R L M := compLieHom M f\nhnp : \u2200 (x : L), IsNilpotent (\u2191(toEndomorphism R L M) x)\nsurj_id : Surjective \u2191LinearMap.id\nthis : LieModule.IsNilpotent R L M\nx\u271d : L\nm\u271d : M\n\u22a2 \u2045\u2191f x\u271d, m\u271d\u2046 = \u2045x\u271d, m\u271d\u2046\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\n\u22a2 \u2203 K', IsEngelian R { x // x \u2208 K' } \u2227 K < K'\n[PROOFSTEP]\nobtain \u27e8x, hx\u2081, hx\u2082\u27e9 := SetLike.exists_of_lt hK\u2082\n[GOAL]\ncase intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\n\u22a2 \u2203 K', IsEngelian R { x // x \u2208 K' } \u2227 K < K'\n[PROOFSTEP]\nlet K' : LieSubalgebra R L :=\n { (R \u2219 x) \u2294 (K : Submodule R L) with lie_mem' := fun {y z} => LieSubalgebra.lie_mem_sup_of_mem_normalizer hx\u2081 }\n[GOAL]\ncase intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\nK' : LieSubalgebra R L :=\n let src := Submodule.span R {x} \u2294 K.toSubmodule;\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {y z : L},\n y \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192\n z \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 K.toSubmodule) }\n\u22a2 \u2203 K', IsEngelian R { x // x \u2208 K' } \u2227 K < K'\n[PROOFSTEP]\nhave hxK' : x \u2208 K' := Submodule.mem_sup_left (Submodule.subset_span (Set.mem_singleton _))\n[GOAL]\ncase intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\nK' : LieSubalgebra R L :=\n let src := Submodule.span R {x} \u2294 K.toSubmodule;\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {y z : L},\n y \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192\n z \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 K.toSubmodule) }\nhxK' : x \u2208 K'\n\u22a2 \u2203 K', IsEngelian R { x // x \u2208 K' } \u2227 K < K'\n[PROOFSTEP]\nhave hKK' : K \u2264 K' := (LieSubalgebra.coe_submodule_le_coe_submodule K K').mp le_sup_right\n[GOAL]\ncase intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\nK' : LieSubalgebra R L :=\n let src := Submodule.span R {x} \u2294 K.toSubmodule;\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {y z : L},\n y \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192\n z \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 K.toSubmodule) }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\n\u22a2 \u2203 K', IsEngelian R { x // x \u2208 K' } \u2227 K < K'\n[PROOFSTEP]\nhave hK' : K' \u2264 K.normalizer := by\n rw [\u2190 LieSubalgebra.coe_submodule_le_coe_submodule]\n exact sup_le ((Submodule.span_singleton_le_iff_mem _ _).mpr hx\u2081) hK\u2082.le\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\nK' : LieSubalgebra R L :=\n let src := Submodule.span R {x} \u2294 K.toSubmodule;\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {y z : L},\n y \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192\n z \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 K.toSubmodule) }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\n\u22a2 K' \u2264 LieSubalgebra.normalizer K\n[PROOFSTEP]\nrw [\u2190 LieSubalgebra.coe_submodule_le_coe_submodule]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\nK' : LieSubalgebra R L :=\n let src := Submodule.span R {x} \u2294 K.toSubmodule;\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {y z : L},\n y \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192\n z \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 K.toSubmodule) }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\n\u22a2 K'.toSubmodule \u2264 (LieSubalgebra.normalizer K).toSubmodule\n[PROOFSTEP]\nexact sup_le ((Submodule.span_singleton_le_iff_mem _ _).mpr hx\u2081) hK\u2082.le\n[GOAL]\ncase intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\nK' : LieSubalgebra R L :=\n let src := Submodule.span R {x} \u2294 K.toSubmodule;\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {y z : L},\n y \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192\n z \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 K.toSubmodule) }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 LieSubalgebra.normalizer K\n\u22a2 \u2203 K', IsEngelian R { x // x \u2208 K' } \u2227 K < K'\n[PROOFSTEP]\nrefine' \u27e8K', _, lt_iff_le_and_ne.mpr \u27e8hKK', fun contra => hx\u2082 (contra.symm \u25b8 hxK')\u27e9\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\nK' : LieSubalgebra R L :=\n let src := Submodule.span R {x} \u2294 K.toSubmodule;\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {y z : L},\n y \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192\n z \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 K.toSubmodule) }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 LieSubalgebra.normalizer K\n\u22a2 IsEngelian R { x // x \u2208 K' }\n[PROOFSTEP]\nintro M _i1 _i2 _i3 _i4 h\n[GOAL]\ncase intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\nK' : LieSubalgebra R L :=\n let src := Submodule.span R {x} \u2294 K.toSubmodule;\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {y z : L},\n y \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192\n z \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 K.toSubmodule) }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 LieSubalgebra.normalizer K\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule { x // x \u2208 K' } M\n_i4 : LieModule R { x // x \u2208 K' } M\nh : \u2200 (x : { x // x \u2208 K' }), _root_.IsNilpotent (\u2191(toEndomorphism R { x // x \u2208 K' } M) x)\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 K' } M\n[PROOFSTEP]\nobtain \u27e8I, hI\u2081 : (I : LieSubalgebra R K') = LieSubalgebra.ofLe hKK'\u27e9 :=\n LieSubalgebra.exists_nested_lieIdeal_ofLe_normalizer hKK' hK'\n[GOAL]\ncase intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\nK' : LieSubalgebra R L :=\n let src := Submodule.span R {x} \u2294 K.toSubmodule;\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {y z : L},\n y \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192\n z \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 K.toSubmodule) }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 LieSubalgebra.normalizer K\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule { x // x \u2208 K' } M\n_i4 : LieModule R { x // x \u2208 K' } M\nh : \u2200 (x : { x // x \u2208 K' }), _root_.IsNilpotent (\u2191(toEndomorphism R { x // x \u2208 K' } M) x)\nI : LieIdeal R { x // x \u2208 K' }\nhI\u2081 : \u2191R { x // x \u2208 K' } I = LieSubalgebra.ofLe hKK'\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 K' } M\n[PROOFSTEP]\nhave hI\u2082 : (R \u2219 (\u27e8x, hxK'\u27e9 : K')) \u2294 (LieSubmodule.toSubmodule I) = \u22a4 :=\n by\n rw [\u2190 LieIdeal.coe_to_lieSubalgebra_to_submodule R K' I, hI\u2081]\n apply Submodule.map_injective_of_injective (K' : Submodule R L).injective_subtype\n simp\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\nK' : LieSubalgebra R L :=\n let src := Submodule.span R {x} \u2294 K.toSubmodule;\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {y z : L},\n y \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192\n z \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 K.toSubmodule) }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 LieSubalgebra.normalizer K\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule { x // x \u2208 K' } M\n_i4 : LieModule R { x // x \u2208 K' } M\nh : \u2200 (x : { x // x \u2208 K' }), _root_.IsNilpotent (\u2191(toEndomorphism R { x // x \u2208 K' } M) x)\nI : LieIdeal R { x // x \u2208 K' }\nhI\u2081 : \u2191R { x // x \u2208 K' } I = LieSubalgebra.ofLe hKK'\n\u22a2 Submodule.span R {{ val := x, property := hxK' }} \u2294 \u2191I = \u22a4\n[PROOFSTEP]\nrw [\u2190 LieIdeal.coe_to_lieSubalgebra_to_submodule R K' I, hI\u2081]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\nK' : LieSubalgebra R L :=\n let src := Submodule.span R {x} \u2294 K.toSubmodule;\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {y z : L},\n y \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192\n z \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 K.toSubmodule) }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 LieSubalgebra.normalizer K\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule { x // x \u2208 K' } M\n_i4 : LieModule R { x // x \u2208 K' } M\nh : \u2200 (x : { x // x \u2208 K' }), _root_.IsNilpotent (\u2191(toEndomorphism R { x // x \u2208 K' } M) x)\nI : LieIdeal R { x // x \u2208 K' }\nhI\u2081 : \u2191R { x // x \u2208 K' } I = LieSubalgebra.ofLe hKK'\n\u22a2 Submodule.span R {{ val := x, property := hxK' }} \u2294 (LieSubalgebra.ofLe hKK').toSubmodule = \u22a4\n[PROOFSTEP]\napply Submodule.map_injective_of_injective (K' : Submodule R L).injective_subtype\n[GOAL]\ncase a\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\nK' : LieSubalgebra R L :=\n let src := Submodule.span R {x} \u2294 K.toSubmodule;\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {y z : L},\n y \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192\n z \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 K.toSubmodule) }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 LieSubalgebra.normalizer K\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule { x // x \u2208 K' } M\n_i4 : LieModule R { x // x \u2208 K' } M\nh : \u2200 (x : { x // x \u2208 K' }), _root_.IsNilpotent (\u2191(toEndomorphism R { x // x \u2208 K' } M) x)\nI : LieIdeal R { x // x \u2208 K' }\nhI\u2081 : \u2191R { x // x \u2208 K' } I = LieSubalgebra.ofLe hKK'\n\u22a2 Submodule.map (Submodule.subtype K'.toSubmodule)\n (Submodule.span R {{ val := x, property := hxK' }} \u2294 (LieSubalgebra.ofLe hKK').toSubmodule) =\n Submodule.map (Submodule.subtype K'.toSubmodule) \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\nK' : LieSubalgebra R L :=\n let src := Submodule.span R {x} \u2294 K.toSubmodule;\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {y z : L},\n y \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192\n z \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 K.toSubmodule) }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 LieSubalgebra.normalizer K\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule { x // x \u2208 K' } M\n_i4 : LieModule R { x // x \u2208 K' } M\nh : \u2200 (x : { x // x \u2208 K' }), _root_.IsNilpotent (\u2191(toEndomorphism R { x // x \u2208 K' } M) x)\nI : LieIdeal R { x // x \u2208 K' }\nhI\u2081 : \u2191R { x // x \u2208 K' } I = LieSubalgebra.ofLe hKK'\nhI\u2082 : Submodule.span R {{ val := x, property := hxK' }} \u2294 \u2191I = \u22a4\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 K' } M\n[PROOFSTEP]\nhave e : K \u2243\u2097\u2045R\u2046 I :=\n (LieSubalgebra.equivOfLe hKK').trans (LieEquiv.ofEq _ _ ((LieSubalgebra.coe_set_eq _ _).mpr hI\u2081.symm))\n[GOAL]\ncase intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\nK' : LieSubalgebra R L :=\n let src := Submodule.span R {x} \u2294 K.toSubmodule;\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {y z : L},\n y \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192\n z \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 K.toSubmodule) }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 LieSubalgebra.normalizer K\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule { x // x \u2208 K' } M\n_i4 : LieModule R { x // x \u2208 K' } M\nh : \u2200 (x : { x // x \u2208 K' }), _root_.IsNilpotent (\u2191(toEndomorphism R { x // x \u2208 K' } M) x)\nI : LieIdeal R { x // x \u2208 K' }\nhI\u2081 : \u2191R { x // x \u2208 K' } I = LieSubalgebra.ofLe hKK'\nhI\u2082 : Submodule.span R {{ val := x, property := hxK' }} \u2294 \u2191I = \u22a4\ne : { x // x \u2208 K } \u2243\u2097\u2045R\u2046 { x // x \u2208 \u2191I }\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 K' } M\n[PROOFSTEP]\nhave hI\u2083 : LieAlgebra.IsEngelian R I := e.isEngelian_iff.mp hK\u2081\n[GOAL]\ncase intro.intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\ninst\u271d\u00b9 : LieRingModule L M\u271d\ninst\u271d : LieModule R L M\u271d\nK : LieSubalgebra R L\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K < LieSubalgebra.normalizer K\nx : L\nhx\u2081 : x \u2208 LieSubalgebra.normalizer K\nhx\u2082 : \u00acx \u2208 K\nK' : LieSubalgebra R L :=\n let src := Submodule.span R {x} \u2294 K.toSubmodule;\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' := (_ : \u2200 (c : R) {x : L}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {y z : L},\n y \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192\n z \u2208 Submodule.span R {x} \u2294 K.toSubmodule \u2192 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 K.toSubmodule) }\nhxK' : x \u2208 K'\nhKK' : K \u2264 K'\nhK' : K' \u2264 LieSubalgebra.normalizer K\nM : Type u\u2084\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule { x // x \u2208 K' } M\n_i4 : LieModule R { x // x \u2208 K' } M\nh : \u2200 (x : { x // x \u2208 K' }), _root_.IsNilpotent (\u2191(toEndomorphism R { x // x \u2208 K' } M) x)\nI : LieIdeal R { x // x \u2208 K' }\nhI\u2081 : \u2191R { x // x \u2208 K' } I = LieSubalgebra.ofLe hKK'\nhI\u2082 : Submodule.span R {{ val := x, property := hxK' }} \u2294 \u2191I = \u22a4\ne : { x // x \u2208 K } \u2243\u2097\u2045R\u2046 { x // x \u2208 \u2191I }\nhI\u2083 : IsEngelian R { x // x \u2208 \u2191I }\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 K' } M\n[PROOFSTEP]\nexact LieSubmodule.isNilpotentOfIsNilpotentSpanSupEqTop hI\u2082 (h _) (hI\u2083 _ fun x => h x)\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : LieRingModule L M\ninst\u271d\u00b9 : LieModule R L M\ninst\u271d : IsNoetherian R L\n\u22a2 IsEngelian R L\n[PROOFSTEP]\nintro M _i1 _i2 _i3 _i4 h\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nh : \u2200 (x : L), _root_.IsNilpotent (\u2191(toEndomorphism R L M) x)\n\u22a2 LieModule.IsNilpotent R L M\n[PROOFSTEP]\nrw [\u2190 isNilpotent_range_toEndomorphism_iff]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nh : \u2200 (x : L), _root_.IsNilpotent (\u2191(toEndomorphism R L M) x)\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 LieHom.range (toEndomorphism R L M) } M\n[PROOFSTEP]\nlet L' := (toEndomorphism R L M).range\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nh : \u2200 (x : L), _root_.IsNilpotent (\u2191(toEndomorphism R L M) x)\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 LieHom.range (toEndomorphism R L M) } M\n[PROOFSTEP]\nreplace h : \u2200 y : L', _root_.IsNilpotent (y : Module.End R M)\n[GOAL]\ncase h\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nh : \u2200 (x : L), _root_.IsNilpotent (\u2191(toEndomorphism R L M) x)\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\n\u22a2 \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\n[PROOFSTEP]\nrintro \u27e8-, \u27e8y, rfl\u27e9\u27e9\n[GOAL]\ncase h.mk.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nh : \u2200 (x : L), _root_.IsNilpotent (\u2191(toEndomorphism R L M) x)\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\ny : L\n\u22a2 _root_.IsNilpotent\n \u2191{ val := \u2191\u2191(toEndomorphism R L M) y,\n property := (_ : \u2203 y_1, \u2191\u2191(toEndomorphism R L M) y_1 = \u2191\u2191(toEndomorphism R L M) y) }\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 LieHom.range (toEndomorphism R L M) } M\n[PROOFSTEP]\nchange LieModule.IsNilpotent R L' M\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 L' } M\n[PROOFSTEP]\nlet s := {K : LieSubalgebra R L' | LieAlgebra.IsEngelian R K}\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 L' } M\n[PROOFSTEP]\nhave hs : s.Nonempty := \u27e8\u22a5, LieAlgebra.isEngelian_of_subsingleton\u27e9\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 L' } M\n[PROOFSTEP]\nsuffices \u22a4 \u2208 s by\n rw [\u2190 isNilpotent_of_top_iff]\n apply this M\n simp [LieSubalgebra.toEndomorphism_eq, h]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nthis : \u22a4 \u2208 s\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 L' } M\n[PROOFSTEP]\nrw [\u2190 isNilpotent_of_top_iff]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nthis : \u22a4 \u2208 s\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 \u22a4 } M\n[PROOFSTEP]\napply this M\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nthis : \u22a4 \u2208 s\n\u22a2 \u2200 (x : { x // x \u2208 \u22a4 }), _root_.IsNilpotent (\u2191(toEndomorphism R { x // x \u2208 \u22a4 } M) x)\n[PROOFSTEP]\nsimp [LieSubalgebra.toEndomorphism_eq, h]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\n\u22a2 \u22a4 \u2208 s\n[PROOFSTEP]\nhave : \u2200 K \u2208 s, K \u2260 \u22a4 \u2192 \u2203 K' \u2208 s, K < K' :=\n by\n rintro K (hK\u2081 : LieAlgebra.IsEngelian R K) hK\u2082\n apply LieAlgebra.exists_engelian_lieSubalgebra_of_lt_normalizer hK\u2081\n apply lt_of_le_of_ne K.le_normalizer\n rw [Ne.def, eq_comm, K.normalizer_eq_self_iff, \u2190 Ne.def, \u2190 LieSubmodule.nontrivial_iff_ne_bot R K]\n have : Nontrivial (L' \u29f8 K.toLieSubmodule) :=\n by\n replace hK\u2082 : K.toLieSubmodule \u2260 \u22a4 := by\n rwa [Ne.def, \u2190 LieSubmodule.coe_toSubmodule_eq_iff, K.coe_toLieSubmodule, LieSubmodule.top_coeSubmodule, \u2190\n LieSubalgebra.top_coe_submodule, K.coe_to_submodule_eq_iff]\n exact Submodule.Quotient.nontrivial_of_lt_top _ hK\u2082.lt_top\n have : LieModule.IsNilpotent R K (L' \u29f8 K.toLieSubmodule) := by\n -- Porting note: was refine' hK\u2081 _ fun x => _\n apply hK\u2081\n intro x\n have hx := LieAlgebra.isNilpotent_ad_of_isNilpotent (h x)\n apply Module.End.IsNilpotent.mapQ ?_ hx\n intro X HX\n simp only [LieSubalgebra.coe_toLieSubmodule, LieSubalgebra.mem_coe_submodule] at HX \n simp only [LieSubalgebra.coe_toLieSubmodule, Submodule.mem_comap, ad_apply, LieSubalgebra.mem_coe_submodule]\n exact LieSubalgebra.lie_mem K x.prop HX\n exact nontrivial_max_triv_of_isNilpotent R K (L' \u29f8 K.toLieSubmodule)\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\n\u22a2 \u2200 (K : LieSubalgebra R { x // x \u2208 L' }), K \u2208 s \u2192 K \u2260 \u22a4 \u2192 \u2203 K', K' \u2208 s \u2227 K < K'\n[PROOFSTEP]\nrintro K (hK\u2081 : LieAlgebra.IsEngelian R K) hK\u2082\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\n\u22a2 \u2203 K', K' \u2208 s \u2227 K < K'\n[PROOFSTEP]\napply LieAlgebra.exists_engelian_lieSubalgebra_of_lt_normalizer hK\u2081\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\n\u22a2 K < LieSubalgebra.normalizer K\n[PROOFSTEP]\napply lt_of_le_of_ne K.le_normalizer\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\n\u22a2 K \u2260 LieSubalgebra.normalizer K\n[PROOFSTEP]\nrw [Ne.def, eq_comm, K.normalizer_eq_self_iff, \u2190 Ne.def, \u2190 LieSubmodule.nontrivial_iff_ne_bot R K]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\n\u22a2 Nontrivial { x // x \u2208 \u2191(maxTrivSubmodule R { x // x \u2208 K } ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)) }\n[PROOFSTEP]\nhave : Nontrivial (L' \u29f8 K.toLieSubmodule) :=\n by\n replace hK\u2082 : K.toLieSubmodule \u2260 \u22a4 := by\n rwa [Ne.def, \u2190 LieSubmodule.coe_toSubmodule_eq_iff, K.coe_toLieSubmodule, LieSubmodule.top_coeSubmodule, \u2190\n LieSubalgebra.top_coe_submodule, K.coe_to_submodule_eq_iff]\n exact Submodule.Quotient.nontrivial_of_lt_top _ hK\u2082.lt_top\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\n\u22a2 Nontrivial ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)\n[PROOFSTEP]\nreplace hK\u2082 : K.toLieSubmodule \u2260 \u22a4 := by\n rwa [Ne.def, \u2190 LieSubmodule.coe_toSubmodule_eq_iff, K.coe_toLieSubmodule, LieSubmodule.top_coeSubmodule, \u2190\n LieSubalgebra.top_coe_submodule, K.coe_to_submodule_eq_iff]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\n\u22a2 LieSubalgebra.toLieSubmodule K \u2260 \u22a4\n[PROOFSTEP]\nrwa [Ne.def, \u2190 LieSubmodule.coe_toSubmodule_eq_iff, K.coe_toLieSubmodule, LieSubmodule.top_coeSubmodule, \u2190\n LieSubalgebra.top_coe_submodule, K.coe_to_submodule_eq_iff]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : LieSubalgebra.toLieSubmodule K \u2260 \u22a4\n\u22a2 Nontrivial ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)\n[PROOFSTEP]\nexact Submodule.Quotient.nontrivial_of_lt_top _ hK\u2082.lt_top\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\nthis : Nontrivial ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)\n\u22a2 Nontrivial { x // x \u2208 \u2191(maxTrivSubmodule R { x // x \u2208 K } ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)) }\n[PROOFSTEP]\nhave : LieModule.IsNilpotent R K (L' \u29f8 K.toLieSubmodule) := by\n -- Porting note: was refine' hK\u2081 _ fun x => _\n apply hK\u2081\n intro x\n have hx := LieAlgebra.isNilpotent_ad_of_isNilpotent (h x)\n apply Module.End.IsNilpotent.mapQ ?_ hx\n intro X HX\n simp only [LieSubalgebra.coe_toLieSubmodule, LieSubalgebra.mem_coe_submodule] at HX \n simp only [LieSubalgebra.coe_toLieSubmodule, Submodule.mem_comap, ad_apply, LieSubalgebra.mem_coe_submodule]\n exact LieSubalgebra.lie_mem K x.prop HX\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\nthis : Nontrivial ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 K } ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)\n[PROOFSTEP]\napply hK\u2081\n[GOAL]\ncase x\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\nthis : Nontrivial ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)\n\u22a2 \u2200 (x : { x // x \u2208 K }),\n _root_.IsNilpotent (\u2191(toEndomorphism R { x // x \u2208 K } ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)) x)\n[PROOFSTEP]\nintro x\n[GOAL]\ncase x\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\nthis : Nontrivial ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)\nx : { x // x \u2208 K }\n\u22a2 _root_.IsNilpotent (\u2191(toEndomorphism R { x // x \u2208 K } ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)) x)\n[PROOFSTEP]\nhave hx := LieAlgebra.isNilpotent_ad_of_isNilpotent (h x)\n[GOAL]\ncase x\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\nthis : Nontrivial ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)\nx : { x // x \u2208 K }\nhx : _root_.IsNilpotent (\u2191(ad R { x // x \u2208 L' }) \u2191x)\n\u22a2 _root_.IsNilpotent (\u2191(toEndomorphism R { x // x \u2208 K } ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)) x)\n[PROOFSTEP]\napply Module.End.IsNilpotent.mapQ ?_ hx\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\nthis : Nontrivial ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)\nx : { x // x \u2208 K }\nhx : _root_.IsNilpotent (\u2191(ad R { x // x \u2208 L' }) \u2191x)\n\u22a2 \u2191(LieSubalgebra.toLieSubmodule K) \u2264 Submodule.comap (\u2191(ad R { x // x \u2208 L' }) \u2191x) \u2191(LieSubalgebra.toLieSubmodule K)\n[PROOFSTEP]\nintro X HX\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\nthis : Nontrivial ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)\nx : { x // x \u2208 K }\nhx : _root_.IsNilpotent (\u2191(ad R { x // x \u2208 L' }) \u2191x)\nX : { x // x \u2208 L' }\nHX : X \u2208 \u2191(LieSubalgebra.toLieSubmodule K)\n\u22a2 X \u2208 Submodule.comap (\u2191(ad R { x // x \u2208 L' }) \u2191x) \u2191(LieSubalgebra.toLieSubmodule K)\n[PROOFSTEP]\nsimp only [LieSubalgebra.coe_toLieSubmodule, LieSubalgebra.mem_coe_submodule] at HX \n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\nthis : Nontrivial ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)\nx : { x // x \u2208 K }\nhx : _root_.IsNilpotent (\u2191(ad R { x // x \u2208 L' }) \u2191x)\nX : { x // x \u2208 L' }\nHX : X \u2208 K\n\u22a2 X \u2208 Submodule.comap (\u2191(ad R { x // x \u2208 L' }) \u2191x) \u2191(LieSubalgebra.toLieSubmodule K)\n[PROOFSTEP]\nsimp only [LieSubalgebra.coe_toLieSubmodule, Submodule.mem_comap, ad_apply, LieSubalgebra.mem_coe_submodule]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\nthis : Nontrivial ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)\nx : { x // x \u2208 K }\nhx : _root_.IsNilpotent (\u2191(ad R { x // x \u2208 L' }) \u2191x)\nX : { x // x \u2208 L' }\nHX : X \u2208 K\n\u22a2 \u2045\u2191x, X\u2046 \u2208 K\n[PROOFSTEP]\nexact LieSubalgebra.lie_mem K x.prop HX\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : IsEngelian R { x // x \u2208 K }\nhK\u2082 : K \u2260 \u22a4\nthis\u271d : Nontrivial ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)\nthis : LieModule.IsNilpotent R { x // x \u2208 K } ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)\n\u22a2 Nontrivial { x // x \u2208 \u2191(maxTrivSubmodule R { x // x \u2208 K } ({ x // x \u2208 L' } \u29f8 LieSubalgebra.toLieSubmodule K)) }\n[PROOFSTEP]\nexact nontrivial_max_triv_of_isNilpotent R K (L' \u29f8 K.toLieSubmodule)\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nthis : \u2200 (K : LieSubalgebra R { x // x \u2208 L' }), K \u2208 s \u2192 K \u2260 \u22a4 \u2192 \u2203 K', K' \u2208 s \u2227 K < K'\n\u22a2 \u22a4 \u2208 s\n[PROOFSTEP]\nhaveI _i5 : IsNoetherian R L' := by\n -- Porting note: was\n -- isNoetherian_of_surjective L _ (LinearMap.range_rangeRestrict (toEndomorphism R L M))\n -- abusing the relation between `LieHom.rangeRestrict` and `LinearMap.rangeRestrict`\n refine isNoetherian_of_surjective L (LieHom.rangeRestrict (toEndomorphism R L M)) ?_\n simp only [LieHom.range_coeSubmodule, LieHom.coe_toLinearMap, LinearMap.range_eq_top]\n exact LieHom.surjective_rangeRestrict (toEndomorphism R L M)\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nthis : \u2200 (K : LieSubalgebra R { x // x \u2208 L' }), K \u2208 s \u2192 K \u2260 \u22a4 \u2192 \u2203 K', K' \u2208 s \u2227 K < K'\n\u22a2 IsNoetherian R { x // x \u2208 L' }\n[PROOFSTEP]\nrefine isNoetherian_of_surjective L (LieHom.rangeRestrict (toEndomorphism R L M)) ?_\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nthis : \u2200 (K : LieSubalgebra R { x // x \u2208 L' }), K \u2208 s \u2192 K \u2260 \u22a4 \u2192 \u2203 K', K' \u2208 s \u2227 K < K'\n\u22a2 LinearMap.range \u2191(LieHom.rangeRestrict (toEndomorphism R L M)) = \u22a4\n[PROOFSTEP]\nsimp only [LieHom.range_coeSubmodule, LieHom.coe_toLinearMap, LinearMap.range_eq_top]\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nthis : \u2200 (K : LieSubalgebra R { x // x \u2208 L' }), K \u2208 s \u2192 K \u2260 \u22a4 \u2192 \u2203 K', K' \u2208 s \u2227 K < K'\n\u22a2 Function.Surjective \u2191(LieHom.rangeRestrict (toEndomorphism R L M))\n[PROOFSTEP]\nexact LieHom.surjective_rangeRestrict (toEndomorphism R L M)\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nthis : \u2200 (K : LieSubalgebra R { x // x \u2208 L' }), K \u2208 s \u2192 K \u2260 \u22a4 \u2192 \u2203 K', K' \u2208 s \u2227 K < K'\n_i5 : IsNoetherian R { x // x \u2208 L' }\n\u22a2 \u22a4 \u2208 s\n[PROOFSTEP]\nobtain \u27e8K, hK\u2081, hK\u2082\u27e9 := (LieSubalgebra.wellFounded_of_noetherian R L').has_min s hs\n[GOAL]\ncase intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nthis : \u2200 (K : LieSubalgebra R { x // x \u2208 L' }), K \u2208 s \u2192 K \u2260 \u22a4 \u2192 \u2203 K', K' \u2208 s \u2227 K < K'\n_i5 : IsNoetherian R { x // x \u2208 L' }\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : K \u2208 s\nhK\u2082 : \u2200 (x : LieSubalgebra R { x // x \u2208 L' }), x \u2208 s \u2192 \u00acx > K\n\u22a2 \u22a4 \u2208 s\n[PROOFSTEP]\nhave hK\u2083 : K = \u22a4 := by\n by_contra contra\n obtain \u27e8K', hK'\u2081, hK'\u2082\u27e9 := this K hK\u2081 contra\n exact hK\u2082 K' hK'\u2081 hK'\u2082\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nthis : \u2200 (K : LieSubalgebra R { x // x \u2208 L' }), K \u2208 s \u2192 K \u2260 \u22a4 \u2192 \u2203 K', K' \u2208 s \u2227 K < K'\n_i5 : IsNoetherian R { x // x \u2208 L' }\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : K \u2208 s\nhK\u2082 : \u2200 (x : LieSubalgebra R { x // x \u2208 L' }), x \u2208 s \u2192 \u00acx > K\n\u22a2 K = \u22a4\n[PROOFSTEP]\nby_contra contra\n[GOAL]\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nthis : \u2200 (K : LieSubalgebra R { x // x \u2208 L' }), K \u2208 s \u2192 K \u2260 \u22a4 \u2192 \u2203 K', K' \u2208 s \u2227 K < K'\n_i5 : IsNoetherian R { x // x \u2208 L' }\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : K \u2208 s\nhK\u2082 : \u2200 (x : LieSubalgebra R { x // x \u2208 L' }), x \u2208 s \u2192 \u00acx > K\ncontra : \u00acK = \u22a4\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8K', hK'\u2081, hK'\u2082\u27e9 := this K hK\u2081 contra\n[GOAL]\ncase intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nthis : \u2200 (K : LieSubalgebra R { x // x \u2208 L' }), K \u2208 s \u2192 K \u2260 \u22a4 \u2192 \u2203 K', K' \u2208 s \u2227 K < K'\n_i5 : IsNoetherian R { x // x \u2208 L' }\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : K \u2208 s\nhK\u2082 : \u2200 (x : LieSubalgebra R { x // x \u2208 L' }), x \u2208 s \u2192 \u00acx > K\ncontra : \u00acK = \u22a4\nK' : LieSubalgebra R { x // x \u2208 L' }\nhK'\u2081 : K' \u2208 s\nhK'\u2082 : K < K'\n\u22a2 False\n[PROOFSTEP]\nexact hK\u2082 K' hK'\u2081 hK'\u2082\n[GOAL]\ncase intro.intro\nR : Type u\u2081\nL : Type u\u2082\nL\u2082 : Type u\u2083\nM\u271d : Type u\u2084\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\ninst\u271d\u2076 : LieRing L\u2082\ninst\u271d\u2075 : LieAlgebra R L\u2082\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\ninst\u271d\u00b2 : LieRingModule L M\u271d\ninst\u271d\u00b9 : LieModule R L M\u271d\ninst\u271d : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)\nh : \u2200 (y : { x // x \u2208 L' }), _root_.IsNilpotent \u2191y\ns : Set (LieSubalgebra R { x // x \u2208 L' }) := {K | IsEngelian R { x // x \u2208 K }}\nhs : Set.Nonempty s\nthis : \u2200 (K : LieSubalgebra R { x // x \u2208 L' }), K \u2208 s \u2192 K \u2260 \u22a4 \u2192 \u2203 K', K' \u2208 s \u2227 K < K'\n_i5 : IsNoetherian R { x // x \u2208 L' }\nK : LieSubalgebra R { x // x \u2208 L' }\nhK\u2081 : K \u2208 s\nhK\u2082 : \u2200 (x : LieSubalgebra R { x // x \u2208 L' }), x \u2208 s \u2192 \u00acx > K\nhK\u2083 : K = \u22a4\n\u22a2 \u22a4 \u2208 s\n[PROOFSTEP]\nexact hK\u2083 \u25b8 hK\u2081\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Lie.Engel", "llama_tokens": 45509, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506526772884, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.4188198519633029}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasFiniteProducts C\n\u03b9 : Type w\ninst\u271d : Finite \u03b9\n\u22a2 HasLimitsOfShape (Discrete \u03b9) C\n[PROOFSTEP]\nrcases Finite.exists_equiv_fin \u03b9 with \u27e8n, \u27e8e\u27e9\u27e9\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasFiniteProducts C\n\u03b9 : Type w\ninst\u271d : Finite \u03b9\nn : \u2115\ne : \u03b9 \u2243 Fin n\n\u22a2 HasLimitsOfShape (Discrete \u03b9) C\n[PROOFSTEP]\nhaveI : HasLimitsOfShape (Discrete (Fin n)) C := HasFiniteProducts.out n\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasFiniteProducts C\n\u03b9 : Type w\ninst\u271d : Finite \u03b9\nn : \u2115\ne : \u03b9 \u2243 Fin n\nthis : HasLimitsOfShape (Discrete (Fin n)) C\n\u22a2 HasLimitsOfShape (Discrete \u03b9) C\n[PROOFSTEP]\nexact hasLimitsOfShape_of_equivalence (Discrete.equivalence e.symm)\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\n\u03b9 : Type w\ninst\u271d : Finite \u03b9\n\u22a2 HasColimitsOfShape (Discrete \u03b9) C\n[PROOFSTEP]\nrcases Finite.exists_equiv_fin \u03b9 with \u27e8n, \u27e8e\u27e9\u27e9\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\n\u03b9 : Type w\ninst\u271d : Finite \u03b9\nn : \u2115\ne : \u03b9 \u2243 Fin n\n\u22a2 HasColimitsOfShape (Discrete \u03b9) C\n[PROOFSTEP]\nhaveI : HasColimitsOfShape (Discrete (Fin n)) C := HasFiniteCoproducts.out n\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\n\u03b9 : Type w\ninst\u271d : Finite \u03b9\nn : \u2115\ne : \u03b9 \u2243 Fin n\nthis : HasColimitsOfShape (Discrete (Fin n)) C\n\u22a2 HasColimitsOfShape (Discrete \u03b9) C\n[PROOFSTEP]\nexact hasColimitsOfShape_of_equivalence (Discrete.equivalence e.symm)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasFiniteColimits C\nJ : \u2115\n\u22a2 HasColimitsOfShape (Discrete (Fin J)) C\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts", "llama_tokens": 789, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802735722128, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.4188021109546442}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhs : s \u2208 K.faces\n\u22a2 \u2191(convexHull \ud835\udd5c) \u2191s \u2286 space K\n[PROOFSTEP]\nconvert subset_biUnion_of_mem hs\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhs : s \u2208 K.faces\n\u22a2 \u2191(convexHull \ud835\udd5c) \u2191s = \u2191(convexHull \ud835\udd5c) \u2191s\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhs : s \u2208 K.faces\nht : t \u2208 K.faces\n\u22a2 Disjoint (\u2191(convexHull \ud835\udd5c) \u2191s) (\u2191(convexHull \ud835\udd5c) \u2191t) \u2228\n \u2203 u, u \u2208 K.faces \u2227 \u2191(convexHull \ud835\udd5c) \u2191s \u2229 \u2191(convexHull \ud835\udd5c) \u2191t = \u2191(convexHull \ud835\udd5c) \u2191u\n[PROOFSTEP]\nclassical\nby_contra' h\nrefine'\n h.2 (s \u2229 t)\n (K.down_closed hs (inter_subset_left _ _) fun hst =>\n h.1 <| disjoint_iff_inf_le.mpr <| (K.inter_subset_convexHull hs ht).trans _)\n _\n\u00b7 rw [\u2190 coe_inter, hst, coe_empty, convexHull_empty]\n rfl\n\u00b7 rw [coe_inter, convexHull_inter_convexHull hs ht]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhs : s \u2208 K.faces\nht : t \u2208 K.faces\n\u22a2 Disjoint (\u2191(convexHull \ud835\udd5c) \u2191s) (\u2191(convexHull \ud835\udd5c) \u2191t) \u2228\n \u2203 u, u \u2208 K.faces \u2227 \u2191(convexHull \ud835\udd5c) \u2191s \u2229 \u2191(convexHull \ud835\udd5c) \u2191t = \u2191(convexHull \ud835\udd5c) \u2191u\n[PROOFSTEP]\nby_contra' h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhs : s \u2208 K.faces\nht : t \u2208 K.faces\nh :\n \u00acDisjoint (\u2191(convexHull \ud835\udd5c) \u2191s) (\u2191(convexHull \ud835\udd5c) \u2191t) \u2227\n \u2200 (u : Finset E), u \u2208 K.faces \u2192 \u2191(convexHull \ud835\udd5c) \u2191s \u2229 \u2191(convexHull \ud835\udd5c) \u2191t \u2260 \u2191(convexHull \ud835\udd5c) \u2191u\n\u22a2 False\n[PROOFSTEP]\nrefine'\n h.2 (s \u2229 t)\n (K.down_closed hs (inter_subset_left _ _) fun hst =>\n h.1 <| disjoint_iff_inf_le.mpr <| (K.inter_subset_convexHull hs ht).trans _)\n _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhs : s \u2208 K.faces\nht : t \u2208 K.faces\nh :\n \u00acDisjoint (\u2191(convexHull \ud835\udd5c) \u2191s) (\u2191(convexHull \ud835\udd5c) \u2191t) \u2227\n \u2200 (u : Finset E), u \u2208 K.faces \u2192 \u2191(convexHull \ud835\udd5c) \u2191s \u2229 \u2191(convexHull \ud835\udd5c) \u2191t \u2260 \u2191(convexHull \ud835\udd5c) \u2191u\nhst : s \u2229 t = \u2205\n\u22a2 \u2191(convexHull \ud835\udd5c) (\u2191s \u2229 \u2191t) \u2286 \u22a5\n[PROOFSTEP]\nrw [\u2190 coe_inter, hst, coe_empty, convexHull_empty]\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhs : s \u2208 K.faces\nht : t \u2208 K.faces\nh :\n \u00acDisjoint (\u2191(convexHull \ud835\udd5c) \u2191s) (\u2191(convexHull \ud835\udd5c) \u2191t) \u2227\n \u2200 (u : Finset E), u \u2208 K.faces \u2192 \u2191(convexHull \ud835\udd5c) \u2191s \u2229 \u2191(convexHull \ud835\udd5c) \u2191t \u2260 \u2191(convexHull \ud835\udd5c) \u2191u\nhst : s \u2229 t = \u2205\n\u22a2 \u2205 \u2286 \u22a5\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhs : s \u2208 K.faces\nht : t \u2208 K.faces\nh :\n \u00acDisjoint (\u2191(convexHull \ud835\udd5c) \u2191s) (\u2191(convexHull \ud835\udd5c) \u2191t) \u2227\n \u2200 (u : Finset E), u \u2208 K.faces \u2192 \u2191(convexHull \ud835\udd5c) \u2191s \u2229 \u2191(convexHull \ud835\udd5c) \u2191t \u2260 \u2191(convexHull \ud835\udd5c) \u2191u\n\u22a2 \u2191(convexHull \ud835\udd5c) \u2191s \u2229 \u2191(convexHull \ud835\udd5c) \u2191t = \u2191(convexHull \ud835\udd5c) \u2191(s \u2229 t)\n[PROOFSTEP]\nrw [coe_inter, convexHull_inter_convexHull hs ht]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\n\u22a2 vertices K = \u22c3 (k : Finset E) (_ : k \u2208 K.faces), \u2191k\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx\u271d x : E\n\u22a2 x \u2208 vertices K \u2194 x \u2208 \u22c3 (k : Finset E) (_ : k \u2208 K.faces), \u2191k\n[PROOFSTEP]\nrefine' \u27e8fun h => mem_biUnion h <| mem_coe.2 <| mem_singleton_self x, fun h => _\u27e9\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx\u271d x : E\nh : x \u2208 \u22c3 (k : Finset E) (_ : k \u2208 K.faces), \u2191k\n\u22a2 x \u2208 vertices K\n[PROOFSTEP]\nobtain \u27e8s, hs, hx\u27e9 := mem_iUnion\u2082.1 h\n[GOAL]\ncase h.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns\u271d t : Finset E\nx\u271d x : E\nh : x \u2208 \u22c3 (k : Finset E) (_ : k \u2208 K.faces), \u2191k\ns : Finset E\nhs : s \u2208 K.faces\nhx : x \u2208 \u2191s\n\u22a2 x \u2208 vertices K\n[PROOFSTEP]\nexact K.down_closed hs (Finset.singleton_subset_iff.2 <| mem_coe.1 hx) (singleton_ne_empty _)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhx : x \u2208 vertices K\nhs : s \u2208 K.faces\n\u22a2 x \u2208 \u2191(convexHull \ud835\udd5c) \u2191s \u2194 x \u2208 s\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => subset_convexHull \ud835\udd5c _ h\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhx : x \u2208 vertices K\nhs : s \u2208 K.faces\nh : x \u2208 \u2191(convexHull \ud835\udd5c) \u2191s\n\u22a2 x \u2208 s\n[PROOFSTEP]\nclassical\nhave h := K.inter_subset_convexHull hx hs \u27e8by simp, h\u27e9\nby_contra H\nrwa [\u2190 coe_inter, Finset.disjoint_iff_inter_eq_empty.1 (Finset.disjoint_singleton_right.2 H).symm, coe_empty,\n convexHull_empty] at h \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhx : x \u2208 vertices K\nhs : s \u2208 K.faces\nh : x \u2208 \u2191(convexHull \ud835\udd5c) \u2191s\n\u22a2 x \u2208 s\n[PROOFSTEP]\nhave h := K.inter_subset_convexHull hx hs \u27e8by simp, h\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhx : x \u2208 vertices K\nhs : s \u2208 K.faces\nh : x \u2208 \u2191(convexHull \ud835\udd5c) \u2191s\n\u22a2 x \u2208 \u2191(convexHull \ud835\udd5c) \u2191{x}\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhx : x \u2208 vertices K\nhs : s \u2208 K.faces\nh\u271d : x \u2208 \u2191(convexHull \ud835\udd5c) \u2191s\nh : x \u2208 \u2191(convexHull \ud835\udd5c) (\u2191{x} \u2229 \u2191s)\n\u22a2 x \u2208 s\n[PROOFSTEP]\nby_contra H\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhx : x \u2208 vertices K\nhs : s \u2208 K.faces\nh\u271d : x \u2208 \u2191(convexHull \ud835\udd5c) \u2191s\nh : x \u2208 \u2191(convexHull \ud835\udd5c) (\u2191{x} \u2229 \u2191s)\nH : \u00acx \u2208 s\n\u22a2 False\n[PROOFSTEP]\nrwa [\u2190 coe_inter, Finset.disjoint_iff_inter_eq_empty.1 (Finset.disjoint_singleton_right.2 H).symm, coe_empty,\n convexHull_empty] at h \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhs : s \u2208 K.faces\n\u22a2 \u00acs \u2208 facets K \u2194 \u2203 t, t \u2208 K.faces \u2227 s \u2282 t\n[PROOFSTEP]\nrefine' \u27e8fun hs' : \u00ac(_ \u2227 _) => _, _\u27e9\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhs : s \u2208 K.faces\nhs' : \u00ac(s \u2208 K.faces \u2227 \u2200 \u2983t : Finset E\u2984, t \u2208 K.faces \u2192 s \u2286 t \u2192 s = t)\n\u22a2 \u2203 t, t \u2208 K.faces \u2227 s \u2282 t\n[PROOFSTEP]\npush_neg at hs' \n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhs : s \u2208 K.faces\nhs' : s \u2208 K.faces \u2192 Exists fun \u2983t\u2984 => t \u2208 K.faces \u2227 s \u2286 t \u2227 s \u2260 t\n\u22a2 \u2203 t, t \u2208 K.faces \u2227 s \u2282 t\n[PROOFSTEP]\nobtain \u27e8t, ht\u27e9 := hs' hs\n[GOAL]\ncase refine'_1.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t\u271d : Finset E\nx : E\nhs : s \u2208 K.faces\nhs' : s \u2208 K.faces \u2192 Exists fun \u2983t\u2984 => t \u2208 K.faces \u2227 s \u2286 t \u2227 s \u2260 t\nt : Finset E\nht : t \u2208 K.faces \u2227 s \u2286 t \u2227 s \u2260 t\n\u22a2 \u2203 t, t \u2208 K.faces \u2227 s \u2282 t\n[PROOFSTEP]\nexact \u27e8t, ht.1, \u27e8ht.2.1, fun hts => ht.2.2 (Subset.antisymm ht.2.1 hts)\u27e9\u27e9\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t : Finset E\nx : E\nhs : s \u2208 K.faces\n\u22a2 (\u2203 t, t \u2208 K.faces \u2227 s \u2282 t) \u2192 \u00acs \u2208 facets K\n[PROOFSTEP]\nrintro \u27e8t, ht\u27e9 \u27e8hs, hs'\u27e9\n[GOAL]\ncase refine'_2.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t\u271d : Finset E\nx : E\nhs\u271d : s \u2208 K.faces\nt : Finset E\nht : t \u2208 K.faces \u2227 s \u2282 t\nhs : s \u2208 K.faces\nhs' : \u2200 \u2983t : Finset E\u2984, t \u2208 K.faces \u2192 s \u2286 t \u2192 s = t\n\u22a2 False\n[PROOFSTEP]\nhave := hs' ht.1 ht.2.1\n[GOAL]\ncase refine'_2.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t\u271d : Finset E\nx : E\nhs\u271d : s \u2208 K.faces\nt : Finset E\nht : t \u2208 K.faces \u2227 s \u2282 t\nhs : s \u2208 K.faces\nhs' : \u2200 \u2983t : Finset E\u2984, t \u2208 K.faces \u2192 s \u2286 t \u2192 s = t\nthis : s = t\n\u22a2 False\n[PROOFSTEP]\nrw [this] at ht \n[GOAL]\ncase refine'_2.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : OrderedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nK : SimplicialComplex \ud835\udd5c E\ns t\u271d : Finset E\nx : E\nhs\u271d : s \u2208 K.faces\nt : Finset E\nht : t \u2208 K.faces \u2227 t \u2282 t\nhs : s \u2208 K.faces\nhs' : \u2200 \u2983t : Finset E\u2984, t \u2208 K.faces \u2192 s \u2286 t \u2192 s = t\nthis : s = t\n\u22a2 False\n[PROOFSTEP]\nexact ht.2.2 (Subset.refl t)\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.SimplicialComplex.Basic", "llama_tokens": 5122, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154240079185319, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.41859541277783463}} {"text": "[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d : CommSemiring R\nh : \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\n\u22a2 \u2203! I, Ideal.IsMaximal I\n[PROOFSTEP]\nrcases h with \u27e8P, \u27e8hPnonzero, hPnot_top, _\u27e9, hPunique\u27e9\n[GOAL]\ncase intro.intro.intro.mk\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d : CommSemiring R\nP : Ideal R\nhPunique : \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 Ideal.IsPrime P) y \u2192 y = P\nhPnonzero : P \u2260 \u22a5\nhPnot_top : P \u2260 \u22a4\nmem_or_mem'\u271d : \u2200 {x y : R}, x * y \u2208 P \u2192 x \u2208 P \u2228 y \u2208 P\n\u22a2 \u2203! I, Ideal.IsMaximal I\n[PROOFSTEP]\nrefine' \u27e8P, \u27e8\u27e8hPnot_top, _\u27e9\u27e9, fun M hM => hPunique _ \u27e8_, Ideal.IsMaximal.isPrime hM\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.mk.refine'_1\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d : CommSemiring R\nP : Ideal R\nhPunique : \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 Ideal.IsPrime P) y \u2192 y = P\nhPnonzero : P \u2260 \u22a5\nhPnot_top : P \u2260 \u22a4\nmem_or_mem'\u271d : \u2200 {x y : R}, x * y \u2208 P \u2192 x \u2208 P \u2228 y \u2208 P\n\u22a2 \u2200 (b : Ideal R), P < b \u2192 b = \u22a4\n[PROOFSTEP]\nrefine' Ideal.maximal_of_no_maximal fun M hPM hM => ne_of_lt hPM _\n[GOAL]\ncase intro.intro.intro.mk.refine'_1\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d : CommSemiring R\nP : Ideal R\nhPunique : \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 Ideal.IsPrime P) y \u2192 y = P\nhPnonzero : P \u2260 \u22a5\nhPnot_top : P \u2260 \u22a4\nmem_or_mem'\u271d : \u2200 {x y : R}, x * y \u2208 P \u2192 x \u2208 P \u2228 y \u2208 P\nM : Ideal R\nhPM : P < M\nhM : Ideal.IsMaximal M\n\u22a2 P = M\n[PROOFSTEP]\nexact (hPunique _ \u27e8ne_bot_of_gt hPM, Ideal.IsMaximal.isPrime hM\u27e9).symm\n[GOAL]\ncase intro.intro.intro.mk.refine'_2\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d : CommSemiring R\nP : Ideal R\nhPunique : \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 Ideal.IsPrime P) y \u2192 y = P\nhPnonzero : P \u2260 \u22a5\nhPnot_top : P \u2260 \u22a4\nmem_or_mem'\u271d : \u2200 {x y : R}, x * y \u2208 P \u2192 x \u2208 P \u2228 y \u2208 P\nM : Ideal R\nhM : (fun I => Ideal.IsMaximal I) M\n\u22a2 M \u2260 \u22a5\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase intro.intro.intro.mk.refine'_2\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d : CommSemiring R\nP : Ideal R\nhPunique : \u2200 (y : Ideal R), (fun P => P \u2260 \u22a5 \u2227 Ideal.IsPrime P) y \u2192 y = P\nhPnonzero : P \u2260 \u22a5\nhPnot_top : P \u2260 \u22a4\nmem_or_mem'\u271d : \u2200 {x y : R}, x * y \u2208 P \u2192 x \u2208 P \u2228 y \u2208 P\nhM : Ideal.IsMaximal \u22a5\n\u22a2 False\n[PROOFSTEP]\nexact hPnot_top (hM.1.2 P (bot_lt_iff_ne_bot.2 hPnonzero))\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\na b : R\nh : IsUnit (a + b)\n\u22a2 IsUnit a \u2228 IsUnit b\n[PROOFSTEP]\nrcases h with \u27e8u, hu\u27e9\n[GOAL]\ncase intro\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\na b : R\nu : R\u02e3\nhu : \u2191u = a + b\n\u22a2 IsUnit a \u2228 IsUnit b\n[PROOFSTEP]\nrw [\u2190 Units.inv_mul_eq_one, mul_add] at hu \n[GOAL]\ncase intro\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\na b : R\nu : R\u02e3\nhu : \u2191u\u207b\u00b9 * a + \u2191u\u207b\u00b9 * b = 1\n\u22a2 IsUnit a \u2228 IsUnit b\n[PROOFSTEP]\napply Or.imp _ _ (isUnit_or_isUnit_of_add_one hu)\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\na b : R\nu : R\u02e3\nhu : \u2191u\u207b\u00b9 * a + \u2191u\u207b\u00b9 * b = 1\n\u22a2 IsUnit (\u2191u\u207b\u00b9 * a) \u2192 IsUnit a\n[PROOFSTEP]\nexact isUnit_of_mul_isUnit_right\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\na b : R\nu : R\u02e3\nhu : \u2191u\u207b\u00b9 * a + \u2191u\u207b\u00b9 * b = 1\n\u22a2 IsUnit (\u2191u\u207b\u00b9 * b) \u2192 IsUnit b\n[PROOFSTEP]\nexact isUnit_of_mul_isUnit_right\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\n\u22a2 Ideal.IsMaximal (maximalIdeal R)\n[PROOFSTEP]\nrw [Ideal.isMaximal_iff]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\n\u22a2 \u00ac1 \u2208 maximalIdeal R \u2227 \u2200 (J : Ideal R) (x : R), maximalIdeal R \u2264 J \u2192 \u00acx \u2208 maximalIdeal R \u2192 x \u2208 J \u2192 1 \u2208 J\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\n\u22a2 \u00ac1 \u2208 maximalIdeal R\n[PROOFSTEP]\nintro h\n[GOAL]\ncase left\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\nh : 1 \u2208 maximalIdeal R\n\u22a2 False\n[PROOFSTEP]\napply h\n[GOAL]\ncase left\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\nh : 1 \u2208 maximalIdeal R\n\u22a2 IsUnit 1\n[PROOFSTEP]\nexact isUnit_one\n[GOAL]\ncase right\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\n\u22a2 \u2200 (J : Ideal R) (x : R), maximalIdeal R \u2264 J \u2192 \u00acx \u2208 maximalIdeal R \u2192 x \u2208 J \u2192 1 \u2208 J\n[PROOFSTEP]\nintro I x _ hx H\n[GOAL]\ncase right\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\nI : Ideal R\nx : R\na\u271d : maximalIdeal R \u2264 I\nhx : \u00acx \u2208 maximalIdeal R\nH : x \u2208 I\n\u22a2 1 \u2208 I\n[PROOFSTEP]\nerw [Classical.not_not] at hx \n[GOAL]\ncase right\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\nI : Ideal R\nx : R\na\u271d : maximalIdeal R \u2264 I\nhx : IsUnit x\nH : x \u2208 I\n\u22a2 1 \u2208 I\n[PROOFSTEP]\nrcases hx with \u27e8u, rfl\u27e9\n[GOAL]\ncase right.intro\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\nI : Ideal R\na\u271d : maximalIdeal R \u2264 I\nu : R\u02e3\nH : \u2191u \u2208 I\n\u22a2 1 \u2208 I\n[PROOFSTEP]\nsimpa using I.mul_mem_left (\u2191u\u207b\u00b9) H\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\nJ : Ideal R\nhJ : J \u2260 \u22a4\n\u22a2 J \u2264 maximalIdeal R\n[PROOFSTEP]\nrcases Ideal.exists_le_maximal J hJ with \u27e8M, hM1, hM2\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : LocalRing R\nJ : Ideal R\nhJ : J \u2260 \u22a4\nM : Ideal R\nhM1 : Ideal.IsMaximal M\nhM2 : J \u2264 M\n\u22a2 J \u2264 maximalIdeal R\n[PROOFSTEP]\nrwa [\u2190 eq_maximalIdeal hM1]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\n\u22a2 \u2200 (a : S), IsUnit a \u2228 IsUnit (1 - a)\n[PROOFSTEP]\nintro b\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nb : S\n\u22a2 IsUnit b \u2228 IsUnit (1 - b)\n[PROOFSTEP]\nobtain \u27e8a, rfl\u27e9 := hf b\n[GOAL]\ncase intro\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\na : R\n\u22a2 IsUnit (\u2191f a) \u2228 IsUnit (1 - \u2191f a)\n[PROOFSTEP]\napply (isUnit_or_isUnit_one_sub_self a).imp <| RingHom.isUnit_map _\n[GOAL]\ncase intro\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\na : R\n\u22a2 IsUnit (1 - a) \u2192 IsUnit (1 - \u2191f a)\n[PROOFSTEP]\nrw [\u2190 f.map_one, \u2190 f.map_sub]\n[GOAL]\ncase intro\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\na : R\n\u22a2 IsUnit (1 - a) \u2192 IsUnit (\u2191f (1 - a))\n[PROOFSTEP]\napply f.isUnit_map\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nI : Ideal R\nh : I \u2260 \u22a4\n\u22a2 Ideal.jacobson I = maximalIdeal R\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nI : Ideal R\nh : I \u2260 \u22a4\n\u22a2 Ideal.jacobson I \u2264 maximalIdeal R\n[PROOFSTEP]\nexact sInf_le \u27e8LocalRing.le_maximalIdeal h, LocalRing.maximalIdeal.isMaximal R\u27e9\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nI : Ideal R\nh : I \u2260 \u22a4\n\u22a2 maximalIdeal R \u2264 Ideal.jacobson I\n[PROOFSTEP]\nexact le_sInf fun J (hJ : I \u2264 J \u2227 J.IsMaximal) => le_of_eq (LocalRing.eq_maximalIdeal hJ.2).symm\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Semiring T\nf : R \u2243+* S\na : R\nha : IsUnit (\u2191\u2191f a)\n\u22a2 IsUnit a\n[PROOFSTEP]\nconvert RingHom.isUnit_map (f.symm : S \u2192+* R) ha\n[GOAL]\ncase h.e'_3\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Semiring T\nf : R \u2243+* S\na : R\nha : IsUnit (\u2191\u2191f a)\n\u22a2 a = \u2191\u2191(RingEquiv.symm f) (\u2191\u2191f a)\n[PROOFSTEP]\nexact (RingEquiv.symm_apply_apply f a).symm\n[GOAL]\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : Semiring S\u271d\ninst\u271d\u00b3 : Semiring T\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\nH : LocalRing S\nf : R \u2192+* S\ninst\u271d : IsLocalRingHom f\n\u22a2 LocalRing R\n[PROOFSTEP]\nhaveI : Nontrivial R := pullback_nonzero f f.map_zero f.map_one\n[GOAL]\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : Semiring S\u271d\ninst\u271d\u00b3 : Semiring T\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\nH : LocalRing S\nf : R \u2192+* S\ninst\u271d : IsLocalRingHom f\nthis : Nontrivial R\n\u22a2 LocalRing R\n[PROOFSTEP]\napply LocalRing.of_nonunits_add\n[GOAL]\ncase h\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : Semiring S\u271d\ninst\u271d\u00b3 : Semiring T\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\nH : LocalRing S\nf : R \u2192+* S\ninst\u271d : IsLocalRingHom f\nthis : Nontrivial R\n\u22a2 \u2200 (a b : R), a \u2208 nonunits R \u2192 b \u2208 nonunits R \u2192 a + b \u2208 nonunits R\n[PROOFSTEP]\nintro a b\n[GOAL]\ncase h\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : Semiring S\u271d\ninst\u271d\u00b3 : Semiring T\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\nH : LocalRing S\nf : R \u2192+* S\ninst\u271d : IsLocalRingHom f\nthis : Nontrivial R\na b : R\n\u22a2 a \u2208 nonunits R \u2192 b \u2208 nonunits R \u2192 a + b \u2208 nonunits R\n[PROOFSTEP]\nsimp_rw [\u2190 map_mem_nonunits_iff f, f.map_add]\n[GOAL]\ncase h\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : Semiring S\u271d\ninst\u271d\u00b3 : Semiring T\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\nH : LocalRing S\nf : R \u2192+* S\ninst\u271d : IsLocalRingHom f\nthis : Nontrivial R\na b : R\n\u22a2 \u2191f a \u2208 nonunits S \u2192 \u2191f b \u2208 nonunits S \u2192 \u2191f a + \u2191f b \u2208 nonunits S\n[PROOFSTEP]\nexact LocalRing.nonunits_add\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\n\u22a2 List.TFAE\n [IsLocalRingHom f, \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S),\n Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S, maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S),\n Ideal.comap f (maximalIdeal S) = maximalIdeal R]\n[PROOFSTEP]\ntfae_have 1 \u2192 2\n[GOAL]\ncase tfae_1_to_2\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\n\u22a2 IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\n[PROOFSTEP]\nrintro _ _ \u27e8a, ha, rfl\u27e9\n[GOAL]\ncase tfae_1_to_2.intro.intro\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\n\u271d : IsLocalRingHom f\na : R\nha : a \u2208 \u2191(maximalIdeal R).toAddSubmonoid\n\u22a2 \u2191f a \u2208 \u2191(maximalIdeal S)\n[PROOFSTEP]\nexact map_nonunit f a ha\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\ntfae_1_to_2 : IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\n\u22a2 List.TFAE\n [IsLocalRingHom f, \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S),\n Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S, maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S),\n Ideal.comap f (maximalIdeal S) = maximalIdeal R]\n[PROOFSTEP]\ntfae_have 2 \u2192 4\n[GOAL]\ncase tfae_2_to_4\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\ntfae_1_to_2 : IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\n\u22a2 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S) \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\n[PROOFSTEP]\nexact Set.image_subset_iff.1\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\ntfae_1_to_2 : IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\ntfae_2_to_4 :\n \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S) \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\n\u22a2 List.TFAE\n [IsLocalRingHom f, \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S),\n Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S, maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S),\n Ideal.comap f (maximalIdeal S) = maximalIdeal R]\n[PROOFSTEP]\ntfae_have 3 \u2194 4\n[GOAL]\ncase tfae_3_iff_4\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\ntfae_1_to_2 : IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\ntfae_2_to_4 :\n \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S) \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\n\u22a2 Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S \u2194 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\n[PROOFSTEP]\nexact Ideal.map_le_iff_le_comap\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\ntfae_1_to_2 : IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\ntfae_2_to_4 :\n \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S) \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_3_iff_4 : Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S \u2194 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\n\u22a2 List.TFAE\n [IsLocalRingHom f, \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S),\n Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S, maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S),\n Ideal.comap f (maximalIdeal S) = maximalIdeal R]\n[PROOFSTEP]\ntfae_have 4 \u2192 1\n[GOAL]\ncase tfae_4_to_1\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\ntfae_1_to_2 : IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\ntfae_2_to_4 :\n \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S) \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_3_iff_4 : Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S \u2194 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\n\u22a2 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S) \u2192 IsLocalRingHom f\n[PROOFSTEP]\nintro h\n[GOAL]\ncase tfae_4_to_1\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\ntfae_1_to_2 : IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\ntfae_2_to_4 :\n \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S) \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_3_iff_4 : Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S \u2194 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\nh : maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\n\u22a2 IsLocalRingHom f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase tfae_4_to_1.map_nonunit\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\ntfae_1_to_2 : IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\ntfae_2_to_4 :\n \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S) \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_3_iff_4 : Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S \u2194 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\nh : maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\n\u22a2 \u2200 (a : R), IsUnit (\u2191f a) \u2192 IsUnit a\n[PROOFSTEP]\nexact fun x => not_imp_not.1 (@h x)\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\ntfae_1_to_2 : IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\ntfae_2_to_4 :\n \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S) \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_3_iff_4 : Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S \u2194 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_4_to_1 : maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S) \u2192 IsLocalRingHom f\n\u22a2 List.TFAE\n [IsLocalRingHom f, \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S),\n Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S, maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S),\n Ideal.comap f (maximalIdeal S) = maximalIdeal R]\n[PROOFSTEP]\ntfae_have 1 \u2192 5\n[GOAL]\ncase tfae_1_to_5\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\ntfae_1_to_2 : IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\ntfae_2_to_4 :\n \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S) \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_3_iff_4 : Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S \u2194 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_4_to_1 : maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S) \u2192 IsLocalRingHom f\n\u22a2 IsLocalRingHom f \u2192 Ideal.comap f (maximalIdeal S) = maximalIdeal R\n[PROOFSTEP]\nintro\n[GOAL]\ncase tfae_1_to_5\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\ntfae_1_to_2 : IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\ntfae_2_to_4 :\n \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S) \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_3_iff_4 : Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S \u2194 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_4_to_1 : maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S) \u2192 IsLocalRingHom f\n\u271d : IsLocalRingHom f\n\u22a2 Ideal.comap f (maximalIdeal S) = maximalIdeal R\n[PROOFSTEP]\next\n[GOAL]\ncase tfae_1_to_5.h\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\ntfae_1_to_2 : IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\ntfae_2_to_4 :\n \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S) \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_3_iff_4 : Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S \u2194 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_4_to_1 : maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S) \u2192 IsLocalRingHom f\n\u271d : IsLocalRingHom f\nx\u271d : R\n\u22a2 x\u271d \u2208 Ideal.comap f (maximalIdeal S) \u2194 x\u271d \u2208 maximalIdeal R\n[PROOFSTEP]\nexact not_iff_not.2 (isUnit_map_iff f _)\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\ntfae_1_to_2 : IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\ntfae_2_to_4 :\n \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S) \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_3_iff_4 : Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S \u2194 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_4_to_1 : maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S) \u2192 IsLocalRingHom f\ntfae_1_to_5 : IsLocalRingHom f \u2192 Ideal.comap f (maximalIdeal S) = maximalIdeal R\n\u22a2 List.TFAE\n [IsLocalRingHom f, \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S),\n Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S, maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S),\n Ideal.comap f (maximalIdeal S) = maximalIdeal R]\n[PROOFSTEP]\ntfae_have 5 \u2192 4\n[GOAL]\ncase tfae_5_to_4\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\ntfae_1_to_2 : IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\ntfae_2_to_4 :\n \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S) \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_3_iff_4 : Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S \u2194 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_4_to_1 : maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S) \u2192 IsLocalRingHom f\ntfae_1_to_5 : IsLocalRingHom f \u2192 Ideal.comap f (maximalIdeal S) = maximalIdeal R\n\u22a2 Ideal.comap f (maximalIdeal S) = maximalIdeal R \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\n[PROOFSTEP]\nexact fun h => le_of_eq h.symm\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : LocalRing S\nf : R \u2192+* S\ntfae_1_to_2 : IsLocalRingHom f \u2192 \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S)\ntfae_2_to_4 :\n \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S) \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_3_iff_4 : Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S \u2194 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\ntfae_4_to_1 : maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S) \u2192 IsLocalRingHom f\ntfae_1_to_5 : IsLocalRingHom f \u2192 Ideal.comap f (maximalIdeal S) = maximalIdeal R\ntfae_5_to_4 : Ideal.comap f (maximalIdeal S) = maximalIdeal R \u2192 maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S)\n\u22a2 List.TFAE\n [IsLocalRingHom f, \u2191f '' \u2191(maximalIdeal R).toAddSubmonoid \u2286 \u2191(maximalIdeal S),\n Ideal.map f (maximalIdeal R) \u2264 maximalIdeal S, maximalIdeal R \u2264 Ideal.comap f (maximalIdeal S),\n Ideal.comap f (maximalIdeal S) = maximalIdeal R]\n[PROOFSTEP]\ntfae_finish\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : LocalRing R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Nontrivial S\nf : R \u2192+* S\ninst\u271d : IsLocalRingHom f\nhf : Function.Surjective \u2191f\n\u22a2 \u2200 (a b : S), IsUnit (a + b) \u2192 IsUnit a \u2228 IsUnit b\n[PROOFSTEP]\nintro a b hab\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : LocalRing R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Nontrivial S\nf : R \u2192+* S\ninst\u271d : IsLocalRingHom f\nhf : Function.Surjective \u2191f\na b : S\nhab : IsUnit (a + b)\n\u22a2 IsUnit a \u2228 IsUnit b\n[PROOFSTEP]\nobtain \u27e8a, rfl\u27e9 := hf a\n[GOAL]\ncase intro\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : LocalRing R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Nontrivial S\nf : R \u2192+* S\ninst\u271d : IsLocalRingHom f\nhf : Function.Surjective \u2191f\nb : S\na : R\nhab : IsUnit (\u2191f a + b)\n\u22a2 IsUnit (\u2191f a) \u2228 IsUnit b\n[PROOFSTEP]\nobtain \u27e8b, rfl\u27e9 := hf b\n[GOAL]\ncase intro.intro\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : LocalRing R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Nontrivial S\nf : R \u2192+* S\ninst\u271d : IsLocalRingHom f\nhf : Function.Surjective \u2191f\na b : R\nhab : IsUnit (\u2191f a + \u2191f b)\n\u22a2 IsUnit (\u2191f a) \u2228 IsUnit (\u2191f b)\n[PROOFSTEP]\nrw [\u2190 map_add] at hab \n[GOAL]\ncase intro.intro\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : LocalRing R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Nontrivial S\nf : R \u2192+* S\ninst\u271d : IsLocalRingHom f\nhf : Function.Surjective \u2191f\na b : R\nhab : IsUnit (\u2191f (a + b))\n\u22a2 IsUnit (\u2191f a) \u2228 IsUnit (\u2191f b)\n[PROOFSTEP]\nexact (isUnit_or_isUnit_of_isUnit_add <| IsLocalRingHom.map_nonunit _ hab).imp f.isUnit_map f.isUnit_map\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : IsLocalRingHom f\n\u22a2 Function.Surjective \u2191(Units.map \u2191f)\n[PROOFSTEP]\nintro a\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : IsLocalRingHom f\na : S\u02e3\n\u22a2 \u2203 a_1, \u2191(Units.map \u2191f) a_1 = a\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 := hf (a : S)\n[GOAL]\ncase intro\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : IsLocalRingHom f\na : S\u02e3\nb : R\nhb : \u2191f b = \u2191a\n\u22a2 \u2203 a_1, \u2191(Units.map \u2191f) a_1 = a\n[PROOFSTEP]\nuse(isUnit_of_map_unit f b (by rw [hb]; exact Units.isUnit _)).unit\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : IsLocalRingHom f\na : S\u02e3\nb : R\nhb : \u2191f b = \u2191a\n\u22a2 IsUnit (\u2191f b)\n[PROOFSTEP]\nrw [hb]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : IsLocalRingHom f\na : S\u02e3\nb : R\nhb : \u2191f b = \u2191a\n\u22a2 IsUnit \u2191a\n[PROOFSTEP]\nexact Units.isUnit _\n[GOAL]\ncase h\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : IsLocalRingHom f\na : S\u02e3\nb : R\nhb : \u2191f b = \u2191a\n\u22a2 \u2191(Units.map \u2191f) (IsUnit.unit (_ : IsUnit b)) = a\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : IsLocalRingHom f\na : S\u02e3\nb : R\nhb : \u2191f b = \u2191a\n\u22a2 \u2191(\u2191(Units.map \u2191f) (IsUnit.unit (_ : IsUnit b))) = \u2191a\n[PROOFSTEP]\nexact hb\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LocalRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : LocalRing S\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : LocalRing T\nf : R \u2192+* S\ninst\u271d : IsLocalRingHom f\na : R\nha : a \u2208 maximalIdeal R\n\u22a2 \u2191(RingHom.comp (Ideal.Quotient.mk (maximalIdeal S)) f) a = 0\n[PROOFSTEP]\nerw [Ideal.Quotient.eq_zero_iff_mem]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LocalRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : LocalRing S\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : LocalRing T\nf : R \u2192+* S\ninst\u271d : IsLocalRingHom f\na : R\nha : a \u2208 maximalIdeal R\n\u22a2 \u2191f a \u2208 maximalIdeal S\n[PROOFSTEP]\nexact map_nonunit f a ha\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : LocalRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : LocalRing S\ninst\u271d\u00b9 : CommRing T\ninst\u271d : LocalRing T\nf : R \u2243+* S\nx : ResidueField R\n\u22a2 \u2191(map \u2191(RingEquiv.symm f)) (\u2191(map \u2191f) x) = x\n[PROOFSTEP]\nsimp only [map_map, RingEquiv.symm_comp, map_id, RingHom.id_apply]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : LocalRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : LocalRing S\ninst\u271d\u00b9 : CommRing T\ninst\u271d : LocalRing T\nf : R \u2243+* S\nx : ResidueField S\n\u22a2 \u2191(map \u2191f) (\u2191(map \u2191(RingEquiv.symm f)) x) = x\n[PROOFSTEP]\nsimp only [map_map, RingEquiv.comp_symm, map_id, RingHom.id_apply]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : LocalRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : LocalRing S\ninst\u271d\u00b9 : CommRing T\ninst\u271d : LocalRing T\n\u22a2 IsLocalRingHom (residue R)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase map_nonunit\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : LocalRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : LocalRing S\ninst\u271d\u00b9 : CommRing T\ninst\u271d : LocalRing T\n\u22a2 \u2200 (a : R), IsUnit (\u2191(residue R) a) \u2192 IsUnit a\n[PROOFSTEP]\nintro a ha\n[GOAL]\ncase map_nonunit\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : LocalRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : LocalRing S\ninst\u271d\u00b9 : CommRing T\ninst\u271d : LocalRing T\na : R\nha : IsUnit (\u2191(residue R) a)\n\u22a2 IsUnit a\n[PROOFSTEP]\nby_contra h\n[GOAL]\ncase map_nonunit\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : LocalRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : LocalRing S\ninst\u271d\u00b9 : CommRing T\ninst\u271d : LocalRing T\na : R\nha : IsUnit (\u2191(residue R) a)\nh : \u00acIsUnit a\n\u22a2 False\n[PROOFSTEP]\nerw [Ideal.Quotient.eq_zero_iff_mem.mpr ((LocalRing.mem_maximalIdeal _).mpr h)] at ha \n[GOAL]\ncase map_nonunit\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : LocalRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : LocalRing S\ninst\u271d\u00b9 : CommRing T\ninst\u271d : LocalRing T\na : R\nha : IsUnit 0\nh : \u00acIsUnit a\n\u22a2 False\n[PROOFSTEP]\nexact ha.ne_zero rfl\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d : Field K\na : K\nh : a = 0\n\u22a2 IsUnit (1 - a)\n[PROOFSTEP]\nrw [h, sub_zero]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nK : Type u'\ninst\u271d : Field K\na : K\nh : a = 0\n\u22a2 IsUnit 1\n[PROOFSTEP]\nexact isUnit_one\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Ideal.LocalRing", "llama_tokens": 13950, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239836484143, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.4185953985773608}} {"text": "[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\nterminated_at_n : TerminatedAt g n\n\u22a2 continuantsAux g (n + 2) = continuantsAux g (n + 1)\n[PROOFSTEP]\nrw [terminatedAt_iff_s_none] at terminated_at_n \n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\nterminated_at_n : Stream'.Seq.get? g.s n = none\n\u22a2 continuantsAux g (n + 2) = continuantsAux g (n + 1)\n[PROOFSTEP]\nsimp only [continuantsAux, Nat.add_eq, add_zero, terminated_at_n]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\nn_lt_m : n < m\nterminated_at_n : TerminatedAt g n\n\u22a2 continuantsAux g m = continuantsAux g (n + 1)\n[PROOFSTEP]\nrefine' Nat.le_induction rfl (fun k hnk hk => _) _ n_lt_m\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\nn_lt_m : n < m\nterminated_at_n : TerminatedAt g n\nk : \u2115\nhnk : Nat.succ n \u2264 k\nhk : continuantsAux g k = continuantsAux g (n + 1)\n\u22a2 continuantsAux g (k + 1) = continuantsAux g (n + 1)\n[PROOFSTEP]\nrcases Nat.exists_eq_add_of_lt hnk with \u27e8k, rfl\u27e9\n[GOAL]\ncase intro\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\nn_lt_m : n < m\nterminated_at_n : TerminatedAt g n\nk : \u2115\nhnk : Nat.succ n \u2264 n + k + 1\nhk : continuantsAux g (n + k + 1) = continuantsAux g (n + 1)\n\u22a2 continuantsAux g (n + k + 1 + 1) = continuantsAux g (n + 1)\n[PROOFSTEP]\nrefine' (continuantsAux_stable_step_of_terminated _).trans hk\n[GOAL]\ncase intro\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\nn_lt_m : n < m\nterminated_at_n : TerminatedAt g n\nk : \u2115\nhnk : Nat.succ n \u2264 n + k + 1\nhk : continuantsAux g (n + k + 1) = continuantsAux g (n + 1)\n\u22a2 TerminatedAt g (n + k)\n[PROOFSTEP]\nexact terminated_stable (Nat.le_add_right _ _) terminated_at_n\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.TerminatedAt s n\n\u22a2 convergents'Aux s (n + 1) = convergents'Aux s n\n[PROOFSTEP]\nchange s.get? n = none at terminated_at_n \n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s n = none\n\u22a2 convergents'Aux s (n + 1) = convergents'Aux s n\n[PROOFSTEP]\ninduction' n with n IH generalizing s\n[GOAL]\ncase zero\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n = none\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s Nat.zero = none\n\u22a2 convergents'Aux s (Nat.zero + 1) = convergents'Aux s Nat.zero\ncase succ\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n\u271d = none\nn : \u2115\nIH : \u2200 {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none \u2192 convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\n\u22a2 convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\n[PROOFSTEP]\ncase zero => simp only [convergents'Aux, terminated_at_n, Stream'.Seq.head]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n = none\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s Nat.zero = none\n\u22a2 convergents'Aux s (Nat.zero + 1) = convergents'Aux s Nat.zero\n[PROOFSTEP]\ncase zero => simp only [convergents'Aux, terminated_at_n, Stream'.Seq.head]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n = none\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s Nat.zero = none\n\u22a2 convergents'Aux s (Nat.zero + 1) = convergents'Aux s Nat.zero\n[PROOFSTEP]\nsimp only [convergents'Aux, terminated_at_n, Stream'.Seq.head]\n[GOAL]\ncase succ\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n\u271d = none\nn : \u2115\nIH : \u2200 {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none \u2192 convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\n\u22a2 convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\n[PROOFSTEP]\ncase succ =>\n cases' s_head_eq : s.head with gp_head\n case none => simp only [convergents'Aux, s_head_eq]\n case\n some =>\n have : s.tail.TerminatedAt n := by simp only [Stream'.Seq.TerminatedAt, s.get?_tail, terminated_at_n]\n have := IH this\n rw [convergents'Aux] at this \n simp [this, Nat.add_eq, add_zero, convergents'Aux, s_head_eq]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n\u271d = none\nn : \u2115\nIH : \u2200 {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none \u2192 convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\n\u22a2 convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\n[PROOFSTEP]\ncase succ =>\n cases' s_head_eq : s.head with gp_head\n case none => simp only [convergents'Aux, s_head_eq]\n case\n some =>\n have : s.tail.TerminatedAt n := by simp only [Stream'.Seq.TerminatedAt, s.get?_tail, terminated_at_n]\n have := IH this\n rw [convergents'Aux] at this \n simp [this, Nat.add_eq, add_zero, convergents'Aux, s_head_eq]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n\u271d = none\nn : \u2115\nIH : \u2200 {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none \u2192 convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\n\u22a2 convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\n[PROOFSTEP]\ncases' s_head_eq : s.head with gp_head\n[GOAL]\ncase none\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n\u271d = none\nn : \u2115\nIH : \u2200 {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none \u2192 convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ns_head_eq : Stream'.Seq.head s = none\n\u22a2 convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\ncase some\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n\u271d = none\nn : \u2115\nIH : \u2200 {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none \u2192 convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n\u22a2 convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\n[PROOFSTEP]\ncase none => simp only [convergents'Aux, s_head_eq]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n\u271d = none\nn : \u2115\nIH : \u2200 {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none \u2192 convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ns_head_eq : Stream'.Seq.head s = none\n\u22a2 convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\n[PROOFSTEP]\ncase none => simp only [convergents'Aux, s_head_eq]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n\u271d = none\nn : \u2115\nIH : \u2200 {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none \u2192 convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ns_head_eq : Stream'.Seq.head s = none\n\u22a2 convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\n[PROOFSTEP]\nsimp only [convergents'Aux, s_head_eq]\n[GOAL]\ncase some\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n\u271d = none\nn : \u2115\nIH : \u2200 {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none \u2192 convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n\u22a2 convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\n[PROOFSTEP]\ncase\n some =>\n have : s.tail.TerminatedAt n := by simp only [Stream'.Seq.TerminatedAt, s.get?_tail, terminated_at_n]\n have := IH this\n rw [convergents'Aux] at this \n simp [this, Nat.add_eq, add_zero, convergents'Aux, s_head_eq]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n\u271d = none\nn : \u2115\nIH : \u2200 {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none \u2192 convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n\u22a2 convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\n[PROOFSTEP]\ncase\n some =>\n have : s.tail.TerminatedAt n := by simp only [Stream'.Seq.TerminatedAt, s.get?_tail, terminated_at_n]\n have := IH this\n rw [convergents'Aux] at this \n simp [this, Nat.add_eq, add_zero, convergents'Aux, s_head_eq]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n\u271d = none\nn : \u2115\nIH : \u2200 {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none \u2192 convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n\u22a2 convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\n[PROOFSTEP]\nhave : s.tail.TerminatedAt n := by simp only [Stream'.Seq.TerminatedAt, s.get?_tail, terminated_at_n]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n\u271d = none\nn : \u2115\nIH : \u2200 {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none \u2192 convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\n\u22a2 Stream'.Seq.TerminatedAt (Stream'.Seq.tail s) n\n[PROOFSTEP]\nsimp only [Stream'.Seq.TerminatedAt, s.get?_tail, terminated_at_n]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n\u271d = none\nn : \u2115\nIH : \u2200 {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none \u2192 convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis : Stream'.Seq.TerminatedAt (Stream'.Seq.tail s) n\n\u22a2 convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\n[PROOFSTEP]\nhave := IH this\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n\u271d = none\nn : \u2115\nIH : \u2200 {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none \u2192 convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis\u271d : Stream'.Seq.TerminatedAt (Stream'.Seq.tail s) n\nthis : convergents'Aux (Stream'.Seq.tail s) (n + 1) = convergents'Aux (Stream'.Seq.tail s) n\n\u22a2 convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\n[PROOFSTEP]\nrw [convergents'Aux] at this \n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d m : \u2115\ninst\u271d : DivisionRing K\ns\u271d : Stream'.Seq (Pair K)\nterminated_at_n\u271d : Stream'.Seq.get? s\u271d n\u271d = none\nn : \u2115\nIH : \u2200 {s : Stream'.Seq (Pair K)}, Stream'.Seq.get? s n = none \u2192 convergents'Aux s (n + 1) = convergents'Aux s n\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.get? s (Nat.succ n) = none\ngp_head : Pair K\ns_head_eq : Stream'.Seq.head s = some gp_head\nthis\u271d : Stream'.Seq.TerminatedAt (Stream'.Seq.tail s) n\nthis :\n (match Stream'.Seq.head (Stream'.Seq.tail s) with\n | none => 0\n | some gp => gp.a / (gp.b + convergents'Aux (Stream'.Seq.tail (Stream'.Seq.tail s)) n)) =\n convergents'Aux (Stream'.Seq.tail s) n\n\u22a2 convergents'Aux s (Nat.succ n + 1) = convergents'Aux s (Nat.succ n)\n[PROOFSTEP]\nsimp [this, Nat.add_eq, add_zero, convergents'Aux, s_head_eq]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\ns : Stream'.Seq (Pair K)\nn_le_m : n \u2264 m\nterminated_at_n : Stream'.Seq.TerminatedAt s n\n\u22a2 convergents'Aux s m = convergents'Aux s n\n[PROOFSTEP]\ninduction' n_le_m with m n_le_m IH\n[GOAL]\ncase refl\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.TerminatedAt s n\n\u22a2 convergents'Aux s n = convergents'Aux s n\n[PROOFSTEP]\nrfl\n[GOAL]\ncase step\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m\u271d : \u2115\ninst\u271d : DivisionRing K\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.TerminatedAt s n\nm : \u2115\nn_le_m : Nat.le n m\nIH : convergents'Aux s m = convergents'Aux s n\n\u22a2 convergents'Aux s (Nat.succ m) = convergents'Aux s n\n[PROOFSTEP]\nrefine' (convergents'Aux_stable_step_of_terminated _).trans IH\n[GOAL]\ncase step\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m\u271d : \u2115\ninst\u271d : DivisionRing K\ns : Stream'.Seq (Pair K)\nterminated_at_n : Stream'.Seq.TerminatedAt s n\nm : \u2115\nn_le_m : Nat.le n m\nIH : convergents'Aux s m = convergents'Aux s n\n\u22a2 Stream'.Seq.TerminatedAt s m\n[PROOFSTEP]\nexact s.terminated_stable n_le_m terminated_at_n\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\nn_le_m : n \u2264 m\nterminated_at_n : TerminatedAt g n\n\u22a2 continuants g m = continuants g n\n[PROOFSTEP]\nsimp only [nth_cont_eq_succ_nth_cont_aux,\n continuantsAux_stable_of_terminated (Nat.pred_le_iff.mp n_le_m) terminated_at_n]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\nn_le_m : n \u2264 m\nterminated_at_n : TerminatedAt g n\n\u22a2 numerators g m = numerators g n\n[PROOFSTEP]\nsimp only [num_eq_conts_a, continuants_stable_of_terminated n_le_m terminated_at_n]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\nn_le_m : n \u2264 m\nterminated_at_n : TerminatedAt g n\n\u22a2 denominators g m = denominators g n\n[PROOFSTEP]\nsimp only [denom_eq_conts_b, continuants_stable_of_terminated n_le_m terminated_at_n]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\nn_le_m : n \u2264 m\nterminated_at_n : TerminatedAt g n\n\u22a2 convergents g m = convergents g n\n[PROOFSTEP]\nsimp only [convergents, denominators_stable_of_terminated n_le_m terminated_at_n,\n numerators_stable_of_terminated n_le_m terminated_at_n]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn m : \u2115\ninst\u271d : DivisionRing K\nn_le_m : n \u2264 m\nterminated_at_n : TerminatedAt g n\n\u22a2 convergents' g m = convergents' g n\n[PROOFSTEP]\nsimp only [convergents', convergents'Aux_stable_of_terminated n_le_m terminated_at_n]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.ContinuedFractions.TerminatedStable", "llama_tokens": 7043, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593452091672, "lm_q2_score": 0.672331699179286, "lm_q1q2_score": 0.4184991492345052}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\n\u22a2 Injective toFun\n[PROOFSTEP]\nrintro \u27e8f\u2081, f\u2082, hf\u27e9 \u27e8g\u2081, g\u2082, hg\u27e9 (rfl : f\u2081 = g\u2081)\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf\u2081 : \u03b1 \u2192 Option \u03b2\nf\u2082 : \u03b2 \u2192 Option \u03b1\nhf : \u2200 (a : \u03b1) (b : \u03b2), a \u2208 f\u2082 b \u2194 b \u2208 f\u2081 a\ng\u2082 : \u03b2 \u2192 Option \u03b1\nhg : \u2200 (a : \u03b1) (b : \u03b2), a \u2208 g\u2082 b \u2194 b \u2208 f\u2081 a\n\u22a2 { toFun := f\u2081, invFun := f\u2082, inv := hf } = { toFun := f\u2081, invFun := g\u2082, inv := hg }\n[PROOFSTEP]\ncongr with y x\n[GOAL]\ncase mk.mk.e_invFun.h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf\u2081 : \u03b1 \u2192 Option \u03b2\nf\u2082 : \u03b2 \u2192 Option \u03b1\nhf : \u2200 (a : \u03b1) (b : \u03b2), a \u2208 f\u2082 b \u2194 b \u2208 f\u2081 a\ng\u2082 : \u03b2 \u2192 Option \u03b1\nhg : \u2200 (a : \u03b1) (b : \u03b2), a \u2208 g\u2082 b \u2194 b \u2208 f\u2081 a\ny : \u03b2\nx : \u03b1\n\u22a2 x \u2208 f\u2082 y \u2194 x \u2208 g\u2082 y\n[PROOFSTEP]\nsimp only [hf, hg]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\ng : \u03b2 \u2243. \u03b3\na : \u03b1\nb : \u03b3\n\u22a2 a \u2208 (fun a => Option.bind (\u2191(PEquiv.symm g) a) \u2191(PEquiv.symm f)) b \u2194 b \u2208 (fun a => Option.bind (\u2191f a) \u2191g) a\n[PROOFSTEP]\nsimp_all [and_comm, eq_some_iff f, eq_some_iff g]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\n\u22a2 PEquiv.symm (PEquiv.symm f) = f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ntoFun\u271d : \u03b1 \u2192 Option \u03b2\ninvFun\u271d : \u03b2 \u2192 Option \u03b1\ninv\u271d : \u2200 (a : \u03b1) (b : \u03b2), a \u2208 invFun\u271d b \u2194 b \u2208 toFun\u271d a\n\u22a2 PEquiv.symm (PEquiv.symm { toFun := toFun\u271d, invFun := invFun\u271d, inv := inv\u271d }) =\n { toFun := toFun\u271d, invFun := invFun\u271d, inv := inv\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\ng : \u03b2 \u2243. \u03b3\na : \u03b1\n\u22a2 \u2191(PEquiv.trans f g) a = none \u2194 \u2200 (b : \u03b2) (c : \u03b3), \u00acb \u2208 \u2191f a \u2228 \u00acc \u2208 \u2191g b\n[PROOFSTEP]\nsimp only [eq_none_iff_forall_not_mem, mem_trans, imp_iff_not_or.symm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\ng : \u03b2 \u2243. \u03b3\na : \u03b1\n\u22a2 (\u2200 (a_1 : \u03b3), \u00ac\u2203 b, b \u2208 \u2191f a \u2227 a_1 \u2208 \u2191g b) \u2194 \u2200 (b : \u03b2) (c : \u03b3), b \u2208 \u2191f a \u2192 \u00acc \u2208 \u2191g b\n[PROOFSTEP]\npush_neg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\ng : \u03b2 \u2243. \u03b3\na : \u03b1\n\u22a2 (\u2200 (a_1 : \u03b3) (b : \u03b2), b \u2208 \u2191f a \u2192 \u00aca_1 \u2208 \u2191g b) \u2194 \u2200 (b : \u03b2) (c : \u03b3), b \u2208 \u2191f a \u2192 \u00acc \u2208 \u2191g b\n[PROOFSTEP]\nexact forall_swap\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\n\u22a2 PEquiv.trans (PEquiv.refl \u03b1) f = f\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d : \u03b1\na\u271d : \u03b2\n\u22a2 a\u271d \u2208 \u2191(PEquiv.trans (PEquiv.refl \u03b1) f) x\u271d \u2194 a\u271d \u2208 \u2191f x\u271d\n[PROOFSTEP]\ndsimp [PEquiv.trans]\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d : \u03b1\na\u271d : \u03b2\n\u22a2 a\u271d \u2208 \u2191f x\u271d \u2194 a\u271d \u2208 \u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\n\u22a2 PEquiv.trans f (PEquiv.refl \u03b2) = f\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d : \u03b1\na\u271d : \u03b2\n\u22a2 a\u271d \u2208 \u2191(PEquiv.trans f (PEquiv.refl \u03b2)) x\u271d \u2194 a\u271d \u2208 \u2191f x\u271d\n[PROOFSTEP]\ndsimp [PEquiv.trans]\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d : \u03b1\na\u271d : \u03b2\n\u22a2 a\u271d \u2208 Option.bind (\u2191f x\u271d) \u2191(PEquiv.refl \u03b2) \u2194 a\u271d \u2208 \u2191f x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\na\u2081 a\u2082 : \u03b1\nb : \u03b2\nh\u2081 : b \u2208 \u2191f a\u2081\nh\u2082 : b \u2208 \u2191f a\u2082\n\u22a2 a\u2081 = a\u2082\n[PROOFSTEP]\nrw [\u2190 mem_iff_mem] at *\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\na\u2081 a\u2082 : \u03b1\nb : \u03b2\nh\u2081 : a\u2081 \u2208 \u2191(PEquiv.symm f) b\nh\u2082 : a\u2082 \u2208 \u2191(PEquiv.symm f) b\n\u22a2 a\u2081 = a\u2082\n[PROOFSTEP]\ncases h : f.symm b\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\na\u2081 a\u2082 : \u03b1\nb : \u03b2\nh\u2081 : a\u2081 \u2208 \u2191(PEquiv.symm f) b\nh\u2082 : a\u2082 \u2208 \u2191(PEquiv.symm f) b\nh : \u2191(PEquiv.symm f) b = none\n\u22a2 a\u2081 = a\u2082\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\na\u2081 a\u2082 : \u03b1\nb : \u03b2\nh\u2081 : a\u2081 \u2208 \u2191(PEquiv.symm f) b\nh\u2082 : a\u2082 \u2208 \u2191(PEquiv.symm f) b\nval\u271d : \u03b1\nh : \u2191(PEquiv.symm f) b = some val\u271d\n\u22a2 a\u2081 = a\u2082\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\na\u2082 : \u03b1\nh : \u2200 (a\u2081 : \u03b1), a\u2081 \u2260 a\u2082 \u2192 isSome (\u2191f a\u2081) = true\nx : \u03b1\n\u22a2 (fun b => Option.recOn b a\u2082 fun b' => Option.recOn (\u2191(PEquiv.symm f) b') a\u2082 id) (\u2191f x) = x\n[PROOFSTEP]\nclassical\ncases hfx : f x\n\u00b7 have : x = a\u2082 := not_imp_comm.1 (h x) (hfx.symm \u25b8 by simp)\n simp [this]\n\u00b7 dsimp only\n rw [(eq_some_iff f).2 hfx]\n rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\na\u2082 : \u03b1\nh : \u2200 (a\u2081 : \u03b1), a\u2081 \u2260 a\u2082 \u2192 isSome (\u2191f a\u2081) = true\nx : \u03b1\n\u22a2 (fun b => Option.recOn b a\u2082 fun b' => Option.recOn (\u2191(PEquiv.symm f) b') a\u2082 id) (\u2191f x) = x\n[PROOFSTEP]\ncases hfx : f x\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\na\u2082 : \u03b1\nh : \u2200 (a\u2081 : \u03b1), a\u2081 \u2260 a\u2082 \u2192 isSome (\u2191f a\u2081) = true\nx : \u03b1\nhfx : \u2191f x = none\n\u22a2 (fun b => Option.recOn b a\u2082 fun b' => Option.recOn (\u2191(PEquiv.symm f) b') a\u2082 id) none = x\n[PROOFSTEP]\nhave : x = a\u2082 := not_imp_comm.1 (h x) (hfx.symm \u25b8 by simp)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\na\u2082 : \u03b1\nh : \u2200 (a\u2081 : \u03b1), a\u2081 \u2260 a\u2082 \u2192 isSome (\u2191f a\u2081) = true\nx : \u03b1\nhfx : \u2191f x = none\n\u22a2 \u00acisSome none = true\n[PROOFSTEP]\nsimp\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\na\u2082 : \u03b1\nh : \u2200 (a\u2081 : \u03b1), a\u2081 \u2260 a\u2082 \u2192 isSome (\u2191f a\u2081) = true\nx : \u03b1\nhfx : \u2191f x = none\nthis : x = a\u2082\n\u22a2 (fun b => Option.recOn b a\u2082 fun b' => Option.recOn (\u2191(PEquiv.symm f) b') a\u2082 id) none = x\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\na\u2082 : \u03b1\nh : \u2200 (a\u2081 : \u03b1), a\u2081 \u2260 a\u2082 \u2192 isSome (\u2191f a\u2081) = true\nx : \u03b1\nval\u271d : \u03b2\nhfx : \u2191f x = some val\u271d\n\u22a2 (fun b => Option.recOn b a\u2082 fun b' => Option.recOn (\u2191(PEquiv.symm f) b') a\u2082 id) (some val\u271d) = x\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\na\u2082 : \u03b1\nh : \u2200 (a\u2081 : \u03b1), a\u2081 \u2260 a\u2082 \u2192 isSome (\u2191f a\u2081) = true\nx : \u03b1\nval\u271d : \u03b2\nhfx : \u2191f x = some val\u271d\n\u22a2 Option.rec a\u2082 id (\u2191(PEquiv.symm f) val\u271d) = x\n[PROOFSTEP]\nrw [(eq_some_iff f).2 hfx]\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\na\u2082 : \u03b1\nh : \u2200 (a\u2081 : \u03b1), a\u2081 \u2260 a\u2082 \u2192 isSome (\u2191f a\u2081) = true\nx : \u03b1\nval\u271d : \u03b2\nhfx : \u2191f x = some val\u271d\n\u22a2 Option.rec a\u2082 id (some x) = x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na b : \u03b1\n\u22a2 a \u2208 (fun a => if a \u2208 s then some a else none) b \u2194 b \u2208 (fun a => if a \u2208 s then some a else none) a\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na b : \u03b1\n\u22a2 (a \u2208 if b \u2208 s then some b else none) \u2194 b \u2208 if a \u2208 s then some a else none\n[PROOFSTEP]\nsplit_ifs with hb ha ha\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na b : \u03b1\nhb : b \u2208 s\nha : a \u2208 s\n\u22a2 a \u2208 some b \u2194 b \u2208 some a\n[PROOFSTEP]\nsimp [eq_comm]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na b : \u03b1\nhb : b \u2208 s\nha : \u00aca \u2208 s\n\u22a2 a \u2208 some b \u2194 b \u2208 none\n[PROOFSTEP]\nsimp [ne_of_mem_of_not_mem hb ha]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na b : \u03b1\nhb : \u00acb \u2208 s\nha : a \u2208 s\n\u22a2 a \u2208 none \u2194 b \u2208 some a\n[PROOFSTEP]\nsimp [ne_of_mem_of_not_mem ha hb]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na b : \u03b1\nhb : \u00acb \u2208 s\nha : \u00aca \u2208 s\n\u22a2 a \u2208 none \u2194 b \u2208 none\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\n\u22a2 a \u2208 \u2191(ofSet s) a \u2194 a \u2208 s\n[PROOFSTEP]\ndsimp [ofSet]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\n\u22a2 (a \u2208 if a \u2208 s then some a else none) \u2194 a \u2208 s\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\nh\u271d : a \u2208 s\n\u22a2 a \u2208 some a \u2194 a \u2208 s\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\nh\u271d : \u00aca \u2208 s\n\u22a2 a \u2208 none \u2194 a \u2208 s\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na b : \u03b1\n\u22a2 a \u2208 \u2191(ofSet s) b \u2194 a = b \u2227 a \u2208 s\n[PROOFSTEP]\ndsimp [ofSet]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na b : \u03b1\n\u22a2 (a \u2208 if b \u2208 s then some b else none) \u2194 a = b \u2227 a \u2208 s\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na b : \u03b1\nh : b \u2208 s\n\u22a2 a \u2208 some b \u2194 a = b \u2227 a \u2208 s\n[PROOFSTEP]\nsimp only [mem_def, eq_comm, some.injEq, iff_self_and]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na b : \u03b1\nh : b \u2208 s\n\u22a2 a = b \u2192 a \u2208 s\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\nh : a \u2208 s\n\u22a2 a \u2208 s\n[PROOFSTEP]\nexact h\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na b : \u03b1\nh : \u00acb \u2208 s\n\u22a2 a \u2208 none \u2194 a = b \u2227 a \u2208 s\n[PROOFSTEP]\nsimp only [mem_def, false_iff, not_and]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na b : \u03b1\nh : \u00acb \u2208 s\n\u22a2 a = b \u2192 \u00aca \u2208 s\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\na : \u03b1\nh : \u00aca \u2208 s\n\u22a2 \u00aca \u2208 s\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nh : ofSet s = PEquiv.refl \u03b1\n\u22a2 s = Set.univ\n[PROOFSTEP]\nrw [Set.eq_univ_iff_forall]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nh : ofSet s = PEquiv.refl \u03b1\n\u22a2 \u2200 (x : \u03b1), x \u2208 s\n[PROOFSTEP]\nintro\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nh : ofSet s = PEquiv.refl \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 s\n[PROOFSTEP]\nrw [\u2190 mem_ofSet_self_iff, h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nh : ofSet s = PEquiv.refl \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(PEquiv.refl \u03b1) x\u271d\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ns\u271d : Set \u03b1\ninst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d\ns : Set \u03b1\ninst\u271d : DecidablePred fun x => x \u2208 s\nh : s = Set.univ\n\u22a2 ofSet s = PEquiv.refl \u03b1\n[PROOFSTEP]\nsimp only [\u2190 ofSet_univ, h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\n\u22a2 PEquiv.trans f (PEquiv.symm f) = ofSet {a | isSome (\u2191f a) = true}\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\n\u22a2 a\u271d \u2208 \u2191(PEquiv.trans f (PEquiv.symm f)) x\u271d \u2194 a\u271d \u2208 \u2191(ofSet {a | isSome (\u2191f a) = true}) x\u271d\n[PROOFSTEP]\ndsimp [PEquiv.trans]\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\n\u22a2 a\u271d \u2208 Option.bind (\u2191f x\u271d) \u2191(PEquiv.symm f) \u2194 a\u271d \u2208 \u2191(ofSet {a | isSome (\u2191f a) = true}) x\u271d\n[PROOFSTEP]\nsimp only [eq_some_iff f, Option.isSome_iff_exists, Option.mem_def, bind_eq_some', ofSet_eq_some_iff]\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\n\u22a2 (\u2203 a, \u2191f x\u271d = some a \u2227 \u2191f a\u271d = some a) \u2194 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.a.mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\n\u22a2 (\u2203 a, \u2191f x\u271d = some a \u2227 \u2191f a\u271d = some a) \u2192 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1}\n[PROOFSTEP]\nrintro \u27e8b, hb\u2081, hb\u2082\u27e9\n[GOAL]\ncase h.a.mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\nb : \u03b2\nhb\u2081 : \u2191f x\u271d = some b\nhb\u2082 : \u2191f a\u271d = some b\n\u22a2 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1}\n[PROOFSTEP]\nexact \u27e8PEquiv.inj _ hb\u2082 hb\u2081, b, hb\u2082\u27e9\n[GOAL]\ncase h.a.mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d a\u271d : \u03b1\n\u22a2 a\u271d = x\u271d \u2227 a\u271d \u2208 {a | \u2203 a_1, \u2191f a = some a_1} \u2192 \u2203 a, \u2191f x\u271d = some a \u2227 \u2191f a\u271d = some a\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\n\u22a2 PEquiv.symm (PEquiv.trans (PEquiv.symm f) f) = PEquiv.symm (ofSet {b | isSome (\u2191(PEquiv.symm f) b) = true})\n[PROOFSTEP]\nsimp [symm_trans_rev, self_trans_symm, -symm_symm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\n\u22a2 PEquiv.trans f (PEquiv.symm f) = PEquiv.refl \u03b1 \u2194 \u2200 (a : \u03b1), isSome (\u2191f a) = true\n[PROOFSTEP]\nrw [self_trans_symm, ofSet_eq_refl, Set.eq_univ_iff_forall]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\n\u22a2 (\u2200 (x : \u03b1), x \u2208 {a | isSome (\u2191f a) = true}) \u2194 \u2200 (a : \u03b1), isSome (\u2191f a) = true\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\n\u22a2 \u2200 (a : \u03b1) (b : \u03b2), a \u2208 (fun x => none) b \u2194 b \u2208 (fun x => none) a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\n\u22a2 PEquiv.trans f \u22a5 = \u22a5\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d : \u03b1\na\u271d : \u03b3\n\u22a2 a\u271d \u2208 \u2191(PEquiv.trans f \u22a5) x\u271d \u2194 a\u271d \u2208 \u2191\u22a5 x\u271d\n[PROOFSTEP]\ndsimp [PEquiv.trans]\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\nx\u271d : \u03b1\na\u271d : \u03b3\n\u22a2 a\u271d \u2208 Option.bind (\u2191f x\u271d) \u2191\u22a5 \u2194 a\u271d \u2208 none\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b2 \u2243. \u03b3\n\u22a2 PEquiv.trans \u22a5 f = \u22a5\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b2 \u2243. \u03b3\nx\u271d : \u03b1\na\u271d : \u03b3\n\u22a2 a\u271d \u2208 \u2191(PEquiv.trans \u22a5 f) x\u271d \u2194 a\u271d \u2208 \u2191\u22a5 x\u271d\n[PROOFSTEP]\ndsimp [PEquiv.trans]\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b2 \u2243. \u03b3\nx\u271d : \u03b1\na\u271d : \u03b3\n\u22a2 a\u271d \u2208 none \u2194 a\u271d \u2208 none\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf : \u03b1 \u2243. \u03b2\na : \u03b1\nh : isSome (\u2191f a) = true\n\u22a2 \u2191(PEquiv.symm f) (Option.get (\u2191f a) h) = some a\n[PROOFSTEP]\nrw [f.eq_some_iff, some_get]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 x \u2208 (fun x => if x = b then some a else none) y \u2194 y \u2208 (fun x => if x = a then some b else none) x\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\n\u22a2 (x \u2208 if y = b then some a else none) \u2194 y \u2208 if x = a then some b else none\n[PROOFSTEP]\nsplit_ifs with h1 h2\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\nh1 : y = b\nh2 : x = a\n\u22a2 x \u2208 some a \u2194 y \u2208 some b\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\nh1 : y = b\nh2 : \u00acx = a\n\u22a2 x \u2208 some a \u2194 y \u2208 none\n[PROOFSTEP]\nsimp only [mem_def, some.injEq, iff_false] at *\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\nh1 : y = b\nh2 : \u00acx = a\n\u22a2 \u00aca = x\n[PROOFSTEP]\nexact Ne.symm h2\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\nh1 : \u00acy = b\nh\u271d : x = a\n\u22a2 x \u2208 none \u2194 y \u2208 some b\n[PROOFSTEP]\nsimp only [mem_def, some.injEq, false_iff] at *\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\nh1 : \u00acy = b\nh\u271d : x = a\n\u22a2 \u00acb = y\n[PROOFSTEP]\nexact Ne.symm h1\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na : \u03b1\nb : \u03b2\nx : \u03b1\ny : \u03b2\nh1 : \u00acy = b\nh\u271d : \u00acx = a\n\u22a2 x \u2208 none \u2194 y \u2208 none\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : \u03b2\n\u22a2 b\u2081 \u2208 \u2191(single a\u2082 b\u2082) a\u2081 \u2194 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082\n[PROOFSTEP]\ndsimp [single]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : \u03b2\n\u22a2 (b\u2081 \u2208 if a\u2081 = a\u2082 then some b\u2082 else none) \u2194 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : \u03b2\nh\u271d : a\u2081 = a\u2082\n\u22a2 b\u2081 \u2208 some b\u2082 \u2194 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082\n[PROOFSTEP]\nsimp [*, eq_comm]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : \u03b2\nh\u271d : \u00aca\u2081 = a\u2082\n\u22a2 b\u2081 \u2208 none \u2194 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082\n[PROOFSTEP]\nsimp [*, eq_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na : \u03b1\nb : \u03b2\nc : \u03b3\nf : \u03b2 \u2243. \u03b3\nh : c \u2208 \u2191f b\n\u22a2 PEquiv.trans (single a b) f = single a c\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na : \u03b1\nb : \u03b2\nc : \u03b3\nf : \u03b2 \u2243. \u03b3\nh : c \u2208 \u2191f b\nx\u271d : \u03b1\na\u271d : \u03b3\n\u22a2 a\u271d \u2208 \u2191(PEquiv.trans (single a b) f) x\u271d \u2194 a\u271d \u2208 \u2191(single a c) x\u271d\n[PROOFSTEP]\ndsimp [single, PEquiv.trans]\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na : \u03b1\nb : \u03b2\nc : \u03b3\nf : \u03b2 \u2243. \u03b3\nh : c \u2208 \u2191f b\nx\u271d : \u03b1\na\u271d : \u03b3\n\u22a2 a\u271d \u2208 Option.bind (if x\u271d = a then some b else none) \u2191f \u2194 a\u271d \u2208 if x\u271d = a then some c else none\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na : \u03b1\nb : \u03b2\nc : \u03b3\nf : \u03b2 \u2243. \u03b3\nh : c \u2208 \u2191f b\nx\u271d : \u03b1\na\u271d : \u03b3\nh\u271d : x\u271d = a\n\u22a2 a\u271d \u2208 Option.bind (some b) \u2191f \u2194 a\u271d \u2208 some c\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na : \u03b1\nb : \u03b2\nc : \u03b3\nf : \u03b2 \u2243. \u03b3\nh : c \u2208 \u2191f b\nx\u271d : \u03b1\na\u271d : \u03b3\nh\u271d : \u00acx\u271d = a\n\u22a2 a\u271d \u2208 Option.bind none \u2191f \u2194 a\u271d \u2208 none\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b3\ninst\u271d : Subsingleton \u03b1\na b : \u03b1\n\u22a2 single a b = PEquiv.refl \u03b1\n[PROOFSTEP]\next i j\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b3\ninst\u271d : Subsingleton \u03b1\na b i j : \u03b1\n\u22a2 j \u2208 \u2191(single a b) i \u2194 j \u2208 \u2191(PEquiv.refl \u03b1) i\n[PROOFSTEP]\ndsimp [single]\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b3\ninst\u271d : Subsingleton \u03b1\na b i j : \u03b1\n\u22a2 (j \u2208 if i = a then some b else none) \u2194 j \u2208 some i\n[PROOFSTEP]\nrw [if_pos (Subsingleton.elim i a), Subsingleton.elim i j, Subsingleton.elim b j]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\nb : \u03b2\nc : \u03b3\nf : \u03b4 \u2243. \u03b2\nh : \u2191(PEquiv.symm f) b = none\n\u22a2 PEquiv.trans f (single b c) = \u22a5\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\nb : \u03b2\nc : \u03b3\nf : \u03b4 \u2243. \u03b2\nh : \u2191(PEquiv.symm f) b = none\nx\u271d : \u03b4\na\u271d : \u03b3\n\u22a2 a\u271d \u2208 \u2191(PEquiv.trans f (single b c)) x\u271d \u2194 a\u271d \u2208 \u2191\u22a5 x\u271d\n[PROOFSTEP]\nsimp only [eq_none_iff_forall_not_mem, Option.mem_def, f.eq_some_iff] at h \n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\nb : \u03b2\nc : \u03b3\nf : \u03b4 \u2243. \u03b2\nx\u271d : \u03b4\na\u271d : \u03b3\nh : \u2200 (a : \u03b4), \u00ac\u2191f a = some b\n\u22a2 a\u271d \u2208 \u2191(PEquiv.trans f (single b c)) x\u271d \u2194 a\u271d \u2208 \u2191\u22a5 x\u271d\n[PROOFSTEP]\ndsimp [PEquiv.trans, single]\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\nb : \u03b2\nc : \u03b3\nf : \u03b4 \u2243. \u03b2\nx\u271d : \u03b4\na\u271d : \u03b3\nh : \u2200 (a : \u03b4), \u00ac\u2191f a = some b\n\u22a2 (a\u271d \u2208 Option.bind (\u2191f x\u271d) fun x => if x = b then some c else none) \u2194 a\u271d \u2208 none\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\nb : \u03b2\nc : \u03b3\nf : \u03b4 \u2243. \u03b2\nx\u271d : \u03b4\na\u271d : \u03b3\nh : \u2200 (a : \u03b4), \u00ac\u2191f a = some b\n\u22a2 \u2200 (x : \u03b2), \u2191f x\u271d = some x \u2192 \u00ac(if x = b then some c else none) = some a\u271d\n[PROOFSTEP]\nintros\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\nb : \u03b2\nc : \u03b3\nf : \u03b4 \u2243. \u03b2\nx\u271d\u00b9 : \u03b4\na\u271d\u00b9 : \u03b3\nh : \u2200 (a : \u03b4), \u00ac\u2191f a = some b\nx\u271d : \u03b2\na\u271d : \u2191f x\u271d\u00b9 = some x\u271d\n\u22a2 \u00ac(if x\u271d = b then some c else none) = some a\u271d\u00b9\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\nb : \u03b2\nc : \u03b3\nf : \u03b4 \u2243. \u03b2\nx\u271d\u00b9 : \u03b4\na\u271d\u00b9 : \u03b3\nh : \u2200 (a : \u03b4), \u00ac\u2191f a = some b\nx\u271d : \u03b2\na\u271d : \u2191f x\u271d\u00b9 = some x\u271d\nh\u271d : x\u271d = b\n\u22a2 \u00acsome c = some a\u271d\u00b9\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\nb : \u03b2\nc : \u03b3\nf : \u03b4 \u2243. \u03b2\nx\u271d\u00b9 : \u03b4\na\u271d\u00b9 : \u03b3\nh : \u2200 (a : \u03b4), \u00ac\u2191f a = some b\nx\u271d : \u03b2\na\u271d : \u2191f x\u271d\u00b9 = some x\u271d\nh\u271d : \u00acx\u271d = b\n\u22a2 \u00acFalse\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf g : \u03b1 \u2243. \u03b2\nfg : f \u2264 g\ngf : g \u2264 f\n\u22a2 \u2200 (x : \u03b1), \u2191f x = \u2191g x\n[PROOFSTEP]\nintro a\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf g : \u03b1 \u2243. \u03b2\nfg : f \u2264 g\ngf : g \u2264 f\na : \u03b1\n\u22a2 \u2191f a = \u2191g a\n[PROOFSTEP]\ncases' h : g a with b\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf g : \u03b1 \u2243. \u03b2\nfg : f \u2264 g\ngf : g \u2264 f\na : \u03b1\nh : \u2191g a = none\n\u22a2 \u2191f a = none\n[PROOFSTEP]\nexact eq_none_iff_forall_not_mem.2 fun b hb => Option.not_mem_none b <| h \u25b8 fg a b hb\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nf g : \u03b1 \u2243. \u03b2\nfg : f \u2264 g\ngf : g \u2264 f\na : \u03b1\nb : \u03b2\nh : \u2191g a = some b\n\u22a2 \u2191f a = some b\n[PROOFSTEP]\nexact gf _ _ h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 a \u2208 (fun b => if \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b then \u2191(PEquiv.symm f) b else none) b \u2194\n b \u2208 (fun a => if \u2191f a = \u2191g a then \u2191f a else none) a\n[PROOFSTEP]\nhave hf := @mem_iff_mem _ _ f a b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : a \u2208 \u2191(PEquiv.symm f) b \u2194 b \u2208 \u2191f a\n\u22a2 a \u2208 (fun b => if \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b then \u2191(PEquiv.symm f) b else none) b \u2194\n b \u2208 (fun a => if \u2191f a = \u2191g a then \u2191f a else none) a\n[PROOFSTEP]\nhave hg := @mem_iff_mem _ _ g a b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : a \u2208 \u2191(PEquiv.symm f) b \u2194 b \u2208 \u2191f a\nhg : a \u2208 \u2191(PEquiv.symm g) b \u2194 b \u2208 \u2191g a\n\u22a2 a \u2208 (fun b => if \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b then \u2191(PEquiv.symm f) b else none) b \u2194\n b \u2208 (fun a => if \u2191f a = \u2191g a then \u2191f a else none) a\n[PROOFSTEP]\nsimp only [Option.mem_def] at *\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 \u2191g a = some b\n\u22a2 (if \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b then \u2191(PEquiv.symm f) b else none) = some a \u2194\n (if \u2191f a = \u2191g a then \u2191f a else none) = some b\n[PROOFSTEP]\nsplit_ifs with h1 h2 h2\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 \u2191g a = some b\nh1 : \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\nh2 : \u2191f a = \u2191g a\n\u22a2 \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\n[PROOFSTEP]\ntry simp [hf]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 \u2191g a = some b\nh1 : \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\nh2 : \u2191f a = \u2191g a\n\u22a2 \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\n[PROOFSTEP]\nsimp [hf]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 \u2191g a = some b\nh1 : \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\nh2 : \u00ac\u2191f a = \u2191g a\n\u22a2 \u2191(PEquiv.symm f) b = some a \u2194 False\n[PROOFSTEP]\ntry simp [hf]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 \u2191g a = some b\nh1 : \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\nh2 : \u00ac\u2191f a = \u2191g a\n\u22a2 \u2191(PEquiv.symm f) b = some a \u2194 False\n[PROOFSTEP]\nsimp [hf]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 \u2191g a = some b\nh1 : \u00ac\u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\nh2 : \u2191f a = \u2191g a\n\u22a2 False \u2194 \u2191f a = some b\n[PROOFSTEP]\ntry simp [hf]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 \u2191g a = some b\nh1 : \u00ac\u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\nh2 : \u2191f a = \u2191g a\n\u22a2 False \u2194 \u2191f a = some b\n[PROOFSTEP]\nsimp [hf]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 \u2191g a = some b\nh1 : \u00ac\u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\nh2 : \u00ac\u2191f a = \u2191g a\n\u22a2 False \u2194 False\n[PROOFSTEP]\ntry simp [hf]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 \u2191g a = some b\nh1 : \u00ac\u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\nh2 : \u00ac\u2191f a = \u2191g a\n\u22a2 False \u2194 False\n[PROOFSTEP]\nsimp [hf]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 \u2191g a = some b\nh1 : \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\nh2 : \u00ac\u2191f a = \u2191g a\n\u22a2 \u00ac\u2191f a = some b\n[PROOFSTEP]\ncontrapose! h2\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 \u2191g a = some b\nh1 : \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\nh2 : \u2191f a = some b\n\u22a2 \u2191f a = \u2191g a\n[PROOFSTEP]\nrw [h2]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 \u2191g a = some b\nh1 : \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\nh2 : \u2191f a = some b\n\u22a2 some b = \u2191g a\n[PROOFSTEP]\nrw [\u2190 h1, hf, h2] at hg \n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nhg : some b = some b \u2194 \u2191g a = some b\nh1 : \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\nh2 : \u2191f a = some b\n\u22a2 some b = \u2191g a\n[PROOFSTEP]\nsimp only [mem_def, true_iff_iff, eq_self_iff_true] at hg \n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nh1 : \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\nh2 : \u2191f a = some b\nhg : \u2191g a = some b\n\u22a2 some b = \u2191g a\n[PROOFSTEP]\nrw [hg]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 \u2191g a = some b\nh1 : \u00ac\u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\nh2 : \u2191f a = \u2191g a\n\u22a2 \u00ac\u2191f a = some b\n[PROOFSTEP]\ncontrapose! h1\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 \u2191f a = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 \u2191g a = some b\nh2 : \u2191f a = \u2191g a\nh1 : \u2191f a = some b\n\u22a2 \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\n[PROOFSTEP]\nrw [h1] at hf h2 \n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 some b = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 \u2191g a = some b\nh2 : some b = \u2191g a\nh1 : \u2191f a = some b\n\u22a2 \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\n[PROOFSTEP]\nrw [\u2190 h2] at hg \n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nhf : \u2191(PEquiv.symm f) b = some a \u2194 some b = some b\nhg : \u2191(PEquiv.symm g) b = some a \u2194 some b = some b\nh2 : some b = \u2191g a\nh1 : \u2191f a = some b\n\u22a2 \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\n[PROOFSTEP]\nsimp only [iff_true] at hf hg \n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g : \u03b1 \u2243. \u03b2\na : \u03b1\nb : \u03b2\nh2 : some b = \u2191g a\nh1 : \u2191f a = some b\nhf : \u2191(PEquiv.symm f) b = some a\nhg : \u2191(PEquiv.symm g) b = some a\n\u22a2 \u2191(PEquiv.symm f) b = \u2191(PEquiv.symm g) b\n[PROOFSTEP]\nrw [hf, hg]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nx\u271d\u00b3 x\u271d\u00b2 : \u03b1 \u2243. \u03b2\nx\u271d\u00b9 : \u03b1\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 \u2191(x\u271d\u00b3 \u2293 x\u271d\u00b2) x\u271d\u00b9 \u2192 x\u271d \u2208 \u2191x\u271d\u00b3 x\u271d\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nx\u271d\u00b3 x\u271d\u00b2 : \u03b1 \u2243. \u03b2\nx\u271d\u00b9 : \u03b1\nx\u271d : \u03b2\n\u22a2 (if \u2191x\u271d\u00b3 x\u271d\u00b9 = \u2191x\u271d\u00b2 x\u271d\u00b9 then \u2191x\u271d\u00b3 x\u271d\u00b9 else none) = some x\u271d \u2192 \u2191x\u271d\u00b3 x\u271d\u00b9 = some x\u271d\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nx\u271d\u00b3 x\u271d\u00b2 : \u03b1 \u2243. \u03b2\nx\u271d\u00b9 : \u03b1\nx\u271d : \u03b2\nh\u271d : \u2191x\u271d\u00b3 x\u271d\u00b9 = \u2191x\u271d\u00b2 x\u271d\u00b9\n\u22a2 \u2191x\u271d\u00b3 x\u271d\u00b9 = some x\u271d \u2192 \u2191x\u271d\u00b3 x\u271d\u00b9 = some x\u271d\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nx\u271d\u00b3 x\u271d\u00b2 : \u03b1 \u2243. \u03b2\nx\u271d\u00b9 : \u03b1\nx\u271d : \u03b2\nh\u271d : \u00ac\u2191x\u271d\u00b3 x\u271d\u00b9 = \u2191x\u271d\u00b2 x\u271d\u00b9\n\u22a2 False \u2192 \u2191x\u271d\u00b3 x\u271d\u00b9 = some x\u271d\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nx\u271d\u00b3 x\u271d\u00b2 : \u03b1 \u2243. \u03b2\nx\u271d\u00b9 : \u03b1\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 \u2191(x\u271d\u00b3 \u2293 x\u271d\u00b2) x\u271d\u00b9 \u2192 x\u271d \u2208 \u2191x\u271d\u00b2 x\u271d\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nx\u271d\u00b3 x\u271d\u00b2 : \u03b1 \u2243. \u03b2\nx\u271d\u00b9 : \u03b1\nx\u271d : \u03b2\n\u22a2 (if \u2191x\u271d\u00b3 x\u271d\u00b9 = \u2191x\u271d\u00b2 x\u271d\u00b9 then \u2191x\u271d\u00b3 x\u271d\u00b9 else none) = some x\u271d \u2192 \u2191x\u271d\u00b2 x\u271d\u00b9 = some x\u271d\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nx\u271d\u00b3 x\u271d\u00b2 : \u03b1 \u2243. \u03b2\nx\u271d\u00b9 : \u03b1\nx\u271d : \u03b2\nh\u271d : \u2191x\u271d\u00b3 x\u271d\u00b9 = \u2191x\u271d\u00b2 x\u271d\u00b9\n\u22a2 \u2191x\u271d\u00b3 x\u271d\u00b9 = some x\u271d \u2192 \u2191x\u271d\u00b2 x\u271d\u00b9 = some x\u271d\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nx\u271d\u00b3 x\u271d\u00b2 : \u03b1 \u2243. \u03b2\nx\u271d\u00b9 : \u03b1\nx\u271d : \u03b2\nh\u271d : \u00ac\u2191x\u271d\u00b3 x\u271d\u00b9 = \u2191x\u271d\u00b2 x\u271d\u00b9\n\u22a2 False \u2192 \u2191x\u271d\u00b2 x\u271d\u00b9 = some x\u271d\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g h : \u03b1 \u2243. \u03b2\nfg : f \u2264 g\ngh : f \u2264 h\na : \u03b1\nb : \u03b2\n\u22a2 b \u2208 \u2191f a \u2192 b \u2208 \u2191(g \u2293 h) a\n[PROOFSTEP]\nintro H\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g h : \u03b1 \u2243. \u03b2\nfg : f \u2264 g\ngh : f \u2264 h\na : \u03b1\nb : \u03b2\nH : b \u2208 \u2191f a\n\u22a2 b \u2208 \u2191(g \u2293 h) a\n[PROOFSTEP]\nhave hf := fg a b H\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g h : \u03b1 \u2243. \u03b2\nfg : f \u2264 g\ngh : f \u2264 h\na : \u03b1\nb : \u03b2\nH : b \u2208 \u2191f a\nhf : b \u2208 \u2191g a\n\u22a2 b \u2208 \u2191(g \u2293 h) a\n[PROOFSTEP]\nhave hg := gh a b H\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g h : \u03b1 \u2243. \u03b2\nfg : f \u2264 g\ngh : f \u2264 h\na : \u03b1\nb : \u03b2\nH : b \u2208 \u2191f a\nhf : b \u2208 \u2191g a\nhg : b \u2208 \u2191h a\n\u22a2 b \u2208 \u2191(g \u2293 h) a\n[PROOFSTEP]\nsimp only [Option.mem_def, PEquiv.coe_mk_apply] at *\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nsrc\u271d : PartialOrder (\u03b1 \u2243. \u03b2) := instPartialOrderPEquiv\nf g h : \u03b1 \u2243. \u03b2\nfg : f \u2264 g\ngh : f \u2264 h\na : \u03b1\nb : \u03b2\nH : \u2191f a = some b\nhf : \u2191g a = some b\nhg : \u2191h a = some b\n\u22a2 (if \u2191g a = \u2191h a then \u2191g a else none) = some b\n[PROOFSTEP]\nrw [hf, hg, if_pos rfl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2243 \u03b2\n\u22a2 \u2200 (a : \u03b1) (b : \u03b2), a \u2208 (some \u2218 \u2191f.symm) b \u2194 b \u2208 (some \u2218 \u2191f) a\n[PROOFSTEP]\nsimp [Equiv.eq_symm_apply, eq_comm]\n", "meta": {"mathlib_filename": "Mathlib.Data.PEquiv", "llama_tokens": 19860, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195385342971, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.41833996548870667}} {"text": "[GOAL]\nM : Type ?u.870\ninst\u271d : Semigroup M\nsrc\u271d : Mul (Ultrafilter M) := mul\nU V W : Ultrafilter M\np : M \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : M) in \u2191(U * V * W), p x) \u2194 \u2200\u1da0 (x : M) in \u2191(U * (V * W)), p x\n[PROOFSTEP]\nsimp_rw [Ultrafilter.eventually_mul, mul_assoc]\n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nm : M\nhm : m \u2208 FP a\n\u22a2 \u2203 n, \u2200 (m' : M), m' \u2208 FP (Stream'.drop n a) \u2192 m * m' \u2208 FP a\n[PROOFSTEP]\ninduction' hm with a a m hm ih a m hm ih\n[GOAL]\ncase head\nM : Type u_1\ninst\u271d : Semigroup M\na\u271d : Stream' M\nm : M\na : Stream' M\n\u22a2 \u2203 n, \u2200 (m' : M), m' \u2208 FP (Stream'.drop n a) \u2192 Stream'.head a * m' \u2208 FP a\n[PROOFSTEP]\nexact \u27e81, fun m hm => FP.cons a m hm\u27e9\n[GOAL]\ncase tail\nM : Type u_1\ninst\u271d : Semigroup M\na\u271d : Stream' M\nm\u271d : M\na : Stream' M\nm : M\nhm : FP (Stream'.tail a) m\nih : \u2203 n, \u2200 (m' : M), m' \u2208 FP (Stream'.drop n (Stream'.tail a)) \u2192 m * m' \u2208 FP (Stream'.tail a)\n\u22a2 \u2203 n, \u2200 (m' : M), m' \u2208 FP (Stream'.drop n a) \u2192 m * m' \u2208 FP a\n[PROOFSTEP]\ncases' ih with n hn\n[GOAL]\ncase tail.intro\nM : Type u_1\ninst\u271d : Semigroup M\na\u271d : Stream' M\nm\u271d : M\na : Stream' M\nm : M\nhm : FP (Stream'.tail a) m\nn : \u2115\nhn : \u2200 (m' : M), m' \u2208 FP (Stream'.drop n (Stream'.tail a)) \u2192 m * m' \u2208 FP (Stream'.tail a)\n\u22a2 \u2203 n, \u2200 (m' : M), m' \u2208 FP (Stream'.drop n a) \u2192 m * m' \u2208 FP a\n[PROOFSTEP]\nuse n + 1\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d : Semigroup M\na\u271d : Stream' M\nm\u271d : M\na : Stream' M\nm : M\nhm : FP (Stream'.tail a) m\nn : \u2115\nhn : \u2200 (m' : M), m' \u2208 FP (Stream'.drop n (Stream'.tail a)) \u2192 m * m' \u2208 FP (Stream'.tail a)\n\u22a2 \u2200 (m' : M), m' \u2208 FP (Stream'.drop (n + 1) a) \u2192 m * m' \u2208 FP a\n[PROOFSTEP]\nintro m' hm'\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d : Semigroup M\na\u271d : Stream' M\nm\u271d : M\na : Stream' M\nm : M\nhm : FP (Stream'.tail a) m\nn : \u2115\nhn : \u2200 (m' : M), m' \u2208 FP (Stream'.drop n (Stream'.tail a)) \u2192 m * m' \u2208 FP (Stream'.tail a)\nm' : M\nhm' : m' \u2208 FP (Stream'.drop (n + 1) a)\n\u22a2 m * m' \u2208 FP a\n[PROOFSTEP]\nexact FP.tail _ _ (hn _ hm')\n[GOAL]\ncase cons\nM : Type u_1\ninst\u271d : Semigroup M\na\u271d : Stream' M\nm\u271d : M\na : Stream' M\nm : M\nhm : FP (Stream'.tail a) m\nih : \u2203 n, \u2200 (m' : M), m' \u2208 FP (Stream'.drop n (Stream'.tail a)) \u2192 m * m' \u2208 FP (Stream'.tail a)\n\u22a2 \u2203 n, \u2200 (m' : M), m' \u2208 FP (Stream'.drop n a) \u2192 Stream'.head a * m * m' \u2208 FP a\n[PROOFSTEP]\ncases' ih with n hn\n[GOAL]\ncase cons.intro\nM : Type u_1\ninst\u271d : Semigroup M\na\u271d : Stream' M\nm\u271d : M\na : Stream' M\nm : M\nhm : FP (Stream'.tail a) m\nn : \u2115\nhn : \u2200 (m' : M), m' \u2208 FP (Stream'.drop n (Stream'.tail a)) \u2192 m * m' \u2208 FP (Stream'.tail a)\n\u22a2 \u2203 n, \u2200 (m' : M), m' \u2208 FP (Stream'.drop n a) \u2192 Stream'.head a * m * m' \u2208 FP a\n[PROOFSTEP]\nuse n + 1\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d : Semigroup M\na\u271d : Stream' M\nm\u271d : M\na : Stream' M\nm : M\nhm : FP (Stream'.tail a) m\nn : \u2115\nhn : \u2200 (m' : M), m' \u2208 FP (Stream'.drop n (Stream'.tail a)) \u2192 m * m' \u2208 FP (Stream'.tail a)\n\u22a2 \u2200 (m' : M), m' \u2208 FP (Stream'.drop (n + 1) a) \u2192 Stream'.head a * m * m' \u2208 FP a\n[PROOFSTEP]\nintro m' hm'\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d : Semigroup M\na\u271d : Stream' M\nm\u271d : M\na : Stream' M\nm : M\nhm : FP (Stream'.tail a) m\nn : \u2115\nhn : \u2200 (m' : M), m' \u2208 FP (Stream'.drop n (Stream'.tail a)) \u2192 m * m' \u2208 FP (Stream'.tail a)\nm' : M\nhm' : m' \u2208 FP (Stream'.drop (n + 1) a)\n\u22a2 Stream'.head a * m * m' \u2208 FP a\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d : Semigroup M\na\u271d : Stream' M\nm\u271d : M\na : Stream' M\nm : M\nhm : FP (Stream'.tail a) m\nn : \u2115\nhn : \u2200 (m' : M), m' \u2208 FP (Stream'.drop n (Stream'.tail a)) \u2192 m * m' \u2208 FP (Stream'.tail a)\nm' : M\nhm' : m' \u2208 FP (Stream'.drop (n + 1) a)\n\u22a2 Stream'.head a * (m * m') \u2208 FP a\n[PROOFSTEP]\nexact FP.cons _ _ (hn _ hm')\n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\n\u22a2 \u2203 U, U * U = U \u2227 \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP a\n[PROOFSTEP]\nlet S : Set (Ultrafilter M) := \u22c2 n, {U | \u2200\u1da0 m in U, m \u2208 FP (a.drop n)}\n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\n\u22a2 \u2203 U, U * U = U \u2227 \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP a\n[PROOFSTEP]\nhave h := exists_idempotent_in_compact_subsemigroup ?_ S ?_ ?_ ?_\n[GOAL]\ncase refine_5\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nh : \u2203 m, m \u2208 S \u2227 m * m = m\n\u22a2 \u2203 U, U * U = U \u2227 \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP a\n[PROOFSTEP]\nrcases h with \u27e8U, hU, U_idem\u27e9\n[GOAL]\ncase refine_5.intro.intro\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nU : Ultrafilter M\nhU : U \u2208 S\nU_idem : U * U = U\n\u22a2 \u2203 U, U * U = U \u2227 \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP a\n[PROOFSTEP]\nrefine' \u27e8U, U_idem, _\u27e9\n[GOAL]\ncase refine_5.intro.intro\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nU : Ultrafilter M\nhU : U \u2208 S\nU_idem : U * U = U\n\u22a2 \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP a\n[PROOFSTEP]\nconvert Set.mem_iInter.mp hU 0\n[GOAL]\ncase refine_1\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\n\u22a2 \u2200 (r : Ultrafilter M), Continuous fun x => x * r\n[PROOFSTEP]\nexact Ultrafilter.continuous_mul_left\n[GOAL]\ncase refine_2\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\n\u22a2 Set.Nonempty S\n[PROOFSTEP]\napply IsCompact.nonempty_iInter_of_sequence_nonempty_compact_closed\n[GOAL]\ncase refine_2.hZd\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\n\u22a2 \u2200 (i : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop (i + 1) a)} \u2286 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\n[PROOFSTEP]\nintro n U hU\n[GOAL]\ncase refine_2.hZd\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nn : \u2115\nU : Ultrafilter M\nhU : U \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop (n + 1) a)}\n\u22a2 U \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\n[PROOFSTEP]\napply Eventually.mono hU\n[GOAL]\ncase refine_2.hZd\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nn : \u2115\nU : Ultrafilter M\nhU : U \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop (n + 1) a)}\n\u22a2 \u2200 (x : M), x \u2208 FP (Stream'.drop (n + 1) a) \u2192 x \u2208 FP (Stream'.drop n a)\n[PROOFSTEP]\nrw [add_comm, \u2190 Stream'.drop_drop, \u2190 Stream'.tail_eq_drop]\n[GOAL]\ncase refine_2.hZd\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nn : \u2115\nU : Ultrafilter M\nhU : U \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop (n + 1) a)}\n\u22a2 \u2200 (x : M), x \u2208 FP (Stream'.tail (Stream'.drop n a)) \u2192 x \u2208 FP (Stream'.drop n a)\n[PROOFSTEP]\nexact FP.tail _\n[GOAL]\ncase refine_2.hZn\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\n\u22a2 \u2200 (i : \u2115), Set.Nonempty {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\n[PROOFSTEP]\nintro n\n[GOAL]\ncase refine_2.hZn\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nn : \u2115\n\u22a2 Set.Nonempty {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\n[PROOFSTEP]\nexact \u27e8pure _, mem_pure.mpr <| FP.head _\u27e9\n[GOAL]\ncase refine_2.hZ0\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\n\u22a2 IsCompact {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop 0 a)}\n[PROOFSTEP]\nexact (ultrafilter_isClosed_basic _).isCompact\n[GOAL]\ncase refine_2.hZcl\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\n\u22a2 \u2200 (i : \u2115), IsClosed {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\n[PROOFSTEP]\nintro n\n[GOAL]\ncase refine_2.hZcl\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nn : \u2115\n\u22a2 IsClosed {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\n[PROOFSTEP]\napply ultrafilter_isClosed_basic\n[GOAL]\ncase refine_3\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\n\u22a2 IsCompact S\n[PROOFSTEP]\nexact IsClosed.isCompact (isClosed_iInter fun i => ultrafilter_isClosed_basic _)\n[GOAL]\ncase refine_4\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\n\u22a2 \u2200 (x : Ultrafilter M), x \u2208 S \u2192 \u2200 (y : Ultrafilter M), y \u2208 S \u2192 x * y \u2208 S\n[PROOFSTEP]\nintro U hU V hV\n[GOAL]\ncase refine_4\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nU : Ultrafilter M\nhU : U \u2208 S\nV : Ultrafilter M\nhV : V \u2208 S\n\u22a2 U * V \u2208 S\n[PROOFSTEP]\nrw [Set.mem_iInter] at *\n[GOAL]\ncase refine_4\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nU : Ultrafilter M\nhU : \u2200 (i : \u2115), U \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nV : Ultrafilter M\nhV : \u2200 (i : \u2115), V \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\n\u22a2 \u2200 (i : \u2115), U * V \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\n[PROOFSTEP]\nintro n\n[GOAL]\ncase refine_4\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nU : Ultrafilter M\nhU : \u2200 (i : \u2115), U \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nV : Ultrafilter M\nhV : \u2200 (i : \u2115), V \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nn : \u2115\n\u22a2 U * V \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\n[PROOFSTEP]\nrw [Set.mem_setOf_eq, Ultrafilter.eventually_mul]\n[GOAL]\ncase refine_4\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nU : Ultrafilter M\nhU : \u2200 (i : \u2115), U \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nV : Ultrafilter M\nhV : \u2200 (i : \u2115), V \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nn : \u2115\n\u22a2 \u2200\u1da0 (m : M) in \u2191U, \u2200\u1da0 (m' : M) in \u2191V, m * m' \u2208 FP (Stream'.drop n a)\n[PROOFSTEP]\napply Eventually.mono (hU n)\n[GOAL]\ncase refine_4\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nU : Ultrafilter M\nhU : \u2200 (i : \u2115), U \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nV : Ultrafilter M\nhV : \u2200 (i : \u2115), V \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nn : \u2115\n\u22a2 \u2200 (x : M), x \u2208 FP (Stream'.drop n a) \u2192 \u2200\u1da0 (m' : M) in \u2191V, x * m' \u2208 FP (Stream'.drop n a)\n[PROOFSTEP]\nintro m hm\n[GOAL]\ncase refine_4\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nU : Ultrafilter M\nhU : \u2200 (i : \u2115), U \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nV : Ultrafilter M\nhV : \u2200 (i : \u2115), V \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nn : \u2115\nm : M\nhm : m \u2208 FP (Stream'.drop n a)\n\u22a2 \u2200\u1da0 (m' : M) in \u2191V, m * m' \u2208 FP (Stream'.drop n a)\n[PROOFSTEP]\nobtain \u27e8n', hn\u27e9 := FP.mul hm\n[GOAL]\ncase refine_4.intro\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nU : Ultrafilter M\nhU : \u2200 (i : \u2115), U \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nV : Ultrafilter M\nhV : \u2200 (i : \u2115), V \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nn : \u2115\nm : M\nhm : m \u2208 FP (Stream'.drop n a)\nn' : \u2115\nhn : \u2200 (m' : M), m' \u2208 FP (Stream'.drop n' (Stream'.drop n a)) \u2192 m * m' \u2208 FP (Stream'.drop n a)\n\u22a2 \u2200\u1da0 (m' : M) in \u2191V, m * m' \u2208 FP (Stream'.drop n a)\n[PROOFSTEP]\napply Eventually.mono (hV (n' + n))\n[GOAL]\ncase refine_4.intro\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nU : Ultrafilter M\nhU : \u2200 (i : \u2115), U \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nV : Ultrafilter M\nhV : \u2200 (i : \u2115), V \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nn : \u2115\nm : M\nhm : m \u2208 FP (Stream'.drop n a)\nn' : \u2115\nhn : \u2200 (m' : M), m' \u2208 FP (Stream'.drop n' (Stream'.drop n a)) \u2192 m * m' \u2208 FP (Stream'.drop n a)\n\u22a2 \u2200 (x : M), x \u2208 FP (Stream'.drop (n' + n) a) \u2192 m * x \u2208 FP (Stream'.drop n a)\n[PROOFSTEP]\nintro m' hm'\n[GOAL]\ncase refine_4.intro\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nU : Ultrafilter M\nhU : \u2200 (i : \u2115), U \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nV : Ultrafilter M\nhV : \u2200 (i : \u2115), V \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nn : \u2115\nm : M\nhm : m \u2208 FP (Stream'.drop n a)\nn' : \u2115\nhn : \u2200 (m' : M), m' \u2208 FP (Stream'.drop n' (Stream'.drop n a)) \u2192 m * m' \u2208 FP (Stream'.drop n a)\nm' : M\nhm' : m' \u2208 FP (Stream'.drop (n' + n) a)\n\u22a2 m * m' \u2208 FP (Stream'.drop n a)\n[PROOFSTEP]\napply hn\n[GOAL]\ncase refine_4.intro.a\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := \u22c2 (n : \u2115), {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop n a)}\nU : Ultrafilter M\nhU : \u2200 (i : \u2115), U \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nV : Ultrafilter M\nhV : \u2200 (i : \u2115), V \u2208 {U | \u2200\u1da0 (m : M) in \u2191U, m \u2208 FP (Stream'.drop i a)}\nn : \u2115\nm : M\nhm : m \u2208 FP (Stream'.drop n a)\nn' : \u2115\nhn : \u2200 (m' : M), m' \u2208 FP (Stream'.drop n' (Stream'.drop n a)) \u2192 m * m' \u2208 FP (Stream'.drop n a)\nm' : M\nhm' : m' \u2208 FP (Stream'.drop (n' + n) a)\n\u22a2 m' \u2208 FP (Stream'.drop n' (Stream'.drop n a))\n[PROOFSTEP]\nsimpa only [Stream'.drop_drop] using hm'\n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\ns\u2080 : Set M\nsU : s\u2080 \u2208 U\n\u22a2 \u2203 a, FP a \u2286 s\u2080\n[PROOFSTEP]\nhave exists_elem : \u2200 {s : Set M} (_hs : s \u2208 U), (s \u2229 {m | \u2200\u1da0 m' in U, m * m' \u2208 s}).Nonempty := fun {s} hs =>\n Ultrafilter.nonempty_of_mem (inter_mem hs <| by rwa [\u2190 U_idem] at hs )\n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\ns\u2080 : Set M\nsU : s\u2080 \u2208 U\ns : Set M\nhs : s \u2208 U\n\u22a2 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s} \u2208 \u2191U\n[PROOFSTEP]\nrwa [\u2190 U_idem] at hs \n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\ns\u2080 : Set M\nsU : s\u2080 \u2208 U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\n\u22a2 \u2203 a, FP a \u2286 s\u2080\n[PROOFSTEP]\nlet elem : { s // s \u2208 U } \u2192 M := fun p => (exists_elem p.property).some\n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\ns\u2080 : Set M\nsU : s\u2080 \u2208 U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\n\u22a2 \u2203 a, FP a \u2286 s\u2080\n[PROOFSTEP]\nlet succ : { s // s \u2208 U } \u2192 { s // s \u2208 U } := fun (p : { s // s \u2208 U }) =>\n \u27e8p.val \u2229 {m : M | elem p * m \u2208 p.val},\n inter_mem p.property\n (show (exists_elem p.property).some \u2208 {m : M | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 p.val} from\n p.val.inter_subset_right {m : M | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 p.val} (exists_elem p.property).some_mem)\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\ns\u2080 : Set M\nsU : s\u2080 \u2208 U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\n\u22a2 \u2203 a, FP a \u2286 s\u2080\n[PROOFSTEP]\nuse Stream'.corec elem succ (Subtype.mk s\u2080 sU)\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\ns\u2080 : Set M\nsU : s\u2080 \u2208 U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\n\u22a2 FP (Stream'.corec elem succ { val := s\u2080, property := sU }) \u2286 s\u2080\n[PROOFSTEP]\nsuffices \u2200 (a : Stream' M), \u2200 m \u2208 FP a, \u2200 p, a = Stream'.corec elem succ p \u2192 m \u2208 p.val\n by\n intro m hm\n exact this _ m hm \u27e8s\u2080, sU\u27e9 rfl\n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\ns\u2080 : Set M\nsU : s\u2080 \u2208 U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\nthis : \u2200 (a : Stream' M) (m : M), m \u2208 FP a \u2192 \u2200 (p : { s // s \u2208 U }), a = Stream'.corec elem succ p \u2192 m \u2208 \u2191p\n\u22a2 FP (Stream'.corec elem succ { val := s\u2080, property := sU }) \u2286 s\u2080\n[PROOFSTEP]\nintro m hm\n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\ns\u2080 : Set M\nsU : s\u2080 \u2208 U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\nthis : \u2200 (a : Stream' M) (m : M), m \u2208 FP a \u2192 \u2200 (p : { s // s \u2208 U }), a = Stream'.corec elem succ p \u2192 m \u2208 \u2191p\nm : M\nhm : m \u2208 FP (Stream'.corec elem succ { val := s\u2080, property := sU })\n\u22a2 m \u2208 s\u2080\n[PROOFSTEP]\nexact this _ m hm \u27e8s\u2080, sU\u27e9 rfl\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\ns\u2080 : Set M\nsU : s\u2080 \u2208 U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\n\u22a2 \u2200 (a : Stream' M) (m : M), m \u2208 FP a \u2192 \u2200 (p : { s // s \u2208 U }), a = Stream'.corec elem succ p \u2192 m \u2208 \u2191p\n[PROOFSTEP]\nclear sU s\u2080\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\n\u22a2 \u2200 (a : Stream' M) (m : M), m \u2208 FP a \u2192 \u2200 (p : { s // s \u2208 U }), a = Stream'.corec elem succ p \u2192 m \u2208 \u2191p\n[PROOFSTEP]\nintro a m h\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\na : Stream' M\nm : M\nh : m \u2208 FP a\n\u22a2 \u2200 (p : { s // s \u2208 U }), a = Stream'.corec elem succ p \u2192 m \u2208 \u2191p\n[PROOFSTEP]\ninduction' h with b b n h ih b n h ih\n[GOAL]\ncase h.head\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\na : Stream' M\nm : M\nb : Stream' M\n\u22a2 \u2200 (p : { s // s \u2208 U }), b = Stream'.corec elem succ p \u2192 Stream'.head b \u2208 \u2191p\n[PROOFSTEP]\nrintro p rfl\n[GOAL]\ncase h.head\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\na : Stream' M\nm : M\np : { s // s \u2208 U }\n\u22a2 Stream'.head (Stream'.corec elem succ p) \u2208 \u2191p\n[PROOFSTEP]\nrw [Stream'.corec_eq, Stream'.head_cons]\n[GOAL]\ncase h.head\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\na : Stream' M\nm : M\np : { s // s \u2208 U }\n\u22a2 elem p \u2208 \u2191p\n[PROOFSTEP]\nexact Set.inter_subset_left _ _ (Set.Nonempty.some_mem _)\n[GOAL]\ncase h.tail\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\na : Stream' M\nm : M\nb : Stream' M\nn : M\nh : FP (Stream'.tail b) n\nih : \u2200 (p : { s // s \u2208 U }), Stream'.tail b = Stream'.corec elem succ p \u2192 n \u2208 \u2191p\n\u22a2 \u2200 (p : { s // s \u2208 U }), b = Stream'.corec elem succ p \u2192 n \u2208 \u2191p\n[PROOFSTEP]\nrintro p rfl\n[GOAL]\ncase h.tail\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\na : Stream' M\nm n : M\np : { s // s \u2208 U }\nh : FP (Stream'.tail (Stream'.corec elem succ p)) n\nih : \u2200 (p_1 : { s // s \u2208 U }), Stream'.tail (Stream'.corec elem succ p) = Stream'.corec elem succ p_1 \u2192 n \u2208 \u2191p_1\n\u22a2 n \u2208 \u2191p\n[PROOFSTEP]\nrefine' Set.inter_subset_left _ _ (ih (succ p) _)\n[GOAL]\ncase h.tail\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\na : Stream' M\nm n : M\np : { s // s \u2208 U }\nh : FP (Stream'.tail (Stream'.corec elem succ p)) n\nih : \u2200 (p_1 : { s // s \u2208 U }), Stream'.tail (Stream'.corec elem succ p) = Stream'.corec elem succ p_1 \u2192 n \u2208 \u2191p_1\n\u22a2 Stream'.tail (Stream'.corec elem succ p) = Stream'.corec elem succ (succ p)\n[PROOFSTEP]\nrw [Stream'.corec_eq, Stream'.tail_cons]\n[GOAL]\ncase h.cons\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\na : Stream' M\nm : M\nb : Stream' M\nn : M\nh : FP (Stream'.tail b) n\nih : \u2200 (p : { s // s \u2208 U }), Stream'.tail b = Stream'.corec elem succ p \u2192 n \u2208 \u2191p\n\u22a2 \u2200 (p : { s // s \u2208 U }), b = Stream'.corec elem succ p \u2192 Stream'.head b * n \u2208 \u2191p\n[PROOFSTEP]\nrintro p rfl\n[GOAL]\ncase h.cons\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\na : Stream' M\nm n : M\np : { s // s \u2208 U }\nh : FP (Stream'.tail (Stream'.corec elem succ p)) n\nih : \u2200 (p_1 : { s // s \u2208 U }), Stream'.tail (Stream'.corec elem succ p) = Stream'.corec elem succ p_1 \u2192 n \u2208 \u2191p_1\n\u22a2 Stream'.head (Stream'.corec elem succ p) * n \u2208 \u2191p\n[PROOFSTEP]\nhave := Set.inter_subset_right _ _ (ih (succ p) ?_)\n[GOAL]\ncase h.cons.refine_2\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\na : Stream' M\nm n : M\np : { s // s \u2208 U }\nh : FP (Stream'.tail (Stream'.corec elem succ p)) n\nih : \u2200 (p_1 : { s // s \u2208 U }), Stream'.tail (Stream'.corec elem succ p) = Stream'.corec elem succ p_1 \u2192 n \u2208 \u2191p_1\nthis : n \u2208 {m | elem p * m \u2208 \u2191p}\n\u22a2 Stream'.head (Stream'.corec elem succ p) * n \u2208 \u2191p\n[PROOFSTEP]\nsimpa only using this\n[GOAL]\ncase h.cons.refine_1\nM : Type u_1\ninst\u271d : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\nexists_elem : \u2200 {s : Set M}, s \u2208 U \u2192 Set.Nonempty (s \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 s})\nelem : { s // s \u2208 U } \u2192 M := fun p => Set.Nonempty.some (_ : Set.Nonempty (\u2191p \u2229 {m | \u2200\u1da0 (m' : M) in \u2191U, m * m' \u2208 \u2191p}))\nsucc : { s // s \u2208 U } \u2192 { s // s \u2208 U } :=\n fun p => { val := \u2191p \u2229 {m | elem p * m \u2208 \u2191p}, property := (_ : \u2191p \u2229 {m | elem p * m \u2208 \u2191p} \u2208 \u2191U) }\na : Stream' M\nm n : M\np : { s // s \u2208 U }\nh : FP (Stream'.tail (Stream'.corec elem succ p)) n\nih : \u2200 (p_1 : { s // s \u2208 U }), Stream'.tail (Stream'.corec elem succ p) = Stream'.corec elem succ p_1 \u2192 n \u2208 \u2191p_1\n\u22a2 Stream'.tail (Stream'.corec elem succ p) = Stream'.corec elem succ (succ p)\n[PROOFSTEP]\nrw [Stream'.corec_eq, Stream'.tail_cons]\n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nn : \u2115\n\u22a2 FP (Stream'.drop n a) \u2286 FP a\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\n\u22a2 FP (Stream'.drop Nat.zero a) \u2286 FP a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nn : \u2115\nih : FP (Stream'.drop n a) \u2286 FP a\n\u22a2 FP (Stream'.drop (Nat.succ n) a) \u2286 FP a\n[PROOFSTEP]\nrw [Nat.succ_eq_one_add, \u2190 Stream'.drop_drop]\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\nn : \u2115\nih : FP (Stream'.drop n a) \u2286 FP a\n\u22a2 FP (Stream'.drop 1 (Stream'.drop n a)) \u2286 FP a\n[PROOFSTEP]\nexact _root_.trans (FP.tail _) ih\n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\ni : \u2115\n\u22a2 Stream'.nth a i \u2208 FP a\n[PROOFSTEP]\ninduction' i with i ih generalizing a\n[GOAL]\ncase zero\nM : Type u_1\ninst\u271d : Semigroup M\na\u271d a : Stream' M\n\u22a2 Stream'.nth a Nat.zero \u2208 FP a\n[PROOFSTEP]\napply FP.head\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d : Semigroup M\na\u271d : Stream' M\ni : \u2115\nih : \u2200 (a : Stream' M), Stream'.nth a i \u2208 FP a\na : Stream' M\n\u22a2 Stream'.nth a (Nat.succ i) \u2208 FP a\n[PROOFSTEP]\napply FP.tail\n[GOAL]\ncase succ.h\nM : Type u_1\ninst\u271d : Semigroup M\na\u271d : Stream' M\ni : \u2115\nih : \u2200 (a : Stream' M), Stream'.nth a i \u2208 FP a\na : Stream' M\n\u22a2 FP (Stream'.tail a) (Stream'.nth a (Nat.succ i))\n[PROOFSTEP]\napply ih\n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\ni j : \u2115\nij : i < j\n\u22a2 Stream'.nth a i * Stream'.nth a j \u2208 FP a\n[PROOFSTEP]\nrefine' FP_drop_subset_FP _ i _\n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\ni j : \u2115\nij : i < j\n\u22a2 Stream'.nth a i * Stream'.nth a j \u2208 FP (Stream'.drop i a)\n[PROOFSTEP]\nrw [\u2190 Stream'.head_drop]\n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\ni j : \u2115\nij : i < j\n\u22a2 Stream'.head (Stream'.drop i a) * Stream'.nth a j \u2208 FP (Stream'.drop i a)\n[PROOFSTEP]\napply FP.cons\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\ni j : \u2115\nij : i < j\n\u22a2 FP (Stream'.tail (Stream'.drop i a)) (Stream'.nth a j)\n[PROOFSTEP]\nrcases le_iff_exists_add.mp (Nat.succ_le_of_lt ij) with\n \u27e8d, hd\u27e9\n -- Porting note: need to fix breakage of Set notation\n[GOAL]\ncase h.intro\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\ni j : \u2115\nij : i < j\nd : \u2115\nhd : j = Nat.succ i + d\n\u22a2 FP (Stream'.tail (Stream'.drop i a)) (Stream'.nth a j)\n[PROOFSTEP]\nchange _ \u2208 FP _\n[GOAL]\ncase h.intro\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\ni j : \u2115\nij : i < j\nd : \u2115\nhd : j = Nat.succ i + d\n\u22a2 Stream'.nth a j \u2208 FP (Stream'.tail (Stream'.drop i a))\n[PROOFSTEP]\nhave := FP.singleton (a.drop i).tail d\n[GOAL]\ncase h.intro\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\ni j : \u2115\nij : i < j\nd : \u2115\nhd : j = Nat.succ i + d\nthis : Stream'.nth (Stream'.tail (Stream'.drop i a)) d \u2208 FP (Stream'.tail (Stream'.drop i a))\n\u22a2 Stream'.nth a j \u2208 FP (Stream'.tail (Stream'.drop i a))\n[PROOFSTEP]\nrw [Stream'.tail_eq_drop, Stream'.nth_drop, Stream'.nth_drop] at this \n[GOAL]\ncase h.intro\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\ni j : \u2115\nij : i < j\nd : \u2115\nhd : j = Nat.succ i + d\nthis : Stream'.nth a (d + 1 + i) \u2208 FP (Stream'.drop 1 (Stream'.drop i a))\n\u22a2 Stream'.nth a j \u2208 FP (Stream'.tail (Stream'.drop i a))\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_4.h.e'_3\nM : Type u_1\ninst\u271d : Semigroup M\na : Stream' M\ni j : \u2115\nij : i < j\nd : \u2115\nhd : j = Nat.succ i + d\nthis : Stream'.nth a (d + 1 + i) \u2208 FP (Stream'.drop 1 (Stream'.drop i a))\n\u22a2 j = d + 1 + i\n[PROOFSTEP]\nrw [hd, add_comm, Nat.succ_add, Nat.add_succ]\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nhs : Finset.Nonempty s\n\u22a2 (Finset.prod s fun i => Stream'.nth a i) \u2208 FP a\n[PROOFSTEP]\nrefine' FP_drop_subset_FP _ (s.min' hs) _\n[GOAL]\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nhs : Finset.Nonempty s\n\u22a2 (Finset.prod s fun i => Stream'.nth a i) \u2208 FP (Stream'.drop (Finset.min' s hs) a)\n[PROOFSTEP]\ninduction' s using Finset.strongInduction with s ih\n[GOAL]\ncase H\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns\u271d : Finset \u2115\nhs\u271d : Finset.Nonempty s\u271d\ns : Finset \u2115\nih :\n \u2200 (t : Finset \u2115),\n t \u2282 s \u2192\n \u2200 (hs : Finset.Nonempty t), (Finset.prod t fun i => Stream'.nth a i) \u2208 FP (Stream'.drop (Finset.min' t hs) a)\nhs : Finset.Nonempty s\n\u22a2 (Finset.prod s fun i => Stream'.nth a i) \u2208 FP (Stream'.drop (Finset.min' s hs) a)\n[PROOFSTEP]\nrw [\u2190 Finset.mul_prod_erase _ _ (s.min'_mem hs), \u2190 Stream'.head_drop]\n[GOAL]\ncase H\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns\u271d : Finset \u2115\nhs\u271d : Finset.Nonempty s\u271d\ns : Finset \u2115\nih :\n \u2200 (t : Finset \u2115),\n t \u2282 s \u2192\n \u2200 (hs : Finset.Nonempty t), (Finset.prod t fun i => Stream'.nth a i) \u2208 FP (Stream'.drop (Finset.min' t hs) a)\nhs : Finset.Nonempty s\n\u22a2 (Stream'.head (Stream'.drop (Finset.min' s hs) a) *\n Finset.prod (Finset.erase s (Finset.min' s hs)) fun x => Stream'.nth a x) \u2208\n FP (Stream'.drop (Finset.min' s hs) a)\n[PROOFSTEP]\ncases' (s.erase (s.min' hs)).eq_empty_or_nonempty with h h\n[GOAL]\ncase H.inl\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns\u271d : Finset \u2115\nhs\u271d : Finset.Nonempty s\u271d\ns : Finset \u2115\nih :\n \u2200 (t : Finset \u2115),\n t \u2282 s \u2192\n \u2200 (hs : Finset.Nonempty t), (Finset.prod t fun i => Stream'.nth a i) \u2208 FP (Stream'.drop (Finset.min' t hs) a)\nhs : Finset.Nonempty s\nh : Finset.erase s (Finset.min' s hs) = \u2205\n\u22a2 (Stream'.head (Stream'.drop (Finset.min' s hs) a) *\n Finset.prod (Finset.erase s (Finset.min' s hs)) fun x => Stream'.nth a x) \u2208\n FP (Stream'.drop (Finset.min' s hs) a)\n[PROOFSTEP]\nrw [h, Finset.prod_empty, mul_one]\n[GOAL]\ncase H.inl\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns\u271d : Finset \u2115\nhs\u271d : Finset.Nonempty s\u271d\ns : Finset \u2115\nih :\n \u2200 (t : Finset \u2115),\n t \u2282 s \u2192\n \u2200 (hs : Finset.Nonempty t), (Finset.prod t fun i => Stream'.nth a i) \u2208 FP (Stream'.drop (Finset.min' t hs) a)\nhs : Finset.Nonempty s\nh : Finset.erase s (Finset.min' s hs) = \u2205\n\u22a2 Stream'.head (Stream'.drop (Finset.min' s hs) a) \u2208 FP (Stream'.drop (Finset.min' s hs) a)\n[PROOFSTEP]\nexact FP.head _\n[GOAL]\ncase H.inr\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns\u271d : Finset \u2115\nhs\u271d : Finset.Nonempty s\u271d\ns : Finset \u2115\nih :\n \u2200 (t : Finset \u2115),\n t \u2282 s \u2192\n \u2200 (hs : Finset.Nonempty t), (Finset.prod t fun i => Stream'.nth a i) \u2208 FP (Stream'.drop (Finset.min' t hs) a)\nhs : Finset.Nonempty s\nh : Finset.Nonempty (Finset.erase s (Finset.min' s hs))\n\u22a2 (Stream'.head (Stream'.drop (Finset.min' s hs) a) *\n Finset.prod (Finset.erase s (Finset.min' s hs)) fun x => Stream'.nth a x) \u2208\n FP (Stream'.drop (Finset.min' s hs) a)\n[PROOFSTEP]\napply FP.cons\n[GOAL]\ncase H.inr.h\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns\u271d : Finset \u2115\nhs\u271d : Finset.Nonempty s\u271d\ns : Finset \u2115\nih :\n \u2200 (t : Finset \u2115),\n t \u2282 s \u2192\n \u2200 (hs : Finset.Nonempty t), (Finset.prod t fun i => Stream'.nth a i) \u2208 FP (Stream'.drop (Finset.min' t hs) a)\nhs : Finset.Nonempty s\nh : Finset.Nonempty (Finset.erase s (Finset.min' s hs))\n\u22a2 FP (Stream'.tail (Stream'.drop (Finset.min' s hs) a))\n (Finset.prod (Finset.erase s (Finset.min' s hs)) fun x => Stream'.nth a x)\n[PROOFSTEP]\nrw [Stream'.tail_eq_drop, Stream'.drop_drop, add_comm]\n[GOAL]\ncase H.inr.h\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns\u271d : Finset \u2115\nhs\u271d : Finset.Nonempty s\u271d\ns : Finset \u2115\nih :\n \u2200 (t : Finset \u2115),\n t \u2282 s \u2192\n \u2200 (hs : Finset.Nonempty t), (Finset.prod t fun i => Stream'.nth a i) \u2208 FP (Stream'.drop (Finset.min' t hs) a)\nhs : Finset.Nonempty s\nh : Finset.Nonempty (Finset.erase s (Finset.min' s hs))\n\u22a2 FP (Stream'.drop (Finset.min' s hs + 1) a) (Finset.prod (Finset.erase s (Finset.min' s hs)) fun x => Stream'.nth a x)\n[PROOFSTEP]\nrefine' Set.mem_of_subset_of_mem _ (ih _ (Finset.erase_ssubset <| s.min'_mem hs) h)\n[GOAL]\ncase H.inr.h\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns\u271d : Finset \u2115\nhs\u271d : Finset.Nonempty s\u271d\ns : Finset \u2115\nih :\n \u2200 (t : Finset \u2115),\n t \u2282 s \u2192\n \u2200 (hs : Finset.Nonempty t), (Finset.prod t fun i => Stream'.nth a i) \u2208 FP (Stream'.drop (Finset.min' t hs) a)\nhs : Finset.Nonempty s\nh : Finset.Nonempty (Finset.erase s (Finset.min' s hs))\n\u22a2 FP (Stream'.drop (Finset.min' (Finset.erase s (Finset.min' s hs)) h) a) \u2286 FP (Stream'.drop (Finset.min' s hs + 1) a)\n[PROOFSTEP]\nhave : s.min' hs + 1 \u2264 (s.erase (s.min' hs)).min' h :=\n Nat.succ_le_of_lt (Finset.min'_lt_of_mem_erase_min' _ _ <| Finset.min'_mem _ _)\n[GOAL]\ncase H.inr.h\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns\u271d : Finset \u2115\nhs\u271d : Finset.Nonempty s\u271d\ns : Finset \u2115\nih :\n \u2200 (t : Finset \u2115),\n t \u2282 s \u2192\n \u2200 (hs : Finset.Nonempty t), (Finset.prod t fun i => Stream'.nth a i) \u2208 FP (Stream'.drop (Finset.min' t hs) a)\nhs : Finset.Nonempty s\nh : Finset.Nonempty (Finset.erase s (Finset.min' s hs))\nthis : Finset.min' s hs + 1 \u2264 Finset.min' (Finset.erase s (Finset.min' s hs)) h\n\u22a2 FP (Stream'.drop (Finset.min' (Finset.erase s (Finset.min' s hs)) h) a) \u2286 FP (Stream'.drop (Finset.min' s hs + 1) a)\n[PROOFSTEP]\ncases' le_iff_exists_add.mp this with d hd\n[GOAL]\ncase H.inr.h.intro\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns\u271d : Finset \u2115\nhs\u271d : Finset.Nonempty s\u271d\ns : Finset \u2115\nih :\n \u2200 (t : Finset \u2115),\n t \u2282 s \u2192\n \u2200 (hs : Finset.Nonempty t), (Finset.prod t fun i => Stream'.nth a i) \u2208 FP (Stream'.drop (Finset.min' t hs) a)\nhs : Finset.Nonempty s\nh : Finset.Nonempty (Finset.erase s (Finset.min' s hs))\nthis : Finset.min' s hs + 1 \u2264 Finset.min' (Finset.erase s (Finset.min' s hs)) h\nd : \u2115\nhd : Finset.min' (Finset.erase s (Finset.min' s hs)) h = Finset.min' s hs + 1 + d\n\u22a2 FP (Stream'.drop (Finset.min' (Finset.erase s (Finset.min' s hs)) h) a) \u2286 FP (Stream'.drop (Finset.min' s hs + 1) a)\n[PROOFSTEP]\nrw [hd, add_comm, \u2190 Stream'.drop_drop]\n[GOAL]\ncase H.inr.h.intro\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns\u271d : Finset \u2115\nhs\u271d : Finset.Nonempty s\u271d\ns : Finset \u2115\nih :\n \u2200 (t : Finset \u2115),\n t \u2282 s \u2192\n \u2200 (hs : Finset.Nonempty t), (Finset.prod t fun i => Stream'.nth a i) \u2208 FP (Stream'.drop (Finset.min' t hs) a)\nhs : Finset.Nonempty s\nh : Finset.Nonempty (Finset.erase s (Finset.min' s hs))\nthis : Finset.min' s hs + 1 \u2264 Finset.min' (Finset.erase s (Finset.min' s hs)) h\nd : \u2115\nhd : Finset.min' (Finset.erase s (Finset.min' s hs)) h = Finset.min' s hs + 1 + d\n\u22a2 FP (Stream'.drop d (Stream'.drop (Finset.min' s hs + 1) a)) \u2286 FP (Stream'.drop (Finset.min' s hs + 1) a)\n[PROOFSTEP]\napply FP_drop_subset_FP\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.Hindman", "llama_tokens": 18843, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.41830854299014547}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na\u271d b\u271d : \u03b1\nhr : Symmetric r\na b : \u03b1\n\u22a2 Pairwise (r on fun c => bif c then a else b) \u2194 r a b\n[PROOFSTEP]\nsimpa [Pairwise, Function.onFun] using @hr a b\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns\u271d t u : Set \u03b1\na b : \u03b1\ninst\u271d : IsEquiv \u03b1 r\ns : Set \u03b9\nhs : Set.Nonempty s\n\u22a2 Set.Pairwise s (r on f) \u2194 \u2203 z, \u2200 (x : \u03b9), x \u2208 s \u2192 r (f x) z\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns\u271d t u : Set \u03b1\na b : \u03b1\ninst\u271d : IsEquiv \u03b1 r\ns : Set \u03b9\nhs : Set.Nonempty s\n\u22a2 Set.Pairwise s (r on f) \u2192 \u2203 z, \u2200 (x : \u03b9), x \u2208 s \u2192 r (f x) z\n[PROOFSTEP]\nrcases hs with \u27e8y, hy\u27e9\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns\u271d t u : Set \u03b1\na b : \u03b1\ninst\u271d : IsEquiv \u03b1 r\ns : Set \u03b9\ny : \u03b9\nhy : y \u2208 s\n\u22a2 Set.Pairwise s (r on f) \u2192 \u2203 z, \u2200 (x : \u03b9), x \u2208 s \u2192 r (f x) z\n[PROOFSTEP]\nrefine' fun H => \u27e8f y, fun x hx => _\u27e9\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns\u271d t u : Set \u03b1\na b : \u03b1\ninst\u271d : IsEquiv \u03b1 r\ns : Set \u03b9\ny : \u03b9\nhy : y \u2208 s\nH : Set.Pairwise s (r on f)\nx : \u03b9\nhx : x \u2208 s\n\u22a2 r (f x) (f y)\n[PROOFSTEP]\nrcases eq_or_ne x y with (rfl | hne)\n[GOAL]\ncase mp.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns\u271d t u : Set \u03b1\na b : \u03b1\ninst\u271d : IsEquiv \u03b1 r\ns : Set \u03b9\nH : Set.Pairwise s (r on f)\nx : \u03b9\nhx hy : x \u2208 s\n\u22a2 r (f x) (f x)\n[PROOFSTEP]\napply IsRefl.refl\n[GOAL]\ncase mp.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns\u271d t u : Set \u03b1\na b : \u03b1\ninst\u271d : IsEquiv \u03b1 r\ns : Set \u03b9\ny : \u03b9\nhy : y \u2208 s\nH : Set.Pairwise s (r on f)\nx : \u03b9\nhx : x \u2208 s\nhne : x \u2260 y\n\u22a2 r (f x) (f y)\n[PROOFSTEP]\nexact H hx hy hne\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns\u271d t u : Set \u03b1\na b : \u03b1\ninst\u271d : IsEquiv \u03b1 r\ns : Set \u03b9\nhs : Set.Nonempty s\n\u22a2 (\u2203 z, \u2200 (x : \u03b9), x \u2208 s \u2192 r (f x) z) \u2192 Set.Pairwise s (r on f)\n[PROOFSTEP]\nrintro \u27e8z, hz\u27e9 x hx y hy _\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns\u271d t u : Set \u03b1\na b : \u03b1\ninst\u271d : IsEquiv \u03b1 r\ns : Set \u03b9\nhs : Set.Nonempty s\nz : \u03b1\nhz : \u2200 (x : \u03b9), x \u2208 s \u2192 r (f x) z\nx : \u03b9\nhx : x \u2208 s\ny : \u03b9\nhy : y \u2208 s\na\u271d : x \u2260 y\n\u22a2 (r on f) x y\n[PROOFSTEP]\nexact @IsTrans.trans \u03b1 r _ (f x) z (f y) (hz _ hx) (IsSymm.symm _ _ <| hz _ hy)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr\u271d p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d g : \u03b9 \u2192 \u03b1\ns\u271d t u : Set \u03b1\na b : \u03b1\ninst\u271d\u00b9 : Nonempty \u03b9\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b9\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsEquiv \u03b9 r\n\u22a2 Set.Pairwise s (r on f) \u2194 \u2203 z, \u2200 (x : \u03b1), x \u2208 s \u2192 r (f x) z\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | hne)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr\u271d p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\ninst\u271d\u00b9 : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b9\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsEquiv \u03b9 r\n\u22a2 Set.Pairwise \u2205 (r on f) \u2194 \u2203 z, \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 r (f x) z\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr\u271d p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d g : \u03b9 \u2192 \u03b1\ns\u271d t u : Set \u03b1\na b : \u03b1\ninst\u271d\u00b9 : Nonempty \u03b9\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b9\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsEquiv \u03b9 r\nhne : Set.Nonempty s\n\u22a2 Set.Pairwise s (r on f) \u2194 \u2203 z, \u2200 (x : \u03b1), x \u2208 s \u2192 r (f x) z\n[PROOFSTEP]\nexact hne.pairwise_iff_exists_forall\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\n\u22a2 Set.Pairwise (s \u222a t) r \u2194\n Set.Pairwise s r \u2227 Set.Pairwise t r \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a \u2260 b \u2192 r a b \u2227 r b a\n[PROOFSTEP]\nsimp only [Set.Pairwise, mem_union, or_imp, forall_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\n\u22a2 (((\u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (x_1 : \u03b1), x_1 \u2208 s \u2192 x \u2260 x_1 \u2192 r x x_1) \u2227\n \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (x_1 : \u03b1), x_1 \u2208 s \u2192 x \u2260 x_1 \u2192 r x x_1) \u2227\n (\u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (x_1 : \u03b1), x_1 \u2208 t \u2192 x \u2260 x_1 \u2192 r x x_1) \u2227\n \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (x_1 : \u03b1), x_1 \u2208 t \u2192 x \u2260 x_1 \u2192 r x x_1) \u2194\n (\u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (x_1 : \u03b1), x_1 \u2208 s \u2192 x \u2260 x_1 \u2192 r x x_1) \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (x_1 : \u03b1), x_1 \u2208 t \u2192 x \u2260 x_1 \u2192 r x x_1) \u2227\n (\u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (x_1 : \u03b1), x_1 \u2208 t \u2192 x \u2260 x_1 \u2192 r x x_1) \u2227\n \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (x_2 : \u03b1), x_2 \u2208 t \u2192 x \u2260 x_2 \u2192 r x_2 x\n[PROOFSTEP]\nexact\n \u27e8fun H => \u27e8H.1.1, H.2.2, H.2.1, fun x hx y hy hne => H.1.2 y hy x hx hne.symm\u27e9, fun H =>\n \u27e8\u27e8H.1, fun x hx y hy hne => H.2.2.2 y hy x hx hne.symm\u27e9, H.2.2.1, H.2.1\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\nhr : Symmetric r\n\u22a2 (Set.Pairwise s r \u2227 Set.Pairwise t r \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a \u2260 b \u2192 r a b \u2227 r b a) \u2194\n Set.Pairwise s r \u2227 Set.Pairwise t r \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a \u2260 b \u2192 r a b\n[PROOFSTEP]\nsimp only [hr.iff, and_self_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\n\u22a2 Set.Pairwise (insert a s) r \u2194 Set.Pairwise s r \u2227 \u2200 (b : \u03b1), b \u2208 s \u2192 a \u2260 b \u2192 r a b \u2227 r b a\n[PROOFSTEP]\nsimp only [insert_eq, pairwise_union, pairwise_singleton, true_and_iff, mem_singleton_iff, forall_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b\u271d : \u03b1\nha : \u00aca \u2208 s\nb : \u03b1\nhb : b \u2208 s\n\u22a2 a \u2260 b \u2192 r a b \u2227 r b a \u2194 r a b \u2227 r b a\n[PROOFSTEP]\nsimp [(ne_of_mem_of_not_mem hb ha).symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\nhr : Symmetric r\n\u22a2 Set.Pairwise (insert a s) r \u2194 Set.Pairwise s r \u2227 \u2200 (b : \u03b1), b \u2208 s \u2192 a \u2260 b \u2192 r a b\n[PROOFSTEP]\nsimp only [pairwise_insert, hr.iff a, and_self_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\nhr : Symmetric r\nha : \u00aca \u2208 s\n\u22a2 Set.Pairwise (insert a s) r \u2194 Set.Pairwise s r \u2227 \u2200 (b : \u03b1), b \u2208 s \u2192 r a b\n[PROOFSTEP]\nsimp only [pairwise_insert_of_not_mem ha, hr.iff a, and_self_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\n\u22a2 Set.Pairwise {a, b} r \u2194 a \u2260 b \u2192 r a b \u2227 r b a\n[PROOFSTEP]\nsimp [pairwise_insert]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\nhr : Symmetric r\n\u22a2 Set.Pairwise {a, b} r \u2194 a \u2260 b \u2192 r a b\n[PROOFSTEP]\nsimp [pairwise_insert_of_symmetric hr]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\n\u22a2 Set.Pairwise univ r \u2194 Pairwise r\n[PROOFSTEP]\nsimp only [Set.Pairwise, Pairwise, mem_univ, forall_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns\u271d t u : Set \u03b1\na b : \u03b1\ns : Set \u03b9\nh : InjOn f s\n\u22a2 Set.Pairwise (f '' s) r \u2194 Set.Pairwise s (r on f)\n[PROOFSTEP]\nsimp (config := { contextual := true }) [h.eq_iff, Set.Pairwise]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr\u271d p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 (Pairwise fun x y => r \u2191x \u2191y) \u2194 Set.Pairwise s r\n[PROOFSTEP]\nsimp only [Pairwise, Set.Pairwise, SetCoe.forall, Ne.def, Subtype.ext_iff, Subtype.coe_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b9\nf g\u271d : \u03b9 \u2192 \u03b1\nhs : PairwiseDisjoint s f\ng : \u03b9 \u2192 \u03b9\nhg : f \u2218 g \u2264 f\n\u22a2 PairwiseDisjoint (g '' s) f\n[PROOFSTEP]\nrintro _ \u27e8a, ha, rfl\u27e9 _ \u27e8b, hb, rfl\u27e9 h\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b9\nf g\u271d : \u03b9 \u2192 \u03b1\nhs : PairwiseDisjoint s f\ng : \u03b9 \u2192 \u03b9\nhg : f \u2218 g \u2264 f\na : \u03b9\nha : a \u2208 s\nb : \u03b9\nhb : b \u2208 s\nh : g a \u2260 g b\n\u22a2 (Disjoint on f) (g a) (g b)\n[PROOFSTEP]\nexact (hs ha hb <| ne_of_apply_ne _ h).mono (hg a) (hg b)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b9\nf g\u271d : \u03b9 \u2192 \u03b1\ng : \u2191s \u2192 \u03b9\nhg : \u2200 (i : \u2191s), f (g i) \u2264 f \u2191i\nht : PairwiseDisjoint s f\n\u22a2 PairwiseDisjoint (Set.range g) f\n[PROOFSTEP]\nrintro _ \u27e8x, rfl\u27e9 _ \u27e8y, rfl\u27e9 hxy\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b9\nf g\u271d : \u03b9 \u2192 \u03b1\ng : \u2191s \u2192 \u03b9\nhg : \u2200 (i : \u2191s), f (g i) \u2264 f \u2191i\nht : PairwiseDisjoint s f\nx y : \u2191s\nhxy : g x \u2260 g y\n\u22a2 (Disjoint on f) (g x) (g y)\n[PROOFSTEP]\nexact ((ht x.2 y.2) fun h => hxy <| congr_arg g <| Subtype.ext h).mono (hg x) (hg y)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\n\u22a2 PairwiseDisjoint (range singleton) id\n[PROOFSTEP]\nrintro _ \u27e8a, rfl\u27e9 _ \u27e8b, rfl\u27e9 h\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\na b : \u03b9\nh : {a} \u2260 {b}\n\u22a2 (Disjoint on id) {a} {b}\n[PROOFSTEP]\nexact disjoint_singleton.2 (ne_of_apply_ne _ h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)\n\u22a2 (PairwiseDisjoint s fun a => f a '' t) \u2194 InjOn (fun p => f p.fst p.snd) (s \u00d7\u02e2 t)\n[PROOFSTEP]\nrefine' \u27e8fun hs x hx y hy (h : f _ _ = _) => _, fun hs x hx y hy h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : \u03b1 \u00d7 \u03b2\nhx : x \u2208 s \u00d7\u02e2 t\ny : \u03b1 \u00d7 \u03b2\nhy : y \u2208 s \u00d7\u02e2 t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n\u22a2 x = y\n[PROOFSTEP]\nsuffices x.1 = y.1 by exact Prod.ext this (hf _ hx.1 <| h.trans <| by rw [this])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : \u03b1 \u00d7 \u03b2\nhx : x \u2208 s \u00d7\u02e2 t\ny : \u03b1 \u00d7 \u03b2\nhy : y \u2208 s \u00d7\u02e2 t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\nthis : x.fst = y.fst\n\u22a2 x = y\n[PROOFSTEP]\nexact Prod.ext this (hf _ hx.1 <| h.trans <| by rw [this])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : \u03b1 \u00d7 \u03b2\nhx : x \u2208 s \u00d7\u02e2 t\ny : \u03b1 \u00d7 \u03b2\nhy : y \u2208 s \u00d7\u02e2 t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\nthis : x.fst = y.fst\n\u22a2 (fun p => f p.fst p.snd) y = f x.fst y.snd\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : \u03b1 \u00d7 \u03b2\nhx : x \u2208 s \u00d7\u02e2 t\ny : \u03b1 \u00d7 \u03b2\nhy : y \u2208 s \u00d7\u02e2 t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n\u22a2 x.fst = y.fst\n[PROOFSTEP]\nrefine' hs.elim hx.1 hy.1 (not_disjoint_iff.2 \u27e8_, mem_image_of_mem _ hx.2, _\u27e9)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : \u03b1 \u00d7 \u03b2\nhx : x \u2208 s \u00d7\u02e2 t\ny : \u03b1 \u00d7 \u03b2\nhy : y \u2208 s \u00d7\u02e2 t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n\u22a2 f x.fst x.snd \u2208 f y.fst '' t\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : \u03b1 \u00d7 \u03b2\nhx : x \u2208 s \u00d7\u02e2 t\ny : \u03b1 \u00d7 \u03b2\nhy : y \u2208 s \u00d7\u02e2 t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n\u22a2 (fun p => f p.fst p.snd) y \u2208 f y.fst '' t\n[PROOFSTEP]\nexact mem_image_of_mem _ hy.2\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)\nhs : InjOn (fun p => f p.fst p.snd) (s \u00d7\u02e2 t)\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nh : x \u2260 y\n\u22a2 (Disjoint on fun a => f a '' t) x y\n[PROOFSTEP]\nrefine' disjoint_iff_inf_le.mpr _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)\nhs : InjOn (fun p => f p.fst p.snd) (s \u00d7\u02e2 t)\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nh : x \u2260 y\n\u22a2 (fun a => f a '' t) x \u2293 (fun a => f a '' t) y \u2264 \u22a5\n[PROOFSTEP]\nrintro _ \u27e8\u27e8a, ha, hab\u27e9, b, hb, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 Injective (f a)\nhs : InjOn (fun p => f p.fst p.snd) (s \u00d7\u02e2 t)\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nh : x \u2260 y\na : \u03b2\nha : a \u2208 t\nb : \u03b2\nhb : b \u2208 t\nhab : f x a = f y b\n\u22a2 f y b \u2208 \u22a5\n[PROOFSTEP]\nexact h (congr_arg Prod.fst <| hs (mk_mem_prod hx ha) (mk_mem_prod hy hb) hab)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (b : \u03b2), b \u2208 t \u2192 Injective fun a => f a b\n\u22a2 (PairwiseDisjoint t fun b => (fun a => f a b) '' s) \u2194 InjOn (fun p => f p.fst p.snd) (s \u00d7\u02e2 t)\n[PROOFSTEP]\nrefine' \u27e8fun ht x hx y hy (h : f _ _ = _) => _, fun ht x hx y hy h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (b : \u03b2), b \u2208 t \u2192 Injective fun a => f a b\nht : PairwiseDisjoint t fun b => (fun a => f a b) '' s\nx : \u03b1 \u00d7 \u03b2\nhx : x \u2208 s \u00d7\u02e2 t\ny : \u03b1 \u00d7 \u03b2\nhy : y \u2208 s \u00d7\u02e2 t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n\u22a2 x = y\n[PROOFSTEP]\nsuffices x.2 = y.2 by exact Prod.ext (hf _ hx.2 <| h.trans <| by rw [this]) this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (b : \u03b2), b \u2208 t \u2192 Injective fun a => f a b\nht : PairwiseDisjoint t fun b => (fun a => f a b) '' s\nx : \u03b1 \u00d7 \u03b2\nhx : x \u2208 s \u00d7\u02e2 t\ny : \u03b1 \u00d7 \u03b2\nhy : y \u2208 s \u00d7\u02e2 t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\nthis : x.snd = y.snd\n\u22a2 x = y\n[PROOFSTEP]\nexact Prod.ext (hf _ hx.2 <| h.trans <| by rw [this]) this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (b : \u03b2), b \u2208 t \u2192 Injective fun a => f a b\nht : PairwiseDisjoint t fun b => (fun a => f a b) '' s\nx : \u03b1 \u00d7 \u03b2\nhx : x \u2208 s \u00d7\u02e2 t\ny : \u03b1 \u00d7 \u03b2\nhy : y \u2208 s \u00d7\u02e2 t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\nthis : x.snd = y.snd\n\u22a2 (fun p => f p.fst p.snd) y = (fun a => f a x.snd) y.fst\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (b : \u03b2), b \u2208 t \u2192 Injective fun a => f a b\nht : PairwiseDisjoint t fun b => (fun a => f a b) '' s\nx : \u03b1 \u00d7 \u03b2\nhx : x \u2208 s \u00d7\u02e2 t\ny : \u03b1 \u00d7 \u03b2\nhy : y \u2208 s \u00d7\u02e2 t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n\u22a2 x.snd = y.snd\n[PROOFSTEP]\nrefine' ht.elim hx.2 hy.2 (not_disjoint_iff.2 \u27e8_, mem_image_of_mem _ hx.1, _\u27e9)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (b : \u03b2), b \u2208 t \u2192 Injective fun a => f a b\nht : PairwiseDisjoint t fun b => (fun a => f a b) '' s\nx : \u03b1 \u00d7 \u03b2\nhx : x \u2208 s \u00d7\u02e2 t\ny : \u03b1 \u00d7 \u03b2\nhy : y \u2208 s \u00d7\u02e2 t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n\u22a2 f x.fst x.snd \u2208 (fun a => f a y.snd) '' s\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (b : \u03b2), b \u2208 t \u2192 Injective fun a => f a b\nht : PairwiseDisjoint t fun b => (fun a => f a b) '' s\nx : \u03b1 \u00d7 \u03b2\nhx : x \u2208 s \u00d7\u02e2 t\ny : \u03b1 \u00d7 \u03b2\nhy : y \u2208 s \u00d7\u02e2 t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n\u22a2 (fun p => f p.fst p.snd) y \u2208 (fun a => f a y.snd) '' s\n[PROOFSTEP]\nexact mem_image_of_mem _ hy.1\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (b : \u03b2), b \u2208 t \u2192 Injective fun a => f a b\nht : InjOn (fun p => f p.fst p.snd) (s \u00d7\u02e2 t)\nx : \u03b2\nhx : x \u2208 t\ny : \u03b2\nhy : y \u2208 t\nh : x \u2260 y\n\u22a2 (Disjoint on fun b => (fun a => f a b) '' s) x y\n[PROOFSTEP]\nrefine' disjoint_iff_inf_le.mpr _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (b : \u03b2), b \u2208 t \u2192 Injective fun a => f a b\nht : InjOn (fun p => f p.fst p.snd) (s \u00d7\u02e2 t)\nx : \u03b2\nhx : x \u2208 t\ny : \u03b2\nhy : y \u2208 t\nh : x \u2260 y\n\u22a2 (fun b => (fun a => f a b) '' s) x \u2293 (fun b => (fun a => f a b) '' s) y \u2264 \u22a5\n[PROOFSTEP]\nrintro _ \u27e8\u27e8a, ha, hab\u27e9, b, hb, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nhf : \u2200 (b : \u03b2), b \u2208 t \u2192 Injective fun a => f a b\nht : InjOn (fun p => f p.fst p.snd) (s \u00d7\u02e2 t)\nx : \u03b2\nhx : x \u2208 t\ny : \u03b2\nhy : y \u2208 t\nh : x \u2260 y\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 s\nhab : (fun a => f a x) a = (fun a => f a y) b\n\u22a2 (fun a => f a y) b \u2208 \u22a5\n[PROOFSTEP]\nexact h (congr_arg Prod.snd <| ht (mk_mem_prod ha hx) (mk_mem_prod hb hy) hab)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acPairwiseDisjoint s f\n\u22a2 \u2203 i, i \u2208 s \u2227 \u2203 j, j \u2208 s \u2227 \u2203 _hij x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\nchange \u00ac\u2200 i, i \u2208 s \u2192 \u2200 j, j \u2208 s \u2192 i \u2260 j \u2192 \u2200 t, t \u2264 f i \u2192 t \u2264 f j \u2192 t \u2264 \u22a5 at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u2192 Set \u03b1\nh : \u00ac\u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (j : \u03b9), j \u2208 s \u2192 i \u2260 j \u2192 \u2200 (t : Set \u03b1), t \u2264 f i \u2192 t \u2264 f j \u2192 t \u2264 \u22a5\n\u22a2 \u2203 i, i \u2208 s \u2227 \u2203 j, j \u2208 s \u2227 \u2203 _hij x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\nsimp only [not_forall] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u2192 Set \u03b1\nh : \u2203 x h x_1 h h x_2 h x, \u00acx_2 \u2264 \u22a5\n\u22a2 \u2203 i, i \u2208 s \u2227 \u2203 j, j \u2208 s \u2227 \u2203 _hij x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\nobtain \u27e8i, hi, j, hj, h_ne, t, hfi, hfj, ht\u27e9 := h\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b9 \u2192 Set \u03b1\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nh_ne : i \u2260 j\nt : Set \u03b1\nhfi : t \u2264 f i\nhfj : t \u2264 f j\nht : \u00act \u2264 \u22a5\n\u22a2 \u2203 i, i \u2208 s \u2227 \u2203 j, j \u2208 s \u2227 \u2203 _hij x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\nreplace ht : t.Nonempty := by rwa [le_bot_iff, bot_eq_empty, \u2190 Ne.def, \u2190 nonempty_iff_ne_empty] at ht \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b9 \u2192 Set \u03b1\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nh_ne : i \u2260 j\nt : Set \u03b1\nhfi : t \u2264 f i\nhfj : t \u2264 f j\nht : \u00act \u2264 \u22a5\n\u22a2 Set.Nonempty t\n[PROOFSTEP]\nrwa [le_bot_iff, bot_eq_empty, \u2190 Ne.def, \u2190 nonempty_iff_ne_empty] at ht \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b9 \u2192 Set \u03b1\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nh_ne : i \u2260 j\nt : Set \u03b1\nhfi : t \u2264 f i\nhfj : t \u2264 f j\nht : Set.Nonempty t\n\u22a2 \u2203 i, i \u2208 s \u2227 \u2203 j, j \u2208 s \u2227 \u2203 _hij x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := ht\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt\u271d : Set \u03b9'\nf : \u03b9 \u2192 Set \u03b1\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nh_ne : i \u2260 j\nt : Set \u03b1\nhfi : t \u2264 f i\nhfj : t \u2264 f j\nx : \u03b1\nhx : x \u2208 t\n\u22a2 \u2203 i, i \u2208 s \u2227 \u2203 j, j \u2208 s \u2227 \u2203 _hij x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\nexact \u27e8i, hi, j, hj, h_ne, x, hfi hx, hfj hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acPairwiseDisjoint s f\n\u22a2 \u2203 i, i \u2208 s \u2227 \u2203 j, j \u2208 s \u2227 \u2203 _hij x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\nobtain \u27e8i, hi, j, hj, hne, x, hx\u2081, hx\u2082\u27e9 := exists_ne_mem_inter_of_not_pairwiseDisjoint h\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acPairwiseDisjoint s f\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nhne : i \u2260 j\nx : \u03b1\nhx\u2081 : x \u2208 f i\nhx\u2082 : x \u2208 f j\n\u22a2 \u2203 i, i \u2208 s \u2227 \u2203 j, j \u2208 s \u2227 \u2203 _hij x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\ncases' lt_or_lt_iff_ne.mpr hne with h_lt h_lt\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acPairwiseDisjoint s f\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nhne : i \u2260 j\nx : \u03b1\nhx\u2081 : x \u2208 f i\nhx\u2082 : x \u2208 f j\nh_lt : i < j\n\u22a2 \u2203 i, i \u2208 s \u2227 \u2203 j, j \u2208 s \u2227 \u2203 _hij x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\nexact \u27e8i, hi, j, hj, h_lt, x, hx\u2081, hx\u2082\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acPairwiseDisjoint s f\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nhne : i \u2260 j\nx : \u03b1\nhx\u2081 : x \u2208 f i\nhx\u2082 : x \u2208 f j\nh_lt : j < i\n\u22a2 \u2203 i, i \u2208 s \u2227 \u2203 j, j \u2208 s \u2227 \u2203 _hij x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\nexact \u27e8j, hj, i, hi, h_lt, x, hx\u2082, hx\u2081\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acPairwise (Disjoint on f)\n\u22a2 \u2203 i j _hij x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\nrw [\u2190 pairwise_univ] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acSet.Pairwise univ (Disjoint on f)\n\u22a2 \u2203 i j _hij x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\nobtain \u27e8i, _hi, j, _hj, h\u27e9 := exists_ne_mem_inter_of_not_pairwiseDisjoint h\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b9 \u2192 Set \u03b1\nh\u271d : \u00acSet.Pairwise univ (Disjoint on f)\ni : \u03b9\n_hi : i \u2208 univ\nj : \u03b9\n_hj : j \u2208 univ\nh : \u2203 _hij x, x \u2208 f i \u2229 f j\n\u22a2 \u2203 i j _hij x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\nexact \u27e8i, j, h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acPairwise (Disjoint on f)\n\u22a2 \u2203 i j x x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\nrw [\u2190 pairwise_univ] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acSet.Pairwise univ (Disjoint on f)\n\u22a2 \u2203 i j x x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\nobtain \u27e8i, _hi, j, _hj, h\u27e9 := exists_lt_mem_inter_of_not_pairwiseDisjoint h\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh\u271d : \u00acSet.Pairwise univ (Disjoint on f)\ni : \u03b9\n_hi : i \u2208 univ\nj : \u03b9\n_hj : j \u2208 univ\nh : \u2203 _hij x, x \u2208 f i \u2229 f j\n\u22a2 \u2203 i j x x, x \u2208 f i \u2229 f j\n[PROOFSTEP]\nexact \u27e8i, j, h\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Pairwise.Basic", "llama_tokens": 13390, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419704455589, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.41830853532674744}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : \u00acDifferentiableWithinAt \ud835\udd5c f s x\n\u22a2 fderivWithin \ud835\udd5c f s x = 0\n[PROOFSTEP]\nhave : \u00ac\u2203 f', HasFDerivWithinAt f f' s x := h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : \u00acDifferentiableWithinAt \ud835\udd5c f s x\nthis : \u00ac\u2203 f', HasFDerivWithinAt f f' s x\n\u22a2 fderivWithin \ud835\udd5c f s x = 0\n[PROOFSTEP]\nsimp [fderivWithin, this]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : \u00acDifferentiableAt \ud835\udd5c f x\n\u22a2 fderiv \ud835\udd5c f x = 0\n[PROOFSTEP]\nhave : \u00ac\u2203 f', HasFDerivAt f f' x := h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : \u00acDifferentiableAt \ud835\udd5c f x\nthis : \u00ac\u2203 f', HasFDerivAt f f' x\n\u22a2 fderiv \ud835\udd5c f x = 0\n[PROOFSTEP]\nsimp [fderiv, this]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\n\u22a2 Tendsto (fun n => c n \u2022 (f (x + d n) - f x)) l (\ud835\udcdd (\u2191f' v))\n[PROOFSTEP]\nhave tendsto_arg : Tendsto (fun n => x + d n) l (\ud835\udcdd[s] x) :=\n by\n conv in \ud835\udcdd[s] x => rw [\u2190 add_zero x]\n rw [nhdsWithin, tendsto_inf]\n constructor\n \u00b7 apply tendsto_const_nhds.add (tangentConeAt.lim_zero l clim cdlim)\n \u00b7 rwa [tendsto_principal]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\n\u22a2 Tendsto (fun n => x + d n) l (\ud835\udcdd[s] x)\n[PROOFSTEP]\nconv in \ud835\udcdd[s] x => rw [\u2190 add_zero x]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\n| \ud835\udcdd[s] x\n[PROOFSTEP]\nrw [\u2190 add_zero x]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\n| \ud835\udcdd[s] x\n[PROOFSTEP]\nrw [\u2190 add_zero x]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\n| \ud835\udcdd[s] x\n[PROOFSTEP]\nrw [\u2190 add_zero x]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\n\u22a2 Tendsto (fun n => x + d n) l (\ud835\udcdd[s] (x + 0))\n[PROOFSTEP]\nrw [nhdsWithin, tendsto_inf]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\n\u22a2 Tendsto (fun n => x + d n) l (\ud835\udcdd (x + 0)) \u2227 Tendsto (fun n => x + d n) l (\ud835\udcdf s)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\n\u22a2 Tendsto (fun n => x + d n) l (\ud835\udcdd (x + 0))\n[PROOFSTEP]\napply tendsto_const_nhds.add (tangentConeAt.lim_zero l clim cdlim)\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\n\u22a2 Tendsto (fun n => x + d n) l (\ud835\udcdf s)\n[PROOFSTEP]\nrwa [tendsto_principal]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\ntendsto_arg : Tendsto (fun n => x + d n) l (\ud835\udcdd[s] x)\n\u22a2 Tendsto (fun n => c n \u2022 (f (x + d n) - f x)) l (\ud835\udcdd (\u2191f' v))\n[PROOFSTEP]\nhave : (fun y => f y - f x - f' (y - x)) =o[\ud835\udcdd[s] x] fun y => y - x := h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\ntendsto_arg : Tendsto (fun n => x + d n) l (\ud835\udcdd[s] x)\nthis : (fun y => f y - f x - \u2191f' (y - x)) =o[\ud835\udcdd[s] x] fun y => y - x\n\u22a2 Tendsto (fun n => c n \u2022 (f (x + d n) - f x)) l (\ud835\udcdd (\u2191f' v))\n[PROOFSTEP]\nhave : (fun n => f (x + d n) - f x - f' (x + d n - x)) =o[l] fun n => x + d n - x := this.comp_tendsto tendsto_arg\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\ntendsto_arg : Tendsto (fun n => x + d n) l (\ud835\udcdd[s] x)\nthis\u271d : (fun y => f y - f x - \u2191f' (y - x)) =o[\ud835\udcdd[s] x] fun y => y - x\nthis : (fun n => f (x + d n) - f x - \u2191f' (x + d n - x)) =o[l] fun n => x + d n - x\n\u22a2 Tendsto (fun n => c n \u2022 (f (x + d n) - f x)) l (\ud835\udcdd (\u2191f' v))\n[PROOFSTEP]\nhave : (fun n => f (x + d n) - f x - f' (d n)) =o[l] d := by simpa only [add_sub_cancel']\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\ntendsto_arg : Tendsto (fun n => x + d n) l (\ud835\udcdd[s] x)\nthis\u271d : (fun y => f y - f x - \u2191f' (y - x)) =o[\ud835\udcdd[s] x] fun y => y - x\nthis : (fun n => f (x + d n) - f x - \u2191f' (x + d n - x)) =o[l] fun n => x + d n - x\n\u22a2 (fun n => f (x + d n) - f x - \u2191f' (d n)) =o[l] d\n[PROOFSTEP]\nsimpa only [add_sub_cancel']\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\ntendsto_arg : Tendsto (fun n => x + d n) l (\ud835\udcdd[s] x)\nthis\u271d\u00b9 : (fun y => f y - f x - \u2191f' (y - x)) =o[\ud835\udcdd[s] x] fun y => y - x\nthis\u271d : (fun n => f (x + d n) - f x - \u2191f' (x + d n - x)) =o[l] fun n => x + d n - x\nthis : (fun n => f (x + d n) - f x - \u2191f' (d n)) =o[l] d\n\u22a2 Tendsto (fun n => c n \u2022 (f (x + d n) - f x)) l (\ud835\udcdd (\u2191f' v))\n[PROOFSTEP]\nhave : (fun n => c n \u2022 (f (x + d n) - f x - f' (d n))) =o[l] fun n => c n \u2022 d n := (isBigO_refl c l).smul_isLittleO this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\ntendsto_arg : Tendsto (fun n => x + d n) l (\ud835\udcdd[s] x)\nthis\u271d\u00b2 : (fun y => f y - f x - \u2191f' (y - x)) =o[\ud835\udcdd[s] x] fun y => y - x\nthis\u271d\u00b9 : (fun n => f (x + d n) - f x - \u2191f' (x + d n - x)) =o[l] fun n => x + d n - x\nthis\u271d : (fun n => f (x + d n) - f x - \u2191f' (d n)) =o[l] d\nthis : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) =o[l] fun n => c n \u2022 d n\n\u22a2 Tendsto (fun n => c n \u2022 (f (x + d n) - f x)) l (\ud835\udcdd (\u2191f' v))\n[PROOFSTEP]\nhave : (fun n => c n \u2022 (f (x + d n) - f x - f' (d n))) =o[l] fun _ => (1 : \u211d) := this.trans_isBigO (cdlim.isBigO_one \u211d)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\ntendsto_arg : Tendsto (fun n => x + d n) l (\ud835\udcdd[s] x)\nthis\u271d\u00b3 : (fun y => f y - f x - \u2191f' (y - x)) =o[\ud835\udcdd[s] x] fun y => y - x\nthis\u271d\u00b2 : (fun n => f (x + d n) - f x - \u2191f' (x + d n - x)) =o[l] fun n => x + d n - x\nthis\u271d\u00b9 : (fun n => f (x + d n) - f x - \u2191f' (d n)) =o[l] d\nthis\u271d : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) =o[l] fun n => c n \u2022 d n\nthis : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) =o[l] fun x => 1\n\u22a2 Tendsto (fun n => c n \u2022 (f (x + d n) - f x)) l (\ud835\udcdd (\u2191f' v))\n[PROOFSTEP]\nhave L1 : Tendsto (fun n => c n \u2022 (f (x + d n) - f x - f' (d n))) l (\ud835\udcdd 0) := (isLittleO_one_iff \u211d).1 this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\ntendsto_arg : Tendsto (fun n => x + d n) l (\ud835\udcdd[s] x)\nthis\u271d\u00b3 : (fun y => f y - f x - \u2191f' (y - x)) =o[\ud835\udcdd[s] x] fun y => y - x\nthis\u271d\u00b2 : (fun n => f (x + d n) - f x - \u2191f' (x + d n - x)) =o[l] fun n => x + d n - x\nthis\u271d\u00b9 : (fun n => f (x + d n) - f x - \u2191f' (d n)) =o[l] d\nthis\u271d : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) =o[l] fun n => c n \u2022 d n\nthis : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) =o[l] fun x => 1\nL1 : Tendsto (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => c n \u2022 (f (x + d n) - f x)) l (\ud835\udcdd (\u2191f' v))\n[PROOFSTEP]\nhave L2 : Tendsto (fun n => f' (c n \u2022 d n)) l (\ud835\udcdd (f' v)) := Tendsto.comp f'.cont.continuousAt cdlim\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\ntendsto_arg : Tendsto (fun n => x + d n) l (\ud835\udcdd[s] x)\nthis\u271d\u00b3 : (fun y => f y - f x - \u2191f' (y - x)) =o[\ud835\udcdd[s] x] fun y => y - x\nthis\u271d\u00b2 : (fun n => f (x + d n) - f x - \u2191f' (x + d n - x)) =o[l] fun n => x + d n - x\nthis\u271d\u00b9 : (fun n => f (x + d n) - f x - \u2191f' (d n)) =o[l] d\nthis\u271d : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) =o[l] fun n => c n \u2022 d n\nthis : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) =o[l] fun x => 1\nL1 : Tendsto (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) l (\ud835\udcdd 0)\nL2 : Tendsto (fun n => \u2191f' (c n \u2022 d n)) l (\ud835\udcdd (\u2191f' v))\n\u22a2 Tendsto (fun n => c n \u2022 (f (x + d n) - f x)) l (\ud835\udcdd (\u2191f' v))\n[PROOFSTEP]\nhave L3 : Tendsto (fun n => c n \u2022 (f (x + d n) - f x - f' (d n)) + f' (c n \u2022 d n)) l (\ud835\udcdd (0 + f' v)) := L1.add L2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\ntendsto_arg : Tendsto (fun n => x + d n) l (\ud835\udcdd[s] x)\nthis\u271d\u00b3 : (fun y => f y - f x - \u2191f' (y - x)) =o[\ud835\udcdd[s] x] fun y => y - x\nthis\u271d\u00b2 : (fun n => f (x + d n) - f x - \u2191f' (x + d n - x)) =o[l] fun n => x + d n - x\nthis\u271d\u00b9 : (fun n => f (x + d n) - f x - \u2191f' (d n)) =o[l] d\nthis\u271d : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) =o[l] fun n => c n \u2022 d n\nthis : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) =o[l] fun x => 1\nL1 : Tendsto (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) l (\ud835\udcdd 0)\nL2 : Tendsto (fun n => \u2191f' (c n \u2022 d n)) l (\ud835\udcdd (\u2191f' v))\nL3 : Tendsto (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n)) + \u2191f' (c n \u2022 d n)) l (\ud835\udcdd (0 + \u2191f' v))\n\u22a2 Tendsto (fun n => c n \u2022 (f (x + d n) - f x)) l (\ud835\udcdd (\u2191f' v))\n[PROOFSTEP]\nhave : (fun n => c n \u2022 (f (x + d n) - f x - f' (d n)) + f' (c n \u2022 d n)) = fun n => c n \u2022 (f (x + d n) - f x) :=\n by\n ext n\n simp [smul_add, smul_sub]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\ntendsto_arg : Tendsto (fun n => x + d n) l (\ud835\udcdd[s] x)\nthis\u271d\u00b3 : (fun y => f y - f x - \u2191f' (y - x)) =o[\ud835\udcdd[s] x] fun y => y - x\nthis\u271d\u00b2 : (fun n => f (x + d n) - f x - \u2191f' (x + d n - x)) =o[l] fun n => x + d n - x\nthis\u271d\u00b9 : (fun n => f (x + d n) - f x - \u2191f' (d n)) =o[l] d\nthis\u271d : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) =o[l] fun n => c n \u2022 d n\nthis : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) =o[l] fun x => 1\nL1 : Tendsto (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) l (\ud835\udcdd 0)\nL2 : Tendsto (fun n => \u2191f' (c n \u2022 d n)) l (\ud835\udcdd (\u2191f' v))\nL3 : Tendsto (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n)) + \u2191f' (c n \u2022 d n)) l (\ud835\udcdd (0 + \u2191f' v))\n\u22a2 (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n)) + \u2191f' (c n \u2022 d n)) = fun n => c n \u2022 (f (x + d n) - f x)\n[PROOFSTEP]\next n\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\ntendsto_arg : Tendsto (fun n => x + d n) l (\ud835\udcdd[s] x)\nthis\u271d\u00b3 : (fun y => f y - f x - \u2191f' (y - x)) =o[\ud835\udcdd[s] x] fun y => y - x\nthis\u271d\u00b2 : (fun n => f (x + d n) - f x - \u2191f' (x + d n - x)) =o[l] fun n => x + d n - x\nthis\u271d\u00b9 : (fun n => f (x + d n) - f x - \u2191f' (d n)) =o[l] d\nthis\u271d : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) =o[l] fun n => c n \u2022 d n\nthis : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) =o[l] fun x => 1\nL1 : Tendsto (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) l (\ud835\udcdd 0)\nL2 : Tendsto (fun n => \u2191f' (c n \u2022 d n)) l (\ud835\udcdd (\u2191f' v))\nL3 : Tendsto (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n)) + \u2191f' (c n \u2022 d n)) l (\ud835\udcdd (0 + \u2191f' v))\nn : \u03b1\n\u22a2 c n \u2022 (f (x + d n) - f x - \u2191f' (d n)) + \u2191f' (c n \u2022 d n) = c n \u2022 (f (x + d n) - f x)\n[PROOFSTEP]\nsimp [smul_add, smul_sub]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\n\u03b1 : Type u_6\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nv : E\ndtop : \u2200\u1da0 (n : \u03b1) in l, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) l atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd v)\ntendsto_arg : Tendsto (fun n => x + d n) l (\ud835\udcdd[s] x)\nthis\u271d\u2074 : (fun y => f y - f x - \u2191f' (y - x)) =o[\ud835\udcdd[s] x] fun y => y - x\nthis\u271d\u00b3 : (fun n => f (x + d n) - f x - \u2191f' (x + d n - x)) =o[l] fun n => x + d n - x\nthis\u271d\u00b2 : (fun n => f (x + d n) - f x - \u2191f' (d n)) =o[l] d\nthis\u271d\u00b9 : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) =o[l] fun n => c n \u2022 d n\nthis\u271d : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) =o[l] fun x => 1\nL1 : Tendsto (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n))) l (\ud835\udcdd 0)\nL2 : Tendsto (fun n => \u2191f' (c n \u2022 d n)) l (\ud835\udcdd (\u2191f' v))\nL3 : Tendsto (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n)) + \u2191f' (c n \u2022 d n)) l (\ud835\udcdd (0 + \u2191f' v))\nthis : (fun n => c n \u2022 (f (x + d n) - f x - \u2191f' (d n)) + \u2191f' (c n \u2022 d n)) = fun n => c n \u2022 (f (x + d n) - f x)\n\u22a2 Tendsto (fun n => c n \u2022 (f (x + d n) - f x)) l (\ud835\udcdd (\u2191f' v))\n[PROOFSTEP]\nrwa [this, zero_add] at L3 \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\u22a2 HasFDerivAtFilter f f' x L \u2194 Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 * \u2016f x' - f x - \u2191f' (x' - x)\u2016) L (\ud835\udcdd 0)\n[PROOFSTEP]\nhave h : \u2200 x', \u2016x' - x\u2016 = 0 \u2192 \u2016f x' - f x - f' (x' - x)\u2016 = 0 := fun x' hx' =>\n by\n rw [sub_eq_zero.1 (norm_eq_zero.1 hx')]\n simp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nx' : E\nhx' : \u2016x' - x\u2016 = 0\n\u22a2 \u2016f x' - f x - \u2191f' (x' - x)\u2016 = 0\n[PROOFSTEP]\nrw [sub_eq_zero.1 (norm_eq_zero.1 hx')]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nx' : E\nhx' : \u2016x' - x\u2016 = 0\n\u22a2 \u2016f x - f x - \u2191f' (x - x)\u2016 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : \u2200 (x' : E), \u2016x' - x\u2016 = 0 \u2192 \u2016f x' - f x - \u2191f' (x' - x)\u2016 = 0\n\u22a2 HasFDerivAtFilter f f' x L \u2194 Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 * \u2016f x' - f x - \u2191f' (x' - x)\u2016) L (\ud835\udcdd 0)\n[PROOFSTEP]\nunfold HasFDerivAtFilter\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : \u2200 (x' : E), \u2016x' - x\u2016 = 0 \u2192 \u2016f x' - f x - \u2191f' (x' - x)\u2016 = 0\n\u22a2 ((fun x' => f x' - f x - \u2191f' (x' - x)) =o[L] fun x' => x' - x) \u2194\n Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 * \u2016f x' - f x - \u2191f' (x' - x)\u2016) L (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 isLittleO_norm_left, \u2190 isLittleO_norm_right, isLittleO_iff_tendsto h]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : \u2200 (x' : E), \u2016x' - x\u2016 = 0 \u2192 \u2016f x' - f x - \u2191f' (x' - x)\u2016 = 0\n\u22a2 Tendsto (fun x_1 => \u2016f x_1 - f x - \u2191f' (x_1 - x)\u2016 / \u2016x_1 - x\u2016) L (\ud835\udcdd 0) \u2194\n Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 * \u2016f x' - f x - \u2191f' (x' - x)\u2016) L (\ud835\udcdd 0)\n[PROOFSTEP]\nexact tendsto_congr fun _ => div_eq_inv_mul _ _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\u22a2 HasFDerivAt f f' x \u2194 (fun h => f (x + h) - f x - \u2191f' h) =o[\ud835\udcdd 0] fun h => h\n[PROOFSTEP]\nrw [HasFDerivAt, HasFDerivAtFilter, \u2190 map_add_left_nhds_zero x, isLittleO_map]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\u22a2 ((fun x' => f x' - f x - \u2191f' (x' - x)) \u2218 (fun x x_1 => x + x_1) x) =o[\ud835\udcdd 0]\n ((fun x' => x' - x) \u2218 (fun x x_1 => x + x_1) x) \u2194\n (fun h => f (x + h) - f x - \u2191f' h) =o[\ud835\udcdd 0] fun h => h\n[PROOFSTEP]\nsimp [(\u00b7 \u2218 \u00b7)]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\nx\u2080 : E\nhf : HasFDerivAt f f' x\u2080\nC : \u211d\nhC\u2080 : 0 \u2264 C\nhlip : \u2200\u1da0 (x : E) in \ud835\udcdd x\u2080, \u2016f x - f x\u2080\u2016 \u2264 C * \u2016x - x\u2080\u2016\n\u22a2 \u2016f'\u2016 \u2264 C\n[PROOFSTEP]\nrefine' le_of_forall_pos_le_add fun \u03b5 \u03b50 => op_norm_le_of_nhds_zero _ _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\nx\u2080 : E\nhf : HasFDerivAt f f' x\u2080\nC : \u211d\nhC\u2080 : 0 \u2264 C\nhlip : \u2200\u1da0 (x : E) in \ud835\udcdd x\u2080, \u2016f x - f x\u2080\u2016 \u2264 C * \u2016x - x\u2080\u2016\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 0 \u2264 C + \u03b5\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\nx\u2080 : E\nhf : HasFDerivAt f f' x\u2080\nC : \u211d\nhC\u2080 : 0 \u2264 C\nhlip : \u2200\u1da0 (x : E) in \ud835\udcdd x\u2080, \u2016f x - f x\u2080\u2016 \u2264 C * \u2016x - x\u2080\u2016\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : E) in \ud835\udcdd 0, \u2016\u2191f' x\u2016 \u2264 (C + \u03b5) * \u2016x\u2016\n[PROOFSTEP]\nexact add_nonneg hC\u2080 \u03b50.le\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\nx\u2080 : E\nhf : HasFDerivAt f f' x\u2080\nC : \u211d\nhC\u2080 : 0 \u2264 C\nhlip : \u2200\u1da0 (x : E) in \ud835\udcdd x\u2080, \u2016f x - f x\u2080\u2016 \u2264 C * \u2016x - x\u2080\u2016\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : E) in \ud835\udcdd 0, \u2016\u2191f' x\u2016 \u2264 (C + \u03b5) * \u2016x\u2016\n[PROOFSTEP]\nrw [\u2190 map_add_left_nhds_zero x\u2080, eventually_map] at hlip \n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\nx\u2080 : E\nhf : HasFDerivAt f f' x\u2080\nC : \u211d\nhC\u2080 : 0 \u2264 C\nhlip : \u2200\u1da0 (a : E) in \ud835\udcdd 0, \u2016f ((fun x x_1 => x + x_1) x\u2080 a) - f x\u2080\u2016 \u2264 C * \u2016(fun x x_1 => x + x_1) x\u2080 a - x\u2080\u2016\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : E) in \ud835\udcdd 0, \u2016\u2191f' x\u2016 \u2264 (C + \u03b5) * \u2016x\u2016\n[PROOFSTEP]\nfilter_upwards [isLittleO_iff.1 (hasFDerivAt_iff_isLittleO_nhds_zero.1 hf) \u03b50, hlip] with y hy hyC\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\nx\u2080 : E\nhf : HasFDerivAt f f' x\u2080\nC : \u211d\nhC\u2080 : 0 \u2264 C\nhlip : \u2200\u1da0 (a : E) in \ud835\udcdd 0, \u2016f ((fun x x_1 => x + x_1) x\u2080 a) - f x\u2080\u2016 \u2264 C * \u2016(fun x x_1 => x + x_1) x\u2080 a - x\u2080\u2016\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\ny : E\nhy : \u2016f (x\u2080 + y) - f x\u2080 - \u2191f' y\u2016 \u2264 \u03b5 * \u2016y\u2016\nhyC : \u2016f (x\u2080 + y) - f x\u2080\u2016 \u2264 C * \u2016x\u2080 + y - x\u2080\u2016\n\u22a2 \u2016\u2191f' y\u2016 \u2264 (C + \u03b5) * \u2016y\u2016\n[PROOFSTEP]\nrw [add_sub_cancel'] at hyC \n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\nx\u2080 : E\nhf : HasFDerivAt f f' x\u2080\nC : \u211d\nhC\u2080 : 0 \u2264 C\nhlip : \u2200\u1da0 (a : E) in \ud835\udcdd 0, \u2016f ((fun x x_1 => x + x_1) x\u2080 a) - f x\u2080\u2016 \u2264 C * \u2016(fun x x_1 => x + x_1) x\u2080 a - x\u2080\u2016\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\ny : E\nhy : \u2016f (x\u2080 + y) - f x\u2080 - \u2191f' y\u2016 \u2264 \u03b5 * \u2016y\u2016\nhyC : \u2016f (x\u2080 + y) - f x\u2080\u2016 \u2264 C * \u2016y\u2016\n\u22a2 \u2016\u2191f' y\u2016 \u2264 (C + \u03b5) * \u2016y\u2016\n[PROOFSTEP]\ncalc\n \u2016f' y\u2016 \u2264 \u2016f (x\u2080 + y) - f x\u2080\u2016 + \u2016f (x\u2080 + y) - f x\u2080 - f' y\u2016 := norm_le_insert _ _\n _ \u2264 C * \u2016y\u2016 + \u03b5 * \u2016y\u2016 := (add_le_add hyC hy)\n _ = (C + \u03b5) * \u2016y\u2016 := (add_mul _ _ _).symm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\nx\u2080 : E\nhf : HasFDerivAt f f' x\u2080\ns : Set E\nhs : s \u2208 \ud835\udcdd x\u2080\nC : \u211d\u22650\nhlip : LipschitzOnWith C f s\n\u22a2 \u2016f'\u2016 \u2264 \u2191C\n[PROOFSTEP]\nrefine' hf.le_of_lip' C.coe_nonneg _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2192L[\ud835\udd5c] F\nx\u2080 : E\nhf : HasFDerivAt f f' x\u2080\ns : Set E\nhs : s \u2208 \ud835\udcdd x\u2080\nC : \u211d\u22650\nhlip : LipschitzOnWith C f s\n\u22a2 \u2200\u1da0 (x : E) in \ud835\udcdd x\u2080, \u2016f x - f x\u2080\u2016 \u2264 \u2191C * \u2016x - x\u2080\u2016\n[PROOFSTEP]\nfilter_upwards [hs] with x hx using hlip.norm_sub_le hx (mem_of_mem_nhds hs)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\u22a2 HasFDerivWithinAt f f' univ x \u2194 HasFDerivAt f f' x\n[PROOFSTEP]\nsimp only [HasFDerivWithinAt, nhdsWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\u22a2 HasFDerivAtFilter f f' x (\ud835\udcdd x) \u2194 HasFDerivAt f f' x\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\ny : E\n\u22a2 HasFDerivWithinAt f f' (insert y s) x \u2194 HasFDerivWithinAt f f' s x\n[PROOFSTEP]\nrcases eq_or_ne x y with (rfl | h)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\u22a2 HasFDerivWithinAt f f' (insert x s) x \u2194 HasFDerivWithinAt f f' s x\n[PROOFSTEP]\nsimp_rw [HasFDerivWithinAt, HasFDerivAtFilter]\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\u22a2 ((fun x' => f x' - f x - \u2191f' (x' - x)) =o[\ud835\udcdd[insert x s] x] fun x' => x' - x) \u2194\n (fun x' => f x' - f x - \u2191f' (x' - x)) =o[\ud835\udcdd[s] x] fun x' => x' - x\n[PROOFSTEP]\napply Asymptotics.isLittleO_insert\n[GOAL]\ncase inl.h\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\u22a2 f x - f x - \u2191f' (x - x) = 0\n[PROOFSTEP]\nsimp only [sub_self, map_zero]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\ny : E\nh : x \u2260 y\n\u22a2 HasFDerivWithinAt f f' (insert y s) x \u2194 HasFDerivWithinAt f f' s x\n[PROOFSTEP]\nrefine' \u27e8fun h => h.mono <| subset_insert y s, fun hf => hf.mono_of_mem _\u27e9\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\ny : E\nh : x \u2260 y\nhf : HasFDerivWithinAt f f' s x\n\u22a2 s \u2208 \ud835\udcdd[insert y s] x\n[PROOFSTEP]\nsimp_rw [nhdsWithin_insert_of_ne h, self_mem_nhdsWithin]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\ny : E\n\u22a2 HasFDerivWithinAt f f' (s \\ {y}) x \u2194 HasFDerivWithinAt f f' s x\n[PROOFSTEP]\nrw [\u2190 hasFDerivWithinAt_insert, insert_diff_singleton, hasFDerivWithinAt_insert]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasStrictFDerivAt f f' x\n\u22a2 HasFDerivAt f f' x\n[PROOFSTEP]\nrw [HasFDerivAt, HasFDerivAtFilter, isLittleO_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasStrictFDerivAt f f' x\n\u22a2 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd x, \u2016f x_1 - f x - \u2191f' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\n[PROOFSTEP]\nexact fun c hc => tendsto_id.prod_mk_nhds tendsto_const_nhds (isLittleO_iff.1 hf hc)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasStrictFDerivAt f f' x\nK : \u211d\u22650\nhK : \u2016f'\u2016\u208a < K\n\u22a2 \u2203 s, s \u2208 \ud835\udcdd x \u2227 LipschitzOnWith K f s\n[PROOFSTEP]\nhave := hf.add_isBigOWith (f'.isBigOWith_comp _ _) hK\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasStrictFDerivAt f f' x\nK : \u211d\u22650\nhK : \u2016f'\u2016\u208a < K\nthis :\n IsBigOWith ((fun a => \u2191a) K) (\ud835\udcdd (x, x)) (fun x => f x.fst - f x.snd - \u2191f' (x.fst - x.snd) + \u2191f' (x.fst - x.snd))\n fun p => p.fst - p.snd\n\u22a2 \u2203 s, s \u2208 \ud835\udcdd x \u2227 LipschitzOnWith K f s\n[PROOFSTEP]\nsimp only [sub_add_cancel, IsBigOWith] at this \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasStrictFDerivAt f f' x\nK : \u211d\u22650\nhK : \u2016f'\u2016\u208a < K\nthis : \u2200\u1da0 (x : E \u00d7 E) in \ud835\udcdd (x, x), \u2016f x.fst - f x.snd\u2016 \u2264 \u2191K * \u2016x.fst - x.snd\u2016\n\u22a2 \u2203 s, s \u2208 \ud835\udcdd x \u2227 LipschitzOnWith K f s\n[PROOFSTEP]\nrcases exists_nhds_square this with \u27e8U, Uo, xU, hU\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasStrictFDerivAt f f' x\nK : \u211d\u22650\nhK : \u2016f'\u2016\u208a < K\nthis : \u2200\u1da0 (x : E \u00d7 E) in \ud835\udcdd (x, x), \u2016f x.fst - f x.snd\u2016 \u2264 \u2191K * \u2016x.fst - x.snd\u2016\nU : Set E\nUo : IsOpen U\nxU : x \u2208 U\nhU : U \u00d7\u02e2 U \u2286 {x | (fun x => \u2016f x.fst - f x.snd\u2016 \u2264 \u2191K * \u2016x.fst - x.snd\u2016) x}\n\u22a2 \u2203 s, s \u2208 \ud835\udcdd x \u2227 LipschitzOnWith K f s\n[PROOFSTEP]\nexact \u27e8U, Uo.mem_nhds xU, lipschitzOnWith_iff_norm_sub_le.2 fun x hx y hy => hU (mk_mem_prod hx hy)\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasFDerivAt f f' x\nv : E\n\u03b1 : Type u_6\nc : \u03b1 \u2192 \ud835\udd5c\nl : Filter \u03b1\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\n\u22a2 Tendsto (fun n => c n \u2022 (f (x + (c n)\u207b\u00b9 \u2022 v) - f x)) l (\ud835\udcdd (\u2191f' v))\n[PROOFSTEP]\nrefine' (hasFDerivWithinAt_univ.2 hf).lim _ univ_mem hc _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasFDerivAt f f' x\nv : E\n\u03b1 : Type u_6\nc : \u03b1 \u2192 \ud835\udd5c\nl : Filter \u03b1\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\n\u22a2 Tendsto (fun n => c n \u2022 (c n)\u207b\u00b9 \u2022 v) l (\ud835\udcdd v)\n[PROOFSTEP]\nintro U hU\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasFDerivAt f f' x\nv : E\n\u03b1 : Type u_6\nc : \u03b1 \u2192 \ud835\udd5c\nl : Filter \u03b1\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\nU : Set E\nhU : U \u2208 \ud835\udcdd v\n\u22a2 U \u2208 map (fun n => c n \u2022 (c n)\u207b\u00b9 \u2022 v) l\n[PROOFSTEP]\nrefine' (eventually_ne_of_tendsto_norm_atTop hc (0 : \ud835\udd5c)).mono fun y hy => _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasFDerivAt f f' x\nv : E\n\u03b1 : Type u_6\nc : \u03b1 \u2192 \ud835\udd5c\nl : Filter \u03b1\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\nU : Set E\nhU : U \u2208 \ud835\udcdd v\ny : \u03b1\nhy : c y \u2260 0\n\u22a2 (fun n => c n \u2022 (c n)\u207b\u00b9 \u2022 v) y \u2208 U\n[PROOFSTEP]\nconvert mem_of_mem_nhds hU\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasFDerivAt f f' x\nv : E\n\u03b1 : Type u_6\nc : \u03b1 \u2192 \ud835\udd5c\nl : Filter \u03b1\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\nU : Set E\nhU : U \u2208 \ud835\udcdd v\ny : \u03b1\nhy : c y \u2260 0\n\u22a2 (fun n => c n \u2022 (c n)\u207b\u00b9 \u2022 v) y = v\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasFDerivAt f f' x\nv : E\n\u03b1 : Type u_6\nc : \u03b1 \u2192 \ud835\udd5c\nl : Filter \u03b1\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\nU : Set E\nhU : U \u2208 \ud835\udcdd v\ny : \u03b1\nhy : c y \u2260 0\n\u22a2 c y \u2022 (c y)\u207b\u00b9 \u2022 v = v\n[PROOFSTEP]\nrw [\u2190 mul_smul, mul_inv_cancel hy, one_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh\u2080 : HasFDerivAt f f\u2080' x\nh\u2081 : HasFDerivAt f f\u2081' x\n\u22a2 f\u2080' = f\u2081'\n[PROOFSTEP]\nrw [\u2190 hasFDerivWithinAt_univ] at h\u2080 h\u2081 \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh\u2080 : HasFDerivWithinAt f f\u2080' univ x\nh\u2081 : HasFDerivWithinAt f f\u2081' univ x\n\u22a2 f\u2080' = f\u2081'\n[PROOFSTEP]\nexact uniqueDiffWithinAt_univ.eq h\u2080 h\u2081\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : t \u2208 \ud835\udcdd[s] x\n\u22a2 HasFDerivWithinAt f f' (s \u2229 t) x \u2194 HasFDerivWithinAt f f' s x\n[PROOFSTEP]\nsimp [HasFDerivWithinAt, nhdsWithin_restrict'' s h]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : t \u2208 \ud835\udcdd x\n\u22a2 HasFDerivWithinAt f f' (s \u2229 t) x \u2194 HasFDerivWithinAt f f' s x\n[PROOFSTEP]\nsimp [HasFDerivWithinAt, nhdsWithin_restrict' s h]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhs : HasFDerivWithinAt f f' s x\nht : HasFDerivWithinAt f f' t x\n\u22a2 HasFDerivWithinAt f f' (s \u222a t) x\n[PROOFSTEP]\nsimp only [HasFDerivWithinAt, nhdsWithin_union]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhs : HasFDerivWithinAt f f' s x\nht : HasFDerivWithinAt f f' t x\n\u22a2 HasFDerivAtFilter f f' x (\ud835\udcdd[s] x \u2294 \ud835\udcdd[t] x)\n[PROOFSTEP]\nexact hs.sup ht\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\nhs : s \u2208 \ud835\udcdd x\n\u22a2 HasFDerivAt f f' x\n[PROOFSTEP]\nrwa [\u2190 univ_inter s, hasFDerivWithinAt_inter hs, hasFDerivWithinAt_univ] at h \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : DifferentiableWithinAt \ud835\udd5c f s x\n\u22a2 HasFDerivWithinAt f (fderivWithin \ud835\udd5c f s x) s x\n[PROOFSTEP]\ndsimp only [fderivWithin]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : DifferentiableWithinAt \ud835\udd5c f s x\n\u22a2 HasFDerivWithinAt f (if h : \u2203 f', HasFDerivWithinAt f f' s x then choose h else 0) s x\n[PROOFSTEP]\ndsimp only [DifferentiableWithinAt] at h \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : \u2203 f', HasFDerivWithinAt f f' s x\n\u22a2 HasFDerivWithinAt f (if h : \u2203 f', HasFDerivWithinAt f f' s x then choose h else 0) s x\n[PROOFSTEP]\nrw [dif_pos h]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : \u2203 f', HasFDerivWithinAt f f' s x\n\u22a2 HasFDerivWithinAt f (choose h) s x\n[PROOFSTEP]\nexact Classical.choose_spec h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : DifferentiableAt \ud835\udd5c f x\n\u22a2 HasFDerivAt f (fderiv \ud835\udd5c f x) x\n[PROOFSTEP]\ndsimp only [fderiv]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : DifferentiableAt \ud835\udd5c f x\n\u22a2 HasFDerivAt f (if h : \u2203 f', HasFDerivAt f f' x then choose h else 0) x\n[PROOFSTEP]\ndsimp only [DifferentiableAt] at h \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : \u2203 f', HasFDerivAt f f' x\n\u22a2 HasFDerivAt f (if h : \u2203 f', HasFDerivAt f f' x then choose h else 0) x\n[PROOFSTEP]\nrw [dif_pos h]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : \u2203 f', HasFDerivAt f f' x\n\u22a2 HasFDerivAt f (choose h) x\n[PROOFSTEP]\nexact Classical.choose_spec h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivAt f f' x\n\u22a2 fderiv \ud835\udd5c f x = f'\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivAt f f' x\nx\u271d : E\n\u22a2 \u2191(fderiv \ud835\udd5c f x) x\u271d = \u2191f' x\u271d\n[PROOFSTEP]\nrw [h.unique h.differentiableAt.hasFDerivAt]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : \u00acx \u2208 closure s\n\u22a2 HasFDerivWithinAt f f' s x\n[PROOFSTEP]\nsimp only [mem_closure_iff_nhdsWithin_neBot, neBot_iff, Ne.def, Classical.not_not] at h \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : \ud835\udcdd[s] x = \u22a5\n\u22a2 HasFDerivWithinAt f f' s x\n[PROOFSTEP]\nsimp [HasFDerivWithinAt, HasFDerivAtFilter, h, IsLittleO, IsBigOWith]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : DifferentiableWithinAt \ud835\udd5c f t x\nst : s \u2286 t\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s x\n[PROOFSTEP]\nrcases h with \u27e8f', hf'\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nst : s \u2286 t\nf' : E \u2192L[\ud835\udd5c] F\nhf' : HasFDerivWithinAt f f' t x\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s x\n[PROOFSTEP]\nexact \u27e8f', hf'.mono st\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\u22a2 DifferentiableWithinAt \ud835\udd5c f univ x \u2194 DifferentiableAt \ud835\udd5c f x\n[PROOFSTEP]\nsimp only [DifferentiableWithinAt, hasFDerivWithinAt_univ, DifferentiableAt]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nht : t \u2208 \ud835\udcdd x\n\u22a2 DifferentiableWithinAt \ud835\udd5c f (s \u2229 t) x \u2194 DifferentiableWithinAt \ud835\udd5c f s x\n[PROOFSTEP]\nsimp only [DifferentiableWithinAt, hasFDerivWithinAt_inter ht]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nht : t \u2208 \ud835\udcdd[s] x\n\u22a2 DifferentiableWithinAt \ud835\udd5c f (s \u2229 t) x \u2194 DifferentiableWithinAt \ud835\udd5c f s x\n[PROOFSTEP]\nsimp only [DifferentiableWithinAt, hasFDerivWithinAt_inter' ht]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\u22a2 DifferentiableOn \ud835\udd5c f univ \u2194 Differentiable \ud835\udd5c f\n[PROOFSTEP]\nsimp only [DifferentiableOn, Differentiable, differentiableWithinAt_univ, mem_univ, forall_true_left]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : \u2200 (x : E), x \u2208 s \u2192 \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 DifferentiableOn \ud835\udd5c f (s \u2229 u)\n\u22a2 DifferentiableOn \ud835\udd5c f s\n[PROOFSTEP]\nintro x xs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : \u2200 (x : E), x \u2208 s \u2192 \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 DifferentiableOn \ud835\udd5c f (s \u2229 u)\nx : E\nxs : x \u2208 s\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s x\n[PROOFSTEP]\nrcases h x xs with \u27e8t, t_open, xt, ht\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t\u271d : Set E\nL L\u2081 L\u2082 : Filter E\nh : \u2200 (x : E), x \u2208 s \u2192 \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 DifferentiableOn \ud835\udd5c f (s \u2229 u)\nx : E\nxs : x \u2208 s\nt : Set E\nt_open : IsOpen t\nxt : x \u2208 t\nht : DifferentiableOn \ud835\udd5c f (s \u2229 t)\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s x\n[PROOFSTEP]\nexact (differentiableWithinAt_inter (IsOpen.mem_nhds t_open xt)).1 (ht x \u27e8xs, xt\u27e9)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nht : t \u2208 \ud835\udcdd x\n\u22a2 fderivWithin \ud835\udd5c f (s \u2229 t) x = fderivWithin \ud835\udd5c f s x\n[PROOFSTEP]\nsimp only [fderivWithin, hasFDerivWithinAt_inter ht]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : s \u2208 \ud835\udcdd x\n\u22a2 fderivWithin \ud835\udd5c f s x = fderiv \ud835\udd5c f x\n[PROOFSTEP]\nsimp only [fderiv, fderivWithin, HasFDerivAt, HasFDerivWithinAt, nhdsWithin_eq_nhds.2 h]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhs : UniqueDiffWithinAt \ud835\udd5c s x\nh : DifferentiableAt \ud835\udd5c f x\n\u22a2 fderivWithin \ud835\udd5c f s x = fderiv \ud835\udd5c f x\n[PROOFSTEP]\nrw [\u2190 fderivWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhs : UniqueDiffWithinAt \ud835\udd5c s x\nh : DifferentiableAt \ud835\udd5c f x\n\u22a2 fderivWithin \ud835\udd5c f s x = fderivWithin \ud835\udd5c f univ x\n[PROOFSTEP]\nexact fderivWithin_subset (subset_univ _) hs h.differentiableWithinAt\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\ns : Set (E \u2192L[\ud835\udd5c] F)\nx : E\n\u22a2 fderiv \ud835\udd5c f x \u2208 s \u2194 DifferentiableAt \ud835\udd5c f x \u2227 fderiv \ud835\udd5c f x \u2208 s \u2228 \u00acDifferentiableAt \ud835\udd5c f x \u2227 0 \u2208 s\n[PROOFSTEP]\nby_cases hx : DifferentiableAt \ud835\udd5c f x\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\ns : Set (E \u2192L[\ud835\udd5c] F)\nx : E\nhx : DifferentiableAt \ud835\udd5c f x\n\u22a2 fderiv \ud835\udd5c f x \u2208 s \u2194 DifferentiableAt \ud835\udd5c f x \u2227 fderiv \ud835\udd5c f x \u2208 s \u2228 \u00acDifferentiableAt \ud835\udd5c f x \u2227 0 \u2208 s\n[PROOFSTEP]\nsimp [fderiv_zero_of_not_differentiableAt, *]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\ns : Set (E \u2192L[\ud835\udd5c] F)\nx : E\nhx : \u00acDifferentiableAt \ud835\udd5c f x\n\u22a2 fderiv \ud835\udd5c f x \u2208 s \u2194 DifferentiableAt \ud835\udd5c f x \u2227 fderiv \ud835\udd5c f x \u2208 s \u2228 \u00acDifferentiableAt \ud835\udd5c f x \u2227 0 \u2208 s\n[PROOFSTEP]\nsimp [fderiv_zero_of_not_differentiableAt, *]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t\u271d : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nt : Set E\ns : Set (E \u2192L[\ud835\udd5c] F)\nx : E\n\u22a2 fderivWithin \ud835\udd5c f t x \u2208 s \u2194\n DifferentiableWithinAt \ud835\udd5c f t x \u2227 fderivWithin \ud835\udd5c f t x \u2208 s \u2228 \u00acDifferentiableWithinAt \ud835\udd5c f t x \u2227 0 \u2208 s\n[PROOFSTEP]\nby_cases hx : DifferentiableWithinAt \ud835\udd5c f t x\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t\u271d : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nt : Set E\ns : Set (E \u2192L[\ud835\udd5c] F)\nx : E\nhx : DifferentiableWithinAt \ud835\udd5c f t x\n\u22a2 fderivWithin \ud835\udd5c f t x \u2208 s \u2194\n DifferentiableWithinAt \ud835\udd5c f t x \u2227 fderivWithin \ud835\udd5c f t x \u2208 s \u2228 \u00acDifferentiableWithinAt \ud835\udd5c f t x \u2227 0 \u2208 s\n[PROOFSTEP]\nsimp [fderivWithin_zero_of_not_differentiableWithinAt, *]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t\u271d : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nt : Set E\ns : Set (E \u2192L[\ud835\udd5c] F)\nx : E\nhx : \u00acDifferentiableWithinAt \ud835\udd5c f t x\n\u22a2 fderivWithin \ud835\udd5c f t x \u2208 s \u2194\n DifferentiableWithinAt \ud835\udd5c f t x \u2227 fderivWithin \ud835\udd5c f t x \u2208 s \u2228 \u00acDifferentiableWithinAt \ud835\udd5c f t x \u2227 0 \u2208 s\n[PROOFSTEP]\nsimp [fderivWithin_zero_of_not_differentiableWithinAt, *]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\ns : Set E\nx\u2080 : E\nn : \u2115\nh : f =O[\ud835\udcdd[s] x\u2080] fun x => \u2016x - x\u2080\u2016 ^ n\nhx\u2080 : x\u2080 \u2208 s\nhn : 1 < n\n\u22a2 HasFDerivWithinAt f 0 s x\u2080\n[PROOFSTEP]\nsimp_rw [HasFDerivWithinAt, HasFDerivAtFilter, h.eq_zero_of_norm_pow_within hx\u2080 <| zero_lt_one.trans hn, zero_apply,\n sub_zero, h.trans_isLittleO ((isLittleO_pow_sub_sub x\u2080 hn).mono nhdsWithin_le_nhds)]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nx\u2080 : E\nn : \u2115\nh : f =O[\ud835\udcdd x\u2080] fun x => \u2016x - x\u2080\u2016 ^ n\nhn : 1 < n\n\u22a2 HasFDerivAt f 0 x\u2080\n[PROOFSTEP]\nrw [\u2190 nhdsWithin_univ] at h \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nx\u2080 : E\nn : \u2115\nh : f =O[\ud835\udcdd[univ] x\u2080] fun x => \u2016x - x\u2080\u2016 ^ n\nhn : 1 < n\n\u22a2 HasFDerivAt f 0 x\u2080\n[PROOFSTEP]\nexact (h.hasFDerivWithinAt (mem_univ _) hn).hasFDerivAt_of_univ\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\ns : Set E\nx\u2080 : E\nf' : E \u2192L[\ud835\udd5c] F\nh : HasFDerivWithinAt f f' s x\u2080\n\u22a2 (fun x => f x - f x\u2080) =O[\ud835\udcdd[s] x\u2080] fun x => x - x\u2080\n[PROOFSTEP]\nsimpa only [sub_add_cancel] using h.isBigO.add (isBigO_sub f' (\ud835\udcdd[s] x\u2080) x\u2080)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nx\u2080 : E\nf' : E \u2192L[\ud835\udd5c] F\nh : HasFDerivAt f f' x\u2080\n\u22a2 (fun x => f x - f x\u2080) =O[\ud835\udcdd x\u2080] fun x => x - x\u2080\n[PROOFSTEP]\nsimpa only [sub_add_cancel] using h.isBigO.add (isBigO_sub f' (\ud835\udcdd x\u2080) x\u2080)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhL : L \u2264 \ud835\udcdd x\nh : HasFDerivAtFilter f f' x L\n\u22a2 Tendsto f L (\ud835\udcdd (f x))\n[PROOFSTEP]\nhave : Tendsto (fun x' => f x' - f x) L (\ud835\udcdd 0) :=\n by\n refine' h.isBigO_sub.trans_tendsto (Tendsto.mono_left _ hL)\n rw [\u2190 sub_self x]\n exact tendsto_id.sub tendsto_const_nhds\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhL : L \u2264 \ud835\udcdd x\nh : HasFDerivAtFilter f f' x L\n\u22a2 Tendsto (fun x' => f x' - f x) L (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' h.isBigO_sub.trans_tendsto (Tendsto.mono_left _ hL)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhL : L \u2264 \ud835\udcdd x\nh : HasFDerivAtFilter f f' x L\n\u22a2 Tendsto (fun x' => x' - x) (\ud835\udcdd x) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 sub_self x]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhL : L \u2264 \ud835\udcdd x\nh : HasFDerivAtFilter f f' x L\n\u22a2 Tendsto (fun x' => x' - x) (\ud835\udcdd x) (\ud835\udcdd (x - x))\n[PROOFSTEP]\nexact tendsto_id.sub tendsto_const_nhds\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhL : L \u2264 \ud835\udcdd x\nh : HasFDerivAtFilter f f' x L\nthis : Tendsto (fun x' => f x' - f x) L (\ud835\udcdd 0)\n\u22a2 Tendsto f L (\ud835\udcdd (f x))\n[PROOFSTEP]\nhave := this.add (@tendsto_const_nhds _ _ _ (f x) _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhL : L \u2264 \ud835\udcdd x\nh : HasFDerivAtFilter f f' x L\nthis\u271d : Tendsto (fun x' => f x' - f x) L (\ud835\udcdd 0)\nthis : Tendsto (fun x_1 => f x_1 - f x + f x) L (\ud835\udcdd (0 + f x))\n\u22a2 Tendsto f L (\ud835\udcdd (f x))\n[PROOFSTEP]\nrw [zero_add (f x)] at this \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhL : L \u2264 \ud835\udcdd x\nh : HasFDerivAtFilter f f' x L\nthis\u271d : Tendsto (fun x' => f x' - f x) L (\ud835\udcdd 0)\nthis : Tendsto (fun x_1 => f x_1 - f x + f x) L (\ud835\udcdd (f x))\n\u22a2 Tendsto f L (\ud835\udcdd (f x))\n[PROOFSTEP]\nexact this.congr (by simp only [sub_add_cancel, eq_self_iff_true, forall_const])\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhL : L \u2264 \ud835\udcdd x\nh : HasFDerivAtFilter f f' x L\nthis\u271d : Tendsto (fun x' => f x' - f x) L (\ud835\udcdd 0)\nthis : Tendsto (fun x_1 => f x_1 - f x + f x) L (\ud835\udcdd (f x))\n\u22a2 \u2200 (x_1 : E), f x_1 - f x + f x = f x_1\n[PROOFSTEP]\nsimp only [sub_add_cancel, eq_self_iff_true, forall_const]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\ny : E\nh : s =\u1da0[\ud835\udcdd[{y}\u1d9c] x] t\n\u22a2 HasFDerivWithinAt f f' (s \\ {y}) x \u2194 HasFDerivWithinAt f f' (t \\ {y}) x\n[PROOFSTEP]\nsuffices \ud835\udcdd[s \\ { y }] x = \ud835\udcdd[t \\ { y }] x by simp only [HasFDerivWithinAt, this]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\ny : E\nh : s =\u1da0[\ud835\udcdd[{y}\u1d9c] x] t\nthis : \ud835\udcdd[s \\ {y}] x = \ud835\udcdd[t \\ {y}] x\n\u22a2 HasFDerivWithinAt f f' (s \\ {y}) x \u2194 HasFDerivWithinAt f f' (t \\ {y}) x\n[PROOFSTEP]\nsimp only [HasFDerivWithinAt, this]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\ny : E\nh : s =\u1da0[\ud835\udcdd[{y}\u1d9c] x] t\n\u22a2 \ud835\udcdd[s \\ {y}] x = \ud835\udcdd[t \\ {y}] x\n[PROOFSTEP]\nsimpa only [set_eventuallyEq_iff_inf_principal, \u2190 nhdsWithin_inter', diff_eq, inter_comm] using h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\ny : E\nh : s =\u1da0[\ud835\udcdd[{y}\u1d9c] x] t\n\u22a2 fderivWithin \ud835\udd5c f s x = fderivWithin \ud835\udd5c f t x\n[PROOFSTEP]\nsimp only [fderivWithin, hasFDerivWithinAt_congr_set' y h]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : f\u2080 =\u1da0[\ud835\udcdd x] f\u2081\nh' : \u2200 (y : E), \u2191f\u2080' y = \u2191f\u2081' y\n\u22a2 HasStrictFDerivAt f\u2080 f\u2080' x \u2194 HasStrictFDerivAt f\u2081 f\u2081' x\n[PROOFSTEP]\nrefine' isLittleO_congr ((h.prod_mk_nhds h).mono _) (eventually_of_forall fun _ => _root_.rfl)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : f\u2080 =\u1da0[\ud835\udcdd x] f\u2081\nh' : \u2200 (y : E), \u2191f\u2080' y = \u2191f\u2081' y\n\u22a2 \u2200 (x : E \u00d7 E),\n f\u2080 x.fst = f\u2081 x.fst \u2227 f\u2080 x.snd = f\u2081 x.snd \u2192\n (fun p => f\u2080 p.fst - f\u2080 p.snd - \u2191f\u2080' (p.fst - p.snd)) x = (fun p => f\u2081 p.fst - f\u2081 p.snd - \u2191f\u2081' (p.fst - p.snd)) x\n[PROOFSTEP]\nrintro p \u27e8hp\u2081, hp\u2082\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : f\u2080 =\u1da0[\ud835\udcdd x] f\u2081\nh' : \u2200 (y : E), \u2191f\u2080' y = \u2191f\u2081' y\np : E \u00d7 E\nhp\u2081 : f\u2080 p.fst = f\u2081 p.fst\nhp\u2082 : f\u2080 p.snd = f\u2081 p.snd\n\u22a2 (fun p => f\u2080 p.fst - f\u2080 p.snd - \u2191f\u2080' (p.fst - p.snd)) p = (fun p => f\u2081 p.fst - f\u2081 p.snd - \u2191f\u2081' (p.fst - p.snd)) p\n[PROOFSTEP]\nsimp only [*]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh\u2080 : f\u2080 =\u1da0[L] f\u2081\nhx : f\u2080 x = f\u2081 x\nh\u2081 : \u2200 (x : E), \u2191f\u2080' x = \u2191f\u2081' x\ny : E\nhy : f\u2080 y = f\u2081 y\n\u22a2 (fun x' => f\u2080 x' - f\u2080 x - \u2191f\u2080' (x' - x)) y = (fun x' => f\u2081 x' - f\u2081 x - \u2191f\u2081' (x' - x)) y\n[PROOFSTEP]\nsimp only [hy, h\u2081, hx]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhs : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\n\u22a2 fderivWithin \ud835\udd5c f\u2081 s x = fderivWithin \ud835\udd5c f s x\n[PROOFSTEP]\nsimp only [fderivWithin, hs.hasFDerivWithinAt_iff hx]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : f\u2081 =\u1da0[\ud835\udcdd x] f\n\u22a2 fderiv \ud835\udd5c f\u2081 x = fderiv \ud835\udd5c f x\n[PROOFSTEP]\nrw [\u2190 fderivWithin_univ, \u2190 fderivWithin_univ, h.fderivWithin_eq_nhds]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nx : E\n\u22a2 \u2200 (x : E \u00d7 E), 0 = _root_.id x.fst - _root_.id x.snd - \u2191(ContinuousLinearMap.id \ud835\udd5c E) (x.fst - x.snd)\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL\u271d L\u2081 L\u2082 : Filter E\nx : E\nL : Filter E\n\u22a2 \u2200 (x_1 : E), 0 = _root_.id x_1 - _root_.id x - \u2191(ContinuousLinearMap.id \ud835\udd5c E) (x_1 - x)\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 fderivWithin \ud835\udd5c _root_.id s x = ContinuousLinearMap.id \ud835\udd5c E\n[PROOFSTEP]\nrw [DifferentiableAt.fderivWithin differentiableAt_id hxs]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 fderiv \ud835\udd5c _root_.id x = ContinuousLinearMap.id \ud835\udd5c E\n[PROOFSTEP]\nexact fderiv_id\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d\u00b9 : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nc : F\nx : E\nx\u271d : E \u00d7 E\n\u22a2 0 = (fun x => c) x\u271d.fst - (fun x => c) x\u271d.snd - \u21910 (x\u271d.fst - x\u271d.snd)\n[PROOFSTEP]\nsimp only [zero_apply, sub_self]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d\u00b9 : E\ns t : Set E\nL\u271d L\u2081 L\u2082 : Filter E\nc : F\nx : E\nL : Filter E\nx\u271d : E\n\u22a2 0 = (fun x => c) x\u271d - (fun x => c) x - \u21910 (x\u271d - x)\n[PROOFSTEP]\nsimp only [zero_apply, sub_self]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nc : F\n\u22a2 (fderiv \ud835\udd5c fun x => c) = 0\n[PROOFSTEP]\next m\n[GOAL]\ncase h.h\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nc : F\nm x\u271d : E\n\u22a2 \u2191(fderiv \ud835\udd5c (fun x => c) m) x\u271d = \u2191(OfNat.ofNat 0 m) x\u271d\n[PROOFSTEP]\nrw [fderiv_const_apply]\n[GOAL]\ncase h.h\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nc : F\nm x\u271d : E\n\u22a2 \u21910 x\u271d = \u2191(OfNat.ofNat 0 m) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nc : F\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 fderivWithin \ud835\udd5c (fun x => c) s x = 0\n[PROOFSTEP]\nrw [DifferentiableAt.fderivWithin (differentiableAt_const _) hxs]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nc : F\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 fderiv \ud835\udd5c (fun x => c) x = 0\n[PROOFSTEP]\nexact fderiv_const_apply _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nx : E\n\u22a2 HasFDerivWithinAt f 0 {x} x\n[PROOFSTEP]\nsimp only [HasFDerivWithinAt, nhdsWithin_singleton, HasFDerivAtFilter, isLittleO_pure, ContinuousLinearMap.zero_apply,\n sub_self]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : Subsingleton E\nf : E \u2192 F\nx : E\n\u22a2 HasFDerivAt f 0 x\n[PROOFSTEP]\nrw [\u2190 hasFDerivWithinAt_univ, subsingleton_univ.eq_singleton_of_mem (mem_univ x)]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : Subsingleton E\nf : E \u2192 F\nx : E\n\u22a2 HasFDerivWithinAt f 0 {x} x\n[PROOFSTEP]\nexact hasFDerivWithinAt_singleton f x\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\nx : E\nh : \u00acx \u2208 tsupport f\n\u22a2 HasStrictFDerivAt f 0 x\n[PROOFSTEP]\nrw [not_mem_tsupport_iff_eventuallyEq] at h \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\nx : E\nh : f =\u1da0[\ud835\udcdd x] 0\n\u22a2 HasStrictFDerivAt f 0 x\n[PROOFSTEP]\nexact (hasStrictFDerivAt_const (0 : F) x).congr_of_eventuallyEq h.symm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\nx\u271d x : E\n\u22a2 x \u2208 support (fderiv \ud835\udd5c f) \u2192 x \u2208 tsupport f\n[PROOFSTEP]\nrw [\u2190 not_imp_not, nmem_support]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\nx\u271d x : E\n\u22a2 \u00acx \u2208 tsupport f \u2192 fderiv \ud835\udd5c f x = 0\n[PROOFSTEP]\nexact fderiv_of_not_mem_tsupport _\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.FDeriv.Basic", "llama_tokens": 49035, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147276, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.4182356121934079}} {"text": "[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf g : \u03b1 \u2192 G\n\u22a2 Continuous fun p => p.fst + p.snd\n[PROOFSTEP]\nrefine' continuous_iff_continuousAt.2 _\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf g : \u03b1 \u2192 G\n\u22a2 \u2200 (x : G \u00d7 G), ContinuousAt (fun p => p.fst + p.snd) x\n[PROOFSTEP]\nrintro \u27e8a, b\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf g : \u03b1 \u2192 G\na b : G\n\u22a2 ContinuousAt (fun p => p.fst + p.snd) (a, b)\n[PROOFSTEP]\nrefine' LinearOrderedAddCommGroup.tendsto_nhds.2 fun \u03b5 \u03b50 => _\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf g : \u03b1 \u2192 G\na b \u03b5 : G\n\u03b50 : \u03b5 > 0\n\u22a2 \u2200\u1da0 (b_1 : G \u00d7 G) in \ud835\udcdd (a, b), |b_1.fst + b_1.snd - (fun p => p.fst + p.snd) (a, b)| < \u03b5\n[PROOFSTEP]\nrcases dense_or_discrete 0 \u03b5 with (\u27e8\u03b4, \u03b40, \u03b4\u03b5\u27e9 | \u27e8_h\u2081, h\u2082\u27e9)\n[GOAL]\ncase mk.inl.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf g : \u03b1 \u2192 G\na b \u03b5 : G\n\u03b50 : \u03b5 > 0\n\u03b4 : G\n\u03b40 : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 < \u03b5\n\u22a2 \u2200\u1da0 (b_1 : G \u00d7 G) in \ud835\udcdd (a, b), |b_1.fst + b_1.snd - (fun p => p.fst + p.snd) (a, b)| < \u03b5\n[PROOFSTEP]\nfilter_upwards [(eventually_abs_sub_lt a \u03b40).prod_nhds (eventually_abs_sub_lt b (sub_pos.2 \u03b4\u03b5))]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf g : \u03b1 \u2192 G\na b \u03b5 : G\n\u03b50 : \u03b5 > 0\n\u03b4 : G\n\u03b40 : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 < \u03b5\n\u22a2 \u2200 (a_1 : G \u00d7 G), |a_1.fst - a| < \u03b4 \u2227 |a_1.snd - b| < \u03b5 - \u03b4 \u2192 |a_1.fst + a_1.snd - (a + b)| < \u03b5\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9 \u27e8hx : |x - a| < \u03b4, hy : |y - b| < \u03b5 - \u03b4\u27e9\n[GOAL]\ncase h.mk.intro\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf g : \u03b1 \u2192 G\na b \u03b5 : G\n\u03b50 : \u03b5 > 0\n\u03b4 : G\n\u03b40 : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 < \u03b5\nx y : G\nhx : |x - a| < \u03b4\nhy : |y - b| < \u03b5 - \u03b4\n\u22a2 |(x, y).fst + (x, y).snd - (a + b)| < \u03b5\n[PROOFSTEP]\nrw [add_sub_add_comm]\n[GOAL]\ncase h.mk.intro\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf g : \u03b1 \u2192 G\na b \u03b5 : G\n\u03b50 : \u03b5 > 0\n\u03b4 : G\n\u03b40 : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 < \u03b5\nx y : G\nhx : |x - a| < \u03b4\nhy : |y - b| < \u03b5 - \u03b4\n\u22a2 |(x, y).fst - a + ((x, y).snd - b)| < \u03b5\n[PROOFSTEP]\ncalc\n |x - a + (y - b)| \u2264 |x - a| + |y - b| := abs_add _ _\n _ < \u03b4 + (\u03b5 - \u03b4) := (add_lt_add hx hy)\n _ = \u03b5 := add_sub_cancel'_right _ _\n[GOAL]\ncase mk.inr.intro\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf g : \u03b1 \u2192 G\na b \u03b5 : G\n\u03b50 : \u03b5 > 0\n_h\u2081 : \u2200 (a : G), 0 < a \u2192 \u03b5 \u2264 a\nh\u2082 : \u2200 (a : G), a < \u03b5 \u2192 a \u2264 0\n\u22a2 \u2200\u1da0 (b_1 : G \u00d7 G) in \ud835\udcdd (a, b), |b_1.fst + b_1.snd - (fun p => p.fst + p.snd) (a, b)| < \u03b5\n[PROOFSTEP]\nhave h\u03b5 : \u2200 {x y}, |x - y| < \u03b5 \u2192 x = y := by\n intro x y h\n simpa [sub_eq_zero] using h\u2082 _ h\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf g : \u03b1 \u2192 G\na b \u03b5 : G\n\u03b50 : \u03b5 > 0\n_h\u2081 : \u2200 (a : G), 0 < a \u2192 \u03b5 \u2264 a\nh\u2082 : \u2200 (a : G), a < \u03b5 \u2192 a \u2264 0\n\u22a2 \u2200 {x y : G}, |x - y| < \u03b5 \u2192 x = y\n[PROOFSTEP]\nintro x y h\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf g : \u03b1 \u2192 G\na b \u03b5 : G\n\u03b50 : \u03b5 > 0\n_h\u2081 : \u2200 (a : G), 0 < a \u2192 \u03b5 \u2264 a\nh\u2082 : \u2200 (a : G), a < \u03b5 \u2192 a \u2264 0\nx y : G\nh : |x - y| < \u03b5\n\u22a2 x = y\n[PROOFSTEP]\nsimpa [sub_eq_zero] using h\u2082 _ h\n[GOAL]\ncase mk.inr.intro\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf g : \u03b1 \u2192 G\na b \u03b5 : G\n\u03b50 : \u03b5 > 0\n_h\u2081 : \u2200 (a : G), 0 < a \u2192 \u03b5 \u2264 a\nh\u2082 : \u2200 (a : G), a < \u03b5 \u2192 a \u2264 0\nh\u03b5 : \u2200 {x y : G}, |x - y| < \u03b5 \u2192 x = y\n\u22a2 \u2200\u1da0 (b_1 : G \u00d7 G) in \ud835\udcdd (a, b), |b_1.fst + b_1.snd - (fun p => p.fst + p.snd) (a, b)| < \u03b5\n[PROOFSTEP]\nfilter_upwards [(eventually_abs_sub_lt a \u03b50).prod_nhds (eventually_abs_sub_lt b \u03b50)]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf g : \u03b1 \u2192 G\na b \u03b5 : G\n\u03b50 : \u03b5 > 0\n_h\u2081 : \u2200 (a : G), 0 < a \u2192 \u03b5 \u2264 a\nh\u2082 : \u2200 (a : G), a < \u03b5 \u2192 a \u2264 0\nh\u03b5 : \u2200 {x y : G}, |x - y| < \u03b5 \u2192 x = y\n\u22a2 \u2200 (a_1 : G \u00d7 G), |a_1.fst - a| < \u03b5 \u2227 |a_1.snd - b| < \u03b5 \u2192 |a_1.fst + a_1.snd - (a + b)| < \u03b5\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9 \u27e8hx : |x - a| < \u03b5, hy : |y - b| < \u03b5\u27e9\n[GOAL]\ncase h.mk.intro\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf g : \u03b1 \u2192 G\na b \u03b5 : G\n\u03b50 : \u03b5 > 0\n_h\u2081 : \u2200 (a : G), 0 < a \u2192 \u03b5 \u2264 a\nh\u2082 : \u2200 (a : G), a < \u03b5 \u2192 a \u2264 0\nh\u03b5 : \u2200 {x y : G}, |x - y| < \u03b5 \u2192 x = y\nx y : G\nhx : |x - a| < \u03b5\nhy : |y - b| < \u03b5\n\u22a2 |(x, y).fst + (x, y).snd - (a + b)| < \u03b5\n[PROOFSTEP]\nsimpa [h\u03b5 hx, h\u03b5 hy]\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf g : \u03b1 \u2192 G\na \u03b5 : G\n\u03b50 : \u03b5 > 0\nx : G\nhx : |x - a| < \u03b5\n\u22a2 |-x - (fun a => -a) a| < \u03b5\n[PROOFSTEP]\nrwa [neg_sub_neg, abs_sub_comm]\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf\u271d g f : \u03b1 \u2192 G\n\u22a2 Tendsto f l (\ud835\udcdd 0) \u2194 Tendsto (abs \u2218 f) l (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' \u27e8fun h => (abs_zero : |(0 : G)| = 0) \u25b8 h.abs, fun h => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf\u271d g f : \u03b1 \u2192 G\nh : Tendsto (abs \u2218 f) l (\ud835\udcdd 0)\n\u22a2 Tendsto f l (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : Tendsto (fun a => -|f a|) l (\ud835\udcdd 0) := (neg_zero : -(0 : G) = 0) \u25b8 h.neg\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : OrderTopology G\nl : Filter \u03b1\nf\u271d g f : \u03b1 \u2192 G\nh : Tendsto (abs \u2218 f) l (\ud835\udcdd 0)\nthis : Tendsto (fun a => -|f a|) l (\ud835\udcdd 0)\n\u22a2 Tendsto f l (\ud835\udcdd 0)\n[PROOFSTEP]\nexact tendsto_of_tendsto_of_tendsto_of_le_of_le this h (fun x => neg_abs_le_self <| f x) fun x => le_abs_self <| f x\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Order.Group", "llama_tokens": 3359, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.743168019989179, "lm_q2_score": 0.5621765008857982, "lm_q1q2_score": 0.4177915970477436}} {"text": "[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\n\u22a2 Set.Nonempty (\u2191\u039b '' {f | \u2200 (x : X), x \u2208 K \u2192 1 \u2264 \u2191f x})\n[PROOFSTEP]\nrw [nonempty_image_iff]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\n\u22a2 Set.Nonempty {f | \u2200 (x : X), x \u2208 K \u2192 1 \u2264 \u2191f x}\n[PROOFSTEP]\nuse(1 : X \u2192\u1d47 \u211d\u22650)\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\n\u22a2 1 \u2208 {f | \u2200 (x : X), x \u2208 K \u2192 1 \u2264 \u2191f x}\n[PROOFSTEP]\nintro x _\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\nx : X\na\u271d : x \u2208 K\n\u22a2 1 \u2264 \u21911 x\n[PROOFSTEP]\nsimp only [BoundedContinuousFunction.coe_one, Pi.one_apply]\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\nx : X\na\u271d : x \u2208 K\n\u22a2 1 \u2264 1\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2203 f, (\u2200 (x : X), x \u2208 K \u2192 1 \u2264 \u2191f x) \u2227 \u2191\u039b f < rieszContentAux \u039b K + \u03b5\n[PROOFSTEP]\nobtain \u27e8\u03b1, \u27e8\u27e8f, f_hyp\u27e9, \u03b1_hyp\u27e9\u27e9 :=\n exists_lt_of_csInf_lt (rieszContentAux_image_nonempty \u039b K) (lt_add_of_pos_right (rieszContentAux \u039b K) \u03b5pos)\n[GOAL]\ncase intro.intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b1 : \u211d\u22650\n\u03b1_hyp : \u03b1 < rieszContentAux \u039b K + \u03b5\nf : X \u2192\u1d47 \u211d\u22650\nf_hyp : f \u2208 {f | \u2200 (x : X), x \u2208 K \u2192 1 \u2264 \u2191f x} \u2227 \u2191\u039b f = \u03b1\n\u22a2 \u2203 f, (\u2200 (x : X), x \u2208 K \u2192 1 \u2264 \u2191f x) \u2227 \u2191\u039b f < rieszContentAux \u039b K + \u03b5\n[PROOFSTEP]\nrefine' \u27e8f, f_hyp.left, _\u27e9\n[GOAL]\ncase intro.intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b1 : \u211d\u22650\n\u03b1_hyp : \u03b1 < rieszContentAux \u039b K + \u03b5\nf : X \u2192\u1d47 \u211d\u22650\nf_hyp : f \u2208 {f | \u2200 (x : X), x \u2208 K \u2192 1 \u2264 \u2191f x} \u2227 \u2191\u039b f = \u03b1\n\u22a2 \u2191\u039b f < rieszContentAux \u039b K + \u03b5\n[PROOFSTEP]\nrw [f_hyp.right]\n[GOAL]\ncase intro.intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b1 : \u211d\u22650\n\u03b1_hyp : \u03b1 < rieszContentAux \u039b K + \u03b5\nf : X \u2192\u1d47 \u211d\u22650\nf_hyp : f \u2208 {f | \u2200 (x : X), x \u2208 K \u2192 1 \u2264 \u2191f x} \u2227 \u2191\u039b f = \u03b1\n\u22a2 \u03b1 < rieszContentAux \u039b K + \u03b5\n[PROOFSTEP]\nexact \u03b1_hyp\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2\n[PROOFSTEP]\napply NNReal.le_of_forall_pos_le_add\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u22a2 \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5\n[PROOFSTEP]\nintro \u03b5 \u03b5pos\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5\n[PROOFSTEP]\nobtain \u27e8f1, f_test_function_K1\u27e9 := exists_lt_rieszContentAux_add_pos \u039b K1 (half_pos \u03b5pos)\n[GOAL]\ncase h.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5\n[PROOFSTEP]\nobtain \u27e8f2, f_test_function_K2\u27e9 :=\n exists_lt_rieszContentAux_add_pos \u039b K2\n (half_pos \u03b5pos)\n --let `f := f1 + f2` test function for the content of `K`\n[GOAL]\ncase h.intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5\n[PROOFSTEP]\nhave f_test_function_union : \u2200 x \u2208 K1 \u2294 K2, (1 : \u211d\u22650) \u2264 (f1 + f2) x :=\n by\n rintro x (x_in_K1 | x_in_K2)\n \u00b7 exact le_add_right (f_test_function_K1.left x x_in_K1)\n \u00b7\n exact\n le_add_left\n (f_test_function_K2.left x x_in_K2)\n --use that `\u039bf` is an upper bound for `\u03bb(K1\u2294K2)`\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\n\u22a2 \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x\n[PROOFSTEP]\nrintro x (x_in_K1 | x_in_K2)\n[GOAL]\ncase inl\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nx : X\nx_in_K1 : x \u2208 \u2191K1\n\u22a2 1 \u2264 \u2191(f1 + f2) x\n[PROOFSTEP]\nexact le_add_right (f_test_function_K1.left x x_in_K1)\n[GOAL]\ncase inr\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nx : X\nx_in_K2 : x \u2208 \u2191K2\n\u22a2 1 \u2264 \u2191(f1 + f2) x\n[PROOFSTEP]\nexact\n le_add_left\n (f_test_function_K2.left x x_in_K2)\n --use that `\u039bf` is an upper bound for `\u03bb(K1\u2294K2)`\n[GOAL]\ncase h.intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5\n[PROOFSTEP]\napply (rieszContentAux_le \u039b f_test_function_union).trans (le_of_lt _)\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x\n\u22a2 \u2191\u039b (f1 + f2) < rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5\n[PROOFSTEP]\nrw [map_add]\n --use that `\u039bfi` are lower bounds for `\u03bb(Ki) + \u03b5/2`\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x\n\u22a2 \u2191\u039b f1 + \u2191\u039b f2 < rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5\n[PROOFSTEP]\napply lt_of_lt_of_le (_root_.add_lt_add f_test_function_K1.right f_test_function_K2.right) (le_of_eq _)\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x\n\u22a2 rieszContentAux \u039b K1 + \u03b5 / 2 + (rieszContentAux \u039b K2 + \u03b5 / 2) = rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5\n[PROOFSTEP]\nrw [add_assoc, add_comm (\u03b5 / 2), add_assoc, add_halves \u03b5, add_assoc]\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Integral.RieszMarkovKakutani", "llama_tokens": 4644, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124811, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.41779159065203125}} {"text": "[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)\n[PROOFSTEP]\nsimp only [\u2190 set_integral_congr_set_ae (Box.coe_ae_eq_Icc _)]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =\n \u2211 x : Fin (n + 1),\n ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -\n \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x)\n[PROOFSTEP]\nhave A := (Hi.mono_set Box.coe_subset_Icc).hasBoxIntegral \u22a5 rfl\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =\n \u2211 x : Fin (n + 1),\n ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -\n \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x)\n[PROOFSTEP]\nhave B :=\n hasIntegral_GP_divergence_of_forall_hasDerivWithinAt I f f' (s \u2229 Box.Icc I) (hs.mono (inter_subset_left _ _))\n (fun x hx => Hc _ hx.2) fun x hx => Hd _ \u27e8hx.1, fun h => hx.2 \u27e8h, hx.1\u27e9\u27e9\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n (\u2211 i : Fin (n + 1),\n (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n BoxAdditiveMap.volume -\n BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n BoxAdditiveMap.volume))\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =\n \u2211 x : Fin (n + 1),\n ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -\n \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x)\n[PROOFSTEP]\nrw [continuousOn_pi] at Hc \n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n (\u2211 i : Fin (n + 1),\n (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n BoxAdditiveMap.volume -\n BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n BoxAdditiveMap.volume))\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1 =\n \u2211 x : Fin (n + 1),\n ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.upper I x) x_1) x) -\n \u222b (x_1 : Fin n \u2192 \u211d) in \u2191(Box.face I x), f (Fin.insertNth x (Box.lower I x) x_1) x)\n[PROOFSTEP]\nrefine' (A.unique B).trans (sum_congr rfl fun i _ => _)\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n (\u2211 i : Fin (n + 1),\n (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n BoxAdditiveMap.volume -\n BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n BoxAdditiveMap.volume -\n BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n BoxAdditiveMap.volume =\n (\u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i\n[PROOFSTEP]\nrefine' congr_arg\u2082 Sub.sub _ _\n[GOAL]\ncase refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n (\u2211 i : Fin (n + 1),\n (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n BoxAdditiveMap.volume -\n BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n BoxAdditiveMap.volume =\n \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i\n[PROOFSTEP]\nhave := Box.continuousOn_face_Icc (Hc i) (Set.right_mem_Icc.2 (I.lower_le_upper i))\n[GOAL]\ncase refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n (\u2211 i : Fin (n + 1),\n (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n BoxAdditiveMap.volume -\n BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\nthis : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.upper I i)) (\u2191Box.Icc (Box.face I i))\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n BoxAdditiveMap.volume =\n \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i\n[PROOFSTEP]\nhave := (this.integrableOn_compact (\u03bc := volume) (Box.isCompact_Icc _)).mono_set Box.coe_subset_Icc\n[GOAL]\ncase refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n (\u2211 i : Fin (n + 1),\n (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n BoxAdditiveMap.volume -\n BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\nthis\u271d : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.upper I i)) (\u2191Box.Icc (Box.face I i))\nthis : IntegrableOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.upper I i)) \u2191(Box.face I i)\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n BoxAdditiveMap.volume =\n \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i\n[PROOFSTEP]\nexact (this.hasBoxIntegral \u22a5 rfl).integral_eq\n[GOAL]\ncase refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n (\u2211 i : Fin (n + 1),\n (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n BoxAdditiveMap.volume -\n BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n BoxAdditiveMap.volume =\n \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i\n[PROOFSTEP]\nhave := Box.continuousOn_face_Icc (Hc i) (Set.left_mem_Icc.2 (I.lower_le_upper i))\n[GOAL]\ncase refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n (\u2211 i : Fin (n + 1),\n (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n BoxAdditiveMap.volume -\n BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\nthis : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.lower I i)) (\u2191Box.Icc (Box.face I i))\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n BoxAdditiveMap.volume =\n \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i\n[PROOFSTEP]\nhave := (this.integrableOn_compact (\u03bc := volume) (Box.isCompact_Icc _)).mono_set Box.coe_subset_Icc\n[GOAL]\ncase refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (fun y => f y i) (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc I \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc I) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nA :\n HasIntegral I \u22a5 (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume))\n (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i)\nB :\n HasIntegral I IntegrationParams.GP (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) BoxAdditiveMap.volume\n (\u2211 i : Fin (n + 1),\n (BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i)\n BoxAdditiveMap.volume -\n BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n BoxAdditiveMap.volume))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\nthis\u271d : ContinuousOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.lower I i)) (\u2191Box.Icc (Box.face I i))\nthis : IntegrableOn ((fun y => f y i) \u2218 Fin.insertNth i (Box.lower I i)) \u2191(Box.face I i)\n\u22a2 BoxIntegral.integral (Box.face I i) IntegrationParams.GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i)\n BoxAdditiveMap.volume =\n \u222b (x : Fin n \u2192 \u211d) in \u2191(Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i\n[PROOFSTEP]\nexact (this.hasBoxIntegral \u22a5 rfl).integral_eq\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)\n[PROOFSTEP]\nrcases I.exists_seq_mono_tendsto with \u27e8J, hJ_sub, hJl, hJu\u27e9\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)\n[PROOFSTEP]\nhave hJ_sub' : \u2200 k, Box.Icc (J k) \u2286 Box.Icc I := fun k => (hJ_sub k).trans I.Ioo_subset_Icc\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)\n[PROOFSTEP]\nhave hJ_le : \u2200 k, J k \u2264 I := fun k => Box.le_iff_Icc.2 (hJ_sub' k)\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)\n[PROOFSTEP]\nhave HcJ : \u2200 k, ContinuousOn f (Box.Icc (J k)) := fun k => Hc.mono (hJ_sub' k)\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)\n[PROOFSTEP]\nhave HdJ : \u2200 (k), \u2200 x \u2208 (Box.Icc (J k)) \\ s, HasFDerivWithinAt f (f' x) (Box.Icc (J k)) x := fun k x hx =>\n (Hd x \u27e8hJ_sub k hx.1, hx.2\u27e9).hasFDerivWithinAt\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)\n[PROOFSTEP]\nhave HiJ : \u2200 k, IntegrableOn (\u2211 i, f' \u00b7 (e i) i) (Box.Icc (J k)) volume := fun k =>\n Hi.mono_set\n (hJ_sub' k)\n -- Apply the previous lemma to `J k`.\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)\n[PROOFSTEP]\nhave HJ_eq := fun k =>\n integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2081 (J k) f f' s hs (HcJ k) (HdJ k)\n (HiJ k)\n -- Note that the LHS of `HJ_eq k` tends to the LHS of the goal as `k \u2192 \u221e`.\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)\n[PROOFSTEP]\nhave hI_tendsto :\n Tendsto (fun k => \u222b x in Box.Icc (J k), \u2211 i, f' x (e i) i) atTop (\ud835\udcdd (\u222b x in Box.Icc I, \u2211 i, f' x (e i) i)) :=\n by\n simp only [IntegrableOn, \u2190 Measure.restrict_congr_set (Box.Ioo_ae_eq_Icc _)] at Hi \u22a2\n rw [\u2190 Box.iUnion_Ioo_of_tendsto J.monotone hJl hJu] at Hi \u22a2\n exact tendsto_set_integral_of_monotone (fun k => (J k).measurableSet_Ioo) (Box.Ioo.comp J).monotone Hi\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\n\u22a2 Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) atTop\n (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i))\n[PROOFSTEP]\nsimp only [IntegrableOn, \u2190 Measure.restrict_congr_set (Box.Ioo_ae_eq_Icc _)] at Hi \u22a2\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nHi : Integrable fun x => \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1\n\u22a2 Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Ioo (\u2191J k), \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1) atTop\n (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Ioo I, \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1))\n[PROOFSTEP]\nrw [\u2190 Box.iUnion_Ioo_of_tendsto J.monotone hJl hJu] at Hi \u22a2\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nHi : Integrable fun x => \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1\n\u22a2 Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Ioo (\u2191J k), \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1) atTop\n (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u22c3 (n_1 : \u2115), \u2191Box.Ioo (\u2191J n_1), \u2211 x_1 : Fin (n + 1), \u2191(f' x) (e x_1) x_1))\n[PROOFSTEP]\nexact tendsto_set_integral_of_monotone (fun k => (J k).measurableSet_Ioo) (Box.Ioo.comp J).monotone Hi\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nhI_tendsto :\n Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) atTop\n (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i))\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)\n[PROOFSTEP]\nrefine' tendsto_nhds_unique_of_eventuallyEq hI_tendsto _ (eventually_of_forall HJ_eq)\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nhI_tendsto :\n Tendsto (fun k => \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) atTop\n (\ud835\udcdd (\u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc I, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i))\n\u22a2 Tendsto\n (fun x =>\n \u2211 i : Fin (n + 1),\n ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -\n \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))\n atTop\n (\ud835\udcdd\n (\u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)))\n[PROOFSTEP]\nclear hI_tendsto\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : Tendsto (Box.lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower)\nhJu : Tendsto (Box.upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\n\u22a2 Tendsto\n (fun x =>\n \u2211 i : Fin (n + 1),\n ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -\n \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))\n atTop\n (\ud835\udcdd\n (\u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)))\n[PROOFSTEP]\nrw [tendsto_pi_nhds] at hJl hJu \n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\n\u22a2 Tendsto\n (fun x =>\n \u2211 i : Fin (n + 1),\n ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -\n \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))\n atTop\n (\ud835\udcdd\n (\u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)))\n[PROOFSTEP]\nsuffices\n \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d),\n (\u2200 k, c k \u2208 Icc (I.lower i) (I.upper i)) \u2192\n Tendsto c atTop (\ud835\udcdd d) \u2192\n Tendsto (fun k => \u222b x in Box.Icc ((J k).face i), f (i.insertNth (c k) x) i) atTop\n (\ud835\udcdd <| \u222b x in Box.Icc (I.face i), f (i.insertNth d x) i)\n by\n rw [Box.Icc_eq_pi] at hJ_sub' \n refine' tendsto_finset_sum _ fun i _ => (this _ _ _ _ (hJu _)).sub (this _ _ _ _ (hJl _))\n exacts [fun k => hJ_sub' k (J k).upper_mem_Icc _ trivial, fun k => hJ_sub' k (J k).lower_mem_Icc _ trivial]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nthis :\n \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),\n (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192\n Tendsto c atTop (\ud835\udcdd d) \u2192\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u22a2 Tendsto\n (fun x =>\n \u2211 i : Fin (n + 1),\n ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -\n \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))\n atTop\n (\ud835\udcdd\n (\u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)))\n[PROOFSTEP]\nrw [Box.Icc_eq_pi] at hJ_sub' \n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 Set.pi Set.univ fun i => Set.Icc (Box.lower I i) (Box.upper I i)\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nthis :\n \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),\n (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192\n Tendsto c atTop (\ud835\udcdd d) \u2192\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u22a2 Tendsto\n (fun x =>\n \u2211 i : Fin (n + 1),\n ((\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.upper (\u2191J x) i) x_1) i) -\n \u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (Box.lower (\u2191J x) i) x_1) i))\n atTop\n (\ud835\udcdd\n (\u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.upper I i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i (Box.lower I i) x) i)))\n[PROOFSTEP]\nrefine' tendsto_finset_sum _ fun i _ => (this _ _ _ _ (hJu _)).sub (this _ _ _ _ (hJl _))\n[GOAL]\ncase refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 Set.pi Set.univ fun i => Set.Icc (Box.lower I i) (Box.upper I i)\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nthis :\n \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),\n (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192\n Tendsto c atTop (\ud835\udcdd d) \u2192\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2200 (k : \u2115), (Box.upper \u2218 \u2191J) k i \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\ncase refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 Set.pi Set.univ fun i => Set.Icc (Box.lower I i) (Box.upper I i)\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\nthis :\n \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),\n (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192\n Tendsto c atTop (\ud835\udcdd d) \u2192\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\ni : Fin (n + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2200 (k : \u2115), Box.lower (\u2191J k) i \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\n[PROOFSTEP]\nexacts [fun k => hJ_sub' k (J k).upper_mem_Icc _ trivial, fun k => hJ_sub' k (J k).lower_mem_Icc _ trivial]\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\n\u22a2 \u2200 (i : Fin (n + 1)) (c : \u2115 \u2192 \u211d) (d : \u211d),\n (\u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)) \u2192\n Tendsto c atTop (\ud835\udcdd d) \u2192\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n[PROOFSTEP]\nintro i c d hc hcd\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n[PROOFSTEP]\nhave hd : d \u2208 Icc (I.lower i) (I.upper i) := isClosed_Icc.mem_of_tendsto hcd (eventually_of_forall hc)\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n[PROOFSTEP]\nhave Hic : \u2200 k, IntegrableOn (fun x => f (i.insertNth (c k) x) i) (Box.Icc (I.face i)) := fun k =>\n (Box.continuousOn_face_Icc ((continuous_apply i).comp_continuousOn Hc) (hc k)).integrableOn_Icc\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n[PROOFSTEP]\nhave Hid : IntegrableOn (fun x => f (i.insertNth d x) i) (Box.Icc (I.face i)) :=\n (Box.continuousOn_face_Icc ((continuous_apply i).comp_continuousOn Hc) hd).integrableOn_Icc\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n[PROOFSTEP]\nhave H :\n Tendsto (fun k => \u222b x in Box.Icc ((J k).face i), f (i.insertNth d x) i) atTop\n (\ud835\udcdd <| \u222b x in Box.Icc (I.face i), f (i.insertNth d x) i) :=\n by\n have hIoo : (\u22c3 k, Box.Ioo ((J k).face i)) = Box.Ioo (I.face i) :=\n Box.iUnion_Ioo_of_tendsto ((Box.monotone_face i).comp J.monotone) (tendsto_pi_nhds.2 fun _ => hJl _)\n (tendsto_pi_nhds.2 fun _ => hJu _)\n simp only [IntegrableOn, \u2190 Measure.restrict_congr_set (Box.Ioo_ae_eq_Icc _), \u2190 hIoo] at Hid \u22a2\n exact\n tendsto_set_integral_of_monotone (fun k => ((J k).face i).measurableSet_Ioo)\n (Box.Ioo.monotone.comp ((Box.monotone_face i).comp J.monotone)) Hid\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n[PROOFSTEP]\nhave hIoo : (\u22c3 k, Box.Ioo ((J k).face i)) = Box.Ioo (I.face i) :=\n Box.iUnion_Ioo_of_tendsto ((Box.monotone_face i).comp J.monotone) (tendsto_pi_nhds.2 fun _ => hJl _)\n (tendsto_pi_nhds.2 fun _ => hJu _)\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nhIoo : \u22c3 (k : \u2115), \u2191Box.Ioo (Box.face (\u2191J k) i) = \u2191Box.Ioo (Box.face I i)\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n[PROOFSTEP]\nsimp only [IntegrableOn, \u2190 Measure.restrict_congr_set (Box.Ioo_ae_eq_Icc _), \u2190 hIoo] at Hid \u22a2\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nhIoo : \u22c3 (k : \u2115), \u2191Box.Ioo (Box.face (\u2191J k) i) = \u2191Box.Ioo (Box.face I i)\nHid : Integrable fun x => f (Fin.insertNth i d x) i\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Ioo (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u22c3 (k : \u2115), \u2191Box.Ioo (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i))\n[PROOFSTEP]\nexact\n tendsto_set_integral_of_monotone (fun k => ((J k).face i).measurableSet_Ioo)\n (Box.Ioo.monotone.comp ((Box.monotone_face i).comp J.monotone)) Hid\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u22a2 Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n[PROOFSTEP]\nrefine' H.congr_dist (Metric.nhds_basis_closedBall.tendsto_right_iff.2 fun \u03b5 \u03b5pos => _)\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop,\n dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i d x) i)\n (\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (c x) x_1) i) \u2208\n Metric.closedBall 0 \u03b5\n[PROOFSTEP]\nhave hvol_pos : \u2200 J : Box (Fin n), 0 < \u220f j, (J.upper j - J.lower j) := fun J =>\n prod_pos fun j hj =>\n sub_pos.2 <|\n J.lower_lt_upper\n _\n /- Choose `\u03b4 > 0` such that for any `x y \u2208 I.Icc` at distance at most `\u03b4`, the distance between\n `f x` and `f y` is at most `\u03b5 / volume (I.face i).Icc`, then the distance between the integrals\n is at most `(\u03b5 / volume (I.face i).Icc) * volume ((J k).face i).Icc \u2264 \u03b5`. -/\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop,\n dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i d x) i)\n (\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (c x) x_1) i) \u2208\n Metric.closedBall 0 \u03b5\n[PROOFSTEP]\nrcases Metric.uniformContinuousOn_iff_le.1 (I.isCompact_Icc.uniformContinuousOn_of_continuous Hc)\n (\u03b5 / \u220f j, ((I.face i).upper j - (I.face i).lower j)) (div_pos \u03b5pos (hvol_pos (I.face i))) with\n \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n \u2200 (x : Fin (n + 1) \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2192\n \u2200 (y : Fin (n + 1) \u2192 \u211d),\n y \u2208 \u2191Box.Icc I \u2192\n dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop,\n dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i d x) i)\n (\u222b (x_1 : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J x) i), f (Fin.insertNth i (c x) x_1) i) \u2208\n Metric.closedBall 0 \u03b5\n[PROOFSTEP]\nrefine' (hcd.eventually (Metric.ball_mem_nhds _ \u03b4pos)).mono fun k hk => _\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n \u2200 (x : Fin (n + 1) \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2192\n \u2200 (y : Fin (n + 1) \u2192 \u211d),\n y \u2208 \u2191Box.Icc I \u2192\n dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\n\u22a2 dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i)\n (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) \u2208\n Metric.closedBall 0 \u03b5\n[PROOFSTEP]\nhave Hsub : Box.Icc ((J k).face i) \u2286 Box.Icc (I.face i) := Box.le_iff_Icc.1 (Box.face_mono (hJ_le _) i)\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n \u2200 (x : Fin (n + 1) \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2192\n \u2200 (y : Fin (n + 1) \u2192 \u211d),\n y \u2208 \u2191Box.Icc I \u2192\n dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\n\u22a2 dist (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i)\n (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (c k) x) i) \u2208\n Metric.closedBall 0 \u03b5\n[PROOFSTEP]\nrw [mem_closedBall_zero_iff, Real.norm_eq_abs, abs_of_nonneg dist_nonneg, dist_eq_norm, \u2190\n integral_sub (Hid.mono_set Hsub) ((Hic _).mono_set Hsub)]\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n \u2200 (x : Fin (n + 1) \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2192\n \u2200 (y : Fin (n + 1) \u2192 \u211d),\n y \u2208 \u2191Box.Icc I \u2192\n dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\n\u22a2 \u2016\u222b (a : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d a) i - f (Fin.insertNth i (c k) a) i\u2016 \u2264 \u03b5\n[PROOFSTEP]\ncalc\n \u2016\u222b x in Box.Icc ((J k).face i), f (i.insertNth d x) i - f (i.insertNth (c k) x) i\u2016 \u2264\n (\u03b5 / \u220f j, ((I.face i).upper j - (I.face i).lower j)) * (volume (Box.Icc ((J k).face i))).toReal :=\n by\n refine\n norm_set_integral_le_of_norm_le_const' (((J k).face i).measure_Icc_lt_top _) ((J k).face i).measurableSet_Icc\n fun x hx => ?_\n rw [\u2190 dist_eq_norm]\n calc\n dist (f (i.insertNth d x) i) (f (i.insertNth (c k) x) i) \u2264 dist (f (i.insertNth d x)) (f (i.insertNth (c k) x)) :=\n dist_le_pi_dist (f (i.insertNth d x)) (f (i.insertNth (c k) x)) i\n _ \u2264 \u03b5 / \u220f j, ((I.face i).upper j - (I.face i).lower j) :=\n h\u03b4 _ (I.mapsTo_insertNth_face_Icc hd <| Hsub hx) _ (I.mapsTo_insertNth_face_Icc (hc _) <| Hsub hx) ?_\n rw [Fin.dist_insertNth_insertNth, dist_self, dist_comm]\n exact max_le hk.le \u03b4pos.lt.le\n _ \u2264 \u03b5 :=\n by\n rw [Box.Icc_def, Real.volume_Icc_pi_toReal ((J k).face i).lower_le_upper, \u2190 le_div_iff (hvol_pos _)]\n refine' div_le_div_of_le_left \u03b5pos.le (hvol_pos _) (prod_le_prod (fun j _ => _) fun j _ => _)\n exacts [sub_nonneg.2 (Box.lower_le_upper _ _),\n sub_le_sub ((hJ_sub' _ (J _).upper_mem_Icc).2 _) ((hJ_sub' _ (J _).lower_mem_Icc).1 _)]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n \u2200 (x : Fin (n + 1) \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2192\n \u2200 (y : Fin (n + 1) \u2192 \u211d),\n y \u2208 \u2191Box.Icc I \u2192\n dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\n\u22a2 \u2016\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i - f (Fin.insertNth i (c k) x) i\u2016 \u2264\n (\u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)) *\n ENNReal.toReal (\u2191\u2191volume (\u2191Box.Icc (Box.face (\u2191J k) i)))\n[PROOFSTEP]\nrefine\n norm_set_integral_le_of_norm_le_const' (((J k).face i).measure_Icc_lt_top _) ((J k).face i).measurableSet_Icc\n fun x hx => ?_\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n \u2200 (x : Fin (n + 1) \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2192\n \u2200 (y : Fin (n + 1) \u2192 \u211d),\n y \u2208 \u2191Box.Icc I \u2192\n dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nx : Fin n \u2192 \u211d\nhx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i)\n\u22a2 \u2016f (Fin.insertNth i d x) i - f (Fin.insertNth i (c k) x) i\u2016 \u2264\n \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\n[PROOFSTEP]\nrw [\u2190 dist_eq_norm]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n \u2200 (x : Fin (n + 1) \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2192\n \u2200 (y : Fin (n + 1) \u2192 \u211d),\n y \u2208 \u2191Box.Icc I \u2192\n dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nx : Fin n \u2192 \u211d\nhx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i)\n\u22a2 dist (f (Fin.insertNth i d x) i) (f (Fin.insertNth i (c k) x) i) \u2264\n \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\n[PROOFSTEP]\ncalc\n dist (f (i.insertNth d x) i) (f (i.insertNth (c k) x) i) \u2264 dist (f (i.insertNth d x)) (f (i.insertNth (c k) x)) :=\n dist_le_pi_dist (f (i.insertNth d x)) (f (i.insertNth (c k) x)) i\n _ \u2264 \u03b5 / \u220f j, ((I.face i).upper j - (I.face i).lower j) :=\n h\u03b4 _ (I.mapsTo_insertNth_face_Icc hd <| Hsub hx) _ (I.mapsTo_insertNth_face_Icc (hc _) <| Hsub hx) ?_\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n \u2200 (x : Fin (n + 1) \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2192\n \u2200 (y : Fin (n + 1) \u2192 \u211d),\n y \u2208 \u2191Box.Icc I \u2192\n dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nx : Fin n \u2192 \u211d\nhx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i)\n\u22a2 dist (Fin.insertNth i d x) (Fin.insertNth i (c k) x) \u2264 \u03b4\n[PROOFSTEP]\nrw [Fin.dist_insertNth_insertNth, dist_self, dist_comm]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n \u2200 (x : Fin (n + 1) \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2192\n \u2200 (y : Fin (n + 1) \u2192 \u211d),\n y \u2208 \u2191Box.Icc I \u2192\n dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nx : Fin n \u2192 \u211d\nhx : x \u2208 \u2191Box.Icc (Box.face (\u2191J k) i)\n\u22a2 max (dist (c k) d) 0 \u2264 \u03b4\n[PROOFSTEP]\nexact max_le hk.le \u03b4pos.lt.le\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n \u2200 (x : Fin (n + 1) \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2192\n \u2200 (y : Fin (n + 1) \u2192 \u211d),\n y \u2208 \u2191Box.Icc I \u2192\n dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\n\u22a2 (\u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)) *\n ENNReal.toReal (\u2191\u2191volume (\u2191Box.Icc (Box.face (\u2191J k) i))) \u2264\n \u03b5\n[PROOFSTEP]\nrw [Box.Icc_def, Real.volume_Icc_pi_toReal ((J k).face i).lower_le_upper, \u2190 le_div_iff (hvol_pos _)]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n \u2200 (x : Fin (n + 1) \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2192\n \u2200 (y : Fin (n + 1) \u2192 \u211d),\n y \u2208 \u2191Box.Icc I \u2192\n dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\n\u22a2 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j) \u2264\n \u03b5 / \u220f j : Fin n, (Box.upper (Box.face (\u2191J k) i) j - Box.lower (Box.face (\u2191J k) i) j)\n[PROOFSTEP]\nrefine' div_le_div_of_le_left \u03b5pos.le (hvol_pos _) (prod_le_prod (fun j _ => _) fun j _ => _)\n[GOAL]\ncase refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n \u2200 (x : Fin (n + 1) \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2192\n \u2200 (y : Fin (n + 1) \u2192 \u211d),\n y \u2208 \u2191Box.Icc I \u2192\n dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nj : Fin n\nx\u271d : j \u2208 Finset.univ\n\u22a2 0 \u2264 Box.upper (Box.face (\u2191J k) i) j - Box.lower (Box.face (\u2191J k) i) j\ncase refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (\u2191Box.Icc I)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Ioo I \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc I)\nJ : \u2115 \u2192o Box (Fin (n + 1))\nhJ_sub : \u2200 (n_1 : \u2115), \u2191Box.Icc (\u2191J n_1) \u2286 \u2191Box.Ioo I\nhJl : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.lower \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.lower I x))\nhJu : \u2200 (x : Fin (n + 1)), Tendsto (fun i => (Box.upper \u2218 \u2191J) i x) atTop (\ud835\udcdd (Box.upper I x))\nhJ_sub' : \u2200 (k : \u2115), \u2191Box.Icc (\u2191J k) \u2286 \u2191Box.Icc I\nhJ_le : \u2200 (k : \u2115), \u2191J k \u2264 I\nHcJ : \u2200 (k : \u2115), ContinuousOn f (\u2191Box.Icc (\u2191J k))\nHdJ : \u2200 (k : \u2115) (x : Fin (n + 1) \u2192 \u211d), x \u2208 \u2191Box.Icc (\u2191J k) \\ s \u2192 HasFDerivWithinAt f (f' x) (\u2191Box.Icc (\u2191J k)) x\nHiJ : \u2200 (k : \u2115), IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (\u2191Box.Icc (\u2191J k))\nHJ_eq :\n \u2200 (k : \u2115),\n \u222b (x : Fin (n + 1) \u2192 \u211d) in \u2191Box.Icc (\u2191J k), \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.upper (\u2191J k) i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i (Box.lower (\u2191J k) i) x) i)\ni : Fin (n + 1)\nc : \u2115 \u2192 \u211d\nd : \u211d\nhc : \u2200 (k : \u2115), c k \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nhcd : Tendsto c atTop (\ud835\udcdd d)\nhd : d \u2208 Set.Icc (Box.lower I i) (Box.upper I i)\nHic : \u2200 (k : \u2115), IntegrableOn (fun x => f (Fin.insertNth i (c k) x) i) (\u2191Box.Icc (Box.face I i))\nHid : IntegrableOn (fun x => f (Fin.insertNth i d x) i) (\u2191Box.Icc (Box.face I i))\nH :\n Tendsto (fun k => \u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face (\u2191J k) i), f (Fin.insertNth i d x) i) atTop\n (\ud835\udcdd (\u222b (x : Fin n \u2192 \u211d) in \u2191Box.Icc (Box.face I i), f (Fin.insertNth i d x) i))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nhvol_pos : \u2200 (J : Box (Fin n)), 0 < \u220f j : Fin n, (Box.upper J j - Box.lower J j)\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 :\n \u2200 (x : Fin (n + 1) \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2192\n \u2200 (y : Fin (n + 1) \u2192 \u211d),\n y \u2208 \u2191Box.Icc I \u2192\n dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 / \u220f j : Fin n, (Box.upper (Box.face I i) j - Box.lower (Box.face I i) j)\nk : \u2115\nhk : dist (c k) d < \u03b4\nHsub : \u2191Box.Icc (Box.face (\u2191J k) i) \u2286 \u2191Box.Icc (Box.face I i)\nj : Fin n\nx\u271d : j \u2208 Finset.univ\n\u22a2 Box.upper (Box.face (\u2191J k) i) j - Box.lower (Box.face (\u2191J k) i) j \u2264\n Box.upper (Box.face I i) j - Box.lower (Box.face I i) j\n[PROOFSTEP]\nexacts [sub_nonneg.2 (Box.lower_le_upper _ _),\n sub_le_sub ((hJ_sub' _ (J _).upper_mem_Icc).2 _) ((hJ_sub' _ (J _).lower_mem_Icc).1 _)]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)\n[PROOFSTEP]\nrcases em (\u2203 i, a i = b i) with (\u27e8i, hi\u27e9 | hne)\n[GOAL]\ncase inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)\n[PROOFSTEP]\nrw [volume_pi, \u2190 set_integral_congr_set_ae Measure.univ_pi_Ioc_ae_eq_Icc]\n[GOAL]\ncase inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\n\u22a2 (\u222b (x : Fin (n + 1) \u2192 \u211d) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),\n \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i \u2202Measure.pi fun x => volume) =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)\n[PROOFSTEP]\nhave hi' : Ioc (a i) (b i) = \u2205 := Ioc_eq_empty hi.not_lt\n[GOAL]\ncase inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\n\u22a2 (\u222b (x : Fin (n + 1) \u2192 \u211d) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),\n \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i \u2202Measure.pi fun x => volume) =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)\n[PROOFSTEP]\nhave : (pi Set.univ fun j => Ioc (a j) (b j)) = \u2205 := univ_pi_eq_empty hi'\n[GOAL]\ncase inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\n\u22a2 (\u222b (x : Fin (n + 1) \u2192 \u211d) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),\n \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i \u2202Measure.pi fun x => volume) =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)\n[PROOFSTEP]\nrw [this, integral_empty, sum_eq_zero]\n[GOAL]\ncase inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\n\u22a2 \u2200 (x : Fin (n + 1)),\n x \u2208 Finset.univ \u2192\n (\u222b (x_1 : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove x) (b \u2218 Fin.succAbove x), f (Fin.insertNth x (b x) x_1) x) -\n \u222b (x_1 : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove x) (b \u2218 Fin.succAbove x), f (Fin.insertNth x (a x) x_1) x =\n 0\n[PROOFSTEP]\nrintro j -\n[GOAL]\ncase inl.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\n\u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -\n \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =\n 0\n[PROOFSTEP]\nrcases eq_or_ne i j with (rfl | hne)\n[GOAL]\ncase inl.intro.inl\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\n\u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i =\n 0\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\ncase inl.intro.inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = \u2205\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\nhne : i \u2260 j\n\u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -\n \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =\n 0\n[PROOFSTEP]\nrcases Fin.exists_succAbove_eq hne with \u27e8i, rfl\u27e9\n[GOAL]\ncase inl.intro.inr.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\ni : Fin n\nhi : a (Fin.succAbove j i) = b (Fin.succAbove j i)\nhi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205\nhne : Fin.succAbove j i \u2260 j\n\u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -\n \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =\n 0\n[PROOFSTEP]\nhave : Icc (a \u2218 j.succAbove) (b \u2218 j.succAbove) =\u1d50[volume] (\u2205 : Set \u211d\u207f)\n[GOAL]\ncase this\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\ni : Fin n\nhi : a (Fin.succAbove j i) = b (Fin.succAbove j i)\nhi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205\nhne : Fin.succAbove j i \u2260 j\n\u22a2 Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j) =\u1d50[volume] \u2205\n[PROOFSTEP]\nrw [ae_eq_empty, Real.volume_Icc_pi, prod_eq_zero (Finset.mem_univ i)]\n[GOAL]\ncase this\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\ni : Fin n\nhi : a (Fin.succAbove j i) = b (Fin.succAbove j i)\nhi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205\nhne : Fin.succAbove j i \u2260 j\n\u22a2 ENNReal.ofReal ((b \u2218 Fin.succAbove j) i - (a \u2218 Fin.succAbove j) i) = 0\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\ncase inl.intro.inr.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nthis\u271d : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205\nj : Fin (n + 1)\ni : Fin n\nhi : a (Fin.succAbove j i) = b (Fin.succAbove j i)\nhi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205\nhne : Fin.succAbove j i \u2260 j\nthis : Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j) =\u1d50[volume] \u2205\n\u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -\n \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =\n 0\n[PROOFSTEP]\nrw [set_integral_congr_set_ae this, set_integral_congr_set_ae this, integral_empty, integral_empty, sub_self]\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nhne : \u00ac\u2203 i, a i = b i\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)\n[PROOFSTEP]\nhave hlt : \u2200 i, a i < b i := fun i => (hle i).lt_of_ne fun hi => hne \u27e8i, hi\u27e9\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nn : \u2115\na b : Fin (n + 1) \u2192 \u211d\nhle : a \u2264 b\nf : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E\nf' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E\ns : Set (Fin (n + 1) \u2192 \u211d)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b)\nhne : \u00ac\u2203 i, a i = b i\nhlt : \u2200 (i : Fin (n + 1)), a i < b i\n\u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -\n \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i)\n[PROOFSTEP]\nexact integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2082 \u27e8a, b, hlt\u27e9 f f' s hs Hc Hd Hi\n[GOAL]\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\n\u22a2 \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nx : F\n\u22a2 x \u2208 \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) \u2194 x \u2208 Set.Icc a b\n[PROOFSTEP]\nsimp only [Set.mem_preimage, Set.mem_Icc, he_ord]\n[GOAL]\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\n\u22a2 Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n[PROOFSTEP]\nrw [\u2190 hIcc, eL.symm_preimage_preimage]\n[GOAL]\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 \u222b (x : F) in Set.Icc a b, DF x =\n \u222b (x : F) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\n[PROOFSTEP]\nsimp only [hDF]\n[GOAL]\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 \u222b (x : F) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) =\n \u222b (x : (fun x => Fin (n + 1) \u2192 \u211d) a) in Set.Icc (\u2191eL a) (\u2191eL b),\n \u2211 i : Fin (n + 1), \u2191(f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\n[PROOFSTEP]\nrw [\u2190 he_vol.set_integral_preimage_emb he_emb]\n[GOAL]\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 \u222b (x : F) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) =\n \u222b (x : F) in \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b),\n \u2211 i : Fin (n + 1), \u2191(f' i (\u2191(ContinuousLinearEquiv.symm eL) (\u2191eL x))) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\n[PROOFSTEP]\nsimp only [hIcc, eL.symm_apply_apply]\n[GOAL]\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 \u222b (x : (fun x => Fin (n + 1) \u2192 \u211d) a) in Set.Icc (\u2191eL a) (\u2191eL b),\n \u2211 i : Fin (n + 1), \u2191(f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) (\u2191(ContinuousLinearEquiv.symm eL) (e i)) =\n \u2211 i : Fin (n + 1),\n ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (\u2191eL a \u2218 Fin.succAbove i) (\u2191eL b \u2218 Fin.succAbove i),\n f i (\u2191(ContinuousLinearEquiv.symm eL) (Fin.insertNth i (\u2191eL b i) x))) -\n \u222b (x : Fin n \u2192 \u211d) in Set.Icc (\u2191eL a \u2218 Fin.succAbove i) (\u2191eL b \u2218 Fin.succAbove i),\n f i (\u2191(ContinuousLinearEquiv.symm eL) (Fin.insertNth i (\u2191eL a i) x)))\n[PROOFSTEP]\nrefine\n integral_divergence_of_hasFDerivWithinAt_off_countable' (eL a) (eL b) ((he_ord _ _).2 hle)\n (fun i x => f i (eL.symm x)) (fun i x => f' i (eL.symm x) \u2218L (eL.symm : \u211d\u207f\u207a\u00b9 \u2192L[\u211d] F)) (eL.symm \u207b\u00b9' s)\n (hs.preimage eL.symm.injective) ?_ ?_ ?_\n[GOAL]\ncase refine_1\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 \u2200 (i : Fin (n + 1)), ContinuousOn ((fun i x => f i (\u2191(ContinuousLinearEquiv.symm eL) x)) i) (Set.Icc (\u2191eL a) (\u2191eL b))\n[PROOFSTEP]\nexact fun i => (Hc i).comp eL.symm.continuousOn hIcc'.subset\n[GOAL]\ncase refine_2\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 \u2200 (x : Fin (n + 1) \u2192 \u211d),\n x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s \u2192\n \u2200 (i : Fin (n + 1)),\n HasFDerivAt ((fun i x => f i (\u2191(ContinuousLinearEquiv.symm eL) x)) i)\n ((fun i x =>\n ContinuousLinearMap.comp (f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) \u2191(ContinuousLinearEquiv.symm eL))\n i x)\n x\n[PROOFSTEP]\nrefine' fun x hx i => (Hd (eL.symm x) \u27e8_, hx.2\u27e9 i).comp x eL.symm.hasFDerivAt\n[GOAL]\ncase refine_2\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\nx : Fin (n + 1) \u2192 \u211d\nhx : x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s\ni : Fin (n + 1)\n\u22a2 \u2191(ContinuousLinearEquiv.symm eL) x \u2208 interior (Set.Icc a b)\n[PROOFSTEP]\nrw [\u2190 hIcc]\n[GOAL]\ncase refine_2\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\nx : Fin (n + 1) \u2192 \u211d\nhx : x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s\ni : Fin (n + 1)\n\u22a2 \u2191(ContinuousLinearEquiv.symm eL) x \u2208 interior (\u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b))\n[PROOFSTEP]\nrefine' preimage_interior_subset_interior_preimage eL.continuous _\n[GOAL]\ncase refine_2\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\nx : Fin (n + 1) \u2192 \u211d\nhx : x \u2208 (Set.pi Set.univ fun i => Set.Ioo (\u2191eL a i) (\u2191eL b i)) \\ \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' s\ni : Fin (n + 1)\n\u22a2 \u2191(ContinuousLinearEquiv.symm eL) x \u2208 \u2191eL \u207b\u00b9' interior (Set.Icc (\u2191eL a) (\u2191eL b))\n[PROOFSTEP]\nsimpa only [Set.mem_preimage, eL.apply_symm_apply, \u2190 pi_univ_Icc, interior_pi_set (@finite_univ (Fin _) _),\n interior_Icc] using hx.1\n[GOAL]\ncase refine_3\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 IntegrableOn\n (fun x =>\n \u2211 i : Fin (n + 1),\n \u2191((fun i x =>\n ContinuousLinearMap.comp (f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) \u2191(ContinuousLinearEquiv.symm eL))\n i x)\n (e i))\n (Set.Icc (\u2191eL a) (\u2191eL b))\n[PROOFSTEP]\nrw [\u2190 he_vol.integrableOn_comp_preimage he_emb, hIcc]\n[GOAL]\ncase refine_3\nE : Type u\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\nn : \u2115\nF : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : PartialOrder F\ninst\u271d\u00b9 : MeasureSpace F\ninst\u271d : BorelSpace F\neL : F \u2243L[\u211d] Fin (n + 1) \u2192 \u211d\nhe_ord : \u2200 (x y : F), \u2191eL x \u2264 \u2191eL y \u2194 x \u2264 y\nhe_vol : MeasurePreserving \u2191eL\nf : Fin (n + 1) \u2192 F \u2192 E\nf' : Fin (n + 1) \u2192 F \u2192 F \u2192L[\u211d] E\ns : Set F\nhs : Set.Countable s\na b : F\nhle : a \u2264 b\nHc : \u2200 (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)\nHd : \u2200 (x : F), x \u2208 interior (Set.Icc a b) \\ s \u2192 \u2200 (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x\nDF : F \u2192 E\nhDF : \u2200 (x : F), DF x = \u2211 i : Fin (n + 1), \u2191(f' i x) (\u2191(ContinuousLinearEquiv.symm eL) (e i))\nHi : IntegrableOn DF (Set.Icc a b)\nhe_emb : MeasurableEmbedding \u2191eL\nhIcc : \u2191eL \u207b\u00b9' Set.Icc (\u2191eL a) (\u2191eL b) = Set.Icc a b\nhIcc' : Set.Icc (\u2191eL a) (\u2191eL b) = \u2191(ContinuousLinearEquiv.symm eL) \u207b\u00b9' Set.Icc a b\n\u22a2 IntegrableOn\n ((fun x =>\n \u2211 i : Fin (n + 1),\n \u2191((fun i x =>\n ContinuousLinearMap.comp (f' i (\u2191(ContinuousLinearEquiv.symm eL) x)) \u2191(ContinuousLinearEquiv.symm eL))\n i x)\n (e i)) \u2218\n \u2191eL)\n (Set.Icc a b)\n[PROOFSTEP]\nsimp [\u2190 hDF, (\u00b7 \u2218 \u00b7), Hi]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo a b \\ s \u2192 HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\n\u22a2 \u222b (x : \u211d) in a..b, f' x = f b - f a\n[PROOFSTEP]\nset e : \u211d \u2243L[\u211d] \u211d\u00b9 := (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d).symm\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo a b \\ s \u2192 HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\n\u22a2 \u222b (x : \u211d) in a..b, f' x = f b - f a\n[PROOFSTEP]\nhave e_symm : \u2200 x, e.symm x = x 0 := fun x => rfl\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo a b \\ s \u2192 HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\ne_symm : \u2200 (x : Fin 1 \u2192 \u211d), \u2191(ContinuousLinearEquiv.symm e) x = x 0\n\u22a2 \u222b (x : \u211d) in a..b, f' x = f b - f a\n[PROOFSTEP]\nset F' : \u211d \u2192 \u211d \u2192L[\u211d] E := fun x => smulRight (1 : \u211d \u2192L[\u211d] \u211d) (f' x)\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo a b \\ s \u2192 HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\ne_symm : \u2200 (x : Fin 1 \u2192 \u211d), \u2191(ContinuousLinearEquiv.symm e) x = x 0\nF' : \u211d \u2192 \u211d \u2192L[\u211d] E := fun x => smulRight 1 (f' x)\n\u22a2 \u222b (x : \u211d) in a..b, f' x = f b - f a\n[PROOFSTEP]\nhave hF' : \u2200 x y, F' x y = y \u2022 f' x := fun x y => rfl\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo a b \\ s \u2192 HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\ne_symm : \u2200 (x : Fin 1 \u2192 \u211d), \u2191(ContinuousLinearEquiv.symm e) x = x 0\nF' : \u211d \u2192 \u211d \u2192L[\u211d] E := fun x => smulRight 1 (f' x)\nhF' : \u2200 (x y : \u211d), \u2191(F' x) y = y \u2022 f' x\n\u22a2 \u222b (x : \u211d) in a..b, f' x = f b - f a\n[PROOFSTEP]\ncalc\n \u222b x in a..b, f' x = \u222b x in Icc a b, f' x := by\n rw [intervalIntegral.integral_of_le hle, set_integral_congr_set_ae Ioc_ae_eq_Icc]\n _ =\n \u2211 i : Fin 1,\n ((\u222b x in Icc (e a \u2218 i.succAbove) (e b \u2218 i.succAbove), f (e.symm <| i.insertNth (e b i) x)) -\n \u222b x in Icc (e a \u2218 i.succAbove) (e b \u2218 i.succAbove), f (e.symm <| i.insertNth (e a i) x)) :=\n by\n simp only [\u2190 interior_Icc] at Hd \n refine'\n integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv e _ _ (fun _ => f) (fun _ => F') s hs a b hle\n (fun _ => Hc) (fun x hx _ => Hd x hx) _ _ _\n \u00b7 exact fun x y => (OrderIso.funUnique (Fin 1) \u211d).symm.le_iff_le\n \u00b7 exact (volume_preserving_funUnique (Fin 1) \u211d).symm _\n \u00b7 intro x; rw [Fin.sum_univ_one, hF', e_symm, Pi.single_eq_same, one_smul]\n \u00b7 rw [intervalIntegrable_iff_integrable_Ioc_of_le hle] at Hi \n exact Hi.congr_set_ae Ioc_ae_eq_Icc.symm\n _ = f b - f a := by\n simp only [Fin.sum_univ_one, e_symm]\n have : \u2200 c : \u211d, const (Fin 0) c = isEmptyElim := fun c => Subsingleton.elim _ _\n simp [this, volume_pi, Measure.pi_of_empty fun _ : Fin 0 => volume]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo a b \\ s \u2192 HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\ne_symm : \u2200 (x : Fin 1 \u2192 \u211d), \u2191(ContinuousLinearEquiv.symm e) x = x 0\nF' : \u211d \u2192 \u211d \u2192L[\u211d] E := fun x => smulRight 1 (f' x)\nhF' : \u2200 (x y : \u211d), \u2191(F' x) y = y \u2022 f' x\n\u22a2 \u222b (x : \u211d) in a..b, f' x = \u222b (x : \u211d) in Set.Icc a b, f' x\n[PROOFSTEP]\nrw [intervalIntegral.integral_of_le hle, set_integral_congr_set_ae Ioc_ae_eq_Icc]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo a b \\ s \u2192 HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\ne_symm : \u2200 (x : Fin 1 \u2192 \u211d), \u2191(ContinuousLinearEquiv.symm e) x = x 0\nF' : \u211d \u2192 \u211d \u2192L[\u211d] E := fun x => smulRight 1 (f' x)\nhF' : \u2200 (x y : \u211d), \u2191(F' x) y = y \u2022 f' x\n\u22a2 \u222b (x : \u211d) in Set.Icc a b, f' x =\n \u2211 i : Fin 1,\n ((\u222b (x : Fin 0 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n f (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e b i) x))) -\n \u222b (x : Fin 0 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n f (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e a i) x)))\n[PROOFSTEP]\nsimp only [\u2190 interior_Icc] at Hd \n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHi : IntervalIntegrable f' volume a b\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\ne_symm : \u2200 (x : Fin 1 \u2192 \u211d), \u2191(ContinuousLinearEquiv.symm e) x = x 0\nF' : \u211d \u2192 \u211d \u2192L[\u211d] E := fun x => smulRight 1 (f' x)\nhF' : \u2200 (x y : \u211d), \u2191(F' x) y = y \u2022 f' x\nHd : \u2200 (x : \u211d), x \u2208 interior (Set.Icc a b) \\ s \u2192 HasDerivAt f (f' x) x\n\u22a2 \u222b (x : \u211d) in Set.Icc a b, f' x =\n \u2211 i : Fin 1,\n ((\u222b (x : Fin 0 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n f (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e b i) x))) -\n \u222b (x : Fin 0 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n f (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e a i) x)))\n[PROOFSTEP]\nrefine'\n integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv e _ _ (fun _ => f) (fun _ => F') s hs a b hle\n (fun _ => Hc) (fun x hx _ => Hd x hx) _ _ _\n[GOAL]\ncase refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHi : IntervalIntegrable f' volume a b\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\ne_symm : \u2200 (x : Fin 1 \u2192 \u211d), \u2191(ContinuousLinearEquiv.symm e) x = x 0\nF' : \u211d \u2192 \u211d \u2192L[\u211d] E := fun x => smulRight 1 (f' x)\nhF' : \u2200 (x y : \u211d), \u2191(F' x) y = y \u2022 f' x\nHd : \u2200 (x : \u211d), x \u2208 interior (Set.Icc a b) \\ s \u2192 HasDerivAt f (f' x) x\n\u22a2 \u2200 (x y : \u211d), \u2191e x \u2264 \u2191e y \u2194 x \u2264 y\n[PROOFSTEP]\nexact fun x y => (OrderIso.funUnique (Fin 1) \u211d).symm.le_iff_le\n[GOAL]\ncase refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHi : IntervalIntegrable f' volume a b\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\ne_symm : \u2200 (x : Fin 1 \u2192 \u211d), \u2191(ContinuousLinearEquiv.symm e) x = x 0\nF' : \u211d \u2192 \u211d \u2192L[\u211d] E := fun x => smulRight 1 (f' x)\nhF' : \u2200 (x y : \u211d), \u2191(F' x) y = y \u2022 f' x\nHd : \u2200 (x : \u211d), x \u2208 interior (Set.Icc a b) \\ s \u2192 HasDerivAt f (f' x) x\n\u22a2 MeasurePreserving \u2191e\n[PROOFSTEP]\nexact (volume_preserving_funUnique (Fin 1) \u211d).symm _\n[GOAL]\ncase refine'_3\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHi : IntervalIntegrable f' volume a b\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\ne_symm : \u2200 (x : Fin 1 \u2192 \u211d), \u2191(ContinuousLinearEquiv.symm e) x = x 0\nF' : \u211d \u2192 \u211d \u2192L[\u211d] E := fun x => smulRight 1 (f' x)\nhF' : \u2200 (x y : \u211d), \u2191(F' x) y = y \u2022 f' x\nHd : \u2200 (x : \u211d), x \u2208 interior (Set.Icc a b) \\ s \u2192 HasDerivAt f (f' x) x\n\u22a2 \u2200 (x : \u211d), f' x = \u2211 i : Fin (0 + 1), \u2191((fun x => F') i x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single i 1))\n[PROOFSTEP]\nintro x\n[GOAL]\ncase refine'_3\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHi : IntervalIntegrable f' volume a b\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\ne_symm : \u2200 (x : Fin 1 \u2192 \u211d), \u2191(ContinuousLinearEquiv.symm e) x = x 0\nF' : \u211d \u2192 \u211d \u2192L[\u211d] E := fun x => smulRight 1 (f' x)\nhF' : \u2200 (x y : \u211d), \u2191(F' x) y = y \u2022 f' x\nHd : \u2200 (x : \u211d), x \u2208 interior (Set.Icc a b) \\ s \u2192 HasDerivAt f (f' x) x\nx : \u211d\n\u22a2 f' x = \u2211 i : Fin (0 + 1), \u2191((fun x => F') i x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single i 1))\n[PROOFSTEP]\nrw [Fin.sum_univ_one, hF', e_symm, Pi.single_eq_same, one_smul]\n[GOAL]\ncase refine'_4\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHi : IntervalIntegrable f' volume a b\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\ne_symm : \u2200 (x : Fin 1 \u2192 \u211d), \u2191(ContinuousLinearEquiv.symm e) x = x 0\nF' : \u211d \u2192 \u211d \u2192L[\u211d] E := fun x => smulRight 1 (f' x)\nhF' : \u2200 (x y : \u211d), \u2191(F' x) y = y \u2022 f' x\nHd : \u2200 (x : \u211d), x \u2208 interior (Set.Icc a b) \\ s \u2192 HasDerivAt f (f' x) x\n\u22a2 IntegrableOn (fun x => f' x) (Set.Icc a b)\n[PROOFSTEP]\nrw [intervalIntegrable_iff_integrable_Ioc_of_le hle] at Hi \n[GOAL]\ncase refine'_4\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHi : IntegrableOn f' (Set.Ioc a b)\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\ne_symm : \u2200 (x : Fin 1 \u2192 \u211d), \u2191(ContinuousLinearEquiv.symm e) x = x 0\nF' : \u211d \u2192 \u211d \u2192L[\u211d] E := fun x => smulRight 1 (f' x)\nhF' : \u2200 (x y : \u211d), \u2191(F' x) y = y \u2022 f' x\nHd : \u2200 (x : \u211d), x \u2208 interior (Set.Icc a b) \\ s \u2192 HasDerivAt f (f' x) x\n\u22a2 IntegrableOn (fun x => f' x) (Set.Icc a b)\n[PROOFSTEP]\nexact Hi.congr_set_ae Ioc_ae_eq_Icc.symm\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo a b \\ s \u2192 HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\ne_symm : \u2200 (x : Fin 1 \u2192 \u211d), \u2191(ContinuousLinearEquiv.symm e) x = x 0\nF' : \u211d \u2192 \u211d \u2192L[\u211d] E := fun x => smulRight 1 (f' x)\nhF' : \u2200 (x y : \u211d), \u2191(F' x) y = y \u2022 f' x\n\u22a2 \u2211 i : Fin 1,\n ((\u222b (x : Fin 0 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n f (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e b i) x))) -\n \u222b (x : Fin 0 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n f (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e a i) x))) =\n f b - f a\n[PROOFSTEP]\nsimp only [Fin.sum_univ_one, e_symm]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo a b \\ s \u2192 HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\ne_symm : \u2200 (x : Fin 1 \u2192 \u211d), \u2191(ContinuousLinearEquiv.symm e) x = x 0\nF' : \u211d \u2192 \u211d \u2192L[\u211d] E := fun x => smulRight 1 (f' x)\nhF' : \u2200 (x y : \u211d), \u2191(F' x) y = y \u2022 f' x\n\u22a2 (\u222b (x : Fin 0 \u2192 \u211d) in\n Set.Icc (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)) a \u2218 Fin.succAbove 0)\n (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)) b \u2218 Fin.succAbove 0),\n f (Fin.insertNth 0 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)) b 0) x 0)) -\n \u222b (x : Fin 0 \u2192 \u211d) in\n Set.Icc (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)) a \u2218 Fin.succAbove 0)\n (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)) b \u2218 Fin.succAbove 0),\n f (Fin.insertNth 0 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)) a 0) x 0) =\n f b - f a\n[PROOFSTEP]\nhave : \u2200 c : \u211d, const (Fin 0) c = isEmptyElim := fun c => Subsingleton.elim _ _\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\nhle : a \u2264 b\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo a b \\ s \u2192 HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\ne : \u211d \u2243L[\u211d] Fin 1 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)\ne_symm : \u2200 (x : Fin 1 \u2192 \u211d), \u2191(ContinuousLinearEquiv.symm e) x = x 0\nF' : \u211d \u2192 \u211d \u2192L[\u211d] E := fun x => smulRight 1 (f' x)\nhF' : \u2200 (x y : \u211d), \u2191(F' x) y = y \u2022 f' x\nthis : \u2200 (c : \u211d), const (Fin 0) c = fun a => isEmptyElim a\n\u22a2 (\u222b (x : Fin 0 \u2192 \u211d) in\n Set.Icc (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)) a \u2218 Fin.succAbove 0)\n (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)) b \u2218 Fin.succAbove 0),\n f (Fin.insertNth 0 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)) b 0) x 0)) -\n \u222b (x : Fin 0 \u2192 \u211d) in\n Set.Icc (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)) a \u2218 Fin.succAbove 0)\n (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)) b \u2218 Fin.succAbove 0),\n f (Fin.insertNth 0 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) \u211d \u211d)) a 0) x 0) =\n f b - f a\n[PROOFSTEP]\nsimp [this, volume_pi, Measure.pi_of_empty fun _ : Fin 0 => volume]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f [[a, b]]\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo (min a b) (max a b) \\ s \u2192 HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\n\u22a2 \u222b (x : \u211d) in a..b, f' x = f b - f a\n[PROOFSTEP]\ncases' le_total a b with hab hab\n[GOAL]\ncase inl\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f [[a, b]]\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo (min a b) (max a b) \\ s \u2192 HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\nhab : a \u2264 b\n\u22a2 \u222b (x : \u211d) in a..b, f' x = f b - f a\n[PROOFSTEP]\nsimp only [uIcc_of_le hab, min_eq_left hab, max_eq_right hab] at *\n[GOAL]\ncase inl\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\ns : Set \u211d\nhs : Set.Countable s\nHi : IntervalIntegrable f' volume a b\nhab : a \u2264 b\nHc : ContinuousOn f (Set.Icc a b)\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo a b \\ s \u2192 HasDerivAt f (f' x) x\n\u22a2 \u222b (x : \u211d) in a..b, f' x = f b - f a\n[PROOFSTEP]\nexact integral_eq_of_hasDerivWithinAt_off_countable_of_le f f' hab hs Hc Hd Hi\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\ns : Set \u211d\nhs : Set.Countable s\nHc : ContinuousOn f [[a, b]]\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo (min a b) (max a b) \\ s \u2192 HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\nhab : b \u2264 a\n\u22a2 \u222b (x : \u211d) in a..b, f' x = f b - f a\n[PROOFSTEP]\nsimp only [uIcc_of_ge hab, min_eq_right hab, max_eq_left hab] at *\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\ns : Set \u211d\nhs : Set.Countable s\nHi : IntervalIntegrable f' volume a b\nhab : b \u2264 a\nHc : ContinuousOn f (Set.Icc b a)\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo b a \\ s \u2192 HasDerivAt f (f' x) x\n\u22a2 \u222b (x : \u211d) in a..b, f' x = f b - f a\n[PROOFSTEP]\nrw [intervalIntegral.integral_symm, neg_eq_iff_eq_neg, neg_sub]\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf f' : \u211d \u2192 E\na b : \u211d\ns : Set \u211d\nhs : Set.Countable s\nHi : IntervalIntegrable f' volume a b\nhab : b \u2264 a\nHc : ContinuousOn f (Set.Icc b a)\nHd : \u2200 (x : \u211d), x \u2208 Set.Ioo b a \\ s \u2192 HasDerivAt f (f' x) x\n\u22a2 \u222b (x : \u211d) in b..a, f' x = f a - f b\n[PROOFSTEP]\nexact integral_eq_of_hasDerivWithinAt_off_countable_of_le f f' hab hs Hc Hd Hi.symm\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 \u222b (x : \u211d \u00d7 \u211d) in Set.Icc a b, \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n \u2211 i : Fin 2,\n ((\u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e b i) x))) -\n \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e a i) x)))\n[PROOFSTEP]\nrefine'\n integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv e _ _ ![f, g] ![f', g'] s hs a b hle _ (fun x hx => _)\n _ _ Hi\n[GOAL]\ncase refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 \u2200 (x y : \u211d \u00d7 \u211d), \u2191e x \u2264 \u2191e y \u2194 x \u2264 y\n[PROOFSTEP]\nexact fun x y => (OrderIso.finTwoArrowIso \u211d).symm.le_iff_le\n[GOAL]\ncase refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 MeasurePreserving \u2191e\n[PROOFSTEP]\nexact (volume_preserving_finTwoArrow \u211d).symm _\n[GOAL]\ncase refine'_3\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 \u2200 (i : Fin (1 + 1)), ContinuousOn (Matrix.vecCons f ![g] i) (Set.Icc a b)\n[PROOFSTEP]\nexact Fin.forall_fin_two.2 \u27e8Hcf, Hcg\u27e9\n[GOAL]\ncase refine'_4\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nx : \u211d \u00d7 \u211d\nhx : x \u2208 interior (Set.Icc a b) \\ s\n\u22a2 \u2200 (i : Fin (1 + 1)), HasFDerivAt (Matrix.vecCons f ![g] i) (Matrix.vecCons f' ![g'] i x) x\n[PROOFSTEP]\nrw [Icc_prod_eq, interior_prod_eq, interior_Icc, interior_Icc] at hx \n[GOAL]\ncase refine'_4\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nx : \u211d \u00d7 \u211d\nhx : x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s\n\u22a2 \u2200 (i : Fin (1 + 1)), HasFDerivAt (Matrix.vecCons f ![g] i) (Matrix.vecCons f' ![g'] i x) x\n[PROOFSTEP]\nexact Fin.forall_fin_two.2 \u27e8Hdf x hx, Hdg x hx\u27e9\n[GOAL]\ncase refine'_5\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 \u2200 (x : \u211d \u00d7 \u211d),\n \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n \u2211 i : Fin (1 + 1), \u2191(Matrix.vecCons f' ![g'] i x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single i 1))\n[PROOFSTEP]\nintro x\n[GOAL]\ncase refine'_5\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nx : \u211d \u00d7 \u211d\n\u22a2 \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n \u2211 i : Fin (1 + 1), \u2191(Matrix.vecCons f' ![g'] i x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single i 1))\n[PROOFSTEP]\nrw [Fin.sum_univ_two]\n[GOAL]\ncase refine'_5\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nx : \u211d \u00d7 \u211d\n\u22a2 \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n \u2191(Matrix.vecCons f' ![g'] 0 x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single 0 1)) +\n \u2191(Matrix.vecCons f' ![g'] 1 x) (\u2191(ContinuousLinearEquiv.symm e) (Pi.single 1 1))\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 \u2211 i : Fin 2,\n ((\u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e b i) x))) -\n \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e a i) x))) =\n ((\u222b (y : \u211d) in Set.Icc a.snd b.snd, f (b.fst, y)) - \u222b (y : \u211d) in Set.Icc a.snd b.snd, f (a.fst, y)) +\n ((\u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, b.snd)) - \u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, a.snd))\n[PROOFSTEP]\nhave : \u2200 (a b : \u211d\u00b9) (f : \u211d\u00b9 \u2192 E), \u222b x in Icc a b, f x = \u222b x in Icc (a 0) (b 0), f fun _ => x := fun a b f \u21a6\n by\n convert\n (((volume_preserving_funUnique (Fin 1) \u211d).symm _).set_integral_preimage_emb (MeasurableEquiv.measurableEmbedding _)\n f _).symm\n exact ((OrderIso.funUnique (Fin 1) \u211d).symm.preimage_Icc a b).symm\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u271d b\u271d : \u211d \u00d7 \u211d\nhle : a\u271d \u2264 b\u271d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f\u271d (Set.Icc a\u271d b\u271d)\nHcg : ContinuousOn g (Set.Icc a\u271d b\u271d)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u271d.fst b\u271d.fst \u00d7\u02e2 Set.Ioo a\u271d.snd b\u271d.snd \\ s \u2192 HasFDerivAt f\u271d (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u271d.fst b\u271d.fst \u00d7\u02e2 Set.Ioo a\u271d.snd b\u271d.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u271d b\u271d)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\na b : Fin 1 \u2192 \u211d\nf : (Fin 1 \u2192 \u211d) \u2192 E\n\u22a2 \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc a b, f x = \u222b (x : \u211d) in Set.Icc (a 0) (b 0), f fun x_1 => x\n[PROOFSTEP]\nconvert\n (((volume_preserving_funUnique (Fin 1) \u211d).symm _).set_integral_preimage_emb (MeasurableEquiv.measurableEmbedding _) f\n _).symm\n[GOAL]\ncase h.e'_3.h.e'_6.h.e'_4\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u271d b\u271d : \u211d \u00d7 \u211d\nhle : a\u271d \u2264 b\u271d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f\u271d (Set.Icc a\u271d b\u271d)\nHcg : ContinuousOn g (Set.Icc a\u271d b\u271d)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u271d.fst b\u271d.fst \u00d7\u02e2 Set.Ioo a\u271d.snd b\u271d.snd \\ s \u2192 HasFDerivAt f\u271d (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u271d.fst b\u271d.fst \u00d7\u02e2 Set.Ioo a\u271d.snd b\u271d.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u271d b\u271d)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\na b : Fin 1 \u2192 \u211d\nf : (Fin 1 \u2192 \u211d) \u2192 E\n\u22a2 Set.Icc (a 0) (b 0) = \u2191(MeasurableEquiv.symm (MeasurableEquiv.funUnique (Fin 1) \u211d)) \u207b\u00b9' Set.Icc a b\n[PROOFSTEP]\nexact ((OrderIso.funUnique (Fin 1) \u211d).symm.preimage_Icc a b).symm\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nthis :\n \u2200 (a b : Fin 1 \u2192 \u211d) (f : (Fin 1 \u2192 \u211d) \u2192 E),\n \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc a b, f x = \u222b (x : \u211d) in Set.Icc (a 0) (b 0), f fun x_1 => x\n\u22a2 \u2211 i : Fin 2,\n ((\u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e b i) x))) -\n \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc (\u2191e a \u2218 Fin.succAbove i) (\u2191e b \u2218 Fin.succAbove i),\n Matrix.vecCons f ![g] i (\u2191(ContinuousLinearEquiv.symm e) (Fin.insertNth i (\u2191e a i) x))) =\n ((\u222b (y : \u211d) in Set.Icc a.snd b.snd, f (b.fst, y)) - \u222b (y : \u211d) in Set.Icc a.snd b.snd, f (a.fst, y)) +\n ((\u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, b.snd)) - \u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, a.snd))\n[PROOFSTEP]\nsimp only [Fin.sum_univ_two, this]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\nthis :\n \u2200 (a b : Fin 1 \u2192 \u211d) (f : (Fin 1 \u2192 \u211d) \u2192 E),\n \u222b (x : Fin 1 \u2192 \u211d) in Set.Icc a b, f x = \u222b (x : \u211d) in Set.Icc (a 0) (b 0), f fun x_1 => x\n\u22a2 ((\u222b (x : \u211d) in\n Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 0) 0)\n ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 0) 0),\n Matrix.vecCons f ![g] 0\n (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))\n (Fin.insertNth 0 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b 0) fun x_1 =>\n x))) -\n \u222b (x : \u211d) in\n Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 0) 0)\n ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 0) 0),\n Matrix.vecCons f ![g] 0\n (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))\n (Fin.insertNth 0 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a 0) fun x_1 =>\n x))) +\n ((\u222b (x : \u211d) in\n Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 1) 0)\n ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 1) 0),\n Matrix.vecCons f ![g] 1\n (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))\n (Fin.insertNth 1 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b 1) fun x_1 =>\n x))) -\n \u222b (x : \u211d) in\n Set.Icc ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a \u2218 Fin.succAbove 1) 0)\n ((\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) b \u2218 Fin.succAbove 1) 0),\n Matrix.vecCons f ![g] 1\n (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)))\n (Fin.insertNth 1 (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)) a 1) fun x_1 =>\n x))) =\n ((\u222b (y : \u211d) in Set.Icc a.snd b.snd, f (b.fst, y)) - \u222b (y : \u211d) in Set.Icc a.snd b.snd, f (a.fst, y)) +\n ((\u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, b.snd)) - \u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, a.snd))\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 ((\u222b (y : \u211d) in Set.Icc a.snd b.snd, f (b.fst, y)) - \u222b (y : \u211d) in Set.Icc a.snd b.snd, f (a.fst, y)) +\n ((\u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, b.snd)) - \u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, a.snd)) =\n (((\u222b (x : \u211d) in a.fst..b.fst, g (x, b.snd)) - \u222b (x : \u211d) in a.fst..b.fst, g (x, a.snd)) +\n \u222b (y : \u211d) in a.snd..b.snd, f (b.fst, y)) -\n \u222b (y : \u211d) in a.snd..b.snd, f (a.fst, y)\n[PROOFSTEP]\nsimp only [intervalIntegral.integral_of_le hle.1, intervalIntegral.integral_of_le hle.2,\n set_integral_congr_set_ae (Ioc_ae_eq_Icc (\u03b1 := \u211d) (\u03bc := volume))]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 ((\u222b (y : \u211d) in Set.Icc a.snd b.snd, f (b.fst, y)) - \u222b (y : \u211d) in Set.Icc a.snd b.snd, f (a.fst, y)) +\n ((\u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, b.snd)) - \u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, a.snd)) =\n (((\u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, b.snd)) - \u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, a.snd)) +\n \u222b (y : \u211d) in Set.Icc a.snd b.snd, f (b.fst, y)) -\n \u222b (y : \u211d) in Set.Icc a.snd b.snd, f (a.fst, y)\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na b : \u211d \u00d7 \u211d\nhle : a \u2264 b\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f (Set.Icc a b)\nHcg : ContinuousOn g (Set.Icc a b)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a.fst b.fst \u00d7\u02e2 Set.Ioo a.snd b.snd \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a b)\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] Fin 2 \u2192 \u211d := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow \u211d \u211d)\n\u22a2 ((\u222b (y : \u211d) in Set.Icc a.snd b.snd, f (b.fst, y)) - \u222b (y : \u211d) in Set.Icc a.snd b.snd, f (a.fst, y)) +\n ((\u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, b.snd)) - \u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, a.snd)) =\n (((\u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, b.snd)) - \u222b (x : \u211d) in Set.Icc a.fst b.fst, g (x, a.snd)) +\n \u222b (y : \u211d) in Set.Icc a.snd b.snd, f (b.fst, y)) -\n \u222b (y : \u211d) in Set.Icc a.snd b.snd, f (a.fst, y)\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n[PROOFSTEP]\nwlog h\u2081 : a\u2081 \u2264 b\u2081 generalizing a\u2081 b\u2081\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nthis :\n \u2200 (a\u2081 b\u2081 : \u211d),\n ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n a\u2081 \u2264 b\u2081 \u2192\n \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) +\n \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n[PROOFSTEP]\nspecialize this b\u2081 a\u2081\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\nthis :\n ContinuousOn f ([[b\u2081, a\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n ContinuousOn g ([[b\u2081, a\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min b\u2081 a\u2081) (max b\u2081 a\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min b\u2081 a\u2081) (max b\u2081 a\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[b\u2081, a\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n b\u2081 \u2264 a\u2081 \u2192\n \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n[PROOFSTEP]\nrw [uIcc_comm b\u2081 a\u2081, min_comm b\u2081 a\u2081, max_comm b\u2081 a\u2081] at this \n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\nthis :\n ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n b\u2081 \u2264 a\u2081 \u2192\n \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n[PROOFSTEP]\nsimp only [intervalIntegral.integral_symm b\u2081 a\u2081]\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\nthis :\n ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n b\u2081 \u2264 a\u2081 \u2192\n \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)\n\u22a2 -\u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((-\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - -\u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n[PROOFSTEP]\nrefine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h\u2081))).trans _\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\nthis :\n ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n b\u2081 \u2264 a\u2081 \u2192\n \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)\n\u22a2 -((((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) =\n (((-\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - -\u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n[PROOFSTEP]\nabel\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2081 a\u2082 b\u2081 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : \u00aca\u2081 \u2264 b\u2081\nthis :\n ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n b\u2081 \u2264 a\u2081 \u2192\n \u222b (x : \u211d) in b\u2081..a\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)\n\u22a2 -((((\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) =\n (((-\u222b (x : \u211d) in b\u2081..a\u2081, g (x, b\u2082)) - -\u222b (x : \u211d) in b\u2081..a\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n[PROOFSTEP]\nwlog h\u2082 : a\u2082 \u2264 b\u2082 generalizing a\u2082 b\u2082\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nthis :\n \u2200 (a\u2082 b\u2082 : \u211d),\n ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n a\u2082 \u2264 b\u2082 \u2192\n \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) +\n \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n[PROOFSTEP]\nspecialize this b\u2082 a\u2082\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\nthis :\n ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[b\u2082, a\u2082]]) \u2192\n ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[b\u2082, a\u2082]]) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min b\u2082 a\u2082) (max b\u2082 a\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min b\u2082 a\u2082) (max b\u2082 a\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[b\u2082, a\u2082]]) \u2192\n b\u2082 \u2264 a\u2082 \u2192\n \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n[PROOFSTEP]\nrw [uIcc_comm b\u2082 a\u2082, min_comm b\u2082 a\u2082, max_comm b\u2082 a\u2082] at this \n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\nthis :\n ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n b\u2082 \u2264 a\u2082 \u2192\n \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n[PROOFSTEP]\nsimp only [intervalIntegral.integral_symm b\u2082 a\u2082, intervalIntegral.integral_neg]\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\nthis :\n ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n b\u2082 \u2264 a\u2082 \u2192\n \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n\u22a2 -\u222b (x : \u211d) in a\u2081..b\u2081, \u222b (x_1 : \u211d) in b\u2082..a\u2082, \u2191(f' (x, x_1)) (1, 0) + \u2191(g' (x, x_1)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + -\u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n -\u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n[PROOFSTEP]\nrefine' (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_le h\u2082))).trans _\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\nthis :\n ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n b\u2082 \u2264 a\u2082 \u2192\n \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n\u22a2 -((((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + -\u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n -\u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n[PROOFSTEP]\nabel\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\na\u2082 b\u2082 : \u211d\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2081 : a\u2081 \u2264 b\u2081\nh\u2082 : \u00aca\u2082 \u2264 b\u2082\nthis :\n ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x) \u2192\n (\u2200 (x : \u211d \u00d7 \u211d),\n x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x) \u2192\n IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]]) \u2192\n b\u2082 \u2264 a\u2082 \u2192\n \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in b\u2082..a\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n\u22a2 -((((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) + \u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + -\u222b (y : \u211d) in b\u2082..a\u2082, f (b\u2081, y)) -\n -\u222b (y : \u211d) in b\u2082..a\u2082, f (a\u2081, y)\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nHcf : ContinuousOn f ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHcg : ContinuousOn g ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (min a\u2081 b\u2081) (max a\u2081 b\u2081) \u00d7\u02e2 Set.Ioo (min a\u2082 b\u2082) (max a\u2082 b\u2082) \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) ([[a\u2081, b\u2081]] \u00d7\u02e2 [[a\u2082, b\u2082]])\nh\u2082 : a\u2082 \u2264 b\u2082\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n[PROOFSTEP]\nsimp only [uIcc_of_le h\u2081, uIcc_of_le h\u2082, min_eq_left, max_eq_right, h\u2081, h\u2082] at Hcf Hcg Hdf Hdg Hi \n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\nHcg : ContinuousOn g (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n[PROOFSTEP]\ncalc\n (\u222b x in a\u2081..b\u2081, \u222b y in a\u2082..b\u2082, f' (x, y) (1, 0) + g' (x, y) (0, 1)) =\n \u222b x in Icc a\u2081 b\u2081, \u222b y in Icc a\u2082 b\u2082, f' (x, y) (1, 0) + g' (x, y) (0, 1) :=\n by\n simp only [intervalIntegral.integral_of_le, h\u2081, h\u2082,\n set_integral_congr_set_ae (Ioc_ae_eq_Icc (\u03b1 := \u211d) (\u03bc := volume))]\n _ = \u222b x in Icc a\u2081 b\u2081 \u00d7\u02e2 Icc a\u2082 b\u2082, f' x (1, 0) + g' x (0, 1) := (set_integral_prod _ Hi).symm\n _ = (((\u222b x in a\u2081..b\u2081, g (x, b\u2082)) - \u222b x in a\u2081..b\u2081, g (x, a\u2082)) + \u222b y in a\u2082..b\u2082, f (b\u2081, y)) - \u222b y in a\u2082..b\u2082, f (a\u2081, y) :=\n by\n rw [Icc_prod_Icc] at *\n apply\n integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le f g f' g' (a\u2081, a\u2082) (b\u2081, b\u2082) \u27e8h\u2081, h\u2082\u27e9 s <;>\n assumption\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\nHcg : ContinuousOn g (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\n\u22a2 \u222b (x : \u211d) in a\u2081..b\u2081, \u222b (y : \u211d) in a\u2082..b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1) =\n \u222b (x : \u211d) in Set.Icc a\u2081 b\u2081, \u222b (y : \u211d) in Set.Icc a\u2082 b\u2082, \u2191(f' (x, y)) (1, 0) + \u2191(g' (x, y)) (0, 1)\n[PROOFSTEP]\nsimp only [intervalIntegral.integral_of_le, h\u2081, h\u2082, set_integral_congr_set_ae (Ioc_ae_eq_Icc (\u03b1 := \u211d) (\u03bc := volume))]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\nHcg : ContinuousOn g (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082)\n\u22a2 \u222b (x : \u211d \u00d7 \u211d) in Set.Icc a\u2081 b\u2081 \u00d7\u02e2 Set.Icc a\u2082 b\u2082, \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n[PROOFSTEP]\nrw [Icc_prod_Icc] at *\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHcg : ContinuousOn g (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\n\u22a2 \u222b (x : \u211d \u00d7 \u211d) in Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082), \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1) =\n (((\u222b (x : \u211d) in a\u2081..b\u2081, g (x, b\u2082)) - \u222b (x : \u211d) in a\u2081..b\u2081, g (x, a\u2082)) + \u222b (y : \u211d) in a\u2082..b\u2082, f (b\u2081, y)) -\n \u222b (y : \u211d) in a\u2082..b\u2082, f (a\u2081, y)\n[PROOFSTEP]\napply integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le f g f' g' (a\u2081, a\u2082) (b\u2081, b\u2082) \u27e8h\u2081, h\u2082\u27e9 s\n[GOAL]\ncase hs\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHcg : ContinuousOn g (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\n\u22a2 Set.Countable s\n[PROOFSTEP]\nassumption\n[GOAL]\ncase Hcf\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHcg : ContinuousOn g (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\n\u22a2 ContinuousOn f (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\n[PROOFSTEP]\nassumption\n[GOAL]\ncase Hcg\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHcg : ContinuousOn g (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\n\u22a2 ContinuousOn g (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\n[PROOFSTEP]\nassumption\n[GOAL]\ncase Hdf\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHcg : ContinuousOn g (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\n\u22a2 \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (a\u2081, a\u2082).fst (b\u2081, b\u2082).fst \u00d7\u02e2 Set.Ioo (a\u2081, a\u2082).snd (b\u2081, b\u2082).snd \\ s \u2192 HasFDerivAt f (f' x) x\n[PROOFSTEP]\nassumption\n[GOAL]\ncase Hdg\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHcg : ContinuousOn g (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\n\u22a2 \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo (a\u2081, a\u2082).fst (b\u2081, b\u2082).fst \u00d7\u02e2 Set.Ioo (a\u2081, a\u2082).snd (b\u2081, b\u2082).snd \\ s \u2192 HasFDerivAt g (g' x) x\n[PROOFSTEP]\nassumption\n[GOAL]\ncase Hi\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u211d \u00d7 \u211d \u2192 E\nf' g' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E\ns : Set (\u211d \u00d7 \u211d)\nhs : Set.Countable s\na\u2081 b\u2081 : \u211d\nh\u2081 : a\u2081 \u2264 b\u2081\na\u2082 b\u2082 : \u211d\nh\u2082 : a\u2082 \u2264 b\u2082\nHcf : ContinuousOn f (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHcg : ContinuousOn g (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\nHdf : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt f (f' x) x\nHdg : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 Set.Ioo a\u2081 b\u2081 \u00d7\u02e2 Set.Ioo a\u2082 b\u2082 \\ s \u2192 HasFDerivAt g (g' x) x\nHi : IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\n\u22a2 IntegrableOn (fun x => \u2191(f' x) (1, 0) + \u2191(g' x) (0, 1)) (Set.Icc (a\u2081, a\u2082) (b\u2081, b\u2082))\n[PROOFSTEP]\nassumption\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "llama_tokens": 114658, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7981867777396211, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.4177872017208326}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b2 : Mul R\ninst\u271d\u00b9 : Add R\ninst\u271d : RightDistribClass R\na b c d : R\n\u22a2 (a + b + c) * d = a * d + b * d + c * d\n[PROOFSTEP]\nsimp [right_distrib]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b2 : Add \u03b1\ninst\u271d\u00b9 : MulOneClass \u03b1\ninst\u271d : RightDistribClass \u03b1\na b : \u03b1\n\u22a2 (a + 1) * b = a * b + b\n[PROOFSTEP]\nrw [add_mul, one_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b2 : Add \u03b1\ninst\u271d\u00b9 : MulOneClass \u03b1\ninst\u271d : LeftDistribClass \u03b1\na b : \u03b1\n\u22a2 a * (b + 1) = a * b + a\n[PROOFSTEP]\nrw [mul_add, mul_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b2 : Add \u03b1\ninst\u271d\u00b9 : MulOneClass \u03b1\ninst\u271d : RightDistribClass \u03b1\na b : \u03b1\n\u22a2 (1 + a) * b = b + a * b\n[PROOFSTEP]\nrw [add_mul, one_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b2 : Add \u03b1\ninst\u271d\u00b9 : MulOneClass \u03b1\ninst\u271d : LeftDistribClass \u03b1\na b : \u03b1\n\u22a2 a * (1 + b) = a + a * b\n[PROOFSTEP]\nrw [mul_add, mul_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonAssocSemiring \u03b1\nn : \u03b1\n\u22a2 1 * n + 1 * n = n + n\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonAssocSemiring \u03b1\nn : \u03b1\n\u22a2 n * 1 + n * 1 = n + n\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Mul \u03b1\nP : Prop\ninst\u271d : Decidable P\na b c : \u03b1\n\u22a2 (a * if P then b else c) = if P then a * b else a * c\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Mul \u03b1\nP : Prop\ninst\u271d : Decidable P\na b c : \u03b1\nh\u271d : P\n\u22a2 a * b = a * b\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Mul \u03b1\nP : Prop\ninst\u271d : Decidable P\na b c : \u03b1\nh\u271d : \u00acP\n\u22a2 a * c = a * c\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Mul \u03b1\nP : Prop\ninst\u271d : Decidable P\na b c : \u03b1\n\u22a2 (if P then a else b) * c = if P then a * c else b * c\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Mul \u03b1\nP : Prop\ninst\u271d : Decidable P\na b c : \u03b1\nh\u271d : P\n\u22a2 a * c = a * c\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Mul \u03b1\nP : Prop\ninst\u271d : Decidable P\na b c : \u03b1\nh\u271d : \u00acP\n\u22a2 b * c = b * c\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MulZeroOneClass \u03b1\nP : Prop\ninst\u271d : Decidable P\na : \u03b1\n\u22a2 (a * if P then 1 else 0) = if P then a else 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MulZeroOneClass \u03b1\nP : Prop\ninst\u271d : Decidable P\na : \u03b1\n\u22a2 (if P then 1 else 0) * a = if P then a else 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MulZeroClass \u03b1\nP : Prop\ninst\u271d : Decidable P\na b : \u03b1\n\u22a2 (if P then a * b else 0) = (if P then a else 0) * b\n[PROOFSTEP]\nby_cases h : P\n[GOAL]\ncase pos\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MulZeroClass \u03b1\nP : Prop\ninst\u271d : Decidable P\na b : \u03b1\nh : P\n\u22a2 (if P then a * b else 0) = (if P then a else 0) * b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MulZeroClass \u03b1\nP : Prop\ninst\u271d : Decidable P\na b : \u03b1\nh : \u00acP\n\u22a2 (if P then a * b else 0) = (if P then a else 0) * b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MulZeroClass \u03b1\nP : Prop\ninst\u271d : Decidable P\na b : \u03b1\n\u22a2 (if P then a * b else 0) = a * if P then b else 0\n[PROOFSTEP]\nby_cases h : P\n[GOAL]\ncase pos\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MulZeroClass \u03b1\nP : Prop\ninst\u271d : Decidable P\na b : \u03b1\nh : P\n\u22a2 (if P then a * b else 0) = a * if P then b else 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MulZeroClass \u03b1\nP : Prop\ninst\u271d : Decidable P\na b : \u03b1\nh : \u00acP\n\u22a2 (if P then a * b else 0) = a * if P then b else 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MulZeroClass \u03b1\nP Q : Prop\ninst\u271d\u00b9 : Decidable P\ninst\u271d : Decidable Q\na b : \u03b1\n\u22a2 (if P \u2227 Q then a * b else 0) = (if P then a else 0) * if Q then b else 0\n[PROOFSTEP]\nsimp only [\u2190 ite_and, ite_mul, mul_ite, mul_zero, zero_mul, and_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : CommSemiring \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 (a + b) * (a + b) = a * a + 2 * a * b + b * b\n[PROOFSTEP]\nsimp only [two_mul, add_mul, mul_add, add_assoc, mul_comm b]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : Mul \u03b1\ninst\u271d : HasDistribNeg \u03b1\na b : \u03b1\n\u22a2 -a * -b = a * b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : Mul \u03b1\ninst\u271d : HasDistribNeg \u03b1\na b : \u03b1\n\u22a2 -a * b = a * -b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : MulOneClass \u03b1\ninst\u271d : HasDistribNeg \u03b1\na : \u03b1\n\u22a2 -a = -1 * a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : MulOneClass \u03b1\ninst\u271d : HasDistribNeg \u03b1\na : \u03b1\n\u22a2 a * -1 = -a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : MulOneClass \u03b1\ninst\u271d : HasDistribNeg \u03b1\na : \u03b1\n\u22a2 -1 * a = -a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : MulZeroClass \u03b1\ninst\u271d : HasDistribNeg \u03b1\nsrc\u271d\u00b9 : Zero \u03b1 := inferInstanceAs (Zero \u03b1)\nsrc\u271d : InvolutiveNeg \u03b1 := inferInstanceAs (InvolutiveNeg \u03b1)\n\u22a2 -0 = 0\n[PROOFSTEP]\nrw [\u2190 zero_mul (0 : \u03b1), \u2190 neg_mul, mul_zero, mul_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing \u03b1\na b : \u03b1\n\u22a2 -a * b + a * b = 0\n[PROOFSTEP]\nrw [\u2190 right_distrib, add_left_neg, zero_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing \u03b1\na b : \u03b1\n\u22a2 a * -b + a * b = 0\n[PROOFSTEP]\nrw [\u2190 left_distrib, add_left_neg, mul_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing \u03b1\na b c : \u03b1\n\u22a2 a * (b - c) = a * b - a * c\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg, neg_mul_eq_mul_neg] using mul_add a b (-c)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing \u03b1\na b c : \u03b1\n\u22a2 (a - b) * c = a * c - b * c\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg, neg_mul_eq_neg_mul] using add_mul a (-b) c\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing \u03b1\na b c d e : \u03b1\n\u22a2 a * e + c = b * e + d \u2194 a * e + c = d + b * e\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing \u03b1\na b c d e : \u03b1\nh : a * e + c = d + b * e\n\u22a2 a * e + c - b * e = d\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing \u03b1\na b c d e : \u03b1\nh : a * e + c = d + b * e\n\u22a2 d + b * e - b * e = d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing \u03b1\na b c d e : \u03b1\nh : a * e + c - b * e = d\n\u22a2 a * e + c = d + b * e\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing \u03b1\na b c d e : \u03b1\nh : a * e + c - b * e = d\n\u22a2 a * e + c = a * e + c - b * e + b * e\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing \u03b1\na b c d e : \u03b1\n\u22a2 a * e + c - b * e = d \u2194 (a - b) * e + c = d\n[PROOFSTEP]\nsimp [sub_mul, sub_add_eq_add_sub]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing \u03b1\na b c d e : \u03b1\nh : a * e + c = b * e + d\n\u22a2 (a - b) * e + c = a * e + c - b * e\n[PROOFSTEP]\nsimp [sub_mul, sub_add_eq_add_sub]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing \u03b1\na b c d e : \u03b1\nh : a * e + c = b * e + d\n\u22a2 a * e + c - b * e = d\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing \u03b1\na b c d e : \u03b1\nh : a * e + c = b * e + d\n\u22a2 b * e + d - b * e = d\n[PROOFSTEP]\nsimp [@add_sub_cancel \u03b1]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonAssocRing \u03b1\na b : \u03b1\n\u22a2 (a - 1) * b = a * b - b\n[PROOFSTEP]\nrw [sub_mul, one_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonAssocRing \u03b1\na b : \u03b1\n\u22a2 a * (b - 1) = a * b - a\n[PROOFSTEP]\nrw [mul_sub, mul_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonAssocRing \u03b1\na b : \u03b1\n\u22a2 (1 - a) * b = b - a * b\n[PROOFSTEP]\nrw [sub_mul, one_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonAssocRing \u03b1\na b : \u03b1\n\u22a2 a * (1 - b) = a - a * b\n[PROOFSTEP]\nrw [mul_sub, mul_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : Ring \u03b1\na\u271d b c d e : \u03b1\nsrc\u271d : Ring \u03b1 := inst\u271d\na : \u03b1\n\u22a2 0 * a + 0 * a = 0 * a + 0\n[PROOFSTEP]\nrw [\u2190 add_mul, zero_add, add_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : Ring \u03b1\na\u271d b c d e : \u03b1\nsrc\u271d : Ring \u03b1 := inst\u271d\na : \u03b1\n\u22a2 a * 0 + a * 0 = a * 0 + 0\n[PROOFSTEP]\nrw [\u2190 mul_add, add_zero, add_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : Ring \u03b1\na\u271d b c d e : \u03b1\nsrc\u271d : Ring \u03b1 := inst\u271d\na : \u03b1\n\u22a2 0 * a + 0 * a = 0 * a + 0\n[PROOFSTEP]\nrw [\u2190 add_mul, zero_add, add_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : Ring \u03b1\na\u271d b c d e : \u03b1\nsrc\u271d : Ring \u03b1 := inst\u271d\na : \u03b1\n\u22a2 a * 0 + a * 0 = a * 0 + 0\n[PROOFSTEP]\nrw [\u2190 mul_add, add_zero, add_zero]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Ring.Defs", "llama_tokens": 4672, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.4177545615994975}} {"text": "[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP : \u2115 \u2192 Prop\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 coeff (select P x) n = \u2191(aeval x.coeff) (selectPoly P n)\n[PROOFSTEP]\ndsimp [select, selectPoly]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP : \u2115 \u2192 Prop\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 coeff (mk p fun n => if P n then coeff x n else 0) n = \u2191(aeval x.coeff) (if P n then X n else 0)\n[PROOFSTEP]\nsplit_ifs with hi\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP : \u2115 \u2192 Prop\nx : \ud835\udd4e R\nn : \u2115\nhi : P n\n\u22a2 coeff (mk p fun n => if P n then coeff x n else 0) n = \u2191(aeval x.coeff) (X n)\n[PROOFSTEP]\nrw [aeval_X, mk]\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP : \u2115 \u2192 Prop\nx : \ud835\udd4e R\nn : \u2115\nhi : P n\n\u22a2 coeff { coeff := fun n => if P n then coeff x n else 0 } n = coeff x n\n[PROOFSTEP]\nsimp only [hi]\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP : \u2115 \u2192 Prop\nx : \ud835\udd4e R\nn : \u2115\nhi : P n\n\u22a2 (if True then coeff x n else 0) = coeff x n\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP : \u2115 \u2192 Prop\nx : \ud835\udd4e R\nn : \u2115\nhi : \u00acP n\n\u22a2 coeff (mk p fun n => if P n then coeff x n else 0) n = \u2191(aeval x.coeff) 0\n[PROOFSTEP]\nrw [AlgHom.map_zero, mk]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP : \u2115 \u2192 Prop\nx : \ud835\udd4e R\nn : \u2115\nhi : \u00acP n\n\u22a2 coeff { coeff := fun n => if P n then coeff x n else 0 } n = 0\n[PROOFSTEP]\nsimp only [hi]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP : \u2115 \u2192 Prop\nx : \ud835\udd4e R\nn : \u2115\nhi : \u00acP n\n\u22a2 (if False then coeff x n else 0) = 0\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d P : \u2115 \u2192 Prop\n\u22a2 IsPoly p fun x x_1 x_2 => select P x_2\n[PROOFSTEP]\nuse selectPoly P\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d P : \u2115 \u2192 Prop\n\u22a2 \u2200 \u2983R : Type ?u.31994\u2984 [inst : CommRing R] (x : \ud835\udd4e R), (select P x).coeff = fun n => \u2191(aeval x.coeff) (selectPoly P n)\n[PROOFSTEP]\nrintro R _Rcr x\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR\u271d : Type u_1\ninst\u271d : CommRing R\u271d\nP\u271d P : \u2115 \u2192 Prop\nR : Type ?u.31994\n_Rcr : CommRing R\nx : \ud835\udd4e R\n\u22a2 (select P x).coeff = fun n => \u2191(aeval x.coeff) (selectPoly P n)\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h.h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR\u271d : Type u_1\ninst\u271d : CommRing R\u271d\nP\u271d P : \u2115 \u2192 Prop\nR : Type ?u.31994\n_Rcr : CommRing R\nx : \ud835\udd4e R\ni : \u2115\n\u22a2 coeff (select P x) i = \u2191(aeval x.coeff) (selectPoly P i)\n[PROOFSTEP]\napply coeff_select\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP : \u2115 \u2192 Prop\n\u22a2 \u2200 (x : \ud835\udd4e R), select P x + select (fun i => \u00acP i) x = x\n[PROOFSTEP]\nhave : IsPoly p fun {R} [CommRing R] x \u21a6 select P x + select (fun i \u21a6 \u00acP i) x := IsPoly\u2082.diag (hf := IsPoly\u2082.comp)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP : \u2115 \u2192 Prop\nthis : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => \u00acP i) x\n\u22a2 \u2200 (x : \ud835\udd4e R), select P x + select (fun i => \u00acP i) x = x\n[PROOFSTEP]\nghost_calc x\n[GOAL]\ncase refine_3\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nP : \u2115 \u2192 Prop\nthis : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => \u00acP i) x\nR : Type u_1\nR._inst : CommRing R\nx : \ud835\udd4e R\n\u22a2 \u2200 (n : \u2115), \u2191(ghostComponent n) (select P x + select (fun i => \u00acP i) x) = \u2191(ghostComponent n) x\n[PROOFSTEP]\nintro n\n[GOAL]\ncase refine_3\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nP : \u2115 \u2192 Prop\nthis : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => \u00acP i) x\nR : Type u_1\nR._inst : CommRing R\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 \u2191(ghostComponent n) (select P x + select (fun i => \u00acP i) x) = \u2191(ghostComponent n) x\n[PROOFSTEP]\nsimp only [RingHom.map_add]\n[GOAL]\ncase refine_3\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nP : \u2115 \u2192 Prop\nthis : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => \u00acP i) x\nR : Type u_1\nR._inst : CommRing R\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 \u2191(ghostComponent n) (select P x) + \u2191(ghostComponent n) (select (fun i => \u00acP i) x) = \u2191(ghostComponent n) x\n[PROOFSTEP]\nsuffices\n (bind\u2081 (selectPoly P)) (wittPolynomial p \u2124 n) + (bind\u2081 (selectPoly fun i => \u00acP i)) (wittPolynomial p \u2124 n) =\n wittPolynomial p \u2124 n\n by\n apply_fun aeval x.coeff at this \n simpa only [AlgHom.map_add, aeval_bind\u2081, \u2190 coeff_select]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nP : \u2115 \u2192 Prop\nthis\u271d : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => \u00acP i) x\nR : Type u_1\nR._inst : CommRing R\nx : \ud835\udd4e R\nn : \u2115\nthis :\n \u2191(bind\u2081 (selectPoly P)) (wittPolynomial p \u2124 n) + \u2191(bind\u2081 (selectPoly fun i => \u00acP i)) (wittPolynomial p \u2124 n) =\n wittPolynomial p \u2124 n\n\u22a2 \u2191(ghostComponent n) (select P x) + \u2191(ghostComponent n) (select (fun i => \u00acP i) x) = \u2191(ghostComponent n) x\n[PROOFSTEP]\napply_fun aeval x.coeff at this \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nP : \u2115 \u2192 Prop\nthis\u271d : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => \u00acP i) x\nR : Type u_1\nR._inst : CommRing R\nx : \ud835\udd4e R\nn : \u2115\nthis :\n \u2191(aeval x.coeff)\n (\u2191(bind\u2081 (selectPoly P)) (wittPolynomial p \u2124 n) + \u2191(bind\u2081 (selectPoly fun i => \u00acP i)) (wittPolynomial p \u2124 n)) =\n \u2191(aeval x.coeff) (wittPolynomial p \u2124 n)\n\u22a2 \u2191(ghostComponent n) (select P x) + \u2191(ghostComponent n) (select (fun i => \u00acP i) x) = \u2191(ghostComponent n) x\n[PROOFSTEP]\nsimpa only [AlgHom.map_add, aeval_bind\u2081, \u2190 coeff_select]\n[GOAL]\ncase refine_3\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nP : \u2115 \u2192 Prop\nthis : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => \u00acP i) x\nR : Type u_1\nR._inst : CommRing R\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 \u2191(bind\u2081 (selectPoly P)) (wittPolynomial p \u2124 n) + \u2191(bind\u2081 (selectPoly fun i => \u00acP i)) (wittPolynomial p \u2124 n) =\n wittPolynomial p \u2124 n\n[PROOFSTEP]\nsimp only [wittPolynomial_eq_sum_C_mul_X_pow, selectPoly, AlgHom.map_sum, AlgHom.map_pow, AlgHom.map_mul, bind\u2081_X_right,\n bind\u2081_C_right, \u2190 Finset.sum_add_distrib, \u2190 mul_add]\n[GOAL]\ncase refine_3\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nP : \u2115 \u2192 Prop\nthis : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => \u00acP i) x\nR : Type u_1\nR._inst : CommRing R\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 (Finset.sum (Finset.range (n + 1)) fun x =>\n \u2191C (\u2191p ^ x) * ((if P x then X x else 0) ^ p ^ (n - x) + (if \u00acP x then X x else 0) ^ p ^ (n - x))) =\n Finset.sum (Finset.range (n + 1)) fun x => \u2191C (\u2191p ^ x) * X x ^ p ^ (n - x)\n[PROOFSTEP]\napply Finset.sum_congr rfl\n[GOAL]\ncase refine_3\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nP : \u2115 \u2192 Prop\nthis : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => \u00acP i) x\nR : Type u_1\nR._inst : CommRing R\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 \u2200 (x : \u2115),\n x \u2208 Finset.range (n + 1) \u2192\n \u2191C (\u2191p ^ x) * ((if P x then X x else 0) ^ p ^ (n - x) + (if \u00acP x then X x else 0) ^ p ^ (n - x)) =\n \u2191C (\u2191p ^ x) * X x ^ p ^ (n - x)\n[PROOFSTEP]\nrefine' fun m _ => mul_eq_mul_left_iff.mpr (Or.inl _)\n[GOAL]\ncase refine_3\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nP : \u2115 \u2192 Prop\nthis : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => \u00acP i) x\nR : Type u_1\nR._inst : CommRing R\nx : \ud835\udd4e R\nn m : \u2115\nx\u271d : m \u2208 Finset.range (n + 1)\n\u22a2 (if P m then X m else 0) ^ p ^ (n - m) + (if \u00acP m then X m else 0) ^ p ^ (n - m) = X m ^ p ^ (n - m)\n[PROOFSTEP]\nrw [ite_pow, zero_pow (pow_pos hp.out.pos _)]\n[GOAL]\ncase refine_3\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nP : \u2115 \u2192 Prop\nthis : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => \u00acP i) x\nR : Type u_1\nR._inst : CommRing R\nx : \ud835\udd4e R\nn m : \u2115\nx\u271d : m \u2208 Finset.range (n + 1)\n\u22a2 (if P m then X m ^ p ^ (n - m) else 0) + (if \u00acP m then X m else 0) ^ p ^ (n - m) = X m ^ p ^ (n - m)\n[PROOFSTEP]\nby_cases Pm : P m\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nP : \u2115 \u2192 Prop\nthis : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => \u00acP i) x\nR : Type u_1\nR._inst : CommRing R\nx : \ud835\udd4e R\nn m : \u2115\nx\u271d : m \u2208 Finset.range (n + 1)\nPm : P m\n\u22a2 (if P m then X m ^ p ^ (n - m) else 0) + (if \u00acP m then X m else 0) ^ p ^ (n - m) = X m ^ p ^ (n - m)\n[PROOFSTEP]\nrw [if_pos Pm, if_neg _, zero_pow, add_zero]\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nP : \u2115 \u2192 Prop\nthis : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => \u00acP i) x\nR : Type u_1\nR._inst : CommRing R\nx : \ud835\udd4e R\nn m : \u2115\nx\u271d : m \u2208 Finset.range (n + 1)\nPm : P m\n\u22a2 0 < p ^ (n - m)\n[PROOFSTEP]\nexact Fin.size_positive'\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nP : \u2115 \u2192 Prop\nthis : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => \u00acP i) x\nR : Type u_1\nR._inst : CommRing R\nx : \ud835\udd4e R\nn m : \u2115\nx\u271d : m \u2208 Finset.range (n + 1)\nPm : P m\n\u22a2 \u00ac\u00acP m\n[PROOFSTEP]\nexact not_not_intro Pm\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nP : \u2115 \u2192 Prop\nthis : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => \u00acP i) x\nR : Type u_1\nR._inst : CommRing R\nx : \ud835\udd4e R\nn m : \u2115\nx\u271d : m \u2208 Finset.range (n + 1)\nPm : \u00acP m\n\u22a2 (if P m then X m ^ p ^ (n - m) else 0) + (if \u00acP m then X m else 0) ^ p ^ (n - m) = X m ^ p ^ (n - m)\n[PROOFSTEP]\nrwa [if_neg Pm, if_pos, zero_add]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\n\u22a2 coeff (x + y) n = coeff x n + coeff y n\n[PROOFSTEP]\nlet P : \u2115 \u2192 Prop := fun n => y.coeff n = 0\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\n\u22a2 coeff (x + y) n = coeff x n + coeff y n\n[PROOFSTEP]\nhaveI : DecidablePred P := Classical.decPred P\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\n\u22a2 coeff (x + y) n = coeff x n + coeff y n\n[PROOFSTEP]\nset z := mk p fun n => if P n then x.coeff n else y.coeff n\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\n\u22a2 coeff (x + y) n = coeff x n + coeff y n\n[PROOFSTEP]\nhave hx : select P z = x := by\n ext1 n; rw [select, coeff_mk, coeff_mk]\n split_ifs with hn\n \u00b7 rfl\n \u00b7 rw [(h n).resolve_right hn]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\n\u22a2 select P z = x\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nn : \u2115\n\u22a2 coeff (select P z) n = coeff x n\n[PROOFSTEP]\nrw [select, coeff_mk, coeff_mk]\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nn : \u2115\n\u22a2 (if P n then if P n then coeff x n else coeff y n else 0) = coeff x n\n[PROOFSTEP]\nsplit_ifs with hn\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nn : \u2115\nhn : P n\n\u22a2 coeff x n = coeff x n\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nn : \u2115\nhn : \u00acP n\n\u22a2 0 = coeff x n\n[PROOFSTEP]\nrw [(h n).resolve_right hn]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\n\u22a2 coeff (x + y) n = coeff x n + coeff y n\n[PROOFSTEP]\nhave hy : select (fun i => \u00acP i) z = y := by\n ext1 n; rw [select, coeff_mk, coeff_mk]\n split_ifs with hn\n \u00b7 exact hn.symm\n \u00b7 rfl\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\n\u22a2 select (fun i => \u00acP i) z = y\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nn : \u2115\n\u22a2 coeff (select (fun i => \u00acP i) z) n = coeff y n\n[PROOFSTEP]\nrw [select, coeff_mk, coeff_mk]\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nn : \u2115\n\u22a2 (if \u00acP n then if P n then coeff x n else coeff y n else 0) = coeff y n\n[PROOFSTEP]\nsplit_ifs with hn\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nn : \u2115\nhn : P n\n\u22a2 0 = coeff y n\n[PROOFSTEP]\nexact hn.symm\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nn : \u2115\nhn : \u00acP n\n\u22a2 coeff y n = coeff y n\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nhy : select (fun i => \u00acP i) z = y\n\u22a2 coeff (x + y) n = coeff x n + coeff y n\n[PROOFSTEP]\ncalc\n (x + y).coeff n = z.coeff n := by rw [\u2190 hx, \u2190 hy, select_add_select_not P z]\n _ = x.coeff n + y.coeff n := by\n simp [mk]\n split_ifs with y0\n \u00b7 rw [y0, add_zero]\n \u00b7 rw [h n |>.resolve_right y0, zero_add]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nhy : select (fun i => \u00acP i) z = y\n\u22a2 coeff (x + y) n = coeff z n\n[PROOFSTEP]\nrw [\u2190 hx, \u2190 hy, select_add_select_not P z]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nhy : select (fun i => \u00acP i) z = y\n\u22a2 coeff z n = coeff x n + coeff y n\n[PROOFSTEP]\nsimp [mk]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nhy : select (fun i => \u00acP i) z = y\n\u22a2 (if coeff y n = 0 then coeff x n else coeff y n) = coeff x n + coeff y n\n[PROOFSTEP]\nsplit_ifs with y0\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nhy : select (fun i => \u00acP i) z = y\ny0 : coeff y n = 0\n\u22a2 coeff x n = coeff x n + coeff y n\n[PROOFSTEP]\nrw [y0, add_zero]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nhy : select (fun i => \u00acP i) z = y\ny0 : \u00accoeff y n = 0\n\u22a2 coeff y n = coeff x n + coeff y n\n[PROOFSTEP]\nrw [h n |>.resolve_right y0, zero_add]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 init n x + tail n x = x\n[PROOFSTEP]\nsimp only [init, tail, \u2190 not_lt, select_add_select_not]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 init n (init n x) = init n x\n[PROOFSTEP]\nrw [ext_iff]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 \u2200 (n_1 : \u2115), coeff (init n (init n x)) n_1 = coeff (init n x) n_1\n[PROOFSTEP]\nintro i\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nn i : \u2115\n\u22a2 coeff (init n (init n x)) i = coeff (init n x) i\n[PROOFSTEP]\nsimp only [WittVector.init, WittVector.select, WittVector.coeff_mk]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nn i : \u2115\n\u22a2 (if i < n then if i < n then coeff x i else 0 else 0) = if i < n then coeff x i else 0\n[PROOFSTEP]\nby_cases hi : i < n\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nn i : \u2115\nhi : i < n\n\u22a2 (if i < n then if i < n then coeff x i else 0 else 0) = if i < n then coeff x i else 0\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nn i : \u2115\nhi : \u00aci < n\n\u22a2 (if i < n then if i < n then coeff x i else 0 else 0) = if i < n then coeff x i else 0\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx y : \ud835\udd4e R\nn : \u2115\n\u22a2 init n (x + y) = init n (init n x + init n y)\n[PROOFSTEP]\ninit_ring using wittAdd_vars\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx y : \ud835\udd4e R\nn : \u2115\n\u22a2 init n (x * y) = init n (init n x * init n y)\n[PROOFSTEP]\ninit_ring using wittMul_vars\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 init n (-x) = init n (-init n x)\n[PROOFSTEP]\ninit_ring using wittNeg_vars\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx y : \ud835\udd4e R\nn : \u2115\n\u22a2 init n (x - y) = init n (init n x - init n y)\n[PROOFSTEP]\ninit_ring using wittSub_vars\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nm : \u2115\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 init n (m \u2022 x) = init n (m \u2022 init n x)\n[PROOFSTEP]\ninit_ring using fun p [Fact (Nat.Prime p)] n => wittNSMul_vars p m n\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nm : \u2124\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 init n (m \u2022 x) = init n (m \u2022 init n x)\n[PROOFSTEP]\ninit_ring using fun p [Fact (Nat.Prime p)] n => wittZSMul_vars p m n\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nm : \u2115\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 init n (x ^ m) = init n (init n x ^ m)\n[PROOFSTEP]\ninit_ring using fun p [Fact (Nat.Prime p)] n => wittPow_vars p m n\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.WittVector.InitTail", "llama_tokens": 10439, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.727975460709318, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4176238029642285}} {"text": "[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\ndist_comm : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ns : Set \u03b1\nhs : s \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\nt : Set \u03b1\nht : t \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\n\u22a2 s \u222a t \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with rfl | \u27e8x, hx\u27e9\n[GOAL]\ncase inl\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\ndist_comm : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\nt : Set \u03b1\nht : t \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\nhs : \u2205 \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\n\u22a2 \u2205 \u222a t \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\n[PROOFSTEP]\nrwa [empty_union]\n[GOAL]\ncase inr.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\ndist_comm : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ns : Set \u03b1\nhs : s \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\nt : Set \u03b1\nht : t \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\nx : \u03b1\nhx : x \u2208 s\n\u22a2 s \u222a t \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\n[PROOFSTEP]\nrcases t.eq_empty_or_nonempty with rfl | \u27e8y, hy\u27e9\n[GOAL]\ncase inr.intro.inl\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\ndist_comm : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ns : Set \u03b1\nhs : s \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\nx : \u03b1\nhx : x \u2208 s\nht : \u2205 \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\n\u22a2 s \u222a \u2205 \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\n[PROOFSTEP]\nrwa [union_empty]\n[GOAL]\ncase inr.intro.inr.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\ndist_comm : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ns : Set \u03b1\nhs : s \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\nt : Set \u03b1\nht : t \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 t\n\u22a2 s \u222a t \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\n[PROOFSTEP]\nrsuffices \u27e8C, hC\u27e9 : \u2203 C, \u2200 z \u2208 s \u222a t, dist x z \u2264 C\n[GOAL]\ncase inr.intro.inr.intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\ndist_comm : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ns : Set \u03b1\nhs : s \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\nt : Set \u03b1\nht : t \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 t\nC : \u211d\nhC : \u2200 (z : \u03b1), z \u2208 s \u222a t \u2192 dist x z \u2264 C\n\u22a2 s \u222a t \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\n[PROOFSTEP]\nrefine \u27e8C + C, fun a ha b hb => (dist_triangle a x b).trans ?_\u27e9\n[GOAL]\ncase inr.intro.inr.intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\ndist_comm : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ns : Set \u03b1\nhs : s \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\nt : Set \u03b1\nht : t \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 t\nC : \u211d\nhC : \u2200 (z : \u03b1), z \u2208 s \u222a t \u2192 dist x z \u2264 C\na : \u03b1\nha : a \u2208 s \u222a t\nb : \u03b1\nhb : b \u2208 s \u222a t\n\u22a2 dist a x + dist x b \u2264 C + C\n[PROOFSTEP]\nsimpa only [dist_comm] using add_le_add (hC _ ha) (hC _ hb)\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\ndist_comm : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ns : Set \u03b1\nhs : s \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\nt : Set \u03b1\nht : t \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 t\n\u22a2 \u2203 C, \u2200 (z : \u03b1), z \u2208 s \u222a t \u2192 dist x z \u2264 C\n[PROOFSTEP]\nrcases hs with \u27e8Cs, hs\u27e9\n[GOAL]\ncase intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\ndist_comm : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ns t : Set \u03b1\nht : t \u2208 {s | \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C}\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 t\nCs : \u211d\nhs : \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 Cs\n\u22a2 \u2203 C, \u2200 (z : \u03b1), z \u2208 s \u222a t \u2192 dist x z \u2264 C\n[PROOFSTEP]\nrcases ht with \u27e8Ct, ht\u27e9\n[GOAL]\ncase intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\ndist_comm : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ns t : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 t\nCs : \u211d\nhs : \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 Cs\nCt : \u211d\nht : \u2200 \u2983x : \u03b1\u2984, x \u2208 t \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 t \u2192 dist x y \u2264 Ct\n\u22a2 \u2203 C, \u2200 (z : \u03b1), z \u2208 s \u222a t \u2192 dist x z \u2264 C\n[PROOFSTEP]\nrefine\n \u27e8max Cs (dist x y + Ct), fun z hz =>\n hz.elim (fun hz => (hs hx hz).trans (le_max_left _ _))\n (fun hz => (dist_triangle x y z).trans <| (add_le_add le_rfl (ht hy hz)).trans (le_max_right _ _))\u27e9\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Sort u_3\nx y : \u03b1\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\ndist_self : \u2200 (x : \u03b1), dist x x = 0\ndist_comm : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\n\u22a2 dist x y + dist y x = 2 * dist x y\n[PROOFSTEP]\nrw [two_mul, dist_comm]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\nm m' : PseudoMetricSpace \u03b1\nh : toDist = toDist\n\u22a2 m = m'\n[PROOFSTEP]\ncases' m with d _ _ _ ed hed U hU B hB\n[GOAL]\ncase mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\nm' : PseudoMetricSpace \u03b1\nd : Dist \u03b1\ndist_self\u271d : \u2200 (x : \u03b1), dist x x = 0\ndist_comm\u271d : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle\u271d : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ned : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhed : \u2200 (x y : \u03b1), ed x y = ENNReal.ofReal (dist x y)\nU : UniformSpace \u03b1\nhU : \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\nB : Bornology \u03b1\nhB : (cobounded \u03b1).sets = {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\nh : toDist = toDist\n\u22a2 mk dist_self\u271d dist_comm\u271d dist_triangle\u271d ed hed U B = m'\n[PROOFSTEP]\ncases' m' with d' _ _ _ ed' hed' U' hU' B' hB'\n[GOAL]\ncase mk.mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\nd : Dist \u03b1\ndist_self\u271d\u00b9 : \u2200 (x : \u03b1), dist x x = 0\ndist_comm\u271d\u00b9 : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle\u271d\u00b9 : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ned : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhed : \u2200 (x y : \u03b1), ed x y = ENNReal.ofReal (dist x y)\nU : UniformSpace \u03b1\nhU : \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\nB : Bornology \u03b1\nhB : (cobounded \u03b1).sets = {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\nd' : Dist \u03b1\ndist_self\u271d : \u2200 (x : \u03b1), dist x x = 0\ndist_comm\u271d : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle\u271d : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ned' : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhed' : \u2200 (x y : \u03b1), ed' x y = ENNReal.ofReal (dist x y)\nU' : UniformSpace \u03b1\nhU' : \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\nB' : Bornology \u03b1\nhB' : (cobounded \u03b1).sets = {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\nh : toDist = toDist\n\u22a2 mk dist_self\u271d\u00b9 dist_comm\u271d\u00b9 dist_triangle\u271d\u00b9 ed hed U B = mk dist_self\u271d dist_comm\u271d dist_triangle\u271d ed' hed' U' B'\n[PROOFSTEP]\nobtain rfl : d = d' := h\n[GOAL]\ncase mk.mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\nd : Dist \u03b1\ndist_self\u271d\u00b9 : \u2200 (x : \u03b1), dist x x = 0\ndist_comm\u271d\u00b9 : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle\u271d\u00b9 : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ned : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhed : \u2200 (x y : \u03b1), ed x y = ENNReal.ofReal (dist x y)\nU : UniformSpace \u03b1\nhU : \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\nB : Bornology \u03b1\nhB : (cobounded \u03b1).sets = {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\ned' : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nU' : UniformSpace \u03b1\nB' : Bornology \u03b1\ndist_self\u271d : \u2200 (x : \u03b1), dist x x = 0\ndist_comm\u271d : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle\u271d : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\nhed' : \u2200 (x y : \u03b1), ed' x y = ENNReal.ofReal (dist x y)\nhU' : \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\nhB' : (cobounded \u03b1).sets = {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\n\u22a2 mk dist_self\u271d\u00b9 dist_comm\u271d\u00b9 dist_triangle\u271d\u00b9 ed hed U B = mk dist_self\u271d dist_comm\u271d dist_triangle\u271d ed' hed' U' B'\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_edist\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\nd : Dist \u03b1\ndist_self\u271d\u00b9 : \u2200 (x : \u03b1), dist x x = 0\ndist_comm\u271d\u00b9 : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle\u271d\u00b9 : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ned : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhed : \u2200 (x y : \u03b1), ed x y = ENNReal.ofReal (dist x y)\nU : UniformSpace \u03b1\nhU : \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\nB : Bornology \u03b1\nhB : (cobounded \u03b1).sets = {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\ned' : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nU' : UniformSpace \u03b1\nB' : Bornology \u03b1\ndist_self\u271d : \u2200 (x : \u03b1), dist x x = 0\ndist_comm\u271d : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle\u271d : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\nhed' : \u2200 (x y : \u03b1), ed' x y = ENNReal.ofReal (dist x y)\nhU' : \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\nhB' : (cobounded \u03b1).sets = {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\n\u22a2 ed = ed'\n[PROOFSTEP]\next x y : 2\n[GOAL]\ncase mk.mk.e_edist.h.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\nd : Dist \u03b1\ndist_self\u271d\u00b9 : \u2200 (x : \u03b1), dist x x = 0\ndist_comm\u271d\u00b9 : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle\u271d\u00b9 : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ned : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhed : \u2200 (x y : \u03b1), ed x y = ENNReal.ofReal (dist x y)\nU : UniformSpace \u03b1\nhU : \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\nB : Bornology \u03b1\nhB : (cobounded \u03b1).sets = {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\ned' : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nU' : UniformSpace \u03b1\nB' : Bornology \u03b1\ndist_self\u271d : \u2200 (x : \u03b1), dist x x = 0\ndist_comm\u271d : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle\u271d : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\nhed' : \u2200 (x y : \u03b1), ed' x y = ENNReal.ofReal (dist x y)\nhU' : \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\nhB' : (cobounded \u03b1).sets = {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\nx y : \u03b1\n\u22a2 ed x y = ed' x y\n[PROOFSTEP]\nrw [hed, hed']\n[GOAL]\ncase mk.mk.e_toUniformSpace\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\nd : Dist \u03b1\ndist_self\u271d\u00b9 : \u2200 (x : \u03b1), dist x x = 0\ndist_comm\u271d\u00b9 : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle\u271d\u00b9 : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ned : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhed : \u2200 (x y : \u03b1), ed x y = ENNReal.ofReal (dist x y)\nU : UniformSpace \u03b1\nhU : \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\nB : Bornology \u03b1\nhB : (cobounded \u03b1).sets = {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\ned' : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nU' : UniformSpace \u03b1\nB' : Bornology \u03b1\ndist_self\u271d : \u2200 (x : \u03b1), dist x x = 0\ndist_comm\u271d : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle\u271d : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\nhed' : \u2200 (x y : \u03b1), ed' x y = ENNReal.ofReal (dist x y)\nhU' : \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\nhB' : (cobounded \u03b1).sets = {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\n\u22a2 U = U'\n[PROOFSTEP]\nexact uniformSpace_eq (hU.trans hU'.symm)\n[GOAL]\ncase mk.mk.e_toBornology\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\nd : Dist \u03b1\ndist_self\u271d\u00b9 : \u2200 (x : \u03b1), dist x x = 0\ndist_comm\u271d\u00b9 : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle\u271d\u00b9 : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ned : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhed : \u2200 (x y : \u03b1), ed x y = ENNReal.ofReal (dist x y)\nU : UniformSpace \u03b1\nhU : \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\nB : Bornology \u03b1\nhB : (cobounded \u03b1).sets = {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\ned' : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nU' : UniformSpace \u03b1\nB' : Bornology \u03b1\ndist_self\u271d : \u2200 (x : \u03b1), dist x x = 0\ndist_comm\u271d : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle\u271d : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\nhed' : \u2200 (x y : \u03b1), ed' x y = ENNReal.ofReal (dist x y)\nhU' : \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\nhB' : (cobounded \u03b1).sets = {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\n\u22a2 B = B'\n[PROOFSTEP]\next : 2\n[GOAL]\ncase mk.mk.e_toBornology.h_cobounded.a\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\nd : Dist \u03b1\ndist_self\u271d\u00b9 : \u2200 (x : \u03b1), dist x x = 0\ndist_comm\u271d\u00b9 : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle\u271d\u00b9 : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\ned : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhed : \u2200 (x y : \u03b1), ed x y = ENNReal.ofReal (dist x y)\nU : UniformSpace \u03b1\nhU : \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\nB : Bornology \u03b1\nhB : (cobounded \u03b1).sets = {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\ned' : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nU' : UniformSpace \u03b1\nB' : Bornology \u03b1\ndist_self\u271d : \u2200 (x : \u03b1), dist x x = 0\ndist_comm\u271d : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle\u271d : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\nhed' : \u2200 (x y : \u03b1), ed' x y = ENNReal.ofReal (dist x y)\nhU' : \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\nhB' : (cobounded \u03b1).sets = {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\ns\u271d : Set \u03b1\n\u22a2 s\u271d \u2208 cobounded \u03b1 \u2194 s\u271d \u2208 cobounded \u03b1\n[PROOFSTEP]\nrw [\u2190 Filter.mem_sets, \u2190 Filter.mem_sets, hB, hB']\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\ndist_self : \u2200 (x : \u03b1), dist x x = 0\ndist_comm : \u2200 (x y : \u03b1), dist x y = dist y x\ndist_triangle : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\nH : \u2200 (s : Set \u03b1), IsOpen s \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : \u03b1), dist x y < \u03b5 \u2192 y \u2208 s\nx y : \u03b1\n\u22a2 (fun x y => \u2191{ val := dist x y, property := (_ : 0 \u2264 dist x y) }) x y = ENNReal.ofReal (Dist.dist x y)\n[PROOFSTEP]\nexact ENNReal.coe_nnreal_eq _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u22a2 dist x y \u2264 dist z x + dist z y\n[PROOFSTEP]\nrw [dist_comm z]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u22a2 dist x y \u2264 dist x z + dist z y\n[PROOFSTEP]\napply dist_triangle\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u22a2 dist x y \u2264 dist x z + dist y z\n[PROOFSTEP]\nrw [dist_comm y]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u22a2 dist x y \u2264 dist x z + dist z y\n[PROOFSTEP]\napply dist_triangle\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u2081 y\u2081 x\u2082 y\u2082 : \u03b1\n\u22a2 dist x\u2082 y\u2082 \u2264 dist x\u2081 y\u2081 + (dist x\u2081 x\u2082 + dist y\u2081 y\u2082)\n[PROOFSTEP]\nrw [add_left_comm, dist_comm x\u2081, \u2190 add_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u2081 y\u2081 x\u2082 y\u2082 : \u03b1\n\u22a2 dist x\u2082 y\u2082 \u2264 dist x\u2082 x\u2081 + dist x\u2081 y\u2081 + dist y\u2081 y\u2082\n[PROOFSTEP]\napply dist_triangle4\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u2081 y\u2081 x\u2082 y\u2082 : \u03b1\n\u22a2 dist x\u2081 y\u2081 \u2264 dist x\u2081 x\u2082 + dist y\u2081 y\u2082 + dist x\u2082 y\u2082\n[PROOFSTEP]\nrw [add_right_comm, dist_comm y\u2081]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u2081 y\u2081 x\u2082 y\u2082 : \u03b1\n\u22a2 dist x\u2081 y\u2081 \u2264 dist x\u2081 x\u2082 + dist x\u2082 y\u2082 + dist y\u2082 y\u2081\n[PROOFSTEP]\napply dist_triangle4\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nm n : \u2115\nh : m \u2264 n\n\u22a2 dist (f m) (f n) \u2264 \u2211 i in Finset.Ico m n, dist (f i) (f (i + 1))\n[PROOFSTEP]\ninduction n, h using Nat.le_induction with\n| base => rw [Finset.Ico_self, Finset.sum_empty, dist_self]\n| succ n hle ihn =>\n calc\n dist (f m) (f (n + 1)) \u2264 dist (f m) (f n) + dist (f n) (f (n + 1)) := dist_triangle _ _ _\n _ \u2264 (\u2211 i in Finset.Ico m n, _) + _ := (add_le_add ihn le_rfl)\n _ = \u2211 i in Finset.Ico m (n + 1), _ := by {rw [Nat.Ico_succ_right_eq_insert_Ico hle, Finset.sum_insert, add_comm];\n simp\n }\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nm n : \u2115\nh : m \u2264 n\n\u22a2 dist (f m) (f n) \u2264 \u2211 i in Finset.Ico m n, dist (f i) (f (i + 1))\n[PROOFSTEP]\ninduction n, h using Nat.le_induction with\n| base => rw [Finset.Ico_self, Finset.sum_empty, dist_self]\n| succ n hle ihn =>\n calc\n dist (f m) (f (n + 1)) \u2264 dist (f m) (f n) + dist (f n) (f (n + 1)) := dist_triangle _ _ _\n _ \u2264 (\u2211 i in Finset.Ico m n, _) + _ := (add_le_add ihn le_rfl)\n _ = \u2211 i in Finset.Ico m (n + 1), _ := by {rw [Nat.Ico_succ_right_eq_insert_Ico hle, Finset.sum_insert, add_comm];\n simp\n }\n[GOAL]\ncase base\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nm n : \u2115\n\u22a2 dist (f m) (f m) \u2264 \u2211 i in Finset.Ico m m, dist (f i) (f (i + 1))\n[PROOFSTEP]\n\n| base => rw [Finset.Ico_self, Finset.sum_empty, dist_self]\n[GOAL]\ncase base\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nm n : \u2115\n\u22a2 dist (f m) (f m) \u2264 \u2211 i in Finset.Ico m m, dist (f i) (f (i + 1))\n[PROOFSTEP]\nrw [Finset.Ico_self, Finset.sum_empty, dist_self]\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nm n\u271d n : \u2115\nhle : m \u2264 n\nihn : dist (f m) (f n) \u2264 \u2211 i in Finset.Ico m n, dist (f i) (f (i + 1))\n\u22a2 dist (f m) (f (n + 1)) \u2264 \u2211 i in Finset.Ico m (n + 1), dist (f i) (f (i + 1))\n[PROOFSTEP]\n\n| succ n hle ihn =>\n calc\n dist (f m) (f (n + 1)) \u2264 dist (f m) (f n) + dist (f n) (f (n + 1)) := dist_triangle _ _ _\n _ \u2264 (\u2211 i in Finset.Ico m n, _) + _ := (add_le_add ihn le_rfl)\n _ = \u2211 i in Finset.Ico m (n + 1), _ := by {rw [Nat.Ico_succ_right_eq_insert_Ico hle, Finset.sum_insert, add_comm];\n simp\n }\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nm n\u271d n : \u2115\nhle : m \u2264 n\nihn : dist (f m) (f n) \u2264 \u2211 i in Finset.Ico m n, dist (f i) (f (i + 1))\n\u22a2 dist (f m) (f (n + 1)) \u2264 \u2211 i in Finset.Ico m (n + 1), dist (f i) (f (i + 1))\n[PROOFSTEP]\ncalc\n dist (f m) (f (n + 1)) \u2264 dist (f m) (f n) + dist (f n) (f (n + 1)) := dist_triangle _ _ _\n _ \u2264 (\u2211 i in Finset.Ico m n, _) + _ := (add_le_add ihn le_rfl)\n _ = \u2211 i in Finset.Ico m (n + 1), _ := by {rw [Nat.Ico_succ_right_eq_insert_Ico hle, Finset.sum_insert, add_comm]; simp\n }\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nm n\u271d n : \u2115\nhle : m \u2264 n\nihn : dist (f m) (f n) \u2264 \u2211 i in Finset.Ico m n, dist (f i) (f (i + 1))\n\u22a2 \u2211 i in Finset.Ico m n, dist (f i) (f (i + 1)) + dist (f n) (f (n + 1)) =\n \u2211 i in Finset.Ico m (n + 1), dist (f i) (f (i + 1))\n[PROOFSTEP]\n{rw [Nat.Ico_succ_right_eq_insert_Ico hle, Finset.sum_insert, add_comm]; simp\n}\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nm n\u271d n : \u2115\nhle : m \u2264 n\nihn : dist (f m) (f n) \u2264 \u2211 i in Finset.Ico m n, dist (f i) (f (i + 1))\n\u22a2 \u2211 i in Finset.Ico m n, dist (f i) (f (i + 1)) + dist (f n) (f (n + 1)) =\n \u2211 i in Finset.Ico m (n + 1), dist (f i) (f (i + 1))\n[PROOFSTEP]\nrw [Nat.Ico_succ_right_eq_insert_Ico hle, Finset.sum_insert, add_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nm n\u271d n : \u2115\nhle : m \u2264 n\nihn : dist (f m) (f n) \u2264 \u2211 i in Finset.Ico m n, dist (f i) (f (i + 1))\n\u22a2 \u00acn \u2208 Finset.Ico m n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u22a2 Function.swap dist = dist\n[PROOFSTEP]\nfunext x y\n[GOAL]\ncase h.h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u03b1\n\u22a2 Function.swap dist x y = dist x y\n[PROOFSTEP]\nexact dist_comm _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u03b1\n\u22a2 0 \u2264 dist x y\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u03b1\n\u22a2 edist x y = \u2191(nndist x y)\n[PROOFSTEP]\nrw [edist_dist, dist_nndist, ENNReal.ofReal_coe_nnreal]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u03b1\n\u22a2 nndist x y = ENNReal.toNNReal (edist x y)\n[PROOFSTEP]\nsimp [edist_nndist]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u03b1\nc : \u211d\u22650\n\u22a2 edist x y < \u2191c \u2194 nndist x y < c\n[PROOFSTEP]\nrw [edist_nndist, ENNReal.coe_lt_coe]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u03b1\nc : \u211d\u22650\n\u22a2 edist x y \u2264 \u2191c \u2194 nndist x y \u2264 c\n[PROOFSTEP]\nrw [edist_nndist, ENNReal.coe_le_coe]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u03b1\nr : \u211d\n\u22a2 edist x y < ENNReal.ofReal r \u2194 dist x y < r\n[PROOFSTEP]\nrw [edist_dist, ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u03b1\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 edist x y \u2264 ENNReal.ofReal r \u2194 dist x y \u2264 r\n[PROOFSTEP]\nrw [edist_dist, ENNReal.ofReal_le_ofReal_iff hr]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u03b1\n\u22a2 nndist x y = Real.toNNReal (dist x y)\n[PROOFSTEP]\nrw [dist_nndist, Real.toNNReal_coe]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u03b1\n\u22a2 dist x y = ENNReal.toReal (edist x y)\n[PROOFSTEP]\nrw [edist_dist, ENNReal.toReal_ofReal dist_nonneg]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 y \u2208 ball x \u03b5 \u2194 dist x y < \u03b5\n[PROOFSTEP]\nrw [dist_comm, mem_ball]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nh : 0 < \u03b5\n\u22a2 x \u2208 ball x \u03b5\n[PROOFSTEP]\nrwa [mem_ball, dist_self]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 ball x \u03b5 = \u2205 \u2194 \u03b5 \u2264 0\n[PROOFSTEP]\nrw [\u2190 not_nonempty_iff_eq_empty, nonempty_ball, not_lt]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 ball x 0 = \u2205\n[PROOFSTEP]\nrw [ball_eq_empty]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nh : x \u2208 ball y \u03b5\n\u22a2 \u2203 \u03b5', \u03b5' < \u03b5 \u2227 x \u2208 ball y \u03b5'\n[PROOFSTEP]\nsimp only [mem_ball] at h \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nh : dist x y < \u03b5\n\u22a2 \u2203 \u03b5', \u03b5' < \u03b5 \u2227 dist x y < \u03b5'\n[PROOFSTEP]\nexact \u27e8(dist x y + \u03b5) / 2, by linarith, by linarith\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nh : dist x y < \u03b5\n\u22a2 (dist x y + \u03b5) / 2 < \u03b5\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nh : dist x y < \u03b5\n\u22a2 dist x y < (dist x y + \u03b5) / 2\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u03b5 : \u211d\nx : \u03b1\n\u22a2 UniformSpace.ball x {p | dist p.fst p.snd < \u03b5} = ball x \u03b5\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d\u00b9 y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u03b5 : \u211d\nx x\u271d : \u03b1\n\u22a2 x\u271d \u2208 UniformSpace.ball x {p | dist p.fst p.snd < \u03b5} \u2194 x\u271d \u2208 ball x \u03b5\n[PROOFSTEP]\nsimp [dist_comm, UniformSpace.ball]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 y \u2208 closedBall x \u03b5 \u2194 dist x y \u2264 \u03b5\n[PROOFSTEP]\nrw [dist_comm, mem_closedBall]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 y \u2208 sphere x \u03b5 \u2194 dist x y = \u03b5\n[PROOFSTEP]\nrw [dist_comm, mem_sphere]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nh : y \u2208 sphere x \u03b5\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u00acx \u2208 sphere x \u03b5\n[PROOFSTEP]\nsimpa using h\u03b5.symm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d : Subsingleton \u03b1\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 IsEmpty \u2191(sphere x \u03b5)\n[PROOFSTEP]\nrw [sphere_eq_empty_of_subsingleton h\u03b5]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d : Subsingleton \u03b1\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 IsEmpty \u2191\u2205\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nh : 0 \u2264 \u03b5\n\u22a2 x \u2208 closedBall x \u03b5\n[PROOFSTEP]\nrwa [mem_closedBall, dist_self]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 closedBall x \u03b5 = \u2205 \u2194 \u03b5 < 0\n[PROOFSTEP]\nrw [\u2190 not_nonempty_iff_eq_empty, nonempty_closedBall, not_le]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nh : \u03b4 + \u03b5 \u2264 dist x y\n\u22a2 \u03b5 + \u03b4 \u2264 dist y x\n[PROOFSTEP]\nrwa [add_comm, dist_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 sphere x \u03b5 \u222a ball x \u03b5 = closedBall x \u03b5\n[PROOFSTEP]\nrw [union_comm, ball_union_sphere]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 closedBall x \u03b5 \\ sphere x \u03b5 = ball x \u03b5\n[PROOFSTEP]\nrw [\u2190 ball_union_sphere, Set.union_diff_cancel_right sphere_disjoint_ball.symm.le_bot]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 closedBall x \u03b5 \\ ball x \u03b5 = sphere x \u03b5\n[PROOFSTEP]\nrw [\u2190 ball_union_sphere, Set.union_diff_cancel_left sphere_disjoint_ball.symm.le_bot]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 x \u2208 ball y \u03b5 \u2194 y \u2208 ball x \u03b5\n[PROOFSTEP]\nrw [mem_ball', mem_ball]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 x \u2208 closedBall y \u03b5 \u2194 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nrw [mem_closedBall', mem_closedBall]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 x \u2208 sphere y \u03b5 \u2194 y \u2208 sphere x \u03b5\n[PROOFSTEP]\nrw [mem_sphere', mem_sphere]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 closedBall x \u03b5 = \u22c2 (\u03b4 : \u211d) (_ : \u03b4 > \u03b5), ball x \u03b4\n[PROOFSTEP]\next y\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y\u271d z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ny : \u03b1\n\u22a2 y \u2208 closedBall x \u03b5 \u2194 y \u2208 \u22c2 (\u03b4 : \u211d) (_ : \u03b4 > \u03b5), ball x \u03b4\n[PROOFSTEP]\nrw [mem_closedBall, \u2190 forall_lt_iff_le', mem_iInter\u2082]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y\u271d z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ny : \u03b1\n\u22a2 (\u2200 \u2983c : \u211d\u2984, \u03b5 < c \u2192 dist y x < c) \u2194 \u2200 (i : \u211d), i > \u03b5 \u2192 y \u2208 ball x i\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nh : Set.Nonempty (ball x \u03b5\u2081 \u2229 closedBall y \u03b5\u2082)\n\u22a2 dist x y < \u03b5\u2081 + \u03b5\u2082\n[PROOFSTEP]\nrw [inter_comm] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nh : Set.Nonempty (closedBall y \u03b5\u2082 \u2229 ball x \u03b5\u2081)\n\u22a2 dist x y < \u03b5\u2081 + \u03b5\u2082\n[PROOFSTEP]\nrw [add_comm, dist_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nh : Set.Nonempty (closedBall y \u03b5\u2082 \u2229 ball x \u03b5\u2081)\n\u22a2 dist y x < \u03b5\u2082 + \u03b5\u2081\n[PROOFSTEP]\nexact dist_lt_add_of_nonempty_closedBall_inter_ball h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nx : \u03b1\n\u22a2 \u22c3 (n : \u2115), s \u2229 closedBall x \u2191n = s\n[PROOFSTEP]\nrw [\u2190 inter_iUnion, iUnion_closedBall_nat, inter_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z\u271d : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nh : dist x y \u2264 \u03b5\u2082 - \u03b5\u2081\nz : \u03b1\nzx : z \u2208 ball x \u03b5\u2081\n\u22a2 z \u2208 ball y \u03b5\u2082\n[PROOFSTEP]\nrw [\u2190 add_sub_cancel'_right \u03b5\u2081 \u03b5\u2082]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z\u271d : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nh : dist x y \u2264 \u03b5\u2082 - \u03b5\u2081\nz : \u03b1\nzx : z \u2208 ball x \u03b5\u2081\n\u22a2 z \u2208 ball y (\u03b5\u2081 + (\u03b5\u2082 - \u03b5\u2081))\n[PROOFSTEP]\nexact lt_of_le_of_lt (dist_triangle z x y) (add_lt_add_of_lt_of_le zx h)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y\u271d z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ny : \u03b1\nh : y \u2208 ball x (\u03b5 / 2)\n\u22a2 dist y x \u2264 \u03b5 - \u03b5 / 2\n[PROOFSTEP]\nrw [sub_self_div_two]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y\u271d z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ny : \u03b1\nh : y \u2208 ball x (\u03b5 / 2)\n\u22a2 dist y x \u2264 \u03b5 / 2\n[PROOFSTEP]\nexact le_of_lt h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nh : y \u2208 ball x \u03b5\n\u22a2 dist y x \u2264 \u03b5 - (\u03b5 - dist y x)\n[PROOFSTEP]\nrw [sub_sub_self]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y\u271d z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\np : \u03b1 \u2192 Prop\nx : \u03b1\nH : \u2203\u1da0 (R : \u211d) in atTop, \u2200 (y : \u03b1), y \u2208 closedBall x R \u2192 p y\ny : \u03b1\n\u22a2 p y\n[PROOFSTEP]\nobtain \u27e8R, hR, h\u27e9 : \u2203 R \u2265 dist y x, \u2200 z : \u03b1, z \u2208 closedBall x R \u2192 p z := frequently_iff.1 H (Ici_mem_atTop (dist y x))\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y\u271d z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\np : \u03b1 \u2192 Prop\nx : \u03b1\nH : \u2203\u1da0 (R : \u211d) in atTop, \u2200 (y : \u03b1), y \u2208 closedBall x R \u2192 p y\ny : \u03b1\nR : \u211d\nhR : R \u2265 dist y x\nh : \u2200 (z : \u03b1), z \u2208 closedBall x R \u2192 p z\n\u22a2 p y\n[PROOFSTEP]\nexact h _ hR\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y\u271d z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\np : \u03b1 \u2192 Prop\nx : \u03b1\nH : \u2203\u1da0 (R : \u211d) in atTop, \u2200 (y : \u03b1), y \u2208 ball x R \u2192 p y\ny : \u03b1\n\u22a2 p y\n[PROOFSTEP]\nobtain \u27e8R, hR, h\u27e9 : \u2203 R > dist y x, \u2200 z : \u03b1, z \u2208 ball x R \u2192 p z := frequently_iff.1 H (Ioi_mem_atTop (dist y x))\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y\u271d z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\np : \u03b1 \u2192 Prop\nx : \u03b1\nH : \u2203\u1da0 (R : \u211d) in atTop, \u2200 (y : \u03b1), y \u2208 ball x R \u2192 p y\ny : \u03b1\nR : \u211d\nhR : R > dist y x\nh : \u2200 (z : \u03b1), z \u2208 ball x R \u2192 p z\n\u22a2 p y\n[PROOFSTEP]\nexact h _ hR\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\n\u22a2 IsBounded s \u2194 \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 dist x y \u2264 C\n[PROOFSTEP]\nrw [isBounded_def, \u2190 Filter.mem_sets, @PseudoMetricSpace.cobounded_sets \u03b1, mem_setOf_eq, compl_compl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\n\u22a2 IsBounded s \u2194 \u2203 C, \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2192 nndist x y \u2264 C\n[PROOFSTEP]\nsimp only [isBounded_iff_exists_ge 0, NNReal.exists, \u2190 NNReal.coe_le_coe, \u2190 dist_nndist, NNReal.coe_mk, exists_prop]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 HasBasis (\ud835\udce4 \u03b1) (fun \u03b5 => 0 < \u03b5) fun \u03b5 => {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nrw [toUniformSpace_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 HasBasis (\ud835\udce4 \u03b1) (fun \u03b5 => 0 < \u03b5) fun \u03b5 => {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nexact UniformSpace.hasBasis_ofFun (exists_gt _) _ _ _ _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u03b2 : Type u_3\np : \u03b2 \u2192 Prop\nf : \u03b2 \u2192 \u211d\nhf\u2080 : \u2200 (i : \u03b2), p i \u2192 0 < f i\nhf : \u2200 \u2983\u03b5 : \u211d\u2984, 0 < \u03b5 \u2192 \u2203 i, p i \u2227 f i \u2264 \u03b5\n\u22a2 HasBasis (\ud835\udce4 \u03b1) p fun i => {p | dist p.fst p.snd < f i}\n[PROOFSTEP]\nrefine' \u27e8fun s => uniformity_basis_dist.mem_iff.trans _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b2 : Type u_3\np : \u03b2 \u2192 Prop\nf : \u03b2 \u2192 \u211d\nhf\u2080 : \u2200 (i : \u03b2), p i \u2192 0 < f i\nhf : \u2200 \u2983\u03b5 : \u211d\u2984, 0 < \u03b5 \u2192 \u2203 i, p i \u2227 f i \u2264 \u03b5\ns : Set (\u03b1 \u00d7 \u03b1)\n\u22a2 (\u2203 i, 0 < i \u2227 {p | dist p.fst p.snd < i} \u2286 s) \u2194 \u2203 i, p i \u2227 {p | dist p.fst p.snd < f i} \u2286 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b2 : Type u_3\np : \u03b2 \u2192 Prop\nf : \u03b2 \u2192 \u211d\nhf\u2080 : \u2200 (i : \u03b2), p i \u2192 0 < f i\nhf : \u2200 \u2983\u03b5 : \u211d\u2984, 0 < \u03b5 \u2192 \u2203 i, p i \u2227 f i \u2264 \u03b5\ns : Set (\u03b1 \u00d7 \u03b1)\n\u22a2 (\u2203 i, 0 < i \u2227 {p | dist p.fst p.snd < i} \u2286 s) \u2192 \u2203 i, p i \u2227 {p | dist p.fst p.snd < f i} \u2286 s\n[PROOFSTEP]\nrintro \u27e8\u03b5, \u03b5\u2080, h\u03b5\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b2 : Type u_3\np : \u03b2 \u2192 Prop\nf : \u03b2 \u2192 \u211d\nhf\u2080 : \u2200 (i : \u03b2), p i \u2192 0 < f i\nhf : \u2200 \u2983\u03b5 : \u211d\u2984, 0 < \u03b5 \u2192 \u2203 i, p i \u2227 f i \u2264 \u03b5\ns : Set (\u03b1 \u00d7 \u03b1)\n\u03b5 : \u211d\n\u03b5\u2080 : 0 < \u03b5\nh\u03b5 : {p | dist p.fst p.snd < \u03b5} \u2286 s\n\u22a2 \u2203 i, p i \u2227 {p | dist p.fst p.snd < f i} \u2286 s\n[PROOFSTEP]\nrcases hf \u03b5\u2080 with \u27e8i, hi, H\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b2 : Type u_3\np : \u03b2 \u2192 Prop\nf : \u03b2 \u2192 \u211d\nhf\u2080 : \u2200 (i : \u03b2), p i \u2192 0 < f i\nhf : \u2200 \u2983\u03b5 : \u211d\u2984, 0 < \u03b5 \u2192 \u2203 i, p i \u2227 f i \u2264 \u03b5\ns : Set (\u03b1 \u00d7 \u03b1)\n\u03b5 : \u211d\n\u03b5\u2080 : 0 < \u03b5\nh\u03b5 : {p | dist p.fst p.snd < \u03b5} \u2286 s\ni : \u03b2\nhi : p i\nH : f i \u2264 \u03b5\n\u22a2 \u2203 i, p i \u2227 {p | dist p.fst p.snd < f i} \u2286 s\n[PROOFSTEP]\nexact \u27e8i, hi, fun x (hx : _ < _) => h\u03b5 <| lt_of_lt_of_le hx H\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b2 : Type u_3\np : \u03b2 \u2192 Prop\nf : \u03b2 \u2192 \u211d\nhf\u2080 : \u2200 (i : \u03b2), p i \u2192 0 < f i\nhf : \u2200 \u2983\u03b5 : \u211d\u2984, 0 < \u03b5 \u2192 \u2203 i, p i \u2227 f i \u2264 \u03b5\ns : Set (\u03b1 \u00d7 \u03b1)\n\u22a2 (\u2203 i, p i \u2227 {p | dist p.fst p.snd < f i} \u2286 s) \u2192 \u2203 i, 0 < i \u2227 {p | dist p.fst p.snd < i} \u2286 s\n[PROOFSTEP]\nexact fun \u27e8i, hi, H\u27e9 => \u27e8f i, hf\u2080 i hi, H\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nn : \u2115\nhn : 1 / (\u2191n + 1) < \u03b5\n\u22a2 1 / \u2191(n + 1) \u2264 \u03b5\n[PROOFSTEP]\nexact_mod_cast hn.le\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u03b2 : Type u_3\np : \u03b2 \u2192 Prop\nf : \u03b2 \u2192 \u211d\nhf\u2080 : \u2200 (x : \u03b2), p x \u2192 0 < f x\nhf : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 x, p x \u2227 f x \u2264 \u03b5\n\u22a2 HasBasis (\ud835\udce4 \u03b1) p fun x => {p | dist p.fst p.snd \u2264 f x}\n[PROOFSTEP]\nrefine' \u27e8fun s => uniformity_basis_dist.mem_iff.trans _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b2 : Type u_3\np : \u03b2 \u2192 Prop\nf : \u03b2 \u2192 \u211d\nhf\u2080 : \u2200 (x : \u03b2), p x \u2192 0 < f x\nhf : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 x, p x \u2227 f x \u2264 \u03b5\ns : Set (\u03b1 \u00d7 \u03b1)\n\u22a2 (\u2203 i, 0 < i \u2227 {p | dist p.fst p.snd < i} \u2286 s) \u2194 \u2203 i, p i \u2227 {p | dist p.fst p.snd \u2264 f i} \u2286 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b2 : Type u_3\np : \u03b2 \u2192 Prop\nf : \u03b2 \u2192 \u211d\nhf\u2080 : \u2200 (x : \u03b2), p x \u2192 0 < f x\nhf : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 x, p x \u2227 f x \u2264 \u03b5\ns : Set (\u03b1 \u00d7 \u03b1)\n\u22a2 (\u2203 i, 0 < i \u2227 {p | dist p.fst p.snd < i} \u2286 s) \u2192 \u2203 i, p i \u2227 {p | dist p.fst p.snd \u2264 f i} \u2286 s\n[PROOFSTEP]\nrintro \u27e8\u03b5, \u03b5\u2080, h\u03b5\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b2 : Type u_3\np : \u03b2 \u2192 Prop\nf : \u03b2 \u2192 \u211d\nhf\u2080 : \u2200 (x : \u03b2), p x \u2192 0 < f x\nhf : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 x, p x \u2227 f x \u2264 \u03b5\ns : Set (\u03b1 \u00d7 \u03b1)\n\u03b5 : \u211d\n\u03b5\u2080 : 0 < \u03b5\nh\u03b5 : {p | dist p.fst p.snd < \u03b5} \u2286 s\n\u22a2 \u2203 i, p i \u2227 {p | dist p.fst p.snd \u2264 f i} \u2286 s\n[PROOFSTEP]\nrcases exists_between \u03b5\u2080 with \u27e8\u03b5', h\u03b5'\u27e9\n[GOAL]\ncase mp.intro.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b2 : Type u_3\np : \u03b2 \u2192 Prop\nf : \u03b2 \u2192 \u211d\nhf\u2080 : \u2200 (x : \u03b2), p x \u2192 0 < f x\nhf : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 x, p x \u2227 f x \u2264 \u03b5\ns : Set (\u03b1 \u00d7 \u03b1)\n\u03b5 : \u211d\n\u03b5\u2080 : 0 < \u03b5\nh\u03b5 : {p | dist p.fst p.snd < \u03b5} \u2286 s\n\u03b5' : \u211d\nh\u03b5' : 0 < \u03b5' \u2227 \u03b5' < \u03b5\n\u22a2 \u2203 i, p i \u2227 {p | dist p.fst p.snd \u2264 f i} \u2286 s\n[PROOFSTEP]\nrcases hf \u03b5' h\u03b5'.1 with \u27e8i, hi, H\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b2 : Type u_3\np : \u03b2 \u2192 Prop\nf : \u03b2 \u2192 \u211d\nhf\u2080 : \u2200 (x : \u03b2), p x \u2192 0 < f x\nhf : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 x, p x \u2227 f x \u2264 \u03b5\ns : Set (\u03b1 \u00d7 \u03b1)\n\u03b5 : \u211d\n\u03b5\u2080 : 0 < \u03b5\nh\u03b5 : {p | dist p.fst p.snd < \u03b5} \u2286 s\n\u03b5' : \u211d\nh\u03b5' : 0 < \u03b5' \u2227 \u03b5' < \u03b5\ni : \u03b2\nhi : p i\nH : f i \u2264 \u03b5'\n\u22a2 \u2203 i, p i \u2227 {p | dist p.fst p.snd \u2264 f i} \u2286 s\n[PROOFSTEP]\nexact \u27e8i, hi, fun x (hx : _ \u2264 _) => h\u03b5 <| lt_of_le_of_lt (le_trans hx H) h\u03b5'.2\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b2 : Type u_3\np : \u03b2 \u2192 Prop\nf : \u03b2 \u2192 \u211d\nhf\u2080 : \u2200 (x : \u03b2), p x \u2192 0 < f x\nhf : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 x, p x \u2227 f x \u2264 \u03b5\ns : Set (\u03b1 \u00d7 \u03b1)\n\u22a2 (\u2203 i, p i \u2227 {p | dist p.fst p.snd \u2264 f i} \u2286 s) \u2192 \u2203 i, 0 < i \u2227 {p | dist p.fst p.snd < i} \u2286 s\n[PROOFSTEP]\nexact fun \u27e8i, hi, H\u27e9 => \u27e8f i, hf\u2080 i hi, fun x (hx : _ < _) => H (mem_setOf.2 hx.le)\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 (i' : \u211d), 0 < i' \u2192 \u2203 i, 0 < i \u2227 Prod.map f f \u207b\u00b9' {p | dist p.fst p.snd < i} \u2286 {p | dist p.fst p.snd < i'}) \u2194\n \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 {a b : \u03b1}, dist (f a) (f b) < \u03b5 \u2192 dist a b < \u03b4\n[PROOFSTEP]\nsimp only [subset_def, Prod.forall, gt_iff_lt, preimage_setOf_eq, Prod_map, mem_setOf]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 UniformEmbedding f \u2194\n Function.Injective f \u2227\n UniformContinuous f \u2227 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 {a b : \u03b1}, dist (f a) (f b) < \u03b5 \u2192 dist a b < \u03b4\n[PROOFSTEP]\nrw [uniformEmbedding_iff, and_comm, uniformInducing_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\n\u22a2 TotallyBounded s\n[PROOFSTEP]\ncases' s.eq_empty_or_nonempty with hs hs\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nhs : s = \u2205\n\u22a2 TotallyBounded s\n[PROOFSTEP]\nrw [hs]\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nhs : s = \u2205\n\u22a2 TotallyBounded \u2205\n[PROOFSTEP]\nexact totallyBounded_empty\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nhs : Set.Nonempty s\n\u22a2 TotallyBounded s\n[PROOFSTEP]\nrcases hs with \u27e8x0, hx0\u27e9\n[GOAL]\ncase inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nx0 : \u03b1\nhx0 : x0 \u2208 s\n\u22a2 TotallyBounded s\n[PROOFSTEP]\nhaveI : Inhabited s := \u27e8\u27e8x0, hx0\u27e9\u27e9\n[GOAL]\ncase inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nx0 : \u03b1\nhx0 : x0 \u2208 s\nthis : Inhabited \u2191s\n\u22a2 TotallyBounded s\n[PROOFSTEP]\nrefine' totallyBounded_iff.2 fun \u03b5 \u03b50 => _\n[GOAL]\ncase inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nx0 : \u03b1\nhx0 : x0 \u2208 s\nthis : Inhabited \u2191s\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), ball y \u03b5\n[PROOFSTEP]\nrcases H \u03b5 \u03b50 with \u27e8\u03b2, f\u03b2, F, hF\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nx0 : \u03b1\nhx0 : x0 \u2208 s\nthis : Inhabited \u2191s\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b2 : Type u\nf\u03b2 : Fintype \u03b2\nF : \u2191s \u2192 \u03b2\nhF : \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\n\u22a2 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), ball y \u03b5\n[PROOFSTEP]\nlet Finv := Function.invFun F\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nx0 : \u03b1\nhx0 : x0 \u2208 s\nthis : Inhabited \u2191s\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b2 : Type u\nf\u03b2 : Fintype \u03b2\nF : \u2191s \u2192 \u03b2\nhF : \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nFinv : \u03b2 \u2192 \u2191s := Function.invFun F\n\u22a2 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), ball y \u03b5\n[PROOFSTEP]\nrefine' \u27e8range (Subtype.val \u2218 Finv), finite_range _, fun x xs => _\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nx0 : \u03b1\nhx0 : x0 \u2208 s\nthis : Inhabited \u2191s\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b2 : Type u\nf\u03b2 : Fintype \u03b2\nF : \u2191s \u2192 \u03b2\nhF : \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nFinv : \u03b2 \u2192 \u2191s := Function.invFun F\nx : \u03b1\nxs : x \u2208 s\n\u22a2 x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 range (Subtype.val \u2218 Finv)), ball y \u03b5\n[PROOFSTEP]\nlet x' := Finv (F \u27e8x, xs\u27e9)\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nx0 : \u03b1\nhx0 : x0 \u2208 s\nthis : Inhabited \u2191s\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b2 : Type u\nf\u03b2 : Fintype \u03b2\nF : \u2191s \u2192 \u03b2\nhF : \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nFinv : \u03b2 \u2192 \u2191s := Function.invFun F\nx : \u03b1\nxs : x \u2208 s\nx' : \u2191s := Finv (F { val := x, property := xs })\n\u22a2 x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 range (Subtype.val \u2218 Finv)), ball y \u03b5\n[PROOFSTEP]\nhave : F x' = F \u27e8x, xs\u27e9 := Function.invFun_eq \u27e8\u27e8x, xs\u27e9, rfl\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nx0 : \u03b1\nhx0 : x0 \u2208 s\nthis\u271d : Inhabited \u2191s\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b2 : Type u\nf\u03b2 : Fintype \u03b2\nF : \u2191s \u2192 \u03b2\nhF : \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nFinv : \u03b2 \u2192 \u2191s := Function.invFun F\nx : \u03b1\nxs : x \u2208 s\nx' : \u2191s := Finv (F { val := x, property := xs })\nthis : F x' = F { val := x, property := xs }\n\u22a2 x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 range (Subtype.val \u2218 Finv)), ball y \u03b5\n[PROOFSTEP]\nsimp only [Set.mem_iUnion, Set.mem_range]\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nx0 : \u03b1\nhx0 : x0 \u2208 s\nthis\u271d : Inhabited \u2191s\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b2 : Type u\nf\u03b2 : Fintype \u03b2\nF : \u2191s \u2192 \u03b2\nhF : \u2200 (x y : \u2191s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5\nFinv : \u03b2 \u2192 \u2191s := Function.invFun F\nx : \u03b1\nxs : x \u2208 s\nx' : \u2191s := Finv (F { val := x, property := xs })\nthis : F x' = F { val := x, property := xs }\n\u22a2 \u2203 i i_1, x \u2208 ball i \u03b5\n[PROOFSTEP]\nexact \u27e8_, \u27e8F \u27e8x, xs\u27e9, rfl\u27e9, hF _ _ this.symm\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nhs : TotallyBounded s\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), ball y \u03b5\n[PROOFSTEP]\nintro \u03b5 \u03b5_pos\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nhs : TotallyBounded s\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), ball y \u03b5\n[PROOFSTEP]\nrw [totallyBounded_iff_subset] at hs \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nhs : \u2200 (d : Set (\u03b1 \u00d7 \u03b1)), d \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), ball y \u03b5\n[PROOFSTEP]\nexact hs _ (dist_mem_uniformity \u03b5_pos)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u03b9 : Type u_3\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\np : Filter \u03b9\np' : Filter \u03b2\n\u22a2 TendstoUniformlyOnFilter F f p p' \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9 \u00d7 \u03b2) in p \u00d7\u02e2 p', dist (f n.snd) (F n.fst n.snd) < \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun H \u03b5 h\u03b5 => H _ (dist_mem_uniformity h\u03b5), fun H u hu => _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u03b9 : Type u_3\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\np : Filter \u03b9\np' : Filter \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9 \u00d7 \u03b2) in p \u00d7\u02e2 p', dist (f n.snd) (F n.fst n.snd) < \u03b5\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2200\u1da0 (n : \u03b9 \u00d7 \u03b2) in p \u00d7\u02e2 p', (f n.snd, F n.fst n.snd) \u2208 u\n[PROOFSTEP]\nrcases mem_uniformity_dist.1 hu with \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u03b9 : Type u_3\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\np : Filter \u03b9\np' : Filter \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9 \u00d7 \u03b2) in p \u00d7\u02e2 p', dist (f n.snd) (F n.fst n.snd) < \u03b5\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 u\n\u22a2 \u2200\u1da0 (n : \u03b9 \u00d7 \u03b2) in p \u00d7\u02e2 p', (f n.snd, F n.fst n.snd) \u2208 u\n[PROOFSTEP]\nrefine' (H \u03b5 \u03b5pos).mono fun n hn => h\u03b5 hn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\np : Filter \u03b9\ns : Set \u03b2\n\u22a2 TendstoLocallyUniformlyOn F f p s \u2194\n \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200 (x : \u03b2), x \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b2), y \u2208 t \u2192 dist (f y) (F n y) < \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun H \u03b5 h\u03b5 => H _ (dist_mem_uniformity h\u03b5), fun H u hu x hx => _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx\u271d y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\np : Filter \u03b9\ns : Set \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200 (x : \u03b2), x \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b2), y \u2208 t \u2192 dist (f y) (F n y) < \u03b5\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\nx : \u03b2\nhx : x \u2208 s\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b2), y \u2208 t \u2192 (f y, F n y) \u2208 u\n[PROOFSTEP]\nrcases mem_uniformity_dist.1 hu with \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx\u271d y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\np : Filter \u03b9\ns : Set \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200 (x : \u03b2), x \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b2), y \u2208 t \u2192 dist (f y) (F n y) < \u03b5\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\nx : \u03b2\nhx : x \u2208 s\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 u\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b2), y \u2208 t \u2192 (f y, F n y) \u2208 u\n[PROOFSTEP]\nrcases H \u03b5 \u03b5pos x hx with \u27e8t, ht, Ht\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx\u271d y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\np : Filter \u03b9\ns : Set \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200 (x : \u03b2), x \u2208 s \u2192 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b2), y \u2208 t \u2192 dist (f y) (F n y) < \u03b5\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\nx : \u03b2\nhx : x \u2208 s\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 u\nt : Set \u03b2\nht : t \u2208 \ud835\udcdd[s] x\nHt : \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b2), y \u2208 t \u2192 dist (f y) (F n y) < \u03b5\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b2), y \u2208 t \u2192 (f y, F n y) \u2208 u\n[PROOFSTEP]\nexact \u27e8t, ht, Ht.mono fun n hs x hx => h\u03b5 (hs x hx)\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b9 : Type u_3\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\np : Filter \u03b9\ns : Set \u03b2\n\u22a2 TendstoUniformlyOn F f p s \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b2), x \u2208 s \u2192 dist (f x) (F n x) < \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun H \u03b5 h\u03b5 => H _ (dist_mem_uniformity h\u03b5), fun H u hu => _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b9 : Type u_3\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\np : Filter \u03b9\ns : Set \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b2), x \u2208 s \u2192 dist (f x) (F n x) < \u03b5\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b2), x \u2208 s \u2192 (f x, F n x) \u2208 u\n[PROOFSTEP]\nrcases mem_uniformity_dist.1 hu with \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\n\u03b9 : Type u_3\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\np : Filter \u03b9\ns : Set \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b2), x \u2208 s \u2192 dist (f x) (F n x) < \u03b5\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 u\n\u22a2 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b2), x \u2208 s \u2192 (f x, F n x) \u2208 u\n[PROOFSTEP]\nexact (H \u03b5 \u03b5pos).mono fun n hs x hx => h\u03b5 (hs x hx)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u03b9 : Type u_3\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\np : Filter \u03b9\n\u22a2 TendstoLocallyUniformly F f p \u2194\n \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200 (x : \u03b2), \u2203 t, t \u2208 \ud835\udcdd x \u2227 \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b2), y \u2208 t \u2192 dist (f y) (F n y) < \u03b5\n[PROOFSTEP]\nsimp only [\u2190 tendstoLocallyUniformlyOn_univ, tendstoLocallyUniformlyOn_iff, nhdsWithin_univ, mem_univ, forall_const,\n exists_prop]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u03b9 : Type u_3\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\np : Filter \u03b9\n\u22a2 TendstoUniformly F f p \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b2), dist (f x) (F n x) < \u03b5\n[PROOFSTEP]\nrw [\u2190 tendstoUniformlyOn_univ, tendstoUniformlyOn_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u03b9 : Type u_3\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nf : \u03b2 \u2192 \u03b1\np : Filter \u03b9\n\u22a2 (\u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b2), x \u2208 univ \u2192 dist (f x) (F n x) < \u03b5) \u2194\n \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b2), dist (f x) (F n x) < \u03b5\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\u271d\nx y z : \u03b1\u271d\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\u271d\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : PseudoMetricSpace \u03b1\nf : Filter \u03b9\nx\u2080 : \u03b1\np : \u03b1 \u00d7 \u03b9 \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : \u03b1 \u00d7 \u03b9) in \ud835\udcdd x\u2080 \u00d7\u02e2 f, p x) \u2194\n \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 pa, (\u2200\u1da0 (i : \u03b9) in f, pa i) \u2227 \u2200 {x : \u03b1}, dist x x\u2080 < \u03b5 \u2192 \u2200 {i : \u03b9}, pa i \u2192 p (x, i)\n[PROOFSTEP]\nrefine (nhds_basis_ball.prod f.basis_sets).eventually_iff.trans ?_\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\u271d\nx y z : \u03b1\u271d\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\u271d\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : PseudoMetricSpace \u03b1\nf : Filter \u03b9\nx\u2080 : \u03b1\np : \u03b1 \u00d7 \u03b9 \u2192 Prop\n\u22a2 (\u2203 i, (0 < i.fst \u2227 i.snd \u2208 f) \u2227 \u2200 \u2983x : \u03b1 \u00d7 \u03b9\u2984, x \u2208 ball x\u2080 i.fst \u00d7\u02e2 id i.snd \u2192 p x) \u2194\n \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 pa, (\u2200\u1da0 (i : \u03b9) in f, pa i) \u2227 \u2200 {x : \u03b1}, dist x x\u2080 < \u03b5 \u2192 \u2200 {i : \u03b9}, pa i \u2192 p (x, i)\n[PROOFSTEP]\nsimp only [Prod.exists, forall_prod_set, id, mem_ball, and_assoc, exists_and_left, and_imp]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\u271d\nx y z : \u03b1\u271d\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\u271d\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : PseudoMetricSpace \u03b1\nf : Filter \u03b9\nx\u2080 : \u03b1\np : \u03b1 \u00d7 \u03b9 \u2192 Prop\n\u22a2 (\u2203 a, 0 < a \u2227 \u2203 x, x \u2208 f \u2227 \u2200 (x_1 : \u03b1), dist x_1 x\u2080 < a \u2192 \u2200 (y : \u03b9), y \u2208 x \u2192 p (x_1, y)) \u2194\n \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 pa, (\u2200\u1da0 (i : \u03b9) in f, pa i) \u2227 \u2200 {x : \u03b1}, dist x x\u2080 < \u03b5 \u2192 \u2200 {i : \u03b9}, pa i \u2192 p (x, i)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nf : Filter \u03b9\nx\u2080 : \u03b1\np : \u03b9 \u00d7 \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : \u03b9 \u00d7 \u03b1) in f \u00d7\u02e2 \ud835\udcdd x\u2080, p x) \u2194\n \u2203 pa, (\u2200\u1da0 (i : \u03b9) in f, pa i) \u2227 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : \u03b1}, dist x x\u2080 < \u03b5 \u2192 p (i, x)\n[PROOFSTEP]\nrw [eventually_swap_iff, Metric.eventually_nhds_prod_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nf : Filter \u03b9\nx\u2080 : \u03b1\np : \u03b9 \u00d7 \u03b1 \u2192 Prop\n\u22a2 (\u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 pa, (\u2200\u1da0 (i : \u03b9) in f, pa i) \u2227 \u2200 {x : \u03b1}, dist x x\u2080 < \u03b5 \u2192 \u2200 {i : \u03b9}, pa i \u2192 p (Prod.swap (x, i))) \u2194\n \u2203 pa, (\u2200\u1da0 (i : \u03b9) in f, pa i) \u2227 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : \u03b1}, dist x x\u2080 < \u03b5 \u2192 p (i, x)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nf : Filter \u03b9\nx\u2080 : \u03b1\np : \u03b9 \u00d7 \u03b1 \u2192 Prop\n\u22a2 (\u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 pa, (\u2200\u1da0 (i : \u03b9) in f, pa i) \u2227 \u2200 {x : \u03b1}, dist x x\u2080 < \u03b5 \u2192 \u2200 {i : \u03b9}, pa i \u2192 p (Prod.swap (x, i))) \u2192\n \u2203 pa, (\u2200\u1da0 (i : \u03b9) in f, pa i) \u2227 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : \u03b1}, dist x x\u2080 < \u03b5 \u2192 p (i, x)\n[PROOFSTEP]\nrintro \u27e8a1, a2, a3, a4, a5\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nf : Filter \u03b9\nx\u2080 : \u03b1\np : \u03b9 \u00d7 \u03b1 \u2192 Prop\na1 : \u211d\na2 : a1 > 0\na3 : \u03b9 \u2192 Prop\na4 : \u2200\u1da0 (i : \u03b9) in f, a3 i\na5 : \u2200 {x : \u03b1}, dist x x\u2080 < a1 \u2192 \u2200 {i : \u03b9}, a3 i \u2192 p (Prod.swap (x, i))\n\u22a2 \u2203 pa, (\u2200\u1da0 (i : \u03b9) in f, pa i) \u2227 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : \u03b1}, dist x x\u2080 < \u03b5 \u2192 p (i, x)\n[PROOFSTEP]\nexact \u27e8a3, a4, a1, a2, fun b1 b2 b3 => a5 b3 b1\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nf : Filter \u03b9\nx\u2080 : \u03b1\np : \u03b9 \u00d7 \u03b1 \u2192 Prop\n\u22a2 (\u2203 pa, (\u2200\u1da0 (i : \u03b9) in f, pa i) \u2227 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : \u03b1}, dist x x\u2080 < \u03b5 \u2192 p (i, x)) \u2192\n \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 pa, (\u2200\u1da0 (i : \u03b9) in f, pa i) \u2227 \u2200 {x : \u03b1}, dist x x\u2080 < \u03b5 \u2192 \u2200 {i : \u03b9}, pa i \u2192 p (Prod.swap (x, i))\n[PROOFSTEP]\nrintro \u27e8a1, a2, a3, a4, a5\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nf : Filter \u03b9\nx\u2080 : \u03b1\np : \u03b9 \u00d7 \u03b1 \u2192 Prop\na1 : \u03b9 \u2192 Prop\na2 : \u2200\u1da0 (i : \u03b9) in f, a1 i\na3 : \u211d\na4 : a3 > 0\na5 : \u2200 {i : \u03b9}, a1 i \u2192 \u2200 {x : \u03b1}, dist x x\u2080 < a3 \u2192 p (i, x)\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 pa, (\u2200\u1da0 (i : \u03b9) in f, pa i) \u2227 \u2200 {x : \u03b1}, dist x x\u2080 < \u03b5 \u2192 \u2200 {i : \u03b9}, pa i \u2192 p (Prod.swap (x, i))\n[PROOFSTEP]\nexact \u27e8a3, a4, a1, a2, fun b1 b2 b3 => a5 b3 b1\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 IsOpen s \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 ball x \u03b5 \u2286 s\n[PROOFSTEP]\nsimp only [isOpen_iff_mem_nhds, mem_nhds_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 (\u2200 (ib : \u211d), 0 < ib \u2192 \u2203 ia, 0 < ia \u2227 \u2200 (x : \u03b1), x \u2208 ball a ia \u2229 s \u2192 f x \u2208 ball b ib \u2229 t) \u2194\n \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 {x : \u03b1}, x \u2208 s \u2192 dist x a < \u03b4 \u2192 f x \u2208 t \u2227 dist (f x) b < \u03b5\n[PROOFSTEP]\nsimp only [inter_comm _ s, inter_comm _ t, mem_inter_iff, and_imp]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 (\u2200 (ib : \u211d), 0 < ib \u2192 \u2203 ia, 0 < ia \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 ball a ia \u2192 f x \u2208 t \u2227 f x \u2208 ball b ib) \u2194\n \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 {x : \u03b1}, x \u2208 s \u2192 dist x a < \u03b4 \u2192 f x \u2208 t \u2227 dist (f x) b < \u03b5\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 Tendsto f (\ud835\udcdd[s] a) (\ud835\udcdd b) \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 {x : \u03b1}, x \u2208 s \u2192 dist x a < \u03b4 \u2192 dist (f x) b < \u03b5\n[PROOFSTEP]\nrw [\u2190 nhdsWithin_univ b, tendsto_nhdsWithin_nhdsWithin]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 (\u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 {x : \u03b1}, x \u2208 s \u2192 dist x a < \u03b4 \u2192 f x \u2208 univ \u2227 dist (f x) b < \u03b5) \u2194\n \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 {x : \u03b1}, x \u2208 s \u2192 dist x a < \u03b4 \u2192 dist (f x) b < \u03b5\n[PROOFSTEP]\nsimp only [mem_univ, true_and_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 ContinuousAt f a \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 {x : \u03b1}, dist x a < \u03b4 \u2192 dist (f x) (f a) < \u03b5\n[PROOFSTEP]\nrw [ContinuousAt, tendsto_nhds_nhds]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Set \u03b1\n\u22a2 ContinuousWithinAt f s a \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 {x : \u03b1}, x \u2208 s \u2192 dist x a < \u03b4 \u2192 dist (f x) (f a) < \u03b5\n[PROOFSTEP]\nrw [ContinuousWithinAt, tendsto_nhdsWithin_nhds]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 ContinuousOn f s \u2194\n \u2200 (b : \u03b1), b \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5\n[PROOFSTEP]\nsimp [ContinuousOn, continuousWithinAt_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\n\u22a2 ContinuousAt f b \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b2) in \ud835\udcdd b, dist (f x) (f b) < \u03b5\n[PROOFSTEP]\nrw [ContinuousAt, tendsto_nhds]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\ns : Set \u03b2\n\u22a2 ContinuousWithinAt f s b \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b2) in \ud835\udcdd[s] b, dist (f x) (f b) < \u03b5\n[PROOFSTEP]\nrw [ContinuousWithinAt, tendsto_nhds]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b2 \u2192 \u03b1\ns : Set \u03b2\n\u22a2 ContinuousOn f s \u2194 \u2200 (b : \u03b2), b \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b2) in \ud835\udcdd[s] b, dist (f x) (f b) < \u03b5\n[PROOFSTEP]\nsimp [ContinuousOn, continuousWithinAt_iff']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200 (ib : \u211d), 0 < ib \u2192 \u2203 ia, True \u2227 \u2200 (x : \u03b2), x \u2208 Ici ia \u2192 u x \u2208 ball a ib) \u2194\n \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b2), n \u2265 N \u2192 dist (u n) a < \u03b5\n[PROOFSTEP]\nsimp only [true_and]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200 (ib : \u211d), 0 < ib \u2192 \u2203 ia, \u2200 (x : \u03b2), x \u2208 Ici ia \u2192 u x \u2208 ball a ib) \u2194\n \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b2), n \u2265 N \u2192 dist (u n) a < \u03b5\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d\u00b2 : Nonempty \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : NoMaxOrder \u03b2\nu : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200 (ib : \u211d), 0 < ib \u2192 \u2203 ia, True \u2227 \u2200 (x : \u03b2), x \u2208 Ioi ia \u2192 u x \u2208 ball a ib) \u2194\n \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b2), n > N \u2192 dist (u n) a < \u03b5\n[PROOFSTEP]\nsimp only [true_and, gt_iff_lt, mem_Ioi, mem_ball]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\u271d\nx\u271d y z : \u03b1\u271d\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u22a2 IsOpen {x} \u2194 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : \u03b1), dist y x < \u03b5 \u2192 y = x\n[PROOFSTEP]\nsimp [isOpen_iff, subset_singleton_iff, mem_ball]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nhs : Dense s\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 y, y \u2208 s \u2227 dist x y < \u03b5\n[PROOFSTEP]\nhave : (ball x \u03b5).Nonempty := by simp [h\u03b5]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nhs : Dense s\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 Set.Nonempty (ball x \u03b5)\n[PROOFSTEP]\nsimp [h\u03b5]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y z : \u03b1\n\u03b4 \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nhs : Dense s\nx : \u03b1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nthis : Set.Nonempty (ball x \u03b5)\n\u22a2 \u2203 y, y \u2208 s \u2227 dist x y < \u03b5\n[PROOFSTEP]\nsimpa only [mem_ball'] using hs.exists_mem_open isOpen_ball this\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nd : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\n\u22a2 \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | \u2191(d p.fst p.snd) < \u03b5} = \u2a05 (\u03b5 : \u211d\u22650\u221e) (_ : \u03b5 > 0), \ud835\udcdf {p | \u2191(d p.fst p.snd) < \u03b5}\n[PROOFSTEP]\nsimp only [le_antisymm_iff, le_iInf_iff, le_principal_iff]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nd : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\n\u22a2 (\u2200 (i : \u211d\u22650\u221e), i > 0 \u2192 {p | \u2191(d p.fst p.snd) < i} \u2208 \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | \u2191(d p.fst p.snd) < \u03b5}) \u2227\n \u2200 (i : \u211d), i > 0 \u2192 {p | \u2191(d p.fst p.snd) < i} \u2208 \u2a05 (\u03b5 : \u211d\u22650\u221e) (_ : \u03b5 > 0), \ud835\udcdf {p | \u2191(d p.fst p.snd) < \u03b5}\n[PROOFSTEP]\nrefine \u27e8fun \u03b5 h\u03b5 => ?_, fun \u03b5 h\u03b5 => ?_\u27e9\n[GOAL]\ncase refine_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nd : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\n\u22a2 {p | \u2191(d p.fst p.snd) < \u03b5} \u2208 \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | \u2191(d p.fst p.snd) < \u03b5}\n[PROOFSTEP]\nrcases ENNReal.lt_iff_exists_nnreal_btwn.1 h\u03b5 with \u27e8\u03b5', \u03b5'0, \u03b5'\u03b5\u27e9\n[GOAL]\ncase refine_1.intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nd : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211d\u22650\n\u03b5'0 : 0 < \u2191\u03b5'\n\u03b5'\u03b5 : \u2191\u03b5' < \u03b5\n\u22a2 {p | \u2191(d p.fst p.snd) < \u03b5} \u2208 \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | \u2191(d p.fst p.snd) < \u03b5}\n[PROOFSTEP]\nrefine mem_iInf_of_mem (\u03b5' : \u211d) (mem_iInf_of_mem (ENNReal.coe_pos.1 \u03b5'0) ?_)\n[GOAL]\ncase refine_1.intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nd : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211d\u22650\n\u03b5'0 : 0 < \u2191\u03b5'\n\u03b5'\u03b5 : \u2191\u03b5' < \u03b5\n\u22a2 {p | \u2191(d p.fst p.snd) < \u03b5} \u2208 \ud835\udcdf {p | \u2191(d p.fst p.snd) < \u2191\u03b5'}\n[PROOFSTEP]\nexact fun x hx => lt_trans (ENNReal.coe_lt_coe.2 hx) \u03b5'\u03b5\n[GOAL]\ncase refine_2\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nd : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 {p | \u2191(d p.fst p.snd) < \u03b5} \u2208 \u2a05 (\u03b5 : \u211d\u22650\u221e) (_ : \u03b5 > 0), \ud835\udcdf {p | \u2191(d p.fst p.snd) < \u03b5}\n[PROOFSTEP]\nlift \u03b5 to \u211d\u22650 using le_of_lt h\u03b5\n[GOAL]\ncase refine_2.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nd : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\n\u03b5 : \u211d\u22650\nh\u03b5 : \u2191\u03b5 > 0\n\u22a2 {p | \u2191(d p.fst p.snd) < \u2191\u03b5} \u2208 \u2a05 (\u03b5 : \u211d\u22650\u221e) (_ : \u03b5 > 0), \ud835\udcdf {p | \u2191(d p.fst p.snd) < \u03b5}\n[PROOFSTEP]\nrefine mem_iInf_of_mem (\u03b5 : \u211d\u22650\u221e) (mem_iInf_of_mem (ENNReal.coe_pos.2 h\u03b5) ?_)\n[GOAL]\ncase refine_2.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nd : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\n\u03b5 : \u211d\u22650\nh\u03b5 : \u2191\u03b5 > 0\n\u22a2 {p | \u2191(d p.fst p.snd) < \u2191\u03b5} \u2208 \ud835\udcdf {p | \u2191(d p.fst p.snd) < \u2191\u03b5}\n[PROOFSTEP]\nexact fun _ => ENNReal.coe_lt_coe.1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u22a2 \ud835\udce4 \u03b1 = \u2a05 (\u03b5 : \u211d\u22650\u221e) (_ : \u03b5 > 0), \ud835\udcdf {p | edist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimp only [PseudoMetricSpace.uniformity_dist, dist_nndist, edist_nndist, Metric.uniformity_edist_aux]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nsrc\u271d : PseudoMetricSpace \u03b1 := inst\u271d\n\u22a2 \u2200 (x : \u03b1), EDist.edist x x = 0\n[PROOFSTEP]\nsimp [edist_dist]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nsrc\u271d : PseudoMetricSpace \u03b1 := inst\u271d\nx\u271d\u00b9 x\u271d : \u03b1\n\u22a2 EDist.edist x\u271d\u00b9 x\u271d = EDist.edist x\u271d x\u271d\u00b9\n[PROOFSTEP]\nsimp only [edist_dist, dist_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nsrc\u271d : PseudoMetricSpace \u03b1 := inst\u271d\nx y z : \u03b1\n\u22a2 EDist.edist x z \u2264 EDist.edist x y + EDist.edist y z\n[PROOFSTEP]\nsimp only [edist_dist, \u2190 ENNReal.ofReal_add, dist_nonneg]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nsrc\u271d : PseudoMetricSpace \u03b1 := inst\u271d\nx y z : \u03b1\n\u22a2 ENNReal.ofReal (dist x z) \u2264 ENNReal.ofReal (dist x y + dist y z)\n[PROOFSTEP]\nrw [ENNReal.ofReal_le_ofReal_iff _]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nsrc\u271d : PseudoMetricSpace \u03b1 := inst\u271d\nx y z : \u03b1\n\u22a2 dist x z \u2264 dist x y + dist y z\n[PROOFSTEP]\nexact dist_triangle _ _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nsrc\u271d : PseudoMetricSpace \u03b1 := inst\u271d\nx y z : \u03b1\n\u22a2 0 \u2264 dist x y + dist y z\n[PROOFSTEP]\nsimpa using add_le_add (dist_nonneg : 0 \u2264 dist x y) dist_nonneg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u03b5 : \u211d\n\u22a2 EMetric.ball x (ENNReal.ofReal \u03b5) = ball x \u03b5\n[PROOFSTEP]\next y\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u03b5 : \u211d\ny : \u03b1\n\u22a2 y \u2208 EMetric.ball x (ENNReal.ofReal \u03b5) \u2194 y \u2208 ball x \u03b5\n[PROOFSTEP]\nsimp only [EMetric.mem_ball, mem_ball, edist_dist]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u03b5 : \u211d\ny : \u03b1\n\u22a2 ENNReal.ofReal (dist y x) < ENNReal.ofReal \u03b5 \u2194 dist y x < \u03b5\n[PROOFSTEP]\nexact ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u03b5 : \u211d\u22650\n\u22a2 EMetric.ball x \u2191\u03b5 = ball x \u2191\u03b5\n[PROOFSTEP]\nrw [\u2190 Metric.emetric_ball]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u03b5 : \u211d\u22650\n\u22a2 EMetric.ball x \u2191\u03b5 = EMetric.ball x (ENNReal.ofReal \u2191\u03b5)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u03b5 : \u211d\nh : 0 \u2264 \u03b5\n\u22a2 EMetric.closedBall x (ENNReal.ofReal \u03b5) = closedBall x \u03b5\n[PROOFSTEP]\next y\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u03b5 : \u211d\nh : 0 \u2264 \u03b5\ny : \u03b1\n\u22a2 y \u2208 EMetric.closedBall x (ENNReal.ofReal \u03b5) \u2194 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nsimp [edist_le_ofReal h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u03b5 : \u211d\u22650\n\u22a2 EMetric.closedBall x \u2191\u03b5 = closedBall x \u2191\u03b5\n[PROOFSTEP]\nrw [\u2190 Metric.emetric_closedBall \u03b5.coe_nonneg, ENNReal.ofReal_coe_nnreal]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u03b1\n\u22a2 Inseparable x y \u2194 dist x y = 0\n[PROOFSTEP]\nrw [EMetric.inseparable_iff, edist_nndist, dist_nndist, ENNReal.coe_eq_zero, NNReal.coe_eq_zero]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nU : UniformSpace \u03b1\nm : PseudoMetricSpace \u03b1\nH : \ud835\udce4 \u03b1 = \ud835\udce4 \u03b1\n\u22a2 replaceUniformity m H = m\n[PROOFSTEP]\next\n[GOAL]\ncase h.dist.h.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nU : UniformSpace \u03b1\nm : PseudoMetricSpace \u03b1\nH : \ud835\udce4 \u03b1 = \ud835\udce4 \u03b1\nx\u271d\u00b9 x\u271d : \u03b1\n\u22a2 dist x\u271d\u00b9 x\u271d = dist x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b3 : Type u_3\nU : TopologicalSpace \u03b3\nm : PseudoMetricSpace \u03b3\nH : U = UniformSpace.toTopologicalSpace\n\u22a2 replaceTopology m H = m\n[PROOFSTEP]\next\n[GOAL]\ncase h.dist.h.h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b3 : Type u_3\nU : TopologicalSpace \u03b3\nm : PseudoMetricSpace \u03b3\nH : U = UniformSpace.toTopologicalSpace\nx\u271d\u00b9 x\u271d : \u03b3\n\u22a2 dist x\u271d\u00b9 x\u271d = dist x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u\ne : PseudoEMetricSpace \u03b1\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\nedist_ne_top : \u2200 (x y : \u03b1), edist x y \u2260 \u22a4\nh : \u2200 (x y : \u03b1), dist x y = ENNReal.toReal (edist x y)\nx : \u03b1\n\u22a2 Dist.dist x x = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u\ne : PseudoEMetricSpace \u03b1\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\nedist_ne_top : \u2200 (x y : \u03b1), edist x y \u2260 \u22a4\nh : \u2200 (x y : \u03b1), dist x y = ENNReal.toReal (edist x y)\nx y : \u03b1\n\u22a2 Dist.dist x y = Dist.dist y x\n[PROOFSTEP]\nsimp [h, edist_comm]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u\ne : PseudoEMetricSpace \u03b1\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\nedist_ne_top : \u2200 (x y : \u03b1), edist x y \u2260 \u22a4\nh : \u2200 (x y : \u03b1), dist x y = ENNReal.toReal (edist x y)\nx y z : \u03b1\n\u22a2 Dist.dist x z \u2264 Dist.dist x y + Dist.dist y z\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u\ne : PseudoEMetricSpace \u03b1\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\nedist_ne_top : \u2200 (x y : \u03b1), edist x y \u2260 \u22a4\nh : \u2200 (x y : \u03b1), dist x y = ENNReal.toReal (edist x y)\nx y z : \u03b1\n\u22a2 ENNReal.toReal (edist x z) \u2264 ENNReal.toReal (edist x y) + ENNReal.toReal (edist y z)\n[PROOFSTEP]\nexact ENNReal.toReal_le_add (edist_triangle _ _ _) (edist_ne_top _ _) (edist_ne_top _ _)\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u\ne : PseudoEMetricSpace \u03b1\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\nedist_ne_top : \u2200 (x y : \u03b1), edist x y \u2260 \u22a4\nh : \u2200 (x y : \u03b1), dist x y = ENNReal.toReal (edist x y)\nx\u271d\u00b9 x\u271d : \u03b1\n\u22a2 edist x\u271d\u00b9 x\u271d = ENNReal.ofReal (Dist.dist x\u271d\u00b9 x\u271d)\n[PROOFSTEP]\nsimp only [h, ENNReal.ofReal_toReal (edist_ne_top _ _)]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u\ne : PseudoEMetricSpace \u03b1\ndist : \u03b1 \u2192 \u03b1 \u2192 \u211d\nedist_ne_top : \u2200 (x y : \u03b1), edist x y \u2260 \u22a4\nh : \u2200 (x y : \u03b1), dist x y = ENNReal.toReal (edist x y)\n\u22a2 \u2a05 (\u03b5 : \u211d\u22650\u221e) (_ : \u03b5 > 0), \ud835\udcdf {p | edist p.fst p.snd < \u03b5} = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | Dist.dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimpa only [ENNReal.coe_toNNReal (edist_ne_top _ _), h] using\n (Metric.uniformity_edist_aux fun x y : \u03b1 => (edist x y).toNNReal).symm\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type ?u.169779\nB : Bornology \u03b1\nm : PseudoMetricSpace \u03b1\nH : \u2200 (s : Set \u03b1), IsBounded s \u2194 IsBounded s\ns : Set \u03b1\n\u22a2 IsBounded s \u2194 s\u1d9c \u2208 {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\n[PROOFSTEP]\nrw [isBounded_iff, mem_setOf_eq, compl_compl]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nm : PseudoMetricSpace \u03b1\nB : Bornology \u03b1\nH : \u2200 (s : Set \u03b1), IsBounded s \u2194 IsBounded s\n\u22a2 replaceBornology m H = m\n[PROOFSTEP]\next\n[GOAL]\ncase h.dist.h.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nm : PseudoMetricSpace \u03b1\nB : Bornology \u03b1\nH : \u2200 (s : Set \u03b1), IsBounded s \u2194 IsBounded s\nx\u271d\u00b9 x\u271d : \u03b1\n\u22a2 dist x\u271d\u00b9 x\u271d = dist x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u22a2 \u2200 (x : \u211d), dist x x = 0\n[PROOFSTEP]\nsimp [abs_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u211d\n\u22a2 (fun x y => \u2191{ val := |x - y|, property := (_ : 0 \u2264 |x - y|) }) x y = ENNReal.ofReal (dist x y)\n[PROOFSTEP]\nexact ENNReal.coe_nnreal_eq _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u211d\n\u22a2 dist x 0 = |x|\n[PROOFSTEP]\nsimp [Real.dist_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u211d\nh : y \u2208 uIcc x z\n\u22a2 dist x y \u2264 dist x z\n[PROOFSTEP]\nsimpa only [dist_comm x] using abs_sub_left_of_mem_uIcc h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u211d\nh : y \u2208 uIcc x z\n\u22a2 dist y z \u2264 dist x z\n[PROOFSTEP]\nsimpa only [dist_comm _ z] using abs_sub_right_of_mem_uIcc h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y x' y' : \u211d\nhx : x \u2208 uIcc x' y'\nhy : y \u2208 uIcc x' y'\n\u22a2 y \u2208 uIcc y' x'\n[PROOFSTEP]\nrwa [uIcc_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y x' y' : \u211d\nhx : x \u2208 uIcc x' y'\nhy : y \u2208 uIcc x' y'\n\u22a2 x \u2208 uIcc y' x'\n[PROOFSTEP]\nrwa [uIcc_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y x' y' : \u211d\nhx : x \u2208 Icc x' y'\nhy : y \u2208 Icc x' y'\n\u22a2 dist x y \u2264 y' - x'\n[PROOFSTEP]\nsimpa only [Real.dist_eq, abs_of_nonpos (sub_nonpos.2 <| hx.1.trans hx.2), neg_sub] using\n Real.dist_le_of_mem_uIcc (Icc_subset_uIcc hx) (Icc_subset_uIcc hy)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u211d\nhx : x \u2208 Icc 0 1\nhy : y \u2208 Icc 0 1\n\u22a2 dist x y \u2264 1\n[PROOFSTEP]\nsimpa only [sub_zero] using Real.dist_le_of_mem_Icc hx hy\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u211d\n\u22a2 \ud835\udcdd x = \u2a05 (r : \u211d) (_ : r > 0), \ud835\udcdf {b | |x - b| < r}\n[PROOFSTEP]\nsimp only [nhds_basis_ball.eq_biInf, ball, Real.dist_eq, abs_sub_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx r y : \u211d\n\u22a2 y \u2208 ball x r \u2194 y \u2208 Ioo (x - r) (x + r)\n[PROOFSTEP]\nrw [mem_ball, dist_comm, Real.dist_eq, abs_sub_lt_iff, mem_Ioo, \u2190 sub_lt_iff_lt_add', sub_lt_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx r : \u211d\n\u22a2 closedBall x r = Icc (x - r) (x + r)\n[PROOFSTEP]\next y\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx r y : \u211d\n\u22a2 y \u2208 closedBall x r \u2194 y \u2208 Icc (x - r) (x + r)\n[PROOFSTEP]\nrw [mem_closedBall, dist_comm, Real.dist_eq, abs_sub_le_iff, mem_Icc, \u2190 sub_le_iff_le_add', sub_le_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u211d\n\u22a2 Ioo x y = ball ((x + y) / 2) ((y - x) / 2)\n[PROOFSTEP]\nrw [Real.ball_eq_Ioo, \u2190 sub_div, add_comm, \u2190 sub_add, add_sub_cancel', add_self_div_two, \u2190 add_div, add_assoc,\n add_sub_cancel'_right, add_self_div_two]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u211d\n\u22a2 Icc x y = closedBall ((x + y) / 2) ((y - x) / 2)\n[PROOFSTEP]\nrw [Real.closedBall_eq_Icc, \u2190 sub_div, add_comm, \u2190 sub_add, add_sub_cancel', add_self_div_two, \u2190 add_div, add_assoc,\n add_sub_cancel'_right, add_self_div_two]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u22a2 \ud835\udce4 \u03b1 = comap (fun p => dist p.fst p.snd) (\ud835\udcdd 0)\n[PROOFSTEP]\next s\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set (\u03b1 \u00d7 \u03b1)\n\u22a2 s \u2208 \ud835\udce4 \u03b1 \u2194 s \u2208 comap (fun p => dist p.fst p.snd) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp only [mem_uniformity_dist, (nhds_basis_ball.comap _).mem_iff]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set (\u03b1 \u00d7 \u03b1)\n\u22a2 (\u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 s) \u2194 \u2203 i, 0 < i \u2227 (fun p => dist p.fst p.snd) \u207b\u00b9' ball 0 i \u2286 s\n[PROOFSTEP]\nsimp [subset_def, Real.dist_0_eq_abs]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\n\u22a2 CauchySeq u \u2194 Tendsto (fun n => dist (u n.fst) (u n.snd)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [cauchySeq_iff_tendsto, Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\n\u22a2 Tendsto ((fun p => dist p.fst p.snd) \u2218 Prod.map u u) atTop (\ud835\udcdd 0) \u2194\n Tendsto (fun n => dist (u n.fst) (u n.snd)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u00d7 \u03b1\np : Filter \u03b9\n\u22a2 Tendsto f p (\ud835\udce4 \u03b1) \u2194 Tendsto (fun x => dist (f x).fst (f x).snd) p (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9\u271d : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u00d7 \u03b1\np : Filter \u03b9\n\u22a2 Tendsto ((fun p => dist p.fst p.snd) \u2218 f) p (\ud835\udcdd 0) \u2194 Tendsto (fun x => dist (f x).fst (f x).snd) p (\ud835\udcdd 0)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\nu : Set \u03b1\nhu : u \u2208 \ud835\udcdd x\n\u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd 0, closedBall x r \u2286 u\n[PROOFSTEP]\nobtain \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9 : \u2203 \u03b5, 0 < \u03b5 \u2227 closedBall x \u03b5 \u2286 u := nhds_basis_closedBall.mem_iff.1 hu\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\nu : Set \u03b1\nhu : u \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : closedBall x \u03b5 \u2286 u\n\u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd 0, closedBall x r \u2286 u\n[PROOFSTEP]\nhave : Iic \u03b5 \u2208 \ud835\udcdd (0 : \u211d) := Iic_mem_nhds \u03b5pos\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\nu : Set \u03b1\nhu : u \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : closedBall x \u03b5 \u2286 u\nthis : Iic \u03b5 \u2208 \ud835\udcdd 0\n\u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd 0, closedBall x r \u2286 u\n[PROOFSTEP]\nfilter_upwards [this] with _ hr using Subset.trans (closedBall_subset_closedBall hr) h\u03b5\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\n\u22a2 UniformCauchySeqOn F atTop s \u2194\n \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\n\u22a2 UniformCauchySeqOn F atTop s \u2192\n \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\n[PROOFSTEP]\nintro h \u03b5 h\u03b5\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\nh : UniformCauchySeqOn F atTop s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\n[PROOFSTEP]\nlet u := {a : \u03b1 \u00d7 \u03b1 | dist a.fst a.snd < \u03b5}\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\nh : UniformCauchySeqOn F atTop s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nu : Set (\u03b1 \u00d7 \u03b1) := {a | dist a.fst a.snd < \u03b5}\n\u22a2 \u2203 N, \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\n[PROOFSTEP]\nhave hu : u \u2208 \ud835\udce4 \u03b1 := Metric.mem_uniformity_dist.mpr \u27e8\u03b5, h\u03b5, by simp\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\nh : UniformCauchySeqOn F atTop s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nu : Set (\u03b1 \u00d7 \u03b1) := {a | dist a.fst a.snd < \u03b5}\n\u22a2 \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 u\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\nh : UniformCauchySeqOn F atTop s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nu : Set (\u03b1 \u00d7 \u03b1) := {a | dist a.fst a.snd < \u03b5}\nhu : u \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2203 N, \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\n[PROOFSTEP]\nrw [\u2190 @Filter.eventually_atTop_prod_self' _ _ _ fun m => \u2200 x \u2208 s, dist (F m.fst x) (F m.snd x) < \u03b5]\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\nh : UniformCauchySeqOn F atTop s\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nu : Set (\u03b1 \u00d7 \u03b1) := {a | dist a.fst a.snd < \u03b5}\nhu : u \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2200\u1da0 (x : \u03b2 \u00d7 \u03b2) in atTop, \u2200 (x_1 : \u03b3), x_1 \u2208 s \u2192 dist (F x.fst x_1) (F x.snd x_1) < \u03b5\n[PROOFSTEP]\nspecialize h u hu\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nu : Set (\u03b1 \u00d7 \u03b1) := {a | dist a.fst a.snd < \u03b5}\nhu : u \u2208 \ud835\udce4 \u03b1\nh : \u2200\u1da0 (m : \u03b2 \u00d7 \u03b2) in atTop \u00d7\u02e2 atTop, \u2200 (x : \u03b3), x \u2208 s \u2192 (F m.fst x, F m.snd x) \u2208 u\n\u22a2 \u2200\u1da0 (x : \u03b2 \u00d7 \u03b2) in atTop, \u2200 (x_1 : \u03b3), x_1 \u2208 s \u2192 dist (F x.fst x_1) (F x.snd x_1) < \u03b5\n[PROOFSTEP]\nrw [prod_atTop_atTop_eq] at h \n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nu : Set (\u03b1 \u00d7 \u03b1) := {a | dist a.fst a.snd < \u03b5}\nhu : u \u2208 \ud835\udce4 \u03b1\nh : \u2200\u1da0 (m : \u03b2 \u00d7 \u03b2) in atTop, \u2200 (x : \u03b3), x \u2208 s \u2192 (F m.fst x, F m.snd x) \u2208 u\n\u22a2 \u2200\u1da0 (x : \u03b2 \u00d7 \u03b2) in atTop, \u2200 (x_1 : \u03b3), x_1 \u2208 s \u2192 dist (F x.fst x_1) (F x.snd x_1) < \u03b5\n[PROOFSTEP]\nexact h.mono fun n h x hx => h x hx\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\n\u22a2 (\u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5) \u2192\n UniformCauchySeqOn F atTop s\n[PROOFSTEP]\nintro h u hu\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\nh : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2200\u1da0 (m : \u03b2 \u00d7 \u03b2) in atTop \u00d7\u02e2 atTop, \u2200 (x : \u03b3), x \u2208 s \u2192 (F m.fst x, F m.snd x) \u2208 u\n[PROOFSTEP]\nrcases Metric.mem_uniformity_dist.mp hu with \u27e8\u03b5, h\u03b5, hab\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\nh : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhab : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 u\n\u22a2 \u2200\u1da0 (m : \u03b2 \u00d7 \u03b2) in atTop \u00d7\u02e2 atTop, \u2200 (x : \u03b3), x \u2208 s \u2192 (F m.fst x, F m.snd x) \u2208 u\n[PROOFSTEP]\nrcases h \u03b5 h\u03b5 with \u27e8N, hN\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\nh : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhab : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 u\nN : \u03b2\nhN : \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\n\u22a2 \u2200\u1da0 (m : \u03b2 \u00d7 \u03b2) in atTop \u00d7\u02e2 atTop, \u2200 (x : \u03b3), x \u2208 s \u2192 (F m.fst x, F m.snd x) \u2208 u\n[PROOFSTEP]\nrw [prod_atTop_atTop_eq, eventually_atTop]\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\nh : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhab : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 u\nN : \u03b2\nhN : \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\n\u22a2 \u2203 a, \u2200 (b : \u03b2 \u00d7 \u03b2), b \u2265 a \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 (F b.fst x, F b.snd x) \u2208 u\n[PROOFSTEP]\nuse(N, N)\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\nh : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhab : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 u\nN : \u03b2\nhN : \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\n\u22a2 \u2200 (b : \u03b2 \u00d7 \u03b2), b \u2265 (N, N) \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 (F b.fst x, F b.snd x) \u2208 u\n[PROOFSTEP]\nintro b hb x hx\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\nh : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhab : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 u\nN : \u03b2\nhN : \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\nb : \u03b2 \u00d7 \u03b2\nhb : b \u2265 (N, N)\nx : \u03b3\nhx : x \u2208 s\n\u22a2 (F b.fst x, F b.snd x) \u2208 u\n[PROOFSTEP]\nrcases hb with \u27e8hbl, hbr\u27e9\n[GOAL]\ncase h.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\n\u03b3 : Type u_3\nF : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b3\nh : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhab : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 u\nN : \u03b2\nhN : \u2200 (m : \u03b2), m \u2265 N \u2192 \u2200 (n : \u03b2), n \u2265 N \u2192 \u2200 (x : \u03b3), x \u2208 s \u2192 dist (F m x) (F n x) < \u03b5\nb : \u03b2 \u00d7 \u03b2\nx : \u03b3\nhx : x \u2208 s\nhbl : (N, N).fst \u2264 b.fst\nhbr : (N, N).snd \u2264 b.snd\n\u22a2 (F b.fst x, F b.snd x) \u2208 u\n[PROOFSTEP]\nexact hab (hN b.fst hbl.ge b.snd hbr.ge x hx)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u2115 \u2192 \u03b1\nhu : CauchySeq u\n\u22a2 \u2203 R, R > 0 \u2227 \u2200 (m n : \u2115), dist (u m) (u n) < R\n[PROOFSTEP]\nrcases Metric.cauchySeq_iff'.1 hu 1 zero_lt_one with \u27e8N, hN\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u2115 \u2192 \u03b1\nhu : CauchySeq u\nN : \u2115\nhN : \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (u N) < 1\n\u22a2 \u2203 R, R > 0 \u2227 \u2200 (m n : \u2115), dist (u m) (u n) < R\n[PROOFSTEP]\nsuffices : \u2203 R > 0, \u2200 n, dist (u n) (u N) < R\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u2115 \u2192 \u03b1\nhu : CauchySeq u\nN : \u2115\nhN : \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (u N) < 1\nthis : \u2203 R, R > 0 \u2227 \u2200 (n : \u2115), dist (u n) (u N) < R\n\u22a2 \u2203 R, R > 0 \u2227 \u2200 (m n : \u2115), dist (u m) (u n) < R\n[PROOFSTEP]\nrcases this with \u27e8R, R0, H\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u2115 \u2192 \u03b1\nhu : CauchySeq u\nN : \u2115\nhN : \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (u N) < 1\nR : \u211d\nR0 : R > 0\nH : \u2200 (n : \u2115), dist (u n) (u N) < R\n\u22a2 \u2203 R, R > 0 \u2227 \u2200 (m n : \u2115), dist (u m) (u n) < R\n[PROOFSTEP]\nexact \u27e8_, add_pos R0 R0, fun m n => lt_of_le_of_lt (dist_triangle_right _ _ _) (add_lt_add (H m) (H n))\u27e9\n[GOAL]\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u2115 \u2192 \u03b1\nhu : CauchySeq u\nN : \u2115\nhN : \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (u N) < 1\n\u22a2 \u2203 R, R > 0 \u2227 \u2200 (n : \u2115), dist (u n) (u N) < R\n[PROOFSTEP]\nlet R := Finset.sup (Finset.range N) fun n => nndist (u n) (u N)\n[GOAL]\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u2115 \u2192 \u03b1\nhu : CauchySeq u\nN : \u2115\nhN : \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (u N) < 1\nR : \u211d\u22650 := Finset.sup (Finset.range N) fun n => nndist (u n) (u N)\n\u22a2 \u2203 R, R > 0 \u2227 \u2200 (n : \u2115), dist (u n) (u N) < R\n[PROOFSTEP]\nrefine' \u27e8\u2191R + 1, add_pos_of_nonneg_of_pos R.2 zero_lt_one, fun n => _\u27e9\n[GOAL]\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u2115 \u2192 \u03b1\nhu : CauchySeq u\nN : \u2115\nhN : \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (u N) < 1\nR : \u211d\u22650 := Finset.sup (Finset.range N) fun n => nndist (u n) (u N)\nn : \u2115\n\u22a2 dist (u n) (u N) < \u2191R + 1\n[PROOFSTEP]\ncases' le_or_lt N n with h h\n[GOAL]\ncase this.inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u2115 \u2192 \u03b1\nhu : CauchySeq u\nN : \u2115\nhN : \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (u N) < 1\nR : \u211d\u22650 := Finset.sup (Finset.range N) fun n => nndist (u n) (u N)\nn : \u2115\nh : N \u2264 n\n\u22a2 dist (u n) (u N) < \u2191R + 1\n[PROOFSTEP]\nexact lt_of_lt_of_le (hN _ h) (le_add_of_nonneg_left R.2)\n[GOAL]\ncase this.inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u2115 \u2192 \u03b1\nhu : CauchySeq u\nN : \u2115\nhN : \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (u N) < 1\nR : \u211d\u22650 := Finset.sup (Finset.range N) fun n => nndist (u n) (u N)\nn : \u2115\nh : n < N\n\u22a2 dist (u n) (u N) < \u2191R + 1\n[PROOFSTEP]\nhave : _ \u2264 R := Finset.le_sup (Finset.mem_range.2 h)\n[GOAL]\ncase this.inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u2115 \u2192 \u03b1\nhu : CauchySeq u\nN : \u2115\nhN : \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (u N) < 1\nR : \u211d\u22650 := Finset.sup (Finset.range N) fun n => nndist (u n) (u N)\nn : \u2115\nh : n < N\nthis : nndist (u n) (u N) \u2264 R\n\u22a2 dist (u n) (u N) < \u2191R + 1\n[PROOFSTEP]\nexact lt_of_le_of_lt this (lt_add_of_pos_right _ zero_lt_one)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\n\u22a2 \u2203 b, (\u2200 (n : \u2115), 0 \u2264 b n) \u2227 (\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (s n) (s m) \u2264 b N) \u2227 Tendsto b atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nlet S N := (fun p : \u2115 \u00d7 \u2115 => dist (s p.1) (s p.2)) '' {p | p.1 \u2265 N \u2227 p.2 \u2265 N}\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\nS : \u2115 \u2192 Set \u211d := fun N => (fun p => dist (s p.fst) (s p.snd)) '' {p | p.fst \u2265 N \u2227 p.snd \u2265 N}\n\u22a2 \u2203 b, (\u2200 (n : \u2115), 0 \u2264 b n) \u2227 (\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (s n) (s m) \u2264 b N) \u2227 Tendsto b atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave hS : \u2200 N, \u2203 x, \u2200 y \u2208 S N, y \u2264 x := by\n rcases cauchySeq_bdd hs with \u27e8R, -, hR\u27e9\n refine' fun N => \u27e8R, _\u27e9\n rintro _ \u27e8\u27e8m, n\u27e9, _, rfl\u27e9\n exact\n le_of_lt\n (hR m n)\n -- Prove that it bounds the distances of points in the Cauchy sequence\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\nS : \u2115 \u2192 Set \u211d := fun N => (fun p => dist (s p.fst) (s p.snd)) '' {p | p.fst \u2265 N \u2227 p.snd \u2265 N}\n\u22a2 \u2200 (N : \u2115), \u2203 x, \u2200 (y : \u211d), y \u2208 S N \u2192 y \u2264 x\n[PROOFSTEP]\nrcases cauchySeq_bdd hs with \u27e8R, -, hR\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\nS : \u2115 \u2192 Set \u211d := fun N => (fun p => dist (s p.fst) (s p.snd)) '' {p | p.fst \u2265 N \u2227 p.snd \u2265 N}\nR : \u211d\nhR : \u2200 (m n : \u2115), dist (s m) (s n) < R\n\u22a2 \u2200 (N : \u2115), \u2203 x, \u2200 (y : \u211d), y \u2208 S N \u2192 y \u2264 x\n[PROOFSTEP]\nrefine' fun N => \u27e8R, _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\nS : \u2115 \u2192 Set \u211d := fun N => (fun p => dist (s p.fst) (s p.snd)) '' {p | p.fst \u2265 N \u2227 p.snd \u2265 N}\nR : \u211d\nhR : \u2200 (m n : \u2115), dist (s m) (s n) < R\nN : \u2115\n\u22a2 \u2200 (y : \u211d), y \u2208 S N \u2192 y \u2264 R\n[PROOFSTEP]\nrintro _ \u27e8\u27e8m, n\u27e9, _, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\nS : \u2115 \u2192 Set \u211d := fun N => (fun p => dist (s p.fst) (s p.snd)) '' {p | p.fst \u2265 N \u2227 p.snd \u2265 N}\nR : \u211d\nhR : \u2200 (m n : \u2115), dist (s m) (s n) < R\nN m n : \u2115\nleft\u271d : (m, n) \u2208 {p | p.fst \u2265 N \u2227 p.snd \u2265 N}\n\u22a2 (fun p => dist (s p.fst) (s p.snd)) (m, n) \u2264 R\n[PROOFSTEP]\nexact\n le_of_lt\n (hR m n)\n -- Prove that it bounds the distances of points in the Cauchy sequence\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\nS : \u2115 \u2192 Set \u211d := fun N => (fun p => dist (s p.fst) (s p.snd)) '' {p | p.fst \u2265 N \u2227 p.snd \u2265 N}\nhS : \u2200 (N : \u2115), \u2203 x, \u2200 (y : \u211d), y \u2208 S N \u2192 y \u2264 x\n\u22a2 \u2203 b, (\u2200 (n : \u2115), 0 \u2264 b n) \u2227 (\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (s n) (s m) \u2264 b N) \u2227 Tendsto b atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave ub : \u2200 m n N, N \u2264 m \u2192 N \u2264 n \u2192 dist (s m) (s n) \u2264 sSup (S N) := fun m n N hm hn =>\n le_csSup (hS N) \u27e8\u27e8_, _\u27e9, \u27e8hm, hn\u27e9, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\nS : \u2115 \u2192 Set \u211d := fun N => (fun p => dist (s p.fst) (s p.snd)) '' {p | p.fst \u2265 N \u2227 p.snd \u2265 N}\nhS : \u2200 (N : \u2115), \u2203 x, \u2200 (y : \u211d), y \u2208 S N \u2192 y \u2264 x\nub : \u2200 (m n N : \u2115), N \u2264 m \u2192 N \u2264 n \u2192 dist (s m) (s n) \u2264 sSup (S N)\n\u22a2 \u2203 b, (\u2200 (n : \u2115), 0 \u2264 b n) \u2227 (\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (s n) (s m) \u2264 b N) \u2227 Tendsto b atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave S0m : \u2200 n, (0 : \u211d) \u2208 S n := fun n => \u27e8\u27e8n, n\u27e9, \u27e8le_rfl, le_rfl\u27e9, dist_self _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\nS : \u2115 \u2192 Set \u211d := fun N => (fun p => dist (s p.fst) (s p.snd)) '' {p | p.fst \u2265 N \u2227 p.snd \u2265 N}\nhS : \u2200 (N : \u2115), \u2203 x, \u2200 (y : \u211d), y \u2208 S N \u2192 y \u2264 x\nub : \u2200 (m n N : \u2115), N \u2264 m \u2192 N \u2264 n \u2192 dist (s m) (s n) \u2264 sSup (S N)\nS0m : \u2200 (n : \u2115), 0 \u2208 S n\n\u22a2 \u2203 b, (\u2200 (n : \u2115), 0 \u2264 b n) \u2227 (\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (s n) (s m) \u2264 b N) \u2227 Tendsto b atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave S0 := fun n =>\n le_csSup (hS n)\n (S0m n)\n -- Prove that it tends to `0`, by using the Cauchy property of `s`\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\nS : \u2115 \u2192 Set \u211d := fun N => (fun p => dist (s p.fst) (s p.snd)) '' {p | p.fst \u2265 N \u2227 p.snd \u2265 N}\nhS : \u2200 (N : \u2115), \u2203 x, \u2200 (y : \u211d), y \u2208 S N \u2192 y \u2264 x\nub : \u2200 (m n N : \u2115), N \u2264 m \u2192 N \u2264 n \u2192 dist (s m) (s n) \u2264 sSup (S N)\nS0m : \u2200 (n : \u2115), 0 \u2208 S n\nS0 : \u2200 (n : \u2115), 0 \u2264 sSup (S n)\n\u22a2 \u2203 b, (\u2200 (n : \u2115), 0 \u2264 b n) \u2227 (\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (s n) (s m) \u2264 b N) \u2227 Tendsto b atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' \u27e8fun N => sSup (S N), S0, ub, Metric.tendsto_atTop.2 fun \u03b5 \u03b50 => _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\nS : \u2115 \u2192 Set \u211d := fun N => (fun p => dist (s p.fst) (s p.snd)) '' {p | p.fst \u2265 N \u2227 p.snd \u2265 N}\nhS : \u2200 (N : \u2115), \u2203 x, \u2200 (y : \u211d), y \u2208 S N \u2192 y \u2264 x\nub : \u2200 (m n N : \u2115), N \u2264 m \u2192 N \u2264 n \u2192 dist (s m) (s n) \u2264 sSup (S N)\nS0m : \u2200 (n : \u2115), 0 \u2208 S n\nS0 : \u2200 (n : \u2115), 0 \u2264 sSup (S n)\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (sSup (S n)) 0 < \u03b5\n[PROOFSTEP]\nrefine' (Metric.cauchySeq_iff.1 hs (\u03b5 / 2) (half_pos \u03b50)).imp fun N hN n hn => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\nS : \u2115 \u2192 Set \u211d := fun N => (fun p => dist (s p.fst) (s p.snd)) '' {p | p.fst \u2265 N \u2227 p.snd \u2265 N}\nhS : \u2200 (N : \u2115), \u2203 x, \u2200 (y : \u211d), y \u2208 S N \u2192 y \u2264 x\nub : \u2200 (m n N : \u2115), N \u2264 m \u2192 N \u2264 n \u2192 dist (s m) (s n) \u2264 sSup (S N)\nS0m : \u2200 (n : \u2115), 0 \u2208 S n\nS0 : \u2200 (n : \u2115), 0 \u2264 sSup (S n)\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 dist (s m) (s n) < \u03b5 / 2\nn : \u2115\nhn : n \u2265 N\n\u22a2 dist (sSup (S n)) 0 < \u03b5\n[PROOFSTEP]\nrw [Real.dist_0_eq_abs, abs_of_nonneg (S0 n)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\nS : \u2115 \u2192 Set \u211d := fun N => (fun p => dist (s p.fst) (s p.snd)) '' {p | p.fst \u2265 N \u2227 p.snd \u2265 N}\nhS : \u2200 (N : \u2115), \u2203 x, \u2200 (y : \u211d), y \u2208 S N \u2192 y \u2264 x\nub : \u2200 (m n N : \u2115), N \u2264 m \u2192 N \u2264 n \u2192 dist (s m) (s n) \u2264 sSup (S N)\nS0m : \u2200 (n : \u2115), 0 \u2208 S n\nS0 : \u2200 (n : \u2115), 0 \u2264 sSup (S n)\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 dist (s m) (s n) < \u03b5 / 2\nn : \u2115\nhn : n \u2265 N\n\u22a2 sSup (S n) < \u03b5\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (csSup_le \u27e8_, S0m _\u27e9 _) (half_lt_self \u03b50)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\nS : \u2115 \u2192 Set \u211d := fun N => (fun p => dist (s p.fst) (s p.snd)) '' {p | p.fst \u2265 N \u2227 p.snd \u2265 N}\nhS : \u2200 (N : \u2115), \u2203 x, \u2200 (y : \u211d), y \u2208 S N \u2192 y \u2264 x\nub : \u2200 (m n N : \u2115), N \u2264 m \u2192 N \u2264 n \u2192 dist (s m) (s n) \u2264 sSup (S N)\nS0m : \u2200 (n : \u2115), 0 \u2208 S n\nS0 : \u2200 (n : \u2115), 0 \u2264 sSup (S n)\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 dist (s m) (s n) < \u03b5 / 2\nn : \u2115\nhn : n \u2265 N\n\u22a2 \u2200 (b : \u211d), b \u2208 S n \u2192 b \u2264 \u03b5 / 2\n[PROOFSTEP]\nrintro _ \u27e8\u27e8m', n'\u27e9, \u27e8hm', hn'\u27e9, rfl\u27e9\n[GOAL]\ncase intro.mk.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\nS : \u2115 \u2192 Set \u211d := fun N => (fun p => dist (s p.fst) (s p.snd)) '' {p | p.fst \u2265 N \u2227 p.snd \u2265 N}\nhS : \u2200 (N : \u2115), \u2203 x, \u2200 (y : \u211d), y \u2208 S N \u2192 y \u2264 x\nub : \u2200 (m n N : \u2115), N \u2264 m \u2192 N \u2264 n \u2192 dist (s m) (s n) \u2264 sSup (S N)\nS0m : \u2200 (n : \u2115), 0 \u2208 S n\nS0 : \u2200 (n : \u2115), 0 \u2264 sSup (S n)\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 dist (s m) (s n) < \u03b5 / 2\nn : \u2115\nhn : n \u2265 N\nm' n' : \u2115\nhm' : (m', n').fst \u2265 n\nhn' : (m', n').snd \u2265 n\n\u22a2 (fun p => dist (s p.fst) (s p.snd)) (m', n') \u2264 \u03b5 / 2\n[PROOFSTEP]\nexact le_of_lt (hN _ (le_trans hn hm') _ (le_trans hn hn'))\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type ?u.236042\n\u03b2 : Type ?u.236039\nf : \u03b1 \u2192 \u03b2\nm : PseudoMetricSpace \u03b2\ns : Set \u03b1\n\u22a2 (f '' s\u1d9c)\u1d9c \u2208 cobounded \u03b2 \u2194 s \u2208 {s | \u2203 C, \u2200 (x : \u03b1), x \u2208 s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 s\u1d9c \u2192 dist x y \u2264 C}\n[PROOFSTEP]\nsimp only [\u2190 isBounded_def, isBounded_iff, ball_image_iff, mem_setOf]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\na b x\u271d : \u211d\u22650\n\u22a2 nndist a b \u2264 x\u271d \u2194 max (a - b) (b - a) \u2264 x\u271d\n[PROOFSTEP]\nsimp only [\u2190 NNReal.coe_le_coe, coe_nndist, dist_eq, max_le_iff, abs_sub_le_iff, tsub_le_iff_right, NNReal.coe_add]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nz : \u211d\u22650\n\u22a2 nndist 0 z = z\n[PROOFSTEP]\nsimp only [NNReal.nndist_eq, max_eq_right, tsub_zero, zero_tsub, zero_le']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nz : \u211d\u22650\n\u22a2 nndist z 0 = z\n[PROOFSTEP]\nrw [nndist_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nz : \u211d\u22650\n\u22a2 nndist 0 z = z\n[PROOFSTEP]\nexact NNReal.nndist_zero_eq_val z\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\na b : \u211d\u22650\n\u22a2 a \u2264 b + nndist a b\n[PROOFSTEP]\nsuffices (a : \u211d) \u2264 (b : \u211d) + dist a b by rwa [\u2190 NNReal.coe_le_coe, NNReal.coe_add, coe_nndist]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\na b : \u211d\u22650\nthis : \u2191a \u2264 \u2191b + dist a b\n\u22a2 a \u2264 b + nndist a b\n[PROOFSTEP]\nrwa [\u2190 NNReal.coe_le_coe, NNReal.coe_add, coe_nndist]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\na b : \u211d\u22650\n\u22a2 \u2191a \u2264 \u2191b + dist a b\n[PROOFSTEP]\nrw [\u2190 sub_le_iff_le_add']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\na b : \u211d\u22650\n\u22a2 \u2191a - \u2191b \u2264 dist a b\n[PROOFSTEP]\nexact le_of_abs_le (dist_eq a b).ge\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 (fun x y => dist x.fst y.fst \u2294 dist x.snd y.snd) x y = ENNReal.toReal (edist x y)\n[PROOFSTEP]\nsimp only [sup_eq_max, dist_edist, \u2190 ENNReal.toReal_max (edist_ne_top _ _) (edist_ne_top _ _), Prod.edist_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ni : PseudoMetricSpace (\u03b1 \u00d7 \u03b2) :=\n PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun x y => dist x.fst y.fst \u2294 dist x.snd y.snd)\n (_ : \u2200 (x y : \u03b1 \u00d7 \u03b2), max (edist x.fst y.fst) (edist x.snd y.snd) \u2260 \u22a4)\n (_ :\n \u2200 (x y : \u03b1 \u00d7 \u03b2),\n dist x.fst y.fst \u2294 dist x.snd y.snd = ENNReal.toReal (max (edist x.fst y.fst) (edist x.snd y.snd)))\ns : Set (\u03b1 \u00d7 \u03b2)\n\u22a2 IsBounded s \u2194 IsBounded s\n[PROOFSTEP]\nsimp only [\u2190 isBounded_image_fst_and_snd, isBounded_iff_eventually, ball_image_iff, \u2190 eventually_and, \u2190 forall_and, \u2190\n max_le_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ni : PseudoMetricSpace (\u03b1 \u00d7 \u03b2) :=\n PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun x y => dist x.fst y.fst \u2294 dist x.snd y.snd)\n (_ : \u2200 (x y : \u03b1 \u00d7 \u03b2), max (edist x.fst y.fst) (edist x.snd y.snd) \u2260 \u22a4)\n (_ :\n \u2200 (x y : \u03b1 \u00d7 \u03b2),\n dist x.fst y.fst \u2294 dist x.snd y.snd = ENNReal.toReal (max (edist x.fst y.fst) (edist x.snd y.snd)))\ns : Set (\u03b1 \u00d7 \u03b2)\n\u22a2 (\u2200\u1da0 (x : \u211d) in atTop,\n \u2200 (x_1 : \u03b1 \u00d7 \u03b2), x_1 \u2208 s \u2192 \u2200 (x_3 : \u03b1 \u00d7 \u03b2), x_3 \u2208 s \u2192 max (dist x_1.fst x_3.fst) (dist x_1.snd x_3.snd) \u2264 x) \u2194\n \u2200\u1da0 (C : \u211d) in atTop, \u2200 \u2983x : \u03b1 \u00d7 \u03b2\u2984, x \u2208 s \u2192 \u2200 \u2983y : \u03b1 \u00d7 \u03b2\u2984, y \u2208 s \u2192 dist x y \u2264 C\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1\ny\u2081 y\u2082 : \u03b2\n\u22a2 dist (x, y\u2081) (x, y\u2082) = dist y\u2081 y\u2082\n[PROOFSTEP]\nsimp [Prod.dist_eq, dist_nonneg]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx\u2081 x\u2082 : \u03b1\ny : \u03b2\n\u22a2 dist (x\u2081, y) (x\u2082, y) = dist x\u2081 x\u2082\n[PROOFSTEP]\nsimp [Prod.dist_eq, dist_nonneg]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1\ny : \u03b2\nr : \u211d\nz : \u03b1 \u00d7 \u03b2\n\u22a2 z \u2208 ball x r \u00d7\u02e2 ball y r \u2194 z \u2208 ball (x, y) r\n[PROOFSTEP]\nsimp [Prod.dist_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1\ny : \u03b2\nr : \u211d\nz : \u03b1 \u00d7 \u03b2\n\u22a2 z \u2208 closedBall x r \u00d7\u02e2 closedBall y r \u2194 z \u2208 closedBall (x, y) r\n[PROOFSTEP]\nsimp [Prod.dist_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\n\u22a2 sphere x r = sphere x.fst r \u00d7\u02e2 closedBall x.snd r \u222a closedBall x.fst r \u00d7\u02e2 sphere x.snd r\n[PROOFSTEP]\nobtain hr | rfl | hr := lt_trichotomy r 0\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : r < 0\n\u22a2 sphere x r = sphere x.fst r \u00d7\u02e2 closedBall x.snd r \u222a closedBall x.fst r \u00d7\u02e2 sphere x.snd r\n[PROOFSTEP]\nsimp [hr]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 sphere x 0 = sphere x.fst 0 \u00d7\u02e2 closedBall x.snd 0 \u222a closedBall x.fst 0 \u00d7\u02e2 sphere x.snd 0\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inr.inl.mk\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nfst\u271d : \u03b1\nsnd\u271d : \u03b2\n\u22a2 sphere (fst\u271d, snd\u271d) 0 =\n sphere (fst\u271d, snd\u271d).fst 0 \u00d7\u02e2 closedBall (fst\u271d, snd\u271d).snd 0 \u222a\n closedBall (fst\u271d, snd\u271d).fst 0 \u00d7\u02e2 sphere (fst\u271d, snd\u271d).snd 0\n[PROOFSTEP]\nsimp_rw [\u2190 closedBall_eq_sphere_of_nonpos le_rfl, union_self, closedBall_prod_same]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\n\u22a2 sphere x r = sphere x.fst r \u00d7\u02e2 closedBall x.snd r \u222a closedBall x.fst r \u00d7\u02e2 sphere x.snd r\n[PROOFSTEP]\next \u27e8x', y'\u27e9\n[GOAL]\ncase inr.inr.h.mk\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\nx' : \u03b1\ny' : \u03b2\n\u22a2 (x', y') \u2208 sphere x r \u2194 (x', y') \u2208 sphere x.fst r \u00d7\u02e2 closedBall x.snd r \u222a closedBall x.fst r \u00d7\u02e2 sphere x.snd r\n[PROOFSTEP]\nsimp_rw [Set.mem_union, Set.mem_prod, Metric.mem_closedBall, Metric.mem_sphere, Prod.dist_eq, max_eq_iff]\n[GOAL]\ncase inr.inr.h.mk\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\nx' : \u03b1\ny' : \u03b2\n\u22a2 dist x' x.fst = r \u2227 dist y' x.snd \u2264 dist x' x.fst \u2228 dist y' x.snd = r \u2227 dist x' x.fst \u2264 dist y' x.snd \u2194\n dist x' x.fst = r \u2227 dist y' x.snd \u2264 r \u2228 dist x' x.fst \u2264 r \u2227 dist y' x.snd = r\n[PROOFSTEP]\nrefine' or_congr (and_congr_right _) (and_comm.trans (and_congr_left _))\n[GOAL]\ncase inr.inr.h.mk.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\nx' : \u03b1\ny' : \u03b2\n\u22a2 dist x' x.fst = r \u2192 (dist y' x.snd \u2264 dist x' x.fst \u2194 dist y' x.snd \u2264 r)\ncase inr.inr.h.mk.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\nx' : \u03b1\ny' : \u03b2\n\u22a2 dist y' x.snd = r \u2192 (dist x' x.fst \u2264 dist y' x.snd \u2194 dist x' x.fst \u2264 r)\n[PROOFSTEP]\nall_goals rintro rfl; rfl\n[GOAL]\ncase inr.inr.h.mk.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\nx' : \u03b1\ny' : \u03b2\n\u22a2 dist x' x.fst = r \u2192 (dist y' x.snd \u2264 dist x' x.fst \u2194 dist y' x.snd \u2264 r)\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase inr.inr.h.mk.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nx' : \u03b1\ny' : \u03b2\nhr : 0 < dist x' x.fst\n\u22a2 dist y' x.snd \u2264 dist x' x.fst \u2194 dist y' x.snd \u2264 dist x' x.fst\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr.inr.h.mk.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\nx' : \u03b1\ny' : \u03b2\n\u22a2 dist y' x.snd = r \u2192 (dist x' x.fst \u2264 dist y' x.snd \u2194 dist x' x.fst \u2264 r)\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase inr.inr.h.mk.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nx' : \u03b1\ny' : \u03b2\nhr : 0 < dist y' x.snd\n\u22a2 dist x' x.fst \u2264 dist y' x.snd \u2194 dist x' x.fst \u2264 dist y' x.snd\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u22a2 dist (dist x y) (dist x z) \u2264 dist y z\n[PROOFSTEP]\nsimpa only [dist_comm x] using dist_dist_dist_le_left y z x\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\na : \u03b1\n\u22a2 comap (fun x => dist x a) (\ud835\udcdd 0) = \ud835\udcdd a\n[PROOFSTEP]\nsimp only [@nhds_eq_comap_uniformity \u03b1, Metric.uniformity_eq_comap_nhds_zero, comap_comap, (\u00b7 \u2218 \u00b7), dist_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b2 \u2192 \u03b1\nx : Filter \u03b2\na : \u03b1\n\u22a2 Tendsto f x (\ud835\udcdd a) \u2194 Tendsto (fun b => dist (f b) a) x (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 nhds_comap_dist a, tendsto_comap_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b2 \u2192 \u03b1\nx : Filter \u03b2\na : \u03b1\n\u22a2 Tendsto ((fun x => dist x a) \u2218 f) x (\ud835\udcdd 0) \u2194 Tendsto (fun b => dist (f b) a) x (\ud835\udcdd 0)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\na : \u03b1\n\u22a2 (\u2200 (i : \u211d), 0 < i \u2192 \u2203 y, y \u2208 s \u2227 y \u2208 ball a i) \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 b, b \u2208 s \u2227 dist a b < \u03b5\n[PROOFSTEP]\nsimp only [mem_ball, dist_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ne : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 a \u2208 closure (range e) \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 k, dist a (e k) < \u03b5\n[PROOFSTEP]\nsimp only [mem_closure_iff, exists_range_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ne : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200 (i : \u2115), True \u2192 \u2203 y, y \u2208 range e \u2227 y \u2208 ball a (1 / (\u2191i + 1))) \u2194 \u2200 (n : \u2115), \u2203 k, dist a (e k) < 1 / (\u2191n + 1)\n[PROOFSTEP]\nsimp only [mem_ball, dist_comm, exists_range_iff, forall_const]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nhs : IsClosed s\na : \u03b1\n\u22a2 a \u2208 s \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 b, b \u2208 s \u2227 dist a b < \u03b5\n[PROOFSTEP]\nsimpa only [hs.closure_eq] using @mem_closure_iff _ _ s a\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d s : Set \u03b1\nx : \u03b1\n\u22a2 x \u2208 closure s \u2194 \u2200 (r : \u211d), r > 0 \u2192 Set.Nonempty (ball x r \u2229 s)\n[PROOFSTEP]\nsimp only [mem_closure_iff, Set.Nonempty, exists_prop, mem_inter_iff, mem_ball', and_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\nf : \u03b2 \u2192 \u03b1\nx : \u03b1\n\u22a2 x \u2208 closure (range f) \u2194 \u2200 (r : \u211d), r > 0 \u2192 \u2203 y, dist x (f y) < r\n[PROOFSTEP]\nsimp only [mem_closure_iff, exists_range_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsSeparable s\n\u22a2 IsSeparable (f \u207b\u00b9' s)\n[PROOFSTEP]\nhave : SeparableSpace s := hs.separableSpace\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsSeparable s\nthis : SeparableSpace \u2191s\n\u22a2 IsSeparable (f \u207b\u00b9' s)\n[PROOFSTEP]\nhave : SecondCountableTopology s := UniformSpace.secondCountable_of_separable _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsSeparable s\nthis\u271d : SeparableSpace \u2191s\nthis : SecondCountableTopology \u2191s\n\u22a2 IsSeparable (f \u207b\u00b9' s)\n[PROOFSTEP]\nhave : Inducing ((mapsTo_preimage f s).restrict _ _ _) := (hf.comp inducing_subtype_val).codRestrict _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsSeparable s\nthis\u271d\u00b9 : SeparableSpace \u2191s\nthis\u271d : SecondCountableTopology \u2191s\nthis : Inducing (MapsTo.restrict f (f \u207b\u00b9' s) s (_ : MapsTo f (f \u207b\u00b9' s) s))\n\u22a2 IsSeparable (f \u207b\u00b9' s)\n[PROOFSTEP]\nhave := this.secondCountableTopology\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsSeparable s\nthis\u271d\u00b2 : SeparableSpace \u2191s\nthis\u271d\u00b9 : SecondCountableTopology \u2191s\nthis\u271d : Inducing (MapsTo.restrict f (f \u207b\u00b9' s) s (_ : MapsTo f (f \u207b\u00b9' s) s))\nthis : SecondCountableTopology \u2191(f \u207b\u00b9' s)\n\u22a2 IsSeparable (f \u207b\u00b9' s)\n[PROOFSTEP]\nexact isSeparable_of_separableSpace_subtype _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhf : ContinuousOn f s\nhs : IsSeparable s\n\u22a2 IsSeparable (f '' s)\n[PROOFSTEP]\nrw [image_eq_range, \u2190 image_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhf : ContinuousOn f s\nhs : IsSeparable s\n\u22a2 IsSeparable ((fun x => f \u2191x) '' univ)\n[PROOFSTEP]\nexact (isSeparable_univ_iff.2 hs.separableSpace).image hf.restrict\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\n\u22a2 PseudoMetricSpace ((b : \u03b2) \u2192 \u03c0 b)\n[PROOFSTEP]\nlet i :=\n PseudoEMetricSpace.toPseudoMetricSpaceOfDist\n (fun f g : \u2200 b, \u03c0 b => ((sup univ fun b => nndist (f b) (g b) : \u211d\u22650) : \u211d))\n (fun f g => ((Finset.sup_lt_iff bot_lt_top).2 fun b _ => edist_lt_top _ _).ne)\n (fun f g => by simp only [edist_pi_def, edist_nndist, \u2190 ENNReal.coe_finset_sup, ENNReal.coe_toReal])\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\n\u22a2 (fun f g => \u2191(sup Finset.univ fun b => nndist (f b) (g b))) f g = ENNReal.toReal (edist f g)\n[PROOFSTEP]\nsimp only [edist_pi_def, edist_nndist, \u2190 ENNReal.coe_finset_sup, ENNReal.coe_toReal]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ni : PseudoMetricSpace ((b : \u03b2) \u2192 \u03c0 b) :=\n PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun f g => \u2191(sup Finset.univ fun b => nndist (f b) (g b)))\n (_ : \u2200 (f g : (b : \u03b2) \u2192 \u03c0 b), (sup Finset.univ fun b => edist (f b) (g b)) \u2260 \u22a4)\n (_ :\n \u2200 (f g : (b : \u03b2) \u2192 \u03c0 b),\n \u2191(sup Finset.univ fun b => nndist (f b) (g b)) = ENNReal.toReal (sup Finset.univ fun x => edist (f x) (g x)))\n\u22a2 PseudoMetricSpace ((b : \u03b2) \u2192 \u03c0 b)\n[PROOFSTEP]\nrefine i.replaceBornology fun s => ?_\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ni : PseudoMetricSpace ((b : \u03b2) \u2192 \u03c0 b) :=\n PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun f g => \u2191(sup Finset.univ fun b => nndist (f b) (g b)))\n (_ : \u2200 (f g : (b : \u03b2) \u2192 \u03c0 b), (sup Finset.univ fun b => edist (f b) (g b)) \u2260 \u22a4)\n (_ :\n \u2200 (f g : (b : \u03b2) \u2192 \u03c0 b),\n \u2191(sup Finset.univ fun b => nndist (f b) (g b)) = ENNReal.toReal (sup Finset.univ fun x => edist (f x) (g x)))\ns : Set ((b : \u03b2) \u2192 \u03c0 b)\n\u22a2 IsBounded s \u2194 IsBounded s\n[PROOFSTEP]\nsimp only [\u2190 isBounded_def, isBounded_iff_eventually, \u2190 forall_isBounded_image_eval_iff, ball_image_iff, \u2190\n Filter.eventually_all, Function.eval_apply, @dist_nndist (\u03c0 _)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ni : PseudoMetricSpace ((b : \u03b2) \u2192 \u03c0 b) :=\n PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun f g => \u2191(sup Finset.univ fun b => nndist (f b) (g b)))\n (_ : \u2200 (f g : (b : \u03b2) \u2192 \u03c0 b), (sup Finset.univ fun b => edist (f b) (g b)) \u2260 \u22a4)\n (_ :\n \u2200 (f g : (b : \u03b2) \u2192 \u03c0 b),\n \u2191(sup Finset.univ fun b => nndist (f b) (g b)) = ENNReal.toReal (sup Finset.univ fun x => edist (f x) (g x)))\ns : Set ((b : \u03b2) \u2192 \u03c0 b)\n\u22a2 (\u2200\u1da0 (x : \u211d) in atTop,\n \u2200 (i : \u03b2) (x_1 : (x : \u03b2) \u2192 \u03c0 x),\n x_1 \u2208 s \u2192 \u2200 (x_2 : (x : \u03b2) \u2192 \u03c0 x), x_2 \u2208 s \u2192 \u2191(nndist (Function.eval i x_1) (Function.eval i x_2)) \u2264 x) \u2194\n \u2200\u1da0 (C : \u211d) in atTop, \u2200 \u2983x : (b : \u03b2) \u2192 \u03c0 b\u2984, x \u2208 s \u2192 \u2200 \u2983y : (b : \u03b2) \u2192 \u03c0 b\u2984, y \u2208 s \u2192 dist x y \u2264 C\n[PROOFSTEP]\nrefine' eventually_congr ((eventually_ge_atTop 0).mono fun C hC => _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ni : PseudoMetricSpace ((b : \u03b2) \u2192 \u03c0 b) :=\n PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun f g => \u2191(sup Finset.univ fun b => nndist (f b) (g b)))\n (_ : \u2200 (f g : (b : \u03b2) \u2192 \u03c0 b), (sup Finset.univ fun b => edist (f b) (g b)) \u2260 \u22a4)\n (_ :\n \u2200 (f g : (b : \u03b2) \u2192 \u03c0 b),\n \u2191(sup Finset.univ fun b => nndist (f b) (g b)) = ENNReal.toReal (sup Finset.univ fun x => edist (f x) (g x)))\ns : Set ((b : \u03b2) \u2192 \u03c0 b)\nC : \u211d\nhC : 0 \u2264 C\n\u22a2 (\u2200 (i : \u03b2) (x : (x : \u03b2) \u2192 \u03c0 x),\n x \u2208 s \u2192 \u2200 (x_1 : (x : \u03b2) \u2192 \u03c0 x), x_1 \u2208 s \u2192 \u2191(nndist (Function.eval i x) (Function.eval i x_1)) \u2264 C) \u2194\n \u2200 \u2983x : (b : \u03b2) \u2192 \u03c0 b\u2984, x \u2208 s \u2192 \u2200 \u2983y : (b : \u03b2) \u2192 \u03c0 b\u2984, y \u2208 s \u2192 dist x y \u2264 C\n[PROOFSTEP]\nlift C to \u211d\u22650 using hC\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ni : PseudoMetricSpace ((b : \u03b2) \u2192 \u03c0 b) :=\n PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun f g => \u2191(sup Finset.univ fun b => nndist (f b) (g b)))\n (_ : \u2200 (f g : (b : \u03b2) \u2192 \u03c0 b), (sup Finset.univ fun b => edist (f b) (g b)) \u2260 \u22a4)\n (_ :\n \u2200 (f g : (b : \u03b2) \u2192 \u03c0 b),\n \u2191(sup Finset.univ fun b => nndist (f b) (g b)) = ENNReal.toReal (sup Finset.univ fun x => edist (f x) (g x)))\ns : Set ((b : \u03b2) \u2192 \u03c0 b)\nC : \u211d\u22650\n\u22a2 (\u2200 (i : \u03b2) (x : (x : \u03b2) \u2192 \u03c0 x),\n x \u2208 s \u2192 \u2200 (x_1 : (x : \u03b2) \u2192 \u03c0 x), x_1 \u2208 s \u2192 \u2191(nndist (Function.eval i x) (Function.eval i x_1)) \u2264 \u2191C) \u2194\n \u2200 \u2983x : (b : \u03b2) \u2192 \u03c0 b\u2984, x \u2208 s \u2192 \u2200 \u2983y : (b : \u03b2) \u2192 \u03c0 b\u2984, y \u2208 s \u2192 dist x y \u2264 \u2191C\n[PROOFSTEP]\nrefine'\n \u27e8fun H x hx y hy => NNReal.coe_le_coe.2 <| Finset.sup_le fun b _ => H b x hx y hy, fun H b x hx y hy =>\n NNReal.coe_le_coe.2 _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ni : PseudoMetricSpace ((b : \u03b2) \u2192 \u03c0 b) :=\n PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun f g => \u2191(sup Finset.univ fun b => nndist (f b) (g b)))\n (_ : \u2200 (f g : (b : \u03b2) \u2192 \u03c0 b), (sup Finset.univ fun b => edist (f b) (g b)) \u2260 \u22a4)\n (_ :\n \u2200 (f g : (b : \u03b2) \u2192 \u03c0 b),\n \u2191(sup Finset.univ fun b => nndist (f b) (g b)) = ENNReal.toReal (sup Finset.univ fun x => edist (f x) (g x)))\ns : Set ((b : \u03b2) \u2192 \u03c0 b)\nC : \u211d\u22650\nH : \u2200 \u2983x : (b : \u03b2) \u2192 \u03c0 b\u2984, x \u2208 s \u2192 \u2200 \u2983y : (b : \u03b2) \u2192 \u03c0 b\u2984, y \u2208 s \u2192 dist x y \u2264 \u2191C\nb : \u03b2\nx : (x : \u03b2) \u2192 \u03c0 x\nhx : x \u2208 s\ny : (x : \u03b2) \u2192 \u03c0 x\nhy : y \u2208 s\n\u22a2 nndist (Function.eval b x) (Function.eval b y) \u2264 C\n[PROOFSTEP]\nsimpa only using Finset.sup_le_iff.1 (NNReal.coe_le_coe.1 <| H hx hy) b (Finset.mem_univ b)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\u22650\n\u22a2 nndist f g \u2264 r \u2194 \u2200 (b : \u03b2), nndist (f b) (g b) \u2264 r\n[PROOFSTEP]\nsimp [nndist_pi_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\u22650\nhr : 0 < r\n\u22a2 nndist f g < r \u2194 \u2200 (b : \u03b2), nndist (f b) (g b) < r\n[PROOFSTEP]\nsimp [nndist_pi_def, Finset.sup_lt_iff (show \u22a5 < r from hr)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\u22650\nhr : 0 < r\n\u22a2 nndist f g = r \u2194 (\u2203 i, nndist (f i) (g i) = r) \u2227 \u2200 (b : \u03b2), nndist (f b) (g b) \u2264 r\n[PROOFSTEP]\nrw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, not_forall, and_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\u22650\nhr : 0 < r\n\u22a2 ((\u2203 x, \u00acnndist (f x) (g x) < r) \u2227 \u2200 (b : \u03b2), nndist (f b) (g b) \u2264 r) \u2194\n (\u2203 i, nndist (f i) (g i) = r) \u2227 \u2200 (b : \u03b2), nndist (f b) (g b) \u2264 r\n[PROOFSTEP]\nsimp_rw [not_lt, and_congr_left_iff, le_antisymm_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\u22650\nhr : 0 < r\n\u22a2 (\u2200 (b : \u03b2), nndist (f b) (g b) \u2264 r) \u2192\n ((\u2203 x, r \u2264 nndist (f x) (g x)) \u2194 \u2203 i, nndist (f i) (g i) \u2264 r \u2227 r \u2264 nndist (f i) (g i))\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\u22650\nhr : 0 < r\nh : \u2200 (b : \u03b2), nndist (f b) (g b) \u2264 r\n\u22a2 (\u2203 x, r \u2264 nndist (f x) (g x)) \u2194 \u2203 i, nndist (f i) (g i) \u2264 r \u2227 r \u2264 nndist (f i) (g i)\n[PROOFSTEP]\nrefine' exists_congr fun b => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\u22650\nhr : 0 < r\nh : \u2200 (b : \u03b2), nndist (f b) (g b) \u2264 r\nb : \u03b2\n\u22a2 r \u2264 nndist (f b) (g b) \u2194 nndist (f b) (g b) \u2264 r \u2227 r \u2264 nndist (f b) (g b)\n[PROOFSTEP]\napply (and_iff_right <| h _).symm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : 0 < r\n\u22a2 dist f g < r \u2194 \u2200 (b : \u03b2), dist (f b) (g b) < r\n[PROOFSTEP]\nlift r to \u211d\u22650 using hr.le\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\u22650\nhr : 0 < \u2191r\n\u22a2 dist f g < \u2191r \u2194 \u2200 (b : \u03b2), dist (f b) (g b) < \u2191r\n[PROOFSTEP]\nexact nndist_pi_lt_iff hr\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 dist f g \u2264 r \u2194 \u2200 (b : \u03b2), dist (f b) (g b) \u2264 r\n[PROOFSTEP]\nlift r to \u211d\u22650 using hr\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\u22650\n\u22a2 dist f g \u2264 \u2191r \u2194 \u2200 (b : \u03b2), dist (f b) (g b) \u2264 \u2191r\n[PROOFSTEP]\nexact nndist_pi_le_iff\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : 0 < r\n\u22a2 dist f g = r \u2194 (\u2203 i, dist (f i) (g i) = r) \u2227 \u2200 (b : \u03b2), dist (f b) (g b) \u2264 r\n[PROOFSTEP]\nlift r to \u211d\u22650 using hr.le\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\u22650\nhr : 0 < \u2191r\n\u22a2 dist f g = \u2191r \u2194 (\u2203 i, dist (f i) (g i) = \u2191r) \u2227 \u2200 (b : \u03b2), dist (f b) (g b) \u2264 \u2191r\n[PROOFSTEP]\nsimp_rw [\u2190 coe_nndist, NNReal.coe_eq, nndist_pi_eq_iff hr, NNReal.coe_le_coe]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ninst\u271d : Nonempty \u03b2\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\n\u22a2 dist f g \u2264 r \u2194 \u2200 (b : \u03b2), dist (f b) (g b) \u2264 r\n[PROOFSTEP]\nby_cases hr : 0 \u2264 r\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ninst\u271d : Nonempty \u03b2\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 dist f g \u2264 r \u2194 \u2200 (b : \u03b2), dist (f b) (g b) \u2264 r\n[PROOFSTEP]\nexact dist_pi_le_iff hr\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ninst\u271d : Nonempty \u03b2\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 dist f g \u2264 r \u2194 \u2200 (b : \u03b2), dist (f b) (g b) \u2264 r\n[PROOFSTEP]\nexact iff_of_false (fun h => hr <| dist_nonneg.trans h) fun h => hr <| dist_nonneg.trans <| h <| Classical.arbitrary _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ninst\u271d : Nonempty \u03b2\na b : \u03b1\n\u22a2 (dist (fun x => a) fun x => b) = dist a b\n[PROOFSTEP]\nsimpa only [dist_edist] using congr_arg ENNReal.toReal (edist_pi_const a b)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nb : \u03b2\n\u22a2 nndist (f b) (g b) \u2264 nndist f g\n[PROOFSTEP]\nrw [\u2190 ENNReal.coe_le_coe, \u2190 edist_nndist, \u2190 edist_nndist]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nb : \u03b2\n\u22a2 edist (f b) (g b) \u2264 edist f g\n[PROOFSTEP]\nexact edist_le_pi_edist f g b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nb : \u03b2\n\u22a2 dist (f b) (g b) \u2264 dist f g\n[PROOFSTEP]\nsimp only [dist_nndist, NNReal.coe_le_coe, nndist_le_pi_nndist f g b]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : 0 < r\n\u22a2 ball x r = Set.pi Set.univ fun b => ball (x b) r\n[PROOFSTEP]\next p\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : 0 < r\np : (b : \u03b2) \u2192 \u03c0 b\n\u22a2 p \u2208 ball x r \u2194 p \u2208 Set.pi Set.univ fun b => ball (x b) r\n[PROOFSTEP]\nsimp [dist_pi_lt_iff hr]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ninst\u271d : Nonempty \u03b2\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : r \u2264 0\n\u22a2 ball x r = Set.pi Set.univ fun b => ball (x b) r\n[PROOFSTEP]\nsimp [ball_eq_empty.2 hr]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 closedBall x r = Set.pi Set.univ fun b => closedBall (x b) r\n[PROOFSTEP]\next p\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : 0 \u2264 r\np : (b : \u03b2) \u2192 \u03c0 b\n\u22a2 p \u2208 closedBall x r \u2194 p \u2208 Set.pi Set.univ fun b => closedBall (x b) r\n[PROOFSTEP]\nsimp [dist_pi_le_iff hr]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ninst\u271d : Nonempty \u03b2\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : r < 0\n\u22a2 closedBall x r = Set.pi Set.univ fun b => closedBall (x b) r\n[PROOFSTEP]\nsimp [closedBall_eq_empty.2 hr]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nh : 0 < r \u2228 Nonempty \u03b2\n\u22a2 sphere x r = (\u22c3 (i : \u03b2), Function.eval i \u207b\u00b9' sphere (x i) r) \u2229 closedBall x r\n[PROOFSTEP]\nobtain hr | rfl | hr := lt_trichotomy r 0\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nh : 0 < r \u2228 Nonempty \u03b2\nhr : r < 0\n\u22a2 sphere x r = (\u22c3 (i : \u03b2), Function.eval i \u207b\u00b9' sphere (x i) r) \u2229 closedBall x r\n[PROOFSTEP]\nsimp [hr]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\n\u22a2 sphere x 0 = (\u22c3 (i : \u03b2), Function.eval i \u207b\u00b9' sphere (x i) 0) \u2229 closedBall x 0\n[PROOFSTEP]\nrw [closedBall_eq_sphere_of_nonpos le_rfl, eq_comm, Set.inter_eq_right_iff_subset]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\n\u22a2 sphere x 0 \u2286 \u22c3 (i : \u03b2), Function.eval i \u207b\u00b9' sphere (x i) 0\n[PROOFSTEP]\nletI := h.resolve_left (lt_irrefl _)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\n\u22a2 sphere x 0 \u2286 \u22c3 (i : \u03b2), Function.eval i \u207b\u00b9' sphere (x i) 0\n[PROOFSTEP]\ninhabit \u03b2\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\ninhabited_h : Inhabited \u03b2\n\u22a2 sphere x 0 \u2286 \u22c3 (i : \u03b2), Function.eval i \u207b\u00b9' sphere (x i) 0\n[PROOFSTEP]\nrefine' subset_iUnion_of_subset default _\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\ninhabited_h : Inhabited \u03b2\n\u22a2 sphere x 0 \u2286 Function.eval default \u207b\u00b9' sphere (x default) 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx\u271d : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\ninhabited_h : Inhabited \u03b2\nx : (x : \u03b2) \u2192 \u03c0 x\nhx : x \u2208 sphere x\u271d 0\n\u22a2 x \u2208 Function.eval default \u207b\u00b9' sphere (x\u271d default) 0\n[PROOFSTEP]\nreplace hx := hx.le\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx\u271d : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\ninhabited_h : Inhabited \u03b2\nx : (x : \u03b2) \u2192 \u03c0 x\nhx : dist x x\u271d \u2264 0\n\u22a2 x \u2208 Function.eval default \u207b\u00b9' sphere (x\u271d default) 0\n[PROOFSTEP]\nrw [dist_pi_le_iff le_rfl] at hx \n[GOAL]\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx\u271d : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\ninhabited_h : Inhabited \u03b2\nx : (x : \u03b2) \u2192 \u03c0 x\nhx : \u2200 (b : \u03b2), dist (x b) (x\u271d b) \u2264 0\n\u22a2 x \u2208 Function.eval default \u207b\u00b9' sphere (x\u271d default) 0\n[PROOFSTEP]\nexact le_antisymm (hx default) dist_nonneg\n[GOAL]\ncase inr.inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nh : 0 < r \u2228 Nonempty \u03b2\nhr : 0 < r\n\u22a2 sphere x r = (\u22c3 (i : \u03b2), Function.eval i \u207b\u00b9' sphere (x i) r) \u2229 closedBall x r\n[PROOFSTEP]\next\n[GOAL]\ncase inr.inr.h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nh : 0 < r \u2228 Nonempty \u03b2\nhr : 0 < r\nx\u271d : (b : \u03b2) \u2192 \u03c0 b\n\u22a2 x\u271d \u2208 sphere x r \u2194 x\u271d \u2208 (\u22c3 (i : \u03b2), Function.eval i \u207b\u00b9' sphere (x i) r) \u2229 closedBall x r\n[PROOFSTEP]\nsimp [dist_pi_eq_iff hr, dist_pi_le_iff hr.le]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\u271d\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nn : \u2115\n\u03b1 : Fin (n + 1) \u2192 Type u_4\ninst\u271d : (i : Fin (n + 1)) \u2192 PseudoMetricSpace (\u03b1 i)\ni : Fin (n + 1)\nx y : \u03b1 i\nf g : (j : Fin n) \u2192 \u03b1 (succAbove i j)\nc : \u211d\u22650\n\u22a2 nndist (insertNth i x f) (insertNth i y g) \u2264 c \u2194 max (nndist x y) (nndist f g) \u2264 c\n[PROOFSTEP]\nsimp [nndist_pi_le_iff, i.forall_iff_succAbove]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\u271d\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nn : \u2115\n\u03b1 : Fin (n + 1) \u2192 Type u_4\ninst\u271d : (i : Fin (n + 1)) \u2192 PseudoMetricSpace (\u03b1 i)\ni : Fin (n + 1)\nx y : \u03b1 i\nf g : (j : Fin n) \u2192 \u03b1 (succAbove i j)\n\u22a2 dist (insertNth i x f) (insertNth i y g) = max (dist x y) (dist f g)\n[PROOFSTEP]\nsimp only [dist_nndist, Fin.nndist_insertNth_insertNth, NNReal.coe_max]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx y x' y' : \u03b2 \u2192 \u211d\nhx : x \u2208 Set.Icc x' y'\nhy : y \u2208 Set.Icc x' y'\n\u22a2 dist x y \u2264 dist x' y'\n[PROOFSTEP]\nrefine' (dist_pi_le_iff dist_nonneg).2 fun b => (Real.dist_le_of_mem_uIcc _ _).trans (dist_le_pi_dist x' y' b)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx y x' y' : \u03b2 \u2192 \u211d\nhx : x \u2208 Set.Icc x' y'\nhy : y \u2208 Set.Icc x' y'\nb : \u03b2\n\u22a2 x b \u2208 Set.uIcc (x' b) (y' b)\n[PROOFSTEP]\nrefine' Icc_subset_uIcc _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx y x' y' : \u03b2 \u2192 \u211d\nhx : x \u2208 Set.Icc x' y'\nhy : y \u2208 Set.Icc x' y'\nb : \u03b2\n\u22a2 y b \u2208 Set.uIcc (x' b) (y' b)\n[PROOFSTEP]\nrefine' Icc_subset_uIcc _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx y x' y' : \u03b2 \u2192 \u211d\nhx : x \u2208 Set.Icc x' y'\nhy : y \u2208 Set.Icc x' y'\nb : \u03b2\n\u22a2 x b \u2208 Set.Icc (x' b) (y' b)\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx y x' y' : \u03b2 \u2192 \u211d\nhx : x \u2208 Set.Icc x' y'\nhy : y \u2208 Set.Icc x' y'\nb : \u03b2\n\u22a2 y b \u2208 Set.Icc (x' b) (y' b)\n[PROOFSTEP]\nexacts [\u27e8hx.1 _, hx.2 _\u27e9, \u27e8hy.1 _, hy.2 _\u27e9]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\n\u22a2 SecondCountableTopology \u03b1\n[PROOFSTEP]\nsuffices SigmaCompactSpace \u03b1 by exact EMetric.secondCountable_of_sigmaCompact \u03b1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nthis : SigmaCompactSpace \u03b1\n\u22a2 SecondCountableTopology \u03b1\n[PROOFSTEP]\nexact EMetric.secondCountable_of_sigmaCompact \u03b1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\n\u22a2 SigmaCompactSpace \u03b1\n[PROOFSTEP]\nrcases em (Nonempty \u03b1) with (\u27e8\u27e8x\u27e9\u27e9 | hn)\n[GOAL]\ncase inl.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\n\u22a2 SigmaCompactSpace \u03b1\n[PROOFSTEP]\nexact \u27e8\u27e8fun n => closedBall x n, fun n => isCompact_closedBall _ _, iUnion_closedBall_nat _\u27e9\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nhn : \u00acNonempty \u03b1\n\u22a2 SigmaCompactSpace \u03b1\n[PROOFSTEP]\nexact \u27e8\u27e8fun _ => \u2205, fun _ => isCompact_empty, iUnion_eq_univ_iff.2 fun x => (hn \u27e8x\u27e9).elim\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\n\u22a2 Tendsto (dist x) (cocompact \u03b1) atTop\n[PROOFSTEP]\nsimpa only [dist_comm] using tendsto_dist_right_cocompact_atTop x\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\n\u22a2 \u2203 x, f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nobtain \u27e8t, t_fset, ht\u27e9 : \u2203 t \u2208 f, \u2200 x \u2208 t, \u2200 y \u2208 t, dist x y < 1 := (Metric.cauchy_iff.1 hf).2 1 zero_lt_one\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nt : Set \u03b1\nt_fset : t \u2208 f\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 dist x y < 1\n\u22a2 \u2203 x, f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nrcases hf.1.nonempty_of_mem t_fset with \u27e8x, xt\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nt : Set \u03b1\nt_fset : t \u2208 f\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 dist x y < 1\nx : \u03b1\nxt : x \u2208 t\n\u22a2 \u2203 x, f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nhave : closedBall x 1 \u2208 f := mem_of_superset t_fset fun y yt => (ht y yt x xt).le\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nt : Set \u03b1\nt_fset : t \u2208 f\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 dist x y < 1\nx : \u03b1\nxt : x \u2208 t\nthis : closedBall x 1 \u2208 f\n\u22a2 \u2203 x, f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nrcases(isCompact_iff_totallyBounded_isComplete.1 (isCompact_closedBall x 1)).2 f hf (le_principal_iff.2 this) with\n \u27e8y, -, hy\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nt : Set \u03b1\nt_fset : t \u2208 f\nht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 dist x y < 1\nx : \u03b1\nxt : x \u2208 t\nthis : closedBall x 1 \u2208 f\ny : \u03b1\nhy : f \u2264 \ud835\udcdd y\n\u22a2 \u2203 x, f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nexact \u27e8y, hy\u27e9\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\ninst\u271d\u00b9 : ProperSpace \u03b1\ninst\u271d : ProperSpace \u03b2\n\u22a2 \u2200 (x : \u03b1 \u00d7 \u03b2) (r : \u211d), IsCompact (closedBall x r)\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9 r\n[GOAL]\ncase mk\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\ninst\u271d\u00b9 : ProperSpace \u03b1\ninst\u271d : ProperSpace \u03b2\nx : \u03b1\ny : \u03b2\nr : \u211d\n\u22a2 IsCompact (closedBall (x, y) r)\n[PROOFSTEP]\nrw [\u2190 closedBall_prod_same x y]\n[GOAL]\ncase mk\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\ninst\u271d\u00b2 : PseudoMetricSpace \u03b2\ninst\u271d\u00b9 : ProperSpace \u03b1\ninst\u271d : ProperSpace \u03b2\nx : \u03b1\ny : \u03b2\nr : \u211d\n\u22a2 IsCompact (closedBall x r \u00d7\u02e2 closedBall y r)\n[PROOFSTEP]\nexact (isCompact_closedBall x r).prod (isCompact_closedBall y r)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nh : \u2200 (b : \u03b2), ProperSpace (\u03c0 b)\n\u22a2 ProperSpace ((b : \u03b2) \u2192 \u03c0 b)\n[PROOFSTEP]\nrefine' properSpace_of_compact_closedBall_of_le 0 fun x r hr => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nh : \u2200 (b : \u03b2), ProperSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 IsCompact (closedBall x r)\n[PROOFSTEP]\nrw [closedBall_pi _ hr]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nh : \u2200 (b : \u03b2), ProperSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 IsCompact (Set.pi univ fun b => closedBall (x b) r)\n[PROOFSTEP]\nexact isCompact_univ_pi fun _ => isCompact_closedBall _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\n\u22a2 \u2203 r', r' \u2208 Ioo 0 r \u2227 s \u2286 ball x r'\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | hne)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\nhr : 0 < r\nhs : IsClosed \u2205\nh : \u2205 \u2286 ball x r\n\u22a2 \u2203 r', r' \u2208 Ioo 0 r \u2227 \u2205 \u2286 ball x r'\n[PROOFSTEP]\nexact \u27e8r / 2, \u27e8half_pos hr, half_lt_self hr\u27e9, empty_subset _\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\nhne : Set.Nonempty s\n\u22a2 \u2203 r', r' \u2208 Ioo 0 r \u2227 s \u2286 ball x r'\n[PROOFSTEP]\nhave : IsCompact s := isCompact_of_isClosed_subset (isCompact_closedBall x r) hs (h.trans ball_subset_closedBall)\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\nhne : Set.Nonempty s\nthis : IsCompact s\n\u22a2 \u2203 r', r' \u2208 Ioo 0 r \u2227 s \u2286 ball x r'\n[PROOFSTEP]\nobtain \u27e8y, hys, hy\u27e9 : \u2203 y \u2208 s, s \u2286 closedBall x (dist y x) :=\n this.exists_forall_ge hne (continuous_id.dist continuous_const).continuousOn\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\nhne : Set.Nonempty s\nthis : IsCompact s\ny : \u03b1\nhys : y \u2208 s\nhy : s \u2286 closedBall x (dist y x)\n\u22a2 \u2203 r', r' \u2208 Ioo 0 r \u2227 s \u2286 ball x r'\n[PROOFSTEP]\nhave hyr : dist y x < r := h hys\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\nhne : Set.Nonempty s\nthis : IsCompact s\ny : \u03b1\nhys : y \u2208 s\nhy : s \u2286 closedBall x (dist y x)\nhyr : dist y x < r\n\u22a2 \u2203 r', r' \u2208 Ioo 0 r \u2227 s \u2286 ball x r'\n[PROOFSTEP]\nrcases exists_between hyr with \u27e8r', hyr', hrr'\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\nhne : Set.Nonempty s\nthis : IsCompact s\ny : \u03b1\nhys : y \u2208 s\nhy : s \u2286 closedBall x (dist y x)\nhyr : dist y x < r\nr' : \u211d\nhyr' : dist y x < r'\nhrr' : r' < r\n\u22a2 \u2203 r', r' \u2208 Ioo 0 r \u2227 s \u2286 ball x r'\n[PROOFSTEP]\nexact \u27e8r', \u27e8dist_nonneg.trans_lt hyr', hrr'\u27e9, hy.trans <| closedBall_subset_ball hyr'\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhs : IsClosed s\nh : s \u2286 ball x r\n\u22a2 \u2203 r', r' < r \u2227 s \u2286 ball x r'\n[PROOFSTEP]\ncases' le_or_lt r 0 with hr hr\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhs : IsClosed s\nh : s \u2286 ball x r\nhr : r \u2264 0\n\u22a2 \u2203 r', r' < r \u2227 s \u2286 ball x r'\n[PROOFSTEP]\nrw [ball_eq_empty.2 hr, subset_empty_iff] at h \n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhs : IsClosed s\nh : s = \u2205\nhr : r \u2264 0\n\u22a2 \u2203 r', r' < r \u2227 s \u2286 ball x r'\n[PROOFSTEP]\nsubst s\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\nhr : r \u2264 0\nhs : IsClosed \u2205\n\u22a2 \u2203 r', r' < r \u2227 \u2205 \u2286 ball x r'\n[PROOFSTEP]\nexact (exists_lt r).imp fun r' hr' => \u27e8hr', empty_subset _\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhs : IsClosed s\nh : s \u2286 ball x r\nhr : 0 < r\n\u22a2 \u2203 r', r' < r \u2227 s \u2286 ball x r'\n[PROOFSTEP]\nexact (exists_pos_lt_subset_ball hr hs h).imp fun r' hr' => \u27e8hr'.1.2, hr'.2\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 s, Set.Countable s \u2227 \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 dist x y \u2264 \u03b5\n\u22a2 SecondCountableTopology \u03b1\n[PROOFSTEP]\nrefine' EMetric.secondCountable_of_almost_dense_set fun \u03b5 \u03b50 => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 s, Set.Countable s \u2227 \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 dist x y \u2264 \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 t, Set.Countable t \u2227 \u22c3 (x : \u03b1) (_ : x \u2208 t), EMetric.closedBall x \u03b5 = univ\n[PROOFSTEP]\nrcases ENNReal.lt_iff_exists_nnreal_btwn.1 \u03b50 with \u27e8\u03b5', \u03b5'0, \u03b5'\u03b5\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 s, Set.Countable s \u2227 \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 dist x y \u2264 \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\u22650\n\u03b5'0 : 0 < \u2191\u03b5'\n\u03b5'\u03b5 : \u2191\u03b5' < \u03b5\n\u22a2 \u2203 t, Set.Countable t \u2227 \u22c3 (x : \u03b1) (_ : x \u2208 t), EMetric.closedBall x \u03b5 = univ\n[PROOFSTEP]\nchoose s hsc y hys hyx using H \u03b5' (by exact_mod_cast \u03b5'0)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 s, Set.Countable s \u2227 \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 dist x y \u2264 \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\u22650\n\u03b5'0 : 0 < \u2191\u03b5'\n\u03b5'\u03b5 : \u2191\u03b5' < \u03b5\n\u22a2 \u2191\u03b5' > 0\n[PROOFSTEP]\nexact_mod_cast \u03b5'0\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 s, Set.Countable s \u2227 \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 dist x y \u2264 \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\u22650\n\u03b5'0 : 0 < \u2191\u03b5'\n\u03b5'\u03b5 : \u2191\u03b5' < \u03b5\ns : Set \u03b1\nhsc : Set.Countable s\ny : \u03b1 \u2192 \u03b1\nhys : \u2200 (x : \u03b1), y x \u2208 s\nhyx : \u2200 (x : \u03b1), dist x (y x) \u2264 \u2191\u03b5'\n\u22a2 \u2203 t, Set.Countable t \u2227 \u22c3 (x : \u03b1) (_ : x \u2208 t), EMetric.closedBall x \u03b5 = univ\n[PROOFSTEP]\nrefine' \u27e8s, hsc, iUnion\u2082_eq_univ_iff.2 fun x => \u27e8y x, hys _, le_trans _ \u03b5'\u03b5.le\u27e9\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 s, Set.Countable s \u2227 \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 dist x y \u2264 \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\u22650\n\u03b5'0 : 0 < \u2191\u03b5'\n\u03b5'\u03b5 : \u2191\u03b5' < \u03b5\ns : Set \u03b1\nhsc : Set.Countable s\ny : \u03b1 \u2192 \u03b1\nhys : \u2200 (x : \u03b1), y x \u2208 s\nhyx : \u2200 (x : \u03b1), dist x (y x) \u2264 \u2191\u03b5'\nx : \u03b1\n\u22a2 edist x (y x) \u2264 \u2191\u03b5'\n[PROOFSTEP]\nexact_mod_cast hyx x\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nc : Set (Set \u03b1)\nhs : IsCompact s\nhc\u2081 : \u2200 (t : Set \u03b1), t \u2208 c \u2192 IsOpen t\nhc\u2082 : s \u2286 \u22c3\u2080 c\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 t, t \u2208 c \u2227 ball x \u03b4 \u2286 t\n[PROOFSTEP]\nrw [sUnion_eq_iUnion] at hc\u2082 \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nc : Set (Set \u03b1)\nhs : IsCompact s\nhc\u2081 : \u2200 (t : Set \u03b1), t \u2208 c \u2192 IsOpen t\nhc\u2082 : s \u2286 \u22c3 (i : \u2191c), \u2191i\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 t, t \u2208 c \u2227 ball x \u03b4 \u2286 t\n[PROOFSTEP]\nsimpa using lebesgue_number_lemma_of_metric hs (by simpa) hc\u2082\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nc : Set (Set \u03b1)\nhs : IsCompact s\nhc\u2081 : \u2200 (t : Set \u03b1), t \u2208 c \u2192 IsOpen t\nhc\u2082 : s \u2286 \u22c3 (i : \u2191c), \u2191i\n\u22a2 \u2200 (i : \u2191c), IsOpen \u2191i\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ns : Set \u03b1\n\u22a2 Bounded s \u2194 IsBounded s\n[PROOFSTEP]\nchange Bounded s \u2194 s\u1d9c \u2208 (cobounded \u03b1).sets\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ns : Set \u03b1\n\u22a2 Bounded s \u2194 s\u1d9c \u2208 (cobounded \u03b1).sets\n[PROOFSTEP]\nsimp [PseudoMetricSpace.cobounded_sets, Metric.Bounded]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\n\u22a2 \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 \u2200 (y : \u03b1), y \u2208 \u2205 \u2192 dist x y \u2264 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\nc : \u03b1\n\u22a2 Bounded s \u2194 \u2203 r, s \u2286 closedBall c r\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\nc : \u03b1\n\u22a2 Bounded s \u2192 \u2203 r, s \u2286 closedBall c r\n[PROOFSTEP]\nrintro \u27e8C, hC\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\nc : \u03b1\n\u22a2 (\u2203 r, s \u2286 closedBall c r) \u2192 Bounded s\n[PROOFSTEP]\nrintro \u27e8C, hC\u27e9\n[GOAL]\ncase mp.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\nc : \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist x y \u2264 C\n\u22a2 \u2203 r, s \u2286 closedBall c r\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | \u27e8x, hx\u27e9)\n[GOAL]\ncase mp.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\nt : Set \u03b1\nr : \u211d\nc : \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 \u2200 (y : \u03b1), y \u2208 \u2205 \u2192 dist x y \u2264 C\n\u22a2 \u2203 r, \u2205 \u2286 closedBall c r\n[PROOFSTEP]\nexact \u27e80, by simp\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\nt : Set \u03b1\nr : \u211d\nc : \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 \u2200 (y : \u03b1), y \u2208 \u2205 \u2192 dist x y \u2264 C\n\u22a2 \u2205 \u2286 closedBall c 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mp.intro.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns t : Set \u03b1\nr : \u211d\nc : \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist x y \u2264 C\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2203 r, s \u2286 closedBall c r\n[PROOFSTEP]\nexact\n \u27e8C + dist x c, fun y hy =>\n calc\n dist y c \u2264 dist y x + dist x c := dist_triangle _ _ _\n _ \u2264 C + dist x c := add_le_add_right (hC y hy x hx) _\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\nc : \u03b1\nC : \u211d\nhC : s \u2286 closedBall c C\n\u22a2 Bounded s\n[PROOFSTEP]\nexact bounded_closedBall.mono hC\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\nh : Bounded s\na : \u211d\nc : \u03b1\n\u22a2 \u2203 r, a < r \u2227 s \u2286 closedBall c r\n[PROOFSTEP]\nrcases h.subset_ball c with \u27e8r, hr\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr\u271d : \u211d\nh : Bounded s\na : \u211d\nc : \u03b1\nr : \u211d\nhr : s \u2286 closedBall c r\n\u22a2 \u2203 r, a < r \u2227 s \u2286 closedBall c r\n[PROOFSTEP]\nrefine' \u27e8max r (a + 1), lt_of_lt_of_le (by linarith) (le_max_right _ _), _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr\u271d : \u211d\nh : Bounded s\na : \u211d\nc : \u03b1\nr : \u211d\nhr : s \u2286 closedBall c r\n\u22a2 a < a + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr\u271d : \u211d\nh : Bounded s\na : \u211d\nc : \u03b1\nr : \u211d\nhr : s \u2286 closedBall c r\n\u22a2 s \u2286 closedBall c (max r (a + 1))\n[PROOFSTEP]\nexact hr.trans (closedBall_subset_closedBall (le_max_left _ _))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\n\u22a2 Bounded (s \u222a t) \u2194 Bounded s \u2227 Bounded t\n[PROOFSTEP]\nsimp only [bounded_iff_isBounded, isBounded_union]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\nI : Set \u03b2\ns : \u03b2 \u2192 Set \u03b1\nH : Set.Finite I\n\u22a2 Bounded (\u22c3 (i : \u03b2) (_ : i \u2208 I), s i) \u2194 \u2200 (i : \u03b2), i \u2208 I \u2192 Bounded (s i)\n[PROOFSTEP]\nsimp only [bounded_iff_isBounded, isBounded_biUnion H]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d t\u271d : Set \u03b1\nr : \u211d\ninst\u271d : PseudoMetricSpace \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Bounded s\nht : Bounded t\n\u22a2 Bounded (s \u00d7\u02e2 t)\n[PROOFSTEP]\nrw [bounded_iff_isBounded] at *\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d t\u271d : Set \u03b1\nr : \u211d\ninst\u271d : PseudoMetricSpace \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : IsBounded s\nht : IsBounded t\n\u22a2 IsBounded (s \u00d7\u02e2 t)\n[PROOFSTEP]\nexact hs.prod ht\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\nf : \u03b2 \u2192 \u03b1\nhf : Tendsto (Prod.map f f) (Filter.cofinite \u00d7\u02e2 Filter.cofinite) (\ud835\udce4 \u03b1)\n\u22a2 Bounded (range f)\n[PROOFSTEP]\nrcases(hasBasis_cofinite.prod_self.tendsto_iff uniformity_basis_dist).1 hf 1 zero_lt_one with \u27e8s, hsf, hs1\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\nf : \u03b2 \u2192 \u03b1\nhf : Tendsto (Prod.map f f) (Filter.cofinite \u00d7\u02e2 Filter.cofinite) (\ud835\udce4 \u03b1)\ns : Set \u03b2\nhsf : Set.Finite s\nhs1 : \u2200 (x : \u03b2 \u00d7 \u03b2), x \u2208 s\u1d9c \u00d7\u02e2 s\u1d9c \u2192 Prod.map f f x \u2208 {p | dist p.fst p.snd < 1}\n\u22a2 Bounded (range f)\n[PROOFSTEP]\nrw [\u2190 image_univ, \u2190 union_compl_self s, image_union, bounded_union]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\nf : \u03b2 \u2192 \u03b1\nhf : Tendsto (Prod.map f f) (Filter.cofinite \u00d7\u02e2 Filter.cofinite) (\ud835\udce4 \u03b1)\ns : Set \u03b2\nhsf : Set.Finite s\nhs1 : \u2200 (x : \u03b2 \u00d7 \u03b2), x \u2208 s\u1d9c \u00d7\u02e2 s\u1d9c \u2192 Prod.map f f x \u2208 {p | dist p.fst p.snd < 1}\n\u22a2 Bounded (f '' s) \u2227 Bounded (f '' s\u1d9c)\n[PROOFSTEP]\nuse(hsf.image f).bounded, 1\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\nf : \u03b2 \u2192 \u03b1\nhf : Tendsto (Prod.map f f) (Filter.cofinite \u00d7\u02e2 Filter.cofinite) (\ud835\udce4 \u03b1)\ns : Set \u03b2\nhsf : Set.Finite s\nhs1 : \u2200 (x : \u03b2 \u00d7 \u03b2), x \u2208 s\u1d9c \u00d7\u02e2 s\u1d9c \u2192 Prod.map f f x \u2208 {p | dist p.fst p.snd < 1}\n\u22a2 \u2200 (x : \u03b1), x \u2208 f '' s\u1d9c \u2192 \u2200 (y : \u03b1), y \u2208 f '' s\u1d9c \u2192 dist x y \u2264 1\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9 _ \u27e8y, hy, rfl\u27e9\n[GOAL]\ncase h.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\nf : \u03b2 \u2192 \u03b1\nhf : Tendsto (Prod.map f f) (Filter.cofinite \u00d7\u02e2 Filter.cofinite) (\ud835\udce4 \u03b1)\ns : Set \u03b2\nhsf : Set.Finite s\nhs1 : \u2200 (x : \u03b2 \u00d7 \u03b2), x \u2208 s\u1d9c \u00d7\u02e2 s\u1d9c \u2192 Prod.map f f x \u2208 {p | dist p.fst p.snd < 1}\nx : \u03b2\nhx : x \u2208 s\u1d9c\ny : \u03b2\nhy : y \u2208 s\u1d9c\n\u22a2 dist (f x) (f y) \u2264 1\n[PROOFSTEP]\nexact le_of_lt (hs1 (x, y) \u27e8hx, hy\u27e9)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\nf : \u2115 \u2192 \u03b1\nhf : CauchySeq f\n\u22a2 Cauchy (map f Filter.cofinite)\n[PROOFSTEP]\nrwa [Nat.cofinite_eq_atTop]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\n\u22a2 \u2203 t, k \u2286 t \u2227 IsOpen t \u2227 Bounded (f '' (t \u2229 s))\n[PROOFSTEP]\nrefine hk.induction_on ?_ ?_ ?_ ?_\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\n\u22a2 \u2203 t, \u2205 \u2286 t \u2227 IsOpen t \u2227 Bounded (f '' (t \u2229 s))\n[PROOFSTEP]\nexact \u27e8\u2205, Subset.rfl, isOpen_empty, by simp only [image_empty, bounded_empty, empty_inter]\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\n\u22a2 Bounded (f '' (\u2205 \u2229 s))\n[PROOFSTEP]\nsimp only [image_empty, bounded_empty, empty_inter]\n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\n\u22a2 \u2200 \u2983s_1 t : Set \u03b2\u2984,\n s_1 \u2286 t \u2192\n (\u2203 t_1, t \u2286 t_1 \u2227 IsOpen t_1 \u2227 Bounded (f '' (t_1 \u2229 s))) \u2192 \u2203 t, s_1 \u2286 t \u2227 IsOpen t \u2227 Bounded (f '' (t \u2229 s))\n[PROOFSTEP]\nrintro s s' hss' \u27e8t, s't, t_open, t_bounded\u27e9\n[GOAL]\ncase refine_2.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d\u00b9 t\u271d : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s\u271d : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s\u271d x\ns s' : Set \u03b2\nhss' : s \u2286 s'\nt : Set \u03b2\ns't : s' \u2286 t\nt_open : IsOpen t\nt_bounded : Bounded (f '' (t \u2229 s\u271d))\n\u22a2 \u2203 t, s \u2286 t \u2227 IsOpen t \u2227 Bounded (f '' (t \u2229 s\u271d))\n[PROOFSTEP]\nexact \u27e8t, hss'.trans s't, t_open, t_bounded\u27e9\n[GOAL]\ncase refine_3\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\n\u22a2 \u2200 \u2983s_1 t : Set \u03b2\u2984,\n (\u2203 t, s_1 \u2286 t \u2227 IsOpen t \u2227 Bounded (f '' (t \u2229 s))) \u2192\n (\u2203 t_1, t \u2286 t_1 \u2227 IsOpen t_1 \u2227 Bounded (f '' (t_1 \u2229 s))) \u2192\n \u2203 t_1, s_1 \u222a t \u2286 t_1 \u2227 IsOpen t_1 \u2227 Bounded (f '' (t_1 \u2229 s))\n[PROOFSTEP]\nrintro s s' \u27e8t, st, t_open, t_bounded\u27e9 \u27e8t', s't', t'_open, t'_bounded\u27e9\n[GOAL]\ncase refine_3.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d\u00b9 t\u271d : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s\u271d : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s\u271d x\ns s' t : Set \u03b2\nst : s \u2286 t\nt_open : IsOpen t\nt_bounded : Bounded (f '' (t \u2229 s\u271d))\nt' : Set \u03b2\ns't' : s' \u2286 t'\nt'_open : IsOpen t'\nt'_bounded : Bounded (f '' (t' \u2229 s\u271d))\n\u22a2 \u2203 t, s \u222a s' \u2286 t \u2227 IsOpen t \u2227 Bounded (f '' (t \u2229 s\u271d))\n[PROOFSTEP]\nrefine' \u27e8t \u222a t', union_subset_union st s't', t_open.union t'_open, _\u27e9\n[GOAL]\ncase refine_3.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d\u00b9 t\u271d : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s\u271d : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s\u271d x\ns s' t : Set \u03b2\nst : s \u2286 t\nt_open : IsOpen t\nt_bounded : Bounded (f '' (t \u2229 s\u271d))\nt' : Set \u03b2\ns't' : s' \u2286 t'\nt'_open : IsOpen t'\nt'_bounded : Bounded (f '' (t' \u2229 s\u271d))\n\u22a2 Bounded (f '' ((t \u222a t') \u2229 s\u271d))\n[PROOFSTEP]\nrw [union_inter_distrib_right, image_union]\n[GOAL]\ncase refine_3.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d\u00b9 t\u271d : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s\u271d : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s\u271d x\ns s' t : Set \u03b2\nst : s \u2286 t\nt_open : IsOpen t\nt_bounded : Bounded (f '' (t \u2229 s\u271d))\nt' : Set \u03b2\ns't' : s' \u2286 t'\nt'_open : IsOpen t'\nt'_bounded : Bounded (f '' (t' \u2229 s\u271d))\n\u22a2 Bounded (f '' (t \u2229 s\u271d) \u222a f '' (t' \u2229 s\u271d))\n[PROOFSTEP]\nexact t_bounded.union t'_bounded\n[GOAL]\ncase refine_4\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\n\u22a2 \u2200 (x : \u03b2), x \u2208 k \u2192 \u2203 t, t \u2208 \ud835\udcdd[k] x \u2227 \u2203 t_1, t \u2286 t_1 \u2227 IsOpen t_1 \u2227 Bounded (f '' (t_1 \u2229 s))\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine_4\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\nx : \u03b2\nhx : x \u2208 k\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[k] x \u2227 \u2203 t_1, t \u2286 t_1 \u2227 IsOpen t_1 \u2227 Bounded (f '' (t_1 \u2229 s))\n[PROOFSTEP]\nhave A : ball (f x) 1 \u2208 \ud835\udcdd (f x) := ball_mem_nhds _ zero_lt_one\n[GOAL]\ncase refine_4\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\nx : \u03b2\nhx : x \u2208 k\nA : ball (f x) 1 \u2208 \ud835\udcdd (f x)\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[k] x \u2227 \u2203 t_1, t \u2286 t_1 \u2227 IsOpen t_1 \u2227 Bounded (f '' (t_1 \u2229 s))\n[PROOFSTEP]\nhave B : f \u207b\u00b9' ball (f x) 1 \u2208 \ud835\udcdd[s] x := hf x hx A\n[GOAL]\ncase refine_4\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\nx : \u03b2\nhx : x \u2208 k\nA : ball (f x) 1 \u2208 \ud835\udcdd (f x)\nB : f \u207b\u00b9' ball (f x) 1 \u2208 \ud835\udcdd[s] x\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[k] x \u2227 \u2203 t_1, t \u2286 t_1 \u2227 IsOpen t_1 \u2227 Bounded (f '' (t_1 \u2229 s))\n[PROOFSTEP]\nobtain \u27e8u, u_open, xu, uf\u27e9 : \u2203 u : Set \u03b2, IsOpen u \u2227 x \u2208 u \u2227 u \u2229 s \u2286 f \u207b\u00b9' ball (f x) 1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\nx : \u03b2\nhx : x \u2208 k\nA : ball (f x) 1 \u2208 \ud835\udcdd (f x)\nB : f \u207b\u00b9' ball (f x) 1 \u2208 \ud835\udcdd[s] x\n\u22a2 \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 u \u2229 s \u2286 f \u207b\u00b9' ball (f x) 1\ncase refine_4.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\nx : \u03b2\nhx : x \u2208 k\nA : ball (f x) 1 \u2208 \ud835\udcdd (f x)\nB : f \u207b\u00b9' ball (f x) 1 \u2208 \ud835\udcdd[s] x\nu : Set \u03b2\nu_open : IsOpen u\nxu : x \u2208 u\nuf : u \u2229 s \u2286 f \u207b\u00b9' ball (f x) 1\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[k] x \u2227 \u2203 t_1, t \u2286 t_1 \u2227 IsOpen t_1 \u2227 Bounded (f '' (t_1 \u2229 s))\n[PROOFSTEP]\nexact _root_.mem_nhdsWithin.1 B\n[GOAL]\ncase refine_4.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\nx : \u03b2\nhx : x \u2208 k\nA : ball (f x) 1 \u2208 \ud835\udcdd (f x)\nB : f \u207b\u00b9' ball (f x) 1 \u2208 \ud835\udcdd[s] x\nu : Set \u03b2\nu_open : IsOpen u\nxu : x \u2208 u\nuf : u \u2229 s \u2286 f \u207b\u00b9' ball (f x) 1\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[k] x \u2227 \u2203 t_1, t \u2286 t_1 \u2227 IsOpen t_1 \u2227 Bounded (f '' (t_1 \u2229 s))\n[PROOFSTEP]\nrefine' \u27e8u, _, u, Subset.rfl, u_open, _\u27e9\n[GOAL]\ncase refine_4.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\nx : \u03b2\nhx : x \u2208 k\nA : ball (f x) 1 \u2208 \ud835\udcdd (f x)\nB : f \u207b\u00b9' ball (f x) 1 \u2208 \ud835\udcdd[s] x\nu : Set \u03b2\nu_open : IsOpen u\nxu : x \u2208 u\nuf : u \u2229 s \u2286 f \u207b\u00b9' ball (f x) 1\n\u22a2 u \u2208 \ud835\udcdd[k] x\n[PROOFSTEP]\napply nhdsWithin_le_nhds\n[GOAL]\ncase refine_4.intro.intro.intro.refine'_1.a\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\nx : \u03b2\nhx : x \u2208 k\nA : ball (f x) 1 \u2208 \ud835\udcdd (f x)\nB : f \u207b\u00b9' ball (f x) 1 \u2208 \ud835\udcdd[s] x\nu : Set \u03b2\nu_open : IsOpen u\nxu : x \u2208 u\nuf : u \u2229 s \u2286 f \u207b\u00b9' ball (f x) 1\n\u22a2 u \u2208 \ud835\udcdd x\n[PROOFSTEP]\nexact u_open.mem_nhds xu\n[GOAL]\ncase refine_4.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\nx : \u03b2\nhx : x \u2208 k\nA : ball (f x) 1 \u2208 \ud835\udcdd (f x)\nB : f \u207b\u00b9' ball (f x) 1 \u2208 \ud835\udcdd[s] x\nu : Set \u03b2\nu_open : IsOpen u\nxu : x \u2208 u\nuf : u \u2229 s \u2286 f \u207b\u00b9' ball (f x) 1\n\u22a2 Bounded (f '' (u \u2229 s))\n[PROOFSTEP]\napply Bounded.mono (image_subset _ uf)\n[GOAL]\ncase refine_4.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns\u271d t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk s : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f s x\nx : \u03b2\nhx : x \u2208 k\nA : ball (f x) 1 \u2208 \ud835\udcdd (f x)\nB : f \u207b\u00b9' ball (f x) 1 \u2208 \ud835\udcdd[s] x\nu : Set \u03b2\nu_open : IsOpen u\nxu : x \u2208 u\nuf : u \u2229 s \u2286 f \u207b\u00b9' ball (f x) 1\n\u22a2 Bounded (f '' (f \u207b\u00b9' ball (f x) 1))\n[PROOFSTEP]\nexact bounded_ball.mono (image_preimage_subset _ _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousAt f x\n\u22a2 \u2203 t, k \u2286 t \u2227 IsOpen t \u2227 Bounded (f '' t)\n[PROOFSTEP]\nsimp_rw [\u2190 continuousWithinAt_univ] at hf \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\ninst\u271d : TopologicalSpace \u03b2\nk : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhk : IsCompact k\nhf : \u2200 (x : \u03b2), x \u2208 k \u2192 ContinuousWithinAt f univ x\n\u22a2 \u2203 t, k \u2286 t \u2227 IsOpen t \u2227 Bounded (f '' t)\n[PROOFSTEP]\nsimpa only [inter_univ] using exists_isOpen_bounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk hf\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\ninst\u271d : ProperSpace \u03b1\nhc : IsClosed s\nhb : Bounded s\n\u22a2 IsCompact s\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | \u27e8x, -\u27e9)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx : \u03b1\nt : Set \u03b1\nr : \u211d\ninst\u271d : ProperSpace \u03b1\nhc : IsClosed \u2205\nhb : Bounded \u2205\n\u22a2 IsCompact \u2205\n[PROOFSTEP]\nexact isCompact_empty\n[GOAL]\ncase inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns t : Set \u03b1\nr : \u211d\ninst\u271d : ProperSpace \u03b1\nhc : IsClosed s\nhb : Bounded s\nx : \u03b1\n\u22a2 IsCompact s\n[PROOFSTEP]\nrcases hb.subset_ball x with \u27e8r, hr\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns t : Set \u03b1\nr\u271d : \u211d\ninst\u271d : ProperSpace \u03b1\nhc : IsClosed s\nhb : Bounded s\nx : \u03b1\nr : \u211d\nhr : s \u2286 closedBall x r\n\u22a2 IsCompact s\n[PROOFSTEP]\nexact isCompact_of_isClosed_subset (isCompact_closedBall x r) hc hr\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\nhs : Set.Subsingleton s\n\u22a2 diam s = 0\n[PROOFSTEP]\nsimp only [diam, EMetric.diam_subsingleton hs, ENNReal.zero_toReal]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\n\u22a2 diam {x, y} = dist x y\n[PROOFSTEP]\nsimp only [diam, EMetric.diam_pair, dist_edist]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\n\u22a2 diam {x, y, z} = max (max (dist x y) (dist x z)) (dist y z)\n[PROOFSTEP]\nsimp only [Metric.diam, EMetric.diam_triple, dist_edist]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\n\u22a2 ENNReal.toReal (max (max (edist x y) (edist x z)) (edist y z)) =\n max (max (ENNReal.toReal (edist x y)) (ENNReal.toReal (edist x z))) (ENNReal.toReal (edist y z))\n[PROOFSTEP]\nrw [ENNReal.toReal_max, ENNReal.toReal_max]\n[GOAL]\ncase hr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\n\u22a2 edist x y \u2260 \u22a4\n[PROOFSTEP]\napply_rules [ne_of_lt, edist_lt_top, max_lt]\n[GOAL]\ncase hp\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\n\u22a2 edist x z \u2260 \u22a4\n[PROOFSTEP]\napply_rules [ne_of_lt, edist_lt_top, max_lt]\n[GOAL]\ncase hr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\n\u22a2 max (edist x y) (edist x z) \u2260 \u22a4\n[PROOFSTEP]\napply_rules [ne_of_lt, edist_lt_top, max_lt]\n[GOAL]\ncase hp\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\n\u22a2 edist y z \u2260 \u22a4\n[PROOFSTEP]\napply_rules [ne_of_lt, edist_lt_top, max_lt]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\nh : EMetric.diam s \u2260 \u22a4\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 dist x y \u2264 diam s\n[PROOFSTEP]\nrw [diam, dist_edist]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\nh : EMetric.diam s \u2260 \u22a4\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 ENNReal.toReal (edist x y) \u2264 ENNReal.toReal (EMetric.diam s)\n[PROOFSTEP]\nrw [ENNReal.toReal_le_toReal (edist_ne_top _ _) h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\nh : EMetric.diam s \u2260 \u22a4\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 edist x y \u2264 EMetric.diam s\n[PROOFSTEP]\nexact EMetric.edist_le_diam_of_mem hx hy\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\ninst\u271d : ProperSpace \u03b1\n\u22a2 EMetric.diam univ = \u22a4 \u2194 NoncompactSpace \u03b1\n[PROOFSTEP]\nrw [\u2190 not_compactSpace_iff, compactSpace_iff_bounded_univ, bounded_iff_ediam_ne_top, Classical.not_not]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\ninst\u271d\u00b9 : ProperSpace \u03b1\ninst\u271d : NoncompactSpace \u03b1\n\u22a2 diam univ = 0\n[PROOFSTEP]\nsimp [diam]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\nh : \u00acBounded s\n\u22a2 diam s = 0\n[PROOFSTEP]\nrw [diam, ediam_of_unbounded h, ENNReal.top_toReal]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\nt : Set \u03b1\nxs : x \u2208 s\nyt : y \u2208 t\n\u22a2 diam (s \u222a t) \u2264 diam s + dist x y + diam t\n[PROOFSTEP]\nsimp only [diam, dist_edist]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\nt : Set \u03b1\nxs : x \u2208 s\nyt : y \u2208 t\n\u22a2 ENNReal.toReal (EMetric.diam (s \u222a t)) \u2264\n ENNReal.toReal (EMetric.diam s) + ENNReal.toReal (edist x y) + ENNReal.toReal (EMetric.diam t)\n[PROOFSTEP]\nrefine (ENNReal.toReal_le_add' (EMetric.diam_union xs yt) ?_ ?_).trans (add_le_add_right ENNReal.toReal_add_le _)\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\nt : Set \u03b1\nxs : x \u2208 s\nyt : y \u2208 t\n\u22a2 EMetric.diam s + edist x y = \u22a4 \u2192 EMetric.diam (s \u222a t) = \u22a4\n[PROOFSTEP]\nsimp only [ENNReal.add_eq_top, edist_ne_top, or_false]\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\nt : Set \u03b1\nxs : x \u2208 s\nyt : y \u2208 t\n\u22a2 EMetric.diam s = \u22a4 \u2192 EMetric.diam (s \u222a t) = \u22a4\n[PROOFSTEP]\nexact fun h \u21a6 top_unique <| h \u25b8 EMetric.diam_mono (subset_union_left _ _)\n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\nt : Set \u03b1\nxs : x \u2208 s\nyt : y \u2208 t\n\u22a2 EMetric.diam t = \u22a4 \u2192 EMetric.diam (s \u222a t) = \u22a4\n[PROOFSTEP]\nexact fun h \u21a6 top_unique <| h \u25b8 EMetric.diam_mono (subset_union_right _ _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\nt : Set \u03b1\nh : Set.Nonempty (s \u2229 t)\n\u22a2 diam (s \u222a t) \u2264 diam s + diam t\n[PROOFSTEP]\nrcases h with \u27e8x, \u27e8xs, xt\u27e9\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx\u271d y z : \u03b1\nt : Set \u03b1\nx : \u03b1\nxs : x \u2208 s\nxt : x \u2208 t\n\u22a2 diam (s \u222a t) \u2264 diam s + diam t\n[PROOFSTEP]\nsimpa using diam_union xs xt\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nx y z : \u03b1\nr : \u211d\nhr : 0 \u2264 r\nh : s \u2286 closedBall x r\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 s\n\u22a2 r + r = 2 * r\n[PROOFSTEP]\nsimp [mul_two, mul_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\n\u22a2 Set.Nonempty (\u22c2 (n : \u2115), s n)\n[PROOFSTEP]\nlet u N := (h N).some\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\nu : \u2115 \u2192 \u03b1 := fun N => Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n))\n\u22a2 Set.Nonempty (\u22c2 (n : \u2115), s n)\n[PROOFSTEP]\nhave I : \u2200 n N, n \u2264 N \u2192 u N \u2208 s n := by\n intro n N hn\n apply mem_of_subset_of_mem _ (h N).choose_spec\n intro x hx\n simp only [mem_iInter] at hx \n exact hx n hn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\nu : \u2115 \u2192 \u03b1 := fun N => Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n))\n\u22a2 \u2200 (n N : \u2115), n \u2264 N \u2192 u N \u2208 s n\n[PROOFSTEP]\nintro n N hn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\nu : \u2115 \u2192 \u03b1 := fun N => Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n))\nn N : \u2115\nhn : n \u2264 N\n\u22a2 u N \u2208 s n\n[PROOFSTEP]\napply mem_of_subset_of_mem _ (h N).choose_spec\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\nu : \u2115 \u2192 \u03b1 := fun N => Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n))\nn N : \u2115\nhn : n \u2264 N\n\u22a2 \u22c2 (n : \u2115) (_ : n \u2264 N), s n \u2286 s n\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx\u271d y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\nu : \u2115 \u2192 \u03b1 := fun N => Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n))\nn N : \u2115\nhn : n \u2264 N\nx : \u03b1\nhx : x \u2208 \u22c2 (n : \u2115) (_ : n \u2264 N), s n\n\u22a2 x \u2208 s n\n[PROOFSTEP]\nsimp only [mem_iInter] at hx \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx\u271d y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\nu : \u2115 \u2192 \u03b1 := fun N => Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n))\nn N : \u2115\nhn : n \u2264 N\nx : \u03b1\nhx : \u2200 (i : \u2115), i \u2264 N \u2192 x \u2208 s i\n\u22a2 x \u2208 s n\n[PROOFSTEP]\nexact hx n hn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\nu : \u2115 \u2192 \u03b1 := fun N => Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n))\nI : \u2200 (n N : \u2115), n \u2264 N \u2192 u N \u2208 s n\n\u22a2 Set.Nonempty (\u22c2 (n : \u2115), s n)\n[PROOFSTEP]\nhave : CauchySeq u := by\n apply cauchySeq_of_le_tendsto_0 _ _ h'\n intro m n N hm hn\n exact dist_le_diam_of_mem (h's N) (I _ _ hm) (I _ _ hn)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\nu : \u2115 \u2192 \u03b1 := fun N => Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n))\nI : \u2200 (n N : \u2115), n \u2264 N \u2192 u N \u2208 s n\n\u22a2 CauchySeq u\n[PROOFSTEP]\napply cauchySeq_of_le_tendsto_0 _ _ h'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\nu : \u2115 \u2192 \u03b1 := fun N => Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n))\nI : \u2200 (n N : \u2115), n \u2264 N \u2192 u N \u2208 s n\n\u22a2 \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) \u2264 diam (s N)\n[PROOFSTEP]\nintro m n N hm hn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\nu : \u2115 \u2192 \u03b1 := fun N => Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n))\nI : \u2200 (n N : \u2115), n \u2264 N \u2192 u N \u2208 s n\nm n N : \u2115\nhm : N \u2264 m\nhn : N \u2264 n\n\u22a2 dist (u m) (u n) \u2264 diam (s N)\n[PROOFSTEP]\nexact dist_le_diam_of_mem (h's N) (I _ _ hm) (I _ _ hn)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\nu : \u2115 \u2192 \u03b1 := fun N => Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n))\nI : \u2200 (n N : \u2115), n \u2264 N \u2192 u N \u2208 s n\nthis : CauchySeq u\n\u22a2 Set.Nonempty (\u22c2 (n : \u2115), s n)\n[PROOFSTEP]\nobtain \u27e8x, -, xlim\u27e9 : \u2203 x \u2208 s 0, Tendsto (fun n : \u2115 => u n) atTop (\ud835\udcdd x) :=\n cauchySeq_tendsto_of_isComplete h0 (fun n => I 0 n (zero_le _)) this\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx\u271d y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\nu : \u2115 \u2192 \u03b1 := fun N => Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n))\nI : \u2200 (n N : \u2115), n \u2264 N \u2192 u N \u2208 s n\nthis : CauchySeq u\nx : \u03b1\nxlim : Tendsto (fun n => u n) atTop (\ud835\udcdd x)\n\u22a2 Set.Nonempty (\u22c2 (n : \u2115), s n)\n[PROOFSTEP]\nrefine' \u27e8x, mem_iInter.2 fun n => _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx\u271d y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\nu : \u2115 \u2192 \u03b1 := fun N => Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n))\nI : \u2200 (n N : \u2115), n \u2264 N \u2192 u N \u2208 s n\nthis : CauchySeq u\nx : \u03b1\nxlim : Tendsto (fun n => u n) atTop (\ud835\udcdd x)\nn : \u2115\n\u22a2 x \u2208 s n\n[PROOFSTEP]\napply (hs n).mem_of_tendsto xlim\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx\u271d y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\nu : \u2115 \u2192 \u03b1 := fun N => Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n))\nI : \u2200 (n N : \u2115), n \u2264 N \u2192 u N \u2208 s n\nthis : CauchySeq u\nx : \u03b1\nxlim : Tendsto (fun n => u n) atTop (\ud835\udcdd x)\nn : \u2115\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop, u x \u2208 s n\n[PROOFSTEP]\nfilter_upwards [Ici_mem_atTop n] with p hp\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns\u271d : Set \u03b1\nx\u271d y z : \u03b1\ns : \u2115 \u2192 Set \u03b1\nh0 : IsComplete (s 0)\nhs : \u2200 (n : \u2115), IsClosed (s n)\nh's : \u2200 (n : \u2115), Bounded (s n)\nh : \u2200 (N : \u2115), Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n)\nh' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)\nu : \u2115 \u2192 \u03b1 := fun N => Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), s n))\nI : \u2200 (n N : \u2115), n \u2264 N \u2192 u N \u2208 s n\nthis : CauchySeq u\nx : \u03b1\nxlim : Tendsto (fun n => u n) atTop (\ud835\udcdd x)\nn p : \u2115\nhp : p \u2208 Ici n\n\u22a2 Set.Nonempty.some (_ : Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 p), s n)) \u2208 s n\n[PROOFSTEP]\nexact I n p hp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : PseudoMetricSpace \u03b1\ninst\u271d\u00b3 : ProperSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\na z : \u03b1\nr : \u211d\nhf : ContinuousOn f (closedBall a r)\nhz : z \u2208 closedBall a r\nhf1 : \u2200 (z' : \u03b1), z' \u2208 sphere a r \u2192 f z < f z'\n\u22a2 \u2203 z, z \u2208 ball a r \u2227 IsLocalMin f z\n[PROOFSTEP]\nsimp_rw [\u2190 closedBall_diff_ball] at hf1 \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u2074 : PseudoMetricSpace \u03b1\ninst\u271d\u00b3 : ProperSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\na z : \u03b1\nr : \u211d\nhf : ContinuousOn f (closedBall a r)\nhz : z \u2208 closedBall a r\nhf1 : \u2200 (z' : \u03b1), z' \u2208 closedBall a r \\ ball a r \u2192 f z < f z'\n\u22a2 \u2203 z, z \u2208 ball a r \u2227 IsLocalMin f z\n[PROOFSTEP]\nexact (isCompact_closedBall a r).exists_isLocalMin_mem_open ball_subset_closedBall hf hz hf1 isOpen_ball\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u22a2 comap (fun y => dist y x) atTop \u2264 cocompact \u03b1\n[PROOFSTEP]\nrefine' Filter.hasBasis_cocompact.ge_iff.2 fun s hs => mem_comap.2 _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns : Set \u03b1\nhs : IsCompact s\n\u22a2 \u2203 t, t \u2208 atTop \u2227 (fun y => dist y x) \u207b\u00b9' t \u2286 s\u1d9c\n[PROOFSTEP]\nrcases hs.bounded.subset_ball x with \u27e8r, hr\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns : Set \u03b1\nhs : IsCompact s\nr : \u211d\nhr : s \u2286 closedBall x r\n\u22a2 \u2203 t, t \u2208 atTop \u2227 (fun y => dist y x) \u207b\u00b9' t \u2286 s\u1d9c\n[PROOFSTEP]\nexact \u27e8Ioi r, Ioi_mem_atTop r, fun y hy hys => (mem_closedBall.1 <| hr hys).not_lt hy\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u22a2 comap (dist x) atTop \u2264 cocompact \u03b1\n[PROOFSTEP]\nsimpa only [dist_comm _ x] using comap_dist_right_atTop_le_cocompact x\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b2 \u2192 \u03b1\nl : Filter \u03b2\nx : \u03b1\nh : Tendsto (fun y => dist (f y) x) l atTop\n\u22a2 Tendsto f l (cocompact \u03b1)\n[PROOFSTEP]\nrefine' Tendsto.mono_right _ (comap_dist_right_atTop_le_cocompact x)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b2 \u2192 \u03b1\nl : Filter \u03b2\nx : \u03b1\nh : Tendsto (fun y => dist (f y) x) l atTop\n\u22a2 Tendsto f l (comap (fun y => dist y x) atTop)\n[PROOFSTEP]\nrwa [tendsto_comap_iff]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nm m' : MetricSpace \u03b1\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n\u22a2 m = m'\n[PROOFSTEP]\ncases m\n[GOAL]\ncase mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\nm' : MetricSpace \u03b1\ntoPseudoMetricSpace\u271d : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n\u22a2 mk eq_of_dist_eq_zero\u271d = m'\n[PROOFSTEP]\ncases m'\n[GOAL]\ncase mk.mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\ntoPseudoMetricSpace\u271d\u00b9 : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d\u00b9 : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\ntoPseudoMetricSpace\u271d : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n\u22a2 mk eq_of_dist_eq_zero\u271d\u00b9 = mk eq_of_dist_eq_zero\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toPseudoMetricSpace\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\ntoPseudoMetricSpace\u271d\u00b9 : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d\u00b9 : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\ntoPseudoMetricSpace\u271d : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n\u22a2 toPseudoMetricSpace\u271d\u00b9 = toPseudoMetricSpace\u271d\n[PROOFSTEP]\next1\n[GOAL]\ncase mk.mk.e_toPseudoMetricSpace.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_3\ntoPseudoMetricSpace\u271d\u00b9 : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d\u00b9 : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\ntoPseudoMetricSpace\u271d : PseudoMetricSpace \u03b1\neq_of_dist_eq_zero\u271d : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\nh : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n\u22a2 PseudoMetricSpace.toDist = PseudoMetricSpace.toDist\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx y : \u03b3\n\u22a2 0 = dist x y \u2194 x = y\n[PROOFSTEP]\nrw [eq_comm, dist_eq_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx y : \u03b3\n\u22a2 dist x y \u2260 0 \u2194 x \u2260 y\n[PROOFSTEP]\nsimpa only [not_iff_not] using dist_eq_zero\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx y : \u03b3\n\u22a2 dist x y \u2264 0 \u2194 x = y\n[PROOFSTEP]\nsimpa [le_antisymm_iff, dist_nonneg] using @dist_eq_zero _ _ x y\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx y : \u03b3\n\u22a2 0 < dist x y \u2194 x \u2260 y\n[PROOFSTEP]\nsimpa only [not_le] using not_congr dist_le_zero\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx y : \u03b3\n\u22a2 nndist x y = 0 \u2192 x = y\n[PROOFSTEP]\nsimp only [\u2190 NNReal.eq_iff, \u2190 dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx y : \u03b3\n\u22a2 nndist x y = 0 \u2194 x = y\n[PROOFSTEP]\nsimp only [\u2190 NNReal.eq_iff, \u2190 dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx y : \u03b3\n\u22a2 0 = nndist x y \u2194 x = y\n[PROOFSTEP]\nsimp only [\u2190 NNReal.eq_iff, \u2190 dist_nndist, imp_self, NNReal.coe_zero, zero_eq_dist]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d : \u03b3\ns : Set \u03b3\nx : \u03b3\nr : \u211d\nhr : r \u2264 0\n\u22a2 Set.Subsingleton (closedBall x r)\n[PROOFSTEP]\nrcases hr.lt_or_eq with (hr | rfl)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d : \u03b3\ns : Set \u03b3\nx : \u03b3\nr : \u211d\nhr\u271d : r \u2264 0\nhr : r < 0\n\u22a2 Set.Subsingleton (closedBall x r)\n[PROOFSTEP]\nrw [closedBall_eq_empty.2 hr]\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d : \u03b3\ns : Set \u03b3\nx : \u03b3\nr : \u211d\nhr\u271d : r \u2264 0\nhr : r < 0\n\u22a2 Set.Subsingleton \u2205\n[PROOFSTEP]\nexact subsingleton_empty\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d : \u03b3\ns : Set \u03b3\nx : \u03b3\nhr : 0 \u2264 0\n\u22a2 Set.Subsingleton (closedBall x 0)\n[PROOFSTEP]\nrw [closedBall_zero]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d : \u03b3\ns : Set \u03b3\nx : \u03b3\nhr : 0 \u2264 0\n\u22a2 Set.Subsingleton {x}\n[PROOFSTEP]\nexact subsingleton_singleton\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d\u00b9 : MetricSpace \u03b3\nx : \u03b3\ns : Set \u03b3\ninst\u271d : MetricSpace \u03b2\nf : \u03b3 \u2192 \u03b2\n\u22a2 UniformEmbedding f \u2194\n (\u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 {a b : \u03b3}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5) \u2227\n \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 {a b : \u03b3}, dist (f a) (f b) < \u03b5 \u2192 dist a b < \u03b4\n[PROOFSTEP]\nrw [uniformEmbedding_iff_uniformInducing, uniformInducing_iff, uniformContinuous_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx : \u03b3\ns\u271d s : Set \u03b3\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhs : Set.Pairwise s fun x y => \u03b5 \u2264 dist x y\n\u22a2 Set.Pairwise s fun x y => \u00ac(x, y) \u2208 {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimpa using hs\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\u271d\n\u03b3 : Type w\ninst\u271d\u00b2 : MetricSpace \u03b3\nx : \u03b3\ns : Set \u03b3\n\u03b1 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : DiscreteTopology \u03b1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nf : \u03b1 \u2192 \u03b3\nhf : Pairwise fun x y => \u03b5 \u2264 dist (f x) (f y)\n\u22a2 Pairwise fun x y => \u00ac(f x, f y) \u2208 {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimpa using hf\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx : \u03b3\ns : Set \u03b3\n\u03b2 : Type u_3\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nf : \u03b2 \u2192 \u03b1\nhf : Pairwise fun x y => \u03b5 \u2264 dist (f x) (f y)\n\u22a2 UniformSpace \u03b1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx : \u03b3\ns : Set \u03b3\n\u03b2 : Type u_3\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nf : \u03b2 \u2192 \u03b1\nhf : Pairwise fun x y => \u03b5 \u2264 dist (f x) (f y)\n\u22a2 Pairwise fun x y => \u00ac(f x, f y) \u2208 {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimpa using hf\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d\u00b2 : MetricSpace \u03b3\nx : \u03b3\ns\u271d : Set \u03b3\ninst\u271d\u00b9 : ProperSpace \u03b1\nK s : Set \u03b1\ninst\u271d : DiscreteTopology \u2191s\nhK : Bounded K\nhs : IsClosed s\n\u22a2 Set.Finite (K \u2229 s)\n[PROOFSTEP]\nrefine Set.Finite.subset (IsCompact.finite ?_ ?_) (Set.inter_subset_inter_left s subset_closure)\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d\u00b2 : MetricSpace \u03b3\nx : \u03b3\ns\u271d : Set \u03b3\ninst\u271d\u00b9 : ProperSpace \u03b1\nK s : Set \u03b1\ninst\u271d : DiscreteTopology \u2191s\nhK : Bounded K\nhs : IsClosed s\n\u22a2 IsCompact (closure K \u2229 s)\n[PROOFSTEP]\nexact IsCompact.inter_right (Metric.Bounded.isCompact_closure hK) hs\n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d\u00b2 : MetricSpace \u03b3\nx : \u03b3\ns\u271d : Set \u03b3\ninst\u271d\u00b9 : ProperSpace \u03b1\nK s : Set \u03b1\ninst\u271d : DiscreteTopology \u2191s\nhK : Bounded K\nhs : IsClosed s\n\u22a2 DiscreteTopology \u2191(closure K \u2229 s)\n[PROOFSTEP]\nexact DiscreteTopology.of_subset inferInstance (Set.inter_subset_right _ s)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3\u271d : Type w\ninst\u271d : MetricSpace \u03b3\u271d\n\u03b3 : Type u_3\nU : UniformSpace \u03b3\nm : MetricSpace \u03b3\nH : \ud835\udce4 \u03b3 = \ud835\udce4 \u03b3\n\u22a2 replaceUniformity m H = m\n[PROOFSTEP]\next\n[GOAL]\ncase h.dist.h.h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3\u271d : Type w\ninst\u271d : MetricSpace \u03b3\u271d\n\u03b3 : Type u_3\nU : UniformSpace \u03b3\nm : MetricSpace \u03b3\nH : \ud835\udce4 \u03b3 = \ud835\udce4 \u03b3\nx\u271d\u00b9 x\u271d : \u03b3\n\u22a2 dist x\u271d\u00b9 x\u271d = dist x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3\u271d : Type w\ninst\u271d : MetricSpace \u03b3\u271d\n\u03b3 : Type u_3\nU : TopologicalSpace \u03b3\nm : MetricSpace \u03b3\nH : U = UniformSpace.toTopologicalSpace\n\u22a2 replaceTopology m H = m\n[PROOFSTEP]\next\n[GOAL]\ncase h.dist.h.h\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3\u271d : Type w\ninst\u271d : MetricSpace \u03b3\u271d\n\u03b3 : Type u_3\nU : TopologicalSpace \u03b3\nm : MetricSpace \u03b3\nH : U = UniformSpace.toTopologicalSpace\nx\u271d\u00b9 x\u271d : \u03b3\n\u22a2 dist x\u271d\u00b9 x\u271d = dist x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\u271d\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\n\u03b1 : Type u_3\nm : MetricSpace \u03b1\nB : Bornology \u03b1\nH : \u2200 (s : Set \u03b1), IsBounded s \u2194 IsBounded s\n\u22a2 replaceBornology m H = m\n[PROOFSTEP]\next\n[GOAL]\ncase h.dist.h.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\u271d\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\n\u03b1 : Type u_3\nm : MetricSpace \u03b1\nB : Bornology \u03b1\nH : \u2200 (s : Set \u03b1), IsBounded s \u2194 IsBounded s\nx\u271d\u00b9 x\u271d : \u03b1\n\u22a2 dist x\u271d\u00b9 x\u271d = dist x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : Empty\n\u22a2 0 \u2264 0 + 0\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : PUnit\n\u22a2 0 \u2264 0 + 0\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\n\u22a2 \ud835\udce4 PUnit = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimp (config := { contextual := true }) [principal_univ, eq_top_of_neBot (\ud835\udce4 PUnit)]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\u271d\n\u03b3 : Type w\ninst\u271d\u00b9 : MetricSpace \u03b3\n\u03b1 : Type u\ninst\u271d : MetricSpace \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u03b1), F x = F y \u2192 dist x y \u2264 \u03b5\n\u22a2 SecondCountableTopology \u03b1\n[PROOFSTEP]\nrefine secondCountable_of_almost_dense_set fun \u03b5 \u03b50 => ?_\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\u271d\n\u03b3 : Type w\ninst\u271d\u00b9 : MetricSpace \u03b3\n\u03b1 : Type u\ninst\u271d : MetricSpace \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u03b1), F x = F y \u2192 dist x y \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 s, Set.Countable s \u2227 \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 dist x y \u2264 \u03b5\n[PROOFSTEP]\nrcases H \u03b5 \u03b50 with \u27e8\u03b2, f\u03b2, F, hF\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\u271d\n\u03b3 : Type w\ninst\u271d\u00b9 : MetricSpace \u03b3\n\u03b1 : Type u\ninst\u271d : MetricSpace \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u03b1), F x = F y \u2192 dist x y \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b2 : Type u_3\nf\u03b2 : Encodable \u03b2\nF : \u03b1 \u2192 \u03b2\nhF : \u2200 (x y : \u03b1), F x = F y \u2192 dist x y \u2264 \u03b5\n\u22a2 \u2203 s, Set.Countable s \u2227 \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 dist x y \u2264 \u03b5\n[PROOFSTEP]\nlet Finv := rangeSplitting F\n[GOAL]\ncase intro.intro.intro\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\u271d\n\u03b3 : Type w\ninst\u271d\u00b9 : MetricSpace \u03b3\n\u03b1 : Type u\ninst\u271d : MetricSpace \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u03b1), F x = F y \u2192 dist x y \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b2 : Type u_3\nf\u03b2 : Encodable \u03b2\nF : \u03b1 \u2192 \u03b2\nhF : \u2200 (x y : \u03b1), F x = F y \u2192 dist x y \u2264 \u03b5\nFinv : \u2191(range F) \u2192 \u03b1 := rangeSplitting F\n\u22a2 \u2203 s, Set.Countable s \u2227 \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 dist x y \u2264 \u03b5\n[PROOFSTEP]\nrefine \u27e8range Finv, \u27e8countable_range _, fun x => ?_\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\u271d\n\u03b3 : Type w\ninst\u271d\u00b9 : MetricSpace \u03b3\n\u03b1 : Type u\ninst\u271d : MetricSpace \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u03b1), F x = F y \u2192 dist x y \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b2 : Type u_3\nf\u03b2 : Encodable \u03b2\nF : \u03b1 \u2192 \u03b2\nhF : \u2200 (x y : \u03b1), F x = F y \u2192 dist x y \u2264 \u03b5\nFinv : \u2191(range F) \u2192 \u03b1 := rangeSplitting F\nx : \u03b1\n\u22a2 \u2203 y, y \u2208 range Finv \u2227 dist x y \u2264 \u03b5\n[PROOFSTEP]\nlet x' := Finv \u27e8F x, mem_range_self _\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\u271d\n\u03b3 : Type w\ninst\u271d\u00b9 : MetricSpace \u03b3\n\u03b1 : Type u\ninst\u271d : MetricSpace \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u03b1), F x = F y \u2192 dist x y \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b2 : Type u_3\nf\u03b2 : Encodable \u03b2\nF : \u03b1 \u2192 \u03b2\nhF : \u2200 (x y : \u03b1), F x = F y \u2192 dist x y \u2264 \u03b5\nFinv : \u2191(range F) \u2192 \u03b1 := rangeSplitting F\nx : \u03b1\nx' : \u03b1 := Finv { val := F x, property := (_ : F x \u2208 range F) }\n\u22a2 \u2203 y, y \u2208 range Finv \u2227 dist x y \u2264 \u03b5\n[PROOFSTEP]\nhave : F x' = F x := apply_rangeSplitting F _\n[GOAL]\ncase intro.intro.intro\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\u271d\n\u03b3 : Type w\ninst\u271d\u00b9 : MetricSpace \u03b3\n\u03b1 : Type u\ninst\u271d : MetricSpace \u03b1\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b2 x F, \u2200 (x y : \u03b1), F x = F y \u2192 dist x y \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b2 : Type u_3\nf\u03b2 : Encodable \u03b2\nF : \u03b1 \u2192 \u03b2\nhF : \u2200 (x y : \u03b1), F x = F y \u2192 dist x y \u2264 \u03b5\nFinv : \u2191(range F) \u2192 \u03b1 := rangeSplitting F\nx : \u03b1\nx' : \u03b1 := Finv { val := F x, property := (_ : F x \u2208 range F) }\nthis : F x' = F x\n\u22a2 \u2203 y, y \u2208 range Finv \u2227 dist x y \u2264 \u03b5\n[PROOFSTEP]\nexact \u27e8x', mem_range_self _, hF _ _ this.symm\u27e9\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\u271d\n\u03b3 : Type w\ninst\u271d\u00b9 : MetricSpace \u03b3\n\u03b1 : Type u\ninst\u271d : PseudoMetricSpace \u03b1\np q : UniformSpace.SeparationQuotient \u03b1\nx y x' y' : \u03b1\nhx : Setoid.r x x'\nhy : Setoid.r y y'\n\u22a2 dist x y = dist x' y'\n[PROOFSTEP]\nrw [dist_edist, dist_edist, \u2190 UniformSpace.SeparationQuotient.edist_mk x, \u2190 UniformSpace.SeparationQuotient.edist_mk x',\n Quot.sound hx, Quot.sound hy]\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.Basic", "llama_tokens": 104597, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.417623796192907}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cone (K \u22d9 F)) \u2192 (t : IsLimit c) \u2192 LiftsToLimit K F c t\nd : Cone K\nhd : IsLimit (F.mapCone d)\n\u22a2 IsLimit d\n[PROOFSTEP]\nlet d' : Cone K := (h (F.mapCone d) hd).toLiftableCone.liftedCone\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cone (K \u22d9 F)) \u2192 (t : IsLimit c) \u2192 LiftsToLimit K F c t\nd : Cone K\nhd : IsLimit (F.mapCone d)\nd' : Cone K := (h (F.mapCone d) hd).liftedCone\n\u22a2 IsLimit d\n[PROOFSTEP]\nlet i : F.mapCone d' \u2245 F.mapCone d := (h (F.mapCone d) hd).toLiftableCone.validLift\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cone (K \u22d9 F)) \u2192 (t : IsLimit c) \u2192 LiftsToLimit K F c t\nd : Cone K\nhd : IsLimit (F.mapCone d)\nd' : Cone K := (h (F.mapCone d) hd).liftedCone\ni : F.mapCone d' \u2245 F.mapCone d := (h (F.mapCone d) hd).validLift\n\u22a2 IsLimit d\n[PROOFSTEP]\nlet hd' : IsLimit d' := (h (F.mapCone d) hd).makesLimit\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cone (K \u22d9 F)) \u2192 (t : IsLimit c) \u2192 LiftsToLimit K F c t\nd : Cone K\nhd : IsLimit (F.mapCone d)\nd' : Cone K := (h (F.mapCone d) hd).liftedCone\ni : F.mapCone d' \u2245 F.mapCone d := (h (F.mapCone d) hd).validLift\nhd' : IsLimit d' := (h (F.mapCone d) hd).makesLimit\n\u22a2 IsLimit d\n[PROOFSTEP]\nlet f : d \u27f6 d' := hd'.liftConeMorphism d\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cone (K \u22d9 F)) \u2192 (t : IsLimit c) \u2192 LiftsToLimit K F c t\nd : Cone K\nhd : IsLimit (F.mapCone d)\nd' : Cone K := (h (F.mapCone d) hd).liftedCone\ni : F.mapCone d' \u2245 F.mapCone d := (h (F.mapCone d) hd).validLift\nhd' : IsLimit d' := (h (F.mapCone d) hd).makesLimit\nf : d \u27f6 d' := IsLimit.liftConeMorphism hd' d\n\u22a2 IsLimit d\n[PROOFSTEP]\nhave : (Cones.functoriality K F).map f = i.inv := (hd.ofIsoLimit i.symm).uniq_cone_morphism\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cone (K \u22d9 F)) \u2192 (t : IsLimit c) \u2192 LiftsToLimit K F c t\nd : Cone K\nhd : IsLimit (F.mapCone d)\nd' : Cone K := (h (F.mapCone d) hd).liftedCone\ni : F.mapCone d' \u2245 F.mapCone d := (h (F.mapCone d) hd).validLift\nhd' : IsLimit d' := (h (F.mapCone d) hd).makesLimit\nf : d \u27f6 d' := IsLimit.liftConeMorphism hd' d\nthis : (Cones.functoriality K F).map f = i.inv\n\u22a2 IsLimit d\n[PROOFSTEP]\nhaveI : IsIso ((Cones.functoriality K F).map f) := by\n rw [this]\n infer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cone (K \u22d9 F)) \u2192 (t : IsLimit c) \u2192 LiftsToLimit K F c t\nd : Cone K\nhd : IsLimit (F.mapCone d)\nd' : Cone K := (h (F.mapCone d) hd).liftedCone\ni : F.mapCone d' \u2245 F.mapCone d := (h (F.mapCone d) hd).validLift\nhd' : IsLimit d' := (h (F.mapCone d) hd).makesLimit\nf : d \u27f6 d' := IsLimit.liftConeMorphism hd' d\nthis : (Cones.functoriality K F).map f = i.inv\n\u22a2 IsIso ((Cones.functoriality K F).map f)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cone (K \u22d9 F)) \u2192 (t : IsLimit c) \u2192 LiftsToLimit K F c t\nd : Cone K\nhd : IsLimit (F.mapCone d)\nd' : Cone K := (h (F.mapCone d) hd).liftedCone\ni : F.mapCone d' \u2245 F.mapCone d := (h (F.mapCone d) hd).validLift\nhd' : IsLimit d' := (h (F.mapCone d) hd).makesLimit\nf : d \u27f6 d' := IsLimit.liftConeMorphism hd' d\nthis : (Cones.functoriality K F).map f = i.inv\n\u22a2 IsIso i.inv\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cone (K \u22d9 F)) \u2192 (t : IsLimit c) \u2192 LiftsToLimit K F c t\nd : Cone K\nhd : IsLimit (F.mapCone d)\nd' : Cone K := (h (F.mapCone d) hd).liftedCone\ni : F.mapCone d' \u2245 F.mapCone d := (h (F.mapCone d) hd).validLift\nhd' : IsLimit d' := (h (F.mapCone d) hd).makesLimit\nf : d \u27f6 d' := IsLimit.liftConeMorphism hd' d\nthis\u271d : (Cones.functoriality K F).map f = i.inv\nthis : IsIso ((Cones.functoriality K F).map f)\n\u22a2 IsLimit d\n[PROOFSTEP]\nhaveI : IsIso f := isIso_of_reflects_iso f (Cones.functoriality K F)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cone (K \u22d9 F)) \u2192 (t : IsLimit c) \u2192 LiftsToLimit K F c t\nd : Cone K\nhd : IsLimit (F.mapCone d)\nd' : Cone K := (h (F.mapCone d) hd).liftedCone\ni : F.mapCone d' \u2245 F.mapCone d := (h (F.mapCone d) hd).validLift\nhd' : IsLimit d' := (h (F.mapCone d) hd).makesLimit\nf : d \u27f6 d' := IsLimit.liftConeMorphism hd' d\nthis\u271d\u00b9 : (Cones.functoriality K F).map f = i.inv\nthis\u271d : IsIso ((Cones.functoriality K F).map f)\nthis : IsIso f\n\u22a2 IsLimit d\n[PROOFSTEP]\nexact IsLimit.ofIsoLimit hd' (asIso f).symm\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b2 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nl : Cone (K \u22d9 F)\nhl : IsLimit l\nX : C\ni : F.obj X \u2245 l.pt\nY Z : J\nf : Y \u27f6 Z\n\u22a2 F.map (((Functor.const J).obj X).map f \u226b (fun j => F.preimage (i.hom \u226b NatTrans.app l.\u03c0 j)) Z) =\n F.map ((fun j => F.preimage (i.hom \u226b NatTrans.app l.\u03c0 j)) Y \u226b K.map f)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b2 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nl : Cone (K \u22d9 F)\nhl : IsLimit l\nX : C\ni : F.obj X \u2245 l.pt\nY Z : J\nf : Y \u27f6 Z\n\u22a2 F.map (\ud835\udfd9 X \u226b F.preimage (i.hom \u226b NatTrans.app l.\u03c0 Z)) = F.map (F.preimage (i.hom \u226b NatTrans.app l.\u03c0 Y) \u226b K.map f)\n[PROOFSTEP]\nsimpa using (l.w f).symm\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b2 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nl : Cone (K \u22d9 F)\nhl : IsLimit l\nX : C\ni : F.obj X \u2245 l.pt\nj : J\n\u22a2 NatTrans.app (F.mapCone { pt := X, \u03c0 := NatTrans.mk fun j => F.preimage (i.hom \u226b NatTrans.app l.\u03c0 j) }).\u03c0 j =\n i.hom \u226b NatTrans.app l.\u03c0 j\n[PROOFSTEP]\nsimp only [Functor.image_preimage, Functor.mapCone_\u03c0_app]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cocone (K \u22d9 F)) \u2192 (t : IsColimit c) \u2192 LiftsToColimit K F c t\nd : Cocone K\nhd : IsColimit (F.mapCocone d)\n\u22a2 IsColimit d\n[PROOFSTEP]\nlet d' : Cocone K := (h (F.mapCocone d) hd).toLiftableCocone.liftedCocone\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cocone (K \u22d9 F)) \u2192 (t : IsColimit c) \u2192 LiftsToColimit K F c t\nd : Cocone K\nhd : IsColimit (F.mapCocone d)\nd' : Cocone K := (h (F.mapCocone d) hd).liftedCocone\n\u22a2 IsColimit d\n[PROOFSTEP]\nlet i : F.mapCocone d' \u2245 F.mapCocone d := (h (F.mapCocone d) hd).toLiftableCocone.validLift\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cocone (K \u22d9 F)) \u2192 (t : IsColimit c) \u2192 LiftsToColimit K F c t\nd : Cocone K\nhd : IsColimit (F.mapCocone d)\nd' : Cocone K := (h (F.mapCocone d) hd).liftedCocone\ni : F.mapCocone d' \u2245 F.mapCocone d := (h (F.mapCocone d) hd).validLift\n\u22a2 IsColimit d\n[PROOFSTEP]\nlet hd' : IsColimit d' := (h (F.mapCocone d) hd).makesColimit\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cocone (K \u22d9 F)) \u2192 (t : IsColimit c) \u2192 LiftsToColimit K F c t\nd : Cocone K\nhd : IsColimit (F.mapCocone d)\nd' : Cocone K := (h (F.mapCocone d) hd).liftedCocone\ni : F.mapCocone d' \u2245 F.mapCocone d := (h (F.mapCocone d) hd).validLift\nhd' : IsColimit d' := (h (F.mapCocone d) hd).makesColimit\n\u22a2 IsColimit d\n[PROOFSTEP]\nlet f : d' \u27f6 d := hd'.descCoconeMorphism d\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cocone (K \u22d9 F)) \u2192 (t : IsColimit c) \u2192 LiftsToColimit K F c t\nd : Cocone K\nhd : IsColimit (F.mapCocone d)\nd' : Cocone K := (h (F.mapCocone d) hd).liftedCocone\ni : F.mapCocone d' \u2245 F.mapCocone d := (h (F.mapCocone d) hd).validLift\nhd' : IsColimit d' := (h (F.mapCocone d) hd).makesColimit\nf : d' \u27f6 d := IsColimit.descCoconeMorphism hd' d\n\u22a2 IsColimit d\n[PROOFSTEP]\nhave : (Cocones.functoriality K F).map f = i.hom := (hd.ofIsoColimit i.symm).uniq_cocone_morphism\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cocone (K \u22d9 F)) \u2192 (t : IsColimit c) \u2192 LiftsToColimit K F c t\nd : Cocone K\nhd : IsColimit (F.mapCocone d)\nd' : Cocone K := (h (F.mapCocone d) hd).liftedCocone\ni : F.mapCocone d' \u2245 F.mapCocone d := (h (F.mapCocone d) hd).validLift\nhd' : IsColimit d' := (h (F.mapCocone d) hd).makesColimit\nf : d' \u27f6 d := IsColimit.descCoconeMorphism hd' d\nthis : (Cocones.functoriality K F).map f = i.hom\n\u22a2 IsColimit d\n[PROOFSTEP]\nhaveI : IsIso ((Cocones.functoriality K F).map f) := by\n rw [this]\n infer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cocone (K \u22d9 F)) \u2192 (t : IsColimit c) \u2192 LiftsToColimit K F c t\nd : Cocone K\nhd : IsColimit (F.mapCocone d)\nd' : Cocone K := (h (F.mapCocone d) hd).liftedCocone\ni : F.mapCocone d' \u2245 F.mapCocone d := (h (F.mapCocone d) hd).validLift\nhd' : IsColimit d' := (h (F.mapCocone d) hd).makesColimit\nf : d' \u27f6 d := IsColimit.descCoconeMorphism hd' d\nthis : (Cocones.functoriality K F).map f = i.hom\n\u22a2 IsIso ((Cocones.functoriality K F).map f)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cocone (K \u22d9 F)) \u2192 (t : IsColimit c) \u2192 LiftsToColimit K F c t\nd : Cocone K\nhd : IsColimit (F.mapCocone d)\nd' : Cocone K := (h (F.mapCocone d) hd).liftedCocone\ni : F.mapCocone d' \u2245 F.mapCocone d := (h (F.mapCocone d) hd).validLift\nhd' : IsColimit d' := (h (F.mapCocone d) hd).makesColimit\nf : d' \u27f6 d := IsColimit.descCoconeMorphism hd' d\nthis : (Cocones.functoriality K F).map f = i.hom\n\u22a2 IsIso i.hom\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cocone (K \u22d9 F)) \u2192 (t : IsColimit c) \u2192 LiftsToColimit K F c t\nd : Cocone K\nhd : IsColimit (F.mapCocone d)\nd' : Cocone K := (h (F.mapCocone d) hd).liftedCocone\ni : F.mapCocone d' \u2245 F.mapCocone d := (h (F.mapCocone d) hd).validLift\nhd' : IsColimit d' := (h (F.mapCocone d) hd).makesColimit\nf : d' \u27f6 d := IsColimit.descCoconeMorphism hd' d\nthis\u271d : (Cocones.functoriality K F).map f = i.hom\nthis : IsIso ((Cocones.functoriality K F).map f)\n\u22a2 IsColimit d\n[PROOFSTEP]\nhaveI := isIso_of_reflects_iso f (Cocones.functoriality K F)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d : ReflectsIsomorphisms F\nh : (c : Cocone (K \u22d9 F)) \u2192 (t : IsColimit c) \u2192 LiftsToColimit K F c t\nd : Cocone K\nhd : IsColimit (F.mapCocone d)\nd' : Cocone K := (h (F.mapCocone d) hd).liftedCocone\ni : F.mapCocone d' \u2245 F.mapCocone d := (h (F.mapCocone d) hd).validLift\nhd' : IsColimit d' := (h (F.mapCocone d) hd).makesColimit\nf : d' \u27f6 d := IsColimit.descCoconeMorphism hd' d\nthis\u271d\u00b9 : (Cocones.functoriality K F).map f = i.hom\nthis\u271d : IsIso ((Cocones.functoriality K F).map f)\nthis : IsIso f\n\u22a2 IsColimit d\n[PROOFSTEP]\nexact IsColimit.ofIsoColimit hd' (asIso f)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b2 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nl : Cocone (K \u22d9 F)\nhl : IsColimit l\nX : C\ni : F.obj X \u2245 l.pt\nY Z : J\nf : Y \u27f6 Z\n\u22a2 F.map (K.map f \u226b (fun j => F.preimage (NatTrans.app l.\u03b9 j \u226b i.inv)) Z) =\n F.map ((fun j => F.preimage (NatTrans.app l.\u03b9 j \u226b i.inv)) Y \u226b ((Functor.const J).obj X).map f)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b2 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nl : Cocone (K \u22d9 F)\nhl : IsColimit l\nX : C\ni : F.obj X \u2245 l.pt\nY Z : J\nf : Y \u27f6 Z\n\u22a2 F.map (K.map f \u226b F.preimage (NatTrans.app l.\u03b9 Z \u226b i.inv)) = F.map (F.preimage (NatTrans.app l.\u03b9 Y \u226b i.inv) \u226b \ud835\udfd9 X)\n[PROOFSTEP]\nsimpa [\u2190 cancel_mono i.hom] using l.w f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b2 : Category.{w', w} J\nK\u271d K : J \u2964 C\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nl : Cocone (K \u22d9 F)\nhl : IsColimit l\nX : C\ni : F.obj X \u2245 l.pt\nj : J\n\u22a2 NatTrans.app (F.mapCocone { pt := X, \u03b9 := NatTrans.mk fun j => F.preimage (NatTrans.app l.\u03b9 j \u226b i.inv) }).\u03b9 j \u226b\n i.hom =\n NatTrans.app l.\u03b9 j\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK K\u2081 K\u2082 : J \u2964 C\nF : C \u2964 D\nh : K\u2081 \u2245 K\u2082\ninst\u271d : CreatesLimit K\u2081 F\nsrc\u271d : ReflectsLimit K\u2082 F := reflectsLimitOfIsoDiagram F h\nc : Cone (K\u2082 \u22d9 F)\nt : IsLimit c\nt' : (fun x => IsLimit ((Cones.postcompose (isoWhiskerRight h F).inv).obj c)) t :=\n \u2191(IsLimit.postcomposeInvEquiv (isoWhiskerRight h F) c).symm t\nj : J\n\u22a2 NatTrans.app\n ((Cones.postcompose (isoWhiskerRight h F).hom).obj ((Cones.postcompose (isoWhiskerRight h F).inv).obj c)).\u03c0 j =\n (Iso.refl\n ((Cones.postcompose (isoWhiskerRight h F).hom).obj\n ((Cones.postcompose (isoWhiskerRight h F).inv).obj c)).pt).hom \u226b\n NatTrans.app c.\u03c0 j\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK K\u2081 K\u2082 : J \u2964 C\nF : C \u2964 D\nh : K\u2081 \u2245 K\u2082\ninst\u271d : CreatesLimit K\u2081 F\nsrc\u271d : ReflectsLimit K\u2082 F := reflectsLimitOfIsoDiagram F h\nc : Cone (K\u2082 \u22d9 F)\nt : IsLimit c\nt' : (fun x => IsLimit ((Cones.postcompose (isoWhiskerRight h F).inv).obj c)) t :=\n \u2191(IsLimit.postcomposeInvEquiv (isoWhiskerRight h F) c).symm t\nj : J\n\u22a2 (NatTrans.app c.\u03c0 j \u226b F.map (NatTrans.app h.inv j)) \u226b F.map (NatTrans.app h.hom j) = \ud835\udfd9 c.pt \u226b NatTrans.app c.\u03c0 j\n[PROOFSTEP]\nrw [Category.assoc, \u2190 F.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK K\u2081 K\u2082 : J \u2964 C\nF : C \u2964 D\nh : K\u2081 \u2245 K\u2082\ninst\u271d : CreatesLimit K\u2081 F\nsrc\u271d : ReflectsLimit K\u2082 F := reflectsLimitOfIsoDiagram F h\nc : Cone (K\u2082 \u22d9 F)\nt : IsLimit c\nt' : (fun x => IsLimit ((Cones.postcompose (isoWhiskerRight h F).inv).obj c)) t :=\n \u2191(IsLimit.postcomposeInvEquiv (isoWhiskerRight h F) c).symm t\nj : J\n\u22a2 NatTrans.app c.\u03c0 j \u226b F.map (NatTrans.app h.inv j \u226b NatTrans.app h.hom j) = \ud835\udfd9 c.pt \u226b NatTrans.app c.\u03c0 j\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK : J \u2964 C\nF G : C \u2964 D\nh : F \u2245 G\ninst\u271d : CreatesLimit K F\nc : Cone (K \u22d9 G)\nt : IsLimit c\n\u22a2 G.mapCone (liftLimit (\u2191(IsLimit.postcomposeInvEquiv (isoWhiskerLeft K h) c).symm t)) \u2245 c\n[PROOFSTEP]\nrefine' (IsLimit.mapConeEquiv h _).uniqueUpToIso t\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK : J \u2964 C\nF G : C \u2964 D\nh : F \u2245 G\ninst\u271d : CreatesLimit K F\nc : Cone (K \u22d9 G)\nt : IsLimit c\n\u22a2 IsLimit (F.mapCone (liftLimit (\u2191(IsLimit.postcomposeInvEquiv (isoWhiskerLeft K h) c).symm t)))\n[PROOFSTEP]\napply IsLimit.ofIsoLimit _ (liftedLimitMapsToOriginal _).symm\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK : J \u2964 C\nF G : C \u2964 D\nh : F \u2245 G\ninst\u271d : CreatesLimit K F\nc : Cone (K \u22d9 G)\nt : IsLimit c\n\u22a2 IsLimit ((Cones.postcompose (isoWhiskerLeft K h).inv).obj c)\n[PROOFSTEP]\napply (IsLimit.postcomposeInvEquiv _ _).symm t\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK K\u2081 K\u2082 : J \u2964 C\nF : C \u2964 D\nh : K\u2081 \u2245 K\u2082\ninst\u271d : CreatesColimit K\u2081 F\nsrc\u271d : ReflectsColimit K\u2082 F := reflectsColimitOfIsoDiagram F h\nc : Cocone (K\u2082 \u22d9 F)\nt : IsColimit c\nt' : (fun x => IsColimit ((Cocones.precompose (isoWhiskerRight h F).hom).obj c)) t :=\n \u2191(IsColimit.precomposeHomEquiv (isoWhiskerRight h F) c).symm t\nj : J\n\u22a2 NatTrans.app\n ((Cocones.precompose (isoWhiskerRight h F).inv).obj ((Cocones.precompose (isoWhiskerRight h F).hom).obj c)).\u03b9\n j \u226b\n (Iso.refl\n ((Cocones.precompose (isoWhiskerRight h F).inv).obj\n ((Cocones.precompose (isoWhiskerRight h F).hom).obj c)).pt).hom =\n NatTrans.app c.\u03b9 j\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK K\u2081 K\u2082 : J \u2964 C\nF : C \u2964 D\nh : K\u2081 \u2245 K\u2082\ninst\u271d : CreatesColimit K\u2081 F\nsrc\u271d : ReflectsColimit K\u2082 F := reflectsColimitOfIsoDiagram F h\nc : Cocone (K\u2082 \u22d9 F)\nt : IsColimit c\nt' : (fun x => IsColimit ((Cocones.precompose (isoWhiskerRight h F).hom).obj c)) t :=\n \u2191(IsColimit.precomposeHomEquiv (isoWhiskerRight h F) c).symm t\nj : J\n\u22a2 (F.map (NatTrans.app h.inv j) \u226b F.map (NatTrans.app h.hom j) \u226b NatTrans.app c.\u03b9 j) \u226b \ud835\udfd9 c.pt = NatTrans.app c.\u03b9 j\n[PROOFSTEP]\nrw [\u2190 F.map_comp_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK K\u2081 K\u2082 : J \u2964 C\nF : C \u2964 D\nh : K\u2081 \u2245 K\u2082\ninst\u271d : CreatesColimit K\u2081 F\nsrc\u271d : ReflectsColimit K\u2082 F := reflectsColimitOfIsoDiagram F h\nc : Cocone (K\u2082 \u22d9 F)\nt : IsColimit c\nt' : (fun x => IsColimit ((Cocones.precompose (isoWhiskerRight h F).hom).obj c)) t :=\n \u2191(IsColimit.precomposeHomEquiv (isoWhiskerRight h F) c).symm t\nj : J\n\u22a2 (F.map (NatTrans.app h.inv j \u226b NatTrans.app h.hom j) \u226b NatTrans.app c.\u03b9 j) \u226b \ud835\udfd9 c.pt = NatTrans.app c.\u03b9 j\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK : J \u2964 C\nF G : C \u2964 D\nh : F \u2245 G\ninst\u271d : CreatesColimit K F\nc : Cocone (K \u22d9 G)\nt : IsColimit c\n\u22a2 G.mapCocone (liftColimit (\u2191(IsColimit.precomposeHomEquiv (isoWhiskerLeft K h) c).symm t)) \u2245 c\n[PROOFSTEP]\nrefine' (IsColimit.mapCoconeEquiv h _).uniqueUpToIso t\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK : J \u2964 C\nF G : C \u2964 D\nh : F \u2245 G\ninst\u271d : CreatesColimit K F\nc : Cocone (K \u22d9 G)\nt : IsColimit c\n\u22a2 IsColimit (F.mapCocone (liftColimit (\u2191(IsColimit.precomposeHomEquiv (isoWhiskerLeft K h) c).symm t)))\n[PROOFSTEP]\napply IsColimit.ofIsoColimit _ (liftedColimitMapsToOriginal _).symm\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b9 : Category.{w', w} J\nK : J \u2964 C\nF G : C \u2964 D\nh : F \u2245 G\ninst\u271d : CreatesColimit K F\nc : Cocone (K \u22d9 G)\nt : IsColimit c\n\u22a2 IsColimit ((Cocones.precompose (isoWhiskerLeft K h).hom).obj c)\n[PROOFSTEP]\napply (IsColimit.precomposeHomEquiv _ _).symm t\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b2 : Category.{w', w} J\nK : J \u2964 C\nE : Type u\u2083\n\u2130 : Category.{v\u2083, u\u2083} E\nF : C \u2964 D\nG : D \u2964 E\ninst\u271d\u00b9 : CreatesLimit K F\ninst\u271d : CreatesLimit (K \u22d9 F) G\nc : Cone (K \u22d9 F \u22d9 G)\nt : IsLimit c\n\u22a2 LiftableCone K (F \u22d9 G) c\n[PROOFSTEP]\nlet c' : Cone ((K \u22d9 F) \u22d9 G) := c\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b2 : Category.{w', w} J\nK : J \u2964 C\nE : Type u\u2083\n\u2130 : Category.{v\u2083, u\u2083} E\nF : C \u2964 D\nG : D \u2964 E\ninst\u271d\u00b9 : CreatesLimit K F\ninst\u271d : CreatesLimit (K \u22d9 F) G\nc : Cone (K \u22d9 F \u22d9 G)\nt : IsLimit c\nc' : Cone ((K \u22d9 F) \u22d9 G) := c\n\u22a2 LiftableCone K (F \u22d9 G) c\n[PROOFSTEP]\nlet t' : IsLimit c' := t\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nJ : Type w\ninst\u271d\u00b2 : Category.{w', w} J\nK : J \u2964 C\nE : Type u\u2083\n\u2130 : Category.{v\u2083, u\u2083} E\nF : C \u2964 D\nG : D \u2964 E\ninst\u271d\u00b9 : CreatesLimit K F\ninst\u271d : CreatesLimit (K \u22d9 F) G\nc : Cone (K \u22d9 F \u22d9 G)\nt : IsLimit c\nc' : Cone ((K \u22d9 F) \u22d9 G) := c\nt' : IsLimit c' := t\n\u22a2 LiftableCone K (F \u22d9 G) c\n[PROOFSTEP]\nexact\n { liftedCone := liftLimit (liftedLimitIsLimit t')\n validLift :=\n (Cones.functoriality (K \u22d9 F) G).mapIso (liftedLimitMapsToOriginal (liftedLimitIsLimit t')) \u226a\u226b\n liftedLimitMapsToOriginal t' }\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Creates", "llama_tokens": 11279, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754371026367, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.41762378942158523}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\ns : Finset R\ndegree_f_lt : degree f < \u2191(Finset.card s)\neval_f : \u2200 (x : R), x \u2208 s \u2192 eval x f = 0\n\u22a2 f = 0\n[PROOFSTEP]\nrw [\u2190 mem_degreeLT] at degree_f_lt \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\ns : Finset R\ndegree_f_lt : f \u2208 degreeLT R (Finset.card s)\neval_f : \u2200 (x : R), x \u2208 s \u2192 eval x f = 0\n\u22a2 f = 0\n[PROOFSTEP]\nsimp_rw [eval_eq_sum_degreeLTEquiv degree_f_lt] at eval_f \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\ns : Finset R\ndegree_f_lt : f \u2208 degreeLT R (Finset.card s)\neval_f :\n \u2200 (x : R),\n x \u2208 s \u2192\n \u2211 x_1 : Fin (Finset.card s),\n \u2191(degreeLTEquiv R (Finset.card s)) { val := f, property := degree_f_lt } x_1 * x ^ \u2191x_1 =\n 0\n\u22a2 f = 0\n[PROOFSTEP]\nrw [\u2190 degreeLTEquiv_eq_zero_iff_eq_zero degree_f_lt]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\ns : Finset R\ndegree_f_lt : f \u2208 degreeLT R (Finset.card s)\neval_f :\n \u2200 (x : R),\n x \u2208 s \u2192\n \u2211 x_1 : Fin (Finset.card s),\n \u2191(degreeLTEquiv R (Finset.card s)) { val := f, property := degree_f_lt } x_1 * x ^ \u2191x_1 =\n 0\n\u22a2 \u2191(degreeLTEquiv R (Finset.card s)) { val := f, property := degree_f_lt } = 0\n[PROOFSTEP]\nexact\n Matrix.eq_zero_of_forall_index_sum_mul_pow_eq_zero\n (Injective.comp (Embedding.subtype _).inj' (equivFinOfCardEq (card_coe _)).symm.injective) fun _ =>\n eval_f _ (Finset.coe_mem _)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\ns : Finset R\ndegree_fg_lt : degree (f - g) < \u2191(Finset.card s)\neval_fg : \u2200 (x : R), x \u2208 s \u2192 eval x f = eval x g\n\u22a2 f = g\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\ns : Finset R\ndegree_fg_lt : degree (f - g) < \u2191(Finset.card s)\neval_fg : \u2200 (x : R), x \u2208 s \u2192 eval x f = eval x g\n\u22a2 f - g = 0\n[PROOFSTEP]\nrefine' eq_zero_of_degree_lt_of_eval_finset_eq_zero _ degree_fg_lt _\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\ns : Finset R\ndegree_fg_lt : degree (f - g) < \u2191(Finset.card s)\neval_fg : \u2200 (x : R), x \u2208 s \u2192 eval x f = eval x g\n\u22a2 \u2200 (x : R), x \u2208 s \u2192 eval x (f - g) = 0\n[PROOFSTEP]\nsimp_rw [eval_sub, sub_eq_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\ns : Finset R\ndegree_fg_lt : degree (f - g) < \u2191(Finset.card s)\neval_fg : \u2200 (x : R), x \u2208 s \u2192 eval x f = eval x g\n\u22a2 \u2200 (x : R), x \u2208 s \u2192 eval x f = eval x g\n[PROOFSTEP]\nexact eval_fg\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\ns : Finset R\ndegree_f_lt : degree f < \u2191(Finset.card s)\ndegree_g_lt : degree g < \u2191(Finset.card s)\neval_fg : \u2200 (x : R), x \u2208 s \u2192 eval x f = eval x g\n\u22a2 f = g\n[PROOFSTEP]\nrw [\u2190 mem_degreeLT] at degree_f_lt degree_g_lt \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\ns : Finset R\ndegree_f_lt : f \u2208 degreeLT R (Finset.card s)\ndegree_g_lt : g \u2208 degreeLT R (Finset.card s)\neval_fg : \u2200 (x : R), x \u2208 s \u2192 eval x f = eval x g\n\u22a2 f = g\n[PROOFSTEP]\nrefine' eq_of_degree_sub_lt_of_eval_finset_eq _ _ eval_fg\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\ns : Finset R\ndegree_f_lt : f \u2208 degreeLT R (Finset.card s)\ndegree_g_lt : g \u2208 degreeLT R (Finset.card s)\neval_fg : \u2200 (x : R), x \u2208 s \u2192 eval x f = eval x g\n\u22a2 degree (f - g) < \u2191(Finset.card s)\n[PROOFSTEP]\nrw [\u2190 mem_degreeLT]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\ns : Finset R\ndegree_f_lt : f \u2208 degreeLT R (Finset.card s)\ndegree_g_lt : g \u2208 degreeLT R (Finset.card s)\neval_fg : \u2200 (x : R), x \u2208 s \u2192 eval x f = eval x g\n\u22a2 f - g \u2208 degreeLT R (Finset.card s)\n[PROOFSTEP]\nexact Submodule.sub_mem _ degree_f_lt degree_g_lt\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\ns : Finset R\nh_deg_le : degree f \u2264 \u2191(Finset.card s)\nh_deg_eq : degree f = degree g\nhlc : leadingCoeff f = leadingCoeff g\nh_eval : \u2200 (x : R), x \u2208 s \u2192 eval x f = eval x g\n\u22a2 f = g\n[PROOFSTEP]\nrcases eq_or_ne f 0 with rfl | hf\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ng : R[X]\ns : Finset R\nh_deg_le : degree 0 \u2264 \u2191(Finset.card s)\nh_deg_eq : degree 0 = degree g\nhlc : leadingCoeff 0 = leadingCoeff g\nh_eval : \u2200 (x : R), x \u2208 s \u2192 eval x 0 = eval x g\n\u22a2 0 = g\n[PROOFSTEP]\nrwa [degree_zero, eq_comm, degree_eq_bot, eq_comm] at h_deg_eq \n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\ns : Finset R\nh_deg_le : degree f \u2264 \u2191(Finset.card s)\nh_deg_eq : degree f = degree g\nhlc : leadingCoeff f = leadingCoeff g\nh_eval : \u2200 (x : R), x \u2208 s \u2192 eval x f = eval x g\nhf : f \u2260 0\n\u22a2 f = g\n[PROOFSTEP]\nexact eq_of_degree_sub_lt_of_eval_finset_eq s (lt_of_lt_of_le (degree_sub_lt h_deg_eq hf hlc) h_deg_le) h_eval\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\ndegree_f_lt : degree f < \u2191(card s)\neval_f : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = 0\n\u22a2 f = 0\n[PROOFSTEP]\nclassical\nrw [\u2190 card_image_of_injOn hvs] at degree_f_lt \nrefine' eq_zero_of_degree_lt_of_eval_finset_eq_zero _ degree_f_lt _\nintro x hx\nrcases mem_image.mp hx with \u27e8_, hj, rfl\u27e9\nexact eval_f _ hj\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\ndegree_f_lt : degree f < \u2191(card s)\neval_f : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = 0\n\u22a2 f = 0\n[PROOFSTEP]\nrw [\u2190 card_image_of_injOn hvs] at degree_f_lt \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\ndegree_f_lt : degree f < \u2191(card (image v s))\neval_f : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = 0\n\u22a2 f = 0\n[PROOFSTEP]\nrefine' eq_zero_of_degree_lt_of_eval_finset_eq_zero _ degree_f_lt _\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\ndegree_f_lt : degree f < \u2191(card (image v s))\neval_f : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = 0\n\u22a2 \u2200 (x : R), x \u2208 image v s \u2192 eval x f = 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\ndegree_f_lt : degree f < \u2191(card (image v s))\neval_f : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = 0\nx : R\nhx : x \u2208 image v s\n\u22a2 eval x f = 0\n[PROOFSTEP]\nrcases mem_image.mp hx with \u27e8_, hj, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\ndegree_f_lt : degree f < \u2191(card (image v s))\neval_f : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = 0\nw\u271d : \u03b9\nhj : w\u271d \u2208 s\nhx : v w\u271d \u2208 image v s\n\u22a2 eval (v w\u271d) f = 0\n[PROOFSTEP]\nexact eval_f _ hj\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\ndegree_fg_lt : degree (f - g) < \u2191(card s)\neval_fg : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = eval (v i) g\n\u22a2 f = g\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\ndegree_fg_lt : degree (f - g) < \u2191(card s)\neval_fg : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = eval (v i) g\n\u22a2 f - g = 0\n[PROOFSTEP]\nrefine' eq_zero_of_degree_lt_of_eval_index_eq_zero _ hvs degree_fg_lt _\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\ndegree_fg_lt : degree (f - g) < \u2191(card s)\neval_fg : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = eval (v i) g\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) (f - g) = 0\n[PROOFSTEP]\nsimp_rw [eval_sub, sub_eq_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\ndegree_fg_lt : degree (f - g) < \u2191(card s)\neval_fg : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = eval (v i) g\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = eval (v i) g\n[PROOFSTEP]\nexact eval_fg\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\ndegree_f_lt : degree f < \u2191(card s)\ndegree_g_lt : degree g < \u2191(card s)\neval_fg : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = eval (v i) g\n\u22a2 f = g\n[PROOFSTEP]\nrefine' eq_of_degree_sub_lt_of_eval_index_eq _ hvs _ eval_fg\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\ndegree_f_lt : degree f < \u2191(card s)\ndegree_g_lt : degree g < \u2191(card s)\neval_fg : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = eval (v i) g\n\u22a2 degree (f - g) < \u2191(card s)\n[PROOFSTEP]\nrw [\u2190 mem_degreeLT] at degree_f_lt degree_g_lt \u22a2\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\ndegree_f_lt : f \u2208 degreeLT R (card s)\ndegree_g_lt : g \u2208 degreeLT R (card s)\neval_fg : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = eval (v i) g\n\u22a2 f - g \u2208 degreeLT R (card s)\n[PROOFSTEP]\nexact Submodule.sub_mem _ degree_f_lt degree_g_lt\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\nh_deg_le : degree f \u2264 \u2191(card s)\nh_deg_eq : degree f = degree g\nhlc : leadingCoeff f = leadingCoeff g\nh_eval : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = eval (v i) g\n\u22a2 f = g\n[PROOFSTEP]\nrcases eq_or_ne f 0 with rfl | hf\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ng : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\nh_deg_le : degree 0 \u2264 \u2191(card s)\nh_deg_eq : degree 0 = degree g\nhlc : leadingCoeff 0 = leadingCoeff g\nh_eval : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) 0 = eval (v i) g\n\u22a2 0 = g\n[PROOFSTEP]\nrwa [degree_zero, eq_comm, degree_eq_bot, eq_comm] at h_deg_eq \n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nf g : R[X]\n\u03b9 : Type u_2\nv : \u03b9 \u2192 R\ns : Finset \u03b9\nhvs : Set.InjOn v \u2191s\nh_deg_le : degree f \u2264 \u2191(card s)\nh_deg_eq : degree f = degree g\nhlc : leadingCoeff f = leadingCoeff g\nh_eval : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = eval (v i) g\nhf : f \u2260 0\n\u22a2 f = g\n[PROOFSTEP]\nexact eq_of_degree_sub_lt_of_eval_index_eq s hvs (lt_of_lt_of_le (degree_sub_lt h_deg_eq hf hlc) h_deg_le) h_eval\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nx y : F\n\u22a2 basisDivisor x x = 0\n[PROOFSTEP]\nsimp only [basisDivisor, sub_self, inv_zero, map_zero, zero_mul]\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nx y : F\nhxy : basisDivisor x y = 0\n\u22a2 x = y\n[PROOFSTEP]\nsimp_rw [basisDivisor, mul_eq_zero, X_sub_C_ne_zero, or_false_iff, C_eq_zero, inv_eq_zero, sub_eq_zero] at hxy \n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nx y : F\nhxy : x = y\n\u22a2 x = y\n[PROOFSTEP]\nexact hxy\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nx y : F\n\u22a2 basisDivisor x y \u2260 0 \u2194 x \u2260 y\n[PROOFSTEP]\nrw [Ne.def, basisDivisor_eq_zero_iff]\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nx y : F\nhxy : x \u2260 y\n\u22a2 degree (basisDivisor x y) = 1\n[PROOFSTEP]\nrw [basisDivisor, degree_mul, degree_X_sub_C, degree_C, zero_add]\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nx y : F\nhxy : x \u2260 y\n\u22a2 (x - y)\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nexact inv_ne_zero (sub_ne_zero_of_ne hxy)\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nx y : F\n\u22a2 degree (basisDivisor x x) = \u22a5\n[PROOFSTEP]\nrw [basisDivisor_self, degree_zero]\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nx y : F\n\u22a2 natDegree (basisDivisor x x) = 0\n[PROOFSTEP]\nrw [basisDivisor_self, natDegree_zero]\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nx y : F\n\u22a2 eval y (basisDivisor x y) = 0\n[PROOFSTEP]\nsimp only [basisDivisor, eval_mul, eval_C, eval_sub, eval_X, sub_self, mul_zero]\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nx y : F\nhxy : x \u2260 y\n\u22a2 eval x (basisDivisor x y) = 1\n[PROOFSTEP]\nsimp only [basisDivisor, eval_mul, eval_C, eval_sub, eval_X]\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nx y : F\nhxy : x \u2260 y\n\u22a2 (x - y)\u207b\u00b9 * (x - y) = 1\n[PROOFSTEP]\nexact inv_mul_cancel (sub_ne_zero_of_ne hxy)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni\u271d j i : \u03b9\n\u22a2 Lagrange.basis {i} v i = 1\n[PROOFSTEP]\nrw [Lagrange.basis, erase_singleton, prod_empty]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 Lagrange.basis {i, j} v i = basisDivisor (v i) (v j)\n[PROOFSTEP]\nsimp only [Lagrange.basis, hij, erase_insert_eq_erase, erase_eq_of_not_mem, mem_singleton, not_false_iff,\n prod_singleton]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 Lagrange.basis {i, j} v j = basisDivisor (v j) (v i)\n[PROOFSTEP]\nrw [pair_comm]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 Lagrange.basis {j, i} v j = basisDivisor (v j) (v i)\n[PROOFSTEP]\nexact basis_pair_left hij.symm\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\n\u22a2 Lagrange.basis s v i \u2260 0\n[PROOFSTEP]\nsimp_rw [Lagrange.basis, prod_ne_zero_iff, Ne.def, mem_erase]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\n\u22a2 \u2200 (a : \u03b9), a \u2260 i \u2227 a \u2208 s \u2192 \u00acbasisDivisor (v i) (v a) = 0\n[PROOFSTEP]\nrintro j \u27e8hij, hj\u27e9\n[GOAL]\ncase intro\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j\u271d : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\nj : \u03b9\nhij : j \u2260 i\nhj : j \u2208 s\n\u22a2 \u00acbasisDivisor (v i) (v j) = 0\n[PROOFSTEP]\nrw [basisDivisor_eq_zero_iff, hvs.eq_iff hi hj]\n[GOAL]\ncase intro\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j\u271d : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\nj : \u03b9\nhij : j \u2260 i\nhj : j \u2208 s\n\u22a2 \u00aci = j\n[PROOFSTEP]\nexact hij.symm\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\n\u22a2 eval (v i) (Lagrange.basis s v i) = 1\n[PROOFSTEP]\nrw [Lagrange.basis, eval_prod]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\n\u22a2 \u220f j in Finset.erase s i, eval (v i) (basisDivisor (v i) (v j)) = 1\n[PROOFSTEP]\nrefine' prod_eq_one fun j H => _\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j\u271d : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\nj : \u03b9\nH : j \u2208 Finset.erase s i\n\u22a2 eval (v i) (basisDivisor (v i) (v j)) = 1\n[PROOFSTEP]\nrw [eval_basisDivisor_left_of_ne]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j\u271d : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\nj : \u03b9\nH : j \u2208 Finset.erase s i\n\u22a2 v i \u2260 v j\n[PROOFSTEP]\nrcases mem_erase.mp H with \u27e8hij, hj\u27e9\n[GOAL]\ncase intro\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j\u271d : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\nj : \u03b9\nH : j \u2208 Finset.erase s i\nhij : j \u2260 i\nhj : j \u2208 s\n\u22a2 v i \u2260 v j\n[PROOFSTEP]\nexact mt (hvs hi hj) hij.symm\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhij : i \u2260 j\nhj : j \u2208 s\n\u22a2 eval (v j) (Lagrange.basis s v i) = 0\n[PROOFSTEP]\nsimp_rw [Lagrange.basis, eval_prod, prod_eq_zero_iff]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhij : i \u2260 j\nhj : j \u2208 s\n\u22a2 \u2203 a, a \u2208 Finset.erase s i \u2227 eval (v j) (basisDivisor (v i) (v a)) = 0\n[PROOFSTEP]\nexact \u27e8j, \u27e8mem_erase.mpr \u27e8hij.symm, hj\u27e9, eval_basisDivisor_right\u27e9\u27e9\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\n\u22a2 natDegree (Lagrange.basis s v i) = card s - 1\n[PROOFSTEP]\nhave H : \u2200 j, j \u2208 s.erase i \u2192 basisDivisor (v i) (v j) \u2260 0 :=\n by\n simp_rw [Ne.def, mem_erase, basisDivisor_eq_zero_iff]\n exact fun j \u27e8hij\u2081, hj\u27e9 hij\u2082 => hij\u2081 (hvs hj hi hij\u2082.symm)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\n\u22a2 \u2200 (j : \u03b9), j \u2208 Finset.erase s i \u2192 basisDivisor (v i) (v j) \u2260 0\n[PROOFSTEP]\nsimp_rw [Ne.def, mem_erase, basisDivisor_eq_zero_iff]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\n\u22a2 \u2200 (j : \u03b9), j \u2260 i \u2227 j \u2208 s \u2192 \u00acv i = v j\n[PROOFSTEP]\nexact fun j \u27e8hij\u2081, hj\u27e9 hij\u2082 => hij\u2081 (hvs hj hi hij\u2082.symm)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\nH : \u2200 (j : \u03b9), j \u2208 Finset.erase s i \u2192 basisDivisor (v i) (v j) \u2260 0\n\u22a2 natDegree (Lagrange.basis s v i) = card s - 1\n[PROOFSTEP]\nrw [\u2190 card_erase_of_mem hi, card_eq_sum_ones]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\nH : \u2200 (j : \u03b9), j \u2208 Finset.erase s i \u2192 basisDivisor (v i) (v j) \u2260 0\n\u22a2 natDegree (Lagrange.basis s v i) = \u2211 x in Finset.erase s i, 1\n[PROOFSTEP]\nconvert natDegree_prod _ _ H using 1\n[GOAL]\ncase h.e'_3\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\nH : \u2200 (j : \u03b9), j \u2208 Finset.erase s i \u2192 basisDivisor (v i) (v j) \u2260 0\n\u22a2 \u2211 x in Finset.erase s i, 1 = \u2211 i_1 in Finset.erase s i, natDegree (basisDivisor (v i) (v i_1))\n[PROOFSTEP]\nrefine' sum_congr rfl fun j hj => (natDegree_basisDivisor_of_ne _).symm\n[GOAL]\ncase h.e'_3\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j\u271d : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\nH : \u2200 (j : \u03b9), j \u2208 Finset.erase s i \u2192 basisDivisor (v i) (v j) \u2260 0\nj : \u03b9\nhj : j \u2208 Finset.erase s i\n\u22a2 v i \u2260 v j\n[PROOFSTEP]\nrw [Ne.def, \u2190 basisDivisor_eq_zero_iff]\n[GOAL]\ncase h.e'_3\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j\u271d : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\nH : \u2200 (j : \u03b9), j \u2208 Finset.erase s i \u2192 basisDivisor (v i) (v j) \u2260 0\nj : \u03b9\nhj : j \u2208 Finset.erase s i\n\u22a2 \u00acbasisDivisor (v i) (v j) = 0\n[PROOFSTEP]\nexact H _ hj\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\n\u22a2 degree (Lagrange.basis s v i) = \u2191(card s - 1)\n[PROOFSTEP]\nrw [degree_eq_natDegree (basis_ne_zero hvs hi), natDegree_basis hvs hi]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\n\u22a2 \u2211 j in s, Lagrange.basis s v j = 1\n[PROOFSTEP]\nrefine' eq_of_degrees_lt_of_eval_index_eq s hvs (lt_of_le_of_lt (degree_sum_le _ _) _) _ _\n[GOAL]\ncase refine'_1\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\n\u22a2 (sup s fun b => degree (Lagrange.basis s v b)) < \u2191(card s)\n[PROOFSTEP]\nrw [Nat.cast_withBot, Finset.sup_lt_iff (WithBot.bot_lt_coe s.card)]\n[GOAL]\ncase refine'_1\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\n\u22a2 \u2200 (b : \u03b9), b \u2208 s \u2192 degree (Lagrange.basis s v b) < \u2191(card s)\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase refine'_1\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni\u271d j : \u03b9\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\ni : \u03b9\nhi : i \u2208 s\n\u22a2 degree (Lagrange.basis s v i) < \u2191(card s)\n[PROOFSTEP]\nrw [degree_basis hvs hi, Nat.cast_withBot, WithBot.coe_lt_coe]\n[GOAL]\ncase refine'_1\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni\u271d j : \u03b9\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\ni : \u03b9\nhi : i \u2208 s\n\u22a2 card s - 1 < card s\n[PROOFSTEP]\nexact Nat.pred_lt (card_ne_zero_of_mem hi)\n[GOAL]\ncase refine'_2\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\n\u22a2 degree 1 < \u2191(card s)\n[PROOFSTEP]\nrw [degree_one, \u2190 WithBot.coe_zero, Nat.cast_withBot, WithBot.coe_lt_coe]\n[GOAL]\ncase refine'_2\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\n\u22a2 0 < card s\n[PROOFSTEP]\nexact Nonempty.card_pos hs\n[GOAL]\ncase refine'_3\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) (\u2211 j in s, Lagrange.basis s v j) = eval (v i) 1\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase refine'_3\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni\u271d j : \u03b9\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\ni : \u03b9\nhi : i \u2208 s\n\u22a2 eval (v i) (\u2211 j in s, Lagrange.basis s v j) = eval (v i) 1\n[PROOFSTEP]\nrw [eval_finset_sum, eval_one, \u2190 add_sum_erase _ _ hi, eval_basis_self hvs hi, add_right_eq_self]\n[GOAL]\ncase refine'_3\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni\u271d j : \u03b9\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2211 x in Finset.erase s i, eval (v i) (Lagrange.basis s v x) = 0\n[PROOFSTEP]\nrefine' sum_eq_zero fun j hj => _\n[GOAL]\ncase refine'_3\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni\u271d j\u271d : \u03b9\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 Finset.erase s i\n\u22a2 eval (v i) (Lagrange.basis s v j) = 0\n[PROOFSTEP]\nrcases mem_erase.mp hj with \u27e8hij, _\u27e9\n[GOAL]\ncase refine'_3.intro\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni\u271d j\u271d : \u03b9\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 Finset.erase s i\nhij : j \u2260 i\nright\u271d : j \u2208 s\n\u22a2 eval (v i) (Lagrange.basis s v j) = 0\n[PROOFSTEP]\nrw [eval_basis_of_ne hij hi]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nx y : F\nhxy : x \u2260 y\n\u22a2 basisDivisor x y + basisDivisor y x = 1\n[PROOFSTEP]\nclassical rw [\u2190 sum_basis (Set.injOn_of_injective Function.injective_id _) \u27e8x, mem_insert_self _ { y }\u27e9,\n sum_insert (not_mem_singleton.mpr hxy), sum_singleton, basis_pair_left hxy, basis_pair_right hxy, id, id]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni j : \u03b9\nx y : F\nhxy : x \u2260 y\n\u22a2 basisDivisor x y + basisDivisor y x = 1\n[PROOFSTEP]\nrw [\u2190 sum_basis (Set.injOn_of_injective Function.injective_id _) \u27e8x, mem_insert_self _ { y }\u27e9,\n sum_insert (not_mem_singleton.mpr hxy), sum_singleton, basis_pair_left hxy, basis_pair_right hxy, id, id]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns\u271d t : Finset \u03b9\ni j : \u03b9\nv\u271d r r' : \u03b9 \u2192 F\ns : Finset \u03b9\nv f g : \u03b9 \u2192 F\n\u22a2 (fun r => \u2211 i in s, \u2191C (r i) * Lagrange.basis s v i) (f + g) =\n (fun r => \u2211 i in s, \u2191C (r i) * Lagrange.basis s v i) f + (fun r => \u2211 i in s, \u2191C (r i) * Lagrange.basis s v i) g\n[PROOFSTEP]\nsimp_rw [\u2190 Finset.sum_add_distrib]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns\u271d t : Finset \u03b9\ni j : \u03b9\nv\u271d r r' : \u03b9 \u2192 F\ns : Finset \u03b9\nv f g : \u03b9 \u2192 F\n\u22a2 \u2211 x in s, \u2191C ((f + g) x) * Lagrange.basis s v x =\n \u2211 x in s, (\u2191C (f x) * Lagrange.basis s v x + \u2191C (g x) * Lagrange.basis s v x)\n[PROOFSTEP]\nhave h :\n (fun x => C (f x) * Lagrange.basis s v x + C (g x) * Lagrange.basis s v x) =\n (fun x => C ((f + g) x) * Lagrange.basis s v x) :=\n by simp_rw [\u2190 add_mul, \u2190 C_add, Pi.add_apply]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns\u271d t : Finset \u03b9\ni j : \u03b9\nv\u271d r r' : \u03b9 \u2192 F\ns : Finset \u03b9\nv f g : \u03b9 \u2192 F\n\u22a2 (fun x => \u2191C (f x) * Lagrange.basis s v x + \u2191C (g x) * Lagrange.basis s v x) = fun x =>\n \u2191C ((f + g) x) * Lagrange.basis s v x\n[PROOFSTEP]\nsimp_rw [\u2190 add_mul, \u2190 C_add, Pi.add_apply]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns\u271d t : Finset \u03b9\ni j : \u03b9\nv\u271d r r' : \u03b9 \u2192 F\ns : Finset \u03b9\nv f g : \u03b9 \u2192 F\nh :\n (fun x => \u2191C (f x) * Lagrange.basis s v x + \u2191C (g x) * Lagrange.basis s v x) = fun x =>\n \u2191C ((f + g) x) * Lagrange.basis s v x\n\u22a2 \u2211 x in s, \u2191C ((f + g) x) * Lagrange.basis s v x =\n \u2211 x in s, (\u2191C (f x) * Lagrange.basis s v x + \u2191C (g x) * Lagrange.basis s v x)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns\u271d t : Finset \u03b9\ni j : \u03b9\nv\u271d r r' : \u03b9 \u2192 F\ns : Finset \u03b9\nv : \u03b9 \u2192 F\nc : F\nf : \u03b9 \u2192 F\n\u22a2 AddHom.toFun\n { toFun := fun r => \u2211 i in s, \u2191C (r i) * Lagrange.basis s v i,\n map_add' :=\n (_ :\n \u2200 (f g : \u03b9 \u2192 F),\n (fun r => \u2211 i in s, \u2191C (r i) * Lagrange.basis s v i) (f + g) =\n (fun r => \u2211 i in s, \u2191C (r i) * Lagrange.basis s v i) f +\n (fun r => \u2211 i in s, \u2191C (r i) * Lagrange.basis s v i) g) }\n (c \u2022 f) =\n \u2191(RingHom.id F) c \u2022\n AddHom.toFun\n { toFun := fun r => \u2211 i in s, \u2191C (r i) * Lagrange.basis s v i,\n map_add' :=\n (_ :\n \u2200 (f g : \u03b9 \u2192 F),\n (fun r => \u2211 i in s, \u2191C (r i) * Lagrange.basis s v i) (f + g) =\n (fun r => \u2211 i in s, \u2191C (r i) * Lagrange.basis s v i) f +\n (fun r => \u2211 i in s, \u2191C (r i) * Lagrange.basis s v i) g) }\n f\n[PROOFSTEP]\nsimp_rw [Finset.smul_sum, C_mul', smul_smul, Pi.smul_apply, RingHom.id_apply, smul_eq_mul]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\n\u22a2 \u2191(interpolate \u2205 v) r = 0\n[PROOFSTEP]\nrw [interpolate_apply, sum_empty]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\n\u22a2 \u2191(interpolate {i} v) r = \u2191C (r i)\n[PROOFSTEP]\nrw [interpolate_apply, sum_singleton, basis_singleton, mul_one]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\n\u22a2 \u2191(interpolate s v) 1 = 1\n[PROOFSTEP]\nsimp_rw [interpolate_apply, Pi.one_apply, map_one, one_mul]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\n\u22a2 \u2211 x in s, Lagrange.basis s v x = 1\n[PROOFSTEP]\nexact sum_basis hvs hs\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\n\u22a2 eval (v i) (\u2191(interpolate s v) r) = r i\n[PROOFSTEP]\nrw [interpolate_apply, eval_finset_sum, \u2190 add_sum_erase _ _ hi]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\n\u22a2 eval (v i) (\u2191C (r i) * Lagrange.basis s v i) + \u2211 x in Finset.erase s i, eval (v i) (\u2191C (r x) * Lagrange.basis s v x) =\n r i\n[PROOFSTEP]\nsimp_rw [eval_mul, eval_C, eval_basis_self hvs hi, mul_one, add_right_eq_self]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\n\u22a2 \u2211 x in Finset.erase s i, r x * eval (v i) (Lagrange.basis s v x) = 0\n[PROOFSTEP]\nrefine' sum_eq_zero fun j H => _\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j\u271d : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\nj : \u03b9\nH : j \u2208 Finset.erase s i\n\u22a2 r j * eval (v i) (Lagrange.basis s v j) = 0\n[PROOFSTEP]\nrw [eval_basis_of_ne (mem_erase.mp H).1 hi, mul_zero]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\n\u22a2 degree (\u2191(interpolate s v) r) \u2264 \u2191(card s - 1)\n[PROOFSTEP]\nrefine' (degree_sum_le _ _).trans _\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\n\u22a2 (sup s fun b => degree (\u2191C (r b) * Lagrange.basis s v b)) \u2264 \u2191(card s - 1)\n[PROOFSTEP]\nrw [Finset.sup_le_iff]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\n\u22a2 \u2200 (b : \u03b9), b \u2208 s \u2192 degree (\u2191C (r b) * Lagrange.basis s v b) \u2264 \u2191(card s - 1)\n[PROOFSTEP]\nintro i hi\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\ni : \u03b9\nhi : i \u2208 s\n\u22a2 degree (\u2191C (r i) * Lagrange.basis s v i) \u2264 \u2191(card s - 1)\n[PROOFSTEP]\nrw [degree_mul, degree_basis hvs hi]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\ni : \u03b9\nhi : i \u2208 s\n\u22a2 degree (\u2191C (r i)) + \u2191(card s - 1) \u2264 \u2191(card s - 1)\n[PROOFSTEP]\nby_cases hr : r i = 0\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\ni : \u03b9\nhi : i \u2208 s\nhr : r i = 0\n\u22a2 degree (\u2191C (r i)) + \u2191(card s - 1) \u2264 \u2191(card s - 1)\n[PROOFSTEP]\nsimpa only [hr, map_zero, degree_zero, WithBot.bot_add] using bot_le\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\ni : \u03b9\nhi : i \u2208 s\nhr : \u00acr i = 0\n\u22a2 degree (\u2191C (r i)) + \u2191(card s - 1) \u2264 \u2191(card s - 1)\n[PROOFSTEP]\nrw [degree_C hr, zero_add]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\n\u22a2 degree (\u2191(interpolate s v) r) < \u2191(card s)\n[PROOFSTEP]\nrw [Nat.cast_withBot]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\n\u22a2 degree (\u2191(interpolate s v) r) < \u2191(card s)\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | h)\n[GOAL]\ncase inl\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\nt : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191\u2205\n\u22a2 degree (\u2191(interpolate \u2205 v) r) < \u2191(card \u2205)\n[PROOFSTEP]\nrw [interpolate_empty, degree_zero, card_empty]\n[GOAL]\ncase inl\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\nt : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191\u2205\n\u22a2 \u22a5 < \u21910\n[PROOFSTEP]\nexact WithBot.bot_lt_coe _\n[GOAL]\ncase inr\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nh : Finset.Nonempty s\n\u22a2 degree (\u2191(interpolate s v) r) < \u2191(card s)\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (degree_interpolate_le _ hvs) _\n[GOAL]\ncase inr\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nh : Finset.Nonempty s\n\u22a2 \u2191(card s - 1) < \u2191(card s)\n[PROOFSTEP]\nrw [Nat.cast_withBot, WithBot.coe_lt_coe]\n[GOAL]\ncase inr\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nh : Finset.Nonempty s\n\u22a2 card s - 1 < card s\n[PROOFSTEP]\nexact Nat.sub_lt (Nonempty.card_pos h) zero_lt_one\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\n\u22a2 degree (\u2191(interpolate (Finset.erase s i) v) r) < \u2191(card s - 1)\n[PROOFSTEP]\nrw [\u2190 Finset.card_erase_of_mem hi]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\n\u22a2 degree (\u2191(interpolate (Finset.erase s i) v) r) < \u2191(card (Finset.erase s i))\n[PROOFSTEP]\nexact degree_interpolate_lt _ (Set.InjOn.mono (coe_subset.mpr (erase_subset _ _)) hvs)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nhrr' : \u2191(interpolate s v) r = \u2191(interpolate s v) r'\nx\u271d : \u03b9\nhi : x\u271d \u2208 s\n\u22a2 r x\u271d = r' x\u271d\n[PROOFSTEP]\nrw [\u2190 eval_interpolate_at_node r hvs hi, hrr', eval_interpolate_at_node r' hvs hi]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhrr' : \u2200 (i : \u03b9), i \u2208 s \u2192 r i = r' i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191C (r i) * Lagrange.basis s v i = \u2191C (r' i) * Lagrange.basis s v i\n[PROOFSTEP]\nrw [hrr' _ hi]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nf : F[X]\nhvs : Set.InjOn v \u2191s\ndegree_f_lt : degree f < \u2191(card s)\neval_f : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = r i\n\u22a2 f = \u2191(interpolate s v) r\n[PROOFSTEP]\nrw [eq_interpolate hvs degree_f_lt]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nf : F[X]\nhvs : Set.InjOn v \u2191s\ndegree_f_lt : degree f < \u2191(card s)\neval_f : \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = r i\n\u22a2 (\u2191(interpolate s v) fun i => eval (v i) f) = \u2191(interpolate s v) r\n[PROOFSTEP]\nexact interpolate_eq_of_values_eq_on _ _ eval_f\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nf : F[X]\nhvs : Set.InjOn v \u2191s\n\u22a2 (degree f < \u2191(card s) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = r i) \u2194 f = \u2191(interpolate s v) r\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nf : F[X]\nhvs : Set.InjOn v \u2191s\n\u22a2 (degree f < \u2191(card s) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = r i) \u2192 f = \u2191(interpolate s v) r\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nf : F[X]\nhvs : Set.InjOn v \u2191s\n\u22a2 f = \u2191(interpolate s v) r \u2192 degree f < \u2191(card s) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = r i\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nf : F[X]\nhvs : Set.InjOn v \u2191s\nh : degree f < \u2191(card s) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = r i\n\u22a2 f = \u2191(interpolate s v) r\n[PROOFSTEP]\nexact eq_interpolate_of_eval_eq _ hvs h.1 h.2\n[GOAL]\ncase mpr\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nf : F[X]\nhvs : Set.InjOn v \u2191s\nh : f = \u2191(interpolate s v) r\n\u22a2 degree f < \u2191(card s) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) f = r i\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nf : F[X]\nhvs : Set.InjOn v \u2191s\nh : f = \u2191(interpolate s v) r\n\u22a2 degree (\u2191(interpolate s v) r) < \u2191(card s) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 eval (v i) (\u2191(interpolate s v) r) = r i\n[PROOFSTEP]\nexact \u27e8degree_interpolate_lt _ hvs, fun _ hi => eval_interpolate_at_node _ hvs hi\u27e9\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nc : F\nf : { x // x \u2208 degreeLT F (card s) }\n\u22a2 \u2200 (x : { x // x \u2208 s }),\n AddHom.toFun\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) = (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) }\n (c \u2022 f) x =\n (\u2191(RingHom.id F) c \u2022\n AddHom.toFun\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) }\n f)\n x\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\n\u22a2 Function.LeftInverse\n (fun r =>\n { val := \u2191(interpolate s v) fun x => if hx : x \u2208 s then r { val := x, property := hx } else 0,\n property :=\n (_ :\n (\u2191(interpolate s v) fun x => if hx : x \u2208 s then r { val := x, property := hx } else 0) \u2208\n degreeLT F (card s)) })\n {\n toAddHom :=\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) },\n map_smul' :=\n (_ :\n \u2200 (c : F) (f : { x // x \u2208 degreeLT F (card s) }),\n AddHom.toFun\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) }\n (c \u2022 f) =\n \u2191(RingHom.id F) c \u2022\n AddHom.toFun\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) }\n f) }.toAddHom.toFun\n[PROOFSTEP]\nrintro \u27e8f, hf\u27e9\n[GOAL]\ncase mk\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nf : F[X]\nhf : f \u2208 degreeLT F (card s)\n\u22a2 (fun r =>\n { val := \u2191(interpolate s v) fun x => if hx : x \u2208 s then r { val := x, property := hx } else 0,\n property :=\n (_ :\n (\u2191(interpolate s v) fun x => if hx : x \u2208 s then r { val := x, property := hx } else 0) \u2208\n degreeLT F (card s)) })\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) },\n map_smul' :=\n (_ :\n \u2200 (c : F) (f : { x // x \u2208 degreeLT F (card s) }),\n AddHom.toFun\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) }\n (c \u2022 f) =\n \u2191(RingHom.id F) c \u2022\n AddHom.toFun\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) }\n f) }.toAddHom\n { val := f, property := hf }) =\n { val := f, property := hf }\n[PROOFSTEP]\nsimp only [Subtype.mk_eq_mk, Subtype.coe_mk, dite_eq_ite]\n[GOAL]\ncase mk\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nf : F[X]\nhf : f \u2208 degreeLT F (card s)\n\u22a2 (\u2191(interpolate s v) fun x => if x \u2208 s then eval (v x) f else 0) = f\n[PROOFSTEP]\nrw [mem_degreeLT] at hf \n[GOAL]\ncase mk\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nf : F[X]\nhf : degree f < \u2191(card s)\n\u22a2 (\u2191(interpolate s v) fun x => if x \u2208 s then eval (v x) f else 0) = f\n[PROOFSTEP]\nconv => rhs; rw [eq_interpolate hvs hf]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nf : F[X]\nhf : degree f < \u2191(card s)\n| (\u2191(interpolate s v) fun x => if x \u2208 s then eval (v x) f else 0) = f\n[PROOFSTEP]\nrhs; rw [eq_interpolate hvs hf]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nf : F[X]\nhf : degree f < \u2191(card s)\n| (\u2191(interpolate s v) fun x => if x \u2208 s then eval (v x) f else 0) = f\n[PROOFSTEP]\nrhs; rw [eq_interpolate hvs hf]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nf : F[X]\nhf : degree f < \u2191(card s)\n| (\u2191(interpolate s v) fun x => if x \u2208 s then eval (v x) f else 0) = f\n[PROOFSTEP]\nrhs\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nf : F[X]\nhf : degree f < \u2191(card s)\n| f\n[PROOFSTEP]\nrw [eq_interpolate hvs hf]\n[GOAL]\ncase mk\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nf : F[X]\nhf : degree f < \u2191(card s)\n\u22a2 (\u2191(interpolate s v) fun x => if x \u2208 s then eval (v x) f else 0) = \u2191(interpolate s v) fun i => eval (v i) f\n[PROOFSTEP]\nexact interpolate_eq_of_values_eq_on _ _ fun _ hi => if_pos hi\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\n\u22a2 Function.RightInverse\n (fun r =>\n { val := \u2191(interpolate s v) fun x => if hx : x \u2208 s then r { val := x, property := hx } else 0,\n property :=\n (_ :\n (\u2191(interpolate s v) fun x => if hx : x \u2208 s then r { val := x, property := hx } else 0) \u2208\n degreeLT F (card s)) })\n {\n toAddHom :=\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) },\n map_smul' :=\n (_ :\n \u2200 (c : F) (f : { x // x \u2208 degreeLT F (card s) }),\n AddHom.toFun\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) }\n (c \u2022 f) =\n \u2191(RingHom.id F) c \u2022\n AddHom.toFun\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) }\n f) }.toAddHom.toFun\n[PROOFSTEP]\nintro f\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nf : { x // x \u2208 s } \u2192 F\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) },\n map_smul' :=\n (_ :\n \u2200 (c : F) (f : { x // x \u2208 degreeLT F (card s) }),\n AddHom.toFun\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) }\n (c \u2022 f) =\n \u2191(RingHom.id F) c \u2022\n AddHom.toFun\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) }\n f) }.toAddHom\n ((fun r =>\n { val := \u2191(interpolate s v) fun x => if hx : x \u2208 s then r { val := x, property := hx } else 0,\n property :=\n (_ :\n (\u2191(interpolate s v) fun x => if hx : x \u2208 s then r { val := x, property := hx } else 0) \u2208\n degreeLT F (card s)) })\n f) =\n f\n[PROOFSTEP]\next \u27e8i, hi\u27e9\n[GOAL]\ncase h.mk\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nf : { x // x \u2208 s } \u2192 F\ni : \u03b9\nhi : i \u2208 s\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) },\n map_smul' :=\n (_ :\n \u2200 (c : F) (f : { x // x \u2208 degreeLT F (card s) }),\n AddHom.toFun\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) }\n (c \u2022 f) =\n \u2191(RingHom.id F) c \u2022\n AddHom.toFun\n { toFun := fun f i => eval (v \u2191i) \u2191f,\n map_add' :=\n (_ :\n \u2200 (f g : { x // x \u2208 degreeLT F (card s) }),\n (fun f i => eval (v \u2191i) \u2191f) (f + g) =\n (fun f i => eval (v \u2191i) \u2191f) f + (fun f i => eval (v \u2191i) \u2191f) g) }\n f) }.toAddHom\n ((fun r =>\n { val := \u2191(interpolate s v) fun x => if hx : x \u2208 s then r { val := x, property := hx } else 0,\n property :=\n (_ :\n (\u2191(interpolate s v) fun x => if hx : x \u2208 s then r { val := x, property := hx } else 0) \u2208\n degreeLT F (card s)) })\n f)\n { val := i, property := hi } =\n f { val := i, property := hi }\n[PROOFSTEP]\nsimp only [Subtype.coe_mk, eval_interpolate_at_node _ hvs hi]\n[GOAL]\ncase h.mk\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nf : { x // x \u2208 s } \u2192 F\ni : \u03b9\nhi : i \u2208 s\n\u22a2 (if hx : i \u2208 s then f { val := i, property := hx } else 0) = f { val := i, property := hi }\n[PROOFSTEP]\nexact dif_pos hi\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\n\u22a2 \u2191(interpolate t v) r = \u2211 i in s, \u2191(interpolate (insert i (t \\ s)) v) r * Lagrange.basis s v i\n[PROOFSTEP]\nsymm\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\n\u22a2 \u2211 i in s, \u2191(interpolate (insert i (t \\ s)) v) r * Lagrange.basis s v i = \u2191(interpolate t v) r\n[PROOFSTEP]\nrefine' eq_interpolate_of_eval_eq _ hvt (lt_of_le_of_lt (degree_sum_le _ _) _) fun i hi => _\n[GOAL]\ncase refine'_1\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\n\u22a2 (sup s fun b => degree (\u2191(interpolate (insert b (t \\ s)) v) r * Lagrange.basis s v b)) < \u2191(card t)\n[PROOFSTEP]\nsimp_rw [Nat.cast_withBot, Finset.sup_lt_iff (WithBot.bot_lt_coe t.card), degree_mul]\n[GOAL]\ncase refine'_1\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\n\u22a2 \u2200 (b : \u03b9), b \u2208 s \u2192 degree (\u2191(interpolate (insert b (t \\ s)) v) r) + degree (Lagrange.basis s v b) < \u2191(card t)\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase refine'_1\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 s\n\u22a2 degree (\u2191(interpolate (insert i (t \\ s)) v) r) + degree (Lagrange.basis s v i) < \u2191(card t)\n[PROOFSTEP]\nhave hs : 1 \u2264 s.card := Nonempty.card_pos \u27e8_, hi\u27e9\n[GOAL]\ncase refine'_1\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs\u271d : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 s\nhs : 1 \u2264 card s\n\u22a2 degree (\u2191(interpolate (insert i (t \\ s)) v) r) + degree (Lagrange.basis s v i) < \u2191(card t)\n[PROOFSTEP]\nhave hst' : s.card \u2264 t.card := card_le_of_subset hst\n[GOAL]\ncase refine'_1\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs\u271d : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 s\nhs : 1 \u2264 card s\nhst' : card s \u2264 card t\n\u22a2 degree (\u2191(interpolate (insert i (t \\ s)) v) r) + degree (Lagrange.basis s v i) < \u2191(card t)\n[PROOFSTEP]\nhave H : t.card = 1 + (t.card - s.card) + (s.card - 1) := by\n rw [add_assoc, tsub_add_tsub_cancel hst' hs, \u2190 add_tsub_assoc_of_le (hs.trans hst'), Nat.succ_add_sub_one, zero_add]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs\u271d : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 s\nhs : 1 \u2264 card s\nhst' : card s \u2264 card t\n\u22a2 card t = 1 + (card t - card s) + (card s - 1)\n[PROOFSTEP]\nrw [add_assoc, tsub_add_tsub_cancel hst' hs, \u2190 add_tsub_assoc_of_le (hs.trans hst'), Nat.succ_add_sub_one, zero_add]\n[GOAL]\ncase refine'_1\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs\u271d : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 s\nhs : 1 \u2264 card s\nhst' : card s \u2264 card t\nH : card t = 1 + (card t - card s) + (card s - 1)\n\u22a2 degree (\u2191(interpolate (insert i (t \\ s)) v) r) + degree (Lagrange.basis s v i) < \u2191(card t)\n[PROOFSTEP]\nrw [degree_basis (Set.InjOn.mono hst hvt) hi, H, WithBot.coe_add, Nat.cast_withBot,\n WithBot.add_lt_add_iff_right (@WithBot.coe_ne_bot _ (s.card - 1))]\n[GOAL]\ncase refine'_1\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs\u271d : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 s\nhs : 1 \u2264 card s\nhst' : card s \u2264 card t\nH : card t = 1 + (card t - card s) + (card s - 1)\n\u22a2 degree (\u2191(interpolate (insert i (t \\ s)) v) r) < \u2191(1 + (card t - card s))\n[PROOFSTEP]\nconvert degree_interpolate_lt _ (hvt.mono (coe_subset.mpr (insert_subset_iff.mpr \u27e8hst hi, sdiff_subset _ _\u27e9)))\n[GOAL]\ncase h.e'_4.h.e'_1\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs\u271d : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 s\nhs : 1 \u2264 card s\nhst' : card s \u2264 card t\nH : card t = 1 + (card t - card s) + (card s - 1)\n\u22a2 1 + (card t - card s) = card (insert i (t \\ s))\n[PROOFSTEP]\nrw [card_insert_of_not_mem (not_mem_sdiff_of_mem_right hi), card_sdiff hst, add_comm]\n[GOAL]\ncase refine'_2\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 t\n\u22a2 eval (v i) (\u2211 i in s, \u2191(interpolate (insert i (t \\ s)) v) r * Lagrange.basis s v i) = r i\n[PROOFSTEP]\nsimp_rw [eval_finset_sum, eval_mul]\n[GOAL]\ncase refine'_2\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 t\n\u22a2 \u2211 x in s, eval (v i) (\u2191(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) = r i\n[PROOFSTEP]\nby_cases hi' : i \u2208 s\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 t\nhi' : i \u2208 s\n\u22a2 \u2211 x in s, eval (v i) (\u2191(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) = r i\n[PROOFSTEP]\nrw [\u2190 add_sum_erase _ _ hi', eval_basis_self (hvt.mono hst) hi',\n eval_interpolate_at_node _ (hvt.mono (coe_subset.mpr (insert_subset_iff.mpr \u27e8hi, sdiff_subset _ _\u27e9)))\n (mem_insert_self _ _),\n mul_one, add_right_eq_self]\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 t\nhi' : i \u2208 s\n\u22a2 \u2211 x in Finset.erase s i, eval (v i) (\u2191(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) = 0\n[PROOFSTEP]\nrefine' sum_eq_zero fun j hj => _\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j\u271d : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 t\nhi' : i \u2208 s\nj : \u03b9\nhj : j \u2208 Finset.erase s i\n\u22a2 eval (v i) (\u2191(interpolate (insert j (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v j) = 0\n[PROOFSTEP]\nrcases mem_erase.mp hj with \u27e8hij, _\u27e9\n[GOAL]\ncase pos.intro\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j\u271d : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 t\nhi' : i \u2208 s\nj : \u03b9\nhj : j \u2208 Finset.erase s i\nhij : j \u2260 i\nright\u271d : j \u2208 s\n\u22a2 eval (v i) (\u2191(interpolate (insert j (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v j) = 0\n[PROOFSTEP]\nrw [eval_basis_of_ne hij hi', mul_zero]\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 t\nhi' : \u00aci \u2208 s\n\u22a2 \u2211 x in s, eval (v i) (\u2191(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) = r i\n[PROOFSTEP]\nhave H : (\u2211 j in s, eval (v i) (Lagrange.basis s v j)) = 1 := by\n rw [\u2190 eval_finset_sum, sum_basis (hvt.mono hst) hs, eval_one]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 t\nhi' : \u00aci \u2208 s\n\u22a2 \u2211 j in s, eval (v i) (Lagrange.basis s v j) = 1\n[PROOFSTEP]\nrw [\u2190 eval_finset_sum, sum_basis (hvt.mono hst) hs, eval_one]\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 t\nhi' : \u00aci \u2208 s\nH : \u2211 j in s, eval (v i) (Lagrange.basis s v j) = 1\n\u22a2 \u2211 x in s, eval (v i) (\u2191(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) = r i\n[PROOFSTEP]\nrw [\u2190 mul_one (r i), \u2190 H, mul_sum]\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 t\nhi' : \u00aci \u2208 s\nH : \u2211 j in s, eval (v i) (Lagrange.basis s v j) = 1\n\u22a2 \u2211 x in s, eval (v i) (\u2191(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) =\n \u2211 x in s, r i * eval (v i) (Lagrange.basis s v x)\n[PROOFSTEP]\nrefine' sum_congr rfl fun j hj => _\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j\u271d : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 t\nhi' : \u00aci \u2208 s\nH : \u2211 j in s, eval (v i) (Lagrange.basis s v j) = 1\nj : \u03b9\nhj : j \u2208 s\n\u22a2 eval (v i) (\u2191(interpolate (insert j (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v j) =\n r i * eval (v i) (Lagrange.basis s v j)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase neg.e_a\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni\u271d j\u271d : \u03b9\nv r r' : \u03b9 \u2192 F\nhvt : Set.InjOn v \u2191t\nhs : Finset.Nonempty s\nhst : s \u2286 t\ni : \u03b9\nhi : i \u2208 t\nhi' : \u00aci \u2208 s\nH : \u2211 j in s, eval (v i) (Lagrange.basis s v j) = 1\nj : \u03b9\nhj : j \u2208 s\n\u22a2 eval (v i) (\u2191(interpolate (insert j (t \\ s)) v) r) = r i\n[PROOFSTEP]\nexact\n eval_interpolate_at_node _ (hvt.mono (insert_subset_iff.mpr \u27e8hst hj, sdiff_subset _ _\u27e9))\n (mem_insert.mpr (Or.inr (mem_sdiff.mpr \u27e8hi, hi'\u27e9)))\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\nhj : j \u2208 s\nhij : i \u2260 j\n\u22a2 \u2191(interpolate s v) r =\n \u2191(interpolate (Finset.erase s j) v) r * basisDivisor (v i) (v j) +\n \u2191(interpolate (Finset.erase s i) v) r * basisDivisor (v j) (v i)\n[PROOFSTEP]\nrw [interpolate_eq_sum_interpolate_insert_sdiff _ hvs \u27e8i, mem_insert_self i { j }\u27e9 _,\n sum_insert (not_mem_singleton.mpr hij), sum_singleton, basis_pair_left hij, basis_pair_right hij,\n sdiff_insert_insert_of_mem_of_not_mem hi (not_mem_singleton.mpr hij), sdiff_singleton_eq_erase, pair_comm,\n sdiff_insert_insert_of_mem_of_not_mem hj (not_mem_singleton.mpr hij.symm), sdiff_singleton_eq_erase]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns t : Finset \u03b9\ni j : \u03b9\nv r r' : \u03b9 \u2192 F\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\nhj : j \u2208 s\nhij : i \u2260 j\n\u22a2 {i, j} \u2286 s\n[PROOFSTEP]\nexact insert_subset_iff.mpr \u27e8hi, singleton_subset_iff.mpr hj\u27e9\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\n\u22a2 nodal \u2205 v = 1\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\n\u22a2 degree (nodal s v) = \u2191(card s)\n[PROOFSTEP]\nsimp_rw [nodal, degree_prod, degree_X_sub_C, sum_const, Nat.smul_one_eq_coe]\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx\u271d x : F\n\u22a2 eval x (nodal s v) = \u220f i in s, (x - v i)\n[PROOFSTEP]\nsimp_rw [nodal, eval_prod, eval_sub, eval_X, eval_C]\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\nhi : i \u2208 s\n\u22a2 eval (v i) (nodal s v) = 0\n[PROOFSTEP]\nrw [eval_nodal, prod_eq_zero_iff]\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\nhi : i \u2208 s\n\u22a2 \u2203 a, a \u2208 s \u2227 v i - v a = 0\n[PROOFSTEP]\nexact \u27e8i, hi, sub_eq_zero_of_eq rfl\u27e9\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 x \u2260 v i\n\u22a2 eval x (nodal s v) \u2260 0\n[PROOFSTEP]\nsimp_rw [nodal, eval_prod, prod_ne_zero_iff, eval_sub, eval_X, eval_C, sub_ne_zero]\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 x \u2260 v i\n\u22a2 \u2200 (a : \u03b9), a \u2208 s \u2192 x \u2260 v a\n[PROOFSTEP]\nexact hx\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\n\u22a2 nodal s v = (X - \u2191C (v i)) * nodal (Finset.erase s i) v\n[PROOFSTEP]\nsimp_rw [nodal, Finset.mul_prod_erase _ (fun x => X - C (v x)) hi]\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv\u271d : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\nv : \u03b9 \u2192 F\nhi : i \u2208 s\n\u22a2 nodal s v = (X - \u2191C (v i)) * ?m.437432 v hi\n[PROOFSTEP]\nclassical exact nodal_eq_mul_nodal_erase hi\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv\u271d : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\nv : \u03b9 \u2192 F\nhi : i \u2208 s\n\u22a2 nodal s v = (X - \u2191C (v i)) * ?m.437432 v hi\n[PROOFSTEP]\nexact nodal_eq_mul_nodal_erase hi\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\n\u22a2 nodal (Finset.erase s i) v = nodal s v / (X - \u2191C (v i))\n[PROOFSTEP]\nrw [nodal_eq_mul_nodal_erase hi, EuclideanDomain.mul_div_cancel_left]\n[GOAL]\ncase a0\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\n\u22a2 X - \u2191C (v i) \u2260 0\n[PROOFSTEP]\nexact X_sub_C_ne_zero _\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhi : \u00aci \u2208 s\n\u22a2 nodal (insert i s) v = (X - \u2191C (v i)) * nodal s v\n[PROOFSTEP]\nsimp_rw [nodal, prod_insert hi]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\n\u22a2 \u2191derivative (nodal s v) = \u2211 i in s, nodal (Finset.erase s i) v\n[PROOFSTEP]\nrefine' Finset.induction_on s _ fun _ _ hit IH => _\n[GOAL]\ncase refine'_1\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\n\u22a2 \u2191derivative (nodal \u2205 v) = \u2211 i in \u2205, nodal (Finset.erase \u2205 i) v\n[PROOFSTEP]\nrw [nodal_empty, derivative_one, sum_empty]\n[GOAL]\ncase refine'_2\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nx\u271d\u00b9 : \u03b9\nx\u271d : Finset \u03b9\nhit : \u00acx\u271d\u00b9 \u2208 x\u271d\nIH : \u2191derivative (nodal x\u271d v) = \u2211 i in x\u271d, nodal (Finset.erase x\u271d i) v\n\u22a2 \u2191derivative (nodal (insert x\u271d\u00b9 x\u271d) v) = \u2211 i in insert x\u271d\u00b9 x\u271d, nodal (Finset.erase (insert x\u271d\u00b9 x\u271d) i) v\n[PROOFSTEP]\nrw [nodal_insert_eq_nodal hit, derivative_mul, IH, derivative_sub, derivative_X, derivative_C, sub_zero, one_mul,\n sum_insert hit, mul_sum, erase_insert hit, add_right_inj]\n[GOAL]\ncase refine'_2\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nx\u271d\u00b9 : \u03b9\nx\u271d : Finset \u03b9\nhit : \u00acx\u271d\u00b9 \u2208 x\u271d\nIH : \u2191derivative (nodal x\u271d v) = \u2211 i in x\u271d, nodal (Finset.erase x\u271d i) v\n\u22a2 \u2211 x in x\u271d, (X - \u2191C (v x\u271d\u00b9)) * nodal (Finset.erase x\u271d x) v = \u2211 x in x\u271d, nodal (Finset.erase (insert x\u271d\u00b9 x\u271d) x) v\n[PROOFSTEP]\nrefine' sum_congr rfl fun j hjt => _\n[GOAL]\ncase refine'_2\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nx\u271d\u00b9 : \u03b9\nx\u271d : Finset \u03b9\nhit : \u00acx\u271d\u00b9 \u2208 x\u271d\nIH : \u2191derivative (nodal x\u271d v) = \u2211 i in x\u271d, nodal (Finset.erase x\u271d i) v\nj : \u03b9\nhjt : j \u2208 x\u271d\n\u22a2 (X - \u2191C (v x\u271d\u00b9)) * nodal (Finset.erase x\u271d j) v = nodal (Finset.erase (insert x\u271d\u00b9 x\u271d) j) v\n[PROOFSTEP]\nrw [nodal_erase_eq_nodal_div (mem_insert_of_mem hjt), nodal_insert_eq_nodal hit,\n EuclideanDomain.mul_div_assoc _ (X_sub_C_dvd_nodal v hjt), nodal_erase_eq_nodal_div hjt]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\n\u22a2 eval (v i) (\u2191derivative (nodal s v)) = eval (v i) (nodal (Finset.erase s i) v)\n[PROOFSTEP]\nrw [derivative_nodal, eval_finset_sum, \u2190 add_sum_erase _ _ hi, add_right_eq_self]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\n\u22a2 \u2211 x in Finset.erase s i, eval (v i) (nodal (Finset.erase s x) v) = 0\n[PROOFSTEP]\nrefine' sum_eq_zero fun j hj => _\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 Finset.erase s i\n\u22a2 eval (v i) (nodal (Finset.erase s j) v) = 0\n[PROOFSTEP]\nsimp_rw [nodal, eval_prod, eval_sub, eval_X, eval_C, prod_eq_zero_iff, mem_erase]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 Finset.erase s i\n\u22a2 \u2203 a, (a \u2260 j \u2227 a \u2208 s) \u2227 v i - v a = 0\n[PROOFSTEP]\nexact \u27e8i, \u27e8(mem_erase.mp hj).1.symm, hi\u27e9, sub_eq_zero_of_eq rfl\u27e9\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\n\u22a2 nodalWeight s v i = (eval (v i) (nodal (Finset.erase s i) v))\u207b\u00b9\n[PROOFSTEP]\nrw [eval_nodal, nodalWeight, prod_inv_distrib]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\n\u22a2 nodalWeight s v i = (eval (v i) (\u2191derivative (nodal s v)))\u207b\u00b9\n[PROOFSTEP]\nrw [eval_nodal_derivative_eval_node_eq hi, nodalWeight_eq_eval_nodal_erase_inv]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\n\u22a2 nodalWeight s v i \u2260 0\n[PROOFSTEP]\nrw [nodalWeight, prod_ne_zero_iff]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\n\u22a2 \u2200 (a : \u03b9), a \u2208 Finset.erase s i \u2192 (v i - v a)\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nintro j hj\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 Finset.erase s i\n\u22a2 (v i - v j)\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nrcases mem_erase.mp hj with \u27e8hij, hj\u27e9\n[GOAL]\ncase intro\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhvs : Set.InjOn v \u2191s\nhi : i \u2208 s\nj : \u03b9\nhj\u271d : j \u2208 Finset.erase s i\nhij : j \u2260 i\nhj : j \u2208 s\n\u22a2 (v i - v j)\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nrefine' inv_ne_zero (sub_ne_zero_of_ne (mt (hvs.eq_iff hi hj).mp hij.symm))\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\n\u22a2 Lagrange.basis s v i = \u2191C (nodalWeight s v i) * (nodal s v / (X - \u2191C (v i)))\n[PROOFSTEP]\nsimp_rw [Lagrange.basis, basisDivisor, nodalWeight, prod_mul_distrib, map_prod, \u2190 nodal_erase_eq_nodal_div hi, nodal]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhi : i \u2208 s\nhxi : x \u2260 v i\n\u22a2 eval x (Lagrange.basis s v i) = eval x (nodal s v) * (nodalWeight s v i * (x - v i)\u207b\u00b9)\n[PROOFSTEP]\nrw [mul_comm, basis_eq_prod_sub_inv_mul_nodal_div hi, eval_mul, eval_C, \u2190 nodal_erase_eq_nodal_div hi, eval_nodal,\n eval_nodal, mul_assoc, \u2190 mul_prod_erase _ _ hi, \u2190 mul_assoc (x - v i)\u207b\u00b9, inv_mul_cancel (sub_ne_zero_of_ne hxi),\n one_mul]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nj : \u03b9\nhj : j \u2208 s\n\u22a2 \u2191C (r j) * Lagrange.basis s v j = \u2191C (nodalWeight s v j) * (nodal s v / (X - \u2191C (v j))) * \u2191C (r j)\n[PROOFSTEP]\nrw [mul_comm, basis_eq_prod_sub_inv_mul_nodal_div hj]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 x \u2260 v i\n\u22a2 eval x (\u2191(interpolate s v) r) = eval x (nodal s v) * \u2211 i in s, nodalWeight s v i * (x - v i)\u207b\u00b9 * r i\n[PROOFSTEP]\nsimp_rw [interpolate_apply, mul_sum, eval_finset_sum, eval_mul, eval_C]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 x \u2260 v i\n\u22a2 \u2211 x_1 in s, r x_1 * eval x (Lagrange.basis s v x_1) =\n \u2211 x_1 in s, eval x (nodal s v) * (nodalWeight s v x_1 * (x - v x_1)\u207b\u00b9 * r x_1)\n[PROOFSTEP]\nrefine' sum_congr rfl fun i hi => _\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni\u271d : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 x \u2260 v i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 r i * eval x (Lagrange.basis s v i) = eval x (nodal s v) * (nodalWeight s v i * (x - v i)\u207b\u00b9 * r i)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, mul_comm, eval_basis_not_at_node hi (hx _ hi)]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhvs : Set.InjOn v \u2191s\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 x \u2260 v i\nhs : Finset.Nonempty s\n\u22a2 eval x (nodal s v) * \u2211 i in s, nodalWeight s v i * (x - v i)\u207b\u00b9 = 1\n[PROOFSTEP]\nsimpa only [Pi.one_apply, interpolate_one hvs hs, eval_one, mul_one] using (eval_interpolate_not_at_node 1 hx).symm\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 x \u2260 v i\n\u22a2 eval x (\u2191(interpolate s v) r) =\n (\u2211 i in s, nodalWeight s v i * (x - v i)\u207b\u00b9 * r i) / \u2211 i in s, nodalWeight s v i * (x - v i)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 div_one (eval x (interpolate s v r)), \u2190 @eval_one _ _ x, \u2190 interpolate_one hvs hs,\n eval_interpolate_not_at_node r hx, eval_interpolate_not_at_node 1 hx]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d : DecidableEq \u03b9\nhvs : Set.InjOn v \u2191s\nhs : Finset.Nonempty s\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 x \u2260 v i\n\u22a2 (eval x (nodal s fun i => v i) * \u2211 i in s, nodalWeight s (fun i => v i) i * (x - v i)\u207b\u00b9 * r i) /\n (eval x (nodal s fun i => v i) * \u2211 i in s, nodalWeight s (fun i => v i) i * (x - v i)\u207b\u00b9 * OfNat.ofNat 1 i) =\n (\u2211 i in s, nodalWeight s v i * (x - v i)\u207b\u00b9 * r i) / \u2211 i in s, nodalWeight s v i * (x - v i)\u207b\u00b9\n[PROOFSTEP]\nsimp only [mul_div_mul_left _ _ (eval_nodal_not_at_node hx), Pi.one_apply, mul_one]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d\u00b9 : DecidableEq \u03b9\nG : Subgroup F\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\n\u22a2 nodal (Set.toFinset \u2191G) Units.val = X ^ Fintype.card { x // x \u2208 G } - 1\n[PROOFSTEP]\nhave h : degree (1 : F[X]) < degree ((X : F[X]) ^ Fintype.card G) := by simp [Fintype.card_pos]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d\u00b9 : DecidableEq \u03b9\nG : Subgroup F\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\n\u22a2 degree 1 < degree (X ^ Fintype.card { x // x \u2208 G })\n[PROOFSTEP]\nsimp [Fintype.card_pos]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d\u00b9 : DecidableEq \u03b9\nG : Subgroup F\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nh : degree 1 < degree (X ^ Fintype.card { x // x \u2208 G })\n\u22a2 nodal (Set.toFinset \u2191G) Units.val = X ^ Fintype.card { x // x \u2208 G } - 1\n[PROOFSTEP]\napply eq_of_degree_le_of_eval_index_eq (v := ((\u2191) : F\u02e3 \u2192 F)) (G : Set F\u02e3).toFinset\n[GOAL]\ncase hvs\nF : Type u_1\ninst\u271d\u00b2 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d\u00b9 : DecidableEq \u03b9\nG : Subgroup F\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nh : degree 1 < degree (X ^ Fintype.card { x // x \u2208 G })\n\u22a2 Set.InjOn Units.val \u2191(Set.toFinset \u2191G)\n[PROOFSTEP]\nexact Set.injOn_of_injective Units.ext _\n[GOAL]\ncase h_deg_le\nF : Type u_1\ninst\u271d\u00b2 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d\u00b9 : DecidableEq \u03b9\nG : Subgroup F\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nh : degree 1 < degree (X ^ Fintype.card { x // x \u2208 G })\n\u22a2 degree (nodal (Set.toFinset \u2191G) Units.val) \u2264 \u2191(card (Set.toFinset \u2191G))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h_deg_eq\nF : Type u_1\ninst\u271d\u00b2 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d\u00b9 : DecidableEq \u03b9\nG : Subgroup F\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nh : degree 1 < degree (X ^ Fintype.card { x // x \u2208 G })\n\u22a2 degree (nodal (Set.toFinset \u2191G) Units.val) = degree (X ^ Fintype.card { x // x \u2208 G } - 1)\n[PROOFSTEP]\nrw [degree_sub_eq_left_of_degree_lt h, degree_nodal, Set.toFinset_card, degree_pow, degree_X, nsmul_eq_mul, mul_one,\n Nat.cast_inj]\n[GOAL]\ncase h_deg_eq\nF : Type u_1\ninst\u271d\u00b2 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d\u00b9 : DecidableEq \u03b9\nG : Subgroup F\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nh : degree 1 < degree (X ^ Fintype.card { x // x \u2208 G })\n\u22a2 Fintype.card \u2191\u2191G = Fintype.card { x // x \u2208 G }\n[PROOFSTEP]\nexact rfl\n[GOAL]\ncase hlc\nF : Type u_1\ninst\u271d\u00b2 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d\u00b9 : DecidableEq \u03b9\nG : Subgroup F\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nh : degree 1 < degree (X ^ Fintype.card { x // x \u2208 G })\n\u22a2 leadingCoeff (nodal (Set.toFinset \u2191G) Units.val) = leadingCoeff (X ^ Fintype.card { x // x \u2208 G } - 1)\n[PROOFSTEP]\nrw [nodal_monic, leadingCoeff_sub_of_degree_lt h, monic_X_pow]\n[GOAL]\ncase h_eval\nF : Type u_1\ninst\u271d\u00b2 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d\u00b9 : DecidableEq \u03b9\nG : Subgroup F\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nh : degree 1 < degree (X ^ Fintype.card { x // x \u2208 G })\n\u22a2 \u2200 (i : F\u02e3),\n i \u2208 Set.toFinset \u2191G \u2192\n eval (\u2191i) (nodal (Set.toFinset \u2191G) Units.val) = eval (\u2191i) (X ^ Fintype.card { x // x \u2208 G } - 1)\n[PROOFSTEP]\nintros i hi\n[GOAL]\ncase h_eval\nF : Type u_1\ninst\u271d\u00b2 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni\u271d : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d\u00b9 : DecidableEq \u03b9\nG : Subgroup F\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nh : degree 1 < degree (X ^ Fintype.card { x // x \u2208 G })\ni : F\u02e3\nhi : i \u2208 Set.toFinset \u2191G\n\u22a2 eval (\u2191i) (nodal (Set.toFinset \u2191G) Units.val) = eval (\u2191i) (X ^ Fintype.card { x // x \u2208 G } - 1)\n[PROOFSTEP]\nrw [eval_nodal_at_node hi]\n[GOAL]\ncase h_eval\nF : Type u_1\ninst\u271d\u00b2 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni\u271d : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d\u00b9 : DecidableEq \u03b9\nG : Subgroup F\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nh : degree 1 < degree (X ^ Fintype.card { x // x \u2208 G })\ni : F\u02e3\nhi : i \u2208 Set.toFinset \u2191G\n\u22a2 0 = eval (\u2191i) (X ^ Fintype.card { x // x \u2208 G } - 1)\n[PROOFSTEP]\nreplace hi : i \u2208 G := by simpa using hi\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni\u271d : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d\u00b9 : DecidableEq \u03b9\nG : Subgroup F\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nh : degree 1 < degree (X ^ Fintype.card { x // x \u2208 G })\ni : F\u02e3\nhi : i \u2208 Set.toFinset \u2191G\n\u22a2 i \u2208 G\n[PROOFSTEP]\nsimpa using hi\n[GOAL]\ncase h_eval\nF : Type u_1\ninst\u271d\u00b2 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni\u271d : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d\u00b9 : DecidableEq \u03b9\nG : Subgroup F\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nh : degree 1 < degree (X ^ Fintype.card { x // x \u2208 G })\ni : F\u02e3\nhi : i \u2208 G\n\u22a2 0 = eval (\u2191i) (X ^ Fintype.card { x // x \u2208 G } - 1)\n[PROOFSTEP]\nobtain \u27e8g, rfl\u27e9 : \u2203 g : G, g.val = i := \u27e8\u27e8i, hi\u27e9, rfl\u27e9\n[GOAL]\ncase h_eval.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni : \u03b9\nr : \u03b9 \u2192 F\nx : F\ninst\u271d\u00b9 : DecidableEq \u03b9\nG : Subgroup F\u02e3\ninst\u271d : Fintype { x // x \u2208 G }\nh : degree 1 < degree (X ^ Fintype.card { x // x \u2208 G })\ng : { x // x \u2208 G }\nhi : \u2191g \u2208 G\n\u22a2 0 = eval (\u2191\u2191g) (X ^ Fintype.card { x // x \u2208 G } - 1)\n[PROOFSTEP]\nsimp [\u2190 Units.val_pow_eq_pow_val, \u2190 Subgroup.coe_pow G]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Lagrange", "llama_tokens": 37701, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850933, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4175336613924648}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\ninst\u271d : HasPullbacks C\n\u22a2 FamilyOfElements.Compatible x \u2194 FamilyOfElements.PullbackCompatible x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\ninst\u271d : HasPullbacks C\n\u22a2 FamilyOfElements.Compatible x \u2192 FamilyOfElements.PullbackCompatible x\n[PROOFSTEP]\nintro t Y\u2081 Y\u2082 f\u2081 f\u2082 hf\u2081 hf\u2082\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\ninst\u271d : HasPullbacks C\nt : FamilyOfElements.Compatible x\nY\u2081 Y\u2082 : C\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : R f\u2081\nhf\u2082 : R f\u2082\n\u22a2 P.map pullback.fst.op (x f\u2081 hf\u2081) = P.map pullback.snd.op (x f\u2082 hf\u2082)\n[PROOFSTEP]\napply t\n[GOAL]\ncase mp.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\ninst\u271d : HasPullbacks C\nt : FamilyOfElements.Compatible x\nY\u2081 Y\u2082 : C\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : R f\u2081\nhf\u2082 : R f\u2082\n\u22a2 pullback.fst \u226b f\u2081 = pullback.snd \u226b f\u2082\n[PROOFSTEP]\napply pullback.condition\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\ninst\u271d : HasPullbacks C\n\u22a2 FamilyOfElements.PullbackCompatible x \u2192 FamilyOfElements.Compatible x\n[PROOFSTEP]\nintro t Y\u2081 Y\u2082 Z g\u2081 g\u2082 f\u2081 f\u2082 hf\u2081 hf\u2082 comm\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\ninst\u271d : HasPullbacks C\nt : FamilyOfElements.PullbackCompatible x\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : R f\u2081\nhf\u2082 : R f\u2082\ncomm : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 P.map g\u2081.op (x f\u2081 hf\u2081) = P.map g\u2082.op (x f\u2082 hf\u2082)\n[PROOFSTEP]\nrw [\u2190 pullback.lift_fst _ _ comm, op_comp, FunctorToTypes.map_comp_apply, t hf\u2081 hf\u2082, \u2190 FunctorToTypes.map_comp_apply, \u2190\n op_comp, pullback.lift_snd]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nhx : Compatible x\n\u22a2 Compatible (FamilyOfElements.sieveExtend x)\n[PROOFSTEP]\nintro _ _ _ _ _ _ _ h\u2081 h\u2082 comm\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nhx : Compatible x\nY\u2081\u271d Y\u2082\u271d Z\u271d : C\ng\u2081\u271d : Z\u271d \u27f6 Y\u2081\u271d\ng\u2082\u271d : Z\u271d \u27f6 Y\u2082\u271d\nf\u2081\u271d : Y\u2081\u271d \u27f6 X\nf\u2082\u271d : Y\u2082\u271d \u27f6 X\nh\u2081 : (generate R).arrows f\u2081\u271d\nh\u2082 : (generate R).arrows f\u2082\u271d\ncomm : g\u2081\u271d \u226b f\u2081\u271d = g\u2082\u271d \u226b f\u2082\u271d\n\u22a2 P.map g\u2081\u271d.op (FamilyOfElements.sieveExtend x f\u2081\u271d h\u2081) = P.map g\u2082\u271d.op (FamilyOfElements.sieveExtend x f\u2082\u271d h\u2082)\n[PROOFSTEP]\niterate 2 erw [\u2190 FunctorToTypes.map_comp_apply]; rw [\u2190 op_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nhx : Compatible x\nY\u2081\u271d Y\u2082\u271d Z\u271d : C\ng\u2081\u271d : Z\u271d \u27f6 Y\u2081\u271d\ng\u2082\u271d : Z\u271d \u27f6 Y\u2082\u271d\nf\u2081\u271d : Y\u2081\u271d \u27f6 X\nf\u2082\u271d : Y\u2082\u271d \u27f6 X\nh\u2081 : (generate R).arrows f\u2081\u271d\nh\u2082 : (generate R).arrows f\u2082\u271d\ncomm : g\u2081\u271d \u226b f\u2081\u271d = g\u2082\u271d \u226b f\u2082\u271d\n\u22a2 P.map g\u2081\u271d.op (FamilyOfElements.sieveExtend x f\u2081\u271d h\u2081) = P.map g\u2082\u271d.op (FamilyOfElements.sieveExtend x f\u2082\u271d h\u2082)\n[PROOFSTEP]\nerw [\u2190 FunctorToTypes.map_comp_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nhx : Compatible x\nY\u2081\u271d Y\u2082\u271d Z\u271d : C\ng\u2081\u271d : Z\u271d \u27f6 Y\u2081\u271d\ng\u2082\u271d : Z\u271d \u27f6 Y\u2082\u271d\nf\u2081\u271d : Y\u2081\u271d \u27f6 X\nf\u2082\u271d : Y\u2082\u271d \u27f6 X\nh\u2081 : (generate R).arrows f\u2081\u271d\nh\u2082 : (generate R).arrows f\u2082\u271d\ncomm : g\u2081\u271d \u226b f\u2081\u271d = g\u2082\u271d \u226b f\u2082\u271d\n\u22a2 P.map ((Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2081\u271d)).op \u226b g\u2081\u271d.op)\n (x (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2081\u271d) \u226b g = f\u2081\u271d))\n (_ : R (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2081\u271d) \u226b g = f\u2081\u271d)))) =\n P.map g\u2082\u271d.op (FamilyOfElements.sieveExtend x f\u2082\u271d h\u2082)\n[PROOFSTEP]\nrw [\u2190 op_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nhx : Compatible x\nY\u2081\u271d Y\u2082\u271d Z\u271d : C\ng\u2081\u271d : Z\u271d \u27f6 Y\u2081\u271d\ng\u2082\u271d : Z\u271d \u27f6 Y\u2082\u271d\nf\u2081\u271d : Y\u2081\u271d \u27f6 X\nf\u2082\u271d : Y\u2082\u271d \u27f6 X\nh\u2081 : (generate R).arrows f\u2081\u271d\nh\u2082 : (generate R).arrows f\u2082\u271d\ncomm : g\u2081\u271d \u226b f\u2081\u271d = g\u2082\u271d \u226b f\u2082\u271d\n\u22a2 P.map (g\u2081\u271d \u226b Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2081\u271d)).op\n (x (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2081\u271d) \u226b g = f\u2081\u271d))\n (_ : R (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2081\u271d) \u226b g = f\u2081\u271d)))) =\n P.map g\u2082\u271d.op (FamilyOfElements.sieveExtend x f\u2082\u271d h\u2082)\n[PROOFSTEP]\nerw [\u2190 FunctorToTypes.map_comp_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nhx : Compatible x\nY\u2081\u271d Y\u2082\u271d Z\u271d : C\ng\u2081\u271d : Z\u271d \u27f6 Y\u2081\u271d\ng\u2082\u271d : Z\u271d \u27f6 Y\u2082\u271d\nf\u2081\u271d : Y\u2081\u271d \u27f6 X\nf\u2082\u271d : Y\u2082\u271d \u27f6 X\nh\u2081 : (generate R).arrows f\u2081\u271d\nh\u2082 : (generate R).arrows f\u2082\u271d\ncomm : g\u2081\u271d \u226b f\u2081\u271d = g\u2082\u271d \u226b f\u2082\u271d\n\u22a2 P.map (g\u2081\u271d \u226b Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2081\u271d)).op\n (x (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2081\u271d) \u226b g = f\u2081\u271d))\n (_ : R (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2081\u271d) \u226b g = f\u2081\u271d)))) =\n P.map ((Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2082\u271d)).op \u226b g\u2082\u271d.op)\n (x (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2082\u271d) \u226b g = f\u2082\u271d))\n (_ : R (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2082\u271d) \u226b g = f\u2082\u271d))))\n[PROOFSTEP]\nrw [\u2190 op_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nhx : Compatible x\nY\u2081\u271d Y\u2082\u271d Z\u271d : C\ng\u2081\u271d : Z\u271d \u27f6 Y\u2081\u271d\ng\u2082\u271d : Z\u271d \u27f6 Y\u2082\u271d\nf\u2081\u271d : Y\u2081\u271d \u27f6 X\nf\u2082\u271d : Y\u2082\u271d \u27f6 X\nh\u2081 : (generate R).arrows f\u2081\u271d\nh\u2082 : (generate R).arrows f\u2082\u271d\ncomm : g\u2081\u271d \u226b f\u2081\u271d = g\u2082\u271d \u226b f\u2082\u271d\n\u22a2 P.map (g\u2081\u271d \u226b Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2081\u271d)).op\n (x (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2081\u271d) \u226b g = f\u2081\u271d))\n (_ : R (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2081\u271d) \u226b g = f\u2081\u271d)))) =\n P.map (g\u2082\u271d \u226b Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2082\u271d)).op\n (x (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2082\u271d) \u226b g = f\u2082\u271d))\n (_ : R (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2082\u271d) \u226b g = f\u2082\u271d))))\n[PROOFSTEP]\napply hx\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nhx : Compatible x\nY\u2081\u271d Y\u2082\u271d Z\u271d : C\ng\u2081\u271d : Z\u271d \u27f6 Y\u2081\u271d\ng\u2082\u271d : Z\u271d \u27f6 Y\u2082\u271d\nf\u2081\u271d : Y\u2081\u271d \u27f6 X\nf\u2082\u271d : Y\u2082\u271d \u27f6 X\nh\u2081 : (generate R).arrows f\u2081\u271d\nh\u2082 : (generate R).arrows f\u2082\u271d\ncomm : g\u2081\u271d \u226b f\u2081\u271d = g\u2082\u271d \u226b f\u2082\u271d\n\u22a2 (g\u2081\u271d \u226b Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2081\u271d)) \u226b\n Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2081\u271d) \u226b g = f\u2081\u271d) =\n (g\u2082\u271d \u226b Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2082\u271d)) \u226b\n Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f\u2082\u271d) \u226b g = f\u2082\u271d)\n[PROOFSTEP]\nsimp [comm, h\u2081.choose_spec.choose_spec.choose_spec.2, h\u2082.choose_spec.choose_spec.choose_spec.2]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\n\u22a2 FamilyOfElements.sieveExtend x f (_ : f \u2208 (generate R).arrows) = x f hf\n[PROOFSTEP]\nhave h := (le_generate R Y hf).choose_spec\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\nh : \u2203 h g, R g \u2227 h \u226b g = f\n\u22a2 FamilyOfElements.sieveExtend x f (_ : f \u2208 (generate R).arrows) = x f hf\n[PROOFSTEP]\nunfold FamilyOfElements.sieveExtend\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\nh : \u2203 h g, R g \u2227 h \u226b g = f\n\u22a2 P.map (Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f)).op\n (x (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f))\n (_ : R (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f)))) =\n x f hf\n[PROOFSTEP]\nrw [t h.choose (\ud835\udfd9 _) _ hf _]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\nh : \u2203 h g, R g \u2227 h \u226b g = f\n\u22a2 P.map (\ud835\udfd9 Y).op (x f hf) = x f hf\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\nh : \u2203 h g, R g \u2227 h \u226b g = f\n\u22a2 Exists.choose h \u226b Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f) = \ud835\udfd9 Y \u226b f\n[PROOFSTEP]\nrw [id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\nh : \u2203 h g, R g \u2227 h \u226b g = f\n\u22a2 Exists.choose h \u226b Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f) = f\n[PROOFSTEP]\nexact h.choose_spec.choose_spec.2\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\n\u22a2 FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) (FamilyOfElements.sieveExtend x) = x\n[PROOFSTEP]\nfunext Y f hf\n[GOAL]\ncase h.h.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nY : C\nf : Y \u27f6 X\nhf : R f\n\u22a2 FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) (FamilyOfElements.sieveExtend x) f hf = x f hf\n[PROOFSTEP]\nexact extend_agrees t hf\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P S.arrows\n\u22a2 FamilyOfElements.Compatible x \u2194 FamilyOfElements.SieveCompatible x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P S.arrows\n\u22a2 FamilyOfElements.Compatible x \u2192 FamilyOfElements.SieveCompatible x\n[PROOFSTEP]\nintro h Y Z f g hf\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P S.arrows\nh : FamilyOfElements.Compatible x\nY Z : C\nf : Y \u27f6 X\ng : Z \u27f6 Y\nhf : S.arrows f\n\u22a2 x (g \u226b f) (_ : S.arrows (g \u226b f)) = P.map g.op (x f hf)\n[PROOFSTEP]\nsimpa using h (\ud835\udfd9 _) g (S.downward_closed hf g) hf (id_comp _)\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P S.arrows\n\u22a2 FamilyOfElements.SieveCompatible x \u2192 FamilyOfElements.Compatible x\n[PROOFSTEP]\nintro h Y\u2081 Y\u2082 Z g\u2081 g\u2082 f\u2081 f\u2082 h\u2081 h\u2082 k\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P S.arrows\nh : FamilyOfElements.SieveCompatible x\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nh\u2081 : S.arrows f\u2081\nh\u2082 : S.arrows f\u2082\nk : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 P.map g\u2081.op (x f\u2081 h\u2081) = P.map g\u2082.op (x f\u2082 h\u2082)\n[PROOFSTEP]\nsimp_rw [\u2190 h f\u2081 g\u2081 h\u2081, \u2190 h f\u2082 g\u2082 h\u2082]\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P S.arrows\nh : FamilyOfElements.SieveCompatible x\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nh\u2081 : S.arrows f\u2081\nh\u2082 : S.arrows f\u2082\nk : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 x (g\u2081 \u226b f\u2081) (_ : S.arrows (g\u2081 \u226b f\u2081)) = x (g\u2082 \u226b f\u2082) (_ : S.arrows (g\u2082 \u226b f\u2082))\n[PROOFSTEP]\ncongr\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P (generate R).arrows\nt : FamilyOfElements.Compatible x\n\u22a2 FamilyOfElements.sieveExtend (FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x) = x\n[PROOFSTEP]\nrw [compatible_iff_sieveCompatible] at t \n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P (generate R).arrows\nt : FamilyOfElements.SieveCompatible x\n\u22a2 FamilyOfElements.sieveExtend (FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x) = x\n[PROOFSTEP]\nfunext _ _ h\n[GOAL]\ncase h.h.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P (generate R).arrows\nt : FamilyOfElements.SieveCompatible x\nx\u271d\u00b9 : C\nx\u271d : x\u271d\u00b9 \u27f6 X\nh : (generate R).arrows x\u271d\n\u22a2 FamilyOfElements.sieveExtend (FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x) x\u271d h = x x\u271d h\n[PROOFSTEP]\napply (t _ _ _).symm.trans\n[GOAL]\ncase h.h.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P (generate R).arrows\nt : FamilyOfElements.SieveCompatible x\nx\u271d\u00b9 : C\nx\u271d : x\u271d\u00b9 \u27f6 X\nh : (generate R).arrows x\u271d\n\u22a2 x\n (Exists.choose (_ : \u2203 h_1 g, R g \u2227 h_1 \u226b g = x\u271d) \u226b\n Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h_1 g, R g \u2227 h_1 \u226b g = x\u271d) \u226b g = x\u271d))\n (_ :\n (generate R).arrows\n (Exists.choose (_ : \u2203 h_1 g, R g \u2227 h_1 \u226b g = x\u271d) \u226b\n Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h_1 g, R g \u2227 h_1 \u226b g = x\u271d) \u226b g = x\u271d))) =\n x x\u271d h\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.h.h.e_f\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P (generate R).arrows\nt : FamilyOfElements.SieveCompatible x\nx\u271d\u00b9 : C\nx\u271d : x\u271d\u00b9 \u27f6 X\nh : (generate R).arrows x\u271d\n\u22a2 Exists.choose (_ : \u2203 h_1 g, R g \u2227 h_1 \u226b g = x\u271d) \u226b\n Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h_1 g, R g \u2227 h_1 \u226b g = x\u271d) \u226b g = x\u271d) =\n x\u271d\n[PROOFSTEP]\nexact h.choose_spec.choose_spec.choose_spec.2\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx\u2081 x\u2082 : FamilyOfElements P (generate R).arrows\nt\u2081 : FamilyOfElements.Compatible x\u2081\nt\u2082 : FamilyOfElements.Compatible x\u2082\nh :\n FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x\u2081 =\n FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x\u2082\n\u22a2 x\u2081 = x\u2082\n[PROOFSTEP]\nrw [\u2190 extend_restrict t\u2081, \u2190 extend_restrict t\u2082]\n -- porting note: congr fails to make progress\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx\u2081 x\u2082 : FamilyOfElements P (generate R).arrows\nt\u2081 : FamilyOfElements.Compatible x\u2081\nt\u2082 : FamilyOfElements.Compatible x\u2082\nh :\n FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x\u2081 =\n FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x\u2082\n\u22a2 FamilyOfElements.sieveExtend (FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x\u2081) =\n FamilyOfElements.sieveExtend (FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x\u2082)\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx\u2081 x\u2082 : FamilyOfElements P (generate R).arrows\nt\u2081 : FamilyOfElements.Compatible x\u2081\nt\u2082 : FamilyOfElements.Compatible x\u2082\nh :\n FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x\u2081 =\n FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x\u2082\n\u22a2 FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x\u2081 =\n FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x\u2082\n[PROOFSTEP]\nexact h\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS\u271d : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nS : Sieve X\nx : FamilyOfElements P S.arrows\nt : Compatible x\nf : Y \u27f6 X\nhf : S.arrows f\nZ : C\ng : Z \u27f6 Y\n\u22a2 x (g \u226b f) (_ : S.arrows (g \u226b f)) = P.map g.op (x f hf)\n[PROOFSTEP]\nsimpa using t (\ud835\udfd9 _) g (S.downward_closed hf g) hf (id_comp _)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : D \u2964 C\nZ : D\nT : Presieve (F.obj Z)\nx : FamilyOfElements P T\nh : Compatible x\n\u22a2 Compatible (FamilyOfElements.functorPullback F x)\n[PROOFSTEP]\nintro Z\u2081 Z\u2082 W g\u2081 g\u2082 f\u2081 f\u2082 h\u2081 h\u2082 eq\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : D \u2964 C\nZ : D\nT : Presieve (F.obj Z)\nx : FamilyOfElements P T\nh : Compatible x\nZ\u2081 Z\u2082 W : D\ng\u2081 : W \u27f6 Z\u2081\ng\u2082 : W \u27f6 Z\u2082\nf\u2081 : Z\u2081 \u27f6 Z\nf\u2082 : Z\u2082 \u27f6 Z\nh\u2081 : Presieve.functorPullback F T f\u2081\nh\u2082 : Presieve.functorPullback F T f\u2082\neq : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 (F.op \u22d9 P).map g\u2081.op (FamilyOfElements.functorPullback F x f\u2081 h\u2081) =\n (F.op \u22d9 P).map g\u2082.op (FamilyOfElements.functorPullback F x f\u2082 h\u2082)\n[PROOFSTEP]\nexact h (F.map g\u2081) (F.map g\u2082) h\u2081 h\u2082 (by simp only [\u2190 F.map_comp, eq])\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : D \u2964 C\nZ : D\nT : Presieve (F.obj Z)\nx : FamilyOfElements P T\nh : Compatible x\nZ\u2081 Z\u2082 W : D\ng\u2081 : W \u27f6 Z\u2081\ng\u2082 : W \u27f6 Z\u2082\nf\u2081 : Z\u2081 \u27f6 Z\nf\u2082 : Z\u2082 \u27f6 Z\nh\u2081 : Presieve.functorPullback F T f\u2081\nh\u2082 : Presieve.functorPullback F T f\u2082\neq : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 F.map g\u2081 \u226b F.map f\u2081 = F.map g\u2082 \u226b F.map f\u2082\n[PROOFSTEP]\nsimp only [\u2190 F.map_comp, eq]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX\u271d Y\u271d : C\nS : Sieve X\u271d\nR : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : D \u2964 C\nX : D\nT : Presieve X\nx : FamilyOfElements (F.op \u22d9 P) T\nY : C\nf : Y \u27f6 F.obj X\nh : Presieve.functorPushforward F T f\n\u22a2 P.obj (op Y)\n[PROOFSTEP]\nobtain \u27e8Z, g, h, h\u2081, _\u27e9 := getFunctorPushforwardStructure h\n[GOAL]\ncase mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX\u271d Y\u271d : C\nS : Sieve X\u271d\nR : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : D \u2964 C\nX : D\nT : Presieve X\nx : FamilyOfElements (F.op \u22d9 P) T\nY : C\nf : Y \u27f6 F.obj X\nh\u271d : Presieve.functorPushforward F T f\nZ : D\ng : Z \u27f6 X\nh : Y \u27f6 F.obj Z\nh\u2081 : T g\nfac\u271d : f = h \u226b F.map g\n\u22a2 P.obj (op Y)\n[PROOFSTEP]\nexact P.map h.op (x g h\u2081)\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nf : Y \u27f6 X\nx : FamilyOfElements P S.arrows\nh : Compatible x\n\u22a2 Compatible (FamilyOfElements.pullback f x)\n[PROOFSTEP]\nsimp only [compatible_iff_sieveCompatible] at h \u22a2\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nf : Y \u27f6 X\nx : FamilyOfElements P S.arrows\nh : SieveCompatible x\n\u22a2 SieveCompatible (FamilyOfElements.pullback f x)\n[PROOFSTEP]\nintro W Z f\u2081 f\u2082 hf\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nf : Y \u27f6 X\nx : FamilyOfElements P S.arrows\nh : SieveCompatible x\nW Z : C\nf\u2081 : W \u27f6 Y\nf\u2082 : Z \u27f6 W\nhf : (Sieve.pullback f S).arrows f\u2081\n\u22a2 FamilyOfElements.pullback f x (f\u2082 \u226b f\u2081) (_ : (Sieve.pullback f S).arrows (f\u2082 \u226b f\u2081)) =\n P.map f\u2082.op (FamilyOfElements.pullback f x f\u2081 hf)\n[PROOFSTEP]\nunfold FamilyOfElements.pullback\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nf : Y \u27f6 X\nx : FamilyOfElements P S.arrows\nh : SieveCompatible x\nW Z : C\nf\u2081 : W \u27f6 Y\nf\u2082 : Z \u27f6 W\nhf : (Sieve.pullback f S).arrows f\u2081\n\u22a2 x ((f\u2082 \u226b f\u2081) \u226b f) (_ : (Sieve.pullback f S).arrows (f\u2082 \u226b f\u2081)) = P.map f\u2082.op (x (f\u2081 \u226b f) hf)\n[PROOFSTEP]\nrw [\u2190 h (f\u2081 \u226b f) f\u2082 hf]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nf : Y \u27f6 X\nx : FamilyOfElements P S.arrows\nh : SieveCompatible x\nW Z : C\nf\u2081 : W \u27f6 Y\nf\u2082 : Z \u27f6 W\nhf : (Sieve.pullback f S).arrows f\u2081\n\u22a2 x ((f\u2082 \u226b f\u2081) \u226b f) (_ : (Sieve.pullback f S).arrows (f\u2082 \u226b f\u2081)) = x (f\u2082 \u226b f\u2081 \u226b f) (_ : S.arrows (f\u2082 \u226b f\u2081 \u226b f))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_f\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nf : Y \u27f6 X\nx : FamilyOfElements P S.arrows\nh : SieveCompatible x\nW Z : C\nf\u2081 : W \u27f6 Y\nf\u2082 : Z \u27f6 W\nhf : (Sieve.pullback f S).arrows f\u2081\n\u22a2 (f\u2082 \u226b f\u2081) \u226b f = f\u2082 \u226b f\u2081 \u226b f\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nf : P \u27f6 Q\nx : FamilyOfElements P R\nh : Compatible x\n\u22a2 Compatible (FamilyOfElements.compPresheafMap f x)\n[PROOFSTEP]\nintro Z\u2081 Z\u2082 W g\u2081 g\u2082 f\u2081 f\u2082 h\u2081 h\u2082 eq\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nf : P \u27f6 Q\nx : FamilyOfElements P R\nh : Compatible x\nZ\u2081 Z\u2082 W : C\ng\u2081 : W \u27f6 Z\u2081\ng\u2082 : W \u27f6 Z\u2082\nf\u2081 : Z\u2081 \u27f6 X\nf\u2082 : Z\u2082 \u27f6 X\nh\u2081 : R f\u2081\nh\u2082 : R f\u2082\neq : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 Q.map g\u2081.op (FamilyOfElements.compPresheafMap f x f\u2081 h\u2081) = Q.map g\u2082.op (FamilyOfElements.compPresheafMap f x f\u2082 h\u2082)\n[PROOFSTEP]\nunfold FamilyOfElements.compPresheafMap\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nf : P \u27f6 Q\nx : FamilyOfElements P R\nh : Compatible x\nZ\u2081 Z\u2082 W : C\ng\u2081 : W \u27f6 Z\u2081\ng\u2082 : W \u27f6 Z\u2082\nf\u2081 : Z\u2081 \u27f6 X\nf\u2082 : Z\u2082 \u27f6 X\nh\u2081 : R f\u2081\nh\u2082 : R f\u2082\neq : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 Q.map g\u2081.op (NatTrans.app f (op Z\u2081) (x f\u2081 h\u2081)) = Q.map g\u2082.op (NatTrans.app f (op Z\u2082) (x f\u2082 h\u2082))\n[PROOFSTEP]\nrwa [\u2190 FunctorToTypes.naturality, \u2190 FunctorToTypes.naturality, h]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : P.obj (op X)\nf : P \u27f6 Q\nh : IsAmalgamation x t\n\u22a2 IsAmalgamation (FamilyOfElements.compPresheafMap f x) (NatTrans.app f (op X) t)\n[PROOFSTEP]\nintro Y g hg\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : P.obj (op X)\nf : P \u27f6 Q\nh : IsAmalgamation x t\nY : C\ng : Y \u27f6 X\nhg : R g\n\u22a2 Q.map g.op (NatTrans.app f (op X) t) = FamilyOfElements.compPresheafMap f x g hg\n[PROOFSTEP]\ndsimp [FamilyOfElements.compPresheafMap]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : P.obj (op X)\nf : P \u27f6 Q\nh : IsAmalgamation x t\nY : C\ng : Y \u27f6 X\nhg : R g\n\u22a2 Q.map g.op (NatTrans.app f (op X) t) = NatTrans.app f (op Y) (x g hg)\n[PROOFSTEP]\nchange (f.app _ \u226b Q.map _) _ = _\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : P.obj (op X)\nf : P \u27f6 Q\nh : IsAmalgamation x t\nY : C\ng : Y \u27f6 X\nhg : R g\n\u22a2 (NatTrans.app f (op X) \u226b Q.map g.op) t = NatTrans.app f (op Y) (x g hg)\n[PROOFSTEP]\nrw [\u2190 f.naturality, types_comp_apply, h g hg]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : \u2203 t, FamilyOfElements.IsAmalgamation x t\n\u22a2 FamilyOfElements.Compatible x\n[PROOFSTEP]\ncases' h with t ht\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : P.obj (op X)\nht : FamilyOfElements.IsAmalgamation x t\n\u22a2 FamilyOfElements.Compatible x\n[PROOFSTEP]\nintro Y\u2081 Y\u2082 Z g\u2081 g\u2082 f\u2081 f\u2082 h\u2081 h\u2082 comm\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : P.obj (op X)\nht : FamilyOfElements.IsAmalgamation x t\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nh\u2081 : R f\u2081\nh\u2082 : R f\u2082\ncomm : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 P.map g\u2081.op (x f\u2081 h\u2081) = P.map g\u2082.op (x f\u2082 h\u2082)\n[PROOFSTEP]\nrw [\u2190 ht _ h\u2081, \u2190 ht _ h\u2082, \u2190 FunctorToTypes.map_comp_apply, \u2190 op_comp, comm]\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : P.obj (op X)\nht : FamilyOfElements.IsAmalgamation x t\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nh\u2081 : R f\u2081\nh\u2082 : R f\u2082\ncomm : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 P.map (g\u2082 \u226b f\u2082).op t = P.map g\u2082.op (P.map f\u2082.op t)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\nx : FamilyOfElements P R\nt : P.obj (op X)\nht : FamilyOfElements.IsAmalgamation x t\n\u22a2 FamilyOfElements.IsAmalgamation (FamilyOfElements.sieveExtend x) t\n[PROOFSTEP]\nintro Y f hf\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\nx : FamilyOfElements P R\nt : P.obj (op X)\nht : FamilyOfElements.IsAmalgamation x t\nY : C\nf : Y \u27f6 X\nhf : (generate R).arrows f\n\u22a2 P.map f.op t = FamilyOfElements.sieveExtend x f hf\n[PROOFSTEP]\ndsimp [FamilyOfElements.sieveExtend]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\nx : FamilyOfElements P R\nt : P.obj (op X)\nht : FamilyOfElements.IsAmalgamation x t\nY : C\nf : Y \u27f6 X\nhf : (generate R).arrows f\n\u22a2 P.map f.op t =\n P.map (Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f)).op\n (x (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f))\n (_ : R (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f))))\n[PROOFSTEP]\nrw [\u2190 ht _, \u2190 FunctorToTypes.map_comp_apply, \u2190 op_comp, hf.choose_spec.choose_spec.choose_spec.2]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 IsSeparatedFor P R \u2194 IsSeparatedFor P (generate R).arrows\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 IsSeparatedFor P R \u2192 IsSeparatedFor P (generate R).arrows\n[PROOFSTEP]\nintro h x t\u2081 t\u2082 ht\u2081 ht\u2082\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nh : IsSeparatedFor P R\nx : FamilyOfElements P (generate R).arrows\nt\u2081 t\u2082 : P.obj (op X)\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 t\u2081 = t\u2082\n[PROOFSTEP]\napply h (x.restrict (le_generate R)) t\u2081 t\u2082 _ _\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nh : IsSeparatedFor P R\nx : FamilyOfElements P (generate R).arrows\nt\u2081 t\u2082 : P.obj (op X)\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 FamilyOfElements.IsAmalgamation (FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x) t\u2081\n[PROOFSTEP]\nexact isAmalgamation_restrict _ x t\u2081 ht\u2081\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nh : IsSeparatedFor P R\nx : FamilyOfElements P (generate R).arrows\nt\u2081 t\u2082 : P.obj (op X)\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 FamilyOfElements.IsAmalgamation (FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x) t\u2082\n[PROOFSTEP]\nexact isAmalgamation_restrict _ x t\u2082 ht\u2082\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 IsSeparatedFor P (generate R).arrows \u2192 IsSeparatedFor P R\n[PROOFSTEP]\nintro h x t\u2081 t\u2082 ht\u2081 ht\u2082\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nh : IsSeparatedFor P (generate R).arrows\nx : FamilyOfElements P R\nt\u2081 t\u2082 : P.obj (op X)\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 t\u2081 = t\u2082\n[PROOFSTEP]\napply h x.sieveExtend\n[GOAL]\ncase mpr.a\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nh : IsSeparatedFor P (generate R).arrows\nx : FamilyOfElements P R\nt\u2081 t\u2082 : P.obj (op X)\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 FamilyOfElements.IsAmalgamation (FamilyOfElements.sieveExtend x) t\u2081\n[PROOFSTEP]\nexact isAmalgamation_sieveExtend x t\u2081 ht\u2081\n[GOAL]\ncase mpr.a\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nh : IsSeparatedFor P (generate R).arrows\nx : FamilyOfElements P R\nt\u2081 t\u2082 : P.obj (op X)\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 FamilyOfElements.IsAmalgamation (FamilyOfElements.sieveExtend x) t\u2082\n[PROOFSTEP]\nexact isAmalgamation_sieveExtend x t\u2082 ht\u2082\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nx : FamilyOfElements P \u22a4\nt\u2081 t\u2082 : P.obj (op X)\nh\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nh\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 t\u2081 = t\u2082\n[PROOFSTEP]\nhave q\u2081 := h\u2081 (\ud835\udfd9 X) (by tauto)\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nx : FamilyOfElements P \u22a4\nt\u2081 t\u2082 : P.obj (op X)\nh\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nh\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 \u22a4 (\ud835\udfd9 X)\n[PROOFSTEP]\ntauto\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nx : FamilyOfElements P \u22a4\nt\u2081 t\u2082 : P.obj (op X)\nh\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nh\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\nq\u2081 : P.map (\ud835\udfd9 X).op t\u2081 = x (\ud835\udfd9 X) (_ : True)\n\u22a2 t\u2081 = t\u2082\n[PROOFSTEP]\nhave q\u2082 := h\u2082 (\ud835\udfd9 X) (by tauto)\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nx : FamilyOfElements P \u22a4\nt\u2081 t\u2082 : P.obj (op X)\nh\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nh\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\nq\u2081 : P.map (\ud835\udfd9 X).op t\u2081 = x (\ud835\udfd9 X) (_ : True)\n\u22a2 \u22a4 (\ud835\udfd9 X)\n[PROOFSTEP]\ntauto\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nx : FamilyOfElements P \u22a4\nt\u2081 t\u2082 : P.obj (op X)\nh\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nh\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\nq\u2081 : P.map (\ud835\udfd9 X).op t\u2081 = x (\ud835\udfd9 X) (_ : True)\nq\u2082 : P.map (\ud835\udfd9 X).op t\u2082 = x (\ud835\udfd9 X) (_ : True)\n\u22a2 t\u2081 = t\u2082\n[PROOFSTEP]\nsimp only [op_id, FunctorToTypes.map_id_apply] at q\u2081 q\u2082 \n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nx : FamilyOfElements P \u22a4\nt\u2081 t\u2082 : P.obj (op X)\nh\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nh\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\nq\u2081 : t\u2081 = x (\ud835\udfd9 X) (_ : True)\nq\u2082 : t\u2082 = x (\ud835\udfd9 X) (_ : True)\n\u22a2 t\u2081 = t\u2082\n[PROOFSTEP]\nrw [q\u2081, q\u2082]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u03b1 : functor S \u27f6 P\n\u22a2 { x // FamilyOfElements.Compatible x }\n[PROOFSTEP]\nrefine' \u27e8fun Y f hf => _, _\u27e9\n[GOAL]\ncase refine'_1\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u03b1 : functor S \u27f6 P\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 P.obj (op Y)\n[PROOFSTEP]\napply \u03b1.app (op Y) \u27e8_, hf\u27e9\n[GOAL]\ncase refine'_2\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u03b1 : functor S \u27f6 P\n\u22a2 FamilyOfElements.Compatible fun Y f hf => NatTrans.app \u03b1 (op Y) { val := f, property := hf }\n[PROOFSTEP]\nrw [compatible_iff_sieveCompatible]\n[GOAL]\ncase refine'_2\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u03b1 : functor S \u27f6 P\n\u22a2 FamilyOfElements.SieveCompatible fun Y f hf => NatTrans.app \u03b1 (op Y) { val := f, property := hf }\n[PROOFSTEP]\nintro Y Z f g hf\n[GOAL]\ncase refine'_2\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u03b1 : functor S \u27f6 P\nY Z : C\nf : Y \u27f6 X\ng : Z \u27f6 Y\nhf : S.arrows f\n\u22a2 (fun Y f hf => NatTrans.app \u03b1 (op Y) { val := f, property := hf }) Z (g \u226b f) (_ : S.arrows (g \u226b f)) =\n P.map g.op ((fun Y f hf => NatTrans.app \u03b1 (op Y) { val := f, property := hf }) Y f hf)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_2\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u03b1 : functor S \u27f6 P\nY Z : C\nf : Y \u27f6 X\ng : Z \u27f6 Y\nhf : S.arrows f\n\u22a2 NatTrans.app \u03b1 (op Z) { val := g \u226b f, property := (_ : S.arrows (g \u226b f)) } =\n P.map g.op (NatTrans.app \u03b1 (op Y) { val := f, property := hf })\n[PROOFSTEP]\nrw [\u2190 FunctorToTypes.naturality _ _ \u03b1 g.op]\n[GOAL]\ncase refine'_2\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u03b1 : functor S \u27f6 P\nY Z : C\nf : Y \u27f6 X\ng : Z \u27f6 Y\nhf : S.arrows f\n\u22a2 NatTrans.app \u03b1 (op Z) { val := g \u226b f, property := (_ : S.arrows (g \u226b f)) } =\n NatTrans.app \u03b1 (op Z) ((functor S).map g.op { val := f, property := hf })\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\nt : { x // FamilyOfElements.Compatible x }\nY Z : C\u1d52\u1d56\ng : Y \u27f6 Z\n\u22a2 (functor S).map g \u226b (fun Y f => \u2191t \u2191f (_ : S.arrows \u2191f)) Z = (fun Y f => \u2191t \u2191f (_ : S.arrows \u2191f)) Y \u226b P.map g\n[PROOFSTEP]\next \u27e8f, hf\u27e9\n[GOAL]\ncase h.mk\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\nt : { x // FamilyOfElements.Compatible x }\nY Z : C\u1d52\u1d56\ng : Y \u27f6 Z\nf : Y.unop \u27f6 X\nhf : S.arrows f\n\u22a2 ((functor S).map g \u226b (fun Y f => \u2191t \u2191f (_ : S.arrows \u2191f)) Z) { val := f, property := hf } =\n ((fun Y f => \u2191t \u2191f (_ : S.arrows \u2191f)) Y \u226b P.map g) { val := f, property := hf }\n[PROOFSTEP]\napply t.2.to_sieveCompatible _\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u03b1 : functor S \u27f6 P\n\u22a2 (fun t => NatTrans.mk fun Y f => \u2191t \u2191f (_ : S.arrows \u2191f))\n ((fun \u03b1 =>\n { val := fun Y f hf => NatTrans.app \u03b1 (op Y) { val := f, property := hf },\n property :=\n (_ : FamilyOfElements.Compatible fun Y f hf => NatTrans.app \u03b1 (op Y) { val := f, property := hf }) })\n \u03b1) =\n \u03b1\n[PROOFSTEP]\next X \u27e8_, _\u27e9\n[GOAL]\ncase w.h.h.mk\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX\u271d Y : C\nS : Sieve X\u271d\nR : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u03b1 : functor S \u27f6 P\nX : C\u1d52\u1d56\nval\u271d : X.unop \u27f6 X\u271d\nproperty\u271d : S.arrows val\u271d\n\u22a2 NatTrans.app\n ((fun t => NatTrans.mk fun Y f => \u2191t \u2191f (_ : S.arrows \u2191f))\n ((fun \u03b1 =>\n { val := fun Y f hf => NatTrans.app \u03b1 (op Y) { val := f, property := hf },\n property :=\n (_ : FamilyOfElements.Compatible fun Y f hf => NatTrans.app \u03b1 (op Y) { val := f, property := hf }) })\n \u03b1))\n X { val := val\u271d, property := property\u271d } =\n NatTrans.app \u03b1 X { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u22a2 Function.RightInverse (fun t => NatTrans.mk fun Y f => \u2191t \u2191f (_ : S.arrows \u2191f)) fun \u03b1 =>\n { val := fun Y f hf => NatTrans.app \u03b1 (op Y) { val := f, property := hf },\n property := (_ : FamilyOfElements.Compatible fun Y f hf => NatTrans.app \u03b1 (op Y) { val := f, property := hf }) }\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase mk\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\nx : FamilyOfElements P S.arrows\nhx : FamilyOfElements.Compatible x\n\u22a2 (fun \u03b1 =>\n { val := fun Y f hf => NatTrans.app \u03b1 (op Y) { val := f, property := hf },\n property :=\n (_ : FamilyOfElements.Compatible fun Y f hf => NatTrans.app \u03b1 (op Y) { val := f, property := hf }) })\n ((fun t => NatTrans.mk fun Y f => \u2191t \u2191f (_ : S.arrows \u2191f)) { val := x, property := hx }) =\n { val := x, property := hx }\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\nx : functor S \u27f6 P\ng : yoneda.obj X \u27f6 P\n\u22a2 functorInclusion S \u226b g = x \u2194 FamilyOfElements.IsAmalgamation (\u2191(\u2191natTransEquivCompatibleFamily x)) (\u2191yonedaEquiv g)\n[PROOFSTEP]\nchange _ \u2194 \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (h : S f), P.map f.op (yonedaEquiv g) = x.app (op Y) \u27e8f, h\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\nx : functor S \u27f6 P\ng : yoneda.obj X \u27f6 P\n\u22a2 functorInclusion S \u226b g = x \u2194\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (h : S.arrows f),\n P.map f.op (\u2191yonedaEquiv g) = NatTrans.app x (op Y) { val := f, property := h }\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\nx : functor S \u27f6 P\ng : yoneda.obj X \u27f6 P\n\u22a2 functorInclusion S \u226b g = x \u2192\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (h : S.arrows f),\n P.map f.op (\u2191yonedaEquiv g) = NatTrans.app x (op Y) { val := f, property := h }\n[PROOFSTEP]\nrintro rfl Y f hf\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\ng : yoneda.obj X \u27f6 P\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 P.map f.op (\u2191yonedaEquiv g) = NatTrans.app (functorInclusion S \u226b g) (op Y) { val := f, property := hf }\n[PROOFSTEP]\nrw [yonedaEquiv_naturality]\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\ng : yoneda.obj X \u27f6 P\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 \u2191yonedaEquiv (yoneda.map f \u226b g) = NatTrans.app (functorInclusion S \u226b g) (op Y) { val := f, property := hf }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\ng : yoneda.obj X \u27f6 P\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 NatTrans.app g (op Y) (\ud835\udfd9 Y \u226b f) = NatTrans.app g (op Y) f\n[PROOFSTEP]\nsimp\n -- See note [dsimp, simp].\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\nx : functor S \u27f6 P\ng : yoneda.obj X \u27f6 P\n\u22a2 (\u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (h : S.arrows f),\n P.map f.op (\u2191yonedaEquiv g) = NatTrans.app x (op Y) { val := f, property := h }) \u2192\n functorInclusion S \u226b g = x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\nx : functor S \u27f6 P\ng : yoneda.obj X \u27f6 P\nh :\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (h : S.arrows f),\n P.map f.op (\u2191yonedaEquiv g) = NatTrans.app x (op Y) { val := f, property := h }\n\u22a2 functorInclusion S \u226b g = x\n[PROOFSTEP]\next Y \u27e8f, hf\u27e9\n[GOAL]\ncase mpr.w.h.h.mk\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\nx : functor S \u27f6 P\ng : yoneda.obj X \u27f6 P\nh :\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (h : S.arrows f),\n P.map f.op (\u2191yonedaEquiv g) = NatTrans.app x (op Y) { val := f, property := h }\nY : C\u1d52\u1d56\nf : Y.unop \u27f6 X\nhf : S.arrows f\n\u22a2 NatTrans.app (functorInclusion S \u226b g) Y { val := f, property := hf } = NatTrans.app x Y { val := f, property := hf }\n[PROOFSTEP]\nconvert h f hf\n[GOAL]\ncase h.e'_2.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\nx : functor S \u27f6 P\ng : yoneda.obj X \u27f6 P\nh :\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (h : S.arrows f),\n P.map f.op (\u2191yonedaEquiv g) = NatTrans.app x (op Y) { val := f, property := h }\nY : C\u1d52\u1d56\nf : Y.unop \u27f6 X\nhf : S.arrows f\ne_1\u271d : P.obj Y = P.obj (op Y.unop)\n\u22a2 NatTrans.app (functorInclusion S \u226b g) Y { val := f, property := hf } = P.map f.op (\u2191yonedaEquiv g)\n[PROOFSTEP]\nrw [yonedaEquiv_naturality]\n[GOAL]\ncase h.e'_2.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\nx : functor S \u27f6 P\ng : yoneda.obj X \u27f6 P\nh :\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (h : S.arrows f),\n P.map f.op (\u2191yonedaEquiv g) = NatTrans.app x (op Y) { val := f, property := h }\nY : C\u1d52\u1d56\nf : Y.unop \u27f6 X\nhf : S.arrows f\ne_1\u271d : P.obj Y = P.obj (op Y.unop)\n\u22a2 NatTrans.app (functorInclusion S \u226b g) Y { val := f, property := hf } = \u2191yonedaEquiv (yoneda.map f \u226b g)\n[PROOFSTEP]\nsimp [yonedaEquiv]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u22a2 IsSheafFor P S.arrows \u2194 YonedaSheafCondition P S\n[PROOFSTEP]\nrw [IsSheafFor, YonedaSheafCondition]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u22a2 (\u2200 (x : FamilyOfElements P S.arrows), FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t) \u2194\n \u2200 (f : functor S \u27f6 P), \u2203! g, functorInclusion S \u226b g = f\n[PROOFSTEP]\nsimp_rw [extension_iff_amalgamation]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u22a2 (\u2200 (x : FamilyOfElements P S.arrows), FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t) \u2194\n \u2200 (f : functor S \u27f6 P), \u2203! g, FamilyOfElements.IsAmalgamation (\u2191(\u2191natTransEquivCompatibleFamily f)) (\u2191yonedaEquiv g)\n[PROOFSTEP]\nrw [Equiv.forall_congr_left' natTransEquivCompatibleFamily]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u22a2 (\u2200 (x : FamilyOfElements P S.arrows), FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t) \u2194\n \u2200 (y : { x // FamilyOfElements.Compatible x }),\n \u2203! g,\n FamilyOfElements.IsAmalgamation (\u2191(\u2191natTransEquivCompatibleFamily (\u2191natTransEquivCompatibleFamily.symm y)))\n (\u2191yonedaEquiv g)\n[PROOFSTEP]\nrw [Subtype.forall]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u22a2 (\u2200 (x : FamilyOfElements P S.arrows), FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t) \u2194\n \u2200 (a : FamilyOfElements P S.arrows) (b : FamilyOfElements.Compatible a),\n \u2203! g,\n FamilyOfElements.IsAmalgamation\n (\u2191(\u2191natTransEquivCompatibleFamily (\u2191natTransEquivCompatibleFamily.symm { val := a, property := b })))\n (\u2191yonedaEquiv g)\n[PROOFSTEP]\napply ball_congr\n[GOAL]\ncase H\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\n\u22a2 \u2200 (x : FamilyOfElements P S.arrows) (h : FamilyOfElements.Compatible x),\n (\u2203! t, FamilyOfElements.IsAmalgamation x t) \u2194\n \u2203! g,\n FamilyOfElements.IsAmalgamation\n (\u2191(\u2191natTransEquivCompatibleFamily (\u2191natTransEquivCompatibleFamily.symm { val := x, property := h })))\n (\u2191yonedaEquiv g)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase H\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\nx : FamilyOfElements P S.arrows\nhx : FamilyOfElements.Compatible x\n\u22a2 (\u2203! t, FamilyOfElements.IsAmalgamation x t) \u2194\n \u2203! g,\n FamilyOfElements.IsAmalgamation\n (\u2191(\u2191natTransEquivCompatibleFamily (\u2191natTransEquivCompatibleFamily.symm { val := x, property := hx })))\n (\u2191yonedaEquiv g)\n[PROOFSTEP]\nrw [Equiv.exists_unique_congr_left _]\n[GOAL]\ncase H\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\u2081\nx : FamilyOfElements P S.arrows\nhx : FamilyOfElements.Compatible x\n\u22a2 (\u2203! t, FamilyOfElements.IsAmalgamation x t) \u2194\n \u2203! y,\n FamilyOfElements.IsAmalgamation\n (\u2191(\u2191natTransEquivCompatibleFamily (\u2191natTransEquivCompatibleFamily.symm { val := x, property := hx }))) y\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 (IsSeparatedFor P R \u2227\n \u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2194\n IsSheafFor P R\n[PROOFSTEP]\nrw [IsSeparatedFor, \u2190 forall_and]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 (\u2200 (x : FamilyOfElements P R),\n (\u2200 (t\u2081 t\u2082 : P.obj (op X)),\n FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)) \u2194\n IsSheafFor P R\n[PROOFSTEP]\napply forall_congr'\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 \u2200 (a : FamilyOfElements P R),\n (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation a t\u2081 \u2192 FamilyOfElements.IsAmalgamation a t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible a \u2192 \u2203 t, FamilyOfElements.IsAmalgamation a t) \u2194\n FamilyOfElements.Compatible a \u2192 \u2203! t, FamilyOfElements.IsAmalgamation a t\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\n\u22a2 (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2194\n FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\n\u22a2 (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2192\n FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nintro z hx\n[GOAL]\ncase h.mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nz :\n (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)\nhx : FamilyOfElements.Compatible x\n\u22a2 \u2203! t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nexact exists_unique_of_exists_of_unique (z.2 hx) z.1\n[GOAL]\ncase h.mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\n\u22a2 (FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t) \u2192\n (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\n\u22a2 (\u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)\n[PROOFSTEP]\nrefine' \u27e8_, ExistsUnique.exists \u2218 h\u27e9\n[GOAL]\ncase h.mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\n\u22a2 \u2200 (t\u2081 t\u2082 : P.obj (op X)), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082\n[PROOFSTEP]\nintro t\u2081 t\u2082 ht\u2081 ht\u2082\n[GOAL]\ncase h.mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\nt\u2081 t\u2082 : P.obj (op X)\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 t\u2081 = t\u2082\n[PROOFSTEP]\napply (h _).unique ht\u2081 ht\u2082\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\nt\u2081 t\u2082 : P.obj (op X)\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 FamilyOfElements.Compatible x\n[PROOFSTEP]\nexact is_compatible_of_exists_amalgamation x \u27e8_, ht\u2082\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nt : IsSeparatedFor P R\n\u22a2 (\u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2192\n IsSheafFor P R\n[PROOFSTEP]\nrw [\u2190 isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nt : IsSeparatedFor P R\n\u22a2 (\u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2192\n IsSeparatedFor P R \u2227\n \u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nexact And.intro t\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\n\u22a2 IsSheafFor P R \u2194 IsSheafFor P (generate R).arrows\n[PROOFSTEP]\nrw [\u2190 isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\n\u22a2 (IsSeparatedFor P R \u2227\n \u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2194\n IsSheafFor P (generate R).arrows\n[PROOFSTEP]\nrw [\u2190 isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\n\u22a2 (IsSeparatedFor P R \u2227\n \u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2194\n IsSeparatedFor P (generate R).arrows \u2227\n \u2200 (x : FamilyOfElements P (generate R).arrows),\n FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nrw [\u2190 isSeparatedFor_iff_generate]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\n\u22a2 (IsSeparatedFor P R \u2227\n \u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2194\n IsSeparatedFor P R \u2227\n \u2200 (x : FamilyOfElements P (generate R).arrows),\n FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\napply and_congr (Iff.refl _)\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\n\u22a2 (\u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2194\n \u2200 (x : FamilyOfElements P (generate R).arrows),\n FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\n\u22a2 (\u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2192\n \u2200 (x : FamilyOfElements P (generate R).arrows),\n FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nintro q x hx\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\nq : \u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t\nx : FamilyOfElements P (generate R).arrows\nhx : FamilyOfElements.Compatible x\n\u22a2 \u2203 t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\napply Exists.imp _ (q _ (hx.restrict (le_generate R)))\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\nq : \u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t\nx : FamilyOfElements P (generate R).arrows\nhx : FamilyOfElements.Compatible x\n\u22a2 \u2200 (a : P.obj (op X)),\n FamilyOfElements.IsAmalgamation (FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x) a \u2192\n FamilyOfElements.IsAmalgamation x a\n[PROOFSTEP]\nintro t ht\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\nq : \u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t\nx : FamilyOfElements P (generate R).arrows\nhx : FamilyOfElements.Compatible x\nt : P.obj (op X)\nht : FamilyOfElements.IsAmalgamation (FamilyOfElements.restrict (_ : R \u2264 (generate R).arrows) x) t\n\u22a2 FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nsimpa [hx] using isAmalgamation_sieveExtend _ _ ht\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\n\u22a2 (\u2200 (x : FamilyOfElements P (generate R).arrows),\n FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2192\n \u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nintro q x hx\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\nq :\n \u2200 (x : FamilyOfElements P (generate R).arrows),\n FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\n\u22a2 \u2203 t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\napply Exists.imp _ (q _ hx.sieveExtend)\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\nq :\n \u2200 (x : FamilyOfElements P (generate R).arrows),\n FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\n\u22a2 \u2200 (a : P.obj (op X)),\n FamilyOfElements.IsAmalgamation (FamilyOfElements.sieveExtend x) a \u2192 FamilyOfElements.IsAmalgamation x a\n[PROOFSTEP]\nintro t ht\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\nq :\n \u2200 (x : FamilyOfElements P (generate R).arrows),\n FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\nt : P.obj (op X)\nht : FamilyOfElements.IsAmalgamation (FamilyOfElements.sieveExtend x) t\n\u22a2 FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nsimpa [hx] using isAmalgamation_restrict (le_generate R) _ _ ht\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\n\u22a2 IsSheafFor P (singleton (\ud835\udfd9 X))\n[PROOFSTEP]\nintro x _\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nx : FamilyOfElements P (singleton (\ud835\udfd9 X))\na\u271d : FamilyOfElements.Compatible x\n\u22a2 \u2203! t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nrefine' \u27e8x _ (Presieve.singleton_self _), _, _\u27e9\n[GOAL]\ncase refine'_1\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nx : FamilyOfElements P (singleton (\ud835\udfd9 X))\na\u271d : FamilyOfElements.Compatible x\n\u22a2 (fun t => FamilyOfElements.IsAmalgamation x t) (x (\ud835\udfd9 X) (_ : singleton (\ud835\udfd9 X) (\ud835\udfd9 X)))\n[PROOFSTEP]\nrintro _ _ \u27e8rfl, rfl\u27e9\n[GOAL]\ncase refine'_1.mk\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nx : FamilyOfElements P (singleton (\ud835\udfd9 X))\na\u271d : FamilyOfElements.Compatible x\nY : C\n\u22a2 P.map (\ud835\udfd9 X).op (x (\ud835\udfd9 X) (_ : singleton (\ud835\udfd9 X) (\ud835\udfd9 X))) = x (\ud835\udfd9 X) (_ : singleton' (\ud835\udfd9 X) (\ud835\udfd9 X))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nx : FamilyOfElements P (singleton (\ud835\udfd9 X))\na\u271d : FamilyOfElements.Compatible x\n\u22a2 \u2200 (y : P.obj (op X)), (fun t => FamilyOfElements.IsAmalgamation x t) y \u2192 y = x (\ud835\udfd9 X) (_ : singleton (\ud835\udfd9 X) (\ud835\udfd9 X))\n[PROOFSTEP]\nintro t ht\n[GOAL]\ncase refine'_2\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nx : FamilyOfElements P (singleton (\ud835\udfd9 X))\na\u271d : FamilyOfElements.Compatible x\nt : P.obj (op X)\nht : FamilyOfElements.IsAmalgamation x t\n\u22a2 t = x (\ud835\udfd9 X) (_ : singleton (\ud835\udfd9 X) (\ud835\udfd9 X))\n[PROOFSTEP]\nsimpa using ht _ (Presieve.singleton_self _)\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\n\u22a2 IsSheafFor P \u22a4.arrows\n[PROOFSTEP]\nrw [\u2190 generate_of_singleton_isSplitEpi (\ud835\udfd9 X)]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\n\u22a2 IsSheafFor P (generate (singleton (\ud835\udfd9 X))).arrows\n[PROOFSTEP]\nrw [\u2190 isSheafFor_iff_generate]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\n\u22a2 IsSheafFor P (singleton (\ud835\udfd9 X))\n[PROOFSTEP]\napply isSheafFor_singleton_iso\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP' : C\u1d52\u1d56 \u2964 Type w\ni : P \u2245 P'\n\u22a2 IsSheafFor P R \u2192 IsSheafFor P' R\n[PROOFSTEP]\nintro h x hx\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP' : C\u1d52\u1d56 \u2964 Type w\ni : P \u2245 P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\n\u22a2 \u2203! t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nlet x' := x.compPresheafMap i.inv\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP' : C\u1d52\u1d56 \u2964 Type w\ni : P \u2245 P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\n\u22a2 \u2203! t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nhave : x'.Compatible := FamilyOfElements.Compatible.compPresheafMap i.inv hx\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP' : C\u1d52\u1d56 \u2964 Type w\ni : P \u2245 P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\n\u22a2 \u2203! t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nobtain \u27e8t, ht1, ht2\u27e9 := h x' this\n[GOAL]\ncase intro.intro\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP' : C\u1d52\u1d56 \u2964 Type w\ni : P \u2245 P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj (op X)\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : \u2200 (y : P.obj (op X)), (fun t => FamilyOfElements.IsAmalgamation x' t) y \u2192 y = t\n\u22a2 \u2203! t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nuse i.hom.app _ t\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP' : C\u1d52\u1d56 \u2964 Type w\ni : P \u2245 P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj (op X)\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : \u2200 (y : P.obj (op X)), (fun t => FamilyOfElements.IsAmalgamation x' t) y \u2192 y = t\n\u22a2 (fun t => FamilyOfElements.IsAmalgamation x t) (NatTrans.app i.hom (op X) t) \u2227\n \u2200 (y : P'.obj (op X)), (fun t => FamilyOfElements.IsAmalgamation x t) y \u2192 y = NatTrans.app i.hom (op X) t\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase h.left\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP' : C\u1d52\u1d56 \u2964 Type w\ni : P \u2245 P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj (op X)\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : \u2200 (y : P.obj (op X)), (fun t => FamilyOfElements.IsAmalgamation x' t) y \u2192 y = t\n\u22a2 (fun t => FamilyOfElements.IsAmalgamation x t) (NatTrans.app i.hom (op X) t)\n[PROOFSTEP]\nconvert FamilyOfElements.IsAmalgamation.compPresheafMap i.hom ht1\n[GOAL]\ncase h.e.h.e'_6.h.h.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP' : C\u1d52\u1d56 \u2964 Type w\ni : P \u2245 P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj (op X)\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : \u2200 (y : P.obj (op X)), (fun t => FamilyOfElements.IsAmalgamation x' t) y \u2192 y = t\n\u22a2 x = FamilyOfElements.compPresheafMap i.hom x'\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.right\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP' : C\u1d52\u1d56 \u2964 Type w\ni : P \u2245 P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj (op X)\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : \u2200 (y : P.obj (op X)), (fun t => FamilyOfElements.IsAmalgamation x' t) y \u2192 y = t\n\u22a2 \u2200 (y : P'.obj (op X)), (fun t => FamilyOfElements.IsAmalgamation x t) y \u2192 y = NatTrans.app i.hom (op X) t\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase h.right\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP' : C\u1d52\u1d56 \u2964 Type w\ni : P \u2245 P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj (op X)\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : \u2200 (y : P.obj (op X)), (fun t => FamilyOfElements.IsAmalgamation x' t) y \u2192 y = t\ny : P'.obj (op X)\nhy : FamilyOfElements.IsAmalgamation x y\n\u22a2 y = NatTrans.app i.hom (op X) t\n[PROOFSTEP]\nrw [show y = (i.inv.app (op X) \u226b i.hom.app (op X)) y by simp]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP' : C\u1d52\u1d56 \u2964 Type w\ni : P \u2245 P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj (op X)\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : \u2200 (y : P.obj (op X)), (fun t => FamilyOfElements.IsAmalgamation x' t) y \u2192 y = t\ny : P'.obj (op X)\nhy : FamilyOfElements.IsAmalgamation x y\n\u22a2 y = (NatTrans.app i.inv (op X) \u226b NatTrans.app i.hom (op X)) y\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.right\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP' : C\u1d52\u1d56 \u2964 Type w\ni : P \u2245 P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj (op X)\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : \u2200 (y : P.obj (op X)), (fun t => FamilyOfElements.IsAmalgamation x' t) y \u2192 y = t\ny : P'.obj (op X)\nhy : FamilyOfElements.IsAmalgamation x y\n\u22a2 (NatTrans.app i.inv (op X) \u226b NatTrans.app i.hom (op X)) y = NatTrans.app i.hom (op X) t\n[PROOFSTEP]\nsimp [ht2 (i.inv.app _ y) (FamilyOfElements.IsAmalgamation.compPresheafMap i.inv hy)]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows \u2264 R\nhS : IsSheafFor P S.arrows\ntrans : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R f \u2192 IsSeparatedFor P (Sieve.pullback f S).arrows\n\u22a2 IsSheafFor P R\n[PROOFSTEP]\nrw [\u2190 isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows \u2264 R\nhS : IsSheafFor P S.arrows\ntrans : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R f \u2192 IsSeparatedFor P (Sieve.pullback f S).arrows\n\u22a2 IsSeparatedFor P R \u2227\n \u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows \u2264 R\nhS : IsSheafFor P S.arrows\ntrans : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R f \u2192 IsSeparatedFor P (Sieve.pullback f S).arrows\n\u22a2 IsSeparatedFor P R\n[PROOFSTEP]\nintro x t\u2081 t\u2082 ht\u2081 ht\u2082\n[GOAL]\ncase left\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows \u2264 R\nhS : IsSheafFor P S.arrows\ntrans : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R f \u2192 IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nt\u2081 t\u2082 : P.obj (op X)\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 t\u2081 = t\u2082\n[PROOFSTEP]\nexact hS.isSeparatedFor _ _ _ (isAmalgamation_restrict h x t\u2081 ht\u2081) (isAmalgamation_restrict h x t\u2082 ht\u2082)\n[GOAL]\ncase right\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows \u2264 R\nhS : IsSheafFor P S.arrows\ntrans : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R f \u2192 IsSeparatedFor P (Sieve.pullback f S).arrows\n\u22a2 \u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase right\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows \u2264 R\nhS : IsSheafFor P S.arrows\ntrans : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R f \u2192 IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\n\u22a2 \u2203 t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nuse hS.amalgamate _ (hx.restrict h)\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows \u2264 R\nhS : IsSheafFor P S.arrows\ntrans : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R f \u2192 IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\n\u22a2 FamilyOfElements.IsAmalgamation x\n (IsSheafFor.amalgamate hS (FamilyOfElements.restrict h x)\n (_ : FamilyOfElements.Compatible (FamilyOfElements.restrict h x)))\n[PROOFSTEP]\nintro W j hj\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows \u2264 R\nhS : IsSheafFor P S.arrows\ntrans : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R f \u2192 IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\nW : C\nj : W \u27f6 X\nhj : R j\n\u22a2 P.map j.op\n (IsSheafFor.amalgamate hS (FamilyOfElements.restrict h x)\n (_ : FamilyOfElements.Compatible (FamilyOfElements.restrict h x))) =\n x j hj\n[PROOFSTEP]\napply (trans hj).ext\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows \u2264 R\nhS : IsSheafFor P S.arrows\ntrans : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R f \u2192 IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\nW : C\nj : W \u27f6 X\nhj : R j\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 W\u2984,\n (Sieve.pullback j S).arrows f \u2192\n P.map f.op\n (P.map j.op\n (IsSheafFor.amalgamate hS (FamilyOfElements.restrict h x)\n (_ : FamilyOfElements.Compatible (FamilyOfElements.restrict h x)))) =\n P.map f.op (x j hj)\n[PROOFSTEP]\nintro Y f hf\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows \u2264 R\nhS : IsSheafFor P S.arrows\ntrans : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R f \u2192 IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\nW : C\nj : W \u27f6 X\nhj : R j\nY : C\nf : Y \u27f6 W\nhf : (Sieve.pullback j S).arrows f\n\u22a2 P.map f.op\n (P.map j.op\n (IsSheafFor.amalgamate hS (FamilyOfElements.restrict h x)\n (_ : FamilyOfElements.Compatible (FamilyOfElements.restrict h x)))) =\n P.map f.op (x j hj)\n[PROOFSTEP]\nrw [\u2190 FunctorToTypes.map_comp_apply, \u2190 op_comp, hS.valid_glue (hx.restrict h) _ hf, FamilyOfElements.restrict, \u2190\n hx (\ud835\udfd9 _) f (h _ hf) _ (id_comp _)]\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y\u271d : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows \u2264 R\nhS : IsSheafFor P S.arrows\ntrans : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R f \u2192 IsSeparatedFor P (Sieve.pullback f S).arrows\nx : FamilyOfElements P R\nhx : FamilyOfElements.Compatible x\nW : C\nj : W \u27f6 X\nhj : R j\nY : C\nf : Y \u27f6 W\nhf : (Sieve.pullback j S).arrows f\n\u22a2 x (f \u226b j) (_ : f \u226b j \u2208 R) = P.map (\ud835\udfd9 Y).op (x (f \u226b j) (_ : f \u226b j \u2208 R))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP\u271d Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows \u2264 R\ntrans : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), IsSheafFor P (Sieve.pullback f S).arrows\n\u22a2 IsSheafFor P S.arrows\n[PROOFSTEP]\nsimpa using trans (\ud835\udfd9 _)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\ninst\u271d : HasPullbacks C\nK : Pretopology C\n\u22a2 IsSheaf (Pretopology.toGrothendieck C K) P \u2194\n \u2200 {X : C} (R : Presieve X), R \u2208 Pretopology.coverings K X \u2192 IsSheafFor P R\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\ninst\u271d : HasPullbacks C\nK : Pretopology C\n\u22a2 IsSheaf (Pretopology.toGrothendieck C K) P \u2192\n \u2200 {X : C} (R : Presieve X), R \u2208 Pretopology.coverings K X \u2192 IsSheafFor P R\n[PROOFSTEP]\nintro PJ X R hR\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX\u271d Y : C\nS : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nPJ : IsSheaf (Pretopology.toGrothendieck C K) P\nX : C\nR : Presieve X\nhR : R \u2208 Pretopology.coverings K X\n\u22a2 IsSheafFor P R\n[PROOFSTEP]\nrw [isSheafFor_iff_generate]\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX\u271d Y : C\nS : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nPJ : IsSheaf (Pretopology.toGrothendieck C K) P\nX : C\nR : Presieve X\nhR : R \u2208 Pretopology.coverings K X\n\u22a2 IsSheafFor P (generate R).arrows\n[PROOFSTEP]\napply PJ (Sieve.generate R) \u27e8_, hR, le_generate R\u27e9\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\ninst\u271d : HasPullbacks C\nK : Pretopology C\n\u22a2 (\u2200 {X : C} (R : Presieve X), R \u2208 Pretopology.coverings K X \u2192 IsSheafFor P R) \u2192\n IsSheaf (Pretopology.toGrothendieck C K) P\n[PROOFSTEP]\nrintro PK X S \u27e8R, hR, RS\u27e9\n[GOAL]\ncase mpr.intro.intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX\u271d Y : C\nS\u271d : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nPK : \u2200 {X : C} (R : Presieve X), R \u2208 Pretopology.coverings K X \u2192 IsSheafFor P R\nX : C\nS : Sieve X\nR : Presieve X\nhR : R \u2208 Pretopology.coverings K X\nRS : R \u2264 S.arrows\n\u22a2 IsSheafFor P S.arrows\n[PROOFSTEP]\nhave gRS : \u21d1(generate R) \u2264 S := by\n apply giGenerate.gc.monotone_u\n rwa [sets_iff_generate]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX\u271d Y : C\nS\u271d : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nPK : \u2200 {X : C} (R : Presieve X), R \u2208 Pretopology.coverings K X \u2192 IsSheafFor P R\nX : C\nS : Sieve X\nR : Presieve X\nhR : R \u2208 Pretopology.coverings K X\nRS : R \u2264 S.arrows\n\u22a2 (generate R).arrows \u2264 S.arrows\n[PROOFSTEP]\napply giGenerate.gc.monotone_u\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX\u271d Y : C\nS\u271d : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nPK : \u2200 {X : C} (R : Presieve X), R \u2208 Pretopology.coverings K X \u2192 IsSheafFor P R\nX : C\nS : Sieve X\nR : Presieve X\nhR : R \u2208 Pretopology.coverings K X\nRS : R \u2264 S.arrows\n\u22a2 generate R \u2264 S\n[PROOFSTEP]\nrwa [sets_iff_generate]\n[GOAL]\ncase mpr.intro.intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX\u271d Y : C\nS\u271d : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nPK : \u2200 {X : C} (R : Presieve X), R \u2208 Pretopology.coverings K X \u2192 IsSheafFor P R\nX : C\nS : Sieve X\nR : Presieve X\nhR : R \u2208 Pretopology.coverings K X\nRS : R \u2264 S.arrows\ngRS : (generate R).arrows \u2264 S.arrows\n\u22a2 IsSheafFor P S.arrows\n[PROOFSTEP]\napply isSheafFor_subsieve P gRS _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX\u271d Y : C\nS\u271d : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nPK : \u2200 {X : C} (R : Presieve X), R \u2208 Pretopology.coverings K X \u2192 IsSheafFor P R\nX : C\nS : Sieve X\nR : Presieve X\nhR : R \u2208 Pretopology.coverings K X\nRS : R \u2264 S.arrows\ngRS : (generate R).arrows \u2264 S.arrows\n\u22a2 \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), IsSheafFor P (Sieve.pullback f (generate R)).arrows\n[PROOFSTEP]\nintro Y f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX\u271d Y\u271d : C\nS\u271d : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nPK : \u2200 {X : C} (R : Presieve X), R \u2208 Pretopology.coverings K X \u2192 IsSheafFor P R\nX : C\nS : Sieve X\nR : Presieve X\nhR : R \u2208 Pretopology.coverings K X\nRS : R \u2264 S.arrows\ngRS : (generate R).arrows \u2264 S.arrows\nY : C\nf : Y \u27f6 X\n\u22a2 IsSheafFor P (Sieve.pullback f (generate R)).arrows\n[PROOFSTEP]\nrw [\u2190 pullbackArrows_comm, \u2190 isSheafFor_iff_generate]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX\u271d Y\u271d : C\nS\u271d : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nPK : \u2200 {X : C} (R : Presieve X), R \u2208 Pretopology.coverings K X \u2192 IsSheafFor P R\nX : C\nS : Sieve X\nR : Presieve X\nhR : R \u2208 Pretopology.coverings K X\nRS : R \u2264 S.arrows\ngRS : (generate R).arrows \u2264 S.arrows\nY : C\nf : Y \u27f6 X\n\u22a2 IsSheafFor P (pullbackArrows f R)\n[PROOFSTEP]\nexact PK (pullbackArrows f R) (K.pullbacks f R hR)\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX\u271d Y : C\nS : Sieve X\u271d\nR : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nX : C\n\u22a2 \u2200 (S : Sieve X), S \u2208 GrothendieckTopology.sieves \u22a5 X \u2192 IsSheafFor P S.arrows\n[PROOFSTEP]\nsimp [isSheafFor_top_sieve]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\nW : C\nx : FamilyOfElements (yoneda.obj W) R\nhx : FamilyOfElements.Compatible x\n\u22a2 \u2200 \u2983X_1 Y : FullSubcategory fun f => R f.hom\u2984 (f : X_1 \u27f6 Y),\n (diagram R).map f \u226b (fun f => x f.obj.hom (_ : R f.obj.hom)) Y =\n (fun f => x f.obj.hom (_ : R f.obj.hom)) X_1 \u226b ((Functor.const (FullSubcategory fun f => R f.hom)).obj W).map f\n[PROOFSTEP]\nintro g\u2081 g\u2082 F\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\nW : C\nx : FamilyOfElements (yoneda.obj W) R\nhx : FamilyOfElements.Compatible x\ng\u2081 g\u2082 : FullSubcategory fun f => R f.hom\nF : g\u2081 \u27f6 g\u2082\n\u22a2 (diagram R).map F \u226b (fun f => x f.obj.hom (_ : R f.obj.hom)) g\u2082 =\n (fun f => x f.obj.hom (_ : R f.obj.hom)) g\u2081 \u226b ((Functor.const (FullSubcategory fun f => R f.hom)).obj W).map F\n[PROOFSTEP]\nsimp only [Functor.id_obj, Functor.comp_obj, fullSubcategoryInclusion.obj, Over.forget_obj, Functor.const_obj_obj,\n Functor.comp_map, fullSubcategoryInclusion.map, Over.forget_map, Functor.const_obj_map, Category.comp_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\nW : C\nx : FamilyOfElements (yoneda.obj W) R\nhx : FamilyOfElements.Compatible x\ng\u2081 g\u2082 : FullSubcategory fun f => R f.hom\nF : g\u2081 \u27f6 g\u2082\n\u22a2 F.left \u226b x g\u2082.obj.hom (_ : R g\u2082.obj.hom) = x g\u2081.obj.hom (_ : R g\u2081.obj.hom)\n[PROOFSTEP]\nrw [\u2190 Category.id_comp (x g\u2081.obj.hom g\u2081.property)]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\nW : C\nx : FamilyOfElements (yoneda.obj W) R\nhx : FamilyOfElements.Compatible x\ng\u2081 g\u2082 : FullSubcategory fun f => R f.hom\nF : g\u2081 \u27f6 g\u2082\n\u22a2 F.left \u226b x g\u2082.obj.hom (_ : R g\u2082.obj.hom) = \ud835\udfd9 (op ((\ud835\udfed C).obj g\u2081.obj.left)).unop \u226b x g\u2081.obj.hom (_ : R g\u2081.obj.hom)\n[PROOFSTEP]\napply hx\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nR : Presieve X\nW : C\nx : FamilyOfElements (yoneda.obj W) R\nhx : FamilyOfElements.Compatible x\ng\u2081 g\u2082 : FullSubcategory fun f => R f.hom\nF : g\u2081 \u27f6 g\u2082\n\u22a2 F.1 \u226b g\u2082.obj.hom = \ud835\udfd9 (op ((\ud835\udfed C).obj g\u2081.obj.left)).unop \u226b g\u2081.obj.hom\n[PROOFSTEP]\nsimp only [Functor.id_obj, Over.w, Opposite.unop_op, Category.id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\ns : Cocone (diagram S.arrows)\n\u22a2 FamilyOfElements.Compatible (yonedaFamilyOfElements_fromCocone S.arrows s)\n[PROOFSTEP]\nintro Y\u2081 Y\u2082 Z g\u2081 g\u2082 f\u2081 f\u2082 hf\u2081 hf\u2082 hgf\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\ns : Cocone (diagram S.arrows)\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : S.arrows f\u2081\nhf\u2082 : S.arrows f\u2082\nhgf : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 (yoneda.obj s.pt).map g\u2081.op (yonedaFamilyOfElements_fromCocone S.arrows s f\u2081 hf\u2081) =\n (yoneda.obj s.pt).map g\u2082.op (yonedaFamilyOfElements_fromCocone S.arrows s f\u2082 hf\u2082)\n[PROOFSTEP]\nhave Hs := s.\u03b9.naturality\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\ns : Cocone (diagram S.arrows)\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : S.arrows f\u2081\nhf\u2082 : S.arrows f\u2082\nhgf : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\nHs :\n \u2200 \u2983X_1 Y : FullSubcategory fun f => S.arrows f.hom\u2984 (f : X_1 \u27f6 Y),\n (diagram S.arrows).map f \u226b NatTrans.app s.\u03b9 Y =\n NatTrans.app s.\u03b9 X_1 \u226b ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f\n\u22a2 (yoneda.obj s.pt).map g\u2081.op (yonedaFamilyOfElements_fromCocone S.arrows s f\u2081 hf\u2081) =\n (yoneda.obj s.pt).map g\u2082.op (yonedaFamilyOfElements_fromCocone S.arrows s f\u2082 hf\u2082)\n[PROOFSTEP]\nsimp only [Functor.id_obj, yoneda_obj_obj, Opposite.unop_op, yoneda_obj_map, Quiver.Hom.unop_op]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\ns : Cocone (diagram S.arrows)\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : S.arrows f\u2081\nhf\u2082 : S.arrows f\u2082\nhgf : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\nHs :\n \u2200 \u2983X_1 Y : FullSubcategory fun f => S.arrows f.hom\u2984 (f : X_1 \u27f6 Y),\n (diagram S.arrows).map f \u226b NatTrans.app s.\u03b9 Y =\n NatTrans.app s.\u03b9 X_1 \u226b ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f\n\u22a2 g\u2081 \u226b yonedaFamilyOfElements_fromCocone S.arrows s f\u2081 hf\u2081 = g\u2082 \u226b yonedaFamilyOfElements_fromCocone S.arrows s f\u2082 hf\u2082\n[PROOFSTEP]\ndsimp [yonedaFamilyOfElements_fromCocone]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\ns : Cocone (diagram S.arrows)\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : S.arrows f\u2081\nhf\u2082 : S.arrows f\u2082\nhgf : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\nHs :\n \u2200 \u2983X_1 Y : FullSubcategory fun f => S.arrows f.hom\u2984 (f : X_1 \u27f6 Y),\n (diagram S.arrows).map f \u226b NatTrans.app s.\u03b9 Y =\n NatTrans.app s.\u03b9 X_1 \u226b ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f\n\u22a2 g\u2081 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2081, property := hf\u2081 } =\n g\u2082 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2082, property := hf\u2082 }\n[PROOFSTEP]\nhave hgf\u2081 : S.arrows (g\u2081 \u226b f\u2081) := by exact Sieve.downward_closed S hf\u2081 g\u2081\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\ns : Cocone (diagram S.arrows)\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : S.arrows f\u2081\nhf\u2082 : S.arrows f\u2082\nhgf : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\nHs :\n \u2200 \u2983X_1 Y : FullSubcategory fun f => S.arrows f.hom\u2984 (f : X_1 \u27f6 Y),\n (diagram S.arrows).map f \u226b NatTrans.app s.\u03b9 Y =\n NatTrans.app s.\u03b9 X_1 \u226b ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f\n\u22a2 S.arrows (g\u2081 \u226b f\u2081)\n[PROOFSTEP]\nexact Sieve.downward_closed S hf\u2081 g\u2081\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\ns : Cocone (diagram S.arrows)\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : S.arrows f\u2081\nhf\u2082 : S.arrows f\u2082\nhgf : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\nHs :\n \u2200 \u2983X_1 Y : FullSubcategory fun f => S.arrows f.hom\u2984 (f : X_1 \u27f6 Y),\n (diagram S.arrows).map f \u226b NatTrans.app s.\u03b9 Y =\n NatTrans.app s.\u03b9 X_1 \u226b ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f\nhgf\u2081 : S.arrows (g\u2081 \u226b f\u2081)\n\u22a2 g\u2081 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2081, property := hf\u2081 } =\n g\u2082 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2082, property := hf\u2082 }\n[PROOFSTEP]\nhave hgf\u2082 : S.arrows (g\u2082 \u226b f\u2082) := by exact Sieve.downward_closed S hf\u2082 g\u2082\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\ns : Cocone (diagram S.arrows)\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : S.arrows f\u2081\nhf\u2082 : S.arrows f\u2082\nhgf : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\nHs :\n \u2200 \u2983X_1 Y : FullSubcategory fun f => S.arrows f.hom\u2984 (f : X_1 \u27f6 Y),\n (diagram S.arrows).map f \u226b NatTrans.app s.\u03b9 Y =\n NatTrans.app s.\u03b9 X_1 \u226b ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f\nhgf\u2081 : S.arrows (g\u2081 \u226b f\u2081)\n\u22a2 S.arrows (g\u2082 \u226b f\u2082)\n[PROOFSTEP]\nexact Sieve.downward_closed S hf\u2082 g\u2082\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\ns : Cocone (diagram S.arrows)\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : S.arrows f\u2081\nhf\u2082 : S.arrows f\u2082\nhgf : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\nHs :\n \u2200 \u2983X_1 Y : FullSubcategory fun f => S.arrows f.hom\u2984 (f : X_1 \u27f6 Y),\n (diagram S.arrows).map f \u226b NatTrans.app s.\u03b9 Y =\n NatTrans.app s.\u03b9 X_1 \u226b ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f\nhgf\u2081 : S.arrows (g\u2081 \u226b f\u2081)\nhgf\u2082 : S.arrows (g\u2082 \u226b f\u2082)\n\u22a2 g\u2081 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2081, property := hf\u2081 } =\n g\u2082 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2082, property := hf\u2082 }\n[PROOFSTEP]\nlet F : (Over.mk (g\u2081 \u226b f\u2081) : Over X) \u27f6 (Over.mk (g\u2082 \u226b f\u2082) : Over X) := (Over.homMk (\ud835\udfd9 Z))\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\ns : Cocone (diagram S.arrows)\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : S.arrows f\u2081\nhf\u2082 : S.arrows f\u2082\nhgf : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\nHs :\n \u2200 \u2983X_1 Y : FullSubcategory fun f => S.arrows f.hom\u2984 (f : X_1 \u27f6 Y),\n (diagram S.arrows).map f \u226b NatTrans.app s.\u03b9 Y =\n NatTrans.app s.\u03b9 X_1 \u226b ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f\nhgf\u2081 : S.arrows (g\u2081 \u226b f\u2081)\nhgf\u2082 : S.arrows (g\u2082 \u226b f\u2082)\nF : Over.mk (g\u2081 \u226b f\u2081) \u27f6 Over.mk (g\u2082 \u226b f\u2082) := Over.homMk (\ud835\udfd9 Z)\n\u22a2 g\u2081 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2081, property := hf\u2081 } =\n g\u2082 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2082, property := hf\u2082 }\n[PROOFSTEP]\nlet F\u2081 : (Over.mk (g\u2081 \u226b f\u2081) : Over X) \u27f6 (Over.mk f\u2081 : Over X) := (Over.homMk g\u2081)\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\ns : Cocone (diagram S.arrows)\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : S.arrows f\u2081\nhf\u2082 : S.arrows f\u2082\nhgf : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\nHs :\n \u2200 \u2983X_1 Y : FullSubcategory fun f => S.arrows f.hom\u2984 (f : X_1 \u27f6 Y),\n (diagram S.arrows).map f \u226b NatTrans.app s.\u03b9 Y =\n NatTrans.app s.\u03b9 X_1 \u226b ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f\nhgf\u2081 : S.arrows (g\u2081 \u226b f\u2081)\nhgf\u2082 : S.arrows (g\u2082 \u226b f\u2082)\nF : Over.mk (g\u2081 \u226b f\u2081) \u27f6 Over.mk (g\u2082 \u226b f\u2082) := Over.homMk (\ud835\udfd9 Z)\nF\u2081 : Over.mk (g\u2081 \u226b f\u2081) \u27f6 Over.mk f\u2081 := Over.homMk g\u2081\n\u22a2 g\u2081 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2081, property := hf\u2081 } =\n g\u2082 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2082, property := hf\u2082 }\n[PROOFSTEP]\nlet F\u2082 : (Over.mk (g\u2082 \u226b f\u2082) : Over X) \u27f6 (Over.mk f\u2082 : Over X) := (Over.homMk g\u2082)\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\ns : Cocone (diagram S.arrows)\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : S.arrows f\u2081\nhf\u2082 : S.arrows f\u2082\nhgf : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\nHs :\n \u2200 \u2983X_1 Y : FullSubcategory fun f => S.arrows f.hom\u2984 (f : X_1 \u27f6 Y),\n (diagram S.arrows).map f \u226b NatTrans.app s.\u03b9 Y =\n NatTrans.app s.\u03b9 X_1 \u226b ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f\nhgf\u2081 : S.arrows (g\u2081 \u226b f\u2081)\nhgf\u2082 : S.arrows (g\u2082 \u226b f\u2082)\nF : Over.mk (g\u2081 \u226b f\u2081) \u27f6 Over.mk (g\u2082 \u226b f\u2082) := Over.homMk (\ud835\udfd9 Z)\nF\u2081 : Over.mk (g\u2081 \u226b f\u2081) \u27f6 Over.mk f\u2081 := Over.homMk g\u2081\nF\u2082 : Over.mk (g\u2082 \u226b f\u2082) \u27f6 Over.mk f\u2082 := Over.homMk g\u2082\n\u22a2 g\u2081 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2081, property := hf\u2081 } =\n g\u2082 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2082, property := hf\u2082 }\n[PROOFSTEP]\nhave hF := @Hs \u27e8Over.mk (g\u2081 \u226b f\u2081), hgf\u2081\u27e9 \u27e8Over.mk (g\u2082 \u226b f\u2082), hgf\u2082\u27e9 F\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\ns : Cocone (diagram S.arrows)\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : S.arrows f\u2081\nhf\u2082 : S.arrows f\u2082\nhgf : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\nHs :\n \u2200 \u2983X_1 Y : FullSubcategory fun f => S.arrows f.hom\u2984 (f : X_1 \u27f6 Y),\n (diagram S.arrows).map f \u226b NatTrans.app s.\u03b9 Y =\n NatTrans.app s.\u03b9 X_1 \u226b ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f\nhgf\u2081 : S.arrows (g\u2081 \u226b f\u2081)\nhgf\u2082 : S.arrows (g\u2082 \u226b f\u2082)\nF : Over.mk (g\u2081 \u226b f\u2081) \u27f6 Over.mk (g\u2082 \u226b f\u2082) := Over.homMk (\ud835\udfd9 Z)\nF\u2081 : Over.mk (g\u2081 \u226b f\u2081) \u27f6 Over.mk f\u2081 := Over.homMk g\u2081\nF\u2082 : Over.mk (g\u2082 \u226b f\u2082) \u27f6 Over.mk f\u2082 := Over.homMk g\u2082\nhF :\n (diagram S.arrows).map F \u226b NatTrans.app s.\u03b9 { obj := Over.mk (g\u2082 \u226b f\u2082), property := hgf\u2082 } =\n NatTrans.app s.\u03b9 { obj := Over.mk (g\u2081 \u226b f\u2081), property := hgf\u2081 } \u226b\n ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F\n\u22a2 g\u2081 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2081, property := hf\u2081 } =\n g\u2082 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2082, property := hf\u2082 }\n[PROOFSTEP]\nhave hF\u2081 := @Hs \u27e8Over.mk (g\u2081 \u226b f\u2081), hgf\u2081\u27e9 \u27e8Over.mk f\u2081, hf\u2081\u27e9 F\u2081\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\ns : Cocone (diagram S.arrows)\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : S.arrows f\u2081\nhf\u2082 : S.arrows f\u2082\nhgf : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\nHs :\n \u2200 \u2983X_1 Y : FullSubcategory fun f => S.arrows f.hom\u2984 (f : X_1 \u27f6 Y),\n (diagram S.arrows).map f \u226b NatTrans.app s.\u03b9 Y =\n NatTrans.app s.\u03b9 X_1 \u226b ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f\nhgf\u2081 : S.arrows (g\u2081 \u226b f\u2081)\nhgf\u2082 : S.arrows (g\u2082 \u226b f\u2082)\nF : Over.mk (g\u2081 \u226b f\u2081) \u27f6 Over.mk (g\u2082 \u226b f\u2082) := Over.homMk (\ud835\udfd9 Z)\nF\u2081 : Over.mk (g\u2081 \u226b f\u2081) \u27f6 Over.mk f\u2081 := Over.homMk g\u2081\nF\u2082 : Over.mk (g\u2082 \u226b f\u2082) \u27f6 Over.mk f\u2082 := Over.homMk g\u2082\nhF :\n (diagram S.arrows).map F \u226b NatTrans.app s.\u03b9 { obj := Over.mk (g\u2082 \u226b f\u2082), property := hgf\u2082 } =\n NatTrans.app s.\u03b9 { obj := Over.mk (g\u2081 \u226b f\u2081), property := hgf\u2081 } \u226b\n ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F\nhF\u2081 :\n (diagram S.arrows).map F\u2081 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2081, property := hf\u2081 } =\n NatTrans.app s.\u03b9 { obj := Over.mk (g\u2081 \u226b f\u2081), property := hgf\u2081 } \u226b\n ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F\u2081\n\u22a2 g\u2081 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2081, property := hf\u2081 } =\n g\u2082 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2082, property := hf\u2082 }\n[PROOFSTEP]\nhave hF\u2082 := @Hs \u27e8Over.mk (g\u2082 \u226b f\u2082), hgf\u2082\u27e9 \u27e8Over.mk f\u2082, hf\u2082\u27e9 F\u2082\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\ns : Cocone (diagram S.arrows)\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 X\nf\u2082 : Y\u2082 \u27f6 X\nhf\u2081 : S.arrows f\u2081\nhf\u2082 : S.arrows f\u2082\nhgf : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\nHs :\n \u2200 \u2983X_1 Y : FullSubcategory fun f => S.arrows f.hom\u2984 (f : X_1 \u27f6 Y),\n (diagram S.arrows).map f \u226b NatTrans.app s.\u03b9 Y =\n NatTrans.app s.\u03b9 X_1 \u226b ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map f\nhgf\u2081 : S.arrows (g\u2081 \u226b f\u2081)\nhgf\u2082 : S.arrows (g\u2082 \u226b f\u2082)\nF : Over.mk (g\u2081 \u226b f\u2081) \u27f6 Over.mk (g\u2082 \u226b f\u2082) := Over.homMk (\ud835\udfd9 Z)\nF\u2081 : Over.mk (g\u2081 \u226b f\u2081) \u27f6 Over.mk f\u2081 := Over.homMk g\u2081\nF\u2082 : Over.mk (g\u2082 \u226b f\u2082) \u27f6 Over.mk f\u2082 := Over.homMk g\u2082\nhF :\n (diagram S.arrows).map F \u226b NatTrans.app s.\u03b9 { obj := Over.mk (g\u2082 \u226b f\u2082), property := hgf\u2082 } =\n NatTrans.app s.\u03b9 { obj := Over.mk (g\u2081 \u226b f\u2081), property := hgf\u2081 } \u226b\n ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F\nhF\u2081 :\n (diagram S.arrows).map F\u2081 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2081, property := hf\u2081 } =\n NatTrans.app s.\u03b9 { obj := Over.mk (g\u2081 \u226b f\u2081), property := hgf\u2081 } \u226b\n ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F\u2081\nhF\u2082 :\n (diagram S.arrows).map F\u2082 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2082, property := hf\u2082 } =\n NatTrans.app s.\u03b9 { obj := Over.mk (g\u2082 \u226b f\u2082), property := hgf\u2082 } \u226b\n ((Functor.const (FullSubcategory fun f => S.arrows f.hom)).obj s.pt).map F\u2082\n\u22a2 g\u2081 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2081, property := hf\u2081 } =\n g\u2082 \u226b NatTrans.app s.\u03b9 { obj := Over.mk f\u2082, property := hf\u2082 }\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\n\u22a2 (\u2200 (W : C), IsSheafFor (yoneda.obj W) S.arrows) \u2194 Nonempty (IsColimit (cocone S.arrows))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\n\u22a2 (\u2200 (W : C), IsSheafFor (yoneda.obj W) S.arrows) \u2192 Nonempty (IsColimit (cocone S.arrows))\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nH : \u2200 (W : C), IsSheafFor (yoneda.obj W) S.arrows\n\u22a2 Nonempty (IsColimit (cocone S.arrows))\n[PROOFSTEP]\nrefine Nonempty.intro ?_\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nH : \u2200 (W : C), IsSheafFor (yoneda.obj W) S.arrows\n\u22a2 IsColimit (cocone S.arrows)\n[PROOFSTEP]\nexact\n { desc := fun s =>\n H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s) (yonedaFamily_fromCocone_compatible S s) |>.choose\n fac := by\n intro s f\n replace H := H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s) (yonedaFamily_fromCocone_compatible S s)\n have ht := H.choose_spec.1 f.obj.hom f.property\n aesop_cat\n uniq := by\n intro s Fs HFs\n replace H := H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s) (yonedaFamily_fromCocone_compatible S s)\n apply H.choose_spec.2 Fs\n exact fun _ f hf => HFs \u27e8Over.mk f, hf\u27e9 }\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nH : \u2200 (W : C), IsSheafFor (yoneda.obj W) S.arrows\n\u22a2 \u2200 (s : Cocone (diagram S.arrows)) (j : FullSubcategory fun f => S.arrows f.hom),\n NatTrans.app (cocone S.arrows).\u03b9 j \u226b\n (fun s =>\n Exists.choose (_ : \u2203! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t))\n s =\n NatTrans.app s.\u03b9 j\n[PROOFSTEP]\nintro s f\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nH : \u2200 (W : C), IsSheafFor (yoneda.obj W) S.arrows\ns : Cocone (diagram S.arrows)\nf : FullSubcategory fun f => S.arrows f.hom\n\u22a2 NatTrans.app (cocone S.arrows).\u03b9 f \u226b\n (fun s =>\n Exists.choose (_ : \u2203! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t))\n s =\n NatTrans.app s.\u03b9 f\n[PROOFSTEP]\nreplace H := H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s) (yonedaFamily_fromCocone_compatible S s)\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nH\u271d : \u2200 (W : C), IsSheafFor (yoneda.obj W) S.arrows\ns : Cocone (diagram S.arrows)\nf : FullSubcategory fun f => S.arrows f.hom\nH : \u2203! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t\n\u22a2 NatTrans.app (cocone S.arrows).\u03b9 f \u226b\n (fun s =>\n Exists.choose (_ : \u2203! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t))\n s =\n NatTrans.app s.\u03b9 f\n[PROOFSTEP]\nhave ht := H.choose_spec.1 f.obj.hom f.property\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nH\u271d : \u2200 (W : C), IsSheafFor (yoneda.obj W) S.arrows\ns : Cocone (diagram S.arrows)\nf : FullSubcategory fun f => S.arrows f.hom\nH : \u2203! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t\nht :\n (yoneda.obj s.pt).map f.obj.hom.op (Exists.choose H) =\n yonedaFamilyOfElements_fromCocone S.arrows s f.obj.hom (_ : S.arrows f.obj.hom)\n\u22a2 NatTrans.app (cocone S.arrows).\u03b9 f \u226b\n (fun s =>\n Exists.choose (_ : \u2203! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t))\n s =\n NatTrans.app s.\u03b9 f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nH : \u2200 (W : C), IsSheafFor (yoneda.obj W) S.arrows\n\u22a2 \u2200 (s : Cocone (diagram S.arrows)) (m : (cocone S.arrows).pt \u27f6 s.pt),\n (\u2200 (j : FullSubcategory fun f => S.arrows f.hom), NatTrans.app (cocone S.arrows).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j) \u2192\n m =\n (fun s =>\n Exists.choose (_ : \u2203! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t))\n s\n[PROOFSTEP]\nintro s Fs HFs\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nH : \u2200 (W : C), IsSheafFor (yoneda.obj W) S.arrows\ns : Cocone (diagram S.arrows)\nFs : (cocone S.arrows).pt \u27f6 s.pt\nHFs : \u2200 (j : FullSubcategory fun f => S.arrows f.hom), NatTrans.app (cocone S.arrows).\u03b9 j \u226b Fs = NatTrans.app s.\u03b9 j\n\u22a2 Fs =\n (fun s =>\n Exists.choose (_ : \u2203! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t))\n s\n[PROOFSTEP]\nreplace H := H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s) (yonedaFamily_fromCocone_compatible S s)\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nH\u271d : \u2200 (W : C), IsSheafFor (yoneda.obj W) S.arrows\ns : Cocone (diagram S.arrows)\nFs : (cocone S.arrows).pt \u27f6 s.pt\nHFs : \u2200 (j : FullSubcategory fun f => S.arrows f.hom), NatTrans.app (cocone S.arrows).\u03b9 j \u226b Fs = NatTrans.app s.\u03b9 j\nH : \u2203! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t\n\u22a2 Fs =\n (fun s =>\n Exists.choose (_ : \u2203! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t))\n s\n[PROOFSTEP]\napply H.choose_spec.2 Fs\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nH\u271d : \u2200 (W : C), IsSheafFor (yoneda.obj W) S.arrows\ns : Cocone (diagram S.arrows)\nFs : (cocone S.arrows).pt \u27f6 s.pt\nHFs : \u2200 (j : FullSubcategory fun f => S.arrows f.hom), NatTrans.app (cocone S.arrows).\u03b9 j \u226b Fs = NatTrans.app s.\u03b9 j\nH : \u2203! t, FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) t\n\u22a2 FamilyOfElements.IsAmalgamation (yonedaFamilyOfElements_fromCocone S.arrows s) Fs\n[PROOFSTEP]\nexact fun _ f hf => HFs \u27e8Over.mk f, hf\u27e9\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\n\u22a2 Nonempty (IsColimit (cocone S.arrows)) \u2192 \u2200 (W : C), IsSheafFor (yoneda.obj W) S.arrows\n[PROOFSTEP]\nintro H W x hx\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nH : Nonempty (IsColimit (cocone S.arrows))\nW : C\nx : FamilyOfElements (yoneda.obj W) S.arrows\nhx : FamilyOfElements.Compatible x\n\u22a2 \u2203! t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nreplace H := Classical.choice H\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nW : C\nx : FamilyOfElements (yoneda.obj W) S.arrows\nhx : FamilyOfElements.Compatible x\nH : IsColimit (cocone S.arrows)\n\u22a2 \u2203! t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nlet s := compatibleYonedaFamily_toCocone S W x hx\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nW : C\nx : FamilyOfElements (yoneda.obj W) S.arrows\nhx : FamilyOfElements.Compatible x\nH : IsColimit (cocone S.arrows)\ns : Cocone (diagram S.arrows) := compatibleYonedaFamily_toCocone S.arrows W x hx\n\u22a2 \u2203! t, FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nuse H.desc s\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nW : C\nx : FamilyOfElements (yoneda.obj W) S.arrows\nhx : FamilyOfElements.Compatible x\nH : IsColimit (cocone S.arrows)\ns : Cocone (diagram S.arrows) := compatibleYonedaFamily_toCocone S.arrows W x hx\n\u22a2 (fun t => FamilyOfElements.IsAmalgamation x t) (IsColimit.desc H s) \u2227\n \u2200 (y : (yoneda.obj W).obj (op X)), (fun t => FamilyOfElements.IsAmalgamation x t) y \u2192 y = IsColimit.desc H s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nW : C\nx : FamilyOfElements (yoneda.obj W) S.arrows\nhx : FamilyOfElements.Compatible x\nH : IsColimit (cocone S.arrows)\ns : Cocone (diagram S.arrows) := compatibleYonedaFamily_toCocone S.arrows W x hx\n\u22a2 (fun t => FamilyOfElements.IsAmalgamation x t) (IsColimit.desc H s)\n[PROOFSTEP]\nexact fun _ f hf => (H.fac s) \u27e8Over.mk f, hf\u27e9\n[GOAL]\ncase h.right\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nS : Sieve X\nW : C\nx : FamilyOfElements (yoneda.obj W) S.arrows\nhx : FamilyOfElements.Compatible x\nH : IsColimit (cocone S.arrows)\ns : Cocone (diagram S.arrows) := compatibleYonedaFamily_toCocone S.arrows W x hx\n\u22a2 \u2200 (y : (yoneda.obj W).obj (op X)), (fun t => FamilyOfElements.IsAmalgamation x t) y \u2192 y = IsColimit.desc H s\n[PROOFSTEP]\nexact fun g hg => H.uniq s g (fun \u27e8\u27e8f, _, hom\u27e9, hf\u27e9 => hg hom hf)\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nz\u2081 z\u2082 : FirstObj P R\nh :\n \u2200 (Y : C) (f : Y \u27f6 X) (hf : R f),\n Pi.\u03c0 (fun f => P.obj (op f.fst)) { fst := Y, snd := { val := f, property := hf } } z\u2081 =\n Pi.\u03c0 (fun f => P.obj (op f.fst)) { fst := Y, snd := { val := f, property := hf } } z\u2082\n\u22a2 z\u2081 = z\u2082\n[PROOFSTEP]\napply Limits.Types.limit_ext\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nz\u2081 z\u2082 : FirstObj P R\nh :\n \u2200 (Y : C) (f : Y \u27f6 X) (hf : R f),\n Pi.\u03c0 (fun f => P.obj (op f.fst)) { fst := Y, snd := { val := f, property := hf } } z\u2081 =\n Pi.\u03c0 (fun f => P.obj (op f.fst)) { fst := Y, snd := { val := f, property := hf } } z\u2082\n\u22a2 \u2200 (j : Discrete ((Y : C) \u00d7 { f // R f })),\n limit.\u03c0 (Discrete.functor fun f => P.obj (op f.fst)) j z\u2081 =\n limit.\u03c0 (Discrete.functor fun f => P.obj (op f.fst)) j z\u2082\n[PROOFSTEP]\nrintro \u27e8\u27e8Y, f, hf\u27e9\u27e9\n[GOAL]\ncase w.mk.mk.mk\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nz\u2081 z\u2082 : FirstObj P R\nh :\n \u2200 (Y : C) (f : Y \u27f6 X) (hf : R f),\n Pi.\u03c0 (fun f => P.obj (op f.fst)) { fst := Y, snd := { val := f, property := hf } } z\u2081 =\n Pi.\u03c0 (fun f => P.obj (op f.fst)) { fst := Y, snd := { val := f, property := hf } } z\u2082\nY : C\nf : Y \u27f6 X\nhf : R f\n\u22a2 limit.\u03c0 (Discrete.functor fun f => P.obj (op f.fst)) { as := { fst := Y, snd := { val := f, property := hf } } } z\u2081 =\n limit.\u03c0 (Discrete.functor fun f => P.obj (op f.fst)) { as := { fst := Y, snd := { val := f, property := hf } } } z\u2082\n[PROOFSTEP]\nexact h Y f hf\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nz\u2081 z\u2082 : SecondObj P S\nh :\n \u2200 (Y Z : C) (g : Z \u27f6 Y) (f : Y \u27f6 X) (hf : S.arrows f),\n Pi.\u03c0 (fun f => P.obj (op f.snd.fst))\n { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } z\u2081 =\n Pi.\u03c0 (fun f => P.obj (op f.snd.fst))\n { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } z\u2082\n\u22a2 z\u2081 = z\u2082\n[PROOFSTEP]\napply Limits.Types.limit_ext\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nz\u2081 z\u2082 : SecondObj P S\nh :\n \u2200 (Y Z : C) (g : Z \u27f6 Y) (f : Y \u27f6 X) (hf : S.arrows f),\n Pi.\u03c0 (fun f => P.obj (op f.snd.fst))\n { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } z\u2081 =\n Pi.\u03c0 (fun f => P.obj (op f.snd.fst))\n { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } z\u2082\n\u22a2 \u2200 (j : Discrete ((Y : C) \u00d7 (Z : C) \u00d7 (_ : Z \u27f6 Y) \u00d7 { f' // S.arrows f' })),\n limit.\u03c0 (Discrete.functor fun f => P.obj (op f.snd.fst)) j z\u2081 =\n limit.\u03c0 (Discrete.functor fun f => P.obj (op f.snd.fst)) j z\u2082\n[PROOFSTEP]\nrintro \u27e8\u27e8Y, Z, g, f, hf\u27e9\u27e9\n[GOAL]\ncase w.mk.mk.mk.mk.mk\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nz\u2081 z\u2082 : SecondObj P S\nh :\n \u2200 (Y Z : C) (g : Z \u27f6 Y) (f : Y \u27f6 X) (hf : S.arrows f),\n Pi.\u03c0 (fun f => P.obj (op f.snd.fst))\n { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } z\u2081 =\n Pi.\u03c0 (fun f => P.obj (op f.snd.fst))\n { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } z\u2082\nY Z : C\ng : Z \u27f6 Y\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 limit.\u03c0 (Discrete.functor fun f => P.obj (op f.snd.fst))\n { as := { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } } z\u2081 =\n limit.\u03c0 (Discrete.functor fun f => P.obj (op f.snd.fst))\n { as := { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } } z\u2082\n[PROOFSTEP]\napply h\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\n\u22a2 forkMap P S.arrows \u226b firstMap P S = forkMap P S.arrows \u226b secondMap P S\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\na\u271d : P.obj (op X)\nY\u271d Z\u271d : C\ng\u271d : Z\u271d \u27f6 Y\u271d\nf\u271d : Y\u271d \u27f6 X\nhf\u271d : S.arrows f\u271d\n\u22a2 Pi.\u03c0 (fun f => P.obj (op f.snd.fst))\n { fst := Y\u271d, snd := { fst := Z\u271d, snd := { fst := g\u271d, snd := { val := f\u271d, property := hf\u271d } } } }\n ((forkMap P S.arrows \u226b firstMap P S) a\u271d) =\n Pi.\u03c0 (fun f => P.obj (op f.snd.fst))\n { fst := Y\u271d, snd := { fst := Z\u271d, snd := { fst := g\u271d, snd := { val := f\u271d, property := hf\u271d } } } }\n ((forkMap P S.arrows \u226b secondMap P S) a\u271d)\n[PROOFSTEP]\nsimp [firstMap, secondMap, forkMap]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : FirstObj P S.arrows\n\u22a2 Presieve.FamilyOfElements.Compatible ((firstObjEqFamily P S.arrows).hom x) \u2194 firstMap P S x = secondMap P S x\n[PROOFSTEP]\nrw [Presieve.compatible_iff_sieveCompatible]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : FirstObj P S.arrows\n\u22a2 Presieve.FamilyOfElements.SieveCompatible ((firstObjEqFamily P S.arrows).hom x) \u2194 firstMap P S x = secondMap P S x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : FirstObj P S.arrows\n\u22a2 Presieve.FamilyOfElements.SieveCompatible ((firstObjEqFamily P S.arrows).hom x) \u2192 firstMap P S x = secondMap P S x\n[PROOFSTEP]\nintro t\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : FirstObj P S.arrows\nt : Presieve.FamilyOfElements.SieveCompatible ((firstObjEqFamily P S.arrows).hom x)\n\u22a2 firstMap P S x = secondMap P S x\n[PROOFSTEP]\napply SecondObj.ext\n[GOAL]\ncase mp.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : FirstObj P S.arrows\nt : Presieve.FamilyOfElements.SieveCompatible ((firstObjEqFamily P S.arrows).hom x)\n\u22a2 \u2200 (Y Z : C) (g : Z \u27f6 Y) (f : Y \u27f6 X) (hf : S.arrows f),\n Pi.\u03c0 (fun f => P.obj (op f.snd.fst))\n { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } (firstMap P S x) =\n Pi.\u03c0 (fun f => P.obj (op f.snd.fst))\n { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } (secondMap P S x)\n[PROOFSTEP]\nintros Y Z g f hf\n[GOAL]\ncase mp.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : FirstObj P S.arrows\nt : Presieve.FamilyOfElements.SieveCompatible ((firstObjEqFamily P S.arrows).hom x)\nY Z : C\ng : Z \u27f6 Y\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 Pi.\u03c0 (fun f => P.obj (op f.snd.fst))\n { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } (firstMap P S x) =\n Pi.\u03c0 (fun f => P.obj (op f.snd.fst))\n { fst := Y, snd := { fst := Z, snd := { fst := g, snd := { val := f, property := hf } } } } (secondMap P S x)\n[PROOFSTEP]\nsimpa [firstMap, secondMap] using t _ g hf\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : FirstObj P S.arrows\n\u22a2 firstMap P S x = secondMap P S x \u2192 Presieve.FamilyOfElements.SieveCompatible ((firstObjEqFamily P S.arrows).hom x)\n[PROOFSTEP]\nintro t Y Z f g hf\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : FirstObj P S.arrows\nt : firstMap P S x = secondMap P S x\nY Z : C\nf : Y \u27f6 X\ng : Z \u27f6 Y\nhf : S.arrows f\n\u22a2 (firstObjEqFamily P S.arrows).hom x (g \u226b f) (_ : S.arrows (g \u226b f)) =\n P.map g.op ((firstObjEqFamily P S.arrows).hom x f hf)\n[PROOFSTEP]\nrw [Types.limit_ext_iff'] at t \n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : FirstObj P S.arrows\nt :\n \u2200 (j : Discrete ((Y : C) \u00d7 (Z : C) \u00d7 (_ : Z \u27f6 Y) \u00d7 { f' // S.arrows f' })),\n limit.\u03c0 (Discrete.functor fun f => P.obj (op f.snd.fst)) j (firstMap P S x) =\n limit.\u03c0 (Discrete.functor fun f => P.obj (op f.snd.fst)) j (secondMap P S x)\nY Z : C\nf : Y \u27f6 X\ng : Z \u27f6 Y\nhf : S.arrows f\n\u22a2 (firstObjEqFamily P S.arrows).hom x (g \u226b f) (_ : S.arrows (g \u226b f)) =\n P.map g.op ((firstObjEqFamily P S.arrows).hom x f hf)\n[PROOFSTEP]\nsimpa [firstMap, secondMap] using t \u27e8\u27e8Y, Z, g, f, hf\u27e9\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\n\u22a2 Presieve.IsSheafFor P S.arrows \u2194\n Nonempty\n (IsLimit\n (Fork.of\u03b9 (forkMap P S.arrows) (_ : forkMap P S.arrows \u226b firstMap P S = forkMap P S.arrows \u226b secondMap P S)))\n[PROOFSTEP]\nrw [Types.type_equalizer_iff_unique, \u2190 Equiv.forall_congr_left (firstObjEqFamily P (S : Presieve X)).toEquiv.symm]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\n\u22a2 Presieve.IsSheafFor P S.arrows \u2194\n \u2200 (x : Presieve.FamilyOfElements P S.arrows),\n firstMap P S (\u2191(firstObjEqFamily P S.arrows).symm x) = secondMap P S (\u2191(firstObjEqFamily P S.arrows).symm x) \u2192\n \u2203! x_1, forkMap P S.arrows x_1 = \u2191(firstObjEqFamily P S.arrows).symm x\n[PROOFSTEP]\nsimp_rw [\u2190 compatible_iff]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\n\u22a2 Presieve.IsSheafFor P S.arrows \u2194\n \u2200 (x : Presieve.FamilyOfElements P S.arrows),\n Presieve.FamilyOfElements.Compatible ((firstObjEqFamily P S.arrows).hom (\u2191(firstObjEqFamily P S.arrows).symm x)) \u2192\n \u2203! x_1, forkMap P S.arrows x_1 = \u2191(firstObjEqFamily P S.arrows).symm x\n[PROOFSTEP]\nsimp only [inv_hom_id_apply, Iso.toEquiv_symm_fun]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\n\u22a2 Presieve.IsSheafFor P S.arrows \u2194\n \u2200 (x : Presieve.FamilyOfElements P S.arrows),\n Presieve.FamilyOfElements.Compatible x \u2192 \u2203! x_1, forkMap P S.arrows x_1 = (firstObjEqFamily P S.arrows).inv x\n[PROOFSTEP]\napply ball_congr\n[GOAL]\ncase H\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\n\u22a2 \u2200 (x : Presieve.FamilyOfElements P S.arrows),\n Presieve.FamilyOfElements.Compatible x \u2192\n ((\u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t) \u2194\n \u2203! x_1, forkMap P S.arrows x_1 = (firstObjEqFamily P S.arrows).inv x)\n[PROOFSTEP]\nintro x _\n[GOAL]\ncase H\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : Presieve.FamilyOfElements P S.arrows\nh\u271d : Presieve.FamilyOfElements.Compatible x\n\u22a2 (\u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t) \u2194\n \u2203! x_1, forkMap P S.arrows x_1 = (firstObjEqFamily P S.arrows).inv x\n[PROOFSTEP]\napply exists_unique_congr\n[GOAL]\ncase H.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : Presieve.FamilyOfElements P S.arrows\nh\u271d : Presieve.FamilyOfElements.Compatible x\n\u22a2 \u2200 (a : P.obj (op X)),\n Presieve.FamilyOfElements.IsAmalgamation x a \u2194 forkMap P S.arrows a = (firstObjEqFamily P S.arrows).inv x\n[PROOFSTEP]\nintro t\n[GOAL]\ncase H.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : Presieve.FamilyOfElements P S.arrows\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\n\u22a2 Presieve.FamilyOfElements.IsAmalgamation x t \u2194 forkMap P S.arrows t = (firstObjEqFamily P S.arrows).inv x\n[PROOFSTEP]\nrw [\u2190 Iso.toEquiv_symm_fun]\n[GOAL]\ncase H.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : Presieve.FamilyOfElements P S.arrows\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\n\u22a2 Presieve.FamilyOfElements.IsAmalgamation x t \u2194 forkMap P S.arrows t = \u2191(firstObjEqFamily P S.arrows).symm x\n[PROOFSTEP]\nrw [Equiv.eq_symm_apply]\n[GOAL]\ncase H.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : Presieve.FamilyOfElements P S.arrows\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\n\u22a2 Presieve.FamilyOfElements.IsAmalgamation x t \u2194 \u2191(firstObjEqFamily P S.arrows).toEquiv (forkMap P S.arrows t) = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase H.h.mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : Presieve.FamilyOfElements P S.arrows\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\n\u22a2 Presieve.FamilyOfElements.IsAmalgamation x t \u2192 \u2191(firstObjEqFamily P S.arrows).toEquiv (forkMap P S.arrows t) = x\n[PROOFSTEP]\nintro q\n[GOAL]\ncase H.h.mp\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : Presieve.FamilyOfElements P S.arrows\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\nq : Presieve.FamilyOfElements.IsAmalgamation x t\n\u22a2 \u2191(firstObjEqFamily P S.arrows).toEquiv (forkMap P S.arrows t) = x\n[PROOFSTEP]\nfunext Y f hf\n[GOAL]\ncase H.h.mp.h.h.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : Presieve.FamilyOfElements P S.arrows\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\nq : Presieve.FamilyOfElements.IsAmalgamation x t\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 \u2191(firstObjEqFamily P S.arrows).toEquiv (forkMap P S.arrows t) f hf = x f hf\n[PROOFSTEP]\nsimpa [firstObjEqFamily, forkMap] using q _ _\n[GOAL]\ncase H.h.mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : Presieve.FamilyOfElements P S.arrows\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\n\u22a2 \u2191(firstObjEqFamily P S.arrows).toEquiv (forkMap P S.arrows t) = x \u2192 Presieve.FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nintro q Y f hf\n[GOAL]\ncase H.h.mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : Presieve.FamilyOfElements P S.arrows\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\nq : \u2191(firstObjEqFamily P S.arrows).toEquiv (forkMap P S.arrows t) = x\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 P.map f.op t = x f hf\n[PROOFSTEP]\nrw [\u2190 q]\n[GOAL]\ncase H.h.mpr\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\nx : Presieve.FamilyOfElements P S.arrows\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\nq : \u2191(firstObjEqFamily P S.arrows).toEquiv (forkMap P S.arrows t) = x\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 P.map f.op t = \u2191(firstObjEqFamily P S.arrows).toEquiv (forkMap P S.arrows t) f hf\n[PROOFSTEP]\nsimp [firstObjEqFamily, forkMap]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\n\u22a2 forkMap P R \u226b firstMap P R = forkMap P R \u226b secondMap P R\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\n\u22a2 forkMap P R \u226b firstMap P R = forkMap P R \u226b secondMap P R\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nb\u271d : ((Y : C) \u00d7 { f // R f }) \u00d7 (Z : C) \u00d7 { g // R g }\na\u271d : P.obj (op X)\n\u22a2 ((forkMap P R \u226b firstMap P R) \u226b Pi.\u03c0 (fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) b\u271d) a\u271d =\n ((forkMap P R \u226b secondMap P R) \u226b Pi.\u03c0 (fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) b\u271d) a\u271d\n[PROOFSTEP]\nsimp only [firstMap, secondMap, forkMap]\n[GOAL]\ncase h.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nb\u271d : ((Y : C) \u00d7 { f // R f }) \u00d7 (Z : C) \u00d7 { g // R g }\na\u271d : P.obj (op X)\n\u22a2 (((Pi.lift fun f => P.map (\u2191f.snd).op) \u226b\n Pi.lift fun x => Pi.\u03c0 (fun f => P.obj (op f.fst)) x.fst \u226b P.map pullback.fst.op) \u226b\n Pi.\u03c0 (fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) b\u271d)\n a\u271d =\n (((Pi.lift fun f => P.map (\u2191f.snd).op) \u226b\n Pi.lift fun x => Pi.\u03c0 (fun f => P.obj (op f.fst)) x.snd \u226b P.map pullback.snd.op) \u226b\n Pi.\u03c0 (fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) b\u271d)\n a\u271d\n[PROOFSTEP]\nsimp only [limit.lift_\u03c0, limit.lift_\u03c0_assoc, assoc, Fan.mk_\u03c0_app]\n[GOAL]\ncase h.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nb\u271d : ((Y : C) \u00d7 { f // R f }) \u00d7 (Z : C) \u00d7 { g // R g }\na\u271d : P.obj (op X)\n\u22a2 (P.map (\u2191b\u271d.fst.snd).op \u226b P.map pullback.fst.op) a\u271d = (P.map (\u2191b\u271d.snd.snd).op \u226b P.map pullback.snd.op) a\u271d\n[PROOFSTEP]\nrw [\u2190 P.map_comp, \u2190 op_comp, pullback.condition]\n[GOAL]\ncase h.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nb\u271d : ((Y : C) \u00d7 { f // R f }) \u00d7 (Z : C) \u00d7 { g // R g }\na\u271d : P.obj (op X)\n\u22a2 P.map (pullback.snd \u226b \u2191b\u271d.snd.snd).op a\u271d = (P.map (\u2191b\u271d.snd.snd).op \u226b P.map pullback.snd.op) a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : FirstObj P R\n\u22a2 Presieve.FamilyOfElements.Compatible ((firstObjEqFamily P R).hom x) \u2194 firstMap P R x = secondMap P R x\n[PROOFSTEP]\nrw [Presieve.pullbackCompatible_iff]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : FirstObj P R\n\u22a2 Presieve.FamilyOfElements.PullbackCompatible ((firstObjEqFamily P R).hom x) \u2194 firstMap P R x = secondMap P R x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : FirstObj P R\n\u22a2 Presieve.FamilyOfElements.PullbackCompatible ((firstObjEqFamily P R).hom x) \u2192 firstMap P R x = secondMap P R x\n[PROOFSTEP]\nintro t\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : FirstObj P R\nt : Presieve.FamilyOfElements.PullbackCompatible ((firstObjEqFamily P R).hom x)\n\u22a2 firstMap P R x = secondMap P R x\n[PROOFSTEP]\napply Limits.Types.limit_ext\n[GOAL]\ncase mp.w\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : FirstObj P R\nt : Presieve.FamilyOfElements.PullbackCompatible ((firstObjEqFamily P R).hom x)\n\u22a2 \u2200 (j : Discrete (((Y : C) \u00d7 { f // R f }) \u00d7 (Z : C) \u00d7 { g // R g })),\n limit.\u03c0 (Discrete.functor fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) j (firstMap P R x) =\n limit.\u03c0 (Discrete.functor fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) j (secondMap P R x)\n[PROOFSTEP]\nrintro \u27e8\u27e8Y, f, hf\u27e9, Z, g, hg\u27e9\n[GOAL]\ncase mp.w.mk.mk.mk.mk.mk.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : FirstObj P R\nt : Presieve.FamilyOfElements.PullbackCompatible ((firstObjEqFamily P R).hom x)\nY : C\nf : Y \u27f6 X\nhf : R f\nZ : C\ng : Z \u27f6 X\nhg : R g\n\u22a2 limit.\u03c0 (Discrete.functor fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))\n { as := ({ fst := Y, snd := { val := f, property := hf } }, { fst := Z, snd := { val := g, property := hg } }) }\n (firstMap P R x) =\n limit.\u03c0 (Discrete.functor fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))\n { as := ({ fst := Y, snd := { val := f, property := hf } }, { fst := Z, snd := { val := g, property := hg } }) }\n (secondMap P R x)\n[PROOFSTEP]\nsimpa [firstMap, secondMap] using t hf hg\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : FirstObj P R\n\u22a2 firstMap P R x = secondMap P R x \u2192 Presieve.FamilyOfElements.PullbackCompatible ((firstObjEqFamily P R).hom x)\n[PROOFSTEP]\nintro t Y Z f g hf hg\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : FirstObj P R\nt : firstMap P R x = secondMap P R x\nY Z : C\nf : Y \u27f6 X\ng : Z \u27f6 X\nhf : R f\nhg : R g\n\u22a2 P.map pullback.fst.op ((firstObjEqFamily P R).hom x f hf) = P.map pullback.snd.op ((firstObjEqFamily P R).hom x g hg)\n[PROOFSTEP]\nrw [Types.limit_ext_iff'] at t \n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : FirstObj P R\nt :\n \u2200 (j : Discrete (((Y : C) \u00d7 { f // R f }) \u00d7 (Z : C) \u00d7 { g // R g })),\n limit.\u03c0 (Discrete.functor fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) j (firstMap P R x) =\n limit.\u03c0 (Discrete.functor fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) j (secondMap P R x)\nY Z : C\nf : Y \u27f6 X\ng : Z \u27f6 X\nhf : R f\nhg : R g\n\u22a2 P.map pullback.fst.op ((firstObjEqFamily P R).hom x f hf) = P.map pullback.snd.op ((firstObjEqFamily P R).hom x g hg)\n[PROOFSTEP]\nsimpa [firstMap, secondMap] using t \u27e8\u27e8\u27e8Y, f, hf\u27e9, Z, g, hg\u27e9\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\n\u22a2 Presieve.IsSheafFor P R \u2194\n Nonempty (IsLimit (Fork.of\u03b9 (forkMap P R) (_ : forkMap P R \u226b firstMap P R = forkMap P R \u226b secondMap P R)))\n[PROOFSTEP]\nrw [Types.type_equalizer_iff_unique]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\n\u22a2 Presieve.IsSheafFor P R \u2194 \u2200 (y : FirstObj P R), firstMap P R y = secondMap P R y \u2192 \u2203! x, forkMap P R x = y\n[PROOFSTEP]\nerw [\u2190 Equiv.forall_congr_left (firstObjEqFamily P R).toEquiv.symm]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\n\u22a2 Presieve.IsSheafFor P R \u2194\n \u2200 (x : Presieve.FamilyOfElements P R),\n firstMap P R (\u2191(firstObjEqFamily P R).symm x) = secondMap P R (\u2191(firstObjEqFamily P R).symm x) \u2192\n \u2203! x_1, forkMap P R x_1 = \u2191(firstObjEqFamily P R).symm x\n[PROOFSTEP]\nsimp_rw [\u2190 compatible_iff, \u2190 Iso.toEquiv_fun, Equiv.apply_symm_apply]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\n\u22a2 Presieve.IsSheafFor P R \u2194\n \u2200 (x : Presieve.FamilyOfElements P R),\n Presieve.FamilyOfElements.Compatible x \u2192 \u2203! x_1, forkMap P R x_1 = \u2191(firstObjEqFamily P R).symm x\n[PROOFSTEP]\napply ball_congr\n[GOAL]\ncase H\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\n\u22a2 \u2200 (x : Presieve.FamilyOfElements P R),\n Presieve.FamilyOfElements.Compatible x \u2192\n ((\u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t) \u2194 \u2203! x_1, forkMap P R x_1 = \u2191(firstObjEqFamily P R).symm x)\n[PROOFSTEP]\nintro x _\n[GOAL]\ncase H\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : Presieve.FamilyOfElements P R\nh\u271d : Presieve.FamilyOfElements.Compatible x\n\u22a2 (\u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t) \u2194 \u2203! x_1, forkMap P R x_1 = \u2191(firstObjEqFamily P R).symm x\n[PROOFSTEP]\napply exists_unique_congr\n[GOAL]\ncase H.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : Presieve.FamilyOfElements P R\nh\u271d : Presieve.FamilyOfElements.Compatible x\n\u22a2 \u2200 (a : P.obj (op X)), Presieve.FamilyOfElements.IsAmalgamation x a \u2194 forkMap P R a = \u2191(firstObjEqFamily P R).symm x\n[PROOFSTEP]\nintro t\n[GOAL]\ncase H.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : Presieve.FamilyOfElements P R\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\n\u22a2 Presieve.FamilyOfElements.IsAmalgamation x t \u2194 forkMap P R t = \u2191(firstObjEqFamily P R).symm x\n[PROOFSTEP]\nrw [Equiv.eq_symm_apply]\n[GOAL]\ncase H.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : Presieve.FamilyOfElements P R\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\n\u22a2 Presieve.FamilyOfElements.IsAmalgamation x t \u2194 \u2191(firstObjEqFamily P R).toEquiv (forkMap P R t) = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase H.h.mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : Presieve.FamilyOfElements P R\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\n\u22a2 Presieve.FamilyOfElements.IsAmalgamation x t \u2192 \u2191(firstObjEqFamily P R).toEquiv (forkMap P R t) = x\n[PROOFSTEP]\nintro q\n[GOAL]\ncase H.h.mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : Presieve.FamilyOfElements P R\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\nq : Presieve.FamilyOfElements.IsAmalgamation x t\n\u22a2 \u2191(firstObjEqFamily P R).toEquiv (forkMap P R t) = x\n[PROOFSTEP]\nfunext Y f hf\n[GOAL]\ncase H.h.mp.h.h.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : Presieve.FamilyOfElements P R\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\nq : Presieve.FamilyOfElements.IsAmalgamation x t\nY : C\nf : Y \u27f6 X\nhf : R f\n\u22a2 \u2191(firstObjEqFamily P R).toEquiv (forkMap P R t) f hf = x f hf\n[PROOFSTEP]\nsimpa [forkMap] using q _ _\n[GOAL]\ncase H.h.mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : Presieve.FamilyOfElements P R\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\n\u22a2 \u2191(firstObjEqFamily P R).toEquiv (forkMap P R t) = x \u2192 Presieve.FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nintro q Y f hf\n[GOAL]\ncase H.h.mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : Presieve.FamilyOfElements P R\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\nq : \u2191(firstObjEqFamily P R).toEquiv (forkMap P R t) = x\nY : C\nf : Y \u27f6 X\nhf : R f\n\u22a2 P.map f.op t = x f hf\n[PROOFSTEP]\nrw [\u2190 q]\n[GOAL]\ncase H.h.mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56 \u2964 Type (max v\u2081 u\u2081)\nX : C\nR : Presieve X\nS : Sieve X\ninst\u271d : HasPullbacks C\nx : Presieve.FamilyOfElements P R\nh\u271d : Presieve.FamilyOfElements.Compatible x\nt : P.obj (op X)\nq : \u2191(firstObjEqFamily P R).toEquiv (forkMap P R t) = x\nY : C\nf : Y \u27f6 X\nhf : R f\n\u22a2 P.map f.op t = \u2191(firstObjEqFamily P R).toEquiv (forkMap P R t) f hf\n[PROOFSTEP]\nsimp [forkMap]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.SheafOfTypes", "llama_tokens": 62851, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850933, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4175336613924648}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\n\u22a2 v \ud835\udcb0 f g i j \u27f6 v \ud835\udcb0 f g j i\n[PROOFSTEP]\nhaveI : HasPullback (pullback.snd \u226b \ud835\udcb0.map i \u226b f) g := hasPullback_assoc_symm (\ud835\udcb0.map j) (\ud835\udcb0.map i) (\ud835\udcb0.map i \u226b f) g\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\n\u22a2 v \ud835\udcb0 f g i j \u27f6 v \ud835\udcb0 f g j i\n[PROOFSTEP]\nhaveI : HasPullback (pullback.snd \u226b \ud835\udcb0.map j \u226b f) g := hasPullback_assoc_symm (\ud835\udcb0.map i) (\ud835\udcb0.map j) (\ud835\udcb0.map j \u226b f) g\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis\u271d : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n\u22a2 v \ud835\udcb0 f g i j \u27f6 v \ud835\udcb0 f g j i\n[PROOFSTEP]\nrefine' (pullbackSymmetry _ _).hom \u226b _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis\u271d : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n\u22a2 pullback (OpenCover.map \ud835\udcb0 j) (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) \u27f6 v \ud835\udcb0 f g j i\n[PROOFSTEP]\nrefine' (pullbackAssoc _ _ _ _).inv \u226b _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis\u271d : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n\u22a2 pullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g \u27f6 v \ud835\udcb0 f g j i\n[PROOFSTEP]\nchange pullback _ _ \u27f6 pullback _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis\u271d : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n\u22a2 pullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g \u27f6 pullback (pullback.fst \u226b OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)\n[PROOFSTEP]\nrefine' _ \u226b (pullbackSymmetry _ _).hom\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis\u271d : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n\u22a2 pullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g \u27f6 pullback (OpenCover.map \ud835\udcb0 i) (pullback.fst \u226b OpenCover.map \ud835\udcb0 j)\n[PROOFSTEP]\nrefine' _ \u226b (pullbackAssoc _ _ _ _).hom\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis\u271d : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n\u22a2 pullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g \u27f6 pullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n[PROOFSTEP]\nrefine' pullback.map _ _ _ _ (pullbackSymmetry _ _).hom (\ud835\udfd9 _) (\ud835\udfd9 _) _ _\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis\u271d : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n\u22a2 (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z =\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom \u226b pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis\u271d : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n\u22a2 g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g\n[PROOFSTEP]\nrw [pullbackSymmetry_hom_comp_snd_assoc, pullback.condition_assoc, Category.comp_id]\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis\u271d : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n\u22a2 g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g\n[PROOFSTEP]\nrw [Category.comp_id, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\n\u22a2 t \ud835\udcb0 f g i j \u226b pullback.fst \u226b pullback.fst = pullback.snd\n[PROOFSTEP]\ndelta t\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\n\u22a2 ((pullbackSymmetry (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 i \u226b f) g).inv \u226b\n let_fun this :=\n (pullback.map (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z =\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 j \u226b f) g).hom) \u226b\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 i) (pullback.fst \u226b OpenCover.map \ud835\udcb0 j)).hom;\n this) \u226b\n pullback.fst \u226b pullback.fst =\n pullback.snd\n[PROOFSTEP]\nhaveI : HasPullback (pullback.snd \u226b \ud835\udcb0.map i \u226b f) g := hasPullback_assoc_symm (\ud835\udcb0.map j) (\ud835\udcb0.map i) (\ud835\udcb0.map i \u226b f) g\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\n\u22a2 ((pullbackSymmetry (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 i \u226b f) g).inv \u226b\n let_fun this :=\n (pullback.map (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z =\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 j \u226b f) g).hom) \u226b\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 i) (pullback.fst \u226b OpenCover.map \ud835\udcb0 j)).hom;\n this) \u226b\n pullback.fst \u226b pullback.fst =\n pullback.snd\n[PROOFSTEP]\nhaveI : HasPullback (pullback.snd \u226b \ud835\udcb0.map j \u226b f) g := hasPullback_assoc_symm (\ud835\udcb0.map i) (\ud835\udcb0.map j) (\ud835\udcb0.map j \u226b f) g\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis\u271d : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n\u22a2 ((pullbackSymmetry (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 i \u226b f) g).inv \u226b\n let_fun this :=\n (pullback.map (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z =\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 j \u226b f) g).hom) \u226b\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 i) (pullback.fst \u226b OpenCover.map \ud835\udcb0 j)).hom;\n this) \u226b\n pullback.fst \u226b pullback.fst =\n pullback.snd\n[PROOFSTEP]\nsimp only [Category.assoc, id.def, pullbackSymmetry_hom_comp_fst_assoc, pullbackAssoc_hom_snd_fst,\n pullback.lift_fst_assoc, pullbackSymmetry_hom_comp_snd, pullbackAssoc_inv_fst_fst, pullbackSymmetry_hom_comp_fst]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\n\u22a2 t \ud835\udcb0 f g i j \u226b pullback.fst \u226b pullback.snd = pullback.fst \u226b pullback.snd\n[PROOFSTEP]\ndelta t\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\n\u22a2 ((pullbackSymmetry (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 i \u226b f) g).inv \u226b\n let_fun this :=\n (pullback.map (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z =\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 j \u226b f) g).hom) \u226b\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 i) (pullback.fst \u226b OpenCover.map \ud835\udcb0 j)).hom;\n this) \u226b\n pullback.fst \u226b pullback.snd =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nhaveI : HasPullback (pullback.snd \u226b \ud835\udcb0.map i \u226b f) g := hasPullback_assoc_symm (\ud835\udcb0.map j) (\ud835\udcb0.map i) (\ud835\udcb0.map i \u226b f) g\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\n\u22a2 ((pullbackSymmetry (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 i \u226b f) g).inv \u226b\n let_fun this :=\n (pullback.map (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z =\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 j \u226b f) g).hom) \u226b\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 i) (pullback.fst \u226b OpenCover.map \ud835\udcb0 j)).hom;\n this) \u226b\n pullback.fst \u226b pullback.snd =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nhaveI : HasPullback (pullback.snd \u226b \ud835\udcb0.map j \u226b f) g := hasPullback_assoc_symm (\ud835\udcb0.map i) (\ud835\udcb0.map j) (\ud835\udcb0.map j \u226b f) g\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis\u271d : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n\u22a2 ((pullbackSymmetry (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 i \u226b f) g).inv \u226b\n let_fun this :=\n (pullback.map (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z =\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 j \u226b f) g).hom) \u226b\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 i) (pullback.fst \u226b OpenCover.map \ud835\udcb0 j)).hom;\n this) \u226b\n pullback.fst \u226b pullback.snd =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nsimp only [pullbackSymmetry_hom_comp_snd_assoc, Category.comp_id, Category.assoc, id.def,\n pullbackSymmetry_hom_comp_fst_assoc, pullbackAssoc_hom_snd_snd, pullback.lift_snd, pullbackAssoc_inv_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\n\u22a2 t \ud835\udcb0 f g i j \u226b pullback.snd = pullback.fst \u226b pullback.fst\n[PROOFSTEP]\ndelta t\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\n\u22a2 ((pullbackSymmetry (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 i \u226b f) g).inv \u226b\n let_fun this :=\n (pullback.map (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z =\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 j \u226b f) g).hom) \u226b\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 i) (pullback.fst \u226b OpenCover.map \ud835\udcb0 j)).hom;\n this) \u226b\n pullback.snd =\n pullback.fst \u226b pullback.fst\n[PROOFSTEP]\nhaveI : HasPullback (pullback.snd \u226b \ud835\udcb0.map i \u226b f) g := hasPullback_assoc_symm (\ud835\udcb0.map j) (\ud835\udcb0.map i) (\ud835\udcb0.map i \u226b f) g\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\n\u22a2 ((pullbackSymmetry (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 i \u226b f) g).inv \u226b\n let_fun this :=\n (pullback.map (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z =\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 j \u226b f) g).hom) \u226b\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 i) (pullback.fst \u226b OpenCover.map \ud835\udcb0 j)).hom;\n this) \u226b\n pullback.snd =\n pullback.fst \u226b pullback.fst\n[PROOFSTEP]\nhaveI : HasPullback (pullback.snd \u226b \ud835\udcb0.map j \u226b f) g := hasPullback_assoc_symm (\ud835\udcb0.map i) (\ud835\udcb0.map j) (\ud835\udcb0.map j \u226b f) g\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : \ud835\udcb0.J\nthis\u271d : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g\nthis : HasPullback (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n\u22a2 ((pullbackSymmetry (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 i \u226b f) g).inv \u226b\n let_fun this :=\n (pullback.map (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) g (pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f) g\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z =\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.snd \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n (pullbackAssoc (OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 j \u226b f) g).hom) \u226b\n (pullbackSymmetry (OpenCover.map \ud835\udcb0 i) (pullback.fst \u226b OpenCover.map \ud835\udcb0 j)).hom;\n this) \u226b\n pullback.snd =\n pullback.fst \u226b pullback.fst\n[PROOFSTEP]\nsimp only [pullbackSymmetry_hom_comp_snd_assoc, Category.assoc, id.def, pullbackSymmetry_hom_comp_snd,\n pullbackAssoc_hom_fst, pullback.lift_fst_assoc, pullbackSymmetry_hom_comp_fst, pullbackAssoc_inv_fst_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni : \ud835\udcb0.J\n\u22a2 t \ud835\udcb0 f g i i = \ud835\udfd9 (v \ud835\udcb0 f g i i)\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni : \ud835\udcb0.J\n\u22a2 t \ud835\udcb0 f g i i \u226b pullback.fst = \ud835\udfd9 (v \ud835\udcb0 f g i i) \u226b pullback.fst\n[PROOFSTEP]\nrw [Category.id_comp]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni : \ud835\udcb0.J\n\u22a2 t \ud835\udcb0 f g i i \u226b pullback.snd = \ud835\udfd9 (v \ud835\udcb0 f g i i) \u226b pullback.snd\n[PROOFSTEP]\nrw [Category.id_comp]\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni : \ud835\udcb0.J\n\u22a2 t \ud835\udcb0 f g i i \u226b pullback.fst = pullback.fst\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni : \ud835\udcb0.J\n\u22a2 t \ud835\udcb0 f g i i \u226b pullback.snd = pullback.snd\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni : \ud835\udcb0.J\n\u22a2 (t \ud835\udcb0 f g i i \u226b pullback.fst) \u226b pullback.fst = pullback.fst \u226b pullback.fst\n[PROOFSTEP]\nrw [\u2190 cancel_mono (\ud835\udcb0.map i)]\n[GOAL]\ncase h\u2080.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni : \ud835\udcb0.J\n\u22a2 ((t \ud835\udcb0 f g i i \u226b pullback.fst) \u226b pullback.fst) \u226b OpenCover.map \ud835\udcb0 i = (pullback.fst \u226b pullback.fst) \u226b OpenCover.map \ud835\udcb0 i\n[PROOFSTEP]\nsimp only [pullback.condition, Category.assoc, t_fst_fst]\n[GOAL]\ncase h\u2080.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni : \ud835\udcb0.J\n\u22a2 (t \ud835\udcb0 f g i i \u226b pullback.fst) \u226b pullback.snd = pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nsimp only [Category.assoc, t_fst_snd]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni : \ud835\udcb0.J\n\u22a2 t \ud835\udcb0 f g i i \u226b pullback.snd = pullback.snd\n[PROOFSTEP]\nrw [\u2190 cancel_mono (\ud835\udcb0.map i)]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni : \ud835\udcb0.J\n\u22a2 (t \ud835\udcb0 f g i i \u226b pullback.snd) \u226b OpenCover.map \ud835\udcb0 i = pullback.snd \u226b OpenCover.map \ud835\udcb0 i\n[PROOFSTEP]\nsimp only [pullback.condition, t_snd, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 pullback (fV \ud835\udcb0 f g i j) (fV \ud835\udcb0 f g i k) \u27f6 pullback (fV \ud835\udcb0 f g j k) (fV \ud835\udcb0 f g j i)\n[PROOFSTEP]\nrefine' (pullbackRightPullbackFstIso _ _ _).hom \u226b _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 pullback (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 k) \u27f6\n pullback (fV \ud835\udcb0 f g j k) (fV \ud835\udcb0 f g j i)\n[PROOFSTEP]\nrefine' _ \u226b (pullbackSymmetry _ _).hom\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 pullback (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 k) \u27f6\n pullback (fV \ud835\udcb0 f g j i) (fV \ud835\udcb0 f g j k)\n[PROOFSTEP]\nrefine' _ \u226b (pullbackRightPullbackFstIso _ _ _).inv\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 pullback (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 k) \u27f6\n pullback (fV \ud835\udcb0 f g j i \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 k)\n[PROOFSTEP]\nrefine' pullback.map _ _ _ _ (t \ud835\udcb0 f g i j) (\ud835\udfd9 _) (\ud835\udfd9 _) _ _\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) \u226b \ud835\udfd9 X =\n t \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 j\n[PROOFSTEP]\nsimp only [\u2190 pullback.condition, Category.comp_id, t_fst_fst_assoc]\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 OpenCover.map \ud835\udcb0 k \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 k) \u226b OpenCover.map \ud835\udcb0 k\n[PROOFSTEP]\nsimp only [Category.comp_id, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b pullback.fst \u226b pullback.fst \u226b pullback.fst = pullback.fst \u226b pullback.snd\n[PROOFSTEP]\ndelta t'\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 ((pullbackRightPullbackFstIso (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 k) (fV \ud835\udcb0 f g i j)).hom \u226b\n (pullback.map (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 k)\n (fV \ud835\udcb0 f g j i \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 k) (t \ud835\udcb0 f g i j)\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 k)) (\ud835\udfd9 X)\n (_ :\n (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) \u226b \ud835\udfd9 X =\n t \ud835\udcb0 f g i j \u226b pullback.fst \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 j)\n (_ : OpenCover.map \ud835\udcb0 k \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 k) \u226b OpenCover.map \ud835\udcb0 k) \u226b\n (pullbackRightPullbackFstIso (pullback.fst \u226b OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 k) (fV \ud835\udcb0 f g j i)).inv) \u226b\n (pullbackSymmetry (fV \ud835\udcb0 f g j i) (fV \ud835\udcb0 f g j k)).hom) \u226b\n pullback.fst \u226b pullback.fst \u226b pullback.fst =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nsimp only [Category.assoc, pullbackSymmetry_hom_comp_fst_assoc, pullbackRightPullbackFstIso_inv_snd_fst_assoc,\n pullback.lift_fst_assoc, t_fst_fst, pullbackRightPullbackFstIso_hom_fst_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b pullback.fst \u226b pullback.fst \u226b pullback.snd = pullback.fst \u226b pullback.fst \u226b pullback.snd\n[PROOFSTEP]\ndelta t'\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 ((pullbackRightPullbackFstIso (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 k) (fV \ud835\udcb0 f g i j)).hom \u226b\n (pullback.map (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 k)\n (fV \ud835\udcb0 f g j i \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 k) (t \ud835\udcb0 f g i j)\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 k)) (\ud835\udfd9 X)\n (_ :\n (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) \u226b \ud835\udfd9 X =\n t \ud835\udcb0 f g i j \u226b pullback.fst \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 j)\n (_ : OpenCover.map \ud835\udcb0 k \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 k) \u226b OpenCover.map \ud835\udcb0 k) \u226b\n (pullbackRightPullbackFstIso (pullback.fst \u226b OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 k) (fV \ud835\udcb0 f g j i)).inv) \u226b\n (pullbackSymmetry (fV \ud835\udcb0 f g j i) (fV \ud835\udcb0 f g j k)).hom) \u226b\n pullback.fst \u226b pullback.fst \u226b pullback.snd =\n pullback.fst \u226b pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nsimp only [Category.assoc, pullbackSymmetry_hom_comp_fst_assoc, pullbackRightPullbackFstIso_inv_snd_fst_assoc,\n pullback.lift_fst_assoc, t_fst_snd, pullbackRightPullbackFstIso_hom_fst_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b pullback.fst \u226b pullback.snd = pullback.snd \u226b pullback.snd\n[PROOFSTEP]\ndelta t'\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 ((pullbackRightPullbackFstIso (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 k) (fV \ud835\udcb0 f g i j)).hom \u226b\n (pullback.map (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 k)\n (fV \ud835\udcb0 f g j i \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 k) (t \ud835\udcb0 f g i j)\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 k)) (\ud835\udfd9 X)\n (_ :\n (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) \u226b \ud835\udfd9 X =\n t \ud835\udcb0 f g i j \u226b pullback.fst \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 j)\n (_ : OpenCover.map \ud835\udcb0 k \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 k) \u226b OpenCover.map \ud835\udcb0 k) \u226b\n (pullbackRightPullbackFstIso (pullback.fst \u226b OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 k) (fV \ud835\udcb0 f g j i)).inv) \u226b\n (pullbackSymmetry (fV \ud835\udcb0 f g j i) (fV \ud835\udcb0 f g j k)).hom) \u226b\n pullback.fst \u226b pullback.snd =\n pullback.snd \u226b pullback.snd\n[PROOFSTEP]\nsimp only [Category.comp_id, Category.assoc, pullbackSymmetry_hom_comp_fst_assoc,\n pullbackRightPullbackFstIso_inv_snd_snd, pullback.lift_snd, pullbackRightPullbackFstIso_hom_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b pullback.snd \u226b pullback.fst \u226b pullback.fst = pullback.fst \u226b pullback.snd\n[PROOFSTEP]\ndelta t'\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 ((pullbackRightPullbackFstIso (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 k) (fV \ud835\udcb0 f g i j)).hom \u226b\n (pullback.map (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 k)\n (fV \ud835\udcb0 f g j i \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 k) (t \ud835\udcb0 f g i j)\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 k)) (\ud835\udfd9 X)\n (_ :\n (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) \u226b \ud835\udfd9 X =\n t \ud835\udcb0 f g i j \u226b pullback.fst \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 j)\n (_ : OpenCover.map \ud835\udcb0 k \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 k) \u226b OpenCover.map \ud835\udcb0 k) \u226b\n (pullbackRightPullbackFstIso (pullback.fst \u226b OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 k) (fV \ud835\udcb0 f g j i)).inv) \u226b\n (pullbackSymmetry (fV \ud835\udcb0 f g j i) (fV \ud835\udcb0 f g j k)).hom) \u226b\n pullback.snd \u226b pullback.fst \u226b pullback.fst =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nsimp only [Category.assoc, pullbackSymmetry_hom_comp_snd_assoc, pullbackRightPullbackFstIso_inv_fst_assoc,\n pullback.lift_fst_assoc, t_fst_fst, pullbackRightPullbackFstIso_hom_fst_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b pullback.snd \u226b pullback.fst \u226b pullback.snd = pullback.fst \u226b pullback.fst \u226b pullback.snd\n[PROOFSTEP]\ndelta t'\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 ((pullbackRightPullbackFstIso (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 k) (fV \ud835\udcb0 f g i j)).hom \u226b\n (pullback.map (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 k)\n (fV \ud835\udcb0 f g j i \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 k) (t \ud835\udcb0 f g i j)\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 k)) (\ud835\udfd9 X)\n (_ :\n (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) \u226b \ud835\udfd9 X =\n t \ud835\udcb0 f g i j \u226b pullback.fst \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 j)\n (_ : OpenCover.map \ud835\udcb0 k \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 k) \u226b OpenCover.map \ud835\udcb0 k) \u226b\n (pullbackRightPullbackFstIso (pullback.fst \u226b OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 k) (fV \ud835\udcb0 f g j i)).inv) \u226b\n (pullbackSymmetry (fV \ud835\udcb0 f g j i) (fV \ud835\udcb0 f g j k)).hom) \u226b\n pullback.snd \u226b pullback.fst \u226b pullback.snd =\n pullback.fst \u226b pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nsimp only [Category.assoc, pullbackSymmetry_hom_comp_snd_assoc, pullbackRightPullbackFstIso_inv_fst_assoc,\n pullback.lift_fst_assoc, t_fst_snd, pullbackRightPullbackFstIso_hom_fst_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b pullback.snd \u226b pullback.snd = pullback.fst \u226b pullback.fst \u226b pullback.fst\n[PROOFSTEP]\ndelta t'\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 ((pullbackRightPullbackFstIso (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 k) (fV \ud835\udcb0 f g i j)).hom \u226b\n (pullback.map (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 k)\n (fV \ud835\udcb0 f g j i \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 k) (t \ud835\udcb0 f g i j)\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 k)) (\ud835\udfd9 X)\n (_ :\n (fV \ud835\udcb0 f g i j \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i) \u226b \ud835\udfd9 X =\n t \ud835\udcb0 f g i j \u226b pullback.fst \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 j)\n (_ : OpenCover.map \ud835\udcb0 k \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 k) \u226b OpenCover.map \ud835\udcb0 k) \u226b\n (pullbackRightPullbackFstIso (pullback.fst \u226b OpenCover.map \ud835\udcb0 j) (OpenCover.map \ud835\udcb0 k) (fV \ud835\udcb0 f g j i)).inv) \u226b\n (pullbackSymmetry (fV \ud835\udcb0 f g j i) (fV \ud835\udcb0 f g j k)).hom) \u226b\n pullback.snd \u226b pullback.snd =\n pullback.fst \u226b pullback.fst \u226b pullback.fst\n[PROOFSTEP]\nsimp only [Category.assoc, pullbackSymmetry_hom_comp_snd_assoc, pullbackRightPullbackFstIso_inv_fst_assoc,\n pullback.lift_fst_assoc, t_snd, pullbackRightPullbackFstIso_hom_fst_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j \u226b pullback.fst \u226b pullback.fst \u226b pullback.fst =\n pullback.fst \u226b pullback.fst \u226b pullback.fst\n[PROOFSTEP]\nsimp only [t'_fst_fst_fst, t'_fst_snd, t'_snd_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j \u226b pullback.fst \u226b pullback.fst \u226b pullback.snd =\n pullback.fst \u226b pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nsimp only [t'_fst_fst_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j \u226b pullback.fst \u226b pullback.snd = pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nsimp only [t'_fst_snd, t'_snd_snd, t'_fst_fst_fst]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j \u226b pullback.snd \u226b pullback.fst \u226b pullback.fst =\n pullback.snd \u226b pullback.fst \u226b pullback.fst\n[PROOFSTEP]\nrw [\u2190 cancel_mono (\ud835\udcb0.map i)]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 (t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j \u226b pullback.snd \u226b pullback.fst \u226b pullback.fst) \u226b OpenCover.map \ud835\udcb0 i =\n (pullback.snd \u226b pullback.fst \u226b pullback.fst) \u226b OpenCover.map \ud835\udcb0 i\n[PROOFSTEP]\nsimp only [pullback.condition_assoc, t'_snd_fst_fst, t'_fst_snd, t'_snd_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j \u226b pullback.snd \u226b pullback.fst \u226b pullback.snd =\n pullback.snd \u226b pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nsimp only [pullback.condition_assoc, t'_snd_fst_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j \u226b pullback.snd \u226b pullback.snd = pullback.snd \u226b pullback.snd\n[PROOFSTEP]\nsimp only [t'_snd_snd, t'_fst_fst_fst, t'_fst_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j = \ud835\udfd9 (pullback (fV \ud835\udcb0 f g i j) (fV \ud835\udcb0 f g i k))\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 (t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j) \u226b pullback.fst =\n \ud835\udfd9 (pullback (fV \ud835\udcb0 f g i j) (fV \ud835\udcb0 f g i k)) \u226b pullback.fst\n[PROOFSTEP]\nrw [Category.id_comp]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 (t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j) \u226b pullback.snd =\n \ud835\udfd9 (pullback (fV \ud835\udcb0 f g i j) (fV \ud835\udcb0 f g i k)) \u226b pullback.snd\n[PROOFSTEP]\nrw [Category.id_comp]\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 (t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j) \u226b pullback.fst = pullback.fst\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 ((t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j) \u226b pullback.fst) \u226b pullback.fst = pullback.fst \u226b pullback.fst\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080.h\u2080.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 (((t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j) \u226b pullback.fst) \u226b pullback.fst) \u226b pullback.fst =\n (pullback.fst \u226b pullback.fst) \u226b pullback.fst\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase h\u2080.h\u2080.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j \u226b pullback.fst \u226b pullback.fst \u226b pullback.fst =\n pullback.fst \u226b pullback.fst \u226b pullback.fst\n[PROOFSTEP]\nexact cocycle_fst_fst_fst \ud835\udcb0 f g i j k\n[GOAL]\ncase h\u2080.h\u2080.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 (((t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j) \u226b pullback.fst) \u226b pullback.fst) \u226b pullback.snd =\n (pullback.fst \u226b pullback.fst) \u226b pullback.snd\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase h\u2080.h\u2080.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j \u226b pullback.fst \u226b pullback.fst \u226b pullback.snd =\n pullback.fst \u226b pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nexact cocycle_fst_fst_snd \ud835\udcb0 f g i j k\n[GOAL]\ncase h\u2080.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 ((t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j) \u226b pullback.fst) \u226b pullback.snd = pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase h\u2080.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j \u226b pullback.fst \u226b pullback.snd = pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nexact cocycle_fst_snd \ud835\udcb0 f g i j k\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 (t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j) \u226b pullback.snd = pullback.snd\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2081.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 ((t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j) \u226b pullback.snd) \u226b pullback.fst = pullback.snd \u226b pullback.fst\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2081.h\u2080.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 (((t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j) \u226b pullback.snd) \u226b pullback.fst) \u226b pullback.fst =\n (pullback.snd \u226b pullback.fst) \u226b pullback.fst\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase h\u2081.h\u2080.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j \u226b pullback.snd \u226b pullback.fst \u226b pullback.fst =\n pullback.snd \u226b pullback.fst \u226b pullback.fst\n[PROOFSTEP]\nexact cocycle_snd_fst_fst \ud835\udcb0 f g i j k\n[GOAL]\ncase h\u2081.h\u2080.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 (((t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j) \u226b pullback.snd) \u226b pullback.fst) \u226b pullback.snd =\n (pullback.snd \u226b pullback.fst) \u226b pullback.snd\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase h\u2081.h\u2080.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j \u226b pullback.snd \u226b pullback.fst \u226b pullback.snd =\n pullback.snd \u226b pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nexact cocycle_snd_fst_snd \ud835\udcb0 f g i j k\n[GOAL]\ncase h\u2081.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 ((t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j) \u226b pullback.snd) \u226b pullback.snd = pullback.snd \u226b pullback.snd\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase h\u2081.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 t' \ud835\udcb0 f g i j k \u226b t' \ud835\udcb0 f g j k i \u226b t' \ud835\udcb0 f g k i j \u226b pullback.snd \u226b pullback.snd = pullback.snd \u226b pullback.snd\n[PROOFSTEP]\nexact cocycle_snd_snd \ud835\udcb0 f g i j k\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 (fun i j k => t' \ud835\udcb0 f g i j k) i j k \u226b pullback.snd = pullback.fst \u226b (fun i j => t \ud835\udcb0 f g i j) i j\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 ((fun i j k => t' \ud835\udcb0 f g i j k) i j k \u226b pullback.snd) \u226b pullback.fst =\n (pullback.fst \u226b (fun i j => t \ud835\udcb0 f g i j) i j) \u226b pullback.fst\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 ((fun i j k => t' \ud835\udcb0 f g i j k) i j k \u226b pullback.snd) \u226b pullback.snd =\n (pullback.fst \u226b (fun i j => t \ud835\udcb0 f g i j) i j) \u226b pullback.snd\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 (((fun i j k => t' \ud835\udcb0 f g i j k) i j k \u226b pullback.snd) \u226b pullback.fst) \u226b pullback.fst =\n ((pullback.fst \u226b (fun i j => t \ud835\udcb0 f g i j) i j) \u226b pullback.fst) \u226b pullback.fst\ncase h\u2080.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 (((fun i j k => t' \ud835\udcb0 f g i j k) i j k \u226b pullback.snd) \u226b pullback.fst) \u226b pullback.snd =\n ((pullback.fst \u226b (fun i j => t \ud835\udcb0 f g i j) i j) \u226b pullback.fst) \u226b pullback.snd\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 ((fun i j k => t' \ud835\udcb0 f g i j k) i j k \u226b pullback.snd) \u226b pullback.snd =\n (pullback.fst \u226b (fun i j => t \ud835\udcb0 f g i j) i j) \u226b pullback.snd\n[PROOFSTEP]\nall_goals simp only [t'_snd_fst_fst, t'_snd_fst_snd, t'_snd_snd, t_fst_fst, t_fst_snd, t_snd, Category.assoc]\n[GOAL]\ncase h\u2080.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 (((fun i j k => t' \ud835\udcb0 f g i j k) i j k \u226b pullback.snd) \u226b pullback.fst) \u226b pullback.fst =\n ((pullback.fst \u226b (fun i j => t \ud835\udcb0 f g i j) i j) \u226b pullback.fst) \u226b pullback.fst\n[PROOFSTEP]\nsimp only [t'_snd_fst_fst, t'_snd_fst_snd, t'_snd_snd, t_fst_fst, t_fst_snd, t_snd, Category.assoc]\n[GOAL]\ncase h\u2080.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 (((fun i j k => t' \ud835\udcb0 f g i j k) i j k \u226b pullback.snd) \u226b pullback.fst) \u226b pullback.snd =\n ((pullback.fst \u226b (fun i j => t \ud835\udcb0 f g i j) i j) \u226b pullback.fst) \u226b pullback.snd\n[PROOFSTEP]\nsimp only [t'_snd_fst_fst, t'_snd_fst_snd, t'_snd_snd, t_fst_fst, t_fst_snd, t_snd, Category.assoc]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j k : \ud835\udcb0.J\n\u22a2 ((fun i j k => t' \ud835\udcb0 f g i j k) i j k \u226b pullback.snd) \u226b pullback.snd =\n (pullback.fst \u226b (fun i j => t \ud835\udcb0 f g i j) i j) \u226b pullback.snd\n[PROOFSTEP]\nsimp only [t'_snd_fst_fst, t'_snd_fst_snd, t'_snd_snd, t_fst_fst, t_fst_snd, t_snd, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\n\u22a2 GlueData.glued (gluing \ud835\udcb0 f g) \u27f6 X\n[PROOFSTEP]\nfapply Multicoequalizer.desc\n[GOAL]\ncase k\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\n\u22a2 (b : (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData).R) \u2192\n MultispanIndex.right (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) b \u27f6 X\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\n\u22a2 \u2200 (a : (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData).L),\n MultispanIndex.fst (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a \u226b\n ?k (MultispanIndex.fstFrom (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a) =\n MultispanIndex.snd (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a \u226b\n ?k (MultispanIndex.sndFrom (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a)\n[PROOFSTEP]\nexact fun i => pullback.fst \u226b \ud835\udcb0.map i\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\n\u22a2 \u2200 (a : (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData).L),\n MultispanIndex.fst (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a \u226b\n pullback.fst \u226b OpenCover.map \ud835\udcb0 (MultispanIndex.fstFrom (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a) =\n MultispanIndex.snd (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a \u226b\n pullback.fst \u226b OpenCover.map \ud835\udcb0 (MultispanIndex.sndFrom (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a)\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9\n[GOAL]\ncase h.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : (gluing \ud835\udcb0 f g).toGlueData.J\n\u22a2 MultispanIndex.fst (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) (i, j) \u226b\n pullback.fst \u226b OpenCover.map \ud835\udcb0 (MultispanIndex.fstFrom (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) (i, j)) =\n MultispanIndex.snd (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) (i, j) \u226b\n pullback.fst \u226b OpenCover.map \ud835\udcb0 (MultispanIndex.sndFrom (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) (i, j))\n[PROOFSTEP]\nchange pullback.fst \u226b _ \u226b \ud835\udcb0.map i = (_ \u226b _) \u226b _ \u226b \ud835\udcb0.map j\n[GOAL]\ncase h.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : (gluing \ud835\udcb0 f g).toGlueData.J\n\u22a2 pullback.fst \u226b pullback.fst \u226b OpenCover.map \ud835\udcb0 i =\n (GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i) \u226b\n pullback.fst \u226b OpenCover.map \ud835\udcb0 j\n[PROOFSTEP]\nerw [pullback.condition]\n[GOAL]\ncase h.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : (gluing \ud835\udcb0 f g).toGlueData.J\n\u22a2 pullback.snd \u226b OpenCover.map \ud835\udcb0 j =\n (GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i) \u226b\n pullback.fst \u226b OpenCover.map \ud835\udcb0 j\n[PROOFSTEP]\nrw [\u2190 Category.assoc]\n[GOAL]\ncase h.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : (gluing \ud835\udcb0 f g).toGlueData.J\n\u22a2 pullback.snd \u226b OpenCover.map \ud835\udcb0 j =\n ((GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i) \u226b pullback.fst) \u226b\n OpenCover.map \ud835\udcb0 j\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : (gluing \ud835\udcb0 f g).toGlueData.J\n\u22a2 pullback.snd = (GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i) \u226b pullback.fst\n[PROOFSTEP]\nrw [Category.assoc]\n[GOAL]\ncase h.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : (gluing \ud835\udcb0 f g).toGlueData.J\n\u22a2 pullback.snd = GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b pullback.fst\n[PROOFSTEP]\nexact (t_fst_fst _ _ _ _ _).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\n\u22a2 GlueData.glued (gluing \ud835\udcb0 f g) \u27f6 Y\n[PROOFSTEP]\nfapply Multicoequalizer.desc\n[GOAL]\ncase k\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\n\u22a2 (b : (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData).R) \u2192\n MultispanIndex.right (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) b \u27f6 Y\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\n\u22a2 \u2200 (a : (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData).L),\n MultispanIndex.fst (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a \u226b\n ?k (MultispanIndex.fstFrom (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a) =\n MultispanIndex.snd (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a \u226b\n ?k (MultispanIndex.sndFrom (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a)\n[PROOFSTEP]\nexact fun i => pullback.snd\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\n\u22a2 \u2200 (a : (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData).L),\n MultispanIndex.fst (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a \u226b pullback.snd =\n MultispanIndex.snd (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a \u226b pullback.snd\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9\n[GOAL]\ncase h.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : (gluing \ud835\udcb0 f g).toGlueData.J\n\u22a2 MultispanIndex.fst (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) (i, j) \u226b pullback.snd =\n MultispanIndex.snd (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) (i, j) \u226b pullback.snd\n[PROOFSTEP]\nchange pullback.fst \u226b _ = (_ \u226b _) \u226b _\n[GOAL]\ncase h.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : (gluing \ud835\udcb0 f g).toGlueData.J\n\u22a2 pullback.fst \u226b pullback.snd =\n (GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i) \u226b pullback.snd\n[PROOFSTEP]\nrw [Category.assoc]\n[GOAL]\ncase h.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni j : (gluing \ud835\udcb0 f g).toGlueData.J\n\u22a2 pullback.fst \u226b pullback.snd =\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b pullback.snd\n[PROOFSTEP]\nexact (t_fst_snd _ _ _ _ _).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\n\u22a2 p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g\n[PROOFSTEP]\napply Multicoequalizer.hom_ext\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\n\u22a2 \u2200 (b : (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData).R),\n Multicoequalizer.\u03c0 (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) b \u226b p1 \ud835\udcb0 f g \u226b f =\n Multicoequalizer.\u03c0 (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) b \u226b p2 \ud835\udcb0 f g \u226b g\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni : (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData).R\n\u22a2 Multicoequalizer.\u03c0 (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) i \u226b p1 \ud835\udcb0 f g \u226b f =\n Multicoequalizer.\u03c0 (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) i \u226b p2 \ud835\udcb0 f g \u226b g\n[PROOFSTEP]\nerw [Multicoequalizer.\u03c0_desc_assoc, Multicoequalizer.\u03c0_desc_assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ni : (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData).R\n\u22a2 (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) \u226b f = pullback.snd \u226b g\n[PROOFSTEP]\nrw [Category.assoc, pullback.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 pullback (OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) i)\n (OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) j) \u27f6\n GlueData.V (gluing \ud835\udcb0 f g).toGlueData (i, j)\n[PROOFSTEP]\nchange pullback pullback.fst pullback.fst \u27f6 pullback _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 pullback pullback.fst pullback.fst \u27f6 pullback (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j)\n[PROOFSTEP]\nrefine' (pullbackRightPullbackFstIso _ _ _).hom \u226b _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 pullback (pullback.fst \u226b PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j) \u27f6\n pullback (pullback.fst \u226b OpenCover.map \ud835\udcb0 i) (OpenCover.map \ud835\udcb0 j)\n[PROOFSTEP]\nrefine' pullback.map _ _ _ _ _ (\ud835\udfd9 _) (\ud835\udfd9 _) _ _\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 pullback (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i) \u27f6 pullback (OpenCover.map \ud835\udcb0 i \u226b f) g\n[PROOFSTEP]\nexact (pullbackSymmetry _ _).hom \u226b pullback.map _ _ _ _ (\ud835\udfd9 _) s.snd f (Category.id_comp _).symm s.condition\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 (pullback.fst \u226b PullbackCone.fst s) \u226b \ud835\udfd9 X =\n ((pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n pullback.fst \u226b OpenCover.map \ud835\udcb0 i\n[PROOFSTEP]\nsimpa using pullback.condition\n[GOAL]\ncase refine'_3\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 OpenCover.map \ud835\udcb0 j \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j\n[PROOFSTEP]\nsimp only [Category.comp_id, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 gluedLiftPullbackMap \ud835\udcb0 f g s i j \u226b pullback.fst =\n pullback.fst \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)\n[PROOFSTEP]\ndelta gluedLiftPullbackMap\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 (let_fun this :=\n (pullbackRightPullbackFstIso (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j) pullback.fst).hom \u226b\n pullback.map (pullback.fst \u226b PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j) (pullback.fst \u226b OpenCover.map \ud835\udcb0 i)\n (OpenCover.map \ud835\udcb0 j)\n ((pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g))\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j)) (\ud835\udfd9 X)\n (_ :\n (pullback.fst \u226b PullbackCone.fst s) \u226b \ud835\udfd9 X =\n ((pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i)) (PullbackCone.snd s) f\n (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n pullback.fst \u226b OpenCover.map \ud835\udcb0 i)\n (_ : OpenCover.map \ud835\udcb0 j \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j);\n this) \u226b\n pullback.fst =\n pullback.fst \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 gluedLiftPullbackMap \ud835\udcb0 f g s i j \u226b pullback.snd = pullback.snd \u226b pullback.snd\n[PROOFSTEP]\ndelta gluedLiftPullbackMap\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 (let_fun this :=\n (pullbackRightPullbackFstIso (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j) pullback.fst).hom \u226b\n pullback.map (pullback.fst \u226b PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j) (pullback.fst \u226b OpenCover.map \ud835\udcb0 i)\n (OpenCover.map \ud835\udcb0 j)\n ((pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g))\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j)) (\ud835\udfd9 X)\n (_ :\n (pullback.fst \u226b PullbackCone.fst s) \u226b \ud835\udfd9 X =\n ((pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i)) (PullbackCone.snd s) f\n (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n pullback.fst \u226b OpenCover.map \ud835\udcb0 i)\n (_ : OpenCover.map \ud835\udcb0 j \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j);\n this) \u226b\n pullback.snd =\n pullback.snd \u226b pullback.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\n\u22a2 s.pt \u27f6 GlueData.glued (gluing \ud835\udcb0 f g)\n[PROOFSTEP]\nfapply (\ud835\udcb0.pullbackCover s.fst).glueMorphisms\n[GOAL]\ncase f\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\n\u22a2 (x : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J) \u2192\n OpenCover.obj (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) x \u27f6 GlueData.glued (gluing \ud835\udcb0 f g)\n[PROOFSTEP]\nexact fun i =>\n (pullbackSymmetry _ _).hom \u226b\n pullback.map _ _ _ _ (\ud835\udfd9 _) s.snd f (Category.id_comp _).symm s.condition \u226b (gluing \ud835\udcb0 f g).\u03b9 i\n[GOAL]\ncase hf\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\n\u22a2 \u2200 (x y : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J),\n pullback.fst \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 x)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 x) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 x \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 x))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 x \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 x) \u226b OpenCover.map \ud835\udcb0 x \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) x =\n pullback.snd \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 y)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 y) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 y \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 y))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 y \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 y) \u226b OpenCover.map \ud835\udcb0 y \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) y\n[PROOFSTEP]\nintro i j\n[GOAL]\ncase hf\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\n\u22a2 pullback.fst \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n pullback.snd \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) j\n[PROOFSTEP]\nrw [\u2190 gluedLiftPullbackMap_fst_assoc]\n[GOAL]\ncase hf\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\n\u22a2 gluedLiftPullbackMap \ud835\udcb0 f g s i j \u226b pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n pullback.snd \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) j\n[PROOFSTEP]\nhave : _ = pullback.fst \u226b _ := (gluing \ud835\udcb0 f g).glue_condition i j\n[GOAL]\ncase hf\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis :\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n\u22a2 gluedLiftPullbackMap \ud835\udcb0 f g s i j \u226b pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n pullback.snd \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) j\n[PROOFSTEP]\nrw [\u2190 this, gluing_t, gluing_f]\n[GOAL]\ncase hf\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis :\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n\u22a2 gluedLiftPullbackMap \ud835\udcb0 f g s i j \u226b t \ud835\udcb0 f g i j \u226b pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.snd \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) j\n[PROOFSTEP]\nsimp_rw [\u2190 Category.assoc]\n[GOAL]\ncase hf\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis :\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n\u22a2 ((gluedLiftPullbackMap \ud835\udcb0 f g s i j \u226b t \ud835\udcb0 f g i j) \u226b pullback.fst) \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n ((pullback.snd \u226b (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom) \u226b\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) j\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase hf.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis :\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n\u22a2 (gluedLiftPullbackMap \ud835\udcb0 f g s i j \u226b t \ud835\udcb0 f g i j) \u226b pullback.fst =\n (pullback.snd \u226b (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom) \u226b\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase hf.e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis :\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n\u22a2 ((gluedLiftPullbackMap \ud835\udcb0 f g s i j \u226b t \ud835\udcb0 f g i j) \u226b pullback.fst) \u226b pullback.fst =\n ((pullback.snd \u226b (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom) \u226b\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n pullback.fst\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase hf.e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis :\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n\u22a2 ((gluedLiftPullbackMap \ud835\udcb0 f g s i j \u226b t \ud835\udcb0 f g i j) \u226b pullback.fst) \u226b pullback.snd =\n ((pullback.snd \u226b (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom) \u226b\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n pullback.snd\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase hf.e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis :\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n\u22a2 gluedLiftPullbackMap \ud835\udcb0 f g s i j \u226b t \ud835\udcb0 f g i j \u226b pullback.fst \u226b pullback.fst =\n pullback.snd \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n pullback.fst\n[PROOFSTEP]\nrw [t_fst_fst, gluedLiftPullbackMap_snd]\n[GOAL]\ncase hf.e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis :\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n\u22a2 pullback.snd \u226b pullback.snd =\n pullback.snd \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n pullback.fst\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase hf.e_a.h\u2080.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis :\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n\u22a2 pullback.snd =\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n pullback.fst\n[PROOFSTEP]\nrw [\u2190 Iso.inv_comp_eq, pullbackSymmetry_inv_comp_snd]\n[GOAL]\ncase hf.e_a.h\u2080.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis :\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n\u22a2 pullback.fst =\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n pullback.fst\n[PROOFSTEP]\nerw [pullback.lift_fst]\n[GOAL]\ncase hf.e_a.h\u2080.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis :\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n\u22a2 pullback.fst = pullback.fst \u226b \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j)\n[PROOFSTEP]\nrw [Category.comp_id]\n[GOAL]\ncase hf.e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis :\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n\u22a2 gluedLiftPullbackMap \ud835\udcb0 f g s i j \u226b t \ud835\udcb0 f g i j \u226b pullback.fst \u226b pullback.snd =\n pullback.snd \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n pullback.snd\n[PROOFSTEP]\nrw [t_fst_snd, gluedLiftPullbackMap_fst_assoc]\n[GOAL]\ncase hf.e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis :\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n\u22a2 pullback.fst \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n pullback.snd =\n pullback.snd \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n pullback.snd\n[PROOFSTEP]\nerw [pullback.lift_snd, pullback.lift_snd]\n[GOAL]\ncase hf.e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis :\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n\u22a2 pullback.fst \u226b (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b pullback.snd \u226b PullbackCone.snd s =\n pullback.snd \u226b (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom \u226b pullback.snd \u226b PullbackCone.snd s\n[PROOFSTEP]\nrw [pullbackSymmetry_hom_comp_snd_assoc, pullbackSymmetry_hom_comp_snd_assoc]\n[GOAL]\ncase hf.e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis :\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.fst \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n\u22a2 pullback.fst \u226b pullback.fst \u226b PullbackCone.snd s = pullback.snd \u226b pullback.fst \u226b PullbackCone.snd s\n[PROOFSTEP]\nexact pullback.condition_assoc _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\n\u22a2 gluedLift \ud835\udcb0 f g s \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\n[PROOFSTEP]\nrw [\u2190 cancel_epi (\ud835\udcb0.pullbackCover s.fst).fromGlued]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\n\u22a2 OpenCover.fromGlued (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) \u226b gluedLift \ud835\udcb0 f g s \u226b p1 \ud835\udcb0 f g =\n OpenCover.fromGlued (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) \u226b PullbackCone.fst s\n[PROOFSTEP]\napply Multicoequalizer.hom_ext\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\n\u22a2 \u2200 (b : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R),\n Multicoequalizer.\u03c0\n (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData) b \u226b\n OpenCover.fromGlued (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) \u226b gluedLift \ud835\udcb0 f g s \u226b p1 \ud835\udcb0 f g =\n Multicoequalizer.\u03c0\n (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData) b \u226b\n OpenCover.fromGlued (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) \u226b PullbackCone.fst s\n[PROOFSTEP]\nintro b\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 Multicoequalizer.\u03c0\n (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData) b \u226b\n OpenCover.fromGlued (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) \u226b gluedLift \ud835\udcb0 f g s \u226b p1 \ud835\udcb0 f g =\n Multicoequalizer.\u03c0\n (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData) b \u226b\n OpenCover.fromGlued (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) \u226b PullbackCone.fst s\n[PROOFSTEP]\nerw [Multicoequalizer.\u03c0_desc_assoc, Multicoequalizer.\u03c0_desc_assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b gluedLift \ud835\udcb0 f g s \u226b p1 \ud835\udcb0 f g =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b PullbackCone.fst s\n[PROOFSTEP]\ndelta gluedLift\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b\n OpenCover.glueMorphisms (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))\n (fun i =>\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i)\n (_ :\n \u2200 (i j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J),\n pullback.fst \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i)) (PullbackCone.snd s) f\n (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n pullback.snd \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j)) (PullbackCone.snd s) f\n (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) j) \u226b\n p1 \ud835\udcb0 f g =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b PullbackCone.fst s\n[PROOFSTEP]\nsimp_rw [\u2190 Category.assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 (OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b\n OpenCover.glueMorphisms (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))\n (fun i =>\n ((pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i)\n (_ :\n \u2200 (x y : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J),\n pullback.fst \u226b\n (fun i =>\n ((pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i)) (PullbackCone.snd s) f\n (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i)\n x =\n pullback.snd \u226b\n (fun i =>\n ((pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i)) (PullbackCone.snd s) f\n (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i)\n y)) \u226b\n p1 \ud835\udcb0 f g =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b PullbackCone.fst s\n[PROOFSTEP]\nrw [(\ud835\udcb0.pullbackCover s.fst).\u03b9_glueMorphisms]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 (((pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 b)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 b) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 b \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 b))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 b \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 b) \u226b OpenCover.map \ud835\udcb0 b \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) b) \u226b\n p1 \ud835\udcb0 f g =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b PullbackCone.fst s\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n -- Porting note : `Category.comp_id` is no longer necessary, don't know where `\ud835\udfd9 _` has gone\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 b)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 b) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 b \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 b))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 b \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 b) \u226b OpenCover.map \ud835\udcb0 b \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) b \u226b p1 \ud835\udcb0 f g =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b PullbackCone.fst s\n[PROOFSTEP]\nerw [Multicoequalizer.\u03c0_desc, pullback.lift_fst_assoc, pullback.condition]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 b)).hom \u226b pullback.snd \u226b PullbackCone.fst s =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b PullbackCone.fst s\n[PROOFSTEP]\nrw [pullbackSymmetry_hom_comp_snd_assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 pullback.fst \u226b PullbackCone.fst s =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b PullbackCone.fst s\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\n\u22a2 gluedLift \ud835\udcb0 f g s \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\n[PROOFSTEP]\nrw [\u2190 cancel_epi (\ud835\udcb0.pullbackCover s.fst).fromGlued]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\n\u22a2 OpenCover.fromGlued (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) \u226b gluedLift \ud835\udcb0 f g s \u226b p2 \ud835\udcb0 f g =\n OpenCover.fromGlued (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) \u226b PullbackCone.snd s\n[PROOFSTEP]\napply Multicoequalizer.hom_ext\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\n\u22a2 \u2200 (b : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R),\n Multicoequalizer.\u03c0\n (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData) b \u226b\n OpenCover.fromGlued (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) \u226b gluedLift \ud835\udcb0 f g s \u226b p2 \ud835\udcb0 f g =\n Multicoequalizer.\u03c0\n (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData) b \u226b\n OpenCover.fromGlued (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) \u226b PullbackCone.snd s\n[PROOFSTEP]\nintro b\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 Multicoequalizer.\u03c0\n (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData) b \u226b\n OpenCover.fromGlued (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) \u226b gluedLift \ud835\udcb0 f g s \u226b p2 \ud835\udcb0 f g =\n Multicoequalizer.\u03c0\n (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData) b \u226b\n OpenCover.fromGlued (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) \u226b PullbackCone.snd s\n[PROOFSTEP]\nerw [Multicoequalizer.\u03c0_desc_assoc, Multicoequalizer.\u03c0_desc_assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b gluedLift \ud835\udcb0 f g s \u226b p2 \ud835\udcb0 f g =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b PullbackCone.snd s\n[PROOFSTEP]\ndelta gluedLift\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b\n OpenCover.glueMorphisms (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))\n (fun i =>\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i)\n (_ :\n \u2200 (i j : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J),\n pullback.fst \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i)) (PullbackCone.snd s) f\n (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n pullback.snd \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 j) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 j \u226b f) g\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 j)) (PullbackCone.snd s) f\n (_ : OpenCover.map \ud835\udcb0 j \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 j) \u226b OpenCover.map \ud835\udcb0 j \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) j) \u226b\n p2 \ud835\udcb0 f g =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b PullbackCone.snd s\n[PROOFSTEP]\nsimp_rw [\u2190 Category.assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 (OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b\n OpenCover.glueMorphisms (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))\n (fun i =>\n ((pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i)\n (_ :\n \u2200 (x y : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J),\n pullback.fst \u226b\n (fun i =>\n ((pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i)) (PullbackCone.snd s) f\n (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i)\n x =\n pullback.snd \u226b\n (fun i =>\n ((pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i)) (PullbackCone.snd s) f\n (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i)\n y)) \u226b\n p2 \ud835\udcb0 f g =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b PullbackCone.snd s\n[PROOFSTEP]\nrw [(\ud835\udcb0.pullbackCover s.fst).\u03b9_glueMorphisms]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 (((pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 b)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 b) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 b \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 b))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 b \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 b) \u226b OpenCover.map \ud835\udcb0 b \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) b) \u226b\n p2 \ud835\udcb0 f g =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b PullbackCone.snd s\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 b)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 b) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 b \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 b))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 b \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 b) \u226b OpenCover.map \ud835\udcb0 b \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) b \u226b p2 \ud835\udcb0 f g =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b PullbackCone.snd s\n[PROOFSTEP]\nerw [Multicoequalizer.\u03c0_desc, pullback.lift_snd]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 b)).hom \u226b pullback.snd \u226b PullbackCone.snd s =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b PullbackCone.snd s\n[PROOFSTEP]\nrw [pullbackSymmetry_hom_comp_snd_assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\nb : (GlueData.diagram (OpenCover.gluedCover (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s))).toGlueData).R\n\u22a2 pullback.fst \u226b PullbackCone.snd s =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) b \u226b PullbackCone.snd s\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b pullback.fst = pullback.snd\n[PROOFSTEP]\ndelta pullbackFst\u03b9ToV\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 ((pullbackSymmetry pullback.fst (GlueData.\u03b9 (gluing \ud835\udcb0 f g) j) \u226a\u226b\n pullbackRightPullbackFstIso (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) (GlueData.\u03b9 (gluing \ud835\udcb0 f g) j)).hom \u226b\n (pullback.congrHom\n (_ :\n Multicoequalizer.\u03c0 (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) j \u226b\n Multicoequalizer.desc (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) X\n (fun i => pullback.fst \u226b OpenCover.map \ud835\udcb0 i)\n (_ :\n \u2200 (a : (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData).L),\n MultispanIndex.fst (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a \u226b\n pullback.fst \u226b\n OpenCover.map \ud835\udcb0 (MultispanIndex.fstFrom (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a) =\n MultispanIndex.snd (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a \u226b\n pullback.fst \u226b\n OpenCover.map \ud835\udcb0 (MultispanIndex.sndFrom (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a)) =\n pullback.fst \u226b OpenCover.map \ud835\udcb0 j)\n (_ : OpenCover.map \ud835\udcb0 i = OpenCover.map \ud835\udcb0 i)).hom) \u226b\n pullback.fst =\n pullback.snd\n[PROOFSTEP]\nsimp only [Iso.trans_hom, pullback.congrHom_hom, Category.assoc, pullback.lift_fst, Category.comp_id]\n -- Porting note : `pullbackRightPullbackFstIso_hom_fst` is not used in `simp` even though\n -- instructed, forcing `pullbackSymmetry_hom_comp_fst` to be manually `rw`ed as well.\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 (pullbackSymmetry pullback.fst (GlueData.\u03b9 (gluing \ud835\udcb0 f g) j)).hom \u226b\n (pullbackRightPullbackFstIso (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) (GlueData.\u03b9 (gluing \ud835\udcb0 f g) j)).hom \u226b pullback.fst =\n pullback.snd\n[PROOFSTEP]\nerw [pullbackRightPullbackFstIso_hom_fst]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 (pullbackSymmetry pullback.fst (GlueData.\u03b9 (gluing \ud835\udcb0 f g) j)).hom \u226b pullback.fst = pullback.snd\n[PROOFSTEP]\nrw [pullbackSymmetry_hom_comp_fst]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b pullback.snd = pullback.fst \u226b pullback.snd\n[PROOFSTEP]\ndelta pullbackFst\u03b9ToV\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 ((pullbackSymmetry pullback.fst (GlueData.\u03b9 (gluing \ud835\udcb0 f g) j) \u226a\u226b\n pullbackRightPullbackFstIso (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) (GlueData.\u03b9 (gluing \ud835\udcb0 f g) j)).hom \u226b\n (pullback.congrHom\n (_ :\n Multicoequalizer.\u03c0 (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) j \u226b\n Multicoequalizer.desc (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) X\n (fun i => pullback.fst \u226b OpenCover.map \ud835\udcb0 i)\n (_ :\n \u2200 (a : (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData).L),\n MultispanIndex.fst (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a \u226b\n pullback.fst \u226b\n OpenCover.map \ud835\udcb0 (MultispanIndex.fstFrom (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a) =\n MultispanIndex.snd (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a \u226b\n pullback.fst \u226b\n OpenCover.map \ud835\udcb0 (MultispanIndex.sndFrom (GlueData.diagram (gluing \ud835\udcb0 f g).toGlueData) a)) =\n pullback.fst \u226b OpenCover.map \ud835\udcb0 j)\n (_ : OpenCover.map \ud835\udcb0 i = OpenCover.map \ud835\udcb0 i)).hom) \u226b\n pullback.snd =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nsimp only [Iso.trans_hom, pullback.congrHom_hom, Category.assoc, pullback.lift_snd, Category.comp_id]\n -- Porting note : `pullbackRightPullbackFstIso_hom_snd` is not used in `simp` even though\n -- instructed, forcing `pullbackSymmetry_hom_comp_snd_assoc` to be manually `rw`ed as well.\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 (pullbackSymmetry pullback.fst (GlueData.\u03b9 (gluing \ud835\udcb0 f g) j)).hom \u226b\n (pullbackRightPullbackFstIso (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) (GlueData.\u03b9 (gluing \ud835\udcb0 f g) j)).hom \u226b pullback.snd =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nerw [pullbackRightPullbackFstIso_hom_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni j : \ud835\udcb0.J\n\u22a2 (pullbackSymmetry pullback.fst (GlueData.\u03b9 (gluing \ud835\udcb0 f g) j)).hom \u226b pullback.snd \u226b pullback.snd =\n pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nrw [pullbackSymmetry_hom_comp_snd_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g\n[PROOFSTEP]\nrw [\u2190 pullback.condition_assoc, Category.assoc, p_comm]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n pullback.fst\n[PROOFSTEP]\napply ((gluing \ud835\udcb0 f g).openCover.pullbackCover pullback.fst).hom_ext\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 \u2200 (x : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J),\n OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) x \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) x \u226b pullback.fst\n[PROOFSTEP]\nintro j\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) j \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) j \u226b pullback.fst\n[PROOFSTEP]\ndsimp only [OpenCover.pullbackCover]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n pullback.fst \u226b pullback.fst\n[PROOFSTEP]\ntrans pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i \u226b (gluing \ud835\udcb0 f g).\u03b9 _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j\n[PROOFSTEP]\nrw [\u2190 show _ = fV \ud835\udcb0 f g j i \u226b _ from (gluing \ud835\udcb0 f g).glue_condition j i]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b\n GlueData.t (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n[PROOFSTEP]\nsimp_rw [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 (pullback.fst \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g)) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n ((pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b GlueData.t (gluing \ud835\udcb0 f g).toGlueData j i) \u226b\n GlueData.f (gluing \ud835\udcb0 f g).toGlueData i j) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) =\n (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b GlueData.t (gluing \ud835\udcb0 f g).toGlueData j i) \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData i j\n[PROOFSTEP]\nrw [gluing_f, gluing_t]\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) =\n (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b t \ud835\udcb0 f g j i) \u226b pullback.fst\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 (pullback.fst \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g)) \u226b\n pullback.fst =\n ((pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b t \ud835\udcb0 f g j i) \u226b pullback.fst) \u226b pullback.fst\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n -- Porting note : in the following two bullet points, `rfl` was not necessary\n[GOAL]\ncase e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 (pullback.fst \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g)) \u226b\n pullback.snd =\n ((pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b t \ud835\udcb0 f g j i) \u226b pullback.fst) \u226b pullback.snd\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n -- Porting note : in the following two bullet points, `rfl` was not necessary\n[GOAL]\ncase e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n pullback.fst =\n pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b t \ud835\udcb0 f g j i \u226b pullback.fst \u226b pullback.fst\n[PROOFSTEP]\nrw [t_fst_fst, pullback.lift_fst, pullbackFst\u03b9ToV_snd]\n[GOAL]\ncase e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b pullback.snd = pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nrfl\n[GOAL]\ncase e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n pullback.snd =\n pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b t \ud835\udcb0 f g j i \u226b pullback.fst \u226b pullback.snd\n[PROOFSTEP]\nrw [t_fst_snd, pullback.lift_snd, pullbackFst\u03b9ToV_fst_assoc, pullback.condition_assoc]\n[GOAL]\ncase e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.snd \u226b OpenCover.map (GlueData.openCover (gluing \ud835\udcb0 f g)) j \u226b p2 \ud835\udcb0 f g = pullback.snd \u226b pullback.snd\n[PROOFSTEP]\nerw [Multicoequalizer.\u03c0_desc]\n[GOAL]\ncase e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.snd \u226b pullback.snd = pullback.snd \u226b pullback.snd\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j = pullback.fst \u226b pullback.fst\n[PROOFSTEP]\nrw [pullback.condition, \u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i) \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n pullback.snd \u226b OpenCover.map (GlueData.openCover (gluing \ud835\udcb0 f g)) j\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i = pullback.snd\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i) \u226b pullback.fst = pullback.snd \u226b pullback.fst\n[PROOFSTEP]\nsimp only [pullbackFst\u03b9ToV_fst]\n[GOAL]\ncase e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.snd \u226b pullback.fst = pullback.snd \u226b pullback.fst\n[PROOFSTEP]\nrfl\n[GOAL]\ncase e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i) \u226b pullback.snd = pullback.snd \u226b pullback.snd\n[PROOFSTEP]\nsimp only [pullbackFst\u03b9ToV_fst]\n[GOAL]\ncase e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.snd \u226b pullback.snd = pullback.snd \u226b pullback.snd\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 pullback (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) \u2245 pullback (OpenCover.map \ud835\udcb0 i \u226b f) g\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase hom\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 pullback (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) \u27f6 pullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ncase inv\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 pullback (OpenCover.map \ud835\udcb0 i \u226b f) g \u27f6 pullback (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i)\ncase hom_inv_id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 autoParam (?hom \u226b ?inv = \ud835\udfd9 (pullback (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i))) _auto\u271d\ncase inv_hom_id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 autoParam (?inv \u226b ?hom = \ud835\udfd9 (pullback (OpenCover.map \ud835\udcb0 i \u226b f) g)) _auto\u271d\n[PROOFSTEP]\nexact pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g) (by rw [\u2190 pullback.condition_assoc, Category.assoc, p_comm])\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g\n[PROOFSTEP]\nrw [\u2190 pullback.condition_assoc, Category.assoc, p_comm]\n[GOAL]\ncase inv\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 pullback (OpenCover.map \ud835\udcb0 i \u226b f) g \u27f6 pullback (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i)\ncase hom_inv_id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 autoParam\n (pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n ?inv =\n \ud835\udfd9 (pullback (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i)))\n _auto\u271d\ncase inv_hom_id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 autoParam\n (?inv \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) =\n \ud835\udfd9 (pullback (OpenCover.map \ud835\udcb0 i \u226b f) g))\n _auto\u271d\n[PROOFSTEP]\nrefine' pullback.lift ((gluing \ud835\udcb0 f g).\u03b9 i) pullback.fst (by erw [Multicoequalizer.\u03c0_desc])\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p1 \ud835\udcb0 f g = pullback.fst \u226b OpenCover.map \ud835\udcb0 i\n[PROOFSTEP]\nerw [Multicoequalizer.\u03c0_desc]\n[GOAL]\ncase hom_inv_id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 autoParam\n (pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n pullback.lift (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i) pullback.fst\n (_ : GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p1 \ud835\udcb0 f g = pullback.fst \u226b OpenCover.map \ud835\udcb0 i) =\n \ud835\udfd9 (pullback (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i)))\n _auto\u271d\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase hom_inv_id.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 (pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n pullback.lift (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i) pullback.fst\n (_ : GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p1 \ud835\udcb0 f g = pullback.fst \u226b OpenCover.map \ud835\udcb0 i)) \u226b\n pullback.fst =\n \ud835\udfd9 (pullback (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i)) \u226b pullback.fst\n[PROOFSTEP]\nsimpa using lift_comp_\u03b9 \ud835\udcb0 f g i\n[GOAL]\ncase hom_inv_id.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 (pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n pullback.lift (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i) pullback.fst\n (_ : GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p1 \ud835\udcb0 f g = pullback.fst \u226b OpenCover.map \ud835\udcb0 i)) \u226b\n pullback.snd =\n \ud835\udfd9 (pullback (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i)) \u226b pullback.snd\n[PROOFSTEP]\nsimp only [Category.assoc, pullback.lift_snd, pullback.lift_fst, Category.id_comp]\n[GOAL]\ncase inv_hom_id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 autoParam\n (pullback.lift (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i) pullback.fst\n (_ : GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p1 \ud835\udcb0 f g = pullback.fst \u226b OpenCover.map \ud835\udcb0 i) \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) =\n \ud835\udfd9 (pullback (OpenCover.map \ud835\udcb0 i \u226b f) g))\n _auto\u271d\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase inv_hom_id.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 (pullback.lift (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i) pullback.fst\n (_ : GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p1 \ud835\udcb0 f g = pullback.fst \u226b OpenCover.map \ud835\udcb0 i) \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g)) \u226b\n pullback.fst =\n \ud835\udfd9 (pullback (OpenCover.map \ud835\udcb0 i \u226b f) g) \u226b pullback.fst\n[PROOFSTEP]\nsimp only [Category.assoc, pullback.lift_fst, pullback.lift_snd, Category.id_comp]\n[GOAL]\ncase inv_hom_id.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 (pullback.lift (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i) pullback.fst\n (_ : GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p1 \ud835\udcb0 f g = pullback.fst \u226b OpenCover.map \ud835\udcb0 i) \u226b\n pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g)) \u226b\n pullback.snd =\n \ud835\udfd9 (pullback (OpenCover.map \ud835\udcb0 i \u226b f) g) \u226b pullback.snd\n[PROOFSTEP]\nsimp only [Category.assoc, pullback.lift_snd, pullback.lift_fst_assoc, Category.id_comp]\n[GOAL]\ncase inv_hom_id.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p2 \ud835\udcb0 f g = pullback.snd\n[PROOFSTEP]\nerw [Multicoequalizer.\u03c0_desc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 (pullbackP1Iso \ud835\udcb0 f g i).hom \u226b pullback.fst = pullback.snd\n[PROOFSTEP]\ndelta pullbackP1Iso\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 (Iso.mk\n (pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g))\n (pullback.lift (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i) pullback.fst\n (_ : GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p1 \ud835\udcb0 f g = pullback.fst \u226b OpenCover.map \ud835\udcb0 i))).hom \u226b\n pullback.fst =\n pullback.snd\n[PROOFSTEP]\nsimp only [pullback.lift_fst]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 (pullbackP1Iso \ud835\udcb0 f g i).hom \u226b pullback.snd = pullback.fst \u226b p2 \ud835\udcb0 f g\n[PROOFSTEP]\ndelta pullbackP1Iso\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 (Iso.mk\n (pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g))\n (pullback.lift (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i) pullback.fst\n (_ : GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p1 \ud835\udcb0 f g = pullback.fst \u226b OpenCover.map \ud835\udcb0 i))).hom \u226b\n pullback.snd =\n pullback.fst \u226b p2 \ud835\udcb0 f g\n[PROOFSTEP]\nsimp only [pullback.lift_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 (pullbackP1Iso \ud835\udcb0 f g i).inv \u226b pullback.fst = GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n[PROOFSTEP]\ndelta pullbackP1Iso\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 (Iso.mk\n (pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g))\n (pullback.lift (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i) pullback.fst\n (_ : GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p1 \ud835\udcb0 f g = pullback.fst \u226b OpenCover.map \ud835\udcb0 i))).inv \u226b\n pullback.fst =\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n[PROOFSTEP]\nsimp only [pullback.lift_fst]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 (pullbackP1Iso \ud835\udcb0 f g i).inv \u226b pullback.snd = pullback.fst\n[PROOFSTEP]\ndelta pullbackP1Iso\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 (Iso.mk\n (pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g))\n (pullback.lift (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i) pullback.fst\n (_ : GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p1 \ud835\udcb0 f g = pullback.fst \u226b OpenCover.map \ud835\udcb0 i))).inv \u226b\n pullback.snd =\n pullback.fst\n[PROOFSTEP]\nsimp only [pullback.lift_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 (pullbackP1Iso \ud835\udcb0 f g i).hom \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i = pullback.fst\n[PROOFSTEP]\nrw [\u2190 pullbackP1Iso_inv_fst, Iso.hom_inv_id_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\n\u22a2 IsLimit (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g))\n[PROOFSTEP]\napply PullbackCone.isLimitAux'\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\n\u22a2 (s : PullbackCone f g) \u2192\n { l //\n l \u226b PullbackCone.fst (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)) =\n PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)) =\n PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)) =\n PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)) =\n PullbackCone.snd s \u2192\n m = l }\n[PROOFSTEP]\nintro s\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\n\u22a2 { l //\n l \u226b PullbackCone.fst (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)) =\n PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)) =\n PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)) =\n PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)) =\n PullbackCone.snd s \u2192\n m = l }\n[PROOFSTEP]\nrefine' \u27e8gluedLift \ud835\udcb0 f g s, gluedLift_p1 \ud835\udcb0 f g s, gluedLift_p2 \ud835\udcb0 f g s, _\u27e9\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\n\u22a2 \u2200 {m : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)) =\n PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)) =\n PullbackCone.snd s \u2192\n m = gluedLift \ud835\udcb0 f g s\n[PROOFSTEP]\nintro m h\u2081 h\u2082\n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b PullbackCone.fst (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)) = PullbackCone.snd s\n\u22a2 m = gluedLift \ud835\udcb0 f g s\n[PROOFSTEP]\nchange m \u226b p1 \ud835\udcb0 f g = _ at h\u2081 \n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2082 : m \u226b PullbackCone.snd (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)) = PullbackCone.snd s\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\n\u22a2 m = gluedLift \ud835\udcb0 f g s\n[PROOFSTEP]\nchange m \u226b p2 \ud835\udcb0 f g = _ at h\u2082 \n[GOAL]\ncase create\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\nh\u2082 : m \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\n\u22a2 m = gluedLift \ud835\udcb0 f g s\n[PROOFSTEP]\napply (\ud835\udcb0.pullbackCover s.fst).hom_ext\n[GOAL]\ncase create.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\nh\u2082 : m \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\n\u22a2 \u2200 (x : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J),\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) x \u226b m =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) x \u226b gluedLift \ud835\udcb0 f g s\n[PROOFSTEP]\nintro i\n[GOAL]\ncase create.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\nh\u2082 : m \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\ni : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\n\u22a2 OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) i \u226b m =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)) i \u226b gluedLift \ud835\udcb0 f g s\n[PROOFSTEP]\nrw [OpenCover.pullbackCover_map]\n[GOAL]\ncase create.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\nh\u2082 : m \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\ni : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\n\u22a2 pullback.fst \u226b m = pullback.fst \u226b gluedLift \ud835\udcb0 f g s\n[PROOFSTEP]\nhave := pullbackRightPullbackFstIso (p1 \ud835\udcb0 f g) (\ud835\udcb0.map i) m \u226a\u226b pullback.congrHom h\u2081 rfl\n[GOAL]\ncase create.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\nh\u2082 : m \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\ni : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis : pullback m pullback.fst \u2245 pullback (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)\n\u22a2 pullback.fst \u226b m = pullback.fst \u226b gluedLift \ud835\udcb0 f g s\n[PROOFSTEP]\nerw [(\ud835\udcb0.pullbackCover s.fst).\u03b9_glueMorphisms]\n[GOAL]\ncase create.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\nh\u2082 : m \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\ni : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis : pullback m pullback.fst \u2245 pullback (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)\n\u22a2 pullback.fst \u226b m =\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n[PROOFSTEP]\nrw [\u2190 cancel_epi (pullbackRightPullbackFstIso (p1 \ud835\udcb0 f g) (\ud835\udcb0.map i) m \u226a\u226b pullback.congrHom h\u2081 rfl).hom, Iso.trans_hom,\n Category.assoc, pullback.congrHom_hom, pullback.lift_fst_assoc, Category.comp_id,\n pullbackRightPullbackFstIso_hom_fst_assoc, pullback.condition]\n[GOAL]\ncase create.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\nh\u2082 : m \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\ni : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis : pullback m pullback.fst \u2245 pullback (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)\n\u22a2 pullback.snd \u226b pullback.fst =\n ((pullbackRightPullbackFstIso (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) m).hom \u226b\n pullback.map (m \u226b p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 s.pt)\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i)) (\ud835\udfd9 X) (_ : (m \u226b p1 \ud835\udcb0 f g) \u226b \ud835\udfd9 X = \ud835\udfd9 s.pt \u226b PullbackCone.fst s)\n (_ : OpenCover.map \ud835\udcb0 i \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i)) \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n[PROOFSTEP]\ntrans pullback.snd \u226b (pullbackP1Iso \ud835\udcb0 f g _).hom \u226b (gluing \ud835\udcb0 f g).\u03b9 _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\nh\u2082 : m \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\ni : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis : pullback m pullback.fst \u2245 pullback (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)\n\u22a2 pullback.snd \u226b pullback.fst = pullback.snd \u226b (pullbackP1Iso \ud835\udcb0 f g i).hom \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\nh\u2082 : m \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\ni : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis : pullback m pullback.fst \u2245 pullback (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)\n\u22a2 pullback.fst = (pullbackP1Iso \ud835\udcb0 f g i).hom \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n[PROOFSTEP]\nrw [\u2190 pullbackP1Iso_hom_\u03b9]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\nh\u2082 : m \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\ni : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis : pullback m pullback.fst \u2245 pullback (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)\n\u22a2 pullback.snd \u226b (pullbackP1Iso \ud835\udcb0 f g i).hom \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n ((pullbackRightPullbackFstIso (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) m).hom \u226b\n pullback.map (m \u226b p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 s.pt)\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i)) (\ud835\udfd9 X) (_ : (m \u226b p1 \ud835\udcb0 f g) \u226b \ud835\udfd9 X = \ud835\udfd9 s.pt \u226b PullbackCone.fst s)\n (_ : OpenCover.map \ud835\udcb0 i \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i)) \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n[PROOFSTEP]\nsimp_rw [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\nh\u2082 : m \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\ni : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis : pullback m pullback.fst \u2245 pullback (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)\n\u22a2 (pullback.snd \u226b (pullbackP1Iso \ud835\udcb0 f g i).hom) \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n ((((pullbackRightPullbackFstIso (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) m).hom \u226b\n pullback.map (m \u226b p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 s.pt)\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i)) (\ud835\udfd9 X) (_ : (m \u226b p1 \ud835\udcb0 f g) \u226b \ud835\udfd9 X = \ud835\udfd9 s.pt \u226b PullbackCone.fst s)\n (_ : OpenCover.map \ud835\udcb0 i \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i)) \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom) \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\nh\u2082 : m \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\ni : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis : pullback m pullback.fst \u2245 pullback (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)\n\u22a2 pullback.snd \u226b (pullbackP1Iso \ud835\udcb0 f g i).hom =\n (((pullbackRightPullbackFstIso (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) m).hom \u226b\n pullback.map (m \u226b p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 s.pt)\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i)) (\ud835\udfd9 X) (_ : (m \u226b p1 \ud835\udcb0 f g) \u226b \ud835\udfd9 X = \ud835\udfd9 s.pt \u226b PullbackCone.fst s)\n (_ : OpenCover.map \ud835\udcb0 i \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i)) \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom) \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\nh\u2082 : m \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\ni : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis : pullback m pullback.fst \u2245 pullback (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)\n\u22a2 (pullback.snd \u226b (pullbackP1Iso \ud835\udcb0 f g i).hom) \u226b pullback.fst =\n ((((pullbackRightPullbackFstIso (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) m).hom \u226b\n pullback.map (m \u226b p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 s.pt)\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i)) (\ud835\udfd9 X) (_ : (m \u226b p1 \ud835\udcb0 f g) \u226b \ud835\udfd9 X = \ud835\udfd9 s.pt \u226b PullbackCone.fst s)\n (_ : OpenCover.map \ud835\udcb0 i \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i)) \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom) \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n pullback.fst\n[PROOFSTEP]\nsimp only [Category.comp_id, pullbackRightPullbackFstIso_hom_snd, Category.assoc, pullbackP1Iso_hom_fst,\n pullback.lift_snd, pullback.lift_fst, pullbackSymmetry_hom_comp_fst]\n[GOAL]\ncase e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\nh\u2082 : m \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\ni : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis : pullback m pullback.fst \u2245 pullback (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)\n\u22a2 (pullback.snd \u226b (pullbackP1Iso \ud835\udcb0 f g i).hom) \u226b pullback.snd =\n ((((pullbackRightPullbackFstIso (p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) m).hom \u226b\n pullback.map (m \u226b p1 \ud835\udcb0 f g) (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 s.pt)\n (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i)) (\ud835\udfd9 X) (_ : (m \u226b p1 \ud835\udcb0 f g) \u226b \ud835\udfd9 X = \ud835\udfd9 s.pt \u226b PullbackCone.fst s)\n (_ : OpenCover.map \ud835\udcb0 i \u226b \ud835\udfd9 X = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i)) \u226b\n (pullbackSymmetry (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)).hom) \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i) (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i \u226b f) g (\ud835\udfd9 (OpenCover.obj \ud835\udcb0 i))\n (PullbackCone.snd s) f (_ : OpenCover.map \ud835\udcb0 i \u226b f = \ud835\udfd9 (OpenCover.obj \ud835\udcb0 i) \u226b OpenCover.map \ud835\udcb0 i \u226b f)\n (_ : PullbackCone.fst s \u226b f = PullbackCone.snd s \u226b g)) \u226b\n pullback.snd\n[PROOFSTEP]\nsimp only [Category.comp_id, pullbackRightPullbackFstIso_hom_fst_assoc, pullbackP1Iso_hom_snd, Category.assoc,\n pullback.lift_fst_assoc, pullbackSymmetry_hom_comp_snd_assoc, pullback.lift_snd]\n[GOAL]\ncase e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns\u271d s : PullbackCone f g\nm : s.pt \u27f6 (PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g)).pt\nh\u2081 : m \u226b p1 \ud835\udcb0 f g = PullbackCone.fst s\nh\u2082 : m \u226b p2 \ud835\udcb0 f g = PullbackCone.snd s\ni : (OpenCover.pullbackCover \ud835\udcb0 (PullbackCone.fst s)).J\nthis : pullback m pullback.fst \u2245 pullback (PullbackCone.fst s) (OpenCover.map \ud835\udcb0 i)\n\u22a2 pullback.snd \u226b pullback.fst \u226b p2 \ud835\udcb0 f g = pullback.fst \u226b PullbackCone.snd s\n[PROOFSTEP]\nrw [\u2190 pullback.condition_assoc, h\u2082]\n[GOAL]\nC\u271d : Type u\ninst\u271d\u00b9 : Category.{v, u} C\u271d\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\nA B C : CommRingCat\nf : Spec.obj (Opposite.op A) \u27f6 Spec.obj (Opposite.op C)\ng : Spec.obj (Opposite.op B) \u27f6 Spec.obj (Opposite.op C)\n\u22a2 HasPullback f g\n[PROOFSTEP]\nrw [\u2190 Spec.image_preimage f, \u2190 Spec.image_preimage g]\n[GOAL]\nC\u271d : Type u\ninst\u271d\u00b9 : Category.{v, u} C\u271d\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\nA B C : CommRingCat\nf : Spec.obj (Opposite.op A) \u27f6 Spec.obj (Opposite.op C)\ng : Spec.obj (Opposite.op B) \u27f6 Spec.obj (Opposite.op C)\n\u22a2 HasPullback (Spec.map (Spec.preimage f)) (Spec.map (Spec.preimage g))\n[PROOFSTEP]\nexact \u27e8\u27e8\u27e8_, isLimitOfHasPullbackOfPreservesLimit Spec (Spec.preimage f) (Spec.preimage g)\u27e9\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (affineCover Z).J\n\u22a2 HasPullback (OpenCover.map (OpenCover.pullbackCover (affineCover Z) f) i \u226b f) g\n[PROOFSTEP]\nlet X\u1d62 := pullback f (Z.affineCover.map i)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (affineCover Z).J\nX\u1d62 : Scheme := pullback f (OpenCover.map (affineCover Z) i)\n\u22a2 HasPullback (OpenCover.map (OpenCover.pullbackCover (affineCover Z) f) i \u226b f) g\n[PROOFSTEP]\nlet Y\u1d62 := pullback g (Z.affineCover.map i)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (affineCover Z).J\nX\u1d62 : Scheme := pullback f (OpenCover.map (affineCover Z) i)\nY\u1d62 : Scheme := pullback g (OpenCover.map (affineCover Z) i)\n\u22a2 HasPullback (OpenCover.map (OpenCover.pullbackCover (affineCover Z) f) i \u226b f) g\n[PROOFSTEP]\nlet W := pullback (pullback.snd : Y\u1d62 \u27f6 _) (pullback.snd : X\u1d62 \u27f6 _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (affineCover Z).J\nX\u1d62 : Scheme := pullback f (OpenCover.map (affineCover Z) i)\nY\u1d62 : Scheme := pullback g (OpenCover.map (affineCover Z) i)\nW : Scheme := pullback pullback.snd pullback.snd\n\u22a2 HasPullback (OpenCover.map (OpenCover.pullbackCover (affineCover Z) f) i \u226b f) g\n[PROOFSTEP]\nhave :=\n bigSquareIsPullback (pullback.fst : W \u27f6 _) (pullback.fst : Y\u1d62 \u27f6 _) (pullback.snd : X\u1d62 \u27f6 _) (Z.affineCover.map i)\n pullback.snd pullback.snd g pullback.condition.symm pullback.condition.symm\n (PullbackCone.flipIsLimit <| pullbackIsPullback _ _) (PullbackCone.flipIsLimit <| pullbackIsPullback _ _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (affineCover Z).J\nX\u1d62 : Scheme := pullback f (OpenCover.map (affineCover Z) i)\nY\u1d62 : Scheme := pullback g (OpenCover.map (affineCover Z) i)\nW : Scheme := pullback pullback.snd pullback.snd\nthis :\n IsLimit\n (PullbackCone.mk pullback.snd (pullback.fst \u226b pullback.fst)\n (_ : pullback.snd \u226b pullback.snd \u226b OpenCover.map (affineCover Z) i = (pullback.fst \u226b pullback.fst) \u226b g))\n\u22a2 HasPullback (OpenCover.map (OpenCover.pullbackCover (affineCover Z) f) i \u226b f) g\n[PROOFSTEP]\nhave : HasPullback (pullback.snd \u226b Z.affineCover.map i : X\u1d62 \u27f6 _) g := \u27e8\u27e8\u27e8_, this\u27e9\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (affineCover Z).J\nX\u1d62 : Scheme := pullback f (OpenCover.map (affineCover Z) i)\nY\u1d62 : Scheme := pullback g (OpenCover.map (affineCover Z) i)\nW : Scheme := pullback pullback.snd pullback.snd\nthis\u271d :\n IsLimit\n (PullbackCone.mk pullback.snd (pullback.fst \u226b pullback.fst)\n (_ : pullback.snd \u226b pullback.snd \u226b OpenCover.map (affineCover Z) i = (pullback.fst \u226b pullback.fst) \u226b g))\nthis : HasPullback (pullback.snd \u226b OpenCover.map (affineCover Z) i) g\n\u22a2 HasPullback (OpenCover.map (OpenCover.pullbackCover (affineCover Z) f) i \u226b f) g\n[PROOFSTEP]\nrw [\u2190 pullback.condition] at this \n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (affineCover Z).J\nX\u1d62 : Scheme := pullback f (OpenCover.map (affineCover Z) i)\nY\u1d62 : Scheme := pullback g (OpenCover.map (affineCover Z) i)\nW : Scheme := pullback pullback.snd pullback.snd\nthis\u271d :\n IsLimit\n (PullbackCone.mk pullback.snd (pullback.fst \u226b pullback.fst)\n (_ : pullback.snd \u226b pullback.snd \u226b OpenCover.map (affineCover Z) i = (pullback.fst \u226b pullback.fst) \u226b g))\nthis : HasPullback (pullback.fst \u226b f) g\n\u22a2 HasPullback (OpenCover.map (OpenCover.pullbackCover (affineCover Z) f) i \u226b f) g\n[PROOFSTEP]\nexact this\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 OpenCover (pullback f g)\n[PROOFSTEP]\nfapply\n ((gluing \ud835\udcb0 f g).openCover.pushforwardIso (limit.isoLimitCone \u27e8_, gluedIsLimit \ud835\udcb0 f g\u27e9).inv).copy \ud835\udcb0.J\n (fun i => pullback (\ud835\udcb0.map i \u226b f) g)\n (fun i => pullback.map _ _ _ _ (\ud835\udcb0.map i) (\ud835\udfd9 _) (\ud835\udfd9 _) (Category.comp_id _) (by simp)) (Equiv.refl \ud835\udcb0.J) fun _ =>\n Iso.refl _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 \u2200 (i : \ud835\udcb0.J),\n pullback.map (OpenCover.map \ud835\udcb0 i \u226b f) g f g (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b f) (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) =\n (Iso.refl ((fun i => pullback (OpenCover.map \ud835\udcb0 i \u226b f) g) i)).hom \u226b\n OpenCover.map\n (OpenCover.pushforwardIso (GlueData.openCover (gluing \ud835\udcb0 f g))\n (limit.isoLimitCone\n { cone := PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g),\n isLimit := gluedIsLimit \ud835\udcb0 f g }).inv)\n (\u2191(Equiv.refl \ud835\udcb0.J) i)\n[PROOFSTEP]\nrintro (i : \ud835\udcb0.J)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map (OpenCover.map \ud835\udcb0 i \u226b f) g f g (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b f) (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) =\n (Iso.refl ((fun i => pullback (OpenCover.map \ud835\udcb0 i \u226b f) g) i)).hom \u226b\n OpenCover.map\n (OpenCover.pushforwardIso (GlueData.openCover (gluing \ud835\udcb0 f g))\n (limit.isoLimitCone\n { cone := PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g),\n isLimit := gluedIsLimit \ud835\udcb0 f g }).inv)\n (\u2191(Equiv.refl \ud835\udcb0.J) i)\n[PROOFSTEP]\nchange pullback.map _ _ _ _ _ _ _ _ _ = \ud835\udfd9 _ \u226b (gluing \ud835\udcb0 f g).\u03b9 i \u226b _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map (OpenCover.map \ud835\udcb0 i \u226b f) g f g (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b f) (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) =\n \ud835\udfd9 (GlueData.U (gluing \ud835\udcb0 f g).toGlueData i) \u226b\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b\n OpenCover.map\n (openCoverOfIsIso\n (limit.isoLimitCone\n { cone := PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g),\n isLimit := gluedIsLimit \ud835\udcb0 f g }).inv)\n (\u2191((Equiv.punitProd (GlueData.openCover (gluing \ud835\udcb0 f g)).J).symm.trans\n (Equiv.sigmaEquivProd PUnit (GlueData.openCover (gluing \ud835\udcb0 f g)).J).symm)\n (\u2191(Equiv.refl \ud835\udcb0.J) i)).fst\n[PROOFSTEP]\nrefine' Eq.trans _ (Category.id_comp _).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map (OpenCover.map \ud835\udcb0 i \u226b f) g f g (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b f) (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) =\n GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b\n OpenCover.map\n (openCoverOfIsIso\n (limit.isoLimitCone\n { cone := PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g),\n isLimit := gluedIsLimit \ud835\udcb0 f g }).inv)\n (\u2191((Equiv.punitProd (GlueData.openCover (gluing \ud835\udcb0 f g)).J).symm.trans\n (Equiv.sigmaEquivProd PUnit (GlueData.openCover (gluing \ud835\udcb0 f g)).J).symm)\n (\u2191(Equiv.refl \ud835\udcb0.J) i)).fst\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map (OpenCover.map \ud835\udcb0 i \u226b f) g f g (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b f) (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n pullback.fst =\n (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b\n OpenCover.map\n (openCoverOfIsIso\n (limit.isoLimitCone\n { cone := PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g),\n isLimit := gluedIsLimit \ud835\udcb0 f g }).inv)\n (\u2191((Equiv.punitProd (GlueData.openCover (gluing \ud835\udcb0 f g)).J).symm.trans\n (Equiv.sigmaEquivProd PUnit (GlueData.openCover (gluing \ud835\udcb0 f g)).J).symm)\n (\u2191(Equiv.refl \ud835\udcb0.J) i)).fst) \u226b\n pullback.fst\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map (OpenCover.map \ud835\udcb0 i \u226b f) g f g (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b f) (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n pullback.snd =\n (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b\n OpenCover.map\n (openCoverOfIsIso\n (limit.isoLimitCone\n { cone := PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g),\n isLimit := gluedIsLimit \ud835\udcb0 f g }).inv)\n (\u2191((Equiv.punitProd (GlueData.openCover (gluing \ud835\udcb0 f g)).J).symm.trans\n (Equiv.sigmaEquivProd PUnit (GlueData.openCover (gluing \ud835\udcb0 f g)).J).symm)\n (\u2191(Equiv.refl \ud835\udcb0.J) i)).fst) \u226b\n pullback.snd\n[PROOFSTEP]\nall_goals\n dsimp\n simp only [limit.isoLimitCone_inv_\u03c0, PullbackCone.mk_\u03c0_app_left, Category.comp_id, PullbackCone.mk_\u03c0_app_right,\n Category.assoc, pullback.lift_fst, pullback.lift_snd]\n symm\n exact Multicoequalizer.\u03c0_desc _ _ _ _ _\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map (OpenCover.map \ud835\udcb0 i \u226b f) g f g (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b f) (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n pullback.fst =\n (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b\n OpenCover.map\n (openCoverOfIsIso\n (limit.isoLimitCone\n { cone := PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g),\n isLimit := gluedIsLimit \ud835\udcb0 f g }).inv)\n (\u2191((Equiv.punitProd (GlueData.openCover (gluing \ud835\udcb0 f g)).J).symm.trans\n (Equiv.sigmaEquivProd PUnit (GlueData.openCover (gluing \ud835\udcb0 f g)).J).symm)\n (\u2191(Equiv.refl \ud835\udcb0.J) i)).fst) \u226b\n pullback.fst\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map (OpenCover.map \ud835\udcb0 i \u226b f) g f g (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b f) (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n pullback.fst =\n (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b\n (limit.isoLimitCone\n { cone := PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g),\n isLimit := gluedIsLimit \ud835\udcb0 f g }).inv) \u226b\n pullback.fst\n[PROOFSTEP]\nsimp only [limit.isoLimitCone_inv_\u03c0, PullbackCone.mk_\u03c0_app_left, Category.comp_id, PullbackCone.mk_\u03c0_app_right,\n Category.assoc, pullback.lift_fst, pullback.lift_snd]\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.fst \u226b OpenCover.map \ud835\udcb0 i = GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p1 \ud835\udcb0 f g\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p1 \ud835\udcb0 f g = pullback.fst \u226b OpenCover.map \ud835\udcb0 i\n[PROOFSTEP]\nexact Multicoequalizer.\u03c0_desc _ _ _ _ _\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map (OpenCover.map \ud835\udcb0 i \u226b f) g f g (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b f) (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n pullback.snd =\n (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b\n OpenCover.map\n (openCoverOfIsIso\n (limit.isoLimitCone\n { cone := PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g),\n isLimit := gluedIsLimit \ud835\udcb0 f g }).inv)\n (\u2191((Equiv.punitProd (GlueData.openCover (gluing \ud835\udcb0 f g)).J).symm.trans\n (Equiv.sigmaEquivProd PUnit (GlueData.openCover (gluing \ud835\udcb0 f g)).J).symm)\n (\u2191(Equiv.refl \ud835\udcb0.J) i)).fst) \u226b\n pullback.snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map (OpenCover.map \ud835\udcb0 i \u226b f) g f g (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b f) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b f) (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n pullback.snd =\n (GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b\n (limit.isoLimitCone\n { cone := PullbackCone.mk (p1 \ud835\udcb0 f g) (p2 \ud835\udcb0 f g) (_ : p1 \ud835\udcb0 f g \u226b f = p2 \ud835\udcb0 f g \u226b g),\n isLimit := gluedIsLimit \ud835\udcb0 f g }).inv) \u226b\n pullback.snd\n[PROOFSTEP]\nsimp only [limit.isoLimitCone_inv_\u03c0, PullbackCone.mk_\u03c0_app_left, Category.comp_id, PullbackCone.mk_\u03c0_app_right,\n Category.assoc, pullback.lift_fst, pullback.lift_snd]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.snd = GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p2 \ud835\udcb0 f g\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 GlueData.\u03b9 (gluing \ud835\udcb0 f g) i \u226b p2 \ud835\udcb0 f g = pullback.snd\n[PROOFSTEP]\nexact Multicoequalizer.\u03c0_desc _ _ _ _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 OpenCover (pullback f g)\n[PROOFSTEP]\nfapply\n ((openCoverOfLeft \ud835\udcb0 g f).pushforwardIso (pullbackSymmetry _ _).hom).copy \ud835\udcb0.J (fun i => pullback f (\ud835\udcb0.map i \u226b g))\n (fun i => pullback.map _ _ _ _ (\ud835\udfd9 _) (\ud835\udcb0.map i) (\ud835\udfd9 _) (by simp) (Category.comp_id _)) (Equiv.refl _) fun i =>\n pullbackSymmetry _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 \u2200 (i : \ud835\udcb0.J),\n pullback.map f (OpenCover.map \ud835\udcb0 i \u226b g) f g (\ud835\udfd9 X) (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b g) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b g) =\n (pullbackSymmetry f (OpenCover.map \ud835\udcb0 i \u226b g)).hom \u226b\n OpenCover.map (OpenCover.pushforwardIso (openCoverOfLeft \ud835\udcb0 g f) (pullbackSymmetry g f).hom)\n (\u2191(Equiv.refl \ud835\udcb0.J) i)\n[PROOFSTEP]\nintro i\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map f (OpenCover.map \ud835\udcb0 i \u226b g) f g (\ud835\udfd9 X) (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b g) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b g) =\n (pullbackSymmetry f (OpenCover.map \ud835\udcb0 i \u226b g)).hom \u226b\n OpenCover.map (OpenCover.pushforwardIso (openCoverOfLeft \ud835\udcb0 g f) (pullbackSymmetry g f).hom) (\u2191(Equiv.refl \ud835\udcb0.J) i)\n[PROOFSTEP]\ndsimp [OpenCover.bind]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map f (OpenCover.map \ud835\udcb0 i \u226b g) f g (\ud835\udfd9 X) (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b g) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b g) =\n (pullbackSymmetry f (OpenCover.map \ud835\udcb0 i \u226b g)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i \u226b g) f g f (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 X) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b g) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b g) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) \u226b\n (pullbackSymmetry g f).hom\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map f (OpenCover.map \ud835\udcb0 i \u226b g) f g (\ud835\udfd9 X) (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b g) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b g) \u226b\n pullback.fst =\n ((pullbackSymmetry f (OpenCover.map \ud835\udcb0 i \u226b g)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i \u226b g) f g f (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 X) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b g) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b g) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) \u226b\n (pullbackSymmetry g f).hom) \u226b\n pullback.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map f (OpenCover.map \ud835\udcb0 i \u226b g) f g (\ud835\udfd9 X) (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b g) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b g) \u226b\n pullback.snd =\n ((pullbackSymmetry f (OpenCover.map \ud835\udcb0 i \u226b g)).hom \u226b\n pullback.map (OpenCover.map \ud835\udcb0 i \u226b g) f g f (OpenCover.map \ud835\udcb0 i) (\ud835\udfd9 X) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0 i \u226b g) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0 i \u226b g) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) \u226b\n (pullbackSymmetry g f).hom) \u226b\n pullback.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0X : OpenCover X\n\ud835\udcb0Y : OpenCover Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 OpenCover (pullback f g)\n[PROOFSTEP]\nfapply\n ((openCoverOfLeft \ud835\udcb0X f g).bind fun x => openCoverOfRight \ud835\udcb0Y (\ud835\udcb0X.map x \u226b f) g).copy (\ud835\udcb0X.J \u00d7 \ud835\udcb0Y.J)\n (fun ij => pullback (\ud835\udcb0X.map ij.1 \u226b f) (\ud835\udcb0Y.map ij.2 \u226b g))\n (fun ij => pullback.map _ _ _ _ (\ud835\udcb0X.map ij.1) (\ud835\udcb0Y.map ij.2) (\ud835\udfd9 _) (Category.comp_id _) (Category.comp_id _))\n (Equiv.sigmaEquivProd _ _).symm fun _ => Iso.refl _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0X : OpenCover X\n\ud835\udcb0Y : OpenCover Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 \u2200 (i : \ud835\udcb0X.J \u00d7 \ud835\udcb0Y.J),\n pullback.map (OpenCover.map \ud835\udcb0X i.fst \u226b f) (OpenCover.map \ud835\udcb0Y i.snd \u226b g) f g (OpenCover.map \ud835\udcb0X i.fst)\n (OpenCover.map \ud835\udcb0Y i.snd) (\ud835\udfd9 Z) (_ : (OpenCover.map \ud835\udcb0X i.fst \u226b f) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0X i.fst \u226b f)\n (_ : (OpenCover.map \ud835\udcb0Y i.snd \u226b g) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0Y i.snd \u226b g) =\n (Iso.refl ((fun ij => pullback (OpenCover.map \ud835\udcb0X ij.fst \u226b f) (OpenCover.map \ud835\udcb0Y ij.snd \u226b g)) i)).hom \u226b\n OpenCover.map (OpenCover.bind (openCoverOfLeft \ud835\udcb0X f g) fun x => openCoverOfRight \ud835\udcb0Y (OpenCover.map \ud835\udcb0X x \u226b f) g)\n (\u2191(Equiv.sigmaEquivProd \ud835\udcb0X.J \ud835\udcb0Y.J).symm i)\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0X : OpenCover X\n\ud835\udcb0Y : OpenCover Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0X.J\nj : \ud835\udcb0Y.J\n\u22a2 pullback.map (OpenCover.map \ud835\udcb0X (i, j).fst \u226b f) (OpenCover.map \ud835\udcb0Y (i, j).snd \u226b g) f g (OpenCover.map \ud835\udcb0X (i, j).fst)\n (OpenCover.map \ud835\udcb0Y (i, j).snd) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0X (i, j).fst \u226b f) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0X (i, j).fst \u226b f)\n (_ : (OpenCover.map \ud835\udcb0Y (i, j).snd \u226b g) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0Y (i, j).snd \u226b g) =\n (Iso.refl ((fun ij => pullback (OpenCover.map \ud835\udcb0X ij.fst \u226b f) (OpenCover.map \ud835\udcb0Y ij.snd \u226b g)) (i, j))).hom \u226b\n OpenCover.map (OpenCover.bind (openCoverOfLeft \ud835\udcb0X f g) fun x => openCoverOfRight \ud835\udcb0Y (OpenCover.map \ud835\udcb0X x \u226b f) g)\n (\u2191(Equiv.sigmaEquivProd \ud835\udcb0X.J \ud835\udcb0Y.J).symm (i, j))\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase mk.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0X : OpenCover X\n\ud835\udcb0Y : OpenCover Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0X.J\nj : \ud835\udcb0Y.J\n\u22a2 pullback.map (OpenCover.map \ud835\udcb0X (i, j).fst \u226b f) (OpenCover.map \ud835\udcb0Y (i, j).snd \u226b g) f g (OpenCover.map \ud835\udcb0X (i, j).fst)\n (OpenCover.map \ud835\udcb0Y (i, j).snd) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0X (i, j).fst \u226b f) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0X (i, j).fst \u226b f)\n (_ : (OpenCover.map \ud835\udcb0Y (i, j).snd \u226b g) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0Y (i, j).snd \u226b g) \u226b\n pullback.fst =\n ((Iso.refl ((fun ij => pullback (OpenCover.map \ud835\udcb0X ij.fst \u226b f) (OpenCover.map \ud835\udcb0Y ij.snd \u226b g)) (i, j))).hom \u226b\n OpenCover.map (OpenCover.bind (openCoverOfLeft \ud835\udcb0X f g) fun x => openCoverOfRight \ud835\udcb0Y (OpenCover.map \ud835\udcb0X x \u226b f) g)\n (\u2191(Equiv.sigmaEquivProd \ud835\udcb0X.J \ud835\udcb0Y.J).symm (i, j))) \u226b\n pullback.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0X : OpenCover X\n\ud835\udcb0Y : OpenCover Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0X.J\nj : \ud835\udcb0Y.J\n\u22a2 pullback.map (OpenCover.map \ud835\udcb0X (i, j).fst \u226b f) (OpenCover.map \ud835\udcb0Y (i, j).snd \u226b g) f g (OpenCover.map \ud835\udcb0X (i, j).fst)\n (OpenCover.map \ud835\udcb0Y (i, j).snd) (\ud835\udfd9 Z)\n (_ : (OpenCover.map \ud835\udcb0X (i, j).fst \u226b f) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0X (i, j).fst \u226b f)\n (_ : (OpenCover.map \ud835\udcb0Y (i, j).snd \u226b g) \u226b \ud835\udfd9 Z = OpenCover.map \ud835\udcb0Y (i, j).snd \u226b g) \u226b\n pullback.snd =\n ((Iso.refl ((fun ij => pullback (OpenCover.map \ud835\udcb0X ij.fst \u226b f) (OpenCover.map \ud835\udcb0Y ij.snd \u226b g)) (i, j))).hom \u226b\n OpenCover.map (OpenCover.bind (openCoverOfLeft \ud835\udcb0X f g) fun x => openCoverOfRight \ud835\udcb0Y (OpenCover.map \ud835\udcb0X x \u226b f) g)\n (\u2191(Equiv.sigmaEquivProd \ud835\udcb0X.J \ud835\udcb0Y.J).symm (i, j))) \u226b\n pullback.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 OpenCover (pullback f g)\n[PROOFSTEP]\napply (openCoverOfLeft (\ud835\udcb0.pullbackCover f) f g).bind\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 (x : (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g).J) \u2192\n OpenCover (OpenCover.obj (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g) x)\n[PROOFSTEP]\nintro i\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g).J\n\u22a2 OpenCover (OpenCover.obj (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g) i)\n[PROOFSTEP]\nlet X\u1d62 := pullback f (\ud835\udcb0.map i)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g).J\nX\u1d62 : Scheme := pullback f (OpenCover.map \ud835\udcb0 i)\n\u22a2 OpenCover (OpenCover.obj (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g) i)\n[PROOFSTEP]\nlet Y\u1d62 := pullback g (\ud835\udcb0.map i)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g).J\nX\u1d62 : Scheme := pullback f (OpenCover.map \ud835\udcb0 i)\nY\u1d62 : Scheme := pullback g (OpenCover.map \ud835\udcb0 i)\n\u22a2 OpenCover (OpenCover.obj (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g) i)\n[PROOFSTEP]\nlet W := pullback (pullback.snd : Y\u1d62 \u27f6 _) (pullback.snd : X\u1d62 \u27f6 _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g).J\nX\u1d62 : Scheme := pullback f (OpenCover.map \ud835\udcb0 i)\nY\u1d62 : Scheme := pullback g (OpenCover.map \ud835\udcb0 i)\nW : Scheme := pullback pullback.snd pullback.snd\n\u22a2 OpenCover (OpenCover.obj (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g) i)\n[PROOFSTEP]\nhave :=\n bigSquareIsPullback (pullback.fst : W \u27f6 _) (pullback.fst : Y\u1d62 \u27f6 _) (pullback.snd : X\u1d62 \u27f6 _) (\ud835\udcb0.map i) pullback.snd\n pullback.snd g pullback.condition.symm pullback.condition.symm (PullbackCone.flipIsLimit <| pullbackIsPullback _ _)\n (PullbackCone.flipIsLimit <| pullbackIsPullback _ _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g).J\nX\u1d62 : Scheme := pullback f (OpenCover.map \ud835\udcb0 i)\nY\u1d62 : Scheme := pullback g (OpenCover.map \ud835\udcb0 i)\nW : Scheme := pullback pullback.snd pullback.snd\nthis :\n IsLimit\n (PullbackCone.mk pullback.snd (pullback.fst \u226b pullback.fst)\n (_ : pullback.snd \u226b pullback.snd \u226b OpenCover.map \ud835\udcb0 i = (pullback.fst \u226b pullback.fst) \u226b g))\n\u22a2 OpenCover (OpenCover.obj (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g) i)\n[PROOFSTEP]\nrefine'\n @openCoverOfIsIso (f :=\n (pullbackSymmetry _ _).hom \u226b (limit.isoLimitCone \u27e8_, this\u27e9).inv \u226b pullback.map _ _ _ _ (\ud835\udfd9 _) (\ud835\udfd9 _) (\ud835\udfd9 _) _ _) ?_\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g).J\nX\u1d62 : Scheme := pullback f (OpenCover.map \ud835\udcb0 i)\nY\u1d62 : Scheme := pullback g (OpenCover.map \ud835\udcb0 i)\nW : Scheme := pullback pullback.snd pullback.snd\nthis :\n IsLimit\n (PullbackCone.mk pullback.snd (pullback.fst \u226b pullback.fst)\n (_ : pullback.snd \u226b pullback.snd \u226b OpenCover.map \ud835\udcb0 i = (pullback.fst \u226b pullback.fst) \u226b g))\n\u22a2 (pullback.snd \u226b OpenCover.map \ud835\udcb0 i) \u226b \ud835\udfd9 Z =\n \ud835\udfd9 (pullback f (OpenCover.map \ud835\udcb0 i)) \u226b OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 f) i \u226b f\n[PROOFSTEP]\nsimp only [Category.comp_id, Category.id_comp, \u2190 pullback.condition]\n -- Porting note : `simpa` failed, but this is indeed `rfl`\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g).J\nX\u1d62 : Scheme := pullback f (OpenCover.map \ud835\udcb0 i)\nY\u1d62 : Scheme := pullback g (OpenCover.map \ud835\udcb0 i)\nW : Scheme := pullback pullback.snd pullback.snd\nthis :\n IsLimit\n (PullbackCone.mk pullback.snd (pullback.fst \u226b pullback.fst)\n (_ : pullback.snd \u226b pullback.snd \u226b OpenCover.map \ud835\udcb0 i = (pullback.fst \u226b pullback.fst) \u226b g))\n\u22a2 pullback.fst \u226b f = OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 f) i \u226b f\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g).J\nX\u1d62 : Scheme := pullback f (OpenCover.map \ud835\udcb0 i)\nY\u1d62 : Scheme := pullback g (OpenCover.map \ud835\udcb0 i)\nW : Scheme := pullback pullback.snd pullback.snd\nthis :\n IsLimit\n (PullbackCone.mk pullback.snd (pullback.fst \u226b pullback.fst)\n (_ : pullback.snd \u226b pullback.snd \u226b OpenCover.map \ud835\udcb0 i = (pullback.fst \u226b pullback.fst) \u226b g))\n\u22a2 g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g\n[PROOFSTEP]\nsimp only [Category.comp_id, Category.id_comp]\n -- Porting note : this `IsIso` instance was `inferInstance`\n[GOAL]\ncase refine'_3\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : (openCoverOfLeft (OpenCover.pullbackCover \ud835\udcb0 f) f g).J\nX\u1d62 : Scheme := pullback f (OpenCover.map \ud835\udcb0 i)\nY\u1d62 : Scheme := pullback g (OpenCover.map \ud835\udcb0 i)\nW : Scheme := pullback pullback.snd pullback.snd\nthis :\n IsLimit\n (PullbackCone.mk pullback.snd (pullback.fst \u226b pullback.fst)\n (_ : pullback.snd \u226b pullback.snd \u226b OpenCover.map \ud835\udcb0 i = (pullback.fst \u226b pullback.fst) \u226b g))\n\u22a2 IsIso\n ((pullbackSymmetry pullback.snd pullback.snd).hom \u226b\n (limit.isoLimitCone\n {\n cone :=\n PullbackCone.mk pullback.snd (pullback.fst \u226b pullback.fst)\n (_ : pullback.snd \u226b pullback.snd \u226b OpenCover.map \ud835\udcb0 i = (pullback.fst \u226b pullback.fst) \u226b g),\n isLimit := this }).inv \u226b\n pullback.map (pullback.snd \u226b OpenCover.map \ud835\udcb0 i) g (OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 f) i \u226b f) g\n (\ud835\udfd9 (pullback f (OpenCover.map \ud835\udcb0 i))) (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (pullback.snd \u226b OpenCover.map \ud835\udcb0 i) \u226b \ud835\udfd9 Z =\n \ud835\udfd9 (pullback f (OpenCover.map \ud835\udcb0 i)) \u226b OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 f) i \u226b f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g))\n[PROOFSTEP]\napply IsIso.comp_isIso\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 OpenCover (pullback f g)\n[PROOFSTEP]\napply\n (openCoverOfBase'.{u, u} \ud835\udcb0 f g).copy \ud835\udcb0.J\n (fun i => pullback (pullback.snd : pullback f (\ud835\udcb0.map i) \u27f6 _) (pullback.snd : pullback g (\ud835\udcb0.map i) \u27f6 _))\n (fun i => pullback.map _ _ _ _ pullback.fst pullback.fst (\ud835\udcb0.map i) pullback.condition.symm pullback.condition.symm)\n ((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) fun _ => Iso.refl _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 \u2200 (i : \ud835\udcb0.J),\n pullback.map pullback.snd pullback.snd f g pullback.fst pullback.fst (OpenCover.map \ud835\udcb0 i)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i = pullback.fst \u226b f)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i = pullback.fst \u226b g) =\n (Iso.refl ((fun i => pullback pullback.snd pullback.snd) i)).hom \u226b\n OpenCover.map (openCoverOfBase' \ud835\udcb0 f g)\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i)\n[PROOFSTEP]\nintro i\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map pullback.snd pullback.snd f g pullback.fst pullback.fst (OpenCover.map \ud835\udcb0 i)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i = pullback.fst \u226b f)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i = pullback.fst \u226b g) =\n (Iso.refl ((fun i => pullback pullback.snd pullback.snd) i)).hom \u226b\n OpenCover.map (openCoverOfBase' \ud835\udcb0 f g)\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i)\n[PROOFSTEP]\nrw [Iso.refl_hom, Category.id_comp, openCoverOfBase'_map]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map pullback.snd pullback.snd f g pullback.fst pullback.fst (OpenCover.map \ud835\udcb0 i)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i = pullback.fst \u226b f)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i = pullback.fst \u226b g) =\n (pullbackSymmetry pullback.snd pullback.snd).hom \u226b\n (limit.isoLimitCone\n {\n cone :=\n PullbackCone.mk pullback.snd (pullback.fst \u226b pullback.fst)\n (_ :\n (pullback.snd \u226b pullback.snd) \u226b\n OpenCover.map \ud835\udcb0\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst =\n (pullback.fst \u226b pullback.fst) \u226b g),\n isLimit :=\n bigSquareIsPullback pullback.fst pullback.fst pullback.snd\n (OpenCover.map \ud835\udcb0 (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst)\n pullback.snd pullback.snd g (_ : pullback.snd \u226b pullback.snd = pullback.fst \u226b pullback.snd)\n (_ :\n pullback.snd \u226b\n OpenCover.map \ud835\udcb0\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst =\n pullback.fst \u226b g)\n (PullbackCone.flipIsLimit\n (pullbackIsPullback g\n (OpenCover.map \ud835\udcb0\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst)))\n (PullbackCone.flipIsLimit (pullbackIsPullback pullback.snd pullback.snd)) }).inv \u226b\n pullback.map\n (pullback.snd \u226b\n OpenCover.map \ud835\udcb0 (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst)\n g (pullback.fst \u226b f) g\n (\ud835\udfd9\n (pullback f\n (OpenCover.map \ud835\udcb0 (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst)))\n (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (pullback.snd \u226b\n OpenCover.map \ud835\udcb0 (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst) \u226b\n \ud835\udfd9 Z =\n \ud835\udfd9\n (pullback f\n (OpenCover.map \ud835\udcb0\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst)) \u226b\n pullback.fst \u226b f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n pullback.map (pullback.fst \u226b f) g f g pullback.fst (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 f)\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst \u226b\n f) \u226b\n \ud835\udfd9 Z =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 f)\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst \u226b\n f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g)\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map pullback.snd pullback.snd f g pullback.fst pullback.fst (OpenCover.map \ud835\udcb0 i)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i = pullback.fst \u226b f)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i = pullback.fst \u226b g) \u226b\n pullback.fst =\n ((pullbackSymmetry pullback.snd pullback.snd).hom \u226b\n (limit.isoLimitCone\n {\n cone :=\n PullbackCone.mk pullback.snd (pullback.fst \u226b pullback.fst)\n (_ :\n (pullback.snd \u226b pullback.snd) \u226b\n OpenCover.map \ud835\udcb0\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst =\n (pullback.fst \u226b pullback.fst) \u226b g),\n isLimit :=\n bigSquareIsPullback pullback.fst pullback.fst pullback.snd\n (OpenCover.map \ud835\udcb0 (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst)\n pullback.snd pullback.snd g (_ : pullback.snd \u226b pullback.snd = pullback.fst \u226b pullback.snd)\n (_ :\n pullback.snd \u226b\n OpenCover.map \ud835\udcb0\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst =\n pullback.fst \u226b g)\n (PullbackCone.flipIsLimit\n (pullbackIsPullback g\n (OpenCover.map \ud835\udcb0\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst)))\n (PullbackCone.flipIsLimit (pullbackIsPullback pullback.snd pullback.snd)) }).inv \u226b\n pullback.map\n (pullback.snd \u226b\n OpenCover.map \ud835\udcb0 (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst)\n g (pullback.fst \u226b f) g\n (\ud835\udfd9\n (pullback f\n (OpenCover.map \ud835\udcb0 (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst)))\n (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (pullback.snd \u226b\n OpenCover.map \ud835\udcb0\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst) \u226b\n \ud835\udfd9 Z =\n \ud835\udfd9\n (pullback f\n (OpenCover.map \ud835\udcb0\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst)) \u226b\n pullback.fst \u226b f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n pullback.map (pullback.fst \u226b f) g f g pullback.fst (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 f)\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst \u226b\n f) \u226b\n \ud835\udfd9 Z =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 f)\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst \u226b\n f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g)) \u226b\n pullback.fst\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map pullback.snd pullback.snd f g pullback.fst pullback.fst (OpenCover.map \ud835\udcb0 i)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i = pullback.fst \u226b f)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i = pullback.fst \u226b g) \u226b\n pullback.snd =\n ((pullbackSymmetry pullback.snd pullback.snd).hom \u226b\n (limit.isoLimitCone\n {\n cone :=\n PullbackCone.mk pullback.snd (pullback.fst \u226b pullback.fst)\n (_ :\n (pullback.snd \u226b pullback.snd) \u226b\n OpenCover.map \ud835\udcb0\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst =\n (pullback.fst \u226b pullback.fst) \u226b g),\n isLimit :=\n bigSquareIsPullback pullback.fst pullback.fst pullback.snd\n (OpenCover.map \ud835\udcb0 (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst)\n pullback.snd pullback.snd g (_ : pullback.snd \u226b pullback.snd = pullback.fst \u226b pullback.snd)\n (_ :\n pullback.snd \u226b\n OpenCover.map \ud835\udcb0\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst =\n pullback.fst \u226b g)\n (PullbackCone.flipIsLimit\n (pullbackIsPullback g\n (OpenCover.map \ud835\udcb0\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst)))\n (PullbackCone.flipIsLimit (pullbackIsPullback pullback.snd pullback.snd)) }).inv \u226b\n pullback.map\n (pullback.snd \u226b\n OpenCover.map \ud835\udcb0 (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst)\n g (pullback.fst \u226b f) g\n (\ud835\udfd9\n (pullback f\n (OpenCover.map \ud835\udcb0 (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst)))\n (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (pullback.snd \u226b\n OpenCover.map \ud835\udcb0\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst) \u226b\n \ud835\udfd9 Z =\n \ud835\udfd9\n (pullback f\n (OpenCover.map \ud835\udcb0\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst)) \u226b\n pullback.fst \u226b f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n pullback.map (pullback.fst \u226b f) g f g pullback.fst (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 f)\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst \u226b\n f) \u226b\n \ud835\udfd9 Z =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 f)\n (\u2191((Equiv.prodPUnit \ud835\udcb0.J).symm.trans (Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm) i).fst \u226b\n f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g)) \u226b\n pullback.snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map pullback.snd pullback.snd f g pullback.fst pullback.fst (OpenCover.map \ud835\udcb0 i)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i = pullback.fst \u226b f)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i = pullback.fst \u226b g) \u226b\n pullback.fst =\n ((pullbackSymmetry pullback.snd pullback.snd).hom \u226b\n (limit.isoLimitCone\n {\n cone :=\n PullbackCone.mk pullback.snd (pullback.fst \u226b pullback.fst)\n (_ :\n (pullback.snd \u226b pullback.snd) \u226b\n OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst =\n (pullback.fst \u226b pullback.fst) \u226b g),\n isLimit :=\n bigSquareIsPullback pullback.fst pullback.fst pullback.snd\n (OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst) pullback.snd\n pullback.snd g (_ : pullback.snd \u226b pullback.snd = pullback.fst \u226b pullback.snd)\n (_ :\n pullback.snd \u226b OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst =\n pullback.fst \u226b g)\n (PullbackCone.flipIsLimit\n (pullbackIsPullback g\n (OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst)))\n (PullbackCone.flipIsLimit (pullbackIsPullback pullback.snd pullback.snd)) }).inv \u226b\n pullback.map (pullback.snd \u226b OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst) g\n (pullback.fst \u226b f) g\n (\ud835\udfd9 (pullback f (OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst))) (\ud835\udfd9 Y)\n (\ud835\udfd9 Z)\n (_ :\n (pullback.snd \u226b OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst) \u226b \ud835\udfd9 Z =\n \ud835\udfd9 (pullback f (OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst)) \u226b\n pullback.fst \u226b f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n pullback.map (pullback.fst \u226b f) g f g pullback.fst (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 f)\n (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst \u226b\n f) \u226b\n \ud835\udfd9 Z =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 f)\n (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst \u226b\n f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g)) \u226b\n pullback.fst\n[PROOFSTEP]\nsimp only [limit.lift_\u03c0, PullbackCone.mk_pt, PullbackCone.mk_\u03c0_app, Category.assoc, limit.lift_\u03c0_assoc, cospan_left,\n Category.comp_id, limit.isoLimitCone_inv_\u03c0, limit.isoLimitCone_inv_\u03c0_assoc, pullbackSymmetry_hom_comp_fst_assoc,\n pullbackSymmetry_hom_comp_snd_assoc]\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.fst \u226b pullback.fst = pullback.fst \u226b pullback.fst\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.map pullback.snd pullback.snd f g pullback.fst pullback.fst (OpenCover.map \ud835\udcb0 i)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i = pullback.fst \u226b f)\n (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i = pullback.fst \u226b g) \u226b\n pullback.snd =\n ((pullbackSymmetry pullback.snd pullback.snd).hom \u226b\n (limit.isoLimitCone\n {\n cone :=\n PullbackCone.mk pullback.snd (pullback.fst \u226b pullback.fst)\n (_ :\n (pullback.snd \u226b pullback.snd) \u226b\n OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst =\n (pullback.fst \u226b pullback.fst) \u226b g),\n isLimit :=\n bigSquareIsPullback pullback.fst pullback.fst pullback.snd\n (OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst) pullback.snd\n pullback.snd g (_ : pullback.snd \u226b pullback.snd = pullback.fst \u226b pullback.snd)\n (_ :\n pullback.snd \u226b OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst =\n pullback.fst \u226b g)\n (PullbackCone.flipIsLimit\n (pullbackIsPullback g\n (OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst)))\n (PullbackCone.flipIsLimit (pullbackIsPullback pullback.snd pullback.snd)) }).inv \u226b\n pullback.map (pullback.snd \u226b OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst) g\n (pullback.fst \u226b f) g\n (\ud835\udfd9 (pullback f (OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst))) (\ud835\udfd9 Y)\n (\ud835\udfd9 Z)\n (_ :\n (pullback.snd \u226b OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst) \u226b \ud835\udfd9 Z =\n \ud835\udfd9 (pullback f (OpenCover.map \ud835\udcb0 (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst)) \u226b\n pullback.fst \u226b f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g) \u226b\n pullback.map (pullback.fst \u226b f) g f g pullback.fst (\ud835\udfd9 Y) (\ud835\udfd9 Z)\n (_ :\n (OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 f)\n (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst \u226b\n f) \u226b\n \ud835\udfd9 Z =\n OpenCover.map (OpenCover.pullbackCover \ud835\udcb0 f)\n (\u2191(Equiv.sigmaEquivProd \ud835\udcb0.J PUnit).symm (i, PUnit.unit)).fst \u226b\n f)\n (_ : g \u226b \ud835\udfd9 Z = \ud835\udfd9 Y \u226b g)) \u226b\n pullback.snd\n[PROOFSTEP]\nsimp only [limit.lift_\u03c0, PullbackCone.mk_pt, PullbackCone.mk_\u03c0_app, Category.assoc, limit.lift_\u03c0_assoc, cospan_left,\n Category.comp_id, limit.isoLimitCone_inv_\u03c0, limit.isoLimitCone_inv_\u03c0_assoc, pullbackSymmetry_hom_comp_fst_assoc,\n pullbackSymmetry_hom_comp_snd_assoc]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0\u271d.J), HasPullback (OpenCover.map \ud835\udcb0\u271d i \u226b f\u271d) g\u271d\ns : PullbackCone f\u271d g\u271d\n\ud835\udcb0 : OpenCover Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\ni : \ud835\udcb0.J\n\u22a2 pullback.snd \u226b pullback.fst = pullback.snd \u226b pullback.fst\n[PROOFSTEP]\nrfl\n[GOAL]\nX Y S X' Y' S' : Scheme\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : X' \u27f6 S'\ng' : Y' \u27f6 S'\ni\u2081 : X \u27f6 X'\ni\u2082 : Y \u27f6 Y'\ni\u2083 : S \u27f6 S'\ne\u2081 : f \u226b i\u2083 = i\u2081 \u226b f'\ne\u2082 : g \u226b i\u2083 = i\u2082 \u226b g'\ninst\u271d\u00b2 : IsOpenImmersion i\u2081\ninst\u271d\u00b9 : IsOpenImmersion i\u2082\ninst\u271d : Mono i\u2083\n\u22a2 IsOpenImmersion (pullback.map f g f' g' i\u2081 i\u2082 i\u2083 e\u2081 e\u2082)\n[PROOFSTEP]\nrw [pullback_map_eq_pullbackFstFstIso_inv]\n -- Porting note : was automatic\n[GOAL]\nX Y S X' Y' S' : Scheme\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : X' \u27f6 S'\ng' : Y' \u27f6 S'\ni\u2081 : X \u27f6 X'\ni\u2082 : Y \u27f6 Y'\ni\u2083 : S \u27f6 S'\ne\u2081 : f \u226b i\u2083 = i\u2081 \u226b f'\ne\u2082 : g \u226b i\u2083 = i\u2082 \u226b g'\ninst\u271d\u00b2 : IsOpenImmersion i\u2081\ninst\u271d\u00b9 : IsOpenImmersion i\u2082\ninst\u271d : Mono i\u2083\n\u22a2 IsOpenImmersion ((pullbackFstFstIso f g f' g' i\u2081 i\u2082 i\u2083 e\u2081 e\u2082).inv \u226b pullback.snd \u226b pullback.fst)\n[PROOFSTEP]\nexact PresheafedSpace.IsOpenImmersion.comp _ (hg := PresheafedSpace.IsOpenImmersion.comp _ _)\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.Pullbacks", "llama_tokens": 109064, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7634837635542925, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.41742570195593176}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\n\u22a2 DifferentiableWithinAt \ud835\udd5c (\u2191iso \u2218 f) s x \u2194 DifferentiableWithinAt \ud835\udd5c f s x\n[PROOFSTEP]\nrefine' \u27e8fun H => _, fun H => iso.differentiable.differentiableAt.comp_differentiableWithinAt x H\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\nH : DifferentiableWithinAt \ud835\udd5c (\u2191iso \u2218 f) s x\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s x\n[PROOFSTEP]\nhave : DifferentiableWithinAt \ud835\udd5c (iso.symm \u2218 iso \u2218 f) s x :=\n iso.symm.differentiable.differentiableAt.comp_differentiableWithinAt x H\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\nH : DifferentiableWithinAt \ud835\udd5c (\u2191iso \u2218 f) s x\nthis : DifferentiableWithinAt \ud835\udd5c (\u2191(ContinuousLinearEquiv.symm iso) \u2218 \u2191iso \u2218 f) s x\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s x\n[PROOFSTEP]\nrwa [\u2190 Function.comp.assoc iso.symm iso f, iso.symm_comp_self] at this \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\nx : G\n\u22a2 DifferentiableAt \ud835\udd5c (\u2191iso \u2218 f) x \u2194 DifferentiableAt \ud835\udd5c f x\n[PROOFSTEP]\nrw [\u2190 differentiableWithinAt_univ, \u2190 differentiableWithinAt_univ, iso.comp_differentiableWithinAt_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\n\u22a2 DifferentiableOn \ud835\udd5c (\u2191iso \u2218 f) s \u2194 DifferentiableOn \ud835\udd5c f s\n[PROOFSTEP]\nrw [DifferentiableOn, DifferentiableOn]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\n\u22a2 (\u2200 (x : G), x \u2208 s \u2192 DifferentiableWithinAt \ud835\udd5c (\u2191iso \u2218 f) s x) \u2194 \u2200 (x : G), x \u2208 s \u2192 DifferentiableWithinAt \ud835\udd5c f s x\n[PROOFSTEP]\nsimp only [iso.comp_differentiableWithinAt_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\n\u22a2 Differentiable \ud835\udd5c (\u2191iso \u2218 f) \u2194 Differentiable \ud835\udd5c f\n[PROOFSTEP]\nrw [\u2190 differentiableOn_univ, \u2190 differentiableOn_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\n\u22a2 DifferentiableOn \ud835\udd5c (\u2191iso \u2218 f) univ \u2194 DifferentiableOn \ud835\udd5c f univ\n[PROOFSTEP]\nexact iso.comp_differentiableOn_iff\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\nf' : G \u2192L[\ud835\udd5c] E\n\u22a2 HasFDerivWithinAt (\u2191iso \u2218 f) (comp (\u2191iso) f') s x \u2194 HasFDerivWithinAt f f' s x\n[PROOFSTEP]\nrefine' \u27e8fun H => _, fun H => iso.hasFDerivAt.comp_hasFDerivWithinAt x H\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\nf' : G \u2192L[\ud835\udd5c] E\nH : HasFDerivWithinAt (\u2191iso \u2218 f) (comp (\u2191iso) f') s x\n\u22a2 HasFDerivWithinAt f f' s x\n[PROOFSTEP]\nhave A : f = iso.symm \u2218 iso \u2218 f := by\n rw [\u2190 Function.comp.assoc, iso.symm_comp_self]\n rfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\nf' : G \u2192L[\ud835\udd5c] E\nH : HasFDerivWithinAt (\u2191iso \u2218 f) (comp (\u2191iso) f') s x\n\u22a2 f = \u2191(ContinuousLinearEquiv.symm iso) \u2218 \u2191iso \u2218 f\n[PROOFSTEP]\nrw [\u2190 Function.comp.assoc, iso.symm_comp_self]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\nf' : G \u2192L[\ud835\udd5c] E\nH : HasFDerivWithinAt (\u2191iso \u2218 f) (comp (\u2191iso) f') s x\n\u22a2 f = _root_.id \u2218 f\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\nf' : G \u2192L[\ud835\udd5c] E\nH : HasFDerivWithinAt (\u2191iso \u2218 f) (comp (\u2191iso) f') s x\nA : f = \u2191(ContinuousLinearEquiv.symm iso) \u2218 \u2191iso \u2218 f\n\u22a2 HasFDerivWithinAt f f' s x\n[PROOFSTEP]\nhave B : f' = (iso.symm : F \u2192L[\ud835\udd5c] E).comp ((iso : E \u2192L[\ud835\udd5c] F).comp f') := by\n rw [\u2190 ContinuousLinearMap.comp_assoc, iso.coe_symm_comp_coe, ContinuousLinearMap.id_comp]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\nf' : G \u2192L[\ud835\udd5c] E\nH : HasFDerivWithinAt (\u2191iso \u2218 f) (comp (\u2191iso) f') s x\nA : f = \u2191(ContinuousLinearEquiv.symm iso) \u2218 \u2191iso \u2218 f\n\u22a2 f' = comp (\u2191(ContinuousLinearEquiv.symm iso)) (comp (\u2191iso) f')\n[PROOFSTEP]\nrw [\u2190 ContinuousLinearMap.comp_assoc, iso.coe_symm_comp_coe, ContinuousLinearMap.id_comp]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\nf' : G \u2192L[\ud835\udd5c] E\nH : HasFDerivWithinAt (\u2191iso \u2218 f) (comp (\u2191iso) f') s x\nA : f = \u2191(ContinuousLinearEquiv.symm iso) \u2218 \u2191iso \u2218 f\nB : f' = comp (\u2191(ContinuousLinearEquiv.symm iso)) (comp (\u2191iso) f')\n\u22a2 HasFDerivWithinAt f f' s x\n[PROOFSTEP]\nrw [A, B]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\nf' : G \u2192L[\ud835\udd5c] E\nH : HasFDerivWithinAt (\u2191iso \u2218 f) (comp (\u2191iso) f') s x\nA : f = \u2191(ContinuousLinearEquiv.symm iso) \u2218 \u2191iso \u2218 f\nB : f' = comp (\u2191(ContinuousLinearEquiv.symm iso)) (comp (\u2191iso) f')\n\u22a2 HasFDerivWithinAt (\u2191(ContinuousLinearEquiv.symm iso) \u2218 \u2191iso \u2218 f)\n (comp (\u2191(ContinuousLinearEquiv.symm iso)) (comp (\u2191iso) f')) s x\n[PROOFSTEP]\nexact iso.symm.hasFDerivAt.comp_hasFDerivWithinAt x H\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\nx : G\nf' : G \u2192L[\ud835\udd5c] E\n\u22a2 HasStrictFDerivAt (\u2191iso \u2218 f) (comp (\u2191iso) f') x \u2194 HasStrictFDerivAt f f' x\n[PROOFSTEP]\nrefine' \u27e8fun H => _, fun H => iso.hasStrictFDerivAt.comp x H\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\nx : G\nf' : G \u2192L[\ud835\udd5c] E\nH : HasStrictFDerivAt (\u2191iso \u2218 f) (comp (\u2191iso) f') x\n\u22a2 HasStrictFDerivAt f f' x\n[PROOFSTEP]\nconvert iso.symm.hasStrictFDerivAt.comp x H using 1\n[GOAL]\ncase h.e'_9\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\nx : G\nf' : G \u2192L[\ud835\udd5c] E\nH : HasStrictFDerivAt (\u2191iso \u2218 f) (comp (\u2191iso) f') x\n\u22a2 f = fun x => \u2191(ContinuousLinearEquiv.symm iso) ((\u2191iso \u2218 f) x)\n[PROOFSTEP]\next z\n[GOAL]\ncase h.e'_10\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\nx : G\nf' : G \u2192L[\ud835\udd5c] E\nH : HasStrictFDerivAt (\u2191iso \u2218 f) (comp (\u2191iso) f') x\n\u22a2 f' = comp (\u2191(ContinuousLinearEquiv.symm iso)) (comp (\u2191iso) f')\n[PROOFSTEP]\next z\n[GOAL]\ncase h.e'_9.h\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\nx : G\nf' : G \u2192L[\ud835\udd5c] E\nH : HasStrictFDerivAt (\u2191iso \u2218 f) (comp (\u2191iso) f') x\nz : G\n\u22a2 f z = \u2191(ContinuousLinearEquiv.symm iso) ((\u2191iso \u2218 f) z)\n[PROOFSTEP]\napply (iso.symm_apply_apply _).symm\n[GOAL]\ncase h.e'_10.h\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\nx : G\nf' : G \u2192L[\ud835\udd5c] E\nH : HasStrictFDerivAt (\u2191iso \u2218 f) (comp (\u2191iso) f') x\nz : G\n\u22a2 \u2191f' z = \u2191(comp (\u2191(ContinuousLinearEquiv.symm iso)) (comp (\u2191iso) f')) z\n[PROOFSTEP]\napply (iso.symm_apply_apply _).symm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\nx : G\nf' : G \u2192L[\ud835\udd5c] E\n\u22a2 HasFDerivAt (\u2191iso \u2218 f) (comp (\u2191iso) f') x \u2194 HasFDerivAt f f' x\n[PROOFSTEP]\nsimp_rw [\u2190 hasFDerivWithinAt_univ, iso.comp_hasFDerivWithinAt_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\nf' : G \u2192L[\ud835\udd5c] F\n\u22a2 HasFDerivWithinAt (\u2191iso \u2218 f) f' s x \u2194 HasFDerivWithinAt f (comp (\u2191(ContinuousLinearEquiv.symm iso)) f') s x\n[PROOFSTEP]\nrw [\u2190 iso.comp_hasFDerivWithinAt_iff, \u2190 ContinuousLinearMap.comp_assoc, iso.coe_comp_coe_symm,\n ContinuousLinearMap.id_comp]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\nx : G\nf' : G \u2192L[\ud835\udd5c] F\n\u22a2 HasFDerivAt (\u2191iso \u2218 f) f' x \u2194 HasFDerivAt f (comp (\u2191(ContinuousLinearEquiv.symm iso)) f') x\n[PROOFSTEP]\nsimp_rw [\u2190 hasFDerivWithinAt_univ, iso.comp_hasFDerivWithinAt_iff']\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 fderivWithin \ud835\udd5c (\u2191iso \u2218 f) s x = comp (\u2191iso) (fderivWithin \ud835\udd5c f s x)\n[PROOFSTEP]\nby_cases h : DifferentiableWithinAt \ud835\udd5c f s x\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\nh : DifferentiableWithinAt \ud835\udd5c f s x\n\u22a2 fderivWithin \ud835\udd5c (\u2191iso \u2218 f) s x = comp (\u2191iso) (fderivWithin \ud835\udd5c f s x)\n[PROOFSTEP]\nrw [fderiv.comp_fderivWithin x iso.differentiableAt h hxs, iso.fderiv]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\nh : \u00acDifferentiableWithinAt \ud835\udd5c f s x\n\u22a2 fderivWithin \ud835\udd5c (\u2191iso \u2218 f) s x = comp (\u2191iso) (fderivWithin \ud835\udd5c f s x)\n[PROOFSTEP]\nhave : \u00acDifferentiableWithinAt \ud835\udd5c (iso \u2218 f) s x := mt iso.comp_differentiableWithinAt_iff.1 h\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\ns : Set G\nx : G\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\nh : \u00acDifferentiableWithinAt \ud835\udd5c f s x\nthis : \u00acDifferentiableWithinAt \ud835\udd5c (\u2191iso \u2218 f) s x\n\u22a2 fderivWithin \ud835\udd5c (\u2191iso \u2218 f) s x = comp (\u2191iso) (fderivWithin \ud835\udd5c f s x)\n[PROOFSTEP]\nrw [fderivWithin_zero_of_not_differentiableWithinAt h, fderivWithin_zero_of_not_differentiableWithinAt this,\n ContinuousLinearMap.comp_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\nx : G\n\u22a2 fderiv \ud835\udd5c (\u2191iso \u2218 f) x = comp (\u2191iso) (fderiv \ud835\udd5c f x)\n[PROOFSTEP]\nrw [\u2190 fderivWithin_univ, \u2190 fderivWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : G \u2192 E\nx : G\n\u22a2 fderivWithin \ud835\udd5c (\u2191iso \u2218 f) univ x = comp (\u2191iso) (fderivWithin \ud835\udd5c f univ x)\n[PROOFSTEP]\nexact iso.comp_fderivWithin uniqueDiffWithinAt_univ\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\n\u22a2 DifferentiableWithinAt \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) x \u2194 DifferentiableWithinAt \ud835\udd5c f s (\u2191iso x)\n[PROOFSTEP]\nrefine' \u27e8fun H => _, fun H => H.comp x iso.differentiableWithinAt (mapsTo_preimage _ s)\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) x\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s (\u2191iso x)\n[PROOFSTEP]\nhave : DifferentiableWithinAt \ud835\udd5c ((f \u2218 iso) \u2218 iso.symm) s (iso x) :=\n by\n rw [\u2190 iso.symm_apply_apply x] at H \n apply H.comp (iso x) iso.symm.differentiableWithinAt\n intro y hy\n simpa only [mem_preimage, apply_symm_apply] using hy\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) x\n\u22a2 DifferentiableWithinAt \ud835\udd5c ((f \u2218 \u2191iso) \u2218 \u2191(ContinuousLinearEquiv.symm iso)) s (\u2191iso x)\n[PROOFSTEP]\nrw [\u2190 iso.symm_apply_apply x] at H \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm iso) (\u2191iso x))\n\u22a2 DifferentiableWithinAt \ud835\udd5c ((f \u2218 \u2191iso) \u2218 \u2191(ContinuousLinearEquiv.symm iso)) s (\u2191iso x)\n[PROOFSTEP]\napply H.comp (iso x) iso.symm.differentiableWithinAt\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm iso) (\u2191iso x))\n\u22a2 MapsTo (\u2191(ContinuousLinearEquiv.symm iso)) s (\u2191iso \u207b\u00b9' s)\n[PROOFSTEP]\nintro y hy\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm iso) (\u2191iso x))\ny : F\nhy : y \u2208 s\n\u22a2 \u2191(ContinuousLinearEquiv.symm iso) y \u2208 \u2191iso \u207b\u00b9' s\n[PROOFSTEP]\nsimpa only [mem_preimage, apply_symm_apply] using hy\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nH : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) x\nthis : DifferentiableWithinAt \ud835\udd5c ((f \u2218 \u2191iso) \u2218 \u2191(ContinuousLinearEquiv.symm iso)) s (\u2191iso x)\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s (\u2191iso x)\n[PROOFSTEP]\nrwa [Function.comp.assoc, iso.self_comp_symm] at this \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\nx : E\n\u22a2 DifferentiableAt \ud835\udd5c (f \u2218 \u2191iso) x \u2194 DifferentiableAt \ud835\udd5c f (\u2191iso x)\n[PROOFSTEP]\nsimp only [\u2190 differentiableWithinAt_univ, \u2190 iso.comp_right_differentiableWithinAt_iff, preimage_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\n\u22a2 DifferentiableOn \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) \u2194 DifferentiableOn \ud835\udd5c f s\n[PROOFSTEP]\nrefine' \u27e8fun H y hy => _, fun H y hy => iso.comp_right_differentiableWithinAt_iff.2 (H _ hy)\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nH : DifferentiableOn \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s)\ny : F\nhy : y \u2208 s\n\u22a2 DifferentiableWithinAt \ud835\udd5c f s y\n[PROOFSTEP]\nrw [\u2190 iso.apply_symm_apply y, \u2190 comp_right_differentiableWithinAt_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nH : DifferentiableOn \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s)\ny : F\nhy : y \u2208 s\n\u22a2 DifferentiableWithinAt \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm iso) y)\n[PROOFSTEP]\napply H\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nH : DifferentiableOn \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s)\ny : F\nhy : y \u2208 s\n\u22a2 \u2191(ContinuousLinearEquiv.symm iso) y \u2208 \u2191iso \u207b\u00b9' s\n[PROOFSTEP]\nsimpa only [mem_preimage, apply_symm_apply] using hy\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\n\u22a2 Differentiable \ud835\udd5c (f \u2218 \u2191iso) \u2194 Differentiable \ud835\udd5c f\n[PROOFSTEP]\nsimp only [\u2190 differentiableOn_univ, \u2190 iso.comp_right_differentiableOn_iff, preimage_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nf' : F \u2192L[\ud835\udd5c] G\n\u22a2 HasFDerivWithinAt (f \u2218 \u2191iso) (comp f' \u2191iso) (\u2191iso \u207b\u00b9' s) x \u2194 HasFDerivWithinAt f f' s (\u2191iso x)\n[PROOFSTEP]\nrefine' \u27e8fun H => _, fun H => H.comp x iso.hasFDerivWithinAt (mapsTo_preimage _ s)\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nf' : F \u2192L[\ud835\udd5c] G\nH : HasFDerivWithinAt (f \u2218 \u2191iso) (comp f' \u2191iso) (\u2191iso \u207b\u00b9' s) x\n\u22a2 HasFDerivWithinAt f f' s (\u2191iso x)\n[PROOFSTEP]\nrw [\u2190 iso.symm_apply_apply x] at H \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nf' : F \u2192L[\ud835\udd5c] G\nH : HasFDerivWithinAt (f \u2218 \u2191iso) (comp f' \u2191iso) (\u2191iso \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm iso) (\u2191iso x))\n\u22a2 HasFDerivWithinAt f f' s (\u2191iso x)\n[PROOFSTEP]\nhave A : f = (f \u2218 iso) \u2218 iso.symm := by\n rw [Function.comp.assoc, iso.self_comp_symm]\n rfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nf' : F \u2192L[\ud835\udd5c] G\nH : HasFDerivWithinAt (f \u2218 \u2191iso) (comp f' \u2191iso) (\u2191iso \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm iso) (\u2191iso x))\n\u22a2 f = (f \u2218 \u2191iso) \u2218 \u2191(ContinuousLinearEquiv.symm iso)\n[PROOFSTEP]\nrw [Function.comp.assoc, iso.self_comp_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nf' : F \u2192L[\ud835\udd5c] G\nH : HasFDerivWithinAt (f \u2218 \u2191iso) (comp f' \u2191iso) (\u2191iso \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm iso) (\u2191iso x))\n\u22a2 f = f \u2218 _root_.id\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nf' : F \u2192L[\ud835\udd5c] G\nH : HasFDerivWithinAt (f \u2218 \u2191iso) (comp f' \u2191iso) (\u2191iso \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm iso) (\u2191iso x))\nA : f = (f \u2218 \u2191iso) \u2218 \u2191(ContinuousLinearEquiv.symm iso)\n\u22a2 HasFDerivWithinAt f f' s (\u2191iso x)\n[PROOFSTEP]\nhave B : f' = (f'.comp (iso : E \u2192L[\ud835\udd5c] F)).comp (iso.symm : F \u2192L[\ud835\udd5c] E) := by\n rw [ContinuousLinearMap.comp_assoc, iso.coe_comp_coe_symm, ContinuousLinearMap.comp_id]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nf' : F \u2192L[\ud835\udd5c] G\nH : HasFDerivWithinAt (f \u2218 \u2191iso) (comp f' \u2191iso) (\u2191iso \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm iso) (\u2191iso x))\nA : f = (f \u2218 \u2191iso) \u2218 \u2191(ContinuousLinearEquiv.symm iso)\n\u22a2 f' = comp (comp f' \u2191iso) \u2191(ContinuousLinearEquiv.symm iso)\n[PROOFSTEP]\nrw [ContinuousLinearMap.comp_assoc, iso.coe_comp_coe_symm, ContinuousLinearMap.comp_id]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nf' : F \u2192L[\ud835\udd5c] G\nH : HasFDerivWithinAt (f \u2218 \u2191iso) (comp f' \u2191iso) (\u2191iso \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm iso) (\u2191iso x))\nA : f = (f \u2218 \u2191iso) \u2218 \u2191(ContinuousLinearEquiv.symm iso)\nB : f' = comp (comp f' \u2191iso) \u2191(ContinuousLinearEquiv.symm iso)\n\u22a2 HasFDerivWithinAt f f' s (\u2191iso x)\n[PROOFSTEP]\nrw [A, B]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nf' : F \u2192L[\ud835\udd5c] G\nH : HasFDerivWithinAt (f \u2218 \u2191iso) (comp f' \u2191iso) (\u2191iso \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm iso) (\u2191iso x))\nA : f = (f \u2218 \u2191iso) \u2218 \u2191(ContinuousLinearEquiv.symm iso)\nB : f' = comp (comp f' \u2191iso) \u2191(ContinuousLinearEquiv.symm iso)\n\u22a2 HasFDerivWithinAt ((f \u2218 \u2191iso) \u2218 \u2191(ContinuousLinearEquiv.symm iso))\n (comp (comp f' \u2191iso) \u2191(ContinuousLinearEquiv.symm iso)) s (\u2191iso x)\n[PROOFSTEP]\napply H.comp (iso x) iso.symm.hasFDerivWithinAt\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nf' : F \u2192L[\ud835\udd5c] G\nH : HasFDerivWithinAt (f \u2218 \u2191iso) (comp f' \u2191iso) (\u2191iso \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm iso) (\u2191iso x))\nA : f = (f \u2218 \u2191iso) \u2218 \u2191(ContinuousLinearEquiv.symm iso)\nB : f' = comp (comp f' \u2191iso) \u2191(ContinuousLinearEquiv.symm iso)\n\u22a2 MapsTo (\u2191(ContinuousLinearEquiv.symm iso)) s (\u2191iso \u207b\u00b9' s)\n[PROOFSTEP]\nintro y hy\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nf' : F \u2192L[\ud835\udd5c] G\nH : HasFDerivWithinAt (f \u2218 \u2191iso) (comp f' \u2191iso) (\u2191iso \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm iso) (\u2191iso x))\nA : f = (f \u2218 \u2191iso) \u2218 \u2191(ContinuousLinearEquiv.symm iso)\nB : f' = comp (comp f' \u2191iso) \u2191(ContinuousLinearEquiv.symm iso)\ny : F\nhy : y \u2208 s\n\u22a2 \u2191(ContinuousLinearEquiv.symm iso) y \u2208 \u2191iso \u207b\u00b9' s\n[PROOFSTEP]\nsimpa only [mem_preimage, apply_symm_apply] using hy\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\nx : E\nf' : F \u2192L[\ud835\udd5c] G\n\u22a2 HasFDerivAt (f \u2218 \u2191iso) (comp f' \u2191iso) x \u2194 HasFDerivAt f f' (\u2191iso x)\n[PROOFSTEP]\nsimp only [\u2190 hasFDerivWithinAt_univ, \u2190 comp_right_hasFDerivWithinAt_iff, preimage_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nf' : E \u2192L[\ud835\udd5c] G\n\u22a2 HasFDerivWithinAt (f \u2218 \u2191iso) f' (\u2191iso \u207b\u00b9' s) x \u2194\n HasFDerivWithinAt f (comp f' \u2191(ContinuousLinearEquiv.symm iso)) s (\u2191iso x)\n[PROOFSTEP]\nrw [\u2190 iso.comp_right_hasFDerivWithinAt_iff, ContinuousLinearMap.comp_assoc, iso.coe_symm_comp_coe,\n ContinuousLinearMap.comp_id]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\nx : E\nf' : E \u2192L[\ud835\udd5c] G\n\u22a2 HasFDerivAt (f \u2218 \u2191iso) f' x \u2194 HasFDerivAt f (comp f' \u2191(ContinuousLinearEquiv.symm iso)) (\u2191iso x)\n[PROOFSTEP]\nsimp only [\u2190 hasFDerivWithinAt_univ, \u2190 iso.comp_right_hasFDerivWithinAt_iff', preimage_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nhxs : UniqueDiffWithinAt \ud835\udd5c (\u2191iso \u207b\u00b9' s) x\n\u22a2 fderivWithin \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) x = comp (fderivWithin \ud835\udd5c f s (\u2191iso x)) \u2191iso\n[PROOFSTEP]\nby_cases h : DifferentiableWithinAt \ud835\udd5c f s (iso x)\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nhxs : UniqueDiffWithinAt \ud835\udd5c (\u2191iso \u207b\u00b9' s) x\nh : DifferentiableWithinAt \ud835\udd5c f s (\u2191iso x)\n\u22a2 fderivWithin \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) x = comp (fderivWithin \ud835\udd5c f s (\u2191iso x)) \u2191iso\n[PROOFSTEP]\nexact (iso.comp_right_hasFDerivWithinAt_iff.2 h.hasFDerivWithinAt).fderivWithin hxs\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nhxs : UniqueDiffWithinAt \ud835\udd5c (\u2191iso \u207b\u00b9' s) x\nh : \u00acDifferentiableWithinAt \ud835\udd5c f s (\u2191iso x)\n\u22a2 fderivWithin \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) x = comp (fderivWithin \ud835\udd5c f s (\u2191iso x)) \u2191iso\n[PROOFSTEP]\nhave : \u00acDifferentiableWithinAt \ud835\udd5c (f \u2218 iso) (iso \u207b\u00b9' s) x :=\n by\n intro h'\n exact h (iso.comp_right_differentiableWithinAt_iff.1 h')\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nhxs : UniqueDiffWithinAt \ud835\udd5c (\u2191iso \u207b\u00b9' s) x\nh : \u00acDifferentiableWithinAt \ud835\udd5c f s (\u2191iso x)\n\u22a2 \u00acDifferentiableWithinAt \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) x\n[PROOFSTEP]\nintro h'\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nhxs : UniqueDiffWithinAt \ud835\udd5c (\u2191iso \u207b\u00b9' s) x\nh : \u00acDifferentiableWithinAt \ud835\udd5c f s (\u2191iso x)\nh' : DifferentiableWithinAt \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) x\n\u22a2 False\n[PROOFSTEP]\nexact h (iso.comp_right_differentiableWithinAt_iff.1 h')\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\ns : Set F\nx : E\nhxs : UniqueDiffWithinAt \ud835\udd5c (\u2191iso \u207b\u00b9' s) x\nh : \u00acDifferentiableWithinAt \ud835\udd5c f s (\u2191iso x)\nthis : \u00acDifferentiableWithinAt \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) x\n\u22a2 fderivWithin \ud835\udd5c (f \u2218 \u2191iso) (\u2191iso \u207b\u00b9' s) x = comp (fderivWithin \ud835\udd5c f s (\u2191iso x)) \u2191iso\n[PROOFSTEP]\nrw [fderivWithin_zero_of_not_differentiableWithinAt h, fderivWithin_zero_of_not_differentiableWithinAt this,\n ContinuousLinearMap.zero_comp]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\nx : E\n\u22a2 fderiv \ud835\udd5c (f \u2218 \u2191iso) x = comp (fderiv \ud835\udd5c f (\u2191iso x)) \u2191iso\n[PROOFSTEP]\nrw [\u2190 fderivWithin_univ, \u2190 fderivWithin_univ, \u2190 iso.comp_right_fderivWithin, preimage_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\nx : E\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (\u2191iso \u207b\u00b9' univ) x\n[PROOFSTEP]\nexact uniqueDiffWithinAt_univ\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhg : ContinuousAt g a\nhf : HasStrictFDerivAt f (\u2191f') (g a)\nhfg : \u2200\u1da0 (y : F) in \ud835\udcdd a, f (g y) = y\n\u22a2 HasStrictFDerivAt g (\u2191(ContinuousLinearEquiv.symm f')) a\n[PROOFSTEP]\nreplace hg := hg.prod_map' hg\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhf : HasStrictFDerivAt f (\u2191f') (g a)\nhfg : \u2200\u1da0 (y : F) in \ud835\udcdd a, f (g y) = y\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\n\u22a2 HasStrictFDerivAt g (\u2191(ContinuousLinearEquiv.symm f')) a\n[PROOFSTEP]\nreplace hfg := hfg.prod_mk_nhds hfg\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhf : HasStrictFDerivAt f (\u2191f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : \u2200\u1da0 (p : F \u00d7 F) in \ud835\udcdd (a, a), f (g p.fst) = p.fst \u2227 f (g p.snd) = p.snd\n\u22a2 HasStrictFDerivAt g (\u2191(ContinuousLinearEquiv.symm f')) a\n[PROOFSTEP]\nhave :\n (fun p : F \u00d7 F => g p.1 - g p.2 - f'.symm (p.1 - p.2)) =O[\ud835\udcdd (a, a)] fun p : F \u00d7 F =>\n f' (g p.1 - g p.2) - (p.1 - p.2) :=\n by\n refine' ((f'.symm : F \u2192L[\ud835\udd5c] E).isBigO_comp _ _).congr (fun x => _) fun _ => rfl\n simp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhf : HasStrictFDerivAt f (\u2191f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : \u2200\u1da0 (p : F \u00d7 F) in \ud835\udcdd (a, a), f (g p.fst) = p.fst \u2227 f (g p.snd) = p.snd\n\u22a2 (fun p => g p.fst - g p.snd - \u2191(ContinuousLinearEquiv.symm f') (p.fst - p.snd)) =O[\ud835\udcdd (a, a)] fun p =>\n \u2191f' (g p.fst - g p.snd) - (p.fst - p.snd)\n[PROOFSTEP]\nrefine' ((f'.symm : F \u2192L[\ud835\udd5c] E).isBigO_comp _ _).congr (fun x => _) fun _ => rfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhf : HasStrictFDerivAt f (\u2191f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : \u2200\u1da0 (p : F \u00d7 F) in \ud835\udcdd (a, a), f (g p.fst) = p.fst \u2227 f (g p.snd) = p.snd\nx : F \u00d7 F\n\u22a2 \u2191\u2191(ContinuousLinearEquiv.symm f') (\u2191f' (g x.fst - g x.snd) - (x.fst - x.snd)) =\n g x.fst - g x.snd - \u2191(ContinuousLinearEquiv.symm f') (x.fst - x.snd)\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhf : HasStrictFDerivAt f (\u2191f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : \u2200\u1da0 (p : F \u00d7 F) in \ud835\udcdd (a, a), f (g p.fst) = p.fst \u2227 f (g p.snd) = p.snd\nthis :\n (fun p => g p.fst - g p.snd - \u2191(ContinuousLinearEquiv.symm f') (p.fst - p.snd)) =O[\ud835\udcdd (a, a)] fun p =>\n \u2191f' (g p.fst - g p.snd) - (p.fst - p.snd)\n\u22a2 HasStrictFDerivAt g (\u2191(ContinuousLinearEquiv.symm f')) a\n[PROOFSTEP]\nrefine' this.trans_isLittleO _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhf : HasStrictFDerivAt f (\u2191f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : \u2200\u1da0 (p : F \u00d7 F) in \ud835\udcdd (a, a), f (g p.fst) = p.fst \u2227 f (g p.snd) = p.snd\nthis :\n (fun p => g p.fst - g p.snd - \u2191(ContinuousLinearEquiv.symm f') (p.fst - p.snd)) =O[\ud835\udcdd (a, a)] fun p =>\n \u2191f' (g p.fst - g p.snd) - (p.fst - p.snd)\n\u22a2 (fun p => \u2191f' (g p.fst - g p.snd) - (p.fst - p.snd)) =o[\ud835\udcdd (a, a)] fun p => p.fst - p.snd\n[PROOFSTEP]\nclear this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhf : HasStrictFDerivAt f (\u2191f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : \u2200\u1da0 (p : F \u00d7 F) in \ud835\udcdd (a, a), f (g p.fst) = p.fst \u2227 f (g p.snd) = p.snd\n\u22a2 (fun p => \u2191f' (g p.fst - g p.snd) - (p.fst - p.snd)) =o[\ud835\udcdd (a, a)] fun p => p.fst - p.snd\n[PROOFSTEP]\nrefine' ((hf.comp_tendsto hg).symm.congr' (hfg.mono _) (eventually_of_forall fun _ => rfl)).trans_isBigO _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhf : HasStrictFDerivAt f (\u2191f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : \u2200\u1da0 (p : F \u00d7 F) in \ud835\udcdd (a, a), f (g p.fst) = p.fst \u2227 f (g p.snd) = p.snd\n\u22a2 \u2200 (x : F \u00d7 F),\n f (g x.fst) = x.fst \u2227 f (g x.snd) = x.snd \u2192\n (fun x =>\n \u2191\u2191f' (((fun p => (g p.fst, g p.snd)) x).fst - ((fun p => (g p.fst, g p.snd)) x).snd) -\n (f ((fun p => (g p.fst, g p.snd)) x).fst - f ((fun p => (g p.fst, g p.snd)) x).snd))\n x =\n \u2191f' (g x.fst - g x.snd) - (x.fst - x.snd)\n[PROOFSTEP]\nrintro p \u27e8hp1, hp2\u27e9\n[GOAL]\ncase refine'_1.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhf : HasStrictFDerivAt f (\u2191f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : \u2200\u1da0 (p : F \u00d7 F) in \ud835\udcdd (a, a), f (g p.fst) = p.fst \u2227 f (g p.snd) = p.snd\np : F \u00d7 F\nhp1 : f (g p.fst) = p.fst\nhp2 : f (g p.snd) = p.snd\n\u22a2 (fun x =>\n \u2191\u2191f' (((fun p => (g p.fst, g p.snd)) x).fst - ((fun p => (g p.fst, g p.snd)) x).snd) -\n (f ((fun p => (g p.fst, g p.snd)) x).fst - f ((fun p => (g p.fst, g p.snd)) x).snd))\n p =\n \u2191f' (g p.fst - g p.snd) - (p.fst - p.snd)\n[PROOFSTEP]\nsimp [hp1, hp2]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhf : HasStrictFDerivAt f (\u2191f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : \u2200\u1da0 (p : F \u00d7 F) in \ud835\udcdd (a, a), f (g p.fst) = p.fst \u2227 f (g p.snd) = p.snd\n\u22a2 (fun x => ((fun p => p.fst - p.snd) \u2218 fun p => (g p.fst, g p.snd)) x) =O[\ud835\udcdd (a, a)] fun p => p.fst - p.snd\n[PROOFSTEP]\nrefine (hf.isBigO_sub_rev.comp_tendsto hg).congr' (eventually_of_forall fun _ => rfl) (hfg.mono ?_)\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhf : HasStrictFDerivAt f (\u2191f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : \u2200\u1da0 (p : F \u00d7 F) in \ud835\udcdd (a, a), f (g p.fst) = p.fst \u2227 f (g p.snd) = p.snd\n\u22a2 \u2200 (x : F \u00d7 F),\n f (g x.fst) = x.fst \u2227 f (g x.snd) = x.snd \u2192\n ((fun p => f p.fst - f p.snd) \u2218 fun p => (g p.fst, g p.snd)) x = (fun p => p.fst - p.snd) x\n[PROOFSTEP]\nrintro p \u27e8hp1, hp2\u27e9\n[GOAL]\ncase refine'_2.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhf : HasStrictFDerivAt f (\u2191f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : \u2200\u1da0 (p : F \u00d7 F) in \ud835\udcdd (a, a), f (g p.fst) = p.fst \u2227 f (g p.snd) = p.snd\np : F \u00d7 F\nhp1 : f (g p.fst) = p.fst\nhp2 : f (g p.snd) = p.snd\n\u22a2 ((fun p => f p.fst - f p.snd) \u2218 fun p => (g p.fst, g p.snd)) p = (fun p => p.fst - p.snd) p\n[PROOFSTEP]\nsimp only [(\u00b7 \u2218 \u00b7), hp1, hp2]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhg : ContinuousAt g a\nhf : HasFDerivAt f (\u2191f') (g a)\nhfg : \u2200\u1da0 (y : F) in \ud835\udcdd a, f (g y) = y\n\u22a2 HasFDerivAt g (\u2191(ContinuousLinearEquiv.symm f')) a\n[PROOFSTEP]\nhave : (fun x : F => g x - g a - f'.symm (x - a)) =O[\ud835\udcdd a] fun x : F => f' (g x - g a) - (x - a) :=\n by\n refine' ((f'.symm : F \u2192L[\ud835\udd5c] E).isBigO_comp _ _).congr (fun x => _) fun _ => rfl\n simp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhg : ContinuousAt g a\nhf : HasFDerivAt f (\u2191f') (g a)\nhfg : \u2200\u1da0 (y : F) in \ud835\udcdd a, f (g y) = y\n\u22a2 (fun x => g x - g a - \u2191(ContinuousLinearEquiv.symm f') (x - a)) =O[\ud835\udcdd a] fun x => \u2191f' (g x - g a) - (x - a)\n[PROOFSTEP]\nrefine' ((f'.symm : F \u2192L[\ud835\udd5c] E).isBigO_comp _ _).congr (fun x => _) fun _ => rfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhg : ContinuousAt g a\nhf : HasFDerivAt f (\u2191f') (g a)\nhfg : \u2200\u1da0 (y : F) in \ud835\udcdd a, f (g y) = y\nx : F\n\u22a2 \u2191\u2191(ContinuousLinearEquiv.symm f') (\u2191f' (g x - g a) - (x - a)) = g x - g a - \u2191(ContinuousLinearEquiv.symm f') (x - a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhg : ContinuousAt g a\nhf : HasFDerivAt f (\u2191f') (g a)\nhfg : \u2200\u1da0 (y : F) in \ud835\udcdd a, f (g y) = y\nthis : (fun x => g x - g a - \u2191(ContinuousLinearEquiv.symm f') (x - a)) =O[\ud835\udcdd a] fun x => \u2191f' (g x - g a) - (x - a)\n\u22a2 HasFDerivAt g (\u2191(ContinuousLinearEquiv.symm f')) a\n[PROOFSTEP]\nrefine' this.trans_isLittleO _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhg : ContinuousAt g a\nhf : HasFDerivAt f (\u2191f') (g a)\nhfg : \u2200\u1da0 (y : F) in \ud835\udcdd a, f (g y) = y\nthis : (fun x => g x - g a - \u2191(ContinuousLinearEquiv.symm f') (x - a)) =O[\ud835\udcdd a] fun x => \u2191f' (g x - g a) - (x - a)\n\u22a2 (fun x => \u2191f' (g x - g a) - (x - a)) =o[\ud835\udcdd a] fun x' => x' - a\n[PROOFSTEP]\nclear this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhg : ContinuousAt g a\nhf : HasFDerivAt f (\u2191f') (g a)\nhfg : \u2200\u1da0 (y : F) in \ud835\udcdd a, f (g y) = y\n\u22a2 (fun x => \u2191f' (g x - g a) - (x - a)) =o[\ud835\udcdd a] fun x' => x' - a\n[PROOFSTEP]\nrefine' ((hf.comp_tendsto hg).symm.congr' (hfg.mono _) (eventually_of_forall fun _ => rfl)).trans_isBigO _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhg : ContinuousAt g a\nhf : HasFDerivAt f (\u2191f') (g a)\nhfg : \u2200\u1da0 (y : F) in \ud835\udcdd a, f (g y) = y\n\u22a2 \u2200 (x : F), f (g x) = x \u2192 (fun x => \u2191\u2191f' (g x - g a) - (f (g x) - f (g a))) x = \u2191f' (g x - g a) - (x - a)\n[PROOFSTEP]\nrintro p hp\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhg : ContinuousAt g a\nhf : HasFDerivAt f (\u2191f') (g a)\nhfg : \u2200\u1da0 (y : F) in \ud835\udcdd a, f (g y) = y\np : F\nhp : f (g p) = p\n\u22a2 (fun x => \u2191\u2191f' (g x - g a) - (f (g x) - f (g a))) p = \u2191f' (g p - g a) - (p - a)\n[PROOFSTEP]\nsimp [hp, hfg.self_of_nhds]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhg : ContinuousAt g a\nhf : HasFDerivAt f (\u2191f') (g a)\nhfg : \u2200\u1da0 (y : F) in \ud835\udcdd a, f (g y) = y\n\u22a2 (fun x => ((fun x' => x' - g a) \u2218 g) x) =O[\ud835\udcdd a] fun x' => x' - a\n[PROOFSTEP]\nrefine' ((hf.isBigO_sub_rev f'.antilipschitz).comp_tendsto hg).congr' (eventually_of_forall fun _ => rfl) (hfg.mono _)\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhg : ContinuousAt g a\nhf : HasFDerivAt f (\u2191f') (g a)\nhfg : \u2200\u1da0 (y : F) in \ud835\udcdd a, f (g y) = y\n\u22a2 \u2200 (x : F), f (g x) = x \u2192 ((fun x' => f x' - f (g a)) \u2218 g) x = (fun x' => x' - a) x\n[PROOFSTEP]\nrintro p hp\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\ng : F \u2192 E\na : F\nhg : ContinuousAt g a\nhf : HasFDerivAt f (\u2191f') (g a)\nhfg : \u2200\u1da0 (y : F) in \ud835\udcdd a, f (g y) = y\np : F\nhp : f (g p) = p\n\u22a2 ((fun x' => f x' - f (g a)) \u2218 g) p = (fun x' => x' - a) p\n[PROOFSTEP]\nsimp only [(\u00b7 \u2218 \u00b7), hp, hfg.self_of_nhds]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\nhf' : \u2203 C, \u2200 (z : E), \u2016z\u2016 \u2264 C * \u2016\u2191f' z\u2016\n\u22a2 \u2200\u1da0 (z : E) in \ud835\udcdd[s \\ {x}] x, f z \u2260 f x\n[PROOFSTEP]\nrw [nhdsWithin, diff_eq, \u2190 inf_principal, \u2190 inf_assoc, eventually_inf_principal]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\nhf' : \u2203 C, \u2200 (z : E), \u2016z\u2016 \u2264 C * \u2016\u2191f' z\u2016\n\u22a2 \u2200\u1da0 (x_1 : E) in \ud835\udcdd x \u2293 \ud835\udcdf s, x_1 \u2208 {x}\u1d9c \u2192 f x_1 \u2260 f x\n[PROOFSTEP]\nhave A : (fun z => z - x) =O[\ud835\udcdd[s] x] fun z => f' (z - x) :=\n isBigO_iff.2 <| hf'.imp fun C hC => eventually_of_forall fun z => hC _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\nhf' : \u2203 C, \u2200 (z : E), \u2016z\u2016 \u2264 C * \u2016\u2191f' z\u2016\nA : (fun z => z - x) =O[\ud835\udcdd[s] x] fun z => \u2191f' (z - x)\n\u22a2 \u2200\u1da0 (x_1 : E) in \ud835\udcdd x \u2293 \ud835\udcdf s, x_1 \u2208 {x}\u1d9c \u2192 f x_1 \u2260 f x\n[PROOFSTEP]\nhave : (fun z => f z - f x) ~[\ud835\udcdd[s] x] fun z => f' (z - x) := h.trans_isBigO A\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivWithinAt f f' s x\nhf' : \u2203 C, \u2200 (z : E), \u2016z\u2016 \u2264 C * \u2016\u2191f' z\u2016\nA : (fun z => z - x) =O[\ud835\udcdd[s] x] fun z => \u2191f' (z - x)\nthis : (fun z => f z - f x) ~[\ud835\udcdd[s] x] fun z => \u2191f' (z - x)\n\u22a2 \u2200\u1da0 (x_1 : E) in \ud835\udcdd x \u2293 \ud835\udcdf s, x_1 \u2208 {x}\u1d9c \u2192 f x_1 \u2260 f x\n[PROOFSTEP]\nsimpa [not_imp_not, sub_eq_zero] using (A.trans this.isBigO_symm).eq_zero_imp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nh : HasFDerivAt f f' x\nhf' : \u2203 C, \u2200 (z : E), \u2016z\u2016 \u2264 C * \u2016\u2191f' z\u2016\n\u22a2 \u2200\u1da0 (z : E) in \ud835\udcdd[{x}\u1d9c] x, f z \u2260 f x\n[PROOFSTEP]\nsimpa only [compl_eq_univ_diff] using (hasFDerivWithinAt_univ.2 h).eventually_ne hf'\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 F\nf' : E \u2192L[\u211d] F\nx : E\nL : Filter E\n\u22a2 Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 * \u2016f x' - f x - \u2191f' (x' - x)\u2016) L (\ud835\udcdd 0) \u2194\n Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 \u2022 (f x' - f x - \u2191f' (x' - x))) L (\ud835\udcdd 0)\n[PROOFSTEP]\nsymm\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 F\nf' : E \u2192L[\u211d] F\nx : E\nL : Filter E\n\u22a2 Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 \u2022 (f x' - f x - \u2191f' (x' - x))) L (\ud835\udcdd 0) \u2194\n Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 * \u2016f x' - f x - \u2191f' (x' - x)\u2016) L (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [tendsto_iff_norm_tendsto_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 F\nf' : E \u2192L[\u211d] F\nx : E\nL : Filter E\n\u22a2 Tendsto (fun e => \u2016\u2016e - x\u2016\u207b\u00b9 \u2022 (f e - f x - \u2191f' (e - x)) - 0\u2016) L (\ud835\udcdd 0) \u2194\n Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 * \u2016f x' - f x - \u2191f' (x' - x)\u2016) L (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' tendsto_congr fun x' => _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 F\nf' : E \u2192L[\u211d] F\nx : E\nL : Filter E\nx' : E\n\u22a2 \u2016\u2016x' - x\u2016\u207b\u00b9 \u2022 (f x' - f x - \u2191f' (x' - x)) - 0\u2016 = \u2016x' - x\u2016\u207b\u00b9 * \u2016f x' - f x - \u2191f' (x' - x)\u2016\n[PROOFSTEP]\nsimp [norm_smul]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 F\nf' : E \u2192L[\u211d] F\nx : E\nhf : HasFDerivAt f f' x\nv : E\n\u22a2 Tendsto (fun c => c \u2022 (f (x + c\u207b\u00b9 \u2022 v) - f x)) atTop (\ud835\udcdd (\u2191f' v))\n[PROOFSTEP]\napply hf.lim v\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 F\nf' : E \u2192L[\u211d] F\nx : E\nhf : HasFDerivAt f f' x\nv : E\n\u22a2 Tendsto (fun n => \u2016n\u2016) atTop atTop\n[PROOFSTEP]\nrw [tendsto_atTop_atTop]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 F\nf' : E \u2192L[\u211d] F\nx : E\nhf : HasFDerivAt f f' x\nv : E\n\u22a2 \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u211d), i \u2264 a \u2192 b \u2264 \u2016a\u2016\n[PROOFSTEP]\nexact fun b => \u27e8b, fun a ha => le_trans ha (le_abs_self _)\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\ns : Set E\nf' : E \u2192L[\ud835\udd5c] F\nx : E\nh : HasFDerivWithinAt f f' s x\n\u22a2 MapsTo (\u2191f') (tangentConeAt \ud835\udd5c s x) (tangentConeAt \ud835\udd5c (f '' s) (f x))\n[PROOFSTEP]\nrintro v \u27e8c, d, dtop, clim, cdlim\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\ns : Set E\nf' : E \u2192L[\ud835\udd5c] F\nx : E\nh : HasFDerivWithinAt f f' s x\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\ndtop : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) atTop atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\n\u22a2 \u2191f' v \u2208 tangentConeAt \ud835\udd5c (f '' s) (f x)\n[PROOFSTEP]\nrefine' \u27e8c, fun n => f (x + d n) - f x, mem_of_superset dtop _, clim, h.lim atTop dtop clim cdlim\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\ns : Set E\nf' : E \u2192L[\ud835\udd5c] F\nx : E\nh : HasFDerivWithinAt f f' s x\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\ndtop : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nclim : Tendsto (fun n => \u2016c n\u2016) atTop atTop\ncdlim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\n\u22a2 {x_1 | (fun n => x + d n \u2208 s) x_1} \u2286 {x_1 | (fun n => f x + (fun n => f (x + d n) - f x) n \u2208 f '' s) x_1}\n[PROOFSTEP]\nsimp (config := { contextual := true }) [-mem_image, mem_image_of_mem]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\ns : Set E\nf' : E \u2192L[\ud835\udd5c] F\nx : E\nh : HasFDerivWithinAt f f' s x\nhs : UniqueDiffWithinAt \ud835\udd5c s x\nh' : DenseRange \u2191f'\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (f '' s) (f x)\n[PROOFSTEP]\nrefine' \u27e8h'.dense_of_mapsTo f'.continuous hs.1 _, h.continuousWithinAt.mem_closure_image hs.2\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\ns : Set E\nf' : E \u2192L[\ud835\udd5c] F\nx : E\nh : HasFDerivWithinAt f f' s x\nhs : UniqueDiffWithinAt \ud835\udd5c s x\nh' : DenseRange \u2191f'\n\u22a2 MapsTo \u2191f' \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c s x)) \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (f '' s) (f x)))\n[PROOFSTEP]\nshow Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c s x) \u2264 (Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (f '' s) (f x))).comap f'\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\ns : Set E\nf' : E \u2192L[\ud835\udd5c] F\nx : E\nh : HasFDerivWithinAt f f' s x\nhs : UniqueDiffWithinAt \ud835\udd5c s x\nh' : DenseRange \u2191f'\n\u22a2 Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c s x) \u2264 Submodule.comap f' (Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (f '' s) (f x)))\n[PROOFSTEP]\nrw [Submodule.span_le]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\ns : Set E\nf' : E \u2192L[\ud835\udd5c] F\nx : E\nh : HasFDerivWithinAt f f' s x\nhs : UniqueDiffWithinAt \ud835\udd5c s x\nh' : DenseRange \u2191f'\n\u22a2 tangentConeAt \ud835\udd5c s x \u2286 \u2191(Submodule.comap f' (Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (f '' s) (f x))))\n[PROOFSTEP]\nexact h.mapsTo_tangent_cone.mono Subset.rfl Submodule.subset_span\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\ns : Set E\nf' : E \u2192L[\ud835\udd5c] F\ne : F \u2243L[\ud835\udd5c] E\n\u22a2 UniqueDiffOn \ud835\udd5c (\u2191e \u207b\u00b9' s) \u2194 UniqueDiffOn \ud835\udd5c s\n[PROOFSTEP]\nrw [\u2190 e.image_symm_eq_preimage, e.symm.uniqueDiffOn_image_iff]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.FDeriv.Equiv", "llama_tokens": 39110, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7634837635542925, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.41742570195593176}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\n\u22a2 \u2203 f, \u2191f x\u2080 = 1 \u2227 \u2200 (x : E), x \u2208 s \u2192 \u2191f x < 1\n[PROOFSTEP]\nlet f : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (ne_of_mem_of_not_mem hs\u2080 hx\u2080).symm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\n\u22a2 \u2203 f, \u2191f x\u2080 = 1 \u2227 \u2200 (x : E), x \u2208 s \u2192 \u2191f x < 1\n[PROOFSTEP]\nhave :=\n exists_extension_of_le_sublinear f (gauge s) (fun c hc => gauge_smul_of_nonneg hc.le)\n (gauge_add_le hs\u2081 <| absorbent_nhds_zero <| hs\u2082.mem_nhds hs\u2080) ?_\n[GOAL]\ncase refine_2\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\nthis : \u2203 g, (\u2200 (x : { x // x \u2208 f.domain }), \u2191g \u2191x = \u2191f x) \u2227 \u2200 (x : E), \u2191g x \u2264 gauge s x\n\u22a2 \u2203 f, \u2191f x\u2080 = 1 \u2227 \u2200 (x : E), x \u2208 s \u2192 \u2191f x < 1\ncase refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\n\u22a2 \u2200 (x : { x // x \u2208 f.domain }), \u2191f x \u2264 gauge s \u2191x\n[PROOFSTEP]\nobtain \u27e8\u03c6, h\u03c6\u2081, h\u03c6\u2082\u27e9 := this\n[GOAL]\ncase refine_2.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\n\u03c6 : E \u2192\u2097[\u211d] \u211d\nh\u03c6\u2081 : \u2200 (x : { x // x \u2208 f.domain }), \u2191\u03c6 \u2191x = \u2191f x\nh\u03c6\u2082 : \u2200 (x : E), \u2191\u03c6 x \u2264 gauge s x\n\u22a2 \u2203 f, \u2191f x\u2080 = 1 \u2227 \u2200 (x : E), x \u2208 s \u2192 \u2191f x < 1\ncase refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\n\u22a2 \u2200 (x : { x // x \u2208 f.domain }), \u2191f x \u2264 gauge s \u2191x\n[PROOFSTEP]\nhave h\u03c6\u2083 : \u03c6 x\u2080 = 1 := by\n rw [\u2190 f.domain.coe_mk x\u2080 (Submodule.mem_span_singleton_self _), h\u03c6\u2081, LinearPMap.mkSpanSingleton'_apply_self]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\n\u03c6 : E \u2192\u2097[\u211d] \u211d\nh\u03c6\u2081 : \u2200 (x : { x // x \u2208 f.domain }), \u2191\u03c6 \u2191x = \u2191f x\nh\u03c6\u2082 : \u2200 (x : E), \u2191\u03c6 x \u2264 gauge s x\n\u22a2 \u2191\u03c6 x\u2080 = 1\n[PROOFSTEP]\nrw [\u2190 f.domain.coe_mk x\u2080 (Submodule.mem_span_singleton_self _), h\u03c6\u2081, LinearPMap.mkSpanSingleton'_apply_self]\n[GOAL]\ncase refine_2.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\n\u03c6 : E \u2192\u2097[\u211d] \u211d\nh\u03c6\u2081 : \u2200 (x : { x // x \u2208 f.domain }), \u2191\u03c6 \u2191x = \u2191f x\nh\u03c6\u2082 : \u2200 (x : E), \u2191\u03c6 x \u2264 gauge s x\nh\u03c6\u2083 : \u2191\u03c6 x\u2080 = 1\n\u22a2 \u2203 f, \u2191f x\u2080 = 1 \u2227 \u2200 (x : E), x \u2208 s \u2192 \u2191f x < 1\ncase refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\n\u22a2 \u2200 (x : { x // x \u2208 f.domain }), \u2191f x \u2264 gauge s \u2191x\n[PROOFSTEP]\nhave h\u03c6\u2084 : \u2200 x \u2208 s, \u03c6 x < 1 := fun x hx => (h\u03c6\u2082 x).trans_lt (gauge_lt_one_of_mem_of_open hs\u2082 hx)\n[GOAL]\ncase refine_2.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\n\u03c6 : E \u2192\u2097[\u211d] \u211d\nh\u03c6\u2081 : \u2200 (x : { x // x \u2208 f.domain }), \u2191\u03c6 \u2191x = \u2191f x\nh\u03c6\u2082 : \u2200 (x : E), \u2191\u03c6 x \u2264 gauge s x\nh\u03c6\u2083 : \u2191\u03c6 x\u2080 = 1\nh\u03c6\u2084 : \u2200 (x : E), x \u2208 s \u2192 \u2191\u03c6 x < 1\n\u22a2 \u2203 f, \u2191f x\u2080 = 1 \u2227 \u2200 (x : E), x \u2208 s \u2192 \u2191f x < 1\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u03c6, _\u27e9, h\u03c6\u2083, h\u03c6\u2084\u27e9\n[GOAL]\ncase refine_2.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\n\u03c6 : E \u2192\u2097[\u211d] \u211d\nh\u03c6\u2081 : \u2200 (x : { x // x \u2208 f.domain }), \u2191\u03c6 \u2191x = \u2191f x\nh\u03c6\u2082 : \u2200 (x : E), \u2191\u03c6 x \u2264 gauge s x\nh\u03c6\u2083 : \u2191\u03c6 x\u2080 = 1\nh\u03c6\u2084 : \u2200 (x : E), x \u2208 s \u2192 \u2191\u03c6 x < 1\n\u22a2 Continuous \u03c6.toFun\n[PROOFSTEP]\nrefine'\n \u03c6.continuous_of_nonzero_on_open _ (hs\u2082.vadd (-x\u2080)) (Nonempty.vadd_set \u27e80, hs\u2080\u27e9)\n (vadd_set_subset_iff.mpr fun x hx => _)\n[GOAL]\ncase refine_2.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\n\u03c6 : E \u2192\u2097[\u211d] \u211d\nh\u03c6\u2081 : \u2200 (x : { x // x \u2208 f.domain }), \u2191\u03c6 \u2191x = \u2191f x\nh\u03c6\u2082 : \u2200 (x : E), \u2191\u03c6 x \u2264 gauge s x\nh\u03c6\u2083 : \u2191\u03c6 x\u2080 = 1\nh\u03c6\u2084 : \u2200 (x : E), x \u2208 s \u2192 \u2191\u03c6 x < 1\nx : E\nhx : x \u2208 s\n\u22a2 -x\u2080 +\u1d65 x \u2208 fun x => \u2191\u03c6 x = 0 \u2192 False\n[PROOFSTEP]\nchange \u03c6 (-x\u2080 + x) \u2260 0\n[GOAL]\ncase refine_2.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\n\u03c6 : E \u2192\u2097[\u211d] \u211d\nh\u03c6\u2081 : \u2200 (x : { x // x \u2208 f.domain }), \u2191\u03c6 \u2191x = \u2191f x\nh\u03c6\u2082 : \u2200 (x : E), \u2191\u03c6 x \u2264 gauge s x\nh\u03c6\u2083 : \u2191\u03c6 x\u2080 = 1\nh\u03c6\u2084 : \u2200 (x : E), x \u2208 s \u2192 \u2191\u03c6 x < 1\nx : E\nhx : x \u2208 s\n\u22a2 \u2191\u03c6 (-x\u2080 + x) \u2260 0\n[PROOFSTEP]\nrw [map_add, map_neg]\n[GOAL]\ncase refine_2.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\n\u03c6 : E \u2192\u2097[\u211d] \u211d\nh\u03c6\u2081 : \u2200 (x : { x // x \u2208 f.domain }), \u2191\u03c6 \u2191x = \u2191f x\nh\u03c6\u2082 : \u2200 (x : E), \u2191\u03c6 x \u2264 gauge s x\nh\u03c6\u2083 : \u2191\u03c6 x\u2080 = 1\nh\u03c6\u2084 : \u2200 (x : E), x \u2208 s \u2192 \u2191\u03c6 x < 1\nx : E\nhx : x \u2208 s\n\u22a2 -\u2191\u03c6 x\u2080 + \u2191\u03c6 x \u2260 0\n[PROOFSTEP]\nspecialize h\u03c6\u2084 x hx\n[GOAL]\ncase refine_2.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\n\u03c6 : E \u2192\u2097[\u211d] \u211d\nh\u03c6\u2081 : \u2200 (x : { x // x \u2208 f.domain }), \u2191\u03c6 \u2191x = \u2191f x\nh\u03c6\u2082 : \u2200 (x : E), \u2191\u03c6 x \u2264 gauge s x\nh\u03c6\u2083 : \u2191\u03c6 x\u2080 = 1\nx : E\nhx : x \u2208 s\nh\u03c6\u2084 : \u2191\u03c6 x < 1\n\u22a2 -\u2191\u03c6 x\u2080 + \u2191\u03c6 x \u2260 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\n\u22a2 \u2200 (x : { x // x \u2208 f.domain }), \u2191f x \u2264 gauge s \u2191x\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase refine_1.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\nx : E\nhx : x \u2208 f.domain\n\u22a2 \u2191f { val := x, property := hx } \u2264 gauge s \u2191{ val := x, property := hx }\n[PROOFSTEP]\nobtain \u27e8y, rfl\u27e9 := Submodule.mem_span_singleton.1 hx\n[GOAL]\ncase refine_1.mk.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\ny : \u211d\nhx : y \u2022 x\u2080 \u2208 f.domain\n\u22a2 \u2191f { val := y \u2022 x\u2080, property := hx } \u2264 gauge s \u2191{ val := y \u2022 x\u2080, property := hx }\n[PROOFSTEP]\nrw [LinearPMap.mkSpanSingleton'_apply]\n[GOAL]\ncase refine_1.mk.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\ny : \u211d\nhx : y \u2022 x\u2080 \u2208 f.domain\n\u22a2 y \u2022 1 \u2264 gauge s \u2191{ val := y \u2022 x\u2080, property := hx }\n[PROOFSTEP]\nsimp only [mul_one, Algebra.id.smul_eq_mul, Submodule.coe_mk]\n[GOAL]\ncase refine_1.mk.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\ny : \u211d\nhx : y \u2022 x\u2080 \u2208 f.domain\n\u22a2 y \u2264 gauge s (y \u2022 x\u2080)\n[PROOFSTEP]\nobtain h | h := le_or_lt y 0\n[GOAL]\ncase refine_1.mk.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\ny : \u211d\nhx : y \u2022 x\u2080 \u2208 f.domain\nh : y \u2264 0\n\u22a2 y \u2264 gauge s (y \u2022 x\u2080)\n[PROOFSTEP]\nexact h.trans (gauge_nonneg _)\n[GOAL]\ncase refine_1.mk.intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\ny : \u211d\nhx : y \u2022 x\u2080 \u2208 f.domain\nh : 0 < y\n\u22a2 y \u2264 gauge s (y \u2022 x\u2080)\n[PROOFSTEP]\nrw [gauge_smul_of_nonneg h.le, smul_eq_mul, le_mul_iff_one_le_right h]\n[GOAL]\ncase refine_1.mk.intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs\u2080 : 0 \u2208 s\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nx\u2080 : E\nhx\u2080 : \u00acx\u2080 \u2208 s\nf : E \u2192\u2097.[\u211d] \u211d := LinearPMap.mkSpanSingleton x\u2080 1 (_ : x\u2080 \u2260 0)\ny : \u211d\nhx : y \u2022 x\u2080 \u2208 f.domain\nh : 0 < y\n\u22a2 1 \u2264 gauge s x\u2080\n[PROOFSTEP]\nexact one_le_gauge_of_not_mem (hs\u2081.starConvex hs\u2080) (absorbent_nhds_zero <| hs\u2082.mem_nhds hs\u2080).absorbs hx\u2080\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b\n[PROOFSTEP]\nobtain rfl | \u27e8a\u2080, ha\u2080\u27e9 := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\nt : Set E\nx y : E\nht : Convex \u211d t\nhs\u2081 : Convex \u211d \u2205\nhs\u2082 : IsOpen \u2205\ndisj : Disjoint \u2205 t\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 \u2205 \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b\n[PROOFSTEP]\nexact \u27e80, 0, by simp, fun b _hb => le_rfl\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\nt : Set E\nx y : E\nht : Convex \u211d t\nhs\u2081 : Convex \u211d \u2205\nhs\u2082 : IsOpen \u2205\ndisj : Disjoint \u2205 t\n\u22a2 \u2200 (a : E), a \u2208 \u2205 \u2192 \u21910 a < 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b\n[PROOFSTEP]\nobtain rfl | \u27e8b\u2080, hb\u2080\u27e9 := t.eq_empty_or_nonempty\n[GOAL]\ncase inr.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nht : Convex \u211d \u2205\ndisj : Disjoint s \u2205\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 \u2205 \u2192 u \u2264 \u2191f b\n[PROOFSTEP]\nexact \u27e80, 1, fun a _ha => zero_lt_one, by simp\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nht : Convex \u211d \u2205\ndisj : Disjoint s \u2205\n\u22a2 \u2200 (b : E), b \u2208 \u2205 \u2192 1 \u2264 \u21910 b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.intro.inr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b\n[PROOFSTEP]\nlet x\u2080 := b\u2080 - a\u2080\n[GOAL]\ncase inr.intro.inr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b\n[PROOFSTEP]\nlet C := x\u2080 +\u1d65 (s - t)\n[GOAL]\ncase inr.intro.inr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b\n[PROOFSTEP]\nhave : (0 : E) \u2208 C := \u27e8a\u2080 - b\u2080, sub_mem_sub ha\u2080 hb\u2080, by simp_rw [vadd_eq_add, sub_add_sub_cancel', sub_self]\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\n\u22a2 (fun x => x\u2080 +\u1d65 x) (a\u2080 - b\u2080) = 0\n[PROOFSTEP]\nsimp_rw [vadd_eq_add, sub_add_sub_cancel', sub_self]\n[GOAL]\ncase inr.intro.inr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis : 0 \u2208 C\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b\n[PROOFSTEP]\nhave : Convex \u211d C := (hs\u2081.sub ht).vadd _\n[GOAL]\ncase inr.intro.inr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d : 0 \u2208 C\nthis : Convex \u211d C\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b\n[PROOFSTEP]\nhave : x\u2080 \u2209 C := by\n intro hx\u2080\n rw [\u2190 add_zero x\u2080] at hx\u2080 \n exact disj.zero_not_mem_sub_set (vadd_mem_vadd_set_iff.1 hx\u2080)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d : 0 \u2208 C\nthis : Convex \u211d C\n\u22a2 \u00acx\u2080 \u2208 C\n[PROOFSTEP]\nintro hx\u2080\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d : 0 \u2208 C\nthis : Convex \u211d C\nhx\u2080 : x\u2080 \u2208 C\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 add_zero x\u2080] at hx\u2080 \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d : 0 \u2208 C\nthis : Convex \u211d C\nhx\u2080 : x\u2080 + 0 \u2208 C\n\u22a2 False\n[PROOFSTEP]\nexact disj.zero_not_mem_sub_set (vadd_mem_vadd_set_iff.1 hx\u2080)\n[GOAL]\ncase inr.intro.inr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d\u00b9 : 0 \u2208 C\nthis\u271d : Convex \u211d C\nthis : \u00acx\u2080 \u2208 C\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b\n[PROOFSTEP]\nobtain \u27e8f, hf\u2081, hf\u2082\u27e9 := separate_convex_open_set \u20390 \u2208 C\u203a \u2039_\u203a (hs\u2082.sub_right.vadd _) \u2039x\u2080 \u2209 C\u203a\n[GOAL]\ncase inr.intro.inr.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d\u00b9 : 0 \u2208 C\nthis\u271d : Convex \u211d C\nthis : \u00acx\u2080 \u2208 C\nf : E \u2192L[\u211d] \u211d\nhf\u2081 : \u2191f x\u2080 = 1\nhf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b\n[PROOFSTEP]\nhave : f b\u2080 = f a\u2080 + 1 := by simp [\u2190 hf\u2081]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d\u00b9 : 0 \u2208 C\nthis\u271d : Convex \u211d C\nthis : \u00acx\u2080 \u2208 C\nf : E \u2192L[\u211d] \u211d\nhf\u2081 : \u2191f x\u2080 = 1\nhf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1\n\u22a2 \u2191f b\u2080 = \u2191f a\u2080 + 1\n[PROOFSTEP]\nsimp [\u2190 hf\u2081]\n[GOAL]\ncase inr.intro.inr.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d\u00b2 : 0 \u2208 C\nthis\u271d\u00b9 : Convex \u211d C\nthis\u271d : \u00acx\u2080 \u2208 C\nf : E \u2192L[\u211d] \u211d\nhf\u2081 : \u2191f x\u2080 = 1\nhf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1\nthis : \u2191f b\u2080 = \u2191f a\u2080 + 1\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b\n[PROOFSTEP]\nhave forall_le : \u2200 a \u2208 s, \u2200 b \u2208 t, f a \u2264 f b := by\n intro a ha b hb\n have := hf\u2082 (x\u2080 + (a - b)) (vadd_mem_vadd_set <| sub_mem_sub ha hb)\n simp only [f.map_add, f.map_sub, hf\u2081] at this \n linarith\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d\u00b2 : 0 \u2208 C\nthis\u271d\u00b9 : Convex \u211d C\nthis\u271d : \u00acx\u2080 \u2208 C\nf : E \u2192L[\u211d] \u211d\nhf\u2081 : \u2191f x\u2080 = 1\nhf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1\nthis : \u2191f b\u2080 = \u2191f a\u2080 + 1\n\u22a2 \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u2191f a \u2264 \u2191f b\n[PROOFSTEP]\nintro a ha b hb\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d\u00b2 : 0 \u2208 C\nthis\u271d\u00b9 : Convex \u211d C\nthis\u271d : \u00acx\u2080 \u2208 C\nf : E \u2192L[\u211d] \u211d\nhf\u2081 : \u2191f x\u2080 = 1\nhf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1\nthis : \u2191f b\u2080 = \u2191f a\u2080 + 1\na : E\nha : a \u2208 s\nb : E\nhb : b \u2208 t\n\u22a2 \u2191f a \u2264 \u2191f b\n[PROOFSTEP]\nhave := hf\u2082 (x\u2080 + (a - b)) (vadd_mem_vadd_set <| sub_mem_sub ha hb)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d\u00b3 : 0 \u2208 C\nthis\u271d\u00b2 : Convex \u211d C\nthis\u271d\u00b9 : \u00acx\u2080 \u2208 C\nf : E \u2192L[\u211d] \u211d\nhf\u2081 : \u2191f x\u2080 = 1\nhf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1\nthis\u271d : \u2191f b\u2080 = \u2191f a\u2080 + 1\na : E\nha : a \u2208 s\nb : E\nhb : b \u2208 t\nthis : \u2191f (x\u2080 + (a - b)) < 1\n\u22a2 \u2191f a \u2264 \u2191f b\n[PROOFSTEP]\nsimp only [f.map_add, f.map_sub, hf\u2081] at this \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d\u00b3 : 0 \u2208 C\nthis\u271d\u00b2 : Convex \u211d C\nthis\u271d\u00b9 : \u00acx\u2080 \u2208 C\nf : E \u2192L[\u211d] \u211d\nhf\u2081 : \u2191f x\u2080 = 1\nhf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1\nthis\u271d : \u2191f b\u2080 = \u2191f a\u2080 + 1\na : E\nha : a \u2208 s\nb : E\nhb : b \u2208 t\nthis : \u2191f b\u2080 - \u2191f a\u2080 + (\u2191f a - \u2191f b) < 1\n\u22a2 \u2191f a \u2264 \u2191f b\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase inr.intro.inr.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d\u00b2 : 0 \u2208 C\nthis\u271d\u00b9 : Convex \u211d C\nthis\u271d : \u00acx\u2080 \u2208 C\nf : E \u2192L[\u211d] \u211d\nhf\u2081 : \u2191f x\u2080 = 1\nhf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1\nthis : \u2191f b\u2080 = \u2191f a\u2080 + 1\nforall_le : \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u2191f a \u2264 \u2191f b\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b\n[PROOFSTEP]\nrefine' \u27e8f, sInf (f '' t), image_subset_iff.1 (_ : f '' s \u2286 Iio (sInf (f '' t))), fun b hb => _\u27e9\n[GOAL]\ncase inr.intro.inr.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d\u00b2 : 0 \u2208 C\nthis\u271d\u00b9 : Convex \u211d C\nthis\u271d : \u00acx\u2080 \u2208 C\nf : E \u2192L[\u211d] \u211d\nhf\u2081 : \u2191f x\u2080 = 1\nhf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1\nthis : \u2191f b\u2080 = \u2191f a\u2080 + 1\nforall_le : \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u2191f a \u2264 \u2191f b\n\u22a2 \u2191f '' s \u2286 Iio (sInf (\u2191f '' t))\n[PROOFSTEP]\nrw [\u2190 interior_Iic]\n[GOAL]\ncase inr.intro.inr.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d\u00b2 : 0 \u2208 C\nthis\u271d\u00b9 : Convex \u211d C\nthis\u271d : \u00acx\u2080 \u2208 C\nf : E \u2192L[\u211d] \u211d\nhf\u2081 : \u2191f x\u2080 = 1\nhf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1\nthis : \u2191f b\u2080 = \u2191f a\u2080 + 1\nforall_le : \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u2191f a \u2264 \u2191f b\n\u22a2 \u2191f '' s \u2286 interior (Iic (sInf (\u2191f '' t)))\n[PROOFSTEP]\nrefine' interior_maximal (image_subset_iff.2 fun a ha => _) (f.isOpenMap_of_ne_zero _ _ hs\u2082)\n[GOAL]\ncase inr.intro.inr.intro.intro.intro.refine'_1.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d\u00b2 : 0 \u2208 C\nthis\u271d\u00b9 : Convex \u211d C\nthis\u271d : \u00acx\u2080 \u2208 C\nf : E \u2192L[\u211d] \u211d\nhf\u2081 : \u2191f x\u2080 = 1\nhf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1\nthis : \u2191f b\u2080 = \u2191f a\u2080 + 1\nforall_le : \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u2191f a \u2264 \u2191f b\na : E\nha : a \u2208 s\n\u22a2 a \u2208 \u2191f \u207b\u00b9' Iic (sInf (\u2191f '' t))\n[PROOFSTEP]\nexact le_csInf (Nonempty.image _ \u27e8_, hb\u2080\u27e9) (ball_image_of_ball <| forall_le _ ha)\n[GOAL]\ncase inr.intro.inr.intro.intro.intro.refine'_1.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d\u00b2 : 0 \u2208 C\nthis\u271d\u00b9 : Convex \u211d C\nthis\u271d : \u00acx\u2080 \u2208 C\nf : E \u2192L[\u211d] \u211d\nhf\u2081 : \u2191f x\u2080 = 1\nhf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1\nthis : \u2191f b\u2080 = \u2191f a\u2080 + 1\nforall_le : \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u2191f a \u2264 \u2191f b\n\u22a2 f \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase inr.intro.inr.intro.intro.intro.refine'_1.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d\u00b2 : 0 \u2208 C\nthis\u271d\u00b9 : Convex \u211d C\nthis\u271d : \u00acx\u2080 \u2208 C\nhf\u2081 : \u21910 x\u2080 = 1\nhf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u21910 x < 1\nthis : \u21910 b\u2080 = \u21910 a\u2080 + 1\nforall_le : \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u21910 a \u2264 \u21910 b\n\u22a2 False\n[PROOFSTEP]\nsimp at hf\u2081 \n[GOAL]\ncase inr.intro.inr.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht : Convex \u211d t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nx\u2080 : E := b\u2080 - a\u2080\nC : Set E := x\u2080 +\u1d65 (s - t)\nthis\u271d\u00b2 : 0 \u2208 C\nthis\u271d\u00b9 : Convex \u211d C\nthis\u271d : \u00acx\u2080 \u2208 C\nf : E \u2192L[\u211d] \u211d\nhf\u2081 : \u2191f x\u2080 = 1\nhf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1\nthis : \u2191f b\u2080 = \u2191f a\u2080 + 1\nforall_le : \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u2191f a \u2264 \u2191f b\nb : E\nhb : b \u2208 t\n\u22a2 sInf (\u2191f '' t) \u2264 \u2191f b\n[PROOFSTEP]\nexact csInf_le \u27e8f a\u2080, ball_image_of_ball <| forall_le _ ha\u2080\u27e9 (mem_image_of_mem _ hb)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nht\u2081 : Convex \u211d t\nht\u2082 : IsOpen t\ndisj : \u00acx \u2208 t\nf : E \u2192L[\u211d] \u211d\nhf : \u2200 (a : E), a \u2208 t \u2192 \u2191f a < \u2191f x\n\u22a2 \u2200 (b : E), b \u2208 t \u2192 \u2191(-f) x < \u2191(-f) b\n[PROOFSTEP]\nsimpa\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\ndisj : Disjoint s t\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u < \u2191f b\n[PROOFSTEP]\nobtain rfl | \u27e8a\u2080, ha\u2080\u27e9 := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\nt : Set E\nx y : E\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\nhs\u2081 : Convex \u211d \u2205\nhs\u2082 : IsOpen \u2205\ndisj : Disjoint \u2205 t\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 \u2205 \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u < \u2191f b\n[PROOFSTEP]\nexact \u27e80, -1, by simp, fun b _hb => by norm_num\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\nt : Set E\nx y : E\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\nhs\u2081 : Convex \u211d \u2205\nhs\u2082 : IsOpen \u2205\ndisj : Disjoint \u2205 t\n\u22a2 \u2200 (a : E), a \u2208 \u2205 \u2192 \u21910 a < -1\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\nt : Set E\nx y : E\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\nhs\u2081 : Convex \u211d \u2205\nhs\u2082 : IsOpen \u2205\ndisj : Disjoint \u2205 t\nb : E\n_hb : b \u2208 t\n\u22a2 -1 < \u21910 b\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u < \u2191f b\n[PROOFSTEP]\nobtain rfl | \u27e8b\u2080, hb\u2080\u27e9 := t.eq_empty_or_nonempty\n[GOAL]\ncase inr.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nht\u2081 : Convex \u211d \u2205\nht\u2083 : IsOpen \u2205\ndisj : Disjoint s \u2205\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 \u2205 \u2192 u < \u2191f b\n[PROOFSTEP]\nexact \u27e80, 1, fun a _ha => by norm_num, by simp\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nht\u2081 : Convex \u211d \u2205\nht\u2083 : IsOpen \u2205\ndisj : Disjoint s \u2205\na : E\n_ha : a \u2208 s\n\u22a2 \u21910 a < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nht\u2081 : Convex \u211d \u2205\nht\u2083 : IsOpen \u2205\ndisj : Disjoint s \u2205\n\u22a2 \u2200 (b : E), b \u2208 \u2205 \u2192 1 < \u21910 b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.intro.inr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsOpen s\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\ndisj : Disjoint s t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u < \u2191f b\n[PROOFSTEP]\nobtain \u27e8f, s, hf\u2081, hf\u2082\u27e9 := geometric_hahn_banach_open hs\u2081 hs\u2082 ht\u2081 disj\n[GOAL]\ncase inr.intro.inr.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns\u271d t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsOpen s\u271d\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\ndisj : Disjoint s\u271d t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\u271d\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nf : E \u2192L[\u211d] \u211d\ns : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 s\u271d \u2192 \u2191f a < s\nhf\u2082 : \u2200 (b : E), b \u2208 t \u2192 s \u2264 \u2191f b\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s\u271d \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u < \u2191f b\n[PROOFSTEP]\nhave hf : IsOpenMap f := by\n refine' f.isOpenMap_of_ne_zero _\n rintro rfl\n simp_rw [ContinuousLinearMap.zero_apply] at hf\u2081 hf\u2082 \n exact (hf\u2081 _ ha\u2080).not_le (hf\u2082 _ hb\u2080)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns\u271d t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsOpen s\u271d\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\ndisj : Disjoint s\u271d t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\u271d\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nf : E \u2192L[\u211d] \u211d\ns : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 s\u271d \u2192 \u2191f a < s\nhf\u2082 : \u2200 (b : E), b \u2208 t \u2192 s \u2264 \u2191f b\n\u22a2 IsOpenMap \u2191f\n[PROOFSTEP]\nrefine' f.isOpenMap_of_ne_zero _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns\u271d t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsOpen s\u271d\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\ndisj : Disjoint s\u271d t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\u271d\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nf : E \u2192L[\u211d] \u211d\ns : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 s\u271d \u2192 \u2191f a < s\nhf\u2082 : \u2200 (b : E), b \u2208 t \u2192 s \u2264 \u2191f b\n\u22a2 f \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns\u271d t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsOpen s\u271d\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\ndisj : Disjoint s\u271d t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\u271d\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\ns : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 s\u271d \u2192 \u21910 a < s\nhf\u2082 : \u2200 (b : E), b \u2208 t \u2192 s \u2264 \u21910 b\n\u22a2 False\n[PROOFSTEP]\nsimp_rw [ContinuousLinearMap.zero_apply] at hf\u2081 hf\u2082 \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns\u271d t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsOpen s\u271d\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\ndisj : Disjoint s\u271d t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\u271d\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\ns : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 s\u271d \u2192 0 < s\nhf\u2082 : \u2200 (b : E), b \u2208 t \u2192 s \u2264 0\n\u22a2 False\n[PROOFSTEP]\nexact (hf\u2081 _ ha\u2080).not_le (hf\u2082 _ hb\u2080)\n[GOAL]\ncase inr.intro.inr.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns\u271d t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsOpen s\u271d\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\ndisj : Disjoint s\u271d t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\u271d\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nf : E \u2192L[\u211d] \u211d\ns : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 s\u271d \u2192 \u2191f a < s\nhf\u2082 : \u2200 (b : E), b \u2208 t \u2192 s \u2264 \u2191f b\nhf : IsOpenMap \u2191f\n\u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s\u271d \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u < \u2191f b\n[PROOFSTEP]\nrefine' \u27e8f, s, hf\u2081, image_subset_iff.1 (_ : f '' t \u2286 Ioi s)\u27e9\n[GOAL]\ncase inr.intro.inr.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns\u271d t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsOpen s\u271d\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\ndisj : Disjoint s\u271d t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\u271d\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nf : E \u2192L[\u211d] \u211d\ns : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 s\u271d \u2192 \u2191f a < s\nhf\u2082 : \u2200 (b : E), b \u2208 t \u2192 s \u2264 \u2191f b\nhf : IsOpenMap \u2191f\n\u22a2 \u2191f '' t \u2286 Ioi s\n[PROOFSTEP]\nrw [\u2190 interior_Ici]\n[GOAL]\ncase inr.intro.inr.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns\u271d t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsOpen s\u271d\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\ndisj : Disjoint s\u271d t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\u271d\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nf : E \u2192L[\u211d] \u211d\ns : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 s\u271d \u2192 \u2191f a < s\nhf\u2082 : \u2200 (b : E), b \u2208 t \u2192 s \u2264 \u2191f b\nhf : IsOpenMap \u2191f\n\u22a2 \u2191f '' t \u2286 interior (Ici s)\n[PROOFSTEP]\nrefine' interior_maximal (image_subset_iff.2 hf\u2082) (f.isOpenMap_of_ne_zero _ _ ht\u2083)\n[GOAL]\ncase inr.intro.inr.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns\u271d t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsOpen s\u271d\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\ndisj : Disjoint s\u271d t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\u271d\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\nf : E \u2192L[\u211d] \u211d\ns : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 s\u271d \u2192 \u2191f a < s\nhf\u2082 : \u2200 (b : E), b \u2208 t \u2192 s \u2264 \u2191f b\nhf : IsOpenMap \u2191f\n\u22a2 f \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase inr.intro.inr.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns\u271d t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsOpen s\u271d\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\ndisj : Disjoint s\u271d t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\u271d\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\ns : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 s\u271d \u2192 \u21910 a < s\nhf\u2082 : \u2200 (b : E), b \u2208 t \u2192 s \u2264 \u21910 b\nhf : IsOpenMap \u21910\n\u22a2 False\n[PROOFSTEP]\nsimp_rw [ContinuousLinearMap.zero_apply] at hf\u2081 hf\u2082 \n[GOAL]\ncase inr.intro.inr.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : Module \u211d E\ninst\u271d : ContinuousSMul \u211d E\ns\u271d t : Set E\nx y : E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsOpen s\u271d\nht\u2081 : Convex \u211d t\nht\u2083 : IsOpen t\ndisj : Disjoint s\u271d t\na\u2080 : E\nha\u2080 : a\u2080 \u2208 s\u271d\nb\u2080 : E\nhb\u2080 : b\u2080 \u2208 t\ns : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 s\u271d \u2192 0 < s\nhf\u2082 : \u2200 (b : E), b \u2208 t \u2192 s \u2264 0\nhf : IsOpenMap \u21910\n\u22a2 False\n[PROOFSTEP]\nexact (hf\u2081 _ ha\u2080).not_le (hf\u2082 _ hb\u2080)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns t : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \u211d t\nht\u2082 : IsClosed t\ndisj : Disjoint s t\n\u22a2 \u2203 f u v, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 u < v \u2227 \u2200 (b : E), b \u2208 t \u2192 v < \u2191f b\n[PROOFSTEP]\nobtain rfl | hs := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\nt : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nht\u2081 : Convex \u211d t\nht\u2082 : IsClosed t\nhs\u2081 : Convex \u211d \u2205\nhs\u2082 : IsCompact \u2205\ndisj : Disjoint \u2205 t\n\u22a2 \u2203 f u v, (\u2200 (a : E), a \u2208 \u2205 \u2192 \u2191f a < u) \u2227 u < v \u2227 \u2200 (b : E), b \u2208 t \u2192 v < \u2191f b\n[PROOFSTEP]\nexact \u27e80, -2, -1, by simp, by norm_num, fun b _hb => by norm_num\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\nt : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nht\u2081 : Convex \u211d t\nht\u2082 : IsClosed t\nhs\u2081 : Convex \u211d \u2205\nhs\u2082 : IsCompact \u2205\ndisj : Disjoint \u2205 t\n\u22a2 \u2200 (a : E), a \u2208 \u2205 \u2192 \u21910 a < -2\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\nt : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nht\u2081 : Convex \u211d t\nht\u2082 : IsClosed t\nhs\u2081 : Convex \u211d \u2205\nhs\u2082 : IsCompact \u2205\ndisj : Disjoint \u2205 t\n\u22a2 -2 < -1\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\nt : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nht\u2081 : Convex \u211d t\nht\u2082 : IsClosed t\nhs\u2081 : Convex \u211d \u2205\nhs\u2082 : IsCompact \u2205\ndisj : Disjoint \u2205 t\nb : E\n_hb : b \u2208 t\n\u22a2 -1 < \u21910 b\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns t : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \u211d t\nht\u2082 : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n\u22a2 \u2203 f u v, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 u < v \u2227 \u2200 (b : E), b \u2208 t \u2192 v < \u2191f b\n[PROOFSTEP]\nobtain rfl | _ht := t.eq_empty_or_nonempty\n[GOAL]\ncase inr.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsCompact s\nhs : Set.Nonempty s\nht\u2081 : Convex \u211d \u2205\nht\u2082 : IsClosed \u2205\ndisj : Disjoint s \u2205\n\u22a2 \u2203 f u v, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 u < v \u2227 \u2200 (b : E), b \u2208 \u2205 \u2192 v < \u2191f b\n[PROOFSTEP]\nexact \u27e80, 1, 2, fun a _ha => by norm_num, by norm_num, by simp\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsCompact s\nhs : Set.Nonempty s\nht\u2081 : Convex \u211d \u2205\nht\u2082 : IsClosed \u2205\ndisj : Disjoint s \u2205\na : E\n_ha : a \u2208 s\n\u22a2 \u21910 a < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsCompact s\nhs : Set.Nonempty s\nht\u2081 : Convex \u211d \u2205\nht\u2082 : IsClosed \u2205\ndisj : Disjoint s \u2205\n\u22a2 1 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsCompact s\nhs : Set.Nonempty s\nht\u2081 : Convex \u211d \u2205\nht\u2082 : IsClosed \u2205\ndisj : Disjoint s \u2205\n\u22a2 \u2200 (b : E), b \u2208 \u2205 \u2192 2 < \u21910 b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns t : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \u211d t\nht\u2082 : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\n\u22a2 \u2203 f u v, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 u < v \u2227 \u2200 (b : E), b \u2208 t \u2192 v < \u2191f b\n[PROOFSTEP]\nobtain \u27e8U, V, hU, hV, hU\u2081, hV\u2081, sU, tV, disj'\u27e9 := disj.exists_open_convexes hs\u2081 hs\u2082 ht\u2081 ht\u2082\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns t : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \u211d t\nht\u2082 : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU\u2081 : Convex \u211d U\nhV\u2081 : Convex \u211d V\nsU : s \u2286 U\ntV : t \u2286 V\ndisj' : Disjoint U V\n\u22a2 \u2203 f u v, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 u < v \u2227 \u2200 (b : E), b \u2208 t \u2192 v < \u2191f b\n[PROOFSTEP]\nobtain \u27e8f, u, hf\u2081, hf\u2082\u27e9 := geometric_hahn_banach_open_open hU\u2081 hU hV\u2081 hV disj'\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns t : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \u211d t\nht\u2082 : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU\u2081 : Convex \u211d U\nhV\u2081 : Convex \u211d V\nsU : s \u2286 U\ntV : t \u2286 V\ndisj' : Disjoint U V\nf : E \u2192L[\u211d] \u211d\nu : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 U \u2192 \u2191f a < u\nhf\u2082 : \u2200 (b : E), b \u2208 V \u2192 u < \u2191f b\n\u22a2 \u2203 f u v, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 u < v \u2227 \u2200 (b : E), b \u2208 t \u2192 v < \u2191f b\n[PROOFSTEP]\nobtain \u27e8x, hx\u2081, hx\u2082\u27e9 := hs\u2082.exists_forall_ge hs f.continuous.continuousOn\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns t : Set E\nx\u271d y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \u211d t\nht\u2082 : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU\u2081 : Convex \u211d U\nhV\u2081 : Convex \u211d V\nsU : s \u2286 U\ntV : t \u2286 V\ndisj' : Disjoint U V\nf : E \u2192L[\u211d] \u211d\nu : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 U \u2192 \u2191f a < u\nhf\u2082 : \u2200 (b : E), b \u2208 V \u2192 u < \u2191f b\nx : E\nhx\u2081 : x \u2208 s\nhx\u2082 : \u2200 (y : E), y \u2208 s \u2192 \u2191f y \u2264 \u2191f x\n\u22a2 \u2203 f u v, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 u < v \u2227 \u2200 (b : E), b \u2208 t \u2192 v < \u2191f b\n[PROOFSTEP]\nhave : f x < u := hf\u2081 x (sU hx\u2081)\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns t : Set E\nx\u271d y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \u211d t\nht\u2082 : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU\u2081 : Convex \u211d U\nhV\u2081 : Convex \u211d V\nsU : s \u2286 U\ntV : t \u2286 V\ndisj' : Disjoint U V\nf : E \u2192L[\u211d] \u211d\nu : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 U \u2192 \u2191f a < u\nhf\u2082 : \u2200 (b : E), b \u2208 V \u2192 u < \u2191f b\nx : E\nhx\u2081 : x \u2208 s\nhx\u2082 : \u2200 (y : E), y \u2208 s \u2192 \u2191f y \u2264 \u2191f x\nthis : \u2191f x < u\n\u22a2 \u2203 f u v, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 u < v \u2227 \u2200 (b : E), b \u2208 t \u2192 v < \u2191f b\n[PROOFSTEP]\nexact \u27e8f, (f x + u) / 2, u, fun a ha => by linarith [hx\u2082 a ha], by linarith, fun b hb => hf\u2082 b (tV hb)\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns t : Set E\nx\u271d y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \u211d t\nht\u2082 : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU\u2081 : Convex \u211d U\nhV\u2081 : Convex \u211d V\nsU : s \u2286 U\ntV : t \u2286 V\ndisj' : Disjoint U V\nf : E \u2192L[\u211d] \u211d\nu : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 U \u2192 \u2191f a < u\nhf\u2082 : \u2200 (b : E), b \u2208 V \u2192 u < \u2191f b\nx : E\nhx\u2081 : x \u2208 s\nhx\u2082 : \u2200 (y : E), y \u2208 s \u2192 \u2191f y \u2264 \u2191f x\nthis : \u2191f x < u\na : E\nha : a \u2208 s\n\u22a2 \u2191f a < (\u2191f x + u) / 2\n[PROOFSTEP]\nlinarith [hx\u2082 a ha]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns t : Set E\nx\u271d y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \u211d t\nht\u2082 : IsClosed t\ndisj : Disjoint s t\nhs : Set.Nonempty s\n_ht : Set.Nonempty t\nU V : Set E\nhU : IsOpen U\nhV : IsOpen V\nhU\u2081 : Convex \u211d U\nhV\u2081 : Convex \u211d V\nsU : s \u2286 U\ntV : t \u2286 V\ndisj' : Disjoint U V\nf : E \u2192L[\u211d] \u211d\nu : \u211d\nhf\u2081 : \u2200 (a : E), a \u2208 U \u2192 \u2191f a < u\nhf\u2082 : \u2200 (b : E), b \u2208 V \u2192 u < \u2191f b\nx : E\nhx\u2081 : x \u2208 s\nhx\u2082 : \u2200 (y : E), y \u2208 s \u2192 \u2191f y \u2264 \u2191f x\nthis : \u2191f x < u\n\u22a2 (\u2191f x + u) / 2 < u\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns\u271d t\u271d : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsClosed s\u271d\nht\u2081 : Convex \u211d t\u271d\nht\u2082 : IsCompact t\u271d\ndisj : Disjoint s\u271d t\u271d\nf : E \u2192L[\u211d] \u211d\ns t : \u211d\nhs : \u2200 (a : E), a \u2208 t\u271d \u2192 \u2191f a < s\nst : s < t\nht : \u2200 (b : E), b \u2208 s\u271d \u2192 t < \u2191f b\n\u22a2 \u2200 (a : E), a \u2208 s\u271d \u2192 \u2191(-f) a < -t\n[PROOFSTEP]\nsimpa using ht\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns\u271d t\u271d : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsClosed s\u271d\nht\u2081 : Convex \u211d t\u271d\nht\u2082 : IsCompact t\u271d\ndisj : Disjoint s\u271d t\u271d\nf : E \u2192L[\u211d] \u211d\ns t : \u211d\nhs : \u2200 (a : E), a \u2208 t\u271d \u2192 \u2191f a < s\nst : s < t\nht : \u2200 (b : E), b \u2208 s\u271d \u2192 t < \u2191f b\n\u22a2 -t < -s\n[PROOFSTEP]\nsimpa using st\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns\u271d t\u271d : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsClosed s\u271d\nht\u2081 : Convex \u211d t\u271d\nht\u2082 : IsCompact t\u271d\ndisj : Disjoint s\u271d t\u271d\nf : E \u2192L[\u211d] \u211d\ns t : \u211d\nhs : \u2200 (a : E), a \u2208 t\u271d \u2192 \u2191f a < s\nst : s < t\nht : \u2200 (b : E), b \u2208 s\u271d \u2192 t < \u2191f b\n\u22a2 \u2200 (b : E), b \u2208 t\u271d \u2192 -s < \u2191(-f) b\n[PROOFSTEP]\nsimpa using hs\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : ContinuousSMul \u211d E\ns t : Set E\nx y : E\ninst\u271d\u00b9 : LocallyConvexSpace \u211d E\ninst\u271d : T1Space E\nhxy : x \u2260 y\n\u22a2 \u2203 f, \u2191f x < \u2191f y\n[PROOFSTEP]\nobtain \u27e8f, s, t, hs, st, ht\u27e9 :=\n geometric_hahn_banach_compact_closed (convex_singleton x) isCompact_singleton (convex_singleton y) isClosed_singleton\n (disjoint_singleton.2 hxy)\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : ContinuousSMul \u211d E\ns\u271d t\u271d : Set E\nx y : E\ninst\u271d\u00b9 : LocallyConvexSpace \u211d E\ninst\u271d : T1Space E\nhxy : x \u2260 y\nf : E \u2192L[\u211d] \u211d\ns t : \u211d\nhs : \u2200 (a : E), a \u2208 {x} \u2192 \u2191f a < s\nst : s < t\nht : \u2200 (b : E), b \u2208 {y} \u2192 t < \u2191f b\n\u22a2 \u2203 f, \u2191f x < \u2191f y\n[PROOFSTEP]\nexact \u27e8f, by linarith [hs x rfl, ht y rfl]\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : ContinuousSMul \u211d E\ns\u271d t\u271d : Set E\nx y : E\ninst\u271d\u00b9 : LocallyConvexSpace \u211d E\ninst\u271d : T1Space E\nhxy : x \u2260 y\nf : E \u2192L[\u211d] \u211d\ns t : \u211d\nhs : \u2200 (a : E), a \u2208 {x} \u2192 \u2191f a < s\nst : s < t\nht : \u2200 (b : E), b \u2208 {y} \u2192 t < \u2191f b\n\u22a2 \u2191f x < \u2191f y\n[PROOFSTEP]\nlinarith [hs x rfl, ht y rfl]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns t : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsClosed s\n\u22a2 \u22c2 (l : E \u2192L[\u211d] \u211d), {x | \u2203 y, y \u2208 s \u2227 \u2191l x \u2264 \u2191l y} = s\n[PROOFSTEP]\nrw [Set.iInter_setOf]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns t : Set E\nx y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsClosed s\n\u22a2 {x | \u2200 (i : E \u2192L[\u211d] \u211d), \u2203 y, y \u2208 s \u2227 \u2191i x \u2264 \u2191i y} = s\n[PROOFSTEP]\nrefine' Set.Subset.antisymm (fun x hx => _) fun x hx l => \u27e8x, hx, le_rfl\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns t : Set E\nx\u271d y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsClosed s\nx : E\nhx : x \u2208 {x | \u2200 (i : E \u2192L[\u211d] \u211d), \u2203 y, y \u2208 s \u2227 \u2191i x \u2264 \u2191i y}\n\u22a2 x \u2208 s\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns t : Set E\nx\u271d y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\nhs\u2082 : IsClosed s\nx : E\nhx : x \u2208 {x | \u2200 (i : E \u2192L[\u211d] \u211d), \u2203 y, y \u2208 s \u2227 \u2191i x \u2264 \u2191i y}\nh : \u00acx \u2208 s\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8l, s, hlA, hl\u27e9 := geometric_hahn_banach_closed_point hs\u2081 hs\u2082 h\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns\u271d t : Set E\nx\u271d y : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsClosed s\u271d\nx : E\nhx : x \u2208 {x | \u2200 (i : E \u2192L[\u211d] \u211d), \u2203 y, y \u2208 s\u271d \u2227 \u2191i x \u2264 \u2191i y}\nh : \u00acx \u2208 s\u271d\nl : E \u2192L[\u211d] \u211d\ns : \u211d\nhlA : \u2200 (a : E), a \u2208 s\u271d \u2192 \u2191l a < s\nhl : s < \u2191l x\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8y, hy, hxy\u27e9 := hx l\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : TopologicalAddGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ns\u271d t : Set E\nx\u271d y\u271d : E\ninst\u271d : LocallyConvexSpace \u211d E\nhs\u2081 : Convex \u211d s\u271d\nhs\u2082 : IsClosed s\u271d\nx : E\nhx : x \u2208 {x | \u2200 (i : E \u2192L[\u211d] \u211d), \u2203 y, y \u2208 s\u271d \u2227 \u2191i x \u2264 \u2191i y}\nh : \u00acx \u2208 s\u271d\nl : E \u2192L[\u211d] \u211d\ns : \u211d\nhlA : \u2200 (a : E), a \u2208 s\u271d \u2192 \u2191l a < s\nhl : s < \u2191l x\ny : E\nhy : y \u2208 s\u271d\nhxy : \u2191l x \u2264 \u2191l y\n\u22a2 False\n[PROOFSTEP]\nexact ((hxy.trans_lt (hlA y hy)).trans hl).not_le le_rfl\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "llama_tokens": 28209, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929002541068, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.4173729752658859}} {"text": "[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 IsExtrOn f (Icc a b) c\n[PROOFSTEP]\nhave ne : (Icc a b).Nonempty :=\n nonempty_Icc.2\n (le_of_lt hab)\n -- Consider absolute min and max points\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 IsExtrOn f (Icc a b) c\n[PROOFSTEP]\nobtain \u27e8c, cmem, cle\u27e9 : \u2203 c \u2208 Icc a b, \u2200 x \u2208 Icc a b, f c \u2264 f x := isCompact_Icc.exists_forall_le ne hfc\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 IsExtrOn f (Icc a b) c\n[PROOFSTEP]\nobtain \u27e8C, Cmem, Cge\u27e9 : \u2203 C \u2208 Icc a b, \u2200 x \u2208 Icc a b, f x \u2264 f C := isCompact_Icc.exists_forall_ge ne hfc\n[GOAL]\ncase intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 IsExtrOn f (Icc a b) c\n[PROOFSTEP]\nby_cases hc : f c = f a\n[GOAL]\ncase pos\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\nhc : f c = f a\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 IsExtrOn f (Icc a b) c\n[PROOFSTEP]\nby_cases hC : f C = f a\n[GOAL]\ncase pos\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\nhc : f c = f a\nhC : f C = f a\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 IsExtrOn f (Icc a b) c\n[PROOFSTEP]\nhave : \u2200 x \u2208 Icc a b, f x = f a := fun x hx =>\n le_antisymm (hC \u25b8 Cge x hx)\n (hc \u25b8 cle x hx)\n -- `f` is a constant, so we can take any point in `Ioo a b`\n[GOAL]\ncase pos\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\nhc : f c = f a\nhC : f C = f a\nthis : \u2200 (x : X), x \u2208 Icc a b \u2192 f x = f a\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 IsExtrOn f (Icc a b) c\n[PROOFSTEP]\nrcases nonempty_Ioo.2 hab with \u27e8c', hc'\u27e9\n[GOAL]\ncase pos.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\nhc : f c = f a\nhC : f C = f a\nthis : \u2200 (x : X), x \u2208 Icc a b \u2192 f x = f a\nc' : X\nhc' : c' \u2208 Ioo a b\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 IsExtrOn f (Icc a b) c\n[PROOFSTEP]\nrefine \u27e8c', hc', Or.inl fun x hx \u21a6 ?_\u27e9\n[GOAL]\ncase pos.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\nhc : f c = f a\nhC : f C = f a\nthis : \u2200 (x : X), x \u2208 Icc a b \u2192 f x = f a\nc' : X\nhc' : c' \u2208 Ioo a b\nx : X\nhx : x \u2208 Icc a b\n\u22a2 x \u2208 {x | (fun x => f c' \u2264 f x) x}\n[PROOFSTEP]\nsimp only [mem_setOf_eq, this x hx, this c' (Ioo_subset_Icc_self hc'), le_rfl]\n[GOAL]\ncase neg\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\nhc : f c = f a\nhC : \u00acf C = f a\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 IsExtrOn f (Icc a b) c\n[PROOFSTEP]\nrefine' \u27e8C, \u27e8lt_of_le_of_ne Cmem.1 <| mt _ hC, lt_of_le_of_ne Cmem.2 <| mt _ hC\u27e9, Or.inr Cge\u27e9\n[GOAL]\ncase neg.refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\nhc : f c = f a\nhC : \u00acf C = f a\n\u22a2 a = C \u2192 f C = f a\ncase neg.refine'_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\nhc : f c = f a\nhC : \u00acf C = f a\n\u22a2 C = b \u2192 f C = f a\n[PROOFSTEP]\nexacts [fun h => by rw [h], fun h => by rw [h, hfI]]\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\nhc : f c = f a\nhC : \u00acf C = f a\nh : a = C\n\u22a2 f C = f a\n[PROOFSTEP]\nrw [h]\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\nhc : f c = f a\nhC : \u00acf C = f a\nh : C = b\n\u22a2 f C = f a\n[PROOFSTEP]\nrw [h, hfI]\n[GOAL]\ncase neg\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\nhc : \u00acf c = f a\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 IsExtrOn f (Icc a b) c\n[PROOFSTEP]\nrefine' \u27e8c, \u27e8lt_of_le_of_ne cmem.1 <| mt _ hc, lt_of_le_of_ne cmem.2 <| mt _ hc\u27e9, Or.inl cle\u27e9\n[GOAL]\ncase neg.refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\nhc : \u00acf c = f a\n\u22a2 a = c \u2192 f c = f a\ncase neg.refine'_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\nhc : \u00acf c = f a\n\u22a2 c = b \u2192 f c = f a\n[PROOFSTEP]\nexacts [fun h => by rw [h], fun h => by rw [h, hfI]]\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\nhc : \u00acf c = f a\nh : a = c\n\u22a2 f c = f a\n[PROOFSTEP]\nrw [h]\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : X\ncmem : c \u2208 Icc a b\ncle : \u2200 (x : X), x \u2208 Icc a b \u2192 f c \u2264 f x\nC : X\nCmem : C \u2208 Icc a b\nCge : \u2200 (x : X), x \u2208 Icc a b \u2192 f x \u2264 f C\nhc : \u00acf c = f a\nh : c = b\n\u22a2 f c = f a\n[PROOFSTEP]\nrw [h, hfI]\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Ioo a b)\nha : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd l)\nhb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd l)\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 IsExtrOn f (Ioo a b) c\n[PROOFSTEP]\nhave h : EqOn (extendFrom (Ioo a b) f) f (Ioo a b) := extendFrom_extends hfc\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc : ContinuousOn f (Ioo a b)\nha : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd l)\nhb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd l)\nh : EqOn (extendFrom (Ioo a b) f) f (Ioo a b)\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 IsExtrOn f (Ioo a b) c\n[PROOFSTEP]\nobtain \u27e8c, hc, hfc\u27e9 : \u2203 c \u2208 Ioo a b, IsExtrOn (extendFrom (Ioo a b) f) (Icc a b) c :=\n exists_Ioo_extr_on_Icc hab (continuousOn_Icc_extendFrom_Ioo hab.ne hfc ha hb)\n ((eq_lim_at_left_extendFrom_Ioo hab ha).trans (eq_lim_at_right_extendFrom_Ioo hab hb).symm)\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder X\ninst\u271d\u2075 : DenselyOrdered X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : OrderTopology X\ninst\u271d\u00b2 : LinearOrder Y\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : OrderTopology Y\nf : X \u2192 Y\na b : X\nl : Y\nhab : a < b\nhfc\u271d : ContinuousOn f (Ioo a b)\nha : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd l)\nhb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd l)\nh : EqOn (extendFrom (Ioo a b) f) f (Ioo a b)\nc : X\nhc : c \u2208 Ioo a b\nhfc : IsExtrOn (extendFrom (Ioo a b) f) (Icc a b) c\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 IsExtrOn f (Ioo a b) c\n[PROOFSTEP]\nexact \u27e8c, hc, (hfc.on_subset Ioo_subset_Icc_self).congr h (h hc)\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Order.Rolle", "llama_tokens": 6651, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581741774411, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.41724518995574783}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Zero E\ninst\u271d : TopologicalSpace E\nq : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set E\nA : Set E\nh : HasBasis (\ud835\udcdd 0) q s\n\u22a2 IsVonNBounded \ud835\udd5c A \u2194 \u2200 (i : \u03b9), q i \u2192 Absorbs \ud835\udd5c (s i) A\n[PROOFSTEP]\nrefine' \u27e8fun hA i hi => hA (h.mem_of_mem hi), fun hA V hV => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Zero E\ninst\u271d : TopologicalSpace E\nq : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set E\nA : Set E\nh : HasBasis (\ud835\udcdd 0) q s\nhA : \u2200 (i : \u03b9), q i \u2192 Absorbs \ud835\udd5c (s i) A\nV : Set E\nhV : V \u2208 \ud835\udcdd 0\n\u22a2 Absorbs \ud835\udd5c V A\n[PROOFSTEP]\nrcases h.mem_iff.mp hV with \u27e8i, hi, hV\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b2 : SMul \ud835\udd5c E\ninst\u271d\u00b9 : Zero E\ninst\u271d : TopologicalSpace E\nq : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set E\nA : Set E\nh : HasBasis (\ud835\udcdd 0) q s\nhA : \u2200 (i : \u03b9), q i \u2192 Absorbs \ud835\udd5c (s i) A\nV : Set E\nhV\u271d : V \u2208 \ud835\udcdd 0\ni : \u03b9\nhi : q i\nhV : s i \u2286 V\n\u22a2 Absorbs \ud835\udd5c V A\n[PROOFSTEP]\nexact (hA i hi).mono_left hV\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)\n[PROOFSTEP]\nlet \u03c3' := RingEquiv.ofBijective \u03c3 \u27e8\u03c3.injective, \u03c3.surjective\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)\n[PROOFSTEP]\nhave \u03c3_iso : Isometry \u03c3 := AddMonoidHomClass.isometry_of_norm \u03c3 fun x => RingHomIsometric.is_iso\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)\n[PROOFSTEP]\nhave \u03c3'_symm_iso : Isometry \u03c3'.symm := \u03c3_iso.right_inv \u03c3'.right_inv\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)\n[PROOFSTEP]\nhave f_tendsto_zero := f.continuous.tendsto 0\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd (\u2191f 0))\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)\n[PROOFSTEP]\nrw [map_zero] at f_tendsto_zero \n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)\n[PROOFSTEP]\nintro V hV\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\n\u22a2 Absorbs \ud835\udd5c\u2082 V (\u2191f '' s)\n[PROOFSTEP]\nrcases hs (f_tendsto_zero hV) with \u27e8r, hrpos, hr\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\n\u22a2 Absorbs \ud835\udd5c\u2082 V (\u2191f '' s)\n[PROOFSTEP]\nrefine' \u27e8r, hrpos, fun a ha => _\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\n\u22a2 \u2191f '' s \u2286 a \u2022 V\n[PROOFSTEP]\nrw [\u2190 \u03c3'.apply_symm_apply a]\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\n\u22a2 \u2191f '' s \u2286 \u2191\u03c3' (\u2191(RingEquiv.symm \u03c3') a) \u2022 V\n[PROOFSTEP]\nhave hanz : a \u2260 0 := norm_pos_iff.mp (hrpos.trans_le ha)\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\nhanz : a \u2260 0\n\u22a2 \u2191f '' s \u2286 \u2191\u03c3' (\u2191(RingEquiv.symm \u03c3') a) \u2022 V\n[PROOFSTEP]\nhave : \u03c3'.symm a \u2260 0 := (map_ne_zero \u03c3'.symm.toRingHom).mpr hanz\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\nhanz : a \u2260 0\nthis : \u2191(RingEquiv.symm \u03c3') a \u2260 0\n\u22a2 \u2191f '' s \u2286 \u2191\u03c3' (\u2191(RingEquiv.symm \u03c3') a) \u2022 V\n[PROOFSTEP]\nchange _ \u2286 \u03c3 _ \u2022 _\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\nhanz : a \u2260 0\nthis : \u2191(RingEquiv.symm \u03c3') a \u2260 0\n\u22a2 \u2191f '' s \u2286 \u2191\u03c3 (\u2191(RingEquiv.symm \u03c3') a) \u2022 V\n[PROOFSTEP]\nrw [Set.image_subset_iff, preimage_smul_set\u209b\u2097 _ _ _ f this.isUnit]\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\nhanz : a \u2260 0\nthis : \u2191(RingEquiv.symm \u03c3') a \u2260 0\n\u22a2 s \u2286 \u2191(RingEquiv.symm \u03c3') a \u2022 \u2191f \u207b\u00b9' V\n[PROOFSTEP]\nrefine' hr (\u03c3'.symm a) _\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\nhanz : a \u2260 0\nthis : \u2191(RingEquiv.symm \u03c3') a \u2260 0\n\u22a2 r \u2264 \u2016\u2191(RingEquiv.symm \u03c3') a\u2016\n[PROOFSTEP]\nrwa [\u03c3'_symm_iso.norm_map_of_map_zero (map_zero _)]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5d E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5d E\nS : Set E\n\u03b5 : \u03b9 \u2192 \ud835\udd5c\nx : \u03b9 \u2192 E\nl : Filter \u03b9\nhS : IsVonNBounded \ud835\udd5c S\nhxS : \u2200\u1da0 (n : \u03b9) in l, x n \u2208 S\nh\u03b5 : Tendsto \u03b5 l (\ud835\udcdd 0)\n\u22a2 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [tendsto_def] at *\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5d E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5d E\nS : Set E\n\u03b5 : \u03b9 \u2192 \ud835\udd5c\nx : \u03b9 \u2192 E\nl : Filter \u03b9\nhS : IsVonNBounded \ud835\udd5c S\nhxS : \u2200\u1da0 (n : \u03b9) in l, x n \u2208 S\nh\u03b5 : \u2200 (s : Set \ud835\udd5c), s \u2208 \ud835\udcdd 0 \u2192 \u03b5 \u207b\u00b9' s \u2208 l\n\u22a2 \u2200 (s : Set E), s \u2208 \ud835\udcdd 0 \u2192 (\u03b5 \u2022 x) \u207b\u00b9' s \u2208 l\n[PROOFSTEP]\nintro V hV\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5d E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5d E\nS : Set E\n\u03b5 : \u03b9 \u2192 \ud835\udd5c\nx : \u03b9 \u2192 E\nl : Filter \u03b9\nhS : IsVonNBounded \ud835\udd5c S\nhxS : \u2200\u1da0 (n : \u03b9) in l, x n \u2208 S\nh\u03b5 : \u2200 (s : Set \ud835\udd5c), s \u2208 \ud835\udcdd 0 \u2192 \u03b5 \u207b\u00b9' s \u2208 l\nV : Set E\nhV : V \u2208 \ud835\udcdd 0\n\u22a2 (\u03b5 \u2022 x) \u207b\u00b9' V \u2208 l\n[PROOFSTEP]\nrcases hS hV with \u27e8r, r_pos, hrS\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5d E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5d E\nS : Set E\n\u03b5 : \u03b9 \u2192 \ud835\udd5c\nx : \u03b9 \u2192 E\nl : Filter \u03b9\nhS : IsVonNBounded \ud835\udd5c S\nhxS : \u2200\u1da0 (n : \u03b9) in l, x n \u2208 S\nh\u03b5 : \u2200 (s : Set \ud835\udd5c), s \u2208 \ud835\udcdd 0 \u2192 \u03b5 \u207b\u00b9' s \u2208 l\nV : Set E\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nr_pos : 0 < r\nhrS : \u2200 (a : \ud835\udd5c), r \u2264 \u2016a\u2016 \u2192 S \u2286 a \u2022 V\n\u22a2 (\u03b5 \u2022 x) \u207b\u00b9' V \u2208 l\n[PROOFSTEP]\nfilter_upwards [hxS, h\u03b5 _ (Metric.ball_mem_nhds 0 <| inv_pos.mpr r_pos)] with n hnS hnr\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5d E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5d E\nS : Set E\n\u03b5 : \u03b9 \u2192 \ud835\udd5c\nx : \u03b9 \u2192 E\nl : Filter \u03b9\nhS : IsVonNBounded \ud835\udd5c S\nhxS : \u2200\u1da0 (n : \u03b9) in l, x n \u2208 S\nh\u03b5 : \u2200 (s : Set \ud835\udd5c), s \u2208 \ud835\udcdd 0 \u2192 \u03b5 \u207b\u00b9' s \u2208 l\nV : Set E\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nr_pos : 0 < r\nhrS : \u2200 (a : \ud835\udd5c), r \u2264 \u2016a\u2016 \u2192 S \u2286 a \u2022 V\nn : \u03b9\nhnS : x n \u2208 S\nhnr : n \u2208 \u03b5 \u207b\u00b9' Metric.ball 0 r\u207b\u00b9\n\u22a2 n \u2208 (\u03b5 \u2022 x) \u207b\u00b9' V\n[PROOFSTEP]\nby_cases h\u03b5 : \u03b5 n = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5d E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5d E\nS : Set E\n\u03b5 : \u03b9 \u2192 \ud835\udd5c\nx : \u03b9 \u2192 E\nl : Filter \u03b9\nhS : IsVonNBounded \ud835\udd5c S\nhxS : \u2200\u1da0 (n : \u03b9) in l, x n \u2208 S\nh\u03b5\u271d : \u2200 (s : Set \ud835\udd5c), s \u2208 \ud835\udcdd 0 \u2192 \u03b5 \u207b\u00b9' s \u2208 l\nV : Set E\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nr_pos : 0 < r\nhrS : \u2200 (a : \ud835\udd5c), r \u2264 \u2016a\u2016 \u2192 S \u2286 a \u2022 V\nn : \u03b9\nhnS : x n \u2208 S\nhnr : n \u2208 \u03b5 \u207b\u00b9' Metric.ball 0 r\u207b\u00b9\nh\u03b5 : \u03b5 n = 0\n\u22a2 n \u2208 (\u03b5 \u2022 x) \u207b\u00b9' V\n[PROOFSTEP]\nsimp [h\u03b5, mem_of_mem_nhds hV]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5d E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5d E\nS : Set E\n\u03b5 : \u03b9 \u2192 \ud835\udd5c\nx : \u03b9 \u2192 E\nl : Filter \u03b9\nhS : IsVonNBounded \ud835\udd5c S\nhxS : \u2200\u1da0 (n : \u03b9) in l, x n \u2208 S\nh\u03b5\u271d : \u2200 (s : Set \ud835\udd5c), s \u2208 \ud835\udcdd 0 \u2192 \u03b5 \u207b\u00b9' s \u2208 l\nV : Set E\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nr_pos : 0 < r\nhrS : \u2200 (a : \ud835\udd5c), r \u2264 \u2016a\u2016 \u2192 S \u2286 a \u2022 V\nn : \u03b9\nhnS : x n \u2208 S\nhnr : n \u2208 \u03b5 \u207b\u00b9' Metric.ball 0 r\u207b\u00b9\nh\u03b5 : \u00ac\u03b5 n = 0\n\u22a2 n \u2208 (\u03b5 \u2022 x) \u207b\u00b9' V\n[PROOFSTEP]\nrw [mem_preimage, mem_ball_zero_iff, lt_inv (norm_pos_iff.mpr h\u03b5) r_pos, \u2190 norm_inv] at hnr \n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5d E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5d E\nS : Set E\n\u03b5 : \u03b9 \u2192 \ud835\udd5c\nx : \u03b9 \u2192 E\nl : Filter \u03b9\nhS : IsVonNBounded \ud835\udd5c S\nhxS : \u2200\u1da0 (n : \u03b9) in l, x n \u2208 S\nh\u03b5\u271d : \u2200 (s : Set \ud835\udd5c), s \u2208 \ud835\udcdd 0 \u2192 \u03b5 \u207b\u00b9' s \u2208 l\nV : Set E\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nr_pos : 0 < r\nhrS : \u2200 (a : \ud835\udd5c), r \u2264 \u2016a\u2016 \u2192 S \u2286 a \u2022 V\nn : \u03b9\nhnS : x n \u2208 S\nhnr : r < \u2016(\u03b5 n)\u207b\u00b9\u2016\nh\u03b5 : \u00ac\u03b5 n = 0\n\u22a2 n \u2208 (\u03b5 \u2022 x) \u207b\u00b9' V\n[PROOFSTEP]\nrw [mem_preimage, Pi.smul_apply', \u2190 Set.mem_inv_smul_set_iff\u2080 h\u03b5]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5d E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : ContinuousSMul \ud835\udd5d E\nS : Set E\n\u03b5 : \u03b9 \u2192 \ud835\udd5c\nx : \u03b9 \u2192 E\nl : Filter \u03b9\nhS : IsVonNBounded \ud835\udd5c S\nhxS : \u2200\u1da0 (n : \u03b9) in l, x n \u2208 S\nh\u03b5\u271d : \u2200 (s : Set \ud835\udd5c), s \u2208 \ud835\udcdd 0 \u2192 \u03b5 \u207b\u00b9' s \u2208 l\nV : Set E\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nr_pos : 0 < r\nhrS : \u2200 (a : \ud835\udd5c), r \u2264 \u2016a\u2016 \u2192 S \u2286 a \u2022 V\nn : \u03b9\nhnS : x n \u2208 S\nhnr : r < \u2016(\u03b5 n)\u207b\u00b9\u2016\nh\u03b5 : \u00ac\u03b5 n = 0\n\u22a2 x n \u2208 (\u03b5 n)\u207b\u00b9 \u2022 V\n[PROOFSTEP]\nexact hrS _ hnr.le hnS\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\nS : Set E\nH : \u2200 (x : \u03b9 \u2192 E), (\u2200 (n : \u03b9), x n \u2208 S) \u2192 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\n\u22a2 IsVonNBounded \ud835\udd5d S\n[PROOFSTEP]\nrw [(nhds_basis_balanced \ud835\udd5d E).isVonNBounded_basis_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\nS : Set E\nH : \u2200 (x : \u03b9 \u2192 E), (\u2200 (n : \u03b9), x n \u2208 S) \u2192 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\n\u22a2 \u2200 (i : Set E), i \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5d i \u2192 Absorbs \ud835\udd5d (id i) S\n[PROOFSTEP]\nby_contra' H'\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\nS : Set E\nH : \u2200 (x : \u03b9 \u2192 E), (\u2200 (n : \u03b9), x n \u2208 S) \u2192 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\nH' : \u2203 i, (i \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5d i) \u2227 \u00acAbsorbs \ud835\udd5d (id i) S\n\u22a2 False\n[PROOFSTEP]\nrcases H' with \u27e8V, \u27e8hV, hVb\u27e9, hVS\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\nS : Set E\nH : \u2200 (x : \u03b9 \u2192 E), (\u2200 (n : \u03b9), x n \u2208 S) \u2192 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\nV : Set E\nhVS : \u00acAbsorbs \ud835\udd5d (id V) S\nhV : V \u2208 \ud835\udcdd 0\nhVb : Balanced \ud835\udd5d V\n\u22a2 False\n[PROOFSTEP]\nhave : \u2200\u1da0 n in l, \u2203 x : S, \u03b5 n \u2022 (x : E) \u2209 V :=\n by\n filter_upwards [h\u03b5] with n hn\n rw [Absorbs] at hVS \n push_neg at hVS \n rcases hVS _ (norm_pos_iff.mpr <| inv_ne_zero hn) with \u27e8a, ha\u03b5, haS\u27e9\n rcases Set.not_subset.mp haS with \u27e8x, hxS, hx\u27e9\n refine' \u27e8\u27e8x, hxS\u27e9, fun hnx => _\u27e9\n rw [\u2190 Set.mem_inv_smul_set_iff\u2080 hn] at hnx \n exact hx (hVb.smul_mono ha\u03b5 hnx)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\nS : Set E\nH : \u2200 (x : \u03b9 \u2192 E), (\u2200 (n : \u03b9), x n \u2208 S) \u2192 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\nV : Set E\nhVS : \u00acAbsorbs \ud835\udd5d (id V) S\nhV : V \u2208 \ud835\udcdd 0\nhVb : Balanced \ud835\udd5d V\n\u22a2 \u2200\u1da0 (n : \u03b9) in l, \u2203 x, \u00ac\u03b5 n \u2022 \u2191x \u2208 V\n[PROOFSTEP]\nfilter_upwards [h\u03b5] with n hn\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\nS : Set E\nH : \u2200 (x : \u03b9 \u2192 E), (\u2200 (n : \u03b9), x n \u2208 S) \u2192 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\nV : Set E\nhVS : \u00acAbsorbs \ud835\udd5d (id V) S\nhV : V \u2208 \ud835\udcdd 0\nhVb : Balanced \ud835\udd5d V\nn : \u03b9\nhn : \u03b5 n \u2260 0\n\u22a2 \u2203 x, \u00ac\u03b5 n \u2022 \u2191x \u2208 V\n[PROOFSTEP]\nrw [Absorbs] at hVS \n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\nS : Set E\nH : \u2200 (x : \u03b9 \u2192 E), (\u2200 (n : \u03b9), x n \u2208 S) \u2192 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\nV : Set E\nhVS : \u00ac\u2203 r, 0 < r \u2227 \u2200 (a : \ud835\udd5d), r \u2264 \u2016a\u2016 \u2192 S \u2286 a \u2022 id V\nhV : V \u2208 \ud835\udcdd 0\nhVb : Balanced \ud835\udd5d V\nn : \u03b9\nhn : \u03b5 n \u2260 0\n\u22a2 \u2203 x, \u00ac\u03b5 n \u2022 \u2191x \u2208 V\n[PROOFSTEP]\npush_neg at hVS \n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\nS : Set E\nH : \u2200 (x : \u03b9 \u2192 E), (\u2200 (n : \u03b9), x n \u2208 S) \u2192 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\nV : Set E\nhV : V \u2208 \ud835\udcdd 0\nhVb : Balanced \ud835\udd5d V\nn : \u03b9\nhn : \u03b5 n \u2260 0\nhVS : \u2200 (r : \u211d), 0 < r \u2192 \u2203 a, r \u2264 \u2016a\u2016 \u2227 \u00acS \u2286 a \u2022 id V\n\u22a2 \u2203 x, \u00ac\u03b5 n \u2022 \u2191x \u2208 V\n[PROOFSTEP]\nrcases hVS _ (norm_pos_iff.mpr <| inv_ne_zero hn) with \u27e8a, ha\u03b5, haS\u27e9\n[GOAL]\ncase h.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\nS : Set E\nH : \u2200 (x : \u03b9 \u2192 E), (\u2200 (n : \u03b9), x n \u2208 S) \u2192 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\nV : Set E\nhV : V \u2208 \ud835\udcdd 0\nhVb : Balanced \ud835\udd5d V\nn : \u03b9\nhn : \u03b5 n \u2260 0\nhVS : \u2200 (r : \u211d), 0 < r \u2192 \u2203 a, r \u2264 \u2016a\u2016 \u2227 \u00acS \u2286 a \u2022 id V\na : \ud835\udd5d\nha\u03b5 : \u2016(\u03b5 n)\u207b\u00b9\u2016 \u2264 \u2016a\u2016\nhaS : \u00acS \u2286 a \u2022 id V\n\u22a2 \u2203 x, \u00ac\u03b5 n \u2022 \u2191x \u2208 V\n[PROOFSTEP]\nrcases Set.not_subset.mp haS with \u27e8x, hxS, hx\u27e9\n[GOAL]\ncase h.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\nS : Set E\nH : \u2200 (x : \u03b9 \u2192 E), (\u2200 (n : \u03b9), x n \u2208 S) \u2192 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\nV : Set E\nhV : V \u2208 \ud835\udcdd 0\nhVb : Balanced \ud835\udd5d V\nn : \u03b9\nhn : \u03b5 n \u2260 0\nhVS : \u2200 (r : \u211d), 0 < r \u2192 \u2203 a, r \u2264 \u2016a\u2016 \u2227 \u00acS \u2286 a \u2022 id V\na : \ud835\udd5d\nha\u03b5 : \u2016(\u03b5 n)\u207b\u00b9\u2016 \u2264 \u2016a\u2016\nhaS : \u00acS \u2286 a \u2022 id V\nx : E\nhxS : x \u2208 S\nhx : \u00acx \u2208 a \u2022 id V\n\u22a2 \u2203 x, \u00ac\u03b5 n \u2022 \u2191x \u2208 V\n[PROOFSTEP]\nrefine' \u27e8\u27e8x, hxS\u27e9, fun hnx => _\u27e9\n[GOAL]\ncase h.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\nS : Set E\nH : \u2200 (x : \u03b9 \u2192 E), (\u2200 (n : \u03b9), x n \u2208 S) \u2192 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\nV : Set E\nhV : V \u2208 \ud835\udcdd 0\nhVb : Balanced \ud835\udd5d V\nn : \u03b9\nhn : \u03b5 n \u2260 0\nhVS : \u2200 (r : \u211d), 0 < r \u2192 \u2203 a, r \u2264 \u2016a\u2016 \u2227 \u00acS \u2286 a \u2022 id V\na : \ud835\udd5d\nha\u03b5 : \u2016(\u03b5 n)\u207b\u00b9\u2016 \u2264 \u2016a\u2016\nhaS : \u00acS \u2286 a \u2022 id V\nx : E\nhxS : x \u2208 S\nhx : \u00acx \u2208 a \u2022 id V\nhnx : \u03b5 n \u2022 \u2191{ val := x, property := hxS } \u2208 V\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 Set.mem_inv_smul_set_iff\u2080 hn] at hnx \n[GOAL]\ncase h.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\nS : Set E\nH : \u2200 (x : \u03b9 \u2192 E), (\u2200 (n : \u03b9), x n \u2208 S) \u2192 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\nV : Set E\nhV : V \u2208 \ud835\udcdd 0\nhVb : Balanced \ud835\udd5d V\nn : \u03b9\nhn : \u03b5 n \u2260 0\nhVS : \u2200 (r : \u211d), 0 < r \u2192 \u2203 a, r \u2264 \u2016a\u2016 \u2227 \u00acS \u2286 a \u2022 id V\na : \ud835\udd5d\nha\u03b5 : \u2016(\u03b5 n)\u207b\u00b9\u2016 \u2264 \u2016a\u2016\nhaS : \u00acS \u2286 a \u2022 id V\nx : E\nhxS : x \u2208 S\nhx : \u00acx \u2208 a \u2022 id V\nhnx : \u2191{ val := x, property := hxS } \u2208 (\u03b5 n)\u207b\u00b9 \u2022 V\n\u22a2 False\n[PROOFSTEP]\nexact hx (hVb.smul_mono ha\u03b5 hnx)\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\nS : Set E\nH : \u2200 (x : \u03b9 \u2192 E), (\u2200 (n : \u03b9), x n \u2208 S) \u2192 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\nV : Set E\nhVS : \u00acAbsorbs \ud835\udd5d (id V) S\nhV : V \u2208 \ud835\udcdd 0\nhVb : Balanced \ud835\udd5d V\nthis : \u2200\u1da0 (n : \u03b9) in l, \u2203 x, \u00ac\u03b5 n \u2022 \u2191x \u2208 V\n\u22a2 False\n[PROOFSTEP]\nrcases this.choice with \u27e8x, hx\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\nS : Set E\nH : \u2200 (x : \u03b9 \u2192 E), (\u2200 (n : \u03b9), x n \u2208 S) \u2192 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\nV : Set E\nhVS : \u00acAbsorbs \ud835\udd5d (id V) S\nhV : V \u2208 \ud835\udcdd 0\nhVb : Balanced \ud835\udd5d V\nthis : \u2200\u1da0 (n : \u03b9) in l, \u2203 x, \u00ac\u03b5 n \u2022 \u2191x \u2208 V\nx : \u03b9 \u2192 \u2191S\nhx : \u2200\u1da0 (x_1 : \u03b9) in l, \u00ac\u03b5 x_1 \u2022 \u2191(x x_1) \u2208 V\n\u22a2 False\n[PROOFSTEP]\nrefine' Filter.frequently_false l (Filter.Eventually.frequently _)\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\nS : Set E\nH : \u2200 (x : \u03b9 \u2192 E), (\u2200 (n : \u03b9), x n \u2208 S) \u2192 Tendsto (\u03b5 \u2022 x) l (\ud835\udcdd 0)\nV : Set E\nhVS : \u00acAbsorbs \ud835\udd5d (id V) S\nhV : V \u2208 \ud835\udcdd 0\nhVb : Balanced \ud835\udd5d V\nthis : \u2200\u1da0 (n : \u03b9) in l, \u2203 x, \u00ac\u03b5 n \u2022 \u2191x \u2208 V\nx : \u03b9 \u2192 \u2191S\nhx : \u2200\u1da0 (x_1 : \u03b9) in l, \u00ac\u03b5 x_1 \u2022 \u2191(x x_1) \u2208 V\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, False\n[PROOFSTEP]\nfilter_upwards [hx, (H (_ \u2218 x) fun n => (x n).2).eventually (eventually_mem_set.mpr hV)] using fun n => id\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5d : Type u_7\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : Module \ud835\udd5d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5d E\n\u03b5 : \u03b9 \u2192 \ud835\udd5d\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u03b5 : Tendsto \u03b5 l (\ud835\udcdd[{0}\u1d9c] 0)\nS : Set E\n\u22a2 \u2200\u1da0 (n : \u03b9) in l, \u03b5 n \u2260 0\n[PROOFSTEP]\nexact h\u03b5 self_mem_nhdsWithin\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs : TotallyBounded s\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s\n[PROOFSTEP]\nrw [totallyBounded_iff_subset_finite_iUnion_nhds_zero] at hs \n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d : TotallyBounded s\nhs : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s\n[PROOFSTEP]\nintro U hU\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d : TotallyBounded s\nhs : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\n\u22a2 Absorbs \ud835\udd5c U s\n[PROOFSTEP]\nhave h : Filter.Tendsto (fun x : E \u00d7 E => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd ((0 : E) + (0 : E))) := tendsto_add\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d : TotallyBounded s\nhs : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd (0 + 0))\n\u22a2 Absorbs \ud835\udd5c U s\n[PROOFSTEP]\nrw [add_zero] at h \n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d : TotallyBounded s\nhs : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\n\u22a2 Absorbs \ud835\udd5c U s\n[PROOFSTEP]\nhave h' := (nhds_basis_balanced \ud835\udd5c E).prod (nhds_basis_balanced \ud835\udd5c E)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d : TotallyBounded s\nhs : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nh' :\n HasBasis (\ud835\udcdd 0 \u00d7\u02e2 \ud835\udcdd 0) (fun i => (i.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.fst) \u2227 i.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.snd) fun i =>\n id i.fst \u00d7\u02e2 id i.snd\n\u22a2 Absorbs \ud835\udd5c U s\n[PROOFSTEP]\nsimp_rw [\u2190 nhds_prod_eq, id.def] at h' \n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d : TotallyBounded s\nhs : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nh' :\n HasBasis (\ud835\udcdd (0, 0)) (fun i => (i.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.fst) \u2227 i.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.snd) fun i =>\n i.fst \u00d7\u02e2 i.snd\n\u22a2 Absorbs \ud835\udd5c U s\n[PROOFSTEP]\nrcases h.basis_left h' U hU with \u27e8x, hx, h''\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d : TotallyBounded s\nhs : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nh' :\n HasBasis (\ud835\udcdd (0, 0)) (fun i => (i.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.fst) \u2227 i.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.snd) fun i =>\n i.fst \u00d7\u02e2 i.snd\nx : Set E \u00d7 Set E\nhx : (x.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.fst) \u2227 x.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst \u00d7\u02e2 x.snd) U\n\u22a2 Absorbs \ud835\udd5c U s\n[PROOFSTEP]\nrcases hs x.snd hx.2.1 with \u27e8t, ht, hs\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d\u00b9 : TotallyBounded s\nhs\u271d : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nh' :\n HasBasis (\ud835\udcdd (0, 0)) (fun i => (i.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.fst) \u2227 i.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.snd) fun i =>\n i.fst \u00d7\u02e2 i.snd\nx : Set E \u00d7 Set E\nhx : (x.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.fst) \u2227 x.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst \u00d7\u02e2 x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 x.snd\n\u22a2 Absorbs \ud835\udd5c U s\n[PROOFSTEP]\nrefine' Absorbs.mono_right _ hs\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d\u00b9 : TotallyBounded s\nhs\u271d : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nh' :\n HasBasis (\ud835\udcdd (0, 0)) (fun i => (i.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.fst) \u2227 i.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.snd) fun i =>\n i.fst \u00d7\u02e2 i.snd\nx : Set E \u00d7 Set E\nhx : (x.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.fst) \u2227 x.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst \u00d7\u02e2 x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 x.snd\n\u22a2 Absorbs \ud835\udd5c U (\u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 x.snd)\n[PROOFSTEP]\nrw [ht.absorbs_iUnion]\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d\u00b9 : TotallyBounded s\nhs\u271d : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nh' :\n HasBasis (\ud835\udcdd (0, 0)) (fun i => (i.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.fst) \u2227 i.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.snd) fun i =>\n i.fst \u00d7\u02e2 i.snd\nx : Set E \u00d7 Set E\nhx : (x.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.fst) \u2227 x.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst \u00d7\u02e2 x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 x.snd\n\u22a2 \u2200 (i : E), i \u2208 t \u2192 Absorbs \ud835\udd5c U (i +\u1d65 x.snd)\n[PROOFSTEP]\nhave hx_fstsnd : x.fst + x.snd \u2286 U := by\n intro z hz\n rcases Set.mem_add.mp hz with \u27e8z1, z2, hz1, hz2, hz\u27e9\n have hz' : (z1, z2) \u2208 x.fst \u00d7\u02e2 x.snd := \u27e8hz1, hz2\u27e9\n simpa only [hz] using h'' hz'\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d\u00b9 : TotallyBounded s\nhs\u271d : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nh' :\n HasBasis (\ud835\udcdd (0, 0)) (fun i => (i.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.fst) \u2227 i.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.snd) fun i =>\n i.fst \u00d7\u02e2 i.snd\nx : Set E \u00d7 Set E\nhx : (x.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.fst) \u2227 x.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst \u00d7\u02e2 x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 x.snd\n\u22a2 x.fst + x.snd \u2286 U\n[PROOFSTEP]\nintro z hz\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d\u00b9 : TotallyBounded s\nhs\u271d : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nh' :\n HasBasis (\ud835\udcdd (0, 0)) (fun i => (i.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.fst) \u2227 i.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.snd) fun i =>\n i.fst \u00d7\u02e2 i.snd\nx : Set E \u00d7 Set E\nhx : (x.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.fst) \u2227 x.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst \u00d7\u02e2 x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 x.snd\nz : E\nhz : z \u2208 x.fst + x.snd\n\u22a2 z \u2208 U\n[PROOFSTEP]\nrcases Set.mem_add.mp hz with \u27e8z1, z2, hz1, hz2, hz\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d\u00b9 : TotallyBounded s\nhs\u271d : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nh' :\n HasBasis (\ud835\udcdd (0, 0)) (fun i => (i.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.fst) \u2227 i.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.snd) fun i =>\n i.fst \u00d7\u02e2 i.snd\nx : Set E \u00d7 Set E\nhx : (x.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.fst) \u2227 x.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst \u00d7\u02e2 x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 x.snd\nz : E\nhz\u271d : z \u2208 x.fst + x.snd\nz1 z2 : E\nhz1 : z1 \u2208 x.fst\nhz2 : z2 \u2208 x.snd\nhz : z1 + z2 = z\n\u22a2 z \u2208 U\n[PROOFSTEP]\nhave hz' : (z1, z2) \u2208 x.fst \u00d7\u02e2 x.snd := \u27e8hz1, hz2\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d\u00b9 : TotallyBounded s\nhs\u271d : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nh' :\n HasBasis (\ud835\udcdd (0, 0)) (fun i => (i.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.fst) \u2227 i.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.snd) fun i =>\n i.fst \u00d7\u02e2 i.snd\nx : Set E \u00d7 Set E\nhx : (x.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.fst) \u2227 x.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst \u00d7\u02e2 x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 x.snd\nz : E\nhz\u271d : z \u2208 x.fst + x.snd\nz1 z2 : E\nhz1 : z1 \u2208 x.fst\nhz2 : z2 \u2208 x.snd\nhz : z1 + z2 = z\nhz' : (z1, z2) \u2208 x.fst \u00d7\u02e2 x.snd\n\u22a2 z \u2208 U\n[PROOFSTEP]\nsimpa only [hz] using h'' hz'\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d\u00b9 : TotallyBounded s\nhs\u271d : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nh' :\n HasBasis (\ud835\udcdd (0, 0)) (fun i => (i.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.fst) \u2227 i.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.snd) fun i =>\n i.fst \u00d7\u02e2 i.snd\nx : Set E \u00d7 Set E\nhx : (x.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.fst) \u2227 x.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst \u00d7\u02e2 x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 x.snd\nhx_fstsnd : x.fst + x.snd \u2286 U\n\u22a2 \u2200 (i : E), i \u2208 t \u2192 Absorbs \ud835\udd5c U (i +\u1d65 x.snd)\n[PROOFSTEP]\nrefine' fun y _ => Absorbs.mono_left _ hx_fstsnd\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d\u00b9 : TotallyBounded s\nhs\u271d : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nh' :\n HasBasis (\ud835\udcdd (0, 0)) (fun i => (i.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.fst) \u2227 i.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.snd) fun i =>\n i.fst \u00d7\u02e2 i.snd\nx : Set E \u00d7 Set E\nhx : (x.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.fst) \u2227 x.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst \u00d7\u02e2 x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 x.snd\nhx_fstsnd : x.fst + x.snd \u2286 U\ny : E\nx\u271d : y \u2208 t\n\u22a2 Absorbs \ud835\udd5c (x.fst + x.snd) (y +\u1d65 x.snd)\n[PROOFSTEP]\nrw [\u2190 Set.singleton_vadd, vadd_eq_add]\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : UniformSpace E\ninst\u271d\u00b9 : UniformAddGroup E\ninst\u271d : ContinuousSMul \ud835\udd5c E\ns : Set E\nhs\u271d\u00b9 : TotallyBounded s\nhs\u271d : \u2200 (U : Set E), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 U\nU : Set E\nhU : U \u2208 \ud835\udcdd 0\nh : Tendsto (fun x => x.fst + x.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nh' :\n HasBasis (\ud835\udcdd (0, 0)) (fun i => (i.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.fst) \u2227 i.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c i.snd) fun i =>\n i.fst \u00d7\u02e2 i.snd\nx : Set E \u00d7 Set E\nhx : (x.fst \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.fst) \u2227 x.snd \u2208 \ud835\udcdd 0 \u2227 Balanced \ud835\udd5c x.snd\nh'' : MapsTo (fun x => x.fst + x.snd) (x.fst \u00d7\u02e2 x.snd) U\nt : Set E\nht : Set.Finite t\nhs : s \u2286 \u22c3 (y : E) (_ : y \u2208 t), y +\u1d65 x.snd\nhx_fstsnd : x.fst + x.snd \u2286 U\ny : E\nx\u271d : y \u2208 t\n\u22a2 Absorbs \ud835\udd5c (x.fst + x.snd) ({y} + x.snd)\n[PROOFSTEP]\nexact (absorbent_nhds_zero hx.1.1).absorbs.add hx.2.2.absorbs_self\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c (Metric.ball 0 r)\n[PROOFSTEP]\nrw [Metric.nhds_basis_ball.isVonNBounded_basis_iff, \u2190 ball_normSeminorm \ud835\udd5c E]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\n\u22a2 \u2200 (i : \u211d), 0 < i \u2192 Absorbs \ud835\udd5c (Seminorm.ball (normSeminorm \ud835\udd5c E) 0 i) (Seminorm.ball (normSeminorm \ud835\udd5c E) 0 r)\n[PROOFSTEP]\nexact fun \u03b5 h\u03b5 => (normSeminorm \ud835\udd5c E).ball_zero_absorbs_ball_zero h\u03b5\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\n\u22a2 r < r + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s \u2194 Bornology.IsBounded s\n[PROOFSTEP]\nrw [\u2190 Metric.bounded_iff_isBounded, Metric.bounded_iff_subset_ball (0 : E)]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s \u2194 \u2203 r, s \u2286 Metric.closedBall 0 r\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s \u2192 \u2203 r, s \u2286 Metric.closedBall 0 r\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r\n[PROOFSTEP]\nrcases h (Metric.ball_mem_nhds 0 zero_lt_one) with \u27e8\u03c1, h\u03c1, h\u03c1ball\u27e9\n[GOAL]\ncase mp.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u03c1 : \u211d\nh\u03c1 : 0 < \u03c1\nh\u03c1ball : \u2200 (a : \ud835\udd5c), \u03c1 \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 Metric.ball 0 1\n\u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r\n[PROOFSTEP]\nrcases NormedField.exists_lt_norm \ud835\udd5c \u03c1 with \u27e8a, ha\u27e9\n[GOAL]\ncase mp.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u03c1 : \u211d\nh\u03c1 : 0 < \u03c1\nh\u03c1ball : \u2200 (a : \ud835\udd5c), \u03c1 \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 Metric.ball 0 1\na : \ud835\udd5c\nha : \u03c1 < \u2016a\u2016\n\u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r\n[PROOFSTEP]\nspecialize h\u03c1ball a ha.le\n[GOAL]\ncase mp.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u03c1 : \u211d\nh\u03c1 : 0 < \u03c1\na : \ud835\udd5c\nha : \u03c1 < \u2016a\u2016\nh\u03c1ball : s \u2286 a \u2022 Metric.ball 0 1\n\u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r\n[PROOFSTEP]\nrw [\u2190 ball_normSeminorm \ud835\udd5c E, Seminorm.smul_ball_zero (norm_pos_iff.1 <| h\u03c1.trans ha), ball_normSeminorm, mul_one] at\n h\u03c1ball \n[GOAL]\ncase mp.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u03c1 : \u211d\nh\u03c1 : 0 < \u03c1\na : \ud835\udd5c\nha : \u03c1 < \u2016a\u2016\nh\u03c1ball : s \u2286 Metric.ball 0 \u2016a\u2016\n\u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r\n[PROOFSTEP]\nexact \u27e8\u2016a\u2016, h\u03c1ball.trans Metric.ball_subset_closedBall\u27e9\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 (\u2203 r, s \u2286 Metric.closedBall 0 r) \u2192 Bornology.IsVonNBounded \ud835\udd5c s\n[PROOFSTEP]\nexact fun \u27e8C, hC\u27e9 => (isVonNBounded_closedBall \ud835\udd5c E C).subset hC\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s \u2194 \u2203 r, \u2200 (x : E), x \u2208 s \u2192 \u2016x\u2016 \u2264 r\n[PROOFSTEP]\nrw [NormedSpace.isVonNBounded_iff, \u2190 Metric.bounded_iff_isBounded, bounded_iff_forall_norm_le]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : E' \u2192 E\ns : Set E'\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c (f '' s) \u2194 \u2203 r, \u2200 (x : E'), x \u2208 s \u2192 \u2016f x\u2016 \u2264 r\n[PROOFSTEP]\nsimp_rw [isVonNBounded_iff', Set.ball_image_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\n\u22a2 Bornology.vonNBornology \ud835\udd5c E = PseudoMetricSpace.toBornology\n[PROOFSTEP]\nrw [Bornology.ext_iff_isBounded]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\n\u22a2 \u2200 (s : Set E), Bornology.IsBounded s \u2194 Bornology.IsBounded s\n[PROOFSTEP]\nintro s\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsBounded s \u2194 Bornology.IsBounded s\n[PROOFSTEP]\nrw [Bornology.isBounded_iff_isVonNBounded]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s \u2194 Bornology.IsBounded s\n[PROOFSTEP]\nexact isVonNBounded_iff \ud835\udd5c E s\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsBounded s \u2194 \u2203 a, s \u2286 a \u2022 Metric.ball 0 1\n[PROOFSTEP]\nrw [\u2190 isVonNBounded_iff \ud835\udd5c]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s \u2194 \u2203 a, s \u2286 a \u2022 Metric.ball 0 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s \u2192 \u2203 a, s \u2286 a \u2022 Metric.ball 0 1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u22a2 \u2203 a, s \u2286 a \u2022 Metric.ball 0 1\n[PROOFSTEP]\nrcases h (Metric.ball_mem_nhds 0 zero_lt_one) with \u27e8\u03c1, _, h\u03c1ball\u27e9\n[GOAL]\ncase mp.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u03c1 : \u211d\nleft\u271d : 0 < \u03c1\nh\u03c1ball : \u2200 (a : \ud835\udd5c), \u03c1 \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 Metric.ball 0 1\n\u22a2 \u2203 a, s \u2286 a \u2022 Metric.ball 0 1\n[PROOFSTEP]\nrcases NormedField.exists_lt_norm \ud835\udd5c \u03c1 with \u27e8a, ha\u27e9\n[GOAL]\ncase mp.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u03c1 : \u211d\nleft\u271d : 0 < \u03c1\nh\u03c1ball : \u2200 (a : \ud835\udd5c), \u03c1 \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 Metric.ball 0 1\na : \ud835\udd5c\nha : \u03c1 < \u2016a\u2016\n\u22a2 \u2203 a, s \u2286 a \u2022 Metric.ball 0 1\n[PROOFSTEP]\nexact \u27e8a, h\u03c1ball a ha.le\u27e9\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 (\u2203 a, s \u2286 a \u2022 Metric.ball 0 1) \u2192 Bornology.IsVonNBounded \ud835\udd5c s\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\na : \ud835\udd5c\nha : s \u2286 a \u2022 Metric.ball 0 1\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s\n[PROOFSTEP]\nexact ((isVonNBounded_ball \ud835\udd5c E 1).image (a \u2022 (1 : E \u2192L[\ud835\udd5c] E))).subset ha\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsBounded s \u2194 \u2203 a, s \u2286 a \u2022 Metric.closedBall 0 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsBounded s \u2192 \u2203 a, s \u2286 a \u2022 Metric.closedBall 0 1\n[PROOFSTEP]\nrw [isBounded_iff_subset_smul_ball \ud835\udd5c]\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 (\u2203 a, s \u2286 a \u2022 Metric.ball 0 1) \u2192 \u2203 a, s \u2286 a \u2022 Metric.closedBall 0 1\n[PROOFSTEP]\nexact Exists.imp fun a ha => ha.trans <| Set.smul_set_mono <| Metric.ball_subset_closedBall\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 (\u2203 a, s \u2286 a \u2022 Metric.closedBall 0 1) \u2192 Bornology.IsBounded s\n[PROOFSTEP]\nrw [\u2190 isVonNBounded_iff \ud835\udd5c]\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 (\u2203 a, s \u2286 a \u2022 Metric.closedBall 0 1) \u2192 Bornology.IsVonNBounded \ud835\udd5c s\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\na : \ud835\udd5c\nha : s \u2286 a \u2022 Metric.closedBall 0 1\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s\n[PROOFSTEP]\nexact ((isVonNBounded_closedBall \ud835\udd5c E 1).image (a \u2022 (1 : E \u2192L[\ud835\udd5c] E))).subset ha\n", "meta": {"mathlib_filename": "Mathlib.Analysis.LocallyConvex.Bounded", "llama_tokens": 29876, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581741774411, "lm_q2_score": 0.5660185351961013, "lm_q1q2_score": 0.4172451899557478}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.8376, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\nn : \u2115\n\u22a2 underlying.obj (objX X (n + 1 + 1)) \u27f6 underlying.obj (objX X (n + 1))\n[PROOFSTEP]\nrefine'\n factorThru _ (arrow _ \u226b X.\u03b4 (0 : Fin (n + 3)))\n _\n -- We now need to show that it factors!\n -- A morphism factors through an intersection of subobjects if it factors through each.\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.8376, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\nn : \u2115\n\u22a2 Factors (objX X (n + 1)) (arrow (objX X (n + 1 + 1)) \u226b SimplicialObject.\u03b4 X 0)\n[PROOFSTEP]\nrefine'\n (finset_inf_factors _).mpr fun i _ =>\n _\n -- A morphism `f` factors through the kernel of `g` exactly if `f \u226b g = 0`.\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.8376, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\nn : \u2115\ni : Fin (Nat.add n 0 + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 Factors (kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ i))) (arrow (objX X (n + 1 + 1)) \u226b SimplicialObject.\u03b4 X 0)\n[PROOFSTEP]\napply kernelSubobject_factors\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.8376, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\nn : \u2115\ni : Fin (Nat.add n 0 + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 (arrow (objX X (n + 1 + 1)) \u226b SimplicialObject.\u03b4 X 0) \u226b SimplicialObject.\u03b4 X (Fin.succ i) = 0\n[PROOFSTEP]\ndsimp [objX]\n -- Use a simplicial identity\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.8376, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\nn : \u2115\ni : Fin (Nat.add n 0 + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n SimplicialObject.\u03b4 X 0) \u226b\n SimplicialObject.\u03b4 X (Fin.succ i) =\n 0\n[PROOFSTEP]\nerw [Category.assoc, \u2190 X.\u03b4_comp_\u03b4 (Fin.zero_le i.succ)]\n -- We can rewrite the arrow out of the intersection of all the kernels as a composition\n -- of a morphism we don't care about with the arrow out of the kernel of `X.\u03b4 i.succ.succ`.\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.8376, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\nn : \u2115\ni : Fin (Nat.add n 0 + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n SimplicialObject.\u03b4 X (Fin.succ (Fin.succ i)) \u226b SimplicialObject.\u03b4 X 0 =\n 0\n[PROOFSTEP]\nrw [\u2190 factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ i.succ (by simp)), Category.assoc,\n kernelSubobject_arrow_comp_assoc, zero_comp, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.8376, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\nn : \u2115\ni : Fin (Nat.add n 0 + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 Fin.succ i \u2208 Finset.univ\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\nn : \u2115\n\u22a2 objD X (n + 1) \u226b objD X n = 0\n[PROOFSTEP]\nrcases n with _ | n\n[GOAL]\ncase zero\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\n\u22a2 objD X (Nat.zero + 1) \u226b objD X Nat.zero = 0\n[PROOFSTEP]\ndsimp [objD]\n[GOAL]\ncase succ\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\nn : \u2115\n\u22a2 objD X (Nat.succ n + 1) \u226b objD X (Nat.succ n) = 0\n[PROOFSTEP]\ndsimp [objD]\n[GOAL]\ncase zero\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\n\u22a2 factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k)))\n (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n SimplicialObject.\u03b4 X 0)\n (_ :\n Factors (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k)))\n (arrow (objX X (0 + 1 + 1)) \u226b SimplicialObject.\u03b4 X 0)) \u226b\n arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n SimplicialObject.\u03b4 X 0 \u226b inv (arrow \u22a4) =\n 0\n[PROOFSTEP]\nerw [Subobject.factorThru_arrow_assoc, Category.assoc, \u2190 X.\u03b4_comp_\u03b4_assoc (Fin.zero_le (0 : Fin 2)), \u2190\n factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ (0 : Fin 2) (by simp)), Category.assoc,\n kernelSubobject_arrow_comp_assoc, zero_comp, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\n\u22a2 0 \u2208 Finset.univ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\nn : \u2115\n\u22a2 factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k)))\n (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n SimplicialObject.\u03b4 X 0)\n (_ :\n Factors (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k)))\n (arrow (objX X (n + 1 + 1 + 1)) \u226b SimplicialObject.\u03b4 X 0)) \u226b\n factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k)))\n (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n SimplicialObject.\u03b4 X 0)\n (_ :\n Factors (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k)))\n (arrow (objX X (n + 1 + 1)) \u226b SimplicialObject.\u03b4 X 0)) =\n 0\n[PROOFSTEP]\nerw [factorThru_right, factorThru_eq_zero, factorThru_arrow_assoc, Category.assoc, \u2190\n X.\u03b4_comp_\u03b4 (Fin.zero_le (0 : Fin (n + 3))), \u2190\n factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ (0 : Fin (n + 3)) (by simp)), Category.assoc,\n kernelSubobject_arrow_comp_assoc, zero_comp, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\nn : \u2115\n\u22a2 0 \u2208 Finset.univ\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.84232, u_1} C\ninst\u271d : Abelian C\nX Y : SimplicialObject C\nf\u271d f : X \u27f6 Y\nn : \u2115\n\u22a2 Factors (objX Y n) (arrow (objX X n) \u226b NatTrans.app f (op (SimplexCategory.mk n)))\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.84232, u_1} C\ninst\u271d : Abelian C\nX Y : SimplicialObject C\nf\u271d f : X \u27f6 Y\n\u22a2 Factors (objX Y Nat.zero) (arrow (objX X Nat.zero) \u226b NatTrans.app f (op (SimplexCategory.mk Nat.zero)))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase succ\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.84232, u_1} C\ninst\u271d : Abelian C\nX Y : SimplicialObject C\nf\u271d f : X \u27f6 Y\nn\u271d : \u2115\n\u22a2 Factors (objX Y (Nat.succ n\u271d)) (arrow (objX X (Nat.succ n\u271d)) \u226b NatTrans.app f (op (SimplexCategory.mk (Nat.succ n\u271d))))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase zero\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.84232, u_1} C\ninst\u271d : Abelian C\nX Y : SimplicialObject C\nf\u271d f : X \u27f6 Y\n\u22a2 Factors \u22a4 (arrow \u22a4 \u226b NatTrans.app f (op (SimplexCategory.mk 0)))\n[PROOFSTEP]\napply top_factors\n[GOAL]\ncase succ\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.84232, u_1} C\ninst\u271d : Abelian C\nX Y : SimplicialObject C\nf\u271d f : X \u27f6 Y\nn\u271d : \u2115\n\u22a2 Factors (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Y (Fin.succ k)))\n (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n NatTrans.app f (op (SimplexCategory.mk (Nat.succ n\u271d))))\n[PROOFSTEP]\nrefine' (finset_inf_factors _).mpr fun i _ => kernelSubobject_factors _ _ _\n[GOAL]\ncase succ\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.84232, u_1} C\ninst\u271d : Abelian C\nX Y : SimplicialObject C\nf\u271d f : X \u27f6 Y\nn\u271d : \u2115\ni : Fin (n\u271d + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n NatTrans.app f (op (SimplexCategory.mk (Nat.succ n\u271d)))) \u226b\n SimplicialObject.\u03b4 Y (Fin.succ i) =\n 0\n[PROOFSTEP]\nerw [Category.assoc, \u2190 f.naturality, \u2190 factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ i (by simp)),\n Category.assoc, kernelSubobject_arrow_comp_assoc, zero_comp, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.84232, u_1} C\ninst\u271d : Abelian C\nX Y : SimplicialObject C\nf\u271d f : X \u27f6 Y\nn\u271d : \u2115\ni : Fin (n\u271d + 1)\nx\u271d : i \u2208 Finset.univ\n\u22a2 i \u2208 Finset.univ\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.84232, u_1} C\ninst\u271d : Abelian C\nX Y : SimplicialObject C\nf\u271d f : X \u27f6 Y\nn : \u2115\n\u22a2 (fun n =>\n factorThru (objX Y n) (arrow (objX X n) \u226b NatTrans.app f (op (SimplexCategory.mk n)))\n (_ : Factors (objX Y n) (arrow (objX X n) \u226b NatTrans.app f (op (SimplexCategory.mk n)))))\n (n + 1) \u226b\n objD Y n =\n objD X n \u226b\n (fun n =>\n factorThru (objX Y n) (arrow (objX X n) \u226b NatTrans.app f (op (SimplexCategory.mk n)))\n (_ : Factors (objX Y n) (arrow (objX X n) \u226b NatTrans.app f (op (SimplexCategory.mk n)))))\n n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.84232, u_1} C\ninst\u271d : Abelian C\nX Y : SimplicialObject C\nf\u271d f : X \u27f6 Y\n\u22a2 (fun n =>\n factorThru (objX Y n) (arrow (objX X n) \u226b NatTrans.app f (op (SimplexCategory.mk n)))\n (_ : Factors (objX Y n) (arrow (objX X n) \u226b NatTrans.app f (op (SimplexCategory.mk n)))))\n (Nat.zero + 1) \u226b\n objD Y Nat.zero =\n objD X Nat.zero \u226b\n (fun n =>\n factorThru (objX Y n) (arrow (objX X n) \u226b NatTrans.app f (op (SimplexCategory.mk n)))\n (_ : Factors (objX Y n) (arrow (objX X n) \u226b NatTrans.app f (op (SimplexCategory.mk n)))))\n Nat.zero\n[PROOFSTEP]\ndsimp [objD, objX]\n[GOAL]\ncase succ\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.84232, u_1} C\ninst\u271d : Abelian C\nX Y : SimplicialObject C\nf\u271d f : X \u27f6 Y\nn\u271d : \u2115\n\u22a2 (fun n =>\n factorThru (objX Y n) (arrow (objX X n) \u226b NatTrans.app f (op (SimplexCategory.mk n)))\n (_ : Factors (objX Y n) (arrow (objX X n) \u226b NatTrans.app f (op (SimplexCategory.mk n)))))\n (Nat.succ n\u271d + 1) \u226b\n objD Y (Nat.succ n\u271d) =\n objD X (Nat.succ n\u271d) \u226b\n (fun n =>\n factorThru (objX Y n) (arrow (objX X n) \u226b NatTrans.app f (op (SimplexCategory.mk n)))\n (_ : Factors (objX Y n) (arrow (objX X n) \u226b NatTrans.app f (op (SimplexCategory.mk n)))))\n (Nat.succ n\u271d)\n[PROOFSTEP]\ndsimp [objD, objX]\n[GOAL]\ncase zero\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.84232, u_1} C\ninst\u271d : Abelian C\nX Y : SimplicialObject C\nf\u271d f : X \u27f6 Y\n\u22a2 factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Y (Fin.succ k)))\n (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n NatTrans.app f (op (SimplexCategory.mk (0 + 1))))\n (_ :\n Factors (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Y (Fin.succ k)))\n (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n NatTrans.app f (op (SimplexCategory.mk (Nat.succ 0))))) \u226b\n arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Y (Fin.succ k))) \u226b\n SimplicialObject.\u03b4 Y 0 \u226b inv (arrow \u22a4) =\n (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n SimplicialObject.\u03b4 X 0 \u226b inv (arrow \u22a4)) \u226b\n factorThru \u22a4 (arrow \u22a4 \u226b NatTrans.app f (op (SimplexCategory.mk 0)))\n (_ : Factors \u22a4 (arrow \u22a4 \u226b NatTrans.app f (op (SimplexCategory.mk 0))))\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase succ\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.84232, u_1} C\ninst\u271d : Abelian C\nX Y : SimplicialObject C\nf\u271d f : X \u27f6 Y\nn\u271d : \u2115\n\u22a2 factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Y (Fin.succ k)))\n (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n NatTrans.app f (op (SimplexCategory.mk (Nat.succ n\u271d + 1))))\n (_ :\n Factors (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Y (Fin.succ k)))\n (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n NatTrans.app f (op (SimplexCategory.mk (Nat.succ (Nat.succ n\u271d)))))) \u226b\n factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Y (Fin.succ k)))\n (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Y (Fin.succ k))) \u226b\n SimplicialObject.\u03b4 Y 0)\n (_ :\n Factors (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Y (Fin.succ k)))\n (arrow (objX Y (n\u271d + 1 + 1)) \u226b SimplicialObject.\u03b4 Y 0)) =\n factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k)))\n (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n SimplicialObject.\u03b4 X 0)\n (_ :\n Factors (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k)))\n (arrow (objX X (n\u271d + 1 + 1)) \u226b SimplicialObject.\u03b4 X 0)) \u226b\n factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Y (Fin.succ k)))\n (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n NatTrans.app f (op (SimplexCategory.mk (Nat.succ n\u271d))))\n (_ :\n Factors (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Y (Fin.succ k)))\n (arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n NatTrans.app f (op (SimplexCategory.mk (Nat.succ n\u271d)))))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.127901, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\n\u22a2 { obj := obj, map := fun {X Y} f => map f }.map (\ud835\udfd9 X) = \ud835\udfd9 ({ obj := obj, map := fun {X Y} f => map f }.obj X)\n[PROOFSTEP]\next (_ | _)\n[GOAL]\ncase h.zero\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.127901, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\n\u22a2 HomologicalComplex.Hom.f ({ obj := obj, map := fun {X Y} f => map f }.map (\ud835\udfd9 X)) Nat.zero =\n HomologicalComplex.Hom.f (\ud835\udfd9 ({ obj := obj, map := fun {X Y} f => map f }.obj X)) Nat.zero\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.succ\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.127901, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\nn\u271d : \u2115\n\u22a2 HomologicalComplex.Hom.f ({ obj := obj, map := fun {X Y} f => map f }.map (\ud835\udfd9 X)) (Nat.succ n\u271d) =\n HomologicalComplex.Hom.f (\ud835\udfd9 ({ obj := obj, map := fun {X Y} f => map f }.obj X)) (Nat.succ n\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.zero\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.127901, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\n\u22a2 Subobject.factorThru \u22a4 (Subobject.arrow \u22a4 \u226b \ud835\udfd9 (X.obj (op (SimplexCategory.mk 0))))\n (_ : Subobject.Factors (objX X 0) (Subobject.arrow (objX X 0) \u226b NatTrans.app (\ud835\udfd9 X) (op (SimplexCategory.mk 0)))) =\n \ud835\udfd9 (Subobject.underlying.obj \u22a4)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase h.succ\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.127901, u_1} C\ninst\u271d : Abelian C\nX : SimplicialObject C\nn\u271d : \u2115\n\u22a2 Subobject.factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k)))\n (Subobject.arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))) \u226b\n \ud835\udfd9 (X.obj (op (SimplexCategory.mk (Nat.succ n\u271d)))))\n (_ :\n Subobject.Factors (objX X (Nat.succ n\u271d))\n (Subobject.arrow (objX X (Nat.succ n\u271d)) \u226b NatTrans.app (\ud835\udfd9 X) (op (SimplexCategory.mk (Nat.succ n\u271d))))) =\n \ud835\udfd9 (Subobject.underlying.obj (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X (Fin.succ k))))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.127901, u_1} C\ninst\u271d : Abelian C\nX\u271d Y\u271d Z\u271d : SimplicialObject C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := obj, map := fun {X Y} f => map f }.map (f \u226b g) =\n { obj := obj, map := fun {X Y} f => map f }.map f \u226b { obj := obj, map := fun {X Y} f => map f }.map g\n[PROOFSTEP]\next (_ | _)\n[GOAL]\ncase h.zero\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.127901, u_1} C\ninst\u271d : Abelian C\nX\u271d Y\u271d Z\u271d : SimplicialObject C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 HomologicalComplex.Hom.f ({ obj := obj, map := fun {X Y} f => map f }.map (f \u226b g)) Nat.zero =\n HomologicalComplex.Hom.f\n ({ obj := obj, map := fun {X Y} f => map f }.map f \u226b { obj := obj, map := fun {X Y} f => map f }.map g) Nat.zero\n[PROOFSTEP]\napply Subobject.eq_of_comp_arrow_eq\n[GOAL]\ncase h.succ\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.127901, u_1} C\ninst\u271d : Abelian C\nX\u271d Y\u271d Z\u271d : SimplicialObject C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\nn\u271d : \u2115\n\u22a2 HomologicalComplex.Hom.f ({ obj := obj, map := fun {X Y} f => map f }.map (f \u226b g)) (Nat.succ n\u271d) =\n HomologicalComplex.Hom.f\n ({ obj := obj, map := fun {X Y} f => map f }.map f \u226b { obj := obj, map := fun {X Y} f => map f }.map g)\n (Nat.succ n\u271d)\n[PROOFSTEP]\napply Subobject.eq_of_comp_arrow_eq\n[GOAL]\ncase h.zero.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.127901, u_1} C\ninst\u271d : Abelian C\nX\u271d Y\u271d Z\u271d : SimplicialObject C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 HomologicalComplex.Hom.f ({ obj := obj, map := fun {X Y} f => map f }.map (f \u226b g)) Nat.zero \u226b\n Subobject.arrow (objX Z\u271d Nat.zero) =\n HomologicalComplex.Hom.f\n ({ obj := obj, map := fun {X Y} f => map f }.map f \u226b { obj := obj, map := fun {X Y} f => map f }.map g)\n Nat.zero \u226b\n Subobject.arrow (objX Z\u271d Nat.zero)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.succ.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.127901, u_1} C\ninst\u271d : Abelian C\nX\u271d Y\u271d Z\u271d : SimplicialObject C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\nn\u271d : \u2115\n\u22a2 HomologicalComplex.Hom.f ({ obj := obj, map := fun {X Y} f => map f }.map (f \u226b g)) (Nat.succ n\u271d) \u226b\n Subobject.arrow (objX Z\u271d (Nat.succ n\u271d)) =\n HomologicalComplex.Hom.f\n ({ obj := obj, map := fun {X Y} f => map f }.map f \u226b { obj := obj, map := fun {X Y} f => map f }.map g)\n (Nat.succ n\u271d) \u226b\n Subobject.arrow (objX Z\u271d (Nat.succ n\u271d))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.zero.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.127901, u_1} C\ninst\u271d : Abelian C\nX\u271d Y\u271d Z\u271d : SimplicialObject C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 Subobject.factorThru \u22a4\n (Subobject.arrow \u22a4 \u226b NatTrans.app f (op (SimplexCategory.mk 0)) \u226b NatTrans.app g (op (SimplexCategory.mk 0)))\n (_ :\n Subobject.Factors (objX Z\u271d 0)\n (Subobject.arrow (objX X\u271d 0) \u226b NatTrans.app (f \u226b g) (op (SimplexCategory.mk 0)))) \u226b\n Subobject.arrow \u22a4 =\n (Subobject.factorThru \u22a4 (Subobject.arrow \u22a4 \u226b NatTrans.app f (op (SimplexCategory.mk 0)))\n (_ :\n Subobject.Factors (objX Y\u271d 0) (Subobject.arrow (objX X\u271d 0) \u226b NatTrans.app f (op (SimplexCategory.mk 0)))) \u226b\n Subobject.factorThru \u22a4 (Subobject.arrow \u22a4 \u226b NatTrans.app g (op (SimplexCategory.mk 0)))\n (_ :\n Subobject.Factors (objX Z\u271d 0) (Subobject.arrow (objX Y\u271d 0) \u226b NatTrans.app g (op (SimplexCategory.mk 0))))) \u226b\n Subobject.arrow \u22a4\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase h.succ.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.127901, u_1} C\ninst\u271d : Abelian C\nX\u271d Y\u271d Z\u271d : SimplicialObject C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\nn\u271d : \u2115\n\u22a2 Subobject.factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Z\u271d (Fin.succ k)))\n (Subobject.arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X\u271d (Fin.succ k))) \u226b\n NatTrans.app f (op (SimplexCategory.mk (Nat.succ n\u271d))) \u226b\n NatTrans.app g (op (SimplexCategory.mk (Nat.succ n\u271d))))\n (_ :\n Subobject.Factors (objX Z\u271d (Nat.succ n\u271d))\n (Subobject.arrow (objX X\u271d (Nat.succ n\u271d)) \u226b NatTrans.app (f \u226b g) (op (SimplexCategory.mk (Nat.succ n\u271d))))) \u226b\n Subobject.arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Z\u271d (Fin.succ k))) =\n (Subobject.factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Y\u271d (Fin.succ k)))\n (Subobject.arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 X\u271d (Fin.succ k))) \u226b\n NatTrans.app f (op (SimplexCategory.mk (Nat.succ n\u271d))))\n (_ :\n Subobject.Factors (objX Y\u271d (Nat.succ n\u271d))\n (Subobject.arrow (objX X\u271d (Nat.succ n\u271d)) \u226b NatTrans.app f (op (SimplexCategory.mk (Nat.succ n\u271d))))) \u226b\n Subobject.factorThru (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Z\u271d (Fin.succ k)))\n (Subobject.arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Y\u271d (Fin.succ k))) \u226b\n NatTrans.app g (op (SimplexCategory.mk (Nat.succ n\u271d))))\n (_ :\n Subobject.Factors (objX Z\u271d (Nat.succ n\u271d))\n (Subobject.arrow (objX Y\u271d (Nat.succ n\u271d)) \u226b NatTrans.app g (op (SimplexCategory.mk (Nat.succ n\u271d)))))) \u226b\n Subobject.arrow (Finset.inf Finset.univ fun k => kernelSubobject (SimplicialObject.\u03b4 Z\u271d (Fin.succ k)))\n[PROOFSTEP]\naesop_cat\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicTopology.MooreComplex", "llama_tokens": 9188, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581626286834, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.4172451834189369}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g\u271d : \u03b9 \u2192 Set X\ng : \u03b9' \u2192 \u03b9\nhf : LocallyFinite f\nhg : InjOn g {i | Set.Nonempty (f (g i))}\nx : X\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd x \u2227 Set.Finite {i | Set.Nonempty ((f \u2218 g) i \u2229 t)}\n[PROOFSTEP]\nlet \u27e8t, htx, htf\u27e9 := hf x\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g\u271d : \u03b9 \u2192 Set X\ng : \u03b9' \u2192 \u03b9\nhf : LocallyFinite f\nhg : InjOn g {i | Set.Nonempty (f (g i))}\nx : X\nt : Set X\nhtx : t \u2208 \ud835\udcdd x\nhtf : Set.Finite {i | Set.Nonempty (f i \u2229 t)}\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd x \u2227 Set.Finite {i | Set.Nonempty ((f \u2218 g) i \u2229 t)}\n[PROOFSTEP]\nrefine \u27e8t, htx, htf.preimage <| ?_\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g\u271d : \u03b9 \u2192 Set X\ng : \u03b9' \u2192 \u03b9\nhf : LocallyFinite f\nhg : InjOn g {i | Set.Nonempty (f (g i))}\nx : X\nt : Set X\nhtx : t \u2208 \ud835\udcdd x\nhtf : Set.Finite {i | Set.Nonempty (f i \u2229 t)}\n\u22a2 InjOn (fun i => g i) ((fun i => g i) \u207b\u00b9' {i | Set.Nonempty (f i \u2229 t)})\n[PROOFSTEP]\nexact hg.mono fun i (hi : Set.Nonempty _) => hi.left\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nhf : LocallyFinite f\na : X\n\u22a2 \ud835\udcdd[\u22c3 (i : \u03b9), f i] a = \u2a06 (i : \u03b9), \ud835\udcdd[f i] a\n[PROOFSTEP]\nrcases hf a with \u27e8U, haU, hfin\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nhf : LocallyFinite f\na : X\nU : Set X\nhaU : U \u2208 \ud835\udcdd a\nhfin : Set.Finite {i | Set.Nonempty (f i \u2229 U)}\n\u22a2 \ud835\udcdd[\u22c3 (i : \u03b9), f i] a = \u2a06 (i : \u03b9), \ud835\udcdd[f i] a\n[PROOFSTEP]\nrefine le_antisymm ?_ (Monotone.le_map_iSup fun _ _ \u21a6 nhdsWithin_mono _)\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nhf : LocallyFinite f\na : X\nU : Set X\nhaU : U \u2208 \ud835\udcdd a\nhfin : Set.Finite {i | Set.Nonempty (f i \u2229 U)}\n\u22a2 \ud835\udcdd[\u22c3 (i : \u03b9), f i] a \u2264 \u2a06 (i : \u03b9), \ud835\udcdd[f i] a\n[PROOFSTEP]\ncalc\n \ud835\udcdd[\u22c3 i, f i] a = \ud835\udcdd[\u22c3 i, f i \u2229 U] a := by rw [\u2190 iUnion_inter, \u2190 nhdsWithin_inter_of_mem' (nhdsWithin_le_nhds haU)]\n _ = \ud835\udcdd[\u22c3 i \u2208 {j | (f j \u2229 U).Nonempty}, (f i \u2229 U)] a := by simp only [mem_setOf_eq, iUnion_nonempty_self]\n _ = \u2a06 i \u2208 {j | (f j \u2229 U).Nonempty}, \ud835\udcdd[f i \u2229 U] a := (nhdsWithin_biUnion hfin _ _)\n _ \u2264 \u2a06 i, \ud835\udcdd[f i \u2229 U] a := (iSup\u2082_le_iSup _ _)\n _ \u2264 \u2a06 i, \ud835\udcdd[f i] a := iSup_mono fun i \u21a6 nhdsWithin_mono _ <| inter_subset_left _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nhf : LocallyFinite f\na : X\nU : Set X\nhaU : U \u2208 \ud835\udcdd a\nhfin : Set.Finite {i | Set.Nonempty (f i \u2229 U)}\n\u22a2 \ud835\udcdd[\u22c3 (i : \u03b9), f i] a = \ud835\udcdd[\u22c3 (i : \u03b9), f i \u2229 U] a\n[PROOFSTEP]\nrw [\u2190 iUnion_inter, \u2190 nhdsWithin_inter_of_mem' (nhdsWithin_le_nhds haU)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nhf : LocallyFinite f\na : X\nU : Set X\nhaU : U \u2208 \ud835\udcdd a\nhfin : Set.Finite {i | Set.Nonempty (f i \u2229 U)}\n\u22a2 \ud835\udcdd[\u22c3 (i : \u03b9), f i \u2229 U] a = \ud835\udcdd[\u22c3 (i : \u03b9) (_ : i \u2208 {j | Set.Nonempty (f j \u2229 U)}), f i \u2229 U] a\n[PROOFSTEP]\nsimp only [mem_setOf_eq, iUnion_nonempty_self]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g\u271d : \u03b9 \u2192 Set X\ng : X \u2192 Y\nhf : LocallyFinite f\nhc : \u2200 (i : \u03b9) (x : X), x \u2208 closure (f i) \u2192 ContinuousWithinAt g (f i) x\n\u22a2 ContinuousOn g (\u22c3 (i : \u03b9), f i)\n[PROOFSTEP]\nrintro x -\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g\u271d : \u03b9 \u2192 Set X\ng : X \u2192 Y\nhf : LocallyFinite f\nhc : \u2200 (i : \u03b9) (x : X), x \u2208 closure (f i) \u2192 ContinuousWithinAt g (f i) x\nx : X\n\u22a2 ContinuousWithinAt g (\u22c3 (i : \u03b9), f i) x\n[PROOFSTEP]\nrw [ContinuousWithinAt, hf.nhdsWithin_iUnion, tendsto_iSup]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g\u271d : \u03b9 \u2192 Set X\ng : X \u2192 Y\nhf : LocallyFinite f\nhc : \u2200 (i : \u03b9) (x : X), x \u2208 closure (f i) \u2192 ContinuousWithinAt g (f i) x\nx : X\n\u22a2 \u2200 (i : \u03b9), Tendsto g (\ud835\udcdd[f i] x) (\ud835\udcdd (g x))\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g\u271d : \u03b9 \u2192 Set X\ng : X \u2192 Y\nhf : LocallyFinite f\nhc : \u2200 (i : \u03b9) (x : X), x \u2208 closure (f i) \u2192 ContinuousWithinAt g (f i) x\nx : X\ni : \u03b9\n\u22a2 Tendsto g (\ud835\udcdd[f i] x) (\ud835\udcdd (g x))\n[PROOFSTEP]\nby_cases hx : x \u2208 closure (f i)\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g\u271d : \u03b9 \u2192 Set X\ng : X \u2192 Y\nhf : LocallyFinite f\nhc : \u2200 (i : \u03b9) (x : X), x \u2208 closure (f i) \u2192 ContinuousWithinAt g (f i) x\nx : X\ni : \u03b9\nhx : x \u2208 closure (f i)\n\u22a2 Tendsto g (\ud835\udcdd[f i] x) (\ud835\udcdd (g x))\n[PROOFSTEP]\nexact hc i _ hx\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g\u271d : \u03b9 \u2192 Set X\ng : X \u2192 Y\nhf : LocallyFinite f\nhc : \u2200 (i : \u03b9) (x : X), x \u2208 closure (f i) \u2192 ContinuousWithinAt g (f i) x\nx : X\ni : \u03b9\nhx : \u00acx \u2208 closure (f i)\n\u22a2 Tendsto g (\ud835\udcdd[f i] x) (\ud835\udcdd (g x))\n[PROOFSTEP]\nrw [mem_closure_iff_nhdsWithin_neBot, not_neBot] at hx \n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g\u271d : \u03b9 \u2192 Set X\ng : X \u2192 Y\nhf : LocallyFinite f\nhc : \u2200 (i : \u03b9) (x : X), x \u2208 closure (f i) \u2192 ContinuousWithinAt g (f i) x\nx : X\ni : \u03b9\nhx : \ud835\udcdd[f i] x = \u22a5\n\u22a2 Tendsto g (\ud835\udcdd[f i] x) (\ud835\udcdd (g x))\n[PROOFSTEP]\nrw [hx]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g\u271d : \u03b9 \u2192 Set X\ng : X \u2192 Y\nhf : LocallyFinite f\nhc : \u2200 (i : \u03b9) (x : X), x \u2208 closure (f i) \u2192 ContinuousWithinAt g (f i) x\nx : X\ni : \u03b9\nhx : \ud835\udcdd[f i] x = \u22a5\n\u22a2 Tendsto g \u22a5 (\ud835\udcdd (g x))\n[PROOFSTEP]\nexact tendsto_bot\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nhf : LocallyFinite f\n\u22a2 LocallyFinite fun i => closure (f i)\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nhf : LocallyFinite f\nx : X\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd x \u2227 Set.Finite {i | Set.Nonempty ((fun i => closure (f i)) i \u2229 t)}\n[PROOFSTEP]\nrcases hf x with \u27e8s, hsx, hsf\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nhf : LocallyFinite f\nx : X\ns : Set X\nhsx : s \u2208 \ud835\udcdd x\nhsf : Set.Finite {i | Set.Nonempty (f i \u2229 s)}\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd x \u2227 Set.Finite {i | Set.Nonempty ((fun i => closure (f i)) i \u2229 t)}\n[PROOFSTEP]\nrefine' \u27e8interior s, interior_mem_nhds.2 hsx, hsf.subset fun i hi => _\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nhf : LocallyFinite f\nx : X\ns : Set X\nhsx : s \u2208 \ud835\udcdd x\nhsf : Set.Finite {i | Set.Nonempty (f i \u2229 s)}\ni : \u03b9\nhi : i \u2208 {i | Set.Nonempty ((fun i => closure (f i)) i \u2229 interior s)}\n\u22a2 i \u2208 {i | Set.Nonempty (f i \u2229 s)}\n[PROOFSTEP]\nexact (hi.mono isOpen_interior.closure_inter).of_closure.mono (inter_subset_inter_right _ interior_subset)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nh : LocallyFinite f\n\u22a2 closure (\u22c3 (i : \u03b9), f i) = \u22c3 (i : \u03b9), closure (f i)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nh : LocallyFinite f\nx : X\n\u22a2 x \u2208 closure (\u22c3 (i : \u03b9), f i) \u2194 x \u2208 \u22c3 (i : \u03b9), closure (f i)\n[PROOFSTEP]\nsimp only [mem_closure_iff_nhdsWithin_neBot, h.nhdsWithin_iUnion, iSup_neBot, mem_iUnion]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nhf : LocallyFinite f\nhc : \u2200 (i : \u03b9), IsClosed (f i)\n\u22a2 IsClosed (\u22c3 (i : \u03b9), f i)\n[PROOFSTEP]\nsimp only [\u2190 closure_eq_iff_isClosed, hf.closure_iUnion, (hc _).closure_eq]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nhf : LocallyFinite f\nhc : \u2200 (i : \u03b9), IsClosed (f i)\nx : X\n\u22a2 \u22c2 (i : \u03b9) (_ : \u00acx \u2208 f i), (f i)\u1d9c \u2208 \ud835\udcdd x\n[PROOFSTEP]\nrefine' IsOpen.mem_nhds _ (mem_iInter\u2082.2 fun i => id)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nhf : LocallyFinite f\nhc : \u2200 (i : \u03b9), IsClosed (f i)\nx : X\n\u22a2 IsOpen (\u22c2 (i : \u03b9) (_ : \u00acx \u2208 f i), (f i)\u1d9c)\n[PROOFSTEP]\nsuffices IsClosed (\u22c3 i : { i // x \u2209 f i }, f i) by rwa [\u2190 isOpen_compl_iff, compl_iUnion, iInter_subtype] at this \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nhf : LocallyFinite f\nhc : \u2200 (i : \u03b9), IsClosed (f i)\nx : X\nthis : IsClosed (\u22c3 (i : { i // \u00acx \u2208 f i }), f \u2191i)\n\u22a2 IsOpen (\u22c2 (i : \u03b9) (_ : \u00acx \u2208 f i), (f i)\u1d9c)\n[PROOFSTEP]\nrwa [\u2190 isOpen_compl_iff, compl_iUnion, iInter_subtype] at this \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\nhf : LocallyFinite f\nhc : \u2200 (i : \u03b9), IsClosed (f i)\nx : X\n\u22a2 IsClosed (\u22c3 (i : { i // \u00acx \u2208 f i }), f \u2191i)\n[PROOFSTEP]\nexact (hf.comp_injective Subtype.val_injective).isClosed_iUnion fun i => hc _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\n\u03c0 : X \u2192 Sort u_6\nf : \u2115 \u2192 (x : X) \u2192 \u03c0 x\nhf : LocallyFinite fun n => {x | f (n + 1) x \u2260 f n x}\n\u22a2 \u2203 F, \u2200 (x : X), \u2200\u1da0 (p : \u2115 \u00d7 X) in atTop \u00d7\u02e2 \ud835\udcdd x, f p.fst p.snd = F p.snd\n[PROOFSTEP]\nchoose U hUx hU using hf\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\n\u03c0 : X \u2192 Sort u_6\nf : \u2115 \u2192 (x : X) \u2192 \u03c0 x\nU : X \u2192 Set X\nhUx : \u2200 (x : X), U x \u2208 \ud835\udcdd x\nhU : \u2200 (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x \u2260 f n x}) i \u2229 U x)}\n\u22a2 \u2203 F, \u2200 (x : X), \u2200\u1da0 (p : \u2115 \u00d7 X) in atTop \u00d7\u02e2 \ud835\udcdd x, f p.fst p.snd = F p.snd\n[PROOFSTEP]\nchoose N hN using fun x => (hU x).bddAbove\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\n\u03c0 : X \u2192 Sort u_6\nf : \u2115 \u2192 (x : X) \u2192 \u03c0 x\nU : X \u2192 Set X\nhUx : \u2200 (x : X), U x \u2208 \ud835\udcdd x\nhU : \u2200 (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x \u2260 f n x}) i \u2229 U x)}\nN : X \u2192 \u2115\nhN : \u2200 (x : X), N x \u2208 upperBounds {i | Set.Nonempty ((fun n => {x | f (n + 1) x \u2260 f n x}) i \u2229 U x)}\n\u22a2 \u2203 F, \u2200 (x : X), \u2200\u1da0 (p : \u2115 \u00d7 X) in atTop \u00d7\u02e2 \ud835\udcdd x, f p.fst p.snd = F p.snd\n[PROOFSTEP]\nreplace hN : \u2200 (x), \u2200 n > N x, \u2200 y \u2208 U x, f (n + 1) y = f n y\n[GOAL]\ncase hN\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\n\u03c0 : X \u2192 Sort u_6\nf : \u2115 \u2192 (x : X) \u2192 \u03c0 x\nU : X \u2192 Set X\nhUx : \u2200 (x : X), U x \u2208 \ud835\udcdd x\nhU : \u2200 (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x \u2260 f n x}) i \u2229 U x)}\nN : X \u2192 \u2115\nhN : \u2200 (x : X), N x \u2208 upperBounds {i | Set.Nonempty ((fun n => {x | f (n + 1) x \u2260 f n x}) i \u2229 U x)}\n\u22a2 \u2200 (x : X) (n : \u2115), n > N x \u2192 \u2200 (y : X), y \u2208 U x \u2192 f (n + 1) y = f n y\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\n\u03c0 : X \u2192 Sort u_6\nf : \u2115 \u2192 (x : X) \u2192 \u03c0 x\nU : X \u2192 Set X\nhUx : \u2200 (x : X), U x \u2208 \ud835\udcdd x\nhU : \u2200 (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x \u2260 f n x}) i \u2229 U x)}\nN : X \u2192 \u2115\nhN : \u2200 (x : X) (n : \u2115), n > N x \u2192 \u2200 (y : X), y \u2208 U x \u2192 f (n + 1) y = f n y\n\u22a2 \u2203 F, \u2200 (x : X), \u2200\u1da0 (p : \u2115 \u00d7 X) in atTop \u00d7\u02e2 \ud835\udcdd x, f p.fst p.snd = F p.snd\n[PROOFSTEP]\nexact fun x n hn y hy => by_contra fun hne => hn.lt.not_le <| hN x \u27e8y, hne, hy\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\n\u03c0 : X \u2192 Sort u_6\nf : \u2115 \u2192 (x : X) \u2192 \u03c0 x\nU : X \u2192 Set X\nhUx : \u2200 (x : X), U x \u2208 \ud835\udcdd x\nhU : \u2200 (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x \u2260 f n x}) i \u2229 U x)}\nN : X \u2192 \u2115\nhN : \u2200 (x : X) (n : \u2115), n > N x \u2192 \u2200 (y : X), y \u2208 U x \u2192 f (n + 1) y = f n y\n\u22a2 \u2203 F, \u2200 (x : X), \u2200\u1da0 (p : \u2115 \u00d7 X) in atTop \u00d7\u02e2 \ud835\udcdd x, f p.fst p.snd = F p.snd\n[PROOFSTEP]\nreplace hN : \u2200 (x), \u2200 n \u2265 N x + 1, \u2200 y \u2208 U x, f n y = f (N x + 1) y\n[GOAL]\ncase hN\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\n\u03c0 : X \u2192 Sort u_6\nf : \u2115 \u2192 (x : X) \u2192 \u03c0 x\nU : X \u2192 Set X\nhUx : \u2200 (x : X), U x \u2208 \ud835\udcdd x\nhU : \u2200 (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x \u2260 f n x}) i \u2229 U x)}\nN : X \u2192 \u2115\nhN : \u2200 (x : X) (n : \u2115), n > N x \u2192 \u2200 (y : X), y \u2208 U x \u2192 f (n + 1) y = f n y\n\u22a2 \u2200 (x : X) (n : \u2115), n \u2265 N x + 1 \u2192 \u2200 (y : X), y \u2208 U x \u2192 f n y = f (N x + 1) y\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\n\u03c0 : X \u2192 Sort u_6\nf : \u2115 \u2192 (x : X) \u2192 \u03c0 x\nU : X \u2192 Set X\nhUx : \u2200 (x : X), U x \u2208 \ud835\udcdd x\nhU : \u2200 (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x \u2260 f n x}) i \u2229 U x)}\nN : X \u2192 \u2115\nhN : \u2200 (x : X) (n : \u2115), n \u2265 N x + 1 \u2192 \u2200 (y : X), y \u2208 U x \u2192 f n y = f (N x + 1) y\n\u22a2 \u2203 F, \u2200 (x : X), \u2200\u1da0 (p : \u2115 \u00d7 X) in atTop \u00d7\u02e2 \ud835\udcdd x, f p.fst p.snd = F p.snd\n[PROOFSTEP]\nexact fun x n hn y hy => Nat.le_induction rfl (fun k hle => (hN x _ hle _ hy).trans) n hn\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\n\u03c0 : X \u2192 Sort u_6\nf : \u2115 \u2192 (x : X) \u2192 \u03c0 x\nU : X \u2192 Set X\nhUx : \u2200 (x : X), U x \u2208 \ud835\udcdd x\nhU : \u2200 (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x \u2260 f n x}) i \u2229 U x)}\nN : X \u2192 \u2115\nhN : \u2200 (x : X) (n : \u2115), n \u2265 N x + 1 \u2192 \u2200 (y : X), y \u2208 U x \u2192 f n y = f (N x + 1) y\n\u22a2 \u2203 F, \u2200 (x : X), \u2200\u1da0 (p : \u2115 \u00d7 X) in atTop \u00d7\u02e2 \ud835\udcdd x, f p.fst p.snd = F p.snd\n[PROOFSTEP]\nrefine \u27e8fun x => f (N x + 1) x, fun x => ?_\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\n\u03c0 : X \u2192 Sort u_6\nf : \u2115 \u2192 (x : X) \u2192 \u03c0 x\nU : X \u2192 Set X\nhUx : \u2200 (x : X), U x \u2208 \ud835\udcdd x\nhU : \u2200 (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x \u2260 f n x}) i \u2229 U x)}\nN : X \u2192 \u2115\nhN : \u2200 (x : X) (n : \u2115), n \u2265 N x + 1 \u2192 \u2200 (y : X), y \u2208 U x \u2192 f n y = f (N x + 1) y\nx : X\n\u22a2 \u2200\u1da0 (p : \u2115 \u00d7 X) in atTop \u00d7\u02e2 \ud835\udcdd x, f p.fst p.snd = (fun x => f (N x + 1) x) p.snd\n[PROOFSTEP]\nfilter_upwards [Filter.prod_mem_prod (eventually_gt_atTop (N x)) (hUx x)]\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\n\u03c0 : X \u2192 Sort u_6\nf : \u2115 \u2192 (x : X) \u2192 \u03c0 x\nU : X \u2192 Set X\nhUx : \u2200 (x : X), U x \u2208 \ud835\udcdd x\nhU : \u2200 (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x \u2260 f n x}) i \u2229 U x)}\nN : X \u2192 \u2115\nhN : \u2200 (x : X) (n : \u2115), n \u2265 N x + 1 \u2192 \u2200 (y : X), y \u2208 U x \u2192 f n y = f (N x + 1) y\nx : X\n\u22a2 \u2200 (a : \u2115 \u00d7 X), a \u2208 {x_1 | N x < x_1} \u00d7\u02e2 U x \u2192 f a.fst a.snd = f (N a.snd + 1) a.snd\n[PROOFSTEP]\nrintro \u27e8n, y\u27e9 \u27e8hn : N x < n, hy : y \u2208 U x\u27e9\n[GOAL]\ncase h.mk.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\n\u03c0 : X \u2192 Sort u_6\nf : \u2115 \u2192 (x : X) \u2192 \u03c0 x\nU : X \u2192 Set X\nhUx : \u2200 (x : X), U x \u2208 \ud835\udcdd x\nhU : \u2200 (x : X), Set.Finite {i | Set.Nonempty ((fun n => {x | f (n + 1) x \u2260 f n x}) i \u2229 U x)}\nN : X \u2192 \u2115\nhN : \u2200 (x : X) (n : \u2115), n \u2265 N x + 1 \u2192 \u2200 (y : X), y \u2208 U x \u2192 f n y = f (N x + 1) y\nx : X\nn : \u2115\ny : X\nhn : N x < n\nhy : y \u2208 U x\n\u22a2 f (n, y).fst (n, y).snd = f (N (n, y).snd + 1) (n, y).snd\n[PROOFSTEP]\ncalc\n f n y = f (N x + 1) y := hN _ _ hn _ hy\n _ = f (max (N x + 1) (N y + 1)) y := (hN _ _ (le_max_left _ _) _ hy).symm\n _ = f (N y + 1) y := hN _ _ (le_max_right _ _) _ (mem_of_mem_nhds <| hUx y)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf g : \u03b9 \u2192 Set X\ne : \u03b9' \u2243 \u03b9\nh : LocallyFinite (f \u2218 \u2191e)\n\u22a2 LocallyFinite f\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), e.apply_symm_apply] using h.comp_injective e.symm.injective\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\nf : \u03b9 \u2295 \u03b9' \u2192 Set X\n\u22a2 LocallyFinite f \u2194 LocallyFinite (f \u2218 Sum.inl) \u2227 LocallyFinite (f \u2218 Sum.inr)\n[PROOFSTEP]\nsimp only [locallyFinite_iff_smallSets, \u2190 forall_and, \u2190 finite_preimage_inl_and_inr, preimage_setOf_eq, (\u00b7 \u2218 \u00b7),\n eventually_and]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\nf : Option \u03b9 \u2192 Set X\n\u22a2 LocallyFinite f \u2194 LocallyFinite (f \u2218 some)\n[PROOFSTEP]\nrw [\u2190 (Equiv.optionEquivSumPUnit.{_, 0} \u03b9).symm.locallyFinite_comp_iff, locallyFinite_sum]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\nf : Option \u03b9 \u2192 Set X\n\u22a2 LocallyFinite ((f \u2218 \u2191(Equiv.optionEquivSumPUnit \u03b9).symm) \u2218 Sum.inl) \u2227\n LocallyFinite ((f \u2218 \u2191(Equiv.optionEquivSumPUnit \u03b9).symm) \u2218 Sum.inr) \u2194\n LocallyFinite (f \u2218 some)\n[PROOFSTEP]\nsimp only [locallyFinite_of_finite, and_true]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf\u271d g : \u03b9 \u2192 Set X\nf : Option \u03b9 \u2192 Set X\n\u22a2 LocallyFinite ((f \u2218 \u2191(Equiv.optionEquivSumPUnit \u03b9).symm) \u2218 Sum.inl) \u2194 LocallyFinite (f \u2218 some)\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Topology.LocallyFinite", "llama_tokens": 9642, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947155710233, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.41718809193549466}} {"text": "[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 p\u2081 : Path x\u2080 x\u2081\nF : Homotopy p\u2080 p\u2081\nt : \u2191I\n\u22a2 ContinuousMap.toFun (ContinuousMap.mk \u2191(\u2191(ContinuousMap.Homotopy.curry F.toHomotopy) t)) 0 = x\u2080\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 p\u2081 : Path x\u2080 x\u2081\nF : Homotopy p\u2080 p\u2081\nt : \u2191I\n\u22a2 ContinuousMap.toFun (ContinuousMap.mk \u2191(\u2191(ContinuousMap.Homotopy.curry F.toHomotopy) t)) 1 = x\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 p\u2081 : Path x\u2080 x\u2081\nF : Homotopy p\u2080 p\u2081\n\u22a2 eval F 0 = p\u2080\n[PROOFSTEP]\next t\n[GOAL]\ncase a.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 p\u2081 : Path x\u2080 x\u2081\nF : Homotopy p\u2080 p\u2081\nt : \u2191I\n\u22a2 \u2191(eval F 0) t = \u2191p\u2080 t\n[PROOFSTEP]\nsimp [eval]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 p\u2081 : Path x\u2080 x\u2081\nF : Homotopy p\u2080 p\u2081\n\u22a2 eval F 1 = p\u2081\n[PROOFSTEP]\next t\n[GOAL]\ncase a.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 p\u2081 : Path x\u2080 x\u2081\nF : Homotopy p\u2080 p\u2081\nt : \u2191I\n\u22a2 \u2191(eval F 1) t = \u2191p\u2081 t\n[PROOFSTEP]\nsimp [eval]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\n\u22a2 Continuous fun x => (x.fst, Set.projIcc 0 1 extend.proof_1 (2 * \u2191x.snd))\n[PROOFSTEP]\ncontinuity\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\n\u22a2 Continuous fun x => (x.fst, Set.projIcc 0 1 extend.proof_1 (2 * \u2191x.snd - 1))\n[PROOFSTEP]\ncontinuity\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nx : \u2191I \u00d7 \u2191I\nhx : \u2191x.snd = 1 / 2\n\u22a2 extend (eval F x.fst) (2 * \u2191x.snd) = extend (eval G x.fst) (2 * \u2191x.snd - 1)\n[PROOFSTEP]\nrw [hx]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nx : \u2191I \u00d7 \u2191I\nhx : \u2191x.snd = 1 / 2\n\u22a2 extend (eval F x.fst) (2 * (1 / 2)) = extend (eval G x.fst) (2 * (1 / 2) - 1)\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nx : \u2191I\n\u22a2 ContinuousMap.toFun\n (ContinuousMap.mk fun x =>\n if \u2191x.snd \u2264 1 / 2 then extend (eval F x.fst) (2 * \u2191x.snd) else extend (eval G x.fst) (2 * \u2191x.snd - 1))\n (0, x) =\n \u2191(Path.trans p\u2080 p\u2081).toContinuousMap x\n[PROOFSTEP]\nsimp [Path.trans]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nx : \u2191I\n\u22a2 ContinuousMap.toFun\n (ContinuousMap.mk fun x =>\n if \u2191x.snd \u2264 1 / 2 then extend (eval F x.fst) (2 * \u2191x.snd) else extend (eval G x.fst) (2 * \u2191x.snd - 1))\n (1, x) =\n \u2191(Path.trans q\u2080 q\u2081).toContinuousMap x\n[PROOFSTEP]\nsimp [Path.trans]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nx t : \u2191I\nht : t \u2208 {0, 1}\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n if \u2191x.snd \u2264 1 / 2 then extend (eval F x.fst) (2 * \u2191x.snd)\n else extend (eval G x.fst) (2 * \u2191x.snd - 1),\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(0, x).snd \u2264 1 / 2 then extend (eval F (0, x).fst) (2 * \u2191(0, x).snd)\n else extend (eval G (0, x).fst) (2 * \u2191(0, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend p\u2080 (2 * t) else extend p\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(1, x).snd \u2264 1 / 2 then extend (eval F (1, x).fst) (2 * \u2191(1, x).snd)\n else extend (eval G (1, x).fst) (2 * \u2191(1, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend q\u2080 (2 * t) else extend q\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x) }.toContinuousMap\n (x, x_1))\n t =\n \u2191(Path.trans p\u2080 p\u2081).toContinuousMap t \u2227\n \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n if \u2191x.snd \u2264 1 / 2 then extend (eval F x.fst) (2 * \u2191x.snd)\n else extend (eval G x.fst) (2 * \u2191x.snd - 1),\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(0, x).snd \u2264 1 / 2 then extend (eval F (0, x).fst) (2 * \u2191(0, x).snd)\n else extend (eval G (0, x).fst) (2 * \u2191(0, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend p\u2080 (2 * t) else extend p\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(1, x).snd \u2264 1 / 2 then extend (eval F (1, x).fst) (2 * \u2191(1, x).snd)\n else extend (eval G (1, x).fst) (2 * \u2191(1, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend q\u2080 (2 * t) else extend q\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x) }.toContinuousMap\n (x, x_1))\n t =\n \u2191(Path.trans q\u2080 q\u2081).toContinuousMap t\n[PROOFSTEP]\ncases' ht with ht ht\n[GOAL]\ncase inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nx t : \u2191I\nht : t = 0\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n if \u2191x.snd \u2264 1 / 2 then extend (eval F x.fst) (2 * \u2191x.snd)\n else extend (eval G x.fst) (2 * \u2191x.snd - 1),\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(0, x).snd \u2264 1 / 2 then extend (eval F (0, x).fst) (2 * \u2191(0, x).snd)\n else extend (eval G (0, x).fst) (2 * \u2191(0, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend p\u2080 (2 * t) else extend p\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(1, x).snd \u2264 1 / 2 then extend (eval F (1, x).fst) (2 * \u2191(1, x).snd)\n else extend (eval G (1, x).fst) (2 * \u2191(1, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend q\u2080 (2 * t) else extend q\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x) }.toContinuousMap\n (x, x_1))\n t =\n \u2191(Path.trans p\u2080 p\u2081).toContinuousMap t \u2227\n \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n if \u2191x.snd \u2264 1 / 2 then extend (eval F x.fst) (2 * \u2191x.snd)\n else extend (eval G x.fst) (2 * \u2191x.snd - 1),\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(0, x).snd \u2264 1 / 2 then extend (eval F (0, x).fst) (2 * \u2191(0, x).snd)\n else extend (eval G (0, x).fst) (2 * \u2191(0, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend p\u2080 (2 * t) else extend p\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(1, x).snd \u2264 1 / 2 then extend (eval F (1, x).fst) (2 * \u2191(1, x).snd)\n else extend (eval G (1, x).fst) (2 * \u2191(1, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend q\u2080 (2 * t) else extend q\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x) }.toContinuousMap\n (x, x_1))\n t =\n \u2191(Path.trans q\u2080 q\u2081).toContinuousMap t\n[PROOFSTEP]\nrw [ht]\n[GOAL]\ncase inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nx t : \u2191I\nht : t = 0\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n if \u2191x.snd \u2264 1 / 2 then extend (eval F x.fst) (2 * \u2191x.snd)\n else extend (eval G x.fst) (2 * \u2191x.snd - 1),\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(0, x).snd \u2264 1 / 2 then extend (eval F (0, x).fst) (2 * \u2191(0, x).snd)\n else extend (eval G (0, x).fst) (2 * \u2191(0, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend p\u2080 (2 * t) else extend p\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(1, x).snd \u2264 1 / 2 then extend (eval F (1, x).fst) (2 * \u2191(1, x).snd)\n else extend (eval G (1, x).fst) (2 * \u2191(1, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend q\u2080 (2 * t) else extend q\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x) }.toContinuousMap\n (x, x_1))\n 0 =\n \u2191(Path.trans p\u2080 p\u2081).toContinuousMap 0 \u2227\n \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n if \u2191x.snd \u2264 1 / 2 then extend (eval F x.fst) (2 * \u2191x.snd)\n else extend (eval G x.fst) (2 * \u2191x.snd - 1),\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(0, x).snd \u2264 1 / 2 then extend (eval F (0, x).fst) (2 * \u2191(0, x).snd)\n else extend (eval G (0, x).fst) (2 * \u2191(0, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend p\u2080 (2 * t) else extend p\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(1, x).snd \u2264 1 / 2 then extend (eval F (1, x).fst) (2 * \u2191(1, x).snd)\n else extend (eval G (1, x).fst) (2 * \u2191(1, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend q\u2080 (2 * t) else extend q\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x) }.toContinuousMap\n (x, x_1))\n 0 =\n \u2191(Path.trans q\u2080 q\u2081).toContinuousMap 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nx t : \u2191I\nht : t \u2208 {1}\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n if \u2191x.snd \u2264 1 / 2 then extend (eval F x.fst) (2 * \u2191x.snd)\n else extend (eval G x.fst) (2 * \u2191x.snd - 1),\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(0, x).snd \u2264 1 / 2 then extend (eval F (0, x).fst) (2 * \u2191(0, x).snd)\n else extend (eval G (0, x).fst) (2 * \u2191(0, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend p\u2080 (2 * t) else extend p\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(1, x).snd \u2264 1 / 2 then extend (eval F (1, x).fst) (2 * \u2191(1, x).snd)\n else extend (eval G (1, x).fst) (2 * \u2191(1, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend q\u2080 (2 * t) else extend q\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x) }.toContinuousMap\n (x, x_1))\n t =\n \u2191(Path.trans p\u2080 p\u2081).toContinuousMap t \u2227\n \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n if \u2191x.snd \u2264 1 / 2 then extend (eval F x.fst) (2 * \u2191x.snd)\n else extend (eval G x.fst) (2 * \u2191x.snd - 1),\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(0, x).snd \u2264 1 / 2 then extend (eval F (0, x).fst) (2 * \u2191(0, x).snd)\n else extend (eval G (0, x).fst) (2 * \u2191(0, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend p\u2080 (2 * t) else extend p\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(1, x).snd \u2264 1 / 2 then extend (eval F (1, x).fst) (2 * \u2191(1, x).snd)\n else extend (eval G (1, x).fst) (2 * \u2191(1, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend q\u2080 (2 * t) else extend q\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x) }.toContinuousMap\n (x, x_1))\n t =\n \u2191(Path.trans q\u2080 q\u2081).toContinuousMap t\n[PROOFSTEP]\nrw [Set.mem_singleton_iff] at ht \n[GOAL]\ncase inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nx t : \u2191I\nht : t = 1\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n if \u2191x.snd \u2264 1 / 2 then extend (eval F x.fst) (2 * \u2191x.snd)\n else extend (eval G x.fst) (2 * \u2191x.snd - 1),\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(0, x).snd \u2264 1 / 2 then extend (eval F (0, x).fst) (2 * \u2191(0, x).snd)\n else extend (eval G (0, x).fst) (2 * \u2191(0, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend p\u2080 (2 * t) else extend p\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(1, x).snd \u2264 1 / 2 then extend (eval F (1, x).fst) (2 * \u2191(1, x).snd)\n else extend (eval G (1, x).fst) (2 * \u2191(1, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend q\u2080 (2 * t) else extend q\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x) }.toContinuousMap\n (x, x_1))\n t =\n \u2191(Path.trans p\u2080 p\u2081).toContinuousMap t \u2227\n \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n if \u2191x.snd \u2264 1 / 2 then extend (eval F x.fst) (2 * \u2191x.snd)\n else extend (eval G x.fst) (2 * \u2191x.snd - 1),\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(0, x).snd \u2264 1 / 2 then extend (eval F (0, x).fst) (2 * \u2191(0, x).snd)\n else extend (eval G (0, x).fst) (2 * \u2191(0, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend p\u2080 (2 * t) else extend p\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(1, x).snd \u2264 1 / 2 then extend (eval F (1, x).fst) (2 * \u2191(1, x).snd)\n else extend (eval G (1, x).fst) (2 * \u2191(1, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend q\u2080 (2 * t) else extend q\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x) }.toContinuousMap\n (x, x_1))\n t =\n \u2191(Path.trans q\u2080 q\u2081).toContinuousMap t\n[PROOFSTEP]\nrw [ht]\n[GOAL]\ncase inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nx t : \u2191I\nht : t = 1\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n if \u2191x.snd \u2264 1 / 2 then extend (eval F x.fst) (2 * \u2191x.snd)\n else extend (eval G x.fst) (2 * \u2191x.snd - 1),\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(0, x).snd \u2264 1 / 2 then extend (eval F (0, x).fst) (2 * \u2191(0, x).snd)\n else extend (eval G (0, x).fst) (2 * \u2191(0, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend p\u2080 (2 * t) else extend p\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(1, x).snd \u2264 1 / 2 then extend (eval F (1, x).fst) (2 * \u2191(1, x).snd)\n else extend (eval G (1, x).fst) (2 * \u2191(1, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend q\u2080 (2 * t) else extend q\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x) }.toContinuousMap\n (x, x_1))\n 1 =\n \u2191(Path.trans p\u2080 p\u2081).toContinuousMap 1 \u2227\n \u2191(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n if \u2191x.snd \u2264 1 / 2 then extend (eval F x.fst) (2 * \u2191x.snd)\n else extend (eval G x.fst) (2 * \u2191x.snd - 1),\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(0, x).snd \u2264 1 / 2 then extend (eval F (0, x).fst) (2 * \u2191(0, x).snd)\n else extend (eval G (0, x).fst) (2 * \u2191(0, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend p\u2080 (2 * t) else extend p\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend p\u2080 (2 * x) else extend p\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n (if \u2191(1, x).snd \u2264 1 / 2 then extend (eval F (1, x).fst) (2 * \u2191(1, x).snd)\n else extend (eval G (1, x).fst) (2 * \u2191(1, x).snd - 1)) =\n \u2191{\n toContinuousMap :=\n ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend q\u2080 (2 * t) else extend q\u2081 (2 * t - 1)) \u2218\n Subtype.val),\n source' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 0 =\n x\u2080),\n target' :=\n (_ :\n ((fun x => if x \u2264 1 / 2 then extend q\u2080 (2 * x) else extend q\u2081 (2 * x - 1)) \u2218\n Subtype.val)\n 1 =\n x\u2082) }.toContinuousMap\n x) }.toContinuousMap\n (x, x_1))\n 1 =\n \u2191(Path.trans q\u2080 q\u2081).toContinuousMap 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nx : \u2191I \u00d7 \u2191I\n\u22a2 (if \u2191x.snd \u2264 1 / 2 then extend (eval F x.fst) (2 * \u2191x.snd) else extend (eval G x.fst) (2 * \u2191x.snd - 1)) =\n if h : \u2191x.snd \u2264 1 / 2 then \u2191(eval F x.fst) { val := 2 * \u2191x.snd, property := (_ : 2 * \u2191x.snd \u2208 I) }\n else \u2191(eval G x.fst) { val := 2 * \u2191x.snd - 1, property := (_ : 2 * \u2191x.snd - 1 \u2208 I) }\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u2191x.snd \u2264 1 / 2\n\u22a2 extend (eval F x.fst) (2 * \u2191x.snd) = \u2191(eval F x.fst) { val := 2 * \u2191x.snd, property := (_ : 2 * \u2191x.snd \u2208 I) }\n[PROOFSTEP]\nexact Path.extend_extends _ _\n[GOAL]\ncase neg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u00ac\u2191x.snd \u2264 1 / 2\n\u22a2 extend (eval G x.fst) (2 * \u2191x.snd - 1) =\n \u2191(eval G x.fst) { val := 2 * \u2191x.snd - 1, property := (_ : 2 * \u2191x.snd - 1 \u2208 I) }\n[PROOFSTEP]\nexact Path.extend_extends _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nt : \u2191I\n\u22a2 0 \u2264 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nt : \u2191I\n\u22a2 1 / 2 \u2264 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2080 q\u2080 : Path x\u2080 x\u2081\np\u2081 q\u2081 : Path x\u2081 x\u2082\nF : Homotopy p\u2080 q\u2080\nG : Homotopy p\u2081 q\u2081\nt : \u2191I\n\u22a2 (if \u2191(t, { val := 1 / 2, property := (_ : 0 \u2264 1 / 2 \u2227 1 / 2 \u2264 1) }).snd \u2264 1 / 2 then\n extend (eval F (t, { val := 1 / 2, property := (_ : 0 \u2264 1 / 2 \u2227 1 / 2 \u2264 1) }).fst)\n (2 * \u2191(t, { val := 1 / 2, property := (_ : 0 \u2264 1 / 2 \u2227 1 / 2 \u2264 1) }).snd)\n else\n extend (eval G (t, { val := 1 / 2, property := (_ : 0 \u2264 1 / 2 \u2227 1 / 2 \u2264 1) }).fst)\n (2 * \u2191(t, { val := 1 / 2, property := (_ : 0 \u2264 1 / 2 \u2227 1 / 2 \u2264 1) }).snd - 1)) =\n x\u2081\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\nx : \u2191I \u00d7 \u2191I\n\u22a2 0 \u2264 \u2191(\u03c3 x.fst)\n[PROOFSTEP]\nunit_interval\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\nx : \u2191I \u00d7 \u2191I\n\u22a2 0 \u2264 \u2191x.fst\n[PROOFSTEP]\nunit_interval\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\nx : \u2191I \u00d7 \u2191I\n\u22a2 \u2191(\u03c3 x.fst) + \u2191x.fst = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\n\u22a2 Continuous fun x =>\n \u2191p\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) }\n[PROOFSTEP]\nrefine continuous_const.path_eval ?_\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\n\u22a2 Continuous fun x =>\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) }\n[PROOFSTEP]\napply Continuous.subtype_mk\n[GOAL]\ncase h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\n\u22a2 Continuous fun x => \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd)\n[PROOFSTEP]\napply Continuous.add\n[GOAL]\ncase h.hf\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\n\u22a2 Continuous fun x => \u2191(\u03c3 x.fst) * \u2191x.snd\n[PROOFSTEP]\napply Continuous.mul\n[GOAL]\ncase h.hg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\n\u22a2 Continuous fun x => \u2191x.fst * \u2191(f x.snd)\n[PROOFSTEP]\napply Continuous.mul\n[GOAL]\ncase h.hf.hf\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\n\u22a2 Continuous fun x => \u2191(\u03c3 x.fst)\n[PROOFSTEP]\nexact continuous_induced_dom.comp (unitInterval.continuous_symm.comp continuous_fst)\n[GOAL]\ncase h.hf.hg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\n\u22a2 Continuous fun x => \u2191x.snd\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase h.hg.hf\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\n\u22a2 Continuous fun x => \u2191x.fst\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase h.hg.hg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\n\u22a2 Continuous fun x => \u2191(f x.snd)\n[PROOFSTEP]\ncontinuity\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\nx : \u2191I\n\u22a2 ContinuousMap.toFun\n (ContinuousMap.mk fun x =>\n \u2191p\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) })\n (0, x) =\n \u2191p.toContinuousMap x\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\nx : \u2191I\n\u22a2 ContinuousMap.toFun\n (ContinuousMap.mk fun x =>\n \u2191p\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) })\n (1, x) =\n \u2191(Path.reparam p f hf hf\u2080 hf\u2081).toContinuousMap x\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\nt x : \u2191I\nhx : x \u2208 {0, 1}\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (0, x).fst) * \u2191(0, x).snd + \u2191(0, x).fst * \u2191(f (0, x).snd),\n property := (_ : \u2191(\u03c3 (0, x).fst) \u2022 \u2191(0, x).snd + \u2191(0, x).fst \u2022 \u2191(f (0, x).snd) \u2208 I) } =\n \u2191p x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (1, x).fst) * \u2191(1, x).snd + \u2191(1, x).fst * \u2191(f (1, x).snd),\n property := (_ : \u2191(\u03c3 (1, x).fst) \u2022 \u2191(1, x).snd + \u2191(1, x).fst \u2022 \u2191(f (1, x).snd) \u2208 I) } =\n \u2191p (f x)) }.toContinuousMap\n (t, x))\n x =\n \u2191p.toContinuousMap x \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (0, x).fst) * \u2191(0, x).snd + \u2191(0, x).fst * \u2191(f (0, x).snd),\n property := (_ : \u2191(\u03c3 (0, x).fst) \u2022 \u2191(0, x).snd + \u2191(0, x).fst \u2022 \u2191(f (0, x).snd) \u2208 I) } =\n \u2191p x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (1, x).fst) * \u2191(1, x).snd + \u2191(1, x).fst * \u2191(f (1, x).snd),\n property := (_ : \u2191(\u03c3 (1, x).fst) \u2022 \u2191(1, x).snd + \u2191(1, x).fst \u2022 \u2191(f (1, x).snd) \u2208 I) } =\n \u2191p (f x)) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.reparam p f hf hf\u2080 hf\u2081).toContinuousMap x\n[PROOFSTEP]\ncases' hx with hx hx\n[GOAL]\ncase inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\nt x : \u2191I\nhx : x = 0\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (0, x).fst) * \u2191(0, x).snd + \u2191(0, x).fst * \u2191(f (0, x).snd),\n property := (_ : \u2191(\u03c3 (0, x).fst) \u2022 \u2191(0, x).snd + \u2191(0, x).fst \u2022 \u2191(f (0, x).snd) \u2208 I) } =\n \u2191p x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (1, x).fst) * \u2191(1, x).snd + \u2191(1, x).fst * \u2191(f (1, x).snd),\n property := (_ : \u2191(\u03c3 (1, x).fst) \u2022 \u2191(1, x).snd + \u2191(1, x).fst \u2022 \u2191(f (1, x).snd) \u2208 I) } =\n \u2191p (f x)) }.toContinuousMap\n (t, x))\n x =\n \u2191p.toContinuousMap x \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (0, x).fst) * \u2191(0, x).snd + \u2191(0, x).fst * \u2191(f (0, x).snd),\n property := (_ : \u2191(\u03c3 (0, x).fst) \u2022 \u2191(0, x).snd + \u2191(0, x).fst \u2022 \u2191(f (0, x).snd) \u2208 I) } =\n \u2191p x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (1, x).fst) * \u2191(1, x).snd + \u2191(1, x).fst * \u2191(f (1, x).snd),\n property := (_ : \u2191(\u03c3 (1, x).fst) \u2022 \u2191(1, x).snd + \u2191(1, x).fst \u2022 \u2191(f (1, x).snd) \u2208 I) } =\n \u2191p (f x)) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.reparam p f hf hf\u2080 hf\u2081).toContinuousMap x\n[PROOFSTEP]\nrw [hx]\n[GOAL]\ncase inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\nt x : \u2191I\nhx : x = 0\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (0, x).fst) * \u2191(0, x).snd + \u2191(0, x).fst * \u2191(f (0, x).snd),\n property := (_ : \u2191(\u03c3 (0, x).fst) \u2022 \u2191(0, x).snd + \u2191(0, x).fst \u2022 \u2191(f (0, x).snd) \u2208 I) } =\n \u2191p x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (1, x).fst) * \u2191(1, x).snd + \u2191(1, x).fst * \u2191(f (1, x).snd),\n property := (_ : \u2191(\u03c3 (1, x).fst) \u2022 \u2191(1, x).snd + \u2191(1, x).fst \u2022 \u2191(f (1, x).snd) \u2208 I) } =\n \u2191p (f x)) }.toContinuousMap\n (t, x))\n 0 =\n \u2191p.toContinuousMap 0 \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (0, x).fst) * \u2191(0, x).snd + \u2191(0, x).fst * \u2191(f (0, x).snd),\n property := (_ : \u2191(\u03c3 (0, x).fst) \u2022 \u2191(0, x).snd + \u2191(0, x).fst \u2022 \u2191(f (0, x).snd) \u2208 I) } =\n \u2191p x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (1, x).fst) * \u2191(1, x).snd + \u2191(1, x).fst * \u2191(f (1, x).snd),\n property := (_ : \u2191(\u03c3 (1, x).fst) \u2022 \u2191(1, x).snd + \u2191(1, x).fst \u2022 \u2191(f (1, x).snd) \u2208 I) } =\n \u2191p (f x)) }.toContinuousMap\n (t, x))\n 0 =\n \u2191(Path.reparam p f hf hf\u2080 hf\u2081).toContinuousMap 0\n[PROOFSTEP]\nsimp [hf\u2080]\n -- Porting note: Originally `norm_num [hf\u2080]`\n[GOAL]\ncase inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\nt x : \u2191I\nhx : x \u2208 {1}\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (0, x).fst) * \u2191(0, x).snd + \u2191(0, x).fst * \u2191(f (0, x).snd),\n property := (_ : \u2191(\u03c3 (0, x).fst) \u2022 \u2191(0, x).snd + \u2191(0, x).fst \u2022 \u2191(f (0, x).snd) \u2208 I) } =\n \u2191p x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (1, x).fst) * \u2191(1, x).snd + \u2191(1, x).fst * \u2191(f (1, x).snd),\n property := (_ : \u2191(\u03c3 (1, x).fst) \u2022 \u2191(1, x).snd + \u2191(1, x).fst \u2022 \u2191(f (1, x).snd) \u2208 I) } =\n \u2191p (f x)) }.toContinuousMap\n (t, x))\n x =\n \u2191p.toContinuousMap x \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (0, x).fst) * \u2191(0, x).snd + \u2191(0, x).fst * \u2191(f (0, x).snd),\n property := (_ : \u2191(\u03c3 (0, x).fst) \u2022 \u2191(0, x).snd + \u2191(0, x).fst \u2022 \u2191(f (0, x).snd) \u2208 I) } =\n \u2191p x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (1, x).fst) * \u2191(1, x).snd + \u2191(1, x).fst * \u2191(f (1, x).snd),\n property := (_ : \u2191(\u03c3 (1, x).fst) \u2022 \u2191(1, x).snd + \u2191(1, x).fst \u2022 \u2191(f (1, x).snd) \u2208 I) } =\n \u2191p (f x)) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.reparam p f hf hf\u2080 hf\u2081).toContinuousMap x\n[PROOFSTEP]\nrw [Set.mem_singleton_iff] at hx \n[GOAL]\ncase inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\nt x : \u2191I\nhx : x = 1\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (0, x).fst) * \u2191(0, x).snd + \u2191(0, x).fst * \u2191(f (0, x).snd),\n property := (_ : \u2191(\u03c3 (0, x).fst) \u2022 \u2191(0, x).snd + \u2191(0, x).fst \u2022 \u2191(f (0, x).snd) \u2208 I) } =\n \u2191p x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (1, x).fst) * \u2191(1, x).snd + \u2191(1, x).fst * \u2191(f (1, x).snd),\n property := (_ : \u2191(\u03c3 (1, x).fst) \u2022 \u2191(1, x).snd + \u2191(1, x).fst \u2022 \u2191(f (1, x).snd) \u2208 I) } =\n \u2191p (f x)) }.toContinuousMap\n (t, x))\n x =\n \u2191p.toContinuousMap x \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (0, x).fst) * \u2191(0, x).snd + \u2191(0, x).fst * \u2191(f (0, x).snd),\n property := (_ : \u2191(\u03c3 (0, x).fst) \u2022 \u2191(0, x).snd + \u2191(0, x).fst \u2022 \u2191(f (0, x).snd) \u2208 I) } =\n \u2191p x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (1, x).fst) * \u2191(1, x).snd + \u2191(1, x).fst * \u2191(f (1, x).snd),\n property := (_ : \u2191(\u03c3 (1, x).fst) \u2022 \u2191(1, x).snd + \u2191(1, x).fst \u2022 \u2191(f (1, x).snd) \u2208 I) } =\n \u2191p (f x)) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.reparam p f hf hf\u2080 hf\u2081).toContinuousMap x\n[PROOFSTEP]\nrw [hx]\n[GOAL]\ncase inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nf : \u2191I \u2192 \u2191I\nhf : Continuous f\nhf\u2080 : f 0 = 0\nhf\u2081 : f 1 = 1\nt x : \u2191I\nhx : x = 1\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (0, x).fst) * \u2191(0, x).snd + \u2191(0, x).fst * \u2191(f (0, x).snd),\n property := (_ : \u2191(\u03c3 (0, x).fst) \u2022 \u2191(0, x).snd + \u2191(0, x).fst \u2022 \u2191(f (0, x).snd) \u2208 I) } =\n \u2191p x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (1, x).fst) * \u2191(1, x).snd + \u2191(1, x).fst * \u2191(f (1, x).snd),\n property := (_ : \u2191(\u03c3 (1, x).fst) \u2022 \u2191(1, x).snd + \u2191(1, x).fst \u2022 \u2191(f (1, x).snd) \u2208 I) } =\n \u2191p (f x)) }.toContinuousMap\n (t, x))\n 1 =\n \u2191p.toContinuousMap 1 \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p\n { val := \u2191(\u03c3 x.fst) * \u2191x.snd + \u2191x.fst * \u2191(f x.snd),\n property := (_ : \u2191(\u03c3 x.fst) \u2022 \u2191x.snd + \u2191x.fst \u2022 \u2191(f x.snd) \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (0, x).fst) * \u2191(0, x).snd + \u2191(0, x).fst * \u2191(f (0, x).snd),\n property := (_ : \u2191(\u03c3 (0, x).fst) \u2022 \u2191(0, x).snd + \u2191(0, x).fst \u2022 \u2191(f (0, x).snd) \u2208 I) } =\n \u2191p x),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n \u2191p\n { val := \u2191(\u03c3 (1, x).fst) * \u2191(1, x).snd + \u2191(1, x).fst * \u2191(f (1, x).snd),\n property := (_ : \u2191(\u03c3 (1, x).fst) \u2022 \u2191(1, x).snd + \u2191(1, x).fst \u2022 \u2191(f (1, x).snd) \u2208 I) } =\n \u2191p (f x)) }.toContinuousMap\n (t, x))\n 1 =\n \u2191(Path.reparam p f hf hf\u2080 hf\u2081).toContinuousMap 1\n[PROOFSTEP]\nsimp [hf\u2081]\n -- Porting note: Originally `norm_num [hf\u2080]`\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np q : Path x\u2080 x\u2081\nF : Homotopy p q\n\u22a2 \u2200 (x : \u2191I),\n ContinuousMap.toFun (ContinuousMap.mk fun x => \u2191F (x.fst, \u03c3 x.snd)) (0, x) = \u2191(Path.symm p).toContinuousMap x\n[PROOFSTEP]\nsimp [Path.symm]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np q : Path x\u2080 x\u2081\nF : Homotopy p q\n\u22a2 \u2200 (x : \u2191I),\n ContinuousMap.toFun (ContinuousMap.mk fun x => \u2191F (x.fst, \u03c3 x.snd)) (1, x) = \u2191(Path.symm q).toContinuousMap x\n[PROOFSTEP]\nsimp [Path.symm]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np q : Path x\u2080 x\u2081\nF : Homotopy p q\nt x : \u2191I\nhx : x \u2208 {0, 1}\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun x => \u2191F (x.fst, \u03c3 x.snd),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191F (0, \u03c3 a) = \u2191p (\u03c3 a)),\n map_one_left := (_ : \u2200 (a : \u2191I), \u2191F (1, \u03c3 a) = \u2191q (\u03c3 a)) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.symm p).toContinuousMap x \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun x => \u2191F (x.fst, \u03c3 x.snd),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191F (0, \u03c3 a) = \u2191p (\u03c3 a)),\n map_one_left := (_ : \u2200 (a : \u2191I), \u2191F (1, \u03c3 a) = \u2191q (\u03c3 a)) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.symm q).toContinuousMap x\n[PROOFSTEP]\ncases' hx with hx hx\n[GOAL]\ncase inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np q : Path x\u2080 x\u2081\nF : Homotopy p q\nt x : \u2191I\nhx : x = 0\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun x => \u2191F (x.fst, \u03c3 x.snd),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191F (0, \u03c3 a) = \u2191p (\u03c3 a)),\n map_one_left := (_ : \u2200 (a : \u2191I), \u2191F (1, \u03c3 a) = \u2191q (\u03c3 a)) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.symm p).toContinuousMap x \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun x => \u2191F (x.fst, \u03c3 x.snd),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191F (0, \u03c3 a) = \u2191p (\u03c3 a)),\n map_one_left := (_ : \u2200 (a : \u2191I), \u2191F (1, \u03c3 a) = \u2191q (\u03c3 a)) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.symm q).toContinuousMap x\n[PROOFSTEP]\nrw [hx]\n[GOAL]\ncase inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np q : Path x\u2080 x\u2081\nF : Homotopy p q\nt x : \u2191I\nhx : x = 0\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun x => \u2191F (x.fst, \u03c3 x.snd),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191F (0, \u03c3 a) = \u2191p (\u03c3 a)),\n map_one_left := (_ : \u2200 (a : \u2191I), \u2191F (1, \u03c3 a) = \u2191q (\u03c3 a)) }.toContinuousMap\n (t, x))\n 0 =\n \u2191(Path.symm p).toContinuousMap 0 \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun x => \u2191F (x.fst, \u03c3 x.snd),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191F (0, \u03c3 a) = \u2191p (\u03c3 a)),\n map_one_left := (_ : \u2200 (a : \u2191I), \u2191F (1, \u03c3 a) = \u2191q (\u03c3 a)) }.toContinuousMap\n (t, x))\n 0 =\n \u2191(Path.symm q).toContinuousMap 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np q : Path x\u2080 x\u2081\nF : Homotopy p q\nt x : \u2191I\nhx : x \u2208 {1}\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun x => \u2191F (x.fst, \u03c3 x.snd),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191F (0, \u03c3 a) = \u2191p (\u03c3 a)),\n map_one_left := (_ : \u2200 (a : \u2191I), \u2191F (1, \u03c3 a) = \u2191q (\u03c3 a)) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.symm p).toContinuousMap x \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun x => \u2191F (x.fst, \u03c3 x.snd),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191F (0, \u03c3 a) = \u2191p (\u03c3 a)),\n map_one_left := (_ : \u2200 (a : \u2191I), \u2191F (1, \u03c3 a) = \u2191q (\u03c3 a)) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.symm q).toContinuousMap x\n[PROOFSTEP]\nrw [Set.mem_singleton_iff] at hx \n[GOAL]\ncase inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np q : Path x\u2080 x\u2081\nF : Homotopy p q\nt x : \u2191I\nhx : x = 1\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun x => \u2191F (x.fst, \u03c3 x.snd),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191F (0, \u03c3 a) = \u2191p (\u03c3 a)),\n map_one_left := (_ : \u2200 (a : \u2191I), \u2191F (1, \u03c3 a) = \u2191q (\u03c3 a)) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.symm p).toContinuousMap x \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun x => \u2191F (x.fst, \u03c3 x.snd),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191F (0, \u03c3 a) = \u2191p (\u03c3 a)),\n map_one_left := (_ : \u2200 (a : \u2191I), \u2191F (1, \u03c3 a) = \u2191q (\u03c3 a)) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.symm q).toContinuousMap x\n[PROOFSTEP]\nrw [hx]\n[GOAL]\ncase inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np q : Path x\u2080 x\u2081\nF : Homotopy p q\nt x : \u2191I\nhx : x = 1\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun x => \u2191F (x.fst, \u03c3 x.snd),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191F (0, \u03c3 a) = \u2191p (\u03c3 a)),\n map_one_left := (_ : \u2200 (a : \u2191I), \u2191F (1, \u03c3 a) = \u2191q (\u03c3 a)) }.toContinuousMap\n (t, x))\n 1 =\n \u2191(Path.symm p).toContinuousMap 1 \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk fun x => \u2191F (x.fst, \u03c3 x.snd),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191F (0, \u03c3 a) = \u2191p (\u03c3 a)),\n map_one_left := (_ : \u2200 (a : \u2191I), \u2191F (1, \u03c3 a) = \u2191q (\u03c3 a)) }.toContinuousMap\n (t, x))\n 1 =\n \u2191(Path.symm q).toContinuousMap 1\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np q : Path x\u2080 x\u2081\nF : Homotopy p q\nf : C(X, Y)\n\u22a2 \u2200 (x : \u2191I),\n ContinuousMap.toFun (ContinuousMap.mk (\u2191f \u2218 \u2191F)) (0, x) = \u2191(Path.map p (_ : Continuous \u2191f)).toContinuousMap x\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np q : Path x\u2080 x\u2081\nF : Homotopy p q\nf : C(X, Y)\n\u22a2 \u2200 (x : \u2191I),\n ContinuousMap.toFun (ContinuousMap.mk (\u2191f \u2218 \u2191F)) (1, x) = \u2191(Path.map q (_ : Continuous \u2191f)).toContinuousMap x\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np q : Path x\u2080 x\u2081\nF : Homotopy p q\nf : C(X, Y)\nt x : \u2191I\nhx : x \u2208 {0, 1}\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk (\u2191f \u2218 \u2191F),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (0, a)) = \u2191(Path.map p (_ : Continuous \u2191f)) a),\n map_one_left :=\n (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (1, a)) = \u2191(Path.map q (_ : Continuous \u2191f)) a) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.map p (_ : Continuous \u2191f)).toContinuousMap x \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk (\u2191f \u2218 \u2191F),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (0, a)) = \u2191(Path.map p (_ : Continuous \u2191f)) a),\n map_one_left :=\n (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (1, a)) = \u2191(Path.map q (_ : Continuous \u2191f)) a) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.map q (_ : Continuous \u2191f)).toContinuousMap x\n[PROOFSTEP]\ncases' hx with hx hx\n[GOAL]\ncase inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np q : Path x\u2080 x\u2081\nF : Homotopy p q\nf : C(X, Y)\nt x : \u2191I\nhx : x = 0\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk (\u2191f \u2218 \u2191F),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (0, a)) = \u2191(Path.map p (_ : Continuous \u2191f)) a),\n map_one_left :=\n (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (1, a)) = \u2191(Path.map q (_ : Continuous \u2191f)) a) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.map p (_ : Continuous \u2191f)).toContinuousMap x \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk (\u2191f \u2218 \u2191F),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (0, a)) = \u2191(Path.map p (_ : Continuous \u2191f)) a),\n map_one_left :=\n (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (1, a)) = \u2191(Path.map q (_ : Continuous \u2191f)) a) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.map q (_ : Continuous \u2191f)).toContinuousMap x\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np q : Path x\u2080 x\u2081\nF : Homotopy p q\nf : C(X, Y)\nt x : \u2191I\nhx : x \u2208 {1}\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk (\u2191f \u2218 \u2191F),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (0, a)) = \u2191(Path.map p (_ : Continuous \u2191f)) a),\n map_one_left :=\n (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (1, a)) = \u2191(Path.map q (_ : Continuous \u2191f)) a) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.map p (_ : Continuous \u2191f)).toContinuousMap x \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk (\u2191f \u2218 \u2191F),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (0, a)) = \u2191(Path.map p (_ : Continuous \u2191f)) a),\n map_one_left :=\n (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (1, a)) = \u2191(Path.map q (_ : Continuous \u2191f)) a) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.map q (_ : Continuous \u2191f)).toContinuousMap x\n[PROOFSTEP]\nrw [Set.mem_singleton_iff] at hx \n[GOAL]\ncase inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np q : Path x\u2080 x\u2081\nF : Homotopy p q\nf : C(X, Y)\nt x : \u2191I\nhx : x = 1\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk (\u2191f \u2218 \u2191F),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (0, a)) = \u2191(Path.map p (_ : Continuous \u2191f)) a),\n map_one_left :=\n (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (1, a)) = \u2191(Path.map q (_ : Continuous \u2191f)) a) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.map p (_ : Continuous \u2191f)).toContinuousMap x \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n { toContinuousMap := ContinuousMap.mk (\u2191f \u2218 \u2191F),\n map_zero_left := (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (0, a)) = \u2191(Path.map p (_ : Continuous \u2191f)) a),\n map_one_left :=\n (_ : \u2200 (a : \u2191I), \u2191f (\u2191F (1, a)) = \u2191(Path.map q (_ : Continuous \u2191f)) a) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.map q (_ : Continuous \u2191f)).toContinuousMap x\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2081 : Path x\u2080 x\u2081\np\u2082 : Path x\u2082 x\u2083\nhp : \u2200 (t : \u2191I), \u2191p\u2081 t = \u2191p\u2082 t\n\u22a2 HEq (Quotient.mk (Homotopic.setoid x\u2080 x\u2081) p\u2081) (Quotient.mk (Homotopic.setoid x\u2082 x\u2083) p\u2082)\n[PROOFSTEP]\nobtain rfl : x\u2080 = x\u2082 := by convert hp 0 <;> simp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2081 : Path x\u2080 x\u2081\np\u2082 : Path x\u2082 x\u2083\nhp : \u2200 (t : \u2191I), \u2191p\u2081 t = \u2191p\u2082 t\n\u22a2 x\u2080 = x\u2082\n[PROOFSTEP]\nconvert hp 0\n[GOAL]\ncase h.e'_2\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2081 : Path x\u2080 x\u2081\np\u2082 : Path x\u2082 x\u2083\nhp : \u2200 (t : \u2191I), \u2191p\u2081 t = \u2191p\u2082 t\n\u22a2 x\u2080 = \u2191p\u2081 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\np\u2081 : Path x\u2080 x\u2081\np\u2082 : Path x\u2082 x\u2083\nhp : \u2200 (t : \u2191I), \u2191p\u2081 t = \u2191p\u2082 t\n\u22a2 x\u2082 = \u2191p\u2082 0\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2083 : X\np\u2081 : Path x\u2080 x\u2081\np\u2082 : Path x\u2080 x\u2083\nhp : \u2200 (t : \u2191I), \u2191p\u2081 t = \u2191p\u2082 t\n\u22a2 HEq (Quotient.mk (Homotopic.setoid x\u2080 x\u2081) p\u2081) (Quotient.mk (Homotopic.setoid x\u2080 x\u2083) p\u2082)\n[PROOFSTEP]\nobtain rfl : x\u2081 = x\u2083 := by convert hp 1 <;> simp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2083 : X\np\u2081 : Path x\u2080 x\u2081\np\u2082 : Path x\u2080 x\u2083\nhp : \u2200 (t : \u2191I), \u2191p\u2081 t = \u2191p\u2082 t\n\u22a2 x\u2081 = x\u2083\n[PROOFSTEP]\nconvert hp 1\n[GOAL]\ncase h.e'_2\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2083 : X\np\u2081 : Path x\u2080 x\u2081\np\u2082 : Path x\u2080 x\u2083\nhp : \u2200 (t : \u2191I), \u2191p\u2081 t = \u2191p\u2082 t\n\u22a2 x\u2081 = \u2191p\u2081 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 x\u2083 : X\np\u2081 : Path x\u2080 x\u2081\np\u2082 : Path x\u2080 x\u2083\nhp : \u2200 (t : \u2191I), \u2191p\u2081 t = \u2191p\u2082 t\n\u22a2 x\u2083 = \u2191p\u2082 1\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np\u2081 p\u2082 : Path x\u2080 x\u2081\nhp : \u2200 (t : \u2191I), \u2191p\u2081 t = \u2191p\u2082 t\n\u22a2 HEq (Quotient.mk (Homotopic.setoid x\u2080 x\u2081) p\u2081) (Quotient.mk (Homotopic.setoid x\u2080 x\u2081) p\u2082)\n[PROOFSTEP]\nrw [heq_iff_eq]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np\u2081 p\u2082 : Path x\u2080 x\u2081\nhp : \u2200 (t : \u2191I), \u2191p\u2081 t = \u2191p\u2082 t\n\u22a2 Quotient.mk (Homotopic.setoid x\u2080 x\u2081) p\u2081 = Quotient.mk (Homotopic.setoid x\u2080 x\u2081) p\u2082\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np\u2081 p\u2082 : Path x\u2080 x\u2081\nhp : \u2200 (t : \u2191I), \u2191p\u2081 t = \u2191p\u2082 t\n\u22a2 p\u2081 = p\u2082\n[PROOFSTEP]\next t\n[GOAL]\ncase e_a.a.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np\u2081 p\u2082 : Path x\u2080 x\u2081\nhp : \u2200 (t : \u2191I), \u2191p\u2081 t = \u2191p\u2082 t\nt : \u2191I\n\u22a2 \u2191p\u2081 t = \u2191p\u2082 t\n[PROOFSTEP]\nexact hp t\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\ninst\u271d : Nonempty Y\n\u22a2 Homotopic (const Y x\u2080) (const Y x\u2081) \u2194 Joined x\u2080 x\u2081\n[PROOFSTEP]\ninhabit Y\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\ninst\u271d : Nonempty Y\ninhabited_h : Inhabited Y\n\u22a2 Homotopic (const Y x\u2080) (const Y x\u2081) \u2194 Joined x\u2080 x\u2081\n[PROOFSTEP]\nrefine \u27e8fun \u27e8H\u27e9 \u21a6 \u27e8\u27e8(H.toContinuousMap.comp .prodSwap).curry default, ?_, ?_\u27e9\u27e9, fun \u27e8p\u27e9 \u21a6 \u27e8p.toHomotopyConst\u27e9\u27e9\n[GOAL]\ncase refine_1\nX : Type u\nY : Type v\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\ninst\u271d : Nonempty Y\ninhabited_h : Inhabited Y\nx\u271d : Homotopic (const Y x\u2080) (const Y x\u2081)\nH : Homotopy (const Y x\u2080) (const Y x\u2081)\n\u22a2 ContinuousMap.toFun (\u2191(curry (comp H.toContinuousMap prodSwap)) default) 0 = x\u2080\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_2\nX : Type u\nY : Type v\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\nx\u2080 x\u2081 x\u2082 x\u2083 : X\ninst\u271d : Nonempty Y\ninhabited_h : Inhabited Y\nx\u271d : Homotopic (const Y x\u2080) (const Y x\u2081)\nH : Homotopy (const Y x\u2080) (const Y x\u2081)\n\u22a2 ContinuousMap.toFun (\u2191(curry (comp H.toContinuousMap prodSwap)) default) 1 = x\u2081\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Topology.Homotopy.Path", "llama_tokens": 29610, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085859124002, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.41716328182755674}} {"text": "[GOAL]\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\n\u22a2 Continuous fun p => infEdist p.fst \u2191p.snd\n[PROOFSTEP]\nrefine' continuous_of_le_add_edist 2 (by simp) _\n[GOAL]\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\n\u22a2 2 \u2260 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\n\u22a2 \u2200 (x y : \u03b1 \u00d7 Closeds \u03b1), infEdist x.fst \u2191x.snd \u2264 infEdist y.fst \u2191y.snd + 2 * edist x y\n[PROOFSTEP]\nrintro \u27e8x, s\u27e9 \u27e8y, t\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns\u271d : Set \u03b1\nx : \u03b1\ns : Closeds \u03b1\ny : \u03b1\nt : Closeds \u03b1\n\u22a2 infEdist (x, s).fst \u2191(x, s).snd \u2264 infEdist (y, t).fst \u2191(y, t).snd + 2 * edist (x, s) (y, t)\n[PROOFSTEP]\ncalc\n infEdist x s \u2264 infEdist x t + hausdorffEdist (t : Set \u03b1) s := infEdist_le_infEdist_add_hausdorffEdist\n _ \u2264 infEdist y t + edist x y + hausdorffEdist (t : Set \u03b1) s := (add_le_add_right infEdist_le_infEdist_add_edist _)\n _ = infEdist y t + (edist x y + hausdorffEdist (s : Set \u03b1) t) := by rw [add_assoc, hausdorffEdist_comm]\n _ \u2264 infEdist y t + (edist (x, s) (y, t) + edist (x, s) (y, t)) :=\n (add_le_add_left (add_le_add (le_max_left _ _) (le_max_right _ _)) _)\n _ = infEdist y t + 2 * edist (x, s) (y, t) := by rw [\u2190 mul_two, mul_comm]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns\u271d : Set \u03b1\nx : \u03b1\ns : Closeds \u03b1\ny : \u03b1\nt : Closeds \u03b1\n\u22a2 infEdist y \u2191t + edist x y + hausdorffEdist \u2191t \u2191s = infEdist y \u2191t + (edist x y + hausdorffEdist \u2191s \u2191t)\n[PROOFSTEP]\nrw [add_assoc, hausdorffEdist_comm]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns\u271d : Set \u03b1\nx : \u03b1\ns : Closeds \u03b1\ny : \u03b1\nt : Closeds \u03b1\n\u22a2 infEdist y \u2191t + (edist (x, s) (y, t) + edist (x, s) (y, t)) = infEdist y \u2191t + 2 * edist (x, s) (y, t)\n[PROOFSTEP]\nrw [\u2190 mul_two, mul_comm]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\n\u22a2 IsClosed {t | \u2191t \u2286 s}\n[PROOFSTEP]\nrefine'\n isClosed_of_closure_subset fun t ht x hx =>\n _\n -- t : Closeds \u03b1, ht : t \u2208 closure {t : Closeds \u03b1 | t \u2286 s},\n -- x : \u03b1, hx : x \u2208 t\n -- goal : x \u2208 s\n[GOAL]\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 \u2191t\n\u22a2 x \u2208 s\n[PROOFSTEP]\nhave : x \u2208 closure s := by\n refine' mem_closure_iff.2 fun \u03b5 \u03b5pos => _\n rcases mem_closure_iff.1 ht \u03b5 \u03b5pos with\n \u27e8u, hu, Dtu\u27e9\n -- u : Closeds \u03b1, hu : u \u2208 {t : Closeds \u03b1 | t \u2286 s}, hu' : edist t u < \u03b5\n rcases exists_edist_lt_of_hausdorffEdist_lt hx Dtu with\n \u27e8y, hy, Dxy\u27e9\n -- y : \u03b1, hy : y \u2208 u, Dxy : edist x y < \u03b5\n exact \u27e8y, hu hy, Dxy\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 \u2191t\n\u22a2 x \u2208 closure s\n[PROOFSTEP]\nrefine' mem_closure_iff.2 fun \u03b5 \u03b5pos => _\n[GOAL]\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 \u2191t\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 y, y \u2208 s \u2227 edist x y < \u03b5\n[PROOFSTEP]\nrcases mem_closure_iff.1 ht \u03b5 \u03b5pos with\n \u27e8u, hu, Dtu\u27e9\n -- u : Closeds \u03b1, hu : u \u2208 {t : Closeds \u03b1 | t \u2286 s}, hu' : edist t u < \u03b5\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 \u2191t\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nu : Closeds \u03b1\nhu : u \u2208 {t | \u2191t \u2286 s}\nDtu : edist t u < \u03b5\n\u22a2 \u2203 y, y \u2208 s \u2227 edist x y < \u03b5\n[PROOFSTEP]\nrcases exists_edist_lt_of_hausdorffEdist_lt hx Dtu with\n \u27e8y, hy, Dxy\u27e9\n -- y : \u03b1, hy : y \u2208 u, Dxy : edist x y < \u03b5\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 \u2191t\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nu : Closeds \u03b1\nhu : u \u2208 {t | \u2191t \u2286 s}\nDtu : edist t u < \u03b5\ny : \u03b1\nhy : y \u2208 \u2191u\nDxy : edist x y < \u03b5\n\u22a2 \u2203 y, y \u2208 s \u2227 edist x y < \u03b5\n[PROOFSTEP]\nexact \u27e8y, hu hy, Dxy\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nt : Closeds \u03b1\nht : t \u2208 closure {t | \u2191t \u2286 s}\nx : \u03b1\nhx : x \u2208 \u2191t\nthis : x \u2208 closure s\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrwa [hs.closure_eq] at this \n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\n\u22a2 CompleteSpace (Closeds \u03b1)\n[PROOFSTEP]\nlet B : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\n\u22a2 CompleteSpace (Closeds \u03b1)\n[PROOFSTEP]\nhave B_pos : \u2200 n, (0 : \u211d\u22650\u221e) < B n := by simp [ENNReal.pow_pos]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\n\u22a2 \u2200 (n : \u2115), 0 < B n\n[PROOFSTEP]\nsimp [ENNReal.pow_pos]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\n\u22a2 CompleteSpace (Closeds \u03b1)\n[PROOFSTEP]\nhave B_ne_top : \u2200 n, B n \u2260 \u22a4 := by\n simp [ENNReal.pow_ne_top]\n /- Consider a sequence of closed sets `s n` with `edist (s n) (s (n+1)) < B n`.\n We will show that it converges. The limit set is `t0 = \u22c2n, closure (\u22c3m\u2265n, s m)`.\n We will have to show that a point in `s n` is close to a point in `t0`, and a point\n in `t0` is close to a point in `s n`. The completeness then follows from a\n standard criterion. -/\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\n\u22a2 \u2200 (n : \u2115), B n \u2260 \u22a4\n[PROOFSTEP]\nsimp [ENNReal.pow_ne_top]\n /- Consider a sequence of closed sets `s n` with `edist (s n) (s (n+1)) < B n`.\n We will show that it converges. The limit set is `t0 = \u22c2n, closure (\u22c3m\u2265n, s m)`.\n We will have to show that a point in `s n` is close to a point in `t0`, and a point\n in `t0` is close to a point in `s n`. The completeness then follows from a\n standard criterion. -/\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\n\u22a2 CompleteSpace (Closeds \u03b1)\n[PROOFSTEP]\nrefine' complete_of_convergent_controlled_sequences B B_pos fun s hs => _\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\n\u22a2 \u2203 x, Tendsto s atTop (\ud835\udcdd x)\n[PROOFSTEP]\nlet t0 := \u22c2 n, closure (\u22c3 m \u2265 n, s m : Set \u03b1)\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\n\u22a2 \u2203 x, Tendsto s atTop (\ud835\udcdd x)\n[PROOFSTEP]\nlet t : Closeds \u03b1 := \u27e8t0, isClosed_iInter fun _ => isClosed_closure\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\n\u22a2 \u2203 x, Tendsto s atTop (\ud835\udcdd x)\n[PROOFSTEP]\nuse t\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\n\u22a2 Tendsto s atTop (\ud835\udcdd t)\n[PROOFSTEP]\nhave I1 : \u2200 n, \u2200 x \u2208 s n, \u2203 y \u2208 t0, edist x y \u2264 2 * B n := by\n /- This is the main difficulty of the proof. Starting from `x \u2208 s n`, we want\n to find a point in `t0` which is close to `x`. Define inductively a sequence of\n points `z m` with `z n = x` and `z m \u2208 s m` and `edist (z m) (z (m+1)) \u2264 B m`. This is\n possible since the Hausdorff distance between `s m` and `s (m+1)` is at most `B m`.\n This sequence is a Cauchy sequence, therefore converging as the space is complete, to\n a limit which satisfies the required properties. -/\n intro n x hx\n obtain \u27e8z, hz\u2080, hz\u27e9 : \u2203 z : \u2200 l, s (n + l), (z 0 : \u03b1) = x \u2227 \u2200 k, edist (z k : \u03b1) (z (k + 1) : \u03b1) \u2264 B n / 2 ^ k := by\n -- We prove existence of the sequence by induction.\n have : \u2200 (l) (z : s (n + l)), \u2203 z' : s (n + l + 1), edist (z : \u03b1) z' \u2264 B n / 2 ^ l :=\n by\n intro l z\n obtain \u27e8z', z'_mem, hz'\u27e9 : \u2203 z' \u2208 s (n + l + 1), edist (z : \u03b1) z' < B n / 2 ^ l :=\n by\n refine' exists_edist_lt_of_hausdorffEdist_lt _ _\n \u00b7 exact (s (n + l) : Set \u03b1)\n \u00b7 exact z.2\n simp only [ENNReal.inv_pow, div_eq_mul_inv]\n rw [\u2190 pow_add]\n apply hs <;> simp\n exact \u27e8\u27e8z', z'_mem\u27e9, le_of_lt hz'\u27e9\n use fun k => Nat.recOn k \u27e8x, hx\u27e9 fun l z => choose (this l z)\n simp only [Nat.add_zero, Nat.zero_eq, Nat.rec_zero, Nat.rec_add_one, true_and]\n exact fun k =>\n choose_spec\n (this k _)\n -- it follows from the previous bound that `z` is a Cauchy sequence\n have : CauchySeq fun k => (z k : \u03b1) := cauchySeq_of_edist_le_geometric_two (B n) (B_ne_top n) hz\n rcases cauchySeq_tendsto_of_complete this with \u27e8y, y_lim\u27e9\n use y\n have : y \u2208 t0 :=\n mem_iInter.2 fun k =>\n mem_closure_of_tendsto y_lim\n (by\n simp only [exists_prop, Set.mem_iUnion, Filter.eventually_atTop, Set.mem_preimage, Set.preimage_iUnion]\n exact \u27e8k, fun m hm => \u27e8n + m, zero_add k \u25b8 add_le_add (zero_le n) hm, (z m).2\u27e9\u27e9)\n use this\n rw [\u2190 hz\u2080]\n exact edist_le_of_edist_le_geometric_two_of_tendsto\u2080 (B n) hz y_lim\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\n\u22a2 \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\n[PROOFSTEP]\nintro n x hx\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\n\u22a2 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\n[PROOFSTEP]\nobtain \u27e8z, hz\u2080, hz\u27e9 : \u2203 z : \u2200 l, s (n + l), (z 0 : \u03b1) = x \u2227 \u2200 k, edist (z k : \u03b1) (z (k + 1) : \u03b1) \u2264 B n / 2 ^ k := by\n -- We prove existence of the sequence by induction.\n have : \u2200 (l) (z : s (n + l)), \u2203 z' : s (n + l + 1), edist (z : \u03b1) z' \u2264 B n / 2 ^ l :=\n by\n intro l z\n obtain \u27e8z', z'_mem, hz'\u27e9 : \u2203 z' \u2208 s (n + l + 1), edist (z : \u03b1) z' < B n / 2 ^ l :=\n by\n refine' exists_edist_lt_of_hausdorffEdist_lt _ _\n \u00b7 exact (s (n + l) : Set \u03b1)\n \u00b7 exact z.2\n simp only [ENNReal.inv_pow, div_eq_mul_inv]\n rw [\u2190 pow_add]\n apply hs <;> simp\n exact \u27e8\u27e8z', z'_mem\u27e9, le_of_lt hz'\u27e9\n use fun k => Nat.recOn k \u27e8x, hx\u27e9 fun l z => choose (this l z)\n simp only [Nat.add_zero, Nat.zero_eq, Nat.rec_zero, Nat.rec_add_one, true_and]\n exact fun k =>\n choose_spec\n (this k _)\n -- it follows from the previous bound that `z` is a Cauchy sequence\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\n\u22a2 \u2203 z, \u2191(z 0) = x \u2227 \u2200 (k : \u2115), edist \u2191(z k) \u2191(z (k + 1)) \u2264 B n / 2 ^ k\n[PROOFSTEP]\nhave : \u2200 (l) (z : s (n + l)), \u2203 z' : s (n + l + 1), edist (z : \u03b1) z' \u2264 B n / 2 ^ l :=\n by\n intro l z\n obtain \u27e8z', z'_mem, hz'\u27e9 : \u2203 z' \u2208 s (n + l + 1), edist (z : \u03b1) z' < B n / 2 ^ l :=\n by\n refine' exists_edist_lt_of_hausdorffEdist_lt _ _\n \u00b7 exact (s (n + l) : Set \u03b1)\n \u00b7 exact z.2\n simp only [ENNReal.inv_pow, div_eq_mul_inv]\n rw [\u2190 pow_add]\n apply hs <;> simp\n exact \u27e8\u27e8z', z'_mem\u27e9, le_of_lt hz'\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\n\u22a2 \u2200 (l : \u2115) (z : { x // x \u2208 s (n + l) }), \u2203 z', edist \u2191z \u2191z' \u2264 B n / 2 ^ l\n[PROOFSTEP]\nintro l z\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nl : \u2115\nz : { x // x \u2208 s (n + l) }\n\u22a2 \u2203 z', edist \u2191z \u2191z' \u2264 B n / 2 ^ l\n[PROOFSTEP]\nobtain \u27e8z', z'_mem, hz'\u27e9 : \u2203 z' \u2208 s (n + l + 1), edist (z : \u03b1) z' < B n / 2 ^ l :=\n by\n refine' exists_edist_lt_of_hausdorffEdist_lt _ _\n \u00b7 exact (s (n + l) : Set \u03b1)\n \u00b7 exact z.2\n simp only [ENNReal.inv_pow, div_eq_mul_inv]\n rw [\u2190 pow_add]\n apply hs <;> simp\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nl : \u2115\nz : { x // x \u2208 s (n + l) }\n\u22a2 \u2203 z', z' \u2208 s (n + l + 1) \u2227 edist (\u2191z) z' < B n / 2 ^ l\n[PROOFSTEP]\nrefine' exists_edist_lt_of_hausdorffEdist_lt _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nl : \u2115\nz : { x // x \u2208 s (n + l) }\n\u22a2 Set \u03b1\n[PROOFSTEP]\nexact (s (n + l) : Set \u03b1)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nl : \u2115\nz : { x // x \u2208 s (n + l) }\n\u22a2 \u2191z \u2208 \u2191(s (n + l))\n[PROOFSTEP]\nexact z.2\n[GOAL]\ncase refine'_3\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nl : \u2115\nz : { x // x \u2208 s (n + l) }\n\u22a2 hausdorffEdist \u2191(s (n + l)) \u2191(s (n + l + 1)) < B n / 2 ^ l\n[PROOFSTEP]\nsimp only [ENNReal.inv_pow, div_eq_mul_inv]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nl : \u2115\nz : { x // x \u2208 s (n + l) }\n\u22a2 hausdorffEdist \u2191(s (n + l)) \u2191(s (n + l + 1)) < 2\u207b\u00b9 ^ n * 2\u207b\u00b9 ^ l\n[PROOFSTEP]\nrw [\u2190 pow_add]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nl : \u2115\nz : { x // x \u2208 s (n + l) }\n\u22a2 hausdorffEdist \u2191(s (n + l)) \u2191(s (n + l + 1)) < 2\u207b\u00b9 ^ (n + l)\n[PROOFSTEP]\napply hs\n[GOAL]\ncase refine'_3.a\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nl : \u2115\nz : { x // x \u2208 s (n + l) }\n\u22a2 n + l \u2264 n + l\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3.a\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nl : \u2115\nz : { x // x \u2208 s (n + l) }\n\u22a2 n + l \u2264 n + l + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nl : \u2115\nz : { x // x \u2208 s (n + l) }\nz' : \u03b1\nz'_mem : z' \u2208 s (n + l + 1)\nhz' : edist (\u2191z) z' < B n / 2 ^ l\n\u22a2 \u2203 z', edist \u2191z \u2191z' \u2264 B n / 2 ^ l\n[PROOFSTEP]\nexact \u27e8\u27e8z', z'_mem\u27e9, le_of_lt hz'\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nthis : \u2200 (l : \u2115) (z : { x // x \u2208 s (n + l) }), \u2203 z', edist \u2191z \u2191z' \u2264 B n / 2 ^ l\n\u22a2 \u2203 z, \u2191(z 0) = x \u2227 \u2200 (k : \u2115), edist \u2191(z k) \u2191(z (k + 1)) \u2264 B n / 2 ^ k\n[PROOFSTEP]\nuse fun k => Nat.recOn k \u27e8x, hx\u27e9 fun l z => choose (this l z)\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nthis : \u2200 (l : \u2115) (z : { x // x \u2208 s (n + l) }), \u2203 z', edist \u2191z \u2191z' \u2264 B n / 2 ^ l\n\u22a2 \u2191(Nat.recOn 0 { val := x, property := hx } fun l z => choose (_ : \u2203 z', edist \u2191z \u2191z' \u2264 B n / 2 ^ l)) = x \u2227\n \u2200 (k : \u2115),\n edist \u2191(Nat.recOn k { val := x, property := hx } fun l z => choose (_ : \u2203 z', edist \u2191z \u2191z' \u2264 B n / 2 ^ l))\n \u2191(Nat.recOn (k + 1) { val := x, property := hx } fun l z => choose (_ : \u2203 z', edist \u2191z \u2191z' \u2264 B n / 2 ^ l)) \u2264\n B n / 2 ^ k\n[PROOFSTEP]\nsimp only [Nat.add_zero, Nat.zero_eq, Nat.rec_zero, Nat.rec_add_one, true_and]\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nthis : \u2200 (l : \u2115) (z : { x // x \u2208 s (n + l) }), \u2203 z', edist \u2191z \u2191z' \u2264 B n / 2 ^ l\n\u22a2 \u2200 (k : \u2115),\n edist \u2191(Nat.rec { val := x, property := hx } (fun l z => choose (_ : \u2203 z', edist \u2191z \u2191z' \u2264 B n / 2 ^ l)) k)\n \u2191(choose\n (_ :\n \u2203 z',\n edist\n \u2191(Nat.rec { val := x, property := hx } (fun l z => choose (_ : \u2203 z', edist \u2191z \u2191z' \u2264 B n / 2 ^ l)) k)\n \u2191z' \u2264\n B n / 2 ^ k)) \u2264\n 2\u207b\u00b9 ^ n / 2 ^ k\n[PROOFSTEP]\nexact fun k =>\n choose_spec\n (this k _)\n -- it follows from the previous bound that `z` is a Cauchy sequence\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nz : (l : \u2115) \u2192 { x // x \u2208 s (n + l) }\nhz\u2080 : \u2191(z 0) = x\nhz : \u2200 (k : \u2115), edist \u2191(z k) \u2191(z (k + 1)) \u2264 B n / 2 ^ k\n\u22a2 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\n[PROOFSTEP]\nhave : CauchySeq fun k => (z k : \u03b1) := cauchySeq_of_edist_le_geometric_two (B n) (B_ne_top n) hz\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nz : (l : \u2115) \u2192 { x // x \u2208 s (n + l) }\nhz\u2080 : \u2191(z 0) = x\nhz : \u2200 (k : \u2115), edist \u2191(z k) \u2191(z (k + 1)) \u2264 B n / 2 ^ k\nthis : CauchySeq fun k => \u2191(z k)\n\u22a2 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\n[PROOFSTEP]\nrcases cauchySeq_tendsto_of_complete this with \u27e8y, y_lim\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nz : (l : \u2115) \u2192 { x // x \u2208 s (n + l) }\nhz\u2080 : \u2191(z 0) = x\nhz : \u2200 (k : \u2115), edist \u2191(z k) \u2191(z (k + 1)) \u2264 B n / 2 ^ k\nthis : CauchySeq fun k => \u2191(z k)\ny : \u03b1\ny_lim : Tendsto (fun k => \u2191(z k)) atTop (\ud835\udcdd y)\n\u22a2 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\n[PROOFSTEP]\nuse y\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nz : (l : \u2115) \u2192 { x // x \u2208 s (n + l) }\nhz\u2080 : \u2191(z 0) = x\nhz : \u2200 (k : \u2115), edist \u2191(z k) \u2191(z (k + 1)) \u2264 B n / 2 ^ k\nthis : CauchySeq fun k => \u2191(z k)\ny : \u03b1\ny_lim : Tendsto (fun k => \u2191(z k)) atTop (\ud835\udcdd y)\n\u22a2 y \u2208 t0 \u2227 edist x y \u2264 2 * B n\n[PROOFSTEP]\nhave : y \u2208 t0 :=\n mem_iInter.2 fun k =>\n mem_closure_of_tendsto y_lim\n (by\n simp only [exists_prop, Set.mem_iUnion, Filter.eventually_atTop, Set.mem_preimage, Set.preimage_iUnion]\n exact \u27e8k, fun m hm => \u27e8n + m, zero_add k \u25b8 add_le_add (zero_le n) hm, (z m).2\u27e9\u27e9)\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nz : (l : \u2115) \u2192 { x // x \u2208 s (n + l) }\nhz\u2080 : \u2191(z 0) = x\nhz : \u2200 (k : \u2115), edist \u2191(z k) \u2191(z (k + 1)) \u2264 B n / 2 ^ k\nthis : CauchySeq fun k => \u2191(z k)\ny : \u03b1\ny_lim : Tendsto (fun k => \u2191(z k)) atTop (\ud835\udcdd y)\nk : \u2115\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop, \u2191(z x) \u2208 \u22c3 (m : \u2115) (_ : m \u2265 k), \u2191(s m)\n[PROOFSTEP]\nsimp only [exists_prop, Set.mem_iUnion, Filter.eventually_atTop, Set.mem_preimage, Set.preimage_iUnion]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nz : (l : \u2115) \u2192 { x // x \u2208 s (n + l) }\nhz\u2080 : \u2191(z 0) = x\nhz : \u2200 (k : \u2115), edist \u2191(z k) \u2191(z (k + 1)) \u2264 B n / 2 ^ k\nthis : CauchySeq fun k => \u2191(z k)\ny : \u03b1\ny_lim : Tendsto (fun k => \u2191(z k)) atTop (\ud835\udcdd y)\nk : \u2115\n\u22a2 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 \u2203 i, i \u2265 k \u2227 \u2191(z b) \u2208 \u2191(s i)\n[PROOFSTEP]\nexact \u27e8k, fun m hm => \u27e8n + m, zero_add k \u25b8 add_le_add (zero_le n) hm, (z m).2\u27e9\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nz : (l : \u2115) \u2192 { x // x \u2208 s (n + l) }\nhz\u2080 : \u2191(z 0) = x\nhz : \u2200 (k : \u2115), edist \u2191(z k) \u2191(z (k + 1)) \u2264 B n / 2 ^ k\nthis\u271d : CauchySeq fun k => \u2191(z k)\ny : \u03b1\ny_lim : Tendsto (fun k => \u2191(z k)) atTop (\ud835\udcdd y)\nthis : y \u2208 t0\n\u22a2 y \u2208 t0 \u2227 edist x y \u2264 2 * B n\n[PROOFSTEP]\nuse this\n[GOAL]\ncase right\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nz : (l : \u2115) \u2192 { x // x \u2208 s (n + l) }\nhz\u2080 : \u2191(z 0) = x\nhz : \u2200 (k : \u2115), edist \u2191(z k) \u2191(z (k + 1)) \u2264 B n / 2 ^ k\nthis\u271d : CauchySeq fun k => \u2191(z k)\ny : \u03b1\ny_lim : Tendsto (fun k => \u2191(z k)) atTop (\ud835\udcdd y)\nthis : y \u2208 t0\n\u22a2 edist x y \u2264 2 * B n\n[PROOFSTEP]\nrw [\u2190 hz\u2080]\n[GOAL]\ncase right\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nn : \u2115\nx : \u03b1\nhx : x \u2208 s n\nz : (l : \u2115) \u2192 { x // x \u2208 s (n + l) }\nhz\u2080 : \u2191(z 0) = x\nhz : \u2200 (k : \u2115), edist \u2191(z k) \u2191(z (k + 1)) \u2264 B n / 2 ^ k\nthis\u271d : CauchySeq fun k => \u2191(z k)\ny : \u03b1\ny_lim : Tendsto (fun k => \u2191(z k)) atTop (\ud835\udcdd y)\nthis : y \u2208 t0\n\u22a2 edist (\u2191(z 0)) y \u2264 2 * B n\n[PROOFSTEP]\nexact edist_le_of_edist_le_geometric_two_of_tendsto\u2080 (B n) hz y_lim\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\n\u22a2 Tendsto s atTop (\ud835\udcdd t)\n[PROOFSTEP]\nhave I2 : \u2200 n, \u2200 x \u2208 t0, \u2203 y \u2208 s n, edist x y \u2264 2 * B n := by\n /- For the (much easier) reverse inequality, we start from a point `x \u2208 t0` and we want\n to find a point `y \u2208 s n` which is close to `x`.\n `x` belongs to `t0`, the intersection of the closures. In particular, it is well\n approximated by a point `z` in `\u22c3m\u2265n, s m`, say in `s m`. Since `s m` and\n `s n` are close, this point is itself well approximated by a point `y` in `s n`,\n as required. -/\n intro n x xt0\n have : x \u2208 closure (\u22c3 m \u2265 n, s m : Set \u03b1) := by apply mem_iInter.1 xt0 n\n rcases mem_closure_iff.1 this (B n) (B_pos n) with\n \u27e8z, hz, Dxz\u27e9\n -- z : \u03b1, Dxz : edist x z < B n,\n simp only [exists_prop, Set.mem_iUnion] at hz \n rcases hz with\n \u27e8m, \u27e8m_ge_n, hm\u27e9\u27e9\n -- m : \u2115, m_ge_n : m \u2265 n, hm : z \u2208 s m\n have : hausdorffEdist (s m : Set \u03b1) (s n) < B n := hs n m n m_ge_n (le_refl n)\n rcases exists_edist_lt_of_hausdorffEdist_lt hm this with\n \u27e8y, hy, Dzy\u27e9\n -- y : \u03b1, hy : y \u2208 s n, Dzy : edist z y < B n\n exact\n \u27e8y, hy,\n calc\n edist x y \u2264 edist x z + edist z y := edist_triangle _ _ _\n _ \u2264 B n + B n := (add_le_add (le_of_lt Dxz) (le_of_lt Dzy))\n _ = 2 * B n := (two_mul _).symm\u27e9\n -- Deduce from the above inequalities that the distance between `s n` and `t0` is at most `2 B n`.\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\n\u22a2 \u2200 (n : \u2115) (x : \u03b1), x \u2208 t0 \u2192 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\n[PROOFSTEP]\nintro n x xt0\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nn : \u2115\nx : \u03b1\nxt0 : x \u2208 t0\n\u22a2 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\n[PROOFSTEP]\nhave : x \u2208 closure (\u22c3 m \u2265 n, s m : Set \u03b1) := by apply mem_iInter.1 xt0 n\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nn : \u2115\nx : \u03b1\nxt0 : x \u2208 t0\n\u22a2 x \u2208 closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\n[PROOFSTEP]\napply mem_iInter.1 xt0 n\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nn : \u2115\nx : \u03b1\nxt0 : x \u2208 t0\nthis : x \u2208 closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\n\u22a2 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\n[PROOFSTEP]\nrcases mem_closure_iff.1 this (B n) (B_pos n) with\n \u27e8z, hz, Dxz\u27e9\n -- z : \u03b1, Dxz : edist x z < B n,\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nn : \u2115\nx : \u03b1\nxt0 : x \u2208 t0\nthis : x \u2208 closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nz : \u03b1\nhz : z \u2208 \u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m)\nDxz : edist x z < B n\n\u22a2 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\n[PROOFSTEP]\nsimp only [exists_prop, Set.mem_iUnion] at hz \n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nn : \u2115\nx : \u03b1\nxt0 : x \u2208 t0\nthis : x \u2208 closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nz : \u03b1\nDxz : edist x z < B n\nhz : \u2203 i, i \u2265 n \u2227 z \u2208 \u2191(s i)\n\u22a2 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\n[PROOFSTEP]\nrcases hz with\n \u27e8m, \u27e8m_ge_n, hm\u27e9\u27e9\n -- m : \u2115, m_ge_n : m \u2265 n, hm : z \u2208 s m\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nn : \u2115\nx : \u03b1\nxt0 : x \u2208 t0\nthis : x \u2208 closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nz : \u03b1\nDxz : edist x z < B n\nm : \u2115\nm_ge_n : m \u2265 n\nhm : z \u2208 \u2191(s m)\n\u22a2 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\n[PROOFSTEP]\nhave : hausdorffEdist (s m : Set \u03b1) (s n) < B n := hs n m n m_ge_n (le_refl n)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nn : \u2115\nx : \u03b1\nxt0 : x \u2208 t0\nthis\u271d : x \u2208 closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nz : \u03b1\nDxz : edist x z < B n\nm : \u2115\nm_ge_n : m \u2265 n\nhm : z \u2208 \u2191(s m)\nthis : hausdorffEdist \u2191(s m) \u2191(s n) < B n\n\u22a2 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\n[PROOFSTEP]\nrcases exists_edist_lt_of_hausdorffEdist_lt hm this with\n \u27e8y, hy, Dzy\u27e9\n -- y : \u03b1, hy : y \u2208 s n, Dzy : edist z y < B n\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nn : \u2115\nx : \u03b1\nxt0 : x \u2208 t0\nthis\u271d : x \u2208 closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nz : \u03b1\nDxz : edist x z < B n\nm : \u2115\nm_ge_n : m \u2265 n\nhm : z \u2208 \u2191(s m)\nthis : hausdorffEdist \u2191(s m) \u2191(s n) < B n\ny : \u03b1\nhy : y \u2208 \u2191(s n)\nDzy : edist z y < B n\n\u22a2 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\n[PROOFSTEP]\nexact\n \u27e8y, hy,\n calc\n edist x y \u2264 edist x z + edist z y := edist_triangle _ _ _\n _ \u2264 B n + B n := (add_le_add (le_of_lt Dxz) (le_of_lt Dzy))\n _ = 2 * B n := (two_mul _).symm\u27e9\n -- Deduce from the above inequalities that the distance between `s n` and `t0` is at most `2 B n`.\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nI2 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t0 \u2192 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\n\u22a2 Tendsto s atTop (\ud835\udcdd t)\n[PROOFSTEP]\nhave main : \u2200 n : \u2115, edist (s n) t \u2264 2 * B n := fun n =>\n hausdorffEdist_le_of_mem_edist (I1 n)\n (I2 n)\n -- from this, the convergence of `s n` to `t0` follows.\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nI2 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t0 \u2192 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\nmain : \u2200 (n : \u2115), edist (s n) t \u2264 2 * B n\n\u22a2 Tendsto s atTop (\ud835\udcdd t)\n[PROOFSTEP]\nrefine' tendsto_atTop.2 fun \u03b5 \u03b5pos => _\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nI2 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t0 \u2192 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\nmain : \u2200 (n : \u2115), edist (s n) t \u2264 2 * B n\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 edist (s n) t < \u03b5\n[PROOFSTEP]\nhave : Tendsto (fun n => 2 * B n) atTop (\ud835\udcdd (2 * 0)) :=\n ENNReal.Tendsto.const_mul (ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1 <| by simp [ENNReal.one_lt_two])\n (Or.inr <| by simp)\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nI2 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t0 \u2192 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\nmain : \u2200 (n : \u2115), edist (s n) t \u2264 2 * B n\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 2\u207b\u00b9 < 1\n[PROOFSTEP]\nsimp [ENNReal.one_lt_two]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nI2 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t0 \u2192 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\nmain : \u2200 (n : \u2115), edist (s n) t \u2264 2 * B n\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 2 \u2260 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nI2 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t0 \u2192 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\nmain : \u2200 (n : \u2115), edist (s n) t \u2264 2 * B n\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nthis : Tendsto (fun n => 2 * B n) atTop (\ud835\udcdd (2 * 0))\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 edist (s n) t < \u03b5\n[PROOFSTEP]\nrw [mul_zero] at this \n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nI2 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t0 \u2192 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\nmain : \u2200 (n : \u2115), edist (s n) t \u2264 2 * B n\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nthis : Tendsto (fun n => 2 * B n) atTop (\ud835\udcdd 0)\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 edist (s n) t < \u03b5\n[PROOFSTEP]\nobtain \u27e8N, hN\u27e9 : \u2203 N, \u2200 b \u2265 N, \u03b5 > 2 * B b := ((tendsto_order.1 this).2 \u03b5 \u03b5pos).exists_forall_of_atTop\n[GOAL]\ncase h.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 2\u207b\u00b9 ^ n\nB_pos : \u2200 (n : \u2115), 0 < B n\nB_ne_top : \u2200 (n : \u2115), B n \u2260 \u22a4\ns : \u2115 \u2192 Closeds \u03b1\nhs : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 edist (s n) (s m) < B N\nt0 : Set \u03b1 := \u22c2 (n : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 n), \u2191(s m))\nt : Closeds \u03b1 := { carrier := t0, closed' := (_ : IsClosed (\u22c2 (i : \u2115), closure (\u22c3 (m : \u2115) (_ : m \u2265 i), \u2191(s m)))) }\nI1 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s n \u2192 \u2203 y, y \u2208 t0 \u2227 edist x y \u2264 2 * B n\nI2 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t0 \u2192 \u2203 y, y \u2208 s n \u2227 edist x y \u2264 2 * B n\nmain : \u2200 (n : \u2115), edist (s n) t \u2264 2 * B n\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\nthis : Tendsto (fun n => 2 * B n) atTop (\ud835\udcdd 0)\nN : \u2115\nhN : \u2200 (b : \u2115), b \u2265 N \u2192 \u03b5 > 2 * B b\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 edist (s n) t < \u03b5\n[PROOFSTEP]\nexact \u27e8N, fun n hn => lt_of_le_of_lt (main n) (hN n hn)\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u22a2 IsCompact univ\n[PROOFSTEP]\nrefine' isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun \u03b5 \u03b5pos => _) isClosed_univ\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 t, Set.Finite t \u2227 univ \u2286 \u22c3 (y : Closeds \u03b1) (_ : y \u2208 t), ball y \u03b5\n[PROOFSTEP]\nrcases exists_between \u03b5pos with \u27e8\u03b4, \u03b4pos, \u03b4lt\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u22a2 \u2203 t, Set.Finite t \u2227 univ \u2286 \u22c3 (y : Closeds \u03b1) (_ : y \u2208 t), ball y \u03b5\n[PROOFSTEP]\nrcases EMetric.totallyBounded_iff.1 (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ \u03b1 _ _)).1 \u03b4 \u03b4pos with\n \u27e8s, fs, hs\u27e9\n -- s : Set \u03b1, fs : s.Finite, hs : univ \u2286 \u22c3 (y : \u03b1) (H : y \u2208 s), eball y \u03b4\n -- we first show that any set is well approximated by a subset of `s`.\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\n\u22a2 \u2203 t, Set.Finite t \u2227 univ \u2286 \u22c3 (y : Closeds \u03b1) (_ : y \u2208 t), ball y \u03b5\n[PROOFSTEP]\nhave main : \u2200 u : Set \u03b1, \u2203 (v : _) (_ : v \u2286 s), hausdorffEdist u v \u2264 \u03b4 :=\n by\n intro u\n let v := {x : \u03b1 | x \u2208 s \u2227 \u2203 y \u2208 u, edist x y < \u03b4}\n exists v, (fun x hx => hx.1 : v \u2286 s)\n refine' hausdorffEdist_le_of_mem_edist _ _\n \u00b7 intro x hx\n have : x \u2208 \u22c3 y \u2208 s, ball y \u03b4 := hs (by simp)\n rcases mem_iUnion\u2082.1 this with \u27e8y, ys, dy\u27e9\n have : edist y x < \u03b4 := by simp at dy ; rwa [edist_comm] at dy \n exact \u27e8y, \u27e8ys, \u27e8x, hx, this\u27e9\u27e9, le_of_lt dy\u27e9\n \u00b7 rintro x \u27e8_, \u27e8y, yu, hy\u27e9\u27e9\n exact\n \u27e8y, yu, le_of_lt hy\u27e9\n -- introduce the set F of all subsets of `s` (seen as members of `Closeds \u03b1`).\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\n\u22a2 \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\n[PROOFSTEP]\nintro u\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nu : Set \u03b1\n\u22a2 \u2203 v x, hausdorffEdist u v \u2264 \u03b4\n[PROOFSTEP]\nlet v := {x : \u03b1 | x \u2208 s \u2227 \u2203 y \u2208 u, edist x y < \u03b4}\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nu : Set \u03b1\nv : Set \u03b1 := {x | x \u2208 s \u2227 \u2203 y, y \u2208 u \u2227 edist x y < \u03b4}\n\u22a2 \u2203 v x, hausdorffEdist u v \u2264 \u03b4\n[PROOFSTEP]\nexists v, (fun x hx => hx.1 : v \u2286 s)\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nu : Set \u03b1\nv : Set \u03b1 := {x | x \u2208 s \u2227 \u2203 y, y \u2208 u \u2227 edist x y < \u03b4}\n\u22a2 hausdorffEdist u v \u2264 \u03b4\n[PROOFSTEP]\nrefine' hausdorffEdist_le_of_mem_edist _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nu : Set \u03b1\nv : Set \u03b1 := {x | x \u2208 s \u2227 \u2203 y, y \u2208 u \u2227 edist x y < \u03b4}\n\u22a2 \u2200 (x : \u03b1), x \u2208 u \u2192 \u2203 y, y \u2208 v \u2227 edist x y \u2264 \u03b4\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nu : Set \u03b1\nv : Set \u03b1 := {x | x \u2208 s \u2227 \u2203 y, y \u2208 u \u2227 edist x y < \u03b4}\nx : \u03b1\nhx : x \u2208 u\n\u22a2 \u2203 y, y \u2208 v \u2227 edist x y \u2264 \u03b4\n[PROOFSTEP]\nhave : x \u2208 \u22c3 y \u2208 s, ball y \u03b4 := hs (by simp)\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nu : Set \u03b1\nv : Set \u03b1 := {x | x \u2208 s \u2227 \u2203 y, y \u2208 u \u2227 edist x y < \u03b4}\nx : \u03b1\nhx : x \u2208 u\n\u22a2 x \u2208 univ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nu : Set \u03b1\nv : Set \u03b1 := {x | x \u2208 s \u2227 \u2203 y, y \u2208 u \u2227 edist x y < \u03b4}\nx : \u03b1\nhx : x \u2208 u\nthis : x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\n\u22a2 \u2203 y, y \u2208 v \u2227 edist x y \u2264 \u03b4\n[PROOFSTEP]\nrcases mem_iUnion\u2082.1 this with \u27e8y, ys, dy\u27e9\n[GOAL]\ncase refine'_1.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nu : Set \u03b1\nv : Set \u03b1 := {x | x \u2208 s \u2227 \u2203 y, y \u2208 u \u2227 edist x y < \u03b4}\nx : \u03b1\nhx : x \u2208 u\nthis : x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\ny : \u03b1\nys : y \u2208 s\ndy : x \u2208 ball y \u03b4\n\u22a2 \u2203 y, y \u2208 v \u2227 edist x y \u2264 \u03b4\n[PROOFSTEP]\nhave : edist y x < \u03b4 := by simp at dy ; rwa [edist_comm] at dy \n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nu : Set \u03b1\nv : Set \u03b1 := {x | x \u2208 s \u2227 \u2203 y, y \u2208 u \u2227 edist x y < \u03b4}\nx : \u03b1\nhx : x \u2208 u\nthis : x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\ny : \u03b1\nys : y \u2208 s\ndy : x \u2208 ball y \u03b4\n\u22a2 edist y x < \u03b4\n[PROOFSTEP]\nsimp at dy \n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nu : Set \u03b1\nv : Set \u03b1 := {x | x \u2208 s \u2227 \u2203 y, y \u2208 u \u2227 edist x y < \u03b4}\nx : \u03b1\nhx : x \u2208 u\nthis : x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\ny : \u03b1\nys : y \u2208 s\ndy : edist x y < \u03b4\n\u22a2 edist y x < \u03b4\n[PROOFSTEP]\nrwa [edist_comm] at dy \n[GOAL]\ncase refine'_1.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nu : Set \u03b1\nv : Set \u03b1 := {x | x \u2208 s \u2227 \u2203 y, y \u2208 u \u2227 edist x y < \u03b4}\nx : \u03b1\nhx : x \u2208 u\nthis\u271d : x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\ny : \u03b1\nys : y \u2208 s\ndy : x \u2208 ball y \u03b4\nthis : edist y x < \u03b4\n\u22a2 \u2203 y, y \u2208 v \u2227 edist x y \u2264 \u03b4\n[PROOFSTEP]\nexact \u27e8y, \u27e8ys, \u27e8x, hx, this\u27e9\u27e9, le_of_lt dy\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nu : Set \u03b1\nv : Set \u03b1 := {x | x \u2208 s \u2227 \u2203 y, y \u2208 u \u2227 edist x y < \u03b4}\n\u22a2 \u2200 (x : \u03b1), x \u2208 v \u2192 \u2203 y, y \u2208 u \u2227 edist x y \u2264 \u03b4\n[PROOFSTEP]\nrintro x \u27e8_, \u27e8y, yu, hy\u27e9\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nu : Set \u03b1\nv : Set \u03b1 := {x | x \u2208 s \u2227 \u2203 y, y \u2208 u \u2227 edist x y < \u03b4}\nx : \u03b1\nleft\u271d : x \u2208 s\ny : \u03b1\nyu : y \u2208 u\nhy : edist x y < \u03b4\n\u22a2 \u2203 y, y \u2208 u \u2227 edist x y \u2264 \u03b4\n[PROOFSTEP]\nexact\n \u27e8y, yu, le_of_lt hy\u27e9\n -- introduce the set F of all subsets of `s` (seen as members of `Closeds \u03b1`).\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\n\u22a2 \u2203 t, Set.Finite t \u2227 univ \u2286 \u22c3 (y : Closeds \u03b1) (_ : y \u2208 t), ball y \u03b5\n[PROOFSTEP]\nlet F := {f : Closeds \u03b1 | (f : Set \u03b1) \u2286 s}\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\n\u22a2 \u2203 t, Set.Finite t \u2227 univ \u2286 \u22c3 (y : Closeds \u03b1) (_ : y \u2208 t), ball y \u03b5\n[PROOFSTEP]\nrefine'\n \u27e8F, _, fun u _ => _\u27e9\n -- `F` is finite\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\n\u22a2 Set.Finite F\n[PROOFSTEP]\napply @Finite.of_finite_image _ _ F _\n[GOAL]\ncase intro.intro.intro.intro.refine'_1.h\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\n\u22a2 Set.Finite (?m.23554 '' F)\n[PROOFSTEP]\napply fs.finite_subsets.subset fun b => _\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\n\u22a2 Closeds \u03b1 \u2192 Set \u03b1\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\n\u22a2 \u2200 (b : Set \u03b1), b \u2208 ?m.23554 '' F \u2192 b \u2208 {b | b \u2286 s}\n[PROOFSTEP]\nexact fun s => (s : Set \u03b1)\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\n\u22a2 \u2200 (b : Set \u03b1), b \u2208 (fun s => \u2191s) '' F \u2192 b \u2208 {b | b \u2286 s}\n[PROOFSTEP]\nsimp only [and_imp, Set.mem_image, Set.mem_setOf_eq, exists_imp]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\n\u22a2 \u2200 (b : Set \u03b1) (x : Closeds \u03b1), \u2191x \u2286 s \u2192 \u2191x = b \u2192 b \u2286 s\n[PROOFSTEP]\nintro _ x hx hx'\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\nb\u271d : Set \u03b1\nx : Closeds \u03b1\nhx : \u2191x \u2286 s\nhx' : \u2191x = b\u271d\n\u22a2 b\u271d \u2286 s\n[PROOFSTEP]\nrwa [hx'] at hx \n[GOAL]\ncase intro.intro.intro.intro.refine'_1.hi\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\n\u22a2 InjOn (fun s => \u2191s) F\n[PROOFSTEP]\nexact SetLike.coe_injective.injOn F\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\nu : Closeds \u03b1\nx\u271d : u \u2208 univ\n\u22a2 u \u2208 \u22c3 (y : Closeds \u03b1) (_ : y \u2208 F), ball y \u03b5\n[PROOFSTEP]\nobtain \u27e8t0, t0s, Dut0\u27e9 := main u\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\nu : Closeds \u03b1\nx\u271d : u \u2208 univ\nt0 : Set \u03b1\nt0s : t0 \u2286 s\nDut0 : hausdorffEdist (\u2191u) t0 \u2264 \u03b4\n\u22a2 u \u2208 \u22c3 (y : Closeds \u03b1) (_ : y \u2208 F), ball y \u03b5\n[PROOFSTEP]\nhave : IsClosed t0 := (fs.subset t0s).isCompact.isClosed\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\nu : Closeds \u03b1\nx\u271d : u \u2208 univ\nt0 : Set \u03b1\nt0s : t0 \u2286 s\nDut0 : hausdorffEdist (\u2191u) t0 \u2264 \u03b4\nthis : IsClosed t0\n\u22a2 u \u2208 \u22c3 (y : Closeds \u03b1) (_ : y \u2208 F), ball y \u03b5\n[PROOFSTEP]\nlet t : Closeds \u03b1 := \u27e8t0, this\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\nu : Closeds \u03b1\nx\u271d : u \u2208 univ\nt0 : Set \u03b1\nt0s : t0 \u2286 s\nDut0 : hausdorffEdist (\u2191u) t0 \u2264 \u03b4\nthis : IsClosed t0\nt : Closeds \u03b1 := { carrier := t0, closed' := this }\n\u22a2 u \u2208 \u22c3 (y : Closeds \u03b1) (_ : y \u2208 F), ball y \u03b5\n[PROOFSTEP]\nhave : t \u2208 F := t0s\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\nu : Closeds \u03b1\nx\u271d : u \u2208 univ\nt0 : Set \u03b1\nt0s : t0 \u2286 s\nDut0 : hausdorffEdist (\u2191u) t0 \u2264 \u03b4\nthis\u271d : IsClosed t0\nt : Closeds \u03b1 := { carrier := t0, closed' := this\u271d }\nthis : t \u2208 F\n\u22a2 u \u2208 \u22c3 (y : Closeds \u03b1) (_ : y \u2208 F), ball y \u03b5\n[PROOFSTEP]\nhave : edist u t < \u03b5 := lt_of_le_of_lt Dut0 \u03b4lt\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\nu : Closeds \u03b1\nx\u271d : u \u2208 univ\nt0 : Set \u03b1\nt0s : t0 \u2286 s\nDut0 : hausdorffEdist (\u2191u) t0 \u2264 \u03b4\nthis\u271d\u00b9 : IsClosed t0\nt : Closeds \u03b1 := { carrier := t0, closed' := this\u271d\u00b9 }\nthis\u271d : t \u2208 F\nthis : edist u t < \u03b5\n\u22a2 u \u2208 \u22c3 (y : Closeds \u03b1) (_ : y \u2208 F), ball y \u03b5\n[PROOFSTEP]\napply mem_iUnion\u2082.2\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\ns : Set \u03b1\nfs : Set.Finite s\nhs : univ \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s), ball y \u03b4\nmain : \u2200 (u : Set \u03b1), \u2203 v x, hausdorffEdist u v \u2264 \u03b4\nF : Set (Closeds \u03b1) := {f | \u2191f \u2286 s}\nu : Closeds \u03b1\nx\u271d : u \u2208 univ\nt0 : Set \u03b1\nt0s : t0 \u2286 s\nDut0 : hausdorffEdist (\u2191u) t0 \u2264 \u03b4\nthis\u271d\u00b9 : IsClosed t0\nt : Closeds \u03b1 := { carrier := t0, closed' := this\u271d\u00b9 }\nthis\u271d : t \u2208 F\nthis : edist u t < \u03b5\n\u22a2 \u2203 i j, u \u2208 ball i \u03b5\n[PROOFSTEP]\nexact \u27e8t, \u2039t \u2208 F\u203a, this\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ns t : NonemptyCompacts \u03b1\nh : edist s t = 0\n\u22a2 \u2191s = \u2191t\n[PROOFSTEP]\nhave : closure (s : Set \u03b1) = closure t := hausdorffEdist_zero_iff_closure_eq_closure.1 h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ns t : NonemptyCompacts \u03b1\nh : edist s t = 0\nthis : closure \u2191s = closure \u2191t\n\u22a2 \u2191s = \u2191t\n[PROOFSTEP]\nrwa [s.isCompact.isClosed.closure_eq, t.isCompact.isClosed.closure_eq] at this \n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\n\u22a2 IsClosed (range NonemptyCompacts.toCloseds)\n[PROOFSTEP]\nhave : range NonemptyCompacts.toCloseds = {s : Closeds \u03b1 | (s : Set \u03b1).Nonempty \u2227 IsCompact (s : Set \u03b1)} :=\n by\n ext s\n refine' \u27e8_, fun h => \u27e8\u27e8\u27e8s, h.2\u27e9, h.1\u27e9, Closeds.ext rfl\u27e9\u27e9\n rintro \u27e8s, hs, rfl\u27e9\n exact \u27e8s.nonempty, s.isCompact\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\n\u22a2 range NonemptyCompacts.toCloseds = {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n[PROOFSTEP]\next s\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\ns : Closeds \u03b1\n\u22a2 s \u2208 range NonemptyCompacts.toCloseds \u2194 s \u2208 {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n[PROOFSTEP]\nrefine' \u27e8_, fun h => \u27e8\u27e8\u27e8s, h.2\u27e9, h.1\u27e9, Closeds.ext rfl\u27e9\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\ns : Closeds \u03b1\n\u22a2 s \u2208 range NonemptyCompacts.toCloseds \u2192 s \u2208 {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n[PROOFSTEP]\nrintro \u27e8s, hs, rfl\u27e9\n[GOAL]\ncase h.intro.refl\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\ns : NonemptyCompacts \u03b1\n\u22a2 NonemptyCompacts.toCloseds s \u2208 {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n[PROOFSTEP]\nexact \u27e8s.nonempty, s.isCompact\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n\u22a2 IsClosed (range NonemptyCompacts.toCloseds)\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n\u22a2 IsClosed {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n[PROOFSTEP]\nrefine' isClosed_of_closure_subset fun s hs => \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\ns : Closeds \u03b1\nhs : s \u2208 closure {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n\u22a2 Set.Nonempty \u2191s\n[PROOFSTEP]\nrcases mem_closure_iff.1 hs \u22a4 ENNReal.coe_lt_top with \u27e8t, ht, Dst\u27e9\n[GOAL]\ncase refine'_1.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\ns : Closeds \u03b1\nhs : s \u2208 closure {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\nt : Closeds \u03b1\nht : t \u2208 {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\nDst : edist s t < \u22a4\n\u22a2 Set.Nonempty \u2191s\n[PROOFSTEP]\nrw [edist_comm] at Dst \n[GOAL]\ncase refine'_1.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\ns : Closeds \u03b1\nhs : s \u2208 closure {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\nt : Closeds \u03b1\nht : t \u2208 {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\nDst : edist t s < \u22a4\n\u22a2 Set.Nonempty \u2191s\n[PROOFSTEP]\nexact nonempty_of_hausdorffEdist_ne_top ht.1 (ne_of_lt Dst)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\ns : Closeds \u03b1\nhs : s \u2208 closure {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n\u22a2 IsCompact \u2191s\n[PROOFSTEP]\nrefine' isCompact_iff_totallyBounded_isComplete.2 \u27e8_, s.closed.isComplete\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\ns : Closeds \u03b1\nhs : s \u2208 closure {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n\u22a2 TotallyBounded \u2191s\n[PROOFSTEP]\nrefine'\n totallyBounded_iff.2 fun \u03b5 (\u03b5pos : 0 < \u03b5) =>\n _\n -- we have to show that s is covered by finitely many eballs of radius \u03b5\n -- pick a nonempty compact set t at distance at most \u03b5/2 of s\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\ns : Closeds \u03b1\nhs : s \u2208 closure {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2203 t, Set.Finite t \u2227 \u2191s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), ball y \u03b5\n[PROOFSTEP]\nrcases mem_closure_iff.1 hs (\u03b5 / 2) (ENNReal.half_pos \u03b5pos.ne') with\n \u27e8t, ht, Dst\u27e9\n -- cover this space with finitely many balls of radius \u03b5/2\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\ns : Closeds \u03b1\nhs : s \u2208 closure {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nt : Closeds \u03b1\nht : t \u2208 {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\nDst : edist s t < \u03b5 / 2\n\u22a2 \u2203 t, Set.Finite t \u2227 \u2191s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), ball y \u03b5\n[PROOFSTEP]\nrcases totallyBounded_iff.1 (isCompact_iff_totallyBounded_isComplete.1 ht.2).1 (\u03b5 / 2) (ENNReal.half_pos \u03b5pos.ne') with\n \u27e8u, fu, ut\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\ns : Closeds \u03b1\nhs : s \u2208 closure {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nt : Closeds \u03b1\nht : t \u2208 {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\nDst : edist s t < \u03b5 / 2\nu : Set \u03b1\nfu : Set.Finite u\nut : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 u), ball y (\u03b5 / 2)\n\u22a2 \u2203 t, Set.Finite t \u2227 \u2191s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), ball y \u03b5\n[PROOFSTEP]\nrefine'\n \u27e8u, \u27e8fu, fun x hx => _\u27e9\u27e9\n -- u : set \u03b1, fu : u.finite, ut : t \u2286 \u22c3 (y : \u03b1) (H : y \u2208 u), eball y (\u03b5 / 2)\n -- then s is covered by the union of the balls centered at u of radius \u03b5\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\ns : Closeds \u03b1\nhs : s \u2208 closure {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nt : Closeds \u03b1\nht : t \u2208 {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\nDst : edist s t < \u03b5 / 2\nu : Set \u03b1\nfu : Set.Finite u\nut : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 u), ball y (\u03b5 / 2)\nx : \u03b1\nhx : x \u2208 \u2191s\n\u22a2 x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 u), ball y \u03b5\n[PROOFSTEP]\nrcases exists_edist_lt_of_hausdorffEdist_lt hx Dst with \u27e8z, hz, Dxz\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\ns : Closeds \u03b1\nhs : s \u2208 closure {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nt : Closeds \u03b1\nht : t \u2208 {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\nDst : edist s t < \u03b5 / 2\nu : Set \u03b1\nfu : Set.Finite u\nut : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 u), ball y (\u03b5 / 2)\nx : \u03b1\nhx : x \u2208 \u2191s\nz : \u03b1\nhz : z \u2208 \u2191t\nDxz : edist x z < \u03b5 / 2\n\u22a2 x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 u), ball y \u03b5\n[PROOFSTEP]\nrcases mem_iUnion\u2082.1 (ut hz) with \u27e8y, hy, Dzy\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\ns : Closeds \u03b1\nhs : s \u2208 closure {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nt : Closeds \u03b1\nht : t \u2208 {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\nDst : edist s t < \u03b5 / 2\nu : Set \u03b1\nfu : Set.Finite u\nut : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 u), ball y (\u03b5 / 2)\nx : \u03b1\nhx : x \u2208 \u2191s\nz : \u03b1\nhz : z \u2208 \u2191t\nDxz : edist x z < \u03b5 / 2\ny : \u03b1\nhy : y \u2208 u\nDzy : z \u2208 ball y (\u03b5 / 2)\n\u22a2 x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 u), ball y \u03b5\n[PROOFSTEP]\nhave : edist x y < \u03b5 :=\n calc\n edist x y \u2264 edist x z + edist z y := edist_triangle _ _ _\n _ < \u03b5 / 2 + \u03b5 / 2 := (ENNReal.add_lt_add Dxz Dzy)\n _ = \u03b5 := ENNReal.add_halves _\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : CompleteSpace \u03b1\nthis\u271d : range NonemptyCompacts.toCloseds = {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\ns : Closeds \u03b1\nhs : s \u2208 closure {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nt : Closeds \u03b1\nht : t \u2208 {s | Set.Nonempty \u2191s \u2227 IsCompact \u2191s}\nDst : edist s t < \u03b5 / 2\nu : Set \u03b1\nfu : Set.Finite u\nut : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 u), ball y (\u03b5 / 2)\nx : \u03b1\nhx : x \u2208 \u2191s\nz : \u03b1\nhz : z \u2208 \u2191t\nDxz : edist x z < \u03b5 / 2\ny : \u03b1\nhy : y \u2208 u\nDzy : z \u2208 ball y (\u03b5 / 2)\nthis : edist x y < \u03b5\n\u22a2 x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 u), ball y \u03b5\n[PROOFSTEP]\nexact mem_biUnion hy this\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u22a2 IsCompact univ\n[PROOFSTEP]\nrw [NonemptyCompacts.ToCloseds.uniformEmbedding.embedding.isCompact_iff_isCompact_image]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u22a2 IsCompact (NonemptyCompacts.toCloseds '' univ)\n[PROOFSTEP]\nrw [image_univ]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : CompactSpace \u03b1\n\u22a2 IsCompact (range NonemptyCompacts.toCloseds)\n[PROOFSTEP]\nexact NonemptyCompacts.isClosed_in_closeds.isCompact\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u22a2 SeparableSpace (NonemptyCompacts \u03b1)\n[PROOFSTEP]\nrcases exists_countable_dense \u03b1 with \u27e8s, cs, s_dense\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\n\u22a2 SeparableSpace (NonemptyCompacts \u03b1)\n[PROOFSTEP]\nlet v0 := {t : Set \u03b1 | t.Finite \u2227 t \u2286 s}\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\n\u22a2 SeparableSpace (NonemptyCompacts \u03b1)\n[PROOFSTEP]\nlet v : Set (NonemptyCompacts \u03b1) := {t : NonemptyCompacts \u03b1 | (t : Set \u03b1) \u2208 v0}\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\n\u22a2 SeparableSpace (NonemptyCompacts \u03b1)\n[PROOFSTEP]\nrefine' \u27e8\u27e8v, _, _\u27e9\u27e9\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\n\u22a2 Set.Countable v\n[PROOFSTEP]\nhave : v0.Countable := countable_setOf_finite_subset cs\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nthis : Set.Countable v0\n\u22a2 Set.Countable v\n[PROOFSTEP]\nexact this.preimage SetLike.coe_injective\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\n\u22a2 Dense v\n[PROOFSTEP]\nrefine' fun t =>\n mem_closure_iff.2 fun \u03b5 \u03b5pos =>\n _\n -- t is a compact nonempty set, that we have to approximate uniformly by a a set in `v`.\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nrcases exists_between \u03b5pos with \u27e8\u03b4, \u03b4pos, \u03b4lt\u27e9\n[GOAL]\ncase intro.intro.refine'_2.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nhave \u03b4pos' : 0 < \u03b4 / 2 := ENNReal.half_pos \u03b4pos.ne'\n[GOAL]\ncase intro.intro.refine'_2.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nhave Exy : \u2200 x, \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2 := by\n intro x\n rcases mem_closure_iff.1 (s_dense x) (\u03b4 / 2) \u03b4pos' with \u27e8y, ys, hy\u27e9\n exact \u27e8y, \u27e8ys, hy\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\n\u22a2 \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nx : \u03b1\n\u22a2 \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\n[PROOFSTEP]\nrcases mem_closure_iff.1 (s_dense x) (\u03b4 / 2) \u03b4pos' with \u27e8y, ys, hy\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nx y : \u03b1\nys : y \u2208 s\nhy : edist x y < \u03b4 / 2\n\u22a2 \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\n[PROOFSTEP]\nexact \u27e8y, \u27e8ys, hy\u27e9\u27e9\n[GOAL]\ncase intro.intro.refine'_2.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nlet F x := choose (Exy x)\n[GOAL]\ncase intro.intro.refine'_2.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nhave Fspec : \u2200 x, F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2 := fun x =>\n choose_spec\n (Exy x)\n -- cover `t` with finitely many balls. Their centers form a set `a`\n[GOAL]\ncase intro.intro.refine'_2.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nhave : TotallyBounded (t : Set \u03b1) := t.isCompact.totallyBounded\n[GOAL]\ncase intro.intro.refine'_2.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis : TotallyBounded \u2191t\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nrcases totallyBounded_iff.1 this (\u03b4 / 2) \u03b4pos' with\n \u27e8a, af, ta\u27e9\n -- a : set \u03b1, af : a.finite, ta : t \u2286 \u22c3 (y : \u03b1) (H : y \u2208 a), eball y (\u03b4 / 2)\n -- replace each center by a nearby approximation in `s`, giving a new set `b`\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nlet b := F '' a\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nhave : b.Finite := af.image _\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis : Set.Finite b\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nhave tb : \u2200 x \u2208 t, \u2203 y \u2208 b, edist x y < \u03b4 := by\n intro x hx\n rcases mem_iUnion\u2082.1 (ta hx) with \u27e8z, za, Dxz\u27e9\n exists F z, mem_image_of_mem _ za\n calc\n edist x (F z) \u2264 edist x z + edist z (F z) := edist_triangle _ _ _\n _ < \u03b4 / 2 + \u03b4 / 2 := (ENNReal.add_lt_add Dxz (Fspec z).2)\n _ = \u03b4 :=\n ENNReal.add_halves\n _\n -- keep only the points in `b` that are close to point in `t`, yielding a new set `c`\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis : Set.Finite b\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis : Set.Finite b\nx : \u03b1\nhx : x \u2208 t\n\u22a2 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\n[PROOFSTEP]\nrcases mem_iUnion\u2082.1 (ta hx) with \u27e8z, za, Dxz\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis : Set.Finite b\nx : \u03b1\nhx : x \u2208 t\nz : \u03b1\nza : z \u2208 a\nDxz : x \u2208 ball z (\u03b4 / 2)\n\u22a2 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\n[PROOFSTEP]\nexists F z, mem_image_of_mem _ za\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis : Set.Finite b\nx : \u03b1\nhx : x \u2208 t\nz : \u03b1\nza : z \u2208 a\nDxz : x \u2208 ball z (\u03b4 / 2)\n\u22a2 edist x (F z) < \u03b4\n[PROOFSTEP]\ncalc\n edist x (F z) \u2264 edist x z + edist z (F z) := edist_triangle _ _ _\n _ < \u03b4 / 2 + \u03b4 / 2 := (ENNReal.add_lt_add Dxz (Fspec z).2)\n _ = \u03b4 :=\n ENNReal.add_halves\n _\n -- keep only the points in `b` that are close to point in `t`, yielding a new set `c`\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nlet c := {y \u2208 b | \u2203 x \u2208 t, edist x y < \u03b4}\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nhave : c.Finite :=\n \u2039b.Finite\u203a.subset fun x hx =>\n hx.1\n -- points in `t` are well approximated by points in `c`\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b9 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis : Set.Finite c\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nhave tc : \u2200 x \u2208 t, \u2203 y \u2208 c, edist x y \u2264 \u03b4 := by\n intro x hx\n rcases tb x hx with \u27e8y, yv, Dxy\u27e9\n have : y \u2208 c := by simp [-mem_image]; exact \u27e8yv, \u27e8x, hx, Dxy\u27e9\u27e9\n exact\n \u27e8y, this, le_of_lt Dxy\u27e9\n -- points in `c` are well approximated by points in `t`\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b9 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis : Set.Finite c\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b9 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis : Set.Finite c\nx : \u03b1\nhx : x \u2208 t\n\u22a2 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\n[PROOFSTEP]\nrcases tb x hx with \u27e8y, yv, Dxy\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b9 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis : Set.Finite c\nx : \u03b1\nhx : x \u2208 t\ny : \u03b1\nyv : y \u2208 b\nDxy : edist x y < \u03b4\n\u22a2 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\n[PROOFSTEP]\nhave : y \u2208 c := by simp [-mem_image]; exact \u27e8yv, \u27e8x, hx, Dxy\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b9 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis : Set.Finite c\nx : \u03b1\nhx : x \u2208 t\ny : \u03b1\nyv : y \u2208 b\nDxy : edist x y < \u03b4\n\u22a2 y \u2208 c\n[PROOFSTEP]\nsimp [-mem_image]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b9 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis : Set.Finite c\nx : \u03b1\nhx : x \u2208 t\ny : \u03b1\nyv : y \u2208 b\nDxy : edist x y < \u03b4\n\u22a2 y \u2208 (fun a => choose (_ : \u2203 y, y \u2208 s \u2227 edist a y < \u03b4 / 2)) '' a \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4\n[PROOFSTEP]\nexact \u27e8yv, \u27e8x, hx, Dxy\u27e9\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b2 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d\u00b9 : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis\u271d : Set.Finite c\nx : \u03b1\nhx : x \u2208 t\ny : \u03b1\nyv : y \u2208 b\nDxy : edist x y < \u03b4\nthis : y \u2208 c\n\u22a2 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\n[PROOFSTEP]\nexact\n \u27e8y, this, le_of_lt Dxy\u27e9\n -- points in `c` are well approximated by points in `t`\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b9 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis : Set.Finite c\ntc : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nhave ct : \u2200 y \u2208 c, \u2203 x \u2208 t, edist y x \u2264 \u03b4 := by\n rintro y \u27e8_, x, xt, Dyx\u27e9\n have : edist y x \u2264 \u03b4 :=\n calc\n edist y x = edist x y := edist_comm _ _\n _ \u2264 \u03b4 := le_of_lt Dyx\n exact\n \u27e8x, xt, this\u27e9\n -- it follows that their Hausdorff distance is small\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b9 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis : Set.Finite c\ntc : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\n\u22a2 \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 x, x \u2208 t \u2227 edist y x \u2264 \u03b4\n[PROOFSTEP]\nrintro y \u27e8_, x, xt, Dyx\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b9 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis : Set.Finite c\ntc : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\ny : \u03b1\nleft\u271d : y \u2208 b\nx : \u03b1\nxt : x \u2208 t\nDyx : edist x y < \u03b4\n\u22a2 \u2203 x, x \u2208 t \u2227 edist y x \u2264 \u03b4\n[PROOFSTEP]\nhave : edist y x \u2264 \u03b4 :=\n calc\n edist y x = edist x y := edist_comm _ _\n _ \u2264 \u03b4 := le_of_lt Dyx\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b2 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d\u00b9 : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis\u271d : Set.Finite c\ntc : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\ny : \u03b1\nleft\u271d : y \u2208 b\nx : \u03b1\nxt : x \u2208 t\nDyx : edist x y < \u03b4\nthis : edist y x \u2264 \u03b4\n\u22a2 \u2203 x, x \u2208 t \u2227 edist y x \u2264 \u03b4\n[PROOFSTEP]\nexact\n \u27e8x, xt, this\u27e9\n -- it follows that their Hausdorff distance is small\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b9 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis : Set.Finite c\ntc : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\nct : \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 x, x \u2208 t \u2227 edist y x \u2264 \u03b4\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nhave : hausdorffEdist (t : Set \u03b1) c \u2264 \u03b4 := hausdorffEdist_le_of_mem_edist tc ct\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b2 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d\u00b9 : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis\u271d : Set.Finite c\ntc : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\nct : \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 x, x \u2208 t \u2227 edist y x \u2264 \u03b4\nthis : hausdorffEdist (\u2191t) c \u2264 \u03b4\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nhave Dtc : hausdorffEdist (t : Set \u03b1) c < \u03b5 := this.trans_lt \u03b4lt\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b2 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d\u00b9 : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis\u271d : Set.Finite c\ntc : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\nct : \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 x, x \u2208 t \u2227 edist y x \u2264 \u03b4\nthis : hausdorffEdist (\u2191t) c \u2264 \u03b4\nDtc : hausdorffEdist (\u2191t) c < \u03b5\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nhave hc : c.Nonempty :=\n nonempty_of_hausdorffEdist_ne_top t.nonempty\n (ne_top_of_lt Dtc)\n -- let `d` be the version of `c` in the type `NonemptyCompacts \u03b1`\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b2 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d\u00b9 : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis\u271d : Set.Finite c\ntc : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\nct : \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 x, x \u2208 t \u2227 edist y x \u2264 \u03b4\nthis : hausdorffEdist (\u2191t) c \u2264 \u03b4\nDtc : hausdorffEdist (\u2191t) c < \u03b5\nhc : Set.Nonempty c\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nlet d : NonemptyCompacts \u03b1 := \u27e8\u27e8c, \u2039c.Finite\u203a.isCompact\u27e9, hc\u27e9\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b2 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d\u00b9 : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis\u271d : Set.Finite c\ntc : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\nct : \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 x, x \u2208 t \u2227 edist y x \u2264 \u03b4\nthis : hausdorffEdist (\u2191t) c \u2264 \u03b4\nDtc : hausdorffEdist (\u2191t) c < \u03b5\nhc : Set.Nonempty c\nd : NonemptyCompacts \u03b1 := { toCompacts := { carrier := c, isCompact' := (_ : IsCompact c) }, nonempty' := hc }\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nhave : c \u2286 s := by\n intro x hx\n rcases(mem_image _ _ _).1 hx.1 with \u27e8y, \u27e8_, yx\u27e9\u27e9\n rw [\u2190 yx]\n exact (Fspec y).1\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b2 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d\u00b9 : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis\u271d : Set.Finite c\ntc : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\nct : \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 x, x \u2208 t \u2227 edist y x \u2264 \u03b4\nthis : hausdorffEdist (\u2191t) c \u2264 \u03b4\nDtc : hausdorffEdist (\u2191t) c < \u03b5\nhc : Set.Nonempty c\nd : NonemptyCompacts \u03b1 := { toCompacts := { carrier := c, isCompact' := (_ : IsCompact c) }, nonempty' := hc }\n\u22a2 c \u2286 s\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b2 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d\u00b9 : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis\u271d : Set.Finite c\ntc : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\nct : \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 x, x \u2208 t \u2227 edist y x \u2264 \u03b4\nthis : hausdorffEdist (\u2191t) c \u2264 \u03b4\nDtc : hausdorffEdist (\u2191t) c < \u03b5\nhc : Set.Nonempty c\nd : NonemptyCompacts \u03b1 := { toCompacts := { carrier := c, isCompact' := (_ : IsCompact c) }, nonempty' := hc }\nx : \u03b1\nhx : x \u2208 c\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrcases(mem_image _ _ _).1 hx.1 with \u27e8y, \u27e8_, yx\u27e9\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b2 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d\u00b9 : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis\u271d : Set.Finite c\ntc : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\nct : \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 x, x \u2208 t \u2227 edist y x \u2264 \u03b4\nthis : hausdorffEdist (\u2191t) c \u2264 \u03b4\nDtc : hausdorffEdist (\u2191t) c < \u03b5\nhc : Set.Nonempty c\nd : NonemptyCompacts \u03b1 := { toCompacts := { carrier := c, isCompact' := (_ : IsCompact c) }, nonempty' := hc }\nx : \u03b1\nhx : x \u2208 c\ny : \u03b1\nleft\u271d : y \u2208 a\nyx : F y = x\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrw [\u2190 yx]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b2 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d\u00b9 : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis\u271d : Set.Finite c\ntc : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\nct : \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 x, x \u2208 t \u2227 edist y x \u2264 \u03b4\nthis : hausdorffEdist (\u2191t) c \u2264 \u03b4\nDtc : hausdorffEdist (\u2191t) c < \u03b5\nhc : Set.Nonempty c\nd : NonemptyCompacts \u03b1 := { toCompacts := { carrier := c, isCompact' := (_ : IsCompact c) }, nonempty' := hc }\nx : \u03b1\nhx : x \u2208 c\ny : \u03b1\nleft\u271d : y \u2208 a\nyx : F y = x\n\u22a2 F y \u2208 s\n[PROOFSTEP]\nexact (Fspec y).1\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u00b3 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d\u00b2 : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis\u271d\u00b9 : Set.Finite c\ntc : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\nct : \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 x, x \u2208 t \u2227 edist y x \u2264 \u03b4\nthis\u271d : hausdorffEdist (\u2191t) c \u2264 \u03b4\nDtc : hausdorffEdist (\u2191t) c < \u03b5\nhc : Set.Nonempty c\nd : NonemptyCompacts \u03b1 := { toCompacts := { carrier := c, isCompact' := (_ : IsCompact c) }, nonempty' := hc }\nthis : c \u2286 s\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nhave : d \u2208 v :=\n \u27e8\u2039c.Finite\u203a, this\u27e9\n -- we have proved that `d` is a good approximation of `t` as requested\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : EMetricSpace \u03b1\ns\u271d : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1\ncs : Set.Countable s\ns_dense : Dense s\nv0 : Set (Set \u03b1) := {t | Set.Finite t \u2227 t \u2286 s}\nv : Set (NonemptyCompacts \u03b1) := {t | \u2191t \u2208 v0}\nt : NonemptyCompacts \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\n\u03b4lt : \u03b4 < \u03b5\n\u03b4pos' : 0 < \u03b4 / 2\nExy : \u2200 (x : \u03b1), \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2\nF : \u03b1 \u2192 \u03b1 := fun x => choose (_ : \u2203 y, y \u2208 s \u2227 edist x y < \u03b4 / 2)\nFspec : \u2200 (x : \u03b1), F x \u2208 s \u2227 edist x (F x) < \u03b4 / 2\nthis\u271d\u2074 : TotallyBounded \u2191t\na : Set \u03b1\naf : Set.Finite a\nta : \u2191t \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 a), ball y (\u03b4 / 2)\nb : Set \u03b1 := F '' a\nthis\u271d\u00b3 : Set.Finite b\ntb : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 b \u2227 edist x y < \u03b4\nc : Set \u03b1 := {y | y \u2208 b \u2227 \u2203 x, x \u2208 t \u2227 edist x y < \u03b4}\nthis\u271d\u00b2 : Set.Finite c\ntc : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 c \u2227 edist x y \u2264 \u03b4\nct : \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 x, x \u2208 t \u2227 edist y x \u2264 \u03b4\nthis\u271d\u00b9 : hausdorffEdist (\u2191t) c \u2264 \u03b4\nDtc : hausdorffEdist (\u2191t) c < \u03b5\nhc : Set.Nonempty c\nd : NonemptyCompacts \u03b1 := { toCompacts := { carrier := c, isCompact' := (_ : IsCompact c) }, nonempty' := hc }\nthis\u271d : c \u2286 s\nthis : d \u2208 v\n\u22a2 \u2203 y, y \u2208 v \u2227 edist t y < \u03b5\n[PROOFSTEP]\nexact \u27e8d, \u2039d \u2208 v\u203a, Dtc\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d : MetricSpace \u03b1\nx : \u03b1\ns t : NonemptyCompacts \u03b1\n\u22a2 infDist x \u2191s \u2264 infDist x \u2191t + dist s t\n[PROOFSTEP]\nrw [dist_comm]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : MetricSpace \u03b1\nx : \u03b1\ns t : NonemptyCompacts \u03b1\n\u22a2 infDist x \u2191s \u2264 infDist x \u2191t + dist t s\n[PROOFSTEP]\nexact infDist_le_infDist_add_hausdorffDist (edist_ne_top t s)\n[GOAL]\n\u03b1 : Type u\ninst\u271d : MetricSpace \u03b1\n\u22a2 LipschitzWith 2 fun p => infDist p.fst \u2191p.snd\n[PROOFSTEP]\nconvert\n @LipschitzWith.uncurry \u03b1 (NonemptyCompacts \u03b1) \u211d _ _ _ (fun (x : \u03b1) (s : NonemptyCompacts \u03b1) => infDist x s) 1 1\n (fun s => lipschitz_infDist_pt \u2191s) lipschitz_infDist_set\n[GOAL]\ncase h.e'_5\n\u03b1 : Type u\ninst\u271d : MetricSpace \u03b1\n\u22a2 2 = 1 + 1\n[PROOFSTEP]\nnorm_num\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.Closeds", "llama_tokens": 58342, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.4169687614286824}} {"text": "[GOAL]\nJ : Type w\nX\u271d Y\u271d : WalkingParallelFamily J\nf : X\u271d \u27f6 Y\u271d\n\u22a2 f \u226b \ud835\udfd9 Y\u271d = f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase id\nJ : Type w\nX\u271d : WalkingParallelFamily J\n\u22a2 Hom.id X\u271d \u226b \ud835\udfd9 X\u271d = Hom.id X\u271d\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase line\nJ : Type w\nx\u271d : J\n\u22a2 line x\u271d \u226b \ud835\udfd9 one = line x\u271d\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nW\u271d X\u271d Y\u271d Z\u271d : WalkingParallelFamily J\nf : W\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 Y\u271d\nh : Y\u271d \u27f6 Z\u271d\n\u22a2 (f \u226b g) \u226b h = f \u226b g \u226b h\n[PROOFSTEP]\ncases f\n[GOAL]\ncase id\nJ : Type w\nW\u271d Y\u271d Z\u271d : WalkingParallelFamily J\nh : Y\u271d \u27f6 Z\u271d\ng : W\u271d \u27f6 Y\u271d\n\u22a2 (Hom.id W\u271d \u226b g) \u226b h = Hom.id W\u271d \u226b g \u226b h\n[PROOFSTEP]\ncases g\n[GOAL]\ncase line\nJ : Type w\nY\u271d Z\u271d : WalkingParallelFamily J\nh : Y\u271d \u27f6 Z\u271d\nx\u271d : J\ng : one \u27f6 Y\u271d\n\u22a2 (line x\u271d \u226b g) \u226b h = line x\u271d \u226b g \u226b h\n[PROOFSTEP]\ncases g\n[GOAL]\ncase id.id\nJ : Type w\nW\u271d Z\u271d : WalkingParallelFamily J\nh : W\u271d \u27f6 Z\u271d\n\u22a2 (Hom.id W\u271d \u226b Hom.id W\u271d) \u226b h = Hom.id W\u271d \u226b Hom.id W\u271d \u226b h\n[PROOFSTEP]\ncases h\n[GOAL]\ncase id.line\nJ : Type w\nZ\u271d : WalkingParallelFamily J\nx\u271d : J\nh : one \u27f6 Z\u271d\n\u22a2 (Hom.id zero \u226b line x\u271d) \u226b h = Hom.id zero \u226b line x\u271d \u226b h\n[PROOFSTEP]\ncases h\n[GOAL]\ncase line.id\nJ : Type w\nZ\u271d : WalkingParallelFamily J\nx\u271d : J\nh : one \u27f6 Z\u271d\n\u22a2 (line x\u271d \u226b Hom.id one) \u226b h = line x\u271d \u226b Hom.id one \u226b h\n[PROOFSTEP]\ncases h\n[GOAL]\ncase id.id.id\nJ : Type w\nW\u271d : WalkingParallelFamily J\n\u22a2 (Hom.id W\u271d \u226b Hom.id W\u271d) \u226b Hom.id W\u271d = Hom.id W\u271d \u226b Hom.id W\u271d \u226b Hom.id W\u271d\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase id.id.line\nJ : Type w\nx\u271d : J\n\u22a2 (Hom.id zero \u226b Hom.id zero) \u226b line x\u271d = Hom.id zero \u226b Hom.id zero \u226b line x\u271d\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase id.line.id\nJ : Type w\nx\u271d : J\n\u22a2 (Hom.id zero \u226b line x\u271d) \u226b Hom.id one = Hom.id zero \u226b line x\u271d \u226b Hom.id one\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase line.id.id\nJ : Type w\nx\u271d : J\n\u22a2 (line x\u271d \u226b Hom.id one) \u226b Hom.id one = line x\u271d \u226b Hom.id one \u226b Hom.id one\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\n\u22a2 \u2200 {X_1 Y_1 Z : WalkingParallelFamily J} (f_1 : X_1 \u27f6 Y_1) (g : Y_1 \u27f6 Z),\n { obj := fun x => WalkingParallelFamily.casesOn x X Y,\n map := fun {x y} h =>\n match x, y, h with\n | x, .(x), Hom.id .(x) => \ud835\udfd9 ((fun x => WalkingParallelFamily.casesOn x X Y) x)\n | .(zero), .(one), line j => f j }.map\n (f_1 \u226b g) =\n { obj := fun x => WalkingParallelFamily.casesOn x X Y,\n map := fun {x y} h =>\n match x, y, h with\n | x, .(x), Hom.id .(x) => \ud835\udfd9 ((fun x => WalkingParallelFamily.casesOn x X Y) x)\n | .(zero), .(one), line j => f j }.map\n f_1 \u226b\n { obj := fun x => WalkingParallelFamily.casesOn x X Y,\n map := fun {x y} h =>\n match x, y, h with\n | x, .(x), Hom.id .(x) => \ud835\udfd9 ((fun x => WalkingParallelFamily.casesOn x X Y) x)\n | .(zero), .(one), line j => f j }.map\n g\n[PROOFSTEP]\nrintro _ _ _ \u27e8\u27e9 \u27e8\u27e9\n[GOAL]\ncase id.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nX\u271d : WalkingParallelFamily J\n\u22a2 { obj := fun x => WalkingParallelFamily.casesOn x X Y,\n map := fun {x y} h =>\n match x, y, h with\n | x, .(x), Hom.id .(x) => \ud835\udfd9 ((fun x => WalkingParallelFamily.casesOn x X Y) x)\n | .(zero), .(one), line j => f j }.map\n (Hom.id X\u271d \u226b Hom.id X\u271d) =\n { obj := fun x => WalkingParallelFamily.casesOn x X Y,\n map := fun {x y} h =>\n match x, y, h with\n | x, .(x), Hom.id .(x) => \ud835\udfd9 ((fun x => WalkingParallelFamily.casesOn x X Y) x)\n | .(zero), .(one), line j => f j }.map\n (Hom.id X\u271d) \u226b\n { obj := fun x => WalkingParallelFamily.casesOn x X Y,\n map := fun {x y} h =>\n match x, y, h with\n | x, .(x), Hom.id .(x) => \ud835\udfd9 ((fun x => WalkingParallelFamily.casesOn x X Y) x)\n | .(zero), .(one), line j => f j }.map\n (Hom.id X\u271d)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase id.line\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nx\u271d : J\n\u22a2 { obj := fun x => WalkingParallelFamily.casesOn x X Y,\n map := fun {x y} h =>\n match x, y, h with\n | x, .(x), Hom.id .(x) => \ud835\udfd9 ((fun x => WalkingParallelFamily.casesOn x X Y) x)\n | .(zero), .(one), line j => f j }.map\n (Hom.id zero \u226b line x\u271d) =\n { obj := fun x => WalkingParallelFamily.casesOn x X Y,\n map := fun {x y} h =>\n match x, y, h with\n | x, .(x), Hom.id .(x) => \ud835\udfd9 ((fun x => WalkingParallelFamily.casesOn x X Y) x)\n | .(zero), .(one), line j => f j }.map\n (Hom.id zero) \u226b\n { obj := fun x => WalkingParallelFamily.casesOn x X Y,\n map := fun {x y} h =>\n match x, y, h with\n | x, .(x), Hom.id .(x) => \ud835\udfd9 ((fun x => WalkingParallelFamily.casesOn x X Y) x)\n | .(zero), .(one), line j => f j }.map\n (line x\u271d)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase line.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nx\u271d : J\n\u22a2 { obj := fun x => WalkingParallelFamily.casesOn x X Y,\n map := fun {x y} h =>\n match x, y, h with\n | x, .(x), Hom.id .(x) => \ud835\udfd9 ((fun x => WalkingParallelFamily.casesOn x X Y) x)\n | .(zero), .(one), line j => f j }.map\n (line x\u271d \u226b Hom.id one) =\n { obj := fun x => WalkingParallelFamily.casesOn x X Y,\n map := fun {x y} h =>\n match x, y, h with\n | x, .(x), Hom.id .(x) => \ud835\udfd9 ((fun x => WalkingParallelFamily.casesOn x X Y) x)\n | .(zero), .(one), line j => f j }.map\n (line x\u271d) \u226b\n { obj := fun x => WalkingParallelFamily.casesOn x X Y,\n map := fun {x y} h =>\n match x, y, h with\n | x, .(x), Hom.id .(x) => \ud835\udfd9 ((fun x => WalkingParallelFamily.casesOn x X Y) x)\n | .(zero), .(one), line j => f j }.map\n (Hom.id one)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nj : WalkingParallelFamily J\n\u22a2 F.obj j = (parallelFamily fun j => F.map (line j)).obj j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase zero\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\n\u22a2 F.obj zero = (parallelFamily fun j => F.map (line j)).obj zero\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase one\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\n\u22a2 F.obj one = (parallelFamily fun j => F.map (line j)).obj one\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\n\u22a2 \u2200 {X Y : WalkingParallelFamily J} (f : X \u27f6 Y),\n F.map f \u226b ((fun j => eqToIso (_ : F.obj j = (parallelFamily fun j => F.map (line j)).obj j)) Y).hom =\n ((fun j => eqToIso (_ : F.obj j = (parallelFamily fun j => F.map (line j)).obj j)) X).hom \u226b\n (parallelFamily fun j => F.map (line j)).map f\n[PROOFSTEP]\nrintro _ _ (_ | _)\n[GOAL]\ncase id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nX\u271d : WalkingParallelFamily J\n\u22a2 F.map (Hom.id X\u271d) \u226b ((fun j => eqToIso (_ : F.obj j = (parallelFamily fun j => F.map (line j)).obj j)) X\u271d).hom =\n ((fun j => eqToIso (_ : F.obj j = (parallelFamily fun j => F.map (line j)).obj j)) X\u271d).hom \u226b\n (parallelFamily fun j => F.map (line j)).map (Hom.id X\u271d)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase line\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nx\u271d : J\n\u22a2 F.map (line x\u271d) \u226b ((fun j => eqToIso (_ : F.obj j = (parallelFamily fun j => F.map (line j)).obj j)) one).hom =\n ((fun j => eqToIso (_ : F.obj j = (parallelFamily fun j => F.map (line j)).obj j)) zero).hom \u226b\n (parallelFamily fun j => F.map (line j)).map (line x\u271d)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y : C\nf : J \u2192 (X\u271d \u27f6 Y)\nX : WalkingParallelFamily (ULift Bool)\n\u22a2 (\ud835\udfed (WalkingParallelFamily (ULift Bool))).obj X =\n ((parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).obj\n X\n[PROOFSTEP]\ncases X\n[GOAL]\ncase zero\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\n\u22a2 (\ud835\udfed (WalkingParallelFamily (ULift Bool))).obj zero =\n ((parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).obj\n zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase one\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\n\u22a2 (\ud835\udfed (WalkingParallelFamily (ULift Bool))).obj one =\n ((parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).obj\n one\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\n\u22a2 \u2200 {X Y : WalkingParallelFamily (ULift Bool)} (f : X \u27f6 Y),\n (\ud835\udfed (WalkingParallelFamily (ULift Bool))).map f \u226b\n ((fun X =>\n eqToIso\n (_ :\n (\ud835\udfed (WalkingParallelFamily (ULift Bool))).obj X =\n ((parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).obj\n X))\n Y).hom =\n ((fun X =>\n eqToIso\n (_ :\n (\ud835\udfed (WalkingParallelFamily (ULift Bool))).obj X =\n ((parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).obj\n X))\n X).hom \u226b\n ((parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).map\n f\n[PROOFSTEP]\nrintro _ _ (_ | \u27e8_ | _\u27e9)\n[GOAL]\ncase id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nX\u271d : WalkingParallelFamily (ULift Bool)\n\u22a2 (\ud835\udfed (WalkingParallelFamily (ULift Bool))).map (Hom.id X\u271d) \u226b\n ((fun X =>\n eqToIso\n (_ :\n (\ud835\udfed (WalkingParallelFamily (ULift Bool))).obj X =\n ((parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).obj\n X))\n X\u271d).hom =\n ((fun X =>\n eqToIso\n (_ :\n (\ud835\udfed (WalkingParallelFamily (ULift Bool))).obj X =\n ((parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).obj\n X))\n X\u271d).hom \u226b\n ((parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).map\n (Hom.id X\u271d)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase line.up.false\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\n\u22a2 (\ud835\udfed (WalkingParallelFamily (ULift Bool))).map (line { down := false }) \u226b\n ((fun X =>\n eqToIso\n (_ :\n (\ud835\udfed (WalkingParallelFamily (ULift Bool))).obj X =\n ((parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).obj\n X))\n one).hom =\n ((fun X =>\n eqToIso\n (_ :\n (\ud835\udfed (WalkingParallelFamily (ULift Bool))).obj X =\n ((parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).obj\n X))\n zero).hom \u226b\n ((parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).map\n (line { down := false })\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase line.up.true\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\n\u22a2 (\ud835\udfed (WalkingParallelFamily (ULift Bool))).map (line { down := true }) \u226b\n ((fun X =>\n eqToIso\n (_ :\n (\ud835\udfed (WalkingParallelFamily (ULift Bool))).obj X =\n ((parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).obj\n X))\n one).hom =\n ((fun X =>\n eqToIso\n (_ :\n (\ud835\udfed (WalkingParallelFamily (ULift Bool))).obj X =\n ((parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).obj\n X))\n zero).hom \u226b\n ((parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).map\n (line { down := true })\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y : C\nf : J \u2192 (X\u271d \u27f6 Y)\nX : WalkingParallelPair\n\u22a2 (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n X =\n (\ud835\udfed WalkingParallelPair).obj X\n[PROOFSTEP]\ncases X\n[GOAL]\ncase zero\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\n\u22a2 (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n WalkingParallelPair.zero =\n (\ud835\udfed WalkingParallelPair).obj WalkingParallelPair.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase one\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\n\u22a2 (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n WalkingParallelPair.one =\n (\ud835\udfed WalkingParallelPair).obj WalkingParallelPair.one\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\n\u22a2 \u2200 {X Y : WalkingParallelPair} (f : X \u27f6 Y),\n (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).map\n f \u226b\n ((fun X =>\n eqToIso\n (_ :\n (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n X =\n (\ud835\udfed WalkingParallelPair).obj X))\n Y).hom =\n ((fun X =>\n eqToIso\n (_ :\n (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n X =\n (\ud835\udfed WalkingParallelPair).obj X))\n X).hom \u226b\n (\ud835\udfed WalkingParallelPair).map f\n[PROOFSTEP]\nrintro _ _ (_ | _ | _)\n[GOAL]\ncase left\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\n\u22a2 (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).map\n WalkingParallelPairHom.left \u226b\n ((fun X =>\n eqToIso\n (_ :\n (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n X =\n (\ud835\udfed WalkingParallelPair).obj X))\n WalkingParallelPair.one).hom =\n ((fun X =>\n eqToIso\n (_ :\n (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n X =\n (\ud835\udfed WalkingParallelPair).obj X))\n WalkingParallelPair.zero).hom \u226b\n (\ud835\udfed WalkingParallelPair).map WalkingParallelPairHom.left\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase right\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\n\u22a2 (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).map\n WalkingParallelPairHom.right \u226b\n ((fun X =>\n eqToIso\n (_ :\n (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n X =\n (\ud835\udfed WalkingParallelPair).obj X))\n WalkingParallelPair.one).hom =\n ((fun X =>\n eqToIso\n (_ :\n (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n X =\n (\ud835\udfed WalkingParallelPair).obj X))\n WalkingParallelPair.zero).hom \u226b\n (\ud835\udfed WalkingParallelPair).map WalkingParallelPairHom.right\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nX\u271d : WalkingParallelPair\n\u22a2 (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).map\n (WalkingParallelPairHom.id X\u271d) \u226b\n ((fun X =>\n eqToIso\n (_ :\n (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n X =\n (\ud835\udfed WalkingParallelPair).obj X))\n X\u271d).hom =\n ((fun X =>\n eqToIso\n (_ :\n (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n X =\n (\ud835\udfed WalkingParallelPair).obj X))\n X\u271d).hom \u226b\n (\ud835\udfed WalkingParallelPair).map (WalkingParallelPairHom.id X\u271d)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\n\u22a2 \u2200 (X : WalkingParallelFamily (ULift Bool)),\n (parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).map\n (NatTrans.app\n (NatIso.ofComponents fun X =>\n eqToIso\n (_ :\n (\ud835\udfed (WalkingParallelFamily (ULift Bool))).obj X =\n ((parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).obj\n X)).hom\n X) \u226b\n NatTrans.app\n (NatIso.ofComponents fun X =>\n eqToIso\n (_ :\n (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n X =\n (\ud835\udfed WalkingParallelPair).obj X)).hom\n ((parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n X) =\n \ud835\udfd9 ((parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj X)\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase zero\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\n\u22a2 (parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).map\n (NatTrans.app\n (NatIso.ofComponents fun X =>\n eqToIso\n (_ :\n (\ud835\udfed (WalkingParallelFamily (ULift Bool))).obj X =\n ((parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).obj\n X)).hom\n zero) \u226b\n NatTrans.app\n (NatIso.ofComponents fun X =>\n eqToIso\n (_ :\n (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n X =\n (\ud835\udfed WalkingParallelPair).obj X)).hom\n ((parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n zero) =\n \ud835\udfd9 ((parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj zero)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase one\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\n\u22a2 (parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).map\n (NatTrans.app\n (NatIso.ofComponents fun X =>\n eqToIso\n (_ :\n (\ud835\udfed (WalkingParallelFamily (ULift Bool))).obj X =\n ((parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right) \u22d9\n parallelPair (line { down := true }) (line { down := false })).obj\n X)).hom\n one) \u226b\n NatTrans.app\n (NatIso.ofComponents fun X =>\n eqToIso\n (_ :\n (parallelPair (line { down := true }) (line { down := false }) \u22d9\n parallelFamily fun p =>\n bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n X =\n (\ud835\udfed WalkingParallelPair).obj X)).hom\n ((parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj\n one) =\n \ud835\udfd9 ((parallelFamily fun p => bif p.down then WalkingParallelPairHom.left else WalkingParallelPairHom.right).obj one)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ns : Trident f\nj : J\n\u22a2 NatTrans.app s.\u03c0 zero \u226b f j = NatTrans.app s.\u03c0 one\n[PROOFSTEP]\nrw [\u2190 s.w (line j), parallelFamily_map_left]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ns : Cotrident f\nj : J\n\u22a2 f j \u226b NatTrans.app s.\u03b9 one = NatTrans.app s.\u03b9 zero\n[PROOFSTEP]\nrw [\u2190 s.w (line j), parallelFamily_map_left]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf\u271d : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\nP : C\n\u03b9 : P \u27f6 X\nw : \u2200 (j\u2081 j\u2082 : J), \u03b9 \u226b f\u271d j\u2081 = \u03b9 \u226b f\u271d j\u2082\ni j : WalkingParallelFamily J\nf : i \u27f6 j\n\u22a2 ((Functor.const (WalkingParallelFamily J)).obj P).map f \u226b\n (fun X_1 => WalkingParallelFamily.casesOn X_1 \u03b9 (\u03b9 \u226b f\u271d (Classical.arbitrary J))) j =\n (fun X_1 => WalkingParallelFamily.casesOn X_1 \u03b9 (\u03b9 \u226b f\u271d (Classical.arbitrary J))) i \u226b (parallelFamily f\u271d).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf\u271d : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\nP : C\n\u03b9 : P \u27f6 X\nw : \u2200 (j\u2081 j\u2082 : J), \u03b9 \u226b f\u271d j\u2081 = \u03b9 \u226b f\u271d j\u2082\ni j : WalkingParallelFamily J\nf : i \u27f6 j\n\u22a2 \ud835\udfd9 P \u226b WalkingParallelFamily.rec \u03b9 (\u03b9 \u226b f\u271d (Classical.arbitrary J)) j =\n WalkingParallelFamily.rec \u03b9 (\u03b9 \u226b f\u271d (Classical.arbitrary J)) i \u226b (parallelFamily f\u271d).map f\n[PROOFSTEP]\ncases' f with _ k\n[GOAL]\ncase id\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\nP : C\n\u03b9 : P \u27f6 X\nw : \u2200 (j\u2081 j\u2082 : J), \u03b9 \u226b f j\u2081 = \u03b9 \u226b f j\u2082\ni : WalkingParallelFamily J\n\u22a2 \ud835\udfd9 P \u226b WalkingParallelFamily.rec \u03b9 (\u03b9 \u226b f (Classical.arbitrary J)) i =\n WalkingParallelFamily.rec \u03b9 (\u03b9 \u226b f (Classical.arbitrary J)) i \u226b (parallelFamily f).map (Hom.id i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase line\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\nP : C\n\u03b9 : P \u27f6 X\nw : \u2200 (j\u2081 j\u2082 : J), \u03b9 \u226b f j\u2081 = \u03b9 \u226b f j\u2082\nk : J\n\u22a2 \ud835\udfd9 P \u226b WalkingParallelFamily.rec \u03b9 (\u03b9 \u226b f (Classical.arbitrary J)) one =\n WalkingParallelFamily.rec \u03b9 (\u03b9 \u226b f (Classical.arbitrary J)) zero \u226b (parallelFamily f).map (line k)\n[PROOFSTEP]\nsimp [w (Classical.arbitrary J) k]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf\u271d : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\nP : C\n\u03c0 : Y \u27f6 P\nw : \u2200 (j\u2081 j\u2082 : J), f\u271d j\u2081 \u226b \u03c0 = f\u271d j\u2082 \u226b \u03c0\ni j : WalkingParallelFamily J\nf : i \u27f6 j\n\u22a2 (parallelFamily f\u271d).map f \u226b (fun X_1 => WalkingParallelFamily.casesOn X_1 (f\u271d (Classical.arbitrary J) \u226b \u03c0) \u03c0) j =\n (fun X_1 => WalkingParallelFamily.casesOn X_1 (f\u271d (Classical.arbitrary J) \u226b \u03c0) \u03c0) i \u226b\n ((Functor.const (WalkingParallelFamily J)).obj P).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf\u271d : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\nP : C\n\u03c0 : Y \u27f6 P\nw : \u2200 (j\u2081 j\u2082 : J), f\u271d j\u2081 \u226b \u03c0 = f\u271d j\u2082 \u226b \u03c0\ni j : WalkingParallelFamily J\nf : i \u27f6 j\n\u22a2 (parallelFamily f\u271d).map f \u226b WalkingParallelFamily.rec (f\u271d (Classical.arbitrary J) \u226b \u03c0) \u03c0 j =\n WalkingParallelFamily.rec (f\u271d (Classical.arbitrary J) \u226b \u03c0) \u03c0 i \u226b \ud835\udfd9 P\n[PROOFSTEP]\ncases' f with _ k\n[GOAL]\ncase id\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\nP : C\n\u03c0 : Y \u27f6 P\nw : \u2200 (j\u2081 j\u2082 : J), f j\u2081 \u226b \u03c0 = f j\u2082 \u226b \u03c0\ni : WalkingParallelFamily J\n\u22a2 (parallelFamily f).map (Hom.id i) \u226b WalkingParallelFamily.rec (f (Classical.arbitrary J) \u226b \u03c0) \u03c0 i =\n WalkingParallelFamily.rec (f (Classical.arbitrary J) \u226b \u03c0) \u03c0 i \u226b \ud835\udfd9 P\n[PROOFSTEP]\nsimp\n[GOAL]\ncase line\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\nP : C\n\u03c0 : Y \u27f6 P\nw : \u2200 (j\u2081 j\u2082 : J), f j\u2081 \u226b \u03c0 = f j\u2082 \u226b \u03c0\nk : J\n\u22a2 (parallelFamily f).map (line k) \u226b WalkingParallelFamily.rec (f (Classical.arbitrary J) \u226b \u03c0) \u03c0 one =\n WalkingParallelFamily.rec (f (Classical.arbitrary J) \u226b \u03c0) \u03c0 zero \u226b \ud835\udfd9 P\n[PROOFSTEP]\nsimp [w (Classical.arbitrary J) k]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nj\u2081 j\u2082 : J\nt : Trident f\n\u22a2 \u03b9 t \u226b f j\u2081 = \u03b9 t \u226b f j\u2082\n[PROOFSTEP]\nrw [t.app_zero, t.app_zero]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nj\u2081 j\u2082 : J\nt : Cotrident f\n\u22a2 f j\u2081 \u226b \u03c0 t = f j\u2082 \u226b \u03c0 t\n[PROOFSTEP]\nrw [t.app_one, t.app_one]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\ns : Trident f\nW : C\nk l : W \u27f6 s.pt\nh : k \u226b \u03b9 s = l \u226b \u03b9 s\n\u22a2 k \u226b NatTrans.app s.\u03c0 one = l \u226b NatTrans.app s.\u03c0 one\n[PROOFSTEP]\nrw [\u2190 s.app_zero (Classical.arbitrary J), reassoc_of% h]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\ns : Cotrident f\nW : C\nk l : s.pt \u27f6 W\nh : \u03c0 s \u226b k = \u03c0 s \u226b l\n\u22a2 NatTrans.app s.\u03b9 zero \u226b k = NatTrans.app s.\u03b9 zero \u226b l\n[PROOFSTEP]\nrw [\u2190 s.app_one (Classical.arbitrary J), Category.assoc, Category.assoc, h]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\nt : Trident f\nlift : (s : Trident f) \u2192 s.pt \u27f6 t.pt\nfac : \u2200 (s : Trident f), lift s \u226b \u03b9 t = \u03b9 s\nuniq :\n \u2200 (s : Trident f) (m : s.pt \u27f6 t.pt),\n (\u2200 (j : WalkingParallelFamily J), m \u226b NatTrans.app t.\u03c0 j = NatTrans.app s.\u03c0 j) \u2192 m = lift s\ns : Cone (parallelFamily f)\nj : WalkingParallelFamily J\n\u22a2 lift s \u226b NatTrans.app t.\u03c0 one = NatTrans.app s.\u03c0 one\n[PROOFSTEP]\nrw [\u2190 t.w (line (Classical.arbitrary J)), reassoc_of% fac, s.w]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\nt : Cotrident f\ndesc : (s : Cotrident f) \u2192 t.pt \u27f6 s.pt\nfac : \u2200 (s : Cotrident f), \u03c0 t \u226b desc s = \u03c0 s\nuniq :\n \u2200 (s : Cotrident f) (m : t.pt \u27f6 s.pt),\n (\u2200 (j : WalkingParallelFamily J), NatTrans.app t.\u03b9 j \u226b m = NatTrans.app s.\u03b9 j) \u2192 m = desc s\ns : Cocone (parallelFamily f)\nj : WalkingParallelFamily J\n\u22a2 NatTrans.app t.\u03b9 zero \u226b desc s = NatTrans.app s.\u03b9 zero\n[PROOFSTEP]\nrw [\u2190 t.w_assoc (line (Classical.arbitrary J)), fac, s.w]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\nt : Trident f\nht : IsLimit t\nZ : C\nk : Z \u27f6 t.pt\n\u22a2 \u2200 (j\u2081 j\u2082 : J), (k \u226b \u03b9 t) \u226b f j\u2081 = (k \u226b \u03b9 t) \u226b f j\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\nt : Cotrident f\nht : IsColimit t\nZ : C\nk : t.pt \u27f6 Z\n\u22a2 \u2200 (j\u2081 j\u2082 : J), f j\u2081 \u226b \u03c0 t \u226b k = f j\u2082 \u226b \u03c0 t \u226b k\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y : C\nf : J \u2192 (X\u271d \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Trident fun j => F.map (line j)\nX : WalkingParallelFamily J\n\u22a2 (parallelFamily fun j => F.map (line j)).obj X = F.obj X\n[PROOFSTEP]\ncases X\n[GOAL]\ncase zero\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Trident fun j => F.map (line j)\n\u22a2 (parallelFamily fun j => F.map (line j)).obj zero = F.obj zero\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase one\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Trident fun j => F.map (line j)\n\u22a2 (parallelFamily fun j => F.map (line j)).obj one = F.obj one\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Trident fun j => F.map (line j)\nj j' : WalkingParallelFamily J\ng : j \u27f6 j'\n\u22a2 ((Functor.const (WalkingParallelFamily J)).obj t.pt).map g \u226b\n (fun X => NatTrans.app t.\u03c0 X \u226b eqToHom (_ : (parallelFamily fun j => F.map (line j)).obj X = F.obj X)) j' =\n (fun X => NatTrans.app t.\u03c0 X \u226b eqToHom (_ : (parallelFamily fun j => F.map (line j)).obj X = F.obj X)) j \u226b F.map g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Trident fun j => F.map (line j)\nj : WalkingParallelFamily J\n\u22a2 ((Functor.const (WalkingParallelFamily J)).obj t.pt).map (Hom.id j) \u226b\n (fun X => NatTrans.app t.\u03c0 X \u226b eqToHom (_ : (parallelFamily fun j => F.map (line j)).obj X = F.obj X)) j =\n (fun X => NatTrans.app t.\u03c0 X \u226b eqToHom (_ : (parallelFamily fun j => F.map (line j)).obj X = F.obj X)) j \u226b\n F.map (Hom.id j)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase line\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Trident fun j => F.map (line j)\nx\u271d : J\n\u22a2 ((Functor.const (WalkingParallelFamily J)).obj t.pt).map (line x\u271d) \u226b\n (fun X => NatTrans.app t.\u03c0 X \u226b eqToHom (_ : (parallelFamily fun j => F.map (line j)).obj X = F.obj X)) one =\n (fun X => NatTrans.app t.\u03c0 X \u226b eqToHom (_ : (parallelFamily fun j => F.map (line j)).obj X = F.obj X)) zero \u226b\n F.map (line x\u271d)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y : C\nf : J \u2192 (X\u271d \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cotrident fun j => F.map (line j)\nX : WalkingParallelFamily J\n\u22a2 F.obj X = (parallelFamily fun j => F.map (line j)).obj X\n[PROOFSTEP]\ncases X\n[GOAL]\ncase zero\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cotrident fun j => F.map (line j)\n\u22a2 F.obj zero = (parallelFamily fun j => F.map (line j)).obj zero\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase one\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cotrident fun j => F.map (line j)\n\u22a2 F.obj one = (parallelFamily fun j => F.map (line j)).obj one\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cotrident fun j => F.map (line j)\nj j' : WalkingParallelFamily J\ng : j \u27f6 j'\n\u22a2 F.map g \u226b (fun X => eqToHom (_ : F.obj X = (parallelFamily fun j => F.map (line j)).obj X) \u226b NatTrans.app t.\u03b9 X) j' =\n (fun X => eqToHom (_ : F.obj X = (parallelFamily fun j => F.map (line j)).obj X) \u226b NatTrans.app t.\u03b9 X) j \u226b\n ((Functor.const (WalkingParallelFamily J)).obj t.pt).map g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cotrident fun j => F.map (line j)\nj : WalkingParallelFamily J\n\u22a2 F.map (Hom.id j) \u226b\n (fun X => eqToHom (_ : F.obj X = (parallelFamily fun j => F.map (line j)).obj X) \u226b NatTrans.app t.\u03b9 X) j =\n (fun X => eqToHom (_ : F.obj X = (parallelFamily fun j => F.map (line j)).obj X) \u226b NatTrans.app t.\u03b9 X) j \u226b\n ((Functor.const (WalkingParallelFamily J)).obj t.pt).map (Hom.id j)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase line\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cotrident fun j => F.map (line j)\nx\u271d : J\n\u22a2 F.map (line x\u271d) \u226b\n (fun X => eqToHom (_ : F.obj X = (parallelFamily fun j => F.map (line j)).obj X) \u226b NatTrans.app t.\u03b9 X) one =\n (fun X => eqToHom (_ : F.obj X = (parallelFamily fun j => F.map (line j)).obj X) \u226b NatTrans.app t.\u03b9 X) zero \u226b\n ((Functor.const (WalkingParallelFamily J)).obj t.pt).map (line x\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cotrident fun j => F.map (line j)\nj : WalkingParallelFamily J\n\u22a2 F.map (\ud835\udfd9 j) \u226b eqToHom (_ : F.obj j = (parallelFamily fun j => F.map (line j)).obj j) \u226b NatTrans.app t.\u03b9 j =\n (eqToHom (_ : F.obj j = (parallelFamily fun j => F.map (line j)).obj j) \u226b NatTrans.app t.\u03b9 j) \u226b \ud835\udfd9 t.pt\n[PROOFSTEP]\nsimp [Cotrident.app_one t]\n[GOAL]\ncase line\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cotrident fun j => F.map (line j)\nx\u271d : J\n\u22a2 F.map (line x\u271d) \u226b \ud835\udfd9 (F.obj one) \u226b NatTrans.app t.\u03b9 one = (\ud835\udfd9 (F.obj zero) \u226b NatTrans.app t.\u03b9 zero) \u226b \ud835\udfd9 t.pt\n[PROOFSTEP]\nsimp [Cotrident.app_one t]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Trident fun j => F.map (line j)\nj : WalkingParallelFamily J\n\u22a2 (parallelFamily fun j => F.map (line j)).obj j = F.obj j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase zero\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Trident fun j => F.map (line j)\n\u22a2 (parallelFamily fun j => F.map (line j)).obj zero = F.obj zero\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase one\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Trident fun j => F.map (line j)\n\u22a2 (parallelFamily fun j => F.map (line j)).obj one = F.obj one\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cotrident fun j => F.map (line j)\nj : WalkingParallelFamily J\n\u22a2 F.obj j = (parallelFamily fun j => F.map (line j)).obj j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase zero\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cotrident fun j => F.map (line j)\n\u22a2 F.obj zero = (parallelFamily fun j => F.map (line j)).obj zero\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase one\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cotrident fun j => F.map (line j)\n\u22a2 F.obj one = (parallelFamily fun j => F.map (line j)).obj one\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y : C\nf : J \u2192 (X\u271d \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cone F\nX : WalkingParallelFamily J\n\u22a2 F.obj X = (parallelFamily fun j => F.map (line j)).obj X\n[PROOFSTEP]\ncases X\n[GOAL]\ncase zero\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cone F\n\u22a2 F.obj zero = (parallelFamily fun j => F.map (line j)).obj zero\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase one\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cone F\n\u22a2 F.obj one = (parallelFamily fun j => F.map (line j)).obj one\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cone F\n\u22a2 \u2200 \u2983X Y : WalkingParallelFamily J\u2984 (f : X \u27f6 Y),\n ((Functor.const (WalkingParallelFamily J)).obj t.pt).map f \u226b\n (fun X => NatTrans.app t.\u03c0 X \u226b eqToHom (_ : F.obj X = (parallelFamily fun j => F.map (line j)).obj X)) Y =\n (fun X => NatTrans.app t.\u03c0 X \u226b eqToHom (_ : F.obj X = (parallelFamily fun j => F.map (line j)).obj X)) X \u226b\n (parallelFamily fun j => F.map (line j)).map f\n[PROOFSTEP]\nrintro _ _ (_ | _)\n[GOAL]\ncase id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cone F\nX\u271d : WalkingParallelFamily J\n\u22a2 ((Functor.const (WalkingParallelFamily J)).obj t.pt).map (Hom.id X\u271d) \u226b\n (fun X => NatTrans.app t.\u03c0 X \u226b eqToHom (_ : F.obj X = (parallelFamily fun j => F.map (line j)).obj X)) X\u271d =\n (fun X => NatTrans.app t.\u03c0 X \u226b eqToHom (_ : F.obj X = (parallelFamily fun j => F.map (line j)).obj X)) X\u271d \u226b\n (parallelFamily fun j => F.map (line j)).map (Hom.id X\u271d)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase line\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cone F\nx\u271d : J\n\u22a2 ((Functor.const (WalkingParallelFamily J)).obj t.pt).map (line x\u271d) \u226b\n (fun X => NatTrans.app t.\u03c0 X \u226b eqToHom (_ : F.obj X = (parallelFamily fun j => F.map (line j)).obj X)) one =\n (fun X => NatTrans.app t.\u03c0 X \u226b eqToHom (_ : F.obj X = (parallelFamily fun j => F.map (line j)).obj X)) zero \u226b\n (parallelFamily fun j => F.map (line j)).map (line x\u271d)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y : C\nf : J \u2192 (X\u271d \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cocone F\nX : WalkingParallelFamily J\n\u22a2 (parallelFamily fun j => F.map (line j)).obj X = F.obj X\n[PROOFSTEP]\ncases X\n[GOAL]\ncase zero\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cocone F\n\u22a2 (parallelFamily fun j => F.map (line j)).obj zero = F.obj zero\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase one\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cocone F\n\u22a2 (parallelFamily fun j => F.map (line j)).obj one = F.obj one\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cocone F\n\u22a2 \u2200 \u2983X Y : WalkingParallelFamily J\u2984 (f : X \u27f6 Y),\n (parallelFamily fun j => F.map (line j)).map f \u226b\n (fun X => eqToHom (_ : (parallelFamily fun j => F.map (line j)).obj X = F.obj X) \u226b NatTrans.app t.\u03b9 X) Y =\n (fun X => eqToHom (_ : (parallelFamily fun j => F.map (line j)).obj X = F.obj X) \u226b NatTrans.app t.\u03b9 X) X \u226b\n ((Functor.const (WalkingParallelFamily J)).obj t.pt).map f\n[PROOFSTEP]\nrintro _ _ (_ | _)\n[GOAL]\ncase id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cocone F\nX\u271d : WalkingParallelFamily J\n\u22a2 (parallelFamily fun j => F.map (line j)).map (Hom.id X\u271d) \u226b\n (fun X => eqToHom (_ : (parallelFamily fun j => F.map (line j)).obj X = F.obj X) \u226b NatTrans.app t.\u03b9 X) X\u271d =\n (fun X => eqToHom (_ : (parallelFamily fun j => F.map (line j)).obj X = F.obj X) \u226b NatTrans.app t.\u03b9 X) X\u271d \u226b\n ((Functor.const (WalkingParallelFamily J)).obj t.pt).map (Hom.id X\u271d)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase line\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cocone F\nx\u271d : J\n\u22a2 (parallelFamily fun j => F.map (line j)).map (line x\u271d) \u226b\n (fun X => eqToHom (_ : (parallelFamily fun j => F.map (line j)).obj X = F.obj X) \u226b NatTrans.app t.\u03b9 X) one =\n (fun X => eqToHom (_ : (parallelFamily fun j => F.map (line j)).obj X = F.obj X) \u226b NatTrans.app t.\u03b9 X) zero \u226b\n ((Functor.const (WalkingParallelFamily J)).obj t.pt).map (line x\u271d)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cone F\nj : WalkingParallelFamily J\n\u22a2 F.obj j = (parallelFamily fun j => F.map (line j)).obj j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase zero\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cone F\n\u22a2 F.obj zero = (parallelFamily fun j => F.map (line j)).obj zero\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase one\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cone F\n\u22a2 F.obj one = (parallelFamily fun j => F.map (line j)).obj one\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cocone F\nj : WalkingParallelFamily J\n\u22a2 (parallelFamily fun j => F.map (line j)).obj j = F.obj j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase zero\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cocone F\n\u22a2 (parallelFamily fun j => F.map (line j)).obj zero = F.obj zero\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase one\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\nF : WalkingParallelFamily J \u2964 C\nt : Cocone F\n\u22a2 (parallelFamily fun j => F.map (line j)).obj one = F.obj one\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\ns t : Trident f\nk : s.pt \u27f6 t.pt\nw : autoParam (k \u226b \u03b9 t = \u03b9 s) _auto\u271d\n\u22a2 \u2200 (j : WalkingParallelFamily J), k \u226b NatTrans.app t.\u03c0 j = NatTrans.app s.\u03c0 j\n[PROOFSTEP]\nrintro \u27e8_ | _\u27e9\n[GOAL]\ncase zero\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\ns t : Trident f\nk : s.pt \u27f6 t.pt\nw : autoParam (k \u226b \u03b9 t = \u03b9 s) _auto\u271d\n\u22a2 k \u226b NatTrans.app t.\u03c0 zero = NatTrans.app s.\u03c0 zero\n[PROOFSTEP]\nexact w\n[GOAL]\ncase one\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\ns t : Trident f\nk : s.pt \u27f6 t.pt\nw : autoParam (k \u226b \u03b9 t = \u03b9 s) _auto\u271d\n\u22a2 k \u226b NatTrans.app t.\u03c0 one = NatTrans.app s.\u03c0 one\n[PROOFSTEP]\nsimpa using w =\u226b f (Classical.arbitrary J)\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\ns t : Trident f\ni : s.pt \u2245 t.pt\nw : autoParam (i.hom \u226b \u03b9 t = \u03b9 s) _auto\u271d\n\u22a2 i.inv \u226b \u03b9 s = \u03b9 t\n[PROOFSTEP]\nrw [\u2190 w, Iso.inv_hom_id_assoc]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\ns t : Cotrident f\nk : s.pt \u27f6 t.pt\nw : autoParam (\u03c0 s \u226b k = \u03c0 t) _auto\u271d\n\u22a2 \u2200 (j : WalkingParallelFamily J), NatTrans.app s.\u03b9 j \u226b k = NatTrans.app t.\u03b9 j\n[PROOFSTEP]\nrintro \u27e8_ | _\u27e9\n[GOAL]\ncase zero\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\ns t : Cotrident f\nk : s.pt \u27f6 t.pt\nw : autoParam (\u03c0 s \u226b k = \u03c0 t) _auto\u271d\n\u22a2 NatTrans.app s.\u03b9 zero \u226b k = NatTrans.app t.\u03b9 zero\n[PROOFSTEP]\nsimpa using f (Classical.arbitrary J) \u226b= w\n[GOAL]\ncase one\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\ns t : Cotrident f\nk : s.pt \u27f6 t.pt\nw : autoParam (\u03c0 s \u226b k = \u03c0 t) _auto\u271d\n\u22a2 NatTrans.app s.\u03b9 one \u226b k = NatTrans.app t.\u03b9 one\n[PROOFSTEP]\nexact w\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : C\nf : J \u2192 (X \u27f6 Y)\ninst\u271d : Nonempty J\ns t : Cotrident f\ni : s.pt \u2245 t.pt\nw : autoParam (\u03c0 s \u226b i.hom = \u03c0 t) _auto\u271d\n\u22a2 \u03c0 t \u226b i.inv = \u03c0 s\n[PROOFSTEP]\nrw [Iso.comp_inv_eq, w]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers", "llama_tokens": 18757, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.752012562644147, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.41696875526538263}} {"text": "[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nd : \u211d\u22650\u221e\n\u22a2 d \u2264 infEdist x s \u2194 \u2200 (y : \u03b1), y \u2208 s \u2192 d \u2264 edist x y\n[PROOFSTEP]\nsimp only [infEdist, le_iInf_iff]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\u22650\u221e\n\u22a2 infEdist x s < r \u2194 \u2203 y, y \u2208 s \u2227 edist x y < r\n[PROOFSTEP]\nsimp_rw [infEdist, iInf_lt_iff, exists_prop]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 \u2a05 (z : \u03b1) (_ : z \u2208 s), edist y z + edist x y = (\u2a05 (z : \u03b1) (_ : z \u2208 s), edist y z) + edist x y\n[PROOFSTEP]\nsimp only [ENNReal.iInf_add]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 infEdist x s \u2264 edist x y + infEdist y s\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 infEdist x s \u2264 infEdist y s + edist x y\n[PROOFSTEP]\nexact infEdist_le_infEdist_add_edist\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhy : y \u2208 s\n\u22a2 edist x y \u2264 infEdist x s + diam s\n[PROOFSTEP]\nsimp_rw [infEdist, ENNReal.iInf_add]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhy : y \u2208 s\n\u22a2 edist x y \u2264 \u2a05 (i : \u03b1) (_ : i \u2208 s), edist x i + diam s\n[PROOFSTEP]\nrefine le_iInf\u2082 fun i hi => ?_\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhy : y \u2208 s\ni : \u03b1\nhi : i \u2208 s\n\u22a2 edist x y \u2264 edist x i + diam s\n[PROOFSTEP]\ncalc\n edist x y \u2264 edist x i + edist i y := edist_triangle _ _ _\n _ \u2264 edist x i + diam s := add_le_add le_rfl (edist_le_diam_of_mem hi hy)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 1 \u2260 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (x y : \u03b1), infEdist x s \u2264 infEdist y s + 1 * edist x y\n[PROOFSTEP]\nsimp only [one_mul, infEdist_le_infEdist_add_edist, forall\u2082_true_iff]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 infEdist x (closure s) = infEdist x s\n[PROOFSTEP]\nrefine' le_antisymm (infEdist_anti subset_closure) _\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 infEdist x s \u2264 infEdist x (closure s)\n[PROOFSTEP]\nrefine' ENNReal.le_of_forall_pos_le_add fun \u03b5 \u03b5pos h => _\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : infEdist x (closure s) < \u22a4\n\u22a2 infEdist x s \u2264 infEdist x (closure s) + \u2191\u03b5\n[PROOFSTEP]\nhave \u03b50 : 0 < (\u03b5 / 2 : \u211d\u22650\u221e) := by simpa [pos_iff_ne_zero] using \u03b5pos\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : infEdist x (closure s) < \u22a4\n\u22a2 0 < \u2191\u03b5 / 2\n[PROOFSTEP]\nsimpa [pos_iff_ne_zero] using \u03b5pos\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : infEdist x (closure s) < \u22a4\n\u03b50 : 0 < \u2191\u03b5 / 2\n\u22a2 infEdist x s \u2264 infEdist x (closure s) + \u2191\u03b5\n[PROOFSTEP]\nhave : infEdist x (closure s) < infEdist x (closure s) + \u03b5 / 2 := ENNReal.lt_add_right h.ne \u03b50.ne'\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : infEdist x (closure s) < \u22a4\n\u03b50 : 0 < \u2191\u03b5 / 2\nthis : infEdist x (closure s) < infEdist x (closure s) + \u2191\u03b5 / 2\n\u22a2 infEdist x s \u2264 infEdist x (closure s) + \u2191\u03b5\n[PROOFSTEP]\nrcases infEdist_lt_iff.mp this with\n \u27e8y, ycs, hy\u27e9\n -- y : \u03b1, ycs : y \u2208 closure s, hy : edist x y < infEdist x (closure s) + \u2191\u03b5 / 2\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y\u271d : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : infEdist x (closure s) < \u22a4\n\u03b50 : 0 < \u2191\u03b5 / 2\nthis : infEdist x (closure s) < infEdist x (closure s) + \u2191\u03b5 / 2\ny : \u03b1\nycs : y \u2208 closure s\nhy : edist x y < infEdist x (closure s) + \u2191\u03b5 / 2\n\u22a2 infEdist x s \u2264 infEdist x (closure s) + \u2191\u03b5\n[PROOFSTEP]\nrcases EMetric.mem_closure_iff.1 ycs (\u03b5 / 2) \u03b50 with\n \u27e8z, zs, dyz\u27e9\n -- z : \u03b1, zs : z \u2208 s, dyz : edist y z < \u2191\u03b5 / 2\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y\u271d : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : infEdist x (closure s) < \u22a4\n\u03b50 : 0 < \u2191\u03b5 / 2\nthis : infEdist x (closure s) < infEdist x (closure s) + \u2191\u03b5 / 2\ny : \u03b1\nycs : y \u2208 closure s\nhy : edist x y < infEdist x (closure s) + \u2191\u03b5 / 2\nz : \u03b1\nzs : z \u2208 s\ndyz : edist y z < \u2191\u03b5 / 2\n\u22a2 infEdist x s \u2264 infEdist x (closure s) + \u2191\u03b5\n[PROOFSTEP]\ncalc\n infEdist x s \u2264 edist x z := infEdist_le_edist_of_mem zs\n _ \u2264 edist x y + edist y z := (edist_triangle _ _ _)\n _ \u2264 infEdist x (closure s) + \u03b5 / 2 + \u03b5 / 2 := (add_le_add (le_of_lt hy) (le_of_lt dyz))\n _ = infEdist x (closure s) + \u2191\u03b5 := by rw [add_assoc, ENNReal.add_halves]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y\u271d : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : infEdist x (closure s) < \u22a4\n\u03b50 : 0 < \u2191\u03b5 / 2\nthis : infEdist x (closure s) < infEdist x (closure s) + \u2191\u03b5 / 2\ny : \u03b1\nycs : y \u2208 closure s\nhy : edist x y < infEdist x (closure s) + \u2191\u03b5 / 2\nz : \u03b1\nzs : z \u2208 s\ndyz : edist y z < \u2191\u03b5 / 2\n\u22a2 infEdist x (closure s) + \u2191\u03b5 / 2 + \u2191\u03b5 / 2 = infEdist x (closure s) + \u2191\u03b5\n[PROOFSTEP]\nrw [add_assoc, ENNReal.add_halves]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : x \u2208 closure s\n\u22a2 infEdist x s = 0\n[PROOFSTEP]\nrw [\u2190 infEdist_closure]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : x \u2208 closure s\n\u22a2 infEdist x (closure s) = 0\n[PROOFSTEP]\nexact infEdist_zero_of_mem h\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : infEdist x s = 0\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 > 0\n\u22a2 infEdist x s < \u03b5\n[PROOFSTEP]\nrwa [h]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : IsClosed s\n\u22a2 x \u2208 s \u2194 infEdist x s = 0\n[PROOFSTEP]\nrw [\u2190 mem_closure_iff_infEdist_zero, h.closure_eq]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nE : Set \u03b1\n\u22a2 0 < infEdist x E \u2194 \u00acx \u2208 closure E\n[PROOFSTEP]\nrw [mem_closure_iff_infEdist_zero, pos_iff_ne_zero]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nE : Set \u03b1\n\u22a2 0 < infEdist x (closure E) \u2194 \u00acx \u2208 closure E\n[PROOFSTEP]\nrw [infEdist_closure, infEdist_pos_iff_not_mem_closure]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nE : Set \u03b1\nh : \u00acx \u2208 closure E\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 ENNReal.ofReal \u03b5 < infEdist x E\n[PROOFSTEP]\nrw [\u2190 infEdist_pos_iff_not_mem_closure, ENNReal.lt_iff_exists_real_btwn] at h \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nE : Set \u03b1\nh : \u2203 r, 0 \u2264 r \u2227 0 < ENNReal.ofReal r \u2227 ENNReal.ofReal r < infEdist x E\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 ENNReal.ofReal \u03b5 < infEdist x E\n[PROOFSTEP]\nrcases h with \u27e8\u03b5, \u27e8_, \u27e8\u03b5_pos, \u03b5_lt\u27e9\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nE : Set \u03b1\n\u03b5 : \u211d\nleft\u271d : 0 \u2264 \u03b5\n\u03b5_pos : 0 < ENNReal.ofReal \u03b5\n\u03b5_lt : ENNReal.ofReal \u03b5 < infEdist x E\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 ENNReal.ofReal \u03b5 < infEdist x E\n[PROOFSTEP]\nexact \u27e8\u03b5, \u27e8ENNReal.ofReal_pos.mp \u03b5_pos, \u03b5_lt\u27e9\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\u22650\u221e\nh : r < infEdist x s\n\u22a2 Disjoint (closedBall x r) s\n[PROOFSTEP]\nrw [disjoint_left]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\u22650\u221e\nh : r < infEdist x s\n\u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 closedBall x r \u2192 \u00aca \u2208 s\n[PROOFSTEP]\nintro y hy h'y\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y\u271d : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\u22650\u221e\nh : r < infEdist x s\ny : \u03b1\nhy : y \u2208 closedBall x r\nh'y : y \u2208 s\n\u22a2 False\n[PROOFSTEP]\napply lt_irrefl (infEdist x s)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y\u271d : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\u22650\u221e\nh : r < infEdist x s\ny : \u03b1\nhy : y \u2208 closedBall x r\nh'y : y \u2208 s\n\u22a2 infEdist x s < infEdist x s\n[PROOFSTEP]\ncalc\n infEdist x s \u2264 edist x y := infEdist_le_edist_of_mem h'y\n _ \u2264 r := by rwa [mem_closedBall, edist_comm] at hy \n _ < infEdist x s := h\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y\u271d : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\u22650\u221e\nh : r < infEdist x s\ny : \u03b1\nhy : y \u2208 closedBall x r\nh'y : y \u2208 s\n\u22a2 edist x y \u2264 r\n[PROOFSTEP]\nrwa [mem_closedBall, edist_comm] at hy \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh\u03a6 : Isometry \u03a6\n\u22a2 infEdist (\u03a6 x) (\u03a6 '' t) = infEdist x t\n[PROOFSTEP]\nsimp only [infEdist, iInf_image, h\u03a6.edist_eq]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\n\u22a2 \u2203 F, (\u2200 (n : \u2115), IsClosed (F n)) \u2227 (\u2200 (n : \u2115), F n \u2286 U) \u2227 \u22c3 (n : \u2115), F n = U \u2227 Monotone F\n[PROOFSTEP]\nobtain \u27e8a, a_pos, a_lt_one\u27e9 : \u2203 a : \u211d\u22650\u221e, 0 < a \u2227 a < 1 := exists_between zero_lt_one\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\n\u22a2 \u2203 F, (\u2200 (n : \u2115), IsClosed (F n)) \u2227 (\u2200 (n : \u2115), F n \u2286 U) \u2227 \u22c3 (n : \u2115), F n = U \u2227 Monotone F\n[PROOFSTEP]\nlet F := fun n : \u2115 => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\n\u22a2 \u2203 F, (\u2200 (n : \u2115), IsClosed (F n)) \u2227 (\u2200 (n : \u2115), F n \u2286 U) \u2227 \u22c3 (n : \u2115), F n = U \u2227 Monotone F\n[PROOFSTEP]\nhave F_subset : \u2200 n, F n \u2286 U := fun n x hx \u21a6 by\n by_contra h\n have : infEdist x U\u1d9c \u2260 0 := ((ENNReal.pow_pos a_pos _).trans_le hx).ne'\n exact this (infEdist_zero_of_mem h)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nn : \u2115\nx : \u03b1\nhx : x \u2208 F n\n\u22a2 x \u2208 U\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nn : \u2115\nx : \u03b1\nhx : x \u2208 F n\nh : \u00acx \u2208 U\n\u22a2 False\n[PROOFSTEP]\nhave : infEdist x U\u1d9c \u2260 0 := ((ENNReal.pow_pos a_pos _).trans_le hx).ne'\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nn : \u2115\nx : \u03b1\nhx : x \u2208 F n\nh : \u00acx \u2208 U\nthis : infEdist x U\u1d9c \u2260 0\n\u22a2 False\n[PROOFSTEP]\nexact this (infEdist_zero_of_mem h)\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\n\u22a2 \u2203 F, (\u2200 (n : \u2115), IsClosed (F n)) \u2227 (\u2200 (n : \u2115), F n \u2286 U) \u2227 \u22c3 (n : \u2115), F n = U \u2227 Monotone F\n[PROOFSTEP]\nrefine \u27e8F, fun n => IsClosed.preimage continuous_infEdist isClosed_Ici, F_subset, ?_, ?_\u27e9\n[GOAL]\ncase intro.intro.refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\n\u22a2 \u22c3 (n : \u2115), F n = U\ncase intro.intro.refine_2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\n\u22a2 Monotone F\n[PROOFSTEP]\nshow \u22c3 n, F n = U\n[GOAL]\ncase intro.intro.refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\n\u22a2 \u22c3 (n : \u2115), F n = U\n[PROOFSTEP]\nrefine' Subset.antisymm (by simp only [iUnion_subset_iff, F_subset, forall_const]) fun x hx => _\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\n\u22a2 \u22c3 (n : \u2115), F n \u2286 U\n[PROOFSTEP]\nsimp only [iUnion_subset_iff, F_subset, forall_const]\n[GOAL]\ncase intro.intro.refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\nx : \u03b1\nhx : x \u2208 U\n\u22a2 x \u2208 \u22c3 (n : \u2115), F n\n[PROOFSTEP]\nhave : \u00acx \u2208 U\u1d9c := by simpa using hx\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\nx : \u03b1\nhx : x \u2208 U\n\u22a2 \u00acx \u2208 U\u1d9c\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\ncase intro.intro.refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\nx : \u03b1\nhx : x \u2208 U\nthis : \u00acx \u2208 U\u1d9c\n\u22a2 x \u2208 \u22c3 (n : \u2115), F n\n[PROOFSTEP]\nrw [mem_iff_infEdist_zero_of_closed hU.isClosed_compl] at this \n[GOAL]\ncase intro.intro.refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\nx : \u03b1\nhx : x \u2208 U\nthis : \u00acinfEdist x U\u1d9c = 0\n\u22a2 x \u2208 \u22c3 (n : \u2115), F n\n[PROOFSTEP]\nhave B : 0 < infEdist x U\u1d9c := by simpa [pos_iff_ne_zero] using this\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\nx : \u03b1\nhx : x \u2208 U\nthis : \u00acinfEdist x U\u1d9c = 0\n\u22a2 0 < infEdist x U\u1d9c\n[PROOFSTEP]\nsimpa [pos_iff_ne_zero] using this\n[GOAL]\ncase intro.intro.refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\nx : \u03b1\nhx : x \u2208 U\nthis : \u00acinfEdist x U\u1d9c = 0\nB : 0 < infEdist x U\u1d9c\n\u22a2 x \u2208 \u22c3 (n : \u2115), F n\n[PROOFSTEP]\nhave : Filter.Tendsto (fun n => a ^ n) atTop (\ud835\udcdd 0) := ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1 a_lt_one\n[GOAL]\ncase intro.intro.refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\nx : \u03b1\nhx : x \u2208 U\nthis\u271d : \u00acinfEdist x U\u1d9c = 0\nB : 0 < infEdist x U\u1d9c\nthis : Tendsto (fun n => a ^ n) atTop (\ud835\udcdd 0)\n\u22a2 x \u2208 \u22c3 (n : \u2115), F n\n[PROOFSTEP]\nrcases((tendsto_order.1 this).2 _ B).exists with \u27e8n, hn\u27e9\n[GOAL]\ncase intro.intro.refine_1.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\nx : \u03b1\nhx : x \u2208 U\nthis\u271d : \u00acinfEdist x U\u1d9c = 0\nB : 0 < infEdist x U\u1d9c\nthis : Tendsto (fun n => a ^ n) atTop (\ud835\udcdd 0)\nn : \u2115\nhn : a ^ n < infEdist x U\u1d9c\n\u22a2 x \u2208 \u22c3 (n : \u2115), F n\n[PROOFSTEP]\nsimp only [mem_iUnion, mem_Ici, mem_preimage]\n[GOAL]\ncase intro.intro.refine_1.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\nx : \u03b1\nhx : x \u2208 U\nthis\u271d : \u00acinfEdist x U\u1d9c = 0\nB : 0 < infEdist x U\u1d9c\nthis : Tendsto (fun n => a ^ n) atTop (\ud835\udcdd 0)\nn : \u2115\nhn : a ^ n < infEdist x U\u1d9c\n\u22a2 \u2203 i, a ^ i \u2264 infEdist x U\u1d9c\n[PROOFSTEP]\nexact \u27e8n, hn.le\u27e9\n[GOAL]\ncase intro.intro.refine_2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\n\u22a2 Monotone F\n[PROOFSTEP]\nshow Monotone F\n[GOAL]\ncase intro.intro.refine_2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\n\u22a2 Monotone F\n[PROOFSTEP]\nintro m n hmn x hx\n[GOAL]\ncase intro.intro.refine_2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\nm n : \u2115\nhmn : m \u2264 n\nx : \u03b1\nhx : x \u2208 F m\n\u22a2 x \u2208 F n\n[PROOFSTEP]\nsimp only [mem_Ici, mem_preimage] at hx \u22a2\n[GOAL]\ncase intro.intro.refine_2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nU : Set \u03b1\nhU : IsOpen U\na : \u211d\u22650\u221e\na_pos : 0 < a\na_lt_one : a < 1\nF : \u2115 \u2192 Set \u03b1 := fun n => (fun x => infEdist x U\u1d9c) \u207b\u00b9' Ici (a ^ n)\nF_subset : \u2200 (n : \u2115), F n \u2286 U\nm n : \u2115\nhmn : m \u2264 n\nx : \u03b1\nhx : a ^ m \u2264 infEdist x U\u1d9c\n\u22a2 a ^ n \u2264 infEdist x U\u1d9c\n[PROOFSTEP]\napply le_trans (pow_le_pow_of_le_one' a_lt_one.le hmn) hx\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhne : Set.Nonempty s\nx : \u03b1\n\u22a2 \u2203 y, y \u2208 s \u2227 infEdist x s = edist x y\n[PROOFSTEP]\nhave A : Continuous fun y => edist x y := continuous_const.edist continuous_id\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhne : Set.Nonempty s\nx : \u03b1\nA : Continuous fun y => edist x y\n\u22a2 \u2203 y, y \u2208 s \u2227 infEdist x s = edist x y\n[PROOFSTEP]\nobtain \u27e8y, ys, hy\u27e9 : \u2203 y \u2208 s, \u2200 z, z \u2208 s \u2192 edist x y \u2264 edist x z := hs.exists_forall_le hne A.continuousOn\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhne : Set.Nonempty s\nx : \u03b1\nA : Continuous fun y => edist x y\ny : \u03b1\nys : y \u2208 s\nhy : \u2200 (z : \u03b1), z \u2208 s \u2192 edist x y \u2264 edist x z\n\u22a2 \u2203 y, y \u2208 s \u2227 infEdist x s = edist x y\n[PROOFSTEP]\nexact \u27e8y, ys, le_antisymm (infEdist_le_edist_of_mem ys) (by rwa [le_infEdist])\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhne : Set.Nonempty s\nx : \u03b1\nA : Continuous fun y => edist x y\ny : \u03b1\nys : y \u2208 s\nhy : \u2200 (z : \u03b1), z \u2208 s \u2192 edist x y \u2264 edist x z\n\u22a2 edist x y \u2264 infEdist x s\n[PROOFSTEP]\nrwa [le_infEdist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nht : IsClosed t\nhst : Disjoint s t\n\u22a2 \u2203 r, 0 < r \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | hne)\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\nt : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nht : IsClosed t\nhs : IsCompact \u2205\nhst : Disjoint \u2205 t\n\u22a2 \u2203 r, 0 < r \u2227 \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase h\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\nt : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nht : IsClosed t\nhs : IsCompact \u2205\nhst : Disjoint \u2205 t\n\u22a2 0 < 1 \u2227 \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u21911 < edist x y\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nht : IsClosed t\nhst : Disjoint s t\nhne : Set.Nonempty s\n\u22a2 \u2203 r, 0 < r \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\n[PROOFSTEP]\nobtain \u27e8x, hx, h\u27e9 : \u2203 x \u2208 s, \u2200 y \u2208 s, infEdist x t \u2264 infEdist y t :=\n hs.exists_forall_le hne continuous_infEdist.continuousOn\n[GOAL]\ncase inr.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nht : IsClosed t\nhst : Disjoint s t\nhne : Set.Nonempty s\nx : \u03b1\nhx : x \u2208 s\nh : \u2200 (y : \u03b1), y \u2208 s \u2192 infEdist x t \u2264 infEdist y t\n\u22a2 \u2203 r, 0 < r \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\n[PROOFSTEP]\nhave : 0 < infEdist x t := pos_iff_ne_zero.2 fun H => hst.le_bot \u27e8hx, (mem_iff_infEdist_zero_of_closed ht).mpr H\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nht : IsClosed t\nhst : Disjoint s t\nhne : Set.Nonempty s\nx : \u03b1\nhx : x \u2208 s\nh : \u2200 (y : \u03b1), y \u2208 s \u2192 infEdist x t \u2264 infEdist y t\nthis : 0 < infEdist x t\n\u22a2 \u2203 r, 0 < r \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\n[PROOFSTEP]\nrcases ENNReal.lt_iff_exists_nnreal_btwn.1 this with \u27e8r, h\u2080, hr\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nht : IsClosed t\nhst : Disjoint s t\nhne : Set.Nonempty s\nx : \u03b1\nhx : x \u2208 s\nh : \u2200 (y : \u03b1), y \u2208 s \u2192 infEdist x t \u2264 infEdist y t\nthis : 0 < infEdist x t\nr : \u211d\u22650\nh\u2080 : 0 < \u2191r\nhr : \u2191r < infEdist x t\n\u22a2 \u2203 r, 0 < r \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\n[PROOFSTEP]\nexact \u27e8r, ENNReal.coe_pos.mp h\u2080, fun y hy z hz => hr.trans_le <| le_infEdist.1 (h y hy) z hz\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffEdist s s = 0\n[PROOFSTEP]\nsimp only [hausdorffEdist_def, sup_idem, ENNReal.iSup_eq_zero]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (i : \u03b1), i \u2208 s \u2192 infEdist i s = 0\n[PROOFSTEP]\nexact fun x hx => infEdist_zero_of_mem hx\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffEdist s t = hausdorffEdist t s\n[PROOFSTEP]\nsimp only [hausdorffEdist_def]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 (\u2a06 (x : \u03b1) (_ : x \u2208 s), infEdist x t) \u2294 \u2a06 (y : \u03b1) (_ : y \u2208 t), infEdist y s =\n (\u2a06 (y : \u03b1) (_ : y \u2208 t), infEdist y s) \u2294 \u2a06 (x : \u03b1) (_ : x \u2208 s), infEdist x t\n[PROOFSTEP]\napply sup_comm\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\u22650\u221e\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infEdist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infEdist x s \u2264 r\n\u22a2 hausdorffEdist s t \u2264 r\n[PROOFSTEP]\nsimp only [hausdorffEdist_def, sup_le_iff, iSup_le_iff]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\u22650\u221e\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infEdist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infEdist x s \u2264 r\n\u22a2 (\u2200 (i : \u03b1), i \u2208 s \u2192 infEdist i t \u2264 r) \u2227 \u2200 (i : \u03b1), i \u2208 t \u2192 infEdist i s \u2264 r\n[PROOFSTEP]\nexact \u27e8H1, H2\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\u22650\u221e\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 edist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 edist x y \u2264 r\n\u22a2 hausdorffEdist s t \u2264 r\n[PROOFSTEP]\nrefine hausdorffEdist_le_of_infEdist (fun x xs \u21a6 ?_) (fun x xt \u21a6 ?_)\n[GOAL]\ncase refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\u22650\u221e\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 edist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 edist x y \u2264 r\nx : \u03b1\nxs : x \u2208 s\n\u22a2 infEdist x t \u2264 r\n[PROOFSTEP]\nrcases H1 x xs with \u27e8y, yt, hy\u27e9\n[GOAL]\ncase refine_1.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\u22650\u221e\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 edist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 edist x y \u2264 r\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\nhy : edist x y \u2264 r\n\u22a2 infEdist x t \u2264 r\n[PROOFSTEP]\nexact le_trans (infEdist_le_edist_of_mem yt) hy\n[GOAL]\ncase refine_2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\u22650\u221e\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 edist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 edist x y \u2264 r\nx : \u03b1\nxt : x \u2208 t\n\u22a2 infEdist x s \u2264 r\n[PROOFSTEP]\nrcases H2 x xt with \u27e8y, ys, hy\u27e9\n[GOAL]\ncase refine_2.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\u22650\u221e\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 edist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 edist x y \u2264 r\nx : \u03b1\nxt : x \u2208 t\ny : \u03b1\nys : y \u2208 s\nhy : edist x y \u2264 r\n\u22a2 infEdist x s \u2264 r\n[PROOFSTEP]\nexact le_trans (infEdist_le_edist_of_mem ys) hy\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : x \u2208 s\n\u22a2 infEdist x t \u2264 hausdorffEdist s t\n[PROOFSTEP]\nrw [hausdorffEdist_def]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : x \u2208 s\n\u22a2 infEdist x t \u2264 (\u2a06 (x : \u03b1) (_ : x \u2208 s), infEdist x t) \u2294 \u2a06 (y : \u03b1) (_ : y \u2208 t), infEdist y s\n[PROOFSTEP]\nrefine le_trans ?_ le_sup_left\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : x \u2208 s\n\u22a2 infEdist x t \u2264 \u2a06 (x : \u03b1) (_ : x \u2208 s), infEdist x t\n[PROOFSTEP]\nexact le_iSup\u2082 (\u03b1 := \u211d\u22650\u221e) x h\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : infEdist x s + hausdorffEdist s t < \u22a4\n\u22a2 infEdist x t \u2264 infEdist x s + hausdorffEdist s t + \u2191\u03b5\n[PROOFSTEP]\nhave \u03b50 : (\u03b5 / 2 : \u211d\u22650\u221e) \u2260 0 := by simpa [pos_iff_ne_zero] using \u03b5pos\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : infEdist x s + hausdorffEdist s t < \u22a4\n\u22a2 \u2191\u03b5 / 2 \u2260 0\n[PROOFSTEP]\nsimpa [pos_iff_ne_zero] using \u03b5pos\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : infEdist x s + hausdorffEdist s t < \u22a4\n\u03b50 : \u2191\u03b5 / 2 \u2260 0\n\u22a2 infEdist x t \u2264 infEdist x s + hausdorffEdist s t + \u2191\u03b5\n[PROOFSTEP]\nhave : infEdist x s < infEdist x s + \u03b5 / 2 := ENNReal.lt_add_right (ENNReal.add_lt_top.1 h).1.ne \u03b50\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : infEdist x s + hausdorffEdist s t < \u22a4\n\u03b50 : \u2191\u03b5 / 2 \u2260 0\nthis : infEdist x s < infEdist x s + \u2191\u03b5 / 2\n\u22a2 infEdist x t \u2264 infEdist x s + hausdorffEdist s t + \u2191\u03b5\n[PROOFSTEP]\nrcases infEdist_lt_iff.mp this with\n \u27e8y, ys, dxy\u27e9\n -- y : \u03b1, ys : y \u2208 s, dxy : edist x y < infEdist x s + \u2191\u03b5 / 2\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : infEdist x s + hausdorffEdist s t < \u22a4\n\u03b50 : \u2191\u03b5 / 2 \u2260 0\nthis : infEdist x s < infEdist x s + \u2191\u03b5 / 2\ny : \u03b1\nys : y \u2208 s\ndxy : edist x y < infEdist x s + \u2191\u03b5 / 2\n\u22a2 infEdist x t \u2264 infEdist x s + hausdorffEdist s t + \u2191\u03b5\n[PROOFSTEP]\nhave : hausdorffEdist s t < hausdorffEdist s t + \u03b5 / 2 := ENNReal.lt_add_right (ENNReal.add_lt_top.1 h).2.ne \u03b50\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : infEdist x s + hausdorffEdist s t < \u22a4\n\u03b50 : \u2191\u03b5 / 2 \u2260 0\nthis\u271d : infEdist x s < infEdist x s + \u2191\u03b5 / 2\ny : \u03b1\nys : y \u2208 s\ndxy : edist x y < infEdist x s + \u2191\u03b5 / 2\nthis : hausdorffEdist s t < hausdorffEdist s t + \u2191\u03b5 / 2\n\u22a2 infEdist x t \u2264 infEdist x s + hausdorffEdist s t + \u2191\u03b5\n[PROOFSTEP]\nrcases exists_edist_lt_of_hausdorffEdist_lt ys this with\n \u27e8z, zt, dyz\u27e9\n -- z : \u03b1, zt : z \u2208 t, dyz : edist y z < Hausdorff_edist s t + \u2191\u03b5 / 2\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : infEdist x s + hausdorffEdist s t < \u22a4\n\u03b50 : \u2191\u03b5 / 2 \u2260 0\nthis\u271d : infEdist x s < infEdist x s + \u2191\u03b5 / 2\ny : \u03b1\nys : y \u2208 s\ndxy : edist x y < infEdist x s + \u2191\u03b5 / 2\nthis : hausdorffEdist s t < hausdorffEdist s t + \u2191\u03b5 / 2\nz : \u03b1\nzt : z \u2208 t\ndyz : edist y z < hausdorffEdist s t + \u2191\u03b5 / 2\n\u22a2 infEdist x t \u2264 infEdist x s + hausdorffEdist s t + \u2191\u03b5\n[PROOFSTEP]\ncalc\n infEdist x t \u2264 edist x z := infEdist_le_edist_of_mem zt\n _ \u2264 edist x y + edist y z := (edist_triangle _ _ _)\n _ \u2264 infEdist x s + \u03b5 / 2 + (hausdorffEdist s t + \u03b5 / 2) := (add_le_add dxy.le dyz.le)\n _ = infEdist x s + hausdorffEdist s t + \u03b5 := by simp [ENNReal.add_halves, add_comm, add_left_comm]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh : infEdist x s + hausdorffEdist s t < \u22a4\n\u03b50 : \u2191\u03b5 / 2 \u2260 0\nthis\u271d : infEdist x s < infEdist x s + \u2191\u03b5 / 2\ny : \u03b1\nys : y \u2208 s\ndxy : edist x y < infEdist x s + \u2191\u03b5 / 2\nthis : hausdorffEdist s t < hausdorffEdist s t + \u2191\u03b5 / 2\nz : \u03b1\nzt : z \u2208 t\ndyz : edist y z < hausdorffEdist s t + \u2191\u03b5 / 2\n\u22a2 infEdist x s + \u2191\u03b5 / 2 + (hausdorffEdist s t + \u2191\u03b5 / 2) = infEdist x s + hausdorffEdist s t + \u2191\u03b5\n[PROOFSTEP]\nsimp [ENNReal.add_halves, add_comm, add_left_comm]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : Isometry \u03a6\n\u22a2 hausdorffEdist (\u03a6 '' s) (\u03a6 '' t) = hausdorffEdist s t\n[PROOFSTEP]\nsimp only [hausdorffEdist_def, iSup_image, infEdist_image h]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : Set.Nonempty s\nht : Set.Nonempty t\n\u22a2 hausdorffEdist s t \u2264 diam (s \u222a t)\n[PROOFSTEP]\nrcases hs with \u27e8x, xs\u27e9\n[GOAL]\ncase intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nht : Set.Nonempty t\nx : \u03b1\nxs : x \u2208 s\n\u22a2 hausdorffEdist s t \u2264 diam (s \u222a t)\n[PROOFSTEP]\nrcases ht with \u27e8y, yt\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\n\u22a2 hausdorffEdist s t \u2264 diam (s \u222a t)\n[PROOFSTEP]\nrefine' hausdorffEdist_le_of_mem_edist _ _\n[GOAL]\ncase intro.intro.refine'_1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 edist x y \u2264 diam (s \u222a t)\n[PROOFSTEP]\nintro z hz\n[GOAL]\ncase intro.intro.refine'_1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\nz : \u03b1\nhz : z \u2208 s\n\u22a2 \u2203 y, y \u2208 t \u2227 edist z y \u2264 diam (s \u222a t)\n[PROOFSTEP]\nexact \u27e8y, yt, edist_le_diam_of_mem (subset_union_left _ _ hz) (subset_union_right _ _ yt)\u27e9\n[GOAL]\ncase intro.intro.refine'_2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 edist x y \u2264 diam (s \u222a t)\n[PROOFSTEP]\nintro z hz\n[GOAL]\ncase intro.intro.refine'_2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\nz : \u03b1\nhz : z \u2208 t\n\u22a2 \u2203 y, y \u2208 s \u2227 edist z y \u2264 diam (s \u222a t)\n[PROOFSTEP]\nexact \u27e8x, xs, edist_le_diam_of_mem (subset_union_right _ _ hz) (subset_union_left _ _ xs)\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffEdist s u \u2264 hausdorffEdist s t + hausdorffEdist t u\n[PROOFSTEP]\nrw [hausdorffEdist_def]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 (\u2a06 (x : \u03b1) (_ : x \u2208 s), infEdist x u) \u2294 \u2a06 (y : \u03b1) (_ : y \u2208 u), infEdist y s \u2264 hausdorffEdist s t + hausdorffEdist t u\n[PROOFSTEP]\nsimp only [sup_le_iff, iSup_le_iff]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 (i : \u03b1), i \u2208 s \u2192 infEdist i u \u2264 hausdorffEdist s t + hausdorffEdist t u) \u2227\n \u2200 (i : \u03b1), i \u2208 u \u2192 infEdist i s \u2264 hausdorffEdist s t + hausdorffEdist t u\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (i : \u03b1), i \u2208 s \u2192 infEdist i u \u2264 hausdorffEdist s t + hausdorffEdist t u\n[PROOFSTEP]\nshow \u2200 x \u2208 s, infEdist x u \u2264 hausdorffEdist s t + hausdorffEdist t u\n[GOAL]\ncase left\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 infEdist x u \u2264 hausdorffEdist s t + hausdorffEdist t u\n[PROOFSTEP]\nexact fun x xs =>\n calc\n infEdist x u \u2264 infEdist x t + hausdorffEdist t u := infEdist_le_infEdist_add_hausdorffEdist\n _ \u2264 hausdorffEdist s t + hausdorffEdist t u := add_le_add_right (infEdist_le_hausdorffEdist_of_mem xs) _\n[GOAL]\ncase right\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (i : \u03b1), i \u2208 u \u2192 infEdist i s \u2264 hausdorffEdist s t + hausdorffEdist t u\n[PROOFSTEP]\nshow \u2200 x \u2208 u, infEdist x s \u2264 hausdorffEdist s t + hausdorffEdist t u\n[GOAL]\ncase right\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (x : \u03b1), x \u2208 u \u2192 infEdist x s \u2264 hausdorffEdist s t + hausdorffEdist t u\n[PROOFSTEP]\nexact fun x xu =>\n calc\n infEdist x s \u2264 infEdist x t + hausdorffEdist t s := infEdist_le_infEdist_add_hausdorffEdist\n _ \u2264 hausdorffEdist u t + hausdorffEdist t s := (add_le_add_right (infEdist_le_hausdorffEdist_of_mem xu) _)\n _ = hausdorffEdist s t + hausdorffEdist t u := by simp [hausdorffEdist_comm, add_comm]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxu : x \u2208 u\n\u22a2 hausdorffEdist u t + hausdorffEdist t s = hausdorffEdist s t + hausdorffEdist t u\n[PROOFSTEP]\nsimp [hausdorffEdist_comm, add_comm]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffEdist s t = 0 \u2194 closure s = closure t\n[PROOFSTEP]\nsimp only [hausdorffEdist_def, ENNReal.sup_eq_zero, ENNReal.iSup_eq_zero, \u2190 subset_def, \u2190 mem_closure_iff_infEdist_zero,\n subset_antisymm_iff, isClosed_closure.closure_subset_iff]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffEdist s (closure s) = 0\n[PROOFSTEP]\nrw [hausdorffEdist_zero_iff_closure_eq_closure, closure_closure]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffEdist (closure s) t = hausdorffEdist s t\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffEdist (closure s) t \u2264 hausdorffEdist s t\n[PROOFSTEP]\ncalc\n _ \u2264 hausdorffEdist (closure s) s + hausdorffEdist s t := hausdorffEdist_triangle\n _ = hausdorffEdist s t := by simp [hausdorffEdist_comm]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffEdist (closure s) s + hausdorffEdist s t = hausdorffEdist s t\n[PROOFSTEP]\nsimp [hausdorffEdist_comm]\n[GOAL]\ncase refine'_2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffEdist s t \u2264 hausdorffEdist (closure s) t\n[PROOFSTEP]\ncalc\n _ \u2264 hausdorffEdist s (closure s) + hausdorffEdist (closure s) t := hausdorffEdist_triangle\n _ = hausdorffEdist (closure s) t := by simp\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffEdist s (closure s) + hausdorffEdist (closure s) t = hausdorffEdist (closure s) t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffEdist s (closure t) = hausdorffEdist s t\n[PROOFSTEP]\nsimp [@hausdorffEdist_comm _ _ s _]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffEdist (closure s) (closure t) = hausdorffEdist s t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : IsClosed s\nht : IsClosed t\n\u22a2 hausdorffEdist s t = 0 \u2194 s = t\n[PROOFSTEP]\nrw [hausdorffEdist_zero_iff_closure_eq_closure, hs.closure_eq, ht.closure_eq]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nne : Set.Nonempty s\n\u22a2 hausdorffEdist s \u2205 = \u22a4\n[PROOFSTEP]\nrcases ne with \u27e8x, xs\u27e9\n[GOAL]\ncase intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\n\u22a2 hausdorffEdist s \u2205 = \u22a4\n[PROOFSTEP]\nhave : infEdist x \u2205 \u2264 hausdorffEdist s \u2205 := infEdist_le_hausdorffEdist_of_mem xs\n[GOAL]\ncase intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\nthis : infEdist x \u2205 \u2264 hausdorffEdist s \u2205\n\u22a2 hausdorffEdist s \u2205 = \u22a4\n[PROOFSTEP]\nsimpa using this\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist s t \u2260 \u22a4\n\u22a2 s = \u2205 \u2227 t = \u2205 \u2228 Set.Nonempty s \u2227 Set.Nonempty t\n[PROOFSTEP]\ncases' s.eq_empty_or_nonempty with hs hs\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist s t \u2260 \u22a4\nhs : s = \u2205\n\u22a2 s = \u2205 \u2227 t = \u2205 \u2228 Set.Nonempty s \u2227 Set.Nonempty t\n[PROOFSTEP]\ncases' t.eq_empty_or_nonempty with ht ht\n[GOAL]\ncase inl.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist s t \u2260 \u22a4\nhs : s = \u2205\nht : t = \u2205\n\u22a2 s = \u2205 \u2227 t = \u2205 \u2228 Set.Nonempty s \u2227 Set.Nonempty t\n[PROOFSTEP]\nexact Or.inl \u27e8hs, ht\u27e9\n[GOAL]\ncase inl.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist s t \u2260 \u22a4\nhs : s = \u2205\nht : Set.Nonempty t\n\u22a2 s = \u2205 \u2227 t = \u2205 \u2228 Set.Nonempty s \u2227 Set.Nonempty t\n[PROOFSTEP]\nrw [hausdorffEdist_comm] at fin \n[GOAL]\ncase inl.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist t s \u2260 \u22a4\nhs : s = \u2205\nht : Set.Nonempty t\n\u22a2 s = \u2205 \u2227 t = \u2205 \u2228 Set.Nonempty s \u2227 Set.Nonempty t\n[PROOFSTEP]\nexact Or.inr \u27e8nonempty_of_hausdorffEdist_ne_top ht fin, ht\u27e9\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist s t \u2260 \u22a4\nhs : Set.Nonempty s\n\u22a2 s = \u2205 \u2227 t = \u2205 \u2228 Set.Nonempty s \u2227 Set.Nonempty t\n[PROOFSTEP]\nexact Or.inr \u27e8hs, nonempty_of_hausdorffEdist_ne_top hs fin\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 infDist x s = \u2a05 (y : \u2191s), dist x \u2191y\n[PROOFSTEP]\nrw [infDist, infEdist, iInf_subtype', ENNReal.toReal_iInf]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 \u2a05 (i : { y // y \u2208 s }), ENNReal.toReal (edist x \u2191i) = \u2a05 (y : \u2191s), dist x \u2191y\n[PROOFSTEP]\nsimp only [dist_edist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (i : { y // y \u2208 s }), edist x \u2191i \u2260 \u22a4\n[PROOFSTEP]\nexact fun _ \u21a6 edist_ne_top _ _\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 infDist x \u2205 = 0\n[PROOFSTEP]\nsimp [infDist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : Set.Nonempty s\n\u22a2 infEdist x s \u2260 \u22a4\n[PROOFSTEP]\nrcases h with \u27e8y, hy\u27e9\n[GOAL]\ncase intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y\u271d : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\ny : \u03b1\nhy : y \u2208 s\n\u22a2 infEdist x s \u2260 \u22a4\n[PROOFSTEP]\nexact ne_top_of_le_ne_top (edist_ne_top _ _) (infEdist_le_edist_of_mem hy)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 infEdist x s = \u22a4 \u2194 s = \u2205\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with rfl | hs\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nt u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 infEdist x \u2205 = \u22a4 \u2194 \u2205 = \u2205\n[PROOFSTEP]\nsimp [*, Nonempty.ne_empty, infEdist_ne_top]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : Set.Nonempty s\n\u22a2 infEdist x s = \u22a4 \u2194 s = \u2205\n[PROOFSTEP]\nsimp [*, Nonempty.ne_empty, infEdist_ne_top]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : x \u2208 s\n\u22a2 infDist x s = 0\n[PROOFSTEP]\nsimp [infEdist_zero_of_mem h, infDist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 infDist x {y} = dist x y\n[PROOFSTEP]\nsimp [infDist, dist_edist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : y \u2208 s\n\u22a2 infDist x s \u2264 dist x y\n[PROOFSTEP]\nrw [dist_edist, infDist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : y \u2208 s\n\u22a2 ENNReal.toReal (infEdist x s) \u2264 ENNReal.toReal (edist x y)\n[PROOFSTEP]\nexact ENNReal.toReal_mono (edist_ne_top _ _) (infEdist_le_edist_of_mem h)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhs : Set.Nonempty s\n\u22a2 infDist x s < r \u2194 \u2203 y, y \u2208 s \u2227 dist x y < r\n[PROOFSTEP]\nsimp_rw [infDist, \u2190 ENNReal.lt_ofReal_iff_toReal_lt (infEdist_ne_top hs), infEdist_lt_iff,\n ENNReal.lt_ofReal_iff_toReal_lt (edist_ne_top _ _), \u2190 dist_edist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 infDist x s \u2264 infDist y s + dist x y\n[PROOFSTEP]\nrw [infDist, infDist, dist_edist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 ENNReal.toReal (infEdist x s) \u2264 ENNReal.toReal (infEdist y s) + ENNReal.toReal (edist x y)\n[PROOFSTEP]\nrefine ENNReal.toReal_le_add' infEdist_le_infEdist_add_edist ?_ (flip absurd (edist_ne_top _ _))\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 infEdist y s = \u22a4 \u2192 infEdist x s = \u22a4\n[PROOFSTEP]\nsimp only [infEdist_eq_top_iff, imp_self]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : Bounded s\nhy : y \u2208 s\n\u22a2 dist x y \u2264 infDist x s + diam s\n[PROOFSTEP]\nrw [infDist, diam, dist_edist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : Bounded s\nhy : y \u2208 s\n\u22a2 ENNReal.toReal (edist x y) \u2264 ENNReal.toReal (infEdist x s) + ENNReal.toReal (EMetric.diam s)\n[PROOFSTEP]\nexact toReal_le_add (edist_le_infEdist_add_ediam hy) (infEdist_ne_top \u27e8y, hy\u27e9) hs.ediam_ne_top\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 infDist x (closure s) = infDist x s\n[PROOFSTEP]\nsimp [infDist, infEdist_closure]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhx : x \u2208 closure s\n\u22a2 infDist x s = 0\n[PROOFSTEP]\nrw [\u2190 infDist_closure]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhx : x \u2208 closure s\n\u22a2 infDist x (closure s) = 0\n[PROOFSTEP]\nexact infDist_zero_of_mem hx\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : Set.Nonempty s\n\u22a2 x \u2208 closure s \u2194 infDist x s = 0\n[PROOFSTEP]\nsimp [mem_closure_iff_infEdist_zero, infDist, ENNReal.toReal_eq_zero_iff, infEdist_ne_top h]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : IsClosed s\nhs : Set.Nonempty s\n\u22a2 x \u2208 s \u2194 infDist x s = 0\n[PROOFSTEP]\nrw [\u2190 mem_closure_iff_infDist_zero hs, h.closure_eq]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : IsClosed s\nhs : Set.Nonempty s\n\u22a2 \u00acx \u2208 s \u2194 0 < infDist x s\n[PROOFSTEP]\nsimp [h.mem_iff_infDist_zero hs, infDist_nonneg.gt_iff_ne]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : \u00acx \u2208 closure s\n\u22a2 ContinuousAt (fun x => (infDist x s)\u207b\u00b9) x\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | hs)\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nt u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : \u00acx \u2208 closure \u2205\n\u22a2 ContinuousAt (fun x => (infDist x \u2205)\u207b\u00b9) x\n[PROOFSTEP]\nsimp only [infDist_empty, continuousAt_const]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : \u00acx \u2208 closure s\nhs : Set.Nonempty s\n\u22a2 ContinuousAt (fun x => (infDist x s)\u207b\u00b9) x\n[PROOFSTEP]\nrefine (continuous_infDist_pt s).continuousAt.inv\u2080 ?_\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : \u00acx \u2208 closure s\nhs : Set.Nonempty s\n\u22a2 infDist x s \u2260 0\n[PROOFSTEP]\nrwa [Ne.def, \u2190 mem_closure_iff_infDist_zero hs]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh\u03a6 : Isometry \u03a6\n\u22a2 infDist (\u03a6 x) (\u03a6 '' t) = infDist x t\n[PROOFSTEP]\nsimp [infDist, infEdist_image h\u03a6]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : y \u2208 s\n\u22a2 infDist x (s \u2229 closedBall x (dist y x)) = infDist x s\n[PROOFSTEP]\nreplace h : y \u2208 s \u2229 closedBall x (dist y x) := \u27e8h, mem_closedBall.2 le_rfl\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : y \u2208 s \u2229 closedBall x (dist y x)\n\u22a2 infDist x (s \u2229 closedBall x (dist y x)) = infDist x s\n[PROOFSTEP]\nrefine le_antisymm ?_ (infDist_le_infDist_of_subset (inter_subset_left _ _) \u27e8y, h\u27e9)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : y \u2208 s \u2229 closedBall x (dist y x)\n\u22a2 infDist x (s \u2229 closedBall x (dist y x)) \u2264 infDist x s\n[PROOFSTEP]\nrefine' not_lt.1 fun hlt => _\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : y \u2208 s \u2229 closedBall x (dist y x)\nhlt : infDist x s < infDist x (s \u2229 closedBall x (dist y x))\n\u22a2 False\n[PROOFSTEP]\nrcases(infDist_lt_iff \u27e8y, h.1\u27e9).mp hlt with \u27e8z, hzs, hz\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : y \u2208 s \u2229 closedBall x (dist y x)\nhlt : infDist x s < infDist x (s \u2229 closedBall x (dist y x))\nz : \u03b1\nhzs : z \u2208 s\nhz : dist x z < infDist x (s \u2229 closedBall x (dist y x))\n\u22a2 False\n[PROOFSTEP]\ncases' le_or_lt (dist z x) (dist y x) with hle hlt\n[GOAL]\ncase intro.intro.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : y \u2208 s \u2229 closedBall x (dist y x)\nhlt : infDist x s < infDist x (s \u2229 closedBall x (dist y x))\nz : \u03b1\nhzs : z \u2208 s\nhz : dist x z < infDist x (s \u2229 closedBall x (dist y x))\nhle : dist z x \u2264 dist y x\n\u22a2 False\n[PROOFSTEP]\nexact hz.not_le (infDist_le_dist_of_mem \u27e8hzs, hle\u27e9)\n[GOAL]\ncase intro.intro.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : y \u2208 s \u2229 closedBall x (dist y x)\nhlt\u271d : infDist x s < infDist x (s \u2229 closedBall x (dist y x))\nz : \u03b1\nhzs : z \u2208 s\nhz : dist x z < infDist x (s \u2229 closedBall x (dist y x))\nhlt : dist y x < dist z x\n\u22a2 False\n[PROOFSTEP]\nrw [dist_comm z, dist_comm y] at hlt \n[GOAL]\ncase intro.intro.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : y \u2208 s \u2229 closedBall x (dist y x)\nhlt\u271d : infDist x s < infDist x (s \u2229 closedBall x (dist y x))\nz : \u03b1\nhzs : z \u2208 s\nhz : dist x z < infDist x (s \u2229 closedBall x (dist y x))\nhlt : dist x y < dist x z\n\u22a2 False\n[PROOFSTEP]\nexact (hlt.trans hz).not_le (infDist_le_dist_of_mem h)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y\u271d : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : IsCompact s\nhne : Set.Nonempty s\nx y : \u03b1\nhys : y \u2208 s\nhy : infEdist x s = edist x y\n\u22a2 infDist x s = dist x y\n[PROOFSTEP]\nrw [infDist, dist_edist, hy]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\ninst\u271d : ProperSpace \u03b1\nh : IsClosed s\nhne : Set.Nonempty s\nx : \u03b1\n\u22a2 \u2203 y, y \u2208 s \u2227 infDist x s = dist x y\n[PROOFSTEP]\nrcases hne with \u27e8z, hz\u27e9\n[GOAL]\ncase intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\ninst\u271d : ProperSpace \u03b1\nh : IsClosed s\nx z : \u03b1\nhz : z \u2208 s\n\u22a2 \u2203 y, y \u2208 s \u2227 infDist x s = dist x y\n[PROOFSTEP]\nrw [\u2190 infDist_inter_closedBall_of_mem hz]\n[GOAL]\ncase intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\ninst\u271d : ProperSpace \u03b1\nh : IsClosed s\nx z : \u03b1\nhz : z \u2208 s\n\u22a2 \u2203 y, y \u2208 s \u2227 infDist x (s \u2229 closedBall x (dist z x)) = dist x y\n[PROOFSTEP]\nset t := s \u2229 closedBall x (dist z x)\n[GOAL]\ncase intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ns t\u271d u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\ninst\u271d : ProperSpace \u03b1\nh : IsClosed s\nx z : \u03b1\nhz : z \u2208 s\nt : Set \u03b1 := s \u2229 closedBall x (dist z x)\n\u22a2 \u2203 y, y \u2208 s \u2227 infDist x t = dist x y\n[PROOFSTEP]\nhave htc : IsCompact t := (isCompact_closedBall x (dist z x)).inter_left h\n[GOAL]\ncase intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ns t\u271d u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\ninst\u271d : ProperSpace \u03b1\nh : IsClosed s\nx z : \u03b1\nhz : z \u2208 s\nt : Set \u03b1 := s \u2229 closedBall x (dist z x)\nhtc : IsCompact t\n\u22a2 \u2203 y, y \u2208 s \u2227 infDist x t = dist x y\n[PROOFSTEP]\nhave htne : t.Nonempty := \u27e8z, hz, mem_closedBall.2 le_rfl\u27e9\n[GOAL]\ncase intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ns t\u271d u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\ninst\u271d : ProperSpace \u03b1\nh : IsClosed s\nx z : \u03b1\nhz : z \u2208 s\nt : Set \u03b1 := s \u2229 closedBall x (dist z x)\nhtc : IsCompact t\nhtne : Set.Nonempty t\n\u22a2 \u2203 y, y \u2208 s \u2227 infDist x t = dist x y\n[PROOFSTEP]\nobtain \u27e8y, \u27e8hys, -\u27e9, hyd\u27e9 : \u2203 y \u2208 t, infDist x t = dist x y := htc.exists_infDist_eq_dist htne x\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ns t\u271d u : Set \u03b1\nx\u271d y\u271d : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\ninst\u271d : ProperSpace \u03b1\nh : IsClosed s\nx z : \u03b1\nhz : z \u2208 s\nt : Set \u03b1 := s \u2229 closedBall x (dist z x)\nhtc : IsCompact t\nhtne : Set.Nonempty t\ny : \u03b1\nhyd : infDist x t = dist x y\nhys : y \u2208 s\n\u22a2 \u2203 y, y \u2208 s \u2227 infDist x t = dist x y\n[PROOFSTEP]\nexact \u27e8y, hys, hyd\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\ninst\u271d : ProperSpace \u03b1\nhne : Set.Nonempty s\nx : \u03b1\n\u22a2 \u2203 y, y \u2208 closure s \u2227 infDist x s = dist x y\n[PROOFSTEP]\nsimpa only [infDist_closure] using isClosed_closure.exists_infDist_eq_dist hne.closure x\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 0 \u2264 hausdorffDist s t\n[PROOFSTEP]\nsimp [hausdorffDist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : Set.Nonempty s\nht : Set.Nonempty t\nbs : Bounded s\nbt : Bounded t\n\u22a2 hausdorffEdist s t \u2260 \u22a4\n[PROOFSTEP]\nrcases hs with \u27e8cs, hcs\u27e9\n[GOAL]\ncase intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nht : Set.Nonempty t\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\n\u22a2 hausdorffEdist s t \u2260 \u22a4\n[PROOFSTEP]\nrcases ht with \u27e8ct, hct\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\nct : \u03b1\nhct : ct \u2208 t\n\u22a2 hausdorffEdist s t \u2260 \u22a4\n[PROOFSTEP]\nrcases(bounded_iff_subset_ball ct).1 bs with \u27e8rs, hrs\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\nct : \u03b1\nhct : ct \u2208 t\nrs : \u211d\nhrs : s \u2286 closedBall ct rs\n\u22a2 hausdorffEdist s t \u2260 \u22a4\n[PROOFSTEP]\nrcases(bounded_iff_subset_ball cs).1 bt with \u27e8rt, hrt\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\nct : \u03b1\nhct : ct \u2208 t\nrs : \u211d\nhrs : s \u2286 closedBall ct rs\nrt : \u211d\nhrt : t \u2286 closedBall cs rt\n\u22a2 hausdorffEdist s t \u2260 \u22a4\n[PROOFSTEP]\nhave : hausdorffEdist s t \u2264 ENNReal.ofReal (max rs rt) :=\n by\n apply hausdorffEdist_le_of_mem_edist\n \u00b7 intro x xs\n exists ct, hct\n have : dist x ct \u2264 max rs rt := le_trans (hrs xs) (le_max_left _ _)\n rwa [edist_dist, ENNReal.ofReal_le_ofReal_iff]\n exact le_trans dist_nonneg this\n \u00b7 intro x xt\n exists cs, hcs\n have : dist x cs \u2264 max rs rt := le_trans (hrt xt) (le_max_right _ _)\n rwa [edist_dist, ENNReal.ofReal_le_ofReal_iff]\n exact le_trans dist_nonneg this\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\nct : \u03b1\nhct : ct \u2208 t\nrs : \u211d\nhrs : s \u2286 closedBall ct rs\nrt : \u211d\nhrt : t \u2286 closedBall cs rt\n\u22a2 hausdorffEdist s t \u2264 ENNReal.ofReal (max rs rt)\n[PROOFSTEP]\napply hausdorffEdist_le_of_mem_edist\n[GOAL]\ncase H1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\nct : \u03b1\nhct : ct \u2208 t\nrs : \u211d\nhrs : s \u2286 closedBall ct rs\nrt : \u211d\nhrt : t \u2286 closedBall cs rt\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 edist x y \u2264 ENNReal.ofReal (max rs rt)\n[PROOFSTEP]\nintro x xs\n[GOAL]\ncase H1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\nct : \u03b1\nhct : ct \u2208 t\nrs : \u211d\nhrs : s \u2286 closedBall ct rs\nrt : \u211d\nhrt : t \u2286 closedBall cs rt\nx : \u03b1\nxs : x \u2208 s\n\u22a2 \u2203 y, y \u2208 t \u2227 edist x y \u2264 ENNReal.ofReal (max rs rt)\n[PROOFSTEP]\nexists ct, hct\n[GOAL]\ncase H1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\nct : \u03b1\nhct : ct \u2208 t\nrs : \u211d\nhrs : s \u2286 closedBall ct rs\nrt : \u211d\nhrt : t \u2286 closedBall cs rt\nx : \u03b1\nxs : x \u2208 s\n\u22a2 edist x ct \u2264 ENNReal.ofReal (max rs rt)\n[PROOFSTEP]\nhave : dist x ct \u2264 max rs rt := le_trans (hrs xs) (le_max_left _ _)\n[GOAL]\ncase H1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\nct : \u03b1\nhct : ct \u2208 t\nrs : \u211d\nhrs : s \u2286 closedBall ct rs\nrt : \u211d\nhrt : t \u2286 closedBall cs rt\nx : \u03b1\nxs : x \u2208 s\nthis : dist x ct \u2264 max rs rt\n\u22a2 edist x ct \u2264 ENNReal.ofReal (max rs rt)\n[PROOFSTEP]\nrwa [edist_dist, ENNReal.ofReal_le_ofReal_iff]\n[GOAL]\ncase H1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\nct : \u03b1\nhct : ct \u2208 t\nrs : \u211d\nhrs : s \u2286 closedBall ct rs\nrt : \u211d\nhrt : t \u2286 closedBall cs rt\nx : \u03b1\nxs : x \u2208 s\nthis : dist x ct \u2264 max rs rt\n\u22a2 0 \u2264 max rs rt\n[PROOFSTEP]\nexact le_trans dist_nonneg this\n[GOAL]\ncase H2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\nct : \u03b1\nhct : ct \u2208 t\nrs : \u211d\nhrs : s \u2286 closedBall ct rs\nrt : \u211d\nhrt : t \u2286 closedBall cs rt\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 edist x y \u2264 ENNReal.ofReal (max rs rt)\n[PROOFSTEP]\nintro x xt\n[GOAL]\ncase H2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\nct : \u03b1\nhct : ct \u2208 t\nrs : \u211d\nhrs : s \u2286 closedBall ct rs\nrt : \u211d\nhrt : t \u2286 closedBall cs rt\nx : \u03b1\nxt : x \u2208 t\n\u22a2 \u2203 y, y \u2208 s \u2227 edist x y \u2264 ENNReal.ofReal (max rs rt)\n[PROOFSTEP]\nexists cs, hcs\n[GOAL]\ncase H2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\nct : \u03b1\nhct : ct \u2208 t\nrs : \u211d\nhrs : s \u2286 closedBall ct rs\nrt : \u211d\nhrt : t \u2286 closedBall cs rt\nx : \u03b1\nxt : x \u2208 t\n\u22a2 edist x cs \u2264 ENNReal.ofReal (max rs rt)\n[PROOFSTEP]\nhave : dist x cs \u2264 max rs rt := le_trans (hrt xt) (le_max_right _ _)\n[GOAL]\ncase H2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\nct : \u03b1\nhct : ct \u2208 t\nrs : \u211d\nhrs : s \u2286 closedBall ct rs\nrt : \u211d\nhrt : t \u2286 closedBall cs rt\nx : \u03b1\nxt : x \u2208 t\nthis : dist x cs \u2264 max rs rt\n\u22a2 edist x cs \u2264 ENNReal.ofReal (max rs rt)\n[PROOFSTEP]\nrwa [edist_dist, ENNReal.ofReal_le_ofReal_iff]\n[GOAL]\ncase H2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\nct : \u03b1\nhct : ct \u2208 t\nrs : \u211d\nhrs : s \u2286 closedBall ct rs\nrt : \u211d\nhrt : t \u2286 closedBall cs rt\nx : \u03b1\nxt : x \u2208 t\nthis : dist x cs \u2264 max rs rt\n\u22a2 0 \u2264 max rs rt\n[PROOFSTEP]\nexact le_trans dist_nonneg this\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\ncs : \u03b1\nhcs : cs \u2208 s\nct : \u03b1\nhct : ct \u2208 t\nrs : \u211d\nhrs : s \u2286 closedBall ct rs\nrt : \u211d\nhrt : t \u2286 closedBall cs rt\nthis : hausdorffEdist s t \u2264 ENNReal.ofReal (max rs rt)\n\u22a2 hausdorffEdist s t \u2260 \u22a4\n[PROOFSTEP]\nexact ne_top_of_le_ne_top ENNReal.ofReal_ne_top this\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffDist s s = 0\n[PROOFSTEP]\nsimp [hausdorffDist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffDist s t = hausdorffDist t s\n[PROOFSTEP]\nsimp [hausdorffDist, hausdorffEdist_comm]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffDist s \u2205 = 0\n[PROOFSTEP]\ncases' s.eq_empty_or_nonempty with h h\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : s = \u2205\n\u22a2 hausdorffDist s \u2205 = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : Set.Nonempty s\n\u22a2 hausdorffDist s \u2205 = 0\n[PROOFSTEP]\nsimp [hausdorffDist, hausdorffEdist_empty h]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffDist \u2205 s = 0\n[PROOFSTEP]\nsimp [hausdorffDist_comm]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\n\u22a2 hausdorffDist s t \u2264 r\n[PROOFSTEP]\nby_cases h1 : hausdorffEdist s t = \u22a4\n[GOAL]\ncase pos\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\nh1 : hausdorffEdist s t = \u22a4\n\u22a2 hausdorffDist s t \u2264 r\n[PROOFSTEP]\nrwa [hausdorffDist, h1, ENNReal.top_toReal]\n[GOAL]\ncase neg\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\nh1 : \u00achausdorffEdist s t = \u22a4\n\u22a2 hausdorffDist s t \u2264 r\n[PROOFSTEP]\ncases' s.eq_empty_or_nonempty with hs hs\n[GOAL]\ncase neg.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\nh1 : \u00achausdorffEdist s t = \u22a4\nhs : s = \u2205\n\u22a2 hausdorffDist s t \u2264 r\n[PROOFSTEP]\nrwa [hs, hausdorffDist_empty']\n[GOAL]\ncase neg.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\nh1 : \u00achausdorffEdist s t = \u22a4\nhs : Set.Nonempty s\n\u22a2 hausdorffDist s t \u2264 r\n[PROOFSTEP]\ncases' t.eq_empty_or_nonempty with ht ht\n[GOAL]\ncase neg.inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\nh1 : \u00achausdorffEdist s t = \u22a4\nhs : Set.Nonempty s\nht : t = \u2205\n\u22a2 hausdorffDist s t \u2264 r\n[PROOFSTEP]\nrwa [ht, hausdorffDist_empty]\n[GOAL]\ncase neg.inr.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\nh1 : \u00achausdorffEdist s t = \u22a4\nhs : Set.Nonempty s\nht : Set.Nonempty t\n\u22a2 hausdorffDist s t \u2264 r\n[PROOFSTEP]\nhave : hausdorffEdist s t \u2264 ENNReal.ofReal r :=\n by\n apply hausdorffEdist_le_of_infEdist _ _\n \u00b7 intro x hx\n have I := H1 x hx\n rwa [infDist, \u2190 ENNReal.toReal_ofReal hr, ENNReal.toReal_le_toReal (infEdist_ne_top ht) ENNReal.ofReal_ne_top] at I \n \u00b7 intro x hx\n have I := H2 x hx\n rwa [infDist, \u2190 ENNReal.toReal_ofReal hr, ENNReal.toReal_le_toReal (infEdist_ne_top hs) ENNReal.ofReal_ne_top] at I \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\nh1 : \u00achausdorffEdist s t = \u22a4\nhs : Set.Nonempty s\nht : Set.Nonempty t\n\u22a2 hausdorffEdist s t \u2264 ENNReal.ofReal r\n[PROOFSTEP]\napply hausdorffEdist_le_of_infEdist _ _\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\nh1 : \u00achausdorffEdist s t = \u22a4\nhs : Set.Nonempty s\nht : Set.Nonempty t\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 infEdist x t \u2264 ENNReal.ofReal r\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\nh1 : \u00achausdorffEdist s t = \u22a4\nhs : Set.Nonempty s\nht : Set.Nonempty t\nx : \u03b1\nhx : x \u2208 s\n\u22a2 infEdist x t \u2264 ENNReal.ofReal r\n[PROOFSTEP]\nhave I := H1 x hx\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\nh1 : \u00achausdorffEdist s t = \u22a4\nhs : Set.Nonempty s\nht : Set.Nonempty t\nx : \u03b1\nhx : x \u2208 s\nI : infDist x t \u2264 r\n\u22a2 infEdist x t \u2264 ENNReal.ofReal r\n[PROOFSTEP]\nrwa [infDist, \u2190 ENNReal.toReal_ofReal hr, ENNReal.toReal_le_toReal (infEdist_ne_top ht) ENNReal.ofReal_ne_top] at I \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\nh1 : \u00achausdorffEdist s t = \u22a4\nhs : Set.Nonempty s\nht : Set.Nonempty t\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 infEdist x s \u2264 ENNReal.ofReal r\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\nh1 : \u00achausdorffEdist s t = \u22a4\nhs : Set.Nonempty s\nht : Set.Nonempty t\nx : \u03b1\nhx : x \u2208 t\n\u22a2 infEdist x s \u2264 ENNReal.ofReal r\n[PROOFSTEP]\nhave I := H2 x hx\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\nh1 : \u00achausdorffEdist s t = \u22a4\nhs : Set.Nonempty s\nht : Set.Nonempty t\nx : \u03b1\nhx : x \u2208 t\nI : infDist x s \u2264 r\n\u22a2 infEdist x s \u2264 ENNReal.ofReal r\n[PROOFSTEP]\nrwa [infDist, \u2190 ENNReal.toReal_ofReal hr, ENNReal.toReal_le_toReal (infEdist_ne_top hs) ENNReal.ofReal_ne_top] at I \n[GOAL]\ncase neg.inr.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\nh1 : \u00achausdorffEdist s t = \u22a4\nhs : Set.Nonempty s\nht : Set.Nonempty t\nthis : hausdorffEdist s t \u2264 ENNReal.ofReal r\n\u22a2 hausdorffDist s t \u2264 r\n[PROOFSTEP]\nrwa [hausdorffDist, \u2190 ENNReal.toReal_ofReal hr, ENNReal.toReal_le_toReal h1 ENNReal.ofReal_ne_top]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\n\u22a2 hausdorffDist s t \u2264 r\n[PROOFSTEP]\napply hausdorffDist_le_of_infDist hr\n[GOAL]\ncase H1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\n[PROOFSTEP]\nintro x xs\n[GOAL]\ncase H1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\nx : \u03b1\nxs : x \u2208 s\n\u22a2 infDist x t \u2264 r\n[PROOFSTEP]\nrcases H1 x xs with \u27e8y, yt, hy\u27e9\n[GOAL]\ncase H1.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y\u271d : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\nhy : dist x y \u2264 r\n\u22a2 infDist x t \u2264 r\n[PROOFSTEP]\nexact le_trans (infDist_le_dist_of_mem yt) hy\n[GOAL]\ncase H2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r\n[PROOFSTEP]\nintro x xt\n[GOAL]\ncase H2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\nx : \u03b1\nxt : x \u2208 t\n\u22a2 infDist x s \u2264 r\n[PROOFSTEP]\nrcases H2 x xt with \u27e8y, ys, hy\u27e9\n[GOAL]\ncase H2.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y\u271d : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\nx : \u03b1\nxt : x \u2208 t\ny : \u03b1\nys : y \u2208 s\nhy : dist x y \u2264 r\n\u22a2 infDist x s \u2264 r\n[PROOFSTEP]\nexact le_trans (infDist_le_dist_of_mem ys) hy\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : Set.Nonempty s\nbs : Bounded s\nht : Set.Nonempty t\nbt : Bounded t\n\u22a2 hausdorffDist s t \u2264 diam (s \u222a t)\n[PROOFSTEP]\nrcases hs with \u27e8x, xs\u27e9\n[GOAL]\ncase intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nht : Set.Nonempty t\nbt : Bounded t\nx : \u03b1\nxs : x \u2208 s\n\u22a2 hausdorffDist s t \u2264 diam (s \u222a t)\n[PROOFSTEP]\nrcases ht with \u27e8y, yt\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y\u271d : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\n\u22a2 hausdorffDist s t \u2264 diam (s \u222a t)\n[PROOFSTEP]\nrefine hausdorffDist_le_of_mem_dist diam_nonneg ?_ ?_\n[GOAL]\ncase intro.intro.refine_1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y\u271d : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 diam (s \u222a t)\n[PROOFSTEP]\nexact fun z hz =>\n \u27e8y, yt, dist_le_diam_of_mem (bounded_union.2 \u27e8bs, bt\u27e9) (subset_union_left _ _ hz) (subset_union_right _ _ yt)\u27e9\n[GOAL]\ncase intro.intro.refine_2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y\u271d : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nbs : Bounded s\nbt : Bounded t\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 diam (s \u222a t)\n[PROOFSTEP]\nexact fun z hz =>\n \u27e8x, xs, dist_le_diam_of_mem (bounded_union.2 \u27e8bs, bt\u27e9) (subset_union_right _ _ hz) (subset_union_left _ _ xs)\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nh : x \u2208 s\nH : hausdorffDist s t < r\nfin : hausdorffEdist s t \u2260 \u22a4\n\u22a2 \u2203 y, y \u2208 t \u2227 dist x y < r\n[PROOFSTEP]\nhave r0 : 0 < r := lt_of_le_of_lt hausdorffDist_nonneg H\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nh : x \u2208 s\nH : hausdorffDist s t < r\nfin : hausdorffEdist s t \u2260 \u22a4\nr0 : 0 < r\n\u22a2 \u2203 y, y \u2208 t \u2227 dist x y < r\n[PROOFSTEP]\nhave : hausdorffEdist s t < ENNReal.ofReal r := by\n rwa [hausdorffDist, \u2190 ENNReal.toReal_ofReal (le_of_lt r0), ENNReal.toReal_lt_toReal fin ENNReal.ofReal_ne_top] at H \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nh : x \u2208 s\nH : hausdorffDist s t < r\nfin : hausdorffEdist s t \u2260 \u22a4\nr0 : 0 < r\n\u22a2 hausdorffEdist s t < ENNReal.ofReal r\n[PROOFSTEP]\nrwa [hausdorffDist, \u2190 ENNReal.toReal_ofReal (le_of_lt r0), ENNReal.toReal_lt_toReal fin ENNReal.ofReal_ne_top] at H \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nh : x \u2208 s\nH : hausdorffDist s t < r\nfin : hausdorffEdist s t \u2260 \u22a4\nr0 : 0 < r\nthis : hausdorffEdist s t < ENNReal.ofReal r\n\u22a2 \u2203 y, y \u2208 t \u2227 dist x y < r\n[PROOFSTEP]\nrcases exists_edist_lt_of_hausdorffEdist_lt h this with \u27e8y, hy, yr\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y\u271d : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nh : x \u2208 s\nH : hausdorffDist s t < r\nfin : hausdorffEdist s t \u2260 \u22a4\nr0 : 0 < r\nthis : hausdorffEdist s t < ENNReal.ofReal r\ny : \u03b1\nhy : y \u2208 t\nyr : edist x y < ENNReal.ofReal r\n\u22a2 \u2203 y, y \u2208 t \u2227 dist x y < r\n[PROOFSTEP]\nrw [edist_dist, ENNReal.ofReal_lt_ofReal_iff r0] at yr \n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y\u271d : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nh : x \u2208 s\nH : hausdorffDist s t < r\nfin : hausdorffEdist s t \u2260 \u22a4\nr0 : 0 < r\nthis : hausdorffEdist s t < ENNReal.ofReal r\ny : \u03b1\nhy : y \u2208 t\nyr : dist x y < r\n\u22a2 \u2203 y, y \u2208 t \u2227 dist x y < r\n[PROOFSTEP]\nexact \u27e8y, hy, yr\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nh : y \u2208 t\nH : hausdorffDist s t < r\nfin : hausdorffEdist s t \u2260 \u22a4\n\u22a2 \u2203 x, x \u2208 s \u2227 dist x y < r\n[PROOFSTEP]\nrw [hausdorffDist_comm] at H \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nh : y \u2208 t\nH : hausdorffDist t s < r\nfin : hausdorffEdist s t \u2260 \u22a4\n\u22a2 \u2203 x, x \u2208 s \u2227 dist x y < r\n[PROOFSTEP]\nrw [hausdorffEdist_comm] at fin \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nh : y \u2208 t\nH : hausdorffDist t s < r\nfin : hausdorffEdist t s \u2260 \u22a4\n\u22a2 \u2203 x, x \u2208 s \u2227 dist x y < r\n[PROOFSTEP]\nsimpa [dist_comm] using exists_dist_lt_of_hausdorffDist_lt h H fin\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist s t \u2260 \u22a4\n\u22a2 infDist x t \u2264 infDist x s + hausdorffDist s t\n[PROOFSTEP]\nrefine toReal_le_add' infEdist_le_infEdist_add_hausdorffEdist (fun h \u21a6 ?_) (flip absurd fin)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist s t \u2260 \u22a4\nh : infEdist x s = \u22a4\n\u22a2 infEdist x t = \u22a4\n[PROOFSTEP]\nrw [infEdist_eq_top_iff, \u2190 not_nonempty_iff_eq_empty] at h \u22a2\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist s t \u2260 \u22a4\nh : \u00acSet.Nonempty s\n\u22a2 \u00acSet.Nonempty t\n[PROOFSTEP]\nrw [hausdorffEdist_comm] at fin \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist t s \u2260 \u22a4\nh : \u00acSet.Nonempty s\n\u22a2 \u00acSet.Nonempty t\n[PROOFSTEP]\nexact mt (nonempty_of_hausdorffEdist_ne_top \u00b7 fin) h\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nh : Isometry \u03a6\n\u22a2 hausdorffDist (\u03a6 '' s) (\u03a6 '' t) = hausdorffDist s t\n[PROOFSTEP]\nsimp [hausdorffDist, hausdorffEdist_image h]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist s t \u2260 \u22a4\n\u22a2 hausdorffDist s u \u2264 hausdorffDist s t + hausdorffDist t u\n[PROOFSTEP]\nrefine toReal_le_add' hausdorffEdist_triangle (flip absurd fin) (not_imp_not.1 fun h \u21a6 ?_)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist s t \u2260 \u22a4\nh : \u00achausdorffEdist s u = \u22a4\n\u22a2 \u00achausdorffEdist t u = \u22a4\n[PROOFSTEP]\nrw [hausdorffEdist_comm] at fin \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist t s \u2260 \u22a4\nh : \u00achausdorffEdist s u = \u22a4\n\u22a2 \u00achausdorffEdist t u = \u22a4\n[PROOFSTEP]\nexact ne_top_of_le_ne_top (add_ne_top.2 \u27e8fin, h\u27e9) hausdorffEdist_triangle\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist t u \u2260 \u22a4\n\u22a2 hausdorffDist s u \u2264 hausdorffDist s t + hausdorffDist t u\n[PROOFSTEP]\nrw [hausdorffEdist_comm] at fin \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist u t \u2260 \u22a4\n\u22a2 hausdorffDist s u \u2264 hausdorffDist s t + hausdorffDist t u\n[PROOFSTEP]\nhave I : hausdorffDist u s \u2264 hausdorffDist u t + hausdorffDist t s := hausdorffDist_triangle fin\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist u t \u2260 \u22a4\nI : hausdorffDist u s \u2264 hausdorffDist u t + hausdorffDist t s\n\u22a2 hausdorffDist s u \u2264 hausdorffDist s t + hausdorffDist t u\n[PROOFSTEP]\nsimpa [add_comm, hausdorffDist_comm] using I\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffDist s (closure s) = 0\n[PROOFSTEP]\nsimp [hausdorffDist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffDist (closure s) t = hausdorffDist s t\n[PROOFSTEP]\nsimp [hausdorffDist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffDist s (closure t) = hausdorffDist s t\n[PROOFSTEP]\nsimp [hausdorffDist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffDist (closure s) (closure t) = hausdorffDist s t\n[PROOFSTEP]\nsimp [hausdorffDist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist s t \u2260 \u22a4\n\u22a2 hausdorffDist s t = 0 \u2194 closure s = closure t\n[PROOFSTEP]\nsimp [\u2190 hausdorffEdist_zero_iff_closure_eq_closure, hausdorffDist, ENNReal.toReal_eq_zero_iff, fin]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : IsClosed s\nht : IsClosed t\nfin : hausdorffEdist s t \u2260 \u22a4\n\u22a2 hausdorffDist s t = 0 \u2194 s = t\n[PROOFSTEP]\nsimp [\u2190 hausdorffEdist_zero_iff_eq_of_closed hs ht, hausdorffDist, ENNReal.toReal_eq_zero_iff, fin]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 : \u211d\ns : Set \u03b1\nx\u271d : \u03b1\nE : Set \u03b1\nx : \u03b1\nh : \u00acx \u2208 closure E\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d) in \ud835\udcdd 0, \u00acx \u2208 thickening \u03b4 E\n[PROOFSTEP]\nobtain \u27e8\u03b5, \u27e8\u03b5_pos, \u03b5_lt\u27e9\u27e9 := exists_real_pos_lt_infEdist_of_not_mem_closure h\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 : \u211d\ns : Set \u03b1\nx\u271d : \u03b1\nE : Set \u03b1\nx : \u03b1\nh : \u00acx \u2208 closure E\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\n\u03b5_lt : ENNReal.ofReal \u03b5 < infEdist x E\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d) in \ud835\udcdd 0, \u00acx \u2208 thickening \u03b4 E\n[PROOFSTEP]\nfilter_upwards [eventually_lt_nhds \u03b5_pos] with \u03b4 h\u03b4\n[GOAL]\ncase h\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx\u271d : \u03b1\nE : Set \u03b1\nx : \u03b1\nh : \u00acx \u2208 closure E\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\n\u03b5_lt : ENNReal.ofReal \u03b5 < infEdist x E\n\u03b4 : \u211d\nh\u03b4 : \u03b4 < \u03b5\n\u22a2 \u00acx \u2208 thickening \u03b4 E\n[PROOFSTEP]\nsimp only [thickening, mem_setOf_eq, not_lt]\n[GOAL]\ncase h\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx\u271d : \u03b1\nE : Set \u03b1\nx : \u03b1\nh : \u00acx \u2208 closure E\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\n\u03b5_lt : ENNReal.ofReal \u03b5 < infEdist x E\n\u03b4 : \u211d\nh\u03b4 : \u03b4 < \u03b5\n\u22a2 ENNReal.ofReal \u03b4 \u2264 infEdist x E\n[PROOFSTEP]\nexact (ENNReal.ofReal_le_ofReal h\u03b4.le).trans \u03b5_lt.le\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\n\u22a2 thickening \u03b4 \u2205 = \u2205\n[PROOFSTEP]\nsimp only [thickening, setOf_false, infEdist_empty, not_top_lt]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 : \u211d\ns : Set \u03b1\nx : \u03b1\nA : Set \u03b1\n\u22a2 Pairwise (Disjoint on fun r => frontier (thickening r A))\n[PROOFSTEP]\nrefine' (pairwise_disjoint_on _).2 fun r\u2081 r\u2082 hr => _\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 : \u211d\ns : Set \u03b1\nx : \u03b1\nA : Set \u03b1\nr\u2081 r\u2082 : \u211d\nhr : r\u2081 < r\u2082\n\u22a2 Disjoint (frontier (thickening r\u2081 A)) (frontier (thickening r\u2082 A))\n[PROOFSTEP]\ncases' le_total r\u2081 0 with h\u2081 h\u2081\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 : \u211d\ns : Set \u03b1\nx : \u03b1\nA : Set \u03b1\nr\u2081 r\u2082 : \u211d\nhr : r\u2081 < r\u2082\nh\u2081 : r\u2081 \u2264 0\n\u22a2 Disjoint (frontier (thickening r\u2081 A)) (frontier (thickening r\u2082 A))\n[PROOFSTEP]\nsimp [thickening_of_nonpos h\u2081]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 : \u211d\ns : Set \u03b1\nx : \u03b1\nA : Set \u03b1\nr\u2081 r\u2082 : \u211d\nhr : r\u2081 < r\u2082\nh\u2081 : 0 \u2264 r\u2081\n\u22a2 Disjoint (frontier (thickening r\u2081 A)) (frontier (thickening r\u2082 A))\n[PROOFSTEP]\nrefine'\n ((disjoint_singleton.2 fun h => hr.ne _).preimage _).mono (frontier_thickening_subset _)\n (frontier_thickening_subset _)\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 : \u211d\ns : Set \u03b1\nx : \u03b1\nA : Set \u03b1\nr\u2081 r\u2082 : \u211d\nhr : r\u2081 < r\u2082\nh\u2081 : 0 \u2264 r\u2081\nh : ENNReal.ofReal r\u2081 = ENNReal.ofReal r\u2082\n\u22a2 r\u2081 = r\u2082\n[PROOFSTEP]\napply_fun ENNReal.toReal at h \n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 : \u211d\ns : Set \u03b1\nx : \u03b1\nA : Set \u03b1\nr\u2081 r\u2082 : \u211d\nhr : r\u2081 < r\u2082\nh\u2081 : 0 \u2264 r\u2081\nh : ENNReal.toReal (ENNReal.ofReal r\u2081) = ENNReal.toReal (ENNReal.ofReal r\u2082)\n\u22a2 r\u2081 = r\u2082\n[PROOFSTEP]\nrwa [ENNReal.toReal_ofReal h\u2081, ENNReal.toReal_ofReal (h\u2081.trans hr.le)] at h \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4 : \u211d\ns : Set \u03b1\nx\u271d : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\nE : Set X\nx : X\n\u22a2 x \u2208 thickening \u03b4 E \u2194 \u2203 z, z \u2208 E \u2227 dist x z < \u03b4\n[PROOFSTEP]\nhave key_iff : \u2200 z : X, edist x z < ENNReal.ofReal \u03b4 \u2194 dist x z < \u03b4 := fun z \u21a6 by\n rw [dist_edist, lt_ofReal_iff_toReal_lt (edist_ne_top _ _)]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4 : \u211d\ns : Set \u03b1\nx\u271d : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\nE : Set X\nx z : X\n\u22a2 edist x z < ENNReal.ofReal \u03b4 \u2194 dist x z < \u03b4\n[PROOFSTEP]\nrw [dist_edist, lt_ofReal_iff_toReal_lt (edist_ne_top _ _)]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4 : \u211d\ns : Set \u03b1\nx\u271d : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\nE : Set X\nx : X\nkey_iff : \u2200 (z : X), edist x z < ENNReal.ofReal \u03b4 \u2194 dist x z < \u03b4\n\u22a2 x \u2208 thickening \u03b4 E \u2194 \u2203 z, z \u2208 E \u2227 dist x z < \u03b4\n[PROOFSTEP]\nsimp_rw [mem_thickening_iff_exists_edist_lt, key_iff]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx\u271d : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\n\u03b4 : \u211d\nx : X\n\u22a2 thickening \u03b4 {x} = ball x \u03b4\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx\u271d\u00b9 : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\n\u03b4 : \u211d\nx x\u271d : X\n\u22a2 x\u271d \u2208 thickening \u03b4 {x} \u2194 x\u271d \u2208 ball x \u03b4\n[PROOFSTEP]\nsimp [mem_thickening_iff]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx\u271d : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\nx : X\nE : Set X\nhx : x \u2208 E\n\u03b4 : \u211d\n\u22a2 ball x \u03b4 \u2286 thickening \u03b4 {x}\n[PROOFSTEP]\nsimp [Subset.rfl]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\n\u03b4 : \u211d\nE : Set X\n\u22a2 thickening \u03b4 E = \u22c3 (x : X) (_ : x \u2208 E), ball x \u03b4\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx\u271d : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\n\u03b4 : \u211d\nE : Set X\nx : X\n\u22a2 x \u2208 thickening \u03b4 E \u2194 x \u2208 \u22c3 (x : X) (_ : x \u2208 E), ball x \u03b4\n[PROOFSTEP]\nsimp only [mem_iUnion\u2082, exists_prop]\n[GOAL]\ncase h\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx\u271d : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\n\u03b4 : \u211d\nE : Set X\nx : X\n\u22a2 x \u2208 thickening \u03b4 E \u2194 \u2203 i, i \u2208 E \u2227 x \u2208 ball i \u03b4\n[PROOFSTEP]\nexact mem_thickening_iff\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\n\u03b4 : \u211d\nE : Set X\nh : Bounded E\n\u22a2 Bounded (thickening \u03b4 E)\n[PROOFSTEP]\nrcases E.eq_empty_or_nonempty with rfl | \u27e8x, hx\u27e9\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\n\u03b4 : \u211d\nh : Bounded \u2205\n\u22a2 Bounded (thickening \u03b4 \u2205)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx\u271d : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\n\u03b4 : \u211d\nE : Set X\nh : Bounded E\nx : X\nhx : x \u2208 E\n\u22a2 Bounded (thickening \u03b4 E)\n[PROOFSTEP]\nrefine (bounded_iff_subset_ball x).2 \u27e8\u03b4 + diam E, fun y hy \u21a6 ?_\u27e9\n[GOAL]\ncase inr.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx\u271d : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\n\u03b4 : \u211d\nE : Set X\nh : Bounded E\nx : X\nhx : x \u2208 E\ny : X\nhy : y \u2208 thickening \u03b4 E\n\u22a2 y \u2208 closedBall x (\u03b4 + diam E)\n[PROOFSTEP]\ncalc\n dist y x \u2264 infDist y E + diam E := dist_le_infDist_add_diam (x := y) h hx\n _ \u2264 \u03b4 + diam E := add_le_add_right ((mem_thickening_iff_infDist_lt \u27e8x, hx\u27e9).1 hy).le _\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx\u271d : \u03b1\nE : Set \u03b1\nx : \u03b1\nh : \u00acx \u2208 closure E\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d) in \ud835\udcdd 0, \u00acx \u2208 cthickening \u03b4 E\n[PROOFSTEP]\nobtain \u27e8\u03b5, \u27e8\u03b5_pos, \u03b5_lt\u27e9\u27e9 := exists_real_pos_lt_infEdist_of_not_mem_closure h\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5\u271d : \u211d\ns t : Set \u03b1\nx\u271d : \u03b1\nE : Set \u03b1\nx : \u03b1\nh : \u00acx \u2208 closure E\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\n\u03b5_lt : ENNReal.ofReal \u03b5 < infEdist x E\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d) in \ud835\udcdd 0, \u00acx \u2208 cthickening \u03b4 E\n[PROOFSTEP]\nfilter_upwards [eventually_lt_nhds \u03b5_pos] with \u03b4 h\u03b4\n[GOAL]\ncase h\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns t : Set \u03b1\nx\u271d : \u03b1\nE : Set \u03b1\nx : \u03b1\nh : \u00acx \u2208 closure E\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\n\u03b5_lt : ENNReal.ofReal \u03b5 < infEdist x E\n\u03b4 : \u211d\nh\u03b4 : \u03b4 < \u03b5\n\u22a2 \u00acx \u2208 cthickening \u03b4 E\n[PROOFSTEP]\nsimp only [cthickening, mem_setOf_eq, not_le]\n[GOAL]\ncase h\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns t : Set \u03b1\nx\u271d : \u03b1\nE : Set \u03b1\nx : \u03b1\nh : \u00acx \u2208 closure E\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\n\u03b5_lt : ENNReal.ofReal \u03b5 < infEdist x E\n\u03b4 : \u211d\nh\u03b4 : \u03b4 < \u03b5\n\u22a2 ENNReal.ofReal \u03b4 < infEdist x E\n[PROOFSTEP]\nexact ((ofReal_lt_ofReal_iff \u03b5_pos).mpr h\u03b4).trans \u03b5_lt\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nh : y \u2208 E\nh' : dist x y \u2264 \u03b4\n\u22a2 x \u2208 cthickening \u03b4 E\n[PROOFSTEP]\napply mem_cthickening_of_edist_le x y \u03b4 E h\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nh : y \u2208 E\nh' : dist x y \u2264 \u03b4\n\u22a2 edist x y \u2264 ENNReal.ofReal \u03b4\n[PROOFSTEP]\nrw [edist_dist]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx y : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nh : y \u2208 E\nh' : dist x y \u2264 \u03b4\n\u22a2 ENNReal.ofReal (dist x y) \u2264 ENNReal.ofReal \u03b4\n[PROOFSTEP]\nexact ENNReal.ofReal_le_ofReal h'\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\n\u22a2 cthickening \u03b4 \u2205 = \u2205\n[PROOFSTEP]\nsimp only [cthickening, ENNReal.ofReal_ne_top, setOf_false, infEdist_empty, top_le_iff]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nh\u03b4 : \u03b4 \u2264 0\nE : Set \u03b1\n\u22a2 cthickening \u03b4 E = closure E\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx\u271d : \u03b1\n\u03b4 : \u211d\nh\u03b4 : \u03b4 \u2264 0\nE : Set \u03b1\nx : \u03b1\n\u22a2 x \u2208 cthickening \u03b4 E \u2194 x \u2208 closure E\n[PROOFSTEP]\nsimp [mem_closure_iff_infEdist_zero, cthickening, ENNReal.ofReal_eq_zero.2 h\u03b4]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\n\u22a2 cthickening (max 0 \u03b4) E = cthickening \u03b4 E\n[PROOFSTEP]\ncases le_total \u03b4 0\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nh\u271d : \u03b4 \u2264 0\n\u22a2 cthickening (max 0 \u03b4) E = cthickening \u03b4 E\n[PROOFSTEP]\nsimp [cthickening_of_nonpos, *]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nh\u271d : 0 \u2264 \u03b4\n\u22a2 cthickening (max 0 \u03b4) E = cthickening \u03b4 E\n[PROOFSTEP]\nsimp [cthickening_of_nonpos, *]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u03b4 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 cthickening \u03b4 {x} = closedBall x \u03b4\n[PROOFSTEP]\next y\n[GOAL]\ncase h\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u03b4 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\ny : \u03b1\n\u22a2 y \u2208 cthickening \u03b4 {x} \u2194 y \u2208 closedBall x \u03b4\n[PROOFSTEP]\nsimp [cthickening, edist_dist, ENNReal.ofReal_le_ofReal_iff h\u03b4]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u03b4 : \u211d\n\u22a2 closedBall x \u03b4 \u2286 cthickening \u03b4 {x}\n[PROOFSTEP]\nrcases lt_or_le \u03b4 0 with (h\u03b4 | h\u03b4)\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u03b4 : \u211d\nh\u03b4 : \u03b4 < 0\n\u22a2 closedBall x \u03b4 \u2286 cthickening \u03b4 {x}\n[PROOFSTEP]\nsimp only [closedBall_eq_empty.mpr h\u03b4, empty_subset]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u03b4 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 closedBall x \u03b4 \u2286 cthickening \u03b4 {x}\n[PROOFSTEP]\nsimp only [cthickening_singleton x h\u03b4, Subset.rfl]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\n\u22a2 thickening \u03b4 E \u2286 cthickening \u03b4 E\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx\u271d : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nx : \u03b1\nhx : x \u2208 thickening \u03b4 E\n\u22a2 x \u2208 cthickening \u03b4 E\n[PROOFSTEP]\nrw [thickening, mem_setOf_eq] at hx \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx\u271d : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nx : \u03b1\nhx : infEdist x E < ENNReal.ofReal \u03b4\n\u22a2 x \u2208 cthickening \u03b4 E\n[PROOFSTEP]\nexact hx.le\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nh : Bounded E\n\u22a2 Bounded (Metric.cthickening \u03b4 E)\n[PROOFSTEP]\nhave : Bounded (thickening (max (\u03b4 + 1) 1) E) := h.thickening\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nh : Bounded E\nthis : Bounded (thickening (max (\u03b4 + 1) 1) E)\n\u22a2 Bounded (Metric.cthickening \u03b4 E)\n[PROOFSTEP]\napply Bounded.mono _ this\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nh : Bounded E\nthis : Bounded (thickening (max (\u03b4 + 1) 1) E)\n\u22a2 Metric.cthickening \u03b4 E \u2286 thickening (max (\u03b4 + 1) 1) E\n[PROOFSTEP]\nexact\n cthickening_subset_thickening' (zero_lt_one.trans_le (le_max_right _ _)) ((lt_add_one _).trans_le (le_max_left _ _)) _\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\n\u22a2 closure E \u2286 cthickening \u03b4 E\n[PROOFSTEP]\nrw [\u2190 cthickening_of_nonpos (min_le_right \u03b4 0)]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\n\u22a2 cthickening (min \u03b4 0) E \u2286 cthickening \u03b4 E\n[PROOFSTEP]\nexact cthickening_mono (min_le_left \u03b4 0) E\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nE : Set \u03b1\n\u22a2 closure E \u2286 thickening \u03b4 E\n[PROOFSTEP]\nrw [\u2190 cthickening_zero]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nE : Set \u03b1\n\u22a2 cthickening 0 E \u2286 thickening \u03b4 E\n[PROOFSTEP]\nexact cthickening_subset_thickening' \u03b4_pos \u03b4_pos E\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t\u271d : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\ns t : Set \u03b1\n\u22a2 thickening \u03b4 (s \u222a t) = thickening \u03b4 s \u222a thickening \u03b4 t\n[PROOFSTEP]\nsimp_rw [thickening, infEdist_union, inf_eq_min, min_lt_iff, setOf_or]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t\u271d : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\ns t : Set \u03b1\n\u22a2 cthickening \u03b4 (s \u222a t) = cthickening \u03b4 s \u222a cthickening \u03b4 t\n[PROOFSTEP]\nsimp_rw [cthickening, infEdist_union, inf_eq_min, min_le_iff, setOf_or]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nf : \u03b9 \u2192 Set \u03b1\n\u22a2 thickening \u03b4 (\u22c3 (i : \u03b9), f i) = \u22c3 (i : \u03b9), thickening \u03b4 (f i)\n[PROOFSTEP]\nsimp_rw [thickening, infEdist_iUnion, iInf_lt_iff, setOf_exists]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5\u271d : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b5 : \u211d\u22650\n\u22a2 EMetric.diam (cthickening (\u2191\u03b5) s) \u2264 EMetric.diam s + 2 * \u2191\u03b5\n[PROOFSTEP]\nrefine' diam_le fun x hx y hy => ENNReal.le_of_forall_pos_le_add fun \u03b4 h\u03b4 _ => _\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns t : Set \u03b1\nx\u271d\u00b9 : \u03b1\n\u03b5 : \u211d\u22650\nx : \u03b1\nhx : x \u2208 cthickening (\u2191\u03b5) s\ny : \u03b1\nhy : y \u2208 cthickening (\u2191\u03b5) s\n\u03b4 : \u211d\u22650\nh\u03b4 : 0 < \u03b4\nx\u271d : EMetric.diam s + 2 * \u2191\u03b5 < \u22a4\n\u22a2 edist x y \u2264 EMetric.diam s + 2 * \u2191\u03b5 + \u2191\u03b4\n[PROOFSTEP]\nrw [mem_cthickening_iff, ENNReal.ofReal_coe_nnreal] at hx hy \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns t : Set \u03b1\nx\u271d\u00b9 : \u03b1\n\u03b5 : \u211d\u22650\nx : \u03b1\nhx : infEdist x s \u2264 \u2191\u03b5\ny : \u03b1\nhy : infEdist y s \u2264 \u2191\u03b5\n\u03b4 : \u211d\u22650\nh\u03b4 : 0 < \u03b4\nx\u271d : EMetric.diam s + 2 * \u2191\u03b5 < \u22a4\n\u22a2 edist x y \u2264 EMetric.diam s + 2 * \u2191\u03b5 + \u2191\u03b4\n[PROOFSTEP]\nhave h\u03b5 : (\u03b5 : \u211d\u22650\u221e) < \u03b5 + \u03b4 := ENNReal.coe_lt_coe.2 (lt_add_of_pos_right _ h\u03b4)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns t : Set \u03b1\nx\u271d\u00b9 : \u03b1\n\u03b5 : \u211d\u22650\nx : \u03b1\nhx : infEdist x s \u2264 \u2191\u03b5\ny : \u03b1\nhy : infEdist y s \u2264 \u2191\u03b5\n\u03b4 : \u211d\u22650\nh\u03b4 : 0 < \u03b4\nx\u271d : EMetric.diam s + 2 * \u2191\u03b5 < \u22a4\nh\u03b5 : \u2191\u03b5 < \u2191\u03b5 + \u2191\u03b4\n\u22a2 edist x y \u2264 EMetric.diam s + 2 * \u2191\u03b5 + \u2191\u03b4\n[PROOFSTEP]\nreplace hx := hx.trans_lt h\u03b5\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns t : Set \u03b1\nx\u271d\u00b9 : \u03b1\n\u03b5 : \u211d\u22650\nx y : \u03b1\nhy : infEdist y s \u2264 \u2191\u03b5\n\u03b4 : \u211d\u22650\nh\u03b4 : 0 < \u03b4\nx\u271d : EMetric.diam s + 2 * \u2191\u03b5 < \u22a4\nh\u03b5 : \u2191\u03b5 < \u2191\u03b5 + \u2191\u03b4\nhx : infEdist x s < \u2191\u03b5 + \u2191\u03b4\n\u22a2 edist x y \u2264 EMetric.diam s + 2 * \u2191\u03b5 + \u2191\u03b4\n[PROOFSTEP]\nobtain \u27e8x', hx', hxx'\u27e9 := infEdist_lt_iff.mp hx\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns t : Set \u03b1\nx\u271d\u00b9 : \u03b1\n\u03b5 : \u211d\u22650\nx y : \u03b1\nhy : infEdist y s \u2264 \u2191\u03b5\n\u03b4 : \u211d\u22650\nh\u03b4 : 0 < \u03b4\nx\u271d : EMetric.diam s + 2 * \u2191\u03b5 < \u22a4\nh\u03b5 : \u2191\u03b5 < \u2191\u03b5 + \u2191\u03b4\nhx : infEdist x s < \u2191\u03b5 + \u2191\u03b4\nx' : \u03b1\nhx' : x' \u2208 s\nhxx' : edist x x' < \u2191\u03b5 + \u2191\u03b4\n\u22a2 edist x y \u2264 EMetric.diam s + 2 * \u2191\u03b5 + \u2191\u03b4\n[PROOFSTEP]\ncalc\n edist x y \u2264 edist x x' + edist y x' := edist_triangle_right _ _ _\n _ \u2264 \u03b5 + \u03b4 + (infEdist y s + EMetric.diam s) := (add_le_add hxx'.le (edist_le_infEdist_add_ediam hx'))\n _ \u2264 \u03b5 + \u03b4 + (\u03b5 + EMetric.diam s) := (add_le_add_left (add_le_add_right hy _) _)\n _ = _ := by rw [two_mul]; ac_rfl\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns t : Set \u03b1\nx\u271d\u00b9 : \u03b1\n\u03b5 : \u211d\u22650\nx y : \u03b1\nhy : infEdist y s \u2264 \u2191\u03b5\n\u03b4 : \u211d\u22650\nh\u03b4 : 0 < \u03b4\nx\u271d : EMetric.diam s + 2 * \u2191\u03b5 < \u22a4\nh\u03b5 : \u2191\u03b5 < \u2191\u03b5 + \u2191\u03b4\nhx : infEdist x s < \u2191\u03b5 + \u2191\u03b4\nx' : \u03b1\nhx' : x' \u2208 s\nhxx' : edist x x' < \u2191\u03b5 + \u2191\u03b4\n\u22a2 \u2191\u03b5 + \u2191\u03b4 + (\u2191\u03b5 + EMetric.diam s) = EMetric.diam s + 2 * \u2191\u03b5 + \u2191\u03b4\n[PROOFSTEP]\nrw [two_mul]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns t : Set \u03b1\nx\u271d\u00b9 : \u03b1\n\u03b5 : \u211d\u22650\nx y : \u03b1\nhy : infEdist y s \u2264 \u2191\u03b5\n\u03b4 : \u211d\u22650\nh\u03b4 : 0 < \u03b4\nx\u271d : EMetric.diam s + 2 * \u2191\u03b5 < \u22a4\nh\u03b5 : \u2191\u03b5 < \u2191\u03b5 + \u2191\u03b4\nhx : infEdist x s < \u2191\u03b5 + \u2191\u03b4\nx' : \u03b1\nhx' : x' \u2208 s\nhxx' : edist x x' < \u2191\u03b5 + \u2191\u03b4\n\u22a2 \u2191\u03b5 + \u2191\u03b4 + (\u2191\u03b5 + EMetric.diam s) = EMetric.diam s + (\u2191\u03b5 + \u2191\u03b5) + \u2191\u03b4\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns\u271d t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nh\u03b5 : 0 \u2264 \u03b5\n\u22a2 diam (cthickening \u03b5 s) \u2264 diam s + 2 * \u03b5\n[PROOFSTEP]\nlift \u03b5 to \u211d\u22650 using h\u03b5\n[GOAL]\ncase intro\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 : \u211d\ns\u271d t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\n\u03b5 : \u211d\u22650\n\u22a2 diam (cthickening (\u2191\u03b5) s) \u2264 diam s + 2 * \u2191\u03b5\n[PROOFSTEP]\nrefine (toReal_le_add' (ediam_cthickening_le _) ?_ ?_).trans_eq ?_\n[GOAL]\ncase intro.refine_1\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 : \u211d\ns\u271d t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\n\u03b5 : \u211d\u22650\n\u22a2 EMetric.diam s = \u22a4 \u2192 EMetric.diam (cthickening (\u2191\u03b5) s) = \u22a4\n[PROOFSTEP]\nexact fun h \u21a6 top_unique <| h \u25b8 EMetric.diam_mono (self_subset_cthickening _)\n[GOAL]\ncase intro.refine_2\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 : \u211d\ns\u271d t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\n\u03b5 : \u211d\u22650\n\u22a2 2 * \u2191\u03b5 = \u22a4 \u2192 EMetric.diam (cthickening (\u2191\u03b5) s) = \u22a4\n[PROOFSTEP]\nsimp [mul_eq_top]\n[GOAL]\ncase intro.refine_3\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 : \u211d\ns\u271d t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\n\u03b5 : \u211d\u22650\n\u22a2 ENNReal.toReal (EMetric.diam s) + ENNReal.toReal (2 * \u2191\u03b5) = diam s + 2 * \u2191\u03b5\n[PROOFSTEP]\nsimp [diam]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns\u271d t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nh\u03b5 : 0 \u2264 \u03b5\n\u22a2 diam (thickening \u03b5 s) \u2264 diam s + 2 * \u03b5\n[PROOFSTEP]\nby_cases hs : Bounded s\n[GOAL]\ncase pos\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns\u271d t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nh\u03b5 : 0 \u2264 \u03b5\nhs : Bounded s\n\u22a2 diam (thickening \u03b5 s) \u2264 diam s + 2 * \u03b5\n[PROOFSTEP]\nexact (diam_mono (thickening_subset_cthickening _ _) hs.cthickening).trans (diam_cthickening_le _ h\u03b5)\n[GOAL]\ncase neg\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns\u271d t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nh\u03b5 : 0 \u2264 \u03b5\nhs : \u00acBounded s\n\u22a2 diam (thickening \u03b5 s) \u2264 diam s + 2 * \u03b5\n[PROOFSTEP]\nobtain rfl | h\u03b5 := h\u03b5.eq_or_lt\n[GOAL]\ncase neg.inl\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 : \u211d\ns\u271d t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nhs : \u00acBounded s\nh\u03b5 : 0 \u2264 0\n\u22a2 diam (thickening 0 s) \u2264 diam s + 2 * 0\n[PROOFSTEP]\nsimp [thickening_of_nonpos, diam_nonneg]\n[GOAL]\ncase neg.inr\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns\u271d t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nh\u03b5\u271d : 0 \u2264 \u03b5\nhs : \u00acBounded s\nh\u03b5 : 0 < \u03b5\n\u22a2 diam (thickening \u03b5 s) \u2264 diam s + 2 * \u03b5\n[PROOFSTEP]\nrw [diam_eq_zero_of_unbounded (mt (Bounded.mono <| self_subset_thickening h\u03b5 _) hs)]\n[GOAL]\ncase neg.inr\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns\u271d t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\ns : Set \u03b1\nh\u03b5\u271d : 0 \u2264 \u03b5\nhs : \u00acBounded s\nh\u03b5 : 0 < \u03b5\n\u22a2 0 \u2264 diam s + 2 * \u03b5\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u22a2 thickening \u03b4 (closure s) = thickening \u03b4 s\n[PROOFSTEP]\nsimp_rw [thickening, infEdist_closure]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u22a2 cthickening \u03b4 (closure s) = cthickening \u03b4 s\n[PROOFSTEP]\nsimp_rw [cthickening, infEdist_closure]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nhst : Disjoint s t\nhs : IsCompact s\nht : IsClosed t\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 Disjoint (thickening \u03b4 s) (thickening \u03b4 t)\n[PROOFSTEP]\nobtain \u27e8r, hr, h\u27e9 := exists_pos_forall_lt_edist hs ht hst\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nhst : Disjoint s t\nhs : IsCompact s\nht : IsClosed t\nr : \u211d\u22650\nhr : 0 < r\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 Disjoint (thickening \u03b4 s) (thickening \u03b4 t)\n[PROOFSTEP]\nrefine' \u27e8r / 2, half_pos (NNReal.coe_pos.2 hr), _\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nhst : Disjoint s t\nhs : IsCompact s\nht : IsClosed t\nr : \u211d\u22650\nhr : 0 < r\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\n\u22a2 Disjoint (thickening (\u2191r / 2) s) (thickening (\u2191r / 2) t)\n[PROOFSTEP]\nrw [disjoint_iff_inf_le]\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nhst : Disjoint s t\nhs : IsCompact s\nht : IsClosed t\nr : \u211d\u22650\nhr : 0 < r\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\n\u22a2 thickening (\u2191r / 2) s \u2293 thickening (\u2191r / 2) t \u2264 \u22a5\n[PROOFSTEP]\nrintro z \u27e8hzs, hzt\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nhst : Disjoint s t\nhs : IsCompact s\nht : IsClosed t\nr : \u211d\u22650\nhr : 0 < r\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\nz : \u03b1\nhzs : z \u2208 thickening (\u2191r / 2) s\nhzt : z \u2208 thickening (\u2191r / 2) t\n\u22a2 z \u2208 \u22a5\n[PROOFSTEP]\nrw [mem_thickening_iff_exists_edist_lt] at hzs hzt \n[GOAL]\ncase intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nhst : Disjoint s t\nhs : IsCompact s\nht : IsClosed t\nr : \u211d\u22650\nhr : 0 < r\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\nz : \u03b1\nhzs : \u2203 z_1, z_1 \u2208 s \u2227 edist z z_1 < ENNReal.ofReal (\u2191r / 2)\nhzt : \u2203 z_1, z_1 \u2208 t \u2227 edist z z_1 < ENNReal.ofReal (\u2191r / 2)\n\u22a2 z \u2208 \u22a5\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_two, \u2190 NNReal.coe_div, ENNReal.ofReal_coe_nnreal] at hzs hzt \n[GOAL]\ncase intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nhst : Disjoint s t\nhs : IsCompact s\nht : IsClosed t\nr : \u211d\u22650\nhr : 0 < r\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\nz : \u03b1\nhzs : \u2203 z_1, z_1 \u2208 s \u2227 edist z z_1 < \u2191(r / 2)\nhzt : \u2203 z_1, z_1 \u2208 t \u2227 edist z z_1 < \u2191(r / 2)\n\u22a2 z \u2208 \u22a5\n[PROOFSTEP]\nobtain \u27e8x, hx, hzx\u27e9 := hzs\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx\u271d : \u03b1\nhst : Disjoint s t\nhs : IsCompact s\nht : IsClosed t\nr : \u211d\u22650\nhr : 0 < r\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\nz : \u03b1\nhzt : \u2203 z_1, z_1 \u2208 t \u2227 edist z z_1 < \u2191(r / 2)\nx : \u03b1\nhx : x \u2208 s\nhzx : edist z x < \u2191(r / 2)\n\u22a2 z \u2208 \u22a5\n[PROOFSTEP]\nobtain \u27e8y, hy, hzy\u27e9 := hzt\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx\u271d : \u03b1\nhst : Disjoint s t\nhs : IsCompact s\nht : IsClosed t\nr : \u211d\u22650\nhr : 0 < r\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\nz x : \u03b1\nhx : x \u2208 s\nhzx : edist z x < \u2191(r / 2)\ny : \u03b1\nhy : y \u2208 t\nhzy : edist z y < \u2191(r / 2)\n\u22a2 z \u2208 \u22a5\n[PROOFSTEP]\nrefine' (h x hx y hy).not_le _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx\u271d : \u03b1\nhst : Disjoint s t\nhs : IsCompact s\nht : IsClosed t\nr : \u211d\u22650\nhr : 0 < r\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\nz x : \u03b1\nhx : x \u2208 s\nhzx : edist z x < \u2191(r / 2)\ny : \u03b1\nhy : y \u2208 t\nhzy : edist z y < \u2191(r / 2)\n\u22a2 edist x y \u2264 \u2191r\n[PROOFSTEP]\ncalc\n edist x y \u2264 edist z x + edist z y := edist_triangle_left _ _ _\n _ \u2264 \u2191(r / 2) + \u2191(r / 2) := (add_le_add hzx.le hzy.le)\n _ = r := by rw [\u2190 ENNReal.coe_add, add_halves]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx\u271d : \u03b1\nhst : Disjoint s t\nhs : IsCompact s\nht : IsClosed t\nr : \u211d\u22650\nhr : 0 < r\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2191r < edist x y\nz x : \u03b1\nhx : x \u2208 s\nhzx : edist z x < \u2191(r / 2)\ny : \u03b1\nhy : y \u2208 t\nhzy : edist z y < \u2191(r / 2)\n\u22a2 \u2191(r / 2) + \u2191(r / 2) = \u2191r\n[PROOFSTEP]\nrw [\u2190 ENNReal.coe_add, add_halves]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nhst : Disjoint s t\nhs : IsCompact s\nht : IsClosed t\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 Disjoint (cthickening \u03b4 s) (cthickening \u03b4 t)\n[PROOFSTEP]\nobtain \u27e8\u03b4, h\u03b4, h\u27e9 := hst.exists_thickenings hs ht\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nhst : Disjoint s t\nhs : IsCompact s\nht : IsClosed t\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : Disjoint (thickening \u03b4 s) (thickening \u03b4 t)\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 Disjoint (cthickening \u03b4 s) (cthickening \u03b4 t)\n[PROOFSTEP]\nrefine' \u27e8\u03b4 / 2, half_pos h\u03b4, h.mono _ _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nhst : Disjoint s t\nhs : IsCompact s\nht : IsClosed t\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : Disjoint (thickening \u03b4 s) (thickening \u03b4 t)\n\u22a2 cthickening (\u03b4 / 2) s \u2264 thickening \u03b4 s\n[PROOFSTEP]\nexact cthickening_subset_thickening' h\u03b4 (half_lt_self h\u03b4) _\n[GOAL]\ncase intro.intro.refine'_2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nhst : Disjoint s t\nhs : IsCompact s\nht : IsClosed t\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : Disjoint (thickening \u03b4 s) (thickening \u03b4 t)\n\u22a2 cthickening (\u03b4 / 2) t \u2264 thickening \u03b4 t\n[PROOFSTEP]\nexact cthickening_subset_thickening' h\u03b4 (half_lt_self h\u03b4) _\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nhs : IsCompact s\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 IsCompact (cthickening \u03b4 s)\n[PROOFSTEP]\nrcases exists_compact_superset hs with \u27e8K, K_compact, hK\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nhs : IsCompact s\nK : Set \u03b1\nK_compact : IsCompact K\nhK : s \u2286 interior K\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 IsCompact (cthickening \u03b4 s)\n[PROOFSTEP]\nrcases hs.exists_cthickening_subset_open isOpen_interior hK with \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nhs : IsCompact s\nK : Set \u03b1\nK_compact : IsCompact K\nhK : s \u2286 interior K\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : cthickening \u03b4 s \u2286 interior K\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 IsCompact (cthickening \u03b4 s)\n[PROOFSTEP]\nrefine \u27e8\u03b4, \u03b4pos, ?_\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nhs : IsCompact s\nK : Set \u03b1\nK_compact : IsCompact K\nhK : s \u2286 interior K\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : cthickening \u03b4 s \u2286 interior K\n\u22a2 IsCompact (cthickening \u03b4 s)\n[PROOFSTEP]\nexact isCompact_of_isClosed_subset K_compact isClosed_cthickening (h\u03b4.trans interior_subset)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\n\u22a2 cthickening \u03b4 E = \u22c2 (\u03b5 : \u211d) (_ : \u03b5 \u2208 s), cthickening \u03b5 E\n[PROOFSTEP]\napply Subset.antisymm\n[GOAL]\ncase h\u2081\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\n\u22a2 cthickening \u03b4 E \u2286 \u22c2 (\u03b5 : \u211d) (_ : \u03b5 \u2208 s), cthickening \u03b5 E\n[PROOFSTEP]\nexact subset_iInter\u2082 fun _ h\u03b5 => cthickening_mono (le_of_lt (hs\u03b4 h\u03b5)) E\n[GOAL]\ncase h\u2082\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\n\u22a2 \u22c2 (\u03b5 : \u211d) (_ : \u03b5 \u2208 s), cthickening \u03b5 E \u2286 cthickening \u03b4 E\n[PROOFSTEP]\nunfold cthickening\n[GOAL]\ncase h\u2082\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\n\u22a2 \u22c2 (\u03b5 : \u211d) (_ : \u03b5 \u2208 s), {x | infEdist x E \u2264 ENNReal.ofReal \u03b5} \u2286 {x | infEdist x E \u2264 ENNReal.ofReal \u03b4}\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\u2082\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\n\u03b4 : \u211d\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\nx : \u03b1\nhx : x \u2208 \u22c2 (\u03b5 : \u211d) (_ : \u03b5 \u2208 s), {x | infEdist x E \u2264 ENNReal.ofReal \u03b5}\n\u22a2 x \u2208 {x | infEdist x E \u2264 ENNReal.ofReal \u03b4}\n[PROOFSTEP]\nsimp only [mem_iInter, mem_setOf_eq] at *\n[GOAL]\ncase h\u2082\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\n\u03b4 : \u211d\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\nx : \u03b1\nhx : \u2200 (i : \u211d), i \u2208 s \u2192 infEdist x E \u2264 ENNReal.ofReal i\n\u22a2 infEdist x E \u2264 ENNReal.ofReal \u03b4\n[PROOFSTEP]\napply ENNReal.le_of_forall_pos_le_add\n[GOAL]\ncase h\u2082.h\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\n\u03b4 : \u211d\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\nx : \u03b1\nhx : \u2200 (i : \u211d), i \u2208 s \u2192 infEdist x E \u2264 ENNReal.ofReal i\n\u22a2 \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 ENNReal.ofReal \u03b4 < \u22a4 \u2192 infEdist x E \u2264 ENNReal.ofReal \u03b4 + \u2191\u03b5\n[PROOFSTEP]\nintro \u03b7 \u03b7_pos _\n[GOAL]\ncase h\u2082.h\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\n\u03b4 : \u211d\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\nx : \u03b1\nhx : \u2200 (i : \u211d), i \u2208 s \u2192 infEdist x E \u2264 ENNReal.ofReal i\n\u03b7 : \u211d\u22650\n\u03b7_pos : 0 < \u03b7\na\u271d : ENNReal.ofReal \u03b4 < \u22a4\n\u22a2 infEdist x E \u2264 ENNReal.ofReal \u03b4 + \u2191\u03b7\n[PROOFSTEP]\nrcases hs (\u03b4 + \u03b7) (lt_add_of_pos_right _ (NNReal.coe_pos.mpr \u03b7_pos)) with \u27e8\u03b5, \u27e8hs\u03b5, h\u03b5\u27e9\u27e9\n[GOAL]\ncase h\u2082.h.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\n\u03b4 : \u211d\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\nx : \u03b1\nhx : \u2200 (i : \u211d), i \u2208 s \u2192 infEdist x E \u2264 ENNReal.ofReal i\n\u03b7 : \u211d\u22650\n\u03b7_pos : 0 < \u03b7\na\u271d : ENNReal.ofReal \u03b4 < \u22a4\n\u03b5 : \u211d\nhs\u03b5 : \u03b5 \u2208 s\nh\u03b5 : \u03b5 \u2208 Ioc \u03b4 (\u03b4 + \u2191\u03b7)\n\u22a2 infEdist x E \u2264 ENNReal.ofReal \u03b4 + \u2191\u03b7\n[PROOFSTEP]\napply ((hx \u03b5 hs\u03b5).trans (ENNReal.ofReal_le_ofReal h\u03b5.2)).trans\n[GOAL]\ncase h\u2082.h.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\n\u03b4 : \u211d\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\nx : \u03b1\nhx : \u2200 (i : \u211d), i \u2208 s \u2192 infEdist x E \u2264 ENNReal.ofReal i\n\u03b7 : \u211d\u22650\n\u03b7_pos : 0 < \u03b7\na\u271d : ENNReal.ofReal \u03b4 < \u22a4\n\u03b5 : \u211d\nhs\u03b5 : \u03b5 \u2208 s\nh\u03b5 : \u03b5 \u2208 Ioc \u03b4 (\u03b4 + \u2191\u03b7)\n\u22a2 ENNReal.ofReal (\u03b4 + \u2191\u03b7) \u2264 ENNReal.ofReal \u03b4 + \u2191\u03b7\n[PROOFSTEP]\nrw [ENNReal.coe_nnreal_eq \u03b7]\n[GOAL]\ncase h\u2082.h.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\n\u03b4 : \u211d\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\nx : \u03b1\nhx : \u2200 (i : \u211d), i \u2208 s \u2192 infEdist x E \u2264 ENNReal.ofReal i\n\u03b7 : \u211d\u22650\n\u03b7_pos : 0 < \u03b7\na\u271d : ENNReal.ofReal \u03b4 < \u22a4\n\u03b5 : \u211d\nhs\u03b5 : \u03b5 \u2208 s\nh\u03b5 : \u03b5 \u2208 Ioc \u03b4 (\u03b4 + \u2191\u03b7)\n\u22a2 ENNReal.ofReal (\u03b4 + \u2191\u03b7) \u2264 ENNReal.ofReal \u03b4 + ENNReal.ofReal \u2191\u03b7\n[PROOFSTEP]\nexact ENNReal.ofReal_add_le\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\n\u22a2 cthickening \u03b4 E = \u22c2 (\u03b5 : \u211d) (_ : \u03b4 < \u03b5), cthickening \u03b5 E\n[PROOFSTEP]\napply cthickening_eq_iInter_cthickening' (Ioi \u03b4) rfl.subset\n[GOAL]\ncase hs\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (Ioi \u03b4 \u2229 Ioc \u03b4 \u03b5)\n[PROOFSTEP]\nsimp_rw [inter_eq_right_iff_subset.mpr Ioc_subset_Ioi_self]\n[GOAL]\ncase hs\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (Ioc \u03b4 \u03b5)\n[PROOFSTEP]\nexact fun _ h\u03b5 => nonempty_Ioc.mpr h\u03b5\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\n\u03b4_nn : 0 \u2264 \u03b4\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\n\u22a2 cthickening \u03b4 E = \u22c2 (\u03b5 : \u211d) (_ : \u03b5 \u2208 s), thickening \u03b5 E\n[PROOFSTEP]\nrefine' (subset_iInter\u2082 fun \u03b5 h\u03b5 => _).antisymm _\n[GOAL]\ncase refine'_1\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\n\u03b4_nn : 0 \u2264 \u03b4\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 \u2208 s\n\u22a2 cthickening \u03b4 E \u2286 thickening \u03b5 E\n[PROOFSTEP]\nobtain \u27e8\u03b5', -, h\u03b5'\u27e9 := hs \u03b5 (hs\u03b4 h\u03b5)\n[GOAL]\ncase refine'_1.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\n\u03b4_nn : 0 \u2264 \u03b4\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 \u2208 s\n\u03b5' : \u211d\nh\u03b5' : \u03b5' \u2208 Ioc \u03b4 \u03b5\n\u22a2 cthickening \u03b4 E \u2286 thickening \u03b5 E\n[PROOFSTEP]\nhave ss := cthickening_subset_thickening' (lt_of_le_of_lt \u03b4_nn h\u03b5'.1) h\u03b5'.1 E\n[GOAL]\ncase refine'_1.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\n\u03b4_nn : 0 \u2264 \u03b4\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 \u2208 s\n\u03b5' : \u211d\nh\u03b5' : \u03b5' \u2208 Ioc \u03b4 \u03b5\nss : cthickening \u03b4 E \u2286 thickening \u03b5' E\n\u22a2 cthickening \u03b4 E \u2286 thickening \u03b5 E\n[PROOFSTEP]\nexact ss.trans (thickening_mono h\u03b5'.2 E)\n[GOAL]\ncase refine'_2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\n\u03b4_nn : 0 \u2264 \u03b4\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\n\u22a2 \u22c2 (i : \u211d) (_ : i \u2208 s), thickening i E \u2286 cthickening \u03b4 E\n[PROOFSTEP]\nrw [cthickening_eq_iInter_cthickening' s hs\u03b4 hs E]\n[GOAL]\ncase refine'_2\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\n\u03b4_nn : 0 \u2264 \u03b4\ns : Set \u211d\nhs\u03b4 : s \u2286 Ioi \u03b4\nhs : \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc \u03b4 \u03b5)\nE : Set \u03b1\n\u22a2 \u22c2 (i : \u211d) (_ : i \u2208 s), thickening i E \u2286 \u22c2 (\u03b5 : \u211d) (_ : \u03b5 \u2208 s), cthickening \u03b5 E\n[PROOFSTEP]\nexact iInter\u2082_mono fun \u03b5 _ => thickening_subset_cthickening \u03b5 E\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\n\u03b4_nn : 0 \u2264 \u03b4\nE : Set \u03b1\n\u22a2 cthickening \u03b4 E = \u22c2 (\u03b5 : \u211d) (_ : \u03b4 < \u03b5), thickening \u03b5 E\n[PROOFSTEP]\napply cthickening_eq_iInter_thickening' \u03b4_nn (Ioi \u03b4) rfl.subset\n[GOAL]\ncase hs\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\n\u03b4_nn : 0 \u2264 \u03b4\nE : Set \u03b1\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (Ioi \u03b4 \u2229 Ioc \u03b4 \u03b5)\n[PROOFSTEP]\nsimp_rw [inter_eq_right_iff_subset.mpr Ioc_subset_Ioi_self]\n[GOAL]\ncase hs\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\n\u03b4_nn : 0 \u2264 \u03b4\nE : Set \u03b1\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b4 < \u03b5 \u2192 Set.Nonempty (Ioc \u03b4 \u03b5)\n[PROOFSTEP]\nexact fun _ h\u03b5 => nonempty_Ioc.mpr h\u03b5\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\n\u22a2 cthickening \u03b4 E = \u22c2 (\u03b5 : \u211d) (_ : max 0 \u03b4 < \u03b5), thickening \u03b5 E\n[PROOFSTEP]\nrw [\u2190 cthickening_max_zero, cthickening_eq_iInter_thickening]\n[GOAL]\ncase \u03b4_nn\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\n\u22a2 0 \u2264 max 0 \u03b4\n[PROOFSTEP]\nexact le_max_left _ _\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\nE : Set \u03b1\ns : Set \u211d\nhs : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc 0 \u03b5)\n\u22a2 closure E = \u22c2 (\u03b4 : \u211d) (_ : \u03b4 \u2208 s), cthickening \u03b4 E\n[PROOFSTEP]\nby_cases hs\u2080 : s \u2286 Ioi 0\n[GOAL]\ncase pos\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\nE : Set \u03b1\ns : Set \u211d\nhs : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc 0 \u03b5)\nhs\u2080 : s \u2286 Ioi 0\n\u22a2 closure E = \u22c2 (\u03b4 : \u211d) (_ : \u03b4 \u2208 s), cthickening \u03b4 E\n[PROOFSTEP]\nrw [\u2190 cthickening_zero]\n[GOAL]\ncase pos\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\nE : Set \u03b1\ns : Set \u211d\nhs : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc 0 \u03b5)\nhs\u2080 : s \u2286 Ioi 0\n\u22a2 cthickening 0 E = \u22c2 (\u03b4 : \u211d) (_ : \u03b4 \u2208 s), cthickening \u03b4 E\n[PROOFSTEP]\napply cthickening_eq_iInter_cthickening' _ hs\u2080 hs\n[GOAL]\ncase neg\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\nE : Set \u03b1\ns : Set \u211d\nhs : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc 0 \u03b5)\nhs\u2080 : \u00acs \u2286 Ioi 0\n\u22a2 closure E = \u22c2 (\u03b4 : \u211d) (_ : \u03b4 \u2208 s), cthickening \u03b4 E\n[PROOFSTEP]\nobtain \u27e8\u03b4, h\u03b4s, \u03b4_nonpos\u27e9 := not_subset.mp hs\u2080\n[GOAL]\ncase neg.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\nE : Set \u03b1\ns : Set \u211d\nhs : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc 0 \u03b5)\nhs\u2080 : \u00acs \u2286 Ioi 0\n\u03b4 : \u211d\nh\u03b4s : \u03b4 \u2208 s\n\u03b4_nonpos : \u00ac\u03b4 \u2208 Ioi 0\n\u22a2 closure E = \u22c2 (\u03b4 : \u211d) (_ : \u03b4 \u2208 s), cthickening \u03b4 E\n[PROOFSTEP]\nrw [Set.mem_Ioi, not_lt] at \u03b4_nonpos \n[GOAL]\ncase neg.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\nE : Set \u03b1\ns : Set \u211d\nhs : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc 0 \u03b5)\nhs\u2080 : \u00acs \u2286 Ioi 0\n\u03b4 : \u211d\nh\u03b4s : \u03b4 \u2208 s\n\u03b4_nonpos : \u03b4 \u2264 0\n\u22a2 closure E = \u22c2 (\u03b4 : \u211d) (_ : \u03b4 \u2208 s), cthickening \u03b4 E\n[PROOFSTEP]\napply Subset.antisymm\n[GOAL]\ncase neg.intro.intro.h\u2081\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\nE : Set \u03b1\ns : Set \u211d\nhs : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc 0 \u03b5)\nhs\u2080 : \u00acs \u2286 Ioi 0\n\u03b4 : \u211d\nh\u03b4s : \u03b4 \u2208 s\n\u03b4_nonpos : \u03b4 \u2264 0\n\u22a2 closure E \u2286 \u22c2 (\u03b4 : \u211d) (_ : \u03b4 \u2208 s), cthickening \u03b4 E\n[PROOFSTEP]\nexact subset_iInter\u2082 fun \u03b5 _ => closure_subset_cthickening \u03b5 E\n[GOAL]\ncase neg.intro.intro.h\u2082\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\nE : Set \u03b1\ns : Set \u211d\nhs : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc 0 \u03b5)\nhs\u2080 : \u00acs \u2286 Ioi 0\n\u03b4 : \u211d\nh\u03b4s : \u03b4 \u2208 s\n\u03b4_nonpos : \u03b4 \u2264 0\n\u22a2 \u22c2 (\u03b4 : \u211d) (_ : \u03b4 \u2208 s), cthickening \u03b4 E \u2286 closure E\n[PROOFSTEP]\nrw [\u2190 cthickening_of_nonpos \u03b4_nonpos E]\n[GOAL]\ncase neg.intro.intro.h\u2082\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\nE : Set \u03b1\ns : Set \u211d\nhs : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc 0 \u03b5)\nhs\u2080 : \u00acs \u2286 Ioi 0\n\u03b4 : \u211d\nh\u03b4s : \u03b4 \u2208 s\n\u03b4_nonpos : \u03b4 \u2264 0\n\u22a2 \u22c2 (\u03b4 : \u211d) (_ : \u03b4 \u2208 s), cthickening \u03b4 E \u2286 cthickening \u03b4 E\n[PROOFSTEP]\nexact biInter_subset_of_mem h\u03b4s\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nE : Set \u03b1\n\u22a2 closure E = \u22c2 (\u03b4 : \u211d) (_ : 0 < \u03b4), cthickening \u03b4 E\n[PROOFSTEP]\nrw [\u2190 cthickening_zero]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nE : Set \u03b1\n\u22a2 cthickening 0 E = \u22c2 (\u03b4 : \u211d) (_ : 0 < \u03b4), cthickening \u03b4 E\n[PROOFSTEP]\nexact cthickening_eq_iInter_cthickening E\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\nE : Set \u03b1\ns : Set \u211d\nhs\u2080 : s \u2286 Ioi 0\nhs : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc 0 \u03b5)\n\u22a2 closure E = \u22c2 (\u03b4 : \u211d) (_ : \u03b4 \u2208 s), thickening \u03b4 E\n[PROOFSTEP]\nrw [\u2190 cthickening_zero]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\nE : Set \u03b1\ns : Set \u211d\nhs\u2080 : s \u2286 Ioi 0\nhs : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 Set.Nonempty (s \u2229 Ioc 0 \u03b5)\n\u22a2 cthickening 0 E = \u22c2 (\u03b4 : \u211d) (_ : \u03b4 \u2208 s), thickening \u03b4 E\n[PROOFSTEP]\napply cthickening_eq_iInter_thickening' le_rfl _ hs\u2080 hs\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nE : Set \u03b1\n\u22a2 closure E = \u22c2 (\u03b4 : \u211d) (_ : 0 < \u03b4), thickening \u03b4 E\n[PROOFSTEP]\nrw [\u2190 cthickening_zero]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nE : Set \u03b1\n\u22a2 cthickening 0 E = \u22c2 (\u03b4 : \u211d) (_ : 0 < \u03b4), thickening \u03b4 E\n[PROOFSTEP]\nexact cthickening_eq_iInter_thickening rfl.ge E\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\nE : Set \u03b1\nhx : x \u2208 E\n\u03b4 : \u211d\n\u22a2 closedBall x \u03b4 \u2286 cthickening \u03b4 E\n[PROOFSTEP]\nrefine' (closedBall_subset_cthickening_singleton _ _).trans (cthickening_subset_of_subset _ _)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\nE : Set \u03b1\nhx : x \u2208 E\n\u03b4 : \u211d\n\u22a2 {x} \u2286 E\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nE : Set \u03b1\n\u03b4 \u03b4' : \u211d\nh\u03b4\u2080 : 0 < \u03b4'\nh\u03b4\u03b4' : \u03b4 < \u03b4'\n\u22a2 cthickening \u03b4 E \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 E), closedBall x \u03b4'\n[PROOFSTEP]\nrefine' (cthickening_subset_thickening' h\u03b4\u2080 h\u03b4\u03b4' E).trans fun x hx => _\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nE : Set \u03b1\n\u03b4 \u03b4' : \u211d\nh\u03b4\u2080 : 0 < \u03b4'\nh\u03b4\u03b4' : \u03b4 < \u03b4'\nx : \u03b1\nhx : x \u2208 thickening \u03b4' E\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 E), closedBall x \u03b4'\n[PROOFSTEP]\nobtain \u27e8y, hy\u2081, hy\u2082\u27e9 := mem_thickening_iff.mp hx\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nE : Set \u03b1\n\u03b4 \u03b4' : \u211d\nh\u03b4\u2080 : 0 < \u03b4'\nh\u03b4\u03b4' : \u03b4 < \u03b4'\nx : \u03b1\nhx : x \u2208 thickening \u03b4' E\ny : \u03b1\nhy\u2081 : y \u2208 E\nhy\u2082 : dist x y < \u03b4'\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 E), closedBall x \u03b4'\n[PROOFSTEP]\nexact mem_iUnion\u2082.mpr \u27e8y, hy\u2081, hy\u2082.le\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nhE : IsCompact E\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 cthickening \u03b4 E = \u22c3 (x : \u03b1) (_ : x \u2208 E), closedBall x \u03b4\n[PROOFSTEP]\nrcases eq_empty_or_nonempty E with (rfl | hne)\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b4 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nhE : IsCompact \u2205\n\u22a2 cthickening \u03b4 \u2205 = \u22c3 (x : \u03b1) (_ : x \u2208 \u2205), closedBall x \u03b4\n[PROOFSTEP]\nsimp only [cthickening_empty, biUnion_empty]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nhE : IsCompact E\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\n\u22a2 cthickening \u03b4 E = \u22c3 (x : \u03b1) (_ : x \u2208 E), closedBall x \u03b4\n[PROOFSTEP]\nrefine Subset.antisymm (fun x hx \u21a6 ?_) (iUnion\u2082_subset fun x hx \u21a6 closedBall_subset_cthickening hx _)\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nhE : IsCompact E\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\nx : \u03b1\nhx : x \u2208 cthickening \u03b4 E\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 E), closedBall x \u03b4\n[PROOFSTEP]\nobtain \u27e8y, yE, hy\u27e9 : \u2203 y \u2208 E, infEdist x E = edist x y := hE.exists_infEdist_eq_edist hne _\n[GOAL]\ncase inr.intro.intro\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nhE : IsCompact E\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\nx : \u03b1\nhx : x \u2208 cthickening \u03b4 E\ny : \u03b1\nyE : y \u2208 E\nhy : infEdist x E = edist x y\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 E), closedBall x \u03b4\n[PROOFSTEP]\nhave D1 : edist x y \u2264 ENNReal.ofReal \u03b4 := (le_of_eq hy.symm).trans hx\n[GOAL]\ncase inr.intro.intro\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nhE : IsCompact E\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\nx : \u03b1\nhx : x \u2208 cthickening \u03b4 E\ny : \u03b1\nyE : y \u2208 E\nhy : infEdist x E = edist x y\nD1 : edist x y \u2264 ENNReal.ofReal \u03b4\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 E), closedBall x \u03b4\n[PROOFSTEP]\nhave D2 : dist x y \u2264 \u03b4 := by\n rw [edist_dist] at D1 \n exact (ENNReal.ofReal_le_ofReal_iff h\u03b4).1 D1\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nhE : IsCompact E\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\nx : \u03b1\nhx : x \u2208 cthickening \u03b4 E\ny : \u03b1\nyE : y \u2208 E\nhy : infEdist x E = edist x y\nD1 : edist x y \u2264 ENNReal.ofReal \u03b4\n\u22a2 dist x y \u2264 \u03b4\n[PROOFSTEP]\nrw [edist_dist] at D1 \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nhE : IsCompact E\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\nx : \u03b1\nhx : x \u2208 cthickening \u03b4 E\ny : \u03b1\nyE : y \u2208 E\nhy : infEdist x E = edist x y\nD1 : ENNReal.ofReal (dist x y) \u2264 ENNReal.ofReal \u03b4\n\u22a2 dist x y \u2264 \u03b4\n[PROOFSTEP]\nexact (ENNReal.ofReal_le_ofReal_iff h\u03b4).1 D1\n[GOAL]\ncase inr.intro.intro\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\u271d\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\nhE : IsCompact E\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\nx : \u03b1\nhx : x \u2208 cthickening \u03b4 E\ny : \u03b1\nyE : y \u2208 E\nhy : infEdist x E = edist x y\nD1 : edist x y \u2264 ENNReal.ofReal \u03b4\nD2 : dist x y \u2264 \u03b4\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 E), closedBall x \u03b4\n[PROOFSTEP]\nexact mem_biUnion yE D2\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 cthickening \u03b4 E = \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4\n[PROOFSTEP]\nrcases eq_empty_or_nonempty E with (rfl | hne)\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 cthickening \u03b4 \u2205 = \u22c3 (x : \u03b1) (_ : x \u2208 closure \u2205), closedBall x \u03b4\n[PROOFSTEP]\nsimp only [cthickening_empty, biUnion_empty, closure_empty]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\n\u22a2 cthickening \u03b4 E = \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4\n[PROOFSTEP]\nrw [\u2190 cthickening_closure]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\n\u22a2 cthickening \u03b4 (closure E) = \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4\n[PROOFSTEP]\nrefine Subset.antisymm (fun x hx \u21a6 ?_) (iUnion\u2082_subset fun x hx \u21a6 closedBall_subset_cthickening hx _)\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\nx : \u03b1\nhx : x \u2208 cthickening \u03b4 (closure E)\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4\n[PROOFSTEP]\nobtain \u27e8y, yE, hy\u27e9 : \u2203 y \u2208 closure E, infDist x (closure E) = dist x y :=\n isClosed_closure.exists_infDist_eq_dist (closure_nonempty_iff.mpr hne) x\n[GOAL]\ncase inr.intro.intro\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\nx : \u03b1\nhx : x \u2208 cthickening \u03b4 (closure E)\ny : \u03b1\nyE : y \u2208 closure E\nhy : infDist x (closure E) = dist x y\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4\n[PROOFSTEP]\nreplace hy : dist x y \u2264 \u03b4 :=\n (ENNReal.ofReal_le_ofReal_iff h\u03b4).mp (((congr_arg ENNReal.ofReal hy.symm).le.trans ENNReal.ofReal_toReal_le).trans hx)\n[GOAL]\ncase inr.intro.intro\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\nx : \u03b1\nhx : x \u2208 cthickening \u03b4 (closure E)\ny : \u03b1\nyE : y \u2208 closure E\nhy : dist x y \u2264 \u03b4\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4\n[PROOFSTEP]\nexact mem_biUnion yE hy\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nhE : IsClosed E\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 cthickening \u03b4 E = \u22c3 (x : \u03b1) (_ : x \u2208 E), closedBall x \u03b4\n[PROOFSTEP]\nrw [cthickening_eq_biUnion_closedBall E h\u03b4, hE.closure_eq]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u22a2 infEdist x s \u2264 infEdist x (cthickening \u03b4 s) + ENNReal.ofReal \u03b4\n[PROOFSTEP]\nrefine' le_of_forall_lt' fun r h => _\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nr : \u211d\u22650\u221e\nh : infEdist x (cthickening \u03b4 s) + ENNReal.ofReal \u03b4 < r\n\u22a2 infEdist x s < r\n[PROOFSTEP]\nsimp_rw [\u2190 lt_tsub_iff_right, infEdist_lt_iff, mem_cthickening_iff] at h \n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nr : \u211d\u22650\u221e\nh : \u2203 y, infEdist y s \u2264 ENNReal.ofReal \u03b4 \u2227 edist x y < r - ENNReal.ofReal \u03b4\n\u22a2 infEdist x s < r\n[PROOFSTEP]\nobtain \u27e8y, hy, hxy\u27e9 := h\n[GOAL]\ncase intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nr : \u211d\u22650\u221e\ny : \u03b1\nhy : infEdist y s \u2264 ENNReal.ofReal \u03b4\nhxy : edist x y < r - ENNReal.ofReal \u03b4\n\u22a2 infEdist x s < r\n[PROOFSTEP]\nexact\n infEdist_le_edist_add_infEdist.trans_lt\n ((ENNReal.add_lt_add_of_lt_of_le (hy.trans_lt ENNReal.ofReal_lt_top).ne hxy hy).trans_eq\n (tsub_add_cancel_of_le <| le_self_add.trans (lt_tsub_iff_left.1 hxy).le))\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b5 \u03b4 : \u211d\ns : Set \u03b1\n\u22a2 thickening \u03b5 (thickening \u03b4 s) \u2286 thickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nobtain h\u03b5 | h\u03b5 := le_total \u03b5 0\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b5 \u03b4 : \u211d\ns : Set \u03b1\nh\u03b5 : \u03b5 \u2264 0\n\u22a2 thickening \u03b5 (thickening \u03b4 s) \u2286 thickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nsimp only [thickening_of_nonpos h\u03b5, empty_subset]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b5 \u03b4 : \u211d\ns : Set \u03b1\nh\u03b5 : 0 \u2264 \u03b5\n\u22a2 thickening \u03b5 (thickening \u03b4 s) \u2286 thickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nobtain h\u03b4 | h\u03b4 := le_total \u03b4 0\n[GOAL]\ncase inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b5 \u03b4 : \u211d\ns : Set \u03b1\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : \u03b4 \u2264 0\n\u22a2 thickening \u03b5 (thickening \u03b4 s) \u2286 thickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nsimp only [thickening_of_nonpos h\u03b4, thickening_empty, empty_subset]\n[GOAL]\ncase inr.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b5 \u03b4 : \u211d\ns : Set \u03b1\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 thickening \u03b5 (thickening \u03b4 s) \u2286 thickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nintro x\n[GOAL]\ncase inr.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\n\u03b5 \u03b4 : \u211d\ns : Set \u03b1\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 \u2264 \u03b4\nx : \u03b1\n\u22a2 x \u2208 thickening \u03b5 (thickening \u03b4 s) \u2192 x \u2208 thickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nsimp_rw [mem_thickening_iff_exists_edist_lt, ENNReal.ofReal_add h\u03b5 h\u03b4]\n[GOAL]\ncase inr.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\n\u03b5 \u03b4 : \u211d\ns : Set \u03b1\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 \u2264 \u03b4\nx : \u03b1\n\u22a2 (\u2203 z, (\u2203 z_1, z_1 \u2208 s \u2227 edist z z_1 < ENNReal.ofReal \u03b4) \u2227 edist x z < ENNReal.ofReal \u03b5) \u2192\n \u2203 z, z \u2208 s \u2227 edist x z < ENNReal.ofReal \u03b5 + ENNReal.ofReal \u03b4\n[PROOFSTEP]\nexact fun \u27e8y, \u27e8z, hz, hy\u27e9, hx\u27e9 => \u27e8z, hz, (edist_triangle _ _ _).trans_lt <| ENNReal.add_lt_add hx hy\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\ns : Set \u03b1\n\u22a2 thickening \u03b5 (cthickening \u03b4 s) \u2286 thickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nobtain h\u03b5 | h\u03b5 := le_total \u03b5 0\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\ns : Set \u03b1\nh\u03b5 : \u03b5 \u2264 0\n\u22a2 thickening \u03b5 (cthickening \u03b4 s) \u2286 thickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nsimp only [thickening_of_nonpos h\u03b5, empty_subset]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\n\u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\ns : Set \u03b1\nh\u03b5 : 0 \u2264 \u03b5\n\u22a2 thickening \u03b5 (cthickening \u03b4 s) \u2286 thickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nintro x\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\n\u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\ns : Set \u03b1\nh\u03b5 : 0 \u2264 \u03b5\nx : \u03b1\n\u22a2 x \u2208 thickening \u03b5 (cthickening \u03b4 s) \u2192 x \u2208 thickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nsimp_rw [mem_thickening_iff_exists_edist_lt, mem_cthickening_iff, \u2190 infEdist_lt_iff, ENNReal.ofReal_add h\u03b5 h\u03b4]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\n\u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\ns : Set \u03b1\nh\u03b5 : 0 \u2264 \u03b5\nx : \u03b1\n\u22a2 (\u2203 z, infEdist z s \u2264 ENNReal.ofReal \u03b4 \u2227 edist x z < ENNReal.ofReal \u03b5) \u2192\n infEdist x s < ENNReal.ofReal \u03b5 + ENNReal.ofReal \u03b4\n[PROOFSTEP]\nrintro \u27e8y, hy, hxy\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5\u271d : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\n\u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\ns : Set \u03b1\nh\u03b5 : 0 \u2264 \u03b5\nx y : \u03b1\nhy : infEdist y s \u2264 ENNReal.ofReal \u03b4\nhxy : edist x y < ENNReal.ofReal \u03b5\n\u22a2 infEdist x s < ENNReal.ofReal \u03b5 + ENNReal.ofReal \u03b4\n[PROOFSTEP]\nexact\n infEdist_le_edist_add_infEdist.trans_lt (ENNReal.add_lt_add_of_lt_of_le (hy.trans_lt ENNReal.ofReal_lt_top).ne hxy hy)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\nh\u03b5 : 0 \u2264 \u03b5\n\u03b4 : \u211d\ns : Set \u03b1\n\u22a2 cthickening \u03b5 (thickening \u03b4 s) \u2286 cthickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nobtain h\u03b4 | h\u03b4 := le_total \u03b4 0\n[GOAL]\ncase inl\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\nh\u03b5 : 0 \u2264 \u03b5\n\u03b4 : \u211d\ns : Set \u03b1\nh\u03b4 : \u03b4 \u2264 0\n\u22a2 cthickening \u03b5 (thickening \u03b4 s) \u2286 cthickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nsimp only [thickening_of_nonpos h\u03b4, cthickening_empty, empty_subset]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\nh\u03b5 : 0 \u2264 \u03b5\n\u03b4 : \u211d\ns : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 cthickening \u03b5 (thickening \u03b4 s) \u2286 cthickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nintro x\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\nh\u03b5 : 0 \u2264 \u03b5\n\u03b4 : \u211d\ns : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nx : \u03b1\n\u22a2 x \u2208 cthickening \u03b5 (thickening \u03b4 s) \u2192 x \u2208 cthickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nsimp_rw [mem_cthickening_iff, ENNReal.ofReal_add h\u03b5 h\u03b4]\n[GOAL]\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\nh\u03b5 : 0 \u2264 \u03b5\n\u03b4 : \u211d\ns : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nx : \u03b1\n\u22a2 infEdist x (thickening \u03b4 s) \u2264 ENNReal.ofReal \u03b5 \u2192 infEdist x s \u2264 ENNReal.ofReal \u03b5 + ENNReal.ofReal \u03b4\n[PROOFSTEP]\nexact fun hx => infEdist_le_infEdist_thickening_add.trans (add_le_add_right hx _)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx : \u03b1\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 \u2264 \u03b4\ns : Set \u03b1\n\u22a2 cthickening \u03b5 (cthickening \u03b4 s) \u2286 cthickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 \u2264 \u03b4\ns : Set \u03b1\nx : \u03b1\n\u22a2 x \u2208 cthickening \u03b5 (cthickening \u03b4 s) \u2192 x \u2208 cthickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nsimp_rw [mem_cthickening_iff, ENNReal.ofReal_add h\u03b5 h\u03b4]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns\u271d t : Set \u03b1\nx\u271d : \u03b1\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 \u2264 \u03b4\ns : Set \u03b1\nx : \u03b1\n\u22a2 infEdist x (cthickening \u03b4 s) \u2264 ENNReal.ofReal \u03b5 \u2192 infEdist x s \u2264 ENNReal.ofReal \u03b5 + ENNReal.ofReal \u03b4\n[PROOFSTEP]\nexact fun hx => infEdist_le_infEdist_cthickening_add.trans (add_le_add_right hx _)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\nA : Set \u03b1\nr\u2081 r\u2082 : \u211d\u22650\nhr : r\u2081 \u2260 r\u2082\n\u22a2 ENNReal.ofReal \u2191r\u2081 \u2260 ENNReal.ofReal \u2191r\u2082\n[PROOFSTEP]\nsimpa\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.HausdorffDistance", "llama_tokens": 74120, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585903489891, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.41659567604946757}} {"text": "[GOAL]\n\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : AffineBasis \u03b9 \u211d E\n\u22a2 interior (\u2191(convexHull \u211d) (range \u2191b)) = {x | \u2200 (i : \u03b9), 0 < \u2191(coord b i) x}\n[PROOFSTEP]\ncases subsingleton_or_nontrivial \u03b9\n[GOAL]\ncase inl\n\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : AffineBasis \u03b9 \u211d E\nh\u271d : Subsingleton \u03b9\n\u22a2 interior (\u2191(convexHull \u211d) (range \u2191b)) = {x | \u2200 (i : \u03b9), 0 < \u2191(coord b i) x}\n[PROOFSTEP]\nhave : range b = univ := AffineSubspace.eq_univ_of_subsingleton_span_eq_top (subsingleton_range _) b.tot\n[GOAL]\ncase inl\n\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : AffineBasis \u03b9 \u211d E\nh\u271d : Subsingleton \u03b9\nthis : range \u2191b = univ\n\u22a2 interior (\u2191(convexHull \u211d) (range \u2191b)) = {x | \u2200 (i : \u03b9), 0 < \u2191(coord b i) x}\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : AffineBasis \u03b9 \u211d E\nh\u271d : Nontrivial \u03b9\n\u22a2 interior (\u2191(convexHull \u211d) (range \u2191b)) = {x | \u2200 (i : \u03b9), 0 < \u2191(coord b i) x}\n[PROOFSTEP]\nhaveI : FiniteDimensional \u211d E := b.finiteDimensional\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : AffineBasis \u03b9 \u211d E\nh\u271d : Nontrivial \u03b9\nthis : FiniteDimensional \u211d E\n\u22a2 interior (\u2191(convexHull \u211d) (range \u2191b)) = {x | \u2200 (i : \u03b9), 0 < \u2191(coord b i) x}\n[PROOFSTEP]\nhave : convexHull \u211d (range b) = \u22c2 i, b.coord i \u207b\u00b9' Ici 0 := by rw [b.convexHull_eq_nonneg_coord, setOf_forall]; rfl\n[GOAL]\n\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : AffineBasis \u03b9 \u211d E\nh\u271d : Nontrivial \u03b9\nthis : FiniteDimensional \u211d E\n\u22a2 \u2191(convexHull \u211d) (range \u2191b) = \u22c2 (i : \u03b9), \u2191(coord b i) \u207b\u00b9' Ici 0\n[PROOFSTEP]\nrw [b.convexHull_eq_nonneg_coord, setOf_forall]\n[GOAL]\n\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : AffineBasis \u03b9 \u211d E\nh\u271d : Nontrivial \u03b9\nthis : FiniteDimensional \u211d E\n\u22a2 \u22c2 (i : \u03b9), {x | 0 \u2264 \u2191(coord b i) x} = \u22c2 (i : \u03b9), \u2191(coord b i) \u207b\u00b9' Ici 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : AffineBasis \u03b9 \u211d E\nh\u271d : Nontrivial \u03b9\nthis\u271d : FiniteDimensional \u211d E\nthis : \u2191(convexHull \u211d) (range \u2191b) = \u22c2 (i : \u03b9), \u2191(coord b i) \u207b\u00b9' Ici 0\n\u22a2 interior (\u2191(convexHull \u211d) (range \u2191b)) = {x | \u2200 (i : \u03b9), 0 < \u2191(coord b i) x}\n[PROOFSTEP]\next\n[GOAL]\ncase inr.h\n\u03b9 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : AffineBasis \u03b9 \u211d E\nh\u271d : Nontrivial \u03b9\nthis\u271d : FiniteDimensional \u211d E\nthis : \u2191(convexHull \u211d) (range \u2191b) = \u22c2 (i : \u03b9), \u2191(coord b i) \u207b\u00b9' Ici 0\nx\u271d : E\n\u22a2 x\u271d \u2208 interior (\u2191(convexHull \u211d) (range \u2191b)) \u2194 x\u271d \u2208 {x | \u2200 (i : \u03b9), 0 < \u2191(coord b i) x}\n[PROOFSTEP]\nsimp only [this, interior_iInter, \u2190\n IsOpenMap.preimage_interior_eq_interior_preimage (isOpenMap_barycentric_coord b _) (continuous_barycentric_coord b _),\n interior_Ici, mem_iInter, mem_setOf_eq, mem_Ioi, mem_preimage]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nhne : Set.Nonempty s\nh : AffineIndependent \u211d Subtype.val\n\u22a2 \u2203 t, s \u2286 t \u2227 t \u2286 u \u2227 AffineIndependent \u211d Subtype.val \u2227 affineSpan \u211d t = \u22a4\n[PROOFSTEP]\nobtain \u27e8q, hq\u27e9 := hne\n[GOAL]\ncase intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u22a2 \u2203 t, s \u2286 t \u2227 t \u2286 u \u2227 AffineIndependent \u211d Subtype.val \u2227 affineSpan \u211d t = \u22a4\n[PROOFSTEP]\nobtain \u27e8\u03b5, \u03b50, h\u03b5u\u27e9 := Metric.nhds_basis_closedBall.mem_iff.1 (hu.mem_nhds <| hsu hq)\n[GOAL]\ncase intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\n\u22a2 \u2203 t, s \u2286 t \u2227 t \u2286 u \u2227 AffineIndependent \u211d Subtype.val \u2227 affineSpan \u211d t = \u22a4\n[PROOFSTEP]\nobtain \u27e8t, ht\u2081, ht\u2082, ht\u2083\u27e9 := exists_subset_affineIndependent_affineSpan_eq_top h\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\n\u22a2 \u2203 t, s \u2286 t \u2227 t \u2286 u \u2227 AffineIndependent \u211d Subtype.val \u2227 affineSpan \u211d t = \u22a4\n[PROOFSTEP]\nlet f : P \u2192 P := fun y => lineMap q y (\u03b5 / dist y q)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\n\u22a2 \u2203 t, s \u2286 t \u2227 t \u2286 u \u2227 AffineIndependent \u211d Subtype.val \u2227 affineSpan \u211d t = \u22a4\n[PROOFSTEP]\nhave hf : \u2200 y, f y \u2208 u := by\n refine' fun y => h\u03b5u _\n simp only\n rw [Metric.mem_closedBall, lineMap_apply, dist_vadd_left, norm_smul, Real.norm_eq_abs, dist_eq_norm_vsub V y q,\n abs_div, abs_of_pos \u03b50, abs_of_nonneg (norm_nonneg _), div_mul_comm]\n exact mul_le_of_le_one_left \u03b50.le (div_self_le_one _)\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\n\u22a2 \u2200 (y : P), f y \u2208 u\n[PROOFSTEP]\nrefine' fun y => h\u03b5u _\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\ny : P\n\u22a2 f y \u2208 Metric.closedBall q \u03b5\n[PROOFSTEP]\nsimp only\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\ny : P\n\u22a2 \u2191(lineMap q y) (\u03b5 / dist y q) \u2208 Metric.closedBall q \u03b5\n[PROOFSTEP]\nrw [Metric.mem_closedBall, lineMap_apply, dist_vadd_left, norm_smul, Real.norm_eq_abs, dist_eq_norm_vsub V y q, abs_div,\n abs_of_pos \u03b50, abs_of_nonneg (norm_nonneg _), div_mul_comm]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\ny : P\n\u22a2 \u2016y -\u1d65 q\u2016 / \u2016y -\u1d65 q\u2016 * \u03b5 \u2264 \u03b5\n[PROOFSTEP]\nexact mul_le_of_le_one_left \u03b50.le (div_self_le_one _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\n\u22a2 \u2203 t, s \u2286 t \u2227 t \u2286 u \u2227 AffineIndependent \u211d Subtype.val \u2227 affineSpan \u211d t = \u22a4\n[PROOFSTEP]\nhave h\u03b5yq : \u2200 (y) (_ : y \u2209 s), \u03b5 / dist y q \u2260 0 := fun y hy =>\n div_ne_zero \u03b50.ne' (dist_ne_zero.2 (ne_of_mem_of_not_mem hq hy).symm)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\nh\u03b5yq : \u2200 (y : P), \u00acy \u2208 s \u2192 \u03b5 / dist y q \u2260 0\n\u22a2 \u2203 t, s \u2286 t \u2227 t \u2286 u \u2227 AffineIndependent \u211d Subtype.val \u2227 affineSpan \u211d t = \u22a4\n[PROOFSTEP]\nclassical\nlet w : t \u2192 \u211d\u02e3 := fun p => if hp : (p : P) \u2208 s then 1 else Units.mk0 _ (h\u03b5yq (\u2191p) hp)\nrefine' \u27e8Set.range fun p : t => lineMap q p (w p : \u211d), _, _, _, _\u27e9\n\u00b7 intro p hp; use\u27e8p, ht\u2081 hp\u27e9; simp [hp]\n\u00b7 rintro y \u27e8\u27e8p, hp\u27e9, rfl\u27e9\n by_cases hps : p \u2208 s <;>\n simp only [hps, lineMap_apply_one, Units.val_mk0, dif_neg, dif_pos, not_false_iff, Units.val_one,\n Subtype.coe_mk] <;>\n [exact hsu hps; exact hf p]\n\u00b7 exact (ht\u2082.units_lineMap \u27e8q, ht\u2081 hq\u27e9 w).range\n\u00b7 rw [affineSpan_eq_affineSpan_lineMap_units (ht\u2081 hq) w, ht\u2083]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\nh\u03b5yq : \u2200 (y : P), \u00acy \u2208 s \u2192 \u03b5 / dist y q \u2260 0\n\u22a2 \u2203 t, s \u2286 t \u2227 t \u2286 u \u2227 AffineIndependent \u211d Subtype.val \u2227 affineSpan \u211d t = \u22a4\n[PROOFSTEP]\nlet w : t \u2192 \u211d\u02e3 := fun p => if hp : (p : P) \u2208 s then 1 else Units.mk0 _ (h\u03b5yq (\u2191p) hp)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\nh\u03b5yq : \u2200 (y : P), \u00acy \u2208 s \u2192 \u03b5 / dist y q \u2260 0\nw : \u2191t \u2192 \u211d\u02e3 := fun p => if hp : \u2191p \u2208 s then 1 else Units.mk0 (\u03b5 / dist (\u2191p) q) (_ : \u03b5 / dist (\u2191p) q \u2260 0)\n\u22a2 \u2203 t, s \u2286 t \u2227 t \u2286 u \u2227 AffineIndependent \u211d Subtype.val \u2227 affineSpan \u211d t = \u22a4\n[PROOFSTEP]\nrefine' \u27e8Set.range fun p : t => lineMap q p (w p : \u211d), _, _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\nh\u03b5yq : \u2200 (y : P), \u00acy \u2208 s \u2192 \u03b5 / dist y q \u2260 0\nw : \u2191t \u2192 \u211d\u02e3 := fun p => if hp : \u2191p \u2208 s then 1 else Units.mk0 (\u03b5 / dist (\u2191p) q) (_ : \u03b5 / dist (\u2191p) q \u2260 0)\n\u22a2 s \u2286 range fun p => \u2191(lineMap q \u2191p) \u2191(w p)\n[PROOFSTEP]\nintro p hp\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\nh\u03b5yq : \u2200 (y : P), \u00acy \u2208 s \u2192 \u03b5 / dist y q \u2260 0\nw : \u2191t \u2192 \u211d\u02e3 := fun p => if hp : \u2191p \u2208 s then 1 else Units.mk0 (\u03b5 / dist (\u2191p) q) (_ : \u03b5 / dist (\u2191p) q \u2260 0)\np : P\nhp : p \u2208 s\n\u22a2 p \u2208 range fun p => \u2191(lineMap q \u2191p) \u2191(w p)\n[PROOFSTEP]\nuse\u27e8p, ht\u2081 hp\u27e9\n[GOAL]\ncase h\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\nh\u03b5yq : \u2200 (y : P), \u00acy \u2208 s \u2192 \u03b5 / dist y q \u2260 0\nw : \u2191t \u2192 \u211d\u02e3 := fun p => if hp : \u2191p \u2208 s then 1 else Units.mk0 (\u03b5 / dist (\u2191p) q) (_ : \u03b5 / dist (\u2191p) q \u2260 0)\np : P\nhp : p \u2208 s\n\u22a2 (fun p => \u2191(lineMap q \u2191p) \u2191(w p)) { val := p, property := (_ : p \u2208 t) } = p\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\nh\u03b5yq : \u2200 (y : P), \u00acy \u2208 s \u2192 \u03b5 / dist y q \u2260 0\nw : \u2191t \u2192 \u211d\u02e3 := fun p => if hp : \u2191p \u2208 s then 1 else Units.mk0 (\u03b5 / dist (\u2191p) q) (_ : \u03b5 / dist (\u2191p) q \u2260 0)\n\u22a2 (range fun p => \u2191(lineMap q \u2191p) \u2191(w p)) \u2286 u\n[PROOFSTEP]\nrintro y \u27e8\u27e8p, hp\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.intro.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\nh\u03b5yq : \u2200 (y : P), \u00acy \u2208 s \u2192 \u03b5 / dist y q \u2260 0\nw : \u2191t \u2192 \u211d\u02e3 := fun p => if hp : \u2191p \u2208 s then 1 else Units.mk0 (\u03b5 / dist (\u2191p) q) (_ : \u03b5 / dist (\u2191p) q \u2260 0)\np : P\nhp : p \u2208 t\n\u22a2 (fun p => \u2191(lineMap q \u2191p) \u2191(w p)) { val := p, property := hp } \u2208 u\n[PROOFSTEP]\nby_cases hps : p \u2208 s <;>\n simp only [hps, lineMap_apply_one, Units.val_mk0, dif_neg, dif_pos, not_false_iff, Units.val_one,\n Subtype.coe_mk] <;>\n [exact hsu hps; exact hf p]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.intro.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\nh\u03b5yq : \u2200 (y : P), \u00acy \u2208 s \u2192 \u03b5 / dist y q \u2260 0\nw : \u2191t \u2192 \u211d\u02e3 := fun p => if hp : \u2191p \u2208 s then 1 else Units.mk0 (\u03b5 / dist (\u2191p) q) (_ : \u03b5 / dist (\u2191p) q \u2260 0)\np : P\nhp : p \u2208 t\n\u22a2 (fun p => \u2191(lineMap q \u2191p) \u2191(w p)) { val := p, property := hp } \u2208 u\n[PROOFSTEP]\nby_cases hps : p \u2208 s\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\nh\u03b5yq : \u2200 (y : P), \u00acy \u2208 s \u2192 \u03b5 / dist y q \u2260 0\nw : \u2191t \u2192 \u211d\u02e3 := fun p => if hp : \u2191p \u2208 s then 1 else Units.mk0 (\u03b5 / dist (\u2191p) q) (_ : \u03b5 / dist (\u2191p) q \u2260 0)\np : P\nhp : p \u2208 t\nhps : p \u2208 s\n\u22a2 (fun p => \u2191(lineMap q \u2191p) \u2191(w p)) { val := p, property := hp } \u2208 u\n[PROOFSTEP]\nsimp only [hps, lineMap_apply_one, Units.val_mk0, dif_neg, dif_pos, not_false_iff, Units.val_one, Subtype.coe_mk]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\nh\u03b5yq : \u2200 (y : P), \u00acy \u2208 s \u2192 \u03b5 / dist y q \u2260 0\nw : \u2191t \u2192 \u211d\u02e3 := fun p => if hp : \u2191p \u2208 s then 1 else Units.mk0 (\u03b5 / dist (\u2191p) q) (_ : \u03b5 / dist (\u2191p) q \u2260 0)\np : P\nhp : p \u2208 t\nhps : \u00acp \u2208 s\n\u22a2 (fun p => \u2191(lineMap q \u2191p) \u2191(w p)) { val := p, property := hp } \u2208 u\n[PROOFSTEP]\nsimp only [hps, lineMap_apply_one, Units.val_mk0, dif_neg, dif_pos, not_false_iff, Units.val_one, Subtype.coe_mk]\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\nh\u03b5yq : \u2200 (y : P), \u00acy \u2208 s \u2192 \u03b5 / dist y q \u2260 0\nw : \u2191t \u2192 \u211d\u02e3 := fun p => if hp : \u2191p \u2208 s then 1 else Units.mk0 (\u03b5 / dist (\u2191p) q) (_ : \u03b5 / dist (\u2191p) q \u2260 0)\np : P\nhp : p \u2208 t\nhps : p \u2208 s\n\u22a2 p \u2208 u\n[PROOFSTEP]\nexact hsu hps\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\nh\u03b5yq : \u2200 (y : P), \u00acy \u2208 s \u2192 \u03b5 / dist y q \u2260 0\nw : \u2191t \u2192 \u211d\u02e3 := fun p => if hp : \u2191p \u2208 s then 1 else Units.mk0 (\u03b5 / dist (\u2191p) q) (_ : \u03b5 / dist (\u2191p) q \u2260 0)\np : P\nhp : p \u2208 t\nhps : \u00acp \u2208 s\n\u22a2 \u2191(lineMap q p) (\u03b5 / dist p q) \u2208 u\n[PROOFSTEP]\nexact hf p\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_3\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\nh\u03b5yq : \u2200 (y : P), \u00acy \u2208 s \u2192 \u03b5 / dist y q \u2260 0\nw : \u2191t \u2192 \u211d\u02e3 := fun p => if hp : \u2191p \u2208 s then 1 else Units.mk0 (\u03b5 / dist (\u2191p) q) (_ : \u03b5 / dist (\u2191p) q \u2260 0)\n\u22a2 AffineIndependent \u211d Subtype.val\n[PROOFSTEP]\nexact (ht\u2082.units_lineMap \u27e8q, ht\u2081 hq\u27e9 w).range\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_4\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s \u2286 u\nh : AffineIndependent \u211d Subtype.val\nq : P\nhq : q \u2208 s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nh\u03b5u : Metric.closedBall q \u03b5 \u2286 u\nt : Set P\nht\u2081 : s \u2286 t\nht\u2082 : AffineIndependent \u211d fun p => \u2191p\nht\u2083 : affineSpan \u211d t = \u22a4\nf : P \u2192 P := fun y => \u2191(lineMap q y) (\u03b5 / dist y q)\nhf : \u2200 (y : P), f y \u2208 u\nh\u03b5yq : \u2200 (y : P), \u00acy \u2208 s \u2192 \u03b5 / dist y q \u2260 0\nw : \u2191t \u2192 \u211d\u02e3 := fun p => if hp : \u2191p \u2208 s then 1 else Units.mk0 (\u03b5 / dist (\u2191p) q) (_ : \u03b5 / dist (\u2191p) q \u2260 0)\n\u22a2 affineSpan \u211d (range fun p => \u2191(lineMap q \u2191p) \u2191(w p)) = \u22a4\n[PROOFSTEP]\nrw [affineSpan_eq_affineSpan_lineMap_units (ht\u2081 hq) w, ht\u2083]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nu : Set P\nhu : IsOpen u\nhne : Set.Nonempty u\n\u22a2 \u2203 s x, AffineIndependent \u211d Subtype.val \u2227 affineSpan \u211d s = \u22a4\n[PROOFSTEP]\nrcases hne with \u27e8x, hx\u27e9\n[GOAL]\ncase intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nu : Set P\nhu : IsOpen u\nx : P\nhx : x \u2208 u\n\u22a2 \u2203 s x, AffineIndependent \u211d Subtype.val \u2227 affineSpan \u211d s = \u22a4\n[PROOFSTEP]\nrcases hu.exists_between_affineIndependent_span_eq_top (singleton_subset_iff.mpr hx) (singleton_nonempty _)\n (affineIndependent_of_subsingleton _ _) with\n \u27e8s, -, hsu, hs\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nu : Set P\nhu : IsOpen u\nx : P\nhx : x \u2208 u\ns : Set P\nhsu : s \u2286 u\nhs : AffineIndependent \u211d Subtype.val \u2227 affineSpan \u211d s = \u22a4\n\u22a2 \u2203 s x, AffineIndependent \u211d Subtype.val \u2227 affineSpan \u211d s = \u22a4\n[PROOFSTEP]\nexact \u27e8s, hsu, hs\u27e9\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\nb : AffineBasis \u03b9 \u211d V\n\u22a2 Finset.centroid \u211d Finset.univ \u2191b \u2208 interior (\u2191(convexHull \u211d) (range \u2191b))\n[PROOFSTEP]\nhaveI := b.nonempty\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\nb : AffineBasis \u03b9 \u211d V\nthis : Nonempty \u03b9\n\u22a2 Finset.centroid \u211d Finset.univ \u2191b \u2208 interior (\u2191(convexHull \u211d) (range \u2191b))\n[PROOFSTEP]\nsimp only [b.interior_convexHull, mem_setOf_eq, b.coord_apply_centroid (Finset.mem_univ _), inv_pos, Nat.cast_pos,\n Finset.card_pos, Finset.univ_nonempty, forall_true_iff]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : FiniteDimensional \u211d V\ns : Set V\n\u22a2 Set.Nonempty (interior (\u2191(convexHull \u211d) s)) \u2194 affineSpan \u211d s = \u22a4\n[PROOFSTEP]\nrefine' \u27e8affineSpan_eq_top_of_nonempty_interior, fun h => _\u27e9\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : FiniteDimensional \u211d V\ns : Set V\nh : affineSpan \u211d s = \u22a4\n\u22a2 Set.Nonempty (interior (\u2191(convexHull \u211d) s))\n[PROOFSTEP]\nobtain \u27e8t, hts, b, hb\u27e9 := AffineBasis.exists_affine_subbasis h\n[GOAL]\ncase intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : FiniteDimensional \u211d V\ns : Set V\nh : affineSpan \u211d s = \u22a4\nt : Set V\nhts : t \u2286 s\nb : AffineBasis \u2191t \u211d V\nhb : \u2191b = Subtype.val\n\u22a2 Set.Nonempty (interior (\u2191(convexHull \u211d) s))\n[PROOFSTEP]\nsuffices (interior (convexHull \u211d (range b))).Nonempty\n by\n rw [hb, Subtype.range_coe_subtype, setOf_mem_eq] at this \n refine' this.mono _\n mono*\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : FiniteDimensional \u211d V\ns : Set V\nh : affineSpan \u211d s = \u22a4\nt : Set V\nhts : t \u2286 s\nb : AffineBasis \u2191t \u211d V\nhb : \u2191b = Subtype.val\nthis : Set.Nonempty (interior (\u2191(convexHull \u211d) (range \u2191b)))\n\u22a2 Set.Nonempty (interior (\u2191(convexHull \u211d) s))\n[PROOFSTEP]\nrw [hb, Subtype.range_coe_subtype, setOf_mem_eq] at this \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : FiniteDimensional \u211d V\ns : Set V\nh : affineSpan \u211d s = \u22a4\nt : Set V\nhts : t \u2286 s\nb : AffineBasis \u2191t \u211d V\nhb : \u2191b = Subtype.val\nthis : Set.Nonempty (interior (\u2191(convexHull \u211d) t))\n\u22a2 Set.Nonempty (interior (\u2191(convexHull \u211d) s))\n[PROOFSTEP]\nrefine' this.mono _\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : FiniteDimensional \u211d V\ns : Set V\nh : affineSpan \u211d s = \u22a4\nt : Set V\nhts : t \u2286 s\nb : AffineBasis \u2191t \u211d V\nhb : \u2191b = Subtype.val\nthis : Set.Nonempty (interior (\u2191(convexHull \u211d) t))\n\u22a2 interior (\u2191(convexHull \u211d) t) \u2286 interior (\u2191(convexHull \u211d) s)\n[PROOFSTEP]\nmono*\n[GOAL]\ncase intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : FiniteDimensional \u211d V\ns : Set V\nh : affineSpan \u211d s = \u22a4\nt : Set V\nhts : t \u2286 s\nb : AffineBasis \u2191t \u211d V\nhb : \u2191b = Subtype.val\n\u22a2 Set.Nonempty (interior (\u2191(convexHull \u211d) (range \u2191b)))\n[PROOFSTEP]\nlift t to Finset V using b.finite_set\n[GOAL]\ncase intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : FiniteDimensional \u211d V\ns : Set V\nh : affineSpan \u211d s = \u22a4\nt : Finset V\nhts : \u2191t \u2286 s\nb : AffineBasis \u2191\u2191t \u211d V\nhb : \u2191b = Subtype.val\n\u22a2 Set.Nonempty (interior (\u2191(convexHull \u211d) (range \u2191b)))\n[PROOFSTEP]\nexact \u27e8_, b.centroid_mem_interior_convexHull\u27e9\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : FiniteDimensional \u211d V\ns : Set V\nhs : Convex \u211d s\n\u22a2 Set.Nonempty (interior s) \u2194 affineSpan \u211d s = \u22a4\n[PROOFSTEP]\nrw [\u2190 interior_convexHull_nonempty_iff_affineSpan_eq_top, hs.convexHull_eq]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.AddTorsorBases", "llama_tokens": 13239, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585786300049, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4165956693713733}} {"text": "[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : MeasurableSpace V\ninst\u271d\u00b2 : BorelSpace V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nw : V\n\u22a2 Integrable f \u2194 Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\n[PROOFSTEP]\nhave hL : Continuous fun p : V \u00d7 V => BilinForm.toLin bilinFormOfRealInner p.1 p.2 := continuous_inner\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : MeasurableSpace V\ninst\u271d\u00b2 : BorelSpace V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nw : V\nhL : Continuous fun p => \u2191(\u2191(\u2191BilinForm.toLin bilinFormOfRealInner) p.fst) p.snd\n\u22a2 Integrable f \u2194 Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\n[PROOFSTEP]\nrw [VectorFourier.fourier_integral_convergent_iff Real.continuous_fourierChar hL w]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : MeasurableSpace V\ninst\u271d\u00b2 : BorelSpace V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : FiniteDimensional \u211d V\nw : V\nhL : Continuous fun p => \u2191(\u2191(\u2191BilinForm.toLin bilinFormOfRealInner) p.fst) p.snd\n\u22a2 (Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191(\u2191BilinForm.toLin bilinFormOfRealInner) v) w))) \u2022 f v) \u2194\n Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\n[PROOFSTEP]\nsimp only [BilinForm.toLin_apply, bilinFormOfRealInner_apply]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\n\u22a2 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) =\n -\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\n[PROOFSTEP]\nhave hiw : \u27eai w, w\u27eb = 1 / 2 :=\n by\n rw [inner_smul_left, inner_self_eq_norm_sq_to_K, IsROrC.ofReal_real_eq_id, id.def, IsROrC.conj_to_real, \u2190 div_div,\n div_mul_cancel]\n rwa [Ne.def, sq_eq_zero_iff, norm_eq_zero]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\n\u22a2 inner ((fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) w = 1 / 2\n[PROOFSTEP]\nrw [inner_smul_left, inner_self_eq_norm_sq_to_K, IsROrC.ofReal_real_eq_id, id.def, IsROrC.conj_to_real, \u2190 div_div,\n div_mul_cancel]\n[GOAL]\ncase h\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\n\u22a2 \u2016w\u2016 ^ 2 \u2260 0\n[PROOFSTEP]\nrwa [Ne.def, sq_eq_zero_iff, norm_eq_zero]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhiw : inner ((fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) w = 1 / 2\n\u22a2 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) =\n -\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\n[PROOFSTEP]\nhave : (fun v : V => e[-\u27eav, w\u27eb] \u2022 f (v + i w)) = fun v : V => (fun x : V => -e[-\u27eax, w\u27eb] \u2022 f x) (v + i w) :=\n by\n ext1 v\n simp_rw [inner_add_left, hiw, Real.fourierChar_apply, neg_add, mul_add, ofReal_add, add_mul, exp_add]\n have : 2 * \u03c0 * -(1 / 2) = -\u03c0 := by field_simp; ring\n rw [this, ofReal_neg, neg_mul, exp_neg, exp_pi_mul_I, inv_neg, inv_one, mul_neg_one, neg_neg]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhiw : inner ((fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) w = 1 / 2\n\u22a2 (fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)) = fun v =>\n (fun x => -\u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner x w))) \u2022 f x) (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\n[PROOFSTEP]\next1 v\n[GOAL]\ncase h\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhiw : inner ((fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) w = 1 / 2\nv : V\n\u22a2 \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) =\n (fun x => -\u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner x w))) \u2022 f x) (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\n[PROOFSTEP]\nsimp_rw [inner_add_left, hiw, Real.fourierChar_apply, neg_add, mul_add, ofReal_add, add_mul, exp_add]\n[GOAL]\ncase h\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhiw : inner ((fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) w = 1 / 2\nv : V\n\u22a2 exp (\u2191(2 * \u03c0 * -inner v w) * I) \u2022 f (v + (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) =\n -(exp (\u2191(2 * \u03c0 * -inner v w) * I) * exp (\u2191(2 * \u03c0 * -(1 / 2)) * I)) \u2022 f (v + (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w)\n[PROOFSTEP]\nhave : 2 * \u03c0 * -(1 / 2) = -\u03c0 := by field_simp; ring\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhiw : inner ((fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) w = 1 / 2\nv : V\n\u22a2 2 * \u03c0 * -(1 / 2) = -\u03c0\n[PROOFSTEP]\nfield_simp\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhiw : inner ((fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) w = 1 / 2\nv : V\n\u22a2 2 * \u03c0 = \u03c0 * 2\n[PROOFSTEP]\nring\n[GOAL]\ncase h\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhiw : inner ((fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) w = 1 / 2\nv : V\nthis : 2 * \u03c0 * -(1 / 2) = -\u03c0\n\u22a2 exp (\u2191(2 * \u03c0 * -inner v w) * I) \u2022 f (v + (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) =\n -(exp (\u2191(2 * \u03c0 * -inner v w) * I) * exp (\u2191(2 * \u03c0 * -(1 / 2)) * I)) \u2022 f (v + (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w)\n[PROOFSTEP]\nrw [this, ofReal_neg, neg_mul, exp_neg, exp_pi_mul_I, inv_neg, inv_one, mul_neg_one, neg_neg]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhiw : inner ((fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) w = 1 / 2\nthis :\n (fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)) = fun v =>\n (fun x => -\u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner x w))) \u2022 f x) (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\n\u22a2 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) =\n -\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\n[PROOFSTEP]\nrw [this]\n -- Porting note:\n -- The next three lines had just been\n -- rw [integral_add_right_eq_self (fun (x : V) \u21a6 -(e[-\u27eax, w\u27eb]) \u2022 f x)\n -- ((fun w \u21a6 (1 / (2 * \u2016w\u2016 ^ (2 : \u2115))) \u2022 w) w)]\n -- Unfortunately now we need to specify `volume`, and call `dsimp`.\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhiw : inner ((fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) w = 1 / 2\nthis :\n (fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)) = fun v =>\n (fun x => -\u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner x w))) \u2022 f x) (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\n\u22a2 \u222b (v : V), (fun x => -\u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner x w))) \u2022 f x) (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) =\n -\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\n[PROOFSTEP]\nhave :=\n @integral_add_right_eq_self _ _ _ _ _ volume _ _ _ (fun (x : V) \u21a6 -(e[-\u27eax, w\u27eb]) \u2022 f x)\n ((fun w \u21a6 (1 / (2 * \u2016w\u2016 ^ (2 : \u2115))) \u2022 w) w)\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhiw : inner ((fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) w = 1 / 2\nthis\u271d :\n (fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)) = fun v =>\n (fun x => -\u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner x w))) \u2022 f x) (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\nthis :\n \u222b (x : V),\n -\u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner (x + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) w))) \u2022\n f (x + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) =\n \u222b (x : V), -\u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner x w))) \u2022 f x\n\u22a2 \u222b (v : V), (fun x => -\u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner x w))) \u2022 f x) (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) =\n -\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\n[PROOFSTEP]\nerw [this]\n -- Porting note, we can avoid `erw` by first calling `dsimp at this \u22a2`.\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhiw : inner ((fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) w = 1 / 2\nthis\u271d :\n (fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)) = fun v =>\n (fun x => -\u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner x w))) \u2022 f x) (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\nthis :\n \u222b (x : V),\n -\u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner (x + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) w))) \u2022\n f (x + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w) =\n \u222b (x : V), -\u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner x w))) \u2022 f x\n\u22a2 \u222b (x : V), -\u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner x w))) \u2022 f x =\n -\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\n[PROOFSTEP]\nsimp only [neg_smul, integral_neg]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhf : Integrable f\n\u22a2 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v =\n (1 / 2) \u2022\n \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 (f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w))\n[PROOFSTEP]\nsimp_rw [smul_sub]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhf : Integrable f\n\u22a2 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v =\n (1 / 2) \u2022\n \u222b (v : V),\n \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v -\n \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f (v + (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w)\n[PROOFSTEP]\nrw [integral_sub, fourier_integral_half_period_translate hw, sub_eq_add_neg, neg_neg, \u2190 two_smul \u2102 _, \u2190\n @smul_assoc _ _ _ _ _ _ (IsScalarTower.left \u2102), smul_eq_mul]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhf : Integrable f\n\u22a2 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v =\n (1 / 2 * 2) \u2022 \u222b (a : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner a w))) \u2022 f a\ncase hf\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhf : Integrable f\n\u22a2 Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\ncase hg\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhf : Integrable f\n\u22a2 Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f (v + (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase hf\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhf : Integrable f\n\u22a2 Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\ncase hg\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nw : V\nhw : w \u2260 0\nhf : Integrable f\n\u22a2 Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f (v + (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w)\n[PROOFSTEP]\nexacts [(fourier_integrand_integrable w).mp hf, (fourier_integrand_integrable w).mp (hf.comp_add_right _)]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) (cocompact V) (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' NormedAddCommGroup.tendsto_nhds_zero.mpr fun \u03b5 h\u03b5 => _\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x : V) in cocompact V, \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v x))) \u2022 f v\u2016 < \u03b5\n[PROOFSTEP]\nsuffices \u2203 T : \u211d, \u2200 w : V, T \u2264 \u2016w\u2016 \u2192 \u2016\u222b v : V, e[-\u27eav, w\u27eb] \u2022 f v\u2016 < \u03b5\n by\n simp_rw [\u2190 comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop, dist_eq_norm', sub_zero]\n exact\n let \u27e8T, hT\u27e9 := this\n \u27e8T, fun b hb v hv => hT v (hv.symm \u25b8 hb)\u27e9\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nthis : \u2203 T, \u2200 (w : V), T \u2264 \u2016w\u2016 \u2192 \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\u2016 < \u03b5\n\u22a2 \u2200\u1da0 (x : V) in cocompact V, \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v x))) \u2022 f v\u2016 < \u03b5\n[PROOFSTEP]\nsimp_rw [\u2190 comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop, dist_eq_norm', sub_zero]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nthis : \u2203 T, \u2200 (w : V), T \u2264 \u2016w\u2016 \u2192 \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\u2016 < \u03b5\n\u22a2 \u2203 a, \u2200 (b : \u211d), b \u2265 a \u2192 \u2200 (a : V), \u2016a\u2016 = b \u2192 \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v a))) \u2022 f v\u2016 < \u03b5\n[PROOFSTEP]\nexact\n let \u27e8T, hT\u27e9 := this\n \u27e8T, fun b hb v hv => hT v (hv.symm \u25b8 hb)\u27e9\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 T, \u2200 (w : V), T \u2264 \u2016w\u2016 \u2192 \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\u2016 < \u03b5\n[PROOFSTEP]\nobtain \u27e8R, -, hR_bd\u27e9 : \u2203 R : \u211d, 0 < R \u2227 \u2200 x : V, R \u2264 \u2016x\u2016 \u2192 f x = 0\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 R, 0 < R \u2227 \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\ncase intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\n\u22a2 \u2203 T, \u2200 (w : V), T \u2264 \u2016w\u2016 \u2192 \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\u2016 < \u03b5\n[PROOFSTEP]\nexact hf2.exists_pos_le_norm\n[GOAL]\ncase intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\n\u22a2 \u2203 T, \u2200 (w : V), T \u2264 \u2016w\u2016 \u2192 \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\u2016 < \u03b5\n[PROOFSTEP]\nlet A := {v : V | \u2016v\u2016 \u2264 R + 1}\n[GOAL]\ncase intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\n\u22a2 \u2203 T, \u2200 (w : V), T \u2264 \u2016w\u2016 \u2192 \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\u2016 < \u03b5\n[PROOFSTEP]\nhave mA : MeasurableSet A :=\n by\n suffices A = Metric.closedBall (0 : V) (R + 1) by\n rw [this]\n exact Metric.isClosed_ball.measurableSet\n simp_rw [Metric.closedBall, dist_eq_norm, sub_zero]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\n\u22a2 MeasurableSet A\n[PROOFSTEP]\nsuffices A = Metric.closedBall (0 : V) (R + 1) by\n rw [this]\n exact Metric.isClosed_ball.measurableSet\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nthis : A = Metric.closedBall 0 (R + 1)\n\u22a2 MeasurableSet A\n[PROOFSTEP]\nrw [this]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nthis : A = Metric.closedBall 0 (R + 1)\n\u22a2 MeasurableSet (Metric.closedBall 0 (R + 1))\n[PROOFSTEP]\nexact Metric.isClosed_ball.measurableSet\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\n\u22a2 A = Metric.closedBall 0 (R + 1)\n[PROOFSTEP]\nsimp_rw [Metric.closedBall, dist_eq_norm, sub_zero]\n[GOAL]\ncase intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\n\u22a2 \u2203 T, \u2200 (w : V), T \u2264 \u2016w\u2016 \u2192 \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\u2016 < \u03b5\n[PROOFSTEP]\nobtain \u27e8B, hB_pos, hB_vol\u27e9 : \u2203 B : \u211d\u22650, 0 < B \u2227 volume A \u2264 B :=\n by\n have hc : IsCompact A := by\n simpa only [Metric.closedBall, dist_eq_norm, sub_zero] using isCompact_closedBall (0 : V) _\n let B\u2080 := volume A\n replace hc : B\u2080 < \u22a4 := hc.measure_lt_top\n refine' \u27e8B\u2080.toNNReal + 1, add_pos_of_nonneg_of_pos B\u2080.toNNReal.coe_nonneg one_pos, _\u27e9\n rw [ENNReal.coe_add, ENNReal.coe_one, ENNReal.coe_toNNReal hc.ne]\n exact le_self_add\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\n\u22a2 \u2203 B, 0 < B \u2227 \u2191\u2191volume A \u2264 \u2191B\n[PROOFSTEP]\nhave hc : IsCompact A := by simpa only [Metric.closedBall, dist_eq_norm, sub_zero] using isCompact_closedBall (0 : V) _\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\n\u22a2 IsCompact A\n[PROOFSTEP]\nsimpa only [Metric.closedBall, dist_eq_norm, sub_zero] using isCompact_closedBall (0 : V) _\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nhc : IsCompact A\n\u22a2 \u2203 B, 0 < B \u2227 \u2191\u2191volume A \u2264 \u2191B\n[PROOFSTEP]\nlet B\u2080 := volume A\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nhc : IsCompact A\nB\u2080 : \u211d\u22650\u221e := \u2191\u2191volume A\n\u22a2 \u2203 B, 0 < B \u2227 \u2191\u2191volume A \u2264 \u2191B\n[PROOFSTEP]\nreplace hc : B\u2080 < \u22a4 := hc.measure_lt_top\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB\u2080 : \u211d\u22650\u221e := \u2191\u2191volume A\nhc : B\u2080 < \u22a4\n\u22a2 \u2203 B, 0 < B \u2227 \u2191\u2191volume A \u2264 \u2191B\n[PROOFSTEP]\nrefine' \u27e8B\u2080.toNNReal + 1, add_pos_of_nonneg_of_pos B\u2080.toNNReal.coe_nonneg one_pos, _\u27e9\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB\u2080 : \u211d\u22650\u221e := \u2191\u2191volume A\nhc : B\u2080 < \u22a4\n\u22a2 \u2191\u2191volume A \u2264 \u2191(ENNReal.toNNReal B\u2080 + 1)\n[PROOFSTEP]\nrw [ENNReal.coe_add, ENNReal.coe_one, ENNReal.coe_toNNReal hc.ne]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB\u2080 : \u211d\u22650\u221e := \u2191\u2191volume A\nhc : B\u2080 < \u22a4\n\u22a2 \u2191\u2191volume A \u2264 B\u2080 + 1\n[PROOFSTEP]\nexact le_self_add\n[GOAL]\ncase intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u22a2 \u2203 T, \u2200 (w : V), T \u2264 \u2016w\u2016 \u2192 \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\u2016 < \u03b5\n[PROOFSTEP]\nobtain \u27e8\u03b4, h\u03b41, h\u03b42\u27e9 :=\n Metric.uniformContinuous_iff.mp (hf2.uniformContinuous_of_continuous hf1) (\u03b5 / B) (div_pos h\u03b5 hB_pos)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\n\u22a2 \u2203 T, \u2200 (w : V), T \u2264 \u2016w\u2016 \u2192 \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\u2016 < \u03b5\n[PROOFSTEP]\nrefine' \u27e81 / 2 + 1 / (2 * \u03b4), fun w hw_bd => _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\n\u22a2 \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\u2016 < \u03b5\n[PROOFSTEP]\nhave hw_ne : w \u2260 0 := by\n contrapose! hw_bd; rw [hw_bd, norm_zero]\n exact add_pos one_half_pos (one_div_pos.mpr <| mul_pos two_pos h\u03b41)\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\n\u22a2 w \u2260 0\n[PROOFSTEP]\ncontrapose! hw_bd\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : w = 0\n\u22a2 \u2016w\u2016 < 1 / 2 + 1 / (2 * \u03b4)\n[PROOFSTEP]\nrw [hw_bd, norm_zero]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : w = 0\n\u22a2 0 < 1 / 2 + 1 / (2 * \u03b4)\n[PROOFSTEP]\nexact add_pos one_half_pos (one_div_pos.mpr <| mul_pos two_pos h\u03b41)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\n\u22a2 \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\u2016 < \u03b5\n[PROOFSTEP]\nhave hw'_nm : \u2016i w\u2016 = 1 / (2 * \u2016w\u2016) := by\n rw [norm_smul, norm_div, Real.norm_of_nonneg (mul_nonneg two_pos.le <| sq_nonneg _), norm_one, sq, \u2190 div_div, \u2190\n div_div, \u2190 div_div, div_mul_cancel _ (norm_eq_zero.not.mpr hw_ne)]\n --* Rewrite integral in terms of `f v - f (v + w')`.\n -- Porting note: this was\n -- rw [norm_eq_abs, \u2190 Complex.ofReal_one, \u2190 ofReal_bit0, \u2190 of_real_div,\n -- Complex.abs_of_nonneg one_half_pos.le]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\n\u22a2 \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\n[PROOFSTEP]\nrw [norm_smul, norm_div, Real.norm_of_nonneg (mul_nonneg two_pos.le <| sq_nonneg _), norm_one, sq, \u2190 div_div, \u2190 div_div,\n \u2190 div_div, div_mul_cancel _ (norm_eq_zero.not.mpr hw_ne)]\n --* Rewrite integral in terms of `f v - f (v + w')`.\n -- Porting note: this was\n -- rw [norm_eq_abs, \u2190 Complex.ofReal_one, \u2190 ofReal_bit0, \u2190 of_real_div,\n -- Complex.abs_of_nonneg one_half_pos.le]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\n\u22a2 \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\u2016 < \u03b5\n[PROOFSTEP]\nhave : \u2016(1 / 2 : \u2102)\u2016 = 1 / 2 := by norm_num\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\n\u22a2 \u20161 / 2\u2016 = 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis : \u20161 / 2\u2016 = 1 / 2\n\u22a2 \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v\u2016 < \u03b5\n[PROOFSTEP]\nrw [fourier_integral_eq_half_sub_half_period_translate hw_ne (hf1.integrable_of_hasCompactSupport hf2), norm_smul, this]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis : \u20161 / 2\u2016 = 1 / 2\n\u22a2 1 / 2 *\n \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 (f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w))\u2016 <\n \u03b5\n[PROOFSTEP]\nhave : \u03b5 = 1 / 2 * (2 * \u03b5) := by field_simp; rw [mul_comm]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis : \u20161 / 2\u2016 = 1 / 2\n\u22a2 \u03b5 = 1 / 2 * (2 * \u03b5)\n[PROOFSTEP]\nfield_simp\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis : \u20161 / 2\u2016 = 1 / 2\n\u22a2 \u03b5 * 2 = 2 * \u03b5\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\n\u22a2 1 / 2 *\n \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 (f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w))\u2016 <\n \u03b5\n[PROOFSTEP]\nrw [this, mul_lt_mul_left (one_half_pos : (0 : \u211d) < 1 / 2)]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\n\u22a2 \u2016\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 (f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w))\u2016 < 2 * \u03b5\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (norm_integral_le_integral_norm _) _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\n\u22a2 \u222b (a : V), \u2016\u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner a w))) \u2022 (f a - f (a + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w))\u2016 < 2 * \u03b5\n[PROOFSTEP]\nsimp_rw [norm_smul, norm_eq_abs, abs_coe_circle, one_mul]\n --* Show integral can be taken over A only.\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\n\u22a2 \u222b (a : V), \u2016f a - f (a + (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w)\u2016 < 2 * \u03b5\n[PROOFSTEP]\nhave int_A : \u222b v : V, \u2016f v - f (v + i w)\u2016 = \u222b v in A, \u2016f v - f (v + i w)\u2016 :=\n by\n refine' (set_integral_eq_integral_of_forall_compl_eq_zero fun v hv => _).symm\n dsimp only at hv \n simp only [mem_setOf, not_le] at hv \n rw [hR_bd v _, hR_bd (v + i w) _, sub_zero, norm_zero]\n \u00b7 rw [\u2190 sub_neg_eq_add]\n refine' le_trans _ (norm_sub_norm_le _ _)\n rw [le_sub_iff_add_le, norm_neg]\n refine' le_trans _ hv.le\n rw [add_le_add_iff_left, hw'_nm, \u2190 div_div]\n refine' (div_le_one <| norm_pos_iff.mpr hw_ne).mpr _\n refine' le_trans (le_add_of_nonneg_right <| one_div_nonneg.mpr <| _) hw_bd\n exact (mul_pos (zero_lt_two' \u211d) h\u03b41).le\n \u00b7 exact ((le_add_iff_nonneg_right _).mpr zero_le_one).trans hv.le\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\n\u22a2 \u222b (v : V), \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 =\n \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\n[PROOFSTEP]\nrefine' (set_integral_eq_integral_of_forall_compl_eq_zero fun v hv => _).symm\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nv : V\nhv : \u00acv \u2208 A\n\u22a2 \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 = 0\n[PROOFSTEP]\ndsimp only at hv \n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nv : V\nhv : \u00acv \u2208 {v | \u2016v\u2016 \u2264 R + 1}\n\u22a2 \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 = 0\n[PROOFSTEP]\nsimp only [mem_setOf, not_le] at hv \n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nv : V\nhv : R + 1 < \u2016v\u2016\n\u22a2 \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 = 0\n[PROOFSTEP]\nrw [hR_bd v _, hR_bd (v + i w) _, sub_zero, norm_zero]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nv : V\nhv : R + 1 < \u2016v\u2016\n\u22a2 R \u2264 \u2016v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016\n[PROOFSTEP]\nrw [\u2190 sub_neg_eq_add]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nv : V\nhv : R + 1 < \u2016v\u2016\n\u22a2 R \u2264 \u2016v - -(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016\n[PROOFSTEP]\nrefine' le_trans _ (norm_sub_norm_le _ _)\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nv : V\nhv : R + 1 < \u2016v\u2016\n\u22a2 R \u2264 \u2016v\u2016 - \u2016-(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016\n[PROOFSTEP]\nrw [le_sub_iff_add_le, norm_neg]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nv : V\nhv : R + 1 < \u2016v\u2016\n\u22a2 R + \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 \u2264 \u2016v\u2016\n[PROOFSTEP]\nrefine' le_trans _ hv.le\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nv : V\nhv : R + 1 < \u2016v\u2016\n\u22a2 R + \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 \u2264 R + 1\n[PROOFSTEP]\nrw [add_le_add_iff_left, hw'_nm, \u2190 div_div]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nv : V\nhv : R + 1 < \u2016v\u2016\n\u22a2 1 / 2 / \u2016w\u2016 \u2264 1\n[PROOFSTEP]\nrefine' (div_le_one <| norm_pos_iff.mpr hw_ne).mpr _\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nv : V\nhv : R + 1 < \u2016v\u2016\n\u22a2 1 / 2 \u2264 \u2016w\u2016\n[PROOFSTEP]\nrefine' le_trans (le_add_of_nonneg_right <| one_div_nonneg.mpr <| _) hw_bd\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nv : V\nhv : R + 1 < \u2016v\u2016\n\u22a2 0 \u2264 2 * \u03b4\n[PROOFSTEP]\nexact (mul_pos (zero_lt_two' \u211d) h\u03b41).le\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nv : V\nhv : R + 1 < \u2016v\u2016\n\u22a2 R \u2264 \u2016v\u2016\n[PROOFSTEP]\nexact ((le_add_iff_nonneg_right _).mpr zero_le_one).trans hv.le\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nint_A :\n \u222b (v : V), \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 =\n \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\n\u22a2 \u222b (a : V), \u2016f a - f (a + (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w)\u2016 < 2 * \u03b5\n[PROOFSTEP]\nrw [int_A]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nint_A :\n \u222b (v : V), \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 =\n \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\n\u22a2 \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 < 2 * \u03b5\n[PROOFSTEP]\nclear int_A\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\n\u22a2 \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 < 2 * \u03b5\n[PROOFSTEP]\nhave bdA : \u2200 v : V, v \u2208 A \u2192 \u2016\u2016f v - f (v + i w)\u2016\u2016 \u2264 \u03b5 / B :=\n by\n simp_rw [norm_norm]\n simp_rw [dist_eq_norm] at h\u03b42 \n refine' fun x _ => (h\u03b42 _).le\n rw [sub_add_cancel', norm_neg, hw'_nm, \u2190 div_div, div_lt_iff (norm_pos_iff.mpr hw_ne), \u2190 div_lt_iff' h\u03b41, div_div]\n refine' (lt_add_of_pos_left _ _).trans_le hw_bd\n exact one_half_pos\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\n\u22a2 \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\n[PROOFSTEP]\nsimp_rw [norm_norm]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\n\u22a2 \u2200 (v : V), v \u2208 {v | \u2016v\u2016 \u2264 R + 1} \u2192 \u2016f v - f (v + (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w)\u2016 \u2264 \u03b5 / \u2191B\n[PROOFSTEP]\nsimp_rw [dist_eq_norm] at h\u03b42 \n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nh\u03b42 : \u2200 {a b : V}, \u2016a - b\u2016 < \u03b4 \u2192 \u2016f a - f b\u2016 < \u03b5 / \u2191B\n\u22a2 \u2200 (v : V), v \u2208 {v | \u2016v\u2016 \u2264 R + 1} \u2192 \u2016f v - f (v + (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w)\u2016 \u2264 \u03b5 / \u2191B\n[PROOFSTEP]\nrefine' fun x _ => (h\u03b42 _).le\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nh\u03b42 : \u2200 {a b : V}, \u2016a - b\u2016 < \u03b4 \u2192 \u2016f a - f b\u2016 < \u03b5 / \u2191B\nx : V\nx\u271d : x \u2208 {v | \u2016v\u2016 \u2264 R + 1}\n\u22a2 \u2016x - (x + (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w)\u2016 < \u03b4\n[PROOFSTEP]\nrw [sub_add_cancel', norm_neg, hw'_nm, \u2190 div_div, div_lt_iff (norm_pos_iff.mpr hw_ne), \u2190 div_lt_iff' h\u03b41, div_div]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nh\u03b42 : \u2200 {a b : V}, \u2016a - b\u2016 < \u03b4 \u2192 \u2016f a - f b\u2016 < \u03b5 / \u2191B\nx : V\nx\u271d : x \u2208 {v | \u2016v\u2016 \u2264 R + 1}\n\u22a2 1 / (2 * \u03b4) < \u2016w\u2016\n[PROOFSTEP]\nrefine' (lt_add_of_pos_left _ _).trans_le hw_bd\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nh\u03b42 : \u2200 {a b : V}, \u2016a - b\u2016 < \u03b4 \u2192 \u2016f a - f b\u2016 < \u03b5 / \u2191B\nx : V\nx\u271d : x \u2208 {v | \u2016v\u2016 \u2264 R + 1}\n\u22a2 0 < 1 / 2\n[PROOFSTEP]\nexact one_half_pos\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\n\u22a2 \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 < 2 * \u03b5\n[PROOFSTEP]\nhave bdA2 := norm_set_integral_le_of_norm_le_const (hB_vol.trans_lt ENNReal.coe_lt_top) bdA ?_\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_2\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\nbdA2 : \u2016\u222b (x : V) in A, \u2016f x - f (x + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B * ENNReal.toReal (\u2191\u2191volume A)\n\u22a2 \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 < 2 * \u03b5\ncase intro.intro.intro.intro.intro.intro.refine_1\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\n\u22a2 AEStronglyMeasurable (fun x => \u2016f x - f (x + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016) (Measure.restrict volume A)\n[PROOFSTEP]\nswap\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_1\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\n\u22a2 AEStronglyMeasurable (fun x => \u2016f x - f (x + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016) (Measure.restrict volume A)\n[PROOFSTEP]\napply Continuous.aestronglyMeasurable\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_1.hf\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\n\u22a2 Continuous fun x => \u2016f x - f (x + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\n[PROOFSTEP]\nexact continuous_norm.comp <| Continuous.sub hf1 <| Continuous.comp hf1 <| continuous_id'.add continuous_const\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_2\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d : \u20161 / 2\u2016 = 1 / 2\nthis : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\nbdA2 : \u2016\u222b (x : V) in A, \u2016f x - f (x + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B * ENNReal.toReal (\u2191\u2191volume A)\n\u22a2 \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 < 2 * \u03b5\n[PROOFSTEP]\nhave : \u2016_\u2016 = \u222b v : V in A, \u2016f v - f (v + i w)\u2016 := Real.norm_of_nonneg (set_integral_nonneg mA fun x _ => norm_nonneg _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_2\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d\u00b9 : \u20161 / 2\u2016 = 1 / 2\nthis\u271d : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\nbdA2 : \u2016\u222b (x : V) in A, \u2016f x - f (x + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B * ENNReal.toReal (\u2191\u2191volume A)\nthis :\n \u2016\u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 =\n \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\n\u22a2 \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 < 2 * \u03b5\n[PROOFSTEP]\nrw [this] at bdA2 \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_2\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d\u00b9 : \u20161 / 2\u2016 = 1 / 2\nthis\u271d : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\nbdA2 : \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 \u2264 \u03b5 / \u2191B * ENNReal.toReal (\u2191\u2191volume A)\nthis :\n \u2016\u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 =\n \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\n\u22a2 \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 < 2 * \u03b5\n[PROOFSTEP]\nrefine' bdA2.trans_lt _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_2\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d\u00b9 : \u20161 / 2\u2016 = 1 / 2\nthis\u271d : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\nbdA2 : \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 \u2264 \u03b5 / \u2191B * ENNReal.toReal (\u2191\u2191volume A)\nthis :\n \u2016\u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 =\n \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\n\u22a2 \u03b5 / \u2191B * ENNReal.toReal (\u2191\u2191volume A) < 2 * \u03b5\n[PROOFSTEP]\nrw [div_mul_eq_mul_div, div_lt_iff (NNReal.coe_pos.mpr hB_pos), mul_comm (2 : \u211d), mul_assoc, mul_lt_mul_left h\u03b5]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_2\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d\u00b9 : \u20161 / 2\u2016 = 1 / 2\nthis\u271d : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\nbdA2 : \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 \u2264 \u03b5 / \u2191B * ENNReal.toReal (\u2191\u2191volume A)\nthis :\n \u2016\u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 =\n \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\n\u22a2 ENNReal.toReal (\u2191\u2191volume A) < 2 * \u2191B\n[PROOFSTEP]\nrw [\u2190 ENNReal.toReal_le_toReal] at hB_vol \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_2\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : ENNReal.toReal (\u2191\u2191volume A) \u2264 ENNReal.toReal \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d\u00b9 : \u20161 / 2\u2016 = 1 / 2\nthis\u271d : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\nbdA2 : \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 \u2264 \u03b5 / \u2191B * ENNReal.toReal (\u2191\u2191volume A)\nthis :\n \u2016\u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 =\n \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\n\u22a2 ENNReal.toReal (\u2191\u2191volume A) < 2 * \u2191B\n[PROOFSTEP]\nrefine' hB_vol.trans_lt _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_2\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : ENNReal.toReal (\u2191\u2191volume A) \u2264 ENNReal.toReal \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d\u00b9 : \u20161 / 2\u2016 = 1 / 2\nthis\u271d : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\nbdA2 : \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 \u2264 \u03b5 / \u2191B * ENNReal.toReal (\u2191\u2191volume A)\nthis :\n \u2016\u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 =\n \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\n\u22a2 ENNReal.toReal \u2191B < 2 * \u2191B\n[PROOFSTEP]\nrw [(by rfl : (\u2191B : ENNReal).toReal = \u2191B), two_mul]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : ENNReal.toReal (\u2191\u2191volume A) \u2264 ENNReal.toReal \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d\u00b9 : \u20161 / 2\u2016 = 1 / 2\nthis\u271d : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\nbdA2 : \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 \u2264 \u03b5 / \u2191B * ENNReal.toReal (\u2191\u2191volume A)\nthis :\n \u2016\u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 =\n \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\n\u22a2 ENNReal.toReal \u2191B = \u2191B\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_2\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : ENNReal.toReal (\u2191\u2191volume A) \u2264 ENNReal.toReal \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d\u00b9 : \u20161 / 2\u2016 = 1 / 2\nthis\u271d : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\nbdA2 : \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 \u2264 \u03b5 / \u2191B * ENNReal.toReal (\u2191\u2191volume A)\nthis :\n \u2016\u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 =\n \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\n\u22a2 \u2191B < \u2191B + \u2191B\n[PROOFSTEP]\nexact lt_add_of_pos_left _ hB_pos\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine_2.ha\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d\u00b9 : \u20161 / 2\u2016 = 1 / 2\nthis\u271d : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\nbdA2 : \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 \u2264 \u03b5 / \u2191B * ENNReal.toReal (\u2191\u2191volume A)\nthis :\n \u2016\u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 =\n \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\n\u22a2 \u2191\u2191volume A \u2260 \u22a4\ncase intro.intro.intro.intro.intro.intro.refine_2.hb\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhf1 : Continuous f\nhf2 : HasCompactSupport f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nR : \u211d\nhR_bd : \u2200 (x : V), R \u2264 \u2016x\u2016 \u2192 f x = 0\nA : Set V := {v | \u2016v\u2016 \u2264 R + 1}\nmA : MeasurableSet A\nB : \u211d\u22650\nhB_pos : 0 < B\nhB_vol : \u2191\u2191volume A \u2264 \u2191B\n\u03b4 : \u211d\nh\u03b41 : \u03b4 > 0\nh\u03b42 : \u2200 {a b : V}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 / \u2191B\nw : V\nhw_bd : 1 / 2 + 1 / (2 * \u03b4) \u2264 \u2016w\u2016\nhw_ne : w \u2260 0\nhw'_nm : \u2016(fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w\u2016 = 1 / (2 * \u2016w\u2016)\nthis\u271d\u00b9 : \u20161 / 2\u2016 = 1 / 2\nthis\u271d : \u03b5 = 1 / 2 * (2 * \u03b5)\nbdA : \u2200 (v : V), v \u2208 A \u2192 \u2016\u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 \u2264 \u03b5 / \u2191B\nbdA2 : \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016 \u2264 \u03b5 / \u2191B * ENNReal.toReal (\u2191\u2191volume A)\nthis :\n \u2016\u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\u2016 =\n \u222b (v : V) in A, \u2016f v - f (v + (fun w => (1 / (2 * \u2016w\u2016 ^ 2)) \u2022 w) w)\u2016\n\u22a2 \u2191B \u2260 \u22a4\n[PROOFSTEP]\nexacts [(hB_vol.trans_lt ENNReal.coe_lt_top).ne, ENNReal.coe_lt_top.ne]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) (cocompact V) (\ud835\udcdd 0)\n[PROOFSTEP]\nby_cases hfi : Integrable f\n[GOAL]\ncase pos\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : Integrable f\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) (cocompact V) (\ud835\udcdd 0)\ncase neg\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : \u00acIntegrable f\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) (cocompact V) (\ud835\udcdd 0)\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : \u00acIntegrable f\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) (cocompact V) (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert tendsto_const_nhds (a := (0 : E)) with w\n[GOAL]\ncase h.e.h.e'_3.h\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : \u00acIntegrable f\nw : V\n\u22a2 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v = 0\n[PROOFSTEP]\napply integral_undef\n[GOAL]\ncase h.e.h.e'_3.h.h\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : \u00acIntegrable f\nw : V\n\u22a2 \u00acIntegrable fun a => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner a w))) \u2022 f a\n[PROOFSTEP]\nrwa [\u2190 fourier_integrand_integrable w]\n[GOAL]\ncase pos\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : Integrable f\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) (cocompact V) (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' Metric.tendsto_nhds.mpr fun \u03b5 h\u03b5 => _\n[GOAL]\ncase pos\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : Integrable f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x : V) in cocompact V, dist (\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v x))) \u2022 f v) 0 < \u03b5\n[PROOFSTEP]\nobtain \u27e8g, hg_supp, hfg, hg_cont, -\u27e9 := hfi.exists_hasCompactSupport_integral_sub_le (div_pos h\u03b5 two_pos)\n[GOAL]\ncase pos.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : Integrable f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ng : V \u2192 E\nhg_supp : HasCompactSupport g\nhfg : \u222b (x : V), \u2016f x - g x\u2016 \u2264 \u03b5 / 2\nhg_cont : Continuous g\n\u22a2 \u2200\u1da0 (x : V) in cocompact V, dist (\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v x))) \u2022 f v) 0 < \u03b5\n[PROOFSTEP]\nrefine'\n ((Metric.tendsto_nhds.mp (tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support hg_cont hg_supp)) _\n (div_pos h\u03b5 two_pos)).mp\n (eventually_of_forall fun w hI => _)\n[GOAL]\ncase pos.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : Integrable f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ng : V \u2192 E\nhg_supp : HasCompactSupport g\nhfg : \u222b (x : V), \u2016f x - g x\u2016 \u2264 \u03b5 / 2\nhg_cont : Continuous g\nw : V\nhI : dist (\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v) 0 < \u03b5 / 2\n\u22a2 dist (\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) 0 < \u03b5\n[PROOFSTEP]\nrw [dist_eq_norm] at hI \u22a2\n[GOAL]\ncase pos.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : Integrable f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ng : V \u2192 E\nhg_supp : HasCompactSupport g\nhfg : \u222b (x : V), \u2016f x - g x\u2016 \u2264 \u03b5 / 2\nhg_cont : Continuous g\nw : V\nhI : \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v) - 0\u2016 < \u03b5 / 2\n\u22a2 \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) - 0\u2016 < \u03b5\n[PROOFSTEP]\nhave : \u2016(\u222b v, e[-\u27eav, w\u27eb] \u2022 f v) - \u222b v, e[-\u27eav, w\u27eb] \u2022 g v\u2016 \u2264 \u03b5 / 2 :=\n by\n refine' le_trans _ hfg\n simp_rw [\u2190\n integral_sub ((fourier_integrand_integrable w).mp hfi)\n ((fourier_integrand_integrable w).mp (hg_cont.integrable_of_hasCompactSupport hg_supp)),\n \u2190 smul_sub, \u2190 Pi.sub_apply]\n exact VectorFourier.norm_fourierIntegral_le_integral_norm e volume (BilinForm.toLin bilinFormOfRealInner) (f - g) w\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : Integrable f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ng : V \u2192 E\nhg_supp : HasCompactSupport g\nhfg : \u222b (x : V), \u2016f x - g x\u2016 \u2264 \u03b5 / 2\nhg_cont : Continuous g\nw : V\nhI : \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v) - 0\u2016 < \u03b5 / 2\n\u22a2 \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) -\n \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v\u2016 \u2264\n \u03b5 / 2\n[PROOFSTEP]\nrefine' le_trans _ hfg\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : Integrable f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ng : V \u2192 E\nhg_supp : HasCompactSupport g\nhfg : \u222b (x : V), \u2016f x - g x\u2016 \u2264 \u03b5 / 2\nhg_cont : Continuous g\nw : V\nhI : \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v) - 0\u2016 < \u03b5 / 2\n\u22a2 \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) -\n \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v\u2016 \u2264\n \u222b (x : V), \u2016f x - g x\u2016\n[PROOFSTEP]\nsimp_rw [\u2190\n integral_sub ((fourier_integrand_integrable w).mp hfi)\n ((fourier_integrand_integrable w).mp (hg_cont.integrable_of_hasCompactSupport hg_supp)),\n \u2190 smul_sub, \u2190 Pi.sub_apply]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : Integrable f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ng : V \u2192 E\nhg_supp : HasCompactSupport g\nhfg : \u222b (x : V), \u2016f x - g x\u2016 \u2264 \u03b5 / 2\nhg_cont : Continuous g\nw : V\nhI : \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v) - 0\u2016 < \u03b5 / 2\n\u22a2 \u2016\u222b (a : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner a w))) \u2022 (f - g) a\u2016 \u2264 \u222b (x : V), \u2016(f - g) x\u2016\n[PROOFSTEP]\nexact VectorFourier.norm_fourierIntegral_le_integral_norm e volume (BilinForm.toLin bilinFormOfRealInner) (f - g) w\n[GOAL]\ncase pos.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : Integrable f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ng : V \u2192 E\nhg_supp : HasCompactSupport g\nhfg : \u222b (x : V), \u2016f x - g x\u2016 \u2264 \u03b5 / 2\nhg_cont : Continuous g\nw : V\nhI : \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v) - 0\u2016 < \u03b5 / 2\nthis :\n \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) -\n \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v\u2016 \u2264\n \u03b5 / 2\n\u22a2 \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) - 0\u2016 < \u03b5\n[PROOFSTEP]\nreplace := add_lt_add_of_le_of_lt this hI\n[GOAL]\ncase pos.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : Integrable f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ng : V \u2192 E\nhg_supp : HasCompactSupport g\nhfg : \u222b (x : V), \u2016f x - g x\u2016 \u2264 \u03b5 / 2\nhg_cont : Continuous g\nw : V\nhI : \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v) - 0\u2016 < \u03b5 / 2\nthis :\n \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) -\n \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v\u2016 +\n \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v) - 0\u2016 <\n \u03b5 / 2 + \u03b5 / 2\n\u22a2 \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) - 0\u2016 < \u03b5\n[PROOFSTEP]\nrw [add_halves] at this \n[GOAL]\ncase pos.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : Integrable f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ng : V \u2192 E\nhg_supp : HasCompactSupport g\nhfg : \u222b (x : V), \u2016f x - g x\u2016 \u2264 \u03b5 / 2\nhg_cont : Continuous g\nw : V\nhI : \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v) - 0\u2016 < \u03b5 / 2\nthis :\n \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) -\n \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v\u2016 +\n \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v) - 0\u2016 <\n \u03b5\n\u22a2 \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) - 0\u2016 < \u03b5\n[PROOFSTEP]\nrefine' ((le_of_eq _).trans (norm_add_le _ _)).trans_lt this\n[GOAL]\ncase pos.intro.intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : MeasurableSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : FiniteDimensional \u211d V\ninst\u271d : CompleteSpace E\nhfi : Integrable f\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ng : V \u2192 E\nhg_supp : HasCompactSupport g\nhfg : \u222b (x : V), \u2016f x - g x\u2016 \u2264 \u03b5 / 2\nhg_cont : Continuous g\nw : V\nhI : \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v) - 0\u2016 < \u03b5 / 2\nthis :\n \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) -\n \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v\u2016 +\n \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v) - 0\u2016 <\n \u03b5\n\u22a2 \u2016(\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) - 0\u2016 =\n \u2016((\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) -\n \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v) +\n ((\u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 g v) - 0)\u2016\n[PROOFSTEP]\nsimp only [sub_zero, sub_add_cancel]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2076 : NormedAddCommGroup V\ninst\u271d\u2075 : MeasurableSpace V\ninst\u271d\u2074 : BorelSpace V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v \u2202\u03bc) (cocompact (V \u2192L[\u211d] \u211d)) (\ud835\udcdd 0)\n[PROOFSTEP]\nobtain \u27e8C, _, _, hC\u27e9 := \u03bc.isAddHaarMeasure_eq_smul_isAddHaarMeasure volume\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2076 : NormedAddCommGroup V\ninst\u271d\u2075 : MeasurableSpace V\ninst\u271d\u2074 : BorelSpace V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nC : \u211d\u22650\u221e\nleft\u271d\u00b9 : C \u2260 0\nleft\u271d : C \u2260 \u22a4\nhC : \u03bc = C \u2022 volume\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v \u2202\u03bc) (cocompact (V \u2192L[\u211d] \u211d)) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [hC]\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2076 : NormedAddCommGroup V\ninst\u271d\u2075 : MeasurableSpace V\ninst\u271d\u2074 : BorelSpace V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nC : \u211d\u22650\u221e\nleft\u271d\u00b9 : C \u2260 0\nleft\u271d : C \u2260 \u22a4\nhC : \u03bc = C \u2022 volume\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v \u2202C \u2022 volume) (cocompact (V \u2192L[\u211d] \u211d)) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp_rw [integral_smul_measure]\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2076 : NormedAddCommGroup V\ninst\u271d\u2075 : MeasurableSpace V\ninst\u271d\u2074 : BorelSpace V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nC : \u211d\u22650\u221e\nleft\u271d\u00b9 : C \u2260 0\nleft\u271d : C \u2260 \u22a4\nhC : \u03bc = C \u2022 volume\n\u22a2 Tendsto (fun w => ENNReal.toReal C \u2022 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v) (cocompact (V \u2192L[\u211d] \u211d))\n (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 (smul_zero _ : C.toReal \u2022 (0 : E) = 0)]\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\nV : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2076 : NormedAddCommGroup V\ninst\u271d\u2075 : MeasurableSpace V\ninst\u271d\u2074 : BorelSpace V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nC : \u211d\u22650\u221e\nleft\u271d\u00b9 : C \u2260 0\nleft\u271d : C \u2260 \u22a4\nhC : \u03bc = C \u2022 volume\n\u22a2 Tendsto (fun w => ENNReal.toReal C \u2022 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v) (cocompact (V \u2192L[\u211d] \u211d))\n (\ud835\udcdd (ENNReal.toReal C \u2022 0))\n[PROOFSTEP]\napply Tendsto.const_smul\n[GOAL]\ncase intro.intro.intro.hf\nE : Type u_1\nV : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2076 : NormedAddCommGroup V\ninst\u271d\u2075 : MeasurableSpace V\ninst\u271d\u2074 : BorelSpace V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nC : \u211d\u22650\u221e\nleft\u271d\u00b9 : C \u2260 0\nleft\u271d : C \u2260 \u22a4\nhC : \u03bc = C \u2022 volume\n\u22a2 Tendsto (fun x => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191x v))) \u2022 f v) (cocompact (V \u2192L[\u211d] \u211d)) (\ud835\udcdd 0)\n[PROOFSTEP]\nlet A := (InnerProductSpace.toDual \u211d V).symm\n[GOAL]\ncase intro.intro.intro.hf\nE : Type u_1\nV : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2076 : NormedAddCommGroup V\ninst\u271d\u2075 : MeasurableSpace V\ninst\u271d\u2074 : BorelSpace V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nC : \u211d\u22650\u221e\nleft\u271d\u00b9 : C \u2260 0\nleft\u271d : C \u2260 \u22a4\nhC : \u03bc = C \u2022 volume\nA : NormedSpace.Dual \u211d V \u2243\u2097\u1d62\u22c6[\u211d] V := LinearIsometryEquiv.symm (InnerProductSpace.toDual \u211d V)\n\u22a2 Tendsto (fun x => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191x v))) \u2022 f v) (cocompact (V \u2192L[\u211d] \u211d)) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : (fun w : V \u2192L[\u211d] \u211d => \u222b v, e[-w v] \u2022 f v) = (fun w : V => \u222b v, e[-\u27eav, w\u27eb] \u2022 f v) \u2218 A :=\n by\n ext1 w\n congr 1 with v : 1\n rw [\u2190 inner_conj_symm, IsROrC.conj_to_real, InnerProductSpace.toDual_symm_apply, Real.fourierChar_apply]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2076 : NormedAddCommGroup V\ninst\u271d\u2075 : MeasurableSpace V\ninst\u271d\u2074 : BorelSpace V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nC : \u211d\u22650\u221e\nleft\u271d\u00b9 : C \u2260 0\nleft\u271d : C \u2260 \u22a4\nhC : \u03bc = C \u2022 volume\nA : NormedSpace.Dual \u211d V \u2243\u2097\u1d62\u22c6[\u211d] V := LinearIsometryEquiv.symm (InnerProductSpace.toDual \u211d V)\n\u22a2 (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v) =\n (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) \u2218 \u2191A\n[PROOFSTEP]\next1 w\n[GOAL]\ncase h\nE : Type u_1\nV : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2076 : NormedAddCommGroup V\ninst\u271d\u2075 : MeasurableSpace V\ninst\u271d\u2074 : BorelSpace V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nC : \u211d\u22650\u221e\nleft\u271d\u00b9 : C \u2260 0\nleft\u271d : C \u2260 \u22a4\nhC : \u03bc = C \u2022 volume\nA : NormedSpace.Dual \u211d V \u2243\u2097\u1d62\u22c6[\u211d] V := LinearIsometryEquiv.symm (InnerProductSpace.toDual \u211d V)\nw : V \u2192L[\u211d] \u211d\n\u22a2 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v =\n ((fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) \u2218 \u2191A) w\n[PROOFSTEP]\ncongr 1 with v : 1\n[GOAL]\ncase h.e_f.h\nE : Type u_1\nV : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2076 : NormedAddCommGroup V\ninst\u271d\u2075 : MeasurableSpace V\ninst\u271d\u2074 : BorelSpace V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nC : \u211d\u22650\u221e\nleft\u271d\u00b9 : C \u2260 0\nleft\u271d : C \u2260 \u22a4\nhC : \u03bc = C \u2022 volume\nA : NormedSpace.Dual \u211d V \u2243\u2097\u1d62\u22c6[\u211d] V := LinearIsometryEquiv.symm (InnerProductSpace.toDual \u211d V)\nw : V \u2192L[\u211d] \u211d\nv : V\n\u22a2 \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v = \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v (\u2191A w)))) \u2022 f v\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, IsROrC.conj_to_real, InnerProductSpace.toDual_symm_apply, Real.fourierChar_apply]\n[GOAL]\ncase intro.intro.intro.hf\nE : Type u_1\nV : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2076 : NormedAddCommGroup V\ninst\u271d\u2075 : MeasurableSpace V\ninst\u271d\u2074 : BorelSpace V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nC : \u211d\u22650\u221e\nleft\u271d\u00b9 : C \u2260 0\nleft\u271d : C \u2260 \u22a4\nhC : \u03bc = C \u2022 volume\nA : NormedSpace.Dual \u211d V \u2243\u2097\u1d62\u22c6[\u211d] V := LinearIsometryEquiv.symm (InnerProductSpace.toDual \u211d V)\nthis :\n (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v) =\n (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) \u2218 \u2191A\n\u22a2 Tendsto (fun x => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191x v))) \u2022 f v) (cocompact (V \u2192L[\u211d] \u211d)) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase intro.intro.intro.hf\nE : Type u_1\nV : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u2076 : NormedAddCommGroup V\ninst\u271d\u2075 : MeasurableSpace V\ninst\u271d\u2074 : BorelSpace V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nC : \u211d\u22650\u221e\nleft\u271d\u00b9 : C \u2260 0\nleft\u271d : C \u2260 \u22a4\nhC : \u03bc = C \u2022 volume\nA : NormedSpace.Dual \u211d V \u2243\u2097\u1d62\u22c6[\u211d] V := LinearIsometryEquiv.symm (InnerProductSpace.toDual \u211d V)\nthis :\n (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v) =\n (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) \u2218 \u2191A\n\u22a2 Tendsto ((fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-inner v w))) \u2022 f v) \u2218 \u2191A) (cocompact (V \u2192L[\u211d] \u211d)) (\ud835\udcdd 0)\n[PROOFSTEP]\nexact (tendsto_integral_exp_inner_smul_cocompact f).comp A.toHomeomorph.toCocompactMap.cocompact_tendsto'\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v \u2202\u03bc) (cocompact (V \u2192L[\u211d] \u211d)) (\ud835\udcdd 0)\n[PROOFSTEP]\nlet V' := EuclideanSpace \u211d (Fin (finrank \u211d V))\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v \u2202\u03bc) (cocompact (V \u2192L[\u211d] \u211d)) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave A : V \u2243L[\u211d] V' := toEuclidean\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v \u2202\u03bc) (cocompact (V \u2192L[\u211d] \u211d)) (\ud835\udcdd 0)\n[PROOFSTEP]\nborelize V'\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v \u2202\u03bc) (cocompact (V \u2192L[\u211d] \u211d)) (\ud835\udcdd 0)\n[PROOFSTEP]\nlet A\u2098 : MeasurableEquiv V V' := A.toHomeomorph.toMeasurableEquiv\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v \u2202\u03bc) (cocompact (V \u2192L[\u211d] \u211d)) (\ud835\udcdd 0)\n[PROOFSTEP]\nlet Adual\u2097 : (V \u2192L[\u211d] \u211d) \u2243\u2097[\u211d] V' \u2192L[\u211d] \u211d :=\n { toFun := fun t => t.comp A.symm.toContinuousLinearMap\n invFun := fun t => t.comp A.toContinuousLinearMap\n map_add' := by\n intro t s\n ext1 v\n simp only [ContinuousLinearMap.coe_comp', Function.comp_apply, ContinuousLinearMap.add_apply]\n map_smul' := by\n intro x f\n ext1 v\n simp only [RingHom.id_apply, ContinuousLinearMap.coe_comp', Function.comp_apply, ContinuousLinearMap.smul_apply]\n left_inv := by\n intro w\n ext1 v\n simp only [ContinuousLinearMap.coe_comp', ContinuousLinearEquiv.coe_coe, Function.comp_apply,\n ContinuousLinearEquiv.symm_apply_apply]\n right_inv := by\n intro w\n ext1 v\n simp only [ContinuousLinearMap.coe_comp', ContinuousLinearEquiv.coe_coe, Function.comp_apply,\n ContinuousLinearEquiv.apply_symm_apply] }\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\n\u22a2 \u2200 (x y : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (x + y) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) x +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) y\n[PROOFSTEP]\nintro t s\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nt s : V \u2192L[\u211d] \u211d\n\u22a2 (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s\n[PROOFSTEP]\next1 v\n[GOAL]\ncase h\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nt s : V \u2192L[\u211d] \u211d\nv : V'\n\u22a2 \u2191((fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s)) v =\n \u2191((fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s)\n v\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.coe_comp', Function.comp_apply, ContinuousLinearMap.add_apply]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\n\u22a2 \u2200 (r : \u211d) (x : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (r \u2022 x) =\n \u2191(RingHom.id \u211d) r \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n x\n[PROOFSTEP]\nintro x f\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf\u271d : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nx : \u211d\nf : V \u2192L[\u211d] \u211d\n\u22a2 AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f\n[PROOFSTEP]\next1 v\n[GOAL]\ncase h\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf\u271d : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nx : \u211d\nf : V \u2192L[\u211d] \u211d\nv : V'\n\u22a2 \u2191(AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f))\n v =\n \u2191(\u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f)\n v\n[PROOFSTEP]\nsimp only [RingHom.id_apply, ContinuousLinearMap.coe_comp', Function.comp_apply, ContinuousLinearMap.smul_apply]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\n\u22a2 Function.LeftInverse (fun t => ContinuousLinearMap.comp t \u2191A)\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom.toFun\n[PROOFSTEP]\nintro w\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nw : V \u2192L[\u211d] \u211d\n\u22a2 (fun t => ContinuousLinearMap.comp t \u2191A)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n w) =\n w\n[PROOFSTEP]\next1 v\n[GOAL]\ncase h\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nw : V \u2192L[\u211d] \u211d\nv : V\n\u22a2 \u2191((fun t => ContinuousLinearMap.comp t \u2191A)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n w))\n v =\n \u2191w v\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.coe_comp', ContinuousLinearEquiv.coe_coe, Function.comp_apply,\n ContinuousLinearEquiv.symm_apply_apply]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\n\u22a2 Function.RightInverse (fun t => ContinuousLinearMap.comp t \u2191A)\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom.toFun\n[PROOFSTEP]\nintro w\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nw : V' \u2192L[\u211d] \u211d\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n ((fun t => ContinuousLinearMap.comp t \u2191A) w) =\n w\n[PROOFSTEP]\next1 v\n[GOAL]\ncase h\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nw : V' \u2192L[\u211d] \u211d\nv : V'\n\u22a2 \u2191(AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n ((fun t => ContinuousLinearMap.comp t \u2191A) w))\n v =\n \u2191w v\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.coe_comp', ContinuousLinearEquiv.coe_coe, Function.comp_apply,\n ContinuousLinearEquiv.apply_symm_apply]\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nAdual\u2097 : (V \u2192L[\u211d] \u211d) \u2243\u2097[\u211d] V' \u2192L[\u211d] \u211d :=\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) },\n invFun := fun t => ContinuousLinearMap.comp t \u2191A,\n left_inv :=\n (_ :\n \u2200 (w : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191A)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n w) =\n w),\n right_inv :=\n (_ :\n \u2200 (w : V' \u2192L[\u211d] \u211d),\n AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n ((fun t => ContinuousLinearMap.comp t \u2191A) w) =\n w) }\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v \u2202\u03bc) (cocompact (V \u2192L[\u211d] \u211d)) (\ud835\udcdd 0)\n[PROOFSTEP]\nlet Adual : (V \u2192L[\u211d] \u211d) \u2243L[\u211d] V' \u2192L[\u211d] \u211d :=\n { Adual\u2097 with\n continuous_toFun := Adual\u2097.toLinearMap.continuous_of_finiteDimensional\n continuous_invFun := Adual\u2097.symm.toLinearMap.continuous_of_finiteDimensional }\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nAdual\u2097 : (V \u2192L[\u211d] \u211d) \u2243\u2097[\u211d] V' \u2192L[\u211d] \u211d :=\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) },\n invFun := fun t => ContinuousLinearMap.comp t \u2191A,\n left_inv :=\n (_ :\n \u2200 (w : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191A)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n w) =\n w),\n right_inv :=\n (_ :\n \u2200 (w : V' \u2192L[\u211d] \u211d),\n AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n ((fun t => ContinuousLinearMap.comp t \u2191A) w) =\n w) }\nAdual : (V \u2192L[\u211d] \u211d) \u2243L[\u211d] V' \u2192L[\u211d] \u211d :=\n ContinuousLinearEquiv.mk\n { toLinearMap := \u2191Adual\u2097, invFun := Adual\u2097.invFun,\n left_inv := (_ : Function.LeftInverse Adual\u2097.invFun Adual\u2097.toFun),\n right_inv := (_ : Function.RightInverse Adual\u2097.invFun Adual\u2097.toFun) }\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v \u2202\u03bc) (cocompact (V \u2192L[\u211d] \u211d)) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : (\u03bc.map A\u2098).IsAddHaarMeasure := Measure.MapContinuousLinearEquiv.isAddHaarMeasure _ A\n[GOAL]\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nAdual\u2097 : (V \u2192L[\u211d] \u211d) \u2243\u2097[\u211d] V' \u2192L[\u211d] \u211d :=\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) },\n invFun := fun t => ContinuousLinearMap.comp t \u2191A,\n left_inv :=\n (_ :\n \u2200 (w : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191A)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n w) =\n w),\n right_inv :=\n (_ :\n \u2200 (w : V' \u2192L[\u211d] \u211d),\n AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n ((fun t => ContinuousLinearMap.comp t \u2191A) w) =\n w) }\nAdual : (V \u2192L[\u211d] \u211d) \u2243L[\u211d] V' \u2192L[\u211d] \u211d :=\n ContinuousLinearEquiv.mk\n { toLinearMap := \u2191Adual\u2097, invFun := Adual\u2097.invFun,\n left_inv := (_ : Function.LeftInverse Adual\u2097.invFun Adual\u2097.toFun),\n right_inv := (_ : Function.RightInverse Adual\u2097.invFun Adual\u2097.toFun) }\nthis : Measure.IsAddHaarMeasure (Measure.map (\u2191A\u2098) \u03bc)\n\u22a2 Tendsto (fun w => \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v \u2202\u03bc) (cocompact (V \u2192L[\u211d] \u211d)) (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert\n (tendsto_integral_exp_smul_cocompact_of_inner_product (f \u2218 A.symm) (\u03bc.map A\u2098)).comp\n Adual.toHomeomorph.toCocompactMap.cocompact_tendsto' with\n w\n[GOAL]\ncase h.e'_3.h\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nAdual\u2097 : (V \u2192L[\u211d] \u211d) \u2243\u2097[\u211d] V' \u2192L[\u211d] \u211d :=\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) },\n invFun := fun t => ContinuousLinearMap.comp t \u2191A,\n left_inv :=\n (_ :\n \u2200 (w : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191A)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n w) =\n w),\n right_inv :=\n (_ :\n \u2200 (w : V' \u2192L[\u211d] \u211d),\n AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n ((fun t => ContinuousLinearMap.comp t \u2191A) w) =\n w) }\nAdual : (V \u2192L[\u211d] \u211d) \u2243L[\u211d] V' \u2192L[\u211d] \u211d :=\n ContinuousLinearEquiv.mk\n { toLinearMap := \u2191Adual\u2097, invFun := Adual\u2097.invFun,\n left_inv := (_ : Function.LeftInverse Adual\u2097.invFun Adual\u2097.toFun),\n right_inv := (_ : Function.RightInverse Adual\u2097.invFun Adual\u2097.toFun) }\nthis : Measure.IsAddHaarMeasure (Measure.map (\u2191A\u2098) \u03bc)\nw : V \u2192L[\u211d] \u211d\n\u22a2 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v \u2202\u03bc =\n ((fun w =>\n \u222b (v : V'),\n \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 (f \u2218 \u2191(ContinuousLinearEquiv.symm A)) v \u2202Measure.map (\u2191A\u2098) \u03bc) \u2218\n (Homeomorph.toCocompactMap (ContinuousLinearEquiv.toHomeomorph Adual)).toContinuousMap.toFun)\n w\n[PROOFSTEP]\nrw [Function.comp_apply, integral_map_equiv]\n[GOAL]\ncase h.e'_3.h\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nAdual\u2097 : (V \u2192L[\u211d] \u211d) \u2243\u2097[\u211d] V' \u2192L[\u211d] \u211d :=\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) },\n invFun := fun t => ContinuousLinearMap.comp t \u2191A,\n left_inv :=\n (_ :\n \u2200 (w : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191A)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n w) =\n w),\n right_inv :=\n (_ :\n \u2200 (w : V' \u2192L[\u211d] \u211d),\n AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n ((fun t => ContinuousLinearMap.comp t \u2191A) w) =\n w) }\nAdual : (V \u2192L[\u211d] \u211d) \u2243L[\u211d] V' \u2192L[\u211d] \u211d :=\n ContinuousLinearEquiv.mk\n { toLinearMap := \u2191Adual\u2097, invFun := Adual\u2097.invFun,\n left_inv := (_ : Function.LeftInverse Adual\u2097.invFun Adual\u2097.toFun),\n right_inv := (_ : Function.RightInverse Adual\u2097.invFun Adual\u2097.toFun) }\nthis : Measure.IsAddHaarMeasure (Measure.map (\u2191A\u2098) \u03bc)\nw : V \u2192L[\u211d] \u211d\n\u22a2 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v \u2202\u03bc =\n \u222b (x : V),\n \u2191(\u2191e\n (\u2191Multiplicative.ofAdd\n (-\u2191(ContinuousMap.toFun\n (Homeomorph.toCocompactMap (ContinuousLinearEquiv.toHomeomorph Adual)).toContinuousMap w)\n (\u2191A\u2098 x)))) \u2022\n (f \u2218 \u2191(ContinuousLinearEquiv.symm A)) (\u2191A\u2098 x) \u2202\u03bc\n[PROOFSTEP]\ncongr 1 with v : 1\n[GOAL]\ncase h.e'_3.h.e_f.h\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nAdual\u2097 : (V \u2192L[\u211d] \u211d) \u2243\u2097[\u211d] V' \u2192L[\u211d] \u211d :=\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) },\n invFun := fun t => ContinuousLinearMap.comp t \u2191A,\n left_inv :=\n (_ :\n \u2200 (w : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191A)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n w) =\n w),\n right_inv :=\n (_ :\n \u2200 (w : V' \u2192L[\u211d] \u211d),\n AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n ((fun t => ContinuousLinearMap.comp t \u2191A) w) =\n w) }\nAdual : (V \u2192L[\u211d] \u211d) \u2243L[\u211d] V' \u2192L[\u211d] \u211d :=\n ContinuousLinearEquiv.mk\n { toLinearMap := \u2191Adual\u2097, invFun := Adual\u2097.invFun,\n left_inv := (_ : Function.LeftInverse Adual\u2097.invFun Adual\u2097.toFun),\n right_inv := (_ : Function.RightInverse Adual\u2097.invFun Adual\u2097.toFun) }\nthis : Measure.IsAddHaarMeasure (Measure.map (\u2191A\u2098) \u03bc)\nw : V \u2192L[\u211d] \u211d\nv : V\n\u22a2 \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191w v))) \u2022 f v =\n \u2191(\u2191e\n (\u2191Multiplicative.ofAdd\n (-\u2191(ContinuousMap.toFun\n (Homeomorph.toCocompactMap (ContinuousLinearEquiv.toHomeomorph Adual)).toContinuousMap w)\n (\u2191A\u2098 v)))) \u2022\n (f \u2218 \u2191(ContinuousLinearEquiv.symm A)) (\u2191A\u2098 v)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e'_3.h.e_f.h.e_a.e_self.h.e_6.h.h.e_6.h.e_a\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nAdual\u2097 : (V \u2192L[\u211d] \u211d) \u2243\u2097[\u211d] V' \u2192L[\u211d] \u211d :=\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) },\n invFun := fun t => ContinuousLinearMap.comp t \u2191A,\n left_inv :=\n (_ :\n \u2200 (w : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191A)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n w) =\n w),\n right_inv :=\n (_ :\n \u2200 (w : V' \u2192L[\u211d] \u211d),\n AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n ((fun t => ContinuousLinearMap.comp t \u2191A) w) =\n w) }\nAdual : (V \u2192L[\u211d] \u211d) \u2243L[\u211d] V' \u2192L[\u211d] \u211d :=\n ContinuousLinearEquiv.mk\n { toLinearMap := \u2191Adual\u2097, invFun := Adual\u2097.invFun,\n left_inv := (_ : Function.LeftInverse Adual\u2097.invFun Adual\u2097.toFun),\n right_inv := (_ : Function.RightInverse Adual\u2097.invFun Adual\u2097.toFun) }\nthis : Measure.IsAddHaarMeasure (Measure.map (\u2191A\u2098) \u03bc)\nw : V \u2192L[\u211d] \u211d\nv : V\n\u22a2 \u2191w v =\n \u2191(ContinuousMap.toFun (Homeomorph.toCocompactMap (ContinuousLinearEquiv.toHomeomorph Adual)).toContinuousMap w)\n (\u2191A\u2098 v)\n[PROOFSTEP]\napply congr_arg w\n[GOAL]\ncase h.e'_3.h.e_f.h.e_a.e_self.h.e_6.h.h.e_6.h.e_a\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nAdual\u2097 : (V \u2192L[\u211d] \u211d) \u2243\u2097[\u211d] V' \u2192L[\u211d] \u211d :=\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) },\n invFun := fun t => ContinuousLinearMap.comp t \u2191A,\n left_inv :=\n (_ :\n \u2200 (w : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191A)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n w) =\n w),\n right_inv :=\n (_ :\n \u2200 (w : V' \u2192L[\u211d] \u211d),\n AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n ((fun t => ContinuousLinearMap.comp t \u2191A) w) =\n w) }\nAdual : (V \u2192L[\u211d] \u211d) \u2243L[\u211d] V' \u2192L[\u211d] \u211d :=\n ContinuousLinearEquiv.mk\n { toLinearMap := \u2191Adual\u2097, invFun := Adual\u2097.invFun,\n left_inv := (_ : Function.LeftInverse Adual\u2097.invFun Adual\u2097.toFun),\n right_inv := (_ : Function.RightInverse Adual\u2097.invFun Adual\u2097.toFun) }\nthis : Measure.IsAddHaarMeasure (Measure.map (\u2191A\u2098) \u03bc)\nw : V \u2192L[\u211d] \u211d\nv : V\n\u22a2 v = \u2191\u2191\u2191(ContinuousLinearEquiv.symm A) (\u2191A\u2098 v)\n[PROOFSTEP]\nexact (ContinuousLinearEquiv.symm_apply_apply A v).symm\n[GOAL]\ncase h.e'_3.h.e_f.h.e_a.e_a\nE : Type u_1\nV : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u2102 E\nf : V \u2192 E\ninst\u271d\u00b9\u2070 : AddCommGroup V\ninst\u271d\u2079 : TopologicalSpace V\ninst\u271d\u2078 : TopologicalAddGroup V\ninst\u271d\u2077 : T2Space V\ninst\u271d\u2076 : MeasurableSpace V\ninst\u271d\u2075 : BorelSpace V\ninst\u271d\u2074 : Module \u211d V\ninst\u271d\u00b3 : ContinuousSMul \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : CompleteSpace E\n\u03bc : Measure V\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\nV' : Type := EuclideanSpace \u211d (Fin (finrank \u211d V))\nA : V \u2243L[\u211d] V'\nthis\u271d\u00b9 : MeasurableSpace V' := borel V'\nthis\u271d : BorelSpace V'\nA\u2098 : V \u2243\u1d50 V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)\nAdual\u2097 : (V \u2192L[\u211d] \u211d) \u2243\u2097[\u211d] V' \u2192L[\u211d] \u211d :=\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) },\n invFun := fun t => ContinuousLinearMap.comp t \u2191A,\n left_inv :=\n (_ :\n \u2200 (w : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191A)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n w) =\n w),\n right_inv :=\n (_ :\n \u2200 (w : V' \u2192L[\u211d] \u211d),\n AddHom.toFun\n {\n toAddHom :=\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) },\n map_smul' :=\n (_ :\n \u2200 (x : \u211d) (f : V \u2192L[\u211d] \u211d),\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n (x \u2022 f) =\n \u2191(RingHom.id \u211d) x \u2022\n AddHom.toFun\n { toFun := fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A),\n map_add' :=\n (_ :\n \u2200 (t s : V \u2192L[\u211d] \u211d),\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) (t + s) =\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) t +\n (fun t => ContinuousLinearMap.comp t \u2191(ContinuousLinearEquiv.symm A)) s) }\n f) }.toAddHom\n ((fun t => ContinuousLinearMap.comp t \u2191A) w) =\n w) }\nAdual : (V \u2192L[\u211d] \u211d) \u2243L[\u211d] V' \u2192L[\u211d] \u211d :=\n ContinuousLinearEquiv.mk\n { toLinearMap := \u2191Adual\u2097, invFun := Adual\u2097.invFun,\n left_inv := (_ : Function.LeftInverse Adual\u2097.invFun Adual\u2097.toFun),\n right_inv := (_ : Function.RightInverse Adual\u2097.invFun Adual\u2097.toFun) }\nthis : Measure.IsAddHaarMeasure (Measure.map (\u2191A\u2098) \u03bc)\nw : V \u2192L[\u211d] \u211d\nv : V\n\u22a2 v = \u2191(ContinuousLinearEquiv.symm A) (\u2191A\u2098 v)\n[PROOFSTEP]\nexact (ContinuousLinearEquiv.symm_apply_apply A v).symm\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "llama_tokens": 87063, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389817407016, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.4165895350317427}} {"text": "[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bd : ProbabilityMeasure \u03a9\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bd s)) univ \u2260 0\n[PROOFSTEP]\nsimp only [coeFn_univ, Ne.def, one_ne_zero, not_false_iff]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bd : ProbabilityMeasure \u03a9\ns : Set \u03a9\n\u22a2 \u2191((fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bd s)) s) = \u2191\u2191\u2191\u03bd s\n[PROOFSTEP]\nrw [\u2190 coeFn_comp_toFiniteMeasure_eq_coeFn, FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure,\n toMeasure_comp_toFiniteMeasure_eq_toMeasure]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\ns\u2081 s\u2082 : Set \u03a9\nh : s\u2081 \u2286 s\u2082\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s\u2081 \u2264 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s\u2082\n[PROOFSTEP]\nrw [\u2190 coeFn_comp_toFiniteMeasure_eq_coeFn]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\ns\u2081 s\u2082 : Set \u03a9\nh : s\u2081 \u2286 s\u2082\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(toFiniteMeasure \u03bc) s)) s\u2081 \u2264 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(toFiniteMeasure \u03bc) s)) s\u2082\n[PROOFSTEP]\nexact MeasureTheory.FiniteMeasure.apply_mono _ h\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 Nonempty \u03a9\n[PROOFSTEP]\nby_contra maybe_empty\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\nmaybe_empty : \u00acNonempty \u03a9\n\u22a2 False\n[PROOFSTEP]\nhave zero : (\u03bc : Measure \u03a9) univ = 0 := by rw [univ_eq_empty_iff.mpr (not_nonempty_iff.mp maybe_empty), measure_empty]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\nmaybe_empty : \u00acNonempty \u03a9\n\u22a2 \u2191\u2191\u2191\u03bc univ = 0\n[PROOFSTEP]\nrw [univ_eq_empty_iff.mpr (not_nonempty_iff.mp maybe_empty), measure_empty]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\nmaybe_empty : \u00acNonempty \u03a9\nzero : \u2191\u2191\u2191\u03bc univ = 0\n\u22a2 False\n[PROOFSTEP]\nrw [measure_univ] at zero \n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\nmaybe_empty : \u00acNonempty \u03a9\nzero : 1 = 0\n\u22a2 False\n[PROOFSTEP]\nexact zero_ne_one zero.symm\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : ProbabilityMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s\n\u22a2 \u03bc = \u03bd\n[PROOFSTEP]\napply toMeasure_injective\n[GOAL]\ncase a\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : ProbabilityMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s\n\u22a2 \u2191\u03bc = \u2191\u03bd\n[PROOFSTEP]\next1 s s_mble\n[GOAL]\ncase a.h\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : ProbabilityMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s\ns : Set \u03a9\ns_mble : MeasurableSet s\n\u22a2 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s\n[PROOFSTEP]\nexact h s s_mble\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : ProbabilityMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bd s)) s\n\u22a2 \u03bc = \u03bd\n[PROOFSTEP]\next1 s s_mble\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : ProbabilityMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bd s)) s\ns : Set \u03a9\ns_mble : MeasurableSet s\n\u22a2 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s\n[PROOFSTEP]\nsimpa [ennreal_coeFn_eq_coeFn_toMeasure] using congr_arg ((\u2191) : \u211d\u22650 \u2192 \u211d\u22650\u221e) (h s s_mble)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 toFiniteMeasure \u03bc \u2260 0\n[PROOFSTEP]\nrw [\u2190 FiniteMeasure.mass_nonzero_iff, \u03bc.mass_toFiniteMeasure]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nexact one_ne_zero\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 ProbabilityMeasure \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2194 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\n[PROOFSTEP]\nrw [tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 ProbabilityMeasure \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 Tendsto (toFiniteMeasure \u2218 \u03bcs) F (\ud835\udcdd (toFiniteMeasure \u03bc)) \u2194\n \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\n[PROOFSTEP]\nexact FiniteMeasure.tendsto_iff_forall_lintegral_tendsto\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 ProbabilityMeasure \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2194 \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191\u03bc))\n[PROOFSTEP]\nrw [tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 ProbabilityMeasure \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 Tendsto (toFiniteMeasure \u2218 \u03bcs) F (\ud835\udcdd (toFiniteMeasure \u03bc)) \u2194\n \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191\u03bc))\n[PROOFSTEP]\nrw [FiniteMeasure.tendsto_iff_forall_integral_tendsto]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 ProbabilityMeasure \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 (\u2200 (f : \u03a9 \u2192\u1d47 \u211d),\n Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191((toFiniteMeasure \u2218 \u03bcs) i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191(toFiniteMeasure \u03bc)))) \u2194\n \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u2191\u03bc))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nzero : \u00acmass \u03bc = 0\n\u22a2 IsProbabilityMeasure \u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc)\n[PROOFSTEP]\nrefine'\n \u27e8_\u27e9\n -- porting note: paying the price that this isn't `simp` lemma now.\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nzero : \u00acmass \u03bc = 0\n\u22a2 \u2191\u2191\u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc) univ = 1\n[PROOFSTEP]\nrw [FiniteMeasure.toMeasure_smul]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nzero : \u00acmass \u03bc = 0\n\u22a2 \u2191\u2191((mass \u03bc)\u207b\u00b9 \u2022 \u2191\u03bc) univ = 1\n[PROOFSTEP]\nsimp only [Measure.smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply, Measure.nnreal_smul_coe_apply, ne_eq,\n mass_zero_iff, ENNReal.coe_inv zero, ennreal_mass]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nzero : \u00acmass \u03bc = 0\n\u22a2 (\u2191\u2191\u2191\u03bc univ)\u207b\u00b9 * \u2191\u2191\u2191\u03bc univ = 1\n[PROOFSTEP]\nrw [\u2190 Ne.def, \u2190 ENNReal.coe_ne_zero, ennreal_mass] at zero \n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nzero : \u2191\u2191\u2191\u03bc univ \u2260 0\n\u22a2 (\u2191\u2191\u2191\u03bc univ)\u207b\u00b9 * \u2191\u2191\u2191\u03bc univ = 1\n[PROOFSTEP]\nexact ENNReal.inv_mul_cancel zero \u03bc.prop.measure_univ_lt_top.ne\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = mass \u03bc * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize \u03bc) s)) s\n[PROOFSTEP]\nobtain rfl | h := eq_or_ne \u03bc 0\n[GOAL]\ncase inl\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\ns : Set \u03a9\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u21910 s)) s = mass 0 * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize 0) s)) s\n[PROOFSTEP]\nsimp only [zero_mass, coeFn_zero, Pi.zero_apply, zero_mul]\n[GOAL]\ncase inl\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\ns : Set \u03a9\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u21910 s) = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\nh : \u03bc \u2260 0\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = mass \u03bc * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize \u03bc) s)) s\n[PROOFSTEP]\nhave mass_nonzero : \u03bc.mass \u2260 0 := by rwa [\u03bc.mass_nonzero_iff]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\nh : \u03bc \u2260 0\n\u22a2 mass \u03bc \u2260 0\n[PROOFSTEP]\nrwa [\u03bc.mass_nonzero_iff]\n[GOAL]\ncase inr\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\nh : \u03bc \u2260 0\nmass_nonzero : mass \u03bc \u2260 0\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = mass \u03bc * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize \u03bc) s)) s\n[PROOFSTEP]\nsimp only [normalize, dif_neg mass_nonzero]\n[GOAL]\ncase inr\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\nh : \u03bc \u2260 0\nmass_nonzero : mass \u03bc \u2260 0\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s) =\n mass \u03bc *\n ENNReal.toNNReal (\u2191\u2191\u2191{ val := \u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc), property := (_ : IsProbabilityMeasure \u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc)) } s)\n[PROOFSTEP]\nchange \u03bc s = mass \u03bc * ((mass \u03bc)\u207b\u00b9 \u2022 \u03bc) s\n[GOAL]\ncase inr\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\nh : \u03bc \u2260 0\nmass_nonzero : mass \u03bc \u2260 0\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = mass \u03bc * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc) s)) s\n[PROOFSTEP]\nsimp only [toMeasure_smul, Measure.smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply,\n Measure.nnreal_smul_coe_apply, ne_eq, mass_zero_iff, ENNReal.toNNReal_mul, ENNReal.toNNReal_coe,\n mul_inv_cancel_left\u2080 mass_nonzero]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 \u03bc = mass \u03bc \u2022 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)\n[PROOFSTEP]\napply eq_of_forall_apply_eq\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 \u2200 (s : Set \u03a9),\n MeasurableSet s \u2192\n (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s =\n (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(mass \u03bc \u2022 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) s)) s\n[PROOFSTEP]\nintro s _s_mble\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\n_s_mble : MeasurableSet s\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s =\n (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(mass \u03bc \u2022 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) s)) s\n[PROOFSTEP]\nrw [\u03bc.self_eq_mass_mul_normalize s, coeFn_smul_apply, smul_eq_mul,\n ProbabilityMeasure.coeFn_comp_toFiniteMeasure_eq_coeFn]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nnonzero : \u03bc \u2260 0\ns : Set \u03a9\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize \u03bc) s)) s = (mass \u03bc)\u207b\u00b9 * (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s\n[PROOFSTEP]\nsimp only [\u03bc.self_eq_mass_mul_normalize, \u03bc.mass_nonzero_iff.mpr nonzero, inv_mul_cancel_left\u2080, Ne.def, not_false_iff]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nnonzero : \u03bc \u2260 0\n\u22a2 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc) = (mass \u03bc)\u207b\u00b9 \u2022 \u03bc\n[PROOFSTEP]\nnth_rw 3 [\u03bc.self_eq_mass_smul_normalize]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nnonzero : \u03bc \u2260 0\n\u22a2 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc) =\n (mass \u03bc)\u207b\u00b9 \u2022 mass \u03bc \u2022 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)\n[PROOFSTEP]\nrw [\u2190 smul_assoc]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nnonzero : \u03bc \u2260 0\n\u22a2 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc) =\n ((mass \u03bc)\u207b\u00b9 \u2022 mass \u03bc) \u2022 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)\n[PROOFSTEP]\nsimp only [\u03bc.mass_nonzero_iff.mpr nonzero, Algebra.id.smul_eq_mul, inv_mul_cancel, Ne.def, not_false_iff, one_smul]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nnonzero : \u03bc \u2260 0\n\u22a2 \u2191(normalize \u03bc) = \u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc)\n[PROOFSTEP]\next1 s _s_mble\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nnonzero : \u03bc \u2260 0\ns : Set \u03a9\n_s_mble : MeasurableSet s\n\u22a2 \u2191\u2191\u2191(normalize \u03bc) s = \u2191\u2191\u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc) s\n[PROOFSTEP]\nrw [\u2190 \u03bc.normalize.ennreal_coeFn_eq_coeFn_toMeasure s, \u03bc.normalize_eq_of_nonzero nonzero s, ENNReal.coe_mul,\n ennreal_coeFn_eq_coeFn_toMeasure]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nnonzero : \u03bc \u2260 0\ns : Set \u03a9\n_s_mble : MeasurableSet s\n\u22a2 \u2191(mass \u03bc)\u207b\u00b9 * \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc) s\n[PROOFSTEP]\nexact Measure.coe_nnreal_smul_apply _ _ _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : FiniteMeasure \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 normalize (ProbabilityMeasure.toFiniteMeasure \u03bc) = \u03bc\n[PROOFSTEP]\napply ProbabilityMeasure.eq_of_forall_apply_eq\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : FiniteMeasure \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u22a2 \u2200 (s : Set \u03a9),\n MeasurableSet s \u2192\n (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize (ProbabilityMeasure.toFiniteMeasure \u03bc)) s)) s =\n (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s\n[PROOFSTEP]\nintro s _s_mble\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : FiniteMeasure \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\ns : Set \u03a9\n_s_mble : MeasurableSet s\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize (ProbabilityMeasure.toFiniteMeasure \u03bc)) s)) s =\n (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s\n[PROOFSTEP]\nrw [\u03bc.toFiniteMeasure.normalize_eq_of_nonzero \u03bc.toFiniteMeasure_nonzero s]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : FiniteMeasure \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\ns : Set \u03a9\n_s_mble : MeasurableSet s\n\u22a2 (mass (ProbabilityMeasure.toFiniteMeasure \u03bc))\u207b\u00b9 *\n (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(ProbabilityMeasure.toFiniteMeasure \u03bc) s)) s =\n (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s\n[PROOFSTEP]\nsimp only [ProbabilityMeasure.mass_toFiniteMeasure, inv_one, one_mul]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d : Nonempty \u03a9\nm0\u271d : MeasurableSpace \u03a9\n\u03bc\u271d : FiniteMeasure \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\ns : Set \u03a9\n_s_mble : MeasurableSet s\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u2191(ProbabilityMeasure.toFiniteMeasure \u03bc) s) = ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192 E\n\u22a2 average (\u2191\u03bc) f = \u222b (\u03c9 : \u03a9), f \u03c9 \u2202\u2191(normalize \u03bc)\n[PROOFSTEP]\nrw [\u03bc.toMeasure_normalize_eq_of_nonzero nonzero, average]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192 E\n\u22a2 \u222b (x : \u03a9), f x \u2202(\u2191\u2191\u2191\u03bc univ)\u207b\u00b9 \u2022 \u2191\u03bc = \u222b (\u03c9 : \u03a9), f \u03c9 \u2202\u2191((mass \u03bc)\u207b\u00b9 \u2022 \u03bc)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_\u03bc.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192 E\n\u22a2 (\u2191\u2191\u2191\u03bc univ)\u207b\u00b9 = \u2191ENNReal.ofNNRealHom (mass \u03bc)\u207b\u00b9\n[PROOFSTEP]\nsimp only [RingHom.toFun_eq_coe, ENNReal.coe_ofNNRealHom, ENNReal.coe_inv (\u03bc.mass_nonzero_iff.mpr nonzero),\n ennreal_mass]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d : TopologicalSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN \u03bc f = mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f\n[PROOFSTEP]\nnth_rw 1 [\u03bc.self_eq_mass_smul_normalize]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d : TopologicalSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN (mass \u03bc \u2022 ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f =\n mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f\n[PROOFSTEP]\nrw [\u03bc.normalize.toFiniteMeasure.smul_testAgainstNN_apply \u03bc.mass f, smul_eq_mul]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d : TopologicalSpace \u03a9\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f = (mass \u03bc)\u207b\u00b9 * testAgainstNN \u03bc f\n[PROOFSTEP]\nsimp [\u03bc.testAgainstNN_eq_mass_mul, inv_mul_cancel_left\u2080 <| \u03bc.mass_nonzero_iff.mpr nonzero]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f))\n[PROOFSTEP]\nby_cases h_mass : \u03bc.mass = 0\n[GOAL]\ncase pos\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : mass \u03bc = 0\n\u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f))\n[PROOFSTEP]\nsimp only [\u03bc.mass_zero_iff.mp h_mass, zero_testAgainstNN_apply, zero_mass, eq_self_iff_true] at *\n[GOAL]\ncase pos\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize 0))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nh_mass : True\n\u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd 0)\n[PROOFSTEP]\nexact tendsto_zero_testAgainstNN_of_tendsto_zero_mass mass_lim f\n[GOAL]\ncase neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : \u00acmass \u03bc = 0\n\u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f))\n[PROOFSTEP]\nsimp_rw [fun i => (\u03bcs i).testAgainstNN_eq_mass_mul f, \u03bc.testAgainstNN_eq_mass_mul f]\n[GOAL]\ncase neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : \u00acmass \u03bc = 0\n\u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\n[PROOFSTEP]\nrw [ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds] at \u03bcs_lim \n[GOAL]\ncase neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim :\n Tendsto (ProbabilityMeasure.toFiniteMeasure \u2218 fun i => normalize (\u03bcs i)) F\n (\ud835\udcdd (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : \u00acmass \u03bc = 0\n\u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\n[PROOFSTEP]\nrw [tendsto_iff_forall_testAgainstNN_tendsto] at \u03bcs_lim \n[GOAL]\ncase neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim :\n \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650),\n Tendsto (fun i => testAgainstNN ((ProbabilityMeasure.toFiniteMeasure \u2218 fun i => normalize (\u03bcs i)) i) f) F\n (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : \u00acmass \u03bc = 0\n\u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\n[PROOFSTEP]\nhave lim_pair :\n Tendsto (fun i => (\u27e8(\u03bcs i).mass, (\u03bcs i).normalize.toFiniteMeasure.testAgainstNN f\u27e9 : \u211d\u22650 \u00d7 \u211d\u22650)) F\n (\ud835\udcdd \u27e8\u03bc.mass, \u03bc.normalize.toFiniteMeasure.testAgainstNN f\u27e9) :=\n (Prod.tendsto_iff _ _).mpr \u27e8mass_lim, \u03bcs_lim f\u27e9\n[GOAL]\ncase neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim :\n \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650),\n Tendsto (fun i => testAgainstNN ((ProbabilityMeasure.toFiniteMeasure \u2218 fun i => normalize (\u03bcs i)) i) f) F\n (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nh_mass : \u00acmass \u03bc = 0\nlim_pair :\n Tendsto (fun i => (mass (\u03bcs i), testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f)) F\n (\ud835\udcdd (mass \u03bc, testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\n\u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\n[PROOFSTEP]\nexact tendsto_mul.comp lim_pair\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\n[PROOFSTEP]\nhave lim_mass := \u03bcs_lim.mass\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\n\u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\n[PROOFSTEP]\nhave aux : {(0 : \u211d\u22650)}\u1d9c \u2208 \ud835\udcdd \u03bc.mass := isOpen_compl_singleton.mem_nhds (\u03bc.mass_nonzero_iff.mpr nonzero)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\n\u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\n[PROOFSTEP]\nhave eventually_nonzero : \u2200\u1da0 i in F, \u03bcs i \u2260 0 :=\n by\n simp_rw [\u2190 mass_nonzero_iff]\n exact lim_mass aux\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\n\u22a2 \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\n[PROOFSTEP]\nsimp_rw [\u2190 mass_nonzero_iff]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\n\u22a2 \u2200\u1da0 (i : \u03b3) in F, mass (\u03bcs i) \u2260 0\n[PROOFSTEP]\nexact lim_mass aux\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\n\u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\n[PROOFSTEP]\nhave eve : \u2200\u1da0 i in F, (\u03bcs i).normalize.toFiniteMeasure.testAgainstNN f = (\u03bcs i).mass\u207b\u00b9 * (\u03bcs i).testAgainstNN f :=\n by\n filter_upwards [eventually_iff.mp eventually_nonzero]\n intro i hi\n apply normalize_testAgainstNN _ hi\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\n\u22a2 \u2200\u1da0 (i : \u03b3) in F,\n testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\n[PROOFSTEP]\nfilter_upwards [eventually_iff.mp eventually_nonzero]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\n\u22a2 \u2200 (a : \u03b3),\n \u03bcs a \u2260 0 \u2192\n testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs a))) f = (mass (\u03bcs a))\u207b\u00b9 * testAgainstNN (\u03bcs a) f\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\ni : \u03b3\nhi : \u03bcs i \u2260 0\n\u22a2 testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\n[PROOFSTEP]\napply normalize_testAgainstNN _ hi\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\neve :\n \u2200\u1da0 (i : \u03b3) in F,\n testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\n\u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F\n (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\n[PROOFSTEP]\nsimp_rw [tendsto_congr' eve, \u03bc.normalize_testAgainstNN nonzero]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\neve :\n \u2200\u1da0 (i : \u03b3) in F,\n testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\n\u22a2 Tendsto (fun x => (mass (\u03bcs x))\u207b\u00b9 * testAgainstNN (\u03bcs x) f) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9 * testAgainstNN \u03bc f))\n[PROOFSTEP]\nhave lim_pair :\n Tendsto (fun i => (\u27e8(\u03bcs i).mass\u207b\u00b9, (\u03bcs i).testAgainstNN f\u27e9 : \u211d\u22650 \u00d7 \u211d\u22650)) F (\ud835\udcdd \u27e8\u03bc.mass\u207b\u00b9, \u03bc.testAgainstNN f\u27e9) :=\n by\n refine' (Prod.tendsto_iff _ _).mpr \u27e8_, _\u27e9\n \u00b7 exact (continuousOn_inv\u2080.continuousAt aux).tendsto.comp lim_mass\n \u00b7 exact tendsto_iff_forall_testAgainstNN_tendsto.mp \u03bcs_lim f\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\neve :\n \u2200\u1da0 (i : \u03b3) in F,\n testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\n\u22a2 Tendsto (fun i => ((mass (\u03bcs i))\u207b\u00b9, testAgainstNN (\u03bcs i) f)) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9, testAgainstNN \u03bc f))\n[PROOFSTEP]\nrefine' (Prod.tendsto_iff _ _).mpr \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\neve :\n \u2200\u1da0 (i : \u03b3) in F,\n testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\n\u22a2 Tendsto (fun n => ((mass (\u03bcs n))\u207b\u00b9, testAgainstNN (\u03bcs n) f).fst) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9, testAgainstNN \u03bc f).fst)\n[PROOFSTEP]\nexact (continuousOn_inv\u2080.continuousAt aux).tendsto.comp lim_mass\n[GOAL]\ncase refine'_2\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\neve :\n \u2200\u1da0 (i : \u03b3) in F,\n testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\n\u22a2 Tendsto (fun n => ((mass (\u03bcs n))\u207b\u00b9, testAgainstNN (\u03bcs n) f).snd) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9, testAgainstNN \u03bc f).snd)\n[PROOFSTEP]\nexact tendsto_iff_forall_testAgainstNN_tendsto.mp \u03bcs_lim f\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nlim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\naux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc)\neventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0\neve :\n \u2200\u1da0 (i : \u03b3) in F,\n testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f\nlim_pair : Tendsto (fun i => ((mass (\u03bcs i))\u207b\u00b9, testAgainstNN (\u03bcs i) f)) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9, testAgainstNN \u03bc f))\n\u22a2 Tendsto (fun x => (mass (\u03bcs x))\u207b\u00b9 * testAgainstNN (\u03bcs x) f) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9 * testAgainstNN \u03bc f))\n[PROOFSTEP]\nexact tendsto_mul.comp lim_pair\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc)\n[PROOFSTEP]\nrw [tendsto_iff_forall_testAgainstNN_tendsto]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\n\u22a2 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f))\n[PROOFSTEP]\nexact fun f => tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass \u03bcs_lim mass_lim f\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\n\u22a2 Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\n[PROOFSTEP]\nrw [ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds, tendsto_iff_forall_testAgainstNN_tendsto]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\nnonzero : \u03bc \u2260 0\n\u22a2 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650),\n Tendsto (fun i => testAgainstNN ((ProbabilityMeasure.toFiniteMeasure \u2218 fun i => normalize (\u03bcs i)) i) f) F\n (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f))\n[PROOFSTEP]\nexact fun f => tendsto_normalize_testAgainstNN_of_tendsto \u03bcs_lim nonzero f\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nnonzero : \u03bc \u2260 0\n\u22a2 Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) \u2227 Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) \u2194\n Tendsto \u03bcs F (\ud835\udcdd \u03bc)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nnonzero : \u03bc \u2260 0\n\u22a2 Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) \u2227 Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) \u2192\n Tendsto \u03bcs F (\ud835\udcdd \u03bc)\n[PROOFSTEP]\nrintro \u27e8normalized_lim, mass_lim\u27e9\n[GOAL]\ncase mp.intro\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nnonzero : \u03bc \u2260 0\nnormalized_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc))\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc)\n[PROOFSTEP]\nexact tendsto_of_tendsto_normalize_testAgainstNN_of_tendsto_mass normalized_lim mass_lim\n[GOAL]\ncase mpr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nnonzero : \u03bc \u2260 0\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2192\n Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) \u2227 Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\n[PROOFSTEP]\nintro \u03bcs_lim\n[GOAL]\ncase mpr\n\u03a9 : Type u_1\ninst\u271d\u00b2 : Nonempty \u03a9\nm0 : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nnonzero : \u03bc \u2260 0\n\u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc)\n\u22a2 Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) \u2227 Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc))\n[PROOFSTEP]\nrefine' \u27e8tendsto_normalize_of_tendsto \u03bcs_lim nonzero, \u03bcs_lim.mass\u27e9\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "llama_tokens": 16838, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300698514778, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.4163661078643705}} {"text": "[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : _root_.Finite K\ninst\u271d : FiniteDimensional K V\n\u22a2 _root_.Finite V\n[PROOFSTEP]\ncases nonempty_fintype K\n[GOAL]\ncase intro\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : _root_.Finite K\ninst\u271d : FiniteDimensional K V\nval\u271d : Fintype K\n\u22a2 _root_.Finite V\n[PROOFSTEP]\nhaveI := fintypeOfFintype K V\n[GOAL]\ncase intro\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : _root_.Finite K\ninst\u271d : FiniteDimensional K V\nval\u271d : Fintype K\nthis : Fintype V\n\u22a2 _root_.Finite V\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninst\u271d : _root_.Finite \u03b9\nh : Basis \u03b9 K V\n\u22a2 FiniteDimensional K V\n[PROOFSTEP]\nclassical\ncases nonempty_fintype \u03b9\nexact\n \u27e8\u27e8Finset.univ.image h, by\n convert h.span_eq\n simp\u27e9\u27e9\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninst\u271d : _root_.Finite \u03b9\nh : Basis \u03b9 K V\n\u22a2 FiniteDimensional K V\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninst\u271d : _root_.Finite \u03b9\nh : Basis \u03b9 K V\nval\u271d : Fintype \u03b9\n\u22a2 FiniteDimensional K V\n[PROOFSTEP]\nexact\n \u27e8\u27e8Finset.univ.image h, by\n convert h.span_eq\n simp\u27e9\u27e9\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninst\u271d : _root_.Finite \u03b9\nh : Basis \u03b9 K V\nval\u271d : Fintype \u03b9\n\u22a2 span K \u2191(Finset.image (\u2191h) Finset.univ) = \u22a4\n[PROOFSTEP]\nconvert h.span_eq\n[GOAL]\ncase h.e'_2.h.e'_6\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninst\u271d : _root_.Finite \u03b9\nh : Basis \u03b9 K V\nval\u271d : Fintype \u03b9\n\u22a2 \u2191(Finset.image (\u2191h) Finset.univ) = Set.range \u2191h\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u22a2 Fintype \u2191(Basis.ofVectorSpaceIndex K V)\n[PROOFSTEP]\nletI : IsNoetherian K V := IsNoetherian.iff_fg.2 inferInstance\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nthis : IsNoetherian K V := Iff.mpr iff_fg inferInstance\n\u22a2 Fintype \u2191(Basis.ofVectorSpaceIndex K V)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\n\u22a2 FiniteDimensional K { x // x \u2208 S }\n[PROOFSTEP]\nletI : IsNoetherian K V := iff_fg.2 ?_\n[GOAL]\ncase refine_2\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nthis : IsNoetherian K V := Iff.mpr iff_fg ?refine_1\n\u22a2 FiniteDimensional K { x // x \u2208 S }\ncase refine_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\n\u22a2 Module.Finite K V\n[PROOFSTEP]\nexact iff_fg.1 (IsNoetherian.iff_rank_lt_aleph0.2 (lt_of_le_of_lt (rank_submodule_le _) (rank_lt_aleph0 K V)))\n[GOAL]\ncase refine_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\n\u22a2 Module.Finite K V\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nh : 0 < finrank K V\n\u22a2 FiniteDimensional K V\n[PROOFSTEP]\ncontrapose h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nh : \u00acFiniteDimensional K V\n\u22a2 \u00ac0 < finrank K V\n[PROOFSTEP]\nsimp [finrank_of_infinite_dimensional h]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nhn : finrank K V = Nat.succ n\n\u22a2 0 < finrank K V\n[PROOFSTEP]\nrw [hn]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nhn : finrank K V = Nat.succ n\n\u22a2 0 < Nat.succ n\n[PROOFSTEP]\nexact n.succ_pos\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\nn : \u2115\ninst\u271d : Fact (finrank K V = n + 1)\n\u22a2 0 < finrank K V\n[PROOFSTEP]\nconvert Nat.succ_pos n\n[GOAL]\ncase h.e'_4\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\nn : \u2115\ninst\u271d : Fact (finrank K V = n + 1)\n\u22a2 finrank K V = Nat.succ n\n[PROOFSTEP]\napply Fact.out\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\nW : Type v\ninst\u271d\u00b9 : AddCommGroup W\ninst\u271d : Module K W\nn : \u2115\nhn : n \u2260 0\nhVW : Module.rank K V = n \u2022 Module.rank K W\n\u22a2 FiniteDimensional K V \u2194 FiniteDimensional K W\n[PROOFSTEP]\nsimp only [FiniteDimensional, \u2190 IsNoetherian.iff_fg, IsNoetherian.iff_rank_lt_aleph0, hVW,\n Cardinal.nsmul_lt_aleph0_iff_of_ne_zero hn]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\nh : Basis \u03b9 K V\n\u22a2 \u2191(finrank K V) = #\u03b9\n[PROOFSTEP]\nhaveI : IsNoetherian K V := iff_fg.2 inferInstance\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\nh : Basis \u03b9 K V\nthis : IsNoetherian K V\n\u22a2 \u2191(finrank K V) = #\u03b9\n[PROOFSTEP]\nhaveI : Fintype \u03b9 := fintypeBasisIndex h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\nh : Basis \u03b9 K V\nthis\u271d : IsNoetherian K V\nthis : Fintype \u03b9\n\u22a2 \u2191(finrank K V) = #\u03b9\n[PROOFSTEP]\nrw [Cardinal.mk_fintype, finrank_eq_card_basis h]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\n\u03b9 : Type u_1\nb : Basis \u03b9 K K\n\u22a2 Unique \u03b9\n[PROOFSTEP]\nhave A : Cardinal.mk \u03b9 = \u2191(FiniteDimensional.finrank K K) := (FiniteDimensional.finrank_eq_card_basis' b).symm\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\n\u03b9 : Type u_1\nb : Basis \u03b9 K K\nA : #\u03b9 = \u2191(finrank K K)\n\u22a2 Unique \u03b9\n[PROOFSTEP]\nsimp only [Cardinal.eq_one_iff_unique, FiniteDimensional.finrank_self, Nat.cast_one] at A \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\n\u03b9 : Type u_1\nb : Basis \u03b9 K K\nA : Subsingleton \u03b9 \u2227 Nonempty \u03b9\n\u22a2 Unique \u03b9\n[PROOFSTEP]\nexact Nonempty.some ((unique_iff_subsingleton_and_nonempty _).2 A)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh : finrank K V = 1\nv : V\ni : \u03b9\nhv : v = 0\n\u22a2 \u2191(\u2191(basisUnique \u03b9 h).repr v) i = 0\n[PROOFSTEP]\nrw [hv, LinearEquiv.map_zero, Finsupp.zero_apply]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\nb : \u03b9 \u2192 V\nh : LinearIndependent K b\n\u22a2 #\u03b9 \u2264 \u2191(finrank K V)\n[PROOFSTEP]\nrw [\u2190 lift_le.{max v w}]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\nb : \u03b9 \u2192 V\nh : LinearIndependent K b\n\u22a2 lift #\u03b9 \u2264 lift \u2191(finrank K V)\n[PROOFSTEP]\nsimpa [\u2190 finrank_eq_rank', -finrank_eq_rank] using cardinal_lift_le_rank_of_linearIndependent h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : FiniteDimensional K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nh : LinearIndependent K b\n\u22a2 Fintype.card \u03b9 \u2264 finrank K V\n[PROOFSTEP]\nsimpa using cardinal_mk_le_finrank_of_linearIndependent h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nb : Finset V\nh : LinearIndependent K fun x => \u2191x\n\u22a2 Finset.card b \u2264 finrank K V\n[PROOFSTEP]\nrw [\u2190 Fintype.card_coe]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nb : Finset V\nh : LinearIndependent K fun x => \u2191x\n\u22a2 Fintype.card { x // x \u2208 b } \u2264 finrank K V\n[PROOFSTEP]\nexact fintype_card_le_finrank_of_linearIndependent h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninst\u271d : FiniteDimensional K V\nv : \u03b9 \u2192 V\nh : LinearIndependent K v\n\u22a2 #\u03b9 < \u2135\u2080\n[PROOFSTEP]\napply Cardinal.lift_lt.1\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninst\u271d : FiniteDimensional K V\nv : \u03b9 \u2192 V\nh : LinearIndependent K v\n\u22a2 lift #\u03b9 < lift \u2135\u2080\n[PROOFSTEP]\napply lt_of_le_of_lt\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninst\u271d : FiniteDimensional K V\nv : \u03b9 \u2192 V\nh : LinearIndependent K v\n\u22a2 lift #\u03b9 \u2264 ?b\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninst\u271d : FiniteDimensional K V\nv : \u03b9 \u2192 V\nh : LinearIndependent K v\n\u22a2 ?b < lift \u2135\u2080\ncase b\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninst\u271d : FiniteDimensional K V\nv : \u03b9 \u2192 V\nh : LinearIndependent K v\n\u22a2 Cardinal.{max w ?u.252424}\n[PROOFSTEP]\napply cardinal_lift_le_rank_of_linearIndependent h\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninst\u271d : FiniteDimensional K V\nv : \u03b9 \u2192 V\nh : LinearIndependent K v\n\u22a2 lift (Module.rank K V) < lift \u2135\u2080\n[PROOFSTEP]\nrw [\u2190 finrank_eq_rank, Cardinal.lift_aleph0, Cardinal.lift_natCast]\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninst\u271d : FiniteDimensional K V\nv : \u03b9 \u2192 V\nh : LinearIndependent K v\n\u22a2 \u2191(finrank K V) < \u2135\u2080\n[PROOFSTEP]\napply Cardinal.nat_lt_aleph0\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninf : Infinite \u03b9\ninst\u271d : FiniteDimensional K V\nv : \u03b9 \u2192 V\n\u22a2 \u00acLinearIndependent K v\n[PROOFSTEP]\nintro h_lin_indep\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninf : Infinite \u03b9\ninst\u271d : FiniteDimensional K V\nv : \u03b9 \u2192 V\nh_lin_indep : LinearIndependent K v\n\u22a2 False\n[PROOFSTEP]\nhave : \u00ac\u2135\u2080 \u2264 #\u03b9 := not_le.mpr (lt_aleph0_of_linearIndependent h_lin_indep)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninf : Infinite \u03b9\ninst\u271d : FiniteDimensional K V\nv : \u03b9 \u2192 V\nh_lin_indep : LinearIndependent K v\nthis : \u00ac\u2135\u2080 \u2264 #\u03b9\n\u22a2 False\n[PROOFSTEP]\nhave : \u2135\u2080 \u2264 #\u03b9 := infinite_iff.mp inf\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type w\ninf : Infinite \u03b9\ninst\u271d : FiniteDimensional K V\nv : \u03b9 \u2192 V\nh_lin_indep : LinearIndependent K v\nthis\u271d : \u00ac\u2135\u2080 \u2264 #\u03b9\nthis : \u2135\u2080 \u2264 #\u03b9\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u22a2 0 < finrank K V \u2194 0 < Module.rank K V\n[PROOFSTEP]\nrw [\u2190 finrank_eq_rank]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u22a2 0 < finrank K V \u2194 0 < \u2191(finrank K V)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u22a2 0 < finrank K V \u2194 0 < Module.rank K V\n[PROOFSTEP]\nrw [\u2190 finrank_eq_rank]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u22a2 0 < finrank K V \u2194 0 < \u2191(finrank K V)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u22a2 finrank K V = 0 \u2194 Module.rank K V = 0\n[PROOFSTEP]\nrw [\u2190 finrank_eq_rank]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u22a2 finrank K V = 0 \u2194 \u2191(finrank K V) = 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\n\u22a2 S = \u22a4\n[PROOFSTEP]\nhaveI : IsNoetherian K V := iff_fg.2 inferInstance\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis : IsNoetherian K V\n\u22a2 S = \u22a4\n[PROOFSTEP]\nset bS := Basis.ofVectorSpace K S with bS_eq\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis : IsNoetherian K V\nbS : Basis (\u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S })) K { x // x \u2208 S } := Basis.ofVectorSpace K { x // x \u2208 S }\nbS_eq : bS = Basis.ofVectorSpace K { x // x \u2208 S }\n\u22a2 S = \u22a4\n[PROOFSTEP]\nhave : LinearIndependent K ((\u2191) : ((\u2191) '' Basis.ofVectorSpaceIndex K S : Set V) \u2192 V) :=\n @LinearIndependent.image_subtype _ _ _ _ _ _ _ _ _ (Submodule.subtype S) (by simpa using bS.linearIndependent)\n (by simp)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis : IsNoetherian K V\nbS : Basis (\u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S })) K { x // x \u2208 S } := Basis.ofVectorSpace K { x // x \u2208 S }\nbS_eq : bS = Basis.ofVectorSpace K { x // x \u2208 S }\n\u22a2 LinearIndependent K fun x => \u2191x\n[PROOFSTEP]\nsimpa using bS.linearIndependent\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis : IsNoetherian K V\nbS : Basis (\u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S })) K { x // x \u2208 S } := Basis.ofVectorSpace K { x // x \u2208 S }\nbS_eq : bS = Basis.ofVectorSpace K { x // x \u2208 S }\n\u22a2 Disjoint (span K (Basis.ofVectorSpaceIndex K { x // x \u2208 S })) (LinearMap.ker (Submodule.subtype S))\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis\u271d : IsNoetherian K V\nbS : Basis (\u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S })) K { x // x \u2208 S } := Basis.ofVectorSpace K { x // x \u2208 S }\nbS_eq : bS = Basis.ofVectorSpace K { x // x \u2208 S }\nthis : LinearIndependent K Subtype.val\n\u22a2 S = \u22a4\n[PROOFSTEP]\nset b := Basis.extend this with b_eq\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis\u271d : IsNoetherian K V\nbS : Basis (\u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S })) K { x // x \u2208 S } := Basis.ofVectorSpace K { x // x \u2208 S }\nbS_eq : bS = Basis.ofVectorSpace K { x // x \u2208 S }\nthis : LinearIndependent K Subtype.val\nb : Basis (\u2191(LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ))) K\n V :=\n Basis.extend this\nb_eq : b = Basis.extend this\n\u22a2 S = \u22a4\n[PROOFSTEP]\nletI i1 : Fintype (this.extend _) := (finite_of_linearIndependent (by simpa using b.linearIndependent)).fintype\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis\u271d : IsNoetherian K V\nbS : Basis (\u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S })) K { x // x \u2208 S } := Basis.ofVectorSpace K { x // x \u2208 S }\nbS_eq : bS = Basis.ofVectorSpace K { x // x \u2208 S }\nthis : LinearIndependent K Subtype.val\nb : Basis (\u2191(LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ))) K\n V :=\n Basis.extend this\nb_eq : b = Basis.extend this\n\u22a2 LinearIndependent ?m.281540 Subtype.val\n[PROOFSTEP]\nsimpa using b.linearIndependent\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis\u271d : IsNoetherian K V\nbS : Basis (\u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S })) K { x // x \u2208 S } := Basis.ofVectorSpace K { x // x \u2208 S }\nbS_eq : bS = Basis.ofVectorSpace K { x // x \u2208 S }\nthis : LinearIndependent K Subtype.val\nb : Basis (\u2191(LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ))) K\n V :=\n Basis.extend this\nb_eq : b = Basis.extend this\ni1 : Fintype\n \u2191(LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)) :=\n Set.Finite.fintype\n (_ :\n Set.Finite\n (LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)))\n\u22a2 S = \u22a4\n[PROOFSTEP]\nletI i2 : Fintype (((\u2191) : S \u2192 V) '' Basis.ofVectorSpaceIndex K S) := (finite_of_linearIndependent this).fintype\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis\u271d : IsNoetherian K V\nbS : Basis (\u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S })) K { x // x \u2208 S } := Basis.ofVectorSpace K { x // x \u2208 S }\nbS_eq : bS = Basis.ofVectorSpace K { x // x \u2208 S }\nthis : LinearIndependent K Subtype.val\nb : Basis (\u2191(LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ))) K\n V :=\n Basis.extend this\nb_eq : b = Basis.extend this\ni1 : Fintype\n \u2191(LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)) :=\n Set.Finite.fintype\n (_ :\n Set.Finite\n (LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)))\ni2 : Fintype \u2191(Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }) :=\n Set.Finite.fintype (_ : Set.Finite (Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\n\u22a2 S = \u22a4\n[PROOFSTEP]\nletI i3 : Fintype (Basis.ofVectorSpaceIndex K S) :=\n (finite_of_linearIndependent (by simpa using bS.linearIndependent)).fintype\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis\u271d : IsNoetherian K V\nbS : Basis (\u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S })) K { x // x \u2208 S } := Basis.ofVectorSpace K { x // x \u2208 S }\nbS_eq : bS = Basis.ofVectorSpace K { x // x \u2208 S }\nthis : LinearIndependent K Subtype.val\nb : Basis (\u2191(LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ))) K\n V :=\n Basis.extend this\nb_eq : b = Basis.extend this\ni1 : Fintype\n \u2191(LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)) :=\n Set.Finite.fintype\n (_ :\n Set.Finite\n (LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)))\ni2 : Fintype \u2191(Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }) :=\n Set.Finite.fintype (_ : Set.Finite (Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\n\u22a2 LinearIndependent ?m.287044 Subtype.val\n[PROOFSTEP]\nsimpa using bS.linearIndependent\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis\u271d : IsNoetherian K V\nbS : Basis (\u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S })) K { x // x \u2208 S } := Basis.ofVectorSpace K { x // x \u2208 S }\nbS_eq : bS = Basis.ofVectorSpace K { x // x \u2208 S }\nthis : LinearIndependent K Subtype.val\nb : Basis (\u2191(LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ))) K\n V :=\n Basis.extend this\nb_eq : b = Basis.extend this\ni1 : Fintype\n \u2191(LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)) :=\n Set.Finite.fintype\n (_ :\n Set.Finite\n (LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)))\ni2 : Fintype \u2191(Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }) :=\n Set.Finite.fintype (_ : Set.Finite (Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\ni3 : Fintype \u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S }) :=\n Set.Finite.fintype (_ : Set.Finite (Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\n\u22a2 S = \u22a4\n[PROOFSTEP]\nhave : (\u2191) '' Basis.ofVectorSpaceIndex K S = this.extend (Set.subset_univ _) :=\n Set.eq_of_subset_of_card_le (this.subset_extend _)\n (by\n rw [Set.card_image_of_injective _ Subtype.coe_injective, \u2190 finrank_eq_card_basis bS, \u2190 finrank_eq_card_basis b,\n h])\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis\u271d : IsNoetherian K V\nbS : Basis (\u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S })) K { x // x \u2208 S } := Basis.ofVectorSpace K { x // x \u2208 S }\nbS_eq : bS = Basis.ofVectorSpace K { x // x \u2208 S }\nthis : LinearIndependent K Subtype.val\nb : Basis (\u2191(LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ))) K\n V :=\n Basis.extend this\nb_eq : b = Basis.extend this\ni1 : Fintype\n \u2191(LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)) :=\n Set.Finite.fintype\n (_ :\n Set.Finite\n (LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)))\ni2 : Fintype \u2191(Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }) :=\n Set.Finite.fintype (_ : Set.Finite (Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\ni3 : Fintype \u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S }) :=\n Set.Finite.fintype (_ : Set.Finite (Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\n\u22a2 Fintype.card\n \u2191(LinearIndependent.extend this (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)) \u2264\n Fintype.card \u2191(Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S })\n[PROOFSTEP]\nrw [Set.card_image_of_injective _ Subtype.coe_injective, \u2190 finrank_eq_card_basis bS, \u2190 finrank_eq_card_basis b, h]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis\u271d\u00b9 : IsNoetherian K V\nbS : Basis (\u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S })) K { x // x \u2208 S } := Basis.ofVectorSpace K { x // x \u2208 S }\nbS_eq : bS = Basis.ofVectorSpace K { x // x \u2208 S }\nthis\u271d : LinearIndependent K Subtype.val\nb : Basis (\u2191(LinearIndependent.extend this\u271d (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)))\n K V :=\n Basis.extend this\u271d\nb_eq : b = Basis.extend this\u271d\ni1 : Fintype\n \u2191(LinearIndependent.extend this\u271d (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)) :=\n Set.Finite.fintype\n (_ :\n Set.Finite\n (LinearIndependent.extend this\u271d (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)))\ni2 : Fintype \u2191(Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }) :=\n Set.Finite.fintype (_ : Set.Finite (Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\ni3 : Fintype \u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S }) :=\n Set.Finite.fintype (_ : Set.Finite (Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\nthis :\n Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } =\n LinearIndependent.extend this\u271d (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)\n\u22a2 S = \u22a4\n[PROOFSTEP]\nrw [\u2190 b.span_eq, b_eq, Basis.coe_extend, Subtype.range_coe, \u2190 this, \u2190 Submodule.coeSubtype, span_image]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis\u271d\u00b9 : IsNoetherian K V\nbS : Basis (\u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S })) K { x // x \u2208 S } := Basis.ofVectorSpace K { x // x \u2208 S }\nbS_eq : bS = Basis.ofVectorSpace K { x // x \u2208 S }\nthis\u271d : LinearIndependent K Subtype.val\nb : Basis (\u2191(LinearIndependent.extend this\u271d (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)))\n K V :=\n Basis.extend this\u271d\nb_eq : b = Basis.extend this\u271d\ni1 : Fintype\n \u2191(LinearIndependent.extend this\u271d (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)) :=\n Set.Finite.fintype\n (_ :\n Set.Finite\n (LinearIndependent.extend this\u271d (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)))\ni2 : Fintype \u2191(Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }) :=\n Set.Finite.fintype (_ : Set.Finite (Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\ni3 : Fintype \u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S }) :=\n Set.Finite.fintype (_ : Set.Finite (Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\nthis :\n Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } =\n LinearIndependent.extend this\u271d (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)\n\u22a2 S = Submodule.map (Submodule.subtype S) (span K (Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\n[PROOFSTEP]\nhave := bS.span_eq\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis\u271d\u00b2 : IsNoetherian K V\nbS : Basis (\u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S })) K { x // x \u2208 S } := Basis.ofVectorSpace K { x // x \u2208 S }\nbS_eq : bS = Basis.ofVectorSpace K { x // x \u2208 S }\nthis\u271d\u00b9 : LinearIndependent K Subtype.val\nb : Basis (\u2191(LinearIndependent.extend this\u271d\u00b9 (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)))\n K V :=\n Basis.extend this\u271d\u00b9\nb_eq : b = Basis.extend this\u271d\u00b9\ni1 : Fintype\n \u2191(LinearIndependent.extend this\u271d\u00b9 (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)) :=\n Set.Finite.fintype\n (_ :\n Set.Finite\n (LinearIndependent.extend this\u271d\u00b9 (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)))\ni2 : Fintype \u2191(Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }) :=\n Set.Finite.fintype (_ : Set.Finite (Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\ni3 : Fintype \u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S }) :=\n Set.Finite.fintype (_ : Set.Finite (Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\nthis\u271d :\n Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } =\n LinearIndependent.extend this\u271d\u00b9 (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)\nthis : span K (Set.range \u2191bS) = \u22a4\n\u22a2 S = Submodule.map (Submodule.subtype S) (span K (Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\n[PROOFSTEP]\nrw [bS_eq, Basis.coe_ofVectorSpace, Subtype.range_coe] at this \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nS : Submodule K V\nh : finrank K { x // x \u2208 S } = finrank K V\nthis\u271d\u00b2 : IsNoetherian K V\nbS : Basis (\u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S })) K { x // x \u2208 S } := Basis.ofVectorSpace K { x // x \u2208 S }\nbS_eq : bS = Basis.ofVectorSpace K { x // x \u2208 S }\nthis\u271d\u00b9 : LinearIndependent K Subtype.val\nb : Basis (\u2191(LinearIndependent.extend this\u271d\u00b9 (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)))\n K V :=\n Basis.extend this\u271d\u00b9\nb_eq : b = Basis.extend this\u271d\u00b9\ni1 : Fintype\n \u2191(LinearIndependent.extend this\u271d\u00b9 (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)) :=\n Set.Finite.fintype\n (_ :\n Set.Finite\n (LinearIndependent.extend this\u271d\u00b9 (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)))\ni2 : Fintype \u2191(Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }) :=\n Set.Finite.fintype (_ : Set.Finite (Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\ni3 : Fintype \u2191(Basis.ofVectorSpaceIndex K { x // x \u2208 S }) :=\n Set.Finite.fintype (_ : Set.Finite (Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\nthis\u271d :\n Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } =\n LinearIndependent.extend this\u271d\u00b9 (_ : Subtype.val '' Basis.ofVectorSpaceIndex K { x // x \u2208 S } \u2286 Set.univ)\nthis : span K (Basis.ofVectorSpaceIndex K { x // x \u2208 S }) = \u22a4\n\u22a2 S = Submodule.map (Submodule.subtype S) (span K (Basis.ofVectorSpaceIndex K { x // x \u2208 S }))\n[PROOFSTEP]\nrw [this, map_top (Submodule.subtype S), range_subtype]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\n\u22a2 FiniteDimensional K K\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\np : \u03b9 \u2192 Submodule K V\nhp : CompleteLattice.Independent p\n\u22a2 #{ i // p i \u2260 \u22a5 } \u2264 \u2191(finrank K V)\n[PROOFSTEP]\nsuffices Cardinal.lift.{v} #{ i // p i \u2260 \u22a5 } \u2264 Cardinal.lift.{v} (finrank K V : Cardinal.{w}) by\n rwa [Cardinal.lift_le] at this \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\np : \u03b9 \u2192 Submodule K V\nhp : CompleteLattice.Independent p\nthis : lift #{ i // p i \u2260 \u22a5 } \u2264 lift \u2191(finrank K V)\n\u22a2 #{ i // p i \u2260 \u22a5 } \u2264 \u2191(finrank K V)\n[PROOFSTEP]\nrwa [Cardinal.lift_le] at this \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\np : \u03b9 \u2192 Submodule K V\nhp : CompleteLattice.Independent p\n\u22a2 lift #{ i // p i \u2260 \u22a5 } \u2264 lift \u2191(finrank K V)\n[PROOFSTEP]\ncalc\n Cardinal.lift.{v} #{ i // p i \u2260 \u22a5 } \u2264 Cardinal.lift.{w} (Module.rank K V) := hp.subtype_ne_bot_le_rank\n _ = Cardinal.lift.{w} (finrank K V : Cardinal.{v}) := by rw [finrank_eq_rank]\n _ = Cardinal.lift.{v} (finrank K V : Cardinal.{w}) := by simp\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\np : \u03b9 \u2192 Submodule K V\nhp : CompleteLattice.Independent p\n\u22a2 lift (Module.rank K V) = lift \u2191(finrank K V)\n[PROOFSTEP]\nrw [finrank_eq_rank]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\np : \u03b9 \u2192 Submodule K V\nhp : CompleteLattice.Independent p\n\u22a2 lift \u2191(finrank K V) = lift \u2191(finrank K V)\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\np : \u03b9 \u2192 Submodule K V\nhp : CompleteLattice.Independent p\n\u22a2 Fintype { i // p i \u2260 \u22a5 }\n[PROOFSTEP]\nsuffices #{ i // p i \u2260 \u22a5 } < (\u2135\u2080 : Cardinal.{w})\n by\n rw [Cardinal.lt_aleph0_iff_fintype] at this \n exact this.some\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\np : \u03b9 \u2192 Submodule K V\nhp : CompleteLattice.Independent p\nthis : #{ i // p i \u2260 \u22a5 } < \u2135\u2080\n\u22a2 Fintype { i // p i \u2260 \u22a5 }\n[PROOFSTEP]\nrw [Cardinal.lt_aleph0_iff_fintype] at this \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\np : \u03b9 \u2192 Submodule K V\nhp : CompleteLattice.Independent p\nthis : Nonempty (Fintype { i // p i \u2260 \u22a5 })\n\u22a2 Fintype { i // p i \u2260 \u22a5 }\n[PROOFSTEP]\nexact this.some\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\np : \u03b9 \u2192 Submodule K V\nhp : CompleteLattice.Independent p\n\u22a2 #{ i // p i \u2260 \u22a5 } < \u2135\u2080\n[PROOFSTEP]\nrefine' lt_of_le_of_lt hp.subtype_ne_bot_le_finrank_aux _\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\np : \u03b9 \u2192 Submodule K V\nhp : CompleteLattice.Independent p\n\u22a2 \u2191(finrank K V) < \u2135\u2080\n[PROOFSTEP]\nsimp [Cardinal.nat_lt_aleph0]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : FiniteDimensional K V\n\u03b9 : Type w\np : \u03b9 \u2192 Submodule K V\nhp : CompleteLattice.Independent p\ninst\u271d : Fintype { i // p i \u2260 \u22a5 }\n\u22a2 Fintype.card { i // p i \u2260 \u22a5 } \u2264 finrank K V\n[PROOFSTEP]\nsimpa using hp.subtype_ne_bot_le_finrank_aux\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nclassical\nhave := mt finset_card_le_finrank_of_linearIndependent (by simpa using h)\nrw [not_linearIndependent_iff] at this \nobtain \u27e8s, g, sum, z, zm, nonzero\u27e9 := this\nlet f : V \u2192 K := fun x =>\n if h : x \u2208 t then if (\u27e8x, h\u27e9 : t) \u2208 s then g \u27e8x, h\u27e9 else 0\n else\n 0\n -- and finally clean up the mess caused by the extension.\nrefine' \u27e8f, _, _\u27e9\n\u00b7 dsimp\n rw [\u2190 (sum)]\n -- porting note: need parens to disambiguate\n fapply sum_bij_ne_zero fun v hvt _ => (\u27e8v, hvt\u27e9 : { v // v \u2208 t })\n \u00b7 intro v hvt H\n dsimp\n rw [dif_pos hvt] at H \n contrapose! H\n rw [if_neg H, zero_smul]\n \u00b7 intro _ _ _ _ _ _\n exact Subtype.mk.inj\n \u00b7 intro b hbs hb\n use b\n simpa only [hbs, exists_prop, dif_pos, Finset.mk_coe, and_true_iff, if_true, Finset.coe_mem, eq_self_iff_true,\n exists_prop_of_true, Ne.def] using hb\n \u00b7 intro a h\u2081\n dsimp\n rw [dif_pos h\u2081]\n intro h\u2082\n rw [if_pos]\n contrapose! h\u2082\n rw [if_neg h\u2082, zero_smul]\n\u00b7 refine' \u27e8z, z.2, _\u27e9\n dsimp only\n erw [dif_pos z.2, if_pos] <;> rwa [Subtype.coe_eta]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nhave := mt finset_card_le_finrank_of_linearIndependent (by simpa using h)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\n\u22a2 \u00accard ?m.337366 \u2264 finrank ?m.337360 ?m.337361\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\nthis : \u00acLinearIndependent K fun x => \u2191x\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nrw [not_linearIndependent_iff] at this \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\nthis : \u2203 s g, \u2211 i in s, g i \u2022 \u2191i = 0 \u2227 \u2203 i, i \u2208 s \u2227 g i \u2260 0\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nobtain \u27e8s, g, sum, z, zm, nonzero\u27e9 := this\n[GOAL]\ncase intro.intro.intro.intro.intro\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nlet f : V \u2192 K := fun x =>\n if h : x \u2208 t then if (\u27e8x, h\u27e9 : t) \u2208 s then g \u27e8x, h\u27e9 else 0\n else\n 0\n -- and finally clean up the mess caused by the extension.\n[GOAL]\ncase intro.intro.intro.intro.intro\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nrefine' \u27e8f, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\n\u22a2 \u2211 e in t, f e \u2022 e = 0\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\n\u22a2 \u2211 e in t,\n (if h : e \u2208 t then if { val := e, property := h } \u2208 s then g { val := e, property := h } else 0 else 0) \u2022 e =\n 0\n[PROOFSTEP]\nrw [\u2190 (sum)]\n -- porting note: need parens to disambiguate\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\n\u22a2 \u2211 e in t,\n (if h : e \u2208 t then if { val := e, property := h } \u2208 s then g { val := e, property := h } else 0 else 0) \u2022 e =\n \u2211 i in s, g i \u2022 \u2191i\n[PROOFSTEP]\nfapply sum_bij_ne_zero fun v hvt _ => (\u27e8v, hvt\u27e9 : { v // v \u2208 t })\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.hi\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\n\u22a2 \u2200 (a : V) (h\u2081 : a \u2208 t),\n (if h : a \u2208 t then if { val := a, property := h } \u2208 s then g { val := a, property := h } else 0 else 0) \u2022 a \u2260 0 \u2192\n { val := a, property := h\u2081 } \u2208 s\n[PROOFSTEP]\nintro v hvt H\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.hi\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\nv : V\nhvt : v \u2208 t\nH : (if h : v \u2208 t then if { val := v, property := h } \u2208 s then g { val := v, property := h } else 0 else 0) \u2022 v \u2260 0\n\u22a2 { val := v, property := hvt } \u2208 s\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.hi\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\nv : V\nhvt : v \u2208 t\nH : (if h : v \u2208 t then if { val := v, property := h } \u2208 s then g { val := v, property := h } else 0 else 0) \u2022 v \u2260 0\n\u22a2 { val := v, property := hvt } \u2208 s\n[PROOFSTEP]\nrw [dif_pos hvt] at H \n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.hi\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\nv : V\nhvt : v \u2208 t\nH : (if { val := v, property := hvt } \u2208 s then g { val := v, property := hvt } else 0) \u2022 v \u2260 0\n\u22a2 { val := v, property := hvt } \u2208 s\n[PROOFSTEP]\ncontrapose! H\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.hi\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\nv : V\nhvt : v \u2208 t\nH : \u00ac{ val := v, property := hvt } \u2208 s\n\u22a2 (if { val := v, property := hvt } \u2208 s then g { val := v, property := hvt } else 0) \u2022 v = 0\n[PROOFSTEP]\nrw [if_neg H, zero_smul]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.i_inj\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\n\u22a2 \u2200 (a\u2081 a\u2082 : V) (h\u2081\u2081 : a\u2081 \u2208 t),\n (if h : a\u2081 \u2208 t then if { val := a\u2081, property := h } \u2208 s then g { val := a\u2081, property := h } else 0 else 0) \u2022 a\u2081 \u2260\n 0 \u2192\n \u2200 (h\u2082\u2081 : a\u2082 \u2208 t),\n (if h : a\u2082 \u2208 t then if { val := a\u2082, property := h } \u2208 s then g { val := a\u2082, property := h } else 0 else 0) \u2022\n a\u2082 \u2260\n 0 \u2192\n { val := a\u2081, property := h\u2081\u2081 } = { val := a\u2082, property := h\u2082\u2081 } \u2192 a\u2081 = a\u2082\n[PROOFSTEP]\nintro _ _ _ _ _ _\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.i_inj\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\na\u2081\u271d a\u2082\u271d : V\nh\u2081\u2081\u271d : a\u2081\u271d \u2208 t\nh\u2081\u2082\u271d :\n (if h : a\u2081\u271d \u2208 t then if { val := a\u2081\u271d, property := h } \u2208 s then g { val := a\u2081\u271d, property := h } else 0 else 0) \u2022 a\u2081\u271d \u2260\n 0\nh\u2082\u2081\u271d : a\u2082\u271d \u2208 t\nh\u2082\u2082\u271d :\n (if h : a\u2082\u271d \u2208 t then if { val := a\u2082\u271d, property := h } \u2208 s then g { val := a\u2082\u271d, property := h } else 0 else 0) \u2022 a\u2082\u271d \u2260\n 0\n\u22a2 { val := a\u2081\u271d, property := h\u2081\u2081\u271d } = { val := a\u2082\u271d, property := h\u2082\u2081\u271d } \u2192 a\u2081\u271d = a\u2082\u271d\n[PROOFSTEP]\nexact Subtype.mk.inj\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.i_surj\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\n\u22a2 \u2200 (b : { v // v \u2208 t }), b \u2208 s \u2192 g b \u2022 \u2191b \u2260 0 \u2192 \u2203 a h\u2081 h\u2082, b = { val := a, property := h\u2081 }\n[PROOFSTEP]\nintro b hbs hb\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.i_surj\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\nb : { v // v \u2208 t }\nhbs : b \u2208 s\nhb : g b \u2022 \u2191b \u2260 0\n\u22a2 \u2203 a h\u2081 h\u2082, b = { val := a, property := h\u2081 }\n[PROOFSTEP]\nuse b\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\nb : { v // v \u2208 t }\nhbs : b \u2208 s\nhb : g b \u2022 \u2191b \u2260 0\n\u22a2 \u2203 h\u2081 h\u2082, b = { val := \u2191b, property := h\u2081 }\n[PROOFSTEP]\nsimpa only [hbs, exists_prop, dif_pos, Finset.mk_coe, and_true_iff, if_true, Finset.coe_mem, eq_self_iff_true,\n exists_prop_of_true, Ne.def] using hb\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.h\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\n\u22a2 \u2200 (a : V) (h\u2081 : a \u2208 t),\n (if h : a \u2208 t then if { val := a, property := h } \u2208 s then g { val := a, property := h } else 0 else 0) \u2022 a \u2260 0 \u2192\n (if h : a \u2208 t then if { val := a, property := h } \u2208 s then g { val := a, property := h } else 0 else 0) \u2022 a =\n g { val := a, property := h\u2081 } \u2022 \u2191{ val := a, property := h\u2081 }\n[PROOFSTEP]\nintro a h\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.h\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\na : V\nh\u2081 : a \u2208 t\n\u22a2 (if h : a \u2208 t then if { val := a, property := h } \u2208 s then g { val := a, property := h } else 0 else 0) \u2022 a \u2260 0 \u2192\n (if h : a \u2208 t then if { val := a, property := h } \u2208 s then g { val := a, property := h } else 0 else 0) \u2022 a =\n g { val := a, property := h\u2081 } \u2022 \u2191{ val := a, property := h\u2081 }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.h\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\na : V\nh\u2081 : a \u2208 t\n\u22a2 \u00ac(if h : a \u2208 t then if { val := a, property := h } \u2208 s then g { val := a, property := h } else 0 else 0) \u2022 a = 0 \u2192\n (if h : a \u2208 t then if { val := a, property := h } \u2208 s then g { val := a, property := h } else 0 else 0) \u2022 a =\n g { val := a, property := h\u2081 } \u2022 a\n[PROOFSTEP]\nrw [dif_pos h\u2081]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.h\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\na : V\nh\u2081 : a \u2208 t\n\u22a2 \u00ac(if { val := a, property := h\u2081 } \u2208 s then g { val := a, property := h\u2081 } else 0) \u2022 a = 0 \u2192\n (if { val := a, property := h\u2081 } \u2208 s then g { val := a, property := h\u2081 } else 0) \u2022 a =\n g { val := a, property := h\u2081 } \u2022 a\n[PROOFSTEP]\nintro h\u2082\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.h\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\na : V\nh\u2081 : a \u2208 t\nh\u2082 : \u00ac(if { val := a, property := h\u2081 } \u2208 s then g { val := a, property := h\u2081 } else 0) \u2022 a = 0\n\u22a2 (if { val := a, property := h\u2081 } \u2208 s then g { val := a, property := h\u2081 } else 0) \u2022 a =\n g { val := a, property := h\u2081 } \u2022 a\n[PROOFSTEP]\nrw [if_pos]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.h.hc\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\na : V\nh\u2081 : a \u2208 t\nh\u2082 : \u00ac(if { val := a, property := h\u2081 } \u2208 s then g { val := a, property := h\u2081 } else 0) \u2022 a = 0\n\u22a2 { val := a, property := h\u2081 } \u2208 s\n[PROOFSTEP]\ncontrapose! h\u2082\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.h.hc\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\na : V\nh\u2081 : a \u2208 t\nh\u2082 : \u00ac{ val := a, property := h\u2081 } \u2208 s\n\u22a2 (if { val := a, property := h\u2081 } \u2208 s then g { val := a, property := h\u2081 } else 0) \u2022 a = 0\n[PROOFSTEP]\nrw [if_neg h\u2082, zero_smul]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\n\u22a2 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nrefine' \u27e8z, z.2, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\n\u22a2 f \u2191z \u2260 0\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\n\u22a2 (if h : \u2191z \u2208 t then if { val := \u2191z, property := h } \u2208 s then g { val := \u2191z, property := h } else 0 else 0) \u2260 0\n[PROOFSTEP]\nerw [dif_pos z.2, if_pos]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\n\u22a2 g { val := \u2191z, property := (_ : \u2191z \u2208 t) } \u2260 0\n[PROOFSTEP]\nrwa [Subtype.coe_eta]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2.hc\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V < card t\ns : Finset { x // x \u2208 t }\ng : { x // x \u2208 t } \u2192 K\nsum : \u2211 i in s, g i \u2022 \u2191i = 0\nz : { x // x \u2208 t }\nzm : z \u2208 s\nnonzero : g z \u2260 0\nf : V \u2192 K :=\n fun x => if h : x \u2208 t then if { val := x, property := h } \u2208 s then g { val := x, property := h } else 0 else 0\n\u22a2 { val := \u2191z, property := (_ : \u2191z \u2208 t) } \u2208 s\n[PROOFSTEP]\nrwa [Subtype.coe_eta]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nclassical\n -- Pick an element x\u2080 \u2208 t,\nhave card_pos : 0 < t.card := lt_trans (Nat.succ_pos _) h\nobtain \u27e8x\u2080, m\u27e9 := (Finset.card_pos.1 card_pos).bex\nlet shift : V \u21aa V := \u27e8fun x => x - x\u2080, sub_left_injective\u27e9\nlet t' := (t.erase x\u2080).map shift\nhave h' : finrank K V < t'.card := by\n simp only [card_map, Finset.card_erase_of_mem m]\n exact Nat.lt_pred_iff.mpr h\nobtain \u27e8g, gsum, x\u2081, x\u2081_mem, nz\u27e9 := exists_nontrivial_relation_of_rank_lt_card h'\nlet f : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in t.erase x\u2080, g (z - x\u2080) else g (z - x\u2080)\nrefine'\n \u27e8f, _, _, _\u27e9\n -- After this, it's a matter of verifying the properties,\n -- based on the corresponding properties for `g`.\n\u00b7 show\n (\u2211 e : V in t, f e \u2022 e) =\n 0\n -- We prove this by splitting off the `x\u2080` term of the sum,\n -- which is itself a sum over `t.erase x\u2080`,\n -- combining the two sums, and\n -- observing that after reindexing we have exactly\n -- \u2211 (x : V) in t', g x \u2022 x = 0.\n simp only\n conv_lhs =>\n apply_congr\n rfl\n rw [ite_smul]\n rw [Finset.sum_ite]\n conv =>\n congr\n congr\n apply_congr\n -- Porting note: the next two steps used to work by `simp [filter_eq', m]`\n erw [filter_eq']\n simp [m]\n conv =>\n congr\n congr\n rfl\n apply_congr\n simp [filter_ne']\n rw [sum_singleton, neg_smul, add_comm, \u2190 sub_eq_add_neg, sum_smul, \u2190 sum_sub_distrib]\n simp only [\u2190 smul_sub]\n -- At the end we have to reindex the sum, so we use `change` to\n -- express the summand using `shift`.\n change\n (\u2211 x : V in t.erase x\u2080, (fun e => g e \u2022 e) (shift x)) =\n 0\n -- porting note: last argument can't be inferred\n rw [\u2190 sum_map _ shift (fun e => g e \u2022 e)]\n exact gsum\n\u00b7 show\n (\u2211 e : V in t, f e) =\n 0\n -- Again we split off the `x\u2080` term,\n -- observing that it exactly cancels the other terms.\n rw [\u2190 insert_erase m, sum_insert (not_mem_erase x\u2080 t)]\n dsimp\n rw [if_pos rfl]\n conv_lhs =>\n congr\n rfl\n apply_congr\n rfl\n rw [if_neg (show _ \u2260 x\u2080 from (mem_erase.mp \u2039_\u203a).1)]\n exact neg_add_self _\n\u00b7 show\n \u2203 (x : V),\n x \u2208 t \u2227\n f x \u2260\n 0\n -- We can use x\u2081 + x\u2080.\n refine' \u27e8x\u2081 + x\u2080, _, _\u27e9\n \u00b7 rw [Finset.mem_map] at x\u2081_mem \n rcases x\u2081_mem with \u27e8x\u2081, x\u2081_mem, rfl\u27e9\n rw [mem_erase] at x\u2081_mem \n simp only [x\u2081_mem, sub_add_cancel, Function.Embedding.coeFn_mk]\n \u00b7 dsimp only\n rwa [if_neg, add_sub_cancel]\n rw [add_left_eq_self]\n rintro rfl\n simp only [sub_eq_zero, exists_prop, Finset.mem_map, Embedding.coeFn_mk, eq_self_iff_true, mem_erase, not_true,\n exists_eq_right, Ne.def, false_and_iff] at x\u2081_mem \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nhave card_pos : 0 < t.card := lt_trans (Nat.succ_pos _) h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nobtain \u27e8x\u2080, m\u27e9 := (Finset.card_pos.1 card_pos).bex\n[GOAL]\ncase intro\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nlet shift : V \u21aa V := \u27e8fun x => x - x\u2080, sub_left_injective\u27e9\n[GOAL]\ncase intro\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nlet t' := (t.erase x\u2080).map shift\n[GOAL]\ncase intro\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nhave h' : finrank K V < t'.card := by\n simp only [card_map, Finset.card_erase_of_mem m]\n exact Nat.lt_pred_iff.mpr h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\n\u22a2 finrank K V < card t'\n[PROOFSTEP]\nsimp only [card_map, Finset.card_erase_of_mem m]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\n\u22a2 finrank K V < card t - 1\n[PROOFSTEP]\nexact Nat.lt_pred_iff.mpr h\n[GOAL]\ncase intro\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nobtain \u27e8g, gsum, x\u2081, x\u2081_mem, nz\u27e9 := exists_nontrivial_relation_of_rank_lt_card h'\n[GOAL]\ncase intro.intro.intro.intro.intro\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nlet f : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in t.erase x\u2080, g (z - x\u2080) else g (z - x\u2080)\n[GOAL]\ncase intro.intro.intro.intro.intro\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nrefine'\n \u27e8f, _, _, _\u27e9\n -- After this, it's a matter of verifying the properties,\n -- based on the corresponding properties for `g`.\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u2211 e in t, f e \u2022 e = 0\n[PROOFSTEP]\nshow\n (\u2211 e : V in t, f e \u2022 e) =\n 0\n -- We prove this by splitting off the `x\u2080` term of the sum,\n -- which is itself a sum over `t.erase x\u2080`,\n -- combining the two sums, and\n -- observing that after reindexing we have exactly\n -- \u2211 (x : V) in t', g x \u2022 x = 0.\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u2211 e in t, f e \u2022 e = 0\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u2211 x in t, (if x = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x - x\u2080)) \u2022 x = 0\n[PROOFSTEP]\nconv_lhs =>\n apply_congr\n rfl\n rw [ite_smul]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in t, (if x = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x - x\u2080)) \u2022 x\n[PROOFSTEP]\n apply_congr\n rfl\n rw [ite_smul]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in t, (if x = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x - x\u2080)) \u2022 x\n[PROOFSTEP]\n apply_congr\n rfl\n rw [ite_smul]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in t, (if x = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x - x\u2080)) \u2022 x\n[PROOFSTEP]\napply_congr\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 t = ?s\u2082\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\nx\u271d : V\na\u271d : x\u271d \u2208 ?s\u2082\n\u22a2 (if x\u271d = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x\u271d - x\u2080)) \u2022 x\u271d = ?g x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\nx\u271d : V\na\u271d : x\u271d \u2208 t\n\u22a2 (if x\u271d = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x\u271d - x\u2080)) \u2022 x\u271d = ?g x\u271d\n[PROOFSTEP]\nrw [ite_smul]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 (\u2211 x in t, if x = x\u2080 then (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x else g (x - x\u2080) \u2022 x) = 0\n[PROOFSTEP]\nrw [Finset.sum_ite]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u2211 x in filter (fun x => x = x\u2080) t, (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x +\n \u2211 x in filter (fun x => \u00acx = x\u2080) t, g (x - x\u2080) \u2022 x =\n 0\n[PROOFSTEP]\nconv =>\n congr\n congr\n apply_congr\n -- Porting note: the next two steps used to work by `simp [filter_eq', m]`\n erw [filter_eq']\n simp [m]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in filter (fun x => x = x\u2080) t, (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x +\n \u2211 x in filter (fun x => \u00acx = x\u2080) t, g (x - x\u2080) \u2022 x =\n 0\n[PROOFSTEP]\n congr\n congr\n apply_congr\n -- Porting note: the next two steps used to work by `simp [filter_eq', m]`\n erw [filter_eq']\n simp [m]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in filter (fun x => x = x\u2080) t, (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x +\n \u2211 x in filter (fun x => \u00acx = x\u2080) t, g (x - x\u2080) \u2022 x =\n 0\n[PROOFSTEP]\n congr\n congr\n apply_congr\n -- Porting note: the next two steps used to work by `simp [filter_eq', m]`\n erw [filter_eq']\n simp [m]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in filter (fun x => x = x\u2080) t, (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x +\n \u2211 x in filter (fun x => \u00acx = x\u2080) t, g (x - x\u2080) \u2022 x =\n 0\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in filter (fun x => x = x\u2080) t, (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x +\n \u2211 x in filter (fun x => \u00acx = x\u2080) t, g (x - x\u2080) \u2022 x\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| 0\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in filter (fun x => x = x\u2080) t, (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x\ncase a.a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in filter (fun x => \u00acx = x\u2080) t, g (x - x\u2080) \u2022 x\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| 0\n[PROOFSTEP]\napply_congr\n -- Porting note: the next two steps used to work by `simp [filter_eq', m]`\n[GOAL]\ncase a.a.h\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 filter (fun x => x = x\u2080) t = ?a.a.s\u2082\ncase a.a.a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\nx\u271d : V\na\u271d : x\u271d \u2208 ?a.a.s\u2082\n\u22a2 (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x\u271d = ?a.a.g x\u271d\ncase a.a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in filter (fun x => \u00acx = x\u2080) t, g (x - x\u2080) \u2022 x\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| 0\n[PROOFSTEP]\nerw [filter_eq']\n[GOAL]\ncase a.a.h\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| if x\u2080 \u2208 t then {x\u2080} else \u2205\ncase a.a.a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\nx\u271d : V\na\u271d : x\u271d \u2208 ?a.a.s\u2082\n\u22a2 (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x\u271d = ?a.a.g x\u271d\ncase a.a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in filter (fun x => \u00acx = x\u2080) t, g (x - x\u2080) \u2022 x\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| 0\n[PROOFSTEP]\nsimp [m]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u2211 x in {x\u2080}, (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x + \u2211 x in filter (fun x => \u00acx = x\u2080) t, g (x - x\u2080) \u2022 x = 0\n[PROOFSTEP]\nconv =>\n congr\n congr\n rfl\n apply_congr\n simp [filter_ne']\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in {x\u2080}, (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x + \u2211 x in filter (fun x => \u00acx = x\u2080) t, g (x - x\u2080) \u2022 x = 0\n[PROOFSTEP]\n congr\n congr\n rfl\n apply_congr\n simp [filter_ne']\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in {x\u2080}, (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x + \u2211 x in filter (fun x => \u00acx = x\u2080) t, g (x - x\u2080) \u2022 x = 0\n[PROOFSTEP]\n congr\n congr\n rfl\n apply_congr\n simp [filter_ne']\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in {x\u2080}, (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x + \u2211 x in filter (fun x => \u00acx = x\u2080) t, g (x - x\u2080) \u2022 x = 0\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in {x\u2080}, (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x + \u2211 x in filter (fun x => \u00acx = x\u2080) t, g (x - x\u2080) \u2022 x\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| 0\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in {x\u2080}, (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x\ncase a.a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in filter (fun x => \u00acx = x\u2080) t, g (x - x\u2080) \u2022 x\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in filter (fun x => \u00acx = x\u2080) t, g (x - x\u2080) \u2022 x\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| 0\n[PROOFSTEP]\napply_congr\n[GOAL]\ncase a.a.h\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 filter (fun x => \u00acx = x\u2080) t = ?a.a.s\u2082\ncase a.a.a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\nx\u271d : V\na\u271d : x\u271d \u2208 ?a.a.s\u2082\n\u22a2 g (x\u271d - x\u2080) \u2022 x\u271d = ?a.a.g x\u271d\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| 0\n[PROOFSTEP]\nsimp [filter_ne']\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u2211 x in {x\u2080}, (-\u2211 z in erase t x\u2080, g (z - x\u2080)) \u2022 x + \u2211 x in erase t x\u2080, g (x - x\u2080) \u2022 x = 0\n[PROOFSTEP]\nrw [sum_singleton, neg_smul, add_comm, \u2190 sub_eq_add_neg, sum_smul, \u2190 sum_sub_distrib]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u2211 x in erase t x\u2080, (g (x - x\u2080) \u2022 x - g (x - x\u2080) \u2022 x\u2080) = 0\n[PROOFSTEP]\nsimp only [\u2190 smul_sub]\n -- At the end we have to reindex the sum, so we use `change` to\n -- express the summand using `shift`.\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u2211 x in erase t x\u2080, g (x - x\u2080) \u2022 (x - x\u2080) = 0\n[PROOFSTEP]\nchange\n (\u2211 x : V in t.erase x\u2080, (fun e => g e \u2022 e) (shift x)) =\n 0\n -- porting note: last argument can't be inferred\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u2211 x in erase t x\u2080, (fun e => g e \u2022 e) (\u2191shift x) = 0\n[PROOFSTEP]\nrw [\u2190 sum_map _ shift (fun e => g e \u2022 e)]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u2211 x in Finset.map shift (erase t x\u2080), g x \u2022 x = 0\n[PROOFSTEP]\nexact gsum\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u2211 e in t, f e = 0\n[PROOFSTEP]\nshow\n (\u2211 e : V in t, f e) =\n 0\n -- Again we split off the `x\u2080` term,\n -- observing that it exactly cancels the other terms.\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u2211 e in t, f e = 0\n[PROOFSTEP]\nrw [\u2190 insert_erase m, sum_insert (not_mem_erase x\u2080 t)]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 f x\u2080 + \u2211 x in erase t x\u2080, f x = 0\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 ((if x\u2080 = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x\u2080 - x\u2080)) +\n \u2211 x in erase t x\u2080, if x = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x - x\u2080)) =\n 0\n[PROOFSTEP]\nrw [if_pos rfl]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 (-\u2211 z in erase t x\u2080, g (z - x\u2080) + \u2211 x in erase t x\u2080, if x = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x - x\u2080)) =\n 0\n[PROOFSTEP]\nconv_lhs =>\n congr\n rfl\n apply_congr\n rfl\n rw [if_neg (show _ \u2260 x\u2080 from (mem_erase.mp \u2039_\u203a).1)]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| -\u2211 z in erase t x\u2080, g (z - x\u2080) + \u2211 x in erase t x\u2080, if x = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x - x\u2080)\n[PROOFSTEP]\n congr\n rfl\n apply_congr\n rfl\n rw [if_neg (show _ \u2260 x\u2080 from (mem_erase.mp \u2039_\u203a).1)]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| -\u2211 z in erase t x\u2080, g (z - x\u2080) + \u2211 x in erase t x\u2080, if x = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x - x\u2080)\n[PROOFSTEP]\n congr\n rfl\n apply_congr\n rfl\n rw [if_neg (show _ \u2260 x\u2080 from (mem_erase.mp \u2039_\u203a).1)]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| -\u2211 z in erase t x\u2080, g (z - x\u2080) + \u2211 x in erase t x\u2080, if x = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x - x\u2080)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| -\u2211 z in erase t x\u2080, g (z - x\u2080)\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in erase t x\u2080, if x = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x - x\u2080)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n| \u2211 x in erase t x\u2080, if x = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x - x\u2080)\n[PROOFSTEP]\napply_congr\n[GOAL]\ncase a.h\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 erase t x\u2080 = ?a.s\u2082\ncase a.a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\nx\u271d : V\na\u271d : x\u271d \u2208 ?a.s\u2082\n\u22a2 (if x\u271d = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x\u271d - x\u2080)) = ?a.g x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.a\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\nx\u271d : V\na\u271d : x\u271d \u2208 erase t x\u2080\n\u22a2 (if x\u271d = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x\u271d - x\u2080)) = ?a.g x\u271d\n[PROOFSTEP]\nrw [if_neg (show _ \u2260 x\u2080 from (mem_erase.mp \u2039_\u203a).1)]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 -\u2211 z in erase t x\u2080, g (z - x\u2080) + \u2211 x in erase t x\u2080, g (x - x\u2080) = 0\n[PROOFSTEP]\nexact neg_add_self _\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nshow\n \u2203 (x : V),\n x \u2208 t \u2227\n f x \u2260\n 0\n -- We can use x\u2081 + x\u2080.\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u2203 x, x \u2208 t \u2227 f x \u2260 0\n[PROOFSTEP]\nrefine' \u27e8x\u2081 + x\u2080, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 x\u2081 + x\u2080 \u2208 t\n[PROOFSTEP]\nrw [Finset.mem_map] at x\u2081_mem \n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3.refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : \u2203 a, a \u2208 erase t x\u2080 \u2227 \u2191shift a = x\u2081\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 x\u2081 + x\u2080 \u2208 t\n[PROOFSTEP]\nrcases x\u2081_mem with \u27e8x\u2081, x\u2081_mem, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3.refine'_1.intro.intro\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 erase t x\u2080\nnz : g (\u2191shift x\u2081) \u2260 0\n\u22a2 \u2191shift x\u2081 + x\u2080 \u2208 t\n[PROOFSTEP]\nrw [mem_erase] at x\u2081_mem \n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3.refine'_1.intro.intro\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2260 x\u2080 \u2227 x\u2081 \u2208 t\nnz : g (\u2191shift x\u2081) \u2260 0\n\u22a2 \u2191shift x\u2081 + x\u2080 \u2208 t\n[PROOFSTEP]\nsimp only [x\u2081_mem, sub_add_cancel, Function.Embedding.coeFn_mk]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3.refine'_2\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 f (x\u2081 + x\u2080) \u2260 0\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3.refine'_2\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 (if x\u2081 + x\u2080 = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (x\u2081 + x\u2080 - x\u2080)) \u2260 0\n[PROOFSTEP]\nrwa [if_neg, add_sub_cancel]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3.refine'_2.hnc\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u00acx\u2081 + x\u2080 = x\u2080\n[PROOFSTEP]\nrw [add_left_eq_self]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3.refine'_2.hnc\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nx\u2081 : V\nx\u2081_mem : x\u2081 \u2208 t'\nnz : g x\u2081 \u2260 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\n\u22a2 \u00acx\u2081 = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3.refine'_2.hnc\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nt : Finset V\nh : finrank K V + 1 < card t\ncard_pos : 0 < card t\nx\u2080 : V\nm : x\u2080 \u2208 t\nshift : V \u21aa V := { toFun := fun x => x - x\u2080, inj' := (_ : Injective fun a => a - x\u2080) }\nt' : Finset V := Finset.map shift (erase t x\u2080)\nh' : finrank K V < card t'\ng : V \u2192 K\ngsum : \u2211 e in t', g e \u2022 e = 0\nf : V \u2192 K := fun z => if z = x\u2080 then -\u2211 z in erase t x\u2080, g (z - x\u2080) else g (z - x\u2080)\nx\u2081_mem : 0 \u2208 t'\nnz : g 0 \u2260 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [sub_eq_zero, exists_prop, Finset.mem_map, Embedding.coeFn_mk, eq_self_iff_true, mem_erase, not_true,\n exists_eq_right, Ne.def, false_and_iff] at x\u2081_mem \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2078 : DivisionRing K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nV\u2082 : Type v'\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module K V\u2082\nL : Type u_1\ninst\u271d\u00b3 : LinearOrderedField L\nW : Type v\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module L W\ninst\u271d : FiniteDimensional L W\nt : Finset W\nh : finrank L W + 1 < card t\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 0 < f x\n[PROOFSTEP]\nobtain \u27e8f, sum, total, nonzero\u27e9 := exists_nontrivial_relation_sum_zero_of_rank_succ_lt_card h\n[GOAL]\ncase intro.intro.intro\nK : Type u\nV : Type v\ninst\u271d\u2078 : DivisionRing K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nV\u2082 : Type v'\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module K V\u2082\nL : Type u_1\ninst\u271d\u00b3 : LinearOrderedField L\nW : Type v\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module L W\ninst\u271d : FiniteDimensional L W\nt : Finset W\nh : finrank L W + 1 < card t\nf : W \u2192 L\nsum : \u2211 e in t, f e \u2022 e = 0\ntotal : \u2211 e in t, f e = 0\nnonzero : \u2203 x, x \u2208 t \u2227 f x \u2260 0\n\u22a2 \u2203 f, \u2211 e in t, f e \u2022 e = 0 \u2227 \u2211 e in t, f e = 0 \u2227 \u2203 x, x \u2208 t \u2227 0 < f x\n[PROOFSTEP]\nexact \u27e8f, sum, total, exists_pos_of_sum_zero_of_exists_nonzero f total nonzero\u27e9\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh\u271d : finrank K V = 1\nv : V\nhv : v \u2260 0\nb : Basis \u03b9 K V := basisUnique \u03b9 h\u271d\nh : \u2191(\u2191b.repr v) default \u2260 0 := mt basisUnique.repr_eq_zero_iff.mp hv\n\u22a2 \u2200 (x y : V),\n (fun w => Finsupp.single default (\u2191(\u2191b.repr w) default / \u2191(\u2191b.repr v) default)) (x + y) =\n (fun w => Finsupp.single default (\u2191(\u2191b.repr w) default / \u2191(\u2191b.repr v) default)) x +\n (fun w => Finsupp.single default (\u2191(\u2191b.repr w) default / \u2191(\u2191b.repr v) default)) y\n[PROOFSTEP]\nsimp [add_div]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh\u271d : finrank K V = 1\nv : V\nhv : v \u2260 0\nb : Basis \u03b9 K V := basisUnique \u03b9 h\u271d\nh : \u2191(\u2191b.repr v) default \u2260 0 := mt basisUnique.repr_eq_zero_iff.mp hv\n\u22a2 \u2200 (r : K) (x : V),\n AddHom.toFun\n { toFun := fun w => Finsupp.single default (\u2191(\u2191b.repr w) default / \u2191(\u2191b.repr v) default),\n map_add' :=\n (_ :\n \u2200 (a a_1 : V),\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr (a + a_1)) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) =\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) +\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a_1) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default)) }\n (r \u2022 x) =\n \u2191(RingHom.id K) r \u2022\n AddHom.toFun\n { toFun := fun w => Finsupp.single default (\u2191(\u2191b.repr w) default / \u2191(\u2191b.repr v) default),\n map_add' :=\n (_ :\n \u2200 (a a_1 : V),\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr (a + a_1)) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) =\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) +\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a_1) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default)) }\n x\n[PROOFSTEP]\nsimp [mul_div]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh\u271d : finrank K V = 1\nv : V\nhv : v \u2260 0\nb : Basis \u03b9 K V := basisUnique \u03b9 h\u271d\nh : \u2191(\u2191b.repr v) default \u2260 0 := mt basisUnique.repr_eq_zero_iff.mp hv\nw : V\n\u22a2 (fun f => \u2191f default \u2022 v)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun w => Finsupp.single default (\u2191(\u2191b.repr w) default / \u2191(\u2191b.repr v) default),\n map_add' :=\n (_ :\n \u2200 (a a_1 : V),\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr (a + a_1)) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) =\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) +\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a_1) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default)) },\n map_smul' :=\n (_ :\n \u2200 (a : K) (a_1 : V),\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr (a \u2022 a_1)) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) =\n a \u2022\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a_1) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default)) }.toAddHom\n w) =\n w\n[PROOFSTEP]\napply_fun b.repr using b.repr.toEquiv.injective\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh\u271d : finrank K V = 1\nv : V\nhv : v \u2260 0\nb : Basis \u03b9 K V := basisUnique \u03b9 h\u271d\nh : \u2191(\u2191b.repr v) default \u2260 0 := mt basisUnique.repr_eq_zero_iff.mp hv\nw : V\n\u22a2 \u2191b.repr\n ((fun f => \u2191f default \u2022 v)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun w => Finsupp.single default (\u2191(\u2191b.repr w) default / \u2191(\u2191b.repr v) default),\n map_add' :=\n (_ :\n \u2200 (a a_1 : V),\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr (a + a_1)) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) =\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) +\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a_1) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default)) },\n map_smul' :=\n (_ :\n \u2200 (a : K) (a_1 : V),\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr (a \u2022 a_1)) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) =\n a \u2022\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a_1) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default)) }.toAddHom\n w)) =\n \u2191b.repr w\n[PROOFSTEP]\napply_fun Equiv.finsuppUnique\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh\u271d : finrank K V = 1\nv : V\nhv : v \u2260 0\nb : Basis \u03b9 K V := basisUnique \u03b9 h\u271d\nh : \u2191(\u2191b.repr v) default \u2260 0 := mt basisUnique.repr_eq_zero_iff.mp hv\nw : V\n\u22a2 \u2191Equiv.finsuppUnique\n (\u2191b.repr\n ((fun f => \u2191f default \u2022 v)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun w => Finsupp.single default (\u2191(\u2191b.repr w) default / \u2191(\u2191b.repr v) default),\n map_add' :=\n (_ :\n \u2200 (a a_1 : V),\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr (a + a_1)) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) =\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) +\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a_1) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default)) },\n map_smul' :=\n (_ :\n \u2200 (a : K) (a_1 : V),\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr (a \u2022 a_1)) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) =\n a \u2022\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a_1) default /\n \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default)) }.toAddHom\n w))) =\n \u2191Equiv.finsuppUnique (\u2191b.repr w)\n[PROOFSTEP]\nsimp only [LinearEquiv.map_smul\u209b\u2097, Finsupp.coe_smul, Finsupp.single_eq_same, RingHom.id_apply, smul_eq_mul,\n Pi.smul_apply, Equiv.finsuppUnique_apply]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh\u271d : finrank K V = 1\nv : V\nhv : v \u2260 0\nb : Basis \u03b9 K V := basisUnique \u03b9 h\u271d\nh : \u2191(\u2191b.repr v) default \u2260 0 := mt basisUnique.repr_eq_zero_iff.mp hv\nw : V\n\u22a2 \u2191(\u2191(basisUnique \u03b9 h\u271d).repr w) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default *\n \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default =\n \u2191(\u2191(basisUnique \u03b9 h\u271d).repr w) default\n[PROOFSTEP]\nexact div_mul_cancel _ h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh\u271d : finrank K V = 1\nv : V\nhv : v \u2260 0\nb : Basis \u03b9 K V := basisUnique \u03b9 h\u271d\nh : \u2191(\u2191b.repr v) default \u2260 0 := mt basisUnique.repr_eq_zero_iff.mp hv\nf : \u03b9 \u2192\u2080 K\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun w => Finsupp.single default (\u2191(\u2191b.repr w) default / \u2191(\u2191b.repr v) default),\n map_add' :=\n (_ :\n \u2200 (a a_1 : V),\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr (a + a_1)) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) =\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) +\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a_1) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default)) },\n map_smul' :=\n (_ :\n \u2200 (a : K) (a_1 : V),\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr (a \u2022 a_1)) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) =\n a \u2022\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a_1) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default)) }.toAddHom\n ((fun f => \u2191f default \u2022 v) f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh\u271d : finrank K V = 1\nv : V\nhv : v \u2260 0\nb : Basis \u03b9 K V := basisUnique \u03b9 h\u271d\nh : \u2191(\u2191b.repr v) default \u2260 0 := mt basisUnique.repr_eq_zero_iff.mp hv\nf : \u03b9 \u2192\u2080 K\n\u22a2 \u2191(AddHom.toFun\n {\n toAddHom :=\n { toFun := fun w => Finsupp.single default (\u2191(\u2191b.repr w) default / \u2191(\u2191b.repr v) default),\n map_add' :=\n (_ :\n \u2200 (a a_1 : V),\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr (a + a_1)) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) =\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) +\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a_1) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default)) },\n map_smul' :=\n (_ :\n \u2200 (a : K) (a_1 : V),\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr (a \u2022 a_1)) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default) =\n a \u2022\n Finsupp.single default\n (\u2191(\u2191(basisUnique \u03b9 h\u271d).repr a_1) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default)) }.toAddHom\n ((fun f => \u2191f default \u2022 v) f))\n default =\n \u2191f default\n[PROOFSTEP]\nsimp only [LinearEquiv.map_smul\u209b\u2097, Finsupp.coe_smul, Finsupp.single_eq_same, RingHom.id_apply, smul_eq_mul,\n Pi.smul_apply]\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh\u271d : finrank K V = 1\nv : V\nhv : v \u2260 0\nb : Basis \u03b9 K V := basisUnique \u03b9 h\u271d\nh : \u2191(\u2191b.repr v) default \u2260 0 := mt basisUnique.repr_eq_zero_iff.mp hv\nf : \u03b9 \u2192\u2080 K\n\u22a2 \u2191f default * \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default / \u2191(\u2191(basisUnique \u03b9 h\u271d).repr v) default = \u2191f default\n[PROOFSTEP]\nexact mul_div_cancel _ h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh : finrank K V = 1\nv : V\nhv : v \u2260 0\ni : \u03b9\n\u22a2 \u2191(basisSingleton \u03b9 h v hv) i = v\n[PROOFSTEP]\ncases Unique.uniq \u2039Unique \u03b9\u203a i\n[GOAL]\ncase refl\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh : finrank K V = 1\nv : V\nhv : v \u2260 0\n\u22a2 \u2191(basisSingleton \u03b9 h v hv) default = v\n[PROOFSTEP]\nsimp [basisSingleton]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh : finrank K V = 1\nv : V\nhv : v \u2260 0\n\u22a2 Set.range \u2191(basisSingleton \u03b9 h v hv) = {v}\n[PROOFSTEP]\nrw [Set.range_unique, basisSingleton_apply]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nn : \u2115\nh : Module.rank K V = \u2191n\n\u22a2 FiniteDimensional K V\n[PROOFSTEP]\nrw [FiniteDimensional, \u2190 IsNoetherian.iff_fg, IsNoetherian.iff_rank_lt_aleph0, h]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nn : \u2115\nh : Module.rank K V = \u2191n\n\u22a2 \u2191n < \u2135\u2080\n[PROOFSTEP]\nexact nat_lt_aleph0 n\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : Module.rank K V = 0\n\u22a2 finrank K V = 0\n[PROOFSTEP]\nconvert finrank_eq_rank K V\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : Module.rank K V = 0\n\u22a2 finrank K V = 0 \u2194 \u2191(finrank K V) = Module.rank K V\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : Module.rank K V = 0\n\u22a2 finrank K V = 0 \u2194 \u2191(finrank K V) = 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\n\u22a2 Module.rank K { x // x \u2208 \u22a5 } = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : Module.rank K V = 0\n\u22a2 \u22a5 = \u22a4\n[PROOFSTEP]\nhaveI : FiniteDimensional _ _ := finiteDimensional_of_rank_eq_zero h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : Module.rank K V = 0\nthis : FiniteDimensional K V\n\u22a2 \u22a5 = \u22a4\n[PROOFSTEP]\napply eq_top_of_finrank_eq\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : Module.rank K V = 0\nthis : FiniteDimensional K V\n\u22a2 finrank K { x // x \u2208 \u22a5 } = finrank K V\n[PROOFSTEP]\nrw [finrank_bot, finrank_eq_zero_of_rank_eq_zero h]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS : Submodule K V\nh : S = \u22a5\n\u22a2 Module.rank K { x // x \u2208 S } = 0\n[PROOFSTEP]\nrw [h, rank_bot]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\nS : Submodule K V\ninst\u271d : FiniteDimensional K { x // x \u2208 S }\n\u22a2 finrank K { x // x \u2208 S } = 0 \u2194 S = \u22a5\n[PROOFSTEP]\nrw [\u2190 rank_eq_zero, \u2190 finrank_eq_rank, \u2190 @Nat.cast_zero Cardinal, Cardinal.natCast_inj]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS\u2081 S\u2082 : Submodule K V\nh\u2081 : FiniteDimensional K { x // x \u2208 S\u2081 }\nh\u2082 : FiniteDimensional K { x // x \u2208 S\u2082 }\n\u22a2 FiniteDimensional K { x // x \u2208 S\u2081 \u2294 S\u2082 }\n[PROOFSTEP]\nunfold FiniteDimensional at *\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS\u2081 S\u2082 : Submodule K V\nh\u2081 : Module.Finite K { x // x \u2208 S\u2081 }\nh\u2082 : Module.Finite K { x // x \u2208 S\u2082 }\n\u22a2 Module.Finite K { x // x \u2208 S\u2081 \u2294 S\u2082 }\n[PROOFSTEP]\nrw [finite_def] at *\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nS\u2081 S\u2082 : Submodule K V\nh\u2081 : FG \u22a4\nh\u2082 : FG \u22a4\n\u22a2 FG \u22a4\n[PROOFSTEP]\nexact (fg_top _).2 (((fg_top S\u2081).1 h\u2081).sup ((fg_top S\u2082).1 h\u2082))\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ns : Finset \u03b9\nS : \u03b9 \u2192 Submodule K V\ninst\u271d : \u2200 (i : \u03b9), FiniteDimensional K { x // x \u2208 S i }\n\u22a2 FiniteDimensional K { x // x \u2208 Finset.sup s S }\n[PROOFSTEP]\nrefine'\n @Finset.sup_induction _ _ _ _ s S (fun i => FiniteDimensional K \u2191i) (finiteDimensional_bot K V) _ fun i _ => by\n infer_instance\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ns : Finset \u03b9\nS : \u03b9 \u2192 Submodule K V\ninst\u271d : \u2200 (i : \u03b9), FiniteDimensional K { x // x \u2208 S i }\ni : \u03b9\nx\u271d : i \u2208 s\n\u22a2 (fun i => FiniteDimensional K { x // x \u2208 i }) (S i)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ns : Finset \u03b9\nS : \u03b9 \u2192 Submodule K V\ninst\u271d : \u2200 (i : \u03b9), FiniteDimensional K { x // x \u2208 S i }\n\u22a2 \u2200 (a\u2081 : Submodule K V),\n (fun i => FiniteDimensional K { x // x \u2208 i }) a\u2081 \u2192\n \u2200 (a\u2082 : Submodule K V),\n (fun i => FiniteDimensional K { x // x \u2208 i }) a\u2082 \u2192 (fun i => FiniteDimensional K { x // x \u2208 i }) (a\u2081 \u2294 a\u2082)\n[PROOFSTEP]\nintro S\u2081 hS\u2081 S\u2082 hS\u2082\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ns : Finset \u03b9\nS : \u03b9 \u2192 Submodule K V\ninst\u271d : \u2200 (i : \u03b9), FiniteDimensional K { x // x \u2208 S i }\nS\u2081 : Submodule K V\nhS\u2081 : FiniteDimensional K { x // x \u2208 S\u2081 }\nS\u2082 : Submodule K V\nhS\u2082 : FiniteDimensional K { x // x \u2208 S\u2082 }\n\u22a2 FiniteDimensional K { x // x \u2208 S\u2081 \u2294 S\u2082 }\n[PROOFSTEP]\nexact Submodule.finiteDimensional_sup S\u2081 S\u2082\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\n\u03b9 : Sort u_1\ninst\u271d\u00b9 : _root_.Finite \u03b9\nS : \u03b9 \u2192 Submodule K V\ninst\u271d : \u2200 (i : \u03b9), FiniteDimensional K { x // x \u2208 S i }\n\u22a2 FiniteDimensional K { x // x \u2208 \u2a06 (i : \u03b9), S i }\n[PROOFSTEP]\ncases nonempty_fintype (PLift \u03b9)\n[GOAL]\ncase intro\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\n\u03b9 : Sort u_1\ninst\u271d\u00b9 : _root_.Finite \u03b9\nS : \u03b9 \u2192 Submodule K V\ninst\u271d : \u2200 (i : \u03b9), FiniteDimensional K { x // x \u2208 S i }\nval\u271d : Fintype (PLift \u03b9)\n\u22a2 FiniteDimensional K { x // x \u2208 \u2a06 (i : \u03b9), S i }\n[PROOFSTEP]\nrw [\u2190 iSup_plift_down, \u2190 Finset.sup_univ_eq_iSup]\n[GOAL]\ncase intro\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\n\u03b9 : Sort u_1\ninst\u271d\u00b9 : _root_.Finite \u03b9\nS : \u03b9 \u2192 Submodule K V\ninst\u271d : \u2200 (i : \u03b9), FiniteDimensional K { x // x \u2208 S i }\nval\u271d : Fintype (PLift \u03b9)\n\u22a2 FiniteDimensional K { x // x \u2208 Finset.sup Finset.univ fun i => S i.down }\n[PROOFSTEP]\nexact Submodule.finiteDimensional_finset_sup _ _\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns : Submodule K V\n\u22a2 finrank K (V \u29f8 s) + finrank K { x // x \u2208 s } = finrank K V\n[PROOFSTEP]\nhave := rank_quotient_add_rank s\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns : Submodule K V\nthis : Module.rank K (V \u29f8 s) + Module.rank K { x // x \u2208 s } = Module.rank K V\n\u22a2 finrank K (V \u29f8 s) + finrank K { x // x \u2208 s } = finrank K V\n[PROOFSTEP]\nrw [\u2190 finrank_eq_rank, \u2190 finrank_eq_rank, \u2190 finrank_eq_rank] at this \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns : Submodule K V\nthis : \u2191(finrank K (V \u29f8 s)) + \u2191(finrank K { x // x \u2208 s }) = \u2191(finrank K V)\n\u22a2 finrank K (V \u29f8 s) + finrank K { x // x \u2208 s } = finrank K V\n[PROOFSTEP]\nexact_mod_cast this\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns : Submodule K V\nh : s < \u22a4\n\u22a2 finrank K { x // x \u2208 s } < finrank K V\n[PROOFSTEP]\nrw [\u2190 s.finrank_quotient_add_finrank, add_comm]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns : Submodule K V\nh : s < \u22a4\n\u22a2 finrank K { x // x \u2208 s } < finrank K { x // x \u2208 s } + finrank K (V \u29f8 s)\n[PROOFSTEP]\nexact Nat.lt_add_of_pos_right (finrank_pos_iff.mpr (Quotient.nontrivial_of_lt_top _ h))\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ns t : Submodule K V\ninst\u271d\u00b9 : FiniteDimensional K { x // x \u2208 s }\ninst\u271d : FiniteDimensional K { x // x \u2208 t }\n\u22a2 finrank K { x // x \u2208 s \u2294 t } + finrank K { x // x \u2208 s \u2293 t } = finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t }\n[PROOFSTEP]\nhave key : Module.rank K \u2191(s \u2294 t) + Module.rank K \u2191(s \u2293 t) = Module.rank K s + Module.rank K t :=\n rank_sup_add_rank_inf_eq s t\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ns t : Submodule K V\ninst\u271d\u00b9 : FiniteDimensional K { x // x \u2208 s }\ninst\u271d : FiniteDimensional K { x // x \u2208 t }\nkey :\n Module.rank K { x // x \u2208 s \u2294 t } + Module.rank K { x // x \u2208 s \u2293 t } =\n Module.rank K { x // x \u2208 s } + Module.rank K { x // x \u2208 t }\n\u22a2 finrank K { x // x \u2208 s \u2294 t } + finrank K { x // x \u2208 s \u2293 t } = finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t }\n[PROOFSTEP]\nrepeat' rw [\u2190 finrank_eq_rank] at key \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ns t : Submodule K V\ninst\u271d\u00b9 : FiniteDimensional K { x // x \u2208 s }\ninst\u271d : FiniteDimensional K { x // x \u2208 t }\nkey :\n Module.rank K { x // x \u2208 s \u2294 t } + Module.rank K { x // x \u2208 s \u2293 t } =\n Module.rank K { x // x \u2208 s } + Module.rank K { x // x \u2208 t }\n\u22a2 finrank K { x // x \u2208 s \u2294 t } + finrank K { x // x \u2208 s \u2293 t } = finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t }\n[PROOFSTEP]\nrw [\u2190 finrank_eq_rank] at key \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ns t : Submodule K V\ninst\u271d\u00b9 : FiniteDimensional K { x // x \u2208 s }\ninst\u271d : FiniteDimensional K { x // x \u2208 t }\nkey :\n \u2191(finrank K { x // x \u2208 s \u2294 t }) + Module.rank K { x // x \u2208 s \u2293 t } =\n Module.rank K { x // x \u2208 s } + Module.rank K { x // x \u2208 t }\n\u22a2 finrank K { x // x \u2208 s \u2294 t } + finrank K { x // x \u2208 s \u2293 t } = finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t }\n[PROOFSTEP]\nrw [\u2190 finrank_eq_rank] at key \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ns t : Submodule K V\ninst\u271d\u00b9 : FiniteDimensional K { x // x \u2208 s }\ninst\u271d : FiniteDimensional K { x // x \u2208 t }\nkey :\n \u2191(finrank K { x // x \u2208 s \u2294 t }) + \u2191(finrank K { x // x \u2208 s \u2293 t }) =\n Module.rank K { x // x \u2208 s } + Module.rank K { x // x \u2208 t }\n\u22a2 finrank K { x // x \u2208 s \u2294 t } + finrank K { x // x \u2208 s \u2293 t } = finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t }\n[PROOFSTEP]\nrw [\u2190 finrank_eq_rank] at key \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ns t : Submodule K V\ninst\u271d\u00b9 : FiniteDimensional K { x // x \u2208 s }\ninst\u271d : FiniteDimensional K { x // x \u2208 t }\nkey :\n \u2191(finrank K { x // x \u2208 s \u2294 t }) + \u2191(finrank K { x // x \u2208 s \u2293 t }) =\n \u2191(finrank K { x // x \u2208 s }) + Module.rank K { x // x \u2208 t }\n\u22a2 finrank K { x // x \u2208 s \u2294 t } + finrank K { x // x \u2208 s \u2293 t } = finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t }\n[PROOFSTEP]\nrw [\u2190 finrank_eq_rank] at key \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ns t : Submodule K V\ninst\u271d\u00b9 : FiniteDimensional K { x // x \u2208 s }\ninst\u271d : FiniteDimensional K { x // x \u2208 t }\nkey :\n \u2191(finrank K { x // x \u2208 s \u2294 t }) + \u2191(finrank K { x // x \u2208 s \u2293 t }) =\n \u2191(finrank K { x // x \u2208 s }) + \u2191(finrank K { x // x \u2208 t })\n\u22a2 finrank K { x // x \u2208 s \u2294 t } + finrank K { x // x \u2208 s \u2293 t } = finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t }\n[PROOFSTEP]\nrw [\u2190 finrank_eq_rank] at key \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ns t : Submodule K V\ninst\u271d\u00b9 : FiniteDimensional K { x // x \u2208 s }\ninst\u271d : FiniteDimensional K { x // x \u2208 t }\nkey :\n \u2191(finrank K { x // x \u2208 s \u2294 t }) + \u2191(finrank K { x // x \u2208 s \u2293 t }) =\n \u2191(finrank K { x // x \u2208 s }) + \u2191(finrank K { x // x \u2208 t })\n\u22a2 finrank K { x // x \u2208 s \u2294 t } + finrank K { x // x \u2208 s \u2293 t } = finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t }\n[PROOFSTEP]\nnorm_cast at key \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ns t : Submodule K V\ninst\u271d\u00b9 : FiniteDimensional K { x // x \u2208 s }\ninst\u271d : FiniteDimensional K { x // x \u2208 t }\n\u22a2 finrank K { x // x \u2208 s \u2294 t } \u2264 finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t }\n[PROOFSTEP]\nrw [\u2190 finrank_sup_add_finrank_inf_eq]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ns t : Submodule K V\ninst\u271d\u00b9 : FiniteDimensional K { x // x \u2208 s }\ninst\u271d : FiniteDimensional K { x // x \u2208 t }\n\u22a2 finrank K { x // x \u2208 s \u2294 t } \u2264 finrank K { x // x \u2208 s \u2294 t } + finrank K { x // x \u2208 s \u2293 t }\n[PROOFSTEP]\nexact self_le_add_right _ _\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns t : Submodule K V\nhdim : finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t } = finrank K V\nhdisjoint : Disjoint s t\n\u22a2 s \u2294 t = \u22a4\n[PROOFSTEP]\nhave h_finrank_inf : finrank K \u2191(s \u2293 t) = 0 :=\n by\n rw [disjoint_iff_inf_le, le_bot_iff] at hdisjoint \n rw [hdisjoint, finrank_bot]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns t : Submodule K V\nhdim : finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t } = finrank K V\nhdisjoint : Disjoint s t\n\u22a2 finrank K { x // x \u2208 s \u2293 t } = 0\n[PROOFSTEP]\nrw [disjoint_iff_inf_le, le_bot_iff] at hdisjoint \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns t : Submodule K V\nhdim : finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t } = finrank K V\nhdisjoint : s \u2293 t = \u22a5\n\u22a2 finrank K { x // x \u2208 s \u2293 t } = 0\n[PROOFSTEP]\nrw [hdisjoint, finrank_bot]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns t : Submodule K V\nhdim : finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t } = finrank K V\nhdisjoint : Disjoint s t\nh_finrank_inf : finrank K { x // x \u2208 s \u2293 t } = 0\n\u22a2 s \u2294 t = \u22a4\n[PROOFSTEP]\napply eq_top_of_finrank_eq\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns t : Submodule K V\nhdim : finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t } = finrank K V\nhdisjoint : Disjoint s t\nh_finrank_inf : finrank K { x // x \u2208 s \u2293 t } = 0\n\u22a2 finrank K { x // x \u2208 s \u2294 t } = finrank K V\n[PROOFSTEP]\nrw [\u2190 hdim]\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns t : Submodule K V\nhdim : finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t } = finrank K V\nhdisjoint : Disjoint s t\nh_finrank_inf : finrank K { x // x \u2208 s \u2293 t } = 0\n\u22a2 finrank K { x // x \u2208 s \u2294 t } = finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t }\n[PROOFSTEP]\nconvert s.finrank_sup_add_finrank_inf_eq t\n[GOAL]\ncase h.e'_2\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns t : Submodule K V\nhdim : finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t } = finrank K V\nhdisjoint : Disjoint s t\nh_finrank_inf : finrank K { x // x \u2208 s \u2293 t } = 0\n\u22a2 finrank K { x // x \u2208 s \u2294 t } = finrank K { x // x \u2208 s \u2294 t } + finrank K { x // x \u2208 s \u2293 t }\n[PROOFSTEP]\nrw [h_finrank_inf]\n[GOAL]\ncase h.e'_2\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns t : Submodule K V\nhdim : finrank K { x // x \u2208 s } + finrank K { x // x \u2208 t } = finrank K V\nhdisjoint : Disjoint s t\nh_finrank_inf : finrank K { x // x \u2208 s \u2293 t } = 0\n\u22a2 finrank K { x // x \u2208 s \u2294 t } = finrank K { x // x \u2208 s \u2294 t } + 0\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\nS\u2081 S\u2082 : Submodule K V\ninst\u271d : FiniteDimensional K { x // x \u2208 S\u2082 }\nhle : S\u2081 \u2264 S\u2082\nhd : finrank K { x // x \u2208 S\u2082 } \u2264 finrank K { x // x \u2208 S\u2081 }\n\u22a2 S\u2081 = S\u2082\n[PROOFSTEP]\nrw [\u2190 LinearEquiv.finrank_eq (Submodule.comapSubtypeEquivOfLe hle)] at hd \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\nS\u2081 S\u2082 : Submodule K V\ninst\u271d : FiniteDimensional K { x // x \u2208 S\u2082 }\nhle : S\u2081 \u2264 S\u2082\nhd : finrank K { x // x \u2208 S\u2082 } \u2264 finrank K { x // x \u2208 comap (Submodule.subtype S\u2082) S\u2081 }\n\u22a2 S\u2081 = S\u2082\n[PROOFSTEP]\nexact\n le_antisymm hle\n (Submodule.comap_subtype_eq_top.1\n (eq_top_of_finrank_eq (le_antisymm (comap (Submodule.subtype S\u2082) S\u2081).finrank_le hd)))\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2082\np : Subspace K V\nq : Subspace K V\u2082\nf\u2081 : { x // x \u2208 p } \u2243\u2097[K] { x // x \u2208 q }\nf\u2082 : V \u2243\u2097[K] V\u2082\n\u22a2 finrank K (V \u29f8 p) = finrank K (V\u2082 \u29f8 q)\n[PROOFSTEP]\nrw [\u2190 @add_right_cancel_iff _ _ _ (finrank K p), Submodule.finrank_quotient_add_finrank, LinearEquiv.finrank_eq f\u2081,\n Submodule.finrank_quotient_add_finrank, LinearEquiv.finrank_eq f\u2082]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2082\np q : Subspace K V\nf : (V \u29f8 p) \u2243\u2097[K] { x // x \u2208 q }\n\u22a2 finrank K (V \u29f8 q) + ?m.798563 f = finrank K { x // x \u2208 p } + ?m.798563 f\n[PROOFSTEP]\nrw [Submodule.finrank_quotient_add_finrank, \u2190 LinearEquiv.finrank_eq f, add_comm,\n Submodule.finrank_quotient_add_finrank]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\nhinj : Injective \u2191f\n\u22a2 Surjective \u2191f\n[PROOFSTEP]\nhave h := rank_range_of_injective _ hinj\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\nhinj : Injective \u2191f\nh : Module.rank K { x // x \u2208 range f } = Module.rank K V\n\u22a2 Surjective \u2191f\n[PROOFSTEP]\nrw [\u2190 finrank_eq_rank, \u2190 finrank_eq_rank, natCast_inj] at h \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\nhinj : Injective \u2191f\nh : finrank K { x // x \u2208 range f } = finrank K V\n\u22a2 Surjective \u2191f\n[PROOFSTEP]\nexact range_eq_top.1 (eq_top_of_finrank_eq h)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\n\u22a2 ker f = \u22a5 \u2194 range f = \u22a4\n[PROOFSTEP]\nrw [range_eq_top, ker_eq_bot, injective_iff_surjective]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf g : V \u2192\u2097[K] V\nhfg : f * g = 1\n\u22a2 g * f = 1\n[PROOFSTEP]\nhave ginj : Injective g := HasLeftInverse.injective \u27e8f, fun x => show (f * g) x = (1 : V \u2192\u2097[K] V) x by rw [hfg]\u27e9\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf g : V \u2192\u2097[K] V\nhfg : f * g = 1\nx : V\n\u22a2 \u2191(f * g) x = \u21911 x\n[PROOFSTEP]\nrw [hfg]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf g : V \u2192\u2097[K] V\nhfg : f * g = 1\nginj : Injective \u2191g\n\u22a2 g * f = 1\n[PROOFSTEP]\nlet \u27e8i, hi\u27e9 := g.exists_rightInverse_of_surjective (range_eq_top.2 (injective_iff_surjective.1 ginj))\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf g : V \u2192\u2097[K] V\nhfg : f * g = 1\nginj : Injective \u2191g\ni : V \u2192\u2097[K] V\nhi : comp g i = id\n\u22a2 g * f = 1\n[PROOFSTEP]\nhave : f * (g * i) = f * 1 := congr_arg _ hi\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf g : V \u2192\u2097[K] V\nhfg : f * g = 1\nginj : Injective \u2191g\ni : V \u2192\u2097[K] V\nhi : comp g i = id\nthis : f * (g * i) = f * 1\n\u22a2 g * f = 1\n[PROOFSTEP]\nrw [\u2190 mul_assoc, hfg, one_mul, mul_one] at this \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf g : V \u2192\u2097[K] V\nhfg : f * g = 1\nginj : Injective \u2191g\ni : V \u2192\u2097[K] V\nhi : comp g i = id\nthis : i = f\n\u22a2 g * f = 1\n[PROOFSTEP]\nrwa [\u2190 this]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\u2082\n\u22a2 finrank K { x // x \u2208 range f } + finrank K { x // x \u2208 ker f } = finrank K V\n[PROOFSTEP]\nrw [\u2190 f.quotKerEquivRange.finrank_eq]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\u2082\n\u22a2 finrank K (V \u29f8 ker f) + finrank K { x // x \u2208 ker f } = finrank K V\n[PROOFSTEP]\nexact Submodule.finrank_quotient_add_finrank _\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\n\u22a2 IsUnit f \u2194 ker f = \u22a5\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\n\u22a2 IsUnit f \u2192 ker f = \u22a5\n[PROOFSTEP]\nrintro \u27e8u, rfl\u27e9\n[GOAL]\ncase mp.intro\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nu : (V \u2192\u2097[K] V)\u02e3\n\u22a2 ker \u2191u = \u22a5\n[PROOFSTEP]\nexact LinearMap.ker_eq_bot_of_inverse u.inv_mul\n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\n\u22a2 ker f = \u22a5 \u2192 IsUnit f\n[PROOFSTEP]\nintro h_inj\n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\nh_inj : ker f = \u22a5\n\u22a2 IsUnit f\n[PROOFSTEP]\nrw [ker_eq_bot] at h_inj \n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\nh_inj : Injective \u2191f\n\u22a2 IsUnit f\n[PROOFSTEP]\nexact\n \u27e8\u27e8f, (LinearEquiv.ofInjectiveEndo f h_inj).symm.toLinearMap, LinearEquiv.ofInjectiveEndo_right_inv f h_inj,\n LinearEquiv.ofInjectiveEndo_left_inv f h_inj\u27e9,\n rfl\u27e9\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\n\u22a2 IsUnit f \u2194 range f = \u22a4\n[PROOFSTEP]\nrw [isUnit_iff_ker_eq_bot, ker_eq_bot_iff_range_eq_top]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2082\nH : finrank K V = finrank K V\u2082\nf : V \u2192\u2097[K] V\u2082\n\u22a2 Injective \u2191f \u2194 Surjective \u2191f\n[PROOFSTEP]\nhave := finrank_range_add_finrank_ker f\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2082\nH : finrank K V = finrank K V\u2082\nf : V \u2192\u2097[K] V\u2082\nthis : finrank K { x // x \u2208 range f } + finrank K { x // x \u2208 ker f } = finrank K V\n\u22a2 Injective \u2191f \u2194 Surjective \u2191f\n[PROOFSTEP]\nrw [\u2190 ker_eq_bot, \u2190 range_eq_top]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2082\nH : finrank K V = finrank K V\u2082\nf : V \u2192\u2097[K] V\u2082\nthis : finrank K { x // x \u2208 range f } + finrank K { x // x \u2208 ker f } = finrank K V\n\u22a2 ker f = \u22a5 \u2194 range f = \u22a4\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2082\nH : finrank K V = finrank K V\u2082\nf : V \u2192\u2097[K] V\u2082\nthis : finrank K { x // x \u2208 range f } + finrank K { x // x \u2208 ker f } = finrank K V\nh : ker f = \u22a5\n\u22a2 range f = \u22a4\n[PROOFSTEP]\nrw [h, finrank_bot, add_zero, H] at this \n[GOAL]\ncase refine'_1\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2082\nH : finrank K V = finrank K V\u2082\nf : V \u2192\u2097[K] V\u2082\nthis : finrank K { x // x \u2208 range f } = finrank K V\u2082\nh : ker f = \u22a5\n\u22a2 range f = \u22a4\n[PROOFSTEP]\nexact eq_top_of_finrank_eq this\n[GOAL]\ncase refine'_2\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2082\nH : finrank K V = finrank K V\u2082\nf : V \u2192\u2097[K] V\u2082\nthis : finrank K { x // x \u2208 range f } + finrank K { x // x \u2208 ker f } = finrank K V\nh : range f = \u22a4\n\u22a2 ker f = \u22a5\n[PROOFSTEP]\nrw [h, finrank_top, H] at this \n[GOAL]\ncase refine'_2\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2082\nH : finrank K V = finrank K V\u2082\nf : V \u2192\u2097[K] V\u2082\nthis : finrank K V\u2082 + finrank K { x // x \u2208 ker f } = finrank K V\u2082\nh : range f = \u22a4\n\u22a2 ker f = \u22a5\n[PROOFSTEP]\nexact finrank_eq_zero.1 (add_right_injective _ this)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2082\nH : finrank K V = finrank K V\u2082\nf : V \u2192\u2097[K] V\u2082\n\u22a2 ker f = \u22a5 \u2194 range f = \u22a4\n[PROOFSTEP]\nrw [range_eq_top, ker_eq_bot, injective_iff_surjective_of_finrank_eq_finrank H]\n[GOAL]\nK\u271d : Type u\nV : Type v\nF : Type u_1\nK : Type u_2\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Ring K\ninst\u271d\u00b2 : IsDomain K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : FiniteDimensional F K\nx : K\nH : x \u2260 0\n\u22a2 \u2203 y, x * y = 1\n[PROOFSTEP]\nhave : Function.Surjective (LinearMap.mulLeft F x) :=\n LinearMap.injective_iff_surjective.1 fun y z => ((mul_right_inj' H).1 : x * y = x * z \u2192 y = z)\n[GOAL]\nK\u271d : Type u\nV : Type v\nF : Type u_1\nK : Type u_2\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Ring K\ninst\u271d\u00b2 : IsDomain K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : FiniteDimensional F K\nx : K\nH : x \u2260 0\nthis : Surjective \u2191(LinearMap.mulLeft F x)\n\u22a2 \u2203 y, x * y = 1\n[PROOFSTEP]\nexact this 1\n[GOAL]\nK\u271d : Type u\nV : Type v\nF : Type u_1\nK : Type u_2\ninst\u271d\u00b3 : Field F\nh : Ring K\ninst\u271d\u00b2 : IsDomain K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : FiniteDimensional F K\nsrc\u271d : IsDomain K := inst\u271d\u00b2\nx : K\nhx : x \u2260 0\n\u22a2 (x * if H : x = 0 then 0 else Classical.choose (_ : \u2203 y, x * y = 1)) = 1\n[PROOFSTEP]\nrw [dif_neg hx]\n[GOAL]\nK\u271d : Type u\nV : Type v\nF : Type u_1\nK : Type u_2\ninst\u271d\u00b3 : Field F\nh : Ring K\ninst\u271d\u00b2 : IsDomain K\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : FiniteDimensional F K\nsrc\u271d : IsDomain K := inst\u271d\u00b2\nx : K\nhx : x \u2260 0\n\u22a2 x * Classical.choose (_ : \u2203 y, x * y = 1) = 1\n[PROOFSTEP]\nexact (Classical.choose_spec (FiniteDimensional.exists_mul_eq_one F hx) :)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nU W : Submodule K V\nh : IsCompl U W\n\u22a2 finrank K { x // x \u2208 U } + finrank K { x // x \u2208 W } = finrank K V\n[PROOFSTEP]\nrw [\u2190 finrank_sup_add_finrank_inf_eq, h.codisjoint.eq_top, h.disjoint.eq_bot, finrank_bot, add_zero]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nU W : Submodule K V\nh : IsCompl U W\n\u22a2 finrank K { x // x \u2208 \u22a4 } = finrank K V\n[PROOFSTEP]\nexact finrank_top _ _\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : V\nhv : v \u2260 0\n\u22a2 finrank K { x // x \u2208 span K {v} } = 1\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : V\nhv : v \u2260 0\n\u22a2 finrank K { x // x \u2208 span K {v} } \u2264 1\n[PROOFSTEP]\nexact finrank_span_le_card ({ v } : Set V)\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : V\nhv : v \u2260 0\n\u22a2 1 \u2264 finrank K { x // x \u2208 span K {v} }\n[PROOFSTEP]\nrw [Nat.succ_le_iff, finrank_pos_iff]\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : V\nhv : v \u2260 0\n\u22a2 Nontrivial { x // x \u2208 span K {v} }\n[PROOFSTEP]\nuse\u27e8v, mem_span_singleton_self v\u27e9, 0\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : V\nhv : v \u2260 0\n\u22a2 { val := v, property := (_ : v \u2208 span K {v}) } \u2260 0\n[PROOFSTEP]\nsimp [hv]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : finrank K V = 1\nx : V\nhx : x \u2260 0\ny : V\n\u22a2 \u2203 c, c \u2022 x = y\n[PROOFSTEP]\nhave : Submodule.span K { x } = \u22a4 :=\n by\n have : FiniteDimensional K V := finiteDimensional_of_finrank (zero_lt_one.trans_le h.symm.le)\n apply eq_top_of_finrank_eq\n rw [h]\n exact finrank_span_singleton hx\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : finrank K V = 1\nx : V\nhx : x \u2260 0\ny : V\n\u22a2 span K {x} = \u22a4\n[PROOFSTEP]\nhave : FiniteDimensional K V := finiteDimensional_of_finrank (zero_lt_one.trans_le h.symm.le)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : finrank K V = 1\nx : V\nhx : x \u2260 0\ny : V\nthis : FiniteDimensional K V\n\u22a2 span K {x} = \u22a4\n[PROOFSTEP]\napply eq_top_of_finrank_eq\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : finrank K V = 1\nx : V\nhx : x \u2260 0\ny : V\nthis : FiniteDimensional K V\n\u22a2 finrank K { x_1 // x_1 \u2208 span K {x} } = finrank K V\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : finrank K V = 1\nx : V\nhx : x \u2260 0\ny : V\nthis : FiniteDimensional K V\n\u22a2 finrank K { x_1 // x_1 \u2208 span K {x} } = 1\n[PROOFSTEP]\nexact finrank_span_singleton hx\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : finrank K V = 1\nx : V\nhx : x \u2260 0\ny : V\nthis : span K {x} = \u22a4\n\u22a2 \u2203 c, c \u2022 x = y\n[PROOFSTEP]\nhave : y \u2208 Submodule.span K { x } := by rw [this]; trivial\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : finrank K V = 1\nx : V\nhx : x \u2260 0\ny : V\nthis : span K {x} = \u22a4\n\u22a2 y \u2208 span K {x}\n[PROOFSTEP]\nrw [this]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : finrank K V = 1\nx : V\nhx : x \u2260 0\ny : V\nthis : span K {x} = \u22a4\n\u22a2 y \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : finrank K V = 1\nx : V\nhx : x \u2260 0\ny : V\nthis\u271d : span K {x} = \u22a4\nthis : y \u2208 span K {x}\n\u22a2 \u2203 c, c \u2022 x = y\n[PROOFSTEP]\nexact mem_span_singleton.1 this\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card \u03b9 = finrank K V\n\u22a2 span K (Set.range b) = \u22a4\n[PROOFSTEP]\nby_cases fin : FiniteDimensional K V\n[GOAL]\ncase pos\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card \u03b9 = finrank K V\nfin : FiniteDimensional K V\n\u22a2 span K (Set.range b) = \u22a4\n[PROOFSTEP]\nreplace fin : FiniteDimensional _ _ := fin\n[GOAL]\ncase pos\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card \u03b9 = finrank K V\nfin : FiniteDimensional K V\n\u22a2 span K (Set.range b) = \u22a4\n[PROOFSTEP]\nby_contra ne_top\n[GOAL]\ncase pos\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card \u03b9 = finrank K V\nfin : FiniteDimensional K V\nne_top : \u00acspan K (Set.range b) = \u22a4\n\u22a2 False\n[PROOFSTEP]\nhave lt_top : span K (Set.range b) < \u22a4 := lt_of_le_of_ne le_top ne_top\n[GOAL]\ncase pos\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card \u03b9 = finrank K V\nfin : FiniteDimensional K V\nne_top : \u00acspan K (Set.range b) = \u22a4\nlt_top : span K (Set.range b) < \u22a4\n\u22a2 False\n[PROOFSTEP]\nexact ne_of_lt (Submodule.finrank_lt lt_top) (_root_.trans (finrank_span_eq_card lin_ind) card_eq)\n[GOAL]\ncase neg\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card \u03b9 = finrank K V\nfin : \u00acFiniteDimensional K V\n\u22a2 span K (Set.range b) = \u22a4\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.h\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card \u03b9 = finrank K V\nfin : \u00acFiniteDimensional K V\n\u22a2 False\n[PROOFSTEP]\napply ne_of_lt (Fintype.card_pos_iff.mpr h\u03b9)\n[GOAL]\ncase neg.h\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card \u03b9 = finrank K V\nfin : \u00acFiniteDimensional K V\n\u22a2 0 = Fintype.card \u03b9\n[PROOFSTEP]\nsymm\n[GOAL]\ncase neg.h\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card \u03b9 = finrank K V\nfin : \u00acFiniteDimensional K V\n\u22a2 Fintype.card \u03b9 = 0\n[PROOFSTEP]\nreplace fin := (not_iff_not.2 IsNoetherian.iff_fg).2 fin\n[GOAL]\ncase neg.h\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nlin_ind : LinearIndependent K b\ncard_eq : Fintype.card \u03b9 = finrank K V\nfin : \u00acIsNoetherian K V\n\u22a2 Fintype.card \u03b9 = 0\n[PROOFSTEP]\ncalc\n Fintype.card \u03b9 = finrank K V := card_eq\n _ = 0 := dif_neg (mt IsNoetherian.iff_rank_lt_aleph0.mpr fin)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\ns : Finset V\nhs : Finset.Nonempty s\nlin_ind : LinearIndependent K Subtype.val\ncard_eq : Finset.card s = finrank K V\n\u22a2 \u2191(finsetBasisOfLinearIndependentOfCardEqFinrank hs lin_ind card_eq) = Subtype.val\n[PROOFSTEP]\nrw [finsetBasisOfLinearIndependentOfCardEqFinrank]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\ns : Finset V\nhs : Finset.Nonempty s\nlin_ind : LinearIndependent K Subtype.val\ncard_eq : Finset.card s = finrank K V\n\u22a2 \u2191(basisOfLinearIndependentOfCardEqFinrank lin_ind (_ : Fintype.card { x // x \u2208 s } = finrank K V)) = Subtype.val\n[PROOFSTEP]\nexact Basis.coe_mk _ _\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ns : Set V\ninst\u271d\u00b9 : Nonempty \u2191s\ninst\u271d : Fintype \u2191s\nlin_ind : LinearIndependent K Subtype.val\ncard_eq : Finset.card (Set.toFinset s) = finrank K V\n\u22a2 \u2191(setBasisOfLinearIndependentOfCardEqFinrank lin_ind card_eq) = Subtype.val\n[PROOFSTEP]\nrw [setBasisOfLinearIndependentOfCardEqFinrank]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ns : Set V\ninst\u271d\u00b9 : Nonempty \u2191s\ninst\u271d : Fintype \u2191s\nlin_ind : LinearIndependent K Subtype.val\ncard_eq : Finset.card (Set.toFinset s) = finrank K V\n\u22a2 \u2191(basisOfLinearIndependentOfCardEqFinrank lin_ind (_ : Fintype.card \u2191s = finrank K V)) = Subtype.val\n[PROOFSTEP]\nexact Basis.coe_mk _ _\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : V\nnz : v \u2260 0\nh : finrank K V = 1\n\u22a2 span K {v} = \u22a4\n[PROOFSTEP]\nsimpa using (basisSingleton PUnit.{u + 1} h v nz).span_eq\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : V\nnz : v \u2260 0\ns : span K {v} = \u22a4\n\u22a2 \u22a4 \u2264 span K (Set.range fun x => \u2191x)\n[PROOFSTEP]\nconvert s.ge\n[GOAL]\ncase h.e'_4.h.e'_6\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : V\nnz : v \u2260 0\ns : span K {v} = \u22a4\n\u22a2 (Set.range fun x => \u2191x) = {v}\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : V\nnz : v \u2260 0\n\u22a2 finrank K V = 1 \u2194 \u2200 (w : V), \u2203 c, c \u2022 v = w\n[PROOFSTEP]\nrw [finrank_eq_one_iff_of_nonzero v nz]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : V\nnz : v \u2260 0\n\u22a2 span K {v} = \u22a4 \u2194 \u2200 (w : V), \u2203 c, c \u2022 v = w\n[PROOFSTEP]\napply span_singleton_eq_top_iff\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\n\u22a2 finrank K V = 1 \u2194 Nonempty (Basis \u03b9 K V)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\n\u22a2 finrank K V = 1 \u2192 Nonempty (Basis \u03b9 K V)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh : finrank K V = 1\n\u22a2 Nonempty (Basis \u03b9 K V)\n[PROOFSTEP]\nhaveI := finiteDimensional_of_finrank (_root_.zero_lt_one.trans_le h.symm.le)\n[GOAL]\ncase mp\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh : finrank K V = 1\nthis : FiniteDimensional K V\n\u22a2 Nonempty (Basis \u03b9 K V)\n[PROOFSTEP]\nexact \u27e8basisUnique \u03b9 h\u27e9\n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\n\u22a2 Nonempty (Basis \u03b9 K V) \u2192 finrank K V = 1\n[PROOFSTEP]\nrintro \u27e8b\u27e9\n[GOAL]\ncase mpr.intro\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nb : Basis \u03b9 K V\n\u22a2 finrank K V = 1\n[PROOFSTEP]\nsimpa using finrank_eq_card_basis b\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\n\u22a2 finrank K V = 1 \u2194 \u2203 v _n, \u2200 (w : V), \u2203 c, c \u2022 v = w\n[PROOFSTEP]\nrw [finrank_eq_one_iff PUnit.{u + 1}, Basis.basis_singleton_iff PUnit]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\n\u22a2 finrank K V \u2264 1 \u2194 \u2203 v, \u2200 (w : V), \u2203 c, c \u2022 v = w\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\n\u22a2 finrank K V \u2264 1 \u2192 \u2203 v, \u2200 (w : V), \u2203 c, c \u2022 v = w\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : finrank K V \u2264 1\n\u22a2 \u2203 v, \u2200 (w : V), \u2203 c, c \u2022 v = w\n[PROOFSTEP]\nby_cases h' : finrank K V = 0\n[GOAL]\ncase pos\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : finrank K V \u2264 1\nh' : finrank K V = 0\n\u22a2 \u2203 v, \u2200 (w : V), \u2203 c, c \u2022 v = w\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : finrank K V \u2264 1\nh' : finrank K V = 0\n\u22a2 \u2200 (w : V), \u2203 c, c \u2022 0 = w\n[PROOFSTEP]\nintro w\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : finrank K V \u2264 1\nh' : finrank K V = 0\nw : V\n\u22a2 \u2203 c, c \u2022 0 = w\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : finrank K V \u2264 1\nh' : finrank K V = 0\nw : V\n\u22a2 0 \u2022 0 = w\n[PROOFSTEP]\nhaveI := finrank_zero_iff.mp h'\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : finrank K V \u2264 1\nh' : finrank K V = 0\nw : V\nthis : Subsingleton V\n\u22a2 0 \u2022 0 = w\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\ncase neg\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : finrank K V \u2264 1\nh' : \u00acfinrank K V = 0\n\u22a2 \u2203 v, \u2200 (w : V), \u2203 c, c \u2022 v = w\n[PROOFSTEP]\nreplace h' := zero_lt_iff.mpr h'\n[GOAL]\ncase neg\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : finrank K V \u2264 1\nh' : 0 < finrank K V\n\u22a2 \u2203 v, \u2200 (w : V), \u2203 c, c \u2022 v = w\n[PROOFSTEP]\nhave : finrank K V = 1 := by linarith\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : finrank K V \u2264 1\nh' : 0 < finrank K V\n\u22a2 finrank K V = 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : finrank K V \u2264 1\nh' : 0 < finrank K V\nthis : finrank K V = 1\n\u22a2 \u2203 v, \u2200 (w : V), \u2203 c, c \u2022 v = w\n[PROOFSTEP]\nobtain \u27e8v, -, p\u27e9 := finrank_eq_one_iff'.mp this\n[GOAL]\ncase neg.intro.intro\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : finrank K V \u2264 1\nh' : 0 < finrank K V\nthis : finrank K V = 1\nv : V\np : \u2200 (w : V), \u2203 c, c \u2022 v = w\n\u22a2 \u2203 v, \u2200 (w : V), \u2203 c, c \u2022 v = w\n[PROOFSTEP]\nuse v, p\n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\n\u22a2 (\u2203 v, \u2200 (w : V), \u2203 c, c \u2022 v = w) \u2192 finrank K V \u2264 1\n[PROOFSTEP]\nrintro \u27e8v, p\u27e9\n[GOAL]\ncase mpr.intro\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nv : V\np : \u2200 (w : V), \u2203 c, c \u2022 v = w\n\u22a2 finrank K V \u2264 1\n[PROOFSTEP]\nexact finrank_le_one v p\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\nW : Submodule K V\ninst\u271d : FiniteDimensional K { x // x \u2208 W }\n\u22a2 finrank K { x // x \u2208 W } \u2264 1 \u2194 IsPrincipal W\n[PROOFSTEP]\nrw [\u2190 W.rank_le_one_iff_isPrincipal, \u2190 finrank_eq_rank, \u2190 Cardinal.natCast_le, Nat.cast_one]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\n\u22a2 finrank K V \u2264 1 \u2194 IsPrincipal \u22a4\n[PROOFSTEP]\nrw [\u2190 Module.rank_le_one_iff_top_isPrincipal, \u2190 finrank_eq_rank, \u2190 Cardinal.natCast_le, Nat.cast_one]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2078 : DivisionRing K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nW : Type u_1\nA : Type u_2\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Module A V\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module K W\ninst\u271d\u00b9 : Module A W\ninst\u271d : LinearMap.CompatibleSMul V W K A\nh : finrank K W = 1\nf : V \u2192\u2097[A] W\nw : f \u2260 0\n\u22a2 Surjective \u2191f\n[PROOFSTEP]\nchange Surjective (f.restrictScalars K)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2078 : DivisionRing K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nW : Type u_1\nA : Type u_2\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Module A V\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module K W\ninst\u271d\u00b9 : Module A W\ninst\u271d : LinearMap.CompatibleSMul V W K A\nh : finrank K W = 1\nf : V \u2192\u2097[A] W\nw : f \u2260 0\n\u22a2 Surjective \u2191(\u2191K f)\n[PROOFSTEP]\nobtain \u27e8v, n\u27e9 := FunLike.ne_iff.mp w\n[GOAL]\ncase intro\nK : Type u\nV : Type v\ninst\u271d\u2078 : DivisionRing K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nW : Type u_1\nA : Type u_2\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Module A V\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module K W\ninst\u271d\u00b9 : Module A W\ninst\u271d : LinearMap.CompatibleSMul V W K A\nh : finrank K W = 1\nf : V \u2192\u2097[A] W\nw : f \u2260 0\nv : V\nn : \u2191f v \u2260 \u21910 v\n\u22a2 Surjective \u2191(\u2191K f)\n[PROOFSTEP]\nintro z\n[GOAL]\ncase intro\nK : Type u\nV : Type v\ninst\u271d\u2078 : DivisionRing K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nW : Type u_1\nA : Type u_2\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Module A V\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module K W\ninst\u271d\u00b9 : Module A W\ninst\u271d : LinearMap.CompatibleSMul V W K A\nh : finrank K W = 1\nf : V \u2192\u2097[A] W\nw : f \u2260 0\nv : V\nn : \u2191f v \u2260 \u21910 v\nz : W\n\u22a2 \u2203 a, \u2191(\u2191K f) a = z\n[PROOFSTEP]\nobtain \u27e8c, rfl\u27e9 := (finrank_eq_one_iff_of_nonzero' (f v) n).mp h z\n[GOAL]\ncase intro.intro\nK : Type u\nV : Type v\ninst\u271d\u2078 : DivisionRing K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nW : Type u_1\nA : Type u_2\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Module A V\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module K W\ninst\u271d\u00b9 : Module A W\ninst\u271d : LinearMap.CompatibleSMul V W K A\nh : finrank K W = 1\nf : V \u2192\u2097[A] W\nw : f \u2260 0\nv : V\nn : \u2191f v \u2260 \u21910 v\nc : K\n\u22a2 \u2203 a, \u2191(\u2191K f) a = c \u2022 \u2191f v\n[PROOFSTEP]\nexact \u27e8c \u2022 v, by simp\u27e9\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2078 : DivisionRing K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nW : Type u_1\nA : Type u_2\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Module A V\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module K W\ninst\u271d\u00b9 : Module A W\ninst\u271d : LinearMap.CompatibleSMul V W K A\nh : finrank K W = 1\nf : V \u2192\u2097[A] W\nw : f \u2260 0\nv : V\nn : \u2191f v \u2260 \u21910 v\nc : K\n\u22a2 \u2191(\u2191K f) (c \u2022 v) = c \u2022 \u2191f v\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\n\u22a2 IsSimpleOrder (Submodule A V)\n[PROOFSTEP]\nhaveI := nontrivial_of_finrank_eq_succ h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\nthis : Nontrivial V\n\u22a2 IsSimpleOrder (Submodule A V)\n[PROOFSTEP]\nrefine' \u27e8fun S => or_iff_not_imp_left.2 fun hn => _\u27e9\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\nthis : Nontrivial V\nS : Submodule A V\nhn : \u00acS = \u22a5\n\u22a2 S = \u22a4\n[PROOFSTEP]\nrw [\u2190 restrictScalars_inj K] at hn \u22a2\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\nthis : Nontrivial V\nS : Submodule A V\nhn : \u00acrestrictScalars K S = restrictScalars K \u22a5\n\u22a2 restrictScalars K S = restrictScalars K \u22a4\n[PROOFSTEP]\nhaveI : FiniteDimensional _ _ := finiteDimensional_of_finrank_eq_succ h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\nthis\u271d : Nontrivial V\nS : Submodule A V\nhn : \u00acrestrictScalars K S = restrictScalars K \u22a5\nthis : FiniteDimensional K V\n\u22a2 restrictScalars K S = restrictScalars K \u22a4\n[PROOFSTEP]\nrefine' eq_top_of_finrank_eq ((Submodule.finrank_le _).antisymm _)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\nthis\u271d : Nontrivial V\nS : Submodule A V\nhn : \u00acrestrictScalars K S = restrictScalars K \u22a5\nthis : FiniteDimensional K V\n\u22a2 finrank K V \u2264 finrank K { x // x \u2208 restrictScalars K S }\n[PROOFSTEP]\nsimpa only [h, finrank_bot] using Submodule.finrank_strictMono (Ne.bot_lt hn)\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\n\u22a2 FiniteDimensional F { x // x \u2208 \u22a5 }\n[PROOFSTEP]\nnontriviality E\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\n\u271d : Nontrivial E\n\u22a2 FiniteDimensional F { x // x \u2208 \u22a5 }\n[PROOFSTEP]\nexact finiteDimensional_of_rank_eq_one Subalgebra.rank_bot\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x \u2208 S } \u2264 1\n\u22a2 S = \u22a5\n[PROOFSTEP]\nnontriviality E\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x \u2208 S } \u2264 1\n\u271d : Nontrivial E\n\u22a2 S = \u22a5\n[PROOFSTEP]\nobtain \u27e8m, _, he\u27e9 :=\n Cardinal.exists_nat_eq_of_le_nat\n (h.trans_eq Nat.cast_one.symm)\n -- porting note: fails without explicit type\n[GOAL]\ncase intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x \u2208 S } \u2264 1\n\u271d : Nontrivial E\nm : \u2115\nleft\u271d : m \u2264 1\nhe : Module.rank F { x // x \u2208 S } = \u2191m\n\u22a2 S = \u22a5\n[PROOFSTEP]\nhaveI : FiniteDimensional F S := finiteDimensional_of_rank_eq_nat he\n[GOAL]\ncase intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x \u2208 S } \u2264 1\n\u271d : Nontrivial E\nm : \u2115\nleft\u271d : m \u2264 1\nhe : Module.rank F { x // x \u2208 S } = \u2191m\nthis : FiniteDimensional F { x // x \u2208 S }\n\u22a2 S = \u22a5\n[PROOFSTEP]\nrw [\u2190 not_bot_lt_iff, \u2190 Subalgebra.toSubmodule.lt_iff_lt]\n -- porting note: fails without explicit type\n[GOAL]\ncase intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x \u2208 S } \u2264 1\n\u271d : Nontrivial E\nm : \u2115\nleft\u271d : m \u2264 1\nhe : Module.rank F { x // x \u2208 S } = \u2191m\nthis : FiniteDimensional F { x // x \u2208 S }\n\u22a2 \u00ac\u2191toSubmodule \u22a5 < \u2191toSubmodule S\n[PROOFSTEP]\nhaveI : FiniteDimensional F (Subalgebra.toSubmodule S) := S.toSubmoduleEquiv.symm.finiteDimensional\n[GOAL]\ncase intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x \u2208 S } \u2264 1\n\u271d : Nontrivial E\nm : \u2115\nleft\u271d : m \u2264 1\nhe : Module.rank F { x // x \u2208 S } = \u2191m\nthis\u271d : FiniteDimensional F { x // x \u2208 S }\nthis : FiniteDimensional F { x // x \u2208 \u2191toSubmodule S }\n\u22a2 \u00ac\u2191toSubmodule \u22a5 < \u2191toSubmodule S\n[PROOFSTEP]\nrefine fun hl => (Submodule.finrank_lt_finrank_of_lt hl).not_le (natCast_le.1 ?_)\n[GOAL]\ncase intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x \u2208 S } \u2264 1\n\u271d : Nontrivial E\nm : \u2115\nleft\u271d : m \u2264 1\nhe : Module.rank F { x // x \u2208 S } = \u2191m\nthis\u271d : FiniteDimensional F { x // x \u2208 S }\nthis : FiniteDimensional F { x // x \u2208 \u2191toSubmodule S }\nhl : \u2191toSubmodule \u22a5 < \u2191toSubmodule S\n\u22a2 \u2191(finrank F { x // x \u2208 \u2191toSubmodule S }) \u2264 \u2191(finrank F { x // x \u2208 \u2191toSubmodule \u22a5 })\n[PROOFSTEP]\niterate 2 rw [Subalgebra.finrank_toSubmodule, finrank_eq_rank]\n[GOAL]\ncase intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x \u2208 S } \u2264 1\n\u271d : Nontrivial E\nm : \u2115\nleft\u271d : m \u2264 1\nhe : Module.rank F { x // x \u2208 S } = \u2191m\nthis\u271d : FiniteDimensional F { x // x \u2208 S }\nthis : FiniteDimensional F { x // x \u2208 \u2191toSubmodule S }\nhl : \u2191toSubmodule \u22a5 < \u2191toSubmodule S\n\u22a2 \u2191(finrank F { x // x \u2208 \u2191toSubmodule S }) \u2264 \u2191(finrank F { x // x \u2208 \u2191toSubmodule \u22a5 })\n[PROOFSTEP]\nrw [Subalgebra.finrank_toSubmodule, finrank_eq_rank]\n[GOAL]\ncase intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x \u2208 S } \u2264 1\n\u271d : Nontrivial E\nm : \u2115\nleft\u271d : m \u2264 1\nhe : Module.rank F { x // x \u2208 S } = \u2191m\nthis\u271d : FiniteDimensional F { x // x \u2208 S }\nthis : FiniteDimensional F { x // x \u2208 \u2191toSubmodule S }\nhl : \u2191toSubmodule \u22a5 < \u2191toSubmodule S\n\u22a2 Module.rank F { x // x \u2208 S } \u2264 \u2191(finrank F { x // x \u2208 \u2191toSubmodule \u22a5 })\n[PROOFSTEP]\nrw [Subalgebra.finrank_toSubmodule, finrank_eq_rank]\n[GOAL]\ncase intro.intro\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nS : Subalgebra F E\nh : Module.rank F { x // x \u2208 S } \u2264 1\n\u271d : Nontrivial E\nm : \u2115\nleft\u271d : m \u2264 1\nhe : Module.rank F { x // x \u2208 S } = \u2191m\nthis\u271d : FiniteDimensional F { x // x \u2208 S }\nthis : FiniteDimensional F { x // x \u2208 \u2191toSubmodule S }\nhl : \u2191toSubmodule \u22a5 < \u2191toSubmodule S\n\u22a2 Module.rank F { x // x \u2208 S } \u2264 Module.rank F { x // x \u2208 \u22a5 }\n[PROOFSTEP]\nexact h.trans_eq Subalgebra.rank_bot.symm\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nS : Subalgebra F E\nh : finrank F { x // x \u2208 S } = 1\n\u22a2 Module.rank F { x // x \u2208 S } \u2264 1\n[PROOFSTEP]\nhaveI : FiniteDimensional F S := finiteDimensional_of_finrank_eq_succ h\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nS : Subalgebra F E\nh : finrank F { x // x \u2208 S } = 1\nthis : FiniteDimensional F { x // x \u2208 S }\n\u22a2 Module.rank F { x // x \u2208 S } \u2264 1\n[PROOFSTEP]\nrw [\u2190 finrank_eq_rank, h, Nat.cast_one]\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Ring E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Nontrivial E\n\u22a2 \u22a5 = \u22a4 \u2194 Module.rank F E = 1\n[PROOFSTEP]\nrw [\u2190 rank_top, Subalgebra.rank_eq_one_iff, eq_comm]\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Ring E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Nontrivial E\n\u22a2 \u22a5 = \u22a4 \u2194 finrank F E = 1\n[PROOFSTEP]\nrw [\u2190 finrank_top, \u2190 subalgebra_top_finrank_eq_submodule_top_finrank, Subalgebra.finrank_eq_one_iff, eq_comm]\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nh : \u22a5 = \u22a4\n\u22a2 False\n[PROOFSTEP]\ncases hr.symm.trans (Subalgebra.bot_eq_top_iff_finrank_eq_one.1 h)\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\n\u22a2 \u2200 (a : Subalgebra F E), a = \u22a5 \u2228 a = \u22a4\n[PROOFSTEP]\nintro S\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\n\u22a2 S = \u22a5 \u2228 S = \u22a4\n[PROOFSTEP]\nhaveI : FiniteDimensional F E := finiteDimensional_of_finrank_eq_succ hr\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis : FiniteDimensional F E\n\u22a2 S = \u22a5 \u2228 S = \u22a4\n[PROOFSTEP]\nhaveI : FiniteDimensional F S := FiniteDimensional.finiteDimensional_submodule (Subalgebra.toSubmodule S)\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis\u271d : FiniteDimensional F E\nthis : FiniteDimensional F { x // x \u2208 S }\n\u22a2 S = \u22a5 \u2228 S = \u22a4\n[PROOFSTEP]\nhave : finrank F S \u2264 2 := hr \u25b8 S.toSubmodule.finrank_le\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis\u271d\u00b9 : FiniteDimensional F E\nthis\u271d : FiniteDimensional F { x // x \u2208 S }\nthis : finrank F { x // x \u2208 S } \u2264 2\n\u22a2 S = \u22a5 \u2228 S = \u22a4\n[PROOFSTEP]\nhave : 0 < finrank F S := finrank_pos_iff.mpr inferInstance\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis\u271d\u00b2 : FiniteDimensional F E\nthis\u271d\u00b9 : FiniteDimensional F { x // x \u2208 S }\nthis\u271d : finrank F { x // x \u2208 S } \u2264 2\nthis : 0 < finrank F { x // x \u2208 S }\n\u22a2 S = \u22a5 \u2228 S = \u22a4\n[PROOFSTEP]\ninterval_cases h : finrank F { x // x \u2208 S }\n[GOAL]\ncase \u00ab1\u00bb\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis\u271d\u00b2 : FiniteDimensional F E\nthis\u271d\u00b9 : FiniteDimensional F { x // x \u2208 S }\nh : finrank F { x // x \u2208 S } = 1\nthis\u271d : 1 \u2264 2\nthis : 0 < 1\n\u22a2 S = \u22a5 \u2228 S = \u22a4\n[PROOFSTEP]\nleft\n[GOAL]\ncase \u00ab1\u00bb.h\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis\u271d\u00b2 : FiniteDimensional F E\nthis\u271d\u00b9 : FiniteDimensional F { x // x \u2208 S }\nh : finrank F { x // x \u2208 S } = 1\nthis\u271d : 1 \u2264 2\nthis : 0 < 1\n\u22a2 S = \u22a5\n[PROOFSTEP]\nexact Subalgebra.eq_bot_of_finrank_one h\n[GOAL]\ncase \u00ab2\u00bb\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis\u271d\u00b2 : FiniteDimensional F E\nthis\u271d\u00b9 : FiniteDimensional F { x // x \u2208 S }\nh : finrank F { x // x \u2208 S } = 2\nthis\u271d : 2 \u2264 2\nthis : 0 < 2\n\u22a2 S = \u22a5 \u2228 S = \u22a4\n[PROOFSTEP]\nright\n[GOAL]\ncase \u00ab2\u00bb.h\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis\u271d\u00b2 : FiniteDimensional F E\nthis\u271d\u00b9 : FiniteDimensional F { x // x \u2208 S }\nh : finrank F { x // x \u2208 S } = 2\nthis\u271d : 2 \u2264 2\nthis : 0 < 2\n\u22a2 S = \u22a4\n[PROOFSTEP]\nrw [\u2190 hr] at h \n[GOAL]\ncase \u00ab2\u00bb.h\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis\u271d\u00b2 : FiniteDimensional F E\nthis\u271d\u00b9 : FiniteDimensional F { x // x \u2208 S }\nh : finrank F { x // x \u2208 S } = finrank F E\nthis\u271d : 2 \u2264 2\nthis : 0 < 2\n\u22a2 S = \u22a4\n[PROOFSTEP]\nrw [\u2190 Algebra.toSubmodule_eq_top]\n[GOAL]\ncase \u00ab2\u00bb.h\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\nhr : finrank F E = 2\ni : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))\nS : Subalgebra F E\nthis\u271d\u00b2 : FiniteDimensional F E\nthis\u271d\u00b9 : FiniteDimensional F { x // x \u2208 S }\nh : finrank F { x // x \u2208 S } = finrank F E\nthis\u271d : 2 \u2264 2\nthis : 0 < 2\n\u22a2 \u2191toSubmodule S = \u22a4\n[PROOFSTEP]\nexact eq_top_of_finrank_eq h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u22a2 \u2203 k, k \u2264 finrank K V \u2227 LinearMap.ker (f ^ k) = LinearMap.ker (f ^ Nat.succ k)\n[PROOFSTEP]\nclassical\nby_contra h_contra\nsimp_rw [not_exists, not_and] at h_contra \nhave h_le_ker_pow : \u2200 n : \u2115, n \u2264 (finrank K V).succ \u2192 n \u2264 finrank K (LinearMap.ker (f ^ n)) :=\n by\n intro n hn\n induction' n with n ih\n \u00b7 exact zero_le (finrank _ _)\n \u00b7 have h_ker_lt_ker : LinearMap.ker (f ^ n) < LinearMap.ker (f ^ n.succ) :=\n by\n refine' lt_of_le_of_ne _ (h_contra n (Nat.le_of_succ_le_succ hn))\n rw [pow_succ]\n apply LinearMap.ker_le_ker_comp\n have h_finrank_lt_finrank : finrank K (LinearMap.ker (f ^ n)) < finrank K (LinearMap.ker (f ^ n.succ)) := by\n apply Submodule.finrank_lt_finrank_of_lt h_ker_lt_ker\n calc\n n.succ \u2264 (finrank K \u2191(LinearMap.ker (f ^ n))).succ := Nat.succ_le_succ (ih (Nat.le_of_succ_le hn))\n _ \u2264 finrank K \u2191(LinearMap.ker (f ^ n.succ)) := Nat.succ_le_of_lt h_finrank_lt_finrank\nhave h_any_n_lt : \u2200 n, n \u2264 (finrank K V).succ \u2192 n \u2264 finrank K V := fun n hn =>\n (h_le_ker_pow n hn).trans (Submodule.finrank_le _)\nshow False\nexact Nat.not_succ_le_self _ (h_any_n_lt (finrank K V).succ (finrank K V).succ.le_refl)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u22a2 \u2203 k, k \u2264 finrank K V \u2227 LinearMap.ker (f ^ k) = LinearMap.ker (f ^ Nat.succ k)\n[PROOFSTEP]\nby_contra h_contra\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_contra : \u00ac\u2203 k, k \u2264 finrank K V \u2227 LinearMap.ker (f ^ k) = LinearMap.ker (f ^ Nat.succ k)\n\u22a2 False\n[PROOFSTEP]\nsimp_rw [not_exists, not_and] at h_contra \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_contra : \u2200 (x : \u2115), x \u2264 finrank K V \u2192 \u00acLinearMap.ker (f ^ x) = LinearMap.ker (f ^ Nat.succ x)\n\u22a2 False\n[PROOFSTEP]\nhave h_le_ker_pow : \u2200 n : \u2115, n \u2264 (finrank K V).succ \u2192 n \u2264 finrank K (LinearMap.ker (f ^ n)) :=\n by\n intro n hn\n induction' n with n ih\n \u00b7 exact zero_le (finrank _ _)\n \u00b7 have h_ker_lt_ker : LinearMap.ker (f ^ n) < LinearMap.ker (f ^ n.succ) :=\n by\n refine' lt_of_le_of_ne _ (h_contra n (Nat.le_of_succ_le_succ hn))\n rw [pow_succ]\n apply LinearMap.ker_le_ker_comp\n have h_finrank_lt_finrank : finrank K (LinearMap.ker (f ^ n)) < finrank K (LinearMap.ker (f ^ n.succ)) := by\n apply Submodule.finrank_lt_finrank_of_lt h_ker_lt_ker\n calc\n n.succ \u2264 (finrank K \u2191(LinearMap.ker (f ^ n))).succ := Nat.succ_le_succ (ih (Nat.le_of_succ_le hn))\n _ \u2264 finrank K \u2191(LinearMap.ker (f ^ n.succ)) := Nat.succ_le_of_lt h_finrank_lt_finrank\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_contra : \u2200 (x : \u2115), x \u2264 finrank K V \u2192 \u00acLinearMap.ker (f ^ x) = LinearMap.ker (f ^ Nat.succ x)\n\u22a2 \u2200 (n : \u2115), n \u2264 Nat.succ (finrank K V) \u2192 n \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ n) }\n[PROOFSTEP]\nintro n hn\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_contra : \u2200 (x : \u2115), x \u2264 finrank K V \u2192 \u00acLinearMap.ker (f ^ x) = LinearMap.ker (f ^ Nat.succ x)\nn : \u2115\nhn : n \u2264 Nat.succ (finrank K V)\n\u22a2 n \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ n) }\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_contra : \u2200 (x : \u2115), x \u2264 finrank K V \u2192 \u00acLinearMap.ker (f ^ x) = LinearMap.ker (f ^ Nat.succ x)\nn : \u2115\nhn\u271d : n \u2264 Nat.succ (finrank K V)\nhn : Nat.zero \u2264 Nat.succ (finrank K V)\n\u22a2 Nat.zero \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ Nat.zero) }\n[PROOFSTEP]\nexact zero_le (finrank _ _)\n[GOAL]\ncase succ\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_contra : \u2200 (x : \u2115), x \u2264 finrank K V \u2192 \u00acLinearMap.ker (f ^ x) = LinearMap.ker (f ^ Nat.succ x)\nn\u271d : \u2115\nhn\u271d : n\u271d \u2264 Nat.succ (finrank K V)\nn : \u2115\nih : n \u2264 Nat.succ (finrank K V) \u2192 n \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ n) }\nhn : Nat.succ n \u2264 Nat.succ (finrank K V)\n\u22a2 Nat.succ n \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ Nat.succ n) }\n[PROOFSTEP]\nhave h_ker_lt_ker : LinearMap.ker (f ^ n) < LinearMap.ker (f ^ n.succ) :=\n by\n refine' lt_of_le_of_ne _ (h_contra n (Nat.le_of_succ_le_succ hn))\n rw [pow_succ]\n apply LinearMap.ker_le_ker_comp\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_contra : \u2200 (x : \u2115), x \u2264 finrank K V \u2192 \u00acLinearMap.ker (f ^ x) = LinearMap.ker (f ^ Nat.succ x)\nn\u271d : \u2115\nhn\u271d : n\u271d \u2264 Nat.succ (finrank K V)\nn : \u2115\nih : n \u2264 Nat.succ (finrank K V) \u2192 n \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ n) }\nhn : Nat.succ n \u2264 Nat.succ (finrank K V)\n\u22a2 LinearMap.ker (f ^ n) < LinearMap.ker (f ^ Nat.succ n)\n[PROOFSTEP]\nrefine' lt_of_le_of_ne _ (h_contra n (Nat.le_of_succ_le_succ hn))\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_contra : \u2200 (x : \u2115), x \u2264 finrank K V \u2192 \u00acLinearMap.ker (f ^ x) = LinearMap.ker (f ^ Nat.succ x)\nn\u271d : \u2115\nhn\u271d : n\u271d \u2264 Nat.succ (finrank K V)\nn : \u2115\nih : n \u2264 Nat.succ (finrank K V) \u2192 n \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ n) }\nhn : Nat.succ n \u2264 Nat.succ (finrank K V)\n\u22a2 LinearMap.ker (f ^ n) \u2264 LinearMap.ker (f ^ Nat.succ n)\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_contra : \u2200 (x : \u2115), x \u2264 finrank K V \u2192 \u00acLinearMap.ker (f ^ x) = LinearMap.ker (f ^ Nat.succ x)\nn\u271d : \u2115\nhn\u271d : n\u271d \u2264 Nat.succ (finrank K V)\nn : \u2115\nih : n \u2264 Nat.succ (finrank K V) \u2192 n \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ n) }\nhn : Nat.succ n \u2264 Nat.succ (finrank K V)\n\u22a2 LinearMap.ker (f ^ n) \u2264 LinearMap.ker (f * f ^ n)\n[PROOFSTEP]\napply LinearMap.ker_le_ker_comp\n[GOAL]\ncase succ\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_contra : \u2200 (x : \u2115), x \u2264 finrank K V \u2192 \u00acLinearMap.ker (f ^ x) = LinearMap.ker (f ^ Nat.succ x)\nn\u271d : \u2115\nhn\u271d : n\u271d \u2264 Nat.succ (finrank K V)\nn : \u2115\nih : n \u2264 Nat.succ (finrank K V) \u2192 n \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ n) }\nhn : Nat.succ n \u2264 Nat.succ (finrank K V)\nh_ker_lt_ker : LinearMap.ker (f ^ n) < LinearMap.ker (f ^ Nat.succ n)\n\u22a2 Nat.succ n \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ Nat.succ n) }\n[PROOFSTEP]\nhave h_finrank_lt_finrank : finrank K (LinearMap.ker (f ^ n)) < finrank K (LinearMap.ker (f ^ n.succ)) := by\n apply Submodule.finrank_lt_finrank_of_lt h_ker_lt_ker\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_contra : \u2200 (x : \u2115), x \u2264 finrank K V \u2192 \u00acLinearMap.ker (f ^ x) = LinearMap.ker (f ^ Nat.succ x)\nn\u271d : \u2115\nhn\u271d : n\u271d \u2264 Nat.succ (finrank K V)\nn : \u2115\nih : n \u2264 Nat.succ (finrank K V) \u2192 n \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ n) }\nhn : Nat.succ n \u2264 Nat.succ (finrank K V)\nh_ker_lt_ker : LinearMap.ker (f ^ n) < LinearMap.ker (f ^ Nat.succ n)\n\u22a2 finrank K { x // x \u2208 LinearMap.ker (f ^ n) } < finrank K { x // x \u2208 LinearMap.ker (f ^ Nat.succ n) }\n[PROOFSTEP]\napply Submodule.finrank_lt_finrank_of_lt h_ker_lt_ker\n[GOAL]\ncase succ\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_contra : \u2200 (x : \u2115), x \u2264 finrank K V \u2192 \u00acLinearMap.ker (f ^ x) = LinearMap.ker (f ^ Nat.succ x)\nn\u271d : \u2115\nhn\u271d : n\u271d \u2264 Nat.succ (finrank K V)\nn : \u2115\nih : n \u2264 Nat.succ (finrank K V) \u2192 n \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ n) }\nhn : Nat.succ n \u2264 Nat.succ (finrank K V)\nh_ker_lt_ker : LinearMap.ker (f ^ n) < LinearMap.ker (f ^ Nat.succ n)\nh_finrank_lt_finrank :\n finrank K { x // x \u2208 LinearMap.ker (f ^ n) } < finrank K { x // x \u2208 LinearMap.ker (f ^ Nat.succ n) }\n\u22a2 Nat.succ n \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ Nat.succ n) }\n[PROOFSTEP]\ncalc\n n.succ \u2264 (finrank K \u2191(LinearMap.ker (f ^ n))).succ := Nat.succ_le_succ (ih (Nat.le_of_succ_le hn))\n _ \u2264 finrank K \u2191(LinearMap.ker (f ^ n.succ)) := Nat.succ_le_of_lt h_finrank_lt_finrank\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_contra : \u2200 (x : \u2115), x \u2264 finrank K V \u2192 \u00acLinearMap.ker (f ^ x) = LinearMap.ker (f ^ Nat.succ x)\nh_le_ker_pow : \u2200 (n : \u2115), n \u2264 Nat.succ (finrank K V) \u2192 n \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ n) }\n\u22a2 False\n[PROOFSTEP]\nhave h_any_n_lt : \u2200 n, n \u2264 (finrank K V).succ \u2192 n \u2264 finrank K V := fun n hn =>\n (h_le_ker_pow n hn).trans (Submodule.finrank_le _)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_contra : \u2200 (x : \u2115), x \u2264 finrank K V \u2192 \u00acLinearMap.ker (f ^ x) = LinearMap.ker (f ^ Nat.succ x)\nh_le_ker_pow : \u2200 (n : \u2115), n \u2264 Nat.succ (finrank K V) \u2192 n \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ n) }\nh_any_n_lt : \u2200 (n : \u2115), n \u2264 Nat.succ (finrank K V) \u2192 n \u2264 finrank K V\n\u22a2 False\n[PROOFSTEP]\nshow False\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_contra : \u2200 (x : \u2115), x \u2264 finrank K V \u2192 \u00acLinearMap.ker (f ^ x) = LinearMap.ker (f ^ Nat.succ x)\nh_le_ker_pow : \u2200 (n : \u2115), n \u2264 Nat.succ (finrank K V) \u2192 n \u2264 finrank K { x // x \u2208 LinearMap.ker (f ^ n) }\nh_any_n_lt : \u2200 (n : \u2115), n \u2264 Nat.succ (finrank K V) \u2192 n \u2264 finrank K V\n\u22a2 False\n[PROOFSTEP]\nexact Nat.not_succ_le_self _ (h_any_n_lt (finrank K V).succ (finrank K V).succ.le_refl)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nk : \u2115\nh : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ Nat.succ k)\n\u22a2 LinearMap.ker (f ^ k) = LinearMap.ker (f ^ (k + 0))\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nk : \u2115\nh : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ Nat.succ k)\nm : \u2115\n\u22a2 LinearMap.ker (f ^ k) = LinearMap.ker (f ^ (k + (m + 1)))\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nk : \u2115\nh : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ Nat.succ k)\nm : \u2115\n\u22a2 LinearMap.ker (f ^ k) \u2264 LinearMap.ker (f ^ (k + (m + 1)))\n[PROOFSTEP]\nrw [add_comm, pow_add]\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nk : \u2115\nh : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ Nat.succ k)\nm : \u2115\n\u22a2 LinearMap.ker (f ^ k) \u2264 LinearMap.ker (f ^ (m + 1) * f ^ k)\n[PROOFSTEP]\napply LinearMap.ker_le_ker_comp\n[GOAL]\ncase a\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End K V\nk : \u2115\nh : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ Nat.succ k)\nm : \u2115\n\u22a2 LinearMap.ker (f ^ (k + (m + 1))) \u2264 LinearMap.ker (f ^ k)\n[PROOFSTEP]\nrw [ker_pow_constant h m, add_comm m 1, \u2190 add_assoc, pow_add, pow_add f k m, LinearMap.mul_eq_comp,\n LinearMap.mul_eq_comp, LinearMap.ker_comp, LinearMap.ker_comp, h, Nat.add_one]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nm : \u2115\nhm : finrank K V \u2264 m\n\u22a2 LinearMap.ker (f ^ m) = LinearMap.ker (f ^ finrank K V)\n[PROOFSTEP]\nobtain \u27e8k, h_k_le, hk\u27e9 : \u2203 k, k \u2264 finrank K V \u2227 LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ) :=\n exists_ker_pow_eq_ker_pow_succ f\n[GOAL]\ncase intro.intro\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nm : \u2115\nhm : finrank K V \u2264 m\nk : \u2115\nh_k_le : k \u2264 finrank K V\nhk : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ Nat.succ k)\n\u22a2 LinearMap.ker (f ^ m) = LinearMap.ker (f ^ finrank K V)\n[PROOFSTEP]\ncalc\n LinearMap.ker (f ^ m) = LinearMap.ker (f ^ (k + (m - k))) := by rw [add_tsub_cancel_of_le (h_k_le.trans hm)]\n _ = LinearMap.ker (f ^ k) := by rw [ker_pow_constant hk _]\n _ = LinearMap.ker (f ^ (k + (finrank K V - k))) := (ker_pow_constant hk (finrank K V - k))\n _ = LinearMap.ker (f ^ finrank K V) := by rw [add_tsub_cancel_of_le h_k_le]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nm : \u2115\nhm : finrank K V \u2264 m\nk : \u2115\nh_k_le : k \u2264 finrank K V\nhk : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ Nat.succ k)\n\u22a2 LinearMap.ker (f ^ m) = LinearMap.ker (f ^ (k + (m - k)))\n[PROOFSTEP]\nrw [add_tsub_cancel_of_le (h_k_le.trans hm)]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nm : \u2115\nhm : finrank K V \u2264 m\nk : \u2115\nh_k_le : k \u2264 finrank K V\nhk : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ Nat.succ k)\n\u22a2 LinearMap.ker (f ^ (k + (m - k))) = LinearMap.ker (f ^ k)\n[PROOFSTEP]\nrw [ker_pow_constant hk _]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nm : \u2115\nhm : finrank K V \u2264 m\nk : \u2115\nh_k_le : k \u2264 finrank K V\nhk : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ Nat.succ k)\n\u22a2 LinearMap.ker (f ^ (k + (finrank K V - k))) = LinearMap.ker (f ^ finrank K V)\n[PROOFSTEP]\nrw [add_tsub_cancel_of_le h_k_le]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nm : \u2115\n\u22a2 LinearMap.ker (f ^ m) \u2264 LinearMap.ker (f ^ finrank K V)\n[PROOFSTEP]\nby_cases h_cases : m < finrank K V\n[GOAL]\ncase pos\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nm : \u2115\nh_cases : m < finrank K V\n\u22a2 LinearMap.ker (f ^ m) \u2264 LinearMap.ker (f ^ finrank K V)\n[PROOFSTEP]\nrw [\u2190 add_tsub_cancel_of_le (Nat.le_of_lt h_cases), add_comm, pow_add]\n[GOAL]\ncase pos\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nm : \u2115\nh_cases : m < finrank K V\n\u22a2 LinearMap.ker (f ^ m) \u2264 LinearMap.ker (f ^ (finrank K V - m) * f ^ m)\n[PROOFSTEP]\napply LinearMap.ker_le_ker_comp\n[GOAL]\ncase neg\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nm : \u2115\nh_cases : \u00acm < finrank K V\n\u22a2 LinearMap.ker (f ^ m) \u2264 LinearMap.ker (f ^ finrank K V)\n[PROOFSTEP]\nrw [ker_pow_eq_ker_pow_finrank_of_le (le_of_not_lt h_cases)]\n[GOAL]\nK\u271d : Type u\nV\u271d : Type v\nK V : Type u\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\n\u22a2 #V = #K ^ Module.rank K V\n[PROOFSTEP]\nlet s := Basis.ofVectorSpaceIndex K V\n[GOAL]\nK\u271d : Type u\nV\u271d : Type v\nK V : Type u\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns : Set V := Basis.ofVectorSpaceIndex K V\n\u22a2 #V = #K ^ Module.rank K V\n[PROOFSTEP]\nlet hs := Basis.ofVectorSpace K V\n[GOAL]\nK\u271d : Type u\nV\u271d : Type v\nK V : Type u\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns : Set V := Basis.ofVectorSpaceIndex K V\nhs : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\n\u22a2 #V = #K ^ Module.rank K V\n[PROOFSTEP]\ncalc\n #V = #(s \u2192\u2080 K) := Quotient.sound \u27e8hs.repr.toEquiv\u27e9\n _ = #(s \u2192 K) := (Quotient.sound \u27e8Finsupp.equivFunOnFinite\u27e9)\n _ = _ := by rw [\u2190 Cardinal.lift_inj.1 hs.mk_eq_rank, Cardinal.power_def]\n[GOAL]\nK\u271d : Type u\nV\u271d : Type v\nK V : Type u\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\ns : Set V := Basis.ofVectorSpaceIndex K V\nhs : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\n\u22a2 #(\u2191s \u2192 K) = #K ^ Module.rank K V\n[PROOFSTEP]\nrw [\u2190 Cardinal.lift_inj.1 hs.mk_eq_rank, Cardinal.power_def]\n[GOAL]\nK\u271d : Type u\nV\u271d : Type v\nK V : Type u\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : _root_.Finite K\ninst\u271d : FiniteDimensional K V\n\u22a2 #V < \u2135\u2080\n[PROOFSTEP]\nletI : IsNoetherian K V := IsNoetherian.iff_fg.2 inferInstance\n[GOAL]\nK\u271d : Type u\nV\u271d : Type v\nK V : Type u\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : _root_.Finite K\ninst\u271d : FiniteDimensional K V\nthis : IsNoetherian K V := Iff.mpr IsNoetherian.iff_fg inferInstance\n\u22a2 #V < \u2135\u2080\n[PROOFSTEP]\nrw [cardinal_mk_eq_cardinal_mk_field_pow_rank K V]\n[GOAL]\nK\u271d : Type u\nV\u271d : Type v\nK V : Type u\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : _root_.Finite K\ninst\u271d : FiniteDimensional K V\nthis : IsNoetherian K V := Iff.mpr IsNoetherian.iff_fg inferInstance\n\u22a2 #K ^ Module.rank K V < \u2135\u2080\n[PROOFSTEP]\nexact Cardinal.power_lt_aleph0 (Cardinal.lt_aleph0_of_finite K) (IsNoetherian.rank_lt_aleph0 K V)\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.FiniteDimensional", "llama_tokens": 99386, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056295505783, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.4162936128521069}} {"text": "[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\n\u22a2 IsClosed \u2191U\n[PROOFSTEP]\napply isOpen_compl_iff.1\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\n\u22a2 IsOpen (\u2191U)\u1d9c\n[PROOFSTEP]\nrefine' isOpen_iff_forall_mem_open.2 fun x hx => \u27e8(fun y => y * x\u207b\u00b9) \u207b\u00b9' U, _, _, _\u27e9\n[GOAL]\ncase refine'_1\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\n\u22a2 (fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U \u2286 (\u2191U)\u1d9c\n[PROOFSTEP]\nrefine' fun u hux hu => hx _\n[GOAL]\ncase refine'_1\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\nu : G\nhux : u \u2208 (fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U\nhu : u \u2208 \u2191U\n\u22a2 x \u2208 \u2191U\n[PROOFSTEP]\nsimp only [Set.mem_preimage, SetLike.mem_coe] at hux hu \u22a2\n[GOAL]\ncase refine'_1\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\nu : G\nhux : u * x\u207b\u00b9 \u2208 U\nhu : u \u2208 U\n\u22a2 x \u2208 U\n[PROOFSTEP]\nconvert U.mul_mem (U.inv_mem hux) hu\n[GOAL]\ncase a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\nu : G\nhux : u * x\u207b\u00b9 \u2208 U\nhu : u \u2208 U\n\u22a2 x \u2208 U \u2194 (u * x\u207b\u00b9)\u207b\u00b9 * u \u2208 \u2191U\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\n\u22a2 IsOpen ((fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U)\n[PROOFSTEP]\nexact U.isOpen.preimage (continuous_mul_right _)\n[GOAL]\ncase refine'_3\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\n\u22a2 x \u2208 (fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U\n[PROOFSTEP]\nsimp [one_mem]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : ContinuousMul G\nH : Subgroup G\ng : G\nhg : \u2191H \u2208 \ud835\udcdd g\n\u22a2 IsOpen \u2191H\n[PROOFSTEP]\nrefine' isOpen_iff_mem_nhds.2 fun x hx => _\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : ContinuousMul G\nH : Subgroup G\ng : G\nhg : \u2191H \u2208 \ud835\udcdd g\nx : G\nhx : x \u2208 \u2191H\n\u22a2 \u2191H \u2208 \ud835\udcdd x\n[PROOFSTEP]\nhave hg' : g \u2208 H := SetLike.mem_coe.1 (mem_of_mem_nhds hg)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : ContinuousMul G\nH : Subgroup G\ng : G\nhg : \u2191H \u2208 \ud835\udcdd g\nx : G\nhx : x \u2208 \u2191H\nhg' : g \u2208 H\n\u22a2 \u2191H \u2208 \ud835\udcdd x\n[PROOFSTEP]\nhave : Filter.Tendsto (fun y => y * (x\u207b\u00b9 * g)) (\ud835\udcdd x) (\ud835\udcdd g) :=\n (continuous_id.mul continuous_const).tendsto' _ _ (mul_inv_cancel_left _ _)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : ContinuousMul G\nH : Subgroup G\ng : G\nhg : \u2191H \u2208 \ud835\udcdd g\nx : G\nhx : x \u2208 \u2191H\nhg' : g \u2208 H\nthis : Filter.Tendsto (fun y => y * (x\u207b\u00b9 * g)) (\ud835\udcdd x) (\ud835\udcdd g)\n\u22a2 \u2191H \u2208 \ud835\udcdd x\n[PROOFSTEP]\nsimpa only [SetLike.mem_coe, Filter.mem_map', H.mul_mem_cancel_right (H.mul_mem (H.inv_mem hx) hg')] using this hg\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.OpenSubgroup", "llama_tokens": 1656, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.689305616785446, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.41629360514283714}} {"text": "[GOAL]\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\nj j' : J\nf : j \u27f6 j'\n\u22a2 ((Functor.const J).obj (limit F)).map f \u226b (fun j => Hom.mk (limit.\u03c0 (F \u22d9 forget C) j)) j' =\n (fun j => Hom.mk (limit.\u03c0 (F \u22d9 forget C) j)) j \u226b F.map f\n[PROOFSTEP]\next\n[GOAL]\ncase w\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\nj j' : J\nf : j \u27f6 j'\n\u22a2 (((Functor.const J).obj (limit F)).map f \u226b (fun j => Hom.mk (limit.\u03c0 (F \u22d9 forget C) j)) j').hom =\n ((fun j => Hom.mk (limit.\u03c0 (F \u22d9 forget C) j)) j \u226b F.map f).hom\n[PROOFSTEP]\nexact (limit.cone (F \u22d9 Mon_.forget C)).\u03c0.naturality f\n[GOAL]\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\n\u22a2 \u2200 (j : J),\n NatTrans.app ((forget C).mapCone (limitCone F)).\u03c0 j =\n (Iso.refl ((forget C).mapCone (limitCone F)).pt).hom \u226b NatTrans.app (limit.cone (F \u22d9 forget C)).\u03c0 j\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\n\u22a2 s.pt.mul \u226b limit.lift (F \u22d9 forget C) ((forget C).mapCone s) =\n MonoidalCategory.tensorHom (limit.lift (F \u22d9 forget C) ((forget C).mapCone s))\n (limit.lift (F \u22d9 forget C) ((forget C).mapCone s)) \u226b\n (limitCone F).pt.mul\n[PROOFSTEP]\ndsimp\n[GOAL]\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\n\u22a2 s.pt.mul \u226b limit.lift (F \u22d9 forget C) ((forget C).mapCone s) =\n MonoidalCategory.tensorHom (limit.lift (F \u22d9 forget C) ((forget C).mapCone s))\n (limit.lift (F \u22d9 forget C) ((forget C).mapCone s)) \u226b\n (limit F).mul\n[PROOFSTEP]\next\n[GOAL]\ncase w\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\nj\u271d : J\n\u22a2 (s.pt.mul \u226b limit.lift (F \u22d9 forget C) ((forget C).mapCone s)) \u226b limit.\u03c0 (F \u22d9 forget C) j\u271d =\n (MonoidalCategory.tensorHom (limit.lift (F \u22d9 forget C) ((forget C).mapCone s))\n (limit.lift (F \u22d9 forget C) ((forget C).mapCone s)) \u226b\n (limit F).mul) \u226b\n limit.\u03c0 (F \u22d9 forget C) j\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase w\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\nj\u271d : J\n\u22a2 MonoidalCategory.tensorHom (NatTrans.app s.\u03c0 j\u271d).hom (NatTrans.app s.\u03c0 j\u271d).hom \u226b (F.obj j\u271d).mul =\n MonoidalCategory.tensorHom (limit.lift (F \u22d9 forget C) ((forget C).mapCone s))\n (limit.lift (F \u22d9 forget C) ((forget C).mapCone s)) \u226b\n MonoidalCategory.tensorHom (limit.\u03c0 (F \u22d9 forget C) j\u271d) (limit.\u03c0 (F \u22d9 forget C) j\u271d) \u226b (F.obj j\u271d).mul\n[PROOFSTEP]\nslice_rhs 1 2 => rw [\u2190 MonoidalCategory.tensor_comp, limit.lift_\u03c0]\n[GOAL]\ncase a\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\nj\u271d : J\n| MonoidalCategory.tensorHom (limit.lift (F \u22d9 forget C) ((forget C).mapCone s))\n (limit.lift (F \u22d9 forget C) ((forget C).mapCone s)) \u226b\n MonoidalCategory.tensorHom (limit.\u03c0 (F \u22d9 forget C) j\u271d) (limit.\u03c0 (F \u22d9 forget C) j\u271d)\ncase a\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\nj\u271d : J\n| (F.obj j\u271d).mul\n[PROOFSTEP]\nrw [\u2190 MonoidalCategory.tensor_comp, limit.lift_\u03c0]\n[GOAL]\ncase a\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\nj\u271d : J\n| MonoidalCategory.tensorHom (limit.lift (F \u22d9 forget C) ((forget C).mapCone s))\n (limit.lift (F \u22d9 forget C) ((forget C).mapCone s)) \u226b\n MonoidalCategory.tensorHom (limit.\u03c0 (F \u22d9 forget C) j\u271d) (limit.\u03c0 (F \u22d9 forget C) j\u271d)\ncase a\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\nj\u271d : J\n| (F.obj j\u271d).mul\n[PROOFSTEP]\nrw [\u2190 MonoidalCategory.tensor_comp, limit.lift_\u03c0]\n[GOAL]\ncase a\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\nj\u271d : J\n| MonoidalCategory.tensorHom (limit.lift (F \u22d9 forget C) ((forget C).mapCone s))\n (limit.lift (F \u22d9 forget C) ((forget C).mapCone s)) \u226b\n MonoidalCategory.tensorHom (limit.\u03c0 (F \u22d9 forget C) j\u271d) (limit.\u03c0 (F \u22d9 forget C) j\u271d)\ncase a\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\nj\u271d : J\n| (F.obj j\u271d).mul\n[PROOFSTEP]\nrw [\u2190 MonoidalCategory.tensor_comp, limit.lift_\u03c0]\n[GOAL]\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\nh : J\n\u22a2 (fun s => Hom.mk (limit.lift (F \u22d9 forget C) ((forget C).mapCone s))) s \u226b NatTrans.app (limitCone F).\u03c0 h =\n NatTrans.app s.\u03c0 h\n[PROOFSTEP]\next\n[GOAL]\ncase w\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\nh : J\n\u22a2 ((fun s => Hom.mk (limit.lift (F \u22d9 forget C) ((forget C).mapCone s))) s \u226b NatTrans.app (limitCone F).\u03c0 h).hom =\n (NatTrans.app s.\u03c0 h).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\nm : s.pt \u27f6 (limitCone F).pt\nw : \u2200 (j : J), m \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m = (fun s => Hom.mk (limit.lift (F \u22d9 forget C) ((forget C).mapCone s))) s\n[PROOFSTEP]\next1\n[GOAL]\ncase w\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\nm : s.pt \u27f6 (limitCone F).pt\nw : \u2200 (j : J), m \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m.hom = ((fun s => Hom.mk (limit.lift (F \u22d9 forget C) ((forget C).mapCone s))) s).hom\n[PROOFSTEP]\nrefine' limit.hom_ext (fun j => _)\n[GOAL]\ncase w\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\nm : s.pt \u27f6 (limitCone F).pt\nw : \u2200 (j : J), m \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app s.\u03c0 j\nj : J\n\u22a2 m.hom \u226b limit.\u03c0 (MonFunctorCategoryEquivalence.inverse.obj F).X j =\n ((fun s => Hom.mk (limit.lift (F \u22d9 forget C) ((forget C).mapCone s))) s).hom \u226b\n limit.\u03c0 (MonFunctorCategoryEquivalence.inverse.obj F).X j\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\nm : s.pt \u27f6 (limitCone F).pt\nw : \u2200 (j : J), m \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app s.\u03c0 j\nj : J\n\u22a2 m.hom \u226b limit.\u03c0 (F \u22d9 forget C) j = limit.lift (F \u22d9 forget C) ((forget C).mapCone s) \u226b limit.\u03c0 (F \u22d9 forget C) j\n[PROOFSTEP]\nsimp only [Mon_.forget_map, limit.lift_\u03c0, Functor.mapCone_\u03c0_app]\n[GOAL]\ncase w\nJ : Type v\ninst\u271d\u00b3 : SmallCategory J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : MonoidalCategory C\nF : J \u2964 Mon_ C\ns : Cone F\nm : s.pt \u27f6 (limitCone F).pt\nw : \u2200 (j : J), m \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app s.\u03c0 j\nj : J\n\u22a2 m.hom \u226b limit.\u03c0 (F \u22d9 forget C) j = (NatTrans.app s.\u03c0 j).hom\n[PROOFSTEP]\nexact congr_arg Mon_.Hom.hom (w j)\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monoidal.Internal.Limits", "llama_tokens": 3719, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.4161086328964721}} {"text": "[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\n\u22a2 List.NodupKeys (List.map Prod.toSigma (Finset.toList (graph f)))\n[PROOFSTEP]\nrw [List.NodupKeys, List.keys, List.map_map, Prod.fst_comp_toSigma, List.nodup_map_iff_inj_on]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\n\u22a2 \u2200 (x : \u03b1 \u00d7 M), x \u2208 Finset.toList (graph f) \u2192 \u2200 (y : \u03b1 \u00d7 M), y \u2208 Finset.toList (graph f) \u2192 x.fst = y.fst \u2192 x = y\n[PROOFSTEP]\nrintro \u27e8b, m\u27e9 hb \u27e8c, n\u27e9 hc (rfl : b = c)\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\nb : \u03b1\nm : M\nhb : (b, m) \u2208 Finset.toList (graph f)\nn : M\nhc : (b, n) \u2208 Finset.toList (graph f)\n\u22a2 (b, m) = (b, n)\n[PROOFSTEP]\nrw [Finset.mem_toList, Finsupp.mem_graph_iff] at hb hc \n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\nb : \u03b1\nm : M\nhb : \u2191f (b, m).fst = (b, m).snd \u2227 (b, m).snd \u2260 0\nn : M\nhc : \u2191f (b, n).fst = (b, n).snd \u2227 (b, n).snd \u2260 0\n\u22a2 (b, m) = (b, n)\n[PROOFSTEP]\ndsimp at hb hc \n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\nb : \u03b1\nm : M\nhb : \u2191f b = m \u2227 \u00acm = 0\nn : M\nhc : \u2191f b = n \u2227 \u00acn = 0\n\u22a2 (b, m) = (b, n)\n[PROOFSTEP]\nrw [\u2190 hc.1, hb.1]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\n\u22a2 List.Nodup (Finset.toList (graph f))\n[PROOFSTEP]\napply Finset.nodup_toList\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192\u2080 M\n\u22a2 List.toFinset (AList.keys (toAList f)) = f.support\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nf : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 List.toFinset (AList.keys (toAList f)) \u2194 a\u271d \u2208 f.support\n[PROOFSTEP]\nsimp [toAList, AList.mem_keys, AList.keys, List.keys]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\nx : \u03b1\n\u22a2 x \u2208 toAList f \u2194 \u2191f x \u2260 0\n[PROOFSTEP]\nclassical rw [AList.mem_keys, \u2190 List.mem_toFinset, toAList_keys_toFinset, mem_support_iff]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\nx : \u03b1\n\u22a2 x \u2208 toAList f \u2194 \u2191f x \u2260 0\n[PROOFSTEP]\nrw [AList.mem_keys, \u2190 List.mem_toFinset, toAList_keys_toFinset, mem_support_iff]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nl : AList fun _x => M\n\u22a2 Finset \u03b1\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b1\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nl : AList fun _x => M\nthis : DecidableEq \u03b1\n\u22a2 Finset \u03b1\n[PROOFSTEP]\nhaveI := Classical.decEq M\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nl : AList fun _x => M\nthis\u271d : DecidableEq \u03b1\nthis : DecidableEq M\n\u22a2 Finset \u03b1\n[PROOFSTEP]\nexact (l.1.filter fun x => Sigma.snd x \u2260 0).keys.toFinset\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nl : AList fun _x => M\na : \u03b1\n\u22a2 a \u2208 toFinset (List.keys (filter (fun x => decide (x.snd \u2260 0)) l.entries)) \u2194\n (fun a => Option.getD (lookup a l) 0) a \u2260 0\n[PROOFSTEP]\nclassical\nsimp_rw [@mem_toFinset _ _, List.mem_keys, List.mem_filter, \u2190 mem_lookup_iff]\ncases lookup a l <;> simp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nl : AList fun _x => M\na : \u03b1\n\u22a2 a \u2208 toFinset (List.keys (filter (fun x => decide (x.snd \u2260 0)) l.entries)) \u2194\n (fun a => Option.getD (lookup a l) 0) a \u2260 0\n[PROOFSTEP]\nsimp_rw [@mem_toFinset _ _, List.mem_keys, List.mem_filter, \u2190 mem_lookup_iff]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nl : AList fun _x => M\na : \u03b1\n\u22a2 (\u2203 b, b \u2208 lookup a l \u2227 decide (b \u2260 0) = true) \u2194 Option.getD (lookup a l) 0 \u2260 0\n[PROOFSTEP]\ncases lookup a l\n[GOAL]\ncase none\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nl : AList fun _x => M\na : \u03b1\n\u22a2 (\u2203 b, b \u2208 none \u2227 decide (b \u2260 0) = true) \u2194 Option.getD none 0 \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nl : AList fun _x => M\na : \u03b1\nval\u271d : M\n\u22a2 (\u2203 b, b \u2208 some val\u271d \u2227 decide (b \u2260 0) = true) \u2194 Option.getD (some val\u271d) 0 \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nl : AList fun _x => M\na : \u03b1\n\u22a2 \u2191(lookupFinsupp l) a = Option.getD (lookup a l) 0\n[PROOFSTEP]\nsimp only [lookupFinsupp, ne_eq, Finsupp.coe_mk]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nl : AList fun _x => M\na : \u03b1\n\u22a2 Option.getD (lookup a l) 0 = Option.getD (lookup a l) 0\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq M\nl : AList fun _x => M\n\u22a2 (lookupFinsupp l).support = toFinset (List.keys (filter (fun x => decide (x.snd \u2260 0)) l.entries))\n[PROOFSTEP]\nsimp only [lookupFinsupp, ne_eq, Finsupp.coe_mk]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq M\nl : AList fun _x => M\n\u22a2 toFinset (List.keys (filter (fun x => decide \u00acx.snd = 0) l.entries)) =\n toFinset (List.keys (filter (fun x => decide \u00acx.snd = 0) l.entries))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e_2.h\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq M\nl : AList fun _x => M\n\u22a2 (fun a b => Classical.decEq \u03b1 a b) = fun a b => inst\u271d\u00b9 a b\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\ncase h.e_3.h.e_a.e_p\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq M\nl : AList fun _x => M\n\u22a2 (fun x => decide \u00acx.snd = 0) = fun x => decide \u00acx.snd = 0\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h.e_3.h.e_a.e_p.h\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq M\nl : AList fun _x => M\nx\u271d : (_ : \u03b1) \u00d7 M\n\u22a2 (decide \u00acx\u271d.snd = 0) = decide \u00acx\u271d.snd = 0\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nl : AList fun _x => M\na : \u03b1\nx : M\nhx : x \u2260 0\n\u22a2 \u2191(lookupFinsupp l) a = x \u2194 x \u2208 lookup a l\n[PROOFSTEP]\nrw [lookupFinsupp_apply]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nl : AList fun _x => M\na : \u03b1\nx : M\nhx : x \u2260 0\n\u22a2 Option.getD (lookup a l) 0 = x \u2194 x \u2208 lookup a l\n[PROOFSTEP]\ncases' lookup a l with m\n[GOAL]\ncase none\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nl : AList fun _x => M\na : \u03b1\nx : M\nhx : x \u2260 0\n\u22a2 Option.getD none 0 = x \u2194 x \u2208 none\n[PROOFSTEP]\nsimp [hx.symm]\n[GOAL]\ncase some\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nl : AList fun _x => M\na : \u03b1\nx : M\nhx : x \u2260 0\nm : M\n\u22a2 Option.getD (some m) 0 = x \u2194 x \u2208 some m\n[PROOFSTEP]\nsimp [hx.symm]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nl : AList fun _x => M\na : \u03b1\n\u22a2 \u2191(lookupFinsupp l) a = 0 \u2194 \u00aca \u2208 l \u2228 0 \u2208 lookup a l\n[PROOFSTEP]\nrw [lookupFinsupp_apply, \u2190 lookup_eq_none]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nl : AList fun _x => M\na : \u03b1\n\u22a2 Option.getD (lookup a l) 0 = 0 \u2194 lookup a l = none \u2228 0 \u2208 lookup a l\n[PROOFSTEP]\ncases' lookup a l with m\n[GOAL]\ncase none\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nl : AList fun _x => M\na : \u03b1\n\u22a2 Option.getD none 0 = 0 \u2194 none = none \u2228 0 \u2208 none\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nl : AList fun _x => M\na : \u03b1\nm : M\n\u22a2 Option.getD (some m) 0 = 0 \u2194 some m = none \u2228 0 \u2208 some m\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\n\u22a2 lookupFinsupp \u2205 = 0\n[PROOFSTEP]\nclassical\next\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\n\u22a2 lookupFinsupp \u2205 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\na\u271d : \u03b1\n\u22a2 \u2191(lookupFinsupp \u2205) a\u271d = \u21910 a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nl : AList fun _x => M\na : \u03b1\nm : M\n\u22a2 lookupFinsupp (insert a m l) = Finsupp.update (lookupFinsupp l) a m\n[PROOFSTEP]\next b\n[GOAL]\ncase h\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nl : AList fun _x => M\na : \u03b1\nm : M\nb : \u03b1\n\u22a2 \u2191(lookupFinsupp (insert a m l)) b = \u2191(Finsupp.update (lookupFinsupp l) a m) b\n[PROOFSTEP]\nby_cases h : b = a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nl : AList fun _x => M\na : \u03b1\nm : M\nb : \u03b1\nh : b = a\n\u22a2 \u2191(lookupFinsupp (insert a m l)) b = \u2191(Finsupp.update (lookupFinsupp l) a m) b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\nl : AList fun _x => M\na : \u03b1\nm : M\nb : \u03b1\nh : \u00acb = a\n\u22a2 \u2191(lookupFinsupp (insert a m l)) b = \u2191(Finsupp.update (lookupFinsupp l) a m) b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\na : \u03b1\nm : M\n\u22a2 lookupFinsupp (singleton a m) = Finsupp.single a m\n[PROOFSTEP]\nclassical\n -- porting note: was `simp [\u2190AList.insert_empty]` but timeout issuessimp only [\u2190 AList.insert_empty,\n insert_lookupFinsupp, empty_lookupFinsupp, Finsupp.zero_update]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\na : \u03b1\nm : M\n\u22a2 lookupFinsupp (singleton a m) = Finsupp.single a m\n[PROOFSTEP]\nsimp only [\u2190 AList.insert_empty, insert_lookupFinsupp, empty_lookupFinsupp, Finsupp.zero_update]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\n\u22a2 lookupFinsupp (Finsupp.toAList f) = f\n[PROOFSTEP]\next a\n[GOAL]\ncase h\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 \u2191(lookupFinsupp (Finsupp.toAList f)) a = \u2191f a\n[PROOFSTEP]\nclassical\nby_cases h : f a = 0\n\u00b7 suffices f.toAList.lookup a = none by simp [h, this]\n \u00b7 simp [lookup_eq_none, h]\n\u00b7 suffices f.toAList.lookup a = some (f a) by simp [h, this]\n \u00b7 apply mem_lookup_iff.2\n simpa using h\n[GOAL]\ncase h\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 \u2191(lookupFinsupp (Finsupp.toAList f)) a = \u2191f a\n[PROOFSTEP]\nby_cases h : f a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : \u2191f a = 0\n\u22a2 \u2191(lookupFinsupp (Finsupp.toAList f)) a = \u2191f a\n[PROOFSTEP]\nsuffices f.toAList.lookup a = none by simp [h, this]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : \u2191f a = 0\nthis : lookup a (Finsupp.toAList f) = none\n\u22a2 \u2191(lookupFinsupp (Finsupp.toAList f)) a = \u2191f a\n[PROOFSTEP]\nsimp [h, this]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : \u2191f a = 0\n\u22a2 lookup a (Finsupp.toAList f) = none\n[PROOFSTEP]\nsimp [lookup_eq_none, h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : \u00ac\u2191f a = 0\n\u22a2 \u2191(lookupFinsupp (Finsupp.toAList f)) a = \u2191f a\n[PROOFSTEP]\nsuffices f.toAList.lookup a = some (f a) by simp [h, this]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : \u00ac\u2191f a = 0\nthis : lookup a (Finsupp.toAList f) = some (\u2191f a)\n\u22a2 \u2191(lookupFinsupp (Finsupp.toAList f)) a = \u2191f a\n[PROOFSTEP]\nsimp [h, this]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : \u00ac\u2191f a = 0\n\u22a2 lookup a (Finsupp.toAList f) = some (\u2191f a)\n[PROOFSTEP]\napply mem_lookup_iff.2\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : \u00ac\u2191f a = 0\n\u22a2 { fst := a, snd := \u2191f a } \u2208 (Finsupp.toAList f).entries\n[PROOFSTEP]\nsimpa using h\n", "meta": {"mathlib_filename": "Mathlib.Data.Finsupp.AList", "llama_tokens": 5400, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.4161086328964721}} {"text": "[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nsrc\u271d : ((n \u2192 R) \u2192\u2097[R] n \u2192 R) \u2243\u2097[R] Matrix n n R := LinearMap.toMatrix'\nT S : Module.End R (n \u2192 R)\n\u22a2 AddHom.toFun src\u271d.toAddHom \u2045T, S\u2046 = \u2045AddHom.toFun src\u271d.toAddHom T, AddHom.toFun src\u271d.toAddHom S\u2046\n[PROOFSTEP]\nlet f := @LinearMap.toMatrix' R _ n n _ _\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nsrc\u271d : ((n \u2192 R) \u2192\u2097[R] n \u2192 R) \u2243\u2097[R] Matrix n n R := LinearMap.toMatrix'\nT S : Module.End R (n \u2192 R)\nf : ((n \u2192 R) \u2192\u2097[R] n \u2192 R) \u2243\u2097[R] Matrix n n R := LinearMap.toMatrix'\n\u22a2 AddHom.toFun src\u271d.toAddHom \u2045T, S\u2046 = \u2045AddHom.toFun src\u271d.toAddHom T, AddHom.toFun src\u271d.toAddHom S\u2046\n[PROOFSTEP]\nchange f (T.comp S - S.comp T) = f T * f S - f S * f T\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nsrc\u271d : ((n \u2192 R) \u2192\u2097[R] n \u2192 R) \u2243\u2097[R] Matrix n n R := LinearMap.toMatrix'\nT S : Module.End R (n \u2192 R)\nf : ((n \u2192 R) \u2192\u2097[R] n \u2192 R) \u2243\u2097[R] Matrix n n R := LinearMap.toMatrix'\n\u22a2 \u2191f (LinearMap.comp T S - LinearMap.comp S T) = \u2191f T * \u2191f S - \u2191f S * \u2191f T\n[PROOFSTEP]\nhave h : \u2200 T S : Module.End R _, f (T.comp S) = f T * f S := LinearMap.toMatrix'_comp\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nsrc\u271d : ((n \u2192 R) \u2192\u2097[R] n \u2192 R) \u2243\u2097[R] Matrix n n R := LinearMap.toMatrix'\nT S : Module.End R (n \u2192 R)\nf : ((n \u2192 R) \u2192\u2097[R] n \u2192 R) \u2243\u2097[R] Matrix n n R := LinearMap.toMatrix'\nh : \u2200 (T S : Module.End R (n \u2192 R)), \u2191f (LinearMap.comp T S) = \u2191f T * \u2191f S\n\u22a2 \u2191f (LinearMap.comp T S - LinearMap.comp S T) = \u2191f T * \u2191f S - \u2191f S * \u2191f T\n[PROOFSTEP]\nrw [LinearEquiv.map_sub, h, h]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nP A : Matrix n n R\nh : Invertible P\n\u22a2 \u2191(lieConj P h) A = P * A * P\u207b\u00b9\n[PROOFSTEP]\nsimp [LinearEquiv.conj_apply, Matrix.lieConj, LinearMap.toMatrix'_comp, LinearMap.toMatrix'_toLin']\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nP A : Matrix n n R\nh : Invertible P\n\u22a2 \u2191(LieEquiv.symm (lieConj P h)) A = P\u207b\u00b9 * A * P\n[PROOFSTEP]\nsimp [LinearEquiv.symm_conj_apply, Matrix.lieConj, LinearMap.toMatrix'_comp, LinearMap.toMatrix'_toLin']\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nn : Type w\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nm : Type w\u2081\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : Fintype m\ne : n \u2243 m\nsrc\u271d : Matrix n n R \u2243\u2097[R] Matrix m m R := reindexLinearEquiv R R e e\nx\u271d\u00b9 x\u271d : Matrix n n R\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := \u2191(reindex e e),\n map_add' :=\n (_ :\n \u2200 (x y : Matrix n n R),\n AddHom.toFun src\u271d.toAddHom (x + y) = AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : Matrix n n R),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n \u2045x\u271d\u00b9, x\u271d\u2046 =\n \u2045AddHom.toFun\n {\n toAddHom :=\n { toFun := \u2191(reindex e e),\n map_add' :=\n (_ :\n \u2200 (x y : Matrix n n R),\n AddHom.toFun src\u271d.toAddHom (x + y) =\n AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : Matrix n n R),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n x\u271d\u00b9,\n AddHom.toFun\n {\n toAddHom :=\n { toFun := \u2191(reindex e e),\n map_add' :=\n (_ :\n \u2200 (x y : Matrix n n R),\n AddHom.toFun src\u271d.toAddHom (x + y) =\n AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : Matrix n n R),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n x\u271d\u2046\n[PROOFSTEP]\nsimp only [LieRing.of_associative_ring_bracket, Matrix.reindex_apply, Matrix.submatrix_mul_equiv, Matrix.submatrix_sub,\n Pi.sub_apply]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Lie.Matrix", "llama_tokens": 1918, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390747, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.41601069433215204}} {"text": "[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng h : b \u27f6 c\n\u03b7 : g \u2245 h\n\u22a2 f \u25c1 \u03b7.hom \u226b f \u25c1 \u03b7.inv = \ud835\udfd9 (f \u226b g)\n[PROOFSTEP]\nrw [\u2190 whiskerLeft_comp, hom_inv_id, whiskerLeft_id]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf g : a \u27f6 b\n\u03b7 : f \u2245 g\nh : b \u27f6 c\n\u22a2 \u03b7.hom \u25b7 h \u226b \u03b7.inv \u25b7 h = \ud835\udfd9 (f \u226b h)\n[PROOFSTEP]\nrw [\u2190 comp_whiskerRight, hom_inv_id, id_whiskerRight]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng h : b \u27f6 c\n\u03b7 : g \u2245 h\n\u22a2 f \u25c1 \u03b7.inv \u226b f \u25c1 \u03b7.hom = \ud835\udfd9 (f \u226b h)\n[PROOFSTEP]\nrw [\u2190 whiskerLeft_comp, inv_hom_id, whiskerLeft_id]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf g : a \u27f6 b\n\u03b7 : f \u2245 g\nh : b \u27f6 c\n\u22a2 \u03b7.inv \u25b7 h \u226b \u03b7.hom \u25b7 h = \ud835\udfd9 (g \u226b h)\n[PROOFSTEP]\nrw [\u2190 comp_whiskerRight, inv_hom_id, id_whiskerRight]\n[GOAL]\nB : Type u\ninst\u271d\u00b9 : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng h : b \u27f6 c\n\u03b7 : g \u27f6 h\ninst\u271d : IsIso \u03b7\n\u22a2 inv (f \u25c1 \u03b7) = f \u25c1 inv \u03b7\n[PROOFSTEP]\naesop_cat_nonterminal\n[GOAL]\ncase hom_inv_id\nB : Type u\ninst : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng h : b \u27f6 c\n\u03b7 : g \u27f6 h\ninst_1 : IsIso \u03b7\n\u22a2 f \u25c1 \u03b7 \u226b f \u25c1 inv \u03b7 = \ud835\udfd9 (f \u226b g)\n[PROOFSTEP]\nsimp only [\u2190 whiskerLeft_comp, whiskerLeft_id, IsIso.hom_inv_id]\n[GOAL]\nB : Type u\ninst\u271d\u00b9 : Bicategory B\na b c d e : B\nf g : a \u27f6 b\n\u03b7 : f \u27f6 g\nh : b \u27f6 c\ninst\u271d : IsIso \u03b7\n\u22a2 inv (\u03b7 \u25b7 h) = inv \u03b7 \u25b7 h\n[PROOFSTEP]\naesop_cat_nonterminal\n[GOAL]\ncase hom_inv_id\nB : Type u\ninst : Bicategory B\na b c d e : B\nf g : a \u27f6 b\n\u03b7 : f \u27f6 g\nh : b \u27f6 c\ninst_1 : IsIso \u03b7\n\u22a2 \u03b7 \u25b7 h \u226b inv \u03b7 \u25b7 h = \ud835\udfd9 (f \u226b h)\n[PROOFSTEP]\nsimp only [\u2190 comp_whiskerRight, id_whiskerRight, IsIso.hom_inv_id]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\nh : c \u27f6 d\ni : d \u27f6 e\n\u22a2 inv (f \u25c1 (\u03b1_ g h i).inv \u226b (\u03b1_ f (g \u226b h) i).inv \u226b (\u03b1_ f g h).inv \u25b7 i) =\n inv ((\u03b1_ f g (h \u226b i)).inv \u226b (\u03b1_ (f \u226b g) h i).inv)\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\nh : c \u27f6 d\ni : d \u27f6 e\n\u22a2 (\u03b1_ f (g \u226b h) i).inv \u226b (\u03b1_ f g h).inv \u25b7 i \u226b (\u03b1_ (f \u226b g) h i).hom = f \u25c1 (\u03b1_ g h i).hom \u226b (\u03b1_ f g (h \u226b i)).inv\n[PROOFSTEP]\nrw [\u2190 cancel_epi (f \u25c1 (\u03b1_ g h i).inv), \u2190 cancel_mono (\u03b1_ (f \u226b g) h i).inv]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\nh : c \u27f6 d\ni : d \u27f6 e\n\u22a2 (f \u25c1 (\u03b1_ g h i).inv \u226b (\u03b1_ f (g \u226b h) i).inv \u226b (\u03b1_ f g h).inv \u25b7 i \u226b (\u03b1_ (f \u226b g) h i).hom) \u226b (\u03b1_ (f \u226b g) h i).inv =\n (f \u25c1 (\u03b1_ g h i).inv \u226b f \u25c1 (\u03b1_ g h i).hom \u226b (\u03b1_ f g (h \u226b i)).inv) \u226b (\u03b1_ (f \u226b g) h i).inv\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\nh : c \u27f6 d\ni : d \u27f6 e\n\u22a2 inv ((\u03b1_ (f \u226b g) h i).inv \u226b (\u03b1_ f g h).hom \u25b7 i \u226b (\u03b1_ f (g \u226b h) i).hom) =\n inv ((\u03b1_ f g (h \u226b i)).hom \u226b f \u25c1 (\u03b1_ g h i).inv)\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\nh : c \u27f6 d\ni : d \u27f6 e\n\u22a2 f \u25c1 (\u03b1_ g h i).hom \u226b (\u03b1_ f g (h \u226b i)).inv \u226b (\u03b1_ (f \u226b g) h i).inv = (\u03b1_ f (g \u226b h) i).inv \u226b (\u03b1_ f g h).inv \u25b7 i\n[PROOFSTEP]\nsimp [\u2190 cancel_epi (f \u25c1 (\u03b1_ g h i).inv)]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\nh : c \u27f6 d\ni : d \u27f6 e\n\u22a2 inv ((\u03b1_ (f \u226b g) h i).hom \u226b (\u03b1_ f g (h \u226b i)).hom \u226b f \u25c1 (\u03b1_ g h i).inv) =\n inv ((\u03b1_ f g h).hom \u25b7 i \u226b (\u03b1_ f (g \u226b h) i).hom)\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\nh : c \u27f6 d\ni : d \u27f6 e\n\u22a2 (\u03b1_ f g (h \u226b i)).hom \u226b f \u25c1 (\u03b1_ g h i).inv \u226b (\u03b1_ f (g \u226b h) i).inv = (\u03b1_ (f \u226b g) h i).inv \u226b (\u03b1_ f g h).hom \u25b7 i\n[PROOFSTEP]\nrw [\u2190 cancel_epi (\u03b1_ f g (h \u226b i)).inv, \u2190 cancel_mono ((\u03b1_ f g h).inv \u25b7 i)]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\nh : c \u27f6 d\ni : d \u27f6 e\n\u22a2 ((\u03b1_ f g (h \u226b i)).inv \u226b (\u03b1_ f g (h \u226b i)).hom \u226b f \u25c1 (\u03b1_ g h i).inv \u226b (\u03b1_ f (g \u226b h) i).inv) \u226b (\u03b1_ f g h).inv \u25b7 i =\n ((\u03b1_ f g (h \u226b i)).inv \u226b (\u03b1_ (f \u226b g) h i).inv \u226b (\u03b1_ f g h).hom \u25b7 i) \u226b (\u03b1_ f g h).inv \u25b7 i\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\nh : c \u27f6 d\ni : d \u27f6 e\n\u22a2 inv ((\u03b1_ f (g \u226b h) i).hom \u226b f \u25c1 (\u03b1_ g h i).hom \u226b (\u03b1_ f g (h \u226b i)).inv) =\n inv ((\u03b1_ f g h).inv \u25b7 i \u226b (\u03b1_ (f \u226b g) h i).hom)\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\nh : c \u27f6 d\ni : d \u27f6 e\n\u22a2 (\u03b1_ f g h).inv \u25b7 i \u226b (\u03b1_ (f \u226b g) h i).hom \u226b (\u03b1_ f g (h \u226b i)).hom = (\u03b1_ f (g \u226b h) i).hom \u226b f \u25c1 (\u03b1_ g h i).hom\n[PROOFSTEP]\nsimp [\u2190 cancel_epi ((\u03b1_ f g h).hom \u25b7 i)]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\nh : c \u27f6 d\ni : d \u27f6 e\n\u22a2 inv ((\u03b1_ f g (h \u226b i)).inv \u226b (\u03b1_ (f \u226b g) h i).inv \u226b (\u03b1_ f g h).hom \u25b7 i) =\n inv (f \u25c1 (\u03b1_ g h i).inv \u226b (\u03b1_ f (g \u226b h) i).inv)\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\n\u22a2 (\u03b1_ f (\ud835\udfd9 b) g).inv \u226b (\u03c1_ f).hom \u25b7 g = f \u25c1 (\u03bb_ g).hom\n[PROOFSTEP]\nrw [\u2190 triangle, inv_hom_id_assoc]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\n\u22a2 (\u03c1_ f).inv \u25b7 g \u226b (\u03b1_ f (\ud835\udfd9 b) g).hom = f \u25c1 (\u03bb_ g).inv\n[PROOFSTEP]\nsimp [\u2190 cancel_mono (f \u25c1 (\u03bb_ g).hom)]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\n\u22a2 f \u25c1 (\u03bb_ g).inv \u226b (\u03b1_ f (\ud835\udfd9 b) g).inv = (\u03c1_ f).inv \u25b7 g\n[PROOFSTEP]\nsimp [\u2190 cancel_mono ((\u03c1_ f).hom \u25b7 g)]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf f' : a \u27f6 b\n\u03b7 : f \u27f6 f'\ng : b \u27f6 c\nh : c \u27f6 d\n\u22a2 \u03b7 \u25b7 g \u25b7 h \u226b (\u03b1_ f' g h).hom = (\u03b1_ f g h).hom \u226b \u03b7 \u25b7 (g \u226b h)\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf f' : a \u27f6 b\n\u03b7 : f \u27f6 f'\ng : b \u27f6 c\nh : c \u27f6 d\n\u22a2 \u03b7 \u25b7 (g \u226b h) \u226b (\u03b1_ f' g h).inv = (\u03b1_ f g h).inv \u226b \u03b7 \u25b7 g \u25b7 h\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf f' : a \u27f6 b\n\u03b7 : f \u27f6 f'\ng : b \u27f6 c\nh : c \u27f6 d\n\u22a2 \u03b7 \u25b7 g \u25b7 h = (\u03b1_ f g h).hom \u226b \u03b7 \u25b7 (g \u226b h) \u226b (\u03b1_ f' g h).inv\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng g' : b \u27f6 c\n\u03b7 : g \u27f6 g'\nh : c \u27f6 d\n\u22a2 (f \u25c1 \u03b7) \u25b7 h \u226b (\u03b1_ f g' h).hom = (\u03b1_ f g h).hom \u226b f \u25c1 \u03b7 \u25b7 h\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng g' : b \u27f6 c\n\u03b7 : g \u27f6 g'\nh : c \u27f6 d\n\u22a2 f \u25c1 \u03b7 \u25b7 h \u226b (\u03b1_ f g' h).inv = (\u03b1_ f g h).inv \u226b (f \u25c1 \u03b7) \u25b7 h\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng g' : b \u27f6 c\n\u03b7 : g \u27f6 g'\nh : c \u27f6 d\n\u22a2 f \u25c1 \u03b7 \u25b7 h = (\u03b1_ f g h).inv \u226b (f \u25c1 \u03b7) \u25b7 h \u226b (\u03b1_ f g' h).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\nh h' : c \u27f6 d\n\u03b7 : h \u27f6 h'\n\u22a2 (f \u226b g) \u25c1 \u03b7 \u226b (\u03b1_ f g h').hom = (\u03b1_ f g h).hom \u226b f \u25c1 g \u25c1 \u03b7\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\nh h' : c \u27f6 d\n\u03b7 : h \u27f6 h'\n\u22a2 f \u25c1 g \u25c1 \u03b7 \u226b (\u03b1_ f g h').inv = (\u03b1_ f g h).inv \u226b (f \u226b g) \u25c1 \u03b7\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\nh h' : c \u27f6 d\n\u03b7 : h \u27f6 h'\n\u22a2 f \u25c1 g \u25c1 \u03b7 = (\u03b1_ f g h).inv \u226b (f \u226b g) \u25c1 \u03b7 \u226b (\u03b1_ f g h').hom\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf g : a \u27f6 b\n\u03b7 : f \u27f6 g\n\u22a2 \ud835\udfd9 a \u25c1 \u03b7 \u226b (\u03bb_ g).hom = (\u03bb_ f).hom \u226b \u03b7\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf g : a \u27f6 b\n\u03b7 : f \u27f6 g\n\u22a2 \u03b7 \u226b (\u03bb_ g).inv = (\u03bb_ f).inv \u226b \ud835\udfd9 a \u25c1 \u03b7\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf g : a \u27f6 b\n\u03b7 : f \u27f6 g\n\u22a2 \u03b7 = (\u03bb_ f).inv \u226b \ud835\udfd9 a \u25c1 \u03b7 \u226b (\u03bb_ g).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf g : a \u27f6 b\n\u03b7 : f \u27f6 g\n\u22a2 \u03b7 \u25b7 \ud835\udfd9 b \u226b (\u03c1_ g).hom = (\u03c1_ f).hom \u226b \u03b7\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf g : a \u27f6 b\n\u03b7 : f \u27f6 g\n\u22a2 \u03b7 \u226b (\u03c1_ g).inv = (\u03c1_ f).inv \u226b \u03b7 \u25b7 \ud835\udfd9 b\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf g : a \u27f6 b\n\u03b7 : f \u27f6 g\n\u22a2 \u03b7 = (\u03c1_ f).inv \u226b \u03b7 \u25b7 \ud835\udfd9 b \u226b (\u03c1_ g).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf g : a \u27f6 b\n\u03b7 \u03b8 : f \u27f6 g\n\u22a2 \ud835\udfd9 a \u25c1 \u03b7 = \ud835\udfd9 a \u25c1 \u03b8 \u2194 \u03b7 = \u03b8\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf g : a \u27f6 b\n\u03b7 \u03b8 : f \u27f6 g\n\u22a2 \u03b7 \u25b7 \ud835\udfd9 b = \u03b8 \u25b7 \ud835\udfd9 b \u2194 \u03b7 = \u03b8\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\n\u22a2 (\u03bb_ f).hom \u25b7 g = (\u03b1_ (\ud835\udfd9 a) f g).hom \u226b (\u03bb_ (f \u226b g)).hom\n[PROOFSTEP]\nrw [\u2190 whiskerLeft_iff, whiskerLeft_comp, \u2190 cancel_epi (\u03b1_ _ _ _).hom, \u2190 cancel_epi ((\u03b1_ _ _ _).hom \u25b7 _), pentagon_assoc,\n triangle, \u2190 associator_naturality_middle, \u2190 comp_whiskerRight_assoc, triangle, associator_naturality_left]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\n\u22a2 inv ((\u03bb_ f).inv \u25b7 g) = inv ((\u03bb_ (f \u226b g)).inv \u226b (\u03b1_ (\ud835\udfd9 a) f g).inv)\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\n\u22a2 f \u25c1 (\u03c1_ g).hom = (\u03b1_ f g (\ud835\udfd9 c)).inv \u226b (\u03c1_ (f \u226b g)).hom\n[PROOFSTEP]\nrw [\u2190 whiskerRight_iff, comp_whiskerRight, \u2190 cancel_epi (\u03b1_ _ _ _).inv, \u2190 cancel_epi (f \u25c1 (\u03b1_ _ _ _).inv),\n pentagon_inv_assoc, triangle_assoc_comp_right, \u2190 associator_inv_naturality_middle, \u2190 whiskerLeft_comp_assoc,\n triangle_assoc_comp_right, associator_inv_naturality_right]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\n\u22a2 inv (f \u25c1 (\u03c1_ g).inv) = inv ((\u03c1_ (f \u226b g)).inv \u226b (\u03b1_ f g (\ud835\udfd9 c)).hom)\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\n\u22a2 (\u03bb_ (f \u226b g)).hom = (\u03b1_ (\ud835\udfd9 a) f g).inv \u226b (\u03bb_ f).hom \u25b7 g\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\n\u22a2 (\u03bb_ (f \u226b g)).inv = (\u03bb_ f).inv \u25b7 g \u226b (\u03b1_ (\ud835\udfd9 a) f g).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\n\u22a2 (\u03c1_ (f \u226b g)).hom = (\u03b1_ f g (\ud835\udfd9 c)).hom \u226b f \u25c1 (\u03c1_ g).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng : b \u27f6 c\n\u22a2 (\u03c1_ (f \u226b g)).inv = f \u25c1 (\u03c1_ g).inv \u226b (\u03b1_ f g (\ud835\udfd9 c)).inv\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\n\u22a2 (\u03bb_ (\ud835\udfd9 a)).hom = (\u03c1_ (\ud835\udfd9 a)).hom\n[PROOFSTEP]\nrw [\u2190 whiskerLeft_iff, \u2190 cancel_epi (\u03b1_ _ _ _).hom, \u2190 cancel_mono (\u03c1_ _).hom, triangle, \u2190 rightUnitor_comp,\n rightUnitor_naturality]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\n\u22a2 (\u03bb_ (\ud835\udfd9 a)).inv = (\u03c1_ (\ud835\udfd9 a)).inv\n[PROOFSTEP]\nsimp [Iso.inv_eq_inv]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Bicategory.Basic", "llama_tokens": 5915, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390747, "lm_q2_score": 0.538983220687684, "lm_q1q2_score": 0.41601069433215193}} {"text": "[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\n\u22a2 Presieve.IsSheafFor P (Sieve.bind U.arrows B).arrows\n[PROOFSTEP]\nintro s hs\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation s t\n[PROOFSTEP]\nlet y : \u2200 \u2983Y\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U f), Presieve.FamilyOfElements P (B hf : Presieve Y) := fun Y f hf Z g hg =>\n s _ (Presieve.bind_comp _ _ hg)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation s t\n[PROOFSTEP]\nhave hy : \u2200 \u2983Y\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U f), (y hf).Compatible :=\n by\n intro Y f H Y\u2081 Y\u2082 Z g\u2081 g\u2082 f\u2081 f\u2082 hf\u2081 hf\u2082 comm\n apply hs\n apply reassoc_of% comm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\n[PROOFSTEP]\nintro Y f H Y\u2081 Y\u2082 Z g\u2081 g\u2082 f\u2081 f\u2082 hf\u2081 hf\u2082 comm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nY : C\nf : Y \u27f6 X\nH : U.arrows f\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 Y\nf\u2082 : Y\u2082 \u27f6 Y\nhf\u2081 : (B H).arrows f\u2081\nhf\u2082 : (B H).arrows f\u2082\ncomm : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 P.map g\u2081.op (y H f\u2081 hf\u2081) = P.map g\u2082.op (y H f\u2082 hf\u2082)\n[PROOFSTEP]\napply hs\n[GOAL]\ncase a\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nY : C\nf : Y \u27f6 X\nH : U.arrows f\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 Y\nf\u2082 : Y\u2082 \u27f6 Y\nhf\u2081 : (B H).arrows f\u2081\nhf\u2082 : (B H).arrows f\u2082\ncomm : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 g\u2081 \u226b f\u2081 \u226b f = g\u2082 \u226b f\u2082 \u226b f\n[PROOFSTEP]\napply reassoc_of% comm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation s t\n[PROOFSTEP]\nlet t : Presieve.FamilyOfElements P (U : Presieve X) := fun Y f hf => (hB hf).amalgamate (y hf) (hy hf)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation s t\n[PROOFSTEP]\nhave ht : \u2200 \u2983Y\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U f), (y hf).IsAmalgamation (t f hf) := fun Y f hf => (hB hf).isAmalgamation _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation s t\n[PROOFSTEP]\nhave hT : t.Compatible := by\n rw [Presieve.compatible_iff_sieveCompatible]\n intro Z W f h hf\n apply (hB (U.downward_closed hf h)).isSeparatedFor.ext\n intro Y l hl\n apply (hB' hf (l \u226b h)).ext\n intro M m hm\n have : bind U B (m \u226b l \u226b h \u226b f) := by\n -- porting note: had to make explicit the parameter `((m \u226b l \u226b h) \u226b f)` and\n -- using `by exact`\n have : bind U B ((m \u226b l \u226b h) \u226b f) := by exact Presieve.bind_comp f hf hm\n simpa using this\n trans s (m \u226b l \u226b h \u226b f) this\n \u00b7 have := ht (U.downward_closed hf h) _ ((B _).downward_closed hl m)\n rw [op_comp, FunctorToTypes.map_comp_apply] at this \n rw [this]\n change\n s _ _ =\n s _\n _\n -- porting note: the proof was `by simp`\n congr 1\n simp only [assoc]\n \u00b7 have h : s _ _ = _ := (ht hf _ hm).symm\n conv_lhs at h => congr; rw [assoc, assoc]\n rw [h]\n simp only [op_comp, assoc, FunctorToTypes.map_comp_apply]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\n\u22a2 Presieve.FamilyOfElements.Compatible t\n[PROOFSTEP]\nrw [Presieve.compatible_iff_sieveCompatible]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\n\u22a2 Presieve.FamilyOfElements.SieveCompatible t\n[PROOFSTEP]\nintro Z W f h hf\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\n\u22a2 t (h \u226b f) (_ : U.arrows (h \u226b f)) = P.map h.op (t f hf)\n[PROOFSTEP]\napply (hB (U.downward_closed hf h)).isSeparatedFor.ext\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f_1 : Y \u27f6 W\u2984,\n (B (_ : U.arrows (h \u226b f))).arrows f_1 \u2192\n P.map f_1.op (t (h \u226b f) (_ : U.arrows (h \u226b f))) = P.map f_1.op (P.map h.op (t f hf))\n[PROOFSTEP]\nintro Y l hl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h \u226b f))).arrows l\n\u22a2 P.map l.op (t (h \u226b f) (_ : U.arrows (h \u226b f))) = P.map l.op (P.map h.op (t f hf))\n[PROOFSTEP]\napply (hB' hf (l \u226b h)).ext\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h \u226b f))).arrows l\n\u22a2 \u2200 \u2983Y_1 : C\u2984 \u2983f_1 : Y_1 \u27f6 Y\u2984,\n (Sieve.pullback (l \u226b h) (B hf)).arrows f_1 \u2192\n P.map f_1.op (P.map l.op (t (h \u226b f) (_ : U.arrows (h \u226b f)))) = P.map f_1.op (P.map l.op (P.map h.op (t f hf)))\n[PROOFSTEP]\nintro M m hm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h) (B hf)).arrows m\n\u22a2 P.map m.op (P.map l.op (t (h \u226b f) (_ : U.arrows (h \u226b f)))) = P.map m.op (P.map l.op (P.map h.op (t f hf)))\n[PROOFSTEP]\nhave : bind U B (m \u226b l \u226b h \u226b f) := by\n -- porting note: had to make explicit the parameter `((m \u226b l \u226b h) \u226b f)` and\n -- using `by exact`\n have : bind U B ((m \u226b l \u226b h) \u226b f) := by exact Presieve.bind_comp f hf hm\n simpa using this\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h) (B hf)).arrows m\n\u22a2 (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h \u226b f)\n[PROOFSTEP]\nhave : bind U B ((m \u226b l \u226b h) \u226b f) := by exact Presieve.bind_comp f hf hm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h) (B hf)).arrows m\n\u22a2 (Sieve.bind U.arrows B).arrows ((m \u226b l \u226b h) \u226b f)\n[PROOFSTEP]\nexact Presieve.bind_comp f hf hm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h) (B hf)).arrows m\nthis : (Sieve.bind U.arrows B).arrows ((m \u226b l \u226b h) \u226b f)\n\u22a2 (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h \u226b f)\n[PROOFSTEP]\nsimpa using this\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h) (B hf)).arrows m\nthis : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h \u226b f)\n\u22a2 P.map m.op (P.map l.op (t (h \u226b f) (_ : U.arrows (h \u226b f)))) = P.map m.op (P.map l.op (P.map h.op (t f hf)))\n[PROOFSTEP]\ntrans s (m \u226b l \u226b h \u226b f) this\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h) (B hf)).arrows m\nthis : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h \u226b f)\n\u22a2 P.map m.op (P.map l.op (t (h \u226b f) (_ : U.arrows (h \u226b f)))) = s (m \u226b l \u226b h \u226b f) this\n[PROOFSTEP]\nhave := ht (U.downward_closed hf h) _ ((B _).downward_closed hl m)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h) (B hf)).arrows m\nthis\u271d : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h \u226b f)\nthis :\n P.map (m \u226b l).op (t (h \u226b f) (_ : U.arrows (h \u226b f))) =\n y (_ : U.arrows (h \u226b f)) (m \u226b l) (_ : (B (_ : U.arrows (h \u226b f))).arrows (m \u226b l))\n\u22a2 P.map m.op (P.map l.op (t (h \u226b f) (_ : U.arrows (h \u226b f)))) = s (m \u226b l \u226b h \u226b f) this\u271d\n[PROOFSTEP]\nrw [op_comp, FunctorToTypes.map_comp_apply] at this \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h) (B hf)).arrows m\nthis\u271d : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h \u226b f)\nthis :\n P.map m.op (P.map l.op (t (h \u226b f) (_ : U.arrows (h \u226b f)))) =\n y (_ : U.arrows (h \u226b f)) (m \u226b l) (_ : (B (_ : U.arrows (h \u226b f))).arrows (m \u226b l))\n\u22a2 P.map m.op (P.map l.op (t (h \u226b f) (_ : U.arrows (h \u226b f)))) = s (m \u226b l \u226b h \u226b f) this\u271d\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h) (B hf)).arrows m\nthis\u271d : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h \u226b f)\nthis :\n P.map m.op (P.map l.op (t (h \u226b f) (_ : U.arrows (h \u226b f)))) =\n y (_ : U.arrows (h \u226b f)) (m \u226b l) (_ : (B (_ : U.arrows (h \u226b f))).arrows (m \u226b l))\n\u22a2 y (_ : U.arrows (h \u226b f)) (m \u226b l) (_ : (B (_ : U.arrows (h \u226b f))).arrows (m \u226b l)) = s (m \u226b l \u226b h \u226b f) this\u271d\n[PROOFSTEP]\nchange\n s _ _ =\n s _\n _\n -- porting note: the proof was `by simp`\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h) (B hf)).arrows m\nthis\u271d : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h \u226b f)\nthis :\n P.map m.op (P.map l.op (t (h \u226b f) (_ : U.arrows (h \u226b f)))) =\n y (_ : U.arrows (h \u226b f)) (m \u226b l) (_ : (B (_ : U.arrows (h \u226b f))).arrows (m \u226b l))\n\u22a2 s ((m \u226b l) \u226b h \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) ((m \u226b l) \u226b h \u226b f)) =\n s (m \u226b l \u226b h \u226b f) this\u271d\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_f\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h) (B hf)).arrows m\nthis\u271d : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h \u226b f)\nthis :\n P.map m.op (P.map l.op (t (h \u226b f) (_ : U.arrows (h \u226b f)))) =\n y (_ : U.arrows (h \u226b f)) (m \u226b l) (_ : (B (_ : U.arrows (h \u226b f))).arrows (m \u226b l))\n\u22a2 (m \u226b l) \u226b h \u226b f = m \u226b l \u226b h \u226b f\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h) (B hf)).arrows m\nthis : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h \u226b f)\n\u22a2 s (m \u226b l \u226b h \u226b f) this = P.map m.op (P.map l.op (P.map h.op (t f hf)))\n[PROOFSTEP]\nhave h : s _ _ = _ := (ht hf _ hm).symm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh\u271d : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h\u271d \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h\u271d) (B hf)).arrows m\nthis : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h\u271d \u226b f)\nh :\n s ((m \u226b l \u226b h\u271d) \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) ((m \u226b l \u226b h\u271d) \u226b f)) =\n P.map (m \u226b l \u226b h\u271d).op (t f hf)\n\u22a2 s (m \u226b l \u226b h\u271d \u226b f) this = P.map m.op (P.map l.op (P.map h\u271d.op (t f hf)))\n[PROOFSTEP]\nconv_lhs at h => congr; rw [assoc, assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh\u271d : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h\u271d \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h\u271d) (B hf)).arrows m\nthis : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h\u271d \u226b f)\nh :\n s ((m \u226b l \u226b h\u271d) \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) ((m \u226b l \u226b h\u271d) \u226b f)) =\n P.map (m \u226b l \u226b h\u271d).op (t f hf)\n| s ((m \u226b l \u226b h\u271d) \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) ((m \u226b l \u226b h\u271d) \u226b f))\n[PROOFSTEP]\ncongr; rw [assoc, assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh\u271d : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h\u271d \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h\u271d) (B hf)).arrows m\nthis : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h\u271d \u226b f)\nh :\n s ((m \u226b l \u226b h\u271d) \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) ((m \u226b l \u226b h\u271d) \u226b f)) =\n P.map (m \u226b l \u226b h\u271d).op (t f hf)\n| s ((m \u226b l \u226b h\u271d) \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) ((m \u226b l \u226b h\u271d) \u226b f))\n[PROOFSTEP]\ncongr; rw [assoc, assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh\u271d : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h\u271d \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h\u271d) (B hf)).arrows m\nthis : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h\u271d \u226b f)\nh :\n s ((m \u226b l \u226b h\u271d) \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) ((m \u226b l \u226b h\u271d) \u226b f)) =\n P.map (m \u226b l \u226b h\u271d).op (t f hf)\n| s ((m \u226b l \u226b h\u271d) \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) ((m \u226b l \u226b h\u271d) \u226b f))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase f\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh\u271d : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h\u271d \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h\u271d) (B hf)).arrows m\nthis : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h\u271d \u226b f)\nh :\n s ((m \u226b l \u226b h\u271d) \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) ((m \u226b l \u226b h\u271d) \u226b f)) =\n P.map (m \u226b l \u226b h\u271d).op (t f hf)\n| (m \u226b l \u226b h\u271d) \u226b f\n[PROOFSTEP]\nrw [assoc, assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh\u271d : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h\u271d \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h\u271d) (B hf)).arrows m\nthis : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h\u271d \u226b f)\nh : s (m \u226b l \u226b h\u271d \u226b f) (_ : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h\u271d \u226b f)) = P.map (m \u226b l \u226b h\u271d).op (t f hf)\n\u22a2 s (m \u226b l \u226b h\u271d \u226b f) this = P.map m.op (P.map l.op (P.map h\u271d.op (t f hf)))\n[PROOFSTEP]\nrw [h]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nZ W : C\nf : Z \u27f6 X\nh\u271d : W \u27f6 Z\nhf : U.arrows f\nY : C\nl : Y \u27f6 W\nhl : (B (_ : U.arrows (h\u271d \u226b f))).arrows l\nM : C\nm : M \u27f6 Y\nhm : (Sieve.pullback (l \u226b h\u271d) (B hf)).arrows m\nthis : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h\u271d \u226b f)\nh : s (m \u226b l \u226b h\u271d \u226b f) (_ : (Sieve.bind U.arrows B).arrows (m \u226b l \u226b h\u271d \u226b f)) = P.map (m \u226b l \u226b h\u271d).op (t f hf)\n\u22a2 P.map (m \u226b l \u226b h\u271d).op (t f hf) = P.map m.op (P.map l.op (P.map h\u271d.op (t f hf)))\n[PROOFSTEP]\nsimp only [op_comp, assoc, FunctorToTypes.map_comp_apply]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nhT : Presieve.FamilyOfElements.Compatible t\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation s t\n[PROOFSTEP]\nrefine' \u27e8hU.amalgamate t hT, _, _\u27e9\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nhT : Presieve.FamilyOfElements.Compatible t\n\u22a2 (fun t => Presieve.FamilyOfElements.IsAmalgamation s t) (Presieve.IsSheafFor.amalgamate hU t hT)\n[PROOFSTEP]\nrintro Z _ \u27e8Y, f, g, hg, hf, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nhT : Presieve.FamilyOfElements.Compatible t\nZ Y : C\nf : Z \u27f6 Y\ng : Y \u27f6 X\nhg : U.arrows g\nhf : (B hg).arrows f\n\u22a2 P.map (f \u226b g).op (Presieve.IsSheafFor.amalgamate hU t hT) =\n s (f \u226b g) (_ : \u2203 Y_1 g_1 f_1 H, (fun Y f h => (B h).arrows) Y_1 f_1 H g_1 \u2227 g_1 \u226b f_1 = f \u226b g)\n[PROOFSTEP]\nrw [op_comp, FunctorToTypes.map_comp_apply, Presieve.IsSheafFor.valid_glue _ _ _ hg]\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nhT : Presieve.FamilyOfElements.Compatible t\nZ Y : C\nf : Z \u27f6 Y\ng : Y \u27f6 X\nhg : U.arrows g\nhf : (B hg).arrows f\n\u22a2 P.map f.op (t g hg) = s (f \u226b g) (_ : \u2203 Y_1 g_1 f_1 H, (fun Y f h => (B h).arrows) Y_1 f_1 H g_1 \u2227 g_1 \u226b f_1 = f \u226b g)\n[PROOFSTEP]\napply ht hg _ hf\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y hf)\n (_ : Presieve.FamilyOfElements.Compatible (y hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y hf) (t f hf)\nhT : Presieve.FamilyOfElements.Compatible t\n\u22a2 \u2200 (y : P.obj (Opposite.op X)),\n (fun t => Presieve.FamilyOfElements.IsAmalgamation s t) y \u2192 y = Presieve.IsSheafFor.amalgamate hU t hT\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny\u271d : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy\u271d : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y\u271d hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y\u271d hf)\n (_ : Presieve.FamilyOfElements.Compatible (y\u271d hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y\u271d hf) (t f hf)\nhT : Presieve.FamilyOfElements.Compatible t\ny : P.obj (Opposite.op X)\nhy : Presieve.FamilyOfElements.IsAmalgamation s y\n\u22a2 y = Presieve.IsSheafFor.amalgamate hU t hT\n[PROOFSTEP]\napply hU.isSeparatedFor.ext\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny\u271d : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy\u271d : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y\u271d hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y\u271d hf)\n (_ : Presieve.FamilyOfElements.Compatible (y\u271d hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y\u271d hf) (t f hf)\nhT : Presieve.FamilyOfElements.Compatible t\ny : P.obj (Opposite.op X)\nhy : Presieve.FamilyOfElements.IsAmalgamation s y\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, U.arrows f \u2192 P.map f.op y = P.map f.op (Presieve.IsSheafFor.amalgamate hU t hT)\n[PROOFSTEP]\nintro Y f hf\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny\u271d : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy\u271d : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y\u271d hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y\u271d hf)\n (_ : Presieve.FamilyOfElements.Compatible (y\u271d hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y\u271d hf) (t f hf)\nhT : Presieve.FamilyOfElements.Compatible t\ny : P.obj (Opposite.op X)\nhy : Presieve.FamilyOfElements.IsAmalgamation s y\nY : C\nf : Y \u27f6 X\nhf : U.arrows f\n\u22a2 P.map f.op y = P.map f.op (Presieve.IsSheafFor.amalgamate hU t hT)\n[PROOFSTEP]\napply (hB hf).isSeparatedFor.ext\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny\u271d : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy\u271d : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y\u271d hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y\u271d hf)\n (_ : Presieve.FamilyOfElements.Compatible (y\u271d hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y\u271d hf) (t f hf)\nhT : Presieve.FamilyOfElements.Compatible t\ny : P.obj (Opposite.op X)\nhy : Presieve.FamilyOfElements.IsAmalgamation s y\nY : C\nf : Y \u27f6 X\nhf : U.arrows f\n\u22a2 \u2200 \u2983Y_1 : C\u2984 \u2983f_1 : Y_1 \u27f6 Y\u2984,\n (B hf).arrows f_1 \u2192 P.map f_1.op (P.map f.op y) = P.map f_1.op (P.map f.op (Presieve.IsSheafFor.amalgamate hU t hT))\n[PROOFSTEP]\nintro Z g hg\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nU : Sieve X\nB : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 U.arrows f \u2192 Sieve Y\nhU : Presieve.IsSheafFor P U.arrows\nhB : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows\nhB' :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (h : U.arrows f) \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (B h)).arrows\ns : Presieve.FamilyOfElements P (Sieve.bind U.arrows B).arrows\nhs : Presieve.FamilyOfElements.Compatible s\ny\u271d : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 (hf : U.arrows f) \u2192 Presieve.FamilyOfElements P (B hf).arrows :=\n fun Y f hf Z g hg => s (g \u226b f) (_ : Presieve.bind (fun Y f => U.arrows f) (fun Y f H => (B H).arrows) (g \u226b f))\nhy\u271d : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.Compatible (y\u271d hf)\nt : Presieve.FamilyOfElements P U.arrows :=\n fun Y f hf =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P (B hf).arrows) (y\u271d hf)\n (_ : Presieve.FamilyOfElements.Compatible (y\u271d hf))\nht : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (hf : U.arrows f), Presieve.FamilyOfElements.IsAmalgamation (y\u271d hf) (t f hf)\nhT : Presieve.FamilyOfElements.Compatible t\ny : P.obj (Opposite.op X)\nhy : Presieve.FamilyOfElements.IsAmalgamation s y\nY : C\nf : Y \u27f6 X\nhf : U.arrows f\nZ : C\ng : Z \u27f6 Y\nhg : (B hf).arrows g\n\u22a2 P.map g.op (P.map f.op y) = P.map g.op (P.map f.op (Presieve.IsSheafFor.amalgamate hU t hT))\n[PROOFSTEP]\nrw [\u2190 FunctorToTypes.map_comp_apply, \u2190 op_comp, hy _ (Presieve.bind_comp _ _ hg), hU.valid_glue _ _ hf, ht hf _ hg]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\n\u22a2 Presieve.IsSheafFor P S.arrows\n[PROOFSTEP]\nhave : (bind R fun Y f _ => S.pullback f : Presieve X) \u2264 S :=\n by\n rintro Z f \u27e8W, f, g, hg, hf : S _, rfl\u27e9\n apply hf\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\n\u22a2 (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\n[PROOFSTEP]\nrintro Z f \u27e8W, f, g, hg, hf : S _, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nZ W : C\nf : Z \u27f6 W\ng : W \u27f6 X\nhg : R.arrows g\nhf : S.arrows (f \u226b g)\n\u22a2 f \u226b g \u2208 S.arrows\n[PROOFSTEP]\napply hf\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\n\u22a2 Presieve.IsSheafFor P S.arrows\n[PROOFSTEP]\napply Presieve.isSheafFor_subsieve_aux P this\n[GOAL]\ncase hS\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\n\u22a2 Presieve.IsSheafFor P (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows\ncase trans\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S)).arrows\n[PROOFSTEP]\napply isSheafFor_bind _ _ _ hR hS\n[GOAL]\ncase hS\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n R.arrows f \u2192 \u2200 \u2983Z : C\u2984 (g : Z \u27f6 Y), Presieve.IsSeparatedFor P (Sieve.pullback g (Sieve.pullback f S)).arrows\n[PROOFSTEP]\nintro Y f hf Z g\n[GOAL]\ncase hS\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\nY : C\nf : Y \u27f6 X\nhf : R.arrows f\nZ : C\ng : Z \u27f6 Y\n\u22a2 Presieve.IsSeparatedFor P (Sieve.pullback g (Sieve.pullback f S)).arrows\n[PROOFSTEP]\nrw [\u2190 pullback_comp]\n[GOAL]\ncase hS\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\nY : C\nf : Y \u27f6 X\nhf : R.arrows f\nZ : C\ng : Z \u27f6 Y\n\u22a2 Presieve.IsSeparatedFor P (Sieve.pullback (g \u226b f) S).arrows\n[PROOFSTEP]\napply (hS (R.downward_closed hf _)).isSeparatedFor\n[GOAL]\ncase trans\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S)).arrows\n[PROOFSTEP]\nintro Y f hf\n[GOAL]\ncase trans\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S)).arrows\n[PROOFSTEP]\nhave : Sieve.pullback f (bind R fun T (k : T \u27f6 X) (_ : R k) => pullback k S) = R.pullback f :=\n by\n ext Z g\n constructor\n \u00b7 rintro \u27e8W, k, l, hl, _, comm\u27e9\n rw [pullback_apply, \u2190 comm]\n simp [hl]\n \u00b7 intro a\n refine' \u27e8Z, \ud835\udfd9 Z, _, a, _\u27e9\n simp [hf]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S) = Sieve.pullback f R\n[PROOFSTEP]\next Z g\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\nZ : C\ng : Z \u27f6 Y\n\u22a2 (Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S)).arrows g \u2194 (Sieve.pullback f R).arrows g\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\nZ : C\ng : Z \u27f6 Y\n\u22a2 (Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S)).arrows g \u2192 (Sieve.pullback f R).arrows g\n[PROOFSTEP]\nrintro \u27e8W, k, l, hl, _, comm\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\nZ : C\ng : Z \u27f6 Y\nW : C\nk : Z \u27f6 W\nl : W \u27f6 X\nhl : R.arrows l\nleft\u271d : ((fun T k x => Sieve.pullback k S) W l hl).arrows k\ncomm : k \u226b l = g \u226b f\n\u22a2 (Sieve.pullback f R).arrows g\n[PROOFSTEP]\nrw [pullback_apply, \u2190 comm]\n[GOAL]\ncase h.mp.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\nZ : C\ng : Z \u27f6 Y\nW : C\nk : Z \u27f6 W\nl : W \u27f6 X\nhl : R.arrows l\nleft\u271d : ((fun T k x => Sieve.pullback k S) W l hl).arrows k\ncomm : k \u226b l = g \u226b f\n\u22a2 R.arrows (k \u226b l)\n[PROOFSTEP]\nsimp [hl]\n[GOAL]\ncase h.mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\nZ : C\ng : Z \u27f6 Y\n\u22a2 (Sieve.pullback f R).arrows g \u2192 (Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S)).arrows g\n[PROOFSTEP]\nintro a\n[GOAL]\ncase h.mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\nZ : C\ng : Z \u27f6 Y\na : (Sieve.pullback f R).arrows g\n\u22a2 (Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S)).arrows g\n[PROOFSTEP]\nrefine' \u27e8Z, \ud835\udfd9 Z, _, a, _\u27e9\n[GOAL]\ncase h.mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\nZ : C\ng : Z \u27f6 Y\na : (Sieve.pullback f R).arrows g\n\u22a2 (fun Y f h => ((fun T k x => Sieve.pullback k S) Y f h).arrows) Z (g \u226b f) a (\ud835\udfd9 Z) \u2227 \ud835\udfd9 Z \u226b g \u226b f = g \u226b f\n[PROOFSTEP]\nsimp [hf]\n[GOAL]\ncase trans\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis\u271d : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\nthis : Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S) = Sieve.pullback f R\n\u22a2 Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S)).arrows\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase trans\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX Y\u271d : C\nS\u271d : Sieve X\nR\u271d : Presieve X\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nR S : Sieve X\nhR : Presieve.IsSheafFor P R.arrows\nhR' : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\nhS : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, R.arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\nthis\u271d : (Sieve.bind R.arrows fun Y f x => Sieve.pullback f S).arrows \u2264 S.arrows\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\nthis : Sieve.pullback f (Sieve.bind R.arrows fun T k x => Sieve.pullback k S) = Sieve.pullback f R\n\u22a2 Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows\n[PROOFSTEP]\napply hR' hf\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y\u271d : C\nS : Sieve X\u271d\nR : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nf : Y \u27f6 X\n\u22a2 Presieve.IsSheafFor P (Sieve.pullback f \u22a4).arrows\n[PROOFSTEP]\nrw [Sieve.pullback_top]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y\u271d : C\nS : Sieve X\u271d\nR : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nf : Y \u27f6 X\n\u22a2 Presieve.IsSheafFor P \u22a4.arrows\n[PROOFSTEP]\nexact Presieve.isSheafFor_top_sieve P\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y\u271d : C\nS\u271d : Sieve X\u271d\nR : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nf : Y \u27f6 X\nhS : S \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) X\nZ : C\ng : Z \u27f6 Y\n\u22a2 Presieve.IsSheafFor P (Sieve.pullback g (Sieve.pullback f S)).arrows\n[PROOFSTEP]\nrw [\u2190 pullback_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y\u271d : C\nS\u271d : Sieve X\u271d\nR : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nf : Y \u27f6 X\nhS : S \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) X\nZ : C\ng : Z \u27f6 Y\n\u22a2 Presieve.IsSheafFor P (Sieve.pullback (g \u226b f) S).arrows\n[PROOFSTEP]\napply hS\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y : C\nS\u271d : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) X\nR : Sieve X\nhR :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f R \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) Y\nZ : C\ng : Z \u27f6 X\n\u22a2 Presieve.IsSheafFor P (Sieve.pullback g R).arrows\n[PROOFSTEP]\nrefine' isSheafFor_trans P (pullback g S) _ (hS Z g) _ _\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y : C\nS\u271d : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) X\nR : Sieve X\nhR :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f R \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) Y\nZ : C\ng : Z \u27f6 X\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 Z\u2984,\n (Sieve.pullback g R).arrows f \u2192 Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.pullback g S)).arrows\n[PROOFSTEP]\nintro Y f _\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y\u271d : C\nS\u271d : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) X\nR : Sieve X\nhR :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f R \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) Y\nZ : C\ng : Z \u27f6 X\nY : C\nf : Y \u27f6 Z\nx\u271d : (Sieve.pullback g R).arrows f\n\u22a2 Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.pullback g S)).arrows\n[PROOFSTEP]\nrw [\u2190 pullback_comp]\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y\u271d : C\nS\u271d : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) X\nR : Sieve X\nhR :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f R \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) Y\nZ : C\ng : Z \u27f6 X\nY : C\nf : Y \u27f6 Z\nx\u271d : (Sieve.pullback g R).arrows f\n\u22a2 Presieve.IsSeparatedFor P (Sieve.pullback (f \u226b g) S).arrows\n[PROOFSTEP]\napply (hS _ _).isSeparatedFor\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y : C\nS\u271d : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) X\nR : Sieve X\nhR :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f R \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) Y\nZ : C\ng : Z \u27f6 X\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 Z\u2984,\n (Sieve.pullback g S).arrows f \u2192 Presieve.IsSheafFor P (Sieve.pullback f (Sieve.pullback g R)).arrows\n[PROOFSTEP]\nintro Y f hf\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y\u271d : C\nS\u271d : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) X\nR : Sieve X\nhR :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f R \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) Y\nZ : C\ng : Z \u27f6 X\nY : C\nf : Y \u27f6 Z\nhf : (Sieve.pullback g S).arrows f\n\u22a2 Presieve.IsSheafFor P (Sieve.pullback f (Sieve.pullback g R)).arrows\n[PROOFSTEP]\nhave := hR hf _ (\ud835\udfd9 _)\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y\u271d : C\nS\u271d : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) X\nR : Sieve X\nhR :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f R \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) Y\nZ : C\ng : Z \u27f6 X\nY : C\nf : Y \u27f6 Z\nhf : (Sieve.pullback g S).arrows f\nthis : Presieve.IsSheafFor P (Sieve.pullback (\ud835\udfd9 Y) (Sieve.pullback (f \u226b g) R)).arrows\n\u22a2 Presieve.IsSheafFor P (Sieve.pullback f (Sieve.pullback g R)).arrows\n[PROOFSTEP]\nrw [pullback_id, pullback_comp] at this \n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y\u271d : C\nS\u271d : Sieve X\u271d\nR\u271d : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 Type v\nX : C\nS : Sieve X\nhS : S \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) X\nR : Sieve X\nhR :\n \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192\n Sieve.pullback f R \u2208 (fun X S => \u2200 (Y : C) (f : Y \u27f6 X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows) Y\nZ : C\ng : Z \u27f6 X\nY : C\nf : Y \u27f6 Z\nhf : (Sieve.pullback g S).arrows f\nthis : Presieve.IsSheafFor P (Sieve.pullback f (Sieve.pullback g R)).arrows\n\u22a2 Presieve.IsSheafFor P (Sieve.pullback f (Sieve.pullback g R)).arrows\n[PROOFSTEP]\napply this\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y : C\nS\u271d : Sieve X\u271d\nR : Presieve X\u271d\nJ J\u2082 : GrothendieckTopology C\nPs : Set (C\u1d52\u1d56 \u2964 Type v)\nh : P \u2208 Ps\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves (finestTopology Ps) X\n\u22a2 Presieve.IsSheafFor P S.arrows\n[PROOFSTEP]\nsimpa using hS _ \u27e8\u27e8_, _, \u27e8_, h, rfl\u27e9, rfl\u27e9, rfl\u27e9 _ (\ud835\udfd9 _)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ\u271d J\u2082 : GrothendieckTopology C\nPs : Set (C\u1d52\u1d56 \u2964 Type v)\nJ : GrothendieckTopology C\nhJ : \u2200 (P : C\u1d52\u1d56 \u2964 Type v), P \u2208 Ps \u2192 Presieve.IsSheaf J P\n\u22a2 J \u2264 finestTopology Ps\n[PROOFSTEP]\nrintro X S hS _ \u27e8\u27e8_, _, \u27e8P, hP, rfl\u27e9, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase intro.mk.intro.intro.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y : C\nS\u271d : Sieve X\u271d\nR : Presieve X\u271d\nJ\u271d J\u2082 : GrothendieckTopology C\nPs : Set (C\u1d52\u1d56 \u2964 Type v)\nJ : GrothendieckTopology C\nhJ : \u2200 (P : C\u1d52\u1d56 \u2964 Type v), P \u2208 Ps \u2192 Presieve.IsSheaf J P\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nP : C\u1d52\u1d56 \u2964 Type v\nhP : P \u2208 Ps\n\u22a2 S \u2208\n (fun f => \u2191f X)\n { val := (finestTopologySingle P).sieves,\n property := (_ : \u2203 a, a \u2208 finestTopologySingle '' Ps \u2227 a.sieves = (finestTopologySingle P).sieves) }\n[PROOFSTEP]\nintro Y f\n[GOAL]\ncase intro.mk.intro.intro.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nP\u271d : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y\u271d : C\nS\u271d : Sieve X\u271d\nR : Presieve X\u271d\nJ\u271d J\u2082 : GrothendieckTopology C\nPs : Set (C\u1d52\u1d56 \u2964 Type v)\nJ : GrothendieckTopology C\nhJ : \u2200 (P : C\u1d52\u1d56 \u2964 Type v), P \u2208 Ps \u2192 Presieve.IsSheaf J P\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nP : C\u1d52\u1d56 \u2964 Type v\nhP : P \u2208 Ps\nY : C\nf : Y \u27f6 X\n\u22a2 Presieve.IsSheafFor P (Sieve.pullback f S).arrows\n[PROOFSTEP]\nexact hJ P hP (S.pullback f) (J.pullback_stable f hS)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP : C\u1d52\u1d56 \u2964 Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ\u271d J\u2082 J : GrothendieckTopology C\nh : \u2200 (X : C), Presieve.IsSheaf J (yoneda.obj X)\n\u22a2 \u2200 (P : C\u1d52\u1d56 \u2964 Type v), P \u2208 Set.range yoneda.toPrefunctor.obj \u2192 Presieve.IsSheaf J P\n[PROOFSTEP]\nrintro P \u27e8X, rfl\u27e9\n[GOAL]\ncase intro\nC : Type u\ninst\u271d : Category.{v, u} C\nP : C\u1d52\u1d56 \u2964 Type v\nX\u271d Y : C\nS : Sieve X\u271d\nR : Presieve X\u271d\nJ\u271d J\u2082 J : GrothendieckTopology C\nh : \u2200 (X : C), Presieve.IsSheaf J (yoneda.obj X)\nX : C\n\u22a2 Presieve.IsSheaf J (yoneda.obj X)\n[PROOFSTEP]\napply h\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.Canonical", "llama_tokens": 45813, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006920020959544, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.4159765674053582}} {"text": "[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2083\ninst\u271d\u00b3 : (i : Fin n) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b2 : (i : Fin n) \u2192 Module R (M i)\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin n \u2192 Type u_6\ne : (i : Fin n) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\n\u22a2 f = g\n[PROOFSTEP]\ninduction' n with m hm\n[GOAL]\ncase zero\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nM : Fin Nat.zero \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin Nat.zero) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin Nat.zero) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin Nat.zero \u2192 Type u_6\ne : (i : Fin Nat.zero) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin Nat.zero) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\n\u22a2 f = g\n[PROOFSTEP]\next x\n[GOAL]\ncase zero.H\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nM : Fin Nat.zero \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin Nat.zero) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin Nat.zero) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin Nat.zero \u2192 Type u_6\ne : (i : Fin Nat.zero) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin Nat.zero) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\nx : (i : Fin Nat.zero) \u2192 M i\n\u22a2 \u2191f x = \u2191g x\n[PROOFSTEP]\nconvert h finZeroElim\n[GOAL]\ncase succ\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\n\u22a2 f = g\n[PROOFSTEP]\napply Function.LeftInverse.injective uncurry_curryLeft\n[GOAL]\ncase succ.a\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\n\u22a2 (fun x => curryLeft x) f = (fun x => curryLeft x) g\n[PROOFSTEP]\nrefine' Basis.ext (e 0) _\n[GOAL]\ncase succ.a\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\n\u22a2 \u2200 (i : \u03b9\u2081 0), \u2191((fun x => curryLeft x) f) (\u2191(e 0) i) = \u2191((fun x => curryLeft x) g) (\u2191(e 0) i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase succ.a\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\n\u22a2 \u2191((fun x => curryLeft x) f) (\u2191(e 0) i) = \u2191((fun x => curryLeft x) g) (\u2191(e 0) i)\n[PROOFSTEP]\napply hm (Fin.tail e)\n[GOAL]\ncase succ.a\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\n\u22a2 \u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)),\n (\u2191(\u2191((fun x => curryLeft x) f) (\u2191(e 0) i)) fun i => \u2191(Fin.tail e i) (v i)) =\n \u2191(\u2191((fun x => curryLeft x) g) (\u2191(e 0) i)) fun i => \u2191(Fin.tail e i) (v i)\n[PROOFSTEP]\nintro j\n[GOAL]\ncase succ.a\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\n\u22a2 (\u2191(\u2191((fun x => curryLeft x) f) (\u2191(e 0) i)) fun i => \u2191(Fin.tail e i) (j i)) =\n \u2191(\u2191((fun x => curryLeft x) g) (\u2191(e 0) i)) fun i => \u2191(Fin.tail e i) (j i)\n[PROOFSTEP]\nconvert h (Fin.cons i j)\n[GOAL]\ncase h.e'_2\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\n\u22a2 (\u2191(\u2191((fun x => curryLeft x) f) (\u2191(e 0) i)) fun i => \u2191(Fin.tail e i) (j i)) = \u2191f fun i_1 => \u2191(e i_1) (Fin.cons i j i_1)\ncase h.e'_3\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\n\u22a2 (\u2191(\u2191((fun x => curryLeft x) g) (\u2191(e 0) i)) fun i => \u2191(Fin.tail e i) (j i)) = \u2191g fun i_1 => \u2191(e i_1) (Fin.cons i j i_1)\n[PROOFSTEP]\niterate 2 \n rw [curryLeft_apply]\n congr 1 with x\n refine' Fin.cases rfl (fun x => _) x\n dsimp [Fin.tail]\n rw [Fin.cons_succ, Fin.cons_succ]\n[GOAL]\ncase h.e'_2\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\n\u22a2 (\u2191(\u2191((fun x => curryLeft x) f) (\u2191(e 0) i)) fun i => \u2191(Fin.tail e i) (j i)) = \u2191f fun i_1 => \u2191(e i_1) (Fin.cons i j i_1)\ncase h.e'_3\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\n\u22a2 (\u2191(\u2191((fun x => curryLeft x) g) (\u2191(e 0) i)) fun i => \u2191(Fin.tail e i) (j i)) = \u2191g fun i_1 => \u2191(e i_1) (Fin.cons i j i_1)\n[PROOFSTEP]\nrw [curryLeft_apply]\n[GOAL]\ncase h.e'_2\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\n\u22a2 \u2191f (Fin.cons (\u2191(e 0) i) fun i => \u2191(Fin.tail e i) (j i)) = \u2191f fun i_1 => \u2191(e i_1) (Fin.cons i j i_1)\ncase h.e'_3\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\n\u22a2 (\u2191(\u2191((fun x => curryLeft x) g) (\u2191(e 0) i)) fun i => \u2191(Fin.tail e i) (j i)) = \u2191g fun i_1 => \u2191(e i_1) (Fin.cons i j i_1)\n[PROOFSTEP]\ncongr 1 with x\n[GOAL]\ncase h.e'_2.h.e_6.h.h\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\nx : Fin (m + 1)\n\u22a2 Fin.cons (\u2191(e 0) i) (fun i => \u2191(Fin.tail e i) (j i)) x = \u2191(e x) (Fin.cons i j x)\ncase h.e'_3\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\n\u22a2 (\u2191(\u2191((fun x => curryLeft x) g) (\u2191(e 0) i)) fun i => \u2191(Fin.tail e i) (j i)) = \u2191g fun i_1 => \u2191(e i_1) (Fin.cons i j i_1)\n[PROOFSTEP]\nrefine' Fin.cases rfl (fun x => _) x\n[GOAL]\ncase h.e'_2.h.e_6.h.h\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\nx\u271d : Fin (m + 1)\nx : Fin m\n\u22a2 Fin.cons (\u2191(e 0) i) (fun i => \u2191(Fin.tail e i) (j i)) (Fin.succ x) = \u2191(e (Fin.succ x)) (Fin.cons i j (Fin.succ x))\ncase h.e'_3\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\n\u22a2 (\u2191(\u2191((fun x => curryLeft x) g) (\u2191(e 0) i)) fun i => \u2191(Fin.tail e i) (j i)) = \u2191g fun i_1 => \u2191(e i_1) (Fin.cons i j i_1)\n[PROOFSTEP]\ndsimp [Fin.tail]\n[GOAL]\ncase h.e'_2.h.e_6.h.h\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\nx\u271d : Fin (m + 1)\nx : Fin m\n\u22a2 Fin.cons (\u2191(e 0) i) (fun i => \u2191(e (Fin.succ i)) (j i)) (Fin.succ x) = \u2191(e (Fin.succ x)) (Fin.cons i j (Fin.succ x))\ncase h.e'_3\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\n\u22a2 (\u2191(\u2191((fun x => curryLeft x) g) (\u2191(e 0) i)) fun i => \u2191(Fin.tail e i) (j i)) = \u2191g fun i_1 => \u2191(e i_1) (Fin.cons i j i_1)\n[PROOFSTEP]\nrw [Fin.cons_succ, Fin.cons_succ]\n[GOAL]\ncase h.e'_3\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\n\u22a2 (\u2191(\u2191((fun x => curryLeft x) g) (\u2191(e 0) i)) fun i => \u2191(Fin.tail e i) (j i)) = \u2191g fun i_1 => \u2191(e i_1) (Fin.cons i j i_1)\n[PROOFSTEP]\nrw [curryLeft_apply]\n[GOAL]\ncase h.e'_3\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\n\u22a2 \u2191g (Fin.cons (\u2191(e 0) i) fun i => \u2191(Fin.tail e i) (j i)) = \u2191g fun i_1 => \u2191(e i_1) (Fin.cons i j i_1)\n[PROOFSTEP]\ncongr 1 with x\n[GOAL]\ncase h.e'_3.h.e_6.h.h\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\nx : Fin (m + 1)\n\u22a2 Fin.cons (\u2191(e 0) i) (fun i => \u2191(Fin.tail e i) (j i)) x = \u2191(e x) (Fin.cons i j x)\n[PROOFSTEP]\nrefine' Fin.cases rfl (fun x => _) x\n[GOAL]\ncase h.e'_3.h.e_6.h.h\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\nx\u271d : Fin (m + 1)\nx : Fin m\n\u22a2 Fin.cons (\u2191(e 0) i) (fun i => \u2191(Fin.tail e i) (j i)) (Fin.succ x) = \u2191(e (Fin.succ x)) (Fin.cons i j (Fin.succ x))\n[PROOFSTEP]\ndsimp [Fin.tail]\n[GOAL]\ncase h.e'_3.h.e_6.h.h\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM\u271d : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : (i : Fin n) \u2192 AddCommMonoid (M\u271d i)\ninst\u271d\u2074 : (i : Fin n) \u2192 Module R (M\u271d i)\ninst\u271d\u00b3 : Module R M\u2082\ninst\u271d\u00b2 : Module R M\u2083\nf\u271d g\u271d : MultilinearMap R M\u271d M\u2082\n\u03b9\u2081\u271d : Fin n \u2192 Type u_6\ne\u271d : (i : Fin n) \u2192 Basis (\u03b9\u2081\u271d i) R (M\u271d i)\nh\u271d : \u2200 (v : (i : Fin n) \u2192 \u03b9\u2081\u271d i), (\u2191f\u271d fun i => \u2191(e\u271d i) (v i)) = \u2191g\u271d fun i => \u2191(e\u271d i) (v i)\nm : \u2115\nhm :\n \u2200 {M : Fin m \u2192 Type u_3} [inst : (i : Fin m) \u2192 AddCommMonoid (M i)] [inst_1 : (i : Fin m) \u2192 Module R (M i)]\n {f g : MultilinearMap R M M\u2082} {\u03b9\u2081 : Fin m \u2192 Type u_6} (e : (i : Fin m) \u2192 Basis (\u03b9\u2081 i) R (M i)),\n (\u2200 (v : (i : Fin m) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)) \u2192 f = g\nM : Fin (Nat.succ m) \u2192 Type u_3\ninst\u271d\u00b9 : (i : Fin (Nat.succ m)) \u2192 AddCommMonoid (M i)\ninst\u271d : (i : Fin (Nat.succ m)) \u2192 Module R (M i)\nf g : MultilinearMap R M M\u2082\n\u03b9\u2081 : Fin (Nat.succ m) \u2192 Type u_6\ne : (i : Fin (Nat.succ m)) \u2192 Basis (\u03b9\u2081 i) R (M i)\nh : \u2200 (v : (i : Fin (Nat.succ m)) \u2192 \u03b9\u2081 i), (\u2191f fun i => \u2191(e i) (v i)) = \u2191g fun i => \u2191(e i) (v i)\ni : \u03b9\u2081 0\nj : (i : Fin m) \u2192 \u03b9\u2081 (Fin.succ i)\nx\u271d : Fin (m + 1)\nx : Fin m\n\u22a2 Fin.cons (\u2191(e 0) i) (fun i => \u2191(e (Fin.succ i)) (j i)) (Fin.succ x) = \u2191(e (Fin.succ x)) (Fin.cons i j (Fin.succ x))\n[PROOFSTEP]\nrw [Fin.cons_succ, Fin.cons_succ]\n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : (i : Fin n) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b3 : (i : Fin n) \u2192 Module R (M i)\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Finite \u03b9\nf g : MultilinearMap R (fun x => M\u2082) M\u2083\n\u03b9\u2081 : Type u_6\ne : Basis \u03b9\u2081 R M\u2082\nh : \u2200 (v : \u03b9 \u2192 \u03b9\u2081), (\u2191f fun i => \u2191e (v i)) = \u2191g fun i => \u2191e (v i)\n\u22a2 f = g\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\nR : Type u_1\n\u03b9 : Type u_2\nn : \u2115\nM : Fin n \u2192 Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2077 : CommSemiring R\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : (i : Fin n) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b3 : (i : Fin n) \u2192 Module R (M i)\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Finite \u03b9\nf g : MultilinearMap R (fun x => M\u2082) M\u2083\n\u03b9\u2081 : Type u_6\ne : Basis \u03b9\u2081 R M\u2082\nh : \u2200 (v : \u03b9 \u2192 \u03b9\u2081), (\u2191f fun i => \u2191e (v i)) = \u2191g fun i => \u2191e (v i)\nval\u271d : Fintype \u03b9\n\u22a2 f = g\n[PROOFSTEP]\nexact (domDomCongr_eq_iff (Fintype.equivFin \u03b9) f g).mp (Basis.ext_multilinear_fin (fun _ => e) fun i => h (i \u2218 _))\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Multilinear.Basis", "llama_tokens": 19483, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673178375735, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.41595770395882475}} {"text": "[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / R\u2081\n[PROOFSTEP]\nhave hR\u2081 : 0 < R\u2081 := nonempty_ball.1 \u27e8z, hz\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / R\u2081\n[PROOFSTEP]\nsuffices \u2200\u1da0 r in \ud835\udcdd[<] R\u2081, \u2016dslope f c z\u2016 \u2264 R\u2082 / r\n by\n refine' ge_of_tendsto _ this\n exact (tendsto_const_nhds.div tendsto_id hR\u2081.ne').mono_left nhdsWithin_le_nhds\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\nthis : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Iio R\u2081] R\u2081, \u2016dslope f c z\u2016 \u2264 R\u2082 / r\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / R\u2081\n[PROOFSTEP]\nrefine' ge_of_tendsto _ this\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\nthis : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Iio R\u2081] R\u2081, \u2016dslope f c z\u2016 \u2264 R\u2082 / r\n\u22a2 Tendsto (fun c => R\u2082 / c) (\ud835\udcdd[Iio R\u2081] R\u2081) (\ud835\udcdd (R\u2082 / R\u2081))\n[PROOFSTEP]\nexact (tendsto_const_nhds.div tendsto_id hR\u2081.ne').mono_left nhdsWithin_le_nhds\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\n\u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Iio R\u2081] R\u2081, \u2016dslope f c z\u2016 \u2264 R\u2082 / r\n[PROOFSTEP]\nrw [mem_ball] at hz \n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : dist z c < R\u2081\nhR\u2081 : 0 < R\u2081\n\u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Iio R\u2081] R\u2081, \u2016dslope f c z\u2016 \u2264 R\u2082 / r\n[PROOFSTEP]\nfilter_upwards [Ioo_mem_nhdsWithin_Iio \u27e8hz, le_rfl\u27e9] with r hr\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : dist z c < R\u2081\nhR\u2081 : 0 < R\u2081\nr : \u211d\nhr : r \u2208 Ioo (dist z c) R\u2081\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / r\n[PROOFSTEP]\nhave hr\u2080 : 0 < r := dist_nonneg.trans_lt hr.1\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : dist z c < R\u2081\nhR\u2081 : 0 < R\u2081\nr : \u211d\nhr : r \u2208 Ioo (dist z c) R\u2081\nhr\u2080 : 0 < r\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / r\n[PROOFSTEP]\nreplace hd : DiffContOnCl \u2102 (dslope f c) (ball c r)\n[GOAL]\ncase hd\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : dist z c < R\u2081\nhR\u2081 : 0 < R\u2081\nr : \u211d\nhr : r \u2208 Ioo (dist z c) R\u2081\nhr\u2080 : 0 < r\n\u22a2 DiffContOnCl \u2102 (dslope f c) (ball c r)\n[PROOFSTEP]\nrefine' DifferentiableOn.diffContOnCl _\n[GOAL]\ncase hd\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : dist z c < R\u2081\nhR\u2081 : 0 < R\u2081\nr : \u211d\nhr : r \u2208 Ioo (dist z c) R\u2081\nhr\u2080 : 0 < r\n\u22a2 DifferentiableOn \u2102 (dslope f c) (closure (ball c r))\n[PROOFSTEP]\nrw [closure_ball c hr\u2080.ne']\n[GOAL]\ncase hd\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : dist z c < R\u2081\nhR\u2081 : 0 < R\u2081\nr : \u211d\nhr : r \u2208 Ioo (dist z c) R\u2081\nhr\u2080 : 0 < r\n\u22a2 DifferentiableOn \u2102 (dslope f c) (closedBall c r)\n[PROOFSTEP]\nexact ((differentiableOn_dslope <| ball_mem_nhds _ hR\u2081).mpr hd).mono (closedBall_subset_ball hr.2)\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : dist z c < R\u2081\nhR\u2081 : 0 < R\u2081\nr : \u211d\nhr : r \u2208 Ioo (dist z c) R\u2081\nhr\u2080 : 0 < r\nhd : DiffContOnCl \u2102 (dslope f c) (ball c r)\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / r\n[PROOFSTEP]\nrefine' norm_le_of_forall_mem_frontier_norm_le bounded_ball hd _ _\n[GOAL]\ncase h.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : dist z c < R\u2081\nhR\u2081 : 0 < R\u2081\nr : \u211d\nhr : r \u2208 Ioo (dist z c) R\u2081\nhr\u2080 : 0 < r\nhd : DiffContOnCl \u2102 (dslope f c) (ball c r)\n\u22a2 \u2200 (z : \u2102), z \u2208 frontier (ball c r) \u2192 \u2016dslope f c z\u2016 \u2264 R\u2082 / r\n[PROOFSTEP]\nrw [frontier_ball c hr\u2080.ne']\n[GOAL]\ncase h.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : dist z c < R\u2081\nhR\u2081 : 0 < R\u2081\nr : \u211d\nhr : r \u2208 Ioo (dist z c) R\u2081\nhr\u2080 : 0 < r\nhd : DiffContOnCl \u2102 (dslope f c) (ball c r)\n\u22a2 \u2200 (z : \u2102), z \u2208 sphere c r \u2192 \u2016dslope f c z\u2016 \u2264 R\u2082 / r\n[PROOFSTEP]\nintro z hz\n[GOAL]\ncase h.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z\u271d z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz\u271d : dist z\u271d c < R\u2081\nhR\u2081 : 0 < R\u2081\nr : \u211d\nhr : r \u2208 Ioo (dist z\u271d c) R\u2081\nhr\u2080 : 0 < r\nhd : DiffContOnCl \u2102 (dslope f c) (ball c r)\nz : \u2102\nhz : z \u2208 sphere c r\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / r\n[PROOFSTEP]\nhave hz' : z \u2260 c := ne_of_mem_sphere hz hr\u2080.ne'\n[GOAL]\ncase h.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z\u271d z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz\u271d : dist z\u271d c < R\u2081\nhR\u2081 : 0 < R\u2081\nr : \u211d\nhr : r \u2208 Ioo (dist z\u271d c) R\u2081\nhr\u2080 : 0 < r\nhd : DiffContOnCl \u2102 (dslope f c) (ball c r)\nz : \u2102\nhz : z \u2208 sphere c r\nhz' : z \u2260 c\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / r\n[PROOFSTEP]\nrw [dslope_of_ne _ hz', slope_def_module, norm_smul, norm_inv, mem_sphere_iff_norm.1 hz, \u2190 div_eq_inv_mul,\n div_le_div_right hr\u2080, \u2190 dist_eq_norm]\n[GOAL]\ncase h.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z\u271d z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz\u271d : dist z\u271d c < R\u2081\nhR\u2081 : 0 < R\u2081\nr : \u211d\nhr : r \u2208 Ioo (dist z\u271d c) R\u2081\nhr\u2080 : 0 < r\nhd : DiffContOnCl \u2102 (dslope f c) (ball c r)\nz : \u2102\nhz : z \u2208 sphere c r\nhz' : z \u2260 c\n\u22a2 dist (f z) (f c) \u2264 R\u2082\n[PROOFSTEP]\nexact le_of_lt (h_maps (mem_ball.2 (by rw [mem_sphere.1 hz]; exact hr.2)))\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z\u271d z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz\u271d : dist z\u271d c < R\u2081\nhR\u2081 : 0 < R\u2081\nr : \u211d\nhr : r \u2208 Ioo (dist z\u271d c) R\u2081\nhr\u2080 : 0 < r\nhd : DiffContOnCl \u2102 (dslope f c) (ball c r)\nz : \u2102\nhz : z \u2208 sphere c r\nhz' : z \u2260 c\n\u22a2 dist z c < R\u2081\n[PROOFSTEP]\nrw [mem_sphere.1 hz]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z\u271d z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz\u271d : dist z\u271d c < R\u2081\nhR\u2081 : 0 < R\u2081\nr : \u211d\nhr : r \u2208 Ioo (dist z\u271d c) R\u2081\nhr\u2080 : 0 < r\nhd : DiffContOnCl \u2102 (dslope f c) (ball c r)\nz : \u2102\nhz : z \u2208 sphere c r\nhz' : z \u2260 c\n\u22a2 r < R\u2081\n[PROOFSTEP]\nexact hr.2\n[GOAL]\ncase h.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : dist z c < R\u2081\nhR\u2081 : 0 < R\u2081\nr : \u211d\nhr : r \u2208 Ioo (dist z c) R\u2081\nhr\u2080 : 0 < r\nhd : DiffContOnCl \u2102 (dslope f c) (ball c r)\n\u22a2 z \u2208 closure (ball c r)\n[PROOFSTEP]\nrw [closure_ball c hr\u2080.ne', mem_closedBall]\n[GOAL]\ncase h.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf\u271d : \u2102 \u2192 E\nc z z\u2080 : \u2102\nf : \u2102 \u2192 \u2102\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : dist z c < R\u2081\nhR\u2081 : 0 < R\u2081\nr : \u211d\nhr : r \u2208 Ioo (dist z c) R\u2081\nhr\u2080 : 0 < r\nhd : DiffContOnCl \u2102 (dslope f c) (ball c r)\n\u22a2 dist z c \u2264 r\n[PROOFSTEP]\nexact hr.1.le\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / R\u2081\n[PROOFSTEP]\nhave hR\u2081 : 0 < R\u2081 := nonempty_ball.1 \u27e8z, hz\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / R\u2081\n[PROOFSTEP]\nhave hR\u2082 : 0 < R\u2082 := nonempty_ball.1 \u27e8f z, h_maps hz\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\nhR\u2082 : 0 < R\u2082\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / R\u2081\n[PROOFSTEP]\ncases' eq_or_ne (dslope f c z) 0 with hc hc\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\nhR\u2082 : 0 < R\u2082\nhc : dslope f c z = 0\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / R\u2081\n[PROOFSTEP]\nrw [hc, norm_zero]\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\nhR\u2082 : 0 < R\u2082\nhc : dslope f c z = 0\n\u22a2 0 \u2264 R\u2082 / R\u2081\n[PROOFSTEP]\nexact div_nonneg hR\u2082.le hR\u2081.le\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\nhR\u2082 : 0 < R\u2082\nhc : dslope f c z \u2260 0\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / R\u2081\n[PROOFSTEP]\nrcases exists_dual_vector \u2102 _ hc with \u27e8g, hg, hgf\u27e9\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\nhR\u2082 : 0 < R\u2082\nhc : dslope f c z \u2260 0\ng : E \u2192L[\u2102] \u2102\nhg : \u2016g\u2016 = 1\nhgf : \u2191g (dslope f c z) = \u2191\u2016dslope f c z\u2016\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / R\u2081\n[PROOFSTEP]\nhave hg' : \u2016g\u2016\u208a = 1 := NNReal.eq hg\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\nhR\u2082 : 0 < R\u2082\nhc : dslope f c z \u2260 0\ng : E \u2192L[\u2102] \u2102\nhg : \u2016g\u2016 = 1\nhgf : \u2191g (dslope f c z) = \u2191\u2016dslope f c z\u2016\nhg' : \u2016g\u2016\u208a = 1\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / R\u2081\n[PROOFSTEP]\nhave hg\u2080 : \u2016g\u2016\u208a \u2260 0 := by simpa only [hg'] using one_ne_zero\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\nhR\u2082 : 0 < R\u2082\nhc : dslope f c z \u2260 0\ng : E \u2192L[\u2102] \u2102\nhg : \u2016g\u2016 = 1\nhgf : \u2191g (dslope f c z) = \u2191\u2016dslope f c z\u2016\nhg' : \u2016g\u2016\u208a = 1\n\u22a2 \u2016g\u2016\u208a \u2260 0\n[PROOFSTEP]\nsimpa only [hg'] using one_ne_zero\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\nhR\u2082 : 0 < R\u2082\nhc : dslope f c z \u2260 0\ng : E \u2192L[\u2102] \u2102\nhg : \u2016g\u2016 = 1\nhgf : \u2191g (dslope f c z) = \u2191\u2016dslope f c z\u2016\nhg' : \u2016g\u2016\u208a = 1\nhg\u2080 : \u2016g\u2016\u208a \u2260 0\n\u22a2 \u2016dslope f c z\u2016 \u2264 R\u2082 / R\u2081\n[PROOFSTEP]\ncalc\n \u2016dslope f c z\u2016 = \u2016dslope (g \u2218 f) c z\u2016 :=\n by\n rw [g.dslope_comp, hgf, IsROrC.norm_ofReal, abs_norm]\n exact fun _ => hd.differentiableAt (ball_mem_nhds _ hR\u2081)\n _ \u2264 R\u2082 / R\u2081 :=\n by\n refine' schwarz_aux (g.differentiable.comp_differentiableOn hd) (MapsTo.comp _ h_maps) hz\n simpa only [hg', NNReal.coe_one, one_mul] using g.lipschitz.mapsTo_ball hg\u2080 (f c) R\u2082\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\nhR\u2082 : 0 < R\u2082\nhc : dslope f c z \u2260 0\ng : E \u2192L[\u2102] \u2102\nhg : \u2016g\u2016 = 1\nhgf : \u2191g (dslope f c z) = \u2191\u2016dslope f c z\u2016\nhg' : \u2016g\u2016\u208a = 1\nhg\u2080 : \u2016g\u2016\u208a \u2260 0\n\u22a2 \u2016dslope f c z\u2016 = \u2016dslope (\u2191g \u2218 f) c z\u2016\n[PROOFSTEP]\nrw [g.dslope_comp, hgf, IsROrC.norm_ofReal, abs_norm]\n[GOAL]\ncase H\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\nhR\u2082 : 0 < R\u2082\nhc : dslope f c z \u2260 0\ng : E \u2192L[\u2102] \u2102\nhg : \u2016g\u2016 = 1\nhgf : \u2191g (dslope f c z) = \u2191\u2016dslope f c z\u2016\nhg' : \u2016g\u2016\u208a = 1\nhg\u2080 : \u2016g\u2016\u208a \u2260 0\n\u22a2 c = z \u2192 DifferentiableAt \u2102 f c\n[PROOFSTEP]\nexact fun _ => hd.differentiableAt (ball_mem_nhds _ hR\u2081)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\nhR\u2082 : 0 < R\u2082\nhc : dslope f c z \u2260 0\ng : E \u2192L[\u2102] \u2102\nhg : \u2016g\u2016 = 1\nhgf : \u2191g (dslope f c z) = \u2191\u2016dslope f c z\u2016\nhg' : \u2016g\u2016\u208a = 1\nhg\u2080 : \u2016g\u2016\u208a \u2260 0\n\u22a2 \u2016dslope (\u2191g \u2218 f) c z\u2016 \u2264 R\u2082 / R\u2081\n[PROOFSTEP]\nrefine' schwarz_aux (g.differentiable.comp_differentiableOn hd) (MapsTo.comp _ h_maps) hz\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhR\u2081 : 0 < R\u2081\nhR\u2082 : 0 < R\u2082\nhc : dslope f c z \u2260 0\ng : E \u2192L[\u2102] \u2102\nhg : \u2016g\u2016 = 1\nhgf : \u2191g (dslope f c z) = \u2191\u2016dslope f c z\u2016\nhg' : \u2016g\u2016\u208a = 1\nhg\u2080 : \u2016g\u2016\u208a \u2260 0\n\u22a2 MapsTo (\u2191g) (ball (f c) R\u2082) (ball ((\u2191g \u2218 f) c) R\u2082)\n[PROOFSTEP]\nsimpa only [hg', NNReal.coe_one, one_mul] using g.lipschitz.mapsTo_ball hg\u2080 (f c) R\u2082\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : StrictConvexSpace \u211d E\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nh_z\u2080 : z\u2080 \u2208 ball c R\u2081\nh_eq : \u2016dslope f c z\u2080\u2016 = R\u2082 / R\u2081\n\u22a2 EqOn f (fun z => f c + (z - c) \u2022 dslope f c z\u2080) (ball c R\u2081)\n[PROOFSTEP]\nset g := dslope f c\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : StrictConvexSpace \u211d E\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nh_z\u2080 : z\u2080 \u2208 ball c R\u2081\ng : \u2102 \u2192 E := dslope f c\nh_eq : \u2016g z\u2080\u2016 = R\u2082 / R\u2081\n\u22a2 EqOn f (fun z => f c + (z - c) \u2022 g z\u2080) (ball c R\u2081)\n[PROOFSTEP]\nrintro z hz\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z\u271d z\u2080 : \u2102\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : StrictConvexSpace \u211d E\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nh_z\u2080 : z\u2080 \u2208 ball c R\u2081\ng : \u2102 \u2192 E := dslope f c\nh_eq : \u2016g z\u2080\u2016 = R\u2082 / R\u2081\nz : \u2102\nhz : z \u2208 ball c R\u2081\n\u22a2 f z = (fun z => f c + (z - c) \u2022 g z\u2080) z\n[PROOFSTEP]\nby_cases z = c\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z\u271d z\u2080 : \u2102\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : StrictConvexSpace \u211d E\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nh_z\u2080 : z\u2080 \u2208 ball c R\u2081\ng : \u2102 \u2192 E := dslope f c\nh_eq : \u2016g z\u2080\u2016 = R\u2082 / R\u2081\nz : \u2102\nhz : z \u2208 ball c R\u2081\n\u22a2 f z = (fun z => f c + (z - c) \u2022 g z\u2080) z\n[PROOFSTEP]\nby_cases z = c\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z\u271d z\u2080 : \u2102\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : StrictConvexSpace \u211d E\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nh_z\u2080 : z\u2080 \u2208 ball c R\u2081\ng : \u2102 \u2192 E := dslope f c\nh_eq : \u2016g z\u2080\u2016 = R\u2082 / R\u2081\nz : \u2102\nhz : z \u2208 ball c R\u2081\nh : z = c\n\u22a2 f z = (fun z => f c + (z - c) \u2022 g z\u2080) z\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z\u271d z\u2080 : \u2102\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : StrictConvexSpace \u211d E\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nh_z\u2080 : z\u2080 \u2208 ball c R\u2081\ng : \u2102 \u2192 E := dslope f c\nh_eq : \u2016g z\u2080\u2016 = R\u2082 / R\u2081\nz : \u2102\nhz : z \u2208 ball c R\u2081\nh : \u00acz = c\n\u22a2 f z = (fun z => f c + (z - c) \u2022 g z\u2080) z\n[PROOFSTEP]\nhave h_R\u2081 : 0 < R\u2081 := nonempty_ball.mp \u27e8_, h_z\u2080\u27e9\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z\u271d z\u2080 : \u2102\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : StrictConvexSpace \u211d E\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nh_z\u2080 : z\u2080 \u2208 ball c R\u2081\ng : \u2102 \u2192 E := dslope f c\nh_eq : \u2016g z\u2080\u2016 = R\u2082 / R\u2081\nz : \u2102\nhz : z \u2208 ball c R\u2081\nh : \u00acz = c\nh_R\u2081 : 0 < R\u2081\n\u22a2 f z = (fun z => f c + (z - c) \u2022 g z\u2080) z\n[PROOFSTEP]\nhave g_le_div : \u2200 z \u2208 ball c R\u2081, \u2016g z\u2016 \u2264 R\u2082 / R\u2081 := fun z hz => norm_dslope_le_div_of_mapsTo_ball hd h_maps hz\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z\u271d z\u2080 : \u2102\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : StrictConvexSpace \u211d E\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nh_z\u2080 : z\u2080 \u2208 ball c R\u2081\ng : \u2102 \u2192 E := dslope f c\nh_eq : \u2016g z\u2080\u2016 = R\u2082 / R\u2081\nz : \u2102\nhz : z \u2208 ball c R\u2081\nh : \u00acz = c\nh_R\u2081 : 0 < R\u2081\ng_le_div : \u2200 (z : \u2102), z \u2208 ball c R\u2081 \u2192 \u2016g z\u2016 \u2264 R\u2082 / R\u2081\n\u22a2 f z = (fun z => f c + (z - c) \u2022 g z\u2080) z\n[PROOFSTEP]\nhave g_max : IsMaxOn (norm \u2218 g) (ball c R\u2081) z\u2080 := isMaxOn_iff.mpr fun z hz => by simpa [h_eq] using g_le_div z hz\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z\u271d\u00b9 z\u2080 : \u2102\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : StrictConvexSpace \u211d E\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nh_z\u2080 : z\u2080 \u2208 ball c R\u2081\ng : \u2102 \u2192 E := dslope f c\nh_eq : \u2016g z\u2080\u2016 = R\u2082 / R\u2081\nz\u271d : \u2102\nhz\u271d : z\u271d \u2208 ball c R\u2081\nh : \u00acz\u271d = c\nh_R\u2081 : 0 < R\u2081\ng_le_div : \u2200 (z : \u2102), z \u2208 ball c R\u2081 \u2192 \u2016g z\u2016 \u2264 R\u2082 / R\u2081\nz : \u2102\nhz : z \u2208 ball c R\u2081\n\u22a2 (norm \u2218 g) z \u2264 (norm \u2218 g) z\u2080\n[PROOFSTEP]\nsimpa [h_eq] using g_le_div z hz\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z\u271d z\u2080 : \u2102\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : StrictConvexSpace \u211d E\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nh_z\u2080 : z\u2080 \u2208 ball c R\u2081\ng : \u2102 \u2192 E := dslope f c\nh_eq : \u2016g z\u2080\u2016 = R\u2082 / R\u2081\nz : \u2102\nhz : z \u2208 ball c R\u2081\nh : \u00acz = c\nh_R\u2081 : 0 < R\u2081\ng_le_div : \u2200 (z : \u2102), z \u2208 ball c R\u2081 \u2192 \u2016g z\u2016 \u2264 R\u2082 / R\u2081\ng_max : IsMaxOn (norm \u2218 g) (ball c R\u2081) z\u2080\n\u22a2 f z = (fun z => f c + (z - c) \u2022 g z\u2080) z\n[PROOFSTEP]\nhave g_diff : DifferentiableOn \u2102 g (ball c R\u2081) :=\n (differentiableOn_dslope (isOpen_ball.mem_nhds (mem_ball_self h_R\u2081))).mpr hd\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z\u271d z\u2080 : \u2102\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : StrictConvexSpace \u211d E\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nh_z\u2080 : z\u2080 \u2208 ball c R\u2081\ng : \u2102 \u2192 E := dslope f c\nh_eq : \u2016g z\u2080\u2016 = R\u2082 / R\u2081\nz : \u2102\nhz : z \u2208 ball c R\u2081\nh : \u00acz = c\nh_R\u2081 : 0 < R\u2081\ng_le_div : \u2200 (z : \u2102), z \u2208 ball c R\u2081 \u2192 \u2016g z\u2016 \u2264 R\u2082 / R\u2081\ng_max : IsMaxOn (norm \u2218 g) (ball c R\u2081) z\u2080\ng_diff : DifferentiableOn \u2102 g (ball c R\u2081)\n\u22a2 f z = (fun z => f c + (z - c) \u2022 g z\u2080) z\n[PROOFSTEP]\nhave : g z = g z\u2080 :=\n eqOn_of_isPreconnected_of_isMaxOn_norm (convex_ball c R\u2081).isPreconnected isOpen_ball g_diff h_z\u2080 g_max hz\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z\u271d z\u2080 : \u2102\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : StrictConvexSpace \u211d E\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nh_z\u2080 : z\u2080 \u2208 ball c R\u2081\ng : \u2102 \u2192 E := dslope f c\nh_eq : \u2016g z\u2080\u2016 = R\u2082 / R\u2081\nz : \u2102\nhz : z \u2208 ball c R\u2081\nh : \u00acz = c\nh_R\u2081 : 0 < R\u2081\ng_le_div : \u2200 (z : \u2102), z \u2208 ball c R\u2081 \u2192 \u2016g z\u2016 \u2264 R\u2082 / R\u2081\ng_max : IsMaxOn (norm \u2218 g) (ball c R\u2081) z\u2080\ng_diff : DifferentiableOn \u2102 g (ball c R\u2081)\nthis : g z = g z\u2080\n\u22a2 f z = (fun z => f c + (z - c) \u2022 g z\u2080) z\n[PROOFSTEP]\nsimp [\u2190 this]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nh\u2080 : 0 < R\u2081\n\u22a2 \u2016deriv f c\u2016 \u2264 R\u2082 / R\u2081\n[PROOFSTEP]\nsimpa only [dslope_same] using norm_dslope_le_div_of_mapsTo_ball hd h_maps (mem_ball_self h\u2080)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\n\u22a2 dist (f z) (f c) \u2264 R\u2082 / R\u2081 * dist z c\n[PROOFSTEP]\nrcases eq_or_ne z c with (rfl | hne)\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nz z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball z R\u2081)\nh_maps : MapsTo f (ball z R\u2081) (ball (f z) R\u2082)\nhz : z \u2208 ball z R\u2081\n\u22a2 dist (f z) (f z) \u2264 R\u2082 / R\u2081 * dist z z\n[PROOFSTEP]\nsimp only [dist_self, mul_zero, le_rfl]\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nhz : z \u2208 ball c R\u2081\nhne : z \u2260 c\n\u22a2 dist (f z) (f c) \u2264 R\u2082 / R\u2081 * dist z c\n[PROOFSTEP]\nsimpa only [dslope_of_ne _ hne, slope_def_module, norm_smul, norm_inv, \u2190 div_eq_inv_mul, \u2190 dist_eq_norm,\n div_le_iff (dist_pos.2 hne)] using norm_dslope_le_div_of_mapsTo_ball hd h_maps hz\n[GOAL]\nf : \u2102 \u2192 \u2102\nc z : \u2102\nR R\u2081 R\u2082 : \u211d\nhd : DifferentiableOn \u2102 f (ball c R)\nh_maps : MapsTo f (ball c R) (ball c R)\nhc : f c = c\nh\u2080 : 0 < R\n\u22a2 MapsTo f (ball c R) (ball (f c) R)\n[PROOFSTEP]\nrwa [hc]\n[GOAL]\nf : \u2102 \u2192 \u2102\nc z : \u2102\nR R\u2081 R\u2082 : \u211d\nhd : DifferentiableOn \u2102 f (ball c R)\nh_maps : MapsTo f (ball c R) (ball c R)\nhc : f c = c\nhz : z \u2208 ball c R\n\u22a2 dist (f z) c \u2264 dist z c\n[PROOFSTEP]\nhave := dist_le_div_mul_dist_of_mapsTo_ball hd (by rwa [hc]) hz\n[GOAL]\nf : \u2102 \u2192 \u2102\nc z : \u2102\nR R\u2081 R\u2082 : \u211d\nhd : DifferentiableOn \u2102 f (ball c R)\nh_maps : MapsTo f (ball c R) (ball c R)\nhc : f c = c\nhz : z \u2208 ball c R\n\u22a2 MapsTo f (ball c R) (ball (f c) ?m.39633)\n[PROOFSTEP]\nrwa [hc]\n[GOAL]\nf : \u2102 \u2192 \u2102\nc z : \u2102\nR R\u2081 R\u2082 : \u211d\nhd : DifferentiableOn \u2102 f (ball c R)\nh_maps : MapsTo f (ball c R) (ball c R)\nhc : f c = c\nhz : z \u2208 ball c R\nthis : dist (f z) (f c) \u2264 R / R * dist z c\n\u22a2 dist (f z) c \u2264 dist z c\n[PROOFSTEP]\nrwa [hc, div_self, one_mul] at this \n[GOAL]\nf : \u2102 \u2192 \u2102\nc z : \u2102\nR R\u2081 R\u2082 : \u211d\nhd : DifferentiableOn \u2102 f (ball c R)\nh_maps : MapsTo f (ball c R) (ball c R)\nhc : f c = c\nhz : z \u2208 ball c R\nthis : dist (f z) c \u2264 R / R * dist z c\n\u22a2 R \u2260 0\n[PROOFSTEP]\nexact (nonempty_ball.1 \u27e8z, hz\u27e9).ne'\n[GOAL]\nf : \u2102 \u2192 \u2102\nc z : \u2102\nR R\u2081 R\u2082 : \u211d\nhd : DifferentiableOn \u2102 f (ball 0 R)\nh_maps : MapsTo f (ball 0 R) (ball 0 R)\nh\u2080 : f 0 = 0\nhz : \u2191abs z < R\n\u22a2 \u2191abs (f z) \u2264 \u2191abs z\n[PROOFSTEP]\nreplace hz : z \u2208 ball (0 : \u2102) R\n[GOAL]\ncase hz\nf : \u2102 \u2192 \u2102\nc z : \u2102\nR R\u2081 R\u2082 : \u211d\nhd : DifferentiableOn \u2102 f (ball 0 R)\nh_maps : MapsTo f (ball 0 R) (ball 0 R)\nh\u2080 : f 0 = 0\nhz : \u2191abs z < R\n\u22a2 z \u2208 ball 0 R\nf : \u2102 \u2192 \u2102\nc z : \u2102\nR R\u2081 R\u2082 : \u211d\nhd : DifferentiableOn \u2102 f (ball 0 R)\nh_maps : MapsTo f (ball 0 R) (ball 0 R)\nh\u2080 : f 0 = 0\nhz : z \u2208 ball 0 R\n\u22a2 \u2191abs (f z) \u2264 \u2191abs z\n[PROOFSTEP]\nexact mem_ball_zero_iff.2 hz\n[GOAL]\nf : \u2102 \u2192 \u2102\nc z : \u2102\nR R\u2081 R\u2082 : \u211d\nhd : DifferentiableOn \u2102 f (ball 0 R)\nh_maps : MapsTo f (ball 0 R) (ball 0 R)\nh\u2080 : f 0 = 0\nhz : z \u2208 ball 0 R\n\u22a2 \u2191abs (f z) \u2264 \u2191abs z\n[PROOFSTEP]\nsimpa only [dist_zero_right] using dist_le_dist_of_mapsTo_ball_self hd h_maps h\u2080 hz\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Complex.Schwarz", "llama_tokens": 13392, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.41587817987003717}} {"text": "[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n\u22a2 M.dest (P F) (corecF g x) = (TypeVec.id ::: corecF g) <$$> repr (g x)\n[PROOFSTEP]\nrw [corecF, M.dest_corec]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\n\u22a2 \u2200 (a b : M (P F) \u03b1), Mcongr a b \u2192 (fun x => Quot.mk Mcongr (g <$$> x)) a = (fun x => Quot.mk Mcongr (g <$$> x)) b\n[PROOFSTEP]\nrintro aa\u2081 aa\u2082 \u27e8r, pr, ra\u2081a\u2082\u27e9\n[GOAL]\ncase intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082 : r aa\u2081 aa\u2082\n\u22a2 (fun x => Quot.mk Mcongr (g <$$> x)) aa\u2081 = (fun x => Quot.mk Mcongr (g <$$> x)) aa\u2082\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase intro.intro.a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082 : r aa\u2081 aa\u2082\n\u22a2 Mcongr (g <$$> aa\u2081) (g <$$> aa\u2082)\n[PROOFSTEP]\nlet r' b\u2081 b\u2082 := \u2203 a\u2081 a\u2082 : q.P.M \u03b1, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\n[GOAL]\ncase intro.intro.a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082 : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\n\u22a2 Mcongr (g <$$> aa\u2081) (g <$$> aa\u2082)\n[PROOFSTEP]\nuse r'\n[GOAL]\ncase h\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082 : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\n\u22a2 IsPrecongr r' \u2227 r' (g <$$> aa\u2081) (g <$$> aa\u2082)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082 : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\n\u22a2 IsPrecongr r'\n[PROOFSTEP]\nshow IsPrecongr r'\n[GOAL]\ncase h.left\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082 : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\n\u22a2 IsPrecongr r'\n[PROOFSTEP]\nrintro b\u2081 b\u2082 \u27e8a\u2081, a\u2082, ra\u2081a\u2082, b\u2081eq, b\u2082eq\u27e9\n[GOAL]\ncase h.left.intro.intro.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082\u271d : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\nb\u2081 b\u2082 : M (P F) \u03b2\na\u2081 a\u2082 : M (P F) \u03b1\nra\u2081a\u2082 : r a\u2081 a\u2082\nb\u2081eq : b\u2081 = g <$$> a\u2081\nb\u2082eq : b\u2082 = g <$$> a\u2082\n\u22a2 abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b\u2081) = abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b\u2082)\n[PROOFSTEP]\nlet u : Quot r \u2192 Quot r' :=\n Quot.lift (fun x : q.P.M \u03b1 => Quot.mk r' (g <$$> x))\n (by\n intro a\u2081 a\u2082 ra\u2081a\u2082\n apply Quot.sound\n exact \u27e8a\u2081, a\u2082, ra\u2081a\u2082, rfl, rfl\u27e9)\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082\u271d : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\nb\u2081 b\u2082 : M (P F) \u03b2\na\u2081 a\u2082 : M (P F) \u03b1\nra\u2081a\u2082 : r a\u2081 a\u2082\nb\u2081eq : b\u2081 = g <$$> a\u2081\nb\u2082eq : b\u2082 = g <$$> a\u2082\n\u22a2 \u2200 (a b : M (P F) \u03b1), r a b \u2192 (fun x => Quot.mk r' (g <$$> x)) a = (fun x => Quot.mk r' (g <$$> x)) b\n[PROOFSTEP]\nintro a\u2081 a\u2082 ra\u2081a\u2082\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082\u271d\u00b9 : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\nb\u2081 b\u2082 : M (P F) \u03b2\na\u2081\u271d a\u2082\u271d : M (P F) \u03b1\nra\u2081a\u2082\u271d : r a\u2081\u271d a\u2082\u271d\nb\u2081eq : b\u2081 = g <$$> a\u2081\u271d\nb\u2082eq : b\u2082 = g <$$> a\u2082\u271d\na\u2081 a\u2082 : M (P F) \u03b1\nra\u2081a\u2082 : r a\u2081 a\u2082\n\u22a2 (fun x => Quot.mk r' (g <$$> x)) a\u2081 = (fun x => Quot.mk r' (g <$$> x)) a\u2082\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082\u271d\u00b9 : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\nb\u2081 b\u2082 : M (P F) \u03b2\na\u2081\u271d a\u2082\u271d : M (P F) \u03b1\nra\u2081a\u2082\u271d : r a\u2081\u271d a\u2082\u271d\nb\u2081eq : b\u2081 = g <$$> a\u2081\u271d\nb\u2082eq : b\u2082 = g <$$> a\u2082\u271d\na\u2081 a\u2082 : M (P F) \u03b1\nra\u2081a\u2082 : r a\u2081 a\u2082\n\u22a2 r' (g <$$> a\u2081) (g <$$> a\u2082)\n[PROOFSTEP]\nexact \u27e8a\u2081, a\u2082, ra\u2081a\u2082, rfl, rfl\u27e9\n[GOAL]\ncase h.left.intro.intro.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082\u271d : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\nb\u2081 b\u2082 : M (P F) \u03b2\na\u2081 a\u2082 : M (P F) \u03b1\nra\u2081a\u2082 : r a\u2081 a\u2082\nb\u2081eq : b\u2081 = g <$$> a\u2081\nb\u2082eq : b\u2082 = g <$$> a\u2082\nu : Quot r \u2192 Quot r' :=\n Quot.lift (fun x => Quot.mk r' (g <$$> x))\n (_ : \u2200 (a\u2081 a\u2082 : M (P F) \u03b1), r a\u2081 a\u2082 \u2192 Quot.mk r' (g <$$> a\u2081) = Quot.mk r' (g <$$> a\u2082))\n\u22a2 abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b\u2081) = abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b\u2082)\n[PROOFSTEP]\nhave hu : (Quot.mk r' \u2218 fun x : q.P.M \u03b1 => g <$$> x) = u \u2218 Quot.mk r :=\n by\n ext x\n rfl\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082\u271d : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\nb\u2081 b\u2082 : M (P F) \u03b2\na\u2081 a\u2082 : M (P F) \u03b1\nra\u2081a\u2082 : r a\u2081 a\u2082\nb\u2081eq : b\u2081 = g <$$> a\u2081\nb\u2082eq : b\u2082 = g <$$> a\u2082\nu : Quot r \u2192 Quot r' :=\n Quot.lift (fun x => Quot.mk r' (g <$$> x))\n (_ : \u2200 (a\u2081 a\u2082 : M (P F) \u03b1), r a\u2081 a\u2082 \u2192 Quot.mk r' (g <$$> a\u2081) = Quot.mk r' (g <$$> a\u2082))\n\u22a2 (Quot.mk r' \u2218 fun x => g <$$> x) = u \u2218 Quot.mk r\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082\u271d : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\nb\u2081 b\u2082 : M (P F) \u03b2\na\u2081 a\u2082 : M (P F) \u03b1\nra\u2081a\u2082 : r a\u2081 a\u2082\nb\u2081eq : b\u2081 = g <$$> a\u2081\nb\u2082eq : b\u2082 = g <$$> a\u2082\nu : Quot r \u2192 Quot r' :=\n Quot.lift (fun x => Quot.mk r' (g <$$> x))\n (_ : \u2200 (a\u2081 a\u2082 : M (P F) \u03b1), r a\u2081 a\u2082 \u2192 Quot.mk r' (g <$$> a\u2081) = Quot.mk r' (g <$$> a\u2082))\nx : M (P F) \u03b1\n\u22a2 (Quot.mk r' \u2218 fun x => g <$$> x) x = (u \u2218 Quot.mk r) x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.left.intro.intro.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082\u271d : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\nb\u2081 b\u2082 : M (P F) \u03b2\na\u2081 a\u2082 : M (P F) \u03b1\nra\u2081a\u2082 : r a\u2081 a\u2082\nb\u2081eq : b\u2081 = g <$$> a\u2081\nb\u2082eq : b\u2082 = g <$$> a\u2082\nu : Quot r \u2192 Quot r' :=\n Quot.lift (fun x => Quot.mk r' (g <$$> x))\n (_ : \u2200 (a\u2081 a\u2082 : M (P F) \u03b1), r a\u2081 a\u2082 \u2192 Quot.mk r' (g <$$> a\u2081) = Quot.mk r' (g <$$> a\u2082))\nhu : (Quot.mk r' \u2218 fun x => g <$$> x) = u \u2218 Quot.mk r\n\u22a2 abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b\u2081) = abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b\u2082)\n[PROOFSTEP]\nrw [b\u2081eq, b\u2082eq, M.dest_map, M.dest_map, \u2190 q.P.comp_map, \u2190 q.P.comp_map]\n[GOAL]\ncase h.left.intro.intro.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082\u271d : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\nb\u2081 b\u2082 : M (P F) \u03b2\na\u2081 a\u2082 : M (P F) \u03b1\nra\u2081a\u2082 : r a\u2081 a\u2082\nb\u2081eq : b\u2081 = g <$$> a\u2081\nb\u2082eq : b\u2082 = g <$$> a\u2082\nu : Quot r \u2192 Quot r' :=\n Quot.lift (fun x => Quot.mk r' (g <$$> x))\n (_ : \u2200 (a\u2081 a\u2082 : M (P F) \u03b1), r a\u2081 a\u2082 \u2192 Quot.mk r' (g <$$> a\u2081) = Quot.mk r' (g <$$> a\u2082))\nhu : (Quot.mk r' \u2218 fun x => g <$$> x) = u \u2218 Quot.mk r\n\u22a2 abs (((TypeVec.id ::: Quot.mk r') \u229a (g ::: fun x => g <$$> x)) <$$> M.dest (P F) a\u2081) =\n abs (((TypeVec.id ::: Quot.mk r') \u229a (g ::: fun x => g <$$> x)) <$$> M.dest (P F) a\u2082)\n[PROOFSTEP]\nrw [\u2190 appendFun_comp, id_comp, hu, \u2190 comp_id g, appendFun_comp]\n[GOAL]\ncase h.left.intro.intro.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082\u271d : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\nb\u2081 b\u2082 : M (P F) \u03b2\na\u2081 a\u2082 : M (P F) \u03b1\nra\u2081a\u2082 : r a\u2081 a\u2082\nb\u2081eq : b\u2081 = g <$$> a\u2081\nb\u2082eq : b\u2082 = g <$$> a\u2082\nu : Quot r \u2192 Quot r' :=\n Quot.lift (fun x => Quot.mk r' (g <$$> x))\n (_ : \u2200 (a\u2081 a\u2082 : M (P F) \u03b1), r a\u2081 a\u2082 \u2192 Quot.mk r' (g <$$> a\u2081) = Quot.mk r' (g <$$> a\u2082))\nhu : (Quot.mk r' \u2218 fun x => g <$$> x) = u \u2218 Quot.mk r\n\u22a2 abs (((g ::: u) \u229a (TypeVec.id ::: Quot.mk r)) <$$> M.dest (P F) a\u2081) =\n abs (((g ::: u) \u229a (TypeVec.id ::: Quot.mk r)) <$$> M.dest (P F) a\u2082)\n[PROOFSTEP]\nrw [q.P.comp_map, q.P.comp_map, abs_map, pr ra\u2081a\u2082, \u2190 abs_map]\n[GOAL]\ncase h.right\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082 : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\n\u22a2 r' (g <$$> aa\u2081) (g <$$> aa\u2082)\n[PROOFSTEP]\nshow r' (g <$$> aa\u2081) (g <$$> aa\u2082)\n[GOAL]\ncase h.right\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\naa\u2081 aa\u2082 : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nra\u2081a\u2082 : r aa\u2081 aa\u2082\nr' : M (P F) \u03b2 \u2192 M (P F) \u03b2 \u2192 Prop := fun b\u2081 b\u2082 => \u2203 a\u2081 a\u2082, r a\u2081 a\u2082 \u2227 b\u2081 = g <$$> a\u2081 \u2227 b\u2082 = g <$$> a\u2082\n\u22a2 r' (g <$$> aa\u2081) (g <$$> aa\u2082)\n[PROOFSTEP]\nexact \u27e8aa\u2081, aa\u2082, ra\u2081a\u2082, rfl, rfl\u27e9\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u22a2 \u2200 (a b : M (P F) \u03b1),\n Mcongr a b \u2192\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> abs (M.dest (P F) x)) a =\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> abs (M.dest (P F) x)) b\n[PROOFSTEP]\nrintro x y \u27e8r, pr, rxy\u27e9\n[GOAL]\ncase intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx y : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\n\u22a2 (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> abs (M.dest (P F) x)) x =\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> abs (M.dest (P F) x)) y\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx y : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\n\u22a2 (TypeVec.id ::: Quot.mk Mcongr) <$$> abs (M.dest (P F) x) = (TypeVec.id ::: Quot.mk Mcongr) <$$> abs (M.dest (P F) y)\n[PROOFSTEP]\nhave : \u2200 x y, r x y \u2192 Mcongr x y := by\n intro x y h\n exact \u27e8r, pr, h\u27e9\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx y : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\n\u22a2 \u2200 (x y : M (P F) \u03b1), r x y \u2192 Mcongr x y\n[PROOFSTEP]\nintro x y h\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d y\u271d : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nrxy : r x\u271d y\u271d\nx y : M (P F) \u03b1\nh : r x y\n\u22a2 Mcongr x y\n[PROOFSTEP]\nexact \u27e8r, pr, h\u27e9\n[GOAL]\ncase intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx y : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\nthis : \u2200 (x y : M (P F) \u03b1), r x y \u2192 Mcongr x y\n\u22a2 (TypeVec.id ::: Quot.mk Mcongr) <$$> abs (M.dest (P F) x) = (TypeVec.id ::: Quot.mk Mcongr) <$$> abs (M.dest (P F) y)\n[PROOFSTEP]\nrw [\u2190 Quot.factor_mk_eq _ _ this]\n[GOAL]\ncase intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx y : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\nthis : \u2200 (x y : M (P F) \u03b1), r x y \u2192 Mcongr x y\n\u22a2 (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$$>\n abs (M.dest (P F) x) =\n (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$$>\n abs (M.dest (P F) y)\n[PROOFSTEP]\nconv =>\n lhs\n rw [appendFun_comp_id, comp_map, \u2190 abs_map, pr rxy, abs_map, \u2190 comp_map, \u2190 appendFun_comp_id]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx y : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\nthis : \u2200 (x y : M (P F) \u03b1), r x y \u2192 Mcongr x y\n| (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$$>\n abs (M.dest (P F) x) =\n (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$$>\n abs (M.dest (P F) y)\n[PROOFSTEP]\n lhs\n rw [appendFun_comp_id, comp_map, \u2190 abs_map, pr rxy, abs_map, \u2190 comp_map, \u2190 appendFun_comp_id]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx y : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\nthis : \u2200 (x y : M (P F) \u03b1), r x y \u2192 Mcongr x y\n| (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$$>\n abs (M.dest (P F) x) =\n (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$$>\n abs (M.dest (P F) y)\n[PROOFSTEP]\n lhs\n rw [appendFun_comp_id, comp_map, \u2190 abs_map, pr rxy, abs_map, \u2190 comp_map, \u2190 appendFun_comp_id]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx y : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\nthis : \u2200 (x y : M (P F) \u03b1), r x y \u2192 Mcongr x y\n| (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$$>\n abs (M.dest (P F) x) =\n (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$$>\n abs (M.dest (P F) y)\n[PROOFSTEP]\nlhs\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx y : M (P F) \u03b1\nr : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\nthis : \u2200 (x y : M (P F) \u03b1), r x y \u2192 Mcongr x y\n| (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$$>\n abs (M.dest (P F) x)\n[PROOFSTEP]\nrw [appendFun_comp_id, comp_map, \u2190 abs_map, pr rxy, abs_map, \u2190 comp_map, \u2190 appendFun_comp_id]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n\u22a2 dest (corec g x) = (TypeVec.id ::: corec g) <$$> g x\n[PROOFSTEP]\nconv =>\n lhs\n rw [Cofix.dest, Cofix.corec];\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n| dest (corec g x) = (TypeVec.id ::: corec g) <$$> g x\n[PROOFSTEP]\n lhs\n rw [Cofix.dest, Cofix.corec];\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n| dest (corec g x) = (TypeVec.id ::: corec g) <$$> g x\n[PROOFSTEP]\n lhs\n rw [Cofix.dest, Cofix.corec];\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n| dest (corec g x) = (TypeVec.id ::: corec g) <$$> g x\n[PROOFSTEP]\nlhs\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n| dest (corec g x)\n[PROOFSTEP]\nrw [Cofix.dest, Cofix.corec]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n\u22a2 Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n (_ :\n \u2200 (x y : M (P F) \u03b1),\n Mcongr x y \u2192\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n (Quot.mk Mcongr (corecF g x)) =\n (TypeVec.id ::: corec g) <$$> g x\n[PROOFSTEP]\ndsimp\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n\u22a2 (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) (corecF g x)) = (TypeVec.id ::: corec g) <$$> g x\n[PROOFSTEP]\nrw [corecF_eq, abs_map, abs_repr, \u2190 comp_map, \u2190 appendFun_comp]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: \u03b2)\nx : \u03b2\n\u22a2 (TypeVec.id \u229a TypeVec.id ::: Quot.mk Mcongr \u2218 corecF g) <$$> g x = (TypeVec.id ::: corec g) <$$> g x\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\n\u22a2 \u2200 (x y : Cofix F \u03b1), r x y \u2192 x = y\n[PROOFSTEP]\nintro x\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx : Cofix F \u03b1\n\u22a2 \u2200 (y : Cofix F \u03b1), r x y \u2192 x = y\n[PROOFSTEP]\nrcases x\n[GOAL]\ncase mk\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx : Cofix F \u03b1\na\u271d : M (P F) \u03b1\n\u22a2 \u2200 (y : Cofix F \u03b1), r (Quot.mk Mcongr a\u271d) y \u2192 Quot.mk Mcongr a\u271d = y\n[PROOFSTEP]\nclear x\n[GOAL]\ncase mk\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\na\u271d : M (P F) \u03b1\n\u22a2 \u2200 (y : Cofix F \u03b1), r (Quot.mk Mcongr a\u271d) y \u2192 Quot.mk Mcongr a\u271d = y\n[PROOFSTEP]\nrename M (P F) \u03b1 => x\n[GOAL]\ncase mk\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx : M (P F) \u03b1\n\u22a2 \u2200 (y : Cofix F \u03b1), r (Quot.mk Mcongr x) y \u2192 Quot.mk Mcongr x = y\n[PROOFSTEP]\nintro y\n[GOAL]\ncase mk\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx : M (P F) \u03b1\ny : Cofix F \u03b1\n\u22a2 r (Quot.mk Mcongr x) y \u2192 Quot.mk Mcongr x = y\n[PROOFSTEP]\nrcases y\n[GOAL]\ncase mk.mk\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx : M (P F) \u03b1\ny : Cofix F \u03b1\na\u271d : M (P F) \u03b1\n\u22a2 r (Quot.mk Mcongr x) (Quot.mk Mcongr a\u271d) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a\u271d\n[PROOFSTEP]\nclear y\n[GOAL]\ncase mk.mk\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx a\u271d : M (P F) \u03b1\n\u22a2 r (Quot.mk Mcongr x) (Quot.mk Mcongr a\u271d) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a\u271d\n[PROOFSTEP]\nrename M (P F) \u03b1 => y\n[GOAL]\ncase mk.mk\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\n\u22a2 r (Quot.mk Mcongr x) (Quot.mk Mcongr y) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr y\n[PROOFSTEP]\nintro rxy\n[GOAL]\ncase mk.mk\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Quot.mk Mcongr x = Quot.mk Mcongr y\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase mk.mk.a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Mcongr x y\n[PROOFSTEP]\nlet r' := fun x y => r (Quot.mk _ x) (Quot.mk _ y)\n[GOAL]\ncase mk.mk.a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Mcongr x y\n[PROOFSTEP]\nhave hr' : r' = fun x y => r (Quot.mk _ x) (Quot.mk _ y) := by rfl\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.mk.a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Mcongr x y\n[PROOFSTEP]\nhave : IsPrecongr r' := by\n intro a b r'ab\n have h\u2080 :\n appendFun id (Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest q.P a) =\n appendFun id (Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest q.P b) :=\n by rw [appendFun_comp_id, comp_map, comp_map]; exact h _ _ r'ab\n have h\u2081 : \u2200 u v : q.P.M \u03b1, Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v :=\n by\n intro u v cuv\n apply Quot.sound\n dsimp [hr']\n rw [Quot.sound cuv]\n apply h'\n let f : Quot r \u2192 Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n (by\n intro c\n apply\n Quot.inductionOn (motive := fun c => \u2200 b, r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b) c\n clear c\n intro c d\n apply\n Quot.inductionOn (motive := fun d =>\n r (Quot.mk Mcongr c) d \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 d) d\n clear d\n intro d rcd; apply Quot.sound; apply rcd)\n have : f \u2218 Quot.mk r \u2218 Quot.mk Mcongr = Quot.mk r' := rfl\n rw [\u2190 this, appendFun_comp_id, q.P.comp_map, q.P.comp_map, abs_map, abs_map, abs_map, abs_map, h\u2080]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 IsPrecongr r'\n[PROOFSTEP]\nintro a b r'ab\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\n\u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =\n MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b)\n[PROOFSTEP]\nhave h\u2080 :\n appendFun id (Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest q.P a) =\n appendFun id (Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest q.P b) :=\n by rw [appendFun_comp_id, comp_map, comp_map]; exact h _ _ r'ab\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\n\u22a2 (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\n[PROOFSTEP]\nrw [appendFun_comp_id, comp_map, comp_map]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\n\u22a2 (TypeVec.id ::: Quot.mk r) <$$> (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r) <$$> (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\n[PROOFSTEP]\nexact h _ _ r'ab\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\n\u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =\n MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b)\n[PROOFSTEP]\nhave h\u2081 : \u2200 u v : q.P.M \u03b1, Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v :=\n by\n intro u v cuv\n apply Quot.sound\n dsimp [hr']\n rw [Quot.sound cuv]\n apply h'\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\n\u22a2 \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n[PROOFSTEP]\nintro u v cuv\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nu v : M (P F) \u03b1\ncuv : Mcongr u v\n\u22a2 Quot.mk r' u = Quot.mk r' v\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nu v : M (P F) \u03b1\ncuv : Mcongr u v\n\u22a2 r' u v\n[PROOFSTEP]\ndsimp [hr']\n[GOAL]\ncase a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nu v : M (P F) \u03b1\ncuv : Mcongr u v\n\u22a2 r (Quot.mk Mcongr u) (Quot.mk Mcongr v)\n[PROOFSTEP]\nrw [Quot.sound cuv]\n[GOAL]\ncase a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nu v : M (P F) \u03b1\ncuv : Mcongr u v\n\u22a2 r (Quot.mk Mcongr v) (Quot.mk Mcongr v)\n[PROOFSTEP]\napply h'\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =\n MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b)\n[PROOFSTEP]\nlet f : Quot r \u2192 Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n (by\n intro c\n apply\n Quot.inductionOn (motive := fun c => \u2200 b, r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b) c\n clear c\n intro c d\n apply\n Quot.inductionOn (motive := fun d =>\n r (Quot.mk Mcongr c) d \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 d) d\n clear d\n intro d rcd; apply Quot.sound; apply rcd)\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 \u2200 (a b : Cofix F \u03b1), r a b \u2192 Quot.lift (Quot.mk r') h\u2081 a = Quot.lift (Quot.mk r') h\u2081 b\n[PROOFSTEP]\nintro c\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : Cofix F \u03b1\n\u22a2 \u2200 (b : Cofix F \u03b1), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b\n[PROOFSTEP]\napply Quot.inductionOn (motive := fun c => \u2200 b, r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b) c\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : Cofix F \u03b1\n\u22a2 \u2200 (a : M (P F) \u03b1) (b : Cofix F \u03b1),\n r (Quot.mk Mcongr a) b \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h\u2081 b\n[PROOFSTEP]\nclear c\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 \u2200 (a : M (P F) \u03b1) (b : Cofix F \u03b1),\n r (Quot.mk Mcongr a) b \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h\u2081 b\n[PROOFSTEP]\nintro c d\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : M (P F) \u03b1\nd : Cofix F \u03b1\n\u22a2 r (Quot.mk Mcongr c) d \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 d\n[PROOFSTEP]\napply\n Quot.inductionOn (motive := fun d =>\n r (Quot.mk Mcongr c) d \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 d) d\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : M (P F) \u03b1\nd : Cofix F \u03b1\n\u22a2 \u2200 (a : M (P F) \u03b1),\n r (Quot.mk Mcongr c) (Quot.mk Mcongr a) \u2192\n Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a)\n[PROOFSTEP]\nclear d\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : M (P F) \u03b1\n\u22a2 \u2200 (a : M (P F) \u03b1),\n r (Quot.mk Mcongr c) (Quot.mk Mcongr a) \u2192\n Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a)\n[PROOFSTEP]\nintro d rcd\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc d : M (P F) \u03b1\nrcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)\n\u22a2 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr d)\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc d : M (P F) \u03b1\nrcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)\n\u22a2 r' c d\n[PROOFSTEP]\napply rcd\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nf : Quot r \u2192 Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n (_ : \u2200 (c b : Cofix F \u03b1), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b)\n\u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =\n MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b)\n[PROOFSTEP]\nhave : f \u2218 Quot.mk r \u2218 Quot.mk Mcongr = Quot.mk r' := rfl\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : M (P F) \u03b1\nr'ab : r' a b\nh\u2080 :\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =\n (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b)\nh\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nf : Quot r \u2192 Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n (_ : \u2200 (c b : Cofix F \u03b1), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b)\nthis : f \u2218 Quot.mk r \u2218 Quot.mk Mcongr = Quot.mk r'\n\u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =\n MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b)\n[PROOFSTEP]\nrw [\u2190 this, appendFun_comp_id, q.P.comp_map, q.P.comp_map, abs_map, abs_map, abs_map, abs_map, h\u2080]\n[GOAL]\ncase mk.mk.a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh' : \u2200 (x : Cofix F \u03b1), r x x\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nx y : M (P F) \u03b1\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nhr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nthis : IsPrecongr r'\n\u22a2 Mcongr x y\n[PROOFSTEP]\nrefine' \u27e8r', this, rxy\u27e9\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\n\u22a2 \u2200 (x y : Cofix F \u03b1), r x y \u2192 x = y\n[PROOFSTEP]\nlet r' (x y) := x = y \u2228 r x y\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\n\u22a2 \u2200 (x y : Cofix F \u03b1), r x y \u2192 x = y\n[PROOFSTEP]\nintro x y rxy\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 x = y\n[PROOFSTEP]\napply Cofix.bisim_aux r'\n[GOAL]\ncase h'\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 \u2200 (x : Cofix F \u03b1), r' x x\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h'\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y : Cofix F \u03b1\nrxy : r x\u271d y\nx : Cofix F \u03b1\n\u22a2 r' x x\n[PROOFSTEP]\nleft\n[GOAL]\ncase h'.h\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y : Cofix F \u03b1\nrxy : r x\u271d y\nx : Cofix F \u03b1\n\u22a2 x = x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 \u2200 (x y : Cofix F \u03b1), r' x y \u2192 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y\n[PROOFSTEP]\nintro x y r'xy\n[GOAL]\ncase h\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r' x y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y\n[PROOFSTEP]\ncases r'xy\n[GOAL]\ncase h.inl\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nh\u271d : x = y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y\ncase h.inr\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nh\u271d : r x y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y\n[PROOFSTEP]\ncase inl h => rw [h]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh\u271d : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nh : x = y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y\n[PROOFSTEP]\ncase inl h => rw [h]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh\u271d : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nh : x = y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase h.inr\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nh\u271d : r x y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y\n[PROOFSTEP]\ncase inr r'xy =>\n have : \u2200 x y, r x y \u2192 r' x y := fun x y h => Or.inr h\n rw [\u2190 Quot.factor_mk_eq _ _ this]\n dsimp\n rw [appendFun_comp_id]\n rw [@comp_map _ _ _ q _ _ _ (appendFun id (Quot.mk r)), @comp_map _ _ _ q _ _ _ (appendFun id (Quot.mk r))]\n rw [h _ _ r'xy]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y\n[PROOFSTEP]\ncase inr r'xy =>\n have : \u2200 x y, r x y \u2192 r' x y := fun x y h => Or.inr h\n rw [\u2190 Quot.factor_mk_eq _ _ this]\n dsimp\n rw [appendFun_comp_id]\n rw [@comp_map _ _ _ q _ _ _ (appendFun id (Quot.mk r)), @comp_map _ _ _ q _ _ _ (appendFun id (Quot.mk r))]\n rw [h _ _ r'xy]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y\n[PROOFSTEP]\nhave : \u2200 x y, r x y \u2192 r' x y := fun x y h => Or.inr h\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\nthis : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y\n\u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y\n[PROOFSTEP]\nrw [\u2190 Quot.factor_mk_eq _ _ this]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\nthis : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y\n\u22a2 (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$$> dest x =\n (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$$> dest y\n[PROOFSTEP]\ndsimp\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\nthis : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y\n\u22a2 (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this \u2218 Quot.mk fun x y => r x y) <$$>\n dest x =\n (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this \u2218 Quot.mk fun x y => r x y) <$$>\n dest y\n[PROOFSTEP]\nrw [appendFun_comp_id]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\nthis : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y\n\u22a2 ((TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) \u229a\n (TypeVec.id ::: Quot.mk fun x y => r x y)) <$$>\n dest x =\n ((TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) \u229a\n (TypeVec.id ::: Quot.mk fun x y => r x y)) <$$>\n dest y\n[PROOFSTEP]\nrw [@comp_map _ _ _ q _ _ _ (appendFun id (Quot.mk r)), @comp_map _ _ _ q _ _ _ (appendFun id (Quot.mk r))]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F \u03b1\nrxy : r x\u271d y\u271d\nx y : Cofix F \u03b1\nr'xy : r x y\nthis : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y\n\u22a2 (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) <$$>\n (TypeVec.id ::: Quot.mk r) <$$> dest x =\n (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) <$$>\n (TypeVec.id ::: Quot.mk r) <$$> dest y\n[PROOFSTEP]\nrw [h _ _ r'xy]\n[GOAL]\ncase a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 r' x y\n[PROOFSTEP]\nright\n[GOAL]\ncase a.h\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y\nr' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 r x y\n[PROOFSTEP]\nexact rxy\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\n\u22a2 \u2200 (x y : Cofix F \u03b1), r x y \u2192 x = y\n[PROOFSTEP]\napply Cofix.bisim_rel\n[GOAL]\ncase h\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\n\u22a2 \u2200 (x y : Cofix F \u03b1),\n r x y \u2192\n (TypeVec.id ::: Quot.mk fun x y => r x y) <$$> dest x = (TypeVec.id ::: Quot.mk fun x y => r x y) <$$> dest y\n[PROOFSTEP]\nintro x y rxy\n[GOAL]\ncase h\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\nx y : Cofix F \u03b1\nrxy : r x y\n\u22a2 (TypeVec.id ::: Quot.mk fun x y => r x y) <$$> dest x = (TypeVec.id ::: Quot.mk fun x y => r x y) <$$> dest y\n[PROOFSTEP]\nrcases(liftR_iff (fun a b => RelLast \u03b1 r a b) (dest x) (dest y)).mp (h x y rxy) with \u27e8a, f\u2080, f\u2081, dxeq, dyeq, h'\u27e9\n[GOAL]\ncase h.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\nx y : Cofix F \u03b1\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : B (P F) a \u27f9 fun {i} => (\u03b1 ::: Cofix F \u03b1) i\ndxeq : dest x = MvQPF.abs { fst := a, snd := f\u2080 }\ndyeq : dest y = MvQPF.abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : Fin2 (n + 1)) (j : B (P F) a i), RelLast \u03b1 r (f\u2080 i j) (f\u2081 i j)\n\u22a2 (TypeVec.id ::: Quot.mk fun x y => r x y) <$$> dest x = (TypeVec.id ::: Quot.mk fun x y => r x y) <$$> dest y\n[PROOFSTEP]\nrw [dxeq, dyeq, \u2190 abs_map, \u2190 abs_map, MvPFunctor.map_eq, MvPFunctor.map_eq]\n[GOAL]\ncase h.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\nx y : Cofix F \u03b1\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : B (P F) a \u27f9 fun {i} => (\u03b1 ::: Cofix F \u03b1) i\ndxeq : dest x = MvQPF.abs { fst := a, snd := f\u2080 }\ndyeq : dest y = MvQPF.abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : Fin2 (n + 1)) (j : B (P F) a i), RelLast \u03b1 r (f\u2080 i j) (f\u2081 i j)\n\u22a2 MvQPF.abs { fst := a, snd := (TypeVec.id ::: Quot.mk fun x y => r x y) \u229a f\u2080 } =\n MvQPF.abs { fst := a, snd := (TypeVec.id ::: Quot.mk fun x y => r x y) \u229a f\u2081 }\n[PROOFSTEP]\nrw [\u2190 split_dropFun_lastFun f\u2080, \u2190 split_dropFun_lastFun f\u2081]\n[GOAL]\ncase h.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\nx y : Cofix F \u03b1\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : B (P F) a \u27f9 fun {i} => (\u03b1 ::: Cofix F \u03b1) i\ndxeq : dest x = MvQPF.abs { fst := a, snd := f\u2080 }\ndyeq : dest y = MvQPF.abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : Fin2 (n + 1)) (j : B (P F) a i), RelLast \u03b1 r (f\u2080 i j) (f\u2081 i j)\n\u22a2 MvQPF.abs { fst := a, snd := (TypeVec.id ::: Quot.mk fun x y => r x y) \u229a splitFun (dropFun f\u2080) (lastFun f\u2080) } =\n MvQPF.abs { fst := a, snd := (TypeVec.id ::: Quot.mk fun x y => r x y) \u229a splitFun (dropFun f\u2081) (lastFun f\u2081) }\n[PROOFSTEP]\nrw [appendFun_comp_splitFun, appendFun_comp_splitFun]\n[GOAL]\ncase h.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\nx y : Cofix F \u03b1\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : B (P F) a \u27f9 fun {i} => (\u03b1 ::: Cofix F \u03b1) i\ndxeq : dest x = MvQPF.abs { fst := a, snd := f\u2080 }\ndyeq : dest y = MvQPF.abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : Fin2 (n + 1)) (j : B (P F) a i), RelLast \u03b1 r (f\u2080 i j) (f\u2081 i j)\n\u22a2 MvQPF.abs { fst := a, snd := splitFun (TypeVec.id \u229a dropFun f\u2080) ((Quot.mk fun x y => r x y) \u2218 lastFun f\u2080) } =\n MvQPF.abs { fst := a, snd := splitFun (TypeVec.id \u229a dropFun f\u2081) ((Quot.mk fun x y => r x y) \u2218 lastFun f\u2081) }\n[PROOFSTEP]\nrw [id_comp, id_comp]\n[GOAL]\ncase h.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\nx y : Cofix F \u03b1\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : B (P F) a \u27f9 fun {i} => (\u03b1 ::: Cofix F \u03b1) i\ndxeq : dest x = MvQPF.abs { fst := a, snd := f\u2080 }\ndyeq : dest y = MvQPF.abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : Fin2 (n + 1)) (j : B (P F) a i), RelLast \u03b1 r (f\u2080 i j) (f\u2081 i j)\n\u22a2 MvQPF.abs { fst := a, snd := splitFun (dropFun f\u2080) ((Quot.mk fun x y => r x y) \u2218 lastFun f\u2080) } =\n MvQPF.abs { fst := a, snd := splitFun (dropFun f\u2081) ((Quot.mk fun x y => r x y) \u2218 lastFun f\u2081) }\n[PROOFSTEP]\ncongr 2 with (i j)\n[GOAL]\ncase h.intro.intro.intro.intro.intro.e_a.e_snd.a.h\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\nx y : Cofix F \u03b1\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : B (P F) a \u27f9 fun {i} => (\u03b1 ::: Cofix F \u03b1) i\ndxeq : dest x = MvQPF.abs { fst := a, snd := f\u2080 }\ndyeq : dest y = MvQPF.abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : Fin2 (n + 1)) (j : B (P F) a i), RelLast \u03b1 r (f\u2080 i j) (f\u2081 i j)\ni : Fin2 (n + 1)\nj : B (P F) a i\n\u22a2 splitFun (dropFun f\u2080) ((Quot.mk fun x y => r x y) \u2218 lastFun f\u2080) i j =\n splitFun (dropFun f\u2081) ((Quot.mk fun x y => r x y) \u2218 lastFun f\u2081) i j\n[PROOFSTEP]\ncases' i with _ i\n[GOAL]\ncase h.intro.intro.intro.intro.intro.e_a.e_snd.a.h.fz\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\nx y : Cofix F \u03b1\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : B (P F) a \u27f9 fun {i} => (\u03b1 ::: Cofix F \u03b1) i\ndxeq : dest x = MvQPF.abs { fst := a, snd := f\u2080 }\ndyeq : dest y = MvQPF.abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : Fin2 (n + 1)) (j : B (P F) a i), RelLast \u03b1 r (f\u2080 i j) (f\u2081 i j)\nj : B (P F) a Fin2.fz\n\u22a2 splitFun (dropFun f\u2080) ((Quot.mk fun x y => r x y) \u2218 lastFun f\u2080) Fin2.fz j =\n splitFun (dropFun f\u2081) ((Quot.mk fun x y => r x y) \u2218 lastFun f\u2081) Fin2.fz j\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.intro.intro.intro.intro.intro.e_a.e_snd.a.h.fs\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\nx y : Cofix F \u03b1\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : B (P F) a \u27f9 fun {i} => (\u03b1 ::: Cofix F \u03b1) i\ndxeq : dest x = MvQPF.abs { fst := a, snd := f\u2080 }\ndyeq : dest y = MvQPF.abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : Fin2 (n + 1)) (j : B (P F) a i), RelLast \u03b1 r (f\u2080 i j) (f\u2081 i j)\na\u271d : Fin2 (Nat.add n 0)\nj : B (P F) a (Fin2.fs a\u271d)\n\u22a2 splitFun (dropFun f\u2080) ((Quot.mk fun x y => r x y) \u2218 lastFun f\u2080) (Fin2.fs a\u271d) j =\n splitFun (dropFun f\u2081) ((Quot.mk fun x y => r x y) \u2218 lastFun f\u2081) (Fin2.fs a\u271d) j\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.intro.intro.intro.intro.intro.e_a.e_snd.a.h.fz\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\nx y : Cofix F \u03b1\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : B (P F) a \u27f9 fun {i} => (\u03b1 ::: Cofix F \u03b1) i\ndxeq : dest x = MvQPF.abs { fst := a, snd := f\u2080 }\ndyeq : dest y = MvQPF.abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : Fin2 (n + 1)) (j : B (P F) a i), RelLast \u03b1 r (f\u2080 i j) (f\u2081 i j)\nj : B (P F) a Fin2.fz\n\u22a2 splitFun (dropFun f\u2080) ((Quot.mk fun x y => r x y) \u2218 lastFun f\u2080) Fin2.fz j =\n splitFun (dropFun f\u2081) ((Quot.mk fun x y => r x y) \u2218 lastFun f\u2081) Fin2.fz j\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase h.intro.intro.intro.intro.intro.e_a.e_snd.a.h.fz.a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\nx y : Cofix F \u03b1\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : B (P F) a \u27f9 fun {i} => (\u03b1 ::: Cofix F \u03b1) i\ndxeq : dest x = MvQPF.abs { fst := a, snd := f\u2080 }\ndyeq : dest y = MvQPF.abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : Fin2 (n + 1)) (j : B (P F) a i), RelLast \u03b1 r (f\u2080 i j) (f\u2081 i j)\nj : B (P F) a Fin2.fz\n\u22a2 r (lastFun f\u2080 j) (lastFun f\u2081 j)\n[PROOFSTEP]\napply h' _ j\n[GOAL]\ncase h.intro.intro.intro.intro.intro.e_a.e_snd.a.h.fs\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\nx y : Cofix F \u03b1\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : B (P F) a \u27f9 fun {i} => (\u03b1 ::: Cofix F \u03b1) i\ndxeq : dest x = MvQPF.abs { fst := a, snd := f\u2080 }\ndyeq : dest y = MvQPF.abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : Fin2 (n + 1)) (j : B (P F) a i), RelLast \u03b1 r (f\u2080 i j) (f\u2081 i j)\na\u271d : Fin2 (Nat.add n 0)\nj : B (P F) a (Fin2.fs a\u271d)\n\u22a2 splitFun (dropFun f\u2080) ((Quot.mk fun x y => r x y) \u2218 lastFun f\u2080) (Fin2.fs a\u271d) j =\n splitFun (dropFun f\u2081) ((Quot.mk fun x y => r x y) \u2218 lastFun f\u2081) (Fin2.fs a\u271d) j\n[PROOFSTEP]\nchange f\u2080 _ j = f\u2081 _ j\n[GOAL]\ncase h.intro.intro.intro.intro.intro.e_a.e_snd.a.h.fs\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\nx y : Cofix F \u03b1\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : B (P F) a \u27f9 fun {i} => (\u03b1 ::: Cofix F \u03b1) i\ndxeq : dest x = MvQPF.abs { fst := a, snd := f\u2080 }\ndyeq : dest y = MvQPF.abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : Fin2 (n + 1)) (j : B (P F) a i), RelLast \u03b1 r (f\u2080 i j) (f\u2081 i j)\na\u271d : Fin2 (Nat.add n 0)\nj : B (P F) a (Fin2.fs a\u271d)\n\u22a2 f\u2080 (Fin2.fs a\u271d) j = f\u2081 (Fin2.fs a\u271d) j\n[PROOFSTEP]\napply h' _ j\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR' (RelLast' \u03b1 r) (dest x) (dest y)\n\u22a2 \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y)\n[PROOFSTEP]\nintros\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR' (RelLast' \u03b1 r) (dest x) (dest y)\nx\u271d y\u271d : Cofix F \u03b1\na\u271d : r x\u271d y\u271d\n\u22a2 LiftR (fun {i} => RelLast \u03b1 r) (dest x\u271d) (dest y\u271d)\n[PROOFSTEP]\nrw [\u2190 LiftR_RelLast_iff]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR' (RelLast' \u03b1 r) (dest x) (dest y)\nx\u271d y\u271d : Cofix F \u03b1\na\u271d : r x\u271d y\u271d\n\u22a2 LiftR' (RelLast' \u03b1 r) (dest x\u271d) (dest y\u271d)\n[PROOFSTEP]\napply h\n[GOAL]\ncase a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nr : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop\nh : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR' (RelLast' \u03b1 r) (dest x) (dest y)\nx\u271d y\u271d : Cofix F \u03b1\na\u271d : r x\u271d y\u271d\n\u22a2 r x\u271d y\u271d\n[PROOFSTEP]\nassumption\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u_1\nQ : \u03b2 \u2192 Prop\nu v : \u03b2 \u2192 Cofix F \u03b1\nh :\n \u2200 (x : \u03b2),\n Q x \u2192\n \u2203 a f' f\u2080 f\u2081,\n dest (u x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2080 } \u2227\n dest (v x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2081 } \u2227\n \u2200 (i : PFunctor.B (MvPFunctor.last (P F)) a), \u2203 x', Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x'\nx\u271d\u00b9 : \u03b2\nQx : Q x\u271d\u00b9\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F \u03b1\nx\u271d : R x y\nx' : \u03b2\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\n\u22a2 LiftR (fun {i} => RelLast \u03b1 R) (dest x) (dest y)\n[PROOFSTEP]\nrcases h x' Qx' with \u27e8a, f', f\u2080, f\u2081, ux'eq, vx'eq, h'\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u_1\nQ : \u03b2 \u2192 Prop\nu v : \u03b2 \u2192 Cofix F \u03b1\nh :\n \u2200 (x : \u03b2),\n Q x \u2192\n \u2203 a f' f\u2080 f\u2081,\n dest (u x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2080 } \u2227\n dest (v x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2081 } \u2227\n \u2200 (i : PFunctor.B (MvPFunctor.last (P F)) a), \u2203 x', Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x'\nx\u271d\u00b9 : \u03b2\nQx : Q x\u271d\u00b9\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F \u03b1\nx\u271d : R x y\nx' : \u03b2\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf' : B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 Cofix F \u03b1\nux'eq : dest (u x') = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2080 }\nvx'eq : dest (v x') = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2081 }\nh' : \u2200 (i : PFunctor.B (MvPFunctor.last (P F)) a), \u2203 x', Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x'\n\u22a2 LiftR (fun {i} => RelLast \u03b1 R) (dest x) (dest y)\n[PROOFSTEP]\nrw [liftR_iff]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u_1\nQ : \u03b2 \u2192 Prop\nu v : \u03b2 \u2192 Cofix F \u03b1\nh :\n \u2200 (x : \u03b2),\n Q x \u2192\n \u2203 a f' f\u2080 f\u2081,\n dest (u x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2080 } \u2227\n dest (v x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2081 } \u2227\n \u2200 (i : PFunctor.B (MvPFunctor.last (P F)) a), \u2203 x', Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x'\nx\u271d\u00b9 : \u03b2\nQx : Q x\u271d\u00b9\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F \u03b1\nx\u271d : R x y\nx' : \u03b2\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf' : B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 Cofix F \u03b1\nux'eq : dest (u x') = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2080 }\nvx'eq : dest (v x') = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2081 }\nh' : \u2200 (i : PFunctor.B (MvPFunctor.last (P F)) a), \u2203 x', Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x'\n\u22a2 \u2203 a f\u2080 f\u2081,\n dest x = MvQPF.abs { fst := a, snd := f\u2080 } \u2227\n dest y = MvQPF.abs { fst := a, snd := f\u2081 } \u2227 \u2200 (i : Fin2 (n + 1)) (j : B (P F) a i), RelLast \u03b1 R (f\u2080 i j) (f\u2081 i j)\n[PROOFSTEP]\nrefine' \u27e8a, q.P.appendContents f' f\u2080, q.P.appendContents f' f\u2081, xeq.symm \u25b8 ux'eq, yeq.symm \u25b8 vx'eq, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u_1\nQ : \u03b2 \u2192 Prop\nu v : \u03b2 \u2192 Cofix F \u03b1\nh :\n \u2200 (x : \u03b2),\n Q x \u2192\n \u2203 a f' f\u2080 f\u2081,\n dest (u x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2080 } \u2227\n dest (v x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2081 } \u2227\n \u2200 (i : PFunctor.B (MvPFunctor.last (P F)) a), \u2203 x', Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x'\nx\u271d\u00b9 : \u03b2\nQx : Q x\u271d\u00b9\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F \u03b1\nx\u271d : R x y\nx' : \u03b2\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf' : B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 Cofix F \u03b1\nux'eq : dest (u x') = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2080 }\nvx'eq : dest (v x') = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2081 }\nh' : \u2200 (i : PFunctor.B (MvPFunctor.last (P F)) a), \u2203 x', Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x'\n\u22a2 \u2200 (i : Fin2 (n + 1)) (j : B (P F) a i), RelLast \u03b1 R (appendContents (P F) f' f\u2080 i j) (appendContents (P F) f' f\u2081 i j)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u_1\nQ : \u03b2 \u2192 Prop\nu v : \u03b2 \u2192 Cofix F \u03b1\nh :\n \u2200 (x : \u03b2),\n Q x \u2192\n \u2203 a f' f\u2080 f\u2081,\n dest (u x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2080 } \u2227\n dest (v x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2081 } \u2227\n \u2200 (i : PFunctor.B (MvPFunctor.last (P F)) a), \u2203 x', Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x'\nx\u271d\u00b9 : \u03b2\nQx : Q x\u271d\u00b9\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F \u03b1\nx\u271d : R x y\nx' : \u03b2\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf' : B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 Cofix F \u03b1\nux'eq : dest (u x') = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2080 }\nvx'eq : dest (v x') = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2081 }\nh' : \u2200 (i : PFunctor.B (MvPFunctor.last (P F)) a), \u2203 x', Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x'\ni : Fin2 (n + 1)\n\u22a2 \u2200 (j : B (P F) a i), RelLast \u03b1 R (appendContents (P F) f' f\u2080 i j) (appendContents (P F) f' f\u2081 i j)\n[PROOFSTEP]\ncases i\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.fz\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u_1\nQ : \u03b2 \u2192 Prop\nu v : \u03b2 \u2192 Cofix F \u03b1\nh :\n \u2200 (x : \u03b2),\n Q x \u2192\n \u2203 a f' f\u2080 f\u2081,\n dest (u x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2080 } \u2227\n dest (v x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2081 } \u2227\n \u2200 (i : PFunctor.B (MvPFunctor.last (P F)) a), \u2203 x', Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x'\nx\u271d\u00b9 : \u03b2\nQx : Q x\u271d\u00b9\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F \u03b1\nx\u271d : R x y\nx' : \u03b2\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf' : B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 Cofix F \u03b1\nux'eq : dest (u x') = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2080 }\nvx'eq : dest (v x') = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2081 }\nh' : \u2200 (i : PFunctor.B (MvPFunctor.last (P F)) a), \u2203 x', Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x'\n\u22a2 \u2200 (j : B (P F) a Fin2.fz), RelLast \u03b1 R (appendContents (P F) f' f\u2080 Fin2.fz j) (appendContents (P F) f' f\u2081 Fin2.fz j)\n[PROOFSTEP]\napply h'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.fs\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u_1\nQ : \u03b2 \u2192 Prop\nu v : \u03b2 \u2192 Cofix F \u03b1\nh :\n \u2200 (x : \u03b2),\n Q x \u2192\n \u2203 a f' f\u2080 f\u2081,\n dest (u x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2080 } \u2227\n dest (v x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2081 } \u2227\n \u2200 (i : PFunctor.B (MvPFunctor.last (P F)) a), \u2203 x', Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x'\nx\u271d\u00b9 : \u03b2\nQx : Q x\u271d\u00b9\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F \u03b1\nx\u271d : R x y\nx' : \u03b2\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf' : B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 Cofix F \u03b1\nux'eq : dest (u x') = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2080 }\nvx'eq : dest (v x') = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2081 }\nh' : \u2200 (i : PFunctor.B (MvPFunctor.last (P F)) a), \u2203 x', Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x'\na\u271d : Fin2 (Nat.add n 0)\n\u22a2 \u2200 (j : B (P F) a (Fin2.fs a\u271d)),\n RelLast \u03b1 R (appendContents (P F) f' f\u2080 (Fin2.fs a\u271d) j) (appendContents (P F) f' f\u2081 (Fin2.fs a\u271d) j)\n[PROOFSTEP]\nintro j\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.fs\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u_1\nQ : \u03b2 \u2192 Prop\nu v : \u03b2 \u2192 Cofix F \u03b1\nh :\n \u2200 (x : \u03b2),\n Q x \u2192\n \u2203 a f' f\u2080 f\u2081,\n dest (u x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2080 } \u2227\n dest (v x) = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2081 } \u2227\n \u2200 (i : PFunctor.B (MvPFunctor.last (P F)) a), \u2203 x', Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x'\nx\u271d\u00b9 : \u03b2\nQx : Q x\u271d\u00b9\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F \u03b1\nx\u271d : R x y\nx' : \u03b2\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf' : B (MvPFunctor.drop (P F)) a \u27f9 \u03b1\nf\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 Cofix F \u03b1\nux'eq : dest (u x') = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2080 }\nvx'eq : dest (v x') = MvQPF.abs { fst := a, snd := appendContents (P F) f' f\u2081 }\nh' : \u2200 (i : PFunctor.B (MvPFunctor.last (P F)) a), \u2203 x', Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x'\na\u271d : Fin2 (Nat.add n 0)\nj : B (P F) a (Fin2.fs a\u271d)\n\u22a2 RelLast \u03b1 R (appendContents (P F) f' f\u2080 (Fin2.fs a\u271d) j) (appendContents (P F) f' f\u2081 (Fin2.fs a\u271d) j)\n[PROOFSTEP]\napply Eq.refl\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx : Cofix F \u03b1\n\u22a2 mk (dest x) = x\n[PROOFSTEP]\napply Cofix.bisim_rel (fun x y : Cofix F \u03b1 => x = Cofix.mk (Cofix.dest y)) _ _ _ rfl\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx : Cofix F \u03b1\n\u22a2 \u2200 (x y : Cofix F \u03b1),\n (fun x y => x = mk (dest y)) x y \u2192\n (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest x =\n (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest y\n[PROOFSTEP]\ndsimp\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx : Cofix F \u03b1\n\u22a2 \u2200 (x y : Cofix F \u03b1),\n x = mk (dest y) \u2192\n (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest x =\n (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest y\n[PROOFSTEP]\nintro x y h\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d x y : Cofix F \u03b1\nh : x = mk (dest y)\n\u22a2 (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest x =\n (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest y\n[PROOFSTEP]\nrw [h]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d x y : Cofix F \u03b1\nh : x = mk (dest y)\n\u22a2 (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest (mk (dest y)) =\n (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest y\n[PROOFSTEP]\nconv =>\n lhs\n congr\n rfl\n rw [Cofix.mk]\n rw [Cofix.dest_corec]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d x y : Cofix F \u03b1\nh : x = mk (dest y)\n| (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest (mk (dest y)) =\n (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest y\n[PROOFSTEP]\n lhs\n congr\n rfl\n rw [Cofix.mk]\n rw [Cofix.dest_corec]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d x y : Cofix F \u03b1\nh : x = mk (dest y)\n| (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest (mk (dest y)) =\n (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest y\n[PROOFSTEP]\n lhs\n congr\n rfl\n rw [Cofix.mk]\n rw [Cofix.dest_corec]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d x y : Cofix F \u03b1\nh : x = mk (dest y)\n| (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest (mk (dest y)) =\n (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest y\n[PROOFSTEP]\nlhs\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d x y : Cofix F \u03b1\nh : x = mk (dest y)\n| (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest (mk (dest y))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d x y : Cofix F \u03b1\nh : x = mk (dest y)\n| TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)\ncase a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d x y : Cofix F \u03b1\nh : x = mk (dest y)\n| dest (mk (dest y))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d x y : Cofix F \u03b1\nh : x = mk (dest y)\n| dest (mk (dest y))\n[PROOFSTEP]\nrw [Cofix.mk]\n[GOAL]\ncase a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d x y : Cofix F \u03b1\nh : x = mk (dest y)\n| dest (corec (fun x => (TypeVec.id ::: fun i => dest i) <$$> x) (dest y))\n[PROOFSTEP]\nrw [Cofix.dest_corec]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d x y : Cofix F \u03b1\nh : x = mk (dest y)\n\u22a2 (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$>\n (TypeVec.id ::: corec fun x => (TypeVec.id ::: fun i => dest i) <$$> x) <$$>\n (TypeVec.id ::: fun i => dest i) <$$> dest y =\n (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest y\n[PROOFSTEP]\nrw [\u2190 comp_map, \u2190 appendFun_comp, id_comp]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d x y : Cofix F \u03b1\nh : x = mk (dest y)\n\u22a2 (TypeVec.id ::: (Quot.mk fun x y => x = mk (dest y)) \u2218 corec fun x => (TypeVec.id ::: fun i => dest i) <$$> x) <$$>\n (TypeVec.id ::: fun i => dest i) <$$> dest y =\n (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest y\n[PROOFSTEP]\nrw [\u2190 comp_map, \u2190 appendFun_comp, id_comp, \u2190 Cofix.mk]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d x y : Cofix F \u03b1\nh : x = mk (dest y)\n\u22a2 (TypeVec.id ::: ((Quot.mk fun x y => x = mk (dest y)) \u2218 mk) \u2218 fun i => dest i) <$$> dest y =\n (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y)) <$$> dest y\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d x y : Cofix F \u03b1\nh : x = mk (dest y)\n\u22a2 (TypeVec.id ::: ((Quot.mk fun x y => x = mk (dest y)) \u2218 mk) \u2218 fun i => dest i) =\n (TypeVec.id ::: Quot.mk fun x y => x = mk (dest y))\n[PROOFSTEP]\napply congrArg\n[GOAL]\ncase e_a.h\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d x y : Cofix F \u03b1\nh : x = mk (dest y)\n\u22a2 (((Quot.mk fun x y => x = mk (dest y)) \u2218 mk) \u2218 fun i => dest i) = Quot.mk fun x y => x = mk (dest y)\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase e_a.h.h\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d\u00b9 x\u271d y : Cofix F \u03b1\nh : x\u271d = mk (dest y)\nx : Cofix F \u03b1\n\u22a2 (((Quot.mk fun x y => x = mk (dest y)) \u2218 mk) \u2218 fun i => dest i) x = Quot.mk (fun x y => x = mk (dest y)) x\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase e_a.h.h.a\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx\u271d\u00b9 x\u271d y : Cofix F \u03b1\nh : x\u271d = mk (dest y)\nx : Cofix F \u03b1\n\u22a2 mk ((fun i => dest i) x) = mk (dest x)\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx : F (\u03b1 ::: Cofix F \u03b1)\n\u22a2 dest (mk x) = x\n[PROOFSTEP]\nhave : Cofix.mk \u2218 Cofix.dest = @_root_.id (Cofix F \u03b1) := funext Cofix.mk_dest\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx : F (\u03b1 ::: Cofix F \u03b1)\nthis : mk \u2218 dest = _root_.id\n\u22a2 dest (mk x) = x\n[PROOFSTEP]\nrw [Cofix.mk, Cofix.dest_corec, \u2190 comp_map, \u2190 Cofix.mk, \u2190 appendFun_comp, this, id_comp, appendFun_id_id,\n MvFunctor.id_map]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx y : Cofix F \u03b1\nh : dest x = dest y\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 Cofix.mk_dest x, h, Cofix.mk_dest]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx y : F (\u03b1 ::: Cofix F \u03b1)\nh : mk x = mk y\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 Cofix.dest_mk x, h, Cofix.dest_mk]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\nF' : TypeVec n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 LiftR' R (f <$$> x) (g <$$> x)\n[PROOFSTEP]\nrw [LiftR_def]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\nF' : TypeVec n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 \u2203 u, (prod.fst \u229a subtypeVal R) <$$> u = f <$$> x \u2227 (prod.snd \u229a subtypeVal R) <$$> u = g <$$> x\n[PROOFSTEP]\nexists h <$$> x\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\nF' : TypeVec n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 (prod.fst \u229a subtypeVal R) <$$> h <$$> x = f <$$> x \u2227 (prod.snd \u229a subtypeVal R) <$$> h <$$> x = g <$$> x\n[PROOFSTEP]\nrw [MvFunctor.map_map, comp_assoc, hh, \u2190 comp_assoc, fst_prod_mk, comp_assoc, fst_diag]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\nF' : TypeVec n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 (f \u229a TypeVec.id) <$$> x = f <$$> x \u2227 (prod.snd \u229a subtypeVal R) <$$> h <$$> x = g <$$> x\n[PROOFSTEP]\nrw [MvFunctor.map_map, comp_assoc, hh, \u2190 comp_assoc, snd_prod_mk, comp_assoc, snd_diag]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\nF' : TypeVec n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 (f \u229a TypeVec.id) <$$> x = f <$$> x \u2227 (g \u229a TypeVec.id) <$$> x = g <$$> x\n[PROOFSTEP]\ndsimp [LiftR']\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\nF' : TypeVec n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 f <$$> x = f <$$> x \u2227 g <$$> x = g <$$> x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\nF' : TypeVec n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 f <$$> x = f <$$> x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 \u03b2 : TypeVec n\nF' : TypeVec n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F'\ninst\u271d : LawfulMvFunctor F'\nR : \u03b2 \u2297 \u03b2 \u27f9 repeat n Prop\nx : F' \u03b1\nf g : \u03b1 \u27f9 \u03b2\nh : \u03b1 \u27f9 Subtype_ R\nhh : subtypeVal R \u229a h = (f \u2297' g) \u229a prod.diag\n\u22a2 g <$$> x = g <$$> x\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 subtypeVal (RelLast' \u03b1 R) \u229a toSubtype (RelLast' \u03b1 R) \u229a fromAppend1DropLast \u229a c \u229a b =\n ((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a prod.diag\n[PROOFSTEP]\ndsimp\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 subtypeVal (RelLast' \u03b1 R) \u229a\n toSubtype (RelLast' \u03b1 R) \u229a\n fromAppend1DropLast \u229a\n (ofSubtype (repeatEq \u03b1) ::: _root_.id) \u229a\n (diagSub ::: fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }) =\n ((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a prod.diag\n[PROOFSTEP]\napply eq_of_drop_last_eq\n[GOAL]\ncase h\u2080\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 dropFun\n (subtypeVal (RelLast' \u03b1 R) \u229a\n toSubtype (RelLast' \u03b1 R) \u229a\n fromAppend1DropLast \u229a\n (ofSubtype (repeatEq \u03b1) ::: _root_.id) \u229a\n (diagSub ::: fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) })) =\n dropFun (((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a prod.diag)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2080\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 subtypeVal (repeatEq \u03b1) \u229a dropFun (toSubtype (RelLast' \u03b1 R)) \u229a ofSubtype (repeatEq \u03b1) \u229a diagSub =\n dropFun ((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a dropFun prod.diag\n[PROOFSTEP]\nsimp only [prod_map_id, dropFun_prod, dropFun_appendFun, dropFun_diag, id_comp, dropFun_toSubtype]\n[GOAL]\ncase h\u2080\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 subtypeVal (repeatEq \u03b1) \u229a (toSubtype fun i x => RelLast' \u03b1 R (Fin2.fs i) x) \u229a ofSubtype (repeatEq \u03b1) \u229a diagSub =\n prod.diag\n[PROOFSTEP]\nerw [toSubtype_of_subtype_assoc, id_comp]\n[GOAL]\ncase h\u2080\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 (fun i x => subtypeVal (repeatEq \u03b1) i (diagSub i x)) = prod.diag\n[PROOFSTEP]\nclear liftR_map_last q mvf lawful F x R f g hh h b c\n[GOAL]\ncase h\u2080\nn : \u2115\n\u03b1 : TypeVec n\n\u03b9 \u03b9' : Type u\n\u22a2 (fun i x => subtypeVal (repeatEq \u03b1) i (diagSub i x)) = prod.diag\n[PROOFSTEP]\next (i x) : 2\n[GOAL]\ncase h\u2080.a.h\nn : \u2115\n\u03b1 : TypeVec n\n\u03b9 \u03b9' : Type u\ni : Fin2 n\nx : TypeVec.drop (\u03b1 ::: \u03b9) i\n\u22a2 subtypeVal (repeatEq \u03b1) i (diagSub i x) = prod.diag i x\n[PROOFSTEP]\ninduction i with\n| fz => rfl\n| fs _ ih => apply ih\n[GOAL]\ncase h\u2080.a.h\nn : \u2115\n\u03b1 : TypeVec n\n\u03b9 \u03b9' : Type u\ni : Fin2 n\nx : TypeVec.drop (\u03b1 ::: \u03b9) i\n\u22a2 subtypeVal (repeatEq \u03b1) i (diagSub i x) = prod.diag i x\n[PROOFSTEP]\ninduction i with\n| fz => rfl\n| fs _ ih => apply ih\n[GOAL]\ncase h\u2080.a.h.fz\nn : \u2115\n\u03b9 \u03b9' : Type u\nn\u271d : \u2115\n\u03b1 : TypeVec (Nat.succ n\u271d)\nx : TypeVec.drop (\u03b1 ::: \u03b9) Fin2.fz\n\u22a2 subtypeVal (repeatEq \u03b1) Fin2.fz (diagSub Fin2.fz x) = prod.diag Fin2.fz x\n[PROOFSTEP]\n\n| fz => rfl\n[GOAL]\ncase h\u2080.a.h.fz\nn : \u2115\n\u03b9 \u03b9' : Type u\nn\u271d : \u2115\n\u03b1 : TypeVec (Nat.succ n\u271d)\nx : TypeVec.drop (\u03b1 ::: \u03b9) Fin2.fz\n\u22a2 subtypeVal (repeatEq \u03b1) Fin2.fz (diagSub Fin2.fz x) = prod.diag Fin2.fz x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2080.a.h.fs\nn : \u2115\n\u03b9 \u03b9' : Type u\nn\u271d : \u2115\na\u271d : Fin2 n\u271d\nih : \u2200 {\u03b1 : TypeVec n\u271d} (x : TypeVec.drop (\u03b1 ::: \u03b9) a\u271d), subtypeVal (repeatEq \u03b1) a\u271d (diagSub a\u271d x) = prod.diag a\u271d x\n\u03b1 : TypeVec (Nat.succ n\u271d)\nx : TypeVec.drop (\u03b1 ::: \u03b9) (Fin2.fs a\u271d)\n\u22a2 subtypeVal (repeatEq \u03b1) (Fin2.fs a\u271d) (diagSub (Fin2.fs a\u271d) x) = prod.diag (Fin2.fs a\u271d) x\n[PROOFSTEP]\n\n| fs _ ih => apply ih\n[GOAL]\ncase h\u2080.a.h.fs\nn : \u2115\n\u03b9 \u03b9' : Type u\nn\u271d : \u2115\na\u271d : Fin2 n\u271d\nih : \u2200 {\u03b1 : TypeVec n\u271d} (x : TypeVec.drop (\u03b1 ::: \u03b9) a\u271d), subtypeVal (repeatEq \u03b1) a\u271d (diagSub a\u271d x) = prod.diag a\u271d x\n\u03b1 : TypeVec (Nat.succ n\u271d)\nx : TypeVec.drop (\u03b1 ::: \u03b9) (Fin2.fs a\u271d)\n\u22a2 subtypeVal (repeatEq \u03b1) (Fin2.fs a\u271d) (diagSub (Fin2.fs a\u271d) x) = prod.diag (Fin2.fs a\u271d) x\n[PROOFSTEP]\napply ih\n[GOAL]\ncase h\u2081\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 lastFun\n (subtypeVal (RelLast' \u03b1 R) \u229a\n toSubtype (RelLast' \u03b1 R) \u229a\n fromAppend1DropLast \u229a\n (ofSubtype (repeatEq \u03b1) ::: _root_.id) \u229a\n (diagSub ::: fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) })) =\n lastFun (((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a prod.diag)\n[PROOFSTEP]\nsimp only [lastFun_from_append1_drop_last, lastFun_toSubtype, lastFun_appendFun, lastFun_subtypeVal, comp.left_id,\n lastFun_comp, lastFun_prod]\n[GOAL]\ncase h\u2081\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n ofSubtype (repeatEq \u03b1) ::: _root_.id\n\u22a2 (Subtype.val \u2218 fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }) = Prod.map f g \u2218 lastFun prod.diag\n[PROOFSTEP]\next1\n[GOAL]\ncase h\u2081.h\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\nlawful : LawfulMvFunctor F\n\u03b1 : TypeVec n\n\u03b9 \u03b9' : Type u\nR : \u03b9' \u2192 \u03b9' \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf g : \u03b9 \u2192 \u03b9'\nhh : \u2200 (x : \u03b9), R (f x) (g x)\nh : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }\nb : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h\nc : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9\n (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=\n ofSubtype (repeatEq \u03b1) ::: _root_.id\nx\u271d : TypeVec.last (\u03b1 ::: \u03b9)\n\u22a2 (Subtype.val \u2218 fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }) x\u271d =\n (Prod.map f g \u2218 lastFun prod.diag) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\ninst\u271d : LawfulMvFunctor F\n\u03b1 : TypeVec n\n\u03b9 : Type u\nR : \u03b9 \u2192 \u03b9 \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf : \u03b9 \u2192 \u03b9\nhh : \u2200 (x : \u03b9), R (f x) x\n\u22a2 LiftR' (RelLast' \u03b1 R) ((TypeVec.id ::: f) <$$> x) x\n[PROOFSTEP]\nhave := liftR_map_last R x f id hh\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\ninst\u271d : LawfulMvFunctor F\n\u03b1 : TypeVec n\n\u03b9 : Type u\nR : \u03b9 \u2192 \u03b9 \u2192 Prop\nx : F (\u03b1 ::: \u03b9)\nf : \u03b9 \u2192 \u03b9\nhh : \u2200 (x : \u03b9), R (f x) x\nthis : LiftR' (RelLast' \u03b1 R) ((TypeVec.id ::: f) <$$> x) ((TypeVec.id ::: _root_.id) <$$> x)\n\u22a2 LiftR' (RelLast' \u03b1 R) ((TypeVec.id ::: f) <$$> x) x\n[PROOFSTEP]\nrwa [appendFun_id_id, MvFunctor.id_map] at this \n[GOAL]\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx : Cofix F \u03b1\n\u22a2 Quot.mk Mcongr (repr x) = x\n[PROOFSTEP]\nlet R := fun x y : Cofix F \u03b1 => abs (repr y) = x\n[GOAL]\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\n\u22a2 Quot.mk Mcongr (repr x) = x\n[PROOFSTEP]\nrefine' Cofix.bisim\u2082 R _ _ _ rfl\n[GOAL]\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nx : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\n\u22a2 \u2200 (x y : Cofix F \u03b1), R x y \u2192 LiftR' (RelLast' \u03b1 R) (dest x) (dest y)\n[PROOFSTEP]\nclear x\n[GOAL]\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\n\u22a2 \u2200 (x y : Cofix F \u03b1), R x y \u2192 LiftR' (RelLast' \u03b1 R) (dest x) (dest y)\n[PROOFSTEP]\nrintro x y h\n[GOAL]\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\nx y : Cofix F \u03b1\nh : R x y\n\u22a2 LiftR' (RelLast' \u03b1 R) (dest x) (dest y)\n[PROOFSTEP]\nsubst h\n[GOAL]\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\ny : Cofix F \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 R) (dest (abs (repr y))) (dest y)\n[PROOFSTEP]\ndsimp [Cofix.dest, Cofix.abs]\n[GOAL]\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\ny : Cofix F \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (repr y) = x)\n ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) (repr y)))\n (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n (_ :\n \u2200 (x y : M (P F) \u03b1),\n Mcongr x y \u2192\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n y)\n[PROOFSTEP]\ninduction y using Quot.ind\n[GOAL]\ncase mk\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (repr y) = x)\n ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) (repr (Quot.mk Mcongr a\u271d))))\n (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n (_ :\n \u2200 (x y : M (P F) \u03b1),\n Mcongr x y \u2192\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n (Quot.mk Mcongr a\u271d))\n[PROOFSTEP]\nsimp only [Cofix.repr, M.dest_corec, abs_map, MvQPF.abs_repr, Function.comp]\n[GOAL]\ncase mk\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x)\n ((TypeVec.id ::: Quot.mk Mcongr) <$$>\n (TypeVec.id ::: M.corec (P F) fun x => MvQPF.repr (dest x)) <$$> dest (Quot.mk Mcongr a\u271d))\n ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d))\n[PROOFSTEP]\nconv =>\n congr\n rfl\n rw [Cofix.dest]\n[GOAL]\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n| LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x)\n ((TypeVec.id ::: Quot.mk Mcongr) <$$>\n (TypeVec.id ::: M.corec (P F) fun x => MvQPF.repr (dest x)) <$$> dest (Quot.mk Mcongr a\u271d))\n ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d))\n[PROOFSTEP]\n congr\n rfl\n rw [Cofix.dest]\n[GOAL]\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n| LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x)\n ((TypeVec.id ::: Quot.mk Mcongr) <$$>\n (TypeVec.id ::: M.corec (P F) fun x => MvQPF.repr (dest x)) <$$> dest (Quot.mk Mcongr a\u271d))\n ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d))\n[PROOFSTEP]\n congr\n rfl\n rw [Cofix.dest]\n[GOAL]\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n| LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x)\n ((TypeVec.id ::: Quot.mk Mcongr) <$$>\n (TypeVec.id ::: M.corec (P F) fun x => MvQPF.repr (dest x)) <$$> dest (Quot.mk Mcongr a\u271d))\n ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase R\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n| RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x\ncase a\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n| (TypeVec.id ::: Quot.mk Mcongr) <$$>\n (TypeVec.id ::: M.corec (P F) fun x => MvQPF.repr (dest x)) <$$> dest (Quot.mk Mcongr a\u271d)\ncase a\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n| (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n| (TypeVec.id ::: Quot.mk Mcongr) <$$>\n (TypeVec.id ::: M.corec (P F) fun x => MvQPF.repr (dest x)) <$$> dest (Quot.mk Mcongr a\u271d)\ncase a\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n| (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d)\n[PROOFSTEP]\nrw [Cofix.dest]\n[GOAL]\ncase mk\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x)\n ((TypeVec.id ::: Quot.mk Mcongr) <$$>\n (TypeVec.id :::\n M.corec (P F) fun x =>\n MvQPF.repr\n (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n (_ :\n \u2200 (x y : M (P F) \u03b1),\n Mcongr x y \u2192\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n x)) <$$>\n Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n (_ :\n \u2200 (x y : M (P F) \u03b1),\n Mcongr x y \u2192\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n (Quot.mk Mcongr a\u271d))\n ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d))\n[PROOFSTEP]\nrw [MvFunctor.map_map, MvFunctor.map_map, \u2190 appendFun_comp_id, \u2190 appendFun_comp_id]\n[GOAL]\ncase mk\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n\u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x)\n ((TypeVec.id :::\n (Quot.mk Mcongr \u2218\n M.corec (P F) fun x =>\n MvQPF.repr\n (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n (_ :\n \u2200 (x y : M (P F) \u03b1),\n Mcongr x y \u2192\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n x)) \u2218\n Quot.mk Mcongr) <$$>\n MvQPF.abs (M.dest (P F) a\u271d))\n ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d))\n[PROOFSTEP]\napply liftR_map_last\n[GOAL]\ncase mk.hh\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d : M (P F) \u03b1\n\u22a2 \u2200 (x : M (P F) \u03b1),\n Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) (Quot.mk Mcongr x)) =\n ((Quot.mk Mcongr \u2218\n M.corec (P F) fun x =>\n MvQPF.repr\n (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n (_ :\n \u2200 (x y : M (P F) \u03b1),\n Mcongr x y \u2192\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n x)) \u2218\n Quot.mk Mcongr)\n x\n[PROOFSTEP]\nintros\n[GOAL]\ncase mk.hh\nn : \u2115\nF\u271d : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\u271d\nq\u271d : MvQPF F\u271d\nF : TypeVec (n + 1) \u2192 Type u\ninst\u271d : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x\na\u271d x\u271d : M (P F) \u03b1\n\u22a2 Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) (Quot.mk Mcongr x\u271d)) =\n ((Quot.mk Mcongr \u2218\n M.corec (P F) fun x =>\n MvQPF.repr\n (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))\n (_ :\n \u2200 (x y : M (P F) \u03b1),\n Mcongr x y \u2192\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =\n (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)\n x)) \u2218\n Quot.mk Mcongr)\n x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nX Y : Type u\nx\u2080 : X\nf : X \u2192 Y\ng : Y \u2192 F (\u03b1 ::: X)\n\u22a2 Cofix.corec (g \u2218 f) x\u2080 = Cofix.corec (MvFunctor.map (TypeVec.id ::: f) \u2218 g) (f x\u2080)\n[PROOFSTEP]\nmv_bisim x\u2080 with R a b x Ha Hb\n[GOAL]\ncase intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nX Y : Type u\nx\u2080 : X\nf : X \u2192 Y\ng : Y \u2192 F (\u03b1 ::: X)\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n fun a b => \u2203 x, a = Cofix.corec (g \u2218 f) x \u2227 b = Cofix.corec (MvFunctor.map (TypeVec.id ::: f) \u2218 g) (f x)\na b : Cofix F \u03b1\nx : X\nHa : a = Cofix.corec (g \u2218 f) x\nHb : b = Cofix.corec (MvFunctor.map (TypeVec.id ::: f) \u2218 g) (f x)\n\u22a2 LiftR' (RelLast' \u03b1 R) (Cofix.dest a) (Cofix.dest b)\n[PROOFSTEP]\nrw [Ha, Hb, Cofix.dest_corec, Cofix.dest_corec, Function.comp_apply, Function.comp_apply]\n[GOAL]\ncase intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nX Y : Type u\nx\u2080 : X\nf : X \u2192 Y\ng : Y \u2192 F (\u03b1 ::: X)\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n fun a b => \u2203 x, a = Cofix.corec (g \u2218 f) x \u2227 b = Cofix.corec (MvFunctor.map (TypeVec.id ::: f) \u2218 g) (f x)\na b : Cofix F \u03b1\nx : X\nHa : a = Cofix.corec (g \u2218 f) x\nHb : b = Cofix.corec (MvFunctor.map (TypeVec.id ::: f) \u2218 g) (f x)\n\u22a2 LiftR' (RelLast' \u03b1 R) ((TypeVec.id ::: Cofix.corec (g \u2218 f)) <$$> g (f x))\n ((TypeVec.id ::: Cofix.corec (MvFunctor.map (TypeVec.id ::: f) \u2218 g)) <$$> (TypeVec.id ::: f) <$$> g (f x))\n[PROOFSTEP]\nrw [MvFunctor.map_map, \u2190 appendFun_comp_id]\n[GOAL]\ncase intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nX Y : Type u\nx\u2080 : X\nf : X \u2192 Y\ng : Y \u2192 F (\u03b1 ::: X)\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n fun a b => \u2203 x, a = Cofix.corec (g \u2218 f) x \u2227 b = Cofix.corec (MvFunctor.map (TypeVec.id ::: f) \u2218 g) (f x)\na b : Cofix F \u03b1\nx : X\nHa : a = Cofix.corec (g \u2218 f) x\nHb : b = Cofix.corec (MvFunctor.map (TypeVec.id ::: f) \u2218 g) (f x)\n\u22a2 LiftR' (RelLast' \u03b1 R) ((TypeVec.id ::: Cofix.corec (g \u2218 f)) <$$> g (f x))\n ((TypeVec.id ::: Cofix.corec (MvFunctor.map (TypeVec.id ::: f) \u2218 g) \u2218 f) <$$> g (f x))\n[PROOFSTEP]\nrefine' liftR_map_last _ _ _ _ _\n[GOAL]\ncase intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nX Y : Type u\nx\u2080 : X\nf : X \u2192 Y\ng : Y \u2192 F (\u03b1 ::: X)\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n fun a b => \u2203 x, a = Cofix.corec (g \u2218 f) x \u2227 b = Cofix.corec (MvFunctor.map (TypeVec.id ::: f) \u2218 g) (f x)\na b : Cofix F \u03b1\nx : X\nHa : a = Cofix.corec (g \u2218 f) x\nHb : b = Cofix.corec (MvFunctor.map (TypeVec.id ::: f) \u2218 g) (f x)\n\u22a2 \u2200 (x : X), R (Cofix.corec (g \u2218 f) x) ((Cofix.corec (MvFunctor.map (TypeVec.id ::: f) \u2218 g) \u2218 f) x)\n[PROOFSTEP]\nintro a\n[GOAL]\ncase intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\nX Y : Type u\nx\u2080 : X\nf : X \u2192 Y\ng : Y \u2192 F (\u03b1 ::: X)\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n fun a b => \u2203 x, a = Cofix.corec (g \u2218 f) x \u2227 b = Cofix.corec (MvFunctor.map (TypeVec.id ::: f) \u2218 g) (f x)\na\u271d b : Cofix F \u03b1\nx : X\nHa : a\u271d = Cofix.corec (g \u2218 f) x\nHb : b = Cofix.corec (MvFunctor.map (TypeVec.id ::: f) \u2218 g) (f x)\na : X\n\u22a2 R (Cofix.corec (g \u2218 f) a) ((Cofix.corec (MvFunctor.map (TypeVec.id ::: f) \u2218 g) \u2218 f) a)\n[PROOFSTEP]\nrefine' \u27e8a, rfl, rfl\u27e9\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\n\u22a2 dest (corec' g x) = (TypeVec.id ::: Sum.elim _root_.id (corec' g)) <$$> g x\n[PROOFSTEP]\nrw [Cofix.corec', Cofix.dest_corec]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\n\u22a2 (TypeVec.id ::: corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g)) <$$>\n Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g (Sum.inr x) =\n (TypeVec.id ::: Sum.elim _root_.id (corec' g)) <$$> g x\n[PROOFSTEP]\ndsimp\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\n\u22a2 (TypeVec.id ::: corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g)) <$$> g x =\n (TypeVec.id ::: Sum.elim _root_.id (corec' g)) <$$> g x\n[PROOFSTEP]\ncongr!\n[GOAL]\ncase h.e'_6.h.e'_7\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\n\u22a2 corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g) = Sum.elim _root_.id (corec' g)\n[PROOFSTEP]\next (i | i)\n[GOAL]\ncase h.e'_6.h.e'_7.h.inl\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\ni : Cofix F \u03b1\n\u22a2 corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g) (Sum.inl i) =\n Sum.elim _root_.id (corec' g) (Sum.inl i)\n[PROOFSTEP]\nerw [corec_roll]\n[GOAL]\ncase h.e'_6.h.e'_7.h.inr\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx i : \u03b2\n\u22a2 corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g) (Sum.inr i) =\n Sum.elim _root_.id (corec' g) (Sum.inr i)\n[PROOFSTEP]\nerw [corec_roll]\n[GOAL]\ncase h.e'_6.h.e'_7.h.inl\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\ni : Cofix F \u03b1\n\u22a2 corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) val) fun val => g val)\n (Sum.inl i) =\n Sum.elim _root_.id (corec' g) (Sum.inl i)\n[PROOFSTEP]\ndsimp [Cofix.corec']\n[GOAL]\ncase h.e'_6.h.e'_7.h.inr\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx i : \u03b2\n\u22a2 corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) val) fun val => g val)\n (Sum.inr i) =\n Sum.elim _root_.id (corec' g) (Sum.inr i)\n[PROOFSTEP]\ndsimp [Cofix.corec']\n[GOAL]\ncase h.e'_6.h.e'_7.h.inl\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx : \u03b2\ni : Cofix F \u03b1\n\u22a2 corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl i) =\n i\n[PROOFSTEP]\nmv_bisim i with R a b x Ha Hb\n[GOAL]\ncase h.e'_6.h.e'_7.h.inl.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n fun a b =>\n \u2203 x,\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x) \u2227\n b = x\na b x : Cofix F \u03b1\nHa :\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x)\nHb : b = x\n\u22a2 LiftR' (RelLast' \u03b1 R) (dest a) (dest b)\n[PROOFSTEP]\nrw [Ha, Hb, Cofix.dest_corec]\n[GOAL]\ncase h.e'_6.h.e'_7.h.inl.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n fun a b =>\n \u2203 x,\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x) \u2227\n b = x\na b x : Cofix F \u03b1\nHa :\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x)\nHb : b = x\n\u22a2 LiftR' (RelLast' \u03b1 R)\n ((TypeVec.id :::\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)) <$$>\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x))\n (dest x)\n[PROOFSTEP]\ndsimp [Function.comp]\n[GOAL]\ncase h.e'_6.h.e'_7.h.inl.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n fun a b =>\n \u2203 x,\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x) \u2227\n b = x\na b x : Cofix F \u03b1\nHa :\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x)\nHb : b = x\n\u22a2 LiftR'\n (RelLast' \u03b1 fun a b =>\n \u2203 x,\n a =\n corec\n (fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n (Sum.inl x) \u2227\n b = x)\n ((TypeVec.id :::\n corec fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) <$$>\n (TypeVec.id ::: fun x => x) <$$> (TypeVec.id ::: Sum.inl) <$$> dest x)\n (dest x)\n[PROOFSTEP]\nrepeat rw [MvFunctor.map_map, \u2190 appendFun_comp_id]\n[GOAL]\ncase h.e'_6.h.e'_7.h.inl.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n fun a b =>\n \u2203 x,\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x) \u2227\n b = x\na b x : Cofix F \u03b1\nHa :\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x)\nHb : b = x\n\u22a2 LiftR'\n (RelLast' \u03b1 fun a b =>\n \u2203 x,\n a =\n corec\n (fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n (Sum.inl x) \u2227\n b = x)\n ((TypeVec.id :::\n corec fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) <$$>\n (TypeVec.id ::: fun x => x) <$$> (TypeVec.id ::: Sum.inl) <$$> dest x)\n (dest x)\n[PROOFSTEP]\nrw [MvFunctor.map_map, \u2190 appendFun_comp_id]\n[GOAL]\ncase h.e'_6.h.e'_7.h.inl.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n fun a b =>\n \u2203 x,\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x) \u2227\n b = x\na b x : Cofix F \u03b1\nHa :\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x)\nHb : b = x\n\u22a2 LiftR'\n (RelLast' \u03b1 fun a b =>\n \u2203 x,\n a =\n corec\n (fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n (Sum.inl x) \u2227\n b = x)\n ((TypeVec.id :::\n (corec fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) \u2218\n fun x => x) <$$>\n (TypeVec.id ::: Sum.inl) <$$> dest x)\n (dest x)\n[PROOFSTEP]\nrw [MvFunctor.map_map, \u2190 appendFun_comp_id]\n[GOAL]\ncase h.e'_6.h.e'_7.h.inl.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n fun a b =>\n \u2203 x,\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x) \u2227\n b = x\na b x : Cofix F \u03b1\nHa :\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x)\nHb : b = x\n\u22a2 LiftR'\n (RelLast' \u03b1 fun a b =>\n \u2203 x,\n a =\n corec\n (fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n (Sum.inl x) \u2227\n b = x)\n ((TypeVec.id :::\n ((corec fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) \u2218\n fun x => x) \u2218\n Sum.inl) <$$>\n dest x)\n (dest x)\n[PROOFSTEP]\nrw [MvFunctor.map_map, \u2190 appendFun_comp_id]\n[GOAL]\ncase h.e'_6.h.e'_7.h.inl.intro.intro\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n fun a b =>\n \u2203 x,\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x) \u2227\n b = x\na b x : Cofix F \u03b1\nHa :\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x)\nHb : b = x\n\u22a2 LiftR'\n (RelLast' \u03b1 fun a b =>\n \u2203 x,\n a =\n corec\n (fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n (Sum.inl x) \u2227\n b = x)\n ((TypeVec.id :::\n ((corec fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) \u2218\n fun x => x) \u2218\n Sum.inl) <$$>\n dest x)\n (dest x)\n[PROOFSTEP]\napply liftR_map_last'\n[GOAL]\ncase h.e'_6.h.e'_7.h.inl.intro.intro.hh\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n fun a b =>\n \u2203 x,\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x) \u2227\n b = x\na b x : Cofix F \u03b1\nHa :\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x)\nHb : b = x\n\u22a2 \u2200 (x : Cofix F \u03b1),\n \u2203 x_1,\n (((corec fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x) \u2218\n fun x => x) \u2218\n Sum.inl)\n x =\n corec\n (fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n (Sum.inl x_1) \u2227\n x = x_1\n[PROOFSTEP]\ndsimp [Function.comp]\n[GOAL]\ncase h.e'_6.h.e'_7.h.inl.intro.intro.hh\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n fun a b =>\n \u2203 x,\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x) \u2227\n b = x\na b x : Cofix F \u03b1\nHa :\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x)\nHb : b = x\n\u22a2 \u2200 (x : Cofix F \u03b1),\n \u2203 x_1,\n corec\n (fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n (Sum.inl x) =\n corec\n (fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n (Sum.inl x_1) \u2227\n x = x_1\n[PROOFSTEP]\nintros\n[GOAL]\ncase h.e'_6.h.e'_7.h.inl.intro.intro.hh\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx\u271d\u00b9 : \u03b2\ni : Cofix F \u03b1\nR : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop :=\n fun a b =>\n \u2203 x,\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x) \u2227\n b = x\na b x : Cofix F \u03b1\nHa :\n a =\n corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inl x)\nHb : b = x\nx\u271d : Cofix F \u03b1\n\u22a2 \u2203 x,\n corec\n (fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n (Sum.inl x\u271d) =\n corec\n (fun x =>\n (TypeVec.id ::: fun x => x) <$$>\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) (fun val => g val) x)\n (Sum.inl x) \u2227\n x\u271d = x\n[PROOFSTEP]\nexact \u27e8_, rfl, rfl\u27e9\n[GOAL]\ncase h.e'_6.h.e'_7.h.inr\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx i : \u03b2\n\u22a2 corec\n (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n (Sum.inr i) =\n corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g) (Sum.inr i)\n[PROOFSTEP]\ncongr with y\n[GOAL]\ncase h.e'_6.h.e'_7.h.inr.e_g.h\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx i : \u03b2\ny : Cofix F \u03b1 \u2295 \u03b2\n\u22a2 (MvFunctor.map (TypeVec.id ::: fun x => x) \u2218\n Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val)\n y =\n Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g y\n[PROOFSTEP]\nerw [appendFun_id_id]\n[GOAL]\ncase h.e'_6.h.e'_7.h.inr.e_g.h\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : \u03b2 \u2192 F (\u03b1 ::: (Cofix F \u03b1 \u2295 \u03b2))\nx i : \u03b2\ny : Cofix F \u03b1 \u2295 \u03b2\n\u22a2 (MvFunctor.map TypeVec.id \u2218 Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> dest val) fun val => g val) y =\n Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) \u2218 dest) g y\n[PROOFSTEP]\nsimp [MvFunctor.id_map, Sum.elim]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : {X : Type u} \u2192 (Cofix F \u03b1 \u2192 X) \u2192 (\u03b2 \u2192 X) \u2192 \u03b2 \u2192 F (\u03b1 ::: X)\nx : \u03b2\nh :\n \u2200 (X Y : Type u) (f : Cofix F \u03b1 \u2192 X) (f' : \u03b2 \u2192 X) (k : X \u2192 Y), g (k \u2218 f) (k \u2218 f') x = (TypeVec.id ::: k) <$$> g f f' x\n\u22a2 dest (corec\u2081 g x) = g _root_.id (corec\u2081 g) x\n[PROOFSTEP]\nrw [Cofix.corec\u2081, Cofix.dest_corec', \u2190 h]\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1 : TypeVec n\n\u03b2 : Type u\ng : {X : Type u} \u2192 (Cofix F \u03b1 \u2192 X) \u2192 (\u03b2 \u2192 X) \u2192 \u03b2 \u2192 F (\u03b1 ::: X)\nx : \u03b2\nh :\n \u2200 (X Y : Type u) (f : Cofix F \u03b1 \u2192 X) (f' : \u03b2 \u2192 X) (k : X \u2192 Y), g (k \u2218 f) (k \u2218 f') x = (TypeVec.id ::: k) <$$> g f f' x\n\u22a2 g (Sum.elim _root_.id (corec' fun x => g Sum.inl Sum.inr x) \u2218 Sum.inl)\n (Sum.elim _root_.id (corec' fun x => g Sum.inl Sum.inr x) \u2218 Sum.inr) x =\n g _root_.id (corec\u2081 g) x\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u22a2 \u2200 {\u03b1 \u03b2 : TypeVec n} (f : \u03b1 \u27f9 \u03b2) (p : Obj (mp (P F)) \u03b1),\n (fun {\u03b1} => Quot.mk Mcongr) (f <$$> p) = f <$$> (fun {\u03b1} => Quot.mk Mcongr) p\n[PROOFSTEP]\nintros\n[GOAL]\nn : \u2115\nF : TypeVec (n + 1) \u2192 Type u\nmvf : MvFunctor F\nq : MvQPF F\n\u03b1\u271d \u03b2\u271d : TypeVec n\nf\u271d : \u03b1\u271d \u27f9 \u03b2\u271d\np\u271d : Obj (mp (P F)) \u03b1\u271d\n\u22a2 (fun {\u03b1} => Quot.mk Mcongr) (f\u271d <$$> p\u271d) = f\u271d <$$> (fun {\u03b1} => Quot.mk Mcongr) p\u271d\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Data.QPF.Multivariate.Constructions.Cofix", "llama_tokens": 66073, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506635289836, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.4158767380837068}} {"text": "[GOAL]\nm n : \u2115\n\u22a2 (fun b => bif b then 1 else 0) (Matrix.vecCons false ![true] 0) = 0 \u2227\n (fun b => bif b then 1 else 0) (Matrix.vecCons false ![true] 1) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nm n : \u2115\n\u22a2 Matrix.vecCons false ![true] ((fun b => bif b then 1 else 0) false) = false \u2227\n Matrix.vecCons false ![true] ((fun b => bif b then 1 else 0) true) = true\n[PROOFSTEP]\nsimp\n[GOAL]\nm n : \u2115\n\u03b1 : Fin 2 \u2192 Type u\ns : Set (\u03b1 0)\nt : Set (\u03b1 1)\n\u22a2 (fun f => (f 0, f 1)) \u207b\u00b9' s \u00d7\u02e2 t = Set.pi Set.univ (cons s (cons t finZeroElim))\n[PROOFSTEP]\next f\n[GOAL]\ncase h\nm n : \u2115\n\u03b1 : Fin 2 \u2192 Type u\ns : Set (\u03b1 0)\nt : Set (\u03b1 1)\nf : (i : Fin 2) \u2192 \u03b1 i\n\u22a2 f \u2208 (fun f => (f 0, f 1)) \u207b\u00b9' s \u00d7\u02e2 t \u2194 f \u2208 Set.pi Set.univ (cons s (cons t finZeroElim))\n[PROOFSTEP]\nsimp [Fin.forall_fin_two]\n[GOAL]\nm n : \u2115\ni x : Fin (n + 1)\n\u22a2 (fun x => Option.casesOn' x i (Fin.succAbove i)) (Fin.insertNth i none some i) = i\n[PROOFSTEP]\nsimp\n[GOAL]\nm n : \u2115\ni x : Fin (n + 1)\nj : Fin n\n\u22a2 (fun x => Option.casesOn' x i (Fin.succAbove i)) (Fin.insertNth i none some (Fin.succAbove i j)) = Fin.succAbove i j\n[PROOFSTEP]\nsimp\n[GOAL]\nm n : \u2115\ni : Fin (n + 1)\nx : Option (Fin n)\n\u22a2 Fin.insertNth i none some ((fun x => Option.casesOn' x i (Fin.succAbove i)) x) = x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase none\nm n : \u2115\ni : Fin (n + 1)\n\u22a2 Fin.insertNth i none some ((fun x => Option.casesOn' x i (Fin.succAbove i)) none) = none\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some\nm n : \u2115\ni : Fin (n + 1)\nval\u271d : Fin n\n\u22a2 Fin.insertNth i none some ((fun x => Option.casesOn' x i (Fin.succAbove i)) (some val\u271d)) = some val\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none\nm n : \u2115\ni : Fin (n + 1)\n\u22a2 Fin.insertNth i none some i = none\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some\nm n : \u2115\ni : Fin (n + 1)\nval\u271d : Fin n\n\u22a2 Fin.insertNth i none some (Fin.succAbove i val\u271d) = some val\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nm n : \u2115\ni : Fin (n + 1)\n\u22a2 \u2191(finSuccEquiv' i) i = none\n[PROOFSTEP]\nsimp [finSuccEquiv']\n[GOAL]\nm\u271d n : \u2115\ni : Fin (n + 1)\nm : Fin n\nh : Fin.castSucc m < i\n\u22a2 \u2191(finSuccEquiv' i) (Fin.castSucc m) = some m\n[PROOFSTEP]\nrw [\u2190 Fin.succAbove_below _ _ h, finSuccEquiv'_succAbove]\n[GOAL]\nm\u271d n : \u2115\ni : Fin (n + 1)\nm : Fin n\nh : i \u2264 Fin.castSucc m\n\u22a2 \u2191(finSuccEquiv' i) (Fin.succ m) = some m\n[PROOFSTEP]\nrw [\u2190 Fin.succAbove_above _ _ h, finSuccEquiv'_succAbove]\n[GOAL]\nm n : \u2115\ni : Fin n\n\u22a2 \u2191(finSuccEquiv' (Fin.last n)) (Fin.castSucc i) = some i\n[PROOFSTEP]\nrw [\u2190 Fin.succAbove_last, finSuccEquiv'_succAbove]\n[GOAL]\nm n : \u2115\ni : Fin (n + 1)\nh : i \u2260 Fin.last n\n\u22a2 \u2191(finSuccEquiv' (Fin.last n)) i = some (Fin.castLT i (_ : \u2191i < n))\n[PROOFSTEP]\nrcases Fin.exists_castSucc_eq.2 h with \u27e8i, rfl\u27e9\n[GOAL]\ncase intro\nm n : \u2115\ni : Fin n\nh : Fin.castSucc i \u2260 Fin.last n\n\u22a2 \u2191(finSuccEquiv' (Fin.last n)) (Fin.castSucc i) = some (Fin.castLT (Fin.castSucc i) (_ : \u2191(Fin.castSucc i) < n))\n[PROOFSTEP]\nrw [finSuccEquiv'_last_apply_castSucc]\n[GOAL]\ncase intro\nm n : \u2115\ni : Fin n\nh : Fin.castSucc i \u2260 Fin.last n\n\u22a2 some i = some (Fin.castLT (Fin.castSucc i) (_ : \u2191(Fin.castSucc i) < n))\n[PROOFSTEP]\nrfl\n[GOAL]\nm n : \u2115\ni j : Fin (n + 1)\nhi : i \u2260 Fin.last n\nhj : j \u2260 i\n\u22a2 \u2191(finSuccEquiv' i) j = some (Fin.predAbove (Fin.castLT i (_ : \u2191i < n)) j)\n[PROOFSTEP]\nrcases Fin.exists_succAbove_eq hj with \u27e8j, rfl\u27e9\n[GOAL]\ncase intro\nm n : \u2115\ni : Fin (n + 1)\nhi : i \u2260 Fin.last n\nj : Fin n\nhj : Fin.succAbove i j \u2260 i\n\u22a2 \u2191(finSuccEquiv' i) (Fin.succAbove i j) = some (Fin.predAbove (Fin.castLT i (_ : \u2191i < n)) (Fin.succAbove i j))\n[PROOFSTEP]\nrcases Fin.exists_castSucc_eq.2 hi with \u27e8i, rfl\u27e9\n[GOAL]\ncase intro.intro\nm n : \u2115\nj i : Fin n\nhi : Fin.castSucc i \u2260 Fin.last n\nhj : Fin.succAbove (Fin.castSucc i) j \u2260 Fin.castSucc i\n\u22a2 \u2191(finSuccEquiv' (Fin.castSucc i)) (Fin.succAbove (Fin.castSucc i) j) =\n some (Fin.predAbove (Fin.castLT (Fin.castSucc i) (_ : \u2191(Fin.castSucc i) < n)) (Fin.succAbove (Fin.castSucc i) j))\n[PROOFSTEP]\nsimp\n[GOAL]\nm n : \u2115\nx : { x // x \u2260 Fin.last n }\n\u22a2 \u2191(OrderIso.symm (finSuccAboveEquiv (Fin.last n))) x = Fin.castLT \u2191x (_ : \u2191\u2191x < n)\n[PROOFSTEP]\nrw [\u2190 Option.some_inj]\n[GOAL]\nm n : \u2115\nx : { x // x \u2260 Fin.last n }\n\u22a2 some (\u2191(OrderIso.symm (finSuccAboveEquiv (Fin.last n))) x) = some (Fin.castLT \u2191x (_ : \u2191\u2191x < n))\n[PROOFSTEP]\nsimpa [finSuccAboveEquiv, OrderIso.symm] using finSuccEquiv'_last_apply x.property\n[GOAL]\nm n : \u2115\np : Fin (n + 1)\nh : p \u2260 Fin.last n\nx : { x // x \u2260 p }\n\u22a2 \u2191(OrderIso.symm (finSuccAboveEquiv p)) x = Fin.predAbove (Fin.castLT p (_ : \u2191p < n)) \u2191x\n[PROOFSTEP]\nrw [\u2190 Option.some_inj]\n[GOAL]\nm n : \u2115\np : Fin (n + 1)\nh : p \u2260 Fin.last n\nx : { x // x \u2260 p }\n\u22a2 some (\u2191(OrderIso.symm (finSuccAboveEquiv p)) x) = some (Fin.predAbove (Fin.castLT p (_ : \u2191p < n)) \u2191x)\n[PROOFSTEP]\nsimpa [finSuccAboveEquiv, OrderIso.symm] using finSuccEquiv'_ne_last_apply h x.property\n[GOAL]\nm n : \u2115\n\u22a2 \u2191finSuccEquivLast (Fin.last n) = none\n[PROOFSTEP]\nsimp [finSuccEquivLast]\n[GOAL]\nm n : \u2115\n\u03b1 : Fin (n + 1) \u2192 Type u\ni : Fin (n + 1)\nf : (j : Fin (n + 1)) \u2192 \u03b1 j\n\u22a2 (fun f => Fin.insertNth i f.fst f.snd) ((fun f => (f i, fun j => f (Fin.succAbove i j))) f) = f\n[PROOFSTEP]\nsimp [Fin.insertNth_eq_iff]\n[GOAL]\nm n : \u2115\n\u03b1 : Fin (n + 1) \u2192 Type u\ni : Fin (n + 1)\nf : \u03b1 i \u00d7 ((j : Fin n) \u2192 \u03b1 (Fin.succAbove i j))\n\u22a2 (fun f => (f i, fun j => f (Fin.succAbove i j))) ((fun f => Fin.insertNth i f.fst f.snd) f) = f\n[PROOFSTEP]\nsimp\n[GOAL]\nm n : \u2115\nx : Fin m \u2295 Fin n\n\u22a2 (fun i => Fin.addCases Sum.inl Sum.inr i) (Sum.elim (Fin.castAdd n) (Fin.natAdd m) x) = x\n[PROOFSTEP]\ncases' x with y y\n[GOAL]\ncase inl\nm n : \u2115\ny : Fin m\n\u22a2 (fun i => Fin.addCases Sum.inl Sum.inr i) (Sum.elim (Fin.castAdd n) (Fin.natAdd m) (Sum.inl y)) = Sum.inl y\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase inr\nm n : \u2115\ny : Fin n\n\u22a2 (fun i => Fin.addCases Sum.inl Sum.inr i) (Sum.elim (Fin.castAdd n) (Fin.natAdd m) (Sum.inr y)) = Sum.inr y\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase inl\nm n : \u2115\ny : Fin m\n\u22a2 Fin.addCases Sum.inl Sum.inr (Fin.castAdd n y) = Sum.inl y\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nm n : \u2115\ny : Fin n\n\u22a2 Fin.addCases Sum.inl Sum.inr (Fin.natAdd m y) = Sum.inr y\n[PROOFSTEP]\nsimp\n[GOAL]\nm n : \u2115\nx : Fin (m + n)\n\u22a2 Sum.elim (Fin.castAdd n) (Fin.natAdd m) ((fun i => Fin.addCases Sum.inl Sum.inr i) x) = x\n[PROOFSTEP]\nrefine' Fin.addCases (fun i => _) (fun i => _) x\n[GOAL]\ncase refine'_1\nm n : \u2115\nx : Fin (m + n)\ni : Fin m\n\u22a2 Sum.elim (Fin.castAdd n) (Fin.natAdd m) ((fun i => Fin.addCases Sum.inl Sum.inr i) (Fin.castAdd n i)) =\n Fin.castAdd n i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nm n : \u2115\nx : Fin (m + n)\ni : Fin n\n\u22a2 Sum.elim (Fin.castAdd n) (Fin.natAdd m) ((fun i => Fin.addCases Sum.inl Sum.inr i) (Fin.natAdd m i)) = Fin.natAdd m i\n[PROOFSTEP]\nsimp\n[GOAL]\nm n\u271d : \u2115\nk : Fin m\nn : \u2115\n\u22a2 \u2191finAddFlip (Fin.castAdd n k) = Fin.natAdd n k\n[PROOFSTEP]\nsimp [finAddFlip]\n[GOAL]\nm\u271d n : \u2115\nk : Fin n\nm : \u2115\n\u22a2 \u2191finAddFlip (Fin.natAdd m k) = Fin.castAdd m k\n[PROOFSTEP]\nsimp [finAddFlip]\n[GOAL]\nm n k : \u2115\nh : k < m\nhk : optParam (k < m + n) (_ : k < m + n)\nhnk : optParam (n + k < n + m) (_ : n + k < n + m)\n\u22a2 \u2191finAddFlip { val := k, isLt := hk } = { val := n + k, isLt := hnk }\n[PROOFSTEP]\nconvert finAddFlip_apply_castAdd \u27e8k, h\u27e9 n\n[GOAL]\nm n k : \u2115\nh\u2081 : m \u2264 k\nh\u2082 : k < m + n\n\u22a2 \u2191finAddFlip { val := k, isLt := h\u2082 } = { val := k - m, isLt := (_ : k - m < n + m) }\n[PROOFSTEP]\nconvert @finAddFlip_apply_natAdd n \u27e8k - m, (tsub_lt_iff_right h\u2081).2 ?_\u27e9 m\n[GOAL]\ncase h.e'_2.h.e'_6.h.e'_2\nm n k : \u2115\nh\u2081 : m \u2264 k\nh\u2082 : k < m + n\n\u22a2 k = \u2191(Fin.natAdd m { val := k - m, isLt := (_ : k - m < n) })\n[PROOFSTEP]\nsimp [add_tsub_cancel_of_le h\u2081]\n[GOAL]\nm n k : \u2115\nh\u2081 : m \u2264 k\nh\u2082 : k < m + n\n\u22a2 k < n + m\n[PROOFSTEP]\nrwa [add_comm]\n[GOAL]\nm n k : \u2115\nh : k < n\n\u22a2 \u2191(finRotate (n + 1)) { val := k, isLt := (_ : k < n + 1) } = { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ n) }\n[PROOFSTEP]\next\n[GOAL]\ncase h\nm n k : \u2115\nh : k < n\n\u22a2 \u2191(\u2191(finRotate (n + 1)) { val := k, isLt := (_ : k < n + 1) }) =\n \u2191{ val := k + 1, isLt := (_ : Nat.succ k < Nat.succ n) }\n[PROOFSTEP]\ndsimp [finRotate_succ]\n[GOAL]\ncase h\nm n k : \u2115\nh : k < n\n\u22a2 \u2191(\u2191finAddFlip { val := k, isLt := (_ : k < n + 1) }) = k + 1\n[PROOFSTEP]\nsimp [finAddFlip_apply_mk_left h, add_comm]\n[GOAL]\nm n : \u2115\n\u22a2 \u2191(finRotate (n + 1)) { val := n, isLt := (_ : n < n + 1) } = { val := 0, isLt := (_ : 0 < Nat.succ n) }\n[PROOFSTEP]\ndsimp [finRotate_succ]\n[GOAL]\nm n : \u2115\n\u22a2 \u2191(finCongr (_ : 1 + n = n + 1)) (\u2191finAddFlip { val := n, isLt := (_ : n < n + 1) }) = 0\n[PROOFSTEP]\nrw [finAddFlip_apply_mk_right le_rfl]\n[GOAL]\nm n : \u2115\n\u22a2 \u2191(finCongr (_ : 1 + n = n + 1)) { val := n - n, isLt := (_ : n - n < 1 + n) } = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nm n : \u2115\n\u03b1 : Type u_1\nv : Fin n \u2192 \u03b1\na : \u03b1\n\u22a2 snoc v a = fun i => cons a v (\u2191(finRotate (n + 1)) i)\n[PROOFSTEP]\next \u27e8i, h\u27e9\n[GOAL]\ncase h.mk\nm n : \u2115\n\u03b1 : Type u_1\nv : Fin n \u2192 \u03b1\na : \u03b1\ni : \u2115\nh : i < n + 1\n\u22a2 snoc v a { val := i, isLt := h } = cons a v (\u2191(finRotate (n + 1)) { val := i, isLt := h })\n[PROOFSTEP]\nby_cases h' : i < n\n[GOAL]\ncase pos\nm n : \u2115\n\u03b1 : Type u_1\nv : Fin n \u2192 \u03b1\na : \u03b1\ni : \u2115\nh : i < n + 1\nh' : i < n\n\u22a2 snoc v a { val := i, isLt := h } = cons a v (\u2191(finRotate (n + 1)) { val := i, isLt := h })\n[PROOFSTEP]\nrw [finRotate_of_lt h', Fin.snoc, Fin.cons, dif_pos h']\n[GOAL]\ncase pos\nm n : \u2115\n\u03b1 : Type u_1\nv : Fin n \u2192 \u03b1\na : \u03b1\ni : \u2115\nh : i < n + 1\nh' : i < n\n\u22a2 _root_.cast (_ : \u03b1 = \u03b1) (v (castLT { val := i, isLt := h } h')) =\n cases a v { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nm n : \u2115\n\u03b1 : Type u_1\nv : Fin n \u2192 \u03b1\na : \u03b1\ni : \u2115\nh : i < n + 1\nh' : \u00aci < n\n\u22a2 snoc v a { val := i, isLt := h } = cons a v (\u2191(finRotate (n + 1)) { val := i, isLt := h })\n[PROOFSTEP]\nhave h'' : n = i := by\n simp only [not_lt] at h' \n exact (Nat.eq_of_le_of_lt_succ h' h).symm\n[GOAL]\nm n : \u2115\n\u03b1 : Type u_1\nv : Fin n \u2192 \u03b1\na : \u03b1\ni : \u2115\nh : i < n + 1\nh' : \u00aci < n\n\u22a2 n = i\n[PROOFSTEP]\nsimp only [not_lt] at h' \n[GOAL]\nm n : \u2115\n\u03b1 : Type u_1\nv : Fin n \u2192 \u03b1\na : \u03b1\ni : \u2115\nh : i < n + 1\nh' : n \u2264 i\n\u22a2 n = i\n[PROOFSTEP]\nexact (Nat.eq_of_le_of_lt_succ h' h).symm\n[GOAL]\ncase neg\nm n : \u2115\n\u03b1 : Type u_1\nv : Fin n \u2192 \u03b1\na : \u03b1\ni : \u2115\nh : i < n + 1\nh' : \u00aci < n\nh'' : n = i\n\u22a2 snoc v a { val := i, isLt := h } = cons a v (\u2191(finRotate (n + 1)) { val := i, isLt := h })\n[PROOFSTEP]\nsubst h''\n[GOAL]\ncase neg\nm n : \u2115\n\u03b1 : Type u_1\nv : Fin n \u2192 \u03b1\na : \u03b1\nh : n < n + 1\nh' : \u00acn < n\n\u22a2 snoc v a { val := n, isLt := h } = cons a v (\u2191(finRotate (n + 1)) { val := n, isLt := h })\n[PROOFSTEP]\nrw [finRotate_last', Fin.snoc, Fin.cons, dif_neg (lt_irrefl _)]\n[GOAL]\ncase neg\nm n : \u2115\n\u03b1 : Type u_1\nv : Fin n \u2192 \u03b1\na : \u03b1\nh : n < n + 1\nh' : \u00acn < n\n\u22a2 _root_.cast (_ : \u03b1 = \u03b1) a = cases a v { val := 0, isLt := (_ : 0 < Nat.succ n) }\n[PROOFSTEP]\nrfl\n[GOAL]\nm n : \u2115\ni : Fin (n + 1)\n\u22a2 \u2191(finRotate (n + 1)) i = i + 1\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nm : \u2115\ni : Fin (Nat.zero + 1)\n\u22a2 \u2191(finRotate (Nat.zero + 1)) i = i + 1\n[PROOFSTEP]\nexact @Subsingleton.elim (Fin 1) _ _ _\n[GOAL]\ncase succ\nm n\u271d : \u2115\ni : Fin (Nat.succ n\u271d + 1)\n\u22a2 \u2191(finRotate (Nat.succ n\u271d + 1)) i = i + 1\n[PROOFSTEP]\nrcases i.le_last.eq_or_lt with (rfl | h)\n[GOAL]\ncase succ.inl\nm n\u271d : \u2115\n\u22a2 \u2191(finRotate (Nat.succ n\u271d + 1)) (Fin.last (n\u271d + 1)) = Fin.last (n\u271d + 1) + 1\n[PROOFSTEP]\nsimp [finRotate_last]\n[GOAL]\ncase succ.inr\nm n\u271d : \u2115\ni : Fin (Nat.succ n\u271d + 1)\nh : i < Fin.last (n\u271d + 1)\n\u22a2 \u2191(finRotate (Nat.succ n\u271d + 1)) i = i + 1\n[PROOFSTEP]\ncases i\n[GOAL]\ncase succ.inr.mk\nm n\u271d val\u271d : \u2115\nisLt\u271d : val\u271d < Nat.succ n\u271d + 1\nh : { val := val\u271d, isLt := isLt\u271d } < Fin.last (n\u271d + 1)\n\u22a2 \u2191(finRotate (Nat.succ n\u271d + 1)) { val := val\u271d, isLt := isLt\u271d } = { val := val\u271d, isLt := isLt\u271d } + 1\n[PROOFSTEP]\nsimp only [Fin.lt_iff_val_lt_val, Fin.val_last, Fin.val_mk] at h \n[GOAL]\ncase succ.inr.mk\nm n\u271d val\u271d : \u2115\nisLt\u271d : val\u271d < Nat.succ n\u271d + 1\nh : val\u271d < n\u271d + 1\n\u22a2 \u2191(finRotate (Nat.succ n\u271d + 1)) { val := val\u271d, isLt := isLt\u271d } = { val := val\u271d, isLt := isLt\u271d } + 1\n[PROOFSTEP]\nsimp [finRotate_of_lt h, Fin.eq_iff_veq, Fin.add_def, Nat.mod_eq_of_lt (Nat.succ_lt_succ h)]\n[GOAL]\nm n : \u2115\n\u22a2 \u2191(finRotate (Nat.succ n)) 0 = 1\n[PROOFSTEP]\nrw [finRotate_succ_apply, zero_add]\n[GOAL]\nm n : \u2115\ni : Fin (Nat.succ n)\nh : i \u2260 Fin.last n\n\u22a2 \u2191(\u2191(finRotate (n + 1)) i) = \u2191i + 1\n[PROOFSTEP]\nrw [finRotate_succ_apply]\n[GOAL]\nm n : \u2115\ni : Fin (Nat.succ n)\nh : i \u2260 Fin.last n\n\u22a2 \u2191(i + 1) = \u2191i + 1\n[PROOFSTEP]\nhave : (i : \u2115) < n := Fin.val_lt_last h\n[GOAL]\nm n : \u2115\ni : Fin (Nat.succ n)\nh : i \u2260 Fin.last n\nthis : \u2191i < n\n\u22a2 \u2191(i + 1) = \u2191i + 1\n[PROOFSTEP]\nexact Fin.val_add_one_of_lt this\n[GOAL]\nm n : \u2115\ni : Fin (Nat.succ n)\n\u22a2 \u2191(\u2191(finRotate (Nat.succ n)) i) = if i = Fin.last n then 0 else \u2191i + 1\n[PROOFSTEP]\nrw [finRotate_succ_apply, Fin.val_add_one i]\n[GOAL]\nm n : \u2115\nx : Fin m \u00d7 Fin n\n\u22a2 \u2191x.snd + n * \u2191x.fst + 1 = \u2191x.fst * n + \u2191x.snd + 1\n[PROOFSTEP]\nac_rfl\n[GOAL]\nm n : \u2115\nx\u271d : Fin m \u00d7 Fin n\nx : Fin m\ny : Fin n\nH : 0 < n\n\u22a2 \u2191y / n + \u2191x = 0 + \u2191x\n[PROOFSTEP]\nrw [Nat.div_eq_of_lt y.2]\n[GOAL]\nm n\u271d n : \u2115\ninst\u271d : NeZero n\np : \u2115 \u00d7 Fin n\n\u22a2 (fun a => (a / n, \u2191a)) ((fun p => p.fst * n + \u2191p.snd) p) = p\n[PROOFSTEP]\nrefine' Prod.ext _ (Fin.ext <| Nat.mul_add_mod_of_lt p.2.is_lt)\n[GOAL]\nm n\u271d n : \u2115\ninst\u271d : NeZero n\np : \u2115 \u00d7 Fin n\n\u22a2 ((fun a => (a / n, \u2191a)) ((fun p => p.fst * n + \u2191p.snd) p)).fst = p.fst\n[PROOFSTEP]\ndsimp only\n[GOAL]\nm n\u271d n : \u2115\ninst\u271d : NeZero n\np : \u2115 \u00d7 Fin n\n\u22a2 (p.fst * n + \u2191p.snd) / n = p.fst\n[PROOFSTEP]\nrw [add_comm, Nat.add_mul_div_right _ _ (NeZero.pos n), Nat.div_eq_of_lt p.2.is_lt, zero_add]\n[GOAL]\nm n\u271d n : \u2115\ninst\u271d : NeZero n\na : \u2124\n\u22a2 (fun p => p.fst * \u2191n + \u2191\u2191p.snd) ((fun a => (a / \u2191n, \u2191(natMod a \u2191n))) a) = a\n[PROOFSTEP]\nsimp_rw [Fin.coe_ofNat_eq_mod, Int.coe_nat_mod, Int.natMod, Int.toNat_of_nonneg (Int.emod_nonneg _ <| NeZero.ne \u2191n),\n Int.emod_emod, Int.ediv_add_emod']\n[GOAL]\nm n\u271d n : \u2115\ninst\u271d : NeZero n\nx\u271d : \u2124 \u00d7 Fin n\nq : \u2124\nr : \u2115\nhrn : r < n\n\u22a2 (fun a => (a / \u2191n, \u2191(natMod a \u2191n))) ((fun p => p.fst * \u2191n + \u2191\u2191p.snd) (q, { val := r, isLt := hrn })) =\n (q, { val := r, isLt := hrn })\n[PROOFSTEP]\nsimp only [Fin.val_mk, Prod.mk.inj_iff, Fin.ext_iff]\n[GOAL]\nm n\u271d n : \u2115\ninst\u271d : NeZero n\nx\u271d : \u2124 \u00d7 Fin n\nq : \u2124\nr : \u2115\nhrn : r < n\n\u22a2 (q * \u2191n + \u2191r) / \u2191n = q \u2227 \u2191\u2191(natMod (q * \u2191n + \u2191r) \u2191n) = r\n[PROOFSTEP]\nobtain \u27e8h1, h2\u27e9 := Int.coe_nat_nonneg r, Int.ofNat_lt.2 hrn\n[GOAL]\nm n\u271d n : \u2115\ninst\u271d : NeZero n\nx\u271d : \u2124 \u00d7 Fin n\nq : \u2124\nr : \u2115\nhrn : r < n\nh1 : 0 \u2264 \u2191r\nh2 : \u2191r < \u2191n\n\u22a2 (q * \u2191n + \u2191r) / \u2191n = q \u2227 \u2191\u2191(natMod (q * \u2191n + \u2191r) \u2191n) = r\n[PROOFSTEP]\nrw [add_comm, Int.add_mul_ediv_right _ _ (NeZero.ne \u2191n), Int.ediv_eq_zero_of_lt h1 h2, Int.natMod,\n Int.add_mul_emod_self, Int.emod_eq_of_lt h1 h2, Int.toNat_coe_nat]\n[GOAL]\nm n\u271d n : \u2115\ninst\u271d : NeZero n\nx\u271d : \u2124 \u00d7 Fin n\nq : \u2124\nr : \u2115\nhrn : r < n\nh1 : 0 \u2264 \u2191r\nh2 : \u2191r < \u2191n\n\u22a2 0 + q = q \u2227 \u2191\u2191r = r\n[PROOFSTEP]\nexact \u27e8zero_add q, Fin.val_cast_of_lt hrn\u27e9\n[GOAL]\nm\u271d n\u271d n m : \u2115\nh : n \u2264 m\ni : Fin n\n\u22a2 \u2191(castLE h i) < n\n[PROOFSTEP]\nsimp\n[GOAL]\nm\u271d n\u271d n m : \u2115\nh : n \u2264 m\nx\u271d : Fin n\n\u22a2 (fun i => { val := \u2191\u2191i, isLt := (_ : \u2191\u2191i < n) }) ((fun i => { val := castLE h i, property := (_ : \u2191i < n) }) x\u271d) = x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nm\u271d n\u271d n m : \u2115\nh : n \u2264 m\nx\u271d : { i // \u2191i < n }\n\u22a2 (fun i => { val := castLE h i, property := (_ : \u2191i < n) }) ((fun i => { val := \u2191\u2191i, isLt := (_ : \u2191\u2191i < n) }) x\u271d) = x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nm\u271d n\u271d n m : \u2115\nh : n \u2264 m\n\u22a2 \u2200 {a b : Fin n},\n \u2191{ toFun := fun i => { val := castLE h i, property := (_ : \u2191i < n) },\n invFun := fun i => { val := \u2191\u2191i, isLt := (_ : \u2191\u2191i < n) },\n left_inv := (_ : \u2200 (x : Fin n), { val := \u2191x, isLt := (_ : \u2191x < n) } = x),\n right_inv :=\n (_ :\n \u2200 (x : { i // \u2191i < n }),\n { val := castLE h ((fun i => { val := \u2191\u2191i, isLt := (_ : \u2191\u2191i < n) }) x),\n property := (_ : \u2191((fun i => { val := \u2191\u2191i, isLt := (_ : \u2191\u2191i < n) }) x) < n) } =\n x) }\n a \u2264\n \u2191{ toFun := fun i => { val := castLE h i, property := (_ : \u2191i < n) },\n invFun := fun i => { val := \u2191\u2191i, isLt := (_ : \u2191\u2191i < n) },\n left_inv := (_ : \u2200 (x : Fin n), { val := \u2191x, isLt := (_ : \u2191x < n) } = x),\n right_inv :=\n (_ :\n \u2200 (x : { i // \u2191i < n }),\n { val := castLE h ((fun i => { val := \u2191\u2191i, isLt := (_ : \u2191\u2191i < n) }) x),\n property := (_ : \u2191((fun i => { val := \u2191\u2191i, isLt := (_ : \u2191\u2191i < n) }) x) < n) } =\n x) }\n b \u2194\n a \u2264 b\n[PROOFSTEP]\nsimp [(strictMono_castLE h).le_iff_le]\n", "meta": {"mathlib_filename": "Mathlib.Logic.Equiv.Fin", "llama_tokens": 8543, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.41565459927447623}} {"text": "[GOAL]\nJ : Type v\u2081\n\u22a2 IsConnected (WidePullbackShape J)\n[PROOFSTEP]\napply IsConnected.of_induct\n[GOAL]\ncase h\nJ : Type v\u2081\n\u22a2 \u2200 (p : Set (WidePullbackShape J)),\n ?j\u2080 \u2208 p \u2192 (\u2200 {j\u2081 j\u2082 : WidePullbackShape J}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)) \u2192 \u2200 (j : WidePullbackShape J), j \u2208 p\ncase j\u2080 J : Type v\u2081 \u22a2 WidePullbackShape J\n[PROOFSTEP]\nintrov hp t\n[GOAL]\ncase h\nJ : Type v\u2081\np : Set (WidePullbackShape J)\nhp : ?j\u2080 \u2208 p\nt : \u2200 {j\u2081 j\u2082 : WidePullbackShape J}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)\nj : WidePullbackShape J\n\u22a2 j \u2208 p\ncase j\u2080 J : Type v\u2081 \u22a2 WidePullbackShape J\n[PROOFSTEP]\ncases j\n[GOAL]\ncase h.none\nJ : Type v\u2081\np : Set (WidePullbackShape J)\nhp : ?j\u2080 \u2208 p\nt : \u2200 {j\u2081 j\u2082 : WidePullbackShape J}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)\n\u22a2 none \u2208 p\n[PROOFSTEP]\nexact hp\n[GOAL]\ncase h.some\nJ : Type v\u2081\np : Set (WidePullbackShape J)\nhp : none \u2208 p\nt : \u2200 {j\u2081 j\u2082 : WidePullbackShape J}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)\nval\u271d : J\n\u22a2 some val\u271d \u2208 p\n[PROOFSTEP]\nrwa [t (WidePullbackShape.Hom.term _)]\n[GOAL]\nJ : Type v\u2081\n\u22a2 IsConnected (WidePushoutShape J)\n[PROOFSTEP]\napply IsConnected.of_induct\n[GOAL]\ncase h\nJ : Type v\u2081\n\u22a2 \u2200 (p : Set (WidePushoutShape J)),\n ?j\u2080 \u2208 p \u2192 (\u2200 {j\u2081 j\u2082 : WidePushoutShape J}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)) \u2192 \u2200 (j : WidePushoutShape J), j \u2208 p\ncase j\u2080 J : Type v\u2081 \u22a2 WidePushoutShape J\n[PROOFSTEP]\nintrov hp t\n[GOAL]\ncase h\nJ : Type v\u2081\np : Set (WidePushoutShape J)\nhp : ?j\u2080 \u2208 p\nt : \u2200 {j\u2081 j\u2082 : WidePushoutShape J}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)\nj : WidePushoutShape J\n\u22a2 j \u2208 p\ncase j\u2080 J : Type v\u2081 \u22a2 WidePushoutShape J\n[PROOFSTEP]\ncases j\n[GOAL]\ncase h.none\nJ : Type v\u2081\np : Set (WidePushoutShape J)\nhp : ?j\u2080 \u2208 p\nt : \u2200 {j\u2081 j\u2082 : WidePushoutShape J}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)\n\u22a2 none \u2208 p\n[PROOFSTEP]\nexact hp\n[GOAL]\ncase h.some\nJ : Type v\u2081\np : Set (WidePushoutShape J)\nhp : none \u2208 p\nt : \u2200 {j\u2081 j\u2082 : WidePushoutShape J}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)\nval\u271d : J\n\u22a2 some val\u271d \u2208 p\n[PROOFSTEP]\nrwa [\u2190 t (WidePushoutShape.Hom.init _)]\n[GOAL]\n\u22a2 IsConnected WalkingParallelPair\n[PROOFSTEP]\napply IsConnected.of_induct\n[GOAL]\ncase h\n\u22a2 \u2200 (p : Set WalkingParallelPair),\n ?j\u2080 \u2208 p \u2192 (\u2200 {j\u2081 j\u2082 : WalkingParallelPair}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)) \u2192 \u2200 (j : WalkingParallelPair), j \u2208 p\ncase j\u2080 \u22a2 WalkingParallelPair\n[PROOFSTEP]\nintrov _ t\n[GOAL]\ncase h\np : Set WalkingParallelPair\na\u271d : ?j\u2080 \u2208 p\nt : \u2200 {j\u2081 j\u2082 : WalkingParallelPair}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)\nj : WalkingParallelPair\n\u22a2 j \u2208 p\ncase j\u2080 \u22a2 WalkingParallelPair\n[PROOFSTEP]\ncases j\n[GOAL]\ncase h.zero\np : Set WalkingParallelPair\na\u271d : ?j\u2080 \u2208 p\nt : \u2200 {j\u2081 j\u2082 : WalkingParallelPair}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)\n\u22a2 WalkingParallelPair.zero \u2208 p\n[PROOFSTEP]\nrwa [t WalkingParallelPairHom.left]\n[GOAL]\ncase h.one\np : Set WalkingParallelPair\na\u271d : WalkingParallelPair.one \u2208 p\nt : \u2200 {j\u2081 j\u2082 : WalkingParallelPair}, (j\u2081 \u27f6 j\u2082) \u2192 (j\u2081 \u2208 p \u2194 j\u2082 \u2208 p)\n\u22a2 WalkingParallelPair.one \u2208 p\n[PROOFSTEP]\nassumption\n[GOAL]\nC : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} C\ninst\u271d\u00b2 : HasBinaryProducts C\nJ : Type v\u2082\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsConnected J\nX : C\nK : J \u2964 C\nc : Cone K\nl : IsLimit c\ns : Cone (K \u22d9 prod.functor.obj X)\nj : J\n\u22a2 (fun s => prod.lift (NatTrans.app s.\u03c0 (Classical.arbitrary J) \u226b prod.fst) (IsLimit.lift l (forgetCone s))) s \u226b\n NatTrans.app ((prod.functor.obj X).mapCone c).\u03c0 j =\n NatTrans.app s.\u03c0 j\n[PROOFSTEP]\napply prod.hom_ext\n[GOAL]\ncase h\u2081\nC : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} C\ninst\u271d\u00b2 : HasBinaryProducts C\nJ : Type v\u2082\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsConnected J\nX : C\nK : J \u2964 C\nc : Cone K\nl : IsLimit c\ns : Cone (K \u22d9 prod.functor.obj X)\nj : J\n\u22a2 ((fun s => prod.lift (NatTrans.app s.\u03c0 (Classical.arbitrary J) \u226b prod.fst) (IsLimit.lift l (forgetCone s))) s \u226b\n NatTrans.app ((prod.functor.obj X).mapCone c).\u03c0 j) \u226b\n prod.fst =\n NatTrans.app s.\u03c0 j \u226b prod.fst\n[PROOFSTEP]\nerw [assoc, limMap_\u03c0, comp_id, limit.lift_\u03c0]\n[GOAL]\ncase h\u2081\nC : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} C\ninst\u271d\u00b2 : HasBinaryProducts C\nJ : Type v\u2082\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsConnected J\nX : C\nK : J \u2964 C\nc : Cone K\nl : IsLimit c\ns : Cone (K \u22d9 prod.functor.obj X)\nj : J\n\u22a2 NatTrans.app (BinaryFan.mk (NatTrans.app s.\u03c0 (Classical.arbitrary J) \u226b prod.fst) (IsLimit.lift l (forgetCone s))).\u03c0\n { as := WalkingPair.left } =\n NatTrans.app s.\u03c0 j \u226b prod.fst\n[PROOFSTEP]\nexact (nat_trans_from_is_connected (s.\u03c0 \u226b \u03b3\u2081 X) j (Classical.arbitrary _)).symm\n[GOAL]\ncase h\u2082\nC : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} C\ninst\u271d\u00b2 : HasBinaryProducts C\nJ : Type v\u2082\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsConnected J\nX : C\nK : J \u2964 C\nc : Cone K\nl : IsLimit c\ns : Cone (K \u22d9 prod.functor.obj X)\nj : J\n\u22a2 ((fun s => prod.lift (NatTrans.app s.\u03c0 (Classical.arbitrary J) \u226b prod.fst) (IsLimit.lift l (forgetCone s))) s \u226b\n NatTrans.app ((prod.functor.obj X).mapCone c).\u03c0 j) \u226b\n prod.snd =\n NatTrans.app s.\u03c0 j \u226b prod.snd\n[PROOFSTEP]\nsimp [\u2190 l.fac (forgetCone s) j]\n[GOAL]\nC : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} C\ninst\u271d\u00b2 : HasBinaryProducts C\nJ : Type v\u2082\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsConnected J\nX : C\nK : J \u2964 C\nc : Cone K\nl : IsLimit c\ns : Cone (K \u22d9 prod.functor.obj X)\nm : s.pt \u27f6 ((prod.functor.obj X).mapCone c).pt\nL : \u2200 (j : J), m \u226b NatTrans.app ((prod.functor.obj X).mapCone c).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m = (fun s => prod.lift (NatTrans.app s.\u03c0 (Classical.arbitrary J) \u226b prod.fst) (IsLimit.lift l (forgetCone s))) s\n[PROOFSTEP]\napply prod.hom_ext\n[GOAL]\ncase h\u2081\nC : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} C\ninst\u271d\u00b2 : HasBinaryProducts C\nJ : Type v\u2082\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsConnected J\nX : C\nK : J \u2964 C\nc : Cone K\nl : IsLimit c\ns : Cone (K \u22d9 prod.functor.obj X)\nm : s.pt \u27f6 ((prod.functor.obj X).mapCone c).pt\nL : \u2200 (j : J), m \u226b NatTrans.app ((prod.functor.obj X).mapCone c).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m \u226b prod.fst =\n (fun s => prod.lift (NatTrans.app s.\u03c0 (Classical.arbitrary J) \u226b prod.fst) (IsLimit.lift l (forgetCone s))) s \u226b\n prod.fst\n[PROOFSTEP]\nerw [limit.lift_\u03c0, \u2190 L (Classical.arbitrary J), assoc, limMap_\u03c0, comp_id]\n[GOAL]\ncase h\u2081\nC : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} C\ninst\u271d\u00b2 : HasBinaryProducts C\nJ : Type v\u2082\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsConnected J\nX : C\nK : J \u2964 C\nc : Cone K\nl : IsLimit c\ns : Cone (K \u22d9 prod.functor.obj X)\nm : s.pt \u27f6 ((prod.functor.obj X).mapCone c).pt\nL : \u2200 (j : J), m \u226b NatTrans.app ((prod.functor.obj X).mapCone c).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m \u226b prod.fst =\n NatTrans.app (BinaryFan.mk (m \u226b limit.\u03c0 (pair X c.pt) { as := WalkingPair.left }) (IsLimit.lift l (forgetCone s))).\u03c0\n { as := WalkingPair.left }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2082\nC : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} C\ninst\u271d\u00b2 : HasBinaryProducts C\nJ : Type v\u2082\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsConnected J\nX : C\nK : J \u2964 C\nc : Cone K\nl : IsLimit c\ns : Cone (K \u22d9 prod.functor.obj X)\nm : s.pt \u27f6 ((prod.functor.obj X).mapCone c).pt\nL : \u2200 (j : J), m \u226b NatTrans.app ((prod.functor.obj X).mapCone c).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m \u226b prod.snd =\n (fun s => prod.lift (NatTrans.app s.\u03c0 (Classical.arbitrary J) \u226b prod.fst) (IsLimit.lift l (forgetCone s))) s \u226b\n prod.snd\n[PROOFSTEP]\nrw [limit.lift_\u03c0]\n[GOAL]\ncase h\u2082\nC : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} C\ninst\u271d\u00b2 : HasBinaryProducts C\nJ : Type v\u2082\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsConnected J\nX : C\nK : J \u2964 C\nc : Cone K\nl : IsLimit c\ns : Cone (K \u22d9 prod.functor.obj X)\nm : s.pt \u27f6 ((prod.functor.obj X).mapCone c).pt\nL : \u2200 (j : J), m \u226b NatTrans.app ((prod.functor.obj X).mapCone c).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m \u226b prod.snd =\n NatTrans.app (BinaryFan.mk (NatTrans.app s.\u03c0 (Classical.arbitrary J) \u226b prod.fst) (IsLimit.lift l (forgetCone s))).\u03c0\n { as := WalkingPair.right }\n[PROOFSTEP]\napply l.uniq (forgetCone s)\n[GOAL]\ncase h\u2082.x\nC : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} C\ninst\u271d\u00b2 : HasBinaryProducts C\nJ : Type v\u2082\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsConnected J\nX : C\nK : J \u2964 C\nc : Cone K\nl : IsLimit c\ns : Cone (K \u22d9 prod.functor.obj X)\nm : s.pt \u27f6 ((prod.functor.obj X).mapCone c).pt\nL : \u2200 (j : J), m \u226b NatTrans.app ((prod.functor.obj X).mapCone c).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 \u2200 (j : J), (m \u226b prod.snd) \u226b NatTrans.app c.\u03c0 j = NatTrans.app (forgetCone s).\u03c0 j\n[PROOFSTEP]\nintro j\n[GOAL]\ncase h\u2082.x\nC : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} C\ninst\u271d\u00b2 : HasBinaryProducts C\nJ : Type v\u2082\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsConnected J\nX : C\nK : J \u2964 C\nc : Cone K\nl : IsLimit c\ns : Cone (K \u22d9 prod.functor.obj X)\nm : s.pt \u27f6 ((prod.functor.obj X).mapCone c).pt\nL : \u2200 (j : J), m \u226b NatTrans.app ((prod.functor.obj X).mapCone c).\u03c0 j = NatTrans.app s.\u03c0 j\nj : J\n\u22a2 (m \u226b prod.snd) \u226b NatTrans.app c.\u03c0 j = NatTrans.app (forgetCone s).\u03c0 j\n[PROOFSTEP]\nsimp [\u2190 L j]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Connected", "llama_tokens": 4156, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4156291315518877}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\nl : List \u03b1\n\u22a2 decodeList (encodeList l) = some l\n[PROOFSTEP]\ninduction' l with a l IH\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\n\u22a2 decodeList (encodeList []) = some []\n[PROOFSTEP]\nsimp [encodeList, decodeList, unpair_pair, encodek, *]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\na : \u03b1\nl : List \u03b1\nIH : decodeList (encodeList l) = some l\n\u22a2 decodeList (encodeList (a :: l)) = some (a :: l)\n[PROOFSTEP]\nsimp [encodeList, decodeList, unpair_pair, encodek, *]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : Encodable \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : Countable \u03b1\n\u22a2 Countable (List \u03b1)\n[PROOFSTEP]\nhaveI := Encodable.ofCountable \u03b1\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : Encodable \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : Countable \u03b1\nthis : Encodable \u03b1\n\u22a2 Countable (List \u03b1)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\n\u22a2 decodeList 0 = some []\n[PROOFSTEP]\nrw [decodeList]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\nv : \u2115\n\u22a2 decodeList (succ v) = Seq.seq ((fun x x_1 => x :: x_1) <$> decode (unpair v).fst) fun x => decode (unpair v).snd\n[PROOFSTEP]\ncases' e : unpair v with v\u2081 v\u2082\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\n\u22a2 decodeList (succ v) = Seq.seq ((fun x x_1 => x :: x_1) <$> decode (v\u2081, v\u2082).fst) fun x => decode (v\u2081, v\u2082).snd\n[PROOFSTEP]\nsimp [decodeList, e]\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\n\u22a2 (Seq.seq (Option.map (fun x x_1 => x :: x_1) (decode v\u2081)) fun x => decodeList v\u2082) =\n Seq.seq (Option.map (fun x x_1 => x :: x_1) (decode v\u2081)) fun x => decode v\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\na b : \u03b1\n\u22a2 Decidable (Encodable.enle a b)\n[PROOFSTEP]\nunfold enle Order.Preimage\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\na b : \u03b1\n\u22a2 Decidable ((fun x x_1 => x \u2264 x_1) (encode a) (encode b))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\ns : Multiset \u03b1\n\u22a2 decodeMultiset (encodeMultiset s) = some s\n[PROOFSTEP]\nsimp [encodeMultiset, decodeMultiset, encodek]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Fintype \u03b1\n\u22a2 Encodable \u03b1\n[PROOFSTEP]\nclassical exact (Fintype.truncEncodable \u03b1).out\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Fintype \u03b1\n\u22a2 Encodable \u03b1\n[PROOFSTEP]\nexact (Fintype.truncEncodable \u03b1).out\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\n\u22a2 \u2203 a, a \u2208 decodeList 0 \u2227 encodeList a = 0\n[PROOFSTEP]\nrw [decodeList]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\n\u22a2 \u2203 a, a \u2208 some [] \u2227 encodeList a = 0\n[PROOFSTEP]\nexact \u27e8_, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv : \u2115\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v\n[PROOFSTEP]\ncases' e : unpair v with v\u2081 v\u2082\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v\n[PROOFSTEP]\nhave h := unpair_right_le v\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\nh : (unpair v).snd \u2264 v\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v\n[PROOFSTEP]\nrw [e] at h \n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\nh : (v\u2081, v\u2082).snd \u2264 v\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v\n[PROOFSTEP]\nrcases have : v\u2082 < succ v := lt_succ_of_le h\n denumerable_list_aux v\u2082 with\n \u27e8a, h\u2081, h\u2082\u27e9\n[GOAL]\ncase mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\nh : (v\u2081, v\u2082).snd \u2264 v\na : List \u03b1\nh\u2081 : a \u2208 decodeList v\u2082\nh\u2082 : encodeList a = v\u2082\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v\n[PROOFSTEP]\nrw [Option.mem_def] at h\u2081 \n[GOAL]\ncase mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\nh : (v\u2081, v\u2082).snd \u2264 v\na : List \u03b1\nh\u2081 : decodeList v\u2082 = some a\nh\u2082 : encodeList a = v\u2082\n\u22a2 \u2203 a, a \u2208 decodeList (succ v) \u2227 encodeList a = succ v\n[PROOFSTEP]\nuse ofNat \u03b1 v\u2081 :: a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\nh : (v\u2081, v\u2082).snd \u2264 v\na : List \u03b1\nh\u2081 : decodeList v\u2082 = some a\nh\u2082 : encodeList a = v\u2082\n\u22a2 ofNat \u03b1 v\u2081 :: a \u2208 decodeList (succ v) \u2227 encodeList (ofNat \u03b1 v\u2081 :: a) = succ v\n[PROOFSTEP]\nsimp [decodeList, e, h\u2082, h\u2081, encodeList, pair_unpair' e]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\n\u22a2 ofNat (List \u03b1) 0 = []\n[PROOFSTEP]\nrw [\u2190 @encode_list_nil \u03b1, ofNat_encode]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv : \u2115\n\u22a2 decodeList (succ v) = some (ofNat \u03b1 (unpair v).fst :: ofNat (List \u03b1) (unpair v).snd)\n[PROOFSTEP]\ncases' e : unpair v with v\u2081 v\u2082\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\n\u22a2 decodeList (succ v) = some (ofNat \u03b1 (v\u2081, v\u2082).fst :: ofNat (List \u03b1) (v\u2081, v\u2082).snd)\n[PROOFSTEP]\nsimp [decodeList, e]\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nv v\u2081 v\u2082 : \u2115\ne : unpair v = (v\u2081, v\u2082)\n\u22a2 (Seq.seq (some fun x => ofNat \u03b1 v\u2081 :: x) fun x => decodeList v\u2082) = some (ofNat \u03b1 v\u2081 :: ofNat (List \u03b1) v\u2082)\n[PROOFSTEP]\nrw [show decodeList v\u2082 = decode (\u03b1 := List \u03b1) v\u2082 from rfl, decode_eq_ofNat, Option.seq_some, Option.some.injEq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nm : \u2115\nl : List \u2115\nn : \u2115\n\u22a2 lower (raise (m :: l) n) n = m :: l\n[PROOFSTEP]\nrw [raise, lower, add_tsub_cancel_right, lower_raise l]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nm : \u2115\nl : List \u2115\nn : \u2115\nh : Sorted (fun x x_1 => x \u2264 x_1) (n :: m :: l)\n\u22a2 raise (lower (m :: l) n) n = m :: l\n[PROOFSTEP]\nhave : n \u2264 m := List.rel_of_sorted_cons h _ (l.mem_cons_self _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nm : \u2115\nl : List \u2115\nn : \u2115\nh : Sorted (fun x x_1 => x \u2264 x_1) (n :: m :: l)\nthis : n \u2264 m\n\u22a2 raise (lower (m :: l) n) n = m :: l\n[PROOFSTEP]\nsimp [raise, lower, tsub_add_cancel_of_le this, raise_lower h.of_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\ns : Multiset \u03b1\n\u22a2 (fun n => Multiset.map (ofNat \u03b1) \u2191(raise (ofNat (List \u2115) n) 0))\n ((fun s => encode (lower (Multiset.sort (fun x x_1 => x \u2264 x_1) (Multiset.map encode s)) 0)) s) =\n s\n[PROOFSTEP]\nhave := raise_lower (List.sorted_cons.2 \u27e8fun n _ => Nat.zero_le n, (s.map encode).sort_sorted _\u27e9)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\ns : Multiset \u03b1\nthis :\n raise (lower (Multiset.sort (fun x x_1 => x \u2264 x_1) (Multiset.map encode s)) 0) 0 =\n Multiset.sort (fun x x_1 => x \u2264 x_1) (Multiset.map encode s)\n\u22a2 (fun n => Multiset.map (ofNat \u03b1) \u2191(raise (ofNat (List \u2115) n) 0))\n ((fun s => encode (lower (Multiset.sort (fun x x_1 => x \u2264 x_1) (Multiset.map encode s)) 0)) s) =\n s\n[PROOFSTEP]\nsimp [-Multiset.coe_map, this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nn : \u2115\n\u22a2 (fun s => encode (lower (Multiset.sort (fun x x_1 => x \u2264 x_1) (Multiset.map encode s)) 0))\n ((fun n => Multiset.map (ofNat \u03b1) \u2191(raise (ofNat (List \u2115) n) 0)) n) =\n n\n[PROOFSTEP]\nsimp [-Multiset.coe_map, List.mergeSort_eq_self _ (raise_sorted _ _), lower_raise]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nm : \u2115\nl : List \u2115\nn : \u2115\n\u22a2 lower' (raise' (m :: l) n) n = m :: l\n[PROOFSTEP]\nsimp [raise', lower', add_tsub_cancel_right, lower_raise']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nm : \u2115\nl : List \u2115\nn : \u2115\nh\u2081 : \u2200 (m_1 : \u2115), m_1 \u2208 m :: l \u2192 n \u2264 m_1\nh\u2082 : Sorted (fun x x_1 => x < x_1) (m :: l)\n\u22a2 raise' (lower' (m :: l) n) n = m :: l\n[PROOFSTEP]\nhave : n \u2264 m := h\u2081 _ (l.mem_cons_self _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nm : \u2115\nl : List \u2115\nn : \u2115\nh\u2081 : \u2200 (m_1 : \u2115), m_1 \u2208 m :: l \u2192 n \u2264 m_1\nh\u2082 : Sorted (fun x x_1 => x < x_1) (m :: l)\nthis : n \u2264 m\n\u22a2 raise' (lower' (m :: l) n) n = m :: l\n[PROOFSTEP]\nsimp [raise', lower', tsub_add_cancel_of_le this, raise_lower' (List.rel_of_sorted_cons h\u2082 : \u2200 a \u2208 l, m < a) h\u2082.of_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\ns : Finset \u03b1\n\u22a2 ((fun n => Finset.map (Equiv.toEmbedding (eqv \u03b1).symm) (raise'Finset (ofNat (List \u2115) n) 0))\n ((fun s => encode (lower' (Finset.sort (fun x x_1 => x \u2264 x_1) (Finset.map (Equiv.toEmbedding (eqv \u03b1)) s)) 0))\n s)).val =\n s.val\n[PROOFSTEP]\nsimp [-Multiset.coe_map, raise'Finset, raise_lower' (fun n _ => Nat.zero_le n) (Finset.sort_sorted_lt _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nn : \u2115\n\u22a2 (fun s => encode (lower' (Finset.sort (fun x x_1 => x \u2264 x_1) (Finset.map (Equiv.toEmbedding (eqv \u03b1)) s)) 0))\n ((fun n => Finset.map (Equiv.toEmbedding (eqv \u03b1).symm) (raise'Finset (ofNat (List \u2115) n) 0)) n) =\n n\n[PROOFSTEP]\nsimp [-Multiset.coe_map, Finset.map, raise'Finset, Finset.sort,\n List.mergeSort_eq_self (\u00b7 \u2264 \u00b7) ((raise'_sorted _ _).imp (@le_of_lt _ _)), lower_raise']\n[GOAL]\nu : List Unit\n\u22a2 length ((fun n => replicate n ()) (length u)) = length u\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Logic.Equiv.List", "llama_tokens": 4509, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195269001831, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.41552525927489664}} {"text": "[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring S\ninst\u271d\u2079 : Semiring R\u2081\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : AddCommMonoid M\u2084\ninst\u271d\u00b9 : AddCommMonoid N\u2081\ninst\u271d : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne\u271d e' e : M \u2243\u209b\u2097[\u03c3] M\u2082\nsrc\u271d\u00b9 : M\u2082 \u2192\u209b\u2097[\u03c3'] M :=\n LinearMap.inverse (\u2191e) e.invFun (_ : LeftInverse e.invFun e.toFun) (_ : Function.RightInverse e.invFun e.toFun)\nsrc\u271d : M\u2082 \u2243 M := (toEquiv e).symm\nr : S\nx : M\u2082\n\u22a2 AddHom.toFun\n {\n toFun :=\n \u2191(LinearMap.inverse (\u2191e) e.invFun (_ : LeftInverse e.invFun e.toFun)\n (_ : Function.RightInverse e.invFun e.toFun)),\n map_add' :=\n (_ :\n \u2200 (x y : M\u2082),\n AddHom.toFun src\u271d\u00b9.toAddHom (x + y) = AddHom.toFun src\u271d\u00b9.toAddHom x + AddHom.toFun src\u271d\u00b9.toAddHom y) }\n (r \u2022 x) =\n \u2191\u03c3' r \u2022\n AddHom.toFun\n {\n toFun :=\n \u2191(LinearMap.inverse (\u2191e) e.invFun (_ : LeftInverse e.invFun e.toFun)\n (_ : Function.RightInverse e.invFun e.toFun)),\n map_add' :=\n (_ :\n \u2200 (x y : M\u2082),\n AddHom.toFun src\u271d\u00b9.toAddHom (x + y) = AddHom.toFun src\u271d\u00b9.toAddHom x + AddHom.toFun src\u271d\u00b9.toAddHom y) }\n x\n[PROOFSTEP]\ndsimp only\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring S\ninst\u271d\u2079 : Semiring R\u2081\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : AddCommMonoid M\u2084\ninst\u271d\u00b9 : AddCommMonoid N\u2081\ninst\u271d : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne\u271d e' e : M \u2243\u209b\u2097[\u03c3] M\u2082\nsrc\u271d\u00b9 : M\u2082 \u2192\u209b\u2097[\u03c3'] M :=\n LinearMap.inverse (\u2191e) e.invFun (_ : LeftInverse e.invFun e.toFun) (_ : Function.RightInverse e.invFun e.toFun)\nsrc\u271d : M\u2082 \u2243 M := (toEquiv e).symm\nr : S\nx : M\u2082\n\u22a2 \u2191(LinearMap.inverse (\u2191e) e.invFun (_ : LeftInverse e.invFun e.toFun) (_ : Function.RightInverse e.invFun e.toFun))\n (r \u2022 x) =\n \u2191\u03c3' r \u2022\n \u2191(LinearMap.inverse (\u2191e) e.invFun (_ : LeftInverse e.invFun e.toFun) (_ : Function.RightInverse e.invFun e.toFun))\n x\n[PROOFSTEP]\nrw [map_smul\u209b\u2097]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ng : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083\n\u22a2 f = LinearMap.comp g \u2191(symm e\u2081\u2082) \u2194 LinearMap.comp f \u2191e\u2081\u2082 = g\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ng : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083\n\u22a2 f = LinearMap.comp g \u2191(symm e\u2081\u2082) \u2192 LinearMap.comp f \u2191e\u2081\u2082 = g\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ng : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083\n\u22a2 LinearMap.comp f \u2191e\u2081\u2082 = g \u2192 f = LinearMap.comp g \u2191(symm e\u2081\u2082)\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ng : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083\nH : f = LinearMap.comp g \u2191(symm e\u2081\u2082)\n\u22a2 LinearMap.comp f \u2191e\u2081\u2082 = g\n[PROOFSTEP]\next\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ng : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083\nH : LinearMap.comp f \u2191e\u2081\u2082 = g\n\u22a2 f = LinearMap.comp g \u2191(symm e\u2081\u2082)\n[PROOFSTEP]\next\n[GOAL]\ncase mp.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ng : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083\nH : f = LinearMap.comp g \u2191(symm e\u2081\u2082)\nx\u271d : M\u2081\n\u22a2 \u2191(LinearMap.comp f \u2191e\u2081\u2082) x\u271d = \u2191g x\u271d\n[PROOFSTEP]\nsimp [H, e\u2081\u2082.toEquiv.eq_comp_symm f g]\n[GOAL]\ncase mpr.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ng : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083\nH : LinearMap.comp f \u2191e\u2081\u2082 = g\nx\u271d : M\u2082\n\u22a2 \u2191f x\u271d = \u2191(LinearMap.comp g \u2191(symm e\u2081\u2082)) x\u271d\n[PROOFSTEP]\nsimp [\u2190 H, \u2190 e\u2081\u2082.toEquiv.eq_comp_symm f g]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ng : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083\n\u22a2 LinearMap.comp g \u2191(symm e\u2081\u2082) = f \u2194 g = LinearMap.comp f \u2191e\u2081\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ng : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083\n\u22a2 LinearMap.comp g \u2191(symm e\u2081\u2082) = f \u2192 g = LinearMap.comp f \u2191e\u2081\u2082\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ng : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083\n\u22a2 g = LinearMap.comp f \u2191e\u2081\u2082 \u2192 LinearMap.comp g \u2191(symm e\u2081\u2082) = f\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ng : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083\nH : LinearMap.comp g \u2191(symm e\u2081\u2082) = f\n\u22a2 g = LinearMap.comp f \u2191e\u2081\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ng : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083\nH : g = LinearMap.comp f \u2191e\u2081\u2082\n\u22a2 LinearMap.comp g \u2191(symm e\u2081\u2082) = f\n[PROOFSTEP]\next\n[GOAL]\ncase mp.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ng : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083\nH : LinearMap.comp g \u2191(symm e\u2081\u2082) = f\nx\u271d : M\u2081\n\u22a2 \u2191g x\u271d = \u2191(LinearMap.comp f \u2191e\u2081\u2082) x\u271d\n[PROOFSTEP]\nsimp [\u2190 H, \u2190 e\u2081\u2082.toEquiv.comp_symm_eq f g]\n[GOAL]\ncase mpr.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ng : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083\nH : g = LinearMap.comp f \u2191e\u2081\u2082\nx\u271d : M\u2082\n\u22a2 \u2191(LinearMap.comp g \u2191(symm e\u2081\u2082)) x\u271d = \u2191f x\u271d\n[PROOFSTEP]\nsimp [H, e\u2081\u2082.toEquiv.comp_symm_eq f g]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2081] M\u2081\ng : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2082] M\u2082\n\u22a2 f = LinearMap.comp (\u2191(symm e\u2081\u2082)) g \u2194 LinearMap.comp (\u2191e\u2081\u2082) f = g\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2081] M\u2081\ng : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2082] M\u2082\n\u22a2 f = LinearMap.comp (\u2191(symm e\u2081\u2082)) g \u2192 LinearMap.comp (\u2191e\u2081\u2082) f = g\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2081] M\u2081\ng : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2082] M\u2082\n\u22a2 LinearMap.comp (\u2191e\u2081\u2082) f = g \u2192 f = LinearMap.comp (\u2191(symm e\u2081\u2082)) g\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2081] M\u2081\ng : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2082] M\u2082\nH : f = LinearMap.comp (\u2191(symm e\u2081\u2082)) g\n\u22a2 LinearMap.comp (\u2191e\u2081\u2082) f = g\n[PROOFSTEP]\next\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2081] M\u2081\ng : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2082] M\u2082\nH : LinearMap.comp (\u2191e\u2081\u2082) f = g\n\u22a2 f = LinearMap.comp (\u2191(symm e\u2081\u2082)) g\n[PROOFSTEP]\next\n[GOAL]\ncase mp.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2081] M\u2081\ng : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2082] M\u2082\nH : f = LinearMap.comp (\u2191(symm e\u2081\u2082)) g\nx\u271d : M\u2083\n\u22a2 \u2191(LinearMap.comp (\u2191e\u2081\u2082) f) x\u271d = \u2191g x\u271d\n[PROOFSTEP]\nsimp [H, e\u2081\u2082.toEquiv.eq_symm_comp f g]\n[GOAL]\ncase mpr.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2081] M\u2081\ng : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2082] M\u2082\nH : LinearMap.comp (\u2191e\u2081\u2082) f = g\nx\u271d : M\u2083\n\u22a2 \u2191f x\u271d = \u2191(LinearMap.comp (\u2191(symm e\u2081\u2082)) g) x\u271d\n[PROOFSTEP]\nsimp [\u2190 H, \u2190 e\u2081\u2082.toEquiv.eq_symm_comp f g]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2081] M\u2081\ng : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2082] M\u2082\n\u22a2 LinearMap.comp (\u2191(symm e\u2081\u2082)) g = f \u2194 g = LinearMap.comp (\u2191e\u2081\u2082) f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2081] M\u2081\ng : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2082] M\u2082\n\u22a2 LinearMap.comp (\u2191(symm e\u2081\u2082)) g = f \u2192 g = LinearMap.comp (\u2191e\u2081\u2082) f\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2081] M\u2081\ng : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2082] M\u2082\n\u22a2 g = LinearMap.comp (\u2191e\u2081\u2082) f \u2192 LinearMap.comp (\u2191(symm e\u2081\u2082)) g = f\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2081] M\u2081\ng : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2082] M\u2082\nH : LinearMap.comp (\u2191(symm e\u2081\u2082)) g = f\n\u22a2 g = LinearMap.comp (\u2191e\u2081\u2082) f\n[PROOFSTEP]\next\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2081] M\u2081\ng : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2082] M\u2082\nH : g = LinearMap.comp (\u2191e\u2081\u2082) f\n\u22a2 LinearMap.comp (\u2191(symm e\u2081\u2082)) g = f\n[PROOFSTEP]\next\n[GOAL]\ncase mp.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2081] M\u2081\ng : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2082] M\u2082\nH : LinearMap.comp (\u2191(symm e\u2081\u2082)) g = f\nx\u271d : M\u2083\n\u22a2 \u2191g x\u271d = \u2191(LinearMap.comp (\u2191e\u2081\u2082) f) x\u271d\n[PROOFSTEP]\nsimp [\u2190 H, \u2190 e\u2081\u2082.toEquiv.symm_comp_eq f g]\n[GOAL]\ncase mpr.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2081] M\u2081\ng : M\u2083 \u2192\u209b\u2097[\u03c3\u2083\u2082] M\u2082\nH : g = LinearMap.comp (\u2191e\u2081\u2082) f\nx\u271d : M\u2083\n\u22a2 \u2191(LinearMap.comp (\u2191(symm e\u2081\u2082)) g) x\u271d = \u2191f x\u271d\n[PROOFSTEP]\nsimp [H, e\u2081\u2082.toEquiv.symm_comp_eq f g]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\n\u22a2 trans f (symm f) = refl R\u2081 M\u2081\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nx : M\u2081\n\u22a2 \u2191(trans f (symm f)) x = \u2191(refl R\u2081 M\u2081) x\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\n\u22a2 trans (symm f) f = refl R\u2082 M\u2082\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\nf : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nx : M\u2082\n\u22a2 \u2191(trans (symm f) f) x = \u2191(refl R\u2082 M\u2082) x\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne\u271d e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\ne : M \u2243\u209b\u2097[\u03c3] M\u2082\n\u22a2 symm (symm e) = e\n[PROOFSTEP]\ncases e\n[GOAL]\ncase mk\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : Semiring R\u2081\ninst\u271d\u00b9\u2074 : Semiring R\u2082\ninst\u271d\u00b9\u00b3 : Semiring R\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid N\u2081\ninst\u271d\u2076 : AddCommMonoid N\u2082\nmodule_M : Module R M\nmodule_S_M\u2082 : Module S M\u2082\n\u03c3 : R \u2192+* S\n\u03c3' : S \u2192+* R\nre\u2081 : RingHomInvPair \u03c3 \u03c3'\nre\u2082 : RingHomInvPair \u03c3' \u03c3\ne e' : M \u2243\u209b\u2097[\u03c3] M\u2082\nmodule_M\u2081 : Module R\u2081 M\u2081\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\nmodule_N\u2081 : Module R\u2081 N\u2081\nmodule_N\u2082 : Module R\u2081 N\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ne\u2081\u2082 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ne\u2082\u2083 : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2083 \u03c3\u2082\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2083\u2081 \u03c3\u2081\u2082 \u03c3\u2083\u2082\ntoLinearMap\u271d : M \u2192\u209b\u2097[\u03c3] M\u2082\ninvFun\u271d : M\u2082 \u2192 M\nleft_inv\u271d : LeftInverse invFun\u271d toLinearMap\u271d.toFun\nright_inv\u271d : Function.RightInverse invFun\u271d toLinearMap\u271d.toFun\n\u22a2 symm (symm { toLinearMap := toLinearMap\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }) =\n { toLinearMap := toLinearMap\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring S\ninst\u271d\u2079 : Semiring R\u2081\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Subsingleton M\ninst\u271d : Subsingleton M\u2082\n\u22a2 ofSubsingleton M M = refl R M\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\nM\u2084 : Type u_16\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring S\ninst\u271d\u2079 : Semiring R\u2081\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Subsingleton M\ninst\u271d : Subsingleton M\u2082\nx\u271d : M\n\u22a2 \u2191(ofSubsingleton M M) x\u271d = \u2191(refl R M) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\ne : M \u2243+ M\u2082\nc : \u2115\na : M\n\u22a2 \u2191e (c \u2022 a) = c \u2022 \u2191e a\n[PROOFSTEP]\nerw [e.toAddMonoidHom.map_nsmul]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\ne : M \u2243+ M\u2082\nc : \u2115\na : M\n\u22a2 c \u2022 \u2191(toAddMonoidHom e) a = c \u2022 \u2191e a\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\ne : M \u2243+ M\u2082\n\u22a2 \u2191(toNatLinearEquiv e) = e\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\ne : M \u2243+ M\u2082\nx\u271d : M\n\u22a2 \u2191\u2191(toNatLinearEquiv e) x\u271d = \u2191e x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : AddCommGroup M\u2082\ninst\u271d : AddCommGroup M\u2083\ne : M \u2243+ M\u2082\n\u22a2 \u2191(toIntLinearEquiv e) = e\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nM : Type u_7\nM\u2081 : Type u_8\nM\u2082 : Type u_9\nM\u2083 : Type u_10\nN\u2081 : Type u_11\nN\u2082 : Type u_12\nN\u2083 : Type u_13\nN\u2084 : Type u_14\n\u03b9 : Type u_15\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : AddCommGroup M\u2082\ninst\u271d : AddCommGroup M\u2083\ne : M \u2243+ M\u2082\nx\u271d : M\n\u22a2 \u2191\u2191(toIntLinearEquiv e) x\u271d = \u2191e x\u271d\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Module.Equiv", "llama_tokens": 31587, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744761936437, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.4151740189429882}} {"text": "[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh0 : comp p q = 0\n\u22a2 natDegree (comp p q) \u2264 natDegree p * natDegree q\n[PROOFSTEP]\nrw [h0, natDegree_zero]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh0 : comp p q = 0\n\u22a2 0 \u2264 natDegree p * natDegree q\n[PROOFSTEP]\nexact Nat.zero_le _\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh0 : \u00accomp p q = 0\nn : \u2115\nhn : n \u2208 support p\n\u22a2 \u2191(natDegree (\u2191C (coeff p n))) + n \u2022 \u2191(natDegree q) = \u2191(n * natDegree q)\n[PROOFSTEP]\nrw [natDegree_C, Nat.cast_zero, zero_add, nsmul_eq_mul]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh0 : \u00accomp p q = 0\nn : \u2115\nhn : n \u2208 support p\n\u22a2 \u2191n * \u2191(natDegree q) = \u2191(n * natDegree q)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nhp : p \u2260 0\nh : IsRoot p a\nhlt : 0 \u2265 degree p\n\u22a2 False\n[PROOFSTEP]\nhave := eq_C_of_degree_le_zero hlt\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nhp : p \u2260 0\nh : IsRoot p a\nhlt : 0 \u2265 degree p\nthis : p = \u2191C (coeff p 0)\n\u22a2 False\n[PROOFSTEP]\nrw [IsRoot, this, eval_C] at h \n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nhp : p \u2260 0\nh : coeff p 0 = 0\nhlt : 0 \u2265 degree p\nthis : p = \u2191C (coeff p 0)\n\u22a2 False\n[PROOFSTEP]\nsimp only [h, RingHom.map_zero] at this \n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nhp : p \u2260 0\nh : coeff p 0 = 0\nhlt : 0 \u2265 degree p\nthis : p = 0\n\u22a2 False\n[PROOFSTEP]\nexact hp this\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 natDegree p \u2264 n \u2194 \u2200 (N : \u2115), n < N \u2192 coeff p N = 0\n[PROOFSTEP]\nsimp_rw [natDegree_le_iff_degree_le, degree_le_iff_coeff_zero, Nat.cast_withBot, WithBot.coe_lt_coe]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r : R[X]\nn : \u2115\np q : R[X]\nqn : natDegree q \u2264 n\n\u22a2 natDegree (p + q) \u2264 n \u2194 natDegree p \u2264 n\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => natDegree_add_le_of_degree_le h qn\u27e9\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r : R[X]\nn : \u2115\np q : R[X]\nqn : natDegree q \u2264 n\nh : natDegree (p + q) \u2264 n\n\u22a2 natDegree p \u2264 n\n[PROOFSTEP]\nrefine' natDegree_le_iff_coeff_eq_zero.mpr fun m hm => _\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm\u271d n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r : R[X]\nn : \u2115\np q : R[X]\nqn : natDegree q \u2264 n\nh : natDegree (p + q) \u2264 n\nm : \u2115\nhm : n < m\n\u22a2 coeff p m = 0\n[PROOFSTEP]\nconvert natDegree_le_iff_coeff_eq_zero.mp h m hm using 1\n[GOAL]\ncase h.e'_2\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm\u271d n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r : R[X]\nn : \u2115\np q : R[X]\nqn : natDegree q \u2264 n\nh : natDegree (p + q) \u2264 n\nm : \u2115\nhm : n < m\n\u22a2 coeff p m = coeff (p + q) m\n[PROOFSTEP]\nrw [coeff_add, natDegree_le_iff_coeff_eq_zero.mp qn _ hm, add_zero]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r : R[X]\nn : \u2115\np q : R[X]\npn : natDegree p \u2264 n\n\u22a2 natDegree (p + q) \u2264 n \u2194 natDegree q \u2264 n\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r : R[X]\nn : \u2115\np q : R[X]\npn : natDegree p \u2264 n\n\u22a2 natDegree (q + p) \u2264 n \u2194 natDegree q \u2264 n\n[PROOFSTEP]\nexact natDegree_add_le_iff_left _ _ pn\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na\u271d b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\na : R\nf : R[X]\n\u22a2 natDegree (\u2191C a) + natDegree f = 0 + natDegree f\n[PROOFSTEP]\nrw [natDegree_C a]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na\u271d b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r f : R[X]\na : R\n\u22a2 natDegree f + natDegree (\u2191C a) = natDegree f + 0\n[PROOFSTEP]\nrw [natDegree_C a]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nai : R\nau : ai * a = 1\n\u22a2 natDegree p = natDegree (1 * p)\n[PROOFSTEP]\nnth_rw 1 [\u2190 one_mul p]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nai : R\nau : ai * a = 1\n\u22a2 natDegree (1 * p) = natDegree (\u2191C ai * (\u2191C a * p))\n[PROOFSTEP]\nrw [\u2190 C_1, \u2190 au, RingHom.map_mul, \u2190 mul_assoc]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nai : R\nau : a * ai = 1\n\u22a2 natDegree p = natDegree (p * 1)\n[PROOFSTEP]\nnth_rw 1 [\u2190 mul_one p]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nai : R\nau : a * ai = 1\n\u22a2 natDegree (p * 1) = natDegree (p * \u2191C a * \u2191C ai)\n[PROOFSTEP]\nrw [\u2190 C_1, \u2190 au, RingHom.map_mul, \u2190 mul_assoc]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : leadingCoeff p * a \u2260 0\n\u22a2 natDegree (p * \u2191C a) = natDegree p\n[PROOFSTEP]\nrefine' eq_natDegree_of_le_mem_support (natDegree_mul_C_le p a) _\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : leadingCoeff p * a \u2260 0\n\u22a2 natDegree p \u2208 support (p * \u2191C a)\n[PROOFSTEP]\nrefine' mem_support_iff.mpr _\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : leadingCoeff p * a \u2260 0\n\u22a2 coeff (p * \u2191C a) (natDegree p) \u2260 0\n[PROOFSTEP]\nrwa [coeff_mul_C]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : a * leadingCoeff p \u2260 0\n\u22a2 natDegree (\u2191C a * p) = natDegree p\n[PROOFSTEP]\nrefine' eq_natDegree_of_le_mem_support (natDegree_C_mul_le a p) _\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : a * leadingCoeff p \u2260 0\n\u22a2 natDegree p \u2208 support (\u2191C a * p)\n[PROOFSTEP]\nrefine' mem_support_iff.mpr _\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : a * leadingCoeff p \u2260 0\n\u22a2 coeff (\u2191C a * p) (natDegree p) \u2260 0\n[PROOFSTEP]\nrwa [coeff_C_mul]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r f g : R[X]\n\u22a2 coeff (f * g) (natDegree f + natDegree g) = coeff f (natDegree f) * coeff g (natDegree g)\n[PROOFSTEP]\nsimp only [coeff_natDegree, coeff_mul_degree_add_degree]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\npm : natDegree p \u2264 m\nqn : natDegree q \u2264 n\n\u22a2 coeff (p * q) (m + n) = coeff p m * coeff q n\n[PROOFSTEP]\nrcases eq_or_lt_of_le pm with (rfl | hm)\n[GOAL]\ncase inl\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nn : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nqn : natDegree q \u2264 n\npm : natDegree p \u2264 natDegree p\n\u22a2 coeff (p * q) (natDegree p + n) = coeff p (natDegree p) * coeff q n\n[PROOFSTEP]\nrcases eq_or_lt_of_le qn with (rfl | hn)\n[GOAL]\ncase inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\npm : natDegree p \u2264 m\nqn : natDegree q \u2264 n\nhm : natDegree p < m\n\u22a2 coeff (p * q) (m + n) = coeff p m * coeff q n\n[PROOFSTEP]\nrcases eq_or_lt_of_le qn with (rfl | hn)\n[GOAL]\ncase inl.inl\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\ninst\u271d : Semiring R\np q r : R[X]\npm : natDegree p \u2264 natDegree p\nqn : natDegree q \u2264 natDegree q\n\u22a2 coeff (p * q) (natDegree p + natDegree q) = coeff p (natDegree p) * coeff q (natDegree q)\n[PROOFSTEP]\nexact natDegree_add_coeff_mul _ _\n[GOAL]\ncase inl.inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nn : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nqn : natDegree q \u2264 n\npm : natDegree p \u2264 natDegree p\nhn : natDegree q < n\n\u22a2 coeff (p * q) (natDegree p + n) = coeff p (natDegree p) * coeff q n\n[PROOFSTEP]\nrw [coeff_eq_zero_of_natDegree_lt hn, mul_zero]\n[GOAL]\ncase inl.inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nn : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nqn : natDegree q \u2264 n\npm : natDegree p \u2264 natDegree p\nhn : natDegree q < n\n\u22a2 coeff (p * q) (natDegree p + n) = 0\n[PROOFSTEP]\nexact natDegree_lt_coeff_mul (add_lt_add_left hn _)\n[GOAL]\ncase inr.inl\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm : \u2115\ninst\u271d : Semiring R\np q r : R[X]\npm : natDegree p \u2264 m\nhm : natDegree p < m\nqn : natDegree q \u2264 natDegree q\n\u22a2 coeff (p * q) (m + natDegree q) = coeff p m * coeff q (natDegree q)\n[PROOFSTEP]\nrw [coeff_eq_zero_of_natDegree_lt hm, zero_mul]\n[GOAL]\ncase inr.inl\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm : \u2115\ninst\u271d : Semiring R\np q r : R[X]\npm : natDegree p \u2264 m\nhm : natDegree p < m\nqn : natDegree q \u2264 natDegree q\n\u22a2 coeff (p * q) (m + natDegree q) = 0\n[PROOFSTEP]\nexact natDegree_lt_coeff_mul (add_lt_add_right hm _)\n[GOAL]\ncase inr.inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\npm : natDegree p \u2264 m\nqn : natDegree q \u2264 n\nhm : natDegree p < m\nhn : natDegree q < n\n\u22a2 coeff (p * q) (m + n) = coeff p m * coeff q n\n[PROOFSTEP]\nrw [coeff_eq_zero_of_natDegree_lt hn, mul_zero]\n[GOAL]\ncase inr.inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\npm : natDegree p \u2264 m\nqn : natDegree q \u2264 n\nhm : natDegree p < m\nhn : natDegree q < n\n\u22a2 coeff (p * q) (m + n) = 0\n[PROOFSTEP]\nexact natDegree_lt_coeff_mul (add_lt_add hm hn)\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\npn : natDegree p \u2264 n\n\u22a2 coeff (p ^ m) (m * n) = coeff p n ^ m\n[PROOFSTEP]\ninduction' m with m hm\n[GOAL]\ncase zero\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\npn : natDegree p \u2264 n\n\u22a2 coeff (p ^ Nat.zero) (Nat.zero * n) = coeff p n ^ Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm\u271d n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\npn : natDegree p \u2264 n\nm : \u2115\nhm : coeff (p ^ m) (m * n) = coeff p n ^ m\n\u22a2 coeff (p ^ Nat.succ m) (Nat.succ m * n) = coeff p n ^ Nat.succ m\n[PROOFSTEP]\nrw [pow_succ', pow_succ', \u2190 hm, Nat.succ_mul, coeff_mul_of_natDegree_le _ pn]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm\u271d n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\npn : natDegree p \u2264 n\nm : \u2115\nhm : coeff (p ^ m) (m * n) = coeff p n ^ m\n\u22a2 natDegree (p ^ m) \u2264 m * n\n[PROOFSTEP]\nrefine' natDegree_pow_le.trans (le_trans _ (le_refl _))\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm\u271d n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\npn : natDegree p \u2264 n\nm : \u2115\nhm : coeff (p ^ m) (m * n) = coeff p n ^ m\n\u22a2 m * natDegree p \u2264 m * n\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_left pn m.zero_le\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\no : \u2115\npn : natDegree p \u2264 n\nmno : m * n \u2264 o\n\u22a2 coeff (p ^ m) o = if o = m * n then coeff p n ^ m else 0\n[PROOFSTEP]\nrcases eq_or_ne o (m * n) with rfl | h\n[GOAL]\ncase inl\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\npn : natDegree p \u2264 n\nmno : m * n \u2264 m * n\n\u22a2 coeff (p ^ m) (m * n) = if m * n = m * n then coeff p n ^ m else 0\n[PROOFSTEP]\nsimpa only [ite_true] using coeff_pow_of_natDegree_le pn\n[GOAL]\ncase inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\no : \u2115\npn : natDegree p \u2264 n\nmno : m * n \u2264 o\nh : o \u2260 m * n\n\u22a2 coeff (p ^ m) o = if o = m * n then coeff p n ^ m else 0\n[PROOFSTEP]\nsimpa only [h, ite_false] using\n coeff_eq_zero_of_natDegree_lt $ lt_of_le_of_lt (natDegree_pow_le_of_le m pn) (lt_of_le_of_ne mno h.symm)\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\npn : natDegree p < n\n\u22a2 coeff (p + q) n = coeff q n\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\npn : natDegree p < n\n\u22a2 coeff (q + p) n = coeff q n\n[PROOFSTEP]\nexact coeff_add_eq_left_of_lt pn\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\n\u22a2 degree (Finset.sum s f) = sup s fun i => degree (f i)\n[PROOFSTEP]\ninduction' s using Finset.induction_on with x s hx IH\n[GOAL]\ncase empty\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nh : Set.Pairwise {i | i \u2208 \u2205 \u2227 f i \u2260 0} (Ne on degree \u2218 f)\n\u22a2 degree (Finset.sum \u2205 f) = sup \u2205 fun i => degree (f i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nIH : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f) \u2192 degree (Finset.sum s f) = sup s fun i => degree (f i)\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\n\u22a2 degree (Finset.sum (insert x s) f) = sup (insert x s) fun i => degree (f i)\n[PROOFSTEP]\nsimp only [hx, Finset.sum_insert, not_false_iff, Finset.sup_insert]\n[GOAL]\ncase insert\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nIH : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f) \u2192 degree (Finset.sum s f) = sup s fun i => degree (f i)\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)\n[PROOFSTEP]\nspecialize IH (h.mono fun _ => by simp (config := { contextual := true }))\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nIH : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f) \u2192 degree (Finset.sum s f) = sup s fun i => degree (f i)\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx\u271d : S\n\u22a2 x\u271d \u2208 {i | i \u2208 s \u2227 f i \u2260 0} \u2192 x\u271d \u2208 {i | i \u2208 insert x s \u2227 f i \u2260 0}\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\ncase insert\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)\n[PROOFSTEP]\nrcases lt_trichotomy (degree (f x)) (degree (s.sum f)) with (H | H | H)\n[GOAL]\ncase insert.inl\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nH : degree (f x) < degree (Finset.sum s f)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)\n[PROOFSTEP]\nrw [\u2190 IH, sup_eq_right.mpr H.le, degree_add_eq_right_of_degree_lt H]\n[GOAL]\ncase insert.inr.inl\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nH : degree (f x) = degree (Finset.sum s f)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | hs)\n[GOAL]\ncase insert.inr.inl.inl\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\nhx : \u00acx \u2208 \u2205\nh : Set.Pairwise {i | i \u2208 insert x \u2205 \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum \u2205 f) = sup \u2205 fun i => degree (f i)\nH : degree (f x) = degree (Finset.sum \u2205 f)\n\u22a2 degree (f x + Finset.sum \u2205 fun x => f x) = degree (f x) \u2294 sup \u2205 fun i => degree (f i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert.inr.inl.inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nH : degree (f x) = degree (Finset.sum s f)\nhs : Finset.Nonempty s\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)\n[PROOFSTEP]\nobtain \u27e8y, hy, hy'\u27e9 := Finset.exists_mem_eq_sup s hs fun i => degree (f i)\n[GOAL]\ncase insert.inr.inl.inr.intro.intro\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nH : degree (f x) = degree (Finset.sum s f)\nhs : Finset.Nonempty s\ny : S\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)\n[PROOFSTEP]\nrw [IH, hy'] at H \n[GOAL]\ncase insert.inr.inl.inr.intro.intro\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)\n[PROOFSTEP]\nby_cases hx0 : f x = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : f x = 0\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)\n[PROOFSTEP]\nsimp [hx0, IH]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)\n[PROOFSTEP]\nhave hy0 : f y \u2260 0 := by\n contrapose! H\n simpa [H, degree_eq_bot] using hx0\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\n\u22a2 f y \u2260 0\n[PROOFSTEP]\ncontrapose! H\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\nH : f y = 0\n\u22a2 degree (f x) \u2260 degree (f y)\n[PROOFSTEP]\nsimpa [H, degree_eq_bot] using hx0\n[GOAL]\ncase neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\nhy0 : f y \u2260 0\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)\n[PROOFSTEP]\nrefine' absurd H (h _ _ fun H => hx _)\n[GOAL]\ncase neg.refine'_1\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\nhy0 : f y \u2260 0\n\u22a2 x \u2208 {i | i \u2208 insert x s \u2227 f i \u2260 0}\n[PROOFSTEP]\nsimp [hx0]\n[GOAL]\ncase neg.refine'_2\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\nhy0 : f y \u2260 0\n\u22a2 y \u2208 {i | i \u2208 insert x s \u2227 f i \u2260 0}\n[PROOFSTEP]\nsimp [hy, hy0]\n[GOAL]\ncase neg.refine'_3\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH\u271d : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\nhy0 : f y \u2260 0\nH : x = y\n\u22a2 x \u2208 s\n[PROOFSTEP]\nexact H.symm \u25b8 hy\n[GOAL]\ncase insert.inr.inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nH : degree (Finset.sum s f) < degree (f x)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)\n[PROOFSTEP]\nrw [\u2190 IH, sup_eq_left.mpr H.le, degree_add_eq_left_of_degree_lt H]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\n\u22a2 natDegree (Finset.sum s f) = sup s fun i => natDegree (f i)\n[PROOFSTEP]\nby_cases H : \u2203 x \u2208 s, f x \u2260 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nH : \u2203 x, x \u2208 s \u2227 f x \u2260 0\n\u22a2 natDegree (Finset.sum s f) = sup s fun i => natDegree (f i)\n[PROOFSTEP]\nobtain \u27e8x, hx, hx'\u27e9 := H\n[GOAL]\ncase pos.intro.intro\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\n\u22a2 natDegree (Finset.sum s f) = sup s fun i => natDegree (f i)\n[PROOFSTEP]\nhave hs : s.Nonempty := \u27e8x, hx\u27e9\n[GOAL]\ncase pos.intro.intro\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\n\u22a2 natDegree (Finset.sum s f) = sup s fun i => natDegree (f i)\n[PROOFSTEP]\nrefine' natDegree_eq_of_degree_eq_some _\n[GOAL]\ncase pos.intro.intro\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\n\u22a2 degree (Finset.sum s f) = \u2191(sup s fun i => natDegree (f i))\n[PROOFSTEP]\nrw [degree_sum_eq_of_disjoint]\n[GOAL]\ncase pos.intro.intro\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\n\u22a2 (sup s fun i => degree (f i)) = \u2191(sup s fun i => natDegree (f i))\n[PROOFSTEP]\nrw [\u2190 Finset.sup'_eq_sup hs, \u2190 Finset.sup'_eq_sup hs, Nat.cast_withBot, Finset.coe_sup' hs, \u2190 Finset.sup'_eq_sup hs]\n[GOAL]\ncase pos.intro.intro\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\n\u22a2 (sup' s hs fun i => degree (f i)) = sup' s hs (WithBot.some \u2218 fun i => natDegree (f i))\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase pos.intro.intro.refine'_1\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\n\u22a2 (sup' s hs fun i => degree (f i)) \u2264 sup' s hs (WithBot.some \u2218 fun i => natDegree (f i))\n[PROOFSTEP]\nrw [Finset.sup'_le_iff]\n[GOAL]\ncase pos.intro.intro.refine'_1\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\n\u22a2 \u2200 (b : S), b \u2208 s \u2192 degree (f b) \u2264 sup' s hs (WithBot.some \u2218 fun i => natDegree (f i))\n[PROOFSTEP]\nintro b hb\n[GOAL]\ncase pos.intro.intro.refine'_1\nR : Type u\nS : Type v\n\u03b9 : Type w\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\nb : S\nhb : b \u2208 s\n\u22a2 degree (f b) \u2264 sup' s hs (WithBot.some \u2218 fun i => natDegree (f i))\n[PROOFSTEP]\nby_cases hb' : f b = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\n\u03b9 : Type w\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\nb : S\nhb : b \u2208 s\nhb' : f b = 0\n\u22a2 degree (f b) \u2264 sup' s hs (WithBot.some \u2218 fun i => natDegree (f i))\n[PROOFSTEP]\nsimpa [hb'] using hs\n[GOAL]\ncase neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\nb : S\nhb : b \u2208 s\nhb' : \u00acf b = 0\n\u22a2 degree (f b) \u2264 sup' s hs (WithBot.some \u2218 fun i => natDegree (f i))\n[PROOFSTEP]\nrw [degree_eq_natDegree hb', Nat.cast_withBot]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\nb : S\nhb : b \u2208 s\nhb' : \u00acf b = 0\n\u22a2 \u2191(natDegree (f b)) \u2264 sup' s hs (WithBot.some \u2218 fun i => natDegree (f i))\n[PROOFSTEP]\nexact Finset.le_sup' (fun i : S => (natDegree (f i) : WithBot \u2115)) hb\n[GOAL]\ncase pos.intro.intro.refine'_2\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\n\u22a2 sup' s hs (WithBot.some \u2218 fun i => natDegree (f i)) \u2264 sup' s hs fun i => degree (f i)\n[PROOFSTEP]\nrw [Finset.sup'_le_iff]\n[GOAL]\ncase pos.intro.intro.refine'_2\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\n\u22a2 \u2200 (b : S), b \u2208 s \u2192 (WithBot.some \u2218 fun i => natDegree (f i)) b \u2264 sup' s hs fun i => degree (f i)\n[PROOFSTEP]\nintro b hb\n[GOAL]\ncase pos.intro.intro.refine'_2\nR : Type u\nS : Type v\n\u03b9 : Type w\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\nb : S\nhb : b \u2208 s\n\u22a2 (WithBot.some \u2218 fun i => natDegree (f i)) b \u2264 sup' s hs fun i => degree (f i)\n[PROOFSTEP]\nsimp only [Finset.le_sup'_iff, exists_prop, Function.comp_apply]\n[GOAL]\ncase pos.intro.intro.refine'_2\nR : Type u\nS : Type v\n\u03b9 : Type w\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\nb : S\nhb : b \u2208 s\n\u22a2 \u2203 b_1, b_1 \u2208 s \u2227 \u2191(natDegree (f b)) \u2264 degree (f b_1)\n[PROOFSTEP]\nby_cases hb' : f b = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\n\u03b9 : Type w\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\nb : S\nhb : b \u2208 s\nhb' : f b = 0\n\u22a2 \u2203 b_1, b_1 \u2208 s \u2227 \u2191(natDegree (f b)) \u2264 degree (f b_1)\n[PROOFSTEP]\nrefine' \u27e8x, hx, _\u27e9\n[GOAL]\ncase pos\nR : Type u\nS : Type v\n\u03b9 : Type w\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\nb : S\nhb : b \u2208 s\nhb' : f b = 0\n\u22a2 \u2191(natDegree (f b)) \u2264 degree (f x)\n[PROOFSTEP]\ncontrapose! hx'\n[GOAL]\ncase pos\nR : Type u\nS : Type v\n\u03b9 : Type w\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhs : Finset.Nonempty s\nb : S\nhb : b \u2208 s\nhb' : f b = 0\nhx' : degree (f x) < \u2191(natDegree (f b))\n\u22a2 f x = 0\n[PROOFSTEP]\nsimpa [\u2190 Nat.cast_withBot, hb', degree_eq_bot] using hx'\n[GOAL]\ncase neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\nb : S\nhb : b \u2208 s\nhb' : \u00acf b = 0\n\u22a2 \u2203 b_1, b_1 \u2208 s \u2227 \u2191(natDegree (f b)) \u2264 degree (f b_1)\n[PROOFSTEP]\nexact \u27e8b, hb, (degree_eq_natDegree hb').ge\u27e9\n[GOAL]\ncase pos.intro.intro.h\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nx : S\nhx : x \u2208 s\nhx' : f x \u2260 0\nhs : Finset.Nonempty s\n\u22a2 Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\n[PROOFSTEP]\nexact h.imp fun x y hxy hxy' => hxy (natDegree_eq_of_degree_eq hxy')\n[GOAL]\ncase neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nH : \u00ac\u2203 x, x \u2208 s \u2227 f x \u2260 0\n\u22a2 natDegree (Finset.sum s f) = sup s fun i => natDegree (f i)\n[PROOFSTEP]\npush_neg at H \n[GOAL]\ncase neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nH : \u2200 (x : S), x \u2208 s \u2192 f x = 0\n\u22a2 natDegree (Finset.sum s f) = sup s fun i => natDegree (f i)\n[PROOFSTEP]\nrw [Finset.sum_eq_zero H, natDegree_zero, eq_comm, show 0 = \u22a5 from rfl, Finset.sup_eq_bot_iff]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nH : \u2200 (x : S), x \u2208 s \u2192 f x = 0\n\u22a2 \u2200 (s_1 : S), s_1 \u2208 s \u2192 natDegree (f s_1) = \u22a5\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on natDegree \u2218 f)\nH : \u2200 (x : S), x \u2208 s \u2192 f x = 0\nx : S\nhx : x \u2208 s\n\u22a2 natDegree (f x) = \u22a5\n[PROOFSTEP]\nsimp [H x hx]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na\u271d b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r a : R[X]\n\u22a2 max (natDegree (bit0 a)) (natDegree 1) \u2264 natDegree a\n[PROOFSTEP]\nsimp [natDegree_bit0]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\np : R[X]\nhp : p \u2260 0\nf : R \u2192+* S\nz : S\nhz : eval\u2082 f z p = 0\ninj : \u2200 (x : R), \u2191f x = 0 \u2192 x = 0\nhlt : 0 \u2265 natDegree p\n\u22a2 False\n[PROOFSTEP]\nhave A : p = C (p.coeff 0) := eq_C_of_natDegree_le_zero hlt\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\np : R[X]\nhp : p \u2260 0\nf : R \u2192+* S\nz : S\nhz : eval\u2082 f z p = 0\ninj : \u2200 (x : R), \u2191f x = 0 \u2192 x = 0\nhlt : 0 \u2265 natDegree p\nA : p = \u2191C (coeff p 0)\n\u22a2 False\n[PROOFSTEP]\nrw [A, eval\u2082_C] at hz \n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\np : R[X]\nhp : p \u2260 0\nf : R \u2192+* S\nz : S\nhz : \u2191f (coeff p 0) = 0\ninj : \u2200 (x : R), \u2191f x = 0 \u2192 x = 0\nhlt : 0 \u2265 natDegree p\nA : p = \u2191C (coeff p 0)\n\u22a2 False\n[PROOFSTEP]\nsimp only [inj (p.coeff 0) hz, RingHom.map_zero] at A \n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\np : R[X]\nhp : p \u2260 0\nf : R \u2192+* S\nz : S\nhz : \u2191f (coeff p 0) = 0\ninj : \u2200 (x : R), \u2191f x = 0 \u2192 x = 0\nhlt : 0 \u2265 natDegree p\nA : p = 0\n\u22a2 False\n[PROOFSTEP]\nexact hp A\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\np : R[X]\nn : \u2115\n\u22a2 \u2191n < degree p \u2194 n < natDegree p\n[PROOFSTEP]\nby_cases h : p = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\np : R[X]\nn : \u2115\nh : p = 0\n\u22a2 \u2191n < degree p \u2194 n < natDegree p\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\np : R[X]\nn : \u2115\nh : \u00acp = 0\n\u22a2 \u2191n < degree p \u2194 n < natDegree p\n[PROOFSTEP]\nsimp [degree_eq_natDegree h, Nat.cast_withBot, WithBot.coe_lt_coe]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Ring R\np q : R[X]\n\u22a2 natDegree (p - q) = natDegree (q - p)\n[PROOFSTEP]\nrw [\u2190 natDegree_neg, neg_sub]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Ring R\np q : R[X]\nqn : natDegree q \u2264 n\n\u22a2 natDegree (p - q) \u2264 n \u2194 natDegree p \u2264 n\n[PROOFSTEP]\nrw [\u2190 natDegree_neg] at qn \n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Ring R\np q : R[X]\nqn : natDegree (-q) \u2264 n\n\u22a2 natDegree (p - q) \u2264 n \u2194 natDegree p \u2264 n\n[PROOFSTEP]\nrw [sub_eq_add_neg, natDegree_add_le_iff_left _ _ qn]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Ring R\np q : R[X]\npn : natDegree p \u2264 n\n\u22a2 natDegree (p - q) \u2264 n \u2194 natDegree q \u2264 n\n[PROOFSTEP]\nrwa [natDegree_sub, natDegree_sub_le_iff_left]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Ring R\np q : R[X]\ndg : natDegree q < n\n\u22a2 coeff (p - q) n = coeff p n\n[PROOFSTEP]\nrw [\u2190 natDegree_neg] at dg \n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Ring R\np q : R[X]\ndg : natDegree (-q) < n\n\u22a2 coeff (p - q) n = coeff p n\n[PROOFSTEP]\nrw [sub_eq_add_neg, coeff_add_eq_left_of_lt dg]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Ring R\np q : R[X]\ndf : natDegree p < n\n\u22a2 coeff (p - q) n = -coeff q n\n[PROOFSTEP]\nrwa [sub_eq_add_neg, coeff_add_eq_right_of_lt, coeff_neg]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\na0 : a \u2260 0\n\u22a2 degree (p * \u2191C a) = degree p\n[PROOFSTEP]\nrw [degree_mul, degree_C a0, add_zero]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\na0 : a \u2260 0\n\u22a2 degree (\u2191C a * p) = degree p\n[PROOFSTEP]\nrw [degree_mul, degree_C a0, zero_add]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\na0 : a \u2260 0\n\u22a2 natDegree (p * \u2191C a) = natDegree p\n[PROOFSTEP]\nsimp only [natDegree, degree_mul_C a0]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\na0 : a \u2260 0\n\u22a2 natDegree (\u2191C a * p) = natDegree p\n[PROOFSTEP]\nsimp only [natDegree, degree_C_mul a0]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\n\u22a2 natDegree (comp p q) = natDegree p * natDegree q\n[PROOFSTEP]\nby_cases q0 : q.natDegree = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nq0 : natDegree q = 0\n\u22a2 natDegree (comp p q) = natDegree p * natDegree q\n[PROOFSTEP]\nrw [degree_le_zero_iff.mp (natDegree_eq_zero_iff_degree_le_zero.mp q0), comp_C, natDegree_C, natDegree_C, mul_zero]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nq0 : \u00acnatDegree q = 0\n\u22a2 natDegree (comp p q) = natDegree p * natDegree q\n[PROOFSTEP]\nby_cases p0 : p = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nq0 : \u00acnatDegree q = 0\np0 : p = 0\n\u22a2 natDegree (comp p q) = natDegree p * natDegree q\n[PROOFSTEP]\nsimp only [p0, zero_comp, natDegree_zero, zero_mul]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nq0 : \u00acnatDegree q = 0\np0 : \u00acp = 0\n\u22a2 natDegree (comp p q) = natDegree p * natDegree q\n[PROOFSTEP]\nrefine' le_antisymm natDegree_comp_le (le_natDegree_of_ne_zero _)\n[GOAL]\ncase neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nq0 : \u00acnatDegree q = 0\np0 : \u00acp = 0\n\u22a2 coeff (comp p q) (natDegree p * natDegree q) \u2260 0\n[PROOFSTEP]\nsimp only [coeff_comp_degree_mul_degree q0, p0, mul_eq_zero, leadingCoeff_eq_zero, or_self_iff,\n ne_zero_of_natDegree_gt (Nat.pos_of_ne_zero q0), pow_ne_zero, Ne.def, not_false_iff]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nk : \u2115\n\u22a2 natDegree ((comp p)^[k] q) = natDegree p ^ k * natDegree q\n[PROOFSTEP]\ninduction' k with k IH\n[GOAL]\ncase zero\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\n\u22a2 natDegree ((comp p)^[Nat.zero] q) = natDegree p ^ Nat.zero * natDegree q\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nk : \u2115\nIH : natDegree ((comp p)^[k] q) = natDegree p ^ k * natDegree q\n\u22a2 natDegree ((comp p)^[Nat.succ k] q) = natDegree p ^ Nat.succ k * natDegree q\n[PROOFSTEP]\nrw [Function.iterate_succ_apply', natDegree_comp, IH, pow_succ, mul_assoc]\n[GOAL]\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhq : natDegree q \u2260 0\n\u22a2 leadingCoeff (comp p q) = leadingCoeff p * leadingCoeff q ^ natDegree p\n[PROOFSTEP]\nrw [\u2190 coeff_comp_degree_mul_degree hq, \u2190 natDegree_comp, coeff_natDegree]\n", "meta": {"mathlib_filename": "Mathlib.Data.Polynomial.Degree.Lemmas", "llama_tokens": 20943, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583376458152, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.4150926209911921}} {"text": "[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ninst\u271d : DecidableEq \u03b9\n\u22a2 toMatrix b \u2191b = 1\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : AffineSpace V P\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : Module k V\nb : AffineBasis \u03b9 k P\ninst\u271d : DecidableEq \u03b9\ni j : \u03b9\n\u22a2 toMatrix b (\u2191b) i j = OfNat.ofNat 1 i j\n[PROOFSTEP]\nrw [toMatrix_apply, coord_apply, Matrix.one_eq_pi_single, Pi.single_apply]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AffineSpace V P\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9'\u271d : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9'\u271d\ninst\u271d : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\n\u03b9' : Type u_2\nq : \u03b9' \u2192 P\ni : \u03b9'\n\u22a2 \u2211 j : \u03b9, toMatrix b q i j = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : AffineSpace V P\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d : DecidableEq \u03b9'\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : toMatrix b p * A = 1\n\u22a2 AffineIndependent k p\n[PROOFSTEP]\nrw [affineIndependent_iff_eq_of_fintype_affineCombination_eq]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : AffineSpace V P\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d : DecidableEq \u03b9'\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : toMatrix b p * A = 1\n\u22a2 \u2200 (w1 w2 : \u03b9' \u2192 k),\n \u2211 i : \u03b9', w1 i = 1 \u2192\n \u2211 i : \u03b9', w2 i = 1 \u2192\n \u2191(Finset.affineCombination k Finset.univ p) w1 = \u2191(Finset.affineCombination k Finset.univ p) w2 \u2192 w1 = w2\n[PROOFSTEP]\nintro w\u2081 w\u2082 hw\u2081 hw\u2082 hweq\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : AffineSpace V P\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d : DecidableEq \u03b9'\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : toMatrix b p * A = 1\nw\u2081 w\u2082 : \u03b9' \u2192 k\nhw\u2081 : \u2211 i : \u03b9', w\u2081 i = 1\nhw\u2082 : \u2211 i : \u03b9', w\u2082 i = 1\nhweq : \u2191(Finset.affineCombination k Finset.univ p) w\u2081 = \u2191(Finset.affineCombination k Finset.univ p) w\u2082\n\u22a2 w\u2081 = w\u2082\n[PROOFSTEP]\nhave hweq' : (b.toMatrix p).vecMul w\u2081 = (b.toMatrix p).vecMul w\u2082 :=\n by\n ext j\n change\n (\u2211 i, w\u2081 i \u2022 b.coord j (p i)) =\n \u2211 i,\n w\u2082 i \u2022\n b.coord j\n (p i)\n -- Porting note: Added `u` because `\u2218` was causing trouble\n have u : (fun i => b.coord j (p i)) = b.coord j \u2218 p := by simp only [(\u00b7 \u2218 \u00b7)]\n rw [\u2190 Finset.univ.affineCombination_eq_linear_combination _ _ hw\u2081, \u2190\n Finset.univ.affineCombination_eq_linear_combination _ _ hw\u2082, u, \u2190 Finset.univ.map_affineCombination p w\u2081 hw\u2081, \u2190\n Finset.univ.map_affineCombination p w\u2082 hw\u2082, hweq]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : AffineSpace V P\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d : DecidableEq \u03b9'\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : toMatrix b p * A = 1\nw\u2081 w\u2082 : \u03b9' \u2192 k\nhw\u2081 : \u2211 i : \u03b9', w\u2081 i = 1\nhw\u2082 : \u2211 i : \u03b9', w\u2082 i = 1\nhweq : \u2191(Finset.affineCombination k Finset.univ p) w\u2081 = \u2191(Finset.affineCombination k Finset.univ p) w\u2082\n\u22a2 vecMul w\u2081 (toMatrix b p) = vecMul w\u2082 (toMatrix b p)\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : AffineSpace V P\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d : DecidableEq \u03b9'\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : toMatrix b p * A = 1\nw\u2081 w\u2082 : \u03b9' \u2192 k\nhw\u2081 : \u2211 i : \u03b9', w\u2081 i = 1\nhw\u2082 : \u2211 i : \u03b9', w\u2082 i = 1\nhweq : \u2191(Finset.affineCombination k Finset.univ p) w\u2081 = \u2191(Finset.affineCombination k Finset.univ p) w\u2082\nj : \u03b9\n\u22a2 vecMul w\u2081 (toMatrix b p) j = vecMul w\u2082 (toMatrix b p) j\n[PROOFSTEP]\nchange\n (\u2211 i, w\u2081 i \u2022 b.coord j (p i)) =\n \u2211 i,\n w\u2082 i \u2022\n b.coord j\n (p i)\n -- Porting note: Added `u` because `\u2218` was causing trouble\n[GOAL]\ncase h\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : AffineSpace V P\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d : DecidableEq \u03b9'\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : toMatrix b p * A = 1\nw\u2081 w\u2082 : \u03b9' \u2192 k\nhw\u2081 : \u2211 i : \u03b9', w\u2081 i = 1\nhw\u2082 : \u2211 i : \u03b9', w\u2082 i = 1\nhweq : \u2191(Finset.affineCombination k Finset.univ p) w\u2081 = \u2191(Finset.affineCombination k Finset.univ p) w\u2082\nj : \u03b9\n\u22a2 \u2211 i : \u03b9', w\u2081 i \u2022 \u2191(coord b j) (p i) = \u2211 i : \u03b9', w\u2082 i \u2022 \u2191(coord b j) (p i)\n[PROOFSTEP]\nhave u : (fun i => b.coord j (p i)) = b.coord j \u2218 p := by simp only [(\u00b7 \u2218 \u00b7)]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : AffineSpace V P\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d : DecidableEq \u03b9'\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : toMatrix b p * A = 1\nw\u2081 w\u2082 : \u03b9' \u2192 k\nhw\u2081 : \u2211 i : \u03b9', w\u2081 i = 1\nhw\u2082 : \u2211 i : \u03b9', w\u2082 i = 1\nhweq : \u2191(Finset.affineCombination k Finset.univ p) w\u2081 = \u2191(Finset.affineCombination k Finset.univ p) w\u2082\nj : \u03b9\n\u22a2 (fun i => \u2191(coord b j) (p i)) = \u2191(coord b j) \u2218 p\n[PROOFSTEP]\nsimp only [(\u00b7 \u2218 \u00b7)]\n[GOAL]\ncase h\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : AffineSpace V P\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d : DecidableEq \u03b9'\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : toMatrix b p * A = 1\nw\u2081 w\u2082 : \u03b9' \u2192 k\nhw\u2081 : \u2211 i : \u03b9', w\u2081 i = 1\nhw\u2082 : \u2211 i : \u03b9', w\u2082 i = 1\nhweq : \u2191(Finset.affineCombination k Finset.univ p) w\u2081 = \u2191(Finset.affineCombination k Finset.univ p) w\u2082\nj : \u03b9\nu : (fun i => \u2191(coord b j) (p i)) = \u2191(coord b j) \u2218 p\n\u22a2 \u2211 i : \u03b9', w\u2081 i \u2022 \u2191(coord b j) (p i) = \u2211 i : \u03b9', w\u2082 i \u2022 \u2191(coord b j) (p i)\n[PROOFSTEP]\nrw [\u2190 Finset.univ.affineCombination_eq_linear_combination _ _ hw\u2081, \u2190\n Finset.univ.affineCombination_eq_linear_combination _ _ hw\u2082, u, \u2190 Finset.univ.map_affineCombination p w\u2081 hw\u2081, \u2190\n Finset.univ.map_affineCombination p w\u2082 hw\u2082, hweq]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : AffineSpace V P\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d : DecidableEq \u03b9'\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : toMatrix b p * A = 1\nw\u2081 w\u2082 : \u03b9' \u2192 k\nhw\u2081 : \u2211 i : \u03b9', w\u2081 i = 1\nhw\u2082 : \u2211 i : \u03b9', w\u2082 i = 1\nhweq : \u2191(Finset.affineCombination k Finset.univ p) w\u2081 = \u2191(Finset.affineCombination k Finset.univ p) w\u2082\nhweq' : vecMul w\u2081 (toMatrix b p) = vecMul w\u2082 (toMatrix b p)\n\u22a2 w\u2081 = w\u2082\n[PROOFSTEP]\nreplace hweq' := congr_arg (fun w => A.vecMul w) hweq'\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : AffineSpace V P\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d : DecidableEq \u03b9'\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : toMatrix b p * A = 1\nw\u2081 w\u2082 : \u03b9' \u2192 k\nhw\u2081 : \u2211 i : \u03b9', w\u2081 i = 1\nhw\u2082 : \u2211 i : \u03b9', w\u2082 i = 1\nhweq : \u2191(Finset.affineCombination k Finset.univ p) w\u2081 = \u2191(Finset.affineCombination k Finset.univ p) w\u2082\nhweq' : (fun w => vecMul w A) (vecMul w\u2081 (toMatrix b p)) = (fun w => vecMul w A) (vecMul w\u2082 (toMatrix b p))\n\u22a2 w\u2081 = w\u2082\n[PROOFSTEP]\nsimpa only [Matrix.vecMul_vecMul, hA, Matrix.vecMul_one] using hweq'\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\n\u22a2 affineSpan k (range p) = \u22a4\n[PROOFSTEP]\nsuffices \u2200 i, b i \u2208 affineSpan k (range p)\n by\n rw [eq_top_iff, \u2190 b.tot, affineSpan_le]\n rintro q \u27e8i, rfl\u27e9\n exact this i\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\nthis : \u2200 (i : \u03b9), \u2191b i \u2208 affineSpan k (range p)\n\u22a2 affineSpan k (range p) = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff, \u2190 b.tot, affineSpan_le]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\nthis : \u2200 (i : \u03b9), \u2191b i \u2208 affineSpan k (range p)\n\u22a2 range \u2191b \u2286 \u2191(affineSpan k (range p))\n[PROOFSTEP]\nrintro q \u27e8i, rfl\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\nthis : \u2200 (i : \u03b9), \u2191b i \u2208 affineSpan k (range p)\ni : \u03b9\n\u22a2 \u2191b i \u2208 \u2191(affineSpan k (range p))\n[PROOFSTEP]\nexact this i\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\n\u22a2 \u2200 (i : \u03b9), \u2191b i \u2208 affineSpan k (range p)\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\ni : \u03b9\n\u22a2 \u2191b i \u2208 affineSpan k (range p)\n[PROOFSTEP]\nhave hAi : \u2211 j, A i j = 1 := by\n calc\n \u2211 j, A i j = \u2211 j, A i j * \u2211 l, b.toMatrix p j l := by simp\n _ = \u2211 j, \u2211 l, A i j * b.toMatrix p j l := by simp_rw [Finset.mul_sum]\n _ = \u2211 l, \u2211 j, A i j * b.toMatrix p j l := by rw [Finset.sum_comm]\n _ = \u2211 l, (A * b.toMatrix p) i l := rfl\n _ = 1 := by simp [hA, Matrix.one_apply, Finset.filter_eq]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\ni : \u03b9\n\u22a2 \u2211 j : \u03b9', A i j = 1\n[PROOFSTEP]\ncalc\n \u2211 j, A i j = \u2211 j, A i j * \u2211 l, b.toMatrix p j l := by simp\n _ = \u2211 j, \u2211 l, A i j * b.toMatrix p j l := by simp_rw [Finset.mul_sum]\n _ = \u2211 l, \u2211 j, A i j * b.toMatrix p j l := by rw [Finset.sum_comm]\n _ = \u2211 l, (A * b.toMatrix p) i l := rfl\n _ = 1 := by simp [hA, Matrix.one_apply, Finset.filter_eq]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\ni : \u03b9\n\u22a2 \u2211 j : \u03b9', A i j = \u2211 j : \u03b9', A i j * \u2211 l : \u03b9, toMatrix b p j l\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\ni : \u03b9\n\u22a2 \u2211 j : \u03b9', A i j * \u2211 l : \u03b9, toMatrix b p j l = \u2211 j : \u03b9', \u2211 l : \u03b9, A i j * toMatrix b p j l\n[PROOFSTEP]\nsimp_rw [Finset.mul_sum]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\ni : \u03b9\n\u22a2 \u2211 j : \u03b9', \u2211 l : \u03b9, A i j * toMatrix b p j l = \u2211 l : \u03b9, \u2211 j : \u03b9', A i j * toMatrix b p j l\n[PROOFSTEP]\nrw [Finset.sum_comm]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\ni : \u03b9\n\u22a2 \u2211 l : \u03b9, (A * toMatrix b p) i l = 1\n[PROOFSTEP]\nsimp [hA, Matrix.one_apply, Finset.filter_eq]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\ni : \u03b9\nhAi : \u2211 j : \u03b9', A i j = 1\n\u22a2 \u2191b i \u2208 affineSpan k (range p)\n[PROOFSTEP]\nhave hbi : b i = Finset.univ.affineCombination k p (A i) :=\n by\n apply b.ext_elem\n intro j\n rw [b.coord_apply, Finset.univ.map_affineCombination _ _ hAi,\n Finset.univ.affineCombination_eq_linear_combination _ _ hAi]\n change _ = (A * b.toMatrix p) i j\n simp_rw [hA, Matrix.one_apply, @eq_comm _ i j]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\ni : \u03b9\nhAi : \u2211 j : \u03b9', A i j = 1\n\u22a2 \u2191b i = \u2191(Finset.affineCombination k Finset.univ p) (A i)\n[PROOFSTEP]\napply b.ext_elem\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\ni : \u03b9\nhAi : \u2211 j : \u03b9', A i j = 1\n\u22a2 \u2200 (i_1 : \u03b9), \u2191(coord b i_1) (\u2191b i) = \u2191(coord b i_1) (\u2191(Finset.affineCombination k Finset.univ p) (A i))\n[PROOFSTEP]\nintro j\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\ni : \u03b9\nhAi : \u2211 j : \u03b9', A i j = 1\nj : \u03b9\n\u22a2 \u2191(coord b j) (\u2191b i) = \u2191(coord b j) (\u2191(Finset.affineCombination k Finset.univ p) (A i))\n[PROOFSTEP]\nrw [b.coord_apply, Finset.univ.map_affineCombination _ _ hAi,\n Finset.univ.affineCombination_eq_linear_combination _ _ hAi]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\ni : \u03b9\nhAi : \u2211 j : \u03b9', A i j = 1\nj : \u03b9\n\u22a2 (if j = i then 1 else 0) = \u2211 i_1 : \u03b9', A i i_1 \u2022 (\u2191(coord b j) \u2218 p) i_1\n[PROOFSTEP]\nchange _ = (A * b.toMatrix p) i j\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\ni : \u03b9\nhAi : \u2211 j : \u03b9', A i j = 1\nj : \u03b9\n\u22a2 (if j = i then 1 else 0) = (A * toMatrix b p) i j\n[PROOFSTEP]\nsimp_rw [hA, Matrix.one_apply, @eq_comm _ i j]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\ni : \u03b9\nhAi : \u2211 j : \u03b9', A i j = 1\nhbi : \u2191b i = \u2191(Finset.affineCombination k Finset.univ p) (A i)\n\u22a2 \u2191b i \u2208 affineSpan k (range p)\n[PROOFSTEP]\nrw [hbi]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9' \u2192 P\nA : Matrix \u03b9 \u03b9' k\nhA : A * toMatrix b p = 1\ni : \u03b9\nhAi : \u2211 j : \u03b9', A i j = 1\nhbi : \u2191b i = \u2191(Finset.affineCombination k Finset.univ p) (A i)\n\u22a2 \u2191(Finset.affineCombination k Finset.univ p) (A i) \u2208 affineSpan k (range p)\n[PROOFSTEP]\nexact affineCombination_mem_affineSpan hAi p\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AffineSpace V P\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9'\ninst\u271d : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\nx : P\n\u22a2 vecMul (\u2191(coords b\u2082) x) (toMatrix b \u2191b\u2082) = \u2191(coords b) x\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AffineSpace V P\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9'\ninst\u271d : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\nx : P\nj : \u03b9\n\u22a2 vecMul (\u2191(coords b\u2082) x) (toMatrix b \u2191b\u2082) j = \u2191(coords b) x j\n[PROOFSTEP]\nchange _ = b.coord j x\n[GOAL]\ncase h\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AffineSpace V P\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9'\ninst\u271d : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\nx : P\nj : \u03b9\n\u22a2 vecMul (\u2191(coords b\u2082) x) (toMatrix b \u2191b\u2082) j = \u2191(coord b j) x\n[PROOFSTEP]\nconv_rhs => rw [\u2190 b\u2082.affineCombination_coord_eq_self x]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AffineSpace V P\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9'\ninst\u271d : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\nx : P\nj : \u03b9\n| \u2191(coord b j) x\n[PROOFSTEP]\nrw [\u2190 b\u2082.affineCombination_coord_eq_self x]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AffineSpace V P\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9'\ninst\u271d : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\nx : P\nj : \u03b9\n| \u2191(coord b j) x\n[PROOFSTEP]\nrw [\u2190 b\u2082.affineCombination_coord_eq_self x]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AffineSpace V P\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9'\ninst\u271d : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\nx : P\nj : \u03b9\n| \u2191(coord b j) x\n[PROOFSTEP]\nrw [\u2190 b\u2082.affineCombination_coord_eq_self x]\n[GOAL]\ncase h\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AffineSpace V P\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9'\ninst\u271d : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\nx : P\nj : \u03b9\n\u22a2 vecMul (\u2191(coords b\u2082) x) (toMatrix b \u2191b\u2082) j =\n \u2191(coord b j) (\u2191(Finset.affineCombination k Finset.univ \u2191b\u2082) fun i => \u2191(coord b\u2082 i) x)\n[PROOFSTEP]\nrw [Finset.map_affineCombination _ _ _ (b\u2082.sum_coord_apply_eq_one x)]\n[GOAL]\ncase h\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AffineSpace V P\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9'\ninst\u271d : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\nx : P\nj : \u03b9\n\u22a2 vecMul (\u2191(coords b\u2082) x) (toMatrix b \u2191b\u2082) j =\n \u2191(Finset.affineCombination ((fun a => k) x) Finset.univ (\u2191(coord b j) \u2218 \u2191b\u2082)) fun i => \u2191(coord b\u2082 i) x\n[PROOFSTEP]\nsimp [Matrix.vecMul, Matrix.dotProduct, toMatrix_apply, coords]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : AffineSpace V P\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d : DecidableEq \u03b9\n\u22a2 toMatrix b \u2191b\u2082 * toMatrix b\u2082 \u2191b = 1\n[PROOFSTEP]\next l m\n[GOAL]\ncase a.h\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : AffineSpace V P\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d : DecidableEq \u03b9\nl m : \u03b9\n\u22a2 (toMatrix b \u2191b\u2082 * toMatrix b\u2082 \u2191b) l m = OfNat.ofNat 1 l m\n[PROOFSTEP]\nchange (b\u2082.toMatrix b).vecMul (b.coords (b\u2082 l)) m = _\n[GOAL]\ncase a.h\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : AffineSpace V P\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d : DecidableEq \u03b9\nl m : \u03b9\n\u22a2 vecMul (\u2191(coords b) (\u2191b\u2082 l)) (toMatrix b\u2082 \u2191b) m = OfNat.ofNat 1 l m\n[PROOFSTEP]\nrw [toMatrix_vecMul_coords, coords_apply, \u2190 toMatrix_apply, toMatrix_self]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\n\u22a2 IsUnit (toMatrix b p) \u2194 AffineIndependent k p \u2227 affineSpan k (range p) = \u22a4\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\n\u22a2 IsUnit (toMatrix b p) \u2192 AffineIndependent k p \u2227 affineSpan k (range p) = \u22a4\n[PROOFSTEP]\nrintro \u27e8\u27e8B, A, hA, hA'\u27e9, rfl : B = b.toMatrix p\u27e9\n[GOAL]\ncase mp.intro.mk\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nA : Matrix \u03b9 \u03b9 k\nhA : toMatrix b p * A = 1\nhA' : A * toMatrix b p = 1\n\u22a2 AffineIndependent k p \u2227 affineSpan k (range p) = \u22a4\n[PROOFSTEP]\nexact \u27e8b.affineIndependent_of_toMatrix_right_inv p hA, b.affineSpan_eq_top_of_toMatrix_left_inv p hA'\u27e9\n[GOAL]\ncase mpr\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\n\u22a2 AffineIndependent k p \u2227 affineSpan k (range p) = \u22a4 \u2192 IsUnit (toMatrix b p)\n[PROOFSTEP]\nrintro \u27e8h_tot, h_ind\u27e9\n[GOAL]\ncase mpr.intro\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nh_tot : AffineIndependent k p\nh_ind : affineSpan k (range p) = \u22a4\n\u22a2 IsUnit (toMatrix b p)\n[PROOFSTEP]\nlet b' : AffineBasis \u03b9 k P := \u27e8p, h_tot, h_ind\u27e9\n[GOAL]\ncase mpr.intro\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nh_tot : AffineIndependent k p\nh_ind : affineSpan k (range p) = \u22a4\nb' : AffineBasis \u03b9 k P := { toFun := p, ind' := h_tot, tot' := h_ind }\n\u22a2 IsUnit (toMatrix b p)\n[PROOFSTEP]\nchange IsUnit (b.toMatrix b')\n[GOAL]\ncase mpr.intro\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : AffineSpace V P\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : Module k V\nb : AffineBasis \u03b9 k P\n\u03b9' : Type u_1\ninst\u271d\u00b3 : Fintype \u03b9'\ninst\u271d\u00b2 : Fintype \u03b9\nb\u2082 : AffineBasis \u03b9 k P\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nh_tot : AffineIndependent k p\nh_ind : affineSpan k (range p) = \u22a4\nb' : AffineBasis \u03b9 k P := { toFun := p, ind' := h_tot, tot' := h_ind }\n\u22a2 IsUnit (toMatrix b \u2191b')\n[PROOFSTEP]\nexact b.isUnit_toMatrix b'\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AffineSpace V P\ninst\u271d\u00b3 : CommRing k\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb b\u2082 : AffineBasis \u03b9 k P\nx : P\n\u22a2 vecMul (\u2191(coords b) x) (toMatrix b \u2191b\u2082)\u207b\u00b9 = \u2191(coords b\u2082) x\n[PROOFSTEP]\nhave hu := b.isUnit_toMatrix b\u2082\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AffineSpace V P\ninst\u271d\u00b3 : CommRing k\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb b\u2082 : AffineBasis \u03b9 k P\nx : P\nhu : IsUnit (toMatrix b \u2191b\u2082)\n\u22a2 vecMul (\u2191(coords b) x) (toMatrix b \u2191b\u2082)\u207b\u00b9 = \u2191(coords b\u2082) x\n[PROOFSTEP]\nrw [Matrix.isUnit_iff_isUnit_det] at hu \n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AffineSpace V P\ninst\u271d\u00b3 : CommRing k\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb b\u2082 : AffineBasis \u03b9 k P\nx : P\nhu : IsUnit (det (toMatrix b \u2191b\u2082))\n\u22a2 vecMul (\u2191(coords b) x) (toMatrix b \u2191b\u2082)\u207b\u00b9 = \u2191(coords b\u2082) x\n[PROOFSTEP]\nrw [\u2190 b.toMatrix_vecMul_coords b\u2082, Matrix.vecMul_vecMul, Matrix.mul_nonsing_inv _ hu, Matrix.vecMul_one]\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AffineSpace V P\ninst\u271d\u00b3 : CommRing k\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb b\u2082 : AffineBasis \u03b9 k P\nx : P\n\u22a2 det (toMatrix b \u2191b\u2082) \u2022 \u2191(coords b\u2082) x = \u2191(cramer (toMatrix b \u2191b\u2082)\u1d40) (\u2191(coords b) x)\n[PROOFSTEP]\nhave hu := b.isUnit_toMatrix b\u2082\n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AffineSpace V P\ninst\u271d\u00b3 : CommRing k\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb b\u2082 : AffineBasis \u03b9 k P\nx : P\nhu : IsUnit (toMatrix b \u2191b\u2082)\n\u22a2 det (toMatrix b \u2191b\u2082) \u2022 \u2191(coords b\u2082) x = \u2191(cramer (toMatrix b \u2191b\u2082)\u1d40) (\u2191(coords b) x)\n[PROOFSTEP]\nrw [Matrix.isUnit_iff_isUnit_det] at hu \n[GOAL]\n\u03b9 : Type u\u2081\nk : Type u\u2082\nV : Type u\u2083\nP : Type u\u2084\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AffineSpace V P\ninst\u271d\u00b3 : CommRing k\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nb b\u2082 : AffineBasis \u03b9 k P\nx : P\nhu : IsUnit (det (toMatrix b \u2191b\u2082))\n\u22a2 det (toMatrix b \u2191b\u2082) \u2022 \u2191(coords b\u2082) x = \u2191(cramer (toMatrix b \u2191b\u2082)\u1d40) (\u2191(coords b) x)\n[PROOFSTEP]\nrw [\u2190 b.toMatrix_inv_vecMul_toMatrix, Matrix.det_smul_inv_vecMul_eq_cramer_transpose _ _ hu]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.AffineSpace.Matrix", "llama_tokens": 14293, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583376458153, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.4150926209911921}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nJ : C\nhP : Injective J\n\u22a2 PreservesFiniteColimits (preadditiveYonedaObj J)\n[PROOFSTEP]\nletI := (injective_iff_preservesEpimorphisms_preadditive_yoneda_obj' J).mp hP\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nJ : C\nhP : Injective J\nthis : Functor.PreservesEpimorphisms (preadditiveYonedaObj J) :=\n Iff.mp (injective_iff_preservesEpimorphisms_preadditive_yoneda_obj' J) hP\n\u22a2 PreservesFiniteColimits (preadditiveYonedaObj J)\n[PROOFSTEP]\napply Functor.preservesFiniteColimitsOfPreservesEpisAndKernels\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nJ : C\nhP : PreservesFiniteColimits (preadditiveYonedaObj J)\n\u22a2 Injective J\n[PROOFSTEP]\nrw [injective_iff_preservesEpimorphisms_preadditive_yoneda_obj']\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nJ : C\nhP : PreservesFiniteColimits (preadditiveYonedaObj J)\n\u22a2 Functor.PreservesEpimorphisms (preadditiveYonedaObj J)\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Abelian.Injective", "llama_tokens": 453, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.743167997235783, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.4149307541828946}} {"text": "[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nv : List \u03b1\nhv : List.length v = n\nw : List \u03b1\nhw : List.length w = n\nh : \u2200 (m : Fin n), get { val := v, property := hv } m = get { val := w, property := hw } m\n\u22a2 List.length \u2191{ val := v, property := hv } = List.length \u2191{ val := w, property := hw }\n[PROOFSTEP]\nrw [hv, hw]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\na : \u03b1\nv : Vector \u03b1 (Nat.succ n)\nv' : Vector \u03b1 n\nh : head v = a \u2227 tail v = v'\n\u22a2 head v ::\u1d65 tail v = a ::\u1d65 v'\n[PROOFSTEP]\nrw [h.1, h.2]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\na : \u03b1\nv : Vector \u03b1 (Nat.succ n)\nv' : Vector \u03b1 n\n\u22a2 v \u2260 a ::\u1d65 v' \u2194 head v \u2260 a \u2228 tail v \u2260 v'\n[PROOFSTEP]\nrw [Ne.def, eq_cons_iff a v v', not_and_or]\n[GOAL]\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1\n\u22a2 toList (ofFn f) = List.ofFn f\n[PROOFSTEP]\nrw [ofFn, List.ofFn_succ, toList_cons, toList_ofFn]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nv : Vector \u03b1 n\nf : \u03b1 \u2192 \u03b2\n\u22a2 toList (map f v) = List.map f (toList v)\n[PROOFSTEP]\ncases v\n[GOAL]\ncase mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 toList (map f { val := val\u271d, property := property\u271d }) = List.map f (toList { val := val\u271d, property := property\u271d })\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nv : Vector \u03b1 (n + 1)\nf : \u03b1 \u2192 \u03b2\n\u22a2 head (map f v) = f (head v)\n[PROOFSTEP]\nobtain \u27e8a, v', h\u27e9 := Vector.exists_eq_cons v\n[GOAL]\ncase intro.intro\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nv : Vector \u03b1 (n + 1)\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nv' : Vector \u03b1 n\nh : v = a ::\u1d65 v'\n\u22a2 head (map f v) = f (head v)\n[PROOFSTEP]\nrw [h, map_cons, head_cons, head_cons]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nv : Vector \u03b1 (n + 1)\nf : \u03b1 \u2192 \u03b2\n\u22a2 tail (map f v) = map f (tail v)\n[PROOFSTEP]\nobtain \u27e8a, v', h\u27e9 := Vector.exists_eq_cons v\n[GOAL]\ncase intro.intro\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nv : Vector \u03b1 (n + 1)\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nv' : Vector \u03b1 n\nh : v = a ::\u1d65 v'\n\u22a2 tail (map f v) = map f (tail v)\n[PROOFSTEP]\nrw [h, map_cons, tail_cons, tail_cons]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 n\ni : Fin n\n\u22a2 \u2191i < List.length (toList v)\n[PROOFSTEP]\nrw [toList_length]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 n\ni : Fin n\n\u22a2 \u2191i < n\n[PROOFSTEP]\nexact i.2\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\na : \u03b1\ni : Fin n\n\u22a2 get (replicate n a) i = a\n[PROOFSTEP]\napply List.get_replicate\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nv : Vector \u03b1 n\nf : \u03b1 \u2192 \u03b2\ni : Fin n\n\u22a2 get (map f v) i = f (get v i)\n[PROOFSTEP]\ncases v\n[GOAL]\ncase mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\ni : Fin n\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 get (map f { val := val\u271d, property := property\u271d }) i = f (get { val := val\u271d, property := property\u271d } i)\n[PROOFSTEP]\nsimp [Vector.map, get_eq_get]\n[GOAL]\ncase mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\ni : Fin n\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 f (List.get val\u271d { val := \u2191i, isLt := (_ : \u2191i < List.length val\u271d) }) =\n f (List.get val\u271d (\u2191(Fin.castIso (_ : n = List.length (toList { val := val\u271d, property := property\u271d }))) i))\n[PROOFSTEP]\nrfl\n[GOAL]\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : Fin n\n\u22a2 get (ofFn f) i = f i\n[PROOFSTEP]\nconv_rhs => erw [\u2190 List.get_ofFn f \u27e8i, by simp\u27e9]\n[GOAL]\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : Fin n\n| f i\n[PROOFSTEP]\nerw [\u2190 List.get_ofFn f \u27e8i, by simp\u27e9]\n[GOAL]\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : Fin n\n| f i\n[PROOFSTEP]\nerw [\u2190 List.get_ofFn f \u27e8i, by simp\u27e9]\n[GOAL]\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : Fin n\n| f i\n[PROOFSTEP]\nerw [\u2190 List.get_ofFn f \u27e8i, by simp\u27e9]\n[GOAL]\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : Fin n\n\u22a2 \u2191i < List.length (List.ofFn f)\n[PROOFSTEP]\nsimp\n[GOAL]\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : Fin n\n\u22a2 get (ofFn f) i = List.get (List.ofFn f) { val := \u2191i, isLt := (_ : \u2191i < List.length (List.ofFn f)) }\n[PROOFSTEP]\nsimp only [get_eq_get]\n[GOAL]\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : Fin n\n\u22a2 List.get (toList (ofFn f)) (\u2191(Fin.castIso (_ : n = List.length (toList (ofFn f)))) i) =\n List.get (List.ofFn f) { val := \u2191i, isLt := (_ : \u2191i < List.length (List.ofFn f)) }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e_2.h\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : Fin n\n\u22a2 toList (ofFn f) = List.ofFn f\n[PROOFSTEP]\nsimp [Fin.heq_ext_iff]\n[GOAL]\ncase h.e_3\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : Fin n\n\u22a2 HEq (\u2191(Fin.castIso (_ : n = List.length (toList (ofFn f)))) i)\n { val := \u2191i, isLt := (_ : \u2191i < List.length (List.ofFn f)) }\n[PROOFSTEP]\nsimp [Fin.heq_ext_iff]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 n\n\u22a2 ofFn (get v) = v\n[PROOFSTEP]\nrcases v with \u27e8l, rfl\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 ofFn (get { val := l, property := (_ : List.length l = List.length l) }) =\n { val := l, property := (_ : List.length l = List.length l) }\n[PROOFSTEP]\napply toList_injective\n[GOAL]\ncase mk.a\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 toList (ofFn (get { val := l, property := (_ : List.length l = List.length l) })) =\n toList { val := l, property := (_ : List.length l = List.length l) }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.a\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 toList (ofFn (get { val := l, property := (_ : List.length l = List.length l) })) = l\n[PROOFSTEP]\nsimpa only [toList_ofFn] using List.ofFn_get _\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nx : Vector \u03b1 n\ni : Fin (n - 1)\n\u22a2 get (tail x) i = get x { val := \u2191i + 1, isLt := (_ : \u2191i + 1 < n) }\n[PROOFSTEP]\ncases' i with i ih\n[GOAL]\ncase mk\nn : \u2115\n\u03b1 : Type u_1\nx : Vector \u03b1 n\ni : \u2115\nih : i < n - 1\n\u22a2 get (tail x) { val := i, isLt := ih } =\n get x { val := \u2191{ val := i, isLt := ih } + 1, isLt := (_ : \u2191{ val := i, isLt := ih } + 1 < n) }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk\nn : \u2115\n\u03b1 : Type u_1\nx : Vector \u03b1 n\ni : \u2115\nih : i < n - 1\n\u22a2 get (tail x) { val := i, isLt := ih } = get x { val := i + 1, isLt := (_ : i + 1 < n) }\n[PROOFSTEP]\nrcases x with \u27e8_ | _, h\u27e9\n[GOAL]\ncase mk.mk.nil\nn : \u2115\n\u03b1 : Type u_1\ni : \u2115\nih : i < n - 1\nh : List.length [] = n\n\u22a2 get (tail { val := [], property := h }) { val := i, isLt := ih } =\n get { val := [], property := h } { val := i + 1, isLt := (_ : i + 1 < n) }\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase mk.mk.nil\nn : \u2115\n\u03b1 : Type u_1\ni : \u2115\nih : i < n - 1\nh : List.length [] = n\n\u22a2 get (tail { val := [], property := h }) { val := i, isLt := ih } =\n get { val := [], property := h } { val := i + 1, isLt := (_ : i + 1 < n) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.mk.cons\nn : \u2115\n\u03b1 : Type u_1\ni : \u2115\nih : i < n - 1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh : List.length (head\u271d :: tail\u271d) = n\n\u22a2 get (tail { val := head\u271d :: tail\u271d, property := h }) { val := i, isLt := ih } =\n get { val := head\u271d :: tail\u271d, property := h } { val := i + 1, isLt := (_ : i + 1 < n) }\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase mk.mk.cons\nn : \u2115\n\u03b1 : Type u_1\ni : \u2115\nih : i < n - 1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh : List.length (head\u271d :: tail\u271d) = n\n\u22a2 get (tail { val := head\u271d :: tail\u271d, property := h }) { val := i, isLt := ih } =\n get { val := head\u271d :: tail\u271d, property := h } { val := i + 1, isLt := (_ : i + 1 < n) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.mk.nil\nn : \u2115\n\u03b1 : Type u_1\ni : \u2115\nih : i < n - 1\nh : List.length [] = n\n\u22a2 get (tail { val := [], property := h }) { val := i, isLt := ih } =\n get { val := [], property := h } { val := i + 1, isLt := (_ : i + 1 < n) }\n[PROOFSTEP]\nrw [List.length] at h \n[GOAL]\ncase mk.mk.nil\nn : \u2115\n\u03b1 : Type u_1\ni : \u2115\nih : i < n - 1\nh\u271d : List.length [] = n\nh : 0 = n\n\u22a2 get (tail { val := [], property := h\u271d }) { val := i, isLt := ih } =\n get { val := [], property := h\u271d } { val := i + 1, isLt := (_ : i + 1 < n) }\n[PROOFSTEP]\nrw [\u2190 h] at ih \n[GOAL]\ncase mk.mk.nil\nn : \u2115\n\u03b1 : Type u_1\ni : \u2115\nih\u271d : i < n - 1\nih : i < 0 - 1\nh\u271d : List.length [] = n\nh : 0 = n\n\u22a2 get (tail { val := [], property := h\u271d }) { val := i, isLt := ih\u271d } =\n get { val := [], property := h\u271d } { val := i + 1, isLt := (_ : i + 1 < n) }\n[PROOFSTEP]\ncontradiction\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\na : \u03b1\nl : List \u03b1\ne : List.length (a :: l) = Nat.succ n\ni : \u2115\nh : i < n\n\u22a2 get (tail { val := a :: l, property := e }) { val := i, isLt := h } =\n get { val := a :: l, property := e } (Fin.succ { val := i, isLt := h })\n[PROOFSTEP]\nsimp [get_eq_get]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\na : \u03b1\nl : List \u03b1\ne : List.length (a :: l) = Nat.succ n\ni : \u2115\nh : i < n\n\u22a2 List.get (toList (tail { val := a :: l, property := e }))\n { val := i, isLt := (_ : i < List.length (toList (tail { val := a :: l, property := e }))) } =\n List.get l { val := i, isLt := (_ : i < List.length l) }\n[PROOFSTEP]\nrfl\n[GOAL]\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin (Nat.succ n) \u2192 \u03b1\n\u22a2 ofFn (get (tail (ofFn f))) = ofFn fun i => f (Fin.succ i)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin (Nat.succ n) \u2192 \u03b1\n\u22a2 get (tail (ofFn f)) = fun i => f (Fin.succ i)\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase e_a.h\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin (Nat.succ n) \u2192 \u03b1\ni : Fin (Nat.succ n - 1)\n\u22a2 get (tail (ofFn f)) i = f (Fin.succ i)\n[PROOFSTEP]\nrw [get_tail, get_ofFn]\n[GOAL]\ncase e_a.h\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin (Nat.succ n) \u2192 \u03b1\ni : Fin (Nat.succ n - 1)\n\u22a2 f { val := \u2191i + 1, isLt := (_ : \u2191i + 1 < Nat.succ n) } = f (Fin.succ i)\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 1\n\u22a2 toList v = [head v]\n[PROOFSTEP]\nrw [\u2190 v.cons_head_tail]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 1\n\u22a2 toList (head v ::\u1d65 tail v) = [head (head v ::\u1d65 tail v)]\n[PROOFSTEP]\nsimp only [toList_cons, toList_nil, head_cons, eq_self_iff_true, and_self_iff, singleton_tail]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 \u00acList.isEmpty (toList v) = true\n[PROOFSTEP]\nsimp only [empty_toList_eq_ff, Bool.coe_sort_false, not_false_iff]\n[GOAL]\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nv : Vector \u03b1 n\n\u22a2 toList (map id v) = toList v\n[PROOFSTEP]\nsimp only [List.map_id, Vector.toList_map]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 n\n\u22a2 List.Nodup (toList v) \u2194 Function.Injective (get v)\n[PROOFSTEP]\ncases' v with l hl\n[GOAL]\ncase mk\nn : \u2115\n\u03b1 : Type u_1\nl : List \u03b1\nhl : List.length l = n\n\u22a2 List.Nodup (toList { val := l, property := hl }) \u2194 Function.Injective (get { val := l, property := hl })\n[PROOFSTEP]\nsubst hl\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 List.Nodup (toList { val := l, property := (_ : List.length l = List.length l) }) \u2194\n Function.Injective (get { val := l, property := (_ : List.length l = List.length l) })\n[PROOFSTEP]\nexact List.nodup_iff_injective_get\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 n\n\u22a2 List.length (List.reverse (toList v)) = n\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 n\n\u22a2 reverse (reverse v) = v\n[PROOFSTEP]\ncases v\n[GOAL]\ncase mk\nn : \u2115\n\u03b1 : Type u_1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 reverse (reverse { val := val\u271d, property := property\u271d }) = { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nsimp [Vector.reverse]\n[GOAL]\nn\u271d : \u2115\n\u03b1 : Type u_1\nn : \u2115\nf : Fin (Nat.succ n) \u2192 \u03b1\n\u22a2 head (ofFn f) = f 0\n[PROOFSTEP]\nrw [\u2190 get_zero, get_ofFn]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\na : \u03b1\nv : Vector \u03b1 n\n\u22a2 get (a ::\u1d65 v) 0 = a\n[PROOFSTEP]\nsimp [get_zero]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\na : \u03b1\nv : Vector \u03b1 n\ni : Fin n\n\u22a2 get (a ::\u1d65 v) (Fin.succ i) = get v i\n[PROOFSTEP]\nrw [\u2190 get_tail_succ, tail_cons]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 head (reverse v) = last v\n[PROOFSTEP]\nrw [\u2190 get_zero, last_def, get_eq_get, get_eq_get]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 List.get (toList (reverse v)) (\u2191(Fin.castIso (_ : Nat.succ n = List.length (toList (reverse v)))) 0) =\n List.get (toList v) (\u2191(Fin.castIso (_ : n + 1 = List.length (toList v))) (Fin.last n))\n[PROOFSTEP]\nsimp_rw [toList_reverse]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 List.get (List.reverse (toList v)) (\u2191(Fin.castIso (_ : Nat.succ n = List.length (toList (reverse v)))) 0) =\n List.get (toList v) (\u2191(Fin.castIso (_ : n + 1 = List.length (toList v))) (Fin.last n))\n[PROOFSTEP]\nrw [\u2190 Option.some_inj, \u2190 List.get?_eq_get, \u2190 List.get?_eq_get, List.get?_reverse]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 List.get? (toList v)\n (List.length (toList v) - 1 - \u2191(\u2191(Fin.castIso (_ : Nat.succ n = List.length (toList (reverse v)))) 0)) =\n List.get? (toList v) \u2191(\u2191(Fin.castIso (_ : n + 1 = List.length (toList v))) (Fin.last n))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 List.length (toList v) - 1 - \u2191(\u2191(Fin.castIso (_ : Nat.succ n = List.length (toList (reverse v)))) 0) =\n \u2191(\u2191(Fin.castIso (_ : n + 1 = List.length (toList v))) (Fin.last n))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 (n + 1)\n\u22a2 \u2191(\u2191(Fin.castIso (_ : Nat.succ n = List.length (toList (reverse v)))) 0) < List.length (toList v)\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 n\n\u22a2 List.length (List.scanl f b (toList v)) = n + 1\n[PROOFSTEP]\nrw [List.length_scanl, toList_length]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 n\nx : \u03b1\n\u22a2 scanl f b (x ::\u1d65 v) = b ::\u1d65 scanl f (f b x) v\n[PROOFSTEP]\nsimp only [scanl, toList_cons, List.scanl]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 n\nx : \u03b1\n\u22a2 { val := b :: List.scanl f (f b x) \u2191v,\n property := (_ : (fun l => List.length l = Nat.succ n + 1) (b :: List.scanl f (f b x) \u2191v)) } =\n b ::\u1d65\n { val := List.scanl f (f b x) (toList v),\n property := (_ : List.length (List.scanl f (f b x) (toList v)) = n + 1) }\n[PROOFSTEP]\ndsimp\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 n\nx : \u03b1\n\u22a2 { val := b :: List.scanl f (f b x) \u2191v,\n property := (_ : List.length (List.scanl f b (toList (x ::\u1d65 v))) = Nat.succ n + 1) } =\n b ::\u1d65\n { val := List.scanl f (f b x) (toList v),\n property := (_ : List.length (List.scanl f (f b x) (toList v)) = n + 1) }\n[PROOFSTEP]\nsimp only [cons]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 n\nx : \u03b1\n\u22a2 { val := b :: List.scanl f (f b x) \u2191v,\n property := (_ : List.length (List.scanl f b (toList (x ::\u1d65 v))) = Nat.succ n + 1) } =\n { val := b :: List.scanl f (f b x) (toList v),\n property := (_ : Nat.succ (List.length (List.scanl f (f b x) (toList v))) = Nat.succ (n + 1)) }\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv\u271d : Vector \u03b1 n\nv : Vector \u03b1 1\n\u22a2 scanl f b v = b ::\u1d65 f b (head v) ::\u1d65 nil\n[PROOFSTEP]\nrw [\u2190 cons_head_tail v]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv\u271d : Vector \u03b1 n\nv : Vector \u03b1 1\n\u22a2 scanl f b (head v ::\u1d65 tail v) = b ::\u1d65 f b (head (head v ::\u1d65 tail v)) ::\u1d65 nil\n[PROOFSTEP]\nsimp only [scanl_cons, scanl_nil, head_cons, singleton_tail]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 n\n\u22a2 head (scanl f b v) = b\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 Nat.zero\n\u22a2 head (scanl f b v) = b\n[PROOFSTEP]\nhave : v = nil := by simp only [Nat.zero_eq, eq_iff_true_of_subsingleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 Nat.zero\n\u22a2 v = nil\n[PROOFSTEP]\nsimp only [Nat.zero_eq, eq_iff_true_of_subsingleton]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 Nat.zero\nthis : v = nil\n\u22a2 head (scanl f b v) = b\n[PROOFSTEP]\nsimp only [this, scanl_nil, head_cons]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nn\u271d : \u2115\nv : Vector \u03b1 (Nat.succ n\u271d)\n\u22a2 head (scanl f b v) = b\n[PROOFSTEP]\nrw [\u2190 cons_head_tail v]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nn\u271d : \u2115\nv : Vector \u03b1 (Nat.succ n\u271d)\n\u22a2 head (scanl f b (head v ::\u1d65 tail v)) = b\n[PROOFSTEP]\nsimp only [\u2190 get_zero, get_eq_get, toList_scanl, toList_cons, List.scanl, Fin.val_zero, List.get]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 n\ni : Fin n\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nv : Vector \u03b1 Nat.zero\ni : Fin Nat.zero\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\n[PROOFSTEP]\nexact i.elim0\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\nn : \u2115\nv : Vector \u03b1 (Nat.succ n)\ni : Fin (Nat.succ n)\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\n[PROOFSTEP]\ninduction' n with n hn generalizing b\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n)\ni\u271d : Fin (Nat.succ n)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ Nat.zero)\ni : Fin (Nat.succ Nat.zero)\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\n[PROOFSTEP]\nhave i0 : i = 0 := Fin.eq_zero _\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n)\ni\u271d : Fin (Nat.succ n)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ Nat.zero)\ni : Fin (Nat.succ Nat.zero)\ni0 : i = 0\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\n[PROOFSTEP]\nsimp [scanl_singleton, i0, get_zero]\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n)\ni\u271d : Fin (Nat.succ n)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ Nat.zero)\ni : Fin (Nat.succ Nat.zero)\ni0 : i = 0\n\u22a2 get (b ::\u1d65 f b (head v) ::\u1d65 nil) 1 = f b (head v)\n[PROOFSTEP]\nsimp [get_eq_get]\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n\u271d)\ni\u271d : Fin (Nat.succ n\u271d)\nn : \u2115\nhn :\n \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),\n get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n\u22a2 get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\n[PROOFSTEP]\nrw [\u2190 cons_head_tail v, scanl_cons, get_cons_succ]\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n\u271d)\ni\u271d : Fin (Nat.succ n\u271d)\nn : \u2115\nhn :\n \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),\n get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n\u22a2 get (scanl f (f b (head v)) (tail v)) i =\n f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc i)) (get (head v ::\u1d65 tail v) i)\n[PROOFSTEP]\nrefine' Fin.cases _ _ i\n[GOAL]\ncase succ.succ.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n\u271d)\ni\u271d : Fin (Nat.succ n\u271d)\nn : \u2115\nhn :\n \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),\n get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n\u22a2 get (scanl f (f b (head v)) (tail v)) 0 =\n f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc 0)) (get (head v ::\u1d65 tail v) 0)\n[PROOFSTEP]\nsimp only [get_zero, scanl_head, Fin.castSucc_zero, head_cons]\n[GOAL]\ncase succ.succ.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n\u271d)\ni\u271d : Fin (Nat.succ n\u271d)\nn : \u2115\nhn :\n \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),\n get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n\u22a2 \u2200 (i : Fin (n + 1)),\n get (scanl f (f b (head v)) (tail v)) (Fin.succ i) =\n f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc (Fin.succ i))) (get (head v ::\u1d65 tail v) (Fin.succ i))\n[PROOFSTEP]\nintro i'\n[GOAL]\ncase succ.succ.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb\u271d : \u03b2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 (Nat.succ n\u271d)\ni\u271d : Fin (Nat.succ n\u271d)\nn : \u2115\nhn :\n \u2200 (b : \u03b2) (v : Vector \u03b1 (Nat.succ n)) (i : Fin (Nat.succ n)),\n get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (Fin.castSucc i)) (get v i)\nb : \u03b2\nv : Vector \u03b1 (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\ni' : Fin (n + 1)\n\u22a2 get (scanl f (f b (head v)) (tail v)) (Fin.succ i') =\n f (get (b ::\u1d65 scanl f (f b (head v)) (tail v)) (Fin.castSucc (Fin.succ i'))) (get (head v ::\u1d65 tail v) (Fin.succ i'))\n[PROOFSTEP]\nsimp only [hn, Fin.castSucc_fin_succ, get_cons_succ]\n[GOAL]\nn\u271d : \u2115\n\u03b1\u271d : Type u_1\nm : Type u_2 \u2192 Type u_3\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1 : Type u_2\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1\n\u22a2 (mOfFn fun i => pure (f i)) = pure (ofFn f)\n[PROOFSTEP]\nrw [mOfFn, @mOfFn_pure m _ _ _ n _, ofFn]\n[GOAL]\nn\u271d : \u2115\n\u03b1\u271d : Type u_1\nm : Type u_2 \u2192 Type u_3\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1 : Type u_2\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1\n\u22a2 (do\n let a \u2190 pure (f 0)\n let v \u2190 pure (ofFn fun i => f (Fin.succ i))\n pure (a ::\u1d65 v)) =\n pure (f 0 ::\u1d65 ofFn fun i => f (Fin.succ i))\n[PROOFSTEP]\nsimp\n[GOAL]\nn\u271d : \u2115\n\u03b1 : Type u_1\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Sort u_2\nn : \u2115\nv : Vector \u03b1 n\nh_nil : C nil\nh_cons : {n : \u2115} \u2192 {x : \u03b1} \u2192 {w : Vector \u03b1 n} \u2192 C w \u2192 C (x ::\u1d65 w)\n\u22a2 C v\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nn\u271d : \u2115\n\u03b1 : Type u_1\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Sort u_2\nn : \u2115\nv\u271d : Vector \u03b1 n\nh_nil : C nil\nh_cons : {n : \u2115} \u2192 {x : \u03b1} \u2192 {w : Vector \u03b1 n} \u2192 C w \u2192 C (x ::\u1d65 w)\nv : Vector \u03b1 Nat.zero\n\u22a2 C v\n[PROOFSTEP]\nrcases v with \u27e8_ | \u27e8-, -\u27e9, - | -\u27e9\n[GOAL]\ncase zero.mk.nil.refl\nn\u271d : \u2115\n\u03b1 : Type u_1\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Sort u_2\nn : \u2115\nv : Vector \u03b1 n\nh_nil : C nil\nh_cons : {n : \u2115} \u2192 {x : \u03b1} \u2192 {w : Vector \u03b1 n} \u2192 C w \u2192 C (x ::\u1d65 w)\n\u22a2 C { val := [], property := (_ : List.length [] = List.length []) }\n[PROOFSTEP]\nexact h_nil\n[GOAL]\ncase succ\nn\u271d\u00b9 : \u2115\n\u03b1 : Type u_1\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Sort u_2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 n\u271d\nh_nil : C nil\nh_cons : {n : \u2115} \u2192 {x : \u03b1} \u2192 {w : Vector \u03b1 n} \u2192 C w \u2192 C (x ::\u1d65 w)\nn : \u2115\nih : (v : Vector \u03b1 n) \u2192 C v\nv : Vector \u03b1 (Nat.succ n)\n\u22a2 C v\n[PROOFSTEP]\nrcases v with \u27e8_ | \u27e8a, v\u27e9, v_property\u27e9\n[GOAL]\ncase succ.mk.nil\nn\u271d\u00b9 : \u2115\n\u03b1 : Type u_1\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Sort u_2\nn\u271d : \u2115\nv : Vector \u03b1 n\u271d\nh_nil : C nil\nh_cons : {n : \u2115} \u2192 {x : \u03b1} \u2192 {w : Vector \u03b1 n} \u2192 C w \u2192 C (x ::\u1d65 w)\nn : \u2115\nih : (v : Vector \u03b1 n) \u2192 C v\nv_property : List.length [] = Nat.succ n\n\u22a2 C { val := [], property := v_property }\ncase succ.mk.cons\nn\u271d\u00b9 : \u2115\n\u03b1 : Type u_1\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Sort u_2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 n\u271d\nh_nil : C nil\nh_cons : {n : \u2115} \u2192 {x : \u03b1} \u2192 {w : Vector \u03b1 n} \u2192 C w \u2192 C (x ::\u1d65 w)\nn : \u2115\nih : (v : Vector \u03b1 n) \u2192 C v\na : \u03b1\nv : List \u03b1\nv_property : List.length (a :: v) = Nat.succ n\n\u22a2 C { val := a :: v, property := v_property }\n[PROOFSTEP]\ncases v_property\n[GOAL]\ncase succ.mk.cons\nn\u271d\u00b9 : \u2115\n\u03b1 : Type u_1\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Sort u_2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 n\u271d\nh_nil : C nil\nh_cons : {n : \u2115} \u2192 {x : \u03b1} \u2192 {w : Vector \u03b1 n} \u2192 C w \u2192 C (x ::\u1d65 w)\nn : \u2115\nih : (v : Vector \u03b1 n) \u2192 C v\na : \u03b1\nv : List \u03b1\nv_property : List.length (a :: v) = Nat.succ n\n\u22a2 C { val := a :: v, property := v_property }\n[PROOFSTEP]\napply @h_cons n _ \u27e8v, (add_left_inj 1).mp v_property\u27e9\n[GOAL]\ncase succ.mk.cons\nn\u271d\u00b9 : \u2115\n\u03b1 : Type u_1\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Sort u_2\nn\u271d : \u2115\nv\u271d : Vector \u03b1 n\u271d\nh_nil : C nil\nh_cons : {n : \u2115} \u2192 {x : \u03b1} \u2192 {w : Vector \u03b1 n} \u2192 C w \u2192 C (x ::\u1d65 w)\nn : \u2115\nih : (v : Vector \u03b1 n) \u2192 C v\na : \u03b1\nv : List \u03b1\nv_property : List.length (a :: v) = Nat.succ n\n\u22a2 C { val := v, property := (_ : List.length v = n) }\n[PROOFSTEP]\napply ih\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\nv : Vector \u03b1 n\n\u22a2 True\n[PROOFSTEP]\ninduction v using Vector.inductionOn\n[GOAL]\ncase h_nil\nn : \u2115\n\u03b1 : Type u_1\n\u22a2 True\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase h_cons\nn : \u2115\n\u03b1 : Type u_1\nn\u271d : \u2115\nx\u271d : \u03b1\nw\u271d : Vector \u03b1 n\u271d\na\u271d : True\n\u22a2 True\n[PROOFSTEP]\ntrivial\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Sort u_4\nv : Vector \u03b1 n\nw : Vector \u03b2 n\nnil : C Vector.nil Vector.nil\ncons : {n : \u2115} \u2192 {a : \u03b1} \u2192 {b : \u03b2} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 C x y \u2192 C (a ::\u1d65 x) (b ::\u1d65 y)\n\u22a2 C v w\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Sort u_4\nv\u271d : Vector \u03b1 n\nw\u271d : Vector \u03b2 n\nnil : C Vector.nil Vector.nil\ncons : {n : \u2115} \u2192 {a : \u03b1} \u2192 {b : \u03b2} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 C x y \u2192 C (a ::\u1d65 x) (b ::\u1d65 y)\nv : Vector \u03b1 Nat.zero\nw : Vector \u03b2 Nat.zero\n\u22a2 C v w\n[PROOFSTEP]\nrcases v with \u27e8_ | \u27e8-, -\u27e9, - | -\u27e9\n[GOAL]\ncase zero.mk.nil.refl\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Sort u_4\nv : Vector \u03b1 n\nw\u271d : Vector \u03b2 n\nnil : C Vector.nil Vector.nil\ncons : {n : \u2115} \u2192 {a : \u03b1} \u2192 {b : \u03b2} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 C x y \u2192 C (a ::\u1d65 x) (b ::\u1d65 y)\nw : Vector \u03b2 Nat.zero\n\u22a2 C { val := [], property := (_ : List.length [] = List.length []) } w\n[PROOFSTEP]\nrcases w with \u27e8_ | \u27e8-, -\u27e9, - | -\u27e9\n[GOAL]\ncase zero.mk.nil.refl.mk.nil.refl\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Sort u_4\nv : Vector \u03b1 n\nw : Vector \u03b2 n\nnil : C Vector.nil Vector.nil\ncons : {n : \u2115} \u2192 {a : \u03b1} \u2192 {b : \u03b2} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 C x y \u2192 C (a ::\u1d65 x) (b ::\u1d65 y)\n\u22a2 C { val := [], property := (_ : List.length [] = List.length []) }\n { val := [], property := (_ : List.length [] = List.length []) }\n[PROOFSTEP]\nexact nil\n[GOAL]\ncase succ\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Sort u_4\nv\u271d : Vector \u03b1 n\u271d\nw\u271d : Vector \u03b2 n\u271d\nnil : C Vector.nil Vector.nil\ncons : {n : \u2115} \u2192 {a : \u03b1} \u2192 {b : \u03b2} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 C x y \u2192 C (a ::\u1d65 x) (b ::\u1d65 y)\nn : \u2115\nih : (v : Vector \u03b1 n) \u2192 (w : Vector \u03b2 n) \u2192 C v w\nv : Vector \u03b1 (Nat.succ n)\nw : Vector \u03b2 (Nat.succ n)\n\u22a2 C v w\n[PROOFSTEP]\nrcases v with \u27e8_ | \u27e8a, v\u27e9, v_property\u27e9\n[GOAL]\ncase succ.mk.nil\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Sort u_4\nv : Vector \u03b1 n\u271d\nw\u271d : Vector \u03b2 n\u271d\nnil : C Vector.nil Vector.nil\ncons : {n : \u2115} \u2192 {a : \u03b1} \u2192 {b : \u03b2} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 C x y \u2192 C (a ::\u1d65 x) (b ::\u1d65 y)\nn : \u2115\nih : (v : Vector \u03b1 n) \u2192 (w : Vector \u03b2 n) \u2192 C v w\nw : Vector \u03b2 (Nat.succ n)\nv_property : List.length [] = Nat.succ n\n\u22a2 C { val := [], property := v_property } w\ncase succ.mk.cons\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Sort u_4\nv\u271d : Vector \u03b1 n\u271d\nw\u271d : Vector \u03b2 n\u271d\nnil : C Vector.nil Vector.nil\ncons : {n : \u2115} \u2192 {a : \u03b1} \u2192 {b : \u03b2} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 C x y \u2192 C (a ::\u1d65 x) (b ::\u1d65 y)\nn : \u2115\nih : (v : Vector \u03b1 n) \u2192 (w : Vector \u03b2 n) \u2192 C v w\nw : Vector \u03b2 (Nat.succ n)\na : \u03b1\nv : List \u03b1\nv_property : List.length (a :: v) = Nat.succ n\n\u22a2 C { val := a :: v, property := v_property } w\n[PROOFSTEP]\ncases v_property\n[GOAL]\ncase succ.mk.cons\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Sort u_4\nv\u271d : Vector \u03b1 n\u271d\nw\u271d : Vector \u03b2 n\u271d\nnil : C Vector.nil Vector.nil\ncons : {n : \u2115} \u2192 {a : \u03b1} \u2192 {b : \u03b2} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 C x y \u2192 C (a ::\u1d65 x) (b ::\u1d65 y)\nn : \u2115\nih : (v : Vector \u03b1 n) \u2192 (w : Vector \u03b2 n) \u2192 C v w\nw : Vector \u03b2 (Nat.succ n)\na : \u03b1\nv : List \u03b1\nv_property : List.length (a :: v) = Nat.succ n\n\u22a2 C { val := a :: v, property := v_property } w\n[PROOFSTEP]\nrcases w with \u27e8_ | \u27e8b, w\u27e9, w_property\u27e9\n[GOAL]\ncase succ.mk.cons.mk.nil\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Sort u_4\nv\u271d : Vector \u03b1 n\u271d\nw : Vector \u03b2 n\u271d\nnil : C Vector.nil Vector.nil\ncons : {n : \u2115} \u2192 {a : \u03b1} \u2192 {b : \u03b2} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 C x y \u2192 C (a ::\u1d65 x) (b ::\u1d65 y)\nn : \u2115\nih : (v : Vector \u03b1 n) \u2192 (w : Vector \u03b2 n) \u2192 C v w\na : \u03b1\nv : List \u03b1\nv_property : List.length (a :: v) = Nat.succ n\nw_property : List.length [] = Nat.succ n\n\u22a2 C { val := a :: v, property := v_property } { val := [], property := w_property }\ncase succ.mk.cons.mk.cons\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Sort u_4\nv\u271d : Vector \u03b1 n\u271d\nw\u271d : Vector \u03b2 n\u271d\nnil : C Vector.nil Vector.nil\ncons : {n : \u2115} \u2192 {a : \u03b1} \u2192 {b : \u03b2} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 C x y \u2192 C (a ::\u1d65 x) (b ::\u1d65 y)\nn : \u2115\nih : (v : Vector \u03b1 n) \u2192 (w : Vector \u03b2 n) \u2192 C v w\na : \u03b1\nv : List \u03b1\nv_property : List.length (a :: v) = Nat.succ n\nb : \u03b2\nw : List \u03b2\nw_property : List.length (b :: w) = Nat.succ n\n\u22a2 C { val := a :: v, property := v_property } { val := b :: w, property := w_property }\n[PROOFSTEP]\ncases w_property\n[GOAL]\ncase succ.mk.cons.mk.cons\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Sort u_4\nv\u271d : Vector \u03b1 n\u271d\nw\u271d : Vector \u03b2 n\u271d\nnil : C Vector.nil Vector.nil\ncons : {n : \u2115} \u2192 {a : \u03b1} \u2192 {b : \u03b2} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 C x y \u2192 C (a ::\u1d65 x) (b ::\u1d65 y)\nn : \u2115\nih : (v : Vector \u03b1 n) \u2192 (w : Vector \u03b2 n) \u2192 C v w\na : \u03b1\nv : List \u03b1\nv_property : List.length (a :: v) = Nat.succ n\nb : \u03b2\nw : List \u03b2\nw_property : List.length (b :: w) = Nat.succ n\n\u22a2 C { val := a :: v, property := v_property } { val := b :: w, property := w_property }\n[PROOFSTEP]\napply @cons n _ _ \u27e8v, (add_left_inj 1).mp v_property\u27e9 \u27e8w, (add_left_inj 1).mp w_property\u27e9\n[GOAL]\ncase succ.mk.cons.mk.cons\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Sort u_4\nv\u271d : Vector \u03b1 n\u271d\nw\u271d : Vector \u03b2 n\u271d\nnil : C Vector.nil Vector.nil\ncons : {n : \u2115} \u2192 {a : \u03b1} \u2192 {b : \u03b2} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 C x y \u2192 C (a ::\u1d65 x) (b ::\u1d65 y)\nn : \u2115\nih : (v : Vector \u03b1 n) \u2192 (w : Vector \u03b2 n) \u2192 C v w\na : \u03b1\nv : List \u03b1\nv_property : List.length (a :: v) = Nat.succ n\nb : \u03b2\nw : List \u03b2\nw_property : List.length (b :: w) = Nat.succ n\n\u22a2 C { val := v, property := (_ : List.length v = n) } { val := w, property := (_ : List.length w = n) }\n[PROOFSTEP]\napply ih\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu : Vector \u03b1 n\nv : Vector \u03b2 n\nw : Vector \u03b3 n\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\n\u22a2 C u v w\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu\u271d : Vector \u03b1 n\nv\u271d : Vector \u03b2 n\nw\u271d : Vector \u03b3 n\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\nu : Vector \u03b1 Nat.zero\nv : Vector \u03b2 Nat.zero\nw : Vector \u03b3 Nat.zero\n\u22a2 C u v w\n[PROOFSTEP]\nrcases u with \u27e8_ | \u27e8-, -\u27e9, - | -\u27e9\n[GOAL]\ncase zero.mk.nil.refl\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu : Vector \u03b1 n\nv\u271d : Vector \u03b2 n\nw\u271d : Vector \u03b3 n\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\nv : Vector \u03b2 Nat.zero\nw : Vector \u03b3 Nat.zero\n\u22a2 C { val := [], property := (_ : List.length [] = List.length []) } v w\n[PROOFSTEP]\nrcases v with \u27e8_ | \u27e8-, -\u27e9, - | -\u27e9\n[GOAL]\ncase zero.mk.nil.refl.mk.nil.refl\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu : Vector \u03b1 n\nv : Vector \u03b2 n\nw\u271d : Vector \u03b3 n\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\nw : Vector \u03b3 Nat.zero\n\u22a2 C { val := [], property := (_ : List.length [] = List.length []) }\n { val := [], property := (_ : List.length [] = List.length []) } w\n[PROOFSTEP]\nrcases w with \u27e8_ | \u27e8-, -\u27e9, - | -\u27e9\n[GOAL]\ncase zero.mk.nil.refl.mk.nil.refl.mk.nil.refl\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu : Vector \u03b1 n\nv : Vector \u03b2 n\nw : Vector \u03b3 n\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\n\u22a2 C { val := [], property := (_ : List.length [] = List.length []) }\n { val := [], property := (_ : List.length [] = List.length []) }\n { val := [], property := (_ : List.length [] = List.length []) }\n[PROOFSTEP]\nexact nil\n[GOAL]\ncase succ\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu\u271d : Vector \u03b1 n\u271d\nv\u271d : Vector \u03b2 n\u271d\nw\u271d : Vector \u03b3 n\u271d\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\nn : \u2115\nih : (u : Vector \u03b1 n) \u2192 (v : Vector \u03b2 n) \u2192 (w : Vector \u03b3 n) \u2192 C u v w\nu : Vector \u03b1 (Nat.succ n)\nv : Vector \u03b2 (Nat.succ n)\nw : Vector \u03b3 (Nat.succ n)\n\u22a2 C u v w\n[PROOFSTEP]\nrcases u with \u27e8_ | \u27e8a, u\u27e9, u_property\u27e9\n[GOAL]\ncase succ.mk.nil\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu : Vector \u03b1 n\u271d\nv\u271d : Vector \u03b2 n\u271d\nw\u271d : Vector \u03b3 n\u271d\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\nn : \u2115\nih : (u : Vector \u03b1 n) \u2192 (v : Vector \u03b2 n) \u2192 (w : Vector \u03b3 n) \u2192 C u v w\nv : Vector \u03b2 (Nat.succ n)\nw : Vector \u03b3 (Nat.succ n)\nu_property : List.length [] = Nat.succ n\n\u22a2 C { val := [], property := u_property } v w\ncase succ.mk.cons\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu\u271d : Vector \u03b1 n\u271d\nv\u271d : Vector \u03b2 n\u271d\nw\u271d : Vector \u03b3 n\u271d\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\nn : \u2115\nih : (u : Vector \u03b1 n) \u2192 (v : Vector \u03b2 n) \u2192 (w : Vector \u03b3 n) \u2192 C u v w\nv : Vector \u03b2 (Nat.succ n)\nw : Vector \u03b3 (Nat.succ n)\na : \u03b1\nu : List \u03b1\nu_property : List.length (a :: u) = Nat.succ n\n\u22a2 C { val := a :: u, property := u_property } v w\n[PROOFSTEP]\ncases u_property\n[GOAL]\ncase succ.mk.cons\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu\u271d : Vector \u03b1 n\u271d\nv\u271d : Vector \u03b2 n\u271d\nw\u271d : Vector \u03b3 n\u271d\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\nn : \u2115\nih : (u : Vector \u03b1 n) \u2192 (v : Vector \u03b2 n) \u2192 (w : Vector \u03b3 n) \u2192 C u v w\nv : Vector \u03b2 (Nat.succ n)\nw : Vector \u03b3 (Nat.succ n)\na : \u03b1\nu : List \u03b1\nu_property : List.length (a :: u) = Nat.succ n\n\u22a2 C { val := a :: u, property := u_property } v w\n[PROOFSTEP]\nrcases v with \u27e8_ | \u27e8b, v\u27e9, v_property\u27e9\n[GOAL]\ncase succ.mk.cons.mk.nil\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu\u271d : Vector \u03b1 n\u271d\nv : Vector \u03b2 n\u271d\nw\u271d : Vector \u03b3 n\u271d\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\nn : \u2115\nih : (u : Vector \u03b1 n) \u2192 (v : Vector \u03b2 n) \u2192 (w : Vector \u03b3 n) \u2192 C u v w\nw : Vector \u03b3 (Nat.succ n)\na : \u03b1\nu : List \u03b1\nu_property : List.length (a :: u) = Nat.succ n\nv_property : List.length [] = Nat.succ n\n\u22a2 C { val := a :: u, property := u_property } { val := [], property := v_property } w\ncase succ.mk.cons.mk.cons\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu\u271d : Vector \u03b1 n\u271d\nv\u271d : Vector \u03b2 n\u271d\nw\u271d : Vector \u03b3 n\u271d\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\nn : \u2115\nih : (u : Vector \u03b1 n) \u2192 (v : Vector \u03b2 n) \u2192 (w : Vector \u03b3 n) \u2192 C u v w\nw : Vector \u03b3 (Nat.succ n)\na : \u03b1\nu : List \u03b1\nu_property : List.length (a :: u) = Nat.succ n\nb : \u03b2\nv : List \u03b2\nv_property : List.length (b :: v) = Nat.succ n\n\u22a2 C { val := a :: u, property := u_property } { val := b :: v, property := v_property } w\n[PROOFSTEP]\ncases v_property\n[GOAL]\ncase succ.mk.cons.mk.cons\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu\u271d : Vector \u03b1 n\u271d\nv\u271d : Vector \u03b2 n\u271d\nw\u271d : Vector \u03b3 n\u271d\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\nn : \u2115\nih : (u : Vector \u03b1 n) \u2192 (v : Vector \u03b2 n) \u2192 (w : Vector \u03b3 n) \u2192 C u v w\nw : Vector \u03b3 (Nat.succ n)\na : \u03b1\nu : List \u03b1\nu_property : List.length (a :: u) = Nat.succ n\nb : \u03b2\nv : List \u03b2\nv_property : List.length (b :: v) = Nat.succ n\n\u22a2 C { val := a :: u, property := u_property } { val := b :: v, property := v_property } w\n[PROOFSTEP]\nrcases w with \u27e8_ | \u27e8c, w\u27e9, w_property\u27e9\n[GOAL]\ncase succ.mk.cons.mk.cons.mk.nil\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu\u271d : Vector \u03b1 n\u271d\nv\u271d : Vector \u03b2 n\u271d\nw : Vector \u03b3 n\u271d\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\nn : \u2115\nih : (u : Vector \u03b1 n) \u2192 (v : Vector \u03b2 n) \u2192 (w : Vector \u03b3 n) \u2192 C u v w\na : \u03b1\nu : List \u03b1\nu_property : List.length (a :: u) = Nat.succ n\nb : \u03b2\nv : List \u03b2\nv_property : List.length (b :: v) = Nat.succ n\nw_property : List.length [] = Nat.succ n\n\u22a2 C { val := a :: u, property := u_property } { val := b :: v, property := v_property }\n { val := [], property := w_property }\ncase succ.mk.cons.mk.cons.mk.cons\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu\u271d : Vector \u03b1 n\u271d\nv\u271d : Vector \u03b2 n\u271d\nw\u271d : Vector \u03b3 n\u271d\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\nn : \u2115\nih : (u : Vector \u03b1 n) \u2192 (v : Vector \u03b2 n) \u2192 (w : Vector \u03b3 n) \u2192 C u v w\na : \u03b1\nu : List \u03b1\nu_property : List.length (a :: u) = Nat.succ n\nb : \u03b2\nv : List \u03b2\nv_property : List.length (b :: v) = Nat.succ n\nc : \u03b3\nw : List \u03b3\nw_property : List.length (c :: w) = Nat.succ n\n\u22a2 C { val := a :: u, property := u_property } { val := b :: v, property := v_property }\n { val := c :: w, property := w_property }\n[PROOFSTEP]\ncases w_property\n[GOAL]\ncase succ.mk.cons.mk.cons.mk.cons\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu\u271d : Vector \u03b1 n\u271d\nv\u271d : Vector \u03b2 n\u271d\nw\u271d : Vector \u03b3 n\u271d\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\nn : \u2115\nih : (u : Vector \u03b1 n) \u2192 (v : Vector \u03b2 n) \u2192 (w : Vector \u03b3 n) \u2192 C u v w\na : \u03b1\nu : List \u03b1\nu_property : List.length (a :: u) = Nat.succ n\nb : \u03b2\nv : List \u03b2\nv_property : List.length (b :: v) = Nat.succ n\nc : \u03b3\nw : List \u03b3\nw_property : List.length (c :: w) = Nat.succ n\n\u22a2 C { val := a :: u, property := u_property } { val := b :: v, property := v_property }\n { val := c :: w, property := w_property }\n[PROOFSTEP]\napply\n @cons n _ _ _ \u27e8u, (add_left_inj 1).mp u_property\u27e9 \u27e8v, (add_left_inj 1).mp v_property\u27e9\n \u27e8w, (add_left_inj 1).mp w_property\u27e9\n[GOAL]\ncase succ.mk.cons.mk.cons.mk.cons\nn\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Vector \u03b3 n \u2192 Sort u_4\nu\u271d : Vector \u03b1 n\u271d\nv\u271d : Vector \u03b2 n\u271d\nw\u271d : Vector \u03b3 n\u271d\nnil : C Vector.nil Vector.nil Vector.nil\ncons :\n {n : \u2115} \u2192\n {a : \u03b1} \u2192\n {b : \u03b2} \u2192\n {c : \u03b3} \u2192 {x : Vector \u03b1 n} \u2192 {y : Vector \u03b2 n} \u2192 {z : Vector \u03b3 n} \u2192 C x y z \u2192 C (a ::\u1d65 x) (b ::\u1d65 y) (c ::\u1d65 z)\nn : \u2115\nih : (u : Vector \u03b1 n) \u2192 (v : Vector \u03b2 n) \u2192 (w : Vector \u03b3 n) \u2192 C u v w\na : \u03b1\nu : List \u03b1\nu_property : List.length (a :: u) = Nat.succ n\nb : \u03b2\nv : List \u03b2\nv_property : List.length (b :: v) = Nat.succ n\nc : \u03b3\nw : List \u03b3\nw_property : List.length (c :: w) = Nat.succ n\n\u22a2 C { val := u, property := (_ : List.length u = n) } { val := v, property := (_ : List.length v = n) }\n { val := w, property := (_ : List.length w = n) }\n[PROOFSTEP]\napply ih\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nxs : List \u03b1\nproperty\u271d : List.length xs = n\n\u22a2 Array \u03b1 = Array \u03b1\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a : \u03b1\ni : Fin (n + 1)\nv : Vector \u03b1 n\n\u22a2 List.length (List.insertNth (\u2191i) a \u2191v) = n + 1\n[PROOFSTEP]\nrw [List.length_insertNth, v.2]\n[GOAL]\ncase a\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a : \u03b1\ni : Fin (n + 1)\nv : Vector \u03b1 n\n\u22a2 \u2191i \u2264 List.length \u2191v\n[PROOFSTEP]\nrw [v.2, \u2190 Nat.succ_le_succ_iff]\n[GOAL]\ncase a\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a : \u03b1\ni : Fin (n + 1)\nv : Vector \u03b1 n\n\u22a2 Nat.succ \u2191i \u2264 Nat.succ n\n[PROOFSTEP]\nexact i.2\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\n\u22a2 removeNth (Fin.succAbove { val := j, isLt := hj } { val := i, isLt := hi }) (insertNth a { val := j, isLt := hj } v) =\n insertNth a (Fin.predAbove { val := i, isLt := hi } { val := j, isLt := hj }) (removeNth { val := i, isLt := hi } v)\n[PROOFSTEP]\ndsimp [insertNth, removeNth, Fin.succAbove, Fin.predAbove]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\n\u22a2 {\n val :=\n List.removeNth (List.insertNth j a \u2191v)\n \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }\n else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) }),\n property :=\n (_ :\n List.length\n (List.removeNth (List.insertNth j a \u2191v)\n \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }\n else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) })) =\n n + 1 + 1 - 1) } =\n {\n val :=\n List.insertNth\n (\u2191(if h : { val := i, isLt := (_ : i < Nat.succ (n + 1)) } < { val := j, isLt := hj } then\n Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)\n else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))\n a\n \u2191(match v with\n | { val := l, property := p } =>\n { val := List.removeNth l i,\n property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) }),\n property :=\n (_ :\n List.length\n (List.insertNth\n (\u2191(if h : { val := i, isLt := (_ : i < Nat.succ (n + 1)) } < { val := j, isLt := hj } then\n Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)\n else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))\n a\n \u2191(match v with\n | { val := l, property := p } =>\n { val := List.removeNth l i,\n property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })) =\n n + 1) }\n[PROOFSTEP]\nrw [Subtype.mk_eq_mk]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\n\u22a2 List.removeNth (List.insertNth j a \u2191v)\n \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }\n else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) }) =\n List.insertNth\n (\u2191(if h : { val := i, isLt := (_ : i < Nat.succ (n + 1)) } < { val := j, isLt := hj } then\n Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)\n else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))\n a\n \u2191(match v with\n | { val := l, property := p } =>\n { val := List.removeNth l i,\n property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })\n[PROOFSTEP]\nsimp only [Fin.lt_iff_val_lt_val]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\n\u22a2 List.removeNth (List.insertNth j a \u2191v)\n \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }\n else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) }) =\n List.insertNth\n (\u2191(if h : i < j then Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)\n else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))\n a\n \u2191(match v with\n | { val := l, property := p } =>\n { val := List.removeNth l i,\n property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })\n[PROOFSTEP]\nsplit_ifs with hij\n[GOAL]\ncase pos\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : i < j\n\u22a2 List.removeNth (List.insertNth j a \u2191v) \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } =\n List.insertNth (\u2191(Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0))) a\n \u2191(match v with\n | { val := l, property := p } =>\n { val := List.removeNth l i,\n property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })\n[PROOFSTEP]\nrcases Nat.exists_eq_succ_of_ne_zero (Nat.pos_iff_ne_zero.1 (lt_of_le_of_lt (Nat.zero_le _) hij)) with \u27e8j, rfl\u27e9\n[GOAL]\ncase pos.intro\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : Nat.succ j < n + 2\nhij : i < Nat.succ j\n\u22a2 List.removeNth (List.insertNth (Nat.succ j) a \u2191v) \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } =\n List.insertNth (\u2191(Fin.pred { val := Nat.succ j, isLt := hj } (_ : { val := Nat.succ j, isLt := hj } \u2260 0))) a\n \u2191(match v with\n | { val := l, property := p } =>\n { val := List.removeNth l i,\n property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })\n[PROOFSTEP]\nrw [\u2190 List.insertNth_removeNth_of_ge]\n[GOAL]\ncase pos.intro\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : Nat.succ j < n + 2\nhij : i < Nat.succ j\n\u22a2 List.insertNth j a (List.removeNth \u2191v \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) }) =\n List.insertNth (\u2191(Fin.pred { val := Nat.succ j, isLt := hj } (_ : { val := Nat.succ j, isLt := hj } \u2260 0))) a\n \u2191(match v with\n | { val := l, property := p } =>\n { val := List.removeNth l i,\n property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos.intro\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : Nat.succ j < n + 2\nhij : i < Nat.succ j\n\u22a2 List.insertNth j a (List.removeNth (\u2191v) i) =\n List.insertNth j a\n \u2191(match v with\n | { val := l, property := p } =>\n { val := List.removeNth l i,\n property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.intro.a\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : Nat.succ j < n + 2\nhij : i < Nat.succ j\n\u22a2 \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } < List.length \u2191v\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase pos.intro.a\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : Nat.succ j < n + 2\nhij : i < Nat.succ j\n\u22a2 \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } \u2264 j\n[PROOFSTEP]\nsimpa [Nat.lt_succ_iff] using hij\n[GOAL]\ncase neg\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : \u00aci < j\n\u22a2 List.removeNth (List.insertNth j a \u2191v) \u2191{ val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) } =\n List.insertNth (\u2191{ val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }) a\n \u2191(match v with\n | { val := l, property := p } =>\n { val := List.removeNth l i,\n property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase neg\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : \u00aci < j\n\u22a2 List.removeNth (List.insertNth j a \u2191v) (i + 1) =\n List.insertNth j a\n \u2191(match v with\n | { val := l, property := p } =>\n { val := List.removeNth l i,\n property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })\n[PROOFSTEP]\nrw [\u2190 List.insertNth_removeNth_of_le i j _ _ _]\n[GOAL]\ncase neg\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : \u00aci < j\n\u22a2 List.insertNth j a (List.removeNth (\u2191v) i) =\n List.insertNth j a\n \u2191(match v with\n | { val := l, property := p } =>\n { val := List.removeNth l i,\n property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : \u00aci < j\n\u22a2 i < List.length \u2191v\n[PROOFSTEP]\nsimpa\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nv : Vector \u03b1 (n + 1)\ni : \u2115\nhi : i < n + 1\nj : \u2115\nhj : j < n + 2\nhij : \u00aci < j\n\u22a2 j \u2264 i\n[PROOFSTEP]\nsimpa [not_lt] using hij\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 insertNth b (Fin.succ j) (insertNth a i { val := l, property := hl }) =\n insertNth a (Fin.castSucc i) (insertNth b j { val := l, property := hl })\n[PROOFSTEP]\nrefine' Subtype.eq _\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 \u2191(insertNth b (Fin.succ j) (insertNth a i { val := l, property := hl })) =\n \u2191(insertNth a (Fin.castSucc i) (insertNth b j { val := l, property := hl }))\n[PROOFSTEP]\nsimp only [insertNth_val, Fin.val_succ, Fin.castSucc, Fin.coe_castAdd]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 List.insertNth (\u2191j + 1) b (List.insertNth (\u2191i) a l) = List.insertNth (\u2191i) a (List.insertNth (\u2191j) b l)\n[PROOFSTEP]\napply List.insertNth_comm\n[GOAL]\ncase x\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 \u2191i \u2264 \u2191j\n[PROOFSTEP]\nassumption\n[GOAL]\ncase x\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 \u2191j \u2264 List.length l\n[PROOFSTEP]\nrw [hl]\n[GOAL]\ncase x\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d a b : \u03b1\ni j : Fin (n + 1)\nh : i \u2264 j\nl : List \u03b1\nhl : List.length l = n\n\u22a2 \u2191j \u2264 n\n[PROOFSTEP]\nexact Nat.le_of_succ_le_succ j.2\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nv : Vector \u03b1 n\ni : Fin n\na : \u03b1\n\u22a2 List.length (List.set (\u2191v) (\u2191i) a) = n\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nv : Vector \u03b1 n\ni : Fin n\na : \u03b1\n\u22a2 get (set v i a) i = a\n[PROOFSTEP]\ncases v\n[GOAL]\ncase mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ni : Fin n\na : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 get (set { val := val\u271d, property := property\u271d } i a) i = a\n[PROOFSTEP]\ncases i\n[GOAL]\ncase mk.mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nval\u271d\u00b9 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b9 = n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\n\u22a2 get (set { val := val\u271d\u00b9, property := property\u271d } { val := val\u271d, isLt := isLt\u271d } a) { val := val\u271d, isLt := isLt\u271d } = a\n[PROOFSTEP]\nsimp [Vector.set, get_eq_get]\n[GOAL]\ncase mk.mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nval\u271d\u00b9 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b9 = n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\n\u22a2 List.get (List.set val\u271d\u00b9 val\u271d a)\n (\u2191(Fin.castIso\n (_ :\n n =\n List.length\n (toList\n { val := List.set val\u271d\u00b9 val\u271d a,\n property :=\n (_ :\n List.length\n (List.set (\u2191{ val := val\u271d\u00b9, property := property\u271d }) (\u2191{ val := val\u271d, isLt := isLt\u271d })\n a) =\n n) })))\n { val := val\u271d, isLt := isLt\u271d }) =\n a\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nval\u271d\u00b9 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b9 = n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\n\u22a2 List.get (List.set val\u271d\u00b9 val\u271d a)\n (\u2191(Fin.castIso (_ : n = List.length (List.set val\u271d\u00b9 val\u271d a))) { val := val\u271d, isLt := isLt\u271d }) =\n a\n[PROOFSTEP]\nexact List.get_set_eq _ _ _ _\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nv : Vector \u03b1 n\ni j : Fin n\nh : i \u2260 j\na : \u03b1\n\u22a2 get (set v i a) j = get v j\n[PROOFSTEP]\ncases v\n[GOAL]\ncase mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ni j : Fin n\nh : i \u2260 j\na : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 get (set { val := val\u271d, property := property\u271d } i a) j = get { val := val\u271d, property := property\u271d } j\n[PROOFSTEP]\ncases i\n[GOAL]\ncase mk.mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nj : Fin n\na : \u03b1\nval\u271d\u00b9 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b9 = n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\nh : { val := val\u271d, isLt := isLt\u271d } \u2260 j\n\u22a2 get (set { val := val\u271d\u00b9, property := property\u271d } { val := val\u271d, isLt := isLt\u271d } a) j =\n get { val := val\u271d\u00b9, property := property\u271d } j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase mk.mk.mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nval\u271d\u00b2 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b2 = n\nval\u271d\u00b9 : \u2115\nisLt\u271d\u00b9 : val\u271d\u00b9 < n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\nh : { val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 } \u2260 { val := val\u271d, isLt := isLt\u271d }\n\u22a2 get (set { val := val\u271d\u00b2, property := property\u271d } { val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 } a) { val := val\u271d, isLt := isLt\u271d } =\n get { val := val\u271d\u00b2, property := property\u271d } { val := val\u271d, isLt := isLt\u271d }\n[PROOFSTEP]\nsimp [Vector.set, Vector.get_eq_get, List.get_set_of_ne (Fin.vne_of_ne h)]\n[GOAL]\ncase mk.mk.mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nval\u271d\u00b2 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b2 = n\nval\u271d\u00b9 : \u2115\nisLt\u271d\u00b9 : val\u271d\u00b9 < n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\nh : { val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 } \u2260 { val := val\u271d, isLt := isLt\u271d }\n\u22a2 List.get (List.set val\u271d\u00b2 val\u271d\u00b9 a)\n (\u2191(Fin.castIso\n (_ :\n n =\n List.length\n (toList\n { val := List.set val\u271d\u00b2 val\u271d\u00b9 a,\n property :=\n (_ :\n List.length\n (List.set (\u2191{ val := val\u271d\u00b2, property := property\u271d }) (\u2191{ val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 })\n a) =\n n) })))\n { val := val\u271d, isLt := isLt\u271d }) =\n List.get val\u271d\u00b2\n (\u2191(Fin.castIso (_ : n = List.length (toList { val := val\u271d\u00b2, property := property\u271d })))\n { val := val\u271d, isLt := isLt\u271d })\n[PROOFSTEP]\nrw [List.get_set_of_ne]\n[GOAL]\ncase mk.mk.mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nval\u271d\u00b2 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b2 = n\nval\u271d\u00b9 : \u2115\nisLt\u271d\u00b9 : val\u271d\u00b9 < n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\nh : { val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 } \u2260 { val := val\u271d, isLt := isLt\u271d }\n\u22a2 List.get val\u271d\u00b2\n {\n val :=\n \u2191(\u2191(Fin.castIso\n (_ :\n n =\n List.length\n (toList\n { val := List.set val\u271d\u00b2 val\u271d\u00b9 a,\n property :=\n (_ :\n List.length\n (List.set (\u2191{ val := val\u271d\u00b2, property := property\u271d })\n (\u2191{ val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 }) a) =\n n) })))\n { val := val\u271d, isLt := isLt\u271d }),\n isLt :=\n (_ :\n \u2191(\u2191(Fin.castIso\n (_ :\n n =\n List.length\n (toList\n { val := List.set val\u271d\u00b2 val\u271d\u00b9 a,\n property :=\n (_ :\n List.length\n (List.set (\u2191{ val := val\u271d\u00b2, property := property\u271d })\n (\u2191{ val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 }) a) =\n n) })))\n { val := val\u271d, isLt := isLt\u271d }) <\n List.length val\u271d\u00b2) } =\n List.get val\u271d\u00b2\n (\u2191(Fin.castIso (_ : n = List.length (toList { val := val\u271d\u00b2, property := property\u271d })))\n { val := val\u271d, isLt := isLt\u271d })\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.mk.mk.h\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nval\u271d\u00b2 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b2 = n\nval\u271d\u00b9 : \u2115\nisLt\u271d\u00b9 : val\u271d\u00b9 < n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\nh : { val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 } \u2260 { val := val\u271d, isLt := isLt\u271d }\n\u22a2 val\u271d\u00b9 \u2260\n \u2191(\u2191(Fin.castIso\n (_ :\n n =\n List.length\n (toList\n { val := List.set val\u271d\u00b2 val\u271d\u00b9 a,\n property :=\n (_ :\n List.length\n (List.set (\u2191{ val := val\u271d\u00b2, property := property\u271d }) (\u2191{ val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 })\n a) =\n n) })))\n { val := val\u271d, isLt := isLt\u271d })\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nv : Vector \u03b1 n\ni j : Fin n\na : \u03b1\n\u22a2 get (set v i a) j = if i = j then a else get v j\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nv : Vector \u03b1 n\ni j : Fin n\na : \u03b1\nh\u271d : i = j\n\u22a2 get (set v i a) j = a\n[PROOFSTEP]\ntry simp [*]\n[GOAL]\ncase pos\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nv : Vector \u03b1 n\ni j : Fin n\na : \u03b1\nh\u271d : i = j\n\u22a2 get (set v i a) j = a\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nv : Vector \u03b1 n\ni j : Fin n\na : \u03b1\nh\u271d : \u00aci = j\n\u22a2 get (set v i a) j = get v j\n[PROOFSTEP]\ntry simp [*]\n[GOAL]\ncase neg\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nv : Vector \u03b1 n\ni j : Fin n\na : \u03b1\nh\u271d : \u00aci = j\n\u22a2 get (set v i a) j = get v j\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nv : Vector \u03b1 n\ni j : Fin n\na : \u03b1\nh\u271d : \u00aci = j\n\u22a2 get (set v i a) j = get v j\n[PROOFSTEP]\nrwa [get_set_of_ne]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Monoid \u03b1\nv : Vector \u03b1 n\ni : Fin n\na : \u03b1\n\u22a2 List.prod (toList (set v i a)) = List.prod (toList (take (\u2191i) v)) * a * List.prod (toList (drop (\u2191i + 1) v))\n[PROOFSTEP]\nrefine' (List.prod_set v.toList i a).trans _\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Monoid \u03b1\nv : Vector \u03b1 n\ni : Fin n\na : \u03b1\n\u22a2 (List.prod (List.take (\u2191i) (toList v)) * if \u2191i < List.length (toList v) then a else 1) *\n List.prod (List.drop (\u2191i + 1) (toList v)) =\n List.prod (toList (take (\u2191i) v)) * a * List.prod (toList (drop (\u2191i + 1) v))\n[PROOFSTEP]\nsimp_all\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommGroup \u03b1\nv : Vector \u03b1 n\ni : Fin n\na : \u03b1\n\u22a2 List.prod (toList (set v i a)) = List.prod (toList v) * (get v i)\u207b\u00b9 * a\n[PROOFSTEP]\nrefine' (List.prod_set' v.toList i a).trans _\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommGroup \u03b1\nv : Vector \u03b1 n\ni : Fin n\na : \u03b1\n\u22a2 (List.prod (toList v) * if hn : \u2191i < List.length (toList v) then (List.nthLe (toList v) (\u2191i) hn)\u207b\u00b9 * a else 1) =\n List.prod (toList v) * (get v i)\u207b\u00b9 * a\n[PROOFSTEP]\nsimp [get_eq_get, mul_assoc]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : CommGroup \u03b1\nv : Vector \u03b1 n\ni : Fin n\na : \u03b1\n\u22a2 List.nthLe (toList v) \u2191i (_ : \u2191i < List.length (toList v)) =\n List.get (toList v) (\u2191(Fin.castIso (_ : n = List.length (toList v))) i)\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nF G : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : Applicative G\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\nv : List \u03b1\nHv : List.length v = n\n\u22a2 F (Vector \u03b2 (List.length v)) = F (Vector \u03b2 n)\n[PROOFSTEP]\nrw [Hv]\n[GOAL]\nn : \u2115\nF G : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : Applicative G\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\nx : \u03b1\n\u22a2 \u2200 (xs : Vector \u03b1 n), Vector.traverse f (x ::\u1d65 xs) = Seq.seq (cons <$> f x) fun x => Vector.traverse f xs\n[PROOFSTEP]\nrintro \u27e8xs, rfl\u27e9\n[GOAL]\ncase mk\nF G : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : Applicative G\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\nx : \u03b1\nxs : List \u03b1\n\u22a2 Vector.traverse f (x ::\u1d65 { val := xs, property := (_ : List.length xs = List.length xs) }) =\n Seq.seq (cons <$> f x) fun x => Vector.traverse f { val := xs, property := (_ : List.length xs = List.length xs) }\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nF G : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : Applicative G\n\u03b1 \u03b2 : Type u\n\u22a2 \u2200 (x : Vector \u03b1 n), Vector.traverse pure x = x\n[PROOFSTEP]\nrintro \u27e8x, rfl\u27e9\n[GOAL]\ncase mk\nF G : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : Applicative G\n\u03b1 \u03b2 : Type u\nx : List \u03b1\n\u22a2 Vector.traverse pure { val := x, property := (_ : List.length x = List.length x) } =\n { val := x, property := (_ : List.length x = List.length x) }\n[PROOFSTEP]\ndsimp [Vector.traverse, cast]\n[GOAL]\ncase mk\nF G : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : Applicative G\n\u03b1 \u03b2 : Type u\nx : List \u03b1\n\u22a2 Vector.traverseAux pure x = { val := x, property := (_ : List.length x = List.length x) }\n[PROOFSTEP]\ninduction' x with x xs IH\n[GOAL]\ncase mk.nil\nF G : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : Applicative G\n\u03b1 \u03b2 : Type u\n\u22a2 Vector.traverseAux pure [] = { val := [], property := (_ : List.length [] = List.length []) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.cons\nF G : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : Applicative G\n\u03b1 \u03b2 : Type u\nx : \u03b1\nxs : List \u03b1\nIH : Vector.traverseAux pure xs = { val := xs, property := (_ : List.length xs = List.length xs) }\n\u22a2 Vector.traverseAux pure (x :: xs) =\n { val := x :: xs, property := (_ : List.length (x :: xs) = List.length (x :: xs)) }\n[PROOFSTEP]\nsimp! [IH]\n[GOAL]\ncase mk.cons\nF G : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : Applicative G\n\u03b1 \u03b2 : Type u\nx : \u03b1\nxs : List \u03b1\nIH : Vector.traverseAux pure xs = { val := xs, property := (_ : List.length xs = List.length xs) }\n\u22a2 (Seq.seq (cons x) fun x => { val := xs, property := (_ : List.length xs = List.length xs) }) =\n { val := x :: xs, property := (_ : List.length (x :: xs) = List.length (x :: xs)) }\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nx : Vector \u03b1 n\n\u22a2 Vector.traverse (Comp.mk \u2218 Functor.map f \u2218 g) x = Comp.mk (Vector.traverse f <$> Vector.traverse g x)\n[PROOFSTEP]\ninduction' x using Vector.inductionOn with n x xs ih\n[GOAL]\ncase h_nil\nn : \u2115\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\n\u22a2 Vector.traverse (Comp.mk \u2218 Functor.map f \u2218 g) nil = Comp.mk (Vector.traverse f <$> Vector.traverse g nil)\ncase h_cons\nn\u271d : \u2115\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nn : \u2115\nx : \u03b1\nxs : Vector \u03b1 n\nih : Vector.traverse (Comp.mk \u2218 Functor.map f \u2218 g) xs = Comp.mk (Vector.traverse f <$> Vector.traverse g xs)\n\u22a2 Vector.traverse (Comp.mk \u2218 Functor.map f \u2218 g) (x ::\u1d65 xs) =\n Comp.mk (Vector.traverse f <$> Vector.traverse g (x ::\u1d65 xs))\n[PROOFSTEP]\nsimp! [cast, *, functor_norm]\n[GOAL]\ncase h_nil\nn : \u2115\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\n\u22a2 pure nil = Comp.mk (pure (pure nil))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_cons\nn\u271d : \u2115\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nn : \u2115\nx : \u03b1\nxs : Vector \u03b1 n\nih : Vector.traverse (Comp.mk \u2218 Functor.map f \u2218 g) xs = Comp.mk (Vector.traverse f <$> Vector.traverse g xs)\n\u22a2 Vector.traverse (Comp.mk \u2218 Functor.map f \u2218 g) (x ::\u1d65 xs) =\n Comp.mk (Vector.traverse f <$> Vector.traverse g (x ::\u1d65 xs))\n[PROOFSTEP]\nrw [Vector.traverse_def, ih]\n[GOAL]\ncase h_cons\nn\u271d : \u2115\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nn : \u2115\nx : \u03b1\nxs : Vector \u03b1 n\nih : Vector.traverse (Comp.mk \u2218 Functor.map f \u2218 g) xs = Comp.mk (Vector.traverse f <$> Vector.traverse g xs)\n\u22a2 (Seq.seq (cons <$> (Comp.mk \u2218 Functor.map f \u2218 g) x) fun x => Comp.mk (Vector.traverse f <$> Vector.traverse g xs)) =\n Comp.mk (Vector.traverse f <$> Vector.traverse g (x ::\u1d65 xs))\n[PROOFSTEP]\nsimp [functor_norm, (\u00b7 \u2218 \u00b7)]\n[GOAL]\nn : \u2115\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u271d \u03b2\u271d \u03b3 : Type u\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (x : Vector \u03b1 n), Vector.traverse (pure \u2218 f) x = pure (map f x)\n[PROOFSTEP]\nrintro \u27e8x, rfl\u27e9\n[GOAL]\ncase mk\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u271d \u03b2\u271d \u03b3 : Type u\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\nx : List \u03b1\n\u22a2 Vector.traverse (pure \u2218 f) { val := x, property := (_ : List.length x = List.length x) } =\n pure (map f { val := x, property := (_ : List.length x = List.length x) })\n[PROOFSTEP]\nsimp!\n[GOAL]\ncase mk\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u271d \u03b2\u271d \u03b3 : Type u\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\nx : List \u03b1\n\u22a2 Vector.traverseAux (pure \u2218 f) x =\n { val := List.map f x, property := (_ : List.length (List.map f x) = List.length x) }\n[PROOFSTEP]\ninduction x\n[GOAL]\ncase mk.nil\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u271d \u03b2\u271d \u03b3 : Type u\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\n\u22a2 Vector.traverseAux (pure \u2218 f) [] =\n { val := List.map f [], property := (_ : List.length (List.map f []) = List.length []) }\n[PROOFSTEP]\nsimp! [*, functor_norm]\n[GOAL]\ncase mk.cons\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u271d \u03b2\u271d \u03b3 : Type u\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d :\n Vector.traverseAux (pure \u2218 f) tail\u271d =\n { val := List.map f tail\u271d, property := (_ : List.length (List.map f tail\u271d) = List.length tail\u271d) }\n\u22a2 Vector.traverseAux (pure \u2218 f) (head\u271d :: tail\u271d) =\n { val := List.map f (head\u271d :: tail\u271d),\n property := (_ : List.length (List.map f (head\u271d :: tail\u271d)) = List.length (head\u271d :: tail\u271d)) }\n[PROOFSTEP]\nsimp! [*, functor_norm]\n[GOAL]\ncase mk.nil\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u271d \u03b2\u271d \u03b3 : Type u\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\n\u22a2 nil = { val := [], property := (_ : (fun l => List.length l = List.length []) []) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.cons\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u271d \u03b2\u271d \u03b3 : Type u\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d :\n Vector.traverseAux (pure \u2218 f) tail\u271d =\n { val := List.map f tail\u271d, property := (_ : List.length (List.map f tail\u271d) = List.length tail\u271d) }\n\u22a2 (Seq.seq (cons (f head\u271d)) fun x =>\n { val := List.map f tail\u271d, property := (_ : List.length (List.map f tail\u271d) = List.length tail\u271d) }) =\n { val := f head\u271d :: List.map f tail\u271d,\n property := (_ : (fun l => List.length l = List.length (head\u271d :: tail\u271d)) (f head\u271d :: List.map f tail\u271d)) }\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u271d \u03b2\u271d \u03b3 : Type u\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\nx : Vector \u03b1 n\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (Vector.traverse f x) =\n Vector.traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) x\n[PROOFSTEP]\ninduction' x using Vector.inductionOn with n x xs ih\n[GOAL]\ncase h_nil\nn : \u2115\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u271d \u03b2\u271d \u03b3 : Type u\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (Vector.traverse f nil) =\n Vector.traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) nil\n[PROOFSTEP]\nsimp! [functor_norm, cast, \u03b7.preserves_pure]\n[GOAL]\ncase h_cons\nn\u271d : \u2115\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u271d \u03b2\u271d \u03b3 : Type u\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\nn : \u2115\nx : \u03b1\nxs : Vector \u03b1 n\nih :\n (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (Vector.traverse f xs) =\n Vector.traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) xs\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (Vector.traverse f (x ::\u1d65 xs)) =\n Vector.traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) (x ::\u1d65 xs)\n[PROOFSTEP]\nrw [Vector.traverse_def, Vector.traverse_def, \u2190 ih, \u03b7.preserves_seq, \u03b7.preserves_map]\n[GOAL]\ncase h_cons\nn\u271d : \u2115\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1\u271d \u03b2\u271d \u03b3 : Type u\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\nn : \u2115\nx : \u03b1\nxs : Vector \u03b1 n\nih :\n (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (Vector.traverse f xs) =\n Vector.traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) xs\n\u22a2 (Seq.seq (cons <$> (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (f x)) fun x =>\n (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (Vector.traverse f xs)) =\n Seq.seq (cons <$> ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) x) fun x =>\n (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (Vector.traverse f xs)\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u22a2 \u2200 {\u03b1 : Type u} (x : flip Vector n \u03b1), id <$> x = x\n[PROOFSTEP]\nintro _ x\n[GOAL]\nn : \u2115\n\u03b1\u271d : Type u\nx : flip Vector n \u03b1\u271d\n\u22a2 id <$> x = x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nn : \u2115\n\u03b1\u271d : Type u\nval\u271d : List \u03b1\u271d\nproperty\u271d : List.length val\u271d = n\n\u22a2 id <$> { val := val\u271d, property := property\u271d } = { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nsimp! [(\u00b7 <$> \u00b7)]\n[GOAL]\nn : \u2115\n\u22a2 \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 \u03b3) (x : flip Vector n \u03b1), (h \u2218 g) <$> x = h <$> g <$> x\n[PROOFSTEP]\nintro _ _ _ _ _ x\n[GOAL]\nn : \u2115\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nh\u271d : \u03b2\u271d \u2192 \u03b3\u271d\nx : flip Vector n \u03b1\u271d\n\u22a2 (h\u271d \u2218 g\u271d) <$> x = h\u271d <$> g\u271d <$> x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nn : \u2115\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nh\u271d : \u03b2\u271d \u2192 \u03b3\u271d\nval\u271d : List \u03b1\u271d\nproperty\u271d : List.length val\u271d = n\n\u22a2 (h\u271d \u2218 g\u271d) <$> { val := val\u271d, property := property\u271d } = h\u271d <$> g\u271d <$> { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nsimp! [(\u00b7 <$> \u00b7)]\n[GOAL]\n\u03b1 : Type ?u.94018\nm n : \u2115\nxs : Vector \u03b1 n\nys : Vector \u03b1 m\nx : \u03b1\n\u22a2 0 < Nat.succ n + m\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nm n : \u2115\nxs : Vector \u03b1 n\nys : Vector \u03b1 m\n\u22a2 append xs nil = xs\n[PROOFSTEP]\ncases xs\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nm n : \u2115\nys : Vector \u03b1 m\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 append { val := val\u271d, property := property\u271d } nil = { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nsimp [append]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 : Type u_1\nv\u2081 : Vector \u03b1 n\nv\u2082 : Vector \u03b2 n\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ni : Fin n\n\u22a2 get (map\u2082 f v\u2081 v\u2082) i = f (get v\u2081 i) (get v\u2082 i)\n[PROOFSTEP]\nclear * - v\u2081 v\u2082\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nv\u2081 : Vector \u03b1 n\nv\u2082 : Vector \u03b2 n\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ni : Fin n\n\u22a2 get (map\u2082 f v\u2081 v\u2082) i = f (get v\u2081 i) (get v\u2082 i)\n[PROOFSTEP]\ninduction v\u2081, v\u2082 using inductionOn\u2082\n[GOAL]\ncase nil\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ni : Fin 0\n\u22a2 get (map\u2082 f nil nil) i = f (get nil i) (get nil i)\ncase cons\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\na\u271d\u00b9 : \u03b1\nb\u271d : \u03b2\nx\u271d : Vector \u03b1 n\u271d\ny\u271d : Vector \u03b2 n\u271d\na\u271d : \u2200 (i : Fin n\u271d), get (map\u2082 f x\u271d y\u271d) i = f (get x\u271d i) (get y\u271d i)\ni : Fin (Nat.succ n\u271d)\n\u22a2 get (map\u2082 f (a\u271d\u00b9 ::\u1d65 x\u271d) (b\u271d ::\u1d65 y\u271d)) i = f (get (a\u271d\u00b9 ::\u1d65 x\u271d) i) (get (b\u271d ::\u1d65 y\u271d) i)\n[PROOFSTEP]\ncase nil => exact Fin.elim0 i\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ni : Fin 0\n\u22a2 get (map\u2082 f nil nil) i = f (get nil i) (get nil i)\n[PROOFSTEP]\ncase nil => exact Fin.elim0 i\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ni : Fin 0\n\u22a2 get (map\u2082 f nil nil) i = f (get nil i) (get nil i)\n[PROOFSTEP]\nexact Fin.elim0 i\n[GOAL]\ncase cons\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\na\u271d\u00b9 : \u03b1\nb\u271d : \u03b2\nx\u271d : Vector \u03b1 n\u271d\ny\u271d : Vector \u03b2 n\u271d\na\u271d : \u2200 (i : Fin n\u271d), get (map\u2082 f x\u271d y\u271d) i = f (get x\u271d i) (get y\u271d i)\ni : Fin (Nat.succ n\u271d)\n\u22a2 get (map\u2082 f (a\u271d\u00b9 ::\u1d65 x\u271d) (b\u271d ::\u1d65 y\u271d)) i = f (get (a\u271d\u00b9 ::\u1d65 x\u271d) i) (get (b\u271d ::\u1d65 y\u271d) i)\n[PROOFSTEP]\ncase cons x xs y ys ih =>\n rw [map\u2082_cons]\n cases i using Fin.cases\n \u00b7 simp only [get_zero, head_cons]\n \u00b7 simp only [get_cons_succ, ih]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\nx : \u03b1\nxs : \u03b2\ny : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i)\ni : Fin (Nat.succ n\u271d)\n\u22a2 get (map\u2082 f (x ::\u1d65 y) (xs ::\u1d65 ys)) i = f (get (x ::\u1d65 y) i) (get (xs ::\u1d65 ys) i)\n[PROOFSTEP]\ncase cons x xs y ys ih =>\n rw [map\u2082_cons]\n cases i using Fin.cases\n \u00b7 simp only [get_zero, head_cons]\n \u00b7 simp only [get_cons_succ, ih]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\nx : \u03b1\nxs : \u03b2\ny : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i)\ni : Fin (Nat.succ n\u271d)\n\u22a2 get (map\u2082 f (x ::\u1d65 y) (xs ::\u1d65 ys)) i = f (get (x ::\u1d65 y) i) (get (xs ::\u1d65 ys) i)\n[PROOFSTEP]\nrw [map\u2082_cons]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\nx : \u03b1\nxs : \u03b2\ny : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i)\ni : Fin (Nat.succ n\u271d)\n\u22a2 get (f x xs ::\u1d65 map\u2082 f y ys) i = f (get (x ::\u1d65 y) i) (get (xs ::\u1d65 ys) i)\n[PROOFSTEP]\ncases i using Fin.cases\n[GOAL]\ncase zero\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\nx : \u03b1\nxs : \u03b2\ny : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i)\n\u22a2 get (f x xs ::\u1d65 map\u2082 f y ys) 0 = f (get (x ::\u1d65 y) 0) (get (xs ::\u1d65 ys) 0)\n[PROOFSTEP]\nsimp only [get_zero, head_cons]\n[GOAL]\ncase succ\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn : \u2115\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u2115\nx : \u03b1\nxs : \u03b2\ny : Vector \u03b1 n\u271d\nys : Vector \u03b2 n\u271d\nih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i)\ni\u271d : Fin n\u271d\n\u22a2 get (f x xs ::\u1d65 map\u2082 f y ys) (Fin.succ i\u271d) = f (get (x ::\u1d65 y) (Fin.succ i\u271d)) (get (xs ::\u1d65 ys) (Fin.succ i\u271d))\n[PROOFSTEP]\nsimp only [get_cons_succ, ih]\n", "meta": {"mathlib_filename": "Mathlib.Data.Vector.Basic", "llama_tokens": 36634, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737214979745, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.4147749112500925}} {"text": "[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2124\n\u22a2 \u2191n / \u21911 = \u2191n\n[PROOFSTEP]\nrw [Nat.cast_one, div_one]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2115\n\u22a2 \u2191\u2191n = \u2191n\n[PROOFSTEP]\nrw [\u2190 Int.cast_ofNat, cast_coe_int, Int.cast_ofNat]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nr : \u211a\na : \u03b1\n\u22a2 Commute (\u2191r) a\n[PROOFSTEP]\nsimpa only [cast_def] using (r.1.cast_commute a).div_left (r.2.cast_commute a)\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\n\u22a2 \u2191(a /. b) = \u2191a / \u2191b\n[PROOFSTEP]\nhave b0' : b \u2260 0 := by\n refine' mt _ b0\n simp (config := { contextual := true })\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\n\u22a2 b \u2260 0\n[PROOFSTEP]\nrefine' mt _ b0\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\n\u22a2 b = 0 \u2192 \u2191b = 0\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\nb0' : b \u2260 0\n\u22a2 \u2191(a /. b) = \u2191a / \u2191b\n[PROOFSTEP]\ncases' e : a /. b with n d h c\n[GOAL]\ncase mk'\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\nb0' : b \u2260 0\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\ne : a /. b = mk' n d\n\u22a2 \u2191(mk' n d) = \u2191a / \u2191b\n[PROOFSTEP]\nhave d0 : (d : \u03b1) \u2260 0 := by\n intro d0\n have dd := den_dvd a b\n cases' show (d : \u2124) \u2223 b by rwa [e] at dd with k ke\n have : (b : \u03b1) = (d : \u03b1) * (k : \u03b1) := by rw [ke, Int.cast_mul, Int.cast_ofNat]\n rw [d0, zero_mul] at this \n contradiction\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\nb0' : b \u2260 0\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\ne : a /. b = mk' n d\n\u22a2 \u2191d \u2260 0\n[PROOFSTEP]\nintro d0\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\nb0' : b \u2260 0\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\ne : a /. b = mk' n d\nd0 : \u2191d = 0\n\u22a2 False\n[PROOFSTEP]\nhave dd := den_dvd a b\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\nb0' : b \u2260 0\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\ne : a /. b = mk' n d\nd0 : \u2191d = 0\ndd : \u2191(a /. b).den \u2223 b\n\u22a2 False\n[PROOFSTEP]\ncases' show (d : \u2124) \u2223 b by rwa [e] at dd with k ke\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\nb0' : b \u2260 0\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\ne : a /. b = mk' n d\nd0 : \u2191d = 0\ndd : \u2191(a /. b).den \u2223 b\n\u22a2 \u2191d \u2223 b\n[PROOFSTEP]\nrwa [e] at dd \n[GOAL]\ncase intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\nb0' : b \u2260 0\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\ne : a /. b = mk' n d\nd0 : \u2191d = 0\ndd : \u2191(a /. b).den \u2223 b\nk : \u2124\nke : b = \u2191d * k\n\u22a2 False\n[PROOFSTEP]\nhave : (b : \u03b1) = (d : \u03b1) * (k : \u03b1) := by rw [ke, Int.cast_mul, Int.cast_ofNat]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\nb0' : b \u2260 0\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\ne : a /. b = mk' n d\nd0 : \u2191d = 0\ndd : \u2191(a /. b).den \u2223 b\nk : \u2124\nke : b = \u2191d * k\n\u22a2 \u2191b = \u2191d * \u2191k\n[PROOFSTEP]\nrw [ke, Int.cast_mul, Int.cast_ofNat]\n[GOAL]\ncase intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\nb0' : b \u2260 0\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\ne : a /. b = mk' n d\nd0 : \u2191d = 0\ndd : \u2191(a /. b).den \u2223 b\nk : \u2124\nke : b = \u2191d * k\nthis : \u2191b = \u2191d * \u2191k\n\u22a2 False\n[PROOFSTEP]\nrw [d0, zero_mul] at this \n[GOAL]\ncase intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\nb0' : b \u2260 0\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\ne : a /. b = mk' n d\nd0 : \u2191d = 0\ndd : \u2191(a /. b).den \u2223 b\nk : \u2124\nke : b = \u2191d * k\nthis : \u2191b = 0\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase mk'\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\nb0' : b \u2260 0\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\ne : a /. b = mk' n d\nd0 : \u2191d \u2260 0\n\u22a2 \u2191(mk' n d) = \u2191a / \u2191b\n[PROOFSTEP]\nrw [num_den'] at e \n[GOAL]\ncase mk'\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\nb0' : b \u2260 0\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\ne : a /. b = n /. \u2191d\nd0 : \u2191d \u2260 0\n\u22a2 \u2191(mk' n d) = \u2191a / \u2191b\n[PROOFSTEP]\nhave := congr_arg ((\u2191) : \u2124 \u2192 \u03b1) ((divInt_eq_iff b0' <| ne_of_gt <| Int.coe_nat_pos.2 h.bot_lt).1 e)\n[GOAL]\ncase mk'\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\nb0' : b \u2260 0\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\ne : a /. b = n /. \u2191d\nd0 : \u2191d \u2260 0\nthis : \u2191(a * \u2191d) = \u2191(n * b)\n\u22a2 \u2191(mk' n d) = \u2191a / \u2191b\n[PROOFSTEP]\nrw [Int.cast_mul, Int.cast_mul, Int.cast_ofNat] at this \n[GOAL]\ncase mk'\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\nb0' : b \u2260 0\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\ne : a /. b = n /. \u2191d\nd0 : \u2191d \u2260 0\nthis : \u2191a * \u2191d = \u2191n * \u2191b\n\u22a2 \u2191(mk' n d) = \u2191a / \u2191b\n[PROOFSTEP]\napply Eq.symm\n[GOAL]\ncase mk'.h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\na b : \u2124\nb0 : \u2191b \u2260 0\nb0' : b \u2260 0\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\ne : a /. b = n /. \u2191d\nd0 : \u2191d \u2260 0\nthis : \u2191a * \u2191d = \u2191n * \u2191b\n\u22a2 \u2191a / \u2191b = \u2191(mk' n d)\n[PROOFSTEP]\nrw [cast_def, div_eq_mul_inv, eq_div_iff_mul_eq d0, mul_assoc, (d.commute_cast _).eq, \u2190 mul_assoc, this, mul_assoc,\n mul_inv_cancel b0, mul_one]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\n\u22a2 \u2191(mk' n\u2081 d\u2081 + mk' n\u2082 d\u2082) = \u2191(mk' n\u2081 d\u2081) + \u2191(mk' n\u2082 d\u2082)\n[PROOFSTEP]\nhave d\u20810' : (d\u2081 : \u2124) \u2260 0 := Int.coe_nat_ne_zero.2 fun e => by rw [e] at d\u20810 ; exact d\u20810 Nat.cast_zero\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\ne : d\u2081 = 0\n\u22a2 False\n[PROOFSTEP]\nrw [e] at d\u20810 \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u21910 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\ne : d\u2081 = 0\n\u22a2 False\n[PROOFSTEP]\nexact d\u20810 Nat.cast_zero\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\n\u22a2 \u2191(mk' n\u2081 d\u2081 + mk' n\u2082 d\u2082) = \u2191(mk' n\u2081 d\u2081) + \u2191(mk' n\u2082 d\u2082)\n[PROOFSTEP]\nhave d\u20820' : (d\u2082 : \u2124) \u2260 0 := Int.coe_nat_ne_zero.2 fun e => by rw [e] at d\u20820 ; exact d\u20820 Nat.cast_zero\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\ne : d\u2082 = 0\n\u22a2 False\n[PROOFSTEP]\nrw [e] at d\u20820 \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u21910 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\ne : d\u2082 = 0\n\u22a2 False\n[PROOFSTEP]\nexact d\u20820 Nat.cast_zero\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\n\u22a2 \u2191(mk' n\u2081 d\u2081 + mk' n\u2082 d\u2082) = \u2191(mk' n\u2081 d\u2081) + \u2191(mk' n\u2082 d\u2082)\n[PROOFSTEP]\nrw [num_den', num_den', add_def'' d\u20810' d\u20820']\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\n\u22a2 \u2191((n\u2081 * \u2191d\u2082 + n\u2082 * \u2191d\u2081) /. (\u2191d\u2081 * \u2191d\u2082)) = \u2191(n\u2081 /. \u2191d\u2081) + \u2191(n\u2082 /. \u2191d\u2082)\n[PROOFSTEP]\nsuffices\n (n\u2081 * (d\u2082 * ((d\u2082 : \u03b1)\u207b\u00b9 * (d\u2081 : \u03b1)\u207b\u00b9)) + n\u2082 * (d\u2081 * (d\u2082 : \u03b1)\u207b\u00b9) * (d\u2081 : \u03b1)\u207b\u00b9 : \u03b1) = n\u2081 * (d\u2081 : \u03b1)\u207b\u00b9 + n\u2082 * (d\u2082 : \u03b1)\u207b\u00b9\n by\n rw [cast_mk_of_ne_zero, cast_mk_of_ne_zero, cast_mk_of_ne_zero]\n \u00b7 simpa [division_def, left_distrib, right_distrib, mul_inv_rev, d\u20810, d\u20820, mul_assoc]\n all_goals simp [d\u20810, d\u20820]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191d\u2082 * ((\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9)) + \u2191n\u2082 * (\u2191d\u2081 * (\u2191d\u2082)\u207b\u00b9) * (\u2191d\u2081)\u207b\u00b9 = \u2191n\u2081 * (\u2191d\u2081)\u207b\u00b9 + \u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9\n\u22a2 \u2191((n\u2081 * \u2191d\u2082 + n\u2082 * \u2191d\u2081) /. (\u2191d\u2081 * \u2191d\u2082)) = \u2191(n\u2081 /. \u2191d\u2081) + \u2191(n\u2082 /. \u2191d\u2082)\n[PROOFSTEP]\nrw [cast_mk_of_ne_zero, cast_mk_of_ne_zero, cast_mk_of_ne_zero]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191d\u2082 * ((\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9)) + \u2191n\u2082 * (\u2191d\u2081 * (\u2191d\u2082)\u207b\u00b9) * (\u2191d\u2081)\u207b\u00b9 = \u2191n\u2081 * (\u2191d\u2081)\u207b\u00b9 + \u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9\n\u22a2 \u2191(n\u2081 * \u2191d\u2082 + n\u2082 * \u2191d\u2081) / \u2191(\u2191d\u2081 * \u2191d\u2082) = \u2191n\u2081 / \u2191\u2191d\u2081 + \u2191n\u2082 / \u2191\u2191d\u2082\n[PROOFSTEP]\nsimpa [division_def, left_distrib, right_distrib, mul_inv_rev, d\u20810, d\u20820, mul_assoc]\n[GOAL]\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191d\u2082 * ((\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9)) + \u2191n\u2082 * (\u2191d\u2081 * (\u2191d\u2082)\u207b\u00b9) * (\u2191d\u2081)\u207b\u00b9 = \u2191n\u2081 * (\u2191d\u2081)\u207b\u00b9 + \u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9\n\u22a2 \u2191\u2191d\u2082 \u2260 0\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191d\u2082 * ((\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9)) + \u2191n\u2082 * (\u2191d\u2081 * (\u2191d\u2082)\u207b\u00b9) * (\u2191d\u2081)\u207b\u00b9 = \u2191n\u2081 * (\u2191d\u2081)\u207b\u00b9 + \u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9\n\u22a2 \u2191\u2191d\u2081 \u2260 0\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191d\u2082 * ((\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9)) + \u2191n\u2082 * (\u2191d\u2081 * (\u2191d\u2082)\u207b\u00b9) * (\u2191d\u2081)\u207b\u00b9 = \u2191n\u2081 * (\u2191d\u2081)\u207b\u00b9 + \u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9\n\u22a2 \u2191(\u2191d\u2081 * \u2191d\u2082) \u2260 0\n[PROOFSTEP]\nall_goals simp [d\u20810, d\u20820]\n[GOAL]\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191d\u2082 * ((\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9)) + \u2191n\u2082 * (\u2191d\u2081 * (\u2191d\u2082)\u207b\u00b9) * (\u2191d\u2081)\u207b\u00b9 = \u2191n\u2081 * (\u2191d\u2081)\u207b\u00b9 + \u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9\n\u22a2 \u2191\u2191d\u2082 \u2260 0\n[PROOFSTEP]\nsimp [d\u20810, d\u20820]\n[GOAL]\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191d\u2082 * ((\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9)) + \u2191n\u2082 * (\u2191d\u2081 * (\u2191d\u2082)\u207b\u00b9) * (\u2191d\u2081)\u207b\u00b9 = \u2191n\u2081 * (\u2191d\u2081)\u207b\u00b9 + \u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9\n\u22a2 \u2191\u2191d\u2081 \u2260 0\n[PROOFSTEP]\nsimp [d\u20810, d\u20820]\n[GOAL]\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191d\u2082 * ((\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9)) + \u2191n\u2082 * (\u2191d\u2081 * (\u2191d\u2082)\u207b\u00b9) * (\u2191d\u2081)\u207b\u00b9 = \u2191n\u2081 * (\u2191d\u2081)\u207b\u00b9 + \u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9\n\u22a2 \u2191(\u2191d\u2081 * \u2191d\u2082) \u2260 0\n[PROOFSTEP]\nsimp [d\u20810, d\u20820]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\n\u22a2 \u2191n\u2081 * (\u2191d\u2082 * ((\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9)) + \u2191n\u2082 * (\u2191d\u2081 * (\u2191d\u2082)\u207b\u00b9) * (\u2191d\u2081)\u207b\u00b9 = \u2191n\u2081 * (\u2191d\u2081)\u207b\u00b9 + \u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 mul_assoc (d\u2082 : \u03b1), mul_inv_cancel d\u20820, one_mul, (Nat.cast_commute _ _).eq]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\n\u22a2 \u2191n\u2081 * (\u2191d\u2081)\u207b\u00b9 + \u2191n\u2082 * ((\u2191d\u2082)\u207b\u00b9 * \u2191d\u2081) * (\u2191d\u2081)\u207b\u00b9 = \u2191n\u2081 * (\u2191d\u2081)\u207b\u00b9 + \u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9\n[PROOFSTEP]\nsimp [d\u20810, mul_assoc]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\n\u22a2 \u2191(-mk' n d) = -\u2191(mk' n d)\n[PROOFSTEP]\nsimpa only [cast_def] using\n show (\u2191(-n) / d : \u03b1) = -(n / d) by rw [div_eq_mul_inv, div_eq_mul_inv, Int.cast_neg, neg_mul_eq_neg_mul]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\n\u22a2 \u2191(-n) / \u2191d = -(\u2191n / \u2191d)\n[PROOFSTEP]\nrw [div_eq_mul_inv, div_eq_mul_inv, Int.cast_neg, neg_mul_eq_neg_mul]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nm n : \u211a\nm0 : \u2191m.den \u2260 0\nn0 : \u2191n.den \u2260 0\n\u22a2 \u2191(m - n) = \u2191m - \u2191n\n[PROOFSTEP]\nhave : ((-n).den : \u03b1) \u2260 0 := by cases n; exact n0\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nm n : \u211a\nm0 : \u2191m.den \u2260 0\nn0 : \u2191n.den \u2260 0\n\u22a2 \u2191(-n).den \u2260 0\n[PROOFSTEP]\ncases n\n[GOAL]\ncase mk'\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nm : \u211a\nm0 : \u2191m.den \u2260 0\nnum\u271d : \u2124\nden\u271d : \u2115\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : Nat.coprime (Int.natAbs num\u271d) den\u271d\nn0 : \u2191(mk' num\u271d den\u271d).den \u2260 0\n\u22a2 \u2191(-mk' num\u271d den\u271d).den \u2260 0\n[PROOFSTEP]\nexact n0\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nm n : \u211a\nm0 : \u2191m.den \u2260 0\nn0 : \u2191n.den \u2260 0\nthis : \u2191(-n).den \u2260 0\n\u22a2 \u2191(m - n) = \u2191m - \u2191n\n[PROOFSTEP]\nsimp [sub_eq_add_neg, cast_add_of_ne_zero m0 this]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\n\u22a2 \u2191(mk' n\u2081 d\u2081 * mk' n\u2082 d\u2082) = \u2191(mk' n\u2081 d\u2081) * \u2191(mk' n\u2082 d\u2082)\n[PROOFSTEP]\nhave d\u20810' : (d\u2081 : \u2124) \u2260 0 := Int.coe_nat_ne_zero.2 fun e => by rw [e] at d\u20810 ; exact d\u20810 Nat.cast_zero\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\ne : d\u2081 = 0\n\u22a2 False\n[PROOFSTEP]\nrw [e] at d\u20810 \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u21910 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\ne : d\u2081 = 0\n\u22a2 False\n[PROOFSTEP]\nexact d\u20810 Nat.cast_zero\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\n\u22a2 \u2191(mk' n\u2081 d\u2081 * mk' n\u2082 d\u2082) = \u2191(mk' n\u2081 d\u2081) * \u2191(mk' n\u2082 d\u2082)\n[PROOFSTEP]\nhave d\u20820' : (d\u2082 : \u2124) \u2260 0 := Int.coe_nat_ne_zero.2 fun e => by rw [e] at d\u20820 ; exact d\u20820 Nat.cast_zero\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\ne : d\u2082 = 0\n\u22a2 False\n[PROOFSTEP]\nrw [e] at d\u20820 \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u21910 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\ne : d\u2082 = 0\n\u22a2 False\n[PROOFSTEP]\nexact d\u20820 Nat.cast_zero\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\n\u22a2 \u2191(mk' n\u2081 d\u2081 * mk' n\u2082 d\u2082) = \u2191(mk' n\u2081 d\u2081) * \u2191(mk' n\u2082 d\u2082)\n[PROOFSTEP]\nrw [num_den', num_den', mul_def' d\u20810' d\u20820']\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\n\u22a2 \u2191(n\u2081 * n\u2082 /. (\u2191d\u2081 * \u2191d\u2082)) = \u2191(n\u2081 /. \u2191d\u2081) * \u2191(n\u2082 /. \u2191d\u2082)\n[PROOFSTEP]\nsuffices (n\u2081 * (n\u2082 * (d\u2082 : \u03b1)\u207b\u00b9 * (d\u2081 : \u03b1)\u207b\u00b9) : \u03b1) = n\u2081 * ((d\u2081 : \u03b1)\u207b\u00b9 * (n\u2082 * (d\u2082 : \u03b1)\u207b\u00b9))\n by\n rw [cast_mk_of_ne_zero, cast_mk_of_ne_zero, cast_mk_of_ne_zero]\n \u00b7 simpa [division_def, mul_inv_rev, d\u20810, d\u20820, mul_assoc]\n all_goals simp [d\u20810, d\u20820]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9) = \u2191n\u2081 * ((\u2191d\u2081)\u207b\u00b9 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9))\n\u22a2 \u2191(n\u2081 * n\u2082 /. (\u2191d\u2081 * \u2191d\u2082)) = \u2191(n\u2081 /. \u2191d\u2081) * \u2191(n\u2082 /. \u2191d\u2082)\n[PROOFSTEP]\nrw [cast_mk_of_ne_zero, cast_mk_of_ne_zero, cast_mk_of_ne_zero]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9) = \u2191n\u2081 * ((\u2191d\u2081)\u207b\u00b9 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9))\n\u22a2 \u2191(n\u2081 * n\u2082) / \u2191(\u2191d\u2081 * \u2191d\u2082) = \u2191n\u2081 / \u2191\u2191d\u2081 * (\u2191n\u2082 / \u2191\u2191d\u2082)\n[PROOFSTEP]\nsimpa [division_def, mul_inv_rev, d\u20810, d\u20820, mul_assoc]\n[GOAL]\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9) = \u2191n\u2081 * ((\u2191d\u2081)\u207b\u00b9 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9))\n\u22a2 \u2191\u2191d\u2082 \u2260 0\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9) = \u2191n\u2081 * ((\u2191d\u2081)\u207b\u00b9 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9))\n\u22a2 \u2191\u2191d\u2081 \u2260 0\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9) = \u2191n\u2081 * ((\u2191d\u2081)\u207b\u00b9 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9))\n\u22a2 \u2191(\u2191d\u2081 * \u2191d\u2082) \u2260 0\n[PROOFSTEP]\nall_goals simp [d\u20810, d\u20820]\n[GOAL]\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9) = \u2191n\u2081 * ((\u2191d\u2081)\u207b\u00b9 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9))\n\u22a2 \u2191\u2191d\u2082 \u2260 0\n[PROOFSTEP]\nsimp [d\u20810, d\u20820]\n[GOAL]\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9) = \u2191n\u2081 * ((\u2191d\u2081)\u207b\u00b9 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9))\n\u22a2 \u2191\u2191d\u2081 \u2260 0\n[PROOFSTEP]\nsimp [d\u20810, d\u20820]\n[GOAL]\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\nthis : \u2191n\u2081 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9) = \u2191n\u2081 * ((\u2191d\u2081)\u207b\u00b9 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9))\n\u22a2 \u2191(\u2191d\u2081 * \u2191d\u2082) \u2260 0\n[PROOFSTEP]\nsimp [d\u20810, d\u20820]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nh\u2081 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nh\u2082 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u20810 : \u2191d\u2081 \u2260 0\nd\u20820 : \u2191d\u2082 \u2260 0\nd\u20810' : \u2191d\u2081 \u2260 0\nd\u20820' : \u2191d\u2082 \u2260 0\n\u22a2 \u2191n\u2081 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9 * (\u2191d\u2081)\u207b\u00b9) = \u2191n\u2081 * ((\u2191d\u2081)\u207b\u00b9 * (\u2191n\u2082 * (\u2191d\u2082)\u207b\u00b9))\n[PROOFSTEP]\nrw [(d\u2081.commute_cast (_ : \u03b1)).inv_right\u2080.eq]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2115\n\u22a2 \u2191(\u2191n)\u207b\u00b9 = (\u2191n)\u207b\u00b9\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\n\u22a2 \u2191(\u2191Nat.zero)\u207b\u00b9 = (\u2191Nat.zero)\u207b\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2115\n\u22a2 \u2191(\u2191(Nat.succ n))\u207b\u00b9 = (\u2191(Nat.succ n))\u207b\u00b9\n[PROOFSTEP]\nrw [cast_def, inv_coe_nat_num, inv_coe_nat_den, if_neg n.succ_ne_zero,\n Int.sign_eq_one_of_pos (Nat.cast_pos.mpr n.succ_pos), Int.cast_one, one_div]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2124\n\u22a2 \u2191(\u2191n)\u207b\u00b9 = (\u2191n)\u207b\u00b9\n[PROOFSTEP]\ncases' n with n n\n[GOAL]\ncase ofNat\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2115\n\u22a2 \u2191(\u2191(Int.ofNat n))\u207b\u00b9 = (\u2191(Int.ofNat n))\u207b\u00b9\n[PROOFSTEP]\nsimp [ofInt_eq_cast, cast_inv_nat]\n[GOAL]\ncase negSucc\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2115\n\u22a2 \u2191(\u2191(Int.negSucc n))\u207b\u00b9 = (\u2191(Int.negSucc n))\u207b\u00b9\n[PROOFSTEP]\nsimp only [ofInt_eq_cast, Int.cast_negSucc, \u2190 Nat.cast_succ, cast_neg, inv_neg, cast_inv_nat]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\nn0 : \u2191n \u2260 0\nd0 : \u2191d \u2260 0\n\u22a2 \u2191(mk' n d)\u207b\u00b9 = (\u2191(mk' n d))\u207b\u00b9\n[PROOFSTEP]\nhave _ : (n : \u2124) \u2260 0 := fun e => by rw [e] at n0 ; exact n0 Int.cast_zero\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\nn0 : \u2191n \u2260 0\nd0 : \u2191d \u2260 0\ne : n = 0\n\u22a2 False\n[PROOFSTEP]\nrw [e] at n0 \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\nn0 : \u21910 \u2260 0\nd0 : \u2191d \u2260 0\ne : n = 0\n\u22a2 False\n[PROOFSTEP]\nexact n0 Int.cast_zero\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\nn0 : \u2191n \u2260 0\nd0 : \u2191d \u2260 0\nx\u271d : n \u2260 0\n\u22a2 \u2191(mk' n d)\u207b\u00b9 = (\u2191(mk' n d))\u207b\u00b9\n[PROOFSTEP]\nhave _ : (d : \u2124) \u2260 0 := Int.coe_nat_ne_zero.2 fun e => by rw [e] at d0 ; exact d0 Nat.cast_zero\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\nn0 : \u2191n \u2260 0\nd0 : \u2191d \u2260 0\nx\u271d : n \u2260 0\ne : d = 0\n\u22a2 False\n[PROOFSTEP]\nrw [e] at d0 \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\nn0 : \u2191n \u2260 0\nd0 : \u21910 \u2260 0\nx\u271d : n \u2260 0\ne : d = 0\n\u22a2 False\n[PROOFSTEP]\nexact d0 Nat.cast_zero\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\nn0 : \u2191n \u2260 0\nd0 : \u2191d \u2260 0\nx\u271d\u00b9 : n \u2260 0\nx\u271d : \u2191d \u2260 0\n\u22a2 \u2191(mk' n d)\u207b\u00b9 = (\u2191(mk' n d))\u207b\u00b9\n[PROOFSTEP]\nrw [num_den', inv_def']\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\nn0 : \u2191n \u2260 0\nd0 : \u2191d \u2260 0\nx\u271d\u00b9 : n \u2260 0\nx\u271d : \u2191d \u2260 0\n\u22a2 \u2191(\u2191d /. n) = (\u2191(n /. \u2191d))\u207b\u00b9\n[PROOFSTEP]\nrw [cast_mk_of_ne_zero, cast_mk_of_ne_zero, inv_div]\n[GOAL]\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\nn0 : \u2191n \u2260 0\nd0 : \u2191d \u2260 0\nx\u271d\u00b9 : n \u2260 0\nx\u271d : \u2191d \u2260 0\n\u22a2 \u2191\u2191d \u2260 0\n[PROOFSTEP]\nsimp [n0, d0]\n[GOAL]\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nn : \u2124\nd : \u2115\nh : d \u2260 0\nc : Nat.coprime (Int.natAbs n) d\nn0 : \u2191n \u2260 0\nd0 : \u2191d \u2260 0\nx\u271d\u00b9 : n \u2260 0\nx\u271d : \u2191d \u2260 0\n\u22a2 \u2191n \u2260 0\n[PROOFSTEP]\nsimp [n0, d0]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nm n : \u211a\nmd : \u2191m.den \u2260 0\nnn : \u2191n.num \u2260 0\nnd : \u2191n.den \u2260 0\n\u22a2 \u2191(m / n) = \u2191m / \u2191n\n[PROOFSTEP]\nhave : (n\u207b\u00b9.den : \u2124) \u2223 n.num := by\n conv in n\u207b\u00b9.den => rw [\u2190 @num_den n, inv_def']\n apply den_dvd\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nm n : \u211a\nmd : \u2191m.den \u2260 0\nnn : \u2191n.num \u2260 0\nnd : \u2191n.den \u2260 0\n\u22a2 \u2191n\u207b\u00b9.den \u2223 n.num\n[PROOFSTEP]\nconv in n\u207b\u00b9.den => rw [\u2190 @num_den n, inv_def']\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nm n : \u211a\nmd : \u2191m.den \u2260 0\nnn : \u2191n.num \u2260 0\nnd : \u2191n.den \u2260 0\n| n\u207b\u00b9.den\n[PROOFSTEP]\nrw [\u2190 @num_den n, inv_def']\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nm n : \u211a\nmd : \u2191m.den \u2260 0\nnn : \u2191n.num \u2260 0\nnd : \u2191n.den \u2260 0\n| n\u207b\u00b9.den\n[PROOFSTEP]\nrw [\u2190 @num_den n, inv_def']\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nm n : \u211a\nmd : \u2191m.den \u2260 0\nnn : \u2191n.num \u2260 0\nnd : \u2191n.den \u2260 0\n| n\u207b\u00b9.den\n[PROOFSTEP]\nrw [\u2190 @num_den n, inv_def']\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nm n : \u211a\nmd : \u2191m.den \u2260 0\nnn : \u2191n.num \u2260 0\nnd : \u2191n.den \u2260 0\n\u22a2 \u2191(\u2191n.den /. n.num).den \u2223 n.num\n[PROOFSTEP]\napply den_dvd\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nm n : \u211a\nmd : \u2191m.den \u2260 0\nnn : \u2191n.num \u2260 0\nnd : \u2191n.den \u2260 0\nthis : \u2191n\u207b\u00b9.den \u2223 n.num\n\u22a2 \u2191(m / n) = \u2191m / \u2191n\n[PROOFSTEP]\nhave : (n\u207b\u00b9.den : \u03b1) = 0 \u2192 (n.num : \u03b1) = 0 := fun h =>\n by\n let \u27e8k, e\u27e9 := this\n have := congr_arg ((\u2191) : \u2124 \u2192 \u03b1) e; rwa [Int.cast_mul, Int.cast_ofNat, h, zero_mul] at this \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nm n : \u211a\nmd : \u2191m.den \u2260 0\nnn : \u2191n.num \u2260 0\nnd : \u2191n.den \u2260 0\nthis : \u2191n\u207b\u00b9.den \u2223 n.num\nh : \u2191n\u207b\u00b9.den = 0\n\u22a2 \u2191n.num = 0\n[PROOFSTEP]\nlet \u27e8k, e\u27e9 := this\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nm n : \u211a\nmd : \u2191m.den \u2260 0\nnn : \u2191n.num \u2260 0\nnd : \u2191n.den \u2260 0\nthis : \u2191n\u207b\u00b9.den \u2223 n.num\nh : \u2191n\u207b\u00b9.den = 0\nk : \u2124\ne : n.num = \u2191n\u207b\u00b9.den * k\n\u22a2 \u2191n.num = 0\n[PROOFSTEP]\nhave := congr_arg ((\u2191) : \u2124 \u2192 \u03b1) e\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nm n : \u211a\nmd : \u2191m.den \u2260 0\nnn : \u2191n.num \u2260 0\nnd : \u2191n.den \u2260 0\nthis\u271d : \u2191n\u207b\u00b9.den \u2223 n.num\nh : \u2191n\u207b\u00b9.den = 0\nk : \u2124\ne : n.num = \u2191n\u207b\u00b9.den * k\nthis : \u2191n.num = \u2191(\u2191n\u207b\u00b9.den * k)\n\u22a2 \u2191n.num = 0\n[PROOFSTEP]\nrwa [Int.cast_mul, Int.cast_ofNat, h, zero_mul] at this \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : DivisionRing \u03b1\nm n : \u211a\nmd : \u2191m.den \u2260 0\nnn : \u2191n.num \u2260 0\nnd : \u2191n.den \u2260 0\nthis\u271d : \u2191n\u207b\u00b9.den \u2223 n.num\nthis : \u2191n\u207b\u00b9.den = 0 \u2192 \u2191n.num = 0\n\u22a2 \u2191(m / n) = \u2191m / \u2191n\n[PROOFSTEP]\nrw [division_def, cast_mul_of_ne_zero md (mt this nn), cast_inv_of_ne_zero nn nd, division_def]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nd\u20810 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nd\u20820 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\n\u22a2 \u2191(mk' n\u2081 d\u2081) = \u2191(mk' n\u2082 d\u2082) \u2194 mk' n\u2081 d\u2081 = mk' n\u2082 d\u2082\n[PROOFSTEP]\nrefine' \u27e8fun h => _, congr_arg _\u27e9\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nd\u20810 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nd\u20820 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nh : \u2191(mk' n\u2081 d\u2081) = \u2191(mk' n\u2082 d\u2082)\n\u22a2 mk' n\u2081 d\u2081 = mk' n\u2082 d\u2082\n[PROOFSTEP]\nhave d\u2081a : (d\u2081 : \u03b1) \u2260 0 := Nat.cast_ne_zero.2 d\u20810\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nd\u20810 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nd\u20820 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nh : \u2191(mk' n\u2081 d\u2081) = \u2191(mk' n\u2082 d\u2082)\nd\u2081a : \u2191d\u2081 \u2260 0\n\u22a2 mk' n\u2081 d\u2081 = mk' n\u2082 d\u2082\n[PROOFSTEP]\nhave d\u2082a : (d\u2082 : \u03b1) \u2260 0 := Nat.cast_ne_zero.2 d\u20820\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nd\u20810 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nd\u20820 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nh : \u2191(mk' n\u2081 d\u2081) = \u2191(mk' n\u2082 d\u2082)\nd\u2081a : \u2191d\u2081 \u2260 0\nd\u2082a : \u2191d\u2082 \u2260 0\n\u22a2 mk' n\u2081 d\u2081 = mk' n\u2082 d\u2082\n[PROOFSTEP]\nrw [num_den', num_den'] at h \u22a2\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nd\u20810 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nd\u20820 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nh : \u2191(n\u2081 /. \u2191d\u2081) = \u2191(n\u2082 /. \u2191d\u2082)\nd\u2081a : \u2191d\u2081 \u2260 0\nd\u2082a : \u2191d\u2082 \u2260 0\n\u22a2 n\u2081 /. \u2191d\u2081 = n\u2082 /. \u2191d\u2082\n[PROOFSTEP]\nrw [cast_mk_of_ne_zero, cast_mk_of_ne_zero] at h \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nd\u20810 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nd\u20820 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nh : \u2191n\u2081 / \u2191\u2191d\u2081 = \u2191n\u2082 / \u2191\u2191d\u2082\nd\u2081a : \u2191d\u2081 \u2260 0\nd\u2082a : \u2191d\u2082 \u2260 0\n\u22a2 n\u2081 /. \u2191d\u2081 = n\u2082 /. \u2191d\u2082\n[PROOFSTEP]\nsimp [d\u20810, d\u20820] at h \u22a2\n[GOAL]\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nd\u20810 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nd\u20820 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nh : \u2191n\u2081 / \u2191\u2191d\u2081 = \u2191(n\u2082 /. \u2191d\u2082)\nd\u2081a : \u2191d\u2081 \u2260 0\nd\u2082a : \u2191d\u2082 \u2260 0\n\u22a2 \u2191\u2191d\u2082 \u2260 0\n[PROOFSTEP]\nsimp [d\u20810, d\u20820] at h \u22a2\n[GOAL]\ncase b0\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nd\u20810 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nd\u20820 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nh : \u2191(n\u2081 /. \u2191d\u2081) = \u2191(n\u2082 /. \u2191d\u2082)\nd\u2081a : \u2191d\u2081 \u2260 0\nd\u2082a : \u2191d\u2082 \u2260 0\n\u22a2 \u2191\u2191d\u2081 \u2260 0\n[PROOFSTEP]\nsimp [d\u20810, d\u20820] at h \u22a2\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn\u2081 : \u2124\nd\u2081 : \u2115\nd\u20810 : d\u2081 \u2260 0\nc\u2081 : Nat.coprime (Int.natAbs n\u2081) d\u2081\nn\u2082 : \u2124\nd\u2082 : \u2115\nd\u20820 : d\u2082 \u2260 0\nc\u2082 : Nat.coprime (Int.natAbs n\u2082) d\u2082\nd\u2081a : \u2191d\u2081 \u2260 0\nd\u2082a : \u2191d\u2082 \u2260 0\nh : \u2191n\u2081 / \u2191d\u2081 = \u2191n\u2082 / \u2191d\u2082\n\u22a2 mkRat n\u2081 d\u2081 = mkRat n\u2082 d\u2082\n[PROOFSTEP]\nrwa [eq_div_iff_mul_eq d\u2082a, division_def, mul_assoc, (d\u2081.cast_commute (d\u2082 : \u03b1)).inv_left\u2080.eq, \u2190 mul_assoc, \u2190\n division_def, eq_comm, eq_div_iff_mul_eq d\u2081a, eq_comm, \u2190 Int.cast_ofNat d\u2081, \u2190 Int.cast_mul, \u2190 Int.cast_ofNat d\u2082, \u2190\n Int.cast_mul, Int.cast_inj, \u2190 mkRat_eq_iff d\u20810 d\u20820] at h \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn : \u211a\n\u22a2 \u2191n = 0 \u2194 n = 0\n[PROOFSTEP]\nrw [\u2190 cast_zero, cast_inj]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn : \u211a\n\u22a2 \u2191(bit1 n) = bit1 \u2191n\n[PROOFSTEP]\nrw [bit1, cast_add, cast_one, cast_bit0]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\nn : \u211a\n\u22a2 bit0 \u2191n + 1 = bit1 \u2191n\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : DivisionRing \u03b1\ninst\u271d : CharZero \u03b1\na b : \u2124\n\u22a2 \u2191(a /. b) = \u2191a / \u2191b\n[PROOFSTEP]\nsimp only [divInt_eq_div, cast_div, cast_coe_int]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\nr : \u211a\nhr : 0 < r\n\u22a2 0 < \u2191r\n[PROOFSTEP]\nrw [Rat.cast_def]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\nr : \u211a\nhr : 0 < r\n\u22a2 0 < \u2191r.num / \u2191r.den\n[PROOFSTEP]\nexact div_pos (Int.cast_pos.2 <| num_pos_iff_pos.2 hr) (Nat.cast_pos.2 r.pos)\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\nm n : \u211a\n\u22a2 m < n \u2192 \u2191m < \u2191n\n[PROOFSTEP]\nsimpa only [sub_pos, cast_sub] using @cast_pos_of_pos K _ (n - m)\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\nn : \u211a\n\u22a2 0 \u2264 \u2191n \u2194 0 \u2264 n\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\nn : \u211a\n\u22a2 \u2191n \u2264 0 \u2194 n \u2264 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\nn : \u211a\n\u22a2 0 < \u2191n \u2194 0 < n\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\nn : \u211a\n\u22a2 \u2191n < 0 \u2194 n < 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\nq : \u211a\n\u22a2 \u2191|q| = |\u2191q|\n[PROOFSTEP]\nsimp [abs_eq_max_neg]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na b : \u211a\n\u22a2 Rat.cast \u207b\u00b9' Icc \u2191a \u2191b = Icc a b\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na b x : \u211a\n\u22a2 x \u2208 Rat.cast \u207b\u00b9' Icc \u2191a \u2191b \u2194 x \u2208 Icc a b\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na b : \u211a\n\u22a2 Rat.cast \u207b\u00b9' Ico \u2191a \u2191b = Ico a b\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na b x : \u211a\n\u22a2 x \u2208 Rat.cast \u207b\u00b9' Ico \u2191a \u2191b \u2194 x \u2208 Ico a b\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na b : \u211a\n\u22a2 Rat.cast \u207b\u00b9' Ioc \u2191a \u2191b = Ioc a b\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na b x : \u211a\n\u22a2 x \u2208 Rat.cast \u207b\u00b9' Ioc \u2191a \u2191b \u2194 x \u2208 Ioc a b\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na b : \u211a\n\u22a2 Rat.cast \u207b\u00b9' Ioo \u2191a \u2191b = Ioo a b\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na b x : \u211a\n\u22a2 x \u2208 Rat.cast \u207b\u00b9' Ioo \u2191a \u2191b \u2194 x \u2208 Ioo a b\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na : \u211a\n\u22a2 Rat.cast \u207b\u00b9' Ici \u2191a = Ici a\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na x : \u211a\n\u22a2 x \u2208 Rat.cast \u207b\u00b9' Ici \u2191a \u2194 x \u2208 Ici a\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na : \u211a\n\u22a2 Rat.cast \u207b\u00b9' Iic \u2191a = Iic a\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na x : \u211a\n\u22a2 x \u2208 Rat.cast \u207b\u00b9' Iic \u2191a \u2194 x \u2208 Iic a\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na : \u211a\n\u22a2 Rat.cast \u207b\u00b9' Ioi \u2191a = Ioi a\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na x : \u211a\n\u22a2 x \u2208 Rat.cast \u207b\u00b9' Ioi \u2191a \u2194 x \u2208 Ioi a\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na : \u211a\n\u22a2 Rat.cast \u207b\u00b9' Iio \u2191a = Iio a\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : LinearOrderedField K\na x : \u211a\n\u22a2 x \u2208 Rat.cast \u207b\u00b9' Iio \u2191a \u2194 x \u2208 Iio a\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b2 : DivisionRing \u03b1\ninst\u271d\u00b9 : DivisionRing \u03b2\ninst\u271d : RingHomClass F \u03b1 \u03b2\nf : F\nq : \u211a\n\u22a2 \u2191f \u2191q = \u2191q\n[PROOFSTEP]\nrw [cast_def, map_div\u2080, map_intCast, map_natCast, cast_def]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nk : Type u_5\ninst\u271d\u00b9 : DivisionRing k\ninst\u271d : RingHomClass F \u211a k\nf : F\nr : \u211a\n\u22a2 \u2191f r = \u2191r\n[PROOFSTEP]\nrw [\u2190 map_ratCast f, Rat.cast_id]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nM\u2080 : Type u_5\ninst\u271d\u00b9 : MonoidWithZero M\u2080\ninst\u271d : MonoidWithZeroHomClass F \u211a M\u2080\nf g : F\nh : \u2200 (m : \u2124), \u2191f \u2191m = \u2191g \u2191m\nr : \u211a\n\u22a2 \u2191f r = \u2191g r\n[PROOFSTEP]\nrw [\u2190 r.num_div_den, div_eq_mul_inv, map_mul, map_mul, h, \u2190 Int.cast_ofNat, eq_on_inv\u2080 f g]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nM\u2080 : Type u_5\ninst\u271d\u00b9 : MonoidWithZero M\u2080\ninst\u271d : MonoidWithZeroHomClass F \u211a M\u2080\nf g : F\nh : \u2200 (m : \u2124), \u2191f \u2191m = \u2191g \u2191m\nr : \u211a\n\u22a2 \u2191f \u2191\u2191r.den = \u2191g \u2191\u2191r.den\n[PROOFSTEP]\napply h\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nM\u2080 : Type u_5\ninst\u271d\u00b9 : MonoidWithZero M\u2080\ninst\u271d : MonoidWithZeroHomClass F \u211a M\u2080\nf g : F\nsame_on_neg_one : \u2191f (-1) = \u2191g (-1)\nsame_on_pnat : \u2200 (n : \u2115), 0 < n \u2192 \u2191f \u2191n = \u2191g \u2191n\n\u22a2 \u2191(comp \u2191f \u2191(Int.castRingHom \u211a)) (-1) = \u2191(comp \u2191g \u2191(Int.castRingHom \u211a)) (-1)\n[PROOFSTEP]\nsimpa\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nM\u2080 : Type u_5\ninst\u271d\u00b9 : MonoidWithZero M\u2080\ninst\u271d : MonoidWithZeroHomClass F \u211a M\u2080\nf g : F\nsame_on_neg_one : \u2191f (-1) = \u2191g (-1)\nsame_on_pnat : \u2200 (n : \u2115), 0 < n \u2192 \u2191f \u2191n = \u2191g \u2191n\n\u22a2 \u2200 (n : \u2115), 0 < n \u2192 \u2191(comp \u2191f \u2191(Int.castRingHom \u211a)) \u2191n = \u2191(comp \u2191g \u2191(Int.castRingHom \u211a)) \u2191n\n[PROOFSTEP]\nsimpa\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : DivisionRing K\na : \u211a\n\u22a2 a \u2022 0 = 0\n[PROOFSTEP]\nrw [smul_def, mul_zero]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : DivisionRing K\na : \u211a\nx y : K\n\u22a2 a \u2022 (x + y) = a \u2022 x + a \u2022 y\n[PROOFSTEP]\nrw [smul_def, smul_def, smul_def, mul_add]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nK : Type u_5\ninst\u271d : DivisionRing K\na : \u211a\nx y : K\n\u22a2 (a \u2022 x) \u2022 y = a \u2022 x \u2022 y\n[PROOFSTEP]\nsimp only [smul_def, smul_eq_mul, mul_assoc]\n", "meta": {"mathlib_filename": "Mathlib.Data.Rat.Cast", "llama_tokens": 23417, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707283, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.41452134314362876}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd5c' : Type u_6\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nc : E \u2192 \ud835\udd5c'\nc' : E \u2192L[\ud835\udd5c] \ud835\udd5c'\nhc : HasStrictFDerivAt c c' x\nf : F\n\u22a2 HasStrictFDerivAt (fun y => c y \u2022 f) (smulRight c' f) x\n[PROOFSTEP]\nsimpa only [smul_zero, zero_add] using hc.smul (hasStrictFDerivAt_const f x)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd5c' : Type u_6\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nc : E \u2192 \ud835\udd5c'\nc' : E \u2192L[\ud835\udd5c] \ud835\udd5c'\nhc : HasFDerivWithinAt c c' s x\nf : F\n\u22a2 HasFDerivWithinAt (fun y => c y \u2022 f) (smulRight c' f) s x\n[PROOFSTEP]\nsimpa only [smul_zero, zero_add] using hc.smul (hasFDerivWithinAt_const f x s)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd5c' : Type u_6\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nc : E \u2192 \ud835\udd5c'\nc' : E \u2192L[\ud835\udd5c] \ud835\udd5c'\nhc : HasFDerivAt c c' x\nf : F\n\u22a2 HasFDerivAt (fun y => c y \u2022 f) (smulRight c' f) x\n[PROOFSTEP]\nsimpa only [smul_zero, zero_add] using hc.smul (hasFDerivAt_const f x)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasStrictFDerivAt c c' x\nhd : HasStrictFDerivAt d d' x\n\u22a2 HasStrictFDerivAt (fun y => c y * d y) (c x \u2022 d' + d x \u2022 c') x\n[PROOFSTEP]\nconvert hc.mul' hd\n[GOAL]\ncase h.e'_10.h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasStrictFDerivAt c c' x\nhd : HasStrictFDerivAt d d' x\n\u22a2 d x \u2022 c' = smulRight c' (d x)\n[PROOFSTEP]\next z\n[GOAL]\ncase h.e'_10.h.e'_6.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasStrictFDerivAt c c' x\nhd : HasStrictFDerivAt d d' x\nz : E\n\u22a2 \u2191(d x \u2022 c') z = \u2191(smulRight c' (d x)) z\n[PROOFSTEP]\napply mul_comm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivWithinAt c c' s x\nhd : HasFDerivWithinAt d d' s x\n\u22a2 HasFDerivWithinAt (fun y => c y * d y) (c x \u2022 d' + d x \u2022 c') s x\n[PROOFSTEP]\nconvert hc.mul' hd\n[GOAL]\ncase h.e'_10.h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivWithinAt c c' s x\nhd : HasFDerivWithinAt d d' s x\n\u22a2 d x \u2022 c' = smulRight c' (d x)\n[PROOFSTEP]\next z\n[GOAL]\ncase h.e'_10.h.e'_6.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivWithinAt c c' s x\nhd : HasFDerivWithinAt d d' s x\nz : E\n\u22a2 \u2191(d x \u2022 c') z = \u2191(smulRight c' (d x)) z\n[PROOFSTEP]\napply mul_comm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivAt c c' x\nhd : HasFDerivAt d d' x\n\u22a2 HasFDerivAt (fun y => c y * d y) (c x \u2022 d' + d x \u2022 c') x\n[PROOFSTEP]\nconvert hc.mul' hd\n[GOAL]\ncase h.e'_10.h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivAt c c' x\nhd : HasFDerivAt d d' x\n\u22a2 d x \u2022 c' = smulRight c' (d x)\n[PROOFSTEP]\next z\n[GOAL]\ncase h.e'_10.h.e'_6.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivAt c c' x\nhd : HasFDerivAt d d' x\nz : E\n\u22a2 \u2191(d x \u2022 c') z = \u2191(smulRight c' (d x)) z\n[PROOFSTEP]\napply mul_comm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nha : DifferentiableWithinAt \ud835\udd5c a s x\n\u22a2 DifferentiableWithinAt \ud835\udd5c (fun x => a x ^ 0) s x\n[PROOFSTEP]\nsimp only [pow_zero, differentiableWithinAt_const]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nha : DifferentiableWithinAt \ud835\udd5c a s x\nn : \u2115\n\u22a2 DifferentiableWithinAt \ud835\udd5c (fun x => a x ^ (n + 1)) s x\n[PROOFSTEP]\nsimp only [pow_succ, DifferentiableWithinAt.pow ha n, ha.mul]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d\u271d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasStrictFDerivAt c c' x\nd : \ud835\udd38'\n\u22a2 HasStrictFDerivAt (fun y => c y * d) (d \u2022 c') x\n[PROOFSTEP]\nconvert hc.mul_const' d\n[GOAL]\ncase h.e'_10\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d\u271d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasStrictFDerivAt c c' x\nd : \ud835\udd38'\n\u22a2 d \u2022 c' = smulRight c' d\n[PROOFSTEP]\next z\n[GOAL]\ncase h.e'_10.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d\u271d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasStrictFDerivAt c c' x\nd : \ud835\udd38'\nz : E\n\u22a2 \u2191(d \u2022 c') z = \u2191(smulRight c' d) z\n[PROOFSTEP]\napply mul_comm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d\u271d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivWithinAt c c' s x\nd : \ud835\udd38'\n\u22a2 HasFDerivWithinAt (fun y => c y * d) (d \u2022 c') s x\n[PROOFSTEP]\nconvert hc.mul_const' d\n[GOAL]\ncase h.e'_10\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d\u271d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivWithinAt c c' s x\nd : \ud835\udd38'\n\u22a2 d \u2022 c' = smulRight c' d\n[PROOFSTEP]\next z\n[GOAL]\ncase h.e'_10.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d\u271d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivWithinAt c c' s x\nd : \ud835\udd38'\nz : E\n\u22a2 \u2191(d \u2022 c') z = \u2191(smulRight c' d) z\n[PROOFSTEP]\napply mul_comm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d\u271d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivAt c c' x\nd : \ud835\udd38'\n\u22a2 HasFDerivAt (fun y => c y * d) (d \u2022 c') x\n[PROOFSTEP]\nconvert hc.mul_const' d\n[GOAL]\ncase h.e'_10\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d\u271d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivAt c c' x\nd : \ud835\udd38'\n\u22a2 d \u2022 c' = smulRight c' d\n[PROOFSTEP]\next z\n[GOAL]\ncase h.e'_10.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\ud835\udd38 : Type u_6\n\ud835\udd38' : Type u_7\ninst\u271d\u00b3 : NormedRing \ud835\udd38\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38'\na b : E \u2192 \ud835\udd38\na' b' : E \u2192L[\ud835\udd5c] \ud835\udd38\nc d\u271d : E \u2192 \ud835\udd38'\nc' d' : E \u2192L[\ud835\udd5c] \ud835\udd38'\nhc : HasFDerivAt c c' x\nd : \ud835\udd38'\nz : E\n\u22a2 \u2191(d \u2022 c') z = \u2191(smulRight c' d) z\n[PROOFSTEP]\napply mul_comm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nR : Type u_6\ninst\u271d\u00b2 : NormedRing R\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c R\ninst\u271d : CompleteSpace R\nx : R\u02e3\nthis : (fun t => Ring.inverse (\u2191x + t) - \u2191x\u207b\u00b9 + \u2191x\u207b\u00b9 * t * \u2191x\u207b\u00b9) =o[\ud835\udcdd 0] _root_.id\n\u22a2 HasFDerivAt Ring.inverse (-\u2191(\u2191(mulLeftRight \ud835\udd5c R) \u2191x\u207b\u00b9) \u2191x\u207b\u00b9) \u2191x\n[PROOFSTEP]\nsimpa [hasFDerivAt_iff_isLittleO_nhds_zero] using this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nR : Type u_6\ninst\u271d\u00b2 : NormedDivisionRing R\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c R\ninst\u271d : CompleteSpace R\nx : R\nhx : x \u2260 0\n\u22a2 HasFDerivAt Inv.inv (-\u2191(\u2191(mulLeftRight \ud835\udd5c R) x\u207b\u00b9) x\u207b\u00b9) x\n[PROOFSTEP]\nsimpa using hasFDerivAt_ring_inverse (Units.mk0 _ hx)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\nR : Type u_6\ninst\u271d\u00b2 : NormedDivisionRing R\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c R\ninst\u271d : CompleteSpace R\ns : Set R\nx : R\nhx : x \u2260 0\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 fderivWithin \ud835\udd5c (fun x => x\u207b\u00b9) s x = -\u2191(\u2191(mulLeftRight \ud835\udd5c R) x\u207b\u00b9) x\u207b\u00b9\n[PROOFSTEP]\nrw [DifferentiableAt.fderivWithin (differentiableAt_inv' hx) hxs]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t : Set E\nL L\u2081 L\u2082 : Filter E\nR : Type u_6\ninst\u271d\u00b2 : NormedDivisionRing R\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c R\ninst\u271d : CompleteSpace R\ns : Set R\nx : R\nhx : x \u2260 0\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 fderiv \ud835\udd5c Inv.inv x = -\u2191(\u2191(mulLeftRight \ud835\udd5c R) x\u207b\u00b9) x\u207b\u00b9\n[PROOFSTEP]\nexact fderiv_inv' hx\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.FDeriv.Mul", "llama_tokens": 11391, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581626286834, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.41441299646599733}} {"text": "[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : M\n\u22a2 (fun m\u2081 m\u2082 =>\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 \u2208 evenOdd Q 0) })\n (x\u271d\u00b2 + x\u271d\u00b9) x\u271d =\n (fun m\u2081 m\u2082 =>\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 \u2208 evenOdd Q 0) })\n x\u271d\u00b2 x\u271d +\n (fun m\u2081 m\u2082 =>\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 \u2208 evenOdd Q 0) })\n x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nsimp only [LinearMap.map_add, add_mul]\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : M\n\u22a2 {\n val :=\n \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b2 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d + \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d,\n property :=\n (_ :\n (fun x => x \u2208 even Q)\n (\u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b2 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d +\n \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d)) } =\n { val := \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b2 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b2 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d \u2208 evenOdd Q 0) } +\n { val := \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d \u2208 evenOdd Q 0) }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nx\u271d\u00b2 : R\nx\u271d\u00b9 x\u271d : M\n\u22a2 (fun m\u2081 m\u2082 =>\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 \u2208 evenOdd Q 0) })\n (x\u271d\u00b2 \u2022 x\u271d\u00b9) x\u271d =\n x\u271d\u00b2 \u2022\n (fun m\u2081 m\u2082 =>\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 \u2208 evenOdd Q 0) })\n x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nsimp only [LinearMap.map_smul, smul_mul_assoc]\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nx\u271d\u00b2 : R\nx\u271d\u00b9 x\u271d : M\n\u22a2 { val := x\u271d\u00b2 \u2022 (\u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d),\n property := (_ : (fun x => x \u2208 even Q) (x\u271d\u00b2 \u2022 (\u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d))) } =\n x\u271d\u00b2 \u2022\n { val := \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d \u2208 evenOdd Q 0) }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : M\n\u22a2 (fun m\u2081 m\u2082 =>\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 \u2208 evenOdd Q 0) })\n x\u271d\u00b2 (x\u271d\u00b9 + x\u271d) =\n (fun m\u2081 m\u2082 =>\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 \u2208 evenOdd Q 0) })\n x\u271d\u00b2 x\u271d\u00b9 +\n (fun m\u2081 m\u2082 =>\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 \u2208 evenOdd Q 0) })\n x\u271d\u00b2 x\u271d\n[PROOFSTEP]\nsimp only [LinearMap.map_add, mul_add]\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : M\n\u22a2 {\n val :=\n \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b2 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9 +\n \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b2 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d,\n property :=\n (_ :\n (fun x => x \u2208 even Q)\n (\u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b2 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9 +\n \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b2 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d)) } =\n { val := \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b2 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b2 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9 \u2208 evenOdd Q 0) } +\n { val := \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b2 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b2 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d \u2208 evenOdd Q 0) }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nx\u271d\u00b2 : R\nx\u271d\u00b9 x\u271d : M\n\u22a2 (fun m\u2081 m\u2082 =>\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 \u2208 evenOdd Q 0) })\n x\u271d\u00b9 (x\u271d\u00b2 \u2022 x\u271d) =\n x\u271d\u00b2 \u2022\n (fun m\u2081 m\u2082 =>\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 \u2208 evenOdd Q 0) })\n x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nsimp only [LinearMap.map_smul, mul_smul_comm]\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nx\u271d\u00b2 : R\nx\u271d\u00b9 x\u271d : M\n\u22a2 { val := x\u271d\u00b2 \u2022 (\u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d),\n property := (_ : (fun x => x \u2208 even Q) (x\u271d\u00b2 \u2022 (\u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d))) } =\n x\u271d\u00b2 \u2022\n { val := \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) x\u271d\u00b9 * \u2191(CliffordAlgebra.\u03b9 Q) x\u271d \u2208 evenOdd Q 0) }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nm\u2081 m\u2082 m\u2083 : M\n\u22a2 \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * (\u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2083) =\n \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * (\u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2083)\n[PROOFSTEP]\nsimp only [mul_assoc]\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nm\u2081 m\u2082 m\u2083 : M\n\u22a2 \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * (\u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2083) =\n \u2191Q m\u2082 \u2022 (\u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2083)\n[PROOFSTEP]\nrw [Algebra.smul_def, \u03b9_sq_scalar, Algebra.left_comm]\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : EvenHom Q A\nf g : { x // x \u2208 even Q } \u2192\u2090[R] A\nh : EvenHom.compr\u2082 (\u03b9 Q) f = EvenHom.compr\u2082 (\u03b9 Q) g\n\u22a2 f = g\n[PROOFSTEP]\nrw [EvenHom.ext_iff] at h \n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : EvenHom Q A\nf g : { x // x \u2208 even Q } \u2192\u2090[R] A\nh : (EvenHom.compr\u2082 (\u03b9 Q) f).bilin = (EvenHom.compr\u2082 (\u03b9 Q) g).bilin\n\u22a2 f = g\n[PROOFSTEP]\next \u27e8x, hx\u27e9\n[GOAL]\ncase H.mk\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : EvenHom Q A\nf g : { x // x \u2208 even Q } \u2192\u2090[R] A\nh : (EvenHom.compr\u2082 (\u03b9 Q) f).bilin = (EvenHom.compr\u2082 (\u03b9 Q) g).bilin\nx : CliffordAlgebra Q\nhx : x \u2208 even Q\n\u22a2 \u2191f { val := x, property := hx } = \u2191g { val := x, property := hx }\n[PROOFSTEP]\nrefine' even_induction _ _ _ _ _ hx\n[GOAL]\ncase H.mk.refine'_1\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : EvenHom Q A\nf g : { x // x \u2208 even Q } \u2192\u2090[R] A\nh : (EvenHom.compr\u2082 (\u03b9 Q) f).bilin = (EvenHom.compr\u2082 (\u03b9 Q) g).bilin\nx : CliffordAlgebra Q\nhx : x \u2208 even Q\n\u22a2 \u2200 (r : R),\n \u2191f\n { val := \u2191(algebraMap R (CliffordAlgebra ?m.107570)) r,\n property := (_ : \u2191(algebraMap R (CliffordAlgebra ?m.107570)) r \u2208 evenOdd ?m.107570 0) } =\n \u2191g\n { val := \u2191(algebraMap R (CliffordAlgebra ?m.107570)) r,\n property := (_ : \u2191(algebraMap R (CliffordAlgebra ?m.107570)) r \u2208 evenOdd ?m.107570 0) }\n[PROOFSTEP]\nintro r\n[GOAL]\ncase H.mk.refine'_1\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : EvenHom Q A\nf g : { x // x \u2208 even Q } \u2192\u2090[R] A\nh : (EvenHom.compr\u2082 (\u03b9 Q) f).bilin = (EvenHom.compr\u2082 (\u03b9 Q) g).bilin\nx : CliffordAlgebra Q\nhx : x \u2208 even Q\nr : R\n\u22a2 \u2191f\n { val := \u2191(algebraMap R (CliffordAlgebra ?m.107570)) r,\n property := (_ : \u2191(algebraMap R (CliffordAlgebra ?m.107570)) r \u2208 evenOdd ?m.107570 0) } =\n \u2191g\n { val := \u2191(algebraMap R (CliffordAlgebra ?m.107570)) r,\n property := (_ : \u2191(algebraMap R (CliffordAlgebra ?m.107570)) r \u2208 evenOdd ?m.107570 0) }\n[PROOFSTEP]\nexact (f.commutes r).trans (g.commutes r).symm\n[GOAL]\ncase H.mk.refine'_2\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : EvenHom Q A\nf g : { x // x \u2208 even Q } \u2192\u2090[R] A\nh : (EvenHom.compr\u2082 (\u03b9 Q) f).bilin = (EvenHom.compr\u2082 (\u03b9 Q) g).bilin\nx : CliffordAlgebra Q\nhx : x \u2208 even Q\n\u22a2 \u2200 {x y : CliffordAlgebra Q} {hx : x \u2208 evenOdd Q 0} {hy : y \u2208 evenOdd Q 0},\n \u2191f { val := x, property := hx } = \u2191g { val := x, property := hx } \u2192\n \u2191f { val := y, property := hy } = \u2191g { val := y, property := hy } \u2192\n \u2191f { val := x + y, property := (_ : x + y \u2208 evenOdd Q 0) } =\n \u2191g { val := x + y, property := (_ : x + y \u2208 evenOdd Q 0) }\n[PROOFSTEP]\nintro x y hx hy ihx ihy\n[GOAL]\ncase H.mk.refine'_2\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : EvenHom Q A\nf g : { x // x \u2208 even Q } \u2192\u2090[R] A\nh : (EvenHom.compr\u2082 (\u03b9 Q) f).bilin = (EvenHom.compr\u2082 (\u03b9 Q) g).bilin\nx\u271d : CliffordAlgebra Q\nhx\u271d : x\u271d \u2208 even Q\nx y : CliffordAlgebra Q\nhx : x \u2208 evenOdd Q 0\nhy : y \u2208 evenOdd Q 0\nihx : \u2191f { val := x, property := hx } = \u2191g { val := x, property := hx }\nihy : \u2191f { val := y, property := hy } = \u2191g { val := y, property := hy }\n\u22a2 \u2191f { val := x + y, property := (_ : x + y \u2208 evenOdd Q 0) } =\n \u2191g { val := x + y, property := (_ : x + y \u2208 evenOdd Q 0) }\n[PROOFSTEP]\nhave := congr_arg\u2082 (\u00b7 + \u00b7) ihx ihy\n[GOAL]\ncase H.mk.refine'_2\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : EvenHom Q A\nf g : { x // x \u2208 even Q } \u2192\u2090[R] A\nh : (EvenHom.compr\u2082 (\u03b9 Q) f).bilin = (EvenHom.compr\u2082 (\u03b9 Q) g).bilin\nx\u271d : CliffordAlgebra Q\nhx\u271d : x\u271d \u2208 even Q\nx y : CliffordAlgebra Q\nhx : x \u2208 evenOdd Q 0\nhy : y \u2208 evenOdd Q 0\nihx : \u2191f { val := x, property := hx } = \u2191g { val := x, property := hx }\nihy : \u2191f { val := y, property := hy } = \u2191g { val := y, property := hy }\nthis :\n (fun x_1 x_2 => x_1 + x_2) (\u2191f { val := x, property := hx }) (\u2191f { val := y, property := hy }) =\n (fun x_1 x_2 => x_1 + x_2) (\u2191g { val := x, property := hx }) (\u2191g { val := y, property := hy })\n\u22a2 \u2191f { val := x + y, property := (_ : x + y \u2208 evenOdd Q 0) } =\n \u2191g { val := x + y, property := (_ : x + y \u2208 evenOdd Q 0) }\n[PROOFSTEP]\nexact (f.map_add _ _).trans (this.trans <| (g.map_add _ _).symm)\n[GOAL]\ncase H.mk.refine'_3\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : EvenHom Q A\nf g : { x // x \u2208 even Q } \u2192\u2090[R] A\nh : (EvenHom.compr\u2082 (\u03b9 Q) f).bilin = (EvenHom.compr\u2082 (\u03b9 Q) g).bilin\nx : CliffordAlgebra Q\nhx : x \u2208 even Q\n\u22a2 \u2200 (m\u2081 m\u2082 : M) {x : CliffordAlgebra Q} {hx : x \u2208 evenOdd Q 0},\n \u2191f { val := x, property := hx } = \u2191g { val := x, property := hx } \u2192\n \u2191f\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x \u2208 evenOdd Q 0) } =\n \u2191g\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x \u2208 evenOdd Q 0) }\n[PROOFSTEP]\nintro m\u2081 m\u2082 x hx ih\n[GOAL]\ncase H.mk.refine'_3\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : EvenHom Q A\nf g : { x // x \u2208 even Q } \u2192\u2090[R] A\nh : (EvenHom.compr\u2082 (\u03b9 Q) f).bilin = (EvenHom.compr\u2082 (\u03b9 Q) g).bilin\nx\u271d : CliffordAlgebra Q\nhx\u271d : x\u271d \u2208 even Q\nm\u2081 m\u2082 : M\nx : CliffordAlgebra Q\nhx : x \u2208 evenOdd Q 0\nih : \u2191f { val := x, property := hx } = \u2191g { val := x, property := hx }\n\u22a2 \u2191f\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x \u2208 evenOdd Q 0) } =\n \u2191g\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x \u2208 evenOdd Q 0) }\n[PROOFSTEP]\nhave := congr_arg\u2082 (\u00b7 * \u00b7) (LinearMap.congr_fun (LinearMap.congr_fun h m\u2081) m\u2082) ih\n[GOAL]\ncase H.mk.refine'_3\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : EvenHom Q A\nf g : { x // x \u2208 even Q } \u2192\u2090[R] A\nh : (EvenHom.compr\u2082 (\u03b9 Q) f).bilin = (EvenHom.compr\u2082 (\u03b9 Q) g).bilin\nx\u271d : CliffordAlgebra Q\nhx\u271d : x\u271d \u2208 even Q\nm\u2081 m\u2082 : M\nx : CliffordAlgebra Q\nhx : x \u2208 evenOdd Q 0\nih : \u2191f { val := x, property := hx } = \u2191g { val := x, property := hx }\nthis :\n (fun x_1 x_2 => x_1 * x_2) (\u2191(\u2191(EvenHom.compr\u2082 (\u03b9 Q) f).bilin m\u2081) m\u2082) (\u2191f { val := x, property := hx }) =\n (fun x_1 x_2 => x_1 * x_2) (\u2191(\u2191(EvenHom.compr\u2082 (\u03b9 Q) g).bilin m\u2081) m\u2082) (\u2191g { val := x, property := hx })\n\u22a2 \u2191f\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x \u2208 evenOdd Q 0) } =\n \u2191g\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x \u2208 evenOdd Q 0) }\n[PROOFSTEP]\nexact (f.map_mul _ _).trans (this.trans <| (g.map_mul _ _).symm)\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm\u2081 m\u2082 : M\na : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }\nm\u2083 : M\n\u22a2 \u2191(\u2191f.bilin m\u2083) (m\u2081 + m\u2082) * a.fst = \u2191(\u2191f.bilin m\u2083) m\u2081 * a.fst + \u2191(\u2191f.bilin m\u2083) m\u2082 * a.fst\n[PROOFSTEP]\nrw [map_add, add_mul]\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nc : R\nm : M\na : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }\nm\u2083 : M\n\u22a2 \u2191(\u2191f.bilin m\u2083) (c \u2022 m) * a.fst = c \u2022 (\u2191(\u2191f.bilin m\u2083) m * a.fst)\n[PROOFSTEP]\nrw [LinearMap.map_smul, smul_mul_assoc]\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\nx : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }\n\u22a2 \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) = \u2191Q m \u2022 x\n[PROOFSTEP]\nobtain \u27e8a, \u27e8g, hg\u27e9\u27e9 := x\n[GOAL]\ncase mk.mk\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\na : A\ng : M \u2192\u2097[R] A\nhg : g \u2208 CliffordAlgebra.even.lift.S f\n\u22a2 \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m)\n (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (a, { val := g, property := hg })) =\n \u2191Q m \u2022 (a, { val := g, property := hg })\n[PROOFSTEP]\next : 2\n[GOAL]\ncase mk.mk.h\u2081\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\na : A\ng : M \u2192\u2097[R] A\nhg : g \u2208 CliffordAlgebra.even.lift.S f\n\u22a2 (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m)\n (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (a, { val := g, property := hg }))).fst =\n (\u2191Q m \u2022 (a, { val := g, property := hg })).fst\n[PROOFSTEP]\nchange f.bilin m m * a = Q m \u2022 a\n[GOAL]\ncase mk.mk.h\u2081\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\na : A\ng : M \u2192\u2097[R] A\nhg : g \u2208 CliffordAlgebra.even.lift.S f\n\u22a2 \u2191(\u2191f.bilin m) m * a = \u2191Q m \u2022 a\n[PROOFSTEP]\nrw [Algebra.smul_def, f.contract]\n[GOAL]\ncase mk.mk.h\u2082.a\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\na : A\ng : M \u2192\u2097[R] A\nhg : g \u2208 CliffordAlgebra.even.lift.S f\n\u22a2 \u2191(\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m)\n (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (a, { val := g, property := hg }))).snd =\n \u2191(\u2191Q m \u2022 (a, { val := g, property := hg })).snd\n[PROOFSTEP]\next m\u2081\n[GOAL]\ncase mk.mk.h\u2082.a.h\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\na : A\ng : M \u2192\u2097[R] A\nhg : g \u2208 CliffordAlgebra.even.lift.S f\nm\u2081 : M\n\u22a2 \u2191\u2191(\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m)\n (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (a, { val := g, property := hg }))).snd\n m\u2081 =\n \u2191\u2191(\u2191Q m \u2022 (a, { val := g, property := hg })).snd m\u2081\n[PROOFSTEP]\nchange f.bilin _ _ * g m = Q m \u2022 g m\u2081\n[GOAL]\ncase mk.mk.h\u2082.a.h\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\na : A\ng : M \u2192\u2097[R] A\nhg : g \u2208 CliffordAlgebra.even.lift.S f\nm\u2081 : M\n\u22a2 \u2191(\u2191f.bilin m\u2081) m * \u2191g m = \u2191Q m \u2022 \u2191g m\u2081\n[PROOFSTEP]\napply Submodule.span_induction' _ _ _ _ hg\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\na : A\ng : M \u2192\u2097[R] A\nhg : g \u2208 CliffordAlgebra.even.lift.S f\nm\u2081 : M\n\u22a2 \u2200 (x : M \u2192\u2097[R] A),\n x \u2208 {f' | \u2203 x m\u2082, f' = \u2191(LinearMap.lcomp R A (\u2191(LinearMap.flip f.bilin) m\u2082)) (LinearMap.mulRight R x)} \u2192\n \u2191(\u2191f.bilin m\u2081) m * \u2191x m = \u2191Q m \u2022 \u2191x m\u2081\n[PROOFSTEP]\nrintro _ \u27e8b, m\u2083, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\na : A\ng : M \u2192\u2097[R] A\nhg : g \u2208 CliffordAlgebra.even.lift.S f\nm\u2081 : M\nb : A\nm\u2083 : M\n\u22a2 \u2191(\u2191f.bilin m\u2081) m * \u2191(\u2191(LinearMap.lcomp R A (\u2191(LinearMap.flip f.bilin) m\u2083)) (LinearMap.mulRight R b)) m =\n \u2191Q m \u2022 \u2191(\u2191(LinearMap.lcomp R A (\u2191(LinearMap.flip f.bilin) m\u2083)) (LinearMap.mulRight R b)) m\u2081\n[PROOFSTEP]\nchange f.bilin _ _ * (f.bilin _ _ * b) = Q m \u2022 (f.bilin _ _ * b)\n[GOAL]\ncase intro.intro\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\na : A\ng : M \u2192\u2097[R] A\nhg : g \u2208 CliffordAlgebra.even.lift.S f\nm\u2081 : M\nb : A\nm\u2083 : M\n\u22a2 \u2191(\u2191f.bilin m\u2081) m * (\u2191(\u2191f.bilin m) m\u2083 * b) = \u2191Q m \u2022 (\u2191(\u2191f.bilin m\u2081) m\u2083 * b)\n[PROOFSTEP]\nrw [\u2190 smul_mul_assoc, \u2190 mul_assoc, f.contract_mid]\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\na : A\ng : M \u2192\u2097[R] A\nhg : g \u2208 CliffordAlgebra.even.lift.S f\nm\u2081 : M\n\u22a2 \u2191(\u2191f.bilin m\u2081) m * \u21910 m = \u2191Q m \u2022 \u21910 m\u2081\n[PROOFSTEP]\nchange\n f.bilin m\u2081 m * 0 =\n Q m \u2022\n (0 : A)\n -- porting note: `\u2022` now needs the type of `0`\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\na : A\ng : M \u2192\u2097[R] A\nhg : g \u2208 CliffordAlgebra.even.lift.S f\nm\u2081 : M\n\u22a2 \u2191(\u2191f.bilin m\u2081) m * 0 = \u2191Q m \u2022 0\n[PROOFSTEP]\nrw [mul_zero, smul_zero]\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\na : A\ng : M \u2192\u2097[R] A\nhg : g \u2208 CliffordAlgebra.even.lift.S f\nm\u2081 : M\n\u22a2 \u2200 (x : M \u2192\u2097[R] A),\n x \u2208\n Submodule.span R\n {f' | \u2203 x m\u2082, f' = \u2191(LinearMap.lcomp R A (\u2191(LinearMap.flip f.bilin) m\u2082)) (LinearMap.mulRight R x)} \u2192\n \u2200 (y : M \u2192\u2097[R] A),\n y \u2208\n Submodule.span R\n {f' | \u2203 x m\u2082, f' = \u2191(LinearMap.lcomp R A (\u2191(LinearMap.flip f.bilin) m\u2082)) (LinearMap.mulRight R x)} \u2192\n \u2191(\u2191f.bilin m\u2081) m * \u2191x m = \u2191Q m \u2022 \u2191x m\u2081 \u2192\n \u2191(\u2191f.bilin m\u2081) m * \u2191y m = \u2191Q m \u2022 \u2191y m\u2081 \u2192 \u2191(\u2191f.bilin m\u2081) m * \u2191(x + y) m = \u2191Q m \u2022 \u2191(x + y) m\u2081\n[PROOFSTEP]\nrintro x hx y hy ihx ihy\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\na : A\ng : M \u2192\u2097[R] A\nhg : g \u2208 CliffordAlgebra.even.lift.S f\nm\u2081 : M\nx : M \u2192\u2097[R] A\nhx :\n x \u2208\n Submodule.span R {f' | \u2203 x m\u2082, f' = \u2191(LinearMap.lcomp R A (\u2191(LinearMap.flip f.bilin) m\u2082)) (LinearMap.mulRight R x)}\ny : M \u2192\u2097[R] A\nhy :\n y \u2208\n Submodule.span R {f' | \u2203 x m\u2082, f' = \u2191(LinearMap.lcomp R A (\u2191(LinearMap.flip f.bilin) m\u2082)) (LinearMap.mulRight R x)}\nihx : \u2191(\u2191f.bilin m\u2081) m * \u2191x m = \u2191Q m \u2022 \u2191x m\u2081\nihy : \u2191(\u2191f.bilin m\u2081) m * \u2191y m = \u2191Q m \u2022 \u2191y m\u2081\n\u22a2 \u2191(\u2191f.bilin m\u2081) m * \u2191(x + y) m = \u2191Q m \u2022 \u2191(x + y) m\u2081\n[PROOFSTEP]\nrw [LinearMap.add_apply, LinearMap.add_apply, mul_add, smul_add, ihx, ihy]\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\na : A\ng : M \u2192\u2097[R] A\nhg : g \u2208 CliffordAlgebra.even.lift.S f\nm\u2081 : M\n\u22a2 \u2200 (a : R) (x : M \u2192\u2097[R] A),\n x \u2208\n Submodule.span R\n {f' | \u2203 x m\u2082, f' = \u2191(LinearMap.lcomp R A (\u2191(LinearMap.flip f.bilin) m\u2082)) (LinearMap.mulRight R x)} \u2192\n \u2191(\u2191f.bilin m\u2081) m * \u2191x m = \u2191Q m \u2022 \u2191x m\u2081 \u2192 \u2191(\u2191f.bilin m\u2081) m * \u2191(a \u2022 x) m = \u2191Q m \u2022 \u2191(a \u2022 x) m\u2081\n[PROOFSTEP]\nrintro x hx c ihx\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm : M\na : A\ng : M \u2192\u2097[R] A\nhg : g \u2208 CliffordAlgebra.even.lift.S f\nm\u2081 : M\nx : R\nhx : M \u2192\u2097[R] A\nc :\n hx \u2208\n Submodule.span R {f' | \u2203 x m\u2082, f' = \u2191(LinearMap.lcomp R A (\u2191(LinearMap.flip f.bilin) m\u2082)) (LinearMap.mulRight R x)}\nihx : \u2191(\u2191f.bilin m\u2081) m * \u2191hx m = \u2191Q m \u2022 \u2191hx m\u2081\n\u22a2 \u2191(\u2191f.bilin m\u2081) m * \u2191(x \u2022 hx) m = \u2191Q m \u2022 \u2191(x \u2022 hx) m\u2081\n[PROOFSTEP]\nrw [LinearMap.smul_apply, LinearMap.smul_apply, mul_smul_comm, ihx, smul_comm]\n -- Porting note: In Lean 3, `aux_apply` isn't a simp lemma. I changed `{ attrs := [] }` to\n -- `{ isSimp := false }`, so that `aux_apply` isn't a simp lemma.\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d f : EvenHom Q A\n\u22a2 { x // x \u2208 even Q } \u2192\u2097[R] A\n[PROOFSTEP]\nrefine ?_ \u2218\u2097 (even Q).val.toLinearMap\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d f : EvenHom Q A\n\u22a2 CliffordAlgebra Q \u2192\u2097[R] A\n[PROOFSTEP]\nletI : AddCommGroup (S f) := AddSubgroupClass.toAddCommGroup _\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d f : EvenHom Q A\nthis : AddCommGroup { x // x \u2208 CliffordAlgebra.even.lift.S f } :=\n AddSubgroupClass.toAddCommGroup (CliffordAlgebra.even.lift.S f)\n\u22a2 CliffordAlgebra Q \u2192\u2097[R] A\n[PROOFSTEP]\nexact LinearMap.fst R _ _ \u2218\u2097 foldr Q (fFold f) (fFold_fFold f) (1, 0)\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm\u2081 m\u2082 : M\n\u22a2 (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) = \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n (\u2191(CliffordAlgebra.\u03b9 Q) m\u2082)))\n (\u2191(CliffordAlgebra.\u03b9 Q) m\u2081)).fst =\n \u2191(\u2191f.bilin m\u2081) m\u2082\n[PROOFSTEP]\nrw [foldr_\u03b9, foldr_\u03b9]\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nm\u2081 m\u2082 : M\n\u22a2 (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m\u2081) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m\u2082) (1, 0))).fst =\n \u2191(\u2191f.bilin m\u2081) m\u2082\n[PROOFSTEP]\nexact mul_one _\n[GOAL]\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nx y : { x // x \u2208 even Q }\n\u22a2 \u2191(aux f) (x * y) = \u2191(aux f) x * \u2191(aux f) y\n[PROOFSTEP]\ncases' x with x x_property\n[GOAL]\ncase mk\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\ny : { x // x \u2208 even Q }\nx : CliffordAlgebra Q\nx_property : x \u2208 even Q\n\u22a2 \u2191(aux f) ({ val := x, property := x_property } * y) = \u2191(aux f) { val := x, property := x_property } * \u2191(aux f) y\n[PROOFSTEP]\ncases y\n[GOAL]\ncase mk.mk\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nx : CliffordAlgebra Q\nx_property : x \u2208 even Q\nval\u271d : CliffordAlgebra Q\nproperty\u271d : val\u271d \u2208 even Q\n\u22a2 \u2191(aux f) ({ val := x, property := x_property } * { val := val\u271d, property := property\u271d }) =\n \u2191(aux f) { val := x, property := x_property } * \u2191(aux f) { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrefine' (congr_arg Prod.fst (foldr_mul _ _ _ _ _ _)).trans _\n[GOAL]\ncase mk.mk\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nx : CliffordAlgebra Q\nx_property : x \u2208 even Q\nval\u271d : CliffordAlgebra Q\nproperty\u271d : val\u271d \u2208 even Q\n\u22a2 (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) = \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n \u2191{ val := val\u271d, property := property\u271d }))\n \u2191{ val := x, property := x_property }).fst =\n \u2191(aux f) { val := x, property := x_property } * \u2191(aux f) { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase mk.mk\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nx : CliffordAlgebra Q\nx_property : x \u2208 even Q\nval\u271d : CliffordAlgebra Q\nproperty\u271d : val\u271d \u2208 even Q\n\u22a2 (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) = \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n x).fst =\n \u2191(aux f) { val := x, property := x_property } * \u2191(aux f) { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrefine' even_induction Q _ _ _ _ x_property\n[GOAL]\ncase mk.mk.refine'_1\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nx : CliffordAlgebra Q\nx_property : x \u2208 even Q\nval\u271d : CliffordAlgebra Q\nproperty\u271d : val\u271d \u2208 even Q\n\u22a2 \u2200 (r : R),\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n (\u2191(algebraMap R (CliffordAlgebra Q)) r)).fst =\n \u2191(aux f)\n { val := \u2191(algebraMap R (CliffordAlgebra Q)) r,\n property := (_ : \u2191(algebraMap R (CliffordAlgebra Q)) r \u2208 evenOdd Q 0) } *\n \u2191(aux f) { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nintro r\n[GOAL]\ncase mk.mk.refine'_1\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nx : CliffordAlgebra Q\nx_property : x \u2208 even Q\nval\u271d : CliffordAlgebra Q\nproperty\u271d : val\u271d \u2208 even Q\nr : R\n\u22a2 (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) = \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n (\u2191(algebraMap R (CliffordAlgebra Q)) r)).fst =\n \u2191(aux f)\n { val := \u2191(algebraMap R (CliffordAlgebra Q)) r,\n property := (_ : \u2191(algebraMap R (CliffordAlgebra Q)) r \u2208 evenOdd Q 0) } *\n \u2191(aux f) { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrw [foldr_algebraMap, aux_algebraMap]\n[GOAL]\ncase mk.mk.refine'_1\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nx : CliffordAlgebra Q\nx_property : x \u2208 even Q\nval\u271d : CliffordAlgebra Q\nproperty\u271d : val\u271d \u2208 even Q\nr : R\n\u22a2 (r \u2022\n \u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d).fst =\n \u2191(algebraMap R A) r * \u2191(aux f) { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nexact Algebra.smul_def r _\n[GOAL]\ncase mk.mk.refine'_2\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nx : CliffordAlgebra Q\nx_property : x \u2208 even Q\nval\u271d : CliffordAlgebra Q\nproperty\u271d : val\u271d \u2208 even Q\n\u22a2 \u2200 {x y : CliffordAlgebra Q} {hx : x \u2208 evenOdd Q 0} {hy : y \u2208 evenOdd Q 0},\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n x).fst =\n \u2191(aux f) { val := x, property := hx } * \u2191(aux f) { val := val\u271d, property := property\u271d } \u2192\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m)\n (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n y).fst =\n \u2191(aux f) { val := y, property := hy } * \u2191(aux f) { val := val\u271d, property := property\u271d } \u2192\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m)\n (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n (x + y)).fst =\n \u2191(aux f) { val := x + y, property := (_ : x + y \u2208 evenOdd Q 0) } *\n \u2191(aux f) { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nintro x y hx hy ihx ihy\n[GOAL]\ncase mk.mk.refine'_2\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nx\u271d : CliffordAlgebra Q\nx_property : x\u271d \u2208 even Q\nval\u271d : CliffordAlgebra Q\nproperty\u271d : val\u271d \u2208 even Q\nx y : CliffordAlgebra Q\nhx : x \u2208 evenOdd Q 0\nhy : y \u2208 evenOdd Q 0\nihx :\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) = \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n x).fst =\n \u2191(aux f) { val := x, property := hx } * \u2191(aux f) { val := val\u271d, property := property\u271d }\nihy :\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) = \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n y).fst =\n \u2191(aux f) { val := y, property := hy } * \u2191(aux f) { val := val\u271d, property := property\u271d }\n\u22a2 (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) = \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n (x + y)).fst =\n \u2191(aux f) { val := x + y, property := (_ : x + y \u2208 evenOdd Q 0) } * \u2191(aux f) { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrw [LinearMap.map_add, Prod.fst_add, ihx, ihy, \u2190 add_mul, \u2190 LinearMap.map_add]\n[GOAL]\ncase mk.mk.refine'_2\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nx\u271d : CliffordAlgebra Q\nx_property : x\u271d \u2208 even Q\nval\u271d : CliffordAlgebra Q\nproperty\u271d : val\u271d \u2208 even Q\nx y : CliffordAlgebra Q\nhx : x \u2208 evenOdd Q 0\nhy : y \u2208 evenOdd Q 0\nihx :\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) = \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n x).fst =\n \u2191(aux f) { val := x, property := hx } * \u2191(aux f) { val := val\u271d, property := property\u271d }\nihy :\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) = \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n y).fst =\n \u2191(aux f) { val := y, property := hy } * \u2191(aux f) { val := val\u271d, property := property\u271d }\n\u22a2 \u2191(aux f) ({ val := x, property := hx } + { val := y, property := hy }) *\n \u2191(aux f) { val := val\u271d, property := property\u271d } =\n \u2191(aux f) { val := x + y, property := (_ : x + y \u2208 evenOdd Q 0) } * \u2191(aux f) { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.mk.refine'_3\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nx : CliffordAlgebra Q\nx_property : x \u2208 even Q\nval\u271d : CliffordAlgebra Q\nproperty\u271d : val\u271d \u2208 even Q\n\u22a2 \u2200 (m\u2081 m\u2082 : M) {x : CliffordAlgebra Q} {hx : x \u2208 evenOdd Q 0},\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n x).fst =\n \u2191(aux f) { val := x, property := hx } * \u2191(aux f) { val := val\u271d, property := property\u271d } \u2192\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n (\u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x)).fst =\n \u2191(aux f)\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x \u2208 evenOdd Q 0) } *\n \u2191(aux f) { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrintro m\u2081 m\u2082 x (hx : x \u2208 even Q) ih\n[GOAL]\ncase mk.mk.refine'_3\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nx\u271d : CliffordAlgebra Q\nx_property : x\u271d \u2208 even Q\nval\u271d : CliffordAlgebra Q\nproperty\u271d : val\u271d \u2208 even Q\nm\u2081 m\u2082 : M\nx : CliffordAlgebra Q\nhx : x \u2208 even Q\nih :\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) = \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n x).fst =\n \u2191(aux f) { val := x, property := hx } * \u2191(aux f) { val := val\u271d, property := property\u271d }\n\u22a2 (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) = \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n (\u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x)).fst =\n \u2191(aux f)\n { val := \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x,\n property := (_ : \u2191(CliffordAlgebra.\u03b9 Q) m\u2081 * \u2191(CliffordAlgebra.\u03b9 Q) m\u2082 * x \u2208 evenOdd Q 0) } *\n \u2191(aux f) { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrw [aux_apply, foldr_mul, foldr_mul, foldr_\u03b9, foldr_\u03b9, fst_fFold_fFold, ih, \u2190 mul_assoc, Subtype.coe_mk, foldr_mul,\n foldr_mul, foldr_\u03b9, foldr_\u03b9, fst_fFold_fFold]\n[GOAL]\ncase mk.mk.refine'_3\nR : Type uR\nM : Type uM\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nA : Type uA\nB : Type uB\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : EvenHom Q A\nx\u271d : CliffordAlgebra Q\nx_property : x\u271d \u2208 even Q\nval\u271d : CliffordAlgebra Q\nproperty\u271d : val\u271d \u2208 even Q\nm\u2081 m\u2082 : M\nx : CliffordAlgebra Q\nhx : x \u2208 even Q\nih :\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) = \u2191Q m \u2022 x))\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n val\u271d))\n x).fst =\n \u2191(aux f) { val := x, property := hx } * \u2191(aux f) { val := val\u271d, property := property\u271d }\n\u22a2 \u2191(\u2191f.bilin m\u2081) m\u2082 * \u2191(aux f) { val := x, property := hx } * \u2191(aux f) { val := val\u271d, property := property\u271d } =\n \u2191(\u2191f.bilin m\u2081) m\u2082 *\n (\u2191(\u2191(foldr Q (CliffordAlgebra.even.lift.fFold f)\n (_ :\n \u2200 (m : M) (x : A \u00d7 { x // x \u2208 CliffordAlgebra.even.lift.S f }),\n \u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) (\u2191(\u2191(CliffordAlgebra.even.lift.fFold f) m) x) =\n \u2191Q m \u2022 x))\n (1, 0))\n x).fst *\n \u2191(aux f) { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.CliffordAlgebra.Even", "llama_tokens": 23761, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943712746407, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.41431509413503936}} {"text": "[GOAL]\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type ?u.1599\ninst\u271d : Category.{?u.1600, ?u.1599} C\n\u03c6 : V \u2964q C\nX : Paths V\n\u22a2 { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n (\ud835\udfd9 X) =\n \ud835\udfd9\n ({ obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.obj\n X)\n[PROOFSTEP]\nrfl\n[GOAL]\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type ?u.1599\ninst\u271d : Category.{?u.1600, ?u.1599} C\n\u03c6 : V \u2964q C\nX\u271d Y\u271d Z\u271d : Paths V\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n (f \u226b g) =\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map f \u226b\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map g\n[PROOFSTEP]\ninduction' g with _ _ g' p ih _ _ _\n[GOAL]\ncase nil\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type ?u.1599\ninst\u271d : Category.{?u.1600, ?u.1599} C\n\u03c6 : V \u2964q C\nX\u271d Y\u271d Z\u271d : Paths V\nf : X\u271d \u27f6 Y\u271d\n\u22a2 { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n (f \u226b Quiver.Path.nil) =\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map f \u226b\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n Quiver.Path.nil\n[PROOFSTEP]\nrw [Category.comp_id]\n[GOAL]\ncase nil\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type ?u.1599\ninst\u271d : Category.{?u.1600, ?u.1599} C\n\u03c6 : V \u2964q C\nX\u271d Y\u271d Z\u271d : Paths V\nf : X\u271d \u27f6 Y\u271d\n\u22a2 { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n (f \u226b Quiver.Path.nil) =\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map f\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type ?u.1599\ninst\u271d : Category.{?u.1600, ?u.1599} C\n\u03c6 : V \u2964q C\nX\u271d Y\u271d Z\u271d : Paths V\nf : X\u271d \u27f6 Y\u271d\nb\u271d c\u271d : Paths V\ng' : Quiver.Path Y\u271d b\u271d\np : b\u271d \u27f6 c\u271d\nih :\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n (f \u226b g') =\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map f \u226b\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n g'\n\u22a2 { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n (f \u226b Quiver.Path.cons g' p) =\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map f \u226b\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n (Quiver.Path.cons g' p)\n[PROOFSTEP]\nhave : f \u226b Quiver.Path.cons g' p = (f \u226b g').cons p := by apply Quiver.Path.comp_cons\n[GOAL]\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type ?u.1599\ninst\u271d : Category.{?u.1600, ?u.1599} C\n\u03c6 : V \u2964q C\nX\u271d Y\u271d Z\u271d : Paths V\nf : X\u271d \u27f6 Y\u271d\nb\u271d c\u271d : Paths V\ng' : Quiver.Path Y\u271d b\u271d\np : b\u271d \u27f6 c\u271d\nih :\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n (f \u226b g') =\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map f \u226b\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n g'\n\u22a2 f \u226b Quiver.Path.cons g' p = Quiver.Path.cons (f \u226b g') p\n[PROOFSTEP]\napply Quiver.Path.comp_cons\n[GOAL]\ncase cons\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type ?u.1599\ninst\u271d : Category.{?u.1600, ?u.1599} C\n\u03c6 : V \u2964q C\nX\u271d Y\u271d Z\u271d : Paths V\nf : X\u271d \u27f6 Y\u271d\nb\u271d c\u271d : Paths V\ng' : Quiver.Path Y\u271d b\u271d\np : b\u271d \u27f6 c\u271d\nih :\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n (f \u226b g') =\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map f \u226b\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n g'\nthis : f \u226b Quiver.Path.cons g' p = Quiver.Path.cons (f \u226b g') p\n\u22a2 { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n (f \u226b Quiver.Path.cons g' p) =\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map f \u226b\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n (Quiver.Path.cons g' p)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase cons\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type ?u.1599\ninst\u271d : Category.{?u.1600, ?u.1599} C\n\u03c6 : V \u2964q C\nX\u271d Y\u271d Z\u271d : Paths V\nf : X\u271d \u27f6 Y\u271d\nb\u271d c\u271d : Paths V\ng' : Quiver.Path Y\u271d b\u271d\np : b\u271d \u27f6 c\u271d\nih :\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n (f \u226b g') =\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map f \u226b\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n g'\nthis : f \u226b Quiver.Path.cons g' p = Quiver.Path.cons (f \u226b g') p\n\u22a2 { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n (Quiver.Path.cons (f \u226b g') p) =\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map f \u226b\n { obj := \u03c6.obj, map := fun {X Y} f => Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f }.map\n (Quiver.Path.cons g' p)\n[PROOFSTEP]\nsimp only at ih \u22a2\n[GOAL]\ncase cons\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type ?u.1599\ninst\u271d : Category.{?u.1600, ?u.1599} C\n\u03c6 : V \u2964q C\nX\u271d Y\u271d Z\u271d : Paths V\nf : X\u271d \u27f6 Y\u271d\nb\u271d c\u271d : Paths V\ng' : Quiver.Path Y\u271d b\u271d\np : b\u271d \u27f6 c\u271d\nih :\n Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X\u271d)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) (f \u226b g') =\n Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X\u271d)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f \u226b\n Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj Y\u271d)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) g'\nthis : f \u226b Quiver.Path.cons g' p = Quiver.Path.cons (f \u226b g') p\n\u22a2 Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X\u271d)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) (f \u226b g') \u226b \u03c6.map p =\n Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj X\u271d)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) f \u226b\n Quiver.Path.rec (\ud835\udfd9 (\u03c6.obj Y\u271d)) (fun {b c} x f ihp => ihp \u226b \u03c6.map f) g' \u226b \u03c6.map p\n[PROOFSTEP]\nrw [ih, Category.assoc]\n[GOAL]\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03c6 : V \u2964q C\nX Y : V\nf : X \u27f6 Y\n\u22a2 (lift \u03c6).map (Quiver.Hom.toPath f) = \u03c6.map f\n[PROOFSTEP]\ndsimp [Quiver.Hom.toPath, lift]\n[GOAL]\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03c6 : V \u2964q C\nX Y : V\nf : X \u27f6 Y\n\u22a2 \ud835\udfd9 (\u03c6.obj X) \u226b \u03c6.map f = \u03c6.map f\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03c6 : V \u2964q C\n\u22a2 of \u22d9q (lift \u03c6).toPrefunctor = \u03c6\n[PROOFSTEP]\nfapply Prefunctor.ext\n[GOAL]\ncase h_obj\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03c6 : V \u2964q C\n\u22a2 \u2200 (X : V), (of \u22d9q (lift \u03c6).toPrefunctor).obj X = \u03c6.obj X\n[PROOFSTEP]\nrintro X\n[GOAL]\ncase h_obj\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03c6 : V \u2964q C\nX : V\n\u22a2 (of \u22d9q (lift \u03c6).toPrefunctor).obj X = \u03c6.obj X\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_map\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03c6 : V \u2964q C\n\u22a2 \u2200 (X Y : V) (f : X \u27f6 Y),\n (of \u22d9q (lift \u03c6).toPrefunctor).map f =\n Eq.recOn (_ : \u03c6.obj Y = (of \u22d9q (lift \u03c6).toPrefunctor).obj Y)\n (Eq.recOn (_ : \u03c6.obj X = (of \u22d9q (lift \u03c6).toPrefunctor).obj X) (\u03c6.map f))\n[PROOFSTEP]\nrintro X Y f\n[GOAL]\ncase h_map\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03c6 : V \u2964q C\nX Y : V\nf : X \u27f6 Y\n\u22a2 (of \u22d9q (lift \u03c6).toPrefunctor).map f =\n Eq.recOn (_ : \u03c6.obj Y = (of \u22d9q (lift \u03c6).toPrefunctor).obj Y)\n (Eq.recOn (_ : \u03c6.obj X = (of \u22d9q (lift \u03c6).toPrefunctor).obj X) (\u03c6.map f))\n[PROOFSTEP]\nrcases \u03c6 with \u27e8\u03c6o, \u03c6m\u27e9\n[GOAL]\ncase h_map.mk\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : V\nf : X \u27f6 Y\n\u03c6o : V \u2192 C\n\u03c6m : {X Y : V} \u2192 (X \u27f6 Y) \u2192 (\u03c6o X \u27f6 \u03c6o Y)\n\u22a2 (of \u22d9q (lift { obj := \u03c6o, map := \u03c6m }).toPrefunctor).map f =\n Eq.recOn (_ : { obj := \u03c6o, map := \u03c6m }.obj Y = (of \u22d9q (lift { obj := \u03c6o, map := \u03c6m }).toPrefunctor).obj Y)\n (Eq.recOn (_ : { obj := \u03c6o, map := \u03c6m }.obj X = (of \u22d9q (lift { obj := \u03c6o, map := \u03c6m }).toPrefunctor).obj X)\n ({ obj := \u03c6o, map := \u03c6m }.map f))\n[PROOFSTEP]\ndsimp [lift, Quiver.Hom.toPath]\n[GOAL]\ncase h_map.mk\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : V\nf : X \u27f6 Y\n\u03c6o : V \u2192 C\n\u03c6m : {X Y : V} \u2192 (X \u27f6 Y) \u2192 (\u03c6o X \u27f6 \u03c6o Y)\n\u22a2 \ud835\udfd9 (\u03c6o X) \u226b \u03c6m f = \u03c6m f\n[PROOFSTEP]\nsimp only [Category.id_comp]\n[GOAL]\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03c6 : V \u2964q C\n\u03a6 : Paths V \u2964 C\nh\u03a6 : of \u22d9q \u03a6.toPrefunctor = \u03c6\n\u22a2 \u03a6 = lift \u03c6\n[PROOFSTEP]\nsubst_vars\n[GOAL]\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03a6 : Paths V \u2964 C\n\u22a2 \u03a6 = lift (of \u22d9q \u03a6.toPrefunctor)\n[PROOFSTEP]\nfapply Functor.ext\n[GOAL]\ncase h_obj\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03a6 : Paths V \u2964 C\n\u22a2 \u2200 (X : Paths V), \u03a6.obj X = (lift (of \u22d9q \u03a6.toPrefunctor)).obj X\n[PROOFSTEP]\nrintro X\n[GOAL]\ncase h_obj\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03a6 : Paths V \u2964 C\nX : Paths V\n\u22a2 \u03a6.obj X = (lift (of \u22d9q \u03a6.toPrefunctor)).obj X\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_map\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03a6 : Paths V \u2964 C\n\u22a2 autoParam\n (\u2200 (X Y : Paths V) (f : X \u27f6 Y),\n \u03a6.map f =\n eqToHom (_ : \u03a6.obj X = \u03a6.obj X) \u226b\n (lift (of \u22d9q \u03a6.toPrefunctor)).map f \u226b eqToHom (_ : (lift (of \u22d9q \u03a6.toPrefunctor)).obj Y = \u03a6.obj Y))\n _auto\u271d\n[PROOFSTEP]\nrintro X Y f\n[GOAL]\ncase h_map\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03a6 : Paths V \u2964 C\nX Y : Paths V\nf : X \u27f6 Y\n\u22a2 \u03a6.map f =\n eqToHom (_ : \u03a6.obj X = \u03a6.obj X) \u226b\n (lift (of \u22d9q \u03a6.toPrefunctor)).map f \u226b eqToHom (_ : (lift (of \u22d9q \u03a6.toPrefunctor)).obj Y = \u03a6.obj Y)\n[PROOFSTEP]\ndsimp [lift]\n[GOAL]\ncase h_map\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03a6 : Paths V \u2964 C\nX Y : Paths V\nf : X \u27f6 Y\n\u22a2 \u03a6.map f =\n \ud835\udfd9 (\u03a6.obj X) \u226b Quiver.Path.rec (\ud835\udfd9 (\u03a6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03a6.map (Quiver.Hom.toPath f)) f \u226b \ud835\udfd9 (\u03a6.obj Y)\n[PROOFSTEP]\ninduction' f with _ _ p f' ih\n[GOAL]\ncase h_map.nil\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03a6 : Paths V \u2964 C\nX Y : Paths V\n\u22a2 \u03a6.map Quiver.Path.nil =\n \ud835\udfd9 (\u03a6.obj X) \u226b\n Quiver.Path.rec (\ud835\udfd9 (\u03a6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03a6.map (Quiver.Hom.toPath f)) Quiver.Path.nil \u226b\n \ud835\udfd9 (\u03a6.obj X)\n[PROOFSTEP]\nsimp only [Category.comp_id]\n[GOAL]\ncase h_map.nil\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03a6 : Paths V \u2964 C\nX Y : Paths V\n\u22a2 \u03a6.map Quiver.Path.nil = \ud835\udfd9 (\u03a6.obj X)\n[PROOFSTEP]\napply Functor.map_id\n[GOAL]\ncase h_map.cons\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03a6 : Paths V \u2964 C\nX Y b\u271d c\u271d : Paths V\np : Quiver.Path X b\u271d\nf' : b\u271d \u27f6 c\u271d\nih :\n \u03a6.map p =\n \ud835\udfd9 (\u03a6.obj X) \u226b\n Quiver.Path.rec (\ud835\udfd9 (\u03a6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03a6.map (Quiver.Hom.toPath f)) p \u226b \ud835\udfd9 (\u03a6.obj b\u271d)\n\u22a2 \u03a6.map (Quiver.Path.cons p f') =\n \ud835\udfd9 (\u03a6.obj X) \u226b\n Quiver.Path.rec (\ud835\udfd9 (\u03a6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03a6.map (Quiver.Hom.toPath f)) (Quiver.Path.cons p f') \u226b\n \ud835\udfd9 (\u03a6.obj c\u271d)\n[PROOFSTEP]\nsimp only [Category.comp_id, Category.id_comp] at ih \u22a2\n -- porting note: Had to do substitute `p.cons f'` and `f'.toPath` by their fully qualified\n -- versions in this `have` clause (elsewhere too).\n[GOAL]\ncase h_map.cons\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03a6 : Paths V \u2964 C\nX Y b\u271d c\u271d : Paths V\np : Quiver.Path X b\u271d\nf' : b\u271d \u27f6 c\u271d\nih : \u03a6.map p = Quiver.Path.rec (\ud835\udfd9 (\u03a6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03a6.map (Quiver.Hom.toPath f)) p\n\u22a2 \u03a6.map (Quiver.Path.cons p f') =\n Quiver.Path.rec (\ud835\udfd9 (\u03a6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03a6.map (Quiver.Hom.toPath f)) p \u226b\n \u03a6.map (Quiver.Hom.toPath f')\n[PROOFSTEP]\nhave : \u03a6.map (Quiver.Path.cons p f') = \u03a6.map p \u226b \u03a6.map (Quiver.Hom.toPath f') := by\n convert Functor.map_comp \u03a6 p (Quiver.Hom.toPath f')\n[GOAL]\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03a6 : Paths V \u2964 C\nX Y b\u271d c\u271d : Paths V\np : Quiver.Path X b\u271d\nf' : b\u271d \u27f6 c\u271d\nih : \u03a6.map p = Quiver.Path.rec (\ud835\udfd9 (\u03a6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03a6.map (Quiver.Hom.toPath f)) p\n\u22a2 \u03a6.map (Quiver.Path.cons p f') = \u03a6.map p \u226b \u03a6.map (Quiver.Hom.toPath f')\n[PROOFSTEP]\nconvert Functor.map_comp \u03a6 p (Quiver.Hom.toPath f')\n[GOAL]\ncase h_map.cons\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03a6 : Paths V \u2964 C\nX Y b\u271d c\u271d : Paths V\np : Quiver.Path X b\u271d\nf' : b\u271d \u27f6 c\u271d\nih : \u03a6.map p = Quiver.Path.rec (\ud835\udfd9 (\u03a6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03a6.map (Quiver.Hom.toPath f)) p\nthis : \u03a6.map (Quiver.Path.cons p f') = \u03a6.map p \u226b \u03a6.map (Quiver.Hom.toPath f')\n\u22a2 \u03a6.map (Quiver.Path.cons p f') =\n Quiver.Path.rec (\ud835\udfd9 (\u03a6.obj X)) (fun {b c} x f ihp => ihp \u226b \u03a6.map (Quiver.Hom.toPath f)) p \u226b\n \u03a6.map (Quiver.Hom.toPath f')\n[PROOFSTEP]\nrw [this, ih]\n[GOAL]\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nF G : Paths V \u2964 C\nh_obj : F.obj = G.obj\nh :\n \u2200 (a b : V) (e : a \u27f6 b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) \u226b G.map (Quiver.Hom.toPath e) \u226b eqToHom (_ : G.obj b = F.obj b)\n\u22a2 F = G\n[PROOFSTEP]\nfapply Functor.ext\n[GOAL]\ncase h_obj\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nF G : Paths V \u2964 C\nh_obj : F.obj = G.obj\nh :\n \u2200 (a b : V) (e : a \u27f6 b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) \u226b G.map (Quiver.Hom.toPath e) \u226b eqToHom (_ : G.obj b = F.obj b)\n\u22a2 \u2200 (X : Paths V), F.obj X = G.obj X\n[PROOFSTEP]\nintro X\n[GOAL]\ncase h_obj\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nF G : Paths V \u2964 C\nh_obj : F.obj = G.obj\nh :\n \u2200 (a b : V) (e : a \u27f6 b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) \u226b G.map (Quiver.Hom.toPath e) \u226b eqToHom (_ : G.obj b = F.obj b)\nX : Paths V\n\u22a2 F.obj X = G.obj X\n[PROOFSTEP]\nrw [h_obj]\n[GOAL]\ncase h_map\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nF G : Paths V \u2964 C\nh_obj : F.obj = G.obj\nh :\n \u2200 (a b : V) (e : a \u27f6 b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) \u226b G.map (Quiver.Hom.toPath e) \u226b eqToHom (_ : G.obj b = F.obj b)\n\u22a2 autoParam\n (\u2200 (X Y : Paths V) (f : X \u27f6 Y),\n F.map f = eqToHom (_ : F.obj X = G.obj X) \u226b G.map f \u226b eqToHom (_ : G.obj Y = F.obj Y))\n _auto\u271d\n[PROOFSTEP]\nintro X Y f\n[GOAL]\ncase h_map\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nF G : Paths V \u2964 C\nh_obj : F.obj = G.obj\nh :\n \u2200 (a b : V) (e : a \u27f6 b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) \u226b G.map (Quiver.Hom.toPath e) \u226b eqToHom (_ : G.obj b = F.obj b)\nX Y : Paths V\nf : X \u27f6 Y\n\u22a2 F.map f = eqToHom (_ : F.obj X = G.obj X) \u226b G.map f \u226b eqToHom (_ : G.obj Y = F.obj Y)\n[PROOFSTEP]\ninduction' f with Y' Z' g e ih\n[GOAL]\ncase h_map.nil\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nF G : Paths V \u2964 C\nh_obj : F.obj = G.obj\nh :\n \u2200 (a b : V) (e : a \u27f6 b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) \u226b G.map (Quiver.Hom.toPath e) \u226b eqToHom (_ : G.obj b = F.obj b)\nX Y : Paths V\n\u22a2 F.map Quiver.Path.nil = eqToHom (_ : F.obj X = G.obj X) \u226b G.map Quiver.Path.nil \u226b eqToHom (_ : G.obj X = F.obj X)\n[PROOFSTEP]\nerw [F.map_id, G.map_id, Category.id_comp, eqToHom_trans, eqToHom_refl]\n[GOAL]\ncase h_map.cons\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nF G : Paths V \u2964 C\nh_obj : F.obj = G.obj\nh :\n \u2200 (a b : V) (e : a \u27f6 b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) \u226b G.map (Quiver.Hom.toPath e) \u226b eqToHom (_ : G.obj b = F.obj b)\nX Y Y' Z' : Paths V\ng : Quiver.Path X Y'\ne : Y' \u27f6 Z'\nih : F.map g = eqToHom (_ : F.obj X = G.obj X) \u226b G.map g \u226b eqToHom (_ : G.obj Y' = F.obj Y')\n\u22a2 F.map (Quiver.Path.cons g e) =\n eqToHom (_ : F.obj X = G.obj X) \u226b G.map (Quiver.Path.cons g e) \u226b eqToHom (_ : G.obj Z' = F.obj Z')\n[PROOFSTEP]\nerw [F.map_comp g (Quiver.Hom.toPath e), G.map_comp g (Quiver.Hom.toPath e), ih, h]\n[GOAL]\ncase h_map.cons\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nF G : Paths V \u2964 C\nh_obj : F.obj = G.obj\nh :\n \u2200 (a b : V) (e : a \u27f6 b),\n F.map (Quiver.Hom.toPath e) =\n eqToHom (_ : F.obj a = G.obj a) \u226b G.map (Quiver.Hom.toPath e) \u226b eqToHom (_ : G.obj b = F.obj b)\nX Y Y' Z' : Paths V\ng : Quiver.Path X Y'\ne : Y' \u27f6 Z'\nih : F.map g = eqToHom (_ : F.obj X = G.obj X) \u226b G.map g \u226b eqToHom (_ : G.obj Y' = F.obj Y')\n\u22a2 (eqToHom (_ : F.obj X = G.obj X) \u226b G.map g \u226b eqToHom (_ : G.obj Y' = F.obj Y')) \u226b\n eqToHom (_ : F.obj Y' = G.obj Y') \u226b G.map (Quiver.Hom.toPath e) \u226b eqToHom (_ : G.obj Z' = F.obj Z') =\n eqToHom (_ : F.obj X = G.obj X) \u226b (G.map g \u226b G.map (Quiver.Hom.toPath e)) \u226b eqToHom (_ : G.obj Z' = F.obj Z')\n[PROOFSTEP]\nsimp only [Category.id_comp, eqToHom_refl, eqToHom_trans_assoc, Category.assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf : Path X Y\ng : Path Y Z\n\u22a2 composePath (Path.comp f g) = composePath f \u226b composePath g\n[PROOFSTEP]\ninduction' g with Y' Z' g e ih\n[GOAL]\ncase nil\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf : Path X Y\n\u22a2 composePath (Path.comp f Path.nil) = composePath f \u226b composePath Path.nil\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y Z : C\nf : Path X Y\nY' Z' : C\ng : Path Y Y'\ne : Y' \u27f6 Z'\nih : composePath (Path.comp f g) = composePath f \u226b composePath g\n\u22a2 composePath (Path.comp f (Path.cons g e)) = composePath f \u226b composePath (Path.cons g e)\n[PROOFSTEP]\nsimp [ih]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\n\u22a2 pathsHomRel C (Hom.toPath (\ud835\udfd9 X))\n (\ud835\udfd9\n ({ obj := fun X => { as := X },\n map := fun {X Y} f => Quot.mk (Quotient.CompClosure (pathsHomRel C)) (Hom.toPath f) }.obj\n X).as)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 pathsHomRel C (Hom.toPath (f \u226b g)) (Hom.toPath f \u226b Hom.toPath g)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : Quotient (pathsHomRel C)\n\u22a2 (\ud835\udfed (Quotient (pathsHomRel C))).obj X \u2245 (quotientPathsTo C \u22d9 toQuotientPaths C).obj X\n[PROOFSTEP]\ncases X\n[GOAL]\ncase mk\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nas\u271d : Paths C\n\u22a2 (\ud835\udfed (Quotient (pathsHomRel C))).obj { as := as\u271d } \u2245 (quotientPathsTo C \u22d9 toQuotientPaths C).obj { as := as\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d : Quotient (pathsHomRel C)\nf : X\u271d.as \u27f6 Y\u271d.as\n\u22a2 (\ud835\udfed (Quotient (pathsHomRel C))).map (Quot.mk (Quotient.CompClosure (pathsHomRel C)) f) \u226b\n ((fun X =>\n Quotient.casesOn (motive := fun t =>\n X = t \u2192 ((\ud835\udfed (Quotient (pathsHomRel C))).obj X \u2245 (quotientPathsTo C \u22d9 toQuotientPaths C).obj X)) X\n (fun as h => (_ : { as := as } = X) \u25b8 Iso.refl ((\ud835\udfed (Quotient (pathsHomRel C))).obj { as := as }))\n (_ : X = X))\n Y\u271d).hom =\n ((fun X =>\n Quotient.casesOn (motive := fun t =>\n X = t \u2192 ((\ud835\udfed (Quotient (pathsHomRel C))).obj X \u2245 (quotientPathsTo C \u22d9 toQuotientPaths C).obj X)) X\n (fun as h => (_ : { as := as } = X) \u25b8 Iso.refl ((\ud835\udfed (Quotient (pathsHomRel C))).obj { as := as }))\n (_ : X = X))\n X\u271d).hom \u226b\n (quotientPathsTo C \u22d9 toQuotientPaths C).map (Quot.mk (Quotient.CompClosure (pathsHomRel C)) f)\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d : Quotient (pathsHomRel C)\nf : X\u271d.as \u27f6 Y\u271d.as\n\u22a2 Quotient.CompClosure (pathsHomRel C) (f \u226b \ud835\udfd9 ((\ud835\udfed (Quotient (pathsHomRel C))).obj { as := Y\u271d.as }).as)\n (\ud835\udfd9 ((\ud835\udfed (Quotient (pathsHomRel C))).obj { as := X\u271d.as }).as \u226b\n Hom.toPath ((quotientPathsTo C).map (Quot.mk (Quotient.CompClosure (pathsHomRel C)) f)))\n[PROOFSTEP]\napply Quotient.CompClosure.of\n[GOAL]\ncase a.h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d : Quotient (pathsHomRel C)\nf : X\u271d.as \u27f6 Y\u271d.as\n\u22a2 pathsHomRel C (f \u226b \ud835\udfd9 ((\ud835\udfed (Quotient (pathsHomRel C))).obj { as := Y\u271d.as }).as)\n (\ud835\udfd9 ((\ud835\udfed (Quotient (pathsHomRel C))).obj { as := X\u271d.as }).as \u226b\n Hom.toPath ((quotientPathsTo C).map (Quot.mk (Quotient.CompClosure (pathsHomRel C)) f)))\n[PROOFSTEP]\nsimp [Category.comp_id, Category.id_comp, pathsHomRel]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (toQuotientPaths C \u22d9 quotientPathsTo C).map f \u226b\n ((fun X => Iso.refl ((toQuotientPaths C \u22d9 quotientPathsTo C).obj X)) Y\u271d).hom =\n ((fun X => Iso.refl ((toQuotientPaths C \u22d9 quotientPathsTo C).obj X)) X\u271d).hom \u226b (\ud835\udfed C).map f\n[PROOFSTEP]\nsimp [Quot.liftOn_mk]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : Quotient (pathsHomRel C)\n\u22a2 (quotientPathsTo C).map\n (NatTrans.app\n (NatIso.ofComponents fun X =>\n Quotient.casesOn (motive := fun t =>\n X = t \u2192 ((\ud835\udfed (Quotient (pathsHomRel C))).obj X \u2245 (quotientPathsTo C \u22d9 toQuotientPaths C).obj X)) X\n (fun as h => (_ : { as := as } = X) \u25b8 Iso.refl ((\ud835\udfed (Quotient (pathsHomRel C))).obj { as := as }))\n (_ : X = X)).hom\n X) \u226b\n NatTrans.app (NatIso.ofComponents fun X => Iso.refl ((toQuotientPaths C \u22d9 quotientPathsTo C).obj X)).hom\n ((quotientPathsTo C).obj X) =\n \ud835\udfd9 ((quotientPathsTo C).obj X)\n[PROOFSTEP]\ncases X\n[GOAL]\ncase mk\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nas\u271d : Paths C\n\u22a2 (quotientPathsTo C).map\n (NatTrans.app\n (NatIso.ofComponents fun X =>\n Quotient.casesOn (motive := fun t =>\n X = t \u2192 ((\ud835\udfed (Quotient (pathsHomRel C))).obj X \u2245 (quotientPathsTo C \u22d9 toQuotientPaths C).obj X)) X\n (fun as h => (_ : { as := as } = X) \u25b8 Iso.refl ((\ud835\udfed (Quotient (pathsHomRel C))).obj { as := as }))\n (_ : X = X)).hom\n { as := as\u271d }) \u226b\n NatTrans.app (NatIso.ofComponents fun X => Iso.refl ((toQuotientPaths C \u22d9 quotientPathsTo C).obj X)).hom\n ((quotientPathsTo C).obj { as := as\u271d }) =\n \ud835\udfd9 ((quotientPathsTo C).obj { as := as\u271d })\n[PROOFSTEP]\nsimp only [pathsHomRel, pathComposition_obj, pathComposition_map, Functor.id_obj, quotientPathsTo_obj, Functor.comp_obj,\n toQuotientPaths_obj_as, NatIso.ofComponents_hom_app, Iso.refl_hom, quotientPathsTo_map, Category.comp_id]\n[GOAL]\ncase mk\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nas\u271d : Paths C\n\u22a2 Quot.liftOn (\ud835\udfd9 { as := as\u271d }) (fun f => composePath f)\n (_ :\n \u2200 (a b : { as := as\u271d }.as \u27f6 ((toQuotientPaths C).obj as\u271d).as),\n Quotient.CompClosure (fun x x_1 p q => composePath p = composePath q) a b \u2192\n (fun f => composePath f) a = (fun f => composePath f) b) =\n \ud835\udfd9 as\u271d\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.PathCategory", "llama_tokens": 12303, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432182679956, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.4140485000456329}} {"text": "[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b9 : Ring M\ninst\u271d : Module M X\nP : M\nh : IsLprojection X P\nx : X\n\u22a2 \u2016x\u2016 = \u2016(1 - P) \u2022 x\u2016 + \u2016(1 - (1 - P)) \u2022 x\u2016\n[PROOFSTEP]\nrw [add_comm, sub_sub_cancel]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b9 : Ring M\ninst\u271d : Module M X\nP : M\nh : IsLprojection X P\nx : X\n\u22a2 \u2016x\u2016 = \u2016P \u2022 x\u2016 + \u2016(1 - P) \u2022 x\u2016\n[PROOFSTEP]\nexact h.Lnorm x\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\n\u22a2 Commute P Q\n[PROOFSTEP]\nhave PR_eq_RPR : \u2200 R : M, IsLprojection X R \u2192 P * R = R * P * R := fun R h\u2083 => by\n --porting note: Needed to fix function, which changes indent of following linesrefine\n @eq_of_smul_eq_smul _ X _ _ _ _ fun x => by\n rw [\u2190 norm_sub_eq_zero_iff]\n have e1 : \u2016R \u2022 x\u2016 \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 :=\n calc\n \u2016R \u2022 x\u2016 =\n \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016(R * R) \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 (1 - P) \u2022 R \u2022 x\u2016) :=\n by rw [h\u2081.Lnorm, h\u2083.Lnorm, h\u2083.Lnorm ((1 - P) \u2022 R \u2022 x), sub_smul 1 P, one_smul, smul_sub, mul_smul]\n _ =\n \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 +\n (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016((1 - R) * R) \u2022 x - (1 - R) \u2022 P \u2022 R \u2022 x\u2016) :=\n by rw [h\u2083.proj.eq, sub_smul 1 P, one_smul, smul_sub, mul_smul]\n _ = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016) := by\n rw [sub_mul, h\u2083.proj.eq, one_mul, sub_self, zero_smul, zero_sub, norm_neg]\n _ = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 := by abel\n _ \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 :=\n by\n rw [GE.ge]\n have := add_le_add_right (norm_le_insert' (R \u2022 x) (R \u2022 P \u2022 R \u2022 x)) (2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016)\n simpa only [mul_smul, sub_smul, one_smul] using this\n rw [GE.ge] at e1 \n nth_rewrite 2 [\u2190 add_zero \u2016R \u2022 x\u2016] at e1 \n rw [add_le_add_iff_left, two_smul, \u2190 two_mul] at e1 \n rw [le_antisymm_iff]\n refine' \u27e8_, norm_nonneg _\u27e9\n rwa [\u2190 mul_zero (2 : \u211d), mul_le_mul_left (show (0 : \u211d) < 2 by norm_num)] at e1 \n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\n\u22a2 P * R = R * P * R\n[PROOFSTEP]\nrefine\n @eq_of_smul_eq_smul _ X _ _ _ _ fun x => by\n rw [\u2190 norm_sub_eq_zero_iff]\n have e1 : \u2016R \u2022 x\u2016 \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 :=\n calc\n \u2016R \u2022 x\u2016 =\n \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016(R * R) \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 (1 - P) \u2022 R \u2022 x\u2016) :=\n by rw [h\u2081.Lnorm, h\u2083.Lnorm, h\u2083.Lnorm ((1 - P) \u2022 R \u2022 x), sub_smul 1 P, one_smul, smul_sub, mul_smul]\n _ =\n \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 +\n (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016((1 - R) * R) \u2022 x - (1 - R) \u2022 P \u2022 R \u2022 x\u2016) :=\n by rw [h\u2083.proj.eq, sub_smul 1 P, one_smul, smul_sub, mul_smul]\n _ = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016) := by\n rw [sub_mul, h\u2083.proj.eq, one_mul, sub_self, zero_smul, zero_sub, norm_neg]\n _ = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 := by abel\n _ \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 :=\n by\n rw [GE.ge]\n have := add_le_add_right (norm_le_insert' (R \u2022 x) (R \u2022 P \u2022 R \u2022 x)) (2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016)\n simpa only [mul_smul, sub_smul, one_smul] using this\n rw [GE.ge] at e1 \n nth_rewrite 2 [\u2190 add_zero \u2016R \u2022 x\u2016] at e1 \n rw [add_le_add_iff_left, two_smul, \u2190 two_mul] at e1 \n rw [le_antisymm_iff]\n refine' \u27e8_, norm_nonneg _\u27e9\n rwa [\u2190 mul_zero (2 : \u211d), mul_le_mul_left (show (0 : \u211d) < 2 by norm_num)] at e1 \n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 (P * R) \u2022 x = (R * P * R) \u2022 x\n[PROOFSTEP]\nrw [\u2190 norm_sub_eq_zero_iff]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 = 0\n[PROOFSTEP]\nhave e1 : \u2016R \u2022 x\u2016 \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 :=\n calc\n \u2016R \u2022 x\u2016 = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016(R * R) \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 (1 - P) \u2022 R \u2022 x\u2016) :=\n by rw [h\u2081.Lnorm, h\u2083.Lnorm, h\u2083.Lnorm ((1 - P) \u2022 R \u2022 x), sub_smul 1 P, one_smul, smul_sub, mul_smul]\n _ =\n \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 +\n (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016((1 - R) * R) \u2022 x - (1 - R) \u2022 P \u2022 R \u2022 x\u2016) :=\n by rw [h\u2083.proj.eq, sub_smul 1 P, one_smul, smul_sub, mul_smul]\n _ = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016) := by\n rw [sub_mul, h\u2083.proj.eq, one_mul, sub_self, zero_smul, zero_sub, norm_neg]\n _ = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 := by abel\n _ \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 :=\n by\n rw [GE.ge]\n have := add_le_add_right (norm_le_insert' (R \u2022 x) (R \u2022 P \u2022 R \u2022 x)) (2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016)\n simpa only [mul_smul, sub_smul, one_smul] using this\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016R \u2022 x\u2016 = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016(R * R) \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 (1 - P) \u2022 R \u2022 x\u2016)\n[PROOFSTEP]\nrw [h\u2081.Lnorm, h\u2083.Lnorm, h\u2083.Lnorm ((1 - P) \u2022 R \u2022 x), sub_smul 1 P, one_smul, smul_sub, mul_smul]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016(R * R) \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 (1 - P) \u2022 R \u2022 x\u2016) =\n \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016((1 - R) * R) \u2022 x - (1 - R) \u2022 P \u2022 R \u2022 x\u2016)\n[PROOFSTEP]\nrw [h\u2083.proj.eq, sub_smul 1 P, one_smul, smul_sub, mul_smul]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016((1 - R) * R) \u2022 x - (1 - R) \u2022 P \u2022 R \u2022 x\u2016) =\n \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016)\n[PROOFSTEP]\nrw [sub_mul, h\u2083.proj.eq, one_mul, sub_self, zero_smul, zero_sub, norm_neg]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016) =\n \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016\n[PROOFSTEP]\nabel\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016) =\n \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016\n[PROOFSTEP]\nabel\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016\n[PROOFSTEP]\nrw [GE.ge]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016\n[PROOFSTEP]\nhave := add_le_add_right (norm_le_insert' (R \u2022 x) (R \u2022 P \u2022 R \u2022 x)) (2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016)\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\nthis : \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 \u2264 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016\n\u22a2 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016\n[PROOFSTEP]\nsimpa only [mul_smul, sub_smul, one_smul] using this\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : \u2016R \u2022 x\u2016 \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 = 0\n[PROOFSTEP]\nrw [GE.ge] at e1 \n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 \u2016R \u2022 x\u2016\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 = 0\n[PROOFSTEP]\nnth_rewrite 2 [\u2190 add_zero \u2016R \u2022 x\u2016] at e1 \n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 \u2016R \u2022 x\u2016 + 0\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 = 0\n[PROOFSTEP]\nrw [add_le_add_iff_left, two_smul, \u2190 two_mul] at e1 \n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : 2 * \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 0\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 = 0\n[PROOFSTEP]\nrw [le_antisymm_iff]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : 2 * \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 0\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 0 \u2227 0 \u2264 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016\n[PROOFSTEP]\nrefine' \u27e8_, norm_nonneg _\u27e9\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : 2 * \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 0\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 0\n[PROOFSTEP]\nrwa [\u2190 mul_zero (2 : \u211d), mul_le_mul_left (show (0 : \u211d) < 2 by norm_num)] at e1 \n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : 2 * \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 2 * 0\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\n\u22a2 Commute P Q\n[PROOFSTEP]\nhave QP_eq_QPQ : Q * P = Q * P * Q :=\n by\n have e1 : P * (1 - Q) = P * (1 - Q) - (Q * P - Q * P * Q) :=\n calc\n P * (1 - Q) = (1 - Q) * P * (1 - Q) := by rw [PR_eq_RPR (1 - Q) h\u2082.Lcomplement]\n _ = P * (1 - Q) - (Q * P - Q * P * Q) := by noncomm_ring\n rwa [eq_sub_iff_add_eq, add_right_eq_self, sub_eq_zero] at e1 \n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\n\u22a2 Q * P = Q * P * Q\n[PROOFSTEP]\nhave e1 : P * (1 - Q) = P * (1 - Q) - (Q * P - Q * P * Q) :=\n calc\n P * (1 - Q) = (1 - Q) * P * (1 - Q) := by rw [PR_eq_RPR (1 - Q) h\u2082.Lcomplement]\n _ = P * (1 - Q) - (Q * P - Q * P * Q) := by noncomm_ring\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\n\u22a2 P * (1 - Q) = (1 - Q) * P * (1 - Q)\n[PROOFSTEP]\nrw [PR_eq_RPR (1 - Q) h\u2082.Lcomplement]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\n\u22a2 (1 - Q) * P * (1 - Q) = P * (1 - Q) - (Q * P - Q * P * Q)\n[PROOFSTEP]\nnoncomm_ring\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\ne1 : P * (1 - Q) = P * (1 - Q) - (Q * P - Q * P * Q)\n\u22a2 Q * P = Q * P * Q\n[PROOFSTEP]\nrwa [eq_sub_iff_add_eq, add_right_eq_self, sub_eq_zero] at e1 \n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\nQP_eq_QPQ : Q * P = Q * P * Q\n\u22a2 Commute P Q\n[PROOFSTEP]\nshow P * Q = Q * P\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\nQP_eq_QPQ : Q * P = Q * P * Q\n\u22a2 P * Q = Q * P\n[PROOFSTEP]\nrw [QP_eq_QPQ, PR_eq_RPR Q h\u2082]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\n\u22a2 IsLprojection X (P * Q)\n[PROOFSTEP]\nrefine' \u27e8IsIdempotentElem.mul_of_commute (h\u2081.commute h\u2082) h\u2081.proj h\u2082.proj, _\u27e9\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\n\u22a2 \u2200 (x : X), \u2016x\u2016 = \u2016(P * Q) \u2022 x\u2016 + \u2016(1 - P * Q) \u2022 x\u2016\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nx : X\n\u22a2 \u2016x\u2016 = \u2016(P * Q) \u2022 x\u2016 + \u2016(1 - P * Q) \u2022 x\u2016\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nx : X\n\u22a2 \u2016x\u2016 \u2264 \u2016(P * Q) \u2022 x\u2016 + \u2016(1 - P * Q) \u2022 x\u2016\n[PROOFSTEP]\ncalc\n \u2016x\u2016 = \u2016(P * Q) \u2022 x + (x - (P * Q) \u2022 x)\u2016 := by rw [add_sub_cancel'_right ((P * Q) \u2022 x) x]\n _ \u2264 \u2016(P * Q) \u2022 x\u2016 + \u2016x - (P * Q) \u2022 x\u2016 := by apply norm_add_le\n _ = \u2016(P * Q) \u2022 x\u2016 + \u2016(1 - P * Q) \u2022 x\u2016 := by rw [sub_smul, one_smul]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nx : X\n\u22a2 \u2016x\u2016 = \u2016(P * Q) \u2022 x + (x - (P * Q) \u2022 x)\u2016\n[PROOFSTEP]\nrw [add_sub_cancel'_right ((P * Q) \u2022 x) x]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nx : X\n\u22a2 \u2016(P * Q) \u2022 x + (x - (P * Q) \u2022 x)\u2016 \u2264 \u2016(P * Q) \u2022 x\u2016 + \u2016x - (P * Q) \u2022 x\u2016\n[PROOFSTEP]\napply norm_add_le\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nx : X\n\u22a2 \u2016(P * Q) \u2022 x\u2016 + \u2016x - (P * Q) \u2022 x\u2016 = \u2016(P * Q) \u2022 x\u2016 + \u2016(1 - P * Q) \u2022 x\u2016\n[PROOFSTEP]\nrw [sub_smul, one_smul]\n[GOAL]\ncase refine'_2\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nx : X\n\u22a2 \u2016(P * Q) \u2022 x\u2016 + \u2016(1 - P * Q) \u2022 x\u2016 \u2264 \u2016x\u2016\n[PROOFSTEP]\ncalc\n \u2016x\u2016 = \u2016P \u2022 Q \u2022 x\u2016 + (\u2016Q \u2022 x - P \u2022 Q \u2022 x\u2016 + \u2016x - Q \u2022 x\u2016) := by\n rw [h\u2082.Lnorm x, h\u2081.Lnorm (Q \u2022 x), sub_smul, one_smul, sub_smul, one_smul, add_assoc]\n _ \u2265 \u2016P \u2022 Q \u2022 x\u2016 + \u2016Q \u2022 x - P \u2022 Q \u2022 x + (x - Q \u2022 x)\u2016 :=\n ((add_le_add_iff_left \u2016P \u2022 Q \u2022 x\u2016).mpr (norm_add_le (Q \u2022 x - P \u2022 Q \u2022 x) (x - Q \u2022 x)))\n _ = \u2016(P * Q) \u2022 x\u2016 + \u2016(1 - P * Q) \u2022 x\u2016 := by rw [sub_add_sub_cancel', sub_smul, one_smul, mul_smul]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nx : X\n\u22a2 \u2016x\u2016 = \u2016P \u2022 Q \u2022 x\u2016 + (\u2016Q \u2022 x - P \u2022 Q \u2022 x\u2016 + \u2016x - Q \u2022 x\u2016)\n[PROOFSTEP]\nrw [h\u2082.Lnorm x, h\u2081.Lnorm (Q \u2022 x), sub_smul, one_smul, sub_smul, one_smul, add_assoc]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nx : X\n\u22a2 \u2016P \u2022 Q \u2022 x\u2016 + \u2016Q \u2022 x - P \u2022 Q \u2022 x + (x - Q \u2022 x)\u2016 = \u2016(P * Q) \u2022 x\u2016 + \u2016(1 - P * Q) \u2022 x\u2016\n[PROOFSTEP]\nrw [sub_add_sub_cancel', sub_smul, one_smul, mul_smul]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\n\u22a2 IsLprojection X (P + Q - P * Q)\n[PROOFSTEP]\nconvert (Lcomplement_iff _).mp (h\u2081.Lcomplement.mul h\u2082.Lcomplement) using 1\n[GOAL]\ncase h.e'_6\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\n\u22a2 P + Q - P * Q = 1 - (1 - P) * (1 - Q)\n[PROOFSTEP]\nnoncomm_ring\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP : { P // IsLprojection X P }\n\u22a2 P \u2264 P\n[PROOFSTEP]\nsimpa only [coe_inf, \u2190 sq] using P.prop.proj.eq.symm\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\nh\u2081 : P \u2264 Q\nh\u2082 : Q \u2264 R\n\u22a2 P \u2264 R\n[PROOFSTEP]\nsimp only [coe_inf] at h\u2081 h\u2082 \u22a2\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\nh\u2081 : \u2191P = \u2191P * \u2191Q\nh\u2082 : \u2191Q = \u2191Q * \u2191R\n\u22a2 \u2191P = \u2191P * \u2191R\n[PROOFSTEP]\nrw [h\u2081, mul_assoc, \u2190 h\u2082]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : { P // IsLprojection X P }\nh\u2081 : P \u2264 Q\nh\u2082 : Q \u2264 P\n\u22a2 \u2191P = \u2191Q\n[PROOFSTEP]\nconvert (P.prop.commute Q.prop).eq\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b9 : Ring M\ninst\u271d : Module M X\n\u22a2 IsIdempotentElem 0\n[PROOFSTEP]\nrw [IsIdempotentElem, zero_mul]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b9 : Ring M\ninst\u271d : Module M X\nx : X\n\u22a2 \u2016x\u2016 = \u20160 \u2022 x\u2016 + \u2016(1 - 0) \u2022 x\u2016\n[PROOFSTEP]\nsimp only [zero_smul, norm_zero, sub_zero, one_smul, zero_add]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b9 : Ring M\ninst\u271d : Module M X\nP : { P // IsLprojection X P }\nQ : M\n\u22a2 \u2191P\u1d9c * Q = Q - \u2191P * Q\n[PROOFSTEP]\nrw [coe_compl, sub_mul, one_mul]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b9 : Ring M\ninst\u271d : Module M X\nP : { P // IsLprojection X P }\n\u22a2 \u2191P * \u2191P\u1d9c = 0\n[PROOFSTEP]\nrw [coe_compl, mul_sub, mul_one, P.prop.proj.eq, sub_self]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n\u22a2 (\u2191P + \u2191P\u1d9c * \u2191R) * (\u2191P + \u2191Q * \u2191R * \u2191P\u1d9c) = \u2191P + \u2191Q * \u2191R * \u2191P\u1d9c\n[PROOFSTEP]\nrw [add_mul, mul_add, mul_add, (mul_assoc _ (R : M) (\u2191Q * \u2191R * \u2191P\u1d9c)), \u2190 mul_assoc (R : M) (\u2191Q * \u2191R) _, \u2190 coe_inf Q,\n (P\u1d9c.prop.commute R.prop).eq, ((Q \u2293 R).prop.commute P\u1d9c.prop).eq, (R.prop.commute (Q \u2293 R).prop).eq, coe_inf Q,\n mul_assoc (Q : M), \u2190 mul_assoc, mul_assoc (R : M), (P\u1d9c.prop.commute P.prop).eq, mul_compl_self, zero_mul, mul_zero,\n zero_add, add_zero, \u2190 mul_assoc, P.prop.proj.eq, R.prop.proj.eq, \u2190 coe_inf Q, mul_assoc,\n ((Q \u2293 R).prop.commute P\u1d9c.prop).eq, \u2190 mul_assoc, P\u1d9c.prop.proj.eq]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : { P // IsLprojection X P }\n\u22a2 P \u2264 P \u2294 Q\n[PROOFSTEP]\nrw [le_def, coe_inf, coe_sup, \u2190 add_sub, mul_add, mul_sub, \u2190 mul_assoc, P.prop.proj.eq, sub_self, add_zero]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : { P // IsLprojection X P }\n\u22a2 Q \u2264 P \u2294 Q\n[PROOFSTEP]\nrw [le_def, coe_inf, coe_sup, \u2190 add_sub, mul_add, mul_sub, (P.prop.commute Q.prop).eq, \u2190 mul_assoc, Q.prop.proj.eq,\n add_sub_cancel'_right]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n\u22a2 P \u2264 R \u2192 Q \u2264 R \u2192 P \u2294 Q \u2264 R\n[PROOFSTEP]\nrw [le_def, le_def, le_def, coe_inf, coe_inf, coe_sup, coe_inf, coe_sup, \u2190 add_sub, add_mul, sub_mul, mul_assoc]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n\u22a2 \u2191P = \u2191P * \u2191R \u2192 \u2191Q = \u2191Q * \u2191R \u2192 \u2191P + (\u2191Q - \u2191P * \u2191Q) = \u2191P * \u2191R + (\u2191Q * \u2191R - \u2191P * (\u2191Q * \u2191R))\n[PROOFSTEP]\nintro h\u2081 h\u2082\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\nh\u2081 : \u2191P = \u2191P * \u2191R\nh\u2082 : \u2191Q = \u2191Q * \u2191R\n\u22a2 \u2191P + (\u2191Q - \u2191P * \u2191Q) = \u2191P * \u2191R + (\u2191Q * \u2191R - \u2191P * (\u2191Q * \u2191R))\n[PROOFSTEP]\nrw [\u2190 h\u2082, \u2190 h\u2081]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : { P // IsLprojection X P }\n\u22a2 P \u2293 Q \u2264 P\n[PROOFSTEP]\nrw [le_def, coe_inf, coe_inf, coe_inf, mul_assoc, (Q.prop.commute P.prop).eq, \u2190 mul_assoc, P.prop.proj.eq]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : { P // IsLprojection X P }\n\u22a2 P \u2293 Q \u2264 Q\n[PROOFSTEP]\nrw [le_def, coe_inf, coe_inf, coe_inf, mul_assoc, Q.prop.proj.eq]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n\u22a2 P \u2264 Q \u2192 P \u2264 R \u2192 P \u2264 Q \u2293 R\n[PROOFSTEP]\nrw [le_def, le_def, le_def, coe_inf, coe_inf, coe_inf, coe_inf, \u2190 mul_assoc]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n\u22a2 \u2191P = \u2191P * \u2191Q \u2192 \u2191P = \u2191P * \u2191R \u2192 \u2191P = \u2191P * \u2191Q * \u2191R\n[PROOFSTEP]\nintro h\u2081 h\u2082\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\nh\u2081 : \u2191P = \u2191P * \u2191Q\nh\u2082 : \u2191P = \u2191P * \u2191R\n\u22a2 \u2191P = \u2191P * \u2191Q * \u2191R\n[PROOFSTEP]\nrw [\u2190 h\u2081, \u2190 h\u2082]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n\u22a2 (P \u2294 Q) \u2293 (P \u2294 R) \u2264 P \u2294 Q \u2293 R\n[PROOFSTEP]\nhave e\u2081 : \u2191((P \u2294 Q) \u2293 (P \u2294 R)) = \u2191P + \u2191Q * (R : M) * \u2191P\u1d9c := by\n rw [coe_inf, coe_sup, coe_sup, \u2190 add_sub, \u2190 add_sub, \u2190 compl_mul, \u2190 compl_mul, add_mul, mul_add,\n (P\u1d9c.prop.commute Q.prop).eq, mul_add, \u2190 mul_assoc, mul_assoc (Q : M), (P\u1d9c.prop.commute P.prop).eq, mul_compl_self,\n zero_mul, mul_zero, zero_add, add_zero, \u2190 mul_assoc, mul_assoc (Q : M), P.prop.proj.eq, P\u1d9c.prop.proj.eq, mul_assoc,\n (P\u1d9c.prop.commute R.prop).eq, \u2190 mul_assoc]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n\u22a2 \u2191((P \u2294 Q) \u2293 (P \u2294 R)) = \u2191P + \u2191Q * \u2191R * \u2191P\u1d9c\n[PROOFSTEP]\nrw [coe_inf, coe_sup, coe_sup, \u2190 add_sub, \u2190 add_sub, \u2190 compl_mul, \u2190 compl_mul, add_mul, mul_add,\n (P\u1d9c.prop.commute Q.prop).eq, mul_add, \u2190 mul_assoc, mul_assoc (Q : M), (P\u1d9c.prop.commute P.prop).eq, mul_compl_self,\n zero_mul, mul_zero, zero_add, add_zero, \u2190 mul_assoc, mul_assoc (Q : M), P.prop.proj.eq, P\u1d9c.prop.proj.eq, mul_assoc,\n (P\u1d9c.prop.commute R.prop).eq, \u2190 mul_assoc]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\ne\u2081 : \u2191((P \u2294 Q) \u2293 (P \u2294 R)) = \u2191P + \u2191Q * \u2191R * \u2191P\u1d9c\n\u22a2 (P \u2294 Q) \u2293 (P \u2294 R) \u2264 P \u2294 Q \u2293 R\n[PROOFSTEP]\nhave e\u2082 : \u2191((P \u2294 Q) \u2293 (P \u2294 R)) * \u2191(P \u2294 Q \u2293 R) = (P : M) + \u2191Q * \u2191R * \u2191P\u1d9c := by\n rw [coe_inf, coe_sup, coe_sup, coe_sup, \u2190 add_sub, \u2190 add_sub, \u2190 add_sub, \u2190 compl_mul, \u2190 compl_mul, \u2190 compl_mul,\n (P\u1d9c.prop.commute (Q \u2293 R).prop).eq, coe_inf, mul_assoc, distrib_lattice_lemma, (Q.prop.commute R.prop).eq,\n distrib_lattice_lemma]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\ne\u2081 : \u2191((P \u2294 Q) \u2293 (P \u2294 R)) = \u2191P + \u2191Q * \u2191R * \u2191P\u1d9c\n\u22a2 \u2191((P \u2294 Q) \u2293 (P \u2294 R)) * \u2191(P \u2294 Q \u2293 R) = \u2191P + \u2191Q * \u2191R * \u2191P\u1d9c\n[PROOFSTEP]\nrw [coe_inf, coe_sup, coe_sup, coe_sup, \u2190 add_sub, \u2190 add_sub, \u2190 add_sub, \u2190 compl_mul, \u2190 compl_mul, \u2190 compl_mul,\n (P\u1d9c.prop.commute (Q \u2293 R).prop).eq, coe_inf, mul_assoc, distrib_lattice_lemma, (Q.prop.commute R.prop).eq,\n distrib_lattice_lemma]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\ne\u2081 : \u2191((P \u2294 Q) \u2293 (P \u2294 R)) = \u2191P + \u2191Q * \u2191R * \u2191P\u1d9c\ne\u2082 : \u2191((P \u2294 Q) \u2293 (P \u2294 R)) * \u2191(P \u2294 Q \u2293 R) = \u2191P + \u2191Q * \u2191R * \u2191P\u1d9c\n\u22a2 (P \u2294 Q) \u2293 (P \u2294 R) \u2264 P \u2294 Q \u2293 R\n[PROOFSTEP]\nrw [le_def, e\u2081, coe_inf, e\u2082]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nsrc\u271d\u00b2 : HasCompl { f // IsLprojection X f } := hasCompl\nsrc\u271d\u00b9 : SDiff { P // IsLprojection X P } := sdiff\nsrc\u271d : BoundedOrder { P // IsLprojection X P } := boundedOrder\nP : { P // IsLprojection X P }\n\u22a2 \u2191(P \u2293 P\u1d9c) = \u2191\u22a5\n[PROOFSTEP]\nrw [coe_inf, coe_compl, coe_bot, \u2190 coe_compl, mul_compl_self]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nsrc\u271d\u00b2 : HasCompl { f // IsLprojection X f } := hasCompl\nsrc\u271d\u00b9 : SDiff { P // IsLprojection X P } := sdiff\nsrc\u271d : BoundedOrder { P // IsLprojection X P } := boundedOrder\nP : { P // IsLprojection X P }\n\u22a2 \u2191\u22a4 = \u2191(P \u2294 P\u1d9c)\n[PROOFSTEP]\nrw [coe_top, coe_sup, coe_compl, add_sub_cancel'_right, \u2190 coe_compl, mul_compl_self, sub_zero]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nsrc\u271d\u00b2 : HasCompl { f // IsLprojection X f } := hasCompl\nsrc\u271d\u00b9 : SDiff { P // IsLprojection X P } := sdiff\nsrc\u271d : BoundedOrder { P // IsLprojection X P } := boundedOrder\nP Q : { P // IsLprojection X P }\n\u22a2 \u2191(P \\ Q) = \u2191(P \u2293 Q\u1d9c)\n[PROOFSTEP]\nrw [coe_sdiff, \u2190 coe_compl, coe_inf]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.MStructure", "llama_tokens": 14815, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432182679956, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.41404850004563276}} {"text": "[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nhg : HasDerivAtFilter g\u2081 g\u2081' (h x) L'\nhh : HasDerivAtFilter h h' x L\nhL : Tendsto h L L'\n\u22a2 HasDerivAtFilter (g\u2081 \u2218 h) (h' \u2022 g\u2081') x L\n[PROOFSTEP]\nsimpa using ((hg.restrictScalars \ud835\udd5c).comp x hh hL).hasDerivAtFilter\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nhg : HasStrictDerivAt g\u2081 g\u2081' (h x)\nhh : HasStrictDerivAt h h' x\n\u22a2 HasStrictDerivAt (g\u2081 \u2218 h) (h' \u2022 g\u2081') x\n[PROOFSTEP]\nsimpa using ((hg.restrictScalars \ud835\udd5c).comp x hh).hasStrictDerivAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nf : E \u2192 \ud835\udd5c'\nf' : E \u2192L[\ud835\udd5c] \ud835\udd5c'\nx : E\nL'' : Filter E\nhh\u2082 : HasDerivAtFilter h\u2082 h\u2082' (f x) L'\nhf : HasFDerivAtFilter f f' x L''\nhL : Tendsto f L'' L'\n\u22a2 HasFDerivAtFilter (h\u2082 \u2218 f) (h\u2082' \u2022 f') x L''\n[PROOFSTEP]\nconvert (hh\u2082.restrictScalars \ud835\udd5c).comp x hf hL\n[GOAL]\ncase h.e'_10.h.e\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nf : E \u2192 \ud835\udd5c'\nf' : E \u2192L[\ud835\udd5c] \ud835\udd5c'\nx : E\nL'' : Filter E\nhh\u2082 : HasDerivAtFilter h\u2082 h\u2082' (f x) L'\nhf : HasFDerivAtFilter f f' x L''\nhL : Tendsto f L'' L'\n\u22a2 HSMul.hSMul h\u2082' = ContinuousLinearMap.comp (ContinuousLinearMap.restrictScalars \ud835\udd5c (smulRight 1 h\u2082'))\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_10.h.e.h.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx\u271d\u00b2 : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nf : E \u2192 \ud835\udd5c'\nf' : E \u2192L[\ud835\udd5c] \ud835\udd5c'\nx\u271d\u00b9 : E\nL'' : Filter E\nhh\u2082 : HasDerivAtFilter h\u2082 h\u2082' (f x\u271d\u00b9) L'\nhf : HasFDerivAtFilter f f' x\u271d\u00b9 L''\nhL : Tendsto f L'' L'\nx : E \u2192L[\ud835\udd5c] \ud835\udd5c'\nx\u271d : E\n\u22a2 \u2191(h\u2082' \u2022 x) x\u271d = \u2191(ContinuousLinearMap.comp (ContinuousLinearMap.restrictScalars \ud835\udd5c (smulRight 1 h\u2082')) x) x\u271d\n[PROOFSTEP]\nsimp [mul_comm]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nf : E \u2192 \ud835\udd5c'\nf' : E \u2192L[\ud835\udd5c] \ud835\udd5c'\nx : E\nhh : HasStrictDerivAt h\u2082 h\u2082' (f x)\nhf : HasStrictFDerivAt f f' x\n\u22a2 HasStrictFDerivAt (h\u2082 \u2218 f) (h\u2082' \u2022 f') x\n[PROOFSTEP]\nrw [HasStrictDerivAt] at hh \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nf : E \u2192 \ud835\udd5c'\nf' : E \u2192L[\ud835\udd5c] \ud835\udd5c'\nx : E\nhh : HasStrictFDerivAt h\u2082 (smulRight 1 h\u2082') (f x)\nhf : HasStrictFDerivAt f f' x\n\u22a2 HasStrictFDerivAt (h\u2082 \u2218 f) (h\u2082' \u2022 f') x\n[PROOFSTEP]\nconvert (hh.restrictScalars \ud835\udd5c).comp x hf\n[GOAL]\ncase h.e'_10.h.e\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nf : E \u2192 \ud835\udd5c'\nf' : E \u2192L[\ud835\udd5c] \ud835\udd5c'\nx : E\nhh : HasStrictFDerivAt h\u2082 (smulRight 1 h\u2082') (f x)\nhf : HasStrictFDerivAt f f' x\n\u22a2 HSMul.hSMul h\u2082' = ContinuousLinearMap.comp (ContinuousLinearMap.restrictScalars \ud835\udd5c (smulRight 1 h\u2082'))\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_10.h.e.h.h\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx\u271d\u00b2 : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nf : E \u2192 \ud835\udd5c'\nf' : E \u2192L[\ud835\udd5c] \ud835\udd5c'\nx\u271d\u00b9 : E\nhh : HasStrictFDerivAt h\u2082 (smulRight 1 h\u2082') (f x\u271d\u00b9)\nhf : HasStrictFDerivAt f f' x\u271d\u00b9\nx : E \u2192L[\ud835\udd5c] \ud835\udd5c'\nx\u271d : E\n\u22a2 \u2191(h\u2082' \u2022 x) x\u271d = \u2191(ContinuousLinearMap.comp (ContinuousLinearMap.restrictScalars \ud835\udd5c (smulRight 1 h\u2082')) x) x\u271d\n[PROOFSTEP]\nsimp [mul_comm]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nhh\u2082 : HasDerivAtFilter h\u2082 h\u2082' (h x) L'\nhh : HasDerivAtFilter h h' x L\nhL : Tendsto h L L'\n\u22a2 HasDerivAtFilter (h\u2082 \u2218 h) (h\u2082' * h') x L\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nhh\u2082 : HasDerivAtFilter h\u2082 h\u2082' (h x) L'\nhh : HasDerivAtFilter h h' x L\nhL : Tendsto h L L'\n\u22a2 HasDerivAtFilter (h\u2082 \u2218 h) (h' * h\u2082') x L\n[PROOFSTEP]\nexact hh\u2082.scomp x hh hL\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nhh\u2082 : HasDerivWithinAt h\u2082 h\u2082' s' (h x)\nhh : HasDerivWithinAt h h' s x\nhst : MapsTo h s s'\n\u22a2 HasDerivWithinAt (h\u2082 \u2218 h) (h\u2082' * h') s x\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nhh\u2082 : HasDerivWithinAt h\u2082 h\u2082' s' (h x)\nhh : HasDerivWithinAt h h' s x\nhst : MapsTo h s s'\n\u22a2 HasDerivWithinAt (h\u2082 \u2218 h) (h' * h\u2082') s x\n[PROOFSTEP]\nexact hh\u2082.scomp x hh hst\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nhh\u2082 : HasStrictDerivAt h\u2082 h\u2082' (h x)\nhh : HasStrictDerivAt h h' x\n\u22a2 HasStrictDerivAt (h\u2082 \u2218 h) (h\u2082' * h') x\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nhh\u2082 : HasStrictDerivAt h\u2082 h\u2082' (h x)\nhh : HasStrictDerivAt h h' x\n\u22a2 HasStrictDerivAt (h\u2082 \u2218 h) (h' * h\u2082') x\n[PROOFSTEP]\nexact hh\u2082.scomp x hh\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nf : \ud835\udd5c \u2192 \ud835\udd5c\nf' : \ud835\udd5c\nhf : HasDerivAtFilter f f' x L\nhL : Tendsto f L L\nhx : f x = x\nn : \u2115\n\u22a2 HasDerivAtFilter f^[n] (f' ^ n) x L\n[PROOFSTEP]\nhave := hf.iterate hL hx n\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nf : \ud835\udd5c \u2192 \ud835\udd5c\nf' : \ud835\udd5c\nhf : HasDerivAtFilter f f' x L\nhL : Tendsto f L L\nhx : f x = x\nn : \u2115\nthis : HasFDerivAtFilter f^[n] (smulRight 1 f' ^ n) x L\n\u22a2 HasDerivAtFilter f^[n] (f' ^ n) x L\n[PROOFSTEP]\nrwa [ContinuousLinearMap.smulRight_one_pow] at this \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nf : \ud835\udd5c \u2192 \ud835\udd5c\nf' : \ud835\udd5c\nhf : HasDerivAt f f' x\nhx : f x = x\nn : \u2115\nthis : Tendsto f (\ud835\udcdd x) (\ud835\udcdd (f x))\n\u22a2 Tendsto f (\ud835\udcdd x) (\ud835\udcdd x)\n[PROOFSTEP]\nrwa [hx] at this \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nf : \ud835\udd5c \u2192 \ud835\udd5c\nf' : \ud835\udd5c\nhf : HasDerivWithinAt f f' s x\nhx : f x = x\nhs : MapsTo f s s\nn : \u2115\n\u22a2 HasDerivWithinAt f^[n] (f' ^ n) s x\n[PROOFSTEP]\nhave := HasFDerivWithinAt.iterate hf hx hs n\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nf : \ud835\udd5c \u2192 \ud835\udd5c\nf' : \ud835\udd5c\nhf : HasDerivWithinAt f f' s x\nhx : f x = x\nhs : MapsTo f s s\nn : \u2115\nthis : HasFDerivWithinAt f^[n] (smulRight 1 f' ^ n) s x\n\u22a2 HasDerivWithinAt f^[n] (f' ^ n) s x\n[PROOFSTEP]\nrwa [ContinuousLinearMap.smulRight_one_pow] at this \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nf : \ud835\udd5c \u2192 \ud835\udd5c\nf' : \ud835\udd5c\nhf : HasStrictDerivAt f f' x\nhx : f x = x\nn : \u2115\n\u22a2 HasStrictDerivAt f^[n] (f' ^ n) x\n[PROOFSTEP]\nhave := hf.iterate hx n\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\nf : \ud835\udd5c \u2192 \ud835\udd5c\nf' : \ud835\udd5c\nhf : HasStrictDerivAt f f' x\nhx : f x = x\nn : \u2115\nthis : HasStrictFDerivAt f^[n] (smulRight 1 f' ^ n) x\n\u22a2 HasStrictDerivAt f^[n] (f' ^ n) x\n[PROOFSTEP]\nrwa [ContinuousLinearMap.smulRight_one_pow] at this \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t\u271d : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nl : F \u2192 E\nl' : F \u2192L[\ud835\udd5c] E\nt : Set F\nhl : HasFDerivWithinAt l l' t (f x)\nhf : HasDerivWithinAt f f' s x\nhst : MapsTo f s t\n\u22a2 HasDerivWithinAt (l \u2218 f) (\u2191l' f') s x\n[PROOFSTEP]\nsimpa only [one_apply, one_smul, smulRight_apply, coe_comp', (\u00b7 \u2218 \u00b7)] using\n (hl.comp x hf.hasFDerivWithinAt hst).hasDerivWithinAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nl : F \u2192 E\nl' : F \u2192L[\ud835\udd5c] E\nhl : HasStrictFDerivAt l l' (f x)\nhf : HasStrictDerivAt f f' x\n\u22a2 HasStrictDerivAt (l \u2218 f) (\u2191l' f') x\n[PROOFSTEP]\nsimpa only [one_apply, one_smul, smulRight_apply, coe_comp', (\u00b7 \u2218 \u00b7)] using\n (hl.comp x hf.hasStrictFDerivAt).hasStrictDerivAt\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.Deriv.Comp", "llama_tokens": 9388, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4140484931784741}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g\u271d : E \u2192 F\nf' f\u2080' f\u2081' g'\u271d e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\ng : F \u2192 G\ng' : F \u2192L[\ud835\udd5c] G\nL' : Filter F\nhg : HasFDerivAtFilter g g' (f x) L'\nhf : HasFDerivAtFilter f f' x L\nhL : Tendsto f L L'\n\u22a2 HasFDerivAtFilter (g \u2218 f) (ContinuousLinearMap.comp g' f') x L\n[PROOFSTEP]\nlet eq\u2081 := (g'.isBigO_comp _ _).trans_isLittleO hf\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g\u271d : E \u2192 F\nf' f\u2080' f\u2081' g'\u271d e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\ng : F \u2192 G\ng' : F \u2192L[\ud835\udd5c] G\nL' : Filter F\nhg : HasFDerivAtFilter g g' (f x) L'\nhf : HasFDerivAtFilter f f' x L\nhL : Tendsto f L L'\neq\u2081 : (fun x' => \u2191g' (f x' - f x - \u2191f' (x' - x))) =o[L] fun x' => x' - x :=\n IsBigO.trans_isLittleO (isBigO_comp g' (fun x' => f x' - f x - \u2191f' (x' - x)) L) hf\n\u22a2 HasFDerivAtFilter (g \u2218 f) (ContinuousLinearMap.comp g' f') x L\n[PROOFSTEP]\nlet eq\u2082 := (hg.comp_tendsto hL).trans_isBigO hf.isBigO_sub\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g\u271d : E \u2192 F\nf' f\u2080' f\u2081' g'\u271d e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\ng : F \u2192 G\ng' : F \u2192L[\ud835\udd5c] G\nL' : Filter F\nhg : HasFDerivAtFilter g g' (f x) L'\nhf : HasFDerivAtFilter f f' x L\nhL : Tendsto f L L'\neq\u2081 : (fun x' => \u2191g' (f x' - f x - \u2191f' (x' - x))) =o[L] fun x' => x' - x :=\n IsBigO.trans_isLittleO (isBigO_comp g' (fun x' => f x' - f x - \u2191f' (x' - x)) L) hf\neq\u2082 : ((fun x' => g x' - g (f x) - \u2191g' (x' - f x)) \u2218 f) =o[L] fun x' => x' - x :=\n IsLittleO.trans_isBigO (IsLittleO.comp_tendsto hg hL) (isBigO_sub hf)\n\u22a2 HasFDerivAtFilter (g \u2218 f) (ContinuousLinearMap.comp g' f') x L\n[PROOFSTEP]\nrefine' eq\u2082.triangle (eq\u2081.congr_left fun x' => _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g\u271d : E \u2192 F\nf' f\u2080' f\u2081' g'\u271d e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\ng : F \u2192 G\ng' : F \u2192L[\ud835\udd5c] G\nL' : Filter F\nhg : HasFDerivAtFilter g g' (f x) L'\nhf : HasFDerivAtFilter f f' x L\nhL : Tendsto f L L'\neq\u2081 : (fun x' => \u2191g' (f x' - f x - \u2191f' (x' - x))) =o[L] fun x' => x' - x :=\n IsBigO.trans_isLittleO (isBigO_comp g' (fun x' => f x' - f x - \u2191f' (x' - x)) L) hf\neq\u2082 : ((fun x' => g x' - g (f x) - \u2191g' (x' - f x)) \u2218 f) =o[L] fun x' => x' - x :=\n IsLittleO.trans_isBigO (IsLittleO.comp_tendsto hg hL) (isBigO_sub hf)\nx' : E\n\u22a2 \u2191g' (f x' - f x - \u2191f' (x' - x)) = \u2191g' (f x' - f x) - \u2191(ContinuousLinearMap.comp g' f') (x' - x)\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t\u271d : Set E\nL L\u2081 L\u2082 : Filter E\ng : F \u2192 G\nf : E \u2192 F\nx : E\ny : F\ns : Set E\nt : Set F\nhg : DifferentiableWithinAt \ud835\udd5c g t y\nhf : DifferentiableWithinAt \ud835\udd5c f s x\nh : MapsTo f s t\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\nhy : f x = y\nv : E\n\u22a2 \u2191(fderivWithin \ud835\udd5c g t y) (\u2191(fderivWithin \ud835\udd5c f s x) v) = \u2191(fderivWithin \ud835\udd5c (g \u2218 f) s x) v\n[PROOFSTEP]\nsubst y\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t\u271d : Set E\nL L\u2081 L\u2082 : Filter E\ng : F \u2192 G\nf : E \u2192 F\nx : E\ns : Set E\nt : Set F\nhf : DifferentiableWithinAt \ud835\udd5c f s x\nh : MapsTo f s t\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\nv : E\nhg : DifferentiableWithinAt \ud835\udd5c g t (f x)\n\u22a2 \u2191(fderivWithin \ud835\udd5c g t (f x)) (\u2191(fderivWithin \ud835\udd5c f s x) v) = \u2191(fderivWithin \ud835\udd5c (g \u2218 f) s x) v\n[PROOFSTEP]\nrw [fderivWithin.comp x hg hf h hxs]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g\u271d : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns\u271d t\u271d : Set E\nL L\u2081 L\u2082 : Filter E\ng : F \u2192 G\nf : E \u2192 F\nx : E\ns : Set E\nt : Set F\nhf : DifferentiableWithinAt \ud835\udd5c f s x\nh : MapsTo f s t\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\nv : E\nhg : DifferentiableWithinAt \ud835\udd5c g t (f x)\n\u22a2 \u2191(fderivWithin \ud835\udd5c g t (f x)) (\u2191(fderivWithin \ud835\udd5c f s x) v) = \u2191(comp (fderivWithin \ud835\udd5c g t (f x)) (fderivWithin \ud835\udd5c f s x)) v\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g\u271d : E \u2192 F\nf' f\u2080' f\u2081' g'\u271d e : E \u2192L[\ud835\udd5c] F\nx : E\ns t\u271d : Set E\nL L\u2081 L\u2082 : Filter E\ng' : G \u2192 G'\ng : F \u2192 G\nt : Set F\nu : Set G\ny : F\ny' : G\nhg' : DifferentiableWithinAt \ud835\udd5c g' u y'\nhg : DifferentiableWithinAt \ud835\udd5c g t y\nhf : DifferentiableWithinAt \ud835\udd5c f s x\nh2g : MapsTo g t u\nh2f : MapsTo f s t\nh3g : g y = y'\nh3f : f x = y\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 fderivWithin \ud835\udd5c (g' \u2218 g \u2218 f) s x =\n ContinuousLinearMap.comp (fderivWithin \ud835\udd5c g' u y')\n (ContinuousLinearMap.comp (fderivWithin \ud835\udd5c g t y) (fderivWithin \ud835\udd5c f s x))\n[PROOFSTEP]\nsubsts h3g h3f\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g\u271d : E \u2192 F\nf' f\u2080' f\u2081' g'\u271d e : E \u2192L[\ud835\udd5c] F\nx : E\ns t\u271d : Set E\nL L\u2081 L\u2082 : Filter E\ng' : G \u2192 G'\ng : F \u2192 G\nt : Set F\nu : Set G\nhf : DifferentiableWithinAt \ud835\udd5c f s x\nh2g : MapsTo g t u\nh2f : MapsTo f s t\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\nhg : DifferentiableWithinAt \ud835\udd5c g t (f x)\nhg' : DifferentiableWithinAt \ud835\udd5c g' u (g (f x))\n\u22a2 fderivWithin \ud835\udd5c (g' \u2218 g \u2218 f) s x =\n ContinuousLinearMap.comp (fderivWithin \ud835\udd5c g' u (g (f x)))\n (ContinuousLinearMap.comp (fderivWithin \ud835\udd5c g t (f x)) (fderivWithin \ud835\udd5c f s x))\n[PROOFSTEP]\nexact\n (hg'.hasFDerivWithinAt.comp x (hg.hasFDerivWithinAt.comp x hf.hasFDerivWithinAt h2f) <| h2g.comp h2f).fderivWithin hxs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g\u271d : E \u2192 F\nf' f\u2080' f\u2081' g'\u271d e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\ng : F \u2192 G\ng' : F \u2192L[\ud835\udd5c] G\nhg : HasStrictFDerivAt g g' (f x)\nhf : HasStrictFDerivAt f f' x\n\u22a2 (fun x =>\n \u2191g' (((fun p => (f p.fst, f p.snd)) x).fst - ((fun p => (f p.fst, f p.snd)) x).snd) -\n \u2191(comp g' f') (x.fst - x.snd)) =o[\ud835\udcdd (x, x)]\n fun p => p.fst - p.snd\n[PROOFSTEP]\nsimpa only [g'.map_sub, f'.coe_comp'] using (g'.isBigO_comp _ _).trans_isLittleO hf\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasFDerivAtFilter f f' x L\nhL : Tendsto f L L\nhx : f x = x\nn : \u2115\n\u22a2 HasFDerivAtFilter f^[n] (f' ^ n) x L\n[PROOFSTEP]\ninduction' n with n ihn\n[GOAL]\ncase zero\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasFDerivAtFilter f f' x L\nhL : Tendsto f L L\nhx : f x = x\n\u22a2 HasFDerivAtFilter f^[Nat.zero] (f' ^ Nat.zero) x L\n[PROOFSTEP]\nexact hasFDerivAtFilter_id x L\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasFDerivAtFilter f f' x L\nhL : Tendsto f L L\nhx : f x = x\nn : \u2115\nihn : HasFDerivAtFilter f^[n] (f' ^ n) x L\n\u22a2 HasFDerivAtFilter f^[Nat.succ n] (f' ^ Nat.succ n) x L\n[PROOFSTEP]\nrw [Function.iterate_succ, pow_succ']\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasFDerivAtFilter f f' x L\nhL : Tendsto f L L\nhx : f x = x\nn : \u2115\nihn : HasFDerivAtFilter f^[n] (f' ^ n) x L\n\u22a2 HasFDerivAtFilter (f^[n] \u2218 f) (f' ^ n * f') x L\n[PROOFSTEP]\nrw [\u2190 hx] at ihn \n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasFDerivAtFilter f f' x L\nhL : Tendsto f L L\nhx : f x = x\nn : \u2115\nihn : HasFDerivAtFilter f^[n] (f' ^ n) (f x) L\n\u22a2 HasFDerivAtFilter (f^[n] \u2218 f) (f' ^ n * f') x L\n[PROOFSTEP]\nexact ihn.comp x hf hL\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasFDerivAt f f' x\nhx : f x = x\nn : \u2115\n\u22a2 HasFDerivAt f^[n] (f' ^ n) x\n[PROOFSTEP]\nrefine' HasFDerivAtFilter.iterate hf _ hx n\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasFDerivAt f f' x\nhx : f x = x\nn : \u2115\n\u22a2 Tendsto f (\ud835\udcdd x) (\ud835\udcdd x)\n[PROOFSTEP]\nhave := hf.continuousAt\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasFDerivAt f f' x\nhx : f x = x\nn : \u2115\nthis : ContinuousAt f x\n\u22a2 Tendsto f (\ud835\udcdd x) (\ud835\udcdd x)\n[PROOFSTEP]\nunfold ContinuousAt at this \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasFDerivAt f f' x\nhx : f x = x\nn : \u2115\nthis : Tendsto f (\ud835\udcdd x) (\ud835\udcdd (f x))\n\u22a2 Tendsto f (\ud835\udcdd x) (\ud835\udcdd x)\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_5.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasFDerivAt f f' x\nhx : f x = x\nn : \u2115\nthis : Tendsto f (\ud835\udcdd x) (\ud835\udcdd (f x))\n\u22a2 x = f x\n[PROOFSTEP]\nexact hx.symm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasFDerivWithinAt f f' s x\nhx : f x = x\nhs : MapsTo f s s\nn : \u2115\n\u22a2 HasFDerivWithinAt f^[n] (f' ^ n) s x\n[PROOFSTEP]\nrefine' HasFDerivAtFilter.iterate hf _ hx n\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasFDerivWithinAt f f' s x\nhx : f x = x\nhs : MapsTo f s s\nn : \u2115\n\u22a2 Tendsto f (\ud835\udcdd[s] x) (\ud835\udcdd[s] x)\n[PROOFSTEP]\nrw [_root_.nhdsWithin]\n -- Porting note: Added `rw` to get rid of an error\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasFDerivWithinAt f f' s x\nhx : f x = x\nhs : MapsTo f s s\nn : \u2115\n\u22a2 Tendsto f (\ud835\udcdd x \u2293 \ud835\udcdf s) (\ud835\udcdd x \u2293 \ud835\udcdf s)\n[PROOFSTEP]\nconvert tendsto_inf.2 \u27e8hf.continuousWithinAt, _\u27e9\n[GOAL]\ncase h.e'_5.h.e'_3.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasFDerivWithinAt f f' s x\nhx : f x = x\nhs : MapsTo f s s\nn : \u2115\n\u22a2 x = f x\ncase convert_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasFDerivWithinAt f f' s x\nhx : f x = x\nhs : MapsTo f s s\nn : \u2115\n\u22a2 Tendsto f (\ud835\udcdd[s] x) (\ud835\udcdf s)\n[PROOFSTEP]\nexacts [hx.symm, (tendsto_principal_principal.2 hs).mono_left inf_le_right]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasStrictFDerivAt f f' x\nhx : f x = x\nn : \u2115\n\u22a2 HasStrictFDerivAt f^[n] (f' ^ n) x\n[PROOFSTEP]\ninduction' n with n ihn\n[GOAL]\ncase zero\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasStrictFDerivAt f f' x\nhx : f x = x\n\u22a2 HasStrictFDerivAt f^[Nat.zero] (f' ^ Nat.zero) x\n[PROOFSTEP]\nexact hasStrictFDerivAt_id x\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasStrictFDerivAt f f' x\nhx : f x = x\nn : \u2115\nihn : HasStrictFDerivAt f^[n] (f' ^ n) x\n\u22a2 HasStrictFDerivAt f^[Nat.succ n] (f' ^ Nat.succ n) x\n[PROOFSTEP]\nrw [Function.iterate_succ, pow_succ']\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasStrictFDerivAt f f' x\nhx : f x = x\nn : \u2115\nihn : HasStrictFDerivAt f^[n] (f' ^ n) x\n\u22a2 HasStrictFDerivAt (f^[n] \u2218 f) (f' ^ n * f') x\n[PROOFSTEP]\nrw [\u2190 hx] at ihn \n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasStrictFDerivAt f f' x\nhx : f x = x\nn : \u2115\nihn : HasStrictFDerivAt f^[n] (f' ^ n) (f x)\n\u22a2 HasStrictFDerivAt (f^[n] \u2218 f) (f' ^ n * f') x\n[PROOFSTEP]\nexact ihn.comp x hf\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.FDeriv.Comp", "llama_tokens": 10810, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4140484931784741}} {"text": "[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI J : Box \u03b9\ni : \u03b9\nf g : \u03b9 \u2192\u1d47\u1d43[I\u2080] M\nh : f.toFun = g.toFun\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI J : Box \u03b9\ni : \u03b9\ng : \u03b9 \u2192\u1d47\u1d43[I\u2080] M\ntoFun\u271d : Box \u03b9 \u2192 M\nsum_partition_boxes'\u271d :\n \u2200 (J : Box \u03b9), \u2191J \u2264 I\u2080 \u2192 \u2200 (\u03c0 : Prepartition J), IsPartition \u03c0 \u2192 \u2211 Ji in \u03c0.boxes, toFun\u271d Ji = toFun\u271d J\nh : { toFun := toFun\u271d, sum_partition_boxes' := sum_partition_boxes'\u271d }.toFun = g.toFun\n\u22a2 { toFun := toFun\u271d, sum_partition_boxes' := sum_partition_boxes'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI J : Box \u03b9\ni : \u03b9\ntoFun\u271d\u00b9 : Box \u03b9 \u2192 M\nsum_partition_boxes'\u271d\u00b9 :\n \u2200 (J : Box \u03b9), \u2191J \u2264 I\u2080 \u2192 \u2200 (\u03c0 : Prepartition J), IsPartition \u03c0 \u2192 \u2211 Ji in \u03c0.boxes, toFun\u271d\u00b9 Ji = toFun\u271d\u00b9 J\ntoFun\u271d : Box \u03b9 \u2192 M\nsum_partition_boxes'\u271d :\n \u2200 (J : Box \u03b9), \u2191J \u2264 I\u2080 \u2192 \u2200 (\u03c0 : Prepartition J), IsPartition \u03c0 \u2192 \u2211 Ji in \u03c0.boxes, toFun\u271d Ji = toFun\u271d J\nh :\n { toFun := toFun\u271d\u00b9, sum_partition_boxes' := sum_partition_boxes'\u271d\u00b9 }.toFun =\n { toFun := toFun\u271d, sum_partition_boxes' := sum_partition_boxes'\u271d }.toFun\n\u22a2 { toFun := toFun\u271d\u00b9, sum_partition_boxes' := sum_partition_boxes'\u271d\u00b9 } =\n { toFun := toFun\u271d, sum_partition_boxes' := sum_partition_boxes'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI\u271d J : Box \u03b9\ni : \u03b9\nf g : \u03b9 \u2192\u1d47\u1d43[I\u2080] M\nI : Box \u03b9\nhI : \u2191I \u2264 I\u2080\n\u03c0 : Prepartition I\nh\u03c0 : IsPartition \u03c0\n\u22a2 \u2211 Ji in \u03c0.boxes, (\u2191f + \u2191g) Ji = (\u2191f + \u2191g) I\n[PROOFSTEP]\nsimp only [Pi.add_apply, sum_add_distrib, sum_partition_boxes _ hI h\u03c0]\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI\u271d J : Box \u03b9\ni : \u03b9\nR : Type ?u.25829\ninst\u271d\u00b9 : Monoid R\ninst\u271d : DistribMulAction R M\nr : R\nf : \u03b9 \u2192\u1d47\u1d43[I\u2080] M\nI : Box \u03b9\nhI : \u2191I \u2264 I\u2080\n\u03c0 : Prepartition I\nh\u03c0 : IsPartition \u03c0\n\u22a2 \u2211 Ji in \u03c0.boxes, (r \u2022 \u2191f) Ji = (r \u2022 \u2191f) I\n[PROOFSTEP]\nsimp only [Pi.smul_apply, \u2190 smul_sum, sum_partition_boxes _ hI h\u03c0]\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI J : Box \u03b9\ni\u271d : \u03b9\nf : \u03b9 \u2192\u1d47\u1d43[I\u2080] M\nhI : \u2191I \u2264 I\u2080\ni : \u03b9\nx : \u211d\n\u22a2 Option.elim' 0 (\u2191f) (splitLower I i x) + Option.elim' 0 (\u2191f) (splitUpper I i x) = \u2191f I\n[PROOFSTEP]\nrw [\u2190 f.sum_partition_boxes hI (isPartitionSplit I i x), sum_split_boxes]\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf :\n \u2200 (I : Box \u03b9),\n \u2191I \u2264 I\u2080 \u2192\n \u2200 {i : \u03b9} {x : \u211d},\n x \u2208 Set.Ioo (lower I i) (upper I i) \u2192\n Option.elim' 0 f (splitLower I i x) + Option.elim' 0 f (splitUpper I i x) = f I\n\u22a2 \u03b9 \u2192\u1d47\u1d43[I\u2080] M\n[PROOFSTEP]\nrefine' \u27e8f, _\u27e9\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf :\n \u2200 (I : Box \u03b9),\n \u2191I \u2264 I\u2080 \u2192\n \u2200 {i : \u03b9} {x : \u211d},\n x \u2208 Set.Ioo (lower I i) (upper I i) \u2192\n Option.elim' 0 f (splitLower I i x) + Option.elim' 0 f (splitUpper I i x) = f I\n\u22a2 \u2200 (J : Box \u03b9), \u2191J \u2264 I\u2080 \u2192 \u2200 (\u03c0 : Prepartition J), IsPartition \u03c0 \u2192 \u2211 Ji in \u03c0.boxes, f Ji = f J\n[PROOFSTEP]\nreplace hf : \u2200 I : Box \u03b9, \u2191I \u2264 I\u2080 \u2192 \u2200 s, (\u2211 J in (splitMany I s).boxes, f J) = f I\n[GOAL]\ncase hf\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf :\n \u2200 (I : Box \u03b9),\n \u2191I \u2264 I\u2080 \u2192\n \u2200 {i : \u03b9} {x : \u211d},\n x \u2208 Set.Ioo (lower I i) (upper I i) \u2192\n Option.elim' 0 f (splitLower I i x) + Option.elim' 0 f (splitUpper I i x) = f I\n\u22a2 \u2200 (I : Box \u03b9), \u2191I \u2264 I\u2080 \u2192 \u2200 (s : Finset (\u03b9 \u00d7 \u211d)), \u2211 J in (splitMany I s).boxes, f J = f I\n[PROOFSTEP]\nintro I hI s\n[GOAL]\ncase hf\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI\u271d J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf :\n \u2200 (I : Box \u03b9),\n \u2191I \u2264 I\u2080 \u2192\n \u2200 {i : \u03b9} {x : \u211d},\n x \u2208 Set.Ioo (lower I i) (upper I i) \u2192\n Option.elim' 0 f (splitLower I i x) + Option.elim' 0 f (splitUpper I i x) = f I\nI : Box \u03b9\nhI : \u2191I \u2264 I\u2080\ns : Finset (\u03b9 \u00d7 \u211d)\n\u22a2 \u2211 J in (splitMany I s).boxes, f J = f I\n[PROOFSTEP]\ninduction' s using Finset.induction_on with a s _ ihs\n[GOAL]\ncase hf.empty\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI\u271d J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf :\n \u2200 (I : Box \u03b9),\n \u2191I \u2264 I\u2080 \u2192\n \u2200 {i : \u03b9} {x : \u211d},\n x \u2208 Set.Ioo (lower I i) (upper I i) \u2192\n Option.elim' 0 f (splitLower I i x) + Option.elim' 0 f (splitUpper I i x) = f I\nI : Box \u03b9\nhI : \u2191I \u2264 I\u2080\n\u22a2 \u2211 J in (splitMany I \u2205).boxes, f J = f I\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hf.insert\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI\u271d J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf :\n \u2200 (I : Box \u03b9),\n \u2191I \u2264 I\u2080 \u2192\n \u2200 {i : \u03b9} {x : \u211d},\n x \u2208 Set.Ioo (lower I i) (upper I i) \u2192\n Option.elim' 0 f (splitLower I i x) + Option.elim' 0 f (splitUpper I i x) = f I\nI : Box \u03b9\nhI : \u2191I \u2264 I\u2080\na : \u03b9 \u00d7 \u211d\ns : Finset (\u03b9 \u00d7 \u211d)\na\u271d : \u00aca \u2208 s\nihs : \u2211 J in (splitMany I s).boxes, f J = f I\n\u22a2 \u2211 J in (splitMany I (insert a s)).boxes, f J = f I\n[PROOFSTEP]\nrw [splitMany_insert, inf_split, \u2190 ihs, biUnion_boxes, sum_biUnion_boxes]\n[GOAL]\ncase hf.insert\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI\u271d J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf :\n \u2200 (I : Box \u03b9),\n \u2191I \u2264 I\u2080 \u2192\n \u2200 {i : \u03b9} {x : \u211d},\n x \u2208 Set.Ioo (lower I i) (upper I i) \u2192\n Option.elim' 0 f (splitLower I i x) + Option.elim' 0 f (splitUpper I i x) = f I\nI : Box \u03b9\nhI : \u2191I \u2264 I\u2080\na : \u03b9 \u00d7 \u211d\ns : Finset (\u03b9 \u00d7 \u211d)\na\u271d : \u00aca \u2208 s\nihs : \u2211 J in (splitMany I s).boxes, f J = f I\n\u22a2 \u2211 J in (splitMany I s).boxes, \u2211 J' in (split J a.fst a.snd).boxes, f J' = \u2211 J in (splitMany I s).boxes, f J\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun J' hJ' => _\n[GOAL]\ncase hf.insert\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI\u271d J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf :\n \u2200 (I : Box \u03b9),\n \u2191I \u2264 I\u2080 \u2192\n \u2200 {i : \u03b9} {x : \u211d},\n x \u2208 Set.Ioo (lower I i) (upper I i) \u2192\n Option.elim' 0 f (splitLower I i x) + Option.elim' 0 f (splitUpper I i x) = f I\nI : Box \u03b9\nhI : \u2191I \u2264 I\u2080\na : \u03b9 \u00d7 \u211d\ns : Finset (\u03b9 \u00d7 \u211d)\na\u271d : \u00aca \u2208 s\nihs : \u2211 J in (splitMany I s).boxes, f J = f I\nJ' : Box \u03b9\nhJ' : J' \u2208 (splitMany I s).boxes\n\u22a2 \u2211 J' in (split J' a.fst a.snd).boxes, f J' = f J'\n[PROOFSTEP]\nby_cases h : a.2 \u2208 Ioo (J'.lower a.1) (J'.upper a.1)\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI\u271d J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf :\n \u2200 (I : Box \u03b9),\n \u2191I \u2264 I\u2080 \u2192\n \u2200 {i : \u03b9} {x : \u211d},\n x \u2208 Set.Ioo (lower I i) (upper I i) \u2192\n Option.elim' 0 f (splitLower I i x) + Option.elim' 0 f (splitUpper I i x) = f I\nI : Box \u03b9\nhI : \u2191I \u2264 I\u2080\na : \u03b9 \u00d7 \u211d\ns : Finset (\u03b9 \u00d7 \u211d)\na\u271d : \u00aca \u2208 s\nihs : \u2211 J in (splitMany I s).boxes, f J = f I\nJ' : Box \u03b9\nhJ' : J' \u2208 (splitMany I s).boxes\nh : a.snd \u2208 Set.Ioo (lower J' a.fst) (upper J' a.fst)\n\u22a2 \u2211 J' in (split J' a.fst a.snd).boxes, f J' = f J'\n[PROOFSTEP]\nrw [sum_split_boxes]\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI\u271d J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf :\n \u2200 (I : Box \u03b9),\n \u2191I \u2264 I\u2080 \u2192\n \u2200 {i : \u03b9} {x : \u211d},\n x \u2208 Set.Ioo (lower I i) (upper I i) \u2192\n Option.elim' 0 f (splitLower I i x) + Option.elim' 0 f (splitUpper I i x) = f I\nI : Box \u03b9\nhI : \u2191I \u2264 I\u2080\na : \u03b9 \u00d7 \u211d\ns : Finset (\u03b9 \u00d7 \u211d)\na\u271d : \u00aca \u2208 s\nihs : \u2211 J in (splitMany I s).boxes, f J = f I\nJ' : Box \u03b9\nhJ' : J' \u2208 (splitMany I s).boxes\nh : a.snd \u2208 Set.Ioo (lower J' a.fst) (upper J' a.fst)\n\u22a2 Option.elim' 0 (fun J' => f J') (splitLower J' a.fst a.snd) +\n Option.elim' 0 (fun J' => f J') (splitUpper J' a.fst a.snd) =\n f J'\n[PROOFSTEP]\nexact hf _ ((WithTop.coe_le_coe.2 <| le_of_mem _ hJ').trans hI) h\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI\u271d J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf :\n \u2200 (I : Box \u03b9),\n \u2191I \u2264 I\u2080 \u2192\n \u2200 {i : \u03b9} {x : \u211d},\n x \u2208 Set.Ioo (lower I i) (upper I i) \u2192\n Option.elim' 0 f (splitLower I i x) + Option.elim' 0 f (splitUpper I i x) = f I\nI : Box \u03b9\nhI : \u2191I \u2264 I\u2080\na : \u03b9 \u00d7 \u211d\ns : Finset (\u03b9 \u00d7 \u211d)\na\u271d : \u00aca \u2208 s\nihs : \u2211 J in (splitMany I s).boxes, f J = f I\nJ' : Box \u03b9\nhJ' : J' \u2208 (splitMany I s).boxes\nh : \u00aca.snd \u2208 Set.Ioo (lower J' a.fst) (upper J' a.fst)\n\u22a2 \u2211 J' in (split J' a.fst a.snd).boxes, f J' = f J'\n[PROOFSTEP]\nrw [split_of_not_mem_Ioo h, top_boxes, Finset.sum_singleton]\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf : \u2200 (I : Box \u03b9), \u2191I \u2264 I\u2080 \u2192 \u2200 (s : Finset (\u03b9 \u00d7 \u211d)), \u2211 J in (splitMany I s).boxes, f J = f I\n\u22a2 \u2200 (J : Box \u03b9), \u2191J \u2264 I\u2080 \u2192 \u2200 (\u03c0 : Prepartition J), IsPartition \u03c0 \u2192 \u2211 Ji in \u03c0.boxes, f Ji = f J\n[PROOFSTEP]\nintro I hI \u03c0 h\u03c0\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI\u271d J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf : \u2200 (I : Box \u03b9), \u2191I \u2264 I\u2080 \u2192 \u2200 (s : Finset (\u03b9 \u00d7 \u211d)), \u2211 J in (splitMany I s).boxes, f J = f I\nI : Box \u03b9\nhI : \u2191I \u2264 I\u2080\n\u03c0 : Prepartition I\nh\u03c0 : IsPartition \u03c0\n\u22a2 \u2211 Ji in \u03c0.boxes, f Ji = f I\n[PROOFSTEP]\nhave Hle : \u2200 J \u2208 \u03c0, \u2191J \u2264 I\u2080 := fun J hJ => (WithTop.coe_le_coe.2 <| \u03c0.le_of_mem hJ).trans hI\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI\u271d J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf : \u2200 (I : Box \u03b9), \u2191I \u2264 I\u2080 \u2192 \u2200 (s : Finset (\u03b9 \u00d7 \u211d)), \u2211 J in (splitMany I s).boxes, f J = f I\nI : Box \u03b9\nhI : \u2191I \u2264 I\u2080\n\u03c0 : Prepartition I\nh\u03c0 : IsPartition \u03c0\nHle : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u2191J \u2264 I\u2080\n\u22a2 \u2211 Ji in \u03c0.boxes, f Ji = f I\n[PROOFSTEP]\nrcases h\u03c0.exists_splitMany_le with \u27e8s, hs\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI\u271d J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf : \u2200 (I : Box \u03b9), \u2191I \u2264 I\u2080 \u2192 \u2200 (s : Finset (\u03b9 \u00d7 \u211d)), \u2211 J in (splitMany I s).boxes, f J = f I\nI : Box \u03b9\nhI : \u2191I \u2264 I\u2080\n\u03c0 : Prepartition I\nh\u03c0 : IsPartition \u03c0\nHle : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u2191J \u2264 I\u2080\ns : Finset (\u03b9 \u00d7 \u211d)\nhs : splitMany I s \u2264 \u03c0\n\u22a2 \u2211 Ji in \u03c0.boxes, f Ji = f I\n[PROOFSTEP]\nrw [\u2190 hf _ hI, \u2190 inf_of_le_right hs, inf_splitMany, biUnion_boxes, sum_biUnion_boxes]\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI\u271d J : Box \u03b9\ni : \u03b9\ninst\u271d : Fintype \u03b9\nf : Box \u03b9 \u2192 M\nI\u2080 : WithTop (Box \u03b9)\nhf : \u2200 (I : Box \u03b9), \u2191I \u2264 I\u2080 \u2192 \u2200 (s : Finset (\u03b9 \u00d7 \u211d)), \u2211 J in (splitMany I s).boxes, f J = f I\nI : Box \u03b9\nhI : \u2191I \u2264 I\u2080\n\u03c0 : Prepartition I\nh\u03c0 : IsPartition \u03c0\nHle : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u2191J \u2264 I\u2080\ns : Finset (\u03b9 \u00d7 \u211d)\nhs : splitMany I s \u2264 \u03c0\n\u22a2 \u2211 Ji in \u03c0.boxes, f Ji = \u2211 J in \u03c0.boxes, \u2211 J' in (splitMany J s).boxes, f J'\n[PROOFSTEP]\nexact Finset.sum_congr rfl fun J hJ => (hf _ (Hle _ hJ) _).symm\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI\u271d J : Box \u03b9\ni : \u03b9\nf : \u03b9 \u2192\u1d47\u1d43[I\u2080] M\ng : M \u2192+ N\nI : Box \u03b9\nhI : \u2191I \u2264 I\u2080\n\u03c0 : Prepartition I\nh\u03c0 : IsPartition \u03c0\n\u22a2 \u2211 Ji in \u03c0.boxes, (\u2191g \u2218 \u2191f) Ji = (\u2191g \u2218 \u2191f) I\n[PROOFSTEP]\nsimp_rw [comp, \u2190 g.map_sum, f.sum_partition_boxes hI h\u03c0]\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI J : Box \u03b9\ni : \u03b9\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192\u1d47\u1d43[I\u2080] M\nhI : \u2191I \u2264 I\u2080\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh : Prepartition.iUnion \u03c0\u2081 = Prepartition.iUnion \u03c0\u2082\n\u22a2 \u2211 J in \u03c0\u2081.boxes, \u2191f J = \u2211 J in \u03c0\u2082.boxes, \u2191f J\n[PROOFSTEP]\nrcases exists_splitMany_inf_eq_filter_of_finite { \u03c0\u2081, \u03c0\u2082 } ((finite_singleton _).insert _) with \u27e8s, hs\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI J : Box \u03b9\ni : \u03b9\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192\u1d47\u1d43[I\u2080] M\nhI : \u2191I \u2264 I\u2080\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh : Prepartition.iUnion \u03c0\u2081 = Prepartition.iUnion \u03c0\u2082\ns : Finset (\u03b9 \u00d7 \u211d)\nhs :\n \u2200 (\u03c0 : Prepartition I),\n \u03c0 \u2208 {\u03c0\u2081, \u03c0\u2082} \u2192 \u03c0 \u2293 splitMany I s = Prepartition.filter (splitMany I s) fun J => \u2191J \u2286 Prepartition.iUnion \u03c0\n\u22a2 \u2211 J in \u03c0\u2081.boxes, \u2191f J = \u2211 J in \u03c0\u2082.boxes, \u2191f J\n[PROOFSTEP]\nsimp only [inf_splitMany] at hs \n[GOAL]\ncase intro\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI J : Box \u03b9\ni : \u03b9\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192\u1d47\u1d43[I\u2080] M\nhI : \u2191I \u2264 I\u2080\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh : Prepartition.iUnion \u03c0\u2081 = Prepartition.iUnion \u03c0\u2082\ns : Finset (\u03b9 \u00d7 \u211d)\nhs :\n \u2200 (\u03c0 : Prepartition I),\n \u03c0 \u2208 {\u03c0\u2081, \u03c0\u2082} \u2192\n (biUnion \u03c0 fun J => splitMany J s) = Prepartition.filter (splitMany I s) fun J => \u2191J \u2286 Prepartition.iUnion \u03c0\n\u22a2 \u2211 J in \u03c0\u2081.boxes, \u2191f J = \u2211 J in \u03c0\u2082.boxes, \u2191f J\n[PROOFSTEP]\nrcases hs _ (Or.inl rfl), hs _ (Or.inr rfl) with \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI J : Box \u03b9\ni : \u03b9\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192\u1d47\u1d43[I\u2080] M\nhI : \u2191I \u2264 I\u2080\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh : Prepartition.iUnion \u03c0\u2081 = Prepartition.iUnion \u03c0\u2082\ns : Finset (\u03b9 \u00d7 \u211d)\nhs :\n \u2200 (\u03c0 : Prepartition I),\n \u03c0 \u2208 {\u03c0\u2081, \u03c0\u2082} \u2192\n (biUnion \u03c0 fun J => splitMany J s) = Prepartition.filter (splitMany I s) fun J => \u2191J \u2286 Prepartition.iUnion \u03c0\nh\u2081 : (biUnion \u03c0\u2081 fun J => splitMany J s) = Prepartition.filter (splitMany I s) fun J => \u2191J \u2286 Prepartition.iUnion \u03c0\u2081\nh\u2082 : (biUnion \u03c0\u2082 fun J => splitMany J s) = Prepartition.filter (splitMany I s) fun J => \u2191J \u2286 Prepartition.iUnion \u03c0\u2082\n\u22a2 \u2211 J in \u03c0\u2081.boxes, \u2191f J = \u2211 J in \u03c0\u2082.boxes, \u2191f J\n[PROOFSTEP]\nclear hs\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI J : Box \u03b9\ni : \u03b9\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192\u1d47\u1d43[I\u2080] M\nhI : \u2191I \u2264 I\u2080\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh : Prepartition.iUnion \u03c0\u2081 = Prepartition.iUnion \u03c0\u2082\ns : Finset (\u03b9 \u00d7 \u211d)\nh\u2081 : (biUnion \u03c0\u2081 fun J => splitMany J s) = Prepartition.filter (splitMany I s) fun J => \u2191J \u2286 Prepartition.iUnion \u03c0\u2081\nh\u2082 : (biUnion \u03c0\u2082 fun J => splitMany J s) = Prepartition.filter (splitMany I s) fun J => \u2191J \u2286 Prepartition.iUnion \u03c0\u2082\n\u22a2 \u2211 J in \u03c0\u2081.boxes, \u2191f J = \u2211 J in \u03c0\u2082.boxes, \u2191f J\n[PROOFSTEP]\nrw [h] at h\u2081 \n[GOAL]\ncase intro\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI J : Box \u03b9\ni : \u03b9\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192\u1d47\u1d43[I\u2080] M\nhI : \u2191I \u2264 I\u2080\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh : Prepartition.iUnion \u03c0\u2081 = Prepartition.iUnion \u03c0\u2082\ns : Finset (\u03b9 \u00d7 \u211d)\nh\u2081 : (biUnion \u03c0\u2081 fun J => splitMany J s) = Prepartition.filter (splitMany I s) fun J => \u2191J \u2286 Prepartition.iUnion \u03c0\u2082\nh\u2082 : (biUnion \u03c0\u2082 fun J => splitMany J s) = Prepartition.filter (splitMany I s) fun J => \u2191J \u2286 Prepartition.iUnion \u03c0\u2082\n\u22a2 \u2211 J in \u03c0\u2081.boxes, \u2191f J = \u2211 J in \u03c0\u2082.boxes, \u2191f J\n[PROOFSTEP]\ncalc\n \u2211 J in \u03c0\u2081.boxes, f J = \u2211 J in \u03c0\u2081.boxes, \u2211 J' in (splitMany J s).boxes, f J' :=\n Finset.sum_congr rfl fun J hJ => (f.sum_partition_boxes ?_ (isPartition_splitMany _ _)).symm\n _ = \u2211 J in (\u03c0\u2081.biUnion fun J => splitMany J s).boxes, f J := (sum_biUnion_boxes _ _ _).symm\n _ = \u2211 J in (\u03c0\u2082.biUnion fun J => splitMany J s).boxes, f J := by rw [h\u2081, h\u2082]\n _ = \u2211 J in \u03c0\u2082.boxes, \u2211 J' in (splitMany J s).boxes, f J' := (sum_biUnion_boxes _ _ _)\n _ = \u2211 J in \u03c0\u2082.boxes, f J := Finset.sum_congr rfl fun J hJ => f.sum_partition_boxes ?_ (isPartition_splitMany _ _)\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI J : Box \u03b9\ni : \u03b9\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192\u1d47\u1d43[I\u2080] M\nhI : \u2191I \u2264 I\u2080\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh : Prepartition.iUnion \u03c0\u2081 = Prepartition.iUnion \u03c0\u2082\ns : Finset (\u03b9 \u00d7 \u211d)\nh\u2081 : (biUnion \u03c0\u2081 fun J => splitMany J s) = Prepartition.filter (splitMany I s) fun J => \u2191J \u2286 Prepartition.iUnion \u03c0\u2082\nh\u2082 : (biUnion \u03c0\u2082 fun J => splitMany J s) = Prepartition.filter (splitMany I s) fun J => \u2191J \u2286 Prepartition.iUnion \u03c0\u2082\n\u22a2 \u2211 J in (biUnion \u03c0\u2081 fun J => splitMany J s).boxes, \u2191f J = \u2211 J in (biUnion \u03c0\u2082 fun J => splitMany J s).boxes, \u2191f J\n[PROOFSTEP]\nrw [h\u2081, h\u2082]\n[GOAL]\ncase intro.calc_1\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI J\u271d : Box \u03b9\ni : \u03b9\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192\u1d47\u1d43[I\u2080] M\nhI : \u2191I \u2264 I\u2080\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh : Prepartition.iUnion \u03c0\u2081 = Prepartition.iUnion \u03c0\u2082\ns : Finset (\u03b9 \u00d7 \u211d)\nh\u2081 : (biUnion \u03c0\u2081 fun J => splitMany J s) = Prepartition.filter (splitMany I s) fun J => \u2191J \u2286 Prepartition.iUnion \u03c0\u2082\nh\u2082 : (biUnion \u03c0\u2082 fun J => splitMany J s) = Prepartition.filter (splitMany I s) fun J => \u2191J \u2286 Prepartition.iUnion \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.boxes\n\u22a2 \u2191J \u2264 I\u2080\ncase intro.calc_2\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080 : WithTop (Box \u03b9)\nI J\u271d : Box \u03b9\ni : \u03b9\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192\u1d47\u1d43[I\u2080] M\nhI : \u2191I \u2264 I\u2080\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh : Prepartition.iUnion \u03c0\u2081 = Prepartition.iUnion \u03c0\u2082\ns : Finset (\u03b9 \u00d7 \u211d)\nh\u2081 : (biUnion \u03c0\u2081 fun J => splitMany J s) = Prepartition.filter (splitMany I s) fun J => \u2191J \u2286 Prepartition.iUnion \u03c0\u2082\nh\u2082 : (biUnion \u03c0\u2082 fun J => splitMany J s) = Prepartition.filter (splitMany I s) fun J => \u2191J \u2286 Prepartition.iUnion \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2082.boxes\n\u22a2 \u2191J \u2264 I\u2080\n[PROOFSTEP]\nexacts [(WithTop.coe_le_coe.2 <| \u03c0\u2081.le_of_mem hJ).trans hI, (WithTop.coe_le_coe.2 <| \u03c0\u2082.le_of_mem hJ).trans hI]\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J : Box \u03b9\ni\u271d : \u03b9\nG : Type u\ninst\u271d : AddCommGroup G\nI\u2080 : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : \u211d \u2192 Box (Fin n) \u2192 G\nfb : \u2191(Set.Icc (lower I\u2080 i) (upper I\u2080 i)) \u2192 Fin n \u2192\u1d47\u1d43[\u2191(face I\u2080 i)] G\nhf :\n \u2200 (x : \u211d) (hx : x \u2208 Set.Icc (lower I\u2080 i) (upper I\u2080 i)) (J : Box (Fin n)), f x J = \u2191(fb { val := x, property := hx }) J\n\u22a2 \u2200 (I : Box (Fin (n + 1))),\n \u2191I \u2264 \u2191I\u2080 \u2192\n \u2200 {i_1 : Fin (n + 1)} {x : \u211d},\n x \u2208 Set.Ioo (lower I i_1) (upper I i_1) \u2192\n Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) (splitLower I i_1 x) +\n Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) (splitUpper I i_1 x) =\n (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) I\n[PROOFSTEP]\nintro J hJ j x\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J\u271d : Box \u03b9\ni\u271d : \u03b9\nG : Type u\ninst\u271d : AddCommGroup G\nI\u2080 : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : \u211d \u2192 Box (Fin n) \u2192 G\nfb : \u2191(Set.Icc (lower I\u2080 i) (upper I\u2080 i)) \u2192 Fin n \u2192\u1d47\u1d43[\u2191(face I\u2080 i)] G\nhf :\n \u2200 (x : \u211d) (hx : x \u2208 Set.Icc (lower I\u2080 i) (upper I\u2080 i)) (J : Box (Fin n)), f x J = \u2191(fb { val := x, property := hx }) J\nJ : Box (Fin (n + 1))\nhJ : \u2191J \u2264 \u2191I\u2080\nj : Fin (n + 1)\nx : \u211d\n\u22a2 x \u2208 Set.Ioo (lower J j) (upper J j) \u2192\n Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) (splitLower J j x) +\n Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) (splitUpper J j x) =\n (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) J\n[PROOFSTEP]\nrw [WithTop.coe_le_coe] at hJ \n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J\u271d : Box \u03b9\ni\u271d : \u03b9\nG : Type u\ninst\u271d : AddCommGroup G\nI\u2080 : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : \u211d \u2192 Box (Fin n) \u2192 G\nfb : \u2191(Set.Icc (lower I\u2080 i) (upper I\u2080 i)) \u2192 Fin n \u2192\u1d47\u1d43[\u2191(face I\u2080 i)] G\nhf :\n \u2200 (x : \u211d) (hx : x \u2208 Set.Icc (lower I\u2080 i) (upper I\u2080 i)) (J : Box (Fin n)), f x J = \u2191(fb { val := x, property := hx }) J\nJ : Box (Fin (n + 1))\nhJ : J \u2264 I\u2080\nj : Fin (n + 1)\nx : \u211d\n\u22a2 x \u2208 Set.Ioo (lower J j) (upper J j) \u2192\n Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) (splitLower J j x) +\n Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) (splitUpper J j x) =\n (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) J\n[PROOFSTEP]\nrefine' i.succAboveCases (fun hx => _) (fun j hx => _) j\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J\u271d : Box \u03b9\ni\u271d : \u03b9\nG : Type u\ninst\u271d : AddCommGroup G\nI\u2080 : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : \u211d \u2192 Box (Fin n) \u2192 G\nfb : \u2191(Set.Icc (lower I\u2080 i) (upper I\u2080 i)) \u2192 Fin n \u2192\u1d47\u1d43[\u2191(face I\u2080 i)] G\nhf :\n \u2200 (x : \u211d) (hx : x \u2208 Set.Icc (lower I\u2080 i) (upper I\u2080 i)) (J : Box (Fin n)), f x J = \u2191(fb { val := x, property := hx }) J\nJ : Box (Fin (n + 1))\nhJ : J \u2264 I\u2080\nj : Fin (n + 1)\nx : \u211d\nhx : x \u2208 Set.Ioo (lower J i) (upper J i)\n\u22a2 Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) (splitLower J i x) +\n Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) (splitUpper J i x) =\n (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) J\n[PROOFSTEP]\nsimp only [Box.splitLower_def hx, Box.splitUpper_def hx, update_same, \u2190 WithBot.some_eq_coe, Option.elim', Box.face,\n (\u00b7 \u2218 \u00b7), update_noteq (Fin.succAbove_ne _ _)]\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J\u271d : Box \u03b9\ni\u271d : \u03b9\nG : Type u\ninst\u271d : AddCommGroup G\nI\u2080 : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : \u211d \u2192 Box (Fin n) \u2192 G\nfb : \u2191(Set.Icc (lower I\u2080 i) (upper I\u2080 i)) \u2192 Fin n \u2192\u1d47\u1d43[\u2191(face I\u2080 i)] G\nhf :\n \u2200 (x : \u211d) (hx : x \u2208 Set.Icc (lower I\u2080 i) (upper I\u2080 i)) (J : Box (Fin n)), f x J = \u2191(fb { val := x, property := hx }) J\nJ : Box (Fin (n + 1))\nhJ : J \u2264 I\u2080\nj : Fin (n + 1)\nx : \u211d\nhx : x \u2208 Set.Ioo (lower J i) (upper J i)\n\u22a2 f x\n { lower := fun x => lower J (Fin.succAbove i x), upper := fun x => upper J (Fin.succAbove i x),\n lower_lt_upper :=\n (_ :\n \u2200 (i_1 : Fin n),\n (fun x_1 =>\n lower\n { lower := J.lower, upper := update J.upper i x,\n lower_lt_upper := (_ : \u2200 (x_2 : Fin (n + 1)), lower J x_2 < update J.upper i x x_2) }\n (Fin.succAbove i x_1))\n i_1 <\n (fun x => upper J (Fin.succAbove i x)) i_1) } -\n f (lower J i)\n { lower := fun x => lower J (Fin.succAbove i x), upper := fun x => upper J (Fin.succAbove i x),\n lower_lt_upper :=\n (_ :\n \u2200 (i_1 : Fin n),\n (fun x_1 =>\n lower\n { lower := J.lower, upper := update J.upper i x,\n lower_lt_upper := (_ : \u2200 (x_2 : Fin (n + 1)), lower J x_2 < update J.upper i x x_2) }\n (Fin.succAbove i x_1))\n i_1 <\n (fun x => upper J (Fin.succAbove i x)) i_1) } +\n (f (upper J i)\n { lower := fun x => lower J (Fin.succAbove i x), upper := fun x => upper J (Fin.succAbove i x),\n lower_lt_upper :=\n (_ :\n \u2200 (i_1 : Fin n),\n (fun x => lower J (Fin.succAbove i x)) i_1 < (fun x => upper J (Fin.succAbove i x)) i_1) } -\n f x\n { lower := fun x => lower J (Fin.succAbove i x), upper := fun x => upper J (Fin.succAbove i x),\n lower_lt_upper :=\n (_ :\n \u2200 (i_1 : Fin n),\n (fun x => lower J (Fin.succAbove i x)) i_1 < (fun x => upper J (Fin.succAbove i x)) i_1) }) =\n f (upper J i)\n { lower := fun x => lower J (Fin.succAbove i x), upper := fun x => upper J (Fin.succAbove i x),\n lower_lt_upper :=\n (_ :\n \u2200 (i_1 : Fin n),\n (fun x => lower J (Fin.succAbove i x)) i_1 < (fun x => upper J (Fin.succAbove i x)) i_1) } -\n f (lower J i)\n { lower := fun x => lower J (Fin.succAbove i x), upper := fun x => upper J (Fin.succAbove i x),\n lower_lt_upper :=\n (_ :\n \u2200 (i_1 : Fin n),\n (fun x => lower J (Fin.succAbove i x)) i_1 < (fun x => upper J (Fin.succAbove i x)) i_1) }\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J\u271d : Box \u03b9\ni\u271d : \u03b9\nG : Type u\ninst\u271d : AddCommGroup G\nI\u2080 : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : \u211d \u2192 Box (Fin n) \u2192 G\nfb : \u2191(Set.Icc (lower I\u2080 i) (upper I\u2080 i)) \u2192 Fin n \u2192\u1d47\u1d43[\u2191(face I\u2080 i)] G\nhf :\n \u2200 (x : \u211d) (hx : x \u2208 Set.Icc (lower I\u2080 i) (upper I\u2080 i)) (J : Box (Fin n)), f x J = \u2191(fb { val := x, property := hx }) J\nJ : Box (Fin (n + 1))\nhJ : J \u2264 I\u2080\nj : Fin (n + 1)\nx : \u211d\nhx : x \u2208 Set.Ioo (lower J i) (upper J i)\n\u22a2 f x\n { lower := fun x => lower J (Fin.succAbove i x), upper := fun x => upper J (Fin.succAbove i x),\n lower_lt_upper :=\n (_ :\n \u2200 (i_1 : Fin n),\n (fun x_1 =>\n lower\n { lower := J.lower, upper := update J.upper i x,\n lower_lt_upper := (_ : \u2200 (x_2 : Fin (n + 1)), lower J x_2 < update J.upper i x x_2) }\n (Fin.succAbove i x_1))\n i_1 <\n (fun x => upper J (Fin.succAbove i x)) i_1) } -\n f (lower J i)\n { lower := fun x => lower J (Fin.succAbove i x), upper := fun x => upper J (Fin.succAbove i x),\n lower_lt_upper :=\n (_ :\n \u2200 (i_1 : Fin n),\n (fun x_1 =>\n lower\n { lower := J.lower, upper := update J.upper i x,\n lower_lt_upper := (_ : \u2200 (x_2 : Fin (n + 1)), lower J x_2 < update J.upper i x x_2) }\n (Fin.succAbove i x_1))\n i_1 <\n (fun x => upper J (Fin.succAbove i x)) i_1) } +\n (f (upper J i)\n { lower := fun x => lower J (Fin.succAbove i x), upper := fun x => upper J (Fin.succAbove i x),\n lower_lt_upper :=\n (_ :\n \u2200 (i_1 : Fin n),\n (fun x => lower J (Fin.succAbove i x)) i_1 < (fun x => upper J (Fin.succAbove i x)) i_1) } -\n f x\n { lower := fun x => lower J (Fin.succAbove i x), upper := fun x => upper J (Fin.succAbove i x),\n lower_lt_upper :=\n (_ :\n \u2200 (i_1 : Fin n),\n (fun x => lower J (Fin.succAbove i x)) i_1 < (fun x => upper J (Fin.succAbove i x)) i_1) }) =\n f (upper J i)\n { lower := fun x => lower J (Fin.succAbove i x), upper := fun x => upper J (Fin.succAbove i x),\n lower_lt_upper :=\n (_ :\n \u2200 (i_1 : Fin n),\n (fun x => lower J (Fin.succAbove i x)) i_1 < (fun x => upper J (Fin.succAbove i x)) i_1) } -\n f (lower J i)\n { lower := fun x => lower J (Fin.succAbove i x), upper := fun x => upper J (Fin.succAbove i x),\n lower_lt_upper :=\n (_ :\n \u2200 (i_1 : Fin n),\n (fun x => lower J (Fin.succAbove i x)) i_1 < (fun x => upper J (Fin.succAbove i x)) i_1) }\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J\u271d : Box \u03b9\ni\u271d : \u03b9\nG : Type u\ninst\u271d : AddCommGroup G\nI\u2080 : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : \u211d \u2192 Box (Fin n) \u2192 G\nfb : \u2191(Set.Icc (lower I\u2080 i) (upper I\u2080 i)) \u2192 Fin n \u2192\u1d47\u1d43[\u2191(face I\u2080 i)] G\nhf :\n \u2200 (x : \u211d) (hx : x \u2208 Set.Icc (lower I\u2080 i) (upper I\u2080 i)) (J : Box (Fin n)), f x J = \u2191(fb { val := x, property := hx }) J\nJ : Box (Fin (n + 1))\nhJ : J \u2264 I\u2080\nj\u271d : Fin (n + 1)\nx : \u211d\nj : Fin n\nhx : x \u2208 Set.Ioo (lower J (Fin.succAbove i j)) (upper J (Fin.succAbove i j))\n\u22a2 Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) (splitLower J (Fin.succAbove i j) x) +\n Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i))\n (splitUpper J (Fin.succAbove i j) x) =\n (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) J\n[PROOFSTEP]\nhave : (J.face i : WithTop (Box (Fin n))) \u2264 I\u2080.face i := WithTop.coe_le_coe.2 (face_mono hJ i)\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J\u271d : Box \u03b9\ni\u271d : \u03b9\nG : Type u\ninst\u271d : AddCommGroup G\nI\u2080 : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : \u211d \u2192 Box (Fin n) \u2192 G\nfb : \u2191(Set.Icc (lower I\u2080 i) (upper I\u2080 i)) \u2192 Fin n \u2192\u1d47\u1d43[\u2191(face I\u2080 i)] G\nhf :\n \u2200 (x : \u211d) (hx : x \u2208 Set.Icc (lower I\u2080 i) (upper I\u2080 i)) (J : Box (Fin n)), f x J = \u2191(fb { val := x, property := hx }) J\nJ : Box (Fin (n + 1))\nhJ : J \u2264 I\u2080\nj\u271d : Fin (n + 1)\nx : \u211d\nj : Fin n\nhx : x \u2208 Set.Ioo (lower J (Fin.succAbove i j)) (upper J (Fin.succAbove i j))\nthis : \u2191(face J i) \u2264 \u2191(face I\u2080 i)\n\u22a2 Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) (splitLower J (Fin.succAbove i j) x) +\n Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i))\n (splitUpper J (Fin.succAbove i j) x) =\n (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) J\n[PROOFSTEP]\nrw [le_iff_Icc, @Box.Icc_eq_pi _ I\u2080] at hJ \n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J\u271d : Box \u03b9\ni\u271d : \u03b9\nG : Type u\ninst\u271d : AddCommGroup G\nI\u2080 : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : \u211d \u2192 Box (Fin n) \u2192 G\nfb : \u2191(Set.Icc (lower I\u2080 i) (upper I\u2080 i)) \u2192 Fin n \u2192\u1d47\u1d43[\u2191(face I\u2080 i)] G\nhf :\n \u2200 (x : \u211d) (hx : x \u2208 Set.Icc (lower I\u2080 i) (upper I\u2080 i)) (J : Box (Fin n)), f x J = \u2191(fb { val := x, property := hx }) J\nJ : Box (Fin (n + 1))\nhJ : \u2191Box.Icc J \u2286 Set.pi Set.univ fun i => Set.Icc (lower I\u2080 i) (upper I\u2080 i)\nj\u271d : Fin (n + 1)\nx : \u211d\nj : Fin n\nhx : x \u2208 Set.Ioo (lower J (Fin.succAbove i j)) (upper J (Fin.succAbove i j))\nthis : \u2191(face J i) \u2264 \u2191(face I\u2080 i)\n\u22a2 Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) (splitLower J (Fin.succAbove i j) x) +\n Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i))\n (splitUpper J (Fin.succAbove i j) x) =\n (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) J\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J\u271d : Box \u03b9\ni\u271d : \u03b9\nG : Type u\ninst\u271d : AddCommGroup G\nI\u2080 : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : \u211d \u2192 Box (Fin n) \u2192 G\nfb : \u2191(Set.Icc (lower I\u2080 i) (upper I\u2080 i)) \u2192 Fin n \u2192\u1d47\u1d43[\u2191(face I\u2080 i)] G\nhf :\n \u2200 (x : \u211d) (hx : x \u2208 Set.Icc (lower I\u2080 i) (upper I\u2080 i)) (J : Box (Fin n)), f x J = \u2191(fb { val := x, property := hx }) J\nJ : Box (Fin (n + 1))\nhJ : \u2191Box.Icc J \u2286 Set.pi Set.univ fun i => Set.Icc (lower I\u2080 i) (upper I\u2080 i)\nj\u271d : Fin (n + 1)\nx : \u211d\nj : Fin n\nhx : x \u2208 Set.Ioo (lower J (Fin.succAbove i j)) (upper J (Fin.succAbove i j))\nthis : \u2191(face J i) \u2264 \u2191(face I\u2080 i)\n\u22a2 Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) (splitLower J (Fin.succAbove i j) x) +\n Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i))\n (splitUpper J (Fin.succAbove i j) x) =\n f (upper J i) (face J i) - f (lower J i) (face J i)\n[PROOFSTEP]\nrw [hf _ (hJ J.upper_mem_Icc _ trivial), hf _ (hJ J.lower_mem_Icc _ trivial), \u2190 (fb _).map_split_add this j x, \u2190\n (fb _).map_split_add this j x]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J\u271d : Box \u03b9\ni\u271d : \u03b9\nG : Type u\ninst\u271d : AddCommGroup G\nI\u2080 : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : \u211d \u2192 Box (Fin n) \u2192 G\nfb : \u2191(Set.Icc (lower I\u2080 i) (upper I\u2080 i)) \u2192 Fin n \u2192\u1d47\u1d43[\u2191(face I\u2080 i)] G\nhf :\n \u2200 (x : \u211d) (hx : x \u2208 Set.Icc (lower I\u2080 i) (upper I\u2080 i)) (J : Box (Fin n)), f x J = \u2191(fb { val := x, property := hx }) J\nJ : Box (Fin (n + 1))\nhJ : \u2191Box.Icc J \u2286 Set.pi Set.univ fun i => Set.Icc (lower I\u2080 i) (upper I\u2080 i)\nj\u271d : Fin (n + 1)\nx : \u211d\nj : Fin n\nhx : x \u2208 Set.Ioo (lower J (Fin.succAbove i j)) (upper J (Fin.succAbove i j))\nthis : \u2191(face J i) \u2264 \u2191(face I\u2080 i)\n\u22a2 Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) (splitLower J (Fin.succAbove i j) x) +\n Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i))\n (splitUpper J (Fin.succAbove i j) x) =\n Option.elim' 0\n (\u2191(fb { val := upper J i, property := (_ : upper J i \u2208 (fun i => Set.Icc (lower I\u2080 i) (upper I\u2080 i)) i) }))\n (splitLower (face J i) j x) +\n Option.elim' 0\n (\u2191(fb { val := upper J i, property := (_ : upper J i \u2208 (fun i => Set.Icc (lower I\u2080 i) (upper I\u2080 i)) i) }))\n (splitUpper (face J i) j x) -\n (Option.elim' 0\n (\u2191(fb { val := lower J i, property := (_ : lower J i \u2208 (fun i => Set.Icc (lower I\u2080 i) (upper I\u2080 i)) i) }))\n (splitLower (face J i) j x) +\n Option.elim' 0\n (\u2191(fb { val := lower J i, property := (_ : lower J i \u2208 (fun i => Set.Icc (lower I\u2080 i) (upper I\u2080 i)) i) }))\n (splitUpper (face J i) j x))\n[PROOFSTEP]\nhave hx' : x \u2208 Ioo ((J.face i).lower j) ((J.face i).upper j) := hx\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J\u271d : Box \u03b9\ni\u271d : \u03b9\nG : Type u\ninst\u271d : AddCommGroup G\nI\u2080 : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : \u211d \u2192 Box (Fin n) \u2192 G\nfb : \u2191(Set.Icc (lower I\u2080 i) (upper I\u2080 i)) \u2192 Fin n \u2192\u1d47\u1d43[\u2191(face I\u2080 i)] G\nhf :\n \u2200 (x : \u211d) (hx : x \u2208 Set.Icc (lower I\u2080 i) (upper I\u2080 i)) (J : Box (Fin n)), f x J = \u2191(fb { val := x, property := hx }) J\nJ : Box (Fin (n + 1))\nhJ : \u2191Box.Icc J \u2286 Set.pi Set.univ fun i => Set.Icc (lower I\u2080 i) (upper I\u2080 i)\nj\u271d : Fin (n + 1)\nx : \u211d\nj : Fin n\nhx : x \u2208 Set.Ioo (lower J (Fin.succAbove i j)) (upper J (Fin.succAbove i j))\nthis : \u2191(face J i) \u2264 \u2191(face I\u2080 i)\nhx' : x \u2208 Set.Ioo (lower (face J i) j) (upper (face J i) j)\n\u22a2 Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i)) (splitLower J (Fin.succAbove i j) x) +\n Option.elim' 0 (fun J => f (upper J i) (face J i) - f (lower J i) (face J i))\n (splitUpper J (Fin.succAbove i j) x) =\n Option.elim' 0\n (\u2191(fb { val := upper J i, property := (_ : upper J i \u2208 (fun i => Set.Icc (lower I\u2080 i) (upper I\u2080 i)) i) }))\n (splitLower (face J i) j x) +\n Option.elim' 0\n (\u2191(fb { val := upper J i, property := (_ : upper J i \u2208 (fun i => Set.Icc (lower I\u2080 i) (upper I\u2080 i)) i) }))\n (splitUpper (face J i) j x) -\n (Option.elim' 0\n (\u2191(fb { val := lower J i, property := (_ : lower J i \u2208 (fun i => Set.Icc (lower I\u2080 i) (upper I\u2080 i)) i) }))\n (splitLower (face J i) j x) +\n Option.elim' 0\n (\u2191(fb { val := lower J i, property := (_ : lower J i \u2208 (fun i => Set.Icc (lower I\u2080 i) (upper I\u2080 i)) i) }))\n (splitUpper (face J i) j x))\n[PROOFSTEP]\nsimp only [Box.splitLower_def hx, Box.splitUpper_def hx, Box.splitLower_def hx', Box.splitUpper_def hx', \u2190\n WithBot.some_eq_coe, Option.elim', Box.face_mk, update_noteq (Fin.succAbove_ne _ _).symm, sub_add_sub_comm,\n update_comp_eq_of_injective _ (Fin.strictMono_succAbove i).injective j x, \u2190 hf]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nM : Type u_2\nn : \u2115\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : AddCommMonoid N\nI\u2080\u271d : WithTop (Box \u03b9)\nI J\u271d : Box \u03b9\ni\u271d : \u03b9\nG : Type u\ninst\u271d : AddCommGroup G\nI\u2080 : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : \u211d \u2192 Box (Fin n) \u2192 G\nfb : \u2191(Set.Icc (lower I\u2080 i) (upper I\u2080 i)) \u2192 Fin n \u2192\u1d47\u1d43[\u2191(face I\u2080 i)] G\nhf :\n \u2200 (x : \u211d) (hx : x \u2208 Set.Icc (lower I\u2080 i) (upper I\u2080 i)) (J : Box (Fin n)), f x J = \u2191(fb { val := x, property := hx }) J\nJ : Box (Fin (n + 1))\nhJ : \u2191Box.Icc J \u2286 Set.pi Set.univ fun i => Set.Icc (lower I\u2080 i) (upper I\u2080 i)\nj\u271d : Fin (n + 1)\nx : \u211d\nj : Fin n\nhx : x \u2208 Set.Ioo (lower J (Fin.succAbove i j)) (upper J (Fin.succAbove i j))\nthis : \u2191(face J i) \u2264 \u2191(face I\u2080 i)\nhx' : x \u2208 Set.Ioo (lower (face J i) j) (upper (face J i) j)\n\u22a2 f (upper J i)\n { lower := J.lower \u2218 Fin.succAbove i, upper := update (J.upper \u2218 Fin.succAbove i) j x,\n lower_lt_upper :=\n (_ : \u2200 (i_1 : Fin n), (J.lower \u2218 Fin.succAbove i) i_1 < update (J.upper \u2218 Fin.succAbove i) j x i_1) } +\n f (upper J i)\n { lower := update (J.lower \u2218 Fin.succAbove i) j x, upper := J.upper \u2218 Fin.succAbove i,\n lower_lt_upper :=\n (_ : \u2200 (i_1 : Fin n), update (J.lower \u2218 Fin.succAbove i) j x i_1 < (J.upper \u2218 Fin.succAbove i) i_1) } -\n (f (lower J i)\n { lower := J.lower \u2218 Fin.succAbove i, upper := update (J.upper \u2218 Fin.succAbove i) j x,\n lower_lt_upper :=\n (_ : \u2200 (i_1 : Fin n), (J.lower \u2218 Fin.succAbove i) i_1 < update (J.upper \u2218 Fin.succAbove i) j x i_1) } +\n f (lower J i)\n { lower := update (J.lower \u2218 Fin.succAbove i) j x, upper := J.upper \u2218 Fin.succAbove i,\n lower_lt_upper :=\n (_ : \u2200 (i_1 : Fin n), update (J.lower \u2218 Fin.succAbove i) j x i_1 < (J.upper \u2218 Fin.succAbove i) i_1) }) =\n f (upper J i)\n { lower := (face J i).lower, upper := update (face J i).upper j x,\n lower_lt_upper := (_ : \u2200 (x_1 : Fin n), lower (face J i) x_1 < update (face J i).upper j x x_1) } +\n f (upper J i)\n { lower := update (face J i).lower j x, upper := (face J i).upper,\n lower_lt_upper := (_ : \u2200 (x_1 : Fin n), update (face J i).lower j x x_1 < upper (face J i) x_1) } -\n (f (lower J i)\n { lower := (face J i).lower, upper := update (face J i).upper j x,\n lower_lt_upper := (_ : \u2200 (x_1 : Fin n), lower (face J i) x_1 < update (face J i).upper j x x_1) } +\n f (lower J i)\n { lower := update (face J i).lower j x, upper := (face J i).upper,\n lower_lt_upper := (_ : \u2200 (x_1 : Fin n), update (face J i).lower j x x_1 < upper (face J i) x_1) })\n[PROOFSTEP]\nsimp only [Box.face]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.BoxIntegral.Partition.Additive", "llama_tokens": 18940, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4140484931784741}} {"text": "[GOAL]\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type v\n\u22a2 #(Quotient.out #\u03b1) = #\u03b1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 \u2200 (a : Cardinal.{u}), a \u2264 a\n[PROOFSTEP]\nrintro \u27e8\u03b1\u27e9\n[GOAL]\ncase mk\n\u03b1\u271d \u03b2 : Type u\na\u271d : Cardinal.{u}\n\u03b1 : Type u\n\u22a2 Quot.mk Setoid.r \u03b1 \u2264 Quot.mk Setoid.r \u03b1\n[PROOFSTEP]\nexact \u27e8Embedding.refl _\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 \u2200 (a b c : Cardinal.{u}), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n[PROOFSTEP]\nrintro \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9 \u27e8\u03b3\u27e9 \u27e8e\u2081\u27e9 \u27e8e\u2082\u27e9\n[GOAL]\ncase mk.mk.mk.intro.intro\n\u03b1\u271d \u03b2\u271d : Type u\na\u271d : Cardinal.{u}\n\u03b1 : Type u\nb\u271d : Cardinal.{u}\n\u03b2 : Type u\nc\u271d : Cardinal.{u}\n\u03b3 : Type u\ne\u2081 : \u03b1 \u21aa \u03b2\ne\u2082 : \u03b2 \u21aa \u03b3\n\u22a2 Quot.mk Setoid.r \u03b1 \u2264 Quot.mk Setoid.r \u03b3\n[PROOFSTEP]\nexact \u27e8e\u2081.trans e\u2082\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 \u2200 (a b : Cardinal.{u}), a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\nrintro \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9 \u27e8e\u2081\u27e9 \u27e8e\u2082\u27e9\n[GOAL]\ncase mk.mk.intro.intro\n\u03b1\u271d \u03b2\u271d : Type u\na\u271d : Cardinal.{u}\n\u03b1 : Type u\nb\u271d : Cardinal.{u}\n\u03b2 : Type u\ne\u2081 : \u03b1 \u21aa \u03b2\ne\u2082 : \u03b2 \u21aa \u03b1\n\u22a2 Quot.mk Setoid.r \u03b1 = Quot.mk Setoid.r \u03b2\n[PROOFSTEP]\nexact Quotient.sound (e\u2081.antisymm e\u2082)\n[GOAL]\n\u03b1 \u03b2 : Type u\nsrc\u271d : PartialOrder Cardinal.{u} := partialOrder\n\u22a2 \u2200 (a b : Cardinal.{u}), a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\nrintro \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9\n[GOAL]\ncase mk.mk\n\u03b1\u271d \u03b2\u271d : Type u\nsrc\u271d : PartialOrder Cardinal.{u} := partialOrder\na\u271d : Cardinal.{u}\n\u03b1 : Type u\nb\u271d : Cardinal.{u}\n\u03b2 : Type u\n\u22a2 Quot.mk Setoid.r \u03b1 \u2264 Quot.mk Setoid.r \u03b2 \u2228 Quot.mk Setoid.r \u03b2 \u2264 Quot.mk Setoid.r \u03b1\n[PROOFSTEP]\napply Embedding.total\n[GOAL]\n\u03b1 \u03b2 : Type u\ns : Set \u03b1\n\u22a2 #\u2191(ULift.down \u207b\u00b9' s) = lift #\u2191s\n[PROOFSTEP]\nrw [\u2190 mk_uLift, Cardinal.eq]\n[GOAL]\n\u03b1 \u03b2 : Type u\ns : Set \u03b1\n\u22a2 Nonempty (\u2191(ULift.down \u207b\u00b9' s) \u2243 ULift \u2191s)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase val\n\u03b1 \u03b2 : Type u\ns : Set \u03b1\n\u22a2 \u2191(ULift.down \u207b\u00b9' s) \u2243 ULift \u2191s\n[PROOFSTEP]\nlet f : ULift.down \u207b\u00b9' s \u2192 ULift s := fun x \u21a6 ULift.up (restrictPreimage s ULift.down x)\n[GOAL]\ncase val\n\u03b1 \u03b2 : Type u\ns : Set \u03b1\nf : \u2191(ULift.down \u207b\u00b9' s) \u2192 ULift \u2191s := fun x => { down := restrictPreimage s ULift.down x }\n\u22a2 \u2191(ULift.down \u207b\u00b9' s) \u2243 ULift \u2191s\n[PROOFSTEP]\nhave : Function.Bijective f := ULift.up_bijective.comp (restrictPreimage_bijective _ (ULift.down_bijective))\n[GOAL]\ncase val\n\u03b1 \u03b2 : Type u\ns : Set \u03b1\nf : \u2191(ULift.down \u207b\u00b9' s) \u2192 ULift \u2191s := fun x => { down := restrictPreimage s ULift.down x }\nthis : Bijective f\n\u22a2 \u2191(ULift.down \u207b\u00b9' s) \u2243 ULift \u2191s\n[PROOFSTEP]\nexact Equiv.ofBijective f this\n[GOAL]\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\n\u22a2 c \u2264 c' \u2194 Nonempty (Quotient.out c \u21aa Quotient.out c')\n[PROOFSTEP]\ntrans\n[GOAL]\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\n\u22a2 c \u2264 c' \u2194 ?m.14684\n[PROOFSTEP]\nrw [\u2190 Quotient.out_eq c, \u2190 Quotient.out_eq c']\n[GOAL]\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\n\u22a2 Quotient.mk isEquivalent (Quotient.out c) \u2264 Quotient.mk isEquivalent (Quotient.out c') \u2194\n Nonempty (Quotient.out c \u21aa Quotient.out c')\n[PROOFSTEP]\nrw [mk'_def, mk'_def, le_def]\n[GOAL]\n\u03b1\u271d \u03b2\u271d : Type u\na b : Cardinal.{v}\n\u03b1 \u03b2 : Type v\n\u22a2 lift #\u03b1 \u2264 lift #\u03b2 \u2194 #\u03b1 \u2264 #\u03b2\n[PROOFSTEP]\nrw [\u2190 lift_umax]\n[GOAL]\n\u03b1\u271d \u03b2\u271d : Type u\na b : Cardinal.{v}\n\u03b1 \u03b2 : Type v\n\u22a2 lift #\u03b1 \u2264 lift #\u03b2 \u2194 #\u03b1 \u2264 #\u03b2\n[PROOFSTEP]\nexact lift_mk_le.{u}\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 #(Option \u03b1) = #\u03b1 + 1\n[PROOFSTEP]\nrw [(Equiv.optionEquivSumPUnit.{u, u} \u03b1).cardinal_eq, mk_sum, mk_eq_one PUnit, lift_id, lift_id]\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nh : Fintype \u03b1\n\u22a2 Fintype.card \u03b1 = Fintype.card (ULift (Fin (Fintype.card \u03b1)))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 \u2191(n + 1) = \u2191n + 1\n[PROOFSTEP]\nchange #(ULift.{u} (Fin (n + 1))) = #(ULift.{u} (Fin n)) + 1\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 #(ULift (Fin (n + 1))) = #(ULift (Fin n)) + 1\n[PROOFSTEP]\nrw [\u2190 mk_option, mk_fintype, mk_fintype]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 \u2191(Fintype.card (ULift (Fin (n + 1)))) = \u2191(Fintype.card (Option (ULift (Fin n))))\n[PROOFSTEP]\nsimp only [Fintype.card_ulift, Fintype.card_fin, Fintype.card_option]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\nc : Cardinal.{u}\n\u22a2 c ^ \u2191(n + 1) = c * c ^ \u2191n\n[PROOFSTEP]\nrw [Cardinal.cast_succ, power_add, power_one, mul_comm']\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\n\u22a2 a ^ bit1 b = a ^ b * a ^ b * a\n[PROOFSTEP]\nrw [bit1, \u2190 power_bit0, power_add, power_one]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 #Bool = 2\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 #Prop = 2\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 \u03b2 : Type u\na b c : Cardinal.{u_1}\n\u22a2 a ^ (b * c) = (a ^ b) ^ c\n[PROOFSTEP]\nrw [mul_comm b c]\n[GOAL]\n\u03b1 \u03b2 : Type u\na b c : Cardinal.{u_1}\n\u22a2 a ^ (c * b) = (a ^ b) ^ c\n[PROOFSTEP]\nexact inductionOn\u2083 a b c fun \u03b1 \u03b2 \u03b3 => mk_congr <| Equiv.curry \u03b3 \u03b2 \u03b1\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\n\u22a2 lift (bit1 a) = bit1 (lift a)\n[PROOFSTEP]\nsimp [bit1]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 lift 2 = 2\n[PROOFSTEP]\nsimp [\u2190 one_add_one_eq_two]\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 #(Set \u03b1) = 2 ^ #\u03b1\n[PROOFSTEP]\nsimp [\u2190 one_add_one_eq_two, Set, mk_arrow]\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\n\u22a2 lift (2 ^ a) = 2 ^ lift a\n[PROOFSTEP]\nsimp [\u2190 one_add_one_eq_two]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 \u2200 (a : Cardinal.{u_1}), 0 \u2264 a\n[PROOFSTEP]\nrintro \u27e8\u03b1\u27e9\n[GOAL]\ncase mk\n\u03b1\u271d \u03b2 : Type u\na\u271d : Cardinal.{u_1}\n\u03b1 : Type u_1\n\u22a2 0 \u2264 Quot.mk Setoid.r \u03b1\n[PROOFSTEP]\nexact \u27e8Embedding.ofIsEmpty\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 \u2200 {a b c d : Cardinal.{u_1}}, a \u2264 b \u2192 c \u2264 d \u2192 a + c \u2264 b + d\n[PROOFSTEP]\nrintro \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9 \u27e8\u03b3\u27e9 \u27e8\u03b4\u27e9 \u27e8e\u2081\u27e9 \u27e8e\u2082\u27e9\n[GOAL]\ncase mk.mk.mk.mk.intro.intro\n\u03b1\u271d \u03b2\u271d : Type u\na\u271d : Cardinal.{u_1}\n\u03b1 : Type u_1\nb\u271d : Cardinal.{u_1}\n\u03b2 : Type u_1\nc\u271d : Cardinal.{u_1}\n\u03b3 : Type u_1\nd\u271d : Cardinal.{u_1}\n\u03b4 : Type u_1\ne\u2081 : \u03b1 \u21aa \u03b2\ne\u2082 : \u03b3 \u21aa \u03b4\n\u22a2 Quot.mk Setoid.r \u03b1 + Quot.mk Setoid.r \u03b3 \u2264 Quot.mk Setoid.r \u03b2 + Quot.mk Setoid.r \u03b4\n[PROOFSTEP]\nexact\n \u27e8e\u2081.sumMap e\u2082\u27e9\n -- #align cardinal.add_le_add' Cardinal.add_le_add'\n[GOAL]\n\u03b1\u271d \u03b2\u271d : Type u\nsrc\u271d\u00b9 : CommSemiring Cardinal.{u} := commSemiring\nsrc\u271d : PartialOrder Cardinal.{u} := partialOrder\na b : Cardinal.{u}\n\u03b1 \u03b2 : Type u\n\u22a2 #\u03b1 * #\u03b2 = 0 \u2192 #\u03b1 = 0 \u2228 #\u03b2 = 0\n[PROOFSTEP]\nsimpa only [mul_def, mk_eq_zero_iff, isEmpty_prod] using id\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u}\n\u22a2 0 ^ c \u2264 1\n[PROOFSTEP]\nby_cases h : c = 0\n[GOAL]\ncase pos\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u}\nh : c = 0\n\u22a2 0 ^ c \u2264 1\n[PROOFSTEP]\nrw [h, power_zero]\n[GOAL]\ncase neg\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u}\nh : \u00acc = 0\n\u22a2 0 ^ c \u2264 1\n[PROOFSTEP]\nrw [zero_power h]\n[GOAL]\ncase neg\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u}\nh : \u00acc = 0\n\u22a2 0 \u2264 1\n[PROOFSTEP]\napply zero_le\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 \u2200 {a b c : Cardinal.{u_1}}, a \u2260 0 \u2192 b \u2264 c \u2192 a ^ b \u2264 a ^ c\n[PROOFSTEP]\nrintro \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9 \u27e8\u03b3\u27e9 h\u03b1 \u27e8e\u27e9\n[GOAL]\ncase mk.mk.mk.intro\n\u03b1\u271d \u03b2\u271d : Type u\na\u271d : Cardinal.{u_1}\n\u03b1 : Type u_1\nb\u271d : Cardinal.{u_1}\n\u03b2 : Type u_1\nc\u271d : Cardinal.{u_1}\n\u03b3 : Type u_1\nh\u03b1 : Quot.mk Setoid.r \u03b1 \u2260 0\ne : \u03b2 \u21aa \u03b3\n\u22a2 Quot.mk Setoid.r \u03b1 ^ Quot.mk Setoid.r \u03b2 \u2264 Quot.mk Setoid.r \u03b1 ^ Quot.mk Setoid.r \u03b3\n[PROOFSTEP]\nlet \u27e8a\u27e9 := mk_ne_zero_iff.1 h\u03b1\n[GOAL]\ncase mk.mk.mk.intro\n\u03b1\u271d \u03b2\u271d : Type u\na\u271d : Cardinal.{u_1}\n\u03b1 : Type u_1\nb\u271d : Cardinal.{u_1}\n\u03b2 : Type u_1\nc\u271d : Cardinal.{u_1}\n\u03b3 : Type u_1\nh\u03b1 : Quot.mk Setoid.r \u03b1 \u2260 0\ne : \u03b2 \u21aa \u03b3\na : \u03b1\n\u22a2 Quot.mk Setoid.r \u03b1 ^ Quot.mk Setoid.r \u03b2 \u2264 Quot.mk Setoid.r \u03b1 ^ Quot.mk Setoid.r \u03b3\n[PROOFSTEP]\nexact \u27e8@Function.Embedding.arrowCongrLeft _ _ _ \u27e8a\u27e9 e\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nhb : 1 \u2264 b\n\u22a2 a \u2264 a ^ b\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha)\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nb : Cardinal.{u_1}\nhb : 1 \u2264 b\n\u22a2 0 \u2264 0 ^ b\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nhb : 1 \u2264 b\nha : a \u2260 0\n\u22a2 a \u2264 a ^ b\n[PROOFSTEP]\nconvert power_le_power_left ha hb\n[GOAL]\ncase h.e'_3\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nhb : 1 \u2264 b\nha : a \u2260 0\n\u22a2 a = a ^ 1\n[PROOFSTEP]\nexact power_one.symm\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\n\u22a2 a < 2 ^ a\n[PROOFSTEP]\ninduction' a using Cardinal.inductionOn with \u03b1\n[GOAL]\ncase h\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 #\u03b1 < 2 ^ #\u03b1\n[PROOFSTEP]\nrw [\u2190 mk_set]\n[GOAL]\ncase h\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 #\u03b1 < #(Set \u03b1)\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u27e8singleton, fun a b => singleton_eq_singleton_iff.1\u27e9\u27e9, _\u27e9\n[GOAL]\ncase h\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 \u00acQuotient.liftOn\u2082 (#(Set \u03b1)) (#\u03b1) (fun \u03b1 \u03b2 => Nonempty (\u03b1 \u21aa \u03b2)) instLECardinal.proof_1\n[PROOFSTEP]\nrintro \u27e8\u27e8f, hf\u27e9\u27e9\n[GOAL]\ncase h.intro.mk\n\u03b1\u271d \u03b2 \u03b1 : Type u\nf : Set \u03b1 \u2192 \u03b1\nhf : Injective f\n\u22a2 False\n[PROOFSTEP]\nexact cantor_injective f hf\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 1 < #\u03b1 \u2194 Nontrivial \u03b1\n[PROOFSTEP]\nrw [\u2190 not_le, le_one_iff_subsingleton, \u2190 not_nontrivial_iff_subsingleton, Classical.not_not]\n[GOAL]\n\u03b1 \u03b2 : Type u\na b c : Cardinal.{u_1}\nh : b \u2264 c\n\u22a2 a ^ b \u2264 max (a ^ c) 1\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\n\u03b1 \u03b2 : Type u\na b c : Cardinal.{u_1}\nh : b \u2264 c\nha : a = 0\n\u22a2 a ^ b \u2264 max (a ^ c) 1\n[PROOFSTEP]\nsimp [ha, zero_power_le]\n[GOAL]\ncase neg\n\u03b1 \u03b2 : Type u\na b c : Cardinal.{u_1}\nh : b \u2264 c\nha : \u00aca = 0\n\u22a2 a ^ b \u2264 max (a ^ c) 1\n[PROOFSTEP]\nexact (power_le_power_left ha h).trans (le_max_left _ _)\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nh : \u00acAcc (fun x x_1 => x < x_1) a\n\u22a2 False\n[PROOFSTEP]\nlet \u03b9 := { c : Cardinal // \u00acAcc (\u00b7 < \u00b7) c }\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nh : \u00acAcc (fun x x_1 => x < x_1) a\n\u03b9 : Type (?u.61092 + 1) := { c // \u00acAcc (fun x x_1 => x < x_1) c }\n\u22a2 False\n[PROOFSTEP]\nlet f : \u03b9 \u2192 Cardinal := Subtype.val\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nh : \u00acAcc (fun x x_1 => x < x_1) a\n\u03b9 : Type (?u.61092 + 1) := { c // \u00acAcc (fun x x_1 => x < x_1) c }\nf : \u03b9 \u2192 Cardinal.{?u.61092} := Subtype.val\n\u22a2 False\n[PROOFSTEP]\nhaveI h\u03b9 : Nonempty \u03b9 := \u27e8\u27e8_, h\u27e9\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nh : \u00acAcc (fun x x_1 => x < x_1) a\n\u03b9 : Type (u + 1) := { c // \u00acAcc (fun x x_1 => x < x_1) c }\nf : \u03b9 \u2192 Cardinal.{u} := Subtype.val\nh\u03b9 : Nonempty \u03b9\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8\u27e8c : Cardinal, hc : \u00acAcc (\u00b7 < \u00b7) c\u27e9, \u27e8h_1 : \u2200 j, (f \u27e8c, hc\u27e9).out \u21aa (f j).out\u27e9\u27e9 :=\n Embedding.min_injective fun i => (f i).out\n[GOAL]\ncase intro.mk.intro\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nh : \u00acAcc (fun x x_1 => x < x_1) a\n\u03b9 : Type (u + 1) := { c // \u00acAcc (fun x x_1 => x < x_1) c }\nf : \u03b9 \u2192 Cardinal.{u} := Subtype.val\nh\u03b9 : Nonempty \u03b9\nc : Cardinal.{u}\nhc : \u00acAcc (fun x x_1 => x < x_1) c\nh_1 : (j : \u03b9) \u2192 Quotient.out (f { val := c, property := hc }) \u21aa Quotient.out (f j)\n\u22a2 False\n[PROOFSTEP]\nrefine hc (Acc.intro _ fun j h' => byContradiction fun hj => h'.2 ?_)\n[GOAL]\ncase intro.mk.intro\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nh : \u00acAcc (fun x x_1 => x < x_1) a\n\u03b9 : Type (u + 1) := { c // \u00acAcc (fun x x_1 => x < x_1) c }\nf : \u03b9 \u2192 Cardinal.{u} := Subtype.val\nh\u03b9 : Nonempty \u03b9\nc : Cardinal.{u}\nhc : \u00acAcc (fun x x_1 => x < x_1) c\nh_1 : (j : \u03b9) \u2192 Quotient.out (f { val := c, property := hc }) \u21aa Quotient.out (f j)\nj : Cardinal.{u}\nh' : j < c\nhj : \u00acAcc (fun x x_1 => x < x_1) j\n\u22a2 Quotient.liftOn\u2082 c j (fun \u03b1 \u03b2 => Nonempty (\u03b1 \u21aa \u03b2)) instLECardinal.proof_1\n[PROOFSTEP]\nhave : #_ \u2264 #_ := \u27e8h_1 \u27e8j, hj\u27e9\u27e9\n[GOAL]\ncase intro.mk.intro\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nh : \u00acAcc (fun x x_1 => x < x_1) a\n\u03b9 : Type (u + 1) := { c // \u00acAcc (fun x x_1 => x < x_1) c }\nf : \u03b9 \u2192 Cardinal.{u} := Subtype.val\nh\u03b9 : Nonempty \u03b9\nc : Cardinal.{u}\nhc : \u00acAcc (fun x x_1 => x < x_1) c\nh_1 : (j : \u03b9) \u2192 Quotient.out (f { val := c, property := hc }) \u21aa Quotient.out (f j)\nj : Cardinal.{u}\nh' : j < c\nhj : \u00acAcc (fun x x_1 => x < x_1) j\nthis : #(Quotient.out (f { val := c, property := hc })) \u2264 #(Quotient.out (f { val := j, property := hj }))\n\u22a2 Quotient.liftOn\u2082 c j (fun \u03b1 \u03b2 => Nonempty (\u03b1 \u21aa \u03b2)) instLECardinal.proof_1\n[PROOFSTEP]\nsimpa only [mk_out] using this\n[GOAL]\n\u03b1 \u03b2 : Type u\na\u271d b\u271d : Cardinal.{?u.64039}\n\u22a2 (fun c => sInf {c' | c < c'}) a\u271d \u2264 b\u271d \u2192 a\u271d < b\u271d\n[PROOFSTEP]\napply lt_of_lt_of_le <| csInf_mem <| exists_gt _\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u}\n\u22a2 c + 1 \u2264 succ c\n[PROOFSTEP]\nhave : Set.Nonempty {c' | c < c'} := exists_gt c\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u}\nthis : Set.Nonempty {c' | c < c'}\n\u22a2 c + 1 \u2264 succ c\n[PROOFSTEP]\nsimp_rw [succ_def, le_csInf_iff'' this, mem_setOf]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u}\nthis : Set.Nonempty {c' | c < c'}\n\u22a2 \u2200 (b : Cardinal.{u}), c < b \u2192 c + 1 \u2264 b\n[PROOFSTEP]\nintro b hlt\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u}\nthis : Set.Nonempty {c' | c < c'}\nb : Cardinal.{u}\nhlt : c < b\n\u22a2 c + 1 \u2264 b\n[PROOFSTEP]\nrcases b, c with \u27e8\u27e8\u03b2\u27e9, \u27e8\u03b3\u27e9\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 \u03b2\u271d : Type u\nc b : Cardinal.{u}\n\u03b2 \u03b3 : Type u\nthis : Set.Nonempty {c' | Quot.mk Setoid.r \u03b3 < c'}\nhlt : Quot.mk Setoid.r \u03b3 < Quot.mk Setoid.r \u03b2\n\u22a2 Quot.mk Setoid.r \u03b3 + 1 \u2264 Quot.mk Setoid.r \u03b2\n[PROOFSTEP]\ncases' le_of_lt hlt with f\n[GOAL]\ncase mk.mk.intro\n\u03b1 \u03b2\u271d : Type u\nc b : Cardinal.{u}\n\u03b2 \u03b3 : Type u\nthis : Set.Nonempty {c' | Quot.mk Setoid.r \u03b3 < c'}\nhlt : Quot.mk Setoid.r \u03b3 < Quot.mk Setoid.r \u03b2\nf : \u03b3 \u21aa \u03b2\n\u22a2 Quot.mk Setoid.r \u03b3 + 1 \u2264 Quot.mk Setoid.r \u03b2\n[PROOFSTEP]\nhave : \u00acSurjective f := fun hn => (not_le_of_lt hlt) (mk_le_of_surjective hn)\n[GOAL]\ncase mk.mk.intro\n\u03b1 \u03b2\u271d : Type u\nc b : Cardinal.{u}\n\u03b2 \u03b3 : Type u\nthis\u271d : Set.Nonempty {c' | Quot.mk Setoid.r \u03b3 < c'}\nhlt : Quot.mk Setoid.r \u03b3 < Quot.mk Setoid.r \u03b2\nf : \u03b3 \u21aa \u03b2\nthis : \u00acSurjective \u2191f\n\u22a2 Quot.mk Setoid.r \u03b3 + 1 \u2264 Quot.mk Setoid.r \u03b2\n[PROOFSTEP]\nsimp only [Surjective, not_forall] at this \n[GOAL]\ncase mk.mk.intro\n\u03b1 \u03b2\u271d : Type u\nc b : Cardinal.{u}\n\u03b2 \u03b3 : Type u\nthis\u271d : Set.Nonempty {c' | Quot.mk Setoid.r \u03b3 < c'}\nhlt : Quot.mk Setoid.r \u03b3 < Quot.mk Setoid.r \u03b2\nf : \u03b3 \u21aa \u03b2\nthis : \u2203 x, \u00ac\u2203 a, \u2191f a = x\n\u22a2 Quot.mk Setoid.r \u03b3 + 1 \u2264 Quot.mk Setoid.r \u03b2\n[PROOFSTEP]\nrcases this with \u27e8b, hb\u27e9\n[GOAL]\ncase mk.mk.intro.intro\n\u03b1 \u03b2\u271d : Type u\nc b\u271d : Cardinal.{u}\n\u03b2 \u03b3 : Type u\nthis : Set.Nonempty {c' | Quot.mk Setoid.r \u03b3 < c'}\nhlt : Quot.mk Setoid.r \u03b3 < Quot.mk Setoid.r \u03b2\nf : \u03b3 \u21aa \u03b2\nb : \u03b2\nhb : \u00ac\u2203 a, \u2191f a = b\n\u22a2 Quot.mk Setoid.r \u03b3 + 1 \u2264 Quot.mk Setoid.r \u03b2\n[PROOFSTEP]\ncalc\n #\u03b3 + 1 = #(Option \u03b3) := mk_option.symm\n _ \u2264 #\u03b2 := (f.optionElim b hb).cardinal_le\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Cardinal.{max u_1 u_2}\ni : \u03b9\n\u22a2 f i \u2264 sum f\n[PROOFSTEP]\nrw [\u2190 Quotient.out_eq (f i)]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Cardinal.{max u_1 u_2}\ni : \u03b9\n\u22a2 Quotient.mk isEquivalent (Quotient.out (f i)) \u2264 sum f\n[PROOFSTEP]\nexact \u27e8\u27e8fun a => \u27e8i, a\u27e9, fun a b h => by injection h\u27e9\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Cardinal.{max u_1 u_2}\ni : \u03b9\na b : Quotient.out (f i)\nh : (fun a => { fst := i, snd := a }) a = (fun a => { fst := i, snd := a }) b\n\u22a2 a = b\n[PROOFSTEP]\ninjection h\n[GOAL]\n\u03b1 \u03b2 \u03b9 : Type u\na : Cardinal.{u}\n\u22a2 (sum fun x => a) = #\u03b9 * a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\n\u22a2 sum (f + g) = sum f + sum g\n[PROOFSTEP]\nhave := mk_congr (Equiv.sigmaSumDistrib (Quotient.out \u2218 f) (Quotient.out \u2218 g))\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nthis :\n #((i : \u03b9) \u00d7 ((Quotient.out \u2218 f) i \u2295 (Quotient.out \u2218 g) i)) =\n #((i : \u03b9) \u00d7 (Quotient.out \u2218 f) i \u2295 (i : \u03b9) \u00d7 (Quotient.out \u2218 g) i)\n\u22a2 sum (f + g) = sum f + sum g\n[PROOFSTEP]\nsimp only [comp_apply, mk_sigma, mk_sum, mk_out, lift_id] at this \n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nthis : (sum fun i => f i + g i) = (sum fun i => f i) + sum fun i => g i\n\u22a2 sum (f + g) = sum f + sum g\n[PROOFSTEP]\nexact this\n[GOAL]\n\u03b1 \u03b2 \u03b9 : Type u\nf : \u03b9 \u2192 Cardinal.{v}\na : \u03b9\n\u22a2 Nonempty (Quotient.out (f a) \u2243 Quotient.out ((fun i => lift (f i)) a))\n[PROOFSTEP]\nrw [\u2190 lift_mk_eq.{_, _, v}, mk_out, mk_out, lift_lift]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i \u2264 g i\ni : \u03b9\n\u22a2 Nonempty (Quotient.out (f i) \u21aa Quotient.out (g (\u2191(Embedding.refl \u03b9) i)))\n[PROOFSTEP]\nhave := H i\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i \u2264 g i\ni : \u03b9\nthis : f i \u2264 g i\n\u22a2 Nonempty (Quotient.out (f i) \u21aa Quotient.out (g (\u2191(Embedding.refl \u03b9) i)))\n[PROOFSTEP]\nrwa [\u2190 Quot.out_eq (f i), \u2190 Quot.out_eq (g i)] at this \n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u}\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (b : \u03b2), #\u2191(f \u207b\u00b9' {b}) \u2264 c\n\u22a2 #\u03b1 \u2264 #\u03b2 * c\n[PROOFSTEP]\nsimpa only [\u2190 mk_congr (@Equiv.sigmaFiberEquiv \u03b1 \u03b2 f), mk_sigma, \u2190 sum_const'] using sum_le_sum _ _ hf\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nc : Cardinal.{max u v}\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (b : \u03b2), lift #\u2191(f \u207b\u00b9' {b}) \u2264 c\nb : \u03b2\n\u22a2 \u2191(\u2191Equiv.ulift '' ((fun x => { down := f x.down }) \u207b\u00b9' {{ down := b }})) \u2243 \u2191(f \u207b\u00b9' {b})\n[PROOFSTEP]\nrw [Equiv.image_eq_preimage]\n /- Porting note: Need to insert the following `have` b/c bad fun coercion\n behaviour for Equivs -/\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nc : Cardinal.{max u v}\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (b : \u03b2), lift #\u2191(f \u207b\u00b9' {b}) \u2264 c\nb : \u03b2\n\u22a2 \u2191(\u2191Equiv.ulift.symm \u207b\u00b9' ((fun x => { down := f x.down }) \u207b\u00b9' {{ down := b }})) \u2243 \u2191(f \u207b\u00b9' {b})\n[PROOFSTEP]\nhave : FunLike.coe (Equiv.symm (Equiv.ulift (\u03b1 := \u03b1))) = ULift.up (\u03b1 := \u03b1) := rfl\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nc : Cardinal.{max u v}\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (b : \u03b2), lift #\u2191(f \u207b\u00b9' {b}) \u2264 c\nb : \u03b2\nthis : \u2191Equiv.ulift.symm = ULift.up\n\u22a2 \u2191(\u2191Equiv.ulift.symm \u207b\u00b9' ((fun x => { down := f x.down }) \u207b\u00b9' {{ down := b }})) \u2243 \u2191(f \u207b\u00b9' {b})\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nc : Cardinal.{max u v}\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (b : \u03b2), lift #\u2191(f \u207b\u00b9' {b}) \u2264 c\nb : \u03b2\nthis : \u2191Equiv.ulift.symm = ULift.up\n\u22a2 \u2191(ULift.up \u207b\u00b9' ((fun x => { down := f x.down }) \u207b\u00b9' {{ down := b }})) \u2243 \u2191(f \u207b\u00b9' {b})\n[PROOFSTEP]\nsimp only [preimage, mem_singleton_iff, ULift.up_inj, mem_setOf_eq, coe_setOf]\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nc : Cardinal.{max u v}\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (b : \u03b2), lift #\u2191(f \u207b\u00b9' {b}) \u2264 c\nb : \u03b2\nthis : \u2191Equiv.ulift.symm = ULift.up\n\u22a2 { x // f x = b } \u2243 { x // f x = b }\n[PROOFSTEP]\nexact Equiv.refl _\n[GOAL]\n\u03b1 \u03b2 \u03b9 : Type u\nf : \u03b9 \u2192 Cardinal.{max u v}\n\u22a2 ?m.81660 f \u2208 upperBounds (range f)\n[PROOFSTEP]\nrintro a\n \u27e8i, rfl\u27e9\n -- Porting note: Added universe reference below\n[GOAL]\ncase intro\n\u03b1 \u03b2 \u03b9 : Type u\nf : \u03b9 \u2192 Cardinal.{max u v}\ni : \u03b9\n\u22a2 f i \u2264 ?m.81660 f\n[PROOFSTEP]\nexact le_sum.{v, u} f i\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\n\u22a2 Small.{u, u + 1} \u2191(Iic a)\n[PROOFSTEP]\nrw [\u2190 mk_out a]\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\n\u22a2 Small.{u, u + 1} \u2191(Iic #(Quotient.out a))\n[PROOFSTEP]\napply @small_of_surjective (Set a.out) (Iic #a.out) _ fun x => \u27e8#x, mk_set_le x\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\n\u22a2 Surjective fun x => { val := #\u2191x, property := (_ : #\u2191x \u2264 #(Quotient.out a)) }\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase mk\n\u03b1 \u03b2 : Type u\na x : Cardinal.{u}\nhx : x \u2208 Iic #(Quotient.out a)\n\u22a2 \u2203 a_1, (fun x => { val := #\u2191x, property := (_ : #\u2191x \u2264 #(Quotient.out a)) }) a_1 = { val := x, property := hx }\n[PROOFSTEP]\nsimpa using le_mk_iff_exists_set.1 hx\n[GOAL]\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u}\n\u22a2 Small.{u, u + 1} \u2191s \u2192 BddAbove s\n[PROOFSTEP]\nrintro \u27e8\u03b9, \u27e8e\u27e9\u27e9\n[GOAL]\ncase mk.intro.intro\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u}\n\u03b9 : Type u\ne : \u2191s \u2243 \u03b9\n\u22a2 BddAbove s\n[PROOFSTEP]\nsuffices (range fun x : \u03b9 => (e.symm x).1) = s by\n rw [\u2190 this]\n apply bddAbove_range.{u, u}\n[GOAL]\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u}\n\u03b9 : Type u\ne : \u2191s \u2243 \u03b9\nthis : (range fun x => \u2191(\u2191e.symm x)) = s\n\u22a2 BddAbove s\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u}\n\u03b9 : Type u\ne : \u2191s \u2243 \u03b9\nthis : (range fun x => \u2191(\u2191e.symm x)) = s\n\u22a2 BddAbove (range fun x => \u2191(\u2191e.symm x))\n[PROOFSTEP]\napply bddAbove_range.{u, u}\n[GOAL]\ncase mk.intro.intro\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u}\n\u03b9 : Type u\ne : \u2191s \u2243 \u03b9\n\u22a2 (range fun x => \u2191(\u2191e.symm x)) = s\n[PROOFSTEP]\next x\n[GOAL]\ncase mk.intro.intro.h\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u}\n\u03b9 : Type u\ne : \u2191s \u2243 \u03b9\nx : Cardinal.{u}\n\u22a2 (x \u2208 range fun x => \u2191(\u2191e.symm x)) \u2194 x \u2208 s\n[PROOFSTEP]\nrefine' \u27e8_, fun hx => \u27e8e \u27e8x, hx\u27e9, _\u27e9\u27e9\n[GOAL]\ncase mk.intro.intro.h.refine'_1\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u}\n\u03b9 : Type u\ne : \u2191s \u2243 \u03b9\nx : Cardinal.{u}\n\u22a2 (x \u2208 range fun x => \u2191(\u2191e.symm x)) \u2192 x \u2208 s\n[PROOFSTEP]\nrintro \u27e8a, rfl\u27e9\n[GOAL]\ncase mk.intro.intro.h.refine'_1.intro\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u}\n\u03b9 : Type u\ne : \u2191s \u2243 \u03b9\na : \u03b9\n\u22a2 (fun x => \u2191(\u2191e.symm x)) a \u2208 s\n[PROOFSTEP]\nexact (e.symm a).2\n[GOAL]\ncase mk.intro.intro.h.refine'_2\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u}\n\u03b9 : Type u\ne : \u2191s \u2243 \u03b9\nx : Cardinal.{u}\nhx : x \u2208 s\n\u22a2 (fun x => \u2191(\u2191e.symm x)) (\u2191e { val := x, property := hx }) = x\n[PROOFSTEP]\nsimp_rw [Equiv.symm_apply_apply]\n[GOAL]\n\u03b1 \u03b2 : Type u\nf : Cardinal.{u} \u2192 Cardinal.{max u v}\ns : Set Cardinal.{u}\nhs : BddAbove s\n\u22a2 BddAbove (f '' s)\n[PROOFSTEP]\nrw [bddAbove_iff_small] at hs \u22a2\n -- Porting note: added universes below\n[GOAL]\n\u03b1 \u03b2 : Type u\nf : Cardinal.{u} \u2192 Cardinal.{max u v}\ns : Set Cardinal.{u}\nhs : Small.{u, u + 1} \u2191s\n\u22a2 Small.{max u v, (max u v) + 1} \u2191(f '' s)\n[PROOFSTEP]\nexact small_lift.{_, v, _} _\n[GOAL]\n\u03b1 \u03b2 \u03b9 : Type u\nf : \u03b9 \u2192 Cardinal.{v}\nhf : BddAbove (range f)\ng : Cardinal.{v} \u2192 Cardinal.{max v w}\n\u22a2 BddAbove (range (g \u2218 f))\n[PROOFSTEP]\nrw [range_comp]\n[GOAL]\n\u03b1 \u03b2 \u03b9 : Type u\nf : \u03b9 \u2192 Cardinal.{v}\nhf : BddAbove (range f)\ng : Cardinal.{v} \u2192 Cardinal.{max v w}\n\u22a2 BddAbove (g '' range f)\n[PROOFSTEP]\nexact bddAbove_image.{v, w} g hf\n[GOAL]\n\u03b1 \u03b2 \u03b9 : Type u\nf : \u03b9 \u2192 Cardinal.{max u v}\n\u22a2 sum f \u2264 lift #\u03b9 * iSup f\n[PROOFSTEP]\nrw [\u2190 (iSup f).lift_id, \u2190 lift_umax, lift_umax.{max u v, u}, \u2190 sum_const]\n[GOAL]\n\u03b1 \u03b2 \u03b9 : Type u\nf : \u03b9 \u2192 Cardinal.{max u v}\n\u22a2 sum f \u2264 sum fun x => iSup f\n[PROOFSTEP]\nexact sum_le_sum _ _ (le_ciSup <| bddAbove_range.{u, v} f)\n[GOAL]\n\u03b1 \u03b2 \u03b9 : Type u\nf : \u03b9 \u2192 Cardinal.{u}\n\u22a2 sum f \u2264 #\u03b9 * iSup f\n[PROOFSTEP]\nrw [\u2190 lift_id #\u03b9]\n[GOAL]\n\u03b1 \u03b2 \u03b9 : Type u\nf : \u03b9 \u2192 Cardinal.{u}\n\u22a2 sum f \u2264 lift #\u03b9 * iSup f\n[PROOFSTEP]\nexact sum_le_iSup_lift f\n[GOAL]\n\u03b1 \u03b2 : Type u\nf : \u2115 \u2192 Cardinal.{u}\n\u22a2 sum f = f 0 + sum fun i => f (i + 1)\n[PROOFSTEP]\nrefine' (Equiv.sigmaNatSucc fun i => Quotient.out (f i)).cardinal_eq.trans _\n[GOAL]\n\u03b1 \u03b2 : Type u\nf : \u2115 \u2192 Cardinal.{u}\n\u22a2 #(Quotient.out (f 0) \u2295 (n : \u2115) \u00d7 Quotient.out (f (n + 1))) = f 0 + sum fun i => f (i + 1)\n[PROOFSTEP]\nsimp only [mk_sum, mk_out, lift_id, mk_sigma]\n[GOAL]\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type (max u v)\ninst\u271d : Small.{v, max u v} \u03b1\n\u22a2 lift #(Shrink \u03b1) = #\u03b1\n[PROOFSTEP]\nrw [\u2190 lift_umax', lift_mk_shrink.{max u v, v, 0} \u03b1, \u2190 lift_umax, lift_id]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i \u2264 g i\ni : \u03b9\n\u22a2 Nonempty (Quotient.out (f i) \u21aa Quotient.out (g i))\n[PROOFSTEP]\nhave := H i\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i \u2264 g i\ni : \u03b9\nthis : f i \u2264 g i\n\u22a2 Nonempty (Quotient.out (f i) \u21aa Quotient.out (g i))\n[PROOFSTEP]\nrwa [\u2190 mk_out (f i), \u2190 mk_out (g i)] at this \n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Cardinal.{u}\n\u22a2 prod f = 0 \u2194 \u2203 i, f i = 0\n[PROOFSTEP]\nlift f to \u03b9 \u2192 Type u using fun _ => trivial\n[GOAL]\ncase intro\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Type u\n\u22a2 (prod fun i => #(f i)) = 0 \u2194 \u2203 i, (fun i => #(f i)) i = 0\n[PROOFSTEP]\nsimp only [mk_eq_zero_iff, \u2190 mk_pi, isEmpty_pi]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Cardinal.{u_2}\n\u22a2 prod f \u2260 0 \u2194 \u2200 (i : \u03b9), f i \u2260 0\n[PROOFSTEP]\nsimp [prod_eq_zero]\n[GOAL]\n\u03b1 \u03b2 \u03b9 : Type u\nc : \u03b9 \u2192 Cardinal.{v}\n\u22a2 lift (prod c) = prod fun i => lift (c i)\n[PROOFSTEP]\nlift c to \u03b9 \u2192 Type v using fun _ => trivial\n[GOAL]\ncase intro\n\u03b1 \u03b2 \u03b9 : Type u\nc : \u03b9 \u2192 Type v\n\u22a2 lift (prod fun i => #(c i)) = prod fun i => lift ((fun i => #(c i)) i)\n[PROOFSTEP]\nsimp only [\u2190 mk_pi, \u2190 mk_uLift]\n[GOAL]\ncase intro\n\u03b1 \u03b2 \u03b9 : Type u\nc : \u03b9 \u2192 Type v\n\u22a2 #(ULift ((i : \u03b9) \u2192 c i)) = #((i : \u03b9) \u2192 ULift (c i))\n[PROOFSTEP]\nexact mk_congr (Equiv.ulift.trans <| Equiv.piCongrRight fun i => Equiv.ulift.symm)\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nh : Fintype \u03b1\nf : \u03b1 \u2192 Cardinal.{v}\n\u22a2 prod f = lift (\u220f i : \u03b1, f i)\n[PROOFSTEP]\nrevert f\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nh : Fintype \u03b1\n\u22a2 \u2200 (f : \u03b1 \u2192 Cardinal.{v}), prod f = lift (\u220f i : \u03b1, f i)\n[PROOFSTEP]\nrefine' Fintype.induction_empty_option _ _ _ \u03b1 (h_fintype := h)\n[GOAL]\ncase refine'_1\n\u03b1\u271d \u03b2 \u03b1 : Type u\nh : Fintype \u03b1\n\u22a2 \u2200 (\u03b1 \u03b2 : Type u) [inst : Fintype \u03b2] (e : \u03b1 \u2243 \u03b2),\n (\u2200 (f : \u03b1 \u2192 Cardinal.{v}), prod f = lift (\u220f i : \u03b1, f i)) \u2192 \u2200 (f : \u03b2 \u2192 Cardinal.{v}), prod f = lift (\u220f i : \u03b2, f i)\n[PROOFSTEP]\nintro \u03b1 \u03b2 h\u03b2 e h f\n[GOAL]\ncase refine'_1\n\u03b1\u271d\u00b9 \u03b2\u271d \u03b1\u271d : Type u\nh\u271d : Fintype \u03b1\u271d\n\u03b1 \u03b2 : Type u\nh\u03b2 : Fintype \u03b2\ne : \u03b1 \u2243 \u03b2\nh : \u2200 (f : \u03b1 \u2192 Cardinal.{v}), prod f = lift (\u220f i : \u03b1, f i)\nf : \u03b2 \u2192 Cardinal.{v}\n\u22a2 prod f = lift (\u220f i : \u03b2, f i)\n[PROOFSTEP]\nletI := Fintype.ofEquiv \u03b2 e.symm\n[GOAL]\ncase refine'_1\n\u03b1\u271d\u00b9 \u03b2\u271d \u03b1\u271d : Type u\nh\u271d : Fintype \u03b1\u271d\n\u03b1 \u03b2 : Type u\nh\u03b2 : Fintype \u03b2\ne : \u03b1 \u2243 \u03b2\nh : \u2200 (f : \u03b1 \u2192 Cardinal.{v}), prod f = lift (\u220f i : \u03b1, f i)\nf : \u03b2 \u2192 Cardinal.{v}\nthis : Fintype \u03b1 := Fintype.ofEquiv \u03b2 e.symm\n\u22a2 prod f = lift (\u220f i : \u03b2, f i)\n[PROOFSTEP]\nrw [\u2190 e.prod_comp f, \u2190 h]\n[GOAL]\ncase refine'_1\n\u03b1\u271d\u00b9 \u03b2\u271d \u03b1\u271d : Type u\nh\u271d : Fintype \u03b1\u271d\n\u03b1 \u03b2 : Type u\nh\u03b2 : Fintype \u03b2\ne : \u03b1 \u2243 \u03b2\nh : \u2200 (f : \u03b1 \u2192 Cardinal.{v}), prod f = lift (\u220f i : \u03b1, f i)\nf : \u03b2 \u2192 Cardinal.{v}\nthis : Fintype \u03b1 := Fintype.ofEquiv \u03b2 e.symm\n\u22a2 prod f = prod fun i => f (\u2191e i)\n[PROOFSTEP]\nexact mk_congr (e.piCongrLeft _).symm\n[GOAL]\ncase refine'_2\n\u03b1\u271d \u03b2 \u03b1 : Type u\nh : Fintype \u03b1\n\u22a2 \u2200 (f : PEmpty \u2192 Cardinal.{v}), prod f = lift (\u220f i : PEmpty, f i)\n[PROOFSTEP]\nintro f\n[GOAL]\ncase refine'_2\n\u03b1\u271d \u03b2 \u03b1 : Type u\nh : Fintype \u03b1\nf : PEmpty \u2192 Cardinal.{v}\n\u22a2 prod f = lift (\u220f i : PEmpty, f i)\n[PROOFSTEP]\nrw [Fintype.univ_pempty, Finset.prod_empty, lift_one, Cardinal.prod, mk_eq_one]\n[GOAL]\ncase refine'_3\n\u03b1\u271d \u03b2 \u03b1 : Type u\nh : Fintype \u03b1\n\u22a2 \u2200 (\u03b1 : Type u) [inst : Fintype \u03b1],\n (\u2200 (f : \u03b1 \u2192 Cardinal.{v}), prod f = lift (\u220f i : \u03b1, f i)) \u2192\n \u2200 (f : Option \u03b1 \u2192 Cardinal.{v}), prod f = lift (\u220f i : Option \u03b1, f i)\n[PROOFSTEP]\nintro \u03b1 h\u03b1 h f\n[GOAL]\ncase refine'_3\n\u03b1\u271d\u00b9 \u03b2 \u03b1\u271d : Type u\nh\u271d : Fintype \u03b1\u271d\n\u03b1 : Type u\nh\u03b1 : Fintype \u03b1\nh : \u2200 (f : \u03b1 \u2192 Cardinal.{v}), prod f = lift (\u220f i : \u03b1, f i)\nf : Option \u03b1 \u2192 Cardinal.{v}\n\u22a2 prod f = lift (\u220f i : Option \u03b1, f i)\n[PROOFSTEP]\nrw [Cardinal.prod, mk_congr Equiv.piOptionEquivProd, mk_prod, lift_umax'.{v, u}, mk_out, \u2190 Cardinal.prod, lift_prod,\n Fintype.prod_option, lift_mul, \u2190 h fun a => f (some a)]\n[GOAL]\ncase refine'_3\n\u03b1\u271d\u00b9 \u03b2 \u03b1\u271d : Type u\nh\u271d : Fintype \u03b1\u271d\n\u03b1 : Type u\nh\u03b1 : Fintype \u03b1\nh : \u2200 (f : \u03b1 \u2192 Cardinal.{v}), prod f = lift (\u220f i : \u03b1, f i)\nf : Option \u03b1 \u2192 Cardinal.{v}\n\u22a2 (lift (f none) * prod fun i => lift (f (some i))) = lift (f none) * prod fun a => f (some a)\n[PROOFSTEP]\nsimp only [lift_id]\n[GOAL]\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{v}\n\u22a2 lift (sInf s) = sInf (lift '' s)\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | hs)\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\n\u22a2 lift (sInf \u2205) = sInf (lift '' \u2205)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{v}\nhs : Set.Nonempty s\n\u22a2 lift (sInf s) = sInf (lift '' s)\n[PROOFSTEP]\nexact lift_monotone.map_csInf hs\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 Cardinal.{v}\n\u22a2 lift (iInf f) = \u2a05 (i : \u03b9), lift (f i)\n[PROOFSTEP]\nunfold iInf\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 Cardinal.{v}\n\u22a2 lift (sInf (range f)) = sInf (range fun i => lift (f i))\n[PROOFSTEP]\nconvert lift_sInf (range f)\n[GOAL]\ncase h.e'_3.h.e'_3\n\u03b1 \u03b2 : Type u\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 Cardinal.{v}\n\u22a2 (range fun i => lift (f i)) = lift '' range f\n[PROOFSTEP]\nsimp_rw [\u2190 comp_apply (f := lift), range_comp]\n[GOAL]\n\u03b1\u271d \u03b2\u271d : Type u\na : Cardinal.{u}\nb : Cardinal.{max u v}\n\u03b1 : Type u\n\u03b2 : Type (max u v)\n\u22a2 #\u03b2 \u2264 lift #\u03b1 \u2192 \u2203 a', lift a' = #\u03b2\n[PROOFSTEP]\nrw [\u2190 lift_id #\u03b2, \u2190 lift_umax, \u2190 lift_umax.{u, v}, lift_mk_le.{v}]\n[GOAL]\n\u03b1\u271d \u03b2\u271d : Type u\na : Cardinal.{u}\nb : Cardinal.{max u v}\n\u03b1 : Type u\n\u03b2 : Type (max u v)\n\u22a2 Nonempty (\u03b2 \u21aa \u03b1) \u2192 \u2203 a', lift a' = lift #\u03b2\n[PROOFSTEP]\nexact fun \u27e8f\u27e9 =>\n \u27e8#(Set.range f),\n Eq.symm <|\n lift_mk_eq.{_, _, v}.2\n \u27e8Function.Embedding.equivOfSurjective (Embedding.codRestrict _ f Set.mem_range_self) fun \u27e8a, \u27e8b, e\u27e9\u27e9 =>\n \u27e8b, Subtype.eq e\u27e9\u27e9\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nh : lift (succ a) > succ (lift a)\n\u22a2 False\n[PROOFSTEP]\nrcases lt_lift_iff.1 h with \u27e8b, e, h\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nh\u271d : lift (succ a) > succ (lift a)\nb : Cardinal.{u}\ne : lift b = succ (lift a)\nh : b < succ a\n\u22a2 False\n[PROOFSTEP]\nrw [lt_succ_iff, \u2190 lift_le, e] at h \n[GOAL]\ncase intro.intro\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nh\u271d : lift (succ a) > succ (lift a)\nb : Cardinal.{u}\ne : lift b = succ (lift a)\nh : succ (lift a) \u2264 lift a\n\u22a2 False\n[PROOFSTEP]\nexact h.not_lt (lt_succ _)\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nb : Cardinal.{v}\n\u22a2 lift a = lift b \u2194 lift a = lift b\n[PROOFSTEP]\nrw [\u2190 lift_lift.{v, w, u}, \u2190 lift_lift.{u, w, v}, lift_inj]\n[GOAL]\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u_1}\nhs : BddAbove s\n\u22a2 lift (sSup s) = sSup (lift '' s)\n[PROOFSTEP]\napply ((le_csSup_iff' (bddAbove_image.{_, u} _ hs)).2 fun c hc => _).antisymm (csSup_le' _)\n[GOAL]\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u_1}\nhs : BddAbove s\n\u22a2 \u2200 (c : Cardinal.{max u_1 u}), c \u2208 upperBounds (lift '' s) \u2192 lift (sSup s) \u2264 c\n[PROOFSTEP]\nintro c hc\n[GOAL]\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u_1}\nhs : BddAbove s\nc : Cardinal.{max u_1 u}\nhc : c \u2208 upperBounds (lift '' s)\n\u22a2 lift (sSup s) \u2264 c\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u_1}\nhs : BddAbove s\nc : Cardinal.{max u_1 u}\nhc : c \u2208 upperBounds (lift '' s)\nh : \u00aclift (sSup s) \u2264 c\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8d, rfl\u27e9 := Cardinal.lift_down (not_le.1 h).le\n[GOAL]\ncase intro\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u_1}\nhs : BddAbove s\nd : Cardinal.{u_1}\nhc : lift d \u2208 upperBounds (lift '' s)\nh : \u00aclift (sSup s) \u2264 lift d\n\u22a2 False\n[PROOFSTEP]\nsimp_rw [lift_le] at h hc \n[GOAL]\ncase intro\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u_1}\nhs : BddAbove s\nd : Cardinal.{u_1}\nhc : lift d \u2208 upperBounds (lift '' s)\nh : \u00acsSup s \u2264 d\n\u22a2 False\n[PROOFSTEP]\nrw [csSup_le_iff' hs] at h \n[GOAL]\ncase intro\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u_1}\nhs : BddAbove s\nd : Cardinal.{u_1}\nhc : lift d \u2208 upperBounds (lift '' s)\nh : \u00ac\u2200 (x : Cardinal.{u_1}), x \u2208 s \u2192 x \u2264 d\n\u22a2 False\n[PROOFSTEP]\nexact h fun a ha => lift_le.1 <| hc (mem_image_of_mem _ ha)\n[GOAL]\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u_1}\nhs : BddAbove s\n\u22a2 lift (sSup s) \u2208 upperBounds (lift '' s)\n[PROOFSTEP]\nrintro i \u27e8j, hj, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 \u03b2 : Type u\ns : Set Cardinal.{u_1}\nhs : BddAbove s\nj : Cardinal.{u_1}\nhj : j \u2208 s\n\u22a2 lift j \u2264 lift (sSup s)\n[PROOFSTEP]\nexact lift_le.2 (le_csSup hs hj)\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type v\nf : \u03b9 \u2192 Cardinal.{w}\nhf : BddAbove (range f)\n\u22a2 lift (iSup f) = \u2a06 (i : \u03b9), lift (f i)\n[PROOFSTEP]\nrw [iSup, iSup, lift_sSup hf, \u2190 range_comp]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type v\nf : \u03b9 \u2192 Cardinal.{w}\nhf : BddAbove (range f)\n\u22a2 sSup (range (lift \u2218 f)) = sSup (range fun i => lift (f i))\n[PROOFSTEP]\nsimp [Function.comp]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type v\nf : \u03b9 \u2192 Cardinal.{w}\nt : Cardinal.{max u w}\nhf : BddAbove (range f)\nw : \u2200 (i : \u03b9), lift (f i) \u2264 t\n\u22a2 lift (iSup f) \u2264 t\n[PROOFSTEP]\nrw [lift_iSup hf]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type v\nf : \u03b9 \u2192 Cardinal.{w}\nt : Cardinal.{max u w}\nhf : BddAbove (range f)\nw : \u2200 (i : \u03b9), lift (f i) \u2264 t\n\u22a2 \u2a06 (i : \u03b9), lift (f i) \u2264 t\n[PROOFSTEP]\nexact ciSup_le' w\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type v\nf : \u03b9 \u2192 Cardinal.{w}\nhf : BddAbove (range f)\nt : Cardinal.{max u w}\n\u22a2 lift (iSup f) \u2264 t \u2194 \u2200 (i : \u03b9), lift (f i) \u2264 t\n[PROOFSTEP]\nrw [lift_iSup hf]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type v\nf : \u03b9 \u2192 Cardinal.{w}\nhf : BddAbove (range f)\nt : Cardinal.{max u w}\n\u22a2 \u2a06 (i : \u03b9), lift (f i) \u2264 t \u2194 \u2200 (i : \u03b9), lift (f i) \u2264 t\n[PROOFSTEP]\nexact ciSup_le_iff' (bddAbove_range_comp.{_, _, u} hf _)\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nf : \u03b9 \u2192 Cardinal.{w}\nf' : \u03b9' \u2192 Cardinal.{w'}\nhf : BddAbove (range f)\nhf' : BddAbove (range f')\ng : \u03b9 \u2192 \u03b9'\nh : \u2200 (i : \u03b9), lift (f i) \u2264 lift (f' (g i))\n\u22a2 lift (iSup f) \u2264 lift (iSup f')\n[PROOFSTEP]\nrw [lift_iSup hf, lift_iSup hf']\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nf : \u03b9 \u2192 Cardinal.{w}\nf' : \u03b9' \u2192 Cardinal.{w'}\nhf : BddAbove (range f)\nhf' : BddAbove (range f')\ng : \u03b9 \u2192 \u03b9'\nh : \u2200 (i : \u03b9), lift (f i) \u2264 lift (f' (g i))\n\u22a2 \u2a06 (i : \u03b9), lift (f i) \u2264 \u2a06 (i : \u03b9'), lift (f' i)\n[PROOFSTEP]\nexact ciSup_mono' (bddAbove_range_comp.{_, _, w} hf' _) fun i => \u27e8_, h i\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u}\n\u22a2 \u2135\u2080 \u2264 lift c \u2194 \u2135\u2080 \u2264 c\n[PROOFSTEP]\nrw [\u2190 lift_aleph0.{u, v}, lift_le]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u}\n\u22a2 lift c \u2264 \u2135\u2080 \u2194 c \u2264 \u2135\u2080\n[PROOFSTEP]\nrw [\u2190 lift_aleph0.{u, v}, lift_le]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u}\n\u22a2 \u2135\u2080 < lift c \u2194 \u2135\u2080 < c\n[PROOFSTEP]\nrw [\u2190 lift_aleph0.{u, v}, lift_lt]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u}\n\u22a2 lift c < \u2135\u2080 \u2194 c < \u2135\u2080\n[PROOFSTEP]\nrw [\u2190 lift_aleph0.{u, v}, lift_lt]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 #(Fin n) = \u2191n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 lift \u2191n = \u2191n\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u03b1 \u03b2 : Type u\n\u22a2 lift \u2191Nat.zero = \u2191Nat.zero\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase succ\n\u03b1 \u03b2 : Type u\nn\u271d : \u2115\nn_ih\u271d : lift \u2191n\u271d = \u2191n\u271d\n\u22a2 lift \u2191(Nat.succ n\u271d) = \u2191(Nat.succ n\u271d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\na : Cardinal.{u}\n\u22a2 \u2191n = lift a \u2194 \u2191n = a\n[PROOFSTEP]\nrw [\u2190 lift_natCast.{v, u} n, lift_inj]\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nn : \u2115\n\u22a2 lift a \u2264 \u2191n \u2194 a \u2264 \u2191n\n[PROOFSTEP]\nrw [\u2190 lift_natCast.{v, u}, lift_le]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\na : Cardinal.{u}\n\u22a2 \u2191n \u2264 lift a \u2194 \u2191n \u2264 a\n[PROOFSTEP]\nrw [\u2190 lift_natCast.{v, u}, lift_le]\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nn : \u2115\n\u22a2 lift a < \u2191n \u2194 a < \u2191n\n[PROOFSTEP]\nrw [\u2190 lift_natCast.{v, u}, lift_lt]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\na : Cardinal.{u}\n\u22a2 \u2191n < lift a \u2194 \u2191n < a\n[PROOFSTEP]\nrw [\u2190 lift_natCast.{v, u}, lift_lt]\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\ns : Finset \u03b1\n\u22a2 #{ x // x \u2208 s } = \u2191(Finset.card s)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : Fintype \u03b1\n\u22a2 #(Finset \u03b1) = 2 ^ Fintype.card \u03b1\n[PROOFSTEP]\nsimp [Pow.pow]\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Zero \u03b2\n\u22a2 #(\u03b1 \u2192\u2080 \u03b2) = lift #\u03b2 ^ Fintype.card \u03b1\n[PROOFSTEP]\nsimpa using (@Finsupp.equivFunOnFinite \u03b1 \u03b2 _ _).cardinal_eq\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b1 \u03b2 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Zero \u03b2\n\u22a2 #(\u03b1 \u2192\u2080 \u03b2) = #\u03b2 ^ Fintype.card \u03b1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 \u03b2 : Type u\nm n : \u2115\n\u22a2 \u2191(m ^ n) = \u2191m ^ \u2191n\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u03b1 \u03b2 : Type u\nm : \u2115\n\u22a2 \u2191(m ^ Nat.zero) = \u2191m ^ \u2191Nat.zero\n[PROOFSTEP]\nsimp [pow_succ', power_add, *, Pow.pow]\n[GOAL]\ncase succ\n\u03b1 \u03b2 : Type u\nm n\u271d : \u2115\nn_ih\u271d : \u2191(m ^ n\u271d) = \u2191m ^ \u2191n\u271d\n\u22a2 \u2191(m ^ Nat.succ n\u271d) = \u2191m ^ \u2191(Nat.succ n\u271d)\n[PROOFSTEP]\nsimp [pow_succ', power_add, *, Pow.pow]\n[GOAL]\n\u03b1 \u03b2 : Type u\nm n : \u2115\n\u22a2 \u2191m \u2264 \u2191n \u2194 m \u2264 n\n[PROOFSTEP]\nrw [\u2190 lift_mk_fin, \u2190 lift_mk_fin, lift_le, le_def, Function.Embedding.nonempty_iff_card_le, Fintype.card_fin,\n Fintype.card_fin]\n[GOAL]\n\u03b1 \u03b2 : Type u\nm n : \u2115\n\u22a2 \u2191m < \u2191n \u2194 m < n\n[PROOFSTEP]\nrw [lt_iff_le_not_le, \u2190 not_le]\n[GOAL]\n\u03b1 \u03b2 : Type u\nm n : \u2115\n\u22a2 \u2191m \u2264 \u2191n \u2227 \u00ac\u2191n \u2264 \u2191m \u2194 \u00acn \u2264 m\n[PROOFSTEP]\nsimp only [natCast_le, not_le, and_iff_right_iff_imp]\n[GOAL]\n\u03b1 \u03b2 : Type u\nm n : \u2115\n\u22a2 m < n \u2192 m \u2264 n\n[PROOFSTEP]\nexact fun h \u21a6 le_of_lt h\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 \u2191(Nat.succ n) = succ \u2191n\n[PROOFSTEP]\nrw [Nat.cast_succ]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 \u2191n + 1 = succ \u2191n\n[PROOFSTEP]\nrefine (add_one_le_succ _).antisymm (succ_le_of_lt ?_)\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 \u2191n < \u2191n + 1\n[PROOFSTEP]\nrw [\u2190 Nat.cast_succ]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 \u2191n < \u2191(Nat.succ n)\n[PROOFSTEP]\nexact natCast_lt.2 (Nat.lt_succ_self _)\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 succ 0 = 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\nH : \u2200 (s : Finset \u03b1), Finset.card s \u2264 n\n\u22a2 #\u03b1 \u2264 \u2191n\n[PROOFSTEP]\nrefine' le_of_lt_succ (lt_of_not_ge fun hn => _)\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\nH : \u2200 (s : Finset \u03b1), Finset.card s \u2264 n\nhn : #\u03b1 \u2265 succ \u2191n\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 Cardinal.nat_succ, \u2190 lift_mk_fin n.succ] at hn \n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\nH : \u2200 (s : Finset \u03b1), Finset.card s \u2264 n\nhn : #\u03b1 \u2265 lift #(Fin (Nat.succ n))\n\u22a2 False\n[PROOFSTEP]\ncases' hn with f\n[GOAL]\ncase intro\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\nH : \u2200 (s : Finset \u03b1), Finset.card s \u2264 n\nf : ULift (Fin (Nat.succ n)) \u21aa \u03b1\n\u22a2 False\n[PROOFSTEP]\nrefine' (H <| Finset.univ.map f).not_lt _\n[GOAL]\ncase intro\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\nH : \u2200 (s : Finset \u03b1), Finset.card s \u2264 n\nf : ULift (Fin (Nat.succ n)) \u21aa \u03b1\n\u22a2 n < Finset.card (Finset.map f Finset.univ)\n[PROOFSTEP]\nrw [Finset.card_map, \u2190 Fintype.card, Fintype.card_ulift, Fintype.card_fin]\n[GOAL]\ncase intro\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\nH : \u2200 (s : Finset \u03b1), Finset.card s \u2264 n\nf : ULift (Fin (Nat.succ n)) \u21aa \u03b1\n\u22a2 n < Nat.succ n\n[PROOFSTEP]\nexact n.lt_succ_self\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nhb : 1 < b\n\u22a2 a < b ^ a\n[PROOFSTEP]\nrw [\u2190 succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb \n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nhb\u271d : 1 < b\nhb : succ 1 \u2264 b\n\u22a2 succ 1 = 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nhb\u271d : 1 < b\nhb : 2 \u2264 b\n\u22a2 a < b ^ a\n[PROOFSTEP]\nexact (cantor a).trans_le (power_le_power_right hb)\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\n\u22a2 1 \u2264 c \u2194 0 < c\n[PROOFSTEP]\nrw [\u2190 succ_zero, succ_le_iff]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\n\u22a2 1 \u2264 c \u2194 c \u2260 0\n[PROOFSTEP]\nrw [one_le_iff_pos, pos_iff_ne_zero]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 succ \u2191n \u2264 \u2135\u2080\n[PROOFSTEP]\nrw [\u2190 nat_succ, \u2190 lift_mk_fin, aleph0, lift_mk_le.{u}]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 Nonempty (Fin (Nat.succ n) \u21aa \u2115)\n[PROOFSTEP]\nexact \u27e8\u27e8(\u2191), fun a b => Fin.ext\u27e9\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 1 < \u2135\u2080\n[PROOFSTEP]\nsimpa using nat_lt_aleph0 1\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\nh : c < \u2135\u2080\n\u22a2 \u2203 n, c = \u2191n\n[PROOFSTEP]\nrcases lt_lift_iff.1 h with \u27e8c, rfl, h'\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 \u03b2 : Type u\nc : Cardinal.{0}\nh' : c < #\u2115\nh : lift c < \u2135\u2080\n\u22a2 \u2203 n, lift c = \u2191n\n[PROOFSTEP]\nrcases le_mk_iff_exists_set.1 h'.1 with \u27e8S, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 \u03b2 : Type u\nS : Set \u2115\nh' : #\u2191S < #\u2115\nh : lift #\u2191S < \u2135\u2080\n\u22a2 \u2203 n, lift #\u2191S = \u2191n\n[PROOFSTEP]\nsuffices S.Finite by\n lift S to Finset \u2115 using this\n simp\n[GOAL]\n\u03b1 \u03b2 : Type u\nS : Set \u2115\nh' : #\u2191S < #\u2115\nh : lift #\u2191S < \u2135\u2080\nthis : Set.Finite S\n\u22a2 \u2203 n, lift #\u2191S = \u2191n\n[PROOFSTEP]\nlift S to Finset \u2115 using this\n[GOAL]\ncase intro\n\u03b1 \u03b2 : Type u\nS : Finset \u2115\nh' : #\u2191\u2191S < #\u2115\nh : lift #\u2191\u2191S < \u2135\u2080\n\u22a2 \u2203 n, lift #\u2191\u2191S = \u2191n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro\n\u03b1 \u03b2 : Type u\nS : Set \u2115\nh' : #\u2191S < #\u2115\nh : lift #\u2191S < \u2135\u2080\n\u22a2 Set.Finite S\n[PROOFSTEP]\ncontrapose! h'\n[GOAL]\ncase intro.intro.intro\n\u03b1 \u03b2 : Type u\nS : Set \u2115\nh : lift #\u2191S < \u2135\u2080\nh' : \u00acSet.Finite S\n\u22a2 #\u2115 \u2264 #\u2191S\n[PROOFSTEP]\nhaveI := Infinite.to_subtype h'\n[GOAL]\ncase intro.intro.intro\n\u03b1 \u03b2 : Type u\nS : Set \u2115\nh : lift #\u2191S < \u2135\u2080\nh' : \u00acSet.Finite S\nthis : Infinite \u2191S\n\u22a2 #\u2115 \u2264 #\u2191S\n[PROOFSTEP]\nexact \u27e8Infinite.natEmbedding S\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\nh : \u2200 (n : \u2115), \u2191n \u2264 c\nhn : c < \u2135\u2080\n\u22a2 False\n[PROOFSTEP]\nrcases lt_aleph0.1 hn with \u27e8n, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 \u03b2 : Type u\nn : \u2115\nh : \u2200 (n_1 : \u2115), \u2191n_1 \u2264 \u2191n\nhn : \u2191n < \u2135\u2080\n\u22a2 False\n[PROOFSTEP]\nexact (Nat.lt_succ_self _).not_le (natCast_le.1 (h (n + 1)))\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\nha : a < \u2135\u2080\n\u22a2 succ a < \u2135\u2080\n[PROOFSTEP]\nrcases lt_aleph0.1 ha with \u27e8n, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 \u03b2 : Type u\nn : \u2115\nha : \u2191n < \u2135\u2080\n\u22a2 succ \u2191n < \u2135\u2080\n[PROOFSTEP]\nrw [\u2190 nat_succ]\n[GOAL]\ncase intro\n\u03b1 \u03b2 : Type u\nn : \u2115\nha : \u2191n < \u2135\u2080\n\u22a2 \u2191(Nat.succ n) < \u2135\u2080\n[PROOFSTEP]\napply nat_lt_aleph0\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\nh : IsLimit c\n\u22a2 \u2135\u2080 \u2264 c\n[PROOFSTEP]\nby_contra' h'\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\nh : IsLimit c\nh' : c < \u2135\u2080\n\u22a2 False\n[PROOFSTEP]\nrcases lt_aleph0.1 h' with \u27e8_ | n, rfl\u27e9\n[GOAL]\ncase intro.zero\n\u03b1 \u03b2 : Type u\nh : IsLimit \u2191Nat.zero\nh' : \u2191Nat.zero < \u2135\u2080\n\u22a2 False\n[PROOFSTEP]\nexact h.ne_zero.irrefl\n[GOAL]\ncase intro.succ\n\u03b1 \u03b2 : Type u\nn : \u2115\nh : IsLimit \u2191(Nat.succ n)\nh' : \u2191(Nat.succ n) < \u2135\u2080\n\u22a2 False\n[PROOFSTEP]\nrw [nat_succ] at h \n[GOAL]\ncase intro.succ\n\u03b1 \u03b2 : Type u\nn : \u2115\nh : IsLimit (succ \u2191n)\nh' : \u2191(Nat.succ n) < \u2135\u2080\n\u22a2 False\n[PROOFSTEP]\nexact not_isSuccLimit_succ _ h.isSuccLimit\n[GOAL]\n\u03b1 \u03b2 : Type u\nx : Cardinal.{u_1}\n\u22a2 x \u2208 range Nat.cast \u2194 x \u2208 Iio \u2135\u2080\n[PROOFSTEP]\nsimp only [mem_Iio, mem_range, eq_comm, lt_aleph0]\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\n\u22a2 #\u03b1 = \u2191n \u2194 Nonempty (\u03b1 \u2243 Fin n)\n[PROOFSTEP]\nrw [\u2190 lift_mk_fin, \u2190 lift_uzero #\u03b1, lift_mk_eq']\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 #\u03b1 < \u2135\u2080 \u2194 Finite \u03b1\n[PROOFSTEP]\nsimp only [lt_aleph0, mk_eq_nat_iff, finite_iff_exists_equiv_fin]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 #\u03b1 \u2264 \u2135\u2080 \u2194 Countable \u03b1\n[PROOFSTEP]\nrw [countable_iff_nonempty_embedding, aleph0, \u2190 lift_uzero #\u03b1, lift_mk_le']\n[GOAL]\n\u03b1 \u03b2 : Type u\ns : Set \u03b1\n\u22a2 #\u2191s \u2264 \u2135\u2080 \u2194 Set.Countable s\n[PROOFSTEP]\nrw [mk_le_aleph0_iff, countable_coe_iff]\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nm n : \u2115\nha : \u2191m < \u2135\u2080\nhb : \u2191n < \u2135\u2080\n\u22a2 \u2191m + \u2191n < \u2135\u2080\n[PROOFSTEP]\nrw [\u2190 Nat.cast_add]\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nm n : \u2115\nha : \u2191m < \u2135\u2080\nhb : \u2191n < \u2135\u2080\n\u22a2 \u2191(m + n) < \u2135\u2080\n[PROOFSTEP]\napply nat_lt_aleph0\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\n\u22a2 \u2135\u2080 \u2264 a + b \u2194 \u2135\u2080 \u2264 a \u2228 \u2135\u2080 \u2264 b\n[PROOFSTEP]\nsimp only [\u2190 not_lt, add_lt_aleph0_iff, not_and_or]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\na : Cardinal.{u_1}\n\u22a2 n \u2022 a < \u2135\u2080 \u2194 n = 0 \u2228 a < \u2135\u2080\n[PROOFSTEP]\ncases n with\n| zero => simpa using nat_lt_aleph0 0\n| succ n =>\n simp only [Nat.succ_ne_zero, false_or_iff]\n induction' n with n ih\n \u00b7 simp\n rw [succ_nsmul, add_lt_aleph0_iff, ih, and_self_iff]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\na : Cardinal.{u_1}\n\u22a2 n \u2022 a < \u2135\u2080 \u2194 n = 0 \u2228 a < \u2135\u2080\n[PROOFSTEP]\ncases n with\n| zero => simpa using nat_lt_aleph0 0\n| succ n =>\n simp only [Nat.succ_ne_zero, false_or_iff]\n induction' n with n ih\n \u00b7 simp\n rw [succ_nsmul, add_lt_aleph0_iff, ih, and_self_iff]\n[GOAL]\ncase zero\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\n\u22a2 Nat.zero \u2022 a < \u2135\u2080 \u2194 Nat.zero = 0 \u2228 a < \u2135\u2080\n[PROOFSTEP]\n\n| zero => simpa using nat_lt_aleph0 0\n[GOAL]\ncase zero\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\n\u22a2 Nat.zero \u2022 a < \u2135\u2080 \u2194 Nat.zero = 0 \u2228 a < \u2135\u2080\n[PROOFSTEP]\nsimpa using nat_lt_aleph0 0\n[GOAL]\ncase succ\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\nn : \u2115\n\u22a2 Nat.succ n \u2022 a < \u2135\u2080 \u2194 Nat.succ n = 0 \u2228 a < \u2135\u2080\n[PROOFSTEP]\n\n| succ n =>\n simp only [Nat.succ_ne_zero, false_or_iff]\n induction' n with n ih\n \u00b7 simp\n rw [succ_nsmul, add_lt_aleph0_iff, ih, and_self_iff]\n[GOAL]\ncase succ\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\nn : \u2115\n\u22a2 Nat.succ n \u2022 a < \u2135\u2080 \u2194 Nat.succ n = 0 \u2228 a < \u2135\u2080\n[PROOFSTEP]\nsimp only [Nat.succ_ne_zero, false_or_iff]\n[GOAL]\ncase succ\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\nn : \u2115\n\u22a2 Nat.succ n \u2022 a < \u2135\u2080 \u2194 a < \u2135\u2080\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase succ.zero\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\n\u22a2 Nat.succ Nat.zero \u2022 a < \u2135\u2080 \u2194 a < \u2135\u2080\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.succ\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\nn : \u2115\nih : Nat.succ n \u2022 a < \u2135\u2080 \u2194 a < \u2135\u2080\n\u22a2 Nat.succ (Nat.succ n) \u2022 a < \u2135\u2080 \u2194 a < \u2135\u2080\n[PROOFSTEP]\nrw [succ_nsmul, add_lt_aleph0_iff, ih, and_self_iff]\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nm n : \u2115\nha : \u2191m < \u2135\u2080\nhb : \u2191n < \u2135\u2080\n\u22a2 \u2191m * \u2191n < \u2135\u2080\n[PROOFSTEP]\nrw [\u2190 Nat.cast_mul]\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nm n : \u2115\nha : \u2191m < \u2135\u2080\nhb : \u2191n < \u2135\u2080\n\u22a2 \u2191(m * n) < \u2135\u2080\n[PROOFSTEP]\napply nat_lt_aleph0\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\n\u22a2 a * b < \u2135\u2080 \u2194 a = 0 \u2228 b = 0 \u2228 a < \u2135\u2080 \u2227 b < \u2135\u2080\n[PROOFSTEP]\nrefine' \u27e8fun h => _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nh : a * b < \u2135\u2080\n\u22a2 a = 0 \u2228 b = 0 \u2228 a < \u2135\u2080 \u2227 b < \u2135\u2080\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nh : a * b < \u2135\u2080\nha : a = 0\n\u22a2 a = 0 \u2228 b = 0 \u2228 a < \u2135\u2080 \u2227 b < \u2135\u2080\n[PROOFSTEP]\nexact Or.inl ha\n[GOAL]\ncase neg\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nh : a * b < \u2135\u2080\nha : \u00aca = 0\n\u22a2 a = 0 \u2228 b = 0 \u2228 a < \u2135\u2080 \u2227 b < \u2135\u2080\n[PROOFSTEP]\nright\n[GOAL]\ncase neg.h\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nh : a * b < \u2135\u2080\nha : \u00aca = 0\n\u22a2 b = 0 \u2228 a < \u2135\u2080 \u2227 b < \u2135\u2080\n[PROOFSTEP]\nby_cases hb : b = 0\n[GOAL]\ncase pos\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nh : a * b < \u2135\u2080\nha : \u00aca = 0\nhb : b = 0\n\u22a2 b = 0 \u2228 a < \u2135\u2080 \u2227 b < \u2135\u2080\n[PROOFSTEP]\nexact Or.inl hb\n[GOAL]\ncase neg\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nh : a * b < \u2135\u2080\nha : \u00aca = 0\nhb : \u00acb = 0\n\u22a2 b = 0 \u2228 a < \u2135\u2080 \u2227 b < \u2135\u2080\n[PROOFSTEP]\nright\n[GOAL]\ncase neg.h\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nh : a * b < \u2135\u2080\nha : \u00aca = 0\nhb : \u00acb = 0\n\u22a2 a < \u2135\u2080 \u2227 b < \u2135\u2080\n[PROOFSTEP]\nrw [\u2190 Ne, \u2190 one_le_iff_ne_zero] at ha hb \n[GOAL]\ncase neg.h\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nh : a * b < \u2135\u2080\nha : 1 \u2264 a\nhb : 1 \u2264 b\n\u22a2 a < \u2135\u2080 \u2227 b < \u2135\u2080\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase neg.h.left\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nh : a * b < \u2135\u2080\nha : 1 \u2264 a\nhb : 1 \u2264 b\n\u22a2 a < \u2135\u2080\n[PROOFSTEP]\nrw [\u2190 mul_one a]\n[GOAL]\ncase neg.h.left\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nh : a * b < \u2135\u2080\nha : 1 \u2264 a\nhb : 1 \u2264 b\n\u22a2 a * 1 < \u2135\u2080\n[PROOFSTEP]\nrefine' (mul_le_mul' le_rfl hb).trans_lt h\n[GOAL]\ncase neg.h.right\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nh : a * b < \u2135\u2080\nha : 1 \u2264 a\nhb : 1 \u2264 b\n\u22a2 b < \u2135\u2080\n[PROOFSTEP]\nrw [\u2190 one_mul b]\n[GOAL]\ncase neg.h.right\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nh : a * b < \u2135\u2080\nha : 1 \u2264 a\nhb : 1 \u2264 b\n\u22a2 1 * b < \u2135\u2080\n[PROOFSTEP]\nrefine' (mul_le_mul' ha le_rfl).trans_lt h\n[GOAL]\ncase refine'_2\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\n\u22a2 a = 0 \u2228 b = 0 \u2228 a < \u2135\u2080 \u2227 b < \u2135\u2080 \u2192 a * b < \u2135\u2080\n[PROOFSTEP]\nrintro (rfl | rfl | \u27e8ha, hb\u27e9)\n[GOAL]\ncase refine'_2.inl\n\u03b1 \u03b2 : Type u\nb : Cardinal.{u_1}\n\u22a2 0 * b < \u2135\u2080\n[PROOFSTEP]\nsimp only [*, mul_lt_aleph0, aleph0_pos, zero_mul, mul_zero]\n[GOAL]\ncase refine'_2.inr.inl\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\n\u22a2 a * 0 < \u2135\u2080\n[PROOFSTEP]\nsimp only [*, mul_lt_aleph0, aleph0_pos, zero_mul, mul_zero]\n[GOAL]\ncase refine'_2.inr.inr.intro\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nha : a < \u2135\u2080\nhb : b < \u2135\u2080\n\u22a2 a * b < \u2135\u2080\n[PROOFSTEP]\nsimp only [*, mul_lt_aleph0, aleph0_pos, zero_mul, mul_zero]\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\n\u22a2 \u2135\u2080 \u2264 a * b \u2194 a \u2260 0 \u2227 b \u2260 0 \u2227 (\u2135\u2080 \u2264 a \u2228 \u2135\u2080 \u2264 b)\n[PROOFSTEP]\nlet h := (@mul_lt_aleph0_iff a b).not\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nh : \u00aca * b < \u2135\u2080 \u2194 \u00ac(a = 0 \u2228 b = 0 \u2228 a < \u2135\u2080 \u2227 b < \u2135\u2080) := Iff.not mul_lt_aleph0_iff\n\u22a2 \u2135\u2080 \u2264 a * b \u2194 a \u2260 0 \u2227 b \u2260 0 \u2227 (\u2135\u2080 \u2264 a \u2228 \u2135\u2080 \u2264 b)\n[PROOFSTEP]\nrwa [not_lt, not_or, not_or, not_and_or, not_lt, not_lt] at h \n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u}\n\u22a2 \u2135\u2080 \u2264 a * b \u2194 a \u2260 0 \u2227 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 a \u2227 b \u2260 0\n[PROOFSTEP]\nhave : \u2200 {a : Cardinal.{u}}, \u2135\u2080 \u2264 a \u2192 a \u2260 0 := fun a => ne_bot_of_le_ne_bot aleph0_ne_zero a\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u}\nthis : \u2200 {a : Cardinal.{u}}, \u2135\u2080 \u2264 a \u2192 a \u2260 0\n\u22a2 \u2135\u2080 \u2264 a * b \u2194 a \u2260 0 \u2227 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 a \u2227 b \u2260 0\n[PROOFSTEP]\nsimp only [aleph0_le_mul_iff, and_or_left, and_iff_right_of_imp this, @and_left_comm (a \u2260 0)]\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u}\nthis : \u2200 {a : Cardinal.{u}}, \u2135\u2080 \u2264 a \u2192 a \u2260 0\n\u22a2 b \u2260 0 \u2227 \u2135\u2080 \u2264 a \u2228 a \u2260 0 \u2227 \u2135\u2080 \u2264 b \u2194 a \u2260 0 \u2227 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 a \u2227 b \u2260 0\n[PROOFSTEP]\nsimp only [and_comm, or_comm]\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 a * b < \u2135\u2080 \u2194 a < \u2135\u2080 \u2227 b < \u2135\u2080\n[PROOFSTEP]\nsimp [mul_lt_aleph0_iff, ha, hb]\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nm n : \u2115\nha : \u2191m < \u2135\u2080\nhb : \u2191n < \u2135\u2080\n\u22a2 \u2191m ^ \u2191n < \u2135\u2080\n[PROOFSTEP]\nrw [\u2190 natCast_pow]\n[GOAL]\n\u03b1 \u03b2 : Type u\na b : Cardinal.{u_1}\nm n : \u2115\nha : \u2191m < \u2135\u2080\nhb : \u2191n < \u2135\u2080\n\u22a2 \u2191(m ^ n) < \u2135\u2080\n[PROOFSTEP]\napply nat_lt_aleph0\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 Infinite \u03b1 \u2194 \u2135\u2080 \u2264 #\u03b1\n[PROOFSTEP]\nrw [\u2190 not_lt, lt_aleph0_iff_finite, not_finite_iff_infinite]\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nh : #\u03b1 = \u2135\u2080\n\u22a2 Nonempty (Denumerable \u03b1)\n[PROOFSTEP]\ncases' Quotient.exact h with f\n[GOAL]\ncase intro\n\u03b1\u271d \u03b2 \u03b1 : Type u\nh : #\u03b1 = \u2135\u2080\nf : \u03b1 \u2243 ULift \u2115\n\u22a2 Nonempty (Denumerable \u03b1)\n[PROOFSTEP]\nexact \u27e8Denumerable.mk' <| f.trans Equiv.ulift\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\nhn : n \u2260 0\n\u22a2 1 \u2264 \u2191n\n[PROOFSTEP]\nrwa [\u2190 Nat.cast_one, natCast_le, Nat.one_le_iff_ne_zero]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u2135\u2080 * \u2191n = \u2135\u2080\n[PROOFSTEP]\nrw [mul_comm, nat_mul_aleph0 hn]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 \u2191n + \u2135\u2080 = \u2135\u2080\n[PROOFSTEP]\nrw [add_comm, aleph0_add_nat]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 (fun c => if h : c < \u2135\u2080 then choose (_ : \u2203 n, c = \u2191n) else 0) 0 = 0\n[PROOFSTEP]\nhave h : 0 < \u2135\u2080 := nat_lt_aleph0 0\n[GOAL]\n\u03b1 \u03b2 : Type u\nh : 0 < \u2135\u2080\n\u22a2 (fun c => if h : c < \u2135\u2080 then choose (_ : \u2203 n, c = \u2191n) else 0) 0 = 0\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 \u03b2 : Type u\nh : 0 < \u2135\u2080\n\u22a2 (if h : 0 < \u2135\u2080 then choose (_ : \u2203 n, 0 = \u2191n) else 0) = 0\n[PROOFSTEP]\nrw [dif_pos h, \u2190 Cardinal.natCast_inj, \u2190 Classical.choose_spec (lt_aleph0.1 h), Nat.cast_zero]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\nh : c < \u2135\u2080\n\u22a2 \u2191(\u2191toNat c) = c\n[PROOFSTEP]\nrw [toNat_apply_of_lt_aleph0 h, \u2190 Classical.choose_spec (lt_aleph0.1 h)]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\nh : \u2135\u2080 \u2264 c\n\u22a2 \u2191(\u2191toNat c) = 0\n[PROOFSTEP]\nrw [toNat_apply_of_aleph0_le h, Nat.cast_zero]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc d : Cardinal.{u_1}\nhc : c < \u2135\u2080\nhd : d < \u2135\u2080\n\u22a2 \u2191toNat c = \u2191toNat d \u2194 c = d\n[PROOFSTEP]\nrw [\u2190 natCast_inj, cast_toNat_of_lt_aleph0 hc, cast_toNat_of_lt_aleph0 hd]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc d : Cardinal.{u_1}\nhc : c < \u2135\u2080\nhd : d < \u2135\u2080\n\u22a2 \u2191toNat c \u2264 \u2191toNat d \u2194 c \u2264 d\n[PROOFSTEP]\nrw [\u2190 natCast_le, cast_toNat_of_lt_aleph0 hc, cast_toNat_of_lt_aleph0 hd]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc d : Cardinal.{u_1}\nhc : c < \u2135\u2080\nhd : d < \u2135\u2080\n\u22a2 \u2191toNat c < \u2191toNat d \u2194 c < d\n[PROOFSTEP]\nrw [\u2190 natCast_lt, cast_toNat_of_lt_aleph0 hc, cast_toNat_of_lt_aleph0 hd]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 \u2191toNat \u2191n = n\n[PROOFSTEP]\nrw [toNat_apply_of_lt_aleph0 (nat_lt_aleph0 n), \u2190 natCast_inj]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 \u2191(choose (_ : \u2203 n_1, \u2191n = \u2191n_1)) = \u2191n\n[PROOFSTEP]\nexact (Classical.choose_spec (lt_aleph0.1 (nat_lt_aleph0 n))).symm\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : Fintype \u03b1\n\u22a2 \u2191toNat #\u03b1 = Fintype.card \u03b1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 \u2191toNat 0 = 0\n[PROOFSTEP]\nrw [\u2190 toNat_cast 0, Nat.cast_zero]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 \u2191toNat 1 = 1\n[PROOFSTEP]\nrw [\u2190 toNat_cast 1, Nat.cast_one]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\n\u22a2 \u2191toNat c = 1 \u2194 c = 1\n[PROOFSTEP]\nrw [toNat_eq_iff one_ne_zero, Nat.cast_one]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{v}\n\u22a2 \u2191toNat (lift c) = \u2191toNat c\n[PROOFSTEP]\napply natCast_injective\n[GOAL]\ncase a\n\u03b1 \u03b2 : Type u\nc : Cardinal.{v}\n\u22a2 \u2191(\u2191toNat (lift c)) = \u2191(\u2191toNat c)\n[PROOFSTEP]\ncases' lt_or_ge c \u2135\u2080 with hc hc\n[GOAL]\ncase a.inl\n\u03b1 \u03b2 : Type u\nc : Cardinal.{v}\nhc : c < \u2135\u2080\n\u22a2 \u2191(\u2191toNat (lift c)) = \u2191(\u2191toNat c)\n[PROOFSTEP]\nrw [cast_toNat_of_lt_aleph0, \u2190 lift_natCast.{u, v}, cast_toNat_of_lt_aleph0 hc]\n[GOAL]\ncase a.inl\n\u03b1 \u03b2 : Type u\nc : Cardinal.{v}\nhc : c < \u2135\u2080\n\u22a2 lift c < \u2135\u2080\n[PROOFSTEP]\nrwa [lift_lt_aleph0]\n[GOAL]\ncase a.inr\n\u03b1 \u03b2 : Type u\nc : Cardinal.{v}\nhc : c \u2265 \u2135\u2080\n\u22a2 \u2191(\u2191toNat (lift c)) = \u2191(\u2191toNat c)\n[PROOFSTEP]\nrw [cast_toNat_of_aleph0_le, \u2190 lift_natCast.{u, v}, cast_toNat_of_aleph0_le hc, lift_zero]\n[GOAL]\ncase a.inr\n\u03b1 \u03b2 : Type u\nc : Cardinal.{v}\nhc : c \u2265 \u2135\u2080\n\u22a2 \u2135\u2080 \u2264 lift c\n[PROOFSTEP]\nrwa [aleph0_le_lift]\n[GOAL]\n\u03b1 \u03b2\u271d : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\n\u22a2 \u2191toNat #\u03b1 = \u2191toNat #\u03b2\n[PROOFSTEP]\nrw [\u2190 toNat_lift, (lift_mk_eq.{_, _, v}).mpr \u27e8e\u27e9, toNat_lift]\n[GOAL]\n\u03b1 \u03b2 : Type u\nx y : Cardinal.{u_1}\n\u22a2 \u2191toNat (x * y) = \u2191toNat x * \u2191toNat y\n[PROOFSTEP]\nrcases eq_or_ne x 0 with (rfl | hx1)\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\ny : Cardinal.{u_1}\n\u22a2 \u2191toNat (0 * y) = \u2191toNat 0 * \u2191toNat y\n[PROOFSTEP]\nrw [zero_mul, zero_toNat, zero_mul]\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\nx y : Cardinal.{u_1}\nhx1 : x \u2260 0\n\u22a2 \u2191toNat (x * y) = \u2191toNat x * \u2191toNat y\n[PROOFSTEP]\nrcases eq_or_ne y 0 with (rfl | hy1)\n[GOAL]\ncase inr.inl\n\u03b1 \u03b2 : Type u\nx : Cardinal.{u_1}\nhx1 : x \u2260 0\n\u22a2 \u2191toNat (x * 0) = \u2191toNat x * \u2191toNat 0\n[PROOFSTEP]\nrw [mul_zero, zero_toNat, mul_zero]\n[GOAL]\ncase inr.inr\n\u03b1 \u03b2 : Type u\nx y : Cardinal.{u_1}\nhx1 : x \u2260 0\nhy1 : y \u2260 0\n\u22a2 \u2191toNat (x * y) = \u2191toNat x * \u2191toNat y\n[PROOFSTEP]\ncases' lt_or_le x \u2135\u2080 with hx2 hx2\n[GOAL]\ncase inr.inr.inl\n\u03b1 \u03b2 : Type u\nx y : Cardinal.{u_1}\nhx1 : x \u2260 0\nhy1 : y \u2260 0\nhx2 : x < \u2135\u2080\n\u22a2 \u2191toNat (x * y) = \u2191toNat x * \u2191toNat y\n[PROOFSTEP]\ncases' lt_or_le y \u2135\u2080 with hy2 hy2\n[GOAL]\ncase inr.inr.inl.inl\n\u03b1 \u03b2 : Type u\nx y : Cardinal.{u_1}\nhx1 : x \u2260 0\nhy1 : y \u2260 0\nhx2 : x < \u2135\u2080\nhy2 : y < \u2135\u2080\n\u22a2 \u2191toNat (x * y) = \u2191toNat x * \u2191toNat y\n[PROOFSTEP]\nlift x to \u2115 using hx2\n[GOAL]\ncase inr.inr.inl.inl.intro\n\u03b1 \u03b2 : Type u\ny : Cardinal.{u_1}\nhy1 : y \u2260 0\nhy2 : y < \u2135\u2080\nx : \u2115\nhx1 : \u2191x \u2260 0\n\u22a2 \u2191toNat (\u2191x * y) = \u2191toNat \u2191x * \u2191toNat y\n[PROOFSTEP]\nlift y to \u2115 using hy2\n[GOAL]\ncase inr.inr.inl.inl.intro.intro\n\u03b1 \u03b2 : Type u\nx : \u2115\nhx1 : \u2191x \u2260 0\ny : \u2115\nhy1 : \u2191y \u2260 0\n\u22a2 \u2191toNat (\u2191x * \u2191y) = \u2191toNat \u2191x * \u2191toNat \u2191y\n[PROOFSTEP]\nrw [\u2190 Nat.cast_mul, toNat_cast, toNat_cast, toNat_cast]\n[GOAL]\ncase inr.inr.inl.inr\n\u03b1 \u03b2 : Type u\nx y : Cardinal.{u_1}\nhx1 : x \u2260 0\nhy1 : y \u2260 0\nhx2 : x < \u2135\u2080\nhy2 : \u2135\u2080 \u2264 y\n\u22a2 \u2191toNat (x * y) = \u2191toNat x * \u2191toNat y\n[PROOFSTEP]\nrw [toNat_apply_of_aleph0_le hy2, mul_zero, toNat_apply_of_aleph0_le]\n[GOAL]\ncase inr.inr.inl.inr\n\u03b1 \u03b2 : Type u\nx y : Cardinal.{u_1}\nhx1 : x \u2260 0\nhy1 : y \u2260 0\nhx2 : x < \u2135\u2080\nhy2 : \u2135\u2080 \u2264 y\n\u22a2 \u2135\u2080 \u2264 x * y\n[PROOFSTEP]\nexact aleph0_le_mul_iff'.2 (Or.inl \u27e8hx1, hy2\u27e9)\n[GOAL]\ncase inr.inr.inr\n\u03b1 \u03b2 : Type u\nx y : Cardinal.{u_1}\nhx1 : x \u2260 0\nhy1 : y \u2260 0\nhx2 : \u2135\u2080 \u2264 x\n\u22a2 \u2191toNat (x * y) = \u2191toNat x * \u2191toNat y\n[PROOFSTEP]\nrw [toNat_apply_of_aleph0_le hx2, zero_mul, toNat_apply_of_aleph0_le]\n[GOAL]\ncase inr.inr.inr\n\u03b1 \u03b2 : Type u\nx y : Cardinal.{u_1}\nhx1 : x \u2260 0\nhy1 : y \u2260 0\nhx2 : \u2135\u2080 \u2264 x\n\u22a2 \u2135\u2080 \u2264 x * y\n[PROOFSTEP]\nexact aleph0_le_mul_iff'.2 (Or.inr \u27e8hx2, hy1\u27e9)\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nb : Cardinal.{v}\nha : a < \u2135\u2080\nhb : b < \u2135\u2080\n\u22a2 \u2191toNat (lift a + lift b) = \u2191toNat a + \u2191toNat b\n[PROOFSTEP]\napply Cardinal.natCast_injective\n[GOAL]\ncase a\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nb : Cardinal.{v}\nha : a < \u2135\u2080\nhb : b < \u2135\u2080\n\u22a2 \u2191(\u2191toNat (lift a + lift b)) = \u2191(\u2191toNat a + \u2191toNat b)\n[PROOFSTEP]\nreplace ha : lift.{v, u} a < \u2135\u2080 := by rwa [lift_lt_aleph0]\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nb : Cardinal.{v}\nha : a < \u2135\u2080\nhb : b < \u2135\u2080\n\u22a2 lift a < \u2135\u2080\n[PROOFSTEP]\nrwa [lift_lt_aleph0]\n[GOAL]\ncase a\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nb : Cardinal.{v}\nhb : b < \u2135\u2080\nha : lift a < \u2135\u2080\n\u22a2 \u2191(\u2191toNat (lift a + lift b)) = \u2191(\u2191toNat a + \u2191toNat b)\n[PROOFSTEP]\nreplace hb : lift.{u, v} b < \u2135\u2080 := by rwa [lift_lt_aleph0]\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nb : Cardinal.{v}\nhb : b < \u2135\u2080\nha : lift a < \u2135\u2080\n\u22a2 lift b < \u2135\u2080\n[PROOFSTEP]\nrwa [lift_lt_aleph0]\n[GOAL]\ncase a\n\u03b1 \u03b2 : Type u\na : Cardinal.{u}\nb : Cardinal.{v}\nha : lift a < \u2135\u2080\nhb : lift b < \u2135\u2080\n\u22a2 \u2191(\u2191toNat (lift a + lift b)) = \u2191(\u2191toNat a + \u2191toNat b)\n[PROOFSTEP]\nrw [Nat.cast_add, \u2190 toNat_lift.{v, u} a, \u2190 toNat_lift.{u, v} b, cast_toNat_of_lt_aleph0 ha, cast_toNat_of_lt_aleph0 hb,\n cast_toNat_of_lt_aleph0 (add_lt_aleph0 ha hb)]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 (fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4) 0 = 0\n[PROOFSTEP]\nsimp [if_pos (zero_lt_one.trans one_lt_aleph0)]\n[GOAL]\n\u03b1 \u03b2 : Type u\nx y : Cardinal.{?u.569475}\n\u22a2 ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n (x + y) =\n ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n x +\n ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n y\n[PROOFSTEP]\nby_cases hx : x < \u2135\u2080\n[GOAL]\ncase pos\n\u03b1 \u03b2 : Type u\nx y : Cardinal.{?u.580039}\nhx : x < \u2135\u2080\n\u22a2 ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n (x + y) =\n ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n x +\n ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n y\n[PROOFSTEP]\nobtain \u27e8x0, rfl\u27e9 := lt_aleph0.1 hx\n[GOAL]\ncase pos.intro\n\u03b1 \u03b2 : Type u\ny : Cardinal.{?u.580039}\nx0 : \u2115\nhx : \u2191x0 < \u2135\u2080\n\u22a2 ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n (\u2191x0 + y) =\n ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n \u2191x0 +\n ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n y\n[PROOFSTEP]\nby_cases hy : y < \u2135\u2080\n[GOAL]\ncase pos\n\u03b1 \u03b2 : Type u\ny : Cardinal.{?u.580319}\nx0 : \u2115\nhx : \u2191x0 < \u2135\u2080\nhy : y < \u2135\u2080\n\u22a2 ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n (\u2191x0 + y) =\n ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n \u2191x0 +\n ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n y\n[PROOFSTEP]\nobtain \u27e8y0, rfl\u27e9 := lt_aleph0.1 hy\n[GOAL]\ncase pos.intro\n\u03b1 \u03b2 : Type u\nx0 : \u2115\nhx : \u2191x0 < \u2135\u2080\ny0 : \u2115\nhy : \u2191y0 < \u2135\u2080\n\u22a2 ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n (\u2191x0 + \u2191y0) =\n ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n \u2191x0 +\n ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n \u2191y0\n[PROOFSTEP]\nsimp only [add_lt_aleph0 hx hy, hx, hy, toNat_cast, if_true]\n[GOAL]\ncase pos.intro\n\u03b1 \u03b2 : Type u\nx0 : \u2115\nhx : \u2191x0 < \u2135\u2080\ny0 : \u2115\nhy : \u2191y0 < \u2135\u2080\n\u22a2 \u2191(\u2191toNat (\u2191x0 + \u2191y0)) = \u2191x0 + \u2191y0\n[PROOFSTEP]\nrw [\u2190 Nat.cast_add, toNat_cast, Nat.cast_add]\n[GOAL]\ncase neg\n\u03b1 \u03b2 : Type u\ny : Cardinal.{?u.581337}\nx0 : \u2115\nhx : \u2191x0 < \u2135\u2080\nhy : \u00acy < \u2135\u2080\n\u22a2 ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n (\u2191x0 + y) =\n ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n \u2191x0 +\n ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n y\n[PROOFSTEP]\nsimp_rw [if_neg hy, PartENat.add_top]\n[GOAL]\ncase neg\n\u03b1 \u03b2 : Type u\ny : Cardinal.{?u.581337}\nx0 : \u2115\nhx : \u2191x0 < \u2135\u2080\nhy : \u00acy < \u2135\u2080\n\u22a2 (if \u2191x0 + y < \u2135\u2080 then \u2191(\u2191toNat (\u2191x0 + y)) else \u22a4) = \u22a4\n[PROOFSTEP]\ncontrapose! hy\n[GOAL]\ncase neg\n\u03b1 \u03b2 : Type u\ny : Cardinal.{?u.581337}\nx0 : \u2115\nhx : \u2191x0 < \u2135\u2080\nhy : (if \u2191x0 + y < \u2135\u2080 then \u2191(\u2191toNat (\u2191x0 + y)) else \u22a4) \u2260 \u22a4\n\u22a2 y < \u2135\u2080\n[PROOFSTEP]\nsimp only [ne_eq, ite_eq_right_iff, PartENat.natCast_ne_top, not_forall, exists_prop, and_true] at hy \n[GOAL]\ncase neg\n\u03b1 \u03b2 : Type u\ny : Cardinal.{?u.581337}\nx0 : \u2115\nhx : \u2191x0 < \u2135\u2080\nhy : \u2191x0 + y < \u2135\u2080\n\u22a2 y < \u2135\u2080\n[PROOFSTEP]\nexact le_add_self.trans_lt hy\n[GOAL]\ncase neg\n\u03b1 \u03b2 : Type u\nx y : Cardinal.{?u.581337}\nhx : \u00acx < \u2135\u2080\n\u22a2 ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n (x + y) =\n ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n x +\n ZeroHom.toFun\n { toFun := fun c => if c < \u2135\u2080 then \u2191(\u2191toNat c) else \u22a4,\n map_zero' := (_ : (if 0 < \u2135\u2080 then \u2191(\u2191toNat 0) else \u22a4) = 0) }\n y\n[PROOFSTEP]\nsimp_rw [if_neg hx, PartENat.top_add]\n[GOAL]\ncase neg\n\u03b1 \u03b2 : Type u\nx y : Cardinal.{?u.581337}\nhx : \u00acx < \u2135\u2080\n\u22a2 (if x + y < \u2135\u2080 then \u2191(\u2191toNat (x + y)) else \u22a4) = \u22a4\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\ncase neg\n\u03b1 \u03b2 : Type u\nx y : Cardinal.{?u.581337}\nhx : (if x + y < \u2135\u2080 then \u2191(\u2191toNat (x + y)) else \u22a4) \u2260 \u22a4\n\u22a2 x < \u2135\u2080\n[PROOFSTEP]\nsimp only [ne_eq, ite_eq_right_iff, PartENat.natCast_ne_top, not_forall, exists_prop, and_true] at hx \n[GOAL]\ncase neg\n\u03b1 \u03b2 : Type u\nx y : Cardinal.{?u.581337}\nhx : x + y < \u2135\u2080\n\u22a2 x < \u2135\u2080\n[PROOFSTEP]\nexact le_self_add.trans_lt hx\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 \u2191toPartENat \u2191n = \u2191n\n[PROOFSTEP]\nrw [toPartENat_apply_of_lt_aleph0 (nat_lt_aleph0 n), toNat_cast]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\n\u22a2 \u2191toPartENat c = \u22a4 \u2194 \u2135\u2080 \u2264 c\n[PROOFSTEP]\ncases lt_or_ge c \u2135\u2080 with\n| inl hc => simp only [toPartENat_apply_of_lt_aleph0 hc, PartENat.natCast_ne_top, false_iff, not_le, hc]\n| inr hc => simp only [toPartENat_apply_of_aleph0_le hc, eq_self_iff_true, true_iff]; exact hc\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\nx\u271d : c < \u2135\u2080 \u2228 c \u2265 \u2135\u2080\n\u22a2 \u2191toPartENat c = \u22a4 \u2194 \u2135\u2080 \u2264 c\n[PROOFSTEP]\ncases lt_or_ge c \u2135\u2080 with\n| inl hc => simp only [toPartENat_apply_of_lt_aleph0 hc, PartENat.natCast_ne_top, false_iff, not_le, hc]\n| inr hc => simp only [toPartENat_apply_of_aleph0_le hc, eq_self_iff_true, true_iff]; exact hc\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\nhc : c < \u2135\u2080\n\u22a2 \u2191toPartENat c = \u22a4 \u2194 \u2135\u2080 \u2264 c\n[PROOFSTEP]\n\n| inl hc => simp only [toPartENat_apply_of_lt_aleph0 hc, PartENat.natCast_ne_top, false_iff, not_le, hc]\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\nhc : c < \u2135\u2080\n\u22a2 \u2191toPartENat c = \u22a4 \u2194 \u2135\u2080 \u2264 c\n[PROOFSTEP]\nsimp only [toPartENat_apply_of_lt_aleph0 hc, PartENat.natCast_ne_top, false_iff, not_le, hc]\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\nhc : c \u2265 \u2135\u2080\n\u22a2 \u2191toPartENat c = \u22a4 \u2194 \u2135\u2080 \u2264 c\n[PROOFSTEP]\n\n| inr hc => simp only [toPartENat_apply_of_aleph0_le hc, eq_self_iff_true, true_iff]; exact hc\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\nhc : c \u2265 \u2135\u2080\n\u22a2 \u2191toPartENat c = \u22a4 \u2194 \u2135\u2080 \u2264 c\n[PROOFSTEP]\nsimp only [toPartENat_apply_of_aleph0_le hc, eq_self_iff_true, true_iff]\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\nc : Cardinal.{u_1}\nhc : c \u2265 \u2135\u2080\n\u22a2 \u2135\u2080 \u2264 c\n[PROOFSTEP]\nexact hc\n[GOAL]\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\ncases lt_or_ge c \u2135\u2080 with\n| inl hc =>\n rw [toPartENat_apply_of_lt_aleph0 hc]\n cases lt_or_ge c' \u2135\u2080 with\n | inl hc' =>\n rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n exact toNat_le_iff_le_of_lt_aleph0 hc hc'\n | inr hc' =>\n simp only [toPartENat_apply_of_aleph0_le hc', le_top, true_iff]\n exact le_trans h hc'\n| inr hc =>\n rw [toPartENat_apply_of_aleph0_le hc]\n simp only [top_le_iff, toPartENat_eq_top_iff_le_aleph0, le_antisymm h hc]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 \u2135\u2080\nx\u271d : c < \u2135\u2080 \u2228 c \u2265 \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\ncases lt_or_ge c \u2135\u2080 with\n| inl hc =>\n rw [toPartENat_apply_of_lt_aleph0 hc]\n cases lt_or_ge c' \u2135\u2080 with\n | inl hc' =>\n rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n exact toNat_le_iff_le_of_lt_aleph0 hc hc'\n | inr hc' =>\n simp only [toPartENat_apply_of_aleph0_le hc', le_top, true_iff]\n exact le_trans h hc'\n| inr hc =>\n rw [toPartENat_apply_of_aleph0_le hc]\n simp only [top_le_iff, toPartENat_eq_top_iff_le_aleph0, le_antisymm h hc]\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 \u2135\u2080\nhc : c < \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\n\n| inl hc =>\n rw [toPartENat_apply_of_lt_aleph0 hc]\n cases lt_or_ge c' \u2135\u2080 with\n | inl hc' =>\n rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n exact toNat_le_iff_le_of_lt_aleph0 hc hc'\n | inr hc' =>\n simp only [toPartENat_apply_of_aleph0_le hc', le_top, true_iff]\n exact le_trans h hc'\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 \u2135\u2080\nhc : c < \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\nrw [toPartENat_apply_of_lt_aleph0 hc]\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 \u2135\u2080\nhc : c < \u2135\u2080\n\u22a2 \u2191(\u2191toNat c) \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\ncases lt_or_ge c' \u2135\u2080 with\n| inl hc' =>\n rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n exact toNat_le_iff_le_of_lt_aleph0 hc hc'\n| inr hc' =>\n simp only [toPartENat_apply_of_aleph0_le hc', le_top, true_iff]\n exact le_trans h hc'\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 \u2135\u2080\nhc : c < \u2135\u2080\nx\u271d : c' < \u2135\u2080 \u2228 c' \u2265 \u2135\u2080\n\u22a2 \u2191(\u2191toNat c) \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\ncases lt_or_ge c' \u2135\u2080 with\n| inl hc' =>\n rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n exact toNat_le_iff_le_of_lt_aleph0 hc hc'\n| inr hc' =>\n simp only [toPartENat_apply_of_aleph0_le hc', le_top, true_iff]\n exact le_trans h hc'\n[GOAL]\ncase inl.inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 \u2135\u2080\nhc : c < \u2135\u2080\nhc' : c' < \u2135\u2080\n\u22a2 \u2191(\u2191toNat c) \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\n\n| inl hc' =>\n rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n exact toNat_le_iff_le_of_lt_aleph0 hc hc'\n[GOAL]\ncase inl.inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 \u2135\u2080\nhc : c < \u2135\u2080\nhc' : c' < \u2135\u2080\n\u22a2 \u2191(\u2191toNat c) \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\nrw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n[GOAL]\ncase inl.inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 \u2135\u2080\nhc : c < \u2135\u2080\nhc' : c' < \u2135\u2080\n\u22a2 \u2191toNat c \u2264 \u2191toNat c' \u2194 c \u2264 c'\n[PROOFSTEP]\nexact toNat_le_iff_le_of_lt_aleph0 hc hc'\n[GOAL]\ncase inl.inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 \u2135\u2080\nhc : c < \u2135\u2080\nhc' : c' \u2265 \u2135\u2080\n\u22a2 \u2191(\u2191toNat c) \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\n\n| inr hc' =>\n simp only [toPartENat_apply_of_aleph0_le hc', le_top, true_iff]\n exact le_trans h hc'\n[GOAL]\ncase inl.inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 \u2135\u2080\nhc : c < \u2135\u2080\nhc' : c' \u2265 \u2135\u2080\n\u22a2 \u2191(\u2191toNat c) \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\nsimp only [toPartENat_apply_of_aleph0_le hc', le_top, true_iff]\n[GOAL]\ncase inl.inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 \u2135\u2080\nhc : c < \u2135\u2080\nhc' : c' \u2265 \u2135\u2080\n\u22a2 c \u2264 c'\n[PROOFSTEP]\nexact le_trans h hc'\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 \u2135\u2080\nhc : c \u2265 \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\n\n| inr hc =>\n rw [toPartENat_apply_of_aleph0_le hc]\n simp only [top_le_iff, toPartENat_eq_top_iff_le_aleph0, le_antisymm h hc]\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 \u2135\u2080\nhc : c \u2265 \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\nrw [toPartENat_apply_of_aleph0_le hc]\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 \u2135\u2080\nhc : c \u2265 \u2135\u2080\n\u22a2 \u22a4 \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\nsimp only [top_le_iff, toPartENat_eq_top_iff_le_aleph0, le_antisymm h hc]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nhc' : c' < \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\ncases lt_or_ge c \u2135\u2080 with\n| inl hc =>\n rw [toPartENat_apply_of_lt_aleph0 hc]\n rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n exact toNat_le_iff_le_of_lt_aleph0 hc hc'\n| inr hc =>\n rw [toPartENat_apply_of_aleph0_le hc]\n simp only [top_le_iff, toPartENat_eq_top_iff_le_aleph0]\n rw [\u2190 not_iff_not, not_le, not_le]\n simp only [hc', lt_of_lt_of_le hc' hc]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nhc' : c' < \u2135\u2080\nx\u271d : c < \u2135\u2080 \u2228 c \u2265 \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\ncases lt_or_ge c \u2135\u2080 with\n| inl hc =>\n rw [toPartENat_apply_of_lt_aleph0 hc]\n rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n exact toNat_le_iff_le_of_lt_aleph0 hc hc'\n| inr hc =>\n rw [toPartENat_apply_of_aleph0_le hc]\n simp only [top_le_iff, toPartENat_eq_top_iff_le_aleph0]\n rw [\u2190 not_iff_not, not_le, not_le]\n simp only [hc', lt_of_lt_of_le hc' hc]\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nhc' : c' < \u2135\u2080\nhc : c < \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\n\n| inl hc =>\n rw [toPartENat_apply_of_lt_aleph0 hc]\n rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n exact toNat_le_iff_le_of_lt_aleph0 hc hc'\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nhc' : c' < \u2135\u2080\nhc : c < \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\nrw [toPartENat_apply_of_lt_aleph0 hc]\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nhc' : c' < \u2135\u2080\nhc : c < \u2135\u2080\n\u22a2 \u2191(\u2191toNat c) \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\nrw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nhc' : c' < \u2135\u2080\nhc : c < \u2135\u2080\n\u22a2 \u2191toNat c \u2264 \u2191toNat c' \u2194 c \u2264 c'\n[PROOFSTEP]\nexact toNat_le_iff_le_of_lt_aleph0 hc hc'\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nhc' : c' < \u2135\u2080\nhc : c \u2265 \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\n\n| inr hc =>\n rw [toPartENat_apply_of_aleph0_le hc]\n simp only [top_le_iff, toPartENat_eq_top_iff_le_aleph0]\n rw [\u2190 not_iff_not, not_le, not_le]\n simp only [hc', lt_of_lt_of_le hc' hc]\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nhc' : c' < \u2135\u2080\nhc : c \u2265 \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\nrw [toPartENat_apply_of_aleph0_le hc]\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nhc' : c' < \u2135\u2080\nhc : c \u2265 \u2135\u2080\n\u22a2 \u22a4 \u2264 \u2191toPartENat c' \u2194 c \u2264 c'\n[PROOFSTEP]\nsimp only [top_le_iff, toPartENat_eq_top_iff_le_aleph0]\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nhc' : c' < \u2135\u2080\nhc : c \u2265 \u2135\u2080\n\u22a2 \u2135\u2080 \u2264 c' \u2194 c \u2264 c'\n[PROOFSTEP]\nrw [\u2190 not_iff_not, not_le, not_le]\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nhc' : c' < \u2135\u2080\nhc : c \u2265 \u2135\u2080\n\u22a2 c' < \u2135\u2080 \u2194 c' < c\n[PROOFSTEP]\nsimp only [hc', lt_of_lt_of_le hc' hc]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nhc : c \u2264 \u2135\u2080\nhc' : c' \u2264 \u2135\u2080\n\u22a2 \u2191toPartENat c = \u2191toPartENat c' \u2194 c = c'\n[PROOFSTEP]\nrw [le_antisymm_iff, le_antisymm_iff, toPartENat_le_iff_of_le_aleph0 hc, toPartENat_le_iff_of_le_aleph0 hc']\n[GOAL]\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 c'\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c'\n[PROOFSTEP]\ncases lt_or_ge c \u2135\u2080 with\n| inl hc =>\n rw [toPartENat_apply_of_lt_aleph0 hc]\n cases lt_or_ge c' \u2135\u2080 with\n | inl hc' =>\n rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n exact toNat_le_of_le_of_lt_aleph0 hc' h\n | inr hc' =>\n rw [toPartENat_apply_of_aleph0_le hc']\n exact le_top\n| inr hc => rw [toPartENat_apply_of_aleph0_le hc, toPartENat_apply_of_aleph0_le (le_trans hc h)]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 c'\nx\u271d : c < \u2135\u2080 \u2228 c \u2265 \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c'\n[PROOFSTEP]\ncases lt_or_ge c \u2135\u2080 with\n| inl hc =>\n rw [toPartENat_apply_of_lt_aleph0 hc]\n cases lt_or_ge c' \u2135\u2080 with\n | inl hc' =>\n rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n exact toNat_le_of_le_of_lt_aleph0 hc' h\n | inr hc' =>\n rw [toPartENat_apply_of_aleph0_le hc']\n exact le_top\n| inr hc => rw [toPartENat_apply_of_aleph0_le hc, toPartENat_apply_of_aleph0_le (le_trans hc h)]\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 c'\nhc : c < \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c'\n[PROOFSTEP]\n\n| inl hc =>\n rw [toPartENat_apply_of_lt_aleph0 hc]\n cases lt_or_ge c' \u2135\u2080 with\n | inl hc' =>\n rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n exact toNat_le_of_le_of_lt_aleph0 hc' h\n | inr hc' =>\n rw [toPartENat_apply_of_aleph0_le hc']\n exact le_top\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 c'\nhc : c < \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c'\n[PROOFSTEP]\nrw [toPartENat_apply_of_lt_aleph0 hc]\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 c'\nhc : c < \u2135\u2080\n\u22a2 \u2191(\u2191toNat c) \u2264 \u2191toPartENat c'\n[PROOFSTEP]\ncases lt_or_ge c' \u2135\u2080 with\n| inl hc' =>\n rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n exact toNat_le_of_le_of_lt_aleph0 hc' h\n| inr hc' =>\n rw [toPartENat_apply_of_aleph0_le hc']\n exact le_top\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 c'\nhc : c < \u2135\u2080\nx\u271d : c' < \u2135\u2080 \u2228 c' \u2265 \u2135\u2080\n\u22a2 \u2191(\u2191toNat c) \u2264 \u2191toPartENat c'\n[PROOFSTEP]\ncases lt_or_ge c' \u2135\u2080 with\n| inl hc' =>\n rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n exact toNat_le_of_le_of_lt_aleph0 hc' h\n| inr hc' =>\n rw [toPartENat_apply_of_aleph0_le hc']\n exact le_top\n[GOAL]\ncase inl.inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 c'\nhc : c < \u2135\u2080\nhc' : c' < \u2135\u2080\n\u22a2 \u2191(\u2191toNat c) \u2264 \u2191toPartENat c'\n[PROOFSTEP]\n\n| inl hc' =>\n rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n exact toNat_le_of_le_of_lt_aleph0 hc' h\n[GOAL]\ncase inl.inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 c'\nhc : c < \u2135\u2080\nhc' : c' < \u2135\u2080\n\u22a2 \u2191(\u2191toNat c) \u2264 \u2191toPartENat c'\n[PROOFSTEP]\nrw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]\n[GOAL]\ncase inl.inl\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 c'\nhc : c < \u2135\u2080\nhc' : c' < \u2135\u2080\n\u22a2 \u2191toNat c \u2264 \u2191toNat c'\n[PROOFSTEP]\nexact toNat_le_of_le_of_lt_aleph0 hc' h\n[GOAL]\ncase inl.inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 c'\nhc : c < \u2135\u2080\nhc' : c' \u2265 \u2135\u2080\n\u22a2 \u2191(\u2191toNat c) \u2264 \u2191toPartENat c'\n[PROOFSTEP]\n\n| inr hc' =>\n rw [toPartENat_apply_of_aleph0_le hc']\n exact le_top\n[GOAL]\ncase inl.inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 c'\nhc : c < \u2135\u2080\nhc' : c' \u2265 \u2135\u2080\n\u22a2 \u2191(\u2191toNat c) \u2264 \u2191toPartENat c'\n[PROOFSTEP]\nrw [toPartENat_apply_of_aleph0_le hc']\n[GOAL]\ncase inl.inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 c'\nhc : c < \u2135\u2080\nhc' : c' \u2265 \u2135\u2080\n\u22a2 \u2191(\u2191toNat c) \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 c'\nhc : c \u2265 \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c'\n[PROOFSTEP]\n\n| inr hc => rw [toPartENat_apply_of_aleph0_le hc, toPartENat_apply_of_aleph0_le (le_trans hc h)]\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\nc c' : Cardinal.{u_1}\nh : c \u2264 c'\nhc : c \u2265 \u2135\u2080\n\u22a2 \u2191toPartENat c \u2264 \u2191toPartENat c'\n[PROOFSTEP]\nrw [toPartENat_apply_of_aleph0_le hc, toPartENat_apply_of_aleph0_le (le_trans hc h)]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{v}\n\u22a2 \u2191toPartENat (lift c) = \u2191toPartENat c\n[PROOFSTEP]\ncases' lt_or_ge c \u2135\u2080 with hc hc\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc : Cardinal.{v}\nhc : c < \u2135\u2080\n\u22a2 \u2191toPartENat (lift c) = \u2191toPartENat c\n[PROOFSTEP]\nrw [toPartENat_apply_of_lt_aleph0 hc, Cardinal.toPartENat_apply_of_lt_aleph0 _]\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\nc : Cardinal.{v}\nhc : c < \u2135\u2080\n\u22a2 \u2191(\u2191toNat (lift c)) = \u2191(\u2191toNat c)\n\u03b1 \u03b2 : Type u c : Cardinal.{v} hc : c < \u2135\u2080 \u22a2 lift c < \u2135\u2080\n[PROOFSTEP]\nsimp only [toNat_lift]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{v}\nhc : c < \u2135\u2080\n\u22a2 lift c < \u2135\u2080\n[PROOFSTEP]\nrw [lift_lt_aleph0]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{v}\nhc : c < \u2135\u2080\n\u22a2 c < \u2135\u2080\n[PROOFSTEP]\nexact hc\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\nc : Cardinal.{v}\nhc : c \u2265 \u2135\u2080\n\u22a2 \u2191toPartENat (lift c) = \u2191toPartENat c\n[PROOFSTEP]\nrw [toPartENat_apply_of_aleph0_le hc, toPartENat_apply_of_aleph0_le _]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{v}\nhc : c \u2265 \u2135\u2080\n\u22a2 \u2135\u2080 \u2264 lift c\n[PROOFSTEP]\nrw [aleph0_le_lift]\n[GOAL]\n\u03b1 \u03b2 : Type u\nc : Cardinal.{v}\nhc : c \u2265 \u2135\u2080\n\u22a2 \u2135\u2080 \u2264 c\n[PROOFSTEP]\nexact hc\n[GOAL]\n\u03b1 \u03b2\u271d : Type u\n\u03b2 : Type v\ne : \u03b1 \u2243 \u03b2\n\u22a2 \u2191toPartENat #\u03b1 = \u2191toPartENat #\u03b2\n[PROOFSTEP]\nrw [\u2190 toPartENat_lift, lift_mk_eq.{_, _, v}.mpr \u27e8e\u27e9, toPartENat_lift]\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : Fintype \u03b1\n\u22a2 \u2191toPartENat #\u03b1 = \u2191(Fintype.card \u03b1)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i < g i\nx\u271d : sum f \u2265 prod g\nF : ((i : \u03b9) \u2192 Quotient.out (g i)) \u21aa (i : \u03b9) \u00d7 Quotient.out (f i)\n\u22a2 False\n[PROOFSTEP]\nhave : Inhabited (\u2200 i : \u03b9, (g i).out) :=\n by\n refine' \u27e8fun i => Classical.choice <| mk_ne_zero_iff.1 _\u27e9\n rw [mk_out]\n exact (H i).ne_bot\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i < g i\nx\u271d : sum f \u2265 prod g\nF : ((i : \u03b9) \u2192 Quotient.out (g i)) \u21aa (i : \u03b9) \u00d7 Quotient.out (f i)\n\u22a2 Inhabited ((i : \u03b9) \u2192 Quotient.out (g i))\n[PROOFSTEP]\nrefine' \u27e8fun i => Classical.choice <| mk_ne_zero_iff.1 _\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i < g i\nx\u271d : sum f \u2265 prod g\nF : ((i : \u03b9) \u2192 Quotient.out (g i)) \u21aa (i : \u03b9) \u00d7 Quotient.out (f i)\ni : \u03b9\n\u22a2 #(Quotient.out (g i)) \u2260 0\n[PROOFSTEP]\nrw [mk_out]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i < g i\nx\u271d : sum f \u2265 prod g\nF : ((i : \u03b9) \u2192 Quotient.out (g i)) \u21aa (i : \u03b9) \u00d7 Quotient.out (f i)\ni : \u03b9\n\u22a2 g i \u2260 0\n[PROOFSTEP]\nexact (H i).ne_bot\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i < g i\nx\u271d : sum f \u2265 prod g\nF : ((i : \u03b9) \u2192 Quotient.out (g i)) \u21aa (i : \u03b9) \u00d7 Quotient.out (f i)\nthis : Inhabited ((i : \u03b9) \u2192 Quotient.out (g i))\n\u22a2 False\n[PROOFSTEP]\nlet G := invFun F\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i < g i\nx\u271d : sum f \u2265 prod g\nF : ((i : \u03b9) \u2192 Quotient.out (g i)) \u21aa (i : \u03b9) \u00d7 Quotient.out (f i)\nthis : Inhabited ((i : \u03b9) \u2192 Quotient.out (g i))\nG : (i : \u03b9) \u00d7 Quotient.out (f i) \u2192 (i : \u03b9) \u2192 Quotient.out (g i) := invFun \u2191F\n\u22a2 False\n[PROOFSTEP]\nhave sG : Surjective G := invFun_surjective F.2\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i < g i\nx\u271d : sum f \u2265 prod g\nF : ((i : \u03b9) \u2192 Quotient.out (g i)) \u21aa (i : \u03b9) \u00d7 Quotient.out (f i)\nthis : Inhabited ((i : \u03b9) \u2192 Quotient.out (g i))\nG : (i : \u03b9) \u00d7 Quotient.out (f i) \u2192 (i : \u03b9) \u2192 Quotient.out (g i) := invFun \u2191F\nsG : Surjective G\n\u22a2 False\n[PROOFSTEP]\nchoose C hc using\n show \u2200 i, \u2203 b, \u2200 a, G \u27e8i, a\u27e9 i \u2260 b by\n intro i\n simp only [not_exists.symm, not_forall.symm]\n refine' fun h => (H i).not_le _\n rw [\u2190 mk_out (f i), \u2190 mk_out (g i)]\n exact \u27e8Embedding.ofSurjective _ h\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i < g i\nx\u271d : sum f \u2265 prod g\nF : ((i : \u03b9) \u2192 Quotient.out (g i)) \u21aa (i : \u03b9) \u00d7 Quotient.out (f i)\nthis : Inhabited ((i : \u03b9) \u2192 Quotient.out (g i))\nG : (i : \u03b9) \u00d7 Quotient.out (f i) \u2192 (i : \u03b9) \u2192 Quotient.out (g i) := invFun \u2191F\nsG : Surjective G\n\u22a2 \u2200 (i : \u03b9), \u2203 b, \u2200 (a : Quotient.out (f i)), G { fst := i, snd := a } i \u2260 b\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i < g i\nx\u271d : sum f \u2265 prod g\nF : ((i : \u03b9) \u2192 Quotient.out (g i)) \u21aa (i : \u03b9) \u00d7 Quotient.out (f i)\nthis : Inhabited ((i : \u03b9) \u2192 Quotient.out (g i))\nG : (i : \u03b9) \u00d7 Quotient.out (f i) \u2192 (i : \u03b9) \u2192 Quotient.out (g i) := invFun \u2191F\nsG : Surjective G\ni : \u03b9\n\u22a2 \u2203 b, \u2200 (a : Quotient.out (f i)), G { fst := i, snd := a } i \u2260 b\n[PROOFSTEP]\nsimp only [not_exists.symm, not_forall.symm]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i < g i\nx\u271d : sum f \u2265 prod g\nF : ((i : \u03b9) \u2192 Quotient.out (g i)) \u21aa (i : \u03b9) \u00d7 Quotient.out (f i)\nthis : Inhabited ((i : \u03b9) \u2192 Quotient.out (g i))\nG : (i : \u03b9) \u00d7 Quotient.out (f i) \u2192 (i : \u03b9) \u2192 Quotient.out (g i) := invFun \u2191F\nsG : Surjective G\ni : \u03b9\n\u22a2 \u00ac\u2200 (x : Quotient.out (g i)), \u2203 x_1, invFun \u2191F { fst := i, snd := x_1 } i = x\n[PROOFSTEP]\nrefine' fun h => (H i).not_le _\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i < g i\nx\u271d : sum f \u2265 prod g\nF : ((i : \u03b9) \u2192 Quotient.out (g i)) \u21aa (i : \u03b9) \u00d7 Quotient.out (f i)\nthis : Inhabited ((i : \u03b9) \u2192 Quotient.out (g i))\nG : (i : \u03b9) \u00d7 Quotient.out (f i) \u2192 (i : \u03b9) \u2192 Quotient.out (g i) := invFun \u2191F\nsG : Surjective G\ni : \u03b9\nh : \u2200 (x : Quotient.out (g i)), \u2203 x_1, invFun \u2191F { fst := i, snd := x_1 } i = x\n\u22a2 g i \u2264 f i\n[PROOFSTEP]\nrw [\u2190 mk_out (f i), \u2190 mk_out (g i)]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i < g i\nx\u271d : sum f \u2265 prod g\nF : ((i : \u03b9) \u2192 Quotient.out (g i)) \u21aa (i : \u03b9) \u00d7 Quotient.out (f i)\nthis : Inhabited ((i : \u03b9) \u2192 Quotient.out (g i))\nG : (i : \u03b9) \u00d7 Quotient.out (f i) \u2192 (i : \u03b9) \u2192 Quotient.out (g i) := invFun \u2191F\nsG : Surjective G\ni : \u03b9\nh : \u2200 (x : Quotient.out (g i)), \u2203 x_1, invFun \u2191F { fst := i, snd := x_1 } i = x\n\u22a2 #(Quotient.out (g i)) \u2264 #(Quotient.out (f i))\n[PROOFSTEP]\nexact \u27e8Embedding.ofSurjective _ h\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i < g i\nx\u271d : sum f \u2265 prod g\nF : ((i : \u03b9) \u2192 Quotient.out (g i)) \u21aa (i : \u03b9) \u00d7 Quotient.out (f i)\nthis : Inhabited ((i : \u03b9) \u2192 Quotient.out (g i))\nG : (i : \u03b9) \u00d7 Quotient.out (f i) \u2192 (i : \u03b9) \u2192 Quotient.out (g i) := invFun \u2191F\nsG : Surjective G\nC : (i : \u03b9) \u2192 Quotient.out (g i)\nhc : \u2200 (i : \u03b9) (a : Quotient.out (f i)), G { fst := i, snd := a } i \u2260 C i\n\u22a2 False\n[PROOFSTEP]\nlet \u27e8\u27e8i, a\u27e9, h\u27e9 := sG C\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal.{u_2}\nH : \u2200 (i : \u03b9), f i < g i\nx\u271d : sum f \u2265 prod g\nF : ((i : \u03b9) \u2192 Quotient.out (g i)) \u21aa (i : \u03b9) \u00d7 Quotient.out (f i)\nthis : Inhabited ((i : \u03b9) \u2192 Quotient.out (g i))\nG : (i : \u03b9) \u00d7 Quotient.out (f i) \u2192 (i : \u03b9) \u2192 Quotient.out (g i) := invFun \u2191F\nsG : Surjective G\nC : (i : \u03b9) \u2192 Quotient.out (g i)\nhc : \u2200 (i : \u03b9) (a : Quotient.out (f i)), G { fst := i, snd := a } i \u2260 C i\ni : \u03b9\na : Quotient.out (f i)\nh : G { fst := i, snd := a } = C\n\u22a2 False\n[PROOFSTEP]\nexact hc i a (congr_fun h _)\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\n\u22a2 #(Fin n \u2192 \u03b1) = #\u03b1 ^ n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 #((n : \u2115) \u00d7 Vector \u03b1 n) = sum fun n => #\u03b1 ^ n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\ns : Set \u03b1\n\u22a2 #\u2191s = 0 \u2194 s = \u2205\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1\u271d \u03b2 \u03b1 : Type u\ns : Set \u03b1\n\u22a2 #\u2191s = 0 \u2192 s = \u2205\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1\u271d \u03b2 \u03b1 : Type u\ns : Set \u03b1\nh : #\u2191s = 0\n\u22a2 s = \u2205\n[PROOFSTEP]\nrw [mk_eq_zero_iff] at h \n[GOAL]\ncase mp\n\u03b1\u271d \u03b2 \u03b1 : Type u\ns : Set \u03b1\nh : IsEmpty \u2191s\n\u22a2 s = \u2205\n[PROOFSTEP]\nexact eq_empty_iff_forall_not_mem.2 fun x hx => h.elim' \u27e8x, hx\u27e9\n[GOAL]\ncase mpr\n\u03b1\u271d \u03b2 \u03b1 : Type u\ns : Set \u03b1\n\u22a2 s = \u2205 \u2192 #\u2191s = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mpr\n\u03b1\u271d \u03b2 \u03b1 : Type u\n\u22a2 #\u2191\u2205 = 0\n[PROOFSTEP]\nexact mk_emptyCollection _\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\n\u22a2 lift #\u03b1 \u2264 lift #\u03b2\n[PROOFSTEP]\nrw [\u2190 Cardinal.mk_range_eq_of_injective hf]\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\n\u22a2 lift #\u2191(range f) \u2264 lift #\u03b2\n[PROOFSTEP]\nexact Cardinal.lift_le.2 (Cardinal.mk_set_le _)\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nA : Set (Set \u03b1)\n\u22a2 #\u2191(\u22c3\u2080 A) \u2264 #\u2191A * \u2a06 (s : \u2191A), #\u2191\u2191s\n[PROOFSTEP]\nrw [sUnion_eq_iUnion]\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nA : Set (Set \u03b1)\n\u22a2 #\u2191(\u22c3 (i : \u2191A), \u2191i) \u2264 #\u2191A * \u2a06 (s : \u2191A), #\u2191\u2191s\n[PROOFSTEP]\napply mk_iUnion_le\n[GOAL]\n\u03b1\u271d \u03b2 \u03b9 \u03b1 : Type u\nA : \u03b9 \u2192 Set \u03b1\ns : Set \u03b9\n\u22a2 #\u2191(\u22c3 (x : \u03b9) (_ : x \u2208 s), A x) \u2264 #\u2191s * \u2a06 (x : \u2191s), #\u2191(A \u2191x)\n[PROOFSTEP]\nrw [biUnion_eq_iUnion]\n[GOAL]\n\u03b1\u271d \u03b2 \u03b9 \u03b1 : Type u\nA : \u03b9 \u2192 Set \u03b1\ns : Set \u03b9\n\u22a2 #\u2191(\u22c3 (x : \u2191s), A \u2191x) \u2264 #\u2191s * \u2a06 (x : \u2191s), #\u2191(A \u2191x)\n[PROOFSTEP]\napply mk_iUnion_le\n[GOAL]\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type u_1\ns : Set \u03b1\nn : \u2115\n\u22a2 #\u2191s = \u2191n \u2194 \u2203 t, \u2191t = s \u2227 Finset.card t = n\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type u_1\ns : Set \u03b1\nn : \u2115\n\u22a2 #\u2191s = \u2191n \u2192 \u2203 t, \u2191t = s \u2227 Finset.card t = n\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type u_1\ns : Set \u03b1\nn : \u2115\nh : #\u2191s = \u2191n\n\u22a2 \u2203 t, \u2191t = s \u2227 Finset.card t = n\n[PROOFSTEP]\nlift s to Finset \u03b1 using lt_aleph0_iff_set_finite.1 (h.symm \u25b8 nat_lt_aleph0 n)\n[GOAL]\ncase mp.intro\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type u_1\nn : \u2115\ns : Finset \u03b1\nh : #\u2191\u2191s = \u2191n\n\u22a2 \u2203 t, \u2191t = \u2191s \u2227 Finset.card t = n\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase mpr\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type u_1\ns : Set \u03b1\nn : \u2115\n\u22a2 (\u2203 t, \u2191t = s \u2227 Finset.card t = n) \u2192 #\u2191s = \u2191n\n[PROOFSTEP]\nrintro \u27e8t, rfl, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type u_1\nt : Finset \u03b1\n\u22a2 #\u2191\u2191t = \u2191(Finset.card t)\n[PROOFSTEP]\nexact mk_coe_finset\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 #\u03b1 = \u2191n \u2194 \u2203 t, \u2191t = univ \u2227 Finset.card t = n\n[PROOFSTEP]\nrw [\u2190 mk_univ, mk_set_eq_nat_iff_finset]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 #\u03b1 = \u2191n \u2194 \u2203 h, Fintype.card \u03b1 = n\n[PROOFSTEP]\nrw [mk_eq_nat_iff_finset]\n[GOAL]\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 (\u2203 t, \u2191t = univ \u2227 Finset.card t = n) \u2194 \u2203 h, Fintype.card \u03b1 = n\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 (\u2203 t, \u2191t = univ \u2227 Finset.card t = n) \u2192 \u2203 h, Fintype.card \u03b1 = n\n[PROOFSTEP]\nrintro \u27e8t, ht, hn\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 \u03b2 : Type u\nn : \u2115\nt : Finset \u03b1\nht : \u2191t = univ\nhn : Finset.card t = n\n\u22a2 \u2203 h, Fintype.card \u03b1 = n\n[PROOFSTEP]\nexact \u27e8\u27e8t, eq_univ_iff_forall.1 ht\u27e9, hn\u27e9\n[GOAL]\ncase mpr\n\u03b1 \u03b2 : Type u\nn : \u2115\n\u22a2 (\u2203 h, Fintype.card \u03b1 = n) \u2192 \u2203 t, \u2191t = univ \u2227 Finset.card t = n\n[PROOFSTEP]\nrintro \u27e8\u27e8t, ht\u27e9, hn\u27e9\n[GOAL]\ncase mpr.intro.mk\n\u03b1 \u03b2 : Type u\nn : \u2115\nt : Finset \u03b1\nht : \u2200 (x : \u03b1), x \u2208 t\nhn : Fintype.card \u03b1 = n\n\u22a2 \u2203 t, \u2191t = univ \u2227 Finset.card t = n\n[PROOFSTEP]\nexact \u27e8t, eq_univ_iff_forall.2 ht, hn\u27e9\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\ns : Set \u03b1\na : \u03b1\nh : \u00aca \u2208 s\n\u22a2 #\u2191(insert a s) = #\u2191s + 1\n[PROOFSTEP]\nrw [\u2190 union_singleton, mk_union_of_disjoint, mk_singleton]\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\ns : Set \u03b1\na : \u03b1\nh : \u00aca \u2208 s\n\u22a2 Disjoint s {a}\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\nt : Set \u03b1\n\u22a2 #\u2191t \u2264 \u2191n \u2194 \u2200 (s : Finset \u03b1), \u2191s \u2286 t \u2192 Finset.card s \u2264 n\n[PROOFSTEP]\nrefine \u27e8fun H s hs \u21a6 by simpa using (mk_le_mk_of_subset hs).trans H, fun H \u21a6 ?_\u27e9\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\nt : Set \u03b1\nH : #\u2191t \u2264 \u2191n\ns : Finset \u03b1\nhs : \u2191s \u2286 t\n\u22a2 Finset.card s \u2264 n\n[PROOFSTEP]\nsimpa using (mk_le_mk_of_subset hs).trans H\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\nt : Set \u03b1\nH : \u2200 (s : Finset \u03b1), \u2191s \u2286 t \u2192 Finset.card s \u2264 n\n\u22a2 #\u2191t \u2264 \u2191n\n[PROOFSTEP]\napply card_le_of (fun s \u21a6 ?_)\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\nt : Set \u03b1\nH : \u2200 (s : Finset \u03b1), \u2191s \u2286 t \u2192 Finset.card s \u2264 n\ns : Finset \u2191t\n\u22a2 Finset.card s \u2264 n\n[PROOFSTEP]\nlet u : Finset \u03b1 := s.image Subtype.val\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\nt : Set \u03b1\nH : \u2200 (s : Finset \u03b1), \u2191s \u2286 t \u2192 Finset.card s \u2264 n\ns : Finset \u2191t\nu : Finset \u03b1 := Finset.image Subtype.val s\n\u22a2 Finset.card s \u2264 n\n[PROOFSTEP]\nhave : u.card = s.card := Finset.card_image_of_injOn (injOn_of_injective Subtype.coe_injective _)\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\nt : Set \u03b1\nH : \u2200 (s : Finset \u03b1), \u2191s \u2286 t \u2192 Finset.card s \u2264 n\ns : Finset \u2191t\nu : Finset \u03b1 := Finset.image Subtype.val s\nthis : Finset.card u = Finset.card s\n\u22a2 Finset.card s \u2264 n\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\nt : Set \u03b1\nH : \u2200 (s : Finset \u03b1), \u2191s \u2286 t \u2192 Finset.card s \u2264 n\ns : Finset \u2191t\nu : Finset \u03b1 := Finset.image Subtype.val s\nthis : Finset.card u = Finset.card s\n\u22a2 Finset.card u \u2264 n\n[PROOFSTEP]\napply H\n[GOAL]\ncase a\n\u03b1\u271d \u03b2 \u03b1 : Type u\nn : \u2115\nt : Set \u03b1\nH : \u2200 (s : Finset \u03b1), \u2191s \u2286 t \u2192 Finset.card s \u2264 n\ns : Finset \u2191t\nu : Finset \u03b1 := Finset.image Subtype.val s\nthis : Finset.card u = Finset.card s\n\u22a2 \u2191u \u2286 t\n[PROOFSTEP]\nsimp only [Finset.coe_image, image_subset_iff, Subtype.coe_preimage_self, subset_univ]\n[GOAL]\n\u03b1 \u03b2 : Type u\nS T : Set \u03b1\nh : T \u2286 S\n\u22a2 #\u2191(S \\ T) + #\u2191T = #\u2191S\n[PROOFSTEP]\nrefine (mk_union_of_disjoint <| ?_).symm.trans <| by rw [diff_union_of_subset h]\n[GOAL]\n\u03b1 \u03b2 : Type u\nS T : Set \u03b1\nh : T \u2286 S\n\u22a2 #\u2191(S \\ T \u222a T) = #\u2191S\n[PROOFSTEP]\nrw [diff_union_of_subset h]\n[GOAL]\n\u03b1 \u03b2 : Type u\nS T : Set \u03b1\nh : T \u2286 S\n\u22a2 Disjoint (S \\ T) T\n[PROOFSTEP]\nexact disjoint_sdiff_self_left\n[GOAL]\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type u_1\nP Q : Set \u03b1\n\u22a2 #\u2191(P \u222a Q) \u2264 \u2135\u2080 \u2194 #\u2191P \u2264 \u2135\u2080 \u2227 #\u2191Q \u2264 \u2135\u2080\n[PROOFSTEP]\nsimp only [le_aleph0_iff_subtype_countable, mem_union, setOf_mem_eq, Set.union_def, \u2190 countable_union]\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : Injective f\n\u22a2 lift #\u2191(f \u207b\u00b9' s) \u2264 lift #\u2191s\n[PROOFSTEP]\nrw [lift_mk_le.{0}]\n -- Porting note: Needed to insert `mem_preimage.mp` below\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : Injective f\n\u22a2 Nonempty (\u2191(f \u207b\u00b9' s) \u21aa \u2191s)\n[PROOFSTEP]\nuse Subtype.coind (fun x => f x.1) fun x => mem_preimage.mp x.2\n[GOAL]\ncase inj'\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : Injective f\n\u22a2 Injective (Subtype.coind (fun x => f \u2191x) (_ : \u2200 (x : \u2191(f \u207b\u00b9' s)), f \u2191x \u2208 s))\n[PROOFSTEP]\napply Subtype.coind_injective\n[GOAL]\ncase inj'.hf\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : Injective f\n\u22a2 Injective fun x => f \u2191x\n[PROOFSTEP]\nexact h.comp Subtype.val_injective\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : s \u2286 range f\n\u22a2 lift #\u2191s \u2264 lift #\u2191(f \u207b\u00b9' s)\n[PROOFSTEP]\nrw [lift_mk_le.{0}]\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : s \u2286 range f\n\u22a2 Nonempty (\u2191s \u21aa \u2191(f \u207b\u00b9' s))\n[PROOFSTEP]\nrefine' \u27e8\u27e8_, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : s \u2286 range f\n\u22a2 \u2191s \u2192 \u2191(f \u207b\u00b9' s)\n[PROOFSTEP]\nrintro \u27e8y, hy\u27e9\n[GOAL]\ncase refine'_1.mk\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : s \u2286 range f\ny : \u03b2\nhy : y \u2208 s\n\u22a2 \u2191(f \u207b\u00b9' s)\n[PROOFSTEP]\nrcases Classical.subtype_of_exists (h hy) with \u27e8x, rfl\u27e9\n[GOAL]\ncase refine'_1.mk.mk\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : s \u2286 range f\nx : \u03b1\nhy : f x \u2208 s\n\u22a2 \u2191(f \u207b\u00b9' s)\n[PROOFSTEP]\nexact \u27e8x, hy\u27e9\n[GOAL]\ncase refine'_2\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : s \u2286 range f\n\u22a2 Injective fun a =>\n Subtype.casesOn a fun y hy =>\n Subtype.casesOn (subtype_of_exists (_ : y \u2208 range f)) fun x property =>\n Eq.ndrec (motive := fun y => y \u2208 s \u2192 \u2191(f \u207b\u00b9' s)) (fun hy => { val := x, property := hy }) property hy\n[PROOFSTEP]\nrintro \u27e8y, hy\u27e9 \u27e8y', hy'\u27e9\n[GOAL]\ncase refine'_2.mk.mk\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : s \u2286 range f\ny : \u03b2\nhy : y \u2208 s\ny' : \u03b2\nhy' : y' \u2208 s\n\u22a2 (fun a =>\n Subtype.casesOn a fun y hy =>\n Subtype.casesOn (subtype_of_exists (_ : y \u2208 range f)) fun x property =>\n Eq.ndrec (motive := fun y => y \u2208 s \u2192 \u2191(f \u207b\u00b9' s)) (fun hy => { val := x, property := hy }) property hy)\n { val := y, property := hy } =\n (fun a =>\n Subtype.casesOn a fun y hy =>\n Subtype.casesOn (subtype_of_exists (_ : y \u2208 range f)) fun x property =>\n Eq.ndrec (motive := fun y => y \u2208 s \u2192 \u2191(f \u207b\u00b9' s)) (fun hy => { val := x, property := hy }) property hy)\n { val := y', property := hy' } \u2192\n { val := y, property := hy } = { val := y', property := hy' }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_2.mk.mk\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : s \u2286 range f\ny : \u03b2\nhy : y \u2208 s\ny' : \u03b2\nhy' : y' \u2208 s\n\u22a2 Eq.rec (motive := fun x x_1 => x \u2208 s \u2192 \u2191(f \u207b\u00b9' s))\n (fun hy_1 => { val := \u2191(subtype_of_exists (_ : y \u2208 range f)), property := hy_1 })\n (_ : f \u2191(subtype_of_exists (_ : y \u2208 range f)) = y) hy =\n Eq.rec (motive := fun x x_1 => x \u2208 s \u2192 \u2191(f \u207b\u00b9' s))\n (fun hy => { val := \u2191(subtype_of_exists (_ : y' \u2208 range f)), property := hy })\n (_ : f \u2191(subtype_of_exists (_ : y' \u2208 range f)) = y') hy' \u2192\n { val := y, property := hy } = { val := y', property := hy' }\n[PROOFSTEP]\nrcases Classical.subtype_of_exists (h hy) with \u27e8x, rfl\u27e9\n[GOAL]\ncase refine'_2.mk.mk.mk\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : s \u2286 range f\ny' : \u03b2\nhy' : y' \u2208 s\nx : \u03b1\nhy : f x \u2208 s\n\u22a2 Eq.rec (motive := fun x_1 x => x_1 \u2208 s \u2192 \u2191(f \u207b\u00b9' s))\n (fun hy => { val := \u2191{ val := x, property := (_ : f x = f x) }, property := hy })\n (_ : f \u2191{ val := x, property := (_ : f x = f x) } = f x) hy =\n Eq.rec (motive := fun x x_1 => x \u2208 s \u2192 \u2191(f \u207b\u00b9' s))\n (fun hy => { val := \u2191(subtype_of_exists (_ : y' \u2208 range f)), property := hy })\n (_ : f \u2191(subtype_of_exists (_ : y' \u2208 range f)) = y') hy' \u2192\n { val := f x, property := hy } = { val := y', property := hy' }\n[PROOFSTEP]\nrcases Classical.subtype_of_exists (h hy') with \u27e8x', rfl\u27e9\n[GOAL]\ncase refine'_2.mk.mk.mk.mk\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : s \u2286 range f\nx : \u03b1\nhy : f x \u2208 s\nx' : \u03b1\nhy' : f x' \u2208 s\n\u22a2 Eq.rec (motive := fun x_1 x => x_1 \u2208 s \u2192 \u2191(f \u207b\u00b9' s))\n (fun hy => { val := \u2191{ val := x, property := (_ : f x = f x) }, property := hy })\n (_ : f \u2191{ val := x, property := (_ : f x = f x) } = f x) hy =\n Eq.rec (motive := fun x x_1 => x \u2208 s \u2192 \u2191(f \u207b\u00b9' s))\n (fun hy => { val := \u2191{ val := x', property := (_ : f x' = f x') }, property := hy })\n (_ : f \u2191{ val := x', property := (_ : f x' = f x') } = f x') hy' \u2192\n { val := f x, property := hy } = { val := f x', property := hy' }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2.mk.mk.mk.mk\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : s \u2286 range f\nx : \u03b1\nhy : f x \u2208 s\nx' : \u03b1\nhy' : f x' \u2208 s\n\u22a2 x = x' \u2192 f x = f x'\n[PROOFSTEP]\nintro hxx'\n[GOAL]\ncase refine'_2.mk.mk.mk.mk\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : s \u2286 range f\nx : \u03b1\nhy : f x \u2208 s\nx' : \u03b1\nhy' : f x' \u2208 s\nhxx' : x = x'\n\u22a2 f x = f x'\n[PROOFSTEP]\nrw [hxx']\n[GOAL]\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : Injective f\n\u22a2 #\u2191(f \u207b\u00b9' s) \u2264 #\u2191s\n[PROOFSTEP]\nrw [\u2190 lift_id #(\u2191(f \u207b\u00b9' s)), \u2190 lift_id #(\u2191s)]\n[GOAL]\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : Injective f\n\u22a2 lift #\u2191(f \u207b\u00b9' s) \u2264 lift #\u2191s\n[PROOFSTEP]\nexact mk_preimage_of_injective_lift f s h\n[GOAL]\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : s \u2286 range f\n\u22a2 #\u2191s \u2264 #\u2191(f \u207b\u00b9' s)\n[PROOFSTEP]\nrw [\u2190 lift_id #(\u2191(f \u207b\u00b9' s)), \u2190 lift_id #(\u2191s)]\n[GOAL]\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : s \u2286 range f\n\u22a2 lift #\u2191s \u2264 lift #\u2191(f \u207b\u00b9' s)\n[PROOFSTEP]\nexact mk_preimage_of_subset_range_lift f s h\n[GOAL]\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : Injective f\nh2 : s \u2286 range f\n\u22a2 #\u2191(f \u207b\u00b9' s) = #\u2191s\n[PROOFSTEP]\nconvert mk_preimage_of_injective_of_subset_range_lift.{u, u} f s h h2 using 1\n[GOAL]\ncase h.e'_2\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : Injective f\nh2 : s \u2286 range f\n\u22a2 #\u2191(f \u207b\u00b9' s) = lift #\u2191(f \u207b\u00b9' s)\n[PROOFSTEP]\nrw [lift_id]\n[GOAL]\ncase h.e'_3\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nh : Injective f\nh2 : s \u2286 range f\n\u22a2 #\u2191s = lift #\u2191s\n[PROOFSTEP]\nrw [lift_id]\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : t \u2286 f '' s\n\u22a2 lift #\u2191t \u2264 lift #\u2191{x | x \u2208 s \u2227 f x \u2208 t}\n[PROOFSTEP]\nrw [image_eq_range] at h \n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : t \u2286 range fun x => f \u2191x\n\u22a2 lift #\u2191t \u2264 lift #\u2191{x | x \u2208 s \u2227 f x \u2208 t}\n[PROOFSTEP]\nconvert mk_preimage_of_subset_range_lift _ _ h using 1\n[GOAL]\ncase h.e'_4\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : t \u2286 range fun x => f \u2191x\n\u22a2 lift #\u2191{x | x \u2208 s \u2227 f x \u2208 t} = lift #\u2191((fun x => f \u2191x) \u207b\u00b9' t)\n[PROOFSTEP]\nrw [mk_sep]\n[GOAL]\ncase h.e'_4\n\u03b1\u271d \u03b2\u271d \u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : t \u2286 range fun x => f \u2191x\n\u22a2 lift #\u2191{x | f \u2191x \u2208 t} = lift #\u2191((fun x => f \u2191x) \u207b\u00b9' t)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : t \u2286 f '' s\n\u22a2 #\u2191t \u2264 #\u2191{x | x \u2208 s \u2227 f x \u2208 t}\n[PROOFSTEP]\nrw [image_eq_range] at h \n[GOAL]\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : t \u2286 range fun x => f \u2191x\n\u22a2 #\u2191t \u2264 #\u2191{x | x \u2208 s \u2227 f x \u2208 t}\n[PROOFSTEP]\nconvert mk_preimage_of_subset_range _ _ h using 1\n[GOAL]\ncase h.e'_4\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : t \u2286 range fun x => f \u2191x\n\u22a2 #\u2191{x | x \u2208 s \u2227 f x \u2208 t} = #\u2191((fun x => f \u2191x) \u207b\u00b9' t)\n[PROOFSTEP]\nrw [mk_sep]\n[GOAL]\ncase h.e'_4\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : t \u2286 range fun x => f \u2191x\n\u22a2 #\u2191{x | f \u2191x \u2208 t} = #\u2191((fun x => f \u2191x) \u207b\u00b9' t)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d \u03b2 : Type u\nc : Cardinal.{u}\n\u03b1 : Type u\ns : Set \u03b1\n\u22a2 c \u2264 #\u2191s \u2194 \u2203 p, p \u2286 s \u2227 #\u2191p = c\n[PROOFSTEP]\nrw [le_mk_iff_exists_set, \u2190 Subtype.exists_set_subtype]\n[GOAL]\n\u03b1\u271d \u03b2 : Type u\nc : Cardinal.{u}\n\u03b1 : Type u\ns : Set \u03b1\n\u22a2 (\u2203 p, #\u2191p = c) \u2194 \u2203 s_1, #\u2191(Subtype.val '' s_1) = c\n[PROOFSTEP]\napply exists_congr\n[GOAL]\ncase h\n\u03b1\u271d \u03b2 : Type u\nc : Cardinal.{u}\n\u03b1 : Type u\ns : Set \u03b1\n\u22a2 \u2200 (a : Set \u2191s), #\u2191a = c \u2194 #\u2191(Subtype.val '' a) = c\n[PROOFSTEP]\nintro t\n[GOAL]\ncase h\n\u03b1\u271d \u03b2 : Type u\nc : Cardinal.{u}\n\u03b1 : Type u\ns : Set \u03b1\nt : Set \u2191s\n\u22a2 #\u2191t = c \u2194 #\u2191(Subtype.val '' t) = c\n[PROOFSTEP]\nrw [mk_image_eq]\n[GOAL]\ncase h\n\u03b1\u271d \u03b2 : Type u\nc : Cardinal.{u}\n\u03b1 : Type u\ns : Set \u03b1\nt : Set \u2191s\n\u22a2 Injective Subtype.val\n[PROOFSTEP]\napply Subtype.val_injective\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 2 \u2264 #\u03b1 \u2194 \u2203 x y, x \u2260 y\n[PROOFSTEP]\nrw [\u2190 Nat.cast_two, nat_succ, succ_le_iff, Nat.cast_one, one_lt_iff_nontrivial, nontrivial_iff]\n[GOAL]\n\u03b1 \u03b2 : Type u\nx : \u03b1\n\u22a2 2 \u2264 #\u03b1 \u2194 \u2203 y, y \u2260 x\n[PROOFSTEP]\nrw [two_le_iff, \u2190 nontrivial_iff, nontrivial_iff_exists_ne x]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 #\u03b1 = 2 \u2194 \u2203 x y, x \u2260 y \u2227 {x, y} = univ\n[PROOFSTEP]\nsimp only [\u2190 @Nat.cast_two Cardinal, mk_eq_nat_iff_finset, Finset.card_eq_two]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u22a2 (\u2203 t, \u2191t = univ \u2227 \u2203 x y, x \u2260 y \u2227 t = {x, y}) \u2194 \u2203 x y, x \u2260 y \u2227 {x, y} = univ\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 \u03b2 : Type u\n\u22a2 (\u2203 t, \u2191t = univ \u2227 \u2203 x y, x \u2260 y \u2227 t = {x, y}) \u2192 \u2203 x y, x \u2260 y \u2227 {x, y} = univ\n[PROOFSTEP]\nrintro \u27e8t, ht, x, y, hne, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 \u03b2 : Type u\nx y : \u03b1\nhne : x \u2260 y\nht : \u2191{x, y} = univ\n\u22a2 \u2203 x y, x \u2260 y \u2227 {x, y} = univ\n[PROOFSTEP]\nexact \u27e8x, y, hne, by simpa using ht\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\nx y : \u03b1\nhne : x \u2260 y\nht : \u2191{x, y} = univ\n\u22a2 {x, y} = univ\n[PROOFSTEP]\nsimpa using ht\n[GOAL]\ncase mpr\n\u03b1 \u03b2 : Type u\n\u22a2 (\u2203 x y, x \u2260 y \u2227 {x, y} = univ) \u2192 \u2203 t, \u2191t = univ \u2227 \u2203 x y, x \u2260 y \u2227 t = {x, y}\n[PROOFSTEP]\nrintro \u27e8x, y, hne, h\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 \u03b2 : Type u\nx y : \u03b1\nhne : x \u2260 y\nh : {x, y} = univ\n\u22a2 \u2203 t, \u2191t = univ \u2227 \u2203 x y, x \u2260 y \u2227 t = {x, y}\n[PROOFSTEP]\nexact \u27e8{ x, y }, by simpa using h, x, y, hne, rfl\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\nx y : \u03b1\nhne : x \u2260 y\nh : {x, y} = univ\n\u22a2 \u2191{x, y} = univ\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1 \u03b2 : Type u\nx : \u03b1\n\u22a2 #\u03b1 = 2 \u2194 \u2203! y, y \u2260 x\n[PROOFSTEP]\nrw [mk_eq_two_iff]\n[GOAL]\n\u03b1 \u03b2 : Type u\nx : \u03b1\n\u22a2 (\u2203 x y, x \u2260 y \u2227 {x, y} = univ) \u2194 \u2203! y, y \u2260 x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 \u03b2 : Type u\nx : \u03b1\n\u22a2 (\u2203 x y, x \u2260 y \u2227 {x, y} = univ) \u2192 \u2203! y, y \u2260 x\n[PROOFSTEP]\nrintro \u27e8a, b, hne, h\u27e9\n[GOAL]\ncase mp.intro.intro.intro\n\u03b1 \u03b2 : Type u\nx a b : \u03b1\nhne : a \u2260 b\nh : {a, b} = univ\n\u22a2 \u2203! y, y \u2260 x\n[PROOFSTEP]\nsimp only [eq_univ_iff_forall, mem_insert_iff, mem_singleton_iff] at h \n[GOAL]\ncase mp.intro.intro.intro\n\u03b1 \u03b2 : Type u\nx a b : \u03b1\nhne : a \u2260 b\nh : \u2200 (x : \u03b1), x = a \u2228 x = b\n\u22a2 \u2203! y, y \u2260 x\n[PROOFSTEP]\nrcases h x with (rfl | rfl)\n[GOAL]\ncase mp.intro.intro.intro.inl\n\u03b1 \u03b2 : Type u\nx b : \u03b1\nhne : x \u2260 b\nh : \u2200 (x_1 : \u03b1), x_1 = x \u2228 x_1 = b\n\u22a2 \u2203! y, y \u2260 x\ncase mp.intro.intro.intro.inr \u03b1 \u03b2 : Type u x a : \u03b1 hne : a \u2260 x h : \u2200 (x_1 : \u03b1), x_1 = a \u2228 x_1 = x \u22a2 \u2203! y, y \u2260 x\n[PROOFSTEP]\nexacts [\u27e8b, hne.symm, fun z => (h z).resolve_left\u27e9, \u27e8a, hne, fun z => (h z).resolve_right\u27e9]\n[GOAL]\ncase mpr\n\u03b1 \u03b2 : Type u\nx : \u03b1\n\u22a2 (\u2203! y, y \u2260 x) \u2192 \u2203 x y, x \u2260 y \u2227 {x, y} = univ\n[PROOFSTEP]\nrintro \u27e8y, hne, hy\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 \u03b2 : Type u\nx y : \u03b1\nhne : y \u2260 x\nhy : \u2200 (y_1 : \u03b1), (fun y => y \u2260 x) y_1 \u2192 y_1 = y\n\u22a2 \u2203 x y, x \u2260 y \u2227 {x, y} = univ\n[PROOFSTEP]\nexact \u27e8x, y, hne.symm, eq_univ_of_forall fun z => or_iff_not_imp_left.2 (hy z)\u27e9\n[GOAL]\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type u_1\nl : List \u03b1\nh : \u2191(List.length l) < #\u03b1\n\u22a2 \u2203 z, \u00acz \u2208 l\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type u_1\nl : List \u03b1\nh : \u2200 (z : \u03b1), z \u2208 l\n\u22a2 #\u03b1 \u2264 \u2191(List.length l)\n[PROOFSTEP]\ncalc\n #\u03b1 = #(Set.univ : Set \u03b1) := mk_univ.symm\n _ \u2264 #l.toFinset := (mk_le_mk_of_subset fun x _ => List.mem_toFinset.mpr (h x))\n _ = l.toFinset.card := Cardinal.mk_coe_finset\n _ \u2264 l.length := Cardinal.natCast_le.mpr (List.toFinset_card_le l)\n[GOAL]\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type u_1\nh : 3 \u2264 #\u03b1\nx y : \u03b1\n\u22a2 \u2203 z, z \u2260 x \u2227 z \u2260 y\n[PROOFSTEP]\nhave : \u2191(3 : \u2115) \u2264 #\u03b1\n[GOAL]\ncase this\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type u_1\nh : 3 \u2264 #\u03b1\nx y : \u03b1\n\u22a2 \u21913 \u2264 #\u03b1\n\u03b1\u271d \u03b2 : Type u \u03b1 : Type u_1 h : 3 \u2264 #\u03b1 x y : \u03b1 this : \u21913 \u2264 #\u03b1 \u22a2 \u2203 z, z \u2260 x \u2227 z \u2260 y\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type u_1\nh : 3 \u2264 #\u03b1\nx y : \u03b1\nthis : \u21913 \u2264 #\u03b1\n\u22a2 \u2203 z, z \u2260 x \u2227 z \u2260 y\n[PROOFSTEP]\nhave : \u2191(2 : \u2115) < #\u03b1\n[GOAL]\ncase this\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type u_1\nh : 3 \u2264 #\u03b1\nx y : \u03b1\nthis : \u21913 \u2264 #\u03b1\n\u22a2 \u21912 < #\u03b1\n\u03b1\u271d \u03b2 : Type u \u03b1 : Type u_1 h : 3 \u2264 #\u03b1 x y : \u03b1 this\u271d : \u21913 \u2264 #\u03b1 this : \u21912 < #\u03b1 \u22a2 \u2203 z, z \u2260 x \u2227 z \u2260 y\n[PROOFSTEP]\nrwa [\u2190 succ_le_iff, \u2190 Cardinal.nat_succ]\n[GOAL]\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type u_1\nh : 3 \u2264 #\u03b1\nx y : \u03b1\nthis\u271d : \u21913 \u2264 #\u03b1\nthis : \u21912 < #\u03b1\n\u22a2 \u2203 z, z \u2260 x \u2227 z \u2260 y\n[PROOFSTEP]\nhave := exists_not_mem_of_length_lt [x, y] this\n[GOAL]\n\u03b1\u271d \u03b2 : Type u\n\u03b1 : Type u_1\nh : 3 \u2264 #\u03b1\nx y : \u03b1\nthis\u271d\u00b9 : \u21913 \u2264 #\u03b1\nthis\u271d : \u21912 < #\u03b1\nthis : \u2203 z, \u00acz \u2208 [x, y]\n\u22a2 \u2203 z, z \u2260 x \u2227 z \u2260 y\n[PROOFSTEP]\nsimpa [not_or] using this\n[GOAL]\n\u03b1 \u03b2 : Type u\nb c a : Cardinal.{u}\nh : c < b\n\u22a2 a ^ c \u2264 a ^< b\n[PROOFSTEP]\nrefine le_ciSup (f := fun y : Iio b => a ^ y) ?_ \u27e8c, h\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\nb c a : Cardinal.{u}\nh : c < b\n\u22a2 BddAbove (range fun y => a ^ \u2191y)\n[PROOFSTEP]\nrw [\u2190 image_eq_range]\n[GOAL]\n\u03b1 \u03b2 : Type u\nb c a : Cardinal.{u}\nh : c < b\n\u22a2 BddAbove (HPow.hPow a '' Iio b)\n[PROOFSTEP]\nexact bddAbove_image.{u, u} _ bddAbove_Iio\n[GOAL]\n\u03b1 \u03b2 : Type u\na b c : Cardinal.{u}\n\u22a2 a ^< b \u2264 c \u2194 \u2200 (x : Cardinal.{u}), x < b \u2192 a ^ x \u2264 c\n[PROOFSTEP]\nrw [powerlt, ciSup_le_iff']\n[GOAL]\n\u03b1 \u03b2 : Type u\na b c : Cardinal.{u}\n\u22a2 (\u2200 (i : \u2191(Iio b)), a ^ \u2191i \u2264 c) \u2194 \u2200 (x : Cardinal.{u}), x < b \u2192 a ^ x \u2264 c\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\n\u03b1 \u03b2 : Type u\na b c : Cardinal.{u}\n\u22a2 BddAbove (range fun c => a ^ \u2191c)\n[PROOFSTEP]\nrw [\u2190 image_eq_range]\n[GOAL]\ncase h\n\u03b1 \u03b2 : Type u\na b c : Cardinal.{u}\n\u22a2 BddAbove (HPow.hPow a '' Iio b)\n[PROOFSTEP]\nexact bddAbove_image.{u, u} _ bddAbove_Iio\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\nh : a \u2260 0\n\u22a2 0 ^< a = 1\n[PROOFSTEP]\napply (powerlt_le.2 fun c _ => zero_power_le _).antisymm\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\nh : a \u2260 0\n\u22a2 1 \u2264 0 ^< a\n[PROOFSTEP]\nrw [\u2190 power_zero]\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\nh : a \u2260 0\n\u22a2 ?m.860481 ^ 0 \u2264 0 ^< a\n\u03b1 \u03b2 : Type u a : Cardinal.{u_1} h : a \u2260 0 \u22a2 Cardinal.{u_1}\n[PROOFSTEP]\nexact le_powerlt 0 (pos_iff_ne_zero.2 h)\n[GOAL]\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\n\u22a2 a ^< 0 = 0\n[PROOFSTEP]\nconvert Cardinal.iSup_of_empty _\n[GOAL]\ncase convert_3\n\u03b1 \u03b2 : Type u\na : Cardinal.{u_1}\n\u22a2 IsEmpty \u2191(Iio 0)\n[PROOFSTEP]\nexact Subtype.isEmpty_of_false fun x => mem_Iio.not.mpr (Cardinal.zero_le x).not_lt\n[GOAL]\n\u03b1 \u03b2 R : Type u\nE : Type v\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Module R E\ninst\u271d\u00b9 : Nontrivial E\ninst\u271d : NoZeroSMulDivisors R E\n\u22a2 lift #R \u2264 lift #E\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 : \u2203 (x : E), x \u2260 0 := exists_ne 0\n[GOAL]\ncase intro\n\u03b1 \u03b2 R : Type u\nE : Type v\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Module R E\ninst\u271d\u00b9 : Nontrivial E\ninst\u271d : NoZeroSMulDivisors R E\nx : E\nhx : x \u2260 0\n\u22a2 lift #R \u2264 lift #E\n[PROOFSTEP]\nhave : Injective (fun k \u21a6 k \u2022 x) := smul_left_injective R hx\n[GOAL]\ncase intro\n\u03b1 \u03b2 R : Type u\nE : Type v\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Module R E\ninst\u271d\u00b9 : Nontrivial E\ninst\u271d : NoZeroSMulDivisors R E\nx : E\nhx : x \u2260 0\nthis : Injective fun k => k \u2022 x\n\u22a2 lift #R \u2264 lift #E\n[PROOFSTEP]\nexact lift_mk_le_lift_mk_of_injective this\n", "meta": {"mathlib_filename": "Mathlib.SetTheory.Cardinal.Basic", "llama_tokens": 55041, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.41402276995444987}} {"text": "[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\na : \u03b1\n\u22a2 Function.Injective (mk a)\n[PROOFSTEP]\nintro b\u2081 b\u2082 h\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\na : \u03b1\nb\u2081 b\u2082 : \u03b2\nh : (a, b\u2081) = (a, b\u2082)\n\u22a2 b\u2081 = b\u2082\n[PROOFSTEP]\nsimpa only [true_and, Prod.mk.inj_iff, eq_self_iff_true] using h\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nb : \u03b2\n\u22a2 Function.Injective fun a => (a, b)\n[PROOFSTEP]\nintro b\u2081 b\u2082 h\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\nb : \u03b2\nb\u2081 b\u2082 : \u03b1\nh : (fun a => (a, b)) b\u2081 = (fun a => (a, b)) b\u2082\n\u22a2 b\u2081 = b\u2082\n[PROOFSTEP]\nsimpa only [and_true, eq_self_iff_true, mk.inj_iff] using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\np q : \u03b1 \u00d7 \u03b2\n\u22a2 p = q \u2194 p.fst = q.fst \u2227 p.snd = q.snd\n[PROOFSTEP]\nrw [\u2190 @mk.eta _ _ p, \u2190 @mk.eta _ _ q, mk.inj_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\np\u2081 : \u03b1\np\u2082 : \u03b2\nq\u2081 : \u03b1\nq\u2082 : \u03b2\n\u22a2 (p\u2081, p\u2082) = (q\u2081, q\u2082) \u2194 (p\u2081, p\u2082).fst = (q\u2081, q\u2082).fst \u2227 (p\u2081, p\u2082).snd = (q\u2081, q\u2082).snd\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\na : \u03b1\nb : \u03b2\nx : \u03b1\n\u22a2 (a, b).fst = x \u2194 (a, b) = (x, (a, b).snd)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\na : \u03b1\nb x : \u03b2\n\u22a2 (a, b).snd = x \u2194 (a, b) = ((a, b).fst, x)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\np q : \u03b1 \u00d7 \u03b2\nh : Prod.Lex r s p q\n\u22a2 r p.fst q.fst \u2228 p.fst = q.fst \u2227 s p.snd q.snd\n[PROOFSTEP]\ncases h\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\na\u2081\u271d : \u03b1\nb\u2081\u271d : \u03b2\na\u2082\u271d : \u03b1\nb\u2082\u271d : \u03b2\nh\u271d : r a\u2081\u271d a\u2082\u271d\n\u22a2 r (a\u2081\u271d, b\u2081\u271d).fst (a\u2082\u271d, b\u2082\u271d).fst \u2228 (a\u2081\u271d, b\u2081\u271d).fst = (a\u2082\u271d, b\u2082\u271d).fst \u2227 s (a\u2081\u271d, b\u2081\u271d).snd (a\u2082\u271d, b\u2082\u271d).snd\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\na\u271d : \u03b1\nb\u2081\u271d b\u2082\u271d : \u03b2\nh\u271d : s b\u2081\u271d b\u2082\u271d\n\u22a2 r (a\u271d, b\u2081\u271d).fst (a\u271d, b\u2082\u271d).fst \u2228 (a\u271d, b\u2081\u271d).fst = (a\u271d, b\u2082\u271d).fst \u2227 s (a\u271d, b\u2081\u271d).snd (a\u271d, b\u2082\u271d).snd\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\np q : \u03b1 \u00d7 \u03b2\nh\u271d : r p.fst q.fst \u2228 p.fst = q.fst \u2227 s p.snd q.snd\na : \u03b1\nb : \u03b2\nc : \u03b1\nd : \u03b2\ne : (a, b).fst = (c, d).fst\nh : s (a, b).snd (c, d).snd\n\u22a2 Prod.Lex r s (a, b) (c, d)\n[PROOFSTEP]\nsubst e\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\np q : \u03b1 \u00d7 \u03b2\nh\u271d : r p.fst q.fst \u2228 p.fst = q.fst \u2227 s p.snd q.snd\na : \u03b1\nb d : \u03b2\nh : s (a, b).snd ((a, b).fst, d).snd\n\u22a2 Prod.Lex r s (a, b) ((a, b).fst, d)\n[PROOFSTEP]\nexact Lex.right _ h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b9 : IsIrrefl \u03b1 r\ninst\u271d : IsIrrefl \u03b2 s\n\u22a2 \u2200 (a : \u03b1 \u00d7 \u03b2), \u00acProd.Lex r s a a\n[PROOFSTEP]\nrintro \u27e8i, a\u27e9 (\u27e8_, _, h\u27e9 | \u27e8_, h\u27e9)\n[GOAL]\ncase mk.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b9 : IsIrrefl \u03b1 r\ninst\u271d : IsIrrefl \u03b2 s\ni : \u03b1\na : \u03b2\nh : r i i\n\u22a2 False\n[PROOFSTEP]\nexact irrefl _ h\n[GOAL]\ncase mk.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b9 : IsIrrefl \u03b1 r\ninst\u271d : IsIrrefl \u03b2 s\ni : \u03b1\na : \u03b2\nh : s a a\n\u22a2 False\n[PROOFSTEP]\nexact irrefl _ h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrichotomous \u03b1 r\ninst\u271d : IsTotal \u03b2 s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 \u03b2\ni : \u03b1\na : \u03b2\nj : \u03b1\nb : \u03b2\n\u22a2 Prod.Lex r s (i, a) (j, b) \u2228 Prod.Lex r s (j, b) (i, a)\n[PROOFSTEP]\nobtain hij | rfl | hji := trichotomous_of r i j\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrichotomous \u03b1 r\ninst\u271d : IsTotal \u03b2 s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 \u03b2\ni : \u03b1\na : \u03b2\nj : \u03b1\nb : \u03b2\nhij : r i j\n\u22a2 Prod.Lex r s (i, a) (j, b) \u2228 Prod.Lex r s (j, b) (i, a)\n[PROOFSTEP]\nexact Or.inl (.left _ _ hij)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrichotomous \u03b1 r\ninst\u271d : IsTotal \u03b2 s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 \u03b2\ni : \u03b1\na b : \u03b2\n\u22a2 Prod.Lex r s (i, a) (i, b) \u2228 Prod.Lex r s (i, b) (i, a)\n[PROOFSTEP]\nexact (total_of s a b).imp (.right _) (.right _)\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrichotomous \u03b1 r\ninst\u271d : IsTotal \u03b2 s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 \u03b2\ni : \u03b1\na : \u03b2\nj : \u03b1\nb : \u03b2\nhji : r j i\n\u22a2 Prod.Lex r s (i, a) (j, b) \u2228 Prod.Lex r s (j, b) (i, a)\n[PROOFSTEP]\nexact Or.inr (.left _ _ hji)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b9 : _root_.IsTrichotomous \u03b1 r\ninst\u271d : _root_.IsTrichotomous \u03b2 s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 \u03b2\ni : \u03b1\na : \u03b2\nj : \u03b1\nb : \u03b2\n\u22a2 Prod.Lex r s (i, a) (j, b) \u2228 (i, a) = (j, b) \u2228 Prod.Lex r s (j, b) (i, a)\n[PROOFSTEP]\nobtain hij | rfl | hji := trichotomous_of r i j\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b9 : _root_.IsTrichotomous \u03b1 r\ninst\u271d : _root_.IsTrichotomous \u03b2 s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 \u03b2\ni : \u03b1\na : \u03b2\nj : \u03b1\nb : \u03b2\nhij : r i j\n\u22a2 Prod.Lex r s (i, a) (j, b) \u2228 (i, a) = (j, b) \u2228 Prod.Lex r s (j, b) (i, a)\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b9 : _root_.IsTrichotomous \u03b1 r\ninst\u271d : _root_.IsTrichotomous \u03b2 s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 \u03b2\ni : \u03b1\na b : \u03b2\n\u22a2 Prod.Lex r s (i, a) (i, b) \u2228 (i, a) = (i, b) \u2228 Prod.Lex r s (i, b) (i, a)\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b9 : _root_.IsTrichotomous \u03b1 r\ninst\u271d : _root_.IsTrichotomous \u03b2 s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 \u03b2\ni : \u03b1\na : \u03b2\nj : \u03b1\nb : \u03b2\nhji : r j i\n\u22a2 Prod.Lex r s (i, a) (j, b) \u2228 (i, a) = (j, b) \u2228 Prod.Lex r s (j, b) (i, a)\n[PROOFSTEP]\n{exact Or.inl (Lex.left _ _ hij)\n}\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b9 : _root_.IsTrichotomous \u03b1 r\ninst\u271d : _root_.IsTrichotomous \u03b2 s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 \u03b2\ni : \u03b1\na : \u03b2\nj : \u03b1\nb : \u03b2\nhij : r i j\n\u22a2 Prod.Lex r s (i, a) (j, b) \u2228 (i, a) = (j, b) \u2228 Prod.Lex r s (j, b) (i, a)\n[PROOFSTEP]\nexact Or.inl (Lex.left _ _ hij)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b9 : _root_.IsTrichotomous \u03b1 r\ninst\u271d : _root_.IsTrichotomous \u03b2 s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 \u03b2\ni : \u03b1\na b : \u03b2\n\u22a2 Prod.Lex r s (i, a) (i, b) \u2228 (i, a) = (i, b) \u2228 Prod.Lex r s (i, b) (i, a)\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b9 : _root_.IsTrichotomous \u03b1 r\ninst\u271d : _root_.IsTrichotomous \u03b2 s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 \u03b2\ni : \u03b1\na : \u03b2\nj : \u03b1\nb : \u03b2\nhji : r j i\n\u22a2 Prod.Lex r s (i, a) (j, b) \u2228 (i, a) = (j, b) \u2228 Prod.Lex r s (j, b) (i, a)\n[PROOFSTEP]\n{exact (trichotomous_of (s) a b).imp3 (Lex.right _) (congr_arg _) (Lex.right _)\n}\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b9 : _root_.IsTrichotomous \u03b1 r\ninst\u271d : _root_.IsTrichotomous \u03b2 s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 \u03b2\ni : \u03b1\na b : \u03b2\n\u22a2 Prod.Lex r s (i, a) (i, b) \u2228 (i, a) = (i, b) \u2228 Prod.Lex r s (i, b) (i, a)\n[PROOFSTEP]\nexact (trichotomous_of (s) a b).imp3 (Lex.right _) (congr_arg _) (Lex.right _)\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b9 : _root_.IsTrichotomous \u03b1 r\ninst\u271d : _root_.IsTrichotomous \u03b2 s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 \u03b2\ni : \u03b1\na : \u03b2\nj : \u03b1\nb : \u03b2\nhji : r j i\n\u22a2 Prod.Lex r s (i, a) (j, b) \u2228 (i, a) = (j, b) \u2228 Prod.Lex r s (j, b) (i, a)\n[PROOFSTEP]\n{exact Or.inr (Or.inr $ Lex.left _ _ hji)\n}\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx y : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b9 : _root_.IsTrichotomous \u03b1 r\ninst\u271d : _root_.IsTrichotomous \u03b2 s\nx\u271d\u00b9 x\u271d : \u03b1 \u00d7 \u03b2\ni : \u03b1\na : \u03b2\nj : \u03b1\nb : \u03b2\nhji : r j i\n\u22a2 Prod.Lex r s (i, a) (j, b) \u2228 (i, a) = (j, b) \u2228 Prod.Lex r s (j, b) (i, a)\n[PROOFSTEP]\nexact Or.inr (Or.inr $ Lex.left _ _ hji)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nf\u2081 : \u03b1 \u2192 \u03b2\ng\u2081 : \u03b3 \u2192 \u03b4\nf\u2082 : \u03b2 \u2192 \u03b1\ng\u2082 : \u03b4 \u2192 \u03b3\nhf : LeftInverse f\u2081 f\u2082\nhg : LeftInverse g\u2081 g\u2082\na : \u03b2 \u00d7 \u03b4\n\u22a2 map f\u2081 g\u2081 (map f\u2082 g\u2082 a) = a\n[PROOFSTEP]\nrw [Prod.map_map, hf.comp_eq_id, hg.comp_eq_id, map_id, id]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh : Injective (map f g)\na\u2081 a\u2082 : \u03b1\nha : f a\u2081 = f a\u2082\n\u22a2 a\u2081 = a\u2082\n[PROOFSTEP]\ninhabit \u03b2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh : Injective (map f g)\na\u2081 a\u2082 : \u03b1\nha : f a\u2081 = f a\u2082\ninhabited_h : Inhabited \u03b2\n\u22a2 a\u2081 = a\u2082\n[PROOFSTEP]\ninjection @h (a\u2081, default) (a\u2082, default) (congr_arg (fun c : \u03b3 => Prod.mk c (g default)) ha : _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh : Injective (map f g)\nb\u2081 b\u2082 : \u03b2\nhb : g b\u2081 = g b\u2082\n\u22a2 b\u2081 = b\u2082\n[PROOFSTEP]\ninhabit \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh : Injective (map f g)\nb\u2081 b\u2082 : \u03b2\nhb : g b\u2081 = g b\u2082\ninhabited_h : Inhabited \u03b1\n\u22a2 b\u2081 = b\u2082\n[PROOFSTEP]\ninjection @h (default, b\u2081) (default, b\u2082) (congr_arg (Prod.mk (f default)) hb : _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b3\ninst\u271d : Nonempty \u03b4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh : Surjective (map f g)\nc : \u03b3\n\u22a2 \u2203 a, f a = c\n[PROOFSTEP]\ninhabit \u03b4\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b3\ninst\u271d : Nonempty \u03b4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh : Surjective (map f g)\nc : \u03b3\ninhabited_h : Inhabited \u03b4\n\u22a2 \u2203 a, f a = c\n[PROOFSTEP]\nobtain \u27e8\u27e8a, b\u27e9, h\u27e9 := h (c, default)\n[GOAL]\ncase intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b3\ninst\u271d : Nonempty \u03b4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh\u271d : Surjective (map f g)\nc : \u03b3\ninhabited_h : Inhabited \u03b4\na : \u03b1\nb : \u03b2\nh : map f g (a, b) = (c, default)\n\u22a2 \u2203 a, f a = c\n[PROOFSTEP]\nexact \u27e8a, congr_arg Prod.fst h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b3\ninst\u271d : Nonempty \u03b4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh : Surjective (map f g)\nd : \u03b4\n\u22a2 \u2203 a, g a = d\n[PROOFSTEP]\ninhabit \u03b3\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b3\ninst\u271d : Nonempty \u03b4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh : Surjective (map f g)\nd : \u03b4\ninhabited_h : Inhabited \u03b3\n\u22a2 \u2203 a, g a = d\n[PROOFSTEP]\nobtain \u27e8\u27e8a, b\u27e9, h\u27e9 := h (default, d)\n[GOAL]\ncase intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b3\ninst\u271d : Nonempty \u03b4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nh\u271d : Surjective (map f g)\nd : \u03b4\ninhabited_h : Inhabited \u03b3\na : \u03b1\nb : \u03b2\nh : map f g (a, b) = (default, d)\n\u22a2 \u2203 a, g a = d\n[PROOFSTEP]\nexact \u27e8b, congr_arg Prod.snd h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\n\u22a2 Bijective (map f g) \u2194 Bijective f \u2227 Bijective g\n[PROOFSTEP]\nhaveI := Nonempty.map f \u2039_\u203a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nthis : Nonempty \u03b3\n\u22a2 Bijective (map f g) \u2194 Bijective f \u2227 Bijective g\n[PROOFSTEP]\nhaveI := Nonempty.map g \u2039_\u203a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Nonempty \u03b2\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nthis\u271d : Nonempty \u03b3\nthis : Nonempty \u03b4\n\u22a2 Bijective (map f g) \u2194 Bijective f \u2227 Bijective g\n[PROOFSTEP]\nexact (map_injective.and map_surjective).trans (and_and_and_comm)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : Nonempty \u03b4\nf\u2081 : \u03b1 \u2192 \u03b2\ng\u2081 : \u03b3 \u2192 \u03b4\nf\u2082 : \u03b2 \u2192 \u03b1\ng\u2082 : \u03b4 \u2192 \u03b3\nh : LeftInverse (map f\u2081 g\u2081) (map f\u2082 g\u2082)\nb : \u03b2\n\u22a2 f\u2081 (f\u2082 b) = b\n[PROOFSTEP]\ninhabit \u03b4\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : Nonempty \u03b4\nf\u2081 : \u03b1 \u2192 \u03b2\ng\u2081 : \u03b3 \u2192 \u03b4\nf\u2082 : \u03b2 \u2192 \u03b1\ng\u2082 : \u03b4 \u2192 \u03b3\nh : LeftInverse (map f\u2081 g\u2081) (map f\u2082 g\u2082)\nb : \u03b2\ninhabited_h : Inhabited \u03b4\n\u22a2 f\u2081 (f\u2082 b) = b\n[PROOFSTEP]\nexact congr_arg Prod.fst (h (b, default))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : Nonempty \u03b4\nf\u2081 : \u03b1 \u2192 \u03b2\ng\u2081 : \u03b3 \u2192 \u03b4\nf\u2082 : \u03b2 \u2192 \u03b1\ng\u2082 : \u03b4 \u2192 \u03b3\nh : LeftInverse (map f\u2081 g\u2081) (map f\u2082 g\u2082)\nd : \u03b4\n\u22a2 g\u2081 (g\u2082 d) = d\n[PROOFSTEP]\ninhabit \u03b2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : Nonempty \u03b4\nf\u2081 : \u03b1 \u2192 \u03b2\ng\u2081 : \u03b3 \u2192 \u03b4\nf\u2082 : \u03b2 \u2192 \u03b1\ng\u2082 : \u03b4 \u2192 \u03b3\nh : LeftInverse (map f\u2081 g\u2081) (map f\u2082 g\u2082)\nd : \u03b4\ninhabited_h : Inhabited \u03b2\n\u22a2 g\u2081 (g\u2082 d) = d\n[PROOFSTEP]\nexact congr_arg Prod.snd (h (default, d))\n", "meta": {"mathlib_filename": "Mathlib.Data.Prod.Basic", "llama_tokens": 7638, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.712232184238947, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4140227699544498}} {"text": "[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\n\u22a2 Commensurable H H\n[PROOFSTEP]\nsimp [Commensurable]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\ng : ConjAct G\na b : { x // x \u2208 K }\n\u22a2 Setoid.r a b \u2194 Setoid.r (\u2191(Subgroup.equivSMul g K).toEquiv a) (\u2191(Subgroup.equivSMul g K).toEquiv b)\n[PROOFSTEP]\ndsimp\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\ng : ConjAct G\na b : { x // x \u2208 K }\n\u22a2 Setoid.r a b \u2194 Setoid.r (\u2191(Subgroup.equivSMul g K) a) (\u2191(Subgroup.equivSMul g K) b)\n[PROOFSTEP]\nrw [\u2190 Quotient.eq'', \u2190 Quotient.eq'', QuotientGroup.eq', QuotientGroup.eq', Subgroup.mem_subgroupOf,\n Subgroup.mem_subgroupOf, \u2190 MulEquiv.map_inv, \u2190 MulEquiv.map_mul, Subgroup.equivSMul_apply_coe]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\ng : ConjAct G\na b : { x // x \u2208 K }\n\u22a2 \u2191(a\u207b\u00b9 * b) \u2208 H \u2194 g \u2022 \u2191(a\u207b\u00b9 * b) \u2208 g \u2022 H\n[PROOFSTEP]\nexact Subgroup.smul_mem_pointwise_smul_iff.symm\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\ng : ConjAct G\n\u22a2 Commensurable (g \u2022 H) H \u2194 Commensurable H (g\u207b\u00b9 \u2022 H)\n[PROOFSTEP]\nrw [commensurable_conj, inv_smul_smul]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\na\u271d b\u271d : ConjAct G\nha : a\u271d \u2208 {g | Commensurable (g \u2022 H) H}\nhb : b\u271d \u2208 {g | Commensurable (g \u2022 H) H}\n\u22a2 a\u271d * b\u271d \u2208 {g | Commensurable (g \u2022 H) H}\n[PROOFSTEP]\nrw [Set.mem_setOf_eq, mul_smul]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\na\u271d b\u271d : ConjAct G\nha : a\u271d \u2208 {g | Commensurable (g \u2022 H) H}\nhb : b\u271d \u2208 {g | Commensurable (g \u2022 H) H}\n\u22a2 Commensurable (a\u271d \u2022 b\u271d \u2022 H) H\n[PROOFSTEP]\nexact trans ((commensurable_conj _).mp hb) ha\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\n\u22a2 1 \u2208\n { carrier := {g | Commensurable (g \u2022 H) H},\n mul_mem' :=\n (_ :\n \u2200 {a b : ConjAct G},\n a \u2208 {g | Commensurable (g \u2022 H) H} \u2192\n b \u2208 {g | Commensurable (g \u2022 H) H} \u2192 a * b \u2208 {g | Commensurable (g \u2022 H) H}) }.carrier\n[PROOFSTEP]\nrw [Set.mem_setOf_eq, one_smul]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nx\u271d\u00b9 : ConjAct G\nx\u271d :\n x\u271d\u00b9 \u2208\n {\n toSubsemigroup :=\n { carrier := {g | Commensurable (g \u2022 H) H},\n mul_mem' :=\n (_ :\n \u2200 {a b : ConjAct G},\n a \u2208 {g | Commensurable (g \u2022 H) H} \u2192\n b \u2208 {g | Commensurable (g \u2022 H) H} \u2192 a * b \u2208 {g | Commensurable (g \u2022 H) H}) },\n one_mem' :=\n (_ :\n 1 \u2208\n { carrier := {g | Commensurable (g \u2022 H) H},\n mul_mem' :=\n (_ :\n \u2200 {a b : ConjAct G},\n a \u2208 {g | Commensurable (g \u2022 H) H} \u2192\n b \u2208 {g | Commensurable (g \u2022 H) H} \u2192\n a * b \u2208 {g | Commensurable (g \u2022 H) H}) }.carrier) }.toSubsemigroup.carrier\n\u22a2 x\u271d\u00b9\u207b\u00b9 \u2208\n {\n toSubsemigroup :=\n { carrier := {g | Commensurable (g \u2022 H) H},\n mul_mem' :=\n (_ :\n \u2200 {a b : ConjAct G},\n a \u2208 {g | Commensurable (g \u2022 H) H} \u2192\n b \u2208 {g | Commensurable (g \u2022 H) H} \u2192 a * b \u2208 {g | Commensurable (g \u2022 H) H}) },\n one_mem' :=\n (_ :\n 1 \u2208\n { carrier := {g | Commensurable (g \u2022 H) H},\n mul_mem' :=\n (_ :\n \u2200 {a b : ConjAct G},\n a \u2208 {g | Commensurable (g \u2022 H) H} \u2192\n b \u2208 {g | Commensurable (g \u2022 H) H} \u2192\n a * b \u2208 {g | Commensurable (g \u2022 H) H}) }.carrier) }.toSubsemigroup.carrier\n[PROOFSTEP]\nrwa [Set.mem_setOf_eq, comm, \u2190 commensurable_inv]\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Commensurable", "llama_tokens": 1535, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.41389001763507205}} {"text": "[GOAL]\nl : Filter \u2102\nhre : Tendsto re l atTop\nr : \u211d\nhr : im =O[l] fun z => z.re ^ r\nn : \u2115\n\u22a2 (fun z => z.im ^ \u2191n) =O[l] fun z => (z.re ^ r) ^ \u2191n\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nl : Filter \u2102\nhre : Tendsto re l atTop\nr : \u211d\nhr : im =O[l] fun z => z.re ^ r\nn : \u2115\n\u22a2 (fun z => z.im ^ n) =O[l] fun z => (z.re ^ r) ^ n\n[PROOFSTEP]\nexact hr.pow n\n[GOAL]\nl : Filter \u2102\nhre : Tendsto re l atTop\nr : \u211d\nhr : im =O[l] fun z => z.re ^ r\nn : \u2115\nz : \u2102\nhz : 0 \u2264 z.re\n\u22a2 (fun z => (z.re ^ r) ^ \u2191n) z = (fun z => z.re ^ (r * \u2191n)) z\n[PROOFSTEP]\nsimp only [Real.rpow_mul hz r n, Real.rpow_nat_cast]\n[GOAL]\nl : Filter \u2102\nhre : Tendsto re l atTop\nn : \u2115\nhr : im =O[l] fun z => z.re ^ \u2191n\n\u22a2 im =O[l] fun z => z.re ^ \u2191n\n[PROOFSTEP]\nnorm_cast at hr \n[GOAL]\nl : Filter \u2102\nhre : Tendsto re l atTop\nn : \u2115\nhr : im =O[l] fun z => z.re ^ n\n\u22a2 im =O[l] fun z => z.re ^ \u2191n\n[PROOFSTEP]\nsimpa only [Real.rpow_nat_cast]\n[GOAL]\nl : Filter \u2102\nhre : Tendsto re l atTop\nhim : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun z => |z.im|\n\u22a2 im =O[l] fun z => z.re ^ \u21910\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nl : Filter \u2102\nhre : Tendsto re l atTop\nhim : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun z => |z.im|\n\u22a2 im =O[l] fun z => z.re ^ 0\n[PROOFSTEP]\nsimpa only [pow_zero] using @IsBoundedUnder.isBigO_const \u2102 \u211d \u211d _ _ _ l him 1 one_ne_zero\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\n\u22a2 \u2200 (x : \u2102), ((fun z => z.im ^ \u2191n) ^ 2) x = (x.im ^ \u2191n) ^ 2\n[PROOFSTEP]\nsimp\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nx\u271d : \u2102\n\u22a2 (x\u271d.im ^ \u2191n) ^ 2 = x\u271d.im ^ (2 * \u2191n)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nx\u271d : \u2102\n\u22a2 (x\u271d.im ^ n) ^ 2 = x\u271d.im ^ (2 * n)\n[PROOFSTEP]\nrw [pow_mul']\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\n\u22a2 (fun z => z.im ^ (2 * \u2191n)) =O[l] fun z => Real.exp z.re\n[PROOFSTEP]\nhave := hl.isBigO_im_pow_re (2 * n)\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nthis : (fun z => z.im ^ \u2191(2 * n)) =O[l] fun z => Real.exp z.re\n\u22a2 (fun z => z.im ^ (2 * \u2191n)) =O[l] fun z => Real.exp z.re\n[PROOFSTEP]\nnorm_cast at *\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nx\u271d : \u2102\n\u22a2 Real.exp x\u271d.re = Real.exp x\u271d.re ^ 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nx\u271d : \u2102\n\u22a2 Real.exp x\u271d.re = Real.exp x\u271d.re ^ 1\n[PROOFSTEP]\nrw [pow_one]\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\n\u22a2 (fun z => Real.exp z.re ^ 1) =o[l] fun z => Real.exp z.re ^ 2\n[PROOFSTEP]\nhave := (isLittleO_pow_pow_atTop_of_lt one_lt_two).comp_tendsto <| Real.tendsto_exp_atTop.comp hl.tendsto_re\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nthis : ((fun x => x ^ 1) \u2218 Real.exp \u2218 re) =o[l] ((fun x => x ^ 2) \u2218 Real.exp \u2218 re)\n\u22a2 (fun z => Real.exp z.re ^ 1) =o[l] fun z => Real.exp z.re ^ 2\n[PROOFSTEP]\nsimpa only [pow_one, Real.rpow_one, Real.rpow_two]\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\n\u22a2 \u2200 (x : \u2102), Real.exp x.re ^ 2 = ((fun z => Real.exp z.re) ^ 2) x\n[PROOFSTEP]\nsimp\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\n\u22a2 (fun z => |z.im| ^ \u2191n) \u2264\u1da0[l] fun z => Real.exp z.re\n[PROOFSTEP]\nsimpa using (hl.isLittleO_im_pow_exp_re n).bound zero_lt_one\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\n\u22a2 \u2016Real.log (\u2191abs z)\u2016 \u2264 1 * \u2016Real.log (Real.sqrt 2) + Real.log (max z.re |z.im|)\u2016\n[PROOFSTEP]\nhave h2 : 0 < Real.sqrt 2 := by simp\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\n\u22a2 0 < Real.sqrt 2\n[PROOFSTEP]\nsimp\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < Real.sqrt 2\n\u22a2 \u2016Real.log (\u2191abs z)\u2016 \u2264 1 * \u2016Real.log (Real.sqrt 2) + Real.log (max z.re |z.im|)\u2016\n[PROOFSTEP]\nhave hz' : 1 \u2264 abs z := hz.trans (re_le_abs z)\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 \u2264 \u2191abs z\n\u22a2 \u2016Real.log (\u2191abs z)\u2016 \u2264 1 * \u2016Real.log (Real.sqrt 2) + Real.log (max z.re |z.im|)\u2016\n[PROOFSTEP]\nhave _ : 0 < abs z := one_pos.trans_le hz'\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 \u2264 \u2191abs z\nx\u271d : 0 < \u2191abs z\n\u22a2 \u2016Real.log (\u2191abs z)\u2016 \u2264 1 * \u2016Real.log (Real.sqrt 2) + Real.log (max z.re |z.im|)\u2016\n[PROOFSTEP]\nhave hm\u2080 : 0 < max z.re |z.im| := lt_max_iff.2 (Or.inl <| one_pos.trans_le hz)\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 \u2264 \u2191abs z\nx\u271d : 0 < \u2191abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 \u2016Real.log (\u2191abs z)\u2016 \u2264 1 * \u2016Real.log (Real.sqrt 2) + Real.log (max z.re |z.im|)\u2016\n[PROOFSTEP]\nrw [one_mul, Real.norm_eq_abs, _root_.abs_of_nonneg (Real.log_nonneg hz')]\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 \u2264 \u2191abs z\nx\u271d : 0 < \u2191abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 Real.log (\u2191abs z) \u2264 \u2016Real.log (Real.sqrt 2) + Real.log (max z.re |z.im|)\u2016\n[PROOFSTEP]\nrefine' le_trans _ (le_abs_self _)\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 \u2264 \u2191abs z\nx\u271d : 0 < \u2191abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 Real.log (\u2191abs z) \u2264 Real.log (Real.sqrt 2) + Real.log (max z.re |z.im|)\n[PROOFSTEP]\nrw [\u2190 Real.log_mul, Real.log_le_log, \u2190 _root_.abs_of_nonneg (le_trans zero_le_one hz)]\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 \u2264 \u2191abs z\nx\u271d : 0 < \u2191abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 \u2191abs z \u2264 Real.sqrt 2 * max |z.re| |z.im|\ncase h\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 \u2264 \u2191abs z\nx\u271d : 0 < \u2191abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 0 < \u2191abs z\ncase h\u2081\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 \u2264 \u2191abs z\nx\u271d : 0 < \u2191abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 0 < Real.sqrt 2 * max z.re |z.im|\ncase hx\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 \u2264 \u2191abs z\nx\u271d : 0 < \u2191abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 Real.sqrt 2 \u2260 0\ncase hy\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < Real.sqrt 2\nhz' : 1 \u2264 \u2191abs z\nx\u271d : 0 < \u2191abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 max z.re |z.im| \u2260 0\n[PROOFSTEP]\nexacts [abs_le_sqrt_two_mul_max z, one_pos.trans_le hz', mul_pos h2 hm\u2080, h2.ne', hm\u2080.ne']\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\n\u22a2 \u2200\u1da0 (x : \u2102) in l, \u2191n * \u2016Real.log (max x.re |x.im|)\u2016 \u2264 \u2016x.re\u2016\n[PROOFSTEP]\nfilter_upwards [isLittleO_iff_nat_mul_le'.1 hl.isLittleO_log_re_re n, hl.abs_im_pow_eventuallyLE_exp_re n,\n hl.tendsto_re.eventually_gt_atTop 1] with z hre him h\u2081\n[GOAL]\ncase h\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nz : \u2102\nhre : \u2191n * \u2016Real.log z.re\u2016 \u2264 \u2016z.re\u2016\nhim : |z.im| ^ \u2191n \u2264 Real.exp z.re\nh\u2081 : 1 < z.re\n\u22a2 \u2191n * \u2016Real.log (max z.re |z.im|)\u2016 \u2264 \u2016z.re\u2016\n[PROOFSTEP]\ncases' le_total |z.im| z.re with hle hle\n[GOAL]\ncase h.inl\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nz : \u2102\nhre : \u2191n * \u2016Real.log z.re\u2016 \u2264 \u2016z.re\u2016\nhim : |z.im| ^ \u2191n \u2264 Real.exp z.re\nh\u2081 : 1 < z.re\nhle : |z.im| \u2264 z.re\n\u22a2 \u2191n * \u2016Real.log (max z.re |z.im|)\u2016 \u2264 \u2016z.re\u2016\n[PROOFSTEP]\nrwa [max_eq_left hle]\n[GOAL]\ncase h.inr\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nz : \u2102\nhre : \u2191n * \u2016Real.log z.re\u2016 \u2264 \u2016z.re\u2016\nhim : |z.im| ^ \u2191n \u2264 Real.exp z.re\nh\u2081 : 1 < z.re\nhle : z.re \u2264 |z.im|\n\u22a2 \u2191n * \u2016Real.log (max z.re |z.im|)\u2016 \u2264 \u2016z.re\u2016\n[PROOFSTEP]\nhave H : 1 < |z.im| := h\u2081.trans_le hle\n[GOAL]\ncase h.inr\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nz : \u2102\nhre : \u2191n * \u2016Real.log z.re\u2016 \u2264 \u2016z.re\u2016\nhim : |z.im| ^ \u2191n \u2264 Real.exp z.re\nh\u2081 : 1 < z.re\nhle : z.re \u2264 |z.im|\nH : 1 < |z.im|\n\u22a2 \u2191n * \u2016Real.log (max z.re |z.im|)\u2016 \u2264 \u2016z.re\u2016\n[PROOFSTEP]\nnorm_cast at *\n[GOAL]\ncase h.inr\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nz : \u2102\nhre : \u2191n * \u2016Real.log z.re\u2016 \u2264 \u2016z.re\u2016\nhle : z.re \u2264 |z.im|\nhim : |z.im| ^ n \u2264 Real.exp z.re\nh\u2081 : 1 < z.re\nH : 1 < |z.im|\n\u22a2 \u2191n * \u2016Real.log (max z.re |z.im|)\u2016 \u2264 \u2016z.re\u2016\n[PROOFSTEP]\nrwa [max_eq_right hle, Real.norm_eq_abs, Real.norm_eq_abs, abs_of_pos (Real.log_pos H), \u2190 Real.log_pow,\n Real.log_le_iff_le_exp (pow_pos (one_pos.trans H) _), abs_of_pos (one_pos.trans h\u2081)]\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\na : \u2102\nb : \u211d\nhb : 0 < b\nz : \u2102\nhz : z \u2260 0\n\u22a2 (let_fun this := fun z => \u2191abs z ^ a.re;\n this)\n z =\n (fun z => Real.exp (a.re * Real.log (\u2191abs z))) z\n[PROOFSTEP]\nsimp only [Real.rpow_def_of_pos, abs.pos hz, mul_comm]\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\na : \u2102\nb : \u211d\nhb : 0 < b\n\u22a2 (fun z => Real.exp (a.re * Real.log (\u2191abs z))) =o[l] fun x => \u2016exp (\u2191b * x)\u2016\n[PROOFSTEP]\nsimp only [norm_eq_abs, abs_exp, ofReal_mul_re, Real.isLittleO_exp_comp_exp_comp]\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\na : \u2102\nb : \u211d\nhb : 0 < b\n\u22a2 Tendsto (fun x => b * x.re - a.re * Real.log (\u2191abs x)) l atTop\n[PROOFSTEP]\nrefine' (IsEquivalent.refl.sub_isLittleO _).symm.tendsto_atTop (hl.tendsto_re.const_mul_atTop hb)\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\na : \u2102\nb : \u211d\nhb : 0 < b\n\u22a2 (fun x => a.re * Real.log (\u2191abs x)) =o[l] fun x => b * x.re\n[PROOFSTEP]\nexact (hl.isLittleO_log_abs_re.const_mul_left _).const_mul_right hb.ne'\n[GOAL]\nl : Filter \u2102\nb\u2081 b\u2082 : \u211d\nhl : IsExpCmpFilter l\nhb : b\u2081 < b\u2082\na\u2081 a\u2082 z : \u2102\nhz : z \u2260 0\n\u22a2 (fun z => z ^ a\u2081 * exp (\u2191b\u2081 * z)) z = (fun z => z ^ a\u2082 * exp (\u2191b\u2081 * z) * z ^ (a\u2081 - a\u2082)) z\n[PROOFSTEP]\nsimp only\n[GOAL]\nl : Filter \u2102\nb\u2081 b\u2082 : \u211d\nhl : IsExpCmpFilter l\nhb : b\u2081 < b\u2082\na\u2081 a\u2082 z : \u2102\nhz : z \u2260 0\n\u22a2 z ^ a\u2081 * exp (\u2191b\u2081 * z) = z ^ a\u2082 * exp (\u2191b\u2081 * z) * z ^ (a\u2081 - a\u2082)\n[PROOFSTEP]\nrw [mul_right_comm, \u2190 cpow_add _ _ hz, add_sub_cancel'_right]\n[GOAL]\nl : Filter \u2102\nb\u2081 b\u2082 : \u211d\nhl : IsExpCmpFilter l\nhb : b\u2081 < b\u2082\na\u2081 a\u2082 : \u2102\n\u22a2 (fun z => z ^ a\u2082 * exp (\u2191b\u2081 * z) * exp (\u2191(b\u2082 - b\u2081) * z)) =\u1da0[l] fun z => z ^ a\u2082 * exp (\u2191b\u2082 * z)\n[PROOFSTEP]\nsimp only [ofReal_sub, sub_mul, mul_assoc, \u2190 exp_add, add_sub_cancel'_right]\n[GOAL]\nl : Filter \u2102\nb\u2081 b\u2082 : \u211d\nhl : IsExpCmpFilter l\nhb : b\u2081 < b\u2082\na\u2081 a\u2082 : \u2102\n\u22a2 (fun z => z ^ a\u2082 * exp (\u2191b\u2082 * z)) =\u1da0[l] fun z => z ^ a\u2082 * exp (\u2191b\u2082 * z)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nl : Filter \u2102\nhl : IsExpCmpFilter l\na : \u2102\nb : \u211d\nhb : b < 0\n\u22a2 (fun z => exp (\u2191b * z)) =o[l] fun z => z ^ a\n[PROOFSTEP]\nsimpa using hl.isLittleO_cpow_mul_exp hb 0 a\n[GOAL]\nl : Filter \u2102\nb\u2081 b\u2082 : \u211d\nhl : IsExpCmpFilter l\nhb : b\u2081 < b\u2082\nm n : \u2115\n\u22a2 (fun z => z ^ \u2191m * exp (\u2191b\u2081 * z)) =o[l] fun z => z ^ \u2191n * exp (\u2191b\u2082 * z)\n[PROOFSTEP]\nsimpa only [cpow_nat_cast] using hl.isLittleO_cpow_mul_exp hb m n\n[GOAL]\nl : Filter \u2102\nb\u2081 b\u2082 : \u211d\nhl : IsExpCmpFilter l\nhb : b\u2081 < b\u2082\nm n : \u2124\n\u22a2 (fun z => z ^ \u2191m * exp (\u2191b\u2081 * z)) =o[l] fun z => z ^ \u2191n * exp (\u2191b\u2082 * z)\n[PROOFSTEP]\nsimpa only [cpow_int_cast] using hl.isLittleO_cpow_mul_exp hb m n\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.CompareExp", "llama_tokens": 5629, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585786300049, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.4137927058186993}} {"text": "[GOAL]\n\u03b1 : Type ?u.3\n\u22a2 (fun {\u03b1 \u03b2} e => \u2191(uniqueCongr e)) (Equiv.refl \u03b1) = id\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2200 {\u03b1 \u03b2 \u03b3 : Type ?u.3} (k : \u03b1 \u2243 \u03b2) (h : \u03b2 \u2243 \u03b3),\n (fun {\u03b1 \u03b2} e => \u2191(uniqueCongr e)) (k.trans h) =\n (fun {\u03b1 \u03b2} e => \u2191(uniqueCongr e)) h \u2218 (fun {\u03b1 \u03b2} e => \u2191(uniqueCongr e)) k\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.3738\n\u22a2 (fun {\u03b1 \u03b2} e p => (e.symm.trans p).trans e) (Equiv.refl \u03b1) = id\n[PROOFSTEP]\next\n[GOAL]\ncase h.H\n\u03b1 : Type ?u.3738\nx\u271d\u00b9 : Perm \u03b1\nx\u271d : \u03b1\n\u22a2 \u2191((fun {\u03b1 \u03b2} e p => (e.symm.trans p).trans e) (Equiv.refl \u03b1) x\u271d\u00b9) x\u271d = \u2191(id x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.3738\nx\u271d\u00b9 : \u03b1\u271d \u2243 \u03b2\u271d\nx\u271d : \u03b2\u271d \u2243 \u03b3\u271d\n\u22a2 (fun {\u03b1 \u03b2} e p => (e.symm.trans p).trans e) (x\u271d\u00b9.trans x\u271d) =\n (fun {\u03b1 \u03b2} e p => (e.symm.trans p).trans e) x\u271d \u2218 (fun {\u03b1 \u03b2} e p => (e.symm.trans p).trans e) x\u271d\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase h.H\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.3738\nx\u271d\u00b3 : \u03b1\u271d \u2243 \u03b2\u271d\nx\u271d\u00b2 : \u03b2\u271d \u2243 \u03b3\u271d\nx\u271d\u00b9 : Perm \u03b1\u271d\nx\u271d : \u03b3\u271d\n\u22a2 \u2191((fun {\u03b1 \u03b2} e p => (e.symm.trans p).trans e) (x\u271d\u00b3.trans x\u271d\u00b2) x\u271d\u00b9) x\u271d =\n \u2191(((fun {\u03b1 \u03b2} e p => (e.symm.trans p).trans e) x\u271d\u00b2 \u2218 (fun {\u03b1 \u03b2} e p => (e.symm.trans p).trans e) x\u271d\u00b3) x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.6374\n\u22a2 (fun {\u03b1 \u03b2} e s => Finset.map (Equiv.toEmbedding e) s) (Equiv.refl \u03b1) = id\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\n\u03b1 : Type ?u.6374\nx\u271d : Finset \u03b1\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 (fun {\u03b1 \u03b2} e s => Finset.map (Equiv.toEmbedding e) s) (Equiv.refl \u03b1) x\u271d \u2194 a\u271d \u2208 id x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.6374\nk : \u03b1\u271d \u2243 \u03b2\u271d\nh : \u03b2\u271d \u2243 \u03b3\u271d\n\u22a2 (fun {\u03b1 \u03b2} e s => Finset.map (Equiv.toEmbedding e) s) (k.trans h) =\n (fun {\u03b1 \u03b2} e s => Finset.map (Equiv.toEmbedding e) s) h \u2218 (fun {\u03b1 \u03b2} e s => Finset.map (Equiv.toEmbedding e) s) k\n[PROOFSTEP]\next _ a\n[GOAL]\ncase h.a\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.6374\nk : \u03b1\u271d \u2243 \u03b2\u271d\nh : \u03b2\u271d \u2243 \u03b3\u271d\nx\u271d : Finset \u03b1\u271d\na : \u03b3\u271d\n\u22a2 a \u2208 (fun {\u03b1 \u03b2} e s => Finset.map (Equiv.toEmbedding e) s) (k.trans h) x\u271d \u2194\n a \u2208\n ((fun {\u03b1 \u03b2} e s => Finset.map (Equiv.toEmbedding e) s) h \u2218\n (fun {\u03b1 \u03b2} e s => Finset.map (Equiv.toEmbedding e) s) k)\n x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.a\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.6374\nk : \u03b1\u271d \u2243 \u03b2\u271d\nh : \u03b2\u271d \u2243 \u03b3\u271d\nx\u271d : Finset \u03b1\u271d\na : \u03b3\u271d\n\u22a2 (\u2203 a_1, a_1 \u2208 x\u271d \u2227 \u2191h (\u2191k a_1) = a) \u2194 \u2191k.symm (\u2191h.symm a) \u2208 x\u271d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.a.mp\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.6374\nk : \u03b1\u271d \u2243 \u03b2\u271d\nh : \u03b2\u271d \u2243 \u03b3\u271d\nx\u271d : Finset \u03b1\u271d\na : \u03b3\u271d\n\u22a2 (\u2203 a_1, a_1 \u2208 x\u271d \u2227 \u2191h (\u2191k a_1) = a) \u2192 \u2191k.symm (\u2191h.symm a) \u2208 x\u271d\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase h.a.mpr\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.6374\nk : \u03b1\u271d \u2243 \u03b2\u271d\nh : \u03b2\u271d \u2243 \u03b3\u271d\nx\u271d : Finset \u03b1\u271d\na : \u03b3\u271d\n\u22a2 \u2191k.symm (\u2191h.symm a) \u2208 x\u271d \u2192 \u2203 a_2, a_2 \u2208 x\u271d \u2227 \u2191h (\u2191k a_2) = a\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase h.a.mp\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.6374\nk : \u03b1\u271d \u2243 \u03b2\u271d\nh : \u03b2\u271d \u2243 \u03b3\u271d\nx\u271d : Finset \u03b1\u271d\na : \u03b3\u271d\nh' : \u2203 a_1, a_1 \u2208 x\u271d \u2227 \u2191h (\u2191k a_1) = a\n\u22a2 \u2191k.symm (\u2191h.symm a) \u2208 x\u271d\n[PROOFSTEP]\nlet \u27e8a, ha\u2081, ha\u2082\u27e9 := h'\n[GOAL]\ncase h.a.mp\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.6374\nk : \u03b1\u271d \u2243 \u03b2\u271d\nh : \u03b2\u271d \u2243 \u03b3\u271d\nx\u271d : Finset \u03b1\u271d\na\u271d : \u03b3\u271d\nh' : \u2203 a, a \u2208 x\u271d \u2227 \u2191h (\u2191k a) = a\u271d\na : \u03b1\u271d\nha\u2081 : a \u2208 x\u271d\nha\u2082 : \u2191h (\u2191k a) = a\u271d\n\u22a2 \u2191k.symm (\u2191h.symm a\u271d) \u2208 x\u271d\n[PROOFSTEP]\nrw [\u2190 ha\u2082]\n[GOAL]\ncase h.a.mp\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.6374\nk : \u03b1\u271d \u2243 \u03b2\u271d\nh : \u03b2\u271d \u2243 \u03b3\u271d\nx\u271d : Finset \u03b1\u271d\na\u271d : \u03b3\u271d\nh' : \u2203 a, a \u2208 x\u271d \u2227 \u2191h (\u2191k a) = a\u271d\na : \u03b1\u271d\nha\u2081 : a \u2208 x\u271d\nha\u2082 : \u2191h (\u2191k a) = a\u271d\n\u22a2 \u2191k.symm (\u2191h.symm (\u2191h (\u2191k a))) \u2208 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.a.mp\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.6374\nk : \u03b1\u271d \u2243 \u03b2\u271d\nh : \u03b2\u271d \u2243 \u03b3\u271d\nx\u271d : Finset \u03b1\u271d\na\u271d : \u03b3\u271d\nh' : \u2203 a, a \u2208 x\u271d \u2227 \u2191h (\u2191k a) = a\u271d\na : \u03b1\u271d\nha\u2081 : a \u2208 x\u271d\nha\u2082 : \u2191h (\u2191k a) = a\u271d\n\u22a2 a \u2208 x\u271d\n[PROOFSTEP]\napply ha\u2081\n[GOAL]\ncase h.a.mpr\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.6374\nk : \u03b1\u271d \u2243 \u03b2\u271d\nh : \u03b2\u271d \u2243 \u03b3\u271d\nx\u271d : Finset \u03b1\u271d\na : \u03b3\u271d\nh' : \u2191k.symm (\u2191h.symm a) \u2208 x\u271d\n\u22a2 \u2203 a_1, a_1 \u2208 x\u271d \u2227 \u2191h (\u2191k a_1) = a\n[PROOFSTEP]\nexists (Equiv.symm k) ((Equiv.symm h) a)\n[GOAL]\ncase h.a.mpr\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.6374\nk : \u03b1\u271d \u2243 \u03b2\u271d\nh : \u03b2\u271d \u2243 \u03b3\u271d\nx\u271d : Finset \u03b1\u271d\na : \u03b3\u271d\nh' : \u2191k.symm (\u2191h.symm a) \u2208 x\u271d\n\u22a2 \u2191k.symm (\u2191h.symm a) \u2208 x\u271d \u2227 \u2191h (\u2191k (\u2191k.symm (\u2191h.symm a))) = a\n[PROOFSTEP]\nsimp [h']\n[GOAL]\n\u03b1 : Type ?u.9577\n\u22a2 (fun {\u03b1 \u03b2} e s => Fintype.ofBijective \u2191e (_ : Function.Bijective \u2191e)) (Equiv.refl \u03b1) = id\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type ?u.9577\nx\u271d : Fintype \u03b1\n\u22a2 (fun {\u03b1 \u03b2} e s => Fintype.ofBijective \u2191e (_ : Function.Bijective \u2191e)) (Equiv.refl \u03b1) x\u271d = id x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2200 {\u03b1 \u03b2 \u03b3 : Type ?u.9577} (k : \u03b1 \u2243 \u03b2) (h : \u03b2 \u2243 \u03b3),\n (fun {\u03b1 \u03b2} e s => Fintype.ofBijective \u2191e (_ : Function.Bijective \u2191e)) (k.trans h) =\n (fun {\u03b1 \u03b2} e s => Fintype.ofBijective \u2191e (_ : Function.Bijective \u2191e)) h \u2218\n (fun {\u03b1 \u03b2} e s => Fintype.ofBijective \u2191e (_ : Function.Bijective \u2191e)) k\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Control.EquivFunctor.Instances", "llama_tokens": 3078, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300698514777, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.413712322173101}} {"text": "[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\n\u03c3 : Perm \u03b1\nm n : \u2124\nx : \u03b1\n\u22a2 \u2191(\u03c3 ^ m) (\u2191(\u03c3 ^ n) x) = \u2191(\u03c3 ^ n) (\u2191(\u03c3 ^ m) x)\n[PROOFSTEP]\nrw [\u2190 Equiv.Perm.mul_apply, \u2190 Equiv.Perm.mul_apply, zpow_mul_comm]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u22a2 Function.Injective \u2191(sumCongrHom \u03b1 \u03b2)\n[PROOFSTEP]\nrintro \u27e8\u27e9 \u27e8\u27e9 h\n[GOAL]\ncase mk.mk\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nfst\u271d\u00b9 : Perm \u03b1\nsnd\u271d\u00b9 : Perm \u03b2\nfst\u271d : Perm \u03b1\nsnd\u271d : Perm \u03b2\nh : \u2191(sumCongrHom \u03b1 \u03b2) (fst\u271d\u00b9, snd\u271d\u00b9) = \u2191(sumCongrHom \u03b1 \u03b2) (fst\u271d, snd\u271d)\n\u22a2 (fst\u271d\u00b9, snd\u271d\u00b9) = (fst\u271d, snd\u271d)\n[PROOFSTEP]\nrw [Prod.mk.inj_iff]\n[GOAL]\ncase mk.mk\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nfst\u271d\u00b9 : Perm \u03b1\nsnd\u271d\u00b9 : Perm \u03b2\nfst\u271d : Perm \u03b1\nsnd\u271d : Perm \u03b2\nh : \u2191(sumCongrHom \u03b1 \u03b2) (fst\u271d\u00b9, snd\u271d\u00b9) = \u2191(sumCongrHom \u03b1 \u03b2) (fst\u271d, snd\u271d)\n\u22a2 fst\u271d\u00b9 = fst\u271d \u2227 snd\u271d\u00b9 = snd\u271d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.mk.left\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nfst\u271d\u00b9 : Perm \u03b1\nsnd\u271d\u00b9 : Perm \u03b2\nfst\u271d : Perm \u03b1\nsnd\u271d : Perm \u03b2\nh : \u2191(sumCongrHom \u03b1 \u03b2) (fst\u271d\u00b9, snd\u271d\u00b9) = \u2191(sumCongrHom \u03b1 \u03b2) (fst\u271d, snd\u271d)\n\u22a2 fst\u271d\u00b9 = fst\u271d\n[PROOFSTEP]\next i\n[GOAL]\ncase mk.mk.right\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nfst\u271d\u00b9 : Perm \u03b1\nsnd\u271d\u00b9 : Perm \u03b2\nfst\u271d : Perm \u03b1\nsnd\u271d : Perm \u03b2\nh : \u2191(sumCongrHom \u03b1 \u03b2) (fst\u271d\u00b9, snd\u271d\u00b9) = \u2191(sumCongrHom \u03b1 \u03b2) (fst\u271d, snd\u271d)\n\u22a2 snd\u271d\u00b9 = snd\u271d\n[PROOFSTEP]\next i\n[GOAL]\ncase mk.mk.left.H\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nfst\u271d\u00b9 : Perm \u03b1\nsnd\u271d\u00b9 : Perm \u03b2\nfst\u271d : Perm \u03b1\nsnd\u271d : Perm \u03b2\nh : \u2191(sumCongrHom \u03b1 \u03b2) (fst\u271d\u00b9, snd\u271d\u00b9) = \u2191(sumCongrHom \u03b1 \u03b2) (fst\u271d, snd\u271d)\ni : \u03b1\n\u22a2 \u2191fst\u271d\u00b9 i = \u2191fst\u271d i\n[PROOFSTEP]\nsimpa using Equiv.congr_fun h (Sum.inl i)\n[GOAL]\ncase mk.mk.right.H\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nfst\u271d\u00b9 : Perm \u03b1\nsnd\u271d\u00b9 : Perm \u03b2\nfst\u271d : Perm \u03b1\nsnd\u271d : Perm \u03b2\nh : \u2191(sumCongrHom \u03b1 \u03b2) (fst\u271d\u00b9, snd\u271d\u00b9) = \u2191(sumCongrHom \u03b1 \u03b2) (fst\u271d, snd\u271d)\ni : \u03b2\n\u22a2 \u2191snd\u271d\u00b9 i = \u2191snd\u271d i\n[PROOFSTEP]\nsimpa using Equiv.congr_fun h (Sum.inr i)\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\n\u22a2 Function.Injective \u2191(sigmaCongrRightHom \u03b2)\n[PROOFSTEP]\nintro x y h\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nx y : (a : \u03b1) \u2192 Perm (\u03b2 a)\nh : \u2191(sigmaCongrRightHom \u03b2) x = \u2191(sigmaCongrRightHom \u03b2) y\n\u22a2 x = y\n[PROOFSTEP]\next a b\n[GOAL]\ncase h.H\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nx y : (a : \u03b1) \u2192 Perm (\u03b2 a)\nh : \u2191(sigmaCongrRightHom \u03b2) x = \u2191(sigmaCongrRightHom \u03b2) y\na : \u03b1\nb : \u03b2 a\n\u22a2 \u2191(x a) b = \u2191(y a) b\n[PROOFSTEP]\nsimpa using Equiv.congr_fun h \u27e8a, b\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 Function.Injective \u2191(subtypeCongrHom p)\n[PROOFSTEP]\nrintro \u27e8\u27e9 \u27e8\u27e9 h\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nfst\u271d\u00b9 : Perm { a // p a }\nsnd\u271d\u00b9 : Perm { a // \u00acp a }\nfst\u271d : Perm { a // p a }\nsnd\u271d : Perm { a // \u00acp a }\nh : \u2191(subtypeCongrHom p) (fst\u271d\u00b9, snd\u271d\u00b9) = \u2191(subtypeCongrHom p) (fst\u271d, snd\u271d)\n\u22a2 (fst\u271d\u00b9, snd\u271d\u00b9) = (fst\u271d, snd\u271d)\n[PROOFSTEP]\nrw [Prod.mk.inj_iff]\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nfst\u271d\u00b9 : Perm { a // p a }\nsnd\u271d\u00b9 : Perm { a // \u00acp a }\nfst\u271d : Perm { a // p a }\nsnd\u271d : Perm { a // \u00acp a }\nh : \u2191(subtypeCongrHom p) (fst\u271d\u00b9, snd\u271d\u00b9) = \u2191(subtypeCongrHom p) (fst\u271d, snd\u271d)\n\u22a2 fst\u271d\u00b9 = fst\u271d \u2227 snd\u271d\u00b9 = snd\u271d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.mk.left\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nfst\u271d\u00b9 : Perm { a // p a }\nsnd\u271d\u00b9 : Perm { a // \u00acp a }\nfst\u271d : Perm { a // p a }\nsnd\u271d : Perm { a // \u00acp a }\nh : \u2191(subtypeCongrHom p) (fst\u271d\u00b9, snd\u271d\u00b9) = \u2191(subtypeCongrHom p) (fst\u271d, snd\u271d)\n\u22a2 fst\u271d\u00b9 = fst\u271d\n[PROOFSTEP]\next i\n[GOAL]\ncase mk.mk.right\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nfst\u271d\u00b9 : Perm { a // p a }\nsnd\u271d\u00b9 : Perm { a // \u00acp a }\nfst\u271d : Perm { a // p a }\nsnd\u271d : Perm { a // \u00acp a }\nh : \u2191(subtypeCongrHom p) (fst\u271d\u00b9, snd\u271d\u00b9) = \u2191(subtypeCongrHom p) (fst\u271d, snd\u271d)\n\u22a2 snd\u271d\u00b9 = snd\u271d\n[PROOFSTEP]\next i\n[GOAL]\ncase mk.mk.left.H.a\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nfst\u271d\u00b9 : Perm { a // p a }\nsnd\u271d\u00b9 : Perm { a // \u00acp a }\nfst\u271d : Perm { a // p a }\nsnd\u271d : Perm { a // \u00acp a }\nh : \u2191(subtypeCongrHom p) (fst\u271d\u00b9, snd\u271d\u00b9) = \u2191(subtypeCongrHom p) (fst\u271d, snd\u271d)\ni : { a // p a }\n\u22a2 \u2191(\u2191fst\u271d\u00b9 i) = \u2191(\u2191fst\u271d i)\n[PROOFSTEP]\nsimpa using Equiv.congr_fun h i\n[GOAL]\ncase mk.mk.right.H.a\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nfst\u271d\u00b9 : Perm { a // p a }\nsnd\u271d\u00b9 : Perm { a // \u00acp a }\nfst\u271d : Perm { a // p a }\nsnd\u271d : Perm { a // \u00acp a }\nh : \u2191(subtypeCongrHom p) (fst\u271d\u00b9, snd\u271d\u00b9) = \u2191(subtypeCongrHom p) (fst\u271d, snd\u271d)\ni : { a // \u00acp a }\n\u22a2 \u2191(\u2191snd\u271d\u00b9 i) = \u2191(\u2191snd\u271d i)\n[PROOFSTEP]\nsimpa using Equiv.congr_fun h i\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d f : Perm \u03b1\nh : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nx : { x // p x }\n\u22a2 p (\u2191f (\u2191f\u207b\u00b9 \u2191x))\n[PROOFSTEP]\nsimpa using x.2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d f : Perm \u03b1\nh : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nx\u271d : { x // p x }\n\u22a2 (fun x => { val := \u2191f\u207b\u00b9 \u2191x, property := (_ : p (\u2191f\u207b\u00b9 \u2191x)) })\n ((fun x => { val := \u2191f \u2191x, property := (_ : p (\u2191f \u2191x)) }) x\u271d) =\n x\u271d\n[PROOFSTEP]\nsimp only [Perm.inv_apply_self, Subtype.coe_eta, Subtype.coe_mk]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d f : Perm \u03b1\nh : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nx\u271d : { x // p x }\n\u22a2 (fun x => { val := \u2191f \u2191x, property := (_ : p (\u2191f \u2191x)) })\n ((fun x => { val := \u2191f\u207b\u00b9 \u2191x, property := (_ : p (\u2191f\u207b\u00b9 \u2191x)) }) x\u271d) =\n x\u271d\n[PROOFSTEP]\nsimp only [Perm.apply_inv_self, Subtype.coe_eta, Subtype.coe_mk]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf : Perm \u03b1\n\u22a2 (\u2200 (x : \u03b1), p (\u2191f\u207b\u00b9 x) \u2194 p (\u2191f (\u2191f\u207b\u00b9 x))) \u2194 \u2200 (x : \u03b1), p x \u2194 p (\u2191f\u207b\u00b9 x)\n[PROOFSTEP]\nsimp_rw [f.apply_inv_self, Iff.comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d f : Perm \u03b1\nn : \u2115\nhf : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\n\u22a2 subtypePerm f hf ^ n = subtypePerm (f ^ n) (_ : \u2200 (x : \u03b1), p x \u2194 p (\u2191(f ^ n) x))\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d f : Perm \u03b1\nhf : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\n\u22a2 subtypePerm f hf ^ Nat.zero = subtypePerm (f ^ Nat.zero) (_ : \u2200 (x : \u03b1), p x \u2194 p (\u2191(f ^ Nat.zero) x))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d f : Perm \u03b1\nhf : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nn : \u2115\nih : subtypePerm f hf ^ n = subtypePerm (f ^ n) (_ : \u2200 (x : \u03b1), p x \u2194 p (\u2191(f ^ n) x))\n\u22a2 subtypePerm f hf ^ Nat.succ n = subtypePerm (f ^ Nat.succ n) (_ : \u2200 (x : \u03b1), p x \u2194 p (\u2191(f ^ Nat.succ n) x))\n[PROOFSTEP]\nsimp_rw [pow_succ', ih, subtypePerm_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf : Perm \u03b1\nhf : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nn : \u2115\n\u22a2 \u2200 (x : \u03b1), p x \u2194 p (\u2191(f ^ Int.negSucc n) x)\n[PROOFSTEP]\nrw [zpow_negSucc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf : Perm \u03b1\nhf : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nn : \u2115\n\u22a2 \u2200 (x : \u03b1), p x \u2194 p (\u2191(f ^ (n + 1))\u207b\u00b9 x)\n[PROOFSTEP]\nexact inv_aux.1 (pow_aux hf)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d f : Perm \u03b1\nn : \u2124\nhf : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\n\u22a2 subtypePerm f hf ^ n = subtypePerm (f ^ n) (_ : \u2200 (x : \u03b1), p x \u2194 p (\u2191(f ^ n) x))\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase ofNat\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d f : Perm \u03b1\nhf : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nn : \u2115\n\u22a2 subtypePerm f hf ^ Int.ofNat n = subtypePerm (f ^ Int.ofNat n) (_ : \u2200 (x : \u03b1), p x \u2194 p (\u2191(f ^ Int.ofNat n) x))\n[PROOFSTEP]\nexact subtypePerm_pow _ _ _\n[GOAL]\ncase negSucc\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d f : Perm \u03b1\nhf : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nih : \u2115\n\u22a2 subtypePerm f hf ^ Int.negSucc ih =\n subtypePerm (f ^ Int.negSucc ih) (_ : \u2200 (x : \u03b1), p x \u2194 p (\u2191(f ^ Int.negSucc ih) x))\n[PROOFSTEP]\nsimp only [zpow_negSucc, subtypePerm_pow, subtypePerm_inv]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\n\u22a2 \u2191(\u2191ofSubtype (subtypePerm f h\u2081)) x = \u2191f x\n[PROOFSTEP]\nby_cases hx : p x\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\nhx : p x\n\u22a2 \u2191(\u2191ofSubtype (subtypePerm f h\u2081)) x = \u2191f x\n[PROOFSTEP]\nexact (subtypePerm f h\u2081).extendDomain_apply_subtype _ hx\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\nhx : \u00acp x\n\u22a2 \u2191(\u2191ofSubtype (subtypePerm f h\u2081)) x = \u2191f x\n[PROOFSTEP]\nrw [ofSubtype, MonoidHom.coe_mk]\n -- Porting note: added `dsimp`\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\nhx : \u00acp x\n\u22a2 \u2191(\u2191{ toFun := fun f => extendDomain f (Equiv.refl { x // p x }),\n map_one' := (_ : extendDomain 1 (Equiv.refl { x // p x }) = 1) }\n (subtypePerm f h\u2081))\n x =\n \u2191f x\n[PROOFSTEP]\ndsimp only [OneHom.coe_mk]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\nhx : \u00acp x\n\u22a2 \u2191(extendDomain (subtypePerm f h\u2081) (Equiv.refl { x // p x })) x = \u2191f x\n[PROOFSTEP]\nrw [Equiv.Perm.extendDomain_apply_not_subtype _ _ hx]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\nhx : \u00acp x\n\u22a2 x = \u2191f x\n[PROOFSTEP]\nexact not_not.mp fun h => hx (h\u2082 x (Ne.symm h))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm (Subtype p)\nx : \u03b1\nh : p x\n\u22a2 p x \u2194 p (\u2191(\u2191ofSubtype f) x)\n[PROOFSTEP]\nsimpa only [h, true_iff_iff, MonoidHom.coe_mk, ofSubtype_apply_of_mem f h] using (f \u27e8x, h\u27e9).2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm (Subtype p)\nx : \u03b1\nh : \u00acp x\n\u22a2 p x \u2194 p (\u2191(\u2191ofSubtype f) x)\n[PROOFSTEP]\nsimp [h, ofSubtype_apply_of_not_mem f h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y z : \u03b1\n\u22a2 \u2191(swap x y * f) z = \u2191(f * swap (\u2191f\u207b\u00b9 x) (\u2191f\u207b\u00b9 y)) z\n[PROOFSTEP]\nsimp only [Perm.mul_apply, swap_apply_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y z : \u03b1\n\u22a2 (if \u2191f z = x then y else if \u2191f z = y then x else \u2191f z) =\n \u2191f (if z = \u2191f\u207b\u00b9 x then \u2191f\u207b\u00b9 y else if z = \u2191f\u207b\u00b9 y then \u2191f\u207b\u00b9 x else z)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y z : \u03b1\nh\u271d\u00b9 : \u2191f z = x\nh\u271d : z = \u2191f\u207b\u00b9 x\n\u22a2 y = \u2191f (\u2191f\u207b\u00b9 y)\n[PROOFSTEP]\nsimp_all only [Perm.apply_inv_self, Perm.eq_inv_iff_eq, eq_self_iff_true, not_true]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y z : \u03b1\nh\u271d\u00b2 : \u2191f z = x\nh\u271d\u00b9 : \u00acz = \u2191f\u207b\u00b9 x\nh\u271d : z = \u2191f\u207b\u00b9 y\n\u22a2 y = \u2191f (\u2191f\u207b\u00b9 x)\n[PROOFSTEP]\nsimp_all only [Perm.apply_inv_self, Perm.eq_inv_iff_eq, eq_self_iff_true, not_true]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y z : \u03b1\nh\u271d\u00b2 : \u2191f z = x\nh\u271d\u00b9 : \u00acz = \u2191f\u207b\u00b9 x\nh\u271d : \u00acz = \u2191f\u207b\u00b9 y\n\u22a2 y = \u2191f z\n[PROOFSTEP]\nsimp_all only [Perm.apply_inv_self, Perm.eq_inv_iff_eq, eq_self_iff_true, not_true]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y z : \u03b1\nh\u271d\u00b2 : \u00ac\u2191f z = x\nh\u271d\u00b9 : \u2191f z = y\nh\u271d : z = \u2191f\u207b\u00b9 x\n\u22a2 x = \u2191f (\u2191f\u207b\u00b9 y)\n[PROOFSTEP]\nsimp_all only [Perm.apply_inv_self, Perm.eq_inv_iff_eq, eq_self_iff_true, not_true]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y z : \u03b1\nh\u271d\u00b3 : \u00ac\u2191f z = x\nh\u271d\u00b2 : \u2191f z = y\nh\u271d\u00b9 : \u00acz = \u2191f\u207b\u00b9 x\nh\u271d : z = \u2191f\u207b\u00b9 y\n\u22a2 x = \u2191f (\u2191f\u207b\u00b9 x)\n[PROOFSTEP]\nsimp_all only [Perm.apply_inv_self, Perm.eq_inv_iff_eq, eq_self_iff_true, not_true]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y z : \u03b1\nh\u271d\u00b3 : \u00ac\u2191f z = x\nh\u271d\u00b2 : \u2191f z = y\nh\u271d\u00b9 : \u00acz = \u2191f\u207b\u00b9 x\nh\u271d : \u00acz = \u2191f\u207b\u00b9 y\n\u22a2 x = \u2191f z\n[PROOFSTEP]\nsimp_all only [Perm.apply_inv_self, Perm.eq_inv_iff_eq, eq_self_iff_true, not_true]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y z : \u03b1\nh\u271d\u00b2 : \u00ac\u2191f z = x\nh\u271d\u00b9 : \u00ac\u2191f z = y\nh\u271d : z = \u2191f\u207b\u00b9 x\n\u22a2 \u2191f z = \u2191f (\u2191f\u207b\u00b9 y)\n[PROOFSTEP]\nsimp_all only [Perm.apply_inv_self, Perm.eq_inv_iff_eq, eq_self_iff_true, not_true]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y z : \u03b1\nh\u271d\u00b3 : \u00ac\u2191f z = x\nh\u271d\u00b2 : \u00ac\u2191f z = y\nh\u271d\u00b9 : \u00acz = \u2191f\u207b\u00b9 x\nh\u271d : z = \u2191f\u207b\u00b9 y\n\u22a2 \u2191f z = \u2191f (\u2191f\u207b\u00b9 x)\n[PROOFSTEP]\nsimp_all only [Perm.apply_inv_self, Perm.eq_inv_iff_eq, eq_self_iff_true, not_true]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y z : \u03b1\nh\u271d\u00b3 : \u00ac\u2191f z = x\nh\u271d\u00b2 : \u00ac\u2191f z = y\nh\u271d\u00b9 : \u00acz = \u2191f\u207b\u00b9 x\nh\u271d : \u00acz = \u2191f\u207b\u00b9 y\n\u22a2 \u2191f z = \u2191f z\n[PROOFSTEP]\nsimp_all only [Perm.apply_inv_self, Perm.eq_inv_iff_eq, eq_self_iff_true, not_true]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\n\u22a2 f * swap x y = swap (\u2191f x) (\u2191f y) * f\n[PROOFSTEP]\nrw [swap_mul_eq_mul_swap, Perm.inv_apply_self, Perm.inv_apply_self]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\n\u22a2 swap (\u2191f x) (\u2191f y) = f * swap x y * f\u207b\u00b9\n[PROOFSTEP]\nrw [mul_swap_eq_swap_mul, mul_inv_cancel_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\ni j : \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 swap i j * (swap i j * \u03c3) = \u03c3\n[PROOFSTEP]\nrw [\u2190 mul_assoc, swap_mul_self, one_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\ni j : \u03b1\n\u03c3 : Perm \u03b1\n\u22a2 \u03c3 * swap i j * swap i j = \u03c3\n[PROOFSTEP]\nrw [mul_assoc, swap_mul_self, mul_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\ni j : \u03b1\n\u03c3 : Perm \u03b1\nh : swap i j * \u03c3 = \u03c3\n\u22a2 i = j\n[PROOFSTEP]\nhave swap_id : swap i j = 1 := mul_right_cancel (_root_.trans h (one_mul \u03c3).symm)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\ni j : \u03b1\n\u03c3 : Perm \u03b1\nh : swap i j * \u03c3 = \u03c3\nswap_id : swap i j = 1\n\u22a2 i = j\n[PROOFSTEP]\nrw [\u2190 swap_apply_right i j, swap_id]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\ni j : \u03b1\n\u03c3 : Perm \u03b1\nh : swap i j * \u03c3 = \u03c3\nswap_id : swap i j = 1\n\u22a2 \u21911 j = j\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\ni j : \u03b1\n\u03c3 : Perm \u03b1\nh : i = j\n\u22a2 swap i j * \u03c3 = \u03c3\n[PROOFSTEP]\nerw [h, swap_self, one_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\ni j : \u03b1\n\u03c3 : Perm \u03b1\nh : \u03c3 * swap i j = \u03c3\n\u22a2 i = j\n[PROOFSTEP]\nhave swap_id : swap i j = 1 := mul_left_cancel (_root_.trans h (one_mul \u03c3).symm)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\ni j : \u03b1\n\u03c3 : Perm \u03b1\nh : \u03c3 * swap i j = \u03c3\nswap_id : swap i j = 1\n\u22a2 i = j\n[PROOFSTEP]\nrw [\u2190 swap_apply_right i j, swap_id]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\ni j : \u03b1\n\u03c3 : Perm \u03b1\nh : \u03c3 * swap i j = \u03c3\nswap_id : swap i j = 1\n\u22a2 \u21911 j = j\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\ni j : \u03b1\n\u03c3 : Perm \u03b1\nh : i = j\n\u22a2 \u03c3 * swap i j = \u03c3\n[PROOFSTEP]\nerw [h, swap_self, mul_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\n\u22a2 \u2191(swap y z * swap x y * swap y z) n = \u2191(swap z x) n\n[PROOFSTEP]\nsimp only [swap_apply_def, Perm.mul_apply]\n -- Porting note: was `cc`\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\n\u22a2 (if\n (if (if n = y then z else if n = z then y else n) = x then y\n else\n if (if n = y then z else if n = z then y else n) = y then x\n else if n = y then z else if n = z then y else n) =\n y then\n z\n else\n if\n (if (if n = y then z else if n = z then y else n) = x then y\n else\n if (if n = y then z else if n = z then y else n) = y then x\n else if n = y then z else if n = z then y else n) =\n z then\n y\n else\n if (if n = y then z else if n = z then y else n) = x then y\n else\n if (if n = y then z else if n = z then y else n) = y then x\n else if n = y then z else if n = z then y else n) =\n if n = z then x else if n = x then z else n\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u00b3 : n = y\nh\u271d\u00b2 : z = x\nh\u271d\u00b9 : y = y\nh\u271d : n = z\n\u22a2 z = x\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2074 : n = y\nh\u271d\u00b3 : z = x\nh\u271d\u00b2 : y = y\nh\u271d\u00b9 : \u00acn = z\nh\u271d : n = x\n\u22a2 z = z\n[PROOFSTEP]\naesop\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2074 : n = y\nh\u271d\u00b3 : z = x\nh\u271d\u00b2 : y = y\nh\u271d\u00b9 : \u00acn = z\nh\u271d : \u00acn = x\n\u22a2 z = n\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2074 : n = y\nh\u271d\u00b3 : z = x\nh\u271d\u00b2 : \u00acy = y\nh\u271d\u00b9 : y = z\nh\u271d : n = z\n\u22a2 y = x\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2075 : n = y\nh\u271d\u2074 : z = x\nh\u271d\u00b3 : \u00acy = y\nh\u271d\u00b2 : y = z\nh\u271d\u00b9 : \u00acn = z\nh\u271d : n = x\n\u22a2 y = z\n[PROOFSTEP]\naesop\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2075 : n = y\nh\u271d\u2074 : z = x\nh\u271d\u00b3 : \u00acy = y\nh\u271d\u00b2 : y = z\nh\u271d\u00b9 : \u00acn = z\nh\u271d : \u00acn = x\n\u22a2 y = n\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2074 : n = y\nh\u271d\u00b3 : z = x\nh\u271d\u00b2 : \u00acy = y\nh\u271d\u00b9 : \u00acy = z\nh\u271d : n = z\n\u22a2 y = x\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2075 : n = y\nh\u271d\u2074 : z = x\nh\u271d\u00b3 : \u00acy = y\nh\u271d\u00b2 : \u00acy = z\nh\u271d\u00b9 : \u00acn = z\nh\u271d : n = x\n\u22a2 y = z\n[PROOFSTEP]\naesop\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2075 : n = y\nh\u271d\u2074 : z = x\nh\u271d\u00b3 : \u00acy = y\nh\u271d\u00b2 : \u00acy = z\nh\u271d\u00b9 : \u00acn = z\nh\u271d : \u00acn = x\n\u22a2 y = n\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u00b3 : n = y\nh\u271d\u00b2 : \u00acz = x\nh\u271d\u00b9 : z = y\nh\u271d : n = z\n\u22a2 x = x\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2074 : n = y\nh\u271d\u00b3 : \u00acz = x\nh\u271d\u00b2 : z = y\nh\u271d\u00b9 : \u00acn = z\nh\u271d : n = x\n\u22a2 x = z\n[PROOFSTEP]\naesop\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2074 : n = y\nh\u271d\u00b3 : \u00acz = x\nh\u271d\u00b2 : z = y\nh\u271d\u00b9 : \u00acn = z\nh\u271d : \u00acn = x\n\u22a2 x = n\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2074 : n = y\nh\u271d\u00b3 : \u00acz = x\nh\u271d\u00b2 : \u00acz = y\nh\u271d\u00b9 : z = z\nh\u271d : n = z\n\u22a2 y = x\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2075 : n = y\nh\u271d\u2074 : \u00acz = x\nh\u271d\u00b3 : \u00acz = y\nh\u271d\u00b2 : z = z\nh\u271d\u00b9 : \u00acn = z\nh\u271d : n = x\n\u22a2 y = z\n[PROOFSTEP]\naesop\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2075 : n = y\nh\u271d\u2074 : \u00acz = x\nh\u271d\u00b3 : \u00acz = y\nh\u271d\u00b2 : z = z\nh\u271d\u00b9 : \u00acn = z\nh\u271d : \u00acn = x\n\u22a2 y = n\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2074 : n = y\nh\u271d\u00b3 : \u00acz = x\nh\u271d\u00b2 : \u00acz = y\nh\u271d\u00b9 : \u00acz = z\nh\u271d : n = z\n\u22a2 z = x\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2075 : n = y\nh\u271d\u2074 : \u00acz = x\nh\u271d\u00b3 : \u00acz = y\nh\u271d\u00b2 : \u00acz = z\nh\u271d\u00b9 : \u00acn = z\nh\u271d : n = x\n\u22a2 z = z\n[PROOFSTEP]\naesop\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2075 : n = y\nh\u271d\u2074 : \u00acz = x\nh\u271d\u00b3 : \u00acz = y\nh\u271d\u00b2 : \u00acz = z\nh\u271d\u00b9 : \u00acn = z\nh\u271d : \u00acn = x\n\u22a2 z = n\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u00b3 : \u00acn = y\nh\u271d\u00b2 : n = z\nh\u271d\u00b9 : y = x\nh\u271d : y = y\n\u22a2 z = x\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2074 : \u00acn = y\nh\u271d\u00b3 : n = z\nh\u271d\u00b2 : y = x\nh\u271d\u00b9 : \u00acy = y\nh\u271d : y = z\n\u22a2 y = x\n[PROOFSTEP]\naesop\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2074 : \u00acn = y\nh\u271d\u00b3 : n = z\nh\u271d\u00b2 : y = x\nh\u271d\u00b9 : \u00acy = y\nh\u271d : \u00acy = z\n\u22a2 y = x\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u00b3 : \u00acn = y\nh\u271d\u00b2 : n = z\nh\u271d\u00b9 : \u00acy = x\nh\u271d : y = y\n\u22a2 x = x\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2074 : \u00acn = y\nh\u271d\u00b3 : n = z\nh\u271d\u00b2 : \u00acy = x\nh\u271d\u00b9 : \u00acy = y\nh\u271d : y = z\n\u22a2 y = x\n[PROOFSTEP]\naesop\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2074 : \u00acn = y\nh\u271d\u00b3 : n = z\nh\u271d\u00b2 : \u00acy = x\nh\u271d\u00b9 : \u00acy = y\nh\u271d : \u00acy = z\n\u22a2 y = x\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u00b3 : \u00acn = y\nh\u271d\u00b2 : \u00acn = z\nh\u271d\u00b9 : n = x\nh\u271d : y = y\n\u22a2 z = z\n[PROOFSTEP]\naesop\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2074 : \u00acn = y\nh\u271d\u00b3 : \u00acn = z\nh\u271d\u00b2 : n = x\nh\u271d\u00b9 : \u00acy = y\nh\u271d : y = z\n\u22a2 y = z\n[PROOFSTEP]\naesop\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u2074 : \u00acn = y\nh\u271d\u00b3 : \u00acn = z\nh\u271d\u00b2 : n = x\nh\u271d\u00b9 : \u00acy = y\nh\u271d : \u00acy = z\n\u22a2 y = z\n[PROOFSTEP]\naesop\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhwz : x \u2260 y\nhxz : x \u2260 z\nn : \u03b1\nh\u271d\u00b2 : \u00acn = y\nh\u271d\u00b9 : \u00acn = z\nh\u271d : \u00acn = x\n\u22a2 n = n\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : AddGroup \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 Equiv.addLeft a ^ n = Equiv.addLeft (n \u2022 a)\n[PROOFSTEP]\next\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : AddGroup \u03b1\na b : \u03b1\nn : \u2115\nx\u271d : \u03b1\n\u22a2 \u2191(Equiv.addLeft a ^ n) x\u271d = \u2191(Equiv.addLeft (n \u2022 a)) x\u271d\n[PROOFSTEP]\nsimp [Perm.coe_pow]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : AddGroup \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 Equiv.addRight a ^ n = Equiv.addRight (n \u2022 a)\n[PROOFSTEP]\next\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : AddGroup \u03b1\na b : \u03b1\nn : \u2115\nx\u271d : \u03b1\n\u22a2 \u2191(Equiv.addRight a ^ n) x\u271d = \u2191(Equiv.addRight (n \u2022 a)) x\u271d\n[PROOFSTEP]\nsimp [Perm.coe_pow]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : AddGroup \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 Equiv.addRight a ^ Int.ofNat n = Equiv.addRight (Int.ofNat n \u2022 a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : AddGroup \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 Equiv.addRight a ^ Int.negSucc n = Equiv.addRight (Int.negSucc n \u2022 a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Group \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 Equiv.mulLeft a ^ n = Equiv.mulLeft (a ^ n)\n[PROOFSTEP]\next\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Group \u03b1\na b : \u03b1\nn : \u2115\nx\u271d : \u03b1\n\u22a2 \u2191(Equiv.mulLeft a ^ n) x\u271d = \u2191(Equiv.mulLeft (a ^ n)) x\u271d\n[PROOFSTEP]\nsimp [Perm.coe_pow]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Group \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 Equiv.mulRight a ^ n = Equiv.mulRight (a ^ n)\n[PROOFSTEP]\next\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Group \u03b1\na b : \u03b1\nn : \u2115\nx\u271d : \u03b1\n\u22a2 \u2191(Equiv.mulRight a ^ n) x\u271d = \u2191(Equiv.mulRight (a ^ n)) x\u271d\n[PROOFSTEP]\nsimp [Perm.coe_pow]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Group \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 Equiv.mulRight a ^ Int.ofNat n = Equiv.mulRight (a ^ Int.ofNat n)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Group \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 Equiv.mulRight a ^ Int.negSucc n = Equiv.mulRight (a ^ Int.negSucc n)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nf : Perm \u03b1\ns : Set \u03b1\n\u22a2 MapsTo (\u2191f) s s \u2192 \u2200 (n : \u2115), MapsTo (\u2191(f ^ n)) s s\n[PROOFSTEP]\nsimp_rw [Equiv.Perm.coe_pow]\n[GOAL]\n\u03b1 : Type u_1\nf : Perm \u03b1\ns : Set \u03b1\n\u22a2 MapsTo (\u2191f) s s \u2192 \u2200 (n : \u2115), MapsTo (\u2191f)^[n] s s\n[PROOFSTEP]\nexact MapsTo.iterate\n[GOAL]\n\u03b1 : Type u_1\nf : Perm \u03b1\ns : Set \u03b1\n\u22a2 SurjOn (\u2191f) s s \u2192 \u2200 (n : \u2115), SurjOn (\u2191(f ^ n)) s s\n[PROOFSTEP]\nsimp_rw [Equiv.Perm.coe_pow]\n[GOAL]\n\u03b1 : Type u_1\nf : Perm \u03b1\ns : Set \u03b1\n\u22a2 SurjOn (\u2191f) s s \u2192 \u2200 (n : \u2115), SurjOn (\u2191f)^[n] s s\n[PROOFSTEP]\nexact SurjOn.iterate\n[GOAL]\n\u03b1 : Type u_1\nf : Perm \u03b1\ns : Set \u03b1\n\u22a2 BijOn (\u2191f) s s \u2192 \u2200 (n : \u2115), BijOn (\u2191(f ^ n)) s s\n[PROOFSTEP]\nsimp_rw [Equiv.Perm.coe_pow]\n[GOAL]\n\u03b1 : Type u_1\nf : Perm \u03b1\ns : Set \u03b1\n\u22a2 BijOn (\u2191f) s s \u2192 \u2200 (n : \u2115), BijOn (\u2191f)^[n] s s\n[PROOFSTEP]\nexact BijOn.iterate\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Perm.Basic", "llama_tokens": 13369, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.41354305496693555}} {"text": "[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Fintype R\np : R[X]\nhp : 0 < degree p\na : R\nx\u271d : a \u2208 image (fun x => eval x p) univ\n\u22a2 filter (fun x => eval x p = a) univ = Multiset.toFinset (roots (p - \u2191C a))\n[PROOFSTEP]\nsimp [Finset.ext_iff, \u2190 mem_roots_sub_C hp]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Fintype R\nf g : R[X]\nhf2 : degree f = 2\nhg2 : degree g = 2\nhR : Fintype.card R % 2 = 1\nthis : DecidableEq R := Classical.decEq R\nhd : Disjoint (image (fun x => eval x f) univ) (image (fun x => eval x (-g)) univ)\n\u22a2 0 < degree f\n[PROOFSTEP]\nrw [hf2]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Fintype R\nf g : R[X]\nhf2 : degree f = 2\nhg2 : degree g = 2\nhR : Fintype.card R % 2 = 1\nthis : DecidableEq R := Classical.decEq R\nhd : Disjoint (image (fun x => eval x f) univ) (image (fun x => eval x (-g)) univ)\n\u22a2 0 < 2\n[PROOFSTEP]\nexact by decide\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Fintype R\nf g : R[X]\nhf2 : degree f = 2\nhg2 : degree g = 2\nhR : Fintype.card R % 2 = 1\nthis : DecidableEq R := Classical.decEq R\nhd : Disjoint (image (fun x => eval x f) univ) (image (fun x => eval x (-g)) univ)\n\u22a2 0 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Fintype R\nf g : R[X]\nhf2 : degree f = 2\nhg2 : degree g = 2\nhR : Fintype.card R % 2 = 1\nthis : DecidableEq R := Classical.decEq R\nhd : Disjoint (image (fun x => eval x f) univ) (image (fun x => eval x (-g)) univ)\n\u22a2 \u00acFintype.card R % 2 = natDegree f * card (image (fun x => eval x f) univ) % 2\n[PROOFSTEP]\nsimp [natDegree_eq_of_degree_eq_some hf2, hR]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Fintype R\nf g : R[X]\nhf2 : degree f = 2\nhg2 : degree g = 2\nhR : Fintype.card R % 2 = 1\nthis : DecidableEq R := Classical.decEq R\nhd : Disjoint (image (fun x => eval x f) univ) (image (fun x => eval x (-g)) univ)\n\u22a2 0 < degree (-g)\n[PROOFSTEP]\nrw [degree_neg, hg2]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Fintype R\nf g : R[X]\nhf2 : degree f = 2\nhg2 : degree g = 2\nhR : Fintype.card R % 2 = 1\nthis : DecidableEq R := Classical.decEq R\nhd : Disjoint (image (fun x => eval x f) univ) (image (fun x => eval x (-g)) univ)\n\u22a2 0 < 2\n[PROOFSTEP]\nexact by decide\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Fintype R\nf g : R[X]\nhf2 : degree f = 2\nhg2 : degree g = 2\nhR : Fintype.card R % 2 = 1\nthis : DecidableEq R := Classical.decEq R\nhd : Disjoint (image (fun x => eval x f) univ) (image (fun x => eval x (-g)) univ)\n\u22a2 0 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Fintype R\nf g : R[X]\nhf2 : degree f = 2\nhg2 : degree g = 2\nhR : Fintype.card R % 2 = 1\nthis : DecidableEq R := Classical.decEq R\nhd : Disjoint (image (fun x => eval x f) univ) (image (fun x => eval x (-g)) univ)\n\u22a2 natDegree f * card (image (fun x => eval x f) univ) + natDegree (-g) * card (image (fun x => eval x (-g)) univ) =\n 2 * card (image (fun x => eval x f) univ \u222a image (fun x => eval x (-g)) univ)\n[PROOFSTEP]\nrw [card_disjoint_union hd]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Fintype R\nf g : R[X]\nhf2 : degree f = 2\nhg2 : degree g = 2\nhR : Fintype.card R % 2 = 1\nthis : DecidableEq R := Classical.decEq R\nhd : Disjoint (image (fun x => eval x f) univ) (image (fun x => eval x (-g)) univ)\n\u22a2 natDegree f * card (image (fun x => eval x f) univ) + natDegree (-g) * card (image (fun x => eval x (-g)) univ) =\n 2 * (card (image (fun x => eval x f) univ) + card (image (fun x => eval x (-g)) univ))\n[PROOFSTEP]\nsimp [natDegree_eq_of_degree_eq_some hf2, natDegree_eq_of_degree_eq_some hg2, mul_add]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Fintype R\nf g : R[X]\nhf2 : degree f = 2\nhg2 : degree g = 2\nhR : Fintype.card R % 2 = 1\nthis\u271d : DecidableEq R := Classical.decEq R\nthis : \u00acDisjoint (image (fun x => eval x f) univ) (image (fun x => eval x (-g)) univ)\n\u22a2 \u2203 a b, eval a f + eval b g = 0\n[PROOFSTEP]\nsimp only [disjoint_left, mem_image] at this \n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Fintype R\nf g : R[X]\nhf2 : degree f = 2\nhg2 : degree g = 2\nhR : Fintype.card R % 2 = 1\nthis\u271d : DecidableEq R := Classical.decEq R\nthis : \u00ac\u2200 \u2983a : R\u2984, (\u2203 a_1, a_1 \u2208 univ \u2227 eval a_1 f = a) \u2192 \u00ac\u2203 a_2, a_2 \u2208 univ \u2227 eval a_2 (-g) = a\n\u22a2 \u2203 a b, eval a f + eval b g = 0\n[PROOFSTEP]\npush_neg at this \n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Fintype R\nf g : R[X]\nhf2 : degree f = 2\nhg2 : degree g = 2\nhR : Fintype.card R % 2 = 1\nthis\u271d : DecidableEq R := Classical.decEq R\nthis : Exists fun \u2983a\u2984 => (\u2203 a_1, a_1 \u2208 univ \u2227 eval a_1 f = a) \u2227 \u2203 a_1, a_1 \u2208 univ \u2227 eval a_1 (-g) = a\n\u22a2 \u2203 a b, eval a f + eval b g = 0\n[PROOFSTEP]\nrcases this with \u27e8x, \u27e8a, _, ha\u27e9, \u27e8b, _, hb\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Fintype R\nf g : R[X]\nhf2 : degree f = 2\nhg2 : degree g = 2\nhR : Fintype.card R % 2 = 1\nthis : DecidableEq R := Classical.decEq R\nx a : R\nleft\u271d\u00b9 : a \u2208 univ\nha : eval a f = x\nb : R\nleft\u271d : b \u2208 univ\nhb : eval b (-g) = x\n\u22a2 \u2203 a b, eval a f + eval b g = 0\n[PROOFSTEP]\nexact \u27e8a, b, by rw [ha, \u2190 hb, eval_neg, neg_add_self]\u27e9\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : Fintype R\nf g : R[X]\nhf2 : degree f = 2\nhg2 : degree g = 2\nhR : Fintype.card R % 2 = 1\nthis : DecidableEq R := Classical.decEq R\nx a : R\nleft\u271d\u00b9 : a \u2208 univ\nha : eval a f = x\nb : R\nleft\u271d : b \u2208 univ\nhb : eval b (-g) = x\n\u22a2 eval a f + eval b g = 0\n[PROOFSTEP]\nrw [ha, \u2190 hb, eval_neg, neg_add_self]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : Fintype K\u02e3\n\u22a2 \u220f x : K\u02e3, x = -1\n[PROOFSTEP]\nclassical\nhave : (\u220f x in (@univ K\u02e3 _).erase (-1), x) = 1 :=\n prod_involution (fun x _ => x\u207b\u00b9) (by simp)\n (fun a => by simp (config := { contextual := true }) [Units.inv_eq_self_iff])\n (fun a => by simp [@inv_eq_iff_eq_inv _ _ a]) (by simp)\nrw [\u2190 insert_erase (mem_univ (-1 : K\u02e3)), prod_insert (not_mem_erase _ _), this, mul_one]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : Fintype K\u02e3\n\u22a2 \u220f x : K\u02e3, x = -1\n[PROOFSTEP]\nhave : (\u220f x in (@univ K\u02e3 _).erase (-1), x) = 1 :=\n prod_involution (fun x _ => x\u207b\u00b9) (by simp)\n (fun a => by simp (config := { contextual := true }) [Units.inv_eq_self_iff])\n (fun a => by simp [@inv_eq_iff_eq_inv _ _ a]) (by simp)\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : Fintype K\u02e3\n\u22a2 \u2200 (a : K\u02e3) (ha : a \u2208 erase univ (-1)), a * (fun x x_1 => x\u207b\u00b9) a ha = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : Fintype K\u02e3\na : K\u02e3\n\u22a2 \u2200 (ha : a \u2208 erase univ (-1)), a \u2260 1 \u2192 (fun x x_1 => x\u207b\u00b9) a ha \u2260 a\n[PROOFSTEP]\nsimp (config := { contextual := true }) [Units.inv_eq_self_iff]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : Fintype K\u02e3\na : K\u02e3\n\u22a2 \u2200 (ha : a \u2208 erase univ (-1)), (fun x x_1 => x\u207b\u00b9) a ha \u2208 erase univ (-1)\n[PROOFSTEP]\nsimp [@inv_eq_iff_eq_inv _ _ a]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : Fintype K\u02e3\n\u22a2 \u2200 (a : K\u02e3) (ha : a \u2208 erase univ (-1)), (fun x x_1 => x\u207b\u00b9) ((fun x x_1 => x\u207b\u00b9) a ha) (_ : a\u207b\u00b9 \u2208 erase univ (-1)) = a\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : Fintype K\u02e3\nthis : \u220f x in erase univ (-1), x = 1\n\u22a2 \u220f x : K\u02e3, x = -1\n[PROOFSTEP]\nrw [\u2190 insert_erase (mem_univ (-1 : K\u02e3)), prod_insert (not_mem_erase _ _), this, mul_one]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nha : a \u2260 0\n\u22a2 a ^ (q - 1) = 1\n[PROOFSTEP]\ncalc\n a ^ (Fintype.card K - 1) = (Units.mk0 a ha ^ (Fintype.card K - 1) : K\u02e3).1 := by\n rw [Units.val_pow_eq_pow_val, Units.val_mk0]\n _ = 1 := by\n classical\n rw [\u2190 Fintype.card_units, pow_card_eq_one]\n rfl\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nha : a \u2260 0\n\u22a2 a ^ (q - 1) = \u2191(Units.mk0 a ha ^ (q - 1))\n[PROOFSTEP]\nrw [Units.val_pow_eq_pow_val, Units.val_mk0]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nha : a \u2260 0\n\u22a2 \u2191(Units.mk0 a ha ^ (q - 1)) = 1\n[PROOFSTEP]\nclassical\nrw [\u2190 Fintype.card_units, pow_card_eq_one]\nrfl\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nha : a \u2260 0\n\u22a2 \u2191(Units.mk0 a ha ^ (q - 1)) = 1\n[PROOFSTEP]\nrw [\u2190 Fintype.card_units, pow_card_eq_one]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nha : a \u2260 0\n\u22a2 \u21911 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\n\u22a2 a ^ q = a\n[PROOFSTEP]\nhave hp : 0 < Fintype.card K := lt_trans zero_lt_one Fintype.one_lt_card\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nhp : 0 < q\n\u22a2 a ^ q = a\n[PROOFSTEP]\nby_cases h : a = 0\n[GOAL]\ncase pos\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nhp : 0 < q\nh : a = 0\n\u22a2 a ^ q = a\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase pos\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nhp : 0 < q\nh : a = 0\n\u22a2 0 ^ q = 0\n[PROOFSTEP]\napply zero_pow hp\n[GOAL]\ncase neg\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nhp : 0 < q\nh : \u00aca = 0\n\u22a2 a ^ q = a\n[PROOFSTEP]\nrw [\u2190 Nat.succ_pred_eq_of_pos hp, pow_succ, Nat.pred_eq_sub_one, pow_card_sub_one_eq_one a h, mul_one]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\nn : \u2115\na : K\n\u22a2 a ^ q ^ n = a\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\n\u22a2 a ^ q ^ Nat.zero = a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : GroupWithZero K\ninst\u271d : Fintype K\na : K\nn : \u2115\nih : a ^ q ^ n = a\n\u22a2 a ^ q ^ Nat.succ n = a\n[PROOFSTEP]\nsimp [pow_succ, pow_mul, ih, pow_card]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\np : \u2115\ninst\u271d : CharP K p\n\u22a2 \u2203 n, Nat.Prime p \u2227 q = p ^ \u2191n\n[PROOFSTEP]\nhaveI hp : Fact p.Prime := \u27e8CharP.char_is_prime K p\u27e9\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\np : \u2115\ninst\u271d : CharP K p\nhp : Fact (Nat.Prime p)\n\u22a2 \u2203 n, Nat.Prime p \u2227 q = p ^ \u2191n\n[PROOFSTEP]\nletI : Module (ZMod p) K := { (ZMod.castHom dvd_rfl K : ZMod p \u2192+* _).toModule with }\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\np : \u2115\ninst\u271d : CharP K p\nhp : Fact (Nat.Prime p)\nthis : Module (ZMod p) K :=\n let src := RingHom.toModule (ZMod.castHom (_ : p \u2223 p) K);\n Module.mk (_ : \u2200 (r s : ZMod p) (x : K), (r + s) \u2022 x = r \u2022 x + s \u2022 x) (_ : \u2200 (x : K), 0 \u2022 x = 0)\n\u22a2 \u2203 n, Nat.Prime p \u2227 q = p ^ \u2191n\n[PROOFSTEP]\nobtain \u27e8n, h\u27e9 := VectorSpace.card_fintype (ZMod p) K\n[GOAL]\ncase intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\np : \u2115\ninst\u271d : CharP K p\nhp : Fact (Nat.Prime p)\nthis : Module (ZMod p) K :=\n let src := RingHom.toModule (ZMod.castHom (_ : p \u2223 p) K);\n Module.mk (_ : \u2200 (r s : ZMod p) (x : K), (r + s) \u2022 x = r \u2022 x + s \u2022 x) (_ : \u2200 (x : K), 0 \u2022 x = 0)\nn : \u2115\nh : q = Fintype.card (ZMod p) ^ n\n\u22a2 \u2203 n, Nat.Prime p \u2227 q = p ^ \u2191n\n[PROOFSTEP]\nrw [ZMod.card] at h \n[GOAL]\ncase intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\np : \u2115\ninst\u271d : CharP K p\nhp : Fact (Nat.Prime p)\nthis : Module (ZMod p) K :=\n let src := RingHom.toModule (ZMod.castHom (_ : p \u2223 p) K);\n Module.mk (_ : \u2200 (r s : ZMod p) (x : K), (r + s) \u2022 x = r \u2022 x + s \u2022 x) (_ : \u2200 (x : K), 0 \u2022 x = 0)\nn : \u2115\nh : q = p ^ n\n\u22a2 \u2203 n, Nat.Prime p \u2227 q = p ^ \u2191n\n[PROOFSTEP]\nrefine' \u27e8\u27e8n, _\u27e9, hp.1, h\u27e9\n[GOAL]\ncase intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\np : \u2115\ninst\u271d : CharP K p\nhp : Fact (Nat.Prime p)\nthis : Module (ZMod p) K :=\n let src := RingHom.toModule (ZMod.castHom (_ : p \u2223 p) K);\n Module.mk (_ : \u2200 (r s : ZMod p) (x : K), (r + s) \u2022 x = r \u2022 x + s \u2022 x) (_ : \u2200 (x : K), 0 \u2022 x = 0)\nn : \u2115\nh : q = p ^ n\n\u22a2 0 < n\n[PROOFSTEP]\napply Or.resolve_left (Nat.eq_zero_or_pos n)\n[GOAL]\ncase intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\np : \u2115\ninst\u271d : CharP K p\nhp : Fact (Nat.Prime p)\nthis : Module (ZMod p) K :=\n let src := RingHom.toModule (ZMod.castHom (_ : p \u2223 p) K);\n Module.mk (_ : \u2200 (r s : ZMod p) (x : K), (r + s) \u2022 x = r \u2022 x + s \u2022 x) (_ : \u2200 (x : K), 0 \u2022 x = 0)\nn : \u2115\nh : q = p ^ n\n\u22a2 \u00acn = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\np : \u2115\ninst\u271d : CharP K p\nhp : Fact (Nat.Prime p)\nthis : Module (ZMod p) K :=\n let src := RingHom.toModule (ZMod.castHom (_ : p \u2223 p) K);\n Module.mk (_ : \u2200 (r s : ZMod p) (x : K), (r + s) \u2022 x = r \u2022 x + s \u2022 x) (_ : \u2200 (x : K), 0 \u2022 x = 0)\nh : q = p ^ 0\n\u22a2 False\n[PROOFSTEP]\nrw [pow_zero] at h \n[GOAL]\ncase intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\np : \u2115\ninst\u271d : CharP K p\nhp : Fact (Nat.Prime p)\nthis : Module (ZMod p) K :=\n let src := RingHom.toModule (ZMod.castHom (_ : p \u2223 p) K);\n Module.mk (_ : \u2200 (r s : ZMod p) (x : K), (r + s) \u2022 x = r \u2022 x + s \u2022 x) (_ : \u2200 (x : K), 0 \u2022 x = 0)\nh : q = 1\n\u22a2 False\n[PROOFSTEP]\nhave : (0 : K) = 1 := by apply Fintype.card_le_one_iff.mp (le_of_eq h)\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\np : \u2115\ninst\u271d : CharP K p\nhp : Fact (Nat.Prime p)\nthis : Module (ZMod p) K :=\n let src := RingHom.toModule (ZMod.castHom (_ : p \u2223 p) K);\n Module.mk (_ : \u2200 (r s : ZMod p) (x : K), (r + s) \u2022 x = r \u2022 x + s \u2022 x) (_ : \u2200 (x : K), 0 \u2022 x = 0)\nh : q = 1\n\u22a2 0 = 1\n[PROOFSTEP]\napply Fintype.card_le_one_iff.mp (le_of_eq h)\n[GOAL]\ncase intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\np : \u2115\ninst\u271d : CharP K p\nhp : Fact (Nat.Prime p)\nthis\u271d : Module (ZMod p) K :=\n let src := RingHom.toModule (ZMod.castHom (_ : p \u2223 p) K);\n Module.mk (_ : \u2200 (r s : ZMod p) (x : K), (r + s) \u2022 x = r \u2022 x + s \u2022 x) (_ : \u2200 (x : K), 0 \u2022 x = 0)\nh : q = 1\nthis : 0 = 1\n\u22a2 False\n[PROOFSTEP]\nexact absurd this zero_ne_one\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\n\u22a2 \u2191q = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\n\u22a2 (\u2200 (x : K\u02e3), x ^ i = 1) \u2194 q - 1 \u2223 i\n[PROOFSTEP]\nclassical\nobtain \u27e8x, hx\u27e9 := IsCyclic.exists_generator (\u03b1 := K\u02e3)\nrw [\u2190 Fintype.card_units, \u2190 orderOf_eq_card_of_forall_mem_zpowers hx, orderOf_dvd_iff_pow_eq_one]\nconstructor\n\u00b7 intro h; apply h\n\u00b7 intro h y\n simp_rw [\u2190 mem_powers_iff_mem_zpowers] at hx \n rcases hx y with \u27e8j, rfl\u27e9\n rw [\u2190 pow_mul, mul_comm, pow_mul, h, one_pow]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\n\u22a2 (\u2200 (x : K\u02e3), x ^ i = 1) \u2194 q - 1 \u2223 i\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := IsCyclic.exists_generator (\u03b1 := K\u02e3)\n[GOAL]\ncase intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nx : K\u02e3\nhx : \u2200 (x_1 : K\u02e3), x_1 \u2208 Subgroup.zpowers x\n\u22a2 (\u2200 (x : K\u02e3), x ^ i = 1) \u2194 q - 1 \u2223 i\n[PROOFSTEP]\nrw [\u2190 Fintype.card_units, \u2190 orderOf_eq_card_of_forall_mem_zpowers hx, orderOf_dvd_iff_pow_eq_one]\n[GOAL]\ncase intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nx : K\u02e3\nhx : \u2200 (x_1 : K\u02e3), x_1 \u2208 Subgroup.zpowers x\n\u22a2 (\u2200 (x : K\u02e3), x ^ i = 1) \u2194 x ^ i = 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.mp\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nx : K\u02e3\nhx : \u2200 (x_1 : K\u02e3), x_1 \u2208 Subgroup.zpowers x\n\u22a2 (\u2200 (x : K\u02e3), x ^ i = 1) \u2192 x ^ i = 1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase intro.mp\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nx : K\u02e3\nhx : \u2200 (x_1 : K\u02e3), x_1 \u2208 Subgroup.zpowers x\nh : \u2200 (x : K\u02e3), x ^ i = 1\n\u22a2 x ^ i = 1\n[PROOFSTEP]\napply h\n[GOAL]\ncase intro.mpr\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nx : K\u02e3\nhx : \u2200 (x_1 : K\u02e3), x_1 \u2208 Subgroup.zpowers x\n\u22a2 x ^ i = 1 \u2192 \u2200 (x : K\u02e3), x ^ i = 1\n[PROOFSTEP]\nintro h y\n[GOAL]\ncase intro.mpr\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nx : K\u02e3\nhx : \u2200 (x_1 : K\u02e3), x_1 \u2208 Subgroup.zpowers x\nh : x ^ i = 1\ny : K\u02e3\n\u22a2 y ^ i = 1\n[PROOFSTEP]\nsimp_rw [\u2190 mem_powers_iff_mem_zpowers] at hx \n[GOAL]\ncase intro.mpr\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nx : K\u02e3\nh : x ^ i = 1\ny : K\u02e3\nhx : \u2200 (x_1 : K\u02e3), x_1 \u2208 Submonoid.powers x\n\u22a2 y ^ i = 1\n[PROOFSTEP]\nrcases hx y with \u27e8j, rfl\u27e9\n[GOAL]\ncase intro.mpr.intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nx : K\u02e3\nh : x ^ i = 1\nhx : \u2200 (x_1 : K\u02e3), x_1 \u2208 Submonoid.powers x\nj : \u2115\n\u22a2 (fun x x_1 => x ^ x_1) x j ^ i = 1\n[PROOFSTEP]\nrw [\u2190 pow_mul, mul_comm, pow_mul, h, one_pow]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u22a2 \u2211 x : K\u02e3, \u2191(x ^ i) = if q - 1 \u2223 i then -1 else 0\n[PROOFSTEP]\nlet \u03c6 : K\u02e3 \u2192* K :=\n { toFun := fun x => x ^ i\n map_one' := by simp\n map_mul' := by intros; simp [mul_pow] }\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u22a2 (fun x => \u2191(x ^ i)) 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u22a2 \u2200 (x y : K\u02e3),\n OneHom.toFun { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) } (x * y) =\n OneHom.toFun { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) } x *\n OneHom.toFun { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) } y\n[PROOFSTEP]\nintros\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\nx\u271d y\u271d : K\u02e3\n\u22a2 OneHom.toFun { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) } (x\u271d * y\u271d) =\n OneHom.toFun { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) } x\u271d *\n OneHom.toFun { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) } y\u271d\n[PROOFSTEP]\nsimp [mul_pow]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\n\u22a2 \u2211 x : K\u02e3, \u2191(x ^ i) = if q - 1 \u2223 i then -1 else 0\n[PROOFSTEP]\nhave : Decidable (\u03c6 = 1) := by classical infer_instance\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\n\u22a2 Decidable (\u03c6 = 1)\n[PROOFSTEP]\nclassical infer_instance\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\n\u22a2 Decidable (\u03c6 = 1)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\nthis : Decidable (\u03c6 = 1)\n\u22a2 \u2211 x : K\u02e3, \u2191(x ^ i) = if q - 1 \u2223 i then -1 else 0\n[PROOFSTEP]\ncalc\n (\u2211 x : K\u02e3, \u03c6 x) = if \u03c6 = 1 then Fintype.card K\u02e3 else 0 := sum_hom_units \u03c6\n _ = if q - 1 \u2223 i then -1 else 0 :=\n by\n suffices q - 1 \u2223 i \u2194 \u03c6 = 1 by\n simp only [this]\n split_ifs; swap\n \u00b7 exact Nat.cast_zero\n \u00b7 rw [Fintype.card_units, Nat.cast_sub, cast_card_eq_zero, Nat.cast_one, zero_sub]\n show 1 \u2264 q; exact Fintype.card_pos_iff.mpr \u27e80\u27e9\n rw [\u2190 forall_pow_eq_one_iff, FunLike.ext_iff]\n apply forall_congr'; intro x; simp [Units.ext_iff]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\nthis : Decidable (\u03c6 = 1)\n\u22a2 \u2191(if \u03c6 = 1 then Fintype.card K\u02e3 else 0) = if q - 1 \u2223 i then -1 else 0\n[PROOFSTEP]\nsuffices q - 1 \u2223 i \u2194 \u03c6 = 1 by\n simp only [this]\n split_ifs; swap\n \u00b7 exact Nat.cast_zero\n \u00b7 rw [Fintype.card_units, Nat.cast_sub, cast_card_eq_zero, Nat.cast_one, zero_sub]\n show 1 \u2264 q; exact Fintype.card_pos_iff.mpr \u27e80\u27e9\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\nthis\u271d : Decidable (\u03c6 = 1)\nthis : q - 1 \u2223 i \u2194 \u03c6 = 1\n\u22a2 \u2191(if \u03c6 = 1 then Fintype.card K\u02e3 else 0) = if q - 1 \u2223 i then -1 else 0\n[PROOFSTEP]\nsimp only [this]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\nthis\u271d : Decidable (\u03c6 = 1)\nthis : q - 1 \u2223 i \u2194 \u03c6 = 1\n\u22a2 \u2191(if\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) } =\n 1 then\n Fintype.card K\u02e3\n else 0) =\n if\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) } =\n 1 then\n -1\n else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\nthis\u271d : Decidable (\u03c6 = 1)\nthis : q - 1 \u2223 i \u2194 \u03c6 = 1\nh\u271d :\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) } =\n 1\n\u22a2 \u2191(Fintype.card K\u02e3) = -1\ncase neg\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\nthis\u271d : Decidable (\u03c6 = 1)\nthis : q - 1 \u2223 i \u2194 \u03c6 = 1\nh\u271d :\n \u00ac{ toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) } =\n 1\n\u22a2 \u21910 = 0\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\nthis\u271d : Decidable (\u03c6 = 1)\nthis : q - 1 \u2223 i \u2194 \u03c6 = 1\nh\u271d :\n \u00ac{ toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) } =\n 1\n\u22a2 \u21910 = 0\n[PROOFSTEP]\nexact Nat.cast_zero\n[GOAL]\ncase pos\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\nthis\u271d : Decidable (\u03c6 = 1)\nthis : q - 1 \u2223 i \u2194 \u03c6 = 1\nh\u271d :\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) } =\n 1\n\u22a2 \u2191(Fintype.card K\u02e3) = -1\n[PROOFSTEP]\nrw [Fintype.card_units, Nat.cast_sub, cast_card_eq_zero, Nat.cast_one, zero_sub]\n[GOAL]\ncase pos\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\nthis\u271d : Decidable (\u03c6 = 1)\nthis : q - 1 \u2223 i \u2194 \u03c6 = 1\nh\u271d :\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) } =\n 1\n\u22a2 1 \u2264 q\n[PROOFSTEP]\nshow 1 \u2264 q\n[GOAL]\ncase pos\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\nthis\u271d : Decidable (\u03c6 = 1)\nthis : q - 1 \u2223 i \u2194 \u03c6 = 1\nh\u271d :\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) } =\n 1\n\u22a2 1 \u2264 q\n[PROOFSTEP]\nexact Fintype.card_pos_iff.mpr \u27e80\u27e9\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\nthis : Decidable (\u03c6 = 1)\n\u22a2 q - 1 \u2223 i \u2194 \u03c6 = 1\n[PROOFSTEP]\nrw [\u2190 forall_pow_eq_one_iff, FunLike.ext_iff]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\nthis : Decidable (\u03c6 = 1)\n\u22a2 (\u2200 (x : K\u02e3), x ^ i = 1) \u2194 \u2200 (x : K\u02e3), \u2191\u03c6 x = \u21911 x\n[PROOFSTEP]\napply forall_congr'\n[GOAL]\ncase h\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\nthis : Decidable (\u03c6 = 1)\n\u22a2 \u2200 (a : K\u02e3), a ^ i = 1 \u2194 \u2191\u03c6 a = \u21911 a\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : DecidableEq K\ni : \u2115\n\u03c6 : K\u02e3 \u2192* K :=\n { toOneHom := { toFun := fun x => \u2191(x ^ i), map_one' := (_ : \u2191(1 ^ i) = 1) },\n map_mul' := (_ : \u2200 (x y : K\u02e3), \u2191((x * y) ^ i) = \u2191(x ^ i) * \u2191(y ^ i)) }\nthis : Decidable (\u03c6 = 1)\nx : K\u02e3\n\u22a2 x ^ i = 1 \u2194 \u2191\u03c6 x = \u21911 x\n[PROOFSTEP]\nsimp [Units.ext_iff]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nh : i < q - 1\n\u22a2 \u2211 x : K, x ^ i = 0\n[PROOFSTEP]\nby_cases hi : i = 0\n[GOAL]\ncase pos\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nh : i < q - 1\nhi : i = 0\n\u22a2 \u2211 x : K, x ^ i = 0\n[PROOFSTEP]\nsimp only [hi, nsmul_one, sum_const, pow_zero, card_univ, cast_card_eq_zero]\n[GOAL]\ncase neg\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nh : i < q - 1\nhi : \u00aci = 0\n\u22a2 \u2211 x : K, x ^ i = 0\n[PROOFSTEP]\nclassical\nhave hiq : \u00acq - 1 \u2223 i := by contrapose! h; exact Nat.le_of_dvd (Nat.pos_of_ne_zero hi) h\nlet \u03c6 : K\u02e3 \u21aa K := \u27e8fun x \u21a6 x, Units.ext\u27e9\nhave : univ.map \u03c6 = univ \\ {0} := by\n ext x\n simp only [true_and_iff, Function.Embedding.coeFn_mk, mem_sdiff, Units.exists_iff_ne_zero, mem_univ, mem_map,\n exists_prop_of_true, mem_singleton]\ncalc\n \u2211 x : K, x ^ i = \u2211 x in univ \\ {(0 : K)}, x ^ i := by\n rw [\u2190 sum_sdiff ({0} : Finset K).subset_univ, sum_singleton, zero_pow (Nat.pos_of_ne_zero hi), add_zero]\n _ = \u2211 x : K\u02e3, (x ^ i : K) := by simp [\u2190 this, univ.sum_map \u03c6]\n _ = 0 := by rw [sum_pow_units K i, if_neg]; exact hiq\n[GOAL]\ncase neg\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nh : i < q - 1\nhi : \u00aci = 0\n\u22a2 \u2211 x : K, x ^ i = 0\n[PROOFSTEP]\nhave hiq : \u00acq - 1 \u2223 i := by contrapose! h; exact Nat.le_of_dvd (Nat.pos_of_ne_zero hi) h\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nh : i < q - 1\nhi : \u00aci = 0\n\u22a2 \u00acq - 1 \u2223 i\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nhi : \u00aci = 0\nh : q - 1 \u2223 i\n\u22a2 q - 1 \u2264 i\n[PROOFSTEP]\nexact Nat.le_of_dvd (Nat.pos_of_ne_zero hi) h\n[GOAL]\ncase neg\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nh : i < q - 1\nhi : \u00aci = 0\nhiq : \u00acq - 1 \u2223 i\n\u22a2 \u2211 x : K, x ^ i = 0\n[PROOFSTEP]\nlet \u03c6 : K\u02e3 \u21aa K := \u27e8fun x \u21a6 x, Units.ext\u27e9\n[GOAL]\ncase neg\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nh : i < q - 1\nhi : \u00aci = 0\nhiq : \u00acq - 1 \u2223 i\n\u03c6 : K\u02e3 \u21aa K := { toFun := fun x => \u2191x, inj' := (_ : Function.Injective Units.val) }\n\u22a2 \u2211 x : K, x ^ i = 0\n[PROOFSTEP]\nhave : univ.map \u03c6 = univ \\ {0} := by\n ext x\n simp only [true_and_iff, Function.Embedding.coeFn_mk, mem_sdiff, Units.exists_iff_ne_zero, mem_univ, mem_map,\n exists_prop_of_true, mem_singleton]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nh : i < q - 1\nhi : \u00aci = 0\nhiq : \u00acq - 1 \u2223 i\n\u03c6 : K\u02e3 \u21aa K := { toFun := fun x => \u2191x, inj' := (_ : Function.Injective Units.val) }\n\u22a2 map \u03c6 univ = univ \\ {0}\n[PROOFSTEP]\next x\n[GOAL]\ncase a\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nh : i < q - 1\nhi : \u00aci = 0\nhiq : \u00acq - 1 \u2223 i\n\u03c6 : K\u02e3 \u21aa K := { toFun := fun x => \u2191x, inj' := (_ : Function.Injective Units.val) }\nx : K\n\u22a2 x \u2208 map \u03c6 univ \u2194 x \u2208 univ \\ {0}\n[PROOFSTEP]\nsimp only [true_and_iff, Function.Embedding.coeFn_mk, mem_sdiff, Units.exists_iff_ne_zero, mem_univ, mem_map,\n exists_prop_of_true, mem_singleton]\n[GOAL]\ncase neg\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nh : i < q - 1\nhi : \u00aci = 0\nhiq : \u00acq - 1 \u2223 i\n\u03c6 : K\u02e3 \u21aa K := { toFun := fun x => \u2191x, inj' := (_ : Function.Injective Units.val) }\nthis : map \u03c6 univ = univ \\ {0}\n\u22a2 \u2211 x : K, x ^ i = 0\n[PROOFSTEP]\ncalc\n \u2211 x : K, x ^ i = \u2211 x in univ \\ {(0 : K)}, x ^ i := by\n rw [\u2190 sum_sdiff ({0} : Finset K).subset_univ, sum_singleton, zero_pow (Nat.pos_of_ne_zero hi), add_zero]\n _ = \u2211 x : K\u02e3, (x ^ i : K) := by simp [\u2190 this, univ.sum_map \u03c6]\n _ = 0 := by rw [sum_pow_units K i, if_neg]; exact hiq\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nh : i < q - 1\nhi : \u00aci = 0\nhiq : \u00acq - 1 \u2223 i\n\u03c6 : K\u02e3 \u21aa K := { toFun := fun x => \u2191x, inj' := (_ : Function.Injective Units.val) }\nthis : map \u03c6 univ = univ \\ {0}\n\u22a2 \u2211 x : K, x ^ i = \u2211 x in univ \\ {0}, x ^ i\n[PROOFSTEP]\nrw [\u2190 sum_sdiff ({0} : Finset K).subset_univ, sum_singleton, zero_pow (Nat.pos_of_ne_zero hi), add_zero]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nh : i < q - 1\nhi : \u00aci = 0\nhiq : \u00acq - 1 \u2223 i\n\u03c6 : K\u02e3 \u21aa K := { toFun := fun x => \u2191x, inj' := (_ : Function.Injective Units.val) }\nthis : map \u03c6 univ = univ \\ {0}\n\u22a2 \u2211 x in univ \\ {0}, x ^ i = \u2211 x : K\u02e3, \u2191(x ^ i)\n[PROOFSTEP]\nsimp [\u2190 this, univ.sum_map \u03c6]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nh : i < q - 1\nhi : \u00aci = 0\nhiq : \u00acq - 1 \u2223 i\n\u03c6 : K\u02e3 \u21aa K := { toFun := fun x => \u2191x, inj' := (_ : Function.Injective Units.val) }\nthis : map \u03c6 univ = univ \\ {0}\n\u22a2 \u2211 x : K\u02e3, \u2191(x ^ i) = 0\n[PROOFSTEP]\nrw [sum_pow_units K i, if_neg]\n[GOAL]\ncase hnc\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype K\ni : \u2115\nh : i < q - 1\nhi : \u00aci = 0\nhiq : \u00acq - 1 \u2223 i\n\u03c6 : K\u02e3 \u21aa K := { toFun := fun x => \u2191x, inj' := (_ : Function.Injective Units.val) }\nthis : map \u03c6 univ = univ \\ {0}\n\u22a2 \u00acq - 1 \u2223 i\n[PROOFSTEP]\nexact hiq\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\nK' : Type u_3\ninst\u271d : Field K'\np n : \u2115\nhp : 1 < p\n\u22a2 natDegree (X ^ p - X) = p\n[PROOFSTEP]\nhave h1 : (X : K'[X]).degree < (X ^ p : K'[X]).degree :=\n by\n rw [degree_X_pow, degree_X]\n -- Porting note: the following line was `exact_mod_cast hp`\n exact WithBot.coe_lt_coe.2 hp\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\nK' : Type u_3\ninst\u271d : Field K'\np n : \u2115\nhp : 1 < p\n\u22a2 degree X < degree (X ^ p)\n[PROOFSTEP]\nrw [degree_X_pow, degree_X]\n -- Porting note: the following line was `exact_mod_cast hp`\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\nK' : Type u_3\ninst\u271d : Field K'\np n : \u2115\nhp : 1 < p\n\u22a2 1 < \u2191p\n[PROOFSTEP]\nexact WithBot.coe_lt_coe.2 hp\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\nK' : Type u_3\ninst\u271d : Field K'\np n : \u2115\nhp : 1 < p\nh1 : degree X < degree (X ^ p)\n\u22a2 natDegree (X ^ p - X) = p\n[PROOFSTEP]\nrw [natDegree_eq_of_degree_eq (degree_sub_eq_left_of_degree_lt h1), natDegree_X_pow]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Algebra (ZMod p) K\n\u22a2 roots (X ^ q - X) = univ.val\n[PROOFSTEP]\nclassical\nhave aux : (X ^ q - X : K[X]) \u2260 0 := X_pow_card_sub_X_ne_zero K Fintype.one_lt_card\nhave : (roots (X ^ q - X : K[X])).toFinset = Finset.univ :=\n by\n rw [eq_univ_iff_forall]\n intro x\n rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X, sub_eq_zero, pow_card]\nrw [\u2190 this, Multiset.toFinset_val, eq_comm, Multiset.dedup_eq_self]\napply nodup_roots\nrw [separable_def]\nconvert isCoprime_one_right.neg_right (R := K[X]) using 1\nrw [derivative_sub, derivative_X, derivative_X_pow, CharP.cast_card_eq_zero K, C_0, zero_mul, zero_sub]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Algebra (ZMod p) K\n\u22a2 roots (X ^ q - X) = univ.val\n[PROOFSTEP]\nhave aux : (X ^ q - X : K[X]) \u2260 0 := X_pow_card_sub_X_ne_zero K Fintype.one_lt_card\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Algebra (ZMod p) K\naux : X ^ q - X \u2260 0\n\u22a2 roots (X ^ q - X) = univ.val\n[PROOFSTEP]\nhave : (roots (X ^ q - X : K[X])).toFinset = Finset.univ :=\n by\n rw [eq_univ_iff_forall]\n intro x\n rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X, sub_eq_zero, pow_card]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Algebra (ZMod p) K\naux : X ^ q - X \u2260 0\n\u22a2 Multiset.toFinset (roots (X ^ q - X)) = univ\n[PROOFSTEP]\nrw [eq_univ_iff_forall]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Algebra (ZMod p) K\naux : X ^ q - X \u2260 0\n\u22a2 \u2200 (x : K), x \u2208 Multiset.toFinset (roots (X ^ q - X))\n[PROOFSTEP]\nintro x\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Algebra (ZMod p) K\naux : X ^ q - X \u2260 0\nx : K\n\u22a2 x \u2208 Multiset.toFinset (roots (X ^ q - X))\n[PROOFSTEP]\nrw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X, sub_eq_zero, pow_card]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Algebra (ZMod p) K\naux : X ^ q - X \u2260 0\nthis : Multiset.toFinset (roots (X ^ q - X)) = univ\n\u22a2 roots (X ^ q - X) = univ.val\n[PROOFSTEP]\nrw [\u2190 this, Multiset.toFinset_val, eq_comm, Multiset.dedup_eq_self]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Algebra (ZMod p) K\naux : X ^ q - X \u2260 0\nthis : Multiset.toFinset (roots (X ^ q - X)) = univ\n\u22a2 Multiset.Nodup (roots (X ^ q - X))\n[PROOFSTEP]\napply nodup_roots\n[GOAL]\ncase hsep\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Algebra (ZMod p) K\naux : X ^ q - X \u2260 0\nthis : Multiset.toFinset (roots (X ^ q - X)) = univ\n\u22a2 Separable (X ^ q - X)\n[PROOFSTEP]\nrw [separable_def]\n[GOAL]\ncase hsep\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Algebra (ZMod p) K\naux : X ^ q - X \u2260 0\nthis : Multiset.toFinset (roots (X ^ q - X)) = univ\n\u22a2 IsCoprime (X ^ q - X) (\u2191derivative (X ^ q - X))\n[PROOFSTEP]\nconvert isCoprime_one_right.neg_right (R := K[X]) using 1\n[GOAL]\ncase h.e'_4\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Algebra (ZMod p) K\naux : X ^ q - X \u2260 0\nthis : Multiset.toFinset (roots (X ^ q - X)) = univ\n\u22a2 \u2191derivative (X ^ q - X) = -1\n[PROOFSTEP]\nrw [derivative_sub, derivative_X, derivative_X_pow, CharP.cast_card_eq_zero K, C_0, zero_mul, zero_sub]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\np\u271d : \u2115\ninst\u271d\u00b3 : Fact (Nat.Prime p\u271d)\ninst\u271d\u00b2 : Algebra (ZMod p\u271d) K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nn : \u2115\nhcard : q = p ^ n\n\u22a2 frobenius K p ^ n = 1\n[PROOFSTEP]\next x\n[GOAL]\ncase a\nK : Type u_1\nR : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\np\u271d : \u2115\ninst\u271d\u00b3 : Fact (Nat.Prime p\u271d)\ninst\u271d\u00b2 : Algebra (ZMod p\u271d) K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nn : \u2115\nhcard : q = p ^ n\nx : K\n\u22a2 \u2191(frobenius K p ^ n) x = \u21911 x\n[PROOFSTEP]\nconv_rhs => rw [RingHom.one_def, RingHom.id_apply, \u2190 pow_card x, hcard]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\np\u271d : \u2115\ninst\u271d\u00b3 : Fact (Nat.Prime p\u271d)\ninst\u271d\u00b2 : Algebra (ZMod p\u271d) K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nn : \u2115\nhcard : q = p ^ n\nx : K\n| \u21911 x\n[PROOFSTEP]\nrw [RingHom.one_def, RingHom.id_apply, \u2190 pow_card x, hcard]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\np\u271d : \u2115\ninst\u271d\u00b3 : Fact (Nat.Prime p\u271d)\ninst\u271d\u00b2 : Algebra (ZMod p\u271d) K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nn : \u2115\nhcard : q = p ^ n\nx : K\n| \u21911 x\n[PROOFSTEP]\nrw [RingHom.one_def, RingHom.id_apply, \u2190 pow_card x, hcard]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\np\u271d : \u2115\ninst\u271d\u00b3 : Fact (Nat.Prime p\u271d)\ninst\u271d\u00b2 : Algebra (ZMod p\u271d) K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nn : \u2115\nhcard : q = p ^ n\nx : K\n| \u21911 x\n[PROOFSTEP]\nrw [RingHom.one_def, RingHom.id_apply, \u2190 pow_card x, hcard]\n[GOAL]\ncase a\nK : Type u_1\nR : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\np\u271d : \u2115\ninst\u271d\u00b3 : Fact (Nat.Prime p\u271d)\ninst\u271d\u00b2 : Algebra (ZMod p\u271d) K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nn : \u2115\nhcard : q = p ^ n\nx : K\n\u22a2 \u2191(frobenius K p ^ n) x = x ^ p ^ n\n[PROOFSTEP]\nclear hcard\n[GOAL]\ncase a\nK : Type u_1\nR : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\np\u271d : \u2115\ninst\u271d\u00b3 : Fact (Nat.Prime p\u271d)\ninst\u271d\u00b2 : Algebra (ZMod p\u271d) K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nn : \u2115\nx : K\n\u22a2 \u2191(frobenius K p ^ n) x = x ^ p ^ n\n[PROOFSTEP]\ninduction' n with n hn\n[GOAL]\ncase a.zero\nK : Type u_1\nR : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\np\u271d : \u2115\ninst\u271d\u00b3 : Fact (Nat.Prime p\u271d)\ninst\u271d\u00b2 : Algebra (ZMod p\u271d) K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nx : K\n\u22a2 \u2191(frobenius K p ^ Nat.zero) x = x ^ p ^ Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.succ\nK : Type u_1\nR : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\np\u271d : \u2115\ninst\u271d\u00b3 : Fact (Nat.Prime p\u271d)\ninst\u271d\u00b2 : Algebra (ZMod p\u271d) K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nx : K\nn : \u2115\nhn : \u2191(frobenius K p ^ n) x = x ^ p ^ n\n\u22a2 \u2191(frobenius K p ^ Nat.succ n) x = x ^ p ^ Nat.succ n\n[PROOFSTEP]\nrw [pow_succ, pow_succ', pow_mul, RingHom.mul_def, RingHom.comp_apply, frobenius_def, hn]\n[GOAL]\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Algebra (ZMod p) K\nf : K[X]\n\u22a2 \u2191(expand K q) f = f ^ q\n[PROOFSTEP]\ncases' CharP.exists K with p hp\n[GOAL]\ncase intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np\u271d : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p\u271d)\ninst\u271d : Algebra (ZMod p\u271d) K\nf : K[X]\np : \u2115\nhp : CharP K p\n\u22a2 \u2191(expand K q) f = f ^ q\n[PROOFSTEP]\nletI := hp\n[GOAL]\ncase intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np\u271d : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p\u271d)\ninst\u271d : Algebra (ZMod p\u271d) K\nf : K[X]\np : \u2115\nhp : CharP K p\nthis : CharP K p := hp\n\u22a2 \u2191(expand K q) f = f ^ q\n[PROOFSTEP]\nrcases FiniteField.card K p with \u27e8\u27e8n, npos\u27e9, \u27e8hp, hn\u27e9\u27e9\n[GOAL]\ncase intro.intro.mk.intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np\u271d : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p\u271d)\ninst\u271d : Algebra (ZMod p\u271d) K\nf : K[X]\np : \u2115\nhp\u271d : CharP K p\nthis : CharP K p := hp\u271d\nn : \u2115\nnpos : 0 < n\nhp : Nat.Prime p\nhn : q = p ^ \u2191{ val := n, property := npos }\n\u22a2 \u2191(expand K q) f = f ^ q\n[PROOFSTEP]\nhaveI : Fact p.Prime := \u27e8hp\u27e9\n[GOAL]\ncase intro.intro.mk.intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np\u271d : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p\u271d)\ninst\u271d : Algebra (ZMod p\u271d) K\nf : K[X]\np : \u2115\nhp\u271d : CharP K p\nthis\u271d : CharP K p := hp\u271d\nn : \u2115\nnpos : 0 < n\nhp : Nat.Prime p\nhn : q = p ^ \u2191{ val := n, property := npos }\nthis : Fact (Nat.Prime p)\n\u22a2 \u2191(expand K q) f = f ^ q\n[PROOFSTEP]\ndsimp at hn \n[GOAL]\ncase intro.intro.mk.intro\nK : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\np\u271d : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p\u271d)\ninst\u271d : Algebra (ZMod p\u271d) K\nf : K[X]\np : \u2115\nhp\u271d : CharP K p\nthis\u271d : CharP K p := hp\u271d\nn : \u2115\nnpos : 0 < n\nhp : Nat.Prime p\nhn : q = p ^ n\nthis : Fact (Nat.Prime p)\n\u22a2 \u2191(expand K q) f = f ^ q\n[PROOFSTEP]\nrw [hn, \u2190 map_expand_pow_char, frobenius_pow hn, RingHom.one_def, map_id]\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\nhp : Fact (Nat.Prime p)\nx : ZMod p\n\u22a2 \u2203 a b, a ^ 2 + b ^ 2 = x\n[PROOFSTEP]\ncases' hp.1.eq_two_or_odd with hp2 hp_odd\n[GOAL]\ncase inl\nK : Type u_1\nR : Type u_2\np : \u2115\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp2 : p = 2\n\u22a2 \u2203 a b, a ^ 2 + b ^ 2 = x\n[PROOFSTEP]\nsubst p\n[GOAL]\ncase inl\nK : Type u_1\nR : Type u_2\nhp : Fact (Nat.Prime 2)\nx : ZMod 2\n\u22a2 \u2203 a b, a ^ 2 + b ^ 2 = x\n[PROOFSTEP]\nchange Fin 2 at x \n[GOAL]\ncase inl\nK : Type u_1\nR : Type u_2\nhp : Fact (Nat.Prime 2)\nx : Fin 2\n\u22a2 \u2203 a b, a ^ 2 + b ^ 2 = x\n[PROOFSTEP]\nfin_cases x\n[GOAL]\ncase inl.head\nK : Type u_1\nR : Type u_2\nhp : Fact (Nat.Prime 2)\n\u22a2 \u2203 a b, a ^ 2 + b ^ 2 = { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nK : Type u_1\nR : Type u_2\nhp : Fact (Nat.Prime 2)\n\u22a2 \u2203 b, 0 ^ 2 + b ^ 2 = { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.tail.head\nK : Type u_1\nR : Type u_2\nhp : Fact (Nat.Prime 2)\n\u22a2 \u2203 a b, a ^ 2 + b ^ 2 = { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nuse 0, 1\n[GOAL]\ncase h\nK : Type u_1\nR : Type u_2\nhp : Fact (Nat.Prime 2)\n\u22a2 0 ^ 2 + 1 ^ 2 = { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nK : Type u_1\nR : Type u_2\np : \u2115\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp_odd : p % 2 = 1\n\u22a2 \u2203 a b, a ^ 2 + b ^ 2 = x\n[PROOFSTEP]\nlet f : (ZMod p)[X] := X ^ 2\n[GOAL]\ncase inr\nK : Type u_1\nR : Type u_2\np : \u2115\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp_odd : p % 2 = 1\nf : (ZMod p)[X] := X ^ 2\n\u22a2 \u2203 a b, a ^ 2 + b ^ 2 = x\n[PROOFSTEP]\nlet g : (ZMod p)[X] := X ^ 2 - C x\n[GOAL]\ncase inr\nK : Type u_1\nR : Type u_2\np : \u2115\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp_odd : p % 2 = 1\nf : (ZMod p)[X] := X ^ 2\ng : (ZMod p)[X] := X ^ 2 - \u2191C x\n\u22a2 \u2203 a b, a ^ 2 + b ^ 2 = x\n[PROOFSTEP]\nobtain \u27e8a, b, hab\u27e9 : \u2203 a b, f.eval a + g.eval b = 0 :=\n @exists_root_sum_quadratic _ _ _ _ f g (degree_X_pow 2) (degree_X_pow_sub_C (by decide) _) (by rw [ZMod.card, hp_odd])\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp_odd : p % 2 = 1\nf : (ZMod p)[X] := X ^ 2\ng : (ZMod p)[X] := X ^ 2 - \u2191C x\n\u22a2 0 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp_odd : p % 2 = 1\nf : (ZMod p)[X] := X ^ 2\ng : (ZMod p)[X] := X ^ 2 - \u2191C x\n\u22a2 Fintype.card (ZMod p) % 2 = 1\n[PROOFSTEP]\nrw [ZMod.card, hp_odd]\n[GOAL]\ncase inr.intro.intro\nK : Type u_1\nR : Type u_2\np : \u2115\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp_odd : p % 2 = 1\nf : (ZMod p)[X] := X ^ 2\ng : (ZMod p)[X] := X ^ 2 - \u2191C x\na b : ZMod p\nhab : eval a f + eval b g = 0\n\u22a2 \u2203 a b, a ^ 2 + b ^ 2 = x\n[PROOFSTEP]\nrefine' \u27e8a, b, _\u27e9\n[GOAL]\ncase inr.intro.intro\nK : Type u_1\nR : Type u_2\np : \u2115\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp_odd : p % 2 = 1\nf : (ZMod p)[X] := X ^ 2\ng : (ZMod p)[X] := X ^ 2 - \u2191C x\na b : ZMod p\nhab : eval a f + eval b g = 0\n\u22a2 a ^ 2 + b ^ 2 = x\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero]\n[GOAL]\ncase inr.intro.intro\nK : Type u_1\nR : Type u_2\np : \u2115\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp_odd : p % 2 = 1\nf : (ZMod p)[X] := X ^ 2\ng : (ZMod p)[X] := X ^ 2 - \u2191C x\na b : ZMod p\nhab : eval a f + eval b g = 0\n\u22a2 a ^ 2 + b ^ 2 - x = 0\n[PROOFSTEP]\nsimpa only [eval_C, eval_X, eval_pow, eval_sub, \u2190 add_sub_assoc] using hab\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Prime p)\nx : \u2124\n\u22a2 \u2203 a b, a \u2264 p / 2 \u2227 b \u2264 p / 2 \u2227 \u2191a ^ 2 + \u2191b ^ 2 \u2261 x [ZMOD \u2191p]\n[PROOFSTEP]\nrcases ZMod.sq_add_sq p x with \u27e8a, b, hx\u27e9\n[GOAL]\ncase intro.intro\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Prime p)\nx : \u2124\na b : ZMod p\nhx : a ^ 2 + b ^ 2 = \u2191x\n\u22a2 \u2203 a b, a \u2264 p / 2 \u2227 b \u2264 p / 2 \u2227 \u2191a ^ 2 + \u2191b ^ 2 \u2261 x [ZMOD \u2191p]\n[PROOFSTEP]\nrefine \u27e8a.valMinAbs.natAbs, b.valMinAbs.natAbs, ZMod.natAbs_valMinAbs_le _, ZMod.natAbs_valMinAbs_le _, ?_\u27e9\n[GOAL]\ncase intro.intro\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Prime p)\nx : \u2124\na b : ZMod p\nhx : a ^ 2 + b ^ 2 = \u2191x\n\u22a2 \u2191(Int.natAbs (ZMod.valMinAbs a)) ^ 2 + \u2191(Int.natAbs (ZMod.valMinAbs b)) ^ 2 \u2261 x [ZMOD \u2191p]\n[PROOFSTEP]\nrw [\u2190 a.coe_valMinAbs, \u2190 b.coe_valMinAbs] at hx \n[GOAL]\ncase intro.intro\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Prime p)\nx : \u2124\na b : ZMod p\nhx : \u2191(ZMod.valMinAbs a) ^ 2 + \u2191(ZMod.valMinAbs b) ^ 2 = \u2191x\n\u22a2 \u2191(Int.natAbs (ZMod.valMinAbs a)) ^ 2 + \u2191(Int.natAbs (ZMod.valMinAbs b)) ^ 2 \u2261 x [ZMOD \u2191p]\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase intro.intro\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Prime p)\nx : \u2124\na b : ZMod p\nhx : \u2191(ZMod.valMinAbs a) ^ 2 + \u2191(ZMod.valMinAbs b) ^ 2 = \u2191x\n\u22a2 |ZMod.valMinAbs a| ^ 2 + |ZMod.valMinAbs b| ^ 2 \u2261 x [ZMOD \u2191p]\n[PROOFSTEP]\nrw [sq_abs, sq_abs, \u2190 ZMod.int_cast_eq_int_cast_iff]\n[GOAL]\ncase intro.intro\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Prime p)\nx : \u2124\na b : ZMod p\nhx : \u2191(ZMod.valMinAbs a) ^ 2 + \u2191(ZMod.valMinAbs b) ^ 2 = \u2191x\n\u22a2 \u2191(ZMod.valMinAbs a ^ 2 + ZMod.valMinAbs b ^ 2) = \u2191x\n[PROOFSTEP]\nexact_mod_cast hx\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Prime p)\nx : \u2115\n\u22a2 \u2203 a b, a \u2264 p / 2 \u2227 b \u2264 p / 2 \u2227 a ^ 2 + b ^ 2 \u2261 x [MOD p]\n[PROOFSTEP]\nsimpa only [\u2190 Int.coe_nat_modEq_iff] using Nat.sq_add_sq_zmodEq p x\n[GOAL]\nK : Type u_1\nR\u271d : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\np : \u2115\ninst\u271d\u00b9 : NeZero p\ninst\u271d : CharP R p\nx : \u2124\n\u22a2 \u2203 a b, \u2191a ^ 2 + \u2191b ^ 2 = \u2191x\n[PROOFSTEP]\nhaveI := char_is_prime_of_pos R p\n[GOAL]\nK : Type u_1\nR\u271d : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\np : \u2115\ninst\u271d\u00b9 : NeZero p\ninst\u271d : CharP R p\nx : \u2124\nthis : Fact (Nat.Prime p)\n\u22a2 \u2203 a b, \u2191a ^ 2 + \u2191b ^ 2 = \u2191x\n[PROOFSTEP]\nobtain \u27e8a, b, hab\u27e9 := ZMod.sq_add_sq p x\n[GOAL]\ncase intro.intro\nK : Type u_1\nR\u271d : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\np : \u2115\ninst\u271d\u00b9 : NeZero p\ninst\u271d : CharP R p\nx : \u2124\nthis : Fact (Nat.Prime p)\na b : ZMod p\nhab : a ^ 2 + b ^ 2 = \u2191x\n\u22a2 \u2203 a b, \u2191a ^ 2 + \u2191b ^ 2 = \u2191x\n[PROOFSTEP]\nrefine' \u27e8a.val, b.val, _\u27e9\n[GOAL]\ncase intro.intro\nK : Type u_1\nR\u271d : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\np : \u2115\ninst\u271d\u00b9 : NeZero p\ninst\u271d : CharP R p\nx : \u2124\nthis : Fact (Nat.Prime p)\na b : ZMod p\nhab : a ^ 2 + b ^ 2 = \u2191x\n\u22a2 \u2191(ZMod.val a) ^ 2 + \u2191(ZMod.val b) ^ 2 = \u2191x\n[PROOFSTEP]\nsimpa using congr_arg (ZMod.castHom dvd_rfl R) hab\n[GOAL]\nK : Type u_1\nR : Type u_2\nn : \u2115\nx : (ZMod n)\u02e3\n\u22a2 x ^ \u03c6 n = 1\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nK : Type u_1\nR : Type u_2\nx : (ZMod Nat.zero)\u02e3\n\u22a2 x ^ \u03c6 Nat.zero = 1\n[PROOFSTEP]\nrw [Nat.totient_zero, pow_zero]\n[GOAL]\ncase succ\nK : Type u_1\nR : Type u_2\nn\u271d : \u2115\nx : (ZMod (Nat.succ n\u271d))\u02e3\n\u22a2 x ^ \u03c6 (Nat.succ n\u271d) = 1\n[PROOFSTEP]\nrw [\u2190 card_units_eq_totient, pow_card_eq_one]\n[GOAL]\nK : Type u_1\nR : Type u_2\nx n : \u2115\nh : coprime x n\n\u22a2 x ^ \u03c6 n \u2261 1 [MOD n]\n[PROOFSTEP]\nrw [\u2190 ZMod.eq_iff_modEq_nat]\n[GOAL]\nK : Type u_1\nR : Type u_2\nx n : \u2115\nh : coprime x n\n\u22a2 \u2191(x ^ \u03c6 n) = \u21911\n[PROOFSTEP]\nlet x' : Units (ZMod n) := ZMod.unitOfCoprime _ h\n[GOAL]\nK : Type u_1\nR : Type u_2\nx n : \u2115\nh : coprime x n\nx' : (ZMod n)\u02e3 := unitOfCoprime x h\n\u22a2 \u2191(x ^ \u03c6 n) = \u21911\n[PROOFSTEP]\nhave := ZMod.pow_totient x'\n[GOAL]\nK : Type u_1\nR : Type u_2\nx n : \u2115\nh : coprime x n\nx' : (ZMod n)\u02e3 := unitOfCoprime x h\nthis : x' ^ \u03c6 n = 1\n\u22a2 \u2191(x ^ \u03c6 n) = \u21911\n[PROOFSTEP]\napply_fun ((fun (x : Units (ZMod n)) => (x : ZMod n)) : Units (ZMod n) \u2192 ZMod n) at this \n[GOAL]\nK : Type u_1\nR : Type u_2\nx n : \u2115\nh : coprime x n\nx' : (ZMod n)\u02e3 := unitOfCoprime x h\nthis : \u2191(x' ^ \u03c6 n) = \u21911\n\u22a2 \u2191(x ^ \u03c6 n) = \u21911\n[PROOFSTEP]\nsimpa only [Nat.succ_eq_add_one, Nat.cast_pow, Units.val_one, Nat.cast_one, coe_unitOfCoprime, Units.val_pow_eq_pow_val]\n[GOAL]\nK : Type u_1\nR : Type u_2\nV : Type u_3\ninst\u271d\u2074 : Fintype K\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : Fintype V\n\u22a2 Fintype.card V = q ^ FiniteDimensional.finrank K V\n[PROOFSTEP]\nlet b := IsNoetherian.finsetBasis K V\n[GOAL]\nK : Type u_1\nR : Type u_2\nV : Type u_3\ninst\u271d\u2074 : Fintype K\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : Fintype V\nb : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex K V } K V := IsNoetherian.finsetBasis K V\n\u22a2 Fintype.card V = q ^ FiniteDimensional.finrank K V\n[PROOFSTEP]\nrw [Module.card_fintype b, \u2190 FiniteDimensional.finrank_eq_card_basis b]\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx : ZMod p\n\u22a2 x ^ p = x\n[PROOFSTEP]\nhave h := FiniteField.pow_card x\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx : ZMod p\nh : x ^ Fintype.card (ZMod p) = x\n\u22a2 x ^ p = x\n[PROOFSTEP]\nrwa [ZMod.card p] at h \n[GOAL]\nK : Type u_1\nR : Type u_2\nn p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx : ZMod p\n\u22a2 x ^ p ^ n = x\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx : ZMod p\n\u22a2 x ^ p ^ Nat.zero = x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx : ZMod p\nn : \u2115\nih : x ^ p ^ n = x\n\u22a2 x ^ p ^ Nat.succ n = x\n[PROOFSTEP]\nsimp [pow_succ, pow_mul, ih, pow_card]\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 frobenius (ZMod p) p = RingHom.id (ZMod p)\n[PROOFSTEP]\next a\n[GOAL]\ncase a\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\n\u22a2 \u2191(frobenius (ZMod p) p) a = \u2191(RingHom.id (ZMod p)) a\n[PROOFSTEP]\nrw [frobenius_def, ZMod.pow_card, RingHom.id_apply]\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 Fintype.card (ZMod p)\u02e3 = p - 1\n[PROOFSTEP]\nrw [Fintype.card_units, card]\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : (ZMod p)\u02e3\n\u22a2 a ^ (p - 1) = 1\n[PROOFSTEP]\nrw [\u2190 card_units p, pow_card_eq_one]\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\n\u22a2 a ^ (p - 1) = 1\n[PROOFSTEP]\nhave h := FiniteField.pow_card_sub_one_eq_one a ha\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\nh : a ^ (Fintype.card (ZMod p) - 1) = 1\n\u22a2 a ^ (p - 1) = 1\n[PROOFSTEP]\nrwa [ZMod.card p] at h \n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : (ZMod p)[X]\n\u22a2 \u2191(expand (ZMod p) p) f = f ^ p\n[PROOFSTEP]\nhave h := FiniteField.expand_card f\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : (ZMod p)[X]\nh : \u2191(expand (ZMod p) (Fintype.card (ZMod p))) f = f ^ Fintype.card (ZMod p)\n\u22a2 \u2191(expand (ZMod p) p) f = f ^ p\n[PROOFSTEP]\nrwa [ZMod.card p] at h \n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\nhp : Nat.Prime p\nn : \u2124\nhpn : IsCoprime n \u2191p\n\u22a2 n ^ (p - 1) \u2261 1 [ZMOD \u2191p]\n[PROOFSTEP]\nhaveI : Fact p.Prime := \u27e8hp\u27e9\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\nhp : Nat.Prime p\nn : \u2124\nhpn : IsCoprime n \u2191p\nthis : Fact (Nat.Prime p)\n\u22a2 n ^ (p - 1) \u2261 1 [ZMOD \u2191p]\n[PROOFSTEP]\nhave : \u00ac(n : ZMod p) = 0 :=\n by\n rw [CharP.int_cast_eq_zero_iff _ p, \u2190 (Nat.prime_iff_prime_int.mp hp).coprime_iff_not_dvd]\n \u00b7 exact hpn.symm\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\nhp : Nat.Prime p\nn : \u2124\nhpn : IsCoprime n \u2191p\nthis : Fact (Nat.Prime p)\n\u22a2 \u00ac\u2191n = 0\n[PROOFSTEP]\nrw [CharP.int_cast_eq_zero_iff _ p, \u2190 (Nat.prime_iff_prime_int.mp hp).coprime_iff_not_dvd]\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\nhp : Nat.Prime p\nn : \u2124\nhpn : IsCoprime n \u2191p\nthis : Fact (Nat.Prime p)\n\u22a2 IsCoprime (\u2191p) n\n[PROOFSTEP]\nexact hpn.symm\n[GOAL]\nK : Type u_1\nR : Type u_2\np : \u2115\nhp : Nat.Prime p\nn : \u2124\nhpn : IsCoprime n \u2191p\nthis\u271d : Fact (Nat.Prime p)\nthis : \u00ac\u2191n = 0\n\u22a2 n ^ (p - 1) \u2261 1 [ZMOD \u2191p]\n[PROOFSTEP]\nsimpa [\u2190 ZMod.int_cast_eq_int_cast_iff] using ZMod.pow_card_sub_one_eq_one this\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\n\u22a2 \u2203 a, \u00acIsSquare a\n[PROOFSTEP]\nlet sq : F \u2192 F := fun x => x ^ 2\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\n\u22a2 \u2203 a, \u00acIsSquare a\n[PROOFSTEP]\nhave h : \u00acFunction.Injective sq :=\n by\n simp only [Function.Injective, not_forall, exists_prop]\n refine' \u27e8-1, 1, _, Ring.neg_one_ne_one_of_char_ne_two hF\u27e9\n simp only [one_pow, neg_one_sq]\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\n\u22a2 \u00acFunction.Injective sq\n[PROOFSTEP]\nsimp only [Function.Injective, not_forall, exists_prop]\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\n\u22a2 \u2203 x x_1, x ^ 2 = x_1 ^ 2 \u2227 \u00acx = x_1\n[PROOFSTEP]\nrefine' \u27e8-1, 1, _, Ring.neg_one_ne_one_of_char_ne_two hF\u27e9\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\n\u22a2 (-1) ^ 2 = 1 ^ 2\n[PROOFSTEP]\nsimp only [one_pow, neg_one_sq]\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\nh : \u00acFunction.Injective sq\n\u22a2 \u2203 a, \u00acIsSquare a\n[PROOFSTEP]\nrw [Finite.injective_iff_surjective] at h \n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\nh : \u00acFunction.Surjective sq\n\u22a2 \u2203 a, \u00acIsSquare a\n[PROOFSTEP]\nsimp_rw [IsSquare, \u2190 pow_two, @eq_comm _ _ (_ ^ 2)]\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\nh : \u00acFunction.Surjective sq\n\u22a2 \u2203 a, \u00ac\u2203 r, r ^ 2 = a\n[PROOFSTEP]\nunfold Function.Surjective at h \n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\nh : \u00ac\u2200 (b : F), \u2203 a, sq a = b\n\u22a2 \u2203 a, \u00ac\u2203 r, r ^ 2 = a\n[PROOFSTEP]\npush_neg at h \u22a2\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\nh : \u2203 b, \u2200 (a : F), (fun x => x ^ 2) a \u2260 b\n\u22a2 \u2203 a, \u2200 (r : F), r ^ 2 \u2260 a\n[PROOFSTEP]\nexact h\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\n\u22a2 ringChar F = 2 \u2194 Fintype.card F % 2 = 0\n[PROOFSTEP]\nrcases FiniteField.card F (ringChar F) with \u27e8n, hp, h\u27e9\n[GOAL]\ncase intro.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nn : \u2115+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ \u2191n\n\u22a2 ringChar F = 2 \u2194 Fintype.card F % 2 = 0\n[PROOFSTEP]\nrw [h, Nat.pow_mod]\n[GOAL]\ncase intro.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nn : \u2115+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ \u2191n\n\u22a2 ringChar F = 2 \u2194 (ringChar F % 2) ^ \u2191n % 2 = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.mp\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nn : \u2115+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ \u2191n\n\u22a2 ringChar F = 2 \u2192 (ringChar F % 2) ^ \u2191n % 2 = 0\n[PROOFSTEP]\nintro hF\n[GOAL]\ncase intro.intro.mp\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nn : \u2115+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ \u2191n\nhF : ringChar F = 2\n\u22a2 (ringChar F % 2) ^ \u2191n % 2 = 0\n[PROOFSTEP]\nsimp [hF]\n[GOAL]\ncase intro.intro.mpr\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nn : \u2115+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ \u2191n\n\u22a2 (ringChar F % 2) ^ \u2191n % 2 = 0 \u2192 ringChar F = 2\n[PROOFSTEP]\nrw [\u2190 Nat.even_iff, Nat.even_pow]\n[GOAL]\ncase intro.intro.mpr\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nn : \u2115+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ \u2191n\n\u22a2 Even (ringChar F % 2) \u2227 \u2191n \u2260 0 \u2192 ringChar F = 2\n[PROOFSTEP]\nrintro \u27e8hev, hnz\u27e9\n[GOAL]\ncase intro.intro.mpr.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nn : \u2115+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ \u2191n\nhev : Even (ringChar F % 2)\nhnz : \u2191n \u2260 0\n\u22a2 ringChar F = 2\n[PROOFSTEP]\nrw [Nat.even_iff, Nat.mod_mod] at hev \n[GOAL]\ncase intro.intro.mpr.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nn : \u2115+\nhp : Nat.Prime (ringChar F)\nh : Fintype.card F = ringChar F ^ \u2191n\nhev : ringChar F % 2 = 0\nhnz : \u2191n \u2260 0\n\u22a2 ringChar F = 2\n[PROOFSTEP]\nexact (Nat.Prime.eq_two_or_odd hp).resolve_right (ne_of_eq_of_ne hev zero_ne_one)\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 a ^ (Fintype.card F / 2) = 1 \u2228 a ^ (Fintype.card F / 2) = -1\n[PROOFSTEP]\nhave h\u2081 := FiniteField.pow_card_sub_one_eq_one a ha\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\nh\u2081 : a ^ (Fintype.card F - 1) = 1\n\u22a2 a ^ (Fintype.card F / 2) = 1 \u2228 a ^ (Fintype.card F / 2) = -1\n[PROOFSTEP]\nrw [\u2190 Nat.two_mul_odd_div_two (FiniteField.odd_card_of_char_ne_two hF), mul_comm, pow_mul, pow_two] at h\u2081 \n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\nh\u2081 : a ^ (Fintype.card F / 2) * a ^ (Fintype.card F / 2) = 1\n\u22a2 a ^ (Fintype.card F / 2) = 1 \u2228 a ^ (Fintype.card F / 2) = -1\n[PROOFSTEP]\nexact mul_self_eq_one_iff.mp h\u2081\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\u02e3\n\u22a2 IsSquare a \u2194 a ^ (Fintype.card F / 2) = 1\n[PROOFSTEP]\nclassical\nobtain \u27e8g, hg\u27e9 := IsCyclic.exists_generator (\u03b1 := F\u02e3)\nobtain \u27e8n, hn\u27e9 : a \u2208 Submonoid.powers g := by rw [mem_powers_iff_mem_zpowers]; apply hg\nhave hodd := Nat.two_mul_odd_div_two (FiniteField.odd_card_of_char_ne_two hF)\nconstructor\n\u00b7 rintro \u27e8y, rfl\u27e9\n rw [\u2190 pow_two, \u2190 pow_mul, hodd]\n apply_fun Units.val using Units.ext (\u03b1 := F)\n \u00b7 push_cast\n exact FiniteField.pow_card_sub_one_eq_one (y : F) (Units.ne_zero y)\n\u00b7 subst a; intro h\n have key : 2 * (Fintype.card F / 2) \u2223 n * (Fintype.card F / 2) :=\n by\n rw [\u2190 pow_mul] at h \n rw [hodd, \u2190 Fintype.card_units, \u2190 orderOf_eq_card_of_forall_mem_zpowers hg]\n apply orderOf_dvd_of_pow_eq_one h\n have : 0 < Fintype.card F / 2 := Nat.div_pos Fintype.one_lt_card (by norm_num)\n obtain \u27e8m, rfl\u27e9 := Nat.dvd_of_mul_dvd_mul_right this key\n refine' \u27e8g ^ m, _\u27e9\n dsimp\n rw [mul_comm, pow_mul, pow_two]\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\u02e3\n\u22a2 IsSquare a \u2194 a ^ (Fintype.card F / 2) = 1\n[PROOFSTEP]\nobtain \u27e8g, hg\u27e9 := IsCyclic.exists_generator (\u03b1 := F\u02e3)\n[GOAL]\ncase intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na g : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\n\u22a2 IsSquare a \u2194 a ^ (Fintype.card F / 2) = 1\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : a \u2208 Submonoid.powers g := by rw [mem_powers_iff_mem_zpowers]; apply hg\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na g : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\n\u22a2 a \u2208 Submonoid.powers g\n[PROOFSTEP]\nrw [mem_powers_iff_mem_zpowers]\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na g : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\n\u22a2 a \u2208 Subgroup.zpowers g\n[PROOFSTEP]\napply hg\n[GOAL]\ncase intro.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na g : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhn : (fun x x_1 => x ^ x_1) g n = a\n\u22a2 IsSquare a \u2194 a ^ (Fintype.card F / 2) = 1\n[PROOFSTEP]\nhave hodd := Nat.two_mul_odd_div_two (FiniteField.odd_card_of_char_ne_two hF)\n[GOAL]\ncase intro.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na g : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhn : (fun x x_1 => x ^ x_1) g n = a\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\n\u22a2 IsSquare a \u2194 a ^ (Fintype.card F / 2) = 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.mp\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na g : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhn : (fun x x_1 => x ^ x_1) g n = a\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\n\u22a2 IsSquare a \u2192 a ^ (Fintype.card F / 2) = 1\n[PROOFSTEP]\nrintro \u27e8y, rfl\u27e9\n[GOAL]\ncase intro.intro.mp.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ng : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\ny : F\u02e3\nhn : (fun x x_1 => x ^ x_1) g n = y * y\n\u22a2 (y * y) ^ (Fintype.card F / 2) = 1\n[PROOFSTEP]\nrw [\u2190 pow_two, \u2190 pow_mul, hodd]\n[GOAL]\ncase intro.intro.mp.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ng : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\ny : F\u02e3\nhn : (fun x x_1 => x ^ x_1) g n = y * y\n\u22a2 y ^ (Fintype.card F - 1) = 1\n[PROOFSTEP]\napply_fun Units.val using Units.ext (\u03b1 := F)\n[GOAL]\ncase intro.intro.mp.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ng : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\ny : F\u02e3\nhn : (fun x x_1 => x ^ x_1) g n = y * y\n\u22a2 \u2191(y ^ (Fintype.card F - 1)) = \u21911\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase intro.intro.mp.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ng : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\ny : F\u02e3\nhn : (fun x x_1 => x ^ x_1) g n = y * y\n\u22a2 \u2191y ^ (Fintype.card F - 1) = 1\n[PROOFSTEP]\nexact FiniteField.pow_card_sub_one_eq_one (y : F) (Units.ne_zero y)\n[GOAL]\ncase intro.intro.mpr\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na g : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhn : (fun x x_1 => x ^ x_1) g n = a\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\n\u22a2 a ^ (Fintype.card F / 2) = 1 \u2192 IsSquare a\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase intro.intro.mpr\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ng : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\n\u22a2 (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1 \u2192 IsSquare ((fun x x_1 => x ^ x_1) g n)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase intro.intro.mpr\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ng : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1\n\u22a2 IsSquare ((fun x x_1 => x ^ x_1) g n)\n[PROOFSTEP]\nhave key : 2 * (Fintype.card F / 2) \u2223 n * (Fintype.card F / 2) :=\n by\n rw [\u2190 pow_mul] at h \n rw [hodd, \u2190 Fintype.card_units, \u2190 orderOf_eq_card_of_forall_mem_zpowers hg]\n apply orderOf_dvd_of_pow_eq_one h\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ng : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1\n\u22a2 2 * (Fintype.card F / 2) \u2223 n * (Fintype.card F / 2)\n[PROOFSTEP]\nrw [\u2190 pow_mul] at h \n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ng : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : g ^ (n * (Fintype.card F / 2)) = 1\n\u22a2 2 * (Fintype.card F / 2) \u2223 n * (Fintype.card F / 2)\n[PROOFSTEP]\nrw [hodd, \u2190 Fintype.card_units, \u2190 orderOf_eq_card_of_forall_mem_zpowers hg]\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ng : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : g ^ (n * (Fintype.card F / 2)) = 1\n\u22a2 orderOf g \u2223 n * (Fintype.card F / 2)\n[PROOFSTEP]\napply orderOf_dvd_of_pow_eq_one h\n[GOAL]\ncase intro.intro.mpr\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ng : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) \u2223 n * (Fintype.card F / 2)\n\u22a2 IsSquare ((fun x x_1 => x ^ x_1) g n)\n[PROOFSTEP]\nhave : 0 < Fintype.card F / 2 := Nat.div_pos Fintype.one_lt_card (by norm_num)\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ng : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) \u2223 n * (Fintype.card F / 2)\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.intro.mpr\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ng : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nn : \u2115\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nh : (fun x x_1 => x ^ x_1) g n ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) \u2223 n * (Fintype.card F / 2)\nthis : 0 < Fintype.card F / 2\n\u22a2 IsSquare ((fun x x_1 => x ^ x_1) g n)\n[PROOFSTEP]\nobtain \u27e8m, rfl\u27e9 := Nat.dvd_of_mul_dvd_mul_right this key\n[GOAL]\ncase intro.intro.mpr.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ng : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nthis : 0 < Fintype.card F / 2\nm : \u2115\nh : (fun x x_1 => x ^ x_1) g (2 * m) ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) \u2223 2 * m * (Fintype.card F / 2)\n\u22a2 IsSquare ((fun x x_1 => x ^ x_1) g (2 * m))\n[PROOFSTEP]\nrefine' \u27e8g ^ m, _\u27e9\n[GOAL]\ncase intro.intro.mpr.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ng : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nthis : 0 < Fintype.card F / 2\nm : \u2115\nh : (fun x x_1 => x ^ x_1) g (2 * m) ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) \u2223 2 * m * (Fintype.card F / 2)\n\u22a2 (fun x x_1 => x ^ x_1) g (2 * m) = g ^ m * g ^ m\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro.mpr.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ng : F\u02e3\nhg : \u2200 (x : F\u02e3), x \u2208 Subgroup.zpowers g\nhodd : 2 * (Fintype.card F / 2) = Fintype.card F - 1\nthis : 0 < Fintype.card F / 2\nm : \u2115\nh : (fun x x_1 => x ^ x_1) g (2 * m) ^ (Fintype.card F / 2) = 1\nkey : 2 * (Fintype.card F / 2) \u2223 2 * m * (Fintype.card F / 2)\n\u22a2 g ^ (2 * m) = g ^ m * g ^ m\n[PROOFSTEP]\nrw [mul_comm, pow_mul, pow_two]\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 IsSquare a \u2194 a ^ (Fintype.card F / 2) = 1\n[PROOFSTEP]\napply (iff_congr _ (by simp [Units.ext_iff])).mp (FiniteField.unit_isSquare_iff hF (Units.mk0 a ha))\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 Units.mk0 a ha ^ (Fintype.card F / 2) = 1 \u2194 a ^ (Fintype.card F / 2) = 1\n[PROOFSTEP]\nsimp [Units.ext_iff]\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 IsSquare (Units.mk0 a ha) \u2194 IsSquare a\n[PROOFSTEP]\nsimp only [IsSquare, Units.ext_iff, Units.val_mk0, Units.val_mul]\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 (\u2203 r, a = \u2191r * \u2191r) \u2194 \u2203 r, a = r * r\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 (\u2203 r, a = \u2191r * \u2191r) \u2192 \u2203 r, a = r * r\n[PROOFSTEP]\nrintro \u27e8y, hy\u27e9\n[GOAL]\ncase mp.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\ny : F\u02e3\nhy : a = \u2191y * \u2191y\n\u22a2 \u2203 r, a = r * r\n[PROOFSTEP]\nexact \u27e8y, hy\u27e9\n[GOAL]\ncase mpr\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\na : F\nha : a \u2260 0\n\u22a2 (\u2203 r, a = r * r) \u2192 \u2203 r, a = \u2191r * \u2191r\n[PROOFSTEP]\nrintro \u27e8y, rfl\u27e9\n[GOAL]\ncase mpr.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ny : F\nha : y * y \u2260 0\n\u22a2 \u2203 r, y * y = \u2191r * \u2191r\n[PROOFSTEP]\nhave hy : y \u2260 0 := by rintro rfl; simp at ha \n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ny : F\nha : y * y \u2260 0\n\u22a2 y \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\nha : 0 * 0 \u2260 0\n\u22a2 False\n[PROOFSTEP]\nsimp at ha \n[GOAL]\ncase mpr.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ny : F\nha : y * y \u2260 0\nhy : y \u2260 0\n\u22a2 \u2203 r, y * y = \u2191r * \u2191r\n[PROOFSTEP]\nrefine' \u27e8Units.mk0 y hy, _\u27e9\n[GOAL]\ncase mpr.intro\nK : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F \u2260 2\ny : F\nha : y * y \u2260 0\nhy : y \u2260 0\n\u22a2 y * y = \u2191(Units.mk0 y hy) * \u2191(Units.mk0 y hy)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.FieldTheory.Finite.Basic", "llama_tokens": 37279, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.41325932684134725}} {"text": "[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\n\u22a2 IsReflexivePair (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\n[PROOFSTEP]\napply IsReflexivePair.mk' ((F).map ((adj).unit.app _)) _ _\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\n\u22a2 F.map (NatTrans.app adj.unit A.A) \u226b F.map A.a = \ud835\udfd9 (F.obj ((\ud835\udfed C).obj A.A))\n[PROOFSTEP]\nrw [\u2190 (F).map_comp, \u2190 (F).map_id]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\n\u22a2 F.map (NatTrans.app adj.unit A.A \u226b A.a) = F.map (\ud835\udfd9 ((\ud835\udfed C).obj A.A))\n[PROOFSTEP]\nexact congr_arg (F).map A.unit\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\n\u22a2 F.map (NatTrans.app adj.unit A.A) \u226b NatTrans.app adj.counit (F.obj A.A) = \ud835\udfd9 (F.obj ((\ud835\udfed C).obj A.A))\n[PROOFSTEP]\nrw [(adj).left_triangle_components]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\n\u22a2 \ud835\udfd9 (F.obj A.A) = \ud835\udfd9 (F.obj ((\ud835\udfed C).obj A.A))\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\nB : D\ninst\u271d : HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\n\u22a2 { f // F.map A.a \u226b f = NatTrans.app adj.counit (F.obj A.A) \u226b f } \u2243\n { g // G.map (F.map g) \u226b G.map (NatTrans.app adj.counit B) = A.a \u226b g }\n[PROOFSTEP]\nrefine' ((adj).homEquiv _ _).subtypeEquiv _\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\nB : D\ninst\u271d : HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\n\u22a2 \u2200 (a : F.obj A.A \u27f6 B),\n F.map A.a \u226b a = NatTrans.app adj.counit (F.obj A.A) \u226b a \u2194\n G.map (F.map (\u2191(Adjunction.homEquiv adj A.A B) a)) \u226b G.map (NatTrans.app adj.counit B) =\n A.a \u226b \u2191(Adjunction.homEquiv adj A.A B) a\n[PROOFSTEP]\nintro f\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\nB : D\ninst\u271d : HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nf : F.obj A.A \u27f6 B\n\u22a2 F.map A.a \u226b f = NatTrans.app adj.counit (F.obj A.A) \u226b f \u2194\n G.map (F.map (\u2191(Adjunction.homEquiv adj A.A B) f)) \u226b G.map (NatTrans.app adj.counit B) =\n A.a \u226b \u2191(Adjunction.homEquiv adj A.A B) f\n[PROOFSTEP]\nrw [\u2190 ((adj).homEquiv _ _).injective.eq_iff, Adjunction.homEquiv_naturality_left, (adj).homEquiv_unit,\n (adj).homEquiv_unit, G.map_comp]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\nB : D\ninst\u271d : HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nf : F.obj A.A \u27f6 B\n\u22a2 A.a \u226b NatTrans.app adj.unit A.A \u226b G.map f =\n NatTrans.app adj.unit ((Adjunction.toMonad adj).toFunctor.obj A.A) \u226b\n G.map (NatTrans.app adj.counit (F.obj A.A)) \u226b G.map f \u2194\n G.map (F.map (NatTrans.app adj.unit A.A \u226b G.map f)) \u226b G.map (NatTrans.app adj.counit B) =\n A.a \u226b NatTrans.app adj.unit A.A \u226b G.map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\nB : D\ninst\u271d : HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nf : F.obj A.A \u27f6 B\n\u22a2 A.a \u226b NatTrans.app adj.unit A.A \u226b G.map f =\n NatTrans.app adj.unit (G.obj (F.obj A.A)) \u226b G.map (NatTrans.app adj.counit (F.obj A.A)) \u226b G.map f \u2194\n G.map (F.map (NatTrans.app adj.unit A.A \u226b G.map f)) \u226b G.map (NatTrans.app adj.counit B) =\n A.a \u226b NatTrans.app adj.unit A.A \u226b G.map f\n[PROOFSTEP]\nrw [(adj).right_triangle_components_assoc, \u2190 G.map_comp, (F).map_comp, Category.assoc, (adj).counit_naturality,\n (adj).left_triangle_components_assoc]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\nB : D\ninst\u271d : HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nf : F.obj A.A \u27f6 B\n\u22a2 A.a \u226b NatTrans.app adj.unit A.A \u226b G.map f = G.map f \u2194 G.map f = A.a \u226b NatTrans.app adj.unit A.A \u226b G.map f\n[PROOFSTEP]\napply eq_comm\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\nB : D\ninst\u271d : HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\ng : { g // G.map (F.map g) \u226b G.map (NatTrans.app adj.counit B) = A.a \u226b g }\n\u22a2 (fun f =>\n { val := f.f,\n property := (_ : (Adjunction.toMonad adj).toFunctor.map f.f \u226b ((comparison adj).obj B).a = A.a \u226b f.f) })\n ((fun g => Algebra.Hom.mk \u2191g) g) =\n g\n[PROOFSTEP]\next\n[GOAL]\ncase a\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\nB : D\ninst\u271d : HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\ng : { g // G.map (F.map g) \u226b G.map (NatTrans.app adj.counit B) = A.a \u226b g }\n\u22a2 \u2191((fun f =>\n { val := f.f,\n property := (_ : (Adjunction.toMonad adj).toFunctor.map f.f \u226b ((comparison adj).obj B).a = A.a \u226b f.f) })\n ((fun g => Algebra.Hom.mk \u2191g) g)) =\n \u2191g\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\nB : D\ninst\u271d : HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nf : A \u27f6 (comparison adj).obj B\n\u22a2 (fun g => Algebra.Hom.mk \u2191g)\n ((fun f =>\n { val := f.f,\n property := (_ : (Adjunction.toMonad adj).toFunctor.map f.f \u226b ((comparison adj).obj B).a = A.a \u226b f.f) })\n f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\nB : D\ninst\u271d : HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nf : A \u27f6 (comparison adj).obj B\n\u22a2 ((fun g => Algebra.Hom.mk \u2191g)\n ((fun f =>\n { val := f.f,\n property := (_ : (Adjunction.toMonad adj).toFunctor.map f.f \u226b ((comparison adj).obj B).a = A.a \u226b f.f) })\n f)).f =\n f.f\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : \u2200 (A : Algebra (Adjunction.toMonad adj)), HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\n\u22a2 Algebra (Adjunction.toMonad adj) \u2964 D\n[PROOFSTEP]\nrefine'\n Adjunction.leftAdjointOfEquiv (G := comparison adj) (F_obj := fun A => comparisonLeftAdjointObj A) (fun A B => _) _\n[GOAL]\ncase refine'_1\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : \u2200 (A : Algebra (Adjunction.toMonad adj)), HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nA : Algebra (Adjunction.toMonad adj)\nB : D\n\u22a2 ((fun A => comparisonLeftAdjointObj A) A \u27f6 B) \u2243 (A \u27f6 (comparison adj).obj B)\n[PROOFSTEP]\napply comparisonLeftAdjointHomEquiv\n[GOAL]\ncase refine'_2\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : \u2200 (A : Algebra (Adjunction.toMonad adj)), HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\n\u22a2 \u2200 (X : Algebra (Adjunction.toMonad adj)) (Y Y' : D) (g : Y \u27f6 Y') (h : (fun A => comparisonLeftAdjointObj A) X \u27f6 Y),\n \u2191((fun A B => comparisonLeftAdjointHomEquiv A B) X Y') (h \u226b g) =\n \u2191((fun A B => comparisonLeftAdjointHomEquiv A B) X Y) h \u226b (comparison adj).map g\n[PROOFSTEP]\nintro A B B' g h\n[GOAL]\ncase refine'_2\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : \u2200 (A : Algebra (Adjunction.toMonad adj)), HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nA : Algebra (Adjunction.toMonad adj)\nB B' : D\ng : B \u27f6 B'\nh : (fun A => comparisonLeftAdjointObj A) A \u27f6 B\n\u22a2 \u2191((fun A B => comparisonLeftAdjointHomEquiv A B) A B') (h \u226b g) =\n \u2191((fun A B => comparisonLeftAdjointHomEquiv A B) A B) h \u226b (comparison adj).map g\n[PROOFSTEP]\next1\n -- Porting note: the goal was previously closed by the following, which succeeds until\n -- `Category.assoc`.\n -- dsimp [comparisonLeftAdjointHomEquiv]\n -- rw [\u2190 (adj).homEquiv_naturality_right, Category.assoc]\n[GOAL]\ncase refine'_2.h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : \u2200 (A : Algebra (Adjunction.toMonad adj)), HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nA : Algebra (Adjunction.toMonad adj)\nB B' : D\ng : B \u27f6 B'\nh : (fun A => comparisonLeftAdjointObj A) A \u27f6 B\n\u22a2 (\u2191((fun A B => comparisonLeftAdjointHomEquiv A B) A B') (h \u226b g)).f =\n (\u2191((fun A B => comparisonLeftAdjointHomEquiv A B) A B) h \u226b (comparison adj).map g).f\n[PROOFSTEP]\nsimp [Cofork.IsColimit.homIso]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\ninst\u271d : HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\n\u22a2 G.map (F.map A.a) \u226b G.map (coequalizer.\u03c0 (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))) =\n G.map (NatTrans.app adj.counit (F.obj A.A)) \u226b\n G.map (coequalizer.\u03c0 (F.map A.a) (NatTrans.app adj.counit (F.obj A.A)))\n[PROOFSTEP]\nchange _ = G.map _ \u226b _\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\nA : Algebra (Adjunction.toMonad adj)\ninst\u271d : HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\n\u22a2 G.map (F.map A.a) \u226b G.map (coequalizer.\u03c0 (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))) =\n G.map (NatTrans.app adj.counit (F.obj A.A)) \u226b\n G.map (coequalizer.\u03c0 (F.map A.a) (NatTrans.app adj.counit (F.obj A.A)))\n[PROOFSTEP]\nrw [\u2190 G.map_comp, coequalizer.condition, G.map_comp]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : \u2200 (A : Algebra (Adjunction.toMonad adj)), HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nA : Algebra (Adjunction.toMonad adj)\n\u22a2 (NatTrans.app comparisonAdjunction.unit A).f = IsColimit.desc (beckCoequalizer A) (unitCofork A)\n[PROOFSTEP]\napply Limits.Cofork.IsColimit.hom_ext (beckCoequalizer A)\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : \u2200 (A : Algebra (Adjunction.toMonad adj)), HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nA : Algebra (Adjunction.toMonad adj)\n\u22a2 Cofork.\u03c0 (beckCofork A) \u226b (NatTrans.app comparisonAdjunction.unit A).f =\n Cofork.\u03c0 (beckCofork A) \u226b IsColimit.desc (beckCoequalizer A) (unitCofork A)\n[PROOFSTEP]\nrw [Cofork.IsColimit.\u03c0_desc]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : \u2200 (A : Algebra (Adjunction.toMonad adj)), HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nA : Algebra (Adjunction.toMonad adj)\n\u22a2 Cofork.\u03c0 (beckCofork A) \u226b (NatTrans.app comparisonAdjunction.unit A).f = Cofork.\u03c0 (unitCofork A)\n[PROOFSTEP]\ndsimp only [beckCofork_\u03c0, unitCofork_\u03c0]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : \u2200 (A : Algebra (Adjunction.toMonad adj)), HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nA : Algebra (Adjunction.toMonad adj)\n\u22a2 A.a \u226b (NatTrans.app comparisonAdjunction.unit A).f =\n G.map (coequalizer.\u03c0 (F.map A.a) (NatTrans.app adj.counit (F.obj A.A)))\n[PROOFSTEP]\nrw [comparisonAdjunction_unit_f_aux, \u2190 (adj).homEquiv_naturality_left A.a, coequalizer.condition,\n (adj).homEquiv_naturality_right, (adj).homEquiv_unit, Category.assoc]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : \u2200 (A : Algebra (Adjunction.toMonad adj)), HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nA : Algebra (Adjunction.toMonad adj)\n\u22a2 NatTrans.app adj.unit ((Adjunction.toMonad adj).toFunctor.obj A.A) \u226b\n G.map (NatTrans.app adj.counit (F.obj A.A)) \u226b\n G.map (coequalizer.\u03c0 (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))) =\n G.map (coequalizer.\u03c0 (F.map A.a) (NatTrans.app adj.counit (F.obj A.A)))\n[PROOFSTEP]\napply (adj).right_triangle_components_assoc\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : \u2200 (A : Algebra (Adjunction.toMonad adj)), HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nB : D\n\u22a2 NatTrans.app comparisonAdjunction.counit B =\n colimit.desc (parallelPair (F.map (G.map (NatTrans.app adj.counit B))) (NatTrans.app adj.counit (F.obj (G.obj B))))\n (counitCofork B)\n[PROOFSTEP]\napply coequalizer.hom_ext\n[GOAL]\ncase h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : \u2200 (A : Algebra (Adjunction.toMonad adj)), HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nB : D\n\u22a2 coequalizer.\u03c0 (F.map ((comparison adj).obj B).a) (NatTrans.app adj.counit (F.obj ((comparison adj).obj B).A)) \u226b\n NatTrans.app comparisonAdjunction.counit B =\n coequalizer.\u03c0 (F.map ((comparison adj).obj B).a) (NatTrans.app adj.counit (F.obj ((comparison adj).obj B).A)) \u226b\n colimit.desc\n (parallelPair (F.map (G.map (NatTrans.app adj.counit B))) (NatTrans.app adj.counit (F.obj (G.obj B))))\n (counitCofork B)\n[PROOFSTEP]\nchange\n coequalizer.\u03c0 _ _ \u226b coequalizer.desc (((adj).homEquiv _ B).symm (\ud835\udfd9 _)) _ = coequalizer.\u03c0 _ _ \u226b coequalizer.desc _ _\n[GOAL]\ncase h\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : \u2200 (A : Algebra (Adjunction.toMonad adj)), HasCoequalizer (F.map A.a) (NatTrans.app adj.counit (F.obj A.A))\nB : D\n\u22a2 coequalizer.\u03c0 (F.map ((comparison adj).obj B).a) (NatTrans.app adj.counit (F.obj ((comparison adj).obj B).A)) \u226b\n coequalizer.desc (\u2191(Adjunction.homEquiv adj (G.obj B) B).symm (\ud835\udfd9 (G.obj B)))\n (_ :\n F.map ((comparison adj).obj B).a \u226b\n \u2191(\u2191(Equiv.subtypeEquiv (Adjunction.homEquiv adj ((comparison adj).obj B).A ((\ud835\udfed D).obj B))\n (_ :\n \u2200 (f : F.obj ((comparison adj).obj B).A \u27f6 (\ud835\udfed D).obj B),\n F.map ((comparison adj).obj B).a \u226b f =\n NatTrans.app adj.counit (F.obj ((comparison adj).obj B).A) \u226b f \u2194\n G.map (F.map (\u2191(Adjunction.homEquiv adj ((comparison adj).obj B).A ((\ud835\udfed D).obj B)) f)) \u226b\n G.map (NatTrans.app adj.counit ((\ud835\udfed D).obj B)) =\n ((comparison adj).obj B).a \u226b\n \u2191(Adjunction.homEquiv adj ((comparison adj).obj B).A ((\ud835\udfed D).obj B)) f)).symm\n (\u2191{ toFun := fun g => Algebra.Hom.mk \u2191g,\n invFun := fun f =>\n { val := f.f,\n property :=\n (_ :\n (Adjunction.toMonad adj).toFunctor.map f.f \u226b ((comparison adj).obj ((\ud835\udfed D).obj B)).a =\n ((comparison adj).obj B).a \u226b f.f) },\n left_inv :=\n (_ :\n \u2200\n (g :\n { g //\n G.map (F.map g) \u226b G.map (NatTrans.app adj.counit ((\ud835\udfed D).obj B)) =\n ((comparison adj).obj B).a \u226b g }),\n (fun f =>\n { val := f.f,\n property :=\n (_ :\n (Adjunction.toMonad adj).toFunctor.map f.f \u226b\n ((comparison adj).obj ((\ud835\udfed D).obj B)).a =\n ((comparison adj).obj B).a \u226b f.f) })\n ((fun g => Algebra.Hom.mk \u2191g) g) =\n g),\n right_inv :=\n (_ :\n \u2200 (f : (comparison adj).obj B \u27f6 (comparison adj).obj ((\ud835\udfed D).obj B)),\n (fun g => Algebra.Hom.mk \u2191g)\n ((fun f =>\n { val := f.f,\n property :=\n (_ :\n (Adjunction.toMonad adj).toFunctor.map f.f \u226b\n ((comparison adj).obj ((\ud835\udfed D).obj B)).a =\n ((comparison adj).obj B).a \u226b f.f) })\n f) =\n f) }.symm\n (\ud835\udfd9 ((comparison adj).obj B)))) =\n NatTrans.app adj.counit (F.obj ((comparison adj).obj B).A) \u226b\n \u2191(\u2191(Equiv.subtypeEquiv (Adjunction.homEquiv adj ((comparison adj).obj B).A ((\ud835\udfed D).obj B))\n (_ :\n \u2200 (f : F.obj ((comparison adj).obj B).A \u27f6 (\ud835\udfed D).obj B),\n F.map ((comparison adj).obj B).a \u226b f =\n NatTrans.app adj.counit (F.obj ((comparison adj).obj B).A) \u226b f \u2194\n G.map (F.map (\u2191(Adjunction.homEquiv adj ((comparison adj).obj B).A ((\ud835\udfed D).obj B)) f)) \u226b\n G.map (NatTrans.app adj.counit ((\ud835\udfed D).obj B)) =\n ((comparison adj).obj B).a \u226b\n \u2191(Adjunction.homEquiv adj ((comparison adj).obj B).A ((\ud835\udfed D).obj B)) f)).symm\n (\u2191{ toFun := fun g => Algebra.Hom.mk \u2191g,\n invFun := fun f =>\n { val := f.f,\n property :=\n (_ :\n (Adjunction.toMonad adj).toFunctor.map f.f \u226b ((comparison adj).obj ((\ud835\udfed D).obj B)).a =\n ((comparison adj).obj B).a \u226b f.f) },\n left_inv :=\n (_ :\n \u2200\n (g :\n { g //\n G.map (F.map g) \u226b G.map (NatTrans.app adj.counit ((\ud835\udfed D).obj B)) =\n ((comparison adj).obj B).a \u226b g }),\n (fun f =>\n { val := f.f,\n property :=\n (_ :\n (Adjunction.toMonad adj).toFunctor.map f.f \u226b\n ((comparison adj).obj ((\ud835\udfed D).obj B)).a =\n ((comparison adj).obj B).a \u226b f.f) })\n ((fun g => Algebra.Hom.mk \u2191g) g) =\n g),\n right_inv :=\n (_ :\n \u2200 (f : (comparison adj).obj B \u27f6 (comparison adj).obj ((\ud835\udfed D).obj B)),\n (fun g => Algebra.Hom.mk \u2191g)\n ((fun f =>\n { val := f.f,\n property :=\n (_ :\n (Adjunction.toMonad adj).toFunctor.map f.f \u226b\n ((comparison adj).obj ((\ud835\udfed D).obj B)).a =\n ((comparison adj).obj B).a \u226b f.f) })\n f) =\n f) }.symm\n (\ud835\udfd9 ((comparison adj).obj B))))) =\n coequalizer.\u03c0 (F.map ((comparison adj).obj B).a) (NatTrans.app adj.counit (F.obj ((comparison adj).obj B).A)) \u226b\n coequalizer.desc (NatTrans.app adj.counit B)\n (_ :\n F.map (G.map (NatTrans.app adj.counit B)) \u226b NatTrans.app adj.counit ((\ud835\udfed D).obj B) =\n NatTrans.app adj.counit ((G \u22d9 F).obj B) \u226b NatTrans.app adj.counit B)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : MonadicRightAdjoint G\nA B : D\nf g : A \u27f6 B\ninst\u271d : Functor.IsSplitPair G f g\n\u22a2 CreatesColimit (parallelPair f g) G\n[PROOFSTEP]\napply (config := { allowSynthFailures := true }) monadicCreatesColimitOfPreservesColimit\n[GOAL]\ncase inst\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : MonadicRightAdjoint G\nA B : D\nf g : A \u27f6 B\ninst\u271d : Functor.IsSplitPair G f g\n\u22a2 PreservesColimit (parallelPair f g \u22d9 G) (F \u22d9 G)\n[PROOFSTEP]\napply @preservesColimitOfIsoDiagram _ _ _ _ _ _ _ _ _ (diagramIsoParallelPair.{v\u2081} _).symm ?_\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : MonadicRightAdjoint G\nA B : D\nf g : A \u27f6 B\ninst\u271d : Functor.IsSplitPair G f g\n\u22a2 PreservesColimit\n (parallelPair ((parallelPair f g \u22d9 G).map WalkingParallelPairHom.left)\n ((parallelPair f g \u22d9 G).map WalkingParallelPairHom.right))\n (F \u22d9 G)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : MonadicRightAdjoint G\nA B : D\nf g : A \u27f6 B\ninst\u271d : Functor.IsSplitPair G f g\n\u22a2 PreservesColimit (parallelPair (G.map f) (G.map g)) (F \u22d9 G)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase inst\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : MonadicRightAdjoint G\nA B : D\nf g : A \u27f6 B\ninst\u271d : Functor.IsSplitPair G f g\n\u22a2 PreservesColimit ((parallelPair f g \u22d9 G) \u22d9 F \u22d9 G) (F \u22d9 G)\n[PROOFSTEP]\napply @preservesColimitOfIsoDiagram _ _ _ _ _ _ _ _ _ (diagramIsoParallelPair.{v\u2081} _).symm ?_\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : MonadicRightAdjoint G\nA B : D\nf g : A \u27f6 B\ninst\u271d : Functor.IsSplitPair G f g\n\u22a2 PreservesColimit\n (parallelPair (((parallelPair f g \u22d9 G) \u22d9 F \u22d9 G).map WalkingParallelPairHom.left)\n (((parallelPair f g \u22d9 G) \u22d9 F \u22d9 G).map WalkingParallelPairHom.right))\n (F \u22d9 G)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : MonadicRightAdjoint G\nA B : D\nf g : A \u27f6 B\ninst\u271d : Functor.IsSplitPair G f g\n\u22a2 PreservesColimit (parallelPair (G.map (F.map (G.map f))) (G.map (F.map (G.map g)))) (F \u22d9 G)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\n\u22a2 MonadicRightAdjoint G\n[PROOFSTEP]\nlet i : IsRightAdjoint (comparison (ofRightAdjoint G)) := \u27e8_, comparisonAdjunction\u27e9\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\n\u22a2 MonadicRightAdjoint G\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase eqv\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\n\u22a2 IsEquivalence (comparison adj)\n[PROOFSTEP]\nlet _ : \u2200 X : (ofRightAdjoint G).toMonad.Algebra, IsIso ((ofRightAdjoint (comparison (ofRightAdjoint G))).unit.app X) :=\n by\n intro X\n apply @isIso_of_reflects_iso _ _ _ _ _ _ _ (Monad.forget (ofRightAdjoint G).toMonad) ?_ _\n \u00b7 change IsIso (comparisonAdjunction.unit.app X).f\n rw [comparisonAdjunction_unit_f]\n change IsIso (IsColimit.coconePointUniqueUpToIso (beckCoequalizer X) (unitColimitOfPreservesCoequalizer X)).hom\n refine' IsIso.of_iso (IsColimit.coconePointUniqueUpToIso _ _)\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\n\u22a2 \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X)\n[PROOFSTEP]\nintro X\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nX : Algebra (toMonad adj)\n\u22a2 IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X)\n[PROOFSTEP]\napply @isIso_of_reflects_iso _ _ _ _ _ _ _ (Monad.forget (ofRightAdjoint G).toMonad) ?_ _\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nX : Algebra (toMonad adj)\n\u22a2 IsIso ((forget (toMonad adj)).map (NatTrans.app (ofRightAdjoint (comparison adj)).unit X))\n[PROOFSTEP]\nchange IsIso (comparisonAdjunction.unit.app X).f\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nX : Algebra (toMonad adj)\n\u22a2 IsIso (NatTrans.app comparisonAdjunction.unit X).f\n[PROOFSTEP]\nrw [comparisonAdjunction_unit_f]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nX : Algebra (toMonad adj)\n\u22a2 IsIso (IsColimit.desc (beckCoequalizer X) (unitCofork X))\n[PROOFSTEP]\nchange IsIso (IsColimit.coconePointUniqueUpToIso (beckCoequalizer X) (unitColimitOfPreservesCoequalizer X)).hom\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nX : Algebra (toMonad adj)\n\u22a2 IsIso (IsColimit.coconePointUniqueUpToIso (beckCoequalizer X) (unitColimitOfPreservesCoequalizer X)).hom\n[PROOFSTEP]\nrefine' IsIso.of_iso (IsColimit.coconePointUniqueUpToIso _ _)\n[GOAL]\ncase eqv\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\n\u22a2 IsEquivalence (comparison adj)\n[PROOFSTEP]\nlet _ : \u2200 Y : D, IsIso ((ofRightAdjoint (comparison (ofRightAdjoint G))).counit.app Y) :=\n by\n intro Y\n change IsIso (comparisonAdjunction.counit.app Y)\n rw [comparisonAdjunction_counit_app]\n -- Porting note: passing instances through\n change IsIso (IsColimit.coconePointUniqueUpToIso _ ?_).hom\n infer_instance\n -- Porting note: passing instances through\n apply @counitCoequalizerOfReflectsCoequalizer _ _ _ _ _ _ _ ?_\n letI _ :\n G.IsSplitPair ((leftAdjoint G).map (G.map ((Adjunction.ofRightAdjoint G).counit.app Y)))\n ((Adjunction.ofRightAdjoint G).counit.app ((leftAdjoint G).obj (G.obj Y))) :=\n MonadicityInternal.main_pair_G_split ((comparison (Adjunction.ofRightAdjoint G)).obj Y)\n infer_instance\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\n\u22a2 \u2200 (Y : D), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).counit Y)\n[PROOFSTEP]\nintro Y\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nY : D\n\u22a2 IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).counit Y)\n[PROOFSTEP]\nchange IsIso (comparisonAdjunction.counit.app Y)\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nY : D\n\u22a2 IsIso (NatTrans.app comparisonAdjunction.counit Y)\n[PROOFSTEP]\nrw [comparisonAdjunction_counit_app]\n -- Porting note: passing instances through\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nY : D\n\u22a2 IsIso\n (colimit.desc (parallelPair (F.map (G.map (NatTrans.app adj.counit Y))) (NatTrans.app adj.counit (F.obj (G.obj Y))))\n (counitCofork Y))\n[PROOFSTEP]\nchange IsIso (IsColimit.coconePointUniqueUpToIso _ ?_).hom\n[GOAL]\ncase refine_2\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nY : D\n\u22a2 IsIso\n (IsColimit.coconePointUniqueUpToIso\n (colimit.isColimit\n (parallelPair (F.map (G.map (NatTrans.app adj.counit Y))) (NatTrans.app adj.counit (F.obj (G.obj Y)))))\n ?refine_1).hom\ncase refine_1\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nY : D\n\u22a2 IsColimit (counitCofork Y)\n[PROOFSTEP]\ninfer_instance\n -- Porting note: passing instances through\n[GOAL]\ncase refine_1\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nY : D\n\u22a2 IsColimit (counitCofork Y)\n[PROOFSTEP]\napply @counitCoequalizerOfReflectsCoequalizer _ _ _ _ _ _ _ ?_\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nY : D\n\u22a2 ReflectsColimit (parallelPair (F.map (G.map (NatTrans.app adj.counit Y))) (NatTrans.app adj.counit (F.obj (G.obj Y))))\n G\n[PROOFSTEP]\nletI _ :\n G.IsSplitPair ((leftAdjoint G).map (G.map ((Adjunction.ofRightAdjoint G).counit.app Y)))\n ((Adjunction.ofRightAdjoint G).counit.app ((leftAdjoint G).obj (G.obj Y))) :=\n MonadicityInternal.main_pair_G_split ((comparison (Adjunction.ofRightAdjoint G)).obj Y)\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d\u00b9 : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nY : D\nx\u271d : Functor.IsSplitPair G (F.map (G.map (NatTrans.app adj.counit Y))) (NatTrans.app adj.counit (F.obj (G.obj Y))) :=\n main_pair_G_split ((comparison adj).obj Y)\n\u22a2 ReflectsColimit (parallelPair (F.map (G.map (NatTrans.app adj.counit Y))) (NatTrans.app adj.counit (F.obj (G.obj Y))))\n G\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase eqv\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\ninst\u271d : ReflectsColimitOfIsSplitPair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d\u00b9 : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nx\u271d : \u2200 (Y : D), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).counit Y) :=\n fun Y =>\n let_fun this :=\n Eq.mpr\n (_root_.id (comparisonAdjunction_counit_app Y \u25b8 Eq.refl (IsIso (NatTrans.app comparisonAdjunction.counit Y))))\n (let_fun this := inferInstance;\n this);\n this\n\u22a2 IsEquivalence (comparison adj)\n[PROOFSTEP]\nexact Adjunction.isRightAdjointToIsEquivalence\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : CreatesColimitOfIsSplitPair G\n\u22a2 MonadicRightAdjoint G\n[PROOFSTEP]\nlet I {A B} (f g : A \u27f6 B) [G.IsSplitPair f g] : HasColimit (parallelPair f g \u22d9 G) :=\n by\n apply @hasColimitOfIso _ _ _ _ _ _ ?_ (diagramIsoParallelPair.{v\u2081} _)\n exact inferInstanceAs <| HasCoequalizer (G.map f) (G.map g)\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b2 : IsRightAdjoint G\ninst\u271d\u00b9 : CreatesColimitOfIsSplitPair G\nA B : D\nf g : A \u27f6 B\ninst\u271d : Functor.IsSplitPair G f g\n\u22a2 HasColimit (parallelPair f g \u22d9 G)\n[PROOFSTEP]\napply @hasColimitOfIso _ _ _ _ _ _ ?_ (diagramIsoParallelPair.{v\u2081} _)\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b2 : IsRightAdjoint G\ninst\u271d\u00b9 : CreatesColimitOfIsSplitPair G\nA B : D\nf g : A \u27f6 B\ninst\u271d : Functor.IsSplitPair G f g\n\u22a2 HasColimit\n (parallelPair ((parallelPair f g \u22d9 G).map WalkingParallelPairHom.left)\n ((parallelPair f g \u22d9 G).map WalkingParallelPairHom.right))\n[PROOFSTEP]\nexact inferInstanceAs <| HasCoequalizer (G.map f) (G.map g)\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : CreatesColimitOfIsSplitPair G\nI : \u2200 {A B : D} (f g : A \u27f6 B) [inst : Functor.IsSplitPair G f g], HasColimit (parallelPair f g \u22d9 G) :=\n fun {A B} f g [Functor.IsSplitPair G f g] => hasColimitOfIso (diagramIsoParallelPair (parallelPair f g \u22d9 G))\n\u22a2 MonadicRightAdjoint G\n[PROOFSTEP]\napply @monadicOfHasPreservesReflectsGSplitCoequalizers _ _ _ _ _ _ ?_ ?_ ?_\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : CreatesColimitOfIsSplitPair G\nI : \u2200 {A B : D} (f g : A \u27f6 B) [inst : Functor.IsSplitPair G f g], HasColimit (parallelPair f g \u22d9 G) :=\n fun {A B} f g [Functor.IsSplitPair G f g] => hasColimitOfIso (diagramIsoParallelPair (parallelPair f g \u22d9 G))\n\u22a2 HasCoequalizerOfIsSplitPair G\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase out\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : CreatesColimitOfIsSplitPair G\nI : \u2200 {A B : D} (f g : A \u27f6 B) [inst : Functor.IsSplitPair G f g], HasColimit (parallelPair f g \u22d9 G) :=\n fun {A B} f g [Functor.IsSplitPair G f g] => hasColimitOfIso (diagramIsoParallelPair (parallelPair f g \u22d9 G))\n\u22a2 \u2200 {A B : D} (f g : A \u27f6 B) [inst : Functor.IsSplitPair G f g], HasCoequalizer f g\n[PROOFSTEP]\nintros\n[GOAL]\ncase out\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b2 : IsRightAdjoint G\ninst\u271d\u00b9 : CreatesColimitOfIsSplitPair G\nI : \u2200 {A B : D} (f g : A \u27f6 B) [inst : Functor.IsSplitPair G f g], HasColimit (parallelPair f g \u22d9 G) :=\n fun {A B} f g [Functor.IsSplitPair G f g] => hasColimitOfIso (diagramIsoParallelPair (parallelPair f g \u22d9 G))\nA\u271d B\u271d : D\nf\u271d g\u271d : A\u271d \u27f6 B\u271d\ninst\u271d : Functor.IsSplitPair G f\u271d g\u271d\n\u22a2 HasCoequalizer f\u271d g\u271d\n[PROOFSTEP]\napply hasColimit_of_created (parallelPair _ _) G\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : CreatesColimitOfIsSplitPair G\nI : \u2200 {A B : D} (f g : A \u27f6 B) [inst : Functor.IsSplitPair G f g], HasColimit (parallelPair f g \u22d9 G) :=\n fun {A B} f g [Functor.IsSplitPair G f g] => hasColimitOfIso (diagramIsoParallelPair (parallelPair f g \u22d9 G))\n\u22a2 PreservesColimitOfIsSplitPair G\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase out\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : CreatesColimitOfIsSplitPair G\nI : \u2200 {A B : D} (f g : A \u27f6 B) [inst : Functor.IsSplitPair G f g], HasColimit (parallelPair f g \u22d9 G) :=\n fun {A B} f g [Functor.IsSplitPair G f g] => hasColimitOfIso (diagramIsoParallelPair (parallelPair f g \u22d9 G))\n\u22a2 {A B : D} \u2192 (f g : A \u27f6 B) \u2192 [inst : Functor.IsSplitPair G f g] \u2192 PreservesColimit (parallelPair f g) G\n[PROOFSTEP]\nintros\n[GOAL]\ncase out\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b2 : IsRightAdjoint G\ninst\u271d\u00b9 : CreatesColimitOfIsSplitPair G\nI : \u2200 {A B : D} (f g : A \u27f6 B) [inst : Functor.IsSplitPair G f g], HasColimit (parallelPair f g \u22d9 G) :=\n fun {A B} f g [Functor.IsSplitPair G f g] => hasColimitOfIso (diagramIsoParallelPair (parallelPair f g \u22d9 G))\nA\u271d B\u271d : D\nf\u271d g\u271d : A\u271d \u27f6 B\u271d\ninst\u271d : Functor.IsSplitPair G f\u271d g\u271d\n\u22a2 PreservesColimit (parallelPair f\u271d g\u271d) G\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : CreatesColimitOfIsSplitPair G\nI : \u2200 {A B : D} (f g : A \u27f6 B) [inst : Functor.IsSplitPair G f g], HasColimit (parallelPair f g \u22d9 G) :=\n fun {A B} f g [Functor.IsSplitPair G f g] => hasColimitOfIso (diagramIsoParallelPair (parallelPair f g \u22d9 G))\n\u22a2 ReflectsColimitOfIsSplitPair G\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase out\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b9 : IsRightAdjoint G\ninst\u271d : CreatesColimitOfIsSplitPair G\nI : \u2200 {A B : D} (f g : A \u27f6 B) [inst : Functor.IsSplitPair G f g], HasColimit (parallelPair f g \u22d9 G) :=\n fun {A B} f g [Functor.IsSplitPair G f g] => hasColimitOfIso (diagramIsoParallelPair (parallelPair f g \u22d9 G))\n\u22a2 {A B : D} \u2192 (f g : A \u27f6 B) \u2192 [inst : Functor.IsSplitPair G f g] \u2192 ReflectsColimit (parallelPair f g) G\n[PROOFSTEP]\nintros\n[GOAL]\ncase out\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b2 : IsRightAdjoint G\ninst\u271d\u00b9 : CreatesColimitOfIsSplitPair G\nI : \u2200 {A B : D} (f g : A \u27f6 B) [inst : Functor.IsSplitPair G f g], HasColimit (parallelPair f g \u22d9 G) :=\n fun {A B} f g [Functor.IsSplitPair G f g] => hasColimitOfIso (diagramIsoParallelPair (parallelPair f g \u22d9 G))\nA\u271d B\u271d : D\nf\u271d g\u271d : A\u271d \u27f6 B\u271d\ninst\u271d : Functor.IsSplitPair G f\u271d g\u271d\n\u22a2 ReflectsColimit (parallelPair f\u271d g\u271d) G\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : ReflectsIsomorphisms G\ninst\u271d\u00b9 : HasCoequalizerOfIsSplitPair G\ninst\u271d : PreservesColimitOfIsSplitPair G\n\u22a2 MonadicRightAdjoint G\n[PROOFSTEP]\napply (config := { allowSynthFailures := true }) @monadicOfHasPreservesReflectsGSplitCoequalizers\n[GOAL]\ncase inst\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : ReflectsIsomorphisms G\ninst\u271d\u00b9 : HasCoequalizerOfIsSplitPair G\ninst\u271d : PreservesColimitOfIsSplitPair G\n\u22a2 ReflectsColimitOfIsSplitPair G\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase inst.out\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : ReflectsIsomorphisms G\ninst\u271d\u00b9 : HasCoequalizerOfIsSplitPair G\ninst\u271d : PreservesColimitOfIsSplitPair G\n\u22a2 {A B : D} \u2192 (f g : A \u27f6 B) \u2192 [inst : Functor.IsSplitPair G f g] \u2192 ReflectsColimit (parallelPair f g) G\n[PROOFSTEP]\nintro _ _ f g _\n[GOAL]\ncase inst.out\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2075 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u2074 : IsRightAdjoint G\ninst\u271d\u00b3 : ReflectsIsomorphisms G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\nA\u271d B\u271d : D\nf g : A\u271d \u27f6 B\u271d\ninst\u271d : Functor.IsSplitPair G f g\n\u22a2 ReflectsColimit (parallelPair f g) G\n[PROOFSTEP]\nhave := HasCoequalizerOfIsSplitPair.out G f g\n[GOAL]\ncase inst.out\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2075 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u2074 : IsRightAdjoint G\ninst\u271d\u00b3 : ReflectsIsomorphisms G\ninst\u271d\u00b2 : HasCoequalizerOfIsSplitPair G\ninst\u271d\u00b9 : PreservesColimitOfIsSplitPair G\nA\u271d B\u271d : D\nf g : A\u271d \u27f6 B\u271d\ninst\u271d : Functor.IsSplitPair G f g\nthis : HasCoequalizer f g\n\u22a2 ReflectsColimit (parallelPair f g) G\n[PROOFSTEP]\napply reflectsColimitOfReflectsIsomorphisms\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\n\u22a2 MonadicRightAdjoint G\n[PROOFSTEP]\nletI i : IsRightAdjoint (comparison (Adjunction.ofRightAdjoint G)) := \u27e8_, comparisonAdjunction\u27e9\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\n\u22a2 MonadicRightAdjoint G\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase eqv\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\n\u22a2 IsEquivalence (comparison adj)\n[PROOFSTEP]\nlet _ :\n \u2200 X : (Adjunction.ofRightAdjoint G).toMonad.Algebra,\n IsIso ((Adjunction.ofRightAdjoint (comparison (Adjunction.ofRightAdjoint G))).unit.app X) :=\n by\n intro X\n apply @isIso_of_reflects_iso _ _ _ _ _ _ _ (Monad.forget (Adjunction.ofRightAdjoint G).toMonad) ?_ _\n \u00b7 change IsIso (comparisonAdjunction.unit.app X).f\n rw [comparisonAdjunction_unit_f]\n change IsIso (IsColimit.coconePointUniqueUpToIso (beckCoequalizer X) (unitColimitOfPreservesCoequalizer X)).hom\n apply IsIso.of_iso (IsColimit.coconePointUniqueUpToIso _ _)\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\n\u22a2 \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X)\n[PROOFSTEP]\nintro X\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nX : Algebra (toMonad adj)\n\u22a2 IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X)\n[PROOFSTEP]\napply @isIso_of_reflects_iso _ _ _ _ _ _ _ (Monad.forget (Adjunction.ofRightAdjoint G).toMonad) ?_ _\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nX : Algebra (toMonad adj)\n\u22a2 IsIso ((forget (toMonad adj)).map (NatTrans.app (ofRightAdjoint (comparison adj)).unit X))\n[PROOFSTEP]\nchange IsIso (comparisonAdjunction.unit.app X).f\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nX : Algebra (toMonad adj)\n\u22a2 IsIso (NatTrans.app comparisonAdjunction.unit X).f\n[PROOFSTEP]\nrw [comparisonAdjunction_unit_f]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nX : Algebra (toMonad adj)\n\u22a2 IsIso (IsColimit.desc (beckCoequalizer X) (unitCofork X))\n[PROOFSTEP]\nchange IsIso (IsColimit.coconePointUniqueUpToIso (beckCoequalizer X) (unitColimitOfPreservesCoequalizer X)).hom\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nX : Algebra (toMonad adj)\n\u22a2 IsIso (IsColimit.coconePointUniqueUpToIso (beckCoequalizer X) (unitColimitOfPreservesCoequalizer X)).hom\n[PROOFSTEP]\napply IsIso.of_iso (IsColimit.coconePointUniqueUpToIso _ _)\n[GOAL]\ncase eqv\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\n\u22a2 IsEquivalence (comparison adj)\n[PROOFSTEP]\nlet _ : \u2200 Y : D, IsIso ((ofRightAdjoint (comparison (Adjunction.ofRightAdjoint G))).counit.app Y) :=\n by\n intro Y\n change IsIso (comparisonAdjunction.counit.app Y)\n rw [comparisonAdjunction_counit_app]\n -- Porting note: passing instances through\n change IsIso (IsColimit.coconePointUniqueUpToIso _ ?_).hom\n infer_instance\n -- Porting note: passing instances through\n apply @counitCoequalizerOfReflectsCoequalizer _ _ _ _ _ _ _ ?_\n apply reflectsColimitOfReflectsIsomorphisms\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\n\u22a2 \u2200 (Y : D), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).counit Y)\n[PROOFSTEP]\nintro Y\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nY : D\n\u22a2 IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).counit Y)\n[PROOFSTEP]\nchange IsIso (comparisonAdjunction.counit.app Y)\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nY : D\n\u22a2 IsIso (NatTrans.app comparisonAdjunction.counit Y)\n[PROOFSTEP]\nrw [comparisonAdjunction_counit_app]\n -- Porting note: passing instances through\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nY : D\n\u22a2 IsIso\n (colimit.desc (parallelPair (F.map (G.map (NatTrans.app adj.counit Y))) (NatTrans.app adj.counit (F.obj (G.obj Y))))\n (counitCofork Y))\n[PROOFSTEP]\nchange IsIso (IsColimit.coconePointUniqueUpToIso _ ?_).hom\n[GOAL]\ncase refine_2\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nY : D\n\u22a2 IsIso\n (IsColimit.coconePointUniqueUpToIso\n (colimit.isColimit\n (parallelPair (F.map (G.map (NatTrans.app adj.counit Y))) (NatTrans.app adj.counit (F.obj (G.obj Y)))))\n ?refine_1).hom\ncase refine_1\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nY : D\n\u22a2 IsColimit (counitCofork Y)\n[PROOFSTEP]\ninfer_instance\n -- Porting note: passing instances through\n[GOAL]\ncase refine_1\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nY : D\n\u22a2 IsColimit (counitCofork Y)\n[PROOFSTEP]\napply @counitCoequalizerOfReflectsCoequalizer _ _ _ _ _ _ _ ?_\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nY : D\n\u22a2 ReflectsColimit (parallelPair (F.map (G.map (NatTrans.app adj.counit Y))) (NatTrans.app adj.counit (F.obj (G.obj Y))))\n G\n[PROOFSTEP]\napply reflectsColimitOfReflectsIsomorphisms\n[GOAL]\ncase eqv\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2081, u\u2082} D\nG : D \u2964 C\ninst\u271d\u00b3 : IsRightAdjoint G\ninst\u271d\u00b2 : HasReflexiveCoequalizers D\ninst\u271d\u00b9 : ReflectsIsomorphisms G\ninst\u271d : PreservesColimitOfIsReflexivePair G\ni : IsRightAdjoint (comparison adj) := { left := leftAdjointComparison, adj := comparisonAdjunction }\nx\u271d\u00b9 : \u2200 (X : Algebra (toMonad adj)), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) :=\n fun X => isIso_of_reflects_iso (NatTrans.app (ofRightAdjoint (comparison adj)).unit X) (forget (toMonad adj))\nx\u271d : \u2200 (Y : D), IsIso (NatTrans.app (ofRightAdjoint (comparison adj)).counit Y) :=\n fun Y =>\n let_fun this :=\n Eq.mpr\n (_root_.id (comparisonAdjunction_counit_app Y \u25b8 Eq.refl (IsIso (NatTrans.app comparisonAdjunction.counit Y))))\n (let_fun this := inferInstance;\n this);\n this\n\u22a2 IsEquivalence (comparison adj)\n[PROOFSTEP]\nexact Adjunction.isRightAdjointToIsEquivalence\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monad.Monadicity", "llama_tokens": 24806, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834734, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.4131540417076696}} {"text": "[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\n\u22a2 Mem\u2113p f 0 \u2194 Set.Finite {i | f i \u2260 0}\n[PROOFSTEP]\ndsimp [Mem\u2113p]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\n\u22a2 (if 0 = 0 then Set.Finite {i | \u00acf i = 0}\n else if 0 = \u22a4 then BddAbove (Set.range fun i => \u2016f i\u2016) else Summable fun i => \u2016f i\u2016 ^ 0) \u2194\n Set.Finite {i | \u00acf i = 0}\n[PROOFSTEP]\nrw [if_pos rfl]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\n\u22a2 Mem\u2113p f \u22a4 \u2194 BddAbove (Set.range fun i => \u2016f i\u2016)\n[PROOFSTEP]\ndsimp [Mem\u2113p]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\n\u22a2 (if \u22a4 = 0 then Set.Finite {i | \u00acf i = 0}\n else if \u22a4 = \u22a4 then BddAbove (Set.range fun i => \u2016f i\u2016) else Summable fun i => \u2016f i\u2016 ^ 0) \u2194\n BddAbove (Set.range fun i => \u2016f i\u2016)\n[PROOFSTEP]\nrw [if_neg ENNReal.top_ne_zero, if_pos rfl]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : (i : \u03b1) \u2192 E i\n\u22a2 Mem\u2113p f p \u2194 Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [ENNReal.toReal_pos_iff] at hp \n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < p \u2227 p < \u22a4\nf : (i : \u03b1) \u2192 E i\n\u22a2 Mem\u2113p f p \u2194 Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\ndsimp [Mem\u2113p]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < p \u2227 p < \u22a4\nf : (i : \u03b1) \u2192 E i\n\u22a2 (if p = 0 then Set.Finite {i | \u00acf i = 0}\n else if p = \u22a4 then BddAbove (Set.range fun i => \u2016f i\u2016) else Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal p) \u2194\n Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [if_neg hp.1.ne', if_neg hp.2.ne]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal p\n\u22a2 Mem\u2113p f p\n[PROOFSTEP]\nrcases p.trichotomy with (rfl | rfl | hp)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal 0\n\u22a2 Mem\u2113p f 0\n[PROOFSTEP]\napply mem\u2113p_zero\n[GOAL]\ncase inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal 0\n\u22a2 Set.Finite {i | f i \u2260 0}\n[PROOFSTEP]\nhave H : Summable fun _ : \u03b1 => (1 : \u211d) := by simpa using hf\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal 0\n\u22a2 Summable fun x => 1\n[PROOFSTEP]\nsimpa using hf\n[GOAL]\ncase inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal 0\nH : Summable fun x => 1\n\u22a2 Set.Finite {i | f i \u2260 0}\n[PROOFSTEP]\nexact (Set.Finite.of_summable_const (by norm_num) H).subset (Set.subset_univ _)\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal 0\nH : Summable fun x => 1\n\u22a2 0 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal \u22a4\n\u22a2 Mem\u2113p f \u22a4\n[PROOFSTEP]\napply mem\u2113p_infty\n[GOAL]\ncase inr.inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal \u22a4\n\u22a2 BddAbove (Set.range fun i => \u2016f i\u2016)\n[PROOFSTEP]\nhave H : Summable fun _ : \u03b1 => (1 : \u211d) := by simpa using hf\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal \u22a4\n\u22a2 Summable fun x => 1\n[PROOFSTEP]\nsimpa using hf\n[GOAL]\ncase inr.inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal \u22a4\nH : Summable fun x => 1\n\u22a2 BddAbove (Set.range fun i => \u2016f i\u2016)\n[PROOFSTEP]\nsimpa using ((Set.Finite.of_summable_const (by norm_num) H).image fun i => \u2016f i\u2016).bddAbove\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal \u22a4\nH : Summable fun x => 1\n\u22a2 0 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal p\nhp : 0 < ENNReal.toReal p\n\u22a2 Mem\u2113p f p\n[PROOFSTEP]\nexact (mem\u2113p_gen_iff hp).2 hf\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nC : \u211d\nf : (i : \u03b1) \u2192 E i\nhf : \u2200 (s : Finset \u03b1), \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p \u2264 C\n\u22a2 Mem\u2113p f p\n[PROOFSTEP]\napply mem\u2113p_gen\n[GOAL]\ncase hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nC : \u211d\nf : (i : \u03b1) \u2192 E i\nhf : \u2200 (s : Finset \u03b1), \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p \u2264 C\n\u22a2 Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nuse\u2a06 s : Finset \u03b1, \u2211 i in s, \u2016f i\u2016 ^ p.toReal\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nC : \u211d\nf : (i : \u03b1) \u2192 E i\nhf : \u2200 (s : Finset \u03b1), \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p \u2264 C\n\u22a2 HasSum (fun i => \u2016f i\u2016 ^ ENNReal.toReal p) (\u2a06 (s : Finset \u03b1), \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\napply hasSum_of_isLUB_of_nonneg\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nC : \u211d\nf : (i : \u03b1) \u2192 E i\nhf : \u2200 (s : Finset \u03b1), \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p \u2264 C\n\u22a2 \u2200 (i : \u03b1), 0 \u2264 \u2016f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nintro b\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nC : \u211d\nf : (i : \u03b1) \u2192 E i\nhf : \u2200 (s : Finset \u03b1), \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p \u2264 C\nb : \u03b1\n\u22a2 0 \u2264 \u2016f b\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nexact Real.rpow_nonneg_of_nonneg (norm_nonneg _) _\n[GOAL]\ncase h.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nC : \u211d\nf : (i : \u03b1) \u2192 E i\nhf : \u2200 (s : Finset \u03b1), \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p \u2264 C\n\u22a2 IsLUB (Set.range fun s => \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p) (\u2a06 (s : Finset \u03b1), \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\napply isLUB_ciSup\n[GOAL]\ncase h.hf.H\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nC : \u211d\nf : (i : \u03b1) \u2192 E i\nhf : \u2200 (s : Finset \u03b1), \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p \u2264 C\n\u22a2 BddAbove (Set.range fun s => \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nuse C\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nC : \u211d\nf : (i : \u03b1) \u2192 E i\nhf : \u2200 (s : Finset \u03b1), \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p \u2264 C\n\u22a2 C \u2208 upperBounds (Set.range fun s => \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nrintro - \u27e8s, rfl\u27e9\n[GOAL]\ncase h.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nC : \u211d\nf : (i : \u03b1) \u2192 E i\nhf : \u2200 (s : Finset \u03b1), \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p \u2264 C\ns : Finset \u03b1\n\u22a2 (fun s => \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p) s \u2264 C\n[PROOFSTEP]\nexact hf s\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u22a2 Mem\u2113p 0 p\n[PROOFSTEP]\nrcases p.trichotomy with (rfl | rfl | hp)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u22a2 Mem\u2113p 0 0\n[PROOFSTEP]\napply mem\u2113p_zero\n[GOAL]\ncase inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u22a2 Set.Finite {i | OfNat.ofNat 0 i \u2260 0}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u22a2 Mem\u2113p 0 \u22a4\n[PROOFSTEP]\napply mem\u2113p_infty\n[GOAL]\ncase inr.inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u22a2 BddAbove (Set.range fun i => \u2016OfNat.ofNat 0 i\u2016)\n[PROOFSTEP]\nsimp only [norm_zero, Pi.zero_apply]\n[GOAL]\ncase inr.inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u22a2 BddAbove (Set.range fun i => 0)\n[PROOFSTEP]\nexact bddAbove_singleton.mono Set.range_const_subset\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\n\u22a2 Mem\u2113p 0 p\n[PROOFSTEP]\napply mem\u2113p_gen\n[GOAL]\ncase inr.inr.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\n\u22a2 Summable fun i => \u2016OfNat.ofNat 0 i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nsimp [Real.zero_rpow hp.ne', summable_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\n\u22a2 Mem\u2113p (-f) p\n[PROOFSTEP]\nrcases p.trichotomy with (rfl | rfl | hp)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f 0\n\u22a2 Mem\u2113p (-f) 0\n[PROOFSTEP]\napply mem\u2113p_zero\n[GOAL]\ncase inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f 0\n\u22a2 Set.Finite {i | (-f) i \u2260 0}\n[PROOFSTEP]\nsimp [hf.finite_dsupport]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f \u22a4\n\u22a2 Mem\u2113p (-f) \u22a4\n[PROOFSTEP]\napply mem\u2113p_infty\n[GOAL]\ncase inr.inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f \u22a4\n\u22a2 BddAbove (Set.range fun i => \u2016(-f) i\u2016)\n[PROOFSTEP]\nsimpa using hf.bddAbove\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhp : 0 < ENNReal.toReal p\n\u22a2 Mem\u2113p (-f) p\n[PROOFSTEP]\napply mem\u2113p_gen\n[GOAL]\ncase inr.inr.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhp : 0 < ENNReal.toReal p\n\u22a2 Summable fun i => \u2016(-f) i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nsimpa using hf.summable hp\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhpq : q \u2264 p\n\u22a2 Mem\u2113p f p\n[PROOFSTEP]\nrcases ENNReal.trichotomy\u2082 hpq with (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9 | \u27e8rfl, hp\u27e9 | \u27e8rfl, rfl\u27e9 | \u27e8hq, rfl\u27e9 | \u27e8hq, _, hpq'\u27e9)\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f 0\nhpq : 0 \u2264 0\n\u22a2 Mem\u2113p f 0\n[PROOFSTEP]\nexact hfq\n[GOAL]\ncase inr.inl.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f 0\nhpq : 0 \u2264 \u22a4\n\u22a2 Mem\u2113p f \u22a4\n[PROOFSTEP]\napply mem\u2113p_infty\n[GOAL]\ncase inr.inl.intro.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f 0\nhpq : 0 \u2264 \u22a4\n\u22a2 BddAbove (Set.range fun i => \u2016f i\u2016)\n[PROOFSTEP]\nobtain \u27e8C, hC\u27e9 := (hfq.finite_dsupport.image fun i => \u2016f i\u2016).bddAbove\n[GOAL]\ncase inr.inl.intro.hf.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f 0\nhpq : 0 \u2264 \u22a4\nC : \u211d\nhC : C \u2208 upperBounds ((fun i => \u2016f i\u2016) '' {i | f i \u2260 0})\n\u22a2 BddAbove (Set.range fun i => \u2016f i\u2016)\n[PROOFSTEP]\nuse max 0 C\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f 0\nhpq : 0 \u2264 \u22a4\nC : \u211d\nhC : C \u2208 upperBounds ((fun i => \u2016f i\u2016) '' {i | f i \u2260 0})\n\u22a2 max 0 C \u2208 upperBounds (Set.range fun i => \u2016f i\u2016)\n[PROOFSTEP]\nrintro x \u27e8i, rfl\u27e9\n[GOAL]\ncase h.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f 0\nhpq : 0 \u2264 \u22a4\nC : \u211d\nhC : C \u2208 upperBounds ((fun i => \u2016f i\u2016) '' {i | f i \u2260 0})\ni : \u03b1\n\u22a2 (fun i => \u2016f i\u2016) i \u2264 max 0 C\n[PROOFSTEP]\nby_cases hi : f i = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f 0\nhpq : 0 \u2264 \u22a4\nC : \u211d\nhC : C \u2208 upperBounds ((fun i => \u2016f i\u2016) '' {i | f i \u2260 0})\ni : \u03b1\nhi : f i = 0\n\u22a2 (fun i => \u2016f i\u2016) i \u2264 max 0 C\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f 0\nhpq : 0 \u2264 \u22a4\nC : \u211d\nhC : C \u2208 upperBounds ((fun i => \u2016f i\u2016) '' {i | f i \u2260 0})\ni : \u03b1\nhi : \u00acf i = 0\n\u22a2 (fun i => \u2016f i\u2016) i \u2264 max 0 C\n[PROOFSTEP]\nexact (hC \u27e8i, hi, rfl\u27e9).trans (le_max_right _ _)\n[GOAL]\ncase inr.inr.inl.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhp : 0 < ENNReal.toReal p\nhfq : Mem\u2113p f 0\nhpq : 0 \u2264 p\n\u22a2 Mem\u2113p f p\n[PROOFSTEP]\napply mem\u2113p_gen\n[GOAL]\ncase inr.inr.inl.intro.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhp : 0 < ENNReal.toReal p\nhfq : Mem\u2113p f 0\nhpq : 0 \u2264 p\n\u22a2 Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nhave : \u2200 (i) (_ : i \u2209 hfq.finite_dsupport.toFinset), \u2016f i\u2016 ^ p.toReal = 0 :=\n by\n intro i hi\n have : f i = 0 := by simpa using hi\n simp [this, Real.zero_rpow hp.ne']\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhp : 0 < ENNReal.toReal p\nhfq : Mem\u2113p f 0\nhpq : 0 \u2264 p\n\u22a2 \u2200 (i : \u03b1), \u00aci \u2208 Set.Finite.toFinset (_ : Set.Finite {i | f i \u2260 0}) \u2192 \u2016f i\u2016 ^ ENNReal.toReal p = 0\n[PROOFSTEP]\nintro i hi\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhp : 0 < ENNReal.toReal p\nhfq : Mem\u2113p f 0\nhpq : 0 \u2264 p\ni : \u03b1\nhi : \u00aci \u2208 Set.Finite.toFinset (_ : Set.Finite {i | f i \u2260 0})\n\u22a2 \u2016f i\u2016 ^ ENNReal.toReal p = 0\n[PROOFSTEP]\nhave : f i = 0 := by simpa using hi\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhp : 0 < ENNReal.toReal p\nhfq : Mem\u2113p f 0\nhpq : 0 \u2264 p\ni : \u03b1\nhi : \u00aci \u2208 Set.Finite.toFinset (_ : Set.Finite {i | f i \u2260 0})\n\u22a2 f i = 0\n[PROOFSTEP]\nsimpa using hi\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhp : 0 < ENNReal.toReal p\nhfq : Mem\u2113p f 0\nhpq : 0 \u2264 p\ni : \u03b1\nhi : \u00aci \u2208 Set.Finite.toFinset (_ : Set.Finite {i | f i \u2260 0})\nthis : f i = 0\n\u22a2 \u2016f i\u2016 ^ ENNReal.toReal p = 0\n[PROOFSTEP]\nsimp [this, Real.zero_rpow hp.ne']\n[GOAL]\ncase inr.inr.inl.intro.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhp : 0 < ENNReal.toReal p\nhfq : Mem\u2113p f 0\nhpq : 0 \u2264 p\nthis : \u2200 (i : \u03b1), \u00aci \u2208 Set.Finite.toFinset (_ : Set.Finite {i | f i \u2260 0}) \u2192 \u2016f i\u2016 ^ ENNReal.toReal p = 0\n\u22a2 Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nexact summable_of_ne_finset_zero this\n[GOAL]\ncase inr.inr.inr.inl.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f \u22a4\nhpq : \u22a4 \u2264 \u22a4\n\u22a2 Mem\u2113p f \u22a4\n[PROOFSTEP]\nexact hfq\n[GOAL]\ncase inr.inr.inr.inr.inl.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nq : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhq : 0 < ENNReal.toReal q\nhpq : q \u2264 \u22a4\n\u22a2 Mem\u2113p f \u22a4\n[PROOFSTEP]\napply mem\u2113p_infty\n[GOAL]\ncase inr.inr.inr.inr.inl.intro.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nq : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhq : 0 < ENNReal.toReal q\nhpq : q \u2264 \u22a4\n\u22a2 BddAbove (Set.range fun i => \u2016f i\u2016)\n[PROOFSTEP]\nobtain \u27e8A, hA\u27e9 := (hfq.summable hq).tendsto_cofinite_zero.bddAbove_range_of_cofinite\n[GOAL]\ncase inr.inr.inr.inr.inl.intro.hf.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nq : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhq : 0 < ENNReal.toReal q\nhpq : q \u2264 \u22a4\nA : \u211d\nhA : A \u2208 upperBounds (Set.range fun i => \u2016f i\u2016 ^ ENNReal.toReal q)\n\u22a2 BddAbove (Set.range fun i => \u2016f i\u2016)\n[PROOFSTEP]\nuse A ^ q.toReal\u207b\u00b9\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nq : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhq : 0 < ENNReal.toReal q\nhpq : q \u2264 \u22a4\nA : \u211d\nhA : A \u2208 upperBounds (Set.range fun i => \u2016f i\u2016 ^ ENNReal.toReal q)\n\u22a2 A ^ (ENNReal.toReal q)\u207b\u00b9 \u2208 upperBounds (Set.range fun i => \u2016f i\u2016)\n[PROOFSTEP]\nrintro x \u27e8i, rfl\u27e9\n[GOAL]\ncase h.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nq : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhq : 0 < ENNReal.toReal q\nhpq : q \u2264 \u22a4\nA : \u211d\nhA : A \u2208 upperBounds (Set.range fun i => \u2016f i\u2016 ^ ENNReal.toReal q)\ni : \u03b1\n\u22a2 (fun i => \u2016f i\u2016) i \u2264 A ^ (ENNReal.toReal q)\u207b\u00b9\n[PROOFSTEP]\nhave : 0 \u2264 \u2016f i\u2016 ^ q.toReal := Real.rpow_nonneg_of_nonneg (norm_nonneg _) _\n[GOAL]\ncase h.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nq : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhq : 0 < ENNReal.toReal q\nhpq : q \u2264 \u22a4\nA : \u211d\nhA : A \u2208 upperBounds (Set.range fun i => \u2016f i\u2016 ^ ENNReal.toReal q)\ni : \u03b1\nthis : 0 \u2264 \u2016f i\u2016 ^ ENNReal.toReal q\n\u22a2 (fun i => \u2016f i\u2016) i \u2264 A ^ (ENNReal.toReal q)\u207b\u00b9\n[PROOFSTEP]\nsimpa [\u2190 Real.rpow_mul, mul_inv_cancel hq.ne'] using Real.rpow_le_rpow this (hA \u27e8i, rfl\u27e9) (inv_nonneg.mpr hq.le)\n[GOAL]\ncase inr.inr.inr.inr.inr.intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhpq : q \u2264 p\nhq : 0 < ENNReal.toReal q\nleft\u271d : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q \u2264 ENNReal.toReal p\n\u22a2 Mem\u2113p f p\n[PROOFSTEP]\napply mem\u2113p_gen\n[GOAL]\ncase inr.inr.inr.inr.inr.intro.intro.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhpq : q \u2264 p\nhq : 0 < ENNReal.toReal q\nleft\u271d : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q \u2264 ENNReal.toReal p\n\u22a2 Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nhave hf' := hfq.summable hq\n[GOAL]\ncase inr.inr.inr.inr.inr.intro.intro.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhpq : q \u2264 p\nhq : 0 < ENNReal.toReal q\nleft\u271d : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q \u2264 ENNReal.toReal p\nhf' : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal q\n\u22a2 Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nrefine' summable_of_norm_bounded_eventually _ hf' (@Set.Finite.subset _ {i | 1 \u2264 \u2016f i\u2016} _ _ _)\n[GOAL]\ncase inr.inr.inr.inr.inr.intro.intro.hf.refine'_1\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhpq : q \u2264 p\nhq : 0 < ENNReal.toReal q\nleft\u271d : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q \u2264 ENNReal.toReal p\nhf' : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal q\n\u22a2 Set.Finite {i | 1 \u2264 \u2016f i\u2016}\n[PROOFSTEP]\nhave H : {x : \u03b1 | 1 \u2264 \u2016f x\u2016 ^ q.toReal}.Finite := by\n simpa using eventually_lt_of_tendsto_lt (by norm_num) hf'.tendsto_cofinite_zero\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhpq : q \u2264 p\nhq : 0 < ENNReal.toReal q\nleft\u271d : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q \u2264 ENNReal.toReal p\nhf' : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal q\n\u22a2 Set.Finite {x | 1 \u2264 \u2016f x\u2016 ^ ENNReal.toReal q}\n[PROOFSTEP]\nsimpa using eventually_lt_of_tendsto_lt (by norm_num) hf'.tendsto_cofinite_zero\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhpq : q \u2264 p\nhq : 0 < ENNReal.toReal q\nleft\u271d : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q \u2264 ENNReal.toReal p\nhf' : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal q\n\u22a2 0 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr.inr.inr.inr.inr.intro.intro.hf.refine'_1\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhpq : q \u2264 p\nhq : 0 < ENNReal.toReal q\nleft\u271d : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q \u2264 ENNReal.toReal p\nhf' : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal q\nH : Set.Finite {x | 1 \u2264 \u2016f x\u2016 ^ ENNReal.toReal q}\n\u22a2 Set.Finite {i | 1 \u2264 \u2016f i\u2016}\n[PROOFSTEP]\nexact H.subset fun i hi => Real.one_le_rpow hi hq.le\n[GOAL]\ncase inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhpq : q \u2264 p\nhq : 0 < ENNReal.toReal q\nleft\u271d : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q \u2264 ENNReal.toReal p\nhf' : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal q\n\u22a2 {x | (fun i => \u2016\u2016f i\u2016 ^ ENNReal.toReal p\u2016 \u2264 \u2016f i\u2016 ^ ENNReal.toReal q) x}\u1d9c \u2286 {i | 1 \u2264 \u2016f i\u2016}\n[PROOFSTEP]\nshow \u2200 i, \u00ac|\u2016f i\u2016 ^ p.toReal| \u2264 \u2016f i\u2016 ^ q.toReal \u2192 1 \u2264 \u2016f i\u2016\n[GOAL]\ncase inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhpq : q \u2264 p\nhq : 0 < ENNReal.toReal q\nleft\u271d : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q \u2264 ENNReal.toReal p\nhf' : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal q\n\u22a2 \u2200 (i : \u03b1), \u00ac|\u2016f i\u2016 ^ ENNReal.toReal p| \u2264 \u2016f i\u2016 ^ ENNReal.toReal q \u2192 1 \u2264 \u2016f i\u2016\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhpq : q \u2264 p\nhq : 0 < ENNReal.toReal q\nleft\u271d : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q \u2264 ENNReal.toReal p\nhf' : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal q\ni : \u03b1\nhi : \u00ac|\u2016f i\u2016 ^ ENNReal.toReal p| \u2264 \u2016f i\u2016 ^ ENNReal.toReal q\n\u22a2 1 \u2264 \u2016f i\u2016\n[PROOFSTEP]\nhave : 0 \u2264 \u2016f i\u2016 ^ p.toReal := Real.rpow_nonneg_of_nonneg (norm_nonneg _) p.toReal\n[GOAL]\ncase inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhpq : q \u2264 p\nhq : 0 < ENNReal.toReal q\nleft\u271d : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q \u2264 ENNReal.toReal p\nhf' : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal q\ni : \u03b1\nhi : \u00ac|\u2016f i\u2016 ^ ENNReal.toReal p| \u2264 \u2016f i\u2016 ^ ENNReal.toReal q\nthis : 0 \u2264 \u2016f i\u2016 ^ ENNReal.toReal p\n\u22a2 1 \u2264 \u2016f i\u2016\n[PROOFSTEP]\nsimp only [abs_of_nonneg, this] at hi \n[GOAL]\ncase inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhpq : q \u2264 p\nhq : 0 < ENNReal.toReal q\nleft\u271d : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q \u2264 ENNReal.toReal p\nhf' : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal q\ni : \u03b1\nthis : 0 \u2264 \u2016f i\u2016 ^ ENNReal.toReal p\nhi : \u00ac\u2016f i\u2016 ^ ENNReal.toReal p \u2264 \u2016f i\u2016 ^ ENNReal.toReal q\n\u22a2 1 \u2264 \u2016f i\u2016\n[PROOFSTEP]\ncontrapose! hi\n[GOAL]\ncase inr.inr.inr.inr.inr.intro.intro.hf.refine'_2\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nf : (i : \u03b1) \u2192 E i\nhfq : Mem\u2113p f q\nhpq : q \u2264 p\nhq : 0 < ENNReal.toReal q\nleft\u271d : 0 < ENNReal.toReal p\nhpq' : ENNReal.toReal q \u2264 ENNReal.toReal p\nhf' : Summable fun i => \u2016f i\u2016 ^ ENNReal.toReal q\ni : \u03b1\nthis : 0 \u2264 \u2016f i\u2016 ^ ENNReal.toReal p\nhi : \u2016f i\u2016 < 1\n\u22a2 \u2016f i\u2016 ^ ENNReal.toReal p \u2264 \u2016f i\u2016 ^ ENNReal.toReal q\n[PROOFSTEP]\nexact Real.rpow_le_rpow_of_exponent_ge' (norm_nonneg _) hi.le hq.le hpq'\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhg : Mem\u2113p g p\n\u22a2 Mem\u2113p (f + g) p\n[PROOFSTEP]\nrcases p.trichotomy with (rfl | rfl | hp)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f 0\nhg : Mem\u2113p g 0\n\u22a2 Mem\u2113p (f + g) 0\n[PROOFSTEP]\napply mem\u2113p_zero\n[GOAL]\ncase inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f 0\nhg : Mem\u2113p g 0\n\u22a2 Set.Finite {i | (f + g) i \u2260 0}\n[PROOFSTEP]\nrefine' (hf.finite_dsupport.union hg.finite_dsupport).subset fun i => _\n[GOAL]\ncase inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f 0\nhg : Mem\u2113p g 0\ni : \u03b1\n\u22a2 i \u2208 {i | (f + g) i \u2260 0} \u2192 i \u2208 {i | f i \u2260 0} \u222a {i | g i \u2260 0}\n[PROOFSTEP]\nsimp only [Pi.add_apply, Ne.def, Set.mem_union, Set.mem_setOf_eq]\n[GOAL]\ncase inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f 0\nhg : Mem\u2113p g 0\ni : \u03b1\n\u22a2 \u00acf i + g i = 0 \u2192 \u00acf i = 0 \u2228 \u00acg i = 0\n[PROOFSTEP]\ncontrapose!\n[GOAL]\ncase inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f 0\nhg : Mem\u2113p g 0\ni : \u03b1\n\u22a2 f i = 0 \u2227 g i = 0 \u2192 f i + g i = 0\n[PROOFSTEP]\nrintro \u27e8hf', hg'\u27e9\n[GOAL]\ncase inl.hf.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f 0\nhg : Mem\u2113p g 0\ni : \u03b1\nhf' : f i = 0\nhg' : g i = 0\n\u22a2 f i + g i = 0\n[PROOFSTEP]\nsimp [hf', hg']\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f \u22a4\nhg : Mem\u2113p g \u22a4\n\u22a2 Mem\u2113p (f + g) \u22a4\n[PROOFSTEP]\napply mem\u2113p_infty\n[GOAL]\ncase inr.inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f \u22a4\nhg : Mem\u2113p g \u22a4\n\u22a2 BddAbove (Set.range fun i => \u2016(f + g) i\u2016)\n[PROOFSTEP]\nobtain \u27e8A, hA\u27e9 := hf.bddAbove\n[GOAL]\ncase inr.inl.hf.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f \u22a4\nhg : Mem\u2113p g \u22a4\nA : \u211d\nhA : A \u2208 upperBounds (Set.range fun i => \u2016f i\u2016)\n\u22a2 BddAbove (Set.range fun i => \u2016(f + g) i\u2016)\n[PROOFSTEP]\nobtain \u27e8B, hB\u27e9 := hg.bddAbove\n[GOAL]\ncase inr.inl.hf.intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f \u22a4\nhg : Mem\u2113p g \u22a4\nA : \u211d\nhA : A \u2208 upperBounds (Set.range fun i => \u2016f i\u2016)\nB : \u211d\nhB : B \u2208 upperBounds (Set.range fun i => \u2016g i\u2016)\n\u22a2 BddAbove (Set.range fun i => \u2016(f + g) i\u2016)\n[PROOFSTEP]\nrefine' \u27e8A + B, _\u27e9\n[GOAL]\ncase inr.inl.hf.intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f \u22a4\nhg : Mem\u2113p g \u22a4\nA : \u211d\nhA : A \u2208 upperBounds (Set.range fun i => \u2016f i\u2016)\nB : \u211d\nhB : B \u2208 upperBounds (Set.range fun i => \u2016g i\u2016)\n\u22a2 A + B \u2208 upperBounds (Set.range fun i => \u2016(f + g) i\u2016)\n[PROOFSTEP]\nrintro a \u27e8i, rfl\u27e9\n[GOAL]\ncase inr.inl.hf.intro.intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f \u22a4\nhg : Mem\u2113p g \u22a4\nA : \u211d\nhA : A \u2208 upperBounds (Set.range fun i => \u2016f i\u2016)\nB : \u211d\nhB : B \u2208 upperBounds (Set.range fun i => \u2016g i\u2016)\ni : \u03b1\n\u22a2 (fun i => \u2016(f + g) i\u2016) i \u2264 A + B\n[PROOFSTEP]\nexact le_trans (norm_add_le _ _) (add_le_add (hA \u27e8i, rfl\u27e9) (hB \u27e8i, rfl\u27e9))\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhg : Mem\u2113p g p\nhp : 0 < ENNReal.toReal p\n\u22a2 Mem\u2113p (f + g) p\n[PROOFSTEP]\napply mem\u2113p_gen\n[GOAL]\ncase inr.inr.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhg : Mem\u2113p g p\nhp : 0 < ENNReal.toReal p\n\u22a2 Summable fun i => \u2016(f + g) i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nlet C : \u211d := if p.toReal < 1 then 1 else (2 : \u211d) ^ (p.toReal - 1)\n[GOAL]\ncase inr.inr.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhg : Mem\u2113p g p\nhp : 0 < ENNReal.toReal p\nC : \u211d := if ENNReal.toReal p < 1 then 1 else 2 ^ (ENNReal.toReal p - 1)\n\u22a2 Summable fun i => \u2016(f + g) i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nrefine' summable_of_nonneg_of_le _ (fun i => _) (((hf.summable hp).add (hg.summable hp)).mul_left C)\n[GOAL]\ncase inr.inr.hf.refine'_1\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhg : Mem\u2113p g p\nhp : 0 < ENNReal.toReal p\nC : \u211d := if ENNReal.toReal p < 1 then 1 else 2 ^ (ENNReal.toReal p - 1)\n\u22a2 \u2200 (b : \u03b1), 0 \u2264 \u2016(f + g) b\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nexact fun b => Real.rpow_nonneg_of_nonneg (norm_nonneg (f b + g b)) p.toReal\n[GOAL]\ncase inr.inr.hf.refine'_2\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhg : Mem\u2113p g p\nhp : 0 < ENNReal.toReal p\nC : \u211d := if ENNReal.toReal p < 1 then 1 else 2 ^ (ENNReal.toReal p - 1)\ni : \u03b1\n\u22a2 \u2016(f + g) i\u2016 ^ ENNReal.toReal p \u2264 C * (\u2016f i\u2016 ^ ENNReal.toReal p + \u2016g i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nrefine' (Real.rpow_le_rpow (norm_nonneg _) (norm_add_le _ _) hp.le).trans _\n[GOAL]\ncase inr.inr.hf.refine'_2\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhg : Mem\u2113p g p\nhp : 0 < ENNReal.toReal p\nC : \u211d := if ENNReal.toReal p < 1 then 1 else 2 ^ (ENNReal.toReal p - 1)\ni : \u03b1\n\u22a2 (\u2016f i\u2016 + \u2016g i\u2016) ^ ENNReal.toReal p \u2264 C * (\u2016f i\u2016 ^ ENNReal.toReal p + \u2016g i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase inr.inr.hf.refine'_2\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhg : Mem\u2113p g p\nhp : 0 < ENNReal.toReal p\nC : \u211d := if ENNReal.toReal p < 1 then 1 else 2 ^ (ENNReal.toReal p - 1)\ni : \u03b1\n\u22a2 (\u2016f i\u2016 + \u2016g i\u2016) ^ ENNReal.toReal p \u2264\n (if ENNReal.toReal p < 1 then 1 else 2 ^ (ENNReal.toReal p - 1)) *\n (\u2016f i\u2016 ^ ENNReal.toReal p + \u2016g i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhg : Mem\u2113p g p\nhp : 0 < ENNReal.toReal p\nC : \u211d := if ENNReal.toReal p < 1 then 1 else 2 ^ (ENNReal.toReal p - 1)\ni : \u03b1\nh : ENNReal.toReal p < 1\n\u22a2 (\u2016f i\u2016 + \u2016g i\u2016) ^ ENNReal.toReal p \u2264 1 * (\u2016f i\u2016 ^ ENNReal.toReal p + \u2016g i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nsimpa using NNReal.coe_le_coe.2 (NNReal.rpow_add_le_add_rpow \u2016f i\u2016\u208a \u2016g i\u2016\u208a hp.le h.le)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhg : Mem\u2113p g p\nhp : 0 < ENNReal.toReal p\nC : \u211d := if ENNReal.toReal p < 1 then 1 else 2 ^ (ENNReal.toReal p - 1)\ni : \u03b1\nh : \u00acENNReal.toReal p < 1\n\u22a2 (\u2016f i\u2016 + \u2016g i\u2016) ^ ENNReal.toReal p \u2264\n 2 ^ (ENNReal.toReal p - 1) * (\u2016f i\u2016 ^ ENNReal.toReal p + \u2016g i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nlet F : Fin 2 \u2192 \u211d\u22650 := ![\u2016f i\u2016\u208a, \u2016g i\u2016\u208a]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhg : Mem\u2113p g p\nhp : 0 < ENNReal.toReal p\nC : \u211d := if ENNReal.toReal p < 1 then 1 else 2 ^ (ENNReal.toReal p - 1)\ni : \u03b1\nh : \u00acENNReal.toReal p < 1\nF : Fin 2 \u2192 \u211d\u22650 := ![\u2016f i\u2016\u208a, \u2016g i\u2016\u208a]\n\u22a2 (\u2016f i\u2016 + \u2016g i\u2016) ^ ENNReal.toReal p \u2264\n 2 ^ (ENNReal.toReal p - 1) * (\u2016f i\u2016 ^ ENNReal.toReal p + \u2016g i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nsimp only [not_lt] at h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhg : Mem\u2113p g p\nhp : 0 < ENNReal.toReal p\nC : \u211d := if ENNReal.toReal p < 1 then 1 else 2 ^ (ENNReal.toReal p - 1)\ni : \u03b1\nF : Fin 2 \u2192 \u211d\u22650 := ![\u2016f i\u2016\u208a, \u2016g i\u2016\u208a]\nh : 1 \u2264 ENNReal.toReal p\n\u22a2 (\u2016f i\u2016 + \u2016g i\u2016) ^ ENNReal.toReal p \u2264\n 2 ^ (ENNReal.toReal p - 1) * (\u2016f i\u2016 ^ ENNReal.toReal p + \u2016g i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nsimpa [Fin.sum_univ_succ] using Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg Finset.univ h fun i _ => (F i).coe_nonneg\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhg : Mem\u2113p g p\n\u22a2 Mem\u2113p (f - g) p\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf g : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhg : Mem\u2113p g p\n\u22a2 Mem\u2113p (f + -g) p\n[PROOFSTEP]\nexact hf.add hg.neg\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ns : Finset \u03b9\nf : \u03b9 \u2192 (i : \u03b1) \u2192 E i\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2113p (f i) p\n\u22a2 Mem\u2113p (fun a => \u2211 i in s, f i a) p\n[PROOFSTEP]\nhaveI : DecidableEq \u03b9 := Classical.decEq _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ns : Finset \u03b9\nf : \u03b9 \u2192 (i : \u03b1) \u2192 E i\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2113p (f i) p\nthis : DecidableEq \u03b9\n\u22a2 Mem\u2113p (fun a => \u2211 i in s, f i a) p\n[PROOFSTEP]\nrevert hf\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ns : Finset \u03b9\nf : \u03b9 \u2192 (i : \u03b1) \u2192 E i\nthis : DecidableEq \u03b9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2113p (f i) p) \u2192 Mem\u2113p (fun a => \u2211 i in s, f i a) p\n[PROOFSTEP]\nrefine' Finset.induction_on s _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ns : Finset \u03b9\nf : \u03b9 \u2192 (i : \u03b1) \u2192 E i\nthis : DecidableEq \u03b9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Mem\u2113p (f i) p) \u2192 Mem\u2113p (fun a => \u2211 i in \u2205, f i a) p\n[PROOFSTEP]\nsimp only [zero_mem_\u2113p', Finset.sum_empty, imp_true_iff]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ns : Finset \u03b9\nf : \u03b9 \u2192 (i : \u03b1) \u2192 E i\nthis : DecidableEq \u03b9\n\u22a2 \u2200 \u2983a : \u03b9\u2984 {s : Finset \u03b9},\n \u00aca \u2208 s \u2192\n ((\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2113p (f i) p) \u2192 Mem\u2113p (fun a => \u2211 i in s, f i a) p) \u2192\n (\u2200 (i : \u03b9), i \u2208 insert a s \u2192 Mem\u2113p (f i) p) \u2192 Mem\u2113p (fun a_3 => \u2211 i in insert a s, f i a_3) p\n[PROOFSTEP]\nintro i s his ih hf\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 (i : \u03b1) \u2192 E i\nthis : DecidableEq \u03b9\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2113p (f i) p) \u2192 Mem\u2113p (fun a => \u2211 i in s, f i a) p\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Mem\u2113p (f i_1) p\n\u22a2 Mem\u2113p (fun a => \u2211 i in insert i s, f i a) p\n[PROOFSTEP]\nsimp only [his, Finset.sum_insert, not_false_iff]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 (i : \u03b1) \u2192 E i\nthis : DecidableEq \u03b9\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2113p (f i) p) \u2192 Mem\u2113p (fun a => \u2211 i in s, f i a) p\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Mem\u2113p (f i_1) p\n\u22a2 Mem\u2113p (fun a => f i a + \u2211 i in s, f i a) p\n[PROOFSTEP]\nexact (hf i (s.mem_insert_self i)).add (ih fun j hj => hf j (Finset.mem_insert_of_mem hj))\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nc : \ud835\udd5c\n\u22a2 Mem\u2113p (c \u2022 f) p\n[PROOFSTEP]\nrcases p.trichotomy with (rfl | rfl | hp)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : (i : \u03b1) \u2192 E i\nc : \ud835\udd5c\nhf : Mem\u2113p f 0\n\u22a2 Mem\u2113p (c \u2022 f) 0\n[PROOFSTEP]\napply mem\u2113p_zero\n[GOAL]\ncase inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : (i : \u03b1) \u2192 E i\nc : \ud835\udd5c\nhf : Mem\u2113p f 0\n\u22a2 Set.Finite {i | (c \u2022 f) i \u2260 0}\n[PROOFSTEP]\nrefine' hf.finite_dsupport.subset fun i => (_ : \u00acc \u2022 f i = 0 \u2192 \u00acf i = 0)\n[GOAL]\ncase inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : (i : \u03b1) \u2192 E i\nc : \ud835\udd5c\nhf : Mem\u2113p f 0\ni : \u03b1\n\u22a2 \u00acc \u2022 f i = 0 \u2192 \u00acf i = 0\n[PROOFSTEP]\nexact not_imp_not.mpr fun hf' => hf'.symm \u25b8 smul_zero c\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : (i : \u03b1) \u2192 E i\nc : \ud835\udd5c\nhf : Mem\u2113p f \u22a4\n\u22a2 Mem\u2113p (c \u2022 f) \u22a4\n[PROOFSTEP]\nobtain \u27e8A, hA\u27e9 := hf.bddAbove\n[GOAL]\ncase inr.inl.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : (i : \u03b1) \u2192 E i\nc : \ud835\udd5c\nhf : Mem\u2113p f \u22a4\nA : \u211d\nhA : A \u2208 upperBounds (Set.range fun i => \u2016f i\u2016)\n\u22a2 Mem\u2113p (c \u2022 f) \u22a4\n[PROOFSTEP]\nrefine' mem\u2113p_infty \u27e8\u2016c\u2016 * A, _\u27e9\n[GOAL]\ncase inr.inl.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : (i : \u03b1) \u2192 E i\nc : \ud835\udd5c\nhf : Mem\u2113p f \u22a4\nA : \u211d\nhA : A \u2208 upperBounds (Set.range fun i => \u2016f i\u2016)\n\u22a2 \u2016c\u2016 * A \u2208 upperBounds (Set.range fun i => \u2016(c \u2022 f) i\u2016)\n[PROOFSTEP]\nrintro a \u27e8i, rfl\u27e9\n[GOAL]\ncase inr.inl.intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : (i : \u03b1) \u2192 E i\nc : \ud835\udd5c\nhf : Mem\u2113p f \u22a4\nA : \u211d\nhA : A \u2208 upperBounds (Set.range fun i => \u2016f i\u2016)\ni : \u03b1\n\u22a2 (fun i => \u2016(c \u2022 f) i\u2016) i \u2264 \u2016c\u2016 * A\n[PROOFSTEP]\nrefine' (norm_smul_le _ _).trans _\n[GOAL]\ncase inr.inl.intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : (i : \u03b1) \u2192 E i\nc : \ud835\udd5c\nhf : Mem\u2113p f \u22a4\nA : \u211d\nhA : A \u2208 upperBounds (Set.range fun i => \u2016f i\u2016)\ni : \u03b1\n\u22a2 \u2016c\u2016 * \u2016f i\u2016 \u2264 \u2016c\u2016 * A\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_left (hA \u27e8i, rfl\u27e9) (norm_nonneg c)\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nc : \ud835\udd5c\nhp : 0 < ENNReal.toReal p\n\u22a2 Mem\u2113p (c \u2022 f) p\n[PROOFSTEP]\napply mem\u2113p_gen\n[GOAL]\ncase inr.inr.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nc : \ud835\udd5c\nhp : 0 < ENNReal.toReal p\n\u22a2 Summable fun i => \u2016(c \u2022 f) i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nhave := (hf.summable hp).mul_left (\u2191(\u2016c\u2016\u208a ^ p.toReal) : \u211d)\n[GOAL]\ncase inr.inr.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nc : \ud835\udd5c\nhp : 0 < ENNReal.toReal p\nthis : Summable fun i => \u2191(\u2016c\u2016\u208a ^ ENNReal.toReal p) * \u2016f i\u2016 ^ ENNReal.toReal p\n\u22a2 Summable fun i => \u2016(c \u2022 f) i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nsimp_rw [\u2190 coe_nnnorm, \u2190 NNReal.coe_rpow, \u2190 NNReal.coe_mul, NNReal.summable_coe, \u2190 NNReal.mul_rpow] at this \u22a2\n[GOAL]\ncase inr.inr.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nc : \ud835\udd5c\nhp : 0 < ENNReal.toReal p\nthis : Summable fun a => (\u2016c\u2016\u208a * \u2016f a\u2016\u208a) ^ ENNReal.toReal p\n\u22a2 Summable fun a => \u2016(c \u2022 f) a\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nrefine' NNReal.summable_of_le _ this\n[GOAL]\ncase inr.inr.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nc : \ud835\udd5c\nhp : 0 < ENNReal.toReal p\nthis : Summable fun a => (\u2016c\u2016\u208a * \u2016f a\u2016\u208a) ^ ENNReal.toReal p\n\u22a2 \u2200 (b : \u03b1), \u2016(c \u2022 f) b\u2016\u208a ^ ENNReal.toReal p \u2264 (\u2016c\u2016\u208a * \u2016f b\u2016\u208a) ^ ENNReal.toReal p\n[PROOFSTEP]\nintro i\n[GOAL]\ncase inr.inr.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nc : \ud835\udd5c\nhp : 0 < ENNReal.toReal p\nthis : Summable fun a => (\u2016c\u2016\u208a * \u2016f a\u2016\u208a) ^ ENNReal.toReal p\ni : \u03b1\n\u22a2 \u2016(c \u2022 f) i\u2016\u208a ^ ENNReal.toReal p \u2264 (\u2016c\u2016\u208a * \u2016f i\u2016\u208a) ^ ENNReal.toReal p\n[PROOFSTEP]\nexact NNReal.rpow_le_rpow (nnnorm_smul_le _ _) ENNReal.toReal_nonneg\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nf : \u03b1 \u2192 \ud835\udd5c\nhf : Mem\u2113p f p\nc : \ud835\udd5c\ni : \u03b1\n\u22a2 BoundedSMul \ud835\udd5c ((fun x => \ud835\udd5c) i)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u22a2 AddCommGroup (PreLp E)\n[PROOFSTEP]\nunfold PreLp\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u22a2 AddCommGroup ((i : \u03b1) \u2192 E i)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nf : \u03b9 \u2192 { x // x \u2208 lp E p }\ns : Finset \u03b9\n\u22a2 \u2191(\u2211 i in s, f i) = \u2211 i in s, \u2191(f i)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nhp : p = 0\n\u22a2 \u211d\n[PROOFSTEP]\nsubst hp\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E 0 }\n\u22a2 \u211d\n[PROOFSTEP]\nexact ((lp.mem\u2113p f).finite_dsupport.toFinset.card : \u211d)\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\n\u22a2 \u2016f\u2016 = \u2a06 (i : \u03b1), \u2016\u2191f i\u2016\n[PROOFSTEP]\ndsimp [norm]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\n\u22a2 (if hp : \u22a4 = 0 then\n Eq.rec (motive := fun x x_1 => { x_2 // x_2 \u2208 lp E x } \u2192 \u211d)\n (fun f => \u2191(Finset.card (Set.Finite.toFinset (_ : Set.Finite {i | \u2191f i \u2260 0})))) (_ : 0 = \u22a4) f\n else if \u22a4 = \u22a4 then \u2a06 (i : \u03b1), \u2016\u2191f i\u2016 else (\u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ 0) ^ (1 / 0)) =\n \u2a06 (i : \u03b1), \u2016\u2191f i\u2016\n[PROOFSTEP]\nrw [dif_neg ENNReal.top_ne_zero, if_pos rfl]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d : Nonempty \u03b1\nf : { x // x \u2208 lp E \u22a4 }\n\u22a2 IsLUB (Set.range fun i => \u2016\u2191f i\u2016) \u2016f\u2016\n[PROOFSTEP]\nrw [lp.norm_eq_ciSup]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d : Nonempty \u03b1\nf : { x // x \u2208 lp E \u22a4 }\n\u22a2 IsLUB (Set.range fun i => \u2016\u2191f i\u2016) (\u2a06 (i : \u03b1), \u2016\u2191f i\u2016)\n[PROOFSTEP]\nexact isLUB_ciSup (lp.mem\u2113p f)\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\n\u22a2 \u2016f\u2016 = (\u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\ndsimp [norm]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\n\u22a2 (if hp : p = 0 then\n Eq.rec (motive := fun x x_1 => { x_2 // x_2 \u2208 lp E x } \u2192 \u211d)\n (fun f => \u2191(Finset.card (Set.Finite.toFinset (_ : Set.Finite {i | \u2191f i \u2260 0})))) (_ : 0 = p) f\n else if p = \u22a4 then \u2a06 (i : \u03b1), \u2016\u2191f i\u2016 else (\u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)) =\n (\u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nrw [ENNReal.toReal_pos_iff] at hp \n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < p \u2227 p < \u22a4\nf : { x // x \u2208 lp E p }\n\u22a2 (if hp : p = 0 then\n Eq.rec (motive := fun x x_1 => { x_2 // x_2 \u2208 lp E x } \u2192 \u211d)\n (fun f => \u2191(Finset.card (Set.Finite.toFinset (_ : Set.Finite {i | \u2191f i \u2260 0})))) (_ : 0 = p) f\n else if p = \u22a4 then \u2a06 (i : \u03b1), \u2016\u2191f i\u2016 else (\u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)) =\n (\u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nrw [dif_neg hp.1.ne', if_neg hp.2.ne]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\n\u22a2 \u2016f\u2016 ^ ENNReal.toReal p = \u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [norm_eq_tsum_rpow hp, \u2190 Real.rpow_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\n\u22a2 (\u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p * ENNReal.toReal p) =\n \u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nfield_simp [hp.ne']\n[GOAL]\ncase hx\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\n\u22a2 0 \u2264 \u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\napply tsum_nonneg\n[GOAL]\ncase hx.h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\n\u22a2 \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nintro i\n[GOAL]\ncase hx.h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ni : \u03b1\n\u22a2 0 \u2264 \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\ncalc\n (0 : \u211d) = (0 : \u211d) ^ p.toReal := by rw [Real.zero_rpow hp.ne']\n _ \u2264 _ := Real.rpow_le_rpow rfl.le (norm_nonneg (f i)) hp.le\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ni : \u03b1\n\u22a2 0 = 0 ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [Real.zero_rpow hp.ne']\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\n\u22a2 HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nrw [norm_rpow_eq_tsum hp]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\n\u22a2 HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nexact ((lp.mem\u2113p f).summable hp).hasSum\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\n\u22a2 0 \u2264 \u2016f\u2016\n[PROOFSTEP]\nrcases p.trichotomy with (rfl | rfl | hp)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E 0 }\n\u22a2 0 \u2264 \u2016f\u2016\n[PROOFSTEP]\nsimp [lp.norm_eq_card_dsupport f]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\n\u22a2 0 \u2264 \u2016f\u2016\n[PROOFSTEP]\ncases' isEmpty_or_nonempty \u03b1 with _i _i\n[GOAL]\ncase inr.inl.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\n_i : IsEmpty \u03b1\n\u22a2 0 \u2264 \u2016f\u2016\n[PROOFSTEP]\nrw [lp.norm_eq_ciSup]\n[GOAL]\ncase inr.inl.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\n_i : IsEmpty \u03b1\n\u22a2 0 \u2264 \u2a06 (i : \u03b1), \u2016\u2191f i\u2016\n[PROOFSTEP]\nsimp [Real.ciSup_empty]\n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\n_i : Nonempty \u03b1\n\u22a2 0 \u2264 \u2016f\u2016\n[PROOFSTEP]\ninhabit \u03b1\n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\n_i : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\n\u22a2 0 \u2264 \u2016f\u2016\n[PROOFSTEP]\nexact (norm_nonneg (f default)).trans ((lp.isLUB_norm f).1 \u27e8default, rfl\u27e9)\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\n\u22a2 0 \u2264 \u2016f\u2016\n[PROOFSTEP]\nrw [lp.norm_eq_tsum_rpow hp f]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\n\u22a2 0 \u2264 (\u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nrefine' Real.rpow_nonneg_of_nonneg (tsum_nonneg _) _\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\n\u22a2 \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nexact fun i => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u22a2 \u20160\u2016 = 0\n[PROOFSTEP]\nrcases p.trichotomy with (rfl | rfl | hp)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u22a2 \u20160\u2016 = 0\n[PROOFSTEP]\nsimp [lp.norm_eq_card_dsupport]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u22a2 \u20160\u2016 = 0\n[PROOFSTEP]\nsimp [lp.norm_eq_ciSup]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\n\u22a2 \u20160\u2016 = 0\n[PROOFSTEP]\nrw [lp.norm_eq_tsum_rpow hp]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\n\u22a2 (\u2211' (i : \u03b1), \u2016\u21910 i\u2016 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) = 0\n[PROOFSTEP]\nhave hp' : 1 / p.toReal \u2260 0 := one_div_ne_zero hp.ne'\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nhp' : 1 / ENNReal.toReal p \u2260 0\n\u22a2 (\u2211' (i : \u03b1), \u2016\u21910 i\u2016 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) = 0\n[PROOFSTEP]\nsimpa [Real.zero_rpow hp.ne'] using Real.zero_rpow hp'\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\n\u22a2 \u2016f\u2016 = 0 \u2194 f = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => _, by rintro rfl; exact norm_zero\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\n\u22a2 f = 0 \u2192 \u2016f\u2016 = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u22a2 \u20160\u2016 = 0\n[PROOFSTEP]\nexact norm_zero\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nh : \u2016f\u2016 = 0\n\u22a2 f = 0\n[PROOFSTEP]\nrcases p.trichotomy with (rfl | rfl | hp)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E 0 }\nh : \u2016f\u2016 = 0\n\u22a2 f = 0\n[PROOFSTEP]\next i\n[GOAL]\ncase inl.h.h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E 0 }\nh : \u2016f\u2016 = 0\ni : \u03b1\n\u22a2 \u2191f i = \u21910 i\n[PROOFSTEP]\nhave : {i : \u03b1 | \u00acf i = 0} = \u2205 := by simpa [lp.norm_eq_card_dsupport f] using h\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E 0 }\nh : \u2016f\u2016 = 0\ni : \u03b1\n\u22a2 {i | \u00ac\u2191f i = 0} = \u2205\n[PROOFSTEP]\nsimpa [lp.norm_eq_card_dsupport f] using h\n[GOAL]\ncase inl.h.h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E 0 }\nh : \u2016f\u2016 = 0\ni : \u03b1\nthis : {i | \u00ac\u2191f i = 0} = \u2205\n\u22a2 \u2191f i = \u21910 i\n[PROOFSTEP]\nhave : (\u00acf i = 0) = False := congr_fun this i\n[GOAL]\ncase inl.h.h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E 0 }\nh : \u2016f\u2016 = 0\ni : \u03b1\nthis\u271d : {i | \u00ac\u2191f i = 0} = \u2205\nthis : (\u00ac\u2191f i = 0) = False\n\u22a2 \u2191f i = \u21910 i\n[PROOFSTEP]\ntauto\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh : \u2016f\u2016 = 0\n\u22a2 f = 0\n[PROOFSTEP]\ncases' isEmpty_or_nonempty \u03b1 with _i _i\n[GOAL]\ncase inr.inl.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh : \u2016f\u2016 = 0\n_i : IsEmpty \u03b1\n\u22a2 f = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh : \u2016f\u2016 = 0\n_i : Nonempty \u03b1\n\u22a2 f = 0\n[PROOFSTEP]\nhave H : IsLUB (Set.range fun i => \u2016f i\u2016) 0 := by simpa [h] using lp.isLUB_norm f\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh : \u2016f\u2016 = 0\n_i : Nonempty \u03b1\n\u22a2 IsLUB (Set.range fun i => \u2016\u2191f i\u2016) 0\n[PROOFSTEP]\nsimpa [h] using lp.isLUB_norm f\n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh : \u2016f\u2016 = 0\n_i : Nonempty \u03b1\nH : IsLUB (Set.range fun i => \u2016\u2191f i\u2016) 0\n\u22a2 f = 0\n[PROOFSTEP]\next i\n[GOAL]\ncase inr.inl.inr.h.h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh : \u2016f\u2016 = 0\n_i : Nonempty \u03b1\nH : IsLUB (Set.range fun i => \u2016\u2191f i\u2016) 0\ni : \u03b1\n\u22a2 \u2191f i = \u21910 i\n[PROOFSTEP]\nhave : \u2016f i\u2016 = 0 := le_antisymm (H.1 \u27e8i, rfl\u27e9) (norm_nonneg _)\n[GOAL]\ncase inr.inl.inr.h.h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh : \u2016f\u2016 = 0\n_i : Nonempty \u03b1\nH : IsLUB (Set.range fun i => \u2016\u2191f i\u2016) 0\ni : \u03b1\nthis : \u2016\u2191f i\u2016 = 0\n\u22a2 \u2191f i = \u21910 i\n[PROOFSTEP]\nsimpa using this\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nh : \u2016f\u2016 = 0\nhp : 0 < ENNReal.toReal p\n\u22a2 f = 0\n[PROOFSTEP]\nhave hf : HasSum (fun i : \u03b1 => \u2016f i\u2016 ^ p.toReal) 0 :=\n by\n have := lp.hasSum_norm hp f\n rwa [h, Real.zero_rpow hp.ne'] at this \n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nh : \u2016f\u2016 = 0\nhp : 0 < ENNReal.toReal p\n\u22a2 HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) 0\n[PROOFSTEP]\nhave := lp.hasSum_norm hp f\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nh : \u2016f\u2016 = 0\nhp : 0 < ENNReal.toReal p\nthis : HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)\n\u22a2 HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) 0\n[PROOFSTEP]\nrwa [h, Real.zero_rpow hp.ne'] at this \n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nh : \u2016f\u2016 = 0\nhp : 0 < ENNReal.toReal p\nhf : HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) 0\n\u22a2 f = 0\n[PROOFSTEP]\nhave : \u2200 i, 0 \u2264 \u2016f i\u2016 ^ p.toReal := fun i => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nh : \u2016f\u2016 = 0\nhp : 0 < ENNReal.toReal p\nhf : HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) 0\nthis : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n\u22a2 f = 0\n[PROOFSTEP]\nrw [hasSum_zero_iff_of_nonneg this] at hf \n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nh : \u2016f\u2016 = 0\nhp : 0 < ENNReal.toReal p\nhf : (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) = 0\nthis : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n\u22a2 f = 0\n[PROOFSTEP]\next i\n[GOAL]\ncase inr.inr.h.h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nh : \u2016f\u2016 = 0\nhp : 0 < ENNReal.toReal p\nhf : (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) = 0\nthis : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016 ^ ENNReal.toReal p\ni : \u03b1\n\u22a2 \u2191f i = \u21910 i\n[PROOFSTEP]\nhave : f i = 0 \u2227 p.toReal \u2260 0 := by simpa [Real.rpow_eq_zero_iff_of_nonneg (norm_nonneg (f i))] using congr_fun hf i\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nh : \u2016f\u2016 = 0\nhp : 0 < ENNReal.toReal p\nhf : (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) = 0\nthis : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016 ^ ENNReal.toReal p\ni : \u03b1\n\u22a2 \u2191f i = 0 \u2227 ENNReal.toReal p \u2260 0\n[PROOFSTEP]\nsimpa [Real.rpow_eq_zero_iff_of_nonneg (norm_nonneg (f i))] using congr_fun hf i\n[GOAL]\ncase inr.inr.h.h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nh : \u2016f\u2016 = 0\nhp : 0 < ENNReal.toReal p\nhf : (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) = 0\nthis\u271d : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016 ^ ENNReal.toReal p\ni : \u03b1\nthis : \u2191f i = 0 \u2227 ENNReal.toReal p \u2260 0\n\u22a2 \u2191f i = \u21910 i\n[PROOFSTEP]\nexact this.1\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\n\u22a2 f = 0 \u2194 \u2191f = 0\n[PROOFSTEP]\nrw [lp.ext_iff, coeFn_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\n\u22a2 \u2016-f\u2016 = \u2016f\u2016\n[PROOFSTEP]\nrcases p.trichotomy with (rfl | rfl | hp)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E 0 }\n\u22a2 \u2016-f\u2016 = \u2016f\u2016\n[PROOFSTEP]\nsimp only [norm_eq_card_dsupport, coeFn_neg, Pi.neg_apply, ne_eq, neg_eq_zero]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\n\u22a2 \u2016-f\u2016 = \u2016f\u2016\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b1\n[GOAL]\ncase inr.inl.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : IsEmpty \u03b1\n\u22a2 \u2016-f\u2016 = \u2016f\u2016\n[PROOFSTEP]\nsimp only [lp.eq_zero' f, neg_zero, norm_zero]\n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\n\u22a2 \u2016-f\u2016 = \u2016f\u2016\n[PROOFSTEP]\napply (lp.isLUB_norm (-f)).unique\n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\n\u22a2 IsLUB (Set.range fun i => \u2016\u2191(-f) i\u2016) \u2016f\u2016\n[PROOFSTEP]\nsimpa only [coeFn_neg, Pi.neg_apply, norm_neg] using lp.isLUB_norm f\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\n\u22a2 \u2016-f\u2016 = \u2016f\u2016\n[PROOFSTEP]\nsuffices \u2016-f\u2016 ^ p.toReal = \u2016f\u2016 ^ p.toReal by exact Real.rpow_left_injOn hp.ne' (norm_nonneg' _) (norm_nonneg' _) this\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\nthis : \u2016-f\u2016 ^ ENNReal.toReal p = \u2016f\u2016 ^ ENNReal.toReal p\n\u22a2 \u2016-f\u2016 = \u2016f\u2016\n[PROOFSTEP]\nexact Real.rpow_left_injOn hp.ne' (norm_nonneg' _) (norm_nonneg' _) this\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\n\u22a2 \u2016-f\u2016 ^ ENNReal.toReal p = \u2016f\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\napply (lp.hasSum_norm hp (-f)).unique\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\n\u22a2 HasSum (fun i => \u2016\u2191(-f) i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nsimpa only [coeFn_neg, Pi.neg_apply, _root_.norm_neg] using lp.hasSum_norm hp f\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 p)\nf g : { x // x \u2208 lp E p }\n\u22a2 \u2016f + g\u2016 \u2264 \u2016f\u2016 + \u2016g\u2016\n[PROOFSTEP]\nrcases p.dichotomy with (rfl | hp')\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 \u22a4)\nf g : { x // x \u2208 lp E \u22a4 }\n\u22a2 \u2016f + g\u2016 \u2264 \u2016f\u2016 + \u2016g\u2016\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b1\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 \u22a4)\nf g : { x // x \u2208 lp E \u22a4 }\nh\u271d : IsEmpty \u03b1\n\u22a2 \u2016f + g\u2016 \u2264 \u2016f\u2016 + \u2016g\u2016\n[PROOFSTEP]\nrw [lp.eq_zero' f]\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 \u22a4)\nf g : { x // x \u2208 lp E \u22a4 }\nh\u271d : IsEmpty \u03b1\n\u22a2 \u20160 + g\u2016 \u2264 \u20160\u2016 + \u2016g\u2016\n[PROOFSTEP]\nsimp only [zero_add, norm_zero, le_refl]\n -- porting note: just `simp` was slow\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 \u22a4)\nf g : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\n\u22a2 \u2016f + g\u2016 \u2264 \u2016f\u2016 + \u2016g\u2016\n[PROOFSTEP]\nrefine' (lp.isLUB_norm (f + g)).2 _\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 \u22a4)\nf g : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\n\u22a2 \u2016f\u2016 + \u2016g\u2016 \u2208 upperBounds (Set.range fun i => \u2016\u2191(f + g) i\u2016)\n[PROOFSTEP]\nrintro x \u27e8i, rfl\u27e9\n[GOAL]\ncase inl.inr.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 \u22a4)\nf g : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\ni : \u03b1\n\u22a2 (fun i => \u2016\u2191(f + g) i\u2016) i \u2264 \u2016f\u2016 + \u2016g\u2016\n[PROOFSTEP]\nrefine' le_trans _ (add_mem_upperBounds_add (lp.isLUB_norm f).1 (lp.isLUB_norm g).1 \u27e8_, _, \u27e8i, rfl\u27e9, \u27e8i, rfl\u27e9, rfl\u27e9)\n[GOAL]\ncase inl.inr.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 \u22a4)\nf g : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\ni : \u03b1\n\u22a2 (fun i => \u2016\u2191(f + g) i\u2016) i \u2264 (fun x x_1 => x + x_1) ((fun i => \u2016\u2191f i\u2016) i) ((fun i => \u2016\u2191g i\u2016) i)\n[PROOFSTEP]\nexact norm_add_le (f i) (g i)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 p)\nf g : { x // x \u2208 lp E p }\nhp' : 1 \u2264 ENNReal.toReal p\n\u22a2 \u2016f + g\u2016 \u2264 \u2016f\u2016 + \u2016g\u2016\n[PROOFSTEP]\nhave hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 p)\nf g : { x // x \u2208 lp E p }\nhp' : 1 \u2264 ENNReal.toReal p\nhp'' : 0 < ENNReal.toReal p\n\u22a2 \u2016f + g\u2016 \u2264 \u2016f\u2016 + \u2016g\u2016\n[PROOFSTEP]\nhave hf\u2081 : \u2200 i, 0 \u2264 \u2016f i\u2016 := fun i => norm_nonneg _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 p)\nf g : { x // x \u2208 lp E p }\nhp' : 1 \u2264 ENNReal.toReal p\nhp'' : 0 < ENNReal.toReal p\nhf\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016\n\u22a2 \u2016f + g\u2016 \u2264 \u2016f\u2016 + \u2016g\u2016\n[PROOFSTEP]\nhave hg\u2081 : \u2200 i, 0 \u2264 \u2016g i\u2016 := fun i => norm_nonneg _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 p)\nf g : { x // x \u2208 lp E p }\nhp' : 1 \u2264 ENNReal.toReal p\nhp'' : 0 < ENNReal.toReal p\nhf\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016\nhg\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191g i\u2016\n\u22a2 \u2016f + g\u2016 \u2264 \u2016f\u2016 + \u2016g\u2016\n[PROOFSTEP]\nhave hf\u2082 := lp.hasSum_norm hp'' f\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 p)\nf g : { x // x \u2208 lp E p }\nhp' : 1 \u2264 ENNReal.toReal p\nhp'' : 0 < ENNReal.toReal p\nhf\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016\nhg\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191g i\u2016\nhf\u2082 : HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)\n\u22a2 \u2016f + g\u2016 \u2264 \u2016f\u2016 + \u2016g\u2016\n[PROOFSTEP]\nhave hg\u2082 := lp.hasSum_norm hp'' g\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 p)\nf g : { x // x \u2208 lp E p }\nhp' : 1 \u2264 ENNReal.toReal p\nhp'' : 0 < ENNReal.toReal p\nhf\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016\nhg\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191g i\u2016\nhf\u2082 : HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)\nhg\u2082 : HasSum (fun i => \u2016\u2191g i\u2016 ^ ENNReal.toReal p) (\u2016g\u2016 ^ ENNReal.toReal p)\n\u22a2 \u2016f + g\u2016 \u2264 \u2016f\u2016 + \u2016g\u2016\n[PROOFSTEP]\nobtain \u27e8C, hC\u2081, hC\u2082, hCfg\u27e9 := Real.Lp_add_le_hasSum_of_nonneg hp' hf\u2081 hg\u2081 (norm_nonneg' _) (norm_nonneg' _) hf\u2082 hg\u2082\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 p)\nf g : { x // x \u2208 lp E p }\nhp' : 1 \u2264 ENNReal.toReal p\nhp'' : 0 < ENNReal.toReal p\nhf\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016\nhg\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191g i\u2016\nhf\u2082 : HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)\nhg\u2082 : HasSum (fun i => \u2016\u2191g i\u2016 ^ ENNReal.toReal p) (\u2016g\u2016 ^ ENNReal.toReal p)\nC : \u211d\nhC\u2081 : 0 \u2264 C\nhC\u2082 : C \u2264 \u2016f\u2016 + \u2016g\u2016\nhCfg : HasSum (fun i => (\u2016\u2191f i\u2016 + \u2016\u2191g i\u2016) ^ ENNReal.toReal p) (C ^ ENNReal.toReal p)\n\u22a2 \u2016f + g\u2016 \u2264 \u2016f\u2016 + \u2016g\u2016\n[PROOFSTEP]\nrefine' le_trans _ hC\u2082\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 p)\nf g : { x // x \u2208 lp E p }\nhp' : 1 \u2264 ENNReal.toReal p\nhp'' : 0 < ENNReal.toReal p\nhf\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016\nhg\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191g i\u2016\nhf\u2082 : HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)\nhg\u2082 : HasSum (fun i => \u2016\u2191g i\u2016 ^ ENNReal.toReal p) (\u2016g\u2016 ^ ENNReal.toReal p)\nC : \u211d\nhC\u2081 : 0 \u2264 C\nhC\u2082 : C \u2264 \u2016f\u2016 + \u2016g\u2016\nhCfg : HasSum (fun i => (\u2016\u2191f i\u2016 + \u2016\u2191g i\u2016) ^ ENNReal.toReal p) (C ^ ENNReal.toReal p)\n\u22a2 \u2016f + g\u2016 \u2264 C\n[PROOFSTEP]\nrw [\u2190 Real.rpow_le_rpow_iff (norm_nonneg' (f + g)) hC\u2081 hp'']\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 p)\nf g : { x // x \u2208 lp E p }\nhp' : 1 \u2264 ENNReal.toReal p\nhp'' : 0 < ENNReal.toReal p\nhf\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016\nhg\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191g i\u2016\nhf\u2082 : HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)\nhg\u2082 : HasSum (fun i => \u2016\u2191g i\u2016 ^ ENNReal.toReal p) (\u2016g\u2016 ^ ENNReal.toReal p)\nC : \u211d\nhC\u2081 : 0 \u2264 C\nhC\u2082 : C \u2264 \u2016f\u2016 + \u2016g\u2016\nhCfg : HasSum (fun i => (\u2016\u2191f i\u2016 + \u2016\u2191g i\u2016) ^ ENNReal.toReal p) (C ^ ENNReal.toReal p)\n\u22a2 \u2016f + g\u2016 ^ ENNReal.toReal p \u2264 C ^ ENNReal.toReal p\n[PROOFSTEP]\nrefine' hasSum_le _ (lp.hasSum_norm hp'' (f + g)) hCfg\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 p)\nf g : { x // x \u2208 lp E p }\nhp' : 1 \u2264 ENNReal.toReal p\nhp'' : 0 < ENNReal.toReal p\nhf\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016\nhg\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191g i\u2016\nhf\u2082 : HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)\nhg\u2082 : HasSum (fun i => \u2016\u2191g i\u2016 ^ ENNReal.toReal p) (\u2016g\u2016 ^ ENNReal.toReal p)\nC : \u211d\nhC\u2081 : 0 \u2264 C\nhC\u2082 : C \u2264 \u2016f\u2016 + \u2016g\u2016\nhCfg : HasSum (fun i => (\u2016\u2191f i\u2016 + \u2016\u2191g i\u2016) ^ ENNReal.toReal p) (C ^ ENNReal.toReal p)\n\u22a2 \u2200 (i : \u03b1), \u2016\u2191(f + g) i\u2016 ^ ENNReal.toReal p \u2264 (\u2016\u2191f i\u2016 + \u2016\u2191g i\u2016) ^ ENNReal.toReal p\n[PROOFSTEP]\nintro i\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : Fact (1 \u2264 p)\nf g : { x // x \u2208 lp E p }\nhp' : 1 \u2264 ENNReal.toReal p\nhp'' : 0 < ENNReal.toReal p\nhf\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016\nhg\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191g i\u2016\nhf\u2082 : HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)\nhg\u2082 : HasSum (fun i => \u2016\u2191g i\u2016 ^ ENNReal.toReal p) (\u2016g\u2016 ^ ENNReal.toReal p)\nC : \u211d\nhC\u2081 : 0 \u2264 C\nhC\u2082 : C \u2264 \u2016f\u2016 + \u2016g\u2016\nhCfg : HasSum (fun i => (\u2016\u2191f i\u2016 + \u2016\u2191g i\u2016) ^ ENNReal.toReal p) (C ^ ENNReal.toReal p)\ni : \u03b1\n\u22a2 \u2016\u2191(f + g) i\u2016 ^ ENNReal.toReal p \u2264 (\u2016\u2191f i\u2016 + \u2016\u2191g i\u2016) ^ ENNReal.toReal p\n[PROOFSTEP]\nexact Real.rpow_le_rpow (norm_nonneg _) (norm_add_le _ _) hp''.le\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nhpq : Real.IsConjugateExponent (ENNReal.toReal p) (ENNReal.toReal q)\nf : { x // x \u2208 lp E p }\ng : { x // x \u2208 lp E q }\n\u22a2 (Summable fun i => \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016) \u2227 \u2211' (i : \u03b1), \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016 \u2264 \u2016f\u2016 * \u2016g\u2016\n[PROOFSTEP]\nhave hf\u2081 : \u2200 i, 0 \u2264 \u2016f i\u2016 := fun i => norm_nonneg _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nhpq : Real.IsConjugateExponent (ENNReal.toReal p) (ENNReal.toReal q)\nf : { x // x \u2208 lp E p }\ng : { x // x \u2208 lp E q }\nhf\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016\n\u22a2 (Summable fun i => \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016) \u2227 \u2211' (i : \u03b1), \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016 \u2264 \u2016f\u2016 * \u2016g\u2016\n[PROOFSTEP]\nhave hg\u2081 : \u2200 i, 0 \u2264 \u2016g i\u2016 := fun i => norm_nonneg _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nhpq : Real.IsConjugateExponent (ENNReal.toReal p) (ENNReal.toReal q)\nf : { x // x \u2208 lp E p }\ng : { x // x \u2208 lp E q }\nhf\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016\nhg\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191g i\u2016\n\u22a2 (Summable fun i => \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016) \u2227 \u2211' (i : \u03b1), \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016 \u2264 \u2016f\u2016 * \u2016g\u2016\n[PROOFSTEP]\nhave hf\u2082 := lp.hasSum_norm hpq.pos f\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nhpq : Real.IsConjugateExponent (ENNReal.toReal p) (ENNReal.toReal q)\nf : { x // x \u2208 lp E p }\ng : { x // x \u2208 lp E q }\nhf\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016\nhg\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191g i\u2016\nhf\u2082 : HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)\n\u22a2 (Summable fun i => \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016) \u2227 \u2211' (i : \u03b1), \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016 \u2264 \u2016f\u2016 * \u2016g\u2016\n[PROOFSTEP]\nhave hg\u2082 := lp.hasSum_norm hpq.symm.pos g\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nhpq : Real.IsConjugateExponent (ENNReal.toReal p) (ENNReal.toReal q)\nf : { x // x \u2208 lp E p }\ng : { x // x \u2208 lp E q }\nhf\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016\nhg\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191g i\u2016\nhf\u2082 : HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)\nhg\u2082 : HasSum (fun i => \u2016\u2191g i\u2016 ^ ENNReal.toReal q) (\u2016g\u2016 ^ ENNReal.toReal q)\n\u22a2 (Summable fun i => \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016) \u2227 \u2211' (i : \u03b1), \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016 \u2264 \u2016f\u2016 * \u2016g\u2016\n[PROOFSTEP]\nobtain \u27e8C, -, hC', hC\u27e9 := Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg hpq (norm_nonneg' _) (norm_nonneg' _) hf\u2081 hg\u2081 hf\u2082 hg\u2082\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nhpq : Real.IsConjugateExponent (ENNReal.toReal p) (ENNReal.toReal q)\nf : { x // x \u2208 lp E p }\ng : { x // x \u2208 lp E q }\nhf\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016\nhg\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191g i\u2016\nhf\u2082 : HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)\nhg\u2082 : HasSum (fun i => \u2016\u2191g i\u2016 ^ ENNReal.toReal q) (\u2016g\u2016 ^ ENNReal.toReal q)\nC : \u211d\nhC' : C \u2264 \u2016f\u2016 * \u2016g\u2016\nhC : HasSum (fun i => \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016) C\n\u22a2 (Summable fun i => \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016) \u2227 \u2211' (i : \u03b1), \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016 \u2264 \u2016f\u2016 * \u2016g\u2016\n[PROOFSTEP]\nrw [\u2190 hC.tsum_eq] at hC' \n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q\u271d : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\np q : \u211d\u22650\u221e\nhpq : Real.IsConjugateExponent (ENNReal.toReal p) (ENNReal.toReal q)\nf : { x // x \u2208 lp E p }\ng : { x // x \u2208 lp E q }\nhf\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016\nhg\u2081 : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191g i\u2016\nhf\u2082 : HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)\nhg\u2082 : HasSum (fun i => \u2016\u2191g i\u2016 ^ ENNReal.toReal q) (\u2016g\u2016 ^ ENNReal.toReal q)\nC : \u211d\nhC' : \u2211' (b : \u03b1), \u2016\u2191f b\u2016 * \u2016\u2191g b\u2016 \u2264 \u2016f\u2016 * \u2016g\u2016\nhC : HasSum (fun i => \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016) C\n\u22a2 (Summable fun i => \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016) \u2227 \u2211' (i : \u03b1), \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016 \u2264 \u2016f\u2016 * \u2016g\u2016\n[PROOFSTEP]\nexact \u27e8hC.summable, hC'\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : p \u2260 0\nf : { x // x \u2208 lp E p }\ni : \u03b1\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016\n[PROOFSTEP]\nrcases eq_or_ne p \u221e with (rfl | hp')\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ni : \u03b1\nhp : \u22a4 \u2260 0\nf : { x // x \u2208 lp E \u22a4 }\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016\n[PROOFSTEP]\nhaveI : Nonempty \u03b1 := \u27e8i\u27e9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ni : \u03b1\nhp : \u22a4 \u2260 0\nf : { x // x \u2208 lp E \u22a4 }\nthis : Nonempty \u03b1\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016\n[PROOFSTEP]\nexact (isLUB_norm f).1 \u27e8i, rfl\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : p \u2260 0\nf : { x // x \u2208 lp E p }\ni : \u03b1\nhp' : p \u2260 \u22a4\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016\n[PROOFSTEP]\nhave hp'' : 0 < p.toReal := ENNReal.toReal_pos hp hp'\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : p \u2260 0\nf : { x // x \u2208 lp E p }\ni : \u03b1\nhp' : p \u2260 \u22a4\nhp'' : 0 < ENNReal.toReal p\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016\n[PROOFSTEP]\nhave : \u2200 i, 0 \u2264 \u2016f i\u2016 ^ p.toReal := fun i => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : p \u2260 0\nf : { x // x \u2208 lp E p }\ni : \u03b1\nhp' : p \u2260 \u22a4\nhp'' : 0 < ENNReal.toReal p\nthis : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n\u22a2 \u2016\u2191f i\u2016 \u2264 \u2016f\u2016\n[PROOFSTEP]\nrw [\u2190 Real.rpow_le_rpow_iff (norm_nonneg _) (norm_nonneg' _) hp'']\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : p \u2260 0\nf : { x // x \u2208 lp E p }\ni : \u03b1\nhp' : p \u2260 \u22a4\nhp'' : 0 < ENNReal.toReal p\nthis : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n\u22a2 \u2016\u2191f i\u2016 ^ ENNReal.toReal p \u2264 \u2016f\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nconvert le_hasSum (hasSum_norm hp'' f) i fun i _ => this i\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\n\u22a2 \u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p \u2264 \u2016f\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [lp.norm_rpow_eq_tsum hp f]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\n\u22a2 \u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p \u2264 \u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nhave : \u2200 i, 0 \u2264 \u2016f i\u2016 ^ p.toReal := fun i => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\nthis : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n\u22a2 \u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p \u2264 \u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nrefine' sum_le_tsum _ (fun i _ => this i) _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\nthis : \u2200 (i : \u03b1), 0 \u2264 \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n\u22a2 Summable fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nexact (lp.mem\u2113p f).summable hp\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d : Nonempty \u03b1\nf : { x // x \u2208 lp E \u22a4 }\nC : \u211d\nhCf : \u2200 (i : \u03b1), \u2016\u2191f i\u2016 \u2264 C\n\u22a2 \u2016f\u2016 \u2264 C\n[PROOFSTEP]\nrefine' (isLUB_norm f).2 _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d : Nonempty \u03b1\nf : { x // x \u2208 lp E \u22a4 }\nC : \u211d\nhCf : \u2200 (i : \u03b1), \u2016\u2191f i\u2016 \u2264 C\n\u22a2 C \u2208 upperBounds (Set.range fun i => \u2016\u2191f i\u2016)\n[PROOFSTEP]\nrintro - \u27e8i, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d : Nonempty \u03b1\nf : { x // x \u2208 lp E \u22a4 }\nC : \u211d\nhCf : \u2200 (i : \u03b1), \u2016\u2191f i\u2016 \u2264 C\ni : \u03b1\n\u22a2 (fun i => \u2016\u2191f i\u2016) i \u2264 C\n[PROOFSTEP]\nexact hCf i\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nC : \u211d\nhC : 0 \u2264 C\nhCf : \u2200 (i : \u03b1), \u2016\u2191f i\u2016 \u2264 C\n\u22a2 \u2016f\u2016 \u2264 C\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b1\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nC : \u211d\nhC : 0 \u2264 C\nhCf : \u2200 (i : \u03b1), \u2016\u2191f i\u2016 \u2264 C\nh\u271d : IsEmpty \u03b1\n\u22a2 \u2016f\u2016 \u2264 C\n[PROOFSTEP]\nsimpa [eq_zero' f] using hC\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nC : \u211d\nhC : 0 \u2264 C\nhCf : \u2200 (i : \u03b1), \u2016\u2191f i\u2016 \u2264 C\nh\u271d : Nonempty \u03b1\n\u22a2 \u2016f\u2016 \u2264 C\n[PROOFSTEP]\nexact norm_le_of_forall_le' C hCf\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nC : \u211d\nhC : 0 \u2264 C\nf : { x // x \u2208 lp E p }\nhf : \u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p \u2264 C ^ ENNReal.toReal p\n\u22a2 \u2016f\u2016 \u2264 C\n[PROOFSTEP]\nrw [\u2190 Real.rpow_le_rpow_iff (norm_nonneg' _) hC hp, norm_rpow_eq_tsum hp]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nC : \u211d\nhC : 0 \u2264 C\nf : { x // x \u2208 lp E p }\nhf : \u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p \u2264 C ^ ENNReal.toReal p\n\u22a2 \u2211' (i : \u03b1), \u2016\u2191f i\u2016 ^ ENNReal.toReal p \u2264 C ^ ENNReal.toReal p\n[PROOFSTEP]\nexact hf\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nsrc\u271d : AddSubgroup (PreLp E) := lp E p\nc : \ud835\udd5c\nf : PreLp E\nhf : f \u2208 src\u271d.carrier\n\u22a2 c \u2022 f \u2208 src\u271d.carrier\n[PROOFSTEP]\nsimpa using mem_lp_const_smul c \u27e8f, hf\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nhp : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nrcases p.trichotomy with (rfl | rfl | hp)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nc : \ud835\udd5c\nhp : 0 \u2260 0\nf : { x // x \u2208 lp E 0 }\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nexact absurd rfl hp\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nc : \ud835\udd5c\nhp : \u22a4 \u2260 0\nf : { x // x \u2208 lp E \u22a4 }\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b1\n[GOAL]\ncase inr.inl.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nc : \ud835\udd5c\nhp : \u22a4 \u2260 0\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : IsEmpty \u03b1\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nsimp [lp.eq_zero' f]\n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nc : \ud835\udd5c\nhp : \u22a4 \u2260 0\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nhave hcf := lp.isLUB_norm (c \u2022 f)\n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nc : \ud835\udd5c\nhp : \u22a4 \u2260 0\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\nhcf : IsLUB (Set.range fun i => \u2016\u2191(c \u2022 f) i\u2016) \u2016c \u2022 f\u2016\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nhave hfc := (lp.isLUB_norm f).mul_left (norm_nonneg c)\n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nc : \ud835\udd5c\nhp : \u22a4 \u2260 0\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\nhcf : IsLUB (Set.range fun i => \u2016\u2191(c \u2022 f) i\u2016) \u2016c \u2022 f\u2016\nhfc : IsLUB ((fun b => \u2016c\u2016 * b) '' Set.range fun i => \u2016\u2191f i\u2016) (\u2016c\u2016 * \u2016f\u2016)\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nsimp_rw [\u2190 Set.range_comp, Function.comp] at hfc \n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nc : \ud835\udd5c\nhp : \u22a4 \u2260 0\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\nhcf : IsLUB (Set.range fun i => \u2016\u2191(c \u2022 f) i\u2016) \u2016c \u2022 f\u2016\nhfc : IsLUB (Set.range fun x => \u2016c\u2016 * \u2016\u2191f x\u2016) (\u2016c\u2016 * \u2016f\u2016)\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nrefine' hcf.right _\n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nc : \ud835\udd5c\nhp : \u22a4 \u2260 0\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\nhcf : IsLUB (Set.range fun i => \u2016\u2191(c \u2022 f) i\u2016) \u2016c \u2022 f\u2016\nhfc : IsLUB (Set.range fun x => \u2016c\u2016 * \u2016\u2191f x\u2016) (\u2016c\u2016 * \u2016f\u2016)\n\u22a2 \u2016c\u2016 * \u2016f\u2016 \u2208 upperBounds (Set.range fun i => \u2016\u2191(c \u2022 f) i\u2016)\n[PROOFSTEP]\nhave := hfc.left\n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nc : \ud835\udd5c\nhp : \u22a4 \u2260 0\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\nhcf : IsLUB (Set.range fun i => \u2016\u2191(c \u2022 f) i\u2016) \u2016c \u2022 f\u2016\nhfc : IsLUB (Set.range fun x => \u2016c\u2016 * \u2016\u2191f x\u2016) (\u2016c\u2016 * \u2016f\u2016)\nthis : \u2016c\u2016 * \u2016f\u2016 \u2208 upperBounds (Set.range fun x => \u2016c\u2016 * \u2016\u2191f x\u2016)\n\u22a2 \u2016c\u2016 * \u2016f\u2016 \u2208 upperBounds (Set.range fun i => \u2016\u2191(c \u2022 f) i\u2016)\n[PROOFSTEP]\nsimp_rw [mem_upperBounds, Set.mem_range, forall_exists_index, forall_apply_eq_imp_iff'] at this \u22a2\n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nc : \ud835\udd5c\nhp : \u22a4 \u2260 0\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\nhcf : IsLUB (Set.range fun i => \u2016\u2191(c \u2022 f) i\u2016) \u2016c \u2022 f\u2016\nhfc : IsLUB (Set.range fun x => \u2016c\u2016 * \u2016\u2191f x\u2016) (\u2016c\u2016 * \u2016f\u2016)\nthis : \u2200 (a : \u03b1), \u2016c\u2016 * \u2016\u2191f a\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n\u22a2 \u2200 (a : \u03b1), \u2016\u2191(c \u2022 f) a\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nintro a\n[GOAL]\ncase inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nc : \ud835\udd5c\nhp : \u22a4 \u2260 0\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\nhcf : IsLUB (Set.range fun i => \u2016\u2191(c \u2022 f) i\u2016) \u2016c \u2022 f\u2016\nhfc : IsLUB (Set.range fun x => \u2016c\u2016 * \u2016\u2191f x\u2016) (\u2016c\u2016 * \u2016f\u2016)\nthis : \u2200 (a : \u03b1), \u2016c\u2016 * \u2016\u2191f a\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\na : \u03b1\n\u22a2 \u2016\u2191(c \u2022 f) a\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nexact (norm_smul_le _ _).trans (this a)\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nhp\u271d : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nletI inst : NNNorm (lp E p) := \u27e8fun f => \u27e8\u2016f\u2016, norm_nonneg' _\u27e9\u27e9\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nhp\u271d : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x \u2208 lp E p } := { nnnorm := fun f => { val := \u2016f\u2016, property := (_ : 0 \u2264 \u2016f\u2016) } }\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nhave coe_nnnorm : \u2200 f : lp E p, \u2191\u2016f\u2016\u208a = \u2016f\u2016 := fun _ => rfl\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nhp\u271d : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x \u2208 lp E p } := { nnnorm := fun f => { val := \u2016f\u2016, property := (_ : 0 \u2264 \u2016f\u2016) } }\ncoe_nnnorm : \u2200 (f : { x // x \u2208 lp E p }), \u2191\u2016f\u2016\u208a = \u2016f\u2016\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nsuffices \u2016c \u2022 f\u2016\u208a ^ p.toReal \u2264 (\u2016c\u2016\u208a * \u2016f\u2016\u208a) ^ p.toReal by rwa [NNReal.rpow_le_rpow_iff hp] at this \n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nhp\u271d : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x \u2208 lp E p } := { nnnorm := fun f => { val := \u2016f\u2016, property := (_ : 0 \u2264 \u2016f\u2016) } }\ncoe_nnnorm : \u2200 (f : { x // x \u2208 lp E p }), \u2191\u2016f\u2016\u208a = \u2016f\u2016\nthis : \u2016c \u2022 f\u2016\u208a ^ ENNReal.toReal p \u2264 (\u2016c\u2016\u208a * \u2016f\u2016\u208a) ^ ENNReal.toReal p\n\u22a2 \u2016c \u2022 f\u2016 \u2264 \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nrwa [NNReal.rpow_le_rpow_iff hp] at this \n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nhp\u271d : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x \u2208 lp E p } := { nnnorm := fun f => { val := \u2016f\u2016, property := (_ : 0 \u2264 \u2016f\u2016) } }\ncoe_nnnorm : \u2200 (f : { x // x \u2208 lp E p }), \u2191\u2016f\u2016\u208a = \u2016f\u2016\n\u22a2 \u2016c \u2022 f\u2016\u208a ^ ENNReal.toReal p \u2264 (\u2016c\u2016\u208a * \u2016f\u2016\u208a) ^ ENNReal.toReal p\n[PROOFSTEP]\nclear_value inst\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nhp\u271d : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x \u2208 lp E p }\ncoe_nnnorm : \u2200 (f : { x // x \u2208 lp E p }), \u2191\u2016f\u2016\u208a = \u2016f\u2016\n\u22a2 \u2016c \u2022 f\u2016\u208a ^ ENNReal.toReal p \u2264 (\u2016c\u2016\u208a * \u2016f\u2016\u208a) ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [NNReal.mul_rpow]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nhp\u271d : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x \u2208 lp E p }\ncoe_nnnorm : \u2200 (f : { x // x \u2208 lp E p }), \u2191\u2016f\u2016\u208a = \u2016f\u2016\n\u22a2 \u2016c \u2022 f\u2016\u208a ^ ENNReal.toReal p \u2264 \u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016f\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nhave hLHS := lp.hasSum_norm hp (c \u2022 f)\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nhp\u271d : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x \u2208 lp E p }\ncoe_nnnorm : \u2200 (f : { x // x \u2208 lp E p }), \u2191\u2016f\u2016\u208a = \u2016f\u2016\nhLHS : HasSum (fun i => \u2016\u2191(c \u2022 f) i\u2016 ^ ENNReal.toReal p) (\u2016c \u2022 f\u2016 ^ ENNReal.toReal p)\n\u22a2 \u2016c \u2022 f\u2016\u208a ^ ENNReal.toReal p \u2264 \u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016f\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nhave hRHS := (lp.hasSum_norm hp f).mul_left (\u2016c\u2016 ^ p.toReal)\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nhp\u271d : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x \u2208 lp E p }\ncoe_nnnorm : \u2200 (f : { x // x \u2208 lp E p }), \u2191\u2016f\u2016\u208a = \u2016f\u2016\nhLHS : HasSum (fun i => \u2016\u2191(c \u2022 f) i\u2016 ^ ENNReal.toReal p) (\u2016c \u2022 f\u2016 ^ ENNReal.toReal p)\nhRHS :\n HasSum (fun i => \u2016c\u2016 ^ ENNReal.toReal p * \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2016c\u2016 ^ ENNReal.toReal p * \u2016f\u2016 ^ ENNReal.toReal p)\n\u22a2 \u2016c \u2022 f\u2016\u208a ^ ENNReal.toReal p \u2264 \u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016f\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nsimp_rw [\u2190 coe_nnnorm, \u2190 _root_.coe_nnnorm, \u2190 NNReal.coe_rpow, \u2190 NNReal.coe_mul, NNReal.hasSum_coe] at hRHS hLHS \n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nhp\u271d : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x \u2208 lp E p }\ncoe_nnnorm : \u2200 (f : { x // x \u2208 lp E p }), \u2191\u2016f\u2016\u208a = \u2016f\u2016\nhRHS :\n HasSum (fun a => \u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016\u2191f a\u2016\u208a ^ ENNReal.toReal p)\n (\u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016f\u2016\u208a ^ ENNReal.toReal p)\nhLHS : HasSum (fun a => \u2016\u2191(c \u2022 f) a\u2016\u208a ^ ENNReal.toReal p) (\u2016c \u2022 f\u2016\u208a ^ ENNReal.toReal p)\n\u22a2 \u2016c \u2022 f\u2016\u208a ^ ENNReal.toReal p \u2264 \u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016f\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nrefine' hasSum_mono hLHS hRHS fun i => _\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nhp\u271d : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x \u2208 lp E p }\ncoe_nnnorm : \u2200 (f : { x // x \u2208 lp E p }), \u2191\u2016f\u2016\u208a = \u2016f\u2016\nhRHS :\n HasSum (fun a => \u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016\u2191f a\u2016\u208a ^ ENNReal.toReal p)\n (\u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016f\u2016\u208a ^ ENNReal.toReal p)\nhLHS : HasSum (fun a => \u2016\u2191(c \u2022 f) a\u2016\u208a ^ ENNReal.toReal p) (\u2016c \u2022 f\u2016\u208a ^ ENNReal.toReal p)\ni : \u03b1\n\u22a2 \u2016\u2191(c \u2022 f) i\u2016\u208a ^ ENNReal.toReal p \u2264 \u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016\u2191f i\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nhp\u271d : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x \u2208 lp E p }\ncoe_nnnorm : \u2200 (f : { x // x \u2208 lp E p }), \u2191\u2016f\u2016\u208a = \u2016f\u2016\nhRHS :\n HasSum (fun a => \u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016\u2191f a\u2016\u208a ^ ENNReal.toReal p)\n (\u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016f\u2016\u208a ^ ENNReal.toReal p)\nhLHS : HasSum (fun a => \u2016\u2191(c \u2022 f) a\u2016\u208a ^ ENNReal.toReal p) (\u2016c \u2022 f\u2016\u208a ^ ENNReal.toReal p)\ni : \u03b1\n\u22a2 \u2016\u2191(c \u2022 f) i\u2016\u208a ^ ENNReal.toReal p \u2264 \u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016\u2191f i\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [\u2190 NNReal.mul_rpow]\n -- Porting note: added\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nhp\u271d : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x \u2208 lp E p }\ncoe_nnnorm : \u2200 (f : { x // x \u2208 lp E p }), \u2191\u2016f\u2016\u208a = \u2016f\u2016\nhRHS :\n HasSum (fun a => \u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016\u2191f a\u2016\u208a ^ ENNReal.toReal p)\n (\u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016f\u2016\u208a ^ ENNReal.toReal p)\nhLHS : HasSum (fun a => \u2016\u2191(c \u2022 f) a\u2016\u208a ^ ENNReal.toReal p) (\u2016c \u2022 f\u2016\u208a ^ ENNReal.toReal p)\ni : \u03b1\n\u22a2 \u2016\u2191(c \u2022 f) i\u2016\u208a ^ ENNReal.toReal p \u2264 (\u2016c\u2016\u208a * \u2016\u2191f i\u2016\u208a) ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [lp.coeFn_smul, Pi.smul_apply]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2076 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\n\ud835\udd5c' : Type u_4\ninst\u271d\u2075 : NormedRing \ud835\udd5c\ninst\u271d\u2074 : NormedRing \ud835\udd5c'\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c' (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c' (E i)\nhp\u271d : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x \u2208 lp E p }\ncoe_nnnorm : \u2200 (f : { x // x \u2208 lp E p }), \u2191\u2016f\u2016\u208a = \u2016f\u2016\nhRHS :\n HasSum (fun a => \u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016\u2191f a\u2016\u208a ^ ENNReal.toReal p)\n (\u2016c\u2016\u208a ^ ENNReal.toReal p * \u2016f\u2016\u208a ^ ENNReal.toReal p)\nhLHS : HasSum (fun a => \u2016\u2191(c \u2022 f) a\u2016\u208a ^ ENNReal.toReal p) (\u2016c \u2022 f\u2016\u208a ^ ENNReal.toReal p)\ni : \u03b1\n\u22a2 \u2016c \u2022 \u2191f i\u2016\u208a ^ ENNReal.toReal p \u2264 (\u2016c\u2016\u208a * \u2016\u2191f i\u2016\u208a) ^ ENNReal.toReal p\n[PROOFSTEP]\nexact NNReal.rpow_le_rpow (nnnorm_smul_le _ _) ENNReal.toReal_nonneg\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nhp : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\n\u22a2 \u2016c \u2022 f\u2016 = \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nobtain rfl | hc := eq_or_ne c 0\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nhp : p \u2260 0\nf : { x // x \u2208 lp E p }\n\u22a2 \u20160 \u2022 f\u2016 = \u20160\u2016 * \u2016f\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nhp : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhc : c \u2260 0\n\u22a2 \u2016c \u2022 f\u2016 = \u2016c\u2016 * \u2016f\u2016\n[PROOFSTEP]\nrefine' le_antisymm (norm_const_smul_le hp c f) _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nhp : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhc : c \u2260 0\n\u22a2 \u2016c\u2016 * \u2016f\u2016 \u2264 \u2016c \u2022 f\u2016\n[PROOFSTEP]\nhave := mul_le_mul_of_nonneg_left (norm_const_smul_le hp c\u207b\u00b9 (c \u2022 f)) (norm_nonneg c)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\nhp : p \u2260 0\nc : \ud835\udd5c\nf : { x // x \u2208 lp E p }\nhc : c \u2260 0\nthis : \u2016c\u2016 * \u2016c\u207b\u00b9 \u2022 c \u2022 f\u2016 \u2264 \u2016c\u2016 * (\u2016c\u207b\u00b9\u2016 * \u2016c \u2022 f\u2016)\n\u22a2 \u2016c\u2016 * \u2016f\u2016 \u2264 \u2016c \u2022 f\u2016\n[PROOFSTEP]\nrwa [inv_smul_smul\u2080 hc, norm_inv, mul_inv_cancel_left\u2080 (norm_ne_zero_iff.mpr hc)] at this \n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9 : (i : \u03b1) \u2192 StarAddMonoid (E i)\ninst\u271d : \u2200 (i : \u03b1), NormedStarGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\n\u22a2 Mem\u2113p (star f) p\n[PROOFSTEP]\nrcases p.trichotomy with (rfl | rfl | hp)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9 : (i : \u03b1) \u2192 StarAddMonoid (E i)\ninst\u271d : \u2200 (i : \u03b1), NormedStarGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f 0\n\u22a2 Mem\u2113p (star f) 0\n[PROOFSTEP]\napply mem\u2113p_zero\n[GOAL]\ncase inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9 : (i : \u03b1) \u2192 StarAddMonoid (E i)\ninst\u271d : \u2200 (i : \u03b1), NormedStarGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f 0\n\u22a2 Set.Finite {i | star f i \u2260 0}\n[PROOFSTEP]\nsimp [hf.finite_dsupport]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9 : (i : \u03b1) \u2192 StarAddMonoid (E i)\ninst\u271d : \u2200 (i : \u03b1), NormedStarGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f \u22a4\n\u22a2 Mem\u2113p (star f) \u22a4\n[PROOFSTEP]\napply mem\u2113p_infty\n[GOAL]\ncase inr.inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9 : (i : \u03b1) \u2192 StarAddMonoid (E i)\ninst\u271d : \u2200 (i : \u03b1), NormedStarGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f \u22a4\n\u22a2 BddAbove (Set.range fun i => \u2016star f i\u2016)\n[PROOFSTEP]\nsimpa using hf.bddAbove\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9 : (i : \u03b1) \u2192 StarAddMonoid (E i)\ninst\u271d : \u2200 (i : \u03b1), NormedStarGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhp : 0 < ENNReal.toReal p\n\u22a2 Mem\u2113p (star f) p\n[PROOFSTEP]\napply mem\u2113p_gen\n[GOAL]\ncase inr.inr.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9 : (i : \u03b1) \u2192 StarAddMonoid (E i)\ninst\u271d : \u2200 (i : \u03b1), NormedStarGroup (E i)\nf : (i : \u03b1) \u2192 E i\nhf : Mem\u2113p f p\nhp : 0 < ENNReal.toReal p\n\u22a2 Summable fun i => \u2016star f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nsimpa using hf.summable hp\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9 : (i : \u03b1) \u2192 StarAddMonoid (E i)\ninst\u271d : \u2200 (i : \u03b1), NormedStarGroup (E i)\nx : { x // x \u2208 lp E p }\n\u22a2 star (star x) = x\n[PROOFSTEP]\nsimp [star]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9 : (i : \u03b1) \u2192 StarAddMonoid (E i)\ninst\u271d : \u2200 (i : \u03b1), NormedStarGroup (E i)\nhp : Fact (1 \u2264 p)\nf : { x // x \u2208 lp E p }\n\u22a2 \u2016star f\u2016 = \u2016f\u2016\n[PROOFSTEP]\nrcases p.trichotomy with (rfl | rfl | h)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9 : (i : \u03b1) \u2192 StarAddMonoid (E i)\ninst\u271d : \u2200 (i : \u03b1), NormedStarGroup (E i)\nhp : Fact (1 \u2264 0)\nf : { x // x \u2208 lp E 0 }\n\u22a2 \u2016star f\u2016 = \u2016f\u2016\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase inl.h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9 : (i : \u03b1) \u2192 StarAddMonoid (E i)\ninst\u271d : \u2200 (i : \u03b1), NormedStarGroup (E i)\nhp : Fact (1 \u2264 0)\nf : { x // x \u2208 lp E 0 }\n\u22a2 False\n[PROOFSTEP]\nhave := ENNReal.toReal_mono ENNReal.zero_ne_top hp.elim\n[GOAL]\ncase inl.h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9 : (i : \u03b1) \u2192 StarAddMonoid (E i)\ninst\u271d : \u2200 (i : \u03b1), NormedStarGroup (E i)\nhp : Fact (1 \u2264 0)\nf : { x // x \u2208 lp E 0 }\nthis : ENNReal.toReal 1 \u2264 ENNReal.toReal 0\n\u22a2 False\n[PROOFSTEP]\nnorm_num at this \n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9 : (i : \u03b1) \u2192 StarAddMonoid (E i)\ninst\u271d : \u2200 (i : \u03b1), NormedStarGroup (E i)\nhp : Fact (1 \u2264 \u22a4)\nf : { x // x \u2208 lp E \u22a4 }\n\u22a2 \u2016star f\u2016 = \u2016f\u2016\n[PROOFSTEP]\nsimp only [lp.norm_eq_ciSup, lp.star_apply, norm_star]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9 : (i : \u03b1) \u2192 StarAddMonoid (E i)\ninst\u271d : \u2200 (i : \u03b1), NormedStarGroup (E i)\nhp : Fact (1 \u2264 p)\nf : { x // x \u2208 lp E p }\nh : 0 < ENNReal.toReal p\n\u22a2 \u2016star f\u2016 = \u2016f\u2016\n[PROOFSTEP]\nsimp only [lp.norm_eq_tsum_rpow h, lp.star_apply, norm_star]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d : (i : I) \u2192 NonUnitalNormedRing (B i)\nf g : (i : I) \u2192 B i\nhf : Mem\u2113p f \u22a4\nhg : Mem\u2113p g \u22a4\n\u22a2 Mem\u2113p (f * g) \u22a4\n[PROOFSTEP]\nrw [mem\u2113p_infty_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d : (i : I) \u2192 NonUnitalNormedRing (B i)\nf g : (i : I) \u2192 B i\nhf : Mem\u2113p f \u22a4\nhg : Mem\u2113p g \u22a4\n\u22a2 BddAbove (Set.range fun i => \u2016(f * g) i\u2016)\n[PROOFSTEP]\nobtain \u27e8\u27e8Cf, hCf\u27e9, \u27e8Cg, hCg\u27e9\u27e9 := hf.bddAbove, hg.bddAbove\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d : (i : I) \u2192 NonUnitalNormedRing (B i)\nf g : (i : I) \u2192 B i\nhf : Mem\u2113p f \u22a4\nhg : Mem\u2113p g \u22a4\nCf : \u211d\nhCf : Cf \u2208 upperBounds (Set.range fun i => \u2016f i\u2016)\nCg : \u211d\nhCg : Cg \u2208 upperBounds (Set.range fun i => \u2016g i\u2016)\n\u22a2 BddAbove (Set.range fun i => \u2016(f * g) i\u2016)\n[PROOFSTEP]\nrefine' \u27e8Cf * Cg, _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d : (i : I) \u2192 NonUnitalNormedRing (B i)\nf g : (i : I) \u2192 B i\nhf : Mem\u2113p f \u22a4\nhg : Mem\u2113p g \u22a4\nCf : \u211d\nhCf : Cf \u2208 upperBounds (Set.range fun i => \u2016f i\u2016)\nCg : \u211d\nhCg : Cg \u2208 upperBounds (Set.range fun i => \u2016g i\u2016)\n\u22a2 Cf * Cg \u2208 upperBounds (Set.range fun i => \u2016(f * g) i\u2016)\n[PROOFSTEP]\nrintro _ \u27e8i, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d : (i : I) \u2192 NonUnitalNormedRing (B i)\nf g : (i : I) \u2192 B i\nhf : Mem\u2113p f \u22a4\nhg : Mem\u2113p g \u22a4\nCf : \u211d\nhCf : Cf \u2208 upperBounds (Set.range fun i => \u2016f i\u2016)\nCg : \u211d\nhCg : Cg \u2208 upperBounds (Set.range fun i => \u2016g i\u2016)\ni : I\n\u22a2 (fun i => \u2016(f * g) i\u2016) i \u2264 Cf * Cg\n[PROOFSTEP]\ncalc\n \u2016(f * g) i\u2016 \u2264 \u2016f i\u2016 * \u2016g i\u2016 := norm_mul_le (f i) (g i)\n _ \u2264 Cf * Cg := mul_le_mul (hCf \u27e8i, rfl\u27e9) (hCg \u27e8i, rfl\u27e9) (norm_nonneg _) ((norm_nonneg _).trans (hCf \u27e8i, rfl\u27e9))\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b3 : (i : I) \u2192 NonUnitalNormedRing (B i)\ninst\u271d\u00b2 : (i : I) \u2192 StarRing (B i)\ninst\u271d\u00b9 : \u2200 (i : I), NormedStarGroup (B i)\ninst\u271d : \u2200 (i : I), CstarRing (B i)\n\u22a2 \u2200 {x : { x // x \u2208 lp B \u22a4 }}, \u2016star x * x\u2016 = \u2016x\u2016 * \u2016x\u2016\n[PROOFSTEP]\nintro f\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b3 : (i : I) \u2192 NonUnitalNormedRing (B i)\ninst\u271d\u00b2 : (i : I) \u2192 StarRing (B i)\ninst\u271d\u00b9 : \u2200 (i : I), NormedStarGroup (B i)\ninst\u271d : \u2200 (i : I), CstarRing (B i)\nf : { x // x \u2208 lp B \u22a4 }\n\u22a2 \u2016star f * f\u2016 = \u2016f\u2016 * \u2016f\u2016\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b3 : (i : I) \u2192 NonUnitalNormedRing (B i)\ninst\u271d\u00b2 : (i : I) \u2192 StarRing (B i)\ninst\u271d\u00b9 : \u2200 (i : I), NormedStarGroup (B i)\ninst\u271d : \u2200 (i : I), CstarRing (B i)\nf : { x // x \u2208 lp B \u22a4 }\n\u22a2 \u2016star f * f\u2016 \u2264 \u2016f\u2016 * \u2016f\u2016\n[PROOFSTEP]\nrw [\u2190 sq]\n[GOAL]\ncase a\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b3 : (i : I) \u2192 NonUnitalNormedRing (B i)\ninst\u271d\u00b2 : (i : I) \u2192 StarRing (B i)\ninst\u271d\u00b9 : \u2200 (i : I), NormedStarGroup (B i)\ninst\u271d : \u2200 (i : I), CstarRing (B i)\nf : { x // x \u2208 lp B \u22a4 }\n\u22a2 \u2016star f * f\u2016 \u2264 \u2016f\u2016 ^ 2\n[PROOFSTEP]\nrefine' lp.norm_le_of_forall_le (sq_nonneg \u2016f\u2016) fun i => _\n[GOAL]\ncase a\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b3 : (i : I) \u2192 NonUnitalNormedRing (B i)\ninst\u271d\u00b2 : (i : I) \u2192 StarRing (B i)\ninst\u271d\u00b9 : \u2200 (i : I), NormedStarGroup (B i)\ninst\u271d : \u2200 (i : I), CstarRing (B i)\nf : { x // x \u2208 lp B \u22a4 }\ni : I\n\u22a2 \u2016\u2191(star f * f) i\u2016 \u2264 \u2016f\u2016 ^ 2\n[PROOFSTEP]\nsimp only [lp.star_apply, CstarRing.norm_star_mul_self, \u2190 sq, infty_coeFn_mul, Pi.mul_apply]\n[GOAL]\ncase a\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b3 : (i : I) \u2192 NonUnitalNormedRing (B i)\ninst\u271d\u00b2 : (i : I) \u2192 StarRing (B i)\ninst\u271d\u00b9 : \u2200 (i : I), NormedStarGroup (B i)\ninst\u271d : \u2200 (i : I), CstarRing (B i)\nf : { x // x \u2208 lp B \u22a4 }\ni : I\n\u22a2 \u2016\u2191f i\u2016 ^ 2 \u2264 \u2016f\u2016 ^ 2\n[PROOFSTEP]\nrefine' sq_le_sq' _ (lp.norm_apply_le_norm ENNReal.top_ne_zero _ _)\n[GOAL]\ncase a\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b3 : (i : I) \u2192 NonUnitalNormedRing (B i)\ninst\u271d\u00b2 : (i : I) \u2192 StarRing (B i)\ninst\u271d\u00b9 : \u2200 (i : I), NormedStarGroup (B i)\ninst\u271d : \u2200 (i : I), CstarRing (B i)\nf : { x // x \u2208 lp B \u22a4 }\ni : I\n\u22a2 -\u2016f\u2016 \u2264 \u2016\u2191f i\u2016\n[PROOFSTEP]\nlinarith [norm_nonneg (f i), norm_nonneg f]\n[GOAL]\ncase a\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b3 : (i : I) \u2192 NonUnitalNormedRing (B i)\ninst\u271d\u00b2 : (i : I) \u2192 StarRing (B i)\ninst\u271d\u00b9 : \u2200 (i : I), NormedStarGroup (B i)\ninst\u271d : \u2200 (i : I), CstarRing (B i)\nf : { x // x \u2208 lp B \u22a4 }\n\u22a2 \u2016f\u2016 * \u2016f\u2016 \u2264 \u2016star f * f\u2016\n[PROOFSTEP]\nrw [\u2190 sq, \u2190 Real.le_sqrt (norm_nonneg _) (norm_nonneg _)]\n[GOAL]\ncase a\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b3 : (i : I) \u2192 NonUnitalNormedRing (B i)\ninst\u271d\u00b2 : (i : I) \u2192 StarRing (B i)\ninst\u271d\u00b9 : \u2200 (i : I), NormedStarGroup (B i)\ninst\u271d : \u2200 (i : I), CstarRing (B i)\nf : { x // x \u2208 lp B \u22a4 }\n\u22a2 \u2016f\u2016 \u2264 Real.sqrt \u2016star f * f\u2016\n[PROOFSTEP]\nrefine' lp.norm_le_of_forall_le \u2016star f * f\u2016.sqrt_nonneg fun i => _\n[GOAL]\ncase a\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b3 : (i : I) \u2192 NonUnitalNormedRing (B i)\ninst\u271d\u00b2 : (i : I) \u2192 StarRing (B i)\ninst\u271d\u00b9 : \u2200 (i : I), NormedStarGroup (B i)\ninst\u271d : \u2200 (i : I), CstarRing (B i)\nf : { x // x \u2208 lp B \u22a4 }\ni : I\n\u22a2 \u2016\u2191f i\u2016 \u2264 Real.sqrt \u2016star f * f\u2016\n[PROOFSTEP]\nrw [Real.le_sqrt (norm_nonneg _) (norm_nonneg _), sq, \u2190 CstarRing.norm_star_mul_self]\n[GOAL]\ncase a\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b3 : (i : I) \u2192 NonUnitalNormedRing (B i)\ninst\u271d\u00b2 : (i : I) \u2192 StarRing (B i)\ninst\u271d\u00b9 : \u2200 (i : I), NormedStarGroup (B i)\ninst\u271d : \u2200 (i : I), CstarRing (B i)\nf : { x // x \u2208 lp B \u22a4 }\ni : I\n\u22a2 \u2016star (\u2191f i) * \u2191f i\u2016 \u2264 \u2016star f * f\u2016\n[PROOFSTEP]\nexact lp.norm_apply_le_norm ENNReal.top_ne_zero (star f * f) i\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b9 : (i : I) \u2192 NormedRing (B i)\ninst\u271d : \u2200 (i : I), NormOneClass (B i)\n\u22a2 1 \u2208 upperBounds (Set.range fun i => \u2016OfNat.ofNat 1 i\u2016)\n[PROOFSTEP]\nrintro i \u27e8i, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b9 : (i : I) \u2192 NormedRing (B i)\ninst\u271d : \u2200 (i : I), NormOneClass (B i)\ni : I\n\u22a2 (fun i => \u2016OfNat.ofNat 1 i\u2016) i \u2264 1\n[PROOFSTEP]\nexact norm_one.le\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b3 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b2 : (i : I) \u2192 NormedRing (B i)\ninst\u271d\u00b9 : \u2200 (i : I), NormOneClass (B i)\ninst\u271d : Nonempty I\n\u22a2 \u20161\u2016 = 1\n[PROOFSTEP]\nsimp_rw [lp.norm_eq_ciSup, infty_coeFn_one, Pi.one_apply, norm_one, ciSup_const]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b9 : (i : I) \u2192 NormedCommRing (B i)\ninst\u271d : \u2200 (i : I), NormOneClass (B i)\nsrc\u271d : Ring { x // x \u2208 lp B \u22a4 } := inftyRing\nf g : { x // x \u2208 lp B \u22a4 }\n\u22a2 f * g = g * f\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\nB : I \u2192 Type u_4\ninst\u271d\u00b9 : (i : I) \u2192 NormedCommRing (B i)\ninst\u271d : \u2200 (i : I), NormOneClass (B i)\nsrc\u271d : Ring { x // x \u2208 lp B \u22a4 } := inftyRing\nf g : { x // x \u2208 lp B \u22a4 }\nx\u271d : I\n\u22a2 \u2191(f * g) x\u271d = \u2191(g * f) x\u271d\n[PROOFSTEP]\nsimp only [lp.infty_coeFn_mul, Pi.mul_apply, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\n\ud835\udd5c : Type u_4\nB : I \u2192 Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : (i : I) \u2192 NormedRing (B i)\ninst\u271d\u00b9 : (i : I) \u2192 NormedAlgebra \ud835\udd5c (B i)\ninst\u271d : \u2200 (i : I), NormOneClass (B i)\nk : \ud835\udd5c\n\u22a2 Mem\u2113p (\u2191(algebraMap \ud835\udd5c ((i : I) \u2192 B i)) k) \u22a4\n[PROOFSTEP]\nrw [Algebra.algebraMap_eq_smul_one]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nI : Type u_3\n\ud835\udd5c : Type u_4\nB : I \u2192 Type u_5\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : (i : I) \u2192 NormedRing (B i)\ninst\u271d\u00b9 : (i : I) \u2192 NormedAlgebra \ud835\udd5c (B i)\ninst\u271d : \u2200 (i : I), NormOneClass (B i)\nk : \ud835\udd5c\n\u22a2 Mem\u2113p (k \u2022 1) \u22a4\n[PROOFSTEP]\nexact (one_mem\u2113p_infty.const_smul k : Mem\u2113p (k \u2022 (1 : \u2200 i, B i)) \u221e)\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\n\u22a2 (fun j => if h : j = i then (_ : i = j) \u25b8 a else 0) \u2208 lp E p\n[PROOFSTEP]\nrefine' (mem\u2113p_zero _).of_exponent_ge (zero_le p)\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\n\u22a2 Set.Finite {i_1 | (fun j => if h : j = i then (_ : i = j) \u25b8 a else 0) i_1 \u2260 0}\n[PROOFSTEP]\nrefine' (Set.finite_singleton i).subset _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\n\u22a2 {i_1 | (fun j => if h : j = i then (_ : i = j) \u25b8 a else 0) i_1 \u2260 0} \u2286 {i}\n[PROOFSTEP]\nintro j\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\nj : \u03b1\n\u22a2 j \u2208 {i_1 | (fun j => if h : j = i then (_ : i = j) \u25b8 a else 0) i_1 \u2260 0} \u2192 j \u2208 {i}\n[PROOFSTEP]\nsimp only [forall_exists_index, Set.mem_singleton_iff, Ne.def, dite_eq_right_iff, Set.mem_setOf_eq, not_forall]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\nj : \u03b1\n\u22a2 \u2200 (x : j = i), \u00ac(_ : i = j) \u25b8 a = 0 \u2192 j = i\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\nj : \u03b1\na : E j\n\u22a2 \u00ac(_ : j = j) \u25b8 a = 0 \u2192 j = j\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\n\u22a2 \u2191(lp.single p i a) i = a\n[PROOFSTEP]\nrw [lp.single_apply, dif_pos rfl]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\nj : \u03b1\nhij : j \u2260 i\n\u22a2 \u2191(lp.single p i a) j = 0\n[PROOFSTEP]\nrw [lp.single_apply, dif_neg hij]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\n\u22a2 lp.single p i (-a) = -lp.single p i a\n[PROOFSTEP]\nrefine' ext (funext (fun (j : \u03b1) => _))\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\nj : \u03b1\n\u22a2 \u2191(lp.single p i (-a)) j = \u2191(-lp.single p i a) j\n[PROOFSTEP]\nby_cases hi : j = i\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\nj : \u03b1\nhi : j = i\n\u22a2 \u2191(lp.single p i (-a)) j = \u2191(-lp.single p i a) j\n[PROOFSTEP]\nsubst hi\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\nj : \u03b1\na : E j\n\u22a2 \u2191(lp.single p j (-a)) j = \u2191(-lp.single p j a) j\n[PROOFSTEP]\nsimp [lp.single_apply_self]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\nj : \u03b1\nhi : \u00acj = i\n\u22a2 \u2191(lp.single p i (-a)) j = \u2191(-lp.single p i a) j\n[PROOFSTEP]\nsimp [lp.single_apply_ne p i _ hi]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\nc : \ud835\udd5c\n\u22a2 lp.single p i (c \u2022 a) = c \u2022 lp.single p i a\n[PROOFSTEP]\nrefine' ext (funext (fun (j : \u03b1) => _))\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\nc : \ud835\udd5c\nj : \u03b1\n\u22a2 \u2191(lp.single p i (c \u2022 a)) j = \u2191(c \u2022 lp.single p i a) j\n[PROOFSTEP]\nby_cases hi : j = i\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\nc : \ud835\udd5c\nj : \u03b1\nhi : j = i\n\u22a2 \u2191(lp.single p i (c \u2022 a)) j = \u2191(c \u2022 lp.single p i a) j\n[PROOFSTEP]\nsubst hi\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\nc : \ud835\udd5c\nj : \u03b1\na : E j\n\u22a2 \u2191(lp.single p j (c \u2022 a)) j = \u2191(c \u2022 lp.single p j a) j\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\nc : \ud835\udd5c\nj : \u03b1\na : E j\n\u22a2 \u2191(lp.single p j (c \u2022 a)) j = c \u2022 \u2191(lp.single p j a) j\n[PROOFSTEP]\nsimp [lp.single_apply_self]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\nc : \ud835\udd5c\nj : \u03b1\nhi : \u00acj = i\n\u22a2 \u2191(lp.single p i (c \u2022 a)) j = \u2191(c \u2022 lp.single p i a) j\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np\u271d q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\np : \u211d\u22650\u221e\ni : \u03b1\na : E i\nc : \ud835\udd5c\nj : \u03b1\nhi : \u00acj = i\n\u22a2 \u2191(lp.single p i (c \u2022 a)) j = c \u2022 \u2191(lp.single p i a) j\n[PROOFSTEP]\nsimp [lp.single_apply_ne p i _ hi]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : (i : \u03b1) \u2192 E i\ns : Finset \u03b1\n\u22a2 \u2016\u2211 i in s, lp.single p i (f i)\u2016 ^ ENNReal.toReal p = \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nrefine' (hasSum_norm hp (\u2211 i in s, lp.single p i (f i))).unique _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : (i : \u03b1) \u2192 E i\ns : Finset \u03b1\n\u22a2 HasSum (fun i => \u2016\u2191(\u2211 i in s, lp.single p i (f i)) i\u2016 ^ ENNReal.toReal p) (\u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nsimp only [lp.single_apply, coeFn_sum, Finset.sum_apply, Finset.sum_dite_eq]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : (i : \u03b1) \u2192 E i\ns : Finset \u03b1\n\u22a2 HasSum (fun i => \u2016if i \u2208 s then f i else 0\u2016 ^ ENNReal.toReal p) (\u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nhave h : \u2200 (i) (_ : i \u2209 s), \u2016ite (i \u2208 s) (f i) 0\u2016 ^ p.toReal = 0 :=\n by\n intro i hi\n simp [if_neg hi, Real.zero_rpow hp.ne']\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : (i : \u03b1) \u2192 E i\ns : Finset \u03b1\n\u22a2 \u2200 (i : \u03b1), \u00aci \u2208 s \u2192 \u2016if i \u2208 s then f i else 0\u2016 ^ ENNReal.toReal p = 0\n[PROOFSTEP]\nintro i hi\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : (i : \u03b1) \u2192 E i\ns : Finset \u03b1\ni : \u03b1\nhi : \u00aci \u2208 s\n\u22a2 \u2016if i \u2208 s then f i else 0\u2016 ^ ENNReal.toReal p = 0\n[PROOFSTEP]\nsimp [if_neg hi, Real.zero_rpow hp.ne']\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : (i : \u03b1) \u2192 E i\ns : Finset \u03b1\nh : \u2200 (i : \u03b1), \u00aci \u2208 s \u2192 \u2016if i \u2208 s then f i else 0\u2016 ^ ENNReal.toReal p = 0\n\u22a2 HasSum (fun i => \u2016if i \u2208 s then f i else 0\u2016 ^ ENNReal.toReal p) (\u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nhave h' : \u2200 i \u2208 s, \u2016f i\u2016 ^ p.toReal = \u2016ite (i \u2208 s) (f i) 0\u2016 ^ p.toReal :=\n by\n intro i hi\n rw [if_pos hi]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : (i : \u03b1) \u2192 E i\ns : Finset \u03b1\nh : \u2200 (i : \u03b1), \u00aci \u2208 s \u2192 \u2016if i \u2208 s then f i else 0\u2016 ^ ENNReal.toReal p = 0\n\u22a2 \u2200 (i : \u03b1), i \u2208 s \u2192 \u2016f i\u2016 ^ ENNReal.toReal p = \u2016if i \u2208 s then f i else 0\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nintro i hi\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : (i : \u03b1) \u2192 E i\ns : Finset \u03b1\nh : \u2200 (i : \u03b1), \u00aci \u2208 s \u2192 \u2016if i \u2208 s then f i else 0\u2016 ^ ENNReal.toReal p = 0\ni : \u03b1\nhi : i \u2208 s\n\u22a2 \u2016f i\u2016 ^ ENNReal.toReal p = \u2016if i \u2208 s then f i else 0\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [if_pos hi]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : (i : \u03b1) \u2192 E i\ns : Finset \u03b1\nh : \u2200 (i : \u03b1), \u00aci \u2208 s \u2192 \u2016if i \u2208 s then f i else 0\u2016 ^ ENNReal.toReal p = 0\nh' : \u2200 (i : \u03b1), i \u2208 s \u2192 \u2016f i\u2016 ^ ENNReal.toReal p = \u2016if i \u2208 s then f i else 0\u2016 ^ ENNReal.toReal p\n\u22a2 HasSum (fun i => \u2016if i \u2208 s then f i else 0\u2016 ^ ENNReal.toReal p) (\u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nsimpa [Finset.sum_congr rfl h'] using hasSum_sum_of_ne_finset_zero h\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : (i : \u03b1) \u2192 E i\ni : \u03b1\n\u22a2 \u2016lp.single p i (f i)\u2016 = \u2016f i\u2016\n[PROOFSTEP]\nrefine' Real.rpow_left_injOn hp.ne' (norm_nonneg' _) (norm_nonneg _) _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : (i : \u03b1) \u2192 E i\ni : \u03b1\n\u22a2 (fun y => y ^ ENNReal.toReal p) \u2016lp.single p i (f i)\u2016 = (fun y => y ^ ENNReal.toReal p) \u2016f i\u2016\n[PROOFSTEP]\nsimpa using lp.norm_sum_single hp f { i }\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\n\u22a2 \u2016f\u2016 ^ ENNReal.toReal p - \u2016f - \u2211 i in s, lp.single p i (\u2191f i)\u2016 ^ ENNReal.toReal p = \u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nrefine' ((hasSum_norm hp f).sub (hasSum_norm hp (f - \u2211 i in s, lp.single p i (f i)))).unique _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\n\u22a2 HasSum (fun b => \u2016\u2191f b\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) b\u2016 ^ ENNReal.toReal p)\n (\u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nlet F : \u03b1 \u2192 \u211d := fun i => \u2016f i\u2016 ^ p.toReal - \u2016(f - \u2211 i in s, lp.single p i (f i)) i\u2016 ^ p.toReal\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\nF : \u03b1 \u2192 \u211d := fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) i\u2016 ^ ENNReal.toReal p\n\u22a2 HasSum (fun b => \u2016\u2191f b\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) b\u2016 ^ ENNReal.toReal p)\n (\u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nhave hF : \u2200 (i) (_ : i \u2209 s), F i = 0 := by\n intro i hi\n suffices \u2016f i\u2016 ^ p.toReal - \u2016f i - ite (i \u2208 s) (f i) 0\u2016 ^ p.toReal = 0 by\n simpa only [coeFn_sum, lp.single_apply, coeFn_sub, Pi.sub_apply, Finset.sum_apply, Finset.sum_dite_eq] using this\n simp only [if_neg hi, sub_zero, sub_self]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\nF : \u03b1 \u2192 \u211d := fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) i\u2016 ^ ENNReal.toReal p\n\u22a2 \u2200 (i : \u03b1), \u00aci \u2208 s \u2192 F i = 0\n[PROOFSTEP]\nintro i hi\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\nF : \u03b1 \u2192 \u211d := fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) i\u2016 ^ ENNReal.toReal p\ni : \u03b1\nhi : \u00aci \u2208 s\n\u22a2 F i = 0\n[PROOFSTEP]\nsuffices \u2016f i\u2016 ^ p.toReal - \u2016f i - ite (i \u2208 s) (f i) 0\u2016 ^ p.toReal = 0 by\n simpa only [coeFn_sum, lp.single_apply, coeFn_sub, Pi.sub_apply, Finset.sum_apply, Finset.sum_dite_eq] using this\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\nF : \u03b1 \u2192 \u211d := fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) i\u2016 ^ ENNReal.toReal p\ni : \u03b1\nhi : \u00aci \u2208 s\nthis : \u2016\u2191f i\u2016 ^ ENNReal.toReal p - \u2016\u2191f i - if i \u2208 s then \u2191f i else 0\u2016 ^ ENNReal.toReal p = 0\n\u22a2 F i = 0\n[PROOFSTEP]\nsimpa only [coeFn_sum, lp.single_apply, coeFn_sub, Pi.sub_apply, Finset.sum_apply, Finset.sum_dite_eq] using this\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\nF : \u03b1 \u2192 \u211d := fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) i\u2016 ^ ENNReal.toReal p\ni : \u03b1\nhi : \u00aci \u2208 s\n\u22a2 \u2016\u2191f i\u2016 ^ ENNReal.toReal p - \u2016\u2191f i - if i \u2208 s then \u2191f i else 0\u2016 ^ ENNReal.toReal p = 0\n[PROOFSTEP]\nsimp only [if_neg hi, sub_zero, sub_self]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\nF : \u03b1 \u2192 \u211d := fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) i\u2016 ^ ENNReal.toReal p\nhF : \u2200 (i : \u03b1), \u00aci \u2208 s \u2192 F i = 0\n\u22a2 HasSum (fun b => \u2016\u2191f b\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) b\u2016 ^ ENNReal.toReal p)\n (\u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nhave hF' : \u2200 i \u2208 s, F i = \u2016f i\u2016 ^ p.toReal := by\n intro i hi\n simp only [coeFn_sum, lp.single_apply, if_pos hi, sub_self, eq_self_iff_true, coeFn_sub, Pi.sub_apply,\n Finset.sum_apply, Finset.sum_dite_eq, sub_eq_self]\n simp [Real.zero_rpow hp.ne']\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\nF : \u03b1 \u2192 \u211d := fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) i\u2016 ^ ENNReal.toReal p\nhF : \u2200 (i : \u03b1), \u00aci \u2208 s \u2192 F i = 0\n\u22a2 \u2200 (i : \u03b1), i \u2208 s \u2192 F i = \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nintro i hi\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\nF : \u03b1 \u2192 \u211d := fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) i\u2016 ^ ENNReal.toReal p\nhF : \u2200 (i : \u03b1), \u00aci \u2208 s \u2192 F i = 0\ni : \u03b1\nhi : i \u2208 s\n\u22a2 F i = \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nsimp only [coeFn_sum, lp.single_apply, if_pos hi, sub_self, eq_self_iff_true, coeFn_sub, Pi.sub_apply, Finset.sum_apply,\n Finset.sum_dite_eq, sub_eq_self]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\nF : \u03b1 \u2192 \u211d := fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) i\u2016 ^ ENNReal.toReal p\nhF : \u2200 (i : \u03b1), \u00aci \u2208 s \u2192 F i = 0\ni : \u03b1\nhi : i \u2208 s\n\u22a2 \u20160\u2016 ^ ENNReal.toReal p = 0\n[PROOFSTEP]\nsimp [Real.zero_rpow hp.ne']\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\nF : \u03b1 \u2192 \u211d := fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) i\u2016 ^ ENNReal.toReal p\nhF : \u2200 (i : \u03b1), \u00aci \u2208 s \u2192 F i = 0\nhF' : \u2200 (i : \u03b1), i \u2208 s \u2192 F i = \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n\u22a2 HasSum (fun b => \u2016\u2191f b\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) b\u2016 ^ ENNReal.toReal p)\n (\u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nhave : HasSum F (\u2211 i in s, F i) := hasSum_sum_of_ne_finset_zero hF\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\nF : \u03b1 \u2192 \u211d := fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) i\u2016 ^ ENNReal.toReal p\nhF : \u2200 (i : \u03b1), \u00aci \u2208 s \u2192 F i = 0\nhF' : \u2200 (i : \u03b1), i \u2208 s \u2192 F i = \u2016\u2191f i\u2016 ^ ENNReal.toReal p\nthis : HasSum F (\u2211 i in s, F i)\n\u22a2 HasSum (fun b => \u2016\u2191f b\u2016 ^ ENNReal.toReal p - \u2016\u2191(f - \u2211 i in s, lp.single p i (\u2191f i)) b\u2016 ^ ENNReal.toReal p)\n (\u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p)\n[PROOFSTEP]\nrwa [Finset.sum_congr rfl hF'] at this \n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2074 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b9 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d : DecidableEq \u03b1\nhp : 0 < ENNReal.toReal p\nf : { x // x \u2208 lp E p }\ns : Finset \u03b1\n\u22a2 \u2016f - \u2211 i in s, lp.single p i (\u2191f i)\u2016 ^ ENNReal.toReal p = \u2016f\u2016 ^ ENNReal.toReal p - \u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nlinarith [lp.norm_sub_norm_compl_sub_single hp f s]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\n\u22a2 HasSum (fun i => lp.single p i (\u2191f i)) f\n[PROOFSTEP]\nhave hp\u2080 : 0 < p := zero_lt_one.trans_le Fact.out\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\nhp\u2080 : 0 < p\n\u22a2 HasSum (fun i => lp.single p i (\u2191f i)) f\n[PROOFSTEP]\nhave hp' : 0 < p.toReal := ENNReal.toReal_pos hp\u2080.ne' hp\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\nhp\u2080 : 0 < p\nhp' : 0 < ENNReal.toReal p\n\u22a2 HasSum (fun i => lp.single p i (\u2191f i)) f\n[PROOFSTEP]\nhave := lp.hasSum_norm hp' f\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\nhp\u2080 : 0 < p\nhp' : 0 < ENNReal.toReal p\nthis : HasSum (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)\n\u22a2 HasSum (fun i => lp.single p i (\u2191f i)) f\n[PROOFSTEP]\nrw [HasSum, Metric.tendsto_nhds] at this \u22a2\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\nhp\u2080 : 0 < p\nhp' : 0 < ENNReal.toReal p\nthis :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192 \u2200\u1da0 (x : Finset \u03b1) in Filter.atTop, dist (\u2211 b in x, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p) < \u03b5\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : Finset \u03b1) in Filter.atTop, dist (\u2211 b in x, lp.single p b (\u2191f b)) f < \u03b5\n[PROOFSTEP]\nintro \u03b5 h\u03b5\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\nhp\u2080 : 0 < p\nhp' : 0 < ENNReal.toReal p\nthis :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192 \u2200\u1da0 (x : Finset \u03b1) in Filter.atTop, dist (\u2211 b in x, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x : Finset \u03b1) in Filter.atTop, dist (\u2211 b in x, lp.single p b (\u2191f b)) f < \u03b5\n[PROOFSTEP]\nrefine' (this _ (Real.rpow_pos_of_pos h\u03b5 p.toReal)).mono _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\nhp\u2080 : 0 < p\nhp' : 0 < ENNReal.toReal p\nthis :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192 \u2200\u1da0 (x : Finset \u03b1) in Filter.atTop, dist (\u2211 b in x, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200 (x : Finset \u03b1),\n dist (\u2211 b in x, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p) < \u03b5 ^ ENNReal.toReal p \u2192\n dist (\u2211 b in x, lp.single p b (\u2191f b)) f < \u03b5\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\nhp\u2080 : 0 < p\nhp' : 0 < ENNReal.toReal p\nthis :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192 \u2200\u1da0 (x : Finset \u03b1) in Filter.atTop, dist (\u2211 b in x, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ns : Finset \u03b1\nhs : dist (\u2211 b in s, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p) < \u03b5 ^ ENNReal.toReal p\n\u22a2 dist (\u2211 b in s, lp.single p b (\u2191f b)) f < \u03b5\n[PROOFSTEP]\nrw [\u2190 Real.rpow_lt_rpow_iff dist_nonneg (le_of_lt h\u03b5) hp']\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\nhp\u2080 : 0 < p\nhp' : 0 < ENNReal.toReal p\nthis :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192 \u2200\u1da0 (x : Finset \u03b1) in Filter.atTop, dist (\u2211 b in x, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ns : Finset \u03b1\nhs : dist (\u2211 b in s, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p) < \u03b5 ^ ENNReal.toReal p\n\u22a2 dist (\u2211 b in s, lp.single p b (\u2191f b)) f ^ ENNReal.toReal p < \u03b5 ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [dist_comm] at hs \n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\nhp\u2080 : 0 < p\nhp' : 0 < ENNReal.toReal p\nthis :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192 \u2200\u1da0 (x : Finset \u03b1) in Filter.atTop, dist (\u2211 b in x, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ns : Finset \u03b1\nhs : dist (\u2016f\u2016 ^ ENNReal.toReal p) (\u2211 b in s, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) < \u03b5 ^ ENNReal.toReal p\n\u22a2 dist (\u2211 b in s, lp.single p b (\u2191f b)) f ^ ENNReal.toReal p < \u03b5 ^ ENNReal.toReal p\n[PROOFSTEP]\nsimp only [dist_eq_norm, Real.norm_eq_abs] at hs \u22a2\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\nhp\u2080 : 0 < p\nhp' : 0 < ENNReal.toReal p\nthis :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192 \u2200\u1da0 (x : Finset \u03b1) in Filter.atTop, dist (\u2211 b in x, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ns : Finset \u03b1\nhs : |\u2016f\u2016 ^ ENNReal.toReal p - \u2211 x in s, \u2016\u2191f x\u2016 ^ ENNReal.toReal p| < \u03b5 ^ ENNReal.toReal p\n\u22a2 \u2016\u2211 x in s, lp.single p x (\u2191f x) - f\u2016 ^ ENNReal.toReal p < \u03b5 ^ ENNReal.toReal p\n[PROOFSTEP]\nhave H : \u2016(\u2211 i in s, lp.single p i (f i : E i)) - f\u2016 ^ p.toReal = \u2016f\u2016 ^ p.toReal - \u2211 i in s, \u2016f i\u2016 ^ p.toReal := by\n simpa only [coeFn_neg, Pi.neg_apply, lp.single_neg, Finset.sum_neg_distrib, neg_sub_neg, norm_neg,\n _root_.norm_neg] using lp.norm_compl_sum_single hp' (-f) s\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\nhp\u2080 : 0 < p\nhp' : 0 < ENNReal.toReal p\nthis :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192 \u2200\u1da0 (x : Finset \u03b1) in Filter.atTop, dist (\u2211 b in x, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ns : Finset \u03b1\nhs : |\u2016f\u2016 ^ ENNReal.toReal p - \u2211 x in s, \u2016\u2191f x\u2016 ^ ENNReal.toReal p| < \u03b5 ^ ENNReal.toReal p\n\u22a2 \u2016\u2211 i in s, lp.single p i (\u2191f i) - f\u2016 ^ ENNReal.toReal p = \u2016f\u2016 ^ ENNReal.toReal p - \u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nsimpa only [coeFn_neg, Pi.neg_apply, lp.single_neg, Finset.sum_neg_distrib, neg_sub_neg, norm_neg,\n _root_.norm_neg] using lp.norm_compl_sum_single hp' (-f) s\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\nhp\u2080 : 0 < p\nhp' : 0 < ENNReal.toReal p\nthis :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192 \u2200\u1da0 (x : Finset \u03b1) in Filter.atTop, dist (\u2211 b in x, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ns : Finset \u03b1\nhs : |\u2016f\u2016 ^ ENNReal.toReal p - \u2211 x in s, \u2016\u2191f x\u2016 ^ ENNReal.toReal p| < \u03b5 ^ ENNReal.toReal p\nH :\n \u2016\u2211 i in s, lp.single p i (\u2191f i) - f\u2016 ^ ENNReal.toReal p = \u2016f\u2016 ^ ENNReal.toReal p - \u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n\u22a2 \u2016\u2211 x in s, lp.single p x (\u2191f x) - f\u2016 ^ ENNReal.toReal p < \u03b5 ^ ENNReal.toReal p\n[PROOFSTEP]\nrw [\u2190 H] at hs \n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\nhp\u2080 : 0 < p\nhp' : 0 < ENNReal.toReal p\nthis :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192 \u2200\u1da0 (x : Finset \u03b1) in Filter.atTop, dist (\u2211 b in x, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ns : Finset \u03b1\nhs : |\u2016\u2211 i in s, lp.single p i (\u2191f i) - f\u2016 ^ ENNReal.toReal p| < \u03b5 ^ ENNReal.toReal p\nH :\n \u2016\u2211 i in s, lp.single p i (\u2191f i) - f\u2016 ^ ENNReal.toReal p = \u2016f\u2016 ^ ENNReal.toReal p - \u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n\u22a2 \u2016\u2211 x in s, lp.single p x (\u2191f x) - f\u2016 ^ ENNReal.toReal p < \u03b5 ^ ENNReal.toReal p\n[PROOFSTEP]\nhave :\n |\u2016(\u2211 i in s, lp.single p i (f i : E i)) - f\u2016 ^ p.toReal| = \u2016(\u2211 i in s, lp.single p i (f i : E i)) - f\u2016 ^ p.toReal :=\n by simp only [Real.abs_rpow_of_nonneg (norm_nonneg _), abs_norm]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\nhp\u2080 : 0 < p\nhp' : 0 < ENNReal.toReal p\nthis :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192 \u2200\u1da0 (x : Finset \u03b1) in Filter.atTop, dist (\u2211 b in x, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ns : Finset \u03b1\nhs : |\u2016\u2211 i in s, lp.single p i (\u2191f i) - f\u2016 ^ ENNReal.toReal p| < \u03b5 ^ ENNReal.toReal p\nH :\n \u2016\u2211 i in s, lp.single p i (\u2191f i) - f\u2016 ^ ENNReal.toReal p = \u2016f\u2016 ^ ENNReal.toReal p - \u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p\n\u22a2 |\u2016\u2211 i in s, lp.single p i (\u2191f i) - f\u2016 ^ ENNReal.toReal p| = \u2016\u2211 i in s, lp.single p i (\u2191f i) - f\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nsimp only [Real.abs_rpow_of_nonneg (norm_nonneg _), abs_norm]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u2075 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedRing \ud835\udd5c\ninst\u271d\u00b3 : (i : \u03b1) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b2 : \u2200 (i : \u03b1), BoundedSMul \ud835\udd5c (E i)\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nf : { x // x \u2208 lp E p }\nhp\u2080 : 0 < p\nhp' : 0 < ENNReal.toReal p\nthis\u271d :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192 \u2200\u1da0 (x : Finset \u03b1) in Filter.atTop, dist (\u2211 b in x, \u2016\u2191f b\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ns : Finset \u03b1\nhs : |\u2016\u2211 i in s, lp.single p i (\u2191f i) - f\u2016 ^ ENNReal.toReal p| < \u03b5 ^ ENNReal.toReal p\nH :\n \u2016\u2211 i in s, lp.single p i (\u2191f i) - f\u2016 ^ ENNReal.toReal p = \u2016f\u2016 ^ ENNReal.toReal p - \u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p\nthis :\n |\u2016\u2211 i in s, lp.single p i (\u2191f i) - f\u2016 ^ ENNReal.toReal p| = \u2016\u2211 i in s, lp.single p i (\u2191f i) - f\u2016 ^ ENNReal.toReal p\n\u22a2 \u2016\u2211 x in s, lp.single p x (\u2191f x) - f\u2016 ^ ENNReal.toReal p < \u03b5 ^ ENNReal.toReal p\n[PROOFSTEP]\nexact this \u25b8 hs\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\n\u22a2 UniformContinuous Subtype.val\n[PROOFSTEP]\nhave hp : p \u2260 0 := (zero_lt_one.trans_le _i.elim).ne'\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\n\u22a2 UniformContinuous Subtype.val\n[PROOFSTEP]\nrw [uniformContinuous_pi]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\n\u22a2 \u2200 (i : \u03b1), UniformContinuous fun x => \u2191x i\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u22a2 UniformContinuous fun x => \u2191x i\n[PROOFSTEP]\nrw [NormedAddCommGroup.uniformity_basis_dist.uniformContinuous_iff NormedAddCommGroup.uniformity_basis_dist]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u22a2 \u2200 (i_1 : \u211d),\n 0 < i_1 \u2192\n \u2203 j,\n 0 < j \u2227\n \u2200 (x y : { x // x \u2208 lp E p }),\n (x, y) \u2208 {p_1 | \u2016p_1.fst - p_1.snd\u2016 < j} \u2192 (\u2191x i, \u2191y i) \u2208 {p | \u2016p.fst - p.snd\u2016 < i_1}\n[PROOFSTEP]\nintro \u03b5 h\u03b5\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 j,\n 0 < j \u2227\n \u2200 (x y : { x // x \u2208 lp E p }), (x, y) \u2208 {p_1 | \u2016p_1.fst - p_1.snd\u2016 < j} \u2192 (\u2191x i, \u2191y i) \u2208 {p | \u2016p.fst - p.snd\u2016 < \u03b5}\n[PROOFSTEP]\nrefine' \u27e8\u03b5, h\u03b5, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2200 (x y : { x // x \u2208 lp E p }), (x, y) \u2208 {p_1 | \u2016p_1.fst - p_1.snd\u2016 < \u03b5} \u2192 (\u2191x i, \u2191y i) \u2208 {p | \u2016p.fst - p.snd\u2016 < \u03b5}\n[PROOFSTEP]\nrintro f g (hfg : \u2016f - g\u2016 < \u03b5)\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nf g : { x // x \u2208 lp E p }\nhfg : \u2016f - g\u2016 < \u03b5\n\u22a2 (\u2191f i, \u2191g i) \u2208 {p | \u2016p.fst - p.snd\u2016 < \u03b5}\n[PROOFSTEP]\nhave : \u2016f i - g i\u2016 \u2264 \u2016f - g\u2016 := norm_apply_le_norm hp (f - g) i\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nf g : { x // x \u2208 lp E p }\nhfg : \u2016f - g\u2016 < \u03b5\nthis : \u2016\u2191f i - \u2191g i\u2016 \u2264 \u2016f - g\u2016\n\u22a2 (\u2191f i, \u2191g i) \u2208 {p | \u2016p.fst - p.snd\u2016 < \u03b5}\n[PROOFSTEP]\nexact this.trans_lt hfg\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\na : \u03b1\n\u22a2 \u2016f a\u2016 \u2264 C\n[PROOFSTEP]\nhave : Tendsto (fun k => \u2016F k a\u2016) l (\ud835\udcdd \u2016f a\u2016) := (Tendsto.comp (continuous_apply a).continuousAt hf).norm\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\na : \u03b1\nthis : Tendsto (fun k => \u2016\u2191(F k) a\u2016) l (\ud835\udcdd \u2016f a\u2016)\n\u22a2 \u2016f a\u2016 \u2264 C\n[PROOFSTEP]\nrefine' le_of_tendsto this (hCF.mono _)\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\na : \u03b1\nthis : Tendsto (fun k => \u2016\u2191(F k) a\u2016) l (\ud835\udcdd \u2016f a\u2016)\n\u22a2 \u2200 (x : \u03b9), \u2016F x\u2016 \u2264 C \u2192 \u2016\u2191(F x) a\u2016 \u2264 C\n[PROOFSTEP]\nintro k hCFk\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\na : \u03b1\nthis : Tendsto (fun k => \u2016\u2191(F k) a\u2016) l (\ud835\udcdd \u2016f a\u2016)\nk : \u03b9\nhCFk : \u2016F k\u2016 \u2264 C\n\u22a2 \u2016\u2191(F k) a\u2016 \u2264 C\n[PROOFSTEP]\nexact (norm_apply_le_norm ENNReal.top_ne_zero (F k) a).trans hCFk\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\ns : Finset \u03b1\n\u22a2 \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p \u2264 C ^ ENNReal.toReal p\n[PROOFSTEP]\nhave hp' : p \u2260 0 := (zero_lt_one.trans_le _i.elim).ne'\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\ns : Finset \u03b1\nhp' : p \u2260 0\n\u22a2 \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p \u2264 C ^ ENNReal.toReal p\n[PROOFSTEP]\nhave hp'' : 0 < p.toReal := ENNReal.toReal_pos hp' hp\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\ns : Finset \u03b1\nhp' : p \u2260 0\nhp'' : 0 < ENNReal.toReal p\n\u22a2 \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p \u2264 C ^ ENNReal.toReal p\n[PROOFSTEP]\nlet G : (\u2200 a, E a) \u2192 \u211d := fun f => \u2211 a in s, \u2016f a\u2016 ^ p.toReal\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\ns : Finset \u03b1\nhp' : p \u2260 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : \u03b1) \u2192 E a) \u2192 \u211d := fun f => \u2211 a in s, \u2016f a\u2016 ^ ENNReal.toReal p\n\u22a2 \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p \u2264 C ^ ENNReal.toReal p\n[PROOFSTEP]\nhave hG : Continuous G := by\n refine' continuous_finset_sum s _\n intro a _\n have : Continuous fun f : \u2200 a, E a => f a := continuous_apply a\n exact this.norm.rpow_const fun _ => Or.inr hp''.le\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\ns : Finset \u03b1\nhp' : p \u2260 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : \u03b1) \u2192 E a) \u2192 \u211d := fun f => \u2211 a in s, \u2016f a\u2016 ^ ENNReal.toReal p\n\u22a2 Continuous G\n[PROOFSTEP]\nrefine' continuous_finset_sum s _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\ns : Finset \u03b1\nhp' : p \u2260 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : \u03b1) \u2192 E a) \u2192 \u211d := fun f => \u2211 a in s, \u2016f a\u2016 ^ ENNReal.toReal p\n\u22a2 \u2200 (i : \u03b1), i \u2208 s \u2192 Continuous fun f => \u2016f i\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nintro a _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\ns : Finset \u03b1\nhp' : p \u2260 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : \u03b1) \u2192 E a) \u2192 \u211d := fun f => \u2211 a in s, \u2016f a\u2016 ^ ENNReal.toReal p\na : \u03b1\na\u271d : a \u2208 s\n\u22a2 Continuous fun f => \u2016f a\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nhave : Continuous fun f : \u2200 a, E a => f a := continuous_apply a\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\ns : Finset \u03b1\nhp' : p \u2260 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : \u03b1) \u2192 E a) \u2192 \u211d := fun f => \u2211 a in s, \u2016f a\u2016 ^ ENNReal.toReal p\na : \u03b1\na\u271d : a \u2208 s\nthis : Continuous fun f => f a\n\u22a2 Continuous fun f => \u2016f a\u2016 ^ ENNReal.toReal p\n[PROOFSTEP]\nexact this.norm.rpow_const fun _ => Or.inr hp''.le\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\ns : Finset \u03b1\nhp' : p \u2260 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : \u03b1) \u2192 E a) \u2192 \u211d := fun f => \u2211 a in s, \u2016f a\u2016 ^ ENNReal.toReal p\nhG : Continuous G\n\u22a2 \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p \u2264 C ^ ENNReal.toReal p\n[PROOFSTEP]\nrefine' le_of_tendsto (hG.continuousAt.tendsto.comp hf) _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\ns : Finset \u03b1\nhp' : p \u2260 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : \u03b1) \u2192 E a) \u2192 \u211d := fun f => \u2211 a in s, \u2016f a\u2016 ^ ENNReal.toReal p\nhG : Continuous G\n\u22a2 \u2200\u1da0 (c : \u03b9) in l, (G \u2218 id fun i => \u2191(F i)) c \u2264 C ^ ENNReal.toReal p\n[PROOFSTEP]\nrefine' hCF.mono _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\ns : Finset \u03b1\nhp' : p \u2260 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : \u03b1) \u2192 E a) \u2192 \u211d := fun f => \u2211 a in s, \u2016f a\u2016 ^ ENNReal.toReal p\nhG : Continuous G\n\u22a2 \u2200 (x : \u03b9), \u2016F x\u2016 \u2264 C \u2192 (G \u2218 id fun i => \u2191(F i)) x \u2264 C ^ ENNReal.toReal p\n[PROOFSTEP]\nintro k hCFk\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\ns : Finset \u03b1\nhp' : p \u2260 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : \u03b1) \u2192 E a) \u2192 \u211d := fun f => \u2211 a in s, \u2016f a\u2016 ^ ENNReal.toReal p\nhG : Continuous G\nk : \u03b9\nhCFk : \u2016F k\u2016 \u2264 C\n\u22a2 (G \u2218 id fun i => \u2191(F i)) k \u2264 C ^ ENNReal.toReal p\n[PROOFSTEP]\nrefine' (lp.sum_rpow_le_norm_rpow hp'' (F k) s).trans _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\ns : Finset \u03b1\nhp' : p \u2260 0\nhp'' : 0 < ENNReal.toReal p\nG : ((a : \u03b1) \u2192 E a) \u2192 \u211d := fun f => \u2211 a in s, \u2016f a\u2016 ^ ENNReal.toReal p\nhG : Continuous G\nk : \u03b9\nhCFk : \u2016F k\u2016 \u2264 C\n\u22a2 \u2016F k\u2016 ^ ENNReal.toReal p \u2264 C ^ ENNReal.toReal p\n[PROOFSTEP]\nexact Real.rpow_le_rpow (norm_nonneg _) hCFk hp''.le\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd \u2191f)\n\u22a2 \u2016f\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := hCF.exists\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd \u2191f)\ni : \u03b9\nhi : \u2016F i\u2016 \u2264 C\n\u22a2 \u2016f\u2016 \u2264 C\n[PROOFSTEP]\nhave hC : 0 \u2264 C := (norm_nonneg _).trans hi\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd \u2191f)\ni : \u03b9\nhi : \u2016F i\u2016 \u2264 C\nhC : 0 \u2264 C\n\u22a2 \u2016f\u2016 \u2264 C\n[PROOFSTEP]\nrcases eq_top_or_lt_top p with (rfl | hp)\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nC : \u211d\ni : \u03b9\nhC : 0 \u2264 C\n_i : Fact (1 \u2264 \u22a4)\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : { x // x \u2208 lp E \u22a4 }\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd \u2191f)\nhi : \u2016F i\u2016 \u2264 C\n\u22a2 \u2016f\u2016 \u2264 C\n[PROOFSTEP]\napply norm_le_of_forall_le hC\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nC : \u211d\ni : \u03b9\nhC : 0 \u2264 C\n_i : Fact (1 \u2264 \u22a4)\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : { x // x \u2208 lp E \u22a4 }\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd \u2191f)\nhi : \u2016F i\u2016 \u2264 C\n\u22a2 \u2200 (i : \u03b1), \u2016\u2191f i\u2016 \u2264 C\n[PROOFSTEP]\nexact norm_apply_le_of_tendsto hCF hf\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd \u2191f)\ni : \u03b9\nhi : \u2016F i\u2016 \u2264 C\nhC : 0 \u2264 C\nhp : p < \u22a4\n\u22a2 \u2016f\u2016 \u2264 C\n[PROOFSTEP]\nhave : 0 < p := zero_lt_one.trans_le _i.elim\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd \u2191f)\ni : \u03b9\nhi : \u2016F i\u2016 \u2264 C\nhC : 0 \u2264 C\nhp : p < \u22a4\nthis : 0 < p\n\u22a2 \u2016f\u2016 \u2264 C\n[PROOFSTEP]\nhave hp' : 0 < p.toReal := ENNReal.toReal_pos this.ne' hp.ne\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd \u2191f)\ni : \u03b9\nhi : \u2016F i\u2016 \u2264 C\nhC : 0 \u2264 C\nhp : p < \u22a4\nthis : 0 < p\nhp' : 0 < ENNReal.toReal p\n\u22a2 \u2016f\u2016 \u2264 C\n[PROOFSTEP]\napply norm_le_of_forall_sum_le hp' hC\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd \u2191f)\ni : \u03b9\nhi : \u2016F i\u2016 \u2264 C\nhC : 0 \u2264 C\nhp : p < \u22a4\nthis : 0 < p\nhp' : 0 < ENNReal.toReal p\n\u22a2 \u2200 (s : Finset \u03b1), \u2211 i in s, \u2016\u2191f i\u2016 ^ ENNReal.toReal p \u2264 C ^ ENNReal.toReal p\n[PROOFSTEP]\nexact sum_rpow_le_of_tendsto hp.ne hCF hf\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\n\u22a2 Mem\u2113p f p\n[PROOFSTEP]\nobtain \u27e8C, _, hCF'\u27e9 := hF.exists_pos_norm_le\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\nC : \u211d\nleft\u271d : C > 0\nhCF' : \u2200 (x : { x // x \u2208 lp E p }), x \u2208 Set.range F \u2192 \u2016x\u2016 \u2264 C\n\u22a2 Mem\u2113p f p\n[PROOFSTEP]\nhave hCF : \u2200 k, \u2016F k\u2016 \u2264 C := fun k => hCF' _ \u27e8k, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\nC : \u211d\nleft\u271d : C > 0\nhCF' : \u2200 (x : { x // x \u2208 lp E p }), x \u2208 Set.range F \u2192 \u2016x\u2016 \u2264 C\nhCF : \u2200 (k : \u03b9), \u2016F k\u2016 \u2264 C\n\u22a2 Mem\u2113p f p\n[PROOFSTEP]\nrcases eq_top_or_lt_top p with (rfl | hp)\n[GOAL]\ncase intro.intro.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nf : (a : \u03b1) \u2192 E a\nC : \u211d\nleft\u271d : C > 0\n_i : Fact (1 \u2264 \u22a4)\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhF : Metric.Bounded (Set.range F)\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\nhCF' : \u2200 (x : { x // x \u2208 lp E \u22a4 }), x \u2208 Set.range F \u2192 \u2016x\u2016 \u2264 C\nhCF : \u2200 (k : \u03b9), \u2016F k\u2016 \u2264 C\n\u22a2 Mem\u2113p f \u22a4\n[PROOFSTEP]\napply mem\u2113p_infty\n[GOAL]\ncase intro.intro.inl.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nf : (a : \u03b1) \u2192 E a\nC : \u211d\nleft\u271d : C > 0\n_i : Fact (1 \u2264 \u22a4)\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhF : Metric.Bounded (Set.range F)\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\nhCF' : \u2200 (x : { x // x \u2208 lp E \u22a4 }), x \u2208 Set.range F \u2192 \u2016x\u2016 \u2264 C\nhCF : \u2200 (k : \u03b9), \u2016F k\u2016 \u2264 C\n\u22a2 BddAbove (Set.range fun i => \u2016f i\u2016)\n[PROOFSTEP]\nuse C\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nf : (a : \u03b1) \u2192 E a\nC : \u211d\nleft\u271d : C > 0\n_i : Fact (1 \u2264 \u22a4)\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhF : Metric.Bounded (Set.range F)\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\nhCF' : \u2200 (x : { x // x \u2208 lp E \u22a4 }), x \u2208 Set.range F \u2192 \u2016x\u2016 \u2264 C\nhCF : \u2200 (k : \u03b9), \u2016F k\u2016 \u2264 C\n\u22a2 C \u2208 upperBounds (Set.range fun i => \u2016f i\u2016)\n[PROOFSTEP]\nrintro _ \u27e8a, rfl\u27e9\n[GOAL]\ncase h.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nf : (a : \u03b1) \u2192 E a\nC : \u211d\nleft\u271d : C > 0\n_i : Fact (1 \u2264 \u22a4)\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhF : Metric.Bounded (Set.range F)\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\nhCF' : \u2200 (x : { x // x \u2208 lp E \u22a4 }), x \u2208 Set.range F \u2192 \u2016x\u2016 \u2264 C\nhCF : \u2200 (k : \u03b9), \u2016F k\u2016 \u2264 C\na : \u03b1\n\u22a2 (fun i => \u2016f i\u2016) a \u2264 C\n[PROOFSTEP]\nrefine' norm_apply_le_of_tendsto (eventually_of_forall hCF) hf a\n[GOAL]\ncase intro.intro.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\nC : \u211d\nleft\u271d : C > 0\nhCF' : \u2200 (x : { x // x \u2208 lp E p }), x \u2208 Set.range F \u2192 \u2016x\u2016 \u2264 C\nhCF : \u2200 (k : \u03b9), \u2016F k\u2016 \u2264 C\nhp : p < \u22a4\n\u22a2 Mem\u2113p f p\n[PROOFSTEP]\napply mem\u2113p_gen'\n[GOAL]\ncase intro.intro.inr.hf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\nC : \u211d\nleft\u271d : C > 0\nhCF' : \u2200 (x : { x // x \u2208 lp E p }), x \u2208 Set.range F \u2192 \u2016x\u2016 \u2264 C\nhCF : \u2200 (k : \u03b9), \u2016F k\u2016 \u2264 C\nhp : p < \u22a4\n\u22a2 \u2200 (s : Finset \u03b1), \u2211 i in s, \u2016f i\u2016 ^ ENNReal.toReal p \u2264 ?intro.intro.inr.C\ncase intro.intro.inr.C\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nF : \u03b9 \u2192 { x // x \u2208 lp E p }\nhF : Metric.Bounded (Set.range F)\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\nC : \u211d\nleft\u271d : C > 0\nhCF' : \u2200 (x : { x // x \u2208 lp E p }), x \u2208 Set.range F \u2192 \u2016x\u2016 \u2264 C\nhCF : \u2200 (k : \u03b9), \u2016F k\u2016 \u2264 C\nhp : p < \u22a4\n\u22a2 \u211d\n[PROOFSTEP]\nexact sum_rpow_le_of_tendsto hp.ne (eventually_of_forall hCF) hf\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nF : \u2115 \u2192 { x // x \u2208 lp E p }\nhF : CauchySeq F\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) atTop (\ud835\udcdd \u2191f)\n\u22a2 Tendsto F atTop (\ud835\udcdd f)\n[PROOFSTEP]\nrw [Metric.nhds_basis_closedBall.tendsto_right_iff]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nF : \u2115 \u2192 { x // x \u2208 lp E p }\nhF : CauchySeq F\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) atTop (\ud835\udcdd \u2191f)\n\u22a2 \u2200 (i : \u211d), 0 < i \u2192 \u2200\u1da0 (x : \u2115) in atTop, F x \u2208 Metric.closedBall f i\n[PROOFSTEP]\nintro \u03b5 h\u03b5\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nF : \u2115 \u2192 { x // x \u2208 lp E p }\nhF : CauchySeq F\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) atTop (\ud835\udcdd \u2191f)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop, F x \u2208 Metric.closedBall f \u03b5\n[PROOFSTEP]\nhave h\u03b5' : {p : lp E p \u00d7 lp E p | \u2016p.1 - p.2\u2016 < \u03b5} \u2208 uniformity (lp E p) :=\n NormedAddCommGroup.uniformity_basis_dist.mem_of_mem h\u03b5\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nF : \u2115 \u2192 { x // x \u2208 lp E p }\nhF : CauchySeq F\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) atTop (\ud835\udcdd \u2191f)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b5' : {p_1 | \u2016p_1.fst - p_1.snd\u2016 < \u03b5} \u2208 uniformity { x // x \u2208 lp E p }\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop, F x \u2208 Metric.closedBall f \u03b5\n[PROOFSTEP]\nrefine' (hF.eventually_eventually h\u03b5').mono _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nF : \u2115 \u2192 { x // x \u2208 lp E p }\nhF : CauchySeq F\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) atTop (\ud835\udcdd \u2191f)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b5' : {p_1 | \u2016p_1.fst - p_1.snd\u2016 < \u03b5} \u2208 uniformity { x // x \u2208 lp E p }\n\u22a2 \u2200 (x : \u2115), (\u2200\u1da0 (l : \u2115) in atTop, (F x, F l) \u2208 {p_1 | \u2016p_1.fst - p_1.snd\u2016 < \u03b5}) \u2192 F x \u2208 Metric.closedBall f \u03b5\n[PROOFSTEP]\nrintro n (hn : \u2200\u1da0 l in atTop, \u2016(fun f => F n - f) (F l)\u2016 < \u03b5)\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nF : \u2115 \u2192 { x // x \u2208 lp E p }\nhF : CauchySeq F\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) atTop (\ud835\udcdd \u2191f)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b5' : {p_1 | \u2016p_1.fst - p_1.snd\u2016 < \u03b5} \u2208 uniformity { x // x \u2208 lp E p }\nn : \u2115\nhn : \u2200\u1da0 (l : \u2115) in atTop, \u2016(fun f => F n - f) (F l)\u2016 < \u03b5\n\u22a2 F n \u2208 Metric.closedBall f \u03b5\n[PROOFSTEP]\nrefine' norm_le_of_tendsto (hn.mono fun k hk => hk.le) _\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nF : \u2115 \u2192 { x // x \u2208 lp E p }\nhF : CauchySeq F\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) atTop (\ud835\udcdd \u2191f)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b5' : {p_1 | \u2016p_1.fst - p_1.snd\u2016 < \u03b5} \u2208 uniformity { x // x \u2208 lp E p }\nn : \u2115\nhn : \u2200\u1da0 (l : \u2115) in atTop, \u2016(fun f => F n - f) (F l)\u2016 < \u03b5\n\u22a2 Tendsto (id fun i => \u2191((fun f => F n - f) (F i))) atTop (\ud835\udcdd \u2191(F n - f))\n[PROOFSTEP]\nrw [tendsto_pi_nhds]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nF : \u2115 \u2192 { x // x \u2208 lp E p }\nhF : CauchySeq F\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) atTop (\ud835\udcdd \u2191f)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b5' : {p_1 | \u2016p_1.fst - p_1.snd\u2016 < \u03b5} \u2208 uniformity { x // x \u2208 lp E p }\nn : \u2115\nhn : \u2200\u1da0 (l : \u2115) in atTop, \u2016(fun f => F n - f) (F l)\u2016 < \u03b5\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun i => id (fun i => \u2191((fun f => F n - f) (F i))) i x) atTop (\ud835\udcdd (\u2191(F n - f) x))\n[PROOFSTEP]\nintro a\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\n_i : Fact (1 \u2264 p)\nF : \u2115 \u2192 { x // x \u2208 lp E p }\nhF : CauchySeq F\nf : { x // x \u2208 lp E p }\nhf : Tendsto (id fun i => \u2191(F i)) atTop (\ud835\udcdd \u2191f)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b5' : {p_1 | \u2016p_1.fst - p_1.snd\u2016 < \u03b5} \u2208 uniformity { x // x \u2208 lp E p }\nn : \u2115\nhn : \u2200\u1da0 (l : \u2115) in atTop, \u2016(fun f => F n - f) (F l)\u2016 < \u03b5\na : \u03b1\n\u22a2 Tendsto (fun i => id (fun i => \u2191((fun f => F n - f) (F i))) i a) atTop (\ud835\udcdd (\u2191(F n - f) a))\n[PROOFSTEP]\nexact (hf.apply a).const_sub (F n a)\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d\u00b9 : NeBot l\n_i : Fact (1 \u2264 p)\ninst\u271d : \u2200 (a : \u03b1), CompleteSpace (E a)\n\u22a2 \u2200 (u : \u2115 \u2192 { x // x \u2208 lp E p }), CauchySeq u \u2192 \u2203 a, Tendsto u atTop (\ud835\udcdd a)\n[PROOFSTEP]\nintro F hF\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d\u00b9 : NeBot l\n_i : Fact (1 \u2264 p)\ninst\u271d : \u2200 (a : \u03b1), CompleteSpace (E a)\nF : \u2115 \u2192 { x // x \u2208 lp E p }\nhF : CauchySeq F\n\u22a2 \u2203 a, Tendsto F atTop (\ud835\udcdd a)\n[PROOFSTEP]\nobtain \u27e8f, hf\u27e9 :=\n cauchySeq_tendsto_of_complete\n ((uniformContinuous_coe (p := p)).comp_cauchySeq hF)\n -- Since the Cauchy sequence is bounded, its pointwise limit `f` is in `lp E p`.\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d\u00b9 : NeBot l\n_i : Fact (1 \u2264 p)\ninst\u271d : \u2200 (a : \u03b1), CompleteSpace (E a)\nF : \u2115 \u2192 { x // x \u2208 lp E p }\nhF : CauchySeq F\nf : (i : \u03b1) \u2192 E i\nhf : Tendsto (Subtype.val \u2218 F) atTop (\ud835\udcdd f)\n\u22a2 \u2203 a, Tendsto F atTop (\ud835\udcdd a)\n[PROOFSTEP]\nhave hf' : Mem\u2113p f p := mem\u2113p_of_tendsto hF.bounded_range hf\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b2 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d\u00b9 : NeBot l\n_i : Fact (1 \u2264 p)\ninst\u271d : \u2200 (a : \u03b1), CompleteSpace (E a)\nF : \u2115 \u2192 { x // x \u2208 lp E p }\nhF : CauchySeq F\nf : (i : \u03b1) \u2192 E i\nhf : Tendsto (Subtype.val \u2218 F) atTop (\ud835\udcdd f)\nhf' : Mem\u2113p f p\n\u22a2 \u2203 a, Tendsto F atTop (\ud835\udcdd a)\n[PROOFSTEP]\nexact \u27e8\u27e8f, hf'\u27e9, tendsto_lp_of_tendsto_pi hF hf\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nK : \u211d\u22650\nhg : \u2200 (i : \u03b9), LipschitzWith K fun x => g x i\na\u2080 : \u03b1\nhga\u2080b : Mem\u2113p (g a\u2080) \u22a4\na : \u03b1\n\u22a2 Mem\u2113p (g a) \u22a4\n[PROOFSTEP]\nrcases hga\u2080b with \u27e8M, hM\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nK : \u211d\u22650\nhg : \u2200 (i : \u03b9), LipschitzWith K fun x => g x i\na\u2080 a : \u03b1\nM : \u211d\nhM : M \u2208 upperBounds (Set.range fun i => \u2016g a\u2080 i\u2016)\n\u22a2 Mem\u2113p (g a) \u22a4\n[PROOFSTEP]\nuse\u2191K * dist a a\u2080 + M\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nK : \u211d\u22650\nhg : \u2200 (i : \u03b9), LipschitzWith K fun x => g x i\na\u2080 a : \u03b1\nM : \u211d\nhM : M \u2208 upperBounds (Set.range fun i => \u2016g a\u2080 i\u2016)\n\u22a2 \u2191K * dist a a\u2080 + M \u2208 upperBounds (Set.range fun i => \u2016g a i\u2016)\n[PROOFSTEP]\nrintro - \u27e8i, rfl\u27e9\n[GOAL]\ncase h.intro\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nK : \u211d\u22650\nhg : \u2200 (i : \u03b9), LipschitzWith K fun x => g x i\na\u2080 a : \u03b1\nM : \u211d\nhM : M \u2208 upperBounds (Set.range fun i => \u2016g a\u2080 i\u2016)\ni : \u03b9\n\u22a2 (fun i => \u2016g a i\u2016) i \u2264 \u2191K * dist a a\u2080 + M\n[PROOFSTEP]\ncalc\n |g a i| = |g a i - g a\u2080 i + g a\u2080 i| := by simp\n _ \u2264 |g a i - g a\u2080 i| + |g a\u2080 i| := (abs_add _ _)\n _ \u2264 \u2191K * dist a a\u2080 + M := by\n gcongr\n \u00b7 exact lipschitzWith_iff_dist_le_mul.1 (hg i) a a\u2080\n \u00b7 exact hM \u27e8i, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nK : \u211d\u22650\nhg : \u2200 (i : \u03b9), LipschitzWith K fun x => g x i\na\u2080 a : \u03b1\nM : \u211d\nhM : M \u2208 upperBounds (Set.range fun i => \u2016g a\u2080 i\u2016)\ni : \u03b9\n\u22a2 |g a i| = |g a i - g a\u2080 i + g a\u2080 i|\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nK : \u211d\u22650\nhg : \u2200 (i : \u03b9), LipschitzWith K fun x => g x i\na\u2080 a : \u03b1\nM : \u211d\nhM : M \u2208 upperBounds (Set.range fun i => \u2016g a\u2080 i\u2016)\ni : \u03b9\n\u22a2 |g a i - g a\u2080 i| + |g a\u2080 i| \u2264 \u2191K * dist a a\u2080 + M\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nK : \u211d\u22650\nhg : \u2200 (i : \u03b9), LipschitzWith K fun x => g x i\na\u2080 a : \u03b1\nM : \u211d\nhM : M \u2208 upperBounds (Set.range fun i => \u2016g a\u2080 i\u2016)\ni : \u03b9\n\u22a2 |g a i - g a\u2080 i| \u2264 \u2191K * dist a a\u2080\n[PROOFSTEP]\nexact lipschitzWith_iff_dist_le_mul.1 (hg i) a a\u2080\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nK : \u211d\u22650\nhg : \u2200 (i : \u03b9), LipschitzWith K fun x => g x i\na\u2080 a : \u03b1\nM : \u211d\nhM : M \u2208 upperBounds (Set.range fun i => \u2016g a\u2080 i\u2016)\ni : \u03b9\n\u22a2 |g a\u2080 i| \u2264 M\n[PROOFSTEP]\nexact hM \u27e8i, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 { x // x \u2208 lp (fun i => \u211d) \u22a4 }\ns : Set \u03b1\nK : \u211d\u22650\n\u22a2 LipschitzOnWith K f s \u2194 \u2200 (i : \u03b9), LipschitzOnWith K (fun a => \u2191(f a) i) s\n[PROOFSTEP]\nsimp_rw [lipschitzOnWith_iff_dist_le_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 { x // x \u2208 lp (fun i => \u211d) \u22a4 }\ns : Set \u03b1\nK : \u211d\u22650\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (f x) (f y) \u2264 \u2191K * dist x y) \u2194\n \u2200 (i : \u03b9) (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (\u2191(f x) i) (\u2191(f y) i) \u2264 \u2191K * dist x y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 { x // x \u2208 lp (fun i => \u211d) \u22a4 }\ns : Set \u03b1\nK : \u211d\u22650\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (f x) (f y) \u2264 \u2191K * dist x y) \u2192\n \u2200 (i : \u03b9) (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (\u2191(f x) i) (\u2191(f y) i) \u2264 \u2191K * dist x y\n[PROOFSTEP]\nintro hfl i x hx y hy\n[GOAL]\ncase mp\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 { x // x \u2208 lp (fun i => \u211d) \u22a4 }\ns : Set \u03b1\nK : \u211d\u22650\nhfl : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (f x) (f y) \u2264 \u2191K * dist x y\ni : \u03b9\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\n\u22a2 dist (\u2191(f x) i) (\u2191(f y) i) \u2264 \u2191K * dist x y\n[PROOFSTEP]\ncalc\n dist (f x i) (f y i) \u2264 dist (f x) (f y) := lp.norm_apply_le_norm top_ne_zero (f x - f y) i\n _ \u2264 K * dist x y := hfl x hx y hy\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 { x // x \u2208 lp (fun i => \u211d) \u22a4 }\ns : Set \u03b1\nK : \u211d\u22650\n\u22a2 (\u2200 (i : \u03b9) (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (\u2191(f x) i) (\u2191(f y) i) \u2264 \u2191K * dist x y) \u2192\n \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (f x) (f y) \u2264 \u2191K * dist x y\n[PROOFSTEP]\nintro hgl x hx y hy\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 { x // x \u2208 lp (fun i => \u211d) \u22a4 }\ns : Set \u03b1\nK : \u211d\u22650\nhgl : \u2200 (i : \u03b9) (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (\u2191(f x) i) (\u2191(f y) i) \u2264 \u2191K * dist x y\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\n\u22a2 dist (f x) (f y) \u2264 \u2191K * dist x y\n[PROOFSTEP]\napply lp.norm_le_of_forall_le\n[GOAL]\ncase mpr.hC\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 { x // x \u2208 lp (fun i => \u211d) \u22a4 }\ns : Set \u03b1\nK : \u211d\u22650\nhgl : \u2200 (i : \u03b9) (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (\u2191(f x) i) (\u2191(f y) i) \u2264 \u2191K * dist x y\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\n\u22a2 0 \u2264 \u2191K * dist x y\ncase mpr.hCf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 { x // x \u2208 lp (fun i => \u211d) \u22a4 }\ns : Set \u03b1\nK : \u211d\u22650\nhgl : \u2200 (i : \u03b9) (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (\u2191(f x) i) (\u2191(f y) i) \u2264 \u2191K * dist x y\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\n\u22a2 \u2200 (i : \u03b9), \u2016\u2191(f x - f y) i\u2016 \u2264 \u2191K * dist x y\n[PROOFSTEP]\npositivity\n[GOAL]\ncase mpr.hCf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 { x // x \u2208 lp (fun i => \u211d) \u22a4 }\ns : Set \u03b1\nK : \u211d\u22650\nhgl : \u2200 (i : \u03b9) (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (\u2191(f x) i) (\u2191(f y) i) \u2264 \u2191K * dist x y\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\n\u22a2 \u2200 (i : \u03b9), \u2016\u2191(f x - f y) i\u2016 \u2264 \u2191K * dist x y\n[PROOFSTEP]\nintro i\n[GOAL]\ncase mpr.hCf\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 { x // x \u2208 lp (fun i => \u211d) \u22a4 }\ns : Set \u03b1\nK : \u211d\u22650\nhgl : \u2200 (i : \u03b9) (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist (\u2191(f x) i) (\u2191(f y) i) \u2264 \u2191K * dist x y\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\ni : \u03b9\n\u22a2 \u2016\u2191(f x - f y) i\u2016 \u2264 \u2191K * dist x y\n[PROOFSTEP]\napply hgl i x hx y hy\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 { x // x \u2208 lp (fun i => \u211d) \u22a4 }\nK : \u211d\u22650\n\u22a2 LipschitzWith K f \u2194 \u2200 (i : \u03b9), LipschitzWith K fun a => \u2191(f a) i\n[PROOFSTEP]\nsimp_rw [\u2190 lipschitz_on_univ]\n[GOAL]\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u03b1 \u2192 { x // x \u2208 lp (fun i => \u211d) \u22a4 }\nK : \u211d\u22650\n\u22a2 LipschitzOnWith K f Set.univ \u2194 \u2200 (i : \u03b9), LipschitzOnWith K (fun a => \u2191(f a) i) Set.univ\n[PROOFSTEP]\napply LipschitzOnWith.coordinate\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.lpSpace", "llama_tokens": 96647, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936430859598, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.41301941574296935}} {"text": "[GOAL]\ns t : \u2124\nr' : \u2115\ns' t' : \u2124\n\u22a2 xgcdAux 0 s t r' s' t' = (r', s', t')\n[PROOFSTEP]\nsimp [xgcdAux]\n[GOAL]\nr : \u2115\ns t : \u2124\nr' : \u2115\ns' t' : \u2124\nh : 0 < r\n\u22a2 xgcdAux r s t r' s' t' = xgcdAux (r' % r) (s' - \u2191r' / \u2191r * s) (t' - \u2191r' / \u2191r * t) r s t\n[PROOFSTEP]\nobtain \u27e8r, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero h.ne'\n[GOAL]\ncase intro\ns t : \u2124\nr' : \u2115\ns' t' : \u2124\nr : \u2115\nh : 0 < succ r\n\u22a2 xgcdAux (succ r) s t r' s' t' =\n xgcdAux (r' % succ r) (s' - \u2191r' / \u2191(succ r) * s) (t' - \u2191r' / \u2191(succ r) * t) (succ r) s t\n[PROOFSTEP]\nrfl\n[GOAL]\ns : \u2115\n\u22a2 gcdA 0 s = 0\n[PROOFSTEP]\nunfold gcdA\n[GOAL]\ns : \u2115\n\u22a2 (xgcd 0 s).fst = 0\n[PROOFSTEP]\nrw [xgcd, xgcd_zero_left]\n[GOAL]\ns : \u2115\n\u22a2 gcdB 0 s = 1\n[PROOFSTEP]\nunfold gcdB\n[GOAL]\ns : \u2115\n\u22a2 (xgcd 0 s).snd = 1\n[PROOFSTEP]\nrw [xgcd, xgcd_zero_left]\n[GOAL]\ns : \u2115\nh : s \u2260 0\n\u22a2 gcdA s 0 = 1\n[PROOFSTEP]\nunfold gcdA xgcd\n[GOAL]\ns : \u2115\nh : s \u2260 0\n\u22a2 (xgcdAux s 1 0 0 0 1).snd.fst = 1\n[PROOFSTEP]\nobtain \u27e8s, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero h\n[GOAL]\ncase intro\ns : \u2115\nh : succ s \u2260 0\n\u22a2 (xgcdAux (succ s) 1 0 0 0 1).snd.fst = 1\n[PROOFSTEP]\nrw [xgcdAux_succ]\n[GOAL]\ncase intro\ns : \u2115\nh : succ s \u2260 0\n\u22a2 (xgcdAux (0 % succ s) (0 - \u21910 / \u2191(succ s) * 1) (1 - \u21910 / \u2191(succ s) * 0) (succ s) 1 0).snd.fst = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ns : \u2115\nh : s \u2260 0\n\u22a2 gcdB s 0 = 0\n[PROOFSTEP]\nunfold gcdB xgcd\n[GOAL]\ns : \u2115\nh : s \u2260 0\n\u22a2 (xgcdAux s 1 0 0 0 1).snd.snd = 0\n[PROOFSTEP]\nobtain \u27e8s, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero h\n[GOAL]\ncase intro\ns : \u2115\nh : succ s \u2260 0\n\u22a2 (xgcdAux (succ s) 1 0 0 0 1).snd.snd = 0\n[PROOFSTEP]\nrw [xgcdAux_succ]\n[GOAL]\ncase intro\ns : \u2115\nh : succ s \u2260 0\n\u22a2 (xgcdAux (0 % succ s) (0 - \u21910 / \u2191(succ s) * 1) (1 - \u21910 / \u2191(succ s) * 0) (succ s) 1 0).snd.snd = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nx y : \u2115\n\u22a2 \u2200 (n : \u2115) (s t s' t' : \u2124), (xgcdAux 0 s t n s' t').fst = gcd 0 n\n[PROOFSTEP]\nsimp\n[GOAL]\nx\u271d y\u271d x y : \u2115\nh : 0 < x\nIH : \u2200 (s t s' t' : \u2124), (xgcdAux (y % x) s t x s' t').fst = gcd (y % x) x\ns t s' t' : \u2124\n\u22a2 (xgcdAux x s t y s' t').fst = gcd x y\n[PROOFSTEP]\nsimp [xgcdAux_rec, h, IH]\n[GOAL]\nx\u271d y\u271d x y : \u2115\nh : 0 < x\nIH : \u2200 (s t s' t' : \u2124), (xgcdAux (y % x) s t x s' t').fst = gcd (y % x) x\ns t s' t' : \u2124\n\u22a2 gcd (y % x) x = gcd x y\n[PROOFSTEP]\nrw [\u2190 gcd_rec]\n[GOAL]\nx y : \u2115\n\u22a2 xgcdAux x 1 0 y 0 1 = (gcd x y, xgcd x y)\n[PROOFSTEP]\nrw [xgcd, \u2190 xgcdAux_fst x y 1 0 0 1]\n[GOAL]\nx y : \u2115\n\u22a2 xgcd x y = (gcdA x y, gcdB x y)\n[PROOFSTEP]\nunfold gcdA gcdB\n[GOAL]\nx y : \u2115\n\u22a2 xgcd x y = ((xgcd x y).fst, (xgcd x y).snd)\n[PROOFSTEP]\ncases xgcd x y\n[GOAL]\ncase mk\nx y : \u2115\nfst\u271d snd\u271d : \u2124\n\u22a2 (fst\u271d, snd\u271d) = ((fst\u271d, snd\u271d).fst, (fst\u271d, snd\u271d).snd)\n[PROOFSTEP]\nrfl\n[GOAL]\nx y r r' : \u2115\n\u22a2 \u2200 {s t s' t' : \u2124}, Nat.P x y (r, s, t) \u2192 Nat.P x y (r', s', t') \u2192 Nat.P x y (xgcdAux r s t r' s' t')\n[PROOFSTEP]\ninduction r, r' using gcd.induction with\n| H0 => simp\n| H1 a b h IH =>\n intro s t s' t' p p'\n rw [xgcdAux_rec h]; refine' IH _ p; dsimp [P] at *\n rw [Int.emod_def]; generalize (b / a : \u2124) = k\n rw [p, p', mul_sub, sub_add_eq_add_sub, mul_sub, add_mul, mul_comm k t, mul_comm k s, \u2190 mul_assoc, \u2190 mul_assoc,\n add_comm (x * s * k), \u2190 add_sub_assoc, sub_sub]\n[GOAL]\nx y r r' : \u2115\n\u22a2 \u2200 {s t s' t' : \u2124}, Nat.P x y (r, s, t) \u2192 Nat.P x y (r', s', t') \u2192 Nat.P x y (xgcdAux r s t r' s' t')\n[PROOFSTEP]\ninduction r, r' using gcd.induction with\n| H0 => simp\n| H1 a b h IH =>\n intro s t s' t' p p'\n rw [xgcdAux_rec h]; refine' IH _ p; dsimp [P] at *\n rw [Int.emod_def]; generalize (b / a : \u2124) = k\n rw [p, p', mul_sub, sub_add_eq_add_sub, mul_sub, add_mul, mul_comm k t, mul_comm k s, \u2190 mul_assoc, \u2190 mul_assoc,\n add_comm (x * s * k), \u2190 add_sub_assoc, sub_sub]\n[GOAL]\ncase H0\nx y n\u271d : \u2115\n\u22a2 \u2200 {s t s' t' : \u2124}, Nat.P x y (0, s, t) \u2192 Nat.P x y (n\u271d, s', t') \u2192 Nat.P x y (xgcdAux 0 s t n\u271d s' t')\n[PROOFSTEP]\n\n| H0 => simp\n[GOAL]\ncase H0\nx y n\u271d : \u2115\n\u22a2 \u2200 {s t s' t' : \u2124}, Nat.P x y (0, s, t) \u2192 Nat.P x y (n\u271d, s', t') \u2192 Nat.P x y (xgcdAux 0 s t n\u271d s' t')\n[PROOFSTEP]\nsimp\n[GOAL]\ncase H1\nx y a b : \u2115\nh : 0 < a\nIH : \u2200 {s t s' t' : \u2124}, Nat.P x y (b % a, s, t) \u2192 Nat.P x y (a, s', t') \u2192 Nat.P x y (xgcdAux (b % a) s t a s' t')\n\u22a2 \u2200 {s t s' t' : \u2124}, Nat.P x y (a, s, t) \u2192 Nat.P x y (b, s', t') \u2192 Nat.P x y (xgcdAux a s t b s' t')\n[PROOFSTEP]\n\n| H1 a b h IH =>\n intro s t s' t' p p'\n rw [xgcdAux_rec h]; refine' IH _ p; dsimp [P] at *\n rw [Int.emod_def]; generalize (b / a : \u2124) = k\n rw [p, p', mul_sub, sub_add_eq_add_sub, mul_sub, add_mul, mul_comm k t, mul_comm k s, \u2190 mul_assoc, \u2190 mul_assoc,\n add_comm (x * s * k), \u2190 add_sub_assoc, sub_sub]\n[GOAL]\ncase H1\nx y a b : \u2115\nh : 0 < a\nIH : \u2200 {s t s' t' : \u2124}, Nat.P x y (b % a, s, t) \u2192 Nat.P x y (a, s', t') \u2192 Nat.P x y (xgcdAux (b % a) s t a s' t')\n\u22a2 \u2200 {s t s' t' : \u2124}, Nat.P x y (a, s, t) \u2192 Nat.P x y (b, s', t') \u2192 Nat.P x y (xgcdAux a s t b s' t')\n[PROOFSTEP]\nintro s t s' t' p p'\n[GOAL]\ncase H1\nx y a b : \u2115\nh : 0 < a\nIH : \u2200 {s t s' t' : \u2124}, Nat.P x y (b % a, s, t) \u2192 Nat.P x y (a, s', t') \u2192 Nat.P x y (xgcdAux (b % a) s t a s' t')\ns t s' t' : \u2124\np : Nat.P x y (a, s, t)\np' : Nat.P x y (b, s', t')\n\u22a2 Nat.P x y (xgcdAux a s t b s' t')\n[PROOFSTEP]\nrw [xgcdAux_rec h]\n[GOAL]\ncase H1\nx y a b : \u2115\nh : 0 < a\nIH : \u2200 {s t s' t' : \u2124}, Nat.P x y (b % a, s, t) \u2192 Nat.P x y (a, s', t') \u2192 Nat.P x y (xgcdAux (b % a) s t a s' t')\ns t s' t' : \u2124\np : Nat.P x y (a, s, t)\np' : Nat.P x y (b, s', t')\n\u22a2 Nat.P x y (xgcdAux (b % a) (s' - \u2191b / \u2191a * s) (t' - \u2191b / \u2191a * t) a s t)\n[PROOFSTEP]\nrefine' IH _ p\n[GOAL]\ncase H1\nx y a b : \u2115\nh : 0 < a\nIH : \u2200 {s t s' t' : \u2124}, Nat.P x y (b % a, s, t) \u2192 Nat.P x y (a, s', t') \u2192 Nat.P x y (xgcdAux (b % a) s t a s' t')\ns t s' t' : \u2124\np : Nat.P x y (a, s, t)\np' : Nat.P x y (b, s', t')\n\u22a2 Nat.P x y (b % a, s' - \u2191b / \u2191a * s, t' - \u2191b / \u2191a * t)\n[PROOFSTEP]\ndsimp [P] at *\n[GOAL]\ncase H1\nx y a b : \u2115\nh : 0 < a\nIH :\n \u2200 {s t s' t' : \u2124},\n \u2191b % \u2191a = \u2191x * s + \u2191y * t \u2192\n \u2191a = \u2191x * s' + \u2191y * t' \u2192\n \u2191(xgcdAux (b % a) s t a s' t').fst =\n \u2191x * (xgcdAux (b % a) s t a s' t').2.fst + \u2191y * (xgcdAux (b % a) s t a s' t').2.snd\ns t s' t' : \u2124\np : \u2191a = \u2191x * s + \u2191y * t\np' : \u2191b = \u2191x * s' + \u2191y * t'\n\u22a2 \u2191b % \u2191a = \u2191x * (s' - \u2191b / \u2191a * s) + \u2191y * (t' - \u2191b / \u2191a * t)\n[PROOFSTEP]\nrw [Int.emod_def]\n[GOAL]\ncase H1\nx y a b : \u2115\nh : 0 < a\nIH :\n \u2200 {s t s' t' : \u2124},\n \u2191b % \u2191a = \u2191x * s + \u2191y * t \u2192\n \u2191a = \u2191x * s' + \u2191y * t' \u2192\n \u2191(xgcdAux (b % a) s t a s' t').fst =\n \u2191x * (xgcdAux (b % a) s t a s' t').2.fst + \u2191y * (xgcdAux (b % a) s t a s' t').2.snd\ns t s' t' : \u2124\np : \u2191a = \u2191x * s + \u2191y * t\np' : \u2191b = \u2191x * s' + \u2191y * t'\n\u22a2 \u2191b - \u2191a * (\u2191b / \u2191a) = \u2191x * (s' - \u2191b / \u2191a * s) + \u2191y * (t' - \u2191b / \u2191a * t)\n[PROOFSTEP]\ngeneralize (b / a : \u2124) = k\n[GOAL]\ncase H1\nx y a b : \u2115\nh : 0 < a\nIH :\n \u2200 {s t s' t' : \u2124},\n \u2191b % \u2191a = \u2191x * s + \u2191y * t \u2192\n \u2191a = \u2191x * s' + \u2191y * t' \u2192\n \u2191(xgcdAux (b % a) s t a s' t').fst =\n \u2191x * (xgcdAux (b % a) s t a s' t').2.fst + \u2191y * (xgcdAux (b % a) s t a s' t').2.snd\ns t s' t' : \u2124\np : \u2191a = \u2191x * s + \u2191y * t\np' : \u2191b = \u2191x * s' + \u2191y * t'\nk : \u2124\n\u22a2 \u2191b - \u2191a * k = \u2191x * (s' - k * s) + \u2191y * (t' - k * t)\n[PROOFSTEP]\nrw [p, p', mul_sub, sub_add_eq_add_sub, mul_sub, add_mul, mul_comm k t, mul_comm k s, \u2190 mul_assoc, \u2190 mul_assoc,\n add_comm (x * s * k), \u2190 add_sub_assoc, sub_sub]\n[GOAL]\nx y : \u2115\n\u22a2 \u2191(gcd x y) = \u2191x * gcdA x y + \u2191y * gcdB x y\n[PROOFSTEP]\nhave := @xgcdAux_P x y x y 1 0 0 1 (by simp [P]) (by simp [P])\n[GOAL]\nx y : \u2115\n\u22a2 Nat.P x y (x, 1, 0)\n[PROOFSTEP]\nsimp [P]\n[GOAL]\nx y : \u2115\n\u22a2 Nat.P x y (y, 0, 1)\n[PROOFSTEP]\nsimp [P]\n[GOAL]\nx y : \u2115\nthis : Nat.P x y (xgcdAux x 1 0 y 0 1)\n\u22a2 \u2191(gcd x y) = \u2191x * gcdA x y + \u2191y * gcdB x y\n[PROOFSTEP]\nrwa [xgcdAux_val, xgcd_val] at this \n[GOAL]\nk n : \u2115\nhk : gcd n k < k\n\u22a2 \u2203 m, n * m % k = gcd n k\n[PROOFSTEP]\nhave hk' := Int.ofNat_ne_zero.2 (ne_of_gt (lt_of_le_of_lt (zero_le (gcd n k)) hk))\n[GOAL]\nk n : \u2115\nhk : gcd n k < k\nhk' : \u2191k \u2260 0\n\u22a2 \u2203 m, n * m % k = gcd n k\n[PROOFSTEP]\nhave key := congr_arg (fun (m : \u2124) => (m % k).toNat) (gcd_eq_gcd_ab n k)\n[GOAL]\nk n : \u2115\nhk : gcd n k < k\nhk' : \u2191k \u2260 0\nkey : (fun m => Int.toNat (m % \u2191k)) \u2191(gcd n k) = (fun m => Int.toNat (m % \u2191k)) (\u2191n * gcdA n k + \u2191k * gcdB n k)\n\u22a2 \u2203 m, n * m % k = gcd n k\n[PROOFSTEP]\nsimp only at key \n[GOAL]\nk n : \u2115\nhk : gcd n k < k\nhk' : \u2191k \u2260 0\nkey : Int.toNat (\u2191(gcd n k) % \u2191k) = Int.toNat ((\u2191n * gcdA n k + \u2191k * gcdB n k) % \u2191k)\n\u22a2 \u2203 m, n * m % k = gcd n k\n[PROOFSTEP]\nrw [Int.add_mul_emod_self_left, \u2190 Int.coe_nat_mod, Int.toNat_coe_nat, mod_eq_of_lt hk] at key \n[GOAL]\nk n : \u2115\nhk : gcd n k < k\nhk' : \u2191k \u2260 0\nkey : gcd n k = Int.toNat (\u2191n * gcdA n k % \u2191k)\n\u22a2 \u2203 m, n * m % k = gcd n k\n[PROOFSTEP]\nrefine' \u27e8(n.gcdA k % k).toNat, Eq.trans (Int.ofNat.inj _) key.symm\u27e9\n[GOAL]\nk n : \u2115\nhk : gcd n k < k\nhk' : \u2191k \u2260 0\nkey : gcd n k = Int.toNat (\u2191n * gcdA n k % \u2191k)\n\u22a2 Int.ofNat (n * Int.toNat (gcdA n k % \u2191k) % k) = Int.ofNat (Int.toNat (\u2191n * gcdA n k % \u2191k))\n[PROOFSTEP]\nrw [Int.ofNat_eq_coe, Int.coe_nat_mod, Int.ofNat_mul, Int.toNat_of_nonneg (Int.emod_nonneg _ hk'), Int.ofNat_eq_coe,\n Int.toNat_of_nonneg (Int.emod_nonneg _ hk'), Int.mul_emod, Int.emod_emod, \u2190 Int.mul_emod]\n[GOAL]\nm n : \u2115\n\u22a2 \u2191(gcd \u2191m -[n+1]) = \u2191m * gcdA \u2191m -[n+1] + -(\u2191n + 1) * -Nat.gcdB (natAbs \u2191m) (Nat.succ n)\n[PROOFSTEP]\nrw [neg_mul_neg]\n[GOAL]\nm n : \u2115\n\u22a2 \u2191(gcd \u2191m -[n+1]) = \u2191m * gcdA \u2191m -[n+1] + (\u2191n + 1) * Nat.gcdB (natAbs \u2191m) (Nat.succ n)\n[PROOFSTEP]\napply Nat.gcd_eq_gcd_ab\n[GOAL]\nm n : \u2115\n\u22a2 \u2191(gcd -[m+1] \u2191n) = -(\u2191m + 1) * -Nat.gcdA (Nat.succ m) (natAbs \u2191n) + \u2191n * gcdB -[m+1] \u2191n\n[PROOFSTEP]\nrw [neg_mul_neg]\n[GOAL]\nm n : \u2115\n\u22a2 \u2191(gcd -[m+1] \u2191n) = (\u2191m + 1) * Nat.gcdA (Nat.succ m) (natAbs \u2191n) + \u2191n * gcdB -[m+1] \u2191n\n[PROOFSTEP]\napply Nat.gcd_eq_gcd_ab\n[GOAL]\nm n : \u2115\n\u22a2 \u2191(gcd -[m+1] -[n+1]) =\n -(\u2191m + 1) * -Nat.gcdA (Nat.succ m) (natAbs -[n+1]) + -(\u2191n + 1) * -Nat.gcdB (natAbs -[m+1]) (Nat.succ n)\n[PROOFSTEP]\nrw [neg_mul_neg, neg_mul_neg]\n[GOAL]\nm n : \u2115\n\u22a2 \u2191(gcd -[m+1] -[n+1]) =\n (\u2191m + 1) * Nat.gcdA (Nat.succ m) (natAbs -[n+1]) + (\u2191n + 1) * Nat.gcdB (natAbs -[m+1]) (Nat.succ n)\n[PROOFSTEP]\napply Nat.gcd_eq_gcd_ab\n[GOAL]\na b : \u2124\nH : b \u2223 a\n\u22a2 natAbs (a / b) = natAbs a / natAbs b\n[PROOFSTEP]\nrcases Nat.eq_zero_or_pos (natAbs b) with (h | h)\n[GOAL]\ncase inl\na b : \u2124\nH : b \u2223 a\nh : natAbs b = 0\n\u22a2 natAbs (a / b) = natAbs a / natAbs b\n[PROOFSTEP]\nrw [natAbs_eq_zero.1 h]\n[GOAL]\ncase inl\na b : \u2124\nH : b \u2223 a\nh : natAbs b = 0\n\u22a2 natAbs (a / 0) = natAbs a / natAbs 0\n[PROOFSTEP]\nsimp [Int.ediv_zero]\n[GOAL]\ncase inr\na b : \u2124\nH : b \u2223 a\nh : natAbs b > 0\n\u22a2 natAbs (a / b) = natAbs a / natAbs b\n[PROOFSTEP]\ncalc\n natAbs (a / b) = natAbs (a / b) * 1 := by rw [mul_one]\n _ = natAbs (a / b) * (natAbs b / natAbs b) := by rw [Nat.div_self h]\n _ = natAbs (a / b) * natAbs b / natAbs b := by rw [Nat.mul_div_assoc _ dvd_rfl]\n _ = natAbs (a / b * b) / natAbs b := by rw [natAbs_mul (a / b) b]\n _ = natAbs a / natAbs b := by rw [Int.ediv_mul_cancel H]\n[GOAL]\na b : \u2124\nH : b \u2223 a\nh : natAbs b > 0\n\u22a2 natAbs (a / b) = natAbs (a / b) * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\na b : \u2124\nH : b \u2223 a\nh : natAbs b > 0\n\u22a2 natAbs (a / b) * 1 = natAbs (a / b) * (natAbs b / natAbs b)\n[PROOFSTEP]\nrw [Nat.div_self h]\n[GOAL]\na b : \u2124\nH : b \u2223 a\nh : natAbs b > 0\n\u22a2 natAbs (a / b) * (natAbs b / natAbs b) = natAbs (a / b) * natAbs b / natAbs b\n[PROOFSTEP]\nrw [Nat.mul_div_assoc _ dvd_rfl]\n[GOAL]\na b : \u2124\nH : b \u2223 a\nh : natAbs b > 0\n\u22a2 natAbs (a / b) * natAbs b / natAbs b = natAbs (a / b * b) / natAbs b\n[PROOFSTEP]\nrw [natAbs_mul (a / b) b]\n[GOAL]\na b : \u2124\nH : b \u2223 a\nh : natAbs b > 0\n\u22a2 natAbs (a / b * b) / natAbs b = natAbs a / natAbs b\n[PROOFSTEP]\nrw [Int.ediv_mul_cancel H]\n[GOAL]\ni j k : \u2124\nk_non_zero : k \u2260 0\nH : k * i \u2223 k * j\nl : \u2124\nH1 : k * j = k * i * l\n\u22a2 i \u2223 j\n[PROOFSTEP]\nrw [mul_assoc] at H1 \n[GOAL]\ni j k : \u2124\nk_non_zero : k \u2260 0\nH : k * i \u2223 k * j\nl : \u2124\nH1 : k * j = k * (i * l)\n\u22a2 i \u2223 j\n[PROOFSTEP]\nexact \u27e8_, mul_left_cancel\u2080 k_non_zero H1\u27e9\n[GOAL]\ni j k : \u2124\nk_non_zero : k \u2260 0\nH : i * k \u2223 j * k\n\u22a2 i \u2223 j\n[PROOFSTEP]\nrw [mul_comm i k, mul_comm j k] at H \n[GOAL]\ni j k : \u2124\nk_non_zero : k \u2260 0\nH : k * i \u2223 k * j\n\u22a2 i \u2223 j\n[PROOFSTEP]\nexact dvd_of_mul_dvd_mul_left k_non_zero H\n[GOAL]\ni j : \u2124\n\u22a2 gcd i j * lcm i j = natAbs (i * j)\n[PROOFSTEP]\nrw [Int.gcd, Int.lcm, Nat.gcd_mul_lcm, natAbs_mul]\n[GOAL]\ni : \u2124\n\u22a2 gcd i i = natAbs i\n[PROOFSTEP]\nsimp [gcd]\n[GOAL]\ni : \u2124\n\u22a2 gcd 0 i = natAbs i\n[PROOFSTEP]\nsimp [gcd]\n[GOAL]\ni : \u2124\n\u22a2 gcd i 0 = natAbs i\n[PROOFSTEP]\nsimp [gcd]\n[GOAL]\nx y : \u2124\n\u22a2 gcd x (-y) = gcd x y\n[PROOFSTEP]\nrw [Int.gcd, Int.gcd, natAbs_neg]\n[GOAL]\nx y : \u2124\n\u22a2 gcd (-x) y = gcd x y\n[PROOFSTEP]\nrw [Int.gcd, Int.gcd, natAbs_neg]\n[GOAL]\ni j k : \u2124\n\u22a2 gcd (i * j) (i * k) = natAbs i * gcd j k\n[PROOFSTEP]\nrw [Int.gcd, Int.gcd, natAbs_mul, natAbs_mul]\n[GOAL]\ni j k : \u2124\n\u22a2 Nat.gcd (natAbs i * natAbs j) (natAbs i * natAbs k) = natAbs i * Nat.gcd (natAbs j) (natAbs k)\n[PROOFSTEP]\napply Nat.gcd_mul_left\n[GOAL]\ni j k : \u2124\n\u22a2 gcd (i * j) (k * j) = gcd i k * natAbs j\n[PROOFSTEP]\nrw [Int.gcd, Int.gcd, natAbs_mul, natAbs_mul]\n[GOAL]\ni j k : \u2124\n\u22a2 Nat.gcd (natAbs i * natAbs j) (natAbs k * natAbs j) = Nat.gcd (natAbs i) (natAbs k) * natAbs j\n[PROOFSTEP]\napply Nat.gcd_mul_right\n[GOAL]\ni j : \u2124\n\u22a2 gcd i j = 0 \u2194 i = 0 \u2227 j = 0\n[PROOFSTEP]\nrw [gcd, Nat.gcd_eq_zero_iff, natAbs_eq_zero, natAbs_eq_zero]\n[GOAL]\ni j k : \u2124\nH1 : k \u2223 i\nH2 : k \u2223 j\n\u22a2 gcd (i / k) (j / k) = gcd i j / natAbs k\n[PROOFSTEP]\nrw [gcd, natAbs_ediv i k H1, natAbs_ediv j k H2]\n[GOAL]\ni j k : \u2124\nH1 : k \u2223 i\nH2 : k \u2223 j\n\u22a2 Nat.gcd (natAbs i / natAbs k) (natAbs j / natAbs k) = gcd i j / natAbs k\n[PROOFSTEP]\nexact Nat.gcd_div (natAbs_dvd_natAbs.mpr H1) (natAbs_dvd_natAbs.mpr H2)\n[GOAL]\ni j : \u2124\nH : 0 < gcd i j\n\u22a2 gcd (i / \u2191(gcd i j)) (j / \u2191(gcd i j)) = 1\n[PROOFSTEP]\nrw [gcd_div (gcd_dvd_left i j) (gcd_dvd_right i j), natAbs_ofNat, Nat.div_self H]\n[GOAL]\ni j : \u2124\nH : j \u2223 i\n\u22a2 gcd i j = natAbs j\n[PROOFSTEP]\nrw [gcd_comm, gcd_eq_left H]\n[GOAL]\nx y : \u2124\nhc : gcd x y \u2260 0\n\u22a2 x \u2260 0 \u2228 y \u2260 0\n[PROOFSTEP]\ncontrapose! hc\n[GOAL]\nx y : \u2124\nhc : x = 0 \u2227 y = 0\n\u22a2 gcd x y = 0\n[PROOFSTEP]\nrw [hc.left, hc.right, gcd_zero_right, natAbs_zero]\n[GOAL]\nm n : \u2124\nk : \u2115\nk0 : 0 < k\n\u22a2 m ^ k \u2223 n ^ k \u2194 m \u2223 n\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => pow_dvd_pow_of_dvd h _\u27e9\n[GOAL]\nm n : \u2124\nk : \u2115\nk0 : 0 < k\nh : m ^ k \u2223 n ^ k\n\u22a2 m \u2223 n\n[PROOFSTEP]\nrwa [\u2190 natAbs_dvd_natAbs, \u2190 Nat.pow_dvd_pow_iff k0, \u2190 Int.natAbs_pow, \u2190 Int.natAbs_pow, natAbs_dvd_natAbs]\n[GOAL]\na b : \u2124\nn : \u2115\n\u22a2 gcd a b \u2223 n \u2194 \u2203 x y, \u2191n = a * x + b * y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\na b : \u2124\nn : \u2115\n\u22a2 gcd a b \u2223 n \u2192 \u2203 x y, \u2191n = a * x + b * y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\na b : \u2124\nn : \u2115\nh : gcd a b \u2223 n\n\u22a2 \u2203 x y, \u2191n = a * x + b * y\n[PROOFSTEP]\nrw [\u2190 Nat.mul_div_cancel' h, Int.ofNat_mul, gcd_eq_gcd_ab, add_mul, mul_assoc, mul_assoc]\n[GOAL]\ncase mp\na b : \u2124\nn : \u2115\nh : gcd a b \u2223 n\n\u22a2 \u2203 x y, a * (gcdA a b * \u2191(n / gcd a b)) + b * (gcdB a b * \u2191(n / gcd a b)) = a * x + b * y\n[PROOFSTEP]\nexact \u27e8_, _, rfl\u27e9\n[GOAL]\ncase mpr\na b : \u2124\nn : \u2115\n\u22a2 (\u2203 x y, \u2191n = a * x + b * y) \u2192 gcd a b \u2223 n\n[PROOFSTEP]\nrintro \u27e8x, y, h\u27e9\n[GOAL]\ncase mpr.intro.intro\na b : \u2124\nn : \u2115\nx y : \u2124\nh : \u2191n = a * x + b * y\n\u22a2 gcd a b \u2223 n\n[PROOFSTEP]\nrw [\u2190 Int.coe_nat_dvd, h]\n[GOAL]\ncase mpr.intro.intro\na b : \u2124\nn : \u2115\nx y : \u2124\nh : \u2191n = a * x + b * y\n\u22a2 \u2191(gcd a b) \u2223 a * x + b * y\n[PROOFSTEP]\nexact dvd_add (dvd_mul_of_dvd_left (gcd_dvd_left a b) _) (dvd_mul_of_dvd_left (gcd_dvd_right a b) y)\n[GOAL]\na b c : \u2124\nhabc : a \u2223 b * c\nhab : gcd a c = 1\n\u22a2 a \u2223 b\n[PROOFSTEP]\nhave := gcd_eq_gcd_ab a c\n[GOAL]\na b c : \u2124\nhabc : a \u2223 b * c\nhab : gcd a c = 1\nthis : \u2191(gcd a c) = a * gcdA a c + c * gcdB a c\n\u22a2 a \u2223 b\n[PROOFSTEP]\nsimp only [hab, Int.ofNat_zero, Int.ofNat_succ, zero_add] at this \n[GOAL]\na b c : \u2124\nhabc : a \u2223 b * c\nhab : gcd a c = 1\nthis : 1 = a * gcdA a c + c * gcdB a c\n\u22a2 a \u2223 b\n[PROOFSTEP]\nhave : b * a * gcdA a c + b * c * gcdB a c = b := by simp [mul_assoc, \u2190 mul_add, \u2190 this]\n[GOAL]\na b c : \u2124\nhabc : a \u2223 b * c\nhab : gcd a c = 1\nthis : 1 = a * gcdA a c + c * gcdB a c\n\u22a2 b * a * gcdA a c + b * c * gcdB a c = b\n[PROOFSTEP]\nsimp [mul_assoc, \u2190 mul_add, \u2190 this]\n[GOAL]\na b c : \u2124\nhabc : a \u2223 b * c\nhab : gcd a c = 1\nthis\u271d : 1 = a * gcdA a c + c * gcdB a c\nthis : b * a * gcdA a c + b * c * gcdB a c = b\n\u22a2 a \u2223 b\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\na b c : \u2124\nhabc : a \u2223 b * c\nhab : gcd a c = 1\nthis\u271d : 1 = a * gcdA a c + c * gcdB a c\nthis : b * a * gcdA a c + b * c * gcdB a c = b\n\u22a2 a \u2223 b * a * gcdA a c + b * c * gcdB a c\n[PROOFSTEP]\nexact dvd_add (dvd_mul_of_dvd_left (dvd_mul_left a b) _) (dvd_mul_of_dvd_left habc _)\n[GOAL]\na b c : \u2124\nhabc : a \u2223 b * c\nhab : gcd a b = 1\n\u22a2 a \u2223 c\n[PROOFSTEP]\nrw [mul_comm] at habc \n[GOAL]\na b c : \u2124\nhabc : a \u2223 c * b\nhab : gcd a b = 1\n\u22a2 a \u2223 c\n[PROOFSTEP]\nexact dvd_of_dvd_mul_left_of_gcd_one habc hab\n[GOAL]\na b : \u2124\nha : a \u2260 0\n\u22a2 IsLeast {n | 0 < n \u2227 \u2203 x y, \u2191n = a * x + b * y} (gcd a b)\n[PROOFSTEP]\nsimp_rw [\u2190 gcd_dvd_iff]\n[GOAL]\na b : \u2124\nha : a \u2260 0\n\u22a2 IsLeast {n | 0 < n \u2227 gcd a b \u2223 n} (gcd a b)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\na b : \u2124\nha : a \u2260 0\n\u22a2 gcd a b \u2208 {n | 0 < n \u2227 gcd a b \u2223 n}\n[PROOFSTEP]\nsimpa [and_true_iff, dvd_refl, Set.mem_setOf_eq] using gcd_pos_of_ne_zero_left b ha\n[GOAL]\ncase right\na b : \u2124\nha : a \u2260 0\n\u22a2 gcd a b \u2208 lowerBounds {n | 0 < n \u2227 gcd a b \u2223 n}\n[PROOFSTEP]\nsimp only [lowerBounds, and_imp, Set.mem_setOf_eq]\n[GOAL]\ncase right\na b : \u2124\nha : a \u2260 0\n\u22a2 \u2200 \u2983a_1 : \u2115\u2984, 0 < a_1 \u2192 gcd a b \u2223 a_1 \u2192 gcd a b \u2264 a_1\n[PROOFSTEP]\nexact fun n hn_pos hn => Nat.le_of_dvd hn_pos hn\n[GOAL]\ni j : \u2124\n\u22a2 lcm i j = lcm j i\n[PROOFSTEP]\nrw [Int.lcm, Int.lcm]\n[GOAL]\ni j : \u2124\n\u22a2 Nat.lcm (natAbs i) (natAbs j) = Nat.lcm (natAbs j) (natAbs i)\n[PROOFSTEP]\nexact Nat.lcm_comm _ _\n[GOAL]\ni j k : \u2124\n\u22a2 lcm (\u2191(lcm i j)) k = lcm i \u2191(lcm j k)\n[PROOFSTEP]\nrw [Int.lcm, Int.lcm, Int.lcm, Int.lcm, natAbs_ofNat, natAbs_ofNat]\n[GOAL]\ni j k : \u2124\n\u22a2 Nat.lcm (Nat.lcm (natAbs i) (natAbs j)) (natAbs k) = Nat.lcm (natAbs i) (Nat.lcm (natAbs j) (natAbs k))\n[PROOFSTEP]\napply Nat.lcm_assoc\n[GOAL]\ni : \u2124\n\u22a2 lcm 0 i = 0\n[PROOFSTEP]\nrw [Int.lcm]\n[GOAL]\ni : \u2124\n\u22a2 Nat.lcm (natAbs 0) (natAbs i) = 0\n[PROOFSTEP]\napply Nat.lcm_zero_left\n[GOAL]\ni : \u2124\n\u22a2 lcm i 0 = 0\n[PROOFSTEP]\nrw [Int.lcm]\n[GOAL]\ni : \u2124\n\u22a2 Nat.lcm (natAbs i) (natAbs 0) = 0\n[PROOFSTEP]\napply Nat.lcm_zero_right\n[GOAL]\ni : \u2124\n\u22a2 lcm 1 i = natAbs i\n[PROOFSTEP]\nrw [Int.lcm]\n[GOAL]\ni : \u2124\n\u22a2 Nat.lcm (natAbs 1) (natAbs i) = natAbs i\n[PROOFSTEP]\napply Nat.lcm_one_left\n[GOAL]\ni : \u2124\n\u22a2 lcm i 1 = natAbs i\n[PROOFSTEP]\nrw [Int.lcm]\n[GOAL]\ni : \u2124\n\u22a2 Nat.lcm (natAbs i) (natAbs 1) = natAbs i\n[PROOFSTEP]\napply Nat.lcm_one_right\n[GOAL]\ni : \u2124\n\u22a2 lcm i i = natAbs i\n[PROOFSTEP]\nrw [Int.lcm]\n[GOAL]\ni : \u2124\n\u22a2 Nat.lcm (natAbs i) (natAbs i) = natAbs i\n[PROOFSTEP]\napply Nat.lcm_self\n[GOAL]\ni j : \u2124\n\u22a2 i \u2223 \u2191(lcm i j)\n[PROOFSTEP]\nrw [Int.lcm]\n[GOAL]\ni j : \u2124\n\u22a2 i \u2223 \u2191(Nat.lcm (natAbs i) (natAbs j))\n[PROOFSTEP]\napply coe_nat_dvd_right.mpr\n[GOAL]\ni j : \u2124\n\u22a2 natAbs i \u2223 Nat.lcm (natAbs i) (natAbs j)\n[PROOFSTEP]\napply Nat.dvd_lcm_left\n[GOAL]\ni j : \u2124\n\u22a2 j \u2223 \u2191(lcm i j)\n[PROOFSTEP]\nrw [Int.lcm]\n[GOAL]\ni j : \u2124\n\u22a2 j \u2223 \u2191(Nat.lcm (natAbs i) (natAbs j))\n[PROOFSTEP]\napply coe_nat_dvd_right.mpr\n[GOAL]\ni j : \u2124\n\u22a2 natAbs j \u2223 Nat.lcm (natAbs i) (natAbs j)\n[PROOFSTEP]\napply Nat.dvd_lcm_right\n[GOAL]\ni j k : \u2124\n\u22a2 i \u2223 k \u2192 j \u2223 k \u2192 \u2191(lcm i j) \u2223 k\n[PROOFSTEP]\nrw [Int.lcm]\n[GOAL]\ni j k : \u2124\n\u22a2 i \u2223 k \u2192 j \u2223 k \u2192 \u2191(Nat.lcm (natAbs i) (natAbs j)) \u2223 k\n[PROOFSTEP]\nintro hi hj\n[GOAL]\ni j k : \u2124\nhi : i \u2223 k\nhj : j \u2223 k\n\u22a2 \u2191(Nat.lcm (natAbs i) (natAbs j)) \u2223 k\n[PROOFSTEP]\nexact coe_nat_dvd_left.mpr (Nat.lcm_dvd (natAbs_dvd_natAbs.mpr hi) (natAbs_dvd_natAbs.mpr hj))\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\nx : M\nm n : \u2115\nhm : x ^ m = 1\nhn : x ^ n = 1\n\u22a2 x ^ Nat.gcd m n = 1\n[PROOFSTEP]\nrcases m with (rfl | m)\n[GOAL]\ncase zero\nM : Type u_1\ninst\u271d : Monoid M\nx : M\nn : \u2115\nhn : x ^ n = 1\nhm : x ^ Nat.zero = 1\n\u22a2 x ^ Nat.gcd Nat.zero n = 1\n[PROOFSTEP]\nsimp [hn]\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d : Monoid M\nx : M\nn : \u2115\nhn : x ^ n = 1\nm : \u2115\nhm : x ^ Nat.succ m = 1\n\u22a2 x ^ Nat.gcd (Nat.succ m) n = 1\n[PROOFSTEP]\nobtain \u27e8y, rfl\u27e9 := isUnit_ofPowEqOne hm m.succ_ne_zero\n[GOAL]\ncase succ.intro\nM : Type u_1\ninst\u271d : Monoid M\nn m : \u2115\ny : M\u02e3\nhn : \u2191y ^ n = 1\nhm : \u2191y ^ Nat.succ m = 1\n\u22a2 \u2191y ^ Nat.gcd (Nat.succ m) n = 1\n[PROOFSTEP]\nsimp only [\u2190 Units.val_pow_eq_pow_val] at *\n[GOAL]\ncase succ.intro\nM : Type u_1\ninst\u271d : Monoid M\nn m : \u2115\ny : M\u02e3\nhn : \u2191(y ^ n) = 1\nhm : \u2191(y ^ Nat.succ m) = 1\n\u22a2 \u2191(y ^ Nat.gcd (Nat.succ m) n) = 1\n[PROOFSTEP]\nrw [\u2190 Units.val_one, \u2190 zpow_coe_nat, \u2190 Units.ext_iff] at *\n[GOAL]\ncase succ.intro\nM : Type u_1\ninst\u271d : Monoid M\nn m : \u2115\ny : M\u02e3\nhn : y ^ \u2191n = 1\nhm : y ^ \u2191(Nat.succ m) = 1\n\u22a2 y ^ \u2191(Nat.gcd (Nat.succ m) n) = 1\n[PROOFSTEP]\nsimp only [Nat.gcd_eq_gcd_ab, zpow_add, zpow_mul, hm, hn, one_zpow, one_mul]\n", "meta": {"mathlib_filename": "Mathlib.Data.Int.GCD", "llama_tokens": 12042, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390747, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.4130122428393563}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d : MonoidWithZero R\na\u271d b\u271d : R\nhx\u2081 : a\u271d \u2208 {x | \u2200 (z : R), z * x = 0 \u2192 z = 0}\nhx\u2082 : b\u271d \u2208 {x | \u2200 (z : R), z * x = 0 \u2192 z = 0}\nx\u271d : R\nhz : x\u271d * (a\u271d * b\u271d) = 0\n\u22a2 x\u271d = 0\n[PROOFSTEP]\nrw [\u2190 mul_assoc] at hz \n[GOAL]\nR : Type u_1\ninst\u271d : MonoidWithZero R\na\u271d b\u271d : R\nhx\u2081 : a\u271d \u2208 {x | \u2200 (z : R), z * x = 0 \u2192 z = 0}\nhx\u2082 : b\u271d \u2208 {x | \u2200 (z : R), z * x = 0 \u2192 z = 0}\nx\u271d : R\nhz : x\u271d * a\u271d * b\u271d = 0\n\u22a2 x\u271d = 0\n[PROOFSTEP]\nexact hx\u2081 _ (hx\u2082 _ hz)\n[GOAL]\nR : Type u_1\ninst\u271d : MonoidWithZero R\nx\u271d : R\nhz : x\u271d * 1 = 0\n\u22a2 x\u271d = 0\n[PROOFSTEP]\nrwa [mul_one] at hz \n[GOAL]\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\nx r : M\nhr : r \u2208 M\u2070\n\u22a2 x = 0 \u2192 x * r = 0\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\nr x : M\u2081\nhr : r \u2208 M\u2081\u2070\n\u22a2 r * x = 0 \u2194 x = 0\n[PROOFSTEP]\nrw [mul_comm, mul_right_mem_nonZeroDivisors_eq_zero_iff hr]\n[GOAL]\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\nx y r : R\nhr : r \u2208 R\u2070\n\u22a2 x * r = y * r \u2194 x = y\n[PROOFSTEP]\nrefine \u27e8fun h \u21a6 ?_, congrArg (\u00b7 * r)\u27e9\n[GOAL]\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\nx y r : R\nhr : r \u2208 R\u2070\nh : x * r = y * r\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 mul_right_mem_nonZeroDivisors_eq_zero_iff hr, sub_mul, h, sub_self]\n[GOAL]\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\nx y r : R'\nhr : r \u2208 R'\u2070\n\u22a2 r * x = r * y \u2194 x = y\n[PROOFSTEP]\nsimp_rw [mul_comm r, mul_cancel_right_mem_nonZeroDivisors hr]\n[GOAL]\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\na b : M\u2081\n\u22a2 a * b \u2208 M\u2081\u2070 \u2194 a \u2208 M\u2081\u2070 \u2227 b \u2208 M\u2081\u2070\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\na b : M\u2081\n\u22a2 a * b \u2208 M\u2081\u2070 \u2192 a \u2208 M\u2081\u2070 \u2227 b \u2208 M\u2081\u2070\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\na b : M\u2081\nh : a * b \u2208 M\u2081\u2070\n\u22a2 a \u2208 M\u2081\u2070 \u2227 b \u2208 M\u2081\u2070\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.left\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\na b : M\u2081\nh : a * b \u2208 M\u2081\u2070\n\u22a2 a \u2208 M\u2081\u2070\n[PROOFSTEP]\nintro x h'\n[GOAL]\ncase mp.right\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\na b : M\u2081\nh : a * b \u2208 M\u2081\u2070\n\u22a2 b \u2208 M\u2081\u2070\n[PROOFSTEP]\nintro x h'\n[GOAL]\ncase mp.left\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\na b : M\u2081\nh : a * b \u2208 M\u2081\u2070\nx : M\u2081\nh' : x * a = 0\n\u22a2 x = 0\n[PROOFSTEP]\napply h\n[GOAL]\ncase mp.right\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\na b : M\u2081\nh : a * b \u2208 M\u2081\u2070\nx : M\u2081\nh' : x * b = 0\n\u22a2 x = 0\n[PROOFSTEP]\napply h\n[GOAL]\ncase mp.left.a\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\na b : M\u2081\nh : a * b \u2208 M\u2081\u2070\nx : M\u2081\nh' : x * a = 0\n\u22a2 x * (a * b) = 0\n[PROOFSTEP]\nrw [\u2190 mul_assoc, h', zero_mul]\n[GOAL]\ncase mp.right.a\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\na b : M\u2081\nh : a * b \u2208 M\u2081\u2070\nx : M\u2081\nh' : x * b = 0\n\u22a2 x * (a * b) = 0\n[PROOFSTEP]\nrw [mul_comm a b, \u2190 mul_assoc, h', zero_mul]\n[GOAL]\ncase mpr\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\na b : M\u2081\n\u22a2 a \u2208 M\u2081\u2070 \u2227 b \u2208 M\u2081\u2070 \u2192 a * b \u2208 M\u2081\u2070\n[PROOFSTEP]\nrintro \u27e8ha, hb\u27e9 x hx\n[GOAL]\ncase mpr.intro\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\na b : M\u2081\nha : a \u2208 M\u2081\u2070\nhb : b \u2208 M\u2081\u2070\nx : M\u2081\nhx : x * (a * b) = 0\n\u22a2 x = 0\n[PROOFSTEP]\napply ha\n[GOAL]\ncase mpr.intro.a\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\na b : M\u2081\nha : a \u2208 M\u2081\u2070\nhb : b \u2208 M\u2081\u2070\nx : M\u2081\nhx : x * (a * b) = 0\n\u22a2 x * a = 0\n[PROOFSTEP]\napply hb\n[GOAL]\ncase mpr.intro.a.a\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2074 : MonoidWithZero M\ninst\u271d\u00b3 : MonoidWithZero M'\ninst\u271d\u00b2 : CommMonoidWithZero M\u2081\ninst\u271d\u00b9 : Ring R\ninst\u271d : CommRing R'\na b : M\u2081\nha : a \u2208 M\u2081\u2070\nhb : b \u2208 M\u2081\u2070\nx : M\u2081\nhx : x * (a * b) = 0\n\u22a2 x * a * b = 0\n[PROOFSTEP]\nrw [mul_assoc, hx]\n[GOAL]\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M'\ninst\u271d : MonoidWithZeroHomClass F M M'\nf : F\nhf : Function.Injective \u2191f\nS : Submonoid M\nhS : S \u2264 M\u2070\n\u22a2 Submonoid.map f S \u2264 M'\u2070\n[PROOFSTEP]\ncases subsingleton_or_nontrivial M\n[GOAL]\ncase inl\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M'\ninst\u271d : MonoidWithZeroHomClass F M M'\nf : F\nhf : Function.Injective \u2191f\nS : Submonoid M\nhS : S \u2264 M\u2070\nh\u271d : Subsingleton M\n\u22a2 Submonoid.map f S \u2264 M'\u2070\n[PROOFSTEP]\nsimp [Subsingleton.elim S \u22a5]\n[GOAL]\ncase inr\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M'\ninst\u271d : MonoidWithZeroHomClass F M M'\nf : F\nhf : Function.Injective \u2191f\nS : Submonoid M\nhS : S \u2264 M\u2070\nh\u271d : Nontrivial M\n\u22a2 Submonoid.map f S \u2264 M'\u2070\n[PROOFSTEP]\nexact\n le_nonZeroDivisors_of_noZeroDivisors fun h \u21a6\n let \u27e8x, hx, hx0\u27e9 := h\n zero_ne_one (hS (hf (hx0.trans (map_zero f).symm) \u25b8 hx : 0 \u2208 S) 1 (mul_zero 1)).symm\n[GOAL]\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\ns : Multiset M\u2081\n\u22a2 Multiset.prod s = 0 \u2194 \u2203 r x, r = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\ns : Multiset M\u2081\n\u22a2 Multiset.prod s = 0 \u2192 \u2203 r x, r = 0\ncase mpr\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\ns : Multiset M\u2081\n\u22a2 (\u2203 r x, r = 0) \u2192 Multiset.prod s = 0\n[PROOFSTEP]\nswap\n[GOAL]\ncase mpr\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\ns : Multiset M\u2081\n\u22a2 (\u2203 r x, r = 0) \u2192 Multiset.prod s = 0\n[PROOFSTEP]\nrintro \u27e8r, hrs, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\ns : Multiset M\u2081\nhrs : 0 \u2208 s\n\u22a2 Multiset.prod s = 0\n[PROOFSTEP]\nexact Multiset.prod_eq_zero hrs\n[GOAL]\ncase mp\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\ns : Multiset M\u2081\n\u22a2 Multiset.prod s = 0 \u2192 \u2203 r x, r = 0\n[PROOFSTEP]\ninduction' s using Multiset.induction_on with a s ih\n[GOAL]\ncase mp.empty\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\n\u22a2 Multiset.prod 0 = 0 \u2192 \u2203 r x, r = 0\n[PROOFSTEP]\nintro habs\n[GOAL]\ncase mp.empty\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\nhabs : Multiset.prod 0 = 0\n\u22a2 \u2203 r x, r = 0\n[PROOFSTEP]\nsimp at habs \n[GOAL]\ncase mp.cons\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\na : M\u2081\ns : Multiset M\u2081\nih : Multiset.prod s = 0 \u2192 \u2203 r x, r = 0\n\u22a2 Multiset.prod (a ::\u2098 s) = 0 \u2192 \u2203 r x, r = 0\n[PROOFSTEP]\nrw [Multiset.prod_cons]\n[GOAL]\ncase mp.cons\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\na : M\u2081\ns : Multiset M\u2081\nih : Multiset.prod s = 0 \u2192 \u2203 r x, r = 0\n\u22a2 a * Multiset.prod s = 0 \u2192 \u2203 r x, r = 0\n[PROOFSTEP]\nintro hprod\n[GOAL]\ncase mp.cons\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\na : M\u2081\ns : Multiset M\u2081\nih : Multiset.prod s = 0 \u2192 \u2203 r x, r = 0\nhprod : a * Multiset.prod s = 0\n\u22a2 \u2203 r x, r = 0\n[PROOFSTEP]\nreplace hprod := eq_zero_or_eq_zero_of_mul_eq_zero hprod\n[GOAL]\ncase mp.cons\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\na : M\u2081\ns : Multiset M\u2081\nih : Multiset.prod s = 0 \u2192 \u2203 r x, r = 0\nhprod : a = 0 \u2228 Multiset.prod s = 0\n\u22a2 \u2203 r x, r = 0\n[PROOFSTEP]\ncases' hprod with ha hb\n[GOAL]\ncase mp.cons.inl\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\na : M\u2081\ns : Multiset M\u2081\nih : Multiset.prod s = 0 \u2192 \u2203 r x, r = 0\nha : a = 0\n\u22a2 \u2203 r x, r = 0\n[PROOFSTEP]\nexact \u27e8a, Multiset.mem_cons_self a s, ha\u27e9\n[GOAL]\ncase mp.cons.inr\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\na : M\u2081\ns : Multiset M\u2081\nih : Multiset.prod s = 0 \u2192 \u2203 r x, r = 0\nhb : Multiset.prod s = 0\n\u22a2 \u2203 r x, r = 0\n[PROOFSTEP]\napply (ih hb).imp _\n[GOAL]\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\na : M\u2081\ns : Multiset M\u2081\nih : Multiset.prod s = 0 \u2192 \u2203 r x, r = 0\nhb : Multiset.prod s = 0\n\u22a2 \u2200 (a_1 : M\u2081), (\u2203 x, a_1 = 0) \u2192 \u2203 x, a_1 = 0\n[PROOFSTEP]\nrintro b \u27e8hb\u2081, hb\u2082\u27e9\n[GOAL]\ncase intro\nM : Type u_1\nM' : Type u_2\nM\u2081 : Type u_3\nR : Type u_4\nR' : Type u_5\nF : Type u_6\ninst\u271d\u2076 : MonoidWithZero M\ninst\u271d\u2075 : MonoidWithZero M'\ninst\u271d\u2074 : CommMonoidWithZero M\u2081\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R'\ninst\u271d\u00b9 : NoZeroDivisors M\u2081\ninst\u271d : Nontrivial M\u2081\na : M\u2081\ns : Multiset M\u2081\nih : Multiset.prod s = 0 \u2192 \u2203 r x, r = 0\nhb : Multiset.prod s = 0\nb : M\u2081\nhb\u2081 : b \u2208 s\nhb\u2082 : b = 0\n\u22a2 \u2203 x, b = 0\n[PROOFSTEP]\nexact \u27e8Multiset.mem_cons_of_mem hb\u2081, hb\u2082\u27e9\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.NonZeroDivisors", "llama_tokens": 7412, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370111, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.41295563440679484}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 E\n\u22a2 AEMeasurable g \u2194 AEMeasurable fun x => \u2191(f x) \u2022 g x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 E\n\u22a2 AEMeasurable g \u2192 AEMeasurable fun x => \u2191(f x) \u2022 g x\n[PROOFSTEP]\nrintro \u27e8g', g'meas, hg'\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : g =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] g'\n\u22a2 AEMeasurable fun x => \u2191(f x) \u2022 g x\n[PROOFSTEP]\nhave A : MeasurableSet {x : \u03b1 | f x \u2260 0} := (hf (measurableSet_singleton 0)).compl\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : g =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 AEMeasurable fun x => \u2191(f x) \u2022 g x\n[PROOFSTEP]\nrefine' \u27e8fun x => (f x : \u211d) \u2022 g' x, hf.coe_nnreal_real.smul g'meas, _\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : g =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 (fun x => \u2191(f x) \u2022 g x) =\u1d50[\u03bc] fun x => \u2191(f x) \u2022 g' x\n[PROOFSTEP]\napply @ae_of_ae_restrict_of_ae_restrict_compl _ _ _ {x | f x \u2260 0}\n[GOAL]\ncase mp.intro.intro.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : g =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f x \u2260 0}, (fun x => \u2191(f x) \u2022 g x) x = (fun x => \u2191(f x) \u2022 g' x) x\n[PROOFSTEP]\nrw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal] at hg' \n[GOAL]\ncase mp.intro.intro.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f x \u2260 0}, (fun x => \u2191(f x) \u2022 g x) x = (fun x => \u2191(f x) \u2022 g' x) x\n[PROOFSTEP]\nrw [ae_restrict_iff' A]\n[GOAL]\ncase mp.intro.intro.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 {x | f x \u2260 0} \u2192 (fun x => \u2191(f x) \u2022 g x) x = (fun x => \u2191(f x) \u2022 g' x) x\n[PROOFSTEP]\nfilter_upwards [hg']\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200 (a : \u03b1), (\u2191(f a) \u2260 0 \u2192 g a = g' a) \u2192 f a \u2260 0 \u2192 \u2191(f a) \u2022 g a = \u2191(f a) \u2022 g' a\n[PROOFSTEP]\nintro a ha h'a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\n\u22a2 \u2191(f a) \u2022 g a = \u2191(f a) \u2022 g' a\n[PROOFSTEP]\nhave : (f a : \u211d\u22650\u221e) \u2260 0 := by simpa only [Ne.def, coe_eq_zero] using h'a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\n\u22a2 \u2191(f a) \u2260 0\n[PROOFSTEP]\nsimpa only [Ne.def, coe_eq_zero] using h'a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\nthis : \u2191(f a) \u2260 0\n\u22a2 \u2191(f a) \u2022 g a = \u2191(f a) \u2022 g' a\n[PROOFSTEP]\nrw [ha this]\n[GOAL]\ncase mp.intro.intro.htc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : g =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f x \u2260 0}\u1d9c, (fun x => \u2191(f x) \u2022 g x) x = (fun x => \u2191(f x) \u2022 g' x) x\n[PROOFSTEP]\nfilter_upwards [ae_restrict_mem A.compl]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : g =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200 (a : \u03b1), a \u2208 {x | f x \u2260 0}\u1d9c \u2192 \u2191(f a) \u2022 g a = \u2191(f a) \u2022 g' a\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : g =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] g'\nA : MeasurableSet {x | f x \u2260 0}\nx : \u03b1\nhx : x \u2208 {x | f x \u2260 0}\u1d9c\n\u22a2 \u2191(f x) \u2022 g x = \u2191(f x) \u2022 g' x\n[PROOFSTEP]\nsimp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx \n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : g =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] g'\nA : MeasurableSet {x | f x \u2260 0}\nx : \u03b1\nhx : f x = 0\n\u22a2 \u2191(f x) \u2022 g x = \u2191(f x) \u2022 g' x\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 E\n\u22a2 (AEMeasurable fun x => \u2191(f x) \u2022 g x) \u2192 AEMeasurable g\n[PROOFSTEP]\nrintro \u27e8g', g'meas, hg'\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1d50[\u03bc] g'\n\u22a2 AEMeasurable g\n[PROOFSTEP]\nrefine' \u27e8fun x => (f x : \u211d)\u207b\u00b9 \u2022 g' x, hf.coe_nnreal_real.inv.smul g'meas, _\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1d50[\u03bc] g'\n\u22a2 g =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] fun x => (\u2191(f x))\u207b\u00b9 \u2022 g' x\n[PROOFSTEP]\nrw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal]\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1d50[\u03bc] g'\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = (\u2191(f x))\u207b\u00b9 \u2022 g' x\n[PROOFSTEP]\nfilter_upwards [hg']\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1d50[\u03bc] g'\n\u22a2 \u2200 (a : \u03b1), \u2191(f a) \u2022 g a = g' a \u2192 \u2191(f a) \u2260 0 \u2192 g a = (\u2191(f a))\u207b\u00b9 \u2022 g' a\n[PROOFSTEP]\nintro x hx h'x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1d50[\u03bc] g'\nx : \u03b1\nhx : \u2191(f x) \u2022 g x = g' x\nh'x : \u2191(f x) \u2260 0\n\u22a2 g x = (\u2191(f x))\u207b\u00b9 \u2022 g' x\n[PROOFSTEP]\nrw [\u2190 hx, smul_smul, _root_.inv_mul_cancel, one_smul]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1d50[\u03bc] g'\nx : \u03b1\nhx : \u2191(f x) \u2022 g x = g' x\nh'x : \u2191(f x) \u2260 0\n\u22a2 \u2191(f x) \u2260 0\n[PROOFSTEP]\nsimp only [Ne.def, coe_eq_zero] at h'x \n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : TopologicalSpace.SecondCountableTopology E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1d50[\u03bc] g'\nx : \u03b1\nhx : \u2191(f x) \u2022 g x = g' x\nh'x : \u00acf x = 0\n\u22a2 \u2191(f x) \u2260 0\n[PROOFSTEP]\nsimpa only [NNReal.coe_eq_zero, Ne.def] using h'x\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Integral.LebesgueNormedSpace", "llama_tokens": 6388, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370111, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.41295563440679484}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\nX : C\n\u22a2 (yoneda.map (NatTrans.app h.unit X) \u226b\n NatTrans.mk fun Y f => L.preimage (\u2191(Adjunction.homEquiv h Y.unop (L.obj X)).symm f)) =\n \ud835\udfd9 (yoneda.obj ((\ud835\udfed C).obj X))\n[PROOFSTEP]\next x\n[GOAL]\ncase w.h.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\nX : C\nx : C\u1d52\u1d56\na\u271d : (yoneda.obj ((\ud835\udfed C).obj X)).obj x\n\u22a2 NatTrans.app\n (yoneda.map (NatTrans.app h.unit X) \u226b\n NatTrans.mk fun Y f => L.preimage (\u2191(Adjunction.homEquiv h Y.unop (L.obj X)).symm f))\n x a\u271d =\n NatTrans.app (\ud835\udfd9 (yoneda.obj ((\ud835\udfed C).obj X))) x a\u271d\n[PROOFSTEP]\napply L.map_injective\n[GOAL]\ncase w.h.h.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\nX : C\nx : C\u1d52\u1d56\na\u271d : (yoneda.obj ((\ud835\udfed C).obj X)).obj x\n\u22a2 L.map\n (NatTrans.app\n (yoneda.map (NatTrans.app h.unit X) \u226b\n NatTrans.mk fun Y f => L.preimage (\u2191(Adjunction.homEquiv h Y.unop (L.obj X)).symm f))\n x a\u271d) =\n L.map (NatTrans.app (\ud835\udfd9 (yoneda.obj ((\ud835\udfed C).obj X))) x a\u271d)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\nX : C\n\u22a2 (NatTrans.mk fun Y f => L.preimage (\u2191(Adjunction.homEquiv h Y.unop (L.obj X)).symm f)) \u226b\n yoneda.map (NatTrans.app h.unit X) =\n \ud835\udfd9 (yoneda.obj ((L \u22d9 R).obj X))\n[PROOFSTEP]\next x\n[GOAL]\ncase w.h.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\nX : C\nx : C\u1d52\u1d56\na\u271d : (yoneda.obj ((L \u22d9 R).obj X)).obj x\n\u22a2 NatTrans.app\n ((NatTrans.mk fun Y f => L.preimage (\u2191(Adjunction.homEquiv h Y.unop (L.obj X)).symm f)) \u226b\n yoneda.map (NatTrans.app h.unit X))\n x a\u271d =\n NatTrans.app (\ud835\udfd9 (yoneda.obj ((L \u22d9 R).obj X))) x a\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\nX : C\nx : C\u1d52\u1d56\na\u271d : (yoneda.obj ((L \u22d9 R).obj X)).obj x\n\u22a2 L.preimage (\u2191(Adjunction.homEquiv h x.unop (L.obj X)).symm a\u271d) \u226b NatTrans.app h.unit X = a\u271d\n[PROOFSTEP]\nsimp only [Adjunction.homEquiv_counit, preimage_comp, preimage_map, Category.assoc]\n[GOAL]\ncase w.h.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\nX : C\nx : C\u1d52\u1d56\na\u271d : (yoneda.obj ((L \u22d9 R).obj X)).obj x\n\u22a2 a\u271d \u226b L.preimage (NatTrans.app h.counit (L.obj X)) \u226b NatTrans.app h.unit X = a\u271d\n[PROOFSTEP]\nrw [\u2190 h.unit_naturality]\n[GOAL]\ncase w.h.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\nX : C\nx : C\u1d52\u1d56\na\u271d : (yoneda.obj ((L \u22d9 R).obj X)).obj x\n\u22a2 a\u271d \u226b NatTrans.app h.unit (R.obj (L.obj X)) \u226b R.map (L.map (L.preimage (NatTrans.app h.counit (L.obj X)))) = a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\nX : D\n\u22a2 (coyoneda.map (NatTrans.app h.counit X).op \u226b\n NatTrans.mk fun Y f => R.preimage (\u2191(Adjunction.homEquiv h (R.obj X) Y) f)) =\n \ud835\udfd9 (coyoneda.obj (op ((\ud835\udfed D).obj X)))\n[PROOFSTEP]\next x\n[GOAL]\ncase w.h.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\nX x : D\na\u271d : (coyoneda.obj (op ((\ud835\udfed D).obj X))).obj x\n\u22a2 NatTrans.app\n (coyoneda.map (NatTrans.app h.counit X).op \u226b\n NatTrans.mk fun Y f => R.preimage (\u2191(Adjunction.homEquiv h (R.obj X) Y) f))\n x a\u271d =\n NatTrans.app (\ud835\udfd9 (coyoneda.obj (op ((\ud835\udfed D).obj X)))) x a\u271d\n[PROOFSTEP]\napply R.map_injective\n[GOAL]\ncase w.h.h.a\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\nX x : D\na\u271d : (coyoneda.obj (op ((\ud835\udfed D).obj X))).obj x\n\u22a2 R.map\n (NatTrans.app\n (coyoneda.map (NatTrans.app h.counit X).op \u226b\n NatTrans.mk fun Y f => R.preimage (\u2191(Adjunction.homEquiv h (R.obj X) Y) f))\n x a\u271d) =\n R.map (NatTrans.app (\ud835\udfd9 (coyoneda.obj (op ((\ud835\udfed D).obj X)))) x a\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\nX : D\n\u22a2 (NatTrans.mk fun Y f => R.preimage (\u2191(Adjunction.homEquiv h (R.obj X) Y) f)) \u226b\n coyoneda.map (NatTrans.app h.counit X).op =\n \ud835\udfd9 (coyoneda.obj (op ((R \u22d9 L).obj X)))\n[PROOFSTEP]\next x\n[GOAL]\ncase w.h.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\nX x : D\na\u271d : (coyoneda.obj (op ((R \u22d9 L).obj X))).obj x\n\u22a2 NatTrans.app\n ((NatTrans.mk fun Y f => R.preimage (\u2191(Adjunction.homEquiv h (R.obj X) Y) f)) \u226b\n coyoneda.map (NatTrans.app h.counit X).op)\n x a\u271d =\n NatTrans.app (\ud835\udfd9 (coyoneda.obj (op ((R \u22d9 L).obj X)))) x a\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\nX x : D\na\u271d : (coyoneda.obj (op ((R \u22d9 L).obj X))).obj x\n\u22a2 NatTrans.app h.counit X \u226b R.preimage (\u2191(Adjunction.homEquiv h (R.obj X) x) a\u271d) = a\u271d\n[PROOFSTEP]\nsimp only [Adjunction.homEquiv_unit, preimage_comp, preimage_map]\n[GOAL]\ncase w.h.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\nX x : D\na\u271d : (coyoneda.obj (op ((R \u22d9 L).obj X))).obj x\n\u22a2 NatTrans.app h.counit X \u226b R.preimage (NatTrans.app h.unit (R.obj X)) \u226b a\u271d = a\u271d\n[PROOFSTEP]\nrw [\u2190 h.counit_naturality]\n[GOAL]\ncase w.h.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\nX x : D\na\u271d : (coyoneda.obj (op ((R \u22d9 L).obj X))).obj x\n\u22a2 L.map (R.map (R.preimage (NatTrans.app h.unit (R.obj X)) \u226b a\u271d)) \u226b NatTrans.app h.counit x = a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d : IsIso h.unit\nX Y : C\nf g : X \u27f6 Y\nH : L.map f = L.map g\n\u22a2 f = g\n[PROOFSTEP]\nrw [\u2190 (h.homEquiv X (L.obj Y)).apply_eq_iff_eq] at H \n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d : IsIso h.unit\nX Y : C\nf g : X \u27f6 Y\nH : \u2191(Adjunction.homEquiv h X (L.obj Y)) (L.map f) = \u2191(Adjunction.homEquiv h X (L.obj Y)) (L.map g)\n\u22a2 f = g\n[PROOFSTEP]\nsimpa using H =\u226b inv (h.unit.app Y)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d : IsIso h.counit\nX Y : D\nf g : X \u27f6 Y\nH : R.map f = R.map g\n\u22a2 f = g\n[PROOFSTEP]\nrw [\u2190 (h.homEquiv (R.obj X) Y).symm.apply_eq_iff_eq] at H \n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d : IsIso h.counit\nX Y : D\nf g : X \u27f6 Y\nH : \u2191(Adjunction.homEquiv h (R.obj X) Y).symm (R.map f) = \u2191(Adjunction.homEquiv h (R.obj X) Y).symm (R.map g)\n\u22a2 f = g\n[PROOFSTEP]\nsimpa using inv (h.counit.app X) \u226b= H\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\n\u22a2 IsIso (whiskerLeft L h.counit)\n[PROOFSTEP]\nhave := h.left_triangle\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\nthis : whiskerRight h.unit L \u226b whiskerLeft L h.counit = \ud835\udfd9 (\ud835\udfed C \u22d9 L)\n\u22a2 IsIso (whiskerLeft L h.counit)\n[PROOFSTEP]\nrw [\u2190 IsIso.eq_inv_comp] at this \n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\nthis : whiskerLeft L h.counit = inv (whiskerRight h.unit L) \u226b \ud835\udfd9 (\ud835\udfed C \u22d9 L)\n\u22a2 IsIso (whiskerLeft L h.counit)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\nthis : whiskerLeft L h.counit = inv (whiskerRight h.unit L) \u226b \ud835\udfd9 (\ud835\udfed C \u22d9 L)\n\u22a2 IsIso (inv (whiskerRight h.unit L) \u226b \ud835\udfd9 (\ud835\udfed C \u22d9 L))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\n\u22a2 IsIso (whiskerRight h.counit R)\n[PROOFSTEP]\nhave := h.right_triangle\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\nthis : whiskerLeft R h.unit \u226b whiskerRight h.counit R = \ud835\udfd9 (R \u22d9 \ud835\udfed C)\n\u22a2 IsIso (whiskerRight h.counit R)\n[PROOFSTEP]\nrw [\u2190 IsIso.eq_inv_comp] at this \n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\nthis : whiskerRight h.counit R = inv (whiskerLeft R h.unit) \u226b \ud835\udfd9 (R \u22d9 \ud835\udfed C)\n\u22a2 IsIso (whiskerRight h.counit R)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full L\ninst\u271d : Faithful L\nthis : whiskerRight h.counit R = inv (whiskerLeft R h.unit) \u226b \ud835\udfd9 (R \u22d9 \ud835\udfed C)\n\u22a2 IsIso (inv (whiskerLeft R h.unit) \u226b \ud835\udfd9 (R \u22d9 \ud835\udfed C))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\n\u22a2 IsIso (whiskerLeft R h.unit)\n[PROOFSTEP]\nhave := h.right_triangle\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\nthis : whiskerLeft R h.unit \u226b whiskerRight h.counit R = \ud835\udfd9 (R \u22d9 \ud835\udfed C)\n\u22a2 IsIso (whiskerLeft R h.unit)\n[PROOFSTEP]\nrw [\u2190 IsIso.eq_comp_inv] at this \n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\nthis : whiskerLeft R h.unit = \ud835\udfd9 (R \u22d9 \ud835\udfed C) \u226b inv (whiskerRight h.counit R)\n\u22a2 IsIso (whiskerLeft R h.unit)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\nthis : whiskerLeft R h.unit = \ud835\udfd9 (R \u22d9 \ud835\udfed C) \u226b inv (whiskerRight h.counit R)\n\u22a2 IsIso (\ud835\udfd9 (R \u22d9 \ud835\udfed C) \u226b inv (whiskerRight h.counit R))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\n\u22a2 IsIso (whiskerRight h.unit L)\n[PROOFSTEP]\nhave := h.left_triangle\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\nthis : whiskerRight h.unit L \u226b whiskerLeft L h.counit = \ud835\udfd9 (\ud835\udfed C \u22d9 L)\n\u22a2 IsIso (whiskerRight h.unit L)\n[PROOFSTEP]\nrw [\u2190 IsIso.eq_comp_inv] at this \n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\nthis : whiskerRight h.unit L = \ud835\udfd9 (\ud835\udfed C \u22d9 L) \u226b inv (whiskerLeft L h.counit)\n\u22a2 IsIso (whiskerRight h.unit L)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nL : C \u2964 D\nR : D \u2964 C\nh : L \u22a3 R\ninst\u271d\u00b9 : Full R\ninst\u271d : Faithful R\nthis : whiskerRight h.unit L = \ud835\udfd9 (\ud835\udfed C \u22d9 L) \u226b inv (whiskerLeft L h.counit)\n\u22a2 IsIso (\ud835\udfd9 (\ud835\udfed C \u22d9 L) \u226b inv (whiskerLeft L h.counit))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nL\u271d : C \u2964 D\nR\u271d : D \u2964 C\nh : L\u271d \u22a3 R\u271d\nC' : Type u\u2083\ninst\u271d\u2075 : Category.{v\u2083, u\u2083} C'\nD' : Type u\u2084\ninst\u271d\u2074 : Category.{v\u2084, u\u2084} D'\niC : C \u2964 C'\niD : D \u2964 D'\nL' : C' \u2964 D'\nR' : D' \u2964 C'\nadj : L' \u22a3 R'\nL : C \u2964 D\nR : D \u2964 C\ncomm1 : iC \u22d9 L' \u2245 L \u22d9 iD\ncomm2 : iD \u22d9 R' \u2245 R \u22d9 iC\ninst\u271d\u00b3 : Full iC\ninst\u271d\u00b2 : Faithful iC\ninst\u271d\u00b9 : Full iD\ninst\u271d : Faithful iD\nX' X : C\nY : D\nf : X' \u27f6 X\ng : X \u27f6 R.obj Y\n\u22a2 \u2191((fun X Y =>\n Trans.trans\n (Trans.trans\n (Trans.trans\n (Trans.trans (equivOfFullyFaithful iD) (Iso.homCongr (comm1.symm.app X) (Iso.refl (iD.obj Y))))\n (homEquiv adj (iC.obj X) (iD.obj Y)))\n (Iso.homCongr (Iso.refl (iC.obj X)) (comm2.app Y)))\n (equivOfFullyFaithful iC).symm)\n X' Y).symm\n (f \u226b g) =\n L.map f \u226b\n \u2191((fun X Y =>\n Trans.trans\n (Trans.trans\n (Trans.trans\n (Trans.trans (equivOfFullyFaithful iD) (Iso.homCongr (comm1.symm.app X) (Iso.refl (iD.obj Y))))\n (homEquiv adj (iC.obj X) (iD.obj Y)))\n (Iso.homCongr (Iso.refl (iC.obj X)) (comm2.app Y)))\n (equivOfFullyFaithful iC).symm)\n X Y).symm\n g\n[PROOFSTEP]\napply iD.map_injective\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nL\u271d : C \u2964 D\nR\u271d : D \u2964 C\nh : L\u271d \u22a3 R\u271d\nC' : Type u\u2083\ninst\u271d\u2075 : Category.{v\u2083, u\u2083} C'\nD' : Type u\u2084\ninst\u271d\u2074 : Category.{v\u2084, u\u2084} D'\niC : C \u2964 C'\niD : D \u2964 D'\nL' : C' \u2964 D'\nR' : D' \u2964 C'\nadj : L' \u22a3 R'\nL : C \u2964 D\nR : D \u2964 C\ncomm1 : iC \u22d9 L' \u2245 L \u22d9 iD\ncomm2 : iD \u22d9 R' \u2245 R \u22d9 iC\ninst\u271d\u00b3 : Full iC\ninst\u271d\u00b2 : Faithful iC\ninst\u271d\u00b9 : Full iD\ninst\u271d : Faithful iD\nX' X : C\nY : D\nf : X' \u27f6 X\ng : X \u27f6 R.obj Y\n\u22a2 iD.map\n (\u2191((fun X Y =>\n Trans.trans\n (Trans.trans\n (Trans.trans\n (Trans.trans (equivOfFullyFaithful iD) (Iso.homCongr (comm1.symm.app X) (Iso.refl (iD.obj Y))))\n (homEquiv adj (iC.obj X) (iD.obj Y)))\n (Iso.homCongr (Iso.refl (iC.obj X)) (comm2.app Y)))\n (equivOfFullyFaithful iC).symm)\n X' Y).symm\n (f \u226b g)) =\n iD.map\n (L.map f \u226b\n \u2191((fun X Y =>\n Trans.trans\n (Trans.trans\n (Trans.trans\n (Trans.trans (equivOfFullyFaithful iD) (Iso.homCongr (comm1.symm.app X) (Iso.refl (iD.obj Y))))\n (homEquiv adj (iC.obj X) (iD.obj Y)))\n (Iso.homCongr (Iso.refl (iC.obj X)) (comm2.app Y)))\n (equivOfFullyFaithful iC).symm)\n X Y).symm\n g)\n[PROOFSTEP]\nsimpa [Trans.trans] using (comm1.inv.naturality_assoc f _).symm\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nL\u271d : C \u2964 D\nR\u271d : D \u2964 C\nh : L\u271d \u22a3 R\u271d\nC' : Type u\u2083\ninst\u271d\u2075 : Category.{v\u2083, u\u2083} C'\nD' : Type u\u2084\ninst\u271d\u2074 : Category.{v\u2084, u\u2084} D'\niC : C \u2964 C'\niD : D \u2964 D'\nL' : C' \u2964 D'\nR' : D' \u2964 C'\nadj : L' \u22a3 R'\nL : C \u2964 D\nR : D \u2964 C\ncomm1 : iC \u22d9 L' \u2245 L \u22d9 iD\ncomm2 : iD \u22d9 R' \u2245 R \u22d9 iC\ninst\u271d\u00b3 : Full iC\ninst\u271d\u00b2 : Faithful iC\ninst\u271d\u00b9 : Full iD\ninst\u271d : Faithful iD\nX : C\nY' Y : D\nf : L.obj X \u27f6 Y'\ng : Y' \u27f6 Y\n\u22a2 \u2191((fun X Y =>\n Trans.trans\n (Trans.trans\n (Trans.trans\n (Trans.trans (equivOfFullyFaithful iD) (Iso.homCongr (comm1.symm.app X) (Iso.refl (iD.obj Y))))\n (homEquiv adj (iC.obj X) (iD.obj Y)))\n (Iso.homCongr (Iso.refl (iC.obj X)) (comm2.app Y)))\n (equivOfFullyFaithful iC).symm)\n X Y)\n (f \u226b g) =\n \u2191((fun X Y =>\n Trans.trans\n (Trans.trans\n (Trans.trans\n (Trans.trans (equivOfFullyFaithful iD) (Iso.homCongr (comm1.symm.app X) (Iso.refl (iD.obj Y))))\n (homEquiv adj (iC.obj X) (iD.obj Y)))\n (Iso.homCongr (Iso.refl (iC.obj X)) (comm2.app Y)))\n (equivOfFullyFaithful iC).symm)\n X Y')\n f \u226b\n R.map g\n[PROOFSTEP]\napply iC.map_injective\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nL\u271d : C \u2964 D\nR\u271d : D \u2964 C\nh : L\u271d \u22a3 R\u271d\nC' : Type u\u2083\ninst\u271d\u2075 : Category.{v\u2083, u\u2083} C'\nD' : Type u\u2084\ninst\u271d\u2074 : Category.{v\u2084, u\u2084} D'\niC : C \u2964 C'\niD : D \u2964 D'\nL' : C' \u2964 D'\nR' : D' \u2964 C'\nadj : L' \u22a3 R'\nL : C \u2964 D\nR : D \u2964 C\ncomm1 : iC \u22d9 L' \u2245 L \u22d9 iD\ncomm2 : iD \u22d9 R' \u2245 R \u22d9 iC\ninst\u271d\u00b3 : Full iC\ninst\u271d\u00b2 : Faithful iC\ninst\u271d\u00b9 : Full iD\ninst\u271d : Faithful iD\nX : C\nY' Y : D\nf : L.obj X \u27f6 Y'\ng : Y' \u27f6 Y\n\u22a2 iC.map\n (\u2191((fun X Y =>\n Trans.trans\n (Trans.trans\n (Trans.trans\n (Trans.trans (equivOfFullyFaithful iD) (Iso.homCongr (comm1.symm.app X) (Iso.refl (iD.obj Y))))\n (homEquiv adj (iC.obj X) (iD.obj Y)))\n (Iso.homCongr (Iso.refl (iC.obj X)) (comm2.app Y)))\n (equivOfFullyFaithful iC).symm)\n X Y)\n (f \u226b g)) =\n iC.map\n (\u2191((fun X Y =>\n Trans.trans\n (Trans.trans\n (Trans.trans\n (Trans.trans (equivOfFullyFaithful iD) (Iso.homCongr (comm1.symm.app X) (Iso.refl (iD.obj Y))))\n (homEquiv adj (iC.obj X) (iD.obj Y)))\n (Iso.homCongr (Iso.refl (iC.obj X)) (comm2.app Y)))\n (equivOfFullyFaithful iC).symm)\n X Y')\n f \u226b\n R.map g)\n[PROOFSTEP]\nsuffices : R'.map (iD.map g) \u226b comm2.hom.app Y = comm2.hom.app Y' \u226b iC.map (R.map g)\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nL\u271d : C \u2964 D\nR\u271d : D \u2964 C\nh : L\u271d \u22a3 R\u271d\nC' : Type u\u2083\ninst\u271d\u2075 : Category.{v\u2083, u\u2083} C'\nD' : Type u\u2084\ninst\u271d\u2074 : Category.{v\u2084, u\u2084} D'\niC : C \u2964 C'\niD : D \u2964 D'\nL' : C' \u2964 D'\nR' : D' \u2964 C'\nadj : L' \u22a3 R'\nL : C \u2964 D\nR : D \u2964 C\ncomm1 : iC \u22d9 L' \u2245 L \u22d9 iD\ncomm2 : iD \u22d9 R' \u2245 R \u22d9 iC\ninst\u271d\u00b3 : Full iC\ninst\u271d\u00b2 : Faithful iC\ninst\u271d\u00b9 : Full iD\ninst\u271d : Faithful iD\nX : C\nY' Y : D\nf : L.obj X \u27f6 Y'\ng : Y' \u27f6 Y\nthis : R'.map (iD.map g) \u226b NatTrans.app comm2.hom Y = NatTrans.app comm2.hom Y' \u226b iC.map (R.map g)\n\u22a2 iC.map\n (\u2191((fun X Y =>\n Trans.trans\n (Trans.trans\n (Trans.trans\n (Trans.trans (equivOfFullyFaithful iD) (Iso.homCongr (comm1.symm.app X) (Iso.refl (iD.obj Y))))\n (homEquiv adj (iC.obj X) (iD.obj Y)))\n (Iso.homCongr (Iso.refl (iC.obj X)) (comm2.app Y)))\n (equivOfFullyFaithful iC).symm)\n X Y)\n (f \u226b g)) =\n iC.map\n (\u2191((fun X Y =>\n Trans.trans\n (Trans.trans\n (Trans.trans\n (Trans.trans (equivOfFullyFaithful iD) (Iso.homCongr (comm1.symm.app X) (Iso.refl (iD.obj Y))))\n (homEquiv adj (iC.obj X) (iD.obj Y)))\n (Iso.homCongr (Iso.refl (iC.obj X)) (comm2.app Y)))\n (equivOfFullyFaithful iC).symm)\n X Y')\n f \u226b\n R.map g)\n[PROOFSTEP]\nsimp [Trans.trans, this]\n[GOAL]\ncase this\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} D\nL\u271d : C \u2964 D\nR\u271d : D \u2964 C\nh : L\u271d \u22a3 R\u271d\nC' : Type u\u2083\ninst\u271d\u2075 : Category.{v\u2083, u\u2083} C'\nD' : Type u\u2084\ninst\u271d\u2074 : Category.{v\u2084, u\u2084} D'\niC : C \u2964 C'\niD : D \u2964 D'\nL' : C' \u2964 D'\nR' : D' \u2964 C'\nadj : L' \u22a3 R'\nL : C \u2964 D\nR : D \u2964 C\ncomm1 : iC \u22d9 L' \u2245 L \u22d9 iD\ncomm2 : iD \u22d9 R' \u2245 R \u22d9 iC\ninst\u271d\u00b3 : Full iC\ninst\u271d\u00b2 : Faithful iC\ninst\u271d\u00b9 : Full iD\ninst\u271d : Faithful iD\nX : C\nY' Y : D\nf : L.obj X \u27f6 Y'\ng : Y' \u27f6 Y\n\u22a2 R'.map (iD.map g) \u226b NatTrans.app comm2.hom Y = NatTrans.app comm2.hom Y' \u226b iC.map (R.map g)\n[PROOFSTEP]\napply comm2.hom.naturality g\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Adjunction.FullyFaithful", "llama_tokens": 10691, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370111, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.41295563440679484}} {"text": "[GOAL]\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nL M N K : ModuleCat R\nf : L \u27f6 M\ng : M \u27f6 N\nw\u271d : f \u226b g = 0\nh k : _root_.homology f g w\u271d \u27f6 K\nw :\n \u2200 (x : { x // x \u2208 LinearMap.ker g }),\n \u2191h (\u2191(cokernel.\u03c0 (imageToKernel f g w\u271d)) (\u2191toKernelSubobject x)) =\n \u2191k (\u2191(cokernel.\u03c0 (imageToKernel f g w\u271d)) (\u2191toKernelSubobject x))\n\u22a2 h = k\n[PROOFSTEP]\nrefine'\n cokernel_funext fun n =>\n _\n -- porting note: as `equiv_rw` was not ported, it was replaced by `Equiv.surjective`\n -- Gosh it would be nice if `equiv_rw` could directly use an isomorphism, or an enriched `\u2243`.\n[GOAL]\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nL M N K : ModuleCat R\nf : L \u27f6 M\ng : M \u27f6 N\nw\u271d : f \u226b g = 0\nh k : _root_.homology f g w\u271d \u27f6 K\nw :\n \u2200 (x : { x // x \u2208 LinearMap.ker g }),\n \u2191h (\u2191(cokernel.\u03c0 (imageToKernel f g w\u271d)) (\u2191toKernelSubobject x)) =\n \u2191k (\u2191(cokernel.\u03c0 (imageToKernel f g w\u271d)) (\u2191toKernelSubobject x))\nn : (CategoryTheory.forget (ModuleCat R)).obj (Subobject.underlying.obj (kernelSubobject g))\n\u22a2 \u2191h (\u2191(cokernel.\u03c0 (imageToKernel f g w\u271d)) n) = \u2191k (\u2191(cokernel.\u03c0 (imageToKernel f g w\u271d)) n)\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := (kernelSubobjectIso g \u226a\u226b ModuleCat.kernelIsoKer g).toLinearEquiv.toEquiv.symm.surjective n\n[GOAL]\ncase intro\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nL M N K : ModuleCat R\nf : L \u27f6 M\ng : M \u27f6 N\nw\u271d : f \u226b g = 0\nh k : _root_.homology f g w\u271d \u27f6 K\nw :\n \u2200 (x : { x // x \u2208 LinearMap.ker g }),\n \u2191h (\u2191(cokernel.\u03c0 (imageToKernel f g w\u271d)) (\u2191toKernelSubobject x)) =\n \u2191k (\u2191(cokernel.\u03c0 (imageToKernel f g w\u271d)) (\u2191toKernelSubobject x))\nn : \u2191(of R { x // x \u2208 LinearMap.ker g })\n\u22a2 \u2191h\n (\u2191(cokernel.\u03c0 (imageToKernel f g w\u271d))\n (\u2191(LinearEquiv.toEquiv (Iso.toLinearEquiv (kernelSubobjectIso g \u226a\u226b kernelIsoKer g))).symm n)) =\n \u2191k\n (\u2191(cokernel.\u03c0 (imageToKernel f g w\u271d))\n (\u2191(LinearEquiv.toEquiv (Iso.toLinearEquiv (kernelSubobjectIso g \u226a\u226b kernelIsoKer g))).symm n))\n[PROOFSTEP]\nexact w n\n[GOAL]\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC\u271d D C : HomologicalComplex (ModuleCat R) c\ni : \u03b9\nx y : \u2191(Subobject.underlying.obj (cycles C i))\nw : \u2191(Subobject.arrow (cycles C i)) x = \u2191(Subobject.arrow (cycles C i)) y\n\u22a2 x = y\n[PROOFSTEP]\napply_fun (C.cycles i).arrow using (ModuleCat.mono_iff_injective _).mp (cycles C i).arrow_mono\n[GOAL]\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC\u271d D C : HomologicalComplex (ModuleCat R) c\ni : \u03b9\nx y : \u2191(Subobject.underlying.obj (cycles C i))\nw : \u2191(Subobject.arrow (cycles C i)) x = \u2191(Subobject.arrow (cycles C i)) y\n\u22a2 \u2191(Subobject.arrow (cycles C i)) x = \u2191(Subobject.arrow (cycles C i)) y\n[PROOFSTEP]\nexact w\n[GOAL]\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nf : C \u27f6 D\ni : \u03b9\nx : { x // x \u2208 LinearMap.ker (dFrom C i) }\n\u22a2 \u2191(Hom.f f i) \u2191x \u2208 LinearMap.ker (dFrom D i)\n[PROOFSTEP]\nrw [LinearMap.mem_ker, Hom.comm_from_apply, x.2, map_zero]\n[GOAL]\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nf : C \u27f6 D\ni : \u03b9\nx : { x // x \u2208 LinearMap.ker (dFrom C i) }\n\u22a2 \u2191(cyclesMap f i) (toCycles x) =\n toCycles { val := \u2191(Hom.f f i) \u2191x, property := (_ : \u2191(Hom.f f i) \u2191x \u2208 LinearMap.ker (dFrom D i)) }\n[PROOFSTEP]\next\n[GOAL]\ncase w\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nf : C \u27f6 D\ni : \u03b9\nx : { x // x \u2208 LinearMap.ker (dFrom C i) }\n\u22a2 \u2191(Subobject.arrow (cycles D i)) (\u2191(cyclesMap f i) (toCycles x)) =\n \u2191(Subobject.arrow (cycles D i))\n (toCycles { val := \u2191(Hom.f f i) \u2191x, property := (_ : \u2191(Hom.f f i) \u2191x \u2208 LinearMap.ker (dFrom D i)) })\n[PROOFSTEP]\nrw [cyclesMap_arrow_apply, toKernelSubobject_arrow, toKernelSubobject_arrow]\n[GOAL]\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nf g : C \u27f6 D\nh : Homotopy f g\ni : \u03b9\n\u22a2 (homologyFunctor (ModuleCat R) c i).map f = (homologyFunctor (ModuleCat R) c i).map g\n[PROOFSTEP]\napply homology_ext\n[GOAL]\ncase w\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nf g : C \u27f6 D\nh : Homotopy f g\ni : \u03b9\n\u22a2 \u2200 (x : { x // x \u2208 LinearMap.ker (dFrom C i) }),\n \u2191((homologyFunctor (ModuleCat R) c i).map f)\n (\u2191(cokernel.\u03c0 (imageToKernel (dTo C i) (dFrom C i) (_ : dTo C i \u226b dFrom C i = 0))) (\u2191toKernelSubobject x)) =\n \u2191((homologyFunctor (ModuleCat R) c i).map g)\n (\u2191(cokernel.\u03c0 (imageToKernel (dTo C i) (dFrom C i) (_ : dTo C i \u226b dFrom C i = 0))) (\u2191toKernelSubobject x))\n[PROOFSTEP]\nintro x\n[GOAL]\ncase w\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nf g : C \u27f6 D\nh : Homotopy f g\ni : \u03b9\nx : { x // x \u2208 LinearMap.ker (dFrom C i) }\n\u22a2 \u2191((homologyFunctor (ModuleCat R) c i).map f)\n (\u2191(cokernel.\u03c0 (imageToKernel (dTo C i) (dFrom C i) (_ : dTo C i \u226b dFrom C i = 0))) (\u2191toKernelSubobject x)) =\n \u2191((homologyFunctor (ModuleCat R) c i).map g)\n (\u2191(cokernel.\u03c0 (imageToKernel (dTo C i) (dFrom C i) (_ : dTo C i \u226b dFrom C i = 0))) (\u2191toKernelSubobject x))\n[PROOFSTEP]\nsimp only [homologyFunctor_map]\n[GOAL]\ncase w\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nf g : C \u27f6 D\nh : Homotopy f g\ni : \u03b9\nx : { x // x \u2208 LinearMap.ker (dFrom C i) }\n\u22a2 \u2191(homology.map (_ : dTo C i \u226b dFrom C i = 0) (_ : dTo D i \u226b dFrom D i = 0) (Hom.sqTo f i) (Hom.sqFrom f i)\n (_ : (Hom.sqTo f i).right = (Hom.sqTo f i).right))\n (\u2191(cokernel.\u03c0 (imageToKernel (dTo C i) (dFrom C i) (_ : dTo C i \u226b dFrom C i = 0))) (\u2191toKernelSubobject x)) =\n \u2191(homology.map (_ : dTo C i \u226b dFrom C i = 0) (_ : dTo D i \u226b dFrom D i = 0) (Hom.sqTo g i) (Hom.sqFrom g i)\n (_ : (Hom.sqTo g i).right = (Hom.sqTo g i).right))\n (\u2191(cokernel.\u03c0 (imageToKernel (dTo C i) (dFrom C i) (_ : dTo C i \u226b dFrom C i = 0))) (\u2191toKernelSubobject x))\n[PROOFSTEP]\nerw [homology.\u03c0_map_apply, homology.\u03c0_map_apply]\n -- To check that two elements are equal mod boundaries, it suffices to exhibit a boundary:\n[GOAL]\ncase w\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nf g : C \u27f6 D\nh : Homotopy f g\ni : \u03b9\nx : { x // x \u2208 LinearMap.ker (dFrom C i) }\n\u22a2 \u2191(homology.\u03c0 (dTo D i) (dFrom D i) (_ : dTo D i \u226b dFrom D i = 0))\n (\u2191(kernelSubobjectMap (Hom.sqFrom f i)) (\u2191toKernelSubobject x)) =\n \u2191(homology.\u03c0 (dTo D i) (dFrom D i) (_ : dTo D i \u226b dFrom D i = 0))\n (\u2191(kernelSubobjectMap (Hom.sqFrom g i)) (\u2191toKernelSubobject x))\n[PROOFSTEP]\nrefine'\n cokernel_\u03c0_imageSubobject_ext _ _ ((toPrev i h.hom) x.1)\n _\n -- Moreover, to check that two cycles are equal, it suffices to check their underlying elements:\n[GOAL]\ncase w\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nf g : C \u27f6 D\nh : Homotopy f g\ni : \u03b9\nx : { x // x \u2208 LinearMap.ker (dFrom C i) }\n\u22a2 \u2191(kernelSubobjectMap (Hom.sqFrom f i)) (\u2191toKernelSubobject x) =\n \u2191(kernelSubobjectMap (Hom.sqFrom g i)) (\u2191toKernelSubobject x) +\n \u2191(imageToKernel (dTo D i) (dFrom D i) (_ : dTo D i \u226b dFrom D i = 0))\n (\u2191(factorThruImageSubobject (dTo D i)) (\u2191(\u2191(toPrev i) h.hom) \u2191x))\n[PROOFSTEP]\next\n[GOAL]\ncase w.w\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nf g : C \u27f6 D\nh : Homotopy f g\ni : \u03b9\nx : { x // x \u2208 LinearMap.ker (dFrom C i) }\n\u22a2 \u2191(Subobject.arrow (cycles D i)) (\u2191(kernelSubobjectMap (Hom.sqFrom f i)) (\u2191toKernelSubobject x)) =\n \u2191(Subobject.arrow (cycles D i))\n (\u2191(kernelSubobjectMap (Hom.sqFrom g i)) (\u2191toKernelSubobject x) +\n \u2191(imageToKernel (dTo D i) (dFrom D i) (_ : dTo D i \u226b dFrom D i = 0))\n (\u2191(factorThruImageSubobject (dTo D i)) (\u2191(\u2191(toPrev i) h.hom) \u2191x)))\n[PROOFSTEP]\nerw [map_add, CategoryTheory.Limits.kernelSubobjectMap_arrow_apply,\n CategoryTheory.Limits.kernelSubobjectMap_arrow_apply, ModuleCat.toKernelSubobject_arrow, imageToKernel_arrow_apply,\n imageSubobject_arrow_comp_apply]\n[GOAL]\ncase w.w\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nf g : C \u27f6 D\nh : Homotopy f g\ni : \u03b9\nx : { x // x \u2208 LinearMap.ker (dFrom C i) }\n\u22a2 \u2191(Hom.sqFrom f i).left \u2191x = \u2191(Hom.sqFrom g i).left \u2191x + \u2191(dTo D i) (\u2191(\u2191(toPrev i) h.hom) \u2191x)\n[PROOFSTEP]\nrw [Hom.sqFrom_left, Hom.sqFrom_left, h.comm i, LinearMap.add_apply, LinearMap.add_apply, prevD_eq_toPrev_dTo,\n dNext_eq_dFrom_fromNext, comp_apply, comp_apply, x.2, map_zero]\n[GOAL]\ncase w.w\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nf g : C \u27f6 D\nh : Homotopy f g\ni : \u03b9\nx : { x // x \u2208 LinearMap.ker (dFrom C i) }\n\u22a2 0 + \u2191(dTo D i) (\u2191(\u2191(toPrev i) h.hom) \u2191x) + \u2191(Hom.f g i) \u2191x = \u2191(Hom.f g i) \u2191x + \u2191(dTo D i) (\u2191(\u2191(toPrev i) h.hom) \u2191x)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.w\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nf g : C \u27f6 D\nh : Homotopy f g\ni : \u03b9\nx : { x // x \u2208 LinearMap.ker (dFrom C i) }\n\u22a2 0 + \u2191(dTo D i) (\u2191(\u2191(toPrev i) h.hom) \u2191x) + \u2191(Hom.f g i) \u2191x = \u2191(Hom.f g i) \u2191x + \u2191(dTo D i) (\u2191(\u2191(toPrev i) h.hom) \u2191x)\n[PROOFSTEP]\nabel\n[GOAL]\ncase w.w\nR : Type v\ninst\u271d : Ring R\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nC D : HomologicalComplex (ModuleCat R) c\nf g : C \u27f6 D\nh : Homotopy f g\ni : \u03b9\nx : { x // x \u2208 LinearMap.ker (dFrom C i) }\n\u22a2 0 + \u2191(dTo D i) (\u2191(\u2191(toPrev i) h.hom) \u2191x) + \u2191(Hom.f g i) \u2191x = \u2191(Hom.f g i) \u2191x + \u2191(dTo D i) (\u2191(\u2191(toPrev i) h.hom) \u2191x)\n[PROOFSTEP]\nabel\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Homology.ModuleCat", "llama_tokens": 4418, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506526772884, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.41292931065970173}} {"text": "[GOAL]\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\n\u22a2 Surjective fun v i => \u2191(MvPolynomial.eval v) (ps i)\n[PROOFSTEP]\nhave is_int : \u2200 x : R, IsIntegral K x := fun x => isAlgebraic_iff_isIntegral.1 (alg x)\n[GOAL]\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\n\u22a2 Surjective fun v i => \u2191(MvPolynomial.eval v) (ps i)\n[PROOFSTEP]\nclassical\nintro v\ncases nonempty_fintype \u03b9\nlet s : Finset R :=\n (Finset.biUnion (univ : Finset \u03b9) fun i => (ps i).support.image fun x => coeff x (ps i)) \u222a (univ : Finset \u03b9).image v\nhave hv : \u2200 i, v i \u2208 Algebra.adjoin K (s : Set R) := fun j =>\n Algebra.subset_adjoin (mem_union_right _ (mem_image.2 \u27e8j, mem_univ _, rfl\u27e9))\nhave hs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115), k \u2208 (ps i).support \u2192 coeff k (ps i) \u2208 Algebra.adjoin K (s : Set R) := fun i k hk =>\n Algebra.subset_adjoin (mem_union_left _ (mem_biUnion.2 \u27e8i, mem_univ _, mem_image_of_mem _ hk\u27e9))\nletI := isNoetherian_adjoin_finset s fun x _ => is_int x\nletI := Module.IsNoetherian.finite K (Algebra.adjoin K (s : Set R))\nletI : Finite (Algebra.adjoin K (s : Set R)) :=\n FiniteDimensional.finite_of_finite K\n (Algebra.adjoin K (s : Set R))\n -- The restriction of the polynomial map, `ps`, to the subalgebra generated by `s`\nlet res : (\u03b9 \u2192 Algebra.adjoin K (s : Set R)) \u2192 \u03b9 \u2192 Algebra.adjoin K (s : Set R) := fun x i =>\n \u27e8eval (fun j : \u03b9 => (x j : R)) (ps i), eval_mem (hs\u2081 _) fun i => (x i).2\u27e9\nhave hres_inj : Injective res := by\n intro x y hxy\n ext i\n simp only [Subtype.ext_iff, funext_iff] at hxy \n exact congr_fun (hinj (funext hxy)) i\nhave hres_surj : Surjective res := Finite.injective_iff_surjective.1 hres_inj\ncases' hres_surj fun i => \u27e8v i, hv i\u27e9 with w hw\nuse fun i => w i\nsimpa only [Subtype.ext_iff, funext_iff] using hw\n[GOAL]\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\n\u22a2 Surjective fun v i => \u2191(MvPolynomial.eval v) (ps i)\n[PROOFSTEP]\nintro v\n[GOAL]\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\n\u22a2 \u2203 a, (fun v i => \u2191(MvPolynomial.eval v) (ps i)) a = v\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 a, (fun v i => \u2191(MvPolynomial.eval v) (ps i)) a = v\n[PROOFSTEP]\nlet s : Finset R :=\n (Finset.biUnion (univ : Finset \u03b9) fun i => (ps i).support.image fun x => coeff x (ps i)) \u222a (univ : Finset \u03b9).image v\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) \u222a image v univ\n\u22a2 \u2203 a, (fun v i => \u2191(MvPolynomial.eval v) (ps i)) a = v\n[PROOFSTEP]\nhave hv : \u2200 i, v i \u2208 Algebra.adjoin K (s : Set R) := fun j =>\n Algebra.subset_adjoin (mem_union_right _ (mem_image.2 \u27e8j, mem_univ _, rfl\u27e9))\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\n\u22a2 \u2203 a, (fun v i => \u2191(MvPolynomial.eval v) (ps i)) a = v\n[PROOFSTEP]\nhave hs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115), k \u2208 (ps i).support \u2192 coeff k (ps i) \u2208 Algebra.adjoin K (s : Set R) := fun i k hk =>\n Algebra.subset_adjoin (mem_union_left _ (mem_biUnion.2 \u27e8i, mem_univ _, mem_image_of_mem _ hk\u27e9))\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115), k \u2208 MvPolynomial.support (ps i) \u2192 coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\n\u22a2 \u2203 a, (fun v i => \u2191(MvPolynomial.eval v) (ps i)) a = v\n[PROOFSTEP]\nletI := isNoetherian_adjoin_finset s fun x _ => is_int x\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115), k \u2208 MvPolynomial.support (ps i) \u2192 coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis : IsNoetherian K { x // x \u2208 Algebra.adjoin K \u2191s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\n\u22a2 \u2203 a, (fun v i => \u2191(MvPolynomial.eval v) (ps i)) a = v\n[PROOFSTEP]\nletI := Module.IsNoetherian.finite K (Algebra.adjoin K (s : Set R))\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115), k \u2208 MvPolynomial.support (ps i) \u2192 coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d : IsNoetherian K { x // x \u2208 Algebra.adjoin K \u2191s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis : Module.Finite K { x // x \u2208 Algebra.adjoin K \u2191s } := Module.IsNoetherian.finite K { x // x \u2208 Algebra.adjoin K \u2191s }\n\u22a2 \u2203 a, (fun v i => \u2191(MvPolynomial.eval v) (ps i)) a = v\n[PROOFSTEP]\nletI : Finite (Algebra.adjoin K (s : Set R)) :=\n FiniteDimensional.finite_of_finite K\n (Algebra.adjoin K (s : Set R))\n -- The restriction of the polynomial map, `ps`, to the subalgebra generated by `s`\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115), k \u2208 MvPolynomial.support (ps i) \u2192 coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K { x // x \u2208 Algebra.adjoin K \u2191s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis\u271d : Module.Finite K { x // x \u2208 Algebra.adjoin K \u2191s } :=\n Module.IsNoetherian.finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nthis : Finite { x // x \u2208 Algebra.adjoin K \u2191s } := FiniteDimensional.finite_of_finite K { x // x \u2208 Algebra.adjoin K \u2191s }\n\u22a2 \u2203 a, (fun v i => \u2191(MvPolynomial.eval v) (ps i)) a = v\n[PROOFSTEP]\nlet res : (\u03b9 \u2192 Algebra.adjoin K (s : Set R)) \u2192 \u03b9 \u2192 Algebra.adjoin K (s : Set R) := fun x i =>\n \u27e8eval (fun j : \u03b9 => (x j : R)) (ps i), eval_mem (hs\u2081 _) fun i => (x i).2\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115), k \u2208 MvPolynomial.support (ps i) \u2192 coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K { x // x \u2208 Algebra.adjoin K \u2191s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis\u271d : Module.Finite K { x // x \u2208 Algebra.adjoin K \u2191s } :=\n Module.IsNoetherian.finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nthis : Finite { x // x \u2208 Algebra.adjoin K \u2191s } := FiniteDimensional.finite_of_finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nres : (\u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s }) \u2192 \u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s } :=\n fun x i =>\n { val := \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i),\n property := (_ : \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i) \u2208 Algebra.adjoin K \u2191s) }\n\u22a2 \u2203 a, (fun v i => \u2191(MvPolynomial.eval v) (ps i)) a = v\n[PROOFSTEP]\nhave hres_inj : Injective res := by\n intro x y hxy\n ext i\n simp only [Subtype.ext_iff, funext_iff] at hxy \n exact congr_fun (hinj (funext hxy)) i\n[GOAL]\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115), k \u2208 MvPolynomial.support (ps i) \u2192 coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K { x // x \u2208 Algebra.adjoin K \u2191s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis\u271d : Module.Finite K { x // x \u2208 Algebra.adjoin K \u2191s } :=\n Module.IsNoetherian.finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nthis : Finite { x // x \u2208 Algebra.adjoin K \u2191s } := FiniteDimensional.finite_of_finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nres : (\u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s }) \u2192 \u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s } :=\n fun x i =>\n { val := \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i),\n property := (_ : \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i) \u2208 Algebra.adjoin K \u2191s) }\n\u22a2 Injective res\n[PROOFSTEP]\nintro x y hxy\n[GOAL]\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115), k \u2208 MvPolynomial.support (ps i) \u2192 coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K { x // x \u2208 Algebra.adjoin K \u2191s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis\u271d : Module.Finite K { x // x \u2208 Algebra.adjoin K \u2191s } :=\n Module.IsNoetherian.finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nthis : Finite { x // x \u2208 Algebra.adjoin K \u2191s } := FiniteDimensional.finite_of_finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nres : (\u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s }) \u2192 \u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s } :=\n fun x i =>\n { val := \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i),\n property := (_ : \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i) \u2208 Algebra.adjoin K \u2191s) }\nx y : \u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s }\nhxy : res x = res y\n\u22a2 x = y\n[PROOFSTEP]\next i\n[GOAL]\ncase h.a\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115), k \u2208 MvPolynomial.support (ps i) \u2192 coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K { x // x \u2208 Algebra.adjoin K \u2191s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis\u271d : Module.Finite K { x // x \u2208 Algebra.adjoin K \u2191s } :=\n Module.IsNoetherian.finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nthis : Finite { x // x \u2208 Algebra.adjoin K \u2191s } := FiniteDimensional.finite_of_finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nres : (\u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s }) \u2192 \u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s } :=\n fun x i =>\n { val := \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i),\n property := (_ : \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i) \u2208 Algebra.adjoin K \u2191s) }\nx y : \u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s }\nhxy : res x = res y\ni : \u03b9\n\u22a2 \u2191(x i) = \u2191(y i)\n[PROOFSTEP]\nsimp only [Subtype.ext_iff, funext_iff] at hxy \n[GOAL]\ncase h.a\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115), k \u2208 MvPolynomial.support (ps i) \u2192 coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K { x // x \u2208 Algebra.adjoin K \u2191s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis\u271d : Module.Finite K { x // x \u2208 Algebra.adjoin K \u2191s } :=\n Module.IsNoetherian.finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nthis : Finite { x // x \u2208 Algebra.adjoin K \u2191s } := FiniteDimensional.finite_of_finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nres : (\u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s }) \u2192 \u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s } :=\n fun x i =>\n { val := \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i),\n property := (_ : \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i) \u2208 Algebra.adjoin K \u2191s) }\nx y : \u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s }\ni : \u03b9\nhxy : \u2200 (a : \u03b9), \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps a) = \u2191(MvPolynomial.eval fun j => \u2191(y j)) (ps a)\n\u22a2 \u2191(x i) = \u2191(y i)\n[PROOFSTEP]\nexact congr_fun (hinj (funext hxy)) i\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115), k \u2208 MvPolynomial.support (ps i) \u2192 coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K { x // x \u2208 Algebra.adjoin K \u2191s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis\u271d : Module.Finite K { x // x \u2208 Algebra.adjoin K \u2191s } :=\n Module.IsNoetherian.finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nthis : Finite { x // x \u2208 Algebra.adjoin K \u2191s } := FiniteDimensional.finite_of_finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nres : (\u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s }) \u2192 \u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s } :=\n fun x i =>\n { val := \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i),\n property := (_ : \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i) \u2208 Algebra.adjoin K \u2191s) }\nhres_inj : Injective res\n\u22a2 \u2203 a, (fun v i => \u2191(MvPolynomial.eval v) (ps i)) a = v\n[PROOFSTEP]\nhave hres_surj : Surjective res := Finite.injective_iff_surjective.1 hres_inj\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115), k \u2208 MvPolynomial.support (ps i) \u2192 coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K { x // x \u2208 Algebra.adjoin K \u2191s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis\u271d : Module.Finite K { x // x \u2208 Algebra.adjoin K \u2191s } :=\n Module.IsNoetherian.finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nthis : Finite { x // x \u2208 Algebra.adjoin K \u2191s } := FiniteDimensional.finite_of_finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nres : (\u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s }) \u2192 \u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s } :=\n fun x i =>\n { val := \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i),\n property := (_ : \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i) \u2208 Algebra.adjoin K \u2191s) }\nhres_inj : Injective res\nhres_surj : Surjective res\n\u22a2 \u2203 a, (fun v i => \u2191(MvPolynomial.eval v) (ps i)) a = v\n[PROOFSTEP]\ncases' hres_surj fun i => \u27e8v i, hv i\u27e9 with w hw\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115), k \u2208 MvPolynomial.support (ps i) \u2192 coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K { x // x \u2208 Algebra.adjoin K \u2191s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis\u271d : Module.Finite K { x // x \u2208 Algebra.adjoin K \u2191s } :=\n Module.IsNoetherian.finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nthis : Finite { x // x \u2208 Algebra.adjoin K \u2191s } := FiniteDimensional.finite_of_finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nres : (\u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s }) \u2192 \u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s } :=\n fun x i =>\n { val := \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i),\n property := (_ : \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i) \u2208 Algebra.adjoin K \u2191s) }\nhres_inj : Injective res\nhres_surj : Surjective res\nw : \u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s }\nhw : res w = fun i => { val := v i, property := (_ : v i \u2208 Algebra.adjoin K \u2191s) }\n\u22a2 \u2203 a, (fun v i => \u2191(MvPolynomial.eval v) (ps i)) a = v\n[PROOFSTEP]\nuse fun i => w i\n[GOAL]\ncase h\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Finite K\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Algebra K R\nalg : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => \u2191(MvPolynomial.eval v) (ps i)\nis_int : \u2200 (x : R), IsIntegral K x\nv : (i : \u03b9) \u2192 (fun x => R) (ps i)\nval\u271d : Fintype \u03b9\ns : Finset R :=\n (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115), k \u2208 MvPolynomial.support (ps i) \u2192 coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K { x // x \u2208 Algebra.adjoin K \u2191s } := isNoetherian_adjoin_finset s fun x x_1 => is_int x\nthis\u271d : Module.Finite K { x // x \u2208 Algebra.adjoin K \u2191s } :=\n Module.IsNoetherian.finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nthis : Finite { x // x \u2208 Algebra.adjoin K \u2191s } := FiniteDimensional.finite_of_finite K { x // x \u2208 Algebra.adjoin K \u2191s }\nres : (\u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s }) \u2192 \u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s } :=\n fun x i =>\n { val := \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i),\n property := (_ : \u2191(MvPolynomial.eval fun j => \u2191(x j)) (ps i) \u2208 Algebra.adjoin K \u2191s) }\nhres_inj : Injective res\nhres_surj : Surjective res\nw : \u03b9 \u2192 { x // x \u2208 Algebra.adjoin K \u2191s }\nhw : res w = fun i => { val := v i, property := (_ : v i \u2208 Algebra.adjoin K \u2191s) }\n\u22a2 ((fun v i => \u2191(MvPolynomial.eval v) (ps i)) fun i => \u2191(w i)) = v\n[PROOFSTEP]\nsimpa only [Subtype.ext_iff, funext_iff] using hw\n", "meta": {"mathlib_filename": "Mathlib.FieldTheory.AxGrothendieck", "llama_tokens": 10160, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149868676283, "lm_q2_score": 0.546738151984614, "lm_q1q2_score": 0.4127408248254962}} {"text": "[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\n\u22a2 AEStronglyMeasurable (ProbabilityTheory.truncation f A) \u03bc\n[PROOFSTEP]\napply AEStronglyMeasurable.comp_aemeasurable _ hf.aemeasurable\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\n\u22a2 AEStronglyMeasurable (indicator (Set.Ioc (-A) A) id) (Measure.map f \u03bc)\n[PROOFSTEP]\nexact (stronglyMeasurable_id.indicator measurableSet_Ioc).aestronglyMeasurable\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nx : \u03b1\n\u22a2 |truncation f A x| \u2264 |A|\n[PROOFSTEP]\nsimp only [truncation, Set.indicator, Set.mem_Icc, id.def, Function.comp_apply]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nx : \u03b1\n\u22a2 |if f x \u2208 Set.Ioc (-A) A then f x else 0| \u2264 |A|\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nx : \u03b1\nh : f x \u2208 Set.Ioc (-A) A\n\u22a2 |f x| \u2264 |A|\n[PROOFSTEP]\nexact abs_le_abs h.2 (neg_le.2 h.1.le)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nx : \u03b1\nh : \u00acf x \u2208 Set.Ioc (-A) A\n\u22a2 |0| \u2264 |A|\n[PROOFSTEP]\nsimp [abs_nonneg]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\n\u22a2 truncation f 0 = 0\n[PROOFSTEP]\nsimp [truncation]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\n\u22a2 (fun x => 0) \u2218 f = 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nx : \u03b1\n\u22a2 |truncation f A x| \u2264 |f x|\n[PROOFSTEP]\nsimp only [truncation, indicator, Set.mem_Icc, id.def, Function.comp_apply]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nx : \u03b1\n\u22a2 |if f x \u2208 Set.Ioc (-A) A then f x else 0| \u2264 |f x|\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nx : \u03b1\nh\u271d : f x \u2208 Set.Ioc (-A) A\n\u22a2 |f x| \u2264 |f x|\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nx : \u03b1\nh\u271d : \u00acf x \u2208 Set.Ioc (-A) A\n\u22a2 |0| \u2264 |f x|\n[PROOFSTEP]\nsimp [abs_nonneg]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nx : \u03b1\nh : |f x| < A\n\u22a2 truncation f A x = f x\n[PROOFSTEP]\nsimp only [truncation, indicator, Set.mem_Icc, id.def, Function.comp_apply, ite_eq_left_iff]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nx : \u03b1\nh : |f x| < A\n\u22a2 \u00acf x \u2208 Set.Ioc (-A) A \u2192 0 = f x\n[PROOFSTEP]\nintro H\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nx : \u03b1\nh : |f x| < A\nH : \u00acf x \u2208 Set.Ioc (-A) A\n\u22a2 0 = f x\n[PROOFSTEP]\napply H.elim\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nx : \u03b1\nh : |f x| < A\nH : \u00acf x \u2208 Set.Ioc (-A) A\n\u22a2 f x \u2208 Set.Ioc (-A) A\n[PROOFSTEP]\nsimp [(abs_lt.1 h).1, (abs_lt.1 h).2.le]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nh : \u2200 (x : \u03b1), 0 \u2264 f x\n\u22a2 truncation f A = indicator (Set.Ioc 0 A) id \u2218 f\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nh : \u2200 (x : \u03b1), 0 \u2264 f x\nx : \u03b1\n\u22a2 truncation f A x = (indicator (Set.Ioc 0 A) id \u2218 f) x\n[PROOFSTEP]\nrcases(h x).lt_or_eq with (hx | hx)\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nh : \u2200 (x : \u03b1), 0 \u2264 f x\nx : \u03b1\nhx : 0 < f x\n\u22a2 truncation f A x = (indicator (Set.Ioc 0 A) id \u2218 f) x\n[PROOFSTEP]\nsimp only [truncation, indicator, hx, Set.mem_Ioc, id.def, Function.comp_apply, true_and_iff]\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nh : \u2200 (x : \u03b1), 0 \u2264 f x\nx : \u03b1\nhx : 0 < f x\n\u22a2 (if -A < f x \u2227 f x \u2264 A then f x else 0) = if f x \u2264 A then f x else 0\n[PROOFSTEP]\nby_cases h'x : f x \u2264 A\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nh : \u2200 (x : \u03b1), 0 \u2264 f x\nx : \u03b1\nhx : 0 < f x\nh'x : f x \u2264 A\n\u22a2 (if -A < f x \u2227 f x \u2264 A then f x else 0) = if f x \u2264 A then f x else 0\n[PROOFSTEP]\nhave : -A < f x := by linarith [h x]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nh : \u2200 (x : \u03b1), 0 \u2264 f x\nx : \u03b1\nhx : 0 < f x\nh'x : f x \u2264 A\n\u22a2 -A < f x\n[PROOFSTEP]\nlinarith [h x]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nh : \u2200 (x : \u03b1), 0 \u2264 f x\nx : \u03b1\nhx : 0 < f x\nh'x : f x \u2264 A\nthis : -A < f x\n\u22a2 (if -A < f x \u2227 f x \u2264 A then f x else 0) = if f x \u2264 A then f x else 0\n[PROOFSTEP]\nsimp only [this, true_and_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nh : \u2200 (x : \u03b1), 0 \u2264 f x\nx : \u03b1\nhx : 0 < f x\nh'x : \u00acf x \u2264 A\n\u22a2 (if -A < f x \u2227 f x \u2264 A then f x else 0) = if f x \u2264 A then f x else 0\n[PROOFSTEP]\nsimp only [h'x, and_false_iff]\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nA : \u211d\nh : \u2200 (x : \u03b1), 0 \u2264 f x\nx : \u03b1\nhx : 0 = f x\n\u22a2 truncation f A x = (indicator (Set.Ioc 0 A) id \u2218 f) x\n[PROOFSTEP]\nsimp only [truncation, indicator, hx, id.def, Function.comp_apply, ite_self]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\n\u22a2 Integrable (truncation f A)\n[PROOFSTEP]\nrw [\u2190 mem\u2112p_one_iff_integrable]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\n\u22a2 Mem\u2112p (truncation f A) 1\n[PROOFSTEP]\nexact hf.mem\u2112p_truncation\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nhA : 0 \u2264 A\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u222b (x : \u03b1), truncation f A x ^ n \u2202\u03bc = \u222b (y : \u211d) in -A..A, y ^ n \u2202Measure.map f \u03bc\n[PROOFSTEP]\nhave M : MeasurableSet (Set.Ioc (-A) A) := measurableSet_Ioc\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nhA : 0 \u2264 A\nn : \u2115\nhn : n \u2260 0\nM : MeasurableSet (Set.Ioc (-A) A)\n\u22a2 \u222b (x : \u03b1), truncation f A x ^ n \u2202\u03bc = \u222b (y : \u211d) in -A..A, y ^ n \u2202Measure.map f \u03bc\n[PROOFSTEP]\nchange \u222b x, (fun z => indicator (Set.Ioc (-A) A) id z ^ n) (f x) \u2202\u03bc = _\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nhA : 0 \u2264 A\nn : \u2115\nhn : n \u2260 0\nM : MeasurableSet (Set.Ioc (-A) A)\n\u22a2 \u222b (x : \u03b1), (fun z => indicator (Set.Ioc (-A) A) id z ^ n) (f x) \u2202\u03bc = \u222b (y : \u211d) in -A..A, y ^ n \u2202Measure.map f \u03bc\n[PROOFSTEP]\nrw [\u2190 integral_map (f := fun z => _ ^ n) hf.aemeasurable, intervalIntegral.integral_of_le, \u2190 integral_indicator M]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nhA : 0 \u2264 A\nn : \u2115\nhn : n \u2260 0\nM : MeasurableSet (Set.Ioc (-A) A)\n\u22a2 \u222b (y : \u211d), indicator (Set.Ioc (-A) A) id y ^ n \u2202Measure.map f \u03bc =\n \u222b (x : \u211d), indicator (Set.Ioc (-A) A) (fun x => x ^ n) x \u2202Measure.map f \u03bc\n[PROOFSTEP]\nsimp only [indicator, zero_pow' _ hn, id.def, ite_pow]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nhA : 0 \u2264 A\nn : \u2115\nhn : n \u2260 0\nM : MeasurableSet (Set.Ioc (-A) A)\n\u22a2 -A \u2264 A\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nhA : 0 \u2264 A\nn : \u2115\nhn : n \u2260 0\nM : MeasurableSet (Set.Ioc (-A) A)\n\u22a2 AEStronglyMeasurable (fun z => indicator (Set.Ioc (-A) A) id z ^ n) (Measure.map f \u03bc)\n[PROOFSTEP]\nexact ((measurable_id.indicator M).pow_const n).aestronglyMeasurable\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\n\u22a2 \u222b (x : \u03b1), truncation f A x ^ n \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc\n[PROOFSTEP]\nhave M : MeasurableSet (Set.Ioc 0 A) := measurableSet_Ioc\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\n\u22a2 \u222b (x : \u03b1), truncation f A x ^ n \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc\n[PROOFSTEP]\nhave M' : MeasurableSet (Set.Ioc A 0) := measurableSet_Ioc\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\n\u22a2 \u222b (x : \u03b1), truncation f A x ^ n \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc\n[PROOFSTEP]\nrw [truncation_eq_of_nonneg h'f]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\n\u22a2 \u222b (x : \u03b1), (indicator (Set.Ioc 0 A) id \u2218 fun x => f x) x ^ n \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc\n[PROOFSTEP]\nchange \u222b x, (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) \u2202\u03bc = _\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\n\u22a2 \u222b (x : \u03b1), (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc\n[PROOFSTEP]\nrcases le_or_lt 0 A with (hA | hA)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : 0 \u2264 A\n\u22a2 \u222b (x : \u03b1), (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc\n[PROOFSTEP]\nrw [\u2190 integral_map (f := fun z => _ ^ n) hf.aemeasurable, intervalIntegral.integral_of_le hA, \u2190 integral_indicator M]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : 0 \u2264 A\n\u22a2 \u222b (y : \u211d), indicator (Set.Ioc 0 A) id y ^ n \u2202Measure.map f \u03bc =\n \u222b (x : \u211d), indicator (Set.Ioc 0 A) (fun x => x ^ n) x \u2202Measure.map f \u03bc\n[PROOFSTEP]\nsimp only [indicator, zero_pow' _ hn, id.def, ite_pow]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : 0 \u2264 A\n\u22a2 AEStronglyMeasurable (fun z => indicator (Set.Ioc 0 A) id z ^ n) (Measure.map f \u03bc)\n[PROOFSTEP]\nexact ((measurable_id.indicator M).pow_const n).aestronglyMeasurable\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n\u22a2 \u222b (x : \u03b1), (fun z => indicator (Set.Ioc 0 A) id z ^ n) (f x) \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y ^ n \u2202Measure.map f \u03bc\n[PROOFSTEP]\nrw [\u2190 integral_map (f := fun z => _ ^ n) hf.aemeasurable, intervalIntegral.integral_of_ge hA.le, \u2190\n integral_indicator M']\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n\u22a2 \u222b (y : \u211d), indicator (Set.Ioc 0 A) id y ^ n \u2202Measure.map f \u03bc =\n -\u222b (x : \u211d), indicator (Set.Ioc A 0) (fun x => x ^ n) x \u2202Measure.map f \u03bc\n[PROOFSTEP]\nsimp only [Set.Ioc_eq_empty_of_le hA.le, zero_pow' _ hn, Set.indicator_empty, integral_zero, zero_eq_neg]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n\u22a2 \u222b (x : \u211d), indicator (Set.Ioc A 0) (fun x => x ^ n) x \u2202Measure.map f \u03bc = 0\n[PROOFSTEP]\napply integral_eq_zero_of_ae\n[GOAL]\ncase inr.hf\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n\u22a2 (fun a => indicator (Set.Ioc A 0) (fun x => x ^ n) a) =\u1d50[Measure.map f \u03bc] 0\n[PROOFSTEP]\nhave : \u2200\u1d50 x \u2202Measure.map f \u03bc, (0 : \u211d) \u2264 x := (ae_map_iff hf.aemeasurable measurableSet_Ici).2 (eventually_of_forall h'f)\n[GOAL]\ncase inr.hf\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\n\u22a2 (fun a => indicator (Set.Ioc A 0) (fun x => x ^ n) a) =\u1d50[Measure.map f \u03bc] 0\n[PROOFSTEP]\nfilter_upwards [this] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 indicator (Set.Ioc A 0) (fun x => x ^ n) x = OfNat.ofNat 0 x\n[PROOFSTEP]\nsimp only [indicator, Set.mem_Ioc, Pi.zero_apply, ite_eq_right_iff, and_imp]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 A < x \u2192 x \u2264 0 \u2192 x ^ n = 0\n[PROOFSTEP]\nintro _ h''x\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\nx : \u211d\nhx : 0 \u2264 x\na\u271d : A < x\nh''x : x \u2264 0\n\u22a2 x ^ n = 0\n[PROOFSTEP]\nhave : x = 0 := by linarith\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\nx : \u211d\nhx : 0 \u2264 x\na\u271d : A < x\nh''x : x \u2264 0\n\u22a2 x = 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis\u271d : \u2200\u1d50 (x : \u211d) \u2202Measure.map f \u03bc, 0 \u2264 x\nx : \u211d\nhx : 0 \u2264 x\na\u271d : A < x\nh''x : x \u2264 0\nthis : x = 0\n\u22a2 x ^ n = 0\n[PROOFSTEP]\nsimp [this, zero_pow' _ hn]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nn : \u2115\nhn : n \u2260 0\nh'f : 0 \u2264 f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n\u22a2 AEStronglyMeasurable (fun z => indicator (Set.Ioc 0 A) id z ^ n) (Measure.map f \u03bc)\n[PROOFSTEP]\nexact ((measurable_id.indicator M).pow_const n).aestronglyMeasurable\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nhA : 0 \u2264 A\n\u22a2 \u222b (x : \u03b1), truncation f A x \u2202\u03bc = \u222b (y : \u211d) in -A..A, y \u2202Measure.map f \u03bc\n[PROOFSTEP]\nsimpa using moment_truncation_eq_intervalIntegral hf hA one_ne_zero\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : AEStronglyMeasurable f \u03bc\nA : \u211d\nh'f : 0 \u2264 f\n\u22a2 \u222b (x : \u03b1), truncation f A x \u2202\u03bc = \u222b (y : \u211d) in 0 ..A, y \u2202Measure.map f \u03bc\n[PROOFSTEP]\nsimpa using moment_truncation_eq_intervalIntegral_of_nonneg hf one_ne_zero h'f\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nh'f : 0 \u2264 f\nA : \u211d\n\u22a2 \u222b (x : \u03b1), truncation f A x \u2202\u03bc \u2264 \u222b (x : \u03b1), f x \u2202\u03bc\n[PROOFSTEP]\napply integral_mono_of_nonneg (eventually_of_forall fun x => ?_) hf (eventually_of_forall fun x => ?_)\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nh'f : 0 \u2264 f\nA : \u211d\nx : \u03b1\n\u22a2 OfNat.ofNat 0 x \u2264 truncation f A x\n[PROOFSTEP]\nexact truncation_nonneg _ (h'f x)\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f\nh'f : 0 \u2264 f\nA : \u211d\nx : \u03b1\n\u22a2 truncation f A x \u2264 f x\n[PROOFSTEP]\ncalc\n truncation f A x \u2264 |truncation f A x| := le_abs_self _\n _ \u2264 |f x| := (abs_truncation_le_abs_self _ _ _)\n _ = f x := abs_of_nonneg (h'f x)\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 Tendsto (fun A => \u222b (x : \u03b1), truncation f A x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1), f x \u2202\u03bc))\n[PROOFSTEP]\nrefine' tendsto_integral_filter_of_dominated_convergence (fun x => abs (f x)) _ _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 \u2200\u1da0 (n : \u211d) in atTop, AEStronglyMeasurable (fun x => truncation f n x) \u03bc\n[PROOFSTEP]\nexact eventually_of_forall fun A => hf.aestronglyMeasurable.truncation\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 \u2200\u1da0 (n : \u211d) in atTop, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016truncation f n a\u2016 \u2264 (fun x => |f x|) a\n[PROOFSTEP]\napply eventually_of_forall fun A => ?_\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nA : \u211d\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016truncation f A a\u2016 \u2264 (fun x => |f x|) a\n[PROOFSTEP]\napply eventually_of_forall fun x => ?_\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nA : \u211d\nx : \u03b1\n\u22a2 \u2016truncation f A x\u2016 \u2264 (fun x => |f x|) x\n[PROOFSTEP]\nrw [Real.norm_eq_abs]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nA : \u211d\nx : \u03b1\n\u22a2 |truncation f A x| \u2264 (fun x => |f x|) x\n[PROOFSTEP]\nexact abs_truncation_le_abs_self _ _ _\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 Integrable fun x => |f x|\n[PROOFSTEP]\napply hf.abs\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => truncation f n a) atTop (\ud835\udcdd (f a))\n[PROOFSTEP]\napply eventually_of_forall fun x => ?_\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nx : \u03b1\n\u22a2 Tendsto (fun n => truncation f n x) atTop (\ud835\udcdd (f x))\n[PROOFSTEP]\napply tendsto_const_nhds.congr' _\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nx : \u03b1\n\u22a2 (fun x_1 => f x) =\u1da0[atTop] fun n => truncation f n x\n[PROOFSTEP]\nfilter_upwards [Ioi_mem_atTop (abs (f x))] with A hA\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\nhf : Integrable f\nx : \u03b1\nA : \u211d\nhA : A \u2208 Set.Ioi |f x|\n\u22a2 f x = truncation f A x\n[PROOFSTEP]\nexact (truncation_eq_self hA).symm\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)\n[PROOFSTEP]\nlet \u03c1 : Measure \u211d := Measure.map X \u2119\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\n\u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)\n[PROOFSTEP]\nhaveI : IsProbabilityMeasure \u03c1 := isProbabilityMeasure_map hint.aemeasurable\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)\n[PROOFSTEP]\nhave A : \u2211 j in range K, \u222b _ in j..N, (1 : \u211d) \u2202\u03c1 \u2264 \ud835\udd3c[X] + 1 :=\n calc\n \u2211 j in range K, \u222b _ in j..N, (1 : \u211d) \u2202\u03c1 = \u2211 j in range K, \u2211 i in Ico j N, \u222b _ in i..(i + 1 : \u2115), (1 : \u211d) \u2202\u03c1 :=\n by\n apply sum_congr rfl fun j hj => ?_\n rw [intervalIntegral.sum_integral_adjacent_intervals_Ico ((mem_range.1 hj).le.trans hKN)]\n intro k _\n exact continuous_const.intervalIntegrable _ _\n _ = \u2211 i in range N, \u2211 j in range (min (i + 1) K), \u222b _ in i..(i + 1 : \u2115), (1 : \u211d) \u2202\u03c1 :=\n by\n simp_rw [sum_sigma']\n refine'\n sum_bij' (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ (fun a _ => rfl)\n (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ _ _\n \u00b7 rintro \u27e8i, j\u27e9 hij\n simp only [mem_sigma, mem_range, mem_Ico] at hij \n simp only [hij, Nat.lt_succ_iff.2 hij.2.1, mem_sigma, mem_range, lt_min_iff, and_self_iff]\n \u00b7 rintro \u27e8i, j\u27e9 hij\n simp only [mem_sigma, mem_range, lt_min_iff] at hij \n simp only [hij, Nat.lt_succ_iff.1 hij.2.1, mem_sigma, mem_range, mem_Ico, and_self_iff]\n \u00b7 rintro \u27e8i, j\u27e9 hij; rfl\n \u00b7 rintro \u27e8i, j\u27e9 hij; rfl\n _ \u2264 \u2211 i in range N, (i + 1) * \u222b _ in i..(i + 1 : \u2115), (1 : \u211d) \u2202\u03c1 :=\n by\n apply sum_le_sum fun i _ => ?_\n simp only [Nat.cast_add, Nat.cast_one, sum_const, card_range, nsmul_eq_mul, Nat.cast_min]\n refine' mul_le_mul_of_nonneg_right (min_le_left _ _) _\n apply intervalIntegral.integral_nonneg\n \u00b7 simp only [le_add_iff_nonneg_right, zero_le_one]\n \u00b7 simp only [zero_le_one, imp_true_iff]\n _ \u2264 \u2211 i in range N, \u222b x in i..(i + 1 : \u2115), x + 1 \u2202\u03c1 :=\n by\n apply sum_le_sum fun i _ => ?_\n have I : (i : \u211d) \u2264 (i + 1 : \u2115) := by simp only [Nat.cast_add, Nat.cast_one, le_add_iff_nonneg_right, zero_le_one]\n simp_rw [intervalIntegral.integral_of_le I, \u2190 integral_mul_left]\n apply set_integral_mono_on\n \u00b7 exact continuous_const.integrableOn_Ioc\n \u00b7 exact (continuous_id.add continuous_const).integrableOn_Ioc\n \u00b7 exact measurableSet_Ioc\n \u00b7 intro x hx\n simp only [Nat.cast_add, Nat.cast_one, Set.mem_Ioc] at hx \n simp [hx.1.le]\n _ = \u222b x in (0)..N, x + 1 \u2202\u03c1 :=\n by\n rw [intervalIntegral.sum_integral_adjacent_intervals fun k _ => ?_]\n \u00b7 norm_cast\n \u00b7 exact (continuous_id.add continuous_const).intervalIntegrable _ _\n _ = \u222b x in (0)..N, x \u2202\u03c1 + \u222b x in (0)..N, 1 \u2202\u03c1 :=\n by\n rw [intervalIntegral.integral_add]\n \u00b7 exact continuous_id.intervalIntegrable _ _\n \u00b7 exact continuous_const.intervalIntegrable _ _\n _ = \ud835\udd3c[truncation X N] + \u222b x in (0)..N, 1 \u2202\u03c1 := by\n rw [integral_truncation_eq_intervalIntegral_of_nonneg hint.1 hnonneg]\n _ \u2264 \ud835\udd3c[X] + \u222b x in (0)..N, 1 \u2202\u03c1 := (add_le_add_right (integral_truncation_le_integral_of_nonneg hint hnonneg) _)\n _ \u2264 \ud835\udd3c[X] + 1 := by\n refine' add_le_add le_rfl _\n rw [intervalIntegral.integral_of_le (Nat.cast_nonneg _)]\n simp only [integral_const, Measure.restrict_apply', measurableSet_Ioc, Set.univ_inter, Algebra.id.smul_eq_mul,\n mul_one]\n rw [\u2190 ENNReal.one_toReal]\n exact ENNReal.toReal_mono ENNReal.one_ne_top prob_le_one\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 = \u2211 j in range K, \u2211 i in Ico j N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1\n[PROOFSTEP]\napply sum_congr rfl fun j hj => ?_\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nj : \u2115\nhj : j \u2208 range K\n\u22a2 \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 = \u2211 i in Ico j N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1\n[PROOFSTEP]\nrw [intervalIntegral.sum_integral_adjacent_intervals_Ico ((mem_range.1 hj).le.trans hKN)]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nj : \u2115\nhj : j \u2208 range K\n\u22a2 \u2200 (k : \u2115), k \u2208 Set.Ico j N \u2192 IntervalIntegrable (fun x => 1) \u03c1 \u2191k \u2191(k + 1)\n[PROOFSTEP]\nintro k _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nj : \u2115\nhj : j \u2208 range K\nk : \u2115\na\u271d : k \u2208 Set.Ico j N\n\u22a2 IntervalIntegrable (fun x => 1) \u03c1 \u2191k \u2191(k + 1)\n[PROOFSTEP]\nexact continuous_const.intervalIntegrable _ _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2211 j in range K, \u2211 i in Ico j N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 =\n \u2211 i in range N, \u2211 j in range (min (i + 1) K), \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1\n[PROOFSTEP]\nsimp_rw [sum_sigma']\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2211 x in Finset.sigma (range K) fun a => Ico a N, \u222b (x : \u211d) in \u2191x.snd..\u2191(x.snd + 1), 1 \u2202Measure.map X \u2119 =\n \u2211 x in Finset.sigma (range N) fun a => range (min (a + 1) K), \u222b (x : \u211d) in \u2191x.fst..\u2191(x.fst + 1), 1 \u2202Measure.map X \u2119\n[PROOFSTEP]\nrefine'\n sum_bij' (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ (fun a _ => rfl)\n (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ _ _\n[GOAL]\ncase refine'_1\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ico a N),\n (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K)\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hij\n[GOAL]\ncase refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ico a N\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n Finset.sigma (range N) fun a => range (min (a + 1) K)\n[PROOFSTEP]\nsimp only [mem_sigma, mem_range, mem_Ico] at hij \n[GOAL]\ncase refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ico a N\nhij : i < K \u2227 i \u2264 j \u2227 j < N\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208\n Finset.sigma (range N) fun a => range (min (a + 1) K)\n[PROOFSTEP]\nsimp only [hij, Nat.lt_succ_iff.2 hij.2.1, mem_sigma, mem_range, lt_min_iff, and_self_iff]\n[GOAL]\ncase refine'_2\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K)),\n (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => Ico a N\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hij\n[GOAL]\ncase refine'_2.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K)\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208 Finset.sigma (range K) fun a => Ico a N\n[PROOFSTEP]\nsimp only [mem_sigma, mem_range, lt_min_iff] at hij \n[GOAL]\ncase refine'_2.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K)\nhij : i < N \u2227 j < i + 1 \u2227 j < K\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208 Finset.sigma (range K) fun a => Ico a N\n[PROOFSTEP]\nsimp only [hij, Nat.lt_succ_iff.1 hij.2.1, mem_sigma, mem_range, mem_Ico, and_self_iff]\n[GOAL]\ncase refine'_3\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ico a N),\n (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)\n (_ : (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K)) =\n a\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hij\n[GOAL]\ncase refine'_3.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ico a N\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)\n (_ :\n (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n Finset.sigma (range N) fun a => range (min (a + 1) K)) =\n { fst := i, snd := j }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_4\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K)),\n (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)\n (_ : (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => Ico a N) =\n a\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hij\n[GOAL]\ncase refine'_4.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range (min (a + 1) K)\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)\n (_ :\n (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n Finset.sigma (range K) fun a => Ico a N) =\n { fst := i, snd := j }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2211 i in range N, \u2211 j in range (min (i + 1) K), \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264\n \u2211 i in range N, (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1\n[PROOFSTEP]\napply sum_le_sum fun i _ => ?_\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u2211 j in range (min (i + 1) K), \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1\n[PROOFSTEP]\nsimp only [Nat.cast_add, Nat.cast_one, sum_const, card_range, nsmul_eq_mul, Nat.cast_min]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 min (\u2191i + 1) \u2191K * \u222b (x : \u211d) in \u2191i..\u2191i + 1, 1 \u2202Measure.map X \u2119 \u2264 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191i + 1, 1 \u2202Measure.map X \u2119\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_right (min_le_left _ _) _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 0 \u2264 \u222b (x : \u211d) in \u2191i..\u2191i + 1, 1 \u2202Measure.map X \u2119\n[PROOFSTEP]\napply intervalIntegral.integral_nonneg\n[GOAL]\ncase hab\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u2191i \u2264 \u2191i + 1\n[PROOFSTEP]\nsimp only [le_add_iff_nonneg_right, zero_le_one]\n[GOAL]\ncase hf\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u2200 (u : \u211d), u \u2208 Set.Icc (\u2191i) (\u2191i + 1) \u2192 0 \u2264 1\n[PROOFSTEP]\nsimp only [zero_le_one, imp_true_iff]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2211 i in range N, (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264 \u2211 i in range N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), x + 1 \u2202\u03c1\n[PROOFSTEP]\napply sum_le_sum fun i _ => ?_\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191i..\u2191(i + 1), x + 1 \u2202\u03c1\n[PROOFSTEP]\nhave I : (i : \u211d) \u2264 (i + 1 : \u2115) := by simp only [Nat.cast_add, Nat.cast_one, le_add_iff_nonneg_right, zero_le_one]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u2191i \u2264 \u2191(i + 1)\n[PROOFSTEP]\nsimp only [Nat.cast_add, Nat.cast_one, le_add_iff_nonneg_right, zero_le_one]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 (\u2191i + 1) * \u222b (x : \u211d) in \u2191i..\u2191(i + 1), 1 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191i..\u2191(i + 1), x + 1 \u2202\u03c1\n[PROOFSTEP]\nsimp_rw [intervalIntegral.integral_of_le I, \u2190 integral_mul_left]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 \u222b (a : \u211d) in Set.Ioc \u2191i \u2191(i + 1), (\u2191i + 1) * 1 \u2202Measure.map X \u2119 \u2264\n \u222b (x : \u211d) in Set.Ioc \u2191i \u2191(i + 1), x + 1 \u2202Measure.map X \u2119\n[PROOFSTEP]\napply set_integral_mono_on\n[GOAL]\ncase hf\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 IntegrableOn (fun a => (\u2191i + 1) * 1) (Set.Ioc \u2191i \u2191(i + 1))\n[PROOFSTEP]\nexact continuous_const.integrableOn_Ioc\n[GOAL]\ncase hg\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 IntegrableOn (fun a => a + 1) (Set.Ioc \u2191i \u2191(i + 1))\n[PROOFSTEP]\nexact (continuous_id.add continuous_const).integrableOn_Ioc\n[GOAL]\ncase hs\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 MeasurableSet (Set.Ioc \u2191i \u2191(i + 1))\n[PROOFSTEP]\nexact measurableSet_Ioc\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\n\u22a2 \u2200 (x : \u211d), x \u2208 Set.Ioc \u2191i \u2191(i + 1) \u2192 (\u2191i + 1) * 1 \u2264 x + 1\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191i \u2191(i + 1)\n\u22a2 (\u2191i + 1) * 1 \u2264 x + 1\n[PROOFSTEP]\nsimp only [Nat.cast_add, Nat.cast_one, Set.mem_Ioc] at hx \n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\ni : \u2115\nx\u271d : i \u2208 range N\nI : \u2191i \u2264 \u2191(i + 1)\nx : \u211d\nhx : \u2191i < x \u2227 x \u2264 \u2191i + 1\n\u22a2 (\u2191i + 1) * 1 \u2264 x + 1\n[PROOFSTEP]\nsimp [hx.1.le]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u2211 i in range N, \u222b (x : \u211d) in \u2191i..\u2191(i + 1), x + 1 \u2202\u03c1 = \u222b (x : \u211d) in 0 ..\u2191N, x + 1 \u2202\u03c1\n[PROOFSTEP]\nrw [intervalIntegral.sum_integral_adjacent_intervals fun k _ => ?_]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u222b (x : \u211d) in \u21910 ..\u2191N, x + 1 \u2202\u03c1 = \u222b (x : \u211d) in 0 ..\u2191N, x + 1 \u2202\u03c1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nk : \u2115\nx\u271d : k < N\n\u22a2 IntervalIntegrable (fun x => x + 1) \u03c1 \u2191k \u2191(k + 1)\n[PROOFSTEP]\nexact (continuous_id.add continuous_const).intervalIntegrable _ _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u222b (x : \u211d) in 0 ..\u2191N, x + 1 \u2202\u03c1 = \u222b (x : \u211d) in 0 ..\u2191N, x \u2202\u03c1 + \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1\n[PROOFSTEP]\nrw [intervalIntegral.integral_add]\n[GOAL]\ncase hf\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 IntervalIntegrable (fun x => x) \u03c1 0 \u2191N\n[PROOFSTEP]\nexact continuous_id.intervalIntegrable _ _\n[GOAL]\ncase hg\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 IntervalIntegrable (fun x => 1) \u03c1 0 \u2191N\n[PROOFSTEP]\nexact continuous_const.intervalIntegrable _ _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u222b (x : \u211d) in 0 ..\u2191N, x \u2202\u03c1 + \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1 = (\u222b (a : \u03a9), truncation X (\u2191N) a) + \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1\n[PROOFSTEP]\nrw [integral_truncation_eq_intervalIntegral_of_nonneg hint.1 hnonneg]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 (\u222b (a : \u03a9), X a) + \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\n[PROOFSTEP]\nrefine' add_le_add le_rfl _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u222b (x : \u211d) in 0 ..\u2191N, 1 \u2202\u03c1 \u2264 1\n[PROOFSTEP]\nrw [intervalIntegral.integral_of_le (Nat.cast_nonneg _)]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 \u222b (x : \u211d) in Set.Ioc 0 \u2191N, 1 \u2202\u03c1 \u2264 1\n[PROOFSTEP]\nsimp only [integral_const, Measure.restrict_apply', measurableSet_Ioc, Set.univ_inter, Algebra.id.smul_eq_mul, mul_one]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 ENNReal.toReal (\u2191\u2191(Measure.map X \u2119) (Set.Ioc 0 \u2191N)) \u2264 1\n[PROOFSTEP]\nrw [\u2190 ENNReal.one_toReal]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\n\u22a2 ENNReal.toReal (\u2191\u2191(Measure.map X \u2119) (Set.Ioc 0 \u2191N)) \u2264 ENNReal.toReal 1\n[PROOFSTEP]\nexact ENNReal.toReal_mono ENNReal.one_ne_top prob_le_one\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\n\u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)\n[PROOFSTEP]\nhave B : \u2200 a b, \u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b _ in Set.Ioc a b, (1 : \u211d) \u2202\u03c1) :=\n by\n intro a b\n rw [ofReal_set_integral_one \u03c1 _, Measure.map_apply_of_aemeasurable hint.aemeasurable measurableSet_Ioc]\n rfl\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\n\u22a2 \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\n[PROOFSTEP]\nintro a b\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\na b : \u211d\n\u22a2 \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\n[PROOFSTEP]\nrw [ofReal_set_integral_one \u03c1 _, Measure.map_apply_of_aemeasurable hint.aemeasurable measurableSet_Ioc]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\na b : \u211d\n\u22a2 \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = \u2191\u2191\u2119 (X \u207b\u00b9' Set.Ioc a b)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\n\u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)\n[PROOFSTEP]\ncalc\n \u2211 j in range K, \u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc (j : \u211d) N} =\n \u2211 j in range K, ENNReal.ofReal (\u222b _ in Set.Ioc (j : \u211d) N, (1 : \u211d) \u2202\u03c1) :=\n by simp_rw [B]\n _ = ENNReal.ofReal (\u2211 j in range K, \u222b _ in Set.Ioc (j : \u211d) N, (1 : \u211d) \u2202\u03c1) :=\n by\n rw [ENNReal.ofReal_sum_of_nonneg]\n simp only [integral_const, Algebra.id.smul_eq_mul, mul_one, ENNReal.toReal_nonneg, imp_true_iff]\n _ = ENNReal.ofReal (\u2211 j in range K, \u222b _ in (j : \u211d)..N, (1 : \u211d) \u2202\u03c1) :=\n by\n congr 1\n refine' sum_congr rfl fun j hj => _\n rw [intervalIntegral.integral_of_le (Nat.cast_le.2 ((mem_range.1 hj).le.trans hKN))]\n _ \u2264 ENNReal.ofReal (\ud835\udd3c[X] + 1) := ENNReal.ofReal_le_ofReal A\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\n\u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N} = \u2211 j in range K, ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1)\n[PROOFSTEP]\nsimp_rw [B]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\n\u22a2 \u2211 j in range K, ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1) =\n ENNReal.ofReal (\u2211 j in range K, \u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1)\n[PROOFSTEP]\nrw [ENNReal.ofReal_sum_of_nonneg]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\n\u22a2 \u2200 (i : \u2115), i \u2208 range K \u2192 0 \u2264 \u222b (x : \u211d) in Set.Ioc \u2191i \u2191N, 1 \u2202\u03c1\n[PROOFSTEP]\nsimp only [integral_const, Algebra.id.smul_eq_mul, mul_one, ENNReal.toReal_nonneg, imp_true_iff]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\n\u22a2 ENNReal.ofReal (\u2211 j in range K, \u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1) =\n ENNReal.ofReal (\u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_r\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\n\u22a2 \u2211 j in range K, \u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1 = \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1\n[PROOFSTEP]\nrefine' sum_congr rfl fun j hj => _\n[GOAL]\ncase e_r\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK N : \u2115\nhKN : K \u2264 N\n\u03c1 : Measure \u211d := Measure.map X \u2119\nthis : IsProbabilityMeasure \u03c1\nA : \u2211 j in range K, \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1 \u2264 (\u222b (a : \u03a9), X a) + 1\nB : \u2200 (a b : \u211d), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc a b} = ENNReal.ofReal (\u222b (x : \u211d) in Set.Ioc a b, 1 \u2202\u03c1)\nj : \u2115\nhj : j \u2208 range K\n\u22a2 \u222b (x : \u211d) in Set.Ioc \u2191j \u2191N, 1 \u2202\u03c1 = \u222b (x : \u211d) in \u2191j..\u2191N, 1 \u2202\u03c1\n[PROOFSTEP]\nrw [intervalIntegral.integral_of_le (Nat.cast_le.2 ((mem_range.1 hj).le.trans hKN))]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\n\u22a2 \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n[PROOFSTEP]\nsuffices : \u2200 K : \u2115, \u2211 j in range K, \u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi (j : \u211d)} \u2264 ENNReal.ofReal (\ud835\udd3c[X] + 1)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nthis : \u2200 (K : \u2115), \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)\n\u22a2 \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\ncase this\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\n\u22a2 \u2200 (K : \u2115), \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)\n[PROOFSTEP]\nexact\n (le_of_tendsto_of_tendsto (ENNReal.tendsto_nat_tsum _) tendsto_const_nhds (eventually_of_forall this)).trans_lt\n ENNReal.ofReal_lt_top\n[GOAL]\ncase this\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\n\u22a2 \u2200 (K : \u2115), \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)\n[PROOFSTEP]\nintro K\n[GOAL]\ncase this\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\n\u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)\n[PROOFSTEP]\nhave A :\n Tendsto (fun N : \u2115 => \u2211 j in range K, \u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc (j : \u211d) N}) atTop\n (\ud835\udcdd (\u2211 j in range K, \u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi (j : \u211d)})) :=\n by\n refine' tendsto_finset_sum _ fun i _ => _\n have : {\u03c9 | X \u03c9 \u2208 Set.Ioi (i : \u211d)} = \u22c3 N : \u2115, {\u03c9 | X \u03c9 \u2208 Set.Ioc (i : \u211d) N} :=\n by\n apply Set.Subset.antisymm _ _\n \u00b7 intro \u03c9 h\u03c9\n obtain \u27e8N, hN\u27e9 : \u2203 N : \u2115, X \u03c9 \u2264 N := exists_nat_ge (X \u03c9)\n exact Set.mem_iUnion.2 \u27e8N, h\u03c9, hN\u27e9\n \u00b7\n simp (config := { contextual := true }) only [Set.mem_Ioc, Set.mem_Ioi, Set.iUnion_subset_iff,\n Set.setOf_subset_setOf, imp_true_iff]\n rw [this]\n apply tendsto_measure_iUnion\n intro m n hmn x hx\n exact \u27e8hx.1, hx.2.trans (Nat.cast_le.2 hmn)\u27e9\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\n\u22a2 Tendsto (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) atTop\n (\ud835\udcdd (\u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j}))\n[PROOFSTEP]\nrefine' tendsto_finset_sum _ fun i _ => _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u22a2 Tendsto (fun N => \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) atTop (\ud835\udcdd (\u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i}))\n[PROOFSTEP]\nhave : {\u03c9 | X \u03c9 \u2208 Set.Ioi (i : \u211d)} = \u22c3 N : \u2115, {\u03c9 | X \u03c9 \u2208 Set.Ioc (i : \u211d) N} :=\n by\n apply Set.Subset.antisymm _ _\n \u00b7 intro \u03c9 h\u03c9\n obtain \u27e8N, hN\u27e9 : \u2203 N : \u2115, X \u03c9 \u2264 N := exists_nat_ge (X \u03c9)\n exact Set.mem_iUnion.2 \u27e8N, h\u03c9, hN\u27e9\n \u00b7\n simp (config := { contextual := true }) only [Set.mem_Ioc, Set.mem_Ioi, Set.iUnion_subset_iff,\n Set.setOf_subset_setOf, imp_true_iff]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u22a2 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 (N : \u2115), {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\n[PROOFSTEP]\napply Set.Subset.antisymm _ _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u22a2 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} \u2286 \u22c3 (N : \u2115), {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\n[PROOFSTEP]\nintro \u03c9 h\u03c9\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u03c9 : \u03a9\nh\u03c9 : \u03c9 \u2208 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i}\n\u22a2 \u03c9 \u2208 \u22c3 (N : \u2115), {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\n[PROOFSTEP]\nobtain \u27e8N, hN\u27e9 : \u2203 N : \u2115, X \u03c9 \u2264 N := exists_nat_ge (X \u03c9)\n[GOAL]\ncase intro\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u03c9 : \u03a9\nh\u03c9 : \u03c9 \u2208 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i}\nN : \u2115\nhN : X \u03c9 \u2264 \u2191N\n\u22a2 \u03c9 \u2208 \u22c3 (N : \u2115), {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\n[PROOFSTEP]\nexact Set.mem_iUnion.2 \u27e8N, h\u03c9, hN\u27e9\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\n\u22a2 \u22c3 (N : \u2115), {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N} \u2286 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i}\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [Set.mem_Ioc, Set.mem_Ioi, Set.iUnion_subset_iff, Set.setOf_subset_setOf,\n imp_true_iff]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\nthis : {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 (N : \u2115), {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\n\u22a2 Tendsto (fun N => \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) atTop (\ud835\udcdd (\u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i}))\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\nthis : {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 (N : \u2115), {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\n\u22a2 Tendsto (fun N => \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) atTop (\ud835\udcdd (\u2191\u2191\u2119 (\u22c3 (N : \u2115), {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N})))\n[PROOFSTEP]\napply tendsto_measure_iUnion\n[GOAL]\ncase hm\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\nthis : {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 (N : \u2115), {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\n\u22a2 Monotone fun N => {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\n[PROOFSTEP]\nintro m n hmn x hx\n[GOAL]\ncase hm\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK i : \u2115\nx\u271d : i \u2208 range K\nthis : {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 (N : \u2115), {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}\nm n : \u2115\nhmn : m \u2264 n\nx : \u03a9\nhx : x \u2208 (fun N => {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) m\n\u22a2 x \u2208 (fun N => {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) n\n[PROOFSTEP]\nexact \u27e8hx.1, hx.2.trans (Nat.cast_le.2 hmn)\u27e9\n[GOAL]\ncase this\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nA :\n Tendsto (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) atTop\n (\ud835\udcdd (\u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j}))\n\u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)\n[PROOFSTEP]\napply le_of_tendsto_of_tendsto A tendsto_const_nhds\n[GOAL]\ncase this\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nA :\n Tendsto (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) atTop\n (\ud835\udcdd (\u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j}))\n\u22a2 (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) \u2264\u1da0[atTop] fun x => ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)\n[PROOFSTEP]\nfilter_upwards [Ici_mem_atTop K] with N hN\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nA :\n Tendsto (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) atTop\n (\ud835\udcdd (\u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j}))\nN : \u2115\nhN : N \u2208 Set.Ici K\n\u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1)\n[PROOFSTEP]\nexact sum_prob_mem_Ioc_le hint hnonneg hN\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation X \u2191j ^ 2) a \u2264 2 * \u222b (a : \u03a9), X a\n[PROOFSTEP]\nset Y := fun n : \u2115 => truncation X n\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation X \u2191j ^ 2) a \u2264 2 * \u222b (a : \u03a9), X a\n[PROOFSTEP]\nlet \u03c1 : Measure \u211d := Measure.map X \u2119\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation X \u2191j ^ 2) a \u2264 2 * \u222b (a : \u03a9), X a\n[PROOFSTEP]\nhave Y2 : \u2200 n, \ud835\udd3c[Y n ^ 2] = \u222b x in (0)..n, x ^ 2 \u2202\u03c1 := by\n intro n\n change \ud835\udd3c[fun x => Y n x ^ 2] = _\n rw [moment_truncation_eq_intervalIntegral_of_nonneg hint.1 two_ne_zero hnonneg]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\n\u22a2 \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nn : \u2115\n\u22a2 \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n[PROOFSTEP]\nchange \ud835\udd3c[fun x => Y n x ^ 2] = _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nn : \u2115\n\u22a2 \u222b (a : \u03a9), (fun x => Y n x ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n[PROOFSTEP]\nrw [moment_truncation_eq_intervalIntegral_of_nonneg hint.1 two_ne_zero hnonneg]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation X \u2191j ^ 2) a \u2264 2 * \u222b (a : \u03a9), X a\n[PROOFSTEP]\ncalc\n \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * \ud835\udd3c[Y j ^ 2] = \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * \u222b x in (0)..j, x ^ 2 \u2202\u03c1 := by\n simp_rw [Y2]\n _ = \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * \u2211 k in range j, \u222b x in k..(k + 1 : \u2115), x ^ 2 \u2202\u03c1 :=\n by\n congr 1 with j\n congr 1\n rw [intervalIntegral.sum_integral_adjacent_intervals]\n \u00b7 norm_cast\n intro k _\n exact (continuous_id.pow _).intervalIntegrable _ _\n _ = \u2211 k in range K, (\u2211 j in Ioo k K, ((j : \u211d) ^ 2)\u207b\u00b9) * \u222b x in k..(k + 1 : \u2115), x ^ 2 \u2202\u03c1 :=\n by\n simp_rw [mul_sum, sum_mul, sum_sigma']\n refine'\n sum_bij' (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ (fun a _ => rfl)\n (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ _ _\n \u00b7 rintro \u27e8i, j\u27e9 hij\n simp only [mem_sigma, mem_range, mem_filter] at hij \n simp [hij, mem_sigma, mem_range, and_self_iff, hij.2.trans hij.1]\n \u00b7 rintro \u27e8i, j\u27e9 hij\n simp only [mem_sigma, mem_range, mem_Ioo] at hij \n simp only [hij, mem_sigma, mem_range, and_self_iff]\n \u00b7 rintro \u27e8i, j\u27e9 hij; rfl\n \u00b7 rintro \u27e8i, j\u27e9 hij; rfl\n _ \u2264 \u2211 k in range K, \u21912 / (k + \u21911) * \u222b x in k..(k + 1 : \u2115), x ^ 2 \u2202\u03c1 :=\n by\n apply sum_le_sum fun k _ => ?_\n refine' mul_le_mul_of_nonneg_right (sum_Ioo_inv_sq_le _ _) _\n refine' intervalIntegral.integral_nonneg_of_forall _ fun u => sq_nonneg _\n simp only [Nat.cast_add, Nat.cast_one, le_add_iff_nonneg_right, zero_le_one]\n _ \u2264 \u2211 k in range K, \u222b x in k..(k + 1 : \u2115), 2 * x \u2202\u03c1 :=\n by\n apply sum_le_sum fun k _ => ?_\n have Ik : (k : \u211d) \u2264 (k + 1 : \u2115) := by simp\n rw [\u2190 intervalIntegral.integral_const_mul, intervalIntegral.integral_of_le Ik, intervalIntegral.integral_of_le Ik]\n refine' set_integral_mono_on _ _ measurableSet_Ioc fun x hx => _\n \u00b7 apply Continuous.integrableOn_Ioc\n exact continuous_const.mul (continuous_pow 2)\n \u00b7 apply Continuous.integrableOn_Ioc\n exact continuous_const.mul continuous_id'\n \u00b7\n calc\n \u21912 / (\u2191k + \u21911) * x ^ 2 = x / (k + 1) * (2 * x) := by ring\n _ \u2264 1 * (2 * x) :=\n (mul_le_mul_of_nonneg_right\n (by\n convert (div_le_one _).2 hx.2; norm_cast\n simp only [Nat.cast_add, Nat.cast_one]\n linarith only [show (0 : \u211d) \u2264 k from Nat.cast_nonneg k])\n (mul_nonneg zero_le_two ((Nat.cast_nonneg k).trans hx.1.le)))\n _ = 2 * x := by rw [one_mul]\n _ = 2 * \u222b x in (0 : \u211d)..K, x \u2202\u03c1 :=\n by\n rw [intervalIntegral.sum_integral_adjacent_intervals fun k _ => ?_]\n swap; \u00b7 exact (continuous_const.mul continuous_id').intervalIntegrable _ _\n rw [intervalIntegral.integral_const_mul]\n norm_cast\n _ \u2264 2 * \ud835\udd3c[X] :=\n mul_le_mul_of_nonneg_left\n (by\n rw [\u2190 integral_truncation_eq_intervalIntegral_of_nonneg hint.1 hnonneg]\n exact integral_truncation_le_integral_of_nonneg hint hnonneg)\n zero_le_two\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (Y j ^ 2) a = \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1\n[PROOFSTEP]\nsimp_rw [Y2]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 =\n \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u2211 k in range j, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1\n[PROOFSTEP]\ncongr 1 with j\n[GOAL]\ncase e_f.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nj : \u2115\n\u22a2 (\u2191j ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 = (\u2191j ^ 2)\u207b\u00b9 * \u2211 k in range j, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_f.h.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nj : \u2115\n\u22a2 \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 = \u2211 k in range j, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1\n[PROOFSTEP]\nrw [intervalIntegral.sum_integral_adjacent_intervals]\n[GOAL]\ncase e_f.h.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nj : \u2115\n\u22a2 \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 = \u222b (x : \u211d) in \u21910 ..\u2191j, x ^ 2 \u2202\u03c1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase e_f.h.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nj : \u2115\n\u22a2 \u2200 (k : \u2115), k < j \u2192 IntervalIntegrable (fun x => x ^ 2) \u03c1 \u2191k \u2191(k + 1)\n[PROOFSTEP]\nintro k _\n[GOAL]\ncase e_f.h.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nj k : \u2115\na\u271d : k < j\n\u22a2 IntervalIntegrable (fun x => x ^ 2) \u03c1 \u2191k \u2191(k + 1)\n[PROOFSTEP]\nexact (continuous_id.pow _).intervalIntegrable _ _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u2211 k in range j, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 =\n \u2211 k in range K, (\u2211 j in Ioo k K, (\u2191j ^ 2)\u207b\u00b9) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1\n[PROOFSTEP]\nsimp_rw [mul_sum, sum_mul, sum_sigma']\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 x in Finset.sigma (range K) fun a => range a,\n (\u2191x.fst ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in \u2191x.snd..\u2191(x.snd + 1), x ^ 2 \u2202Measure.map X \u2119 =\n \u2211 x in Finset.sigma (range K) fun a => Ioo a K,\n (\u2191x.snd ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in \u2191x.fst..\u2191(x.fst + 1), x ^ 2 \u2202Measure.map X \u2119\n[PROOFSTEP]\nrefine'\n sum_bij' (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ (fun a _ => rfl)\n (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ _ _\n[GOAL]\ncase refine'_1\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => range a),\n (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => Ioo a K\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hij\n[GOAL]\ncase refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => range a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208 Finset.sigma (range K) fun a => Ioo a K\n[PROOFSTEP]\nsimp only [mem_sigma, mem_range, mem_filter] at hij \n[GOAL]\ncase refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => range a\nhij : i < K \u2227 j < i\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208 Finset.sigma (range K) fun a => Ioo a K\n[PROOFSTEP]\nsimp [hij, mem_sigma, mem_range, and_self_iff, hij.2.trans hij.1]\n[GOAL]\ncase refine'_2\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ioo a K),\n (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => range a\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hij\n[GOAL]\ncase refine'_2.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ioo a K\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208 Finset.sigma (range K) fun a => range a\n[PROOFSTEP]\nsimp only [mem_sigma, mem_range, mem_Ioo] at hij \n[GOAL]\ncase refine'_2.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ioo a K\nhij : i < K \u2227 i < j \u2227 j < K\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208 Finset.sigma (range K) fun a => range a\n[PROOFSTEP]\nsimp only [hij, mem_sigma, mem_range, and_self_iff]\n[GOAL]\ncase refine'_3\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => range a),\n (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)\n (_ : (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => Ioo a K) =\n a\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hij\n[GOAL]\ncase refine'_3.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => range a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)\n (_ :\n (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n Finset.sigma (range K) fun a => Ioo a K) =\n { fst := i, snd := j }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_4\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ioo a K),\n (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)\n (_ : (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => range a) =\n a\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hij\n[GOAL]\ncase refine'_4.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\ni j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ioo a K\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)\n (_ :\n (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n Finset.sigma (range K) fun a => range a) =\n { fst := i, snd := j }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 k in range K, (\u2211 j in Ioo k K, (\u2191j ^ 2)\u207b\u00b9) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264\n \u2211 k in range K, 2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1\n[PROOFSTEP]\napply sum_le_sum fun k _ => ?_\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\n\u22a2 (\u2211 j in Ioo k K, (\u2191j ^ 2)\u207b\u00b9) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264\n 2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_right (sum_Ioo_inv_sq_le _ _) _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\n\u22a2 0 \u2264 \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1\n[PROOFSTEP]\nrefine' intervalIntegral.integral_nonneg_of_forall _ fun u => sq_nonneg _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\n\u22a2 \u2191k \u2264 \u2191(k + 1)\n[PROOFSTEP]\nsimp only [Nat.cast_add, Nat.cast_one, le_add_iff_nonneg_right, zero_le_one]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 k in range K, 2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264\n \u2211 k in range K, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), 2 * x \u2202\u03c1\n[PROOFSTEP]\napply sum_le_sum fun k _ => ?_\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\n\u22a2 2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191k..\u2191(k + 1), 2 * x \u2202\u03c1\n[PROOFSTEP]\nhave Ik : (k : \u211d) \u2264 (k + 1 : \u2115) := by simp\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\n\u22a2 \u2191k \u2264 \u2191(k + 1)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191k..\u2191(k + 1), 2 * x \u2202\u03c1\n[PROOFSTEP]\nrw [\u2190 intervalIntegral.integral_const_mul, intervalIntegral.integral_of_le Ik, intervalIntegral.integral_of_le Ik]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 \u222b (x : \u211d) in Set.Ioc \u2191k \u2191(k + 1), 2 / (\u2191k + 1) * x ^ 2 \u2202\u03c1 \u2264 \u222b (x : \u211d) in Set.Ioc \u2191k \u2191(k + 1), 2 * x \u2202\u03c1\n[PROOFSTEP]\nrefine' set_integral_mono_on _ _ measurableSet_Ioc fun x hx => _\n[GOAL]\ncase refine'_1\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 IntegrableOn (fun x => 2 / (\u2191k + 1) * x ^ 2) (Set.Ioc \u2191k \u2191(k + 1))\n[PROOFSTEP]\napply Continuous.integrableOn_Ioc\n[GOAL]\ncase refine'_1.hf\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 Continuous fun x => 2 / (\u2191k + 1) * x ^ 2\n[PROOFSTEP]\nexact continuous_const.mul (continuous_pow 2)\n[GOAL]\ncase refine'_2\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 IntegrableOn (fun x => 2 * x) (Set.Ioc \u2191k \u2191(k + 1))\n[PROOFSTEP]\napply Continuous.integrableOn_Ioc\n[GOAL]\ncase refine'_2.hf\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\n\u22a2 Continuous fun x => 2 * x\n[PROOFSTEP]\nexact continuous_const.mul continuous_id'\n[GOAL]\ncase refine'_3\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 2 / (\u2191k + 1) * x ^ 2 \u2264 2 * x\n[PROOFSTEP]\ncalc\n \u21912 / (\u2191k + \u21911) * x ^ 2 = x / (k + 1) * (2 * x) := by ring\n _ \u2264 1 * (2 * x) :=\n (mul_le_mul_of_nonneg_right\n (by\n convert (div_le_one _).2 hx.2; norm_cast\n simp only [Nat.cast_add, Nat.cast_one]\n linarith only [show (0 : \u211d) \u2264 k from Nat.cast_nonneg k])\n (mul_nonneg zero_le_two ((Nat.cast_nonneg k).trans hx.1.le)))\n _ = 2 * x := by rw [one_mul]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 2 / (\u2191k + 1) * x ^ 2 = x / (\u2191k + 1) * (2 * x)\n[PROOFSTEP]\nring\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 x / (\u2191k + 1) \u2264 1\n[PROOFSTEP]\nconvert (div_le_one _).2 hx.2\n[GOAL]\ncase h.e'_3.h.e'_6\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 \u2191k + 1 = \u2191(k + 1)\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 0 < \u2191(k + 1)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 0 < \u2191(k + 1)\n[PROOFSTEP]\nsimp only [Nat.cast_add, Nat.cast_one]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 0 < \u2191k + 1\n[PROOFSTEP]\nlinarith only [show (0 : \u211d) \u2264 k from Nat.cast_nonneg k]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k \u2208 range K\nIk : \u2191k \u2264 \u2191(k + 1)\nx : \u211d\nhx : x \u2208 Set.Ioc \u2191k \u2191(k + 1)\n\u22a2 1 * (2 * x) = 2 * x\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u2211 k in range K, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), 2 * x \u2202\u03c1 = 2 * \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1\n[PROOFSTEP]\nrw [intervalIntegral.sum_integral_adjacent_intervals fun k _ => ?_]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u222b (x : \u211d) in \u21910 ..\u2191K, 2 * x \u2202\u03c1 = 2 * \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k < K\n\u22a2 IntervalIntegrable (fun x => 2 * x) \u03c1 \u2191k \u2191(k + 1)\n[PROOFSTEP]\nswap\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\nk : \u2115\nx\u271d : k < K\n\u22a2 IntervalIntegrable (fun x => 2 * x) \u03c1 \u2191k \u2191(k + 1)\n[PROOFSTEP]\nexact (continuous_const.mul continuous_id').intervalIntegrable _ _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u222b (x : \u211d) in \u21910 ..\u2191K, 2 * x \u2202\u03c1 = 2 * \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1\n[PROOFSTEP]\nrw [intervalIntegral.integral_const_mul]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 2 * \u222b (x : \u211d) in \u21910 ..\u2191K, x \u2202\u03c1 = 2 * \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1 \u2264 \u222b (a : \u03a9), X a\n[PROOFSTEP]\nrw [\u2190 integral_truncation_eq_intervalIntegral_of_nonneg hint.1 hnonneg]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u03a9 \u2192 \u211d\nhint : Integrable X\nhnonneg : 0 \u2264 X\nK : \u2115\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n\n\u03c1 : Measure \u211d := Measure.map X \u2119\nY2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1\n\u22a2 \u222b (x : \u03a9), truncation X (\u2191K) x \u2264 \u222b (a : \u03a9), X a\n[PROOFSTEP]\nexact integral_truncation_le_integral_of_nonneg hint hnonneg\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nhave c_pos : 0 < c := zero_lt_one.trans c_one\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nhave hX : \u2200 i, AEStronglyMeasurable (X i) \u2119 := fun i => (hident i).symm.aestronglyMeasurable_snd hint.1\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nhave A : \u2200 i, StronglyMeasurable (indicator (Set.Ioc (-i : \u211d) i) id) := fun i =>\n stronglyMeasurable_id.indicator measurableSet_Ioc\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nset Y := fun n : \u2115 => truncation (X n) n\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nset S := fun n => \u2211 i in range n, Y i with hS\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nlet u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nhave u_mono : Monotone u := fun i j hij => Nat.floor_mono (pow_le_pow c_one.le hij)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nhave I1 : \u2200 K, \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * Var[Y j] \u2264 2 * \ud835\udd3c[X 0] :=\n by\n intro K\n calc\n \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * Var[Y j] \u2264 \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * \ud835\udd3c[truncation (X 0) j ^ 2] :=\n by\n apply sum_le_sum fun j _ => ?_\n refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (sq_nonneg _))\n rw [(hident j).truncation.variance_eq]\n exact variance_le_expectation_sq (hX 0).truncation\n _ \u2264 2 * \ud835\udd3c[X 0] := sum_variance_truncation_le hint (hnonneg 0) K\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\n\u22a2 \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\n[PROOFSTEP]\nintro K\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nK : \u2115\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\n[PROOFSTEP]\ncalc\n \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * Var[Y j] \u2264 \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * \ud835\udd3c[truncation (X 0) j ^ 2] :=\n by\n apply sum_le_sum fun j _ => ?_\n refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (sq_nonneg _))\n rw [(hident j).truncation.variance_eq]\n exact variance_le_expectation_sq (hX 0).truncation\n _ \u2264 2 * \ud835\udd3c[X 0] := sum_variance_truncation_le hint (hnonneg 0) K\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nK : \u2115\n\u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation (X 0) \u2191j ^ 2) a\n[PROOFSTEP]\napply sum_le_sum fun j _ => ?_\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nK j : \u2115\nx\u271d : j \u2208 range K\n\u22a2 (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation (X 0) \u2191j ^ 2) a\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (sq_nonneg _))\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nK j : \u2115\nx\u271d : j \u2208 range K\n\u22a2 variance (Y j) \u2119 \u2264 \u222b (a : \u03a9), (truncation (X 0) \u2191j ^ 2) a\n[PROOFSTEP]\nrw [(hident j).truncation.variance_eq]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nK j : \u2115\nx\u271d : j \u2208 range K\n\u22a2 variance (truncation (X 0) \u2191j) \u2119 \u2264 \u222b (a : \u03a9), (truncation (X 0) \u2191j ^ 2) a\n[PROOFSTEP]\nexact variance_le_expectation_sq (hX 0).truncation\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nlet C := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \ud835\udd3c[X 0])\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nhave I2 : \u2200 N, \u2211 i in range N, ((u i : \u211d) ^ 2)\u207b\u00b9 * Var[S (u i)] \u2264 C :=\n by\n intro N\n calc\n \u2211 i in range N, ((u i : \u211d) ^ 2)\u207b\u00b9 * Var[S (u i)] =\n \u2211 i in range N, ((u i : \u211d) ^ 2)\u207b\u00b9 * \u2211 j in range (u i), Var[Y j] :=\n by\n congr 1 with i\n congr 1\n rw [hS, IndepFun.variance_sum]\n \u00b7 intro j _\n exact (hident j).aestronglyMeasurable_fst.mem\u2112p_truncation\n \u00b7 intro k _ l _ hkl\n exact (hindep hkl).comp (A k).measurable (A l).measurable\n _ = \u2211 j in range (u (N - 1)), (\u2211 i in (range N).filter fun i => j < u i, ((u i : \u211d) ^ 2)\u207b\u00b9) * Var[Y j] :=\n by\n simp_rw [mul_sum, sum_mul, sum_sigma']\n refine'\n sum_bij' (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ (fun a _ => rfl)\n (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ _ _\n \u00b7 rintro \u27e8i, j\u27e9 hij\n simp only [mem_sigma, mem_range] at hij \n simp only [hij.1, hij.2, mem_sigma, mem_range, mem_filter, and_true_iff]\n exact hij.2.trans_le (u_mono (Nat.le_pred_of_lt hij.1))\n \u00b7 rintro \u27e8i, j\u27e9 hij\n simp only [mem_sigma, mem_range, mem_filter] at hij \n simp only [hij.2.1, hij.2.2, mem_sigma, mem_range, and_self_iff]\n \u00b7 rintro \u27e8i, j\u27e9 hij; rfl\n \u00b7 rintro \u27e8i, j\u27e9 hij; rfl\n _ \u2264 \u2211 j in range (u (N - 1)), c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * Var[Y j] :=\n by\n apply sum_le_sum fun j hj => ?_\n rcases@eq_zero_or_pos _ _ j with (rfl | hj)\n \u00b7 simp only [Nat.cast_zero, zero_pow', Ne.def, bit0_eq_zero, Nat.one_ne_zero, not_false_iff, div_zero, zero_mul]\n simp only [Nat.cast_zero, truncation_zero, variance_zero, mul_zero, le_rfl]\n apply mul_le_mul_of_nonneg_right _ (variance_nonneg _ _)\n convert sum_div_nat_floor_pow_sq_le_div_sq N (Nat.cast_pos.2 hj) c_one using 2\n \u00b7 simp only [Nat.cast_lt]\n \u00b7 simp only [one_div]\n _ = c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * \u2211 j in range (u (N - 1)), ((j : \u211d) ^ 2)\u207b\u00b9 * Var[Y j] := by\n simp_rw [mul_sum, div_eq_mul_inv, mul_assoc]\n _ \u2264 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \ud835\udd3c[X 0]) :=\n by\n apply mul_le_mul_of_nonneg_left (I1 _)\n apply mul_nonneg (pow_nonneg c_pos.le _)\n exact pow_nonneg (inv_nonneg.2 (sub_nonneg.2 c_one.le)) _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\n\u22a2 \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\n[PROOFSTEP]\nintro N\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\n[PROOFSTEP]\ncalc\n \u2211 i in range N, ((u i : \u211d) ^ 2)\u207b\u00b9 * Var[S (u i)] = \u2211 i in range N, ((u i : \u211d) ^ 2)\u207b\u00b9 * \u2211 j in range (u i), Var[Y j] :=\n by\n congr 1 with i\n congr 1\n rw [hS, IndepFun.variance_sum]\n \u00b7 intro j _\n exact (hident j).aestronglyMeasurable_fst.mem\u2112p_truncation\n \u00b7 intro k _ l _ hkl\n exact (hindep hkl).comp (A k).measurable (A l).measurable\n _ = \u2211 j in range (u (N - 1)), (\u2211 i in (range N).filter fun i => j < u i, ((u i : \u211d) ^ 2)\u207b\u00b9) * Var[Y j] :=\n by\n simp_rw [mul_sum, sum_mul, sum_sigma']\n refine'\n sum_bij' (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ (fun a _ => rfl)\n (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ _ _\n \u00b7 rintro \u27e8i, j\u27e9 hij\n simp only [mem_sigma, mem_range] at hij \n simp only [hij.1, hij.2, mem_sigma, mem_range, mem_filter, and_true_iff]\n exact hij.2.trans_le (u_mono (Nat.le_pred_of_lt hij.1))\n \u00b7 rintro \u27e8i, j\u27e9 hij\n simp only [mem_sigma, mem_range, mem_filter] at hij \n simp only [hij.2.1, hij.2.2, mem_sigma, mem_range, and_self_iff]\n \u00b7 rintro \u27e8i, j\u27e9 hij; rfl\n \u00b7 rintro \u27e8i, j\u27e9 hij; rfl\n _ \u2264 \u2211 j in range (u (N - 1)), c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * Var[Y j] :=\n by\n apply sum_le_sum fun j hj => ?_\n rcases@eq_zero_or_pos _ _ j with (rfl | hj)\n \u00b7 simp only [Nat.cast_zero, zero_pow', Ne.def, bit0_eq_zero, Nat.one_ne_zero, not_false_iff, div_zero, zero_mul]\n simp only [Nat.cast_zero, truncation_zero, variance_zero, mul_zero, le_rfl]\n apply mul_le_mul_of_nonneg_right _ (variance_nonneg _ _)\n convert sum_div_nat_floor_pow_sq_le_div_sq N (Nat.cast_pos.2 hj) c_one using 2\n \u00b7 simp only [Nat.cast_lt]\n \u00b7 simp only [one_div]\n _ = c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * \u2211 j in range (u (N - 1)), ((j : \u211d) ^ 2)\u207b\u00b9 * Var[Y j] := by\n simp_rw [mul_sum, div_eq_mul_inv, mul_assoc]\n _ \u2264 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \ud835\udd3c[X 0]) :=\n by\n apply mul_le_mul_of_nonneg_left (I1 _)\n apply mul_nonneg (pow_nonneg c_pos.le _)\n exact pow_nonneg (inv_nonneg.2 (sub_nonneg.2 c_one.le)) _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 =\n \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * \u2211 j in range (u i), variance (Y j) \u2119\n[PROOFSTEP]\ncongr 1 with i\n[GOAL]\ncase e_f.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i : \u2115\n\u22a2 (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 = (\u2191(u i) ^ 2)\u207b\u00b9 * \u2211 j in range (u i), variance (Y j) \u2119\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_f.h.e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i : \u2115\n\u22a2 variance (S (u i)) \u2119 = \u2211 j in range (u i), variance (Y j) \u2119\n[PROOFSTEP]\nrw [hS, IndepFun.variance_sum]\n[GOAL]\ncase e_f.h.e_a.hs\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i : \u2115\n\u22a2 \u2200 (i_1 : \u2115), i_1 \u2208 range (u i) \u2192 Mem\u2112p (Y i_1) 2\n[PROOFSTEP]\nintro j _\n[GOAL]\ncase e_f.h.e_a.hs\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\na\u271d : j \u2208 range (u i)\n\u22a2 Mem\u2112p (Y j) 2\n[PROOFSTEP]\nexact (hident j).aestronglyMeasurable_fst.mem\u2112p_truncation\n[GOAL]\ncase e_f.h.e_a.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i : \u2115\n\u22a2 Set.Pairwise \u2191(range (u i)) fun i j => IndepFun (Y i) (Y j)\n[PROOFSTEP]\nintro k _ l _ hkl\n[GOAL]\ncase e_f.h.e_a.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i k : \u2115\na\u271d\u00b9 : k \u2208 \u2191(range (u i))\nl : \u2115\na\u271d : l \u2208 \u2191(range (u i))\nhkl : k \u2260 l\n\u22a2 IndepFun (Y k) (Y l)\n[PROOFSTEP]\nexact (hindep hkl).comp (A k).measurable (A l).measurable\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * \u2211 j in range (u i), variance (Y j) \u2119 =\n \u2211 j in range (u (N - 1)), (\u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y j) \u2119\n[PROOFSTEP]\nsimp_rw [mul_sum, sum_mul, sum_sigma']\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2211 x in Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a,\n (\u2191\u230ac ^ x.fst\u230b\u208a ^ 2)\u207b\u00b9 * variance (truncation (X x.snd) \u2191x.snd) \u2119 =\n \u2211 x in Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N),\n (\u2191\u230ac ^ x.snd\u230b\u208a ^ 2)\u207b\u00b9 * variance (truncation (X x.fst) \u2191x.fst) \u2119\n[PROOFSTEP]\nrefine'\n sum_bij' (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ (fun a _ => rfl)\n (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ _ _\n[GOAL]\ncase refine'_1\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a),\n (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208\n Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hij\n[GOAL]\ncase refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)\n[PROOFSTEP]\nsimp only [mem_sigma, mem_range] at hij \n[GOAL]\ncase refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\nhij : i < N \u2227 j < \u230ac ^ i\u230b\u208a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208\n Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)\n[PROOFSTEP]\nsimp only [hij.1, hij.2, mem_sigma, mem_range, mem_filter, and_true_iff]\n[GOAL]\ncase refine'_1.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\nhij : i < N \u2227 j < \u230ac ^ i\u230b\u208a\n\u22a2 j < \u230ac ^ (N - 1)\u230b\u208a\n[PROOFSTEP]\nexact hij.2.trans_le (u_mono (Nat.le_pred_of_lt hij.1))\n[GOAL]\ncase refine'_2\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)),\n (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hij\n[GOAL]\ncase refine'_2.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\n[PROOFSTEP]\nsimp only [mem_sigma, mem_range, mem_filter] at hij \n[GOAL]\ncase refine'_2.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)\nhij : i < \u230ac ^ (N - 1)\u230b\u208a \u2227 j < N \u2227 i < \u230ac ^ j\u230b\u208a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208\n Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\n[PROOFSTEP]\nsimp only [hij.2.1, hij.2.2, mem_sigma, mem_range, and_self_iff]\n[GOAL]\ncase refine'_3\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a),\n (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)\n (_ :\n (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208\n Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)) =\n a\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hij\n[GOAL]\ncase refine'_3.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)\n (_ :\n (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)) =\n { fst := i, snd := j }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_4\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)),\n (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)\n (_ : (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a) =\n a\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hij\n[GOAL]\ncase refine'_4.mk\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN i j : \u2115\nhij : { fst := i, snd := j } \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)\n\u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)\n (_ :\n (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208\n Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a) =\n { fst := i, snd := j }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2211 j in range (u (N - 1)), (\u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y j) \u2119 \u2264\n \u2211 j in range (u (N - 1)), c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * variance (Y j) \u2119\n[PROOFSTEP]\napply sum_le_sum fun j hj => ?_\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN j : \u2115\nhj : j \u2208 range (u (N - 1))\n\u22a2 (\u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y j) \u2119 \u2264\n c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * variance (Y j) \u2119\n[PROOFSTEP]\nrcases@eq_zero_or_pos _ _ j with (rfl | hj)\n[GOAL]\ncase inl\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\nhj : 0 \u2208 range (u (N - 1))\n\u22a2 (\u2211 i in filter (fun i => 0 < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y 0) \u2119 \u2264\n c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u21910 ^ 2 * variance (Y 0) \u2119\n[PROOFSTEP]\nsimp only [Nat.cast_zero, zero_pow', Ne.def, bit0_eq_zero, Nat.one_ne_zero, not_false_iff, div_zero, zero_mul]\n[GOAL]\ncase inl\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\nhj : 0 \u2208 range (u (N - 1))\n\u22a2 (\u2211 x in filter (fun i => 0 < \u230ac ^ i\u230b\u208a) (range N), (\u2191\u230ac ^ x\u230b\u208a ^ 2)\u207b\u00b9) * variance (truncation (X 0) 0) \u2119 \u2264 0\n[PROOFSTEP]\nsimp only [Nat.cast_zero, truncation_zero, variance_zero, mul_zero, le_rfl]\n[GOAL]\ncase inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN j : \u2115\nhj\u271d : j \u2208 range (u (N - 1))\nhj : 0 < j\n\u22a2 (\u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y j) \u2119 \u2264\n c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * variance (Y j) \u2119\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_right _ (variance_nonneg _ _)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN j : \u2115\nhj\u271d : j \u2208 range (u (N - 1))\nhj : 0 < j\n\u22a2 \u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9 \u2264 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2\n[PROOFSTEP]\nconvert sum_div_nat_floor_pow_sq_le_div_sq N (Nat.cast_pos.2 hj) c_one using 2\n[GOAL]\ncase h.e'_3.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN j : \u2115\nhj\u271d : j \u2208 range (u (N - 1))\nhj : 0 < j\n\u22a2 filter (fun i => j < u i) (range N) = filter (fun x => \u2191j < \u2191\u230ac ^ x\u230b\u208a) (range N)\n[PROOFSTEP]\nsimp only [Nat.cast_lt]\n[GOAL]\ncase h.e'_3.a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN j : \u2115\nhj\u271d : j \u2208 range (u (N - 1))\nhj : 0 < j\nx\u271d : \u2115\na\u271d : x\u271d \u2208 filter (fun x => \u2191j < \u2191\u230ac ^ x\u230b\u208a) (range N)\n\u22a2 (\u2191(u x\u271d) ^ 2)\u207b\u00b9 = 1 / \u2191\u230ac ^ x\u271d\u230b\u208a ^ 2\n[PROOFSTEP]\nsimp only [one_div]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 \u2211 j in range (u (N - 1)), c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * variance (Y j) \u2119 =\n c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * \u2211 j in range (u (N - 1)), (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119\n[PROOFSTEP]\nsimp_rw [mul_sum, div_eq_mul_inv, mul_assoc]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * \u2211 j in range (u (N - 1)), (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264\n c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_left (I1 _)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 0 \u2264 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3\n[PROOFSTEP]\napply mul_nonneg (pow_nonneg c_pos.le _)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nN : \u2115\n\u22a2 0 \u2264 (c - 1)\u207b\u00b9 ^ 3\n[PROOFSTEP]\nexact pow_nonneg (inv_nonneg.2 (sub_nonneg.2 c_one.le)) _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nhave I3 : \u2200 N, \u2211 i in range N, \u2119 {\u03c9 | (u i * \u03b5 : \u211d) \u2264 |S (u i) \u03c9 - \ud835\udd3c[S (u i)]|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C) :=\n by\n intro N\n calc\n \u2211 i in range N, \u2119 {\u03c9 | (u i * \u03b5 : \u211d) \u2264 |S (u i) \u03c9 - \ud835\udd3c[S (u i)]|} \u2264\n \u2211 i in range N, ENNReal.ofReal (Var[S (u i)] / (u i * \u03b5) ^ 2) :=\n by\n refine' sum_le_sum fun i _ => _\n apply meas_ge_le_variance_div_sq\n \u00b7 exact mem\u2112p_finset_sum' _ fun j _ => (hident j).aestronglyMeasurable_fst.mem\u2112p_truncation\n \u00b7 apply mul_pos (Nat.cast_pos.2 _) \u03b5pos\n refine' zero_lt_one.trans_le _\n apply Nat.le_floor\n rw [Nat.cast_one]\n apply one_le_pow_of_one_le c_one.le\n _ = ENNReal.ofReal (\u2211 i in range N, Var[S (u i)] / (u i * \u03b5) ^ 2) :=\n by\n rw [ENNReal.ofReal_sum_of_nonneg fun i _ => ?_]\n exact div_nonneg (variance_nonneg _ _) (sq_nonneg _)\n _ \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C) := by\n apply ENNReal.ofReal_le_ofReal\n conv_lhs =>\n enter [2, i]\n rw [div_eq_inv_mul, \u2190 inv_pow, mul_inv, mul_comm _ \u03b5\u207b\u00b9, mul_pow, mul_assoc]\n rw [\u2190 mul_sum]\n refine' mul_le_mul_of_nonneg_left _ (sq_nonneg _)\n conv_lhs => enter [2, i]; rw [inv_pow]\n exact I2 N\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\n\u22a2 \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\n[PROOFSTEP]\nintro N\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\n[PROOFSTEP]\ncalc\n \u2211 i in range N, \u2119 {\u03c9 | (u i * \u03b5 : \u211d) \u2264 |S (u i) \u03c9 - \ud835\udd3c[S (u i)]|} \u2264\n \u2211 i in range N, ENNReal.ofReal (Var[S (u i)] / (u i * \u03b5) ^ 2) :=\n by\n refine' sum_le_sum fun i _ => _\n apply meas_ge_le_variance_div_sq\n \u00b7 exact mem\u2112p_finset_sum' _ fun j _ => (hident j).aestronglyMeasurable_fst.mem\u2112p_truncation\n \u00b7 apply mul_pos (Nat.cast_pos.2 _) \u03b5pos\n refine' zero_lt_one.trans_le _\n apply Nat.le_floor\n rw [Nat.cast_one]\n apply one_le_pow_of_one_le c_one.le\n _ = ENNReal.ofReal (\u2211 i in range N, Var[S (u i)] / (u i * \u03b5) ^ 2) :=\n by\n rw [ENNReal.ofReal_sum_of_nonneg fun i _ => ?_]\n exact div_nonneg (variance_nonneg _ _) (sq_nonneg _)\n _ \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C) := by\n apply ENNReal.ofReal_le_ofReal\n conv_lhs =>\n enter [2, i]\n rw [div_eq_inv_mul, \u2190 inv_pow, mul_inv, mul_comm _ \u03b5\u207b\u00b9, mul_pow, mul_assoc]\n rw [\u2190 mul_sum]\n refine' mul_le_mul_of_nonneg_left _ (sq_nonneg _)\n conv_lhs => enter [2, i]; rw [inv_pow]\n exact I2 N\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264\n \u2211 i in range N, ENNReal.ofReal (variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2)\n[PROOFSTEP]\nrefine' sum_le_sum fun i _ => _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2)\n[PROOFSTEP]\napply meas_ge_le_variance_div_sq\n[GOAL]\ncase hX\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 Mem\u2112p (fun \u03c9 => S (u i) \u03c9) 2\n[PROOFSTEP]\nexact mem\u2112p_finset_sum' _ fun j _ => (hident j).aestronglyMeasurable_fst.mem\u2112p_truncation\n[GOAL]\ncase hc\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 0 < \u2191(u i) * \u03b5\n[PROOFSTEP]\napply mul_pos (Nat.cast_pos.2 _) \u03b5pos\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 0 < u i\n[PROOFSTEP]\nrefine' zero_lt_one.trans_le _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 1 \u2264 u i\n[PROOFSTEP]\napply Nat.le_floor\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 \u21911 \u2264 c ^ i\n[PROOFSTEP]\nrw [Nat.cast_one]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 1 \u2264 c ^ i\n[PROOFSTEP]\napply one_le_pow_of_one_le c_one.le\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 i in range N, ENNReal.ofReal (variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2) =\n ENNReal.ofReal (\u2211 i in range N, variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2)\n[PROOFSTEP]\nrw [ENNReal.ofReal_sum_of_nonneg fun i _ => ?_]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\nx\u271d : i \u2208 range N\n\u22a2 0 \u2264 variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2\n[PROOFSTEP]\nexact div_nonneg (variance_nonneg _ _) (sq_nonneg _)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 ENNReal.ofReal (\u2211 i in range N, variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2) \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\n[PROOFSTEP]\napply ENNReal.ofReal_le_ofReal\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 i in range N, variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2 \u2264 \u03b5\u207b\u00b9 ^ 2 * C\n[PROOFSTEP]\nconv_lhs =>\n enter [2, i]\n rw [div_eq_inv_mul, \u2190 inv_pow, mul_inv, mul_comm _ \u03b5\u207b\u00b9, mul_pow, mul_assoc]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n| \u2211 i in range N, variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2\n[PROOFSTEP]\n enter [2, i]\n rw [div_eq_inv_mul, \u2190 inv_pow, mul_inv, mul_comm _ \u03b5\u207b\u00b9, mul_pow, mul_assoc]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n| \u2211 i in range N, variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2\n[PROOFSTEP]\n enter [2, i]\n rw [div_eq_inv_mul, \u2190 inv_pow, mul_inv, mul_comm _ \u03b5\u207b\u00b9, mul_pow, mul_assoc]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n| \u2211 i in range N, variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2\n[PROOFSTEP]\nenter [2, i]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\n| variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2\n[PROOFSTEP]\nrw [div_eq_inv_mul, \u2190 inv_pow, mul_inv, mul_comm _ \u03b5\u207b\u00b9, mul_pow, mul_assoc]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 i in range N, \u03b5\u207b\u00b9 ^ 2 * ((\u2191(u i))\u207b\u00b9 ^ 2 * variance (S (u i)) \u2119) \u2264 \u03b5\u207b\u00b9 ^ 2 * C\n[PROOFSTEP]\nrw [\u2190 mul_sum]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u03b5\u207b\u00b9 ^ 2 * \u2211 x in range N, (\u2191(u x))\u207b\u00b9 ^ 2 * variance (S (u x)) \u2119 \u2264 \u03b5\u207b\u00b9 ^ 2 * C\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left _ (sq_nonneg _)\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 x in range N, (\u2191(u x))\u207b\u00b9 ^ 2 * variance (S (u x)) \u2119 \u2264 C\n[PROOFSTEP]\nconv_lhs => enter [2, i]; rw [inv_pow]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n| \u2211 x in range N, (\u2191(u x))\u207b\u00b9 ^ 2 * variance (S (u x)) \u2119\n[PROOFSTEP]\nenter [2, i]; rw [inv_pow]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n| \u2211 x in range N, (\u2191(u x))\u207b\u00b9 ^ 2 * variance (S (u x)) \u2119\n[PROOFSTEP]\nenter [2, i]; rw [inv_pow]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n| \u2211 x in range N, (\u2191(u x))\u207b\u00b9 ^ 2 * variance (S (u x)) \u2119\n[PROOFSTEP]\nenter [2, i]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN i : \u2115\n| (\u2191(u i))\u207b\u00b9 ^ 2 * variance (S (u i)) \u2119\n[PROOFSTEP]\nrw [inv_pow]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nN : \u2115\n\u22a2 \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\n[PROOFSTEP]\nexact I2 N\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nI3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nhave I4 : (\u2211' i, \u2119 {\u03c9 | (u i * \u03b5 : \u211d) \u2264 |S (u i) \u03c9 - \ud835\udd3c[S (u i)]|}) < \u221e :=\n (le_of_tendsto_of_tendsto' (ENNReal.tendsto_nat_tsum _) tendsto_const_nhds I3).trans_lt ENNReal.ofReal_lt_top\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nI3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\nI4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nfilter_upwards [ae_eventually_not_mem I4.ne] with \u03c9 h\u03c9\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nI3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\nI4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 :\n \u2200\u1da0 (n : \u2115) in atTop,\n \u00ac\u2191\u230ac ^ n\u230b\u208a * \u03b5 \u2264\n |Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a|\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun n => truncation (X n) \u2191n) a| <\n \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nsimp_rw [not_le, mul_comm, sum_apply] at h\u03c9 \n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nI3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\nI4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 :\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) \u03c9 - \u222b (a : \u03a9), \u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) a| <\n \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun n => truncation (X n) \u2191n) a| <\n \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nconvert h\u03c9\n[GOAL]\ncase h.e'_2.h.h.e'_3.h.e'_3.h.e'_6.h.e'_7.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc_pos : 0 < c\nhX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nA : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id)\nY : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n\nS : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i\nhS : S = fun n => \u2211 i in range n, Y i\nu : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a\nu_mono : Monotone u\nI1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a\nC : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a)\nI2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C\nI3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C)\nI4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 :\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) \u03c9 - \u222b (a : \u03a9), \u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) a| <\n \u03b5 * \u2191\u230ac ^ n\u230b\u208a\nx\u271d\u00b9 : \u2115\nx\u271d : \u03a9\n\u22a2 Finset.sum (range \u230ac ^ x\u271d\u00b9\u230b\u208a) (fun n => truncation (X n) \u2191n) x\u271d = \u2211 c in range \u230ac ^ x\u271d\u00b9\u230b\u208a, truncation (X c) (\u2191c) x\u271d\n[PROOFSTEP]\nsimp only [sum_apply]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n (fun n =>\n \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a) =o[atTop]\n fun n => \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nobtain \u27e8v, -, v_pos, v_lim\u27e9 : \u2203 v : \u2115 \u2192 \u211d, StrictAnti v \u2227 (\u2200 n : \u2115, 0 < v n) \u2227 Tendsto v atTop (\ud835\udcdd 0) :=\n exists_seq_strictAnti_tendsto (0 : \u211d)\n[GOAL]\ncase intro.intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n (fun n =>\n \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a) =o[atTop]\n fun n => \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nhave := fun i => strong_law_aux1 X hint hindep hident hnonneg c_one (v_pos i)\n[GOAL]\ncase intro.intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n \u2200 (i : \u2115),\n \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n v i * \u2191\u230ac ^ n\u230b\u208a\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n (fun n =>\n \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a) =o[atTop]\n fun n => \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nfilter_upwards [ae_all_iff.2 this] with \u03c9 h\u03c9\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n \u2200 (i : \u2115),\n \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n v i * \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n \u2200 (i : \u2115),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n v i * \u2191\u230ac ^ n\u230b\u208a\n\u22a2 (fun n =>\n \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a) =o[atTop]\n fun n => \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\napply Asymptotics.isLittleO_iff.2 fun \u03b5 \u03b5pos => ?_\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n \u2200 (i : \u2115),\n \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n v i * \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n \u2200 (i : \u2115),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n v i * \u2191\u230ac ^ n\u230b\u208a\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop,\n \u2016\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ x\u230b\u208a) (fun i => truncation (X i) \u2191i) a\u2016 \u2264\n \u03b5 * \u2016\u2191\u230ac ^ x\u230b\u208a\u2016\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 : \u2203 i, v i < \u03b5 := ((tendsto_order.1 v_lim).2 \u03b5 \u03b5pos).exists\n[GOAL]\ncase intro\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n \u2200 (i : \u2115),\n \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n v i * \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n \u2200 (i : \u2115),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n v i * \u2191\u230ac ^ n\u230b\u208a\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ni : \u2115\nhi : v i < \u03b5\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop,\n \u2016\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ x\u230b\u208a) (fun i => truncation (X i) \u2191i) a\u2016 \u2264\n \u03b5 * \u2016\u2191\u230ac ^ x\u230b\u208a\u2016\n[PROOFSTEP]\nfilter_upwards [h\u03c9 i] with n hn\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n \u2200 (i : \u2115),\n \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n v i * \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n \u2200 (i : \u2115),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n v i * \u2191\u230ac ^ n\u230b\u208a\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ni : \u2115\nhi : v i < \u03b5\nn : \u2115\nhn :\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n v i * \u2191\u230ac ^ n\u230b\u208a\n\u22a2 \u2016\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a\u2016 \u2264\n \u03b5 * \u2016\u2191\u230ac ^ n\u230b\u208a\u2016\n[PROOFSTEP]\nsimp only [Real.norm_eq_abs, LatticeOrderedGroup.abs_abs, Nat.abs_cast]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nv : \u2115 \u2192 \u211d\nv_pos : \u2200 (n : \u2115), 0 < v n\nv_lim : Tendsto v atTop (\ud835\udcdd 0)\nthis :\n \u2200 (i : \u2115),\n \u2200\u1d50 (\u03c9 : \u03a9),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n v i * \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n \u2200 (i : \u2115),\n \u2200\u1da0 (n : \u2115) in atTop,\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n v i * \u2191\u230ac ^ n\u230b\u208a\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\ni : \u2115\nhi : v i < \u03b5\nn : \u2115\nhn :\n |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <\n v i * \u2191\u230ac ^ n\u230b\u208a\n\u22a2 |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| \u2264\n \u03b5 * \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nexact hn.le.trans (mul_le_mul_of_nonneg_right hi.le (Nat.cast_nonneg _))\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\n\u22a2 (fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) =o[atTop]\n Nat.cast\n[PROOFSTEP]\nhave A : Tendsto (fun i => \ud835\udd3c[truncation (X i) i]) atTop (\ud835\udcdd \ud835\udd3c[X 0]) :=\n by\n convert (tendsto_integral_truncation hint).comp tendsto_nat_cast_atTop_atTop using 1\n ext i\n exact (hident i).truncation.integral_eq\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\n\u22a2 Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nconvert (tendsto_integral_truncation hint).comp tendsto_nat_cast_atTop_atTop using 1\n[GOAL]\ncase h.e'_3\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\n\u22a2 (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) = (fun A => \u222b (x : \u03a9), truncation (X 0) A x) \u2218 Nat.cast\n[PROOFSTEP]\next i\n[GOAL]\ncase h.e'_3.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\ni : \u2115\n\u22a2 \u222b (a : \u03a9), truncation (X i) (\u2191i) a = ((fun A => \u222b (x : \u03a9), truncation (X 0) A x) \u2218 Nat.cast) i\n[PROOFSTEP]\nexact (hident i).truncation.integral_eq\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 (fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) =o[atTop]\n Nat.cast\n[PROOFSTEP]\nconvert Asymptotics.isLittleO_sum_range_of_tendsto_zero (tendsto_sub_nhds_zero_iff.2 A) using 1\n[GOAL]\ncase h.e'_7\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 (fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) = fun n =>\n \u2211 i in range n, ((\u222b (a : \u03a9), truncation (X i) (\u2191i) a) - \u222b (a : \u03a9), X 0 a)\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h.e'_7.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\nn : \u2115\n\u22a2 (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a =\n \u2211 i in range n, ((\u222b (a : \u03a9), truncation (X i) (\u2191i) a) - \u222b (a : \u03a9), X 0 a)\n[PROOFSTEP]\nsimp only [sum_sub_distrib, sum_const, card_range, nsmul_eq_mul, sum_apply, sub_left_inj]\n[GOAL]\ncase h.e'_7.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\nn : \u2115\n\u22a2 \u222b (a : \u03a9), \u2211 c in range n, truncation (X c) (\u2191c) a = \u2211 x in range n, \u222b (a : \u03a9), truncation (X x) (\u2191x) a\n[PROOFSTEP]\nrw [integral_finset_sum _ fun i _ => ?_]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\nn i : \u2115\nx\u271d : i \u2208 range n\n\u22a2 Integrable fun a => truncation (X i) (\u2191i) a\n[PROOFSTEP]\nexact ((hident i).symm.integrable_snd hint).1.integrable_truncation\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9),\n (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n =>\n \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nfilter_upwards [strong_law_aux2 X hint hindep hident hnonneg c_one] with \u03c9 h\u03c9\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03c9 : \u03a9\nh\u03c9 :\n (fun n =>\n \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a) =o[atTop]\n fun n => \u2191\u230ac ^ n\u230b\u208a\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nhave A : Tendsto (fun n : \u2115 => \u230ac ^ n\u230b\u208a) atTop atTop :=\n tendsto_nat_floor_atTop.comp (tendsto_pow_atTop_atTop_of_one_lt c_one)\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03c9 : \u03a9\nh\u03c9 :\n (fun n =>\n \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a) =o[atTop]\n fun n => \u2191\u230ac ^ n\u230b\u208a\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nconvert h\u03c9.add ((strong_law_aux3 X hint hident).comp_tendsto A) using 1\n[GOAL]\ncase h.e'_7\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03c9 : \u03a9\nh\u03c9 :\n (fun n =>\n \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a) =o[atTop]\n fun n => \u2191\u230ac ^ n\u230b\u208a\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) = fun x =>\n (\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ x\u230b\u208a) (fun i => truncation (X i) \u2191i) a) +\n ((fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) \u2218 fun n =>\n \u230ac ^ n\u230b\u208a)\n x\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h.e'_7.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u03c9 : \u03a9\nh\u03c9 :\n (fun n =>\n \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a) =o[atTop]\n fun n => \u2191\u230ac ^ n\u230b\u208a\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\nn : \u2115\n\u22a2 \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a =\n (\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -\n \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a) +\n ((fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) \u2218 fun n =>\n \u230ac ^ n\u230b\u208a)\n n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\n[PROOFSTEP]\nhave A : (\u2211' j : \u2115, \u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi (j : \u211d)}) < \u221e :=\n by\n convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2\n ext1 j\n exact (hident j).measure_mem_eq measurableSet_Ioi\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\n\u22a2 \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n[PROOFSTEP]\nconvert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\n\u22a2 (fun j => \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j}) = fun j => \u2191\u2191\u2119 {\u03c9 | X 0 \u03c9 \u2208 Set.Ioi \u2191j}\n[PROOFSTEP]\next1 j\n[GOAL]\ncase h.e'_3.h.e'_5.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nj : \u2115\n\u22a2 \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} = \u2191\u2191\u2119 {\u03c9 | X 0 \u03c9 \u2208 Set.Ioi \u2191j}\n[PROOFSTEP]\nexact (hident j).measure_mem_eq measurableSet_Ioi\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\n[PROOFSTEP]\nhave B : \u2200\u1d50 \u03c9, Tendsto (fun n : \u2115 => truncation (X n) n \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0) :=\n by\n filter_upwards [ae_eventually_not_mem A.ne] with \u03c9 h\u03c9\n apply tendsto_const_nhds.congr' _\n filter_upwards [h\u03c9, Ioi_mem_atTop 0] with n hn npos\n simp only [truncation, indicator, Set.mem_Ioc, id.def, Function.comp_apply]\n split_ifs with h\n \u00b7 exact (sub_self _).symm\n \u00b7 have : -(n : \u211d) < X n \u03c9 := by\n apply lt_of_lt_of_le _ (hnonneg n \u03c9)\n simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos\n simp only [this, true_and_iff, not_le] at h \n exact (hn h).elim\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nfilter_upwards [ae_eventually_not_mem A.ne] with \u03c9 h\u03c9\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\n\u22a2 Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\napply tendsto_const_nhds.congr' _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\n\u22a2 (fun x => 0) =\u1da0[atTop] fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9\n[PROOFSTEP]\nfilter_upwards [h\u03c9, Ioi_mem_atTop 0] with n hn npos\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 0 = truncation (X n) (\u2191n) \u03c9 - X n \u03c9\n[PROOFSTEP]\nsimp only [truncation, indicator, Set.mem_Ioc, id.def, Function.comp_apply]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 0 = (if -\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n then X n \u03c9 else 0) - X n \u03c9\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nh : -\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n\n\u22a2 0 = X n \u03c9 - X n \u03c9\n[PROOFSTEP]\nexact (sub_self _).symm\n[GOAL]\ncase neg\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nh : \u00ac(-\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n)\n\u22a2 0 = 0 - X n \u03c9\n[PROOFSTEP]\nhave : -(n : \u211d) < X n \u03c9 := by\n apply lt_of_lt_of_le _ (hnonneg n \u03c9)\n simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nh : \u00ac(-\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n)\n\u22a2 -\u2191n < X n \u03c9\n[PROOFSTEP]\napply lt_of_lt_of_le _ (hnonneg n \u03c9)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nh : \u00ac(-\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n)\n\u22a2 -\u2191n < 0\n[PROOFSTEP]\nsimpa only [Right.neg_neg_iff, Nat.cast_pos] using npos\n[GOAL]\ncase neg\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nh : \u00ac(-\u2191n < X n \u03c9 \u2227 X n \u03c9 \u2264 \u2191n)\nthis : -\u2191n < X n \u03c9\n\u22a2 0 = 0 - X n \u03c9\n[PROOFSTEP]\nsimp only [this, true_and_iff, not_le] at h \n[GOAL]\ncase neg\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\n\u03c9 : \u03a9\nh\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nn : \u2115\nhn : \u00acX n \u03c9 \u2208 Set.Ioi \u2191n\nnpos : n \u2208 Set.Ioi 0\nthis : -\u2191n < X n \u03c9\nh : \u2191n < X n \u03c9\n\u22a2 0 = 0 - X n \u03c9\n[PROOFSTEP]\nexact (hn h).elim\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\nB : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\n[PROOFSTEP]\nfilter_upwards [B] with \u03c9 h\u03c9\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\nB : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u03c9 : \u03a9\nh\u03c9 : Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u22a2 (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\n[PROOFSTEP]\nconvert isLittleO_sum_range_of_tendsto_zero h\u03c9 using 1\n[GOAL]\ncase h.e'_7\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\nB : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u03c9 : \u03a9\nh\u03c9 : Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u22a2 (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) = fun n =>\n \u2211 i in range n, (truncation (X i) (\u2191i) \u03c9 - X i \u03c9)\n[PROOFSTEP]\next n\n[GOAL]\ncase h.e'_7.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nA : \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X j \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4\nB : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\n\u03c9 : \u03a9\nh\u03c9 : Tendsto (fun n => truncation (X n) (\u2191n) \u03c9 - X n \u03c9) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9 = \u2211 i in range n, (truncation (X i) (\u2191i) \u03c9 - X i \u03c9)\n[PROOFSTEP]\nrw [sum_sub_distrib]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nhave H : \u2200 n : \u2115, (0 : \u211d) < \u230ac ^ n\u230b\u208a := by\n intro n\n refine' zero_lt_one.trans_le _\n simp only [Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le c_one.le n]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\n\u22a2 \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nn : \u2115\n\u22a2 0 < \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nrefine' zero_lt_one.trans_le _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nn : \u2115\n\u22a2 1 \u2264 \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nsimp only [Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le c_one.le n]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nfilter_upwards [strong_law_aux4 X hint hindep hident hnonneg c_one, strong_law_aux5 X hint hident hnonneg] with \u03c9 h\u03c9 h'\u03c9\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\n\u22a2 Tendsto (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nrw [\u2190 tendsto_sub_nhds_zero_iff, \u2190 Asymptotics.isLittleO_one_iff \u211d]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\n\u22a2 (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a - \u222b (a : \u03a9), X 0 a) =o[atTop] fun _x => 1\n[PROOFSTEP]\nhave L : (fun n : \u2115 => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u230ac ^ n\u230b\u208a * \ud835\udd3c[X 0]) =o[atTop] fun n => (\u230ac ^ n\u230b\u208a : \u211d) :=\n by\n have A : Tendsto (fun n : \u2115 => \u230ac ^ n\u230b\u208a) atTop atTop :=\n tendsto_nat_floor_atTop.comp (tendsto_pow_atTop_atTop_of_one_lt c_one)\n convert h\u03c9.sub (h'\u03c9.comp_tendsto A) using 1\n ext1 n\n simp only [Function.comp_apply, sub_sub_sub_cancel_left]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nhave A : Tendsto (fun n : \u2115 => \u230ac ^ n\u230b\u208a) atTop atTop :=\n tendsto_nat_floor_atTop.comp (tendsto_pow_atTop_atTop_of_one_lt c_one)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\n[PROOFSTEP]\nconvert h\u03c9.sub (h'\u03c9.comp_tendsto A) using 1\n[GOAL]\ncase h.e'_7\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\n\u22a2 (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) = fun x =>\n (\u2211 i in range \u230ac ^ x\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ x\u230b\u208a * \u222b (a : \u03a9), X 0 a) -\n ((fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) \u2218 fun n => \u230ac ^ n\u230b\u208a) x\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h.e'_7.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nA : Tendsto (fun n => \u230ac ^ n\u230b\u208a) atTop atTop\nn : \u2115\n\u22a2 \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a =\n (\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) -\n ((fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) \u2218 fun n => \u230ac ^ n\u230b\u208a) n\n[PROOFSTEP]\nsimp only [Function.comp_apply, sub_sub_sub_cancel_left]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nL : (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\n\u22a2 (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a - \u222b (a : \u03a9), X 0 a) =o[atTop] fun _x => 1\n[PROOFSTEP]\nconvert L.mul_isBigO (isBigO_refl (fun n : \u2115 => (\u230ac ^ n\u230b\u208a : \u211d)\u207b\u00b9) atTop) using 1\n[GOAL]\ncase h.e'_7\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nL : (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\n\u22a2 (fun n => (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a - \u222b (a : \u03a9), X 0 a) = fun x =>\n (\u2211 i in range \u230ac ^ x\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ x\u230b\u208a * \u222b (a : \u03a9), X 0 a) * (\u2191\u230ac ^ x\u230b\u208a)\u207b\u00b9\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h.e'_7.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nL : (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nn : \u2115\n\u22a2 (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac ^ n\u230b\u208a - \u222b (a : \u03a9), X 0 a =\n (\u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) * (\u2191\u230ac ^ n\u230b\u208a)\u207b\u00b9\n[PROOFSTEP]\nfield_simp [(H n).ne']\n[GOAL]\ncase h.e'_8\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nL : (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\n\u22a2 (fun _x => 1) = fun x => \u2191\u230ac ^ x\u230b\u208a * (\u2191\u230ac ^ x\u230b\u208a)\u207b\u00b9\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h.e'_8.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u211d\nc_one : 1 < c\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac ^ n\u230b\u208a\n\u03c9 : \u03a9\nh\u03c9 :\n (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nh'\u03c9 : (fun n => \u2211 i in range n, truncation (X i) (\u2191i) \u03c9 - \u2211 i in range n, X i \u03c9) =o[atTop] fun n => \u2191n\nL : (fun n => \u2211 i in range \u230ac ^ n\u230b\u208a, X i \u03c9 - \u2191\u230ac ^ n\u230b\u208a * \u222b (a : \u03a9), X 0 a) =o[atTop] fun n => \u2191\u230ac ^ n\u230b\u208a\nn : \u2115\n\u22a2 1 = \u2191\u230ac ^ n\u230b\u208a * (\u2191\u230ac ^ n\u230b\u208a)\u207b\u00b9\n[PROOFSTEP]\nfield_simp [(H n).ne']\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nobtain \u27e8c, -, cone, clim\u27e9 : \u2203 c : \u2115 \u2192 \u211d, StrictAnti c \u2227 (\u2200 n : \u2115, 1 < c n) \u2227 Tendsto c atTop (\ud835\udcdd 1) :=\n exists_seq_strictAnti_tendsto (1 : \u211d)\n[GOAL]\ncase intro.intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nhave : \u2200 k, \u2200\u1d50 \u03c9, Tendsto (fun n : \u2115 => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd \ud835\udd3c[X 0]) := fun k =>\n strong_law_aux6 X hint hindep hident hnonneg (cone k)\n[GOAL]\ncase intro.intro.intro\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\nthis :\n \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nfilter_upwards [ae_all_iff.2 this] with \u03c9 h\u03c9\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\nthis :\n \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\napply tendsto_div_of_monotone_of_tendsto_div_floor_pow _ _ _ c cone clim _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\nthis :\n \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 Monotone fun n => \u2211 i in range n, X i \u03c9\n[PROOFSTEP]\nintro m n hmn\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\nthis :\n \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\nm n : \u2115\nhmn : m \u2264 n\n\u22a2 (fun n => \u2211 i in range n, X i \u03c9) m \u2264 (fun n => \u2211 i in range n, X i \u03c9) n\n[PROOFSTEP]\nexact sum_le_sum_of_subset_of_nonneg (range_mono hmn) fun i _ _ => hnonneg i \u03c9\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9\nc : \u2115 \u2192 \u211d\ncone : \u2200 (n : \u2115), 1 < c n\nclim : Tendsto c atTop (\ud835\udcdd 1)\nthis :\n \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n\u22a2 \u2200 (k : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nexact h\u03c9\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nlet pos : \u211d \u2192 \u211d := fun x => max x 0\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\npos : \u211d \u2192 \u211d := fun x => max x 0\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nlet neg : \u211d \u2192 \u211d := fun x => max (-x) 0\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\npos : \u211d \u2192 \u211d := fun x => max x 0\nneg : \u211d \u2192 \u211d := fun x => max (-x) 0\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nhave posm : Measurable pos := measurable_id'.max measurable_const\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\npos : \u211d \u2192 \u211d := fun x => max x 0\nneg : \u211d \u2192 \u211d := fun x => max (-x) 0\nposm : Measurable pos\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nhave negm : Measurable neg := measurable_id'.neg.max measurable_const\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\npos : \u211d \u2192 \u211d := fun x => max x 0\nneg : \u211d \u2192 \u211d := fun x => max (-x) 0\nposm : Measurable pos\nnegm : Measurable neg\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nhave A : \u2200\u1d50 \u03c9, Tendsto (fun n : \u2115 => (\u2211 i in range n, (pos \u2218 X i) \u03c9) / n) atTop (\ud835\udcdd \ud835\udd3c[pos \u2218 X 0]) :=\n strong_law_aux7 _ hint.pos_part (fun i j hij => (hindep hij).comp posm posm) (fun i => (hident i).comp posm)\n fun i \u03c9 => le_max_right _ _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\npos : \u211d \u2192 \u211d := fun x => max x 0\nneg : \u211d \u2192 \u211d := fun x => max (-x) 0\nposm : Measurable pos\nnegm : Measurable neg\nA : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, (pos \u2218 X i) \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), (pos \u2218 X 0) a))\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nhave B : \u2200\u1d50 \u03c9, Tendsto (fun n : \u2115 => (\u2211 i in range n, (neg \u2218 X i) \u03c9) / n) atTop (\ud835\udcdd \ud835\udd3c[neg \u2218 X 0]) :=\n strong_law_aux7 _ hint.neg_part (fun i j hij => (hindep hij).comp negm negm) (fun i => (hident i).comp negm)\n fun i \u03c9 => le_max_right _ _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\npos : \u211d \u2192 \u211d := fun x => max x 0\nneg : \u211d \u2192 \u211d := fun x => max (-x) 0\nposm : Measurable pos\nnegm : Measurable neg\nA : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, (pos \u2218 X i) \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), (pos \u2218 X 0) a))\nB : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, (neg \u2218 X i) \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), (neg \u2218 X 0) a))\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nfilter_upwards [A, B] with \u03c9 h\u03c9pos h\u03c9neg\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\npos : \u211d \u2192 \u211d := fun x => max x 0\nneg : \u211d \u2192 \u211d := fun x => max (-x) 0\nposm : Measurable pos\nnegm : Measurable neg\nA : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, (pos \u2218 X i) \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), (pos \u2218 X 0) a))\nB : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, (neg \u2218 X i) \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), (neg \u2218 X 0) a))\n\u03c9 : \u03a9\nh\u03c9pos :\n Tendsto (fun n => (\u2211 i in range n, ((fun x => max x 0) \u2218 X i) \u03c9) / \u2191n) atTop\n (\ud835\udcdd (\u222b (a : \u03a9), ((fun x => max x 0) \u2218 X 0) a))\nh\u03c9neg :\n Tendsto (fun n => (\u2211 i in range n, ((fun x => max (-x) 0) \u2218 X i) \u03c9) / \u2191n) atTop\n (\ud835\udcdd (\u222b (a : \u03a9), ((fun x => max (-x) 0) \u2218 X 0) a))\n\u22a2 Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a))\n[PROOFSTEP]\nconvert h\u03c9pos.sub h\u03c9neg using 1\n[GOAL]\ncase h.e'_3\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\npos : \u211d \u2192 \u211d := fun x => max x 0\nneg : \u211d \u2192 \u211d := fun x => max (-x) 0\nposm : Measurable pos\nnegm : Measurable neg\nA : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, (pos \u2218 X i) \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), (pos \u2218 X 0) a))\nB : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, (neg \u2218 X i) \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), (neg \u2218 X 0) a))\n\u03c9 : \u03a9\nh\u03c9pos :\n Tendsto (fun n => (\u2211 i in range n, ((fun x => max x 0) \u2218 X i) \u03c9) / \u2191n) atTop\n (\ud835\udcdd (\u222b (a : \u03a9), ((fun x => max x 0) \u2218 X 0) a))\nh\u03c9neg :\n Tendsto (fun n => (\u2211 i in range n, ((fun x => max (-x) 0) \u2218 X i) \u03c9) / \u2191n) atTop\n (\ud835\udcdd (\u222b (a : \u03a9), ((fun x => max (-x) 0) \u2218 X 0) a))\n\u22a2 (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) = fun x =>\n (\u2211 i in range x, ((fun x => max x 0) \u2218 X i) \u03c9) / \u2191x - (\u2211 i in range x, ((fun x => max (-x) 0) \u2218 X i) \u03c9) / \u2191x\n[PROOFSTEP]\nsimp only [\u2190 sub_div, \u2190 sum_sub_distrib, max_zero_sub_max_neg_zero_eq_self, Function.comp_apply]\n[GOAL]\ncase h.e'_5\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nhint : Integrable (X 0)\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\npos : \u211d \u2192 \u211d := fun x => max x 0\nneg : \u211d \u2192 \u211d := fun x => max (-x) 0\nposm : Measurable pos\nnegm : Measurable neg\nA : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, (pos \u2218 X i) \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), (pos \u2218 X 0) a))\nB : \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, (neg \u2218 X i) \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), (neg \u2218 X 0) a))\n\u03c9 : \u03a9\nh\u03c9pos :\n Tendsto (fun n => (\u2211 i in range n, ((fun x => max x 0) \u2218 X i) \u03c9) / \u2191n) atTop\n (\ud835\udcdd (\u222b (a : \u03a9), ((fun x => max x 0) \u2218 X 0) a))\nh\u03c9neg :\n Tendsto (fun n => (\u2211 i in range n, ((fun x => max (-x) 0) \u2218 X i) \u03c9) / \u2191n) atTop\n (\ud835\udcdd (\u222b (a : \u03a9), ((fun x => max (-x) 0) \u2218 X 0) a))\n\u22a2 \ud835\udcdd (\u222b (a : \u03a9), X 0 a) = \ud835\udcdd ((\u222b (a : \u03a9), ((fun x => max x 0) \u2218 X 0) a) - \u222b (a : \u03a9), ((fun x => max (-x) 0) \u2218 X 0) a)\n[PROOFSTEP]\nsimp only [\u2190 integral_sub hint.pos_part hint.neg_part, max_zero_sub_max_neg_zero_eq_self, Function.comp_apply]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\n\u22a2 Tendsto (fun n => snorm (fun \u03c9 => (\u2211 i in range n, X i \u03c9) / \u2191n - \u222b (a : \u03a9), X 0 a) p \u2119) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave hmeas : \u2200 i, AEStronglyMeasurable (X i) \u2119 := fun i => (hident i).aestronglyMeasurable_iff.2 h\u2112p.1\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\n\u22a2 Tendsto (fun n => snorm (fun \u03c9 => (\u2211 i in range n, X i \u03c9) / \u2191n - \u222b (a : \u03a9), X 0 a) p \u2119) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave hint : Integrable (X 0) \u2119 := h\u2112p.integrable hp\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\n\u22a2 Tendsto (fun n => snorm (fun \u03c9 => (\u2211 i in range n, X i \u03c9) / \u2191n - \u222b (a : \u03a9), X 0 a) p \u2119) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave havg : \u2200 n, AEStronglyMeasurable (fun \u03c9 => (\u2211 i in range n, X i \u03c9) / n) \u2119 :=\n by\n intro n\n simp_rw [div_eq_mul_inv]\n exact AEStronglyMeasurable.mul_const (aestronglyMeasurable_sum _ fun i _ => hmeas i) _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\n\u22a2 \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2211 i in range n, X i \u03c9) / \u2191n) \u2119\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nn : \u2115\n\u22a2 AEStronglyMeasurable (fun \u03c9 => (\u2211 i in range n, X i \u03c9) / \u2191n) \u2119\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nn : \u2115\n\u22a2 AEStronglyMeasurable (fun \u03c9 => (\u2211 i in range n, X i \u03c9) * (\u2191n)\u207b\u00b9) \u2119\n[PROOFSTEP]\nexact AEStronglyMeasurable.mul_const (aestronglyMeasurable_sum _ fun i _ => hmeas i) _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2211 i in range n, X i \u03c9) / \u2191n) \u2119\n\u22a2 Tendsto (fun n => snorm (fun \u03c9 => (\u2211 i in range n, X i \u03c9) / \u2191n - \u222b (a : \u03a9), X 0 a) p \u2119) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine'\n tendsto_Lp_of_tendstoInMeasure _ hp hp' havg (mem\u2112p_const _) _\n (tendstoInMeasure_of_tendsto_ae havg (strong_law_ae _ hint hindep hident))\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2211 i in range n, X i \u03c9) / \u2191n) \u2119\n\u22a2 UnifIntegrable (fun n \u03c9 => (\u2211 i in range n, X i \u03c9) / \u2191n) p \u2119\n[PROOFSTEP]\nrw [(_ : (fun n \u03c9 => (\u2211 i in range n, X i \u03c9) / \u2191n) = fun n => (\u2211 i in range n, X i) / \u2191n)]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2211 i in range n, X i \u03c9) / \u2191n) \u2119\n\u22a2 UnifIntegrable (fun n => (\u2211 i in range n, X i) / \u2191n) p \u2119\n[PROOFSTEP]\nexact (uniformIntegrable_average hp <| Mem\u2112p.uniformIntegrable_of_identDistrib hp hp' h\u2112p hident).2.1\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2211 i in range n, X i \u03c9) / \u2191n) \u2119\n\u22a2 (fun n \u03c9 => (\u2211 i in range n, X i \u03c9) / \u2191n) = fun n => (\u2211 i in range n, X i) / \u2191n\n[PROOFSTEP]\next n \u03c9\n[GOAL]\ncase h.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasureSpace \u03a9\ninst\u271d : IsProbabilityMeasure \u2119\np : \u211d\u22650\u221e\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nX : \u2115 \u2192 \u03a9 \u2192 \u211d\nh\u2112p : Mem\u2112p (X 0) p\nhindep : Pairwise fun i j => IndepFun (X i) (X j)\nhident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0)\nhmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119\nhint : Integrable (X 0)\nhavg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2211 i in range n, X i \u03c9) / \u2191n) \u2119\nn : \u2115\n\u03c9 : \u03a9\n\u22a2 (\u2211 i in range n, X i \u03c9) / \u2191n = ((\u2211 i in range n, X i) / \u2191n) \u03c9\n[PROOFSTEP]\nsimp only [Pi.coe_nat, Pi.div_apply, sum_apply]\n", "meta": {"mathlib_filename": "Mathlib.Probability.StrongLaw", "llama_tokens": 125319, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.41271188122389924}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : Add \u03b1\ninst\u271d\u2075 : Sub \u03b1\ninst\u271d\u2074 : OrderedSub \u03b1\na\u271d b\u271d c d : \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : Add \u03b2\ninst\u271d\u00b9 : Sub \u03b2\ninst\u271d : OrderedSub \u03b2\nf : AddHom \u03b1 \u03b2\nhf : Monotone \u2191f\na b : \u03b1\n\u22a2 \u2191f a - \u2191f b \u2264 \u2191f (a - b)\n[PROOFSTEP]\nrw [tsub_le_iff_right, \u2190 f.map_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : Add \u03b1\ninst\u271d\u2075 : Sub \u03b1\ninst\u271d\u2074 : OrderedSub \u03b1\na\u271d b\u271d c d : \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : Add \u03b2\ninst\u271d\u00b9 : Sub \u03b2\ninst\u271d : OrderedSub \u03b2\nf : AddHom \u03b1 \u03b2\nhf : Monotone \u2191f\na b : \u03b1\n\u22a2 \u2191f a \u2264 \u2191f (a - b + b)\n[PROOFSTEP]\nexact hf le_tsub_add\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2078 : Preorder \u03b1\ninst\u271d\u2077 : Add \u03b1\ninst\u271d\u2076 : Sub \u03b1\ninst\u271d\u2075 : OrderedSub \u03b1\na\u271d b\u271d c\u271d d : \u03b1\nR : Type u_3\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Preorder R\ninst\u271d\u00b2 : Sub R\ninst\u271d\u00b9 : OrderedSub R\ninst\u271d : CovariantClass R R (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : R\n\u22a2 a * c - b * c \u2264 (a - b) * c\n[PROOFSTEP]\nsimpa only [mul_comm _ c] using le_mul_tsub\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\ninst\u271d\u2077 : Preorder M\ninst\u271d\u2076 : Add M\ninst\u271d\u2075 : Sub M\ninst\u271d\u2074 : OrderedSub M\ninst\u271d\u00b3 : PartialOrder N\ninst\u271d\u00b2 : Add N\ninst\u271d\u00b9 : Sub N\ninst\u271d : OrderedSub N\ne : M \u2243o N\nh_add : \u2200 (a b : M), \u2191e (a + b) = \u2191e a + \u2191e b\na b : M\n\u22a2 \u2191e (a - b) = \u2191e a - \u2191e b\n[PROOFSTEP]\nlet e_add : M \u2243+ N := { e with map_add' := h_add }\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\ninst\u271d\u2077 : Preorder M\ninst\u271d\u2076 : Add M\ninst\u271d\u2075 : Sub M\ninst\u271d\u2074 : OrderedSub M\ninst\u271d\u00b3 : PartialOrder N\ninst\u271d\u00b2 : Add N\ninst\u271d\u00b9 : Sub N\ninst\u271d : OrderedSub N\ne : M \u2243o N\nh_add : \u2200 (a b : M), \u2191e (a + b) = \u2191e a + \u2191e b\na b : M\ne_add : M \u2243+ N := { toEquiv := e.toEquiv, map_add' := h_add }\n\u22a2 \u2191e (a - b) = \u2191e a - \u2191e b\n[PROOFSTEP]\nrefine' le_antisymm _ (e_add.toAddHom.le_map_tsub e.monotone a b)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\ninst\u271d\u2077 : Preorder M\ninst\u271d\u2076 : Add M\ninst\u271d\u2075 : Sub M\ninst\u271d\u2074 : OrderedSub M\ninst\u271d\u00b3 : PartialOrder N\ninst\u271d\u00b2 : Add N\ninst\u271d\u00b9 : Sub N\ninst\u271d : OrderedSub N\ne : M \u2243o N\nh_add : \u2200 (a b : M), \u2191e (a + b) = \u2191e a + \u2191e b\na b : M\ne_add : M \u2243+ N := { toEquiv := e.toEquiv, map_add' := h_add }\n\u22a2 \u2191e (a - b) \u2264 \u2191e a - \u2191e b\n[PROOFSTEP]\nsuffices e (e.symm (e a) - e.symm (e b)) \u2264 e (e.symm (e a - e b)) by simpa\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\ninst\u271d\u2077 : Preorder M\ninst\u271d\u2076 : Add M\ninst\u271d\u2075 : Sub M\ninst\u271d\u2074 : OrderedSub M\ninst\u271d\u00b3 : PartialOrder N\ninst\u271d\u00b2 : Add N\ninst\u271d\u00b9 : Sub N\ninst\u271d : OrderedSub N\ne : M \u2243o N\nh_add : \u2200 (a b : M), \u2191e (a + b) = \u2191e a + \u2191e b\na b : M\ne_add : M \u2243+ N := { toEquiv := e.toEquiv, map_add' := h_add }\nthis : \u2191e (\u2191(symm e) (\u2191e a) - \u2191(symm e) (\u2191e b)) \u2264 \u2191e (\u2191(symm e) (\u2191e a - \u2191e b))\n\u22a2 \u2191e (a - b) \u2264 \u2191e a - \u2191e b\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nN : Type u_4\ninst\u271d\u2077 : Preorder M\ninst\u271d\u2076 : Add M\ninst\u271d\u2075 : Sub M\ninst\u271d\u2074 : OrderedSub M\ninst\u271d\u00b3 : PartialOrder N\ninst\u271d\u00b2 : Add N\ninst\u271d\u00b9 : Sub N\ninst\u271d : OrderedSub N\ne : M \u2243o N\nh_add : \u2200 (a b : M), \u2191e (a + b) = \u2191e a + \u2191e b\na b : M\ne_add : M \u2243+ N := { toEquiv := e.toEquiv, map_add' := h_add }\n\u22a2 \u2191e (\u2191(symm e) (\u2191e a) - \u2191(symm e) (\u2191e b)) \u2264 \u2191e (\u2191(symm e) (\u2191e a - \u2191e b))\n[PROOFSTEP]\nexact e.monotone (e_add.symm.toAddHom.le_map_tsub e.symm.monotone _ _)\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Sub.Basic", "llama_tokens": 1823, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943925708562, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4122420867343523}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)\n\u22a2 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\n[PROOFSTEP]\nhave : DecidableEq \u03b9 := by classical infer_instance\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)\n\u22a2 DecidableEq \u03b9\n[PROOFSTEP]\nclassical infer_instance\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)\n\u22a2 DecidableEq \u03b9\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)\nthis : DecidableEq \u03b9\n\u22a2 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\n[PROOFSTEP]\nobtain hs | hs := le_total (Fintype.card \u03b1) s.card\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)\nthis : DecidableEq \u03b9\nhs : Fintype.card \u03b1 \u2264 card s\n\u22a2 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\n[PROOFSTEP]\nrw [tsub_eq_zero_of_le hs, pow_zero]\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)\nthis : DecidableEq \u03b9\nhs : Fintype.card \u03b1 \u2264 card s\n\u22a2 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 1\n[PROOFSTEP]\nrefine'\n (card_le_of_subset <|\n biUnion_subset.2 fun i hi a ha \u21a6 mem_compl.2 <| not_mem_singleton.2 <| (hf _ hi).ne_bot ha).trans_eq\n _\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)\nthis : DecidableEq \u03b9\nhs : Fintype.card \u03b1 \u2264 card s\n\u22a2 card {\u22a5}\u1d9c = 2 ^ Fintype.card \u03b1 - 1\n[PROOFSTEP]\nrw [card_compl, Fintype.card_finset, card_singleton]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)\nthis : DecidableEq \u03b9\nhs : card s \u2264 Fintype.card \u03b1\n\u22a2 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\n[PROOFSTEP]\ninduction' s using Finset.cons_induction with i s hi ih generalizing f\n[GOAL]\ncase inr.empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s \u2264 Fintype.card \u03b1\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Set.Intersecting \u2191(f i)\nhs : card \u2205 \u2264 Fintype.card \u03b1\n\u22a2 card (Finset.biUnion \u2205 f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card \u2205)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\n\u22a2 card (Finset.biUnion (cons i s hi) f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card (cons i s hi))\n[PROOFSTEP]\nset f' : \u03b9 \u2192 Finset (Finset \u03b1) := fun j \u21a6 if hj : j \u2208 cons i s hi then (hf j hj).exists_card_eq.choose else \u2205\n[GOAL]\ncase inr.cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\n\u22a2 card (Finset.biUnion (cons i s hi) f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card (cons i s hi))\n[PROOFSTEP]\nhave hf\u2081 :\n \u2200 j, j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * (f' j).card = 2 ^ Fintype.card \u03b1 \u2227 (f' j : Set (Finset \u03b1)).Intersecting :=\n by\n rintro j hj\n simp_rw [dif_pos hj, \u2190 Fintype.card_finset]\n exact Classical.choose_spec (hf j hj).exists_card_eq\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\n\u22a2 \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\n[PROOFSTEP]\nrintro j hj\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nj : \u03b9\nhj : j \u2208 cons i s hi\n\u22a2 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\n[PROOFSTEP]\nsimp_rw [dif_pos hj, \u2190 Fintype.card_finset]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nj : \u03b9\nhj : j \u2208 cons i s hi\n\u22a2 f j \u2286 Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t) \u2227\n 2 * card (Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)) =\n Fintype.card (Finset \u03b1) \u2227\n Set.Intersecting \u2191(Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t))\n[PROOFSTEP]\nexact Classical.choose_spec (hf j hj).exists_card_eq\n[GOAL]\ncase inr.cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\n\u22a2 card (Finset.biUnion (cons i s hi) f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card (cons i s hi))\n[PROOFSTEP]\nhave hf\u2082 : \u2200 j, j \u2208 cons i s hi \u2192 IsUpperSet (f' j : Set (Finset \u03b1)) :=\n by\n refine' fun j hj \u21a6 (hf\u2081 _ hj).2.2.isUpperSet' ((hf\u2081 _ hj).2.2.is_max_iff_card_eq.2 _)\n rw [Fintype.card_finset]\n exact (hf\u2081 _ hj).2.1\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\n\u22a2 \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n[PROOFSTEP]\nrefine' fun j hj \u21a6 (hf\u2081 _ hj).2.2.isUpperSet' ((hf\u2081 _ hj).2.2.is_max_iff_card_eq.2 _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nj : \u03b9\nhj : j \u2208 cons i s hi\n\u22a2 2 * card (f' j) = Fintype.card (Finset \u03b1)\n[PROOFSTEP]\nrw [Fintype.card_finset]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nj : \u03b9\nhj : j \u2208 cons i s hi\n\u22a2 2 * card (f' j) = 2 ^ Fintype.card \u03b1\n[PROOFSTEP]\nexact (hf\u2081 _ hj).2.1\n[GOAL]\ncase inr.cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 card (Finset.biUnion (cons i s hi) f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card (cons i s hi))\n[PROOFSTEP]\nrefine' (card_le_of_subset <| biUnion_mono fun j hj \u21a6 (hf\u2081 _ hj).1).trans _\n[GOAL]\ncase inr.cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 card (Finset.biUnion (cons i s hi) fun j => f' j) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card (cons i s hi))\n[PROOFSTEP]\nnth_rw 1 [cons_eq_insert i]\n[GOAL]\ncase inr.cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 card (Finset.biUnion (insert i s) fun j => f' j) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card (cons i s hi))\n[PROOFSTEP]\nrw [biUnion_insert]\n[GOAL]\ncase inr.cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 card (f' i \u222a Finset.biUnion s fun j => f' j) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card (cons i s hi))\n[PROOFSTEP]\nrefine' (card_mono <| @le_sup_sdiff _ _ _ <| f' i).trans ((card_union_le _ _).trans _)\n[GOAL]\ncase inr.cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 card (f' i) + card ((f' i \u222a Finset.biUnion s fun j => f' j) \\ f' i) \u2264\n 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card (cons i s hi))\n[PROOFSTEP]\nrw [union_sdiff_left, sdiff_eq_inter_compl]\n[GOAL]\ncase inr.cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 card (f' i) + card ((Finset.biUnion s fun j => f' j) \u2229 (f' i)\u1d9c) \u2264\n 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card (cons i s hi))\n[PROOFSTEP]\nrefine' le_of_mul_le_mul_left _ (pow_pos (zero_lt_two' \u2115) <| Fintype.card \u03b1 + 1)\n[GOAL]\ncase inr.cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 2 ^ (Fintype.card \u03b1 + 1) * (card (f' i) + card ((Finset.biUnion s fun j => f' j) \u2229 (f' i)\u1d9c)) \u2264\n 2 ^ (Fintype.card \u03b1 + 1) * (2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card (cons i s hi)))\n[PROOFSTEP]\nrw [pow_succ', mul_add, mul_assoc, mul_comm _ 2, mul_assoc]\n[GOAL]\ncase inr.cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 2 ^ Fintype.card \u03b1 * (2 * card (f' i)) +\n 2 * (2 ^ Fintype.card \u03b1 * card ((Finset.biUnion s fun j => f' j) \u2229 (f' i)\u1d9c)) \u2264\n 2 * 2 ^ Fintype.card \u03b1 * (2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card (cons i s hi)))\n[PROOFSTEP]\nrefine'\n (add_le_add ((mul_le_mul_left <| pow_pos (zero_lt_two' \u2115) _).2 (hf\u2081 _ <| mem_cons_self _ _).2.2.card_le) <|\n (mul_le_mul_left <| zero_lt_two' \u2115).2 <| IsUpperSet.card_inter_le_finset _ _).trans\n _\n[GOAL]\ncase inr.cons.refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 IsUpperSet \u2191(Finset.biUnion s fun j => f' j)\n[PROOFSTEP]\nrw [coe_biUnion]\n[GOAL]\ncase inr.cons.refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 IsUpperSet (\u22c3 (x : \u03b9) (_ : x \u2208 \u2191s), \u2191(f' x))\n[PROOFSTEP]\nexact isUpperSet_iUnion\u2082 fun i hi \u21a6 hf\u2082 _ <| subset_cons _ hi\n[GOAL]\ncase inr.cons.refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 IsLowerSet \u2191(f' i)\u1d9c\n[PROOFSTEP]\nrw [coe_compl]\n[GOAL]\ncase inr.cons.refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 IsLowerSet (\u2191(f' i))\u1d9c\n[PROOFSTEP]\nexact (hf\u2082 _ <| mem_cons_self _ _).compl\n[GOAL]\ncase inr.cons.refine'_3\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 2 ^ Fintype.card \u03b1 * Fintype.card (Finset \u03b1) + 2 * (card (Finset.biUnion s fun j => f' j) * card (f' i)\u1d9c) \u2264\n 2 * 2 ^ Fintype.card \u03b1 * (2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card (cons i s hi)))\n[PROOFSTEP]\nrw [mul_tsub, card_compl, Fintype.card_finset, mul_left_comm, mul_tsub, (hf\u2081 _ <| mem_cons_self _ _).2.1, two_mul,\n add_tsub_cancel_left, \u2190 mul_tsub, \u2190 mul_two, mul_assoc, \u2190 add_mul, mul_comm]\n[GOAL]\ncase inr.cons.refine'_3\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 2 ^ Fintype.card \u03b1 * (2 ^ Fintype.card \u03b1 + card (Finset.biUnion s fun j => f' j)) \u2264\n 2 ^ Fintype.card \u03b1 * (2 * (2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card (cons i s hi))))\n[PROOFSTEP]\nrefine' mul_le_mul_left' _ _\n[GOAL]\ncase inr.cons.refine'_3\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 2 ^ Fintype.card \u03b1 + card (Finset.biUnion s fun j => f' j) \u2264\n 2 * (2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card (cons i s hi)))\n[PROOFSTEP]\nrefine'\n (add_le_add_left (ih _ (fun i hi \u21a6 (hf\u2081 _ <| subset_cons _ hi).2.2) ((card_le_of_subset <| subset_cons _).trans hs))\n _).trans\n _\n[GOAL]\ncase inr.cons.refine'_3\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Nonempty \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 Finset (Finset \u03b1)\nhf\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Set.Intersecting \u2191(f\u271d i)\nthis : DecidableEq \u03b9\nhs\u271d : card s\u271d \u2264 Fintype.card \u03b1\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nih :\n \u2200 (f : \u03b9 \u2192 Finset (Finset \u03b1)),\n (\u2200 (i : \u03b9), i \u2208 s \u2192 Set.Intersecting \u2191(f i)) \u2192\n card s \u2264 Fintype.card \u03b1 \u2192 card (Finset.biUnion s f) \u2264 2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)\nf : \u03b9 \u2192 Finset (Finset \u03b1)\nhf : \u2200 (i_1 : \u03b9), i_1 \u2208 cons i s hi \u2192 Set.Intersecting \u2191(f i_1)\nhs : card (cons i s hi) \u2264 Fintype.card \u03b1\nf' : \u03b9 \u2192 Finset (Finset \u03b1) :=\n fun j =>\n if hj : j \u2208 cons i s hi then\n Exists.choose (_ : \u2203 t, f j \u2286 t \u2227 2 * card t = Fintype.card (Finset \u03b1) \u2227 Set.Intersecting \u2191t)\n else \u2205\nhf\u2081 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 f j \u2286 f' j \u2227 2 * card (f' j) = 2 ^ Fintype.card \u03b1 \u2227 Set.Intersecting \u2191(f' j)\nhf\u2082 : \u2200 (j : \u03b9), j \u2208 cons i s hi \u2192 IsUpperSet \u2191(f' j)\n\u22a2 2 ^ Fintype.card \u03b1 + (2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card s)) \u2264\n 2 * (2 ^ Fintype.card \u03b1 - 2 ^ (Fintype.card \u03b1 - card (cons i s hi)))\n[PROOFSTEP]\nrw [mul_tsub, two_mul, \u2190 pow_succ, \u2190 add_tsub_assoc_of_le (pow_le_pow' (one_le_two : (1 : \u2115) \u2264 2) tsub_le_self),\n tsub_add_eq_add_tsub hs, card_cons, add_tsub_add_eq_tsub_right]\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SetFamily.Kleitman", "llama_tokens": 16402, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943925708562, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4122420867343523}} {"text": "[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS x : Submonoid M\n\u22a2 (fun S =>\n {\n toSubsemigroup :=\n { carrier := \u2191Additive.ofMul \u207b\u00b9' \u2191S,\n mul_mem' :=\n (_ : \u2200 {a b : M}, a \u2208 \u2191Additive.ofMul \u207b\u00b9' \u2191S \u2192 b \u2208 \u2191Additive.ofMul \u207b\u00b9' \u2191S \u2192 a + b \u2208 S.carrier) },\n one_mem' := (_ : 0 \u2208 S.carrier) })\n ((fun S =>\n {\n toAddSubsemigroup :=\n { carrier := \u2191Additive.toMul \u207b\u00b9' \u2191S,\n add_mem' :=\n (_ :\n \u2200 {a b : Additive M},\n a \u2208 \u2191Additive.toMul \u207b\u00b9' \u2191S \u2192\n b \u2208 \u2191Additive.toMul \u207b\u00b9' \u2191S \u2192 \u2191Additive.toMul a * \u2191Additive.toMul b \u2208 S.carrier) },\n zero_mem' := (_ : 1 \u2208 S.carrier) })\n x) =\n x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS : Submonoid M\ntoSubsemigroup\u271d : Subsemigroup M\none_mem'\u271d : 1 \u2208 toSubsemigroup\u271d.carrier\n\u22a2 (fun S =>\n {\n toSubsemigroup :=\n { carrier := \u2191Additive.ofMul \u207b\u00b9' \u2191S,\n mul_mem' :=\n (_ : \u2200 {a b : M}, a \u2208 \u2191Additive.ofMul \u207b\u00b9' \u2191S \u2192 b \u2208 \u2191Additive.ofMul \u207b\u00b9' \u2191S \u2192 a + b \u2208 S.carrier) },\n one_mem' := (_ : 0 \u2208 S.carrier) })\n ((fun S =>\n {\n toAddSubsemigroup :=\n { carrier := \u2191Additive.toMul \u207b\u00b9' \u2191S,\n add_mem' :=\n (_ :\n \u2200 {a b : Additive M},\n a \u2208 \u2191Additive.toMul \u207b\u00b9' \u2191S \u2192\n b \u2208 \u2191Additive.toMul \u207b\u00b9' \u2191S \u2192 \u2191Additive.toMul a * \u2191Additive.toMul b \u2208 S.carrier) },\n zero_mem' := (_ : 1 \u2208 S.carrier) })\n { toSubsemigroup := toSubsemigroup\u271d, one_mem' := one_mem'\u271d }) =\n { toSubsemigroup := toSubsemigroup\u271d, one_mem' := one_mem'\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS : Submonoid M\nx : AddSubmonoid (Additive M)\n\u22a2 (fun S =>\n {\n toAddSubsemigroup :=\n { carrier := \u2191Additive.toMul \u207b\u00b9' \u2191S,\n add_mem' :=\n (_ :\n \u2200 {a b : Additive M},\n a \u2208 \u2191Additive.toMul \u207b\u00b9' \u2191S \u2192\n b \u2208 \u2191Additive.toMul \u207b\u00b9' \u2191S \u2192 \u2191Additive.toMul a * \u2191Additive.toMul b \u2208 S.carrier) },\n zero_mem' := (_ : 1 \u2208 S.carrier) })\n ((fun S =>\n {\n toSubsemigroup :=\n { carrier := \u2191Additive.ofMul \u207b\u00b9' \u2191S,\n mul_mem' :=\n (_ : \u2200 {a b : M}, a \u2208 \u2191Additive.ofMul \u207b\u00b9' \u2191S \u2192 b \u2208 \u2191Additive.ofMul \u207b\u00b9' \u2191S \u2192 a + b \u2208 S.carrier) },\n one_mem' := (_ : 0 \u2208 S.carrier) })\n x) =\n x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS : Submonoid M\ntoAddSubsemigroup\u271d : AddSubsemigroup (Additive M)\nzero_mem'\u271d : 0 \u2208 toAddSubsemigroup\u271d.carrier\n\u22a2 (fun S =>\n {\n toAddSubsemigroup :=\n { carrier := \u2191Additive.toMul \u207b\u00b9' \u2191S,\n add_mem' :=\n (_ :\n \u2200 {a b : Additive M},\n a \u2208 \u2191Additive.toMul \u207b\u00b9' \u2191S \u2192\n b \u2208 \u2191Additive.toMul \u207b\u00b9' \u2191S \u2192 \u2191Additive.toMul a * \u2191Additive.toMul b \u2208 S.carrier) },\n zero_mem' := (_ : 1 \u2208 S.carrier) })\n ((fun S =>\n {\n toSubsemigroup :=\n { carrier := \u2191Additive.ofMul \u207b\u00b9' \u2191S,\n mul_mem' :=\n (_ : \u2200 {a b : M}, a \u2208 \u2191Additive.ofMul \u207b\u00b9' \u2191S \u2192 b \u2208 \u2191Additive.ofMul \u207b\u00b9' \u2191S \u2192 a + b \u2208 S.carrier) },\n one_mem' := (_ : 0 \u2208 S.carrier) })\n { toAddSubsemigroup := toAddSubsemigroup\u271d, zero_mem' := zero_mem'\u271d }) =\n { toAddSubsemigroup := toAddSubsemigroup\u271d, zero_mem' := zero_mem'\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : AddZeroClass A\nx : AddSubmonoid A\n\u22a2 (fun S =>\n {\n toAddSubsemigroup :=\n { carrier := \u2191Multiplicative.ofAdd \u207b\u00b9' \u2191S,\n add_mem' :=\n (_ :\n \u2200 {a b : A},\n a \u2208 \u2191Multiplicative.ofAdd \u207b\u00b9' \u2191S \u2192 b \u2208 \u2191Multiplicative.ofAdd \u207b\u00b9' \u2191S \u2192 a * b \u2208 S.carrier) },\n zero_mem' := (_ : 1 \u2208 S.carrier) })\n ((fun S =>\n {\n toSubsemigroup :=\n { carrier := \u2191Multiplicative.toAdd \u207b\u00b9' \u2191S,\n mul_mem' :=\n (_ :\n \u2200 {a b : Multiplicative A},\n a \u2208 \u2191Multiplicative.toAdd \u207b\u00b9' \u2191S \u2192\n b \u2208 \u2191Multiplicative.toAdd \u207b\u00b9' \u2191S \u2192\n \u2191Multiplicative.toAdd a + \u2191Multiplicative.toAdd b \u2208 S.carrier) },\n one_mem' := (_ : 0 \u2208 S.carrier) })\n x) =\n x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : AddZeroClass A\ntoAddSubsemigroup\u271d : AddSubsemigroup A\nzero_mem'\u271d : 0 \u2208 toAddSubsemigroup\u271d.carrier\n\u22a2 (fun S =>\n {\n toAddSubsemigroup :=\n { carrier := \u2191Multiplicative.ofAdd \u207b\u00b9' \u2191S,\n add_mem' :=\n (_ :\n \u2200 {a b : A},\n a \u2208 \u2191Multiplicative.ofAdd \u207b\u00b9' \u2191S \u2192 b \u2208 \u2191Multiplicative.ofAdd \u207b\u00b9' \u2191S \u2192 a * b \u2208 S.carrier) },\n zero_mem' := (_ : 1 \u2208 S.carrier) })\n ((fun S =>\n {\n toSubsemigroup :=\n { carrier := \u2191Multiplicative.toAdd \u207b\u00b9' \u2191S,\n mul_mem' :=\n (_ :\n \u2200 {a b : Multiplicative A},\n a \u2208 \u2191Multiplicative.toAdd \u207b\u00b9' \u2191S \u2192\n b \u2208 \u2191Multiplicative.toAdd \u207b\u00b9' \u2191S \u2192\n \u2191Multiplicative.toAdd a + \u2191Multiplicative.toAdd b \u2208 S.carrier) },\n one_mem' := (_ : 0 \u2208 S.carrier) })\n { toAddSubsemigroup := toAddSubsemigroup\u271d, zero_mem' := zero_mem'\u271d }) =\n { toAddSubsemigroup := toAddSubsemigroup\u271d, zero_mem' := zero_mem'\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : AddZeroClass A\nx : Submonoid (Multiplicative A)\n\u22a2 (fun S =>\n {\n toSubsemigroup :=\n { carrier := \u2191Multiplicative.toAdd \u207b\u00b9' \u2191S,\n mul_mem' :=\n (_ :\n \u2200 {a b : Multiplicative A},\n a \u2208 \u2191Multiplicative.toAdd \u207b\u00b9' \u2191S \u2192\n b \u2208 \u2191Multiplicative.toAdd \u207b\u00b9' \u2191S \u2192\n \u2191Multiplicative.toAdd a + \u2191Multiplicative.toAdd b \u2208 S.carrier) },\n one_mem' := (_ : 0 \u2208 S.carrier) })\n ((fun S =>\n {\n toAddSubsemigroup :=\n { carrier := \u2191Multiplicative.ofAdd \u207b\u00b9' \u2191S,\n add_mem' :=\n (_ :\n \u2200 {a b : A},\n a \u2208 \u2191Multiplicative.ofAdd \u207b\u00b9' \u2191S \u2192 b \u2208 \u2191Multiplicative.ofAdd \u207b\u00b9' \u2191S \u2192 a * b \u2208 S.carrier) },\n zero_mem' := (_ : 1 \u2208 S.carrier) })\n x) =\n x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : AddZeroClass A\ntoSubsemigroup\u271d : Subsemigroup (Multiplicative A)\none_mem'\u271d : 1 \u2208 toSubsemigroup\u271d.carrier\n\u22a2 (fun S =>\n {\n toSubsemigroup :=\n { carrier := \u2191Multiplicative.toAdd \u207b\u00b9' \u2191S,\n mul_mem' :=\n (_ :\n \u2200 {a b : Multiplicative A},\n a \u2208 \u2191Multiplicative.toAdd \u207b\u00b9' \u2191S \u2192\n b \u2208 \u2191Multiplicative.toAdd \u207b\u00b9' \u2191S \u2192\n \u2191Multiplicative.toAdd a + \u2191Multiplicative.toAdd b \u2208 S.carrier) },\n one_mem' := (_ : 0 \u2208 S.carrier) })\n ((fun S =>\n {\n toAddSubsemigroup :=\n { carrier := \u2191Multiplicative.ofAdd \u207b\u00b9' \u2191S,\n add_mem' :=\n (_ :\n \u2200 {a b : A},\n a \u2208 \u2191Multiplicative.ofAdd \u207b\u00b9' \u2191S \u2192 b \u2208 \u2191Multiplicative.ofAdd \u207b\u00b9' \u2191S \u2192 a * b \u2208 S.carrier) },\n zero_mem' := (_ : 1 \u2208 S.carrier) })\n { toSubsemigroup := toSubsemigroup\u271d, one_mem' := one_mem'\u271d }) =\n { toSubsemigroup := toSubsemigroup\u271d, one_mem' := one_mem'\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS\u271d : Submonoid M\nF : Type u_4\nmc : MonoidHomClass F M N\nf : F\nS : Submonoid N\na\u271d b\u271d : M\nha : a\u271d \u2208 \u2191f \u207b\u00b9' \u2191S\nhb : b\u271d \u2208 \u2191f \u207b\u00b9' \u2191S\n\u22a2 \u2191f (a\u271d * b\u271d) \u2208 S\n[PROOFSTEP]\nrw [map_mul]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS\u271d : Submonoid M\nF : Type u_4\nmc : MonoidHomClass F M N\nf : F\nS : Submonoid N\na\u271d b\u271d : M\nha : a\u271d \u2208 \u2191f \u207b\u00b9' \u2191S\nhb : b\u271d \u2208 \u2191f \u207b\u00b9' \u2191S\n\u22a2 \u2191f a\u271d * \u2191f b\u271d \u2208 S\n[PROOFSTEP]\nexact S.mul_mem ha hb\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS\u271d : Submonoid M\nF : Type u_4\nmc : MonoidHomClass F M N\nf : F\nS : Submonoid N\n\u22a2 \u2191f 1 \u2208 S\n[PROOFSTEP]\nrw [map_one]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS\u271d : Submonoid M\nF : Type u_4\nmc : MonoidHomClass F M N\nf : F\nS : Submonoid N\n\u22a2 1 \u2208 S\n[PROOFSTEP]\nexact S.one_mem\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS\u271d : Submonoid M\nF : Type u_4\nmc : MonoidHomClass F M N\nS : Submonoid P\n\u22a2 \u2200 (x : P), x \u2208 comap (MonoidHom.id P) S \u2194 x \u2208 S\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS\u271d : Submonoid M\nF : Type u_4\nmc : MonoidHomClass F M N\nf : F\nS : Submonoid M\n\u22a2 \u2200 {a b : N}, a \u2208 \u2191f '' \u2191S \u2192 b \u2208 \u2191f '' \u2191S \u2192 a * b \u2208 \u2191f '' \u2191S\n[PROOFSTEP]\nrintro _ _ \u27e8x, hx, rfl\u27e9 \u27e8y, hy, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS\u271d : Submonoid M\nF : Type u_4\nmc : MonoidHomClass F M N\nf : F\nS : Submonoid M\nx : M\nhx : x \u2208 \u2191S\ny : M\nhy : y \u2208 \u2191S\n\u22a2 \u2191f x * \u2191f y \u2208 \u2191f '' \u2191S\n[PROOFSTEP]\nexact \u27e8x * y, S.mul_mem hx hy, by rw [map_mul]\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS\u271d : Submonoid M\nF : Type u_4\nmc : MonoidHomClass F M N\nf : F\nS : Submonoid M\nx : M\nhx : x \u2208 \u2191S\ny : M\nhy : y \u2208 \u2191S\n\u22a2 \u2191f (x * y) = \u2191f x * \u2191f y\n[PROOFSTEP]\nrw [map_mul]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS\u271d : Submonoid M\nF : Type u_4\nmc : MonoidHomClass F M N\nf : F\nS : Submonoid M\ny : N\n\u22a2 y \u2208 map f S \u2194 \u2203 x, x \u2208 S \u2227 \u2191f x = y\n[PROOFSTEP]\nrw [\u2190 bex_def]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS\u271d : Submonoid M\nF : Type u_4\nmc : MonoidHomClass F M N\nf : F\nS : Submonoid M\ny : N\n\u22a2 y \u2208 map f S \u2194 \u2203 x x_1, \u2191f x = y\n[PROOFSTEP]\nexact mem_image_iff_bex\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS\u271d : Submonoid M\nF : Type u_4\nmc : MonoidHomClass F M N\n\u03b9 : Type u_5\nf : F\nhf : Function.Injective \u2191f\nS : Submonoid M\nx : M\n\u22a2 x \u2208 comap f (map f S) \u2192 x \u2208 S\n[PROOFSTEP]\nsimp [mem_comap, mem_map, hf.eq_iff]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : MulOneClass P\nS\u271d : Submonoid M\nF : Type u_4\nmc : MonoidHomClass F M N\n\u03b9 : Type u_5\nf : F\nhf : Function.Surjective \u2191f\nS : Submonoid N\nx : N\nh : x \u2208 S\ny : M\nhy : \u2191f y = x\n\u22a2 y \u2208 comap f S \u2227 \u2191f y = x\n[PROOFSTEP]\nsimp [hy, h]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\n\u22a2 \u2200 (x y : { x // x \u2208 S' }),\n OneHom.toFun { toFun := Subtype.val, map_one' := (_ : \u21911 = \u21911) } (x * y) =\n OneHom.toFun { toFun := Subtype.val, map_one' := (_ : \u21911 = \u21911) } x *\n OneHom.toFun { toFun := Subtype.val, map_one' := (_ : \u21911 = \u21911) } y\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\n\u22a2 \u2200 (x y : { x // x \u2208 S }),\n OneHom.toFun { toFun := Subtype.val, map_one' := (_ : \u21911 = \u21911) } (x * y) =\n OneHom.toFun { toFun := Subtype.val, map_one' := (_ : \u21911 = \u21911) } x *\n OneHom.toFun { toFun := Subtype.val, map_one' := (_ : \u21911 = \u21911) } y\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Set M\nx\u271d\u00b9 : { x // x \u2208 \u2191(closure s) }\nx : M\nhx : x \u2208 \u2191(closure s)\nx\u271d : { val := x, property := hx } \u2208 \u22a4\n\u22a2 { val := x, property := hx } \u2208 closure (Subtype.val \u207b\u00b9' s)\n[PROOFSTEP]\nrefine' closure_induction' _ (fun g hg => subset_closure hg) _ (fun g\u2081 g\u2082 hg\u2081 hg\u2082 => _) hx\n[GOAL]\ncase refine'_1\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Set M\nx\u271d\u00b9 : { x // x \u2208 \u2191(closure s) }\nx : M\nhx : x \u2208 \u2191(closure s)\nx\u271d : { val := x, property := hx } \u2208 \u22a4\n\u22a2 { val := 1, property := (_ : 1 \u2208 closure s) } \u2208 closure (Subtype.val \u207b\u00b9' s)\n[PROOFSTEP]\nexact Submonoid.one_mem _\n[GOAL]\ncase refine'_2\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Set M\nx\u271d\u00b9 : { x // x \u2208 \u2191(closure s) }\nx : M\nhx : x \u2208 \u2191(closure s)\nx\u271d : { val := x, property := hx } \u2208 \u22a4\ng\u2081 : M\ng\u2082 : g\u2081 \u2208 closure s\nhg\u2081 : M\nhg\u2082 : hg\u2081 \u2208 closure s\n\u22a2 { val := g\u2081, property := g\u2082 } \u2208 closure (Subtype.val \u207b\u00b9' s) \u2192\n { val := hg\u2081, property := hg\u2082 } \u2208 closure (Subtype.val \u207b\u00b9' s) \u2192\n { val := g\u2081 * hg\u2081, property := (_ : g\u2081 * hg\u2081 \u2208 closure s) } \u2208 closure (Subtype.val \u207b\u00b9' s)\n[PROOFSTEP]\nexact Submonoid.mul_mem _\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nx : M \u00d7 N\n\u22a2 x \u2208 prod s \u22a4 \u2194 x \u2208 comap (MonoidHom.fst M N) s\n[PROOFSTEP]\nsimp [mem_prod, MonoidHom.coe_fst]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid N\nx : M \u00d7 N\n\u22a2 x \u2208 prod \u22a4 s \u2194 x \u2208 comap (MonoidHom.snd M N) s\n[PROOFSTEP]\nsimp [mem_prod, MonoidHom.coe_snd]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\n\u22a2 \u2191(prod \u22a5 \u22a5) = \u2191\u22a5\n[PROOFSTEP]\nsimp [coe_prod, Prod.one_eq_mk]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\n\u22a2 u \u2264 prod s t \u2194 map (fst M N) u \u2264 s \u2227 map (snd M N) u \u2264 t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\n\u22a2 u \u2264 prod s t \u2192 map (fst M N) u \u2264 s \u2227 map (snd M N) u \u2264 t\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nh : u \u2264 prod s t\n\u22a2 map (fst M N) u \u2264 s \u2227 map (snd M N) u \u2264 t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.left\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nh : u \u2264 prod s t\n\u22a2 map (fst M N) u \u2264 s\n[PROOFSTEP]\nrintro x \u27e8\u27e8y1, y2\u27e9, \u27e8hy1, rfl\u27e9\u27e9\n[GOAL]\ncase mp.left.intro.mk.intro\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nh : u \u2264 prod s t\ny1 : M\ny2 : N\nhy1 : (y1, y2) \u2208 \u2191u\n\u22a2 \u2191(fst M N) (y1, y2) \u2208 s\n[PROOFSTEP]\nexact (h hy1).1\n[GOAL]\ncase mp.right\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nh : u \u2264 prod s t\n\u22a2 map (snd M N) u \u2264 t\n[PROOFSTEP]\nrintro x \u27e8\u27e8y1, y2\u27e9, \u27e8hy1, rfl\u27e9\u27e9\n[GOAL]\ncase mp.right.intro.mk.intro\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nh : u \u2264 prod s t\ny1 : M\ny2 : N\nhy1 : (y1, y2) \u2208 \u2191u\n\u22a2 \u2191(snd M N) (y1, y2) \u2208 t\n[PROOFSTEP]\nexact (h hy1).2\n[GOAL]\ncase mpr\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\n\u22a2 map (fst M N) u \u2264 s \u2227 map (snd M N) u \u2264 t \u2192 u \u2264 prod s t\n[PROOFSTEP]\nrintro \u27e8hH, hK\u27e9 \u27e8x1, x2\u27e9 h\n[GOAL]\ncase mpr.intro.mk\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nhH : map (fst M N) u \u2264 s\nhK : map (snd M N) u \u2264 t\nx1 : M\nx2 : N\nh : (x1, x2) \u2208 u\n\u22a2 (x1, x2) \u2208 prod s t\n[PROOFSTEP]\nexact \u27e8hH \u27e8_, h, rfl\u27e9, hK \u27e8_, h, rfl\u27e9\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\n\u22a2 prod s t \u2264 u \u2194 map (inl M N) s \u2264 u \u2227 map (inr M N) t \u2264 u\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\n\u22a2 prod s t \u2264 u \u2192 map (inl M N) s \u2264 u \u2227 map (inr M N) t \u2264 u\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nh : prod s t \u2264 u\n\u22a2 map (inl M N) s \u2264 u \u2227 map (inr M N) t \u2264 u\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.left\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nh : prod s t \u2264 u\n\u22a2 map (inl M N) s \u2264 u\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase mp.left.intro.intro\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nh : prod s t \u2264 u\nx : M\nhx : x \u2208 \u2191s\n\u22a2 \u2191(inl M N) x \u2208 u\n[PROOFSTEP]\napply h\n[GOAL]\ncase mp.left.intro.intro.a\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nh : prod s t \u2264 u\nx : M\nhx : x \u2208 \u2191s\n\u22a2 \u2191(inl M N) x \u2208 prod s t\n[PROOFSTEP]\nexact \u27e8hx, Submonoid.one_mem _\u27e9\n[GOAL]\ncase mp.right\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nh : prod s t \u2264 u\n\u22a2 map (inr M N) t \u2264 u\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase mp.right.intro.intro\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nh : prod s t \u2264 u\nx : N\nhx : x \u2208 \u2191t\n\u22a2 \u2191(inr M N) x \u2208 u\n[PROOFSTEP]\napply h\n[GOAL]\ncase mp.right.intro.intro.a\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nh : prod s t \u2264 u\nx : N\nhx : x \u2208 \u2191t\n\u22a2 \u2191(inr M N) x \u2208 prod s t\n[PROOFSTEP]\nexact \u27e8Submonoid.one_mem _, hx\u27e9\n[GOAL]\ncase mpr\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\n\u22a2 map (inl M N) s \u2264 u \u2227 map (inr M N) t \u2264 u \u2192 prod s t \u2264 u\n[PROOFSTEP]\nrintro \u27e8hH, hK\u27e9 \u27e8x1, x2\u27e9 \u27e8h1, h2\u27e9\n[GOAL]\ncase mpr.intro.mk.intro\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nhH : map (inl M N) s \u2264 u\nhK : map (inr M N) t \u2264 u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst \u2208 \u2191s\nh2 : (x1, x2).snd \u2208 \u2191t\n\u22a2 (x1, x2) \u2208 u\n[PROOFSTEP]\nhave h1' : inl M N x1 \u2208 u := by\n apply hH\n simpa using h1\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nhH : map (inl M N) s \u2264 u\nhK : map (inr M N) t \u2264 u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst \u2208 \u2191s\nh2 : (x1, x2).snd \u2208 \u2191t\n\u22a2 \u2191(inl M N) x1 \u2208 u\n[PROOFSTEP]\napply hH\n[GOAL]\ncase a\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nhH : map (inl M N) s \u2264 u\nhK : map (inr M N) t \u2264 u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst \u2208 \u2191s\nh2 : (x1, x2).snd \u2208 \u2191t\n\u22a2 \u2191(inl M N) x1 \u2208 map (inl M N) s\n[PROOFSTEP]\nsimpa using h1\n[GOAL]\ncase mpr.intro.mk.intro\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nhH : map (inl M N) s \u2264 u\nhK : map (inr M N) t \u2264 u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst \u2208 \u2191s\nh2 : (x1, x2).snd \u2208 \u2191t\nh1' : \u2191(inl M N) x1 \u2208 u\n\u22a2 (x1, x2) \u2208 u\n[PROOFSTEP]\nhave h2' : inr M N x2 \u2208 u := by\n apply hK\n simpa using h2\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nhH : map (inl M N) s \u2264 u\nhK : map (inr M N) t \u2264 u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst \u2208 \u2191s\nh2 : (x1, x2).snd \u2208 \u2191t\nh1' : \u2191(inl M N) x1 \u2208 u\n\u22a2 \u2191(inr M N) x2 \u2208 u\n[PROOFSTEP]\napply hK\n[GOAL]\ncase a\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nhH : map (inl M N) s \u2264 u\nhK : map (inr M N) t \u2264 u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst \u2208 \u2191s\nh2 : (x1, x2).snd \u2208 \u2191t\nh1' : \u2191(inl M N) x1 \u2208 u\n\u22a2 \u2191(inr M N) x2 \u2208 map (inr M N) t\n[PROOFSTEP]\nsimpa using h2\n[GOAL]\ncase mpr.intro.mk.intro\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M \u00d7 N)\nhH : map (inl M N) s \u2264 u\nhK : map (inr M N) t \u2264 u\nx1 : M\nx2 : N\nh1 : (x1, x2).fst \u2208 \u2191s\nh2 : (x1, x2).snd \u2208 \u2191t\nh1' : \u2191(inl M N) x1 \u2208 u\nh2' : \u2191(inr M N) x2 \u2208 u\n\u22a2 (x1, x2) \u2208 u\n[PROOFSTEP]\nsimpa using Submonoid.mul_mem _ h1' h2'\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\nmc : MonoidHomClass F M N\ng : N \u2192* P\nf : M \u2192* N\n\u22a2 map g (mrange f) = mrange (comp g f)\n[PROOFSTEP]\nsimpa only [mrange_eq_map] using (\u22a4 : Submonoid M).map_map g f\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\nmc : MonoidHomClass F M N\nf : F\n\u22a2 \u2191(mrange f) = \u2191\u22a4 \u2194 Set.range \u2191f = Set.univ\n[PROOFSTEP]\nrw [coe_mrange, coe_top]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\nmc : MonoidHomClass F M N\nf : M \u2192* N\n\u22a2 mrange (restrict f S) = map f S\n[PROOFSTEP]\nsimp [SetLike.ext_iff]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\nmc : MonoidHomClass F M N\nf : M \u2192* N\n\u22a2 mker (mrangeRestrict f) = mker f\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\nmc : MonoidHomClass F M N\nf : M \u2192* N\nx : M\n\u22a2 x \u2208 mker (mrangeRestrict f) \u2194 x \u2208 mker f\n[PROOFSTEP]\nchange (\u27e8f x, _\u27e9 : mrange f) = \u27e81, _\u27e9 \u2194 f x = 1\n[GOAL]\ncase h\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\nmc : MonoidHomClass F M N\nf : M \u2192* N\nx : M\n\u22a2 { val := \u2191f x, property := (_ : \u2203 y, \u2191f y = \u2191f x) } = { val := 1, property := (_ : 1 \u2208 mrange f) } \u2194 \u2191f x = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\nmc : MonoidHomClass F M N\n\u22a2 mker 1 = \u22a4\n[PROOFSTEP]\next\n[GOAL]\ncase h\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\nmc : MonoidHomClass F M N\nx\u271d : M\n\u22a2 x\u271d \u2208 mker 1 \u2194 x\u271d \u2208 \u22a4\n[PROOFSTEP]\nsimp [mem_mker]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u2075 : MulOneClass M\ninst\u271d\u2074 : MulOneClass N\ninst\u271d\u00b3 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d\u00b2 : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\nmc : MonoidHomClass F M N\nM' : Type u_6\nN' : Type u_7\ninst\u271d\u00b9 : MulOneClass M'\ninst\u271d : MulOneClass N'\nf : M \u2192* N\ng : M' \u2192* N'\n\u22a2 mker (prodMap f g) = prod (mker f) (mker g)\n[PROOFSTEP]\nrw [\u2190 comap_bot', \u2190 comap_bot', \u2190 comap_bot', \u2190 prod_map_comap_prod', bot_prod_bot]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\nmc : MonoidHomClass F M N\n\u22a2 mker (inl M N) = \u22a5\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\nmc : MonoidHomClass F M N\nx : M\n\u22a2 x \u2208 mker (inl M N) \u2194 x \u2208 \u22a5\n[PROOFSTEP]\nsimp [mem_mker]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\nmc : MonoidHomClass F M N\n\u22a2 mker (inr M N) = \u22a5\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\nmc : MonoidHomClass F M N\nx : N\n\u22a2 x \u2208 mker (inr M N) \u2194 x \u2208 \u22a5\n[PROOFSTEP]\nsimp [mem_mker]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\nmc : MonoidHomClass F M N\nf : M \u2192* N\nM' : Submonoid M\n\u22a2 Function.Surjective \u2191(submonoidMap f M')\n[PROOFSTEP]\nrintro \u27e8_, x, hx, rfl\u27e9\n[GOAL]\ncase mk.intro.intro\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\nmc : MonoidHomClass F M N\nf : M \u2192* N\nM' : Submonoid M\nx : M\nhx : x \u2208 \u2191M'\n\u22a2 \u2203 a, \u2191(submonoidMap f M') a = { val := \u2191f x, property := (_ : \u2203 a, a \u2208 \u2191M' \u2227 \u2191f a = \u2191f x) }\n[PROOFSTEP]\nexact \u27e8\u27e8x, hx\u27e9, rfl\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\n\u22a2 mrange (inl M N) = prod \u22a4 \u22a5\n[PROOFSTEP]\nsimpa only [mrange_eq_map] using map_inl \u22a4\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\n\u22a2 mrange (inr M N) = prod \u22a5 \u22a4\n[PROOFSTEP]\nsimpa only [mrange_eq_map] using map_inr \u22a4\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\n\u22a2 prod s t = \u22a5 \u2194 s = \u22a5 \u2227 t = \u22a5\n[PROOFSTEP]\nsimp only [eq_bot_iff, prod_le_iff, (gc_map_comap _).le_iff_le, comap_bot', mker_inl, mker_inr]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nt : Submonoid N\n\u22a2 prod s t = \u22a4 \u2194 s = \u22a4 \u2227 t = \u22a4\n[PROOFSTEP]\nsimp only [eq_top_iff, le_prod_iff, \u2190 (gc_map_comap _).le_iff_le, \u2190 mrange_eq_map, mrange_fst, mrange_snd]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\n\u22a2 mrange (inl M N) \u2294 mrange (inr M N) = \u22a4\n[PROOFSTEP]\nsimp only [mrange_inl, mrange_inr, prod_bot_sup_bot_prod, top_prod_top]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\n\u22a2 (\u2200 (x : M), x \u2208 S \u2194 x \u2208 \u22a5) \u2194 \u2200 (x : M), x \u2208 S \u2192 x = 1\n[PROOFSTEP]\nsimp (config := { contextual := true }) [iff_def, S.one_mem]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS\u271d : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nS : Submonoid M\n\u22a2 (\u2203 x hx, { val := x, property := hx } \u2260 { val := 1, property := (_ : 1 \u2208 S) }) \u2194 \u2203 x, x \u2208 S \u2227 x \u2260 1\n[PROOFSTEP]\nsimp [Ne.def]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS\u271d : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nS : Submonoid M\n\u22a2 S = \u22a5 \u2228 Nontrivial { x // x \u2208 S }\n[PROOFSTEP]\nsimp only [eq_bot_iff_forall, nontrivial_iff_exists_ne_one, \u2190 not_forall, \u2190 not_imp, Classical.em]\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nT : Submonoid M\nf : M \u2192* N\ng : N \u2192 M\nh : Function.LeftInverse g \u2191f\nsrc\u271d : M \u2192* { x // x \u2208 MonoidHom.mrange f } := MonoidHom.mrangeRestrict f\nx : { x // x \u2208 MonoidHom.mrange f }\nx' : M\nhx' : \u2191f x' = \u2191x\n\u22a2 \u2191f (g \u2191x) = \u2191x\n[PROOFSTEP]\nrw [\u2190 hx', h x']\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ne : M \u2243* N\n\u22a2 equivMapOfInjective S \u2191e (_ : Function.Injective \u2191e) = MulEquiv.submonoidMap e S\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : MulOneClass M\ninst\u271d\u00b2 : MulOneClass N\ninst\u271d\u00b9 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ne : M \u2243* N\nx\u271d : { x // x \u2208 S }\n\u22a2 \u2191(\u2191(equivMapOfInjective S \u2191e (_ : Function.Injective \u2191e)) x\u271d) = \u2191(\u2191(MulEquiv.submonoidMap e S) x\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u2074 : MulOneClass M\ninst\u271d\u00b3 : MulOneClass N\ninst\u271d\u00b2 : MulOneClass P\nS\u271d : Submonoid M\nA : Type u_4\ninst\u271d\u00b9 : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nM' : Type u_5\n\u03b1 : Type u_6\n\u03b2 : Type u_7\ninst\u271d : Monoid M'\nS : Submonoid M'\n\u22a2 IsScalarTower { x // x \u2208 S } M' M'\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Submonoid.Operations", "llama_tokens": 17183, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754371026367, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.4120475905675761}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\na\u2081 a\u2082 : \u03b1\nb\u2081 : \u03b2 a\u2081\nb\u2082 : \u03b2 a\u2082\nh : { fst := a\u2081, snd := b\u2081 } = { fst := a\u2082, snd := b\u2082 }\n\u22a2 a\u2081 = a\u2082 \u2227 HEq b\u2081 b\u2082\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\na\u2081 : \u03b1\nb\u2081 : \u03b2 a\u2081\n\u22a2 a\u2081 = a\u2081 \u2227 HEq b\u2081 b\u2081\n[PROOFSTEP]\nexact \u27e8rfl, heq_of_eq rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\na\u2081 a\u2082 : \u03b1\nb\u2081 : \u03b2 a\u2081\nb\u2082 : \u03b2 a\u2082\nx\u271d : a\u2081 = a\u2082 \u2227 HEq b\u2081 b\u2082\nh\u2081 : a\u2081 = a\u2082\nh\u2082 : HEq b\u2081 b\u2082\n\u22a2 { fst := a\u2081, snd := b\u2081 } = { fst := a\u2082, snd := b\u2082 }\n[PROOFSTEP]\nsubst h\u2081\n[GOAL]\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2 a\u2081\nx\u271d : a\u2081 = a\u2081 \u2227 HEq b\u2081 b\u2082\nh\u2082 : HEq b\u2081 b\u2082\n\u22a2 { fst := a\u2081, snd := b\u2081 } = { fst := a\u2081, snd := b\u2082 }\n[PROOFSTEP]\nrw [eq_of_heq h\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nx\u2080 x\u2081 : Sigma \u03b2\nh\u2080 : x\u2080.fst = x\u2081.fst\nh\u2081 : HEq x\u2080.snd x\u2081.snd\n\u22a2 x\u2080 = x\u2081\n[PROOFSTEP]\ncases x\u2080\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nx\u2081 : Sigma \u03b2\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\nh\u2080 : { fst := fst\u271d, snd := snd\u271d }.fst = x\u2081.fst\nh\u2081 : HEq { fst := fst\u271d, snd := snd\u271d }.snd x\u2081.snd\n\u22a2 { fst := fst\u271d, snd := snd\u271d } = x\u2081\n[PROOFSTEP]\ncases x\u2081\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nfst\u271d\u00b9 : \u03b1\nsnd\u271d\u00b9 : \u03b2 fst\u271d\u00b9\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\nh\u2080 : { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst = { fst := fst\u271d, snd := snd\u271d }.fst\nh\u2081 : HEq { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.snd { fst := fst\u271d, snd := snd\u271d }.snd\n\u22a2 { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 } = { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\ncases h\u2080\n[GOAL]\ncase mk.mk.refl\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nfst\u271d : \u03b1\nsnd\u271d\u00b9 snd\u271d : \u03b2 fst\u271d\nh\u2081 : HEq { fst := fst\u271d, snd := snd\u271d\u00b9 }.snd { fst := fst\u271d, snd := snd\u271d }.snd\n\u22a2 { fst := fst\u271d, snd := snd\u271d\u00b9 } = { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\ncases h\u2081\n[GOAL]\ncase mk.mk.refl.refl\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\n\u22a2 { fst := fst\u271d, snd := snd\u271d } = { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nx\u2080 x\u2081 : Sigma \u03b2\n\u22a2 x\u2080 = x\u2081 \u2194 x\u2080.fst = x\u2081.fst \u2227 HEq x\u2080.snd x\u2081.snd\n[PROOFSTEP]\ncases x\u2080\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nx\u2081 : Sigma \u03b2\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\n\u22a2 { fst := fst\u271d, snd := snd\u271d } = x\u2081 \u2194\n { fst := fst\u271d, snd := snd\u271d }.fst = x\u2081.fst \u2227 HEq { fst := fst\u271d, snd := snd\u271d }.snd x\u2081.snd\n[PROOFSTEP]\ncases x\u2081\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nfst\u271d\u00b9 : \u03b1\nsnd\u271d\u00b9 : \u03b2 fst\u271d\u00b9\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\n\u22a2 { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 } = { fst := fst\u271d, snd := snd\u271d } \u2194\n { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst = { fst := fst\u271d, snd := snd\u271d }.fst \u2227\n HEq { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.snd { fst := fst\u271d, snd := snd\u271d }.snd\n[PROOFSTEP]\nexact Sigma.mk.inj_iff\n[GOAL]\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\n\u03b3 : Type u_7\ninst\u271d : Nonempty \u03b3\na b : \u03b1\nf : \u03b3 \u2192 \u03b2 a\ng : \u03b3 \u2192 \u03b2 b\nh : mk a \u2218 f = mk b \u2218 g\n\u22a2 a = b \u2227 HEq f g\n[PROOFSTEP]\nrcases\u2039Nonempty \u03b3\u203a with \u27e8i\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\n\u03b3 : Type u_7\ninst\u271d : Nonempty \u03b3\na b : \u03b1\nf : \u03b3 \u2192 \u03b2 a\ng : \u03b3 \u2192 \u03b2 b\nh : mk a \u2218 f = mk b \u2218 g\ni : \u03b3\n\u22a2 a = b \u2227 HEq f g\n[PROOFSTEP]\nobtain rfl : a = b := congr_arg Sigma.fst (congr_fun h i)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\n\u03b3 : Type u_7\ninst\u271d : Nonempty \u03b3\na : \u03b1\nf : \u03b3 \u2192 \u03b2 a\ni : \u03b3\ng : \u03b3 \u2192 \u03b2 a\nh : mk a \u2218 f = mk a \u2218 g\n\u22a2 a = a \u2227 HEq f g\n[PROOFSTEP]\nsimpa [Function.funext_iff] using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nf\u2081 : \u03b1\u2081 \u2192 \u03b1\u2082\nf\u2082 : (a : \u03b1\u2081) \u2192 \u03b2\u2081 a \u2192 \u03b2\u2082 (f\u2081 a)\nh\u2081 : Injective f\u2081\nh\u2082 : \u2200 (a : \u03b1\u2081), Injective (f\u2082 a)\ni : \u03b1\u2081\nx : \u03b2\u2081 i\nj : \u03b1\u2081\ny : \u03b2\u2081 j\nh : Sigma.map f\u2081 f\u2082 { fst := i, snd := x } = Sigma.map f\u2081 f\u2082 { fst := j, snd := y }\n\u22a2 { fst := i, snd := x } = { fst := j, snd := y }\n[PROOFSTEP]\nobtain rfl : i = j := h\u2081 (Sigma.mk.inj_iff.mp h).1\n[GOAL]\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nf\u2081 : \u03b1\u2081 \u2192 \u03b1\u2082\nf\u2082 : (a : \u03b1\u2081) \u2192 \u03b2\u2081 a \u2192 \u03b2\u2082 (f\u2081 a)\nh\u2081 : Injective f\u2081\nh\u2082 : \u2200 (a : \u03b1\u2081), Injective (f\u2082 a)\ni : \u03b1\u2081\nx y : \u03b2\u2081 i\nh : Sigma.map f\u2081 f\u2082 { fst := i, snd := x } = Sigma.map f\u2081 f\u2082 { fst := i, snd := y }\n\u22a2 { fst := i, snd := x } = { fst := i, snd := y }\n[PROOFSTEP]\nobtain rfl : x = y := h\u2082 i (sigma_mk_injective h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nf\u2081 : \u03b1\u2081 \u2192 \u03b1\u2082\nf\u2082 : (a : \u03b1\u2081) \u2192 \u03b2\u2081 a \u2192 \u03b2\u2082 (f\u2081 a)\nh\u2081 : Injective f\u2081\nh\u2082 : \u2200 (a : \u03b1\u2081), Injective (f\u2082 a)\ni : \u03b1\u2081\nx : \u03b2\u2081 i\nh : Sigma.map f\u2081 f\u2082 { fst := i, snd := x } = Sigma.map f\u2081 f\u2082 { fst := i, snd := x }\n\u22a2 { fst := i, snd := x } = { fst := i, snd := x }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nf\u2081 : \u03b1\u2081 \u2192 \u03b1\u2082\nf\u2082 : (a : \u03b1\u2081) \u2192 \u03b2\u2081 a \u2192 \u03b2\u2082 (f\u2081 a)\nh\u2081 : Surjective f\u2081\nh\u2082 : \u2200 (a : \u03b1\u2081), Surjective (f\u2082 a)\n\u22a2 Surjective (Sigma.map f\u2081 f\u2082)\n[PROOFSTEP]\nsimp only [Function.Surjective, Sigma.forall, h\u2081.forall]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nf\u2081 : \u03b1\u2081 \u2192 \u03b1\u2082\nf\u2082 : (a : \u03b1\u2081) \u2192 \u03b2\u2081 a \u2192 \u03b2\u2082 (f\u2081 a)\nh\u2081 : Surjective f\u2081\nh\u2082 : \u2200 (a : \u03b1\u2081), Surjective (f\u2082 a)\n\u22a2 \u2200 (x : \u03b1\u2081) (b : \u03b2\u2082 (f\u2081 x)), \u2203 a, Sigma.map f\u2081 f\u2082 a = { fst := f\u2081 x, snd := b }\n[PROOFSTEP]\nexact fun i \u21a6 (h\u2082 _).forall.2 fun x \u21a6 \u27e8\u27e8i, x\u27e9, rfl\u27e9\n[GOAL]\n\u03b1 : Sort u_1\n\u03b2 : \u03b1 \u2192 Sort u_2\nx\u2080 x\u2081 : PSigma \u03b2\nh\u2080 : x\u2080.fst = x\u2081.fst\nh\u2081 : HEq x\u2080.snd x\u2081.snd\n\u22a2 x\u2080 = x\u2081\n[PROOFSTEP]\ncases x\u2080\n[GOAL]\ncase mk\n\u03b1 : Sort u_1\n\u03b2 : \u03b1 \u2192 Sort u_2\nx\u2081 : PSigma \u03b2\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\nh\u2080 : { fst := fst\u271d, snd := snd\u271d }.fst = x\u2081.fst\nh\u2081 : HEq { fst := fst\u271d, snd := snd\u271d }.snd x\u2081.snd\n\u22a2 { fst := fst\u271d, snd := snd\u271d } = x\u2081\n[PROOFSTEP]\ncases x\u2081\n[GOAL]\ncase mk.mk\n\u03b1 : Sort u_1\n\u03b2 : \u03b1 \u2192 Sort u_2\nfst\u271d\u00b9 : \u03b1\nsnd\u271d\u00b9 : \u03b2 fst\u271d\u00b9\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\nh\u2080 : { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst = { fst := fst\u271d, snd := snd\u271d }.fst\nh\u2081 : HEq { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.snd { fst := fst\u271d, snd := snd\u271d }.snd\n\u22a2 { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 } = { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\ncases h\u2080\n[GOAL]\ncase mk.mk.refl\n\u03b1 : Sort u_1\n\u03b2 : \u03b1 \u2192 Sort u_2\nfst\u271d : \u03b1\nsnd\u271d\u00b9 snd\u271d : \u03b2 fst\u271d\nh\u2081 : HEq { fst := fst\u271d, snd := snd\u271d\u00b9 }.snd { fst := fst\u271d, snd := snd\u271d }.snd\n\u22a2 { fst := fst\u271d, snd := snd\u271d\u00b9 } = { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\ncases h\u2081\n[GOAL]\ncase mk.mk.refl.refl\n\u03b1 : Sort u_1\n\u03b2 : \u03b1 \u2192 Sort u_2\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\n\u22a2 { fst := fst\u271d, snd := snd\u271d } = { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Sort u_1\n\u03b2 : \u03b1 \u2192 Sort u_2\nx\u2080 x\u2081 : PSigma \u03b2\n\u22a2 x\u2080 = x\u2081 \u2194 x\u2080.fst = x\u2081.fst \u2227 HEq x\u2080.snd x\u2081.snd\n[PROOFSTEP]\ncases x\u2080\n[GOAL]\ncase mk\n\u03b1 : Sort u_1\n\u03b2 : \u03b1 \u2192 Sort u_2\nx\u2081 : PSigma \u03b2\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\n\u22a2 { fst := fst\u271d, snd := snd\u271d } = x\u2081 \u2194\n { fst := fst\u271d, snd := snd\u271d }.fst = x\u2081.fst \u2227 HEq { fst := fst\u271d, snd := snd\u271d }.snd x\u2081.snd\n[PROOFSTEP]\ncases x\u2081\n[GOAL]\ncase mk.mk\n\u03b1 : Sort u_1\n\u03b2 : \u03b1 \u2192 Sort u_2\nfst\u271d\u00b9 : \u03b1\nsnd\u271d\u00b9 : \u03b2 fst\u271d\u00b9\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\n\u22a2 { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 } = { fst := fst\u271d, snd := snd\u271d } \u2194\n { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst = { fst := fst\u271d, snd := snd\u271d }.fst \u2227\n HEq { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.snd { fst := fst\u271d, snd := snd\u271d }.snd\n[PROOFSTEP]\nexact PSigma.mk.inj_iff\n", "meta": {"mathlib_filename": "Mathlib.Data.Sigma.Basic", "llama_tokens": 4766, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228891883799, "lm_q2_score": 0.6224593241981982, "lm_q1q2_score": 0.4120200742755178}} {"text": "[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\nx : \u211a\n\u22a2 \u2191x \u2208 s\n[PROOFSTEP]\nsimpa only [Rat.cast_def] using div_mem (coe_int_mem s x.num) (coe_nat_mem s x.den)\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\na : \u211a\nx : { x // x \u2208 s }\n\u22a2 a \u2022 \u2191x \u2208 s\n[PROOFSTEP]\nsimpa only [Rat.smul_def] using mul_mem (coe_rat_mem s a) x.prop\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u21910 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u21911 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u2200 (x y : { x // x \u2208 s }), \u2191(x + y) = \u2191x + \u2191y\n[PROOFSTEP]\nintros _ _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\nx\u271d y\u271d : { x // x \u2208 s }\n\u22a2 \u2191(x\u271d + y\u271d) = \u2191x\u271d + \u2191y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u2200 (x y : { x // x \u2208 s }), \u2191(x * y) = \u2191x * \u2191y\n[PROOFSTEP]\nintros _ _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\nx\u271d y\u271d : { x // x \u2208 s }\n\u22a2 \u2191(x\u271d * y\u271d) = \u2191x\u271d * \u2191y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u2200 (x : { x // x \u2208 s }), \u2191(-x) = -\u2191x\n[PROOFSTEP]\nintros _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\nx\u271d : { x // x \u2208 s }\n\u22a2 \u2191(-x\u271d) = -\u2191x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u2200 (x y : { x // x \u2208 s }), \u2191(x - y) = \u2191x - \u2191y\n[PROOFSTEP]\nintros _ _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\nx\u271d y\u271d : { x // x \u2208 s }\n\u22a2 \u2191(x\u271d - y\u271d) = \u2191x\u271d - \u2191y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u2200 (x : { x // x \u2208 s }), \u2191x\u207b\u00b9 = (\u2191x)\u207b\u00b9\n[PROOFSTEP]\nintros _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\nx\u271d : { x // x \u2208 s }\n\u22a2 \u2191x\u271d\u207b\u00b9 = (\u2191x\u271d)\u207b\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u2200 (x y : { x // x \u2208 s }), \u2191(x / y) = \u2191x / \u2191y\n[PROOFSTEP]\nintros _ _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\nx\u271d y\u271d : { x // x \u2208 s }\n\u22a2 \u2191(x\u271d / y\u271d) = \u2191x\u271d / \u2191y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u2200 (x : { x // x \u2208 s }) (n : \u2115), \u2191(n \u2022 x) = n \u2022 \u2191x\n[PROOFSTEP]\nintros _ _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\nx\u271d : { x // x \u2208 s }\nn\u271d : \u2115\n\u22a2 \u2191(n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u2200 (x : { x // x \u2208 s }) (n : \u2124), \u2191(n \u2022 x) = n \u2022 \u2191x\n[PROOFSTEP]\nintros _ _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\nx\u271d : { x // x \u2208 s }\nn\u271d : \u2124\n\u22a2 \u2191(n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u2200 (x : { x // x \u2208 s }) (n : \u211a), \u2191(n \u2022 x) = n \u2022 \u2191x\n[PROOFSTEP]\nintros _ _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\nx\u271d : { x // x \u2208 s }\nn\u271d : \u211a\n\u22a2 \u2191(n\u271d \u2022 x\u271d) = n\u271d \u2022 \u2191x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u2200 (x : { x // x \u2208 s }) (n : \u2115), \u2191(x ^ n) = \u2191x ^ n\n[PROOFSTEP]\nintros _ _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\nx\u271d : { x // x \u2208 s }\nn\u271d : \u2115\n\u22a2 \u2191(x\u271d ^ n\u271d) = \u2191x\u271d ^ n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u2200 (x : { x // x \u2208 s }) (n : \u2124), \u2191(x ^ n) = \u2191x ^ n\n[PROOFSTEP]\nintros _ _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\nx\u271d : { x // x \u2208 s }\nn\u271d : \u2124\n\u22a2 \u2191(x\u271d ^ n\u271d) = \u2191x\u271d ^ n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u2200 (n : \u2115), \u2191\u2191n = \u2191n\n[PROOFSTEP]\nintros _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\nn\u271d : \u2115\n\u22a2 \u2191\u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u2200 (n : \u2124), \u2191\u2191n = \u2191n\n[PROOFSTEP]\nintros _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\nn\u271d : \u2124\n\u22a2 \u2191\u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\n\u22a2 \u2200 (n : \u211a), \u2191\u2191n = \u2191n\n[PROOFSTEP]\nintros _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field M\nS : Type u_1\ninst\u271d : SetLike S K\nh : SubfieldClass S K\ns : S\nn\u271d : \u211a\n\u22a2 \u2191\u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\np q : Subfield K\nh : (fun s => s.carrier) p = (fun s => s.carrier) q\n\u22a2 p = q\n[PROOFSTEP]\ncases p\n[GOAL]\ncase mk\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nq : Subfield K\ntoSubring\u271d : Subring K\ninv_mem'\u271d : \u2200 (x : K), x \u2208 toSubring\u271d.carrier \u2192 x\u207b\u00b9 \u2208 toSubring\u271d.carrier\nh : (fun s => s.carrier) { toSubring := toSubring\u271d, inv_mem' := inv_mem'\u271d } = (fun s => s.carrier) q\n\u22a2 { toSubring := toSubring\u271d, inv_mem' := inv_mem'\u271d } = q\n[PROOFSTEP]\ncases q\n[GOAL]\ncase mk.mk\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ntoSubring\u271d\u00b9 : Subring K\ninv_mem'\u271d\u00b9 : \u2200 (x : K), x \u2208 toSubring\u271d\u00b9.carrier \u2192 x\u207b\u00b9 \u2208 toSubring\u271d\u00b9.carrier\ntoSubring\u271d : Subring K\ninv_mem'\u271d : \u2200 (x : K), x \u2208 toSubring\u271d.carrier \u2192 x\u207b\u00b9 \u2208 toSubring\u271d.carrier\nh :\n (fun s => s.carrier) { toSubring := toSubring\u271d\u00b9, inv_mem' := inv_mem'\u271d\u00b9 } =\n (fun s => s.carrier) { toSubring := toSubring\u271d, inv_mem' := inv_mem'\u271d }\n\u22a2 { toSubring := toSubring\u271d\u00b9, inv_mem' := inv_mem'\u271d\u00b9 } = { toSubring := toSubring\u271d, inv_mem' := inv_mem'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toSubring\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ntoSubring\u271d\u00b9 : Subring K\ninv_mem'\u271d\u00b9 : \u2200 (x : K), x \u2208 toSubring\u271d\u00b9.carrier \u2192 x\u207b\u00b9 \u2208 toSubring\u271d\u00b9.carrier\ntoSubring\u271d : Subring K\ninv_mem'\u271d : \u2200 (x : K), x \u2208 toSubring\u271d.carrier \u2192 x\u207b\u00b9 \u2208 toSubring\u271d.carrier\nh :\n (fun s => s.carrier) { toSubring := toSubring\u271d\u00b9, inv_mem' := inv_mem'\u271d\u00b9 } =\n (fun s => s.carrier) { toSubring := toSubring\u271d, inv_mem' := inv_mem'\u271d }\n\u22a2 toSubring\u271d\u00b9 = toSubring\u271d\n[PROOFSTEP]\nexact SetLike.ext' h\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns t : Subfield K\nx : K\nhx : x \u2208 s\nn : \u2124\n\u22a2 x ^ n \u2208 s\n[PROOFSTEP]\ncases n\n[GOAL]\ncase ofNat\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns t : Subfield K\nx : K\nhx : x \u2208 s\na\u271d : \u2115\n\u22a2 x ^ Int.ofNat a\u271d \u2208 s\n[PROOFSTEP]\nsimpa using s.pow_mem hx _\n[GOAL]\ncase negSucc\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns t : Subfield K\nx : K\nhx : x \u2208 s\na\u271d : \u2115\n\u22a2 x ^ Int.negSucc a\u271d \u2208 s\n[PROOFSTEP]\nsimpa [pow_succ] using s.inv_mem (s.mul_mem hx (s.pow_mem hx _))\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns\u271d t : Subfield K\nf : K \u2192+* L\ns : Subfield L\nsrc\u271d : Subring K := Subring.comap f s.toSubring\nx : K\nhx :\n x \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n neg_mem' :=\n (_ : \u2200 {x : K}, x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n\u22a2 \u2191f x\u207b\u00b9 \u2208 s\n[PROOFSTEP]\nrw [map_inv\u2080 f]\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns\u271d t : Subfield K\nf : K \u2192+* L\ns : Subfield L\nsrc\u271d : Subring K := Subring.comap f s.toSubring\nx : K\nhx :\n x \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n neg_mem' :=\n (_ : \u2200 {x : K}, x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n\u22a2 (\u2191f x)\u207b\u00b9 \u2208 s\n[PROOFSTEP]\nexact s.inv_mem hx\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns\u271d t : Subfield K\nf : K \u2192+* L\ns : Subfield K\nsrc\u271d : Subring L := Subring.map f s.toSubring\n\u22a2 \u2200 (x : L),\n x \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n neg_mem' :=\n (_ :\n \u2200 {x : L},\n x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier \u2192\n x\u207b\u00b9 \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n neg_mem' :=\n (_ :\n \u2200 {x : L},\n x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase intro.intro\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns\u271d t : Subfield K\nf : K \u2192+* L\ns : Subfield K\nsrc\u271d : Subring L := Subring.map f s.toSubring\nx : K\nhx : x \u2208 s.carrier\n\u22a2 (\u2191f x)\u207b\u00b9 \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n neg_mem' :=\n (_ : \u2200 {x : L}, x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nexact \u27e8x\u207b\u00b9, s.inv_mem hx, map_inv\u2080 f x\u27e9\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns\u271d t : Subfield K\nf\u271d f : K \u2192+* L\ns : Subfield K\ny : L\n\u22a2 y \u2208 map f s \u2194 \u2203 x, x \u2208 s \u2227 \u2191f x = y\n[PROOFSTEP]\nunfold map\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns\u271d t : Subfield K\nf\u271d f : K \u2192+* L\ns : Subfield K\ny : L\n\u22a2 (y \u2208\n let src := Subring.map f s.toSubring;\n {\n toSubring :=\n { toSubsemiring := src.toSubsemiring,\n neg_mem' :=\n (_ :\n \u2200 {x : L},\n x \u2208 (Subring.map f s.toSubring).toSubsemiring.toSubmonoid.toSubsemigroup.carrier \u2192\n -x \u2208 (Subring.map f s.toSubring).toSubsemiring.toSubmonoid.toSubsemigroup.carrier) },\n inv_mem' :=\n (_ :\n \u2200 (x : L),\n x \u2208\n { toSubsemiring := (Subring.map f s.toSubring).toSubsemiring,\n neg_mem' :=\n (_ :\n \u2200 {x : L},\n x \u2208 (Subring.map f s.toSubring).toSubsemiring.toSubmonoid.toSubsemigroup.carrier \u2192\n -x \u2208\n (Subring.map f\n s.toSubring).toSubsemiring.toSubmonoid.toSubsemigroup.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier \u2192\n x\u207b\u00b9 \u2208\n { toSubsemiring := (Subring.map f s.toSubring).toSubsemiring,\n neg_mem' :=\n (_ :\n \u2200 {x : L},\n x \u2208 (Subring.map f s.toSubring).toSubsemiring.toSubmonoid.toSubsemigroup.carrier \u2192\n -x \u2208\n (Subring.map f\n s.toSubring).toSubsemiring.toSubmonoid.toSubsemigroup.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier) }) \u2194\n \u2203 x, x \u2208 s \u2227 \u2191f x = y\n[PROOFSTEP]\nsimp only [mem_mk, Subring.mem_mk, Subring.mem_toSubsemiring, Subring.mem_map, mem_toSubring]\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ng : L \u2192+* M\nf : K \u2192+* L\n\u22a2 fieldRange f = Subfield.map f \u22a4\n[PROOFSTEP]\next\n[GOAL]\ncase h\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ng : L \u2192+* M\nf : K \u2192+* L\nx\u271d : L\n\u22a2 x\u271d \u2208 fieldRange f \u2194 x\u271d \u2208 Subfield.map f \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ng : L \u2192+* M\nf : K \u2192+* L\n\u22a2 Subfield.map g (fieldRange f) = fieldRange (comp g f)\n[PROOFSTEP]\nsimpa only [fieldRange_eq_map] using (\u22a4 : Subfield K).map_map g f\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\nsrc\u271d : Subring K := sInf (toSubring '' S)\n\u22a2 \u2200 (x : K),\n x \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n neg_mem' :=\n (_ :\n \u2200 {x : K},\n x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier \u2192\n x\u207b\u00b9 \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n neg_mem' :=\n (_ :\n \u2200 {x : K},\n x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nrintro x hx\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\nsrc\u271d : Subring K := sInf (toSubring '' S)\nx : K\nhx :\n x \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n neg_mem' :=\n (_ : \u2200 {x : K}, x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n\u22a2 x\u207b\u00b9 \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n neg_mem' :=\n (_ : \u2200 {x : K}, x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\napply Subring.mem_sInf.mpr\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\nsrc\u271d : Subring K := sInf (toSubring '' S)\nx : K\nhx :\n x \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n neg_mem' :=\n (_ : \u2200 {x : K}, x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n\u22a2 \u2200 (p : Subring K), p \u2208 toSubring '' S \u2192 x\u207b\u00b9 \u2208 p\n[PROOFSTEP]\nrintro _ \u27e8p, p_mem, rfl\u27e9\n[GOAL]\ncase intro.intro\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\nsrc\u271d : Subring K := sInf (toSubring '' S)\nx : K\nhx :\n x \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n neg_mem' :=\n (_ : \u2200 {x : K}, x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\np : Subfield K\np_mem : p \u2208 S\n\u22a2 x\u207b\u00b9 \u2208 p.toSubring\n[PROOFSTEP]\nexact p.inv_mem (Subring.mem_sInf.mp hx p.toSubring \u27e8p, p_mem, rfl\u27e9)\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\n\u22a2 \u2191(sInf (toSubring '' S)) = \u22c2 (s : Subfield K) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\nx : K\n\u22a2 x \u2208 \u2191(sInf (toSubring '' S)) \u2194 x \u2208 \u22c2 (s : Subfield K) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\nrw [Subring.coe_sInf, Set.mem_iInter, Set.mem_iInter]\n[GOAL]\ncase h\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\nx : K\n\u22a2 (\u2200 (i : Subring K), x \u2208 \u22c2 (_ : i \u2208 toSubring '' S), \u2191i) \u2194 \u2200 (i : Subfield K), x \u2208 \u22c2 (_ : i \u2208 S), \u2191i\n[PROOFSTEP]\nexact\n \u27e8fun h s s' \u27e8s_mem, s'_eq\u27e9 => h s.toSubring _ \u27e8\u27e8s, s_mem, rfl\u27e9, s'_eq\u27e9,\n fun h s s' \u27e8\u27e8s'', s''_mem, s_eq\u27e9, (s'_eq : \u2191s = s')\u27e9 => h s'' _ \u27e8s''_mem, by simp [\u2190 s_eq, \u2190 s'_eq]\u27e9\u27e9\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\nx : K\nh : \u2200 (i : Subfield K), x \u2208 \u22c2 (_ : i \u2208 S), \u2191i\ns : Subring K\ns' : Set K\nx\u271d : s' \u2208 Set.range fun h => \u2191s\ns'' : Subfield K\ns''_mem : s'' \u2208 S\ns_eq : s''.toSubring = s\ns'_eq : \u2191s = s'\n\u22a2 (fun h => \u2191s'') s''_mem = s'\n[PROOFSTEP]\nsimp [\u2190 s_eq, \u2190 s'_eq]\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set (Subfield K)\n\u22a2 (sInf s).toSubring = \u2a05 (t : Subfield K) (_ : t \u2208 s), t.toSubring\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set (Subfield K)\nx : K\n\u22a2 x \u2208 (sInf s).toSubring \u2194 x \u2208 \u2a05 (t : Subfield K) (_ : t \u2208 s), t.toSubring\n[PROOFSTEP]\nrw [mem_toSubring, mem_sInf]\n[GOAL]\ncase h\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set (Subfield K)\nx : K\n\u22a2 (\u2200 (p : Subfield K), p \u2208 s \u2192 x \u2208 p) \u2194 x \u2208 \u2a05 (t : Subfield K) (_ : t \u2208 s), t.toSubring\n[PROOFSTEP]\nerw [Subring.mem_sInf]\n[GOAL]\ncase h\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set (Subfield K)\nx : K\n\u22a2 (\u2200 (p : Subfield K), p \u2208 s \u2192 x \u2208 p) \u2194 \u2200 (p : Subring K), (p \u2208 Set.range fun t => \u2a05 (_ : t \u2208 s), t.toSubring) \u2192 x \u2208 p\n[PROOFSTEP]\nexact\n \u27e8fun h p \u27e8p', hp\u27e9 => hp \u25b8 Subring.mem_sInf.mpr fun p \u27e8hp', hp\u27e9 => hp \u25b8 h _ hp', fun h p hp =>\n h p.toSubring\n \u27e8p,\n Subring.ext fun x =>\n \u27e8fun hx => Subring.mem_sInf.mp hx _ \u27e8hp, rfl\u27e9, fun hx =>\n Subring.mem_sInf.mpr fun p' \u27e8_, p'_eq\u27e9 => p'_eq \u25b8 hx\u27e9\u27e9\u27e9\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\n\u22a2 IsGLB S (sInf S)\n[PROOFSTEP]\nhave : \u2200 {s t : Subfield K}, (s : Set K) \u2264 t \u2194 s \u2264 t := by simp [SetLike.coe_subset_coe]\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\n\u22a2 \u2200 {s t : Subfield K}, \u2191s \u2264 \u2191t \u2194 s \u2264 t\n[PROOFSTEP]\nsimp [SetLike.coe_subset_coe]\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\nthis : \u2200 {s t : Subfield K}, \u2191s \u2264 \u2191t \u2194 s \u2264 t\n\u22a2 IsGLB S (sInf S)\n[PROOFSTEP]\nrefine' IsGLB.of_image this _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\nthis : \u2200 {s t : Subfield K}, \u2191s \u2264 \u2191t \u2194 s \u2264 t\n\u22a2 IsGLB ((fun {x} => \u2191x) '' S) \u2191(sInf S)\n[PROOFSTEP]\nconvert isGLB_biInf (s := S) (f := SetLike.coe)\n[GOAL]\ncase h.e'_4\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\nthis : \u2200 {s t : Subfield K}, \u2191s \u2264 \u2191t \u2194 s \u2264 t\n\u22a2 \u2191(sInf S) = \u2a05 (x : Subfield K) (_ : x \u2208 S), \u2191x\n[PROOFSTEP]\nexact coe_sInf _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\na\u271d b\u271d : K\nx_mem : a\u271d \u2208 {z | \u2203 x x_1 y x_2, x / y = z}\ny_mem : b\u271d \u2208 {z | \u2203 x x_1 y x_2, x / y = z}\n\u22a2 a\u271d * b\u271d \u2208 {z | \u2203 x x_1 y x_2, x / y = z}\n[PROOFSTEP]\nobtain \u27e8nx, hnx, dx, hdx, rfl\u27e9 := id x_mem\n[GOAL]\ncase intro.intro.intro.intro\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\nb\u271d : K\ny_mem : b\u271d \u2208 {z | \u2203 x x_1 y x_2, x / y = z}\nnx : K\nhnx : nx \u2208 Subring.closure s\ndx : K\nhdx : dx \u2208 Subring.closure s\nx_mem : nx / dx \u2208 {z | \u2203 x x_1 y x_2, x / y = z}\n\u22a2 nx / dx * b\u271d \u2208 {z | \u2203 x x_1 y x_2, x / y = z}\n[PROOFSTEP]\nobtain \u27e8ny, hny, dy, hdy, rfl\u27e9 := id y_mem\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\nnx : K\nhnx : nx \u2208 Subring.closure s\ndx : K\nhdx : dx \u2208 Subring.closure s\nx_mem : nx / dx \u2208 {z | \u2203 x x_1 y x_2, x / y = z}\nny : K\nhny : ny \u2208 Subring.closure s\ndy : K\nhdy : dy \u2208 Subring.closure s\ny_mem : ny / dy \u2208 {z | \u2203 x x_1 y x_2, x / y = z}\n\u22a2 nx / dx * (ny / dy) \u2208 {z | \u2203 x x_1 y x_2, x / y = z}\n[PROOFSTEP]\nexact \u27e8nx * ny, Subring.mul_mem _ hnx hny, dx * dy, Subring.mul_mem _ hdx hdy, (div_mul_div_comm _ _ _ _).symm\u27e9\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\na\u271d b\u271d : K\nx_mem :\n a\u271d \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\ny_mem :\n b\u271d \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\n\u22a2 a\u271d + b\u271d \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\n[PROOFSTEP]\nobtain \u27e8nx, hnx, dx, hdx, rfl\u27e9 := id x_mem\n[GOAL]\ncase intro.intro.intro.intro\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\nb\u271d : K\ny_mem :\n b\u271d \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\nnx : K\nhnx : nx \u2208 Subring.closure s\ndx : K\nhdx : dx \u2208 Subring.closure s\nx_mem :\n nx / dx \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\n\u22a2 nx / dx + b\u271d \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\n[PROOFSTEP]\nobtain \u27e8ny, hny, dy, hdy, rfl\u27e9 := id y_mem\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\nnx : K\nhnx : nx \u2208 Subring.closure s\ndx : K\nhdx : dx \u2208 Subring.closure s\nx_mem :\n nx / dx \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\nny : K\nhny : ny \u2208 Subring.closure s\ndy : K\nhdy : dy \u2208 Subring.closure s\ny_mem :\n ny / dy \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\n\u22a2 nx / dx + ny / dy \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\n[PROOFSTEP]\nby_cases hx0 : dx = 0\n[GOAL]\ncase pos\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\nnx : K\nhnx : nx \u2208 Subring.closure s\ndx : K\nhdx : dx \u2208 Subring.closure s\nx_mem :\n nx / dx \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\nny : K\nhny : ny \u2208 Subring.closure s\ndy : K\nhdy : dy \u2208 Subring.closure s\ny_mem :\n ny / dy \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\nhx0 : dx = 0\n\u22a2 nx / dx + ny / dy \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\n[PROOFSTEP]\nrwa [hx0, div_zero, zero_add]\n[GOAL]\ncase neg\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\nnx : K\nhnx : nx \u2208 Subring.closure s\ndx : K\nhdx : dx \u2208 Subring.closure s\nx_mem :\n nx / dx \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\nny : K\nhny : ny \u2208 Subring.closure s\ndy : K\nhdy : dy \u2208 Subring.closure s\ny_mem :\n ny / dy \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\nhx0 : \u00acdx = 0\n\u22a2 nx / dx + ny / dy \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\n[PROOFSTEP]\nby_cases hy0 : dy = 0\n[GOAL]\ncase pos\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\nnx : K\nhnx : nx \u2208 Subring.closure s\ndx : K\nhdx : dx \u2208 Subring.closure s\nx_mem :\n nx / dx \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\nny : K\nhny : ny \u2208 Subring.closure s\ndy : K\nhdy : dy \u2208 Subring.closure s\ny_mem :\n ny / dy \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\nhx0 : \u00acdx = 0\nhy0 : dy = 0\n\u22a2 nx / dx + ny / dy \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\n[PROOFSTEP]\nrwa [hy0, div_zero, add_zero]\n[GOAL]\ncase neg\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\nnx : K\nhnx : nx \u2208 Subring.closure s\ndx : K\nhdx : dx \u2208 Subring.closure s\nx_mem :\n nx / dx \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\nny : K\nhny : ny \u2208 Subring.closure s\ndy : K\nhdy : dy \u2208 Subring.closure s\ny_mem :\n ny / dy \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\nhx0 : \u00acdx = 0\nhy0 : \u00acdy = 0\n\u22a2 nx / dx + ny / dy \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier\n[PROOFSTEP]\nexact\n \u27e8nx * dy + dx * ny, Subring.add_mem _ (Subring.mul_mem _ hnx hdy) (Subring.mul_mem _ hdx hny), dx * dy,\n Subring.mul_mem _ hdx hdy, (div_add_div nx ny hx0 hy0).symm\u27e9\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\nx : K\n\u22a2 x \u2208\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) },\n add_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n a + b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier),\n zero_mem' := (_ : \u2203 x x_1 y x_2, x / y = 0) }.toSubmonoid.toSubsemigroup.carrier \u2192\n -x \u2208\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) },\n add_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n a + b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier),\n zero_mem' := (_ : \u2203 x x_1 y x_2, x / y = 0) }.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nrintro \u27e8y, hy, z, hz, x_eq\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\nx y : K\nhy : y \u2208 Subring.closure s\nz : K\nhz : z \u2208 Subring.closure s\nx_eq : y / z = x\n\u22a2 -x \u2208\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) },\n add_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n a + b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier),\n zero_mem' := (_ : \u2203 x x_1 y x_2, x / y = 0) }.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nexact \u27e8-y, Subring.neg_mem _ hy, z, hz, x_eq \u25b8 neg_div _ _\u27e9\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\nx : K\n\u22a2 x \u2208\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) },\n add_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n a + b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier),\n zero_mem' := (_ : \u2203 x x_1 y x_2, x / y = 0) },\n neg_mem' :=\n (_ :\n \u2200 {x : K},\n x \u2208\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) },\n add_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' :=\n (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n a + b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' :=\n (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier),\n zero_mem' := (_ : \u2203 x x_1 y x_2, x / y = 0) }.toSubmonoid.toSubsemigroup.carrier \u2192\n -x \u2208\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) },\n add_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' :=\n (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n a + b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' :=\n (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier),\n zero_mem' :=\n (_ :\n \u2203 x x_1 y x_2,\n x / y =\n 0) }.toSubmonoid.toSubsemigroup.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier \u2192\n x\u207b\u00b9 \u2208\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) },\n add_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n a + b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier),\n zero_mem' := (_ : \u2203 x x_1 y x_2, x / y = 0) },\n neg_mem' :=\n (_ :\n \u2200 {x : K},\n x \u2208\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) },\n add_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' :=\n (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n a + b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' :=\n (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier),\n zero_mem' := (_ : \u2203 x x_1 y x_2, x / y = 0) }.toSubmonoid.toSubsemigroup.carrier \u2192\n -x \u2208\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) },\n add_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' :=\n (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n a + b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' :=\n (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier),\n zero_mem' :=\n (_ :\n \u2203 x x_1 y x_2,\n x / y =\n 0) }.toSubmonoid.toSubsemigroup.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nrintro \u27e8y, hy, z, hz, x_eq\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\nx y : K\nhy : y \u2208 Subring.closure s\nz : K\nhz : z \u2208 Subring.closure s\nx_eq : y / z = x\n\u22a2 x\u207b\u00b9 \u2208\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192 a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) },\n add_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n a + b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier),\n zero_mem' := (_ : \u2203 x x_1 y x_2, x / y = 0) },\n neg_mem' :=\n (_ :\n \u2200 {x : K},\n x \u2208\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) },\n add_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n a + b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier),\n zero_mem' := (_ : \u2203 x x_1 y x_2, x / y = 0) }.toSubmonoid.toSubsemigroup.carrier \u2192\n -x \u2208\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) },\n add_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier \u2192\n a + b \u2208\n {\n toSubsemigroup :=\n { carrier := {z | \u2203 x x_1 y x_2, x / y = z},\n mul_mem' :=\n (_ :\n \u2200 {a b : K},\n a \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n b \u2208 {z | \u2203 x x_1 y x_2, x / y = z} \u2192\n a * b \u2208 {z | \u2203 x x_1 y x_2, x / y = z}) },\n one_mem' := (_ : \u2203 x x_1 y x_2, x / y = 1) }.toSubsemigroup.carrier),\n zero_mem' :=\n (_ :\n \u2203 x x_1 y x_2,\n x / y =\n 0) }.toSubmonoid.toSubsemigroup.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nexact \u27e8z, hz, y, hy, x_eq \u25b8 (inv_div _ _).symm\u27e9\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\nx : K\n\u22a2 x \u2208 closure s \u2194 \u2203 y, y \u2208 Subring.closure s \u2227 \u2203 z, z \u2208 Subring.closure s \u2227 y / z = x\n[PROOFSTEP]\nchange x \u2208 (closure s).carrier \u2194 \u2203 y \u2208 Subring.closure s, \u2203 z \u2208 Subring.closure s, y / z = x\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\nx : K\n\u22a2 x \u2208 (closure s).toSubring.toSubsemiring.toSubmonoid.toSubsemigroup.carrier \u2194\n \u2203 y, y \u2208 Subring.closure s \u2227 \u2203 z, z \u2208 Subring.closure s \u2227 y / z = x\n[PROOFSTEP]\nsimp only [closure, exists_prop, Set.mem_setOf_eq]\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\np : K \u2192 Prop\nx : K\nh : x \u2208 closure s\nHs : \u2200 (x : K), x \u2208 s \u2192 p x\nH1 : p 1\nHadd : \u2200 (x y : K), p x \u2192 p y \u2192 p (x + y)\nHneg : \u2200 (x : K), p x \u2192 p (-x)\nHinv : \u2200 (x : K), p x \u2192 p x\u207b\u00b9\nHmul : \u2200 (x y : K), p x \u2192 p y \u2192 p (x * y)\n\u22a2 p x\n[PROOFSTEP]\nletI : Subfield K :=\n \u27e8\u27e8\u27e8\u27e8\u27e8p, by intro _ _; exact Hmul _ _\u27e9, H1\u27e9, by intro _ _; exact Hadd _ _,\n @add_neg_self K _ 1 \u25b8 Hadd _ _ H1 (Hneg _ H1)\u27e9,\n by intro _; exact Hneg _\u27e9,\n Hinv\u27e9\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\np : K \u2192 Prop\nx : K\nh : x \u2208 closure s\nHs : \u2200 (x : K), x \u2208 s \u2192 p x\nH1 : p 1\nHadd : \u2200 (x y : K), p x \u2192 p y \u2192 p (x + y)\nHneg : \u2200 (x : K), p x \u2192 p (-x)\nHinv : \u2200 (x : K), p x \u2192 p x\u207b\u00b9\nHmul : \u2200 (x y : K), p x \u2192 p y \u2192 p (x * y)\n\u22a2 \u2200 {a b : K}, a \u2208 p \u2192 b \u2208 p \u2192 a * b \u2208 p\n[PROOFSTEP]\nintro _ _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\np : K \u2192 Prop\nx : K\nh : x \u2208 closure s\nHs : \u2200 (x : K), x \u2208 s \u2192 p x\nH1 : p 1\nHadd : \u2200 (x y : K), p x \u2192 p y \u2192 p (x + y)\nHneg : \u2200 (x : K), p x \u2192 p (-x)\nHinv : \u2200 (x : K), p x \u2192 p x\u207b\u00b9\nHmul : \u2200 (x y : K), p x \u2192 p y \u2192 p (x * y)\na\u271d b\u271d : K\n\u22a2 a\u271d \u2208 p \u2192 b\u271d \u2208 p \u2192 a\u271d * b\u271d \u2208 p\n[PROOFSTEP]\nexact Hmul _ _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\np : K \u2192 Prop\nx : K\nh : x \u2208 closure s\nHs : \u2200 (x : K), x \u2208 s \u2192 p x\nH1 : p 1\nHadd : \u2200 (x y : K), p x \u2192 p y \u2192 p (x + y)\nHneg : \u2200 (x : K), p x \u2192 p (-x)\nHinv : \u2200 (x : K), p x \u2192 p x\u207b\u00b9\nHmul : \u2200 (x y : K), p x \u2192 p y \u2192 p (x * y)\n\u22a2 \u2200 {a b : K},\n a \u2208\n { toSubsemigroup := { carrier := p, mul_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a * b)) },\n one_mem' := H1 }.toSubsemigroup.carrier \u2192\n b \u2208\n { toSubsemigroup := { carrier := p, mul_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a * b)) },\n one_mem' := H1 }.toSubsemigroup.carrier \u2192\n a + b \u2208\n { toSubsemigroup := { carrier := p, mul_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a * b)) },\n one_mem' := H1 }.toSubsemigroup.carrier\n[PROOFSTEP]\nintro _ _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\np : K \u2192 Prop\nx : K\nh : x \u2208 closure s\nHs : \u2200 (x : K), x \u2208 s \u2192 p x\nH1 : p 1\nHadd : \u2200 (x y : K), p x \u2192 p y \u2192 p (x + y)\nHneg : \u2200 (x : K), p x \u2192 p (-x)\nHinv : \u2200 (x : K), p x \u2192 p x\u207b\u00b9\nHmul : \u2200 (x y : K), p x \u2192 p y \u2192 p (x * y)\na\u271d b\u271d : K\n\u22a2 a\u271d \u2208\n { toSubsemigroup := { carrier := p, mul_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a * b)) },\n one_mem' := H1 }.toSubsemigroup.carrier \u2192\n b\u271d \u2208\n { toSubsemigroup := { carrier := p, mul_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a * b)) },\n one_mem' := H1 }.toSubsemigroup.carrier \u2192\n a\u271d + b\u271d \u2208\n { toSubsemigroup := { carrier := p, mul_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a * b)) },\n one_mem' := H1 }.toSubsemigroup.carrier\n[PROOFSTEP]\nexact Hadd _ _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\np : K \u2192 Prop\nx : K\nh : x \u2208 closure s\nHs : \u2200 (x : K), x \u2208 s \u2192 p x\nH1 : p 1\nHadd : \u2200 (x y : K), p x \u2192 p y \u2192 p (x + y)\nHneg : \u2200 (x : K), p x \u2192 p (-x)\nHinv : \u2200 (x : K), p x \u2192 p x\u207b\u00b9\nHmul : \u2200 (x y : K), p x \u2192 p y \u2192 p (x * y)\n\u22a2 \u2200 {x : K},\n x \u2208\n {\n toSubmonoid :=\n { toSubsemigroup := { carrier := p, mul_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a * b)) },\n one_mem' := H1 },\n add_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a + b)),\n zero_mem' := (_ : p 0) }.toSubmonoid.toSubsemigroup.carrier \u2192\n -x \u2208\n {\n toSubmonoid :=\n { toSubsemigroup := { carrier := p, mul_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a * b)) },\n one_mem' := H1 },\n add_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a + b)),\n zero_mem' := (_ : p 0) }.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nintro _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\np : K \u2192 Prop\nx : K\nh : x \u2208 closure s\nHs : \u2200 (x : K), x \u2208 s \u2192 p x\nH1 : p 1\nHadd : \u2200 (x y : K), p x \u2192 p y \u2192 p (x + y)\nHneg : \u2200 (x : K), p x \u2192 p (-x)\nHinv : \u2200 (x : K), p x \u2192 p x\u207b\u00b9\nHmul : \u2200 (x y : K), p x \u2192 p y \u2192 p (x * y)\nx\u271d : K\n\u22a2 x\u271d \u2208\n {\n toSubmonoid :=\n { toSubsemigroup := { carrier := p, mul_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a * b)) },\n one_mem' := H1 },\n add_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a + b)),\n zero_mem' := (_ : p 0) }.toSubmonoid.toSubsemigroup.carrier \u2192\n -x\u271d \u2208\n {\n toSubmonoid :=\n { toSubsemigroup := { carrier := p, mul_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a * b)) },\n one_mem' := H1 },\n add_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a + b)),\n zero_mem' := (_ : p 0) }.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nexact Hneg _\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Set K\np : K \u2192 Prop\nx : K\nh : x \u2208 closure s\nHs : \u2200 (x : K), x \u2208 s \u2192 p x\nH1 : p 1\nHadd : \u2200 (x y : K), p x \u2192 p y \u2192 p (x + y)\nHneg : \u2200 (x : K), p x \u2192 p (-x)\nHinv : \u2200 (x : K), p x \u2192 p x\u207b\u00b9\nHmul : \u2200 (x y : K), p x \u2192 p y \u2192 p (x * y)\nthis : Subfield K :=\n {\n toSubring :=\n {\n toSubsemiring :=\n {\n toSubmonoid :=\n { toSubsemigroup := { carrier := p, mul_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a * b)) },\n one_mem' := H1 },\n add_mem' := (_ : \u2200 {a b : K}, p a \u2192 p b \u2192 p (a + b)), zero_mem' := (_ : p 0) },\n neg_mem' := (_ : \u2200 {x : K}, p x \u2192 p (-x)) },\n inv_mem' := Hinv }\n\u22a2 p x\n[PROOFSTEP]\nexact (closure_le (t := this)).2 Hs h\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : K\n\u22a2 x \u2208 \u2a06 (i : \u03b9), S i \u2194 \u2203 i, x \u2208 S i\n[PROOFSTEP]\nrefine' \u27e8_, fun \u27e8i, hi\u27e9 => (SetLike.le_def.1 <| le_iSup S i) hi\u27e9\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : K\n\u22a2 x \u2208 \u2a06 (i : \u03b9), S i \u2192 \u2203 i, x \u2208 S i\n[PROOFSTEP]\nsuffices x \u2208 closure (\u22c3 i, (S i : Set K)) \u2192 \u2203 i, x \u2208 S i by simpa only [closure_iUnion, closure_eq]\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : K\nthis : x \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i)) \u2192 \u2203 i, x \u2208 S i\n\u22a2 x \u2208 \u2a06 (i : \u03b9), S i \u2192 \u2203 i, x \u2208 S i\n[PROOFSTEP]\nsimpa only [closure_iUnion, closure_eq]\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : K\n\u22a2 x \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i)) \u2192 \u2203 i, x \u2208 S i\n[PROOFSTEP]\nrefine' fun hx => closure_induction hx (fun x => Set.mem_iUnion.mp) _ _ _ _ _\n[GOAL]\ncase refine'_1\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : K\nhx : x \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\n\u22a2 \u2203 i, 1 \u2208 S i\n[PROOFSTEP]\nexact h\u03b9.elim fun i => \u27e8i, (S i).one_mem\u27e9\n[GOAL]\ncase refine'_2\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : K\nhx : x \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\n\u22a2 \u2200 (x y : K), (\u2203 i, x \u2208 S i) \u2192 (\u2203 i, y \u2208 S i) \u2192 \u2203 i, x + y \u2208 S i\n[PROOFSTEP]\nrintro x y \u27e8i, hi\u27e9 \u27e8j, hj\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx\u271d : K\nhx : x\u271d \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nx y : K\ni : \u03b9\nhi : x \u2208 S i\nj : \u03b9\nhj : y \u2208 S j\n\u22a2 \u2203 i, x + y \u2208 S i\n[PROOFSTEP]\nobtain \u27e8k, hki, hkj\u27e9 := hS i j\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx\u271d : K\nhx : x\u271d \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nx y : K\ni : \u03b9\nhi : x \u2208 S i\nj : \u03b9\nhj : y \u2208 S j\nk : \u03b9\nhki : S i \u2264 S k\nhkj : S j \u2264 S k\n\u22a2 \u2203 i, x + y \u2208 S i\n[PROOFSTEP]\nexact \u27e8k, (S k).add_mem (hki hi) (hkj hj)\u27e9\n[GOAL]\ncase refine'_3\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : K\nhx : x \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\n\u22a2 \u2200 (x : K), (\u2203 i, x \u2208 S i) \u2192 \u2203 i, -x \u2208 S i\n[PROOFSTEP]\nrintro x \u27e8i, hi\u27e9\n[GOAL]\ncase refine'_3.intro\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx\u271d : K\nhx : x\u271d \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nx : K\ni : \u03b9\nhi : x \u2208 S i\n\u22a2 \u2203 i, -x \u2208 S i\n[PROOFSTEP]\nexact \u27e8i, (S i).neg_mem hi\u27e9\n[GOAL]\ncase refine'_4\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : K\nhx : x \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\n\u22a2 \u2200 (x : K), (\u2203 i, x \u2208 S i) \u2192 \u2203 i, x\u207b\u00b9 \u2208 S i\n[PROOFSTEP]\nrintro x \u27e8i, hi\u27e9\n[GOAL]\ncase refine'_4.intro\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx\u271d : K\nhx : x\u271d \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nx : K\ni : \u03b9\nhi : x \u2208 S i\n\u22a2 \u2203 i, x\u207b\u00b9 \u2208 S i\n[PROOFSTEP]\nexact \u27e8i, (S i).inv_mem hi\u27e9\n[GOAL]\ncase refine'_5\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : K\nhx : x \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\n\u22a2 \u2200 (x y : K), (\u2203 i, x \u2208 S i) \u2192 (\u2203 i, y \u2208 S i) \u2192 \u2203 i, x * y \u2208 S i\n[PROOFSTEP]\nrintro x y \u27e8i, hi\u27e9 \u27e8j, hj\u27e9\n[GOAL]\ncase refine'_5.intro.intro\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx\u271d : K\nhx : x\u271d \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nx y : K\ni : \u03b9\nhi : x \u2208 S i\nj : \u03b9\nhj : y \u2208 S j\n\u22a2 \u2203 i, x * y \u2208 S i\n[PROOFSTEP]\nobtain \u27e8k, hki, hkj\u27e9 := hS i j\n[GOAL]\ncase refine'_5.intro.intro.intro.intro\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx\u271d : K\nhx : x\u271d \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nx y : K\ni : \u03b9\nhi : x \u2208 S i\nj : \u03b9\nhj : y \u2208 S j\nk : \u03b9\nhki : S i \u2264 S k\nhkj : S j \u2264 S k\n\u22a2 \u2203 i, x * y \u2208 S i\n[PROOFSTEP]\nexact \u27e8k, (S k).mul_mem (hki hi) (hkj hj)\u27e9\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subfield K\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : K\n\u22a2 x \u2208 \u2191(\u2a06 (i : \u03b9), S i) \u2194 x \u2208 \u22c3 (i : \u03b9), \u2191(S i)\n[PROOFSTEP]\nsimp [mem_iSup_of_directed hS]\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x \u2264 x_1) S\nx : K\n\u22a2 x \u2208 sSup S \u2194 \u2203 s, s \u2208 S \u2227 x \u2208 s\n[PROOFSTEP]\nhaveI : Nonempty S := Sne.to_subtype\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x \u2264 x_1) S\nx : K\nthis : Nonempty \u2191S\n\u22a2 x \u2208 sSup S \u2194 \u2203 s, s \u2208 S \u2227 x \u2208 s\n[PROOFSTEP]\nsimp only [sSup_eq_iSup', mem_iSup_of_directed hS.directed_val, Subtype.exists, exists_prop]\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\nS : Set (Subfield K)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x \u2264 x_1) S\nx : K\n\u22a2 x \u2208 \u2191(sSup S) \u2194 x \u2208 \u22c3 (s : Subfield K) (_ : s \u2208 S), \u2191s\n[PROOFSTEP]\nsimp [mem_sSup_of_directedOn Sne hS]\n[GOAL]\nK : Type u\nL : Type v\nM : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field M\ns : Subfield K\nf g : K \u2192+* L\nsrc\u271d : Subring K := eqLocus f g\nx : K\nhx : \u2191f x = \u2191g x\n\u22a2 \u2191f x\u207b\u00b9 = \u2191g x\u207b\u00b9\n[PROOFSTEP]\nrw [map_inv\u2080 f, map_inv\u2080 g, hx]\n", "meta": {"mathlib_filename": "Mathlib.FieldTheory.Subfield", "llama_tokens": 33698, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410572017153, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.41174304237051734}} {"text": "[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Zero N\nf\u271d g\u271d f g : \u03b1 \u2192\u2080 N\na : \u03b1\n\u22a2 a \u2208 neLocus f g \u2194 \u2191f a \u2260 \u2191g a\n[PROOFSTEP]\nsimpa only [neLocus, Finset.mem_filter, Finset.mem_union, mem_support_iff, and_iff_right_iff_imp] using Ne.ne_or_ne _\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Zero N\nf g : \u03b1 \u2192\u2080 N\n\u22a2 \u2191(neLocus f g) = {x | \u2191f x \u2260 \u2191g x}\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Zero N\nf g : \u03b1 \u2192\u2080 N\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(neLocus f g) \u2194 x\u271d \u2208 {x | \u2191f x \u2260 \u2191g x}\n[PROOFSTEP]\nexact mem_neLocus\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Zero N\nf\u271d g\u271d f g : \u03b1 \u2192\u2080 N\nh : f = g\n\u22a2 neLocus f f = \u2205\n[PROOFSTEP]\nsimp only [neLocus, Ne.def, eq_self_iff_true, not_true, Finset.filter_False]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Zero N\nf g : \u03b1 \u2192\u2080 N\n\u22a2 neLocus f g = neLocus g f\n[PROOFSTEP]\nsimp_rw [neLocus, Finset.union_comm, ne_comm]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Zero N\nf g : \u03b1 \u2192\u2080 N\n\u22a2 neLocus f 0 = f.support\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : Zero N\nf g : \u03b1 \u2192\u2080 N\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 neLocus f 0 \u2194 a\u271d \u2208 f.support\n[PROOFSTEP]\nrw [mem_neLocus, mem_support_iff, coe_zero, Pi.zero_apply]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : DecidableEq N\ninst\u271d\u00b2 : Zero N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : Zero M\nf g : \u03b1 \u2192\u2080 N\nF : N \u2192 M\nF0 : F 0 = 0\nx : \u03b1\n\u22a2 x \u2208 neLocus (mapRange F F0 f) (mapRange F F0 g) \u2192 x \u2208 neLocus f g\n[PROOFSTEP]\nsimpa only [mem_neLocus, mapRange_apply, not_imp_not] using congr_arg F\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b1\ninst\u271d\u2074 : DecidableEq N\ninst\u271d\u00b3 : Zero M\ninst\u271d\u00b2 : DecidableEq P\ninst\u271d\u00b9 : Zero P\ninst\u271d : Zero N\nF : M \u2192 N \u2192 P\nF0 : F 0 0 = 0\nf : \u03b1 \u2192\u2080 M\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 N\nhF : \u2200 (f : M), Function.Injective fun g => F f g\n\u22a2 neLocus (zipWith F F0 f g\u2081) (zipWith F F0 f g\u2082) = neLocus g\u2081 g\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b1\ninst\u271d\u2074 : DecidableEq N\ninst\u271d\u00b3 : Zero M\ninst\u271d\u00b2 : DecidableEq P\ninst\u271d\u00b9 : Zero P\ninst\u271d : Zero N\nF : M \u2192 N \u2192 P\nF0 : F 0 0 = 0\nf : \u03b1 \u2192\u2080 M\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 N\nhF : \u2200 (f : M), Function.Injective fun g => F f g\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 neLocus (zipWith F F0 f g\u2081) (zipWith F F0 f g\u2082) \u2194 a\u271d \u2208 neLocus g\u2081 g\u2082\n[PROOFSTEP]\nsimpa only [mem_neLocus] using (hF _).ne_iff\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b1\ninst\u271d\u2074 : DecidableEq M\ninst\u271d\u00b3 : Zero M\ninst\u271d\u00b2 : DecidableEq P\ninst\u271d\u00b9 : Zero P\ninst\u271d : Zero N\nF : M \u2192 N \u2192 P\nF0 : F 0 0 = 0\nf\u2081 f\u2082 : \u03b1 \u2192\u2080 M\ng : \u03b1 \u2192\u2080 N\nhF : \u2200 (g : N), Function.Injective fun f => F f g\n\u22a2 neLocus (zipWith F F0 f\u2081 g) (zipWith F F0 f\u2082 g) = neLocus f\u2081 f\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b1\ninst\u271d\u2074 : DecidableEq M\ninst\u271d\u00b3 : Zero M\ninst\u271d\u00b2 : DecidableEq P\ninst\u271d\u00b9 : Zero P\ninst\u271d : Zero N\nF : M \u2192 N \u2192 P\nF0 : F 0 0 = 0\nf\u2081 f\u2082 : \u03b1 \u2192\u2080 M\ng : \u03b1 \u2192\u2080 N\nhF : \u2200 (g : N), Function.Injective fun f => F f g\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 neLocus (zipWith F F0 f\u2081 g) (zipWith F F0 f\u2082 g) \u2194 a\u271d \u2208 neLocus f\u2081 f\u2082\n[PROOFSTEP]\nsimpa only [mem_neLocus] using (hF _).ne_iff\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : DecidableEq N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf g : \u03b1 \u2192\u2080 N\nF : N \u2192 M\nF0 : F 0 = 0\nhF : Function.Injective F\n\u22a2 neLocus (mapRange F F0 f) (mapRange F F0 g) = neLocus f g\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : DecidableEq N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf g : \u03b1 \u2192\u2080 N\nF : N \u2192 M\nF0 : F 0 = 0\nhF : Function.Injective F\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 neLocus (mapRange F F0 f) (mapRange F F0 g) \u2194 a\u271d \u2208 neLocus f g\n[PROOFSTEP]\nsimpa only [mem_neLocus] using hF.ne_iff\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : AddGroup N\nf f\u2081 f\u2082 g g\u2081 g\u2082 : \u03b1 \u2192\u2080 N\n\u22a2 neLocus (-f) g = neLocus f (-g)\n[PROOFSTEP]\nrw [\u2190 neLocus_neg_neg, neg_neg]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : AddGroup N\nf f\u2081 f\u2082 g g\u2081 g\u2082 : \u03b1 \u2192\u2080 N\n\u22a2 neLocus f g = (f - g).support\n[PROOFSTEP]\nrw [\u2190 neLocus_add_right _ _ (-g), add_right_neg, neLocus_zero_right, sub_eq_add_neg]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : AddGroup N\nf f\u2081 f\u2082 g g\u2081 g\u2082 : \u03b1 \u2192\u2080 N\n\u22a2 neLocus (f - g\u2081) (f - g\u2082) = neLocus g\u2081 g\u2082\n[PROOFSTEP]\nsimp only [sub_eq_add_neg, neLocus_add_left, neLocus_neg_neg]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : AddGroup N\nf f\u2081 f\u2082 g g\u2081 g\u2082 : \u03b1 \u2192\u2080 N\n\u22a2 neLocus (f\u2081 - g) (f\u2082 - g) = neLocus f\u2081 f\u2082\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using neLocus_add_right _ _ _\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : AddGroup N\nf f\u2081 f\u2082 g g\u2081 g\u2082 : \u03b1 \u2192\u2080 N\n\u22a2 neLocus f (f + g) = g.support\n[PROOFSTEP]\nrw [\u2190 neLocus_zero_left, \u2190 neLocus_add_left f 0 g, add_zero]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : AddGroup N\nf f\u2081 f\u2082 g g\u2081 g\u2082 : \u03b1 \u2192\u2080 N\n\u22a2 neLocus (f + g) f = g.support\n[PROOFSTEP]\nrw [neLocus_comm, neLocus_self_add_right]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : AddGroup N\nf f\u2081 f\u2082 g g\u2081 g\u2082 : \u03b1 \u2192\u2080 N\n\u22a2 neLocus f (f - g) = g.support\n[PROOFSTEP]\nrw [sub_eq_add_neg, neLocus_self_add_right, support_neg]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq N\ninst\u271d : AddGroup N\nf f\u2081 f\u2082 g g\u2081 g\u2082 : \u03b1 \u2192\u2080 N\n\u22a2 neLocus (f - g) f = g.support\n[PROOFSTEP]\nrw [neLocus_comm, neLocus_self_sub_right]\n", "meta": {"mathlib_filename": "Mathlib.Data.Finsupp.NeLocus", "llama_tokens": 3335, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105454764746, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.4117430259476648}} {"text": "[GOAL]\n\u22a2 \u03c6\u207b\u00b9 = -\u03c8\n[PROOFSTEP]\nhave : 1 + Real.sqrt 5 \u2260 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))\n[GOAL]\n\u22a2 0 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 0 < 5\n[PROOFSTEP]\nnorm_num\n[GOAL]\nthis : 1 + sqrt 5 \u2260 0\n\u22a2 \u03c6\u207b\u00b9 = -\u03c8\n[PROOFSTEP]\nfield_simp [sub_mul, mul_add]\n[GOAL]\nthis : 1 + sqrt 5 \u2260 0\n\u22a2 2 * 2 = sqrt 5 * sqrt 5 - 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 \u03c8\u207b\u00b9 = -\u03c6\n[PROOFSTEP]\nrw [inv_eq_iff_eq_inv, \u2190 neg_inv, \u2190 neg_eq_iff_eq_neg]\n[GOAL]\n\u22a2 -\u03c8 = \u03c6\u207b\u00b9\n[PROOFSTEP]\nexact inv_gold.symm\n[GOAL]\n\u22a2 \u03c6 * \u03c8 = -1\n[PROOFSTEP]\nfield_simp\n[GOAL]\n\u22a2 (1 + sqrt 5) * (1 - sqrt 5) = -(2 * 2)\n[PROOFSTEP]\nrw [\u2190 sq_sub_sq]\n[GOAL]\n\u22a2 1 ^ 2 - sqrt 5 ^ 2 = -(2 * 2)\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 \u03c8 * \u03c6 = -1\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\n\u22a2 \u03c6 * \u03c8 = -1\n[PROOFSTEP]\nexact gold_mul_goldConj\n[GOAL]\n\u22a2 \u03c6 + \u03c8 = 1\n[PROOFSTEP]\nrw [goldenRatio, goldenConj]\n[GOAL]\n\u22a2 (1 + sqrt 5) / 2 + (1 - sqrt 5) / 2 = 1\n[PROOFSTEP]\nring\n[GOAL]\n\u22a2 1 - \u03c6 = \u03c8\n[PROOFSTEP]\nlinarith [gold_add_goldConj]\n[GOAL]\n\u22a2 1 - \u03c8 = \u03c6\n[PROOFSTEP]\nlinarith [gold_add_goldConj]\n[GOAL]\n\u22a2 \u03c6 - \u03c8 = sqrt 5\n[PROOFSTEP]\nrw [goldenRatio, goldenConj]\n[GOAL]\n\u22a2 (1 + sqrt 5) / 2 - (1 - sqrt 5) / 2 = sqrt 5\n[PROOFSTEP]\nring\n[GOAL]\n\u22a2 \u03c6 ^ 2 = \u03c6 + 1\n[PROOFSTEP]\nrw [goldenRatio, \u2190 sub_eq_zero]\n[GOAL]\n\u22a2 ((1 + sqrt 5) / 2) ^ 2 - ((1 + sqrt 5) / 2 + 1) = 0\n[PROOFSTEP]\nring_nf\n[GOAL]\n\u22a2 sqrt 5 ^ 2 * (\u2191(Int.ofNat 1) / \u21914) + \u2191(Int.negOfNat 5) * (\u2191(Int.ofNat 1) / \u21914) = 0\n[PROOFSTEP]\nrw [Real.sq_sqrt]\n[GOAL]\n\u22a2 5 * (\u2191(Int.ofNat 1) / \u21914) + \u2191(Int.negOfNat 5) * (\u2191(Int.ofNat 1) / \u21914) = 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 0 \u2264 5\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 \u03c8 ^ 2 = \u03c8 + 1\n[PROOFSTEP]\nrw [goldenConj, \u2190 sub_eq_zero]\n[GOAL]\n\u22a2 ((1 - sqrt 5) / 2) ^ 2 - ((1 - sqrt 5) / 2 + 1) = 0\n[PROOFSTEP]\nring_nf\n[GOAL]\n\u22a2 sqrt 5 ^ 2 * (\u2191(Int.ofNat 1) / \u21914) + \u2191(Int.negOfNat 5) * (\u2191(Int.ofNat 1) / \u21914) = 0\n[PROOFSTEP]\nrw [Real.sq_sqrt]\n[GOAL]\n\u22a2 5 * (\u2191(Int.ofNat 1) / \u21914) + \u2191(Int.negOfNat 5) * (\u2191(Int.ofNat 1) / \u21914) = 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 0 \u2264 5\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 0 < 1 + sqrt 5\n[PROOFSTEP]\napply add_pos\n[GOAL]\ncase ha\n\u22a2 0 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase hb\n\u22a2 0 < sqrt 5\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 1 < \u03c6\n[PROOFSTEP]\nrefine' lt_of_mul_lt_mul_left _ (le_of_lt gold_pos)\n[GOAL]\n\u22a2 \u03c6 * 1 < \u03c6 * \u03c6\n[PROOFSTEP]\nsimp [\u2190 sq, gold_pos, zero_lt_one, -div_pow]\n -- Porting note: Added `- div_pow`\n[GOAL]\n\u22a2 \u03c8 < 0\n[PROOFSTEP]\nlinarith [one_sub_goldConj, one_lt_gold]\n[GOAL]\n\u22a2 -1 < \u03c8\n[PROOFSTEP]\nrw [neg_lt, \u2190 inv_gold]\n[GOAL]\n\u22a2 \u03c6\u207b\u00b9 < 1\n[PROOFSTEP]\nexact inv_lt_one one_lt_gold\n[GOAL]\n\u22a2 Irrational \u03c6\n[PROOFSTEP]\nhave := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)\n[GOAL]\n\u22a2 Nat.Prime 5\n[PROOFSTEP]\nnorm_num\n[GOAL]\nthis : Irrational (sqrt \u21915)\n\u22a2 Irrational \u03c6\n[PROOFSTEP]\nhave := this.rat_add 1\n[GOAL]\nthis\u271d : Irrational (sqrt \u21915)\nthis : Irrational (\u21911 + sqrt \u21915)\n\u22a2 Irrational \u03c6\n[PROOFSTEP]\nhave := this.rat_mul (show (0.5 : \u211a) \u2260 0 by norm_num)\n[GOAL]\nthis\u271d : Irrational (sqrt \u21915)\nthis : Irrational (\u21911 + sqrt \u21915)\n\u22a2 0.5 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\nthis\u271d\u00b9 : Irrational (sqrt \u21915)\nthis\u271d : Irrational (\u21911 + sqrt \u21915)\nthis : Irrational (\u21910.5 * (\u21911 + sqrt \u21915))\n\u22a2 Irrational \u03c6\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_1\nthis\u271d\u00b9 : Irrational (sqrt \u21915)\nthis\u271d : Irrational (\u21911 + sqrt \u21915)\nthis : Irrational (\u21910.5 * (\u21911 + sqrt \u21915))\n\u22a2 \u03c6 = \u21910.5 * (\u21911 + sqrt \u21915)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_1\nthis\u271d\u00b9 : Irrational (sqrt \u21915)\nthis\u271d : Irrational (\u21911 + sqrt \u21915)\nthis : Irrational (\u21910.5 * (\u21911 + sqrt \u21915))\n\u22a2 \u03c6 = 1 / 2 * (1 + sqrt 5)\n[PROOFSTEP]\nfield_simp\n[GOAL]\n\u22a2 Irrational \u03c8\n[PROOFSTEP]\nhave := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)\n[GOAL]\n\u22a2 Nat.Prime 5\n[PROOFSTEP]\nnorm_num\n[GOAL]\nthis : Irrational (sqrt \u21915)\n\u22a2 Irrational \u03c8\n[PROOFSTEP]\nhave := this.rat_sub 1\n[GOAL]\nthis\u271d : Irrational (sqrt \u21915)\nthis : Irrational (\u21911 - sqrt \u21915)\n\u22a2 Irrational \u03c8\n[PROOFSTEP]\nhave := this.rat_mul (show (0.5 : \u211a) \u2260 0 by norm_num)\n[GOAL]\nthis\u271d : Irrational (sqrt \u21915)\nthis : Irrational (\u21911 - sqrt \u21915)\n\u22a2 0.5 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\nthis\u271d\u00b9 : Irrational (sqrt \u21915)\nthis\u271d : Irrational (\u21911 - sqrt \u21915)\nthis : Irrational (\u21910.5 * (\u21911 - sqrt \u21915))\n\u22a2 Irrational \u03c8\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_1\nthis\u271d\u00b9 : Irrational (sqrt \u21915)\nthis\u271d : Irrational (\u21911 - sqrt \u21915)\nthis : Irrational (\u21910.5 * (\u21911 - sqrt \u21915))\n\u22a2 \u03c8 = \u21910.5 * (\u21911 - sqrt \u21915)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_1\nthis\u271d\u00b9 : Irrational (sqrt \u21915)\nthis\u271d : Irrational (\u21911 - sqrt \u21915)\nthis : Irrational (\u21910.5 * (\u21911 - sqrt \u21915))\n\u22a2 \u03c8 = 1 / 2 * (1 - sqrt 5)\n[PROOFSTEP]\nfield_simp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommSemiring \u03b1\n\u03b2 : Type u_2\ninst\u271d : CommRing \u03b2\n\u22a2 LinearRecurrence.charPoly fibRec = X ^ 2 - (X + 1)\n[PROOFSTEP]\nrw [fibRec, LinearRecurrence.charPoly]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommSemiring \u03b1\n\u03b2 : Type u_2\ninst\u271d : CommRing \u03b2\n\u22a2 (\u2191(monomial { order := 2, coeffs := ![1, 1] }.order) 1 -\n Finset.sum Finset.univ fun i => \u2191(monomial \u2191i) (LinearRecurrence.coeffs { order := 2, coeffs := ![1, 1] } i)) =\n X ^ 2 - (X + 1)\n[PROOFSTEP]\nsimp [Finset.sum_fin_eq_sum_range, Finset.sum_range_succ', \u2190 smul_X_eq_monomial]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\n\u22a2 LinearRecurrence.IsSolution fibRec fun x => \u2191(Nat.fib x)\n[PROOFSTEP]\nrw [fibRec]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\n\u22a2 LinearRecurrence.IsSolution { order := 2, coeffs := ![1, 1] } fun x => \u2191(Nat.fib x)\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nn : \u2115\n\u22a2 (fun x => \u2191(Nat.fib x)) (n + { order := 2, coeffs := ![1, 1] }.order) =\n Finset.sum Finset.univ fun i =>\n LinearRecurrence.coeffs { order := 2, coeffs := ![1, 1] } i * (fun x => \u2191(Nat.fib x)) (n + \u2191i)\n[PROOFSTEP]\nsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nn : \u2115\n\u22a2 \u2191(Nat.fib (n + 2)) = Finset.sum Finset.univ fun x => Matrix.vecCons 1 ![1] x * \u2191(Nat.fib (n + \u2191x))\n[PROOFSTEP]\nrw [Nat.fib_add_two, add_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nn : \u2115\n\u22a2 \u2191(Nat.fib (n + 1) + Nat.fib n) = Finset.sum Finset.univ fun x => Matrix.vecCons 1 ![1] x * \u2191(Nat.fib (n + \u2191x))\n[PROOFSTEP]\nsimp [Finset.sum_fin_eq_sum_range, Finset.sum_range_succ']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\n\u22a2 LinearRecurrence.IsSolution fibRec fun x => \u03c6 ^ x\n[PROOFSTEP]\nrw [fibRec.geom_sol_iff_root_charPoly, fibRec_charPoly_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\n\u22a2 IsRoot (X ^ 2 - (X + 1)) \u03c6\n[PROOFSTEP]\nsimp [sub_eq_zero, -div_pow]\n -- Porting note: Added `- div_pow`\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\n\u22a2 LinearRecurrence.IsSolution fibRec fun x => \u03c8 ^ x\n[PROOFSTEP]\nrw [fibRec.geom_sol_iff_root_charPoly, fibRec_charPoly_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\n\u22a2 IsRoot (X ^ 2 - (X + 1)) \u03c8\n[PROOFSTEP]\nsimp [sub_eq_zero, -div_pow]\n -- Porting note: Added `- div_pow`\n[GOAL]\n\u22a2 (fun n => \u2191(Nat.fib n)) = fun n => (\u03c6 ^ n - \u03c8 ^ n) / sqrt 5\n[PROOFSTEP]\nrw [fibRec.sol_eq_of_eq_init]\n[GOAL]\n\u22a2 Set.EqOn (fun n => \u2191(Nat.fib n)) (fun n => (\u03c6 ^ n - \u03c8 ^ n) / sqrt 5) \u2191(Finset.range fibRec.order)\n[PROOFSTEP]\nintro i hi\n[GOAL]\ni : \u2115\nhi : i \u2208 \u2191(Finset.range fibRec.order)\n\u22a2 (fun n => \u2191(Nat.fib n)) i = (fun n => (\u03c6 ^ n - \u03c8 ^ n) / sqrt 5) i\n[PROOFSTEP]\nnorm_cast at hi \n[GOAL]\ni : \u2115\nhi : i \u2208 Finset.range fibRec.order\n\u22a2 (fun n => \u2191(Nat.fib n)) i = (fun n => (\u03c6 ^ n - \u03c8 ^ n) / sqrt 5) i\n[PROOFSTEP]\nfin_cases hi\n[GOAL]\ncase head\n\u22a2 (fun n => \u2191(Nat.fib n)) 0 = (fun n => (\u03c6 ^ n - \u03c8 ^ n) / sqrt 5) 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase tail.head\n\u22a2 (fun n => \u2191(Nat.fib n)) 1 = (fun n => (\u03c6 ^ n - \u03c8 ^ n) / sqrt 5) 1\n[PROOFSTEP]\nsimp only [goldenRatio, goldenConj]\n[GOAL]\ncase tail.head\n\u22a2 \u2191(Nat.fib 1) = (((1 + sqrt 5) / 2) ^ 1 - ((1 - sqrt 5) / 2) ^ 1) / sqrt 5\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase tail.head\n\u22a2 1 = sqrt 5 * (sqrt 5)\u207b\u00b9\n[PROOFSTEP]\nrw [mul_inv_cancel]\n[GOAL]\ncase tail.head\n\u22a2 sqrt 5 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase hu\n\u22a2 LinearRecurrence.IsSolution fibRec fun n => \u2191(Nat.fib n)\n[PROOFSTEP]\nexact fib_isSol_fibRec\n[GOAL]\ncase hv\n\u22a2 LinearRecurrence.IsSolution fibRec fun n => (\u03c6 ^ n - \u03c8 ^ n) / sqrt 5\n[PROOFSTEP]\nsuffices LinearRecurrence.IsSolution fibRec ((fun n \u21a6 (sqrt 5)\u207b\u00b9 * \u03c6 ^ n) - (fun n \u21a6 (sqrt 5)\u207b\u00b9 * \u03c8 ^ n))\n by\n convert this\n rw [Pi.sub_apply]\n ring\n[GOAL]\nthis : LinearRecurrence.IsSolution fibRec ((fun n => (sqrt 5)\u207b\u00b9 * \u03c6 ^ n) - fun n => (sqrt 5)\u207b\u00b9 * \u03c8 ^ n)\n\u22a2 LinearRecurrence.IsSolution fibRec fun n => (\u03c6 ^ n - \u03c8 ^ n) / sqrt 5\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_4.h\nthis : LinearRecurrence.IsSolution fibRec ((fun n => (sqrt 5)\u207b\u00b9 * \u03c6 ^ n) - fun n => (sqrt 5)\u207b\u00b9 * \u03c8 ^ n)\nx\u271d : \u2115\n\u22a2 (\u03c6 ^ x\u271d - \u03c8 ^ x\u271d) / sqrt 5 = ((fun n => (sqrt 5)\u207b\u00b9 * \u03c6 ^ n) - fun n => (sqrt 5)\u207b\u00b9 * \u03c8 ^ n) x\u271d\n[PROOFSTEP]\nrw [Pi.sub_apply]\n[GOAL]\ncase h.e'_4.h\nthis : LinearRecurrence.IsSolution fibRec ((fun n => (sqrt 5)\u207b\u00b9 * \u03c6 ^ n) - fun n => (sqrt 5)\u207b\u00b9 * \u03c8 ^ n)\nx\u271d : \u2115\n\u22a2 (\u03c6 ^ x\u271d - \u03c8 ^ x\u271d) / sqrt 5 = (sqrt 5)\u207b\u00b9 * \u03c6 ^ x\u271d - (sqrt 5)\u207b\u00b9 * \u03c8 ^ x\u271d\n[PROOFSTEP]\nring\n[GOAL]\ncase hv\n\u22a2 LinearRecurrence.IsSolution fibRec ((fun n => (sqrt 5)\u207b\u00b9 * \u03c6 ^ n) - fun n => (sqrt 5)\u207b\u00b9 * \u03c8 ^ n)\n[PROOFSTEP]\napply (@fibRec \u211d _).solSpace.sub_mem\n[GOAL]\ncase hv.a\n\u22a2 (fun n => (sqrt 5)\u207b\u00b9 * \u03c6 ^ n) \u2208 LinearRecurrence.solSpace fibRec\n[PROOFSTEP]\nexact Submodule.smul_mem fibRec.solSpace (Real.sqrt 5)\u207b\u00b9 geom_gold_isSol_fibRec\n[GOAL]\ncase hv.a\n\u22a2 (fun n => (sqrt 5)\u207b\u00b9 * \u03c8 ^ n) \u2208 LinearRecurrence.solSpace fibRec\n[PROOFSTEP]\nexact Submodule.smul_mem fibRec.solSpace (Real.sqrt 5)\u207b\u00b9 geom_goldConj_isSol_fibRec\n[GOAL]\n\u22a2 \u2200 (n : \u2115), \u2191(Nat.fib n) = (\u03c6 ^ n - \u03c8 ^ n) / sqrt 5\n[PROOFSTEP]\nrw [\u2190 Function.funext_iff, Real.coe_fib_eq']\n", "meta": {"mathlib_filename": "Mathlib.Data.Real.GoldenRatio", "llama_tokens": 5067, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581626286834, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.4115753012902599}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr\u271d : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\nhwf : WellFounded r\nx y : (i : \u03b9) \u2192 \u03b2 i\nhlt : x < y\n\u22a2 Pi.Lex r (fun i x x_1 => x < x_1) x y\n[PROOFSTEP]\nsimp_rw [Pi.Lex, le_antisymm_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr\u271d : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\nhwf : WellFounded r\nx y : (i : \u03b9) \u2192 \u03b2 i\nhlt : x < y\n\u22a2 \u2203 i, (\u2200 (j : \u03b9), r j i \u2192 x j \u2264 y j \u2227 y j \u2264 x j) \u2227 x i < y i\n[PROOFSTEP]\nexact lex_lt_of_lt_of_preorder hwf hlt\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b2 i) s\nwf : WellFounded r\na b : (i : \u03b9) \u2192 \u03b2 i\n\u22a2 Pi.Lex r s a b \u2228 a = b \u2228 Pi.Lex r s b a\n[PROOFSTEP]\ncases' eq_or_ne a b with hab hab\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b2 i) s\nwf : WellFounded r\na b : (i : \u03b9) \u2192 \u03b2 i\nhab : a = b\n\u22a2 Pi.Lex r s a b \u2228 a = b \u2228 Pi.Lex r s b a\n[PROOFSTEP]\nexact Or.inr (Or.inl hab)\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b2 i) s\nwf : WellFounded r\na b : (i : \u03b9) \u2192 \u03b2 i\nhab : a \u2260 b\n\u22a2 Pi.Lex r s a b \u2228 a = b \u2228 Pi.Lex r s b a\n[PROOFSTEP]\nrw [Function.ne_iff] at hab \n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b2 i) s\nwf : WellFounded r\na b : (i : \u03b9) \u2192 \u03b2 i\nhab : \u2203 a_1, a a_1 \u2260 b a_1\n\u22a2 Pi.Lex r s a b \u2228 a = b \u2228 Pi.Lex r s b a\n[PROOFSTEP]\nlet i := wf.min _ hab\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b2 i) s\nwf : WellFounded r\na b : (i : \u03b9) \u2192 \u03b2 i\nhab : \u2203 a_1, a a_1 \u2260 b a_1\ni : \u03b9 := WellFounded.min wf (fun x => a x = b x \u2192 False) hab\n\u22a2 Pi.Lex r s a b \u2228 a = b \u2228 Pi.Lex r s b a\n[PROOFSTEP]\nhave hri : \u2200 j, r j i \u2192 a j = b j := by\n intro j\n rw [\u2190 not_imp_not]\n exact fun h' => wf.not_lt_min _ _ h'\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b2 i) s\nwf : WellFounded r\na b : (i : \u03b9) \u2192 \u03b2 i\nhab : \u2203 a_1, a a_1 \u2260 b a_1\ni : \u03b9 := WellFounded.min wf (fun x => a x = b x \u2192 False) hab\n\u22a2 \u2200 (j : \u03b9), r j i \u2192 a j = b j\n[PROOFSTEP]\nintro j\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b2 i) s\nwf : WellFounded r\na b : (i : \u03b9) \u2192 \u03b2 i\nhab : \u2203 a_1, a a_1 \u2260 b a_1\ni : \u03b9 := WellFounded.min wf (fun x => a x = b x \u2192 False) hab\nj : \u03b9\n\u22a2 r j i \u2192 a j = b j\n[PROOFSTEP]\nrw [\u2190 not_imp_not]\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b2 i) s\nwf : WellFounded r\na b : (i : \u03b9) \u2192 \u03b2 i\nhab : \u2203 a_1, a a_1 \u2260 b a_1\ni : \u03b9 := WellFounded.min wf (fun x => a x = b x \u2192 False) hab\nj : \u03b9\n\u22a2 \u00aca j = b j \u2192 \u00acr j i\n[PROOFSTEP]\nexact fun h' => wf.not_lt_min _ _ h'\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b2 i) s\nwf : WellFounded r\na b : (i : \u03b9) \u2192 \u03b2 i\nhab : \u2203 a_1, a a_1 \u2260 b a_1\ni : \u03b9 := WellFounded.min wf (fun x => a x = b x \u2192 False) hab\nhri : \u2200 (j : \u03b9), r j i \u2192 a j = b j\n\u22a2 Pi.Lex r s a b \u2228 a = b \u2228 Pi.Lex r s b a\n[PROOFSTEP]\nhave hne : a i \u2260 b i := wf.min_mem _ hab\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b2 i) s\nwf : WellFounded r\na b : (i : \u03b9) \u2192 \u03b2 i\nhab : \u2203 a_1, a a_1 \u2260 b a_1\ni : \u03b9 := WellFounded.min wf (fun x => a x = b x \u2192 False) hab\nhri : \u2200 (j : \u03b9), r j i \u2192 a j = b j\nhne : a i \u2260 b i\n\u22a2 Pi.Lex r s a b \u2228 a = b \u2228 Pi.Lex r s b a\n[PROOFSTEP]\ncases' trichotomous_of s (a i) (b i) with hi hi\n[GOAL]\ncase inr.inl\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b2 i) s\nwf : WellFounded r\na b : (i : \u03b9) \u2192 \u03b2 i\nhab : \u2203 a_1, a a_1 \u2260 b a_1\ni : \u03b9 := WellFounded.min wf (fun x => a x = b x \u2192 False) hab\nhri : \u2200 (j : \u03b9), r j i \u2192 a j = b j\nhne : a i \u2260 b i\nhi : s (a i) (b i)\n\u22a2 Pi.Lex r s a b \u2228 a = b \u2228 Pi.Lex r s b a\ncase inr.inr\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b2 i) s\nwf : WellFounded r\na b : (i : \u03b9) \u2192 \u03b2 i\nhab : \u2203 a_1, a a_1 \u2260 b a_1\ni : \u03b9 := WellFounded.min wf (fun x => a x = b x \u2192 False) hab\nhri : \u2200 (j : \u03b9), r j i \u2192 a j = b j\nhne : a i \u2260 b i\nhi : a i = b i \u2228 s (b i) (a i)\n\u22a2 Pi.Lex r s a b \u2228 a = b \u2228 Pi.Lex r s b a\n[PROOFSTEP]\nexacts [Or.inl \u27e8i, hri, hi\u27e9, Or.inr <| Or.inr <| \u27e8i, fun j hj => (hri j hj).symm, hi.resolve_left hne\u27e9]\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (a : \u03b9) \u2192 PartialOrder (\u03b2 a)\n\u22a2 \u2200 (a b c : Lex ((i : \u03b9) \u2192 \u03b2 i)), a < b \u2192 b < c \u2192 a < c\n[PROOFSTEP]\nrintro a b c \u27e8N\u2081, lt_N\u2081, a_lt_b\u27e9 \u27e8N\u2082, lt_N\u2082, b_lt_c\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (a : \u03b9) \u2192 PartialOrder (\u03b2 a)\na b c : Lex ((i : \u03b9) \u2192 \u03b2 i)\nN\u2081 : \u03b9\nlt_N\u2081 : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j N\u2081 \u2192 a j = b j\na_lt_b : a N\u2081 < b N\u2081\nN\u2082 : \u03b9\nlt_N\u2082 : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j N\u2082 \u2192 b j = c j\nb_lt_c : b N\u2082 < c N\u2082\n\u22a2 a < c\n[PROOFSTEP]\nrcases lt_trichotomy N\u2081 N\u2082 with (H | rfl | H)\n[GOAL]\ncase intro.intro.intro.intro.inl\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (a : \u03b9) \u2192 PartialOrder (\u03b2 a)\na b c : Lex ((i : \u03b9) \u2192 \u03b2 i)\nN\u2081 : \u03b9\nlt_N\u2081 : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j N\u2081 \u2192 a j = b j\na_lt_b : a N\u2081 < b N\u2081\nN\u2082 : \u03b9\nlt_N\u2082 : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j N\u2082 \u2192 b j = c j\nb_lt_c : b N\u2082 < c N\u2082\nH : N\u2081 < N\u2082\n\u22a2 a < c\ncase intro.intro.intro.intro.inr.inl\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (a : \u03b9) \u2192 PartialOrder (\u03b2 a)\na b c : Lex ((i : \u03b9) \u2192 \u03b2 i)\nN\u2081 : \u03b9\nlt_N\u2081 : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j N\u2081 \u2192 a j = b j\na_lt_b : a N\u2081 < b N\u2081\nlt_N\u2082 : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j N\u2081 \u2192 b j = c j\nb_lt_c : b N\u2081 < c N\u2081\n\u22a2 a < c\ncase intro.intro.intro.intro.inr.inr\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (a : \u03b9) \u2192 PartialOrder (\u03b2 a)\na b c : Lex ((i : \u03b9) \u2192 \u03b2 i)\nN\u2081 : \u03b9\nlt_N\u2081 : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j N\u2081 \u2192 a j = b j\na_lt_b : a N\u2081 < b N\u2081\nN\u2082 : \u03b9\nlt_N\u2082 : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j N\u2082 \u2192 b j = c j\nb_lt_c : b N\u2082 < c N\u2082\nH : N\u2082 < N\u2081\n\u22a2 a < c\n[PROOFSTEP]\nexacts [\u27e8N\u2081, fun j hj => (lt_N\u2081 _ hj).trans (lt_N\u2082 _ <| hj.trans H), lt_N\u2082 _ H \u25b8 a_lt_b\u27e9,\n \u27e8N\u2081, fun j hj => (lt_N\u2081 _ hj).trans (lt_N\u2082 _ hj), a_lt_b.trans b_lt_c\u27e9,\n \u27e8N\u2082, fun j hj => (lt_N\u2081 _ (hj.trans H)).trans (lt_N\u2082 _ hj), (lt_N\u2081 _ H).symm \u25b8 b_lt_c\u27e9]\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\na\u271d : \u03b2 i\u271d\na b : (i : \u03b9) \u2192 \u03b2 i\nh : a \u2264 b\nhne : \u00ac\u2191toLex a = \u2191toLex b\ni : \u03b9\nhi : i \u2208 {i | a i \u2260 b i}\nhl : \u2200 (x : \u03b9), x \u2208 {i | a i \u2260 b i} \u2192 \u00acx < i\nj : \u03b9\nhj : (fun x x_1 => x < x_1) j i\n\u22a2 \u2191toLex a j = \u2191toLex b j\n[PROOFSTEP]\ncontrapose! hl\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\na\u271d : \u03b2 i\u271d\na b : (i : \u03b9) \u2192 \u03b2 i\nh : a \u2264 b\nhne : \u00ac\u2191toLex a = \u2191toLex b\ni : \u03b9\nhi : i \u2208 {i | a i \u2260 b i}\nj : \u03b9\nhj : (fun x x_1 => x < x_1) j i\nhl : \u2191toLex a j \u2260 \u2191toLex b j\n\u22a2 \u2203 x, x \u2208 {i | a i \u2260 b i} \u2227 x < i\n[PROOFSTEP]\nexact \u27e8j, hl, hj\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\na\u271d : \u03b2 i\u271d\na b : (i : \u03b9) \u2192 \u03b2 i\nh : a < b\ni : \u03b9\nhi : i \u2208 {i | a i \u2260 b i}\nhl : \u2200 (x : \u03b9), x \u2208 {i | a i \u2260 b i} \u2192 \u00acx < i\nj : \u03b9\nhj : (fun x x_1 => x < x_1) j i\n\u22a2 \u2191toLex a j = \u2191toLex b j\n[PROOFSTEP]\ncontrapose! hl\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\na\u271d : \u03b2 i\u271d\na b : (i : \u03b9) \u2192 \u03b2 i\nh : a < b\ni : \u03b9\nhi : i \u2208 {i | a i \u2260 b i}\nj : \u03b9\nhj : (fun x x_1 => x < x_1) j i\nhl : \u2191toLex a j \u2260 \u2191toLex b j\n\u22a2 \u2203 x, x \u2208 {i | a i \u2260 b i} \u2227 x < i\n[PROOFSTEP]\nexact \u27e8j, hl, hj\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\n\u22a2 \u2191toLex x < \u2191toLex (update x i a) \u2194 x i < a\n[PROOFSTEP]\nrefine' \u27e8_, fun h => toLex_strictMono <| lt_update_self_iff.2 h\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\n\u22a2 \u2191toLex x < \u2191toLex (update x i a) \u2192 x i < a\n[PROOFSTEP]\nrintro \u27e8j, hj, h\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\nj : \u03b9\nhj : \u2200 (j_1 : \u03b9), (fun x x_1 => x < x_1) j_1 j \u2192 \u2191toLex x j_1 = \u2191toLex (update x i a) j_1\nh : \u2191toLex x j < \u2191toLex (update x i a) j\n\u22a2 x i < a\n[PROOFSTEP]\ndsimp at h \n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\nj : \u03b9\nhj : \u2200 (j_1 : \u03b9), (fun x x_1 => x < x_1) j_1 j \u2192 \u2191toLex x j_1 = \u2191toLex (update x i a) j_1\nh : x j < update x i a j\n\u22a2 x i < a\n[PROOFSTEP]\nobtain rfl : j = i := by\n by_contra H\n rw [update_noteq H] at h \n exact h.false\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\nj : \u03b9\nhj : \u2200 (j_1 : \u03b9), (fun x x_1 => x < x_1) j_1 j \u2192 \u2191toLex x j_1 = \u2191toLex (update x i a) j_1\nh : x j < update x i a j\n\u22a2 j = i\n[PROOFSTEP]\nby_contra H\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\nj : \u03b9\nhj : \u2200 (j_1 : \u03b9), (fun x x_1 => x < x_1) j_1 j \u2192 \u2191toLex x j_1 = \u2191toLex (update x i a) j_1\nh : x j < update x i a j\nH : \u00acj = i\n\u22a2 False\n[PROOFSTEP]\nrw [update_noteq H] at h \n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\nj : \u03b9\nhj : \u2200 (j_1 : \u03b9), (fun x x_1 => x < x_1) j_1 j \u2192 \u2191toLex x j_1 = \u2191toLex (update x i a) j_1\nh : x j < x j\nH : \u00acj = i\n\u22a2 False\n[PROOFSTEP]\nexact h.false\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\nj : \u03b9\na : \u03b2 j\nhj : \u2200 (j_1 : \u03b9), (fun x x_1 => x < x_1) j_1 j \u2192 \u2191toLex x j_1 = \u2191toLex (update x j a) j_1\nh : x j < update x j a j\n\u22a2 x j < a\n[PROOFSTEP]\nrwa [update_same] at h \n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\n\u22a2 \u2191toLex (update x i a) < \u2191toLex x \u2194 a < x i\n[PROOFSTEP]\nrefine' \u27e8_, fun h => toLex_strictMono <| update_lt_self_iff.2 h\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\n\u22a2 \u2191toLex (update x i a) < \u2191toLex x \u2192 a < x i\n[PROOFSTEP]\nrintro \u27e8j, hj, h\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\nj : \u03b9\nhj : \u2200 (j_1 : \u03b9), (fun x x_1 => x < x_1) j_1 j \u2192 \u2191toLex (update x i a) j_1 = \u2191toLex x j_1\nh : \u2191toLex (update x i a) j < \u2191toLex x j\n\u22a2 a < x i\n[PROOFSTEP]\ndsimp at h \n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\nj : \u03b9\nhj : \u2200 (j_1 : \u03b9), (fun x x_1 => x < x_1) j_1 j \u2192 \u2191toLex (update x i a) j_1 = \u2191toLex x j_1\nh : update x i a j < x j\n\u22a2 a < x i\n[PROOFSTEP]\nobtain rfl : j = i := by\n by_contra H\n rw [update_noteq H] at h \n exact h.false\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\nj : \u03b9\nhj : \u2200 (j_1 : \u03b9), (fun x x_1 => x < x_1) j_1 j \u2192 \u2191toLex (update x i a) j_1 = \u2191toLex x j_1\nh : update x i a j < x j\n\u22a2 j = i\n[PROOFSTEP]\nby_contra H\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\nj : \u03b9\nhj : \u2200 (j_1 : \u03b9), (fun x x_1 => x < x_1) j_1 j \u2192 \u2191toLex (update x i a) j_1 = \u2191toLex x j_1\nh : update x i a j < x j\nH : \u00acj = i\n\u22a2 False\n[PROOFSTEP]\nrw [update_noteq H] at h \n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\nj : \u03b9\nhj : \u2200 (j_1 : \u03b9), (fun x x_1 => x < x_1) j_1 j \u2192 \u2191toLex (update x i a) j_1 = \u2191toLex x j_1\nh : x j < x j\nH : \u00acj = i\n\u22a2 False\n[PROOFSTEP]\nexact h.false\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\nj : \u03b9\na : \u03b2 j\nhj : \u2200 (j_1 : \u03b9), (fun x x_1 => x < x_1) j_1 j \u2192 \u2191toLex (update x j a) j_1 = \u2191toLex x j_1\nh : update x j a j < x j\n\u22a2 a < x j\n[PROOFSTEP]\nrwa [update_same] at h \n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\n\u22a2 \u2191toLex x \u2264 \u2191toLex (update x i a) \u2194 x i \u2264 a\n[PROOFSTEP]\nsimp_rw [le_iff_lt_or_eq, lt_toLex_update_self_iff, toLex_inj, eq_update_self_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : IsWellOrder \u03b9 fun x x_1 => x < x_1\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b2 i)\nx y : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\na : \u03b2 i\n\u22a2 \u2191toLex (update x i a) \u2264 \u2191toLex x \u2194 a \u2264 x i\n[PROOFSTEP]\nsimp_rw [le_iff_lt_or_eq, toLex_update_lt_self_iff, toLex_inj, update_eq_self_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\n\u22a2 \u2200 (a\u2081 a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)), a\u2081 < a\u2082 \u2192 \u2203 a, a\u2081 < a \u2227 a < a\u2082\n[PROOFSTEP]\nrintro _ a\u2082 \u27e8i, h, hi\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\na\u2081\u271d a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)\ni : \u03b9\nh : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j i \u2192 a\u2081\u271d j = a\u2082 j\nhi : a\u2081\u271d i < a\u2082 i\n\u22a2 \u2203 a, a\u2081\u271d < a \u2227 a < a\u2082\n[PROOFSTEP]\nobtain \u27e8a, ha\u2081, ha\u2082\u27e9 := exists_between hi\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\na\u2081\u271d a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)\ni : \u03b9\nh : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j i \u2192 a\u2081\u271d j = a\u2082 j\nhi : a\u2081\u271d i < a\u2082 i\na : \u03b2 i\nha\u2081 : a\u2081\u271d i < a\nha\u2082 : a < a\u2082 i\n\u22a2 \u2203 a, a\u2081\u271d < a \u2227 a < a\u2082\n[PROOFSTEP]\nclassical\nrefine' \u27e8Function.update a\u2082 _ a, \u27e8i, fun j hj => _, _\u27e9, i, fun j hj => _, _\u27e9\nrw [h j hj]\ndsimp only at hj \n\u00b7 rw [Function.update_noteq hj.ne a]\n\u00b7 rwa [Function.update_same i a]\n\u00b7 rw [Function.update_noteq hj.ne a]\n\u00b7 rwa [Function.update_same i a]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\na\u2081\u271d a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)\ni : \u03b9\nh : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j i \u2192 a\u2081\u271d j = a\u2082 j\nhi : a\u2081\u271d i < a\u2082 i\na : \u03b2 i\nha\u2081 : a\u2081\u271d i < a\nha\u2082 : a < a\u2082 i\n\u22a2 \u2203 a, a\u2081\u271d < a \u2227 a < a\u2082\n[PROOFSTEP]\nrefine' \u27e8Function.update a\u2082 _ a, \u27e8i, fun j hj => _, _\u27e9, i, fun j hj => _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\na\u2081\u271d a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)\ni : \u03b9\nh : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j i \u2192 a\u2081\u271d j = a\u2082 j\nhi : a\u2081\u271d i < a\u2082 i\na : \u03b2 i\nha\u2081 : a\u2081\u271d i < a\nha\u2082 : a < a\u2082 i\nj : \u03b9\nhj : (fun x x_1 => x < x_1) j i\n\u22a2 a\u2081\u271d j = Function.update a\u2082 i a j\ncase intro.intro.intro.intro.refine'_2\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\na\u2081\u271d a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)\ni : \u03b9\nh : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j i \u2192 a\u2081\u271d j = a\u2082 j\nhi : a\u2081\u271d i < a\u2082 i\na : \u03b2 i\nha\u2081 : a\u2081\u271d i < a\nha\u2082 : a < a\u2082 i\n\u22a2 (fun x x_1 x_2 => x_1 < x_2) i (a\u2081\u271d i) (Function.update a\u2082 i a i)\ncase intro.intro.intro.intro.refine'_3\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\na\u2081\u271d a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)\ni : \u03b9\nh : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j i \u2192 a\u2081\u271d j = a\u2082 j\nhi : a\u2081\u271d i < a\u2082 i\na : \u03b2 i\nha\u2081 : a\u2081\u271d i < a\nha\u2082 : a < a\u2082 i\nj : \u03b9\nhj : (fun x x_1 => x < x_1) j i\n\u22a2 Function.update a\u2082 i a j = a\u2082 j\ncase intro.intro.intro.intro.refine'_4\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\na\u2081\u271d a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)\ni : \u03b9\nh : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j i \u2192 a\u2081\u271d j = a\u2082 j\nhi : a\u2081\u271d i < a\u2082 i\na : \u03b2 i\nha\u2081 : a\u2081\u271d i < a\nha\u2082 : a < a\u2082 i\n\u22a2 (fun x x_1 x_2 => x_1 < x_2) i (Function.update a\u2082 i a i) (a\u2082 i)\n[PROOFSTEP]\nrw [h j hj]\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\na\u2081\u271d a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)\ni : \u03b9\nh : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j i \u2192 a\u2081\u271d j = a\u2082 j\nhi : a\u2081\u271d i < a\u2082 i\na : \u03b2 i\nha\u2081 : a\u2081\u271d i < a\nha\u2082 : a < a\u2082 i\nj : \u03b9\nhj : (fun x x_1 => x < x_1) j i\n\u22a2 a\u2082 j = Function.update a\u2082 i a j\ncase intro.intro.intro.intro.refine'_2\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\na\u2081\u271d a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)\ni : \u03b9\nh : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j i \u2192 a\u2081\u271d j = a\u2082 j\nhi : a\u2081\u271d i < a\u2082 i\na : \u03b2 i\nha\u2081 : a\u2081\u271d i < a\nha\u2082 : a < a\u2082 i\n\u22a2 (fun x x_1 x_2 => x_1 < x_2) i (a\u2081\u271d i) (Function.update a\u2082 i a i)\ncase intro.intro.intro.intro.refine'_3\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\na\u2081\u271d a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)\ni : \u03b9\nh : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j i \u2192 a\u2081\u271d j = a\u2082 j\nhi : a\u2081\u271d i < a\u2082 i\na : \u03b2 i\nha\u2081 : a\u2081\u271d i < a\nha\u2082 : a < a\u2082 i\nj : \u03b9\nhj : (fun x x_1 => x < x_1) j i\n\u22a2 Function.update a\u2082 i a j = a\u2082 j\ncase intro.intro.intro.intro.refine'_4\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\na\u2081\u271d a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)\ni : \u03b9\nh : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j i \u2192 a\u2081\u271d j = a\u2082 j\nhi : a\u2081\u271d i < a\u2082 i\na : \u03b2 i\nha\u2081 : a\u2081\u271d i < a\nha\u2082 : a < a\u2082 i\n\u22a2 (fun x x_1 x_2 => x_1 < x_2) i (Function.update a\u2082 i a i) (a\u2082 i)\n[PROOFSTEP]\ndsimp only at hj \n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\na\u2081\u271d a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)\ni : \u03b9\nh : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j i \u2192 a\u2081\u271d j = a\u2082 j\nhi : a\u2081\u271d i < a\u2082 i\na : \u03b2 i\nha\u2081 : a\u2081\u271d i < a\nha\u2082 : a < a\u2082 i\nj : \u03b9\nhj : j < i\n\u22a2 a\u2082 j = Function.update a\u2082 i a j\n[PROOFSTEP]\nrw [Function.update_noteq hj.ne a]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\na\u2081\u271d a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)\ni : \u03b9\nh : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j i \u2192 a\u2081\u271d j = a\u2082 j\nhi : a\u2081\u271d i < a\u2082 i\na : \u03b2 i\nha\u2081 : a\u2081\u271d i < a\nha\u2082 : a < a\u2082 i\n\u22a2 (fun x x_1 x_2 => x_1 < x_2) i (a\u2081\u271d i) (Function.update a\u2082 i a i)\n[PROOFSTEP]\nrwa [Function.update_same i a]\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\na\u2081\u271d a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)\ni : \u03b9\nh : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j i \u2192 a\u2081\u271d j = a\u2082 j\nhi : a\u2081\u271d i < a\u2082 i\na : \u03b2 i\nha\u2081 : a\u2081\u271d i < a\nha\u2082 : a < a\u2082 i\nj : \u03b9\nhj : (fun x x_1 => x < x_1) j i\n\u22a2 Function.update a\u2082 i a j = a\u2082 j\n[PROOFSTEP]\nrw [Function.update_noteq hj.ne a]\n[GOAL]\ncase intro.intro.intro.intro.refine'_4\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b2 i)\na\u2081\u271d a\u2082 : Lex ((i : \u03b9) \u2192 \u03b2 i)\ni : \u03b9\nh : \u2200 (j : \u03b9), (fun x x_1 => x < x_1) j i \u2192 a\u2081\u271d j = a\u2082 j\nhi : a\u2081\u271d i < a\u2082 i\na : \u03b2 i\nha\u2081 : a\u2081\u271d i < a\nha\u2082 : a < a\u2082 i\n\u22a2 (fun x x_1 x_2 => x_1 < x_2) i (Function.update a\u2082 i a i) (a\u2082 i)\n[PROOFSTEP]\nrwa [Function.update_same i a]\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ni : \u03b9\ninst\u271d : NoMaxOrder (\u03b2 i)\na : Lex ((i : \u03b9) \u2192 \u03b2 i)\n\u22a2 \u2203 b, a < b\n[PROOFSTEP]\nlet \u27e8b, hb\u27e9 := exists_gt (a i)\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ni : \u03b9\ninst\u271d : NoMaxOrder (\u03b2 i)\na : Lex ((i : \u03b9) \u2192 \u03b2 i)\nb : \u03b2 i\nhb : a i < b\n\u22a2 \u2203 b, a < b\n[PROOFSTEP]\nclassical exact\n \u27e8Function.update a i b, i, fun j hj => (Function.update_noteq hj.ne b a).symm, by rwa [Function.update_same i b]\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ni : \u03b9\ninst\u271d : NoMaxOrder (\u03b2 i)\na : Lex ((i : \u03b9) \u2192 \u03b2 i)\nb : \u03b2 i\nhb : a i < b\n\u22a2 \u2203 b, a < b\n[PROOFSTEP]\nexact \u27e8Function.update a i b, i, fun j hj => (Function.update_noteq hj.ne b a).symm, by rwa [Function.update_same i b]\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 LT (\u03b2 i)\ni : \u03b9\ninst\u271d : NoMaxOrder (\u03b2 i)\na : Lex ((i : \u03b9) \u2192 \u03b2 i)\nb : \u03b2 i\nhb : a i < b\n\u22a2 (fun x x_1 x_2 => x_1 < x_2) i (a i) (Function.update a i b i)\n[PROOFSTEP]\nrwa [Function.update_same i b]\n[GOAL]\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ns : {i : \u03b9} \u2192 \u03b2 i \u2192 \u03b2 i \u2192 Prop\n\u03b1 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Preorder \u03b1\nf : \u03b9 \u2192 \u03b1\ni j : \u03b9\nh\u2081 : i \u2264 j\nh\u2082 : f j < f i\n\u22a2 (fun x x_1 x_2 => x_1 < x_2) i (\u2191toLex (f \u2218 \u2191(Equiv.swap i j)) i) (\u2191toLex f i)\n[PROOFSTEP]\nsimpa only [Pi.toLex_apply, Function.comp_apply, Equiv.swap_apply_left] using h\u2082\n", "meta": {"mathlib_filename": "Mathlib.Data.Pi.Lex", "llama_tokens": 14146, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7634837635542925, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.4115049378232469}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nx : \u03b3\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b3\n\u22a2 x \u2208 \u22c3 (i : \u03b9) (j : \u03ba i), s i j \u2194 \u2203 i j, x \u2208 s i j\n[PROOFSTEP]\nsimp_rw [mem_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nx : \u03b3\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b3\n\u22a2 x \u2208 \u22c2 (i : \u03b9) (j : \u03ba i), s i j \u2194 \u2200 (i : \u03b9) (j : \u03ba i), x \u2208 s i j\n[PROOFSTEP]\nsimp_rw [mem_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nsrc\u271d : BooleanAlgebra (Set \u03b1) := instBooleanAlgebraSet\n\u22a2 \u2200 {\u03b9 : Type u_1} {\u03ba : \u03b9 \u2192 Type u_1} (f : (a : \u03b9) \u2192 \u03ba a \u2192 Set \u03b1),\n \u2a05 (a : \u03b9), \u2a06 (b : \u03ba a), f a b = \u2a06 (g : (a : \u03b9) \u2192 \u03ba a), \u2a05 (a : \u03b9), f a (g a)\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nsrc\u271d : BooleanAlgebra (Set \u03b1) := instBooleanAlgebraSet\n\u03b9\u271d : Type u_1\n\u03ba\u271d : \u03b9\u271d \u2192 Type u_1\nf\u271d : (a : \u03b9\u271d) \u2192 \u03ba\u271d a \u2192 Set \u03b1\n\u22a2 \u2a05 (a : \u03b9\u271d), \u2a06 (b : \u03ba\u271d a), f\u271d a b = \u2a06 (g : (a : \u03b9\u271d) \u2192 \u03ba\u271d a), \u2a05 (a : \u03b9\u271d), f\u271d a (g a)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nsrc\u271d : BooleanAlgebra (Set \u03b1) := instBooleanAlgebraSet\n\u03b9\u271d : Type u_1\n\u03ba\u271d : \u03b9\u271d \u2192 Type u_1\nf\u271d : (a : \u03b9\u271d) \u2192 \u03ba\u271d a \u2192 Set \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2a05 (a : \u03b9\u271d), \u2a06 (b : \u03ba\u271d a), f\u271d a b \u2194 x\u271d \u2208 \u2a06 (g : (a : \u03b9\u271d) \u2192 \u03ba\u271d a), \u2a05 (a : \u03b9\u271d), f\u271d a (g a)\n[PROOFSTEP]\nsimp [Classical.skolem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 kernImage f s\u1d9c = (f '' s)\u1d9c\n[PROOFSTEP]\nrw [kernImage_eq_compl, compl_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\n\u22a2 kernImage f \u2205 = (range f)\u1d9c\n[PROOFSTEP]\nrw [kernImage_eq_compl, compl_empty, image_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u22a2 kernImage f (f \u207b\u00b9' s) = s \u2194 (range f)\u1d9c \u2286 s\n[PROOFSTEP]\nrw [kernImage_eq_compl, \u2190 preimage_compl, compl_eq_comm, eq_comm, image_preimage_eq_iff, compl_subset_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 (range f)\u1d9c \u2286 kernImage f s\n[PROOFSTEP]\nrw [\u2190 kernImage_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 kernImage f \u2205 \u2286 kernImage f s\n[PROOFSTEP]\nexact kernImage_mono (empty_subset _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 kernImage f (s \u222a f \u207b\u00b9' t) = kernImage f s \u222a t\n[PROOFSTEP]\nrw [kernImage_eq_compl, kernImage_eq_compl, compl_union, \u2190 preimage_compl, image_inter_preimage, compl_inter,\n compl_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 kernImage f (f \u207b\u00b9' t \u222a s) = t \u222a kernImage f s\n[PROOFSTEP]\nrw [union_comm, kernImage_union_preimage, union_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u22a2 \u2200 (a : Set \u03b1), a \u2264 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b2\nw : \u22c3 (i : \u03b9) (_ : i \u2208 t), s i = \u22a4\nx : \u03b2\n\u22a2 \u2203 i, i \u2208 t \u2227 x \u2208 s i\n[PROOFSTEP]\nhave p : x \u2208 \u22a4 := Set.mem_univ x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b2\nw : \u22c3 (i : \u03b9) (_ : i \u2208 t), s i = \u22a4\nx : \u03b2\np : x \u2208 \u22a4\n\u22a2 \u2203 i, i \u2208 t \u2227 x \u2208 s i\n[PROOFSTEP]\nrw [\u2190 w, Set.mem_iUnion] at p \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b2\nw : \u22c3 (i : \u03b9) (_ : i \u2208 t), s i = \u22a4\nx : \u03b2\np : \u2203 i, x \u2208 \u22c3 (_ : i \u2208 t), s i\n\u22a2 \u2203 i, i \u2208 t \u2227 x \u2208 s i\n[PROOFSTEP]\nsimpa using p\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : Nonempty \u03b1\nw : \u22c3 (i : \u03b9) (_ : i \u2208 t), s i = \u22a4\n\u22a2 Set.Nonempty t\n[PROOFSTEP]\nobtain \u27e8x, m, -\u27e9 := exists_set_mem_of_union_eq_top t s w H.some\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : Nonempty \u03b1\nw : \u22c3 (i : \u03b9) (_ : i \u2208 t), s i = \u22a4\nx : \u03b9\nm : x \u2208 t\n\u22a2 Set.Nonempty t\n[PROOFSTEP]\nexact \u27e8x, m\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b1\nh_Union : Set.Nonempty (\u22c3 (i : \u03b9), s i)\n\u22a2 Nonempty \u03b9\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := h_Union\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b1\nx : \u03b1\nhx : x \u2208 \u22c3 (i : \u03b9), s i\n\u22a2 Nonempty \u03b9\n[PROOFSTEP]\nexact \u27e8Classical.choose $ mem_iUnion.mp hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b1\ninst\u271d : Nonempty \u03b1\nh_Union : \u22c3 (i : \u03b9), s i = univ\n\u22a2 Set.Nonempty (\u22c3 (i : \u03b9), s i)\n[PROOFSTEP]\nsimpa only [h_Union] using univ_nonempty\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\nt : Set \u03b1\n\u22a2 \u22c3 (i : \u03b9) (j : \u03ba i), s i j \u2286 t \u2194 \u2200 (i : \u03b9) (j : \u03ba i), s i j \u2286 t\n[PROOFSTEP]\nsimp_rw [iUnion_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\n\u22a2 s \u2286 \u22c2 (i : \u03b9) (j : \u03ba i), t i j \u2194 \u2200 (i : \u03b9) (j : \u03ba i), s \u2286 t i j\n[PROOFSTEP]\nsimp_rw [subset_iInter_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nP : \u03b9 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u22c3 (i : \u03b9), {x | P i x} = {x | \u2203 i, P i x}\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nP : \u03b9 \u2192 \u03b1 \u2192 Prop\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u22c3 (i : \u03b9), {x | P i x} \u2194 x\u271d \u2208 {x | \u2203 i, P i x}\n[PROOFSTEP]\nexact mem_iUnion\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nP : \u03b9 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u22c2 (i : \u03b9), {x | P i x} = {x | \u2200 (i : \u03b9), P i x}\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nP : \u03b9 \u2192 \u03b1 \u2192 Prop\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u22c2 (i : \u03b9), {x | P i x} \u2194 x\u271d \u2208 {x | \u2200 (i : \u03b9), P i x}\n[PROOFSTEP]\nexact mem_iInter\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\n\u22a2 (\u22c3 (i : \u03b9) (j : \u03ba i), s i j)\u1d9c = \u22c2 (i : \u03b9) (j : \u03ba i), (s i j)\u1d9c\n[PROOFSTEP]\nsimp_rw [compl_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\n\u22a2 (\u22c2 (i : \u03b9) (j : \u03ba i), s i j)\u1d9c = \u22c3 (i : \u03b9) (j : \u03ba i), (s i j)\u1d9c\n[PROOFSTEP]\nsimp_rw [compl_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b2\n\u22a2 \u22c3 (i : \u03b9), s i = (\u22c2 (i : \u03b9), (s i)\u1d9c)\u1d9c\n[PROOFSTEP]\nsimp only [compl_iInter, compl_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b2\n\u22a2 \u22c2 (i : \u03b9), s i = (\u22c3 (i : \u03b9), (s i)\u1d9c)\u1d9c\n[PROOFSTEP]\nsimp only [compl_iUnion, compl_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : Set \u03b2\nt : \u03b9 \u2192 Set \u03b2\n\u22a2 s \\ \u22c3 (i : \u03b9), t i = \u22c2 (i : \u03b9), s \\ t i\n[PROOFSTEP]\nrw [diff_eq, compl_iUnion, inter_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : Set \u03b2\nt : \u03b9 \u2192 Set \u03b2\n\u22a2 \u22c2 (i : \u03b9), s \u2229 (t i)\u1d9c = \u22c2 (i : \u03b9), s \\ t i\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b2\nt : \u03b9 \u2192 Set \u03b2\n\u22a2 s \\ \u22c2 (i : \u03b9), t i = \u22c3 (i : \u03b9), s \\ t i\n[PROOFSTEP]\nrw [diff_eq, compl_iInter, inter_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b2\nt : \u03b9 \u2192 Set \u03b2\n\u22a2 \u22c3 (i : \u03b9), s \u2229 (t i)\u1d9c = \u22c3 (i : \u03b9), s \\ t i\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) f\nh : \u2200 (x : \u03b9), DirectedOn r (f x)\n\u22a2 DirectedOn r (\u22c3 (x : \u03b9), f x)\n[PROOFSTEP]\nsimp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) f\nh : \u2200 (x : \u03b9), DirectedOn r (f x)\n\u22a2 \u2200 (x : \u03b1) (x_1 : \u03b9), x \u2208 f x_1 \u2192 \u2200 (y : \u03b1) (x_2 : \u03b9), y \u2208 f x_2 \u2192 \u2203 z, (\u2203 i, z \u2208 f i) \u2227 r x z \u2227 r y z\n[PROOFSTEP]\nexact fun a\u2081 b\u2081 fb\u2081 a\u2082 b\u2082 fb\u2082 =>\n let \u27e8z, zb\u2081, zb\u2082\u27e9 := hd b\u2081 b\u2082\n let \u27e8x, xf, xa\u2081, xa\u2082\u27e9 := h z a\u2081 (zb\u2081 fb\u2081) a\u2082 (zb\u2082 fb\u2082)\n \u27e8x, \u27e8z, xf\u27e9, xa\u2081, xa\u2082\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b1 : \u03b9 \u2192 Type u_12\nv : (i : \u03b9) \u2192 Set (\u03b1 i)\nhv : Set.Nonempty (pi univ v)\ni : \u03b9\n\u22a2 (fun x => x i) '' \u22c2 (k : \u03b9), (fun x => x k) \u207b\u00b9' v k = v i\n[PROOFSTEP]\nclassical\napply Subset.antisymm\n\u00b7 simp [iInter_subset]\n\u00b7 intro y y_in\n simp only [mem_image, mem_iInter, mem_preimage]\n rcases hv with \u27e8z, hz\u27e9\n refine' \u27e8Function.update z i y, _, update_same i y z\u27e9\n rw [@forall_update_iff \u03b9 \u03b1 _ z i y fun i t => t \u2208 v i]\n exact \u27e8y_in, fun j _ => by simpa using hz j\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b1 : \u03b9 \u2192 Type u_12\nv : (i : \u03b9) \u2192 Set (\u03b1 i)\nhv : Set.Nonempty (pi univ v)\ni : \u03b9\n\u22a2 (fun x => x i) '' \u22c2 (k : \u03b9), (fun x => x k) \u207b\u00b9' v k = v i\n[PROOFSTEP]\napply Subset.antisymm\n[GOAL]\ncase h\u2081\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b1 : \u03b9 \u2192 Type u_12\nv : (i : \u03b9) \u2192 Set (\u03b1 i)\nhv : Set.Nonempty (pi univ v)\ni : \u03b9\n\u22a2 (fun x => x i) '' \u22c2 (k : \u03b9), (fun x => x k) \u207b\u00b9' v k \u2286 v i\n[PROOFSTEP]\nsimp [iInter_subset]\n[GOAL]\ncase h\u2082\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b1 : \u03b9 \u2192 Type u_12\nv : (i : \u03b9) \u2192 Set (\u03b1 i)\nhv : Set.Nonempty (pi univ v)\ni : \u03b9\n\u22a2 v i \u2286 (fun x => x i) '' \u22c2 (k : \u03b9), (fun x => x k) \u207b\u00b9' v k\n[PROOFSTEP]\nintro y y_in\n[GOAL]\ncase h\u2082\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b1 : \u03b9 \u2192 Type u_12\nv : (i : \u03b9) \u2192 Set (\u03b1 i)\nhv : Set.Nonempty (pi univ v)\ni : \u03b9\ny : \u03b1 i\ny_in : y \u2208 v i\n\u22a2 y \u2208 (fun x => x i) '' \u22c2 (k : \u03b9), (fun x => x k) \u207b\u00b9' v k\n[PROOFSTEP]\nsimp only [mem_image, mem_iInter, mem_preimage]\n[GOAL]\ncase h\u2082\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b1 : \u03b9 \u2192 Type u_12\nv : (i : \u03b9) \u2192 Set (\u03b1 i)\nhv : Set.Nonempty (pi univ v)\ni : \u03b9\ny : \u03b1 i\ny_in : y \u2208 v i\n\u22a2 \u2203 x, (\u2200 (i : \u03b9), x i \u2208 v i) \u2227 x i = y\n[PROOFSTEP]\nrcases hv with \u27e8z, hz\u27e9\n[GOAL]\ncase h\u2082.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b1 : \u03b9 \u2192 Type u_12\nv : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ny : \u03b1 i\ny_in : y \u2208 v i\nz : (i : \u03b9) \u2192 \u03b1 i\nhz : z \u2208 pi univ v\n\u22a2 \u2203 x, (\u2200 (i : \u03b9), x i \u2208 v i) \u2227 x i = y\n[PROOFSTEP]\nrefine' \u27e8Function.update z i y, _, update_same i y z\u27e9\n[GOAL]\ncase h\u2082.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b1 : \u03b9 \u2192 Type u_12\nv : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ny : \u03b1 i\ny_in : y \u2208 v i\nz : (i : \u03b9) \u2192 \u03b1 i\nhz : z \u2208 pi univ v\n\u22a2 \u2200 (i_1 : \u03b9), update z i y i_1 \u2208 v i_1\n[PROOFSTEP]\nrw [@forall_update_iff \u03b9 \u03b1 _ z i y fun i t => t \u2208 v i]\n[GOAL]\ncase h\u2082.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b1 : \u03b9 \u2192 Type u_12\nv : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ny : \u03b1 i\ny_in : y \u2208 v i\nz : (i : \u03b9) \u2192 \u03b1 i\nhz : z \u2208 pi univ v\n\u22a2 y \u2208 v i \u2227 \u2200 (x : \u03b9), x \u2260 i \u2192 z x \u2208 v x\n[PROOFSTEP]\nexact \u27e8y_in, fun j _ => by simpa using hz j\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b1 : \u03b9 \u2192 Type u_12\nv : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ny : \u03b1 i\ny_in : y \u2208 v i\nz : (i : \u03b9) \u2192 \u03b1 i\nhz : z \u2208 pi univ v\nj : \u03b9\nx\u271d : j \u2260 i\n\u22a2 z j \u2208 v j\n[PROOFSTEP]\nsimpa using hz j\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b1\n\u22a2 Set.Nonempty (\u22c3 (i : \u03b9), s i) \u2194 \u2203 i, Set.Nonempty (s i)\n[PROOFSTEP]\nsimp [nonempty_iff_ne_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns\u271d : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ns : \u03b1 \u2192 Set \u03b2\n\u22a2 Set.Nonempty (\u22c3 (i : \u03b1) (_ : i \u2208 t), s i) \u2194 \u2203 i, i \u2208 t \u2227 Set.Nonempty (s i)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\np : \u03b9 \u2192 Prop\nq : \u03b9 \u2192 \u03b9' \u2192 Prop\ns : (x : \u03b9) \u2192 (y : \u03b9') \u2192 p x \u2227 q x y \u2192 Set \u03b1\n\u22a2 \u22c3 (x : \u03b9) (y : \u03b9') (h : p x \u2227 q x y), s x y h = \u22c3 (x : \u03b9) (hx : p x) (y : \u03b9') (hy : q x y), s x y (_ : p x \u2227 q x y)\n[PROOFSTEP]\nsimp only [iUnion_and, @iUnion_comm _ \u03b9']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\np : \u03b9' \u2192 Prop\nq : \u03b9 \u2192 \u03b9' \u2192 Prop\ns : (x : \u03b9) \u2192 (y : \u03b9') \u2192 p y \u2227 q x y \u2192 Set \u03b1\n\u22a2 \u22c3 (x : \u03b9) (y : \u03b9') (h : p y \u2227 q x y), s x y h = \u22c3 (y : \u03b9') (hy : p y) (x : \u03b9) (hx : q x y), s x y (_ : p y \u2227 q x y)\n[PROOFSTEP]\nsimp only [iUnion_and, @iUnion_comm _ \u03b9]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\np : \u03b9 \u2192 Prop\nq : \u03b9 \u2192 \u03b9' \u2192 Prop\ns : (x : \u03b9) \u2192 (y : \u03b9') \u2192 p x \u2227 q x y \u2192 Set \u03b1\n\u22a2 \u22c2 (x : \u03b9) (y : \u03b9') (h : p x \u2227 q x y), s x y h = \u22c2 (x : \u03b9) (hx : p x) (y : \u03b9') (hy : q x y), s x y (_ : p x \u2227 q x y)\n[PROOFSTEP]\nsimp only [iInter_and, @iInter_comm _ \u03b9']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\np : \u03b9' \u2192 Prop\nq : \u03b9 \u2192 \u03b9' \u2192 Prop\ns : (x : \u03b9) \u2192 (y : \u03b9') \u2192 p y \u2227 q x y \u2192 Set \u03b1\n\u22a2 \u22c2 (x : \u03b9) (y : \u03b9') (h : p y \u2227 q x y), s x y h = \u22c2 (y : \u03b9') (hy : p y) (x : \u03b9) (hx : q x y), s x y (_ : p y \u2227 q x y)\n[PROOFSTEP]\nsimp only [iInter_and, @iInter_comm _ \u03b9]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nb : \u03b2\np : \u03b2 \u2192 Prop\ns : (x : \u03b2) \u2192 x = b \u2228 p x \u2192 Set \u03b1\n\u22a2 \u22c3 (x : \u03b2) (h : x = b \u2228 p x), s x h = s b (_ : b = b \u2228 p b) \u222a \u22c3 (x : \u03b2) (h : p x), s x (_ : x = b \u2228 p x)\n[PROOFSTEP]\nsimp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nb : \u03b2\np : \u03b2 \u2192 Prop\ns : (x : \u03b2) \u2192 x = b \u2228 p x \u2192 Set \u03b1\n\u22a2 \u22c2 (x : \u03b2) (h : x = b \u2228 p x), s x h = s b (_ : b = b \u2228 p b) \u2229 \u22c2 (x : \u03b2) (h : p x), s x (_ : x = b \u2228 p x)\n[PROOFSTEP]\nsimp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\n\u22a2 \u22c3 (_ : Set.Nonempty s), s = s\n[PROOFSTEP]\nrw [iUnion_nonempty_index, biUnion_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\na : \u03b1\ns : Set \u03b1\nt : \u03b1 \u2192 Set \u03b2\n\u22a2 \u22c2 (x : \u03b1) (_ : x \u2208 insert a s), t x = t a \u2229 \u22c2 (x : \u03b1) (_ : x \u2208 s), t x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\na b : \u03b1\ns : \u03b1 \u2192 Set \u03b2\n\u22a2 \u22c2 (x : \u03b1) (_ : x \u2208 {a, b}), s x = s a \u2229 s b\n[PROOFSTEP]\nrw [biInter_insert, biInter_singleton]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b1 : Type u_12\ns : Set \u03b9\nhs : Set.Nonempty s\nf : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\n\u22a2 \u22c2 (i : \u03b9) (_ : i \u2208 s), f i \u2229 t = (\u22c2 (i : \u03b9) (_ : i \u2208 s), f i) \u2229 t\n[PROOFSTEP]\nhaveI : Nonempty s := hs.to_subtype\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b1 : Type u_12\ns : Set \u03b9\nhs : Set.Nonempty s\nf : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\nthis : Nonempty \u2191s\n\u22a2 \u22c2 (i : \u03b9) (_ : i \u2208 s), f i \u2229 t = (\u22c2 (i : \u03b9) (_ : i \u2208 s), f i) \u2229 t\n[PROOFSTEP]\nsimp [biInter_eq_iInter, \u2190 iInter_inter]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b1 : Type u_12\ns : Set \u03b9\nhs : Set.Nonempty s\nf : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\n\u22a2 \u22c2 (i : \u03b9) (_ : i \u2208 s), t \u2229 f i = t \u2229 \u22c2 (i : \u03b9) (_ : i \u2208 s), f i\n[PROOFSTEP]\nrw [inter_comm, \u2190 biInter_inter hs]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b1 : Type u_12\ns : Set \u03b9\nhs : Set.Nonempty s\nf : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\n\u22a2 \u22c2 (i : \u03b9) (_ : i \u2208 s), t \u2229 f i = \u22c2 (i : \u03b9) (_ : i \u2208 s), f i \u2229 t\n[PROOFSTEP]\nsimp [inter_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\n\u22a2 \u2200 (x : \u03b1), x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 s), {x} \u2194 x \u2208 s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\na : \u03b1\ns : Set \u03b1\nt : \u03b1 \u2192 Set \u03b2\n\u22a2 \u22c3 (x : \u03b1) (_ : x \u2208 insert a s), t x = t a \u222a \u22c3 (x : \u03b1) (_ : x \u2208 s), t x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\na b : \u03b1\ns : \u03b1 \u2192 Set \u03b2\n\u22a2 \u22c3 (x : \u03b1) (_ : x \u2208 {a, b}), s x = s a \u222a s b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\n\u22a2 s \u2229 \u22c3 (i : \u03b9) (j : \u03ba i), t i j = \u22c3 (i : \u03b9) (j : \u03ba i), s \u2229 t i j\n[PROOFSTEP]\nsimp only [inter_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\nt : Set \u03b1\n\u22a2 (\u22c3 (i : \u03b9) (j : \u03ba i), s i j) \u2229 t = \u22c3 (i : \u03b9) (j : \u03ba i), s i j \u2229 t\n[PROOFSTEP]\nsimp_rw [iUnion_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\n\u22a2 s \u222a \u22c2 (i : \u03b9) (j : \u03ba i), t i j = \u22c2 (i : \u03b9) (j : \u03ba i), s \u222a t i j\n[PROOFSTEP]\nsimp_rw [union_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\nt : Set \u03b1\n\u22a2 (\u22c2 (i : \u03b9) (j : \u03ba i), s i j) \u222a t = \u22c2 (i : \u03b9) (j : \u03ba i), s i j \u222a t\n[PROOFSTEP]\nsimp_rw [iInter_union]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nh : S \u2286 {\u2205, univ}\n\u22a2 \u22c3\u2080 S \u2208 {\u2205, univ}\n[PROOFSTEP]\nsimp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nh : S \u2286 {\u2205, univ}\n\u22a2 (\u2203 x x_1, \u00acx = \u2205) \u2192 \u22c3\u2080 S = univ\n[PROOFSTEP]\nrintro \u27e8s, hs, hne\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nh : S \u2286 {\u2205, univ}\ns : Set \u03b1\nhs : s \u2208 S\nhne : \u00acs = \u2205\n\u22a2 \u22c3\u2080 S = univ\n[PROOFSTEP]\nobtain rfl : s = univ := (h hs).resolve_left hne\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nh : S \u2286 {\u2205, univ}\nhs : univ \u2208 S\nhne : \u00acuniv = \u2205\n\u22a2 \u22c3\u2080 S = univ\n[PROOFSTEP]\nexact univ_subset_iff.1 <| subset_sUnion_of_mem hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\n\u22a2 Set.Nonempty (\u22c3\u2080 S) \u2194 \u2203 s, s \u2208 S \u2227 Set.Nonempty s\n[PROOFSTEP]\nsimp [nonempty_iff_ne_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b9 \u2192 Set \u03b1\n\u22a2 \u22c3 (i : \u03b9), f i = univ \u2194 \u2200 (x : \u03b1), \u2203 i, x \u2208 f i\n[PROOFSTEP]\nsimp only [eq_univ_iff_forall, mem_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\n\u22a2 \u22c3 (i : \u03b9) (j : \u03ba i), s i j = univ \u2194 \u2200 (a : \u03b1), \u2203 i j, a \u2208 s i j\n[PROOFSTEP]\nsimp only [iUnion_eq_univ_iff, mem_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nc : Set (Set \u03b1)\n\u22a2 \u22c3\u2080 c = univ \u2194 \u2200 (a : \u03b1), \u2203 b, b \u2208 c \u2227 a \u2208 b\n[PROOFSTEP]\nsimp only [eq_univ_iff_forall, mem_sUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b9 \u2192 Set \u03b1\n\u22a2 \u22c2 (i : \u03b9), f i = \u2205 \u2194 \u2200 (x : \u03b1), \u2203 i, \u00acx \u2208 f i\n[PROOFSTEP]\nsimp [Set.eq_empty_iff_forall_not_mem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\n\u22a2 \u22c2 (i : \u03b9) (j : \u03ba i), s i j = \u2205 \u2194 \u2200 (a : \u03b1), \u2203 i j, \u00aca \u2208 s i j\n[PROOFSTEP]\nsimp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nc : Set (Set \u03b1)\n\u22a2 \u22c2\u2080 c = \u2205 \u2194 \u2200 (a : \u03b1), \u2203 b, b \u2208 c \u2227 \u00aca \u2208 b\n[PROOFSTEP]\nsimp [Set.eq_empty_iff_forall_not_mem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b9 \u2192 Set \u03b1\n\u22a2 Set.Nonempty (\u22c2 (i : \u03b9), f i) \u2194 \u2203 x, \u2200 (i : \u03b9), x \u2208 f i\n[PROOFSTEP]\nsimp [nonempty_iff_ne_empty, iInter_eq_empty_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\n\u22a2 Set.Nonempty (\u22c2 (i : \u03b9) (j : \u03ba i), s i j) \u2194 \u2203 a, \u2200 (i : \u03b9) (j : \u03ba i), a \u2208 s i j\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nc : Set (Set \u03b1)\n\u22a2 Set.Nonempty (\u22c2\u2080 c) \u2194 \u2203 a, \u2200 (b : Set \u03b1), b \u2208 c \u2192 a \u2208 b\n[PROOFSTEP]\nsimp [nonempty_iff_ne_empty, sInter_eq_empty_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nx : \u03b1\n\u22a2 x \u2208 (\u22c3\u2080 S)\u1d9c \u2194 x \u2208 \u22c2\u2080 (compl '' S)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\n\u22a2 \u22c3\u2080 S = (\u22c2\u2080 (compl '' S))\u1d9c\n[PROOFSTEP]\nrw [\u2190 compl_compl (\u22c3\u2080 S), compl_sUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\n\u22a2 (\u22c2\u2080 S)\u1d9c = \u22c3\u2080 (compl '' S)\n[PROOFSTEP]\nrw [sUnion_eq_compl_sInter_compl, compl_compl_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\n\u22a2 \u22c2\u2080 S = (\u22c3\u2080 (compl '' S))\u1d9c\n[PROOFSTEP]\nrw [\u2190 compl_compl (\u22c2\u2080 S), compl_sInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns t : Set \u03b1\nS : Set (Set \u03b1)\nhs : t \u2208 S\nh : s \u2229 \u22c3\u2080 S = \u2205\n\u22a2 s \u2229 t \u2286 \u2205\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns t : Set \u03b1\nS : Set (Set \u03b1)\nhs : t \u2208 S\nh : s \u2229 \u22c3\u2080 S = \u2205\n\u22a2 s \u2229 t \u2286 s \u2229 \u22c3\u2080 S\n[PROOFSTEP]\nexact inter_subset_inter_right _ (subset_sUnion_of_mem hs)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3\u271d : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b3 : \u03b1 \u2192 Type u_11\nf : Sigma \u03b3 \u2192 \u03b2\n\u22a2 \u2200 (x : \u03b2), x \u2208 range f \u2194 x \u2208 \u22c3 (a : \u03b1), range fun b => f { fst := a, snd := b }\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b1 \u2192 Set \u03b2\n\u22a2 \u22c3 (i : \u03b1), s i = range fun a => \u2191a.snd\n[PROOFSTEP]\nsimp [Set.ext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b2\n\u22a2 \u22c3 (i : \u03b9), s i = range fun a => \u2191a.snd\n[PROOFSTEP]\nsimp [Set.ext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03c3 : \u03b9 \u2192 Type u_12\ns : Set (Sigma \u03c3)\n\u22a2 \u22c3 (i : \u03b9), Sigma.mk i '' (Sigma.mk i \u207b\u00b9' s) = s\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03c3 : \u03b9 \u2192 Type u_12\ns : Set (Sigma \u03c3)\nx : (i : \u03b9) \u00d7 \u03c3 i\n\u22a2 x \u2208 \u22c3 (i : \u03b9), Sigma.mk i '' (Sigma.mk i \u207b\u00b9' s) \u2194 x \u2208 s\n[PROOFSTEP]\nsimp only [mem_iUnion, mem_image, mem_preimage]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03c3 : \u03b9 \u2192 Type u_12\ns : Set (Sigma \u03c3)\nx : (i : \u03b9) \u00d7 \u03c3 i\n\u22a2 (\u2203 i x_1, { fst := i, snd := x_1 } \u2208 s \u2227 { fst := i, snd := x_1 } = x) \u2194 x \u2208 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03c3 : \u03b9 \u2192 Type u_12\ns : Set (Sigma \u03c3)\nx : (i : \u03b9) \u00d7 \u03c3 i\n\u22a2 (\u2203 i x_1, { fst := i, snd := x_1 } \u2208 s \u2227 { fst := i, snd := x_1 } = x) \u2192 x \u2208 s\n[PROOFSTEP]\nrintro \u27e8i, a, h, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03c3 : \u03b9 \u2192 Type u_12\ns : Set (Sigma \u03c3)\ni : \u03b9\na : \u03c3 i\nh : { fst := i, snd := a } \u2208 s\n\u22a2 { fst := i, snd := a } \u2208 s\n[PROOFSTEP]\nexact h\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03c3 : \u03b9 \u2192 Type u_12\ns : Set (Sigma \u03c3)\nx : (i : \u03b9) \u00d7 \u03c3 i\n\u22a2 x \u2208 s \u2192 \u2203 i x_1, { fst := i, snd := x_1 } \u2208 s \u2227 { fst := i, snd := x_1 } = x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03c3 : \u03b9 \u2192 Type u_12\ns : Set (Sigma \u03c3)\nx : (i : \u03b9) \u00d7 \u03c3 i\nh : x \u2208 s\n\u22a2 \u2203 i x_1, { fst := i, snd := x_1 } \u2208 s \u2227 { fst := i, snd := x_1 } = x\n[PROOFSTEP]\ncases' x with i a\n[GOAL]\ncase h.mpr.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03c3 : \u03b9 \u2192 Type u_12\ns : Set (Sigma \u03c3)\ni : \u03b9\na : \u03c3 i\nh : { fst := i, snd := a } \u2208 s\n\u22a2 \u2203 i_1 x, { fst := i_1, snd := x } \u2208 s \u2227 { fst := i_1, snd := x } = { fst := i, snd := a }\n[PROOFSTEP]\nexact \u27e8i, a, h, rfl\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b1 : Type u_11\n\u03b2 : Type u_12\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u22c3 (x : \u03b1), {f x} = range f\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b1 : Type u_11\n\u03b2 : Type u_12\nf : \u03b1 \u2192 \u03b2\nx : \u03b2\n\u22a2 x \u2208 \u22c3 (x : \u03b1), {f x} \u2194 x \u2208 range f\n[PROOFSTEP]\nsimp [@eq_comm _ x]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b1 : Type u_11\n\u22a2 \u22c3 (x : \u03b1), {x} = univ\n[PROOFSTEP]\nsimp [Set.ext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\n\u22a2 \u22c3 (i : \u2191s), {\u2191i} = s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set (Set \u03b1)\n\u22a2 \u22c3\u2080 s = \u22c3 (i : Set \u03b1) (_ : i \u2208 s), i\n[PROOFSTEP]\nrw [\u2190 sUnion_image, image_id']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set (Set \u03b1)\n\u22a2 \u22c2\u2080 s = \u22c2 (i : Set \u03b1) (_ : i \u2208 s), i\n[PROOFSTEP]\nrw [\u2190 sInter_image, image_id']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set (Set \u03b1)\n\u22a2 \u22c3\u2080 s = \u22c3 (i : \u2191s), \u2191i\n[PROOFSTEP]\nsimp only [\u2190 sUnion_range, Subtype.range_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set (Set \u03b1)\n\u22a2 \u22c2\u2080 s = \u22c2 (i : \u2191s), \u2191i\n[PROOFSTEP]\nsimp only [\u2190 sInter_range, Subtype.range_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b1\nt : \u03b1 \u2192 Set \u03b2\n\u22a2 \u22c3 (x : \u03b1) (_ : x \u2208 \u22c3 (i : \u03b9), s i), t x = \u22c3 (i : \u03b9) (x : \u03b1) (_ : x \u2208 s i), t x\n[PROOFSTEP]\nsimp [@iUnion_comm _ \u03b9]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b1\nt : \u03b1 \u2192 Set \u03b2\n\u22a2 \u22c2 (x : \u03b1) (_ : x \u2208 \u22c3 (i : \u03b9), s i), t x = \u22c2 (i : \u03b9) (x : \u03b1) (_ : x \u2208 s i), t x\n[PROOFSTEP]\nsimp [@iInter_comm _ \u03b9]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set (Set \u03b1)\n\u22a2 \u22c3\u2080 \u22c3 (i : \u03b9), s i = \u22c3 (i : \u03b9), \u22c3\u2080 s i\n[PROOFSTEP]\nsimp only [sUnion_eq_biUnion, biUnion_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set (Set \u03b1)\n\u22a2 \u22c2\u2080 \u22c3 (i : \u03b9), s i = \u22c2 (i : \u03b9), \u22c2\u2080 s i\n[PROOFSTEP]\nsimp only [sInter_eq_biInter, biInter_iUnion]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b1 : Type u_11\n\u03b2 : Type u_12\nC : Set (Set \u03b1)\nf : (s : \u2191C) \u2192 \u03b2 \u2192 \u2191\u2191s\nhf : \u2200 (s : \u2191C), Surjective (f s)\n\u22a2 (\u22c3 (y : \u03b2), range fun s => \u2191(f s y)) = \u22c3\u2080 C\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b1 : Type u_11\n\u03b2 : Type u_12\nC : Set (Set \u03b1)\nf : (s : \u2191C) \u2192 \u03b2 \u2192 \u2191\u2191s\nhf : \u2200 (s : \u2191C), Surjective (f s)\nx : \u03b1\n\u22a2 (x \u2208 \u22c3 (y : \u03b2), range fun s => \u2191(f s y)) \u2194 x \u2208 \u22c3\u2080 C\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b1 : Type u_11\n\u03b2 : Type u_12\nC : Set (Set \u03b1)\nf : (s : \u2191C) \u2192 \u03b2 \u2192 \u2191\u2191s\nhf : \u2200 (s : \u2191C), Surjective (f s)\nx : \u03b1\n\u22a2 (x \u2208 \u22c3 (y : \u03b2), range fun s => \u2191(f s y)) \u2192 x \u2208 \u22c3\u2080 C\n[PROOFSTEP]\nrintro \u27e8s, \u27e8y, rfl\u27e9, \u27e8s, hs\u27e9, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.mk\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b1 : Type u_11\n\u03b2 : Type u_12\nC : Set (Set \u03b1)\nf : (s : \u2191C) \u2192 \u03b2 \u2192 \u2191\u2191s\nhf : \u2200 (s : \u2191C), Surjective (f s)\ny : \u03b2\ns : Set \u03b1\nhs : s \u2208 C\n\u22a2 (fun s => \u2191(f s y)) { val := s, property := hs } \u2208 \u22c3\u2080 C\n[PROOFSTEP]\nrefine' \u27e8_, hs, _\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.mk\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b1 : Type u_11\n\u03b2 : Type u_12\nC : Set (Set \u03b1)\nf : (s : \u2191C) \u2192 \u03b2 \u2192 \u2191\u2191s\nhf : \u2200 (s : \u2191C), Surjective (f s)\ny : \u03b2\ns : Set \u03b1\nhs : s \u2208 C\n\u22a2 (fun s => \u2191(f s y)) { val := s, property := hs } \u2208 s\n[PROOFSTEP]\nexact (f \u27e8s, hs\u27e9 y).2\n[GOAL]\ncase h.mpr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b1 : Type u_11\n\u03b2 : Type u_12\nC : Set (Set \u03b1)\nf : (s : \u2191C) \u2192 \u03b2 \u2192 \u2191\u2191s\nhf : \u2200 (s : \u2191C), Surjective (f s)\nx : \u03b1\n\u22a2 x \u2208 \u22c3\u2080 C \u2192 x \u2208 \u22c3 (y : \u03b2), range fun s => \u2191(f s y)\n[PROOFSTEP]\nrintro \u27e8s, hs, hx\u27e9\n[GOAL]\ncase h.mpr.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b1 : Type u_11\n\u03b2 : Type u_12\nC : Set (Set \u03b1)\nf : (s : \u2191C) \u2192 \u03b2 \u2192 \u2191\u2191s\nhf : \u2200 (s : \u2191C), Surjective (f s)\nx : \u03b1\ns : Set \u03b1\nhs : s \u2208 C\nhx : x \u2208 s\n\u22a2 x \u2208 \u22c3 (y : \u03b2), range fun s => \u2191(f s y)\n[PROOFSTEP]\ncases' hf \u27e8s, hs\u27e9 \u27e8x, hx\u27e9 with y hy\n[GOAL]\ncase h.mpr.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b1 : Type u_11\n\u03b2 : Type u_12\nC : Set (Set \u03b1)\nf : (s : \u2191C) \u2192 \u03b2 \u2192 \u2191\u2191s\nhf : \u2200 (s : \u2191C), Surjective (f s)\nx : \u03b1\ns : Set \u03b1\nhs : s \u2208 C\nhx : x \u2208 s\ny : \u03b2\nhy : f { val := s, property := hs } y = { val := x, property := hx }\n\u22a2 x \u2208 \u22c3 (y : \u03b2), range fun s => \u2191(f s y)\n[PROOFSTEP]\nrefine' \u27e8_, \u27e8y, rfl\u27e9, \u27e8s, hs\u27e9, _\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b1 : Type u_11\n\u03b2 : Type u_12\nC : Set (Set \u03b1)\nf : (s : \u2191C) \u2192 \u03b2 \u2192 \u2191\u2191s\nhf : \u2200 (s : \u2191C), Surjective (f s)\nx : \u03b1\ns : Set \u03b1\nhs : s \u2208 C\nhx : x \u2208 s\ny : \u03b2\nhy : f { val := s, property := hs } y = { val := x, property := hx }\n\u22a2 (fun s => \u2191(f s y)) { val := s, property := hs } = x\n[PROOFSTEP]\nexact congr_arg Subtype.val hy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nC : \u03b9 \u2192 Set \u03b1\nf : (x : \u03b9) \u2192 \u03b2 \u2192 \u2191(C x)\nhf : \u2200 (x : \u03b9), Surjective (f x)\n\u22a2 (\u22c3 (y : \u03b2), range fun x => \u2191(f x y)) = \u22c3 (x : \u03b9), C x\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nC : \u03b9 \u2192 Set \u03b1\nf : (x : \u03b9) \u2192 \u03b2 \u2192 \u2191(C x)\nhf : \u2200 (x : \u03b9), Surjective (f x)\nx : \u03b1\n\u22a2 (x \u2208 \u22c3 (y : \u03b2), range fun x => \u2191(f x y)) \u2194 x \u2208 \u22c3 (x : \u03b9), C x\n[PROOFSTEP]\nrw [mem_iUnion, mem_iUnion]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nC : \u03b9 \u2192 Set \u03b1\nf : (x : \u03b9) \u2192 \u03b2 \u2192 \u2191(C x)\nhf : \u2200 (x : \u03b9), Surjective (f x)\nx : \u03b1\n\u22a2 (\u2203 i, x \u2208 range fun x => \u2191(f x i)) \u2194 \u2203 i, x \u2208 C i\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nC : \u03b9 \u2192 Set \u03b1\nf : (x : \u03b9) \u2192 \u03b2 \u2192 \u2191(C x)\nhf : \u2200 (x : \u03b9), Surjective (f x)\nx : \u03b1\n\u22a2 (\u2203 i, x \u2208 range fun x => \u2191(f x i)) \u2192 \u2203 i, x \u2208 C i\n[PROOFSTEP]\nrintro \u27e8y, i, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nC : \u03b9 \u2192 Set \u03b1\nf : (x : \u03b9) \u2192 \u03b2 \u2192 \u2191(C x)\nhf : \u2200 (x : \u03b9), Surjective (f x)\ny : \u03b2\ni : \u03b9\n\u22a2 \u2203 i_1, (fun x => \u2191(f x y)) i \u2208 C i_1\n[PROOFSTEP]\nexact \u27e8i, (f i y).2\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nC : \u03b9 \u2192 Set \u03b1\nf : (x : \u03b9) \u2192 \u03b2 \u2192 \u2191(C x)\nhf : \u2200 (x : \u03b9), Surjective (f x)\nx : \u03b1\n\u22a2 (\u2203 i, x \u2208 C i) \u2192 \u2203 i, x \u2208 range fun x => \u2191(f x i)\n[PROOFSTEP]\nrintro \u27e8i, hx\u27e9\n[GOAL]\ncase h.mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nC : \u03b9 \u2192 Set \u03b1\nf : (x : \u03b9) \u2192 \u03b2 \u2192 \u2191(C x)\nhf : \u2200 (x : \u03b9), Surjective (f x)\nx : \u03b1\ni : \u03b9\nhx : x \u2208 C i\n\u22a2 \u2203 i, x \u2208 range fun x => \u2191(f x i)\n[PROOFSTEP]\ncases' hf i \u27e8x, hx\u27e9 with y hy\n[GOAL]\ncase h.mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nC : \u03b9 \u2192 Set \u03b1\nf : (x : \u03b9) \u2192 \u03b2 \u2192 \u2191(C x)\nhf : \u2200 (x : \u03b9), Surjective (f x)\nx : \u03b1\ni : \u03b9\nhx : x \u2208 C i\ny : \u03b2\nhy : f i y = { val := x, property := hx }\n\u22a2 \u2203 i, x \u2208 range fun x => \u2191(f x i)\n[PROOFSTEP]\nexact \u27e8y, i, congr_arg Subtype.val hy\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\n\u22a2 s \u222a \u22c2 (i : \u03b9) (j : \u03ba i), t i j = \u22c2 (i : \u03b9) (j : \u03ba i), s \u222a t i j\n[PROOFSTEP]\nsimp_rw [union_distrib_iInter_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\nt : Set \u03b1\n\u22a2 (\u22c2 (i : \u03b9) (j : \u03ba i), s i j) \u222a t = \u22c2 (i : \u03b9) (j : \u03ba i), s i j \u222a t\n[PROOFSTEP]\nsimp_rw [union_distrib_iInter_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nf : \u03b1 \u2192 \u03b2\n\u22a2 f '' \u22c2\u2080 S \u2286 \u22c2 (s : Set \u03b1) (_ : s \u2208 S), f '' s\n[PROOFSTEP]\nrw [sInter_eq_biInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nf : \u03b1 \u2192 \u03b2\n\u22a2 f '' \u22c2 (i : Set \u03b1) (_ : i \u2208 S), i \u2286 \u22c2 (s : Set \u03b1) (_ : s \u2208 S), f '' s\n[PROOFSTEP]\napply image_iInter\u2082_subset\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nU : \u03b9 \u2192 Set \u03b2\nhU : iUnion U = univ\n\u22a2 Injective f \u2194 \u2200 (i : \u03b9), Injective (restrictPreimage (U i) f)\n[PROOFSTEP]\nrefine' \u27e8fun H i => (U i).restrictPreimage_injective H, fun H x y e => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nU : \u03b9 \u2192 Set \u03b2\nhU : iUnion U = univ\nH : \u2200 (i : \u03b9), Injective (restrictPreimage (U i) f)\nx y : \u03b1\ne : f x = f y\n\u22a2 x = y\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := Set.mem_iUnion.mp (show f x \u2208 Set.iUnion U by rw [hU]; triv)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nU : \u03b9 \u2192 Set \u03b2\nhU : iUnion U = univ\nH : \u2200 (i : \u03b9), Injective (restrictPreimage (U i) f)\nx y : \u03b1\ne : f x = f y\n\u22a2 f x \u2208 iUnion U\n[PROOFSTEP]\nrw [hU]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nU : \u03b9 \u2192 Set \u03b2\nhU : iUnion U = univ\nH : \u2200 (i : \u03b9), Injective (restrictPreimage (U i) f)\nx y : \u03b1\ne : f x = f y\n\u22a2 f x \u2208 univ\n[PROOFSTEP]\ntriv\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nU : \u03b9 \u2192 Set \u03b2\nhU : iUnion U = univ\nH : \u2200 (i : \u03b9), Injective (restrictPreimage (U i) f)\nx y : \u03b1\ne : f x = f y\ni : \u03b9\nhi : f x \u2208 U i\n\u22a2 x = y\n[PROOFSTEP]\ninjection @H i \u27e8x, hi\u27e9 \u27e8y, show f y \u2208 U i from e \u25b8 hi\u27e9 (Subtype.ext e)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nU : \u03b9 \u2192 Set \u03b2\nhU : iUnion U = univ\n\u22a2 Surjective f \u2194 \u2200 (i : \u03b9), Surjective (restrictPreimage (U i) f)\n[PROOFSTEP]\nrefine' \u27e8fun H i => (U i).restrictPreimage_surjective H, fun H x => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nU : \u03b9 \u2192 Set \u03b2\nhU : iUnion U = univ\nH : \u2200 (i : \u03b9), Surjective (restrictPreimage (U i) f)\nx : \u03b2\n\u22a2 \u2203 a, f a = x\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := Set.mem_iUnion.mp (show x \u2208 Set.iUnion U by rw [hU]; triv)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nU : \u03b9 \u2192 Set \u03b2\nhU : iUnion U = univ\nH : \u2200 (i : \u03b9), Surjective (restrictPreimage (U i) f)\nx : \u03b2\n\u22a2 x \u2208 iUnion U\n[PROOFSTEP]\nrw [hU]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nU : \u03b9 \u2192 Set \u03b2\nhU : iUnion U = univ\nH : \u2200 (i : \u03b9), Surjective (restrictPreimage (U i) f)\nx : \u03b2\n\u22a2 x \u2208 univ\n[PROOFSTEP]\ntriv\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nU : \u03b9 \u2192 Set \u03b2\nhU : iUnion U = univ\nH : \u2200 (i : \u03b9), Surjective (restrictPreimage (U i) f)\nx : \u03b2\ni : \u03b9\nhi : x \u2208 U i\n\u22a2 \u2203 a, f a = x\n[PROOFSTEP]\nexact \u27e8_, congr_arg Subtype.val (H i \u27e8x, hi\u27e9).choose_spec\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nU : \u03b9 \u2192 Set \u03b2\nhU : iUnion U = univ\n\u22a2 Bijective f \u2194 \u2200 (i : \u03b9), Bijective (restrictPreimage (U i) f)\n[PROOFSTEP]\nrw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU, surjective_iff_surjective_of_iUnion_eq_univ hU]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nU : \u03b9 \u2192 Set \u03b2\nhU : iUnion U = univ\n\u22a2 ((\u2200 (i : \u03b9), Injective (restrictPreimage (U i) f)) \u2227 \u2200 (i : \u03b9), Surjective (restrictPreimage (U i) f)) \u2194\n \u2200 (i : \u03b9), Bijective (restrictPreimage (U i) f)\n[PROOFSTEP]\nsimp [Bijective, forall_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\nh : InjOn f (\u22c3 (i : \u03b9), s i)\n\u22a2 f '' \u22c2 (i : \u03b9), s i = \u22c2 (i : \u03b9), f '' s i\n[PROOFSTEP]\ninhabit \u03b9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\nh : InjOn f (\u22c3 (i : \u03b9), s i)\ninhabited_h : Inhabited \u03b9\n\u22a2 f '' \u22c2 (i : \u03b9), s i = \u22c2 (i : \u03b9), f '' s i\n[PROOFSTEP]\nrefine' Subset.antisymm (image_iInter_subset s f) fun y hy => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\nh : InjOn f (\u22c3 (i : \u03b9), s i)\ninhabited_h : Inhabited \u03b9\ny : \u03b2\nhy : y \u2208 \u22c2 (i : \u03b9), f '' s i\n\u22a2 y \u2208 f '' \u22c2 (i : \u03b9), s i\n[PROOFSTEP]\nsimp only [mem_iInter, mem_image_iff_bex] at hy \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\nh : InjOn f (\u22c3 (i : \u03b9), s i)\ninhabited_h : Inhabited \u03b9\ny : \u03b2\nhy : \u2200 (i : \u03b9), \u2203 x x_1, f x = y\n\u22a2 y \u2208 f '' \u22c2 (i : \u03b9), s i\n[PROOFSTEP]\nchoose x hx hy using hy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\nh : InjOn f (\u22c3 (i : \u03b9), s i)\ninhabited_h : Inhabited \u03b9\ny : \u03b2\nx : \u03b9 \u2192 \u03b1\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nhy : \u2200 (i : \u03b9), f (x i) = y\n\u22a2 y \u2208 f '' \u22c2 (i : \u03b9), s i\n[PROOFSTEP]\nrefine' \u27e8x default, mem_iInter.2 fun i => _, hy _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\nh : InjOn f (\u22c3 (i : \u03b9), s i)\ninhabited_h : Inhabited \u03b9\ny : \u03b2\nx : \u03b9 \u2192 \u03b1\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nhy : \u2200 (i : \u03b9), f (x i) = y\ni : \u03b9\n\u22a2 x default \u2208 s i\n[PROOFSTEP]\nsuffices x default = x i by\n rw [this]\n apply hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\nh : InjOn f (\u22c3 (i : \u03b9), s i)\ninhabited_h : Inhabited \u03b9\ny : \u03b2\nx : \u03b9 \u2192 \u03b1\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nhy : \u2200 (i : \u03b9), f (x i) = y\ni : \u03b9\nthis : x default = x i\n\u22a2 x default \u2208 s i\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\nh : InjOn f (\u22c3 (i : \u03b9), s i)\ninhabited_h : Inhabited \u03b9\ny : \u03b2\nx : \u03b9 \u2192 \u03b1\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nhy : \u2200 (i : \u03b9), f (x i) = y\ni : \u03b9\nthis : x default = x i\n\u22a2 x i \u2208 s i\n[PROOFSTEP]\napply hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\nh : InjOn f (\u22c3 (i : \u03b9), s i)\ninhabited_h : Inhabited \u03b9\ny : \u03b2\nx : \u03b9 \u2192 \u03b1\nhx : \u2200 (i : \u03b9), x i \u2208 s i\nhy : \u2200 (i : \u03b9), f (x i) = y\ni : \u03b9\n\u22a2 x default = x i\n[PROOFSTEP]\nreplace hx : \u2200 i, x i \u2208 \u22c3 j, s j := fun i => (subset_iUnion _ _) (hx i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\nh : InjOn f (\u22c3 (i : \u03b9), s i)\ninhabited_h : Inhabited \u03b9\ny : \u03b2\nx : \u03b9 \u2192 \u03b1\nhy : \u2200 (i : \u03b9), f (x i) = y\ni : \u03b9\nhx : \u2200 (i : \u03b9), x i \u2208 \u22c3 (j : \u03b9), s j\n\u22a2 x default = x i\n[PROOFSTEP]\napply h (hx _) (hx _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\nh : InjOn f (\u22c3 (i : \u03b9), s i)\ninhabited_h : Inhabited \u03b9\ny : \u03b2\nx : \u03b9 \u2192 \u03b1\nhy : \u2200 (i : \u03b9), f (x i) = y\ni : \u03b9\nhx : \u2200 (i : \u03b9), x i \u2208 \u22c3 (j : \u03b9), s j\n\u22a2 f (x default) = f (x i)\n[PROOFSTEP]\nsimp only [hy]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\np : \u03b9 \u2192 Prop\ns : (i : \u03b9) \u2192 p i \u2192 Set \u03b1\nhp : \u2203 i, p i\nf : \u03b1 \u2192 \u03b2\nh : InjOn f (\u22c3 (i : \u03b9) (hi : p i), s i hi)\n\u22a2 f '' \u22c2 (i : \u03b9) (hi : p i), s i hi = \u22c2 (i : \u03b9) (hi : p i), f '' s i hi\n[PROOFSTEP]\nsimp only [iInter, iInf_subtype']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\np : \u03b9 \u2192 Prop\ns : (i : \u03b9) \u2192 p i \u2192 Set \u03b1\nhp : \u2203 i, p i\nf : \u03b1 \u2192 \u03b2\nh : InjOn f (\u22c3 (i : \u03b9) (hi : p i), s i hi)\n\u22a2 f '' \u2a05 (x : { i // p i }), s \u2191x (_ : p \u2191x) = \u2a05 (x : { i // p i }), f '' s \u2191x (_ : p \u2191x)\n[PROOFSTEP]\nhaveI : Nonempty { i // p i } := nonempty_subtype.2 hp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\np : \u03b9 \u2192 Prop\ns : (i : \u03b9) \u2192 p i \u2192 Set \u03b1\nhp : \u2203 i, p i\nf : \u03b1 \u2192 \u03b2\nh : InjOn f (\u22c3 (i : \u03b9) (hi : p i), s i hi)\nthis : Nonempty { i // p i }\n\u22a2 f '' \u2a05 (x : { i // p i }), s \u2191x (_ : p \u2191x) = \u2a05 (x : { i // p i }), f '' s \u2191x (_ : p \u2191x)\n[PROOFSTEP]\napply InjOn.image_iInter_eq\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\np : \u03b9 \u2192 Prop\ns : (i : \u03b9) \u2192 p i \u2192 Set \u03b1\nhp : \u2203 i, p i\nf : \u03b1 \u2192 \u03b2\nh : InjOn f (\u22c3 (i : \u03b9) (hi : p i), s i hi)\nthis : Nonempty { i // p i }\n\u22a2 InjOn f (\u22c3 (i : { i // p i }), s \u2191i (_ : p \u2191i))\n[PROOFSTEP]\nsimpa only [iUnion, iSup_subtype'] using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\nhf : Bijective f\ns : \u03b9 \u2192 Set \u03b1\n\u22a2 f '' \u22c2 (i : \u03b9), s i = \u22c2 (i : \u03b9), f '' s i\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\nhf : Bijective f\ns : \u03b9 \u2192 Set \u03b1\nh\u271d : IsEmpty \u03b9\n\u22a2 f '' \u22c2 (i : \u03b9), s i = \u22c2 (i : \u03b9), f '' s i\n[PROOFSTEP]\nsimp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\nhf : Bijective f\ns : \u03b9 \u2192 Set \u03b1\nh\u271d : Nonempty \u03b9\n\u22a2 f '' \u22c2 (i : \u03b9), s i = \u22c2 (i : \u03b9), f '' s i\n[PROOFSTEP]\nexact (hf.injective.injOn _).image_iInter_eq\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\nhf : Bijective f\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\n\u22a2 f '' \u22c2 (i : \u03b9) (j : \u03ba i), s i j = \u22c2 (i : \u03b9) (j : \u03ba i), f '' s i j\n[PROOFSTEP]\nsimp_rw [image_iInter hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b1\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), InjOn f (s i)\n\u22a2 InjOn f (\u22c3 (i : \u03b9), s i)\n[PROOFSTEP]\nintro x hx y hy hxy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b1\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), InjOn f (s i)\nx : \u03b1\nhx : x \u2208 \u22c3 (i : \u03b9), s i\ny : \u03b1\nhy : y \u2208 \u22c3 (i : \u03b9), s i\nhxy : f x = f y\n\u22a2 x = y\n[PROOFSTEP]\nrcases mem_iUnion.1 hx with \u27e8i, hx\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b1\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), InjOn f (s i)\nx : \u03b1\nhx\u271d : x \u2208 \u22c3 (i : \u03b9), s i\ny : \u03b1\nhy : y \u2208 \u22c3 (i : \u03b9), s i\nhxy : f x = f y\ni : \u03b9\nhx : x \u2208 s i\n\u22a2 x = y\n[PROOFSTEP]\nrcases mem_iUnion.1 hy with \u27e8j, hy\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b1\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), InjOn f (s i)\nx : \u03b1\nhx\u271d : x \u2208 \u22c3 (i : \u03b9), s i\ny : \u03b1\nhy\u271d : y \u2208 \u22c3 (i : \u03b9), s i\nhxy : f x = f y\ni : \u03b9\nhx : x \u2208 s i\nj : \u03b9\nhy : y \u2208 s j\n\u22a2 x = y\n[PROOFSTEP]\nrcases hs i j with \u27e8k, hi, hj\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b1\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), InjOn f (s i)\nx : \u03b1\nhx\u271d : x \u2208 \u22c3 (i : \u03b9), s i\ny : \u03b1\nhy\u271d : y \u2208 \u22c3 (i : \u03b9), s i\nhxy : f x = f y\ni : \u03b9\nhx : x \u2208 s i\nj : \u03b9\nhy : y \u2208 s j\nk : \u03b9\nhi : s i \u2286 s k\nhj : s j \u2286 s k\n\u22a2 x = y\n[PROOFSTEP]\nexact hf k (hi hx) (hj hy) hxy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nH : \u2200 (i : \u03b9), SurjOn f (s i) t\nHinj : InjOn f (\u22c3 (i : \u03b9), s i)\n\u22a2 SurjOn f (\u22c2 (i : \u03b9), s i) t\n[PROOFSTEP]\nintro y hy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nH : \u2200 (i : \u03b9), SurjOn f (s i) t\nHinj : InjOn f (\u22c3 (i : \u03b9), s i)\ny : \u03b2\nhy : y \u2208 t\n\u22a2 y \u2208 f '' \u22c2 (i : \u03b9), s i\n[PROOFSTEP]\nrw [Hinj.image_iInter_eq, mem_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : Nonempty \u03b9\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nH : \u2200 (i : \u03b9), SurjOn f (s i) t\nHinj : InjOn f (\u22c3 (i : \u03b9), s i)\ny : \u03b2\nhy : y \u2208 t\n\u22a2 \u2200 (i : \u03b9), y \u2208 f '' s i\n[PROOFSTEP]\nexact fun i => H i hy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : \u03b9 \u2192 Set \u03b1\n\u22a2 f '' \u22c3 (i : \u03b9), s i = \u22c3 (i : \u03b9), f '' s i\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : \u03b9 \u2192 Set \u03b1\nx : \u03b2\n\u22a2 x \u2208 f '' \u22c3 (i : \u03b9), s i \u2194 x \u2208 \u22c3 (i : \u03b9), f '' s i\n[PROOFSTEP]\nsimp only [mem_image, mem_iUnion, \u2190 exists_and_right, \u2190 exists_and_left]\n --Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : \u03b9 \u2192 Set \u03b1\nx : \u03b2\n\u22a2 (\u2203 x_1 x_2, x_1 \u2208 s x_2 \u2227 f x_1 = x) \u2194 \u2203 i x_1, x_1 \u2208 s i \u2227 f x_1 = x\n[PROOFSTEP]\nrw [exists_swap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\n\u22a2 f '' \u22c3 (i : \u03b9) (j : \u03ba i), s i j = \u22c3 (i : \u03b9) (j : \u03ba i), f '' s i j\n[PROOFSTEP]\nsimp_rw [image_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\np : \u03b1 \u2192 Prop\nx\u271d : Subtype p\nx : \u03b1\nh : p x\n\u22a2 { val := x, property := h } \u2208 univ \u2194 { val := x, property := h } \u2208 \u22c3 (x : \u03b1) (h : p x), {{ val := x, property := h }}\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03b9 : Sort u_11\nf : \u03b9 \u2192 \u03b1\na : \u03b1\n\u22a2 a \u2208 range f \u2194 a \u2208 \u22c3 (i : \u03b9), {f i}\n[PROOFSTEP]\nsimp [@eq_comm \u03b1 a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nb : \u03b2\n\u22a2 b \u2208 f '' s \u2194 b \u2208 \u22c3 (i : \u03b1) (_ : i \u2208 s), {f i}\n[PROOFSTEP]\nsimp [@eq_comm \u03b2 b]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b9 \u2192 \u03b1\ng : \u03b1 \u2192 Set \u03b2\n\u22a2 \u22c3 (x : \u03b1) (y : \u03b9) (_ : f y = x), g x = \u22c3 (y : \u03b9), g (f y)\n[PROOFSTEP]\nsimpa using biUnion_range\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b9 \u2192 \u03b1\ng : \u03b1 \u2192 Set \u03b2\n\u22a2 \u22c2 (x : \u03b1) (y : \u03b9) (_ : f y = x), g x = \u22c2 (y : \u03b9), g (f y)\n[PROOFSTEP]\nsimpa using biInter_range\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : \u03b9 \u2192 Set \u03b2\n\u22a2 \u2200 (x : \u03b1), x \u2208 f \u207b\u00b9' \u22c3 (i : \u03b9), s i \u2194 x \u2208 \u22c3 (i : \u03b9), f \u207b\u00b9' s i\n[PROOFSTEP]\nsimp [preimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b2\n\u22a2 f \u207b\u00b9' \u22c3 (i : \u03b9) (j : \u03ba i), s i j = \u22c3 (i : \u03b9) (j : \u03ba i), f \u207b\u00b9' s i j\n[PROOFSTEP]\nsimp_rw [preimage_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : Set (Set \u03b2)\n\u22a2 f \u207b\u00b9' \u22c3\u2080 s = \u22c3 (t : Set \u03b2) (_ : t \u2208 s), f \u207b\u00b9' t\n[PROOFSTEP]\nrw [sUnion_eq_biUnion, preimage_iUnion\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : \u03b9 \u2192 Set \u03b2\n\u22a2 f \u207b\u00b9' \u22c2 (i : \u03b9), s i = \u22c2 (i : \u03b9), f \u207b\u00b9' s i\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : \u03b9 \u2192 Set \u03b2\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 f \u207b\u00b9' \u22c2 (i : \u03b9), s i \u2194 x\u271d \u2208 \u22c2 (i : \u03b9), f \u207b\u00b9' s i\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b2\n\u22a2 f \u207b\u00b9' \u22c2 (i : \u03b9) (j : \u03ba i), s i j = \u22c2 (i : \u03b9) (j : \u03ba i), f \u207b\u00b9' s i j\n[PROOFSTEP]\nsimp_rw [preimage_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : Set (Set \u03b2)\n\u22a2 f \u207b\u00b9' \u22c2\u2080 s = \u22c2 (t : Set \u03b2) (_ : t \u2208 s), f \u207b\u00b9' t\n[PROOFSTEP]\nrw [sInter_eq_biInter, preimage_iInter\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u22a2 \u22c3 (y : \u03b2) (_ : y \u2208 s), f \u207b\u00b9' {y} = f \u207b\u00b9' s\n[PROOFSTEP]\nrw [\u2190 preimage_iUnion\u2082, biUnion_of_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u22c3 (y : \u03b2) (_ : y \u2208 range f), f \u207b\u00b9' {y} = univ\n[PROOFSTEP]\nrw [biUnion_preimage_singleton, preimage_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : \u03b9 \u2192 Set \u03b2\n\u22a2 s \u00d7\u02e2 \u22c3 (i : \u03b9), t i = \u22c3 (i : \u03b9), s \u00d7\u02e2 t i\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : \u03b9 \u2192 Set \u03b2\nx\u271d : \u03b1 \u00d7 \u03b2\n\u22a2 x\u271d \u2208 s \u00d7\u02e2 \u22c3 (i : \u03b9), t i \u2194 x\u271d \u2208 \u22c3 (i : \u03b9), s \u00d7\u02e2 t i\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b2\n\u22a2 s \u00d7\u02e2 \u22c3 (i : \u03b9) (j : \u03ba i), t i j = \u22c3 (i : \u03b9) (j : \u03ba i), s \u00d7\u02e2 t i j\n[PROOFSTEP]\nsimp_rw [prod_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nC : Set (Set \u03b2)\n\u22a2 s \u00d7\u02e2 \u22c3\u2080 C = \u22c3\u2080 ((fun t => s \u00d7\u02e2 t) '' C)\n[PROOFSTEP]\nsimp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b2\n\u22a2 (\u22c3 (i : \u03b9), s i) \u00d7\u02e2 t = \u22c3 (i : \u03b9), s i \u00d7\u02e2 t\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b2\nx\u271d : \u03b1 \u00d7 \u03b2\n\u22a2 x\u271d \u2208 (\u22c3 (i : \u03b9), s i) \u00d7\u02e2 t \u2194 x\u271d \u2208 \u22c3 (i : \u03b9), s i \u00d7\u02e2 t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\nt : Set \u03b2\n\u22a2 (\u22c3 (i : \u03b9) (j : \u03ba i), s i j) \u00d7\u02e2 t = \u22c3 (i : \u03b9) (j : \u03ba i), s i j \u00d7\u02e2 t\n[PROOFSTEP]\nsimp_rw [iUnion_prod_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nC : Set (Set \u03b1)\nt : Set \u03b2\n\u22a2 \u22c3\u2080 C \u00d7\u02e2 t = \u22c3\u2080 ((fun s => s \u00d7\u02e2 t) '' C)\n[PROOFSTEP]\nsimp only [sUnion_eq_biUnion, iUnion\u2082_prod_const, biUnion_image]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9'\u271d : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9'\u271d \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b9' : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\ns : \u03b9 \u2192 Set \u03b1\nt : \u03b9' \u2192 Set \u03b2\n\u22a2 \u22c3 (x : \u03b9 \u00d7 \u03b9'), s x.fst \u00d7\u02e2 t x.snd = (\u22c3 (i : \u03b9), s i) \u00d7\u02e2 \u22c3 (i : \u03b9'), t i\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9'\u271d : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort u_8\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort u_9\n\u03ba' : \u03b9'\u271d \u2192 Sort u_10\n\u03b9 : Type u_11\n\u03b9' : Type u_12\n\u03b1 : Type u_13\n\u03b2 : Type u_14\ns : \u03b9 \u2192 Set \u03b1\nt : \u03b9' \u2192 Set \u03b2\nx\u271d : \u03b1 \u00d7 \u03b2\n\u22a2 x\u271d \u2208 \u22c3 (x : \u03b9 \u00d7 \u03b9'), s x.fst \u00d7\u02e2 t x.snd \u2194 x\u271d \u2208 (\u22c3 (i : \u03b9), s i) \u00d7\u02e2 \u22c3 (i : \u03b9'), t i\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : SemilatticeSup \u03b1\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\nhs : Monotone s\nht : Monotone t\n\u22a2 \u22c3 (x : \u03b1), s x \u00d7\u02e2 t x = (\u22c3 (x : \u03b1), s x) \u00d7\u02e2 \u22c3 (x : \u03b1), t x\n[PROOFSTEP]\next \u27e8z, w\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : SemilatticeSup \u03b1\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\nhs : Monotone s\nht : Monotone t\nz : \u03b2\nw : \u03b3\n\u22a2 (z, w) \u2208 \u22c3 (x : \u03b1), s x \u00d7\u02e2 t x \u2194 (z, w) \u2208 (\u22c3 (x : \u03b1), s x) \u00d7\u02e2 \u22c3 (x : \u03b1), t x\n[PROOFSTEP]\nsimp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : SemilatticeSup \u03b1\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\nhs : Monotone s\nht : Monotone t\nz : \u03b2\nw : \u03b3\n\u22a2 (\u2200 (x : \u03b1), z \u2208 s x \u2192 w \u2208 t x \u2192 (\u2203 i, z \u2208 s i) \u2227 \u2203 i, w \u2208 t i) \u2227\n \u2200 (x : \u03b1), z \u2208 s x \u2192 \u2200 (x : \u03b1), w \u2208 t x \u2192 \u2203 i, z \u2208 s i \u2227 w \u2208 t i\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mk.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : SemilatticeSup \u03b1\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\nhs : Monotone s\nht : Monotone t\nz : \u03b2\nw : \u03b3\n\u22a2 \u2200 (x : \u03b1), z \u2208 s x \u2192 w \u2208 t x \u2192 (\u2203 i, z \u2208 s i) \u2227 \u2203 i, w \u2208 t i\n[PROOFSTEP]\nintro x hz hw\n[GOAL]\ncase h.mk.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : SemilatticeSup \u03b1\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\nhs : Monotone s\nht : Monotone t\nz : \u03b2\nw : \u03b3\nx : \u03b1\nhz : z \u2208 s x\nhw : w \u2208 t x\n\u22a2 (\u2203 i, z \u2208 s i) \u2227 \u2203 i, w \u2208 t i\n[PROOFSTEP]\nexact \u27e8\u27e8x, hz\u27e9, x, hw\u27e9\n[GOAL]\ncase h.mk.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : SemilatticeSup \u03b1\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\nhs : Monotone s\nht : Monotone t\nz : \u03b2\nw : \u03b3\n\u22a2 \u2200 (x : \u03b1), z \u2208 s x \u2192 \u2200 (x : \u03b1), w \u2208 t x \u2192 \u2203 i, z \u2208 s i \u2227 w \u2208 t i\n[PROOFSTEP]\nintro x hz x' hw\n[GOAL]\ncase h.mk.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : SemilatticeSup \u03b1\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\nhs : Monotone s\nht : Monotone t\nz : \u03b2\nw : \u03b3\nx : \u03b1\nhz : z \u2208 s x\nx' : \u03b1\nhw : w \u2208 t x'\n\u22a2 \u2203 i, z \u2208 s i \u2227 w \u2208 t i\n[PROOFSTEP]\nexact \u27e8x \u2294 x', hs le_sup_left hz, ht le_sup_right hw\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nT : Set (Set \u03b2)\nhS : Set.Nonempty S\nhT : Set.Nonempty T\n\u22a2 \u22c2\u2080 S \u00d7\u02e2 \u22c2\u2080 T = \u22c2 (r : Set \u03b1 \u00d7 Set \u03b2) (_ : r \u2208 S \u00d7\u02e2 T), r.fst \u00d7\u02e2 r.snd\n[PROOFSTEP]\nobtain \u27e8s\u2081, h\u2081\u27e9 := hS\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nT : Set (Set \u03b2)\nhT : Set.Nonempty T\ns\u2081 : Set \u03b1\nh\u2081 : s\u2081 \u2208 S\n\u22a2 \u22c2\u2080 S \u00d7\u02e2 \u22c2\u2080 T = \u22c2 (r : Set \u03b1 \u00d7 Set \u03b2) (_ : r \u2208 S \u00d7\u02e2 T), r.fst \u00d7\u02e2 r.snd\n[PROOFSTEP]\nobtain \u27e8s\u2082, h\u2082\u27e9 := hT\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nT : Set (Set \u03b2)\ns\u2081 : Set \u03b1\nh\u2081 : s\u2081 \u2208 S\ns\u2082 : Set \u03b2\nh\u2082 : s\u2082 \u2208 T\n\u22a2 \u22c2\u2080 S \u00d7\u02e2 \u22c2\u2080 T = \u22c2 (r : Set \u03b1 \u00d7 Set \u03b2) (_ : r \u2208 S \u00d7\u02e2 T), r.fst \u00d7\u02e2 r.snd\n[PROOFSTEP]\nrefine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nT : Set (Set \u03b2)\ns\u2081 : Set \u03b1\nh\u2081 : s\u2081 \u2208 S\ns\u2082 : Set \u03b2\nh\u2082 : s\u2082 \u2208 T\nx : \u03b1 \u00d7 \u03b2\nhx : x \u2208 \u22c2 (r : Set \u03b1 \u00d7 Set \u03b2) (_ : r \u2208 S \u00d7\u02e2 T), r.fst \u00d7\u02e2 r.snd\n\u22a2 x \u2208 \u22c2\u2080 S \u00d7\u02e2 \u22c2\u2080 T\n[PROOFSTEP]\nrw [mem_iInter\u2082] at hx \n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nT : Set (Set \u03b2)\ns\u2081 : Set \u03b1\nh\u2081 : s\u2081 \u2208 S\ns\u2082 : Set \u03b2\nh\u2082 : s\u2082 \u2208 T\nx : \u03b1 \u00d7 \u03b2\nhx : \u2200 (i : Set \u03b1 \u00d7 Set \u03b2), i \u2208 S \u00d7\u02e2 T \u2192 x \u2208 i.fst \u00d7\u02e2 i.snd\n\u22a2 x \u2208 \u22c2\u2080 S \u00d7\u02e2 \u22c2\u2080 T\n[PROOFSTEP]\nexact \u27e8fun s\u2080 h\u2080 => (hx (s\u2080, s\u2082) \u27e8h\u2080, h\u2082\u27e9).1, fun s\u2080 h\u2080 => (hx (s\u2081, s\u2080) \u27e8h\u2081, h\u2080\u27e9).2\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nhS : Set.Nonempty S\nt : Set \u03b2\n\u22a2 \u22c2\u2080 S \u00d7\u02e2 t = \u22c2 (s : Set \u03b1) (_ : s \u2208 S), s \u00d7\u02e2 t\n[PROOFSTEP]\nrw [\u2190 sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nS : Set (Set \u03b1)\nhS : Set.Nonempty S\nt : Set \u03b2\n\u22a2 \u22c2 (r : Set \u03b1 \u00d7 Set \u03b2) (_ : r \u2208 S \u00d7\u02e2 {t}), r.fst \u00d7\u02e2 r.snd = \u22c2 (s : Set \u03b1) (_ : s \u2208 S), s \u00d7\u02e2 t\n[PROOFSTEP]\nsimp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nT : Set (Set \u03b2)\nhT : Set.Nonempty T\ns : Set \u03b1\n\u22a2 s \u00d7\u02e2 \u22c2\u2080 T = \u22c2 (t : Set \u03b2) (_ : t \u2208 T), s \u00d7\u02e2 t\n[PROOFSTEP]\nrw [\u2190 sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nT : Set (Set \u03b2)\nhT : Set.Nonempty T\ns : Set \u03b1\n\u22a2 \u22c2 (r : Set \u03b1 \u00d7 Set \u03b2) (_ : r \u2208 {s} \u00d7\u02e2 T), r.fst \u00d7\u02e2 r.snd = \u22c2 (t : Set \u03b2) (_ : t \u2208 T), s \u00d7\u02e2 t\n[PROOFSTEP]\nsimp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 \u22c3 (a : \u03b1) (_ : a \u2208 s), f a '' t = image2 f s t\n[PROOFSTEP]\next y\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\ny : \u03b3\n\u22a2 y \u2208 \u22c3 (a : \u03b1) (_ : a \u2208 s), f a '' t \u2194 y \u2208 image2 f s t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\ny : \u03b3\n\u22a2 y \u2208 \u22c3 (a : \u03b1) (_ : a \u2208 s), f a '' t \u2192 y \u2208 image2 f s t\n[PROOFSTEP]\nsimp only [mem_iUnion]\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\ny : \u03b3\n\u22a2 y \u2208 image2 f s t \u2192 y \u2208 \u22c3 (a : \u03b1) (_ : a \u2208 s), f a '' t\n[PROOFSTEP]\nsimp only [mem_iUnion]\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\ny : \u03b3\n\u22a2 (\u2203 i i_1, y \u2208 f i '' t) \u2192 y \u2208 image2 f s t\n[PROOFSTEP]\nrintro \u27e8a, ha, x, hx, ax\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\ny : \u03b3\n\u22a2 y \u2208 image2 f s t \u2192 \u2203 i i_1, y \u2208 f i '' t\n[PROOFSTEP]\nrintro \u27e8a, ha, x, hx, ax\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\ny : \u03b3\na : \u03b1\nha : a \u2208 s\nx : \u03b2\nhx : x \u2208 t\nax : f a x = y\n\u22a2 y \u2208 image2 f s t\n[PROOFSTEP]\nexact \u27e8a, x, ha, hx, ax\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\ny : \u03b3\na : \u03b1\nha : \u03b2\nx : a \u2208 s\nhx : ha \u2208 t\nax : f a ha = y\n\u22a2 \u2203 i i_1, y \u2208 f i '' t\n[PROOFSTEP]\nexact \u27e8a, x, ha, hx, ax\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 \u22c3 (b : \u03b2) (_ : b \u2208 t), (fun a => f a b) '' s = image2 f s t\n[PROOFSTEP]\next y\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\ny : \u03b3\n\u22a2 y \u2208 \u22c3 (b : \u03b2) (_ : b \u2208 t), (fun a => f a b) '' s \u2194 y \u2208 image2 f s t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\ny : \u03b3\n\u22a2 y \u2208 \u22c3 (b : \u03b2) (_ : b \u2208 t), (fun a => f a b) '' s \u2192 y \u2208 image2 f s t\n[PROOFSTEP]\nsimp only [mem_iUnion]\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\ny : \u03b3\n\u22a2 y \u2208 image2 f s t \u2192 y \u2208 \u22c3 (b : \u03b2) (_ : b \u2208 t), (fun a => f a b) '' s\n[PROOFSTEP]\nsimp only [mem_iUnion]\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\ny : \u03b3\n\u22a2 (\u2203 i i_1, y \u2208 (fun a => f a i) '' s) \u2192 y \u2208 image2 f s t\n[PROOFSTEP]\nrintro \u27e8a, b, c, d, e\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\ny : \u03b3\n\u22a2 y \u2208 image2 f s t \u2192 \u2203 i i_1, y \u2208 (fun a => f a i) '' s\n[PROOFSTEP]\nrintro \u27e8a, b, c, d, e\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\ny : \u03b3\na : \u03b2\nb : a \u2208 t\nc : \u03b1\nd : c \u2208 s\ne : (fun a_1 => f a_1 a) c = y\n\u22a2 y \u2208 image2 f s t\ncase h.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\ny : \u03b3\na : \u03b1\nb : \u03b2\nc : a \u2208 s\nd : b \u2208 t\ne : f a b = y\n\u22a2 \u2203 i i_1, y \u2208 (fun a => f a i) '' s\n[PROOFSTEP]\nexact \u27e8c, a, d, b, e\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\ny : \u03b3\na : \u03b1\nb : \u03b2\nc : a \u2208 s\nd : b \u2208 t\ne : f a b = y\n\u22a2 \u2203 i i_1, y \u2208 (fun a => f a i) '' s\n[PROOFSTEP]\nexact \u27e8b, d, a, c, e\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b2\n\u22a2 image2 f (\u22c3 (i : \u03b9), s i) t = \u22c3 (i : \u03b9), image2 f (s i) t\n[PROOFSTEP]\nsimp only [\u2190 image_prod, iUnion_prod_const, image_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : Set \u03b1\nt : \u03b9 \u2192 Set \u03b2\n\u22a2 image2 f s (\u22c3 (i : \u03b9), t i) = \u22c3 (i : \u03b9), image2 f s (t i)\n[PROOFSTEP]\nsimp only [\u2190 image_prod, prod_iUnion, image_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\nt : Set \u03b2\n\u22a2 image2 f (\u22c3 (i : \u03b9) (j : \u03ba i), s i j) t = \u22c3 (i : \u03b9) (j : \u03ba i), image2 f (s i j) t\n[PROOFSTEP]\nsimp_rw [image2_iUnion_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : Set \u03b1\nt : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b2\n\u22a2 image2 f s (\u22c3 (i : \u03b9) (j : \u03ba i), t i j) = \u22c3 (i : \u03b9) (j : \u03ba i), image2 f s (t i j)\n[PROOFSTEP]\nsimp_rw [image2_iUnion_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b2\n\u22a2 image2 f (\u22c2 (i : \u03b9), s i) t \u2286 \u22c2 (i : \u03b9), image2 f (s i) t\n[PROOFSTEP]\nsimp_rw [image2_subset_iff, mem_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b2\n\u22a2 \u2200 (x : \u03b1), (\u2200 (i : \u03b9), x \u2208 s i) \u2192 \u2200 (y : \u03b2), y \u2208 t \u2192 \u2200 (i : \u03b9), f x y \u2208 image2 f (s i) t\n[PROOFSTEP]\nexact fun x hx y hy i => mem_image2_of_mem (hx _) hy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : Set \u03b1\nt : \u03b9 \u2192 Set \u03b2\n\u22a2 image2 f s (\u22c2 (i : \u03b9), t i) \u2286 \u22c2 (i : \u03b9), image2 f s (t i)\n[PROOFSTEP]\nsimp_rw [image2_subset_iff, mem_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : Set \u03b1\nt : \u03b9 \u2192 Set \u03b2\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b2), (\u2200 (i : \u03b9), y \u2208 t i) \u2192 \u2200 (i : \u03b9), f x y \u2208 image2 f s (t i)\n[PROOFSTEP]\nexact fun x hx y hy i => mem_image2_of_mem hx (hy _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\nt : Set \u03b2\n\u22a2 image2 f (\u22c2 (i : \u03b9) (j : \u03ba i), s i j) t \u2286 \u22c2 (i : \u03b9) (j : \u03ba i), image2 f (s i j) t\n[PROOFSTEP]\nsimp_rw [image2_subset_iff, mem_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b1\nt : Set \u03b2\n\u22a2 \u2200 (x : \u03b1),\n (\u2200 (i : \u03b9) (i_1 : \u03ba i), x \u2208 s i i_1) \u2192 \u2200 (y : \u03b2), y \u2208 t \u2192 \u2200 (i : \u03b9) (i_1 : \u03ba i), f x y \u2208 image2 f (s i i_1) t\n[PROOFSTEP]\nexact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : Set \u03b1\nt : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b2\n\u22a2 image2 f s (\u22c2 (i : \u03b9) (j : \u03ba i), t i j) \u2286 \u22c2 (i : \u03b9) (j : \u03ba i), image2 f s (t i j)\n[PROOFSTEP]\nsimp_rw [image2_subset_iff, mem_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : Set \u03b1\nt : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b2\n\u22a2 \u2200 (x : \u03b1),\n x \u2208 s \u2192 \u2200 (y : \u03b2), (\u2200 (i : \u03b9) (i_1 : \u03ba i), y \u2208 t i i_1) \u2192 \u2200 (i : \u03b9) (i_1 : \u03ba i), f x y \u2208 image2 f s (t i i_1)\n[PROOFSTEP]\nexact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 image2 f s t = \u22c3 (i : \u03b1) (_ : i \u2208 s) (j : \u03b2) (_ : j \u2208 t), {f i j}\n[PROOFSTEP]\nsimp_rw [\u2190 image_eq_iUnion, iUnion_image_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 s \u00d7\u02e2 t = \u22c3 (a : \u03b1) (_ : a \u2208 s), (fun b => (a, b)) '' t\n[PROOFSTEP]\nrw [iUnion_image_left, image2_mk_eq_prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 s \u00d7\u02e2 t = \u22c3 (b : \u03b2) (_ : b \u2208 t), (fun a => (a, b)) '' s\n[PROOFSTEP]\nrw [iUnion_image_right, image2_mk_eq_prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set (\u03b1 \u2192 \u03b2)\nt : Set \u03b1\n\u22a2 \u2200 (x : \u03b2), x \u2208 seq s t \u2194 x \u2208 \u22c3 (f : \u03b1 \u2192 \u03b2) (_ : f \u2208 s), f '' t\n[PROOFSTEP]\nsimp [seq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2\nt : Set \u03b1\n\u22a2 \u2200 (x : \u03b2), x \u2208 seq {f} t \u2194 x \u2208 f '' t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set (\u03b1 \u2192 \u03b2)\na : \u03b1\n\u22a2 \u2200 (x : \u03b2), x \u2208 seq s {a} \u2194 x \u2208 (fun f => f a) '' s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set (\u03b2 \u2192 \u03b3)\nt : Set (\u03b1 \u2192 \u03b2)\nu : Set \u03b1\n\u22a2 seq s (seq t u) = seq (seq ((fun x x_1 => x \u2218 x_1) '' s) t) u\n[PROOFSTEP]\nrefine' Set.ext fun c => Iff.intro _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set (\u03b2 \u2192 \u03b3)\nt : Set (\u03b1 \u2192 \u03b2)\nu : Set \u03b1\nc : \u03b3\n\u22a2 c \u2208 seq s (seq t u) \u2192 c \u2208 seq (seq ((fun x x_1 => x \u2218 x_1) '' s) t) u\n[PROOFSTEP]\nrintro \u27e8f, hfs, b, \u27e8g, hg, a, hau, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set (\u03b2 \u2192 \u03b3)\nt : Set (\u03b1 \u2192 \u03b2)\nu : Set \u03b1\nf : \u03b2 \u2192 \u03b3\nhfs : f \u2208 s\ng : \u03b1 \u2192 \u03b2\nhg : g \u2208 t\na : \u03b1\nhau : a \u2208 u\n\u22a2 f (g a) \u2208 seq (seq ((fun x x_1 => x \u2218 x_1) '' s) t) u\n[PROOFSTEP]\nexact \u27e8f \u2218 g, \u27e8(\u00b7 \u2218 \u00b7) f, mem_image_of_mem _ hfs, g, hg, rfl\u27e9, a, hau, rfl\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set (\u03b2 \u2192 \u03b3)\nt : Set (\u03b1 \u2192 \u03b2)\nu : Set \u03b1\nc : \u03b3\n\u22a2 c \u2208 seq (seq ((fun x x_1 => x \u2218 x_1) '' s) t) u \u2192 c \u2208 seq s (seq t u)\n[PROOFSTEP]\nrintro \u27e8fg, \u27e8fc, \u27e8f, hfs, rfl\u27e9, g, hgt, rfl\u27e9, a, ha, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set (\u03b2 \u2192 \u03b3)\nt : Set (\u03b1 \u2192 \u03b2)\nu : Set \u03b1\nf : \u03b2 \u2192 \u03b3\nhfs : f \u2208 s\ng : \u03b1 \u2192 \u03b2\nhgt : g \u2208 t\na : \u03b1\nha : a \u2208 u\n\u22a2 (fun x x_1 => x \u2218 x_1) f g a \u2208 seq s (seq t u)\n[PROOFSTEP]\nexact \u27e8f, hfs, g a, \u27e8g, hgt, a, ha, rfl\u27e9, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b2 \u2192 \u03b3\ns : Set (\u03b1 \u2192 \u03b2)\nt : Set \u03b1\n\u22a2 f '' seq s t = seq ((fun x x_1 => x \u2218 x_1) f '' s) t\n[PROOFSTEP]\nrw [\u2190 singleton_seq, \u2190 singleton_seq, seq_seq, image_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 s \u00d7\u02e2 t = seq (Prod.mk '' s) t\n[PROOFSTEP]\next \u27e8a, b\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 (a, b) \u2208 s \u00d7\u02e2 t \u2194 (a, b) \u2208 seq (Prod.mk '' s) t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mk.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 (a, b) \u2208 s \u00d7\u02e2 t \u2192 (a, b) \u2208 seq (Prod.mk '' s) t\n[PROOFSTEP]\nrintro \u27e8ha, hb\u27e9\n[GOAL]\ncase h.mk.mp.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nha : (a, b).fst \u2208 s\nhb : (a, b).snd \u2208 t\n\u22a2 (a, b) \u2208 seq (Prod.mk '' s) t\n[PROOFSTEP]\nexact \u27e8Prod.mk a, \u27e8a, ha, rfl\u27e9, b, hb, rfl\u27e9\n[GOAL]\ncase h.mk.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 (a, b) \u2208 seq (Prod.mk '' s) t \u2192 (a, b) \u2208 s \u00d7\u02e2 t\n[PROOFSTEP]\nrintro \u27e8f, \u27e8x, hx, rfl\u27e9, y, hy, eq\u27e9\n[GOAL]\ncase h.mk.mpr.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\nhx : x \u2208 s\ny : \u03b2\nhy : y \u2208 t\neq : (x, y) = (a, b)\n\u22a2 (a, b) \u2208 s \u00d7\u02e2 t\n[PROOFSTEP]\nrw [\u2190 eq]\n[GOAL]\ncase h.mk.mpr.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1\nhx : x \u2208 s\ny : \u03b2\nhy : y \u2208 t\neq : (x, y) = (a, b)\n\u22a2 (x, y) \u2208 s \u00d7\u02e2 t\n[PROOFSTEP]\nexact \u27e8hx, hy\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 seq (Prod.mk '' s) t = seq ((fun b a => (a, b)) '' t) s\n[PROOFSTEP]\nrw [\u2190 prod_eq_seq, \u2190 image_swap_prod, prod_eq_seq, image_seq, \u2190 image_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 seq ((fun x x_1 => x \u2218 x_1) Prod.swap \u2218 Prod.mk '' t) s = seq ((fun b a => (a, b)) '' t) s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 image2 f s t = seq (f '' s) t\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nx\u271d : \u03b3\n\u22a2 x\u271d \u2208 image2 f s t \u2194 x\u271d \u2208 seq (f '' s) t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03c0 : \u03b1 \u2192 Type u_11\ni : Set \u03b1\ns : (a : \u03b1) \u2192 Set (\u03c0 a)\n\u22a2 pi i s = \u22c2 (a : \u03b1) (_ : a \u2208 i), eval a \u207b\u00b9' s a\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03c0 : \u03b1 \u2192 Type u_11\ni : Set \u03b1\ns : (a : \u03b1) \u2192 Set (\u03c0 a)\nx\u271d : (i : \u03b1) \u2192 \u03c0 i\n\u22a2 x\u271d \u2208 pi i s \u2194 x\u271d \u2208 \u22c2 (a : \u03b1) (_ : a \u2208 i), eval a \u207b\u00b9' s a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03c0 : \u03b1 \u2192 Type u_11\nt : (i : \u03b1) \u2192 Set (\u03c0 i)\n\u22a2 pi univ t = \u22c2 (i : \u03b1), eval i \u207b\u00b9' t i\n[PROOFSTEP]\nsimp only [pi_def, iInter_true, mem_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03c0 : \u03b1 \u2192 Type u_11\ni : Set \u03b1\ns t : (a : \u03b1) \u2192 Set (\u03c0 a)\n\u22a2 pi i s \\ pi i t \u2286 \u22c3 (a : \u03b1) (_ : a \u2208 i), eval a \u207b\u00b9' (s a \\ t a)\n[PROOFSTEP]\nrefine' diff_subset_comm.2 fun x hx a ha => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03c0 : \u03b1 \u2192 Type u_11\ni : Set \u03b1\ns t : (a : \u03b1) \u2192 Set (\u03c0 a)\nx : (i : \u03b1) \u2192 \u03c0 i\nhx : x \u2208 pi i s \\ \u22c3 (a : \u03b1) (_ : a \u2208 i), eval a \u207b\u00b9' (s a \\ t a)\na : \u03b1\nha : a \u2208 i\n\u22a2 x a \u2208 t a\n[PROOFSTEP]\nsimp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not, eval_apply] at hx \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03c0 : \u03b1 \u2192 Type u_11\ni : Set \u03b1\ns t : (a : \u03b1) \u2192 Set (\u03c0 a)\nx : (i : \u03b1) \u2192 \u03c0 i\na : \u03b1\nha : a \u2208 i\nhx : (\u2200 (i_1 : \u03b1), i_1 \u2208 i \u2192 x i_1 \u2208 s i_1) \u2227 \u2200 (x_1 : \u03b1), x_1 \u2208 i \u2192 eval x_1 x \u2208 s x_1 \u2192 eval x_1 x \u2208 t x_1\n\u22a2 x a \u2208 t a\n[PROOFSTEP]\nexact hx.2 _ ha (hx.1 _ ha)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03c0 : \u03b1 \u2192 Type u_11\nt : (i : \u03b1) \u2192 \u03b9 \u2192 Set (\u03c0 i)\n\u22a2 (\u22c3 (x : \u03b1 \u2192 \u03b9), pi univ fun i => t i (x i)) = pi univ fun i => \u22c3 (j : \u03b9), t i j\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\n\u03c0 : \u03b1 \u2192 Type u_11\nt : (i : \u03b1) \u2192 \u03b9 \u2192 Set (\u03c0 i)\nx\u271d : (i : \u03b1) \u2192 \u03c0 i\n\u22a2 (x\u271d \u2208 \u22c3 (x : \u03b1 \u2192 \u03b9), pi univ fun i => t i (x i)) \u2194 x\u271d \u2208 pi univ fun i => \u22c3 (j : \u03b9), t i j\n[PROOFSTEP]\nsimp [Classical.skolem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 Ici (\u2a06 (i : \u03b9), f i) \u2194 x\u271d \u2208 \u22c2 (i : \u03b9), Ici (f i)\n[PROOFSTEP]\nsimp only [mem_Ici, iSup_le_iff, mem_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 Iic (\u2a05 (i : \u03b9), f i) \u2194 x\u271d \u2208 \u22c2 (i : \u03b9), Iic (f i)\n[PROOFSTEP]\nsimp only [mem_Iic, le_iInf_iff, mem_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : CompleteLattice \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 \u03b1\n\u22a2 Ici (\u2a06 (i : \u03b9) (j : \u03ba i), f i j) = \u22c2 (i : \u03b9) (j : \u03ba i), Ici (f i j)\n[PROOFSTEP]\nsimp_rw [Ici_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : CompleteLattice \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 \u03b1\n\u22a2 Iic (\u2a05 (i : \u03b9) (j : \u03ba i), f i j) = \u22c2 (i : \u03b9) (j : \u03ba i), Iic (f i j)\n[PROOFSTEP]\nsimp_rw [Iic_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : CompleteLattice \u03b1\ns : Set \u03b1\n\u22a2 Ici (sSup s) = \u22c2 (a : \u03b1) (_ : a \u2208 s), Ici a\n[PROOFSTEP]\nrw [sSup_eq_iSup, Ici_iSup\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : CompleteLattice \u03b1\ns : Set \u03b1\n\u22a2 Iic (sInf s) = \u22c2 (a : \u03b1) (_ : a \u2208 s), Iic a\n[PROOFSTEP]\nrw [sInf_eq_iInf, Iic_iInf\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nt : \u03b1 \u2192 Set \u03b2\ns\u2081 s\u2082 : Set \u03b1\n\u22a2 (\u22c3 (x : \u03b1) (_ : x \u2208 s\u2081), t x) \\ \u22c3 (x : \u03b1) (_ : x \u2208 s\u2082), t x \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 s\u2081 \\ s\u2082), t x\n[PROOFSTEP]\nsimp only [diff_subset_iff, \u2190 biUnion_union]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nt : \u03b1 \u2192 Set \u03b2\ns\u2081 s\u2082 : Set \u03b1\n\u22a2 \u22c3 (x : \u03b1) (_ : x \u2208 s\u2081), t x \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 s\u2082 \u222a s\u2081 \\ s\u2082), t x\n[PROOFSTEP]\napply biUnion_subset_biUnion_left\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nt : \u03b1 \u2192 Set \u03b2\ns\u2081 s\u2082 : Set \u03b1\n\u22a2 s\u2081 \u2286 s\u2082 \u222a s\u2081 \\ s\u2082\n[PROOFSTEP]\nrw [union_diff_self]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nt : \u03b1 \u2192 Set \u03b2\ns\u2081 s\u2082 : Set \u03b1\n\u22a2 s\u2081 \u2286 s\u2082 \u222a s\u2081\n[PROOFSTEP]\napply subset_union_right\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nt : \u03b1 \u2192 Set \u03b2\nb : \u03b2\nhb : b \u2208 \u22c3 (i : \u03b1), t i\n\u22a2 \u2203 a, b \u2208 t a\n[PROOFSTEP]\nsimpa using hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nt : \u03b1 \u2192 Set \u03b2\nh : \u2200 (i j : \u03b1), i \u2260 j \u2192 Disjoint (t i) (t j)\na\u2081 : \u03b1\nb\u2081 : \u03b2\nh\u2081 : b\u2081 \u2208 t a\u2081\na\u2082 : \u03b1\nb\u2082 : \u03b2\nh\u2082 : b\u2082 \u2208 t a\u2082\neq :\n sigmaToiUnion t { fst := a\u2081, snd := { val := b\u2081, property := h\u2081 } } =\n sigmaToiUnion t { fst := a\u2082, snd := { val := b\u2082, property := h\u2082 } }\nb_eq : b\u2081 = b\u2082\na_eq : a\u2081 = a\u2082\n\u22a2 \u2191(Eq.recOn a_eq { fst := a\u2081, snd := { val := b\u2081, property := h\u2081 } }.snd) =\n \u2191{ fst := a\u2082, snd := { val := b\u2082, property := h\u2082 } }.snd\n[PROOFSTEP]\nsubst b_eq\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nt : \u03b1 \u2192 Set \u03b2\nh : \u2200 (i j : \u03b1), i \u2260 j \u2192 Disjoint (t i) (t j)\na\u2081 : \u03b1\nb\u2081 : \u03b2\nh\u2081 : b\u2081 \u2208 t a\u2081\na\u2082 : \u03b1\na_eq : a\u2081 = a\u2082\nh\u2082 : b\u2081 \u2208 t a\u2082\neq :\n sigmaToiUnion t { fst := a\u2081, snd := { val := b\u2081, property := h\u2081 } } =\n sigmaToiUnion t { fst := a\u2082, snd := { val := b\u2081, property := h\u2082 } }\n\u22a2 \u2191(Eq.recOn a_eq { fst := a\u2081, snd := { val := b\u2081, property := h\u2081 } }.snd) =\n \u2191{ fst := a\u2082, snd := { val := b\u2081, property := h\u2082 } }.snd\n[PROOFSTEP]\nsubst a_eq\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nt : \u03b1 \u2192 Set \u03b2\nh : \u2200 (i j : \u03b1), i \u2260 j \u2192 Disjoint (t i) (t j)\na\u2081 : \u03b1\nb\u2081 : \u03b2\nh\u2081 h\u2082 : b\u2081 \u2208 t a\u2081\neq :\n sigmaToiUnion t { fst := a\u2081, snd := { val := b\u2081, property := h\u2081 } } =\n sigmaToiUnion t { fst := a\u2081, snd := { val := b\u2081, property := h\u2082 } }\n\u22a2 \u2191(Eq.recOn (_ : a\u2081 = a\u2081) { fst := a\u2081, snd := { val := b\u2081, property := h\u2081 } }.snd) =\n \u2191{ fst := a\u2081, snd := { val := b\u2081, property := h\u2082 } }.snd\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : CompleteLattice \u03b2\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2a06 (a : \u03b1) (_ : a \u2208 \u22c3 (i : \u03b9), s i), f a = \u2a06 (i : \u03b9) (a : \u03b1) (_ : a \u2208 s i), f a\n[PROOFSTEP]\nrw [iSup_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : CompleteLattice \u03b2\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2a06 (a : \u03b1) (_ : a \u2208 \u22c3 (i : \u03b9), s i), f a = \u2a06 (j : \u03b1) (i : \u03b9) (_ : j \u2208 s i), f j\n[PROOFSTEP]\nsimp_rw [mem_iUnion, iSup_exists]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : CompleteLattice \u03b2\ns : Set (Set \u03b2)\n\u22a2 sSup (\u22c3\u2080 s) = \u2a06 (t : Set \u03b2) (_ : t \u2208 s), sSup t\n[PROOFSTEP]\nsimp only [sUnion_eq_biUnion, sSup_eq_iSup, iSup_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : CompleteLattice \u03b2\nS : Set (Set \u03b1)\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2a06 (x : \u03b1) (_ : x \u2208 \u22c3\u2080 S), f x = \u2a06 (s : Set \u03b1) (_ : s \u2208 S) (x : \u03b1) (_ : x \u2208 s), f x\n[PROOFSTEP]\nrw [sUnion_eq_iUnion, iSup_iUnion, \u2190 iSup_subtype'']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : CompleteLattice \u03b2\nS : Set (Set \u03b1)\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2a05 (x : \u03b1) (_ : x \u2208 \u22c3\u2080 S), f x = \u2a05 (s : Set \u03b1) (_ : s \u2208 S) (x : \u03b1) (_ : x \u2208 s), f x\n[PROOFSTEP]\nrw [sUnion_eq_iUnion, iInf_iUnion, \u2190 iInf_subtype'']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : CompleteLattice \u03b2\nS : Set (Set \u03b1)\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200 (x : \u03b1), x \u2208 \u22c3\u2080 S \u2192 p x) \u2194 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 p x\n[PROOFSTEP]\nsimp_rw [\u2190 iInf_Prop_eq, iInf_sUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\ninst\u271d : CompleteLattice \u03b2\nS : Set (Set \u03b1)\np : \u03b1 \u2192 Prop\n\u22a2 (\u2203 x, x \u2208 \u22c3\u2080 S \u2227 p x) \u2194 \u2203 s, s \u2208 S \u2227 \u2203 x, x \u2208 s \u2227 p x\n[PROOFSTEP]\nsimp_rw [\u2190 exists_prop, \u2190 iSup_Prop_eq, iSup_sUnion]\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Lattice", "llama_tokens": 58072, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.6150878555160666, "lm_q1q2_score": 0.4114201249486319}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 \ud835\udcdd 0 = \u2a05 (\u03b3 : \u0393\u2080) (_ : \u03b3 \u2260 0), \ud835\udcdf (Iio \u03b3)\n[PROOFSTEP]\nrw [nhds_eq_update, update_same]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 HasBasis (\ud835\udcdd 0) (fun \u03b3 => \u03b3 \u2260 0) Iio\n[PROOFSTEP]\nrw [nhds_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 HasBasis (\u2a05 (\u03b3 : \u0393\u2080) (_ : \u03b3 \u2260 0), \ud835\udcdf (Iio \u03b3)) (fun \u03b3 => \u03b3 \u2260 0) Iio\n[PROOFSTEP]\nrefine' hasBasis_biInf_principal _ \u27e81, one_ne_zero\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 DirectedOn ((fun \u03b3 => Iio \u03b3) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) fun \u03b3 => \u03b3 = 0 \u2192 False\n[PROOFSTEP]\nexact directedOn_iff_directed.2 (directed_of_inf fun a b hab => Iio_subset_Iio hab)\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 Tendsto f l (\ud835\udcdd 0) \u2194 \u2200 (\u03b3\u2080 : \u0393\u2080), \u03b3\u2080 \u2260 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, f x < \u03b3\u2080\n[PROOFSTEP]\nsimp [nhds_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3\u271d \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u03b3 : \u0393\u2080\nh\u2080 : \u03b3 \u2260 0\n\u22a2 \ud835\udcdd \u03b3 = pure \u03b3\n[PROOFSTEP]\nrw [nhds_eq_update, update_noteq h\u2080]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3\u271d \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\n\u22a2 {\u2191\u03b3} \u2208 \ud835\udcdd \u2191\u03b3\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nh : \u03b3 \u2260 0\n\u22a2 {\u03b3} \u2208 \ud835\udcdd \u03b3\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nx : \u0393\u2080\nh : x \u2260 0\n\u22a2 HasBasis (\ud835\udcdd x) (fun x => True) fun x_1 => {x}\n[PROOFSTEP]\nrw [nhds_of_ne_zero h]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nx : \u0393\u2080\nh : x \u2260 0\n\u22a2 HasBasis (pure x) (fun x => True) fun x_1 => {x}\n[PROOFSTEP]\nexact hasBasis_pure _\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3\u271d \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u03b3 : \u0393\u2080\nh : \u03b3 \u2260 0\n\u22a2 Tendsto f l (\ud835\udcdd \u03b3) \u2194 \u2200\u1da0 (x : \u03b1) in l, f x = \u03b3\n[PROOFSTEP]\nrw [nhds_of_ne_zero h, tendsto_pure]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nh : \u03b3\u2081 < \u03b3\u2082\n\u22a2 Iio \u03b3\u2082 \u2208 \ud835\udcdd \u03b3\u2081\n[PROOFSTEP]\nrcases eq_or_ne \u03b3\u2081 0 with (rfl | h\u2080)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nh : 0 < \u03b3\u2082\n\u22a2 Iio \u03b3\u2082 \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nsimp [*, h.ne', Iio_mem_nhds_zero]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nh : \u03b3\u2081 < \u03b3\u2082\nh\u2080 : \u03b3\u2081 \u2260 0\n\u22a2 Iio \u03b3\u2082 \u2208 \ud835\udcdd \u03b3\u2081\n[PROOFSTEP]\nsimp [*, h.ne', Iio_mem_nhds_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\ns : Set \u0393\u2080\n\u22a2 IsOpen s \u2194 \u00ac0 \u2208 s \u2228 \u2203 \u03b3, \u03b3 \u2260 0 \u2227 Iio \u03b3 \u2286 s\n[PROOFSTEP]\nrw [isOpen_iff_mem_nhds, \u2190 and_forall_ne (0 : \u0393\u2080)]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\ns : Set \u0393\u2080\n\u22a2 ((0 \u2208 s \u2192 s \u2208 \ud835\udcdd 0) \u2227 \u2200 (b : \u0393\u2080), b \u2260 0 \u2192 b \u2208 s \u2192 s \u2208 \ud835\udcdd b) \u2194 \u00ac0 \u2208 s \u2228 \u2203 \u03b3, \u03b3 \u2260 0 \u2227 Iio \u03b3 \u2286 s\n[PROOFSTEP]\nsimp (config := { contextual := true }) [nhds_of_ne_zero, imp_iff_not_or, hasBasis_nhds_zero.mem_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\ns : Set \u0393\u2080\n\u22a2 IsClosed s \u2194 0 \u2208 s \u2228 \u2203 \u03b3, \u03b3 \u2260 0 \u2227 s \u2286 Ici \u03b3\n[PROOFSTEP]\nsimp only [\u2190 isOpen_compl_iff, isOpen_iff, mem_compl_iff, not_not, \u2190 compl_Ici, compl_subset_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 IsClosed {p | p.fst \u2264 p.snd}\n[PROOFSTEP]\nsimp only [\u2190 isOpen_compl_iff, compl_setOf, not_le, isOpen_iff_mem_nhds]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 \u2200 (a : \u0393\u2080 \u00d7 \u0393\u2080), a \u2208 {a | a.snd < a.fst} \u2192 {a | a.snd < a.fst} \u2208 \ud835\udcdd a\n[PROOFSTEP]\nrintro \u27e8a, b\u27e9 (hab : b < a)\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\na b : \u0393\u2080\nhab : b < a\n\u22a2 {a | a.snd < a.fst} \u2208 \ud835\udcdd (a, b)\n[PROOFSTEP]\nrw [nhds_prod_eq, nhds_of_ne_zero (zero_le'.trans_lt hab).ne', pure_prod]\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\na b : \u0393\u2080\nhab : b < a\n\u22a2 {a | a.snd < a.fst} \u2208 map (Prod.mk a) (\ud835\udcdd b)\n[PROOFSTEP]\nexact Iio_mem_nhds hab\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\ns t : Set \u0393\u2080\nh\u2081 : Disjoint (closure s) t\nh\u2082 : Disjoint s (closure t)\n\u22a2 Disjoint (\ud835\udcdd\u02e2 s) (\ud835\udcdd\u02e2 t)\n[PROOFSTEP]\nby_cases hs : 0 \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\ns t : Set \u0393\u2080\nh\u2081 : Disjoint (closure s) t\nh\u2082 : Disjoint s (closure t)\nhs : 0 \u2208 s\n\u22a2 Disjoint (\ud835\udcdd\u02e2 s) (\ud835\udcdd\u02e2 t)\n[PROOFSTEP]\nhave ht : 0 \u2209 t := fun ht => disjoint_left.1 h\u2081 (subset_closure hs) ht\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\ns t : Set \u0393\u2080\nh\u2081 : Disjoint (closure s) t\nh\u2082 : Disjoint s (closure t)\nhs : 0 \u2208 s\nht : \u00ac0 \u2208 t\n\u22a2 Disjoint (\ud835\udcdd\u02e2 s) (\ud835\udcdd\u02e2 t)\n[PROOFSTEP]\nrwa [(isOpen_iff.2 (.inl ht)).nhdsSet_eq, disjoint_nhdsSet_principal]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\ns t : Set \u0393\u2080\nh\u2081 : Disjoint (closure s) t\nh\u2082 : Disjoint s (closure t)\nhs : \u00ac0 \u2208 s\n\u22a2 Disjoint (\ud835\udcdd\u02e2 s) (\ud835\udcdd\u02e2 t)\n[PROOFSTEP]\nrwa [(isOpen_iff.2 (.inl hs)).nhdsSet_eq, disjoint_principal_nhdsSet]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 Continuous fun p => p.fst * p.snd\n[PROOFSTEP]\nsimp only [continuous_iff_continuousAt, ContinuousAt]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 \u2200 (x : \u0393\u2080 \u00d7 \u0393\u2080), Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd x) (\ud835\udcdd (x.fst * x.snd))\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nx y : \u0393\u2080\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (x, y)) (\ud835\udcdd ((x, y).fst * (x, y).snd))\n[PROOFSTEP]\nwlog hle : x \u2264 y generalizing x y\n[GOAL]\ncase mk.inr\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nx y : \u0393\u2080\nthis : \u2200 (x y : \u0393\u2080), x \u2264 y \u2192 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (x, y)) (\ud835\udcdd ((x, y).fst * (x, y).snd))\nhle : \u00acx \u2264 y\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (x, y)) (\ud835\udcdd ((x, y).fst * (x, y).snd))\n[PROOFSTEP]\nhave := (this y x (le_of_not_le hle)).comp (continuous_swap.tendsto (x, y))\n[GOAL]\ncase mk.inr\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nx y : \u0393\u2080\nthis\u271d : \u2200 (x y : \u0393\u2080), x \u2264 y \u2192 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (x, y)) (\ud835\udcdd ((x, y).fst * (x, y).snd))\nhle : \u00acx \u2264 y\nthis : Tendsto ((fun p => p.fst * p.snd) \u2218 Prod.swap) (\ud835\udcdd (x, y)) (\ud835\udcdd ((y, x).fst * (y, x).snd))\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (x, y)) (\ud835\udcdd ((x, y).fst * (x, y).snd))\n[PROOFSTEP]\nsimpa only [mul_comm, Function.comp, Prod.swap] using this\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nx y : \u0393\u2080\nhle : x \u2264 y\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (x, y)) (\ud835\udcdd ((x, y).fst * (x, y).snd))\n[PROOFSTEP]\nrcases eq_or_ne x 0 with (rfl | hx) <;> [rcases eq_or_ne y 0 with (rfl | hy); skip]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nx y : \u0393\u2080\nhle : x \u2264 y\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (x, y)) (\ud835\udcdd ((x, y).fst * (x, y).snd))\n[PROOFSTEP]\nrcases eq_or_ne x 0 with (rfl | hx)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\ny : \u0393\u2080\nhle : 0 \u2264 y\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (0, y)) (\ud835\udcdd ((0, y).fst * (0, y).snd))\n[PROOFSTEP]\nrcases eq_or_ne y 0 with (rfl | hy)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nx y : \u0393\u2080\nhle : x \u2264 y\nhx : x \u2260 0\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (x, y)) (\ud835\udcdd ((x, y).fst * (x, y).snd))\n[PROOFSTEP]\nskip\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nhle : 0 \u2264 0\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd ((0, 0).fst * (0, 0).snd))\n[PROOFSTEP]\nrw [zero_mul]\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nhle : 0 \u2264 0\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine\n ((hasBasis_nhds_zero.prod_nhds hasBasis_nhds_zero).tendsto_iff hasBasis_nhds_zero).2 fun \u03b3 h\u03b3 =>\n \u27e8(\u03b3, 1), \u27e8h\u03b3, one_ne_zero\u27e9, ?_\u27e9\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3\u271d \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nhle : 0 \u2264 0\n\u03b3 : \u0393\u2080\nh\u03b3 : \u03b3 \u2260 0\n\u22a2 \u2200 (x : \u0393\u2080 \u00d7 \u0393\u2080), x \u2208 Iio (\u03b3, 1).fst \u00d7\u02e2 Iio (\u03b3, 1).snd \u2192 x.fst * x.snd \u2208 Iio \u03b3\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9 \u27e8hx : x < \u03b3, hy : y < 1\u27e9\n[GOAL]\ncase inl.inl.mk.intro\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3\u271d \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nhle : 0 \u2264 0\n\u03b3 : \u0393\u2080\nh\u03b3 : \u03b3 \u2260 0\nx y : \u0393\u2080\nhx : x < \u03b3\nhy : y < 1\n\u22a2 (x, y).fst * (x, y).snd \u2208 Iio \u03b3\n[PROOFSTEP]\nexact (mul_lt_mul\u2080 hx hy).trans_eq (mul_one \u03b3)\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\ny : \u0393\u2080\nhle : 0 \u2264 y\nhy : y \u2260 0\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (0, y)) (\ud835\udcdd ((0, y).fst * (0, y).snd))\n[PROOFSTEP]\nrw [zero_mul, nhds_prod_eq, nhds_of_ne_zero hy, prod_pure, tendsto_map'_iff]\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\ny : \u0393\u2080\nhle : 0 \u2264 y\nhy : y \u2260 0\n\u22a2 Tendsto ((fun p => p.fst * p.snd) \u2218 fun a => (a, y)) (\ud835\udcdd 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' (hasBasis_nhds_zero.tendsto_iff hasBasis_nhds_zero).2 fun \u03b3 h\u03b3 => _\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3\u271d \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\ny : \u0393\u2080\nhle : 0 \u2264 y\nhy : y \u2260 0\n\u03b3 : \u0393\u2080\nh\u03b3 : \u03b3 \u2260 0\n\u22a2 \u2203 ia, ia \u2260 0 \u2227 \u2200 (x : \u0393\u2080), x \u2208 Iio ia \u2192 ((fun p => p.fst * p.snd) \u2218 fun a => (a, y)) x \u2208 Iio \u03b3\n[PROOFSTEP]\nrefine' \u27e8\u03b3 / y, div_ne_zero h\u03b3 hy, fun x hx => _\u27e9\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3\u271d \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\ny : \u0393\u2080\nhle : 0 \u2264 y\nhy : y \u2260 0\n\u03b3 : \u0393\u2080\nh\u03b3 : \u03b3 \u2260 0\nx : \u0393\u2080\nhx : x \u2208 Iio (\u03b3 / y)\n\u22a2 ((fun p => p.fst * p.snd) \u2218 fun a => (a, y)) x \u2208 Iio \u03b3\n[PROOFSTEP]\ncalc\n x * y < \u03b3 / y * y := mul_lt_right\u2080 _ hx hy\n _ = \u03b3 := div_mul_cancel _ hy\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nx y : \u0393\u2080\nhle : x \u2264 y\nhx : x \u2260 0\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (x, y)) (\ud835\udcdd ((x, y).fst * (x, y).snd))\n[PROOFSTEP]\nhave hy : y \u2260 0 := ((zero_lt_iff.mpr hx).trans_le hle).ne'\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nx y : \u0393\u2080\nhle : x \u2264 y\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (x, y)) (\ud835\udcdd ((x, y).fst * (x, y).snd))\n[PROOFSTEP]\nrw [nhds_prod_eq, nhds_of_ne_zero hx, nhds_of_ne_zero hy, prod_pure_pure]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nx y : \u0393\u2080\nhle : x \u2264 y\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 Tendsto (fun p => p.fst * p.snd) (pure (x, y)) (\ud835\udcdd ((x, y).fst * (x, y).snd))\n[PROOFSTEP]\nexact pure_le_nhds (x * y)\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3\u271d \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u03b3 : \u0393\u2080\nh : \u03b3 \u2260 0\n\u22a2 ContinuousAt Inv.inv \u03b3\n[PROOFSTEP]\nrw [ContinuousAt, nhds_of_ne_zero h]\n[GOAL]\n\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3\u271d \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u03b3 : \u0393\u2080\nh : \u03b3 \u2260 0\n\u22a2 Tendsto Inv.inv (pure \u03b3) (\ud835\udcdd \u03b3\u207b\u00b9)\n[PROOFSTEP]\nexact pure_le_nhds \u03b3\u207b\u00b9\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.WithZeroTopology", "llama_tokens": 6977, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.41142012494863184}} {"text": "[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\na : L\nh : map i f = \u2191C a\nha : \u00aca = 0\ng\u271d : L[X]\nhg : Irreducible g\u271d\nx\u271d : g\u271d \u2223 map i f\np : L[X]\nhp : map i f = g\u271d * p\n\u22a2 degree g\u271d = 0\n[PROOFSTEP]\nhave := congr_arg degree hp\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\na : L\nh : map i f = \u2191C a\nha : \u00aca = 0\ng\u271d : L[X]\nhg : Irreducible g\u271d\nx\u271d : g\u271d \u2223 map i f\np : L[X]\nhp : map i f = g\u271d * p\nthis : degree (map i f) = degree (g\u271d * p)\n\u22a2 degree g\u271d = 0\n[PROOFSTEP]\nrw [h, degree_C ha, degree_mul, @eq_comm (WithBot \u2115) 0, Nat.WithBot.add_eq_zero_iff] at this \n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\na : L\nh : map i f = \u2191C a\nha : \u00aca = 0\ng\u271d : L[X]\nhg : Irreducible g\u271d\nx\u271d : g\u271d \u2223 map i f\np : L[X]\nhp : map i f = g\u271d * p\nthis : degree g\u271d = 0 \u2227 degree p = 0\n\u22a2 degree g\u271d = 0\n[PROOFSTEP]\nexact this.1\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhf : degree (map i f) = 1\ng\u271d : L[X]\nhg : Irreducible g\u271d\nx\u271d : g\u271d \u2223 map i f\np : L[X]\nhp : map i f = g\u271d * p\n\u22a2 degree g\u271d = 1\n[PROOFSTEP]\nhave := congr_arg degree hp\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhf : degree (map i f) = 1\ng\u271d : L[X]\nhg : Irreducible g\u271d\nx\u271d : g\u271d \u2223 map i f\np : L[X]\nhp : map i f = g\u271d * p\nthis : degree (map i f) = degree (g\u271d * p)\n\u22a2 degree g\u271d = 1\n[PROOFSTEP]\nsimp [Nat.WithBot.add_eq_one_iff, hf, @eq_comm (WithBot \u2115) 1, mt isUnit_iff_degree_eq_zero.2 hg.1] at this \n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhf : degree (map i f) = 1\ng\u271d : L[X]\nhg : Irreducible g\u271d\nx\u271d : g\u271d \u2223 map i f\np : L[X]\nhp : map i f = g\u271d * p\nthis : degree g\u271d = 1 \u2227 degree p = 0\n\u22a2 degree g\u271d = 1\n[PROOFSTEP]\ntauto\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhf : degree f \u2264 1\nhif : \u00acdegree (map i f) \u2264 0\n\u22a2 Splits i f\n[PROOFSTEP]\npush_neg at hif \n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhf : degree f \u2264 1\nhif : 0 < degree (map i f)\n\u22a2 Splits i f\n[PROOFSTEP]\nrw [\u2190 Order.succ_le_iff, \u2190 WithBot.coe_zero, WithBot.succ_coe, Nat.succ_eq_succ] at hif \n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhf : degree f \u2264 1\nhif\u271d : 0 < degree (map i f)\nhif : \u2191(Nat.succ 0) \u2264 degree (map i f)\n\u22a2 Splits i f\n[PROOFSTEP]\nexact splits_of_map_degree_eq_one i (le_antisymm ((degree_map_le i _).trans hf) hif)\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf g : K[X]\nhf : Splits i f\nhg : Splits i g\nh : \u00acmap i (f * g) = 0\np : L[X]\nhp : Irreducible p\nhpf : p \u2223 map i (f * g)\n\u22a2 p \u2223 map i f * map i g\n[PROOFSTEP]\nconvert hpf\n[GOAL]\ncase h.e'_4\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf g : K[X]\nhf : Splits i f\nhg : Splits i g\nh : \u00acmap i (f * g) = 0\np : L[X]\nhp : Irreducible p\nhpf : p \u2223 map i (f * g)\n\u22a2 map i f * map i g = map i (f * g)\n[PROOFSTEP]\nrw [Polynomial.map_mul]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf g : K[X]\nhf\u271d : Splits i f\nhg : Splits i g\nh : \u00acmap i (f * g) = 0\np : L[X]\nhp : Irreducible p\nhpf : p \u2223 map i (f * g)\nhf : map i f = 0\n\u22a2 False\n[PROOFSTEP]\nsimp [hf] at h \n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf g : K[X]\nhf : Splits i f\nhg\u271d : Splits i g\nh : \u00acmap i (f * g) = 0\np : L[X]\nhp : Irreducible p\nhpf : p \u2223 map i (f * g)\nhg : map i g = 0\n\u22a2 False\n[PROOFSTEP]\nsimp [hg] at h \n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf g\u271d : K[X]\nhfg : map i (f * g\u271d) \u2260 0\nh : Splits i (f * g\u271d)\ng : L[X]\nhgi : Irreducible g\nhg : g \u2223 map i f\n\u22a2 g \u2223 map i (f * g\u271d)\n[PROOFSTEP]\nrw [Polynomial.map_mul]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf g\u271d : K[X]\nhfg : map i (f * g\u271d) \u2260 0\nh : Splits i (f * g\u271d)\ng : L[X]\nhgi : Irreducible g\nhg : g \u2223 map i f\n\u22a2 g \u2223 map i f * map i g\u271d\n[PROOFSTEP]\nexact hg.trans (dvd_mul_right _ _)\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf g\u271d : K[X]\nhfg : map i (f * g\u271d) \u2260 0\nh : Splits i (f * g\u271d)\ng : L[X]\nhgi : Irreducible g\nhg : g \u2223 map i g\u271d\n\u22a2 g \u2223 map i (f * g\u271d)\n[PROOFSTEP]\nrw [Polynomial.map_mul]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf g\u271d : K[X]\nhfg : map i (f * g\u271d) \u2260 0\nh : Splits i (f * g\u271d)\ng : L[X]\nhgi : Irreducible g\nhg : g \u2223 map i g\u271d\n\u22a2 g \u2223 map i f * map i g\u271d\n[PROOFSTEP]\nexact hg.trans (dvd_mul_left _ _)\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nj : L \u2192+* F\nf : K[X]\n\u22a2 Splits j (map i f) \u2194 Splits (RingHom.comp j i) f\n[PROOFSTEP]\nsimp [Splits, Polynomial.map_map]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\n\u03b9 : Type u\ns : \u03b9 \u2192 K[X]\nt : Finset \u03b9\n\u22a2 (\u2200 (j : \u03b9), j \u2208 t \u2192 Splits i (s j)) \u2192 Splits i (\u220f x in t, s x)\n[PROOFSTEP]\nrefine' Finset.induction_on t (fun _ => splits_one i) fun a t hat ih ht => _\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\n\u03b9 : Type u\ns : \u03b9 \u2192 K[X]\nt\u271d : Finset \u03b9\na : \u03b9\nt : Finset \u03b9\nhat : \u00aca \u2208 t\nih : (\u2200 (j : \u03b9), j \u2208 t \u2192 Splits i (s j)) \u2192 Splits i (\u220f x in t, s x)\nht : \u2200 (j : \u03b9), j \u2208 insert a t \u2192 Splits i (s j)\n\u22a2 Splits i (\u220f x in insert a t, s x)\n[PROOFSTEP]\nrw [Finset.forall_mem_insert] at ht \n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\n\u03b9 : Type u\ns : \u03b9 \u2192 K[X]\nt\u271d : Finset \u03b9\na : \u03b9\nt : Finset \u03b9\nhat : \u00aca \u2208 t\nih : (\u2200 (j : \u03b9), j \u2208 t \u2192 Splits i (s j)) \u2192 Splits i (\u220f x in t, s x)\nht : Splits i (s a) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 Splits i (s x)\n\u22a2 Splits i (\u220f x in insert a t, s x)\n[PROOFSTEP]\nrw [Finset.prod_insert hat]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\n\u03b9 : Type u\ns : \u03b9 \u2192 K[X]\nt\u271d : Finset \u03b9\na : \u03b9\nt : Finset \u03b9\nhat : \u00aca \u2208 t\nih : (\u2200 (j : \u03b9), j \u2208 t \u2192 Splits i (s j)) \u2192 Splits i (\u220f x in t, s x)\nht : Splits i (s a) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 Splits i (s x)\n\u22a2 Splits i (s a * \u220f x in t, s x)\n[PROOFSTEP]\nexact splits_mul i ht.1 (ih ht.2)\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhf : Splits i f\nn : \u2115\n\u22a2 Splits i (f ^ n)\n[PROOFSTEP]\nrw [\u2190 Finset.card_range n, \u2190 Finset.prod_const]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhf : Splits i f\nn : \u2115\n\u22a2 Splits i (\u220f _x in Finset.range n, f)\n[PROOFSTEP]\nexact splits_prod i fun j _ => hf\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\n\u22a2 Splits (RingHom.id L) (map i f) \u2194 Splits i f\n[PROOFSTEP]\nrw [splits_map_iff, RingHom.id_comp]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhs : Splits i f\nhf0 : degree (map i f) \u2260 0\nhf0' : map i f = 0\n\u22a2 \u2203 x, eval\u2082 i x f = 0\n[PROOFSTEP]\nsimp [eval\u2082_eq_eval_map, hf0']\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\nf : K[X]\nhs : Splits i\u271d f\nhf0 : degree (map i\u271d f) \u2260 0\nhf0' : \u00acmap i\u271d f = 0\ng : L[X]\nhg : Irreducible g \u2227 g \u2223 map i\u271d f\nx : L\nhx : IsRoot g x\ni : L[X]\nhi : map i\u271d f = g * i\n\u22a2 eval\u2082 i\u271d x f = 0\n[PROOFSTEP]\nrw [\u2190 eval_map, hi, eval_mul, show _ = _ from hx, zero_mul]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhs : Splits i f\nhf0 : natDegree (map i f) \u2260 0\nx : L\nhx : eval\u2082 i x f = 0\nh : roots (map i f) = 0\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 eval_map] at hx \n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhs : Splits i f\nhf0 : natDegree (map i f) \u2260 0\nx : L\nhx : eval x (map i f) = 0\nh : roots (map i f) = 0\n\u22a2 False\n[PROOFSTEP]\nhave : f.map i \u2260 0 := by intro; simp_all\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhs : Splits i f\nhf0 : natDegree (map i f) \u2260 0\nx : L\nhx : eval x (map i f) = 0\nh : roots (map i f) = 0\n\u22a2 map i f \u2260 0\n[PROOFSTEP]\nintro\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhs : Splits i f\nhf0 : natDegree (map i f) \u2260 0\nx : L\nhx : eval x (map i f) = 0\nh : roots (map i f) = 0\na\u271d : map i f = 0\n\u22a2 False\n[PROOFSTEP]\nsimp_all\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhs : Splits i f\nhf0 : natDegree (map i f) \u2260 0\nx : L\nhx : eval x (map i f) = 0\nh : roots (map i f) = 0\nthis : map i f \u2260 0\n\u22a2 False\n[PROOFSTEP]\ncases h.subst ((mem_roots this).2 hx)\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nhsplit : Splits i p\n\u22a2 natDegree (map i p) = \u2191Multiset.card (roots (map i p))\n[PROOFSTEP]\nby_cases hp : p.map i = 0\n[GOAL]\ncase pos\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nhsplit : Splits i p\nhp : map i p = 0\n\u22a2 natDegree (map i p) = \u2191Multiset.card (roots (map i p))\n[PROOFSTEP]\nrw [hp, natDegree_zero, roots_zero, Multiset.card_zero]\n[GOAL]\ncase neg\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nhsplit : Splits i p\nhp : \u00acmap i p = 0\n\u22a2 natDegree (map i p) = \u2191Multiset.card (roots (map i p))\n[PROOFSTEP]\nobtain \u27e8q, he, hd, hr\u27e9 := exists_prod_multiset_X_sub_C_mul (p.map i)\n[GOAL]\ncase neg.intro.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nhsplit : Splits i p\nhp : \u00acmap i p = 0\nq : L[X]\nhe : Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = map i p\nhd : \u2191Multiset.card (roots (map i p)) + natDegree q = natDegree (map i p)\nhr : roots q = 0\n\u22a2 natDegree (map i p) = \u2191Multiset.card (roots (map i p))\n[PROOFSTEP]\nrw [\u2190 splits_id_iff_splits, \u2190 he] at hsplit \n[GOAL]\ncase neg.intro.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nhp : \u00acmap i p = 0\nq : L[X]\nhsplit : Splits (RingHom.id L) (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q)\nhe : Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = map i p\nhd : \u2191Multiset.card (roots (map i p)) + natDegree q = natDegree (map i p)\nhr : roots q = 0\n\u22a2 natDegree (map i p) = \u2191Multiset.card (roots (map i p))\n[PROOFSTEP]\nrw [\u2190 he] at hp \n[GOAL]\ncase neg.intro.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nq : L[X]\nhp : \u00acMultiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = 0\nhsplit : Splits (RingHom.id L) (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q)\nhe : Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = map i p\nhd : \u2191Multiset.card (roots (map i p)) + natDegree q = natDegree (map i p)\nhr : roots q = 0\n\u22a2 natDegree (map i p) = \u2191Multiset.card (roots (map i p))\n[PROOFSTEP]\nhave hq : q \u2260 0 := fun h => hp (by rw [h, mul_zero])\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nq : L[X]\nhp : \u00acMultiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = 0\nhsplit : Splits (RingHom.id L) (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q)\nhe : Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = map i p\nhd : \u2191Multiset.card (roots (map i p)) + natDegree q = natDegree (map i p)\nhr : roots q = 0\nh : q = 0\n\u22a2 Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = 0\n[PROOFSTEP]\nrw [h, mul_zero]\n[GOAL]\ncase neg.intro.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nq : L[X]\nhp : \u00acMultiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = 0\nhsplit : Splits (RingHom.id L) (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q)\nhe : Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = map i p\nhd : \u2191Multiset.card (roots (map i p)) + natDegree q = natDegree (map i p)\nhr : roots q = 0\nhq : q \u2260 0\n\u22a2 natDegree (map i p) = \u2191Multiset.card (roots (map i p))\n[PROOFSTEP]\nrw [\u2190 hd, add_right_eq_self]\n[GOAL]\ncase neg.intro.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nq : L[X]\nhp : \u00acMultiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = 0\nhsplit : Splits (RingHom.id L) (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q)\nhe : Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = map i p\nhd : \u2191Multiset.card (roots (map i p)) + natDegree q = natDegree (map i p)\nhr : roots q = 0\nhq : q \u2260 0\n\u22a2 natDegree q = 0\n[PROOFSTEP]\nby_contra h\n[GOAL]\ncase neg.intro.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nq : L[X]\nhp : \u00acMultiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = 0\nhsplit : Splits (RingHom.id L) (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q)\nhe : Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = map i p\nhd : \u2191Multiset.card (roots (map i p)) + natDegree q = natDegree (map i p)\nhr : roots q = 0\nhq : q \u2260 0\nh : \u00acnatDegree q = 0\n\u22a2 False\n[PROOFSTEP]\nhave h' : (map (RingHom.id L) q).natDegree \u2260 0 := by simp [h]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nq : L[X]\nhp : \u00acMultiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = 0\nhsplit : Splits (RingHom.id L) (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q)\nhe : Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = map i p\nhd : \u2191Multiset.card (roots (map i p)) + natDegree q = natDegree (map i p)\nhr : roots q = 0\nhq : q \u2260 0\nh : \u00acnatDegree q = 0\n\u22a2 natDegree (map (RingHom.id L) q) \u2260 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg.intro.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nq : L[X]\nhp : \u00acMultiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = 0\nhsplit : Splits (RingHom.id L) (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q)\nhe : Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = map i p\nhd : \u2191Multiset.card (roots (map i p)) + natDegree q = natDegree (map i p)\nhr : roots q = 0\nhq : q \u2260 0\nh : \u00acnatDegree q = 0\nh' : natDegree (map (RingHom.id L) q) \u2260 0\n\u22a2 False\n[PROOFSTEP]\nhave := roots_ne_zero_of_splits' (RingHom.id L) (splits_of_splits_mul' _ ?_ hsplit).2 h'\n[GOAL]\ncase neg.intro.intro.intro.refine_2\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nq : L[X]\nhp : \u00acMultiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = 0\nhsplit : Splits (RingHom.id L) (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q)\nhe : Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = map i p\nhd : \u2191Multiset.card (roots (map i p)) + natDegree q = natDegree (map i p)\nhr : roots q = 0\nhq : q \u2260 0\nh : \u00acnatDegree q = 0\nh' : natDegree (map (RingHom.id L) q) \u2260 0\nthis : roots (map (RingHom.id L) q) \u2260 0\n\u22a2 False\n[PROOFSTEP]\nrw [map_id] at this \n[GOAL]\ncase neg.intro.intro.intro.refine_2\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nq : L[X]\nhp : \u00acMultiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = 0\nhsplit : Splits (RingHom.id L) (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q)\nhe : Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = map i p\nhd : \u2191Multiset.card (roots (map i p)) + natDegree q = natDegree (map i p)\nhr : roots q = 0\nhq : q \u2260 0\nh : \u00acnatDegree q = 0\nh' : natDegree (map (RingHom.id L) q) \u2260 0\nthis : roots q \u2260 0\n\u22a2 False\n[PROOFSTEP]\nexact this hr\n[GOAL]\ncase neg.intro.intro.intro.refine_1\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nq : L[X]\nhp : \u00acMultiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = 0\nhsplit : Splits (RingHom.id L) (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q)\nhe : Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = map i p\nhd : \u2191Multiset.card (roots (map i p)) + natDegree q = natDegree (map i p)\nhr : roots q = 0\nhq : q \u2260 0\nh : \u00acnatDegree q = 0\nh' : natDegree (map (RingHom.id L) q) \u2260 0\n\u22a2 map (RingHom.id L) (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q) \u2260 0\n[PROOFSTEP]\nrw [map_id]\n[GOAL]\ncase neg.intro.intro.intro.refine_1\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nq : L[X]\nhp : \u00acMultiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = 0\nhsplit : Splits (RingHom.id L) (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q)\nhe : Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q = map i p\nhd : \u2191Multiset.card (roots (map i p)) + natDegree q = natDegree (map i p)\nhr : roots q = 0\nhq : q \u2260 0\nh : \u00acnatDegree q = 0\nh' : natDegree (map (RingHom.id L) q) \u2260 0\n\u22a2 Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) * q \u2260 0\n[PROOFSTEP]\nexact mul_ne_zero monic_prod_multiset_X_sub_C.ne_zero hq\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\np_ne_zero : map i p \u2260 0\nhsplit : Splits i p\n\u22a2 degree (map i p) = \u2191(\u2191Multiset.card (roots (map i p)))\n[PROOFSTEP]\nsimp [degree_eq_natDegree p_ne_zero, natDegree_eq_card_roots' hsplit]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\n\u22a2 Splits i f \u2194 f = 0 \u2228 \u2200 {g : L[X]}, Irreducible g \u2192 g \u2223 map i f \u2192 degree g = 1\n[PROOFSTEP]\nrw [Splits, map_eq_zero]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf g : K[X]\nhf0 : f \u2260 0\nhf : Splits i f\nhgf : g \u2223 f\n\u22a2 Splits i g\n[PROOFSTEP]\nobtain \u27e8f, rfl\u27e9 := hgf\n[GOAL]\ncase intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\ng f : K[X]\nhf0 : g * f \u2260 0\nhf : Splits i (g * f)\n\u22a2 Splits i g\n[PROOFSTEP]\nexact (splits_of_splits_mul i hf0 hf).1\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\n\u03b9 : Type u\ns : \u03b9 \u2192 K[X]\nt : Finset \u03b9\n\u22a2 (\u2200 (j : \u03b9), j \u2208 t \u2192 s j \u2260 0) \u2192 (Splits i (\u220f x in t, s x) \u2194 \u2200 (j : \u03b9), j \u2208 t \u2192 Splits i (s j))\n[PROOFSTEP]\nrefine'\n Finset.induction_on t (fun _ => \u27e8fun _ _ h => by simp only [Finset.not_mem_empty] at h , fun _ => splits_one i\u27e9)\n fun a t hat ih ht => _\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\n\u03b9 : Type u\ns : \u03b9 \u2192 K[X]\nt : Finset \u03b9\nx\u271d\u00b2 : \u2200 (j : \u03b9), j \u2208 \u2205 \u2192 s j \u2260 0\nx\u271d\u00b9 : Splits i (\u220f x in \u2205, s x)\nx\u271d : \u03b9\nh : x\u271d \u2208 \u2205\n\u22a2 Splits i (s x\u271d)\n[PROOFSTEP]\nsimp only [Finset.not_mem_empty] at h \n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\n\u03b9 : Type u\ns : \u03b9 \u2192 K[X]\nt\u271d : Finset \u03b9\na : \u03b9\nt : Finset \u03b9\nhat : \u00aca \u2208 t\nih : (\u2200 (j : \u03b9), j \u2208 t \u2192 s j \u2260 0) \u2192 (Splits i (\u220f x in t, s x) \u2194 \u2200 (j : \u03b9), j \u2208 t \u2192 Splits i (s j))\nht : \u2200 (j : \u03b9), j \u2208 insert a t \u2192 s j \u2260 0\n\u22a2 Splits i (\u220f x in insert a t, s x) \u2194 \u2200 (j : \u03b9), j \u2208 insert a t \u2192 Splits i (s j)\n[PROOFSTEP]\nrw [Finset.forall_mem_insert] at ht \u22a2\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\n\u03b9 : Type u\ns : \u03b9 \u2192 K[X]\nt\u271d : Finset \u03b9\na : \u03b9\nt : Finset \u03b9\nhat : \u00aca \u2208 t\nih : (\u2200 (j : \u03b9), j \u2208 t \u2192 s j \u2260 0) \u2192 (Splits i (\u220f x in t, s x) \u2194 \u2200 (j : \u03b9), j \u2208 t \u2192 Splits i (s j))\nht : s a \u2260 0 \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 s x \u2260 0\n\u22a2 Splits i (\u220f x in insert a t, s x) \u2194 Splits i (s a) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 Splits i (s x)\n[PROOFSTEP]\nrw [Finset.prod_insert hat, splits_mul_iff i ht.1 (Finset.prod_ne_zero_iff.2 ht.2), ih ht.2]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\nhp : Irreducible p\nhp_splits : Splits (RingHom.id K) p\n\u22a2 degree p = 1\n[PROOFSTEP]\nrcases hp_splits with \u27e8\u27e9 | hp_splits\n[GOAL]\ncase inl\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\nhp : Irreducible p\nh\u271d : map (RingHom.id K) p = 0\n\u22a2 degree p = 1\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase inl.h\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\nhp : Irreducible p\nh\u271d : map (RingHom.id K) p = 0\n\u22a2 False\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase inr\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\nhp : Irreducible p\nhp_splits : \u2200 {g : K[X]}, Irreducible g \u2192 g \u2223 map (RingHom.id K) p \u2192 degree g = 1\n\u22a2 degree p = 1\n[PROOFSTEP]\napply hp_splits hp\n[GOAL]\ncase inr\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\nhp : Irreducible p\nhp_splits : \u2200 {g : K[X]}, Irreducible g \u2192 g \u2223 map (RingHom.id K) p \u2192 degree g = 1\n\u22a2 p \u2223 map (RingHom.id K) p\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\np_ne_zero : p \u2260 0\nhsplit : Splits i p\n\u22a2 degree p = \u2191(\u2191Multiset.card (roots (map i p)))\n[PROOFSTEP]\nrw [degree_eq_natDegree p_ne_zero, natDegree_eq_card_roots hsplit]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhf : Splits (RingHom.id K) f\n\u22a2 \u2191Multiset.card (roots f) = natDegree f\n[PROOFSTEP]\nconvert (natDegree_eq_card_roots hf).symm\n[GOAL]\ncase h.e'_2.h.e'_6.h.e'_4\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\nhf : Splits (RingHom.id K) f\n\u22a2 f = map (RingHom.id K) f\n[PROOFSTEP]\nrw [map_id]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ni : K \u2192+* L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\np : F[X]\nh : Splits (algebraMap F K) p\nf : K \u2192\u2090[F] L\n\u22a2 \u2191f '' rootSet p K = rootSet p L\n[PROOFSTEP]\nclassical rw [rootSet, \u2190 Finset.coe_image, \u2190 Multiset.toFinset_map, \u2190 f.coe_toRingHom, \u2190\n roots_map _ ((splits_id_iff_splits (algebraMap F K)).mpr h), map_map, f.comp_algebraMap, \u2190 rootSet]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ni : K \u2192+* L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\np : F[X]\nh : Splits (algebraMap F K) p\nf : K \u2192\u2090[F] L\n\u22a2 \u2191f '' rootSet p K = rootSet p L\n[PROOFSTEP]\nrw [rootSet, \u2190 Finset.coe_image, \u2190 Multiset.toFinset_map, \u2190 f.coe_toRingHom, \u2190\n roots_map _ ((splits_id_iff_splits (algebraMap F K)).mpr h), map_map, f.comp_algebraMap, \u2190 rootSet]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ni : K \u2192+* L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\np : F[X]\nh : Splits (algebraMap F K) p\nf : K \u2192\u2090[F] L\n\u22a2 Algebra.adjoin F (rootSet p L) = AlgHom.range f \u2194 Algebra.adjoin F (rootSet p K) = \u22a4\n[PROOFSTEP]\nrw [\u2190 image_rootSet h f, Algebra.adjoin_image, \u2190 Algebra.map_top]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ni : K \u2192+* L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\np : F[X]\nh : Splits (algebraMap F K) p\nf : K \u2192\u2090[F] L\n\u22a2 Subalgebra.map f (Algebra.adjoin F (rootSet p K)) = Subalgebra.map f \u22a4 \u2194 Algebra.adjoin F (rootSet p K) = \u22a4\n[PROOFSTEP]\nexact (Subalgebra.map_injective f.toRingHom.injective).eq_iff\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nhsplit : Splits i p\n\u22a2 map i p = \u2191C (\u2191i (leadingCoeff p)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p)))\n[PROOFSTEP]\nrw [\u2190 leadingCoeff_map]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nhsplit : Splits i p\n\u22a2 map i p = \u2191C (leadingCoeff (map i p)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p)))\n[PROOFSTEP]\nsymm\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nhsplit : Splits i p\n\u22a2 \u2191C (leadingCoeff (map i p)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map i p))) = map i p\n[PROOFSTEP]\napply C_leadingCoeff_mul_prod_multiset_X_sub_C\n[GOAL]\ncase hroots\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nhsplit : Splits i p\n\u22a2 \u2191Multiset.card (roots (map i p)) = natDegree (map i p)\n[PROOFSTEP]\nrw [natDegree_map]\n[GOAL]\ncase hroots\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni\u271d : K \u2192+* L\np : K[X]\ni : K \u2192+* L\nhsplit : Splits i p\n\u22a2 \u2191Multiset.card (roots (map i p)) = natDegree p\n[PROOFSTEP]\nexact (natDegree_eq_card_roots hsplit).symm\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\nhsplit : Splits (RingHom.id K) p\n\u22a2 p = \u2191C (leadingCoeff p) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p))\n[PROOFSTEP]\nsimpa using eq_prod_roots_of_splits hsplit\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\nm : Monic p\nhsplit : Splits (RingHom.id K) p\n\u22a2 p = Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p))\n[PROOFSTEP]\nconvert eq_prod_roots_of_splits_id hsplit\n[GOAL]\ncase h.e'_3\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\nm : Monic p\nhsplit : Splits (RingHom.id K) p\n\u22a2 Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) =\n \u2191C (leadingCoeff p) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p))\n[PROOFSTEP]\nsimp [m]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nx : K\nh : K[X]\nh_splits : Splits i h\nh_roots : roots (map i h) = {\u2191i x}\n\u22a2 h = \u2191C (leadingCoeff h) * (X - \u2191C x)\n[PROOFSTEP]\napply Polynomial.map_injective _ i.injective\n[GOAL]\ncase a\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nx : K\nh : K[X]\nh_splits : Splits i h\nh_roots : roots (map i h) = {\u2191i x}\n\u22a2 map i h = map i (\u2191C (leadingCoeff h) * (X - \u2191C x))\n[PROOFSTEP]\nrw [eq_prod_roots_of_splits h_splits, h_roots]\n[GOAL]\ncase a\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nx : K\nh : K[X]\nh_splits : Splits i h\nh_roots : roots (map i h) = {\u2191i x}\n\u22a2 \u2191C (\u2191i (leadingCoeff h)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) {\u2191i x}) =\n map i (\u2191C (leadingCoeff h) * (X - \u2191C x))\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ni : K \u2192+* L\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : Monic f\nhr : \u2200 (a : K), a \u2208 roots f \u2192 a \u2208 RingHom.range (algebraMap R K)\n\u22a2 f \u2208 lifts (algebraMap R K)\n[PROOFSTEP]\nrw [eq_prod_roots_of_monic_of_splits_id hm hs, lifts_iff_liftsRing]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ni : K \u2192+* L\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : Monic f\nhr : \u2200 (a : K), a \u2208 roots f \u2192 a \u2208 RingHom.range (algebraMap R K)\n\u22a2 Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots f)) \u2208 liftsRing (algebraMap R K)\n[PROOFSTEP]\nrefine' Subring.multiset_prod_mem _ _ fun P hP => _\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ni : K \u2192+* L\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : Monic f\nhr : \u2200 (a : K), a \u2208 roots f \u2192 a \u2208 RingHom.range (algebraMap R K)\nP : K[X]\nhP : P \u2208 Multiset.map (fun a => X - \u2191C a) (roots f)\n\u22a2 P \u2208 liftsRing (algebraMap R K)\n[PROOFSTEP]\nobtain \u27e8b, hb, rfl\u27e9 := Multiset.mem_map.1 hP\n[GOAL]\ncase intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ni : K \u2192+* L\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : Monic f\nhr : \u2200 (a : K), a \u2208 roots f \u2192 a \u2208 RingHom.range (algebraMap R K)\nb : K\nhb : b \u2208 roots f\nhP : X - \u2191C b \u2208 Multiset.map (fun a => X - \u2191C a) (roots f)\n\u22a2 X - \u2191C b \u2208 liftsRing (algebraMap R K)\n[PROOFSTEP]\nexact Subring.sub_mem _ (X_mem_lifts _) (C'_mem_lifts (hr _ hb))\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Irreducible p\nhdp : p \u2223 map i f\n\u22a2 degree p = 1\n[PROOFSTEP]\nrw [irreducible_iff_prime] at hp \n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 map i f\n\u22a2 degree p = 1\n[PROOFSTEP]\nrw [hs, \u2190 Multiset.prod_toList] at hdp \n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\n\u22a2 degree p = 1\n[PROOFSTEP]\nobtain hd | hd := hp.2.2 _ _ hdp\n[GOAL]\ncase inl\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd : p \u2223 \u2191C (\u2191i (leadingCoeff f))\n\u22a2 degree p = 1\n[PROOFSTEP]\nrefine' (hp.2.1 <| isUnit_of_dvd_unit hd _).elim\n[GOAL]\ncase inl\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd : p \u2223 \u2191C (\u2191i (leadingCoeff f))\n\u22a2 IsUnit (\u2191C (\u2191i (leadingCoeff f)))\n[PROOFSTEP]\nexact isUnit_C.2 ((leadingCoeff_ne_zero.2 hf0).isUnit.map i)\n[GOAL]\ncase inr\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd : p \u2223 List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\n\u22a2 degree p = 1\n[PROOFSTEP]\nobtain \u27e8q, hq, hd\u27e9 := hp.dvd_prod_iff.1 hd\n[GOAL]\ncase inr.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd\u271d : p \u2223 List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nq : L[X]\nhq : q \u2208 Multiset.toList (Multiset.map (fun a => X - \u2191C a) s)\nhd : p \u2223 q\n\u22a2 degree p = 1\n[PROOFSTEP]\nobtain \u27e8a, _, rfl\u27e9 := Multiset.mem_map.1 (Multiset.mem_toList.1 hq)\n[GOAL]\ncase inr.intro.intro.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd\u271d : p \u2223 List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\na : L\nleft\u271d : a \u2208 s\nhq : X - \u2191C a \u2208 Multiset.toList (Multiset.map (fun a => X - \u2191C a) s)\nhd : p \u2223 X - \u2191C a\n\u22a2 degree p = 1\n[PROOFSTEP]\nrw [degree_eq_degree_of_associated ((hp.dvd_prime_iff_associated <| prime_X_sub_C a).1 hd)]\n[GOAL]\ncase inr.intro.intro.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd\u271d : p \u2223 List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\na : L\nleft\u271d : a \u2208 s\nhq : X - \u2191C a \u2208 Multiset.toList (Multiset.map (fun a => X - \u2191C a) s)\nhd : p \u2223 X - \u2191C a\n\u22a2 degree (X - \u2191C a) = 1\n[PROOFSTEP]\nexact degree_X_sub_C a\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf x\u271d\u00b9 : K[X]\nhu : IsUnit x\u271d\u00b9\nx\u271d : Splits (RingHom.id K) x\u271d\u00b9\n\u22a2 0 \u2264 1\n[PROOFSTEP]\ndecide\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf a p : K[X]\nha0 : a \u2260 0\nhp : Prime p\nih : Splits (RingHom.id K) a \u2192 Splits i a\nhfi : Splits (RingHom.id K) (p * a)\n\u22a2 p \u2223 map (RingHom.id K) p\n[PROOFSTEP]\nrw [map_id]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nj : L \u2192+* F\nf : K[X]\nh : Splits i f\n\u22a2 Splits (RingHom.comp j i) f\n[PROOFSTEP]\nrw [\u2190 splits_map_iff]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nj : L \u2192+* F\nf : K[X]\nh : Splits i f\n\u22a2 Splits j (map i f)\n[PROOFSTEP]\nrw [\u2190 RingHom.id_comp i, \u2190 splits_map_iff i] at h \n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nj : L \u2192+* F\nf : K[X]\nh : Splits (RingHom.id L) (map i f)\n\u22a2 Splits j (map i f)\n[PROOFSTEP]\nexact splits_of_splits_id _ h\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\n\u22a2 Splits (RingHom.id K) p \u2194 \u2191Multiset.card (roots p) = natDegree p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\n\u22a2 Splits (RingHom.id K) p \u2192 \u2191Multiset.card (roots p) = natDegree p\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\nH : Splits (RingHom.id K) p\n\u22a2 \u2191Multiset.card (roots p) = natDegree p\n[PROOFSTEP]\nrw [natDegree_eq_card_roots H, map_id]\n[GOAL]\ncase mpr\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\n\u22a2 \u2191Multiset.card (roots p) = natDegree p \u2192 Splits (RingHom.id K) p\n[PROOFSTEP]\nintro hroots\n[GOAL]\ncase mpr\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\nhroots : \u2191Multiset.card (roots p) = natDegree p\n\u22a2 Splits (RingHom.id K) p\n[PROOFSTEP]\nrw [splits_iff_exists_multiset (RingHom.id K)]\n[GOAL]\ncase mpr\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\nhroots : \u2191Multiset.card (roots p) = natDegree p\n\u22a2 \u2203 s, map (RingHom.id K) p = \u2191C (\u2191(RingHom.id K) (leadingCoeff p)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\n[PROOFSTEP]\nuse p.roots\n[GOAL]\ncase h\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\nhroots : \u2191Multiset.card (roots p) = natDegree p\n\u22a2 map (RingHom.id K) p =\n \u2191C (\u2191(RingHom.id K) (leadingCoeff p)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p))\n[PROOFSTEP]\nsimp only [RingHom.id_apply, map_id]\n[GOAL]\ncase h\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\np : K[X]\nhroots : \u2191Multiset.card (roots p) = natDegree p\n\u22a2 p = \u2191C (leadingCoeff p) * Multiset.prod (Multiset.map (fun x => X - \u2191C x) (roots p))\n[PROOFSTEP]\nexact (C_leadingCoeff_mul_prod_multiset_X_sub_C hroots).symm\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\ni : K \u2192+* L\ninst\u271d : Algebra K L\nP : K[X]\nhmo : Monic P\nhP : Splits (algebraMap K L) P\nr : L\nhr : r \u2208 roots (map (algebraMap K L) P)\n\u22a2 \u2191(aeval r) (\u2191derivative P) =\n Multiset.prod (Multiset.map (fun a => r - a) (Multiset.erase (roots (map (algebraMap K L) P)) r))\n[PROOFSTEP]\nreplace hmo := hmo.map (algebraMap K L)\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\ni : K \u2192+* L\ninst\u271d : Algebra K L\nP : K[X]\nhP : Splits (algebraMap K L) P\nr : L\nhr : r \u2208 roots (map (algebraMap K L) P)\nhmo : Monic (map (algebraMap K L) P)\n\u22a2 \u2191(aeval r) (\u2191derivative P) =\n Multiset.prod (Multiset.map (fun a => r - a) (Multiset.erase (roots (map (algebraMap K L) P)) r))\n[PROOFSTEP]\nreplace hP := (splits_id_iff_splits (algebraMap K L)).2 hP\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\ni : K \u2192+* L\ninst\u271d : Algebra K L\nP : K[X]\nr : L\nhr : r \u2208 roots (map (algebraMap K L) P)\nhmo : Monic (map (algebraMap K L) P)\nhP : Splits (RingHom.id L) (map (algebraMap K L) P)\n\u22a2 \u2191(aeval r) (\u2191derivative P) =\n Multiset.prod (Multiset.map (fun a => r - a) (Multiset.erase (roots (map (algebraMap K L) P)) r))\n[PROOFSTEP]\nrw [aeval_def, \u2190 eval_map, \u2190 derivative_map]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\ni : K \u2192+* L\ninst\u271d : Algebra K L\nP : K[X]\nr : L\nhr : r \u2208 roots (map (algebraMap K L) P)\nhmo : Monic (map (algebraMap K L) P)\nhP : Splits (RingHom.id L) (map (algebraMap K L) P)\n\u22a2 eval r (\u2191derivative (map (algebraMap K L) P)) =\n Multiset.prod (Multiset.map (fun a => r - a) (Multiset.erase (roots (map (algebraMap K L) P)) r))\n[PROOFSTEP]\nnth_rw 1 [eq_prod_roots_of_monic_of_splits_id hmo hP]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\ni : K \u2192+* L\ninst\u271d : Algebra K L\nP : K[X]\nr : L\nhr : r \u2208 roots (map (algebraMap K L) P)\nhmo : Monic (map (algebraMap K L) P)\nhP : Splits (RingHom.id L) (map (algebraMap K L) P)\n\u22a2 eval r (\u2191derivative (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots (map (algebraMap K L) P))))) =\n Multiset.prod (Multiset.map (fun a => r - a) (Multiset.erase (roots (map (algebraMap K L) P)) r))\n[PROOFSTEP]\nrw [eval_multiset_prod_X_sub_C_derivative hr]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n\u22a2 coeff P 0 = (-1) ^ natDegree P * Multiset.prod (roots P)\n[PROOFSTEP]\nnth_rw 1 [eq_prod_roots_of_monic_of_splits_id hmo hP]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n\u22a2 coeff (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots P))) 0 = (-1) ^ natDegree P * Multiset.prod (roots P)\n[PROOFSTEP]\nrw [coeff_zero_eq_eval_zero, eval_multiset_prod, Multiset.map_map]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n\u22a2 Multiset.prod (Multiset.map (eval 0 \u2218 fun a => X - \u2191C a) (roots P)) = (-1) ^ natDegree P * Multiset.prod (roots P)\n[PROOFSTEP]\nsimp_rw [Function.comp_apply, eval_sub, eval_X, zero_sub, eval_C]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n\u22a2 Multiset.prod (Multiset.map (fun x => -x) (roots P)) = (-1) ^ natDegree P * Multiset.prod (roots P)\n[PROOFSTEP]\nconv_lhs =>\n congr\n congr\n ext\n rw [neg_eq_neg_one_mul]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n| Multiset.prod (Multiset.map (fun x => -x) (roots P))\n[PROOFSTEP]\n congr\n congr\n ext\n rw [neg_eq_neg_one_mul]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n| Multiset.prod (Multiset.map (fun x => -x) (roots P))\n[PROOFSTEP]\n congr\n congr\n ext\n rw [neg_eq_neg_one_mul]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n| Multiset.prod (Multiset.map (fun x => -x) (roots P))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n| Multiset.map (fun x => -x) (roots P)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.f\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n| fun x => -x\ncase a.s\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n| roots P\n[PROOFSTEP]\next\n[GOAL]\ncase a.f.h\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\nx\u271d : K\n| -x\u271d\ncase a.s\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n| roots P\n[PROOFSTEP]\nrw [neg_eq_neg_one_mul]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n\u22a2 Multiset.prod (Multiset.map (fun x => -1 * x) (roots P)) = (-1) ^ natDegree P * Multiset.prod (roots P)\n[PROOFSTEP]\nsimp only [splits_iff_card_roots.1 hP, neg_mul, one_mul, Multiset.prod_map_neg]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n\u22a2 nextCoeff P = -Multiset.sum (roots P)\n[PROOFSTEP]\nnth_rw 1 [eq_prod_roots_of_monic_of_splits_id hmo hP]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n\u22a2 nextCoeff (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots P))) = -Multiset.sum (roots P)\n[PROOFSTEP]\nrw [Monic.nextCoeff_multiset_prod _ _ fun a ha => _]\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n\u22a2 Multiset.sum (Multiset.map (fun i => nextCoeff (X - \u2191C i)) (roots P)) = -Multiset.sum (roots P)\n[PROOFSTEP]\nsimp_rw [nextCoeff_X_sub_C, Multiset.sum_map_neg']\n[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nP : K[X]\nhmo : Monic P\nhP : Splits (RingHom.id K) P\n\u22a2 \u2200 (a : K), a \u2208 roots P \u2192 Monic (X - \u2191C a)\n[PROOFSTEP]\nsimp only [monic_X_sub_C, implies_true]\n", "meta": {"mathlib_filename": "Mathlib.Data.Polynomial.Splits", "llama_tokens": 23118, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929002541068, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.41130314442274113}} {"text": "[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhfi : iIndepFun (fun x => m\u03b2) f\nhij : i < j\n\u22a2 Indep (MeasurableSpace.comap (f j) m\u03b2) (\u2191(Filtration.natural f hf) i)\n[PROOFSTEP]\nsuffices\n Indep (\u2a06 k \u2208 ({ j } : Set \u03b9), MeasurableSpace.comap (f k) m\u03b2) (\u2a06 k \u2208 {k | k \u2264 i}, MeasurableSpace.comap (f k) m\u03b2) \u03bc by\n rwa [iSup_singleton] at this \n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhfi : iIndepFun (fun x => m\u03b2) f\nhij : i < j\nthis :\n Indep (\u2a06 (k : \u03b9) (_ : k \u2208 {j}), MeasurableSpace.comap (f k) m\u03b2)\n (\u2a06 (k : \u03b9) (_ : k \u2208 {k | k \u2264 i}), MeasurableSpace.comap (f k) m\u03b2)\n\u22a2 Indep (MeasurableSpace.comap (f j) m\u03b2) (\u2191(Filtration.natural f hf) i)\n[PROOFSTEP]\nrwa [iSup_singleton] at this \n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhfi : iIndepFun (fun x => m\u03b2) f\nhij : i < j\n\u22a2 Indep (\u2a06 (k : \u03b9) (_ : k \u2208 {j}), MeasurableSpace.comap (f k) m\u03b2)\n (\u2a06 (k : \u03b9) (_ : k \u2208 {k | k \u2264 i}), MeasurableSpace.comap (f k) m\u03b2)\n[PROOFSTEP]\nexact indep_iSup_of_disjoint (fun k => (hf k).measurable.comap_le) hfi (by simpa)\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhfi : iIndepFun (fun x => m\u03b2) f\nhij : i < j\n\u22a2 Disjoint {j} {k | k \u2264 i}\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (n : \u03b9), MeasurableSet (s n)\nhs : iIndepSet s\nhij : i < j\n\u22a2 \u03bc[Set.indicator (s j) fun x => 1|\u2191(filtrationOfSet hsm) i] =\u1d50[\u03bc] fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j))\n[PROOFSTEP]\nrw [Filtration.filtrationOfSet_eq_natural (\u03b2 := \u211d) hsm]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (n : \u03b9), MeasurableSet (s n)\nhs : iIndepSet s\nhij : i < j\n\u22a2 \u03bc[Set.indicator (s j) fun x =>\n 1|\u2191(Filtration.natural (fun i => Set.indicator (s i) fun x => 1)\n (_ : \u2200 (i : \u03b9), StronglyMeasurable (Set.indicator (s i) 1)))\n i] =\u1d50[\u03bc]\n fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j))\n[PROOFSTEP]\nrefine' (iIndepFun.condexp_natural_ae_eq_of_lt _ hs.iIndepFun_indicator hij).trans _\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (n : \u03b9), MeasurableSet (s n)\nhs : iIndepSet s\nhij : i < j\n\u22a2 (fun x => \u222b (x : \u03a9), Set.indicator (s j) (fun _\u03c9 => 1) x \u2202\u03bc) =\u1d50[\u03bc] fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j))\n[PROOFSTEP]\nsimp only [integral_indicator_const _ (hsm _), Algebra.id.smul_eq_mul, mul_one]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhsm : \u2200 (n : \u03b9), MeasurableSet (s n)\nhs : iIndepSet s\nhij : i < j\n\u22a2 (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j))) =\u1d50[\u03bc] fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\n\u22a2 \u2191\u2191\u03bc (limsup s atTop) = 1\n[PROOFSTEP]\nrw [measure_congr\n (eventuallyEq_set.2 (ae_mem_limsup_atTop_iff \u03bc <| measurableSet_filtrationOfSet' hsm) :\n (limsup s atTop : Set \u03a9) =\u1d50[\u03bc]\n {\u03c9 |\n Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[(s (k + 1)).indicator (1 : \u03a9 \u2192 \u211d)|filtrationOfSet hsm k]) \u03c9) atTop\n atTop})]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\n\u22a2 \u2191\u2191\u03bc\n {\u03c9 |\n Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n atTop} =\n 1\n[PROOFSTEP]\nsuffices\n {\u03c9 |\n Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[(s (k + 1)).indicator (1 : \u03a9 \u2192 \u211d)|filtrationOfSet hsm k]) \u03c9) atTop\n atTop} =\u1d50[\u03bc]\n Set.univ\n by rw [measure_congr this, measure_univ]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n {\u03c9 |\n Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n atTop} =\u1d50[\u03bc]\n Set.univ\n\u22a2 \u2191\u2191\u03bc\n {\u03c9 |\n Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n atTop} =\n 1\n[PROOFSTEP]\nrw [measure_congr this, measure_univ]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\n\u22a2 {\u03c9 |\n Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n atTop} =\u1d50[\u03bc]\n Set.univ\n[PROOFSTEP]\nhave : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 n, (\u03bc[(s (n + 1)).indicator (1 : \u03a9 \u2192 \u211d)|filtrationOfSet hsm n]) \u03c9 = _ :=\n ae_all_iff.2 fun n => hs.condexp_indicator_filtrationOfSet_ae_eq hsm n.lt_succ_self\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u22a2 {\u03c9 |\n Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n atTop} =\u1d50[\u03bc]\n Set.univ\n[PROOFSTEP]\nfilter_upwards [this] with \u03c9 h\u03c9\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\n\u22a2 setOf\n (fun \u03c9 =>\n Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n atTop)\n \u03c9 =\n Set.univ \u03c9\n[PROOFSTEP]\nrefine' eq_true (_ : Tendsto _ _ _)\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\n\u22a2 Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop atTop\n[PROOFSTEP]\nsimp_rw [h\u03c9]\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\n\u22a2 Tendsto (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) atTop atTop\n[PROOFSTEP]\nhave htends : Tendsto (fun n => \u2211 k in Finset.range n, \u03bc (s (k + 1))) atTop (\ud835\udcdd \u221e) :=\n by\n rw [\u2190 ENNReal.tsum_add_one_eq_top hs' (measure_ne_top _ _)]\n exact ENNReal.tendsto_nat_tsum _\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\n\u22a2 Tendsto (fun n => \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) atTop (\ud835\udcdd \u22a4)\n[PROOFSTEP]\nrw [\u2190 ENNReal.tsum_add_one_eq_top hs' (measure_ne_top _ _)]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\n\u22a2 Tendsto (fun n => \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) atTop (\ud835\udcdd (\u2211' (n : \u2115), \u2191\u2191\u03bc (s (n + 1))))\n[PROOFSTEP]\nexact ENNReal.tendsto_nat_tsum _\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : Tendsto (fun n => \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) atTop (\ud835\udcdd \u22a4)\n\u22a2 Tendsto (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) atTop atTop\n[PROOFSTEP]\nrw [ENNReal.tendsto_nhds_top_iff_nnreal] at htends \n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\n\u22a2 Tendsto (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) atTop atTop\n[PROOFSTEP]\nrefine' tendsto_atTop_atTop_of_monotone' _ _\n[GOAL]\ncase h.refine'_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\n\u22a2 Monotone fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))\n[PROOFSTEP]\nrefine' monotone_nat_of_le_succ fun n => _\n[GOAL]\ncase h.refine'_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nn : \u2115\n\u22a2 \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))) \u2264\n \u2211 x in Finset.range (n + 1), ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))\n[PROOFSTEP]\nrw [\u2190 sub_nonneg, Finset.sum_range_succ_sub_sum]\n[GOAL]\ncase h.refine'_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nn : \u2115\n\u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\n[PROOFSTEP]\nexact ENNReal.toReal_nonneg\n[GOAL]\ncase h.refine'_2\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\n\u22a2 \u00acBddAbove (Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))))\n[PROOFSTEP]\nrintro \u27e8B, hB\u27e9\n[GOAL]\ncase h.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : B \u2208 upperBounds (Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))))\n\u22a2 False\n[PROOFSTEP]\nrefine' not_eventually.2 (frequently_of_forall fun n => _) (htends B.toNNReal)\n[GOAL]\ncase h.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : B \u2208 upperBounds (Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1))))\nn : \u2115\n\u22a2 \u00ac\u2191(Real.toNNReal B) < \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))\n[PROOFSTEP]\nrw [mem_upperBounds] at hB \n[GOAL]\ncase h.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B\nn : \u2115\n\u22a2 \u00ac\u2191(Real.toNNReal B) < \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))\n[PROOFSTEP]\nspecialize hB (\u2211 k : \u2115 in Finset.range n, \u03bc (s (k + 1))).toReal _\n[GOAL]\ncase h.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B\nn : \u2115\n\u22a2 ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2208\n Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))\n[PROOFSTEP]\nrefine' \u27e8n, _\u27e9\n[GOAL]\ncase h.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B\nn : \u2115\n\u22a2 (fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) n =\n ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1)))\n[PROOFSTEP]\nrw [ENNReal.toReal_sum]\n[GOAL]\ncase h.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 (x : \u211d), (x \u2208 Set.range fun n => \u2211 x in Finset.range n, ENNReal.toReal (\u2191\u2191\u03bc (s (x + 1)))) \u2192 x \u2264 B\nn : \u2115\n\u22a2 \u2200 (a : \u2115), a \u2208 Finset.range n \u2192 \u2191\u2191\u03bc (s (a + 1)) \u2260 \u22a4\n[PROOFSTEP]\nexact fun _ _ => measure_ne_top _ _\n[GOAL]\ncase h.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2264 B\n\u22a2 \u00ac\u2191(Real.toNNReal B) < \u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))\n[PROOFSTEP]\nrw [not_lt, \u2190 ENNReal.toReal_le_toReal (ENNReal.sum_lt_top _).ne ENNReal.coe_ne_top]\n[GOAL]\ncase h.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2264 B\n\u22a2 ENNReal.toReal (\u2211 a in Finset.range n, \u2191\u2191\u03bc (s (a + 1))) \u2264 ENNReal.toReal \u2191(Real.toNNReal B)\n[PROOFSTEP]\nexact hB.trans (by simp)\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2264 B\n\u22a2 B \u2264 ENNReal.toReal \u2191(Real.toNNReal B)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s\nhs' : \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = \u22a4\nthis :\n \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n \u2200 (n : \u2115),\n (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[Set.indicator (s (n + 1)) 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = ENNReal.toReal (\u2191\u2191\u03bc (s (n + 1)))\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k in Finset.range a, \u2191\u2191\u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : ENNReal.toReal (\u2211 k in Finset.range n, \u2191\u2191\u03bc (s (k + 1))) \u2264 B\n\u22a2 \u2200 (a : \u2115), a \u2208 Finset.range n \u2192 \u2191\u2191\u03bc (s (a + 1)) \u2260 \u22a4\n[PROOFSTEP]\nexact fun _ _ => measure_ne_top _ _\n", "meta": {"mathlib_filename": "Mathlib.Probability.BorelCantelli", "llama_tokens": 14358, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432182679957, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.411287286570204}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\n\u22a2 (fun x => x * \u2191(normUnit x)) 0 = 0\n[PROOFSTEP]\nsimp only [normUnit_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\n\u22a2 0 * \u21911 = 0\n[PROOFSTEP]\nexact mul_one (0 : \u03b1)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\n\u22a2 ZeroHom.toFun { toFun := fun x => x * \u2191(normUnit x), map_zero' := (_ : (fun x => x * \u2191(normUnit x)) 0 = 0) } 1 = 1\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\n\u22a2 1 * \u2191(normUnit 1) = 1\n[PROOFSTEP]\nrw [normUnit_one, one_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\n\u22a2 \u21911 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\nx y : \u03b1\nhx : x = 0\n\u22a2 ZeroHom.toFun { toFun := fun x => x * \u2191(normUnit x), map_zero' := (_ : (fun x => x * \u2191(normUnit x)) 0 = 0) } (x * y) =\n ZeroHom.toFun { toFun := fun x => x * \u2191(normUnit x), map_zero' := (_ : (fun x => x * \u2191(normUnit x)) 0 = 0) } x *\n ZeroHom.toFun { toFun := fun x => x * \u2191(normUnit x), map_zero' := (_ : (fun x => x * \u2191(normUnit x)) 0 = 0) } y\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\nx y : \u03b1\nhx : x = 0\n\u22a2 x * y * \u2191(normUnit (x * y)) = x * \u2191(normUnit x) * (y * \u2191(normUnit y))\n[PROOFSTEP]\nrw [hx, zero_mul, zero_mul, zero_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\nx y : \u03b1\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 ZeroHom.toFun { toFun := fun x => x * \u2191(normUnit x), map_zero' := (_ : (fun x => x * \u2191(normUnit x)) 0 = 0) } (x * y) =\n ZeroHom.toFun { toFun := fun x => x * \u2191(normUnit x), map_zero' := (_ : (fun x => x * \u2191(normUnit x)) 0 = 0) } x *\n ZeroHom.toFun { toFun := fun x => x * \u2191(normUnit x), map_zero' := (_ : (fun x => x * \u2191(normUnit x)) 0 = 0) } y\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\nx y : \u03b1\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 x * y * \u2191(normUnit (x * y)) = x * \u2191(normUnit x) * (y * \u2191(normUnit y))\n[PROOFSTEP]\nrw [hy, mul_zero, zero_mul, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\nx y : \u03b1\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 ZeroHom.toFun { toFun := fun x => x * \u2191(normUnit x), map_zero' := (_ : (fun x => x * \u2191(normUnit x)) 0 = 0) } (x * y) =\n ZeroHom.toFun { toFun := fun x => x * \u2191(normUnit x), map_zero' := (_ : (fun x => x * \u2191(normUnit x)) 0 = 0) } x *\n ZeroHom.toFun { toFun := fun x => x * \u2191(normUnit x), map_zero' := (_ : (fun x => x * \u2191(normUnit x)) 0 = 0) } y\n[PROOFSTEP]\nsimp only [normUnit_mul hx hy, Units.val_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\nx y : \u03b1\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 x * y * (\u2191(normUnit x) * \u2191(normUnit y)) = x * \u2191(normUnit x) * (y * \u2191(normUnit y))\n[PROOFSTEP]\nsimp only [mul_assoc, mul_left_comm y]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\nu : \u03b1\u02e3\n\u22a2 \u2191normalize \u2191u = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\nx : \u03b1\n\u22a2 x = 0 \u2192 \u2191normalize x = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\n\u22a2 \u2191normalize 0 = 0\n[PROOFSTEP]\nexact normalize_zero\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\na : \u03b1\n\u22a2 normUnit (a * \u2191(normUnit a)) = 1\n[PROOFSTEP]\nnontriviality \u03b1 using Subsingleton.elim a 0\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\na : \u03b1\n\u271d : Nontrivial \u03b1\n\u22a2 normUnit (a * \u2191(normUnit a)) = 1\n[PROOFSTEP]\nobtain rfl | h := eq_or_ne a 0\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\n\u271d : Nontrivial \u03b1\n\u22a2 normUnit (0 * \u2191(normUnit 0)) = 1\n[PROOFSTEP]\nrw [normUnit_zero, zero_mul, normUnit_zero]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\na : \u03b1\n\u271d : Nontrivial \u03b1\nh : a \u2260 0\n\u22a2 normUnit (a * \u2191(normUnit a)) = 1\n[PROOFSTEP]\nrw [normUnit_mul h (Units.ne_zero _), normUnit_coe_units, mul_inv_eq_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\nx : \u03b1\n\u22a2 \u2191normalize (\u2191normalize x) = \u2191normalize x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\na b : \u03b1\nhab : a \u2223 b\nhba : b \u2223 a\n\u22a2 \u2191normalize a = \u2191normalize b\n[PROOFSTEP]\nnontriviality \u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\na b : \u03b1\nhab : a \u2223 b\nhba : b \u2223 a\n\u271d : Nontrivial \u03b1\n\u22a2 \u2191normalize a = \u2191normalize b\n[PROOFSTEP]\nrcases associated_of_dvd_dvd hab hba with \u27e8u, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\na : \u03b1\n\u271d : Nontrivial \u03b1\nu : \u03b1\u02e3\nhab : a \u2223 a * \u2191u\nhba : a * \u2191u \u2223 a\n\u22a2 \u2191normalize a = \u2191normalize (a * \u2191u)\n[PROOFSTEP]\nrefine' by_cases (by rintro rfl; simp only [zero_mul]) fun ha : a \u2260 0 => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\na : \u03b1\n\u271d : Nontrivial \u03b1\nu : \u03b1\u02e3\nhab : a \u2223 a * \u2191u\nhba : a * \u2191u \u2223 a\n\u22a2 a = 0 \u2192 \u2191normalize a = \u2191normalize (a * \u2191u)\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\n\u271d : Nontrivial \u03b1\nu : \u03b1\u02e3\nhab : 0 \u2223 0 * \u2191u\nhba : 0 * \u2191u \u2223 0\n\u22a2 \u2191normalize 0 = \u2191normalize (0 * \u2191u)\n[PROOFSTEP]\nsimp only [zero_mul]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\na : \u03b1\n\u271d : Nontrivial \u03b1\nu : \u03b1\u02e3\nhab : a \u2223 a * \u2191u\nhba : a * \u2191u \u2223 a\nha : a \u2260 0\n\u22a2 \u2191normalize a = \u2191normalize (a * \u2191u)\n[PROOFSTEP]\nsuffices a * \u2191(normUnit a) = a * \u2191u * \u2191(normUnit a) * \u2191u\u207b\u00b9 by\n simpa only [normalize_apply, mul_assoc, normUnit_mul ha u.ne_zero, normUnit_coe_units]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\na : \u03b1\n\u271d : Nontrivial \u03b1\nu : \u03b1\u02e3\nhab : a \u2223 a * \u2191u\nhba : a * \u2191u \u2223 a\nha : a \u2260 0\nthis : a * \u2191(normUnit a) = a * \u2191u * \u2191(normUnit a) * \u2191u\u207b\u00b9\n\u22a2 \u2191normalize a = \u2191normalize (a * \u2191u)\n[PROOFSTEP]\nsimpa only [normalize_apply, mul_assoc, normUnit_mul ha u.ne_zero, normUnit_coe_units]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\na : \u03b1\n\u271d : Nontrivial \u03b1\nu : \u03b1\u02e3\nhab : a \u2223 a * \u2191u\nhba : a * \u2191u \u2223 a\nha : a \u2260 0\n\u22a2 a * \u2191(normUnit a) = a * \u2191u * \u2191(normUnit a) * \u2191u\u207b\u00b9\n[PROOFSTEP]\ncalc\n a * \u2191(normUnit a) = a * \u2191(normUnit a) * \u2191u * \u2191u\u207b\u00b9 := (Units.mul_inv_cancel_right _ _).symm\n _ = a * \u2191u * \u2191(normUnit a) * \u2191u\u207b\u00b9 := by rw [mul_right_comm a]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\na : \u03b1\n\u271d : Nontrivial \u03b1\nu : \u03b1\u02e3\nhab : a \u2223 a * \u2191u\nhba : a * \u2191u \u2223 a\nha : a \u2260 0\n\u22a2 a * \u2191(normUnit a) * \u2191u * \u2191u\u207b\u00b9 = a * \u2191u * \u2191(normUnit a) * \u2191u\u207b\u00b9\n[PROOFSTEP]\nrw [mul_right_comm a]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\na b : Associates \u03b1\nx\u271d\u00b9 x\u271d : \u03b1\n\u22a2 Associates.out (Quotient.mk (Associated.setoid \u03b1) x\u271d\u00b9 * Quotient.mk (Associated.setoid \u03b1) x\u271d) =\n Associates.out (Quotient.mk (Associated.setoid \u03b1) x\u271d\u00b9) * Associates.out (Quotient.mk (Associated.setoid \u03b1) x\u271d)\n[PROOFSTEP]\nsimp only [Associates.quotient_mk_eq_mk, out_mk, mk_mul_mk, normalize.map_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\na : \u03b1\nb : Associates \u03b1\n\u22a2 \u2200 (a_1 : \u03b1),\n a \u2223 Associates.out (Quotient.mk (Associated.setoid \u03b1) a_1) \u2194 Associates.mk a \u2264 Quotient.mk (Associated.setoid \u03b1) a_1\n[PROOFSTEP]\nsimp [Associates.out_mk, Associates.quotient_mk_eq_mk, mk_le_mk_iff_dvd_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizationMonoid \u03b1\na : \u03b1\nb : Associates \u03b1\n\u22a2 \u2200 (a_1 : \u03b1),\n Associates.out (Quotient.mk (Associated.setoid \u03b1) a_1) \u2223 a \u2194 Quotient.mk (Associated.setoid \u03b1) a_1 \u2264 Associates.mk a\n[PROOFSTEP]\nsimp [Associates.out_mk, Associates.quotient_mk_eq_mk, mk_le_mk_iff_dvd_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nh : gcd a b = 0\n\u22a2 a = 0 \u2227 b = 0\n[PROOFSTEP]\nlet \u27e8ca, ha\u27e9 := gcd_dvd_left a b\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nh : gcd a b = 0\nca : \u03b1\nha : a = gcd a b * ca\n\u22a2 a = 0 \u2227 b = 0\n[PROOFSTEP]\nlet \u27e8cb, hb\u27e9 := gcd_dvd_right a b\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nh : gcd a b = 0\nca : \u03b1\nha : a = gcd a b * ca\ncb : \u03b1\nhb : b = gcd a b * cb\n\u22a2 a = 0 \u2227 b = 0\n[PROOFSTEP]\nrw [h, zero_mul] at ha hb \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nh : gcd a b = 0\nca : \u03b1\nha : a = 0\ncb : \u03b1\nhb : b = 0\n\u22a2 a = 0 \u2227 b = 0\n[PROOFSTEP]\nexact \u27e8ha, hb\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nx\u271d : a = 0 \u2227 b = 0\nha : a = 0\nhb : b = 0\n\u22a2 gcd a b = 0\n[PROOFSTEP]\nrw [ha, hb, \u2190 zero_dvd_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nx\u271d : a = 0 \u2227 b = 0\nha : a = 0\nhb : b = 0\n\u22a2 0 \u2223 gcd 0 0\n[PROOFSTEP]\napply dvd_gcd\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nx\u271d : a = 0 \u2227 b = 0\nha : a = 0\nhb : b = 0\n\u22a2 0 \u2223 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nx\u271d : a = 0 \u2227 b = 0\nha : a = 0\nhb : b = 0\n\u22a2 0 \u2223 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\n\u22a2 a = 0 \u2192 gcd (a * b) (a * c) = \u2191normalize a * gcd b c\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\nb c : \u03b1\n\u22a2 gcd (0 * b) (0 * c) = \u2191normalize 0 * gcd b c\n[PROOFSTEP]\nsimp only [zero_mul, gcd_zero_left, normalize_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nthis : gcd (a * b) (a * c) = \u2191normalize (a * gcd b c)\n\u22a2 gcd (a * b) (a * c) = \u2191normalize a * gcd b c\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\n\u22a2 Associated (gcd (a * b) (a * c)) (a * gcd b c)\n[PROOFSTEP]\nobtain rfl | ha := eq_or_ne a 0\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nb c : \u03b1\n\u22a2 Associated (gcd (0 * b) (0 * c)) (0 * gcd b c)\n[PROOFSTEP]\nsimp only [zero_mul, gcd_zero_left']\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\n\u22a2 Associated (gcd (a * b) (a * c)) (a * gcd b c)\n[PROOFSTEP]\nobtain \u27e8d, eq\u27e9 := dvd_gcd (dvd_mul_right a b) (dvd_mul_right a c)\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nd : \u03b1\neq : gcd (a * b) (a * c) = a * d\n\u22a2 Associated (gcd (a * b) (a * c)) (a * gcd b c)\n[PROOFSTEP]\napply associated_of_dvd_dvd\n[GOAL]\ncase inr.intro.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nd : \u03b1\neq : gcd (a * b) (a * c) = a * d\n\u22a2 gcd (a * b) (a * c) \u2223 a * gcd b c\n[PROOFSTEP]\nrw [eq]\n[GOAL]\ncase inr.intro.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nd : \u03b1\neq : gcd (a * b) (a * c) = a * d\n\u22a2 a * d \u2223 a * gcd b c\n[PROOFSTEP]\napply mul_dvd_mul_left\n[GOAL]\ncase inr.intro.hab.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nd : \u03b1\neq : gcd (a * b) (a * c) = a * d\n\u22a2 d \u2223 gcd b c\n[PROOFSTEP]\nexact\n dvd_gcd ((mul_dvd_mul_iff_left ha).1 <| eq \u25b8 gcd_dvd_left _ _) ((mul_dvd_mul_iff_left ha).1 <| eq \u25b8 gcd_dvd_right _ _)\n[GOAL]\ncase inr.intro.hba\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nd : \u03b1\neq : gcd (a * b) (a * c) = a * d\n\u22a2 a * gcd b c \u2223 gcd (a * b) (a * c)\n[PROOFSTEP]\nexact dvd_gcd (mul_dvd_mul_left a <| gcd_dvd_left _ _) (mul_dvd_mul_left a <| gcd_dvd_right _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\n\u22a2 gcd (b * a) (c * a) = gcd b c * \u2191normalize a\n[PROOFSTEP]\nsimp only [mul_comm, gcd_mul_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\n\u22a2 Associated (gcd (b * a) (c * a)) (gcd b c * a)\n[PROOFSTEP]\nsimp only [mul_comm, gcd_mul_left']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b : \u03b1\nh : \u2191normalize b = b\n\u22a2 gcd a b = b \u2194 b \u2223 a\n[PROOFSTEP]\nsimpa only [gcd_comm a b] using gcd_eq_left_iff b a h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nm n k : \u03b1\nH : k \u2223 m * n\n\u22a2 k \u2223 m * gcd k n\n[PROOFSTEP]\nrw [mul_comm] at H \u22a2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nm n k : \u03b1\nH : k \u2223 n * m\n\u22a2 k \u2223 gcd k n * m\n[PROOFSTEP]\nexact dvd_gcd_mul_of_dvd_mul H\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nm n k : \u03b1\nH : k \u2223 m * n\n\u22a2 \u2203 d\u2081 d\u2082, d\u2081 \u2223 m \u2227 d\u2082 \u2223 n \u2227 k = d\u2081 * d\u2082\n[PROOFSTEP]\nby_cases h0 : gcd k m = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nm n k : \u03b1\nH : k \u2223 m * n\nh0 : gcd k m = 0\n\u22a2 \u2203 d\u2081 d\u2082, d\u2081 \u2223 m \u2227 d\u2082 \u2223 n \u2227 k = d\u2081 * d\u2082\n[PROOFSTEP]\nrw [gcd_eq_zero_iff] at h0 \n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nm n k : \u03b1\nH : k \u2223 m * n\nh0 : k = 0 \u2227 m = 0\n\u22a2 \u2203 d\u2081 d\u2082, d\u2081 \u2223 m \u2227 d\u2082 \u2223 n \u2227 k = d\u2081 * d\u2082\n[PROOFSTEP]\nrcases h0 with \u27e8rfl, rfl\u27e9\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nn : \u03b1\nH : 0 \u2223 0 * n\n\u22a2 \u2203 d\u2081 d\u2082, d\u2081 \u2223 0 \u2227 d\u2082 \u2223 n \u2227 0 = d\u2081 * d\u2082\n[PROOFSTEP]\nrefine' \u27e80, n, dvd_refl 0, dvd_refl n, _\u27e9\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nn : \u03b1\nH : 0 \u2223 0 * n\n\u22a2 0 = 0 * n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nm n k : \u03b1\nH : k \u2223 m * n\nh0 : \u00acgcd k m = 0\n\u22a2 \u2203 d\u2081 d\u2082, d\u2081 \u2223 m \u2227 d\u2082 \u2223 n \u2227 k = d\u2081 * d\u2082\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := gcd_dvd_left k m\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nm n k : \u03b1\nH : k \u2223 m * n\nh0 : \u00acgcd k m = 0\na : \u03b1\nha : k = gcd k m * a\n\u22a2 \u2203 d\u2081 d\u2082, d\u2081 \u2223 m \u2227 d\u2082 \u2223 n \u2227 k = d\u2081 * d\u2082\n[PROOFSTEP]\nrefine' \u27e8gcd k m, a, gcd_dvd_right _ _, _, ha\u27e9\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nm n k : \u03b1\nH : k \u2223 m * n\nh0 : \u00acgcd k m = 0\na : \u03b1\nha : k = gcd k m * a\n\u22a2 a \u2223 n\n[PROOFSTEP]\nsuffices h : gcd k m * a \u2223 gcd k m * n\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nm n k : \u03b1\nH : k \u2223 m * n\nh0 : \u00acgcd k m = 0\na : \u03b1\nha : k = gcd k m * a\nh : gcd k m * a \u2223 gcd k m * n\n\u22a2 a \u2223 n\n[PROOFSTEP]\ncases' h with b hb\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nm n k : \u03b1\nH : k \u2223 m * n\nh0 : \u00acgcd k m = 0\na : \u03b1\nha : k = gcd k m * a\nb : \u03b1\nhb : gcd k m * n = gcd k m * a * b\n\u22a2 a \u2223 n\n[PROOFSTEP]\nuse b\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nm n k : \u03b1\nH : k \u2223 m * n\nh0 : \u00acgcd k m = 0\na : \u03b1\nha : k = gcd k m * a\nb : \u03b1\nhb : gcd k m * n = gcd k m * a * b\n\u22a2 n = a * b\n[PROOFSTEP]\nrw [mul_assoc] at hb \n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nm n k : \u03b1\nH : k \u2223 m * n\nh0 : \u00acgcd k m = 0\na : \u03b1\nha : k = gcd k m * a\nb : \u03b1\nhb : gcd k m * n = gcd k m * (a * b)\n\u22a2 n = a * b\n[PROOFSTEP]\napply mul_left_cancel\u2080 h0 hb\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nm n k : \u03b1\nH : k \u2223 m * n\nh0 : \u00acgcd k m = 0\na : \u03b1\nha : k = gcd k m * a\n\u22a2 gcd k m * a \u2223 gcd k m * n\n[PROOFSTEP]\nrw [\u2190 ha]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nm n k : \u03b1\nH : k \u2223 m * n\nh0 : \u00acgcd k m = 0\na : \u03b1\nha : k = gcd k m * a\n\u22a2 k \u2223 gcd k m * n\n[PROOFSTEP]\nexact dvd_gcd_mul_of_dvd_mul H\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nk m n : \u03b1\n\u22a2 k \u2223 m * n \u2194 \u2203 d\u2081 d\u2082, d\u2081 \u2223 m \u2227 d\u2082 \u2223 n \u2227 k = d\u2081 * d\u2082\n[PROOFSTEP]\nrefine' \u27e8exists_dvd_and_dvd_of_dvd_mul, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nk m n : \u03b1\n\u22a2 (\u2203 d\u2081 d\u2082, d\u2081 \u2223 m \u2227 d\u2082 \u2223 n \u2227 k = d\u2081 * d\u2082) \u2192 k \u2223 m * n\n[PROOFSTEP]\nrintro \u27e8d\u2081, d\u2082, hy, hz, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nm n d\u2081 d\u2082 : \u03b1\nhy : d\u2081 \u2223 m\nhz : d\u2082 \u2223 n\n\u22a2 d\u2081 * d\u2082 \u2223 m * n\n[PROOFSTEP]\nexact mul_dvd_mul hy hz\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nk m n : \u03b1\n\u22a2 gcd k (m * n) \u2223 gcd k m * gcd k n\n[PROOFSTEP]\nobtain \u27e8m', n', hm', hn', h\u27e9 := exists_dvd_and_dvd_of_dvd_mul (gcd_dvd_right k (m * n))\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nk m n m' n' : \u03b1\nhm' : m' \u2223 m\nhn' : n' \u2223 n\nh : gcd k (m * n) = m' * n'\n\u22a2 gcd k (m * n) \u2223 gcd k m * gcd k n\n[PROOFSTEP]\nreplace h : gcd k (m * n) = m' * n' := h\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nk m n m' n' : \u03b1\nhm' : m' \u2223 m\nhn' : n' \u2223 n\nh : gcd k (m * n) = m' * n'\n\u22a2 gcd k (m * n) \u2223 gcd k m * gcd k n\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nk m n m' n' : \u03b1\nhm' : m' \u2223 m\nhn' : n' \u2223 n\nh : gcd k (m * n) = m' * n'\n\u22a2 m' * n' \u2223 gcd k m * gcd k n\n[PROOFSTEP]\nhave hm'n' : m' * n' \u2223 k := h \u25b8 gcd_dvd_left _ _\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nk m n m' n' : \u03b1\nhm' : m' \u2223 m\nhn' : n' \u2223 n\nh : gcd k (m * n) = m' * n'\nhm'n' : m' * n' \u2223 k\n\u22a2 m' * n' \u2223 gcd k m * gcd k n\n[PROOFSTEP]\napply mul_dvd_mul\n[GOAL]\ncase intro.intro.intro.intro.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nk m n m' n' : \u03b1\nhm' : m' \u2223 m\nhn' : n' \u2223 n\nh : gcd k (m * n) = m' * n'\nhm'n' : m' * n' \u2223 k\n\u22a2 m' \u2223 gcd k m\n[PROOFSTEP]\nhave hm'k : m' \u2223 k := (dvd_mul_right m' n').trans hm'n'\n[GOAL]\ncase intro.intro.intro.intro.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nk m n m' n' : \u03b1\nhm' : m' \u2223 m\nhn' : n' \u2223 n\nh : gcd k (m * n) = m' * n'\nhm'n' : m' * n' \u2223 k\nhm'k : m' \u2223 k\n\u22a2 m' \u2223 gcd k m\n[PROOFSTEP]\nexact dvd_gcd hm'k hm'\n[GOAL]\ncase intro.intro.intro.intro.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nk m n m' n' : \u03b1\nhm' : m' \u2223 m\nhn' : n' \u2223 n\nh : gcd k (m * n) = m' * n'\nhm'n' : m' * n' \u2223 k\n\u22a2 n' \u2223 gcd k n\n[PROOFSTEP]\nhave hn'k : n' \u2223 k := (dvd_mul_left n' m').trans hm'n'\n[GOAL]\ncase intro.intro.intro.intro.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nk m n m' n' : \u03b1\nhm' : m' \u2223 m\nhn' : n' \u2223 n\nh : gcd k (m * n) = m' * n'\nhm'n' : m' * n' \u2223 k\nhn'k : n' \u2223 k\n\u22a2 n' \u2223 gcd k n\n[PROOFSTEP]\nexact dvd_gcd hn'k hn'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nk : \u2115\n\u22a2 gcd a (b ^ k) \u2223 gcd a b ^ k\n[PROOFSTEP]\nby_cases hg : gcd a b = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nk : \u2115\nhg : gcd a b = 0\n\u22a2 gcd a (b ^ k) \u2223 gcd a b ^ k\n[PROOFSTEP]\nrw [gcd_eq_zero_iff] at hg \n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nk : \u2115\nhg : a = 0 \u2227 b = 0\n\u22a2 gcd a (b ^ k) \u2223 gcd a b ^ k\n[PROOFSTEP]\nrcases hg with \u27e8rfl, rfl\u27e9\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nk : \u2115\n\u22a2 gcd 0 (0 ^ k) \u2223 gcd 0 0 ^ k\n[PROOFSTEP]\nexact (gcd_zero_left' (0 ^ k : \u03b1)).dvd.trans (pow_dvd_pow_of_dvd (gcd_zero_left' (0 : \u03b1)).symm.dvd _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nk : \u2115\nhg : \u00acgcd a b = 0\n\u22a2 gcd a (b ^ k) \u2223 gcd a b ^ k\n[PROOFSTEP]\ninduction' k with k hk\n[GOAL]\ncase neg.zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nhg : \u00acgcd a b = 0\n\u22a2 gcd a (b ^ Nat.zero) \u2223 gcd a b ^ Nat.zero\n[PROOFSTEP]\nrw [pow_zero, pow_zero]\n[GOAL]\ncase neg.zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nhg : \u00acgcd a b = 0\n\u22a2 gcd a 1 \u2223 1\n[PROOFSTEP]\nexact (gcd_one_right' a).dvd\n[GOAL]\ncase neg.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nhg : \u00acgcd a b = 0\nk : \u2115\nhk : gcd a (b ^ k) \u2223 gcd a b ^ k\n\u22a2 gcd a (b ^ Nat.succ k) \u2223 gcd a b ^ Nat.succ k\n[PROOFSTEP]\nrw [pow_succ, pow_succ]\n[GOAL]\ncase neg.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nhg : \u00acgcd a b = 0\nk : \u2115\nhk : gcd a (b ^ k) \u2223 gcd a b ^ k\n\u22a2 gcd a (b * b ^ k) \u2223 gcd a b * gcd a b ^ k\n[PROOFSTEP]\ntrans gcd a b * gcd a (b ^ k)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nhg : \u00acgcd a b = 0\nk : \u2115\nhk : gcd a (b ^ k) \u2223 gcd a b ^ k\n\u22a2 gcd a (b * b ^ k) \u2223 gcd a b * gcd a (b ^ k)\n[PROOFSTEP]\nexact gcd_mul_dvd_mul_gcd a b (b ^ k)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nhg : \u00acgcd a b = 0\nk : \u2115\nhk : gcd a (b ^ k) \u2223 gcd a b ^ k\n\u22a2 gcd a b * gcd a (b ^ k) \u2223 gcd a b * gcd a b ^ k\n[PROOFSTEP]\nexact (mul_dvd_mul_iff_left hg).mpr hk\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\n\u22a2 d\u2081 ^ k \u2260 0 \u2227 d\u2081 ^ k \u2223 a\n[PROOFSTEP]\nhave h1 : IsUnit (gcd (d\u2081 ^ k) b) := by\n apply isUnit_of_dvd_one\n trans gcd d\u2081 b ^ k\n \u00b7 exact gcd_pow_left_dvd_pow_gcd\n \u00b7 apply IsUnit.dvd\n apply IsUnit.pow\n apply isUnit_of_dvd_one\n apply dvd_trans _ hab.dvd\n apply gcd_dvd_gcd hd\u2081 (dvd_refl b)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\n\u22a2 IsUnit (gcd (d\u2081 ^ k) b)\n[PROOFSTEP]\napply isUnit_of_dvd_one\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\n\u22a2 gcd (d\u2081 ^ k) b \u2223 1\n[PROOFSTEP]\ntrans gcd d\u2081 b ^ k\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\n\u22a2 gcd (d\u2081 ^ k) b \u2223 gcd d\u2081 b ^ k\n[PROOFSTEP]\nexact gcd_pow_left_dvd_pow_gcd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\n\u22a2 gcd d\u2081 b ^ k \u2223 1\n[PROOFSTEP]\napply IsUnit.dvd\n[GOAL]\ncase hu\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\n\u22a2 IsUnit (gcd d\u2081 b ^ k)\n[PROOFSTEP]\napply IsUnit.pow\n[GOAL]\ncase hu.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\n\u22a2 IsUnit (gcd d\u2081 b)\n[PROOFSTEP]\napply isUnit_of_dvd_one\n[GOAL]\ncase hu.a.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\n\u22a2 gcd d\u2081 b \u2223 1\n[PROOFSTEP]\napply dvd_trans _ hab.dvd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\n\u22a2 gcd d\u2081 b \u2223 gcd a b\n[PROOFSTEP]\napply gcd_dvd_gcd hd\u2081 (dvd_refl b)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\nh1 : IsUnit (gcd (d\u2081 ^ k) b)\n\u22a2 d\u2081 ^ k \u2260 0 \u2227 d\u2081 ^ k \u2223 a\n[PROOFSTEP]\nhave h2 : d\u2081 ^ k \u2223 a * b := by\n use d\u2082 ^ k\n rw [h, hc]\n exact mul_pow d\u2081 d\u2082 k\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\nh1 : IsUnit (gcd (d\u2081 ^ k) b)\n\u22a2 d\u2081 ^ k \u2223 a * b\n[PROOFSTEP]\nuse d\u2082 ^ k\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\nh1 : IsUnit (gcd (d\u2081 ^ k) b)\n\u22a2 a * b = d\u2081 ^ k * d\u2082 ^ k\n[PROOFSTEP]\nrw [h, hc]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\nh1 : IsUnit (gcd (d\u2081 ^ k) b)\n\u22a2 (d\u2081 * d\u2082) ^ k = d\u2081 ^ k * d\u2082 ^ k\n[PROOFSTEP]\nexact mul_pow d\u2081 d\u2082 k\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\nh1 : IsUnit (gcd (d\u2081 ^ k) b)\nh2 : d\u2081 ^ k \u2223 a * b\n\u22a2 d\u2081 ^ k \u2260 0 \u2227 d\u2081 ^ k \u2223 a\n[PROOFSTEP]\nrw [mul_comm] at h2 \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\nh1 : IsUnit (gcd (d\u2081 ^ k) b)\nh2 : d\u2081 ^ k \u2223 b * a\n\u22a2 d\u2081 ^ k \u2260 0 \u2227 d\u2081 ^ k \u2223 a\n[PROOFSTEP]\nhave h3 : d\u2081 ^ k \u2223 a := by\n apply (dvd_gcd_mul_of_dvd_mul h2).trans\n rw [IsUnit.mul_left_dvd _ _ _ h1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\nh1 : IsUnit (gcd (d\u2081 ^ k) b)\nh2 : d\u2081 ^ k \u2223 b * a\n\u22a2 d\u2081 ^ k \u2223 a\n[PROOFSTEP]\napply (dvd_gcd_mul_of_dvd_mul h2).trans\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\nh1 : IsUnit (gcd (d\u2081 ^ k) b)\nh2 : d\u2081 ^ k \u2223 b * a\n\u22a2 gcd (d\u2081 ^ k) b * a \u2223 a\n[PROOFSTEP]\nrw [IsUnit.mul_left_dvd _ _ _ h1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\nh1 : IsUnit (gcd (d\u2081 ^ k) b)\nh2 : d\u2081 ^ k \u2223 b * a\nh3 : d\u2081 ^ k \u2223 a\n\u22a2 d\u2081 ^ k \u2260 0 \u2227 d\u2081 ^ k \u2223 a\n[PROOFSTEP]\nhave h4 : d\u2081 ^ k \u2260 0 := by\n intro hdk\n rw [hdk] at h3 \n apply absurd (zero_dvd_iff.mp h3) ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\nh1 : IsUnit (gcd (d\u2081 ^ k) b)\nh2 : d\u2081 ^ k \u2223 b * a\nh3 : d\u2081 ^ k \u2223 a\n\u22a2 d\u2081 ^ k \u2260 0\n[PROOFSTEP]\nintro hdk\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\nh1 : IsUnit (gcd (d\u2081 ^ k) b)\nh2 : d\u2081 ^ k \u2223 b * a\nh3 : d\u2081 ^ k \u2223 a\nhdk : d\u2081 ^ k = 0\n\u22a2 False\n[PROOFSTEP]\nrw [hdk] at h3 \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\nh1 : IsUnit (gcd (d\u2081 ^ k) b)\nh2 : d\u2081 ^ k \u2223 b * a\nh3 : 0 \u2223 a\nhdk : d\u2081 ^ k = 0\n\u22a2 False\n[PROOFSTEP]\napply absurd (zero_dvd_iff.mp h3) ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c d\u2081 d\u2082 : \u03b1\nha : a \u2260 0\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nhc : c = d\u2081 * d\u2082\nhd\u2081 : d\u2081 \u2223 a\nh1 : IsUnit (gcd (d\u2081 ^ k) b)\nh2 : d\u2081 ^ k \u2223 b * a\nh3 : d\u2081 ^ k \u2223 a\nh4 : d\u2081 ^ k \u2260 0\n\u22a2 d\u2081 ^ k \u2260 0 \u2227 d\u2081 ^ k \u2223 a\n[PROOFSTEP]\nexact \u27e8h4, h3\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\n\u22a2 \u2203 d, Associated (d ^ k) a\n[PROOFSTEP]\ncases subsingleton_or_nontrivial \u03b1\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Subsingleton \u03b1\n\u22a2 \u2203 d, Associated (d ^ k) a\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Subsingleton \u03b1\n\u22a2 Associated (0 ^ k) a\n[PROOFSTEP]\nrw [Subsingleton.elim a (0 ^ k)]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\n\u22a2 \u2203 d, Associated (d ^ k) a\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : a = 0\n\u22a2 \u2203 d, Associated (d ^ k) a\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : a = 0\n\u22a2 Associated (0 ^ k) a\n[PROOFSTEP]\nrw [ha]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : a = 0\n\u22a2 Associated (0 ^ k) 0\n[PROOFSTEP]\nobtain rfl | hk := k.eq_zero_or_pos\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nh\u271d : Nontrivial \u03b1\nha : a = 0\nh : a * b = c ^ 0\n\u22a2 Associated (0 ^ 0) 0\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h.inl.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nh\u271d : Nontrivial \u03b1\nha : a = 0\nh : a * b = c ^ 0\n\u22a2 False\n[PROOFSTEP]\nrevert h\n[GOAL]\ncase h.inl.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nh\u271d : Nontrivial \u03b1\nha : a = 0\n\u22a2 a * b = c ^ 0 \u2192 False\n[PROOFSTEP]\nrw [ha, zero_mul, pow_zero]\n[GOAL]\ncase h.inl.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nh\u271d : Nontrivial \u03b1\nha : a = 0\n\u22a2 0 = 1 \u2192 False\n[PROOFSTEP]\napply zero_ne_one\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : a = 0\nhk : k > 0\n\u22a2 Associated (0 ^ k) 0\n[PROOFSTEP]\nrw [zero_pow hk]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\n\u22a2 \u2203 d, Associated (d ^ k) a\n[PROOFSTEP]\nby_cases hb : b = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : b = 0\n\u22a2 \u2203 d, Associated (d ^ k) a\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : b = 0\n\u22a2 Associated (1 ^ k) a\n[PROOFSTEP]\nrw [one_pow]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : b = 0\n\u22a2 Associated 1 a\n[PROOFSTEP]\napply (associated_one_iff_isUnit.mpr hab).symm.trans\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : b = 0\n\u22a2 Associated (gcd a b) a\n[PROOFSTEP]\nrw [hb]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : b = 0\n\u22a2 Associated (gcd a 0) a\n[PROOFSTEP]\nexact gcd_zero_right' a\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\n\u22a2 \u2203 d, Associated (d ^ k) a\n[PROOFSTEP]\nobtain rfl | hk := k.eq_zero_or_pos\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nh : a * b = c ^ 0\n\u22a2 \u2203 d, Associated (d ^ 0) a\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nh : a * b = c ^ 0\n\u22a2 Associated (1 ^ 0) a\n[PROOFSTEP]\nrw [pow_zero] at h \u22a2\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nh : a * b = 1\n\u22a2 Associated 1 a\n[PROOFSTEP]\nuse Units.mkOfMulEqOne _ _ h\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nh : a * b = 1\n\u22a2 1 * \u2191(Units.mkOfMulEqOne a b h) = a\n[PROOFSTEP]\nrw [Units.val_mkOfMulEqOne, one_mul]\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\n\u22a2 \u2203 d, Associated (d ^ k) a\n[PROOFSTEP]\nhave hc : c \u2223 a * b := by\n rw [h]\n exact dvd_pow_self _ hk.ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\n\u22a2 c \u2223 a * b\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\n\u22a2 c \u2223 c ^ k\n[PROOFSTEP]\nexact dvd_pow_self _ hk.ne'\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\nhc : c \u2223 a * b\n\u22a2 \u2203 d, Associated (d ^ k) a\n[PROOFSTEP]\nobtain \u27e8d\u2081, d\u2082, hd\u2081, hd\u2082, hc\u27e9 := exists_dvd_and_dvd_of_dvd_mul hc\n[GOAL]\ncase neg.inr.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\nhc\u271d : c \u2223 a * b\nd\u2081 d\u2082 : \u03b1\nhd\u2081 : d\u2081 \u2223 a\nhd\u2082 : d\u2082 \u2223 b\nhc : c = d\u2081 * d\u2082\n\u22a2 \u2203 d, Associated (d ^ k) a\n[PROOFSTEP]\nuse d\u2081\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\nhc\u271d : c \u2223 a * b\nd\u2081 d\u2082 : \u03b1\nhd\u2081 : d\u2081 \u2223 a\nhd\u2082 : d\u2082 \u2223 b\nhc : c = d\u2081 * d\u2082\n\u22a2 Associated (d\u2081 ^ k) a\n[PROOFSTEP]\nobtain \u27e8h0\u2081, \u27e8a', ha'\u27e9\u27e9 := pow_dvd_of_mul_eq_pow ha hab h hc hd\u2081\n[GOAL]\ncase h.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : a * b = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\nhc\u271d : c \u2223 a * b\nd\u2081 d\u2082 : \u03b1\nhd\u2081 : d\u2081 \u2223 a\nhd\u2082 : d\u2082 \u2223 b\nhc : c = d\u2081 * d\u2082\nh0\u2081 : d\u2081 ^ k \u2260 0\na' : \u03b1\nha' : a = d\u2081 ^ k * a'\n\u22a2 Associated (d\u2081 ^ k) a\n[PROOFSTEP]\nrw [mul_comm] at h hc \n[GOAL]\ncase h.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd a b)\nk : \u2115\nh : b * a = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\nhc\u271d : c \u2223 a * b\nd\u2081 d\u2082 : \u03b1\nhd\u2081 : d\u2081 \u2223 a\nhd\u2082 : d\u2082 \u2223 b\nhc : c = d\u2082 * d\u2081\nh0\u2081 : d\u2081 ^ k \u2260 0\na' : \u03b1\nha' : a = d\u2081 ^ k * a'\n\u22a2 Associated (d\u2081 ^ k) a\n[PROOFSTEP]\nrw [(gcd_comm' a b).isUnit_iff] at hab \n[GOAL]\ncase h.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd b a)\nk : \u2115\nh : b * a = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\nhc\u271d : c \u2223 a * b\nd\u2081 d\u2082 : \u03b1\nhd\u2081 : d\u2081 \u2223 a\nhd\u2082 : d\u2082 \u2223 b\nhc : c = d\u2082 * d\u2081\nh0\u2081 : d\u2081 ^ k \u2260 0\na' : \u03b1\nha' : a = d\u2081 ^ k * a'\n\u22a2 Associated (d\u2081 ^ k) a\n[PROOFSTEP]\nobtain \u27e8h0\u2082, \u27e8b', hb'\u27e9\u27e9 := pow_dvd_of_mul_eq_pow hb hab h hc hd\u2082\n[GOAL]\ncase h.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd b a)\nk : \u2115\nh : b * a = c ^ k\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\nhc\u271d : c \u2223 a * b\nd\u2081 d\u2082 : \u03b1\nhd\u2081 : d\u2081 \u2223 a\nhd\u2082 : d\u2082 \u2223 b\nhc : c = d\u2082 * d\u2081\nh0\u2081 : d\u2081 ^ k \u2260 0\na' : \u03b1\nha' : a = d\u2081 ^ k * a'\nh0\u2082 : d\u2082 ^ k \u2260 0\nb' : \u03b1\nhb' : b = d\u2082 ^ k * b'\n\u22a2 Associated (d\u2081 ^ k) a\n[PROOFSTEP]\nrw [ha', hb', hc, mul_pow] at h \n[GOAL]\ncase h.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd b a)\nk : \u2115\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\nhc\u271d : c \u2223 a * b\nd\u2081 d\u2082 : \u03b1\nhd\u2081 : d\u2081 \u2223 a\nhd\u2082 : d\u2082 \u2223 b\nhc : c = d\u2082 * d\u2081\nh0\u2081 : d\u2081 ^ k \u2260 0\na' : \u03b1\nha' : a = d\u2081 ^ k * a'\nh0\u2082 : d\u2082 ^ k \u2260 0\nb' : \u03b1\nh : d\u2082 ^ k * b' * (d\u2081 ^ k * a') = d\u2082 ^ k * d\u2081 ^ k\nhb' : b = d\u2082 ^ k * b'\n\u22a2 Associated (d\u2081 ^ k) a\n[PROOFSTEP]\nhave h' : a' * b' = 1 := by\n apply (mul_right_inj' h0\u2081).mp\n rw [mul_one]\n apply (mul_right_inj' h0\u2082).mp\n rw [\u2190 h]\n rw [mul_assoc, mul_comm a', \u2190 mul_assoc _ b', \u2190 mul_assoc b', mul_comm b']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd b a)\nk : \u2115\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\nhc\u271d : c \u2223 a * b\nd\u2081 d\u2082 : \u03b1\nhd\u2081 : d\u2081 \u2223 a\nhd\u2082 : d\u2082 \u2223 b\nhc : c = d\u2082 * d\u2081\nh0\u2081 : d\u2081 ^ k \u2260 0\na' : \u03b1\nha' : a = d\u2081 ^ k * a'\nh0\u2082 : d\u2082 ^ k \u2260 0\nb' : \u03b1\nh : d\u2082 ^ k * b' * (d\u2081 ^ k * a') = d\u2082 ^ k * d\u2081 ^ k\nhb' : b = d\u2082 ^ k * b'\n\u22a2 a' * b' = 1\n[PROOFSTEP]\napply (mul_right_inj' h0\u2081).mp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd b a)\nk : \u2115\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\nhc\u271d : c \u2223 a * b\nd\u2081 d\u2082 : \u03b1\nhd\u2081 : d\u2081 \u2223 a\nhd\u2082 : d\u2082 \u2223 b\nhc : c = d\u2082 * d\u2081\nh0\u2081 : d\u2081 ^ k \u2260 0\na' : \u03b1\nha' : a = d\u2081 ^ k * a'\nh0\u2082 : d\u2082 ^ k \u2260 0\nb' : \u03b1\nh : d\u2082 ^ k * b' * (d\u2081 ^ k * a') = d\u2082 ^ k * d\u2081 ^ k\nhb' : b = d\u2082 ^ k * b'\n\u22a2 d\u2081 ^ k * (a' * b') = d\u2081 ^ k * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd b a)\nk : \u2115\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\nhc\u271d : c \u2223 a * b\nd\u2081 d\u2082 : \u03b1\nhd\u2081 : d\u2081 \u2223 a\nhd\u2082 : d\u2082 \u2223 b\nhc : c = d\u2082 * d\u2081\nh0\u2081 : d\u2081 ^ k \u2260 0\na' : \u03b1\nha' : a = d\u2081 ^ k * a'\nh0\u2082 : d\u2082 ^ k \u2260 0\nb' : \u03b1\nh : d\u2082 ^ k * b' * (d\u2081 ^ k * a') = d\u2082 ^ k * d\u2081 ^ k\nhb' : b = d\u2082 ^ k * b'\n\u22a2 d\u2081 ^ k * (a' * b') = d\u2081 ^ k\n[PROOFSTEP]\napply (mul_right_inj' h0\u2082).mp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd b a)\nk : \u2115\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\nhc\u271d : c \u2223 a * b\nd\u2081 d\u2082 : \u03b1\nhd\u2081 : d\u2081 \u2223 a\nhd\u2082 : d\u2082 \u2223 b\nhc : c = d\u2082 * d\u2081\nh0\u2081 : d\u2081 ^ k \u2260 0\na' : \u03b1\nha' : a = d\u2081 ^ k * a'\nh0\u2082 : d\u2082 ^ k \u2260 0\nb' : \u03b1\nh : d\u2082 ^ k * b' * (d\u2081 ^ k * a') = d\u2082 ^ k * d\u2081 ^ k\nhb' : b = d\u2082 ^ k * b'\n\u22a2 d\u2082 ^ k * (d\u2081 ^ k * (a' * b')) = d\u2082 ^ k * d\u2081 ^ k\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd b a)\nk : \u2115\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\nhc\u271d : c \u2223 a * b\nd\u2081 d\u2082 : \u03b1\nhd\u2081 : d\u2081 \u2223 a\nhd\u2082 : d\u2082 \u2223 b\nhc : c = d\u2082 * d\u2081\nh0\u2081 : d\u2081 ^ k \u2260 0\na' : \u03b1\nha' : a = d\u2081 ^ k * a'\nh0\u2082 : d\u2082 ^ k \u2260 0\nb' : \u03b1\nh : d\u2082 ^ k * b' * (d\u2081 ^ k * a') = d\u2082 ^ k * d\u2081 ^ k\nhb' : b = d\u2082 ^ k * b'\n\u22a2 d\u2082 ^ k * (d\u2081 ^ k * (a' * b')) = d\u2082 ^ k * b' * (d\u2081 ^ k * a')\n[PROOFSTEP]\nrw [mul_assoc, mul_comm a', \u2190 mul_assoc _ b', \u2190 mul_assoc b', mul_comm b']\n[GOAL]\ncase h.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd b a)\nk : \u2115\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\nhc\u271d : c \u2223 a * b\nd\u2081 d\u2082 : \u03b1\nhd\u2081 : d\u2081 \u2223 a\nhd\u2082 : d\u2082 \u2223 b\nhc : c = d\u2082 * d\u2081\nh0\u2081 : d\u2081 ^ k \u2260 0\na' : \u03b1\nha' : a = d\u2081 ^ k * a'\nh0\u2082 : d\u2082 ^ k \u2260 0\nb' : \u03b1\nh : d\u2082 ^ k * b' * (d\u2081 ^ k * a') = d\u2082 ^ k * d\u2081 ^ k\nhb' : b = d\u2082 ^ k * b'\nh' : a' * b' = 1\n\u22a2 Associated (d\u2081 ^ k) a\n[PROOFSTEP]\nuse Units.mkOfMulEqOne _ _ h'\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nhab : IsUnit (gcd b a)\nk : \u2115\nh\u271d : Nontrivial \u03b1\nha : \u00aca = 0\nhb : \u00acb = 0\nhk : k > 0\nhc\u271d : c \u2223 a * b\nd\u2081 d\u2082 : \u03b1\nhd\u2081 : d\u2081 \u2223 a\nhd\u2082 : d\u2082 \u2223 b\nhc : c = d\u2082 * d\u2081\nh0\u2081 : d\u2081 ^ k \u2260 0\na' : \u03b1\nha' : a = d\u2081 ^ k * a'\nh0\u2082 : d\u2082 ^ k \u2260 0\nb' : \u03b1\nh : d\u2082 ^ k * b' * (d\u2081 ^ k * a') = d\u2082 ^ k * d\u2081 ^ k\nhb' : b = d\u2082 ^ k * b'\nh' : a' * b' = 1\n\u22a2 d\u2081 ^ k * \u2191(Units.mkOfMulEqOne a' b' h') = a\n[PROOFSTEP]\nrw [Units.val_mkOfMulEqOne, ha']\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx y x' y' : \u03b1\nex : x = gcd x y * x'\ney : y = gcd x y * y'\nh : gcd x y \u2260 0\n\u22a2 IsUnit (gcd x' y')\n[PROOFSTEP]\nrw [\u2190 associated_one_iff_isUnit]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx y x' y' : \u03b1\nex : x = gcd x y * x'\ney : y = gcd x y * y'\nh : gcd x y \u2260 0\n\u22a2 Associated (gcd x' y') 1\n[PROOFSTEP]\nrefine' Associated.of_mul_left _ (Associated.refl <| gcd x y) h\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx y x' y' : \u03b1\nex : x = gcd x y * x'\ney : y = gcd x y * y'\nh : gcd x y \u2260 0\n\u22a2 Associated (gcd x y * gcd x' y') (gcd x y * 1)\n[PROOFSTEP]\nconvert (gcd_mul_left' (gcd x y) x' y').symm using 1\n[GOAL]\ncase h.e'_4\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx y x' y' : \u03b1\nex : x = gcd x y * x'\ney : y = gcd x y * y'\nh : gcd x y \u2260 0\n\u22a2 gcd x y * 1 = gcd (gcd x y * x') (gcd x y * y')\n[PROOFSTEP]\nrw [\u2190 ex, \u2190 ey, mul_one]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx y : \u03b1\n\u22a2 \u2203 x' y', x = gcd x y * x' \u2227 y = gcd x y * y' \u2227 IsUnit (gcd x' y')\n[PROOFSTEP]\nby_cases h : gcd x y = 0\n[GOAL]\ncase pos\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx y : \u03b1\nh : gcd x y = 0\n\u22a2 \u2203 x' y', x = gcd x y * x' \u2227 y = gcd x y * y' \u2227 IsUnit (gcd x' y')\n[PROOFSTEP]\nobtain \u27e8rfl, rfl\u27e9 := (gcd_eq_zero_iff x y).1 h\n[GOAL]\ncase pos.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nh : gcd 0 0 = 0\n\u22a2 \u2203 x' y', 0 = gcd 0 0 * x' \u2227 0 = gcd 0 0 * y' \u2227 IsUnit (gcd x' y')\n[PROOFSTEP]\nsimp_rw [\u2190 associated_one_iff_isUnit]\n[GOAL]\ncase pos.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nh : gcd 0 0 = 0\n\u22a2 \u2203 x' y', 0 = gcd 0 0 * x' \u2227 0 = gcd 0 0 * y' \u2227 Associated (gcd x' y') 1\n[PROOFSTEP]\nexact \u27e81, 1, by rw [h, zero_mul], by rw [h, zero_mul], gcd_one_left' 1\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nh : gcd 0 0 = 0\n\u22a2 0 = gcd 0 0 * 1\n[PROOFSTEP]\nrw [h, zero_mul]\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nh : gcd 0 0 = 0\n\u22a2 0 = gcd 0 0 * 1\n[PROOFSTEP]\nrw [h, zero_mul]\n[GOAL]\ncase neg\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx y : \u03b1\nh : \u00acgcd x y = 0\n\u22a2 \u2203 x' y', x = gcd x y * x' \u2227 y = gcd x y * y' \u2227 IsUnit (gcd x' y')\n[PROOFSTEP]\nobtain \u27e8x', ex\u27e9 := gcd_dvd_left x y\n[GOAL]\ncase neg.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx y : \u03b1\nh : \u00acgcd x y = 0\nx' : \u03b1\nex : x = gcd x y * x'\n\u22a2 \u2203 x' y', x = gcd x y * x' \u2227 y = gcd x y * y' \u2227 IsUnit (gcd x' y')\n[PROOFSTEP]\nobtain \u27e8y', ey\u27e9 := gcd_dvd_right x y\n[GOAL]\ncase neg.intro.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx y : \u03b1\nh : \u00acgcd x y = 0\nx' : \u03b1\nex : x = gcd x y * x'\ny' : \u03b1\ney : y = gcd x y * y'\n\u22a2 \u2203 x' y', x = gcd x y * x' \u2227 y = gcd x y * y' \u2227 IsUnit (gcd x' y')\n[PROOFSTEP]\nexact \u27e8x', y', ex, ey, isUnit_gcd_of_eq_mul_gcd ex ey h\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\n\u22a2 lcm a b \u2223 c \u2194 a \u2223 c \u2227 b \u2223 c\n[PROOFSTEP]\nby_cases h : a = 0 \u2228 b = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nh : a = 0 \u2228 b = 0\n\u22a2 lcm a b \u2223 c \u2194 a \u2223 c \u2227 b \u2223 c\n[PROOFSTEP]\nrcases h with (rfl | rfl)\n[GOAL]\ncase pos.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nb c : \u03b1\n\u22a2 lcm 0 b \u2223 c \u2194 0 \u2223 c \u2227 b \u2223 c\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [iff_def, lcm_zero_left, lcm_zero_right, zero_dvd_iff, dvd_zero,\n eq_self_iff_true, and_true_iff, imp_true_iff]\n[GOAL]\ncase pos.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na c : \u03b1\n\u22a2 lcm a 0 \u2223 c \u2194 a \u2223 c \u2227 0 \u2223 c\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [iff_def, lcm_zero_left, lcm_zero_right, zero_dvd_iff, dvd_zero,\n eq_self_iff_true, and_true_iff, imp_true_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nh : \u00ac(a = 0 \u2228 b = 0)\n\u22a2 lcm a b \u2223 c \u2194 a \u2223 c \u2227 b \u2223 c\n[PROOFSTEP]\nobtain \u27e8h1, h2\u27e9 := not_or.1 h\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nh : \u00ac(a = 0 \u2228 b = 0)\nh1 : \u00aca = 0\nh2 : \u00acb = 0\n\u22a2 lcm a b \u2223 c \u2194 a \u2223 c \u2227 b \u2223 c\n[PROOFSTEP]\nhave h : gcd a b \u2260 0 := fun H => h1 ((gcd_eq_zero_iff _ _).1 H).1\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nh\u271d : \u00ac(a = 0 \u2228 b = 0)\nh1 : \u00aca = 0\nh2 : \u00acb = 0\nh : gcd a b \u2260 0\n\u22a2 lcm a b \u2223 c \u2194 a \u2223 c \u2227 b \u2223 c\n[PROOFSTEP]\nrw [\u2190 mul_dvd_mul_iff_left h, (gcd_mul_lcm a b).dvd_iff_dvd_left, \u2190 (gcd_mul_right' c a b).dvd_iff_dvd_right,\n dvd_gcd_iff, mul_comm b c, mul_dvd_mul_iff_left h1, mul_dvd_mul_iff_right h2, and_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nh : lcm a b = 0\n\u22a2 a = 0 \u2228 b = 0\n[PROOFSTEP]\nhave : Associated (a * b) 0 := (gcd_mul_lcm a b).symm.trans <| by rw [h, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nh : lcm a b = 0\n\u22a2 Associated (gcd a b * lcm a b) 0\n[PROOFSTEP]\nrw [h, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\nh : lcm a b = 0\nthis : Associated (a * b) 0\n\u22a2 a = 0 \u2228 b = 0\n[PROOFSTEP]\nrwa [\u2190 mul_eq_zero, \u2190 associated_zero_iff_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\n\u22a2 a = 0 \u2228 b = 0 \u2192 lcm a b = 0\n[PROOFSTEP]\nrintro (rfl | rfl) <;> [apply lcm_zero_left; apply lcm_zero_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b : \u03b1\n\u22a2 a = 0 \u2228 b = 0 \u2192 lcm a b = 0\n[PROOFSTEP]\nrintro (rfl | rfl)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nb : \u03b1\n\u22a2 lcm 0 b = 0\n[PROOFSTEP]\napply lcm_zero_left\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na : \u03b1\n\u22a2 lcm a 0 = 0\n[PROOFSTEP]\napply lcm_zero_right\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b : \u03b1\nx\u271d : a \u2223 1 \u2227 b \u2223 1\nc : \u03b1\nhc : 1 = a * c\nd : \u03b1\nhd : 1 = b * d\n\u22a2 lcm \u2191(Units.mkOfMulEqOne a c (_ : a * c = 1)) \u2191(Units.mkOfMulEqOne b d (_ : b * d = 1)) = 1\n[PROOFSTEP]\nrw [lcm_units_coe_left, normalize_coe_units]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\n\u22a2 a = 0 \u2192 lcm (a * b) (a * c) = \u2191normalize a * lcm b c\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\nb c : \u03b1\n\u22a2 lcm (0 * b) (0 * c) = \u2191normalize 0 * lcm b c\n[PROOFSTEP]\nsimp only [zero_mul, lcm_zero_left, normalize_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nthis : lcm (a * b) (a * c) = \u2191normalize (a * lcm b c)\n\u22a2 lcm (a * b) (a * c) = \u2191normalize a * lcm b c\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\n\u22a2 lcm (b * a) (c * a) = lcm b c * \u2191normalize a\n[PROOFSTEP]\nsimp only [mul_comm, lcm_mul_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b : \u03b1\nh : \u2191normalize b = b\n\u22a2 lcm a b = b \u2194 a \u2223 b\n[PROOFSTEP]\nsimpa only [lcm_comm b a] using lcm_eq_left_iff b a h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx : \u03b1\nhi : Irreducible x\na b : \u03b1\nh : x \u2223 a * b\n\u22a2 x \u2223 a \u2228 x \u2223 b\n[PROOFSTEP]\ncases' gcd_dvd_left x a with y hy\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx : \u03b1\nhi : Irreducible x\na b : \u03b1\nh : x \u2223 a * b\ny : \u03b1\nhy : x = gcd x a * y\n\u22a2 x \u2223 a \u2228 x \u2223 b\n[PROOFSTEP]\ncases' hi.isUnit_or_isUnit hy with hu hu\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx : \u03b1\nhi : Irreducible x\na b : \u03b1\nh : x \u2223 a * b\ny : \u03b1\nhy : x = gcd x a * y\nhu : IsUnit (gcd x a)\n\u22a2 x \u2223 a \u2228 x \u2223 b\n[PROOFSTEP]\nright\n[GOAL]\ncase intro.inl.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx : \u03b1\nhi : Irreducible x\na b : \u03b1\nh : x \u2223 a * b\ny : \u03b1\nhy : x = gcd x a * y\nhu : IsUnit (gcd x a)\n\u22a2 x \u2223 b\n[PROOFSTEP]\ntrans gcd (x * b) (a * b)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx : \u03b1\nhi : Irreducible x\na b : \u03b1\nh : x \u2223 a * b\ny : \u03b1\nhy : x = gcd x a * y\nhu : IsUnit (gcd x a)\n\u22a2 x \u2223 gcd (x * b) (a * b)\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx : \u03b1\nhi : Irreducible x\na b : \u03b1\nh : x \u2223 a * b\ny : \u03b1\nhy : x = gcd x a * y\nhu : IsUnit (gcd x a)\n\u22a2 gcd (x * b) (a * b) \u2223 b\n[PROOFSTEP]\napply dvd_gcd (dvd_mul_right x b) h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx : \u03b1\nhi : Irreducible x\na b : \u03b1\nh : x \u2223 a * b\ny : \u03b1\nhy : x = gcd x a * y\nhu : IsUnit (gcd x a)\n\u22a2 gcd (x * b) (a * b) \u2223 b\n[PROOFSTEP]\nrw [(gcd_mul_right' b x a).dvd_iff_dvd_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx : \u03b1\nhi : Irreducible x\na b : \u03b1\nh : x \u2223 a * b\ny : \u03b1\nhy : x = gcd x a * y\nhu : IsUnit (gcd x a)\n\u22a2 gcd x a * b \u2223 b\n[PROOFSTEP]\nexact (associated_unit_mul_left _ _ hu).dvd\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx : \u03b1\nhi : Irreducible x\na b : \u03b1\nh : x \u2223 a * b\ny : \u03b1\nhy : x = gcd x a * y\nhu : IsUnit y\n\u22a2 x \u2223 a \u2228 x \u2223 b\n[PROOFSTEP]\nleft\n[GOAL]\ncase intro.inr.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx : \u03b1\nhi : Irreducible x\na b : \u03b1\nh : x \u2223 a * b\ny : \u03b1\nhy : x = gcd x a * y\nhu : IsUnit y\n\u22a2 x \u2223 a\n[PROOFSTEP]\nrw [hy]\n[GOAL]\ncase intro.inr.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nx : \u03b1\nhi : Irreducible x\na b : \u03b1\nh : x \u2223 a * b\ny : \u03b1\nhy : x = gcd x a * y\nhu : IsUnit y\n\u22a2 gcd x a * y \u2223 a\n[PROOFSTEP]\nexact dvd_trans (associated_mul_unit_left _ _ hu).dvd (gcd_dvd_right x a)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\nx\u271d : NormalizationMonoid \u03b1\nu : \u03b1 \u2192 \u03b1\u02e3\nnormUnit_zero\u271d : u 0 = 1\nnormUnit_mul\u271d : \u2200 {a b : \u03b1}, a \u2260 0 \u2192 b \u2260 0 \u2192 u (a * b) = u a * u b\nnormUnit_coe_units\u271d : \u2200 (u_1 : \u03b1\u02e3), u \u2191u_1 = u_1\u207b\u00b9\n\u22a2 { normUnit := u, normUnit_zero := normUnit_zero\u271d, normUnit_mul := normUnit_mul\u271d,\n normUnit_coe_units := normUnit_coe_units\u271d } =\n default\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_normUnit\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\nx\u271d : NormalizationMonoid \u03b1\nu : \u03b1 \u2192 \u03b1\u02e3\nnormUnit_zero\u271d : u 0 = 1\nnormUnit_mul\u271d : \u2200 {a b : \u03b1}, a \u2260 0 \u2192 b \u2260 0 \u2192 u (a * b) = u a * u b\nnormUnit_coe_units\u271d : \u2200 (u_1 : \u03b1\u02e3), u \u2191u_1 = u_1\u207b\u00b9\n\u22a2 u = fun x => 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\ng\u2081 g\u2082 : GCDMonoid \u03b1\n\u22a2 g\u2081 = g\u2082\n[PROOFSTEP]\nhave hgcd : g\u2081.gcd = g\u2082.gcd := by\n ext a b\n refine'\n associated_iff_eq.mp\n (associated_of_dvd_dvd _ _)\n -- Porting note: Lean4 seems to need help specifying `g\u2081` and `g\u2082`\n \u00b7 exact dvd_gcd (@gcd_dvd_left _ _ g\u2081 _ _) (@gcd_dvd_right _ _ g\u2081 _ _)\n \u00b7 exact @dvd_gcd _ _ g\u2081 _ _ _ (@gcd_dvd_left _ _ g\u2082 _ _) (@gcd_dvd_right _ _ g\u2082 _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\ng\u2081 g\u2082 : GCDMonoid \u03b1\n\u22a2 gcd = gcd\n[PROOFSTEP]\next a b\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\ng\u2081 g\u2082 : GCDMonoid \u03b1\na b : \u03b1\n\u22a2 gcd a b = gcd a b\n[PROOFSTEP]\nrefine'\n associated_iff_eq.mp\n (associated_of_dvd_dvd _ _)\n -- Porting note: Lean4 seems to need help specifying `g\u2081` and `g\u2082`\n[GOAL]\ncase h.h.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\ng\u2081 g\u2082 : GCDMonoid \u03b1\na b : \u03b1\n\u22a2 gcd a b \u2223 gcd a b\n[PROOFSTEP]\nexact dvd_gcd (@gcd_dvd_left _ _ g\u2081 _ _) (@gcd_dvd_right _ _ g\u2081 _ _)\n[GOAL]\ncase h.h.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\ng\u2081 g\u2082 : GCDMonoid \u03b1\na b : \u03b1\n\u22a2 gcd a b \u2223 gcd a b\n[PROOFSTEP]\nexact @dvd_gcd _ _ g\u2081 _ _ _ (@gcd_dvd_left _ _ g\u2082 _ _) (@gcd_dvd_right _ _ g\u2082 _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\ng\u2081 g\u2082 : GCDMonoid \u03b1\nhgcd : gcd = gcd\n\u22a2 g\u2081 = g\u2082\n[PROOFSTEP]\nhave hlcm : g\u2081.lcm = g\u2082.lcm := by\n ext a b\n refine' associated_iff_eq.mp (associated_of_dvd_dvd _ _)\n \u00b7 exact (@lcm_dvd_iff _ _ g\u2081 ..).mpr \u27e8@dvd_lcm_left _ _ g\u2082 _ _, @dvd_lcm_right _ _ g\u2082 _ _\u27e9\n \u00b7 exact lcm_dvd_iff.mpr \u27e8@dvd_lcm_left _ _ g\u2081 _ _, @dvd_lcm_right _ _ g\u2081 _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\ng\u2081 g\u2082 : GCDMonoid \u03b1\nhgcd : gcd = gcd\n\u22a2 lcm = lcm\n[PROOFSTEP]\next a b\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\ng\u2081 g\u2082 : GCDMonoid \u03b1\nhgcd : gcd = gcd\na b : \u03b1\n\u22a2 lcm a b = lcm a b\n[PROOFSTEP]\nrefine' associated_iff_eq.mp (associated_of_dvd_dvd _ _)\n[GOAL]\ncase h.h.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\ng\u2081 g\u2082 : GCDMonoid \u03b1\nhgcd : gcd = gcd\na b : \u03b1\n\u22a2 lcm a b \u2223 lcm a b\n[PROOFSTEP]\nexact (@lcm_dvd_iff _ _ g\u2081 ..).mpr \u27e8@dvd_lcm_left _ _ g\u2082 _ _, @dvd_lcm_right _ _ g\u2082 _ _\u27e9\n[GOAL]\ncase h.h.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\ng\u2081 g\u2082 : GCDMonoid \u03b1\nhgcd : gcd = gcd\na b : \u03b1\n\u22a2 lcm a b \u2223 lcm a b\n[PROOFSTEP]\nexact lcm_dvd_iff.mpr \u27e8@dvd_lcm_left _ _ g\u2081 _ _, @dvd_lcm_right _ _ g\u2081 _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\ng\u2081 g\u2082 : GCDMonoid \u03b1\nhgcd : gcd = gcd\nhlcm : lcm = lcm\n\u22a2 g\u2081 = g\u2082\n[PROOFSTEP]\ncases g\u2081\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\ng\u2082 : GCDMonoid \u03b1\ngcd\u271d lcm\u271d : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left\u271d : \u2200 (a b : \u03b1), gcd\u271d a b \u2223 a\ngcd_dvd_right\u271d : \u2200 (a b : \u03b1), gcd\u271d a b \u2223 b\ndvd_gcd\u271d : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd\u271d c b\ngcd_mul_lcm\u271d : \u2200 (a b : \u03b1), Associated (gcd\u271d a b * lcm\u271d a b) (a * b)\nlcm_zero_left\u271d : \u2200 (a : \u03b1), lcm\u271d 0 a = 0\nlcm_zero_right\u271d : \u2200 (a : \u03b1), lcm\u271d a 0 = 0\nhgcd : gcd = gcd\nhlcm : lcm = lcm\n\u22a2 { gcd := gcd\u271d, lcm := lcm\u271d, gcd_dvd_left := gcd_dvd_left\u271d, gcd_dvd_right := gcd_dvd_right\u271d, dvd_gcd := dvd_gcd\u271d,\n gcd_mul_lcm := gcd_mul_lcm\u271d, lcm_zero_left := lcm_zero_left\u271d, lcm_zero_right := lcm_zero_right\u271d } =\n g\u2082\n[PROOFSTEP]\ncases g\u2082\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\ngcd\u271d\u00b9 lcm\u271d\u00b9 : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left\u271d\u00b9 : \u2200 (a b : \u03b1), gcd\u271d\u00b9 a b \u2223 a\ngcd_dvd_right\u271d\u00b9 : \u2200 (a b : \u03b1), gcd\u271d\u00b9 a b \u2223 b\ndvd_gcd\u271d\u00b9 : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd\u271d\u00b9 c b\ngcd_mul_lcm\u271d\u00b9 : \u2200 (a b : \u03b1), Associated (gcd\u271d\u00b9 a b * lcm\u271d\u00b9 a b) (a * b)\nlcm_zero_left\u271d\u00b9 : \u2200 (a : \u03b1), lcm\u271d\u00b9 0 a = 0\nlcm_zero_right\u271d\u00b9 : \u2200 (a : \u03b1), lcm\u271d\u00b9 a 0 = 0\ngcd\u271d lcm\u271d : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left\u271d : \u2200 (a b : \u03b1), gcd\u271d a b \u2223 a\ngcd_dvd_right\u271d : \u2200 (a b : \u03b1), gcd\u271d a b \u2223 b\ndvd_gcd\u271d : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd\u271d c b\ngcd_mul_lcm\u271d : \u2200 (a b : \u03b1), Associated (gcd\u271d a b * lcm\u271d a b) (a * b)\nlcm_zero_left\u271d : \u2200 (a : \u03b1), lcm\u271d 0 a = 0\nlcm_zero_right\u271d : \u2200 (a : \u03b1), lcm\u271d a 0 = 0\nhgcd : gcd = gcd\nhlcm : lcm = lcm\n\u22a2 { gcd := gcd\u271d\u00b9, lcm := lcm\u271d\u00b9, gcd_dvd_left := gcd_dvd_left\u271d\u00b9, gcd_dvd_right := gcd_dvd_right\u271d\u00b9, dvd_gcd := dvd_gcd\u271d\u00b9,\n gcd_mul_lcm := gcd_mul_lcm\u271d\u00b9, lcm_zero_left := lcm_zero_left\u271d\u00b9, lcm_zero_right := lcm_zero_right\u271d\u00b9 } =\n { gcd := gcd\u271d, lcm := lcm\u271d, gcd_dvd_left := gcd_dvd_left\u271d, gcd_dvd_right := gcd_dvd_right\u271d, dvd_gcd := dvd_gcd\u271d,\n gcd_mul_lcm := gcd_mul_lcm\u271d, lcm_zero_left := lcm_zero_left\u271d, lcm_zero_right := lcm_zero_right\u271d }\n[PROOFSTEP]\ndsimp only at hgcd hlcm \n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\ngcd\u271d\u00b9 lcm\u271d\u00b9 : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left\u271d\u00b9 : \u2200 (a b : \u03b1), gcd\u271d\u00b9 a b \u2223 a\ngcd_dvd_right\u271d\u00b9 : \u2200 (a b : \u03b1), gcd\u271d\u00b9 a b \u2223 b\ndvd_gcd\u271d\u00b9 : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd\u271d\u00b9 c b\ngcd_mul_lcm\u271d\u00b9 : \u2200 (a b : \u03b1), Associated (gcd\u271d\u00b9 a b * lcm\u271d\u00b9 a b) (a * b)\nlcm_zero_left\u271d\u00b9 : \u2200 (a : \u03b1), lcm\u271d\u00b9 0 a = 0\nlcm_zero_right\u271d\u00b9 : \u2200 (a : \u03b1), lcm\u271d\u00b9 a 0 = 0\ngcd\u271d lcm\u271d : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left\u271d : \u2200 (a b : \u03b1), gcd\u271d a b \u2223 a\ngcd_dvd_right\u271d : \u2200 (a b : \u03b1), gcd\u271d a b \u2223 b\ndvd_gcd\u271d : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd\u271d c b\ngcd_mul_lcm\u271d : \u2200 (a b : \u03b1), Associated (gcd\u271d a b * lcm\u271d a b) (a * b)\nlcm_zero_left\u271d : \u2200 (a : \u03b1), lcm\u271d 0 a = 0\nlcm_zero_right\u271d : \u2200 (a : \u03b1), lcm\u271d a 0 = 0\nhgcd : gcd\u271d\u00b9 = gcd\u271d\nhlcm : lcm\u271d\u00b9 = lcm\u271d\n\u22a2 { gcd := gcd\u271d\u00b9, lcm := lcm\u271d\u00b9, gcd_dvd_left := gcd_dvd_left\u271d\u00b9, gcd_dvd_right := gcd_dvd_right\u271d\u00b9, dvd_gcd := dvd_gcd\u271d\u00b9,\n gcd_mul_lcm := gcd_mul_lcm\u271d\u00b9, lcm_zero_left := lcm_zero_left\u271d\u00b9, lcm_zero_right := lcm_zero_right\u271d\u00b9 } =\n { gcd := gcd\u271d, lcm := lcm\u271d, gcd_dvd_left := gcd_dvd_left\u271d, gcd_dvd_right := gcd_dvd_right\u271d, dvd_gcd := dvd_gcd\u271d,\n gcd_mul_lcm := gcd_mul_lcm\u271d, lcm_zero_left := lcm_zero_left\u271d, lcm_zero_right := lcm_zero_right\u271d }\n[PROOFSTEP]\nsimp only [hgcd, hlcm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\n\u22a2 \u2200 (a b : NormalizedGCDMonoid \u03b1), a = b\n[PROOFSTEP]\nintro a b\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\na b : NormalizedGCDMonoid \u03b1\n\u22a2 a = b\n[PROOFSTEP]\ncases' a with a_norm a_gcd\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\nb : NormalizedGCDMonoid \u03b1\na_norm : NormalizationMonoid \u03b1\na_gcd : GCDMonoid \u03b1\nnormalize_gcd\u271d : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nnormalize_lcm\u271d : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\n\u22a2 NormalizedGCDMonoid.mk normalize_gcd\u271d normalize_lcm\u271d = b\n[PROOFSTEP]\ncases' b with b_norm b_gcd\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\na_norm : NormalizationMonoid \u03b1\na_gcd : GCDMonoid \u03b1\nnormalize_gcd\u271d\u00b9 : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nnormalize_lcm\u271d\u00b9 : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nb_norm : NormalizationMonoid \u03b1\nb_gcd : GCDMonoid \u03b1\nnormalize_gcd\u271d : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nnormalize_lcm\u271d : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\n\u22a2 NormalizedGCDMonoid.mk normalize_gcd\u271d\u00b9 normalize_lcm\u271d\u00b9 = NormalizedGCDMonoid.mk normalize_gcd\u271d normalize_lcm\u271d\n[PROOFSTEP]\nhave := Subsingleton.elim a_gcd b_gcd\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\na_norm : NormalizationMonoid \u03b1\na_gcd : GCDMonoid \u03b1\nnormalize_gcd\u271d\u00b9 : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nnormalize_lcm\u271d\u00b9 : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nb_norm : NormalizationMonoid \u03b1\nb_gcd : GCDMonoid \u03b1\nnormalize_gcd\u271d : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nnormalize_lcm\u271d : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nthis : a_gcd = b_gcd\n\u22a2 NormalizedGCDMonoid.mk normalize_gcd\u271d\u00b9 normalize_lcm\u271d\u00b9 = NormalizedGCDMonoid.mk normalize_gcd\u271d normalize_lcm\u271d\n[PROOFSTEP]\nsubst this\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\na_norm : NormalizationMonoid \u03b1\na_gcd : GCDMonoid \u03b1\nnormalize_gcd\u271d\u00b9 : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nnormalize_lcm\u271d\u00b9 : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nb_norm : NormalizationMonoid \u03b1\nnormalize_gcd\u271d : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nnormalize_lcm\u271d : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\n\u22a2 NormalizedGCDMonoid.mk normalize_gcd\u271d\u00b9 normalize_lcm\u271d\u00b9 = NormalizedGCDMonoid.mk normalize_gcd\u271d normalize_lcm\u271d\n[PROOFSTEP]\nhave := Subsingleton.elim a_norm b_norm\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\na_norm : NormalizationMonoid \u03b1\na_gcd : GCDMonoid \u03b1\nnormalize_gcd\u271d\u00b9 : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nnormalize_lcm\u271d\u00b9 : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nb_norm : NormalizationMonoid \u03b1\nnormalize_gcd\u271d : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nnormalize_lcm\u271d : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nthis : a_norm = b_norm\n\u22a2 NormalizedGCDMonoid.mk normalize_gcd\u271d\u00b9 normalize_lcm\u271d\u00b9 = NormalizedGCDMonoid.mk normalize_gcd\u271d normalize_lcm\u271d\n[PROOFSTEP]\nsubst this\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Unique \u03b1\u02e3\na_norm : NormalizationMonoid \u03b1\na_gcd : GCDMonoid \u03b1\nnormalize_gcd\u271d\u00b9 : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nnormalize_lcm\u271d\u00b9 : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nnormalize_gcd\u271d : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nnormalize_lcm\u271d : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\n\u22a2 NormalizedGCDMonoid.mk normalize_gcd\u271d\u00b9 normalize_lcm\u271d\u00b9 = NormalizedGCDMonoid.mk normalize_gcd\u271d normalize_lcm\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\nh : a \u2223 b - c\n\u22a2 gcd a b = gcd a c\n[PROOFSTEP]\napply dvd_antisymm_of_normalize_eq (normalize_gcd _ _) (normalize_gcd _ _)\n[GOAL]\ncase hab\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\nh : a \u2223 b - c\n\u22a2 gcd a b \u2223 gcd a c\n[PROOFSTEP]\nrw [dvd_gcd_iff]\n[GOAL]\ncase hba\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\nh : a \u2223 b - c\n\u22a2 gcd a c \u2223 gcd a b\n[PROOFSTEP]\nrw [dvd_gcd_iff]\n[GOAL]\ncase hab\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\nh : a \u2223 b - c\n\u22a2 gcd a b \u2223 a \u2227 gcd a b \u2223 c\n[PROOFSTEP]\nrefine' \u27e8gcd_dvd_left _ _, _\u27e9\n[GOAL]\ncase hba\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\nh : a \u2223 b - c\n\u22a2 gcd a c \u2223 a \u2227 gcd a c \u2223 b\n[PROOFSTEP]\nrefine' \u27e8gcd_dvd_left _ _, _\u27e9\n[GOAL]\ncase hab\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\nh : a \u2223 b - c\n\u22a2 gcd a b \u2223 c\n[PROOFSTEP]\nrcases h with \u27e8d, hd\u27e9\n[GOAL]\ncase hab.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c d : \u03b1\nhd : b - c = a * d\n\u22a2 gcd a b \u2223 c\n[PROOFSTEP]\nrcases gcd_dvd_right a b with \u27e8e, he\u27e9\n[GOAL]\ncase hab.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c d : \u03b1\nhd : b - c = a * d\ne : \u03b1\nhe : b = gcd a b * e\n\u22a2 gcd a b \u2223 c\n[PROOFSTEP]\nrcases gcd_dvd_left a b with \u27e8f, hf\u27e9\n[GOAL]\ncase hab.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c d : \u03b1\nhd : b - c = a * d\ne : \u03b1\nhe : b = gcd a b * e\nf : \u03b1\nhf : a = gcd a b * f\n\u22a2 gcd a b \u2223 c\n[PROOFSTEP]\nuse e - f * d\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c d : \u03b1\nhd : b - c = a * d\ne : \u03b1\nhe : b = gcd a b * e\nf : \u03b1\nhf : a = gcd a b * f\n\u22a2 c = gcd a b * (e - f * d)\n[PROOFSTEP]\nrw [mul_sub, \u2190 he, \u2190 mul_assoc, \u2190 hf, \u2190 hd, sub_sub_cancel]\n[GOAL]\ncase hba\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\nh : a \u2223 b - c\n\u22a2 gcd a c \u2223 b\n[PROOFSTEP]\nrcases h with \u27e8d, hd\u27e9\n[GOAL]\ncase hba.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c d : \u03b1\nhd : b - c = a * d\n\u22a2 gcd a c \u2223 b\n[PROOFSTEP]\nrcases gcd_dvd_right a c with \u27e8e, he\u27e9\n[GOAL]\ncase hba.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c d : \u03b1\nhd : b - c = a * d\ne : \u03b1\nhe : c = gcd a c * e\n\u22a2 gcd a c \u2223 b\n[PROOFSTEP]\nrcases gcd_dvd_left a c with \u27e8f, hf\u27e9\n[GOAL]\ncase hba.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c d : \u03b1\nhd : b - c = a * d\ne : \u03b1\nhe : c = gcd a c * e\nf : \u03b1\nhf : a = gcd a c * f\n\u22a2 gcd a c \u2223 b\n[PROOFSTEP]\nuse e + f * d\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c d : \u03b1\nhd : b - c = a * d\ne : \u03b1\nhe : c = gcd a c * e\nf : \u03b1\nhf : a = gcd a c * f\n\u22a2 b = gcd a c * (e + f * d)\n[PROOFSTEP]\nrw [mul_add, \u2190 he, \u2190 mul_assoc, \u2190 hf, \u2190 hd, \u2190 add_sub_assoc, add_comm c b, add_sub_cancel]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\nh : a \u2223 b - c\n\u22a2 gcd b a = gcd c a\n[PROOFSTEP]\nrw [gcd_comm _ a, gcd_comm _ a, gcd_eq_of_dvd_sub_right h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Associates \u03b1 \u2192* \u03b1\nhinv : Function.RightInverse (\u2191f) Associates.mk\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 (fun a => if a = 0 then 1 else Classical.choose (_ : Associated a (\u2191f (Associates.mk a)))) (a * b) =\n (fun a => if a = 0 then 1 else Classical.choose (_ : Associated a (\u2191f (Associates.mk a)))) a *\n (fun a => if a = 0 then 1 else Classical.choose (_ : Associated a (\u2191f (Associates.mk a)))) b\n[PROOFSTEP]\nsimp_rw [if_neg (mul_ne_zero ha hb), if_neg ha, if_neg hb, Units.ext_iff, Units.val_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Associates \u03b1 \u2192* \u03b1\nhinv : Function.RightInverse (\u2191f) Associates.mk\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 \u2191(Classical.choose (_ : Associated (a * b) (\u2191f (Associates.mk (a * b))))) =\n \u2191(Classical.choose (_ : Associated a (\u2191f (Associates.mk a)))) *\n \u2191(Classical.choose (_ : Associated b (\u2191f (Associates.mk b))))\n[PROOFSTEP]\nsuffices\n a * b * \u2191(Classical.choose (associated_map_mk hinv (a * b))) =\n a * \u2191(Classical.choose (associated_map_mk hinv a)) * (b * \u2191(Classical.choose (associated_map_mk hinv b)))\n by\n apply\n mul_left_cancel\u2080 (mul_ne_zero ha hb)\n _\n -- Porting note: original `simpa` fails with `unexpected bound variable #1`\n -- simpa only [mul_assoc, mul_comm, mul_left_comm] using this\n rw [this, mul_assoc, \u2190 mul_assoc _ b, mul_comm _ b, \u2190 mul_assoc, \u2190 mul_assoc, mul_assoc (a * b)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Associates \u03b1 \u2192* \u03b1\nhinv : Function.RightInverse (\u2191f) Associates.mk\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nthis :\n a * b * \u2191(Classical.choose (_ : Associated (a * b) (\u2191f (Associates.mk (a * b))))) =\n a * \u2191(Classical.choose (_ : Associated a (\u2191f (Associates.mk a)))) *\n (b * \u2191(Classical.choose (_ : Associated b (\u2191f (Associates.mk b)))))\n\u22a2 \u2191(Classical.choose (_ : Associated (a * b) (\u2191f (Associates.mk (a * b))))) =\n \u2191(Classical.choose (_ : Associated a (\u2191f (Associates.mk a)))) *\n \u2191(Classical.choose (_ : Associated b (\u2191f (Associates.mk b))))\n[PROOFSTEP]\napply\n mul_left_cancel\u2080 (mul_ne_zero ha hb)\n _\n -- Porting note: original `simpa` fails with `unexpected bound variable #1`\n -- simpa only [mul_assoc, mul_comm, mul_left_comm] using this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Associates \u03b1 \u2192* \u03b1\nhinv : Function.RightInverse (\u2191f) Associates.mk\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nthis :\n a * b * \u2191(Classical.choose (_ : Associated (a * b) (\u2191f (Associates.mk (a * b))))) =\n a * \u2191(Classical.choose (_ : Associated a (\u2191f (Associates.mk a)))) *\n (b * \u2191(Classical.choose (_ : Associated b (\u2191f (Associates.mk b)))))\n\u22a2 a * b * \u2191(Classical.choose (_ : Associated (a * b) (\u2191f (Associates.mk (a * b))))) =\n a * b *\n (\u2191(Classical.choose (_ : Associated a (\u2191f (Associates.mk a)))) *\n \u2191(Classical.choose (_ : Associated b (\u2191f (Associates.mk b)))))\n[PROOFSTEP]\nrw [this, mul_assoc, \u2190 mul_assoc _ b, mul_comm _ b, \u2190 mul_assoc, \u2190 mul_assoc, mul_assoc (a * b)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Associates \u03b1 \u2192* \u03b1\nhinv : Function.RightInverse (\u2191f) Associates.mk\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 a * b * \u2191(Classical.choose (_ : Associated (a * b) (\u2191f (Associates.mk (a * b))))) =\n a * \u2191(Classical.choose (_ : Associated a (\u2191f (Associates.mk a)))) *\n (b * \u2191(Classical.choose (_ : Associated b (\u2191f (Associates.mk b)))))\n[PROOFSTEP]\nrw [map_mk_unit_aux hinv a, map_mk_unit_aux hinv (a * b), map_mk_unit_aux hinv b, \u2190 MonoidHom.map_mul,\n Associates.mk_mul_mk]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Associates \u03b1 \u2192* \u03b1\nhinv : Function.RightInverse (\u2191f) Associates.mk\nu : \u03b1\u02e3\n\u22a2 (fun a => if a = 0 then 1 else Classical.choose (_ : Associated a (\u2191f (Associates.mk a)))) \u2191u = u\u207b\u00b9\n[PROOFSTEP]\nnontriviality \u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Associates \u03b1 \u2192* \u03b1\nhinv : Function.RightInverse (\u2191f) Associates.mk\nu : \u03b1\u02e3\n\u271d : Nontrivial \u03b1\n\u22a2 (fun a => if a = 0 then 1 else Classical.choose (_ : Associated a (\u2191f (Associates.mk a)))) \u2191u = u\u207b\u00b9\n[PROOFSTEP]\nsimp_rw [if_neg (Units.ne_zero u), Units.ext_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Associates \u03b1 \u2192* \u03b1\nhinv : Function.RightInverse (\u2191f) Associates.mk\nu : \u03b1\u02e3\n\u271d : Nontrivial \u03b1\n\u22a2 \u2191(Classical.choose (_ : Associated (\u2191u) (\u2191f (Associates.mk \u2191u)))) = \u2191u\u207b\u00b9\n[PROOFSTEP]\napply mul_left_cancel\u2080 (Units.ne_zero u)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nf : Associates \u03b1 \u2192* \u03b1\nhinv : Function.RightInverse (\u2191f) Associates.mk\nu : \u03b1\u02e3\n\u271d : Nontrivial \u03b1\n\u22a2 \u2191u * \u2191(Classical.choose (_ : Associated (\u2191u) (\u2191f (Associates.mk \u2191u)))) = \u2191u * \u2191u\u207b\u00b9\n[PROOFSTEP]\nrw [Units.mul_inv, map_mk_unit_aux hinv u, Associates.mk_eq_mk_iff_associated.2 (associated_one_iff_isUnit.2 \u27e8u, rfl\u27e9),\n Associates.mk_one, MonoidHom.map_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\na b : \u03b1\n\u22a2 Associated (gcd a b * (fun a b => if a = 0 then 0 else Classical.choose (_ : gcd a b \u2223 a * b)) a b) (a * b)\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\na b : \u03b1\n\u22a2 Associated (gcd a b * if a = 0 then 0 else Classical.choose (_ : gcd a b \u2223 a * b)) (a * b)\n[PROOFSTEP]\nsplit_ifs with a0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\na b : \u03b1\na0 : a = 0\n\u22a2 Associated (gcd a b * 0) (a * b)\n[PROOFSTEP]\nrw [mul_zero, a0, zero_mul]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\na b : \u03b1\na0 : \u00aca = 0\n\u22a2 Associated (gcd a b * Classical.choose (_ : gcd a b \u2223 a * b)) (a * b)\n[PROOFSTEP]\nrw [\u2190 Classical.choose_spec ((gcd_dvd_left a b).trans (Dvd.intro b rfl))]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\na : \u03b1\n\u22a2 (fun a b => if a = 0 then 0 else Classical.choose (_ : gcd a b \u2223 a * b)) a 0 = 0\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\na : \u03b1\n\u22a2 (if a = 0 then 0 else Classical.choose (_ : gcd a 0 \u2223 a * 0)) = 0\n[PROOFSTEP]\nsplit_ifs with a0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\na : \u03b1\na0 : a = 0\n\u22a2 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\na : \u03b1\na0 : \u00aca = 0\n\u22a2 Classical.choose (_ : gcd a 0 \u2223 a * 0) = 0\n[PROOFSTEP]\nhave h := (Classical.choose_spec ((gcd_dvd_left a 0).trans (Dvd.intro 0 rfl))).symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\na : \u03b1\na0 : \u00aca = 0\nh : gcd a 0 * Classical.choose (_ : gcd a 0 \u2223 a * 0) = a * 0\n\u22a2 Classical.choose (_ : gcd a 0 \u2223 a * 0) = 0\n[PROOFSTEP]\nhave a0' : gcd a 0 \u2260 0 := by\n contrapose! a0\n rw [\u2190 associated_zero_iff_eq_zero, \u2190 a0]\n exact associated_of_dvd_dvd (dvd_gcd (dvd_refl a) (dvd_zero a)) (gcd_dvd_left _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\na : \u03b1\na0 : \u00aca = 0\nh : gcd a 0 * Classical.choose (_ : gcd a 0 \u2223 a * 0) = a * 0\n\u22a2 gcd a 0 \u2260 0\n[PROOFSTEP]\ncontrapose! a0\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\na : \u03b1\nh : gcd a 0 * Classical.choose (_ : gcd a 0 \u2223 a * 0) = a * 0\na0 : gcd a 0 = 0\n\u22a2 a = 0\n[PROOFSTEP]\nrw [\u2190 associated_zero_iff_eq_zero, \u2190 a0]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\na : \u03b1\nh : gcd a 0 * Classical.choose (_ : gcd a 0 \u2223 a * 0) = a * 0\na0 : gcd a 0 = 0\n\u22a2 Associated a (gcd a 0)\n[PROOFSTEP]\nexact associated_of_dvd_dvd (dvd_gcd (dvd_refl a) (dvd_zero a)) (gcd_dvd_left _ _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\na : \u03b1\na0 : \u00aca = 0\nh : gcd a 0 * Classical.choose (_ : gcd a 0 \u2223 a * 0) = a * 0\na0' : gcd a 0 \u2260 0\n\u22a2 Classical.choose (_ : gcd a 0 \u2223 a * 0) = 0\n[PROOFSTEP]\napply Or.resolve_left (mul_eq_zero.1 _) a0'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\na : \u03b1\na0 : \u00aca = 0\nh : gcd a 0 * Classical.choose (_ : gcd a 0 \u2223 a * 0) = a * 0\na0' : gcd a 0 \u2260 0\n\u22a2 gcd a 0 * Classical.choose (_ : gcd a 0 \u2223 a * 0) = 0\n[PROOFSTEP]\nrw [h, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\n\u22a2 Associated (gcd a b * (fun a b => if a = 0 then 0 else Classical.choose (_ : gcd a b \u2223 \u2191normalize (a * b))) a b)\n (a * b)\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\n\u22a2 Associated (gcd a b * if a = 0 then 0 else Classical.choose (_ : gcd a b \u2223 \u2191normalize (a * b))) (a * b)\n[PROOFSTEP]\nsplit_ifs with a0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\na0 : a = 0\n\u22a2 Associated (gcd a b * 0) (a * b)\n[PROOFSTEP]\nrw [mul_zero, a0, zero_mul]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\na0 : \u00aca = 0\n\u22a2 Associated (gcd a b * Classical.choose (_ : gcd a b \u2223 \u2191normalize (a * b))) (a * b)\n[PROOFSTEP]\nrw [\u2190 Classical.choose_spec (dvd_normalize_iff.2 ((gcd_dvd_left a b).trans (Dvd.intro b rfl)))]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\na0 : \u00aca = 0\n\u22a2 Associated (\u2191normalize (a * b)) (a * b)\n[PROOFSTEP]\nexact normalize_associated (a * b)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\n\u22a2 (fun a b => if a = 0 then 0 else Classical.choose (_ : gcd a b \u2223 \u2191normalize (a * b))) a 0 = 0\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\n\u22a2 (if a = 0 then 0 else Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0))) = 0\n[PROOFSTEP]\nsplit_ifs with a0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\na0 : a = 0\n\u22a2 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\na0 : \u00aca = 0\n\u22a2 Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0)) = 0\n[PROOFSTEP]\nrw [\u2190 normalize_eq_zero] at a0 \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\na0\u271d : \u00aca = 0\na0 : \u00ac\u2191normalize a = 0\n\u22a2 Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0)) = 0\n[PROOFSTEP]\nhave h := (Classical.choose_spec (dvd_normalize_iff.2 ((gcd_dvd_left a 0).trans (Dvd.intro 0 rfl)))).symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\na0\u271d : \u00aca = 0\na0 : \u00ac\u2191normalize a = 0\nh : gcd a 0 * Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0)) = \u2191normalize (a * 0)\n\u22a2 Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0)) = 0\n[PROOFSTEP]\nhave gcd0 : gcd a 0 = normalize a := by\n rw [\u2190 normalize_gcd]\n exact normalize_eq_normalize (gcd_dvd_left _ _) (dvd_gcd (dvd_refl a) (dvd_zero a))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\na0\u271d : \u00aca = 0\na0 : \u00ac\u2191normalize a = 0\nh : gcd a 0 * Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0)) = \u2191normalize (a * 0)\n\u22a2 gcd a 0 = \u2191normalize a\n[PROOFSTEP]\nrw [\u2190 normalize_gcd]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\na0\u271d : \u00aca = 0\na0 : \u00ac\u2191normalize a = 0\nh : gcd a 0 * Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0)) = \u2191normalize (a * 0)\n\u22a2 \u2191normalize (gcd a 0) = \u2191normalize a\n[PROOFSTEP]\nexact normalize_eq_normalize (gcd_dvd_left _ _) (dvd_gcd (dvd_refl a) (dvd_zero a))\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\na0\u271d : \u00aca = 0\na0 : \u00ac\u2191normalize a = 0\nh : gcd a 0 * Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0)) = \u2191normalize (a * 0)\ngcd0 : gcd a 0 = \u2191normalize a\n\u22a2 Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0)) = 0\n[PROOFSTEP]\nrw [\u2190 gcd0] at a0 \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\na0\u271d : \u00aca = 0\na0 : \u00acgcd a 0 = 0\nh : gcd a 0 * Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0)) = \u2191normalize (a * 0)\ngcd0 : gcd a 0 = \u2191normalize a\n\u22a2 Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0)) = 0\n[PROOFSTEP]\napply Or.resolve_left (mul_eq_zero.1 _) a0\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\na0\u271d : \u00aca = 0\na0 : \u00acgcd a 0 = 0\nh : gcd a 0 * Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0)) = \u2191normalize (a * 0)\ngcd0 : gcd a 0 = \u2191normalize a\n\u22a2 gcd a 0 * Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0)) = 0\n[PROOFSTEP]\nrw [h, mul_zero, normalize_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\n\u22a2 \u2191normalize (lcm a b) = lcm a b\n[PROOFSTEP]\ndsimp [normalize]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\n\u22a2 (if a = 0 then 0 else Classical.choose (_ : gcd a b \u2223 a * b * \u2191(normUnit (a * b)))) *\n \u2191(normUnit (if a = 0 then 0 else Classical.choose (_ : gcd a b \u2223 a * b * \u2191(normUnit (a * b))))) =\n if a = 0 then 0 else Classical.choose (_ : gcd a b \u2223 a * b * \u2191(normUnit (a * b)))\n[PROOFSTEP]\nsplit_ifs with a0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\na0 : a = 0\n\u22a2 0 * \u2191(normUnit 0) = 0\n[PROOFSTEP]\nexact @normalize_zero \u03b1 _ _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\na0 : \u00aca = 0\n\u22a2 Classical.choose (_ : gcd a b \u2223 a * b * \u2191(normUnit (a * b))) *\n \u2191(normUnit (Classical.choose (_ : gcd a b \u2223 a * b * \u2191(normUnit (a * b))))) =\n Classical.choose (_ : gcd a b \u2223 a * b * \u2191(normUnit (a * b)))\n[PROOFSTEP]\nhave := (Classical.choose_spec (dvd_normalize_iff.2 ((gcd_dvd_left a b).trans (Dvd.intro b rfl)))).symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\na0 : \u00aca = 0\nthis : gcd a b * Classical.choose (_ : gcd a b \u2223 \u2191normalize (a * b)) = \u2191normalize (a * b)\n\u22a2 Classical.choose (_ : gcd a b \u2223 a * b * \u2191(normUnit (a * b))) *\n \u2191(normUnit (Classical.choose (_ : gcd a b \u2223 a * b * \u2191(normUnit (a * b))))) =\n Classical.choose (_ : gcd a b \u2223 a * b * \u2191(normUnit (a * b)))\n[PROOFSTEP]\nset l := Classical.choose (dvd_normalize_iff.2 ((gcd_dvd_left a b).trans (Dvd.intro b rfl)))\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\na0 : \u00aca = 0\nl : \u03b1 := Classical.choose (_ : gcd a b \u2223 \u2191normalize (a * b))\nthis : gcd a b * l = \u2191normalize (a * b)\n\u22a2 l * \u2191(normUnit l) = l\n[PROOFSTEP]\nobtain rfl | hb :=\n eq_or_ne b\n 0\n -- Porting note: using `simp only` causes the propositions inside `Classical.choose` to\n -- differ, so `set` is unable to produce `l = 0` inside `this`. See\n -- https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/\n -- Classical.2Echoose/near/317491179\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\na0 : \u00aca = 0\nl : \u03b1 := Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0))\nthis : gcd a 0 * l = \u2191normalize (a * 0)\n\u22a2 l * \u2191(normUnit l) = l\n[PROOFSTEP]\nrw [mul_zero a, normalize_zero, mul_eq_zero] at this \n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\na0 : \u00aca = 0\nl : \u03b1 := Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0))\nthis : gcd a 0 = 0 \u2228 l = 0\n\u22a2 l * \u2191(normUnit l) = l\n[PROOFSTEP]\nobtain ha | hl := this\n[GOAL]\ncase neg.inl.inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\na0 : \u00aca = 0\nl : \u03b1 := Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0))\nha : gcd a 0 = 0\n\u22a2 l * \u2191(normUnit l) = l\n[PROOFSTEP]\napply (a0 _).elim\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\na0 : \u00aca = 0\nl : \u03b1 := Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0))\nha : gcd a 0 = 0\n\u22a2 a = 0\n[PROOFSTEP]\nrw [\u2190 zero_dvd_iff, \u2190 ha]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\na0 : \u00aca = 0\nl : \u03b1 := Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0))\nha : gcd a 0 = 0\n\u22a2 gcd a 0 \u2223 a\n[PROOFSTEP]\nexact gcd_dvd_left _ _\n[GOAL]\ncase neg.inl.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na : \u03b1\na0 : \u00aca = 0\nl : \u03b1 := Classical.choose (_ : gcd a 0 \u2223 \u2191normalize (a * 0))\nhl : l = 0\n\u22a2 l * \u2191(normUnit l) = l\n[PROOFSTEP]\nrw [hl, zero_mul]\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\na0 : \u00aca = 0\nl : \u03b1 := Classical.choose (_ : gcd a b \u2223 \u2191normalize (a * b))\nthis : gcd a b * l = \u2191normalize (a * b)\nhb : b \u2260 0\n\u22a2 l * \u2191(normUnit l) = l\n[PROOFSTEP]\nhave h1 : gcd a b \u2260 0 := by\n have hab : a * b \u2260 0 := mul_ne_zero a0 hb\n contrapose! hab\n rw [\u2190 normalize_eq_zero, \u2190 this, hab, zero_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\na0 : \u00aca = 0\nl : \u03b1 := Classical.choose (_ : gcd a b \u2223 \u2191normalize (a * b))\nthis : gcd a b * l = \u2191normalize (a * b)\nhb : b \u2260 0\n\u22a2 gcd a b \u2260 0\n[PROOFSTEP]\nhave hab : a * b \u2260 0 := mul_ne_zero a0 hb\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\na0 : \u00aca = 0\nl : \u03b1 := Classical.choose (_ : gcd a b \u2223 \u2191normalize (a * b))\nthis : gcd a b * l = \u2191normalize (a * b)\nhb : b \u2260 0\nhab : a * b \u2260 0\n\u22a2 gcd a b \u2260 0\n[PROOFSTEP]\ncontrapose! hab\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\na0 : \u00aca = 0\nl : \u03b1 := Classical.choose (_ : gcd a b \u2223 \u2191normalize (a * b))\nthis : gcd a b * l = \u2191normalize (a * b)\nhb : b \u2260 0\nhab : gcd a b = 0\n\u22a2 a * b = 0\n[PROOFSTEP]\nrw [\u2190 normalize_eq_zero, \u2190 this, hab, zero_mul]\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\na0 : \u00aca = 0\nl : \u03b1 := Classical.choose (_ : gcd a b \u2223 \u2191normalize (a * b))\nthis : gcd a b * l = \u2191normalize (a * b)\nhb : b \u2260 0\nh1 : gcd a b \u2260 0\n\u22a2 l * \u2191(normUnit l) = l\n[PROOFSTEP]\nhave h2 : normalize (gcd a b * l) = gcd a b * l := by rw [this, normalize_idem]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\na0 : \u00aca = 0\nl : \u03b1 := Classical.choose (_ : gcd a b \u2223 \u2191normalize (a * b))\nthis : gcd a b * l = \u2191normalize (a * b)\nhb : b \u2260 0\nh1 : gcd a b \u2260 0\n\u22a2 \u2191normalize (gcd a b * l) = gcd a b * l\n[PROOFSTEP]\nrw [this, normalize_idem]\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\na0 : \u00aca = 0\nl : \u03b1 := Classical.choose (_ : gcd a b \u2223 \u2191normalize (a * b))\nthis : gcd a b * l = \u2191normalize (a * b)\nhb : b \u2260 0\nh1 : gcd a b \u2260 0\nh2 : \u2191normalize (gcd a b * l) = gcd a b * l\n\u22a2 l * \u2191(normUnit l) = l\n[PROOFSTEP]\nrw [\u2190 normalize_gcd] at this \n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\ngcd : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngcd_dvd_left : \u2200 (a b : \u03b1), gcd a b \u2223 a\ngcd_dvd_right : \u2200 (a b : \u03b1), gcd a b \u2223 b\ndvd_gcd : \u2200 {a b c : \u03b1}, a \u2223 c \u2192 a \u2223 b \u2192 a \u2223 gcd c b\nnormalize_gcd : \u2200 (a b : \u03b1), \u2191normalize (gcd a b) = gcd a b\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\na0 : \u00aca = 0\nl : \u03b1 := Classical.choose (_ : gcd a b \u2223 \u2191normalize (a * b))\nthis : \u2191normalize (gcd a b) * l = \u2191normalize (a * b)\nhb : b \u2260 0\nh1 : gcd a b \u2260 0\nh2 : \u2191normalize (gcd a b * l) = gcd a b * l\n\u22a2 l * \u2191(normUnit l) = l\n[PROOFSTEP]\nrwa [normalize.map_mul, normalize_gcd, mul_right_inj' h1] at h2 \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\n\u22a2 (fun a b => if a = 0 then b else if b = 0 then a else Classical.choose (_ : lcm a b \u2223 a * b)) a b \u2223 a\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\n\u22a2 (if a = 0 then b else if b = 0 then a else Classical.choose (_ : lcm a b \u2223 a * b)) \u2223 a\n[PROOFSTEP]\nsplit_ifs with h h_1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : a = 0\n\u22a2 b \u2223 a\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : a = 0\n\u22a2 b \u2223 0\n[PROOFSTEP]\napply dvd_zero\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : b = 0\n\u22a2 a \u2223 a\n[PROOFSTEP]\nexact dvd_rfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\n\u22a2 Classical.choose (_ : lcm a b \u2223 a * b) \u2223 a\n[PROOFSTEP]\nhave h0 : lcm a b \u2260 0 := by\n intro con\n have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\n rw [con, zero_dvd_iff, mul_eq_zero] at h \n cases h\n \u00b7 exact absurd \u2039a = 0\u203a h\n \u00b7 exact absurd \u2039b = 0\u203a h_1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\n\u22a2 lcm a b \u2260 0\n[PROOFSTEP]\nintro con\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\n\u22a2 False\n[PROOFSTEP]\nhave h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh\u271d : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh : lcm a b \u2223 a * b\n\u22a2 False\n[PROOFSTEP]\nrw [con, zero_dvd_iff, mul_eq_zero] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh\u271d : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh : a = 0 \u2228 b = 0\n\u22a2 False\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh\u271d : a = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd \u2039a = 0\u203a h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh\u271d : b = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd \u2039b = 0\u203a h_1\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n\u22a2 Classical.choose (_ : lcm a b \u2223 a * b) \u2223 a\n[PROOFSTEP]\nrw [\u2190 mul_dvd_mul_iff_left h0, \u2190 Classical.choose_spec (exists_gcd a b), mul_comm, mul_dvd_mul_iff_right h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n\u22a2 b \u2223 lcm a b\n[PROOFSTEP]\napply dvd_lcm_right\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\n\u22a2 (fun a b => if a = 0 then b else if b = 0 then a else Classical.choose (_ : lcm a b \u2223 a * b)) a b \u2223 b\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\n\u22a2 (if a = 0 then b else if b = 0 then a else Classical.choose (_ : lcm a b \u2223 a * b)) \u2223 b\n[PROOFSTEP]\nsplit_ifs with h h_1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : a = 0\n\u22a2 b \u2223 b\n[PROOFSTEP]\nexact dvd_rfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : b = 0\n\u22a2 a \u2223 b\n[PROOFSTEP]\nrw [h_1]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : b = 0\n\u22a2 a \u2223 0\n[PROOFSTEP]\napply dvd_zero\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\n\u22a2 Classical.choose (_ : lcm a b \u2223 a * b) \u2223 b\n[PROOFSTEP]\nhave h0 : lcm a b \u2260 0 := by\n intro con\n have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\n rw [con, zero_dvd_iff, mul_eq_zero] at h \n cases h\n \u00b7 exact absurd \u2039a = 0\u203a h\n \u00b7 exact absurd \u2039b = 0\u203a h_1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\n\u22a2 lcm a b \u2260 0\n[PROOFSTEP]\nintro con\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\n\u22a2 False\n[PROOFSTEP]\nhave h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh\u271d : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh : lcm a b \u2223 a * b\n\u22a2 False\n[PROOFSTEP]\nrw [con, zero_dvd_iff, mul_eq_zero] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh\u271d : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh : a = 0 \u2228 b = 0\n\u22a2 False\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh\u271d : a = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd \u2039a = 0\u203a h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh\u271d : b = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd \u2039b = 0\u203a h_1\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n\u22a2 Classical.choose (_ : lcm a b \u2223 a * b) \u2223 b\n[PROOFSTEP]\nrw [\u2190 mul_dvd_mul_iff_left h0, \u2190 Classical.choose_spec (exists_gcd a b), mul_dvd_mul_iff_right h_1]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n\u22a2 a \u2223 lcm a b\n[PROOFSTEP]\napply dvd_lcm_left\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\n\u22a2 a \u2223 (fun a b => if a = 0 then b else if b = 0 then a else Classical.choose (_ : lcm a b \u2223 a * b)) c b\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\n\u22a2 a \u2223 if c = 0 then b else if b = 0 then c else Classical.choose (_ : lcm c b \u2223 c * b)\n[PROOFSTEP]\nsplit_ifs with h h_1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : c = 0\n\u22a2 a \u2223 b\n[PROOFSTEP]\nexact ab\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : b = 0\n\u22a2 a \u2223 c\n[PROOFSTEP]\nexact ac\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : \u00acb = 0\n\u22a2 a \u2223 Classical.choose (_ : lcm c b \u2223 c * b)\n[PROOFSTEP]\nhave h0 : lcm c b \u2260 0 := by\n intro con\n have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left c rfl)\n rw [con, zero_dvd_iff, mul_eq_zero] at h \n cases h\n \u00b7 exact absurd \u2039c = 0\u203a h\n \u00b7 exact absurd \u2039b = 0\u203a h_1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : \u00acb = 0\n\u22a2 lcm c b \u2260 0\n[PROOFSTEP]\nintro con\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : \u00acb = 0\ncon : lcm c b = 0\n\u22a2 False\n[PROOFSTEP]\nhave h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left c rfl)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh\u271d : \u00acc = 0\nh_1 : \u00acb = 0\ncon : lcm c b = 0\nh : lcm c b \u2223 c * b\n\u22a2 False\n[PROOFSTEP]\nrw [con, zero_dvd_iff, mul_eq_zero] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh\u271d : \u00acc = 0\nh_1 : \u00acb = 0\ncon : lcm c b = 0\nh : c = 0 \u2228 b = 0\n\u22a2 False\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : \u00acb = 0\ncon : lcm c b = 0\nh\u271d : c = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd \u2039c = 0\u203a h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : \u00acb = 0\ncon : lcm c b = 0\nh\u271d : b = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd \u2039b = 0\u203a h_1\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : \u00acb = 0\nh0 : lcm c b \u2260 0\n\u22a2 a \u2223 Classical.choose (_ : lcm c b \u2223 c * b)\n[PROOFSTEP]\nrw [\u2190 mul_dvd_mul_iff_left h0, \u2190 Classical.choose_spec (exists_gcd c b)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : \u00acb = 0\nh0 : lcm c b \u2260 0\n\u22a2 lcm c b * a \u2223 c * b\n[PROOFSTEP]\nrcases ab with \u27e8d, rfl\u27e9\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na c : \u03b1\nac : a \u2223 c\nh : \u00acc = 0\nd : \u03b1\nh_1 : \u00aca * d = 0\nh0 : lcm c (a * d) \u2260 0\n\u22a2 lcm c (a * d) * a \u2223 c * (a * d)\n[PROOFSTEP]\nrw [mul_eq_zero] at \u2039a * d \u2260 0\u203a \n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na c : \u03b1\nac : a \u2223 c\nh : \u00acc = 0\nd : \u03b1\nh_1 : \u00ac(a = 0 \u2228 d = 0)\nh0 : lcm c (a * d) \u2260 0\n\u22a2 lcm c (a * d) * a \u2223 c * (a * d)\n[PROOFSTEP]\npush_neg at h_1 \n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na c : \u03b1\nac : a \u2223 c\nh : \u00acc = 0\nd : \u03b1\nh0 : lcm c (a * d) \u2260 0\nh_1 : a \u2260 0 \u2227 d \u2260 0\n\u22a2 lcm c (a * d) * a \u2223 c * (a * d)\n[PROOFSTEP]\nrw [mul_comm a, \u2190 mul_assoc, mul_dvd_mul_iff_right h_1.1]\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na c : \u03b1\nac : a \u2223 c\nh : \u00acc = 0\nd : \u03b1\nh0 : lcm c (a * d) \u2260 0\nh_1 : a \u2260 0 \u2227 d \u2260 0\n\u22a2 lcm c (d * a) \u2223 c * d\n[PROOFSTEP]\napply lcm_dvd (Dvd.intro d rfl)\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na c : \u03b1\nac : a \u2223 c\nh : \u00acc = 0\nd : \u03b1\nh0 : lcm c (a * d) \u2260 0\nh_1 : a \u2260 0 \u2227 d \u2260 0\n\u22a2 d * a \u2223 c * d\n[PROOFSTEP]\nrw [mul_comm, mul_dvd_mul_iff_right h_1.2]\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na c : \u03b1\nac : a \u2223 c\nh : \u00acc = 0\nd : \u03b1\nh0 : lcm c (a * d) \u2260 0\nh_1 : a \u2260 0 \u2227 d \u2260 0\n\u22a2 a \u2223 c\n[PROOFSTEP]\napply ac\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\n\u22a2 Associated\n ((fun a b => if a = 0 then b else if b = 0 then a else Classical.choose (_ : lcm a b \u2223 a * b)) a b * lcm a b)\n (a * b)\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\n\u22a2 Associated ((if a = 0 then b else if b = 0 then a else Classical.choose (_ : lcm a b \u2223 a * b)) * lcm a b) (a * b)\n[PROOFSTEP]\nsplit_ifs with h h_1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : a = 0\n\u22a2 Associated (b * lcm a b) (a * b)\n[PROOFSTEP]\nrw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), mul_zero, zero_mul]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : b = 0\n\u22a2 Associated (a * lcm a b) (a * b)\n[PROOFSTEP]\nrw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), mul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 a * b := fun a b => lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\n\u22a2 Associated (Classical.choose (_ : lcm a b \u2223 a * b) * lcm a b) (a * b)\n[PROOFSTEP]\nrw [mul_comm, \u2190 Classical.choose_spec (exists_gcd a b)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\n\u22a2 (fun a b =>\n if a = 0 then \u2191normalize b\n else if b = 0 then \u2191normalize a else Classical.choose (_ : lcm a b \u2223 \u2191normalize (a * b)))\n a b \u2223\n a\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\n\u22a2 (if a = 0 then \u2191normalize b\n else if b = 0 then \u2191normalize a else Classical.choose (_ : lcm a b \u2223 \u2191normalize (a * b))) \u2223\n a\n[PROOFSTEP]\nsplit_ifs with h h_1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : a = 0\n\u22a2 \u2191normalize b \u2223 a\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : a = 0\n\u22a2 \u2191normalize b \u2223 0\n[PROOFSTEP]\napply dvd_zero\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : b = 0\n\u22a2 \u2191normalize a \u2223 a\n[PROOFSTEP]\nexact (normalize_associated _).dvd\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\n\u22a2 Classical.choose (_ : lcm a b \u2223 \u2191normalize (a * b)) \u2223 a\n[PROOFSTEP]\nhave h0 : lcm a b \u2260 0 := by\n intro con\n have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\n rw [con, zero_dvd_iff, mul_eq_zero] at h \n cases h\n \u00b7 exact absurd \u2039a = 0\u203a h\n \u00b7 exact absurd \u2039b = 0\u203a h_1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\n\u22a2 lcm a b \u2260 0\n[PROOFSTEP]\nintro con\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\n\u22a2 False\n[PROOFSTEP]\nhave h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh\u271d : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh : lcm a b \u2223 a * b\n\u22a2 False\n[PROOFSTEP]\nrw [con, zero_dvd_iff, mul_eq_zero] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh\u271d : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh : a = 0 \u2228 b = 0\n\u22a2 False\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh\u271d : a = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd \u2039a = 0\u203a h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh\u271d : b = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd \u2039b = 0\u203a h_1\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n\u22a2 Classical.choose (_ : lcm a b \u2223 \u2191normalize (a * b)) \u2223 a\n[PROOFSTEP]\nrw [\u2190 mul_dvd_mul_iff_left h0, \u2190 Classical.choose_spec (exists_gcd a b), normalize_dvd_iff, mul_comm,\n mul_dvd_mul_iff_right h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n\u22a2 b \u2223 lcm a b\n[PROOFSTEP]\napply dvd_lcm_right\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\n\u22a2 (fun a b =>\n if a = 0 then \u2191normalize b\n else if b = 0 then \u2191normalize a else Classical.choose (_ : lcm a b \u2223 \u2191normalize (a * b)))\n a b \u2223\n b\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\n\u22a2 (if a = 0 then \u2191normalize b\n else if b = 0 then \u2191normalize a else Classical.choose (_ : lcm a b \u2223 \u2191normalize (a * b))) \u2223\n b\n[PROOFSTEP]\nsplit_ifs with h h_1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : a = 0\n\u22a2 \u2191normalize b \u2223 b\n[PROOFSTEP]\nexact (normalize_associated _).dvd\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : b = 0\n\u22a2 \u2191normalize a \u2223 b\n[PROOFSTEP]\nrw [h_1]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : b = 0\n\u22a2 \u2191normalize a \u2223 0\n[PROOFSTEP]\napply dvd_zero\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\n\u22a2 Classical.choose (_ : lcm a b \u2223 \u2191normalize (a * b)) \u2223 b\n[PROOFSTEP]\nhave h0 : lcm a b \u2260 0 := by\n intro con\n have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\n rw [con, zero_dvd_iff, mul_eq_zero] at h \n cases h\n \u00b7 exact absurd \u2039a = 0\u203a h\n \u00b7 exact absurd \u2039b = 0\u203a h_1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\n\u22a2 lcm a b \u2260 0\n[PROOFSTEP]\nintro con\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\n\u22a2 False\n[PROOFSTEP]\nhave h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh\u271d : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh : lcm a b \u2223 a * b\n\u22a2 False\n[PROOFSTEP]\nrw [con, zero_dvd_iff, mul_eq_zero] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh\u271d : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh : a = 0 \u2228 b = 0\n\u22a2 False\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh\u271d : a = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd \u2039a = 0\u203a h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh\u271d : b = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd \u2039b = 0\u203a h_1\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n\u22a2 Classical.choose (_ : lcm a b \u2223 \u2191normalize (a * b)) \u2223 b\n[PROOFSTEP]\nrw [\u2190 mul_dvd_mul_iff_left h0, \u2190 Classical.choose_spec (exists_gcd a b), normalize_dvd_iff, mul_dvd_mul_iff_right h_1]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n\u22a2 a \u2223 lcm a b\n[PROOFSTEP]\napply dvd_lcm_left\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\n\u22a2 a \u2223\n (fun a b =>\n if a = 0 then \u2191normalize b\n else if b = 0 then \u2191normalize a else Classical.choose (_ : lcm a b \u2223 \u2191normalize (a * b)))\n c b\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\n\u22a2 a \u2223\n if c = 0 then \u2191normalize b else if b = 0 then \u2191normalize c else Classical.choose (_ : lcm c b \u2223 \u2191normalize (c * b))\n[PROOFSTEP]\nsplit_ifs with h h_1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : c = 0\n\u22a2 a \u2223 \u2191normalize b\n[PROOFSTEP]\napply dvd_normalize_iff.2 ab\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : b = 0\n\u22a2 a \u2223 \u2191normalize c\n[PROOFSTEP]\napply dvd_normalize_iff.2 ac\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : \u00acb = 0\n\u22a2 a \u2223 Classical.choose (_ : lcm c b \u2223 \u2191normalize (c * b))\n[PROOFSTEP]\nhave h0 : lcm c b \u2260 0 := by\n intro con\n have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left c rfl)\n rw [con, zero_dvd_iff, mul_eq_zero] at h \n cases h\n \u00b7 exact absurd \u2039c = 0\u203a h\n \u00b7 exact absurd \u2039b = 0\u203a h_1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : \u00acb = 0\n\u22a2 lcm c b \u2260 0\n[PROOFSTEP]\nintro con\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : \u00acb = 0\ncon : lcm c b = 0\n\u22a2 False\n[PROOFSTEP]\nhave h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left c rfl)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh\u271d : \u00acc = 0\nh_1 : \u00acb = 0\ncon : lcm c b = 0\nh : lcm c b \u2223 c * b\n\u22a2 False\n[PROOFSTEP]\nrw [con, zero_dvd_iff, mul_eq_zero] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh\u271d : \u00acc = 0\nh_1 : \u00acb = 0\ncon : lcm c b = 0\nh : c = 0 \u2228 b = 0\n\u22a2 False\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : \u00acb = 0\ncon : lcm c b = 0\nh\u271d : c = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd \u2039c = 0\u203a h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : \u00acb = 0\ncon : lcm c b = 0\nh\u271d : b = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd \u2039b = 0\u203a h_1\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : \u00acb = 0\nh0 : lcm c b \u2260 0\n\u22a2 a \u2223 Classical.choose (_ : lcm c b \u2223 \u2191normalize (c * b))\n[PROOFSTEP]\nrw [\u2190 mul_dvd_mul_iff_left h0, \u2190\n Classical.choose_spec (dvd_normalize_iff.2 (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left c rfl))), dvd_normalize_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b c : \u03b1\nac : a \u2223 c\nab : a \u2223 b\nh : \u00acc = 0\nh_1 : \u00acb = 0\nh0 : lcm c b \u2260 0\n\u22a2 lcm c b * a \u2223 c * b\n[PROOFSTEP]\nrcases ab with \u27e8d, rfl\u27e9\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na c : \u03b1\nac : a \u2223 c\nh : \u00acc = 0\nd : \u03b1\nh_1 : \u00aca * d = 0\nh0 : lcm c (a * d) \u2260 0\n\u22a2 lcm c (a * d) * a \u2223 c * (a * d)\n[PROOFSTEP]\nrw [mul_eq_zero] at h_1 \n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na c : \u03b1\nac : a \u2223 c\nh : \u00acc = 0\nd : \u03b1\nh_1 : \u00ac(a = 0 \u2228 d = 0)\nh0 : lcm c (a * d) \u2260 0\n\u22a2 lcm c (a * d) * a \u2223 c * (a * d)\n[PROOFSTEP]\npush_neg at h_1 \n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na c : \u03b1\nac : a \u2223 c\nh : \u00acc = 0\nd : \u03b1\nh0 : lcm c (a * d) \u2260 0\nh_1 : a \u2260 0 \u2227 d \u2260 0\n\u22a2 lcm c (a * d) * a \u2223 c * (a * d)\n[PROOFSTEP]\nrw [mul_comm a, \u2190 mul_assoc, mul_dvd_mul_iff_right h_1.1]\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na c : \u03b1\nac : a \u2223 c\nh : \u00acc = 0\nd : \u03b1\nh0 : lcm c (a * d) \u2260 0\nh_1 : a \u2260 0 \u2227 d \u2260 0\n\u22a2 lcm c (d * a) \u2223 c * d\n[PROOFSTEP]\napply lcm_dvd (Dvd.intro d rfl)\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na c : \u03b1\nac : a \u2223 c\nh : \u00acc = 0\nd : \u03b1\nh0 : lcm c (a * d) \u2260 0\nh_1 : a \u2260 0 \u2227 d \u2260 0\n\u22a2 d * a \u2223 c * d\n[PROOFSTEP]\nrw [mul_comm, mul_dvd_mul_iff_right h_1.2]\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na c : \u03b1\nac : a \u2223 c\nh : \u00acc = 0\nd : \u03b1\nh0 : lcm c (a * d) \u2260 0\nh_1 : a \u2260 0 \u2227 d \u2260 0\n\u22a2 a \u2223 c\n[PROOFSTEP]\napply ac\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\n\u22a2 Associated\n ((fun a b =>\n if a = 0 then \u2191normalize b\n else if b = 0 then \u2191normalize a else Classical.choose (_ : lcm a b \u2223 \u2191normalize (a * b)))\n a b *\n lcm a b)\n (a * b)\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\n\u22a2 Associated\n ((if a = 0 then \u2191normalize b\n else if b = 0 then \u2191normalize a else Classical.choose (_ : lcm a b \u2223 \u2191normalize (a * b))) *\n lcm a b)\n (a * b)\n[PROOFSTEP]\nsplit_ifs with h h_1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : a = 0\n\u22a2 Associated (\u2191normalize b * lcm a b) (a * b)\n[PROOFSTEP]\nrw [h, eq_zero_of_zero_dvd (dvd_lcm_left _ _), mul_zero, zero_mul]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : b = 0\n\u22a2 Associated (\u2191normalize a * lcm a b) (a * b)\n[PROOFSTEP]\nrw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), mul_zero, mul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\n\u22a2 Associated (Classical.choose (_ : lcm a b \u2223 \u2191normalize (a * b)) * lcm a b) (a * b)\n[PROOFSTEP]\nrw [mul_comm, \u2190 Classical.choose_spec (exists_gcd a b)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\n\u22a2 Associated (\u2191normalize (a * b)) (a * b)\n[PROOFSTEP]\nexact normalize_associated (a * b)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\n\u22a2 \u2191normalize (gcd a b) = gcd a b\n[PROOFSTEP]\ndsimp [normalize]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\n\u22a2 (if a = 0 then b * \u2191(normUnit b)\n else if b = 0 then a * \u2191(normUnit a) else Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b)))) *\n \u2191(normUnit\n (if a = 0 then b * \u2191(normUnit b)\n else if b = 0 then a * \u2191(normUnit a) else Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b))))) =\n if a = 0 then b * \u2191(normUnit b)\n else if b = 0 then a * \u2191(normUnit a) else Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b)))\n[PROOFSTEP]\nsplit_ifs with h h_1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : a = 0\n\u22a2 b * \u2191(normUnit b) * \u2191(normUnit (b * \u2191(normUnit b))) = b * \u2191(normUnit b)\n[PROOFSTEP]\napply normalize_idem\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : b = 0\n\u22a2 a * \u2191(normUnit a) * \u2191(normUnit (a * \u2191(normUnit a))) = a * \u2191(normUnit a)\n[PROOFSTEP]\napply normalize_idem\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\n\u22a2 Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b))) *\n \u2191(normUnit (Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b))))) =\n Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b)))\n[PROOFSTEP]\nhave h0 : lcm a b \u2260 0 := by\n intro con\n have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\n rw [con, zero_dvd_iff, mul_eq_zero] at h \n cases h\n \u00b7 exact absurd \u2039a = 0\u203a h\n \u00b7 exact absurd \u2039b = 0\u203a h_1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\n\u22a2 lcm a b \u2260 0\n[PROOFSTEP]\nintro con\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\n\u22a2 False\n[PROOFSTEP]\nhave h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh\u271d : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh : lcm a b \u2223 a * b\n\u22a2 False\n[PROOFSTEP]\nrw [con, zero_dvd_iff, mul_eq_zero] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh\u271d : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh : a = 0 \u2228 b = 0\n\u22a2 False\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh\u271d : a = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd \u2039a = 0\u203a h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\ncon : lcm a b = 0\nh\u271d : b = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd \u2039b = 0\u203a h_1\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n\u22a2 Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b))) *\n \u2191(normUnit (Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b))))) =\n Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b)))\n[PROOFSTEP]\napply mul_left_cancel\u2080 h0\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n\u22a2 lcm a b *\n (Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b))) *\n \u2191(normUnit (Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b)))))) =\n lcm a b * Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b)))\n[PROOFSTEP]\nrefine' _root_.trans _ (Classical.choose_spec (exists_gcd a b))\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n\u22a2 lcm a b *\n (Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b))) *\n \u2191(normUnit (Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b)))))) =\n \u2191normalize (a * b)\n[PROOFSTEP]\nconv_lhs =>\n congr\n rw [\u2190 normalize_lcm a b]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n| lcm a b *\n (Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b))) *\n \u2191(normUnit (Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b))))))\n[PROOFSTEP]\n congr\n rw [\u2190 normalize_lcm a b]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n| lcm a b *\n (Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b))) *\n \u2191(normUnit (Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b))))))\n[PROOFSTEP]\n congr\n rw [\u2190 normalize_lcm a b]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n| lcm a b *\n (Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b))) *\n \u2191(normUnit (Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b))))))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n| lcm a b\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n| Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b))) *\n \u2191(normUnit (Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b)))))\n[PROOFSTEP]\nrw [\u2190 normalize_lcm a b]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nlcm : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ndvd_lcm_left : \u2200 (a b : \u03b1), a \u2223 lcm a b\ndvd_lcm_right : \u2200 (a b : \u03b1), b \u2223 lcm a b\nlcm_dvd : \u2200 {a b c : \u03b1}, c \u2223 a \u2192 b \u2223 a \u2192 lcm c b \u2223 a\nnormalize_lcm : \u2200 (a b : \u03b1), \u2191normalize (lcm a b) = lcm a b\nexists_gcd : \u2200 (a b : \u03b1), lcm a b \u2223 \u2191normalize (a * b) :=\n fun a b => Iff.mpr dvd_normalize_iff (lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl))\nsrc\u271d : NormalizationMonoid \u03b1 := inferInstance\na b : \u03b1\nh : \u00aca = 0\nh_1 : \u00acb = 0\nh0 : lcm a b \u2260 0\n\u22a2 \u2191normalize (lcm a b) *\n (Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b))) *\n \u2191(normUnit (Classical.choose (_ : lcm a b \u2223 a * b * \u2191(normUnit (a * b)))))) =\n \u2191normalize (a * b)\n[PROOFSTEP]\nerw [\u2190 normalize.map_mul, \u2190 Classical.choose_spec (exists_gcd a b), normalize_idem]\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\nx y : G\u2080\nx0 : x \u2260 0\ny0 : y \u2260 0\n\u22a2 \u2191((fun x => if h : x = 0 then 1 else (Units.mk0 x h)\u207b\u00b9) (x * y)) =\n \u2191((fun x => if h : x = 0 then 1 else (Units.mk0 x h)\u207b\u00b9) x * (fun x => if h : x = 0 then 1 else (Units.mk0 x h)\u207b\u00b9) y)\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\nx y : G\u2080\nx0 : x \u2260 0\ny0 : y \u2260 0\n\u22a2 \u2191(if h : x * y = 0 then 1 else (Units.mk0 (x * y) h)\u207b\u00b9) =\n \u2191((if h : x = 0 then 1 else (Units.mk0 x h)\u207b\u00b9) * if h : y = 0 then 1 else (Units.mk0 y h)\u207b\u00b9)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\nx y : G\u2080\nx0 : x \u2260 0\ny0 : y \u2260 0\nh : x * y = 0\n\u22a2 \u21911 = \u2191((Units.mk0 x x0)\u207b\u00b9 * (Units.mk0 y y0)\u207b\u00b9)\n[PROOFSTEP]\nrw [mul_eq_zero] at h \n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\nx y : G\u2080\nx0 : x \u2260 0\ny0 : y \u2260 0\nh : x = 0 \u2228 y = 0\n\u22a2 \u21911 = \u2191((Units.mk0 x x0)\u207b\u00b9 * (Units.mk0 y y0)\u207b\u00b9)\n[PROOFSTEP]\ncases h\n[GOAL]\ncase pos.inl\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\nx y : G\u2080\nx0 : x \u2260 0\ny0 : y \u2260 0\nh\u271d : x = 0\n\u22a2 \u21911 = \u2191((Units.mk0 x x0)\u207b\u00b9 * (Units.mk0 y y0)\u207b\u00b9)\n[PROOFSTEP]\nexact absurd \u2039x = 0\u203a x0\n[GOAL]\ncase pos.inr\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\nx y : G\u2080\nx0 : x \u2260 0\ny0 : y \u2260 0\nh\u271d : y = 0\n\u22a2 \u21911 = \u2191((Units.mk0 x x0)\u207b\u00b9 * (Units.mk0 y y0)\u207b\u00b9)\n[PROOFSTEP]\nexact absurd \u2039y = 0\u203a y0\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\nx y : G\u2080\nx0 : x \u2260 0\ny0 : y \u2260 0\nh : \u00acx * y = 0\n\u22a2 \u2191(Units.mk0 (x * y) h)\u207b\u00b9 = \u2191((Units.mk0 x x0)\u207b\u00b9 * (Units.mk0 y y0)\u207b\u00b9)\n[PROOFSTEP]\nrw [Units.mk0_mul, mul_inv_rev, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\nu : G\u2080\u02e3\n\u22a2 (fun x => if h : x = 0 then 1 else (Units.mk0 x h)\u207b\u00b9) \u2191u = u\u207b\u00b9\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\nu : G\u2080\u02e3\n\u22a2 (if h : \u2191u = 0 then 1 else (Units.mk0 (\u2191u) h)\u207b\u00b9) = u\u207b\u00b9\n[PROOFSTEP]\nrw [dif_neg (Units.ne_zero _), Units.mk0_val]\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\n\u22a2 (fun a b => if a = 0 \u2227 b = 0 then 0 else 1) a b \u2223 a\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\n\u22a2 (if a = 0 \u2227 b = 0 then 0 else 1) \u2223 a\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nh : a = 0 \u2227 b = 0\n\u22a2 0 \u2223 a\n[PROOFSTEP]\nrw [h.1]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nh : \u00ac(a = 0 \u2227 b = 0)\n\u22a2 1 \u2223 a\n[PROOFSTEP]\nexact one_dvd _\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\n\u22a2 (fun a b => if a = 0 \u2227 b = 0 then 0 else 1) a b \u2223 b\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\n\u22a2 (if a = 0 \u2227 b = 0 then 0 else 1) \u2223 b\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nh : a = 0 \u2227 b = 0\n\u22a2 0 \u2223 b\n[PROOFSTEP]\nrw [h.2]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nh : \u00ac(a = 0 \u2227 b = 0)\n\u22a2 1 \u2223 b\n[PROOFSTEP]\nexact one_dvd _\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b c : G\u2080\nhac : a \u2223 c\nhab : a \u2223 b\n\u22a2 a \u2223 (fun a b => if a = 0 \u2227 b = 0 then 0 else 1) c b\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b c : G\u2080\nhac : a \u2223 c\nhab : a \u2223 b\n\u22a2 a \u2223 if c = 0 \u2227 b = 0 then 0 else 1\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b c : G\u2080\nhac : a \u2223 c\nhab : a \u2223 b\nh : c = 0 \u2227 b = 0\n\u22a2 a \u2223 0\n[PROOFSTEP]\napply dvd_zero\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b c : G\u2080\nhac : a \u2223 c\nhab : a \u2223 b\nh : \u00ac(c = 0 \u2227 b = 0)\n\u22a2 a \u2223 1\n[PROOFSTEP]\nrw [not_and_or] at h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b c : G\u2080\nhac : a \u2223 c\nhab : a \u2223 b\nh : \u00acc = 0 \u2228 \u00acb = 0\n\u22a2 a \u2223 1\n[PROOFSTEP]\ncases h\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b c : G\u2080\nhac : a \u2223 c\nhab : a \u2223 b\nh\u271d : \u00acc = 0\n\u22a2 a \u2223 1\n[PROOFSTEP]\nrefine' isUnit_iff_dvd_one.mp (isUnit_of_dvd_unit _ (IsUnit.mk0 _ \u2039c \u2260 0\u203a))\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b c : G\u2080\nhac : a \u2223 c\nhab : a \u2223 b\nh\u271d : \u00acc = 0\n\u22a2 a \u2223 c\n[PROOFSTEP]\nexact hac\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b c : G\u2080\nhac : a \u2223 c\nhab : a \u2223 b\nh\u271d : \u00acb = 0\n\u22a2 a \u2223 1\n[PROOFSTEP]\nrefine' isUnit_iff_dvd_one.mp (isUnit_of_dvd_unit _ (IsUnit.mk0 _ \u2039b \u2260 0\u203a))\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b c : G\u2080\nhac : a \u2223 c\nhab : a \u2223 b\nh\u271d : \u00acb = 0\n\u22a2 a \u2223 b\n[PROOFSTEP]\nexact hab\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\n\u22a2 Associated ((fun a b => if a = 0 \u2227 b = 0 then 0 else 1) a b * (fun a b => if a = 0 \u2228 b = 0 then 0 else 1) a b) (a * b)\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nha : a = 0\n\u22a2 Associated ((fun a b => if a = 0 \u2227 b = 0 then 0 else 1) a b * (fun a b => if a = 0 \u2228 b = 0 then 0 else 1) a b) (a * b)\n[PROOFSTEP]\nsimp only [ha, true_and, true_or, ite_true, mul_zero, zero_mul]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nha : a = 0\n\u22a2 Associated 0 0\n[PROOFSTEP]\nexact Associated.refl _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nha : \u00aca = 0\n\u22a2 Associated ((fun a b => if a = 0 \u2227 b = 0 then 0 else 1) a b * (fun a b => if a = 0 \u2228 b = 0 then 0 else 1) a b) (a * b)\n[PROOFSTEP]\nby_cases hb : b = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nha : \u00aca = 0\nhb : b = 0\n\u22a2 Associated ((fun a b => if a = 0 \u2227 b = 0 then 0 else 1) a b * (fun a b => if a = 0 \u2228 b = 0 then 0 else 1) a b) (a * b)\n[PROOFSTEP]\nsimp only [hb, and_true, or_true, ite_true, mul_zero]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nha : \u00aca = 0\nhb : b = 0\n\u22a2 Associated 0 0\n[PROOFSTEP]\nexact\n Associated.refl\n _\n -- Porting note: need `dsimp only`\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nha : \u00aca = 0\nhb : \u00acb = 0\n\u22a2 Associated ((fun a b => if a = 0 \u2227 b = 0 then 0 else 1) a b * (fun a b => if a = 0 \u2228 b = 0 then 0 else 1) a b) (a * b)\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nha : \u00aca = 0\nhb : \u00acb = 0\n\u22a2 Associated ((if a = 0 \u2227 b = 0 then 0 else 1) * if a = 0 \u2228 b = 0 then 0 else 1) (a * b)\n[PROOFSTEP]\nrw [if_neg (not_and_of_not_left _ ha), one_mul, if_neg (not_or_of_not ha hb)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nha : \u00aca = 0\nhb : \u00acb = 0\n\u22a2 Associated 1 (a * b)\n[PROOFSTEP]\nexact (associated_one_iff_isUnit.mpr ((IsUnit.mk0 _ ha).mul (IsUnit.mk0 _ hb))).symm\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nh : a = 0 \u2227 b = 0\n\u22a2 \u2191normalize (gcd a b) = gcd a b\n[PROOFSTEP]\nsimp [if_pos h]\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nh : \u00ac(a = 0 \u2227 b = 0)\n\u22a2 \u2191normalize (gcd a b) = gcd a b\n[PROOFSTEP]\nsimp [if_neg h]\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nh : a = 0 \u2228 b = 0\n\u22a2 \u2191normalize (lcm a b) = lcm a b\n[PROOFSTEP]\nsimp [if_pos h]\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na b : G\u2080\nh : \u00ac(a = 0 \u2228 b = 0)\n\u22a2 \u2191normalize (lcm a b) = lcm a b\n[PROOFSTEP]\nsimp [if_neg h]\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na : G\u2080\nh0 : a \u2260 0\n\u22a2 \u2191(normUnit a) = a\u207b\u00b9\n[PROOFSTEP]\nsimp [normUnit, h0]\n[GOAL]\n\u03b1 : Type u_1\nG\u2080 : Type u_2\ninst\u271d\u00b9 : CommGroupWithZero G\u2080\ninst\u271d : DecidableEq G\u2080\na : G\u2080\nh0 : a \u2260 0\n\u22a2 \u2191normalize a = 1\n[PROOFSTEP]\nsimp [normalize_apply, h0]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.GCDMonoid.Basic", "llama_tokens": 95177, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4112872797488411}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Bornology \u03b1\ninst\u271d\u00b9 : Bornology \u03b2\ninst\u271d : (i : \u03b9) \u2192 Bornology (\u03c0 i)\ns : Set \u03b1\nt : Set \u03b2\nS : (i : \u03b9) \u2192 Set (\u03c0 i)\n\u22a2 IsBounded (s \u00d7\u02e2 t) \u2194 s = \u2205 \u2228 t = \u2205 \u2228 IsBounded s \u2227 IsBounded t\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | hs)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Bornology \u03b1\ninst\u271d\u00b9 : Bornology \u03b2\ninst\u271d : (i : \u03b9) \u2192 Bornology (\u03c0 i)\nt : Set \u03b2\nS : (i : \u03b9) \u2192 Set (\u03c0 i)\n\u22a2 IsBounded (\u2205 \u00d7\u02e2 t) \u2194 \u2205 = \u2205 \u2228 t = \u2205 \u2228 IsBounded \u2205 \u2227 IsBounded t\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Bornology \u03b1\ninst\u271d\u00b9 : Bornology \u03b2\ninst\u271d : (i : \u03b9) \u2192 Bornology (\u03c0 i)\ns : Set \u03b1\nt : Set \u03b2\nS : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : Set.Nonempty s\n\u22a2 IsBounded (s \u00d7\u02e2 t) \u2194 s = \u2205 \u2228 t = \u2205 \u2228 IsBounded s \u2227 IsBounded t\n[PROOFSTEP]\nrcases t.eq_empty_or_nonempty with (rfl | ht)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Bornology \u03b1\ninst\u271d\u00b9 : Bornology \u03b2\ninst\u271d : (i : \u03b9) \u2192 Bornology (\u03c0 i)\ns : Set \u03b1\nS : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : Set.Nonempty s\n\u22a2 IsBounded (s \u00d7\u02e2 \u2205) \u2194 s = \u2205 \u2228 \u2205 = \u2205 \u2228 IsBounded s \u2227 IsBounded \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Bornology \u03b1\ninst\u271d\u00b9 : Bornology \u03b2\ninst\u271d : (i : \u03b9) \u2192 Bornology (\u03c0 i)\ns : Set \u03b1\nt : Set \u03b2\nS : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : Set.Nonempty s\nht : Set.Nonempty t\n\u22a2 IsBounded (s \u00d7\u02e2 t) \u2194 s = \u2205 \u2228 t = \u2205 \u2228 IsBounded s \u2227 IsBounded t\n[PROOFSTEP]\nsimp only [hs.ne_empty, ht.ne_empty, isBounded_prod_of_nonempty (hs.prod ht), false_or_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Bornology \u03b1\ninst\u271d\u00b9 : Bornology \u03b2\ninst\u271d : (i : \u03b9) \u2192 Bornology (\u03c0 i)\ns : Set \u03b1\nt : Set \u03b2\nS : (i : \u03b9) \u2192 Set (\u03c0 i)\n\u22a2 IsBounded (s \u00d7\u02e2 s) \u2194 IsBounded s\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | hs)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Bornology \u03b1\ninst\u271d\u00b9 : Bornology \u03b2\ninst\u271d : (i : \u03b9) \u2192 Bornology (\u03c0 i)\nt : Set \u03b2\nS : (i : \u03b9) \u2192 Set (\u03c0 i)\n\u22a2 IsBounded (\u2205 \u00d7\u02e2 \u2205) \u2194 IsBounded \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Bornology \u03b1\ninst\u271d\u00b9 : Bornology \u03b2\ninst\u271d : (i : \u03b9) \u2192 Bornology (\u03c0 i)\ns : Set \u03b1\nt : Set \u03b2\nS : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : Set.Nonempty s\n\u22a2 IsBounded (s \u00d7\u02e2 s) \u2194 IsBounded s\n[PROOFSTEP]\nexact (isBounded_prod_of_nonempty (hs.prod hs)).trans (and_self_iff _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Bornology \u03b1\ninst\u271d\u00b9 : Bornology \u03b2\ninst\u271d : (i : \u03b9) \u2192 Bornology (\u03c0 i)\ns : Set \u03b1\nt : Set \u03b2\nS : (i : \u03b9) \u2192 Set (\u03c0 i)\n\u22a2 IsBounded (Set.pi univ S) \u2194 (\u2203 i, S i = \u2205) \u2228 \u2200 (i : \u03b9), IsBounded (S i)\n[PROOFSTEP]\nby_cases hne : \u2203 i, S i = \u2205\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Bornology \u03b1\ninst\u271d\u00b9 : Bornology \u03b2\ninst\u271d : (i : \u03b9) \u2192 Bornology (\u03c0 i)\ns : Set \u03b1\nt : Set \u03b2\nS : (i : \u03b9) \u2192 Set (\u03c0 i)\nhne : \u2203 i, S i = \u2205\n\u22a2 IsBounded (Set.pi univ S) \u2194 (\u2203 i, S i = \u2205) \u2228 \u2200 (i : \u03b9), IsBounded (S i)\n[PROOFSTEP]\nsimp [hne, univ_pi_eq_empty_iff.2 hne]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Bornology \u03b1\ninst\u271d\u00b9 : Bornology \u03b2\ninst\u271d : (i : \u03b9) \u2192 Bornology (\u03c0 i)\ns : Set \u03b1\nt : Set \u03b2\nS : (i : \u03b9) \u2192 Set (\u03c0 i)\nhne : \u00ac\u2203 i, S i = \u2205\n\u22a2 IsBounded (Set.pi univ S) \u2194 (\u2203 i, S i = \u2205) \u2228 \u2200 (i : \u03b9), IsBounded (S i)\n[PROOFSTEP]\nsimp only [hne, false_or_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Bornology \u03b1\ninst\u271d\u00b9 : Bornology \u03b2\ninst\u271d : (i : \u03b9) \u2192 Bornology (\u03c0 i)\ns : Set \u03b1\nt : Set \u03b2\nS : (i : \u03b9) \u2192 Set (\u03c0 i)\nhne : \u00ac\u2203 i, S i = \u2205\n\u22a2 IsBounded (Set.pi univ S) \u2194 \u2200 (i : \u03b9), IsBounded (S i)\n[PROOFSTEP]\nsimp only [not_exists, \u2190 Ne.def, \u2190 nonempty_iff_ne_empty, \u2190 univ_pi_nonempty_iff] at hne \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Bornology \u03b1\ninst\u271d\u00b9 : Bornology \u03b2\ninst\u271d : (i : \u03b9) \u2192 Bornology (\u03c0 i)\ns : Set \u03b1\nt : Set \u03b2\nS : (i : \u03b9) \u2192 Set (\u03c0 i)\nhne : Set.Nonempty (Set.pi univ fun i => S i)\n\u22a2 IsBounded (Set.pi univ S) \u2194 \u2200 (i : \u03b9), IsBounded (S i)\n[PROOFSTEP]\nexact isBounded_pi_of_nonempty hne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Bornology \u03b1\ninst\u271d\u00b3 : Bornology \u03b2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Bornology (\u03c0 i)\ninst\u271d\u00b9 : BoundedSpace \u03b1\ninst\u271d : BoundedSpace \u03b2\n\u22a2 BoundedSpace (\u03b1 \u00d7 \u03b2)\n[PROOFSTEP]\nsimp [\u2190 cobounded_eq_bot_iff, cobounded_prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : Bornology \u03b1\ninst\u271d\u00b2 : Bornology \u03b2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Bornology (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), BoundedSpace (\u03c0 i)\n\u22a2 BoundedSpace ((i : \u03b9) \u2192 \u03c0 i)\n[PROOFSTEP]\nsimp [\u2190 cobounded_eq_bot_iff, cobounded_pi]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : Bornology \u03b1\u271d\ninst\u271d\u00b2 : Bornology \u03b2\u271d\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Bornology (\u03c0 i)\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d : Bornology \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 BoundedSpace \u03b1 \u2194 IsBounded (range f)\n[PROOFSTEP]\nrw [\u2190 @isBounded_univ _ (Bornology.induced f), isBounded_induced, image_univ]\n -- porting note: had to explicitly provided the bornology to `isBounded_univ`.\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Bornology \u03b1\ninst\u271d\u00b9 : Bornology \u03b2\ninst\u271d : (i : \u03b9) \u2192 Bornology (\u03c0 i)\np : \u03b1 \u2192 Prop\n\u22a2 BoundedSpace (Subtype p) \u2194 IsBounded {x | p x}\n[PROOFSTEP]\nrw [boundedSpace_induced_iff, Subtype.range_coe_subtype]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Bornology.Constructions", "llama_tokens": 3027, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.689305616785446, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.4111247177572218}} {"text": "[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nx\u271d\u00b9 : \u211d\nx\u271d : E\n\u22a2 (fun R x => smoothTransition ((R - \u2016x\u2016) / (R - 1))) x\u271d\u00b9 (-x\u271d) =\n (fun R x => smoothTransition ((R - \u2016x\u2016) / (R - 1))) x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nsimp only [norm_neg]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u22a2 ContDiffOn \u211d \u22a4 (uncurry fun R x => smoothTransition ((R - \u2016x\u2016) / (R - 1))) (Set.Ioi 1 \u00d7\u02e2 Set.univ)\n[PROOFSTEP]\nrintro \u27e8R, x\u27e9 \u27e8hR : 1 < R, -\u27e9\n[GOAL]\ncase mk.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nR : \u211d\nx : E\nhR : 1 < R\n\u22a2 ContDiffWithinAt \u211d \u22a4 (uncurry fun R x => smoothTransition ((R - \u2016x\u2016) / (R - 1))) (Set.Ioi 1 \u00d7\u02e2 Set.univ) (R, x)\n[PROOFSTEP]\napply ContDiffAt.contDiffWithinAt\n[GOAL]\ncase mk.intro.h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nR : \u211d\nx : E\nhR : 1 < R\n\u22a2 ContDiffAt \u211d \u22a4 (uncurry fun R x => smoothTransition ((R - \u2016x\u2016) / (R - 1))) (R, x)\n[PROOFSTEP]\nrw [\u2190 sub_pos] at hR \n[GOAL]\ncase mk.intro.h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nR : \u211d\nx : E\nhR\u271d : 1 < R\nhR : 0 < R - 1\n\u22a2 ContDiffAt \u211d \u22a4 (uncurry fun R x => smoothTransition ((R - \u2016x\u2016) / (R - 1))) (R, x)\n[PROOFSTEP]\nrcases eq_or_ne x 0 with rfl | hx\n[GOAL]\ncase mk.intro.h.inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nR : \u211d\nhR\u271d : 1 < R\nhR : 0 < R - 1\n\u22a2 ContDiffAt \u211d \u22a4 (uncurry fun R x => smoothTransition ((R - \u2016x\u2016) / (R - 1))) (R, 0)\n[PROOFSTEP]\nhave A : ContinuousAt (fun p : \u211d \u00d7 E \u21a6 (p.1 - \u2016p.2\u2016) / (p.1 - 1)) (R, 0) :=\n (continuousAt_fst.sub continuousAt_snd.norm).div (continuousAt_fst.sub continuousAt_const) hR.ne'\n[GOAL]\ncase mk.intro.h.inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nR : \u211d\nhR\u271d : 1 < R\nhR : 0 < R - 1\nA : ContinuousAt (fun p => (p.fst - \u2016p.snd\u2016) / (p.fst - 1)) (R, 0)\n\u22a2 ContDiffAt \u211d \u22a4 (uncurry fun R x => smoothTransition ((R - \u2016x\u2016) / (R - 1))) (R, 0)\n[PROOFSTEP]\nhave B : \u2200\u1da0 p in \ud835\udcdd (R, (0 : E)), 1 \u2264 (p.1 - \u2016p.2\u2016) / (p.1 - 1) :=\n A.eventually <| le_mem_nhds <| (one_lt_div hR).2 <| sub_lt_sub_left (by simp) _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nR : \u211d\nhR\u271d : 1 < R\nhR : 0 < R - 1\nA : ContinuousAt (fun p => (p.fst - \u2016p.snd\u2016) / (p.fst - 1)) (R, 0)\n\u22a2 \u2016(R, 0).snd\u2016 < 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.intro.h.inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nR : \u211d\nhR\u271d : 1 < R\nhR : 0 < R - 1\nA : ContinuousAt (fun p => (p.fst - \u2016p.snd\u2016) / (p.fst - 1)) (R, 0)\nB : \u2200\u1da0 (p : \u211d \u00d7 E) in \ud835\udcdd (R, 0), 1 \u2264 (p.fst - \u2016p.snd\u2016) / (p.fst - 1)\n\u22a2 ContDiffAt \u211d \u22a4 (uncurry fun R x => smoothTransition ((R - \u2016x\u2016) / (R - 1))) (R, 0)\n[PROOFSTEP]\nrefine (contDiffAt_const (c := 1)).congr_of_eventuallyEq <| B.mono fun _ \u21a6 smoothTransition.one_of_one_le\n[GOAL]\ncase mk.intro.h.inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nR : \u211d\nx : E\nhR\u271d : 1 < R\nhR : 0 < R - 1\nhx : x \u2260 0\n\u22a2 ContDiffAt \u211d \u22a4 (uncurry fun R x => smoothTransition ((R - \u2016x\u2016) / (R - 1))) (R, x)\n[PROOFSTEP]\nrefine smoothTransition.contDiffAt.comp _ (ContDiffAt.div ?_ ?_ hR.ne')\n[GOAL]\ncase mk.intro.h.inr.refine_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nR : \u211d\nx : E\nhR\u271d : 1 < R\nhR : 0 < R - 1\nhx : x \u2260 0\n\u22a2 ContDiffAt \u211d \u22a4 (fun a => a.fst - \u2016a.snd\u2016) (R, x)\n[PROOFSTEP]\nexact contDiffAt_fst.sub (contDiffAt_snd.norm \u211d hx)\n[GOAL]\ncase mk.intro.h.inr.refine_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nR : \u211d\nx : E\nhR\u271d : 1 < R\nhR : 0 < R - 1\nhx : x \u2260 0\n\u22a2 ContDiffAt \u211d \u22a4 (fun a => a.fst - 1) (R, x)\n[PROOFSTEP]\nexact contDiffAt_fst.sub contDiffAt_const\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nR : \u211d\nhR : 1 < R\n\u22a2 Function.support ((fun R x => smoothTransition ((R - \u2016x\u2016) / (R - 1))) R) = Metric.ball 0 R\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nR : \u211d\nhR : 1 < R\nx : E\n\u22a2 x \u2208 Function.support ((fun R x => smoothTransition ((R - \u2016x\u2016) / (R - 1))) R) \u2194 x \u2208 Metric.ball 0 R\n[PROOFSTEP]\nrw [mem_support, Ne.def, smoothTransition.zero_iff_nonpos, not_le, mem_ball_zero_iff]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nR : \u211d\nhR : 1 < R\nx : E\n\u22a2 0 < (R - \u2016x\u2016) / (R - 1) \u2194 \u2016x\u2016 < R\n[PROOFSTEP]\nsimp [div_pos_iff, sq_lt_sq, abs_of_pos (one_pos.trans hR), hR, hR.not_lt]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "llama_tokens": 2329, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056040203135, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.4111247101436737}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : TopologicalAddGroup E\n\u03b9 : Type u_3\nb : \u03b9 \u2192 Set E\np : \u03b9 \u2192 Prop\nhbasis : HasBasis (\ud835\udcdd 0) p b\nhconvex : \u2200 (i : \u03b9), p i \u2192 Convex \ud835\udd5c (b i)\n\u22a2 LocallyConvexSpace \ud835\udd5c E\n[PROOFSTEP]\nrefine'\n LocallyConvexSpace.ofBases \ud835\udd5c E (fun (x : E) (i : \u03b9) => (\u00b7 + \u00b7) x '' b i) (fun _ => p) (fun x => _) fun x i hi =>\n (hconvex i hi).translate x\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : TopologicalAddGroup E\n\u03b9 : Type u_3\nb : \u03b9 \u2192 Set E\np : \u03b9 \u2192 Prop\nhbasis : HasBasis (\ud835\udcdd 0) p b\nhconvex : \u2200 (i : \u03b9), p i \u2192 Convex \ud835\udd5c (b i)\nx : E\n\u22a2 HasBasis (\ud835\udcdd x) ((fun x => p) x) ((fun x i => (fun x x_1 => x + x_1) x '' b i) x)\n[PROOFSTEP]\nrw [\u2190 map_add_left_nhds_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : TopologicalAddGroup E\n\u03b9 : Type u_3\nb : \u03b9 \u2192 Set E\np : \u03b9 \u2192 Prop\nhbasis : HasBasis (\ud835\udcdd 0) p b\nhconvex : \u2200 (i : \u03b9), p i \u2192 Convex \ud835\udd5c (b i)\nx : E\n\u22a2 HasBasis (map ((fun x x_1 => x + x_1) x) (\ud835\udcdd 0)) ((fun x => p) x) ((fun x i => (fun x x_1 => x + x_1) x '' b i) x)\n[PROOFSTEP]\nexact hbasis.map _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2076 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c E\ninst\u271d : LocallyConvexSpace \ud835\udd5c E\ns t : Set E\ndisj : Disjoint s t\nhs\u2081 : Convex \ud835\udd5c s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \ud835\udd5c t\nht\u2082 : IsClosed t\n\u22a2 \u2203 u v, IsOpen u \u2227 IsOpen v \u2227 Convex \ud835\udd5c u \u2227 Convex \ud835\udd5c v \u2227 s \u2286 u \u2227 t \u2286 v \u2227 Disjoint u v\n[PROOFSTEP]\nletI : UniformSpace E := TopologicalAddGroup.toUniformSpace E\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2076 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c E\ninst\u271d : LocallyConvexSpace \ud835\udd5c E\ns t : Set E\ndisj : Disjoint s t\nhs\u2081 : Convex \ud835\udd5c s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \ud835\udd5c t\nht\u2082 : IsClosed t\nthis : UniformSpace E := TopologicalAddGroup.toUniformSpace E\n\u22a2 \u2203 u v, IsOpen u \u2227 IsOpen v \u2227 Convex \ud835\udd5c u \u2227 Convex \ud835\udd5c v \u2227 s \u2286 u \u2227 t \u2286 v \u2227 Disjoint u v\n[PROOFSTEP]\nhaveI : UniformAddGroup E := comm_topologicalAddGroup_is_uniform\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2076 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c E\ninst\u271d : LocallyConvexSpace \ud835\udd5c E\ns t : Set E\ndisj : Disjoint s t\nhs\u2081 : Convex \ud835\udd5c s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \ud835\udd5c t\nht\u2082 : IsClosed t\nthis\u271d : UniformSpace E := TopologicalAddGroup.toUniformSpace E\nthis : UniformAddGroup E\n\u22a2 \u2203 u v, IsOpen u \u2227 IsOpen v \u2227 Convex \ud835\udd5c u \u2227 Convex \ud835\udd5c v \u2227 s \u2286 u \u2227 t \u2286 v \u2227 Disjoint u v\n[PROOFSTEP]\nhave := (LocallyConvexSpace.convex_open_basis_zero \ud835\udd5c E).comap fun x : E \u00d7 E => x.2 - x.1\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2076 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c E\ninst\u271d : LocallyConvexSpace \ud835\udd5c E\ns t : Set E\ndisj : Disjoint s t\nhs\u2081 : Convex \ud835\udd5c s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \ud835\udd5c t\nht\u2082 : IsClosed t\nthis\u271d\u00b9 : UniformSpace E := TopologicalAddGroup.toUniformSpace E\nthis\u271d : UniformAddGroup E\nthis :\n HasBasis (comap (fun x => x.snd - x.fst) (\ud835\udcdd 0)) (fun s => 0 \u2208 s \u2227 IsOpen s \u2227 Convex \ud835\udd5c s) fun i =>\n (fun x => x.snd - x.fst) \u207b\u00b9' id i\n\u22a2 \u2203 u v, IsOpen u \u2227 IsOpen v \u2227 Convex \ud835\udd5c u \u2227 Convex \ud835\udd5c v \u2227 s \u2286 u \u2227 t \u2286 v \u2227 Disjoint u v\n[PROOFSTEP]\nrw [\u2190 uniformity_eq_comap_nhds_zero] at this \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2076 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c E\ninst\u271d : LocallyConvexSpace \ud835\udd5c E\ns t : Set E\ndisj : Disjoint s t\nhs\u2081 : Convex \ud835\udd5c s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \ud835\udd5c t\nht\u2082 : IsClosed t\nthis\u271d\u00b9 : UniformSpace E := TopologicalAddGroup.toUniformSpace E\nthis\u271d : UniformAddGroup E\nthis : HasBasis (uniformity E) (fun s => 0 \u2208 s \u2227 IsOpen s \u2227 Convex \ud835\udd5c s) fun i => (fun x => x.snd - x.fst) \u207b\u00b9' id i\n\u22a2 \u2203 u v, IsOpen u \u2227 IsOpen v \u2227 Convex \ud835\udd5c u \u2227 Convex \ud835\udd5c v \u2227 s \u2286 u \u2227 t \u2286 v \u2227 Disjoint u v\n[PROOFSTEP]\nrcases disj.exists_uniform_thickening_of_basis this hs\u2082 ht\u2082 with \u27e8V, \u27e8hV0, hVopen, hVconvex\u27e9, hV\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2076 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c E\ninst\u271d : LocallyConvexSpace \ud835\udd5c E\ns t : Set E\ndisj : Disjoint s t\nhs\u2081 : Convex \ud835\udd5c s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \ud835\udd5c t\nht\u2082 : IsClosed t\nthis\u271d\u00b9 : UniformSpace E := TopologicalAddGroup.toUniformSpace E\nthis\u271d : UniformAddGroup E\nthis : HasBasis (uniformity E) (fun s => 0 \u2208 s \u2227 IsOpen s \u2227 Convex \ud835\udd5c s) fun i => (fun x => x.snd - x.fst) \u207b\u00b9' id i\nV : Set E\nhV :\n Disjoint (\u22c3 (x : E) (_ : x \u2208 s), UniformSpace.ball x ((fun x => x.snd - x.fst) \u207b\u00b9' id V))\n (\u22c3 (x : E) (_ : x \u2208 t), UniformSpace.ball x ((fun x => x.snd - x.fst) \u207b\u00b9' id V))\nhV0 : 0 \u2208 V\nhVopen : IsOpen V\nhVconvex : Convex \ud835\udd5c V\n\u22a2 \u2203 u v, IsOpen u \u2227 IsOpen v \u2227 Convex \ud835\udd5c u \u2227 Convex \ud835\udd5c v \u2227 s \u2286 u \u2227 t \u2286 v \u2227 Disjoint u v\n[PROOFSTEP]\nrefine'\n \u27e8s + V, t + V, hVopen.add_left, hVopen.add_left, hs\u2081.add hVconvex, ht\u2081.add hVconvex, subset_add_left _ hV0,\n subset_add_left _ hV0, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2076 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c E\ninst\u271d : LocallyConvexSpace \ud835\udd5c E\ns t : Set E\ndisj : Disjoint s t\nhs\u2081 : Convex \ud835\udd5c s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \ud835\udd5c t\nht\u2082 : IsClosed t\nthis\u271d\u00b9 : UniformSpace E := TopologicalAddGroup.toUniformSpace E\nthis\u271d : UniformAddGroup E\nthis : HasBasis (uniformity E) (fun s => 0 \u2208 s \u2227 IsOpen s \u2227 Convex \ud835\udd5c s) fun i => (fun x => x.snd - x.fst) \u207b\u00b9' id i\nV : Set E\nhV :\n Disjoint (\u22c3 (x : E) (_ : x \u2208 s), UniformSpace.ball x ((fun x => x.snd - x.fst) \u207b\u00b9' id V))\n (\u22c3 (x : E) (_ : x \u2208 t), UniformSpace.ball x ((fun x => x.snd - x.fst) \u207b\u00b9' id V))\nhV0 : 0 \u2208 V\nhVopen : IsOpen V\nhVconvex : Convex \ud835\udd5c V\n\u22a2 Disjoint (s + V) (t + V)\n[PROOFSTEP]\nsimp_rw [\u2190 iUnion_add_left_image, image_add_left]\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2076 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c E\ninst\u271d : LocallyConvexSpace \ud835\udd5c E\ns t : Set E\ndisj : Disjoint s t\nhs\u2081 : Convex \ud835\udd5c s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \ud835\udd5c t\nht\u2082 : IsClosed t\nthis\u271d\u00b9 : UniformSpace E := TopologicalAddGroup.toUniformSpace E\nthis\u271d : UniformAddGroup E\nthis : HasBasis (uniformity E) (fun s => 0 \u2208 s \u2227 IsOpen s \u2227 Convex \ud835\udd5c s) fun i => (fun x => x.snd - x.fst) \u207b\u00b9' id i\nV : Set E\nhV :\n Disjoint (\u22c3 (x : E) (_ : x \u2208 s), UniformSpace.ball x ((fun x => x.snd - x.fst) \u207b\u00b9' id V))\n (\u22c3 (x : E) (_ : x \u2208 t), UniformSpace.ball x ((fun x => x.snd - x.fst) \u207b\u00b9' id V))\nhV0 : 0 \u2208 V\nhVopen : IsOpen V\nhVconvex : Convex \ud835\udd5c V\n\u22a2 Disjoint (\u22c3 (a : E) (_ : a \u2208 s), (fun x => -a + x) \u207b\u00b9' V) (\u22c3 (a : E) (_ : a \u2208 t), (fun x => -a + x) \u207b\u00b9' V)\n[PROOFSTEP]\nsimp_rw [UniformSpace.ball, \u2190 preimage_comp, sub_eq_neg_add] at hV \n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2076 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalAddGroup E\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c E\ninst\u271d : LocallyConvexSpace \ud835\udd5c E\ns t : Set E\ndisj : Disjoint s t\nhs\u2081 : Convex \ud835\udd5c s\nhs\u2082 : IsCompact s\nht\u2081 : Convex \ud835\udd5c t\nht\u2082 : IsClosed t\nthis\u271d\u00b9 : UniformSpace E := TopologicalAddGroup.toUniformSpace E\nthis\u271d : UniformAddGroup E\nthis : HasBasis (uniformity E) (fun s => 0 \u2208 s \u2227 IsOpen s \u2227 Convex \ud835\udd5c s) fun i => (fun x => x.snd - x.fst) \u207b\u00b9' id i\nV : Set E\nhV0 : 0 \u2208 V\nhVopen : IsOpen V\nhVconvex : Convex \ud835\udd5c V\nhV :\n Disjoint (\u22c3 (x : E) (_ : x \u2208 s), (fun x => -x.fst + x.snd) \u2218 Prod.mk x \u207b\u00b9' id V)\n (\u22c3 (x : E) (_ : x \u2208 t), (fun x => -x.fst + x.snd) \u2218 Prod.mk x \u207b\u00b9' id V)\n\u22a2 Disjoint (\u22c3 (a : E) (_ : a \u2208 s), (fun x => -a + x) \u207b\u00b9' V) (\u22c3 (a : E) (_ : a \u2208 t), (fun x => -a + x) \u207b\u00b9' V)\n[PROOFSTEP]\nexact hV\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : AddCommMonoid F\ninst\u271d : Module \ud835\udd5c F\nts : Set (TopologicalSpace E)\nh : \u2200 (t : TopologicalSpace E), t \u2208 ts \u2192 LocallyConvexSpace \ud835\udd5c E\n\u22a2 LocallyConvexSpace \ud835\udd5c E\n[PROOFSTEP]\nletI : TopologicalSpace E := sInf ts\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : AddCommMonoid F\ninst\u271d : Module \ud835\udd5c F\nts : Set (TopologicalSpace E)\nh : \u2200 (t : TopologicalSpace E), t \u2208 ts \u2192 LocallyConvexSpace \ud835\udd5c E\nthis : TopologicalSpace E := sInf ts\n\u22a2 LocallyConvexSpace \ud835\udd5c E\n[PROOFSTEP]\nrefine'\n LocallyConvexSpace.ofBases \ud835\udd5c E (fun _ => fun If : Set ts \u00d7 (ts \u2192 Set E) => \u22c2 i \u2208 If.1, If.2 i)\n (fun x => fun If : Set ts \u00d7 (ts \u2192 Set E) => If.1.Finite \u2227 \u2200 i \u2208 If.1, If.2 i \u2208 @nhds _ (\u2191i) x \u2227 Convex \ud835\udd5c (If.2 i))\n (fun x => _) fun x If hif => convex_iInter fun i => convex_iInter fun hi => (hif.2 i hi).2\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : AddCommMonoid F\ninst\u271d : Module \ud835\udd5c F\nts : Set (TopologicalSpace E)\nh : \u2200 (t : TopologicalSpace E), t \u2208 ts \u2192 LocallyConvexSpace \ud835\udd5c E\nthis : TopologicalSpace E := sInf ts\nx : E\n\u22a2 HasBasis (\ud835\udcdd x)\n ((fun x If => Set.Finite If.fst \u2227 \u2200 (i : \u2191ts), i \u2208 If.fst \u2192 Prod.snd If i \u2208 \ud835\udcdd x \u2227 Convex \ud835\udd5c (Prod.snd If i)) x)\n ((fun x If => \u22c2 (i : \u2191ts) (_ : i \u2208 If.fst), Prod.snd If i) x)\n[PROOFSTEP]\nrw [nhds_sInf, \u2190 iInf_subtype'']\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : AddCommMonoid F\ninst\u271d : Module \ud835\udd5c F\nts : Set (TopologicalSpace E)\nh : \u2200 (t : TopologicalSpace E), t \u2208 ts \u2192 LocallyConvexSpace \ud835\udd5c E\nthis : TopologicalSpace E := sInf ts\nx : E\n\u22a2 HasBasis (\u2a05 (i : \u2191ts), \ud835\udcdd x)\n ((fun x If => Set.Finite If.fst \u2227 \u2200 (i : \u2191ts), i \u2208 If.fst \u2192 Prod.snd If i \u2208 \ud835\udcdd x \u2227 Convex \ud835\udd5c (Prod.snd If i)) x)\n ((fun x If => \u22c2 (i : \u2191ts) (_ : i \u2208 If.fst), Prod.snd If i) x)\n[PROOFSTEP]\nexact hasBasis_iInf' fun i : ts => (@locallyConvexSpace_iff \ud835\udd5c E _ _ _ \u2191i).mp (h (\u2191i) i.2) x\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : AddCommMonoid F\ninst\u271d : Module \ud835\udd5c F\nts' : \u03b9 \u2192 TopologicalSpace E\nh' : \u2200 (i : \u03b9), LocallyConvexSpace \ud835\udd5c E\n\u22a2 LocallyConvexSpace \ud835\udd5c E\n[PROOFSTEP]\nrefine' locallyConvexSpace_sInf _\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : AddCommMonoid F\ninst\u271d : Module \ud835\udd5c F\nts' : \u03b9 \u2192 TopologicalSpace E\nh' : \u2200 (i : \u03b9), LocallyConvexSpace \ud835\udd5c E\n\u22a2 \u2200 (t : TopologicalSpace E), (t \u2208 range fun i => ts' i) \u2192 LocallyConvexSpace \ud835\udd5c E\n[PROOFSTEP]\nrwa [forall_range_iff]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : AddCommMonoid F\ninst\u271d : Module \ud835\udd5c F\nt\u2081 t\u2082 : TopologicalSpace E\nh\u2081 : LocallyConvexSpace \ud835\udd5c E\nh\u2082 : LocallyConvexSpace \ud835\udd5c E\n\u22a2 LocallyConvexSpace \ud835\udd5c E\n[PROOFSTEP]\nrw [inf_eq_iInf]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : AddCommMonoid F\ninst\u271d : Module \ud835\udd5c F\nt\u2081 t\u2082 : TopologicalSpace E\nh\u2081 : LocallyConvexSpace \ud835\udd5c E\nh\u2082 : LocallyConvexSpace \ud835\udd5c E\n\u22a2 LocallyConvexSpace \ud835\udd5c E\n[PROOFSTEP]\nrefine' locallyConvexSpace_iInf fun b => _\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : AddCommMonoid F\ninst\u271d : Module \ud835\udd5c F\nt\u2081 t\u2082 : TopologicalSpace E\nh\u2081 : LocallyConvexSpace \ud835\udd5c E\nh\u2082 : LocallyConvexSpace \ud835\udd5c E\nb : Bool\n\u22a2 LocallyConvexSpace \ud835\udd5c E\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : AddCommMonoid F\ninst\u271d : Module \ud835\udd5c F\nt\u2081 t\u2082 : TopologicalSpace E\nh\u2081 : LocallyConvexSpace \ud835\udd5c E\nh\u2082 : LocallyConvexSpace \ud835\udd5c E\n\u22a2 LocallyConvexSpace \ud835\udd5c E\n[PROOFSTEP]\nassumption\n[GOAL]\ncase true\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : AddCommMonoid F\ninst\u271d : Module \ud835\udd5c F\nt\u2081 t\u2082 : TopologicalSpace E\nh\u2081 : LocallyConvexSpace \ud835\udd5c E\nh\u2082 : LocallyConvexSpace \ud835\udd5c E\n\u22a2 LocallyConvexSpace \ud835\udd5c E\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c F\nt : TopologicalSpace F\ninst\u271d : LocallyConvexSpace \ud835\udd5c F\nf : E \u2192\u2097[\ud835\udd5c] F\n\u22a2 LocallyConvexSpace \ud835\udd5c E\n[PROOFSTEP]\nletI : TopologicalSpace E := t.induced f\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c F\nt : TopologicalSpace F\ninst\u271d : LocallyConvexSpace \ud835\udd5c F\nf : E \u2192\u2097[\ud835\udd5c] F\nthis : TopologicalSpace E := induced (\u2191f) t\n\u22a2 LocallyConvexSpace \ud835\udd5c E\n[PROOFSTEP]\nrefine'\n LocallyConvexSpace.ofBases \ud835\udd5c E (fun _ => preimage f) (fun x => fun s : Set F => s \u2208 \ud835\udcdd (f x) \u2227 Convex \ud835\udd5c s) (fun x => _)\n fun x s \u27e8_, hs\u27e9 => hs.linear_preimage f\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c F\nt : TopologicalSpace F\ninst\u271d : LocallyConvexSpace \ud835\udd5c F\nf : E \u2192\u2097[\ud835\udd5c] F\nthis : TopologicalSpace E := induced (\u2191f) t\nx : E\n\u22a2 HasBasis (\ud835\udcdd x) ((fun x s => s \u2208 \ud835\udcdd (\u2191f x) \u2227 Convex \ud835\udd5c s) x) ((fun x => preimage \u2191f) x)\n[PROOFSTEP]\nrw [nhds_induced]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : OrderedSemiring \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c F\nt : TopologicalSpace F\ninst\u271d : LocallyConvexSpace \ud835\udd5c F\nf : E \u2192\u2097[\ud835\udd5c] F\nthis : TopologicalSpace E := induced (\u2191f) t\nx : E\n\u22a2 HasBasis (comap (\u2191f) (\ud835\udcdd (\u2191f x))) ((fun x s => s \u2208 \ud835\udcdd (\u2191f x) \u2227 Convex \ud835\udd5c s) x) ((fun x => preimage \u2191f) x)\n[PROOFSTEP]\nexact (LocallyConvexSpace.convex_basis <| f x).comap f\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Module.LocallyConvex", "llama_tokens": 6951, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7905303285397349, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.4106973613261855}} {"text": "[GOAL]\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\nh : IsMatching M\nv w : V\nhv : v \u2208 M.verts\nhvw : Adj M v w\n\u22a2 toEdge h { val := v, property := hv } = { val := Quotient.mk (Sym2.Rel.setoid V) (v, w), property := hvw }\n[PROOFSTEP]\nsimp only [IsMatching.toEdge, Subtype.mk_eq_mk]\n[GOAL]\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\nh : IsMatching M\nv w : V\nhv : v \u2208 M.verts\nhvw : Adj M v w\n\u22a2 Quotient.mk (Sym2.Rel.setoid V) (v, Exists.choose (_ : \u2203! w, Adj M (\u2191{ val := v, property := hv }) w)) =\n Quotient.mk (Sym2.Rel.setoid V) (v, w)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_snd\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\nh : IsMatching M\nv w : V\nhv : v \u2208 M.verts\nhvw : Adj M v w\n\u22a2 Exists.choose (_ : \u2203! w, Adj M (\u2191{ val := v, property := hv }) w) = w\n[PROOFSTEP]\nexact ((h (M.edge_vert hvw)).choose_spec.2 w hvw).symm\n[GOAL]\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\nh : IsMatching M\n\u22a2 Function.Surjective (toEdge h)\n[PROOFSTEP]\nrintro \u27e8e, he\u27e9\n[GOAL]\ncase mk\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\nh : IsMatching M\ne : Sym2 V\nhe : e \u2208 edgeSet M\n\u22a2 \u2203 a, toEdge h a = { val := e, property := he }\n[PROOFSTEP]\nrefine Sym2.ind (fun x y he => ?_) e he\n[GOAL]\ncase mk\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\nh : IsMatching M\ne : Sym2 V\nhe\u271d : e \u2208 edgeSet M\nx y : V\nhe : Quotient.mk (Sym2.Rel.setoid V) (x, y) \u2208 edgeSet M\n\u22a2 \u2203 a, toEdge h a = { val := Quotient.mk (Sym2.Rel.setoid V) (x, y), property := he }\n[PROOFSTEP]\nexact \u27e8\u27e8x, M.edge_vert he\u27e9, h.toEdge_eq_of_adj _ he\u27e9\n[GOAL]\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\nv w : V\nh : IsMatching M\nhv : v \u2208 M.verts\nhw : w \u2208 M.verts\nha : Adj M v w\n\u22a2 toEdge h { val := v, property := hv } = toEdge h { val := w, property := hw }\n[PROOFSTEP]\nrw [h.toEdge_eq_of_adj hv ha, h.toEdge_eq_of_adj hw (M.symm ha), Subtype.mk_eq_mk, Sym2.eq_swap]\n[GOAL]\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\nh : IsMatching M\n\u22a2 support M = M.verts\n[PROOFSTEP]\nrefine M.support_subset_verts.antisymm fun v hv => ?_\n[GOAL]\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\nh : IsMatching M\nv : V\nhv : v \u2208 M.verts\n\u22a2 v \u2208 support M\n[PROOFSTEP]\nobtain \u27e8w, hvw, -\u27e9 := h hv\n[GOAL]\ncase intro.intro\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\nh : IsMatching M\nv : V\nhv : v \u2208 M.verts\nw : V\nhvw : Adj M v w\n\u22a2 v \u2208 support M\n[PROOFSTEP]\nexact \u27e8_, hvw\u27e9\n[GOAL]\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\ninst\u271d : (v : V) \u2192 Fintype \u2191(neighborSet M v)\n\u22a2 IsMatching M \u2194 \u2200 (v : V), v \u2208 M.verts \u2192 degree M v = 1\n[PROOFSTEP]\nsimp only [degree_eq_one_iff_unique_adj, IsMatching]\n[GOAL]\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\ninst\u271d : Fintype \u2191M.verts\nh : IsMatching M\n\u22a2 Even (Finset.card (Set.toFinset M.verts))\n[PROOFSTEP]\nclassical\nrw [isMatching_iff_forall_degree] at h \nuse M.coe.edgeFinset.card\nrw [\u2190 two_mul, \u2190 M.coe.sum_degrees_eq_twice_card_edges]\n -- Porting note: `SimpleGraph.Subgraph.coe_degree` does not trigger because it uses\n -- instance arguments instead of implicit arguments for the first `Fintype` argument.\n -- Using a `convert_to` to swap out the `Fintype` instance to the \"right\" one.\nconvert_to _ = Finset.sum Finset.univ fun v => SimpleGraph.degree (Subgraph.coe M) v using 3\nsimp [h, Finset.card_univ]\n[GOAL]\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\ninst\u271d : Fintype \u2191M.verts\nh : IsMatching M\n\u22a2 Even (Finset.card (Set.toFinset M.verts))\n[PROOFSTEP]\nrw [isMatching_iff_forall_degree] at h \n[GOAL]\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\ninst\u271d : Fintype \u2191M.verts\nh\u271d : IsMatching M\nh : \u2200 (v : V), v \u2208 M.verts \u2192 degree M v = 1\n\u22a2 Even (Finset.card (Set.toFinset M.verts))\n[PROOFSTEP]\nuse M.coe.edgeFinset.card\n[GOAL]\ncase h\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\ninst\u271d : Fintype \u2191M.verts\nh\u271d : IsMatching M\nh : \u2200 (v : V), v \u2208 M.verts \u2192 degree M v = 1\n\u22a2 Finset.card (Set.toFinset M.verts) =\n Finset.card (edgeFinset (Subgraph.coe M)) + Finset.card (edgeFinset (Subgraph.coe M))\n[PROOFSTEP]\nrw [\u2190 two_mul, \u2190 M.coe.sum_degrees_eq_twice_card_edges]\n -- Porting note: `SimpleGraph.Subgraph.coe_degree` does not trigger because it uses\n -- instance arguments instead of implicit arguments for the first `Fintype` argument.\n -- Using a `convert_to` to swap out the `Fintype` instance to the \"right\" one.\n[GOAL]\ncase h\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\ninst\u271d : Fintype \u2191M.verts\nh\u271d : IsMatching M\nh : \u2200 (v : V), v \u2208 M.verts \u2192 degree M v = 1\n\u22a2 Finset.card (Set.toFinset M.verts) = Finset.sum Finset.univ fun v => SimpleGraph.degree (Subgraph.coe M) v\n[PROOFSTEP]\nconvert_to _ = Finset.sum Finset.univ fun v => SimpleGraph.degree (Subgraph.coe M) v using 3\n[GOAL]\ncase h.convert_2\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\ninst\u271d : Fintype \u2191M.verts\nh\u271d : IsMatching M\nh : \u2200 (v : V), v \u2208 M.verts \u2192 degree M v = 1\n\u22a2 Finset.card (Set.toFinset M.verts) = Finset.sum Finset.univ fun v => SimpleGraph.degree (Subgraph.coe M) v\n[PROOFSTEP]\nsimp [h, Finset.card_univ]\n[GOAL]\nV : Type u\nG : SimpleGraph V\nM : Subgraph G\n\u22a2 IsPerfectMatching M \u2194 \u2200 (v : V), \u2203! w, Adj M v w\n[PROOFSTEP]\nrefine' \u27e8_, fun hm => \u27e8fun v _ => hm v, fun v => _\u27e9\u27e9\n[GOAL]\ncase refine'_1\nV : Type u\nG : SimpleGraph V\nM : Subgraph G\n\u22a2 IsPerfectMatching M \u2192 \u2200 (v : V), \u2203! w, Adj M v w\n[PROOFSTEP]\nrintro \u27e8hm, hs\u27e9 v\n[GOAL]\ncase refine'_1.intro\nV : Type u\nG : SimpleGraph V\nM : Subgraph G\nhm : IsMatching M\nhs : IsSpanning M\nv : V\n\u22a2 \u2203! w, Adj M v w\n[PROOFSTEP]\nexact hm (hs v)\n[GOAL]\ncase refine'_2\nV : Type u\nG : SimpleGraph V\nM : Subgraph G\nhm : \u2200 (v : V), \u2203! w, Adj M v w\nv : V\n\u22a2 v \u2208 M.verts\n[PROOFSTEP]\nobtain \u27e8w, hw, -\u27e9 := hm v\n[GOAL]\ncase refine'_2.intro.intro\nV : Type u\nG : SimpleGraph V\nM : Subgraph G\nhm : \u2200 (v : V), \u2203! w, Adj M v w\nv w : V\nhw : Adj M v w\n\u22a2 v \u2208 M.verts\n[PROOFSTEP]\nexact M.edge_vert hw\n[GOAL]\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\ninst\u271d : (v : V) \u2192 Fintype \u2191(neighborSet M v)\n\u22a2 IsPerfectMatching M \u2194 \u2200 (v : V), degree M v = 1\n[PROOFSTEP]\nsimp [degree_eq_one_iff_unique_adj, isPerfectMatching_iff]\n[GOAL]\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\ninst\u271d : Fintype V\nh : IsPerfectMatching M\n\u22a2 Even (Fintype.card V)\n[PROOFSTEP]\nclassical simpa only [h.2.card_verts] using IsMatching.even_card h.1\n[GOAL]\nV : Type u\nG : SimpleGraph V\nM\u271d M : Subgraph G\ninst\u271d : Fintype V\nh : IsPerfectMatching M\n\u22a2 Even (Fintype.card V)\n[PROOFSTEP]\nsimpa only [h.2.card_verts] using IsMatching.even_card h.1\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SimpleGraph.Matching", "llama_tokens": 2928, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.640635854839898, "lm_q1q2_score": 0.4104143072974276}} {"text": "[GOAL]\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Free R M\ninst\u271d\u2074 : AddCommGroup N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Finite R M\ninst\u271d : Finite R N\n\u22a2 Free R (M \u2192\u2097[R] N)\n[PROOFSTEP]\ncases subsingleton_or_nontrivial R\n[GOAL]\ncase inl\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Free R M\ninst\u271d\u2074 : AddCommGroup N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Finite R M\ninst\u271d : Finite R N\nh\u271d : Subsingleton R\n\u22a2 Free R (M \u2192\u2097[R] N)\n[PROOFSTEP]\napply Module.Free.of_subsingleton'\n[GOAL]\ncase inr\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Free R M\ninst\u271d\u2074 : AddCommGroup N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Finite R M\ninst\u271d : Finite R N\nh\u271d : Nontrivial R\n\u22a2 Free R (M \u2192\u2097[R] N)\n[PROOFSTEP]\nclassical exact Module.Free.of_equiv (LinearMap.toMatrix (chooseBasis R M) (chooseBasis R N)).symm\n[GOAL]\ncase inr\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Free R M\ninst\u271d\u2074 : AddCommGroup N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Finite R M\ninst\u271d : Finite R N\nh\u271d : Nontrivial R\n\u22a2 Free R (M \u2192\u2097[R] N)\n[PROOFSTEP]\nexact Module.Free.of_equiv (LinearMap.toMatrix (chooseBasis R M) (chooseBasis R N)).symm\n[GOAL]\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Free R M\ninst\u271d\u2074 : AddCommGroup N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Finite R M\ninst\u271d : Finite R N\n\u22a2 Finite R (M \u2192\u2097[R] N)\n[PROOFSTEP]\ncases subsingleton_or_nontrivial R\n[GOAL]\ncase inl\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Free R M\ninst\u271d\u2074 : AddCommGroup N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Finite R M\ninst\u271d : Finite R N\nh\u271d : Subsingleton R\n\u22a2 Finite R (M \u2192\u2097[R] N)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase inr\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Free R M\ninst\u271d\u2074 : AddCommGroup N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Finite R M\ninst\u271d : Finite R N\nh\u271d : Nontrivial R\n\u22a2 Finite R (M \u2192\u2097[R] N)\n[PROOFSTEP]\nclassical\nhave f := (LinearMap.toMatrix (chooseBasis R M) (chooseBasis R N)).symm\nexact Module.Finite.of_surjective f.toLinearMap (LinearEquiv.surjective f)\n[GOAL]\ncase inr\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Free R M\ninst\u271d\u2074 : AddCommGroup N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Finite R M\ninst\u271d : Finite R N\nh\u271d : Nontrivial R\n\u22a2 Finite R (M \u2192\u2097[R] N)\n[PROOFSTEP]\nhave f := (LinearMap.toMatrix (chooseBasis R M) (chooseBasis R N)).symm\n[GOAL]\ncase inr\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Free R M\ninst\u271d\u2074 : AddCommGroup N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Finite R M\ninst\u271d : Finite R N\nh\u271d : Nontrivial R\nf : Matrix (Free.ChooseBasisIndex R N) (Free.ChooseBasisIndex R M) R \u2243\u2097[R] M \u2192\u2097[R] N\n\u22a2 Finite R (M \u2192\u2097[R] N)\n[PROOFSTEP]\nexact Module.Finite.of_surjective f.toLinearMap (LinearEquiv.surjective f)\n[GOAL]\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : StrongRankCondition R\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module.Free R M\ninst\u271d\u2074 : Module.Finite R M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module.Free R N\ninst\u271d : Module.Finite R N\n\u22a2 finrank R (M \u2192\u2097[R] N) = finrank R M * finrank R N\n[PROOFSTEP]\nclassical\nletI := nontrivial_of_invariantBasisNumber R\nhave h := LinearMap.toMatrix (chooseBasis R M) (chooseBasis R N)\nsimp_rw [h.finrank_eq, FiniteDimensional.finrank_matrix, FiniteDimensional.finrank_eq_card_chooseBasisIndex, mul_comm]\n[GOAL]\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : StrongRankCondition R\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module.Free R M\ninst\u271d\u2074 : Module.Finite R M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module.Free R N\ninst\u271d : Module.Finite R N\n\u22a2 finrank R (M \u2192\u2097[R] N) = finrank R M * finrank R N\n[PROOFSTEP]\nletI := nontrivial_of_invariantBasisNumber R\n[GOAL]\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : StrongRankCondition R\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module.Free R M\ninst\u271d\u2074 : Module.Finite R M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module.Free R N\ninst\u271d : Module.Finite R N\nthis : Nontrivial R := nontrivial_of_invariantBasisNumber R\n\u22a2 finrank R (M \u2192\u2097[R] N) = finrank R M * finrank R N\n[PROOFSTEP]\nhave h := LinearMap.toMatrix (chooseBasis R M) (chooseBasis R N)\n[GOAL]\nR : Type u\nM : Type v\nN : Type w\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : StrongRankCondition R\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module.Free R M\ninst\u271d\u2074 : Module.Finite R M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module.Free R N\ninst\u271d : Module.Finite R N\nthis : Nontrivial R := nontrivial_of_invariantBasisNumber R\nh : (M \u2192\u2097[R] N) \u2243\u2097[R] Matrix (Module.Free.ChooseBasisIndex R N) (Module.Free.ChooseBasisIndex R M) R\n\u22a2 finrank R (M \u2192\u2097[R] N) = finrank R M * finrank R N\n[PROOFSTEP]\nsimp_rw [h.finrank_eq, FiniteDimensional.finrank_matrix, FiniteDimensional.finrank_eq_card_chooseBasisIndex, mul_comm]\n[GOAL]\nR : Type u\nM : Type v\nN : Type w\nK m n : Type u\ninst\u271d\u00b3 : CommRing K\ninst\u271d\u00b2 : StrongRankCondition K\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nw : m \u2192 K\nv : n \u2192 K\n\u22a2 LinearMap.rank (\u2191toLin' (vecMulVec w v)) \u2264 1\n[PROOFSTEP]\nrw [Matrix.vecMulVec_eq, Matrix.toLin'_mul]\n[GOAL]\nR : Type u\nM : Type v\nN : Type w\nK m n : Type u\ninst\u271d\u00b3 : CommRing K\ninst\u271d\u00b2 : StrongRankCondition K\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nw : m \u2192 K\nv : n \u2192 K\n\u22a2 LinearMap.rank (LinearMap.comp (\u2191toLin' (col w)) (\u2191toLin' (row v))) \u2264 1\n[PROOFSTEP]\nrefine' le_trans (LinearMap.rank_comp_le_left _ _) _\n[GOAL]\nR : Type u\nM : Type v\nN : Type w\nK m n : Type u\ninst\u271d\u00b3 : CommRing K\ninst\u271d\u00b2 : StrongRankCondition K\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nw : m \u2192 K\nv : n \u2192 K\n\u22a2 LinearMap.rank (\u2191toLin' (col w)) \u2264 1\n[PROOFSTEP]\nrefine' (LinearMap.rank_le_domain _).trans_eq _\n[GOAL]\nR : Type u\nM : Type v\nN : Type w\nK m n : Type u\ninst\u271d\u00b3 : CommRing K\ninst\u271d\u00b2 : StrongRankCondition K\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nw : m \u2192 K\nv : n \u2192 K\n\u22a2 Module.rank K (Unit \u2192 K) = 1\n[PROOFSTEP]\nrw [rank_fun', Fintype.card_unit, Nat.cast_one]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "llama_tokens": 3016, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.41040482267156103}} {"text": "[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d\u00b9 : IsEmpty \u03b9\ninst\u271d : InfSet \u03b1\nf : \u03b9 \u2192 WithTop \u03b1\n\u22a2 \u2a05 (i : \u03b9), f i = \u22a4\n[PROOFSTEP]\nrw [iInf, range_eq_empty, WithTop.sInf_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : InfSet \u03b1\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2191(sInf s) = sInf ((fun a => \u2191a) '' s)\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := hs\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : InfSet \u03b1\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2191(sInf s) = sInf ((fun a => \u2191a) '' s)\n[PROOFSTEP]\nchange _ = ite _ _ _\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : InfSet \u03b1\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2191(sInf s) = if (fun a => \u2191a) '' s \u2286 {\u22a4} then \u22a4 else \u2191(sInf ((fun a => \u2191a) \u207b\u00b9' ((fun a => \u2191a) '' s)))\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : InfSet \u03b1\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nh : (fun a => \u2191a) '' s \u2286 {\u22a4}\n\u22a2 \u2191(sInf s) = \u22a4\n[PROOFSTEP]\ncases h (mem_image_of_mem _ hx)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : InfSet \u03b1\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nh : \u00ac(fun a => \u2191a) '' s \u2286 {\u22a4}\n\u22a2 \u2191(sInf s) = \u2191(sInf ((fun a => \u2191a) \u207b\u00b9' ((fun a => \u2191a) '' s)))\n[PROOFSTEP]\nrw [preimage_image_eq]\n[GOAL]\ncase neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : InfSet \u03b1\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 s\nh : \u00ac(fun a => \u2191a) '' s \u2286 {\u22a4}\n\u22a2 Injective fun a => \u2191a\n[PROOFSTEP]\nexact Option.some_injective _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : InfSet \u03b1\nf : \u03b9 \u2192 \u03b1\n\u22a2 \u2191(\u2a05 (i : \u03b9), f i) = \u2a05 (i : \u03b9), \u2191(f i)\n[PROOFSTEP]\nrw [iInf, iInf, WithTop.coe_sInf' (range_nonempty f), \u2190 range_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : InfSet \u03b1\nf : \u03b9 \u2192 \u03b1\n\u22a2 sInf (range ((fun a => \u2191a) \u2218 f)) = sInf (range fun i => \u2191(f i))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SupSet \u03b1\ns : Set \u03b1\nhs : BddAbove s\n\u22a2 \u2191(sSup s) = sSup ((fun a => \u2191a) '' s)\n[PROOFSTEP]\nchange _ = ite _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SupSet \u03b1\ns : Set \u03b1\nhs : BddAbove s\n\u22a2 \u2191(sSup s) =\n if \u22a4 \u2208 (fun a => \u2191a) '' s then \u22a4\n else\n if BddAbove ((fun a => \u2191a) \u207b\u00b9' ((fun a => \u2191a) '' s)) then \u2191(sSup ((fun a => \u2191a) \u207b\u00b9' ((fun a => \u2191a) '' s))) else \u22a4\n[PROOFSTEP]\nrw [if_neg, preimage_image_eq, if_pos hs]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SupSet \u03b1\ns : Set \u03b1\nhs : BddAbove s\n\u22a2 Injective fun a => \u2191a\n[PROOFSTEP]\nexact Option.some_injective _\n[GOAL]\ncase hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SupSet \u03b1\ns : Set \u03b1\nhs : BddAbove s\n\u22a2 \u00ac\u22a4 \u2208 (fun a => \u2191a) '' s\n[PROOFSTEP]\nrintro \u27e8x, _, \u27e8\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SupSet \u03b1\nf : \u03b9 \u2192 \u03b1\nh : BddAbove (range f)\n\u22a2 \u2191(\u2a06 (i : \u03b9), f i) = \u2a06 (i : \u03b9), \u2191(f i)\n[PROOFSTEP]\nrw [iSup, iSup, WithTop.coe_sSup' h, \u2190 range_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SupSet \u03b1\nf : \u03b9 \u2192 \u03b1\nh : BddAbove (range f)\n\u22a2 sSup (range ((fun a => \u2191a) \u2218 f)) = sSup (range fun i => \u2191(f i))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\nsrc\u271d : ConditionallyCompleteLinearOrder \u03b1 := inst\u271d\na b : \u03b1\n\u22a2 min a b = if a \u2264 b then a else b\n[PROOFSTEP]\nby_cases hab : a = b\n[GOAL]\ncase pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\nsrc\u271d : ConditionallyCompleteLinearOrder \u03b1 := inst\u271d\na b : \u03b1\nhab : a = b\n\u22a2 min a b = if a \u2264 b then a else b\n[PROOFSTEP]\nsimp [hab]\n[GOAL]\ncase neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\nsrc\u271d : ConditionallyCompleteLinearOrder \u03b1 := inst\u271d\na b : \u03b1\nhab : \u00aca = b\n\u22a2 min a b = if a \u2264 b then a else b\n[PROOFSTEP]\nrcases ConditionallyCompleteLinearOrder.le_total a b with (h\u2081 | h\u2082)\n[GOAL]\ncase neg.inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\nsrc\u271d : ConditionallyCompleteLinearOrder \u03b1 := inst\u271d\na b : \u03b1\nhab : \u00aca = b\nh\u2081 : a \u2264 b\n\u22a2 min a b = if a \u2264 b then a else b\n[PROOFSTEP]\nsimp [h\u2081]\n[GOAL]\ncase neg.inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\nsrc\u271d : ConditionallyCompleteLinearOrder \u03b1 := inst\u271d\na b : \u03b1\nhab : \u00aca = b\nh\u2082 : b \u2264 a\n\u22a2 min a b = if a \u2264 b then a else b\n[PROOFSTEP]\nsimp [show \u00ac(a \u2264 b) from fun h => hab (le_antisymm h h\u2082), h\u2082]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\nsrc\u271d : ConditionallyCompleteLinearOrder \u03b1 := inst\u271d\na b : \u03b1\n\u22a2 max a b = if a \u2264 b then b else a\n[PROOFSTEP]\nby_cases hab : a = b\n[GOAL]\ncase pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\nsrc\u271d : ConditionallyCompleteLinearOrder \u03b1 := inst\u271d\na b : \u03b1\nhab : a = b\n\u22a2 max a b = if a \u2264 b then b else a\n[PROOFSTEP]\nsimp [hab]\n[GOAL]\ncase neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\nsrc\u271d : ConditionallyCompleteLinearOrder \u03b1 := inst\u271d\na b : \u03b1\nhab : \u00aca = b\n\u22a2 max a b = if a \u2264 b then b else a\n[PROOFSTEP]\nrcases ConditionallyCompleteLinearOrder.le_total a b with (h\u2081 | h\u2082)\n[GOAL]\ncase neg.inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\nsrc\u271d : ConditionallyCompleteLinearOrder \u03b1 := inst\u271d\na b : \u03b1\nhab : \u00aca = b\nh\u2081 : a \u2264 b\n\u22a2 max a b = if a \u2264 b then b else a\n[PROOFSTEP]\nsimp [h\u2081]\n[GOAL]\ncase neg.inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\nsrc\u271d : ConditionallyCompleteLinearOrder \u03b1 := inst\u271d\na b : \u03b1\nhab : \u00aca = b\nh\u2082 : b \u2264 a\n\u22a2 max a b = if a \u2264 b then b else a\n[PROOFSTEP]\nsimp [show \u00ac(a \u2264 b) from fun h => hab (le_antisymm h h\u2082), h\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : CompleteLattice \u03b1\nsrc\u271d : CompleteLattice \u03b1 := inst\u271d\n\u22a2 \u2200 (s : Set \u03b1) (a : \u03b1), BddAbove s \u2192 a \u2208 s \u2192 a \u2264 sSup s\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : CompleteLattice \u03b1\nsrc\u271d : CompleteLattice \u03b1 := inst\u271d\ns\u271d : Set \u03b1\na\u271d\u00b2 : \u03b1\na\u271d\u00b9 : BddAbove s\u271d\na\u271d : a\u271d\u00b2 \u2208 s\u271d\n\u22a2 a\u271d\u00b2 \u2264 sSup s\u271d\n[PROOFSTEP]\napply le_sSup\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : CompleteLattice \u03b1\nsrc\u271d : CompleteLattice \u03b1 := inst\u271d\ns\u271d : Set \u03b1\na\u271d\u00b2 : \u03b1\na\u271d\u00b9 : BddAbove s\u271d\na\u271d : a\u271d\u00b2 \u2208 s\u271d\n\u22a2 a\u271d\u00b2 \u2208 s\u271d\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : CompleteLattice \u03b1\nsrc\u271d : CompleteLattice \u03b1 := inst\u271d\n\u22a2 \u2200 (s : Set \u03b1) (a : \u03b1), Set.Nonempty s \u2192 a \u2208 upperBounds s \u2192 sSup s \u2264 a\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : CompleteLattice \u03b1\nsrc\u271d : CompleteLattice \u03b1 := inst\u271d\ns\u271d : Set \u03b1\na\u271d\u00b2 : \u03b1\na\u271d\u00b9 : Set.Nonempty s\u271d\na\u271d : a\u271d\u00b2 \u2208 upperBounds s\u271d\n\u22a2 sSup s\u271d \u2264 a\u271d\u00b2\n[PROOFSTEP]\napply sSup_le\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : CompleteLattice \u03b1\nsrc\u271d : CompleteLattice \u03b1 := inst\u271d\ns\u271d : Set \u03b1\na\u271d\u00b2 : \u03b1\na\u271d\u00b9 : Set.Nonempty s\u271d\na\u271d : a\u271d\u00b2 \u2208 upperBounds s\u271d\n\u22a2 \u2200 (b : \u03b1), b \u2208 s\u271d \u2192 b \u2264 a\u271d\u00b2\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : CompleteLattice \u03b1\nsrc\u271d : CompleteLattice \u03b1 := inst\u271d\n\u22a2 \u2200 (s : Set \u03b1) (a : \u03b1), BddBelow s \u2192 a \u2208 s \u2192 sInf s \u2264 a\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : CompleteLattice \u03b1\nsrc\u271d : CompleteLattice \u03b1 := inst\u271d\ns\u271d : Set \u03b1\na\u271d\u00b2 : \u03b1\na\u271d\u00b9 : BddBelow s\u271d\na\u271d : a\u271d\u00b2 \u2208 s\u271d\n\u22a2 sInf s\u271d \u2264 a\u271d\u00b2\n[PROOFSTEP]\napply sInf_le\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : CompleteLattice \u03b1\nsrc\u271d : CompleteLattice \u03b1 := inst\u271d\ns\u271d : Set \u03b1\na\u271d\u00b2 : \u03b1\na\u271d\u00b9 : BddBelow s\u271d\na\u271d : a\u271d\u00b2 \u2208 s\u271d\n\u22a2 a\u271d\u00b2 \u2208 s\u271d\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : CompleteLattice \u03b1\nsrc\u271d : CompleteLattice \u03b1 := inst\u271d\n\u22a2 \u2200 (s : Set \u03b1) (a : \u03b1), Set.Nonempty s \u2192 a \u2208 lowerBounds s \u2192 a \u2264 sInf s\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : CompleteLattice \u03b1\nsrc\u271d : CompleteLattice \u03b1 := inst\u271d\ns\u271d : Set \u03b1\na\u271d\u00b2 : \u03b1\na\u271d\u00b9 : Set.Nonempty s\u271d\na\u271d : a\u271d\u00b2 \u2208 lowerBounds s\u271d\n\u22a2 a\u271d\u00b2 \u2264 sInf s\u271d\n[PROOFSTEP]\napply le_sInf\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : CompleteLattice \u03b1\nsrc\u271d : CompleteLattice \u03b1 := inst\u271d\ns\u271d : Set \u03b1\na\u271d\u00b2 : \u03b1\na\u271d\u00b9 : Set.Nonempty s\u271d\na\u271d : a\u271d\u00b2 \u2208 lowerBounds s\u271d\n\u22a2 \u2200 (b : \u03b1), b \u2208 s\u271d \u2192 a\u271d\u00b2 \u2264 b\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\na : \u03b1\nhs : BddAbove s\nhas : a \u2208 s\n\u22a2 a \u2264 sSup s\n[PROOFSTEP]\nhave h's : (upperBounds s).Nonempty := hs\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\na : \u03b1\nhs : BddAbove s\nhas : a \u2208 s\nh's : Set.Nonempty (upperBounds s)\n\u22a2 a \u2264 sSup s\n[PROOFSTEP]\nsimp only [h's, dif_pos]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\na : \u03b1\nhs : BddAbove s\nhas : a \u2208 s\nh's : Set.Nonempty (upperBounds s)\n\u22a2 a \u2264 WellFounded.min (_ : WellFounded fun x x_1 => x < x_1) (upperBounds s) (_ : Set.Nonempty (upperBounds s))\n[PROOFSTEP]\nexact h.wf.min_mem _ h's has\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\na : \u03b1\nx\u271d : Set.Nonempty s\nhas : a \u2208 upperBounds s\n\u22a2 sSup s \u2264 a\n[PROOFSTEP]\nhave h's : (upperBounds s).Nonempty := \u27e8a, has\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\na : \u03b1\nx\u271d : Set.Nonempty s\nhas : a \u2208 upperBounds s\nh's : Set.Nonempty (upperBounds s)\n\u22a2 sSup s \u2264 a\n[PROOFSTEP]\nsimp only [h's, dif_pos]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\na : \u03b1\nx\u271d : Set.Nonempty s\nhas : a \u2208 upperBounds s\nh's : Set.Nonempty (upperBounds s)\n\u22a2 WellFounded.min (_ : WellFounded fun x x_1 => x < x_1) (upperBounds s) (_ : Set.Nonempty (upperBounds s)) \u2264 a\n[PROOFSTEP]\nsimpa using h.wf.not_lt_min _ h's has\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\na : \u03b1\nx\u271d : BddBelow s\nhas : a \u2208 s\n\u22a2 sInf s \u2264 a\n[PROOFSTEP]\nhave s_ne : s.Nonempty := \u27e8a, has\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\na : \u03b1\nx\u271d : BddBelow s\nhas : a \u2208 s\ns_ne : Set.Nonempty s\n\u22a2 sInf s \u2264 a\n[PROOFSTEP]\nsimpa [s_ne] using not_lt.1 (h.wf.not_lt_min s s_ne has)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\na : \u03b1\nhs : Set.Nonempty s\nhas : a \u2208 lowerBounds s\n\u22a2 a \u2264 sInf s\n[PROOFSTEP]\nsimp only [hs, dif_pos]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\na : \u03b1\nhs : Set.Nonempty s\nhas : a \u2208 lowerBounds s\n\u22a2 a \u2264 WellFounded.min (_ : WellFounded fun x x_1 => x < x_1) s (_ : Set.Nonempty s)\n[PROOFSTEP]\nexact has (h.wf.min_mem s hs)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\n\u22a2 \u2200 (s : Set \u03b1), \u00acBddAbove s \u2192 sSup s = sSup univ\n[PROOFSTEP]\nintro s H\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\nH : \u00acBddAbove s\n\u22a2 sSup s = sSup univ\n[PROOFSTEP]\nhave A : \u00ac(BddAbove (univ : Set \u03b1)) := by contrapose! H; exact H.mono (subset_univ _)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\nH : \u00acBddAbove s\n\u22a2 \u00acBddAbove univ\n[PROOFSTEP]\ncontrapose! H\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\nH : BddAbove univ\n\u22a2 BddAbove s\n[PROOFSTEP]\nexact H.mono (subset_univ _)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\nH : \u00acBddAbove s\nA : \u00acBddAbove univ\n\u22a2 sSup s = sSup univ\n[PROOFSTEP]\nhave B : \u00ac((upperBounds s).Nonempty) := H\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\nH : \u00acBddAbove s\nA : \u00acBddAbove univ\nB : \u00acSet.Nonempty (upperBounds s)\n\u22a2 sSup s = sSup univ\n[PROOFSTEP]\nhave C : \u00ac((upperBounds (univ : Set \u03b1)).Nonempty) := A\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\ns : Set \u03b1\nH : \u00acBddAbove s\nA : \u00acBddAbove univ\nB : \u00acSet.Nonempty (upperBounds s)\nC : \u00acSet.Nonempty (upperBounds univ)\n\u22a2 sSup s = sSup univ\n[PROOFSTEP]\nsimp [sSup, B, C]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ni\u2081 : LinearOrder \u03b1\ni\u2082 : OrderBot \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\nsrc\u271d : Lattice \u03b1 := LinearOrder.toLattice\n\u22a2 sSup \u2205 = \u22a5\n[PROOFSTEP]\nsimpa using eq_bot_iff.2 (not_lt.1 <| h.wf.not_lt_min _ _ <| mem_univ \u22a5)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLattice \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\ns : Set \u03b2\nH : BddAbove (f '' s)\nHne : Set.Nonempty s\n\u22a2 IsLUB (f '' s) (\u2a06 (i : \u2191s), f \u2191i)\n[PROOFSTEP]\nrw [\u2190 sSup_image']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLattice \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\ns : Set \u03b2\nH : BddAbove (f '' s)\nHne : Set.Nonempty s\n\u22a2 IsLUB (f '' s) (sSup (f '' s))\n[PROOFSTEP]\nexact isLUB_csSup (Hne.image _) H\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b2 : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : DenselyOrdered \u03b1\nw : \u03b1\nhw : a < w\n\u22a2 \u2203 a_1, a_1 \u2208 Ioi a \u2227 a_1 < w\n[PROOFSTEP]\nsimpa using exists_between hw\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b2 : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : DenselyOrdered \u03b1\nw : \u03b1\nhw : w < a\n\u22a2 \u2203 a_1, a_1 \u2208 Iio a \u2227 w < a_1\n[PROOFSTEP]\nsimpa [and_comm] using exists_between hw\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 \u03b1\nc : \u03b1\nH : \u2200 (x : \u03b9), f x \u2264 c\n\u22a2 \u2200 (b : \u03b1), b \u2208 range f \u2192 b \u2264 c\n[PROOFSTEP]\nrwa [forall_range_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\nf g : \u03b9 \u2192 \u03b1\nB : BddAbove (range g)\nH : \u2200 (x : \u03b9), f x \u2264 g x\n\u22a2 iSup f \u2264 iSup g\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\nf g : \u03b9 \u2192 \u03b1\nB : BddAbove (range g)\nH : \u2200 (x : \u03b9), f x \u2264 g x\nh\u271d : IsEmpty \u03b9\n\u22a2 iSup f \u2264 iSup g\n[PROOFSTEP]\nrw [iSup_of_empty', iSup_of_empty']\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\nf g : \u03b9 \u2192 \u03b1\nB : BddAbove (range g)\nH : \u2200 (x : \u03b9), f x \u2264 g x\nh\u271d : Nonempty \u03b9\n\u22a2 iSup f \u2264 iSup g\n[PROOFSTEP]\nexact ciSup_le fun x => le_ciSup_of_le B x (H x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na\u271d b : \u03b1\nh\u03b9 : Nonempty \u03b9\na : \u03b1\n\u22a2 \u2a06 (x : \u03b9), a = a\n[PROOFSTEP]\nrw [iSup, range_const, csSup_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\ninst\u271d : Unique \u03b9\ns : \u03b9 \u2192 \u03b1\n\u22a2 \u2a06 (i : \u03b9), s i = s default\n[PROOFSTEP]\nhave : \u2200 i, s i = s default := fun i => congr_arg s (Unique.eq_default i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\ninst\u271d : Unique \u03b9\ns : \u03b9 \u2192 \u03b1\nthis : \u2200 (i : \u03b9), s i = s default\n\u22a2 \u2a06 (i : \u03b9), s i = s default\n[PROOFSTEP]\nsimp only [this, ciSup_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 \u03b1\nh\u2081 : \u2200 (i : \u03b9), b \u2264 f i\nh\u2082 : \u2200 (w : \u03b1), b < w \u2192 \u2203 i, f i < w\n\u22a2 \u2a05 (i : \u03b9), f i = b\n[PROOFSTEP]\nexact @ciSup_eq_of_forall_le_of_forall_lt_exists_gt \u03b1\u1d52\u1d48 _ _ _ _ \u2039_\u203a \u2039_\u203a \u2039_\u203a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf g : \u03b2 \u2192 \u03b1\nhf : Monotone f\nhg : Antitone g\nh : f \u2264 g\n\u22a2 \u2a06 (n : \u03b2), f n \u2208 \u22c2 (n : \u03b2), Icc (f n) (g n)\n[PROOFSTEP]\nrefine' mem_iInter.2 fun n => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf g : \u03b2 \u2192 \u03b1\nhf : Monotone f\nhg : Antitone g\nh : f \u2264 g\nn : \u03b2\n\u22a2 \u2a06 (n : \u03b2), f n \u2208 Icc (f n) (g n)\n[PROOFSTEP]\nhaveI : Nonempty \u03b2 := \u27e8n\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf g : \u03b2 \u2192 \u03b1\nhf : Monotone f\nhg : Antitone g\nh : f \u2264 g\nn : \u03b2\nthis : Nonempty \u03b2\n\u22a2 \u2a06 (n : \u03b2), f n \u2208 Icc (f n) (g n)\n[PROOFSTEP]\nhave : \u2200 m, f m \u2264 g n := fun m => hf.forall_le_of_antitone hg h m n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf g : \u03b2 \u2192 \u03b1\nhf : Monotone f\nhg : Antitone g\nh : f \u2264 g\nn : \u03b2\nthis\u271d : Nonempty \u03b2\nthis : \u2200 (m : \u03b2), f m \u2264 g n\n\u22a2 \u2a06 (n : \u03b2), f n \u2208 Icc (f n) (g n)\n[PROOFSTEP]\nexact \u27e8le_ciSup \u27e8g <| n, forall_range_iff.2 this\u27e9 _, ciSup_le this\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9 : Type u_5\n\u03b1 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 ConditionallyCompleteLattice (\u03b1 i)\nsrc\u271d\u00b2 : Lattice ((i : \u03b9) \u2192 \u03b1 i) := lattice\nsrc\u271d\u00b9 : SupSet ((i : \u03b9) \u2192 \u03b1 i) := supSet\nsrc\u271d : InfSet ((i : \u03b9) \u2192 \u03b1 i) := infSet\ns : Set ((i : \u03b9) \u2192 \u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\nhs : Set.Nonempty s\nhf : f \u2208 upperBounds s\ni : \u03b9\n\u22a2 Set.Nonempty (range fun f => \u2191f i)\n[PROOFSTEP]\nhaveI := hs.to_subtype\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9 : Type u_5\n\u03b1 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 ConditionallyCompleteLattice (\u03b1 i)\nsrc\u271d\u00b2 : Lattice ((i : \u03b9) \u2192 \u03b1 i) := lattice\nsrc\u271d\u00b9 : SupSet ((i : \u03b9) \u2192 \u03b1 i) := supSet\nsrc\u271d : InfSet ((i : \u03b9) \u2192 \u03b1 i) := infSet\ns : Set ((i : \u03b9) \u2192 \u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\nhs : Set.Nonempty s\nhf : f \u2208 upperBounds s\ni : \u03b9\nthis : Nonempty \u2191s\n\u22a2 Set.Nonempty (range fun f => \u2191f i)\n[PROOFSTEP]\napply range_nonempty\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9 : Type u_5\n\u03b1 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 ConditionallyCompleteLattice (\u03b1 i)\nsrc\u271d\u00b2 : Lattice ((i : \u03b9) \u2192 \u03b1 i) := lattice\nsrc\u271d\u00b9 : SupSet ((i : \u03b9) \u2192 \u03b1 i) := supSet\nsrc\u271d : InfSet ((i : \u03b9) \u2192 \u03b1 i) := infSet\ns : Set ((i : \u03b9) \u2192 \u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\nhs : Set.Nonempty s\nhf : f \u2208 lowerBounds s\ni : \u03b9\n\u22a2 Set.Nonempty (range fun f => \u2191f i)\n[PROOFSTEP]\nhaveI := hs.to_subtype\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9 : Type u_5\n\u03b1 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 ConditionallyCompleteLattice (\u03b1 i)\nsrc\u271d\u00b2 : Lattice ((i : \u03b9) \u2192 \u03b1 i) := lattice\nsrc\u271d\u00b9 : SupSet ((i : \u03b9) \u2192 \u03b1 i) := supSet\nsrc\u271d : InfSet ((i : \u03b9) \u2192 \u03b1 i) := infSet\ns : Set ((i : \u03b9) \u2192 \u03b1 i)\nf : (i : \u03b9) \u2192 \u03b1 i\nhs : Set.Nonempty s\nhf : f \u2208 lowerBounds s\ni : \u03b9\nthis : Nonempty \u2191s\n\u22a2 Set.Nonempty (range fun f => \u2191f i)\n[PROOFSTEP]\napply range_nonempty\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns t : Set \u03b1\na b : \u03b1\nhs : Set.Nonempty s\nhb : b < sSup s\n\u22a2 \u2203 a, a \u2208 s \u2227 b < a\n[PROOFSTEP]\ncontrapose! hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns t : Set \u03b1\na b : \u03b1\nhs : Set.Nonempty s\nhb : \u2200 (a : \u03b1), a \u2208 s \u2192 a \u2264 b\n\u22a2 sSup s \u2264 b\n[PROOFSTEP]\nexact csSup_le hs hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\n\u22a2 sSup s = sSup t\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with rfl | s_ne\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns t\u271d : Set \u03b1\na b : \u03b1\nt : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 \u2205 \u2227 y \u2264 x\n\u22a2 sSup \u2205 = sSup t\n[PROOFSTEP]\nhave : t = \u2205 := eq_empty_of_forall_not_mem (fun y yt \u21a6 by simpa using ht y yt)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns t\u271d : Set \u03b1\na b : \u03b1\nt : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 \u2205 \u2227 y \u2264 x\ny : \u03b1\nyt : y \u2208 t\n\u22a2 False\n[PROOFSTEP]\nsimpa using ht y yt\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns t\u271d : Set \u03b1\na b : \u03b1\nt : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 \u2205 \u2227 y \u2264 x\nthis : t = \u2205\n\u22a2 sSup \u2205 = sSup t\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\n\u22a2 sSup s = sSup t\n[PROOFSTEP]\nrcases eq_empty_or_nonempty t with rfl | t_ne\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\ns : Set \u03b1\ns_ne : Set.Nonempty s\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 \u2205 \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 \u2205 \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\n\u22a2 sSup s = sSup \u2205\n[PROOFSTEP]\nhave : s = \u2205 := eq_empty_of_forall_not_mem (fun x xs \u21a6 by simpa using hs x xs)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\ns : Set \u03b1\ns_ne : Set.Nonempty s\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 \u2205 \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 \u2205 \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\nx : \u03b1\nxs : x \u2208 s\n\u22a2 False\n[PROOFSTEP]\nsimpa using hs x xs\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\ns : Set \u03b1\ns_ne : Set.Nonempty s\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 \u2205 \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 \u2205 \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\nthis : s = \u2205\n\u22a2 sSup s = sSup \u2205\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\n\u22a2 sSup s = sSup t\n[PROOFSTEP]\nby_cases B : BddAbove s \u2228 BddAbove t\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nB : BddAbove s \u2228 BddAbove t\n\u22a2 sSup s = sSup t\n[PROOFSTEP]\nhave Bs : BddAbove s := by\n rcases B with hB | \u27e8b, hb\u27e9\n \u00b7 exact hB\n \u00b7 refine \u27e8b, fun x hx \u21a6 ?_\u27e9\n rcases hs x hx with \u27e8y, hy, hxy\u27e9\n exact hxy.trans (hb hy)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nB : BddAbove s \u2228 BddAbove t\n\u22a2 BddAbove s\n[PROOFSTEP]\nrcases B with hB | \u27e8b, hb\u27e9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nhB : BddAbove s\n\u22a2 BddAbove s\n[PROOFSTEP]\nexact hB\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b\u271d : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nb : \u03b1\nhb : b \u2208 upperBounds t\n\u22a2 BddAbove s\n[PROOFSTEP]\nrefine \u27e8b, fun x hx \u21a6 ?_\u27e9\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b\u271d : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nb : \u03b1\nhb : b \u2208 upperBounds t\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2264 b\n[PROOFSTEP]\nrcases hs x hx with \u27e8y, hy, hxy\u27e9\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b\u271d : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nb : \u03b1\nhb : b \u2208 upperBounds t\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 t\nhxy : x \u2264 y\n\u22a2 x \u2264 b\n[PROOFSTEP]\nexact hxy.trans (hb hy)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nB : BddAbove s \u2228 BddAbove t\nBs : BddAbove s\n\u22a2 sSup s = sSup t\n[PROOFSTEP]\nhave Bt : BddAbove t := by\n rcases B with \u27e8b, hb\u27e9 | hB\n \u00b7 refine \u27e8b, fun y hy \u21a6 ?_\u27e9\n rcases ht y hy with \u27e8x, hx, hyx\u27e9\n exact hyx.trans (hb hx)\n \u00b7 exact hB\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nB : BddAbove s \u2228 BddAbove t\nBs : BddAbove s\n\u22a2 BddAbove t\n[PROOFSTEP]\nrcases B with \u27e8b, hb\u27e9 | hB\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b\u271d : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nBs : BddAbove s\nb : \u03b1\nhb : b \u2208 upperBounds s\n\u22a2 BddAbove t\n[PROOFSTEP]\nrefine \u27e8b, fun y hy \u21a6 ?_\u27e9\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b\u271d : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nBs : BddAbove s\nb : \u03b1\nhb : b \u2208 upperBounds s\ny : \u03b1\nhy : y \u2208 t\n\u22a2 y \u2264 b\n[PROOFSTEP]\nrcases ht y hy with \u27e8x, hx, hyx\u27e9\n[GOAL]\ncase inl.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b\u271d : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nBs : BddAbove s\nb : \u03b1\nhb : b \u2208 upperBounds s\ny : \u03b1\nhy : y \u2208 t\nx : \u03b1\nhx : x \u2208 s\nhyx : y \u2264 x\n\u22a2 y \u2264 b\n[PROOFSTEP]\nexact hyx.trans (hb hx)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nBs : BddAbove s\nhB : BddAbove t\n\u22a2 BddAbove t\n[PROOFSTEP]\nexact hB\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nB : BddAbove s \u2228 BddAbove t\nBs : BddAbove s\nBt : BddAbove t\n\u22a2 sSup s = sSup t\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase pos.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nB : BddAbove s \u2228 BddAbove t\nBs : BddAbove s\nBt : BddAbove t\n\u22a2 sSup s \u2264 sSup t\n[PROOFSTEP]\napply csSup_le s_ne (fun x hx \u21a6 ?_)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nB : BddAbove s \u2228 BddAbove t\nBs : BddAbove s\nBt : BddAbove t\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2264 sSup t\n[PROOFSTEP]\nrcases hs x hx with \u27e8y, yt, hxy\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nB : BddAbove s \u2228 BddAbove t\nBs : BddAbove s\nBt : BddAbove t\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\nhxy : x \u2264 y\n\u22a2 x \u2264 sSup t\n[PROOFSTEP]\nexact hxy.trans (le_csSup Bt yt)\n[GOAL]\ncase pos.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nB : BddAbove s \u2228 BddAbove t\nBs : BddAbove s\nBt : BddAbove t\n\u22a2 sSup t \u2264 sSup s\n[PROOFSTEP]\napply csSup_le t_ne (fun y hy \u21a6 ?_)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nB : BddAbove s \u2228 BddAbove t\nBs : BddAbove s\nBt : BddAbove t\ny : \u03b1\nhy : y \u2208 t\n\u22a2 y \u2264 sSup s\n[PROOFSTEP]\nrcases ht y hy with \u27e8x, xs, hyx\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nB : BddAbove s \u2228 BddAbove t\nBs : BddAbove s\nBt : BddAbove t\ny : \u03b1\nhy : y \u2208 t\nx : \u03b1\nxs : x \u2208 s\nhyx : y \u2264 x\n\u22a2 y \u2264 sSup s\n[PROOFSTEP]\nexact hyx.trans (le_csSup Bs xs)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t\u271d : Set \u03b1\na b : \u03b1\ns t : Set \u03b1\nhs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 x \u2264 y\nht : \u2200 (y : \u03b1), y \u2208 t \u2192 \u2203 x, x \u2208 s \u2227 y \u2264 x\ns_ne : Set.Nonempty s\nt_ne : Set.Nonempty t\nB : \u00ac(BddAbove s \u2228 BddAbove t)\n\u22a2 sSup s = sSup t\n[PROOFSTEP]\nsimp [csSup_of_not_bddAbove, (not_or.1 B).1, (not_or.1 B).2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b1\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : IsWellOrder \u03b1 fun x x_1 => x < x_1\nhs : Set.Nonempty s\n\u22a2 IsLeast s (sInf s)\n[PROOFSTEP]\nrw [sInf_eq_argmin_on hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b1\ns t : Set \u03b1\na b : \u03b1\ninst\u271d : IsWellOrder \u03b1 fun x x_1 => x < x_1\nhs : Set.Nonempty s\n\u22a2 IsLeast s (argminOn id (_ : WellFounded fun x x_1 => x < x_1) s hs)\n[PROOFSTEP]\nexact \u27e8argminOn_mem _ _ _ _, fun a ha => argminOn_le id _ _ ha\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\ninst\u271d : IsEmpty \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 \u2a06 (i : \u03b9), f i = \u22a5\n[PROOFSTEP]\nrw [iSup_of_empty', csSup_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set \u03b1\nhs : BddAbove s\n\u22a2 IsLUB s (sSup s)\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | hne)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\nhs : BddAbove \u2205\n\u22a2 IsLUB \u2205 (sSup \u2205)\n[PROOFSTEP]\nsimp only [csSup_empty, isLUB_empty]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set \u03b1\nhs : BddAbove s\nhne : Set.Nonempty s\n\u22a2 IsLUB s (sSup s)\n[PROOFSTEP]\nexact isLUB_csSup hne hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : \u03b9 \u2192 \u03b1\na : \u03b1\nh : BddAbove (range s)\n\u22a2 a \u2264 iSup s \u2194 \u2200 (b : \u03b1), (\u2200 (i : \u03b9), s i \u2264 b) \u2192 a \u2264 b\n[PROOFSTEP]\nsimp [iSup, h, le_csSup_iff', upperBounds]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set \u03b1\na : \u03b1\nh : a < sSup s\n\u22a2 \u2203 b, b \u2208 s \u2227 a < b\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set \u03b1\na : \u03b1\nh : \u2200 (b : \u03b1), b \u2208 s \u2192 b \u2264 a\n\u22a2 sSup s \u2264 a\n[PROOFSTEP]\nexact csSup_le' h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\nf : \u03b9 \u2192 \u03b1\na : \u03b1\nh : a < \u2a06 (i : \u03b9), f i\n\u22a2 \u2203 i, a < f i\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\nf : \u03b9 \u2192 \u03b1\na : \u03b1\nh : \u2200 (i : \u03b9), f i \u2264 a\n\u22a2 \u2a06 (i : \u03b9), f i \u2264 a\n[PROOFSTEP]\nexact ciSup_le' h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\n\u22a2 IsLUB s (sSup s)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\n\u22a2 sSup s \u2208 upperBounds s\n[PROOFSTEP]\nshow ite _ _ _ \u2208 _\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\n\u22a2 (if \u22a4 \u2208 s then \u22a4 else if BddAbove ((fun a => \u2191a) \u207b\u00b9' s) then \u2191(sSup ((fun a => \u2191a) \u207b\u00b9' s)) else \u22a4) \u2208 upperBounds s\n[PROOFSTEP]\nsplit_ifs with h\u2081 h\u2082\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u22a4 \u2208 s\n\u22a2 \u22a4 \u2208 upperBounds s\n[PROOFSTEP]\nintro _ _\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u22a4 \u2208 s\na\u271d\u00b9 : WithTop \u03b2\na\u271d : a\u271d\u00b9 \u2208 s\n\u22a2 a\u271d\u00b9 \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' s)\n\u22a2 \u2191(sSup ((fun a => \u2191a) \u207b\u00b9' s)) \u2208 upperBounds s\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | a) ha\n[GOAL]\ncase pos.none\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' s)\nha : none \u2208 s\n\u22a2 none \u2264 \u2191(sSup ((fun a => \u2191a) \u207b\u00b9' s))\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos.some\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' s)\na : \u03b2\nha : Option.some a \u2208 s\n\u22a2 Option.some a \u2264 \u2191(sSup ((fun a => \u2191a) \u207b\u00b9' s))\n[PROOFSTEP]\napply some_le_some.2\n[GOAL]\ncase pos.some\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' s)\na : \u03b2\nha : Option.some a \u2208 s\n\u22a2 a \u2264 sSup ((fun a => \u2191a) \u207b\u00b9' s)\n[PROOFSTEP]\nexact le_csSup h\u2082 ha\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' s)\n\u22a2 \u22a4 \u2208 upperBounds s\n[PROOFSTEP]\nintro _ _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' s)\na\u271d\u00b9 : WithTop \u03b2\na\u271d : a\u271d\u00b9 \u2208 s\n\u22a2 a\u271d\u00b9 \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\n\u22a2 sSup s \u2208 lowerBounds (upperBounds s)\n[PROOFSTEP]\nshow ite _ _ _ \u2208 _\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\n\u22a2 (if \u22a4 \u2208 s then \u22a4 else if BddAbove ((fun a => \u2191a) \u207b\u00b9' s) then \u2191(sSup ((fun a => \u2191a) \u207b\u00b9' s)) else \u22a4) \u2208\n lowerBounds (upperBounds s)\n[PROOFSTEP]\nsplit_ifs with h\u2081 h\u2082\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u22a4 \u2208 s\n\u22a2 \u22a4 \u2208 lowerBounds (upperBounds s)\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | a) ha\n[GOAL]\ncase pos.none\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u22a4 \u2208 s\nha : none \u2208 upperBounds s\n\u22a2 \u22a4 \u2264 none\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase pos.some\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u22a4 \u2208 s\na : \u03b2\nha : Option.some a \u2208 upperBounds s\n\u22a2 \u22a4 \u2264 Option.some a\n[PROOFSTEP]\nexact False.elim (not_top_le_coe a (ha h\u2081))\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' s)\n\u22a2 \u2191(sSup ((fun a => \u2191a) \u207b\u00b9' s)) \u2208 lowerBounds (upperBounds s)\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | b) hb\n[GOAL]\ncase pos.none\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' s)\nhb : none \u2208 upperBounds s\n\u22a2 \u2191(sSup ((fun a => \u2191a) \u207b\u00b9' s)) \u2264 none\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase pos.some\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' s)\nb : \u03b2\nhb : Option.some b \u2208 upperBounds s\n\u22a2 \u2191(sSup ((fun a => \u2191a) \u207b\u00b9' s)) \u2264 Option.some b\n[PROOFSTEP]\nrefine' some_le_some.2 (csSup_le _ _)\n[GOAL]\ncase pos.some.refine'_1\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' s)\nb : \u03b2\nhb : Option.some b \u2208 upperBounds s\n\u22a2 Set.Nonempty ((fun a => \u2191a) \u207b\u00b9' s)\n[PROOFSTEP]\nrcases hs with \u27e8\u27e8\u27e9 | b, hb\u27e9\n[GOAL]\ncase pos.some.refine'_1.intro.none\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' s)\nb : \u03b2\nhb\u271d : Option.some b \u2208 upperBounds s\nhb : none \u2208 s\n\u22a2 Set.Nonempty ((fun a => \u2191a) \u207b\u00b9' s)\n[PROOFSTEP]\nexact absurd hb h\u2081\n[GOAL]\ncase pos.some.refine'_1.intro.some\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' s)\nb\u271d : \u03b2\nhb\u271d : Option.some b\u271d \u2208 upperBounds s\nb : \u03b2\nhb : Option.some b \u2208 s\n\u22a2 Set.Nonempty ((fun a => \u2191a) \u207b\u00b9' s)\n[PROOFSTEP]\nexact \u27e8b, hb\u27e9\n[GOAL]\ncase pos.some.refine'_2\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' s)\nb : \u03b2\nhb : Option.some b \u2208 upperBounds s\n\u22a2 \u2200 (b_1 : \u03b2), b_1 \u2208 (fun a => \u2191a) \u207b\u00b9' s \u2192 b_1 \u2264 b\n[PROOFSTEP]\nintro a ha\n[GOAL]\ncase pos.some.refine'_2\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' s)\nb : \u03b2\nhb : Option.some b \u2208 upperBounds s\na : \u03b2\nha : a \u2208 (fun a => \u2191a) \u207b\u00b9' s\n\u22a2 a \u2264 b\n[PROOFSTEP]\nexact some_le_some.1 (hb ha)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' s)\n\u22a2 \u22a4 \u2208 lowerBounds (upperBounds s)\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | b) hb\n[GOAL]\ncase neg.none\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' s)\nhb : none \u2208 upperBounds s\n\u22a2 \u22a4 \u2264 none\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase neg.some\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' s)\nb : \u03b2\nhb : Option.some b \u2208 upperBounds s\n\u22a2 \u22a4 \u2264 Option.some b\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.some.h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' s)\nb : \u03b2\nhb : Option.some b \u2208 upperBounds s\n\u22a2 False\n[PROOFSTEP]\napply h\u2082\n[GOAL]\ncase neg.some.h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' s)\nb : \u03b2\nhb : Option.some b \u2208 upperBounds s\n\u22a2 BddAbove ((fun a => \u2191a) \u207b\u00b9' s)\n[PROOFSTEP]\nuse b\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' s)\nb : \u03b2\nhb : Option.some b \u2208 upperBounds s\n\u22a2 b \u2208 upperBounds ((fun a => \u2191a) \u207b\u00b9' s)\n[PROOFSTEP]\nintro a ha\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : Set.Nonempty s\nh\u2081 : \u00ac\u22a4 \u2208 s\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' s)\nb : \u03b2\nhb : Option.some b \u2208 upperBounds s\na : \u03b2\nha : a \u2208 (fun a => \u2191a) \u207b\u00b9' s\n\u22a2 a \u2264 b\n[PROOFSTEP]\nexact some_le_some.1 (hb ha)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\n\u22a2 IsLUB s (sSup s)\n[PROOFSTEP]\ncases' s.eq_empty_or_nonempty with hs hs\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : s = \u2205\n\u22a2 IsLUB s (sSup s)\n[PROOFSTEP]\nrw [hs]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : s = \u2205\n\u22a2 IsLUB \u2205 (sSup \u2205)\n[PROOFSTEP]\ndsimp only [sSup]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : s = \u2205\n\u22a2 IsLUB \u2205 (if \u22a4 \u2208 \u2205 then \u22a4 else if BddAbove ((fun a => \u2191a) \u207b\u00b9' \u2205) then \u2191(sSup ((fun a => \u2191a) \u207b\u00b9' \u2205)) else \u22a4)\n[PROOFSTEP]\nshow IsLUB \u2205 _\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : s = \u2205\n\u22a2 IsLUB \u2205 (if \u22a4 \u2208 \u2205 then \u22a4 else if BddAbove ((fun a => \u2191a) \u207b\u00b9' \u2205) then \u2191(sSup ((fun a => \u2191a) \u207b\u00b9' \u2205)) else \u22a4)\n[PROOFSTEP]\nsplit_ifs with h\u2081 h\u2082\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : s = \u2205\nh\u2081 : \u22a4 \u2208 \u2205\n\u22a2 IsLUB \u2205 \u22a4\n[PROOFSTEP]\ncases h\u2081\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : s = \u2205\nh\u2081 : \u00ac\u22a4 \u2208 \u2205\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' \u2205)\n\u22a2 IsLUB \u2205 \u2191(sSup ((fun a => \u2191a) \u207b\u00b9' \u2205))\n[PROOFSTEP]\nrw [preimage_empty, csSup_empty]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : s = \u2205\nh\u2081 : \u00ac\u22a4 \u2208 \u2205\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' \u2205)\n\u22a2 IsLUB \u2205 \u2191\u22a5\n[PROOFSTEP]\nexact isLUB_empty\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : s = \u2205\nh\u2081 : \u00ac\u22a4 \u2208 \u2205\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' \u2205)\n\u22a2 IsLUB \u2205 \u22a4\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : s = \u2205\nh\u2081 : \u00ac\u22a4 \u2208 \u2205\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' \u2205)\n\u22a2 False\n[PROOFSTEP]\napply h\u2082\n[GOAL]\ncase neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : s = \u2205\nh\u2081 : \u00ac\u22a4 \u2208 \u2205\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' \u2205)\n\u22a2 BddAbove ((fun a => \u2191a) \u207b\u00b9' \u2205)\n[PROOFSTEP]\nuse\u22a5\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : s = \u2205\nh\u2081 : \u00ac\u22a4 \u2208 \u2205\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' \u2205)\n\u22a2 \u22a5 \u2208 upperBounds ((fun a => \u2191a) \u207b\u00b9' \u2205)\n[PROOFSTEP]\nrintro a \u27e8\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : Set.Nonempty s\n\u22a2 IsLUB s (sSup s)\n[PROOFSTEP]\nexact isLUB_sSup' hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\n\u22a2 IsGLB s (sInf s)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\n\u22a2 sInf s \u2208 lowerBounds s\n[PROOFSTEP]\nshow ite _ _ _ \u2208 _\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\n\u22a2 (if s \u2286 {\u22a4} then \u22a4 else \u2191(sInf ((fun a => \u2191a) \u207b\u00b9' s))) \u2208 lowerBounds s\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : s \u2286 {\u22a4}\n\u22a2 \u22a4 \u2208 lowerBounds s\n[PROOFSTEP]\nintro a ha\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : s \u2286 {\u22a4}\na : WithTop \u03b2\nha : a \u2208 s\n\u22a2 \u22a4 \u2264 a\n[PROOFSTEP]\nexact top_le_iff.2 (Set.mem_singleton_iff.1 (h ha))\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : \u00acs \u2286 {\u22a4}\n\u22a2 \u2191(sInf ((fun a => \u2191a) \u207b\u00b9' s)) \u2208 lowerBounds s\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | a) ha\n[GOAL]\ncase neg.none\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : \u00acs \u2286 {\u22a4}\nha : none \u2208 s\n\u22a2 \u2191(sInf ((fun a => \u2191a) \u207b\u00b9' s)) \u2264 none\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase neg.some\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 s\n\u22a2 \u2191(sInf ((fun a => \u2191a) \u207b\u00b9' s)) \u2264 Option.some a\n[PROOFSTEP]\nrefine' some_le_some.2 (csInf_le _ ha)\n[GOAL]\ncase neg.some\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 s\n\u22a2 BddBelow ((fun a => \u2191a) \u207b\u00b9' s)\n[PROOFSTEP]\nrcases hs with \u27e8\u27e8\u27e9 | b, hb\u27e9\n[GOAL]\ncase neg.some.intro.none\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 s\nhb : none \u2208 lowerBounds s\n\u22a2 BddBelow ((fun a => \u2191a) \u207b\u00b9' s)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.some.intro.none.h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 s\nhb : none \u2208 lowerBounds s\n\u22a2 False\n[PROOFSTEP]\napply h\n[GOAL]\ncase neg.some.intro.none.h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 s\nhb : none \u2208 lowerBounds s\n\u22a2 s \u2286 {\u22a4}\n[PROOFSTEP]\nintro c hc\n[GOAL]\ncase neg.some.intro.none.h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 s\nhb : none \u2208 lowerBounds s\nc : WithTop \u03b2\nhc : c \u2208 s\n\u22a2 c \u2208 {\u22a4}\n[PROOFSTEP]\nrw [mem_singleton_iff, \u2190 top_le_iff]\n[GOAL]\ncase neg.some.intro.none.h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 s\nhb : none \u2208 lowerBounds s\nc : WithTop \u03b2\nhc : c \u2208 s\n\u22a2 \u22a4 \u2264 c\n[PROOFSTEP]\nexact hb hc\n[GOAL]\ncase neg.some.intro.some\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 s\nb : \u03b2\nhb : Option.some b \u2208 lowerBounds s\n\u22a2 BddBelow ((fun a => \u2191a) \u207b\u00b9' s)\n[PROOFSTEP]\nuse b\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 s\nb : \u03b2\nhb : Option.some b \u2208 lowerBounds s\n\u22a2 b \u2208 lowerBounds ((fun a => \u2191a) \u207b\u00b9' s)\n[PROOFSTEP]\nintro c hc\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 s\nb : \u03b2\nhb : Option.some b \u2208 lowerBounds s\nc : \u03b2\nhc : c \u2208 (fun a => \u2191a) \u207b\u00b9' s\n\u22a2 b \u2264 c\n[PROOFSTEP]\nexact some_le_some.1 (hb hc)\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\n\u22a2 sInf s \u2208 upperBounds (lowerBounds s)\n[PROOFSTEP]\nshow ite _ _ _ \u2208 _\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\n\u22a2 (if s \u2286 {\u22a4} then \u22a4 else \u2191(sInf ((fun a => \u2191a) \u207b\u00b9' s))) \u2208 upperBounds (lowerBounds s)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : s \u2286 {\u22a4}\n\u22a2 \u22a4 \u2208 upperBounds (lowerBounds s)\n[PROOFSTEP]\nintro _ _\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : s \u2286 {\u22a4}\na\u271d\u00b9 : WithTop \u03b2\na\u271d : a\u271d\u00b9 \u2208 lowerBounds s\n\u22a2 a\u271d\u00b9 \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : \u00acs \u2286 {\u22a4}\n\u22a2 \u2191(sInf ((fun a => \u2191a) \u207b\u00b9' s)) \u2208 upperBounds (lowerBounds s)\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | a) ha\n[GOAL]\ncase neg.none\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : \u00acs \u2286 {\u22a4}\nha : none \u2208 lowerBounds s\n\u22a2 none \u2264 \u2191(sInf ((fun a => \u2191a) \u207b\u00b9' s))\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.none.h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : \u00acs \u2286 {\u22a4}\nha : none \u2208 lowerBounds s\n\u22a2 False\n[PROOFSTEP]\napply h\n[GOAL]\ncase neg.none.h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : \u00acs \u2286 {\u22a4}\nha : none \u2208 lowerBounds s\n\u22a2 s \u2286 {\u22a4}\n[PROOFSTEP]\nintro b hb\n[GOAL]\ncase neg.none.h\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : \u00acs \u2286 {\u22a4}\nha : none \u2208 lowerBounds s\nb : WithTop \u03b2\nhb : b \u2208 s\n\u22a2 b \u2208 {\u22a4}\n[PROOFSTEP]\nexact Set.mem_singleton_iff.2 (top_le_iff.1 (ha hb))\n[GOAL]\ncase neg.some\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 lowerBounds s\n\u22a2 Option.some a \u2264 \u2191(sInf ((fun a => \u2191a) \u207b\u00b9' s))\n[PROOFSTEP]\nrefine' some_le_some.2 (le_csInf _ _)\n[GOAL]\ncase neg.some.refine'_1\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 lowerBounds s\n\u22a2 Set.Nonempty ((fun a => \u2191a) \u207b\u00b9' s)\n[PROOFSTEP]\nclassical\ncontrapose! h\nrintro (\u27e8\u27e9 | a) ha\n\u00b7 exact mem_singleton \u22a4\n\u00b7 exact (h \u27e8a, ha\u27e9).elim\n[GOAL]\ncase neg.some.refine'_1\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 lowerBounds s\n\u22a2 Set.Nonempty ((fun a => \u2191a) \u207b\u00b9' s)\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase neg.some.refine'_1\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\na : \u03b2\nha : Option.some a \u2208 lowerBounds s\nh : \u00acSet.Nonempty ((fun a => \u2191a) \u207b\u00b9' s)\n\u22a2 s \u2286 {\u22a4}\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | a) ha\n[GOAL]\ncase neg.some.refine'_1.none\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\na : \u03b2\nha\u271d : Option.some a \u2208 lowerBounds s\nh : \u00acSet.Nonempty ((fun a => \u2191a) \u207b\u00b9' s)\nha : none \u2208 s\n\u22a2 none \u2208 {\u22a4}\n[PROOFSTEP]\nexact mem_singleton \u22a4\n[GOAL]\ncase neg.some.refine'_1.some\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\na\u271d : \u03b2\nha\u271d : Option.some a\u271d \u2208 lowerBounds s\nh : \u00acSet.Nonempty ((fun a => \u2191a) \u207b\u00b9' s)\na : \u03b2\nha : Option.some a \u2208 s\n\u22a2 Option.some a \u2208 {\u22a4}\n[PROOFSTEP]\nexact (h \u27e8a, ha\u27e9).elim\n[GOAL]\ncase neg.some.refine'_2\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 lowerBounds s\n\u22a2 \u2200 (b : \u03b2), b \u2208 (fun a => \u2191a) \u207b\u00b9' s \u2192 a \u2264 b\n[PROOFSTEP]\nintro b hb\n[GOAL]\ncase neg.some.refine'_2\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 lowerBounds s\nb : \u03b2\nhb : b \u2208 (fun a => \u2191a) \u207b\u00b9' s\n\u22a2 a \u2264 b\n[PROOFSTEP]\nrw [\u2190 some_le_some]\n[GOAL]\ncase neg.some.refine'_2\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrderBot \u03b1\n\u03b2 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Set (WithTop \u03b2)\nhs : BddBelow s\nh : \u00acs \u2286 {\u22a4}\na : \u03b2\nha : Option.some a \u2208 lowerBounds s\nb : \u03b2\nhb : b \u2208 (fun a => \u2191a) \u207b\u00b9' s\n\u22a2 Option.some a \u2264 Option.some b\n[PROOFSTEP]\nexact ha hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\n\u22a2 IsGLB s (sInf s)\n[PROOFSTEP]\nby_cases hs : BddBelow s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : BddBelow s\n\u22a2 IsGLB s (sInf s)\n[PROOFSTEP]\nexact isGLB_sInf' hs\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : \u00acBddBelow s\n\u22a2 IsGLB s (sInf s)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : \u00acBddBelow s\n\u22a2 False\n[PROOFSTEP]\napply hs\n[GOAL]\ncase neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : \u00acBddBelow s\n\u22a2 BddBelow s\n[PROOFSTEP]\nuse\u22a5\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : \u00acBddBelow s\n\u22a2 \u22a5 \u2208 lowerBounds s\n[PROOFSTEP]\nintro _ _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set (WithTop \u03b1)\nhs : \u00acBddBelow s\na\u271d\u00b9 : WithTop \u03b1\na\u271d : a\u271d\u00b9 \u2208 s\n\u22a2 \u22a5 \u2264 a\u271d\u00b9\n[PROOFSTEP]\nexact bot_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set \u03b1\nhb : BddAbove s\n\u22a2 \u2191(sSup s) = \u2a06 (a : \u03b1) (_ : a \u2208 s), \u2191a\n[PROOFSTEP]\nrw [coe_sSup' hb, sSup_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\ns : Set \u03b1\nhs : Set.Nonempty s\n\u22a2 \u2191(sInf s) = \u2a05 (a : \u03b1) (_ : a \u2208 s), \u2191a\n[PROOFSTEP]\nrw [coe_sInf' hs, sInf_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh_mono : Monotone f\ns : Set \u03b1\nc : \u03b1\nhcs : c \u2208 s\nh_bdd : BddBelow s\n\u22a2 sInf (f '' s) \u2264 f c\n[PROOFSTEP]\nlet f' : \u03b1\u1d52\u1d48 \u2192 \u03b2\u1d52\u1d48 := f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh_mono : Monotone f\ns : Set \u03b1\nc : \u03b1\nhcs : c \u2208 s\nh_bdd : BddBelow s\nf' : \u03b1\u1d52\u1d48 \u2192 \u03b2\u1d52\u1d48 := f\n\u22a2 sInf (f '' s) \u2264 f c\n[PROOFSTEP]\nexact @le_csSup_image \u03b1\u1d52\u1d48 \u03b2\u1d52\u1d48 _ _ _ (show Monotone f' from fun x y hxy => h_mono hxy) _ _ hcs h_bdd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh_mono : Monotone f\ns : Set \u03b1\nhs : Set.Nonempty s\nB : \u03b1\nhB : B \u2208 lowerBounds s\n\u22a2 f B \u2264 sInf (f '' s)\n[PROOFSTEP]\nlet f' : \u03b1\u1d52\u1d48 \u2192 \u03b2\u1d52\u1d48 := f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : ConditionallyCompleteLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nh_mono : Monotone f\ns : Set \u03b1\nhs : Set.Nonempty s\nB : \u03b1\nhB : B \u2208 lowerBounds s\nf' : \u03b1\u1d52\u1d48 \u2192 \u03b2\u1d52\u1d48 := f\n\u22a2 f B \u2264 sInf (f '' s)\n[PROOFSTEP]\nexact @csSup_image_le \u03b1\u1d52\u1d48 \u03b2\u1d52\u1d48 _ _ _ (show Monotone f' from fun x y hxy => h_mono hxy) _ hs _ hB\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b2 : ConditionallyCompleteLattice \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b2\ninst\u271d : Nonempty \u03b9\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\ns : Set \u03b1\nhne : Set.Nonempty s\nhbdd : BddAbove s\n\u22a2 l (sSup s) = sSup (l '' s)\n[PROOFSTEP]\nrw [gc.l_csSup hne hbdd, sSup_image']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b2 : ConditionallyCompleteLattice \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b2\ninst\u271d : Nonempty \u03b9\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nf : \u03b9 \u2192 \u03b1\nhf : BddAbove (range f)\n\u22a2 l (\u2a06 (i : \u03b9), f i) = \u2a06 (i : \u03b9), l (f i)\n[PROOFSTEP]\nrw [iSup, gc.l_csSup (range_nonempty _) hf, iSup_range']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b2 : ConditionallyCompleteLattice \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b2\ninst\u271d : Nonempty \u03b9\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b1\nhf : BddAbove (f '' s)\nhne : Set.Nonempty s\n\u22a2 l (\u2a06 (i : \u2191s), f \u2191i) = \u2a06 (i : \u2191s), l (f \u2191i)\n[PROOFSTEP]\nhaveI := hne.to_subtype\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b2 : ConditionallyCompleteLattice \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b2\ninst\u271d : Nonempty \u03b9\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b1\nhf : BddAbove (f '' s)\nhne : Set.Nonempty s\nthis : Nonempty \u2191s\n\u22a2 l (\u2a06 (i : \u2191s), f \u2191i) = \u2a06 (i : \u2191s), l (f \u2191i)\n[PROOFSTEP]\nrw [image_eq_range] at hf \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b2 : ConditionallyCompleteLattice \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b2\ninst\u271d : Nonempty \u03b9\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b1\nhf : BddAbove (range fun x => f \u2191x)\nhne : Set.Nonempty s\nthis : Nonempty \u2191s\n\u22a2 l (\u2a06 (i : \u2191s), f \u2191i) = \u2a06 (i : \u2191s), l (f \u2191i)\n[PROOFSTEP]\nexact gc.l_ciSup hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b2 : ConditionallyCompleteLattice \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b2\ninst\u271d : ConditionallyCompleteLattice \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nl u : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2081 u\u2081 : \u03b2 \u2192 \u03b3 \u2192 \u03b1\nl\u2082 u\u2082 : \u03b1 \u2192 \u03b3 \u2192 \u03b2\nh\u2081 : \u2200 (b : \u03b2), GaloisConnection (swap l b) (u\u2081 b)\nh\u2082 : \u2200 (a : \u03b1), GaloisConnection (l a) (u\u2082 a)\nhs\u2080 : Set.Nonempty s\nhs\u2081 : BddAbove s\nht\u2080 : Set.Nonempty t\nht\u2081 : BddAbove t\n\u22a2 sSup (image2 l s t) = l (sSup s) (sSup t)\n[PROOFSTEP]\nrefine' eq_of_forall_ge_iff fun c => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b2 : ConditionallyCompleteLattice \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b2\ninst\u271d : ConditionallyCompleteLattice \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nl u : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2081 u\u2081 : \u03b2 \u2192 \u03b3 \u2192 \u03b1\nl\u2082 u\u2082 : \u03b1 \u2192 \u03b3 \u2192 \u03b2\nh\u2081 : \u2200 (b : \u03b2), GaloisConnection (swap l b) (u\u2081 b)\nh\u2082 : \u2200 (a : \u03b1), GaloisConnection (l a) (u\u2082 a)\nhs\u2080 : Set.Nonempty s\nhs\u2081 : BddAbove s\nht\u2080 : Set.Nonempty t\nht\u2081 : BddAbove t\nc : \u03b3\n\u22a2 sSup (image2 l s t) \u2264 c \u2194 l (sSup s) (sSup t) \u2264 c\n[PROOFSTEP]\nrw [csSup_le_iff (hs\u2081.image2 (fun _ => (h\u2081 _).monotone_l) (fun _ => (h\u2082 _).monotone_l) ht\u2081) (hs\u2080.image2 ht\u2080),\n forall_image2_iff, forall\u2082_swap, (h\u2082 _).le_iff_le, csSup_le_iff ht\u2081 ht\u2080]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d\u00b2 : ConditionallyCompleteLattice \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b2\ninst\u271d : ConditionallyCompleteLattice \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nl u : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2081 u\u2081 : \u03b2 \u2192 \u03b3 \u2192 \u03b1\nl\u2082 u\u2082 : \u03b1 \u2192 \u03b3 \u2192 \u03b2\nh\u2081 : \u2200 (b : \u03b2), GaloisConnection (swap l b) (u\u2081 b)\nh\u2082 : \u2200 (a : \u03b1), GaloisConnection (l a) (u\u2082 a)\nhs\u2080 : Set.Nonempty s\nhs\u2081 : BddAbove s\nht\u2080 : Set.Nonempty t\nht\u2081 : BddAbove t\nc : \u03b3\n\u22a2 (\u2200 (i\u2082 : \u03b2), i\u2082 \u2208 t \u2192 \u2200 (i\u2081 : \u03b1), i\u2081 \u2208 s \u2192 l i\u2081 i\u2082 \u2264 c) \u2194 \u2200 (b : \u03b2), b \u2208 t \u2192 b \u2264 u\u2082 (sSup s) c\n[PROOFSTEP]\nsimp_rw [\u2190 (h\u2082 _).le_iff_le, (h\u2081 _).le_iff_le, csSup_le_iff hs\u2081 hs\u2080]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nha : \u2200 (b : WithTop (WithBot \u03b1)), b \u2208 S \u2192 b \u2264 a\n\u22a2 sSup S \u2264 a\n[PROOFSTEP]\ncases' S.eq_empty_or_nonempty with h h\n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nha : \u2200 (b : WithTop (WithBot \u03b1)), b \u2208 S \u2192 b \u2264 a\nh : S = \u2205\n\u22a2 sSup S \u2264 a\n[PROOFSTEP]\nshow ite _ _ _ \u2264 a\n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nha : \u2200 (b : WithTop (WithBot \u03b1)), b \u2208 S \u2192 b \u2264 a\nh : S = \u2205\n\u22a2 (if \u22a4 \u2208 S then \u22a4 else if BddAbove ((fun a => \u2191a) \u207b\u00b9' S) then \u2191(sSup ((fun a => \u2191a) \u207b\u00b9' S)) else \u22a4) \u2264 a\n[PROOFSTEP]\nsplit_ifs with h\u2081 h\u2082\n[GOAL]\ncase pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nha : \u2200 (b : WithTop (WithBot \u03b1)), b \u2208 S \u2192 b \u2264 a\nh : S = \u2205\nh\u2081 : \u22a4 \u2208 S\n\u22a2 \u22a4 \u2264 a\n[PROOFSTEP]\nrw [h] at h\u2081 \n[GOAL]\ncase pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nha : \u2200 (b : WithTop (WithBot \u03b1)), b \u2208 S \u2192 b \u2264 a\nh : S = \u2205\nh\u2081 : \u22a4 \u2208 \u2205\n\u22a2 \u22a4 \u2264 a\n[PROOFSTEP]\ncases h\u2081\n[GOAL]\ncase pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nha : \u2200 (b : WithTop (WithBot \u03b1)), b \u2208 S \u2192 b \u2264 a\nh : S = \u2205\nh\u2081 : \u00ac\u22a4 \u2208 S\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' S)\n\u22a2 \u2191(sSup ((fun a => \u2191a) \u207b\u00b9' S)) \u2264 a\n[PROOFSTEP]\nconvert @bot_le _ _ _ a\n[GOAL]\ncase h.e'_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nha : \u2200 (b : WithTop (WithBot \u03b1)), b \u2208 S \u2192 b \u2264 a\nh : S = \u2205\nh\u2081 : \u00ac\u22a4 \u2208 S\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' S)\n\u22a2 \u2191(sSup ((fun a => \u2191a) \u207b\u00b9' S)) = \u22a5\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h.e'_3.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nha : \u2200 (b : WithTop (WithBot \u03b1)), b \u2208 S \u2192 b \u2264 a\nh : S = \u2205\nh\u2081 : \u00ac\u22a4 \u2208 S\nh\u2082 : BddAbove ((fun a => \u2191a) \u207b\u00b9' S)\n\u22a2 sSup ((fun a => \u2191a) \u207b\u00b9' S) = \u22a5\n[PROOFSTEP]\nsimp only [h, preimage_empty, WithBot.csSup_empty]\n[GOAL]\ncase neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nha : \u2200 (b : WithTop (WithBot \u03b1)), b \u2208 S \u2192 b \u2264 a\nh : S = \u2205\nh\u2081 : \u00ac\u22a4 \u2208 S\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' S)\n\u22a2 \u22a4 \u2264 a\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nha : \u2200 (b : WithTop (WithBot \u03b1)), b \u2208 S \u2192 b \u2264 a\nh : S = \u2205\nh\u2081 : \u00ac\u22a4 \u2208 S\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' S)\n\u22a2 False\n[PROOFSTEP]\napply h\u2082\n[GOAL]\ncase neg.h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nha : \u2200 (b : WithTop (WithBot \u03b1)), b \u2208 S \u2192 b \u2264 a\nh : S = \u2205\nh\u2081 : \u00ac\u22a4 \u2208 S\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' S)\n\u22a2 BddAbove ((fun a => \u2191a) \u207b\u00b9' S)\n[PROOFSTEP]\nuse\u22a5\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nha : \u2200 (b : WithTop (WithBot \u03b1)), b \u2208 S \u2192 b \u2264 a\nh : S = \u2205\nh\u2081 : \u00ac\u22a4 \u2208 S\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' S)\n\u22a2 \u22a5 \u2208 upperBounds ((fun a => \u2191a) \u207b\u00b9' S)\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nha : \u2200 (b : WithTop (WithBot \u03b1)), b \u2208 S \u2192 b \u2264 a\nh : S = \u2205\nh\u2081 : \u00ac\u22a4 \u2208 S\nh\u2082 : \u00acBddAbove ((fun a => \u2191a) \u207b\u00b9' S)\n\u22a2 \u22a5 \u2208 upperBounds ((fun a => \u2191a) \u207b\u00b9' \u2205)\n[PROOFSTEP]\nrintro b \u27e8\u27e9\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nha : \u2200 (b : WithTop (WithBot \u03b1)), b \u2208 S \u2192 b \u2264 a\nh : Set.Nonempty S\n\u22a2 sSup S \u2264 a\n[PROOFSTEP]\nrefine' (WithTop.isLUB_sSup' h).2 ha\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nhaS : a \u2208 S\n\u22a2 (if S \u2286 {\u22a4} then \u22a4 else \u2191(sInf ((fun a => \u2191a) \u207b\u00b9' S))) \u2264 a\n[PROOFSTEP]\nsplit_ifs with h\u2081\n[GOAL]\ncase pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nhaS : a \u2208 S\nh\u2081 : S \u2286 {\u22a4}\n\u22a2 \u22a4 \u2264 a\n[PROOFSTEP]\ncases' a with a\n[GOAL]\ncase pos.none\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\nh\u2081 : S \u2286 {\u22a4}\nhaS : none \u2208 S\n\u22a2 \u22a4 \u2264 none\ncase pos.some\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\nh\u2081 : S \u2286 {\u22a4}\na : WithBot \u03b1\nhaS : Option.some a \u2208 S\n\u22a2 \u22a4 \u2264 Option.some a\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase pos.some\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\nh\u2081 : S \u2286 {\u22a4}\na : WithBot \u03b1\nhaS : Option.some a \u2208 S\n\u22a2 \u22a4 \u2264 Option.some a\n[PROOFSTEP]\ncases h\u2081 haS\n[GOAL]\ncase neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\na : WithTop (WithBot \u03b1)\nhaS : a \u2208 S\nh\u2081 : \u00acS \u2286 {\u22a4}\n\u22a2 \u2191(sInf ((fun a => \u2191a) \u207b\u00b9' S)) \u2264 a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase neg.none\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\nh\u2081 : \u00acS \u2286 {\u22a4}\nhaS : none \u2208 S\n\u22a2 \u2191(sInf ((fun a => \u2191a) \u207b\u00b9' S)) \u2264 none\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase neg.some\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\nh\u2081 : \u00acS \u2286 {\u22a4}\nval\u271d : WithBot \u03b1\nhaS : Option.some val\u271d \u2208 S\n\u22a2 \u2191(sInf ((fun a => \u2191a) \u207b\u00b9' S)) \u2264 Option.some val\u271d\n[PROOFSTEP]\napply WithTop.some_le_some.2\n[GOAL]\ncase neg.some\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\nh\u2081 : \u00acS \u2286 {\u22a4}\nval\u271d : WithBot \u03b1\nhaS : Option.some val\u271d \u2208 S\n\u22a2 sInf ((fun a => \u2191a) \u207b\u00b9' S) \u2264 val\u271d\n[PROOFSTEP]\nrefine' csInf_le _ haS\n[GOAL]\ncase neg.some\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\nh\u2081 : \u00acS \u2286 {\u22a4}\nval\u271d : WithBot \u03b1\nhaS : Option.some val\u271d \u2208 S\n\u22a2 BddBelow ((fun a => \u2191a) \u207b\u00b9' S)\n[PROOFSTEP]\nuse\u22a5\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\nh\u2081 : \u00acS \u2286 {\u22a4}\nval\u271d : WithBot \u03b1\nhaS : Option.some val\u271d \u2208 S\n\u22a2 \u22a5 \u2208 lowerBounds ((fun a => \u2191a) \u207b\u00b9' S)\n[PROOFSTEP]\nintro b _\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b1 : Type u_5\ninst\u271d : ConditionallyCompleteLattice \u03b1\nsrc\u271d\u00b3 : InfSet (WithTop (WithBot \u03b1)) := instInfSet\nsrc\u271d\u00b2 : SupSet (WithTop (WithBot \u03b1)) := instSupSet\nsrc\u271d\u00b9 : BoundedOrder (WithTop (WithBot \u03b1)) := boundedOrder\nsrc\u271d : Lattice (WithTop (WithBot \u03b1)) := lattice\nS : Set (WithTop (WithBot \u03b1))\nh\u2081 : \u00acS \u2286 {\u22a4}\nval\u271d : WithBot \u03b1\nhaS : Option.some val\u271d \u2208 S\nb : WithBot \u03b1\na\u271d : b \u2208 (fun a => \u2191a) \u207b\u00b9' S\n\u22a2 \u22a5 \u2264 b\n[PROOFSTEP]\nexact bot_le\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9 : Sort u_5\n\u03b1 : Type u_6\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\nf : \u03b9 \u2192 \u03b1\n\u22a2 \u2a06 (x : \u03b9), \u2191(f x) = \u22a4 \u2194 \u00acBddAbove (range f)\n[PROOFSTEP]\nrw [iSup_eq_top, not_bddAbove_iff]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9 : Sort u_5\n\u03b1 : Type u_6\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\nf : \u03b9 \u2192 \u03b1\n\u22a2 (\u2200 (b : WithTop \u03b1), b < \u22a4 \u2192 \u2203 i, b < \u2191(f i)) \u2194 \u2200 (x : \u03b1), \u2203 y, y \u2208 range f \u2227 x < y\n[PROOFSTEP]\nrefine' \u27e8fun hf r => _, fun hf a ha => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9 : Sort u_5\n\u03b1 : Type u_6\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\nf : \u03b9 \u2192 \u03b1\nhf : \u2200 (b : WithTop \u03b1), b < \u22a4 \u2192 \u2203 i, b < \u2191(f i)\nr : \u03b1\n\u22a2 \u2203 y, y \u2208 range f \u2227 r < y\n[PROOFSTEP]\nrcases hf r (WithTop.coe_lt_top r) with \u27e8i, hi\u27e9\n[GOAL]\ncase refine'_1.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9 : Sort u_5\n\u03b1 : Type u_6\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\nf : \u03b9 \u2192 \u03b1\nhf : \u2200 (b : WithTop \u03b1), b < \u22a4 \u2192 \u2203 i, b < \u2191(f i)\nr : \u03b1\ni : \u03b9\nhi : \u2191r < \u2191(f i)\n\u22a2 \u2203 y, y \u2208 range f \u2227 r < y\n[PROOFSTEP]\nexact \u27e8f i, \u27e8i, rfl\u27e9, WithTop.coe_lt_coe.mp hi\u27e9\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9 : Sort u_5\n\u03b1 : Type u_6\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\nf : \u03b9 \u2192 \u03b1\nhf : \u2200 (x : \u03b1), \u2203 y, y \u2208 range f \u2227 x < y\na : WithTop \u03b1\nha : a < \u22a4\n\u22a2 \u2203 i, a < \u2191(f i)\n[PROOFSTEP]\nrcases hf (a.untop ha.ne) with \u27e8-, \u27e8i, rfl\u27e9, hi\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9 : Sort u_5\n\u03b1 : Type u_6\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\nf : \u03b9 \u2192 \u03b1\nhf : \u2200 (x : \u03b1), \u2203 y, y \u2208 range f \u2227 x < y\na : WithTop \u03b1\nha : a < \u22a4\ni : \u03b9\nhi : untop a (_ : a \u2260 \u22a4) < f i\n\u22a2 \u2203 i, a < \u2191(f i)\n[PROOFSTEP]\nexact \u27e8i, by simpa only [WithTop.coe_untop _ ha.ne] using WithTop.coe_lt_coe.mpr hi\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9 : Sort u_5\n\u03b1 : Type u_6\ninst\u271d : ConditionallyCompleteLinearOrderBot \u03b1\nf : \u03b9 \u2192 \u03b1\nhf : \u2200 (x : \u03b1), \u2203 y, y \u2208 range f \u2227 x < y\na : WithTop \u03b1\nha : a < \u22a4\ni : \u03b9\nhi : untop a (_ : a \u2260 \u22a4) < f i\n\u22a2 a < \u2191(f i)\n[PROOFSTEP]\nsimpa only [WithTop.coe_untop _ ha.ne] using WithTop.coe_lt_coe.mpr hi\n", "meta": {"mathlib_filename": "Mathlib.Order.ConditionallyCompleteLattice.Basic", "llama_tokens": 41089, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.41040482267156103}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\nx\u271d : (WithZero \u03b1)\u02e3\n\u22a2 \u2191((fun a => Units.mk0 \u2191a (_ : \u2191a \u2260 0)) ((fun a => unzero (_ : \u2191a \u2260 0)) x\u271d)) = \u2191x\u271d\n[PROOFSTEP]\nsimp only [coe_unzero, Units.mk0_val]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\nx\u271d\u00b9 x\u271d : (WithZero \u03b1)\u02e3\n\u22a2 \u2191(Equiv.toFun\n { toFun := fun a => unzero (_ : \u2191a \u2260 0), invFun := fun a => Units.mk0 \u2191a (_ : \u2191a \u2260 0),\n left_inv :=\n (_ : \u2200 (x : (WithZero \u03b1)\u02e3), (fun a => Units.mk0 \u2191a (_ : \u2191a \u2260 0)) ((fun a => unzero (_ : \u2191a \u2260 0)) x) = x),\n right_inv :=\n (_ :\n \u2200 (x : \u03b1),\n (fun a => unzero (_ : \u2191a \u2260 0)) ((fun a => Units.mk0 \u2191a (_ : \u2191a \u2260 0)) x) =\n (fun a => unzero (_ : \u2191a \u2260 0)) ((fun a => Units.mk0 \u2191a (_ : \u2191a \u2260 0)) x)) }\n (x\u271d\u00b9 * x\u271d)) =\n \u2191(Equiv.toFun\n { toFun := fun a => unzero (_ : \u2191a \u2260 0), invFun := fun a => Units.mk0 \u2191a (_ : \u2191a \u2260 0),\n left_inv :=\n (_ : \u2200 (x : (WithZero \u03b1)\u02e3), (fun a => Units.mk0 \u2191a (_ : \u2191a \u2260 0)) ((fun a => unzero (_ : \u2191a \u2260 0)) x) = x),\n right_inv :=\n (_ :\n \u2200 (x : \u03b1),\n (fun a => unzero (_ : \u2191a \u2260 0)) ((fun a => Units.mk0 \u2191a (_ : \u2191a \u2260 0)) x) =\n (fun a => unzero (_ : \u2191a \u2260 0)) ((fun a => Units.mk0 \u2191a (_ : \u2191a \u2260 0)) x)) }\n x\u271d\u00b9 *\n Equiv.toFun\n { toFun := fun a => unzero (_ : \u2191a \u2260 0), invFun := fun a => Units.mk0 \u2191a (_ : \u2191a \u2260 0),\n left_inv :=\n (_ : \u2200 (x : (WithZero \u03b1)\u02e3), (fun a => Units.mk0 \u2191a (_ : \u2191a \u2260 0)) ((fun a => unzero (_ : \u2191a \u2260 0)) x) = x),\n right_inv :=\n (_ :\n \u2200 (x : \u03b1),\n (fun a => unzero (_ : \u2191a \u2260 0)) ((fun a => Units.mk0 \u2191a (_ : \u2191a \u2260 0)) x) =\n (fun a => unzero (_ : \u2191a \u2260 0)) ((fun a => Units.mk0 \u2191a (_ : \u2191a \u2260 0)) x)) }\n x\u271d)\n[PROOFSTEP]\nsimp only [Units.val_mul, coe_unzero, coe_mul]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Group.WithOne.Units", "llama_tokens": 871, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.41040482267156103}} {"text": "[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx a\u271d b\u271d : S\nha : a\u271d \u2208 {a | \u2200 (b : S), a * b \u2208 adjoin R {x}}\nhb : b\u271d \u2208 {a | \u2200 (b : S), a * b \u2208 adjoin R {x}}\nc : S\n\u22a2 (a\u271d + b\u271d) * c \u2208 adjoin R {x}\n[PROOFSTEP]\nsimpa only [add_mul] using Subalgebra.add_mem _ (ha c) (hb c)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx b : S\n\u22a2 0 * b \u2208 adjoin R {x}\n[PROOFSTEP]\nsimpa only [zero_mul] using Subalgebra.zero_mem _\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx c a : S\nha :\n a \u2208\n {\n toAddSubsemigroup :=\n { carrier := {a | \u2200 (b : S), a * b \u2208 adjoin R {x}},\n add_mem' :=\n (_ :\n \u2200 {a b : S},\n a \u2208 {a | \u2200 (b : S), a * b \u2208 adjoin R {x}} \u2192\n b \u2208 {a | \u2200 (b : S), a * b \u2208 adjoin R {x}} \u2192 \u2200 (c : S), (a + b) * c \u2208 adjoin R {x}) },\n zero_mem' := (_ : \u2200 (b : S), 0 * b \u2208 adjoin R {x}) }.toAddSubsemigroup.carrier\nb : S\n\u22a2 c \u2022 a * b \u2208 adjoin R {x}\n[PROOFSTEP]\nsimpa only [smul_eq_mul, mul_left_comm, mul_assoc] using ha (c * b)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx y : S\nhy : y \u2208 \u2191(conductor R x)\n\u22a2 y \u2208 \u2191(adjoin R {x})\n[PROOFSTEP]\nsimpa only [mul_one] using hy 1\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nh : adjoin R {x} = \u22a4\n\u22a2 conductor R x = \u22a4\n[PROOFSTEP]\nsimp only [Ideal.eq_top_iff_one, mem_conductor_iff, h, mem_top, forall_const]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nhz' : z \u2208 Ideal.map (algebraMap R S) I\n\u22a2 \u2191(algebraMap R S) p * z \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n[PROOFSTEP]\nrw [Ideal.map, Ideal.span, Finsupp.mem_span_image_iff_total] at hz' \n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nhz' : \u2203 l, l \u2208 Finsupp.supported S S \u2191I \u2227 \u2191(Finsupp.total R S S \u2191(algebraMap R S)) l = z\n\u22a2 \u2191(algebraMap R S) p * z \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n[PROOFSTEP]\nobtain \u27e8l, H, H'\u27e9 := hz'\n[GOAL]\ncase intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : \u2191(Finsupp.total R S S \u2191(algebraMap R S)) l = z\n\u22a2 \u2191(algebraMap R S) p * z \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n[PROOFSTEP]\nrw [Finsupp.total_apply] at H' \n[GOAL]\ncase intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\n\u22a2 \u2191(algebraMap R S) p * z \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n[PROOFSTEP]\nrw [\u2190 H', mul_comm, Finsupp.sum_mul]\n[GOAL]\ncase intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\n\u22a2 (Finsupp.sum l fun a c => c \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p) \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n[PROOFSTEP]\nhave lem :\n \u2200 {a : R}, a \u2208 I \u2192 l a \u2022 algebraMap R S a * algebraMap R S p \u2208 algebraMap R S '' I.map (algebraMap R R) :=\n by\n intro a ha\n rw [Algebra.id.smul_eq_mul, mul_assoc, mul_comm, mul_assoc, Set.mem_image]\n refine Exists.intro (algebraMap R R a * \u27e8l a * algebraMap R S p, show l a * algebraMap R S p \u2208 R from ?_\u27e9) ?_\n \u00b7 refine \u27e8?_, ?_\u27e9\n \u00b7 rw [mul_comm]\n exact mem_conductor_iff.mp (Ideal.mem_comap.mp hp) _\n \u00b7 rw [mul_comm]\n apply Ideal.mul_mem_left (I.map (algebraMap R R)) _ (Ideal.mem_map_of_mem _ ha)\n \u00b7 simp only [RingHom.map_mul, mul_comm (algebraMap R S p) (l a)]\n rfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\n\u22a2 \u2200 {a : R},\n a \u2208 I \u2192\n \u2191l a \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n[PROOFSTEP]\nintro a ha\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\na : R\nha : a \u2208 I\n\u22a2 \u2191l a \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n[PROOFSTEP]\nrw [Algebra.id.smul_eq_mul, mul_assoc, mul_comm, mul_assoc, Set.mem_image]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\na : R\nha : a \u2208 I\n\u22a2 \u2203 x_1,\n x_1 \u2208 \u2191(Ideal.map (algebraMap R { x_2 // x_2 \u2208 adjoin R {x} }) I) \u2227\n \u2191(algebraMap { x_2 // x_2 \u2208 adjoin R {x} } S) x_1 = \u2191(algebraMap R S) a * (\u2191(algebraMap R S) p * \u2191l a)\n[PROOFSTEP]\nrefine Exists.intro (algebraMap R R a * \u27e8l a * algebraMap R S p, show l a * algebraMap R S p \u2208 R from ?_\u27e9) ?_\n[GOAL]\ncase refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\na : R\nha : a \u2208 I\n\u22a2 \u2191(algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) a *\n { val := \u2191l a * \u2191(algebraMap R S) p, property := (_ : \u2191l a * \u2191(algebraMap R S) p \u2208 adjoin R {x}) } \u2208\n \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2227\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\n (\u2191(algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) a *\n { val := \u2191l a * \u2191(algebraMap R S) p, property := (_ : \u2191l a * \u2191(algebraMap R S) p \u2208 adjoin R {x}) }) =\n \u2191(algebraMap R S) a * (\u2191(algebraMap R S) p * \u2191l a)\n[PROOFSTEP]\nrefine \u27e8?_, ?_\u27e9\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\na : R\nha : a \u2208 I\n\u22a2 \u2191l a * \u2191(algebraMap R S) p \u2208 adjoin R {x}\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\na : R\nha : a \u2208 I\n\u22a2 \u2191(algebraMap R S) p * \u2191l a \u2208 adjoin R {x}\n[PROOFSTEP]\nexact mem_conductor_iff.mp (Ideal.mem_comap.mp hp) _\n[GOAL]\ncase refine_1.refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\na : R\nha : a \u2208 I\n\u22a2 \u2191(algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) a *\n { val := \u2191l a * \u2191(algebraMap R S) p, property := (_ : \u2191l a * \u2191(algebraMap R S) p \u2208 adjoin R {x}) } \u2208\n \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase refine_1.refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\na : R\nha : a \u2208 I\n\u22a2 { val := \u2191l a * \u2191(algebraMap R S) p, property := (_ : \u2191l a * \u2191(algebraMap R S) p \u2208 adjoin R {x}) } *\n \u2191(algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) a \u2208\n \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n[PROOFSTEP]\napply Ideal.mul_mem_left (I.map (algebraMap R R)) _ (Ideal.mem_map_of_mem _ ha)\n[GOAL]\ncase refine_1.refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\na : R\nha : a \u2208 I\n\u22a2 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\n (\u2191(algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) a *\n { val := \u2191l a * \u2191(algebraMap R S) p, property := (_ : \u2191l a * \u2191(algebraMap R S) p \u2208 adjoin R {x}) }) =\n \u2191(algebraMap R S) a * (\u2191(algebraMap R S) p * \u2191l a)\n[PROOFSTEP]\nsimp only [RingHom.map_mul, mul_comm (algebraMap R S p) (l a)]\n[GOAL]\ncase refine_1.refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\na : R\nha : a \u2208 I\n\u22a2 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (\u2191(algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) a) *\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\n { val := \u2191l a * \u2191(algebraMap R S) p, property := (_ : \u2191l a * \u2191(algebraMap R S) p \u2208 adjoin R {x}) } =\n \u2191(algebraMap R S) a * (\u2191l a * \u2191(algebraMap R S) p)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\nlem :\n \u2200 {a : R},\n a \u2208 I \u2192\n \u2191l a \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n\u22a2 (Finsupp.sum l fun a c => c \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p) \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n[PROOFSTEP]\nrefine Finset.sum_induction _ (fun u => u \u2208 algebraMap R S '' I.map (algebraMap R R)) (fun a b => ?_) ?_ ?_\n[GOAL]\ncase intro.intro.refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\nlem :\n \u2200 {a : R},\n a \u2208 I \u2192\n \u2191l a \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\na b : S\n\u22a2 (fun u =>\n u \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I))\n a \u2192\n (fun u =>\n u \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) ''\n \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I))\n b \u2192\n (fun u =>\n u \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) ''\n \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I))\n (a + b)\ncase intro.intro.refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\nlem :\n \u2200 {a : R},\n a \u2208 I \u2192\n \u2191l a \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n\u22a2 (fun u =>\n u \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I))\n 0\ncase intro.intro.refine_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\nlem :\n \u2200 {a : R},\n a \u2208 I \u2192\n \u2191l a \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n\u22a2 \u2200 (x_1 : R),\n x_1 \u2208 l.support \u2192\n (fun u =>\n u \u2208\n \u2191(algebraMap { x_2 // x_2 \u2208 adjoin R {x} } S) ''\n \u2191(Ideal.map (algebraMap R { x_2 // x_2 \u2208 adjoin R {x} }) I))\n ((fun a c => c \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p) x_1 (\u2191l x_1))\n[PROOFSTEP]\nrintro \u27e8z, hz, rfl\u27e9 \u27e8y, hy, rfl\u27e9\n[GOAL]\ncase intro.intro.refine_1.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz\u271d : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\u271d\nlem :\n \u2200 {a : R},\n a \u2208 I \u2192\n \u2191l a \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\nz : { x_1 // x_1 \u2208 adjoin R {x} }\nhz : z \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\ny : { x_1 // x_1 \u2208 adjoin R {x} }\nhy : y \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n\u22a2 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) z + \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) y \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\ncase intro.intro.refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\nlem :\n \u2200 {a : R},\n a \u2208 I \u2192\n \u2191l a \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n\u22a2 (fun u =>\n u \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I))\n 0\ncase intro.intro.refine_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\nlem :\n \u2200 {a : R},\n a \u2208 I \u2192\n \u2191l a \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n\u22a2 \u2200 (x_1 : R),\n x_1 \u2208 l.support \u2192\n (fun u =>\n u \u2208\n \u2191(algebraMap { x_2 // x_2 \u2208 adjoin R {x} } S) ''\n \u2191(Ideal.map (algebraMap R { x_2 // x_2 \u2208 adjoin R {x} }) I))\n ((fun a c => c \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p) x_1 (\u2191l x_1))\n[PROOFSTEP]\nrw [\u2190 RingHom.map_add]\n[GOAL]\ncase intro.intro.refine_1.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz\u271d : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\u271d\nlem :\n \u2200 {a : R},\n a \u2208 I \u2192\n \u2191l a \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\nz : { x_1 // x_1 \u2208 adjoin R {x} }\nhz : z \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\ny : { x_1 // x_1 \u2208 adjoin R {x} }\nhy : y \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n\u22a2 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (z + y) \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\ncase intro.intro.refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\nlem :\n \u2200 {a : R},\n a \u2208 I \u2192\n \u2191l a \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n\u22a2 (fun u =>\n u \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I))\n 0\ncase intro.intro.refine_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\nlem :\n \u2200 {a : R},\n a \u2208 I \u2192\n \u2191l a \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n\u22a2 \u2200 (x_1 : R),\n x_1 \u2208 l.support \u2192\n (fun u =>\n u \u2208\n \u2191(algebraMap { x_2 // x_2 \u2208 adjoin R {x} } S) ''\n \u2191(Ideal.map (algebraMap R { x_2 // x_2 \u2208 adjoin R {x} }) I))\n ((fun a c => c \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p) x_1 (\u2191l x_1))\n[PROOFSTEP]\nexact \u27e8z + y, Ideal.add_mem _ (SetLike.mem_coe.mp hz) hy, rfl\u27e9\n[GOAL]\ncase intro.intro.refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\nlem :\n \u2200 {a : R},\n a \u2208 I \u2192\n \u2191l a \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n\u22a2 (fun u =>\n u \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I))\n 0\n[PROOFSTEP]\nexact \u27e80, SetLike.mem_coe.mpr <| Ideal.zero_mem _, RingHom.map_zero _\u27e9\n[GOAL]\ncase intro.intro.refine_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\nlem :\n \u2200 {a : R},\n a \u2208 I \u2192\n \u2191l a \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n\u22a2 \u2200 (x_1 : R),\n x_1 \u2208 l.support \u2192\n (fun u =>\n u \u2208\n \u2191(algebraMap { x_2 // x_2 \u2208 adjoin R {x} } S) ''\n \u2191(Ideal.map (algebraMap R { x_2 // x_2 \u2208 adjoin R {x} }) I))\n ((fun a c => c \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p) x_1 (\u2191l x_1))\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase intro.intro.refine_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nl : R \u2192\u2080 S\nH : l \u2208 Finsupp.supported S S \u2191I\nH' : (Finsupp.sum l fun i a => a \u2022 \u2191(algebraMap R S) i) = z\nlem :\n \u2200 {a : R},\n a \u2208 I \u2192\n \u2191l a \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\ny : R\nhy : y \u2208 l.support\n\u22a2 (fun a c => c \u2022 \u2191(algebraMap R S) a * \u2191(algebraMap R S) p) y (\u2191l y) \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n[PROOFSTEP]\nexact lem ((Finsupp.mem_supported _ l).mp H hy)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\n\u22a2 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I) =\n Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\n\u22a2 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I) \u2264\n Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\ny : { x_1 // x_1 \u2208 adjoin R {x} }\nhy : y \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\n\u22a2 y \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n[PROOFSTEP]\nobtain \u27e8z, hz\u27e9 := y\n[GOAL]\ncase a.mk\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\n\u22a2 { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n[PROOFSTEP]\nobtain \u27e8p, hp, q, hq, hpq\u27e9 := Submodule.mem_sup.mp ((Ideal.eq_top_iff_one _).mp hx)\n[GOAL]\ncase a.mk.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\n\u22a2 { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n[PROOFSTEP]\nhave temp : algebraMap R S p * z + algebraMap R S q * z = z := by\n simp only [\u2190 add_mul, \u2190 RingHom.map_add (algebraMap R S), hpq, map_one, one_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\n\u22a2 \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\n[PROOFSTEP]\nsimp only [\u2190 add_mul, \u2190 RingHom.map_add (algebraMap R S), hpq, map_one, one_mul]\n[GOAL]\ncase a.mk.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\n\u22a2 { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n[PROOFSTEP]\nsuffices z \u2208 algebraMap R S '' I.map (algebraMap R R) \u2194 (\u27e8z, hz\u27e9 : R) \u2208 I.map (algebraMap R R)\n by\n rw [\u2190 this, \u2190 temp]\n obtain \u27e8a, ha\u27e9 :=\n (Set.mem_image _ _ _).mp\n (prod_mem_ideal_map_of_mem_conductor hp (show z \u2208 I.map (algebraMap R S) by rwa [Ideal.mem_comap] at hy ))\n use a + algebraMap R R q * \u27e8z, hz\u27e9\n refine\n \u27e8Ideal.add_mem (I.map (algebraMap R R)) ha.left ?_, by\n simp only [ha.right, map_add, AlgHom.map_mul, add_right_inj]; rfl\u27e9\n rw [mul_comm]\n exact Ideal.mul_mem_left (I.map (algebraMap R R)) _ (Ideal.mem_map_of_mem _ hq)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nthis :\n z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2194\n { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n\u22a2 { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n[PROOFSTEP]\nrw [\u2190 this, \u2190 temp]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nthis :\n z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2194\n { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n\u22a2 \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 :=\n (Set.mem_image _ _ _).mp\n (prod_mem_ideal_map_of_mem_conductor hp (show z \u2208 I.map (algebraMap R S) by rwa [Ideal.mem_comap] at hy ))\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nthis :\n z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2194\n { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n\u22a2 z \u2208 Ideal.map (algebraMap R S) I\n[PROOFSTEP]\nrwa [Ideal.mem_comap] at hy \n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nthis :\n z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2194\n { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\na : { x_1 // x_1 \u2208 adjoin R {x} }\nha :\n a \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2227\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) a = \u2191(algebraMap R S) p * z\n\u22a2 \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z \u2208\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n[PROOFSTEP]\nuse a + algebraMap R R q * \u27e8z, hz\u27e9\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nthis :\n z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2194\n { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\na : { x_1 // x_1 \u2208 adjoin R {x} }\nha :\n a \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2227\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) a = \u2191(algebraMap R S) p * z\n\u22a2 a + \u2191(algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) q * { val := z, property := hz } \u2208\n \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2227\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\n (a + \u2191(algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) q * { val := z, property := hz }) =\n \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z\n[PROOFSTEP]\nrefine\n \u27e8Ideal.add_mem (I.map (algebraMap R R)) ha.left ?_, by\n simp only [ha.right, map_add, AlgHom.map_mul, add_right_inj]; rfl\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nthis :\n z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2194\n { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\na : { x_1 // x_1 \u2208 adjoin R {x} }\nha :\n a \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2227\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) a = \u2191(algebraMap R S) p * z\n\u22a2 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\n (a + \u2191(algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) q * { val := z, property := hz }) =\n \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z\n[PROOFSTEP]\nsimp only [ha.right, map_add, AlgHom.map_mul, add_right_inj]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nthis :\n z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2194\n { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\na : { x_1 // x_1 \u2208 adjoin R {x} }\nha :\n a \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2227\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) a = \u2191(algebraMap R S) p * z\n\u22a2 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\n (\u2191(algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) q * { val := z, property := hz }) =\n \u2191(algebraMap R S) q * z\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nthis :\n z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2194\n { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\na : { x_1 // x_1 \u2208 adjoin R {x} }\nha :\n a \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2227\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) a = \u2191(algebraMap R S) p * z\n\u22a2 \u2191(algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) q * { val := z, property := hz } \u2208\n Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nthis :\n z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2194\n { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\na : { x_1 // x_1 \u2208 adjoin R {x} }\nha :\n a \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2227\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) a = \u2191(algebraMap R S) p * z\n\u22a2 { val := z, property := hz } * \u2191(algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) q \u2208\n Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n[PROOFSTEP]\nexact Ideal.mul_mem_left (I.map (algebraMap R R)) _ (Ideal.mem_map_of_mem _ hq)\n[GOAL]\ncase a.mk.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\n\u22a2 z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I) \u2194\n { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n[PROOFSTEP]\nrefine \u27e8fun h => ?_, fun h => (Set.mem_image _ _ _).mpr (Exists.intro \u27e8z, hz\u27e9 \u27e8by simp [h], rfl\u27e9)\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nh : { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n\u22a2 { val := z, property := hz } \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase a.mk.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nh : z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n\u22a2 { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n[PROOFSTEP]\nobtain \u27e8x\u2081, hx\u2081, hx\u2082\u27e9 := (Set.mem_image _ _ _).mp h\n[GOAL]\ncase a.mk.intro.intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nh : z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\nx\u2081 : { x_1 // x_1 \u2208 adjoin R {x} }\nhx\u2081 : x\u2081 \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\nhx\u2082 : \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) x\u2081 = z\n\u22a2 { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n[PROOFSTEP]\nhave : x\u2081 = \u27e8z, hz\u27e9 := by\n apply h_alg\n simp [hx\u2082]\n rfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nh : z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\nx\u2081 : { x_1 // x_1 \u2208 adjoin R {x} }\nhx\u2081 : x\u2081 \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\nhx\u2082 : \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) x\u2081 = z\n\u22a2 x\u2081 = { val := z, property := hz }\n[PROOFSTEP]\napply h_alg\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nh : z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\nx\u2081 : { x_1 // x_1 \u2208 adjoin R {x} }\nhx\u2081 : x\u2081 \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\nhx\u2082 : \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) x\u2081 = z\n\u22a2 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) x\u2081 =\n \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) { val := z, property := hz }\n[PROOFSTEP]\nsimp [hx\u2082]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nh : z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\nx\u2081 : { x_1 // x_1 \u2208 adjoin R {x} }\nhx\u2081 : x\u2081 \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\nhx\u2082 : \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) x\u2081 = z\n\u22a2 z = \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) { val := z, property := hz }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.mk.intro.intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nz : S\nhz : z \u2208 adjoin R {x}\nhy : { val := z, property := hz } \u2208 comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\np : R\nhp : p \u2208 comap (algebraMap R S) (conductor R x)\nq : R\nhq : q \u2208 I\nhpq : p + q = 1\ntemp : \u2191(algebraMap R S) p * z + \u2191(algebraMap R S) q * z = z\nh : z \u2208 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) '' \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\nx\u2081 : { x_1 // x_1 \u2208 adjoin R {x} }\nhx\u2081 : x\u2081 \u2208 \u2191(Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\nhx\u2082 : \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) x\u2081 = z\nthis : x\u2081 = { val := z, property := hz }\n\u22a2 { val := z, property := hz } \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n[PROOFSTEP]\nrwa [\u2190 this]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\n\u22a2 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2264\n comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\n[PROOFSTEP]\nhave : algebraMap R S = (algebraMap _ S).comp (algebraMap R R) := by ext; rfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\n\u22a2 algebraMap R S =\n RingHom.comp (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} })\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nx\u271d : R\n\u22a2 \u2191(algebraMap R S) x\u271d =\n \u2191(RingHom.comp (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} })) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nthis :\n algebraMap R S =\n RingHom.comp (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} })\n\u22a2 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2264\n comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R S) I)\n[PROOFSTEP]\nrw [this, \u2190 Ideal.map_map]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nthis :\n algebraMap R S =\n RingHom.comp (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} })\n\u22a2 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2264\n comap (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\n (Ideal.map (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\n (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I))\n[PROOFSTEP]\napply Ideal.le_comap_map\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\n\u22a2 { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2243+*\n S \u29f8 Ideal.map (algebraMap R S) I\n[PROOFSTEP]\nlet f : R \u29f8 I.map (algebraMap R R) \u2192+* S \u29f8 I.map (algebraMap R S) :=\n (Ideal.Quotient.lift (I.map (algebraMap R R))\n ((Ideal.Quotient.mk (I.map (algebraMap R S))).comp (algebraMap R S))\n (fun r hr => by\n have : algebraMap R S = (algebraMap R S).comp (algebraMap R R) := by ext; rfl\n rw [RingHom.comp_apply, Ideal.Quotient.eq_zero_iff_mem, this, \u2190 Ideal.map_map]\n exact Ideal.mem_map_of_mem _ hr))\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nr : { x_1 // x_1 \u2208 adjoin R {x} }\nhr : r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n\u22a2 \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)) r = 0\n[PROOFSTEP]\nhave : algebraMap R S = (algebraMap R S).comp (algebraMap R R) := by ext; rfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nr : { x_1 // x_1 \u2208 adjoin R {x} }\nhr : r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n\u22a2 algebraMap R S =\n RingHom.comp (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} })\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nr : { x_1 // x_1 \u2208 adjoin R {x} }\nhr : r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\nx\u271d : R\n\u22a2 \u2191(algebraMap R S) x\u271d =\n \u2191(RingHom.comp (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} })) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nr : { x_1 // x_1 \u2208 adjoin R {x} }\nhr : r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\nthis :\n algebraMap R S =\n RingHom.comp (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} })\n\u22a2 \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)) r = 0\n[PROOFSTEP]\nrw [RingHom.comp_apply, Ideal.Quotient.eq_zero_iff_mem, this, \u2190 Ideal.map_map]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nr : { x_1 // x_1 \u2208 adjoin R {x} }\nhr : r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\nthis :\n algebraMap R S =\n RingHom.comp (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} })\n\u22a2 \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) r \u2208\n Ideal.map (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S) (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n[PROOFSTEP]\nexact Ideal.mem_map_of_mem _ hr\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\n\u22a2 { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2243+*\n S \u29f8 Ideal.map (algebraMap R S) I\n[PROOFSTEP]\nrefine RingEquiv.ofBijective f \u27e8?_, ?_\u27e9\n[GOAL]\ncase refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\n\u22a2 Function.Injective \u2191f\n[PROOFSTEP]\nrefine RingHom.lift_injective_of_ker_le_ideal _ _ fun u hu => ?_\n[GOAL]\ncase refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\nu : { x_1 // x_1 \u2208 adjoin R {x} }\nhu :\n u \u2208 ker (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n\u22a2 u \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I\n[PROOFSTEP]\nrwa [RingHom.mem_ker, RingHom.comp_apply, Ideal.Quotient.eq_zero_iff_mem, \u2190 Ideal.mem_comap,\n comap_map_eq_map_adjoin_of_coprime_conductor hx h_alg] at hu \n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\n\u22a2 Function.Surjective \u2191f\n[PROOFSTEP]\nrefine Ideal.Quotient.lift_surjective_of_surjective _ _ fun y => ?_\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\n\u22a2 \u2203 a,\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)) a =\n y\n[PROOFSTEP]\nobtain \u27e8z, hz\u27e9 := Ideal.Quotient.mk_surjective y\n[GOAL]\ncase refine_2.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\n\u22a2 \u2203 a,\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)) a =\n y\n[PROOFSTEP]\nhave : z \u2208 conductor R x \u2294 I.map (algebraMap R S) :=\n by\n suffices conductor R x \u2294 I.map (algebraMap R S) = \u22a4 by simp only [this, Submodule.mem_top]\n rw [Ideal.eq_top_iff_one] at hx \u22a2\n replace hx := Ideal.mem_map_of_mem (algebraMap R S) hx\n rw [Ideal.map_sup, RingHom.map_one] at hx \n exact\n (sup_le_sup\n (show ((conductor R x).comap (algebraMap R S)).map (algebraMap R S) \u2264 conductor R x from Ideal.map_comap_le)\n (le_refl (I.map (algebraMap R S))))\n hx\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\n\u22a2 z \u2208 conductor R x \u2294 Ideal.map (algebraMap R S) I\n[PROOFSTEP]\nsuffices conductor R x \u2294 I.map (algebraMap R S) = \u22a4 by simp only [this, Submodule.mem_top]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\nthis : conductor R x \u2294 Ideal.map (algebraMap R S) I = \u22a4\n\u22a2 z \u2208 conductor R x \u2294 Ideal.map (algebraMap R S) I\n[PROOFSTEP]\nsimp only [this, Submodule.mem_top]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\n\u22a2 conductor R x \u2294 Ideal.map (algebraMap R S) I = \u22a4\n[PROOFSTEP]\nrw [Ideal.eq_top_iff_one] at hx \u22a2\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : 1 \u2208 comap (algebraMap R S) (conductor R x) \u2294 I\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\n\u22a2 1 \u2208 conductor R x \u2294 Ideal.map (algebraMap R S) I\n[PROOFSTEP]\nreplace hx := Ideal.mem_map_of_mem (algebraMap R S) hx\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\nhx : \u2191(algebraMap R S) 1 \u2208 Ideal.map (algebraMap R S) (comap (algebraMap R S) (conductor R x) \u2294 I)\n\u22a2 1 \u2208 conductor R x \u2294 Ideal.map (algebraMap R S) I\n[PROOFSTEP]\nrw [Ideal.map_sup, RingHom.map_one] at hx \n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\nhx : 1 \u2208 Ideal.map (algebraMap R S) (comap (algebraMap R S) (conductor R x)) \u2294 Ideal.map (algebraMap R S) I\n\u22a2 1 \u2208 conductor R x \u2294 Ideal.map (algebraMap R S) I\n[PROOFSTEP]\nexact\n (sup_le_sup\n (show ((conductor R x).comap (algebraMap R S)).map (algebraMap R S) \u2264 conductor R x from Ideal.map_comap_le)\n (le_refl (I.map (algebraMap R S))))\n hx\n[GOAL]\ncase refine_2.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\nthis : z \u2208 conductor R x \u2294 Ideal.map (algebraMap R S) I\n\u22a2 \u2203 a,\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)) a =\n y\n[PROOFSTEP]\nrw [\u2190 Ideal.mem_quotient_iff_mem_sup, hz, Ideal.mem_map_iff_of_surjective] at this \n[GOAL]\ncase refine_2.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\nthis : \u2203 x_1, x_1 \u2208 conductor R x \u2227 \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) x_1 = y\n\u22a2 \u2203 a,\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)) a =\n y\ncase refine_2.intro.hf\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\nthis : y \u2208 Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (conductor R x)\n\u22a2 Function.Surjective \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n[PROOFSTEP]\nobtain \u27e8u, hu, hu'\u27e9 := this\n[GOAL]\ncase refine_2.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\nu : S\nhu : u \u2208 conductor R x\nhu' : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) u = y\n\u22a2 \u2203 a,\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)) a =\n y\ncase refine_2.intro.hf\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\nthis : y \u2208 Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (conductor R x)\n\u22a2 Function.Surjective \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n[PROOFSTEP]\nuse\u27e8u, conductor_subset_adjoin hu\u27e9\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\nu : S\nhu : u \u2208 conductor R x\nhu' : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) u = y\n\u22a2 \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n { val := u, property := (_ : u \u2208 \u2191(adjoin R {x})) } =\n y\ncase refine_2.intro.hf\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\nthis : y \u2208 Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (conductor R x)\n\u22a2 Function.Surjective \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n[PROOFSTEP]\nsimp only [\u2190 hu']\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\nu : S\nhu : u \u2208 conductor R x\nhu' : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) u = y\n\u22a2 \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n { val := u, property := (_ : u \u2208 \u2191(adjoin R {x})) } =\n \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) u\ncase refine_2.intro.hf\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\nthis : y \u2208 Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (conductor R x)\n\u22a2 Function.Surjective \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine_2.intro.hf\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nh_alg : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\nf : { x_1 // x_1 \u2208 adjoin R {x} } \u29f8 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192+*\n S \u29f8 Ideal.map (algebraMap R S) I :=\n Ideal.Quotient.lift (Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I)\n (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n (_ :\n \u2200 (r : { x_1 // x_1 \u2208 adjoin R {x} }),\n r \u2208 Ideal.map (algebraMap R { x_1 // x_1 \u2208 adjoin R {x} }) I \u2192\n \u2191(RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n (algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))\n r =\n 0)\ny : S \u29f8 Ideal.map (algebraMap R S) I\nz : S\nhz : \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y\nthis : y \u2208 Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (conductor R x)\n\u22a2 Function.Surjective \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I))\n[PROOFSTEP]\nexact Ideal.Quotient.mk_surjective\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\n\u22a2 \u2191{J | J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)} \u2243\n \u2191{d | d \u2208 normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}\n[PROOFSTEP]\nhave : IsPrincipalIdealRing (R \u29f8 I)[X] := inferInstance\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nthis : IsPrincipalIdealRing (R \u29f8 I)[X]\n\u22a2 \u2191{J | J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)} \u2243\n \u2191{d | d \u2208 normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}\n[PROOFSTEP]\nlet f : S \u29f8 map (algebraMap R S) I \u2243+* (R \u29f8 I)[X] \u29f8 span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} :=\n by\n refine\n (quotAdjoinEquivQuotMap hx ?_).symm.trans\n (((Algebra.adjoin.powerBasis' hx').quotientEquivQuotientMinpolyMap I).toRingEquiv.trans (quotEquivOfEq ?_))\n \u00b7 exact NoZeroSMulDivisors.algebraMap_injective (Algebra.adjoin R { x }) S\n \u00b7 rw [Algebra.adjoin.powerBasis'_minpoly_gen hx']\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nthis : IsPrincipalIdealRing (R \u29f8 I)[X]\n\u22a2 S \u29f8 Ideal.map (algebraMap R S) I \u2243+* (R \u29f8 I)[X] \u29f8 span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)}\n[PROOFSTEP]\nrefine\n (quotAdjoinEquivQuotMap hx ?_).symm.trans\n (((Algebra.adjoin.powerBasis' hx').quotientEquivQuotientMinpolyMap I).toRingEquiv.trans (quotEquivOfEq ?_))\n[GOAL]\ncase refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nthis : IsPrincipalIdealRing (R \u29f8 I)[X]\n\u22a2 Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S)\n[PROOFSTEP]\nexact NoZeroSMulDivisors.algebraMap_injective (Algebra.adjoin R { x }) S\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nthis : IsPrincipalIdealRing (R \u29f8 I)[X]\n\u22a2 span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R (adjoin.powerBasis' hx').gen)} =\n span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)}\n[PROOFSTEP]\nrw [Algebra.adjoin.powerBasis'_minpoly_gen hx']\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nthis : IsPrincipalIdealRing (R \u29f8 I)[X]\nf : S \u29f8 Ideal.map (algebraMap R S) I \u2243+* (R \u29f8 I)[X] \u29f8 span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} :=\n RingEquiv.trans\n (RingEquiv.symm (quotAdjoinEquivQuotMap hx (_ : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))))\n (RingEquiv.trans (AlgEquiv.toRingEquiv (PowerBasis.quotientEquivQuotientMinpolyMap (adjoin.powerBasis' hx') I))\n (quotEquivOfEq\n (_ :\n span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R (adjoin.powerBasis' hx').gen)} =\n span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)})))\n\u22a2 \u2191{J | J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)} \u2243\n \u2191{d | d \u2208 normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}\n[PROOFSTEP]\nrefine (normalizedFactorsEquivOfQuotEquiv f ?_ ?_).trans ?_\n[GOAL]\ncase refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nthis : IsPrincipalIdealRing (R \u29f8 I)[X]\nf : S \u29f8 Ideal.map (algebraMap R S) I \u2243+* (R \u29f8 I)[X] \u29f8 span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} :=\n RingEquiv.trans\n (RingEquiv.symm (quotAdjoinEquivQuotMap hx (_ : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))))\n (RingEquiv.trans (AlgEquiv.toRingEquiv (PowerBasis.quotientEquivQuotientMinpolyMap (adjoin.powerBasis' hx') I))\n (quotEquivOfEq\n (_ :\n span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R (adjoin.powerBasis' hx').gen)} =\n span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)})))\n\u22a2 Ideal.map (algebraMap R S) I \u2260 \u22a5\n[PROOFSTEP]\nrwa [Ne.def, map_eq_bot_iff_of_injective (NoZeroSMulDivisors.algebraMap_injective R S), \u2190 Ne.def]\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nthis : IsPrincipalIdealRing (R \u29f8 I)[X]\nf : S \u29f8 Ideal.map (algebraMap R S) I \u2243+* (R \u29f8 I)[X] \u29f8 span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} :=\n RingEquiv.trans\n (RingEquiv.symm (quotAdjoinEquivQuotMap hx (_ : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))))\n (RingEquiv.trans (AlgEquiv.toRingEquiv (PowerBasis.quotientEquivQuotientMinpolyMap (adjoin.powerBasis' hx') I))\n (quotEquivOfEq\n (_ :\n span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R (adjoin.powerBasis' hx').gen)} =\n span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)})))\n\u22a2 span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} \u2260 \u22a5\n[PROOFSTEP]\nby_contra h\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nthis : IsPrincipalIdealRing (R \u29f8 I)[X]\nf : S \u29f8 Ideal.map (algebraMap R S) I \u2243+* (R \u29f8 I)[X] \u29f8 span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} :=\n RingEquiv.trans\n (RingEquiv.symm (quotAdjoinEquivQuotMap hx (_ : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))))\n (RingEquiv.trans (AlgEquiv.toRingEquiv (PowerBasis.quotientEquivQuotientMinpolyMap (adjoin.powerBasis' hx') I))\n (quotEquivOfEq\n (_ :\n span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R (adjoin.powerBasis' hx').gen)} =\n span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)})))\nh : span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} = \u22a5\n\u22a2 False\n[PROOFSTEP]\nexact\n (show Polynomial.map (Ideal.Quotient.mk I) (minpoly R x) \u2260 0 from Polynomial.map_monic_ne_zero (minpoly.monic hx'))\n (span_singleton_eq_bot.mp h)\n[GOAL]\ncase refine_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nthis : IsPrincipalIdealRing (R \u29f8 I)[X]\nf : S \u29f8 Ideal.map (algebraMap R S) I \u2243+* (R \u29f8 I)[X] \u29f8 span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} :=\n RingEquiv.trans\n (RingEquiv.symm (quotAdjoinEquivQuotMap hx (_ : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))))\n (RingEquiv.trans (AlgEquiv.toRingEquiv (PowerBasis.quotientEquivQuotientMinpolyMap (adjoin.powerBasis' hx') I))\n (quotEquivOfEq\n (_ :\n span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R (adjoin.powerBasis' hx').gen)} =\n span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)})))\n\u22a2 \u2191{M | M \u2208 normalizedFactors (span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)})} \u2243\n \u2191{d | d \u2208 normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}\n[PROOFSTEP]\nrefine (normalizedFactorsEquivSpanNormalizedFactors ?_).symm\n[GOAL]\ncase refine_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nthis : IsPrincipalIdealRing (R \u29f8 I)[X]\nf : S \u29f8 Ideal.map (algebraMap R S) I \u2243+* (R \u29f8 I)[X] \u29f8 span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} :=\n RingEquiv.trans\n (RingEquiv.symm (quotAdjoinEquivQuotMap hx (_ : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin R {x} } S))))\n (RingEquiv.trans (AlgEquiv.toRingEquiv (PowerBasis.quotientEquivQuotientMinpolyMap (adjoin.powerBasis' hx') I))\n (quotEquivOfEq\n (_ :\n span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R (adjoin.powerBasis' hx').gen)} =\n span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)})))\n\u22a2 Polynomial.map (Ideal.Quotient.mk I) (minpoly R x) \u2260 0\n[PROOFSTEP]\nexact Polynomial.map_monic_ne_zero (minpoly.monic hx')\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\n\u22a2 multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n[PROOFSTEP]\nrw [normalizedFactorsMapEquivNormalizedFactorsMinPolyMk, Equiv.coe_trans, Function.comp_apply,\n multiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_multiplicity,\n normalizedFactorsEquivOfQuotEquiv_multiplicity_eq_multiplicity]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\n\u22a2 normalizedFactors (Ideal.map (algebraMap R S) I) =\n Multiset.map (fun f => \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))\n (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))\n[PROOFSTEP]\next J\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\n\u22a2 Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =\n Multiset.count J\n (Multiset.map (fun f => \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))\n (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))\n[PROOFSTEP]\nby_cases hJ : J \u2208 normalizedFactors (I.map (algebraMap R S))\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\n\u22a2 Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =\n Multiset.count J\n (Multiset.map (fun f => \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))\n (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : \u00acJ \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\n\u22a2 Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =\n Multiset.count J\n (Multiset.map (fun f => \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))\n (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : \u00acJ \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\n\u22a2 Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =\n Multiset.count J\n (Multiset.map (fun f => \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))\n (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))\n[PROOFSTEP]\nrw [Multiset.count_eq_zero.mpr hJ, eq_comm, Multiset.count_eq_zero, Multiset.mem_map]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : \u00acJ \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\n\u22a2 \u00ac\u2203 a,\n a \u2208 Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) \u2227\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm a) = J\n[PROOFSTEP]\nsimp only [Multiset.mem_attach, true_and_iff, not_exists]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : \u00acJ \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\n\u22a2 \u2200 (x_1 : \u2191{d | d \u2208 normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}),\n \u00ac\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm x_1) = J\n[PROOFSTEP]\nrintro J' rfl\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ' : \u2191{d | d \u2208 normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}\nhJ :\n \u00ac\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') \u2208\n normalizedFactors (Ideal.map (algebraMap R S) I)\n\u22a2 False\n[PROOFSTEP]\nexact hJ ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J').prop\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\n\u22a2 Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =\n Multiset.count J\n (Multiset.map (fun f => \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))\n (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))\n[PROOFSTEP]\nhave := multiplicity_factors_map_eq_multiplicity hI hI' hx hx' hJ\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =\n Multiset.count J\n (Multiset.map (fun f => \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))\n (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))\n[PROOFSTEP]\nrw [multiplicity_eq_count_normalizedFactors, multiplicity_eq_count_normalizedFactors,\n UniqueFactorizationMonoid.normalize_normalized_factor _ hJ, UniqueFactorizationMonoid.normalize_normalized_factor,\n PartENat.natCast_inj] at this \n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =\n Multiset.count\n (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))\n\u22a2 Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =\n Multiset.count J\n (Multiset.map (fun f => \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))\n (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))\ncase pos.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n \u2191(Multiset.count\n (\u2191normalize\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))\n\u22a2 \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }) \u2208\n normalizedFactors ?m.1024627\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n \u2191(Multiset.count\n (\u2191normalize\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))\n\u22a2 (R \u29f8 I)[X]\ncase pos.ha\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Irreducible \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ })\ncase pos.hb\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Polynomial.map (Ideal.Quotient.mk I) (minpoly R x) \u2260 0\ncase pos.ha\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Irreducible J\ncase pos.hb\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Ideal.map (algebraMap R S) I \u2260 0\n[PROOFSTEP]\nrefine'\n this.trans\n _\n -- Get rid of the `map` by applying the equiv to both sides.\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =\n Multiset.count\n (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))\n\u22a2 Multiset.count (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) =\n Multiset.count J\n (Multiset.map (fun f => \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))\n (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))\ncase pos.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n \u2191(Multiset.count\n (\u2191normalize\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))\n\u22a2 \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }) \u2208\n normalizedFactors ?m.1024627\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n \u2191(Multiset.count\n (\u2191normalize\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))\n\u22a2 (R \u29f8 I)[X]\ncase pos.ha\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Irreducible \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ })\ncase pos.hb\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Polynomial.map (Ideal.Quotient.mk I) (minpoly R x) \u2260 0\ncase pos.ha\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Irreducible J\ncase pos.hb\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Ideal.map (algebraMap R S) I \u2260 0\n[PROOFSTEP]\ngeneralize hJ' : (normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') \u27e8J, hJ\u27e9 = J'\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =\n Multiset.count\n (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))\nJ' : \u2191{d | d \u2208 normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}\nhJ' : \u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ } = J'\n\u22a2 Multiset.count (\u2191J') (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) =\n Multiset.count J\n (Multiset.map (fun f => \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))\n (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))\ncase pos.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n \u2191(Multiset.count\n (\u2191normalize\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))\n\u22a2 \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }) \u2208\n normalizedFactors ?m.1024627\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n \u2191(Multiset.count\n (\u2191normalize\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))\n\u22a2 (R \u29f8 I)[X]\ncase pos.ha\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Irreducible \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ })\ncase pos.hb\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Polynomial.map (Ideal.Quotient.mk I) (minpoly R x) \u2260 0\ncase pos.ha\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Irreducible J\ncase pos.hb\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Ideal.map (algebraMap R S) I \u2260 0\n[PROOFSTEP]\nhave : ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J' : Ideal S) = J := by\n rw [\u2190 hJ', Equiv.symm_apply_apply _ _, Subtype.coe_mk]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =\n Multiset.count\n (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))\nJ' : \u2191{d | d \u2208 normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}\nhJ' : \u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ } = J'\n\u22a2 \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') = J\n[PROOFSTEP]\nrw [\u2190 hJ', Equiv.symm_apply_apply _ _, Subtype.coe_mk]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b2 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d\u00b9 :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =\n Multiset.count\n (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))\nJ' : \u2191{d | d \u2208 normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}\nhJ' : \u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ } = J'\nthis : \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') = J\n\u22a2 Multiset.count (\u2191J') (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) =\n Multiset.count J\n (Multiset.map (fun f => \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))\n (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))\ncase pos.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n \u2191(Multiset.count\n (\u2191normalize\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))\n\u22a2 \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }) \u2208\n normalizedFactors ?m.1024627\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n \u2191(Multiset.count\n (\u2191normalize\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))\n\u22a2 (R \u29f8 I)[X]\ncase pos.ha\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Irreducible \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ })\ncase pos.hb\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Polynomial.map (Ideal.Quotient.mk I) (minpoly R x) \u2260 0\ncase pos.ha\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Irreducible J\ncase pos.hb\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Ideal.map (algebraMap R S) I \u2260 0\n[PROOFSTEP]\nsubst this\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ' : \u2191{d | d \u2208 normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}\nhJ :\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') \u2208\n normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'))\n (Ideal.map (algebraMap R S) I) =\n multiplicity\n (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')\n { val := \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'))\n (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity\n (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')\n { val := \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n Multiset.count (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'))\n (normalizedFactors (Ideal.map (algebraMap R S) I)) =\n Multiset.count\n (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')\n { val := \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))\nhJ' :\n \u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')\n { val := \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), property := hJ } =\n J'\n\u22a2 Multiset.count (\u2191J') (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) =\n Multiset.count (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'))\n (Multiset.map (fun f => \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))\n (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))\ncase pos.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n \u2191(Multiset.count\n (\u2191normalize\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))\n\u22a2 \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }) \u2208\n normalizedFactors ?m.1024627\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n \u2191(Multiset.count\n (\u2191normalize\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))\n\u22a2 (R \u29f8 I)[X]\ncase pos.ha\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Irreducible \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ })\ncase pos.hb\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Polynomial.map (Ideal.Quotient.mk I) (minpoly R x) \u2260 0\ncase pos.ha\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Irreducible J\ncase pos.hb\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Ideal.map (algebraMap R S) I \u2260 0\n[PROOFSTEP]\nrw [Multiset.count_map_eq_count' fun f =>\n ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f : Ideal S),\n Multiset.attach_count_eq_count_coe]\n[GOAL]\ncase pos.hf\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ' : \u2191{d | d \u2208 normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}\nhJ :\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') \u2208\n normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'))\n (Ideal.map (algebraMap R S) I) =\n multiplicity\n (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')\n { val := \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'))\n (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity\n (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')\n { val := \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n Multiset.count (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'))\n (normalizedFactors (Ideal.map (algebraMap R S) I)) =\n Multiset.count\n (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')\n { val := \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))\nhJ' :\n \u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx')\n { val := \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J'), property := hJ } =\n J'\n\u22a2 Function.Injective fun f => \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f)\n[PROOFSTEP]\nexact Subtype.coe_injective.comp (Equiv.injective _)\n[GOAL]\ncase pos.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d\u00b9 :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis\u271d :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n \u2191(Multiset.count\n (\u2191normalize\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))\n\u22a2 \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }) \u2208\n normalizedFactors ?m.1024627\n[PROOFSTEP]\nexact (normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx' _).prop\n[GOAL]\ncase pos.ha\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Irreducible \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ })\n[PROOFSTEP]\nexact irreducible_of_normalized_factor _ (normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx' _).prop\n[GOAL]\ncase pos.hb\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis\u271d :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis :\n \u2191(Multiset.count (\u2191normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Polynomial.map (Ideal.Quotient.mk I) (minpoly R x) \u2260 0\n[PROOFSTEP]\nexact Polynomial.map_monic_ne_zero (minpoly.monic hx')\n[GOAL]\ncase pos.ha\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Irreducible J\n[PROOFSTEP]\nexact irreducible_of_normalized_factor _ hJ\n[GOAL]\ncase pos.hb\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nJ : Ideal S\nhJ : J \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\nthis :\n multiplicity J (Ideal.map (algebraMap R S) I) =\n multiplicity (\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))\n (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Ideal.map (algebraMap R S) I \u2260 0\n[PROOFSTEP]\nrwa [\u2190 bot_eq_zero, Ne.def, map_eq_bot_iff_of_injective (NoZeroSMulDivisors.algebraMap_injective R S)]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Irreducible (Ideal.map (algebraMap R S) I)\n[PROOFSTEP]\nhave mem_norm_factors :\n normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) :=\n by simp [normalizedFactors_irreducible hf]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n[PROOFSTEP]\nsimp [normalizedFactors_irreducible hf]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Irreducible (Ideal.map (algebraMap R S) I)\n[PROOFSTEP]\nsuffices \u2203 y, normalizedFactors (I.map (algebraMap R S)) = { y }\n by\n obtain \u27e8y, hy\u27e9 := this\n have h :=\n normalizedFactors_prod\n (show I.map (algebraMap R S) \u2260 0 by\n rwa [\u2190 bot_eq_zero, Ne.def, map_eq_bot_iff_of_injective (NoZeroSMulDivisors.algebraMap_injective R S)])\n rw [associated_iff_eq, hy, Multiset.prod_singleton] at h \n rw [\u2190 h]\n exact irreducible_of_normalized_factor y (show y \u2208 normalizedFactors (I.map (algebraMap R S)) by simp [hy])\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nthis : \u2203 y, normalizedFactors (Ideal.map (algebraMap R S) I) = {y}\n\u22a2 Irreducible (Ideal.map (algebraMap R S) I)\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := this\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\ny : Ideal S\nhy : normalizedFactors (Ideal.map (algebraMap R S) I) = {y}\n\u22a2 Irreducible (Ideal.map (algebraMap R S) I)\n[PROOFSTEP]\nhave h :=\n normalizedFactors_prod\n (show I.map (algebraMap R S) \u2260 0 by\n rwa [\u2190 bot_eq_zero, Ne.def, map_eq_bot_iff_of_injective (NoZeroSMulDivisors.algebraMap_injective R S)])\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\ny : Ideal S\nhy : normalizedFactors (Ideal.map (algebraMap R S) I) = {y}\n\u22a2 Ideal.map (algebraMap R S) I \u2260 0\n[PROOFSTEP]\nrwa [\u2190 bot_eq_zero, Ne.def, map_eq_bot_iff_of_injective (NoZeroSMulDivisors.algebraMap_injective R S)]\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\ny : Ideal S\nhy : normalizedFactors (Ideal.map (algebraMap R S) I) = {y}\nh : Associated (Multiset.prod (normalizedFactors (Ideal.map (algebraMap R S) I))) (Ideal.map (algebraMap R S) I)\n\u22a2 Irreducible (Ideal.map (algebraMap R S) I)\n[PROOFSTEP]\nrw [associated_iff_eq, hy, Multiset.prod_singleton] at h \n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\ny : Ideal S\nhy : normalizedFactors (Ideal.map (algebraMap R S) I) = {y}\nh : y = Ideal.map (algebraMap R S) I\n\u22a2 Irreducible (Ideal.map (algebraMap R S) I)\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\ny : Ideal S\nhy : normalizedFactors (Ideal.map (algebraMap R S) I) = {y}\nh : y = Ideal.map (algebraMap R S) I\n\u22a2 Irreducible y\n[PROOFSTEP]\nexact irreducible_of_normalized_factor y (show y \u2208 normalizedFactors (I.map (algebraMap R S)) by simp [hy])\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\ny : Ideal S\nhy : normalizedFactors (Ideal.map (algebraMap R S) I) = {y}\nh : y = Ideal.map (algebraMap R S) I\n\u22a2 y \u2208 normalizedFactors (Ideal.map (algebraMap R S) I)\n[PROOFSTEP]\nsimp [hy]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 \u2203 y, normalizedFactors (Ideal.map (algebraMap R S) I) = {y}\n[PROOFSTEP]\nrw [normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map hI hI' hx hx']\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 \u2203 y,\n Multiset.map (fun f => \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))\n (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))) =\n {y}\n[PROOFSTEP]\nuse((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm\n \u27e8normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)), mem_norm_factors\u27e9 :\n Ideal S)\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Multiset.map (fun f => \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))\n (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))) =\n {\u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm\n { val := \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)), property := mem_norm_factors })}\n[PROOFSTEP]\nrw [Multiset.map_eq_singleton]\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 \u2203 a,\n Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) = {a} \u2227\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm a) =\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm\n { val := \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)), property := mem_norm_factors })\n[PROOFSTEP]\nuse\u27e8normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)), mem_norm_factors\u27e9\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) =\n {{ val := \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)), property := mem_norm_factors }} \u2227\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm\n { val := \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)), property := mem_norm_factors }) =\n \u2191(\u2191(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm\n { val := \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)), property := mem_norm_factors })\n[PROOFSTEP]\nrefine \u27e8?_, rfl\u27e9\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) =\n {{ val := \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)), property := mem_norm_factors }}\n[PROOFSTEP]\napply Multiset.map_injective Subtype.coe_injective\n[GOAL]\ncase h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 Multiset.map (fun a => \u2191a)\n (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))) =\n Multiset.map (fun a => \u2191a)\n {{ val := \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)), property := mem_norm_factors }}\n[PROOFSTEP]\nrw [Multiset.attach_map_val, Multiset.map_singleton, Subtype.coe_mk]\n[GOAL]\ncase h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nx : S\nI : Ideal R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsIntegrallyClosed R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : IsDedekindDomain S\ninst\u271d : NoZeroSMulDivisors R S\nhI : IsMaximal I\nhI' : I \u2260 \u22a5\nhx : comap (algebraMap R S) (conductor R x) \u2294 I = \u22a4\nhx' : _root_.IsIntegral R x\nhf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\nmem_norm_factors :\n \u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) \u2208\n normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))\n\u22a2 normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) =\n {\u2191normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}\n[PROOFSTEP]\nexact normalizedFactors_irreducible hf\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.KummerDedekind", "llama_tokens": 74554, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702761768249, "lm_q2_score": 0.5660185351961013, "lm_q1q2_score": 0.4102900119287998}} {"text": "[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx : A\n\u22a2 \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A\n[PROOFSTEP]\nsimp only [selfAdjoint.mem_iff, star_smul, add_comm, StarAddMonoid.star_add, star_inv', star_bit0, star_one, star_star,\n star_invOf (2 : R), star_trivial]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx y : A\n\u22a2 (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) (x + y) =\n (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) x +\n (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) y\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx y : A\n\u22a2 \u2191((fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) (x + y)) =\n \u2191((fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) x +\n (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) y)\n[PROOFSTEP]\nsimp [add_add_add_comm]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nr : R\nx : A\n\u22a2 AddHom.toFun\n { toFun := fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) },\n map_add' :=\n (_ :\n \u2200 (x y : A),\n (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) (x + y) =\n (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) x +\n (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) y) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) },\n map_add' :=\n (_ :\n \u2200 (x y : A),\n (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) (x + y) =\n (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) x +\n (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) y) }\n x\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nr : R\nx : A\n\u22a2 \u2191(AddHom.toFun\n { toFun := fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) },\n map_add' :=\n (_ :\n \u2200 (x y : A),\n (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) (x + y) =\n (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) x +\n (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) y) }\n (r \u2022 x)) =\n \u2191(\u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) },\n map_add' :=\n (_ :\n \u2200 (x y : A),\n (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) (x + y) =\n (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) x +\n (fun x => { val := \u215f2 \u2022 (x + star x), property := (_ : \u215f2 \u2022 (x + star x) \u2208 selfAdjoint A) }) y) }\n x)\n[PROOFSTEP]\nsimp [\u2190 mul_smul, show \u215f2 * r = r * \u215f2 from Commute.invOf_left <| (2 : \u2115).cast_commute r]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx : A\n\u22a2 \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A\n[PROOFSTEP]\nsimp only [skewAdjoint.mem_iff, star_smul, star_sub, star_star, star_trivial, \u2190 smul_neg, neg_sub]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx y : A\n\u22a2 (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) (x + y) =\n (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) x +\n (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) y\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx y : A\n\u22a2 \u2191((fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) (x + y)) =\n \u2191((fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) x +\n (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) y)\n[PROOFSTEP]\nsimp only [sub_add, \u2190 smul_add, sub_sub_eq_add_sub, star_add, AddSubgroup.coe_mk, AddSubgroup.coe_add]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nr : R\nx : A\n\u22a2 AddHom.toFun\n { toFun := fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) },\n map_add' :=\n (_ :\n \u2200 (x y : A),\n (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) (x + y) =\n (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) x +\n (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) y) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) },\n map_add' :=\n (_ :\n \u2200 (x y : A),\n (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) (x + y) =\n (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) x +\n (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) y) }\n x\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nr : R\nx : A\n\u22a2 \u2191(AddHom.toFun\n { toFun := fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) },\n map_add' :=\n (_ :\n \u2200 (x y : A),\n (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) (x + y) =\n (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) x +\n (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) y) }\n (r \u2022 x)) =\n \u2191(\u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) },\n map_add' :=\n (_ :\n \u2200 (x y : A),\n (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) (x + y) =\n (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) x +\n (fun x => { val := \u215f2 \u2022 (x - star x), property := (_ : \u215f2 \u2022 (x - star x) \u2208 skewAdjoint A) }) y) }\n x)\n[PROOFSTEP]\nsimp [\u2190 mul_smul, \u2190 smul_sub, show r * \u215f2 = \u215f2 * r from Commute.invOf_right <| (2 : \u2115).commute_cast r]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx : A\n\u22a2 \u2191(\u2191(selfAdjointPart R) x) + \u2191(\u2191(skewAdjointPart R) x) = x\n[PROOFSTEP]\nsimp only [smul_sub, selfAdjointPart_apply_coe, smul_add, skewAdjointPart_apply_coe, add_add_sub_cancel,\n invOf_two_smul_add_invOf_two_smul]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx : A\nhx : IsSelfAdjoint x\n\u22a2 \u2191(\u2191(selfAdjointPart R) x) = x\n[PROOFSTEP]\nrw [selfAdjointPart_apply_coe, hx.star_eq, smul_add, invOf_two_smul_add_invOf_two_smul]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx : A\nhx : IsSelfAdjoint x\n\u22a2 \u2191(\u2191(skewAdjointPart R) x) = \u21910\n[PROOFSTEP]\nrw [skewAdjointPart_apply_coe, hx.star_eq, sub_self, smul_zero, ZeroMemClass.coe_zero]\n -- porting note: todo: make it a `simp`\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx\u271d : { x // x \u2208 skewAdjoint.submodule R A }\nx : A\nhx : star x = -x\n\u22a2 \u2191(\u2191(LinearMap.comp (selfAdjointPart R) (Submodule.subtype (skewAdjoint.submodule R A)))\n { val := x, property := hx }) =\n \u2191(\u21910 { val := x, property := hx })\n[PROOFSTEP]\nsimp [hx]\n -- porting note: todo: make it a `simp`\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx\u271d : { x // x \u2208 skewAdjoint.submodule R A }\nx : A\nhx : star x = -x\n\u22a2 \u2191(\u2191(LinearMap.comp (skewAdjointPart R) (Submodule.subtype (skewAdjoint.submodule R A)))\n { val := x, property := hx }) =\n \u2191(\u2191LinearMap.id { val := x, property := hx })\n[PROOFSTEP]\nsimp only [LinearMap.comp_apply, Submodule.subtype_apply, skewAdjointPart_apply_coe, hx, sub_neg_eq_add, smul_add,\n invOf_two_smul_add_invOf_two_smul]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx\u271d : { x // x \u2208 skewAdjoint.submodule R A }\nx : A\nhx : star x = -x\n\u22a2 x = \u2191(\u2191LinearMap.id { val := x, property := hx })\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\n\u22a2 A \u2243\u2097[R] { x // x \u2208 selfAdjoint A } \u00d7 { x // x \u2208 skewAdjoint A }\n[PROOFSTEP]\nrefine\n LinearEquiv.ofLinear ((selfAdjointPart R).prod (skewAdjointPart R))\n (LinearMap.coprod ((selfAdjoint.submodule R A).subtype) (skewAdjoint.submodule R A).subtype) ?_\n (LinearMap.ext <| StarModule.selfAdjointPart_add_skewAdjointPart R)\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\n\u22a2 LinearMap.comp (LinearMap.prod (selfAdjointPart R) (skewAdjointPart R))\n (LinearMap.coprod (Submodule.subtype (selfAdjoint.submodule R A))\n (Submodule.subtype (skewAdjoint.submodule R A))) =\n LinearMap.id\n[PROOFSTEP]\next\n[GOAL]\ncase hl.h.h\u2081.a\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx\u271d : { x // x \u2208 selfAdjoint A }\n\u22a2 \u2191(\u2191(LinearMap.comp\n (LinearMap.comp (LinearMap.prod (selfAdjointPart R) (skewAdjointPart R))\n (LinearMap.coprod (Submodule.subtype (selfAdjoint.submodule R A))\n (Submodule.subtype (skewAdjoint.submodule R A))))\n (LinearMap.inl R { x // x \u2208 selfAdjoint A } { x // x \u2208 skewAdjoint A }))\n x\u271d).fst =\n \u2191(\u2191(LinearMap.comp LinearMap.id (LinearMap.inl R { x // x \u2208 selfAdjoint A } { x // x \u2208 skewAdjoint A })) x\u271d).fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hl.h.h\u2082.a\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx\u271d : { x // x \u2208 selfAdjoint A }\n\u22a2 \u2191(\u2191(LinearMap.comp\n (LinearMap.comp (LinearMap.prod (selfAdjointPart R) (skewAdjointPart R))\n (LinearMap.coprod (Submodule.subtype (selfAdjoint.submodule R A))\n (Submodule.subtype (skewAdjoint.submodule R A))))\n (LinearMap.inl R { x // x \u2208 selfAdjoint A } { x // x \u2208 skewAdjoint A }))\n x\u271d).snd =\n \u2191(\u2191(LinearMap.comp LinearMap.id (LinearMap.inl R { x // x \u2208 selfAdjoint A } { x // x \u2208 skewAdjoint A })) x\u271d).snd\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hr.h.h\u2081.a\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx\u271d : { x // x \u2208 skewAdjoint A }\n\u22a2 \u2191(\u2191(LinearMap.comp\n (LinearMap.comp (LinearMap.prod (selfAdjointPart R) (skewAdjointPart R))\n (LinearMap.coprod (Submodule.subtype (selfAdjoint.submodule R A))\n (Submodule.subtype (skewAdjoint.submodule R A))))\n (LinearMap.inr R { x // x \u2208 selfAdjoint A } { x // x \u2208 skewAdjoint A }))\n x\u271d).fst =\n \u2191(\u2191(LinearMap.comp LinearMap.id (LinearMap.inr R { x // x \u2208 selfAdjoint A } { x // x \u2208 skewAdjoint A })) x\u271d).fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hr.h.h\u2082.a\nR : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarSemigroup R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx\u271d : { x // x \u2208 skewAdjoint A }\n\u22a2 \u2191(\u2191(LinearMap.comp\n (LinearMap.comp (LinearMap.prod (selfAdjointPart R) (skewAdjointPart R))\n (LinearMap.coprod (Submodule.subtype (selfAdjoint.submodule R A))\n (Submodule.subtype (skewAdjoint.submodule R A))))\n (LinearMap.inr R { x // x \u2208 selfAdjoint A } { x // x \u2208 skewAdjoint A }))\n x\u271d).snd =\n \u2191(\u2191(LinearMap.comp LinearMap.id (LinearMap.inr R { x // x \u2208 selfAdjoint A } { x // x \u2208 skewAdjoint A })) x\u271d).snd\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u00b9\u00b3 : Semiring R\u271d\ninst\u271d\u00b9\u00b2 : StarSemigroup R\u271d\ninst\u271d\u00b9\u00b9 : TrivialStar R\u271d\ninst\u271d\u00b9\u2070 : AddCommGroup A\u271d\ninst\u271d\u2079 : Module R\u271d A\u271d\ninst\u271d\u2078 : StarAddMonoid A\u271d\ninst\u271d\u2077 : StarModule R\u271d A\u271d\ninst\u271d\u2076 : Invertible 2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : StarRing R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : StarSemigroup A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : StarModule R A\nr : R\n\u22a2 \u2191(algebraMap R A) (star r) = star (\u2191(algebraMap R A) r)\n[PROOFSTEP]\nsimp only [Algebra.algebraMap_eq_smul_one, star_smul, star_one]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Star.Module", "llama_tokens": 7339, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8198933447152498, "lm_q2_score": 0.5, "lm_q1q2_score": 0.4099466723576249}} {"text": "[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u22a2 IsCyclotomicExtension {n} A B \u2194 (\u2203 r, IsPrimitiveRoot r \u2191n) \u2227 \u2200 (x : B), x \u2208 adjoin A {b | b ^ \u2191n = 1}\n[PROOFSTEP]\nsimp [IsCyclotomicExtension_iff]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension \u2205 A B\n\u22a2 \u22a5 = \u22a4\n[PROOFSTEP]\nsimpa [Algebra.eq_top_iff, IsCyclotomicExtension_iff] using h\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension {1} A B\nx : B\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nsimpa [adjoin_singleton_one] using ((IsCyclotomicExtension_iff _ _ _).1 h).2 x\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u22a5 = \u22a4\n\u22a2 IsCyclotomicExtension \u2205 A B\n[PROOFSTEP]\nrefine' (iff_adjoin_eq_top _ A _).2 \u27e8fun s hs => by simp at hs , _root_.eq_top_iff.2 fun x hx => _\u27e9\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u22a5 = \u22a4\ns : \u2115+\nhs : s \u2208 \u2205\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191s\n[PROOFSTEP]\nsimp at hs \n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u22a5 = \u22a4\nx : B\nhx : x \u2208 \u22a4\n\u22a2 x \u2208 adjoin A {b | \u2203 n, n \u2208 \u2205 \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nrw [\u2190 h] at hx \n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u22a5 = \u22a4\nx : B\nhx : x \u2208 \u22a5\n\u22a2 x \u2208 adjoin A {b | \u2203 n, n \u2208 \u2205 \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nC : Type w\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nhS : IsCyclotomicExtension S A B\nhT : IsCyclotomicExtension T B C\nh : Function.Injective \u2191(algebraMap B C)\n\u22a2 IsCyclotomicExtension (S \u222a T) A C\n[PROOFSTEP]\nrefine' \u27e8fun hn => _, fun x => _\u27e9\n[GOAL]\ncase refine'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nC : Type w\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nhS : IsCyclotomicExtension S A B\nhT : IsCyclotomicExtension T B C\nh : Function.Injective \u2191(algebraMap B C)\nn\u271d : \u2115+\nhn : n\u271d \u2208 S \u222a T\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191n\u271d\n[PROOFSTEP]\ncases' hn with hn hn\n[GOAL]\ncase refine'_1.inl\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nC : Type w\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nhS : IsCyclotomicExtension S A B\nhT : IsCyclotomicExtension T B C\nh : Function.Injective \u2191(algebraMap B C)\nn\u271d : \u2115+\nhn : n\u271d \u2208 S\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191n\u271d\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 := ((IsCyclotomicExtension_iff _ _ _).1 hS).1 hn\n[GOAL]\ncase refine'_1.inl.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nC : Type w\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nhS : IsCyclotomicExtension S A B\nhT : IsCyclotomicExtension T B C\nh : Function.Injective \u2191(algebraMap B C)\nn\u271d : \u2115+\nhn : n\u271d \u2208 S\nb : B\nhb : IsPrimitiveRoot b \u2191n\u271d\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191n\u271d\n[PROOFSTEP]\nrefine' \u27e8algebraMap B C b, _\u27e9\n[GOAL]\ncase refine'_1.inl.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nC : Type w\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nhS : IsCyclotomicExtension S A B\nhT : IsCyclotomicExtension T B C\nh : Function.Injective \u2191(algebraMap B C)\nn\u271d : \u2115+\nhn : n\u271d \u2208 S\nb : B\nhb : IsPrimitiveRoot b \u2191n\u271d\n\u22a2 IsPrimitiveRoot (\u2191(algebraMap B C) b) \u2191n\u271d\n[PROOFSTEP]\nexact hb.map_of_injective h\n[GOAL]\ncase refine'_1.inr\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nC : Type w\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nhS : IsCyclotomicExtension S A B\nhT : IsCyclotomicExtension T B C\nh : Function.Injective \u2191(algebraMap B C)\nn\u271d : \u2115+\nhn : n\u271d \u2208 T\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191n\u271d\n[PROOFSTEP]\nexact ((IsCyclotomicExtension_iff _ _ _).1 hT).1 hn\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nC : Type w\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nhS : IsCyclotomicExtension S A B\nhT : IsCyclotomicExtension T B C\nh : Function.Injective \u2191(algebraMap B C)\nx : C\n\u22a2 x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nrefine'\n adjoin_induction (((IsCyclotomicExtension_iff T B _).1 hT).2 x)\n (fun c \u27e8n, hn\u27e9 => subset_adjoin \u27e8n, Or.inr hn.1, hn.2\u27e9) (fun b => _) (fun x y hx hy => Subalgebra.add_mem _ hx hy)\n fun x y hx hy => Subalgebra.mul_mem _ hx hy\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nC : Type w\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nhS : IsCyclotomicExtension S A B\nhT : IsCyclotomicExtension T B C\nh : Function.Injective \u2191(algebraMap B C)\nx : C\nb : B\n\u22a2 \u2191(algebraMap B C) b \u2208 adjoin A {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nlet f := IsScalarTower.toAlgHom A B C\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nC : Type w\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nhS : IsCyclotomicExtension S A B\nhT : IsCyclotomicExtension T B C\nh : Function.Injective \u2191(algebraMap B C)\nx : C\nb : B\nf : B \u2192\u2090[A] C := IsScalarTower.toAlgHom A B C\n\u22a2 \u2191(algebraMap B C) b \u2208 adjoin A {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nhave hb : f b \u2208 (adjoin A {b : B | \u2203 a : \u2115+, a \u2208 S \u2227 b ^ (a : \u2115) = 1}).map f :=\n \u27e8b, ((IsCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl\u27e9\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nC : Type w\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nhS : IsCyclotomicExtension S A B\nhT : IsCyclotomicExtension T B C\nh : Function.Injective \u2191(algebraMap B C)\nx : C\nb : B\nf : B \u2192\u2090[A] C := IsScalarTower.toAlgHom A B C\nhb : \u2191f b \u2208 Subalgebra.map f (adjoin A {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1})\n\u22a2 \u2191(algebraMap B C) b \u2208 adjoin A {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nrw [IsScalarTower.toAlgHom_apply, \u2190 adjoin_image] at hb \n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nC : Type w\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nhS : IsCyclotomicExtension S A B\nhT : IsCyclotomicExtension T B C\nh : Function.Injective \u2191(algebraMap B C)\nx : C\nb : B\nf : B \u2192\u2090[A] C := IsScalarTower.toAlgHom A B C\nhb : \u2191(algebraMap B C) b \u2208 adjoin A (\u2191f '' {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1})\n\u22a2 \u2191(algebraMap B C) b \u2208 adjoin A {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nrefine' adjoin_mono (fun y hy => _) hb\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nC : Type w\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nhS : IsCyclotomicExtension S A B\nhT : IsCyclotomicExtension T B C\nh : Function.Injective \u2191(algebraMap B C)\nx : C\nb : B\nf : B \u2192\u2090[A] C := IsScalarTower.toAlgHom A B C\nhb : \u2191(algebraMap B C) b \u2208 adjoin A (\u2191f '' {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1})\ny : C\nhy : y \u2208 \u2191f '' {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1}\n\u22a2 y \u2208 {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nobtain \u27e8b\u2081, \u27e8\u27e8n, hn\u27e9, h\u2081\u27e9\u27e9 := hy\n[GOAL]\ncase refine'_2.intro.intro.intro\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nC : Type w\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nhS : IsCyclotomicExtension S A B\nhT : IsCyclotomicExtension T B C\nh : Function.Injective \u2191(algebraMap B C)\nx : C\nb : B\nf : B \u2192\u2090[A] C := IsScalarTower.toAlgHom A B C\nhb : \u2191(algebraMap B C) b \u2208 adjoin A (\u2191f '' {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1})\ny : C\nb\u2081 : B\nh\u2081 : \u2191f b\u2081 = y\nn : \u2115+\nhn : n \u2208 S \u2227 b\u2081 ^ \u2191n = 1\n\u22a2 y \u2208 {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nexact \u27e8n, \u27e8mem_union_left T hn.1, by rw [\u2190 h\u2081, \u2190 AlgHom.map_pow, hn.2, AlgHom.map_one]\u27e9\u27e9\n[GOAL]\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nC : Type w\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra B C\ninst\u271d : IsScalarTower A B C\nhS : IsCyclotomicExtension S A B\nhT : IsCyclotomicExtension T B C\nh : Function.Injective \u2191(algebraMap B C)\nx : C\nb : B\nf : B \u2192\u2090[A] C := IsScalarTower.toAlgHom A B C\nhb : \u2191(algebraMap B C) b \u2208 adjoin A (\u2191f '' {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1})\ny : C\nb\u2081 : B\nh\u2081 : \u2191f b\u2081 = y\nn : \u2115+\nhn : n \u2208 S \u2227 b\u2081 ^ \u2191n = 1\n\u22a2 y ^ \u2191n = 1\n[PROOFSTEP]\nrw [\u2190 h\u2081, \u2190 AlgHom.map_pow, hn.2, AlgHom.map_one]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 IsCyclotomicExtension S A B \u2194 S = \u2205 \u2228 S = {1}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 IsCyclotomicExtension S A B \u2192 S = \u2205 \u2228 S = {1}\n[PROOFSTEP]\nrintro \u27e8hprim, -\u27e9\n[GOAL]\ncase mp.mk\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nhprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n\n\u22a2 S = \u2205 \u2228 S = {1}\n[PROOFSTEP]\nrw [\u2190 subset_singleton_iff_eq]\n[GOAL]\ncase mp.mk\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nhprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n\n\u22a2 S \u2286 {1}\n[PROOFSTEP]\nintro t ht\n[GOAL]\ncase mp.mk\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nhprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n\nt : \u2115+\nht : t \u2208 S\n\u22a2 t \u2208 {1}\n[PROOFSTEP]\nobtain \u27e8\u03b6, h\u03b6\u27e9 := hprim ht\n[GOAL]\ncase mp.mk.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nhprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n\nt : \u2115+\nht : t \u2208 S\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191t\n\u22a2 t \u2208 {1}\n[PROOFSTEP]\nrw [mem_singleton_iff, \u2190 PNat.coe_eq_one_iff]\n[GOAL]\ncase mp.mk.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nhprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n\nt : \u2115+\nht : t \u2208 S\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191t\n\u22a2 \u2191t = 1\n[PROOFSTEP]\nexact_mod_cast h\u03b6.unique (IsPrimitiveRoot.of_subsingleton \u03b6)\n[GOAL]\ncase mpr\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 S = \u2205 \u2228 S = {1} \u2192 IsCyclotomicExtension S A B\n[PROOFSTEP]\nrintro\n (rfl | rfl)\n -- Porting note: `R := A` was not needed.\n[GOAL]\ncase mpr.inl\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 IsCyclotomicExtension \u2205 A B\n[PROOFSTEP]\nexact \u27e8fun h => h.elim, fun x => by convert (mem_top (R := A) : x \u2208 \u22a4)\u27e9\n[GOAL]\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nx : B\n\u22a2 x \u2208 adjoin A {b | \u2203 n, n \u2208 \u2205 \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nconvert (mem_top (R := A) : x \u2208 \u22a4)\n[GOAL]\ncase mpr.inr\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 IsCyclotomicExtension {1} A B\n[PROOFSTEP]\nrw [iff_singleton]\n[GOAL]\ncase mpr.inr\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 (\u2203 r, IsPrimitiveRoot r \u21911) \u2227 \u2200 (x : B), x \u2208 adjoin A {b | b ^ \u21911 = 1}\n[PROOFSTEP]\nexact \u27e8\u27e80, IsPrimitiveRoot.of_subsingleton 0\u27e9, fun x => by convert (mem_top (R := A) : x \u2208 \u22a4)\u27e9\n[GOAL]\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nx : B\n\u22a2 x \u2208 adjoin A {b | b ^ \u21911 = 1}\n[PROOFSTEP]\nconvert (mem_top (R := A) : x \u2208 \u22a4)\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension (S \u222a T) A B\n\u22a2 IsCyclotomicExtension T { x // x \u2208 adjoin A {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1} } B\n[PROOFSTEP]\nhave :\n {b : B | \u2203 n : \u2115+, n \u2208 S \u222a T \u2227 b ^ (n : \u2115) = 1} =\n {b : B | \u2203 n : \u2115+, n \u2208 S \u2227 b ^ (n : \u2115) = 1} \u222a {b : B | \u2203 n : \u2115+, n \u2208 T \u2227 b ^ (n : \u2115) = 1} :=\n by\n refine' le_antisymm _ _\n \u00b7 rintro x \u27e8n, hn\u2081 | hn\u2082, hnpow\u27e9\n \u00b7 left; exact \u27e8n, hn\u2081, hnpow\u27e9\n \u00b7 right; exact \u27e8n, hn\u2082, hnpow\u27e9\n \u00b7 rintro x (\u27e8n, hn\u27e9 | \u27e8n, hn\u27e9)\n \u00b7 exact \u27e8n, Or.inl hn.1, hn.2\u27e9\n \u00b7 exact \u27e8n, Or.inr hn.1, hn.2\u27e9\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension (S \u222a T) A B\n\u22a2 {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1} = {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} \u222a {b | \u2203 n, n \u2208 T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension (S \u222a T) A B\n\u22a2 {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1} \u2264 {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} \u222a {b | \u2203 n, n \u2208 T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nrintro x \u27e8n, hn\u2081 | hn\u2082, hnpow\u27e9\n[GOAL]\ncase refine'_1.intro.intro.inl\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension (S \u222a T) A B\nx : B\nn : \u2115+\nhnpow : x ^ \u2191n = 1\nhn\u2081 : n \u2208 S\n\u22a2 x \u2208 {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} \u222a {b | \u2203 n, n \u2208 T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nleft\n[GOAL]\ncase refine'_1.intro.intro.inl.h\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension (S \u222a T) A B\nx : B\nn : \u2115+\nhnpow : x ^ \u2191n = 1\nhn\u2081 : n \u2208 S\n\u22a2 x \u2208 {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nexact \u27e8n, hn\u2081, hnpow\u27e9\n[GOAL]\ncase refine'_1.intro.intro.inr\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension (S \u222a T) A B\nx : B\nn : \u2115+\nhnpow : x ^ \u2191n = 1\nhn\u2082 : n \u2208 T\n\u22a2 x \u2208 {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} \u222a {b | \u2203 n, n \u2208 T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nright\n[GOAL]\ncase refine'_1.intro.intro.inr.h\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension (S \u222a T) A B\nx : B\nn : \u2115+\nhnpow : x ^ \u2191n = 1\nhn\u2082 : n \u2208 T\n\u22a2 x \u2208 {b | \u2203 n, n \u2208 T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nexact \u27e8n, hn\u2082, hnpow\u27e9\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension (S \u222a T) A B\n\u22a2 {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} \u222a {b | \u2203 n, n \u2208 T \u2227 b ^ \u2191n = 1} \u2264 {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nrintro x (\u27e8n, hn\u27e9 | \u27e8n, hn\u27e9)\n[GOAL]\ncase refine'_2.inl.intro\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension (S \u222a T) A B\nx : B\nn : \u2115+\nhn : n \u2208 S \u2227 x ^ \u2191n = 1\n\u22a2 x \u2208 {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nexact \u27e8n, Or.inl hn.1, hn.2\u27e9\n[GOAL]\ncase refine'_2.inr.intro\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension (S \u222a T) A B\nx : B\nn : \u2115+\nhn : n \u2208 T \u2227 x ^ \u2191n = 1\n\u22a2 x \u2208 {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nexact \u27e8n, Or.inr hn.1, hn.2\u27e9\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension (S \u222a T) A B\nthis : {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1} = {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} \u222a {b | \u2203 n, n \u2208 T \u2227 b ^ \u2191n = 1}\n\u22a2 IsCyclotomicExtension T { x // x \u2208 adjoin A {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1} } B\n[PROOFSTEP]\nrefine' \u27e8fun hn => ((IsCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => _\u27e9\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension (S \u222a T) A B\nthis : {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1} = {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} \u222a {b | \u2203 n, n \u2208 T \u2227 b ^ \u2191n = 1}\nb : B\n\u22a2 b \u2208 adjoin { x // x \u2208 adjoin A {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1} } {b | \u2203 n, n \u2208 T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nreplace h := ((IsCyclotomicExtension_iff _ _ _).1 h).2 b\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nthis : {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1} = {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} \u222a {b | \u2203 n, n \u2208 T \u2227 b ^ \u2191n = 1}\nb : B\nh : b \u2208 adjoin A {b | \u2203 n, n \u2208 S \u222a T \u2227 b ^ \u2191n = 1}\n\u22a2 b \u2208 adjoin { x // x \u2208 adjoin A {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1} } {b | \u2203 n, n \u2208 T \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nrwa [this, adjoin_union_eq_adjoin_adjoin, Subalgebra.mem_restrictScalars] at h \n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension T A B\nhS : S \u2286 T\n\u22a2 IsCyclotomicExtension S A { x // x \u2208 adjoin A {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1} }\n[PROOFSTEP]\nrefine' \u27e8@fun n hn => _, fun b => _\u27e9\n[GOAL]\ncase refine'_1\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension T A B\nhS : S \u2286 T\nn : \u2115+\nhn : n \u2208 S\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191n\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 := ((IsCyclotomicExtension_iff _ _ _).1 h).1 (hS hn)\n[GOAL]\ncase refine'_1.intro\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension T A B\nhS : S \u2286 T\nn : \u2115+\nhn : n \u2208 S\nb : B\nhb : IsPrimitiveRoot b \u2191n\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191n\n[PROOFSTEP]\nrefine' \u27e8\u27e8b, subset_adjoin \u27e8n, hn, hb.pow_eq_one\u27e9\u27e9, _\u27e9\n[GOAL]\ncase refine'_1.intro\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension T A B\nhS : S \u2286 T\nn : \u2115+\nhn : n \u2208 S\nb : B\nhb : IsPrimitiveRoot b \u2191n\n\u22a2 IsPrimitiveRoot { val := b, property := (_ : b \u2208 \u2191(adjoin A {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1})) } \u2191n\n[PROOFSTEP]\nrwa [\u2190 IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension T A B\nhS : S \u2286 T\nb : { x // x \u2208 adjoin A {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1} }\n\u22a2 b \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nconvert mem_top (R := A) (x := b)\n[GOAL]\ncase h.e'_5\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension T A B\nhS : S \u2286 T\nb : { x // x \u2208 adjoin A {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1} }\n\u22a2 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} = \u22a4\n[PROOFSTEP]\nrw [\u2190 adjoin_adjoin_coe_preimage, preimage_setOf_eq]\n[GOAL]\ncase h.e'_5\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : IsCyclotomicExtension T A B\nhS : S \u2286 T\nb : { x // x \u2208 adjoin A {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1} }\n\u22a2 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} = adjoin A {a | \u2203 a_1, a_1 \u2208 S \u2227 \u2191a ^ \u2191a_1 = 1}\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension S A B\n\u22a2 IsCyclotomicExtension (S \u222a {n}) A B\n[PROOFSTEP]\nrefine' (iff_adjoin_eq_top _ A _).2 \u27e8fun s hs => _, _\u27e9\n[GOAL]\ncase refine'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension S A B\ns : \u2115+\nhs : s \u2208 S \u222a {n}\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191s\n[PROOFSTEP]\nrw [mem_union, mem_singleton_iff] at hs \n[GOAL]\ncase refine'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension S A B\ns : \u2115+\nhs : s \u2208 S \u2228 s = n\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191s\n[PROOFSTEP]\nobtain hs | rfl := hs\n[GOAL]\ncase refine'_1.inl\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension S A B\ns : \u2115+\nhs : s \u2208 S\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191s\n[PROOFSTEP]\nexact H.exists_prim_root hs\n[GOAL]\ncase refine'_1.inr\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhS : Set.Nonempty S\nH : IsCyclotomicExtension S A B\ns : \u2115+\nh : \u2200 (s_1 : \u2115+), s_1 \u2208 S \u2192 s \u2223 s_1\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191s\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := hS\n[GOAL]\ncase refine'_1.inr.intro\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nH : IsCyclotomicExtension S A B\ns : \u2115+\nh : \u2200 (s_1 : \u2115+), s_1 \u2208 S \u2192 s \u2223 s_1\nm : \u2115+\nhm : m \u2208 S\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191s\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 := h m hm\n[GOAL]\ncase refine'_1.inr.intro.intro\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nH : IsCyclotomicExtension S A B\ns : \u2115+\nh : \u2200 (s_1 : \u2115+), s_1 \u2208 S \u2192 s \u2223 s_1\nx : \u2115+\nhm : s * x \u2208 S\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191s\n[PROOFSTEP]\nobtain \u27e8\u03b6, h\u03b6\u27e9 := H.exists_prim_root hm\n[GOAL]\ncase refine'_1.inr.intro.intro.intro\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nH : IsCyclotomicExtension S A B\ns : \u2115+\nh : \u2200 (s_1 : \u2115+), s_1 \u2208 S \u2192 s \u2223 s_1\nx : \u2115+\nhm : s * x \u2208 S\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(s * x)\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191s\n[PROOFSTEP]\nrefine' \u27e8\u03b6 ^ (x : \u2115), _\u27e9\n[GOAL]\ncase refine'_1.inr.intro.intro.intro\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nH : IsCyclotomicExtension S A B\ns : \u2115+\nh : \u2200 (s_1 : \u2115+), s_1 \u2208 S \u2192 s \u2223 s_1\nx : \u2115+\nhm : s * x \u2208 S\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(s * x)\n\u22a2 IsPrimitiveRoot (\u03b6 ^ \u2191x) \u2191s\n[PROOFSTEP]\nconvert h\u03b6.pow_of_dvd x.ne_zero (dvd_mul_left (x : \u2115) s)\n[GOAL]\ncase h.e'_4\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nH : IsCyclotomicExtension S A B\ns : \u2115+\nh : \u2200 (s_1 : \u2115+), s_1 \u2208 S \u2192 s \u2223 s_1\nx : \u2115+\nhm : s * x \u2208 S\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(s * x)\n\u22a2 \u2191s = \u2191(s * x) / \u2191x\n[PROOFSTEP]\nsimp only [PNat.mul_coe, Nat.mul_div_left, PNat.pos]\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension S A B\n\u22a2 adjoin A {b | \u2203 n_1, n_1 \u2208 S \u222a {n} \u2227 b ^ \u2191n_1 = 1} = \u22a4\n[PROOFSTEP]\nrefine' _root_.eq_top_iff.2 _\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension S A B\n\u22a2 \u22a4 \u2264 adjoin A {b | \u2203 n_1, n_1 \u2208 S \u222a {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nrw [\u2190 ((iff_adjoin_eq_top S A B).1 H).2]\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension S A B\n\u22a2 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} \u2264 adjoin A {b | \u2203 n_1, n_1 \u2208 S \u222a {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nrefine' adjoin_mono fun x hx => _\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension S A B\nx : B\nhx : x \u2208 {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1}\n\u22a2 x \u2208 {b | \u2203 n_1, n_1 \u2208 S \u222a {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nsimp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx \u22a2\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension S A B\nx : B\nhx : \u2203 n, n \u2208 S \u2227 x ^ \u2191n = 1\n\u22a2 \u2203 n_1, (n_1 = n \u2228 n_1 \u2208 S) \u2227 x ^ \u2191n_1 = 1\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := hx\n[GOAL]\ncase refine'_2.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension S A B\nx : B\nm : \u2115+\nhm : m \u2208 S \u2227 x ^ \u2191m = 1\n\u22a2 \u2203 n_1, (n_1 = n \u2228 n_1 \u2208 S) \u2227 x ^ \u2191n_1 = 1\n[PROOFSTEP]\nexact \u27e8m, \u27e8Or.inr hm.1, hm.2\u27e9\u27e9\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\n\u22a2 IsCyclotomicExtension S A B \u2194 IsCyclotomicExtension (S \u222a {n}) A B\n[PROOFSTEP]\nrefine' \u27e8fun H => of_union_of_dvd A B h hS, fun H => (iff_adjoin_eq_top _ A _).2 \u27e8fun s hs => _, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension (S \u222a {n}) A B\ns : \u2115+\nhs : s \u2208 S\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191s\n[PROOFSTEP]\nexact H.exists_prim_root (subset_union_left _ _ hs)\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension (S \u222a {n}) A B\n\u22a2 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} = \u22a4\n[PROOFSTEP]\nrw [_root_.eq_top_iff, \u2190 ((iff_adjoin_eq_top _ A B).1 H).2]\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension (S \u222a {n}) A B\n\u22a2 adjoin A {b | \u2203 n_1, n_1 \u2208 S \u222a {n} \u2227 b ^ \u2191n_1 = 1} \u2264 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nrefine' adjoin_mono fun x hx => _\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension (S \u222a {n}) A B\nx : B\nhx : x \u2208 {b | \u2203 n_1, n_1 \u2208 S \u222a {n} \u2227 b ^ \u2191n_1 = 1}\n\u22a2 x \u2208 {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1}\n[PROOFSTEP]\nsimp only [union_singleton, mem_insert_iff, mem_setOf_eq] at hx \u22a2\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension (S \u222a {n}) A B\nx : B\nhx : \u2203 n_1, (n_1 = n \u2228 n_1 \u2208 S) \u2227 x ^ \u2191n_1 = 1\n\u22a2 \u2203 n, n \u2208 S \u2227 x ^ \u2191n = 1\n[PROOFSTEP]\nobtain \u27e8m, rfl | hm, hxpow\u27e9 := hx\n[GOAL]\ncase refine'_2.intro.intro.inl\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhS : Set.Nonempty S\nx : B\nm : \u2115+\nhxpow : x ^ \u2191m = 1\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 m \u2223 s\nH : IsCyclotomicExtension (S \u222a {m}) A B\n\u22a2 \u2203 n, n \u2208 S \u2227 x ^ \u2191n = 1\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := hS\n[GOAL]\ncase refine'_2.intro.intro.inl.intro\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nx : B\nm : \u2115+\nhxpow : x ^ \u2191m = 1\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 m \u2223 s\nH : IsCyclotomicExtension (S \u222a {m}) A B\ny : \u2115+\nhy : y \u2208 S\n\u22a2 \u2203 n, n \u2208 S \u2227 x ^ \u2191n = 1\n[PROOFSTEP]\nrefine' \u27e8y, \u27e8hy, _\u27e9\u27e9\n[GOAL]\ncase refine'_2.intro.intro.inl.intro\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nx : B\nm : \u2115+\nhxpow : x ^ \u2191m = 1\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 m \u2223 s\nH : IsCyclotomicExtension (S \u222a {m}) A B\ny : \u2115+\nhy : y \u2208 S\n\u22a2 x ^ \u2191y = 1\n[PROOFSTEP]\nobtain \u27e8z, rfl\u27e9 := h y hy\n[GOAL]\ncase refine'_2.intro.intro.inl.intro.intro\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nx : B\nm : \u2115+\nhxpow : x ^ \u2191m = 1\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 m \u2223 s\nH : IsCyclotomicExtension (S \u222a {m}) A B\nz : \u2115+\nhy : m * z \u2208 S\n\u22a2 x ^ \u2191(m * z) = 1\n[PROOFSTEP]\nsimp only [PNat.mul_coe, pow_mul, hxpow, one_pow]\n[GOAL]\ncase refine'_2.intro.intro.inr\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u2200 (s : \u2115+), s \u2208 S \u2192 n \u2223 s\nhS : Set.Nonempty S\nH : IsCyclotomicExtension (S \u222a {n}) A B\nx : B\nm : \u2115+\nhxpow : x ^ \u2191m = 1\nhm : m \u2208 S\n\u22a2 \u2203 n, n \u2208 S \u2227 x ^ \u2191n = 1\n[PROOFSTEP]\nexact \u27e8m, \u27e8hm, hxpow\u27e9\u27e9\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u22a2 IsCyclotomicExtension S A B \u2194 IsCyclotomicExtension (S \u222a {1}) A B\n[PROOFSTEP]\nobtain hS | rfl := S.eq_empty_or_nonempty.symm\n[GOAL]\ncase inl\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhS : Set.Nonempty S\n\u22a2 IsCyclotomicExtension S A B \u2194 IsCyclotomicExtension (S \u222a {1}) A B\n[PROOFSTEP]\nexact iff_union_of_dvd _ _ (fun s _ => one_dvd _) hS\n[GOAL]\ncase inr\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u22a2 IsCyclotomicExtension \u2205 A B \u2194 IsCyclotomicExtension (\u2205 \u222a {1}) A B\n[PROOFSTEP]\nrw [empty_union]\n[GOAL]\ncase inr\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u22a2 IsCyclotomicExtension \u2205 A B \u2194 IsCyclotomicExtension {1} A B\n[PROOFSTEP]\nrefine' \u27e8fun H => _, fun H => _\u27e9\n[GOAL]\ncase inr.refine'_1\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nH : IsCyclotomicExtension \u2205 A B\n\u22a2 IsCyclotomicExtension {1} A B\n[PROOFSTEP]\nrefine' (iff_adjoin_eq_top _ A _).2 \u27e8fun s hs => \u27e81, by simp [mem_singleton_iff.1 hs]\u27e9, _\u27e9\n[GOAL]\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nH : IsCyclotomicExtension \u2205 A B\ns : \u2115+\nhs : s \u2208 {1}\n\u22a2 IsPrimitiveRoot 1 \u2191s\n[PROOFSTEP]\nsimp [mem_singleton_iff.1 hs]\n[GOAL]\ncase inr.refine'_1\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nH : IsCyclotomicExtension \u2205 A B\n\u22a2 adjoin A {b | \u2203 n, n \u2208 {1} \u2227 b ^ \u2191n = 1} = \u22a4\n[PROOFSTEP]\nsimp [adjoin_singleton_one, empty]\n[GOAL]\ncase inr.refine'_2\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nH : IsCyclotomicExtension {1} A B\n\u22a2 IsCyclotomicExtension \u2205 A B\n[PROOFSTEP]\nrefine' (iff_adjoin_eq_top _ A _).2 \u27e8fun s hs => (not_mem_empty s hs).elim, _\u27e9\n[GOAL]\ncase inr.refine'_2\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nH : IsCyclotomicExtension {1} A B\n\u22a2 adjoin A {b | \u2203 n, n \u2208 \u2205 \u2227 b ^ \u2191n = 1} = \u22a4\n[PROOFSTEP]\nsimp [@singleton_one A B _ _ _ H]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u22a5 = \u22a4\n\u22a2 IsCyclotomicExtension {1} A B\n[PROOFSTEP]\nconvert (iff_union_singleton_one _ A _).1 (singleton_zero_of_bot_eq_top h)\n[GOAL]\ncase h.e'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh : \u22a5 = \u22a4\n\u22a2 {1} = \u2205 \u222a {1}\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\nC : Type u_1\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nh : IsCyclotomicExtension S A B\nf : B \u2243\u2090[A] C\n\u22a2 IsCyclotomicExtension S A C\n[PROOFSTEP]\nletI : Algebra B C := f.toAlgHom.toRingHom.toAlgebra\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\nC : Type u_1\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nh : IsCyclotomicExtension S A B\nf : B \u2243\u2090[A] C\nthis : Algebra B C := RingHom.toAlgebra \u2191\u2191f\n\u22a2 IsCyclotomicExtension S A C\n[PROOFSTEP]\nhaveI : IsCyclotomicExtension {1} B C := singleton_one_of_algebraMap_bijective f.surjective\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\nC : Type u_1\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nh : IsCyclotomicExtension S A B\nf : B \u2243\u2090[A] C\nthis\u271d : Algebra B C := RingHom.toAlgebra \u2191\u2191f\nthis : IsCyclotomicExtension {1} B C\n\u22a2 IsCyclotomicExtension S A C\n[PROOFSTEP]\nhaveI : IsScalarTower A B C := IsScalarTower.of_ring_hom f.toAlgHom\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\nC : Type u_1\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nh : IsCyclotomicExtension S A B\nf : B \u2243\u2090[A] C\nthis\u271d\u00b9 : Algebra B C := RingHom.toAlgebra \u2191\u2191f\nthis\u271d : IsCyclotomicExtension {1} B C\nthis : IsScalarTower A B C\n\u22a2 IsCyclotomicExtension S A C\n[PROOFSTEP]\nexact (iff_union_singleton_one _ _ _).2 (trans S {1} A B C f.injective)\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nh : IsCyclotomicExtension {n} A B\ninst\u271d : IsDomain B\n\u22a2 NeZero \u2191\u2191n\n[PROOFSTEP]\nobtain \u27e8\u27e8r, hr\u27e9, -\u27e9 := (iff_singleton n A B).1 h\n[GOAL]\ncase intro.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nh : IsCyclotomicExtension {n} A B\ninst\u271d : IsDomain B\nr : B\nhr : IsPrimitiveRoot r \u2191n\n\u22a2 NeZero \u2191\u2191n\n[PROOFSTEP]\nexact hr.neZero'\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsCyclotomicExtension {n} A B\ninst\u271d : IsDomain B\n\u22a2 NeZero \u2191\u2191n\n[PROOFSTEP]\nhaveI := IsCyclotomicExtension.neZero n A B\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsCyclotomicExtension {n} A B\ninst\u271d : IsDomain B\nthis : NeZero \u2191\u2191n\n\u22a2 NeZero \u2191\u2191n\n[PROOFSTEP]\nexact NeZero.nat_of_neZero (algebraMap A B)\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension {n} A B\n\u22a2 Module.Finite A B\n[PROOFSTEP]\nclassical\nrw [Module.finite_def, \u2190 top_toSubmodule, \u2190 ((iff_adjoin_eq_top _ _ _).1 h).2]\nrefine' FG_adjoin_of_finite _ fun b hb => _\n\u00b7 simp only [mem_singleton_iff, exists_eq_left]\n have : {b : B | b ^ (n : \u2115) = 1} = (nthRoots n (1 : B)).toFinset :=\n Set.ext fun x => \u27e8fun h => by simpa using h, fun h => by simpa using h\u27e9\n rw [this]\n exact (nthRoots (\u2191n) 1).toFinset.finite_toSet\n\u00b7 simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb \n refine' \u27e8X ^ (n : \u2115) - 1, \u27e8monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]\u27e9\u27e9\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension {n} A B\n\u22a2 Module.Finite A B\n[PROOFSTEP]\nrw [Module.finite_def, \u2190 top_toSubmodule, \u2190 ((iff_adjoin_eq_top _ _ _).1 h).2]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension {n} A B\n\u22a2 Submodule.FG (\u2191Subalgebra.toSubmodule (adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}))\n[PROOFSTEP]\nrefine' FG_adjoin_of_finite _ fun b hb => _\n[GOAL]\ncase refine'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension {n} A B\n\u22a2 Set.Finite {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nsimp only [mem_singleton_iff, exists_eq_left]\n[GOAL]\ncase refine'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension {n} A B\n\u22a2 Set.Finite {b | b ^ \u2191n = 1}\n[PROOFSTEP]\nhave : {b : B | b ^ (n : \u2115) = 1} = (nthRoots n (1 : B)).toFinset :=\n Set.ext fun x => \u27e8fun h => by simpa using h, fun h => by simpa using h\u27e9\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\nh\u271d : IsCyclotomicExtension {n} A B\nx : B\nh : x \u2208 {b | b ^ \u2191n = 1}\n\u22a2 x \u2208 \u2191(Multiset.toFinset (nthRoots (\u2191n) 1))\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\nh\u271d : IsCyclotomicExtension {n} A B\nx : B\nh : x \u2208 \u2191(Multiset.toFinset (nthRoots (\u2191n) 1))\n\u22a2 x \u2208 {b | b ^ \u2191n = 1}\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase refine'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension {n} A B\nthis : {b | b ^ \u2191n = 1} = \u2191(Multiset.toFinset (nthRoots (\u2191n) 1))\n\u22a2 Set.Finite {b | b ^ \u2191n = 1}\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase refine'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension {n} A B\nthis : {b | b ^ \u2191n = 1} = \u2191(Multiset.toFinset (nthRoots (\u2191n) 1))\n\u22a2 Set.Finite \u2191(Multiset.toFinset (nthRoots (\u2191n) 1))\n[PROOFSTEP]\nexact (nthRoots (\u2191n) 1).toFinset.finite_toSet\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension {n} A B\nb : B\nhb : b \u2208 {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n\u22a2 IsIntegral A b\n[PROOFSTEP]\nsimp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hb \n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension {n} A B\nb : B\nhb : b ^ \u2191n = 1\n\u22a2 IsIntegral A b\n[PROOFSTEP]\nrefine' \u27e8X ^ (n : \u2115) - 1, \u27e8monic_X_pow_sub_C _ n.pos.ne.symm, by simp [hb]\u27e9\u27e9\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension {n} A B\nb : B\nhb : b ^ \u2191n = 1\n\u22a2 eval\u2082 (algebraMap A B) b (X ^ \u2191n - 1) = 0\n[PROOFSTEP]\nsimp [hb]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\nh\u2081 : Finite \u2191S\nh\u2082 : IsCyclotomicExtension S A B\n\u22a2 Module.Finite A B\n[PROOFSTEP]\ncases' nonempty_fintype S with h\n[GOAL]\ncase intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\nh\u2081 : Finite \u2191S\nh\u2082 : IsCyclotomicExtension S A B\nh : Fintype \u2191S\n\u22a2 Module.Finite A B\n[PROOFSTEP]\nrevert h\u2082 A B\n[GOAL]\ncase intro\nn : \u2115+\nS T : Set \u2115+\nK : Type w\nL : Type z\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh\u2081 : Finite \u2191S\nh : Fintype \u2191S\n\u22a2 \u2200 (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]\n [h\u2082 : IsCyclotomicExtension S A B], Module.Finite A B\n[PROOFSTEP]\nrefine' Set.Finite.induction_on (Set.Finite.intro h) (fun A B => _) @fun n S _ _ H A B => _\n[GOAL]\ncase intro.refine'_1\nn : \u2115+\nS T : Set \u2115+\nK : Type w\nL : Type z\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh\u2081 : Finite \u2191S\nh : Fintype \u2191S\nA : Type u\nB : Type v\n\u22a2 \u2200 [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]\n [h\u2082 : IsCyclotomicExtension \u2205 A B], Module.Finite A B\n[PROOFSTEP]\nintro _ _ _ _ _\n[GOAL]\ncase intro.refine'_1\nn : \u2115+\nS T : Set \u2115+\nK : Type w\nL : Type z\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nh\u2081 : Finite \u2191S\nh : Fintype \u2191S\nA : Type u\nB : Type v\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsDomain B\nh\u2082\u271d : IsCyclotomicExtension \u2205 A B\n\u22a2 Module.Finite A B\n[PROOFSTEP]\nrefine' Module.finite_def.2 \u27e8({1} : Finset B), _\u27e9\n[GOAL]\ncase intro.refine'_1\nn : \u2115+\nS T : Set \u2115+\nK : Type w\nL : Type z\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nh\u2081 : Finite \u2191S\nh : Fintype \u2191S\nA : Type u\nB : Type v\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsDomain B\nh\u2082\u271d : IsCyclotomicExtension \u2205 A B\n\u22a2 Submodule.span A \u2191{1} = \u22a4\n[PROOFSTEP]\nsimp [\u2190 top_toSubmodule, \u2190 empty, toSubmodule_bot]\n[GOAL]\ncase intro.refine'_2\nn\u271d : \u2115+\nS\u271d T : Set \u2115+\nK : Type w\nL : Type z\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nh\u2081 : Finite \u2191S\u271d\nh : Fintype \u2191S\u271d\nn : \u2115+\nS : Set \u2115+\nx\u271d\u00b9 : \u00acn \u2208 S\nx\u271d : Set.Finite S\nH :\n \u2200 (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]\n [h\u2082 : IsCyclotomicExtension S A B], Module.Finite A B\nA : Type u\nB : Type v\n\u22a2 \u2200 [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]\n [h\u2082 : IsCyclotomicExtension (insert n S) A B], Module.Finite A B\n[PROOFSTEP]\nintro _ _ _ _ h\n[GOAL]\ncase intro.refine'_2\nn\u271d : \u2115+\nS\u271d T : Set \u2115+\nK : Type w\nL : Type z\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nh\u2081 : Finite \u2191S\u271d\nh\u271d : Fintype \u2191S\u271d\nn : \u2115+\nS : Set \u2115+\nx\u271d\u00b9 : \u00acn \u2208 S\nx\u271d : Set.Finite S\nH :\n \u2200 (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]\n [h\u2082 : IsCyclotomicExtension S A B], Module.Finite A B\nA : Type u\nB : Type v\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension (insert n S) A B\n\u22a2 Module.Finite A B\n[PROOFSTEP]\nhaveI : IsCyclotomicExtension S A (adjoin A {b : B | \u2203 n : \u2115+, n \u2208 S \u2227 b ^ (n : \u2115) = 1}) :=\n union_left _ (insert n S) _ _ (subset_insert n S)\n[GOAL]\ncase intro.refine'_2\nn\u271d : \u2115+\nS\u271d T : Set \u2115+\nK : Type w\nL : Type z\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nh\u2081 : Finite \u2191S\u271d\nh\u271d : Fintype \u2191S\u271d\nn : \u2115+\nS : Set \u2115+\nx\u271d\u00b9 : \u00acn \u2208 S\nx\u271d : Set.Finite S\nH :\n \u2200 (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]\n [h\u2082 : IsCyclotomicExtension S A B], Module.Finite A B\nA : Type u\nB : Type v\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension (insert n S) A B\nthis : IsCyclotomicExtension S A { x // x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} }\n\u22a2 Module.Finite A B\n[PROOFSTEP]\nhaveI := H A (adjoin A {b : B | \u2203 n : \u2115+, n \u2208 S \u2227 b ^ (n : \u2115) = 1})\n[GOAL]\ncase intro.refine'_2\nn\u271d : \u2115+\nS\u271d T : Set \u2115+\nK : Type w\nL : Type z\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nh\u2081 : Finite \u2191S\u271d\nh\u271d : Fintype \u2191S\u271d\nn : \u2115+\nS : Set \u2115+\nx\u271d\u00b9 : \u00acn \u2208 S\nx\u271d : Set.Finite S\nH :\n \u2200 (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]\n [h\u2082 : IsCyclotomicExtension S A B], Module.Finite A B\nA : Type u\nB : Type v\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension (insert n S) A B\nthis\u271d : IsCyclotomicExtension S A { x // x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} }\nthis : Module.Finite A { x // x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} }\n\u22a2 Module.Finite A B\n[PROOFSTEP]\nhave : Module.Finite (adjoin A {b : B | \u2203 n : \u2115+, n \u2208 S \u2227 b ^ (n : \u2115) = 1}) B :=\n by\n rw [\u2190 union_singleton] at h \n letI := @union_right S { n } A B _ _ _ h\n exact finite_of_singleton n _ _\n[GOAL]\nn\u271d : \u2115+\nS\u271d T : Set \u2115+\nK : Type w\nL : Type z\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nh\u2081 : Finite \u2191S\u271d\nh\u271d : Fintype \u2191S\u271d\nn : \u2115+\nS : Set \u2115+\nx\u271d\u00b9 : \u00acn \u2208 S\nx\u271d : Set.Finite S\nH :\n \u2200 (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]\n [h\u2082 : IsCyclotomicExtension S A B], Module.Finite A B\nA : Type u\nB : Type v\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension (insert n S) A B\nthis\u271d : IsCyclotomicExtension S A { x // x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} }\nthis : Module.Finite A { x // x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} }\n\u22a2 Module.Finite { x // x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} } B\n[PROOFSTEP]\nrw [\u2190 union_singleton] at h \n[GOAL]\nn\u271d : \u2115+\nS\u271d T : Set \u2115+\nK : Type w\nL : Type z\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nh\u2081 : Finite \u2191S\u271d\nh\u271d : Fintype \u2191S\u271d\nn : \u2115+\nS : Set \u2115+\nx\u271d\u00b9 : \u00acn \u2208 S\nx\u271d : Set.Finite S\nH :\n \u2200 (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]\n [h\u2082 : IsCyclotomicExtension S A B], Module.Finite A B\nA : Type u\nB : Type v\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension (S \u222a {n}) A B\nthis\u271d : IsCyclotomicExtension S A { x // x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} }\nthis : Module.Finite A { x // x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} }\n\u22a2 Module.Finite { x // x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} } B\n[PROOFSTEP]\nletI := @union_right S { n } A B _ _ _ h\n[GOAL]\nn\u271d : \u2115+\nS\u271d T : Set \u2115+\nK : Type w\nL : Type z\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nh\u2081 : Finite \u2191S\u271d\nh\u271d : Fintype \u2191S\u271d\nn : \u2115+\nS : Set \u2115+\nx\u271d\u00b9 : \u00acn \u2208 S\nx\u271d : Set.Finite S\nH :\n \u2200 (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]\n [h\u2082 : IsCyclotomicExtension S A B], Module.Finite A B\nA : Type u\nB : Type v\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension (S \u222a {n}) A B\nthis\u271d\u00b9 : IsCyclotomicExtension S A { x // x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} }\nthis\u271d : Module.Finite A { x // x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} }\nthis : IsCyclotomicExtension {n} { x // x \u2208 adjoin A {b | \u2203 a, a \u2208 S \u2227 b ^ \u2191a = 1} } B := union_right S {n} A B\n\u22a2 Module.Finite { x // x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} } B\n[PROOFSTEP]\nexact finite_of_singleton n _ _\n[GOAL]\ncase intro.refine'_2\nn\u271d : \u2115+\nS\u271d T : Set \u2115+\nK : Type w\nL : Type z\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\nh\u2081 : Finite \u2191S\u271d\nh\u271d : Fintype \u2191S\u271d\nn : \u2115+\nS : Set \u2115+\nx\u271d\u00b9 : \u00acn \u2208 S\nx\u271d : Set.Finite S\nH :\n \u2200 (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]\n [h\u2082 : IsCyclotomicExtension S A B], Module.Finite A B\nA : Type u\nB : Type v\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension (insert n S) A B\nthis\u271d\u00b9 : IsCyclotomicExtension S A { x // x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} }\nthis\u271d : Module.Finite A { x // x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} }\nthis : Module.Finite { x // x \u2208 adjoin A {b | \u2203 n, n \u2208 S \u2227 b ^ \u2191n = 1} } B\n\u22a2 Module.Finite A B\n[PROOFSTEP]\nexact Module.Finite.trans (adjoin A {b : B | \u2203 n : \u2115+, n \u2208 S \u2227 b ^ (n : \u2115) = 1}) _\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\nh : NumberField K\ninst\u271d\u00b9 : Finite \u2191S\ninst\u271d : IsCyclotomicExtension S K L\n\u22a2 FiniteDimensional \u211a L\n[PROOFSTEP]\nhaveI := charZero_of_injective_algebraMap (algebraMap K L).injective\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\nh : NumberField K\ninst\u271d\u00b9 : Finite \u2191S\ninst\u271d : IsCyclotomicExtension S K L\nthis : CharZero L\n\u22a2 FiniteDimensional \u211a L\n[PROOFSTEP]\nhaveI := IsCyclotomicExtension.finite S K L\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\nh : NumberField K\ninst\u271d\u00b9 : Finite \u2191S\ninst\u271d : IsCyclotomicExtension S K L\nthis\u271d : CharZero L\nthis : Module.Finite K L\n\u22a2 FiniteDimensional \u211a L\n[PROOFSTEP]\nexact Module.Finite.trans K _\n[GOAL]\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\n\u03b6 : B\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 adjoin A (rootSet (cyclotomic (\u2191n) A) B) = adjoin A {b | \u2203 a, a \u2208 {n} \u2227 b ^ \u2191a = 1}\n[PROOFSTEP]\nsimp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]\n[GOAL]\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\n\u03b6 : B\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 adjoin A (rootSet (cyclotomic (\u2191n) A) B) = adjoin A {b | b ^ \u2191n = 1}\n[PROOFSTEP]\nrefine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _)\n[GOAL]\ncase refine'_1\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\n\u03b6 : B\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : x \u2208 rootSet (cyclotomic (\u2191n) A) B\n\u22a2 x \u2208 {b | b ^ \u2191n = 1}\n[PROOFSTEP]\nrw [mem_rootSet'] at hx \n[GOAL]\ncase refine'_1\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\n\u03b6 : B\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : map (algebraMap A B) (cyclotomic (\u2191n) A) \u2260 0 \u2227 \u2191(aeval x) (cyclotomic (\u2191n) A) = 0\n\u22a2 x \u2208 {b | b ^ \u2191n = 1}\n[PROOFSTEP]\nsimp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]\n[GOAL]\ncase refine'_1\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\n\u03b6 : B\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : map (algebraMap A B) (cyclotomic (\u2191n) A) \u2260 0 \u2227 \u2191(aeval x) (cyclotomic (\u2191n) A) = 0\n\u22a2 x ^ \u2191n = 1\n[PROOFSTEP]\nrw [isRoot_of_unity_iff n.pos]\n[GOAL]\ncase refine'_1\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\n\u03b6 : B\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : map (algebraMap A B) (cyclotomic (\u2191n) A) \u2260 0 \u2227 \u2191(aeval x) (cyclotomic (\u2191n) A) = 0\n\u22a2 \u2203 i, i \u2208 Nat.divisors \u2191n \u2227 IsRoot (cyclotomic i B) x\n[PROOFSTEP]\nrefine' \u27e8n, Nat.mem_divisors_self n n.ne_zero, _\u27e9\n[GOAL]\ncase refine'_1\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\n\u03b6 : B\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : map (algebraMap A B) (cyclotomic (\u2191n) A) \u2260 0 \u2227 \u2191(aeval x) (cyclotomic (\u2191n) A) = 0\n\u22a2 IsRoot (cyclotomic (\u2191n) B) x\n[PROOFSTEP]\nrw [IsRoot.def, \u2190 map_cyclotomic n (algebraMap A B), eval_map, \u2190 aeval_def]\n[GOAL]\ncase refine'_1\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\n\u03b6 : B\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : map (algebraMap A B) (cyclotomic (\u2191n) A) \u2260 0 \u2227 \u2191(aeval x) (cyclotomic (\u2191n) A) = 0\n\u22a2 \u2191(aeval x) (cyclotomic (\u2191n) A) = 0\n[PROOFSTEP]\nexact hx.2\n[GOAL]\ncase refine'_2\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\n\u03b6 : B\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : x \u2208 {b | b ^ \u2191n = 1}\n\u22a2 x \u2208 \u2191(adjoin A (rootSet (cyclotomic (\u2191n) A) B))\n[PROOFSTEP]\nsimp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx \n[GOAL]\ncase refine'_2\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\n\u03b6 : B\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : x ^ \u2191n = 1\n\u22a2 x \u2208 \u2191(adjoin A (rootSet (cyclotomic (\u2191n) A) B))\n[PROOFSTEP]\nobtain \u27e8i, _, rfl\u27e9 := h\u03b6.eq_pow_of_pow_eq_one hx n.pos\n[GOAL]\ncase refine'_2.intro.intro\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\n\u03b6 : B\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ni : \u2115\nleft\u271d : i < \u2191n\nhx : (\u03b6 ^ i) ^ \u2191n = 1\n\u22a2 \u03b6 ^ i \u2208 \u2191(adjoin A (rootSet (cyclotomic (\u2191n) A) B))\n[PROOFSTEP]\nrefine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _)\n[GOAL]\ncase refine'_2.intro.intro\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\n\u03b6 : B\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ni : \u2115\nleft\u271d : i < \u2191n\nhx : (\u03b6 ^ i) ^ \u2191n = 1\n\u22a2 \u03b6 \u2208 rootSet (cyclotomic (\u2191n) A) B\n[PROOFSTEP]\nrw [mem_rootSet', map_cyclotomic, aeval_def, \u2190 eval_map, map_cyclotomic, \u2190 IsRoot]\n[GOAL]\ncase refine'_2.intro.intro\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsDomain B\n\u03b6 : B\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ni : \u2115\nleft\u271d : i < \u2191n\nhx : (\u03b6 ^ i) ^ \u2191n = 1\n\u22a2 cyclotomic (\u2191n) B \u2260 0 \u2227 IsRoot (cyclotomic (\u2191n) B) \u03b6\n[PROOFSTEP]\nrefine' \u27e8cyclotomic_ne_zero n B, h\u03b6.isRoot_cyclotomic n.pos\u27e9\n[GOAL]\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 adjoin A (rootSet (cyclotomic (\u2191n) A) B) = adjoin A {\u03b6}\n[PROOFSTEP]\nrefine' le_antisymm (adjoin_le fun x hx => _) (adjoin_mono fun x hx => _)\n[GOAL]\ncase refine'_1\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : x \u2208 rootSet (cyclotomic (\u2191n) A) B\n\u22a2 x \u2208 \u2191(adjoin A {\u03b6})\n[PROOFSTEP]\nsuffices hx : x ^ n.1 = 1\n[GOAL]\ncase refine'_1\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx\u271d : x \u2208 rootSet (cyclotomic (\u2191n) A) B\nhx : x ^ \u2191n = 1\n\u22a2 x \u2208 \u2191(adjoin A {\u03b6})\ncase hx\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : x \u2208 rootSet (cyclotomic (\u2191n) A) B\n\u22a2 x ^ \u2191n = 1\n[PROOFSTEP]\nobtain \u27e8i, _, rfl\u27e9 := h\u03b6.eq_pow_of_pow_eq_one hx n.pos\n[GOAL]\ncase refine'_1.intro.intro\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ni : \u2115\nleft\u271d : i < \u2191n\nhx\u271d : \u03b6 ^ i \u2208 rootSet (cyclotomic (\u2191n) A) B\nhx : (\u03b6 ^ i) ^ \u2191n = 1\n\u22a2 \u03b6 ^ i \u2208 \u2191(adjoin A {\u03b6})\ncase hx\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : x \u2208 rootSet (cyclotomic (\u2191n) A) B\n\u22a2 x ^ \u2191n = 1\n[PROOFSTEP]\nexact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton \u03b6) _)\n[GOAL]\ncase hx\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : x \u2208 rootSet (cyclotomic (\u2191n) A) B\n\u22a2 x ^ \u2191n = 1\n[PROOFSTEP]\nrefine' (isRoot_of_unity_iff n.pos B).2 _\n[GOAL]\ncase hx\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : x \u2208 rootSet (cyclotomic (\u2191n) A) B\n\u22a2 \u2203 i, i \u2208 Nat.divisors \u2191n \u2227 IsRoot (cyclotomic i B) x\n[PROOFSTEP]\nrefine' \u27e8n, Nat.mem_divisors_self n n.ne_zero, _\u27e9\n[GOAL]\ncase hx\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : x \u2208 rootSet (cyclotomic (\u2191n) A) B\n\u22a2 IsRoot (cyclotomic (\u2191n) B) x\n[PROOFSTEP]\nrw [mem_rootSet', aeval_def, \u2190 eval_map, map_cyclotomic, \u2190 IsRoot] at hx \n[GOAL]\ncase hx\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : cyclotomic (\u2191n) B \u2260 0 \u2227 IsRoot (cyclotomic (\u2191n) B) x\n\u22a2 IsRoot (cyclotomic (\u2191n) B) x\n[PROOFSTEP]\nexact hx.2\n[GOAL]\ncase refine'_2\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : x \u2208 {\u03b6}\n\u22a2 x \u2208 rootSet (cyclotomic (\u2191n) A) B\n[PROOFSTEP]\nsimp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx \n[GOAL]\ncase refine'_2\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nx : B\nhx : x = \u03b6\n\u22a2 x \u2208 rootSet (cyclotomic (\u2191n) A) B\n[PROOFSTEP]\nsimpa only [hx, mem_rootSet', map_cyclotomic, aeval_def, \u2190 eval_map, IsRoot] using\n And.intro (cyclotomic_ne_zero n B) (h\u03b6.isRoot_cyclotomic n.pos)\n[GOAL]\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension {n} A B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 adjoin A {\u03b6} = \u22a4\n[PROOFSTEP]\nclassical\nrw [\u2190 adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic h\u03b6]\nrw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots h\u03b6]\nexact ((iff_adjoin_eq_top { n } A B).mp h).2\n[GOAL]\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension {n} A B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 adjoin A {\u03b6} = \u22a4\n[PROOFSTEP]\nrw [\u2190 adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic h\u03b6]\n[GOAL]\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension {n} A B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 adjoin A (rootSet (cyclotomic (\u2191n) A) B) = \u22a4\n[PROOFSTEP]\nrw [adjoin_roots_cyclotomic_eq_adjoin_nth_roots h\u03b6]\n[GOAL]\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nn : \u2115+\ninst\u271d : IsDomain B\nh : IsCyclotomicExtension {n} A B\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 adjoin A {b | \u2203 a, a \u2208 {n} \u2227 b ^ \u2191a = 1} = \u22a4\n[PROOFSTEP]\nexact ((iff_adjoin_eq_top { n } A B).mp h).2\n[GOAL]\nn\u271d\u00b9 : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u03b6 : B\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nn\u271d : \u2115+\nhi : n\u271d \u2208 {n}\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191n\u271d\n[PROOFSTEP]\nrw [Set.mem_singleton_iff] at hi \n[GOAL]\nn\u271d\u00b9 : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u03b6 : B\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nn\u271d : \u2115+\nhi : n\u271d = n\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191n\u271d\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u03b6, subset_adjoin <| Set.mem_singleton \u03b6\u27e9, _\u27e9\n[GOAL]\nn\u271d\u00b9 : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u03b6 : B\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nn\u271d : \u2115+\nhi : n\u271d = n\n\u22a2 IsPrimitiveRoot { val := \u03b6, property := (_ : \u03b6 \u2208 \u2191(adjoin A {\u03b6})) } \u2191n\u271d\n[PROOFSTEP]\nrwa [\u2190 IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk, hi]\n[GOAL]\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u03b6 : B\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx : { x // x \u2208 adjoin A {\u03b6} }\n\u22a2 x \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nrefine adjoin_induction' (x := x) (fun b hb => ?_) (fun a => ?_) (fun b\u2081 b\u2082 hb\u2081 hb\u2082 => ?_) (fun b\u2081 b\u2082 hb\u2081 hb\u2082 => ?_)\n[GOAL]\ncase refine_1\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u03b6 : B\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx : { x // x \u2208 adjoin A {\u03b6} }\nb : B\nhb : b \u2208 {\u03b6}\n\u22a2 { val := b, property := (_ : b \u2208 \u2191(adjoin A {\u03b6})) } \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nrw [Set.mem_singleton_iff] at hb \n[GOAL]\ncase refine_1\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u03b6 : B\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx : { x // x \u2208 adjoin A {\u03b6} }\nb : B\nhb\u271d : b \u2208 {\u03b6}\nhb : b = \u03b6\n\u22a2 { val := b, property := (_ : b \u2208 \u2191(adjoin A {\u03b6})) } \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nrefine' subset_adjoin _\n[GOAL]\ncase refine_1\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u03b6 : B\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx : { x // x \u2208 adjoin A {\u03b6} }\nb : B\nhb\u271d : b \u2208 {\u03b6}\nhb : b = \u03b6\n\u22a2 { val := b, property := (_ : b \u2208 \u2191(adjoin A {\u03b6})) } \u2208 {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nsimp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq, hb]\n[GOAL]\ncase refine_1\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u03b6 : B\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx : { x // x \u2208 adjoin A {\u03b6} }\nb : B\nhb\u271d : b \u2208 {\u03b6}\nhb : b = \u03b6\n\u22a2 { val := \u03b6, property := (_ : (fun x => x \u2208 adjoin A {\u03b6}) \u03b6) } ^ \u2191n = 1\n[PROOFSTEP]\nrw [\u2190 Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]\n[GOAL]\ncase refine_1\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u03b6 : B\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx : { x // x \u2208 adjoin A {\u03b6} }\nb : B\nhb\u271d : b \u2208 {\u03b6}\nhb : b = \u03b6\n\u22a2 \u03b6 ^ \u2191n = 1\n[PROOFSTEP]\nexact ((IsPrimitiveRoot.iff_def \u03b6 n).1 h).1\n[GOAL]\ncase refine_2\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u03b6 : B\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx : { x // x \u2208 adjoin A {\u03b6} }\na : A\n\u22a2 \u2191(algebraMap A { x // x \u2208 adjoin A {\u03b6} }) a \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nexact Subalgebra.algebraMap_mem _ _\n[GOAL]\ncase refine_3\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u03b6 : B\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx b\u2081 b\u2082 : { x // x \u2208 adjoin A {\u03b6} }\nhb\u2081 : b\u2081 \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\nhb\u2082 : b\u2082 \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n\u22a2 b\u2081 + b\u2082 \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nexact Subalgebra.add_mem _ hb\u2081 hb\u2082\n[GOAL]\ncase refine_4\nn\u271d : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u03b6 : B\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx b\u2081 b\u2082 : { x // x \u2208 adjoin A {\u03b6} }\nhb\u2081 : b\u2081 \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\nhb\u2082 : b\u2082 \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n\u22a2 b\u2081 * b\u2082 \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nexact Subalgebra.mul_mem _ hb\u2081 hb\u2082\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nH : IsCyclotomicExtension S K L\nhS : n \u2208 S\n\u22a2 Splits (algebraMap K L) (X ^ \u2191n - 1)\n[PROOFSTEP]\nrw [\u2190 splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow, Polynomial.map_X]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nH : IsCyclotomicExtension S K L\nhS : n \u2208 S\n\u22a2 Splits (RingHom.id L) (X ^ \u2191n - 1)\n[PROOFSTEP]\nobtain \u27e8z, hz\u27e9 := ((IsCyclotomicExtension_iff _ _ _).1 H).1 hS\n[GOAL]\ncase intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nH : IsCyclotomicExtension S K L\nhS : n \u2208 S\nz : L\nhz : IsPrimitiveRoot z \u2191n\n\u22a2 Splits (RingHom.id L) (X ^ \u2191n - 1)\n[PROOFSTEP]\nexact X_pow_sub_one_splits hz\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension S K L\nhS : n \u2208 S\n\u22a2 Splits (algebraMap K L) (cyclotomic (\u2191n) K)\n[PROOFSTEP]\nrefine' splits_of_splits_of_dvd _ (X_pow_sub_C_ne_zero n.pos _) (splits_X_pow_sub_one K L hS) _\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension S K L\nhS : n \u2208 S\n\u22a2 cyclotomic (\u2191n) K \u2223 X ^ \u2191n - \u2191C 1\n[PROOFSTEP]\nuse\u220f i : \u2115 in (n : \u2115).properDivisors, Polynomial.cyclotomic i K\n[GOAL]\ncase h\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension S K L\nhS : n \u2208 S\n\u22a2 X ^ \u2191n - \u2191C 1 = cyclotomic (\u2191n) K * \u220f i in Nat.properDivisors \u2191n, cyclotomic i K\n[PROOFSTEP]\nrw [(eq_cyclotomic_iff n.pos _).1 rfl, RingHom.map_one]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\n\u22a2 adjoin K (rootSet (X ^ \u2191n - 1) L) = \u22a4\n[PROOFSTEP]\nrw [\u2190 ((iff_adjoin_eq_top { n } K L).1 inferInstance).2]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\n\u22a2 adjoin K (rootSet (X ^ \u2191n - 1) L) = adjoin K {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\n\u22a2 rootSet (X ^ \u2191n - 1) L = {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nrefine' Set.ext fun x => _\n[GOAL]\ncase e_s\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nx : L\n\u22a2 x \u2208 rootSet (X ^ \u2191n - 1) L \u2194 x \u2208 {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nsimp only [Polynomial.map_pow, mem_singleton_iff, Multiset.mem_toFinset, exists_eq_left, mem_setOf_eq, Polynomial.map_X,\n Polynomial.map_one, Finset.mem_coe, Polynomial.map_sub]\n[GOAL]\ncase e_s\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nx : L\n\u22a2 x \u2208 rootSet (X ^ \u2191n - 1) L \u2194 x ^ \u2191n = 1\n[PROOFSTEP]\nsimp only [mem_rootSet', map_sub, map_pow, aeval_one, aeval_X, sub_eq_zero, map_X, and_iff_right_iff_imp,\n Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one]\n[GOAL]\ncase e_s\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nx : L\n\u22a2 x ^ \u2191n = 1 \u2192 X ^ \u2191n - 1 \u2260 0\n[PROOFSTEP]\nexact fun _ => X_pow_sub_C_ne_zero n.pos (1 : L)\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\n\u22a2 adjoin K (rootSet (cyclotomic (\u2191n) K) L) = \u22a4\n[PROOFSTEP]\nrw [\u2190 ((iff_adjoin_eq_top { n } K L).1 inferInstance).2]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\n\u22a2 adjoin K (rootSet (cyclotomic (\u2191n) K) L) = adjoin K {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nletI := Classical.decEq L\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nthis : DecidableEq L := Classical.decEq L\n\u22a2 adjoin K (rootSet (cyclotomic (\u2191n) K) L) = adjoin K {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nobtain \u27e8\u03b6 : L, h\u03b6\u27e9 := IsCyclotomicExtension.exists_prim_root K (B := L) (mem_singleton n)\n[GOAL]\ncase intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nthis : DecidableEq L := Classical.decEq L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 adjoin K (rootSet (cyclotomic (\u2191n) K) L) = adjoin K {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nexact adjoin_roots_cyclotomic_eq_adjoin_nth_roots h\u03b6\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u22a2 Field (CyclotomicField n K)\n[PROOFSTEP]\ndelta CyclotomicField\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u22a2 Field (SplittingField (cyclotomic (\u2191n) K))\n[PROOFSTEP]\ninfer_instance\n --Porting note: could not be derived\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u22a2 Algebra K (CyclotomicField n K)\n[PROOFSTEP]\ndelta CyclotomicField\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u22a2 Algebra K (SplittingField (cyclotomic (\u2191n) K))\n[PROOFSTEP]\ninfer_instance\n --Porting note: could not be derived\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u22a2 Inhabited (CyclotomicField n K)\n[PROOFSTEP]\ndelta CyclotomicField\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\n\u22a2 Inhabited (SplittingField (cyclotomic (\u2191n) K))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : NeZero \u2191\u2191n\n\u22a2 IsCyclotomicExtension {n} K (CyclotomicField n K)\n[PROOFSTEP]\nhaveI : NeZero ((n : \u2115) : CyclotomicField n K) := NeZero.nat_of_injective (algebraMap K _).injective\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : NeZero \u2191\u2191n\nthis : NeZero \u2191\u2191n\n\u22a2 IsCyclotomicExtension {n} K (CyclotomicField n K)\n[PROOFSTEP]\nletI := Classical.decEq (CyclotomicField n K)\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : NeZero \u2191\u2191n\nthis\u271d : NeZero \u2191\u2191n\nthis : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)\n\u22a2 IsCyclotomicExtension {n} K (CyclotomicField n K)\n[PROOFSTEP]\nobtain \u27e8\u03b6, h\u03b6\u27e9 :=\n exists_root_of_splits (algebraMap K (CyclotomicField n K)) (SplittingField.splits _)\n (degree_cyclotomic_pos n K n.pos).ne'\n[GOAL]\ncase intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : NeZero \u2191\u2191n\nthis\u271d : NeZero \u2191\u2191n\nthis : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)\n\u03b6 : CyclotomicField n K\nh\u03b6 : eval\u2082 (algebraMap K (CyclotomicField n K)) \u03b6 (cyclotomic (\u2191n) K) = 0\n\u22a2 IsCyclotomicExtension {n} K (CyclotomicField n K)\n[PROOFSTEP]\nrw [\u2190 eval_map, \u2190 IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff] at h\u03b6 \n[GOAL]\ncase intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : NeZero \u2191\u2191n\nthis\u271d : NeZero \u2191\u2191n\nthis : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)\n\u03b6 : CyclotomicField n K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 IsCyclotomicExtension {n} K (CyclotomicField n K)\n[PROOFSTEP]\nrefine \u27e8?_, ?_\u27e9\n[GOAL]\ncase intro.refine_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : NeZero \u2191\u2191n\nthis\u271d : NeZero \u2191\u2191n\nthis : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)\n\u03b6 : CyclotomicField n K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 \u2200 {n_1 : \u2115+}, n_1 \u2208 {n} \u2192 \u2203 r, IsPrimitiveRoot r \u2191n_1\n[PROOFSTEP]\nsimp only [mem_singleton_iff, forall_eq]\n[GOAL]\ncase intro.refine_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : NeZero \u2191\u2191n\nthis\u271d : NeZero \u2191\u2191n\nthis : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)\n\u03b6 : CyclotomicField n K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191n\n[PROOFSTEP]\nexact \u27e8\u03b6, h\u03b6\u27e9\n[GOAL]\ncase intro.refine_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : NeZero \u2191\u2191n\nthis\u271d : NeZero \u2191\u2191n\nthis : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)\n\u03b6 : CyclotomicField n K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 \u2200 (x : CyclotomicField n K), x \u2208 adjoin K {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nrw [\u2190 Algebra.eq_top_iff, \u2190 SplittingField.adjoin_rootSet, eq_comm]\n[GOAL]\ncase intro.refine_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : NeZero \u2191\u2191n\nthis\u271d : NeZero \u2191\u2191n\nthis : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)\n\u03b6 : CyclotomicField n K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 adjoin K (rootSet (cyclotomic (\u2191n) K) (SplittingField (cyclotomic (\u2191n) K))) =\n adjoin K {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nexact IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots h\u03b6\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 NoZeroSMulDivisors A (CyclotomicField n K)\n[PROOFSTEP]\nrefine' NoZeroSMulDivisors.of_algebraMap_injective _\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 Function.Injective \u2191(algebraMap A (CyclotomicField n K))\n[PROOFSTEP]\nrw [IsScalarTower.algebraMap_eq A K (CyclotomicField n K)]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 Function.Injective \u2191(RingHom.comp (algebraMap K (CyclotomicField n K)) (algebraMap A K))\n[PROOFSTEP]\nexact\n (Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective K (CyclotomicField n K))\n (IsFractionRing.injective A K) :\n _)\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 CommRing (CyclotomicRing n A K)\n[PROOFSTEP]\ndelta CyclotomicRing\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 CommRing { x // x \u2208 adjoin A {b | b ^ \u2191n = 1} }\n[PROOFSTEP]\ninfer_instance\n --Porting note: could not be derived\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 IsDomain (CyclotomicRing n A K)\n[PROOFSTEP]\ndelta CyclotomicRing\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 IsDomain { x // x \u2208 adjoin A {b | b ^ \u2191n = 1} }\n[PROOFSTEP]\ninfer_instance\n --Porting note: could not be derived\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 Inhabited (CyclotomicRing n A K)\n[PROOFSTEP]\ndelta CyclotomicRing\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u22a2 Inhabited { x // x \u2208 adjoin A {b | b ^ \u2191n = 1} }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\na : \u2115+\nhan : a \u2208 {n}\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191a\n[PROOFSTEP]\nrw [mem_singleton_iff] at han \n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\na : \u2115+\nhan : a = n\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191a\n[PROOFSTEP]\nsubst a\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191n\n[PROOFSTEP]\nhaveI := NeZero.of_noZeroSMulDivisors A K n\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\nthis : NeZero \u2191\u2191n\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191n\n[PROOFSTEP]\nhaveI := NeZero.of_noZeroSMulDivisors A (CyclotomicField n K) n\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\nthis\u271d : NeZero \u2191\u2191n\nthis : NeZero \u2191\u2191n\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191n\n[PROOFSTEP]\nobtain \u27e8\u03bc, h\u03bc\u27e9 := (CyclotomicField.isCyclotomicExtension n K).exists_prim_root (mem_singleton n)\n[GOAL]\ncase intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\nthis\u271d : NeZero \u2191\u2191n\nthis : NeZero \u2191\u2191n\n\u03bc : CyclotomicField n K\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191n\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u03bc, subset_adjoin _\u27e9, _\u27e9\n[GOAL]\ncase intro.refine'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\nthis\u271d : NeZero \u2191\u2191n\nthis : NeZero \u2191\u2191n\n\u03bc : CyclotomicField n K\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\n\u22a2 \u03bc \u2208 {b | b ^ \u2191n = 1}\n[PROOFSTEP]\napply (isRoot_of_unity_iff n.pos (CyclotomicField n K)).mpr\n[GOAL]\ncase intro.refine'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\nthis\u271d : NeZero \u2191\u2191n\nthis : NeZero \u2191\u2191n\n\u03bc : CyclotomicField n K\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\n\u22a2 \u2203 i, i \u2208 Nat.divisors \u2191n \u2227 IsRoot (cyclotomic i (CyclotomicField n K)) \u03bc\n[PROOFSTEP]\nrefine' \u27e8n, Nat.mem_divisors_self _ n.ne_zero, _\u27e9\n[GOAL]\ncase intro.refine'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\nthis\u271d : NeZero \u2191\u2191n\nthis : NeZero \u2191\u2191n\n\u03bc : CyclotomicField n K\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\n\u22a2 IsRoot (cyclotomic (\u2191n) (CyclotomicField n K)) \u03bc\n[PROOFSTEP]\nrwa [\u2190 isRoot_cyclotomic_iff] at h\u03bc \n[GOAL]\ncase intro.refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\nthis\u271d : NeZero \u2191\u2191n\nthis : NeZero \u2191\u2191n\n\u03bc : CyclotomicField n K\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\n\u22a2 IsPrimitiveRoot { val := \u03bc, property := (_ : \u03bc \u2208 \u2191(adjoin A {b | b ^ \u2191n = 1})) } \u2191n\n[PROOFSTEP]\nrwa [\u2190 IsPrimitiveRoot.coe_submonoidClass_iff, Subtype.coe_mk]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicRing n A K\n\u22a2 x \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nrefine' adjoin_induction' (fun y hy => _) (fun a => _) (fun y z hy hz => _) (fun y z hy hz => _) x\n[GOAL]\ncase refine'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicRing n A K\ny : CyclotomicField n K\nhy : y \u2208 {b | b ^ \u2191n = 1}\n\u22a2 { val := y, property := (_ : y \u2208 \u2191(adjoin A {b | b ^ \u2191n = 1})) } \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nrefine' subset_adjoin _\n[GOAL]\ncase refine'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicRing n A K\ny : CyclotomicField n K\nhy : y \u2208 {b | b ^ \u2191n = 1}\n\u22a2 { val := y, property := (_ : y \u2208 \u2191(adjoin A {b | b ^ \u2191n = 1})) } \u2208 {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nsimp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq]\n[GOAL]\ncase refine'_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicRing n A K\ny : CyclotomicField n K\nhy : y \u2208 {b | b ^ \u2191n = 1}\n\u22a2 { val := y, property := (_ : y \u2208 \u2191(adjoin A {b | b ^ \u2191n = 1})) } ^ \u2191n = 1\n[PROOFSTEP]\nrwa [\u2190 Subalgebra.coe_eq_one, Subalgebra.coe_pow, Subtype.coe_mk]\n[GOAL]\ncase refine'_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicRing n A K\na : A\n\u22a2 \u2191(algebraMap A { x // x \u2208 adjoin A {b | b ^ \u2191n = 1} }) a \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nexact Subalgebra.algebraMap_mem _ a\n[GOAL]\ncase refine'_3\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicRing n A K\ny z : { x // x \u2208 adjoin A {b | b ^ \u2191n = 1} }\nhy : y \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\nhz : z \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n\u22a2 y + z \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nexact Subalgebra.add_mem _ hy hz\n[GOAL]\ncase refine'_4\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicRing n A K\ny z : { x // x \u2208 adjoin A {b | b ^ \u2191n = 1} }\nhy : y \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\nhz : z \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n\u22a2 y * z \u2208 adjoin A {b | \u2203 n_1, n_1 \u2208 {n} \u2227 b ^ \u2191n_1 = 1}\n[PROOFSTEP]\nexact Subalgebra.mul_mem _ hy hz\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx\u271d : { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nx : CyclotomicRing n A K\nhx : x \u2208 nonZeroDivisors (CyclotomicRing n A K)\n\u22a2 IsUnit (\u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191{ val := x, property := hx })\n[PROOFSTEP]\nrw [isUnit_iff_ne_zero]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx\u271d : { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nx : CyclotomicRing n A K\nhx : x \u2208 nonZeroDivisors (CyclotomicRing n A K)\n\u22a2 \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191{ val := x, property := hx } \u2260 0\n[PROOFSTEP]\napply map_ne_zero_of_mem_nonZeroDivisors\n[GOAL]\ncase hg\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx\u271d : { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nx : CyclotomicRing n A K\nhx : x \u2208 nonZeroDivisors (CyclotomicRing n A K)\n\u22a2 Function.Injective \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))\ncase h\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx\u271d : { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nx : CyclotomicRing n A K\nhx : x \u2208 nonZeroDivisors (CyclotomicRing n A K)\n\u22a2 \u2191{ val := x, property := hx } \u2208 nonZeroDivisors (CyclotomicRing n A K)\n[PROOFSTEP]\napply adjoin_algebra_injective\n[GOAL]\ncase h\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx\u271d : { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nx : CyclotomicRing n A K\nhx : x \u2208 nonZeroDivisors (CyclotomicRing n A K)\n\u22a2 \u2191{ val := x, property := hx } \u2208 nonZeroDivisors (CyclotomicRing n A K)\n[PROOFSTEP]\nexact hx\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\n\u22a2 \u2203 x_1,\n x * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191x_1.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x_1.fst\n[PROOFSTEP]\nletI : NeZero ((n : \u2115) : K) := NeZero.nat_of_injective (IsFractionRing.injective A K)\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\n\u22a2 \u2203 x_1,\n x * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191x_1.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x_1.fst\n[PROOFSTEP]\nrefine\n Algebra.adjoin_induction\n (((IsCyclotomicExtension.iff_singleton n K (CyclotomicField n K)).1 (CyclotomicField.isCyclotomicExtension n K)).2\n x)\n (fun y hy => ?_) (fun k => ?_) ?_\n ?_\n -- Porting note: the last goal was `by simpa` that now fails.\n[GOAL]\ncase refine_1\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\ny : CyclotomicField n K\nhy : y \u2208 {b | b ^ \u2191n = 1}\n\u22a2 \u2203 x,\n y * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191x.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.fst\n[PROOFSTEP]\nexact \u27e8\u27e8\u27e8y, subset_adjoin hy\u27e9, 1\u27e9, by simp; rfl\u27e9\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\ny : CyclotomicField n K\nhy : y \u2208 {b | b ^ \u2191n = 1}\n\u22a2 y *\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))\n \u2191({ val := y, property := (_ : y \u2208 \u2191(adjoin A {b | b ^ \u2191n = 1})) }, 1).snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))\n ({ val := y, property := (_ : y \u2208 \u2191(adjoin A {b | b ^ \u2191n = 1})) }, 1).fst\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\ny : CyclotomicField n K\nhy : y \u2208 {b | b ^ \u2191n = 1}\n\u22a2 y =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))\n { val := y, property := (_ : y \u2208 \u2191(adjoin A {b | b ^ \u2191n = 1})) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\nk : K\n\u22a2 \u2203 x,\n \u2191(algebraMap K (CyclotomicField n K)) k * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191x.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.fst\n[PROOFSTEP]\nhave : IsLocalization (nonZeroDivisors A) K := inferInstance\n[GOAL]\ncase refine_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis\u271d : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\nk : K\nthis : IsLocalization (nonZeroDivisors A) K\n\u22a2 \u2203 x,\n \u2191(algebraMap K (CyclotomicField n K)) k * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191x.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.fst\n[PROOFSTEP]\nreplace := this.surj\n[GOAL]\ncase refine_2\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis\u271d : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\nk : K\nthis : \u2200 (z : K), \u2203 x, z * \u2191(algebraMap A K) \u2191x.snd = \u2191(algebraMap A K) x.fst\n\u22a2 \u2203 x,\n \u2191(algebraMap K (CyclotomicField n K)) k * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191x.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.fst\n[PROOFSTEP]\nobtain \u27e8\u27e8z, w\u27e9, hw\u27e9 := this k\n[GOAL]\ncase refine_2.intro.mk\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis\u271d : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\nk : K\nthis : \u2200 (z : K), \u2203 x, z * \u2191(algebraMap A K) \u2191x.snd = \u2191(algebraMap A K) x.fst\nz : A\nw : { x // x \u2208 nonZeroDivisors A }\nhw : k * \u2191(algebraMap A K) \u2191(z, w).snd = \u2191(algebraMap A K) (z, w).fst\n\u22a2 \u2203 x,\n \u2191(algebraMap K (CyclotomicField n K)) k * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191x.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.fst\n[PROOFSTEP]\nrefine'\n \u27e8\u27e8algebraMap A (CyclotomicRing n A K) z, algebraMap A (CyclotomicRing n A K) w,\n map_mem_nonZeroDivisors _ (algebraBase_injective n A K) w.2\u27e9,\n _\u27e9\n[GOAL]\ncase refine_2.intro.mk\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis\u271d : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\nk : K\nthis : \u2200 (z : K), \u2203 x, z * \u2191(algebraMap A K) \u2191x.snd = \u2191(algebraMap A K) x.fst\nz : A\nw : { x // x \u2208 nonZeroDivisors A }\nhw : k * \u2191(algebraMap A K) \u2191(z, w).snd = \u2191(algebraMap A K) (z, w).fst\n\u22a2 \u2191(algebraMap K (CyclotomicField n K)) k *\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))\n \u2191(\u2191(algebraMap A (CyclotomicRing n A K)) z,\n { val := \u2191(algebraMap A (CyclotomicRing n A K)) \u2191w,\n property :=\n (_ : \u2191(algebraMap A (CyclotomicRing n A K)) \u2191w \u2208 nonZeroDivisors (CyclotomicRing n A K)) }).snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))\n (\u2191(algebraMap A (CyclotomicRing n A K)) z,\n { val := \u2191(algebraMap A (CyclotomicRing n A K)) \u2191w,\n property := (_ : \u2191(algebraMap A (CyclotomicRing n A K)) \u2191w \u2208 nonZeroDivisors (CyclotomicRing n A K)) }).fst\n[PROOFSTEP]\nletI : IsScalarTower A K (CyclotomicField n K) := IsScalarTower.of_algebraMap_eq (congr_fun rfl)\n[GOAL]\ncase refine_2.intro.mk\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis\u271d\u00b9 : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\nk : K\nthis\u271d : \u2200 (z : K), \u2203 x, z * \u2191(algebraMap A K) \u2191x.snd = \u2191(algebraMap A K) x.fst\nz : A\nw : { x // x \u2208 nonZeroDivisors A }\nhw : k * \u2191(algebraMap A K) \u2191(z, w).snd = \u2191(algebraMap A K) (z, w).fst\nthis : IsScalarTower A K (CyclotomicField n K) := IsScalarTower.of_algebraMap_eq (congr_fun rfl)\n\u22a2 \u2191(algebraMap K (CyclotomicField n K)) k *\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))\n \u2191(\u2191(algebraMap A (CyclotomicRing n A K)) z,\n { val := \u2191(algebraMap A (CyclotomicRing n A K)) \u2191w,\n property :=\n (_ : \u2191(algebraMap A (CyclotomicRing n A K)) \u2191w \u2208 nonZeroDivisors (CyclotomicRing n A K)) }).snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))\n (\u2191(algebraMap A (CyclotomicRing n A K)) z,\n { val := \u2191(algebraMap A (CyclotomicRing n A K)) \u2191w,\n property := (_ : \u2191(algebraMap A (CyclotomicRing n A K)) \u2191w \u2208 nonZeroDivisors (CyclotomicRing n A K)) }).fst\n[PROOFSTEP]\nrw [\u2190 IsScalarTower.algebraMap_apply, \u2190 IsScalarTower.algebraMap_apply,\n @IsScalarTower.algebraMap_apply A K _ _ _ _ _ (_root_.CyclotomicField.algebra n K) _ _ w, \u2190 RingHom.map_mul, hw, \u2190\n IsScalarTower.algebraMap_apply]\n[GOAL]\ncase refine_3\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\n\u22a2 \u2200 (x y : CyclotomicField n K),\n (\u2203 x_1,\n x * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191x_1.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x_1.fst) \u2192\n (\u2203 x,\n y * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191x.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.fst) \u2192\n \u2203 x_1,\n (x + y) * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191x_1.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x_1.fst\n[PROOFSTEP]\nrintro y z \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase refine_3.intro.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\ny z : CyclotomicField n K\na : CyclotomicRing n A K \u00d7 { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nha :\n y * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191a.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.fst\nb : CyclotomicRing n A K \u00d7 { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nhb :\n z * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191b.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.fst\n\u22a2 \u2203 x,\n (y + z) * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191x.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.fst\n[PROOFSTEP]\nrefine' \u27e8\u27e8a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 \u27e8a.2.2, b.2.2\u27e9\u27e9, _\u27e9\n[GOAL]\ncase refine_3.intro.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\ny z : CyclotomicField n K\na : CyclotomicRing n A K \u00d7 { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nha :\n y * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191a.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.fst\nb : CyclotomicRing n A K \u00d7 { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nhb :\n z * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191b.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.fst\n\u22a2 (y + z) *\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))\n \u2191(a.fst * \u2191b.snd + b.fst * \u2191a.snd,\n { val := \u2191a.snd * \u2191b.snd,\n property := (_ : \u2191a.snd * \u2191b.snd \u2208 nonZeroDivisors (CyclotomicRing n A K)) }).snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))\n (a.fst * \u2191b.snd + b.fst * \u2191a.snd,\n { val := \u2191a.snd * \u2191b.snd, property := (_ : \u2191a.snd * \u2191b.snd \u2208 nonZeroDivisors (CyclotomicRing n A K)) }).fst\n[PROOFSTEP]\nrw [RingHom.map_mul, add_mul, \u2190 mul_assoc, ha, mul_comm ((algebraMap (CyclotomicRing n A K) _) \u2191a.2), \u2190 mul_assoc, hb]\n[GOAL]\ncase refine_3.intro.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\ny z : CyclotomicField n K\na : CyclotomicRing n A K \u00d7 { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nha :\n y * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191a.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.fst\nb : CyclotomicRing n A K \u00d7 { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nhb :\n z * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191b.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.fst\n\u22a2 \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.fst *\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191b.snd +\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.fst *\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191a.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))\n (a.fst * \u2191b.snd + b.fst * \u2191a.snd,\n { val := \u2191a.snd * \u2191b.snd, property := (_ : \u2191a.snd * \u2191b.snd \u2208 nonZeroDivisors (CyclotomicRing n A K)) }).fst\n[PROOFSTEP]\nsimp only [map_add, map_mul]\n[GOAL]\ncase refine_4\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\n\u22a2 \u2200 (x y : CyclotomicField n K),\n (\u2203 x_1,\n x * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191x_1.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x_1.fst) \u2192\n (\u2203 x,\n y * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191x.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.fst) \u2192\n \u2203 x_1,\n x * y * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191x_1.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x_1.fst\n[PROOFSTEP]\nrintro y z \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase refine_4.intro.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\ny z : CyclotomicField n K\na : CyclotomicRing n A K \u00d7 { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nha :\n y * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191a.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.fst\nb : CyclotomicRing n A K \u00d7 { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nhb :\n z * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191b.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.fst\n\u22a2 \u2203 x,\n y * z * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191x.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.fst\n[PROOFSTEP]\nrefine' \u27e8\u27e8a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 \u27e8a.2.2, b.2.2\u27e9\u27e9, _\u27e9\n[GOAL]\ncase refine_4.intro.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\ny z : CyclotomicField n K\na : CyclotomicRing n A K \u00d7 { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nha :\n y * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191a.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.fst\nb : CyclotomicRing n A K \u00d7 { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nhb :\n z * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191b.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.fst\n\u22a2 y * z *\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))\n \u2191(a.fst * b.fst,\n { val := \u2191a.snd * \u2191b.snd,\n property := (_ : \u2191a.snd * \u2191b.snd \u2208 nonZeroDivisors (CyclotomicRing n A K)) }).snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))\n (a.fst * b.fst,\n { val := \u2191a.snd * \u2191b.snd, property := (_ : \u2191a.snd * \u2191b.snd \u2208 nonZeroDivisors (CyclotomicRing n A K)) }).fst\n[PROOFSTEP]\nrw [RingHom.map_mul, mul_comm ((algebraMap (CyclotomicRing n A K) _) \u2191a.2), mul_assoc, \u2190 mul_assoc z, hb, \u2190\n mul_comm ((algebraMap (CyclotomicRing n A K) _) \u2191a.2), \u2190 mul_assoc, ha]\n[GOAL]\ncase refine_4.intro.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx : CyclotomicField n K\nthis : NeZero \u2191\u2191n := NeZero.nat_of_injective (IsFractionRing.injective A K)\ny z : CyclotomicField n K\na : CyclotomicRing n A K \u00d7 { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nha :\n y * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191a.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.fst\nb : CyclotomicRing n A K \u00d7 { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nhb :\n z * \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) \u2191b.snd =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.fst\n\u22a2 \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.fst *\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.fst =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K))\n (a.fst * b.fst,\n { val := \u2191a.snd * \u2191b.snd, property := (_ : \u2191a.snd * \u2191b.snd \u2208 nonZeroDivisors (CyclotomicRing n A K)) }).fst\n[PROOFSTEP]\nsimp only [map_mul]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx y : CyclotomicRing n A K\nh :\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) y\n\u22a2 \u21911 * x = \u21911 * y\n[PROOFSTEP]\nrw [adjoin_algebra_injective n A K h]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : NeZero \u2191\u2191n\nx y : CyclotomicRing n A K\nx\u271d : \u2203 c, \u2191c * x = \u2191c * y\nc : { x // x \u2208 nonZeroDivisors (CyclotomicRing n A K) }\nhc : \u2191c * x = \u2191c * y\n\u22a2 \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x =\n \u2191(algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) y\n[PROOFSTEP]\nrw [mul_left_cancel\u2080 (nonZeroDivisors.ne_zero c.prop) hc]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u03bc : CyclotomicField n K\nh : IsPrimitiveRoot \u03bc \u2191n\n\u22a2 CyclotomicRing n A K = { x // x \u2208 adjoin A {\u03bc} }\n[PROOFSTEP]\nrw [\u2190 IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic h,\n IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots h]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\n\u03bc : CyclotomicField n K\nh : IsPrimitiveRoot \u03bc \u2191n\n\u22a2 CyclotomicRing n A K = { x // x \u2208 adjoin A {b | \u2203 a, a \u2208 {n} \u2227 b ^ \u2191a = 1} }\n[PROOFSTEP]\nsimp [CyclotomicRing]\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsAlgClosed K\nh : \u2200 (a : \u2115+), a \u2208 S \u2192 NeZero \u2191\u2191a\n\u22a2 IsCyclotomicExtension S K K\n[PROOFSTEP]\nrefine' \u27e8@fun a ha => _, Algebra.eq_top_iff.mp <| Subsingleton.elim _ _\u27e9\n[GOAL]\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsAlgClosed K\nh : \u2200 (a : \u2115+), a \u2208 S \u2192 NeZero \u2191\u2191a\na : \u2115+\nha : a \u2208 S\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191a\n[PROOFSTEP]\nobtain \u27e8r, hr\u27e9 := IsAlgClosed.exists_aeval_eq_zero K _ (degree_cyclotomic_pos a K a.pos).ne'\n[GOAL]\ncase intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsAlgClosed K\nh : \u2200 (a : \u2115+), a \u2208 S \u2192 NeZero \u2191\u2191a\na : \u2115+\nha : a \u2208 S\nr : K\nhr : \u2191(aeval r) (cyclotomic (\u2191a) K) = 0\n\u22a2 \u2203 r, IsPrimitiveRoot r \u2191a\n[PROOFSTEP]\nrefine' \u27e8r, _\u27e9\n[GOAL]\ncase intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsAlgClosed K\nh : \u2200 (a : \u2115+), a \u2208 S \u2192 NeZero \u2191\u2191a\na : \u2115+\nha : a \u2208 S\nr : K\nhr : \u2191(aeval r) (cyclotomic (\u2191a) K) = 0\n\u22a2 IsPrimitiveRoot r \u2191a\n[PROOFSTEP]\nhaveI := h a ha\n[GOAL]\ncase intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsAlgClosed K\nh : \u2200 (a : \u2115+), a \u2208 S \u2192 NeZero \u2191\u2191a\na : \u2115+\nha : a \u2208 S\nr : K\nhr : \u2191(aeval r) (cyclotomic (\u2191a) K) = 0\nthis : NeZero \u2191\u2191a\n\u22a2 IsPrimitiveRoot r \u2191a\n[PROOFSTEP]\nrwa [coe_aeval_eq_eval, \u2190 IsRoot.def, isRoot_cyclotomic_iff] at hr \n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.Cyclotomic.Basic", "llama_tokens": 63165, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7772998611746911, "lm_q2_score": 0.5273165233795672, "lm_q1q2_score": 0.4098830604180584}} {"text": "[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type u_2\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_3\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\n\u22a2 MetrizableSpace M\n[PROOFSTEP]\nhaveI := I.locally_compact\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type u_2\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_3\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nthis : LocallyCompactSpace H\n\u22a2 MetrizableSpace M\n[PROOFSTEP]\nhaveI := ChartedSpace.locallyCompact H M\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type u_2\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_3\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nthis\u271d : LocallyCompactSpace H\nthis : LocallyCompactSpace M\n\u22a2 MetrizableSpace M\n[PROOFSTEP]\nhaveI : NormalSpace M := normal_of_paracompact_t2\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type u_2\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_3\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nthis\u271d\u00b9 : LocallyCompactSpace H\nthis\u271d : LocallyCompactSpace M\nthis : NormalSpace M\n\u22a2 MetrizableSpace M\n[PROOFSTEP]\nhaveI := I.secondCountableTopology\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type u_2\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_3\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nthis\u271d\u00b2 : LocallyCompactSpace H\nthis\u271d\u00b9 : LocallyCompactSpace M\nthis\u271d : NormalSpace M\nthis : SecondCountableTopology H\n\u22a2 MetrizableSpace M\n[PROOFSTEP]\nhaveI := ChartedSpace.secondCountable_of_sigma_compact H M\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type u_2\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_3\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : NormalSpace M\nthis\u271d : SecondCountableTopology H\nthis : SecondCountableTopology M\n\u22a2 MetrizableSpace M\n[PROOFSTEP]\nexact metrizableSpace_of_t3_second_countable M\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Manifold.Metrizable", "llama_tokens": 1148, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583376458152, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.40984230050686943}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf g : \u03b1 \u2192 \u03b2\nhf : AEStronglyMeasurable' m f \u03bc\nhfg : f =\u1d50[\u03bc] g\n\u22a2 AEStronglyMeasurable' m g \u03bc\n[PROOFSTEP]\nobtain \u27e8f', hf'_meas, hff'\u27e9 := hf\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf g : \u03b1 \u2192 \u03b2\nhfg : f =\u1d50[\u03bc] g\nf' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhff' : f =\u1d50[\u03bc] f'\n\u22a2 AEStronglyMeasurable' m g \u03bc\n[PROOFSTEP]\nexact \u27e8f', hf'_meas, hfg.symm.trans hff'\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf g : \u03b1 \u2192 \u03b2\nm' : MeasurableSpace \u03b1\nhf : AEStronglyMeasurable' m f \u03bc\nhm : m \u2264 m'\n\u22a2 AEStronglyMeasurable' m' f \u03bc\n[PROOFSTEP]\nobtain \u27e8f', hf'_meas, hff'\u27e9 := hf\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf g : \u03b1 \u2192 \u03b2\nm' : MeasurableSpace \u03b1\nhm : m \u2264 m'\nf' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhff' : f =\u1d50[\u03bc] f'\n\u22a2 AEStronglyMeasurable' m' f \u03bc\n[PROOFSTEP]\nexact \u27e8f', hf'_meas.mono hm, hff'\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : ContinuousAdd \u03b2\nhf : AEStronglyMeasurable' m f \u03bc\nhg : AEStronglyMeasurable' m g \u03bc\n\u22a2 AEStronglyMeasurable' m (f + g) \u03bc\n[PROOFSTEP]\nrcases hf with \u27e8f', h_f'_meas, hff'\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : ContinuousAdd \u03b2\nhg : AEStronglyMeasurable' m g \u03bc\nf' : \u03b1 \u2192 \u03b2\nh_f'_meas : StronglyMeasurable f'\nhff' : f =\u1d50[\u03bc] f'\n\u22a2 AEStronglyMeasurable' m (f + g) \u03bc\n[PROOFSTEP]\nrcases hg with \u27e8g', h_g'_meas, hgg'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : ContinuousAdd \u03b2\nf' : \u03b1 \u2192 \u03b2\nh_f'_meas : StronglyMeasurable f'\nhff' : f =\u1d50[\u03bc] f'\ng' : \u03b1 \u2192 \u03b2\nh_g'_meas : StronglyMeasurable g'\nhgg' : g =\u1d50[\u03bc] g'\n\u22a2 AEStronglyMeasurable' m (f + g) \u03bc\n[PROOFSTEP]\nexact \u27e8f' + g', h_f'_meas.add h_g'_meas, hff'.add hgg'\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nf : \u03b1 \u2192 \u03b2\nhfm : AEStronglyMeasurable' m f \u03bc\n\u22a2 AEStronglyMeasurable' m (-f) \u03bc\n[PROOFSTEP]\nrcases hfm with \u27e8f', hf'_meas, hf_ae\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nf f' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\n\u22a2 AEStronglyMeasurable' m (-f) \u03bc\n[PROOFSTEP]\nrefine' \u27e8-f', hf'_meas.neg, hf_ae.mono fun x hx => _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nf f' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nx : \u03b1\nhx : f x = f' x\n\u22a2 (-f) x = (-f') x\n[PROOFSTEP]\nsimp_rw [Pi.neg_apply]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nf f' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nx : \u03b1\nhx : f x = f' x\n\u22a2 -f x = -f' x\n[PROOFSTEP]\nrw [hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nf g : \u03b1 \u2192 \u03b2\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\n\u22a2 AEStronglyMeasurable' m (f - g) \u03bc\n[PROOFSTEP]\nrcases hfm with \u27e8f', hf'_meas, hf_ae\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nf g : \u03b1 \u2192 \u03b2\nhgm : AEStronglyMeasurable' m g \u03bc\nf' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\n\u22a2 AEStronglyMeasurable' m (f - g) \u03bc\n[PROOFSTEP]\nrcases hgm with \u27e8g', hg'_meas, hg_ae\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nf g f' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\ng' : \u03b1 \u2192 \u03b2\nhg'_meas : StronglyMeasurable g'\nhg_ae : g =\u1d50[\u03bc] g'\n\u22a2 AEStronglyMeasurable' m (f - g) \u03bc\n[PROOFSTEP]\nrefine' \u27e8f' - g', hf'_meas.sub hg'_meas, hf_ae.mp (hg_ae.mono fun x hx1 hx2 => _)\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nf g f' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\ng' : \u03b1 \u2192 \u03b2\nhg'_meas : StronglyMeasurable g'\nhg_ae : g =\u1d50[\u03bc] g'\nx : \u03b1\nhx1 : g x = g' x\nhx2 : f x = f' x\n\u22a2 (f - g) x = (f' - g') x\n[PROOFSTEP]\nsimp_rw [Pi.sub_apply]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : TopologicalAddGroup \u03b2\nf g f' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\ng' : \u03b1 \u2192 \u03b2\nhg'_meas : StronglyMeasurable g'\nhg_ae : g =\u1d50[\u03bc] g'\nx : \u03b1\nhx1 : g x = g' x\nhx2 : f x = f' x\n\u22a2 f x - g x = f' x - g' x\n[PROOFSTEP]\nrw [hx1, hx2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c \u03b2\ninst\u271d : ContinuousConstSMul \ud835\udd5c \u03b2\nc : \ud835\udd5c\nhf : AEStronglyMeasurable' m f \u03bc\n\u22a2 AEStronglyMeasurable' m (c \u2022 f) \u03bc\n[PROOFSTEP]\nrcases hf with \u27e8f', h_f'_meas, hff'\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c \u03b2\ninst\u271d : ContinuousConstSMul \ud835\udd5c \u03b2\nc : \ud835\udd5c\nf' : \u03b1 \u2192 \u03b2\nh_f'_meas : StronglyMeasurable f'\nhff' : f =\u1d50[\u03bc] f'\n\u22a2 AEStronglyMeasurable' m (c \u2022 f) \u03bc\n[PROOFSTEP]\nrefine' \u27e8c \u2022 f', h_f'_meas.const_smul c, _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nf g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SMul \ud835\udd5c \u03b2\ninst\u271d : ContinuousConstSMul \ud835\udd5c \u03b2\nc : \ud835\udd5c\nf' : \u03b1 \u2192 \u03b2\nh_f'_meas : StronglyMeasurable f'\nhff' : f =\u1d50[\u03bc] f'\n\u22a2 c \u2022 f =\u1d50[\u03bc] c \u2022 f'\n[PROOFSTEP]\nexact EventuallyEq.fun_comp hff' fun x => c \u2022 x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\ud835\udd5c\u271d : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\u271d\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\n\ud835\udd5c : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : InnerProductSpace \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nhfm : AEStronglyMeasurable' m f \u03bc\nc : \u03b2\n\u22a2 AEStronglyMeasurable' m (fun x => inner c (f x)) \u03bc\n[PROOFSTEP]\nrcases hfm with \u27e8f', hf'_meas, hf_ae\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\ud835\udd5c\u271d : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\u271d\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\n\ud835\udd5c : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : InnerProductSpace \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nf' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\n\u22a2 AEStronglyMeasurable' m (fun x => inner c (f x)) \u03bc\n[PROOFSTEP]\nrefine' \u27e8fun x => (inner c (f' x) : \ud835\udd5c), (@stronglyMeasurable_const _ _ m _ c).inner hf'_meas, hf_ae.mono fun x hx => _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\ud835\udd5c\u271d : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\u271d\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\n\ud835\udd5c : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : InnerProductSpace \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nf' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nx : \u03b1\nhx : f x = f' x\n\u22a2 (fun x => inner c (f x)) x = (fun x => inner c (f' x)) x\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\ud835\udd5c\u271d : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\u271d\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\n\ud835\udd5c : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : InnerProductSpace \ud835\udd5c \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nf' : \u03b1 \u2192 \u03b2\nhf'_meas : StronglyMeasurable f'\nhf_ae : f =\u1d50[\u03bc] f'\nx : \u03b1\nhx : f x = f' x\n\u22a2 inner c (f x) = inner c (f' x)\n[PROOFSTEP]\nrw [hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\n\u03b3 : Type u_4\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhg : Continuous g\nhf : AEStronglyMeasurable' m f \u03bc\nx : \u03b1\nhx : f x = mk f hf x\n\u22a2 (g \u2218 f) x = (fun x => g (mk f hf x)) x\n[PROOFSTEP]\nrw [Function.comp_apply, hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm m0 m0' : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhm0 : m0 \u2264 m0'\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : AEStronglyMeasurable' m f (Measure.trim \u03bc hm0)\n\u22a2 AEStronglyMeasurable' m f \u03bc\n[PROOFSTEP]\nobtain \u27e8g, hg_meas, hfg\u27e9 := hf\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm m0 m0' : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhm0 : m0 \u2264 m0'\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u03b2\nhg_meas : StronglyMeasurable g\nhfg : f =\u1d50[Measure.trim \u03bc hm0] g\n\u22a2 AEStronglyMeasurable' m f \u03bc\n[PROOFSTEP]\nexact \u27e8g, hg_meas, ae_eq_of_ae_eq_trim hfg\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\nhm : m \u2264 m0\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : AEStronglyMeasurable' m f \u03bc\nhf_zero : f =\u1d50[Measure.restrict \u03bc s\u1d9c] 0\n\u22a2 AEStronglyMeasurable' m\u2082 f \u03bc\n[PROOFSTEP]\nhave h_ind_eq : s.indicator (hf.mk f) =\u1d50[\u03bc] f :=\n by\n refine' Filter.EventuallyEq.trans _ (indicator_ae_eq_of_restrict_compl_ae_eq_zero (hm _ hs_m) hf_zero)\n filter_upwards [hf.ae_eq_mk] with x hx\n by_cases hxs : x \u2208 s\n \u00b7 simp [hxs, hx]\n \u00b7 simp [hxs]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\nhm : m \u2264 m0\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : AEStronglyMeasurable' m f \u03bc\nhf_zero : f =\u1d50[Measure.restrict \u03bc s\u1d9c] 0\n\u22a2 Set.indicator s (mk f hf) =\u1d50[\u03bc] f\n[PROOFSTEP]\nrefine' Filter.EventuallyEq.trans _ (indicator_ae_eq_of_restrict_compl_ae_eq_zero (hm _ hs_m) hf_zero)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\nhm : m \u2264 m0\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : AEStronglyMeasurable' m f \u03bc\nhf_zero : f =\u1d50[Measure.restrict \u03bc s\u1d9c] 0\n\u22a2 Set.indicator s (mk f hf) =\u1d50[\u03bc] Set.indicator s f\n[PROOFSTEP]\nfilter_upwards [hf.ae_eq_mk] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\nhm : m \u2264 m0\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : AEStronglyMeasurable' m f \u03bc\nhf_zero : f =\u1d50[Measure.restrict \u03bc s\u1d9c] 0\nx : \u03b1\nhx : f x = mk f hf x\n\u22a2 Set.indicator s (mk f hf) x = Set.indicator s f x\n[PROOFSTEP]\nby_cases hxs : x \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\nhm : m \u2264 m0\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : AEStronglyMeasurable' m f \u03bc\nhf_zero : f =\u1d50[Measure.restrict \u03bc s\u1d9c] 0\nx : \u03b1\nhx : f x = mk f hf x\nhxs : x \u2208 s\n\u22a2 Set.indicator s (mk f hf) x = Set.indicator s f x\n[PROOFSTEP]\nsimp [hxs, hx]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\nhm : m \u2264 m0\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : AEStronglyMeasurable' m f \u03bc\nhf_zero : f =\u1d50[Measure.restrict \u03bc s\u1d9c] 0\nx : \u03b1\nhx : f x = mk f hf x\nhxs : \u00acx \u2208 s\n\u22a2 Set.indicator s (mk f hf) x = Set.indicator s f x\n[PROOFSTEP]\nsimp [hxs]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\nhm : m \u2264 m0\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : AEStronglyMeasurable' m f \u03bc\nhf_zero : f =\u1d50[Measure.restrict \u03bc s\u1d9c] 0\nh_ind_eq : Set.indicator s (mk f hf) =\u1d50[\u03bc] f\n\u22a2 AEStronglyMeasurable' m\u2082 f \u03bc\n[PROOFSTEP]\nsuffices : StronglyMeasurable[m\u2082] (s.indicator (hf.mk f))\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\nhm : m \u2264 m0\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : AEStronglyMeasurable' m f \u03bc\nhf_zero : f =\u1d50[Measure.restrict \u03bc s\u1d9c] 0\nh_ind_eq : Set.indicator s (mk f hf) =\u1d50[\u03bc] f\nthis : StronglyMeasurable (Set.indicator s (mk f hf))\n\u22a2 AEStronglyMeasurable' m\u2082 f \u03bc\ncase this\n\u03b1 : Type u_1\nE : Type u_2\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\nhm : m \u2264 m0\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : AEStronglyMeasurable' m f \u03bc\nhf_zero : f =\u1d50[Measure.restrict \u03bc s\u1d9c] 0\nh_ind_eq : Set.indicator s (mk f hf) =\u1d50[\u03bc] f\n\u22a2 StronglyMeasurable (Set.indicator s (mk f hf))\n[PROOFSTEP]\nexact AEStronglyMeasurable'.congr this.aeStronglyMeasurable' h_ind_eq\n[GOAL]\ncase this\n\u03b1 : Type u_1\nE : Type u_2\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\nhm : m \u2264 m0\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : AEStronglyMeasurable' m f \u03bc\nhf_zero : f =\u1d50[Measure.restrict \u03bc s\u1d9c] 0\nh_ind_eq : Set.indicator s (mk f hf) =\u1d50[\u03bc] f\n\u22a2 StronglyMeasurable (Set.indicator s (mk f hf))\n[PROOFSTEP]\nhave hf_ind : StronglyMeasurable[m] (s.indicator (hf.mk f)) := hf.stronglyMeasurable_mk.indicator hs_m\n[GOAL]\ncase this\n\u03b1 : Type u_1\nE : Type u_2\nm m\u2082 m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : Zero E\nhm : m \u2264 m0\ns : Set \u03b1\nf : \u03b1 \u2192 E\nhs_m : MeasurableSet s\nhs : \u2200 (t : Set \u03b1), MeasurableSet (s \u2229 t) \u2192 MeasurableSet (s \u2229 t)\nhf : AEStronglyMeasurable' m f \u03bc\nhf_zero : f =\u1d50[Measure.restrict \u03bc s\u1d9c] 0\nh_ind_eq : Set.indicator s (mk f hf) =\u1d50[\u03bc] f\nhf_ind : StronglyMeasurable (Set.indicator s (mk f hf))\n\u22a2 StronglyMeasurable (Set.indicator s (mk f hf))\n[PROOFSTEP]\nexact hf_ind.stronglyMeasurable_of_measurableSpace_le_on hs_m hs fun x hxs => Set.indicator_of_not_mem hxs _\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 Lp F p }\n\u22a2 f \u2208 lpMeasSubgroup F m p \u03bc \u2194 AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n[PROOFSTEP]\nrw [\u2190 AddSubgroup.mem_carrier, lpMeasSubgroup, Set.mem_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : { x // x \u2208 Lp F p }\n\u22a2 f \u2208 lpMeas F \ud835\udd5c m p \u03bc \u2194 AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, \u2190 Submodule.mem_carrier, lpMeas, Set.mem_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\nhf_meas : f \u2208 lpMeasSubgroup F m p \u03bc\n\u22a2 Mem\u2112p (Exists.choose (_ : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc)) p\n[PROOFSTEP]\nhave hf : AEStronglyMeasurable' m f \u03bc := mem_lpMeasSubgroup_iff_aeStronglyMeasurable'.mp hf_meas\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\nhf_meas : f \u2208 lpMeasSubgroup F m p \u03bc\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n\u22a2 Mem\u2112p (Exists.choose (_ : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc)) p\n[PROOFSTEP]\nlet g := hf.choose\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\nhf_meas : f \u2208 lpMeasSubgroup F m p \u03bc\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\ng : \u03b1 \u2192 F := Exists.choose hf\n\u22a2 Mem\u2112p (Exists.choose (_ : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc)) p\n[PROOFSTEP]\nobtain \u27e8hg, hfg\u27e9 := hf.choose_spec\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\nhf_meas : f \u2208 lpMeasSubgroup F m p \u03bc\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\ng : \u03b1 \u2192 F := Exists.choose hf\nhg : StronglyMeasurable (Exists.choose hf)\nhfg : \u2191\u2191f =\u1d50[\u03bc] Exists.choose hf\n\u22a2 Mem\u2112p (Exists.choose (_ : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc)) p\n[PROOFSTEP]\nchange Mem\u2112p g p (\u03bc.trim hm)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\nhf_meas : f \u2208 lpMeasSubgroup F m p \u03bc\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\ng : \u03b1 \u2192 F := Exists.choose hf\nhg : StronglyMeasurable (Exists.choose hf)\nhfg : \u2191\u2191f =\u1d50[\u03bc] Exists.choose hf\n\u22a2 Mem\u2112p g p\n[PROOFSTEP]\nrefine' \u27e8hg.aestronglyMeasurable, _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\nhf_meas : f \u2208 lpMeasSubgroup F m p \u03bc\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\ng : \u03b1 \u2192 F := Exists.choose hf\nhg : StronglyMeasurable (Exists.choose hf)\nhfg : \u2191\u2191f =\u1d50[\u03bc] Exists.choose hf\n\u22a2 snorm g p (Measure.trim \u03bc hm) < \u22a4\n[PROOFSTEP]\nhave h_snorm_fg : snorm g p (\u03bc.trim hm) = snorm f p \u03bc :=\n by\n rw [snorm_trim hm hg]\n exact snorm_congr_ae hfg.symm\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\nhf_meas : f \u2208 lpMeasSubgroup F m p \u03bc\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\ng : \u03b1 \u2192 F := Exists.choose hf\nhg : StronglyMeasurable (Exists.choose hf)\nhfg : \u2191\u2191f =\u1d50[\u03bc] Exists.choose hf\n\u22a2 snorm g p (Measure.trim \u03bc hm) = snorm (\u2191\u2191f) p \u03bc\n[PROOFSTEP]\nrw [snorm_trim hm hg]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\nhf_meas : f \u2208 lpMeasSubgroup F m p \u03bc\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\ng : \u03b1 \u2192 F := Exists.choose hf\nhg : StronglyMeasurable (Exists.choose hf)\nhfg : \u2191\u2191f =\u1d50[\u03bc] Exists.choose hf\n\u22a2 snorm (Exists.choose hf) p \u03bc = snorm (\u2191\u2191f) p \u03bc\n[PROOFSTEP]\nexact snorm_congr_ae hfg.symm\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\nhf_meas : f \u2208 lpMeasSubgroup F m p \u03bc\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\ng : \u03b1 \u2192 F := Exists.choose hf\nhg : StronglyMeasurable (Exists.choose hf)\nhfg : \u2191\u2191f =\u1d50[\u03bc] Exists.choose hf\nh_snorm_fg : snorm g p (Measure.trim \u03bc hm) = snorm (\u2191\u2191f) p \u03bc\n\u22a2 snorm g p (Measure.trim \u03bc hm) < \u22a4\n[PROOFSTEP]\nrw [h_snorm_fg]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\nhf_meas : f \u2208 lpMeasSubgroup F m p \u03bc\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\ng : \u03b1 \u2192 F := Exists.choose hf\nhg : StronglyMeasurable (Exists.choose hf)\nhfg : \u2191\u2191f =\u1d50[\u03bc] Exists.choose hf\nh_snorm_fg : snorm g p (Measure.trim \u03bc hm) = snorm (\u2191\u2191f) p \u03bc\n\u22a2 snorm (\u2191\u2191f) p \u03bc < \u22a4\n[PROOFSTEP]\nexact Lp.snorm_lt_top f\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\n\u22a2 Mem\u2112p.toLp \u2191\u2191f (_ : Mem\u2112p (\u2191\u2191f) p) \u2208 lpMeasSubgroup F m p \u03bc\n[PROOFSTEP]\nlet hf_mem_\u2112p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f)\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\nhf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f)\n\u22a2 Mem\u2112p.toLp \u2191\u2191f (_ : Mem\u2112p (\u2191\u2191f) p) \u2208 lpMeasSubgroup F m p \u03bc\n[PROOFSTEP]\nrw [mem_lpMeasSubgroup_iff_aeStronglyMeasurable']\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\nhf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(Mem\u2112p.toLp \u2191\u2191f (_ : Mem\u2112p (\u2191\u2191f) p))) \u03bc\n[PROOFSTEP]\nrefine' AEStronglyMeasurable'.congr _ (Mem\u2112p.coeFn_toLp hf_mem_\u2112p).symm\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\nhf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n[PROOFSTEP]\nrefine' aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim hm _\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\nhf_mem_\u2112p : Mem\u2112p (\u2191\u2191f) p := mem\u2112p_of_mem\u2112p_trim hm (Lp.mem\u2112p f)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191f) (Measure.trim \u03bc hm)\n[PROOFSTEP]\nexact Lp.aestronglyMeasurable f\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\n\u22a2 Function.RightInverse (lpTrimToLpMeasSubgroup F p \u03bc hm) (lpMeasSubgroupToLpTrim F p \u03bc hm)\n[PROOFSTEP]\nintro f\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\n\u22a2 lpMeasSubgroupToLpTrim F p \u03bc hm (lpTrimToLpMeasSubgroup F p \u03bc hm f) = f\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (lpTrimToLpMeasSubgroup F p \u03bc hm f)) =\u1d50[Measure.trim \u03bc hm] \u2191\u2191f\n[PROOFSTEP]\nrefine' ae_eq_trim_of_stronglyMeasurable hm (Lp.stronglyMeasurable _) (Lp.stronglyMeasurable _) _\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp F p }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (lpTrimToLpMeasSubgroup F p \u03bc hm f)) =\u1d50[\u03bc] \u2191\u2191f\n[PROOFSTEP]\nexact (lpMeasSubgroupToLpTrim_ae_eq hm _).trans (lpTrimToLpMeasSubgroup_ae_eq hm _)\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\n\u22a2 Function.LeftInverse (lpTrimToLpMeasSubgroup F p \u03bc hm) (lpMeasSubgroupToLpTrim F p \u03bc hm)\n[PROOFSTEP]\nintro f\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 lpTrimToLpMeasSubgroup F p \u03bc hm (lpMeasSubgroupToLpTrim F p \u03bc hm f) = f\n[PROOFSTEP]\next1\n[GOAL]\ncase a\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191(lpTrimToLpMeasSubgroup F p \u03bc hm (lpMeasSubgroupToLpTrim F p \u03bc hm f)) = \u2191f\n[PROOFSTEP]\next1\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191(lpTrimToLpMeasSubgroup F p \u03bc hm (lpMeasSubgroupToLpTrim F p \u03bc hm f)) =\u1d50[\u03bc] \u2191\u2191\u2191f\n[PROOFSTEP]\nrw [\u2190 lpMeasSubgroup_coe]\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191(lpTrimToLpMeasSubgroup F p \u03bc hm (lpMeasSubgroupToLpTrim F p \u03bc hm f)) =\u1d50[\u03bc] \u2191\u2191\u2191f\n[PROOFSTEP]\nexact (lpTrimToLpMeasSubgroup_ae_eq hm _).trans (lpMeasSubgroupToLpTrim_ae_eq hm _)\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 lpMeasSubgroupToLpTrim F p \u03bc hm (f + g) = lpMeasSubgroupToLpTrim F p \u03bc hm f + lpMeasSubgroupToLpTrim F p \u03bc hm g\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (f + g)) =\u1d50[Measure.trim \u03bc hm]\n \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f + lpMeasSubgroupToLpTrim F p \u03bc hm g)\n[PROOFSTEP]\nrefine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (f + g)) =\u1d50[Measure.trim \u03bc hm]\n \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g)\n[PROOFSTEP]\nrefine' ae_eq_trim_of_stronglyMeasurable hm (Lp.stronglyMeasurable _) _ _\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 StronglyMeasurable (\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g))\n[PROOFSTEP]\nexact (Lp.stronglyMeasurable _).add (Lp.stronglyMeasurable _)\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (f + g)) =\u1d50[\u03bc]\n \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g)\n[PROOFSTEP]\nrefine' (lpMeasSubgroupToLpTrim_ae_eq hm _).trans _\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191(f + g) =\u1d50[\u03bc] \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g)\n[PROOFSTEP]\nrefine'\n EventuallyEq.trans _\n (EventuallyEq.add (lpMeasSubgroupToLpTrim_ae_eq hm f).symm (lpMeasSubgroupToLpTrim_ae_eq hm g).symm)\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191(f + g) =\u1d50[\u03bc] fun x => \u2191\u2191\u2191f x + \u2191\u2191\u2191g x\n[PROOFSTEP]\nrefine' (Lp.coeFn_add _ _).trans _\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191f + \u2191\u2191\u2191g =\u1d50[\u03bc] fun x => \u2191\u2191\u2191f x + \u2191\u2191\u2191g x\n[PROOFSTEP]\nsimp_rw [lpMeasSubgroup_coe]\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191f + \u2191\u2191\u2191g =\u1d50[\u03bc] fun x => \u2191\u2191\u2191f x + \u2191\u2191\u2191g x\n[PROOFSTEP]\nexact eventually_of_forall fun x => by rfl\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\nx : \u03b1\n\u22a2 (\u2191\u2191\u2191f + \u2191\u2191\u2191g) x = (fun x => \u2191\u2191\u2191f x + \u2191\u2191\u2191g x) x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 lpMeasSubgroupToLpTrim F p \u03bc hm (-f) = -lpMeasSubgroupToLpTrim F p \u03bc hm f\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (-f)) =\u1d50[Measure.trim \u03bc hm] \u2191\u2191(-lpMeasSubgroupToLpTrim F p \u03bc hm f)\n[PROOFSTEP]\nrefine' EventuallyEq.trans _ (Lp.coeFn_neg _).symm\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (-f)) =\u1d50[Measure.trim \u03bc hm] -\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f)\n[PROOFSTEP]\nrefine' ae_eq_trim_of_stronglyMeasurable hm (Lp.stronglyMeasurable _) _ _\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 StronglyMeasurable (-\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f))\n[PROOFSTEP]\nexact @StronglyMeasurable.neg _ _ _ m _ _ _ (Lp.stronglyMeasurable _)\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (-f)) =\u1d50[\u03bc] -\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f)\n[PROOFSTEP]\nrefine' (lpMeasSubgroupToLpTrim_ae_eq hm _).trans _\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191(-f) =\u1d50[\u03bc] -\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f)\n[PROOFSTEP]\nrefine' EventuallyEq.trans _ (EventuallyEq.neg (lpMeasSubgroupToLpTrim_ae_eq hm f).symm)\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2191\u2191\u2191(-f) =\u1d50[\u03bc] fun x => -\u2191\u2191\u2191f x\n[PROOFSTEP]\nrefine' (Lp.coeFn_neg _).trans _\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 -\u2191\u2191\u2191f =\u1d50[\u03bc] fun x => -\u2191\u2191\u2191f x\n[PROOFSTEP]\nsimp_rw [lpMeasSubgroup_coe]\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 -\u2191\u2191\u2191f =\u1d50[\u03bc] fun x => -\u2191\u2191\u2191f x\n[PROOFSTEP]\nexact eventually_of_forall fun x => by rfl\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\nx : \u03b1\n\u22a2 (-\u2191\u2191\u2191f) x = (fun x => -\u2191\u2191\u2191f x) x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 lpMeasSubgroupToLpTrim F p \u03bc hm (f - g) = lpMeasSubgroupToLpTrim F p \u03bc hm f - lpMeasSubgroupToLpTrim F p \u03bc hm g\n[PROOFSTEP]\nrw [sub_eq_add_neg, sub_eq_add_neg, lpMeasSubgroupToLpTrim_add, lpMeasSubgroupToLpTrim_neg]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nc : \ud835\udd5c\nf : { x // x \u2208 lpMeas F \ud835\udd5c m p \u03bc }\n\u22a2 lpMeasToLpTrim F \ud835\udd5c p \u03bc hm (c \u2022 f) = c \u2022 lpMeasToLpTrim F \ud835\udd5c p \u03bc hm f\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nc : \ud835\udd5c\nf : { x // x \u2208 lpMeas F \ud835\udd5c m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasToLpTrim F \ud835\udd5c p \u03bc hm (c \u2022 f)) =\u1d50[Measure.trim \u03bc hm] \u2191\u2191(c \u2022 lpMeasToLpTrim F \ud835\udd5c p \u03bc hm f)\n[PROOFSTEP]\nrefine' EventuallyEq.trans _ (Lp.coeFn_smul _ _).symm\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nc : \ud835\udd5c\nf : { x // x \u2208 lpMeas F \ud835\udd5c m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasToLpTrim F \ud835\udd5c p \u03bc hm (c \u2022 f)) =\u1d50[Measure.trim \u03bc hm] c \u2022 \u2191\u2191(lpMeasToLpTrim F \ud835\udd5c p \u03bc hm f)\n[PROOFSTEP]\nrefine' ae_eq_trim_of_stronglyMeasurable hm (Lp.stronglyMeasurable _) _ _\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nc : \ud835\udd5c\nf : { x // x \u2208 lpMeas F \ud835\udd5c m p \u03bc }\n\u22a2 StronglyMeasurable (c \u2022 \u2191\u2191(lpMeasToLpTrim F \ud835\udd5c p \u03bc hm f))\n[PROOFSTEP]\nexact (Lp.stronglyMeasurable _).const_smul c\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nc : \ud835\udd5c\nf : { x // x \u2208 lpMeas F \ud835\udd5c m p \u03bc }\n\u22a2 \u2191\u2191(lpMeasToLpTrim F \ud835\udd5c p \u03bc hm (c \u2022 f)) =\u1d50[\u03bc] c \u2022 \u2191\u2191(lpMeasToLpTrim F \ud835\udd5c p \u03bc hm f)\n[PROOFSTEP]\nrefine' (lpMeasToLpTrim_ae_eq hm _).trans _\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nc : \ud835\udd5c\nf : { x // x \u2208 lpMeas F \ud835\udd5c m p \u03bc }\n\u22a2 \u2191\u2191\u2191(c \u2022 f) =\u1d50[\u03bc] c \u2022 \u2191\u2191(lpMeasToLpTrim F \ud835\udd5c p \u03bc hm f)\n[PROOFSTEP]\nrefine' (Lp.coeFn_smul _ _).trans _\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nc : \ud835\udd5c\nf : { x // x \u2208 lpMeas F \ud835\udd5c m p \u03bc }\n\u22a2 c \u2022 \u2191\u2191\u2191f =\u1d50[\u03bc] c \u2022 \u2191\u2191(lpMeasToLpTrim F \ud835\udd5c p \u03bc hm f)\n[PROOFSTEP]\nrefine' (lpMeasToLpTrim_ae_eq hm f).mono fun x hx => _\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nc : \ud835\udd5c\nf : { x // x \u2208 lpMeas F \ud835\udd5c m p \u03bc }\nx : \u03b1\nhx : \u2191\u2191(lpMeasToLpTrim F \ud835\udd5c p \u03bc hm f) x = \u2191\u2191\u2191f x\n\u22a2 (c \u2022 \u2191\u2191\u2191f) x = (c \u2022 \u2191\u2191(lpMeasToLpTrim F \ud835\udd5c p \u03bc hm f)) x\n[PROOFSTEP]\nrw [Pi.smul_apply, Pi.smul_apply, hx]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhp : Fact (1 \u2264 p)\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2016lpMeasSubgroupToLpTrim F p \u03bc hm f\u2016 = \u2016f\u2016\n[PROOFSTEP]\nrw [Lp.norm_def, snorm_trim hm (Lp.stronglyMeasurable _), snorm_congr_ae (lpMeasSubgroupToLpTrim_ae_eq hm _),\n lpMeasSubgroup_coe, \u2190 Lp.norm_def]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhp : Fact (1 \u2264 p)\nhm : m \u2264 m0\nf : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 \u2016\u2191f\u2016 = \u2016f\u2016\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhp : Fact (1 \u2264 p)\nhm : m \u2264 m0\nf g : { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n\u22a2 dist (lpMeasSubgroupToLpTrim F p \u03bc hm f) (lpMeasSubgroupToLpTrim F p \u03bc hm g) = dist f g\n[PROOFSTEP]\nrw [dist_eq_norm, \u2190 lpMeasSubgroupToLpTrim_sub, lpMeasSubgroupToLpTrim_norm_map, dist_eq_norm]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : Fact (m \u2264 m0)\ninst\u271d : CompleteSpace F\nhp : Fact (1 \u2264 p)\n\u22a2 CompleteSpace { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n[PROOFSTEP]\nrw [(lpMeasSubgroupToLpTrimIso F p \u03bc hm.elim).completeSpace_iff]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : Fact (m \u2264 m0)\ninst\u271d : CompleteSpace F\nhp : Fact (1 \u2264 p)\n\u22a2 CompleteSpace { x // x \u2208 Lp F p }\n[PROOFSTEP]\ninfer_instance\n -- For now just no-lint this; lean4's tree-based logging will make this easier to debug.\n -- One possible change might be to generalize `\ud835\udd5c` from `IsROrC` to `NormedField`, as this\n -- result may well hold there.\n -- Porting note: removed @[nolint fails_quickly]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : Fact (m \u2264 m0)\ninst\u271d : CompleteSpace F\nhp : Fact (1 \u2264 p)\n\u22a2 CompleteSpace { x // x \u2208 lpMeas F \ud835\udd5c m p \u03bc }\n[PROOFSTEP]\nrw [(lpMeasSubgroupToLpMeasIso F \ud835\udd5c p \u03bc).symm.completeSpace_iff]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : Fact (m \u2264 m0)\ninst\u271d : CompleteSpace F\nhp : Fact (1 \u2264 p)\n\u22a2 CompleteSpace { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhp : Fact (1 \u2264 p)\ninst\u271d : CompleteSpace F\nhm : m \u2264 m0\n\u22a2 IsComplete {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc}\n[PROOFSTEP]\nrw [\u2190 completeSpace_coe_iff_isComplete]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhp : Fact (1 \u2264 p)\ninst\u271d : CompleteSpace F\nhm : m \u2264 m0\n\u22a2 CompleteSpace \u2191{f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc}\n[PROOFSTEP]\nhaveI : Fact (m \u2264 m0) := \u27e8hm\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhp : Fact (1 \u2264 p)\ninst\u271d : CompleteSpace F\nhm : m \u2264 m0\nthis : Fact (m \u2264 m0)\n\u22a2 CompleteSpace \u2191{f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc}\n[PROOFSTEP]\nchange CompleteSpace (lpMeasSubgroup F m p \u03bc)\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\n\u03b9 : Type u_6\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhp : Fact (1 \u2264 p)\ninst\u271d : CompleteSpace F\nhm : m \u2264 m0\nthis : Fact (m \u2264 m0)\n\u22a2 CompleteSpace { x // x \u2208 lpMeasSubgroup F m p \u03bc }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\none_le_p : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\ns : Set \u03b1\n\u03bc : Measure \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s \u2260 \u22a4\nc : F\n\u22a2 \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (indicatorConstLp p hs h\u03bcs c)) =\n indicatorConstLp p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\none_le_p : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\ns : Set \u03b1\n\u03bc : Measure \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s \u2260 \u22a4\nc : F\n\u22a2 \u2191\u2191\u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (indicatorConstLp p hs h\u03bcs c)) =\u1d50[\u03bc]\n \u2191\u2191(indicatorConstLp p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\n[PROOFSTEP]\nrw [\u2190 lpMeas_coe]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\none_le_p : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\ns : Set \u03b1\n\u03bc : Measure \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s \u2260 \u22a4\nc : F\n\u22a2 \u2191\u2191\u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (indicatorConstLp p hs h\u03bcs c)) =\u1d50[\u03bc]\n \u2191\u2191(indicatorConstLp p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\n[PROOFSTEP]\nchange lpTrimToLpMeas F \u211d p \u03bc hm (indicatorConstLp p hs h\u03bcs c) =\u1d50[\u03bc] (indicatorConstLp p _ _ c : \u03b1 \u2192 F)\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\none_le_p : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\ns : Set \u03b1\n\u03bc : Measure \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s \u2260 \u22a4\nc : F\n\u22a2 \u2191\u2191\u2191(lpTrimToLpMeas F \u211d p \u03bc hm (indicatorConstLp p hs h\u03bcs c)) =\u1d50[\u03bc]\n \u2191\u2191(indicatorConstLp p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\n[PROOFSTEP]\nrefine' (lpTrimToLpMeas_ae_eq hm _).trans _\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\none_le_p : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\ns : Set \u03b1\n\u03bc : Measure \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s \u2260 \u22a4\nc : F\n\u22a2 \u2191\u2191(indicatorConstLp p hs h\u03bcs c) =\u1d50[\u03bc] \u2191\u2191(indicatorConstLp p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\n[PROOFSTEP]\nexact (ae_eq_of_ae_eq_trim indicatorConstLp_coeFn).trans indicatorConstLp_coeFn.symm\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\none_le_p : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\n\u22a2 \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) = Mem\u2112p.toLp f (_ : Mem\u2112p f p)\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\none_le_p : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\n\u22a2 \u2191\u2191\u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) =\u1d50[\u03bc]\n \u2191\u2191(Mem\u2112p.toLp f (_ : Mem\u2112p f p))\n[PROOFSTEP]\nrw [\u2190 lpMeas_coe]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\none_le_p : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\n\u22a2 \u2191\u2191\u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) =\u1d50[\u03bc]\n \u2191\u2191(Mem\u2112p.toLp f (_ : Mem\u2112p f p))\n[PROOFSTEP]\nrefine' (lpTrimToLpMeas_ae_eq hm _).trans _\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2078 : CompleteSpace E'\ninst\u271d\u2077 : NormedSpace \u211d E'\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\none_le_p : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\n\u22a2 \u2191\u2191(Mem\u2112p.toLp f hf) =\u1d50[\u03bc] \u2191\u2191(Mem\u2112p.toLp f (_ : Mem\u2112p f p))\n[PROOFSTEP]\nexact (ae_eq_of_ae_eq_trim (Mem\u2112p.coeFn_toLp hf)).trans (Mem\u2112p.coeFn_toLp _).symm\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\n\u22a2 \u2200 (f : { x // x \u2208 Lp F p }), AEStronglyMeasurable' m (\u2191\u2191f) \u03bc \u2192 P f\n[PROOFSTEP]\nintro f hf\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n\u22a2 P f\n[PROOFSTEP]\nlet f' := (\u27e8f, hf\u27e9 : lpMeas F \u211d m p \u03bc)\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\n\u22a2 P f\n[PROOFSTEP]\nlet g := lpMeasToLpTrimLie F \u211d p \u03bc hm f'\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\n\u22a2 P f\n[PROOFSTEP]\nhave hfg : f' = (lpMeasToLpTrimLie F \u211d p \u03bc hm).symm g := by simp only [LinearIsometryEquiv.symm_apply_apply]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\n\u22a2 f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n[PROOFSTEP]\nsimp only [LinearIsometryEquiv.symm_apply_apply]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 P f\n[PROOFSTEP]\nchange P \u2191f'\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 P \u2191f'\n[PROOFSTEP]\nrw [hfg]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g)\n[PROOFSTEP]\nrefine' @Lp.induction \u03b1 F m _ p (\u03bc.trim hm) _ hp_ne_top (fun g => P ((lpMeasToLpTrimLie F \u211d p \u03bc hm).symm g)) _ _ _ g\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191(Measure.trim \u03bc hm) s < \u22a4),\n (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g))\n \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191(Measure.trim \u03bc hm) s \u2260 \u22a4) c)\n[PROOFSTEP]\nintro b t ht h\u03bct\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\nb : F\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\n\u22a2 P\n \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm))\n \u2191(simpleFunc.indicatorConst p ht (_ : \u2191\u2191(Measure.trim \u03bc hm) t \u2260 \u22a4) b))\n[PROOFSTEP]\nrw [@Lp.simpleFunc.coe_indicatorConst _ _ m, lpMeasToLpTrimLie_symm_indicator ht h\u03bct.ne b]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\nb : F\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\n\u22a2 P (indicatorConstLp p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b)\n[PROOFSTEP]\nhave h\u03bct' : \u03bc t < \u221e := (le_trim hm).trans_lt h\u03bct\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\nb : F\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\nh\u03bct' : \u2191\u2191\u03bc t < \u22a4\n\u22a2 P (indicatorConstLp p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b)\n[PROOFSTEP]\nspecialize h_ind b ht h\u03bct'\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\nb : F\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\nh\u03bct' : \u2191\u2191\u03bc t < \u22a4\nh_ind : P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b)\n\u22a2 P (indicatorConstLp p (_ : MeasurableSet t) (_ : \u2191\u2191\u03bc t \u2260 \u22a4) b)\n[PROOFSTEP]\nrwa [Lp.simpleFunc.coe_indicatorConst] at h_ind \n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n Disjoint (Function.support f) (Function.support g) \u2192\n (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g)) (Mem\u2112p.toLp f hf) \u2192\n (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g)) (Mem\u2112p.toLp g hg) \u2192\n (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g))\n (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nintro f g hf hg h_disj hfP hgP\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf))\nhgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\n\u22a2 P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg))\n[PROOFSTEP]\nrw [LinearIsometryEquiv.map_add]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf))\nhgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\n\u22a2 P\n \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf) +\n \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf))\nhgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\n\u22a2 P\n (\u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) +\n \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg)))\n[PROOFSTEP]\nhave h_eq :\n \u2200 (f : \u03b1 \u2192 F) (hf : Mem\u2112p f p (\u03bc.trim hm)),\n ((lpMeasToLpTrimLie F \u211d p \u03bc hm).symm (Mem\u2112p.toLp f hf) : Lp F p \u03bc) = (mem\u2112p_of_mem\u2112p_trim hm hf).toLp f :=\n lpMeasToLpTrimLie_symm_toLp hm\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf))\nhgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\nh_eq :\n \u2200 (f : \u03b1 \u2192 F) (hf : Mem\u2112p f p),\n \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) = Mem\u2112p.toLp f (_ : Mem\u2112p f p)\n\u22a2 P\n (\u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) +\n \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg)))\n[PROOFSTEP]\nrw [h_eq f hf] at hfP \u22a2\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P (Mem\u2112p.toLp f (_ : Mem\u2112p f p))\nhgP : P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg))\nh_eq :\n \u2200 (f : \u03b1 \u2192 F) (hf : Mem\u2112p f p),\n \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) = Mem\u2112p.toLp f (_ : Mem\u2112p f p)\n\u22a2 P (Mem\u2112p.toLp f (_ : Mem\u2112p f p) + \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp g hg)))\n[PROOFSTEP]\nrw [h_eq g hg] at hgP \u22a2\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f\u271d, property := hf\u271d }\ng\u271d : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\u271d\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P (Mem\u2112p.toLp f (_ : Mem\u2112p f p))\nhgP : P (Mem\u2112p.toLp g (_ : Mem\u2112p g p))\nh_eq :\n \u2200 (f : \u03b1 \u2192 F) (hf : Mem\u2112p f p),\n \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) (Mem\u2112p.toLp f hf)) = Mem\u2112p.toLp f (_ : Mem\u2112p f p)\n\u22a2 P (Mem\u2112p.toLp f (_ : Mem\u2112p f p) + Mem\u2112p.toLp g (_ : Mem\u2112p g p))\n[PROOFSTEP]\nexact\n h_add (mem\u2112p_of_mem\u2112p_trim hm hf) (mem\u2112p_of_mem\u2112p_trim hm hg)\n (aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim hm hf.aestronglyMeasurable)\n (aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim hm hg.aestronglyMeasurable) h_disj hfP hgP\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 IsClosed {f | (fun g => P \u2191(\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g)) f}\n[PROOFSTEP]\nchange IsClosed ((lpMeasToLpTrimLie F \u211d p \u03bc hm).symm \u207b\u00b9' {g : lpMeas F \u211d m p \u03bc | P \u2191g})\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nf' : { x // x \u2208 lpMeas F \u211d m p \u03bc } := { val := f, property := hf }\ng : { x // x \u2208 Lp F p } := \u2191(lpMeasToLpTrimLie F \u211d p \u03bc hm) f'\nhfg : f' = \u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) g\n\u22a2 IsClosed (\u2191(LinearIsometryEquiv.symm (lpMeasToLpTrimLie F \u211d p \u03bc hm)) \u207b\u00b9' {g | P \u2191g})\n[PROOFSTEP]\nexact IsClosed.preimage (LinearIsometryEquiv.continuous _) h_closed\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\n\u22a2 \u2200 (f : { x // x \u2208 Lp F p }), AEStronglyMeasurable' m (\u2191\u2191f) \u03bc \u2192 P f\n[PROOFSTEP]\nintro f hf\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n\u22a2 P f\n[PROOFSTEP]\nsuffices h_add_ae :\n \u2200 \u2983f g\u2984,\n \u2200 hf : Mem\u2112p f p \u03bc,\n \u2200 hg : Mem\u2112p g p \u03bc,\n \u2200 _ : AEStronglyMeasurable' m f \u03bc,\n \u2200 _ : AEStronglyMeasurable' m g \u03bc,\n Disjoint (Function.support f) (Function.support g) \u2192\n P (hf.toLp f) \u2192\n P (hg.toLp g) \u2192\n P\n (hf.toLp f + hg.toLp g)\n -- Porting note: `P` should be an explicit argument to `Lp.induction_stronglyMeasurable_aux`, but\n -- it isn't?\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\nh_add_ae :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n\u22a2 P f\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nexact Lp.induction_stronglyMeasurable_aux hm hp_ne_top h_ind h_add_ae h_closed f hf\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf : { x // x \u2208 Lp F p }\nhf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n AEStronglyMeasurable' m f \u03bc \u2192\n AEStronglyMeasurable' m g \u03bc \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nintro f g hf hg hfm hgm h_disj hPf hPg\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nlet s_f : Set \u03b1 := Function.support (hfm.mk f)\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nhave hs_f : MeasurableSet[m] s_f := hfm.stronglyMeasurable_mk.measurableSet_support\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nhave hs_f_eq : s_f =\u1d50[\u03bc] Function.support f := hfm.ae_eq_mk.symm.support\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nlet s_g : Set \u03b1 := Function.support (hgm.mk g)\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nhave hs_g : MeasurableSet[m] s_g := hgm.stronglyMeasurable_mk.measurableSet_support\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nhave hs_g_eq : s_g =\u1d50[\u03bc] Function.support g := hgm.ae_eq_mk.symm.support\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nhave h_inter_empty : (s_f \u2229 s_g : Set \u03b1) =\u1d50[\u03bc] (\u2205 : Set \u03b1) :=\n by\n refine' (hs_f_eq.inter hs_g_eq).trans _\n suffices Function.support f \u2229 Function.support g = \u2205 by rw [this]\n exact Set.disjoint_iff_inter_eq_empty.mp h_disj\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\n\u22a2 s_f \u2229 s_g =\u1d50[\u03bc] \u2205\n[PROOFSTEP]\nrefine' (hs_f_eq.inter hs_g_eq).trans _\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\n\u22a2 Function.support f \u2229 Function.support g =\u1d50[\u03bc] \u2205\n[PROOFSTEP]\nsuffices Function.support f \u2229 Function.support g = \u2205 by rw [this]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nthis : Function.support f \u2229 Function.support g = \u2205\n\u22a2 Function.support f \u2229 Function.support g =\u1d50[\u03bc] \u2205\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\n\u22a2 Function.support f \u2229 Function.support g = \u2205\n[PROOFSTEP]\nexact Set.disjoint_iff_inter_eq_empty.mp h_disj\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nlet f' := (s_f \\ s_g).indicator (hfm.mk f)\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nhave hff' : f =\u1d50[\u03bc] f' :=\n by\n have : s_f \\ s_g =\u1d50[\u03bc] s_f := by\n rw [\u2190 Set.diff_inter_self_eq_diff, Set.inter_comm]\n refine' ((ae_eq_refl s_f).diff h_inter_empty).trans _\n rw [Set.diff_empty]\n refine' ((indicator_ae_eq_of_ae_eq_set this).trans _).symm\n rw [Set.indicator_support]\n exact hfm.ae_eq_mk.symm\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\n\u22a2 f =\u1d50[\u03bc] f'\n[PROOFSTEP]\nhave : s_f \\ s_g =\u1d50[\u03bc] s_f := by\n rw [\u2190 Set.diff_inter_self_eq_diff, Set.inter_comm]\n refine' ((ae_eq_refl s_f).diff h_inter_empty).trans _\n rw [Set.diff_empty]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\n\u22a2 s_f \\ s_g =\u1d50[\u03bc] s_f\n[PROOFSTEP]\nrw [\u2190 Set.diff_inter_self_eq_diff, Set.inter_comm]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\n\u22a2 s_f \\ (s_f \u2229 s_g) =\u1d50[\u03bc] s_f\n[PROOFSTEP]\nrefine' ((ae_eq_refl s_f).diff h_inter_empty).trans _\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\n\u22a2 s_f \\ \u2205 =\u1d50[\u03bc] s_f\n[PROOFSTEP]\nrw [Set.diff_empty]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nthis : s_f \\ s_g =\u1d50[\u03bc] s_f\n\u22a2 f =\u1d50[\u03bc] f'\n[PROOFSTEP]\nrefine' ((indicator_ae_eq_of_ae_eq_set this).trans _).symm\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nthis : s_f \\ s_g =\u1d50[\u03bc] s_f\n\u22a2 Set.indicator s_f (AEStronglyMeasurable'.mk f hfm) =\u1d50[\u03bc] f\n[PROOFSTEP]\nrw [Set.indicator_support]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nthis : s_f \\ s_g =\u1d50[\u03bc] s_f\n\u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[\u03bc] f\n[PROOFSTEP]\nexact hfm.ae_eq_mk.symm\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nhave hf'_meas : StronglyMeasurable[m] f' := hfm.stronglyMeasurable_mk.indicator (hs_f.diff hs_g)\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nhave hf'_Lp : Mem\u2112p f' p \u03bc := hf.ae_eq hff'\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nlet g' := (s_g \\ s_f).indicator (hgm.mk g)\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nhave hgg' : g =\u1d50[\u03bc] g' :=\n by\n have : s_g \\ s_f =\u1d50[\u03bc] s_g := by\n rw [\u2190 Set.diff_inter_self_eq_diff]\n refine' ((ae_eq_refl s_g).diff h_inter_empty).trans _\n rw [Set.diff_empty]\n refine' ((indicator_ae_eq_of_ae_eq_set this).trans _).symm\n rw [Set.indicator_support]\n exact hgm.ae_eq_mk.symm\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\n\u22a2 g =\u1d50[\u03bc] g'\n[PROOFSTEP]\nhave : s_g \\ s_f =\u1d50[\u03bc] s_g := by\n rw [\u2190 Set.diff_inter_self_eq_diff]\n refine' ((ae_eq_refl s_g).diff h_inter_empty).trans _\n rw [Set.diff_empty]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\n\u22a2 s_g \\ s_f =\u1d50[\u03bc] s_g\n[PROOFSTEP]\nrw [\u2190 Set.diff_inter_self_eq_diff]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\n\u22a2 s_g \\ (s_f \u2229 s_g) =\u1d50[\u03bc] s_g\n[PROOFSTEP]\nrefine' ((ae_eq_refl s_g).diff h_inter_empty).trans _\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\n\u22a2 s_g \\ \u2205 =\u1d50[\u03bc] s_g\n[PROOFSTEP]\nrw [Set.diff_empty]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nthis : s_g \\ s_f =\u1d50[\u03bc] s_g\n\u22a2 g =\u1d50[\u03bc] g'\n[PROOFSTEP]\nrefine' ((indicator_ae_eq_of_ae_eq_set this).trans _).symm\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nthis : s_g \\ s_f =\u1d50[\u03bc] s_g\n\u22a2 Set.indicator s_g (AEStronglyMeasurable'.mk g hgm) =\u1d50[\u03bc] g\n[PROOFSTEP]\nrw [Set.indicator_support]\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nthis : s_g \\ s_f =\u1d50[\u03bc] s_g\n\u22a2 AEStronglyMeasurable'.mk g hgm =\u1d50[\u03bc] g\n[PROOFSTEP]\nexact hgm.ae_eq_mk.symm\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nhave hg'_meas : StronglyMeasurable[m] g' := hgm.stronglyMeasurable_mk.indicator (hs_g.diff hs_f)\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\nhg'_meas : StronglyMeasurable g'\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nhave hg'_Lp : Mem\u2112p g' p \u03bc := hg.ae_eq hgg'\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\nhg'_meas : StronglyMeasurable g'\nhg'_Lp : Mem\u2112p g' p\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nhave h_disj : Disjoint (Function.support f') (Function.support g') :=\n haveI : Disjoint (s_f \\ s_g) (s_g \\ s_f) := disjoint_sdiff_sdiff\n this.mono Set.support_indicator_subset Set.support_indicator_subset\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj\u271d : Disjoint (Function.support f) (Function.support g)\nhPf : P (Mem\u2112p.toLp f hf)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\nhg'_meas : StronglyMeasurable g'\nhg'_Lp : Mem\u2112p g' p\nh_disj : Disjoint (Function.support f') (Function.support g')\n\u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nrw [\u2190 Mem\u2112p.toLp_congr hf'_Lp hf hff'.symm] at hPf \u22a2\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj\u271d : Disjoint (Function.support f) (Function.support g)\nhPg : P (Mem\u2112p.toLp g hg)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\nhPf : P (Mem\u2112p.toLp f' hf'_Lp)\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\nhg'_meas : StronglyMeasurable g'\nhg'_Lp : Mem\u2112p g' p\nh_disj : Disjoint (Function.support f') (Function.support g')\n\u22a2 P (Mem\u2112p.toLp f' hf'_Lp + Mem\u2112p.toLp g hg)\n[PROOFSTEP]\nrw [\u2190 Mem\u2112p.toLp_congr hg'_Lp hg hgg'.symm] at hPg \u22a2\n[GOAL]\ncase h_add_ae\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp F p } \u2192 Prop\nh_ind :\n \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P \u2191f}\nf\u271d : { x // x \u2208 Lp F p }\nhf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc\nf g : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhg : Mem\u2112p g p\nhfm : AEStronglyMeasurable' m f \u03bc\nhgm : AEStronglyMeasurable' m g \u03bc\nh_disj\u271d : Disjoint (Function.support f) (Function.support g)\ns_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm)\nhs_f : MeasurableSet s_f\nhs_f_eq : s_f =\u1d50[\u03bc] Function.support f\ns_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm)\nhs_g : MeasurableSet s_g\nhs_g_eq : s_g =\u1d50[\u03bc] Function.support g\nh_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205\nf' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm)\nhff' : f =\u1d50[\u03bc] f'\nhf'_meas : StronglyMeasurable f'\nhf'_Lp : Mem\u2112p f' p\nhPf : P (Mem\u2112p.toLp f' hf'_Lp)\ng' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm)\nhgg' : g =\u1d50[\u03bc] g'\nhg'_meas : StronglyMeasurable g'\nhg'_Lp : Mem\u2112p g' p\nhPg : P (Mem\u2112p.toLp g' hg'_Lp)\nh_disj : Disjoint (Function.support f') (Function.support g')\n\u22a2 P (Mem\u2112p.toLp f' hf'_Lp + Mem\u2112p.toLp g' hg'_Lp)\n[PROOFSTEP]\nexact h_add hf'_Lp hg'_Lp hf'_meas hg'_meas h_disj hPf hPg\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n Disjoint (Function.support f) (Function.support g) \u2192\n Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\n\u22a2 \u2200 \u2983f : \u03b1 \u2192 F\u2984, Mem\u2112p f p \u2192 AEStronglyMeasurable' m f \u03bc \u2192 P f\n[PROOFSTEP]\nintro f hf hfm\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n Disjoint (Function.support f) (Function.support g) \u2192\n Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\n\u22a2 P f\n[PROOFSTEP]\nlet f_Lp := hf.toLp f\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n Disjoint (Function.support f) (Function.support g) \u2192\n Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\n\u22a2 P f\n[PROOFSTEP]\nhave hfm_Lp : AEStronglyMeasurable' m f_Lp \u03bc := hfm.congr hf.coeFn_toLp.symm\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n Disjoint (Function.support f) (Function.support g) \u2192\n Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\n\u22a2 P f\n[PROOFSTEP]\nrefine' h_ae hf.coeFn_toLp (Lp.mem\u2112p _) _\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n Disjoint (Function.support f) (Function.support g) \u2192\n Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\n\u22a2 P \u2191\u2191(toLp f hf)\n[PROOFSTEP]\nchange P f_Lp\n[GOAL]\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n Disjoint (Function.support f) (Function.support g) \u2192\n Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\n\u22a2 P \u2191\u2191f_Lp\n[PROOFSTEP]\nrefine' Lp.induction_stronglyMeasurable hm hp_ne_top (P := fun f => P f) _ _ h_closed f_Lp hfm_Lp\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n Disjoint (Function.support f) (Function.support g) \u2192\n Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\n\u22a2 \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),\n (fun f => P \u2191\u2191f) \u2191(Lp.simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\n[PROOFSTEP]\nintro c s hs h\u03bcs\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n Disjoint (Function.support f) (Function.support g) \u2192\n Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nc : F\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 P \u2191\u2191\u2191(Lp.simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\n[PROOFSTEP]\nrw [Lp.simpleFunc.coe_indicatorConst]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n Disjoint (Function.support f) (Function.support g) \u2192\n Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nc : F\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 P \u2191\u2191(indicatorConstLp p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\n[PROOFSTEP]\nrefine' h_ae indicatorConstLp_coeFn.symm _ (h_ind c hs h\u03bcs)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n Disjoint (Function.support f) (Function.support g) \u2192\n Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nc : F\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 Mem\u2112p (Set.indicator s fun x => c) p\n[PROOFSTEP]\nexact mem\u2112p_indicator_const p (hm s hs) c (Or.inr h\u03bcs.ne)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n Disjoint (Function.support f) (Function.support g) \u2192\n Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf : \u03b1 \u2192 F\nhf : Mem\u2112p f p\nhfm : AEStronglyMeasurable' m f \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n StronglyMeasurable f \u2192\n StronglyMeasurable g \u2192\n Disjoint (Function.support f) (Function.support g) \u2192\n (fun f => P \u2191\u2191f) (toLp f hf) \u2192 (fun f => P \u2191\u2191f) (toLp g hg) \u2192 (fun f => P \u2191\u2191f) (toLp f hf + toLp g hg)\n[PROOFSTEP]\nintro f g hf_mem hg_mem hfm hgm h_disj hfP hgP\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n Disjoint (Function.support f) (Function.support g) \u2192\n Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf\u271d : \u03b1 \u2192 F\nhf : Mem\u2112p f\u271d p\nhfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nf g : \u03b1 \u2192 F\nhf_mem : Mem\u2112p f p\nhg_mem : Mem\u2112p g p\nhfm : StronglyMeasurable f\nhgm : StronglyMeasurable g\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191\u2191(toLp f hf_mem)\nhgP : P \u2191\u2191(toLp g hg_mem)\n\u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem)\n[PROOFSTEP]\nhave hfP' : P f := h_ae hf_mem.coeFn_toLp (Lp.mem\u2112p _) hfP\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n Disjoint (Function.support f) (Function.support g) \u2192\n Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf\u271d : \u03b1 \u2192 F\nhf : Mem\u2112p f\u271d p\nhfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nf g : \u03b1 \u2192 F\nhf_mem : Mem\u2112p f p\nhg_mem : Mem\u2112p g p\nhfm : StronglyMeasurable f\nhgm : StronglyMeasurable g\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191\u2191(toLp f hf_mem)\nhgP : P \u2191\u2191(toLp g hg_mem)\nhfP' : P f\n\u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem)\n[PROOFSTEP]\nhave hgP' : P g := h_ae hg_mem.coeFn_toLp (Lp.mem\u2112p _) hgP\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add :\n \u2200 \u2983f g : \u03b1 \u2192 F\u2984,\n Disjoint (Function.support f) (Function.support g) \u2192\n Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 StronglyMeasurable f \u2192 StronglyMeasurable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf\u271d : \u03b1 \u2192 F\nhf : Mem\u2112p f\u271d p\nhfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nf g : \u03b1 \u2192 F\nhf_mem : Mem\u2112p f p\nhg_mem : Mem\u2112p g p\nhfm : StronglyMeasurable f\nhgm : StronglyMeasurable g\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191\u2191(toLp f hf_mem)\nhgP : P \u2191\u2191(toLp g hg_mem)\nhfP' : P f\nhgP' : P g\n\u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem)\n[PROOFSTEP]\nspecialize h_add h_disj hf_mem hg_mem hfm hgm hfP' hgP'\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf\u271d : \u03b1 \u2192 F\nhf : Mem\u2112p f\u271d p\nhfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nf g : \u03b1 \u2192 F\nhf_mem : Mem\u2112p f p\nhg_mem : Mem\u2112p g p\nhfm : StronglyMeasurable f\nhgm : StronglyMeasurable g\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191\u2191(toLp f hf_mem)\nhgP : P \u2191\u2191(toLp g hg_mem)\nhfP' : P f\nhgP' : P g\nh_add : P (f + g)\n\u22a2 P \u2191\u2191(toLp f hf_mem + toLp g hg_mem)\n[PROOFSTEP]\nrefine' h_ae _ (hf_mem.add hg_mem) h_add\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE' : Type u_2\nF : Type u_3\nF' : Type u_4\n\ud835\udd5c : Type u_5\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2079 : CompleteSpace E'\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b2 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : Fact (1 \u2264 p)\ninst\u271d : NormedSpace \u211d F\nhm : m \u2264 m0\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 F) \u2192 Prop\nh_ind : \u2200 (c : F) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_closed : IsClosed {f | P \u2191\u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 F\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf\u271d : \u03b1 \u2192 F\nhf : Mem\u2112p f\u271d p\nhfm\u271d : AEStronglyMeasurable' m f\u271d \u03bc\nf_Lp : { x // x \u2208 Lp F p } := toLp f\u271d hf\nhfm_Lp : AEStronglyMeasurable' m (\u2191\u2191f_Lp) \u03bc\nf g : \u03b1 \u2192 F\nhf_mem : Mem\u2112p f p\nhg_mem : Mem\u2112p g p\nhfm : StronglyMeasurable f\nhgm : StronglyMeasurable g\nh_disj : Disjoint (Function.support f) (Function.support g)\nhfP : P \u2191\u2191(toLp f hf_mem)\nhgP : P \u2191\u2191(toLp g hg_mem)\nhfP' : P f\nhgP' : P g\nh_add : P (f + g)\n\u22a2 f + g =\u1d50[\u03bc] \u2191\u2191(toLp f hf_mem + toLp g hg_mem)\n[PROOFSTEP]\nexact (hf_mem.coeFn_toLp.symm.add hg_mem.coeFn_toLp.symm).trans (Lp.coeFn_add _ _).symm\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "llama_tokens": 100313, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4098422930795862}} {"text": "[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nx y : R\nhle : x \u2264 y\nz : R\n\u22a2 z + x \u2264 z + y\n[PROOFSTEP]\nrw [StarOrderedRing.le_iff] at hle \u22a2\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nx y : R\nhle\u271d : x \u2264 y\nhle : \u2203 p, p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 y = x + p\nz : R\n\u22a2 \u2203 p, p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 z + y = z + x + p\n[PROOFSTEP]\nrefine hle.imp fun s hs \u21a6 ?_\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nx y : R\nhle\u271d : x \u2264 y\nhle : \u2203 p, p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 y = x + p\nz s : R\nhs : s \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 y = x + s\n\u22a2 s \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 z + y = z + x + s\n[PROOFSTEP]\nrw [hs.2, add_assoc]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nx y : R\nhle\u271d : x \u2264 y\nhle : \u2203 p, p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 y = x + p\nz s : R\nhs : s \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 y = x + s\n\u22a2 s \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 z + (x + s) = z + (x + s)\n[PROOFSTEP]\nexact \u27e8hs.1, rfl\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarRing R\nh_le_iff : \u2200 (x y : R), x \u2264 y \u2194 \u2203 s, y = x + star s * s\nx y : R\n\u22a2 x \u2264 y \u2194 \u2203 p, p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 y = x + p\n[PROOFSTEP]\nrefine' \u27e8fun h => _, _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarRing R\nh_le_iff : \u2200 (x y : R), x \u2264 y \u2194 \u2203 s, y = x + star s * s\nx y : R\nh : x \u2264 y\n\u22a2 \u2203 p, p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 y = x + p\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := (h_le_iff x y).mp h\n[GOAL]\ncase refine'_1.intro\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarRing R\nh_le_iff : \u2200 (x y : R), x \u2264 y \u2194 \u2203 s, y = x + star s * s\nx y : R\nh : x \u2264 y\np : R\nhp : y = x + star p * p\n\u22a2 \u2203 p, p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 y = x + p\n[PROOFSTEP]\nexact \u27e8star p * p, AddSubmonoid.subset_closure \u27e8p, rfl\u27e9, hp\u27e9\n[GOAL]\ncase refine'_2\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarRing R\nh_le_iff : \u2200 (x y : R), x \u2264 y \u2194 \u2203 s, y = x + star s * s\nx y : R\n\u22a2 (\u2203 p, p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 y = x + p) \u2192 x \u2264 y\n[PROOFSTEP]\nrintro \u27e8p, hp, hpxy\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarRing R\nh_le_iff : \u2200 (x y : R), x \u2264 y \u2194 \u2203 s, y = x + star s * s\nx y p : R\nhp : p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\nhpxy : y = x + p\n\u22a2 x \u2264 y\n[PROOFSTEP]\nrevert x y hpxy\n[GOAL]\ncase refine'_2.intro.intro\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarRing R\nh_le_iff : \u2200 (x y : R), x \u2264 y \u2194 \u2203 s, y = x + star s * s\np : R\nhp : p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\n\u22a2 \u2200 (x y : R), y = x + p \u2192 x \u2264 y\n[PROOFSTEP]\nrefine' AddSubmonoid.closure_induction hp _ (fun x y h => add_zero x \u25b8 h.ge) _\n[GOAL]\ncase refine'_2.intro.intro.refine'_1\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarRing R\nh_le_iff : \u2200 (x y : R), x \u2264 y \u2194 \u2203 s, y = x + star s * s\np : R\nhp : p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\n\u22a2 \u2200 (x : R), (x \u2208 Set.range fun s => star s * s) \u2192 \u2200 (x_1 y : R), y = x_1 + x \u2192 x_1 \u2264 y\n[PROOFSTEP]\nrintro _ \u27e8s, rfl\u27e9 x y rfl\n[GOAL]\ncase refine'_2.intro.intro.refine'_1.intro\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarRing R\nh_le_iff : \u2200 (x y : R), x \u2264 y \u2194 \u2203 s, y = x + star s * s\np : R\nhp : p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\ns x : R\n\u22a2 x \u2264 x + (fun s => star s * s) s\n[PROOFSTEP]\nexact (h_le_iff _ _).mpr \u27e8s, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.refine'_2\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarRing R\nh_le_iff : \u2200 (x y : R), x \u2264 y \u2194 \u2203 s, y = x + star s * s\np : R\nhp : p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\n\u22a2 \u2200 (x y : R),\n (\u2200 (x_1 y : R), y = x_1 + x \u2192 x_1 \u2264 y) \u2192\n (\u2200 (x y_1 : R), y_1 = x + y \u2192 x \u2264 y_1) \u2192 \u2200 (x_1 y_1 : R), y_1 = x_1 + (x + y) \u2192 x_1 \u2264 y_1\n[PROOFSTEP]\nrintro a b ha hb x y rfl\n[GOAL]\ncase refine'_2.intro.intro.refine'_2\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarRing R\nh_le_iff : \u2200 (x y : R), x \u2264 y \u2194 \u2203 s, y = x + star s * s\np : R\nhp : p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\na b : R\nha : \u2200 (x y : R), y = x + a \u2192 x \u2264 y\nhb : \u2200 (x y : R), y = x + b \u2192 x \u2264 y\nx : R\n\u22a2 x \u2264 x + (a + b)\n[PROOFSTEP]\nrw [\u2190 add_assoc]\n[GOAL]\ncase refine'_2.intro.intro.refine'_2\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarRing R\nh_le_iff : \u2200 (x y : R), x \u2264 y \u2194 \u2203 s, y = x + star s * s\np : R\nhp : p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\na b : R\nha : \u2200 (x y : R), y = x + a \u2192 x \u2264 y\nhb : \u2200 (x y : R), y = x + b \u2192 x \u2264 y\nx : R\n\u22a2 x \u2264 x + a + b\n[PROOFSTEP]\nexact (ha _ _ rfl).trans (hb _ _ rfl)\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalRing R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarRing R\nh_add : \u2200 {x y : R}, x \u2264 y \u2192 \u2200 (z : R), z + x \u2264 z + y\nh_nonneg_iff : \u2200 (x : R), 0 \u2264 x \u2194 x \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\nx y : R\n\u22a2 x \u2264 y \u2194 \u2203 p, p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 y = x + p\n[PROOFSTEP]\nhaveI : CovariantClass R R (\u00b7 + \u00b7) (\u00b7 \u2264 \u00b7) := \u27e8fun _ _ _ h => h_add h _\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalRing R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarRing R\nh_add : \u2200 {x y : R}, x \u2264 y \u2192 \u2200 (z : R), z + x \u2264 z + y\nh_nonneg_iff : \u2200 (x : R), 0 \u2264 x \u2194 x \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\nx y : R\nthis : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 x \u2264 y \u2194 \u2203 p, p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 y = x + p\n[PROOFSTEP]\nsimpa only [\u2190 sub_eq_iff_eq_add', sub_nonneg, exists_eq_right'] using h_nonneg_iff (y - x)\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalRing R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarRing R\nh_add : \u2200 {x y : R}, x \u2264 y \u2192 \u2200 (z : R), z + x \u2264 z + y\nh_nonneg_iff : \u2200 (x : R), 0 \u2264 x \u2194 \u2203 s, x = star s * s\n\u22a2 \u2200 (x y : R), x \u2264 y \u2194 \u2203 s, y = x + star s * s\n[PROOFSTEP]\nhaveI : CovariantClass R R (\u00b7 + \u00b7) (\u00b7 \u2264 \u00b7) := \u27e8fun _ _ _ h => h_add h _\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalRing R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarRing R\nh_add : \u2200 {x y : R}, x \u2264 y \u2192 \u2200 (z : R), z + x \u2264 z + y\nh_nonneg_iff : \u2200 (x : R), 0 \u2264 x \u2194 \u2203 s, x = star s * s\nthis : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 \u2200 (x y : R), x \u2264 y \u2194 \u2203 s, y = x + star s * s\n[PROOFSTEP]\nsimpa [sub_eq_iff_eq_add', sub_nonneg] using fun x y => h_nonneg_iff (y - x)\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nx : R\n\u22a2 0 \u2264 x \u2194 x \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\n[PROOFSTEP]\nsimp only [le_iff, zero_add, exists_eq_right']\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nr : R\n\u22a2 0 \u2264 r * star r\n[PROOFSTEP]\nsimpa only [star_star] using star_mul_self_nonneg (star r)\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na : R\nha : 0 \u2264 a\nc : R\n\u22a2 0 \u2264 star c * a * c\n[PROOFSTEP]\nrw [StarOrderedRing.nonneg_iff] at ha \n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na : R\nha\u271d : 0 \u2264 a\nha : a \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\nc : R\n\u22a2 0 \u2264 star c * a * c\n[PROOFSTEP]\nrefine' AddSubmonoid.closure_induction ha (fun x hx => _) (by rw [mul_zero, zero_mul]) fun x y hx hy => _\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na : R\nha\u271d : 0 \u2264 a\nha : a \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\nc : R\n\u22a2 0 \u2264 star c * 0 * c\n[PROOFSTEP]\nrw [mul_zero, zero_mul]\n[GOAL]\ncase refine'_1\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na : R\nha\u271d : 0 \u2264 a\nha : a \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\nc x : R\nhx : x \u2208 Set.range fun s => star s * s\n\u22a2 0 \u2264 star c * x * c\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 := hx\n[GOAL]\ncase refine'_1.intro\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na : R\nha\u271d : 0 \u2264 a\nha : a \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\nc x : R\n\u22a2 0 \u2264 star c * (fun s => star s * s) x * c\n[PROOFSTEP]\nconvert star_mul_self_nonneg (x * c) using 1\n[GOAL]\ncase h.e'_4\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na : R\nha\u271d : 0 \u2264 a\nha : a \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\nc x : R\n\u22a2 star c * (fun s => star s * s) x * c = star (x * c) * (x * c)\n[PROOFSTEP]\nrw [star_mul, \u2190 mul_assoc, mul_assoc _ _ c]\n[GOAL]\ncase refine'_2\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na : R\nha\u271d : 0 \u2264 a\nha : a \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\nc x y : R\nhx : 0 \u2264 star c * x * c\nhy : 0 \u2264 star c * y * c\n\u22a2 0 \u2264 star c * (x + y) * c\n[PROOFSTEP]\ncalc\n 0 \u2264 star c * x * c + 0 := by rw [add_zero]; exact hx\n _ \u2264 star c * x * c + star c * y * c := (add_le_add_left hy _)\n _ \u2264 _ := by rw [mul_add, add_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na : R\nha\u271d : 0 \u2264 a\nha : a \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\nc x y : R\nhx : 0 \u2264 star c * x * c\nhy : 0 \u2264 star c * y * c\n\u22a2 0 \u2264 star c * x * c + 0\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na : R\nha\u271d : 0 \u2264 a\nha : a \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\nc x y : R\nhx : 0 \u2264 star c * x * c\nhy : 0 \u2264 star c * y * c\n\u22a2 0 \u2264 star c * x * c\n[PROOFSTEP]\nexact hx\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na : R\nha\u271d : 0 \u2264 a\nha : a \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\nc x y : R\nhx : 0 \u2264 star c * x * c\nhy : 0 \u2264 star c * y * c\n\u22a2 star c * x * c + star c * y * c \u2264 star c * (x + y) * c\n[PROOFSTEP]\nrw [mul_add, add_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na : R\nha : 0 \u2264 a\nc : R\n\u22a2 0 \u2264 c * a * star c\n[PROOFSTEP]\nsimpa only [star_star] using conjugate_nonneg ha (star c)\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na b : R\nhab : a \u2264 b\nc : R\n\u22a2 star c * a * c \u2264 star c * b * c\n[PROOFSTEP]\nrw [StarOrderedRing.le_iff] at hab \u22a2\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na b : R\nhab\u271d : a \u2264 b\nhab : \u2203 p, p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 b = a + p\nc : R\n\u22a2 \u2203 p, p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 star c * b * c = star c * a * c + p\n[PROOFSTEP]\nobtain \u27e8p, hp, rfl\u27e9 := hab\n[GOAL]\ncase intro.intro\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na c p : R\nhp : p \u2208 AddSubmonoid.closure (Set.range fun s => star s * s)\nhab : a \u2264 a + p\n\u22a2 \u2203 p_1, p_1 \u2208 AddSubmonoid.closure (Set.range fun s => star s * s) \u2227 star c * (a + p) * c = star c * a * c + p_1\n[PROOFSTEP]\nsimp_rw [\u2190 StarOrderedRing.nonneg_iff] at hp \u22a2\n[GOAL]\ncase intro.intro\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na c p : R\nhab : a \u2264 a + p\nhp : 0 \u2264 p\n\u22a2 \u2203 p_1, 0 \u2264 p_1 \u2227 star c * (a + p) * c = star c * a * c + p_1\n[PROOFSTEP]\nexact \u27e8star c * p * c, conjugate_nonneg hp c, by simp only [add_mul, mul_add]\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na c p : R\nhab : a \u2264 a + p\nhp : 0 \u2264 p\n\u22a2 star c * (a + p) * c = star c * a * c + star c * p * c\n[PROOFSTEP]\nsimp only [add_mul, mul_add]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : NonUnitalSemiring R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\na b : R\nhab : a \u2264 b\nc : R\n\u22a2 c * a * star c \u2264 c * b * star c\n[PROOFSTEP]\nsimpa only [star_star] using conjugate_le_conjugate hab (star c)\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Star.Order", "llama_tokens": 6101, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.6224593241981982, "lm_q1q2_score": 0.409838112439892}} {"text": "[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n[PROOFSTEP]\nobtain \u27e8u, u_open, x\u2080u, hu\u27e9 : \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 x \u2208 u \u2229 s, g x \u2208 ball (g x\u2080) 1\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n[PROOFSTEP]\nexact mem_nhdsWithin.1 (hcg (ball_mem_nhds _ zero_lt_one))\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n[PROOFSTEP]\nfilter_upwards [tendstoUniformlyOn_iff.1 (hl\u03c6 u u_open x\u2080u) 1 zero_lt_one, hi\u03c6] with i hi h'i\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n[PROOFSTEP]\nhave A : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u) \u03bc :=\n by\n refine' Integrable.smul_of_top_right (hmg.mono (diff_subset _ _) le_rfl) _\n apply mem\u2112p_top_of_bound ((integrable_of_integral_eq_one h'i).aestronglyMeasurable.mono_set (diff_subset _ _)) 1\n filter_upwards [self_mem_ae_restrict (hs.diff u_open.measurableSet)] with x hx\n simpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).le\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n[PROOFSTEP]\nrefine' Integrable.smul_of_top_right (hmg.mono (diff_subset _ _) le_rfl) _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\n\u22a2 Mem\u2112p (fun x => \u03c6 i x) \u22a4\n[PROOFSTEP]\napply mem\u2112p_top_of_bound ((integrable_of_integral_eq_one h'i).aestronglyMeasurable.mono_set (diff_subset _ _)) 1\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s \\ u), \u2016\u03c6 i x\u2016 \u2264 1\n[PROOFSTEP]\nfilter_upwards [self_mem_ae_restrict (hs.diff u_open.measurableSet)] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2016\u03c6 i x\u2016 \u2264 1\n[PROOFSTEP]\nsimpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).le\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n[PROOFSTEP]\nhave B : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \u2229 u) \u03bc :=\n by\n apply Integrable.smul_of_top_left\n \u00b7 exact IntegrableOn.mono_set (integrable_of_integral_eq_one h'i) (inter_subset_left _ _)\n \u00b7 apply mem\u2112p_top_of_bound (hmg.mono_set (inter_subset_left _ _)).aestronglyMeasurable (\u2016g x\u2080\u2016 + 1)\n filter_upwards [self_mem_ae_restrict (hs.inter u_open.measurableSet)] with x hx\n rw [inter_comm] at hx \n exact (norm_lt_of_mem_ball (hu x hx)).le\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \u2229 u)\n[PROOFSTEP]\napply Integrable.smul_of_top_left\n[GOAL]\ncase h\u03c6\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 Integrable fun x => \u03c6 i x\n[PROOFSTEP]\nexact IntegrableOn.mono_set (integrable_of_integral_eq_one h'i) (inter_subset_left _ _)\n[GOAL]\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 Mem\u2112p (fun x => g x) \u22a4\n[PROOFSTEP]\napply mem\u2112p_top_of_bound (hmg.mono_set (inter_subset_left _ _)).aestronglyMeasurable (\u2016g x\u2080\u2016 + 1)\n[GOAL]\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s \u2229 u), \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1\n[PROOFSTEP]\nfilter_upwards [self_mem_ae_restrict (hs.inter u_open.measurableSet)] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1\n[PROOFSTEP]\nrw [inter_comm] at hx \n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nx : \u03b1\nhx : x \u2208 u \u2229 s\n\u22a2 \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1\n[PROOFSTEP]\nexact (norm_lt_of_mem_ball (hu x hx)).le\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nB : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \u2229 u)\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n[PROOFSTEP]\nconvert A.union B\n[GOAL]\ncase h.e'_6\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nB : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \u2229 u)\n\u22a2 s = s \\ u \u222a s \u2229 u\n[PROOFSTEP]\nsimp only [diff_union_inter]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) l (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' Metric.tendsto_nhds.2 fun \u03b5 \u03b5pos => _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u222b (x_1 : \u03b1) in s, \u03c6 x x_1 \u2022 g x_1 \u2202\u03bc) 0 < \u03b5\n[PROOFSTEP]\nobtain \u27e8\u03b4, h\u03b4, \u03b4pos\u27e9 : \u2203 \u03b4, (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4 < \u03b5 \u2227 0 < \u03b4 :=\n by\n have A : Tendsto (fun \u03b4 => (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4) (\ud835\udcdd[>] 0) (\ud835\udcdd ((0 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + 0)) :=\n by\n apply Tendsto.mono_left _ nhdsWithin_le_nhds\n exact (tendsto_id.mul tendsto_const_nhds).add tendsto_id\n rw [zero_mul, zero_add] at A \n exact (((tendsto_order.1 A).2 \u03b5 \u03b5pos).and self_mem_nhdsWithin).exists\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 \u03b4, \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u2227 0 < \u03b4\n[PROOFSTEP]\nhave A : Tendsto (fun \u03b4 => (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4) (\ud835\udcdd[>] 0) (\ud835\udcdd ((0 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + 0)) :=\n by\n apply Tendsto.mono_left _ nhdsWithin_le_nhds\n exact (tendsto_id.mul tendsto_const_nhds).add tendsto_id\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 Tendsto (fun \u03b4 => \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + 0))\n[PROOFSTEP]\napply Tendsto.mono_left _ nhdsWithin_le_nhds\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 Tendsto (fun \u03b4 => \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4) (\ud835\udcdd 0) (\ud835\udcdd (0 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + 0))\n[PROOFSTEP]\nexact (tendsto_id.mul tendsto_const_nhds).add tendsto_id\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nA : Tendsto (fun \u03b4 => \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + 0))\n\u22a2 \u2203 \u03b4, \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u2227 0 < \u03b4\n[PROOFSTEP]\nrw [zero_mul, zero_add] at A \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nA : Tendsto (fun \u03b4 => \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2203 \u03b4, \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5 \u2227 0 < \u03b4\n[PROOFSTEP]\nexact (((tendsto_order.1 A).2 \u03b5 \u03b5pos).and self_mem_nhdsWithin).exists\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u222b (x_1 : \u03b1) in s, \u03c6 x x_1 \u2022 g x_1 \u2202\u03bc) 0 < \u03b5\n[PROOFSTEP]\nsuffices \u2200\u1da0 i in l, \u2016\u222b x in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4\n by\n filter_upwards [this] with i hi\n simp only [dist_zero_right]\n exact hi.trans_lt h\u03b4\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nthis : \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u222b (x_1 : \u03b1) in s, \u03c6 x x_1 \u2022 g x_1 \u2202\u03bc) 0 < \u03b5\n[PROOFSTEP]\nfilter_upwards [this] with i hi\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nthis : \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\ni : \u03b9\nhi : \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\n\u22a2 dist (\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) 0 < \u03b5\n[PROOFSTEP]\nsimp only [dist_zero_right]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nthis : \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\ni : \u03b9\nhi : \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\n\u22a2 \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 < \u03b5\n[PROOFSTEP]\nexact hi.trans_lt h\u03b4\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\n[PROOFSTEP]\nobtain \u27e8u, u_open, x\u2080u, hu\u27e9 : \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 x \u2208 u \u2229 s, g x \u2208 ball (g x\u2080) \u03b4\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\n[PROOFSTEP]\nexact mem_nhdsWithin.1 (hcg (ball_mem_nhds _ \u03b4pos))\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\n[PROOFSTEP]\nfilter_upwards [tendstoUniformlyOn_iff.1 (hl\u03c6 u u_open x\u2080u) \u03b4 \u03b4pos, hi\u03c6, hn\u03c6,\n integrableOn_peak_smul_of_integrableOn_of_continuousWithinAt hs hl\u03c6 hi\u03c6 hmg hcg] with i hi h'i h\u03c6pos h''i\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\n[PROOFSTEP]\nhave B : \u2016\u222b x in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 :=\n calc\n \u2016\u222b x in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u222b x in s \u2229 u, \u2016\u03c6 i x \u2022 g x\u2016 \u2202\u03bc := norm_integral_le_integral_norm _\n _ \u2264 \u222b x in s \u2229 u, \u2016\u03c6 i x\u2016 * \u03b4 \u2202\u03bc :=\n by\n refine' set_integral_mono_on _ _ (hs.inter u_open.measurableSet) fun x hx => _\n \u00b7 exact IntegrableOn.mono_set h''i.norm (inter_subset_left _ _)\n \u00b7 exact IntegrableOn.mono_set ((integrable_of_integral_eq_one h'i).norm.mul_const _) (inter_subset_left _ _)\n rw [norm_smul]\n apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)\n rw [inter_comm, h'g] at hu \n exact (mem_ball_zero_iff.1 (hu x hx)).le\n _ \u2264 \u222b x in s, \u2016\u03c6 i x\u2016 * \u03b4 \u2202\u03bc := by\n apply set_integral_mono_set\n \u00b7 exact (integrable_of_integral_eq_one h'i).norm.mul_const _\n \u00b7 exact eventually_of_forall fun x => mul_nonneg (norm_nonneg _) \u03b4pos.le\n \u00b7 apply eventually_of_forall; exact inter_subset_left s u\n _ = \u222b x in s, \u03c6 i x * \u03b4 \u2202\u03bc := by\n apply set_integral_congr hs fun x hx => ?_\n rw [Real.norm_of_nonneg (h\u03c6pos _ hx)]\n _ = \u03b4 := by rw [integral_mul_right, h'i, one_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 \u222b (x : \u03b1) in s \u2229 u, \u2016\u03c6 i x \u2022 g x\u2016 \u2202\u03bc \u2264 \u222b (x : \u03b1) in s \u2229 u, \u2016\u03c6 i x\u2016 * \u03b4 \u2202\u03bc\n[PROOFSTEP]\nrefine' set_integral_mono_on _ _ (hs.inter u_open.measurableSet) fun x hx => _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 IntegrableOn (fun x => \u2016\u03c6 i x \u2022 g x\u2016) (s \u2229 u)\n[PROOFSTEP]\nexact IntegrableOn.mono_set h''i.norm (inter_subset_left _ _)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 IntegrableOn (fun x => \u2016\u03c6 i x\u2016 * \u03b4) (s \u2229 u)\n[PROOFSTEP]\nexact IntegrableOn.mono_set ((integrable_of_integral_eq_one h'i).norm.mul_const _) (inter_subset_left _ _)\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016\u03c6 i x \u2022 g x\u2016 \u2264 \u2016\u03c6 i x\u2016 * \u03b4\n[PROOFSTEP]\nrw [norm_smul]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016\u03c6 i x\u2016 * \u2016g x\u2016 \u2264 \u2016\u03c6 i x\u2016 * \u03b4\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_left _ (norm_nonneg _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016g x\u2016 \u2264 \u03b4\n[PROOFSTEP]\nrw [inter_comm, h'g] at hu \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 s \u2229 u \u2192 g x \u2208 ball 0 \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016g x\u2016 \u2264 \u03b4\n[PROOFSTEP]\nexact (mem_ball_zero_iff.1 (hu x hx)).le\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 \u222b (x : \u03b1) in s \u2229 u, \u2016\u03c6 i x\u2016 * \u03b4 \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, \u2016\u03c6 i x\u2016 * \u03b4 \u2202\u03bc\n[PROOFSTEP]\napply set_integral_mono_set\n[GOAL]\ncase hfi\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 IntegrableOn (fun x => \u2016\u03c6 i x\u2016 * \u03b4) s\n[PROOFSTEP]\nexact (integrable_of_integral_eq_one h'i).norm.mul_const _\n[GOAL]\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => \u2016\u03c6 i x\u2016 * \u03b4\n[PROOFSTEP]\nexact eventually_of_forall fun x => mul_nonneg (norm_nonneg _) \u03b4pos.le\n[GOAL]\ncase hst\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 s \u2229 u \u2264\u1da0[ae \u03bc] s\n[PROOFSTEP]\napply eventually_of_forall\n[GOAL]\ncase hst.hp\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 \u2200 (x : \u03b1), (s \u2229 u) x \u2264 s x\n[PROOFSTEP]\nexact inter_subset_left s u\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 \u222b (x : \u03b1) in s, \u2016\u03c6 i x\u2016 * \u03b4 \u2202\u03bc = \u222b (x : \u03b1) in s, \u03c6 i x * \u03b4 \u2202\u03bc\n[PROOFSTEP]\napply set_integral_congr hs fun x hx => ?_\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2016\u03c6 i x\u2016 * \u03b4 = \u03c6 i x * \u03b4\n[PROOFSTEP]\nrw [Real.norm_of_nonneg (h\u03c6pos _ hx)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\n\u22a2 \u222b (x : \u03b1) in s, \u03c6 i x * \u03b4 \u2202\u03bc = \u03b4\n[PROOFSTEP]\nrw [integral_mul_right, h'i, one_mul]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\n[PROOFSTEP]\nhave C : \u2016\u222b x in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc :=\n calc\n \u2016\u222b x in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u222b x in s \\ u, \u2016\u03c6 i x \u2022 g x\u2016 \u2202\u03bc := norm_integral_le_integral_norm _\n _ \u2264 \u222b x in s \\ u, \u03b4 * \u2016g x\u2016 \u2202\u03bc :=\n by\n refine' set_integral_mono_on _ _ (hs.diff u_open.measurableSet) fun x hx => _\n \u00b7 exact IntegrableOn.mono_set h''i.norm (diff_subset _ _)\n \u00b7 exact IntegrableOn.mono_set (hmg.norm.const_mul _) (diff_subset _ _)\n rw [norm_smul]\n apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)\n simpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).le\n _ \u2264 \u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc := by\n rw [integral_mul_left]\n apply mul_le_mul_of_nonneg_left (set_integral_mono_set hmg.norm _ _) \u03b4pos.le\n \u00b7 exact eventually_of_forall fun x => norm_nonneg _\n \u00b7 apply eventually_of_forall; exact diff_subset s u\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 \u222b (x : \u03b1) in s \\ u, \u2016\u03c6 i x \u2022 g x\u2016 \u2202\u03bc \u2264 \u222b (x : \u03b1) in s \\ u, \u03b4 * \u2016g x\u2016 \u2202\u03bc\n[PROOFSTEP]\nrefine' set_integral_mono_on _ _ (hs.diff u_open.measurableSet) fun x hx => _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 IntegrableOn (fun x => \u2016\u03c6 i x \u2022 g x\u2016) (s \\ u)\n[PROOFSTEP]\nexact IntegrableOn.mono_set h''i.norm (diff_subset _ _)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 IntegrableOn (fun x => \u03b4 * \u2016g x\u2016) (s \\ u)\n[PROOFSTEP]\nexact IntegrableOn.mono_set (hmg.norm.const_mul _) (diff_subset _ _)\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2016\u03c6 i x \u2022 g x\u2016 \u2264 \u03b4 * \u2016g x\u2016\n[PROOFSTEP]\nrw [norm_smul]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2016\u03c6 i x\u2016 * \u2016g x\u2016 \u2264 \u03b4 * \u2016g x\u2016\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_right _ (norm_nonneg _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2016\u03c6 i x\u2016 \u2264 \u03b4\n[PROOFSTEP]\nsimpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).le\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 \u222b (x : \u03b1) in s \\ u, \u03b4 * \u2016g x\u2016 \u2202\u03bc \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc\n[PROOFSTEP]\nrw [integral_mul_left]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 \u03b4 * \u222b (a : \u03b1) in s \\ u, \u2016g a\u2016 \u2202\u03bc \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_left (set_integral_mono_set hmg.norm _ _) \u03b4pos.le\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun a => \u2016g a\u2016\n[PROOFSTEP]\nexact eventually_of_forall fun x => norm_nonneg _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 s \\ u \u2264\u1da0[ae \u03bc] s\n[PROOFSTEP]\napply eventually_of_forall\n[GOAL]\ncase hp\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\n\u22a2 \u2200 (x : \u03b1), (s \\ u) x \u2264 s x\n[PROOFSTEP]\nexact diff_subset s u\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nC : \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc\n\u22a2 \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4\n[PROOFSTEP]\ncalc\n \u2016\u222b x in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 = \u2016(\u222b x in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc) + \u222b x in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 :=\n by\n conv_lhs => rw [\u2190 diff_union_inter s u]\n rw [integral_union (disjoint_sdiff_inter _ _) (hs.inter u_open.measurableSet) (h''i.mono_set (diff_subset _ _))\n (h''i.mono_set (inter_subset_left _ _))]\n _ \u2264 \u2016\u222b x in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 + \u2016\u222b x in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 := (norm_add_le _ _)\n _ \u2264 (\u03b4 * \u222b x in s, \u2016g x\u2016 \u2202\u03bc) + \u03b4 := add_le_add C B\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nC : \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc\n\u22a2 \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 = \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc + \u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016\n[PROOFSTEP]\nconv_lhs => rw [\u2190 diff_union_inter s u]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nC : \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc\n| \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016\n[PROOFSTEP]\nrw [\u2190 diff_union_inter s u]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nC : \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc\n| \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016\n[PROOFSTEP]\nrw [\u2190 diff_union_inter s u]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nC : \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc\n| \u2016\u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\u2016\n[PROOFSTEP]\nrw [\u2190 diff_union_inter s u]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nhmg : IntegrableOn g s\nh'g : g x\u2080 = 0\nhcg : ContinuousWithinAt g s x\u2080\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\nh\u03b4 : \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc + \u03b4 < \u03b5\n\u03b4pos : 0 < \u03b4\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) \u03b4\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < \u03b4\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = 1\nh\u03c6pos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nh''i : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nB : \u2016\u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4\nC : \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 \u2264 \u03b4 * \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc\n\u22a2 \u2016\u222b (x : \u03b1) in s \\ u \u222a s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016 =\n \u2016\u222b (x : \u03b1) in s \\ u, \u03c6 i x \u2022 g x \u2202\u03bc + \u222b (x : \u03b1) in s \u2229 u, \u03c6 i x \u2022 g x \u2202\u03bc\u2016\n[PROOFSTEP]\nrw [integral_union (disjoint_sdiff_inter _ _) (hs.inter u_open.measurableSet) (h''i.mono_set (diff_subset _ _))\n (h''i.mono_set (inter_subset_left _ _))]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) l (\ud835\udcdd (g x\u2080))\n[PROOFSTEP]\nlet h := g - fun _ => g x\u2080\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) l (\ud835\udcdd (g x\u2080))\n[PROOFSTEP]\nhave A : Tendsto (fun i : \u03b9 => (\u222b x in s, \u03c6 i x \u2022 h x \u2202\u03bc) + (\u222b x in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (0 + (1 : \u211d) \u2022 g x\u2080)) :=\n by\n refine' Tendsto.add _ (Tendsto.smul (tendsto_const_nhds.congr' hi\u03c6.symm) tendsto_const_nhds)\n apply tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt_aux hs hn\u03c6 hl\u03c6 hi\u03c6\n \u00b7 apply Integrable.sub hmg\n apply integrableOn_const.2\n simp only [h's.lt_top, or_true_iff]\n \u00b7 simp only [Pi.sub_apply, sub_self]\n \u00b7 exact hcg.sub continuousWithinAt_const\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 h x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (0 + 1 \u2022 g x\u2080))\n[PROOFSTEP]\nrefine' Tendsto.add _ (Tendsto.smul (tendsto_const_nhds.congr' hi\u03c6.symm) tendsto_const_nhds)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 h x \u2202\u03bc) l (\ud835\udcdd 0)\n[PROOFSTEP]\napply tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt_aux hs hn\u03c6 hl\u03c6 hi\u03c6\n[GOAL]\ncase hmg\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 IntegrableOn (fun x => h x) s\n[PROOFSTEP]\napply Integrable.sub hmg\n[GOAL]\ncase hmg\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 Integrable fun x => g x\u2080\n[PROOFSTEP]\napply integrableOn_const.2\n[GOAL]\ncase hmg\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 g x\u2080 = 0 \u2228 \u2191\u2191\u03bc s < \u22a4\n[PROOFSTEP]\nsimp only [h's.lt_top, or_true_iff]\n[GOAL]\ncase h'g\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 h x\u2080 = 0\n[PROOFSTEP]\nsimp only [Pi.sub_apply, sub_self]\n[GOAL]\ncase hcg\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\n\u22a2 ContinuousWithinAt (fun x => h x) s x\u2080\n[PROOFSTEP]\nexact hcg.sub continuousWithinAt_const\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 h x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (0 + 1 \u2022 g x\u2080))\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) l (\ud835\udcdd (g x\u2080))\n[PROOFSTEP]\nsimp only [one_smul, zero_add] at A \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080))\n\u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) l (\ud835\udcdd (g x\u2080))\n[PROOFSTEP]\nrefine' Tendsto.congr' _ A\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080))\n\u22a2 (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) =\u1da0[l] fun i =>\n \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\n[PROOFSTEP]\nfilter_upwards [integrableOn_peak_smul_of_integrableOn_of_continuousWithinAt hs hl\u03c6 hi\u03c6 hmg hcg, hi\u03c6] with i hi h'i\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080))\ni : \u03b9\nhi : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = OfNat.ofNat 1 i\n\u22a2 \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080 = \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\n[PROOFSTEP]\nsimp only [Pi.sub_apply, smul_sub]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080))\ni : \u03b9\nhi : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = OfNat.ofNat 1 i\n\u22a2 \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x - \u03c6 i x \u2022 g x\u2080 \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080 = \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc\n[PROOFSTEP]\nrw [integral_sub hi, integral_smul_const, sub_add_cancel]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d : CompleteSpace E\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nh : \u03b1 \u2192 E := g - fun x => g x\u2080\nA : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080))\ni : \u03b9\nhi : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s\nh'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = OfNat.ofNat 1 i\n\u22a2 Integrable fun x => \u03c6 i x \u2022 g x\u2080\n[PROOFSTEP]\nexact Integrable.smul_const (integrable_of_integral_eq_one h'i) _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\n[PROOFSTEP]\nlet \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b x in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\n[PROOFSTEP]\nhave hn\u03c6 : \u2200 n, \u2200 x \u2208 s, 0 \u2264 \u03c6 n x := by\n intro n x hx\n apply mul_nonneg (inv_nonneg.2 _) (pow_nonneg (hnc x hx) _)\n exact set_integral_nonneg hs.measurableSet fun x hx => pow_nonneg (hnc x hx) _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\n\u22a2 \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\n[PROOFSTEP]\nintro n x hx\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nn : \u2115\nx : \u03b1\nhx : x \u2208 s\n\u22a2 0 \u2264 \u03c6 n x\n[PROOFSTEP]\napply mul_nonneg (inv_nonneg.2 _) (pow_nonneg (hnc x hx) _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nn : \u2115\nx : \u03b1\nhx : x \u2208 s\n\u22a2 0 \u2264 \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\n[PROOFSTEP]\nexact set_integral_nonneg hs.measurableSet fun x hx => pow_nonneg (hnc x hx) _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\n[PROOFSTEP]\nhave I : \u2200 n, IntegrableOn (fun x => c x ^ n) s \u03bc := fun n => ContinuousOn.integrableOn_compact hs (hc.pow n)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\n[PROOFSTEP]\nhave J : \u2200 n, 0 \u2264\u1d50[\u03bc.restrict s] fun x : \u03b1 => c x ^ n :=\n by\n intro n\n filter_upwards [ae_restrict_mem hs.measurableSet] with x hx\n exact pow_nonneg (hnc x hx) n\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\n\u22a2 \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nn : \u2115\n\u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\n[PROOFSTEP]\nfilter_upwards [ae_restrict_mem hs.measurableSet] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nn : \u2115\nx : \u03b1\nhx : x \u2208 s\n\u22a2 OfNat.ofNat 0 x \u2264 c x ^ n\n[PROOFSTEP]\nexact pow_nonneg (hnc x hx) n\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\n[PROOFSTEP]\nhave P : \u2200 n, (0 : \u211d) < \u222b x in s, c x ^ n \u2202\u03bc := by\n intro n\n refine' (set_integral_pos_iff_support_of_nonneg_ae (J n) (I n)).2 _\n obtain \u27e8u, u_open, x\u2080_u, hu\u27e9 : \u2203 u : Set \u03b1, IsOpen u \u2227 x\u2080 \u2208 u \u2227 u \u2229 s \u2286 c \u207b\u00b9' Ioi 0 :=\n _root_.continuousOn_iff.1 hc x\u2080 h\u2080 (Ioi (0 : \u211d)) isOpen_Ioi hnc\u2080\n apply (h\u03bc u u_open x\u2080_u).trans_le\n exact measure_mono fun x hx => \u27e8ne_of_gt (pow_pos (a := c x) (hu hx) _), hx.2\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\n\u22a2 \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nn : \u2115\n\u22a2 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\n[PROOFSTEP]\nrefine' (set_integral_pos_iff_support_of_nonneg_ae (J n) (I n)).2 _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nn : \u2115\n\u22a2 0 < \u2191\u2191\u03bc ((Function.support fun x => c x ^ n) \u2229 s)\n[PROOFSTEP]\nobtain \u27e8u, u_open, x\u2080_u, hu\u27e9 : \u2203 u : Set \u03b1, IsOpen u \u2227 x\u2080 \u2208 u \u2227 u \u2229 s \u2286 c \u207b\u00b9' Ioi 0 :=\n _root_.continuousOn_iff.1 hc x\u2080 h\u2080 (Ioi (0 : \u211d)) isOpen_Ioi hnc\u2080\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nn : \u2115\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080_u : x\u2080 \u2208 u\nhu : u \u2229 s \u2286 c \u207b\u00b9' Ioi 0\n\u22a2 0 < \u2191\u2191\u03bc ((Function.support fun x => c x ^ n) \u2229 s)\n[PROOFSTEP]\napply (h\u03bc u u_open x\u2080_u).trans_le\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nn : \u2115\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080_u : x\u2080 \u2208 u\nhu : u \u2229 s \u2286 c \u207b\u00b9' Ioi 0\n\u22a2 \u2191\u2191\u03bc (u \u2229 s) \u2264 \u2191\u2191\u03bc ((Function.support fun x => c x ^ n) \u2229 s)\n[PROOFSTEP]\nexact measure_mono fun x hx => \u27e8ne_of_gt (pow_pos (a := c x) (hu hx) _), hx.2\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\n[PROOFSTEP]\nhave hi\u03c6 : \u2200 n, \u222b x in s, \u03c6 n x \u2202\u03bc = 1 := fun n => by rw [integral_mul_left, inv_mul_cancel (P n).ne']\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nn : \u2115\n\u22a2 \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\n[PROOFSTEP]\nrw [integral_mul_left, inv_mul_cancel (P n).ne']\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\n[PROOFSTEP]\nhave A : \u2200 u : Set \u03b1, IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 atTop (s \\ u) :=\n by\n intro u u_open x\u2080u\n obtain \u27e8t, t_pos, tx\u2080, ht\u27e9 : \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 x \u2208 s \\ u, c x \u2264 t :=\n by\n rcases eq_empty_or_nonempty (s \\ u) with (h | h)\n \u00b7 exact \u27e80, le_rfl, hnc\u2080, by simp only [h, mem_empty_iff_false, IsEmpty.forall_iff, imp_true_iff]\u27e9\n obtain \u27e8x, hx, h'x\u27e9 : \u2203 x \u2208 s \\ u, \u2200 y \u2208 s \\ u, c y \u2264 c x :=\n IsCompact.exists_isMaxOn (hs.diff u_open) h (hc.mono (diff_subset _ _))\n refine' \u27e8c x, hnc x hx.1, h'c x hx.1 _, h'x\u27e9\n rintro rfl\n exact hx.2 x\u2080u\n obtain \u27e8t', tt', t'x\u2080\u27e9 : \u2203 t', t < t' \u2227 t' < c x\u2080 := exists_between tx\u2080\n have t'_pos : 0 < t' := t_pos.trans_lt tt'\n obtain \u27e8v, v_open, x\u2080_v, hv\u27e9 : \u2203 v : Set \u03b1, IsOpen v \u2227 x\u2080 \u2208 v \u2227 v \u2229 s \u2286 c \u207b\u00b9' Ioi t' :=\n _root_.continuousOn_iff.1 hc x\u2080 h\u2080 (Ioi t') isOpen_Ioi t'x\u2080\n have M : \u2200 n, \u2200 x \u2208 s \\ u, \u03c6 n x \u2264 (\u03bc (v \u2229 s)).toReal\u207b\u00b9 * (t / t') ^ n :=\n by\n intro n x hx\n have B : t' ^ n * (\u03bc (v \u2229 s)).toReal \u2264 \u222b y in s, c y ^ n \u2202\u03bc :=\n calc\n t' ^ n * (\u03bc (v \u2229 s)).toReal = \u222b _ in v \u2229 s, t' ^ n \u2202\u03bc := by\n simp only [integral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter, Algebra.id.smul_eq_mul,\n mul_comm]\n _ \u2264 \u222b y in v \u2229 s, c y ^ n \u2202\u03bc :=\n by\n apply set_integral_mono_on _ _ (v_open.measurableSet.inter hs.measurableSet) _\n \u00b7 apply integrableOn_const.2 (Or.inr _)\n exact lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) hs.measure_lt_top\n \u00b7 exact (I n).mono (inter_subset_right _ _) le_rfl\n \u00b7 intro x hx\n exact pow_le_pow_of_le_left t'_pos.le (le_of_lt (hv hx)) _\n _ \u2264 \u222b y in s, c y ^ n \u2202\u03bc := set_integral_mono_set (I n) (J n) (eventually_of_forall (inter_subset_right _ _))\n simp_rw [\u2190 div_eq_inv_mul, div_pow, div_div]\n apply div_le_div (pow_nonneg t_pos n) _ _ B\n \u00b7 exact pow_le_pow_of_le_left (hnc _ hx.1) (ht x hx) _\n \u00b7 apply mul_pos (pow_pos (t_pos.trans_lt tt') _) (ENNReal.toReal_pos (h\u03bc v v_open x\u2080_v).ne' _)\n have : \u03bc (v \u2229 s) \u2264 \u03bc s := measure_mono (inter_subset_right _ _)\n exact ne_of_lt (lt_of_le_of_lt this hs.measure_lt_top)\n have N : Tendsto (fun n => (\u03bc (v \u2229 s)).toReal\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd ((\u03bc (v \u2229 s)).toReal\u207b\u00b9 * 0)) :=\n by\n apply Tendsto.mul tendsto_const_nhds _\n apply tendsto_pow_atTop_nhds_0_of_lt_1 (div_nonneg t_pos t'_pos.le)\n exact (div_lt_one t'_pos).2 tt'\n rw [mul_zero] at N \n refine' tendstoUniformlyOn_iff.2 fun \u03b5 \u03b5pos => _\n filter_upwards [(tendsto_order.1 N).2 \u03b5 \u03b5pos] with n hn x hx\n simp only [Pi.zero_apply, dist_zero_left, Real.norm_of_nonneg (hn\u03c6 n x hx.1)]\n exact (M n x hx).trans_lt hn\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\n\u22a2 \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\n[PROOFSTEP]\nintro u u_open x\u2080u\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\n[PROOFSTEP]\nobtain \u27e8t, t_pos, tx\u2080, ht\u27e9 : \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 x \u2208 s \\ u, c x \u2264 t :=\n by\n rcases eq_empty_or_nonempty (s \\ u) with (h | h)\n \u00b7 exact \u27e80, le_rfl, hnc\u2080, by simp only [h, mem_empty_iff_false, IsEmpty.forall_iff, imp_true_iff]\u27e9\n obtain \u27e8x, hx, h'x\u27e9 : \u2203 x \u2208 s \\ u, \u2200 y \u2208 s \\ u, c y \u2264 c x :=\n IsCompact.exists_isMaxOn (hs.diff u_open) h (hc.mono (diff_subset _ _))\n refine' \u27e8c x, hnc x hx.1, h'c x hx.1 _, h'x\u27e9\n rintro rfl\n exact hx.2 x\u2080u\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\n\u22a2 \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\n[PROOFSTEP]\nrcases eq_empty_or_nonempty (s \\ u) with (h | h)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nh : s \\ u = \u2205\n\u22a2 \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\n[PROOFSTEP]\nexact \u27e80, le_rfl, hnc\u2080, by simp only [h, mem_empty_iff_false, IsEmpty.forall_iff, imp_true_iff]\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nh : s \\ u = \u2205\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 0\n[PROOFSTEP]\nsimp only [h, mem_empty_iff_false, IsEmpty.forall_iff, imp_true_iff]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nh : Set.Nonempty (s \\ u)\n\u22a2 \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\n[PROOFSTEP]\nobtain \u27e8x, hx, h'x\u27e9 : \u2203 x \u2208 s \\ u, \u2200 y \u2208 s \\ u, c y \u2264 c x :=\n IsCompact.exists_isMaxOn (hs.diff u_open) h (hc.mono (diff_subset _ _))\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nh : Set.Nonempty (s \\ u)\nx : \u03b1\nhx : x \u2208 s \\ u\nh'x : \u2200 (y : \u03b1), y \u2208 s \\ u \u2192 c y \u2264 c x\n\u22a2 \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\n[PROOFSTEP]\nrefine' \u27e8c x, hnc x hx.1, h'c x hx.1 _, h'x\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nh : Set.Nonempty (s \\ u)\nx : \u03b1\nhx : x \u2208 s \\ u\nh'x : \u2200 (y : \u03b1), y \u2208 s \\ u \u2192 c y \u2264 c x\n\u22a2 x \u2260 x\u2080\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhmg : IntegrableOn g s\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nh : Set.Nonempty (s \\ u)\nx : \u03b1\nhx : x \u2208 s \\ u\nh'x : \u2200 (y : \u03b1), y \u2208 s \\ u \u2192 c y \u2264 c x\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x \u2192 c y < c x\nhnc\u2080 : 0 < c x\nh\u2080 : x \u2208 s\nhcg : ContinuousWithinAt g s x\nx\u2080u : x \u2208 u\n\u22a2 False\n[PROOFSTEP]\nexact hx.2 x\u2080u\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\n[PROOFSTEP]\nobtain \u27e8t', tt', t'x\u2080\u27e9 : \u2203 t', t < t' \u2227 t' < c x\u2080 := exists_between tx\u2080\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\n[PROOFSTEP]\nhave t'_pos : 0 < t' := t_pos.trans_lt tt'\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\n[PROOFSTEP]\nobtain \u27e8v, v_open, x\u2080_v, hv\u27e9 : \u2203 v : Set \u03b1, IsOpen v \u2227 x\u2080 \u2208 v \u2227 v \u2229 s \u2286 c \u207b\u00b9' Ioi t' :=\n _root_.continuousOn_iff.1 hc x\u2080 h\u2080 (Ioi t') isOpen_Ioi t'x\u2080\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\n[PROOFSTEP]\nhave M : \u2200 n, \u2200 x \u2208 s \\ u, \u03c6 n x \u2264 (\u03bc (v \u2229 s)).toReal\u207b\u00b9 * (t / t') ^ n :=\n by\n intro n x hx\n have B : t' ^ n * (\u03bc (v \u2229 s)).toReal \u2264 \u222b y in s, c y ^ n \u2202\u03bc :=\n calc\n t' ^ n * (\u03bc (v \u2229 s)).toReal = \u222b _ in v \u2229 s, t' ^ n \u2202\u03bc := by\n simp only [integral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter, Algebra.id.smul_eq_mul,\n mul_comm]\n _ \u2264 \u222b y in v \u2229 s, c y ^ n \u2202\u03bc :=\n by\n apply set_integral_mono_on _ _ (v_open.measurableSet.inter hs.measurableSet) _\n \u00b7 apply integrableOn_const.2 (Or.inr _)\n exact lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) hs.measure_lt_top\n \u00b7 exact (I n).mono (inter_subset_right _ _) le_rfl\n \u00b7 intro x hx\n exact pow_le_pow_of_le_left t'_pos.le (le_of_lt (hv hx)) _\n _ \u2264 \u222b y in s, c y ^ n \u2202\u03bc := set_integral_mono_set (I n) (J n) (eventually_of_forall (inter_subset_right _ _))\n simp_rw [\u2190 div_eq_inv_mul, div_pow, div_div]\n apply div_le_div (pow_nonneg t_pos n) _ _ B\n \u00b7 exact pow_le_pow_of_le_left (hnc _ hx.1) (ht x hx) _\n \u00b7 apply mul_pos (pow_pos (t_pos.trans_lt tt') _) (ENNReal.toReal_pos (h\u03bc v v_open x\u2080_v).ne' _)\n have : \u03bc (v \u2229 s) \u2264 \u03bc s := measure_mono (inter_subset_right _ _)\n exact ne_of_lt (lt_of_le_of_lt this hs.measure_lt_top)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\n\u22a2 \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\n[PROOFSTEP]\nintro n x hx\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\n[PROOFSTEP]\nhave B : t' ^ n * (\u03bc (v \u2229 s)).toReal \u2264 \u222b y in s, c y ^ n \u2202\u03bc :=\n calc\n t' ^ n * (\u03bc (v \u2229 s)).toReal = \u222b _ in v \u2229 s, t' ^ n \u2202\u03bc := by\n simp only [integral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter, Algebra.id.smul_eq_mul,\n mul_comm]\n _ \u2264 \u222b y in v \u2229 s, c y ^ n \u2202\u03bc :=\n by\n apply set_integral_mono_on _ _ (v_open.measurableSet.inter hs.measurableSet) _\n \u00b7 apply integrableOn_const.2 (Or.inr _)\n exact lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) hs.measure_lt_top\n \u00b7 exact (I n).mono (inter_subset_right _ _) le_rfl\n \u00b7 intro x hx\n exact pow_le_pow_of_le_left t'_pos.le (le_of_lt (hv hx)) _\n _ \u2264 \u222b y in s, c y ^ n \u2202\u03bc := set_integral_mono_set (I n) (J n) (eventually_of_forall (inter_subset_right _ _))\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) = \u222b (x : \u03b1) in v \u2229 s, t' ^ n \u2202\u03bc\n[PROOFSTEP]\nsimp only [integral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter, Algebra.id.smul_eq_mul, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u222b (x : \u03b1) in v \u2229 s, t' ^ n \u2202\u03bc \u2264 \u222b (y : \u03b1) in v \u2229 s, c y ^ n \u2202\u03bc\n[PROOFSTEP]\napply set_integral_mono_on _ _ (v_open.measurableSet.inter hs.measurableSet) _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 IntegrableOn (fun a => t' ^ n) (v \u2229 s)\n[PROOFSTEP]\napply integrableOn_const.2 (Or.inr _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2191\u2191\u03bc (v \u2229 s) < \u22a4\n[PROOFSTEP]\nexact lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) hs.measure_lt_top\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 IntegrableOn (fun a => c a ^ n) (v \u2229 s)\n[PROOFSTEP]\nexact (I n).mono (inter_subset_right _ _) le_rfl\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2200 (x : \u03b1), x \u2208 v \u2229 s \u2192 t' ^ n \u2264 c x ^ n\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx\u271d : \u03b1\nhx\u271d : x\u271d \u2208 s \\ u\nx : \u03b1\nhx : x \u2208 v \u2229 s\n\u22a2 t' ^ n \u2264 c x ^ n\n[PROOFSTEP]\nexact pow_le_pow_of_le_left t'_pos.le (le_of_lt (hv hx)) _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\n\u22a2 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\n[PROOFSTEP]\nsimp_rw [\u2190 div_eq_inv_mul, div_pow, div_div]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\n\u22a2 c x ^ n / \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc \u2264 t ^ n / (t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\n[PROOFSTEP]\napply div_le_div (pow_nonneg t_pos n) _ _ B\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\n\u22a2 c x ^ n \u2264 t ^ n\n[PROOFSTEP]\nexact pow_le_pow_of_le_left (hnc _ hx.1) (ht x hx) _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\n\u22a2 0 < t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s))\n[PROOFSTEP]\napply mul_pos (pow_pos (t_pos.trans_lt tt') _) (ENNReal.toReal_pos (h\u03bc v v_open x\u2080_v).ne' _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\n\u22a2 \u2191\u2191\u03bc (v \u2229 s) \u2260 \u22a4\n[PROOFSTEP]\nhave : \u03bc (v \u2229 s) \u2264 \u03bc s := measure_mono (inter_subset_right _ _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nn : \u2115\nx : \u03b1\nhx : x \u2208 s \\ u\nB : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc\nthis : \u2191\u2191\u03bc (v \u2229 s) \u2264 \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (v \u2229 s) \u2260 \u22a4\n[PROOFSTEP]\nexact ne_of_lt (lt_of_le_of_lt this hs.measure_lt_top)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\n[PROOFSTEP]\nhave N : Tendsto (fun n => (\u03bc (v \u2229 s)).toReal\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd ((\u03bc (v \u2229 s)).toReal\u207b\u00b9 * 0)) :=\n by\n apply Tendsto.mul tendsto_const_nhds _\n apply tendsto_pow_atTop_nhds_0_of_lt_1 (div_nonneg t_pos t'_pos.le)\n exact (div_lt_one t'_pos).2 tt'\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\n\u22a2 Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd ((ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * 0))\n[PROOFSTEP]\napply Tendsto.mul tendsto_const_nhds _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\n\u22a2 Tendsto (fun x => (t / t') ^ x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\napply tendsto_pow_atTop_nhds_0_of_lt_1 (div_nonneg t_pos t'_pos.le)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\n\u22a2 t / t' < 1\n[PROOFSTEP]\nexact (div_lt_one t'_pos).2 tt'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\nN : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd ((ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * 0))\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\n[PROOFSTEP]\nrw [mul_zero] at N \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\nN : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0)\n\u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\n[PROOFSTEP]\nrefine' tendstoUniformlyOn_iff.2 fun \u03b5 \u03b5pos => _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\nN : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0)\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 n x) < \u03b5\n[PROOFSTEP]\nfilter_upwards [(tendsto_order.1 N).2 \u03b5 \u03b5pos] with n hn x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\nN : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0)\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nn : \u2115\nhn : (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n < \u03b5\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 dist (OfNat.ofNat 0 x) ((\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n) < \u03b5\n[PROOFSTEP]\nsimp only [Pi.zero_apply, dist_zero_left, Real.norm_of_nonneg (hn\u03c6 n x hx.1)]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nt : \u211d\nt_pos : 0 \u2264 t\ntx\u2080 : t < c x\u2080\nht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t\nt' : \u211d\ntt' : t < t'\nt'x\u2080 : t' < c x\u2080\nt'_pos : 0 < t'\nv : Set \u03b1\nv_open : IsOpen v\nx\u2080_v : x\u2080 \u2208 v\nhv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t'\nM : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n\nN : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0)\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nn : \u2115\nhn : (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n < \u03b5\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n < \u03b5\n[PROOFSTEP]\nexact (M n x hx).trans_lt hn\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nA : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\n[PROOFSTEP]\nhave : Tendsto (fun i : \u2115 => \u222b x : \u03b1 in s, \u03c6 i x \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080)) :=\n tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt hs.measurableSet hs.measure_lt_top.ne\n (eventually_of_forall hn\u03c6) A (eventually_of_forall hi\u03c6) hmg hcg\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nA : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\nthis : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : BorelSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : MetrizableSpace \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhs : IsCompact s\nh\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 s\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n\nhn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x\nI : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s\nJ : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n\nP : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc\nhi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1\nA : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 atTop (s \\ u)\nthis : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\nx\u271d : \u2115\n\u22a2 (\u222b (x : \u03b1) in s, c x ^ x\u271d \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ x\u271d \u2022 g x \u2202\u03bc = \u222b (x : \u03b1) in s, \u03c6 x\u271d x \u2022 g x \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [\u2190 smul_smul, integral_smul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\n[PROOFSTEP]\nhave : x\u2080 \u2208 s := by rw [\u2190 hs.isClosed.closure_eq]; exact closure_mono interior_subset h\u2080\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 x\u2080 \u2208 s\n[PROOFSTEP]\nrw [\u2190 hs.isClosed.closure_eq]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 x\u2080 \u2208 closure s\n[PROOFSTEP]\nexact closure_mono interior_subset h\u2080\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nthis : x\u2080 \u2208 s\n\u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080))\n[PROOFSTEP]\napply\n tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_measure_nhdsWithin_pos hs _ hc h'c hnc hnc\u2080 this hmg\n hcg\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nthis : x\u2080 \u2208 s\n\u22a2 \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s)\n[PROOFSTEP]\nintro u u_open x\u2080_u\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nthis : x\u2080 \u2208 s\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080_u : x\u2080 \u2208 u\n\u22a2 0 < \u2191\u2191\u03bc (u \u2229 s)\n[PROOFSTEP]\ncalc\n 0 < \u03bc (u \u2229 interior s) := (u_open.inter isOpen_interior).measure_pos \u03bc (_root_.mem_closure_iff.1 h\u2080 u u_open x\u2080_u)\n _ \u2264 \u03bc (u \u2229 s) := measure_mono (inter_subset_inter_right _ interior_subset)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : ContinuousOn g s\n\u22a2 x\u2080 \u2208 s\n[PROOFSTEP]\nrw [\u2190 hs.isClosed.closure_eq]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\nhm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : BorelSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : MetrizableSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nhs : IsCompact s\nc : \u03b1 \u2192 \u211d\nhc : ContinuousOn c s\nh'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080\nhnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x\nhnc\u2080 : 0 < c x\u2080\nh\u2080 : x\u2080 \u2208 closure (interior s)\nhmg : ContinuousOn g s\n\u22a2 x\u2080 \u2208 closure s\n[PROOFSTEP]\nexact closure_mono interior_subset h\u2080\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Integral.PeakFunction", "llama_tokens": 87435, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149758396752, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.4098156223004185}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ntf : TotalFunction \u03b1 \u03b2\n\u22a2 \u2200 (a : \u03b1), a \u2208 zeroDefaultSupp tf \u2194 apply (zeroDefault tf) a \u2260 0\n[PROOFSTEP]\nintro a\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ntf : TotalFunction \u03b1 \u03b2\na : \u03b1\n\u22a2 a \u2208 zeroDefaultSupp tf \u2194 apply (zeroDefault tf) a \u2260 0\n[PROOFSTEP]\nrcases tf with \u27e8A, y\u27e9\n[GOAL]\ncase withDefault\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nA : List ((_ : \u03b1) \u00d7 \u03b2)\ny : \u03b2\n\u22a2 a \u2208 zeroDefaultSupp (withDefault A y) \u2194 apply (zeroDefault (withDefault A y)) a \u2260 0\n[PROOFSTEP]\nsimp only [apply, zeroDefaultSupp, List.mem_map, List.mem_filter, exists_and_right, List.mem_toFinset, exists_eq_right,\n Sigma.exists, Ne.def, zeroDefault]\n[GOAL]\ncase withDefault\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nA : List ((_ : \u03b1) \u00d7 \u03b2)\ny : \u03b2\n\u22a2 (\u2203 x, { fst := a, snd := x } \u2208 List.dedupKeys A \u2227 (decide \u00acx = 0) = true) \u2194 \u00acOption.getD (List.dlookup a A) 0 = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase withDefault.mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nA : List ((_ : \u03b1) \u00d7 \u03b2)\ny : \u03b2\n\u22a2 (\u2203 x, { fst := a, snd := x } \u2208 List.dedupKeys A \u2227 (decide \u00acx = 0) = true) \u2192 \u00acOption.getD (List.dlookup a A) 0 = 0\n[PROOFSTEP]\nrintro \u27e8od, hval, hod\u27e9\n[GOAL]\ncase withDefault.mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nA : List ((_ : \u03b1) \u00d7 \u03b2)\ny od : \u03b2\nhval : { fst := a, snd := od } \u2208 List.dedupKeys A\nhod : (decide \u00acod = 0) = true\n\u22a2 \u00acOption.getD (List.dlookup a A) 0 = 0\n[PROOFSTEP]\nhave := List.mem_dlookup (List.nodupKeys_dedupKeys A) hval\n[GOAL]\ncase withDefault.mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nA : List ((_ : \u03b1) \u00d7 \u03b2)\ny od : \u03b2\nhval : { fst := a, snd := od } \u2208 List.dedupKeys A\nhod : (decide \u00acod = 0) = true\nthis : od \u2208 List.dlookup a (List.dedupKeys A)\n\u22a2 \u00acOption.getD (List.dlookup a A) 0 = 0\n[PROOFSTEP]\nrw [(_ : List.dlookup a A = od)]\n[GOAL]\ncase withDefault.mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nA : List ((_ : \u03b1) \u00d7 \u03b2)\ny od : \u03b2\nhval : { fst := a, snd := od } \u2208 List.dedupKeys A\nhod : (decide \u00acod = 0) = true\nthis : od \u2208 List.dlookup a (List.dedupKeys A)\n\u22a2 \u00acOption.getD (some od) 0 = 0\n[PROOFSTEP]\nsimpa using hod\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nA : List ((_ : \u03b1) \u00d7 \u03b2)\ny od : \u03b2\nhval : { fst := a, snd := od } \u2208 List.dedupKeys A\nhod : (decide \u00acod = 0) = true\nthis : od \u2208 List.dlookup a (List.dedupKeys A)\n\u22a2 List.dlookup a A = some od\n[PROOFSTEP]\nsimpa [List.dlookup_dedupKeys, WithTop.some_eq_coe]\n[GOAL]\ncase withDefault.mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nA : List ((_ : \u03b1) \u00d7 \u03b2)\ny : \u03b2\n\u22a2 \u00acOption.getD (List.dlookup a A) 0 = 0 \u2192 \u2203 x, { fst := a, snd := x } \u2208 List.dedupKeys A \u2227 (decide \u00acx = 0) = true\n[PROOFSTEP]\nintro h\n[GOAL]\ncase withDefault.mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nA : List ((_ : \u03b1) \u00d7 \u03b2)\ny : \u03b2\nh : \u00acOption.getD (List.dlookup a A) 0 = 0\n\u22a2 \u2203 x, { fst := a, snd := x } \u2208 List.dedupKeys A \u2227 (decide \u00acx = 0) = true\n[PROOFSTEP]\nuse(A.dlookup a).getD (0 : \u03b2)\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nA : List ((_ : \u03b1) \u00d7 \u03b2)\ny : \u03b2\nh : \u00acOption.getD (List.dlookup a A) 0 = 0\n\u22a2 { fst := a, snd := Option.getD (List.dlookup a A) 0 } \u2208 List.dedupKeys A \u2227\n (decide \u00acOption.getD (List.dlookup a A) 0 = 0) = true\n[PROOFSTEP]\nrw [\u2190 List.dlookup_dedupKeys] at h \u22a2\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nA : List ((_ : \u03b1) \u00d7 \u03b2)\ny : \u03b2\nh : \u00acOption.getD (List.dlookup a (List.dedupKeys A)) 0 = 0\n\u22a2 { fst := a, snd := Option.getD (List.dlookup a (List.dedupKeys A)) 0 } \u2208 List.dedupKeys A \u2227\n (decide \u00acOption.getD (List.dlookup a (List.dedupKeys A)) 0 = 0) = true\n[PROOFSTEP]\nsimp only [h, \u2190 List.mem_dlookup_iff A.nodupKeys_dedupKeys, and_true_iff, not_false_iff, Option.mem_def]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nA : List ((_ : \u03b1) \u00d7 \u03b2)\ny : \u03b2\nh : \u00acOption.getD (List.dlookup a (List.dedupKeys A)) 0 = 0\n\u22a2 List.dlookup a (List.dedupKeys A) = some (Option.getD (List.dlookup a (List.dedupKeys A)) 0)\n[PROOFSTEP]\ncases haA : List.dlookup a A.dedupKeys\n[GOAL]\ncase h.none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nA : List ((_ : \u03b1) \u00d7 \u03b2)\ny : \u03b2\nh : \u00acOption.getD (List.dlookup a (List.dedupKeys A)) 0 = 0\nhaA : List.dlookup a (List.dedupKeys A) = none\n\u22a2 none = some (Option.getD none 0)\n[PROOFSTEP]\nsimp [haA] at h \n[GOAL]\ncase h.some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nA : List ((_ : \u03b1) \u00d7 \u03b2)\ny : \u03b2\nh : \u00acOption.getD (List.dlookup a (List.dedupKeys A)) 0 = 0\nval\u271d : \u03b2\nhaA : List.dlookup a (List.dedupKeys A) = some val\u271d\n\u22a2 some val\u271d = some (Option.getD (some val\u271d) 0)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List (\u03b1 \u00d7 \u03b1)\nx y z : \u03b1\n\u22a2 applyId ((y, z) :: xs) x = if y = x then z else applyId xs x\n[PROOFSTEP]\nsimp only [List.applyId, List.dlookup, eq_rec_constant, Prod.toSigma, List.map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List (\u03b1 \u00d7 \u03b1)\nx y z : \u03b1\n\u22a2 Option.getD (if h : y = x then some z else List.dlookup x (List.map (fun p => { fst := p.fst, snd := p.snd }) xs)) x =\n if y = x then z else Option.getD (List.dlookup x (List.map (fun p => { fst := p.fst, snd := p.snd }) xs)) x\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List (\u03b1 \u00d7 \u03b1)\nx y z : \u03b1\nh\u271d : y = x\n\u22a2 Option.getD (some z) x = z\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List (\u03b1 \u00d7 \u03b1)\nx y z : \u03b1\nh\u271d : \u00acy = x\n\u22a2 Option.getD (List.dlookup x (List.map (fun p => { fst := p.fst, snd := p.snd }) xs)) x =\n Option.getD (List.dlookup x (List.map (fun p => { fst := p.fst, snd := p.snd }) xs)) x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : List.length xs = List.length ys\nx y : \u03b1\ni : \u2115\nh\u2082 : List.get? xs i = some x\n\u22a2 applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y\n[PROOFSTEP]\ninduction xs generalizing ys i\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nys : List \u03b1\nh\u2080 : List.Nodup []\nh\u2081 : List.length [] = List.length ys\ni : \u2115\nh\u2082 : List.get? [] i = some x\n\u22a2 applyId (List.zip [] ys) x = y \u2194 List.get? ys i = some y\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y head\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d :\n \u2200 {ys : List \u03b1},\n List.Nodup tail\u271d \u2192\n List.length tail\u271d = List.length ys \u2192\n \u2200 (i : \u2115), List.get? tail\u271d i = some x \u2192 (applyId (List.zip tail\u271d ys) x = y \u2194 List.get? ys i = some y)\nys : List \u03b1\nh\u2080 : List.Nodup (head\u271d :: tail\u271d)\nh\u2081 : List.length (head\u271d :: tail\u271d) = List.length ys\ni : \u2115\nh\u2082 : List.get? (head\u271d :: tail\u271d) i = some x\n\u22a2 applyId (List.zip (head\u271d :: tail\u271d) ys) x = y \u2194 List.get? ys i = some y\n[PROOFSTEP]\ncase nil => cases h\u2082\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nys : List \u03b1\nh\u2080 : List.Nodup []\nh\u2081 : List.length [] = List.length ys\ni : \u2115\nh\u2082 : List.get? [] i = some x\n\u22a2 applyId (List.zip [] ys) x = y \u2194 List.get? ys i = some y\n[PROOFSTEP]\ncase nil => cases h\u2082\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nys : List \u03b1\nh\u2080 : List.Nodup []\nh\u2081 : List.length [] = List.length ys\ni : \u2115\nh\u2082 : List.get? [] i = some x\n\u22a2 applyId (List.zip [] ys) x = y \u2194 List.get? ys i = some y\n[PROOFSTEP]\ncases h\u2082\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y head\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d :\n \u2200 {ys : List \u03b1},\n List.Nodup tail\u271d \u2192\n List.length tail\u271d = List.length ys \u2192\n \u2200 (i : \u2115), List.get? tail\u271d i = some x \u2192 (applyId (List.zip tail\u271d ys) x = y \u2194 List.get? ys i = some y)\nys : List \u03b1\nh\u2080 : List.Nodup (head\u271d :: tail\u271d)\nh\u2081 : List.length (head\u271d :: tail\u271d) = List.length ys\ni : \u2115\nh\u2082 : List.get? (head\u271d :: tail\u271d) i = some x\n\u22a2 applyId (List.zip (head\u271d :: tail\u271d) ys) x = y \u2194 List.get? ys i = some y\n[PROOFSTEP]\ncase cons x' xs xs_ih =>\n cases i\n \u00b7 injection h\u2082 with h\u2080; subst h\u2080\n cases ys\n \u00b7 cases h\u2081\n \u00b7\n -- porting note: `open List` no longer makes `zip_cons_cons` visiblesimp only [List.applyId, Prod.toSigma,\n Option.getD_some, List.get?, List.dlookup_cons_eq, List.zip_cons_cons, List.map, Option.some_inj]\n \u00b7 cases ys\n \u00b7 cases h\u2081\n \u00b7 cases' h\u2080 with _ _ h\u2080 h\u2081\n simp only [List.get?, List.zip_cons_cons, List.applyId_cons] at h\u2082 \u22a2\n rw [if_neg]\n \u00b7 apply xs_ih <;> solve_by_elim [Nat.succ.inj]\n \u00b7 apply h\u2080; apply List.get?_mem h\u2082\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x \u2192 (applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y)\nys : List \u03b1\nh\u2080 : List.Nodup (x' :: xs)\nh\u2081 : List.length (x' :: xs) = List.length ys\ni : \u2115\nh\u2082 : List.get? (x' :: xs) i = some x\n\u22a2 applyId (List.zip (x' :: xs) ys) x = y \u2194 List.get? ys i = some y\n[PROOFSTEP]\ncase cons x' xs xs_ih =>\n cases i\n \u00b7 injection h\u2082 with h\u2080; subst h\u2080\n cases ys\n \u00b7 cases h\u2081\n \u00b7\n -- porting note: `open List` no longer makes `zip_cons_cons` visiblesimp only [List.applyId, Prod.toSigma,\n Option.getD_some, List.get?, List.dlookup_cons_eq, List.zip_cons_cons, List.map, Option.some_inj]\n \u00b7 cases ys\n \u00b7 cases h\u2081\n \u00b7 cases' h\u2080 with _ _ h\u2080 h\u2081\n simp only [List.get?, List.zip_cons_cons, List.applyId_cons] at h\u2082 \u22a2\n rw [if_neg]\n \u00b7 apply xs_ih <;> solve_by_elim [Nat.succ.inj]\n \u00b7 apply h\u2080; apply List.get?_mem h\u2082\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x \u2192 (applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y)\nys : List \u03b1\nh\u2080 : List.Nodup (x' :: xs)\nh\u2081 : List.length (x' :: xs) = List.length ys\ni : \u2115\nh\u2082 : List.get? (x' :: xs) i = some x\n\u22a2 applyId (List.zip (x' :: xs) ys) x = y \u2194 List.get? ys i = some y\n[PROOFSTEP]\ncases i\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x \u2192 (applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y)\nys : List \u03b1\nh\u2080 : List.Nodup (x' :: xs)\nh\u2081 : List.length (x' :: xs) = List.length ys\nh\u2082 : List.get? (x' :: xs) Nat.zero = some x\n\u22a2 applyId (List.zip (x' :: xs) ys) x = y \u2194 List.get? ys Nat.zero = some y\n[PROOFSTEP]\ninjection h\u2082 with h\u2080\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x \u2192 (applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y)\nys : List \u03b1\nh\u2080\u271d : List.Nodup (x' :: xs)\nh\u2081 : List.length (x' :: xs) = List.length ys\nh\u2080 : x' = x\n\u22a2 applyId (List.zip (x' :: xs) ys) x = y \u2194 List.get? ys Nat.zero = some y\n[PROOFSTEP]\nsubst h\u2080\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\ny x' : \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup (x' :: xs)\nh\u2081 : List.length (x' :: xs) = List.length ys\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x' \u2192 (applyId (List.zip xs ys) x' = y \u2194 List.get? ys i = some y)\n\u22a2 applyId (List.zip (x' :: xs) ys) x' = y \u2194 List.get? ys Nat.zero = some y\n[PROOFSTEP]\ncases ys\n[GOAL]\ncase zero.nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\ny x' : \u03b1\nxs : List \u03b1\nh\u2080 : List.Nodup (x' :: xs)\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x' \u2192 (applyId (List.zip xs ys) x' = y \u2194 List.get? ys i = some y)\nh\u2081 : List.length (x' :: xs) = List.length []\n\u22a2 applyId (List.zip (x' :: xs) []) x' = y \u2194 List.get? [] Nat.zero = some y\n[PROOFSTEP]\ncases h\u2081\n[GOAL]\ncase zero.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\ny x' : \u03b1\nxs : List \u03b1\nh\u2080 : List.Nodup (x' :: xs)\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x' \u2192 (applyId (List.zip xs ys) x' = y \u2194 List.get? ys i = some y)\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (head\u271d :: tail\u271d)\n\u22a2 applyId (List.zip (x' :: xs) (head\u271d :: tail\u271d)) x' = y \u2194 List.get? (head\u271d :: tail\u271d) Nat.zero = some y\n[PROOFSTEP]\nsimp only [List.applyId, Prod.toSigma, Option.getD_some, List.get?, List.dlookup_cons_eq, List.zip_cons_cons, List.map,\n Option.some_inj]\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x \u2192 (applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y)\nys : List \u03b1\nh\u2080 : List.Nodup (x' :: xs)\nh\u2081 : List.length (x' :: xs) = List.length ys\nn\u271d : \u2115\nh\u2082 : List.get? (x' :: xs) (Nat.succ n\u271d) = some x\n\u22a2 applyId (List.zip (x' :: xs) ys) x = y \u2194 List.get? ys (Nat.succ n\u271d) = some y\n[PROOFSTEP]\ncases ys\n[GOAL]\ncase succ.nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x \u2192 (applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y)\nh\u2080 : List.Nodup (x' :: xs)\nn\u271d : \u2115\nh\u2082 : List.get? (x' :: xs) (Nat.succ n\u271d) = some x\nh\u2081 : List.length (x' :: xs) = List.length []\n\u22a2 applyId (List.zip (x' :: xs) []) x = y \u2194 List.get? [] (Nat.succ n\u271d) = some y\n[PROOFSTEP]\ncases h\u2081\n[GOAL]\ncase succ.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x \u2192 (applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y)\nh\u2080 : List.Nodup (x' :: xs)\nn\u271d : \u2115\nh\u2082 : List.get? (x' :: xs) (Nat.succ n\u271d) = some x\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (head\u271d :: tail\u271d)\n\u22a2 applyId (List.zip (x' :: xs) (head\u271d :: tail\u271d)) x = y \u2194 List.get? (head\u271d :: tail\u271d) (Nat.succ n\u271d) = some y\n[PROOFSTEP]\ncases' h\u2080 with _ _ h\u2080 h\u2081\n[GOAL]\ncase succ.cons.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x \u2192 (applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y)\nn\u271d : \u2115\nh\u2082 : List.get? (x' :: xs) (Nat.succ n\u271d) = some x\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh\u2081\u271d : List.length (x' :: xs) = List.length (head\u271d :: tail\u271d)\nh\u2081 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\n\u22a2 applyId (List.zip (x' :: xs) (head\u271d :: tail\u271d)) x = y \u2194 List.get? (head\u271d :: tail\u271d) (Nat.succ n\u271d) = some y\n[PROOFSTEP]\nsimp only [List.get?, List.zip_cons_cons, List.applyId_cons] at h\u2082 \u22a2\n[GOAL]\ncase succ.cons.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x \u2192 (applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y)\nn\u271d : \u2115\nh\u2082 : List.get? xs n\u271d = some x\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh\u2081\u271d : List.length (x' :: xs) = List.length (head\u271d :: tail\u271d)\nh\u2081 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\n\u22a2 (if x' = x then head\u271d else applyId (List.zip xs tail\u271d) x) = y \u2194 List.get? tail\u271d n\u271d = some y\n[PROOFSTEP]\nrw [if_neg]\n[GOAL]\ncase succ.cons.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x \u2192 (applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y)\nn\u271d : \u2115\nh\u2082 : List.get? xs n\u271d = some x\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh\u2081\u271d : List.length (x' :: xs) = List.length (head\u271d :: tail\u271d)\nh\u2081 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\n\u22a2 applyId (List.zip xs tail\u271d) x = y \u2194 List.get? tail\u271d n\u271d = some y\n[PROOFSTEP]\napply xs_ih\n[GOAL]\ncase succ.cons.cons.h\u2080\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x \u2192 (applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y)\nn\u271d : \u2115\nh\u2082 : List.get? xs n\u271d = some x\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh\u2081\u271d : List.length (x' :: xs) = List.length (head\u271d :: tail\u271d)\nh\u2081 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\n\u22a2 List.Nodup xs\n[PROOFSTEP]\nsolve_by_elim [Nat.succ.inj]\n[GOAL]\ncase succ.cons.cons.h\u2081\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x \u2192 (applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y)\nn\u271d : \u2115\nh\u2082 : List.get? xs n\u271d = some x\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh\u2081\u271d : List.length (x' :: xs) = List.length (head\u271d :: tail\u271d)\nh\u2081 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\n\u22a2 List.length xs = List.length tail\u271d\n[PROOFSTEP]\nsolve_by_elim [Nat.succ.inj]\n[GOAL]\ncase succ.cons.cons.h\u2082\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x \u2192 (applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y)\nn\u271d : \u2115\nh\u2082 : List.get? xs n\u271d = some x\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh\u2081\u271d : List.length (x' :: xs) = List.length (head\u271d :: tail\u271d)\nh\u2081 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\n\u22a2 List.get? xs n\u271d = some x\n[PROOFSTEP]\nsolve_by_elim [Nat.succ.inj]\n[GOAL]\ncase succ.cons.cons.hnc\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x \u2192 (applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y)\nn\u271d : \u2115\nh\u2082 : List.get? xs n\u271d = some x\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh\u2081\u271d : List.length (x' :: xs) = List.length (head\u271d :: tail\u271d)\nh\u2081 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\n\u22a2 \u00acx' = x\n[PROOFSTEP]\napply h\u2080\n[GOAL]\ncase succ.cons.cons.hnc.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx y x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.length xs = List.length ys \u2192\n \u2200 (i : \u2115), List.get? xs i = some x \u2192 (applyId (List.zip xs ys) x = y \u2194 List.get? ys i = some y)\nn\u271d : \u2115\nh\u2082 : List.get? xs n\u271d = some x\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh\u2081\u271d : List.length (x' :: xs) = List.length (head\u271d :: tail\u271d)\nh\u2081 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\n\u22a2 x \u2208 xs\n[PROOFSTEP]\napply List.get?_mem h\u2082\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx : \u03b1\n\u22a2 applyId (List.zip xs ys) x \u2208 ys \u2194 x \u2208 xs\n[PROOFSTEP]\nsimp only [List.applyId]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx : \u03b1\n\u22a2 Option.getD (List.dlookup x (List.map Prod.toSigma (List.zip xs ys))) x \u2208 ys \u2194 x \u2208 xs\n[PROOFSTEP]\ncases h\u2083 : List.dlookup x (List.map Prod.toSigma (xs.zip ys)) with\n| none =>\n dsimp [Option.getD]\n rw [h\u2081.mem_iff]\n| some val =>\n have h\u2082 : ys.Nodup := h\u2081.nodup_iff.1 h\u2080\n replace h\u2081 : xs.length = ys.length := h\u2081.length_eq\n dsimp\n induction xs generalizing ys with\n | nil => contradiction\n | cons x' xs xs_ih =>\n cases' ys with y ys\n \u00b7 cases h\u2083\n dsimp [List.dlookup] at h\u2083 ; split_ifs at h\u2083 with h\n \u00b7 rw [Option.some_inj] at h\u2083 \n subst x'; subst val\n simp only [List.mem_cons, true_or_iff, eq_self_iff_true]\n \u00b7 cases' h\u2080 with _ _ h\u2080 h\u2085\n cases' h\u2082 with _ _ h\u2082 h\u2084\n have h\u2086 := Nat.succ.inj h\u2081\n specialize xs_ih h\u2085 h\u2083 h\u2084 h\u2086\n simp only [Ne.symm h, xs_ih, List.mem_cons, false_or_iff]\n suffices : val \u2208 ys; tauto\n erw [\u2190 Option.mem_def, List.mem_dlookup_iff] at h\u2083 \n simp only [Prod.toSigma, List.mem_map, heq_iff_eq, Prod.exists] at h\u2083 \n rcases h\u2083 with \u27e8a, b, h\u2083, h\u2084, h\u2085\u27e9\n apply (List.mem_zip h\u2083).2\n simp only [List.NodupKeys, List.keys, comp, Prod.fst_toSigma, List.map_map]\n rwa [List.map_fst_zip _ _ (le_of_eq h\u2086)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx : \u03b1\nx\u271d : Option \u03b1\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = x\u271d\n\u22a2 Option.getD x\u271d x \u2208 ys \u2194 x \u2208 xs\n[PROOFSTEP]\ncases h\u2083 : List.dlookup x (List.map Prod.toSigma (xs.zip ys)) with\n| none =>\n dsimp [Option.getD]\n rw [h\u2081.mem_iff]\n| some val =>\n have h\u2082 : ys.Nodup := h\u2081.nodup_iff.1 h\u2080\n replace h\u2081 : xs.length = ys.length := h\u2081.length_eq\n dsimp\n induction xs generalizing ys with\n | nil => contradiction\n | cons x' xs xs_ih =>\n cases' ys with y ys\n \u00b7 cases h\u2083\n dsimp [List.dlookup] at h\u2083 ; split_ifs at h\u2083 with h\n \u00b7 rw [Option.some_inj] at h\u2083 \n subst x'; subst val\n simp only [List.mem_cons, true_or_iff, eq_self_iff_true]\n \u00b7 cases' h\u2080 with _ _ h\u2080 h\u2085\n cases' h\u2082 with _ _ h\u2082 h\u2084\n have h\u2086 := Nat.succ.inj h\u2081\n specialize xs_ih h\u2085 h\u2083 h\u2084 h\u2086\n simp only [Ne.symm h, xs_ih, List.mem_cons, false_or_iff]\n suffices : val \u2208 ys; tauto\n erw [\u2190 Option.mem_def, List.mem_dlookup_iff] at h\u2083 \n simp only [Prod.toSigma, List.mem_map, heq_iff_eq, Prod.exists] at h\u2083 \n rcases h\u2083 with \u27e8a, b, h\u2083, h\u2084, h\u2085\u27e9\n apply (List.mem_zip h\u2083).2\n simp only [List.NodupKeys, List.keys, comp, Prod.fst_toSigma, List.map_map]\n rwa [List.map_fst_zip _ _ (le_of_eq h\u2086)]\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx : \u03b1\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = none\n\u22a2 Option.getD none x \u2208 ys \u2194 x \u2208 xs\n[PROOFSTEP]\n\n| none =>\n dsimp [Option.getD]\n rw [h\u2081.mem_iff]\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx : \u03b1\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = none\n\u22a2 Option.getD none x \u2208 ys \u2194 x \u2208 xs\n[PROOFSTEP]\ndsimp [Option.getD]\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx : \u03b1\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = none\n\u22a2 x \u2208 ys \u2194 x \u2208 xs\n[PROOFSTEP]\nrw [h\u2081.mem_iff]\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx val : \u03b1\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val\n\u22a2 Option.getD (some val) x \u2208 ys \u2194 x \u2208 xs\n[PROOFSTEP]\n\n| some val =>\n have h\u2082 : ys.Nodup := h\u2081.nodup_iff.1 h\u2080\n replace h\u2081 : xs.length = ys.length := h\u2081.length_eq\n dsimp\n induction xs generalizing ys with\n | nil => contradiction\n | cons x' xs xs_ih =>\n cases' ys with y ys\n \u00b7 cases h\u2083\n dsimp [List.dlookup] at h\u2083 ; split_ifs at h\u2083 with h\n \u00b7 rw [Option.some_inj] at h\u2083 \n subst x'; subst val\n simp only [List.mem_cons, true_or_iff, eq_self_iff_true]\n \u00b7 cases' h\u2080 with _ _ h\u2080 h\u2085\n cases' h\u2082 with _ _ h\u2082 h\u2084\n have h\u2086 := Nat.succ.inj h\u2081\n specialize xs_ih h\u2085 h\u2083 h\u2084 h\u2086\n simp only [Ne.symm h, xs_ih, List.mem_cons, false_or_iff]\n suffices : val \u2208 ys; tauto\n erw [\u2190 Option.mem_def, List.mem_dlookup_iff] at h\u2083 \n simp only [Prod.toSigma, List.mem_map, heq_iff_eq, Prod.exists] at h\u2083 \n rcases h\u2083 with \u27e8a, b, h\u2083, h\u2084, h\u2085\u27e9\n apply (List.mem_zip h\u2083).2\n simp only [List.NodupKeys, List.keys, comp, Prod.fst_toSigma, List.map_map]\n rwa [List.map_fst_zip _ _ (le_of_eq h\u2086)]\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx val : \u03b1\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val\n\u22a2 Option.getD (some val) x \u2208 ys \u2194 x \u2208 xs\n[PROOFSTEP]\nhave h\u2082 : ys.Nodup := h\u2081.nodup_iff.1 h\u2080\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx val : \u03b1\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val\nh\u2082 : List.Nodup ys\n\u22a2 Option.getD (some val) x \u2208 ys \u2194 x \u2208 xs\n[PROOFSTEP]\nreplace h\u2081 : xs.length = ys.length := h\u2081.length_eq\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nx val : \u03b1\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val\nh\u2082 : List.Nodup ys\nh\u2081 : List.length xs = List.length ys\n\u22a2 Option.getD (some val) x \u2208 ys \u2194 x \u2208 xs\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nx val : \u03b1\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val\nh\u2082 : List.Nodup ys\nh\u2081 : List.length xs = List.length ys\n\u22a2 val \u2208 ys \u2194 x \u2208 xs\n[PROOFSTEP]\ninduction xs generalizing ys with\n| nil => contradiction\n| cons x' xs xs_ih =>\n cases' ys with y ys\n \u00b7 cases h\u2083\n dsimp [List.dlookup] at h\u2083 ; split_ifs at h\u2083 with h\n \u00b7 rw [Option.some_inj] at h\u2083 \n subst x'; subst val\n simp only [List.mem_cons, true_or_iff, eq_self_iff_true]\n \u00b7 cases' h\u2080 with _ _ h\u2080 h\u2085\n cases' h\u2082 with _ _ h\u2082 h\u2084\n have h\u2086 := Nat.succ.inj h\u2081\n specialize xs_ih h\u2085 h\u2083 h\u2084 h\u2086\n simp only [Ne.symm h, xs_ih, List.mem_cons, false_or_iff]\n suffices : val \u2208 ys; tauto\n erw [\u2190 Option.mem_def, List.mem_dlookup_iff] at h\u2083 \n simp only [Prod.toSigma, List.mem_map, heq_iff_eq, Prod.exists] at h\u2083 \n rcases h\u2083 with \u27e8a, b, h\u2083, h\u2084, h\u2085\u27e9\n apply (List.mem_zip h\u2083).2\n simp only [List.NodupKeys, List.keys, comp, Prod.fst_toSigma, List.map_map]\n rwa [List.map_fst_zip _ _ (le_of_eq h\u2086)]\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nx val : \u03b1\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val\nh\u2082 : List.Nodup ys\nh\u2081 : List.length xs = List.length ys\n\u22a2 val \u2208 ys \u2194 x \u2208 xs\n[PROOFSTEP]\ninduction xs generalizing ys with\n| nil => contradiction\n| cons x' xs xs_ih =>\n cases' ys with y ys\n \u00b7 cases h\u2083\n dsimp [List.dlookup] at h\u2083 ; split_ifs at h\u2083 with h\n \u00b7 rw [Option.some_inj] at h\u2083 \n subst x'; subst val\n simp only [List.mem_cons, true_or_iff, eq_self_iff_true]\n \u00b7 cases' h\u2080 with _ _ h\u2080 h\u2085\n cases' h\u2082 with _ _ h\u2082 h\u2084\n have h\u2086 := Nat.succ.inj h\u2081\n specialize xs_ih h\u2085 h\u2083 h\u2084 h\u2086\n simp only [Ne.symm h, xs_ih, List.mem_cons, false_or_iff]\n suffices : val \u2208 ys; tauto\n erw [\u2190 Option.mem_def, List.mem_dlookup_iff] at h\u2083 \n simp only [Prod.toSigma, List.mem_map, heq_iff_eq, Prod.exists] at h\u2083 \n rcases h\u2083 with \u27e8a, b, h\u2083, h\u2084, h\u2085\u27e9\n apply (List.mem_zip h\u2083).2\n simp only [List.NodupKeys, List.keys, comp, Prod.fst_toSigma, List.map_map]\n rwa [List.map_fst_zip _ _ (le_of_eq h\u2086)]\n[GOAL]\ncase some.nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val : \u03b1\nys : List \u03b1\nh\u2080 : List.Nodup []\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip [] ys)) = some val\nh\u2082 : List.Nodup ys\nh\u2081 : List.length [] = List.length ys\n\u22a2 val \u2208 ys \u2194 x \u2208 []\n[PROOFSTEP]\n\n| nil => contradiction\n[GOAL]\ncase some.nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val : \u03b1\nys : List \u03b1\nh\u2080 : List.Nodup []\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip [] ys)) = some val\nh\u2082 : List.Nodup ys\nh\u2081 : List.length [] = List.length ys\n\u22a2 val \u2208 ys \u2194 x \u2208 []\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase some.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val \u2192\n List.Nodup ys \u2192 List.length xs = List.length ys \u2192 (val \u2208 ys \u2194 x \u2208 xs)\nys : List \u03b1\nh\u2080 : List.Nodup (x' :: xs)\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip (x' :: xs) ys)) = some val\nh\u2082 : List.Nodup ys\nh\u2081 : List.length (x' :: xs) = List.length ys\n\u22a2 val \u2208 ys \u2194 x \u2208 x' :: xs\n[PROOFSTEP]\n\n| cons x' xs xs_ih =>\n cases' ys with y ys\n \u00b7 cases h\u2083\n dsimp [List.dlookup] at h\u2083 ; split_ifs at h\u2083 with h\n \u00b7 rw [Option.some_inj] at h\u2083 \n subst x'; subst val\n simp only [List.mem_cons, true_or_iff, eq_self_iff_true]\n \u00b7 cases' h\u2080 with _ _ h\u2080 h\u2085\n cases' h\u2082 with _ _ h\u2082 h\u2084\n have h\u2086 := Nat.succ.inj h\u2081\n specialize xs_ih h\u2085 h\u2083 h\u2084 h\u2086\n simp only [Ne.symm h, xs_ih, List.mem_cons, false_or_iff]\n suffices : val \u2208 ys; tauto\n erw [\u2190 Option.mem_def, List.mem_dlookup_iff] at h\u2083 \n simp only [Prod.toSigma, List.mem_map, heq_iff_eq, Prod.exists] at h\u2083 \n rcases h\u2083 with \u27e8a, b, h\u2083, h\u2084, h\u2085\u27e9\n apply (List.mem_zip h\u2083).2\n simp only [List.NodupKeys, List.keys, comp, Prod.fst_toSigma, List.map_map]\n rwa [List.map_fst_zip _ _ (le_of_eq h\u2086)]\n[GOAL]\ncase some.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val \u2192\n List.Nodup ys \u2192 List.length xs = List.length ys \u2192 (val \u2208 ys \u2194 x \u2208 xs)\nys : List \u03b1\nh\u2080 : List.Nodup (x' :: xs)\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip (x' :: xs) ys)) = some val\nh\u2082 : List.Nodup ys\nh\u2081 : List.length (x' :: xs) = List.length ys\n\u22a2 val \u2208 ys \u2194 x \u2208 x' :: xs\n[PROOFSTEP]\ncases' ys with y ys\n[GOAL]\ncase some.cons.nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val \u2192\n List.Nodup ys \u2192 List.length xs = List.length ys \u2192 (val \u2208 ys \u2194 x \u2208 xs)\nh\u2080 : List.Nodup (x' :: xs)\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip (x' :: xs) [])) = some val\nh\u2082 : List.Nodup []\nh\u2081 : List.length (x' :: xs) = List.length []\n\u22a2 val \u2208 [] \u2194 x \u2208 x' :: xs\n[PROOFSTEP]\ncases h\u2083\n[GOAL]\ncase some.cons.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val \u2192\n List.Nodup ys \u2192 List.length xs = List.length ys \u2192 (val \u2208 ys \u2194 x \u2208 xs)\nh\u2080 : List.Nodup (x' :: xs)\ny : \u03b1\nys : List \u03b1\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip (x' :: xs) (y :: ys))) = some val\nh\u2082 : List.Nodup (y :: ys)\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\n\u22a2 val \u2208 y :: ys \u2194 x \u2208 x' :: xs\n[PROOFSTEP]\ndsimp [List.dlookup] at h\u2083 \n[GOAL]\ncase some.cons.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val \u2192\n List.Nodup ys \u2192 List.length xs = List.length ys \u2192 (val \u2208 ys \u2194 x \u2208 xs)\nh\u2080 : List.Nodup (x' :: xs)\ny : \u03b1\nys : List \u03b1\nh\u2083 : (if h : x' = x then some (h \u25b8 y) else List.dlookup x (List.map Prod.toSigma (List.zip xs ys))) = some val\nh\u2082 : List.Nodup (y :: ys)\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\n\u22a2 val \u2208 y :: ys \u2194 x \u2208 x' :: xs\n[PROOFSTEP]\nsplit_ifs at h\u2083 with h\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val \u2192\n List.Nodup ys \u2192 List.length xs = List.length ys \u2192 (val \u2208 ys \u2194 x \u2208 xs)\nh\u2080 : List.Nodup (x' :: xs)\ny : \u03b1\nys : List \u03b1\nh\u2082 : List.Nodup (y :: ys)\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : x' = x\nh\u2083 : some (h \u25b8 y) = some val\n\u22a2 val \u2208 y :: ys \u2194 x \u2208 x' :: xs\n[PROOFSTEP]\nrw [Option.some_inj] at h\u2083 \n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val \u2192\n List.Nodup ys \u2192 List.length xs = List.length ys \u2192 (val \u2208 ys \u2194 x \u2208 xs)\nh\u2080 : List.Nodup (x' :: xs)\ny : \u03b1\nys : List \u03b1\nh\u2082 : List.Nodup (y :: ys)\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : x' = x\nh\u2083 : h \u25b8 y = val\n\u22a2 val \u2208 y :: ys \u2194 x \u2208 x' :: xs\n[PROOFSTEP]\nsubst x'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val \u2192\n List.Nodup ys \u2192 List.length xs = List.length ys \u2192 (val \u2208 ys \u2194 x \u2208 xs)\ny : \u03b1\nys : List \u03b1\nh\u2082 : List.Nodup (y :: ys)\nh\u2080 : List.Nodup (x :: xs)\nh\u2081 : List.length (x :: xs) = List.length (y :: ys)\nh\u2083 : (_ : x = x) \u25b8 y = val\n\u22a2 val \u2208 y :: ys \u2194 x \u2208 x :: xs\n[PROOFSTEP]\nsubst val\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\nh\u2082 : List.Nodup (y :: ys)\nh\u2080 : List.Nodup (x :: xs)\nh\u2081 : List.length (x :: xs) = List.length (y :: ys)\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some ((_ : x = x) \u25b8 y) \u2192\n List.Nodup ys \u2192 List.length xs = List.length ys \u2192 ((_ : x = x) \u25b8 y \u2208 ys \u2194 x \u2208 xs)\n\u22a2 (_ : x = x) \u25b8 y \u2208 y :: ys \u2194 x \u2208 x :: xs\n[PROOFSTEP]\nsimp only [List.mem_cons, true_or_iff, eq_self_iff_true]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val \u2192\n List.Nodup ys \u2192 List.length xs = List.length ys \u2192 (val \u2208 ys \u2194 x \u2208 xs)\nh\u2080 : List.Nodup (x' :: xs)\ny : \u03b1\nys : List \u03b1\nh\u2082 : List.Nodup (y :: ys)\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val\n\u22a2 val \u2208 y :: ys \u2194 x \u2208 x' :: xs\n[PROOFSTEP]\ncases' h\u2080 with _ _ h\u2080 h\u2085\n[GOAL]\ncase neg.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val \u2192\n List.Nodup ys \u2192 List.length xs = List.length ys \u2192 (val \u2208 ys \u2194 x \u2208 xs)\ny : \u03b1\nys : List \u03b1\nh\u2082 : List.Nodup (y :: ys)\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\n\u22a2 val \u2208 y :: ys \u2194 x \u2208 x' :: xs\n[PROOFSTEP]\ncases' h\u2082 with _ _ h\u2082 h\u2084\n[GOAL]\ncase neg.cons.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val \u2192\n List.Nodup ys \u2192 List.length xs = List.length ys \u2192 (val \u2208 ys \u2194 x \u2208 xs)\ny : \u03b1\nys : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\nh\u2084 : List.Pairwise (fun x x_1 => x \u2260 x_1) ys\nh\u2082 : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y \u2260 a'\n\u22a2 val \u2208 y :: ys \u2194 x \u2208 x' :: xs\n[PROOFSTEP]\nhave h\u2086 := Nat.succ.inj h\u2081\n[GOAL]\ncase neg.cons.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\nxs_ih :\n \u2200 {ys : List \u03b1},\n List.Nodup xs \u2192\n List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val \u2192\n List.Nodup ys \u2192 List.length xs = List.length ys \u2192 (val \u2208 ys \u2194 x \u2208 xs)\ny : \u03b1\nys : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\nh\u2084 : List.Pairwise (fun x x_1 => x \u2260 x_1) ys\nh\u2082 : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y \u2260 a'\nh\u2086 : List.length xs = List.length ys\n\u22a2 val \u2208 y :: ys \u2194 x \u2208 x' :: xs\n[PROOFSTEP]\nspecialize xs_ih h\u2085 h\u2083 h\u2084 h\u2086\n[GOAL]\ncase neg.cons.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\nh\u2084 : List.Pairwise (fun x x_1 => x \u2260 x_1) ys\nh\u2082 : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y \u2260 a'\nh\u2086 : List.length xs = List.length ys\nxs_ih : val \u2208 ys \u2194 x \u2208 xs\n\u22a2 val \u2208 y :: ys \u2194 x \u2208 x' :: xs\n[PROOFSTEP]\nsimp only [Ne.symm h, xs_ih, List.mem_cons, false_or_iff]\n[GOAL]\ncase neg.cons.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\nh\u2084 : List.Pairwise (fun x x_1 => x \u2260 x_1) ys\nh\u2082 : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y \u2260 a'\nh\u2086 : List.length xs = List.length ys\nxs_ih : val \u2208 ys \u2194 x \u2208 xs\n\u22a2 val = y \u2228 x \u2208 xs \u2194 x \u2208 xs\n[PROOFSTEP]\nsuffices : val \u2208 ys\n[GOAL]\ncase neg.cons.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\nh\u2084 : List.Pairwise (fun x x_1 => x \u2260 x_1) ys\nh\u2082 : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y \u2260 a'\nh\u2086 : List.length xs = List.length ys\nxs_ih : val \u2208 ys \u2194 x \u2208 xs\nthis : val \u2208 ys\n\u22a2 val = y \u2228 x \u2208 xs \u2194 x \u2208 xs\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\nh\u2084 : List.Pairwise (fun x x_1 => x \u2260 x_1) ys\nh\u2082 : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y \u2260 a'\nh\u2086 : List.length xs = List.length ys\nxs_ih : val \u2208 ys \u2194 x \u2208 xs\n\u22a2 val \u2208 ys\n[PROOFSTEP]\ntauto\n[GOAL]\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2083 : List.dlookup x (List.map Prod.toSigma (List.zip xs ys)) = some val\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\nh\u2084 : List.Pairwise (fun x x_1 => x \u2260 x_1) ys\nh\u2082 : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y \u2260 a'\nh\u2086 : List.length xs = List.length ys\nxs_ih : val \u2208 ys \u2194 x \u2208 xs\n\u22a2 val \u2208 ys\n[PROOFSTEP]\nerw [\u2190 Option.mem_def, List.mem_dlookup_iff] at h\u2083 \n[GOAL]\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2083 : { fst := x, snd := val } \u2208 List.map Prod.toSigma (List.zip xs ys)\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\nh\u2084 : List.Pairwise (fun x x_1 => x \u2260 x_1) ys\nh\u2082 : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y \u2260 a'\nh\u2086 : List.length xs = List.length ys\nxs_ih : val \u2208 ys \u2194 x \u2208 xs\n\u22a2 val \u2208 ys\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2083 : val \u2208 List.dlookup x (List.map Prod.toSigma (List.zip xs ys))\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\nh\u2084 : List.Pairwise (fun x x_1 => x \u2260 x_1) ys\nh\u2082 : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y \u2260 a'\nh\u2086 : List.length xs = List.length ys\nxs_ih : val \u2208 ys \u2194 x \u2208 xs\n\u22a2 List.NodupKeys (List.map Prod.toSigma (List.zip xs ys))\n[PROOFSTEP]\nsimp only [Prod.toSigma, List.mem_map, heq_iff_eq, Prod.exists] at h\u2083 \n[GOAL]\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\nh\u2084 : List.Pairwise (fun x x_1 => x \u2260 x_1) ys\nh\u2082 : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y \u2260 a'\nh\u2086 : List.length xs = List.length ys\nxs_ih : val \u2208 ys \u2194 x \u2208 xs\nh\u2083 : \u2203 a b, (a, b) \u2208 List.zip xs ys \u2227 { fst := a, snd := b } = { fst := x, snd := val }\n\u22a2 val \u2208 ys\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2083 : val \u2208 List.dlookup x (List.map Prod.toSigma (List.zip xs ys))\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\nh\u2084 : List.Pairwise (fun x x_1 => x \u2260 x_1) ys\nh\u2082 : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y \u2260 a'\nh\u2086 : List.length xs = List.length ys\nxs_ih : val \u2208 ys \u2194 x \u2208 xs\n\u22a2 List.NodupKeys (List.map Prod.toSigma (List.zip xs ys))\n[PROOFSTEP]\nrcases h\u2083 with \u27e8a, b, h\u2083, h\u2084, h\u2085\u27e9\n[GOAL]\ncase this.intro.intro.intro.refl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\nh\u2084 : List.Pairwise (fun x x_1 => x \u2260 x_1) ys\nh\u2082 : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y \u2260 a'\nh\u2086 : List.length xs = List.length ys\nxs_ih : val \u2208 ys \u2194 x \u2208 xs\nh\u2083 : (x, val) \u2208 List.zip xs ys\n\u22a2 val \u2208 ys\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2083 : val \u2208 List.dlookup x (List.map Prod.toSigma (List.zip xs ys))\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\nh\u2084 : List.Pairwise (fun x x_1 => x \u2260 x_1) ys\nh\u2082 : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y \u2260 a'\nh\u2086 : List.length xs = List.length ys\nxs_ih : val \u2208 ys \u2194 x \u2208 xs\n\u22a2 List.NodupKeys (List.map Prod.toSigma (List.zip xs ys))\n[PROOFSTEP]\napply (List.mem_zip h\u2083).2\n[GOAL]\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2083 : val \u2208 List.dlookup x (List.map Prod.toSigma (List.zip xs ys))\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\nh\u2084 : List.Pairwise (fun x x_1 => x \u2260 x_1) ys\nh\u2082 : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y \u2260 a'\nh\u2086 : List.length xs = List.length ys\nxs_ih : val \u2208 ys \u2194 x \u2208 xs\n\u22a2 List.NodupKeys (List.map Prod.toSigma (List.zip xs ys))\n[PROOFSTEP]\nsimp only [List.NodupKeys, List.keys, comp, Prod.fst_toSigma, List.map_map]\n[GOAL]\ncase this\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nx val x' : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\nh\u2081 : List.length (x' :: xs) = List.length (y :: ys)\nh : \u00acx' = x\nh\u2083 : val \u2208 List.dlookup x (List.map Prod.toSigma (List.zip xs ys))\nh\u2085 : List.Pairwise (fun x x_1 => x \u2260 x_1) xs\nh\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x' \u2260 a'\nh\u2084 : List.Pairwise (fun x x_1 => x \u2260 x_1) ys\nh\u2082 : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y \u2260 a'\nh\u2086 : List.length xs = List.length ys\nxs_ih : val \u2208 ys \u2194 x \u2208 xs\n\u22a2 List.Nodup (List.map (fun x => x.fst) (List.zip xs ys))\n[PROOFSTEP]\nrwa [List.map_fst_zip _ _ (le_of_eq h\u2086)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nx : \u03b1\n\u22a2 \u00acx \u2208 xs \u2192 applyId (List.zip xs ys) x = x\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nx : \u03b1\nh : \u00acx \u2208 xs\n\u22a2 applyId (List.zip xs ys) x = x\n[PROOFSTEP]\ndsimp [List.applyId]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nx : \u03b1\nh : \u00acx \u2208 xs\n\u22a2 Option.getD (List.dlookup x (List.map Prod.toSigma (List.zip xs ys))) x = x\n[PROOFSTEP]\nrw [List.dlookup_eq_none.2]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nx : \u03b1\nh : \u00acx \u2208 xs\n\u22a2 Option.getD none x = x\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nx : \u03b1\nh : \u00acx \u2208 xs\n\u22a2 \u00acx \u2208 List.keys (List.map Prod.toSigma (List.zip xs ys))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nx : \u03b1\nh : \u00acx \u2208 xs\n\u22a2 \u00acx \u2208 List.keys (List.map Prod.toSigma (List.zip xs ys))\n[PROOFSTEP]\nsimp only [List.keys, not_exists, Prod.toSigma, exists_and_right, exists_eq_right, List.mem_map, Function.comp_apply,\n List.map_map, Prod.exists]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nx : \u03b1\nh : \u00acx \u2208 xs\n\u22a2 \u2200 (x_1 : \u03b1), \u00ac(x, x_1) \u2208 List.zip xs ys\n[PROOFSTEP]\nintro y hy\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nx : \u03b1\nh : \u00acx \u2208 xs\ny : \u03b1\nhy : (x, y) \u2208 List.zip xs ys\n\u22a2 False\n[PROOFSTEP]\nexact h (List.mem_zip hy).1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\n\u22a2 Injective (applyId (List.zip xs ys))\n[PROOFSTEP]\nintro x y h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\n\u22a2 x = y\n[PROOFSTEP]\nby_cases hx : x \u2208 xs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : x \u2208 xs\n\u22a2 x = y\n[PROOFSTEP]\nby_cases hy : y \u2208 xs\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : \u00acx \u2208 xs\n\u22a2 x = y\n[PROOFSTEP]\nby_cases hy : y \u2208 xs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : x \u2208 xs\nhy : y \u2208 xs\n\u22a2 x = y\n[PROOFSTEP]\nrw [List.mem_iff_get?] at hx hy \n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : \u2203 n, List.get? xs n = some x\nhy : \u2203 n, List.get? xs n = some y\n\u22a2 x = y\n[PROOFSTEP]\ncases' hx with i hx\n[GOAL]\ncase pos.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhy : \u2203 n, List.get? xs n = some y\ni : \u2115\nhx : List.get? xs i = some x\n\u22a2 x = y\n[PROOFSTEP]\ncases' hy with j hy\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\ni : \u2115\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\n\u22a2 x = y\n[PROOFSTEP]\nsuffices some x = some y by injection this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\ni : \u2115\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nthis : some x = some y\n\u22a2 x = y\n[PROOFSTEP]\ninjection this\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\ni : \u2115\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\n\u22a2 some x = some y\n[PROOFSTEP]\nhave h\u2082 := h\u2081.length_eq\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\ni : \u2115\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 some x = some y\n[PROOFSTEP]\nrw [List.applyId_zip_eq h\u2080 h\u2082 _ _ _ hx] at h \n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 some x = some y\n[PROOFSTEP]\nrw [\u2190 hx, \u2190 hy]\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 List.get? xs i = List.get? xs j\n[PROOFSTEP]\ncongr\n[GOAL]\ncase pos.intro.intro.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 i = j\n[PROOFSTEP]\napply List.get?_injective _ (h\u2081.nodup_iff.1 h\u2080)\n[GOAL]\ncase pos.intro.intro.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 List.get? ys i = List.get? ys j\n[PROOFSTEP]\nsymm\n[GOAL]\ncase pos.intro.intro.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 List.get? ys j = List.get? ys i\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase pos.intro.intro.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 List.get? ys j = some (applyId (List.zip xs ys) y)\n[PROOFSTEP]\nrw [\u2190 List.applyId_zip_eq]\n[GOAL]\ncase pos.intro.intro.e_a.h\u2080\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 List.Nodup xs\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.intro.intro.e_a.h\u2081\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 List.length xs = List.length ys\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.intro.intro.e_a.h\u2082\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 List.get? xs j = some y\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 i < List.length ys\n[PROOFSTEP]\nrw [\u2190 h\u2081.length_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 i < List.length xs\n[PROOFSTEP]\nrw [List.get?_eq_some] at hx \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : \u2203 h, List.get xs { val := i, isLt := h } = x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 i < List.length xs\n[PROOFSTEP]\ncases' hx with hx hx'\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\nhx : i < List.length xs\nhx' : List.get xs { val := i, isLt := hx } = x\n\u22a2 i < List.length xs\n[PROOFSTEP]\nexact hx\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : x \u2208 xs\nhy : \u00acy \u2208 xs\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 applyId_mem_iff h\u2080 h\u2081] at hx hy \n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : applyId (List.zip xs ys) x \u2208 ys\nhy : \u00acapplyId (List.zip xs ys) y \u2208 ys\n\u22a2 x = y\n[PROOFSTEP]\nrw [h] at hx \n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : applyId (List.zip xs ys) y \u2208 ys\nhy : \u00acapplyId (List.zip xs ys) y \u2208 ys\n\u22a2 x = y\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : \u00acx \u2208 xs\nhy : y \u2208 xs\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 applyId_mem_iff h\u2080 h\u2081] at hx hy \n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : \u00acapplyId (List.zip xs ys) x \u2208 ys\nhy : applyId (List.zip xs ys) y \u2208 ys\n\u22a2 x = y\n[PROOFSTEP]\nrw [h] at hx \n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : \u00acapplyId (List.zip xs ys) y \u2208 ys\nhy : applyId (List.zip xs ys) y \u2208 ys\n\u22a2 x = y\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : \u00acx \u2208 xs\nhy : \u00acy \u2208 xs\n\u22a2 x = y\n[PROOFSTEP]\nrwa [List.applyId_eq_self, List.applyId_eq_self] at h \n[GOAL]\ncase neg.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : x = applyId (List.zip xs ys) y\nhx : \u00acx \u2208 xs\nhy : \u00acy \u2208 xs\n\u22a2 \u00acy \u2208 xs\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : \u00acx \u2208 xs\nhy : \u00acy \u2208 xs\n\u22a2 \u00acx \u2208 xs\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\nx\u271d : \u2115\nn : \u2115 := x\u271d\nh : 0 < n\n\u22a2 1 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\n\u03b1 : Type\ninst\u271d : DecidableEq \u03b1\nxs : List ((_ : \u03b1) \u00d7 \u03b1)\nh\u2080\u271d : List.map Sigma.fst xs ~ List.map Sigma.snd xs\nh\u2081\u271d : List.Nodup (List.map Sigma.snd xs)\nxs' ys' : List \u03b1\nh\u2080 : xs' ~ ys'\nh\u2081 : List.Nodup ys'\nh\u2083 : List.length xs' \u2264 List.length ys'\nh\u2084 : List.length ys' \u2264 List.length xs'\n\u22a2 List.map Sigma.fst (List.map Prod.toSigma (List.zip xs' ys')) ~\n List.map Sigma.snd (List.map Prod.toSigma (List.zip xs' ys'))\n[PROOFSTEP]\nsimp only [comp, List.map_fst_zip, List.map_snd_zip, *, Prod.fst_toSigma, Prod.snd_toSigma, List.map_map]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\n\u03b1 : Type\ninst\u271d : DecidableEq \u03b1\nxs : List ((_ : \u03b1) \u00d7 \u03b1)\nh\u2080\u271d : List.map Sigma.fst xs ~ List.map Sigma.snd xs\nh\u2081\u271d : List.Nodup (List.map Sigma.snd xs)\nxs' ys' : List \u03b1\nh\u2080 : xs' ~ ys'\nh\u2081 : List.Nodup ys'\nh\u2083 : List.length xs' \u2264 List.length ys'\nh\u2084 : List.length ys' \u2264 List.length xs'\n\u22a2 List.Nodup (List.map Sigma.snd (List.map Prod.toSigma (List.zip xs' ys')))\n[PROOFSTEP]\nsimp only [comp, List.map_snd_zip, *, Prod.snd_toSigma, List.map_map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\nxs ys : List \u03b1\nh : xs ~ ys\nh' : List.Nodup ys\nh\u2080 : List.length xs \u2264 List.length ys\nh\u2081 : List.length ys \u2264 List.length xs\n\u22a2 List.map Sigma.fst (List.toFinmap' (List.zip xs ys)) ~ List.map Sigma.snd (List.toFinmap' (List.zip xs ys))\n[PROOFSTEP]\nsimp only [List.toFinmap', comp, List.map_fst_zip, List.map_snd_zip, *, Prod.fst_toSigma, Prod.snd_toSigma,\n List.map_map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\nxs ys : List \u03b1\nh : xs ~ ys\nh' : List.Nodup ys\nh\u2080 : List.length xs \u2264 List.length ys\nh\u2081 : List.length ys \u2264 List.length xs\n\u22a2 List.Nodup (List.map Sigma.snd (List.toFinmap' (List.zip xs ys)))\n[PROOFSTEP]\nsimp only [List.toFinmap', comp, List.map_snd_zip, *, Prod.snd_toSigma, List.map_map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nf : InjectiveFunction \u03b1\n\u22a2 Injective (apply f)\n[PROOFSTEP]\ncases' f with xs hperm hnodup\n[GOAL]\ncase mapToSelf\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List ((_ : \u03b1) \u00d7 \u03b1)\nhperm : List.map Sigma.fst xs ~ List.map Sigma.snd xs\nhnodup : List.Nodup (List.map Sigma.snd xs)\n\u22a2 Injective (apply (mapToSelf xs hperm hnodup))\n[PROOFSTEP]\ngeneralize h\u2080 : List.map Sigma.fst xs = xs\u2080\n[GOAL]\ncase mapToSelf\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List ((_ : \u03b1) \u00d7 \u03b1)\nhperm : List.map Sigma.fst xs ~ List.map Sigma.snd xs\nhnodup : List.Nodup (List.map Sigma.snd xs)\nxs\u2080 : List \u03b1\nh\u2080 : List.map Sigma.fst xs = xs\u2080\n\u22a2 Injective (apply (mapToSelf xs hperm hnodup))\n[PROOFSTEP]\ngeneralize h\u2081 : xs.map (@id ((\u03a3 _ : \u03b1, \u03b1) \u2192 \u03b1) <| @Sigma.snd \u03b1 fun _ : \u03b1 => \u03b1) = xs\u2081\n[GOAL]\ncase mapToSelf\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List ((_ : \u03b1) \u00d7 \u03b1)\nhperm : List.map Sigma.fst xs ~ List.map Sigma.snd xs\nhnodup : List.Nodup (List.map Sigma.snd xs)\nxs\u2080 : List \u03b1\nh\u2080 : List.map Sigma.fst xs = xs\u2080\nxs\u2081 : List \u03b1\nh\u2081 : List.map (id Sigma.snd) xs = xs\u2081\n\u22a2 Injective (apply (mapToSelf xs hperm hnodup))\n[PROOFSTEP]\ndsimp [id] at h\u2081 \n[GOAL]\ncase mapToSelf\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List ((_ : \u03b1) \u00d7 \u03b1)\nhperm : List.map Sigma.fst xs ~ List.map Sigma.snd xs\nhnodup : List.Nodup (List.map Sigma.snd xs)\nxs\u2080 : List \u03b1\nh\u2080 : List.map Sigma.fst xs = xs\u2080\nxs\u2081 : List \u03b1\nh\u2081 : List.map Sigma.snd xs = xs\u2081\n\u22a2 Injective (apply (mapToSelf xs hperm hnodup))\n[PROOFSTEP]\nhave hxs : xs = TotalFunction.List.toFinmap' (xs\u2080.zip xs\u2081) :=\n by\n rw [\u2190 h\u2080, \u2190 h\u2081, List.toFinmap']; clear h\u2080 h\u2081 xs\u2080 xs\u2081 hperm hnodup\n induction xs\n case nil => simp only [List.zip_nil_right, List.map_nil]\n case cons xs_hd xs_tl\n xs_ih =>\n simp only [true_and_iff, Prod.toSigma, eq_self_iff_true, Sigma.eta, List.zip_cons_cons, List.map, List.cons_inj]\n exact xs_ih\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List ((_ : \u03b1) \u00d7 \u03b1)\nhperm : List.map Sigma.fst xs ~ List.map Sigma.snd xs\nhnodup : List.Nodup (List.map Sigma.snd xs)\nxs\u2080 : List \u03b1\nh\u2080 : List.map Sigma.fst xs = xs\u2080\nxs\u2081 : List \u03b1\nh\u2081 : List.map Sigma.snd xs = xs\u2081\n\u22a2 xs = List.toFinmap' (List.zip xs\u2080 xs\u2081)\n[PROOFSTEP]\nrw [\u2190 h\u2080, \u2190 h\u2081, List.toFinmap']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List ((_ : \u03b1) \u00d7 \u03b1)\nhperm : List.map Sigma.fst xs ~ List.map Sigma.snd xs\nhnodup : List.Nodup (List.map Sigma.snd xs)\nxs\u2080 : List \u03b1\nh\u2080 : List.map Sigma.fst xs = xs\u2080\nxs\u2081 : List \u03b1\nh\u2081 : List.map Sigma.snd xs = xs\u2081\n\u22a2 xs = List.map Prod.toSigma (List.zip (List.map Sigma.fst xs) (List.map Sigma.snd xs))\n[PROOFSTEP]\nclear h\u2080 h\u2081 xs\u2080 xs\u2081 hperm hnodup\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List ((_ : \u03b1) \u00d7 \u03b1)\n\u22a2 xs = List.map Prod.toSigma (List.zip (List.map Sigma.fst xs) (List.map Sigma.snd xs))\n[PROOFSTEP]\ninduction xs\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\n\u22a2 [] = List.map Prod.toSigma (List.zip (List.map Sigma.fst []) (List.map Sigma.snd []))\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nhead\u271d : (_ : \u03b1) \u00d7 \u03b1\ntail\u271d : List ((_ : \u03b1) \u00d7 \u03b1)\ntail_ih\u271d : tail\u271d = List.map Prod.toSigma (List.zip (List.map Sigma.fst tail\u271d) (List.map Sigma.snd tail\u271d))\n\u22a2 head\u271d :: tail\u271d =\n List.map Prod.toSigma (List.zip (List.map Sigma.fst (head\u271d :: tail\u271d)) (List.map Sigma.snd (head\u271d :: tail\u271d)))\n[PROOFSTEP]\ncase nil => simp only [List.zip_nil_right, List.map_nil]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\n\u22a2 [] = List.map Prod.toSigma (List.zip (List.map Sigma.fst []) (List.map Sigma.snd []))\n[PROOFSTEP]\ncase nil => simp only [List.zip_nil_right, List.map_nil]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\n\u22a2 [] = List.map Prod.toSigma (List.zip (List.map Sigma.fst []) (List.map Sigma.snd []))\n[PROOFSTEP]\nsimp only [List.zip_nil_right, List.map_nil]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nhead\u271d : (_ : \u03b1) \u00d7 \u03b1\ntail\u271d : List ((_ : \u03b1) \u00d7 \u03b1)\ntail_ih\u271d : tail\u271d = List.map Prod.toSigma (List.zip (List.map Sigma.fst tail\u271d) (List.map Sigma.snd tail\u271d))\n\u22a2 head\u271d :: tail\u271d =\n List.map Prod.toSigma (List.zip (List.map Sigma.fst (head\u271d :: tail\u271d)) (List.map Sigma.snd (head\u271d :: tail\u271d)))\n[PROOFSTEP]\ncase cons xs_hd xs_tl\n xs_ih =>\n simp only [true_and_iff, Prod.toSigma, eq_self_iff_true, Sigma.eta, List.zip_cons_cons, List.map, List.cons_inj]\n exact xs_ih\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs_hd : (_ : \u03b1) \u00d7 \u03b1\nxs_tl : List ((_ : \u03b1) \u00d7 \u03b1)\nxs_ih : xs_tl = List.map Prod.toSigma (List.zip (List.map Sigma.fst xs_tl) (List.map Sigma.snd xs_tl))\n\u22a2 xs_hd :: xs_tl =\n List.map Prod.toSigma (List.zip (List.map Sigma.fst (xs_hd :: xs_tl)) (List.map Sigma.snd (xs_hd :: xs_tl)))\n[PROOFSTEP]\ncase cons xs_hd xs_tl\n xs_ih =>\n simp only [true_and_iff, Prod.toSigma, eq_self_iff_true, Sigma.eta, List.zip_cons_cons, List.map, List.cons_inj]\n exact xs_ih\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs_hd : (_ : \u03b1) \u00d7 \u03b1\nxs_tl : List ((_ : \u03b1) \u00d7 \u03b1)\nxs_ih : xs_tl = List.map Prod.toSigma (List.zip (List.map Sigma.fst xs_tl) (List.map Sigma.snd xs_tl))\n\u22a2 xs_hd :: xs_tl =\n List.map Prod.toSigma (List.zip (List.map Sigma.fst (xs_hd :: xs_tl)) (List.map Sigma.snd (xs_hd :: xs_tl)))\n[PROOFSTEP]\nsimp only [true_and_iff, Prod.toSigma, eq_self_iff_true, Sigma.eta, List.zip_cons_cons, List.map, List.cons_inj]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs_hd : (_ : \u03b1) \u00d7 \u03b1\nxs_tl : List ((_ : \u03b1) \u00d7 \u03b1)\nxs_ih : xs_tl = List.map Prod.toSigma (List.zip (List.map Sigma.fst xs_tl) (List.map Sigma.snd xs_tl))\n\u22a2 xs_tl =\n List.map (fun p => { fst := p.fst, snd := p.snd })\n (List.zipWith Prod.mk (List.map Sigma.fst xs_tl) (List.map Sigma.snd xs_tl))\n[PROOFSTEP]\nexact xs_ih\n[GOAL]\ncase mapToSelf\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List ((_ : \u03b1) \u00d7 \u03b1)\nhperm : List.map Sigma.fst xs ~ List.map Sigma.snd xs\nhnodup : List.Nodup (List.map Sigma.snd xs)\nxs\u2080 : List \u03b1\nh\u2080 : List.map Sigma.fst xs = xs\u2080\nxs\u2081 : List \u03b1\nh\u2081 : List.map Sigma.snd xs = xs\u2081\nhxs : xs = List.toFinmap' (List.zip xs\u2080 xs\u2081)\n\u22a2 Injective (apply (mapToSelf xs hperm hnodup))\n[PROOFSTEP]\nrevert hperm hnodup\n[GOAL]\ncase mapToSelf\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List ((_ : \u03b1) \u00d7 \u03b1)\nxs\u2080 : List \u03b1\nh\u2080 : List.map Sigma.fst xs = xs\u2080\nxs\u2081 : List \u03b1\nh\u2081 : List.map Sigma.snd xs = xs\u2081\nhxs : xs = List.toFinmap' (List.zip xs\u2080 xs\u2081)\n\u22a2 \u2200 (hperm : List.map Sigma.fst xs ~ List.map Sigma.snd xs) (hnodup : List.Nodup (List.map Sigma.snd xs)),\n Injective (apply (mapToSelf xs hperm hnodup))\n[PROOFSTEP]\nrw [hxs]\n[GOAL]\ncase mapToSelf\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List ((_ : \u03b1) \u00d7 \u03b1)\nxs\u2080 : List \u03b1\nh\u2080 : List.map Sigma.fst xs = xs\u2080\nxs\u2081 : List \u03b1\nh\u2081 : List.map Sigma.snd xs = xs\u2081\nhxs : xs = List.toFinmap' (List.zip xs\u2080 xs\u2081)\n\u22a2 \u2200\n (hperm :\n List.map Sigma.fst (List.toFinmap' (List.zip xs\u2080 xs\u2081)) ~ List.map Sigma.snd (List.toFinmap' (List.zip xs\u2080 xs\u2081)))\n (hnodup : List.Nodup (List.map Sigma.snd (List.toFinmap' (List.zip xs\u2080 xs\u2081)))),\n Injective (apply (mapToSelf (List.toFinmap' (List.zip xs\u2080 xs\u2081)) hperm hnodup))\n[PROOFSTEP]\nintros hperm hnodup\n[GOAL]\ncase mapToSelf\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List ((_ : \u03b1) \u00d7 \u03b1)\nxs\u2080 : List \u03b1\nh\u2080 : List.map Sigma.fst xs = xs\u2080\nxs\u2081 : List \u03b1\nh\u2081 : List.map Sigma.snd xs = xs\u2081\nhxs : xs = List.toFinmap' (List.zip xs\u2080 xs\u2081)\nhperm : List.map Sigma.fst (List.toFinmap' (List.zip xs\u2080 xs\u2081)) ~ List.map Sigma.snd (List.toFinmap' (List.zip xs\u2080 xs\u2081))\nhnodup : List.Nodup (List.map Sigma.snd (List.toFinmap' (List.zip xs\u2080 xs\u2081)))\n\u22a2 Injective (apply (mapToSelf (List.toFinmap' (List.zip xs\u2080 xs\u2081)) hperm hnodup))\n[PROOFSTEP]\napply InjectiveFunction.applyId_injective\n[GOAL]\ncase mapToSelf.h\u2080\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List ((_ : \u03b1) \u00d7 \u03b1)\nxs\u2080 : List \u03b1\nh\u2080 : List.map Sigma.fst xs = xs\u2080\nxs\u2081 : List \u03b1\nh\u2081 : List.map Sigma.snd xs = xs\u2081\nhxs : xs = List.toFinmap' (List.zip xs\u2080 xs\u2081)\nhperm : List.map Sigma.fst (List.toFinmap' (List.zip xs\u2080 xs\u2081)) ~ List.map Sigma.snd (List.toFinmap' (List.zip xs\u2080 xs\u2081))\nhnodup : List.Nodup (List.map Sigma.snd (List.toFinmap' (List.zip xs\u2080 xs\u2081)))\n\u22a2 List.Nodup xs\u2080\n[PROOFSTEP]\nrwa [\u2190 h\u2080, hxs, hperm.nodup_iff]\n[GOAL]\ncase mapToSelf.h\u2081\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs : List ((_ : \u03b1) \u00d7 \u03b1)\nxs\u2080 : List \u03b1\nh\u2080 : List.map Sigma.fst xs = xs\u2080\nxs\u2081 : List \u03b1\nh\u2081 : List.map Sigma.snd xs = xs\u2081\nhxs : xs = List.toFinmap' (List.zip xs\u2080 xs\u2081)\nhperm : List.map Sigma.fst (List.toFinmap' (List.zip xs\u2080 xs\u2081)) ~ List.map Sigma.snd (List.toFinmap' (List.zip xs\u2080 xs\u2081))\nhnodup : List.Nodup (List.map Sigma.snd (List.toFinmap' (List.zip xs\u2080 xs\u2081)))\n\u22a2 xs\u2080 ~ xs\u2081\n[PROOFSTEP]\nrwa [\u2190 hxs, h\u2080, h\u2081] at hperm \n", "meta": {"mathlib_filename": "Mathlib.Testing.SlimCheck.Functions", "llama_tokens": 31510, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737214979745, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.4096728019800014}} {"text": "[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp ppred pred : Pair K\nnth_s_eq : Stream'.Seq.get? g.s n = some gp\nnth_conts_aux_eq : continuantsAux g n = ppred\nsucc_nth_conts_aux_eq : continuantsAux g (n + 1) = pred\n\u22a2 continuantsAux g (n + 2) = { a := gp.b * pred.a + gp.a * ppred.a, b := gp.b * pred.b + gp.a * ppred.b }\n[PROOFSTEP]\nsimp [*, continuantsAux, nextContinuants, nextDenominator, nextNumerator]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp ppred pred : Pair K\nnth_s_eq : Stream'.Seq.get? g.s n = some gp\nnth_conts_aux_eq : continuantsAux g n = ppred\nsucc_nth_conts_aux_eq : continuantsAux g (n + 1) = pred\n\u22a2 continuants g (n + 1) = { a := gp.b * pred.a + gp.a * ppred.a, b := gp.b * pred.b + gp.a * ppred.b }\n[PROOFSTEP]\nsimp [nth_cont_eq_succ_nth_cont_aux, continuantsAux_recurrence nth_s_eq nth_conts_aux_eq succ_nth_conts_aux_eq]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp ppred pred : Pair K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nnth_conts_eq : continuants g n = ppred\nsucc_nth_conts_eq : continuants g (n + 1) = pred\n\u22a2 continuants g (n + 2) = { a := gp.b * pred.a + gp.a * ppred.a, b := gp.b * pred.b + gp.a * ppred.b }\n[PROOFSTEP]\nrw [nth_cont_eq_succ_nth_cont_aux] at nth_conts_eq succ_nth_conts_eq \n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp ppred pred : Pair K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nnth_conts_eq : continuantsAux g (n + 1) = ppred\nsucc_nth_conts_eq : continuantsAux g (n + 1 + 1) = pred\n\u22a2 continuants g (n + 2) = { a := gp.b * pred.a + gp.a * ppred.a, b := gp.b * pred.b + gp.a * ppred.b }\n[PROOFSTEP]\nexact continuants_recurrenceAux succ_nth_s_eq nth_conts_eq succ_nth_conts_eq\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\nppredA predA : K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nnth_num_eq : numerators g n = ppredA\nsucc_nth_num_eq : numerators g (n + 1) = predA\n\u22a2 numerators g (n + 2) = gp.b * predA + gp.a * ppredA\n[PROOFSTEP]\nobtain \u27e8ppredConts, nth_conts_eq, \u27e8rfl\u27e9\u27e9 : \u2203 conts, g.continuants n = conts \u2227 conts.a = ppredA\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\nppredA predA : K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nnth_num_eq : numerators g n = ppredA\nsucc_nth_num_eq : numerators g (n + 1) = predA\n\u22a2 \u2203 conts, continuants g n = conts \u2227 conts.a = ppredA\ncase intro.intro.refl\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\npredA : K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nsucc_nth_num_eq : numerators g (n + 1) = predA\nppredConts : Pair K\nnth_conts_eq : continuants g n = ppredConts\nnth_num_eq : numerators g n = ppredConts.a\n\u22a2 numerators g (n + 2) = gp.b * predA + gp.a * ppredConts.a\n[PROOFSTEP]\nexact exists_conts_a_of_num nth_num_eq\n[GOAL]\ncase intro.intro.refl\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\npredA : K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nsucc_nth_num_eq : numerators g (n + 1) = predA\nppredConts : Pair K\nnth_conts_eq : continuants g n = ppredConts\nnth_num_eq : numerators g n = ppredConts.a\n\u22a2 numerators g (n + 2) = gp.b * predA + gp.a * ppredConts.a\n[PROOFSTEP]\nobtain \u27e8predConts, succ_nth_conts_eq, \u27e8rfl\u27e9\u27e9 : \u2203 conts, g.continuants (n + 1) = conts \u2227 conts.a = predA\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\npredA : K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nsucc_nth_num_eq : numerators g (n + 1) = predA\nppredConts : Pair K\nnth_conts_eq : continuants g n = ppredConts\nnth_num_eq : numerators g n = ppredConts.a\n\u22a2 \u2203 conts, continuants g (n + 1) = conts \u2227 conts.a = predA\ncase intro.intro.refl.intro.intro.refl\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nppredConts : Pair K\nnth_conts_eq : continuants g n = ppredConts\nnth_num_eq : numerators g n = ppredConts.a\npredConts : Pair K\nsucc_nth_conts_eq : continuants g (n + 1) = predConts\nsucc_nth_num_eq : numerators g (n + 1) = predConts.a\n\u22a2 numerators g (n + 2) = gp.b * predConts.a + gp.a * ppredConts.a\n[PROOFSTEP]\nexact exists_conts_a_of_num succ_nth_num_eq\n[GOAL]\ncase intro.intro.refl.intro.intro.refl\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nppredConts : Pair K\nnth_conts_eq : continuants g n = ppredConts\nnth_num_eq : numerators g n = ppredConts.a\npredConts : Pair K\nsucc_nth_conts_eq : continuants g (n + 1) = predConts\nsucc_nth_num_eq : numerators g (n + 1) = predConts.a\n\u22a2 numerators g (n + 2) = gp.b * predConts.a + gp.a * ppredConts.a\n[PROOFSTEP]\nrw [num_eq_conts_a, continuants_recurrence succ_nth_s_eq nth_conts_eq succ_nth_conts_eq]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\nppredB predB : K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nnth_denom_eq : denominators g n = ppredB\nsucc_nth_denom_eq : denominators g (n + 1) = predB\n\u22a2 denominators g (n + 2) = gp.b * predB + gp.a * ppredB\n[PROOFSTEP]\nobtain \u27e8ppredConts, nth_conts_eq, \u27e8rfl\u27e9\u27e9 : \u2203 conts, g.continuants n = conts \u2227 conts.b = ppredB\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\nppredB predB : K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nnth_denom_eq : denominators g n = ppredB\nsucc_nth_denom_eq : denominators g (n + 1) = predB\n\u22a2 \u2203 conts, continuants g n = conts \u2227 conts.b = ppredB\ncase intro.intro.refl\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\npredB : K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nsucc_nth_denom_eq : denominators g (n + 1) = predB\nppredConts : Pair K\nnth_conts_eq : continuants g n = ppredConts\nnth_denom_eq : denominators g n = ppredConts.b\n\u22a2 denominators g (n + 2) = gp.b * predB + gp.a * ppredConts.b\n[PROOFSTEP]\nexact exists_conts_b_of_denom nth_denom_eq\n[GOAL]\ncase intro.intro.refl\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\npredB : K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nsucc_nth_denom_eq : denominators g (n + 1) = predB\nppredConts : Pair K\nnth_conts_eq : continuants g n = ppredConts\nnth_denom_eq : denominators g n = ppredConts.b\n\u22a2 denominators g (n + 2) = gp.b * predB + gp.a * ppredConts.b\n[PROOFSTEP]\nobtain \u27e8predConts, succ_nth_conts_eq, \u27e8rfl\u27e9\u27e9 : \u2203 conts, g.continuants (n + 1) = conts \u2227 conts.b = predB\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\npredB : K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nsucc_nth_denom_eq : denominators g (n + 1) = predB\nppredConts : Pair K\nnth_conts_eq : continuants g n = ppredConts\nnth_denom_eq : denominators g n = ppredConts.b\n\u22a2 \u2203 conts, continuants g (n + 1) = conts \u2227 conts.b = predB\ncase intro.intro.refl.intro.intro.refl\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nppredConts : Pair K\nnth_conts_eq : continuants g n = ppredConts\nnth_denom_eq : denominators g n = ppredConts.b\npredConts : Pair K\nsucc_nth_conts_eq : continuants g (n + 1) = predConts\nsucc_nth_denom_eq : denominators g (n + 1) = predConts.b\n\u22a2 denominators g (n + 2) = gp.b * predConts.b + gp.a * ppredConts.b\n[PROOFSTEP]\nexact exists_conts_b_of_denom succ_nth_denom_eq\n[GOAL]\ncase intro.intro.refl.intro.intro.refl\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nppredConts : Pair K\nnth_conts_eq : continuants g n = ppredConts\nnth_denom_eq : denominators g n = ppredConts.b\npredConts : Pair K\nsucc_nth_conts_eq : continuants g (n + 1) = predConts\nsucc_nth_denom_eq : denominators g (n + 1) = predConts.b\n\u22a2 denominators g (n + 2) = gp.b * predConts.b + gp.a * ppredConts.b\n[PROOFSTEP]\nrw [denom_eq_conts_b, continuants_recurrence succ_nth_s_eq nth_conts_eq succ_nth_conts_eq]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence", "llama_tokens": 3845, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943805178139, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.40949291302938995}} {"text": "[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L1 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\n\u22a2 List.length (L1 ++ L2) + 2 = List.length (L1 ++ (x, b) :: (x, !b) :: L2)\n[PROOFSTEP]\nrw [List.length_append, List.length_append]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L1 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\n\u22a2 List.length L1 + List.length L2 + 2 = List.length L1 + List.length ((x, b) :: (x, !b) :: L2)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\n\u22a2 Step (L\u2081 ++ (x, !b) :: (x, b) :: L\u2082) (L\u2081 ++ L\u2082)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\n\u22a2 Step (L\u2081 ++ (x, !false) :: (x, false) :: L\u2082) (L\u2081 ++ L\u2082)\n[PROOFSTEP]\nexact Step.not\n[GOAL]\ncase true\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\n\u22a2 Step (L\u2081 ++ (x, !true) :: (x, true) :: L\u2082) (L\u2081 ++ L\u2082)\n[PROOFSTEP]\nexact Step.not\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 x\u271d\u00b9 L\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1\nb\u271d : Bool\n\u22a2 Step (x\u271d\u00b9 ++ (L\u2081\u271d ++ (x\u271d, b\u271d) :: (x\u271d, !b\u271d) :: L\u2082\u271d)) (x\u271d\u00b9 ++ (L\u2081\u271d ++ L\u2082\u271d))\n[PROOFSTEP]\nrw [\u2190 List.append_assoc, \u2190 List.append_assoc]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 x\u271d\u00b9 L\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1\nb\u271d : Bool\n\u22a2 Step (x\u271d\u00b9 ++ L\u2081\u271d ++ (x\u271d, b\u271d) :: (x\u271d, !b\u271d) :: L\u2082\u271d) (x\u271d\u00b9 ++ L\u2081\u271d ++ L\u2082\u271d)\n[PROOFSTEP]\nconstructor\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 x\u271d\u00b9 L\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1\nb\u271d : Bool\n\u22a2 Step (L\u2081\u271d ++ (x\u271d, b\u271d) :: (x\u271d, !b\u271d) :: L\u2082\u271d ++ x\u271d\u00b9) (L\u2081\u271d ++ L\u2082\u271d ++ x\u271d\u00b9)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 \u00acStep [] L\n[PROOFSTEP]\ngeneralize h' : [] = L'\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L' : List (\u03b1 \u00d7 Bool)\nh' : [] = L'\n\u22a2 \u00acStep L' L\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L' : List (\u03b1 \u00d7 Bool)\nh' : [] = L'\nh : Step L' L\n\u22a2 False\n[PROOFSTEP]\ncases' h with L\u2081 L\u2082\n[GOAL]\ncase not\n\u03b1 : Type u\nL\u2081\u271d L\u2082\u271d L\u2083 L\u2084 L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1\nb\u271d : Bool\nh' : [] = L\u2081 ++ (x\u271d, b\u271d) :: (x\u271d, !b\u271d) :: L\u2082\n\u22a2 False\n[PROOFSTEP]\nsimp [List.nil_eq_append] at h' \n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\n\u22a2 Step ((a, b) :: L\u2081) L\u2082 \u2194 (\u2203 L, Step L\u2081 L \u2227 L\u2082 = (a, b) :: L) \u2228 L\u2081 = (a, !b) :: L\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\n\u22a2 Step ((a, b) :: L\u2081) L\u2082 \u2192 (\u2203 L, Step L\u2081 L \u2227 L\u2082 = (a, b) :: L) \u2228 L\u2081 = (a, !b) :: L\u2082\n[PROOFSTEP]\ngeneralize hL : ((a, b) :: L\u2081 : List _) = L\n[GOAL]\ncase mp\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\nL : List (\u03b1 \u00d7 Bool)\nhL : (a, b) :: L\u2081 = L\n\u22a2 Step L L\u2082 \u2192 (\u2203 L, Step L\u2081 L \u2227 L\u2082 = (a, b) :: L) \u2228 L\u2081 = (a, !b) :: L\u2082\n[PROOFSTEP]\nrintro @\u27e8_ | \u27e8p, s'\u27e9, e, a', b'\u27e9\n[GOAL]\ncase mp.not.nil\n\u03b1 : Type u\nL L\u2081 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\ne : List (\u03b1 \u00d7 Bool)\na' : \u03b1\nb' : Bool\nhL : (a, b) :: L\u2081 = [] ++ (a', b') :: (a', !b') :: e\n\u22a2 (\u2203 L, Step L\u2081 L \u2227 [] ++ e = (a, b) :: L) \u2228 L\u2081 = (a, !b) :: ([] ++ e)\n[PROOFSTEP]\nsimp at hL \n[GOAL]\ncase mp.not.nil\n\u03b1 : Type u\nL L\u2081 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\ne : List (\u03b1 \u00d7 Bool)\na' : \u03b1\nb' : Bool\nhL : (a = a' \u2227 b = b') \u2227 L\u2081 = (a', !b') :: e\n\u22a2 (\u2203 L, Step L\u2081 L \u2227 [] ++ e = (a, b) :: L) \u2228 L\u2081 = (a, !b) :: ([] ++ e)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase mp.not.cons\n\u03b1 : Type u\nL L\u2081 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\ne : List (\u03b1 \u00d7 Bool)\na' : \u03b1\nb' : Bool\np : \u03b1 \u00d7 Bool\ns' : List (\u03b1 \u00d7 Bool)\nhL : (a, b) :: L\u2081 = p :: s' ++ (a', b') :: (a', !b') :: e\n\u22a2 (\u2203 L, Step L\u2081 L \u2227 p :: s' ++ e = (a, b) :: L) \u2228 L\u2081 = (a, !b) :: (p :: s' ++ e)\n[PROOFSTEP]\nsimp at hL \n[GOAL]\ncase mp.not.cons\n\u03b1 : Type u\nL L\u2081 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\ne : List (\u03b1 \u00d7 Bool)\na' : \u03b1\nb' : Bool\np : \u03b1 \u00d7 Bool\ns' : List (\u03b1 \u00d7 Bool)\nhL : (a, b) = p \u2227 L\u2081 = s' ++ (a', b') :: (a', !b') :: e\n\u22a2 (\u2203 L, Step L\u2081 L \u2227 p :: s' ++ e = (a, b) :: L) \u2228 L\u2081 = (a, !b) :: (p :: s' ++ e)\n[PROOFSTEP]\nrcases hL with \u27e8rfl, rfl\u27e9\n[GOAL]\ncase mp.not.cons.intro\n\u03b1 : Type u\nL L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\ne : List (\u03b1 \u00d7 Bool)\na' : \u03b1\nb' : Bool\ns' : List (\u03b1 \u00d7 Bool)\n\u22a2 (\u2203 L, Step (s' ++ (a', b') :: (a', !b') :: e) L \u2227 (a, b) :: s' ++ e = (a, b) :: L) \u2228\n s' ++ (a', b') :: (a', !b') :: e = (a, !b) :: ((a, b) :: s' ++ e)\n[PROOFSTEP]\nrefine' Or.inl \u27e8s' ++ e, Step.not, _\u27e9\n[GOAL]\ncase mp.not.cons.intro\n\u03b1 : Type u\nL L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\ne : List (\u03b1 \u00d7 Bool)\na' : \u03b1\nb' : Bool\ns' : List (\u03b1 \u00d7 Bool)\n\u22a2 (a, b) :: s' ++ e = (a, b) :: (s' ++ e)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\n\u22a2 (\u2203 L, Step L\u2081 L \u2227 L\u2082 = (a, b) :: L) \u2228 L\u2081 = (a, !b) :: L\u2082 \u2192 Step ((a, b) :: L\u2081) L\u2082\n[PROOFSTEP]\nrintro (\u27e8L, h, rfl\u27e9 | rfl)\n[GOAL]\ncase mpr.inl.intro.intro\n\u03b1 : Type u\nL\u271d L\u2081 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\nL : List (\u03b1 \u00d7 Bool)\nh : Step L\u2081 L\n\u22a2 Step ((a, b) :: L\u2081) ((a, b) :: L)\n[PROOFSTEP]\nexact Step.cons h\n[GOAL]\ncase mpr.inr\n\u03b1 : Type u\nL L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\n\u22a2 Step ((a, b) :: (a, !b) :: L\u2082) L\u2082\n[PROOFSTEP]\nexact Step.cons_not\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\n\u22a2 \u00acStep [(a, b)] L\n[PROOFSTEP]\nsimp [Step.cons_left_iff, not_step_nil]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 \u2200 {p : \u03b1 \u00d7 Bool}, Step (p :: L\u2081) (p :: L\u2082) \u2194 Step L\u2081 L\u2082\n[PROOFSTEP]\nsimp (config := { contextual := true }) [Step.cons_left_iff, iff_def, or_imp]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 Step ([] ++ L\u2081) ([] ++ L\u2082) \u2194 Step L\u2081 L\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nl : List (\u03b1 \u00d7 Bool)\n\u22a2 Step (p :: l ++ L\u2081) (p :: l ++ L\u2082) \u2194 Step L\u2081 L\u2082\n[PROOFSTEP]\nsimp [Step.append_left_iff l, Step.cons_cons_iff]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 x\u271d\u2075 x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nH : [] ++ (x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2075 = [] ++ (x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074\n\u22a2 [] ++ x\u271d\u2075 = [] ++ x\u271d\u2074 \u2228 \u2203 L\u2085, Step ([] ++ x\u271d\u2075) L\u2085 \u2227 Step ([] ++ x\u271d\u2074) L\u2085\n[PROOFSTEP]\ninjections\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 x\u271d\u2075 x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nfst_eq\u271d\u00b9 : x\u271d\u00b3 = x\u271d\u00b9\nsnd_eq\u271d\u00b9 : x\u271d\u00b2 = x\u271d\ntail_eq\u271d : x\u271d\u2075 = x\u271d\u2074\nfst_eq\u271d : x\u271d\u00b3 = x\u271d\u00b9\nsnd_eq\u271d : (!x\u271d\u00b2) = !x\u271d\n\u22a2 [] ++ x\u271d\u2075 = [] ++ x\u271d\u2074 \u2228 \u2203 L\u2085, Step ([] ++ x\u271d\u2075) L\u2085 \u2227 Step ([] ++ x\u271d\u2074) L\u2085\n[PROOFSTEP]\nsubst_vars\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 x\u271d\u00b2 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nfst_eq\u271d : x\u271d\u00b9 = x\u271d\u00b9\nsnd_eq\u271d : (!x\u271d) = !x\u271d\n\u22a2 [] ++ x\u271d\u00b2 = [] ++ x\u271d\u00b2 \u2228 \u2203 L\u2085, Step ([] ++ x\u271d\u00b2) L\u2085 \u2227 Step ([] ++ x\u271d\u00b2) L\u2085\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 x\u271d\u2075 : List (\u03b1 \u00d7 Bool)\nx3 : \u03b1\nb3 : Bool\nx\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nH : [] ++ (x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2075 = [(x3, b3)] ++ (x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074\n\u22a2 [] ++ x\u271d\u2075 = [(x3, b3)] ++ x\u271d\u2074 \u2228 \u2203 L\u2085, Step ([] ++ x\u271d\u2075) L\u2085 \u2227 Step ([(x3, b3)] ++ x\u271d\u2074) L\u2085\n[PROOFSTEP]\ninjections\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 x\u271d\u2075 : List (\u03b1 \u00d7 Bool)\nx3 : \u03b1\nb3 : Bool\nx\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nfst_eq\u271d\u00b9 : x\u271d\u00b3 = x3\nsnd_eq\u271d\u00b9 : x\u271d\u00b2 = b3\ntail_eq\u271d : x\u271d\u2075 = (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074\nfst_eq\u271d : x\u271d\u00b3 = x\u271d\u00b9\nsnd_eq\u271d : (!x\u271d\u00b2) = x\u271d\n\u22a2 [] ++ x\u271d\u2075 = [(x3, b3)] ++ x\u271d\u2074 \u2228 \u2203 L\u2085, Step ([] ++ x\u271d\u2075) L\u2085 \u2227 Step ([(x3, b3)] ++ x\u271d\u2074) L\u2085\n[PROOFSTEP]\nsubst_vars\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 x\u271d\u00b2 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 : Bool\nx\u271d : \u03b1\n\u22a2 [] ++ (x\u271d, !!x\u271d\u00b9) :: x\u271d\u00b2 = [(x\u271d, x\u271d\u00b9)] ++ x\u271d\u00b2 \u2228\n \u2203 L\u2085, Step ([] ++ (x\u271d, !!x\u271d\u00b9) :: x\u271d\u00b2) L\u2085 \u2227 Step ([(x\u271d, x\u271d\u00b9)] ++ x\u271d\u00b2) L\u2085\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx3 : \u03b1\nb3 : Bool\nx\u271d\u2075 x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nH : [(x3, b3)] ++ (x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2075 = [] ++ (x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074\n\u22a2 [(x3, b3)] ++ x\u271d\u2075 = [] ++ x\u271d\u2074 \u2228 \u2203 L\u2085, Step ([(x3, b3)] ++ x\u271d\u2075) L\u2085 \u2227 Step ([] ++ x\u271d\u2074) L\u2085\n[PROOFSTEP]\ninjections\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx3 : \u03b1\nb3 : Bool\nx\u271d\u2075 x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nfst_eq\u271d\u00b9 : x3 = x\u271d\u00b9\nsnd_eq\u271d\u00b9 : b3 = x\u271d\ntail_eq\u271d : (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2075 = x\u271d\u2074\nfst_eq\u271d : x\u271d\u00b3 = x\u271d\u00b9\nsnd_eq\u271d : x\u271d\u00b2 = !x\u271d\n\u22a2 [(x3, b3)] ++ x\u271d\u2075 = [] ++ x\u271d\u2074 \u2228 \u2203 L\u2085, Step ([(x3, b3)] ++ x\u271d\u2075) L\u2085 \u2227 Step ([] ++ x\u271d\u2074) L\u2085\n[PROOFSTEP]\nsubst_vars\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 x\u271d\u00b2 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\n\u22a2 [(x\u271d\u00b9, x\u271d)] ++ x\u271d\u00b2 = [] ++ (x\u271d\u00b9, !!x\u271d) :: x\u271d\u00b2 \u2228\n \u2203 L\u2085, Step ([(x\u271d\u00b9, x\u271d)] ++ x\u271d\u00b2) L\u2085 \u2227 Step ([] ++ (x\u271d\u00b9, !!x\u271d) :: x\u271d\u00b2) L\u2085\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 x\u271d\u2075 : List (\u03b1 \u00d7 Bool)\nx3 : \u03b1\nb3 : Bool\nx4 : \u03b1\nb4 : Bool\ntl x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nH : [] ++ (x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2075 = (x3, b3) :: (x4, b4) :: tl ++ (x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074\n\u22a2 [] ++ x\u271d\u2075 = (x3, b3) :: (x4, b4) :: tl ++ x\u271d\u2074 \u2228\n \u2203 L\u2085, Step ([] ++ x\u271d\u2075) L\u2085 \u2227 Step ((x3, b3) :: (x4, b4) :: tl ++ x\u271d\u2074) L\u2085\n[PROOFSTEP]\ninjections\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 x\u271d\u2075 : List (\u03b1 \u00d7 Bool)\nx3 : \u03b1\nb3 : Bool\nx4 : \u03b1\nb4 : Bool\ntl x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nfst_eq\u271d\u00b9 : x\u271d\u00b3 = x3\nsnd_eq\u271d\u00b9 : x\u271d\u00b2 = b3\ntail_eq\u271d : x\u271d\u2075 = List.append tl ((x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074)\nfst_eq\u271d : x\u271d\u00b3 = x4\nsnd_eq\u271d : (!x\u271d\u00b2) = b4\n\u22a2 [] ++ x\u271d\u2075 = (x3, b3) :: (x4, b4) :: tl ++ x\u271d\u2074 \u2228\n \u2203 L\u2085, Step ([] ++ x\u271d\u2075) L\u2085 \u2227 Step ((x3, b3) :: (x4, b4) :: tl ++ x\u271d\u2074) L\u2085\n[PROOFSTEP]\nsubst_vars\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 tl x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\n\u22a2 [] ++ List.append tl ((x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074) = (x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: tl ++ x\u271d\u2074 \u2228\n \u2203 L\u2085,\n Step ([] ++ List.append tl ((x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074)) L\u2085 \u2227 Step ((x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: tl ++ x\u271d\u2074) L\u2085\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 tl x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\n\u22a2 tl ++ (x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074 = (x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: (tl ++ x\u271d\u2074) \u2228\n \u2203 L\u2085, Step (tl ++ (x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074) L\u2085 \u2227 Step ((x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: (tl ++ x\u271d\u2074)) L\u2085\n[PROOFSTEP]\nright\n[GOAL]\ncase h\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 tl x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\n\u22a2 \u2203 L\u2085, Step (tl ++ (x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074) L\u2085 \u2227 Step ((x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: (tl ++ x\u271d\u2074)) L\u2085\n[PROOFSTEP]\nexact \u27e8_, Red.Step.not, Red.Step.cons_not\u27e9\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx3 : \u03b1\nb3 : Bool\nx4 : \u03b1\nb4 : Bool\ntl x\u271d\u2075 x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nH : (x3, b3) :: (x4, b4) :: tl ++ (x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2075 = [] ++ (x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074\n\u22a2 (x3, b3) :: (x4, b4) :: tl ++ x\u271d\u2075 = [] ++ x\u271d\u2074 \u2228\n \u2203 L\u2085, Step ((x3, b3) :: (x4, b4) :: tl ++ x\u271d\u2075) L\u2085 \u2227 Step ([] ++ x\u271d\u2074) L\u2085\n[PROOFSTEP]\ninjections\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx3 : \u03b1\nb3 : Bool\nx4 : \u03b1\nb4 : Bool\ntl x\u271d\u2075 x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nfst_eq\u271d\u00b9 : x3 = x\u271d\u00b9\nsnd_eq\u271d\u00b9 : b3 = x\u271d\ntail_eq\u271d : List.append tl ((x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2075) = x\u271d\u2074\nfst_eq\u271d : x4 = x\u271d\u00b9\nsnd_eq\u271d : b4 = !x\u271d\n\u22a2 (x3, b3) :: (x4, b4) :: tl ++ x\u271d\u2075 = [] ++ x\u271d\u2074 \u2228\n \u2203 L\u2085, Step ((x3, b3) :: (x4, b4) :: tl ++ x\u271d\u2075) L\u2085 \u2227 Step ([] ++ x\u271d\u2074) L\u2085\n[PROOFSTEP]\nsubst_vars\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 tl x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\n\u22a2 (x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: tl ++ x\u271d\u2074 = [] ++ List.append tl ((x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2074) \u2228\n \u2203 L\u2085,\n Step ((x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: tl ++ x\u271d\u2074) L\u2085 \u2227 Step ([] ++ List.append tl ((x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2074)) L\u2085\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 tl x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\n\u22a2 (x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: (tl ++ x\u271d\u2074) = tl ++ (x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2074 \u2228\n \u2203 L\u2085, Step ((x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: (tl ++ x\u271d\u2074)) L\u2085 \u2227 Step (tl ++ (x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2074) L\u2085\n[PROOFSTEP]\nright\n[GOAL]\ncase h\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 tl x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\n\u22a2 \u2203 L\u2085, Step ((x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: (tl ++ x\u271d\u2074)) L\u2085 \u2227 Step (tl ++ (x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2074) L\u2085\n[PROOFSTEP]\nexact \u27e8_, Red.Step.cons_not, Red.Step.not\u27e9\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx3 : \u03b1\nb3 : Bool\ntl x\u271d\u2075 : List (\u03b1 \u00d7 Bool)\nx4 : \u03b1\nb4 : Bool\ntl2 x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nH : (x3, b3) :: tl ++ (x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2075 = (x4, b4) :: tl2 ++ (x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074\nH1 : (x3, b3) = (x4, b4)\nH2 : List.append tl ((x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2075) = List.append tl2 ((x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074)\nH3 : tl ++ x\u271d\u2075 = tl2 ++ x\u271d\u2074\n\u22a2 (x3, b3) :: tl ++ x\u271d\u2075 = (x4, b4) :: tl2 ++ x\u271d\u2074\n[PROOFSTEP]\nsimp [H1, H3]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx3 : \u03b1\nb3 : Bool\ntl x\u271d\u2075 : List (\u03b1 \u00d7 Bool)\nx4 : \u03b1\nb4 : Bool\ntl2 x\u271d\u2074 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 : Bool\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nH : (x3, b3) :: tl ++ (x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2075 = (x4, b4) :: tl2 ++ (x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074\nH1 : (x3, b3) = (x4, b4)\nH2 : List.append tl ((x\u271d\u00b3, x\u271d\u00b2) :: (x\u271d\u00b3, !x\u271d\u00b2) :: x\u271d\u2075) = List.append tl2 ((x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: x\u271d\u2074)\nL\u2085 : List (\u03b1 \u00d7 Bool)\nH3 : Step (tl ++ x\u271d\u2075) L\u2085\nH4 : Step (tl2 ++ x\u271d\u2074) L\u2085\n\u22a2 Step ((x4, b4) :: tl2 ++ x\u271d\u2074) ((x3, b3) :: L\u2085)\n[PROOFSTEP]\nsimpa [H1] using Step.cons H4\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 a b\u271d c b : List (\u03b1 \u00d7 Bool)\nhab hac : Step a b\n\u22a2 ReflGen Step b b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 a b\u271d c b : List (\u03b1 \u00d7 Bool)\nhab hac : Step a b\n\u22a2 ReflTransGen Step b b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\n\u22a2 Red (p :: L\u2081) (p :: L\u2082) \u2192 Red L\u2081 L\u2082\n[PROOFSTEP]\ngeneralize eq\u2081 : (p :: L\u2081 : List _) = LL\u2081\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nLL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = LL\u2081\n\u22a2 Red LL\u2081 (p :: L\u2082) \u2192 Red L\u2081 L\u2082\n[PROOFSTEP]\ngeneralize eq\u2082 : (p :: L\u2082 : List _) = LL\u2082\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nLL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = LL\u2081\nLL\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2082 : p :: L\u2082 = LL\u2082\n\u22a2 Red LL\u2081 LL\u2082 \u2192 Red L\u2081 L\u2082\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nLL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = LL\u2081\nLL\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2082 : p :: L\u2082 = LL\u2082\nh : Red LL\u2081 LL\u2082\n\u22a2 Red L\u2081 L\u2082\n[PROOFSTEP]\ninduction' h using Relation.ReflTransGen.head_induction_on with L\u2081 L\u2082 h\u2081\u2082 h ih generalizing L\u2081 L\u2082\n[GOAL]\ncase refl\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nLL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081\u271d : p :: L\u2081\u271d = LL\u2081\nLL\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2082\u271d : p :: L\u2082\u271d = LL\u2082\nL\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = LL\u2082\neq\u2082 : p :: L\u2082 = LL\u2082\n\u22a2 Red L\u2081 L\u2082\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase refl\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nL\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = p :: L\u2082\u271d\neq\u2082 : p :: L\u2082 = p :: L\u2082\u271d\n\u22a2 Red L\u2081 L\u2082\n[PROOFSTEP]\ncases eq\u2082\n[GOAL]\ncase refl.refl\n\u03b1 : Type u\nL L\u2081\u271d L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = p :: L\u2082\n\u22a2 Red L\u2081 L\u2082\n[PROOFSTEP]\ncases eq\u2081\n[GOAL]\ncase refl.refl.refl\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\n\u22a2 Red L\u2082 L\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase head\n\u03b1 : Type u\nL L\u2081\u271d\u00b9 L\u2082\u271d\u00b9 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nLL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081\u271d : p :: L\u2081\u271d\u00b9 = LL\u2081\nLL\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2082\u271d : p :: L\u2082\u271d\u00b9 = LL\u2082\nL\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nh\u2081\u2082 : Step L\u2081\u271d L\u2082\u271d\nh : ReflTransGen Step L\u2082\u271d LL\u2082\nih : \u2200 {L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)}, p :: L\u2081 = L\u2082\u271d \u2192 p :: L\u2082 = LL\u2082 \u2192 Red L\u2081 L\u2082\nL\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = L\u2081\u271d\neq\u2082 : p :: L\u2082 = LL\u2082\n\u22a2 Red L\u2081 L\u2082\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase head\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d\u00b9 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nL\u2082\u271d L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\nh : ReflTransGen Step L\u2082\u271d (p :: L\u2082\u271d\u00b9)\nih : \u2200 {L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)}, p :: L\u2081 = L\u2082\u271d \u2192 p :: L\u2082 = p :: L\u2082\u271d\u00b9 \u2192 Red L\u2081 L\u2082\neq\u2082 : p :: L\u2082 = p :: L\u2082\u271d\u00b9\nh\u2081\u2082 : Step (p :: L\u2081) L\u2082\u271d\n\u22a2 Red L\u2081 L\u2082\n[PROOFSTEP]\ncases eq\u2082\n[GOAL]\ncase head.refl\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nL\u2082 L\u2081 : List (\u03b1 \u00d7 Bool)\nh : ReflTransGen Step L\u2082 (p :: L\u2082\u271d)\nih : \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)}, p :: L\u2081 = L\u2082 \u2192 p :: L\u2082_1 = p :: L\u2082\u271d \u2192 Red L\u2081 L\u2082_1\nh\u2081\u2082 : Step (p :: L\u2081) L\u2082\n\u22a2 Red L\u2081 L\u2082\u271d\n[PROOFSTEP]\ncases' p with a b\n[GOAL]\ncase head.refl.mk\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 L\u2082 L\u2081 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\nh : ReflTransGen Step L\u2082 ((a, b) :: L\u2082\u271d)\nih : \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)}, (a, b) :: L\u2081 = L\u2082 \u2192 (a, b) :: L\u2082_1 = (a, b) :: L\u2082\u271d \u2192 Red L\u2081 L\u2082_1\nh\u2081\u2082 : Step ((a, b) :: L\u2081) L\u2082\n\u22a2 Red L\u2081 L\u2082\u271d\n[PROOFSTEP]\nrw [Step.cons_left_iff] at h\u2081\u2082 \n[GOAL]\ncase head.refl.mk\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 L\u2082 L\u2081 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\nh : ReflTransGen Step L\u2082 ((a, b) :: L\u2082\u271d)\nih : \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)}, (a, b) :: L\u2081 = L\u2082 \u2192 (a, b) :: L\u2082_1 = (a, b) :: L\u2082\u271d \u2192 Red L\u2081 L\u2082_1\nh\u2081\u2082 : (\u2203 L, Step L\u2081 L \u2227 L\u2082 = (a, b) :: L) \u2228 L\u2081 = (a, !b) :: L\u2082\n\u22a2 Red L\u2081 L\u2082\u271d\n[PROOFSTEP]\nrcases h\u2081\u2082 with (\u27e8L, h\u2081\u2082, rfl\u27e9 | rfl)\n[GOAL]\ncase head.refl.mk.inl.intro.intro\n\u03b1 : Type u\nL\u271d L\u2081\u271d L\u2082 L\u2083 L\u2084 L\u2081 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\nL : List (\u03b1 \u00d7 Bool)\nh\u2081\u2082 : Step L\u2081 L\nh : ReflTransGen Step ((a, b) :: L) ((a, b) :: L\u2082)\nih : \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)}, (a, b) :: L\u2081 = (a, b) :: L \u2192 (a, b) :: L\u2082_1 = (a, b) :: L\u2082 \u2192 Red L\u2081 L\u2082_1\n\u22a2 Red L\u2081 L\u2082\n[PROOFSTEP]\nexact (ih rfl rfl).head h\u2081\u2082\n[GOAL]\ncase head.refl.mk.inr\n\u03b1 : Type u\nL L\u2081 L\u2082\u271d L\u2083 L\u2084 L\u2082 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\nh : ReflTransGen Step L\u2082 ((a, b) :: L\u2082\u271d)\nih : \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)}, (a, b) :: L\u2081 = L\u2082 \u2192 (a, b) :: L\u2082_1 = (a, b) :: L\u2082\u271d \u2192 Red L\u2081 L\u2082_1\n\u22a2 Red ((a, !b) :: L\u2082) L\u2082\u271d\n[PROOFSTEP]\nexact (cons_cons h).tail Step.cons_not_rev\n[GOAL]\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nL : List (\u03b1 \u00d7 Bool)\n\u22a2 Red (p :: L ++ L\u2081) (p :: L ++ L\u2082) \u2194 Red L\u2081 L\u2082\n[PROOFSTEP]\nsimp [append_append_left_iff L, cons_cons_iff]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 Red L (L\u2081 ++ L\u2082) \u2192 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 L\u2082\n[PROOFSTEP]\ngeneralize eq : L\u2081 ++ L\u2082 = L\u2081\u2082\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L\u2081\u2082 : List (\u03b1 \u00d7 Bool)\neq : L\u2081 ++ L\u2082 = L\u2081\u2082\n\u22a2 Red L L\u2081\u2082 \u2192 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 L\u2082\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L\u2081\u2082 : List (\u03b1 \u00d7 Bool)\neq : L\u2081 ++ L\u2082 = L\u2081\u2082\nh : Red L L\u2081\u2082\n\u22a2 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 L\u2082\n[PROOFSTEP]\ninduction' h with L' L\u2081\u2082 hLL' h ih generalizing L\u2081 L\u2082\n[GOAL]\ncase refl\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 L\u2081\u2082 : List (\u03b1 \u00d7 Bool)\neq\u271d : L\u2081\u271d ++ L\u2082\u271d = L\u2081\u2082\nL\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\neq : L\u2081 ++ L\u2082 = L\n\u22a2 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 L\u2082\n[PROOFSTEP]\nexact \u27e8_, _, eq.symm, by rfl, by rfl\u27e9\n[GOAL]\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 L\u2081\u2082 : List (\u03b1 \u00d7 Bool)\neq\u271d : L\u2081\u271d ++ L\u2082\u271d = L\u2081\u2082\nL\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\neq : L\u2081 ++ L\u2082 = L\n\u22a2 Red L\u2081 L\u2081\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 L\u2081\u2082 : List (\u03b1 \u00d7 Bool)\neq\u271d : L\u2081\u271d ++ L\u2082\u271d = L\u2081\u2082\nL\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\neq : L\u2081 ++ L\u2082 = L\n\u22a2 Red L\u2082 L\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\ncase tail\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 L\u2081\u2082\u271d : List (\u03b1 \u00d7 Bool)\neq\u271d : L\u2081\u271d ++ L\u2082\u271d = L\u2081\u2082\u271d\nL' L\u2081\u2082 : List (\u03b1 \u00d7 Bool)\nhLL' : ReflTransGen Step L L'\nh : Step L' L\u2081\u2082\nih : \u2200 {L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)}, L\u2081 ++ L\u2082 = L' \u2192 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 L\u2082\nL\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\neq : L\u2081 ++ L\u2082 = L\u2081\u2082\n\u22a2 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 L\u2082\n[PROOFSTEP]\ncases' h with s e a b\n[GOAL]\ncase tail.not\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 L\u2081\u2082 : List (\u03b1 \u00d7 Bool)\neq\u271d : L\u2081\u271d ++ L\u2082\u271d = L\u2081\u2082\nL\u2081 L\u2082 s e : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\nhLL' : ReflTransGen Step L (s ++ (a, b) :: (a, !b) :: e)\nih : \u2200 {L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)}, L\u2081 ++ L\u2082 = s ++ (a, b) :: (a, !b) :: e \u2192 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 L\u2082\neq : L\u2081 ++ L\u2082 = s ++ e\n\u22a2 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 L\u2082\n[PROOFSTEP]\nrcases List.append_eq_append_iff.1 eq with (\u27e8s', rfl, rfl\u27e9 | \u27e8e', rfl, rfl\u27e9)\n[GOAL]\ncase tail.not.inl.intro.intro\n\u03b1 : Type u\nL L\u2081\u271d L\u2082 L\u2083 L\u2084 L\u2081\u2082 : List (\u03b1 \u00d7 Bool)\neq\u271d : L\u2081\u271d ++ L\u2082 = L\u2081\u2082\nL\u2081 e : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\ns' : List (\u03b1 \u00d7 Bool)\nhLL' : ReflTransGen Step L (L\u2081 ++ s' ++ (a, b) :: (a, !b) :: e)\nih :\n \u2200 {L\u2081_1 L\u2082 : List (\u03b1 \u00d7 Bool)},\n L\u2081_1 ++ L\u2082 = L\u2081 ++ s' ++ (a, b) :: (a, !b) :: e \u2192 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081_1 \u2227 Red L\u2084 L\u2082\neq : L\u2081 ++ (s' ++ e) = L\u2081 ++ s' ++ e\n\u22a2 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 (s' ++ e)\n[PROOFSTEP]\nhave : L\u2081 ++ (s' ++ (a, b) :: (a, not b) :: e) = L\u2081 ++ s' ++ (a, b) :: (a, not b) :: e := by simp\n[GOAL]\n\u03b1 : Type u\nL L\u2081\u271d L\u2082 L\u2083 L\u2084 L\u2081\u2082 : List (\u03b1 \u00d7 Bool)\neq\u271d : L\u2081\u271d ++ L\u2082 = L\u2081\u2082\nL\u2081 e : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\ns' : List (\u03b1 \u00d7 Bool)\nhLL' : ReflTransGen Step L (L\u2081 ++ s' ++ (a, b) :: (a, !b) :: e)\nih :\n \u2200 {L\u2081_1 L\u2082 : List (\u03b1 \u00d7 Bool)},\n L\u2081_1 ++ L\u2082 = L\u2081 ++ s' ++ (a, b) :: (a, !b) :: e \u2192 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081_1 \u2227 Red L\u2084 L\u2082\neq : L\u2081 ++ (s' ++ e) = L\u2081 ++ s' ++ e\n\u22a2 L\u2081 ++ (s' ++ (a, b) :: (a, !b) :: e) = L\u2081 ++ s' ++ (a, b) :: (a, !b) :: e\n[PROOFSTEP]\nsimp\n[GOAL]\ncase tail.not.inl.intro.intro\n\u03b1 : Type u\nL L\u2081\u271d L\u2082 L\u2083 L\u2084 L\u2081\u2082 : List (\u03b1 \u00d7 Bool)\neq\u271d : L\u2081\u271d ++ L\u2082 = L\u2081\u2082\nL\u2081 e : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\ns' : List (\u03b1 \u00d7 Bool)\nhLL' : ReflTransGen Step L (L\u2081 ++ s' ++ (a, b) :: (a, !b) :: e)\nih :\n \u2200 {L\u2081_1 L\u2082 : List (\u03b1 \u00d7 Bool)},\n L\u2081_1 ++ L\u2082 = L\u2081 ++ s' ++ (a, b) :: (a, !b) :: e \u2192 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081_1 \u2227 Red L\u2084 L\u2082\neq : L\u2081 ++ (s' ++ e) = L\u2081 ++ s' ++ e\nthis : L\u2081 ++ (s' ++ (a, b) :: (a, !b) :: e) = L\u2081 ++ s' ++ (a, b) :: (a, !b) :: e\n\u22a2 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 (s' ++ e)\n[PROOFSTEP]\nrcases ih this with \u27e8w\u2081, w\u2082, rfl, h\u2081, h\u2082\u27e9\n[GOAL]\ncase tail.not.inl.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\nL\u2081\u271d L\u2082 L\u2083 L\u2084 L\u2081\u2082 : List (\u03b1 \u00d7 Bool)\neq\u271d : L\u2081\u271d ++ L\u2082 = L\u2081\u2082\nL\u2081 e : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\ns' : List (\u03b1 \u00d7 Bool)\neq : L\u2081 ++ (s' ++ e) = L\u2081 ++ s' ++ e\nthis : L\u2081 ++ (s' ++ (a, b) :: (a, !b) :: e) = L\u2081 ++ s' ++ (a, b) :: (a, !b) :: e\nw\u2081 w\u2082 : List (\u03b1 \u00d7 Bool)\nhLL' : ReflTransGen Step (w\u2081 ++ w\u2082) (L\u2081 ++ s' ++ (a, b) :: (a, !b) :: e)\nih :\n \u2200 {L\u2081_1 L\u2082 : List (\u03b1 \u00d7 Bool)},\n L\u2081_1 ++ L\u2082 = L\u2081 ++ s' ++ (a, b) :: (a, !b) :: e \u2192 \u2203 L\u2083 L\u2084, w\u2081 ++ w\u2082 = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081_1 \u2227 Red L\u2084 L\u2082\nh\u2081 : Red w\u2081 L\u2081\nh\u2082 : Red w\u2082 (s' ++ (a, b) :: (a, !b) :: e)\n\u22a2 \u2203 L\u2083 L\u2084, w\u2081 ++ w\u2082 = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 (s' ++ e)\n[PROOFSTEP]\nexact \u27e8w\u2081, w\u2082, rfl, h\u2081, h\u2082.tail Step.not\u27e9\n[GOAL]\ncase tail.not.inr.intro.intro\n\u03b1 : Type u\nL L\u2081 L\u2082\u271d L\u2083 L\u2084 L\u2081\u2082 : List (\u03b1 \u00d7 Bool)\neq\u271d : L\u2081 ++ L\u2082\u271d = L\u2081\u2082\nL\u2082 s : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\ne' : List (\u03b1 \u00d7 Bool)\nhLL' : ReflTransGen Step L (s ++ (a, b) :: (a, !b) :: (e' ++ L\u2082))\nih :\n \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)},\n L\u2081 ++ L\u2082_1 = s ++ (a, b) :: (a, !b) :: (e' ++ L\u2082) \u2192 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 L\u2082_1\neq : s ++ e' ++ L\u2082 = s ++ (e' ++ L\u2082)\n\u22a2 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 (s ++ e') \u2227 Red L\u2084 L\u2082\n[PROOFSTEP]\nhave : s ++ (a, b) :: (a, not b) :: e' ++ L\u2082 = s ++ (a, b) :: (a, not b) :: (e' ++ L\u2082) := by simp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082\u271d L\u2083 L\u2084 L\u2081\u2082 : List (\u03b1 \u00d7 Bool)\neq\u271d : L\u2081 ++ L\u2082\u271d = L\u2081\u2082\nL\u2082 s : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\ne' : List (\u03b1 \u00d7 Bool)\nhLL' : ReflTransGen Step L (s ++ (a, b) :: (a, !b) :: (e' ++ L\u2082))\nih :\n \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)},\n L\u2081 ++ L\u2082_1 = s ++ (a, b) :: (a, !b) :: (e' ++ L\u2082) \u2192 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 L\u2082_1\neq : s ++ e' ++ L\u2082 = s ++ (e' ++ L\u2082)\n\u22a2 s ++ (a, b) :: (a, !b) :: e' ++ L\u2082 = s ++ (a, b) :: (a, !b) :: (e' ++ L\u2082)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase tail.not.inr.intro.intro\n\u03b1 : Type u\nL L\u2081 L\u2082\u271d L\u2083 L\u2084 L\u2081\u2082 : List (\u03b1 \u00d7 Bool)\neq\u271d : L\u2081 ++ L\u2082\u271d = L\u2081\u2082\nL\u2082 s : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\ne' : List (\u03b1 \u00d7 Bool)\nhLL' : ReflTransGen Step L (s ++ (a, b) :: (a, !b) :: (e' ++ L\u2082))\nih :\n \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)},\n L\u2081 ++ L\u2082_1 = s ++ (a, b) :: (a, !b) :: (e' ++ L\u2082) \u2192 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 L\u2082_1\neq : s ++ e' ++ L\u2082 = s ++ (e' ++ L\u2082)\nthis : s ++ (a, b) :: (a, !b) :: e' ++ L\u2082 = s ++ (a, b) :: (a, !b) :: (e' ++ L\u2082)\n\u22a2 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 (s ++ e') \u2227 Red L\u2084 L\u2082\n[PROOFSTEP]\nrcases ih this with \u27e8w\u2081, w\u2082, rfl, h\u2081, h\u2082\u27e9\n[GOAL]\ncase tail.not.inr.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\nL\u2081 L\u2082\u271d L\u2083 L\u2084 L\u2081\u2082 : List (\u03b1 \u00d7 Bool)\neq\u271d : L\u2081 ++ L\u2082\u271d = L\u2081\u2082\nL\u2082 s : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\ne' : List (\u03b1 \u00d7 Bool)\neq : s ++ e' ++ L\u2082 = s ++ (e' ++ L\u2082)\nthis : s ++ (a, b) :: (a, !b) :: e' ++ L\u2082 = s ++ (a, b) :: (a, !b) :: (e' ++ L\u2082)\nw\u2081 w\u2082 : List (\u03b1 \u00d7 Bool)\nhLL' : ReflTransGen Step (w\u2081 ++ w\u2082) (s ++ (a, b) :: (a, !b) :: (e' ++ L\u2082))\nih :\n \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)},\n L\u2081 ++ L\u2082_1 = s ++ (a, b) :: (a, !b) :: (e' ++ L\u2082) \u2192 \u2203 L\u2083 L\u2084, w\u2081 ++ w\u2082 = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 L\u2082_1\nh\u2081 : Red w\u2081 (s ++ (a, b) :: (a, !b) :: e')\nh\u2082 : Red w\u2082 L\u2082\n\u22a2 \u2203 L\u2083 L\u2084, w\u2081 ++ w\u2082 = L\u2083 ++ L\u2084 \u2227 Red L\u2083 (s ++ e') \u2227 Red L\u2084 L\u2082\n[PROOFSTEP]\nexact \u27e8w\u2081, w\u2082, rfl, h\u2081.tail Step.not, h\u2082\u27e9\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nh : Red ((x, b) :: L) []\n\u22a2 Red L [(x, !b)]\n[PROOFSTEP]\nhave h\u2081 : Red ((x, not b) :: (x, b) :: L) [(x, not b)] := cons_cons h\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nh : Red ((x, b) :: L) []\nh\u2081 : Red ((x, !b) :: (x, b) :: L) [(x, !b)]\n\u22a2 Red L [(x, !b)]\n[PROOFSTEP]\nhave h\u2082 : Red ((x, not b) :: (x, b) :: L) L := ReflTransGen.single Step.cons_not_rev\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nh : Red ((x, b) :: L) []\nh\u2081 : Red ((x, !b) :: (x, b) :: L) [(x, !b)]\nh\u2082 : Red ((x, !b) :: (x, b) :: L) L\n\u22a2 Red L [(x, !b)]\n[PROOFSTEP]\nlet \u27e8L', h\u2081, h\u2082\u27e9 := church_rosser h\u2081 h\u2082\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nh : Red ((x, b) :: L) []\nh\u2081\u271d : Red ((x, !b) :: (x, b) :: L) [(x, !b)]\nh\u2082\u271d : Red ((x, !b) :: (x, b) :: L) L\nL' : List (\u03b1 \u00d7 Bool)\nh\u2081 : Red [(x, !b)] L'\nh\u2082 : Red L L'\n\u22a2 Red L [(x, !b)]\n[PROOFSTEP]\nrw [singleton_iff] at h\u2081 \n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nh : Red ((x, b) :: L) []\nh\u2081\u271d : Red ((x, !b) :: (x, b) :: L) [(x, !b)]\nh\u2082\u271d : Red ((x, !b) :: (x, b) :: L) L\nL' : List (\u03b1 \u00d7 Bool)\nh\u2081 : L' = [(x, !b)]\nh\u2082 : Red L L'\n\u22a2 Red L [(x, !b)]\n[PROOFSTEP]\nsubst L'\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nh : Red ((x, b) :: L) []\nh\u2081 : Red ((x, !b) :: (x, b) :: L) [(x, !b)]\nh\u2082\u271d : Red ((x, !b) :: (x, b) :: L) L\nh\u2082 : Red L [(x, !b)]\n\u22a2 Red L [(x, !b)]\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nh : (x1, b1) \u2260 (x2, b2)\n\u22a2 Red [(x1, !b1), (x2, b2)] L \u2194 L = [(x1, !b1), (x2, b2)]\n[PROOFSTEP]\napply reflTransGen_iff_eq\n[GOAL]\ncase h\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nh : (x1, b1) \u2260 (x2, b2)\n\u22a2 \u2200 (b : List (\u03b1 \u00d7 Bool)), \u00acStep [(x1, !b1), (x2, b2)] b\n[PROOFSTEP]\ngeneralize eq : [(x1, not b1), (x2, b2)] = L'\n[GOAL]\ncase h\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nh : (x1, b1) \u2260 (x2, b2)\nL' : List (\u03b1 \u00d7 Bool)\neq : [(x1, !b1), (x2, b2)] = L'\n\u22a2 \u2200 (b : List (\u03b1 \u00d7 Bool)), \u00acStep L' b\n[PROOFSTEP]\nintro L h'\n[GOAL]\ncase h\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nh : (x1, b1) \u2260 (x2, b2)\nL' : List (\u03b1 \u00d7 Bool)\neq : [(x1, !b1), (x2, b2)] = L'\nL : List (\u03b1 \u00d7 Bool)\nh' : Step L' L\n\u22a2 False\n[PROOFSTEP]\ncases h'\n[GOAL]\ncase h.not\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nh : (x1, b1) \u2260 (x2, b2)\nL\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1\nb\u271d : Bool\neq : [(x1, !b1), (x2, b2)] = L\u2081\u271d ++ (x\u271d, b\u271d) :: (x\u271d, !b\u271d) :: L\u2082\u271d\n\u22a2 False\n[PROOFSTEP]\nsimp [List.cons_eq_append_iff, List.nil_eq_append] at eq \n[GOAL]\ncase h.not\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nh : (x1, b1) \u2260 (x2, b2)\nL\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1\nb\u271d : Bool\neq : L\u2081\u271d = [] \u2227 (x\u271d = x1 \u2227 b\u271d = !b1) \u2227 (x\u271d = x2 \u2227 (!b\u271d) = b2) \u2227 L\u2082\u271d = []\n\u22a2 False\n[PROOFSTEP]\nrcases eq with \u27e8rfl, \u27e8rfl, rfl\u27e9, \u27e8rfl, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h.not.intro.intro.intro.intro.intro\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nb1 : Bool\nx\u271d : \u03b1\nh : (x\u271d, b1) \u2260 (x\u271d, !!b1)\n\u22a2 False\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase h.not.intro.intro.intro.intro.intro\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nb1 : Bool\nx\u271d : \u03b1\nh : (x\u271d, b1) \u2260 (x\u271d, !!b1)\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nH1 : (x1, b1) \u2260 (x2, b2)\nH2 : Red ((x1, b1) :: L\u2081) ((x2, b2) :: L\u2082)\n\u22a2 Red L\u2081 ((x1, !b1) :: (x2, b2) :: L\u2082)\n[PROOFSTEP]\nhave : Red ((x1, b1) :: L\u2081) ([(x2, b2)] ++ L\u2082) := H2\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nH1 : (x1, b1) \u2260 (x2, b2)\nH2 : Red ((x1, b1) :: L\u2081) ((x2, b2) :: L\u2082)\nthis : Red ((x1, b1) :: L\u2081) ([(x2, b2)] ++ L\u2082)\n\u22a2 Red L\u2081 ((x1, !b1) :: (x2, b2) :: L\u2082)\n[PROOFSTEP]\nrcases to_append_iff.1 this with \u27e8_ | \u27e8p, L\u2083\u27e9, L\u2084, eq, h\u2081, h\u2082\u27e9\n[GOAL]\ncase intro.nil.intro.intro.intro\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084\u271d : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nH1 : (x1, b1) \u2260 (x2, b2)\nH2 : Red ((x1, b1) :: L\u2081) ((x2, b2) :: L\u2082)\nthis : Red ((x1, b1) :: L\u2081) ([(x2, b2)] ++ L\u2082)\nL\u2084 : List (\u03b1 \u00d7 Bool)\neq : (x1, b1) :: L\u2081 = [] ++ L\u2084\nh\u2081 : Red [] [(x2, b2)]\nh\u2082 : Red L\u2084 L\u2082\n\u22a2 Red L\u2081 ((x1, !b1) :: (x2, b2) :: L\u2082)\n[PROOFSTEP]\nsimp [nil_iff] at h\u2081 \n[GOAL]\ncase intro.cons.intro.intro.intro\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083\u271d L\u2084\u271d : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nH1 : (x1, b1) \u2260 (x2, b2)\nH2 : Red ((x1, b1) :: L\u2081) ((x2, b2) :: L\u2082)\nthis : Red ((x1, b1) :: L\u2081) ([(x2, b2)] ++ L\u2082)\np : \u03b1 \u00d7 Bool\nL\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\neq : (x1, b1) :: L\u2081 = p :: L\u2083 ++ L\u2084\nh\u2081 : Red (p :: L\u2083) [(x2, b2)]\nh\u2082 : Red L\u2084 L\u2082\n\u22a2 Red L\u2081 ((x1, !b1) :: (x2, b2) :: L\u2082)\n[PROOFSTEP]\ncases eq\n[GOAL]\ncase intro.cons.intro.intro.intro.refl\n\u03b1 : Type u\nL L\u2082 L\u2083\u271d L\u2084\u271d : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nH1 : (x1, b1) \u2260 (x2, b2)\nL\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nh\u2082 : Red L\u2084 L\u2082\nh\u2081 : Red ((x1, b1) :: L\u2083) [(x2, b2)]\nH2 : Red ((x1, b1) :: List.append L\u2083 L\u2084) ((x2, b2) :: L\u2082)\nthis : Red ((x1, b1) :: List.append L\u2083 L\u2084) ([(x2, b2)] ++ L\u2082)\n\u22a2 Red (List.append L\u2083 L\u2084) ((x1, !b1) :: (x2, b2) :: L\u2082)\n[PROOFSTEP]\nshow Red (L\u2083 ++ L\u2084) ([(x1, not b1), (x2, b2)] ++ L\u2082)\n[GOAL]\ncase intro.cons.intro.intro.intro.refl\n\u03b1 : Type u\nL L\u2082 L\u2083\u271d L\u2084\u271d : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nH1 : (x1, b1) \u2260 (x2, b2)\nL\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nh\u2082 : Red L\u2084 L\u2082\nh\u2081 : Red ((x1, b1) :: L\u2083) [(x2, b2)]\nH2 : Red ((x1, b1) :: List.append L\u2083 L\u2084) ((x2, b2) :: L\u2082)\nthis : Red ((x1, b1) :: List.append L\u2083 L\u2084) ([(x2, b2)] ++ L\u2082)\n\u22a2 Red (L\u2083 ++ L\u2084) ([(x1, !b1), (x2, b2)] ++ L\u2082)\n[PROOFSTEP]\napply append_append _ h\u2082\n[GOAL]\n\u03b1 : Type u\nL L\u2082 L\u2083\u271d L\u2084\u271d : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nH1 : (x1, b1) \u2260 (x2, b2)\nL\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nh\u2082 : Red L\u2084 L\u2082\nh\u2081 : Red ((x1, b1) :: L\u2083) [(x2, b2)]\nH2 : Red ((x1, b1) :: List.append L\u2083 L\u2084) ((x2, b2) :: L\u2082)\nthis : Red ((x1, b1) :: List.append L\u2083 L\u2084) ([(x2, b2)] ++ L\u2082)\n\u22a2 Red L\u2083 [(x1, !b1), (x2, b2)]\n[PROOFSTEP]\nhave h\u2081 : Red ((x1, not b1) :: (x1, b1) :: L\u2083) [(x1, not b1), (x2, b2)] := cons_cons h\u2081\n[GOAL]\n\u03b1 : Type u\nL L\u2082 L\u2083\u271d L\u2084\u271d : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nH1 : (x1, b1) \u2260 (x2, b2)\nL\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nh\u2082 : Red L\u2084 L\u2082\nh\u2081\u271d : Red ((x1, b1) :: L\u2083) [(x2, b2)]\nH2 : Red ((x1, b1) :: List.append L\u2083 L\u2084) ((x2, b2) :: L\u2082)\nthis : Red ((x1, b1) :: List.append L\u2083 L\u2084) ([(x2, b2)] ++ L\u2082)\nh\u2081 : Red ((x1, !b1) :: (x1, b1) :: L\u2083) [(x1, !b1), (x2, b2)]\n\u22a2 Red L\u2083 [(x1, !b1), (x2, b2)]\n[PROOFSTEP]\nhave h\u2082 : Red ((x1, not b1) :: (x1, b1) :: L\u2083) L\u2083 := Step.cons_not_rev.to_red\n[GOAL]\n\u03b1 : Type u\nL L\u2082 L\u2083\u271d L\u2084\u271d : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nH1 : (x1, b1) \u2260 (x2, b2)\nL\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nh\u2082\u271d : Red L\u2084 L\u2082\nh\u2081\u271d : Red ((x1, b1) :: L\u2083) [(x2, b2)]\nH2 : Red ((x1, b1) :: List.append L\u2083 L\u2084) ((x2, b2) :: L\u2082)\nthis : Red ((x1, b1) :: List.append L\u2083 L\u2084) ([(x2, b2)] ++ L\u2082)\nh\u2081 : Red ((x1, !b1) :: (x1, b1) :: L\u2083) [(x1, !b1), (x2, b2)]\nh\u2082 : Red ((x1, !b1) :: (x1, b1) :: L\u2083) L\u2083\n\u22a2 Red L\u2083 [(x1, !b1), (x2, b2)]\n[PROOFSTEP]\nrcases church_rosser h\u2081 h\u2082 with \u27e8L', h\u2081, h\u2082\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nL L\u2082 L\u2083\u271d L\u2084\u271d : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nH1 : (x1, b1) \u2260 (x2, b2)\nL\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nh\u2082\u271d\u00b9 : Red L\u2084 L\u2082\nh\u2081\u271d\u00b9 : Red ((x1, b1) :: L\u2083) [(x2, b2)]\nH2 : Red ((x1, b1) :: List.append L\u2083 L\u2084) ((x2, b2) :: L\u2082)\nthis : Red ((x1, b1) :: List.append L\u2083 L\u2084) ([(x2, b2)] ++ L\u2082)\nh\u2081\u271d : Red ((x1, !b1) :: (x1, b1) :: L\u2083) [(x1, !b1), (x2, b2)]\nh\u2082\u271d : Red ((x1, !b1) :: (x1, b1) :: L\u2083) L\u2083\nL' : List (\u03b1 \u00d7 Bool)\nh\u2081 : Red [(x1, !b1), (x2, b2)] L'\nh\u2082 : Red L\u2083 L'\n\u22a2 Red L\u2083 [(x1, !b1), (x2, b2)]\n[PROOFSTEP]\nrw [red_iff_irreducible H1] at h\u2081 \n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nL L\u2082 L\u2083\u271d L\u2084\u271d : List (\u03b1 \u00d7 Bool)\nx1 : \u03b1\nb1 : Bool\nx2 : \u03b1\nb2 : Bool\nH1 : (x1, b1) \u2260 (x2, b2)\nL\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nh\u2082\u271d\u00b9 : Red L\u2084 L\u2082\nh\u2081\u271d\u00b9 : Red ((x1, b1) :: L\u2083) [(x2, b2)]\nH2 : Red ((x1, b1) :: List.append L\u2083 L\u2084) ((x2, b2) :: L\u2082)\nthis : Red ((x1, b1) :: List.append L\u2083 L\u2084) ([(x2, b2)] ++ L\u2082)\nh\u2081\u271d : Red ((x1, !b1) :: (x1, b1) :: L\u2083) [(x1, !b1), (x2, b2)]\nh\u2082\u271d : Red ((x1, !b1) :: (x1, b1) :: L\u2083) L\u2083\nL' : List (\u03b1 \u00d7 Bool)\nh\u2081 : L' = [(x1, !b1), (x2, b2)]\nh\u2082 : Red L\u2083 L'\n\u22a2 Red L\u2083 [(x1, !b1), (x2, b2)]\n[PROOFSTEP]\nrwa [h\u2081] at h\u2082 \n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nH : Step L\u2081 L\u2082\n\u22a2 L\u2082 <+ L\u2081\n[PROOFSTEP]\ncases H\n[GOAL]\ncase not\n\u03b1 : Type u\nL L\u2083 L\u2084 L\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1\nb\u271d : Bool\n\u22a2 L\u2081\u271d ++ L\u2082\u271d <+ L\u2081\u271d ++ (x\u271d, b\u271d) :: (x\u271d, !b\u271d) :: L\u2082\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase not\n\u03b1 : Type u\nL L\u2083 L\u2084 L\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1\nb\u271d : Bool\n\u22a2 L\u2082\u271d <+ (x\u271d, b\u271d) :: (x\u271d, !b\u271d) :: L\u2082\u271d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase not.a\n\u03b1 : Type u\nL L\u2083 L\u2084 L\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1\nb\u271d : Bool\n\u22a2 L\u2082\u271d <+ (x\u271d, !b\u271d) :: L\u2082\u271d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase not.a.a\n\u03b1 : Type u\nL L\u2083 L\u2084 L\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1\nb\u271d : Bool\n\u22a2 L\u2082\u271d <+ L\u2082\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L1 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\n\u22a2 sizeOf (L1 ++ L2) < sizeOf (L1 ++ (x, b) :: (x, !b) :: L2)\n[PROOFSTEP]\ninduction' L1 with hd tl ih\n[GOAL]\ncase nil\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\n\u22a2 sizeOf ([] ++ L2) < sizeOf ([] ++ (x, b) :: (x, !b) :: L2)\ncase cons\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nhd : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : sizeOf (tl ++ L2) < sizeOf (tl ++ (x, b) :: (x, !b) :: L2)\n\u22a2 sizeOf (hd :: tl ++ L2) < sizeOf (hd :: tl ++ (x, b) :: (x, !b) :: L2)\n[PROOFSTEP]\ncase nil =>\n -- dsimp [sizeOf]\n dsimp\n simp only [Bool.sizeOf_eq_one]\n have H : 1 + (1 + 1) + (1 + (1 + 1) + sizeOf L2) = sizeOf L2 + (1 + ((1 + 1) + (1 + 1) + 1)) := by ac_rfl\n rw [H]\n apply Nat.lt_add_of_pos_right\n apply Nat.lt_add_right\n apply Nat.zero_lt_one\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\n\u22a2 sizeOf ([] ++ L2) < sizeOf ([] ++ (x, b) :: (x, !b) :: L2)\n[PROOFSTEP]\ncase nil =>\n -- dsimp [sizeOf]\n dsimp\n simp only [Bool.sizeOf_eq_one]\n have H : 1 + (1 + 1) + (1 + (1 + 1) + sizeOf L2) = sizeOf L2 + (1 + ((1 + 1) + (1 + 1) + 1)) := by ac_rfl\n rw [H]\n apply Nat.lt_add_of_pos_right\n apply Nat.lt_add_right\n apply Nat.zero_lt_one\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\n\u22a2 sizeOf ([] ++ L2) < sizeOf ([] ++ (x, b) :: (x, !b) :: L2)\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\n\u22a2 sizeOf L2 < 1 + (1 + sizeOf b) + (1 + (1 + sizeOf !b) + sizeOf L2)\n[PROOFSTEP]\nsimp only [Bool.sizeOf_eq_one]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\n\u22a2 sizeOf L2 < 1 + (1 + 1) + (1 + (1 + 1) + sizeOf L2)\n[PROOFSTEP]\nhave H : 1 + (1 + 1) + (1 + (1 + 1) + sizeOf L2) = sizeOf L2 + (1 + ((1 + 1) + (1 + 1) + 1)) := by ac_rfl\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\n\u22a2 1 + (1 + 1) + (1 + (1 + 1) + sizeOf L2) = sizeOf L2 + (1 + (1 + 1 + (1 + 1) + 1))\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nH : 1 + (1 + 1) + (1 + (1 + 1) + sizeOf L2) = sizeOf L2 + (1 + (1 + 1 + (1 + 1) + 1))\n\u22a2 sizeOf L2 < 1 + (1 + 1) + (1 + (1 + 1) + sizeOf L2)\n[PROOFSTEP]\nrw [H]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nH : 1 + (1 + 1) + (1 + (1 + 1) + sizeOf L2) = sizeOf L2 + (1 + (1 + 1 + (1 + 1) + 1))\n\u22a2 sizeOf L2 < sizeOf L2 + (1 + (1 + 1 + (1 + 1) + 1))\n[PROOFSTEP]\napply Nat.lt_add_of_pos_right\n[GOAL]\ncase h\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nH : 1 + (1 + 1) + (1 + (1 + 1) + sizeOf L2) = sizeOf L2 + (1 + (1 + 1 + (1 + 1) + 1))\n\u22a2 0 < 1 + (1 + 1 + (1 + 1) + 1)\n[PROOFSTEP]\napply Nat.lt_add_right\n[GOAL]\ncase h.h\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nH : 1 + (1 + 1) + (1 + (1 + 1) + sizeOf L2) = sizeOf L2 + (1 + (1 + 1 + (1 + 1) + 1))\n\u22a2 0 < 1\n[PROOFSTEP]\napply Nat.zero_lt_one\n[GOAL]\ncase cons\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nhd : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : sizeOf (tl ++ L2) < sizeOf (tl ++ (x, b) :: (x, !b) :: L2)\n\u22a2 sizeOf (hd :: tl ++ L2) < sizeOf (hd :: tl ++ (x, b) :: (x, !b) :: L2)\n[PROOFSTEP]\ncase cons =>\n dsimp\n exact Nat.add_lt_add_left ih _\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nhd : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : sizeOf (tl ++ L2) < sizeOf (tl ++ (x, b) :: (x, !b) :: L2)\n\u22a2 sizeOf (hd :: tl ++ L2) < sizeOf (hd :: tl ++ (x, b) :: (x, !b) :: L2)\n[PROOFSTEP]\ncase cons =>\n dsimp\n exact Nat.add_lt_add_left ih _\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nhd : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : sizeOf (tl ++ L2) < sizeOf (tl ++ (x, b) :: (x, !b) :: L2)\n\u22a2 sizeOf (hd :: tl ++ L2) < sizeOf (hd :: tl ++ (x, b) :: (x, !b) :: L2)\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L2 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nhd : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : sizeOf (tl ++ L2) < sizeOf (tl ++ (x, b) :: (x, !b) :: L2)\n\u22a2 1 + sizeOf hd + sizeOf (tl ++ L2) < 1 + sizeOf hd + sizeOf (tl ++ (x, b) :: (x, !b) :: L2)\n[PROOFSTEP]\nexact Nat.add_lt_add_left ih _\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nh : Red L\u2081 L\u2082\n\u22a2 \u2203 n, List.length L\u2081 = List.length L\u2082 + 2 * n\n[PROOFSTEP]\ninduction' h with L\u2082 L\u2083 _h\u2081\u2082 h\u2082\u2083 ih\n[GOAL]\ncase refl\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 \u2203 n, List.length L\u2081 = List.length L\u2081 + 2 * n\n[PROOFSTEP]\nexact \u27e80, rfl\u27e9\n[GOAL]\ncase tail\n\u03b1 : Type u\nL L\u2081 L\u2082\u271d L\u2083\u271d L\u2084 L\u2082 L\u2083 : List (\u03b1 \u00d7 Bool)\n_h\u2081\u2082 : ReflTransGen Step L\u2081 L\u2082\nh\u2082\u2083 : Step L\u2082 L\u2083\nih : \u2203 n, List.length L\u2081 = List.length L\u2082 + 2 * n\n\u22a2 \u2203 n, List.length L\u2081 = List.length L\u2083 + 2 * n\n[PROOFSTEP]\nrcases ih with \u27e8n, eq\u27e9\n[GOAL]\ncase tail.intro\n\u03b1 : Type u\nL L\u2081 L\u2082\u271d L\u2083\u271d L\u2084 L\u2082 L\u2083 : List (\u03b1 \u00d7 Bool)\n_h\u2081\u2082 : ReflTransGen Step L\u2081 L\u2082\nh\u2082\u2083 : Step L\u2082 L\u2083\nn : \u2115\neq : List.length L\u2081 = List.length L\u2082 + 2 * n\n\u22a2 \u2203 n, List.length L\u2081 = List.length L\u2083 + 2 * n\n[PROOFSTEP]\nexists 1 + n\n[GOAL]\ncase tail.intro\n\u03b1 : Type u\nL L\u2081 L\u2082\u271d L\u2083\u271d L\u2084 L\u2082 L\u2083 : List (\u03b1 \u00d7 Bool)\n_h\u2081\u2082 : ReflTransGen Step L\u2081 L\u2082\nh\u2082\u2083 : Step L\u2082 L\u2083\nn : \u2115\neq : List.length L\u2081 = List.length L\u2082 + 2 * n\n\u22a2 List.length L\u2081 = List.length L\u2083 + 2 * (1 + n)\n[PROOFSTEP]\nsimp [mul_add, eq, (Step.length h\u2082\u2083).symm, add_assoc]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 a b\u271d c b : List (\u03b1 \u00d7 Bool)\nhab hac : Red.Step a b\n\u22a2 ReflGen Red.Step b b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 a b\u271d c b : List (\u03b1 \u00d7 Bool)\nhab hac : Red.Step a b\n\u22a2 ReflTransGen Red.Step b b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 List.length (invRev L\u2081) = List.length L\u2081\n[PROOFSTEP]\nsimp [invRev]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 invRev (invRev L\u2081) = L\u2081\n[PROOFSTEP]\nsimp [invRev, List.map_reverse, (\u00b7 \u2218 \u00b7)]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 (Red.Step \u21d2 Red.Step) invRev invRev\n[PROOFSTEP]\nintro a b h\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 a b : List (\u03b1 \u00d7 Bool)\nh : Red.Step a b\n\u22a2 Red.Step (invRev a) (invRev b)\n[PROOFSTEP]\ncases h\n[GOAL]\ncase not\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 L\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1\nb\u271d : Bool\n\u22a2 Red.Step (invRev (L\u2081\u271d ++ (x\u271d, b\u271d) :: (x\u271d, !b\u271d) :: L\u2082\u271d)) (invRev (L\u2081\u271d ++ L\u2082\u271d))\n[PROOFSTEP]\nsimp [invRev]\n[GOAL]\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\nh : Step L\u2081 L\u2082\n\u22a2 Step (FreeGroup.invRev L\u2081) (FreeGroup.invRev L\u2082)\n[PROOFSTEP]\ncases' h with a b x y\n[GOAL]\ncase not\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 a b : List (\u03b1 \u00d7 Bool)\nx : \u03b1\ny : Bool\n\u22a2 Step (FreeGroup.invRev (a ++ (x, y) :: (x, !y) :: b)) (FreeGroup.invRev (a ++ b))\n[PROOFSTEP]\nsimp [FreeGroup.invRev]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nh : Step (FreeGroup.invRev L\u2081) (FreeGroup.invRev L\u2082)\n\u22a2 Step L\u2081 L\u2082\n[PROOFSTEP]\nsimpa only [invRev_invRev] using h.invRev\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nh : Red (invRev L\u2081) (invRev L\u2082)\n\u22a2 Red L\u2081 L\u2082\n[PROOFSTEP]\nsimpa only [invRev_invRev] using h.invRev\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 \u2200 (a b c : FreeGroup \u03b1), a * b * c = a * (b * c)\n[PROOFSTEP]\nrintro \u27e8L\u2081\u27e9 \u27e8L\u2082\u27e9 \u27e8L\u2083\u27e9\n[GOAL]\ncase mk.mk.mk\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083\u271d L\u2084 : List (\u03b1 \u00d7 Bool)\na\u271d : FreeGroup \u03b1\nL\u2081 : List (\u03b1 \u00d7 Bool)\nb\u271d : FreeGroup \u03b1\nL\u2082 : List (\u03b1 \u00d7 Bool)\nc\u271d : FreeGroup \u03b1\nL\u2083 : List (\u03b1 \u00d7 Bool)\n\u22a2 Quot.mk Red.Step L\u2081 * Quot.mk Red.Step L\u2082 * Quot.mk Red.Step L\u2083 =\n Quot.mk Red.Step L\u2081 * (Quot.mk Red.Step L\u2082 * Quot.mk Red.Step L\u2083)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 \u2200 (a : FreeGroup \u03b1), 1 * a = a\n[PROOFSTEP]\nrintro \u27e8L\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\n\u22a2 1 * Quot.mk Red.Step L = Quot.mk Red.Step L\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 \u2200 (a : FreeGroup \u03b1), a * 1 = a\n[PROOFSTEP]\nrintro \u27e8L\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\n\u22a2 Quot.mk Red.Step L * 1 = Quot.mk Red.Step L\n[PROOFSTEP]\nsimp [one_eq_mk]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 \u2200 (a : FreeGroup \u03b1), a\u207b\u00b9 * a = 1\n[PROOFSTEP]\nrintro \u27e8L\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\n\u22a2 (Quot.mk Red.Step L)\u207b\u00b9 * Quot.mk Red.Step L = 1\n[PROOFSTEP]\nexact List.recOn L rfl fun \u27e8x, b\u27e9 tl ih => Eq.trans (Quot.sound <| by simp [invRev, one_eq_mk]) ih\n[GOAL]\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\na\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : (Quot.mk Red.Step tl)\u207b\u00b9 * Quot.mk Red.Step tl = 1\nx : \u03b1\nb : Bool\n\u22a2 Red.Step (invRev ((x, b) :: tl) ++ (x, b) :: tl) (invRev tl ++ tl)\n[PROOFSTEP]\nsimp [invRev, one_eq_mk]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 x\u271d : \u03b1\nH : of x\u271d\u00b9 = of x\u271d\n\u22a2 x\u271d\u00b9 = x\u271d\n[PROOFSTEP]\nlet \u27e8L\u2081, hx, hy\u27e9 := Red.exact.1 H\n[GOAL]\n\u03b1 : Type u\nL L\u2081\u271d L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 x\u271d : \u03b1\nH : of x\u271d\u00b9 = of x\u271d\nL\u2081 : List (\u03b1 \u00d7 Bool)\nhx : Red [(x\u271d\u00b9, true)] L\u2081\nhy : Red [(x\u271d, true)] L\u2081\n\u22a2 x\u271d\u00b9 = x\u271d\n[PROOFSTEP]\nsimp [Red.singleton_iff] at hx hy \n[GOAL]\n\u03b1 : Type u\nL L\u2081\u271d L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 x\u271d : \u03b1\nH : of x\u271d\u00b9 = of x\u271d\nL\u2081 : List (\u03b1 \u00d7 Bool)\nhx : L\u2081 = [(x\u271d\u00b9, true)]\nhy : L\u2081 = [(x\u271d, true)]\n\u22a2 x\u271d\u00b9 = x\u271d\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\nf : \u03b1 \u2192 \u03b2\nH : Step L\u2081 L\u2082\n\u22a2 Lift.aux f L\u2081 = Lift.aux f L\u2082\n[PROOFSTEP]\ncases' H with _ _ _ b\n[GOAL]\ncase not\n\u03b1 : Type u\nL L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\nf : \u03b1 \u2192 \u03b2\nL\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1\nb : Bool\n\u22a2 Lift.aux f (L\u2081\u271d ++ (x\u271d, b) :: (x\u271d, !b) :: L\u2082\u271d) = Lift.aux f (L\u2081\u271d ++ L\u2082\u271d)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase not.false\n\u03b1 : Type u\nL L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\nf : \u03b1 \u2192 \u03b2\nL\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1\n\u22a2 Lift.aux f (L\u2081\u271d ++ (x\u271d, false) :: (x\u271d, !false) :: L\u2082\u271d) = Lift.aux f (L\u2081\u271d ++ L\u2082\u271d)\n[PROOFSTEP]\nsimp [Lift.aux]\n[GOAL]\ncase not.true\n\u03b1 : Type u\nL L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\nf : \u03b1 \u2192 \u03b2\nL\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1\n\u22a2 Lift.aux f (L\u2081\u271d ++ (x\u271d, true) :: (x\u271d, !true) :: L\u2082\u271d) = Lift.aux f (L\u2081\u271d ++ L\u2082\u271d)\n[PROOFSTEP]\nsimp [Lift.aux]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (a b : FreeGroup \u03b1),\n Quot.lift (Lift.aux f) (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) (a * b) =\n Quot.lift (Lift.aux f) (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) a *\n Quot.lift (Lift.aux f) (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) b\n[PROOFSTEP]\nrintro \u27e8L\u2081\u27e9 \u27e8L\u2082\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\nf : \u03b1 \u2192 \u03b2\na\u271d : FreeGroup \u03b1\nL\u2081 : List (\u03b1 \u00d7 Bool)\nb\u271d : FreeGroup \u03b1\nL\u2082 : List (\u03b1 \u00d7 Bool)\n\u22a2 Quot.lift (Lift.aux f) (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082)\n (Quot.mk Red.Step L\u2081 * Quot.mk Red.Step L\u2082) =\n Quot.lift (Lift.aux f) (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082)\n (Quot.mk Red.Step L\u2081) *\n Quot.lift (Lift.aux f) (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082)\n (Quot.mk Red.Step L\u2082)\n[PROOFSTEP]\nsimp [Lift.aux]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\ng : FreeGroup \u03b1 \u2192* \u03b2\n\u22a2 \u2200 (x : FreeGroup \u03b1),\n \u2191((fun f =>\n MonoidHom.mk'\n (Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082))\n (_ :\n \u2200 (a b : FreeGroup \u03b1),\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) (a * b) =\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) a *\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) b))\n ((fun g => \u2191g \u2218 of) g))\n x =\n \u2191g x\n[PROOFSTEP]\nrintro \u27e8L\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\ng : FreeGroup \u03b1 \u2192* \u03b2\nx\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\n\u22a2 \u2191((fun f =>\n MonoidHom.mk'\n (Quot.lift (Lift.aux f) (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082))\n (_ :\n \u2200 (a b : FreeGroup \u03b1),\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) (a * b) =\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) a *\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) b))\n ((fun g => \u2191g \u2218 of) g))\n (Quot.mk Red.Step L) =\n \u2191g (Quot.mk Red.Step L)\n[PROOFSTEP]\nexact\n List.recOn L (g.map_one.symm)\n (by\n rintro \u27e8x, _ | _\u27e9 t (ih : _ = g (mk t))\n \u00b7 show _ = g ((of x)\u207b\u00b9 * mk t)\n simpa [Lift.aux] using ih\n \u00b7 show _ = g (of x * mk t)\n simpa [Lift.aux] using ih)\n[GOAL]\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\ng : FreeGroup \u03b1 \u2192* \u03b2\nx\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\n\u22a2 \u2200 (head : \u03b1 \u00d7 Bool) (tail : List (\u03b1 \u00d7 Bool)),\n \u2191((fun f =>\n MonoidHom.mk'\n (Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082))\n (_ :\n \u2200 (a b : FreeGroup \u03b1),\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) (a * b) =\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) a *\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) b))\n ((fun g => \u2191g \u2218 of) g))\n (Quot.mk Red.Step tail) =\n \u2191g (Quot.mk Red.Step tail) \u2192\n \u2191((fun f =>\n MonoidHom.mk'\n (Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082))\n (_ :\n \u2200 (a b : FreeGroup \u03b1),\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) (a * b) =\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) a *\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) b))\n ((fun g => \u2191g \u2218 of) g))\n (Quot.mk Red.Step (head :: tail)) =\n \u2191g (Quot.mk Red.Step (head :: tail))\n[PROOFSTEP]\nrintro \u27e8x, _ | _\u27e9 t (ih : _ = g (mk t))\n[GOAL]\ncase mk.false\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b9 y : FreeGroup \u03b1\ng : FreeGroup \u03b1 \u2192* \u03b2\nx\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nt : List (\u03b1 \u00d7 Bool)\nih :\n \u2191((fun f =>\n MonoidHom.mk'\n (Quot.lift (Lift.aux f) (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082))\n (_ :\n \u2200 (a b : FreeGroup \u03b1),\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) (a * b) =\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) a *\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) b))\n ((fun g => \u2191g \u2218 of) g))\n (Quot.mk Red.Step t) =\n \u2191g (mk t)\n\u22a2 \u2191((fun f =>\n MonoidHom.mk'\n (Quot.lift (Lift.aux f) (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082))\n (_ :\n \u2200 (a b : FreeGroup \u03b1),\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) (a * b) =\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) a *\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) b))\n ((fun g => \u2191g \u2218 of) g))\n (Quot.mk Red.Step ((x, false) :: t)) =\n \u2191g (Quot.mk Red.Step ((x, false) :: t))\n[PROOFSTEP]\nshow _ = g ((of x)\u207b\u00b9 * mk t)\n[GOAL]\ncase mk.false\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b9 y : FreeGroup \u03b1\ng : FreeGroup \u03b1 \u2192* \u03b2\nx\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nt : List (\u03b1 \u00d7 Bool)\nih :\n \u2191((fun f =>\n MonoidHom.mk'\n (Quot.lift (Lift.aux f) (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082))\n (_ :\n \u2200 (a b : FreeGroup \u03b1),\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) (a * b) =\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) a *\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) b))\n ((fun g => \u2191g \u2218 of) g))\n (Quot.mk Red.Step t) =\n \u2191g (mk t)\n\u22a2 \u2191((fun f =>\n MonoidHom.mk'\n (Quot.lift (Lift.aux f) (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082))\n (_ :\n \u2200 (a b : FreeGroup \u03b1),\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) (a * b) =\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) a *\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) b))\n ((fun g => \u2191g \u2218 of) g))\n (Quot.mk Red.Step ((x, false) :: t)) =\n \u2191g ((of x)\u207b\u00b9 * mk t)\n[PROOFSTEP]\nsimpa [Lift.aux] using ih\n[GOAL]\ncase mk.true\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b9 y : FreeGroup \u03b1\ng : FreeGroup \u03b1 \u2192* \u03b2\nx\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nt : List (\u03b1 \u00d7 Bool)\nih :\n \u2191((fun f =>\n MonoidHom.mk'\n (Quot.lift (Lift.aux f) (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082))\n (_ :\n \u2200 (a b : FreeGroup \u03b1),\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) (a * b) =\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) a *\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) b))\n ((fun g => \u2191g \u2218 of) g))\n (Quot.mk Red.Step t) =\n \u2191g (mk t)\n\u22a2 \u2191((fun f =>\n MonoidHom.mk'\n (Quot.lift (Lift.aux f) (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082))\n (_ :\n \u2200 (a b : FreeGroup \u03b1),\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) (a * b) =\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) a *\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) b))\n ((fun g => \u2191g \u2218 of) g))\n (Quot.mk Red.Step ((x, true) :: t)) =\n \u2191g (Quot.mk Red.Step ((x, true) :: t))\n[PROOFSTEP]\nshow _ = g (of x * mk t)\n[GOAL]\ncase mk.true\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b9 y : FreeGroup \u03b1\ng : FreeGroup \u03b1 \u2192* \u03b2\nx\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nt : List (\u03b1 \u00d7 Bool)\nih :\n \u2191((fun f =>\n MonoidHom.mk'\n (Quot.lift (Lift.aux f) (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082))\n (_ :\n \u2200 (a b : FreeGroup \u03b1),\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) (a * b) =\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) a *\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) b))\n ((fun g => \u2191g \u2218 of) g))\n (Quot.mk Red.Step t) =\n \u2191g (mk t)\n\u22a2 \u2191((fun f =>\n MonoidHom.mk'\n (Quot.lift (Lift.aux f) (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082))\n (_ :\n \u2200 (a b : FreeGroup \u03b1),\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) (a * b) =\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) a *\n Quot.lift (Lift.aux f)\n (_ : \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)), Red.Step L\u2081 L\u2082 \u2192 Lift.aux f L\u2081 = Lift.aux f L\u2082) b))\n ((fun g => \u2191g \u2218 of) g))\n (Quot.mk Red.Step ((x, true) :: t)) =\n \u2191g (of x * mk t)\n[PROOFSTEP]\nsimpa [Lift.aux] using ih\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\ns : Subgroup \u03b2\nH : Set.range f \u2286 \u2191s\n\u22a2 MonoidHom.range (\u2191lift f) \u2264 s\n[PROOFSTEP]\nrintro _ \u27e8\u27e8L\u27e9, rfl\u27e9\n[GOAL]\ncase intro.mk\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\ns : Subgroup \u03b2\nH : Set.range f \u2286 \u2191s\nw\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\n\u22a2 \u2191(\u2191lift f) (Quot.mk Red.Step L) \u2208 s\n[PROOFSTEP]\nexact\n List.recOn L s.one_mem fun \u27e8x, b\u27e9 tl ih =>\n Bool.recOn b (by simp at ih \u22a2; exact s.mul_mem (s.inv_mem <| H \u27e8x, rfl\u27e9) ih)\n (by simp at ih \u22a2; exact s.mul_mem (H \u27e8x, rfl\u27e9) ih)\n[GOAL]\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b9 y : FreeGroup \u03b1\ns : Subgroup \u03b2\nH : Set.range f \u2286 \u2191s\nw\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : \u2191(\u2191lift f) (Quot.mk Red.Step tl) \u2208 s\nx : \u03b1\nb : Bool\n\u22a2 \u2191(\u2191lift f) (Quot.mk Red.Step ((x, false) :: tl)) \u2208 s\n[PROOFSTEP]\nsimp at ih \u22a2\n[GOAL]\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b9 y : FreeGroup \u03b1\ns : Subgroup \u03b2\nH : Set.range f \u2286 \u2191s\nw\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : List.prod (List.map (fun x => bif x.snd then f x.fst else (f x.fst)\u207b\u00b9) tl) \u2208 s\nx : \u03b1\nb : Bool\n\u22a2 (f x)\u207b\u00b9 * List.prod (List.map (fun x => bif x.snd then f x.fst else (f x.fst)\u207b\u00b9) tl) \u2208 s\n[PROOFSTEP]\nexact s.mul_mem (s.inv_mem <| H \u27e8x, rfl\u27e9) ih\n[GOAL]\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b9 y : FreeGroup \u03b1\ns : Subgroup \u03b2\nH : Set.range f \u2286 \u2191s\nw\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : \u2191(\u2191lift f) (Quot.mk Red.Step tl) \u2208 s\nx : \u03b1\nb : Bool\n\u22a2 \u2191(\u2191lift f) (Quot.mk Red.Step ((x, true) :: tl)) \u2208 s\n[PROOFSTEP]\nsimp at ih \u22a2\n[GOAL]\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b9 y : FreeGroup \u03b1\ns : Subgroup \u03b2\nH : Set.range f \u2286 \u2191s\nw\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : List.prod (List.map (fun x => bif x.snd then f x.fst else (f x.fst)\u207b\u00b9) tl) \u2208 s\nx : \u03b1\nb : Bool\n\u22a2 f x * List.prod (List.map (fun x => bif x.snd then f x.fst else (f x.fst)\u207b\u00b9) tl) \u2208 s\n[PROOFSTEP]\nexact s.mul_mem (H \u27e8x, rfl\u27e9) ih\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\n\u22a2 MonoidHom.range (\u2191lift f) = Subgroup.closure (Set.range f)\n[PROOFSTEP]\napply le_antisymm (lift.range_le Subgroup.subset_closure)\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\n\u22a2 Subgroup.closure (Set.range f) \u2264 MonoidHom.range (\u2191lift f)\n[PROOFSTEP]\nrw [Subgroup.closure_le]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\n\u22a2 Set.range f \u2286 \u2191(MonoidHom.range (\u2191lift f))\n[PROOFSTEP]\nrintro _ \u27e8a, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\na : \u03b1\n\u22a2 f a \u2208 \u2191(MonoidHom.range (\u2191lift f))\n[PROOFSTEP]\nexact \u27e8FreeGroup.of a, by simp only [lift.of]\u27e9\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\na : \u03b1\n\u22a2 \u2191(\u2191lift f) (FreeGroup.of a) = f a\n[PROOFSTEP]\nsimp only [lift.of]\n[GOAL]\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\nL\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\nH : Red.Step L\u2081 L\u2082\n\u22a2 Red.Step (List.map (fun x => (f x.fst, x.snd)) L\u2081) (List.map (fun x => (f x.fst, x.snd)) L\u2082)\n[PROOFSTEP]\ncases H\n[GOAL]\ncase not\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\nL\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1\nb\u271d : Bool\n\u22a2 Red.Step (List.map (fun x => (f x.fst, x.snd)) (L\u2081\u271d ++ (x\u271d, b\u271d) :: (x\u271d, !b\u271d) :: L\u2082\u271d))\n (List.map (fun x => (f x.fst, x.snd)) (L\u2081\u271d ++ L\u2082\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\n\u22a2 \u2200 (a b : FreeGroup \u03b1),\n Quot.map (List.map fun x => (f x.fst, x.snd))\n (_ :\n \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)),\n Red.Step L\u2081 L\u2082 \u2192\n Red.Step (List.map (fun x => (f x.fst, x.snd)) L\u2081) (List.map (fun x => (f x.fst, x.snd)) L\u2082))\n (a * b) =\n Quot.map (List.map fun x => (f x.fst, x.snd))\n (_ :\n \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)),\n Red.Step L\u2081 L\u2082 \u2192\n Red.Step (List.map (fun x => (f x.fst, x.snd)) L\u2081) (List.map (fun x => (f x.fst, x.snd)) L\u2082))\n a *\n Quot.map (List.map fun x => (f x.fst, x.snd))\n (_ :\n \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)),\n Red.Step L\u2081 L\u2082 \u2192\n Red.Step (List.map (fun x => (f x.fst, x.snd)) L\u2081) (List.map (fun x => (f x.fst, x.snd)) L\u2082))\n b\n[PROOFSTEP]\nrintro \u27e8L\u2081\u27e9 \u27e8L\u2082\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nx y a\u271d : FreeGroup \u03b1\nL\u2081 : List (\u03b1 \u00d7 Bool)\nb\u271d : FreeGroup \u03b1\nL\u2082 : List (\u03b1 \u00d7 Bool)\n\u22a2 Quot.map (List.map fun x => (f x.fst, x.snd))\n (_ :\n \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)),\n Red.Step L\u2081 L\u2082 \u2192 Red.Step (List.map (fun x => (f x.fst, x.snd)) L\u2081) (List.map (fun x => (f x.fst, x.snd)) L\u2082))\n (Quot.mk Red.Step L\u2081 * Quot.mk Red.Step L\u2082) =\n Quot.map (List.map fun x => (f x.fst, x.snd))\n (_ :\n \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)),\n Red.Step L\u2081 L\u2082 \u2192\n Red.Step (List.map (fun x => (f x.fst, x.snd)) L\u2081) (List.map (fun x => (f x.fst, x.snd)) L\u2082))\n (Quot.mk Red.Step L\u2081) *\n Quot.map (List.map fun x => (f x.fst, x.snd))\n (_ :\n \u2200 (L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)),\n Red.Step L\u2081 L\u2082 \u2192\n Red.Step (List.map (fun x => (f x.fst, x.snd)) L\u2081) (List.map (fun x => (f x.fst, x.snd)) L\u2082))\n (Quot.mk Red.Step L\u2082)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nx\u271d y x : FreeGroup \u03b1\n\u22a2 \u2191(map _root_.id) x = x\n[PROOFSTEP]\nrcases x with \u27e8L\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nx\u271d y x : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\n\u22a2 \u2191(map _root_.id) (Quot.mk Red.Step L) = Quot.mk Red.Step L\n[PROOFSTEP]\nsimp [List.map_id']\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\nf\u271d : \u03b1 \u2192 \u03b2\nx\u271d y : FreeGroup \u03b1\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nx : FreeGroup \u03b1\n\u22a2 \u2191(map g) (\u2191(map f) x) = \u2191(map (g \u2218 f)) x\n[PROOFSTEP]\nrcases x with \u27e8L\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\nf\u271d : \u03b1 \u2192 \u03b2\nx\u271d y : FreeGroup \u03b1\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nx : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\n\u22a2 \u2191(map g) (\u2191(map f) (Quot.mk Red.Step L)) = \u2191(map (g \u2218 f)) (Quot.mk Red.Step L)\n[PROOFSTEP]\nsimp [(\u00b7 \u2218 \u00b7)]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\ng : FreeGroup \u03b1 \u2192* FreeGroup \u03b2\nhg : \u2200 (x : \u03b1), \u2191g (FreeGroup.of x) = FreeGroup.of (f x)\n\u22a2 \u2200 {x : FreeGroup \u03b1}, \u2191g x = \u2191(map f) x\n[PROOFSTEP]\nrintro \u27e8L\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\ng : FreeGroup \u03b1 \u2192* FreeGroup \u03b2\nhg : \u2200 (x : \u03b1), \u2191g (FreeGroup.of x) = FreeGroup.of (f x)\nx\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\n\u22a2 \u2191g (Quot.mk Red.Step L) = \u2191(map f) (Quot.mk Red.Step L)\n[PROOFSTEP]\nexact\n List.recOn L g.map_one fun \u27e8x, b\u27e9 t (ih : g (FreeGroup.mk t) = map f (FreeGroup.mk t)) =>\n Bool.recOn b\n (show g ((FreeGroup.of x)\u207b\u00b9 * FreeGroup.mk t) = FreeGroup.map f ((FreeGroup.of x)\u207b\u00b9 * FreeGroup.mk t) by\n simp [g.map_mul, g.map_inv, hg, ih])\n (show g (FreeGroup.of x * FreeGroup.mk t) = FreeGroup.map f (FreeGroup.of x * FreeGroup.mk t) by\n simp [g.map_mul, hg, ih])\n[GOAL]\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b2 y : FreeGroup \u03b1\ng : FreeGroup \u03b1 \u2192* FreeGroup \u03b2\nhg : \u2200 (x : \u03b1), \u2191g (FreeGroup.of x) = FreeGroup.of (f x)\nx\u271d\u00b9 : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1 \u00d7 Bool\nt : List (\u03b1 \u00d7 Bool)\nih : \u2191g (FreeGroup.mk t) = \u2191(map f) (FreeGroup.mk t)\nx : \u03b1\nb : Bool\n\u22a2 \u2191g ((FreeGroup.of x)\u207b\u00b9 * FreeGroup.mk t) = \u2191(map f) ((FreeGroup.of x)\u207b\u00b9 * FreeGroup.mk t)\n[PROOFSTEP]\nsimp [g.map_mul, g.map_inv, hg, ih]\n[GOAL]\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b2 y : FreeGroup \u03b1\ng : FreeGroup \u03b1 \u2192* FreeGroup \u03b2\nhg : \u2200 (x : \u03b1), \u2191g (FreeGroup.of x) = FreeGroup.of (f x)\nx\u271d\u00b9 : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1 \u00d7 Bool\nt : List (\u03b1 \u00d7 Bool)\nih : \u2191g (FreeGroup.mk t) = \u2191(map f) (FreeGroup.mk t)\nx : \u03b1\nb : Bool\n\u22a2 \u2191g (FreeGroup.of x * FreeGroup.mk t) = \u2191(map f) (FreeGroup.of x * FreeGroup.mk t)\n[PROOFSTEP]\nsimp [g.map_mul, hg, ih]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nx\u271d y : FreeGroup \u03b1\nx : \u03b1\n\u22a2 \u2191(\u2191lift (of \u2218 f)) (of x) = of (f x)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1\u271d \u00d7 Bool)\n\u03b2\u271d : Type v\nf : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d y : FreeGroup \u03b1\u271d\n\u03b1 : Type ?u.99493\n\u03b2 : Type ?u.99494\ne : \u03b1 \u2243 \u03b2\nx : FreeGroup \u03b1\n\u22a2 \u2191(map \u2191e.symm) (\u2191(map \u2191e) x) = x\n[PROOFSTEP]\nsimp [Function.comp, map.comp]\n[GOAL]\n\u03b1\u271d : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1\u271d \u00d7 Bool)\n\u03b2\u271d : Type v\nf : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d y : FreeGroup \u03b1\u271d\n\u03b1 : Type ?u.99493\n\u03b2 : Type ?u.99494\ne : \u03b1 \u2243 \u03b2\nx : FreeGroup \u03b2\n\u22a2 \u2191(map \u2191e) (\u2191(map \u2191e.symm) x) = x\n[PROOFSTEP]\nsimp [Function.comp, map.comp]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx : FreeGroup \u03b1\n\u22a2 \u2191(\u2191lift f) x = \u2191prod (\u2191(map f) x)\n[PROOFSTEP]\nrw [\u2190 lift.unique (prod.comp (map f))]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx : FreeGroup \u03b1\n\u22a2 \u2191(MonoidHom.comp prod (map f)) x = \u2191prod (\u2191(map f) x)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase hg\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx : FreeGroup \u03b1\n\u22a2 \u2200 (x : \u03b1), \u2191(MonoidHom.comp prod (map f)) (of x) = f x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 Function.LeftInverse (fun x => 1) fun x => ()\n[PROOFSTEP]\nrintro \u27e8_ | \u27e8\u27e8\u27e8\u27e9, _\u27e9, _\u27e9\u27e9\n[GOAL]\ncase mk.nil\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx\u271d : FreeGroup Empty\n\u22a2 (fun x => 1) ((fun x => ()) (Quot.mk Red.Step [])) = Quot.mk Red.Step []\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx : FreeGroup Unit\n\u22a2 FreeGroup \u2124\n[PROOFSTEP]\nrevert x\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 FreeGroup Unit \u2192 FreeGroup \u2124\n[PROOFSTEP]\nchange (FreeGroup Unit \u2192* FreeGroup \u2124)\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 FreeGroup Unit \u2192* FreeGroup \u2124\n[PROOFSTEP]\napply map fun _ => (1 : \u2124)\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u22a2 Function.LeftInverse (fun x => of () ^ x) fun x =>\n sum\n ((let_fun this := map fun x => 1;\n \u2191this)\n x)\n[PROOFSTEP]\nrintro \u27e8L\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx\u271d : FreeGroup Unit\nL : List (Unit \u00d7 Bool)\n\u22a2 (fun x => of () ^ x)\n ((fun x =>\n sum\n ((let_fun this := map fun x => 1;\n \u2191this)\n x))\n (Quot.mk Red.Step L)) =\n Quot.mk Red.Step L\n[PROOFSTEP]\nsimp only [quot_mk_eq_mk, map.mk, sum_mk, List.map_map]\n[GOAL]\ncase mk\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx\u271d : FreeGroup Unit\nL : List (Unit \u00d7 Bool)\n\u22a2 of () ^ List.sum (List.map ((fun x => bif x.snd then x.fst else -x.fst) \u2218 fun x => (1, x.snd)) L) = mk L\n[PROOFSTEP]\nexact List.recOn L (by rfl) (fun \u27e8\u27e8\u27e9, b\u27e9 tl ih => by cases b <;> simp [zpow_add] at ih \u22a2 <;> rw [ih] <;> rfl)\n[GOAL]\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx\u271d : FreeGroup Unit\nL : List (Unit \u00d7 Bool)\n\u22a2 of () ^ List.sum (List.map ((fun x => bif x.snd then x.fst else -x.fst) \u2218 fun x => (1, x.snd)) []) = mk []\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 : FreeGroup Unit\nL : List (Unit \u00d7 Bool)\nx\u271d : Unit \u00d7 Bool\ntl : List (Unit \u00d7 Bool)\nih : of () ^ List.sum (List.map ((fun x => bif x.snd then x.fst else -x.fst) \u2218 fun x => (1, x.snd)) tl) = mk tl\nb : Bool\n\u22a2 of () ^\n List.sum (List.map ((fun x => bif x.snd then x.fst else -x.fst) \u2218 fun x => (1, x.snd)) ((PUnit.unit, b) :: tl)) =\n mk ((PUnit.unit, b) :: tl)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 : FreeGroup Unit\nL : List (Unit \u00d7 Bool)\nx\u271d : Unit \u00d7 Bool\ntl : List (Unit \u00d7 Bool)\nih : of () ^ List.sum (List.map ((fun x => bif x.snd then x.fst else -x.fst) \u2218 fun x => (1, x.snd)) tl) = mk tl\n\u22a2 of () ^\n List.sum\n (List.map ((fun x => bif x.snd then x.fst else -x.fst) \u2218 fun x => (1, x.snd)) ((PUnit.unit, false) :: tl)) =\n mk ((PUnit.unit, false) :: tl)\n[PROOFSTEP]\nsimp [zpow_add] at ih \u22a2\n[GOAL]\ncase true\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 : FreeGroup Unit\nL : List (Unit \u00d7 Bool)\nx\u271d : Unit \u00d7 Bool\ntl : List (Unit \u00d7 Bool)\nih : of () ^ List.sum (List.map ((fun x => bif x.snd then x.fst else -x.fst) \u2218 fun x => (1, x.snd)) tl) = mk tl\n\u22a2 of () ^\n List.sum\n (List.map ((fun x => bif x.snd then x.fst else -x.fst) \u2218 fun x => (1, x.snd)) ((PUnit.unit, true) :: tl)) =\n mk ((PUnit.unit, true) :: tl)\n[PROOFSTEP]\nsimp [zpow_add] at ih \u22a2\n[GOAL]\ncase false\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 : FreeGroup Unit\nL : List (Unit \u00d7 Bool)\nx\u271d : Unit \u00d7 Bool\ntl : List (Unit \u00d7 Bool)\nih : of () ^ List.sum (List.map ((fun x => bif x.snd then x.fst else -x.fst) \u2218 fun x => (1, x.snd)) tl) = mk tl\n\u22a2 (of ())\u207b\u00b9 * of () ^ List.sum (List.map ((fun x => bif x.snd then x.fst else -x.fst) \u2218 fun x => (1, x.snd)) tl) =\n mk ((PUnit.unit, false) :: tl)\n[PROOFSTEP]\nrw [ih]\n[GOAL]\ncase true\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 : FreeGroup Unit\nL : List (Unit \u00d7 Bool)\nx\u271d : Unit \u00d7 Bool\ntl : List (Unit \u00d7 Bool)\nih : of () ^ List.sum (List.map ((fun x => bif x.snd then x.fst else -x.fst) \u2218 fun x => (1, x.snd)) tl) = mk tl\n\u22a2 of () * of () ^ List.sum (List.map ((fun x => bif x.snd then x.fst else -x.fst) \u2218 fun x => (1, x.snd)) tl) =\n mk ((PUnit.unit, true) :: tl)\n[PROOFSTEP]\nrw [ih]\n[GOAL]\ncase false\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 : FreeGroup Unit\nL : List (Unit \u00d7 Bool)\nx\u271d : Unit \u00d7 Bool\ntl : List (Unit \u00d7 Bool)\nih : of () ^ List.sum (List.map ((fun x => bif x.snd then x.fst else -x.fst) \u2218 fun x => (1, x.snd)) tl) = mk tl\n\u22a2 (of ())\u207b\u00b9 * mk tl = mk ((PUnit.unit, false) :: tl)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 : FreeGroup Unit\nL : List (Unit \u00d7 Bool)\nx\u271d : Unit \u00d7 Bool\ntl : List (Unit \u00d7 Bool)\nih : of () ^ List.sum (List.map ((fun x => bif x.snd then x.fst else -x.fst) \u2218 fun x => (1, x.snd)) tl) = mk tl\n\u22a2 of () * mk tl = mk ((PUnit.unit, true) :: tl)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx : \u2124\n\u22a2 (fun x =>\n sum\n ((let_fun this := map fun x => 1;\n \u2191this)\n x))\n ((fun x => of () ^ x) 0) =\n 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx : \u2124\ni : \u2115\nih :\n (fun x =>\n sum\n ((let_fun this := map fun x => 1;\n \u2191this)\n x))\n ((fun x => of () ^ x) \u2191i) =\n \u2191i\n\u22a2 (fun x =>\n sum\n ((let_fun this := map fun x => 1;\n \u2191this)\n x))\n ((fun x => of () ^ x) (\u2191i + 1)) =\n \u2191i + 1\n[PROOFSTEP]\nsimp only [zpow_coe_nat, map_pow, map.of] at ih \n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx : \u2124\ni : \u2115\nih : sum (of 1 ^ i) = \u2191i\n\u22a2 (fun x =>\n sum\n ((let_fun this := map fun x => 1;\n \u2191this)\n x))\n ((fun x => of () ^ x) (\u2191i + 1)) =\n \u2191i + 1\n[PROOFSTEP]\nsimp [zpow_add, ih]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx : \u2124\ni : \u2115\nih :\n (fun x =>\n sum\n ((let_fun this := map fun x => 1;\n \u2191this)\n x))\n ((fun x => of () ^ x) (-\u2191i)) =\n -\u2191i\n\u22a2 (fun x =>\n sum\n ((let_fun this := map fun x => 1;\n \u2191this)\n x))\n ((fun x => of () ^ x) (-\u2191i - 1)) =\n -\u2191i - 1\n[PROOFSTEP]\nsimp only [zpow_neg, zpow_coe_nat, map_inv, map_pow, map.of, sum.map_inv, neg_inj] at ih \n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\nx : \u2124\ni : \u2115\nih : sum (of 1 ^ i) = \u2191i\n\u22a2 (fun x =>\n sum\n ((let_fun this := map fun x => 1;\n \u2191this)\n x))\n ((fun x => of () ^ x) (-\u2191i - 1)) =\n -\u2191i - 1\n[PROOFSTEP]\nsimp [zpow_add, ih, sub_eq_add_neg]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d : Type u\nx\u271d : FreeGroup \u03b1\u271d\nx : \u03b1\u271d\nih : id <$> pure x = pure x\n\u22a2 id <$> (pure x)\u207b\u00b9 = (pure x)\u207b\u00b9\n[PROOFSTEP]\nrw [map_inv, ih]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d : Type u\nx\u271d x y : FreeGroup \u03b1\u271d\nihx : id <$> x = x\nihy : id <$> y = y\n\u22a2 id <$> (x * y) = x * y\n[PROOFSTEP]\nrw [map_mul, ihx, ihy]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx : FreeGroup \u03b1\u271d\n\u22a2 \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), 1 >>= f >>= g = 1 >>= fun x => f x >>= g\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx : FreeGroup \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 1 >>= f\u271d >>= g\u271d = 1 >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\niterate 3 rw [one_bind]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx : FreeGroup \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 1 >>= f\u271d >>= g\u271d = 1 >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\nrw [one_bind]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx : FreeGroup \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 1 >>= g\u271d = 1 >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\nrw [one_bind]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx : FreeGroup \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 1 = 1 >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\nrw [one_bind]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : FreeGroup \u03b1\u271d\nx : \u03b1\u271d\n\u22a2 \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), pure x >>= f >>= g = pure x >>= fun x => f x >>= g\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : FreeGroup \u03b1\u271d\nx : \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 pure x >>= f\u271d >>= g\u271d = pure x >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\niterate 2 rw [pure_bind]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : FreeGroup \u03b1\u271d\nx : \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 pure x >>= f\u271d >>= g\u271d = pure x >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\nrw [pure_bind]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : FreeGroup \u03b1\u271d\nx : \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 f\u271d x >>= g\u271d = pure x >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\nrw [pure_bind]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : FreeGroup \u03b1\u271d\nx : \u03b1\u271d\nih : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), pure x >>= f >>= g = pure x >>= fun x => f x >>= g\n\u22a2 \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), (pure x)\u207b\u00b9 >>= f >>= g = (pure x)\u207b\u00b9 >>= fun x => f x >>= g\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : FreeGroup \u03b1\u271d\nx : \u03b1\u271d\nih : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), pure x >>= f >>= g = pure x >>= fun x => f x >>= g\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 (pure x)\u207b\u00b9 >>= f\u271d >>= g\u271d = (pure x)\u207b\u00b9 >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\niterate 3 rw [inv_bind]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : FreeGroup \u03b1\u271d\nx : \u03b1\u271d\nih : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), pure x >>= f >>= g = pure x >>= fun x => f x >>= g\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 (pure x)\u207b\u00b9 >>= f\u271d >>= g\u271d = (pure x)\u207b\u00b9 >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\nrw [inv_bind]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : FreeGroup \u03b1\u271d\nx : \u03b1\u271d\nih : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), pure x >>= f >>= g = pure x >>= fun x => f x >>= g\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 (pure x >>= f\u271d)\u207b\u00b9 >>= g\u271d = (pure x)\u207b\u00b9 >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\nrw [inv_bind]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : FreeGroup \u03b1\u271d\nx : \u03b1\u271d\nih : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), pure x >>= f >>= g = pure x >>= fun x => f x >>= g\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 (pure x >>= f\u271d >>= g\u271d)\u207b\u00b9 = (pure x)\u207b\u00b9 >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\nrw [inv_bind]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : FreeGroup \u03b1\u271d\nx : \u03b1\u271d\nih : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), pure x >>= f >>= g = pure x >>= fun x => f x >>= g\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 (pure x >>= f\u271d >>= g\u271d)\u207b\u00b9 = (pure x >>= fun x => f\u271d x >>= g\u271d)\u207b\u00b9\n[PROOFSTEP]\nrw [ih]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d x y : FreeGroup \u03b1\u271d\nihx : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), x >>= f >>= g = x >>= fun x => f x >>= g\nihy : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), y >>= f >>= g = y >>= fun x => f x >>= g\n\u22a2 \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), x * y >>= f >>= g = x * y >>= fun x => f x >>= g\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d x y : FreeGroup \u03b1\u271d\nihx : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), x >>= f >>= g = x >>= fun x => f x >>= g\nihy : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), y >>= f >>= g = y >>= fun x => f x >>= g\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 x * y >>= f\u271d >>= g\u271d = x * y >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\niterate 3 rw [mul_bind]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d x y : FreeGroup \u03b1\u271d\nihx : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), x >>= f >>= g = x >>= fun x => f x >>= g\nihy : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), y >>= f >>= g = y >>= fun x => f x >>= g\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 x * y >>= f\u271d >>= g\u271d = x * y >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\nrw [mul_bind]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d x y : FreeGroup \u03b1\u271d\nihx : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), x >>= f >>= g = x >>= fun x => f x >>= g\nihy : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), y >>= f >>= g = y >>= fun x => f x >>= g\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 (x >>= f\u271d) * (y >>= f\u271d) >>= g\u271d = x * y >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\nrw [mul_bind]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d x y : FreeGroup \u03b1\u271d\nihx : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), x >>= f >>= g = x >>= fun x => f x >>= g\nihy : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), y >>= f >>= g = y >>= fun x => f x >>= g\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 (x >>= f\u271d >>= g\u271d) * (y >>= f\u271d >>= g\u271d) = x * y >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\nrw [mul_bind]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d x y : FreeGroup \u03b1\u271d\nihx : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), x >>= f >>= g = x >>= fun x => f x >>= g\nihy : \u2200 (f : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d) (g : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d), y >>= f >>= g = y >>= fun x => f x >>= g\nf\u271d : \u03b1\u271d \u2192 FreeGroup \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 FreeGroup \u03b3\u271d\n\u22a2 (x >>= f\u271d >>= g\u271d) * (y >>= f\u271d >>= g\u271d) = (x >>= fun x => f\u271d x >>= g\u271d) * (y >>= fun x => f\u271d x >>= g\u271d)\n[PROOFSTEP]\nrw [ihx, ihy]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d : Type u\nf : \u03b1\u271d \u2192 \u03b2\u271d\nx : FreeGroup \u03b1\u271d\n\u22a2 (do\n let y \u2190 1\n pure (f y)) =\n f <$> 1\n[PROOFSTEP]\nrw [one_bind, map_one]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d : Type u\nf : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : FreeGroup \u03b1\u271d\nx : \u03b1\u271d\n\u22a2 (do\n let y \u2190 pure x\n pure (f y)) =\n f <$> pure x\n[PROOFSTEP]\nrw [pure_bind, map_pure]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d : Type u\nf : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : FreeGroup \u03b1\u271d\nx : \u03b1\u271d\nih :\n (do\n let y \u2190 pure x\n pure (f y)) =\n f <$> pure x\n\u22a2 (do\n let y \u2190 (pure x)\u207b\u00b9\n pure (f y)) =\n f <$> (pure x)\u207b\u00b9\n[PROOFSTEP]\nrw [inv_bind, map_inv, ih]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 \u03b1\u271d \u03b2\u271d : Type u\nf : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d x y : FreeGroup \u03b1\u271d\nihx :\n (do\n let y \u2190 x\n pure (f y)) =\n f <$> x\nihy :\n (do\n let y \u2190 y\n pure (f y)) =\n f <$> y\n\u22a2 (do\n let y \u2190 x * y\n pure (f y)) =\n f <$> (x * y)\n[PROOFSTEP]\nrw [mul_bind, map_mul, ihx, ihy]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\n\u22a2 Red L (reduce L)\n[PROOFSTEP]\ninduction' L with hd1 tl1 ih\n[GOAL]\ncase nil\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\n\u22a2 Red [] (reduce [])\ncase cons\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nih : Red tl1 (reduce tl1)\n\u22a2 Red (hd1 :: tl1) (reduce (hd1 :: tl1))\n[PROOFSTEP]\ncase nil => constructor\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\n\u22a2 Red [] (reduce [])\n[PROOFSTEP]\ncase nil => constructor\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\n\u22a2 Red [] (reduce [])\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nih : Red tl1 (reduce tl1)\n\u22a2 Red (hd1 :: tl1) (reduce (hd1 :: tl1))\n[PROOFSTEP]\ncase cons =>\n dsimp\n revert ih\n generalize htl : reduce tl1 = TL\n intro ih\n cases' TL with hd2 tl2\n case nil => exact Red.cons_cons ih\n case cons =>\n dsimp only\n split_ifs with h\n \u00b7 trans\n \u00b7 cases hd1\n cases hd2\n cases h\n dsimp at *\n subst_vars\n apply Red.trans (Red.cons_cons ih)\n exact Red.Step.cons_not_rev.to_red\n \u00b7 exact Red.cons_cons ih\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nih : Red tl1 (reduce tl1)\n\u22a2 Red (hd1 :: tl1) (reduce (hd1 :: tl1))\n[PROOFSTEP]\ncase cons =>\n dsimp\n revert ih\n generalize htl : reduce tl1 = TL\n intro ih\n cases' TL with hd2 tl2\n case nil => exact Red.cons_cons ih\n case cons =>\n dsimp only\n split_ifs with h\n \u00b7 trans\n \u00b7 cases hd1\n cases hd2\n cases h\n dsimp at *\n subst_vars\n apply Red.trans (Red.cons_cons ih)\n exact Red.Step.cons_not_rev.to_red\n \u00b7 exact Red.cons_cons ih\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nih : Red tl1 (reduce tl1)\n\u22a2 Red (hd1 :: tl1) (reduce (hd1 :: tl1))\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nih : Red tl1 (reduce tl1)\n\u22a2 Red (hd1 :: tl1)\n (List.rec [hd1]\n (fun head tail tail_ih => if hd1.fst = head.fst \u2227 hd1.snd = !head.snd then tail else hd1 :: head :: tail)\n (reduce tl1))\n[PROOFSTEP]\nrevert ih\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\n\u22a2 Red tl1 (reduce tl1) \u2192\n Red (hd1 :: tl1)\n (List.rec [hd1]\n (fun head tail tail_ih => if hd1.fst = head.fst \u2227 hd1.snd = !head.snd then tail else hd1 :: head :: tail)\n (reduce tl1))\n[PROOFSTEP]\ngeneralize htl : reduce tl1 = TL\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 TL : List (\u03b1 \u00d7 Bool)\nhtl : reduce tl1 = TL\n\u22a2 Red tl1 TL \u2192\n Red (hd1 :: tl1)\n (List.rec [hd1]\n (fun head tail tail_ih => if hd1.fst = head.fst \u2227 hd1.snd = !head.snd then tail else hd1 :: head :: tail) TL)\n[PROOFSTEP]\nintro ih\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 TL : List (\u03b1 \u00d7 Bool)\nhtl : reduce tl1 = TL\nih : Red tl1 TL\n\u22a2 Red (hd1 :: tl1)\n (List.rec [hd1]\n (fun head tail tail_ih => if hd1.fst = head.fst \u2227 hd1.snd = !head.snd then tail else hd1 :: head :: tail) TL)\n[PROOFSTEP]\ncases' TL with hd2 tl2\n[GOAL]\ncase nil\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nhtl : reduce tl1 = []\nih : Red tl1 []\n\u22a2 Red (hd1 :: tl1)\n (List.rec [hd1]\n (fun head tail tail_ih => if hd1.fst = head.fst \u2227 hd1.snd = !head.snd then tail else hd1 :: head :: tail) [])\ncase cons\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nhd2 : \u03b1 \u00d7 Bool\ntl2 : List (\u03b1 \u00d7 Bool)\nhtl : reduce tl1 = hd2 :: tl2\nih : Red tl1 (hd2 :: tl2)\n\u22a2 Red (hd1 :: tl1)\n (List.rec [hd1]\n (fun head tail tail_ih => if hd1.fst = head.fst \u2227 hd1.snd = !head.snd then tail else hd1 :: head :: tail)\n (hd2 :: tl2))\n[PROOFSTEP]\ncase nil => exact Red.cons_cons ih\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nhtl : reduce tl1 = []\nih : Red tl1 []\n\u22a2 Red (hd1 :: tl1)\n (List.rec [hd1]\n (fun head tail tail_ih => if hd1.fst = head.fst \u2227 hd1.snd = !head.snd then tail else hd1 :: head :: tail) [])\n[PROOFSTEP]\ncase nil => exact Red.cons_cons ih\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nhtl : reduce tl1 = []\nih : Red tl1 []\n\u22a2 Red (hd1 :: tl1)\n (List.rec [hd1]\n (fun head tail tail_ih => if hd1.fst = head.fst \u2227 hd1.snd = !head.snd then tail else hd1 :: head :: tail) [])\n[PROOFSTEP]\nexact Red.cons_cons ih\n[GOAL]\ncase cons\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nhd2 : \u03b1 \u00d7 Bool\ntl2 : List (\u03b1 \u00d7 Bool)\nhtl : reduce tl1 = hd2 :: tl2\nih : Red tl1 (hd2 :: tl2)\n\u22a2 Red (hd1 :: tl1)\n (List.rec [hd1]\n (fun head tail tail_ih => if hd1.fst = head.fst \u2227 hd1.snd = !head.snd then tail else hd1 :: head :: tail)\n (hd2 :: tl2))\n[PROOFSTEP]\ncase cons =>\n dsimp only\n split_ifs with h\n \u00b7 trans\n \u00b7 cases hd1\n cases hd2\n cases h\n dsimp at *\n subst_vars\n apply Red.trans (Red.cons_cons ih)\n exact Red.Step.cons_not_rev.to_red\n \u00b7 exact Red.cons_cons ih\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nhd2 : \u03b1 \u00d7 Bool\ntl2 : List (\u03b1 \u00d7 Bool)\nhtl : reduce tl1 = hd2 :: tl2\nih : Red tl1 (hd2 :: tl2)\n\u22a2 Red (hd1 :: tl1)\n (List.rec [hd1]\n (fun head tail tail_ih => if hd1.fst = head.fst \u2227 hd1.snd = !head.snd then tail else hd1 :: head :: tail)\n (hd2 :: tl2))\n[PROOFSTEP]\ncase cons =>\n dsimp only\n split_ifs with h\n \u00b7 trans\n \u00b7 cases hd1\n cases hd2\n cases h\n dsimp at *\n subst_vars\n apply Red.trans (Red.cons_cons ih)\n exact Red.Step.cons_not_rev.to_red\n \u00b7 exact Red.cons_cons ih\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nhd2 : \u03b1 \u00d7 Bool\ntl2 : List (\u03b1 \u00d7 Bool)\nhtl : reduce tl1 = hd2 :: tl2\nih : Red tl1 (hd2 :: tl2)\n\u22a2 Red (hd1 :: tl1)\n (List.rec [hd1]\n (fun head tail tail_ih => if hd1.fst = head.fst \u2227 hd1.snd = !head.snd then tail else hd1 :: head :: tail)\n (hd2 :: tl2))\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nhd2 : \u03b1 \u00d7 Bool\ntl2 : List (\u03b1 \u00d7 Bool)\nhtl : reduce tl1 = hd2 :: tl2\nih : Red tl1 (hd2 :: tl2)\n\u22a2 Red (hd1 :: tl1) (if hd1.fst = hd2.fst \u2227 hd1.snd = !hd2.snd then tl2 else hd1 :: hd2 :: tl2)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nhd2 : \u03b1 \u00d7 Bool\ntl2 : List (\u03b1 \u00d7 Bool)\nhtl : reduce tl1 = hd2 :: tl2\nih : Red tl1 (hd2 :: tl2)\nh : hd1.fst = hd2.fst \u2227 hd1.snd = !hd2.snd\n\u22a2 Red (hd1 :: tl1) tl2\n[PROOFSTEP]\ntrans\n[GOAL]\ncase pos\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nhd2 : \u03b1 \u00d7 Bool\ntl2 : List (\u03b1 \u00d7 Bool)\nhtl : reduce tl1 = hd2 :: tl2\nih : Red tl1 (hd2 :: tl2)\nh : hd1.fst = hd2.fst \u2227 hd1.snd = !hd2.snd\n\u22a2 Red (hd1 :: tl1) tl2\n[PROOFSTEP]\ncases hd1\n[GOAL]\ncase pos.mk\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\ntl1 : List (\u03b1 \u00d7 Bool)\nhd2 : \u03b1 \u00d7 Bool\ntl2 : List (\u03b1 \u00d7 Bool)\nhtl : reduce tl1 = hd2 :: tl2\nih : Red tl1 (hd2 :: tl2)\nfst\u271d : \u03b1\nsnd\u271d : Bool\nh : (fst\u271d, snd\u271d).fst = hd2.fst \u2227 (fst\u271d, snd\u271d).snd = !hd2.snd\n\u22a2 Red ((fst\u271d, snd\u271d) :: tl1) tl2\n[PROOFSTEP]\ncases hd2\n[GOAL]\ncase pos.mk.mk\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\ntl1 tl2 : List (\u03b1 \u00d7 Bool)\nfst\u271d\u00b9 : \u03b1\nsnd\u271d\u00b9 : Bool\nfst\u271d : \u03b1\nsnd\u271d : Bool\nhtl : reduce tl1 = (fst\u271d, snd\u271d) :: tl2\nih : Red tl1 ((fst\u271d, snd\u271d) :: tl2)\nh : (fst\u271d\u00b9, snd\u271d\u00b9).fst = (fst\u271d, snd\u271d).fst \u2227 (fst\u271d\u00b9, snd\u271d\u00b9).snd = !(fst\u271d, snd\u271d).snd\n\u22a2 Red ((fst\u271d\u00b9, snd\u271d\u00b9) :: tl1) tl2\n[PROOFSTEP]\ncases h\n[GOAL]\ncase pos.mk.mk.intro\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\ntl1 tl2 : List (\u03b1 \u00d7 Bool)\nfst\u271d\u00b9 : \u03b1\nsnd\u271d\u00b9 : Bool\nfst\u271d : \u03b1\nsnd\u271d : Bool\nhtl : reduce tl1 = (fst\u271d, snd\u271d) :: tl2\nih : Red tl1 ((fst\u271d, snd\u271d) :: tl2)\nleft\u271d : (fst\u271d\u00b9, snd\u271d\u00b9).fst = (fst\u271d, snd\u271d).fst\nright\u271d : (fst\u271d\u00b9, snd\u271d\u00b9).snd = !(fst\u271d, snd\u271d).snd\n\u22a2 Red ((fst\u271d\u00b9, snd\u271d\u00b9) :: tl1) tl2\n[PROOFSTEP]\ndsimp at *\n[GOAL]\ncase pos.mk.mk.intro\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\ntl1 tl2 : List (\u03b1 \u00d7 Bool)\nfst\u271d\u00b9 : \u03b1\nsnd\u271d\u00b9 : Bool\nfst\u271d : \u03b1\nsnd\u271d : Bool\nhtl : reduce tl1 = (fst\u271d, snd\u271d) :: tl2\nih : Red tl1 ((fst\u271d, snd\u271d) :: tl2)\nleft\u271d : fst\u271d\u00b9 = fst\u271d\nright\u271d : snd\u271d\u00b9 = !snd\u271d\n\u22a2 Red ((fst\u271d\u00b9, snd\u271d\u00b9) :: tl1) tl2\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase pos.mk.mk.intro\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\ntl1 tl2 : List (\u03b1 \u00d7 Bool)\nfst\u271d : \u03b1\nsnd\u271d : Bool\nhtl : reduce tl1 = (fst\u271d, snd\u271d) :: tl2\nih : Red tl1 ((fst\u271d, snd\u271d) :: tl2)\n\u22a2 Red ((fst\u271d, !snd\u271d) :: tl1) tl2\n[PROOFSTEP]\napply Red.trans (Red.cons_cons ih)\n[GOAL]\ncase pos.mk.mk.intro\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\ntl1 tl2 : List (\u03b1 \u00d7 Bool)\nfst\u271d : \u03b1\nsnd\u271d : Bool\nhtl : reduce tl1 = (fst\u271d, snd\u271d) :: tl2\nih : Red tl1 ((fst\u271d, snd\u271d) :: tl2)\n\u22a2 Red ((fst\u271d, !snd\u271d) :: (fst\u271d, snd\u271d) :: tl2) tl2\n[PROOFSTEP]\nexact Red.Step.cons_not_rev.to_red\n[GOAL]\ncase neg\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nhd1 : \u03b1 \u00d7 Bool\ntl1 : List (\u03b1 \u00d7 Bool)\nhd2 : \u03b1 \u00d7 Bool\ntl2 : List (\u03b1 \u00d7 Bool)\nhtl : reduce tl1 = hd2 :: tl2\nih : Red tl1 (hd2 :: tl2)\nh : \u00ac(hd1.fst = hd2.fst \u2227 hd1.snd = !hd2.snd)\n\u22a2 Red (hd1 :: tl1) (hd1 :: hd2 :: tl2)\n[PROOFSTEP]\nexact Red.cons_cons ih\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nL2 L3 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nh : reduce [] = L2 ++ (x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: L3\n\u22a2 p\n[PROOFSTEP]\ncases L2\n[GOAL]\ncase nil\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nL3 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nh : reduce [] = [] ++ (x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: L3\n\u22a2 p\n[PROOFSTEP]\ninjections\n[GOAL]\ncase cons\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nL3 : List (\u03b1 \u00d7 Bool)\nx\u271d\u00b9 : \u03b1\nx\u271d : Bool\nhead\u271d : \u03b1 \u00d7 Bool\ntail\u271d : List (\u03b1 \u00d7 Bool)\nh : reduce [] = head\u271d :: tail\u271d ++ (x\u271d\u00b9, x\u271d) :: (x\u271d\u00b9, !x\u271d) :: L3\n\u22a2 p\n[PROOFSTEP]\ninjections\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\n\u22a2 reduce ((x, b) :: L1) = L2 ++ (x', b') :: (x', !b') :: L3 \u2192 p\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\n\u22a2 List.rec [(x, b)] (fun head tail tail_ih => if x = head.fst \u2227 b = !head.snd then tail else (x, b) :: head :: tail)\n (reduce L1) =\n L2 ++ (x', b') :: (x', !b') :: L3 \u2192\n p\n[PROOFSTEP]\ncases r : reduce L1 with\n| nil =>\n dsimp\n intro h\n exfalso\n have := congr_arg List.length h\n simp [List.length] at this \n rw [add_comm, add_assoc, add_assoc, add_comm, <- add_assoc] at this \n simp [Nat.one_eq_succ_zero, Nat.succ_add] at this \n cases this\n| cons hd tail =>\n cases' hd with y c\n dsimp only\n split_ifs with h <;> intro H\n \u00b7 rw [H] at r \n exact @reduce.not _ L1 ((y, c) :: L2) L3 x' b' r\n \u00b7 rcases L2 with (_ | \u27e8a, L2\u27e9)\n \u00b7 injections\n subst_vars\n simp at h \n \u00b7 refine' @reduce.not _ L1 L2 L3 x' b' _\n injection H with _ H\n rw [r, H]\n rfl\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\nx\u271d : List (\u03b1 \u00d7 Bool)\nr : reduce L1 = x\u271d\n\u22a2 List.rec [(x, b)] (fun head tail tail_ih => if x = head.fst \u2227 b = !head.snd then tail else (x, b) :: head :: tail)\n x\u271d =\n L2 ++ (x', b') :: (x', !b') :: L3 \u2192\n p\n[PROOFSTEP]\ncases r : reduce L1 with\n| nil =>\n dsimp\n intro h\n exfalso\n have := congr_arg List.length h\n simp [List.length] at this \n rw [add_comm, add_assoc, add_assoc, add_comm, <- add_assoc] at this \n simp [Nat.one_eq_succ_zero, Nat.succ_add] at this \n cases this\n| cons hd tail =>\n cases' hd with y c\n dsimp only\n split_ifs with h <;> intro H\n \u00b7 rw [H] at r \n exact @reduce.not _ L1 ((y, c) :: L2) L3 x' b' r\n \u00b7 rcases L2 with (_ | \u27e8a, L2\u27e9)\n \u00b7 injections\n subst_vars\n simp at h \n \u00b7 refine' @reduce.not _ L1 L2 L3 x' b' _\n injection H with _ H\n rw [r, H]\n rfl\n[GOAL]\ncase nil\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\nr : reduce L1 = []\n\u22a2 List.rec [(x, b)] (fun head tail tail_ih => if x = head.fst \u2227 b = !head.snd then tail else (x, b) :: head :: tail)\n [] =\n L2 ++ (x', b') :: (x', !b') :: L3 \u2192\n p\n[PROOFSTEP]\n\n| nil =>\n dsimp\n intro h\n exfalso\n have := congr_arg List.length h\n simp [List.length] at this \n rw [add_comm, add_assoc, add_assoc, add_comm, <- add_assoc] at this \n simp [Nat.one_eq_succ_zero, Nat.succ_add] at this \n cases this\n[GOAL]\ncase nil\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\nr : reduce L1 = []\n\u22a2 List.rec [(x, b)] (fun head tail tail_ih => if x = head.fst \u2227 b = !head.snd then tail else (x, b) :: head :: tail)\n [] =\n L2 ++ (x', b') :: (x', !b') :: L3 \u2192\n p\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase nil\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\nr : reduce L1 = []\n\u22a2 [(x, b)] = L2 ++ (x', b') :: (x', !b') :: L3 \u2192 p\n[PROOFSTEP]\nintro h\n[GOAL]\ncase nil\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\nr : reduce L1 = []\nh : [(x, b)] = L2 ++ (x', b') :: (x', !b') :: L3\n\u22a2 p\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase nil.h\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\nr : reduce L1 = []\nh : [(x, b)] = L2 ++ (x', b') :: (x', !b') :: L3\n\u22a2 False\n[PROOFSTEP]\nhave := congr_arg List.length h\n[GOAL]\ncase nil.h\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\nr : reduce L1 = []\nh : [(x, b)] = L2 ++ (x', b') :: (x', !b') :: L3\nthis : List.length [(x, b)] = List.length (L2 ++ (x', b') :: (x', !b') :: L3)\n\u22a2 False\n[PROOFSTEP]\nsimp [List.length] at this \n[GOAL]\ncase nil.h\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\nr : reduce L1 = []\nh : [(x, b)] = L2 ++ (x', b') :: (x', !b') :: L3\nthis : 1 = List.length L2 + (List.length L3 + 1 + 1)\n\u22a2 False\n[PROOFSTEP]\nrw [add_comm, add_assoc, add_assoc, add_comm, <- add_assoc] at this \n[GOAL]\ncase nil.h\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\nr : reduce L1 = []\nh : [(x, b)] = L2 ++ (x', b') :: (x', !b') :: L3\nthis : 1 = 1 + 1 + List.length L2 + List.length L3\n\u22a2 False\n[PROOFSTEP]\nsimp [Nat.one_eq_succ_zero, Nat.succ_add] at this \n[GOAL]\ncase nil.h\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\nr : reduce L1 = []\nh : [(x, b)] = L2 ++ (x', b') :: (x', !b') :: L3\nthis : 0 = List.length L2 + List.length L3 + Nat.succ 0\n\u22a2 False\n[PROOFSTEP]\ncases this\n[GOAL]\ncase cons\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\nhd : \u03b1 \u00d7 Bool\ntail : List (\u03b1 \u00d7 Bool)\nr : reduce L1 = hd :: tail\n\u22a2 List.rec [(x, b)] (fun head tail tail_ih => if x = head.fst \u2227 b = !head.snd then tail else (x, b) :: head :: tail)\n (hd :: tail) =\n L2 ++ (x', b') :: (x', !b') :: L3 \u2192\n p\n[PROOFSTEP]\n\n| cons hd tail =>\n cases' hd with y c\n dsimp only\n split_ifs with h <;> intro H\n \u00b7 rw [H] at r \n exact @reduce.not _ L1 ((y, c) :: L2) L3 x' b' r\n \u00b7 rcases L2 with (_ | \u27e8a, L2\u27e9)\n \u00b7 injections\n subst_vars\n simp at h \n \u00b7 refine' @reduce.not _ L1 L2 L3 x' b' _\n injection H with _ H\n rw [r, H]\n rfl\n[GOAL]\ncase cons\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\nhd : \u03b1 \u00d7 Bool\ntail : List (\u03b1 \u00d7 Bool)\nr : reduce L1 = hd :: tail\n\u22a2 List.rec [(x, b)] (fun head tail tail_ih => if x = head.fst \u2227 b = !head.snd then tail else (x, b) :: head :: tail)\n (hd :: tail) =\n L2 ++ (x', b') :: (x', !b') :: L3 \u2192\n p\n[PROOFSTEP]\ncases' hd with y c\n[GOAL]\ncase cons.mk\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\ntail : List (\u03b1 \u00d7 Bool)\ny : \u03b1\nc : Bool\nr : reduce L1 = (y, c) :: tail\n\u22a2 List.rec [(x, b)] (fun head tail tail_ih => if x = head.fst \u2227 b = !head.snd then tail else (x, b) :: head :: tail)\n ((y, c) :: tail) =\n L2 ++ (x', b') :: (x', !b') :: L3 \u2192\n p\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase cons.mk\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\ntail : List (\u03b1 \u00d7 Bool)\ny : \u03b1\nc : Bool\nr : reduce L1 = (y, c) :: tail\n\u22a2 (if x = y \u2227 b = !c then tail else (x, b) :: (y, c) :: tail) = L2 ++ (x', b') :: (x', !b') :: L3 \u2192 p\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\ntail : List (\u03b1 \u00d7 Bool)\ny : \u03b1\nc : Bool\nr : reduce L1 = (y, c) :: tail\nh : x = y \u2227 b = !c\n\u22a2 tail = L2 ++ (x', b') :: (x', !b') :: L3 \u2192 p\n[PROOFSTEP]\nintro H\n[GOAL]\ncase neg\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\ntail : List (\u03b1 \u00d7 Bool)\ny : \u03b1\nc : Bool\nr : reduce L1 = (y, c) :: tail\nh : \u00ac(x = y \u2227 b = !c)\n\u22a2 (x, b) :: (y, c) :: tail = L2 ++ (x', b') :: (x', !b') :: L3 \u2192 p\n[PROOFSTEP]\nintro H\n[GOAL]\ncase pos\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\ntail : List (\u03b1 \u00d7 Bool)\ny : \u03b1\nc : Bool\nr : reduce L1 = (y, c) :: tail\nh : x = y \u2227 b = !c\nH : tail = L2 ++ (x', b') :: (x', !b') :: L3\n\u22a2 p\n[PROOFSTEP]\nrw [H] at r \n[GOAL]\ncase pos\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\ntail : List (\u03b1 \u00d7 Bool)\ny : \u03b1\nc : Bool\nr : reduce L1 = (y, c) :: (L2 ++ (x', b') :: (x', !b') :: L3)\nh : x = y \u2227 b = !c\nH : tail = L2 ++ (x', b') :: (x', !b') :: L3\n\u22a2 p\n[PROOFSTEP]\nexact @reduce.not _ L1 ((y, c) :: L2) L3 x' b' r\n[GOAL]\ncase neg\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L2 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\ntail : List (\u03b1 \u00d7 Bool)\ny : \u03b1\nc : Bool\nr : reduce L1 = (y, c) :: tail\nh : \u00ac(x = y \u2227 b = !c)\nH : (x, b) :: (y, c) :: tail = L2 ++ (x', b') :: (x', !b') :: L3\n\u22a2 p\n[PROOFSTEP]\nrcases L2 with (_ | \u27e8a, L2\u27e9)\n[GOAL]\ncase neg.nil\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\ntail : List (\u03b1 \u00d7 Bool)\ny : \u03b1\nc : Bool\nr : reduce L1 = (y, c) :: tail\nh : \u00ac(x = y \u2227 b = !c)\nH : (x, b) :: (y, c) :: tail = [] ++ (x', b') :: (x', !b') :: L3\n\u22a2 p\n[PROOFSTEP]\ninjections\n[GOAL]\ncase neg.nil\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\ntail : List (\u03b1 \u00d7 Bool)\ny : \u03b1\nc : Bool\nr : reduce L1 = (y, c) :: tail\nh : \u00ac(x = y \u2227 b = !c)\nfst_eq\u271d\u00b9 : x = x'\nsnd_eq\u271d\u00b9 : b = b'\ntail_eq\u271d : tail = L3\nfst_eq\u271d : y = x'\nsnd_eq\u271d : c = !b'\n\u22a2 p\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase neg.nil\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nL1 : List (\u03b1 \u00d7 Bool)\nb' : Bool\ntail : List (\u03b1 \u00d7 Bool)\ny : \u03b1\nr : reduce L1 = (y, !b') :: tail\nh : \u00ac(y = y \u2227 b' = !!b')\n\u22a2 p\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase neg.cons\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\ntail : List (\u03b1 \u00d7 Bool)\ny : \u03b1\nc : Bool\nr : reduce L1 = (y, c) :: tail\nh : \u00ac(x = y \u2227 b = !c)\na : \u03b1 \u00d7 Bool\nL2 : List (\u03b1 \u00d7 Bool)\nH : (x, b) :: (y, c) :: tail = a :: L2 ++ (x', b') :: (x', !b') :: L3\n\u22a2 p\n[PROOFSTEP]\nrefine' @reduce.not _ L1 L2 L3 x' b' _\n[GOAL]\ncase neg.cons\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\ntail : List (\u03b1 \u00d7 Bool)\ny : \u03b1\nc : Bool\nr : reduce L1 = (y, c) :: tail\nh : \u00ac(x = y \u2227 b = !c)\na : \u03b1 \u00d7 Bool\nL2 : List (\u03b1 \u00d7 Bool)\nH : (x, b) :: (y, c) :: tail = a :: L2 ++ (x', b') :: (x', !b') :: L3\n\u22a2 reduce L1 = L2 ++ (x', b') :: (x', !b') :: L3\n[PROOFSTEP]\ninjection H with _ H\n[GOAL]\ncase neg.cons\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\ntail : List (\u03b1 \u00d7 Bool)\ny : \u03b1\nc : Bool\nr : reduce L1 = (y, c) :: tail\nh : \u00ac(x = y \u2227 b = !c)\na : \u03b1 \u00d7 Bool\nL2 : List (\u03b1 \u00d7 Bool)\nhead_eq\u271d : (x, b) = a\nH : (y, c) :: tail = List.append L2 ((x', b') :: (x', !b') :: L3)\n\u22a2 reduce L1 = L2 ++ (x', b') :: (x', !b') :: L3\n[PROOFSTEP]\nrw [r, H]\n[GOAL]\ncase neg.cons\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\np : Prop\nx : \u03b1\nb : Bool\nL1 L3 : List (\u03b1 \u00d7 Bool)\nx' : \u03b1\nb' : Bool\ntail : List (\u03b1 \u00d7 Bool)\ny : \u03b1\nc : Bool\nr : reduce L1 = (y, c) :: tail\nh : \u00ac(x = y \u2227 b = !c)\na : \u03b1 \u00d7 Bool\nL2 : List (\u03b1 \u00d7 Bool)\nhead_eq\u271d : (x, b) = a\nH : (y, c) :: tail = List.append L2 ((x', b') :: (x', !b') :: L3)\n\u22a2 List.append L2 ((x', b') :: (x', !b') :: L3) = L2 ++ (x', b') :: (x', !b') :: L3\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nH : Red (reduce L\u2081) L\u2082\n\u22a2 reduce L\u2081 = L\u2082\n[PROOFSTEP]\ninduction' H with L1 L' L2 H1 H2 ih\n[GOAL]\ncase refl\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\n\u22a2 reduce L\u2081 = reduce L\u2081\n[PROOFSTEP]\nrfl\n[GOAL]\ncase tail\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nL1 L' : List (\u03b1 \u00d7 Bool)\nL2 : ReflTransGen Red.Step (reduce L\u2081) L1\nH1 : Red.Step L1 L'\nH2 : reduce L\u2081 = L1\n\u22a2 reduce L\u2081 = L'\n[PROOFSTEP]\ncases' H1 with L4 L5 x b\n[GOAL]\ncase tail.not\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nL4 L5 : List (\u03b1 \u00d7 Bool)\nx : \u03b1\nb : Bool\nL2 : ReflTransGen Red.Step (reduce L\u2081) (L4 ++ (x, b) :: (x, !b) :: L5)\nH2 : reduce L\u2081 = L4 ++ (x, b) :: (x, !b) :: L5\n\u22a2 reduce L\u2081 = L4 ++ L5\n[PROOFSTEP]\nexact reduce.not H2\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\n\u22a2 \u2200 {x : FreeGroup \u03b1}, mk (toWord x) = x\n[PROOFSTEP]\nrintro \u27e8L\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nx\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\n\u22a2 mk (toWord (Quot.mk Red.Step L)) = Quot.mk Red.Step L\n[PROOFSTEP]\nexact reduce.self\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\n\u22a2 Function.Injective toWord\n[PROOFSTEP]\nrintro \u27e8L\u2081\u27e9 \u27e8L\u2082\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\na\u2081\u271d : FreeGroup \u03b1\nL\u2081 : List (\u03b1 \u00d7 Bool)\na\u2082\u271d : FreeGroup \u03b1\nL\u2082 : List (\u03b1 \u00d7 Bool)\n\u22a2 toWord (Quot.mk Red.Step L\u2081) = toWord (Quot.mk Red.Step L\u2082) \u2192 Quot.mk Red.Step L\u2081 = Quot.mk Red.Step L\u2082\n[PROOFSTEP]\nexact reduce.exact\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\n\u22a2 \u2200 (x : FreeGroup \u03b1), reduce (toWord x) = toWord x\n[PROOFSTEP]\nrintro \u27e8L\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nx\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\n\u22a2 reduce (toWord (Quot.mk Red.Step L)) = toWord (Quot.mk Red.Step L)\n[PROOFSTEP]\nexact reduce.idem\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nw : List (\u03b1 \u00d7 Bool)\n\u22a2 reduce (invRev w) = invRev (reduce w)\n[PROOFSTEP]\napply reduce.min\n[GOAL]\ncase H\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nw : List (\u03b1 \u00d7 Bool)\n\u22a2 Red (reduce (invRev w)) (invRev (reduce w))\n[PROOFSTEP]\nrw [\u2190 red_invRev_iff, invRev_invRev]\n[GOAL]\ncase H\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nw : List (\u03b1 \u00d7 Bool)\n\u22a2 Red (invRev (reduce (invRev w))) (reduce w)\n[PROOFSTEP]\napply Red.reduce_left\n[GOAL]\ncase H.h\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nw : List (\u03b1 \u00d7 Bool)\n\u22a2 Red w (invRev (reduce (invRev w)))\n[PROOFSTEP]\nhave : Red (invRev (invRev w)) (invRev (reduce (invRev w))) := reduce.red.invRev\n[GOAL]\ncase H.h\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nw : List (\u03b1 \u00d7 Bool)\nthis : Red (invRev (invRev w)) (invRev (reduce (invRev w)))\n\u22a2 Red w (invRev (reduce (invRev w)))\n[PROOFSTEP]\nrwa [invRev_invRev] at this \n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nx : FreeGroup \u03b1\n\u22a2 toWord x\u207b\u00b9 = invRev (toWord x)\n[PROOFSTEP]\nrcases x with \u27e8L\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nx : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\n\u22a2 toWord (Quot.mk Red.Step L)\u207b\u00b9 = invRev (toWord (Quot.mk Red.Step L))\n[PROOFSTEP]\nrw [quot_mk_eq_mk, inv_mk, toWord_mk, toWord_mk, reduce_invRev]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nx : FreeGroup \u03b1\n\u22a2 norm x\u207b\u00b9 = norm x\n[PROOFSTEP]\nsimp only [norm, toWord_inv, invRev_length]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nx : FreeGroup \u03b1\n\u22a2 norm x = 0 \u2194 x = 1\n[PROOFSTEP]\nsimp only [norm, List.length_eq_zero, toWord_eq_nil_iff]\n[GOAL]\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\ninst\u271d : DecidableEq \u03b1\nx y : FreeGroup \u03b1\n\u22a2 norm (x * y) = norm (mk (toWord x ++ toWord y))\n[PROOFSTEP]\nrw [\u2190 mul_mk, mk_toWord, mk_toWord]\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.FreeGroup", "llama_tokens": 56005, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850933, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.4094552635889504}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsPiSystem C\nhD : IsPiSystem D\n\u22a2 IsPiSystem (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D)\n[PROOFSTEP]\nrintro _ \u27e8s\u2081, t\u2081, hs\u2081, ht\u2081, rfl\u27e9 _ \u27e8s\u2082, t\u2082, hs\u2082, ht\u2082, rfl\u27e9 hst\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsPiSystem C\nhD : IsPiSystem D\ns\u2081 : Set \u03b1\nt\u2081 : Set \u03b2\nhs\u2081 : s\u2081 \u2208 C\nht\u2081 : t\u2081 \u2208 D\ns\u2082 : Set \u03b1\nt\u2082 : Set \u03b2\nhs\u2082 : s\u2082 \u2208 C\nht\u2082 : t\u2082 \u2208 D\nhst : Set.Nonempty ((fun x x_1 => x \u00d7\u02e2 x_1) s\u2081 t\u2081 \u2229 (fun x x_1 => x \u00d7\u02e2 x_1) s\u2082 t\u2082)\n\u22a2 (fun x x_1 => x \u00d7\u02e2 x_1) s\u2081 t\u2081 \u2229 (fun x x_1 => x \u00d7\u02e2 x_1) s\u2082 t\u2082 \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D\n[PROOFSTEP]\nrw [prod_inter_prod] at hst \u22a2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsPiSystem C\nhD : IsPiSystem D\ns\u2081 : Set \u03b1\nt\u2081 : Set \u03b2\nhs\u2081 : s\u2081 \u2208 C\nht\u2081 : t\u2081 \u2208 D\ns\u2082 : Set \u03b1\nt\u2082 : Set \u03b2\nhs\u2082 : s\u2082 \u2208 C\nht\u2082 : t\u2082 \u2208 D\nhst : Set.Nonempty ((s\u2081 \u2229 s\u2082) \u00d7\u02e2 (t\u2081 \u2229 t\u2082))\n\u22a2 (s\u2081 \u2229 s\u2082) \u00d7\u02e2 (t\u2081 \u2229 t\u2082) \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D\n[PROOFSTEP]\nrw [prod_nonempty_iff] at hst \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsPiSystem C\nhD : IsPiSystem D\ns\u2081 : Set \u03b1\nt\u2081 : Set \u03b2\nhs\u2081 : s\u2081 \u2208 C\nht\u2081 : t\u2081 \u2208 D\ns\u2082 : Set \u03b1\nt\u2082 : Set \u03b2\nhs\u2082 : s\u2082 \u2208 C\nht\u2082 : t\u2082 \u2208 D\nhst : Set.Nonempty (s\u2081 \u2229 s\u2082) \u2227 Set.Nonempty (t\u2081 \u2229 t\u2082)\n\u22a2 (s\u2081 \u2229 s\u2082) \u00d7\u02e2 (t\u2081 \u2229 t\u2082) \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D\n[PROOFSTEP]\nexact mem_image2_of_mem (hC _ hs\u2081 _ hs\u2082 hst.1) (hD _ ht\u2081 _ ht\u2082 hst.2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\n\u22a2 IsCountablySpanning (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D)\n[PROOFSTEP]\nrcases hC, hD with \u27e8\u27e8s, h1s, h2s\u27e9, t, h1t, h2t\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\ns : \u2115 \u2192 Set \u03b1\nh1s : \u2200 (n : \u2115), s n \u2208 C\nh2s : \u22c3 (n : \u2115), s n = univ\nt : \u2115 \u2192 Set \u03b2\nh1t : \u2200 (n : \u2115), t n \u2208 D\nh2t : \u22c3 (n : \u2115), t n = univ\n\u22a2 IsCountablySpanning (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D)\n[PROOFSTEP]\nrefine' \u27e8fun n => s n.unpair.1 \u00d7\u02e2 t n.unpair.2, fun n => mem_image2_of_mem (h1s _) (h1t _), _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\ns : \u2115 \u2192 Set \u03b1\nh1s : \u2200 (n : \u2115), s n \u2208 C\nh2s : \u22c3 (n : \u2115), s n = univ\nt : \u2115 \u2192 Set \u03b2\nh1t : \u2200 (n : \u2115), t n \u2208 D\nh2t : \u22c3 (n : \u2115), t n = univ\n\u22a2 \u22c3 (n : \u2115), (fun n => s (Nat.unpair n).fst \u00d7\u02e2 t (Nat.unpair n).snd) n = univ\n[PROOFSTEP]\nrw [iUnion_unpair_prod, h2s, h2t, univ_prod_univ]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\n\u22a2 Prod.instMeasurableSpace = generateFrom (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\n\u22a2 Prod.instMeasurableSpace \u2264 generateFrom (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D)\n[PROOFSTEP]\nrefine' sup_le _ _\n[GOAL]\ncase a.refine'_1\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\n\u22a2 MeasurableSpace.comap Prod.fst (generateFrom C) \u2264 generateFrom (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D)\n[PROOFSTEP]\nrw [comap_generateFrom]\n[GOAL]\ncase a.refine'_2\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\n\u22a2 MeasurableSpace.comap Prod.snd (generateFrom D) \u2264 generateFrom (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D)\n[PROOFSTEP]\nrw [comap_generateFrom]\n[GOAL]\ncase a.refine'_1\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\n\u22a2 generateFrom (preimage Prod.fst '' C) \u2264 generateFrom (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D)\n[PROOFSTEP]\napply generateFrom_le\n[GOAL]\ncase a.refine'_2\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\n\u22a2 generateFrom (preimage Prod.snd '' D) \u2264 generateFrom (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D)\n[PROOFSTEP]\napply generateFrom_le\n[GOAL]\ncase a.refine'_1.h\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\n\u22a2 \u2200 (t : Set (\u03b1 \u00d7 \u03b2)), t \u2208 preimage Prod.fst '' C \u2192 MeasurableSet t\n[PROOFSTEP]\nrintro _ \u27e8s, hs, rfl\u27e9\n[GOAL]\ncase a.refine'_2.h\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\n\u22a2 \u2200 (t : Set (\u03b1 \u00d7 \u03b2)), t \u2208 preimage Prod.snd '' D \u2192 MeasurableSet t\n[PROOFSTEP]\nrintro _ \u27e8s, hs, rfl\u27e9\n[GOAL]\ncase a.refine'_1.h.intro.intro\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\ns : Set \u03b1\nhs : s \u2208 C\n\u22a2 MeasurableSet (Prod.fst \u207b\u00b9' s)\n[PROOFSTEP]\nrcases hD with \u27e8t, h1t, h2t\u27e9\n[GOAL]\ncase a.refine'_1.h.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\ns : Set \u03b1\nhs : s \u2208 C\nt : \u2115 \u2192 Set \u03b2\nh1t : \u2200 (n : \u2115), t n \u2208 D\nh2t : \u22c3 (n : \u2115), t n = univ\n\u22a2 MeasurableSet (Prod.fst \u207b\u00b9' s)\n[PROOFSTEP]\nrw [\u2190 prod_univ, \u2190 h2t, prod_iUnion]\n[GOAL]\ncase a.refine'_1.h.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\ns : Set \u03b1\nhs : s \u2208 C\nt : \u2115 \u2192 Set \u03b2\nh1t : \u2200 (n : \u2115), t n \u2208 D\nh2t : \u22c3 (n : \u2115), t n = univ\n\u22a2 MeasurableSet (\u22c3 (i : \u2115), s \u00d7\u02e2 t i)\n[PROOFSTEP]\napply MeasurableSet.iUnion\n[GOAL]\ncase a.refine'_1.h.intro.intro.intro.intro.h\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\ns : Set \u03b1\nhs : s \u2208 C\nt : \u2115 \u2192 Set \u03b2\nh1t : \u2200 (n : \u2115), t n \u2208 D\nh2t : \u22c3 (n : \u2115), t n = univ\n\u22a2 \u2200 (b : \u2115), MeasurableSet (s \u00d7\u02e2 t b)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase a.refine'_1.h.intro.intro.intro.intro.h\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\ns : Set \u03b1\nhs : s \u2208 C\nt : \u2115 \u2192 Set \u03b2\nh1t : \u2200 (n : \u2115), t n \u2208 D\nh2t : \u22c3 (n : \u2115), t n = univ\nn : \u2115\n\u22a2 MeasurableSet (s \u00d7\u02e2 t n)\n[PROOFSTEP]\napply measurableSet_generateFrom\n[GOAL]\ncase a.refine'_1.h.intro.intro.intro.intro.h.ht\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\ns : Set \u03b1\nhs : s \u2208 C\nt : \u2115 \u2192 Set \u03b2\nh1t : \u2200 (n : \u2115), t n \u2208 D\nh2t : \u22c3 (n : \u2115), t n = univ\nn : \u2115\n\u22a2 s \u00d7\u02e2 t n \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D\n[PROOFSTEP]\nexact \u27e8s, t n, hs, h1t n, rfl\u27e9\n[GOAL]\ncase a.refine'_2.h.intro.intro\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\ns : Set \u03b2\nhs : s \u2208 D\n\u22a2 MeasurableSet (Prod.snd \u207b\u00b9' s)\n[PROOFSTEP]\nrcases hC with \u27e8t, h1t, h2t\u27e9\n[GOAL]\ncase a.refine'_2.h.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhD : IsCountablySpanning D\ns : Set \u03b2\nhs : s \u2208 D\nt : \u2115 \u2192 Set \u03b1\nh1t : \u2200 (n : \u2115), t n \u2208 C\nh2t : \u22c3 (n : \u2115), t n = univ\n\u22a2 MeasurableSet (Prod.snd \u207b\u00b9' s)\n[PROOFSTEP]\nrw [\u2190 univ_prod, \u2190 h2t, iUnion_prod_const]\n[GOAL]\ncase a.refine'_2.h.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhD : IsCountablySpanning D\ns : Set \u03b2\nhs : s \u2208 D\nt : \u2115 \u2192 Set \u03b1\nh1t : \u2200 (n : \u2115), t n \u2208 C\nh2t : \u22c3 (n : \u2115), t n = univ\n\u22a2 MeasurableSet (\u22c3 (i : \u2115), t i \u00d7\u02e2 s)\n[PROOFSTEP]\napply MeasurableSet.iUnion\n[GOAL]\ncase a.refine'_2.h.intro.intro.intro.intro.h\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhD : IsCountablySpanning D\ns : Set \u03b2\nhs : s \u2208 D\nt : \u2115 \u2192 Set \u03b1\nh1t : \u2200 (n : \u2115), t n \u2208 C\nh2t : \u22c3 (n : \u2115), t n = univ\n\u22a2 \u2200 (b : \u2115), MeasurableSet (t b \u00d7\u02e2 s)\n[PROOFSTEP]\nrintro n\n[GOAL]\ncase a.refine'_2.h.intro.intro.intro.intro.h\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhD : IsCountablySpanning D\ns : Set \u03b2\nhs : s \u2208 D\nt : \u2115 \u2192 Set \u03b1\nh1t : \u2200 (n : \u2115), t n \u2208 C\nh2t : \u22c3 (n : \u2115), t n = univ\nn : \u2115\n\u22a2 MeasurableSet (t n \u00d7\u02e2 s)\n[PROOFSTEP]\napply measurableSet_generateFrom\n[GOAL]\ncase a.refine'_2.h.intro.intro.intro.intro.h.ht\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhD : IsCountablySpanning D\ns : Set \u03b2\nhs : s \u2208 D\nt : \u2115 \u2192 Set \u03b1\nh1t : \u2200 (n : \u2115), t n \u2208 C\nh2t : \u22c3 (n : \u2115), t n = univ\nn : \u2115\n\u22a2 t n \u00d7\u02e2 s \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D\n[PROOFSTEP]\nexact mem_image2_of_mem (h1t n) hs\n[GOAL]\ncase a\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\n\u22a2 generateFrom (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D) \u2264 Prod.instMeasurableSpace\n[PROOFSTEP]\napply generateFrom_le\n[GOAL]\ncase a.h\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\n\u22a2 \u2200 (t : Set (\u03b1 \u00d7 \u03b2)), t \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D \u2192 MeasurableSet t\n[PROOFSTEP]\nrintro _ \u27e8s, t, hs, ht, rfl\u27e9\n[GOAL]\ncase a.h.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\ns : Set \u03b1\nt : Set \u03b2\nhs : s \u2208 C\nht : t \u2208 D\n\u22a2 MeasurableSet ((fun x x_1 => x \u00d7\u02e2 x_1) s t)\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase a.h.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\ns : Set \u03b1\nt : Set \u03b2\nhs : s \u2208 C\nht : t \u2208 D\n\u22a2 MeasurableSet (s \u00d7\u02e2 t)\n[PROOFSTEP]\nrw [prod_eq]\n[GOAL]\ncase a.h.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\ns : Set \u03b1\nt : Set \u03b2\nhs : s \u2208 C\nht : t \u2208 D\n\u22a2 MeasurableSet (Prod.fst \u207b\u00b9' s \u2229 Prod.snd \u207b\u00b9' t)\n[PROOFSTEP]\napply (measurable_fst _).inter (measurable_snd _)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\ns : Set \u03b1\nt : Set \u03b2\nhs : s \u2208 C\nht : t \u2208 D\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nexact measurableSet_generateFrom hs\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\ns : Set \u03b1\nt : Set \u03b2\nhs : s \u2208 C\nht : t \u2208 D\n\u22a2 MeasurableSet t\n[PROOFSTEP]\nexact measurableSet_generateFrom ht\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d : NormedAddCommGroup E\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : generateFrom C = inst\u271d\u2075\nhD : generateFrom D = inst\u271d\u00b3\nh2C : IsCountablySpanning C\nh2D : IsCountablySpanning D\n\u22a2 generateFrom (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D) = Prod.instMeasurableSpace\n[PROOFSTEP]\nrw [\u2190 hC, \u2190 hD, generateFrom_prod_eq h2C h2D]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : IsFiniteMeasure \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)\n[PROOFSTEP]\nrefine'\n induction_on_inter (C := fun s => Measurable fun x => \u03bd (Prod.mk x \u207b\u00b9' s)) generateFrom_prod.symm isPiSystem_prod _ _\n _ _ hs\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : IsFiniteMeasure \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 (fun s => Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) \u2205\n[PROOFSTEP]\nsimp [measurable_zero, const_def]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : IsFiniteMeasure \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set (\u03b1 \u00d7 \u03b2)),\n t \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) {s | MeasurableSet s} {t | MeasurableSet t} \u2192\n (fun s => Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) t\n[PROOFSTEP]\nrintro _ \u27e8s, t, hs, _, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : IsFiniteMeasure \u03bd\ns\u271d : Set (\u03b1 \u00d7 \u03b2)\nhs\u271d : MeasurableSet s\u271d\ns : Set \u03b1\nt : Set \u03b2\nhs : s \u2208 {s | MeasurableSet s}\nleft\u271d : t \u2208 {t | MeasurableSet t}\n\u22a2 Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' (fun x x_1 => x \u00d7\u02e2 x_1) s t)\n[PROOFSTEP]\nsimp only [mk_preimage_prod_right_eq_if, measure_if]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : IsFiniteMeasure \u03bd\ns\u271d : Set (\u03b1 \u00d7 \u03b2)\nhs\u271d : MeasurableSet s\u271d\ns : Set \u03b1\nt : Set \u03b2\nhs : s \u2208 {s | MeasurableSet s}\nleft\u271d : t \u2208 {t | MeasurableSet t}\n\u22a2 Measurable fun x => indicator s (fun x => \u2191\u2191\u03bd t) x\n[PROOFSTEP]\nexact measurable_const.indicator hs\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : IsFiniteMeasure \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set (\u03b1 \u00d7 \u03b2)),\n MeasurableSet t \u2192\n (fun s => Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) t \u2192 (fun s => Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) t\u1d9c\n[PROOFSTEP]\nintro t ht h2t\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : IsFiniteMeasure \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u03b2)\nht : MeasurableSet t\nh2t : Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' t)\n\u22a2 Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' t\u1d9c)\n[PROOFSTEP]\nsimp_rw [preimage_compl, measure_compl (measurable_prod_mk_left ht) (measure_ne_top \u03bd _)]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : IsFiniteMeasure \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nt : Set (\u03b1 \u00d7 \u03b2)\nht : MeasurableSet t\nh2t : Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' t)\n\u22a2 Measurable fun x => \u2191\u2191\u03bd univ - \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' t)\n[PROOFSTEP]\nexact h2t.const_sub _\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : IsFiniteMeasure \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u2200 (f : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b2)),\n Pairwise (Disjoint on f) \u2192\n (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n (\u2200 (i : \u2115), (fun s => Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) (f i)) \u2192\n (fun s => Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nintro f h1f h2f h3f\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : IsFiniteMeasure \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b2)\nh1f : Pairwise (Disjoint on f)\nh2f : \u2200 (i : \u2115), MeasurableSet (f i)\nh3f : \u2200 (i : \u2115), (fun s => Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) (f i)\n\u22a2 Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' \u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nsimp_rw [preimage_iUnion]\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : IsFiniteMeasure \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b2)\nh1f : Pairwise (Disjoint on f)\nh2f : \u2200 (i : \u2115), MeasurableSet (f i)\nh3f : \u2200 (i : \u2115), (fun s => Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) (f i)\n\u22a2 Measurable fun x => \u2191\u2191\u03bd (\u22c3 (i : \u2115), Prod.mk x \u207b\u00b9' f i)\n[PROOFSTEP]\nhave : \u2200 b, \u03bd (\u22c3 i, Prod.mk b \u207b\u00b9' f i) = \u2211' i, \u03bd (Prod.mk b \u207b\u00b9' f i) := fun b =>\n measure_iUnion (fun i j hij => Disjoint.preimage _ (h1f hij)) fun i => measurable_prod_mk_left (h2f i)\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : IsFiniteMeasure \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b2)\nh1f : Pairwise (Disjoint on f)\nh2f : \u2200 (i : \u2115), MeasurableSet (f i)\nh3f : \u2200 (i : \u2115), (fun s => Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) (f i)\nthis : \u2200 (b : \u03b1), \u2191\u2191\u03bd (\u22c3 (i : \u2115), Prod.mk b \u207b\u00b9' f i) = \u2211' (i : \u2115), \u2191\u2191\u03bd (Prod.mk b \u207b\u00b9' f i)\n\u22a2 Measurable fun x => \u2191\u2191\u03bd (\u22c3 (i : \u2115), Prod.mk x \u207b\u00b9' f i)\n[PROOFSTEP]\nsimp_rw [this]\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : IsFiniteMeasure \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nf : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b2)\nh1f : Pairwise (Disjoint on f)\nh2f : \u2200 (i : \u2115), MeasurableSet (f i)\nh3f : \u2200 (i : \u2115), (fun s => Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) (f i)\nthis : \u2200 (b : \u03b1), \u2191\u2191\u03bd (\u22c3 (i : \u2115), Prod.mk b \u207b\u00b9' f i) = \u2211' (i : \u2115), \u2191\u2191\u03bd (Prod.mk b \u207b\u00b9' f i)\n\u22a2 Measurable fun x => \u2211' (i : \u2115), \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' f i)\n[PROOFSTEP]\napply Measurable.ennreal_tsum h3f\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)\n[PROOFSTEP]\nhave : \u2200 x, MeasurableSet (Prod.mk x \u207b\u00b9' s) := fun x => measurable_prod_mk_left hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s)\n\u22a2 Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)\n[PROOFSTEP]\nsimp only [\u2190 @iSup_restrict_spanningSets _ _ \u03bd, this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s)\n\u22a2 Measurable fun x => \u2a06 (i : \u2115), \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk x \u207b\u00b9' s)\n[PROOFSTEP]\napply measurable_iSup\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s)\n\u22a2 \u2200 (i : \u2115), Measurable fun b => \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk b \u207b\u00b9' s)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s)\ni : \u2115\n\u22a2 Measurable fun b => \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk b \u207b\u00b9' s)\n[PROOFSTEP]\nhaveI := Fact.mk (measure_spanningSets_lt_top \u03bd i)\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis\u271d : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s)\ni : \u2115\nthis : Fact (\u2191\u2191\u03bd (spanningSets \u03bd i) < \u22a4)\n\u22a2 Measurable fun b => \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk b \u207b\u00b9' s)\n[PROOFSTEP]\nexact measurable_measure_prod_mk_left_finite hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u22a2 Measurable fun x => map (Prod.mk x) \u03bd\n[PROOFSTEP]\napply measurable_of_measurable_coe\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u22a2 \u2200 (s : Set (\u03b1 \u00d7 \u03b2)), MeasurableSet s \u2192 Measurable fun b => \u2191\u2191(map (Prod.mk b) \u03bd) s\n[PROOFSTEP]\nintro s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 Measurable fun b => \u2191\u2191(map (Prod.mk b) \u03bd) s\n[PROOFSTEP]\nsimp_rw [map_apply measurable_prod_mk_left hs]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 Measurable fun b => \u2191\u2191\u03bd (Prod.mk b \u207b\u00b9' s)\n[PROOFSTEP]\nexact measurable_measure_prod_mk_left hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 Measurable fun y => map (fun x => (x, y)) \u03bc\n[PROOFSTEP]\napply measurable_of_measurable_coe\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 \u2200 (s : Set (\u03b1 \u00d7 \u03b2)), MeasurableSet s \u2192 Measurable fun b => \u2191\u2191(map (fun x => (x, b)) \u03bc) s\n[PROOFSTEP]\nintro s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 Measurable fun b => \u2191\u2191(map (fun x => (x, b)) \u03bc) s\n[PROOFSTEP]\nsimp_rw [map_apply measurable_prod_mk_right hs]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 Measurable fun b => \u2191\u2191\u03bc ((fun x => (x, b)) \u207b\u00b9' s)\n[PROOFSTEP]\nexact measurable_measure_prod_mk_right hs\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\n\u22a2 MeasurableEmbedding fun x => (g x.fst, f x.snd)\n[PROOFSTEP]\nhave h_inj : Function.Injective fun x : \u03b3 \u00d7 \u03b1 => (g x.fst, f x.snd) :=\n by\n intro x y hxy\n rw [\u2190 @Prod.mk.eta _ _ x, \u2190 @Prod.mk.eta _ _ y]\n simp only [Prod.mk.inj_iff] at hxy \u22a2\n exact \u27e8hg.injective hxy.1, hf.injective hxy.2\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\n\u22a2 Injective fun x => (g x.fst, f x.snd)\n[PROOFSTEP]\nintro x y hxy\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\nx y : \u03b3 \u00d7 \u03b1\nhxy : (fun x => (g x.fst, f x.snd)) x = (fun x => (g x.fst, f x.snd)) y\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 @Prod.mk.eta _ _ x, \u2190 @Prod.mk.eta _ _ y]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\nx y : \u03b3 \u00d7 \u03b1\nhxy : (fun x => (g x.fst, f x.snd)) x = (fun x => (g x.fst, f x.snd)) y\n\u22a2 (x.fst, x.snd) = (y.fst, y.snd)\n[PROOFSTEP]\nsimp only [Prod.mk.inj_iff] at hxy \u22a2\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\nx y : \u03b3 \u00d7 \u03b1\nhxy : g x.fst = g y.fst \u2227 f x.snd = f y.snd\n\u22a2 x.fst = y.fst \u2227 x.snd = y.snd\n[PROOFSTEP]\nexact \u27e8hg.injective hxy.1, hf.injective hxy.2\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\nh_inj : Injective fun x => (g x.fst, f x.snd)\n\u22a2 MeasurableEmbedding fun x => (g x.fst, f x.snd)\n[PROOFSTEP]\nrefine' \u27e8h_inj, _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\nh_inj : Injective fun x => (g x.fst, f x.snd)\n\u22a2 Measurable fun x => (g x.fst, f x.snd)\n[PROOFSTEP]\nexact (hg.measurable.comp measurable_fst).prod_mk (hf.measurable.comp measurable_snd)\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\nh_inj : Injective fun x => (g x.fst, f x.snd)\n\u22a2 \u2200 \u2983s : Set (\u03b3 \u00d7 \u03b1)\u2984, MeasurableSet s \u2192 MeasurableSet ((fun x => (g x.fst, f x.snd)) '' s)\n[PROOFSTEP]\nrefine' fun s hs =>\n @MeasurableSpace.induction_on_inter _ (fun s => MeasurableSet ((fun x : \u03b3 \u00d7 \u03b1 => (g x.fst, f x.snd)) '' s)) _ _\n generateFrom_prod.symm isPiSystem_prod _ _ _ _ _ hs\n[GOAL]\ncase refine'_2.refine'_1\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\nh_inj : Injective fun x => (g x.fst, f x.snd)\ns : Set (\u03b3 \u00d7 \u03b1)\nhs : MeasurableSet s\n\u22a2 (fun s => MeasurableSet ((fun x => (g x.fst, f x.snd)) '' s)) \u2205\n[PROOFSTEP]\nsimp only [Set.image_empty, MeasurableSet.empty]\n[GOAL]\ncase refine'_2.refine'_2\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\nh_inj : Injective fun x => (g x.fst, f x.snd)\ns : Set (\u03b3 \u00d7 \u03b1)\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set (\u03b3 \u00d7 \u03b1)),\n t \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) {s | MeasurableSet s} {t | MeasurableSet t} \u2192\n (fun s => MeasurableSet ((fun x => (g x.fst, f x.snd)) '' s)) t\n[PROOFSTEP]\nrintro t \u27e8t\u2081, t\u2082, ht\u2081, ht\u2082, rfl\u27e9\n[GOAL]\ncase refine'_2.refine'_2.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\nh_inj : Injective fun x => (g x.fst, f x.snd)\ns : Set (\u03b3 \u00d7 \u03b1)\nhs : MeasurableSet s\nt\u2081 : Set \u03b3\nt\u2082 : Set \u03b1\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\n\u22a2 MeasurableSet ((fun x => (g x.fst, f x.snd)) '' (fun x x_1 => x \u00d7\u02e2 x_1) t\u2081 t\u2082)\n[PROOFSTEP]\nrw [\u2190 Set.prod_image_image_eq]\n[GOAL]\ncase refine'_2.refine'_2.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\nh_inj : Injective fun x => (g x.fst, f x.snd)\ns : Set (\u03b3 \u00d7 \u03b1)\nhs : MeasurableSet s\nt\u2081 : Set \u03b3\nt\u2082 : Set \u03b1\nht\u2081 : t\u2081 \u2208 {s | MeasurableSet s}\nht\u2082 : t\u2082 \u2208 {t | MeasurableSet t}\n\u22a2 MeasurableSet ((g '' t\u2081) \u00d7\u02e2 (f '' t\u2082))\n[PROOFSTEP]\nexact (hg.measurableSet_image.mpr ht\u2081).prod (hf.measurableSet_image.mpr ht\u2082)\n[GOAL]\ncase refine'_2.refine'_3\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\nh_inj : Injective fun x => (g x.fst, f x.snd)\ns : Set (\u03b3 \u00d7 \u03b1)\nhs : MeasurableSet s\n\u22a2 \u2200 (t : Set (\u03b3 \u00d7 \u03b1)),\n MeasurableSet t \u2192\n (fun s => MeasurableSet ((fun x => (g x.fst, f x.snd)) '' s)) t \u2192\n (fun s => MeasurableSet ((fun x => (g x.fst, f x.snd)) '' s)) t\u1d9c\n[PROOFSTEP]\nintro t _ ht_m\n[GOAL]\ncase refine'_2.refine'_3\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\nh_inj : Injective fun x => (g x.fst, f x.snd)\ns : Set (\u03b3 \u00d7 \u03b1)\nhs : MeasurableSet s\nt : Set (\u03b3 \u00d7 \u03b1)\na\u271d : MeasurableSet t\nht_m : MeasurableSet ((fun x => (g x.fst, f x.snd)) '' t)\n\u22a2 MeasurableSet ((fun x => (g x.fst, f x.snd)) '' t\u1d9c)\n[PROOFSTEP]\nrw [\u2190 Set.range_diff_image h_inj, \u2190 Set.prod_range_range_eq]\n[GOAL]\ncase refine'_2.refine'_3\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\nh_inj : Injective fun x => (g x.fst, f x.snd)\ns : Set (\u03b3 \u00d7 \u03b1)\nhs : MeasurableSet s\nt : Set (\u03b3 \u00d7 \u03b1)\na\u271d : MeasurableSet t\nht_m : MeasurableSet ((fun x => (g x.fst, f x.snd)) '' t)\n\u22a2 MeasurableSet (range g \u00d7\u02e2 range f \\ (fun x => (g x.fst, f x.snd)) '' t)\n[PROOFSTEP]\nexact MeasurableSet.diff (MeasurableSet.prod hg.measurableSet_range hf.measurableSet_range) ht_m\n[GOAL]\ncase refine'_2.refine'_4\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\nhg : MeasurableEmbedding g\nhf : MeasurableEmbedding f\nh_inj : Injective fun x => (g x.fst, f x.snd)\ns : Set (\u03b3 \u00d7 \u03b1)\nhs : MeasurableSet s\n\u22a2 \u2200 (f_1 : \u2115 \u2192 Set (\u03b3 \u00d7 \u03b1)),\n Pairwise (Disjoint on f_1) \u2192\n (\u2200 (i : \u2115), MeasurableSet (f_1 i)) \u2192\n (\u2200 (i : \u2115), (fun s => MeasurableSet ((fun x => (g x.fst, f x.snd)) '' s)) (f_1 i)) \u2192\n (fun s => MeasurableSet ((fun x => (g x.fst, f x.snd)) '' s)) (\u22c3 (i : \u2115), f_1 i)\n[PROOFSTEP]\nintro g _ _ hg\n[GOAL]\ncase refine'_2.refine'_4\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b3 \u2192 \u03b4\nhg\u271d : MeasurableEmbedding g\u271d\nhf : MeasurableEmbedding f\nh_inj : Injective fun x => (g\u271d x.fst, f x.snd)\ns : Set (\u03b3 \u00d7 \u03b1)\nhs : MeasurableSet s\ng : \u2115 \u2192 Set (\u03b3 \u00d7 \u03b1)\na\u271d\u00b9 : Pairwise (Disjoint on g)\na\u271d : \u2200 (i : \u2115), MeasurableSet (g i)\nhg : \u2200 (i : \u2115), (fun s => MeasurableSet ((fun x => (g\u271d x.fst, f x.snd)) '' s)) (g i)\n\u22a2 MeasurableSet ((fun x => (g\u271d x.fst, f x.snd)) '' \u22c3 (i : \u2115), g i)\n[PROOFSTEP]\nsimp_rw [Set.image_iUnion]\n[GOAL]\ncase refine'_2.refine'_4\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b1'\ninst\u271d\u00b3 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b2 : MeasurableSpace \u03b2'\ninst\u271d\u00b9 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\u271d\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\n\u03b4 : Type u_10\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nm\u03b4 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b3 \u2192 \u03b4\nhg\u271d : MeasurableEmbedding g\u271d\nhf : MeasurableEmbedding f\nh_inj : Injective fun x => (g\u271d x.fst, f x.snd)\ns : Set (\u03b3 \u00d7 \u03b1)\nhs : MeasurableSet s\ng : \u2115 \u2192 Set (\u03b3 \u00d7 \u03b1)\na\u271d\u00b9 : Pairwise (Disjoint on g)\na\u271d : \u2200 (i : \u2115), MeasurableSet (g i)\nhg : \u2200 (i : \u2115), (fun s => MeasurableSet ((fun x => (g\u271d x.fst, f x.snd)) '' s)) (g i)\n\u22a2 MeasurableSet (\u22c3 (i : \u2115), (fun x => (g\u271d x.fst, f x.snd)) '' g i)\n[PROOFSTEP]\nexact MeasurableSet.iUnion hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u22a2 \u2200 {f : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e}, Measurable f \u2192 Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd\n[PROOFSTEP]\nhave m := @measurable_prod_mk_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.13769} {\u03b2 : Type ?u.13770} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\n\u22a2 \u2200 {f : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e}, Measurable f \u2192 Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd\n[PROOFSTEP]\nrefine' Measurable.ennreal_induction (P := fun f => Measurable fun (x : \u03b1) => \u222b\u207b y, f (x, y) \u2202\u03bd) _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.13769} {\u03b2 : Type ?u.13770} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\n\u22a2 \u2200 (c : \u211d\u22650\u221e) \u2983s : Set (\u03b1 \u00d7 \u03b2)\u2984,\n MeasurableSet s \u2192 (fun f => Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd) (Set.indicator s fun x => c)\n[PROOFSTEP]\nintro c s hs\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.13769} {\u03b2 : Type ?u.13770} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 Measurable fun x => \u222b\u207b (y : \u03b2), Set.indicator s (fun x => c) (x, y) \u2202\u03bd\n[PROOFSTEP]\nsimp only [\u2190 indicator_comp_right]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.13769} {\u03b2 : Type ?u.13770} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 Measurable fun x => \u222b\u207b (y : \u03b2), Set.indicator (Prod.mk x \u207b\u00b9' s) ((fun x => c) \u2218 Prod.mk x) y \u2202\u03bd\n[PROOFSTEP]\nsuffices Measurable fun x => c * \u03bd (Prod.mk x \u207b\u00b9' s) by simpa [lintegral_indicator _ (m hs)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.13769} {\u03b2 : Type ?u.13770} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nthis : Measurable fun x => c * \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)\n\u22a2 Measurable fun x => \u222b\u207b (y : \u03b2), Set.indicator (Prod.mk x \u207b\u00b9' s) ((fun x => c) \u2218 Prod.mk x) y \u2202\u03bd\n[PROOFSTEP]\nsimpa [lintegral_indicator _ (m hs)]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.13769} {\u03b2 : Type ?u.13770} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 Measurable fun x => c * \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)\n[PROOFSTEP]\nexact (measurable_measure_prod_mk_left hs).const_mul _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.13769} {\u03b2 : Type ?u.13770} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\n\u22a2 \u2200 \u2983f g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\u2984,\n Disjoint (support f) (support g) \u2192\n Measurable f \u2192\n Measurable g \u2192\n (fun f => Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd) f \u2192\n (fun f => Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd) g \u2192\n (fun f => Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd) (f + g)\n[PROOFSTEP]\nrintro f g - hf - h2f h2g\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.13769} {\u03b2 : Type ?u.13770} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd\nh2g : Measurable fun x => \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd\n\u22a2 Measurable fun x => \u222b\u207b (y : \u03b2), (f + g) (x, y) \u2202\u03bd\n[PROOFSTEP]\nsimp only [Pi.add_apply]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.13769} {\u03b2 : Type ?u.13770} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd\nh2g : Measurable fun x => \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd\n\u22a2 Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd\n[PROOFSTEP]\nconv => enter [1, x]; erw [lintegral_add_left (hf.comp m)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.13769} {\u03b2 : Type ?u.13770} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd\nh2g : Measurable fun x => \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd\n| Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd\n[PROOFSTEP]\nenter [1, x]; erw [lintegral_add_left (hf.comp m)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.13769} {\u03b2 : Type ?u.13770} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd\nh2g : Measurable fun x => \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd\n| Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd\n[PROOFSTEP]\nenter [1, x]; erw [lintegral_add_left (hf.comp m)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.13769} {\u03b2 : Type ?u.13770} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd\nh2g : Measurable fun x => \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd\n| Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd\n[PROOFSTEP]\nenter [1, x]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.13769} {\u03b2 : Type ?u.13770} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd\nh2g : Measurable fun x => \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd\nx : \u03b1\n| \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd\n[PROOFSTEP]\nerw [lintegral_add_left (hf.comp m)]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nh2f : Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd\nh2g : Measurable fun x => \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd\n\u22a2 Measurable fun x => \u222b\u207b (a : \u03b2), (f \u2218 Prod.mk x) a \u2202\u03bd + \u222b\u207b (a : \u03b2), g (x, a) \u2202\u03bd\n[PROOFSTEP]\nexact h2f.add h2g\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\n\u22a2 \u2200 \u2983f : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\u2984,\n (\u2200 (n : \u2115), Measurable (f n)) \u2192\n Monotone f \u2192\n (\u2200 (n : \u2115), (fun f => Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd) (f n)) \u2192\n (fun f => Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd) fun x => \u2a06 (n : \u2115), f n x\n[PROOFSTEP]\nintro f hf h2f h3f\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd) (f n)\n\u22a2 Measurable fun x => \u222b\u207b (y : \u03b2), (fun x => \u2a06 (n : \u2115), f n x) (x, y) \u2202\u03bd\n[PROOFSTEP]\nhave := measurable_iSup h3f\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd) (f n)\nthis : Measurable fun b => \u2a06 (i : \u2115), \u222b\u207b (y : \u03b2), f i (b, y) \u2202\u03bd\n\u22a2 Measurable fun x => \u222b\u207b (y : \u03b2), (fun x => \u2a06 (n : \u2115), f n x) (x, y) \u2202\u03bd\n[PROOFSTEP]\nhave : \u2200 x, Monotone fun n y => f n (x, y) := fun x i j hij y => h2f hij (x, y)\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd) (f n)\nthis\u271d : Measurable fun b => \u2a06 (i : \u2115), \u222b\u207b (y : \u03b2), f i (b, y) \u2202\u03bd\nthis : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y)\n\u22a2 Measurable fun x => \u222b\u207b (y : \u03b2), (fun x => \u2a06 (n : \u2115), f n x) (x, y) \u2202\u03bd\n[PROOFSTEP]\nconv => enter [1, x]; erw [lintegral_iSup (fun n => (hf n).comp m) (this x)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd) (f n)\nthis\u271d : Measurable fun b => \u2a06 (i : \u2115), \u222b\u207b (y : \u03b2), f i (b, y) \u2202\u03bd\nthis : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y)\n| Measurable fun x => \u222b\u207b (y : \u03b2), (fun x => \u2a06 (n : \u2115), f n x) (x, y) \u2202\u03bd\n[PROOFSTEP]\nenter [1, x]; erw [lintegral_iSup (fun n => (hf n).comp m) (this x)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd) (f n)\nthis\u271d : Measurable fun b => \u2a06 (i : \u2115), \u222b\u207b (y : \u03b2), f i (b, y) \u2202\u03bd\nthis : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y)\n| Measurable fun x => \u222b\u207b (y : \u03b2), (fun x => \u2a06 (n : \u2115), f n x) (x, y) \u2202\u03bd\n[PROOFSTEP]\nenter [1, x]; erw [lintegral_iSup (fun n => (hf n).comp m) (this x)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd) (f n)\nthis\u271d : Measurable fun b => \u2a06 (i : \u2115), \u222b\u207b (y : \u03b2), f i (b, y) \u2202\u03bd\nthis : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y)\n| Measurable fun x => \u222b\u207b (y : \u03b2), (fun x => \u2a06 (n : \u2115), f n x) (x, y) \u2202\u03bd\n[PROOFSTEP]\nenter [1, x]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd) (f n)\nthis\u271d : Measurable fun b => \u2a06 (i : \u2115), \u222b\u207b (y : \u03b2), f i (b, y) \u2202\u03bd\nthis : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y)\nx : \u03b1\n| \u222b\u207b (y : \u03b2), (fun x => \u2a06 (n : \u2115), f n x) (x, y) \u2202\u03bd\n[PROOFSTEP]\nerw [lintegral_iSup (fun n => (hf n).comp m) (this x)]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => Measurable fun x => \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd) (f n)\nthis\u271d : Measurable fun b => \u2a06 (i : \u2115), \u222b\u207b (y : \u03b2), f i (b, y) \u2202\u03bd\nthis : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y)\n\u22a2 Measurable fun x => \u2a06 (n : \u2115), \u222b\u207b (a : \u03b2), (f n \u2218 Prod.mk x) a \u2202\u03bd\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) s = \u222b\u207b (x : \u03b1), \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [Measure.prod, bind_apply hs (Measurable.map_prod_mk_left (\u03bd := \u03bd)), map_apply measurable_prod_mk_left hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t) = \u2191\u2191\u03bc s * \u2191\u2191\u03bd t\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t) \u2264 \u2191\u2191\u03bc s * \u2191\u2191\u03bd t\n[PROOFSTEP]\nset ST := toMeasurable \u03bc s \u00d7\u02e2 toMeasurable \u03bd t\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nST : Set (\u03b1 \u00d7 \u03b2) := toMeasurable \u03bc s \u00d7\u02e2 toMeasurable \u03bd t\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t) \u2264 \u2191\u2191\u03bc s * \u2191\u2191\u03bd t\n[PROOFSTEP]\nhave hSTm : MeasurableSet ST := (measurableSet_toMeasurable _ _).prod (measurableSet_toMeasurable _ _)\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nST : Set (\u03b1 \u00d7 \u03b2) := toMeasurable \u03bc s \u00d7\u02e2 toMeasurable \u03bd t\nhSTm : MeasurableSet ST\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t) \u2264 \u2191\u2191\u03bc s * \u2191\u2191\u03bd t\n[PROOFSTEP]\ncalc\n \u03bc.prod \u03bd (s \u00d7\u02e2 t) \u2264 \u03bc.prod \u03bd ST := measure_mono <| Set.prod_mono (subset_toMeasurable _ _) (subset_toMeasurable _ _)\n _ = \u03bc (toMeasurable \u03bc s) * \u03bd (toMeasurable \u03bd t) :=\n by\n rw [prod_apply hSTm]\n simp_rw [mk_preimage_prod_right_eq_if, measure_if, lintegral_indicator _ (measurableSet_toMeasurable _ _),\n lintegral_const, restrict_apply_univ, mul_comm]\n _ = \u03bc s * \u03bd t := by rw [measure_toMeasurable, measure_toMeasurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nST : Set (\u03b1 \u00d7 \u03b2) := toMeasurable \u03bc s \u00d7\u02e2 toMeasurable \u03bd t\nhSTm : MeasurableSet ST\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) ST = \u2191\u2191\u03bc (toMeasurable \u03bc s) * \u2191\u2191\u03bd (toMeasurable \u03bd t)\n[PROOFSTEP]\nrw [prod_apply hSTm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nST : Set (\u03b1 \u00d7 \u03b2) := toMeasurable \u03bc s \u00d7\u02e2 toMeasurable \u03bd t\nhSTm : MeasurableSet ST\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' ST) \u2202\u03bc = \u2191\u2191\u03bc (toMeasurable \u03bc s) * \u2191\u2191\u03bd (toMeasurable \u03bd t)\n[PROOFSTEP]\nsimp_rw [mk_preimage_prod_right_eq_if, measure_if, lintegral_indicator _ (measurableSet_toMeasurable _ _),\n lintegral_const, restrict_apply_univ, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nST : Set (\u03b1 \u00d7 \u03b2) := toMeasurable \u03bc s \u00d7\u02e2 toMeasurable \u03bd t\nhSTm : MeasurableSet ST\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s) * \u2191\u2191\u03bd (toMeasurable \u03bd t) = \u2191\u2191\u03bc s * \u2191\u2191\u03bd t\n[PROOFSTEP]\nrw [measure_toMeasurable, measure_toMeasurable]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd t \u2264 \u2191\u2191(Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\n[PROOFSTEP]\nset ST := toMeasurable (\u03bc.prod \u03bd) (s \u00d7\u02e2 t)\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nST : Set (\u03b1 \u00d7 \u03b2) := toMeasurable (Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\n\u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd t \u2264 \u2191\u2191(Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\n[PROOFSTEP]\nhave hSTm : MeasurableSet ST := measurableSet_toMeasurable _ _\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nST : Set (\u03b1 \u00d7 \u03b2) := toMeasurable (Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\nhSTm : MeasurableSet ST\n\u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd t \u2264 \u2191\u2191(Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\n[PROOFSTEP]\nhave hST : s \u00d7\u02e2 t \u2286 ST := subset_toMeasurable _ _\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nST : Set (\u03b1 \u00d7 \u03b2) := toMeasurable (Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\nhSTm : MeasurableSet ST\nhST : s \u00d7\u02e2 t \u2286 ST\n\u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd t \u2264 \u2191\u2191(Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\n[PROOFSTEP]\nset f : \u03b1 \u2192 \u211d\u22650\u221e := fun x => \u03bd (Prod.mk x \u207b\u00b9' ST)\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nST : Set (\u03b1 \u00d7 \u03b2) := toMeasurable (Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\nhSTm : MeasurableSet ST\nhST : s \u00d7\u02e2 t \u2286 ST\nf : \u03b1 \u2192 \u211d\u22650\u221e := fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' ST)\n\u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd t \u2264 \u2191\u2191(Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\n[PROOFSTEP]\nhave hfm : Measurable f := measurable_measure_prod_mk_left hSTm\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nST : Set (\u03b1 \u00d7 \u03b2) := toMeasurable (Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\nhSTm : MeasurableSet ST\nhST : s \u00d7\u02e2 t \u2286 ST\nf : \u03b1 \u2192 \u211d\u22650\u221e := fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' ST)\nhfm : Measurable f\n\u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd t \u2264 \u2191\u2191(Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\n[PROOFSTEP]\nset s' : Set \u03b1 := {x | \u03bd t \u2264 f x}\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nST : Set (\u03b1 \u00d7 \u03b2) := toMeasurable (Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\nhSTm : MeasurableSet ST\nhST : s \u00d7\u02e2 t \u2286 ST\nf : \u03b1 \u2192 \u211d\u22650\u221e := fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' ST)\nhfm : Measurable f\ns' : Set \u03b1 := {x | \u2191\u2191\u03bd t \u2264 f x}\n\u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd t \u2264 \u2191\u2191(Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\n[PROOFSTEP]\nhave hss' : s \u2286 s' := fun x hx => measure_mono fun y hy => hST <| mk_mem_prod hx hy\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nST : Set (\u03b1 \u00d7 \u03b2) := toMeasurable (Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\nhSTm : MeasurableSet ST\nhST : s \u00d7\u02e2 t \u2286 ST\nf : \u03b1 \u2192 \u211d\u22650\u221e := fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' ST)\nhfm : Measurable f\ns' : Set \u03b1 := {x | \u2191\u2191\u03bd t \u2264 f x}\nhss' : s \u2286 s'\n\u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd t \u2264 \u2191\u2191(Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\n[PROOFSTEP]\ncalc\n \u03bc s * \u03bd t \u2264 \u03bc s' * \u03bd t := mul_le_mul_right' (measure_mono hss') _\n _ = \u222b\u207b _ in s', \u03bd t \u2202\u03bc := by rw [set_lintegral_const, mul_comm]\n _ \u2264 \u222b\u207b x in s', f x \u2202\u03bc := (set_lintegral_mono measurable_const hfm fun x => id)\n _ \u2264 \u222b\u207b x, f x \u2202\u03bc := (lintegral_mono' restrict_le_self le_rfl)\n _ = \u03bc.prod \u03bd ST := (prod_apply hSTm).symm\n _ = \u03bc.prod \u03bd (s \u00d7\u02e2 t) := measure_toMeasurable _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nST : Set (\u03b1 \u00d7 \u03b2) := toMeasurable (Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\nhSTm : MeasurableSet ST\nhST : s \u00d7\u02e2 t \u2286 ST\nf : \u03b1 \u2192 \u211d\u22650\u221e := fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' ST)\nhfm : Measurable f\ns' : Set \u03b1 := {x | \u2191\u2191\u03bd t \u2264 f x}\nhss' : s \u2286 s'\n\u22a2 \u2191\u2191\u03bc s' * \u2191\u2191\u03bd t = \u222b\u207b (x : \u03b1) in s', \u2191\u2191\u03bd t \u2202\u03bc\n[PROOFSTEP]\nrw [set_lintegral_const, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\nX : Type u_7\nY : Type u_8\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\nm : MeasurableSpace X\n\u03bc : Measure X\ninst\u271d\u00b2 : IsOpenPosMeasure \u03bc\nm' : MeasurableSpace Y\n\u03bd : Measure Y\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bd\ninst\u271d : SigmaFinite \u03bd\n\u22a2 IsOpenPosMeasure (Measure.prod \u03bc \u03bd)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase open_pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\nX : Type u_7\nY : Type u_8\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\nm : MeasurableSpace X\n\u03bc : Measure X\ninst\u271d\u00b2 : IsOpenPosMeasure \u03bc\nm' : MeasurableSpace Y\n\u03bd : Measure Y\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bd\ninst\u271d : SigmaFinite \u03bd\n\u22a2 \u2200 (U : Set (X \u00d7 Y)), IsOpen U \u2192 Set.Nonempty U \u2192 \u2191\u2191(Measure.prod \u03bc \u03bd) U \u2260 0\n[PROOFSTEP]\nrintro U U_open \u27e8\u27e8x, y\u27e9, hxy\u27e9\n[GOAL]\ncase open_pos.intro.mk\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\nX : Type u_7\nY : Type u_8\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\nm : MeasurableSpace X\n\u03bc : Measure X\ninst\u271d\u00b2 : IsOpenPosMeasure \u03bc\nm' : MeasurableSpace Y\n\u03bd : Measure Y\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bd\ninst\u271d : SigmaFinite \u03bd\nU : Set (X \u00d7 Y)\nU_open : IsOpen U\nx : X\ny : Y\nhxy : (x, y) \u2208 U\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) U \u2260 0\n[PROOFSTEP]\nrcases isOpen_prod_iff.1 U_open x y hxy with \u27e8u, v, u_open, v_open, xu, yv, huv\u27e9\n[GOAL]\ncase open_pos.intro.mk.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\nX : Type u_7\nY : Type u_8\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\nm : MeasurableSpace X\n\u03bc : Measure X\ninst\u271d\u00b2 : IsOpenPosMeasure \u03bc\nm' : MeasurableSpace Y\n\u03bd : Measure Y\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bd\ninst\u271d : SigmaFinite \u03bd\nU : Set (X \u00d7 Y)\nU_open : IsOpen U\nx : X\ny : Y\nhxy : (x, y) \u2208 U\nu : Set X\nv : Set Y\nu_open : IsOpen u\nv_open : IsOpen v\nxu : x \u2208 u\nyv : y \u2208 v\nhuv : u \u00d7\u02e2 v \u2286 U\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) U \u2260 0\n[PROOFSTEP]\nrefine' ne_of_gt (lt_of_lt_of_le _ (measure_mono huv))\n[GOAL]\ncase open_pos.intro.mk.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\nX : Type u_7\nY : Type u_8\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\nm : MeasurableSpace X\n\u03bc : Measure X\ninst\u271d\u00b2 : IsOpenPosMeasure \u03bc\nm' : MeasurableSpace Y\n\u03bd : Measure Y\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bd\ninst\u271d : SigmaFinite \u03bd\nU : Set (X \u00d7 Y)\nU_open : IsOpen U\nx : X\ny : Y\nhxy : (x, y) \u2208 U\nu : Set X\nv : Set Y\nu_open : IsOpen u\nv_open : IsOpen v\nxu : x \u2208 u\nyv : y \u2208 v\nhuv : u \u00d7\u02e2 v \u2286 U\n\u22a2 0 < \u2191\u2191(Measure.prod \u03bc \u03bd) (u \u00d7\u02e2 v)\n[PROOFSTEP]\nsimp only [prod_prod, CanonicallyOrderedCommSemiring.mul_pos]\n[GOAL]\ncase open_pos.intro.mk.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\nX : Type u_7\nY : Type u_8\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\nm : MeasurableSpace X\n\u03bc : Measure X\ninst\u271d\u00b2 : IsOpenPosMeasure \u03bc\nm' : MeasurableSpace Y\n\u03bd : Measure Y\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bd\ninst\u271d : SigmaFinite \u03bd\nU : Set (X \u00d7 Y)\nU_open : IsOpen U\nx : X\ny : Y\nhxy : (x, y) \u2208 U\nu : Set X\nv : Set Y\nu_open : IsOpen u\nv_open : IsOpen v\nxu : x \u2208 u\nyv : y \u2208 v\nhuv : u \u00d7\u02e2 v \u2286 U\n\u22a2 0 < \u2191\u2191\u03bc u \u2227 0 < \u2191\u2191\u03bd v\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase open_pos.intro.mk.intro.intro.intro.intro.intro.intro.left\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\nX : Type u_7\nY : Type u_8\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\nm : MeasurableSpace X\n\u03bc : Measure X\ninst\u271d\u00b2 : IsOpenPosMeasure \u03bc\nm' : MeasurableSpace Y\n\u03bd : Measure Y\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bd\ninst\u271d : SigmaFinite \u03bd\nU : Set (X \u00d7 Y)\nU_open : IsOpen U\nx : X\ny : Y\nhxy : (x, y) \u2208 U\nu : Set X\nv : Set Y\nu_open : IsOpen u\nv_open : IsOpen v\nxu : x \u2208 u\nyv : y \u2208 v\nhuv : u \u00d7\u02e2 v \u2286 U\n\u22a2 0 < \u2191\u2191\u03bc u\n[PROOFSTEP]\nexact u_open.measure_pos \u03bc \u27e8x, xu\u27e9\n[GOAL]\ncase open_pos.intro.mk.intro.intro.intro.intro.intro.intro.right\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\nX : Type u_7\nY : Type u_8\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\nm : MeasurableSpace X\n\u03bc : Measure X\ninst\u271d\u00b2 : IsOpenPosMeasure \u03bc\nm' : MeasurableSpace Y\n\u03bd : Measure Y\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bd\ninst\u271d : SigmaFinite \u03bd\nU : Set (X \u00d7 Y)\nU_open : IsOpen U\nx : X\ny : Y\nhxy : (x, y) \u2208 U\nu : Set X\nv : Set Y\nu_open : IsOpen u\nv_open : IsOpen v\nxu : x \u2208 u\nyv : y \u2208 v\nhuv : u \u00d7\u02e2 v \u2286 U\n\u22a2 0 < \u2191\u2191\u03bd v\n[PROOFSTEP]\nexact v_open.measure_pos \u03bd \u27e8y, yv\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\u271d\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\n\u22a2 IsFiniteMeasure (Measure.prod \u03bc \u03bd)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase measure_univ_lt_top\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\u271d\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) univ < \u22a4\n[PROOFSTEP]\nrw [\u2190 univ_prod_univ, prod_prod]\n[GOAL]\ncase measure_univ_lt_top\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\u271d\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\n\u22a2 \u2191\u2191\u03bc univ * \u2191\u2191\u03bd univ < \u22a4\n[PROOFSTEP]\nexact mul_lt_top (measure_lt_top _ _).ne (measure_lt_top _ _).ne\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\u271d\n\u03b1 : Type u_7\n\u03b2 : Type u_8\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\ninst\u271d : IsProbabilityMeasure \u03bd\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) univ = 1\n[PROOFSTEP]\nrw [\u2190 univ_prod_univ, prod_prod, measure_univ, measure_univ, mul_one]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\n\u03b1 : Type u_7\n\u03b2 : Type u_8\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b9 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d : SigmaFinite \u03bd\n\u22a2 IsFiniteMeasureOnCompacts (Measure.prod \u03bc \u03bd)\n[PROOFSTEP]\nrefine' \u27e8fun K hK => _\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\n\u03b1 : Type u_7\n\u03b2 : Type u_8\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b9 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d : SigmaFinite \u03bd\nK : Set (\u03b1 \u00d7 \u03b2)\nhK : IsCompact K\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) K < \u22a4\n[PROOFSTEP]\nset L := (Prod.fst '' K) \u00d7\u02e2 (Prod.snd '' K) with hL\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\n\u03b1 : Type u_7\n\u03b2 : Type u_8\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b9 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d : SigmaFinite \u03bd\nK : Set (\u03b1 \u00d7 \u03b2)\nhK : IsCompact K\nL : Set (\u03b1 \u00d7 \u03b2) := (Prod.fst '' K) \u00d7\u02e2 (Prod.snd '' K)\nhL : L = (Prod.fst '' K) \u00d7\u02e2 (Prod.snd '' K)\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) K < \u22a4\n[PROOFSTEP]\nhave : K \u2286 L := by\n rintro \u27e8x, y\u27e9 hxy\n simp only [prod_mk_mem_set_prod_eq, mem_image, Prod.exists, exists_and_right, exists_eq_right]\n exact \u27e8\u27e8y, hxy\u27e9, \u27e8x, hxy\u27e9\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\n\u03b1 : Type u_7\n\u03b2 : Type u_8\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b9 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d : SigmaFinite \u03bd\nK : Set (\u03b1 \u00d7 \u03b2)\nhK : IsCompact K\nL : Set (\u03b1 \u00d7 \u03b2) := (Prod.fst '' K) \u00d7\u02e2 (Prod.snd '' K)\nhL : L = (Prod.fst '' K) \u00d7\u02e2 (Prod.snd '' K)\n\u22a2 K \u2286 L\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9 hxy\n[GOAL]\ncase mk\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\n\u03b1 : Type u_7\n\u03b2 : Type u_8\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b9 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d : SigmaFinite \u03bd\nK : Set (\u03b1 \u00d7 \u03b2)\nhK : IsCompact K\nL : Set (\u03b1 \u00d7 \u03b2) := (Prod.fst '' K) \u00d7\u02e2 (Prod.snd '' K)\nhL : L = (Prod.fst '' K) \u00d7\u02e2 (Prod.snd '' K)\nx : \u03b1\ny : \u03b2\nhxy : (x, y) \u2208 K\n\u22a2 (x, y) \u2208 L\n[PROOFSTEP]\nsimp only [prod_mk_mem_set_prod_eq, mem_image, Prod.exists, exists_and_right, exists_eq_right]\n[GOAL]\ncase mk\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\n\u03b1 : Type u_7\n\u03b2 : Type u_8\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b9 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d : SigmaFinite \u03bd\nK : Set (\u03b1 \u00d7 \u03b2)\nhK : IsCompact K\nL : Set (\u03b1 \u00d7 \u03b2) := (Prod.fst '' K) \u00d7\u02e2 (Prod.snd '' K)\nhL : L = (Prod.fst '' K) \u00d7\u02e2 (Prod.snd '' K)\nx : \u03b1\ny : \u03b2\nhxy : (x, y) \u2208 K\n\u22a2 (\u2203 x_1, (x, x_1) \u2208 K) \u2227 \u2203 a, (a, y) \u2208 K\n[PROOFSTEP]\nexact \u27e8\u27e8y, hxy\u27e9, \u27e8x, hxy\u27e9\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\n\u03b1 : Type u_7\n\u03b2 : Type u_8\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b9 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d : SigmaFinite \u03bd\nK : Set (\u03b1 \u00d7 \u03b2)\nhK : IsCompact K\nL : Set (\u03b1 \u00d7 \u03b2) := (Prod.fst '' K) \u00d7\u02e2 (Prod.snd '' K)\nhL : L = (Prod.fst '' K) \u00d7\u02e2 (Prod.snd '' K)\nthis : K \u2286 L\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) K < \u22a4\n[PROOFSTEP]\napply lt_of_le_of_lt (measure_mono this)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\n\u03b1 : Type u_7\n\u03b2 : Type u_8\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b9 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d : SigmaFinite \u03bd\nK : Set (\u03b1 \u00d7 \u03b2)\nhK : IsCompact K\nL : Set (\u03b1 \u00d7 \u03b2) := (Prod.fst '' K) \u00d7\u02e2 (Prod.snd '' K)\nhL : L = (Prod.fst '' K) \u00d7\u02e2 (Prod.snd '' K)\nthis : K \u2286 L\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) L < \u22a4\n[PROOFSTEP]\nrw [hL, prod_prod]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1'\ninst\u271d\u2079 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2078 : MeasurableSpace \u03b2'\ninst\u271d\u2077 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SigmaFinite \u03bd\u271d\n\u03b1 : Type u_7\n\u03b2 : Type u_8\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b9 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d : SigmaFinite \u03bd\nK : Set (\u03b1 \u00d7 \u03b2)\nhK : IsCompact K\nL : Set (\u03b1 \u00d7 \u03b2) := (Prod.fst '' K) \u00d7\u02e2 (Prod.snd '' K)\nhL : L = (Prod.fst '' K) \u00d7\u02e2 (Prod.snd '' K)\nthis : K \u2286 L\n\u22a2 \u2191\u2191\u03bc (Prod.fst '' K) * \u2191\u2191\u03bd (Prod.snd '' K) < \u22a4\n[PROOFSTEP]\nexact\n mul_lt_top (IsCompact.measure_lt_top (hK.image continuous_fst)).ne\n (IsCompact.measure_lt_top (hK.image continuous_snd)).ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) < \u22a4\n[PROOFSTEP]\nrw [prod_apply hs] at h2s \n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u222b\u207b (x : \u03b1), \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) < \u22a4\n[PROOFSTEP]\nrefine' ae_lt_top (measurable_measure_prod_mk_left hs) h2s\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) s = 0 \u2194 (fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) =\u1da0[ae \u03bc] 0\n[PROOFSTEP]\nrw [prod_apply hs, lintegral_eq_zero_iff (measurable_measure_prod_mk_left hs)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nh : \u2191\u2191(Measure.prod \u03bc \u03bd) s = 0\n\u22a2 (fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) =\u1da0[ae \u03bc] 0\n[PROOFSTEP]\nobtain \u27e8t, hst, mt, ht\u27e9 := exists_measurable_superset_of_null h\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nh : \u2191\u2191(Measure.prod \u03bc \u03bd) s = 0\nt : Set (\u03b1 \u00d7 \u03b2)\nhst : s \u2286 t\nmt : MeasurableSet t\nht : \u2191\u2191(Measure.prod \u03bc \u03bd) t = 0\n\u22a2 (fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) =\u1da0[ae \u03bc] 0\n[PROOFSTEP]\nrw [measure_prod_null mt] at ht \n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nh : \u2191\u2191(Measure.prod \u03bc \u03bd) s = 0\nt : Set (\u03b1 \u00d7 \u03b2)\nhst : s \u2286 t\nmt : MeasurableSet t\nht : (fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' t)) =\u1da0[ae \u03bc] 0\n\u22a2 (fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) =\u1da0[ae \u03bc] 0\n[PROOFSTEP]\nrw [eventuallyLE_antisymm_iff]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nh : \u2191\u2191(Measure.prod \u03bc \u03bd) s = 0\nt : Set (\u03b1 \u00d7 \u03b2)\nhst : s \u2286 t\nmt : MeasurableSet t\nht : (fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' t)) =\u1da0[ae \u03bc] 0\n\u22a2 (fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) \u2264\u1da0[ae \u03bc] 0 \u2227 0 \u2264\u1da0[ae \u03bc] fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)\n[PROOFSTEP]\nexact\n \u27e8EventuallyLE.trans_eq (eventually_of_forall fun x => (measure_mono (preimage_mono hst) : _)) ht,\n eventually_of_forall fun x => zero_le _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bd'\nh1 : \u03bc \u226a \u03bc'\nh2 : \u03bd \u226a \u03bd'\n\u22a2 Measure.prod \u03bc \u03bd \u226a Measure.prod \u03bc' \u03bd'\n[PROOFSTEP]\nrefine' AbsolutelyContinuous.mk fun s hs h2s => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bd'\nh1 : \u03bc \u226a \u03bc'\nh2 : \u03bd \u226a \u03bd'\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc' \u03bd') s = 0\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) s = 0\n[PROOFSTEP]\nrw [measure_prod_null hs] at h2s \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bd'\nh1 : \u03bc \u226a \u03bc'\nh2 : \u03bd \u226a \u03bd'\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : (fun x => \u2191\u2191\u03bd' (Prod.mk x \u207b\u00b9' s)) =\u1da0[ae \u03bc'] 0\n\u22a2 (fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s)) =\u1da0[ae \u03bc] 0\n[PROOFSTEP]\nexact (h2s.filter_mono h1.ae_le).mono fun _ h => h2 h\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\u271d\n\u03bd : Measure \u03b2\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nh\u03bc : FiniteSpanningSetsIn \u03bc C\nh\u03bd : FiniteSpanningSetsIn \u03bd D\n\u22a2 FiniteSpanningSetsIn (Measure.prod \u03bc \u03bd) (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D)\n[PROOFSTEP]\nhaveI := h\u03bd.sigmaFinite\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\u271d\n\u03bd : Measure \u03b2\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nh\u03bc : FiniteSpanningSetsIn \u03bc C\nh\u03bd : FiniteSpanningSetsIn \u03bd D\nthis : SigmaFinite \u03bd\n\u22a2 FiniteSpanningSetsIn (Measure.prod \u03bc \u03bd) (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D)\n[PROOFSTEP]\nrefine'\n \u27e8fun n => h\u03bc.set n.unpair.1 \u00d7\u02e2 h\u03bd.set n.unpair.2, fun n => mem_image2_of_mem (h\u03bc.set_mem _) (h\u03bd.set_mem _), fun n =>\n _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\u271d\n\u03bd : Measure \u03b2\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nh\u03bc : FiniteSpanningSetsIn \u03bc C\nh\u03bd : FiniteSpanningSetsIn \u03bd D\nthis : SigmaFinite \u03bd\nn : \u2115\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd)\n ((fun n => FiniteSpanningSetsIn.set h\u03bc (Nat.unpair n).fst \u00d7\u02e2 FiniteSpanningSetsIn.set h\u03bd (Nat.unpair n).snd) n) <\n \u22a4\n[PROOFSTEP]\nrw [prod_prod]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\u271d\n\u03bd : Measure \u03b2\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nh\u03bc : FiniteSpanningSetsIn \u03bc C\nh\u03bd : FiniteSpanningSetsIn \u03bd D\nthis : SigmaFinite \u03bd\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (FiniteSpanningSetsIn.set h\u03bc (Nat.unpair n).fst) * \u2191\u2191\u03bd (FiniteSpanningSetsIn.set h\u03bd (Nat.unpair n).snd) < \u22a4\n[PROOFSTEP]\nexact mul_lt_top (h\u03bc.finite _).ne (h\u03bd.finite _).ne\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\u271d\n\u03bd : Measure \u03b2\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nh\u03bc : FiniteSpanningSetsIn \u03bc C\nh\u03bd : FiniteSpanningSetsIn \u03bd D\nthis : SigmaFinite \u03bd\n\u22a2 \u22c3 (i : \u2115),\n (fun n => FiniteSpanningSetsIn.set h\u03bc (Nat.unpair n).fst \u00d7\u02e2 FiniteSpanningSetsIn.set h\u03bd (Nat.unpair n).snd) i =\n univ\n[PROOFSTEP]\nsimp_rw [iUnion_unpair_prod, h\u03bc.spanning, h\u03bd.spanning, univ_prod_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u22a2 QuasiMeasurePreserving Prod.fst\n[PROOFSTEP]\nrefine' \u27e8measurable_fst, AbsolutelyContinuous.mk fun s hs h2s => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b1\nhs : MeasurableSet s\nh2s : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191(map Prod.fst (Measure.prod \u03bc \u03bd)) s = 0\n[PROOFSTEP]\nrw [map_apply measurable_fst hs, \u2190 prod_univ, prod_prod, h2s, zero_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u22a2 QuasiMeasurePreserving Prod.snd\n[PROOFSTEP]\nrefine' \u27e8measurable_snd, AbsolutelyContinuous.mk fun s hs h2s => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set \u03b2\nhs : MeasurableSet s\nh2s : \u2191\u2191\u03bd s = 0\n\u22a2 \u2191\u2191(map Prod.snd (Measure.prod \u03bc \u03bd)) s = 0\n[PROOFSTEP]\nrw [map_apply measurable_snd hs, \u2190 univ_prod, prod_prod, h2s, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\u271d\ninst\u271d : SigmaFinite \u03bc\u271d\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : generateFrom C = inst\u271d\u2077\nhD : generateFrom D = inst\u271d\u2075\nh2C : IsPiSystem C\nh2D : IsPiSystem D\nh3C : FiniteSpanningSetsIn \u03bc C\nh3D : FiniteSpanningSetsIn \u03bd D\n\u03bc\u03bd : Measure (\u03b1 \u00d7 \u03b2)\nh\u2081 : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2200 (t : Set \u03b2), t \u2208 D \u2192 \u2191\u2191\u03bc\u03bd (s \u00d7\u02e2 t) = \u2191\u2191\u03bc s * \u2191\u2191\u03bd t\n\u22a2 Measure.prod \u03bc \u03bd = \u03bc\u03bd\n[PROOFSTEP]\nrefine'\n (h3C.prod h3D).ext (generateFrom_eq_prod hC hD h3C.isCountablySpanning h3D.isCountablySpanning).symm (h2C.prod h2D) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\u271d\ninst\u271d : SigmaFinite \u03bc\u271d\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : generateFrom C = inst\u271d\u2077\nhD : generateFrom D = inst\u271d\u2075\nh2C : IsPiSystem C\nh2D : IsPiSystem D\nh3C : FiniteSpanningSetsIn \u03bc C\nh3D : FiniteSpanningSetsIn \u03bd D\n\u03bc\u03bd : Measure (\u03b1 \u00d7 \u03b2)\nh\u2081 : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2200 (t : Set \u03b2), t \u2208 D \u2192 \u2191\u2191\u03bc\u03bd (s \u00d7\u02e2 t) = \u2191\u2191\u03bc s * \u2191\u2191\u03bd t\n\u22a2 \u2200 (s : Set (\u03b1 \u00d7 \u03b2)), s \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D \u2192 \u2191\u2191(Measure.prod \u03bc \u03bd) s = \u2191\u2191\u03bc\u03bd s\n[PROOFSTEP]\nrintro _ \u27e8s, t, hs, ht, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\u271d\ninst\u271d : SigmaFinite \u03bc\u271d\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : generateFrom C = inst\u271d\u2077\nhD : generateFrom D = inst\u271d\u2075\nh2C : IsPiSystem C\nh2D : IsPiSystem D\nh3C : FiniteSpanningSetsIn \u03bc C\nh3D : FiniteSpanningSetsIn \u03bd D\n\u03bc\u03bd : Measure (\u03b1 \u00d7 \u03b2)\nh\u2081 : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2200 (t : Set \u03b2), t \u2208 D \u2192 \u2191\u2191\u03bc\u03bd (s \u00d7\u02e2 t) = \u2191\u2191\u03bc s * \u2191\u2191\u03bd t\ns : Set \u03b1\nt : Set \u03b2\nhs : s \u2208 C\nht : t \u2208 D\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) ((fun x x_1 => x \u00d7\u02e2 x_1) s t) = \u2191\u2191\u03bc\u03bd ((fun x x_1 => x \u00d7\u02e2 x_1) s t)\n[PROOFSTEP]\nhaveI := h3D.sigmaFinite\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\u271d\ninst\u271d : SigmaFinite \u03bc\u271d\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\nC : Set (Set \u03b1)\nD : Set (Set \u03b2)\nhC : generateFrom C = inst\u271d\u2077\nhD : generateFrom D = inst\u271d\u2075\nh2C : IsPiSystem C\nh2D : IsPiSystem D\nh3C : FiniteSpanningSetsIn \u03bc C\nh3D : FiniteSpanningSetsIn \u03bd D\n\u03bc\u03bd : Measure (\u03b1 \u00d7 \u03b2)\nh\u2081 : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2200 (t : Set \u03b2), t \u2208 D \u2192 \u2191\u2191\u03bc\u03bd (s \u00d7\u02e2 t) = \u2191\u2191\u03bc s * \u2191\u2191\u03bd t\ns : Set \u03b1\nt : Set \u03b2\nhs : s \u2208 C\nht : t \u2208 D\nthis : SigmaFinite \u03bd\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) ((fun x x_1 => x \u00d7\u02e2 x_1) s t) = \u2191\u2191\u03bc\u03bd ((fun x x_1 => x \u00d7\u02e2 x_1) s t)\n[PROOFSTEP]\nrw [h\u2081 s hs t ht, prod_prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\n\u22a2 map Prod.swap (Measure.prod \u03bc \u03bd) = Measure.prod \u03bd \u03bc\n[PROOFSTEP]\nrefine' (prod_eq _).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\n\u22a2 \u2200 (s : Set \u03b2) (t : Set \u03b1),\n MeasurableSet s \u2192 MeasurableSet t \u2192 \u2191\u2191(map Prod.swap (Measure.prod \u03bc \u03bd)) (s \u00d7\u02e2 t) = \u2191\u2191\u03bd s * \u2191\u2191\u03bc t\n[PROOFSTEP]\nintro s t hs ht\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b2\nt : Set \u03b1\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(map Prod.swap (Measure.prod \u03bc \u03bd)) (s \u00d7\u02e2 t) = \u2191\u2191\u03bd s * \u2191\u2191\u03bc t\n[PROOFSTEP]\nsimp_rw [map_apply measurable_swap (hs.prod ht), preimage_swap_prod, prod_prod, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd) s = \u222b\u207b (y : \u03b2), \u2191\u2191\u03bc ((fun x => (x, y)) \u207b\u00b9' s) \u2202\u03bd\n[PROOFSTEP]\nrw [\u2190 prod_swap, map_apply measurable_swap hs, prod_apply (measurable_swap hs)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (x : \u03b2), \u2191\u2191\u03bc (Prod.mk x \u207b\u00b9' (Prod.swap \u207b\u00b9' s)) \u2202\u03bd = \u222b\u207b (y : \u03b2), \u2191\u2191\u03bc ((fun x => (x, y)) \u207b\u00b9' s) \u2202\u03bd\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03c4\n\u22a2 map (\u2191MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod \u03bc \u03bd) \u03c4) = Measure.prod \u03bc (Measure.prod \u03bd \u03c4)\n[PROOFSTEP]\nrefine'\n (prod_eq_generateFrom generateFrom_measurableSet generateFrom_prod isPiSystem_measurableSet isPiSystem_prod\n \u03bc.toFiniteSpanningSetsIn (\u03bd.toFiniteSpanningSetsIn.prod \u03c4.toFiniteSpanningSetsIn) _).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03c4\n\u22a2 \u2200 (s : Set \u03b1),\n s \u2208 {s | MeasurableSet s} \u2192\n \u2200 (t : Set (\u03b2 \u00d7 \u03b3)),\n t \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) {s | MeasurableSet s} {t | MeasurableSet t} \u2192\n \u2191\u2191(map (\u2191MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod \u03bc \u03bd) \u03c4)) (s \u00d7\u02e2 t) =\n \u2191\u2191\u03bc s * \u2191\u2191(Measure.prod \u03bd \u03c4) t\n[PROOFSTEP]\nrintro s hs _ \u27e8t, u, ht, hu, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03c4\ns : Set \u03b1\nhs : s \u2208 {s | MeasurableSet s}\nt : Set \u03b2\nu : Set \u03b3\nht : t \u2208 {s | MeasurableSet s}\nhu : u \u2208 {t | MeasurableSet t}\n\u22a2 \u2191\u2191(map (\u2191MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod \u03bc \u03bd) \u03c4)) (s \u00d7\u02e2 (fun x x_1 => x \u00d7\u02e2 x_1) t u) =\n \u2191\u2191\u03bc s * \u2191\u2191(Measure.prod \u03bd \u03c4) ((fun x x_1 => x \u00d7\u02e2 x_1) t u)\n[PROOFSTEP]\nrw [mem_setOf_eq] at hs ht hu \n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03c4\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b2\nu : Set \u03b3\nht : MeasurableSet t\nhu : MeasurableSet u\n\u22a2 \u2191\u2191(map (\u2191MeasurableEquiv.prodAssoc) (Measure.prod (Measure.prod \u03bc \u03bd) \u03c4)) (s \u00d7\u02e2 (fun x x_1 => x \u00d7\u02e2 x_1) t u) =\n \u2191\u2191\u03bc s * \u2191\u2191(Measure.prod \u03bd \u03c4) ((fun x x_1 => x \u00d7\u02e2 x_1) t u)\n[PROOFSTEP]\nsimp_rw [map_apply (MeasurableEquiv.measurable _) (hs.prod (ht.prod hu)), MeasurableEquiv.prodAssoc,\n MeasurableEquiv.coe_mk, Equiv.prod_assoc_preimage, prod_prod, mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 Measure.prod (restrict \u03bc s) (restrict \u03bd t) = restrict (Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)\n[PROOFSTEP]\nrefine' prod_eq fun s' t' hs' ht' => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nt : Set \u03b2\ns' : Set \u03b1\nt' : Set \u03b2\nhs' : MeasurableSet s'\nht' : MeasurableSet t'\n\u22a2 \u2191\u2191(restrict (Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 t)) (s' \u00d7\u02e2 t') = \u2191\u2191(restrict \u03bc s) s' * \u2191\u2191(restrict \u03bd t) t'\n[PROOFSTEP]\nrw [restrict_apply (hs'.prod ht'), prod_inter_prod, prod_prod, restrict_apply hs', restrict_apply ht']\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\n\u22a2 Measure.prod (restrict \u03bc s) \u03bd = restrict (Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave : \u03bd = \u03bd.restrict Set.univ := Measure.restrict_univ.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nthis : \u03bd = restrict \u03bd univ\n\u22a2 Measure.prod (restrict \u03bc s) \u03bd = restrict (Measure.prod \u03bc \u03bd) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nrw [this, Measure.prod_restrict, \u2190 this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ny : \u03b2\n\u22a2 Measure.prod \u03bc (dirac y) = map (fun x => (x, y)) \u03bc\n[PROOFSTEP]\nrefine' prod_eq fun s t hs ht => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ny : \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(map (fun x => (x, y)) \u03bc) (s \u00d7\u02e2 t) = \u2191\u2191\u03bc s * \u2191\u2191(dirac y) t\n[PROOFSTEP]\nsimp_rw [map_apply measurable_prod_mk_right (hs.prod ht), mk_preimage_prod_left_eq_if, measure_if, dirac_apply' _ ht, \u2190\n indicator_mul_right _ fun _ => \u03bc s, Pi.one_apply, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\n\u22a2 Measure.prod (dirac x) \u03bd = map (Prod.mk x) \u03bd\n[PROOFSTEP]\nrefine' prod_eq fun s t hs ht => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(map (Prod.mk x) \u03bd) (s \u00d7\u02e2 t) = \u2191\u2191(dirac x) s * \u2191\u2191\u03bd t\n[PROOFSTEP]\nsimp_rw [map_apply measurable_prod_mk_left (hs.prod ht), mk_preimage_prod_right_eq_if, measure_if, dirac_apply' _ hs, \u2190\n indicator_mul_left _ _ fun _ => \u03bd t, Pi.one_apply, one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\ny : \u03b2\n\u22a2 Measure.prod (dirac x) (dirac y) = dirac (x, y)\n[PROOFSTEP]\nrw [prod_dirac, map_dirac measurable_prod_mk_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1'\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : MeasurableSpace \u03b2'\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SigmaFinite \u03bd\u271d\ninst\u271d\u00b2 : SigmaFinite \u03bc\n\u03b9 : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\n\u03bd : \u03b9 \u2192 Measure \u03b2\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bd i)\n\u22a2 Measure.prod \u03bc (sum \u03bd) = sum fun i => Measure.prod \u03bc (\u03bd i)\n[PROOFSTEP]\nrefine' prod_eq fun s t hs ht => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1'\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : MeasurableSpace \u03b2'\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SigmaFinite \u03bd\u271d\ninst\u271d\u00b2 : SigmaFinite \u03bc\n\u03b9 : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\n\u03bd : \u03b9 \u2192 Measure \u03b2\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bd i)\ns : Set \u03b1\nt : Set \u03b2\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(sum fun i => Measure.prod \u03bc (\u03bd i)) (s \u00d7\u02e2 t) = \u2191\u2191\u03bc s * \u2191\u2191(sum \u03bd) t\n[PROOFSTEP]\nsimp_rw [sum_apply _ (hs.prod ht), sum_apply _ ht, prod_prod, ENNReal.tsum_mul_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1'\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : MeasurableSpace \u03b2'\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\u271d\n\u03b9 : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\n\u22a2 Measure.prod (sum \u03bc) \u03bd = sum fun i => Measure.prod (\u03bc i) \u03bd\n[PROOFSTEP]\nrefine' prod_eq fun s t hs ht => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1'\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : MeasurableSpace \u03b2'\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\u271d\n\u03b9 : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ninst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ns : Set \u03b1\nt : Set \u03b2\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(sum fun i => Measure.prod (\u03bc i) \u03bd) (s \u00d7\u02e2 t) = \u2191\u2191(sum \u03bc) s * \u2191\u2191\u03bd t\n[PROOFSTEP]\nsimp_rw [sum_apply _ (hs.prod ht), sum_apply _ hs, prod_prod, ENNReal.tsum_mul_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd'\u271d : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\n\u03bd' : Measure \u03b2\ninst\u271d : SigmaFinite \u03bd'\n\u22a2 Measure.prod \u03bc (\u03bd + \u03bd') = Measure.prod \u03bc \u03bd + Measure.prod \u03bc \u03bd'\n[PROOFSTEP]\nrefine' prod_eq fun s t _ _ => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd'\u271d : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\n\u03bd' : Measure \u03b2\ninst\u271d : SigmaFinite \u03bd'\ns : Set \u03b1\nt : Set \u03b2\nx\u271d\u00b9 : MeasurableSet s\nx\u271d : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd + Measure.prod \u03bc \u03bd') (s \u00d7\u02e2 t) = \u2191\u2191\u03bc s * \u2191\u2191(\u03bd + \u03bd') t\n[PROOFSTEP]\nsimp_rw [add_apply, prod_prod, left_distrib]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc'\u271d : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\n\u03bc' : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc'\n\u22a2 Measure.prod (\u03bc + \u03bc') \u03bd = Measure.prod \u03bc \u03bd + Measure.prod \u03bc' \u03bd\n[PROOFSTEP]\nrefine' prod_eq fun s t _ _ => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc'\u271d : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\n\u03bc' : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc'\ns : Set \u03b1\nt : Set \u03b2\nx\u271d\u00b9 : MeasurableSet s\nx\u271d : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.prod \u03bc \u03bd + Measure.prod \u03bc' \u03bd) (s \u00d7\u02e2 t) = \u2191\u2191(\u03bc + \u03bc') s * \u2191\u2191\u03bd t\n[PROOFSTEP]\nsimp_rw [add_apply, prod_prod, right_distrib]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\u271d\ninst\u271d : SigmaFinite \u03bc\n\u03bd : Measure \u03b2\n\u22a2 Measure.prod 0 \u03bd = 0\n[PROOFSTEP]\nrw [Measure.prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\u271d\ninst\u271d : SigmaFinite \u03bc\n\u03bd : Measure \u03b2\n\u22a2 (bind 0 fun x => map (Prod.mk x) \u03bd) = 0\n[PROOFSTEP]\nexact bind_zero_left _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\u271d\n\u03bc : Measure \u03b1\n\u22a2 Measure.prod \u03bc 0 = 0\n[PROOFSTEP]\nsimp [Measure.prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\nhfa : SigmaFinite (map f \u03bca)\nhgc : SigmaFinite (map g \u03bcc)\nhf : Measurable f\nhg : Measurable g\n\u22a2 Measure.prod (map f \u03bca) (map g \u03bcc) = map (Prod.map f g) (Measure.prod \u03bca \u03bcc)\n[PROOFSTEP]\nhaveI := hgc.of_map \u03bcc hg.aemeasurable\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\nhfa : SigmaFinite (map f \u03bca)\nhgc : SigmaFinite (map g \u03bcc)\nhf : Measurable f\nhg : Measurable g\nthis : SigmaFinite \u03bcc\n\u22a2 Measure.prod (map f \u03bca) (map g \u03bcc) = map (Prod.map f g) (Measure.prod \u03bca \u03bcc)\n[PROOFSTEP]\nrefine' prod_eq fun s t hs ht => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\nhfa : SigmaFinite (map f \u03bca)\nhgc : SigmaFinite (map g \u03bcc)\nhf : Measurable f\nhg : Measurable g\nthis : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(map (Prod.map f g) (Measure.prod \u03bca \u03bcc)) (s \u00d7\u02e2 t) = \u2191\u2191(map f \u03bca) s * \u2191\u2191(map g \u03bcc) t\n[PROOFSTEP]\nrw [map_apply (hf.prod_map hg) (hs.prod ht), map_apply hf hs, map_apply hg ht]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\nhfa : SigmaFinite (map f \u03bca)\nhgc : SigmaFinite (map g \u03bcc)\nhf : Measurable f\nhg : Measurable g\nthis : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.prod \u03bca \u03bcc) (Prod.map f g \u207b\u00b9' s \u00d7\u02e2 t) = \u2191\u2191\u03bca (f \u207b\u00b9' s) * \u2191\u2191\u03bcc (g \u207b\u00b9' t)\n[PROOFSTEP]\nexact prod_prod (f \u207b\u00b9' s) (g \u207b\u00b9' t)\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\n\u22a2 MeasurePreserving fun p => (f p.fst, g p.fst p.snd)\n[PROOFSTEP]\nclassical\nhave : Measurable fun p : \u03b1 \u00d7 \u03b3 => (f p.1, g p.1 p.2) := (hf.1.comp measurable_fst).prod_mk hgm\nrcases eq_or_ne \u03bca 0 with (rfl | ha)\n\u00b7 rw [\u2190 hf.map_eq, zero_prod, Measure.map_zero, zero_prod]\n exact \u27e8this, by simp only [Measure.map_zero]\u27e9\nhave sf : SigmaFinite \u03bcc :=\n by\n rcases(ae_neBot.2 ha).nonempty_of_mem hg with \u27e8x, hx : map (g x) \u03bcc = \u03bcd\u27e9\n exact\n SigmaFinite.of_map _ hgm.of_uncurry_left.aemeasurable\n (by rwa [hx])\n -- Thus we can apply `Measure.prod_eq` to prove equality of measures.\nrefine' \u27e8this, (prod_eq fun s t hs ht => _).symm\u27e9\nrw [map_apply this (hs.prod ht)]\nrefine' (prod_apply (this <| hs.prod ht)).trans _\nhave : \u2200\u1d50 x \u2202\u03bca, \u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun _ => \u03bcd t) x :=\n by\n refine' hg.mono fun x hx => _\n subst hx\n simp only [mk_preimage_prod_right_fn_eq_if, indicator_apply, mem_preimage]\n split_ifs\n exacts [(map_apply hgm.of_uncurry_left ht).symm, measure_empty]\nsimp only [preimage_preimage]\nrw [lintegral_congr_ae this, lintegral_indicator _ (hf.1 hs), set_lintegral_const, hf.measure_preimage hs, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\n\u22a2 MeasurePreserving fun p => (f p.fst, g p.fst p.snd)\n[PROOFSTEP]\nhave : Measurable fun p : \u03b1 \u00d7 \u03b3 => (f p.1, g p.1 p.2) := (hf.1.comp measurable_fst).prod_mk hgm\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\n\u22a2 MeasurePreserving fun p => (f p.fst, g p.fst p.snd)\n[PROOFSTEP]\nrcases eq_or_ne \u03bca 0 with (rfl | ha)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nhf : MeasurePreserving f\nhg : \u2200\u1d50 (x : \u03b1) \u22020, map (g x) \u03bcc = \u03bcd\n\u22a2 MeasurePreserving fun p => (f p.fst, g p.fst p.snd)\n[PROOFSTEP]\nrw [\u2190 hf.map_eq, zero_prod, Measure.map_zero, zero_prod]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nhf : MeasurePreserving f\nhg : \u2200\u1d50 (x : \u03b1) \u22020, map (g x) \u03bcc = \u03bcd\n\u22a2 MeasurePreserving fun p => (f p.fst, g p.fst p.snd)\n[PROOFSTEP]\nexact \u27e8this, by simp only [Measure.map_zero]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nhf : MeasurePreserving f\nhg : \u2200\u1d50 (x : \u03b1) \u22020, map (g x) \u03bcc = \u03bcd\n\u22a2 map (fun p => (f p.fst, g p.fst p.snd)) 0 = 0\n[PROOFSTEP]\nsimp only [Measure.map_zero]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\n\u22a2 MeasurePreserving fun p => (f p.fst, g p.fst p.snd)\n[PROOFSTEP]\nhave sf : SigmaFinite \u03bcc :=\n by\n rcases(ae_neBot.2 ha).nonempty_of_mem hg with \u27e8x, hx : map (g x) \u03bcc = \u03bcd\u27e9\n exact\n SigmaFinite.of_map _ hgm.of_uncurry_left.aemeasurable\n (by rwa [hx])\n -- Thus we can apply `Measure.prod_eq` to prove equality of measures.\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\n\u22a2 SigmaFinite \u03bcc\n[PROOFSTEP]\nrcases(ae_neBot.2 ha).nonempty_of_mem hg with \u27e8x, hx : map (g x) \u03bcc = \u03bcd\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\nx : \u03b1\nhx : map (g x) \u03bcc = \u03bcd\n\u22a2 SigmaFinite \u03bcc\n[PROOFSTEP]\nexact\n SigmaFinite.of_map _ hgm.of_uncurry_left.aemeasurable\n (by rwa [hx])\n -- Thus we can apply `Measure.prod_eq` to prove equality of measures.\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\nx : \u03b1\nhx : map (g x) \u03bcc = \u03bcd\n\u22a2 SigmaFinite (map (g ?m.96015) \u03bcc)\n[PROOFSTEP]\nrwa [hx]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\n\u22a2 MeasurePreserving fun p => (f p.fst, g p.fst p.snd)\n[PROOFSTEP]\nrefine' \u27e8this, (prod_eq fun s t hs ht => _).symm\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(map (fun p => (f p.fst, g p.fst p.snd)) (Measure.prod \u03bca \u03bcc)) (s \u00d7\u02e2 t) = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t\n[PROOFSTEP]\nrw [map_apply this (hs.prod ht)]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.prod \u03bca \u03bcc) ((fun p => (f p.fst, g p.fst p.snd)) \u207b\u00b9' s \u00d7\u02e2 t) = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t\n[PROOFSTEP]\nrefine' (prod_apply (this <| hs.prod ht)).trans _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191\u03bcc (Prod.mk x \u207b\u00b9' ((fun p => (f p.fst, g p.fst p.snd)) \u207b\u00b9' s \u00d7\u02e2 t)) \u2202\u03bca = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t\n[PROOFSTEP]\nhave : \u2200\u1d50 x \u2202\u03bca, \u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun _ => \u03bcd t) x :=\n by\n refine' hg.mono fun x hx => _\n subst hx\n simp only [mk_preimage_prod_right_fn_eq_if, indicator_apply, mem_preimage]\n split_ifs\n exacts [(map_apply hgm.of_uncurry_left ht).symm, measure_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bca, \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x => \u2191\u2191\u03bcd t) x\n[PROOFSTEP]\nrefine' hg.mono fun x hx => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nx : \u03b1\nhx : map (g x) \u03bcc = \u03bcd\n\u22a2 \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x => \u2191\u2191\u03bcd t) x\n[PROOFSTEP]\nsubst hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SigmaFinite \u03bcb\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nx : \u03b1\ninst\u271d : SigmaFinite (map (g x) \u03bcc)\nhg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc\n\u22a2 \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x_1 => \u2191\u2191(map (g x) \u03bcc) t) x\n[PROOFSTEP]\nsimp only [mk_preimage_prod_right_fn_eq_if, indicator_apply, mem_preimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SigmaFinite \u03bcb\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nx : \u03b1\ninst\u271d : SigmaFinite (map (g x) \u03bcc)\nhg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc\n\u22a2 \u2191\u2191\u03bcc (if f x \u2208 s then (fun b => g x b) \u207b\u00b9' t else \u2205) = if f x \u2208 s then \u2191\u2191(map (g x) \u03bcc) t else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SigmaFinite \u03bcb\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nx : \u03b1\ninst\u271d : SigmaFinite (map (g x) \u03bcc)\nhg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc\nh\u271d : f x \u2208 s\n\u22a2 \u2191\u2191\u03bcc ((fun b => g x b) \u207b\u00b9' t) = \u2191\u2191(map (g x) \u03bcc) t\ncase neg\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SigmaFinite \u03bcb\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nthis : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nx : \u03b1\ninst\u271d : SigmaFinite (map (g x) \u03bcc)\nhg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc\nh\u271d : \u00acf x \u2208 s\n\u22a2 \u2191\u2191\u03bcc \u2205 = 0\n[PROOFSTEP]\nexacts [(map_apply hgm.of_uncurry_left ht).symm, measure_empty]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis\u271d : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x => \u2191\u2191\u03bcd t) x\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191\u03bcc (Prod.mk x \u207b\u00b9' ((fun p => (f p.fst, g p.fst p.snd)) \u207b\u00b9' s \u00d7\u02e2 t)) \u2202\u03bca = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t\n[PROOFSTEP]\nsimp only [preimage_preimage]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\ninst\u271d\u00b9 : SigmaFinite \u03bcb\ninst\u271d : SigmaFinite \u03bcd\nf : \u03b1 \u2192 \u03b2\nhf : MeasurePreserving f\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b4\nhgm : Measurable (uncurry g)\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd\nthis\u271d : Measurable fun p => (f p.fst, g p.fst p.snd)\nha : \u03bca \u2260 0\nsf : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x => \u2191\u2191\u03bcd t) x\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191\u03bcc ((fun x_1 => (f x, g x x_1)) \u207b\u00b9' s \u00d7\u02e2 t) \u2202\u03bca = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t\n[PROOFSTEP]\nrw [lintegral_congr_ae this, lintegral_indicator _ (hf.1 hs), set_lintegral_const, hf.measure_preimage hs, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4\u271d : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\n\u03c4 : Measure \u03b3\nhf : Measurable f\ninst\u271d : SigmaFinite \u03bd\nh2f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, QuasiMeasurePreserving fun y => f (x, y)\n\u22a2 QuasiMeasurePreserving f\n[PROOFSTEP]\nrefine' \u27e8hf, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4\u271d : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\n\u03c4 : Measure \u03b3\nhf : Measurable f\ninst\u271d : SigmaFinite \u03bd\nh2f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, QuasiMeasurePreserving fun y => f (x, y)\n\u22a2 map f (Measure.prod \u03bc \u03bd) \u226a \u03c4\n[PROOFSTEP]\nrefine' AbsolutelyContinuous.mk fun s hs h2s => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4\u271d : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\n\u03c4 : Measure \u03b3\nhf : Measurable f\ninst\u271d : SigmaFinite \u03bd\nh2f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, QuasiMeasurePreserving fun y => f (x, y)\ns : Set \u03b3\nhs : MeasurableSet s\nh2s : \u2191\u2191\u03c4 s = 0\n\u22a2 \u2191\u2191(map f (Measure.prod \u03bc \u03bd)) s = 0\n[PROOFSTEP]\nrw [map_apply hf hs, prod_apply (hf hs)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4\u271d : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\n\u03c4 : Measure \u03b3\nhf : Measurable f\ninst\u271d : SigmaFinite \u03bd\nh2f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, QuasiMeasurePreserving fun y => f (x, y)\ns : Set \u03b3\nhs : MeasurableSet s\nh2s : \u2191\u2191\u03c4 s = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' (f \u207b\u00b9' s)) \u2202\u03bc = 0\n[PROOFSTEP]\nsimp_rw [preimage_preimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd\u271d \u03bd' : Measure \u03b2\n\u03c4\u271d : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\n\u03c4 : Measure \u03b3\nhf : Measurable f\ninst\u271d : SigmaFinite \u03bd\nh2f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, QuasiMeasurePreserving fun y => f (x, y)\ns : Set \u03b3\nhs : MeasurableSet s\nh2s : \u2191\u2191\u03c4 s = 0\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191\u03bd ((fun x_1 => f (x, x_1)) \u207b\u00b9' s) \u2202\u03bc = 0\n[PROOFSTEP]\nrw [lintegral_congr_ae (h2f.mono fun x hx => hx.preimage_null h2s), lintegral_zero]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\u271d\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4\u271d : Measure \u03b3\u271d\ninst\u271d\u2075 : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\n\u03c4 : Measure \u03b3\nhf : Measurable f\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nh2f : \u2200\u1d50 (y : \u03b2) \u2202\u03bd, QuasiMeasurePreserving fun x => f (x, y)\n\u22a2 QuasiMeasurePreserving f\n[PROOFSTEP]\nrw [\u2190 prod_swap]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1' : Type u_2\n\u03b2\u271d : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : MeasurableSpace \u03b1'\ninst\u271d\u2078 : MeasurableSpace \u03b2\u271d\ninst\u271d\u2077 : MeasurableSpace \u03b2'\ninst\u271d\u2076 : MeasurableSpace \u03b3\u271d\n\u03bc\u271d \u03bc' : Measure \u03b1\u271d\n\u03bd\u271d \u03bd' : Measure \u03b2\u271d\n\u03c4\u271d : Measure \u03b3\u271d\ninst\u271d\u2075 : NormedAddCommGroup E\n\u03b1 : Type u_7\n\u03b2 : Type u_8\n\u03b3 : Type u_9\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\n\u03bc : Measure \u03b1\n\u03bd : Measure \u03b2\n\u03c4 : Measure \u03b3\nhf : Measurable f\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nh2f : \u2200\u1d50 (y : \u03b2) \u2202\u03bd, QuasiMeasurePreserving fun x => f (x, y)\n\u22a2 QuasiMeasurePreserving f\n[PROOFSTEP]\nconvert\n (QuasiMeasurePreserving.prod_of_right (hf.comp measurable_swap) h2f).comp\n ((measurable_swap.measurePreserving (\u03bd.prod \u03bc)).symm MeasurableEquiv.prodComm).quasiMeasurePreserving\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nf : \u03b2 \u00d7 \u03b1 \u2192 \u03b3\nhf : AEMeasurable f\n\u22a2 AEMeasurable fun z => f (Prod.swap z)\n[PROOFSTEP]\nrw [\u2190 Measure.prod_swap] at hf \n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nf : \u03b2 \u00d7 \u03b1 \u2192 \u03b3\nhf : AEMeasurable f\n\u22a2 AEMeasurable fun z => f (Prod.swap z)\n[PROOFSTEP]\nexact hf.comp_measurable measurable_swap\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u22a2 \u222b\u207b (z : \u03b2 \u00d7 \u03b1), f (Prod.swap z) \u2202Measure.prod \u03bd \u03bc = \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd\n[PROOFSTEP]\nrw [\u2190 prod_swap] at hf \n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u22a2 \u222b\u207b (z : \u03b2 \u00d7 \u03b1), f (Prod.swap z) \u2202Measure.prod \u03bd \u03bc = \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd\n[PROOFSTEP]\nrw [\u2190 lintegral_map' hf measurable_swap.aemeasurable, prod_swap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u22a2 \u2200 (f : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e), Measurable f \u2192 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nhave m := @measurable_prod_mk_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.142685} {\u03b2 : Type ?u.142686} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\n\u22a2 \u2200 (f : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e), Measurable f \u2192 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nrefine' Measurable.ennreal_induction (P := fun f => \u222b\u207b z, f z \u2202\u03bc.prod \u03bd = \u222b\u207b x, \u222b\u207b y, f (x, y) \u2202\u03bd \u2202\u03bc) _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.142685} {\u03b2 : Type ?u.142686} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\n\u22a2 \u2200 (c : \u211d\u22650\u221e) \u2983s : Set (\u03b1 \u00d7 \u03b2)\u2984,\n MeasurableSet s \u2192\n (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (indicator s fun x => c)\n[PROOFSTEP]\nintro c s hs\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.142685} {\u03b2 : Type ?u.142686} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =\n \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), indicator s (fun x => c) (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nconv_rhs =>\n enter [2, x, 2, y]\n rw [\u2190 indicator_comp_right, const_def, const_comp, \u2190 const_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.142685} {\u03b2 : Type ?u.142686} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n| \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), indicator s (fun x => c) (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\n enter [2, x, 2, y]\n rw [\u2190 indicator_comp_right, const_def, const_comp, \u2190 const_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.142685} {\u03b2 : Type ?u.142686} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n| \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), indicator s (fun x => c) (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\n enter [2, x, 2, y]\n rw [\u2190 indicator_comp_right, const_def, const_comp, \u2190 const_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.142685} {\u03b2 : Type ?u.142686} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n| \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), indicator s (fun x => c) (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nenter [2, x, 2, y]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.142685} {\u03b2 : Type ?u.142686} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nx : \u03b1\ny : \u03b2\n| indicator s (fun x => c) (x, y)\n[PROOFSTEP]\nrw [\u2190 indicator_comp_right, const_def, const_comp, \u2190 const_def]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.142685} {\u03b2 : Type ?u.142686} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =\n \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), indicator (Prod.mk x \u207b\u00b9' s) (fun x => c) y \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nconv_rhs =>\n enter [2, x]\n rw [lintegral_indicator _ (m (x := x) hs), lintegral_const, Measure.restrict_apply MeasurableSet.univ, univ_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.142685} {\u03b2 : Type ?u.142686} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n| \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), indicator (Prod.mk x \u207b\u00b9' s) (fun x => c) y \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\n enter [2, x]\n rw [lintegral_indicator _ (m (x := x) hs), lintegral_const, Measure.restrict_apply MeasurableSet.univ, univ_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.142685} {\u03b2 : Type ?u.142686} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n| \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), indicator (Prod.mk x \u207b\u00b9' s) (fun x => c) y \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\n enter [2, x]\n rw [lintegral_indicator _ (m (x := x) hs), lintegral_const, Measure.restrict_apply MeasurableSet.univ, univ_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.142685} {\u03b2 : Type ?u.142686} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n| \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), indicator (Prod.mk x \u207b\u00b9' s) (fun x => c) y \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nenter [2, x]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm :\n \u2200 {\u03b1 : Type ?u.142685} {\u03b2 : Type ?u.142686} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1},\n Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nx : \u03b1\n| \u222b\u207b (y : \u03b2), indicator (Prod.mk x \u207b\u00b9' s) (fun x => c) y \u2202\u03bd\n[PROOFSTEP]\nrw [lintegral_indicator _ (m (x := x) hs), lintegral_const, Measure.restrict_apply MeasurableSet.univ, univ_inter]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nc : \u211d\u22650\u221e\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), c * \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) \u2202\u03bc\n[PROOFSTEP]\nsimp [hs, lintegral_const_mul, measurable_measure_prod_mk_left (\u03bd := \u03bd) hs, prod_apply]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\n\u22a2 \u2200 \u2983f g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\u2984,\n Disjoint (support f) (support g) \u2192\n Measurable f \u2192\n Measurable g \u2192\n (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) f \u2192\n (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) g \u2192\n (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f + g)\n[PROOFSTEP]\nrintro f g - hf _ h2f h2g\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (f + g) z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (f + g) (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nsimp only [Pi.add_apply]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z + g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nconv_lhs => rw [lintegral_add_left hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n| \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z + g z \u2202Measure.prod \u03bc \u03bd\n[PROOFSTEP]\nrw [lintegral_add_left hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n| \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z + g z \u2202Measure.prod \u03bc \u03bd\n[PROOFSTEP]\nrw [lintegral_add_left hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n| \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z + g z \u2202Measure.prod \u03bc \u03bd\n[PROOFSTEP]\nrw [lintegral_add_left hf]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1 \u00d7 \u03b2), f a \u2202Measure.prod \u03bc \u03bd + \u222b\u207b (a : \u03b1 \u00d7 \u03b2), g a \u2202Measure.prod \u03bc \u03bd =\n \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nconv_rhs => enter [2, x]; erw [lintegral_add_left (hf.comp (m (x := x)))]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n| \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nenter [2, x]; erw [lintegral_add_left (hf.comp (m (x := x)))]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n| \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nenter [2, x]; erw [lintegral_add_left (hf.comp (m (x := x)))]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n| \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nenter [2, x]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\nx : \u03b1\n| \u222b\u207b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd\n[PROOFSTEP]\nerw [lintegral_add_left (hf.comp (m (x := x)))]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable f\na\u271d : Measurable g\nh2f : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nh2g : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1 \u00d7 \u03b2), f a \u2202Measure.prod \u03bc \u03bd + \u222b\u207b (a : \u03b1 \u00d7 \u03b2), g a \u2202Measure.prod \u03bc \u03bd =\n \u222b\u207b (x : \u03b1), \u222b\u207b (a : \u03b2), (f \u2218 Prod.mk x) a \u2202\u03bd + \u222b\u207b (a : \u03b2), g (x, a) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nsimp [lintegral_add_left, Measurable.lintegral_prod_right', hf, h2f, h2g]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\n\u22a2 \u2200 \u2983f : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\u2984,\n (\u2200 (n : \u2115), Measurable (f n)) \u2192\n Monotone f \u2192\n (\u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)) \u2192\n (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) fun x =>\n \u2a06 (n : \u2115), f n x\n[PROOFSTEP]\nintro f hf h2f h3f\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 (n : \u2115), f n x) z \u2202Measure.prod \u03bc \u03bd =\n \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 (n : \u2115), f n x) (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nhave kf : \u2200 x n, Measurable fun y => f n (x, y) := fun x n => (hf n).comp m\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)\nkf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y)\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 (n : \u2115), f n x) z \u2202Measure.prod \u03bc \u03bd =\n \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 (n : \u2115), f n x) (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nhave k2f : \u2200 x, Monotone fun n y => f n (x, y) := fun x i j hij y => h2f hij (x, y)\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)\nkf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y)\nk2f : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y)\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 (n : \u2115), f n x) z \u2202Measure.prod \u03bc \u03bd =\n \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 (n : \u2115), f n x) (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nhave lf : \u2200 n, Measurable fun x => \u222b\u207b y, f n (x, y) \u2202\u03bd := fun n => (hf n).lintegral_prod_right'\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)\nkf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y)\nk2f : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y)\nlf : \u2200 (n : \u2115), Measurable fun x => \u222b\u207b (y : \u03b2), f n (x, y) \u2202\u03bd\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 (n : \u2115), f n x) z \u2202Measure.prod \u03bc \u03bd =\n \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 (n : \u2115), f n x) (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nhave l2f : Monotone fun n x => \u222b\u207b y, f n (x, y) \u2202\u03bd := fun i j hij x => lintegral_mono (k2f x hij)\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nm : \u2200 {\u03b1 : Type u_1} {\u03b2 : Type u_3} {m : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {x : \u03b1}, Measurable (Prod.mk x)\nf : \u2115 \u2192 \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh2f : Monotone f\nh3f : \u2200 (n : \u2115), (fun f => \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc) (f n)\nkf : \u2200 (x : \u03b1) (n : \u2115), Measurable fun y => f n (x, y)\nk2f : \u2200 (x : \u03b1), Monotone fun n y => f n (x, y)\nlf : \u2200 (n : \u2115), Measurable fun x => \u222b\u207b (y : \u03b2), f n (x, y) \u2202\u03bd\nl2f : Monotone fun n x => \u222b\u207b (y : \u03b2), f n (x, y) \u2202\u03bd\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), (fun x => \u2a06 (n : \u2115), f n x) z \u2202Measure.prod \u03bc \u03bd =\n \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), (fun x => \u2a06 (n : \u2115), f n x) (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nsimp only [lintegral_iSup hf h2f, lintegral_iSup (kf _), k2f, lintegral_iSup lf l2f, h3f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nhave A : \u222b\u207b z, f z \u2202\u03bc.prod \u03bd = \u222b\u207b z, hf.mk f z \u2202\u03bc.prod \u03bd := lintegral_congr_ae hf.ae_eq_mk\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nA : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (z : \u03b1 \u00d7 \u03b2), AEMeasurable.mk f hf z \u2202Measure.prod \u03bc \u03bd\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nhave B : (\u222b\u207b x, \u222b\u207b y, f (x, y) \u2202\u03bd \u2202\u03bc) = \u222b\u207b x, \u222b\u207b y, hf.mk f (x, y) \u2202\u03bd \u2202\u03bc :=\n by\n apply lintegral_congr_ae\n filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with _ ha using lintegral_congr_ae ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nA : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (z : \u03b1 \u00d7 \u03b2), AEMeasurable.mk f hf z \u2202Measure.prod \u03bc \u03bd\n\u22a2 \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), AEMeasurable.mk f hf (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\napply lintegral_congr_ae\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nA : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (z : \u03b1 \u00d7 \u03b2), AEMeasurable.mk f hf z \u2202Measure.prod \u03bc \u03bd\n\u22a2 (fun a => \u222b\u207b (y : \u03b2), f (a, y) \u2202\u03bd) =\u1da0[ae \u03bc] fun a => \u222b\u207b (y : \u03b2), AEMeasurable.mk f hf (a, y) \u2202\u03bd\n[PROOFSTEP]\nfilter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with _ ha using lintegral_congr_ae ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nA : \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (z : \u03b1 \u00d7 \u03b2), AEMeasurable.mk f hf z \u2202Measure.prod \u03bc \u03bd\nB : \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), AEMeasurable.mk f hf (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nrw [A, B, lintegral_prod_of_measurable _ hf.measurable_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (y : \u03b2), \u222b\u207b (x : \u03b1), f (x, y) \u2202\u03bc \u2202\u03bd\n[PROOFSTEP]\nsimp_rw [\u2190 lintegral_prod_swap f hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u22a2 \u222b\u207b (z : \u03b2 \u00d7 \u03b1), f (Prod.swap z) \u2202Measure.prod \u03bd \u03bc = \u222b\u207b (y : \u03b2), \u222b\u207b (x : \u03b1), f (x, y) \u2202\u03bc \u2202\u03bd\n[PROOFSTEP]\nexact lintegral_prod _ hf.prod_swap\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u211d\u22650\u221e\ng : \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z.fst * g z.snd \u2202Measure.prod \u03bc \u03bd = (\u222b\u207b (x : \u03b1), f x \u2202\u03bc) * \u222b\u207b (y : \u03b2), g y \u2202\u03bd\n[PROOFSTEP]\nsimp [lintegral_prod _ (hf.fst.mul hg.snd), lintegral_lintegral_mul hf hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(fst \u03c1) s = \u2191\u2191\u03c1 (Prod.fst \u207b\u00b9' s)\n[PROOFSTEP]\nrw [fst, Measure.map_apply measurable_fst hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\n\u22a2 \u2191\u2191(fst \u03c1) univ = \u2191\u2191\u03c1 univ\n[PROOFSTEP]\nrw [fst_apply MeasurableSet.univ, preimage_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\n\u22a2 fst 0 = 0\n[PROOFSTEP]\nsimp [fst]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 IsFiniteMeasure (fst \u03c1)\n[PROOFSTEP]\nrw [fst]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 IsFiniteMeasure (map Prod.fst \u03c1)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ninst\u271d : IsProbabilityMeasure \u03c1\n\u22a2 \u2191\u2191(fst \u03c1) univ = 1\n[PROOFSTEP]\nrw [fst_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ninst\u271d : IsProbabilityMeasure \u03c1\n\u22a2 \u2191\u2191\u03c1 univ = 1\n[PROOFSTEP]\nexact measure_univ\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ninst\u271d : IsProbabilityMeasure \u03bd\n\u22a2 fst (Measure.prod \u03bc \u03bd) = \u03bc\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ninst\u271d : IsProbabilityMeasure \u03bd\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(fst (Measure.prod \u03bc \u03bd)) s = \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [fst_apply hs, \u2190 prod_univ, prod_prod, measure_univ, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03b3\n\u03bc : Measure \u03b1\nhY : AEMeasurable Y\n\u22a2 fst (map (fun a => (X a, Y a)) \u03bc) = map X \u03bc\n[PROOFSTEP]\nby_cases hX : AEMeasurable X \u03bc\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03b3\n\u03bc : Measure \u03b1\nhY : AEMeasurable Y\nhX : AEMeasurable X\n\u22a2 fst (map (fun a => (X a, Y a)) \u03bc) = map X \u03bc\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase pos.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03b3\n\u03bc : Measure \u03b1\nhY : AEMeasurable Y\nhX : AEMeasurable X\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(fst (map (fun a => (X a, Y a)) \u03bc)) s = \u2191\u2191(map X \u03bc) s\n[PROOFSTEP]\nrw [Measure.fst_apply hs, Measure.map_apply_of_aemeasurable (hX.prod_mk hY) (measurable_fst hs),\n Measure.map_apply_of_aemeasurable hX hs, \u2190 prod_univ, mk_preimage_prod, preimage_univ, inter_univ]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03b3\n\u03bc : Measure \u03b1\nhY : AEMeasurable Y\nhX : \u00acAEMeasurable X\n\u22a2 fst (map (fun a => (X a, Y a)) \u03bc) = map X \u03bc\n[PROOFSTEP]\nhave : \u00acAEMeasurable (fun x \u21a6 (X x, Y x)) \u03bc := by contrapose! hX; exact measurable_fst.comp_aemeasurable hX\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03b3\n\u03bc : Measure \u03b1\nhY : AEMeasurable Y\nhX : \u00acAEMeasurable X\n\u22a2 \u00acAEMeasurable fun x => (X x, Y x)\n[PROOFSTEP]\ncontrapose! hX\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03b3\n\u03bc : Measure \u03b1\nhY : AEMeasurable Y\nhX : AEMeasurable fun x => (X x, Y x)\n\u22a2 AEMeasurable X\n[PROOFSTEP]\nexact measurable_fst.comp_aemeasurable hX\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03b3\n\u03bc : Measure \u03b1\nhY : AEMeasurable Y\nhX : \u00acAEMeasurable X\nthis : \u00acAEMeasurable fun x => (X x, Y x)\n\u22a2 fst (map (fun a => (X a, Y a)) \u03bc) = map X \u03bc\n[PROOFSTEP]\nsimp [map_of_not_aemeasurable, hX, this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(snd \u03c1) s = \u2191\u2191\u03c1 (Prod.snd \u207b\u00b9' s)\n[PROOFSTEP]\nrw [snd, Measure.map_apply measurable_snd hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\n\u22a2 \u2191\u2191(snd \u03c1) univ = \u2191\u2191\u03c1 univ\n[PROOFSTEP]\nrw [snd_apply MeasurableSet.univ, preimage_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\n\u22a2 snd 0 = 0\n[PROOFSTEP]\nsimp [snd]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 IsFiniteMeasure (snd \u03c1)\n[PROOFSTEP]\nrw [snd]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ninst\u271d : IsFiniteMeasure \u03c1\n\u22a2 IsFiniteMeasure (map Prod.snd \u03c1)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ninst\u271d : IsProbabilityMeasure \u03c1\n\u22a2 \u2191\u2191(snd \u03c1) univ = 1\n[PROOFSTEP]\nrw [snd_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ninst\u271d : IsProbabilityMeasure \u03c1\n\u22a2 \u2191\u2191\u03c1 univ = 1\n[PROOFSTEP]\nexact measure_univ\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ninst\u271d : IsProbabilityMeasure \u03bc\n\u22a2 snd (Measure.prod \u03bc \u03bd) = \u03bd\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ninst\u271d : IsProbabilityMeasure \u03bc\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(snd (Measure.prod \u03bc \u03bd)) s = \u2191\u2191\u03bd s\n[PROOFSTEP]\nrw [snd_apply hs, \u2190 univ_prod, prod_prod, measure_univ, one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03b3\n\u03bc : Measure \u03b1\nhX : AEMeasurable X\n\u22a2 snd (map (fun a => (X a, Y a)) \u03bc) = map Y \u03bc\n[PROOFSTEP]\nby_cases hY : AEMeasurable Y \u03bc\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03b3\n\u03bc : Measure \u03b1\nhX : AEMeasurable X\nhY : AEMeasurable Y\n\u22a2 snd (map (fun a => (X a, Y a)) \u03bc) = map Y \u03bc\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase pos.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03b3\n\u03bc : Measure \u03b1\nhX : AEMeasurable X\nhY : AEMeasurable Y\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(snd (map (fun a => (X a, Y a)) \u03bc)) s = \u2191\u2191(map Y \u03bc) s\n[PROOFSTEP]\nrw [Measure.snd_apply hs, Measure.map_apply_of_aemeasurable (hX.prod_mk hY) (measurable_snd hs),\n Measure.map_apply_of_aemeasurable hY hs, \u2190 univ_prod, mk_preimage_prod, preimage_univ, univ_inter]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03b3\n\u03bc : Measure \u03b1\nhX : AEMeasurable X\nhY : \u00acAEMeasurable Y\n\u22a2 snd (map (fun a => (X a, Y a)) \u03bc) = map Y \u03bc\n[PROOFSTEP]\nhave : \u00acAEMeasurable (fun x \u21a6 (X x, Y x)) \u03bc := by contrapose! hY; exact measurable_snd.comp_aemeasurable hY\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03b3\n\u03bc : Measure \u03b1\nhX : AEMeasurable X\nhY : \u00acAEMeasurable Y\n\u22a2 \u00acAEMeasurable fun x => (X x, Y x)\n[PROOFSTEP]\ncontrapose! hY\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03b3\n\u03bc : Measure \u03b1\nhX : AEMeasurable X\nhY : AEMeasurable fun x => (X x, Y x)\n\u22a2 AEMeasurable Y\n[PROOFSTEP]\nexact measurable_snd.comp_aemeasurable hY\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\nX : \u03b1 \u2192 \u03b2\nY : \u03b1 \u2192 \u03b3\n\u03bc : Measure \u03b1\nhX : AEMeasurable X\nhY : \u00acAEMeasurable Y\nthis : \u00acAEMeasurable fun x => (X x, Y x)\n\u22a2 snd (map (fun a => (X a, Y a)) \u03bc) = map Y \u03bc\n[PROOFSTEP]\nsimp [map_of_not_aemeasurable, hY, this]\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Constructions.Prod.Basic", "llama_tokens": 103708, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.40925069059673047}} {"text": "[GOAL]\nx : { x // x \u2208 \ud835\udcde \u211a }\n\u22a2 IsUnit x \u2194 \u2191x = 1 \u2228 \u2191x = -1\n[PROOFSTEP]\nsimp_rw [(isUnit_map_iff (Rat.ringOfIntegersEquiv : \ud835\udcde \u211a \u2192+* \u2124) x).symm, Int.isUnit_iff, RingEquiv.coe_toRingHom,\n RingEquiv.map_eq_one_iff, RingEquiv.map_eq_neg_one_iff, \u2190 Subtype.coe_injective.eq_iff]\n[GOAL]\nx : { x // x \u2208 \ud835\udcde \u211a }\n\u22a2 \u2191x = \u21911 \u2228 \u2191x = \u2191(-1) \u2194 \u2191x = 1 \u2228 \u2191x = -1\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : { x // x \u2208 \ud835\udcde K }\n\u22a2 IsUnit x \u2194 |\u2191(\u2191(RingOfIntegers.norm \u211a) x)| = 1\n[PROOFSTEP]\nconvert (RingOfIntegers.isUnit_norm \u211a (F := K)).symm\n[GOAL]\ncase h.e'_2.a\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : { x // x \u2208 \ud835\udcde K }\n\u22a2 |\u2191(\u2191(RingOfIntegers.norm \u211a) x)| = 1 \u2194 IsUnit (\u2191(RingOfIntegers.norm \u211a) x)\n[PROOFSTEP]\nrw [\u2190 abs_one, abs_eq_abs, \u2190 Rat.RingOfIntegers.isUnit_iff]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nx\u271d\u00b9 x\u271d : { x // x \u2208 \ud835\udcde K }\u02e3\nh : (fun x => \u2191\u2191x) x\u271d\u00b9 = (fun x => \u2191\u2191x) x\u271d\n\u22a2 x\u271d\u00b9 = x\u271d\n[PROOFSTEP]\nrwa [SetLike.coe_eq_coe, Units.eq_iff] at h \n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nx : { x // x \u2208 \ud835\udcde K }\u02e3\nn : \u2115\n\u22a2 \u2191\u2191(x ^ n) = \u2191\u2191x ^ n\n[PROOFSTEP]\nrw [\u2190 SubmonoidClass.coe_pow, \u2190 val_pow_eq_pow_val]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nx : { x // x \u2208 \ud835\udcde K }\u02e3\nn : \u2124\n\u22a2 \u2191\u2191(x ^ n) = \u2191\u2191x ^ n\n[PROOFSTEP]\nchange ((Units.coeHom K).comp (map (algebraMap (\ud835\udcde K) K))) (x ^ n) = _\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nx : { x // x \u2208 \ud835\udcde K }\u02e3\nn : \u2124\n\u22a2 \u2191(MonoidHom.comp (coeHom K) (map \u2191(algebraMap { x // x \u2208 \ud835\udcde K } K))) (x ^ n) = \u2191\u2191x ^ n\n[PROOFSTEP]\nexact map_zpow _ x n\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\nx : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\n\u22a2 x \u2208 torsion K \u2194 \u2200 (w : InfinitePlace K), \u2191w \u2191\u2191x = 1\n[PROOFSTEP]\nrw [eq_iff_eq (x : K) 1, torsion, CommGroup.mem_torsion, isOfFinOrder_iff_pow_eq_one]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\nx : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\n\u22a2 (\u2203 n, 0 < n \u2227 x ^ n = 1) \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 \u2191\u2191x\u2016 = 1\n[PROOFSTEP]\nrefine \u27e8fun \u27e8n, h_pos, h_eq\u27e9 \u03c6 => ?_, fun h => ?_\u27e9\n[GOAL]\ncase refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\nx : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nx\u271d : \u2203 n, 0 < n \u2227 x ^ n = 1\n\u03c6 : K \u2192+* \u2102\nn : \u2115\nh_pos : 0 < n\nh_eq : x ^ n = 1\n\u22a2 \u2016\u2191\u03c6 \u2191\u2191x\u2016 = 1\n[PROOFSTEP]\nrefine norm_map_one_of_pow_eq_one \u03c6.toMonoidHom (k := \u27e8n, h_pos\u27e9) ?_\n[GOAL]\ncase refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\nx : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nx\u271d : \u2203 n, 0 < n \u2227 x ^ n = 1\n\u03c6 : K \u2192+* \u2102\nn : \u2115\nh_pos : 0 < n\nh_eq : x ^ n = 1\n\u22a2 \u2191\u2191x ^ \u2191{ val := n, property := h_pos } = 1\n[PROOFSTEP]\nrw [PNat.mk_coe, \u2190 coe_pow, h_eq, coe_one]\n[GOAL]\ncase refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\nx : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nh : \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 \u2191\u2191x\u2016 = 1\n\u22a2 \u2203 n, 0 < n \u2227 x ^ n = 1\n[PROOFSTEP]\nobtain \u27e8n, hn, hx\u27e9 := Embeddings.pow_eq_one_of_norm_eq_one K \u2102 x.val.prop h\n[GOAL]\ncase refine_2.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\nx : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nh : \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 \u2191\u2191x\u2016 = 1\nn : \u2115\nhn : 0 < n\nhx : \u2191\u2191x ^ n = 1\n\u22a2 \u2203 n, 0 < n \u2227 x ^ n = 1\n[PROOFSTEP]\nexact \u27e8n, hn, by ext; rw [coe_pow, hx, coe_one]\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\nx : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nh : \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 \u2191\u2191x\u2016 = 1\nn : \u2115\nhn : 0 < n\nhx : \u2191\u2191x ^ n = 1\n\u22a2 x ^ n = 1\n[PROOFSTEP]\next\n[GOAL]\ncase a.a\nK : Type u_1\ninst\u271d\u00b9 : Field K\nx : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nh : \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 \u2191\u2191x\u2016 = 1\nn : \u2115\nhn : 0 < n\nhx : \u2191\u2191x ^ n = 1\n\u22a2 \u2191\u2191(x ^ n) = \u2191\u21911\n[PROOFSTEP]\nrw [coe_pow, hx, coe_one]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u22a2 Fintype { x // x \u2208 torsion K }\n[PROOFSTEP]\nrefine @Fintype.ofFinite _ (Set.finite_coe_iff.mpr ?_)\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u22a2 Set.Finite \u2191(torsion K)\n[PROOFSTEP]\nrefine Set.Finite.of_finite_image ?_ ((coe_injective K).injOn _)\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u22a2 Set.Finite ((fun x => \u2191\u2191x) '' \u2191(torsion K))\n[PROOFSTEP]\nrefine (Embeddings.finite_of_norm_le K \u2102 1).subset (fun a \u27e8u, \u27e8h_tors, h_ua\u27e9\u27e9 => \u27e8?_, fun \u03c6 => ?_\u27e9)\n[GOAL]\ncase refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\na : K\nx\u271d : a \u2208 (fun x => \u2191\u2191x) '' \u2191(torsion K)\nu : { x // x \u2208 \ud835\udcde K }\u02e3\nh_tors : u \u2208 \u2191(torsion K)\nh_ua : (fun x => \u2191\u2191x) u = a\n\u22a2 IsIntegral \u2124 a\n[PROOFSTEP]\nrw [\u2190 h_ua]\n[GOAL]\ncase refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\na : K\nx\u271d : a \u2208 (fun x => \u2191\u2191x) '' \u2191(torsion K)\nu : { x // x \u2208 \ud835\udcde K }\u02e3\nh_tors : u \u2208 \u2191(torsion K)\nh_ua : (fun x => \u2191\u2191x) u = a\n\u22a2 IsIntegral \u2124 ((fun x => \u2191\u2191x) u)\n[PROOFSTEP]\nexact u.val.prop\n[GOAL]\ncase refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\na : K\nx\u271d : a \u2208 (fun x => \u2191\u2191x) '' \u2191(torsion K)\nu : { x // x \u2208 \ud835\udcde K }\u02e3\nh_tors : u \u2208 \u2191(torsion K)\nh_ua : (fun x => \u2191\u2191x) u = a\n\u03c6 : K \u2192+* \u2102\n\u22a2 \u2016\u2191\u03c6 a\u2016 \u2264 1\n[PROOFSTEP]\nrw [\u2190 h_ua]\n[GOAL]\ncase refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\na : K\nx\u271d : a \u2208 (fun x => \u2191\u2191x) '' \u2191(torsion K)\nu : { x // x \u2208 \ud835\udcde K }\u02e3\nh_tors : u \u2208 \u2191(torsion K)\nh_ua : (fun x => \u2191\u2191x) u = a\n\u03c6 : K \u2192+* \u2102\n\u22a2 \u2016\u2191\u03c6 ((fun x => \u2191\u2191x) u)\u2016 \u2264 1\n[PROOFSTEP]\nexact\n le_of_eq\n ((eq_iff_eq _ 1).mp ((mem_torsion K).mp h_tors) \u03c6)\n -- a shortcut instance to stop the next instance from timing out\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u03b6 : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nk : \u2115+\nhc : Nat.coprime \u2191k \u2191(torsion_order K)\n\u22a2 \u03b6 \u2208 rootsOfUnity k { x // x \u2208 \ud835\udcde K } \u2194 \u03b6 = 1\n[PROOFSTEP]\nrw [mem_rootsOfUnity]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u03b6 : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nk : \u2115+\nhc : Nat.coprime \u2191k \u2191(torsion_order K)\n\u22a2 \u03b6 ^ \u2191k = 1 \u2194 \u03b6 = 1\n[PROOFSTEP]\nrefine \u27e8fun h => ?_, fun h => by rw [h, one_pow]\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u03b6 : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nk : \u2115+\nhc : Nat.coprime \u2191k \u2191(torsion_order K)\nh : \u03b6 = 1\n\u22a2 \u03b6 ^ \u2191k = 1\n[PROOFSTEP]\nrw [h, one_pow]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u03b6 : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nk : \u2115+\nhc : Nat.coprime \u2191k \u2191(torsion_order K)\nh : \u03b6 ^ \u2191k = 1\n\u22a2 \u03b6 = 1\n[PROOFSTEP]\nrefine orderOf_eq_one_iff.mp (Nat.eq_one_of_dvd_coprimes hc ?_ ?_)\n[GOAL]\ncase refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u03b6 : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nk : \u2115+\nhc : Nat.coprime \u2191k \u2191(torsion_order K)\nh : \u03b6 ^ \u2191k = 1\n\u22a2 orderOf \u03b6 \u2223 \u2191k\n[PROOFSTEP]\nexact orderOf_dvd_of_pow_eq_one h\n[GOAL]\ncase refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u03b6 : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nk : \u2115+\nhc : Nat.coprime \u2191k \u2191(torsion_order K)\nh : \u03b6 ^ \u2191k = 1\n\u22a2 orderOf \u03b6 \u2223 \u2191(torsion_order K)\n[PROOFSTEP]\nhave h\u03b6 : \u03b6 \u2208 torsion K := by\n rw [torsion, CommGroup.mem_torsion, isOfFinOrder_iff_pow_eq_one]\n exact \u27e8k, k.prop, h\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u03b6 : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nk : \u2115+\nhc : Nat.coprime \u2191k \u2191(torsion_order K)\nh : \u03b6 ^ \u2191k = 1\n\u22a2 \u03b6 \u2208 torsion K\n[PROOFSTEP]\nrw [torsion, CommGroup.mem_torsion, isOfFinOrder_iff_pow_eq_one]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u03b6 : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nk : \u2115+\nhc : Nat.coprime \u2191k \u2191(torsion_order K)\nh : \u03b6 ^ \u2191k = 1\n\u22a2 \u2203 n, 0 < n \u2227 \u03b6 ^ n = 1\n[PROOFSTEP]\nexact \u27e8k, k.prop, h\u27e9\n[GOAL]\ncase refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u03b6 : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nk : \u2115+\nhc : Nat.coprime \u2191k \u2191(torsion_order K)\nh : \u03b6 ^ \u2191k = 1\nh\u03b6 : \u03b6 \u2208 torsion K\n\u22a2 orderOf \u03b6 \u2223 \u2191(torsion_order K)\n[PROOFSTEP]\nrw [orderOf_submonoid (\u27e8\u03b6, h\u03b6\u27e9 : torsion K)]\n[GOAL]\ncase refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u03b6 : { x // x \u2208 \ud835\udcde K }\u02e3\ninst\u271d : NumberField K\nk : \u2115+\nhc : Nat.coprime \u2191k \u2191(torsion_order K)\nh : \u03b6 ^ \u2191k = 1\nh\u03b6 : \u03b6 \u2208 torsion K\n\u22a2 orderOf { val := \u03b6, property := h\u03b6 } \u2223 \u2191(torsion_order K)\n[PROOFSTEP]\nexact orderOf_dvd_card_univ\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u22a2 rootsOfUnity (torsion_order K) { x // x \u2208 \ud835\udcde K } = torsion K\n[PROOFSTEP]\next \u03b6\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u03b6 : { x // x \u2208 \ud835\udcde K }\u02e3\n\u22a2 \u03b6 \u2208 rootsOfUnity (torsion_order K) { x // x \u2208 \ud835\udcde K } \u2194 \u03b6 \u2208 torsion K\n[PROOFSTEP]\nrw [torsion, mem_rootsOfUnity]\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u03b6 : { x // x \u2208 \ud835\udcde K }\u02e3\n\u22a2 \u03b6 ^ \u2191(torsion_order K) = 1 \u2194 \u03b6 \u2208 CommGroup.torsion { x // x \u2208 \ud835\udcde K }\u02e3\n[PROOFSTEP]\nrefine \u27e8fun h => ?_, fun h => ?_\u27e9\n[GOAL]\ncase h.refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u03b6 : { x // x \u2208 \ud835\udcde K }\u02e3\nh : \u03b6 ^ \u2191(torsion_order K) = 1\n\u22a2 \u03b6 \u2208 CommGroup.torsion { x // x \u2208 \ud835\udcde K }\u02e3\n[PROOFSTEP]\nrw [CommGroup.mem_torsion, isOfFinOrder_iff_pow_eq_one]\n[GOAL]\ncase h.refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u03b6 : { x // x \u2208 \ud835\udcde K }\u02e3\nh : \u03b6 ^ \u2191(torsion_order K) = 1\n\u22a2 \u2203 n, 0 < n \u2227 \u03b6 ^ n = 1\n[PROOFSTEP]\nexact \u27e8\u2191(torsion_order K), (torsion_order K).prop, h\u27e9\n[GOAL]\ncase h.refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u03b6 : { x // x \u2208 \ud835\udcde K }\u02e3\nh : \u03b6 \u2208 CommGroup.torsion { x // x \u2208 \ud835\udcde K }\u02e3\n\u22a2 \u03b6 ^ \u2191(torsion_order K) = 1\n[PROOFSTEP]\nexact Subtype.ext_iff.mp (@pow_card_eq_one (torsion K) _ \u27e8\u03b6, h\u27e9 _)\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.NumberField.Units", "llama_tokens": 5065, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.40925069059673047}} {"text": "[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF : C \u2964 C\nA A\u2080 A\u2081 A\u2082 : Algebra F\nf : A\u2080 \u27f6 A\u2081\ng : A\u2081 \u27f6 A\u2082\nh : A\u2080.a \u2245 A\u2081.a\nw : autoParam (F.map h.hom \u226b A\u2081.str = A\u2080.str \u226b h.hom) _auto\u271d\n\u22a2 F.map h.inv \u226b A\u2080.str = A\u2081.str \u226b h.inv\n[PROOFSTEP]\nrw [h.eq_comp_inv, Category.assoc, \u2190 w, \u2190 Functor.map_comp_assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF : C \u2964 C\nA A\u2080 A\u2081 A\u2082 : Algebra F\nf : A\u2080 \u27f6 A\u2081\ng : A\u2081 \u27f6 A\u2082\nh : A\u2080.a \u2245 A\u2081.a\nw : autoParam (F.map h.hom \u226b A\u2081.str = A\u2080.str \u226b h.hom) _auto\u271d\n\u22a2 F.map (h.inv \u226b h.hom) \u226b A\u2081.str = A\u2081.str\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : C \u2964 C\nA A\u2080 A\u2081 A\u2082 : Algebra F\nf\u271d : A\u2080 \u27f6 A\u2081\ng : A\u2081 \u27f6 A\u2082\nf : A\u2080 \u27f6 A\u2081\ninst\u271d : IsIso f.f\n\u22a2 F.map (inv f.f) \u226b A\u2080.str = A\u2081.str \u226b inv f.f\n[PROOFSTEP]\nrw [IsIso.eq_comp_inv f.1, Category.assoc, \u2190 f.h]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : C \u2964 C\nA A\u2080 A\u2081 A\u2082 : Algebra F\nf\u271d : A\u2080 \u27f6 A\u2081\ng : A\u2081 \u27f6 A\u2082\nf : A\u2080 \u27f6 A\u2081\ninst\u271d : IsIso f.f\n\u22a2 F.map (inv f.f) \u226b F.map f.f \u226b A\u2081.str = A\u2081.str\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : C \u2964 C\nA A\u2080 A\u2081 A\u2082 : Algebra F\nf\u271d : A\u2080 \u27f6 A\u2081\ng : A\u2081 \u27f6 A\u2082\nf : A\u2080 \u27f6 A\u2081\ninst\u271d : IsIso f.f\n\u22a2 f \u226b Hom.mk (inv f.f) = \ud835\udfd9 A\u2080\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : C \u2964 C\nA A\u2080 A\u2081 A\u2082 : Algebra F\nf\u271d : A\u2080 \u27f6 A\u2081\ng : A\u2081 \u27f6 A\u2082\nf : A\u2080 \u27f6 A\u2081\ninst\u271d : IsIso f.f\n\u22a2 Hom.mk (inv f.f) \u226b f = \ud835\udfd9 A\u2081\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF\u271d : C \u2964 C\nA A\u2080 A\u2081 A\u2082 : Algebra F\u271d\nf : A\u2080 \u27f6 A\u2081\ng : A\u2081 \u27f6 A\u2082\nF G : C \u2964 C\n\u03b1 : F \u2245 G\n\u22a2 \ud835\udfd9 F = \u03b1.hom \u226b \u03b1.inv\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF\u271d : C \u2964 C\nA A\u2080 A\u2081 A\u2082 : Algebra F\u271d\nf : A\u2080 \u27f6 A\u2081\ng : A\u2081 \u27f6 A\u2082\nF G : C \u2964 C\n\u03b1 : F \u2245 G\n\u22a2 \u03b1.inv \u226b \u03b1.hom = \ud835\udfd9 G\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF : C \u2964 C\nA A\u2080 A\u2081 A\u2082 : Algebra F\nf : A\u2080 \u27f6 A\u2081\ng : A\u2081 \u27f6 A\u2082\nh : Limits.IsInitial A\n\u22a2 F.map (strInv h \u226b A.str) \u226b A.str = A.str \u226b strInv h \u226b A.str\n[PROOFSTEP]\nrw [\u2190 Category.assoc, F.map_comp, strInv, \u2190 Hom.h]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF : C \u2964 C\nA A\u2080 A\u2081 A\u2082 : Algebra F\nf : A\u2080 \u27f6 A\u2081\ng : A\u2081 \u27f6 A\u2082\nh : Limits.IsInitial A\n\u22a2 A.str \u226b strInv h = \ud835\udfd9 (F.obj A.a)\n[PROOFSTEP]\nrw [strInv, \u2190 (h.to \u27e8F.obj A.1, F.map A.str\u27e9).h, \u2190 F.map_id, \u2190 F.map_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF : C \u2964 C\nA A\u2080 A\u2081 A\u2082 : Algebra F\nf : A\u2080 \u27f6 A\u2081\ng : A\u2081 \u27f6 A\u2082\nh : Limits.IsInitial A\n\u22a2 F.map ((Limits.IsInitial.to h { a := F.obj A.a, str := F.map A.str }).f \u226b A.str) = F.map (\ud835\udfd9 A.a)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d : Category.{v, u} C\nF : C \u2964 C\nA A\u2080 A\u2081 A\u2082 : Algebra F\nf : A\u2080 \u27f6 A\u2081\ng : A\u2081 \u27f6 A\u2082\nh : Limits.IsInitial A\n\u22a2 (Limits.IsInitial.to h { a := F.obj A.a, str := F.map A.str }).f \u226b A.str = \ud835\udfd9 A.a\n[PROOFSTEP]\nexact left_inv h\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF : C \u2964 C\nV V\u2080 V\u2081 V\u2082 : Coalgebra F\nf : V\u2080 \u27f6 V\u2081\ng : V\u2081 \u27f6 V\u2082\nh : V\u2080.V \u2245 V\u2081.V\nw : autoParam (V\u2080.str \u226b F.map h.hom = h.hom \u226b V\u2081.str) _auto\u271d\n\u22a2 V\u2081.str \u226b F.map h.inv = h.inv \u226b V\u2080.str\n[PROOFSTEP]\nrw [h.eq_inv_comp, \u2190 Category.assoc, \u2190 w, Category.assoc, \u2190 F.map_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF : C \u2964 C\nV V\u2080 V\u2081 V\u2082 : Coalgebra F\nf : V\u2080 \u27f6 V\u2081\ng : V\u2081 \u27f6 V\u2082\nh : V\u2080.V \u2245 V\u2081.V\nw : autoParam (V\u2080.str \u226b F.map h.hom = h.hom \u226b V\u2081.str) _auto\u271d\n\u22a2 V\u2080.str \u226b F.map (h.hom \u226b h.inv) = V\u2080.str\n[PROOFSTEP]\nsimp only [Iso.hom_inv_id, Functor.map_id, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : C \u2964 C\nV V\u2080 V\u2081 V\u2082 : Coalgebra F\nf\u271d : V\u2080 \u27f6 V\u2081\ng : V\u2081 \u27f6 V\u2082\nf : V\u2080 \u27f6 V\u2081\ninst\u271d : IsIso f.f\n\u22a2 V\u2081.str \u226b F.map (inv f.f) = inv f.f \u226b V\u2080.str\n[PROOFSTEP]\nrw [IsIso.eq_inv_comp f.1, \u2190 Category.assoc, \u2190 f.h, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : C \u2964 C\nV V\u2080 V\u2081 V\u2082 : Coalgebra F\nf\u271d : V\u2080 \u27f6 V\u2081\ng : V\u2081 \u27f6 V\u2082\nf : V\u2080 \u27f6 V\u2081\ninst\u271d : IsIso f.f\n\u22a2 V\u2080.str \u226b F.map f.f \u226b F.map (inv f.f) = V\u2080.str\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : C \u2964 C\nV V\u2080 V\u2081 V\u2082 : Coalgebra F\nf\u271d : V\u2080 \u27f6 V\u2081\ng : V\u2081 \u27f6 V\u2082\nf : V\u2080 \u27f6 V\u2081\ninst\u271d : IsIso f.f\n\u22a2 f \u226b Hom.mk (inv f.f) = \ud835\udfd9 V\u2080\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : C \u2964 C\nV V\u2080 V\u2081 V\u2082 : Coalgebra F\nf\u271d : V\u2080 \u27f6 V\u2081\ng : V\u2081 \u27f6 V\u2082\nf : V\u2080 \u27f6 V\u2081\ninst\u271d : IsIso f.f\n\u22a2 Hom.mk (inv f.f) \u226b f = \ud835\udfd9 V\u2081\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF\u271d : C \u2964 C\nV V\u2080 V\u2081 V\u2082 : Coalgebra F\u271d\nf\u271d : V\u2080 \u27f6 V\u2081\ng : V\u2081 \u27f6 V\u2082\nF G : C \u2964 C\n\u03b1 : F \u27f6 G\nX\u271d Y\u271d : Coalgebra F\nf : X\u271d \u27f6 Y\u271d\n\u22a2 ((fun V => { V := V.V, str := V.str \u226b NatTrans.app \u03b1 V.V }) X\u271d).str \u226b G.map f.f =\n f.f \u226b ((fun V => { V := V.V, str := V.str \u226b NatTrans.app \u03b1 V.V }) Y\u271d).str\n[PROOFSTEP]\nrw [Category.assoc, \u2190 \u03b1.naturality, \u2190 Category.assoc, f.h, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF\u271d : C \u2964 C\nV V\u2080 V\u2081 V\u2082 : Coalgebra F\u271d\nf : V\u2080 \u27f6 V\u2081\ng : V\u2081 \u27f6 V\u2082\nF G : C \u2964 C\n\u03b1 : F \u2245 G\n\u22a2 \ud835\udfd9 F = \u03b1.hom \u226b \u03b1.inv\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF\u271d : C \u2964 C\nV V\u2080 V\u2081 V\u2082 : Coalgebra F\u271d\nf : V\u2080 \u27f6 V\u2081\ng : V\u2081 \u27f6 V\u2082\nF G : C \u2964 C\n\u03b1 : F \u2245 G\n\u22a2 \u03b1.inv \u226b \u03b1.hom = \ud835\udfd9 G\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nA\u2081 A\u2082 : Algebra F\nf : A\u2081 \u27f6 A\u2082\n\u22a2 \u2191(Adjunction.homEquiv adj A\u2081.a A\u2081.a) A\u2081.str \u226b G.map f.f = f.f \u226b \u2191(Adjunction.homEquiv adj A\u2082.a A\u2082.a) A\u2082.str\n[PROOFSTEP]\nrw [\u2190 Adjunction.homEquiv_naturality_right, \u2190 Adjunction.homEquiv_naturality_left, f.h]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nV\u2081 V\u2082 : Coalgebra G\nf : V\u2081 \u27f6 V\u2082\n\u22a2 F.map f.f \u226b \u2191(Adjunction.homEquiv adj V\u2082.V V\u2082.V).symm V\u2082.str = \u2191(Adjunction.homEquiv adj V\u2081.V V\u2081.V).symm V\u2081.str \u226b f.f\n[PROOFSTEP]\nrw [\u2190 Adjunction.homEquiv_naturality_left_symm, \u2190 Adjunction.homEquiv_naturality_right_symm, f.h]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nA : Algebra F\n\u22a2 F.map (\ud835\udfd9 A.a) \u226b ((Algebra.toCoalgebraOf adj \u22d9 Coalgebra.toAlgebraOf adj).obj A).str =\n ((\ud835\udfed (Algebra F)).obj A).str \u226b \ud835\udfd9 A.a\n[PROOFSTEP]\nerw [F.map_id, Category.id_comp, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nA : Algebra F\n\u22a2 ((Algebra.toCoalgebraOf adj \u22d9 Coalgebra.toAlgebraOf adj).obj A).str = ((\ud835\udfed (Algebra F)).obj A).str\n[PROOFSTEP]\napply (adj.homEquiv _ _).left_inv A.str\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nA : Algebra F\n\u22a2 F.map (\ud835\udfd9 A.a) \u226b ((\ud835\udfed (Algebra F)).obj A).str =\n ((Algebra.toCoalgebraOf adj \u22d9 Coalgebra.toAlgebraOf adj).obj A).str \u226b \ud835\udfd9 A.a\n[PROOFSTEP]\nerw [F.map_id, Category.id_comp, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nA : Algebra F\n\u22a2 ((\ud835\udfed (Algebra F)).obj A).str = ((Algebra.toCoalgebraOf adj \u22d9 Coalgebra.toAlgebraOf adj).obj A).str\n[PROOFSTEP]\napply ((adj.homEquiv _ _).left_inv A.str).symm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nA\u2081 A\u2082 : Algebra F\nf : A\u2081 \u27f6 A\u2082\n\u22a2 (Algebra.toCoalgebraOf adj \u22d9 Coalgebra.toAlgebraOf adj).map f \u226b (fun A => Algebra.Hom.mk (\ud835\udfd9 A.a)) A\u2082 =\n (fun A => Algebra.Hom.mk (\ud835\udfd9 A.a)) A\u2081 \u226b (\ud835\udfed (Algebra F)).map f\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nA\u2081 A\u2082 : Algebra F\nf : A\u2081 \u27f6 A\u2082\n\u22a2 autoParam\n (((Algebra.toCoalgebraOf adj \u22d9 Coalgebra.toAlgebraOf adj).map f \u226b (fun A => Algebra.Hom.mk (\ud835\udfd9 A.a)) A\u2082).f =\n ((fun A => Algebra.Hom.mk (\ud835\udfd9 A.a)) A\u2081 \u226b (\ud835\udfed (Algebra F)).map f).f)\n _auto\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nA\u2081 A\u2082 : Algebra F\nf : A\u2081 \u27f6 A\u2082\n\u22a2 autoParam (((Coalgebra.toAlgebraOf adj).map ((Algebra.toCoalgebraOf adj).map f)).f \u226b \ud835\udfd9 A\u2082.a = \ud835\udfd9 A\u2081.a \u226b f.f) _auto\u271d\n[PROOFSTEP]\nerw [Category.comp_id, Category.id_comp]\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nA\u2081 A\u2082 : Algebra F\nf : A\u2081 \u27f6 A\u2082\n\u22a2 autoParam (((Coalgebra.toAlgebraOf adj).map ((Algebra.toCoalgebraOf adj).map f)).f = f.f) _auto\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nV : Coalgebra G\n\u22a2 ((Coalgebra.toAlgebraOf adj \u22d9 Algebra.toCoalgebraOf adj).obj V).str \u226b G.map (\ud835\udfd9 V.V) =\n \ud835\udfd9 V.V \u226b ((\ud835\udfed (Coalgebra G)).obj V).str\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nV : Coalgebra G\n\u22a2 ((Algebra.toCoalgebraOf adj).obj ((Coalgebra.toAlgebraOf adj).obj V)).str \u226b G.map (\ud835\udfd9 V.V) = \ud835\udfd9 V.V \u226b V.str\n[PROOFSTEP]\nerw [G.map_id, Category.id_comp, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nV : Coalgebra G\n\u22a2 ((Algebra.toCoalgebraOf adj).obj ((Coalgebra.toAlgebraOf adj).obj V)).str = V.str\n[PROOFSTEP]\napply (adj.homEquiv _ _).right_inv V.str\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nV\u2081 V\u2082 : Coalgebra G\nf : V\u2081 \u27f6 V\u2082\n\u22a2 (Coalgebra.toAlgebraOf adj \u22d9 Algebra.toCoalgebraOf adj).map f \u226b (fun V => Coalgebra.Hom.mk (\ud835\udfd9 V.V)) V\u2082 =\n (fun V => Coalgebra.Hom.mk (\ud835\udfd9 V.V)) V\u2081 \u226b (\ud835\udfed (Coalgebra G)).map f\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nV\u2081 V\u2082 : Coalgebra G\nf : V\u2081 \u27f6 V\u2082\n\u22a2 autoParam\n (((Coalgebra.toAlgebraOf adj \u22d9 Algebra.toCoalgebraOf adj).map f \u226b (fun V => Coalgebra.Hom.mk (\ud835\udfd9 V.V)) V\u2082).f =\n ((fun V => Coalgebra.Hom.mk (\ud835\udfd9 V.V)) V\u2081 \u226b (\ud835\udfed (Coalgebra G)).map f).f)\n _auto\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nV\u2081 V\u2082 : Coalgebra G\nf : V\u2081 \u27f6 V\u2082\n\u22a2 autoParam (((Algebra.toCoalgebraOf adj).map ((Coalgebra.toAlgebraOf adj).map f)).f \u226b \ud835\udfd9 V\u2082.V = \ud835\udfd9 V\u2081.V \u226b f.f) _auto\u271d\n[PROOFSTEP]\nerw [Category.comp_id, Category.id_comp]\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nV\u2081 V\u2082 : Coalgebra G\nf : V\u2081 \u27f6 V\u2082\n\u22a2 autoParam (((Algebra.toCoalgebraOf adj).map ((Coalgebra.toAlgebraOf adj).map f)).f = f.f) _auto\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nV : Coalgebra G\n\u22a2 ((\ud835\udfed (Coalgebra G)).obj V).str \u226b G.map (\ud835\udfd9 V.V) =\n \ud835\udfd9 V.V \u226b ((Coalgebra.toAlgebraOf adj \u22d9 Algebra.toCoalgebraOf adj).obj V).str\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nV : Coalgebra G\n\u22a2 V.str \u226b G.map (\ud835\udfd9 V.V) = \ud835\udfd9 V.V \u226b ((Algebra.toCoalgebraOf adj).obj ((Coalgebra.toAlgebraOf adj).obj V)).str\n[PROOFSTEP]\nrw [G.map_id, Category.comp_id, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nV : Coalgebra G\n\u22a2 V.str = ((Algebra.toCoalgebraOf adj).obj ((Coalgebra.toAlgebraOf adj).obj V)).str\n[PROOFSTEP]\napply ((adj.homEquiv _ _).right_inv V.str).symm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nA : Algebra F\n\u22a2 (Algebra.toCoalgebraOf adj).map (NatTrans.app (AlgCoalgEquiv.unitIso adj).hom A) \u226b\n NatTrans.app (AlgCoalgEquiv.counitIso adj).hom ((Algebra.toCoalgebraOf adj).obj A) =\n \ud835\udfd9 ((Algebra.toCoalgebraOf adj).obj A)\n[PROOFSTEP]\next\n -- Porting note: why doesn't `simp` work here?\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nF G : C \u2964 C\nadj : F \u22a3 G\nA : Algebra F\n\u22a2 autoParam\n (((Algebra.toCoalgebraOf adj).map (NatTrans.app (AlgCoalgEquiv.unitIso adj).hom A) \u226b\n NatTrans.app (AlgCoalgEquiv.counitIso adj).hom ((Algebra.toCoalgebraOf adj).obj A)).f =\n (\ud835\udfd9 ((Algebra.toCoalgebraOf adj).obj A)).f)\n _auto\u271d\n[PROOFSTEP]\nexact Category.comp_id _\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Endofunctor.Algebra", "llama_tokens": 6011, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754371026367, "lm_q2_score": 0.5621765008857982, "lm_q1q2_score": 0.40925068396116976}} {"text": "[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF\u271d : Presheaf C X\n\u03b9 : Type v\nU\u271d : \u03b9 \u2192 Opens \u2191X\nF : Presheaf (Discrete Unit) X\nx\u271d\u00b2 : Discrete Unit\nU : Opens \u2191X\nS : Sieve U\nx\u271d\u00b9 : S \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nx : Presieve.FamilyOfElements (F \u22d9 coyoneda.obj (op x\u271d\u00b2)) S.arrows\nx\u271d : Presieve.FamilyOfElements.Compatible x\n\u22a2 (fun t => Presieve.FamilyOfElements.IsAmalgamation x t) (eqToHom (_ : (op x\u271d\u00b2).unop = F.obj (op U)))\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF\u271d : Presheaf C X\n\u03b9 : Type v\nU\u271d : \u03b9 \u2192 Opens \u2191X\nF : Presheaf (Discrete Unit) X\nx\u271d\u00b3 : Discrete Unit\nU : Opens \u2191X\nS : Sieve U\nx\u271d\u00b2 : S \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nx : Presieve.FamilyOfElements (F \u22d9 coyoneda.obj (op x\u271d\u00b3)) S.arrows\nx\u271d\u00b9 : Presieve.FamilyOfElements.Compatible x\nx\u271d : (F \u22d9 coyoneda.obj (op x\u271d\u00b3)).obj (op U)\n\u22a2 (fun t => Presieve.FamilyOfElements.IsAmalgamation x t) x\u271d \u2192 x\u271d = eqToHom (_ : (op x\u271d\u00b3).unop = F.obj (op U))\n[PROOFSTEP]\naesop_cat\n", "meta": {"mathlib_filename": "Mathlib.Topology.Sheaves.Sheaf", "llama_tokens": 512, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7431680086124811, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.40919377937218904}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y Z : C\nr : \u2115\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 (r \u2022 f) \u226b g = r \u2022 f \u226b g\n[PROOFSTEP]\nexact (Preadditive.rightComp X g).map_nsmul f r\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y Z : C\nf : X \u27f6 Y\nr : \u2115\ng : Y \u27f6 Z\n\u22a2 f \u226b (r \u2022 g) = r \u2022 f \u226b g\n[PROOFSTEP]\nexact (Preadditive.leftComp Z f).map_nsmul g r\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y Z : C\nr : \u2124\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 (r \u2022 f) \u226b g = r \u2022 f \u226b g\n[PROOFSTEP]\nexact (Preadditive.rightComp X g).map_zsmul f r\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y Z : C\nf : X \u27f6 Y\nr : \u2124\ng : Y \u27f6 Z\n\u22a2 f \u226b (r \u2022 g) = r \u2022 f \u226b g\n[PROOFSTEP]\nexact (Preadditive.leftComp Z f).map_zsmul g r\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type w\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R C\nX : C\n\u22a2 Module R (End X)\n[PROOFSTEP]\ndsimp [End]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type w\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R C\nX : C\n\u22a2 Module R (X \u27f6 X)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type w\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R C\nX Y Z : C\nf : X \u27f6 Y\n\u22a2 \u2200 (x y : Y \u27f6 Z), (fun g => f \u226b g) (x + y) = (fun g => f \u226b g) x + (fun g => f \u226b g) y\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type w\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R C\nX Y Z : C\nf : X \u27f6 Y\n\u22a2 \u2200 (r : R) (x : Y \u27f6 Z),\n AddHom.toFun { toFun := fun g => f \u226b g, map_add' := (_ : \u2200 (a a_1 : Y \u27f6 Z), f \u226b (a + a_1) = f \u226b a + f \u226b a_1) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun { toFun := fun g => f \u226b g, map_add' := (_ : \u2200 (a a_1 : Y \u27f6 Z), f \u226b (a + a_1) = f \u226b a + f \u226b a_1) } x\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type w\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R C\nX Y Z : C\ng : Y \u27f6 Z\n\u22a2 \u2200 (x y : X \u27f6 Y), (fun f => f \u226b g) (x + y) = (fun f => f \u226b g) x + (fun f => f \u226b g) y\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type w\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Linear R C\nX Y Z : C\ng : Y \u27f6 Z\n\u22a2 \u2200 (r : R) (x : X \u27f6 Y),\n AddHom.toFun { toFun := fun f => f \u226b g, map_add' := (_ : \u2200 (a a_1 : X \u27f6 Y), (a + a_1) \u226b g = a \u226b g + a_1 \u226b g) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun { toFun := fun f => f \u226b g, map_add' := (_ : \u2200 (a a_1 : X \u27f6 Y), (a + a_1) \u226b g = a \u226b g + a_1 \u226b g) } x\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\nR : Type w\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Linear R C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Epi f\nr : R\ninst\u271d : Invertible r\nZ\u271d : C\ng g' : Y \u27f6 Z\u271d\nH : (r \u2022 f) \u226b g = (r \u2022 f) \u226b g'\n\u22a2 g = g'\n[PROOFSTEP]\nrw [smul_comp, smul_comp, \u2190 comp_smul, \u2190 comp_smul, cancel_epi] at H \n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\nR : Type w\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Linear R C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Epi f\nr : R\ninst\u271d : Invertible r\nZ\u271d : C\ng g' : Y \u27f6 Z\u271d\nH : r \u2022 g = r \u2022 g'\n\u22a2 g = g'\n[PROOFSTEP]\nsimpa [smul_smul] using congr_arg (fun f => \u215fr \u2022 f) H\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\nR : Type w\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Linear R C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Mono f\nr : R\ninst\u271d : Invertible r\nZ\u271d : C\ng g' : Z\u271d \u27f6 X\nH : g \u226b (r \u2022 f) = g' \u226b (r \u2022 f)\n\u22a2 g = g'\n[PROOFSTEP]\nrw [comp_smul, comp_smul, \u2190 smul_comp, \u2190 smul_comp, cancel_mono] at H \n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\nR : Type w\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Linear R C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Mono f\nr : R\ninst\u271d : Invertible r\nZ\u271d : C\ng g' : Z\u271d \u27f6 X\nH : r \u2022 g = r \u2022 g'\n\u22a2 g = g'\n[PROOFSTEP]\nsimpa [smul_smul] using congr_arg (fun f => \u215fr \u2022 f) H\n[GOAL]\nC\u271d : Type u\ninst\u271d\u2077 : Category.{v, u} C\u271d\ninst\u271d\u2076 : Preadditive C\u271d\nR : Type w\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Linear R C\u271d\nk : Type u_1\nC : Type u_2\ninst\u271d\u00b3 : Category.{?u.35990, u_2} C\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : Linear k C\nX Y W Z : C\nf\u2081 : X \u2245 Y\nf\u2082 : W \u2245 Z\nsrc\u271d : (X \u27f6 W) \u2192\u2097[k] Y \u27f6 Z := comp (rightComp k Y f\u2082.hom) (leftComp k W f\u2081.symm.hom)\nx : X \u27f6 W\n\u22a2 \u2191(comp (leftComp k W f\u2081.hom) (rightComp k Y f\u2082.symm.hom))\n (AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : k) (x : X \u27f6 W),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id k) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n x) =\n x\n[PROOFSTEP]\nsimp only [Iso.symm_hom, LinearMap.toFun_eq_coe, LinearMap.coe_comp, Function.comp_apply, leftComp_apply,\n rightComp_apply, Category.assoc, Iso.hom_inv_id, Category.comp_id, Iso.hom_inv_id_assoc]\n[GOAL]\nC\u271d : Type u\ninst\u271d\u2077 : Category.{v, u} C\u271d\ninst\u271d\u2076 : Preadditive C\u271d\nR : Type w\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Linear R C\u271d\nk : Type u_1\nC : Type u_2\ninst\u271d\u00b3 : Category.{?u.35990, u_2} C\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : Linear k C\nX Y W Z : C\nf\u2081 : X \u2245 Y\nf\u2082 : W \u2245 Z\nsrc\u271d : (X \u27f6 W) \u2192\u2097[k] Y \u27f6 Z := comp (rightComp k Y f\u2082.hom) (leftComp k W f\u2081.symm.hom)\nx : Y \u27f6 Z\n\u22a2 AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : k) (x : X \u27f6 W),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id k) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n (\u2191(comp (leftComp k W f\u2081.hom) (rightComp k Y f\u2082.symm.hom)) x) =\n x\n[PROOFSTEP]\nsimp only [Iso.symm_hom, LinearMap.coe_comp, Function.comp_apply, rightComp_apply, leftComp_apply,\n LinearMap.toFun_eq_coe, Iso.inv_hom_id_assoc, Category.assoc, Iso.inv_hom_id, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nS : Type w\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Linear S C\nX Y Z : C\n\u22a2 \u2200 (x y : X \u27f6 Y), (fun f => leftComp S Z f) (x + y) = (fun f => leftComp S Z f) x + (fun f => leftComp S Z f) y\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nS : Type w\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Linear S C\nX Y Z : C\nx\u271d y\u271d : X \u27f6 Y\n\u22a2 (fun f => leftComp S Z f) (x\u271d + y\u271d) = (fun f => leftComp S Z f) x\u271d + (fun f => leftComp S Z f) y\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nS : Type w\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Linear S C\nX Y Z : C\nx\u271d\u00b9 y\u271d : X \u27f6 Y\nx\u271d : Y \u27f6 Z\n\u22a2 \u2191((fun f => leftComp S Z f) (x\u271d\u00b9 + y\u271d)) x\u271d = \u2191((fun f => leftComp S Z f) x\u271d\u00b9 + (fun f => leftComp S Z f) y\u271d) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nS : Type w\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Linear S C\nX Y Z : C\n\u22a2 \u2200 (r : S) (x : X \u27f6 Y),\n AddHom.toFun\n { toFun := fun f => leftComp S Z f,\n map_add' :=\n (_ :\n \u2200 (x y : X \u27f6 Y),\n (fun f => leftComp S Z f) (x + y) = (fun f => leftComp S Z f) x + (fun f => leftComp S Z f) y) }\n (r \u2022 x) =\n \u2191(RingHom.id S) r \u2022\n AddHom.toFun\n { toFun := fun f => leftComp S Z f,\n map_add' :=\n (_ :\n \u2200 (x y : X \u27f6 Y),\n (fun f => leftComp S Z f) (x + y) = (fun f => leftComp S Z f) x + (fun f => leftComp S Z f) y) }\n x\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nS : Type w\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Linear S C\nX Y Z : C\nr\u271d : S\nx\u271d : X \u27f6 Y\n\u22a2 AddHom.toFun\n { toFun := fun f => leftComp S Z f,\n map_add' :=\n (_ :\n \u2200 (x y : X \u27f6 Y),\n (fun f => leftComp S Z f) (x + y) = (fun f => leftComp S Z f) x + (fun f => leftComp S Z f) y) }\n (r\u271d \u2022 x\u271d) =\n \u2191(RingHom.id S) r\u271d \u2022\n AddHom.toFun\n { toFun := fun f => leftComp S Z f,\n map_add' :=\n (_ :\n \u2200 (x y : X \u27f6 Y),\n (fun f => leftComp S Z f) (x + y) = (fun f => leftComp S Z f) x + (fun f => leftComp S Z f) y) }\n x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nS : Type w\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Linear S C\nX Y Z : C\nr\u271d : S\nx\u271d\u00b9 : X \u27f6 Y\nx\u271d : Y \u27f6 Z\n\u22a2 \u2191(AddHom.toFun\n { toFun := fun f => leftComp S Z f,\n map_add' :=\n (_ :\n \u2200 (x y : X \u27f6 Y),\n (fun f => leftComp S Z f) (x + y) = (fun f => leftComp S Z f) x + (fun f => leftComp S Z f) y) }\n (r\u271d \u2022 x\u271d\u00b9))\n x\u271d =\n \u2191(\u2191(RingHom.id S) r\u271d \u2022\n AddHom.toFun\n { toFun := fun f => leftComp S Z f,\n map_add' :=\n (_ :\n \u2200 (x y : X \u27f6 Y),\n (fun f => leftComp S Z f) (x + y) = (fun f => leftComp S Z f) x + (fun f => leftComp S Z f) y) }\n x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Linear.Basic", "llama_tokens": 4357, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494678483918, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.40912299928736184}} {"text": "[GOAL]\nf g : \u2115 \u2192. \u2115\nhf : Partrec f\nhg : Partrec g\n\u22a2 \u2203 h, Partrec h \u2227 \u2200 (a : \u2115), (\u2200 (x : \u2115), x \u2208 h a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((h a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\n[PROOFSTEP]\nobtain \u27e8cf, rfl\u27e9 := Code.exists_code.1 hf\n[GOAL]\ncase intro\ng : \u2115 \u2192. \u2115\nhg : Partrec g\ncf : Code\nhf : Partrec (Code.eval cf)\n\u22a2 \u2203 h,\n Partrec h \u2227\n \u2200 (a : \u2115), (\u2200 (x : \u2115), x \u2208 h a \u2192 x \u2208 Code.eval cf a \u2228 x \u2208 g a) \u2227 ((h a).Dom \u2194 (Code.eval cf a).Dom \u2228 (g a).Dom)\n[PROOFSTEP]\nobtain \u27e8cg, rfl\u27e9 := Code.exists_code.1 hg\n[GOAL]\ncase intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\n\u22a2 \u2203 h,\n Partrec h \u2227\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115), x \u2208 h a \u2192 x \u2208 Code.eval cf a \u2228 x \u2208 Code.eval cg a) \u2227\n ((h a).Dom \u2194 (Code.eval cf a).Dom \u2228 (Code.eval cg a).Dom)\n[PROOFSTEP]\nhave : Nat.Partrec fun n => Nat.rfindOpt fun k => cf.evaln k n <|> cg.evaln k n :=\n Partrec.nat_iff.1\n (Partrec.rfindOpt <|\n Primrec.option_orElse.to_comp.comp (Code.evaln_prim.to_comp.comp <| (snd.pair (const cf)).pair fst)\n (Code.evaln_prim.to_comp.comp <| (snd.pair (const cg)).pair fst))\n[GOAL]\ncase intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\n\u22a2 \u2203 h,\n Partrec h \u2227\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115), x \u2208 h a \u2192 x \u2208 Code.eval cf a \u2228 x \u2208 Code.eval cg a) \u2227\n ((h a).Dom \u2194 (Code.eval cf a).Dom \u2228 (Code.eval cg a).Dom)\n[PROOFSTEP]\nrefine' \u27e8_, this, fun n => _\u27e9\n[GOAL]\ncase intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\n\u22a2 (\u2200 (x : \u2115),\n (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n) \u2227\n ((rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom \u2194\n (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom)\n[PROOFSTEP]\nsuffices\n[GOAL]\ncase intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis : ?m.104784\n\u22a2 (\u2200 (x : \u2115),\n (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n) \u2227\n ((rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom \u2194\n (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom)\ncase this\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\n\u22a2 ?m.104784\n[PROOFSTEP]\nrefine' \u27e8this, \u27e8fun h => (this _ \u27e8h, rfl\u27e9).imp Exists.fst Exists.fst, _\u27e9\u27e9\n[GOAL]\ncase intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n \u2200 (x : \u2115),\n (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n\u22a2 (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom \u2192\n (rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom\n[PROOFSTEP]\nintro h\n[GOAL]\ncase intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n \u2200 (x : \u2115),\n (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nh : (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom\n\u22a2 (rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom\n[PROOFSTEP]\nrw [Nat.rfindOpt_dom]\n[GOAL]\ncase intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n \u2200 (x : \u2115),\n (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nh : (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom\n\u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n\n[PROOFSTEP]\nsimp only [dom_iff_mem, Code.evaln_complete, Option.mem_def] at h \n[GOAL]\ncase intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n \u2200 (x : \u2115),\n (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nh : (\u2203 y k, Code.evaln k cf n = Option.some y) \u2228 \u2203 y k, Code.evaln k cg n = Option.some y\n\u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n\n[PROOFSTEP]\nobtain \u27e8x, k, e\u27e9 | \u27e8x, k, e\u27e9 := h\n[GOAL]\ncase intro.intro.inl.intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n \u2200 (x : \u2115),\n (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cf n = Option.some x\n\u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n\n[PROOFSTEP]\nrefine' \u27e8k, x, _\u27e9\n[GOAL]\ncase intro.intro.inl.intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n \u2200 (x : \u2115),\n (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cf n = Option.some x\n\u22a2 x \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\n[PROOFSTEP]\nsimp only [e, Option.some_orElse, Option.mem_def]\n[GOAL]\ncase intro.intro.inr.intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n \u2200 (x : \u2115),\n (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\n\u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n\n[PROOFSTEP]\nrefine' \u27e8k, _\u27e9\n[GOAL]\ncase intro.intro.inr.intro.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n \u2200 (x : \u2115),\n (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\n\u22a2 \u2203 a, a \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\n[PROOFSTEP]\ncases' cf.evaln k n with y\n[GOAL]\ncase intro.intro.inr.intro.intro.none\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n \u2200 (x : \u2115),\n (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\n\u22a2 \u2203 a, a \u2208 HOrElse.hOrElse Option.none fun x => Code.evaln k cg n\n[PROOFSTEP]\nexact \u27e8x, by simp only [e, Option.mem_def, Option.none_orElse]\u27e9\n[GOAL]\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n \u2200 (x : \u2115),\n (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\n\u22a2 x \u2208 HOrElse.hOrElse Option.none fun x => Code.evaln k cg n\n[PROOFSTEP]\nsimp only [e, Option.mem_def, Option.none_orElse]\n[GOAL]\ncase intro.intro.inr.intro.intro.some\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n \u2200 (x : \u2115),\n (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\ny : \u2115\n\u22a2 \u2203 a, a \u2208 HOrElse.hOrElse (Option.some y) fun x => Code.evaln k cg n\n[PROOFSTEP]\nexact \u27e8y, by simp only [Option.some_orElse, Option.mem_def]\u27e9\n[GOAL]\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\nthis :\n \u2200 (x : \u2115),\n (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\nx k : \u2115\ne : Code.evaln k cg n = Option.some x\ny : \u2115\n\u22a2 y \u2208 HOrElse.hOrElse (Option.some y) fun x => Code.evaln k cg n\n[PROOFSTEP]\nsimp only [Option.some_orElse, Option.mem_def]\n[GOAL]\ncase this\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn : \u2115\n\u22a2 \u2200 (x : \u2115),\n (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n[PROOFSTEP]\nintro x h\n[GOAL]\ncase this\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\n\u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n[PROOFSTEP]\nobtain \u27e8k, e\u27e9 := Nat.rfindOpt_spec h\n[GOAL]\ncase this.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\ne : x \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\n\u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n[PROOFSTEP]\nrevert e\n[GOAL]\ncase this.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\n\u22a2 (x \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n[PROOFSTEP]\nsimp only [Option.mem_def]\n[GOAL]\ncase this.intro\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\n\u22a2 (HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) = Option.some x \u2192\n x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n[PROOFSTEP]\ncases' e' : cf.evaln k n with y\n[GOAL]\ncase this.intro.none\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\ne' : Code.evaln k cf n = Option.none\n\u22a2 (HOrElse.hOrElse Option.none fun x => Code.evaln k cg n) = Option.some x \u2192 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase this.intro.some\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk y : \u2115\ne' : Code.evaln k cf n = Option.some y\n\u22a2 (HOrElse.hOrElse (Option.some y) fun x => Code.evaln k cg n) = Option.some x \u2192 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase this.intro.none\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\ne' : Code.evaln k cf n = Option.none\n\u22a2 Code.evaln k cg n = Option.some x \u2192 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n[PROOFSTEP]\nintro e\n[GOAL]\ncase this.intro.some\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk y : \u2115\ne' : Code.evaln k cf n = Option.some y\n\u22a2 y = x \u2192 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n[PROOFSTEP]\nintro e\n[GOAL]\ncase this.intro.none\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\ne' : Code.evaln k cf n = Option.none\ne : Code.evaln k cg n = Option.some x\n\u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n[PROOFSTEP]\nexact Or.inr (Code.evaln_sound e)\n[GOAL]\ncase this.intro.some\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk y : \u2115\ne' : Code.evaln k cf n = Option.some y\ne : y = x\n\u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n[PROOFSTEP]\nsubst y\n[GOAL]\ncase this.intro.some\ncf : Code\nhf : Partrec (Code.eval cf)\ncg : Code\nhg : Partrec (Code.eval cg)\nthis : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nn x : \u2115\nh : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n\nk : \u2115\ne' : Code.evaln k cf n = Option.some x\n\u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n\n[PROOFSTEP]\nexact Or.inl (Code.evaln_sound e')\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\n\u22a2 \u2203 k, Partrec k \u2227 \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\n[PROOFSTEP]\nlet \u27e8k, hk, H\u27e9 := Nat.Partrec.merge' (bind_decode\u2082_iff.1 hf) (bind_decode\u2082_iff.1 hg)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\n\u22a2 \u2203 k, Partrec k \u2227 \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\n[PROOFSTEP]\nlet k' (a : \u03b1) := (k (encode a)).bind fun n => (decode (\u03b1 := \u03c3) n : Part \u03c3)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\n\u22a2 \u2203 k, Partrec k \u2227 \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\n[PROOFSTEP]\nrefine' \u27e8k', ((nat_iff.2 hk).comp Computable.encode).bind (Computable.decode.ofOption.comp snd).to\u2082, fun a => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\n\u22a2 (\u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k' a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\n[PROOFSTEP]\nsuffices\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : ?m.108653\n\u22a2 (\u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k' a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\n\u22a2 ?m.108653\n[PROOFSTEP]\nrefine' \u27e8this, \u27e8fun h => (this _ \u27e8h, rfl\u27e9).imp Exists.fst Exists.fst, _\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\n\u22a2 (f a).Dom \u2228 (g a).Dom \u2192 (k' a).Dom\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\n\u22a2 (k' a).Dom\n[PROOFSTEP]\nrw [bind_dom]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\n\u22a2 \u2203 h, (\u2191(decode (Part.get (k (encode a)) h))).Dom\n[PROOFSTEP]\nhave hk : (k (encode a)).Dom := (H _).2.2 (by simpa only [encodek\u2082, bind_some, coe_some] using h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\n\u22a2 (Part.bind \u2191(decode\u2082 \u03b1 (encode a)) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 (encode a)) fun a => Part.map encode (g a)).Dom\n[PROOFSTEP]\nsimpa only [encodek\u2082, bind_some, coe_some] using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk\u271d : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\nhk : (k (encode a)).Dom\n\u22a2 \u2203 h, (\u2191(decode (Part.get (k (encode a)) h))).Dom\n[PROOFSTEP]\nexists hk\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk\u271d : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\nhk : (k (encode a)).Dom\n\u22a2 (\u2191(decode (Part.get (k (encode a)) hk))).Dom\n[PROOFSTEP]\nsimp only [exists_prop, mem_map_iff, mem_coe, mem_bind_iff, Option.mem_def] at H \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk\u271d : Nat.Partrec k\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\nhk : (k (encode a)).Dom\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (\u2203 a_2, decode\u2082 \u03b1 a = Option.some a_2 \u2227 \u2203 a, a \u2208 f a_2 \u2227 encode a = x) \u2228\n \u2203 a_2, decode\u2082 \u03b1 a = Option.some a_2 \u2227 \u2203 a, a \u2208 g a_2 \u2227 encode a = x) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\n\u22a2 (\u2191(decode (Part.get (k (encode a)) hk))).Dom\n[PROOFSTEP]\nobtain \u27e8a', _, y, _, e\u27e9 | \u27e8a', _, y, _, e\u27e9 := (H _).1 _ \u27e8hk, rfl\u27e9\n[GOAL]\ncase inl.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk\u271d : Nat.Partrec k\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\nhk : (k (encode a)).Dom\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (\u2203 a_2, decode\u2082 \u03b1 a = Option.some a_2 \u2227 \u2203 a, a \u2208 f a_2 \u2227 encode a = x) \u2228\n \u2203 a_2, decode\u2082 \u03b1 a = Option.some a_2 \u2227 \u2203 a, a \u2208 g a_2 \u2227 encode a = x) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\na' : \u03b1\nleft\u271d\u00b9 : decode\u2082 \u03b1 (encode a) = Option.some a'\ny : \u03c3\nleft\u271d : y \u2208 f a'\ne : encode y = Part.get (k (encode a)) hk\n\u22a2 (\u2191(decode (Part.get (k (encode a)) hk))).Dom\n[PROOFSTEP]\nsimp only [e.symm, encodek, coe_some, some_dom]\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk\u271d : Nat.Partrec k\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nthis : \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\nh : (f a).Dom \u2228 (g a).Dom\nhk : (k (encode a)).Dom\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (\u2203 a_2, decode\u2082 \u03b1 a = Option.some a_2 \u2227 \u2203 a, a \u2208 f a_2 \u2227 encode a = x) \u2228\n \u2203 a_2, decode\u2082 \u03b1 a = Option.some a_2 \u2227 \u2203 a, a \u2208 g a_2 \u2227 encode a = x) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\na' : \u03b1\nleft\u271d\u00b9 : decode\u2082 \u03b1 (encode a) = Option.some a'\ny : \u03c3\nleft\u271d : y \u2208 g a'\ne : encode y = Part.get (k (encode a)) hk\n\u22a2 (\u2191(decode (Part.get (k (encode a)) hk))).Dom\n[PROOFSTEP]\nsimp only [e.symm, encodek, coe_some, some_dom]\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\n\u22a2 \u2200 (x : \u03c3), x \u2208 k' a \u2192 x \u2208 f a \u2228 x \u2208 g a\n[PROOFSTEP]\nintro x h'\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx : \u03c3\nh' : x \u2208 k' a\n\u22a2 x \u2208 f a \u2228 x \u2208 g a\n[PROOFSTEP]\nsimp only [exists_prop, mem_coe, mem_bind_iff, Option.mem_def] at h' \n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx : \u03c3\nh' : \u2203 a_1, a_1 \u2208 k (encode a) \u2227 decode a_1 = Option.some x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a\n[PROOFSTEP]\nobtain \u27e8n, hn, hx\u27e9 := h'\n[GOAL]\ncase this.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx : \u03c3\nn : \u2115\nhn : n \u2208 k (encode a)\nhx : decode n = Option.some x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a\n[PROOFSTEP]\nhave := (H _).1 _ hn\n[GOAL]\ncase this.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx : \u03c3\nn : \u2115\nhn : n \u2208 k (encode a)\nhx : decode n = Option.some x\nthis :\n (n \u2208 Part.bind \u2191(decode\u2082 \u03b1 (encode a)) fun a => Part.map encode (f a)) \u2228\n n \u2208 Part.bind \u2191(decode\u2082 \u03b1 (encode a)) fun a => Part.map encode (g a)\n\u22a2 x \u2208 f a \u2228 x \u2208 g a\n[PROOFSTEP]\nsimp [mem_decode\u2082, encode_injective.eq_iff] at this \n[GOAL]\ncase this.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx : \u03c3\nn : \u2115\nhn : n \u2208 k (encode a)\nhx : decode n = Option.some x\nthis : (\u2203 a_1, a_1 \u2208 f a \u2227 encode a_1 = n) \u2228 \u2203 a_1, a_1 \u2208 g a \u2227 encode a_1 = n\n\u22a2 x \u2208 f a \u2228 x \u2208 g a\n[PROOFSTEP]\nobtain \u27e8a', ha, rfl\u27e9 | \u27e8a', ha, rfl\u27e9 := this\n[GOAL]\ncase this.intro.intro.inl.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx a' : \u03c3\nha : a' \u2208 f a\nhn : encode a' \u2208 k (encode a)\nhx : decode (encode a') = Option.some x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a\n[PROOFSTEP]\nsimp only [encodek, Option.some_inj] at hx \n[GOAL]\ncase this.intro.intro.inr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx a' : \u03c3\nha : a' \u2208 g a\nhn : encode a' \u2208 k (encode a)\nhx : decode (encode a') = Option.some x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a\n[PROOFSTEP]\nsimp only [encodek, Option.some_inj] at hx \n[GOAL]\ncase this.intro.intro.inl.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx a' : \u03c3\nha : a' \u2208 f a\nhn : encode a' \u2208 k (encode a)\nhx : a' = x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a\n[PROOFSTEP]\nrw [hx] at ha \n[GOAL]\ncase this.intro.intro.inr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx a' : \u03c3\nha : a' \u2208 g a\nhn : encode a' \u2208 k (encode a)\nhx : a' = x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a\n[PROOFSTEP]\nrw [hx] at ha \n[GOAL]\ncase this.intro.intro.inl.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx a' : \u03c3\nha : x \u2208 f a\nhn : encode a' \u2208 k (encode a)\nhx : a' = x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a\n[PROOFSTEP]\nexact Or.inl ha\n[GOAL]\ncase this.intro.intro.inr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nk : \u2115 \u2192. \u2115\nhk : Nat.Partrec k\nH :\n \u2200 (a : \u2115),\n (\u2200 (x : \u2115),\n x \u2208 k a \u2192\n (x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)) \u2228\n x \u2208 Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)) \u2227\n ((k a).Dom \u2194\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (f a)).Dom \u2228\n (Part.bind \u2191(decode\u2082 \u03b1 a) fun a => Part.map encode (g a)).Dom)\nk' : \u03b1 \u2192 Part \u03c3 := fun a => Part.bind (k (encode a)) fun n => \u2191(decode n)\na : \u03b1\nx a' : \u03c3\nha : x \u2208 g a\nhn : encode a' \u2208 k (encode a)\nhx : a' = x\n\u22a2 x \u2208 f a \u2228 x \u2208 g a\n[PROOFSTEP]\nexact Or.inr ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nh : x \u2208 f a \u2228 x \u2208 g a\n\u22a2 x \u2208 k a\n[PROOFSTEP]\nhave : (k a).Dom := (K _).2.2 (h.imp Exists.fst Exists.fst)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nh : x \u2208 f a \u2228 x \u2208 g a\nthis : (k a).Dom\n\u22a2 x \u2208 k a\n[PROOFSTEP]\nrefine' \u27e8this, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nh : x \u2208 f a \u2228 x \u2208 g a\nthis : (k a).Dom\n\u22a2 Part.get (k a) this = x\n[PROOFSTEP]\ncases' h with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nthis : (k a).Dom\nh : x \u2208 f a\n\u22a2 Part.get (k a) this = x\n[PROOFSTEP]\ncases' (K _).1 _ \u27e8this, rfl\u27e9 with h' h'\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nthis : (k a).Dom\nh : x \u2208 g a\n\u22a2 Part.get (k a) this = x\n[PROOFSTEP]\ncases' (K _).1 _ \u27e8this, rfl\u27e9 with h' h'\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nthis : (k a).Dom\nh : x \u2208 f a\nh' : Part.get (k a) this \u2208 f a\n\u22a2 Part.get (k a) this = x\n[PROOFSTEP]\nexact mem_unique h' h\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nthis : (k a).Dom\nh : x \u2208 f a\nh' : Part.get (k a) this \u2208 g a\n\u22a2 Part.get (k a) this = x\n[PROOFSTEP]\nexact (H _ _ h _ h').symm\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nthis : (k a).Dom\nh : x \u2208 g a\nh' : Part.get (k a) this \u2208 f a\n\u22a2 Part.get (k a) this = x\n[PROOFSTEP]\nexact H _ _ h' _ h\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf g : \u03b1 \u2192. \u03c3\nhf : Partrec f\nhg : Partrec g\nH : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y\nk : \u03b1 \u2192. \u03c3\nhk : Partrec k\nK : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom)\na : \u03b1\nx : \u03c3\nthis : (k a).Dom\nh : x \u2208 g a\nh' : Part.get (k a) this \u2208 g a\n\u22a2 Part.get (k a) this = x\n[PROOFSTEP]\nexact mem_unique h' h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Bool\nf g : \u03b1 \u2192. \u03c3\nhc : Computable c\nhf : Partrec f\nhg : Partrec g\ncf : Code\nef : eval cf = fun n => Part.bind \u2191(decode n) fun a => Part.map encode (f a)\ncg : Code\neg : eval cg = fun n => Part.bind \u2191(decode n) fun a => Part.map encode (g a)\na : \u03b1\n\u22a2 (Part.bind (eval (bif c a then cf else cg) (encode a)) fun b => \u2191(decode (a, b).snd)) = bif c a then f a else g a\n[PROOFSTEP]\ncases c a\n[GOAL]\ncase false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Bool\nf g : \u03b1 \u2192. \u03c3\nhc : Computable c\nhf : Partrec f\nhg : Partrec g\ncf : Code\nef : eval cf = fun n => Part.bind \u2191(decode n) fun a => Part.map encode (f a)\ncg : Code\neg : eval cg = fun n => Part.bind \u2191(decode n) fun a => Part.map encode (g a)\na : \u03b1\n\u22a2 (Part.bind (eval (bif false then cf else cg) (encode a)) fun b => \u2191(decode (a, b).snd)) = bif false then f a else g a\n[PROOFSTEP]\nsimp [ef, eg, encodek]\n[GOAL]\ncase true\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Bool\nf g : \u03b1 \u2192. \u03c3\nhc : Computable c\nhf : Partrec f\nhg : Partrec g\ncf : Code\nef : eval cf = fun n => Part.bind \u2191(decode n) fun a => Part.map encode (f a)\ncg : Code\neg : eval cg = fun n => Part.bind \u2191(decode n) fun a => Part.map encode (g a)\na : \u03b1\n\u22a2 (Part.bind (eval (bif true then cf else cg) (encode a)) fun b => \u2191(decode (a, b).snd)) = bif true then f a else g a\n[PROOFSTEP]\nsimp [ef, eg, encodek]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\ng : \u03b1 \u2192 \u03b2 \u2192. \u03c3\nh : \u03b1 \u2192 \u03b3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec\u2082 g\nhh : Partrec\u2082 h\na : \u03b1\n\u22a2 (bif Sum.casesOn (f a) (fun b => true) fun b => false then\n Sum.casesOn (f a) (fun b => Part.map Option.some (g (a, b).fst (a, b).snd)) fun c => Part.some Option.none\n else Sum.casesOn (f a) (fun b => Part.some Option.none) fun b => Part.map Option.some (h (a, b).fst (a, b).snd)) =\n Part.map Option.some (Sum.casesOn (f a) (g a) (h a))\n[PROOFSTEP]\ncases f a\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\ng : \u03b1 \u2192 \u03b2 \u2192. \u03c3\nh : \u03b1 \u2192 \u03b3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec\u2082 g\nhh : Partrec\u2082 h\na : \u03b1\nval\u271d : \u03b2\n\u22a2 (bif Sum.casesOn (Sum.inl val\u271d) (fun b => true) fun b => false then\n Sum.casesOn (Sum.inl val\u271d) (fun b => Part.map Option.some (g (a, b).fst (a, b).snd)) fun c =>\n Part.some Option.none\n else\n Sum.casesOn (Sum.inl val\u271d) (fun b => Part.some Option.none) fun b =>\n Part.map Option.some (h (a, b).fst (a, b).snd)) =\n Part.map Option.some (Sum.casesOn (Sum.inl val\u271d) (g a) (h a))\n[PROOFSTEP]\nsimp only [Bool.cond_true, Bool.cond_false]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\ng : \u03b1 \u2192 \u03b2 \u2192. \u03c3\nh : \u03b1 \u2192 \u03b3 \u2192. \u03c3\nhf : Computable f\nhg : Partrec\u2082 g\nhh : Partrec\u2082 h\na : \u03b1\nval\u271d : \u03b3\n\u22a2 (bif Sum.casesOn (Sum.inr val\u271d) (fun b => true) fun b => false then\n Sum.casesOn (Sum.inr val\u271d) (fun b => Part.map Option.some (g (a, b).fst (a, b).snd)) fun c =>\n Part.some Option.none\n else\n Sum.casesOn (Sum.inr val\u271d) (fun b => Part.some Option.none) fun b =>\n Part.map Option.some (h (a, b).fst (a, b).snd)) =\n Part.map Option.some (Sum.casesOn (Sum.inr val\u271d) (g a) (h a))\n[PROOFSTEP]\nsimp only [Bool.cond_true, Bool.cond_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03b2\nf : \u03b1 \u2192. \u03b2\nh : Partrec f\nn : \u03b1\nx\u271d : Unit\n\u22a2 x\u271d \u2208 Part.map (fun b => ()) (f n) \u2194 x\u271d \u2208 Part.assert ((fun a => (f a).Dom) n) fun x => Part.some ()\n[PROOFSTEP]\nsimp [Part.dom_iff_mem]\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 Prop\n\u22a2 (\u2203 f, Computable f \u2227 p = fun a => f a = true) \u2192 ComputablePred p\n[PROOFSTEP]\nrintro \u27e8f, h, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nh : Computable f\n\u22a2 ComputablePred fun a => f a = true\n[PROOFSTEP]\nexact \u27e8by infer_instance, by simpa using h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nh : Computable f\n\u22a2 DecidablePred fun a => f a = true\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nh : Computable f\n\u22a2 Computable fun a => decide ((fun a => f a = true) a)\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 Prop\nhp : ComputablePred p\n\u22a2 ComputablePred fun a => \u00acp a\n[PROOFSTEP]\nobtain \u27e8f, hf, rfl\u27e9 := computable_iff.1 hp\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\n\u22a2 ComputablePred fun a => \u00ac(fun a => f a = true) a\n[PROOFSTEP]\nexact\n \u27e8by infer_instance,\n (cond hf (const false) (const true)).of_eq fun n => by\n simp\n cases f n <;> rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\n\u22a2 DecidablePred fun a => \u00ac(fun a => f a = true) a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\nn : \u03b1\n\u22a2 (bif f n then false else true) = decide ((fun a => \u00ac(fun a => f a = true) a) n)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\nn : \u03b1\n\u22a2 (bif f n then false else true) = decide (f n = false)\n[PROOFSTEP]\ncases f n\n[GOAL]\ncase false\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\nn : \u03b1\n\u22a2 (bif false then false else true) = decide (false = false)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\nn : \u03b1\n\u22a2 (bif true then false else true) = decide (true = false)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 Prop\nhp : ComputablePred p\n\u22a2 RePred p\n[PROOFSTEP]\nobtain \u27e8f, hf, rfl\u27e9 := computable_iff.1 hp\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\n\u22a2 RePred fun a => f a = true\n[PROOFSTEP]\nunfold RePred\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\n\u22a2 Partrec fun a => Part.assert ((fun a => f a = true) a) fun x => Part.some ()\n[PROOFSTEP]\ndsimp only []\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\n\u22a2 Partrec fun a => Part.assert (f a = true) fun x => Part.some ()\n[PROOFSTEP]\nrefine' (Partrec.cond hf (Decidable.Partrec.const' (Part.some ())) Partrec.none).of_eq fun n => Part.ext fun a => _\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\nn : \u03b1\na : Unit\n\u22a2 (a \u2208 bif f n then Part.some () else Part.none) \u2194 a \u2208 Part.assert (f n = true) fun x => Part.some ()\n[PROOFSTEP]\ncases a\n[GOAL]\ncase intro.intro.unit\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\nn : \u03b1\n\u22a2 (PUnit.unit \u2208 bif f n then Part.some () else Part.none) \u2194 PUnit.unit \u2208 Part.assert (f n = true) fun x => Part.some ()\n[PROOFSTEP]\ncases f n\n[GOAL]\ncase intro.intro.unit.false\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\nn : \u03b1\n\u22a2 (PUnit.unit \u2208 bif false then Part.some () else Part.none) \u2194\n PUnit.unit \u2208 Part.assert (false = true) fun x => Part.some ()\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.unit.true\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 Bool\nhf : Computable f\nhp : ComputablePred fun a => f a = true\nn : \u03b1\n\u22a2 (PUnit.unit \u2208 bif true then Part.some () else Part.none) \u2194\n PUnit.unit \u2208 Part.assert (true = true) fun x => Part.some ()\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nh : ComputablePred fun c => eval c \u2208 C\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\n\u22a2 g \u2208 C\n[PROOFSTEP]\ncases' h with _ h\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\n\u22a2 g \u2208 C\n[PROOFSTEP]\nskip\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\n\u22a2 g \u2208 C\n[PROOFSTEP]\nobtain \u27e8c, e\u27e9 :=\n fixed_point\u2082\n (Partrec.cond (h.comp fst) ((Partrec.nat_iff.2 hg).comp snd).to\u2082 ((Partrec.nat_iff.2 hf).comp snd).to\u2082).to\u2082\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne :\n eval c = fun b =>\n bif decide ((fun c => eval c \u2208 C) (c, b).fst) then (fun a b => g (a, b).snd) (c, b).fst (c, b).snd\n else (fun a b => f (a, b).snd) (c, b).fst (c, b).snd\n\u22a2 g \u2208 C\n[PROOFSTEP]\nsimp at e \n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne : eval c = fun b => if eval c \u2208 C then g b else f b\n\u22a2 g \u2208 C\n[PROOFSTEP]\nby_cases H : eval c \u2208 C\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne : eval c = fun b => if eval c \u2208 C then g b else f b\nH : eval c \u2208 C\n\u22a2 g \u2208 C\n[PROOFSTEP]\nsimp only [H, if_true] at e \n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : eval c \u2208 C\ne : eval c = fun b => g b\n\u22a2 g \u2208 C\n[PROOFSTEP]\nchange (fun b => g b) \u2208 C\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : eval c \u2208 C\ne : eval c = fun b => g b\n\u22a2 (fun b => g b) \u2208 C\n[PROOFSTEP]\nrwa [\u2190 e]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\ne : eval c = fun b => if eval c \u2208 C then g b else f b\nH : \u00aceval c \u2208 C\n\u22a2 g \u2208 C\n[PROOFSTEP]\nsimp only [H, if_false] at e \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : \u00aceval c \u2208 C\ne : eval c = fun b => f b\n\u22a2 g \u2208 C\n[PROOFSTEP]\nrw [e] at H \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set (\u2115 \u2192. \u2115)\nf g : \u2115 \u2192. \u2115\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nfC : f \u2208 C\nw\u271d : DecidablePred fun c => eval c \u2208 C\nh : Computable fun a => decide ((fun c => eval c \u2208 C) a)\nc : Code\nH : \u00ac(fun b => f b) \u2208 C\ne : eval c = fun b => f b\n\u22a2 g \u2208 C\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set Code\nH : \u2200 (cf cg : Code), eval cf = eval cg \u2192 (cf \u2208 C \u2194 cg \u2208 C)\n\u22a2 (ComputablePred fun c => c \u2208 C) \u2194 C = \u2205 \u2228 C = Set.univ\n[PROOFSTEP]\nclassical exact\n have hC : \u2200 f, f \u2208 C \u2194 eval f \u2208 eval '' C := fun f => \u27e8Set.mem_image_of_mem _, fun \u27e8g, hg, e\u27e9 => (H _ _ e).1 hg\u27e9\n \u27e8fun h =>\n or_iff_not_imp_left.2 fun C0 =>\n Set.eq_univ_of_forall fun cg =>\n let \u27e8cf, fC\u27e9 := Set.nonempty_iff_ne_empty.2 C0\n (hC _).2 <|\n rice (eval '' C) (h.of_eq hC) (Partrec.nat_iff.1 <| eval_part.comp (const cf) Computable.id)\n (Partrec.nat_iff.1 <| eval_part.comp (const cg) Computable.id) ((hC _).1 fC),\n fun h => by\n {obtain rfl | rfl := h <;> simp [ComputablePred, Set.mem_empty_iff_false] <;>\n exact \u27e8by infer_instance, Computable.const _\u27e9\n }\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set Code\nH : \u2200 (cf cg : Code), eval cf = eval cg \u2192 (cf \u2208 C \u2194 cg \u2208 C)\n\u22a2 (ComputablePred fun c => c \u2208 C) \u2194 C = \u2205 \u2228 C = Set.univ\n[PROOFSTEP]\nexact\n have hC : \u2200 f, f \u2208 C \u2194 eval f \u2208 eval '' C := fun f => \u27e8Set.mem_image_of_mem _, fun \u27e8g, hg, e\u27e9 => (H _ _ e).1 hg\u27e9\n \u27e8fun h =>\n or_iff_not_imp_left.2 fun C0 =>\n Set.eq_univ_of_forall fun cg =>\n let \u27e8cf, fC\u27e9 := Set.nonempty_iff_ne_empty.2 C0\n (hC _).2 <|\n rice (eval '' C) (h.of_eq hC) (Partrec.nat_iff.1 <| eval_part.comp (const cf) Computable.id)\n (Partrec.nat_iff.1 <| eval_part.comp (const cg) Computable.id) ((hC _).1 fC),\n fun h => by\n {obtain rfl | rfl := h <;> simp [ComputablePred, Set.mem_empty_iff_false] <;>\n exact \u27e8by infer_instance, Computable.const _\u27e9\n }\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set Code\nH : \u2200 (cf cg : Code), eval cf = eval cg \u2192 (cf \u2208 C \u2194 cg \u2208 C)\nhC : \u2200 (f : Code), f \u2208 C \u2194 eval f \u2208 eval '' C\nh : C = \u2205 \u2228 C = Set.univ\n\u22a2 ComputablePred fun c => c \u2208 C\n[PROOFSTEP]\n{obtain rfl | rfl := h <;> simp [ComputablePred, Set.mem_empty_iff_false] <;>\n exact \u27e8by infer_instance, Computable.const _\u27e9\n}\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nC : Set Code\nH : \u2200 (cf cg : Code), eval cf = eval cg \u2192 (cf \u2208 C \u2194 cg \u2208 C)\nhC : \u2200 (f : Code), f \u2208 C \u2194 eval f \u2208 eval '' C\nh : C = \u2205 \u2228 C = Set.univ\n\u22a2 ComputablePred fun c => c \u2208 C\n[PROOFSTEP]\nobtain rfl | rfl := h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nH : \u2200 (cf cg : Code), eval cf = eval cg \u2192 (cf \u2208 \u2205 \u2194 cg \u2208 \u2205)\nhC : \u2200 (f : Code), f \u2208 \u2205 \u2194 eval f \u2208 eval '' \u2205\n\u22a2 ComputablePred fun c => c \u2208 \u2205\n[PROOFSTEP]\nsimp [ComputablePred, Set.mem_empty_iff_false]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nH : \u2200 (cf cg : Code), eval cf = eval cg \u2192 (cf \u2208 Set.univ \u2194 cg \u2208 Set.univ)\nhC : \u2200 (f : Code), f \u2208 Set.univ \u2194 eval f \u2208 eval '' Set.univ\n\u22a2 ComputablePred fun c => c \u2208 Set.univ\n[PROOFSTEP]\nsimp [ComputablePred, Set.mem_empty_iff_false]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nH : \u2200 (cf cg : Code), eval cf = eval cg \u2192 (cf \u2208 \u2205 \u2194 cg \u2208 \u2205)\nhC : \u2200 (f : Code), f \u2208 \u2205 \u2194 eval f \u2208 eval '' \u2205\n\u22a2 \u2203 x, Computable fun a => false\n[PROOFSTEP]\nexact \u27e8by infer_instance, Computable.const _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nH : \u2200 (cf cg : Code), eval cf = eval cg \u2192 (cf \u2208 \u2205 \u2194 cg \u2208 \u2205)\nhC : \u2200 (f : Code), f \u2208 \u2205 \u2194 eval f \u2208 eval '' \u2205\n\u22a2 DecidablePred fun c => c \u2208 \u2205\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nH : \u2200 (cf cg : Code), eval cf = eval cg \u2192 (cf \u2208 Set.univ \u2194 cg \u2208 Set.univ)\nhC : \u2200 (f : Code), f \u2208 Set.univ \u2194 eval f \u2208 eval '' Set.univ\n\u22a2 \u2203 x, Computable fun a => true\n[PROOFSTEP]\nexact \u27e8by infer_instance, Computable.const _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nH : \u2200 (cf cg : Code), eval cf = eval cg \u2192 (cf \u2208 Set.univ \u2194 cg \u2208 Set.univ)\nhC : \u2200 (f : Code), f \u2208 Set.univ \u2194 eval f \u2208 eval '' Set.univ\n\u22a2 DecidablePred fun c => c \u2208 Set.univ\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\n\u22a2 Computable fun a => decide (p a)\n[PROOFSTEP]\nobtain \u27e8k, pk, hk\u27e9 :=\n Partrec.merge (h\u2081.map (Computable.const true).to\u2082) (h\u2082.map (Computable.const false).to\u2082)\n (by\n intro a x hx y hy\n simp at hx hy \n cases hy.1 hx.1)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\n\u22a2 \u2200 (a : \u03b1) (x : Bool),\n x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2192\n \u2200 (y : Bool), y \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ()) \u2192 x = y\n[PROOFSTEP]\nintro a x hx y hy\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\na : \u03b1\nx : Bool\nhx : x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ())\ny : Bool\nhy : y \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ())\n\u22a2 x = y\n[PROOFSTEP]\nsimp at hx hy \n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\na : \u03b1\nx y : Bool\nhx : p a \u2227 true = x\nhy : \u00acp a \u2227 false = y\n\u22a2 x = y\n[PROOFSTEP]\ncases hy.1 hx.1\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\nk : \u03b1 \u2192. Bool\npk : Partrec k\nhk :\n \u2200 (a : \u03b1) (x : Bool),\n x \u2208 k a \u2194\n x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228\n x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ())\n\u22a2 Computable fun a => decide (p a)\n[PROOFSTEP]\nrefine' Partrec.of_eq pk fun n => Part.eq_some_iff.2 _\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\nk : \u03b1 \u2192. Bool\npk : Partrec k\nhk :\n \u2200 (a : \u03b1) (x : Bool),\n x \u2208 k a \u2194\n x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228\n x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ())\nn : \u03b1\n\u22a2 (fun a => decide (p a)) n \u2208 k n\n[PROOFSTEP]\nrw [hk]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\nk : \u03b1 \u2192. Bool\npk : Partrec k\nhk :\n \u2200 (a : \u03b1) (x : Bool),\n x \u2208 k a \u2194\n x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228\n x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ())\nn : \u03b1\n\u22a2 (fun a => decide (p a)) n \u2208 Part.map (fun b => true) (Part.assert (p n) fun x => Part.some ()) \u2228\n (fun a => decide (p a)) n \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) n) fun x => Part.some ())\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx\u271d : RePred p \u2227 RePred fun a => \u00acp a\nh\u2081 : RePred p\nh\u2082 : RePred fun a => \u00acp a\nk : \u03b1 \u2192. Bool\npk : Partrec k\nhk :\n \u2200 (a : \u03b1) (x : Bool),\n x \u2208 k a \u2194\n x \u2208 Part.map (fun b => true) (Part.assert (p a) fun x => Part.some ()) \u2228\n x \u2208 Part.map (fun b => false) (Part.assert ((fun a => \u00acp a) a) fun x => Part.some ())\nn : \u03b1\n\u22a2 p n \u2228 \u00acp n\n[PROOFSTEP]\napply Decidable.em\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 Prop\n\u22a2 ComputablePred p \u2194 RePred p \u2227 RePred fun a => \u00acp a\n[PROOFSTEP]\nclassical exact computable_iff_re_compl_re\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 Prop\n\u22a2 ComputablePred p \u2194 RePred p \u2227 RePred fun a => \u00acp a\n[PROOFSTEP]\nexact computable_iff_re_compl_re\n[GOAL]\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\npf : Partrec' f\n\u22a2 _root_.Partrec f\n[PROOFSTEP]\ninduction pf\n[GOAL]\ncase prim\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192 \u2115\na\u271d : Primrec' f\u271d\n\u22a2 _root_.Partrec \u2191f\u271d\ncase comp\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115\na\u271d\u00b9 : Partrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Partrec' (g\u271d i)\na_ih\u271d\u00b9 : _root_.Partrec f\u271d\na_ih\u271d : \u2200 (i : Fin n\u271d), _root_.Partrec (g\u271d i)\n\u22a2 _root_.Partrec fun v => (mOfFn fun i => g\u271d i v) >>= f\u271d\ncase rfind\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Partrec' \u2191f\u271d\na_ih\u271d : _root_.Partrec \u2191f\u271d\n\u22a2 _root_.Partrec fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))\n[PROOFSTEP]\ncase prim n f hf => exact hf.to_prim.to_comp\n[GOAL]\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nhf : Primrec' f\n\u22a2 _root_.Partrec \u2191f\n[PROOFSTEP]\ncase prim n f hf => exact hf.to_prim.to_comp\n[GOAL]\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 n \u2192 \u2115\nhf : Primrec' f\n\u22a2 _root_.Partrec \u2191f\n[PROOFSTEP]\nexact hf.to_prim.to_comp\n[GOAL]\ncase comp\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nm\u271d n\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\ng\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115\na\u271d\u00b9 : Partrec' f\u271d\na\u271d : \u2200 (i : Fin n\u271d), Partrec' (g\u271d i)\na_ih\u271d\u00b9 : _root_.Partrec f\u271d\na_ih\u271d : \u2200 (i : Fin n\u271d), _root_.Partrec (g\u271d i)\n\u22a2 _root_.Partrec fun v => (mOfFn fun i => g\u271d i v) >>= f\u271d\ncase rfind\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Partrec' \u2191f\u271d\na_ih\u271d : _root_.Partrec \u2191f\u271d\n\u22a2 _root_.Partrec fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))\n[PROOFSTEP]\ncase comp m n f g _ _ hf hg => exact (Partrec.vector_mOfFn fun i => hg i).bind (hf.comp snd)\n[GOAL]\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nm n : \u2115\nf : Vector \u2115 n \u2192. \u2115\ng : Fin n \u2192 Vector \u2115 m \u2192. \u2115\na\u271d\u00b9 : Partrec' f\na\u271d : \u2200 (i : Fin n), Partrec' (g i)\nhf : _root_.Partrec f\nhg : \u2200 (i : Fin n), _root_.Partrec (g i)\n\u22a2 _root_.Partrec fun v => (mOfFn fun i => g i v) >>= f\n[PROOFSTEP]\ncase comp m n f g _ _ hf hg => exact (Partrec.vector_mOfFn fun i => hg i).bind (hf.comp snd)\n[GOAL]\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nm n : \u2115\nf : Vector \u2115 n \u2192. \u2115\ng : Fin n \u2192 Vector \u2115 m \u2192. \u2115\na\u271d\u00b9 : Partrec' f\na\u271d : \u2200 (i : Fin n), Partrec' (g i)\nhf : _root_.Partrec f\nhg : \u2200 (i : Fin n), _root_.Partrec (g i)\n\u22a2 _root_.Partrec fun v => (mOfFn fun i => g i v) >>= f\n[PROOFSTEP]\nexact (Partrec.vector_mOfFn fun i => hg i).bind (hf.comp snd)\n[GOAL]\ncase rfind\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nn\u271d : \u2115\nf\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115\na\u271d : Partrec' \u2191f\u271d\na_ih\u271d : _root_.Partrec \u2191f\u271d\n\u22a2 _root_.Partrec fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))\n[PROOFSTEP]\ncase rfind n f _ hf =>\n have := hf.comp (vector_cons.comp snd fst)\n have := ((Primrec.eq.comp _root_.Primrec.id (_root_.Primrec.const 0)).to_comp.comp this).to\u2082.partrec\u2082\n exact _root_.Partrec.rfind this\n[GOAL]\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Partrec' \u2191f\nhf : _root_.Partrec \u2191f\n\u22a2 _root_.Partrec fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))\n[PROOFSTEP]\ncase rfind n f _ hf =>\n have := hf.comp (vector_cons.comp snd fst)\n have := ((Primrec.eq.comp _root_.Primrec.id (_root_.Primrec.const 0)).to_comp.comp this).to\u2082.partrec\u2082\n exact _root_.Partrec.rfind this\n[GOAL]\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Partrec' \u2191f\nhf : _root_.Partrec \u2191f\n\u22a2 _root_.Partrec fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))\n[PROOFSTEP]\nhave := hf.comp (vector_cons.comp snd fst)\n[GOAL]\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Partrec' \u2191f\nhf : _root_.Partrec \u2191f\nthis : _root_.Partrec fun a => \u2191f (a.snd ::\u1d65 a.fst)\n\u22a2 _root_.Partrec fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))\n[PROOFSTEP]\nhave := ((Primrec.eq.comp _root_.Primrec.id (_root_.Primrec.const 0)).to_comp.comp this).to\u2082.partrec\u2082\n[GOAL]\nn\u271d : \u2115\nf\u271d : Vector \u2115 n\u271d \u2192. \u2115\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\na\u271d : Partrec' \u2191f\nhf : _root_.Partrec \u2191f\nthis\u271d : _root_.Partrec fun a => \u2191f (a.snd ::\u1d65 a.fst)\nthis : Partrec\u2082 fun a => \u2191fun b => decide ((fun a => id a = 0) (f ((a, b).snd ::\u1d65 (a, b).fst)))\n\u22a2 _root_.Partrec fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))\n[PROOFSTEP]\nexact _root_.Partrec.rfind this\n[GOAL]\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : Partrec' f\nv : Vector \u2115 (succ n)\n\u22a2 (mOfFn fun i => (\u2191fun v => Vector.get v (Fin.succ i)) v) >>= f = f (Vector.tail v)\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : Partrec' f\nv : Vector \u2115 (succ n)\n\u22a2 f (ofFn fun i => Vector.get v (Fin.succ i)) = f (Vector.tail v)\n[PROOFSTEP]\nrw [\u2190 ofFn_get v.tail]\n[GOAL]\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : Partrec' f\nv : Vector \u2115 (succ n)\n\u22a2 f (ofFn fun i => Vector.get v (Fin.succ i)) = f (ofFn (Vector.get (Vector.tail v)))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_a\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : Partrec' f\nv : Vector \u2115 (succ n)\n\u22a2 (fun i => Vector.get v (Fin.succ i)) = Vector.get (Vector.tail v)\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase e_a.e_a.h\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : Partrec' f\nv : Vector \u2115 (succ n)\ni : Fin n\n\u22a2 Vector.get v (Fin.succ i) = Vector.get (Vector.tail v) i\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\ng : Vector \u2115 (n + 1) \u2192. \u2115\nhf : Partrec' f\nhg : Partrec' g\ni : Fin (n + 1)\n\u22a2 Partrec' ((fun i => Fin.cases f (fun i v => \u2191(some (Vector.get v i))) i) i)\n[PROOFSTEP]\nrefine' Fin.cases _ (fun i => _) i\n[GOAL]\ncase refine'_1\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\ng : Vector \u2115 (n + 1) \u2192. \u2115\nhf : Partrec' f\nhg : Partrec' g\ni : Fin (n + 1)\n\u22a2 Partrec' ((fun i => Fin.cases f (fun i v => \u2191(some (Vector.get v i))) i) 0)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase refine'_2\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\ng : Vector \u2115 (n + 1) \u2192. \u2115\nhf : Partrec' f\nhg : Partrec' g\ni\u271d : Fin (n + 1)\ni : Fin n\n\u22a2 Partrec' ((fun i => Fin.cases f (fun i v => \u2191(some (Vector.get v i))) i) (Fin.succ i))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase refine'_2\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\ng : Vector \u2115 (n + 1) \u2192. \u2115\nhf : Partrec' f\nhg : Partrec' g\ni\u271d : Fin (n + 1)\ni : Fin n\n\u22a2 Partrec' fun v => Part.some (Vector.get v i)\n[PROOFSTEP]\nexact prim (Nat.Primrec'.get _)\n[GOAL]\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\ng : Vector \u2115 (n + 1) \u2192. \u2115\nhf : Partrec' f\nhg : Partrec' g\nv : Vector \u2115 n\n\u22a2 (mOfFn fun i => Fin.cases f (fun i v => \u2191(some (Vector.get v i))) i v) >>= g = Part.bind (f v) fun a => g (a ::\u1d65 v)\n[PROOFSTEP]\nsimp [mOfFn, Part.bind_assoc, pure]\n[GOAL]\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\ng : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' f\nhg : Partrec' \u2191g\n\u22a2 Partrec' fun v => Part.map (fun a => g (a ::\u1d65 v)) (f v)\n[PROOFSTEP]\nsimp [(Part.bind_some_eq_map _ _).symm]\n[GOAL]\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\ng : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' f\nhg : Partrec' \u2191g\n\u22a2 Partrec' fun v => Part.bind (f v) (Part.some \u2218 fun a => g (a ::\u1d65 v))\n[PROOFSTEP]\nexact hf.bind hg\n[GOAL]\nn m : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 n \u2192 Vector \u2115 m\nhf : Partrec' \u2191f\nhg : Vec g\ni : Fin (succ m)\n\u22a2 Partrec' fun v => \u2191(some (Vector.get ((fun v => f v ::\u1d65 g v) v) 0))\n[PROOFSTEP]\nsimpa using hf\n[GOAL]\nn m : \u2115\nf : Vector \u2115 n \u2192 \u2115\ng : Vector \u2115 n \u2192 Vector \u2115 m\nhf : Partrec' \u2191f\nhg : Vec g\ni\u271d : Fin (succ m)\ni : Fin m\n\u22a2 Partrec' fun v => \u2191(some (Vector.get ((fun v => f v ::\u1d65 g v) v) (Fin.succ i)))\n[PROOFSTEP]\nsimp only [hg i, get_cons_succ]\n[GOAL]\nn m : \u2115\nf : Vector \u2115 m \u2192. \u2115\ng : Vector \u2115 n \u2192 Vector \u2115 m\nhf : Partrec' f\nhg : Vec g\nv : Vector \u2115 n\n\u22a2 (mOfFn fun i => \u2191(some (Vector.get (g v) i))) >>= f = f (g v)\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\nf : \u2115 \u2192. \u2115\ng : Vector \u2115 n \u2192 \u2115\nhf : Partrec' fun v => f (Vector.head v)\nhg : Partrec' \u2191g\n\u22a2 Partrec' fun v => f (g v)\n[PROOFSTEP]\nsimpa using hf.comp' (Partrec'.cons hg Partrec'.nil)\n[GOAL]\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb : \u2115\n\u22a2 (b \u2208 Part.bind (Nat.rfind fun n_1 => Part.some (decide (1 - f (n_1 ::\u1d65 v) = 0))) fun a => \u2191pred (f (a ::\u1d65 v))) \u2194\n b \u2208 Nat.rfindOpt fun a => ofNat (Option \u2115) (f (a ::\u1d65 v))\n[PROOFSTEP]\nsimp only [Nat.rfindOpt, exists_prop, tsub_eq_zero_iff_le, PFun.coe_val, Part.mem_bind_iff, Part.mem_some_iff,\n Option.mem_def, Part.mem_coe]\n[GOAL]\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb : \u2115\n\u22a2 (\u2203 a, (a \u2208 Nat.rfind fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) \u2227 b = pred (f (a ::\u1d65 v))) \u2194\n \u2203 a,\n (a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))) \u2227\n ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b\n[PROOFSTEP]\nrefine' exists_congr fun a => (and_congr (iff_of_eq _) Iff.rfl).trans (and_congr_right fun h => _)\n[GOAL]\ncase refine'_1\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\n\u22a2 (a \u2208 Nat.rfind fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) =\n (a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase refine'_1.e_a.e_p\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\n\u22a2 (fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) = fun n_1 =>\n \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\n[PROOFSTEP]\nfunext n\n[GOAL]\ncase refine'_1.e_a.e_p.h\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\u271d\nb a n : \u2115\n\u22a2 Part.some (decide (1 \u2264 f (n ::\u1d65 v))) = \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n ::\u1d65 v)))))\n[PROOFSTEP]\ncases f (n ::\u1d65 v)\n[GOAL]\ncase refine'_1.e_a.e_p.h.zero\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\u271d\nb a n : \u2115\n\u22a2 Part.some (decide (1 \u2264 zero)) = \u2191(some (Option.isSome (ofNat (Option \u2115) zero)))\n[PROOFSTEP]\nsimp [Nat.succ_le_succ]\n[GOAL]\ncase refine'_1.e_a.e_p.h.succ\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 (n\u271d\u00b9 + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\u271d\u00b9\nb a n n\u271d : \u2115\n\u22a2 Part.some (decide (1 \u2264 succ n\u271d)) = \u2191(some (Option.isSome (ofNat (Option \u2115) (succ n\u271d))))\n[PROOFSTEP]\nsimp [Nat.succ_le_succ]\n[GOAL]\ncase refine'_1.e_a.e_p.h.succ\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 (n\u271d\u00b9 + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\u271d\u00b9\nb a n n\u271d : \u2115\n\u22a2 true = Option.isSome (ofNat (Option \u2115) (succ n\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\n\u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b\n[PROOFSTEP]\nhave := Nat.rfind_spec h\n[GOAL]\ncase refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nthis : true \u2208 \u2191(some (Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v)))))\n\u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b\n[PROOFSTEP]\nsimp only [Part.coe_some, Part.mem_some_iff] at this \n[GOAL]\ncase refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nthis : true = Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v)))\n\u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b\n[PROOFSTEP]\nrevert this\n[GOAL]\ncase refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\n\u22a2 true = Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v))) \u2192\n (b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b)\n[PROOFSTEP]\ncases' f (a ::\u1d65 v) with c\n[GOAL]\ncase refine'_2.zero\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\n\u22a2 true = Option.isSome (ofNat (Option \u2115) zero) \u2192 (b = pred zero \u2194 ofNat (Option \u2115) zero = some b)\n[PROOFSTEP]\nintro this\n[GOAL]\ncase refine'_2.succ\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nc : \u2115\n\u22a2 true = Option.isSome (ofNat (Option \u2115) (succ c)) \u2192 (b = pred (succ c) \u2194 ofNat (Option \u2115) (succ c) = some b)\n[PROOFSTEP]\nintro this\n[GOAL]\ncase refine'_2.zero\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nthis : true = Option.isSome (ofNat (Option \u2115) zero)\n\u22a2 b = pred zero \u2194 ofNat (Option \u2115) zero = some b\n[PROOFSTEP]\ncases this\n[GOAL]\ncase refine'_2.succ\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nc : \u2115\nthis : true = Option.isSome (ofNat (Option \u2115) (succ c))\n\u22a2 b = pred (succ c) \u2194 ofNat (Option \u2115) (succ c) = some b\n[PROOFSTEP]\nrw [\u2190 Option.some_inj, eq_comm]\n[GOAL]\ncase refine'_2.succ\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nc : \u2115\nthis : true = Option.isSome (ofNat (Option \u2115) (succ c))\n\u22a2 some (pred (succ c)) = some b \u2194 ofNat (Option \u2115) (succ c) = some b\n[PROOFSTEP]\nrfl\n[GOAL]\nf : \u2115 \u2192. \u2115\nhf : Partrec f\n\u22a2 Partrec' fun v => f (Vector.head v)\n[PROOFSTEP]\nobtain \u27e8c, rfl\u27e9 := exists_code.1 hf\n[GOAL]\ncase intro\nc : Code\nhf : Partrec (eval c)\n\u22a2 Partrec' fun v => eval c (Vector.head v)\n[PROOFSTEP]\nsimpa [eval_eq_rfindOpt] using\n rfindOpt <|\n of_prim <|\n Primrec.encode_iff.2 <|\n evaln_prim.comp <|\n (Primrec.vector_head.pair (_root_.Primrec.const c)).pair <| Primrec.vector_head.comp Primrec.vector_tail\n[GOAL]\nthis : \u2200 (f : \u2115 \u2192. \u2115), Partrec f \u2192 Partrec' fun v => f (Vector.head v)\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : _root_.Partrec f\n\u22a2 Partrec' f\n[PROOFSTEP]\nlet g := fun n\u2081 => (Part.ofOption (decode (\u03b1 := Vector \u2115 n) n\u2081)).bind (fun a => Part.map encode (f a))\n[GOAL]\nthis : \u2200 (f : \u2115 \u2192. \u2115), Partrec f \u2192 Partrec' fun v => f (Vector.head v)\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : _root_.Partrec f\ng : \u2115 \u2192 Part \u2115 := fun n\u2081 => Part.bind \u2191(decode n\u2081) fun a => Part.map encode (f a)\n\u22a2 Partrec' f\n[PROOFSTEP]\nexact (comp\u2081 g (this g hf) (prim Nat.Primrec'.encode)).of_eq fun i => by dsimp only; simp [encodek, Part.map_id']\n[GOAL]\nthis : \u2200 (f : \u2115 \u2192. \u2115), Partrec f \u2192 Partrec' fun v => f (Vector.head v)\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : _root_.Partrec f\ng : \u2115 \u2192 Part \u2115 := fun n\u2081 => Part.bind \u2191(decode n\u2081) fun a => Part.map encode (f a)\ni : Vector \u2115 n\n\u22a2 g (encode i) = f i\n[PROOFSTEP]\ndsimp only\n[GOAL]\nthis : \u2200 (f : \u2115 \u2192. \u2115), Partrec f \u2192 Partrec' fun v => f (Vector.head v)\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : _root_.Partrec f\ng : \u2115 \u2192 Part \u2115 := fun n\u2081 => Part.bind \u2191(decode n\u2081) fun a => Part.map encode (f a)\ni : Vector \u2115 n\n\u22a2 (Part.bind \u2191(decode (encode i)) fun a => Part.map encode (f a)) = f i\n[PROOFSTEP]\nsimp [encodek, Part.map_id']\n[GOAL]\nf : \u2115 \u2192. \u2115\nh : _root_.Partrec fun v => f (Vector.head v)\nv : \u2115\n\u22a2 f (Vector.head (ofFn fun i => id v)) = f v\n[PROOFSTEP]\nsimp only [id.def, head_ofFn]\n[GOAL]\nf : \u2115 \u2192 \u2115 \u2192. \u2115\nh : _root_.Partrec fun v => f (Vector.head v) (Vector.head (Vector.tail v))\nv : \u2115 \u00d7 \u2115\n\u22a2 f (Vector.head (v.fst ::\u1d65 v.snd ::\u1d65 nil)) (Vector.head (Vector.tail (v.fst ::\u1d65 v.snd ::\u1d65 nil))) = f v.fst v.snd\n[PROOFSTEP]\nsimp only [head_cons, tail_cons]\n[GOAL]\nm n : \u2115\nf : Vector \u2115 m \u2192 Vector \u2115 n\nh : Vec f\n\u22a2 Computable f\n[PROOFSTEP]\nsimpa only [ofFn_get] using vector_ofFn fun i => to_part (h i)\n", "meta": {"mathlib_filename": "Mathlib.Computability.Halting", "llama_tokens": 38070, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.4089413334831328}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Mul \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\na b c d : \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\nh : a * b \u2264 c * d\n\u22a2 min a b \u2264 max c d\n[PROOFSTEP]\nsimp_rw [min_le_iff, le_max_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Mul \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\na b c d : \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\nh : a * b \u2264 c * d\n\u22a2 (a \u2264 c \u2228 a \u2264 d) \u2228 b \u2264 c \u2228 b \u2264 d\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Mul \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\na b c d : \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\nh : (c < a \u2227 d < a) \u2227 c < b \u2227 d < b\n\u22a2 c * d < a * b\n[PROOFSTEP]\nexact mul_lt_mul_of_lt_of_lt h.1.1 h.2.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nh : a \u2264 a * b\n\u22a2 ?m.15077 h * 1 \u2264 ?m.15077 h * b\n[PROOFSTEP]\nsimpa only [mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nh : a * b \u2264 a\n\u22a2 ?m.15584 h * b \u2264 ?m.15584 h * 1\n[PROOFSTEP]\nsimpa only [mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nh : b \u2264 a * b\n\u22a2 1 * ?m.16128 h \u2264 a * ?m.16128 h\n[PROOFSTEP]\nsimpa only [one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nh : a * b \u2264 b\n\u22a2 a * ?m.16669 h \u2264 1 * ?m.16669 h\n[PROOFSTEP]\nsimpa only [one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a \u2264 a * b \u2194 a * 1 \u2264 a * b\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a \u2264 b * a \u2194 1 * a \u2264 b * a\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a * b \u2264 a \u2194 a * b \u2264 a * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a * b \u2264 b \u2194 a * b \u2264 1 * b\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : \u03b1\nh : a < a * b\n\u22a2 ?m.21985 h * 1 < ?m.21985 h * b\n[PROOFSTEP]\nsimpa only [mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : \u03b1\nh : a * b < a\n\u22a2 ?m.22488 h * b < ?m.22488 h * 1\n[PROOFSTEP]\nsimpa only [mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : \u03b1\nh : b < a * b\n\u22a2 1 * ?m.23024 h < a * ?m.23024 h\n[PROOFSTEP]\nsimpa only [one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : \u03b1\nh : a * b < b\n\u22a2 a * ?m.23557 h < 1 * ?m.23557 h\n[PROOFSTEP]\nsimpa only [one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : LT \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : \u03b1\n\u22a2 a < a * b \u2194 a * 1 < a * b\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : LT \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : \u03b1\n\u22a2 a < b * a \u2194 1 * a < b * a\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : LT \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : \u03b1\n\u22a2 a * b < a \u2194 a * b < a * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : LT \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : \u03b1\n\u22a2 a * b < b \u2194 a * b < 1 * b\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nha : 1 \u2264 a\nhb : 1 \u2264 b\n\u22a2 a = 1 \u2227 b = 1 \u2192 a * b = 1\n[PROOFSTEP]\nrintro \u27e8rfl, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nha hb : 1 \u2264 1\n\u22a2 1 * 1 = 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MulOneClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\nha : a\u2081 \u2264 a\u2082\nhb : b\u2081 \u2264 b\u2082\n\u22a2 a\u2082 * b\u2082 \u2264 a\u2081 * b\u2081 \u2194 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082\n[PROOFSTEP]\nrefine' \u27e8fun h \u21a6 _, by rintro \u27e8rfl, rfl\u27e9; rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MulOneClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\nha : a\u2081 \u2264 a\u2082\nhb : b\u2081 \u2264 b\u2082\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082 \u2192 a\u2082 * b\u2082 \u2264 a\u2081 * b\u2081\n[PROOFSTEP]\nrintro \u27e8rfl, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MulOneClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 b\u2081 : \u03b1\nha : a\u2081 \u2264 a\u2081\nhb : b\u2081 \u2264 b\u2081\n\u22a2 a\u2081 * b\u2081 \u2264 a\u2081 * b\u2081\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MulOneClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\nha : a\u2081 \u2264 a\u2082\nhb : b\u2081 \u2264 b\u2082\nh : a\u2082 * b\u2082 \u2264 a\u2081 * b\u2081\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082\n[PROOFSTEP]\nsimp only [eq_iff_le_not_lt, ha, hb, true_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MulOneClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\nha : a\u2081 \u2264 a\u2082\nhb : b\u2081 \u2264 b\u2082\nh : a\u2082 * b\u2082 \u2264 a\u2081 * b\u2081\n\u22a2 \u00aca\u2081 < a\u2082 \u2227 \u00acb\u2081 < b\u2082\n[PROOFSTEP]\nrefine' \u27e8fun ha \u21a6 h.not_lt _, fun hb \u21a6 h.not_lt _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MulOneClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\nha\u271d : a\u2081 \u2264 a\u2082\nhb : b\u2081 \u2264 b\u2082\nh : a\u2082 * b\u2082 \u2264 a\u2081 * b\u2081\nha : a\u2081 < a\u2082\n\u22a2 a\u2081 * b\u2081 < a\u2082 * b\u2082\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MulOneClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\nha : a\u2081 \u2264 a\u2082\nhb\u271d : b\u2081 \u2264 b\u2082\nh : a\u2082 * b\u2082 \u2264 a\u2081 * b\u2081\nhb : b\u2081 < b\u2082\n\u22a2 a\u2081 * b\u2081 < a\u2082 * b\u2082\n[PROOFSTEP]\n{exact mul_lt_mul_of_lt_of_le ha hb\n}\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MulOneClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\nha\u271d : a\u2081 \u2264 a\u2082\nhb : b\u2081 \u2264 b\u2082\nh : a\u2082 * b\u2082 \u2264 a\u2081 * b\u2081\nha : a\u2081 < a\u2082\n\u22a2 a\u2081 * b\u2081 < a\u2082 * b\u2082\n[PROOFSTEP]\nexact mul_lt_mul_of_lt_of_le ha hb\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MulOneClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\nha : a\u2081 \u2264 a\u2082\nhb\u271d : b\u2081 \u2264 b\u2082\nh : a\u2082 * b\u2082 \u2264 a\u2081 * b\u2081\nhb : b\u2081 < b\u2082\n\u22a2 a\u2081 * b\u2081 < a\u2082 * b\u2082\n[PROOFSTEP]\n{exact mul_lt_mul_of_le_of_lt ha hb\n}\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : MulOneClass \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\nha : a\u2081 \u2264 a\u2082\nhb\u271d : b\u2081 \u2264 b\u2082\nh : a\u2082 * b\u2082 \u2264 a\u2081 * b\u2081\nhb : b\u2081 < b\u2082\n\u22a2 a\u2081 * b\u2081 < a\u2082 * b\u2082\n[PROOFSTEP]\nexact mul_lt_mul_of_le_of_lt ha hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : \u03b1\n\u22a2 \u2203 b, b * b \u2264 a\n[PROOFSTEP]\nby_cases h : a < 1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : \u03b1\nh : a < 1\n\u22a2 \u2203 b, b * b \u2264 a\n[PROOFSTEP]\nuse a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : \u03b1\nh : a < 1\n\u22a2 a * a \u2264 a\n[PROOFSTEP]\nhave : a * a < a * 1 := mul_lt_mul_left' h a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : \u03b1\nh : a < 1\nthis : a * a < a * 1\n\u22a2 a * a \u2264 a\n[PROOFSTEP]\nrw [mul_one] at this \n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : \u03b1\nh : a < 1\nthis : a * a < a\n\u22a2 a * a \u2264 a\n[PROOFSTEP]\nexact le_of_lt this\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : \u03b1\nh : \u00aca < 1\n\u22a2 \u2203 b, b * b \u2264 a\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : \u03b1\nh : \u00aca < 1\n\u22a2 1 * 1 \u2264 a\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : \u03b1\nh : 1 \u2264 a\n\u22a2 1 * 1 \u2264 a\n[PROOFSTEP]\nrwa [mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\nhac : a \u2264 c\nhbd : b \u2264 d\n\u22a2 a * b = c * d \u2194 a = c \u2227 b = d\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => congr_arg\u2082 (\u00b7 * \u00b7) h.1 h.2\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\nhac : a \u2264 c\nhbd : b \u2264 d\nh : a * b = c * d\n\u22a2 a = c \u2227 b = d\n[PROOFSTEP]\nrcases hac.eq_or_lt with (rfl | hac)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b d : \u03b1\nhbd : b \u2264 d\nhac : a \u2264 a\nh : a * b = a * d\n\u22a2 a = a \u2227 b = d\n[PROOFSTEP]\nexact \u27e8rfl, mul_left_cancel'' h\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\nhac\u271d : a \u2264 c\nhbd : b \u2264 d\nh : a * b = c * d\nhac : a < c\n\u22a2 a = c \u2227 b = d\n[PROOFSTEP]\nrcases eq_or_lt_of_le hbd with (rfl | hbd)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nhac\u271d : a \u2264 c\nhac : a < c\nhbd : b \u2264 b\nh : a * b = c * b\n\u22a2 a = c \u2227 b = b\n[PROOFSTEP]\nexact \u27e8mul_right_cancel'' h, rfl\u27e9\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\nhac\u271d : a \u2264 c\nhbd\u271d : b \u2264 d\nh : a * b = c * d\nhac : a < c\nhbd : b < d\n\u22a2 a = c \u2227 b = d\n[PROOFSTEP]\nexact ((Left.mul_lt_mul hac hbd).ne h).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\nhac : a \u2264 c\nhbd : b \u2264 d\n\u22a2 a * b = c * d \u2194 a = c \u2227 b = d\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => congr_arg\u2082 (\u00b7 * \u00b7) h.1 h.2\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\nhac : a \u2264 c\nhbd : b \u2264 d\nh : a * b = c * d\n\u22a2 a = c \u2227 b = d\n[PROOFSTEP]\nrcases hac.eq_or_lt with (rfl | hac)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b d : \u03b1\nhbd : b \u2264 d\nhac : a \u2264 a\nh : a * b = a * d\n\u22a2 a = a \u2227 b = d\n[PROOFSTEP]\nexact \u27e8rfl, mul_left_cancel'' h\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\nhac\u271d : a \u2264 c\nhbd : b \u2264 d\nh : a * b = c * d\nhac : a < c\n\u22a2 a = c \u2227 b = d\n[PROOFSTEP]\nrcases eq_or_lt_of_le hbd with (rfl | hbd)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nhac\u271d : a \u2264 c\nhac : a < c\nhbd : b \u2264 b\nh : a * b = c * b\n\u22a2 a = c \u2227 b = b\n[PROOFSTEP]\nexact \u27e8mul_right_cancel'' h, rfl\u27e9\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\nhac\u271d : a \u2264 c\nhbd\u271d : b \u2264 d\nh : a * b = c * d\nhac : a < c\nhbd : b < d\n\u22a2 a = c \u2227 b = d\n[PROOFSTEP]\nexact ((Right.mul_lt_mul hac hbd).ne h).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : LE \u03b1\na b : \u03b1\n\u22a2 1 * a \u2264 1 * b \u2192 a \u2264 b\n[PROOFSTEP]\nsimpa only [one_mul] using id\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : CommSemigroup \u03b1\ninst\u271d : PartialOrder \u03b1\na : \u03b1\nha : MulLECancellable a\nb c : \u03b1\nh : (fun x => x * a) b = (fun x => x * a) c\n\u22a2 (fun x x_1 => x * x_1) a b = (fun x x_1 => x * x_1) a c\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : CommSemigroup \u03b1\ninst\u271d : PartialOrder \u03b1\na : \u03b1\nha : MulLECancellable a\nb c : \u03b1\nh : (fun x => x * a) b = (fun x => x * a) c\n\u22a2 a * b = a * c\n[PROOFSTEP]\nrwa [mul_comm a, mul_comm a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : CommSemigroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nha : MulLECancellable a\n\u22a2 b * a \u2264 c * a \u2194 b \u2264 c\n[PROOFSTEP]\nrw [mul_comm b, mul_comm c, ha.mul_le_mul_iff_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : MulOneClass \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nha : MulLECancellable a\n\u22a2 a \u2264 a * b \u2194 a * 1 \u2264 a * b\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : MulOneClass \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nha : MulLECancellable a\n\u22a2 a * b \u2264 a \u2194 a * b \u2264 a * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nha : MulLECancellable a\n\u22a2 a \u2264 b * a \u2194 1 \u2264 b\n[PROOFSTEP]\nrw [mul_comm, ha.le_mul_iff_one_le_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nha : MulLECancellable a\n\u22a2 b * a \u2264 a \u2194 b \u2264 1\n[PROOFSTEP]\nrw [mul_comm, ha.mul_le_iff_le_one_right]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Monoid.Lemmas", "llama_tokens": 11166, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581741774411, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.4087324355855454}} {"text": "[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\ninst\u271d : DecidableEq (Away f)\nz : Away f\nhz : z \u2208 span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\n\u22a2 \u2203 c N acd,\n f ^ N \u2022 z =\n \u2191(algebraMap A (Away f))\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))\n[PROOFSTEP]\nrw [\u2190 submodule_span_eq, Finsupp.span_eq_range_total, LinearMap.mem_range] at hz \n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\ninst\u271d : DecidableEq (Away f)\nz : Away f\nhz : \u2203 y, \u2191(Finsupp.total (\u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)) (Away f) (Away f) Subtype.val) y = z\n\u22a2 \u2203 c N acd,\n f ^ N \u2022 z =\n \u2191(algebraMap A (Away f))\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))\n[PROOFSTEP]\nrcases hz with \u27e8c, eq1\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\ninst\u271d : DecidableEq (Away f)\nz : Away f\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\neq1 : \u2191(Finsupp.total (\u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)) (Away f) (Away f) Subtype.val) c = z\n\u22a2 \u2203 c N acd,\n f ^ N \u2022 z =\n \u2191(algebraMap A (Away f))\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))\n[PROOFSTEP]\nrw [Finsupp.total_apply, Finsupp.sum] at eq1 \n[GOAL]\ncase intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\ninst\u271d : DecidableEq (Away f)\nz : Away f\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\neq1 : \u2211 a in c.support, \u2191c a \u2022 \u2191a = z\n\u22a2 \u2203 c N acd,\n f ^ N \u2022 z =\n \u2191(algebraMap A (Away f))\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))\n[PROOFSTEP]\nobtain \u27e8\u27e8_, N, rfl\u27e9, hN\u27e9 := IsLocalization.exist_integer_multiples_of_finset (Submonoid.powers f) (c.support.image c)\n[GOAL]\ncase intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\ninst\u271d : DecidableEq (Away f)\nz : Away f\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\neq1 : \u2211 a in c.support, \u2191c a \u2022 \u2191a = z\nN : \u2115\nhN :\n \u2200 (a : Away f),\n a \u2208 image (\u2191c) c.support \u2192\n IsLocalization.IsInteger A\n (\u2191{ val := (fun x x_1 => x ^ x_1) f N,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f N) } \u2022\n a)\n\u22a2 \u2203 c N acd,\n f ^ N \u2022 z =\n \u2191(algebraMap A (Away f))\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))\n[PROOFSTEP]\nchoose acd hacd using hN\n[GOAL]\ncase intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\ninst\u271d : DecidableEq (Away f)\nz : Away f\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\neq1 : \u2211 a in c.support, \u2191c a \u2022 \u2191a = z\nN : \u2115\nacd : (a : Away f) \u2192 a \u2208 image (\u2191c) c.support \u2192 A\nhacd :\n \u2200 (a : Away f) (a_1 : a \u2208 image (\u2191c) c.support),\n \u2191(algebraMap A (Away f)) (acd a a_1) =\n \u2191{ val := (fun x x_1 => x ^ x_1) f N,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f N) } \u2022\n a\n\u22a2 \u2203 c N acd,\n f ^ N \u2022 z =\n \u2191(algebraMap A (Away f))\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))\n[PROOFSTEP]\nrefine' \u27e8c, N, acd, _\u27e9\n[GOAL]\ncase intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\ninst\u271d : DecidableEq (Away f)\nz : Away f\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\neq1 : \u2211 a in c.support, \u2191c a \u2022 \u2191a = z\nN : \u2115\nacd : (a : Away f) \u2192 a \u2208 image (\u2191c) c.support \u2192 A\nhacd :\n \u2200 (a : Away f) (a_1 : a \u2208 image (\u2191c) c.support),\n \u2191(algebraMap A (Away f)) (acd a a_1) =\n \u2191{ val := (fun x x_1 => x ^ x_1) f N,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f N) } \u2022\n a\n\u22a2 f ^ N \u2022 z =\n \u2191(algebraMap A (Away f))\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))\n[PROOFSTEP]\nrw [\u2190 eq1, smul_sum, map_sum, \u2190 sum_attach]\n[GOAL]\ncase intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\ninst\u271d : DecidableEq (Away f)\nz : Away f\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\neq1 : \u2211 a in c.support, \u2191c a \u2022 \u2191a = z\nN : \u2115\nacd : (a : Away f) \u2192 a \u2208 image (\u2191c) c.support \u2192 A\nhacd :\n \u2200 (a : Away f) (a_1 : a \u2208 image (\u2191c) c.support),\n \u2191(algebraMap A (Away f)) (acd a a_1) =\n \u2191{ val := (fun x x_1 => x ^ x_1) f N,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f N) } \u2022\n a\n\u22a2 \u2211 x_1 in attach c.support, f ^ N \u2022 \u2191c \u2191x_1 \u2022 \u2191\u2191x_1 =\n \u2211 x_1 in attach c.support,\n \u2191(algebraMap A (Away f))\n (acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase intro.intro.mk.intro.e_f\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\ninst\u271d : DecidableEq (Away f)\nz : Away f\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\neq1 : \u2211 a in c.support, \u2191c a \u2022 \u2191a = z\nN : \u2115\nacd : (a : Away f) \u2192 a \u2208 image (\u2191c) c.support \u2192 A\nhacd :\n \u2200 (a : Away f) (a_1 : a \u2208 image (\u2191c) c.support),\n \u2191(algebraMap A (Away f)) (acd a a_1) =\n \u2191{ val := (fun x x_1 => x ^ x_1) f N,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f N) } \u2022\n a\n\u22a2 (fun x_1 => f ^ N \u2022 \u2191c \u2191x_1 \u2022 \u2191\u2191x_1) = fun x_1 =>\n \u2191(algebraMap A (Away f))\n (acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))\n[PROOFSTEP]\next i\n[GOAL]\ncase intro.intro.mk.intro.e_f.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d\u00b9 : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\ninst\u271d : DecidableEq (Away f)\nz : Away f\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\neq1 : \u2211 a in c.support, \u2191c a \u2022 \u2191a = z\nN : \u2115\nacd : (a : Away f) \u2192 a \u2208 image (\u2191c) c.support \u2192 A\nhacd :\n \u2200 (a : Away f) (a_1 : a \u2208 image (\u2191c) c.support),\n \u2191(algebraMap A (Away f)) (acd a a_1) =\n \u2191{ val := (fun x x_1 => x ^ x_1) f N,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f N) } \u2022\n a\ni : { x_1 // x_1 \u2208 c.support }\n\u22a2 f ^ N \u2022 \u2191c \u2191i \u2022 \u2191\u2191i =\n \u2191(algebraMap A (Away f))\n (acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))\n[PROOFSTEP]\nrw [_root_.map_mul, hacd, (Classical.choose_spec i.1.2).2, smul_eq_mul, smul_mul_assoc]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\n\u22a2 Disjoint \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2191(Submonoid.powers f)\n[PROOFSTEP]\nby_contra rid\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nrid : \u00acDisjoint \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2191(Submonoid.powers f)\n\u22a2 False\n[PROOFSTEP]\nrw [Set.not_disjoint_iff] at rid \n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nrid : \u2203 x_1, x_1 \u2208 \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2227 x_1 \u2208 \u2191(Submonoid.powers f)\n\u22a2 False\n[PROOFSTEP]\nchoose g hg using rid\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\ng : A\nhg : g \u2208 \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2227 g \u2208 \u2191(Submonoid.powers f)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8hg1, \u27e8k, rfl\u27e9\u27e9 := hg\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nk : \u2115\nhg1 : (fun x x_1 => x ^ x_1) f k \u2208 \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal)\n\u22a2 False\n[PROOFSTEP]\nby_cases k_ineq : 0 < k\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nk : \u2115\nhg1 : (fun x x_1 => x ^ x_1) f k \u2208 \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal)\nk_ineq : 0 < k\n\u22a2 False\n[PROOFSTEP]\nerw [x.1.isPrime.pow_mem_iff_mem _ k_ineq] at hg1 \n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nk : \u2115\nhg1 : f \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nk_ineq : 0 < k\n\u22a2 False\n[PROOFSTEP]\nexact x.2 hg1\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nk : \u2115\nhg1 : (fun x x_1 => x ^ x_1) f k \u2208 \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal)\nk_ineq : \u00ac0 < k\n\u22a2 False\n[PROOFSTEP]\ndsimp at hg1 \n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nk : \u2115\nhg1 : f ^ k \u2208 \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal)\nk_ineq : \u00ac0 < k\n\u22a2 False\n[PROOFSTEP]\nerw [show k = 0 by linarith, pow_zero, \u2190 Ideal.eq_top_iff_one] at hg1 \n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nk : \u2115\nhg1 : f ^ k \u2208 \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal)\nk_ineq : \u00ac0 < k\n\u22a2 k = 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nk : \u2115\nhg1 : HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal = \u22a4\nk_ineq : \u00ac0 < k\n\u22a2 False\n[PROOFSTEP]\napply x.1.isPrime.1\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nk : \u2115\nhg1 : HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal = \u22a4\nk_ineq : \u00ac0 < k\n\u22a2 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal = \u22a4\n[PROOFSTEP]\nexact hg1\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\n\u22a2 carrier x \u2260 \u22a4\n[PROOFSTEP]\nhave eq_top := disjoint x\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : Disjoint \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2191(Submonoid.powers f)\n\u22a2 carrier x \u2260 \u22a4\n[PROOFSTEP]\nclassical\ncontrapose! eq_top\nobtain \u27e8c, N, acd, eq1\u27e9 := MemCarrier.clear_denominator x ((Ideal.eq_top_iff_one _).mp eq_top)\nrw [Algebra.smul_def, HomogeneousLocalization.one_val, mul_one] at eq1 \nchange Localization.mk (f ^ N) 1 = Localization.mk _ 1 at eq1 \nsimp only [mk_eq_mk', IsLocalization.eq] at eq1 \nrcases eq1 with \u27e8\u27e8_, \u27e8M, rfl\u27e9\u27e9, eq1\u27e9\ndsimp at eq1 \nerw [one_mul, one_mul] at eq1 \nchange f ^ _ * f ^ _ = f ^ _ * _ at eq1 \nrw [Set.not_disjoint_iff_nonempty_inter]\nrefine'\n \u27e8f ^ M * f ^ N, eq1.symm \u25b8 mul_mem_left _ _ (sum_mem _ fun i _ => mul_mem_left _ _ _),\n \u27e8M + N, by dsimp; rw [pow_add]\u27e9\u27e9\ngeneralize_proofs h\u2081 h\u2082\nexact (Classical.choose_spec h\u2082).1\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : Disjoint \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2191(Submonoid.powers f)\n\u22a2 carrier x \u2260 \u22a4\n[PROOFSTEP]\ncontrapose! eq_top\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : carrier x = \u22a4\n\u22a2 \u00acDisjoint \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2191(Submonoid.powers f)\n[PROOFSTEP]\nobtain \u27e8c, N, acd, eq1\u27e9 := MemCarrier.clear_denominator x ((Ideal.eq_top_iff_one _).mp eq_top)\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : carrier x = \u22a4\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\neq1 :\n f ^ N \u2022 HomogeneousLocalization.val 1 =\n \u2191(algebraMap A (Away f))\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))\n\u22a2 \u00acDisjoint \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2191(Submonoid.powers f)\n[PROOFSTEP]\nrw [Algebra.smul_def, HomogeneousLocalization.one_val, mul_one] at eq1 \n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : carrier x = \u22a4\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\neq1 :\n \u2191(algebraMap A (Localization (Submonoid.powers f))) (f ^ N) =\n \u2191(algebraMap A (Away f))\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))\n\u22a2 \u00acDisjoint \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2191(Submonoid.powers f)\n[PROOFSTEP]\nchange Localization.mk (f ^ N) 1 = Localization.mk _ 1 at eq1 \n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : carrier x = \u22a4\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\neq1 :\n Localization.mk (f ^ N) 1 =\n Localization.mk\n (\u2191(algebraMap A A)\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\n 1\n\u22a2 \u00acDisjoint \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2191(Submonoid.powers f)\n[PROOFSTEP]\nsimp only [mk_eq_mk', IsLocalization.eq] at eq1 \n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : carrier x = \u22a4\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\neq1 :\n \u2203 c_1,\n \u2191c_1 * (\u21911 * f ^ N) =\n \u2191c_1 *\n (\u21911 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\n\u22a2 \u00acDisjoint \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2191(Submonoid.powers f)\n[PROOFSTEP]\nrcases eq1 with \u27e8\u27e8_, \u27e8M, rfl\u27e9\u27e9, eq1\u27e9\n[GOAL]\ncase intro.intro.intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : carrier x = \u22a4\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n \u2191{ val := (fun x x_1 => x ^ x_1) f M,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f M) } *\n (\u21911 * f ^ N) =\n \u2191{ val := (fun x x_1 => x ^ x_1) f M,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f M) } *\n (\u21911 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\n\u22a2 \u00acDisjoint \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2191(Submonoid.powers f)\n[PROOFSTEP]\ndsimp at eq1 \n[GOAL]\ncase intro.intro.intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : carrier x = \u22a4\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (1 * f ^ N) =\n f ^ M *\n (1 *\n \u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))\n\u22a2 \u00acDisjoint \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2191(Submonoid.powers f)\n[PROOFSTEP]\nerw [one_mul, one_mul] at eq1 \n[GOAL]\ncase intro.intro.intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : carrier x = \u22a4\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * f ^ N =\n f ^ M *\n \u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\n\u22a2 \u00acDisjoint \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2191(Submonoid.powers f)\n[PROOFSTEP]\nchange f ^ _ * f ^ _ = f ^ _ * _ at eq1 \n[GOAL]\ncase intro.intro.intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : carrier x = \u22a4\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * f ^ N =\n f ^ M *\n \u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\n\u22a2 \u00acDisjoint \u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2191(Submonoid.powers f)\n[PROOFSTEP]\nrw [Set.not_disjoint_iff_nonempty_inter]\n[GOAL]\ncase intro.intro.intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : carrier x = \u22a4\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * f ^ N =\n f ^ M *\n \u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\n\u22a2 Set.Nonempty (\u2191(HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal) \u2229 \u2191(Submonoid.powers f))\n[PROOFSTEP]\nrefine'\n \u27e8f ^ M * f ^ N, eq1.symm \u25b8 mul_mem_left _ _ (sum_mem _ fun i _ => mul_mem_left _ _ _),\n \u27e8M + N, by dsimp; rw [pow_add]\u27e9\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : carrier x = \u22a4\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * f ^ N =\n f ^ M *\n \u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\n\u22a2 (fun x x_1 => x ^ x_1) f (M + N) = f ^ M * f ^ N\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : carrier x = \u22a4\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * f ^ N =\n f ^ M *\n \u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\n\u22a2 f ^ (M + N) = f ^ M * f ^ N\n[PROOFSTEP]\nrw [pow_add]\n[GOAL]\ncase intro.intro.intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : carrier x = \u22a4\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * f ^ N =\n f ^ M *\n \u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\ni : { x_1 // x_1 \u2208 c.support }\nx\u271d : i \u2208 attach c.support\n\u22a2 Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2208\n HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n[PROOFSTEP]\ngeneralize_proofs h\u2081 h\u2082\n[GOAL]\ncase intro.intro.intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\neq_top : carrier x = \u22a4\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * f ^ N =\n f ^ M *\n \u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\ni : { x_1 // x_1 \u2208 c.support }\nx\u271d : i \u2208 attach c.support\nh\u2081 : AddSubmonoidClass (Submodule R A) A\nh\u2082 : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal\n\u22a2 Exists.choose h\u2082 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n[PROOFSTEP]\nexact (Classical.choose_spec h\u2082).1\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nx1 x2 : \u2191(CommRingCat.of (A\u2070_ f))\nhx12 : x1 * x2 \u2208 carrier x\n\u22a2 x1 \u2208 carrier x \u2228 x2 \u2208 carrier x\n[PROOFSTEP]\nclassical\nrw [mem_carrier_iff] at hx12 \nlet J := span ((algebraMap A (Away f) : A \u2192 (Away f)) '' x.val.asHomogeneousIdeal)\nsuffices h : \u2200 x y : Localization.Away f, x * y \u2208 J \u2192 x \u2208 J \u2228 y \u2208 J\n\u00b7 rw [HomogeneousLocalization.mul_val] at hx12 ; exact h x1.val x2.val hx12\nclear x1 x2 hx12\nintro x1 x2 hx12\ninduction' x1 using Localization.induction_on with data_x1\ninduction' x2 using Localization.induction_on with data_x2\nrcases data_x1, data_x2 with \u27e8\u27e8a1, _, \u27e8n1, rfl\u27e9\u27e9, \u27e8a2, _, \u27e8n2, rfl\u27e9\u27e9\u27e9\nrcases MemCarrier.clear_denominator' x hx12 with \u27e8c, N, acd, eq1\u27e9\nsimp only [Algebra.smul_def] at eq1 \nchange Localization.mk (f ^ N) 1 * (Localization.mk _ _ * Localization.mk _ _) = Localization.mk _ _ at eq1 \nsimp only [Localization.mk_mul, one_mul] at eq1 \nsimp only [mk_eq_mk', IsLocalization.eq] at eq1 \nrcases eq1 with \u27e8\u27e8_, \u27e8M, rfl\u27e9\u27e9, eq1\u27e9\nrw [Submonoid.coe_one, one_mul] at eq1 \nchange f ^ _ * (_ * _) = f ^ _ * (f ^ _ * f ^ _ * _) at eq1 \nhave that : a1 * a2 * f ^ N * f ^ M \u2208 x.val.asHomogeneousIdeal.toIdeal := ?_\nrcases x.1.isPrime.mem_or_mem (show a1 * a2 * f ^ N * f ^ M \u2208 _ from that) with (h1 | rid2)\nrcases x.1.isPrime.mem_or_mem h1 with (h1 | rid1)\nrcases x.1.isPrime.mem_or_mem h1 with (h1 | h2)\n\u00b7 left;\n simp only [show\n (Localization.mk a1 \u27e8f ^ n1, _\u27e9 : Away f) =\n Localization.mk a1 1 * Localization.mk 1 (\u27e8f ^ n1, \u27e8n1, rfl\u27e9\u27e9 : Submonoid.powers f)\n by rw [Localization.mk_mul, mul_one, one_mul]]\n exact Ideal.mul_mem_right _ _ (Ideal.subset_span \u27e8_, h1, rfl\u27e9)\n\u00b7 right;\n simp only [show (mk a2 \u27e8f ^ n2, _\u27e9 : Away f) = mk a2 1 * Localization.mk 1 (\u27e8f ^ n2, \u27e8n2, rfl\u27e9\u27e9 : Submonoid.powers f)\n by rw [Localization.mk_mul, mul_one, one_mul]]\n exact Ideal.mul_mem_right _ _ (Ideal.subset_span \u27e8_, h2, rfl\u27e9)\n\u00b7 exact False.elim (x.2 (x.1.isPrime.mem_of_pow_mem N rid1))\n\u00b7 exact False.elim (x.2 (x.1.isPrime.mem_of_pow_mem M rid2))\n\u00b7 rw [\u2190 mul_comm (f ^ M), \u2190 mul_comm (f ^ N), eq1]\n refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i _ => mul_mem_left _ _ _))\n generalize_proofs h\u2081 h\u2082; exact (Classical.choose_spec h\u2082).1\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nx1 x2 : \u2191(CommRingCat.of (A\u2070_ f))\nhx12 : x1 * x2 \u2208 carrier x\n\u22a2 x1 \u2208 carrier x \u2228 x2 \u2208 carrier x\n[PROOFSTEP]\nrw [mem_carrier_iff] at hx12 \n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nx1 x2 : \u2191(CommRingCat.of (A\u2070_ f))\nhx12 : HomogeneousLocalization.val (x1 * x2) \u2208 span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\n\u22a2 x1 \u2208 carrier x \u2228 x2 \u2208 carrier x\n[PROOFSTEP]\nlet J := span ((algebraMap A (Away f) : A \u2192 (Away f)) '' x.val.asHomogeneousIdeal)\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nx1 x2 : \u2191(CommRingCat.of (A\u2070_ f))\nhx12 : HomogeneousLocalization.val (x1 * x2) \u2208 span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\n\u22a2 x1 \u2208 carrier x \u2228 x2 \u2208 carrier x\n[PROOFSTEP]\nsuffices h : \u2200 x y : Localization.Away f, x * y \u2208 J \u2192 x \u2208 J \u2228 y \u2208 J\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nx1 x2 : \u2191(CommRingCat.of (A\u2070_ f))\nhx12 : HomogeneousLocalization.val (x1 * x2) \u2208 span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nh : \u2200 (x y : Away f), x * y \u2208 J \u2192 x \u2208 J \u2228 y \u2208 J\n\u22a2 x1 \u2208 carrier x \u2228 x2 \u2208 carrier x\n[PROOFSTEP]\nrw [HomogeneousLocalization.mul_val] at hx12 \n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nx1 x2 : \u2191(CommRingCat.of (A\u2070_ f))\nhx12 :\n HomogeneousLocalization.val x1 * HomogeneousLocalization.val x2 \u2208\n span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nh : \u2200 (x y : Away f), x * y \u2208 J \u2192 x \u2208 J \u2228 y \u2208 J\n\u22a2 x1 \u2208 carrier x \u2228 x2 \u2208 carrier x\n[PROOFSTEP]\nexact h x1.val x2.val hx12\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nx1 x2 : \u2191(CommRingCat.of (A\u2070_ f))\nhx12 : HomogeneousLocalization.val (x1 * x2) \u2208 span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\n\u22a2 \u2200 (x y : Away f), x * y \u2208 J \u2192 x \u2208 J \u2228 y \u2208 J\n[PROOFSTEP]\nclear x1 x2 hx12\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\n\u22a2 \u2200 (x y : Away f), x * y \u2208 J \u2192 x \u2208 J \u2228 y \u2208 J\n[PROOFSTEP]\nintro x1 x2 hx12\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12 : x1 * x2 \u2208 J\n\u22a2 x1 \u2208 J \u2228 x2 \u2208 J\n[PROOFSTEP]\ninduction' x1 using Localization.induction_on with data_x1\n[GOAL]\ncase h.H\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d : x1 * x2 \u2208 J\ndata_x1 : A \u00d7 { x // x \u2208 Submonoid.powers f }\nhx12 : Localization.mk data_x1.fst data_x1.snd * x2 \u2208 J\n\u22a2 Localization.mk data_x1.fst data_x1.snd \u2208 J \u2228 x2 \u2208 J\n[PROOFSTEP]\ninduction' x2 using Localization.induction_on with data_x2\n[GOAL]\ncase h.H.H\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\ndata_x1 : A \u00d7 { x // x \u2208 Submonoid.powers f }\nhx12\u271d\u00b9 : Localization.mk data_x1.fst data_x1.snd * x2 \u2208 J\ndata_x2 : A \u00d7 { x // x \u2208 Submonoid.powers f }\nhx12\u271d : x1 * Localization.mk data_x2.fst data_x2.snd \u2208 J\nhx12 : Localization.mk data_x1.fst data_x1.snd * Localization.mk data_x2.fst data_x2.snd \u2208 J\n\u22a2 Localization.mk data_x1.fst data_x1.snd \u2208 J \u2228 Localization.mk data_x2.fst data_x2.snd \u2208 J\n[PROOFSTEP]\nrcases data_x1, data_x2 with \u27e8\u27e8a1, _, \u27e8n1, rfl\u27e9\u27e9, \u27e8a2, _, \u27e8n2, rfl\u27e9\u27e9\u27e9\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\n[PROOFSTEP]\nrcases MemCarrier.clear_denominator' x hx12 with \u27e8c, N, acd, eq1\u27e9\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\neq1 :\n f ^ N \u2022\n (Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd) =\n \u2191(algebraMap A (Away f))\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\n[PROOFSTEP]\nsimp only [Algebra.smul_def] at eq1 \n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\neq1 :\n \u2191(algebraMap A (Away f)) (f ^ N) *\n (Localization.mk a1\n { val := f ^ n1, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) } *\n Localization.mk a2\n { val := f ^ n2, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }) =\n \u2191(algebraMap A (Away f))\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\n[PROOFSTEP]\nchange Localization.mk (f ^ N) 1 * (Localization.mk _ _ * Localization.mk _ _) = Localization.mk _ _ at eq1 \n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\neq1 :\n Localization.mk (f ^ N) 1 *\n (Localization.mk a1\n { val := f ^ n1, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) } *\n Localization.mk a2\n { val := f ^ n2, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }) =\n Localization.mk\n (\u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\n 1\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\n[PROOFSTEP]\nsimp only [Localization.mk_mul, one_mul] at eq1 \n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\neq1 :\n Localization.mk (f ^ N * (a1 * a2))\n ({ val := f ^ n1, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) } *\n { val := f ^ n2, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }) =\n Localization.mk\n (\u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\n 1\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\n[PROOFSTEP]\nsimp only [mk_eq_mk', IsLocalization.eq] at eq1 \n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\neq1 :\n \u2203 c_1,\n \u2191c_1 * (\u21911 * (f ^ N * (a1 * a2))) =\n \u2191c_1 *\n (\u2191({ val := f ^ n1, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) } *\n { val := f ^ n2, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }) *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\n[PROOFSTEP]\nrcases eq1 with \u27e8\u27e8_, \u27e8M, rfl\u27e9\u27e9, eq1\u27e9\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n \u2191{ val := (fun x x_1 => x ^ x_1) f M,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f M) } *\n (\u21911 * (f ^ N * (a1 * a2))) =\n \u2191{ val := (fun x x_1 => x ^ x_1) f M,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f M) } *\n (\u2191({ val := f ^ n1, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) } *\n { val := f ^ n2, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }) *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\n[PROOFSTEP]\nrw [Submonoid.coe_one, one_mul] at eq1 \n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n \u2191{ val := (fun x x_1 => x ^ x_1) f M,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f M) } *\n (f ^ N * (a1 * a2)) =\n \u2191{ val := (fun x x_1 => x ^ x_1) f M,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f M) } *\n (\u2191({ val := f ^ n1, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) } *\n { val := f ^ n2, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }) *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\n[PROOFSTEP]\nchange f ^ _ * (_ * _) = f ^ _ * (f ^ _ * f ^ _ * _) at eq1 \n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\n[PROOFSTEP]\nhave that : a1 * a2 * f ^ N * f ^ M \u2208 x.val.asHomogeneousIdeal.toIdeal := ?_\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\n\u22a2 a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n[PROOFSTEP]\nrcases x.1.isPrime.mem_or_mem (show a1 * a2 * f ^ N * f ^ M \u2208 _ from that) with (h1 | rid2)\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_2.inl\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1 : a1 * a2 * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_2.inr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nrid2 : f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\n\u22a2 a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n[PROOFSTEP]\nrcases x.1.isPrime.mem_or_mem h1 with (h1 | rid1)\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_2.inl.inl\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1\u271d : a1 * a2 * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1 : a1 * a2 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_2.inl.inr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1 : a1 * a2 * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nrid1 : f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_2.inr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nrid2 : f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\n\u22a2 a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n[PROOFSTEP]\nrcases x.1.isPrime.mem_or_mem h1 with (h1 | h2)\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_2.inl.inl.inl\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1\u271d\u00b9 : a1 * a2 * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1\u271d : a1 * a2 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1 : a1 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\n[PROOFSTEP]\nleft\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_2.inl.inl.inl.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1\u271d\u00b9 : a1 * a2 * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1\u271d : a1 * a2 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1 : a1 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J\n[PROOFSTEP]\nsimp only [show\n (Localization.mk a1 \u27e8f ^ n1, _\u27e9 : Away f) =\n Localization.mk a1 1 * Localization.mk 1 (\u27e8f ^ n1, \u27e8n1, rfl\u27e9\u27e9 : Submonoid.powers f)\n by rw [Localization.mk_mul, mul_one, one_mul]]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1\u271d\u00b9 : a1 * a2 * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1\u271d : a1 * a2 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1 : a1 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk a1 { val := f ^ n1, property := ?m.499245 } =\n Localization.mk a1 1 *\n Localization.mk 1 { val := f ^ n1, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ n1) }\n[PROOFSTEP]\nrw [Localization.mk_mul, mul_one, one_mul]\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_2.inl.inl.inl.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1\u271d\u00b9 : a1 * a2 * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1\u271d : a1 * a2 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1 : a1 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk a1 1 *\n Localization.mk 1 { val := f ^ n1, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ n1) } \u2208\n span ((fun a => \u2191(algebraMap A (Away f)) a) '' \u2191(\u2191x).asHomogeneousIdeal)\n[PROOFSTEP]\nexact Ideal.mul_mem_right _ _ (Ideal.subset_span \u27e8_, h1, rfl\u27e9)\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_2.inl.inl.inr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1\u271d : a1 * a2 * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1 : a1 * a2 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh2 : a2 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\n[PROOFSTEP]\nright\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_2.inl.inl.inr.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1\u271d : a1 * a2 * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1 : a1 * a2 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh2 : a2 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\n[PROOFSTEP]\nsimp only [show (mk a2 \u27e8f ^ n2, _\u27e9 : Away f) = mk a2 1 * Localization.mk 1 (\u27e8f ^ n2, \u27e8n2, rfl\u27e9\u27e9 : Submonoid.powers f) by\n rw [Localization.mk_mul, mul_one, one_mul]]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1\u271d : a1 * a2 * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1 : a1 * a2 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh2 : a2 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk a2 { val := f ^ n2, property := ?m.654160 } =\n Localization.mk a2 1 *\n Localization.mk 1 { val := f ^ n2, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ n2) }\n[PROOFSTEP]\nrw [Localization.mk_mul, mul_one, one_mul]\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_2.inl.inl.inr.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1\u271d : a1 * a2 * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1 : a1 * a2 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh2 : a2 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk a2 1 *\n Localization.mk 1 { val := f ^ n2, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ n2) } \u2208\n span ((fun a => \u2191(algebraMap A (Away f)) a) '' \u2191(\u2191x).asHomogeneousIdeal)\n[PROOFSTEP]\nexact Ideal.mul_mem_right _ _ (Ideal.subset_span \u27e8_, h2, rfl\u27e9)\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_2.inl.inr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nh1 : a1 * a2 * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nrid1 : f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\n[PROOFSTEP]\nexact False.elim (x.2 (x.1.isPrime.mem_of_pow_mem N rid1))\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_2.inr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\nthat : a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\nrid2 : f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n\u22a2 Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd \u2208\n J \u2228\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\n[PROOFSTEP]\nexact False.elim (x.2 (x.1.isPrime.mem_of_pow_mem M rid2))\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\n\u22a2 a1 * a2 * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n[PROOFSTEP]\nrw [\u2190 mul_comm (f ^ M), \u2190 mul_comm (f ^ N), eq1]\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\n\u22a2 f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal))) \u2208\n HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n[PROOFSTEP]\nrefine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i _ => mul_mem_left _ _ _))\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d\u00b9 : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\ni : { x_1 // x_1 \u2208 c.support }\nx\u271d : i \u2208 attach c.support\n\u22a2 Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2208\n HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n[PROOFSTEP]\ngeneralize_proofs h\u2081 h\u2082\n[GOAL]\ncase h.H.H.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.mk.intro.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx\u271d\u00b9 : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\nx : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nJ : Ideal (Away f) := span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)\nx1 x2 : Away f\nhx12\u271d\u00b2 : x1 * x2 \u2208 J\na1 : A\nn1 : \u2115\nhx12\u271d\u00b9 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n x2 \u2208\n J\na2 : A\nn2 : \u2115\nhx12\u271d :\n x1 *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nhx12 :\n Localization.mk\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).fst\n (a1,\n { val := (fun x x_1 => x ^ x_1) f n1,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n1) }).snd *\n Localization.mk\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).fst\n (a2,\n { val := (fun x x_1 => x ^ x_1) f n2,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f n2) }).snd \u2208\n J\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y : Away f) \u2192 y \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * (a1 * a2)) =\n f ^ M *\n (f ^ n1 * f ^ n2 *\n \u2191(algebraMap A A)\n (\u2211 x_1 in attach c.support,\n acd (\u2191c \u2191x_1) (_ : \u2191c \u2191x_1 \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x_1 \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal)))\ni : { x_1 // x_1 \u2208 c.support }\nx\u271d : i \u2208 attach c.support\nh\u2081 : AddSubmonoidClass (Submodule R A) A\nh\u2082 : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191x).asHomogeneousIdeal\n\u22a2 Exists.choose h\u2082 \u2208 HomogeneousIdeal.toIdeal (\u2191x).asHomogeneousIdeal\n[PROOFSTEP]\nexact (Classical.choose_spec h\u2082).1\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\n\u22a2 toFun f \u207b\u00b9'\n \u2191(sbo\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }) =\n {x | \u2191x \u2208 (pbo f) \u2293 pbo a}\n[PROOFSTEP]\nclassical\next1 y\nconstructor <;> intro hy\n\u00b7 refine' \u27e8y.2, _\u27e9\n rw [Set.mem_preimage, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen] at hy \n rw [ProjectiveSpectrum.mem_coe_basicOpen]\n intro a_mem_y\n apply hy\n rw [toFun, mem_carrier_iff, HomogeneousLocalization.val_mk'', Subtype.coe_mk]\n dsimp; rcases b_mem2 with \u27e8k, hk\u27e9\n dsimp at hk \n simp only [show (mk a \u27e8b, \u27e8k, hk\u27e9\u27e9 : Away f) = Localization.mk 1 (\u27e8f ^ k, \u27e8_, rfl\u27e9\u27e9 : Submonoid.powers f) * mk a 1 by\n rw [mk_mul, one_mul, mul_one]; congr; rw [hk]]\n exact Ideal.mul_mem_left _ _ (Ideal.subset_span \u27e8_, a_mem_y, rfl\u27e9)\n\u00b7 change y.1 \u2208 ProjectiveSpectrum.basicOpen \ud835\udc9c f \u2293 ProjectiveSpectrum.basicOpen \ud835\udc9c a at hy \n rcases hy with \u27e8hy1, hy2\u27e9\n rw [ProjectiveSpectrum.mem_coe_basicOpen] at hy1 hy2 \n rw [Set.mem_preimage, toFun, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen]\n intro rid; dsimp at rid \n rcases MemCarrier.clear_denominator _ rid with \u27e8c, N, acd, eq1\u27e9\n rw [Algebra.smul_def] at eq1 \n change Localization.mk (f ^ N) 1 * Localization.mk _ _ = Localization.mk _ _ at eq1 \n rw [mk_mul, one_mul, mk_eq_mk', IsLocalization.eq] at eq1 \n rcases eq1 with \u27e8\u27e8_, \u27e8M, rfl\u27e9\u27e9, eq1\u27e9\n rw [Submonoid.coe_one, one_mul] at eq1 \n simp only [Subtype.coe_mk] at eq1 \n have : a * f ^ N * f ^ M \u2208 y.val.asHomogeneousIdeal.toIdeal :=\n by\n rw [mul_comm _ (f ^ N), mul_comm _ (f ^ M), eq1]\n refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i _ => mul_mem_left _ _ _))\n generalize_proofs h\u2081 h\u2082; exact (Classical.choose_spec h\u2082).1\n rcases y.1.isPrime.mem_or_mem this with (H1 | H3)\n rcases y.1.isPrime.mem_or_mem H1 with (H1 | H2)\n \u00b7 exact hy2 H1\n \u00b7 exact y.2 (y.1.isPrime.mem_of_pow_mem N H2)\n \u00b7 exact y.2 (y.1.isPrime.mem_of_pow_mem M H3)\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\n\u22a2 toFun f \u207b\u00b9'\n \u2191(sbo\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }) =\n {x | \u2191x \u2208 (pbo f) \u2293 pbo a}\n[PROOFSTEP]\next1 y\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\n\u22a2 y \u2208\n toFun f \u207b\u00b9'\n \u2191(sbo\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }) \u2194\n y \u2208 {x | \u2191x \u2208 (pbo f) \u2293 pbo a}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\n\u22a2 y \u2208\n toFun f \u207b\u00b9'\n \u2191(sbo\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }) \u2192\n y \u2208 {x | \u2191x \u2208 (pbo f) \u2293 pbo a}\n[PROOFSTEP]\nintro hy\n[GOAL]\ncase h.mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\n\u22a2 y \u2208 {x | \u2191x \u2208 (pbo f) \u2293 pbo a} \u2192\n y \u2208\n toFun f \u207b\u00b9'\n \u2191(sbo\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 })\n[PROOFSTEP]\nintro hy\n[GOAL]\ncase h.mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy :\n y \u2208\n toFun f \u207b\u00b9'\n \u2191(sbo\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 })\n\u22a2 y \u2208 {x | \u2191x \u2208 (pbo f) \u2293 pbo a}\n[PROOFSTEP]\nrefine' \u27e8y.2, _\u27e9\n[GOAL]\ncase h.mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy :\n y \u2208\n toFun f \u207b\u00b9'\n \u2191(sbo\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 })\n\u22a2 \u2191y \u2208 \u2191(pbo a)\n[PROOFSTEP]\nrw [Set.mem_preimage, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen] at hy \n[GOAL]\ncase h.mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy :\n \u00acQuotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 } \u2208\n (toFun f y).asIdeal\n\u22a2 \u2191y \u2208 \u2191(pbo a)\n[PROOFSTEP]\nrw [ProjectiveSpectrum.mem_coe_basicOpen]\n[GOAL]\ncase h.mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy :\n \u00acQuotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 } \u2208\n (toFun f y).asIdeal\n\u22a2 \u00aca \u2208 (\u2191y).asHomogeneousIdeal\n[PROOFSTEP]\nintro a_mem_y\n[GOAL]\ncase h.mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy :\n \u00acQuotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 } \u2208\n (toFun f y).asIdeal\na_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal\n\u22a2 False\n[PROOFSTEP]\napply hy\n[GOAL]\ncase h.mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy :\n \u00acQuotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 } \u2208\n (toFun f y).asIdeal\na_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal\n\u22a2 Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n (toFun f y).asIdeal\n[PROOFSTEP]\nrw [toFun, mem_carrier_iff, HomogeneousLocalization.val_mk'', Subtype.coe_mk]\n[GOAL]\ncase h.mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy :\n \u00acQuotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 } \u2208\n (toFun f y).asIdeal\na_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal\n\u22a2 Localization.mk a\n {\n val :=\n \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }.den,\n property :=\n (_ :\n \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }.den \u2208\n Submonoid.powers f) } \u2208\n span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy :\n \u00acQuotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 } \u2208\n (toFun f y).asIdeal\na_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal\n\u22a2 Localization.mk a { val := b, property := b_mem2 } \u2208 span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)\n[PROOFSTEP]\nrcases b_mem2 with \u27e8k, hk\u27e9\n[GOAL]\ncase h.mp.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk\u271d : \u2115\na_mem : a \u2208 \ud835\udc9c k\u271d\nb_mem1 : b \u2208 \ud835\udc9c k\u271d\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\na_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal\nk : \u2115\nhk : (fun x x_1 => x ^ x_1) f k = b\nhy :\n \u00acQuotient.mk''\n { deg := k\u271d, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208\n (toFun f y).asIdeal\n\u22a2 Localization.mk a { val := b, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208\n span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)\n[PROOFSTEP]\ndsimp at hk \n[GOAL]\ncase h.mp.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk\u271d : \u2115\na_mem : a \u2208 \ud835\udc9c k\u271d\nb_mem1 : b \u2208 \ud835\udc9c k\u271d\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\na_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal\nk : \u2115\nhk : f ^ k = b\nhy :\n \u00acQuotient.mk''\n { deg := k\u271d, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208\n (toFun f y).asIdeal\n\u22a2 Localization.mk a { val := b, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208\n span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)\n[PROOFSTEP]\nsimp only [show (mk a \u27e8b, \u27e8k, hk\u27e9\u27e9 : Away f) = Localization.mk 1 (\u27e8f ^ k, \u27e8_, rfl\u27e9\u27e9 : Submonoid.powers f) * mk a 1 by\n rw [mk_mul, one_mul, mul_one]; congr; rw [hk]]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk\u271d : \u2115\na_mem : a \u2208 \ud835\udc9c k\u271d\nb_mem1 : b \u2208 \ud835\udc9c k\u271d\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\na_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal\nk : \u2115\nhk : f ^ k = b\nhy :\n \u00acQuotient.mk''\n { deg := k\u271d, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208\n (toFun f y).asIdeal\n\u22a2 Localization.mk a { val := b, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } =\n Localization.mk 1 { val := f ^ k, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ k) } * Localization.mk a 1\n[PROOFSTEP]\nrw [mk_mul, one_mul, mul_one]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk\u271d : \u2115\na_mem : a \u2208 \ud835\udc9c k\u271d\nb_mem1 : b \u2208 \ud835\udc9c k\u271d\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\na_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal\nk : \u2115\nhk : f ^ k = b\nhy :\n \u00acQuotient.mk''\n { deg := k\u271d, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208\n (toFun f y).asIdeal\n\u22a2 Localization.mk a { val := b, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } =\n Localization.mk a { val := f ^ k, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ k) }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_y.e_val\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk\u271d : \u2115\na_mem : a \u2208 \ud835\udc9c k\u271d\nb_mem1 : b \u2208 \ud835\udc9c k\u271d\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\na_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal\nk : \u2115\nhk : f ^ k = b\nhy :\n \u00acQuotient.mk''\n { deg := k\u271d, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208\n (toFun f y).asIdeal\n\u22a2 b = f ^ k\n[PROOFSTEP]\nrw [hk]\n[GOAL]\ncase h.mp.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk\u271d : \u2115\na_mem : a \u2208 \ud835\udc9c k\u271d\nb_mem1 : b \u2208 \ud835\udc9c k\u271d\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\na_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal\nk : \u2115\nhk : f ^ k = b\nhy :\n \u00acQuotient.mk''\n { deg := k\u271d, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208\n (toFun f y).asIdeal\n\u22a2 Localization.mk 1 { val := f ^ k, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ k) } * Localization.mk a 1 \u2208\n span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)\n[PROOFSTEP]\nexact Ideal.mul_mem_left _ _ (Ideal.subset_span \u27e8_, a_mem_y, rfl\u27e9)\n[GOAL]\ncase h.mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy : y \u2208 {x | \u2191x \u2208 (pbo f) \u2293 pbo a}\n\u22a2 y \u2208\n toFun f \u207b\u00b9'\n \u2191(sbo\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 })\n[PROOFSTEP]\nchange y.1 \u2208 ProjectiveSpectrum.basicOpen \ud835\udc9c f \u2293 ProjectiveSpectrum.basicOpen \ud835\udc9c a at hy \n[GOAL]\ncase h.mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy : \u2191y \u2208 (pbo f) \u2293 pbo a\n\u22a2 y \u2208\n toFun f \u207b\u00b9'\n \u2191(sbo\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 })\n[PROOFSTEP]\nrcases hy with \u27e8hy1, hy2\u27e9\n[GOAL]\ncase h.mpr.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u2191y \u2208 \u2191(pbo f)\nhy2 : \u2191y \u2208 \u2191(pbo a)\n\u22a2 y \u2208\n toFun f \u207b\u00b9'\n \u2191(sbo\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 })\n[PROOFSTEP]\nrw [ProjectiveSpectrum.mem_coe_basicOpen] at hy1 hy2 \n[GOAL]\ncase h.mpr.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\n\u22a2 y \u2208\n toFun f \u207b\u00b9'\n \u2191(sbo\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 })\n[PROOFSTEP]\nrw [Set.mem_preimage, toFun, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen]\n[GOAL]\ncase h.mpr.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\n\u22a2 \u00acQuotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 } \u2208\n { asIdeal := carrier y, IsPrime := (_ : IsPrime (carrier y)) }.asIdeal\n[PROOFSTEP]\nintro rid\n[GOAL]\ncase h.mpr.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n { asIdeal := carrier y, IsPrime := (_ : IsPrime (carrier y)) }.asIdeal\n\u22a2 False\n[PROOFSTEP]\ndsimp at rid \n[GOAL]\ncase h.mpr.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\n\u22a2 False\n[PROOFSTEP]\nrcases MemCarrier.clear_denominator _ rid with \u27e8c, N, acd, eq1\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\neq1 :\n f ^ N \u2022\n HomogeneousLocalization.val\n (Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }) =\n \u2191(algebraMap A (Away f))\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))\n\u22a2 False\n[PROOFSTEP]\nrw [Algebra.smul_def] at eq1 \n[GOAL]\ncase h.mpr.intro.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\neq1 :\n \u2191(algebraMap A (Localization (Submonoid.powers f))) (f ^ N) *\n HomogeneousLocalization.val\n (Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }) =\n \u2191(algebraMap A (Away f))\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))\n\u22a2 False\n[PROOFSTEP]\nchange Localization.mk (f ^ N) 1 * Localization.mk _ _ = Localization.mk _ _ at eq1 \n[GOAL]\ncase h.mpr.intro.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\neq1 :\n Localization.mk (f ^ N) 1 *\n Localization.mk\n \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }.num\n {\n val :=\n \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }.den,\n property :=\n (_ :\n \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }.den \u2208\n Submonoid.powers f) } =\n Localization.mk\n (\u2191(algebraMap A A)\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)))\n 1\n\u22a2 False\n[PROOFSTEP]\nrw [mk_mul, one_mul, mk_eq_mk', IsLocalization.eq] at eq1 \n[GOAL]\ncase h.mpr.intro.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\neq1 :\n \u2203 c_1,\n \u2191c_1 *\n (\u21911 *\n (f ^ N *\n \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }.num)) =\n \u2191c_1 *\n (\u2191{\n val :=\n \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }.den,\n property :=\n (_ :\n \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }.den \u2208\n Submonoid.powers f) } *\n \u2191(algebraMap A A)\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)))\n\u22a2 False\n[PROOFSTEP]\nrcases eq1 with \u27e8\u27e8_, \u27e8M, rfl\u27e9\u27e9, eq1\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n \u2191{ val := (fun x x_1 => x ^ x_1) f M,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f M) } *\n (\u21911 *\n (f ^ N *\n \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }.num)) =\n \u2191{ val := (fun x x_1 => x ^ x_1) f M,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f M) } *\n (\u2191{\n val :=\n \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }.den,\n property :=\n (_ :\n \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }.den \u2208\n Submonoid.powers f) } *\n \u2191(algebraMap A A)\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)))\n\u22a2 False\n[PROOFSTEP]\nrw [Submonoid.coe_one, one_mul] at eq1 \n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n \u2191{ val := (fun x x_1 => x ^ x_1) f M,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f M) } *\n (f ^ N *\n \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }.num) =\n \u2191{ val := (fun x x_1 => x ^ x_1) f M,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f M) } *\n (\u2191{\n val :=\n \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }.den,\n property :=\n (_ :\n \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 },\n den_mem := b_mem2 }.den \u2208\n Submonoid.powers f) } *\n \u2191(algebraMap A A)\n (\u2211 i in attach c.support,\n acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)))\n\u22a2 False\n[PROOFSTEP]\nsimp only [Subtype.coe_mk] at eq1 \n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * a) =\n f ^ M *\n (b *\n \u2191(algebraMap A A)\n (\u2211 x in attach c.support,\n acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)))\n\u22a2 False\n[PROOFSTEP]\nhave : a * f ^ N * f ^ M \u2208 y.val.asHomogeneousIdeal.toIdeal :=\n by\n rw [mul_comm _ (f ^ N), mul_comm _ (f ^ M), eq1]\n refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i _ => mul_mem_left _ _ _))\n generalize_proofs h\u2081 h\u2082; exact (Classical.choose_spec h\u2082).1\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * a) =\n f ^ M *\n (b *\n \u2191(algebraMap A A)\n (\u2211 x in attach c.support,\n acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)))\n\u22a2 a * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\n[PROOFSTEP]\nrw [mul_comm _ (f ^ N), mul_comm _ (f ^ M), eq1]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * a) =\n f ^ M *\n (b *\n \u2191(algebraMap A A)\n (\u2211 x in attach c.support,\n acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)))\n\u22a2 f ^ M *\n (b *\n \u2191(algebraMap A A)\n (\u2211 x in attach c.support,\n acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) \u2208\n HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\n[PROOFSTEP]\nrefine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i _ => mul_mem_left _ _ _))\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * a) =\n f ^ M *\n (b *\n \u2191(algebraMap A A)\n (\u2211 x in attach c.support,\n acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)))\ni : { x // x \u2208 c.support }\nx\u271d : i \u2208 attach c.support\n\u22a2 Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2208\n HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\n[PROOFSTEP]\ngeneralize_proofs h\u2081 h\u2082\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * a) =\n f ^ M *\n (b *\n \u2191(algebraMap A A)\n (\u2211 x in attach c.support,\n acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)))\ni : { x // x \u2208 c.support }\nx\u271d : i \u2208 attach c.support\nh\u2081 : AddSubmonoidClass (Submodule R A) A\nh\u2082 : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal\n\u22a2 Exists.choose h\u2082 \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\n[PROOFSTEP]\nexact (Classical.choose_spec h\u2082).1\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * a) =\n f ^ M *\n (b *\n \u2191(algebraMap A A)\n (\u2211 x in attach c.support,\n acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)))\nthis : a * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\n\u22a2 False\n[PROOFSTEP]\nrcases y.1.isPrime.mem_or_mem this with (H1 | H3)\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro.mk.intro.inl\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * a) =\n f ^ M *\n (b *\n \u2191(algebraMap A A)\n (\u2211 x in attach c.support,\n acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)))\nthis : a * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\nH1 : a * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\n\u22a2 False\ncase h.mpr.intro.intro.intro.intro.intro.mk.intro.inr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * a) =\n f ^ M *\n (b *\n \u2191(algebraMap A A)\n (\u2211 x in attach c.support,\n acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)))\nthis : a * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\nH3 : f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\n\u22a2 False\n[PROOFSTEP]\nrcases y.1.isPrime.mem_or_mem H1 with (H1 | H2)\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro.mk.intro.inl.inl\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * a) =\n f ^ M *\n (b *\n \u2191(algebraMap A A)\n (\u2211 x in attach c.support,\n acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)))\nthis : a * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\nH1\u271d : a * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\nH1 : a \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\n\u22a2 False\n[PROOFSTEP]\nexact hy2 H1\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro.mk.intro.inl.inr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * a) =\n f ^ M *\n (b *\n \u2191(algebraMap A A)\n (\u2211 x in attach c.support,\n acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)))\nthis : a * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\nH1 : a * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\nH2 : f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\n\u22a2 False\n[PROOFSTEP]\nexact y.2 (y.1.isPrime.mem_of_pow_mem N H2)\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro.mk.intro.inr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nx : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))))\na b : A\nk : \u2115\na_mem : a \u2208 \ud835\udc9c k\nb_mem1 : b \u2208 \ud835\udc9c k\nb_mem2 : b \u2208 Submonoid.powers f\ny : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace\nhy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal\nhy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal\nrid :\n Quotient.mk''\n { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208\n carrier y\nc : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f\nN : \u2115\nacd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A\nM : \u2115\neq1 :\n f ^ M * (f ^ N * a) =\n f ^ M *\n (b *\n \u2191(algebraMap A A)\n (\u2211 x in attach c.support,\n acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) *\n Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)))\nthis : a * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\nH3 : f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal\n\u22a2 False\n[PROOFSTEP]\nexact y.2 (y.1.isPrime.mem_of_pow_mem M H3)\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\n\u22a2 Continuous (ToSpec.toFun f)\n[PROOFSTEP]\napply IsTopologicalBasis.continuous PrimeSpectrum.isTopologicalBasis_basic_opens\n[GOAL]\ncase hf\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\n\u22a2 \u2200 (s : Set (PrimeSpectrum \u2191(CommRingCat.of (A\u2070_ f)))),\n (s \u2208 Set.range fun r => \u2191(sbo r)) \u2192 IsOpen (ToSpec.toFun f \u207b\u00b9' s)\n[PROOFSTEP]\nrintro _ \u27e8\u27e8k, \u27e8a, ha\u27e9, \u27e8b, hb1\u27e9, \u27e8k', hb2\u27e9\u27e9, rfl\u27e9\n[GOAL]\ncase hf.intro.mk.mk.mk.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nw\u271d : \u2191(CommRingCat.of (A\u2070_ f))\nk : \u2115\na : A\nha : a \u2208 \ud835\udc9c k\nb : A\nhb1 : b \u2208 \ud835\udc9c k\nk' : \u2115\nhb2 : (fun x x_1 => x ^ x_1) f k' = \u2191{ val := b, property := hb1 }\n\u22a2 IsOpen\n (ToSpec.toFun f \u207b\u00b9'\n (fun r => \u2191(sbo r))\n (Quot.mk Setoid.r\n { deg := k, num := { val := a, property := ha }, den := { val := b, property := hb1 },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := b, property := hb1 }) }))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase hf.intro.mk.mk.mk.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nw\u271d : \u2191(CommRingCat.of (A\u2070_ f))\nk : \u2115\na : A\nha : a \u2208 \ud835\udc9c k\nb : A\nhb1 : b \u2208 \ud835\udc9c k\nk' : \u2115\nhb2 : (fun x x_1 => x ^ x_1) f k' = \u2191{ val := b, property := hb1 }\n\u22a2 IsOpen\n (ToSpec.toFun f \u207b\u00b9'\n \u2191(sbo\n Quot.mk Setoid.r\n { deg := k, num := { val := a, property := ha }, den := { val := b, property := hb1 },\n den_mem := (_ : \u2203 y, f ^ y = b) }))\n[PROOFSTEP]\nerw [ToSpec.preimage_eq f a b k ha hb1 \u27e8k', hb2\u27e9]\n[GOAL]\ncase hf.intro.mk.mk.mk.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nw\u271d : \u2191(CommRingCat.of (A\u2070_ f))\nk : \u2115\na : A\nha : a \u2208 \ud835\udc9c k\nb : A\nhb1 : b \u2208 \ud835\udc9c k\nk' : \u2115\nhb2 : (fun x x_1 => x ^ x_1) f k' = \u2191{ val := b, property := hb1 }\n\u22a2 IsOpen {x | \u2191x \u2208 (pbo f) \u2293 pbo a}\n[PROOFSTEP]\nrefine' isOpen_induced_iff.mpr \u27e8(pbo f).1 \u2293 (pbo a).1, IsOpen.inter (pbo f).2 (pbo a).2, _\u27e9\n[GOAL]\ncase hf.intro.mk.mk.mk.mk.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nw\u271d : \u2191(CommRingCat.of (A\u2070_ f))\nk : \u2115\na : A\nha : a \u2208 \ud835\udc9c k\nb : A\nhb1 : b \u2208 \ud835\udc9c k\nk' : \u2115\nhb2 : (fun x x_1 => x ^ x_1) f k' = \u2191{ val := b, property := hb1 }\n\u22a2 Subtype.val \u207b\u00b9' ((pbo f).carrier \u2293 (pbo a).carrier) = {x | \u2191x \u2208 (pbo f) \u2293 pbo a}\n[PROOFSTEP]\next z\n[GOAL]\ncase hf.intro.mk.mk.mk.mk.intro.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nw\u271d : \u2191(CommRingCat.of (A\u2070_ f))\nk : \u2115\na : A\nha : a \u2208 \ud835\udc9c k\nb : A\nhb1 : b \u2208 \ud835\udc9c k\nk' : \u2115\nhb2 : (fun x x_1 => x ^ x_1) f k' = \u2191{ val := b, property := hb1 }\nz : \u2191((Opens.toTopCat Proj.T).obj (pbo f))\n\u22a2 z \u2208 Subtype.val \u207b\u00b9' ((pbo f).carrier \u2293 (pbo a).carrier) \u2194 z \u2208 {x | \u2191x \u2208 (pbo f) \u2293 pbo a}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase hf.intro.mk.mk.mk.mk.intro.h.mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nw\u271d : \u2191(CommRingCat.of (A\u2070_ f))\nk : \u2115\na : A\nha : a \u2208 \ud835\udc9c k\nb : A\nhb1 : b \u2208 \ud835\udc9c k\nk' : \u2115\nhb2 : (fun x x_1 => x ^ x_1) f k' = \u2191{ val := b, property := hb1 }\nz : \u2191((Opens.toTopCat Proj.T).obj (pbo f))\n\u22a2 z \u2208 Subtype.val \u207b\u00b9' ((pbo f).carrier \u2293 (pbo a).carrier) \u2192 z \u2208 {x | \u2191x \u2208 (pbo f) \u2293 pbo a}\n[PROOFSTEP]\nintro hz\n[GOAL]\ncase hf.intro.mk.mk.mk.mk.intro.h.mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nw\u271d : \u2191(CommRingCat.of (A\u2070_ f))\nk : \u2115\na : A\nha : a \u2208 \ud835\udc9c k\nb : A\nhb1 : b \u2208 \ud835\udc9c k\nk' : \u2115\nhb2 : (fun x x_1 => x ^ x_1) f k' = \u2191{ val := b, property := hb1 }\nz : \u2191((Opens.toTopCat Proj.T).obj (pbo f))\nhz : z \u2208 Subtype.val \u207b\u00b9' ((pbo f).carrier \u2293 (pbo a).carrier)\n\u22a2 z \u2208 {x | \u2191x \u2208 (pbo f) \u2293 pbo a}\n[PROOFSTEP]\nsimpa [Set.mem_preimage]\n[GOAL]\ncase hf.intro.mk.mk.mk.mk.intro.h.mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nw\u271d : \u2191(CommRingCat.of (A\u2070_ f))\nk : \u2115\na : A\nha : a \u2208 \ud835\udc9c k\nb : A\nhb1 : b \u2208 \ud835\udc9c k\nk' : \u2115\nhb2 : (fun x x_1 => x ^ x_1) f k' = \u2191{ val := b, property := hb1 }\nz : \u2191((Opens.toTopCat Proj.T).obj (pbo f))\n\u22a2 z \u2208 {x | \u2191x \u2208 (pbo f) \u2293 pbo a} \u2192 z \u2208 Subtype.val \u207b\u00b9' ((pbo f).carrier \u2293 (pbo a).carrier)\n[PROOFSTEP]\nintro hz\n[GOAL]\ncase hf.intro.mk.mk.mk.mk.intro.h.mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nw\u271d : \u2191(CommRingCat.of (A\u2070_ f))\nk : \u2115\na : A\nha : a \u2208 \ud835\udc9c k\nb : A\nhb1 : b \u2208 \ud835\udc9c k\nk' : \u2115\nhb2 : (fun x x_1 => x ^ x_1) f k' = \u2191{ val := b, property := hb1 }\nz : \u2191((Opens.toTopCat Proj.T).obj (pbo f))\nhz : z \u2208 {x | \u2191x \u2208 (pbo f) \u2293 pbo a}\n\u22a2 z \u2208 Subtype.val \u207b\u00b9' ((pbo f).carrier \u2293 (pbo a).carrier)\n[PROOFSTEP]\nsimpa only [Set.inf_eq_inter, Set.mem_inter_iff, Set.mem_preimage]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\n\u22a2 \u2191(proj \ud835\udc9c i) a ^ m \u2208 \ud835\udc9c (m * i)\n[PROOFSTEP]\nmem_tac\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\n\u22a2 f ^ i \u2208 \ud835\udc9c (m * i)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\n\u22a2 f ^ i \u2208 \ud835\udc9c (i * m)\n[PROOFSTEP]\nmem_tac\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\n\u22a2 \u2191(proj \ud835\udc9c i) a ^ m \u2208 \ud835\udc9c (m * i)\n[PROOFSTEP]\nmem_tac\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\n\u22a2 f ^ i \u2208 \ud835\udc9c (m * i)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\n\u22a2 f ^ i \u2208 \ud835\udc9c (i * m)\n[PROOFSTEP]\nmem_tac\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\n\u22a2 (\u2200 (i : \u2115),\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal) \u2194\n \u2200 (i : \u2115),\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ i) } \u2208\n \u2191(algebraMap (A\u2070_ f) (Away f)) '' {s | s \u2208 q.asIdeal}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\n\u22a2 (\u2200 (i : \u2115),\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal) \u2192\n \u2200 (i : \u2115),\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ i) } \u2208\n \u2191(algebraMap (A\u2070_ f) (Away f)) '' {s | s \u2208 q.asIdeal}\n[PROOFSTEP]\nintro h i\n[GOAL]\ncase mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\n\u22a2 (\u2200 (i : \u2115),\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ i) } \u2208\n \u2191(algebraMap (A\u2070_ f) (Away f)) '' {s | s \u2208 q.asIdeal}) \u2192\n \u2200 (i : \u2115),\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nintro h i\n[GOAL]\ncase mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\nh :\n \u2200 (i : \u2115),\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\ni : \u2115\n\u22a2 Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ i) } \u2208\n \u2191(algebraMap (A\u2070_ f) (Away f)) '' {s | s \u2208 q.asIdeal}\n[PROOFSTEP]\nspecialize h i\n[GOAL]\ncase mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\nh :\n \u2200 (i : \u2115),\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ i) } \u2208\n \u2191(algebraMap (A\u2070_ f) (Away f)) '' {s | s \u2208 q.asIdeal}\ni : \u2115\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nspecialize h i\n[GOAL]\ncase mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\nh :\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n\u22a2 Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ i) } \u2208\n \u2191(algebraMap (A\u2070_ f) (Away f)) '' {s | s \u2208 q.asIdeal}\n[PROOFSTEP]\nrw [Set.mem_image]\n[GOAL]\ncase mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\nh :\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n\u22a2 \u2203 x,\n x \u2208 {s | s \u2208 q.asIdeal} \u2227\n \u2191(algebraMap (A\u2070_ f) (Away f)) x =\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ i) }\n[PROOFSTEP]\nrefine' \u27e8_, h, rfl\u27e9\n[GOAL]\ncase mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\nh :\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ i) } \u2208\n \u2191(algebraMap (A\u2070_ f) (Away f)) '' {s | s \u2208 q.asIdeal}\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nrw [Set.mem_image] at h \n[GOAL]\ncase mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\nh :\n \u2203 x,\n x \u2208 {s | s \u2208 q.asIdeal} \u2227\n \u2191(algebraMap (A\u2070_ f) (Away f)) x =\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ i) }\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nrcases h with \u27e8x, h, hx\u27e9\n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\nx : A\u2070_ f\nh : x \u2208 {s | s \u2208 q.asIdeal}\nhx :\n \u2191(algebraMap (A\u2070_ f) (Away f)) x =\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ i) }\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nchange x \u2208 q.asIdeal at h \n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\nx : A\u2070_ f\nhx :\n \u2191(algebraMap (A\u2070_ f) (Away f)) x =\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ i) }\nh : x \u2208 q.asIdeal\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nconvert h\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\nx : A\u2070_ f\nhx :\n \u2191(algebraMap (A\u2070_ f) (Away f)) x =\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ i) }\nh : x \u2208 q.asIdeal\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } =\n x\n[PROOFSTEP]\nrw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'']\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\nx : A\u2070_ f\nhx :\n \u2191(algebraMap (A\u2070_ f) (Away f)) x =\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ i) }\nh : x \u2208 q.asIdeal\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 Localization.mk\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.num\n {\n val :=\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den \u2208\n Submonoid.powers f) } =\n HomogeneousLocalization.val x\n[PROOFSTEP]\ndsimp only [Subtype.coe_mk]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\nx : A\u2070_ f\nhx :\n \u2191(algebraMap (A\u2070_ f) (Away f)) x =\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ i) }\nh : x \u2208 q.asIdeal\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, f ^ y = f ^ i) } =\n HomogeneousLocalization.val x\n[PROOFSTEP]\nrw [\u2190 hx]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na : A\ni : \u2115\nx : A\u2070_ f\nhx :\n \u2191(algebraMap (A\u2070_ f) (Away f)) x =\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m) { val := f ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ i) }\nh : x \u2208 q.asIdeal\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 \u2191(algebraMap (A\u2070_ f) (Away f)) x = HomogeneousLocalization.val x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\n\u22a2 a + b \u2208 carrier f_deg q\n[PROOFSTEP]\nrefine' fun i => (q.2.mem_or_mem _).elim id id\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) (a + b)) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) (a + b)) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nchange (Quotient.mk'' \u27e8_, _, _, _\u27e9 : A\u2070_ f) \u2208 q.1\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\n\u22a2 Quotient.mk''\n {\n deg :=\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.deg +\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.deg,\n num :=\n {\n val :=\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.num *\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.num,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.num *\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.num \u2208\n \ud835\udc9c\n ({ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.deg +\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.deg)) },\n den :=\n {\n val :=\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den *\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den *\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den \u2208\n \ud835\udc9c\n ({ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.deg +\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.deg)) },\n den_mem :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den *\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den \u2208\n Submonoid.powers f) } \u2208\n q.asIdeal\n[PROOFSTEP]\ndsimp only [Subtype.coe_mk]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\n\u22a2 Quotient.mk''\n { deg := m * i + m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m * \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.num *\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.num \u2208\n \ud835\udc9c\n ({ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.deg +\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.deg)) },\n den :=\n { val := f ^ i * f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den *\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den \u2208\n \ud835\udc9c\n ({ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.deg +\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.deg)) },\n den_mem :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den *\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (a + b) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (a + b)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den \u2208\n Submonoid.powers f) } \u2208\n q.asIdeal\n[PROOFSTEP]\nsimp_rw [\u2190 pow_add, map_add, add_pow, mul_comm, \u2190 nsmul_eq_mul]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\n\u22a2 Quotient.mk''\n { deg := m * i + m * i,\n num :=\n { val := \u2211 x in range (m + m + 1), Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)),\n property :=\n (_ :\n (fun x => x \u2208 \ud835\udc9c (m * i + m * i))\n (\u2211 x in range (m + m + 1),\n Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)))) },\n den := { val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) },\n den_mem :=\n (_ :\n \u2191{ val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) } \u2208\n Submonoid.powers f) } \u2208\n q.asIdeal\n[PROOFSTEP]\nlet g : \u2115 \u2192 A\u2070_ f := fun j =>\n (m + m).choose j \u2022\n if h2 : m + m < j then (0 : A\u2070_ f)\n else\n -- Porting note: inlining `l`, `r` causes a \"can't synth HMul A\u2070_ f A\u2070_ f ?\" errorif h1 : j \u2264 m then\n letI l : A\u2070_ f :=\n Quotient.mk'' \u27e8m * i, \u27e8proj \ud835\udc9c i a ^ j * proj \ud835\udc9c i b ^ (m - j), ?_\u27e9, \u27e8_, by rw [mul_comm]; mem_tac\u27e9, \u27e8i, rfl\u27e9\u27e9\n letI r : A\u2070_ f := Quotient.mk'' \u27e8m * i, \u27e8proj \ud835\udc9c i b ^ m, by mem_tac\u27e9, \u27e8_, by rw [mul_comm]; mem_tac\u27e9, \u27e8i, rfl\u27e9\u27e9\n l * r\n else\n letI l : A\u2070_ f := Quotient.mk'' \u27e8m * i, \u27e8proj \ud835\udc9c i a ^ m, by mem_tac\u27e9, \u27e8_, by rw [mul_comm]; mem_tac\u27e9, \u27e8i, rfl\u27e9\u27e9\n letI r : A\u2070_ f :=\n Quotient.mk''\n \u27e8m * i, \u27e8proj \ud835\udc9c i a ^ (j - m) * proj \ud835\udc9c i b ^ (m + m - j), ?_\u27e9, \u27e8_, by rw [mul_comm]; mem_tac\u27e9, \u27e8i, rfl\u27e9\u27e9\n l * r\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 f ^ i \u2208 \ud835\udc9c (m * i)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 f ^ i \u2208 \ud835\udc9c (i * m)\n[PROOFSTEP]\nmem_tac\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : j \u2264 m\nl : A\u2070_ f :=\n Quotient.mk''\n { deg := m * i, num := { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j), property := ?m.1574888 },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 \u2191(proj \ud835\udc9c i) b ^ m \u2208 \ud835\udc9c (m * i)\n[PROOFSTEP]\nmem_tac\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : j \u2264 m\nl : A\u2070_ f :=\n Quotient.mk''\n { deg := m * i, num := { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j), property := ?m.1574888 },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 f ^ i \u2208 \ud835\udc9c (m * i)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : j \u2264 m\nl : A\u2070_ f :=\n Quotient.mk''\n { deg := m * i, num := { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j), property := ?m.1574888 },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 f ^ i \u2208 \ud835\udc9c (i * m)\n[PROOFSTEP]\nmem_tac\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\n\u22a2 \u2191(proj \ud835\udc9c i) a ^ m \u2208 \ud835\udc9c (m * i)\n[PROOFSTEP]\nmem_tac\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\n\u22a2 f ^ i \u2208 \ud835\udc9c (m * i)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\n\u22a2 f ^ i \u2208 \ud835\udc9c (i * m)\n[PROOFSTEP]\nmem_tac\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\nl : A\u2070_ f :=\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208 \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 f ^ i \u2208 \ud835\udc9c (m * i)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\nl : A\u2070_ f :=\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208 \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 f ^ i \u2208 \ud835\udc9c (i * m)\n[PROOFSTEP]\nmem_tac\n[GOAL]\ncase refine_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property :=\n (_ :\n let l :=\n Quotient.mk''\n { deg := m * i,\n num := { val := \u2191(proj \ud835\udc9c i) a ^ m, property := (_ : \u2191(proj \ud835\udc9c i) a ^ m \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) };\n \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 Quotient.mk''\n { deg := m * i + m * i,\n num :=\n { val := \u2211 x in range (m + m + 1), Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)),\n property :=\n (_ :\n (fun x => x \u2208 \ud835\udc9c (m * i + m * i))\n (\u2211 x in range (m + m + 1),\n Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)))) },\n den := { val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) },\n den_mem :=\n (_ :\n \u2191{ val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) } \u2208\n Submonoid.powers f) } \u2208\n q.asIdeal\ncase refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\nl : A\u2070_ f :=\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208 \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)\n[PROOFSTEP]\nrotate_left\n[GOAL]\ncase refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)\n[PROOFSTEP]\nrw [(_ : m * i = _)]\n -- Porting note: it seems unification with mul_mem is more fiddly reducing value of mem_tac\n[GOAL]\ncase refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c ?m.1610121\n[PROOFSTEP]\napply GradedMonoid.toGradedMul.mul_mem (i := j \u2022 i) (j := (m - j) \u2022 i)\n[GOAL]\ncase refine_1.a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 \u2191(proj \ud835\udc9c i) a ^ j \u2208 \ud835\udc9c (j \u2022 i)\n[PROOFSTEP]\nmem_tac_aux\n[GOAL]\ncase refine_1.a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c ((m - j) \u2022 i)\n[PROOFSTEP]\nmem_tac_aux\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 m * i = j \u2022 i + (m - j) \u2022 i\n[PROOFSTEP]\nrw [\u2190 add_smul, Nat.add_sub_of_le h1]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 m * i = m \u2022 i\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\nl : A\u2070_ f :=\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208 \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)\n[PROOFSTEP]\nrw [(_ : m * i = _)]\n[GOAL]\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\nl : A\u2070_ f :=\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208 \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c ?m.1612661\n[PROOFSTEP]\napply GradedMonoid.toGradedMul.mul_mem (i := (j - m) \u2022 i) (j := (m + m - j) \u2022 i)\n[GOAL]\ncase refine_2.a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\nl : A\u2070_ f :=\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208 \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 \u2191(proj \ud835\udc9c i) a ^ (j - m) \u2208 \ud835\udc9c ((j - m) \u2022 i)\n[PROOFSTEP]\nmem_tac_aux\n[GOAL]\ncase refine_2.a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\nl : A\u2070_ f :=\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208 \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c ((m + m - j) \u2022 i)\n[PROOFSTEP]\nmem_tac_aux\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\nl : A\u2070_ f :=\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208 \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 m * i = (j - m) \u2022 i + (m + m - j) \u2022 i\n[PROOFSTEP]\nrw [\u2190 add_smul]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\nl : A\u2070_ f :=\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208 \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 m * i = (j - m + (m + m - j)) \u2022 i\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\nl : A\u2070_ f :=\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208 \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 m = j - m + (m + m - j)\n[PROOFSTEP]\nzify [le_of_not_lt h2, le_of_not_le h1]\n[GOAL]\ncase e_a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\nl : A\u2070_ f :=\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208 \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 \u2191m = \u2191j - \u2191m + (\u2191m + \u2191m - \u2191j)\n[PROOFSTEP]\nabel\n[GOAL]\ncase e_a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni j : \u2115\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\nl : A\u2070_ f :=\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208 \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 \u2191m = \u2191j - \u2191m + (\u2191m + \u2191m - \u2191j)\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 Quotient.mk''\n { deg := m * i + m * i,\n num :=\n { val := \u2211 x in range (m + m + 1), Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)),\n property :=\n (_ :\n (fun x => x \u2208 \ud835\udc9c (m * i + m * i))\n (\u2211 x in range (m + m + 1),\n Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)))) },\n den := { val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) },\n den_mem :=\n (_ :\n \u2191{ val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) } \u2208\n Submonoid.powers f) } \u2208\n q.asIdeal\n[PROOFSTEP]\nconvert_to \u2211 i in range (m + m + 1), g i \u2208 q.1\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 Quotient.mk''\n { deg := m * i + m * i,\n num :=\n { val := \u2211 x in range (m + m + 1), Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)),\n property :=\n (_ :\n (fun x => x \u2208 \ud835\udc9c (m * i + m * i))\n (\u2211 x in range (m + m + 1),\n Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)))) },\n den := { val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) },\n den_mem :=\n (_ :\n \u2191{ val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) } \u2208\n Submonoid.powers f) } =\n \u2211 i in range (m + m + 1), g i\ncase refine_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 \u2211 i in range (m + m + 1), g i \u2208 q.asIdeal\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n\u22a2 \u2211 i in range (m + m + 1), g i \u2208 q.asIdeal\n[PROOFSTEP]\nrefine' q.1.sum_mem fun j _ => nsmul_mem _ _\n[GOAL]\ncase refine_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\nj : \u2115\nx\u271d : j \u2208 range (m + m + 1)\n\u22a2 (if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }) \u2208\n q.asIdeal\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\nj : \u2115\nx\u271d : j \u2208 range (m + m + 1)\nh\u271d : m + m < j\n\u22a2 0 \u2208 q.asIdeal\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\nj : \u2115\nx\u271d : j \u2208 range (m + m + 1)\nh\u271d\u00b9 : \u00acm + m < j\nh\u271d : j \u2264 m\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\nj : \u2115\nx\u271d : j \u2208 range (m + m + 1)\nh\u271d\u00b9 : \u00acm + m < j\nh\u271d : \u00acj \u2264 m\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nexacts [q.1.zero_mem, q.1.mul_mem_left _ (hb i), q.1.mul_mem_right _ (ha i)]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 Quotient.mk''\n { deg := m * i + m * i,\n num :=\n { val := \u2211 x in range (m + m + 1), Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)),\n property :=\n (_ :\n (fun x => x \u2208 \ud835\udc9c (m * i + m * i))\n (\u2211 x in range (m + m + 1),\n Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)))) },\n den := { val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) },\n den_mem :=\n (_ :\n \u2191{ val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) } \u2208\n Submonoid.powers f) } =\n \u2211 i in range (m + m + 1), g i\n[PROOFSTEP]\nrw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'']\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 Localization.mk\n \u2191{ deg := m * i + m * i,\n num :=\n {\n val :=\n \u2211 x in range (m + m + 1), Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)),\n property :=\n (_ :\n (fun x => x \u2208 \ud835\udc9c (m * i + m * i))\n (\u2211 x in range (m + m + 1),\n Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)))) },\n den := { val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) },\n den_mem :=\n (_ :\n \u2191{ val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) } \u2208\n Submonoid.powers f) }.num\n {\n val :=\n \u2191{ deg := m * i + m * i,\n num :=\n {\n val :=\n \u2211 x in range (m + m + 1),\n Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)),\n property :=\n (_ :\n (fun x => x \u2208 \ud835\udc9c (m * i + m * i))\n (\u2211 x in range (m + m + 1),\n Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)))) },\n den := { val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) },\n den_mem :=\n (_ :\n \u2191{ val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) } \u2208\n Submonoid.powers f) }.den,\n property :=\n (_ :\n \u2191{ deg := m * i + m * i,\n num :=\n {\n val :=\n \u2211 x in range (m + m + 1),\n Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)),\n property :=\n (_ :\n (fun x => x \u2208 \ud835\udc9c (m * i + m * i))\n (\u2211 x in range (m + m + 1),\n Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)))) },\n den := { val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) },\n den_mem :=\n (_ :\n \u2191{ val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) } \u2208\n Submonoid.powers f) }.den \u2208\n Submonoid.powers f) } =\n HomogeneousLocalization.val (\u2211 i in range (m + m + 1), g i)\n[PROOFSTEP]\nchange _ = (algebraMap (HomogeneousLocalization.Away \ud835\udc9c f) (Localization.Away f)) _\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 Localization.mk\n \u2191{ deg := m * i + m * i,\n num :=\n {\n val :=\n \u2211 x in range (m + m + 1), Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)),\n property :=\n (_ :\n (fun x => x \u2208 \ud835\udc9c (m * i + m * i))\n (\u2211 x in range (m + m + 1),\n Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)))) },\n den := { val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) },\n den_mem :=\n (_ :\n \u2191{ val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) } \u2208\n Submonoid.powers f) }.num\n {\n val :=\n \u2191{ deg := m * i + m * i,\n num :=\n {\n val :=\n \u2211 x in range (m + m + 1),\n Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)),\n property :=\n (_ :\n (fun x => x \u2208 \ud835\udc9c (m * i + m * i))\n (\u2211 x in range (m + m + 1),\n Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)))) },\n den := { val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) },\n den_mem :=\n (_ :\n \u2191{ val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) } \u2208\n Submonoid.powers f) }.den,\n property :=\n (_ :\n \u2191{ deg := m * i + m * i,\n num :=\n {\n val :=\n \u2211 x in range (m + m + 1),\n Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)),\n property :=\n (_ :\n (fun x => x \u2208 \ud835\udc9c (m * i + m * i))\n (\u2211 x in range (m + m + 1),\n Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)))) },\n den := { val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) },\n den_mem :=\n (_ :\n \u2191{ val := f ^ (i + i), property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i + m * i)) (f ^ (i + i))) } \u2208\n Submonoid.powers f) }.den \u2208\n Submonoid.powers f) } =\n \u2191(algebraMap (A\u2070_ f) (Away f)) (\u2211 i in range (m + m + 1), g i)\n[PROOFSTEP]\ndsimp only [Subtype.coe_mk]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 Localization.mk (\u2211 x in range (m + m + 1), Nat.choose (m + m) x \u2022 (\u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m + m - x)))\n { val := f ^ (i + i),\n property :=\n (_ : \u2191{ val := f ^ (i + i), property := (_ : f ^ (i + i) \u2208 \ud835\udc9c (m * i + m * i)) } \u2208 Submonoid.powers f) } =\n \u2191(algebraMap (A\u2070_ f) (Away f))\n (\u2211 i_1 in range (m + m + 1),\n Nat.choose (m + m) i_1 \u2022\n if h2 : m + m < i_1 then 0\n else\n if h1 : i_1 \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ i_1 * \u2191(proj \ud835\udc9c i) b ^ (m - i_1),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ i_1 * \u2191(proj \ud835\udc9c i) b ^ (m - i_1) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (i_1 - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - i_1),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (i_1 - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - i_1) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) })\n[PROOFSTEP]\nrw [map_sum, mk_sum]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 \u2211 i_1 in range (m + m + 1),\n Localization.mk (Nat.choose (m + m) i_1 \u2022 (\u2191(proj \ud835\udc9c i) a ^ i_1 * \u2191(proj \ud835\udc9c i) b ^ (m + m - i_1)))\n { val := f ^ (i + i),\n property :=\n (_ : \u2191{ val := f ^ (i + i), property := (_ : f ^ (i + i) \u2208 \ud835\udc9c (m * i + m * i)) } \u2208 Submonoid.powers f) } =\n \u2211 x in range (m + m + 1),\n \u2191(algebraMap (A\u2070_ f) (Away f))\n (Nat.choose (m + m) x \u2022\n if h2 : m + m < x then 0\n else\n if h1 : x \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m - x),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ x * \u2191(proj \ud835\udc9c i) b ^ (m - x) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (x - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - x),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (x - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - x) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) })\n[PROOFSTEP]\napply Finset.sum_congr rfl fun j hj => _\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 \u2200 (j : \u2115),\n j \u2208 range (m + m + 1) \u2192\n Localization.mk (Nat.choose (m + m) j \u2022 (\u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m + m - j)))\n { val := f ^ (i + i),\n property :=\n (_ : \u2191{ val := f ^ (i + i), property := (_ : f ^ (i + i) \u2208 \ud835\udc9c (m * i + m * i)) } \u2208 Submonoid.powers f) } =\n \u2191(algebraMap (A\u2070_ f) (Away f))\n (Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) })\n[PROOFSTEP]\nintro j hj\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\n\u22a2 Localization.mk (Nat.choose (m + m) j \u2022 (\u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m + m - j)))\n { val := f ^ (i + i),\n property :=\n (_ : \u2191{ val := f ^ (i + i), property := (_ : f ^ (i + i) \u2208 \ud835\udc9c (m * i + m * i)) } \u2208 Submonoid.powers f) } =\n \u2191(algebraMap (A\u2070_ f) (Away f))\n (Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) })\n[PROOFSTEP]\nchange _ = HomogeneousLocalization.val _\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\n\u22a2 Localization.mk (Nat.choose (m + m) j \u2022 (\u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m + m - j)))\n { val := f ^ (i + i),\n property :=\n (_ : \u2191{ val := f ^ (i + i), property := (_ : f ^ (i + i) \u2208 \ud835\udc9c (m * i + m * i)) } \u2208 Submonoid.powers f) } =\n HomogeneousLocalization.val\n (Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) })\n[PROOFSTEP]\nrw [HomogeneousLocalization.smul_val]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\n\u22a2 Localization.mk (Nat.choose (m + m) j \u2022 (\u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m + m - j)))\n { val := f ^ (i + i),\n property :=\n (_ : \u2191{ val := f ^ (i + i), property := (_ : f ^ (i + i) \u2208 \ud835\udc9c (m * i + m * i)) } \u2208 Submonoid.powers f) } =\n Nat.choose (m + m) j \u2022\n HomogeneousLocalization.val\n (if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) })\n[PROOFSTEP]\nsplit_ifs with h2 h1\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\nh2 : m + m < j\n\u22a2 Localization.mk (Nat.choose (m + m) j \u2022 (\u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m + m - j)))\n { val := f ^ (i + i),\n property :=\n (_ : \u2191{ val := f ^ (i + i), property := (_ : f ^ (i + i) \u2208 \ud835\udc9c (m * i + m * i)) } \u2208 Submonoid.powers f) } =\n Nat.choose (m + m) j \u2022 HomogeneousLocalization.val 0\n[PROOFSTEP]\nexact ((Finset.mem_range.1 hj).not_le h2).elim\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 Localization.mk (Nat.choose (m + m) j \u2022 (\u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m + m - j)))\n { val := f ^ (i + i),\n property :=\n (_ : \u2191{ val := f ^ (i + i), property := (_ : f ^ (i + i) \u2208 \ud835\udc9c (m * i + m * i)) } \u2208 Submonoid.powers f) } =\n Nat.choose (m + m) j \u2022\n HomogeneousLocalization.val\n (Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) })\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\n\u22a2 Localization.mk (Nat.choose (m + m) j \u2022 (\u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m + m - j)))\n { val := f ^ (i + i),\n property :=\n (_ : \u2191{ val := f ^ (i + i), property := (_ : f ^ (i + i) \u2208 \ud835\udc9c (m * i + m * i)) } \u2208 Submonoid.powers f) } =\n Nat.choose (m + m) j \u2022\n HomogeneousLocalization.val\n (Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) })\n[PROOFSTEP]\nall_goals\n simp only [HomogeneousLocalization.mul_val, HomogeneousLocalization.zero_val, HomogeneousLocalization.val_mk'',\n Subtype.coe_mk, mk_mul, \u2190 smul_mk];\n congr 2\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 Localization.mk (Nat.choose (m + m) j \u2022 (\u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m + m - j)))\n { val := f ^ (i + i),\n property :=\n (_ : \u2191{ val := f ^ (i + i), property := (_ : f ^ (i + i) \u2208 \ud835\udc9c (m * i + m * i)) } \u2208 Submonoid.powers f) } =\n Nat.choose (m + m) j \u2022\n HomogeneousLocalization.val\n (Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) })\n[PROOFSTEP]\nsimp only [HomogeneousLocalization.mul_val, HomogeneousLocalization.zero_val, HomogeneousLocalization.val_mk'',\n Subtype.coe_mk, mk_mul, \u2190 smul_mk]\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 Nat.choose (m + m) j \u2022\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m + m - j))\n { val := f ^ (i + i),\n property :=\n (_ : \u2191{ val := f ^ (i + i), property := (_ : f ^ (i + i) \u2208 \ud835\udc9c (m * i + m * i)) } \u2208 Submonoid.powers f) } =\n Nat.choose (m + m) j \u2022\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) * \u2191(proj \ud835\udc9c i) b ^ m)\n ({ val := f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) }.den \u2208\n Submonoid.powers f) } *\n { val := f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) }.den \u2208\n Submonoid.powers f) })\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\n\u22a2 Localization.mk (Nat.choose (m + m) j \u2022 (\u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m + m - j)))\n { val := f ^ (i + i),\n property :=\n (_ : \u2191{ val := f ^ (i + i), property := (_ : f ^ (i + i) \u2208 \ud835\udc9c (m * i + m * i)) } \u2208 Submonoid.powers f) } =\n Nat.choose (m + m) j \u2022\n HomogeneousLocalization.val\n (Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }, den_mem := (_ : \u2203 y, f ^ y = f ^ i) })\n[PROOFSTEP]\nsimp only [HomogeneousLocalization.mul_val, HomogeneousLocalization.zero_val, HomogeneousLocalization.val_mk'',\n Subtype.coe_mk, mk_mul, \u2190 smul_mk]\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\n\u22a2 Nat.choose (m + m) j \u2022\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m + m - j))\n { val := f ^ (i + i),\n property :=\n (_ : \u2191{ val := f ^ (i + i), property := (_ : f ^ (i + i) \u2208 \ud835\udc9c (m * i + m * i)) } \u2208 Submonoid.powers f) } =\n Nat.choose (m + m) j \u2022\n Localization.mk (\u2191(proj \ud835\udc9c i) a ^ m * (\u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j)))\n ({ val := f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) }.den \u2208\n Submonoid.powers f) } *\n { val := f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) }.den \u2208\n Submonoid.powers f) })\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase pos.e_a.e_x\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) = \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) * \u2191(proj \ud835\udc9c i) b ^ m\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase pos.e_a.e_x\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) a) i) ^ j * \u2191(\u2191(\u2191(decompose \ud835\udc9c) b) i) ^ (m + m - j) =\n \u2191(\u2191(\u2191(decompose \ud835\udc9c) a) i) ^ j * \u2191(\u2191(\u2191(decompose \ud835\udc9c) b) i) ^ (m - j) * \u2191(\u2191(\u2191(decompose \ud835\udc9c) b) i) ^ m\n[PROOFSTEP]\nrw [mul_assoc, \u2190 pow_add, add_comm (m - j), Nat.add_sub_assoc h1]\n[GOAL]\ncase pos.e_a.e_y\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 { val := f ^ (i + i),\n property :=\n (_ : \u2191{ val := f ^ (i + i), property := (_ : f ^ (i + i) \u2208 \ud835\udc9c (m * i + m * i)) } \u2208 Submonoid.powers f) } =\n { val := f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) }.den \u2208\n Submonoid.powers f) } *\n { val := f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) }.den \u2208\n Submonoid.powers f) }\n[PROOFSTEP]\nsimp_rw [pow_add]\n[GOAL]\ncase pos.e_a.e_y\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\nh2 : \u00acm + m < j\nh1 : j \u2264 m\n\u22a2 { val := f ^ i * f ^ i, property := (_ : (fun x => x \u2208 Submonoid.powers f) (f ^ i * f ^ i)) } =\n { val := f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) }.den \u2208\n Submonoid.powers f) } *\n { val := f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) }.den \u2208\n Submonoid.powers f) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg.e_a.e_x\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\n\u22a2 \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) =\n \u2191(proj \ud835\udc9c i) a ^ m * (\u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase neg.e_a.e_x\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) a) i) ^ j * \u2191(\u2191(\u2191(decompose \ud835\udc9c) b) i) ^ (m + m - j) =\n \u2191(\u2191(\u2191(decompose \ud835\udc9c) a) i) ^ m * (\u2191(\u2191(\u2191(decompose \ud835\udc9c) a) i) ^ (j - m) * \u2191(\u2191(\u2191(decompose \ud835\udc9c) b) i) ^ (m + m - j))\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 pow_add, Nat.add_sub_of_le (le_of_not_le h1)]\n[GOAL]\ncase neg.e_a.e_y\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\n\u22a2 { val := f ^ (i + i),\n property :=\n (_ : \u2191{ val := f ^ (i + i), property := (_ : f ^ (i + i) \u2208 \ud835\udc9c (m * i + m * i)) } \u2208 Submonoid.powers f) } =\n { val := f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) }.den \u2208\n Submonoid.powers f) } *\n { val := f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) }.den \u2208\n Submonoid.powers f) }\n[PROOFSTEP]\nsimp_rw [pow_add]\n[GOAL]\ncase neg.e_a.e_y\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\na b : A\nha : a \u2208 carrier f_deg q\nhb : b \u2208 carrier f_deg q\ni : \u2115\ng : \u2115 \u2192 A\u2070_ f :=\n fun j =>\n Nat.choose (m + m) j \u2022\n if h2 : m + m < j then 0\n else\n if h1 : j \u2264 m then\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ j * \u2191(proj \ud835\udc9c i) b ^ (m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) b ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) b) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\n else\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } *\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\nj : \u2115\nhj : j \u2208 range (m + m + 1)\nh2 : \u00acm + m < j\nh1 : \u00acj \u2264 m\n\u22a2 { val := f ^ i * f ^ i, property := (_ : (fun x => x \u2208 Submonoid.powers f) (f ^ i * f ^ i)) } =\n { val := f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) a) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) }.den \u2208\n Submonoid.powers f) } *\n { val := f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j),\n property := (_ : \u2191(proj \ud835\udc9c i) a ^ (j - m) * \u2191(proj \ud835\udc9c i) b ^ (m + m - j) \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, f ^ y = f ^ i) }.den \u2208\n Submonoid.powers f) }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\ni : \u2115\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) 0 ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) 0) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nconvert Submodule.zero_mem q.1 using 1\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\ni : \u2115\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) 0 ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) 0) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } =\n 0\n[PROOFSTEP]\nrw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'', HomogeneousLocalization.zero_val]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\ni : \u2115\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 Localization.mk\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) 0 ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) 0) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.num\n {\n val :=\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) 0 ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) 0) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) 0 ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) 0) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den \u2208\n Submonoid.powers f) } =\n 0\n[PROOFSTEP]\nsimp_rw [map_zero, zero_pow hm]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\ni : \u2115\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 Localization.mk 0\n { val := f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) 0 ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) 0) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den \u2208\n Submonoid.powers f) } =\n 0\n[PROOFSTEP]\nconvert Localization.mk_zero (S := Submonoid.powers f) _ using 1\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nc x : A\nhx : x \u2208 carrier f_deg q\n\u22a2 c \u2022 x \u2208 carrier f_deg q\n[PROOFSTEP]\nrevert c\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\n\u22a2 \u2200 (c : A), c \u2022 x \u2208 carrier f_deg q\n[PROOFSTEP]\nrefine' DirectSum.Decomposition.inductionOn \ud835\udc9c _ _ _\n[GOAL]\ncase refine'_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\n\u22a2 0 \u2022 x \u2208 carrier f_deg q\n[PROOFSTEP]\nrw [zero_smul]\n[GOAL]\ncase refine'_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\n\u22a2 0 \u2208 carrier f_deg q\n[PROOFSTEP]\nexact carrier.zero_mem f_deg hm _\n[GOAL]\ncase refine'_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\n\u22a2 \u2200 {i : \u2115} (m_1 : { x // x \u2208 \ud835\udc9c i }), \u2191m_1 \u2022 x \u2208 carrier f_deg q\n[PROOFSTEP]\nrintro n \u27e8a, ha\u27e9 i\n[GOAL]\ncase refine'_2.mk\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) (\u2191{ val := a, property := ha } \u2022 x) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n (\u2191{ val := a, property := ha } \u2022 x)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nsimp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem \ud835\udc9c i ha]\n -- Porting note: having trouble with Mul instance\n[GOAL]\ncase refine'_2.mk\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := (if n \u2264 i then a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) else 0) ^ m,\n property :=\n (_ : (fun x => x \u2208 \ud835\udc9c (m * i)) ((if n \u2264 i then a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) else 0) ^ m)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nlet product : A\u2070_ f :=\n Mul.mul (Quotient.mk'' \u27e8_, \u27e8a ^ m, pow_mem_graded m ha\u27e9, \u27e8_, ?_\u27e9, \u27e8n, rfl\u27e9\u27e9 : A\u2070_ f)\n (Quotient.mk'' \u27e8_, \u27e8proj \ud835\udc9c (i - n) x ^ m, by mem_tac\u27e9, \u27e8_, ?_\u27e9, \u27e8i - n, rfl\u27e9\u27e9 : A\u2070_ f)\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 \u2191(proj \ud835\udc9c (i - n)) x ^ m \u2208 \ud835\udc9c ?m.1735966\n[PROOFSTEP]\nmem_tac\n[GOAL]\ncase refine'_2.mk.refine_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\nproduct : A\u2070_ f :=\n Mul.mul\n (Quotient.mk''\n { deg := m \u2022 n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := ?refine'_2.mk.refine_1 }) })\n (Quotient.mk''\n { deg := m \u2022 (i - n),\n num := { val := \u2191(proj \ud835\udc9c (i - n)) x ^ m, property := (_ : \u2191(proj \ud835\udc9c (i - n)) x ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ (i - n), property := ?refine'_2.mk.refine_2 }) })\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := (if n \u2264 i then a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) else 0) ^ m,\n property :=\n (_ : (fun x => x \u2208 \ud835\udc9c (m * i)) ((if n \u2264 i then a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) else 0) ^ m)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\nproduct : A\u2070_ f :=\n Mul.mul\n (Quotient.mk''\n { deg := m \u2022 n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := ?refine'_2.mk.refine_1 }) })\n (Quotient.mk''\n { deg := m \u2022 (i - n),\n num := { val := \u2191(proj \ud835\udc9c (i - n)) x ^ m, property := (_ : \u2191(proj \ud835\udc9c (i - n)) x ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ (i - n), property := ?refine'_2.mk.refine_2 }) })\nh : n \u2264 i\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := (a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m,\n property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i)) ((a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nconvert_to product \u2208 q.1\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\nproduct : A\u2070_ f :=\n Mul.mul\n (Quotient.mk''\n { deg := m \u2022 n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := ?refine'_2.mk.refine_1 }) })\n (Quotient.mk''\n { deg := m \u2022 (i - n),\n num := { val := \u2191(proj \ud835\udc9c (i - n)) x ^ m, property := (_ : \u2191(proj \ud835\udc9c (i - n)) x ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ (i - n), property := ?refine'_2.mk.refine_2 }) })\nh : n \u2264 i\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := (a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m,\n property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i)) ((a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } =\n product\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\nproduct : A\u2070_ f :=\n Mul.mul\n (Quotient.mk''\n { deg := m \u2022 n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := ?refine'_2.mk.refine_1 }) })\n (Quotient.mk''\n { deg := m \u2022 (i - n),\n num := { val := \u2191(proj \ud835\udc9c (i - n)) x ^ m, property := (_ : \u2191(proj \ud835\udc9c (i - n)) x ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ (i - n), property := ?refine'_2.mk.refine_2 }) })\nh : n \u2264 i\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := (a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m,\n property := (_ : (a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } =\n Mul.mul\n (Quotient.mk''\n { deg := m * n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := ?refine'_2.mk.refine_1 }, den_mem := (_ : \u2203 y, f ^ y = f ^ n) })\n (Quotient.mk''\n { deg := m * (i - n),\n num :=\n { val := \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m,\n property := (_ : \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },\n den_mem := (_ : \u2203 y, f ^ y = f ^ (i - n)) })\n[PROOFSTEP]\nerw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'', HomogeneousLocalization.mul_val,\n HomogeneousLocalization.val_mk'', HomogeneousLocalization.val_mk'']\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\nproduct : A\u2070_ f :=\n Mul.mul\n (Quotient.mk''\n { deg := m \u2022 n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := ?refine'_2.mk.refine_1 }) })\n (Quotient.mk''\n { deg := m \u2022 (i - n),\n num := { val := \u2191(proj \ud835\udc9c (i - n)) x ^ m, property := (_ : \u2191(proj \ud835\udc9c (i - n)) x ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ (i - n), property := ?refine'_2.mk.refine_2 }) })\nh : n \u2264 i\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 Localization.mk\n \u2191{ deg := m * i,\n num :=\n { val := (a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m,\n property := (_ : (a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.num\n {\n val :=\n \u2191{ deg := m * i,\n num :=\n { val := (a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m,\n property := (_ : (a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := (a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m,\n property := (_ : (a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den \u2208\n Submonoid.powers f) } =\n Localization.mk\n \u2191{ deg := m * n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := ?refine'_2.mk.refine_1 }, den_mem := (_ : \u2203 y, f ^ y = f ^ n) }.num\n {\n val :=\n \u2191{ deg := m * n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },\n den_mem := (_ : \u2203 y, f ^ y = f ^ n) }.den,\n property :=\n (_ :\n \u2191{ deg := m * n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },\n den_mem := (_ : \u2203 y, f ^ y = f ^ n) }.den \u2208\n Submonoid.powers f) } *\n Localization.mk\n \u2191{ deg := m * (i - n),\n num :=\n { val := \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m,\n property := (_ : \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },\n den_mem := (_ : \u2203 y, f ^ y = f ^ (i - n)) }.num\n {\n val :=\n \u2191{ deg := m * (i - n),\n num :=\n { val := \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m,\n property := (_ : \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },\n den_mem := (_ : \u2203 y, f ^ y = f ^ (i - n)) }.den,\n property :=\n (_ :\n \u2191{ deg := m * (i - n),\n num :=\n { val := \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m,\n property := (_ : \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },\n den_mem := (_ : \u2203 y, f ^ y = f ^ (i - n)) }.den \u2208\n Submonoid.powers f) }\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\n[PROOFSTEP]\nsimp_rw [mul_pow]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\nproduct : A\u2070_ f :=\n Mul.mul\n (Quotient.mk''\n { deg := m \u2022 n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := ?refine'_2.mk.refine_1 }) })\n (Quotient.mk''\n { deg := m \u2022 (i - n),\n num := { val := \u2191(proj \ud835\udc9c (i - n)) x ^ m, property := (_ : \u2191(proj \ud835\udc9c (i - n)) x ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ (i - n), property := ?refine'_2.mk.refine_2 }) })\nh : n \u2264 i\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 Localization.mk (a ^ m * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m)\n { val := f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := (a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m,\n property := (_ : (a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den \u2208\n Submonoid.powers f) } =\n Localization.mk (a ^ m)\n { val := f ^ n,\n property :=\n (_ :\n \u2191{ deg := m * n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },\n den_mem := (_ : \u2203 y, f ^ y = f ^ n) }.den \u2208\n Submonoid.powers f) } *\n Localization.mk (\u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m)\n { val := f ^ (i - n),\n property :=\n (_ :\n \u2191{ deg := m * (i - n),\n num :=\n { val := \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m,\n property := (_ : \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },\n den_mem := (_ : \u2203 y, f ^ y = f ^ (i - n)) }.den \u2208\n Submonoid.powers f) }\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\n[PROOFSTEP]\nrw [Localization.mk_mul]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\nproduct : A\u2070_ f :=\n Mul.mul\n (Quotient.mk''\n { deg := m \u2022 n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := ?refine'_2.mk.refine_1 }) })\n (Quotient.mk''\n { deg := m \u2022 (i - n),\n num := { val := \u2191(proj \ud835\udc9c (i - n)) x ^ m, property := (_ : \u2191(proj \ud835\udc9c (i - n)) x ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ (i - n), property := ?refine'_2.mk.refine_2 }) })\nh : n \u2264 i\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 Localization.mk (a ^ m * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m)\n { val := f ^ i,\n property :=\n (_ :\n \u2191{ deg := m * i,\n num :=\n { val := (a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m,\n property := (_ : (a * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n))) ^ m \u2208 \ud835\udc9c (m * i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) }.den \u2208\n Submonoid.powers f) } =\n Localization.mk (a ^ m * \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m)\n ({ val := f ^ n,\n property :=\n (_ :\n \u2191{ deg := m * n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },\n den_mem := (_ : \u2203 y, f ^ y = f ^ n) }.den \u2208\n Submonoid.powers f) } *\n { val := f ^ (i - n),\n property :=\n (_ :\n \u2191{ deg := m * (i - n),\n num :=\n { val := \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m,\n property := (_ : \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },\n den_mem := (_ : \u2203 y, f ^ y = f ^ (i - n)) }.den \u2208\n Submonoid.powers f) })\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e'_4.h.e_y.e_val\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\nproduct : A\u2070_ f :=\n Mul.mul\n (Quotient.mk''\n { deg := m \u2022 n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := ?refine'_2.mk.refine_1 }) })\n (Quotient.mk''\n { deg := m \u2022 (i - n),\n num := { val := \u2191(proj \ud835\udc9c (i - n)) x ^ m, property := (_ : \u2191(proj \ud835\udc9c (i - n)) x ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ (i - n), property := ?refine'_2.mk.refine_2 }) })\nh : n \u2264 i\ne_1\u271d : Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) = A\u2070_ f\n\u22a2 f ^ i =\n \u2191{ val := f ^ n,\n property :=\n (_ :\n \u2191{ deg := m * n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := ?refine'_2.mk.refine_1 },\n den_mem := (_ : \u2203 y, f ^ y = f ^ n) }.den \u2208\n Submonoid.powers f) } *\n \u2191{ val := f ^ (i - n),\n property :=\n (_ :\n \u2191{ deg := m * (i - n),\n num :=\n { val := \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m,\n property := (_ : \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) (i - n)) ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := ?refine'_2.mk.refine_2 },\n den_mem := (_ : \u2203 y, f ^ y = f ^ (i - n)) }.den \u2208\n Submonoid.powers f) }\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\n[PROOFSTEP]\nerw [\u2190 pow_add, Nat.add_sub_of_le h]\n[GOAL]\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c (m \u2022 n)\n[PROOFSTEP]\nrw [(_ : m \u2022 n = _)]\n[GOAL]\ncase refine'_2.mk.refine_1\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ n \u2208 \ud835\udc9c ?m.1761702\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 \u2115\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 m \u2022 n = ?m.1761702\n[PROOFSTEP]\nmem_tac\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 m \u2022 n = n \u2022 m\n[PROOFSTEP]\nsimp only [smul_eq_mul, mul_comm]\n[GOAL]\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))\n[PROOFSTEP]\nrw [(_ : m \u2022 (i - n) = _)]\n[GOAL]\ncase refine'_2.mk.refine_2\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 f ^ (i - n) \u2208 \ud835\udc9c ?m.1764150\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 \u2115\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 m \u2022 (i - n) = ?m.1764150\n[PROOFSTEP]\nmem_tac\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\n\u22a2 m \u2022 (i - n) = (i - n) \u2022 m\n[PROOFSTEP]\nsimp only [smul_eq_mul, mul_comm]\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\nproduct : A\u2070_ f :=\n Mul.mul\n (Quotient.mk''\n { deg := m \u2022 n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m \u2022 n)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m \u2022 n)) }) })\n (Quotient.mk''\n { deg := m \u2022 (i - n),\n num := { val := \u2191(proj \ud835\udc9c (i - n)) x ^ m, property := (_ : \u2191(proj \ud835\udc9c (i - n)) x ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := (_ : f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ (i - n), property := (_ : f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))) }) })\nh : n \u2264 i\n\u22a2 product \u2208 q.asIdeal\n[PROOFSTEP]\napply Ideal.mul_mem_left (\u03b1 := A\u2070_ f) _ _ (hx _)\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\nproduct : A\u2070_ f :=\n Mul.mul\n (Quotient.mk''\n { deg := m \u2022 n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m \u2022 n)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m \u2022 n)) }) })\n (Quotient.mk''\n { deg := m \u2022 (i - n),\n num := { val := \u2191(proj \ud835\udc9c (i - n)) x ^ m, property := (_ : \u2191(proj \ud835\udc9c (i - n)) x ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := (_ : f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ (i - n), property := (_ : f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))) }) })\nh : \u00acn \u2264 i\n\u22a2 Quotient.mk''\n { deg := m * i, num := { val := 0 ^ m, property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i)) (0 ^ m)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nsimp_rw [zero_pow hm]\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\nproduct : A\u2070_ f :=\n Mul.mul\n (Quotient.mk''\n { deg := m \u2022 n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m \u2022 n)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m \u2022 n)) }) })\n (Quotient.mk''\n { deg := m \u2022 (i - n),\n num := { val := \u2191(proj \ud835\udc9c (i - n)) x ^ m, property := (_ : \u2191(proj \ud835\udc9c (i - n)) x ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := (_ : f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ (i - n), property := (_ : f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))) }) })\nh : \u00acn \u2264 i\n\u22a2 Quotient.mk''\n { deg := m * i, num := { val := 0, property := (_ : (fun x => x \u2208 \ud835\udc9c (m * i)) 0) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nconvert carrier.zero_mem f_deg hm q i\n[GOAL]\ncase h.e'_4.h.e'_3.h.e'_10.h.e'_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\nn : \u2115\na : A\nha : a \u2208 \ud835\udc9c n\ni : \u2115\nproduct : A\u2070_ f :=\n Mul.mul\n (Quotient.mk''\n { deg := m \u2022 n, num := { val := a ^ m, property := (_ : a ^ m \u2208 \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m \u2022 n)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m \u2022 n)) }) })\n (Quotient.mk''\n { deg := m \u2022 (i - n),\n num := { val := \u2191(proj \ud835\udc9c (i - n)) x ^ m, property := (_ : \u2191(proj \ud835\udc9c (i - n)) x ^ m \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den := { val := f ^ (i - n), property := (_ : f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ (i - n), property := (_ : f ^ (i - n) \u2208 \ud835\udc9c (m \u2022 (i - n))) }) })\nh : \u00acn \u2264 i\n\u22a2 0 = \u2191(proj \ud835\udc9c i) 0 ^ m\n[PROOFSTEP]\nrw [map_zero, zero_pow hm]\n[GOAL]\ncase refine'_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\n\u22a2 \u2200 (m_1 m' : A), m_1 \u2022 x \u2208 carrier f_deg q \u2192 m' \u2022 x \u2208 carrier f_deg q \u2192 (m_1 + m') \u2022 x \u2208 carrier f_deg q\n[PROOFSTEP]\nsimp_rw [add_smul]\n[GOAL]\ncase refine'_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx : A\nhx : x \u2208 carrier f_deg q\n\u22a2 \u2200 (m_1 m' : A), m_1 \u2022 x \u2208 carrier f_deg q \u2192 m' \u2022 x \u2208 carrier f_deg q \u2192 m_1 \u2022 x + m' \u2022 x \u2208 carrier f_deg q\n[PROOFSTEP]\nexact fun _ _ => carrier.add_mem f_deg q\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\ni : \u2115\na : A\nha : a \u2208 asIdeal f_deg hm q\nj : \u2115\nh : i = j\n\u22a2 Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) \u2191(\u2191(\u2191(decompose \ud835\udc9c) a) i) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n \u2191(\u2191(\u2191(decompose \ud835\udc9c) a) i)) ^\n m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nsimpa only [proj_apply, decompose_coe, of_eq_same] using ha _\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\ni : \u2115\na : A\nha : a \u2208 asIdeal f_deg hm q\nj : \u2115\nh : \u00aci = j\n\u22a2 Quotient.mk''\n { deg := m * j,\n num :=\n { val := \u2191(proj \ud835\udc9c j) \u2191(\u2191(\u2191(decompose \ud835\udc9c) a) i) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply j) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c)))\n \u2191(\u2191(\u2191(decompose \ud835\udc9c) a) i)) ^\n m \u2208\n \ud835\udc9c (m \u2022 j)) },\n den := { val := f ^ j, property := (_ : f ^ j \u2208 \ud835\udc9c (m * j)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ j, property := (_ : f ^ j \u2208 \ud835\udc9c (m * j)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nsimp only [proj_apply, decompose_of_mem_ne \ud835\udc9c (Submodule.coe_mem (decompose \ud835\udc9c a i)) h, zero_pow hm]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\ni : \u2115\na : A\nha : a \u2208 asIdeal f_deg hm q\nj : \u2115\nh : \u00aci = j\n\u22a2 Quotient.mk''\n { deg := m * j, num := { val := 0, property := (_ : (fun x => x \u2208 \ud835\udc9c (m * j)) 0) },\n den := { val := f ^ j, property := (_ : f ^ j \u2208 \ud835\udc9c (m * j)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ j, property := (_ : f ^ j \u2208 \ud835\udc9c (m * j)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nconvert carrier.zero_mem f_deg hm q j\n[GOAL]\ncase h.e'_4.h.e'_3.h.e'_10.h.e'_3\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\ni : \u2115\na : A\nha : a \u2208 asIdeal f_deg hm q\nj : \u2115\nh : \u00aci = j\n\u22a2 0 = \u2191(proj \ud835\udc9c j) 0 ^ m\n[PROOFSTEP]\nrw [map_zero, zero_pow hm]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nrid : f \u2208 asIdeal f_deg hm q\n\u22a2 1 \u2208 q.asIdeal\n[PROOFSTEP]\nconvert rid m\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nrid : f \u2208 asIdeal f_deg hm q\ne_1\u271d :\n \u2191(CommRingCat.of (A\u2070_ f)) =\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f)))\n\u22a2 1 =\n Quotient.mk''\n { deg := m * m,\n num :=\n { val := \u2191(proj \ud835\udc9c m) f ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply m) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) f) ^ m \u2208\n \ud835\udc9c (m \u2022 m)) },\n den := { val := f ^ m, property := (_ : f ^ m \u2208 \ud835\udc9c (m * m)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ m, property := (_ : f ^ m \u2208 \ud835\udc9c (m * m)) }) }\n[PROOFSTEP]\nrw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.one_val, HomogeneousLocalization.val_mk'']\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nrid : f \u2208 asIdeal f_deg hm q\ne_1\u271d :\n \u2191(CommRingCat.of (A\u2070_ f)) =\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f)))\n\u22a2 1 =\n Localization.mk\n \u2191{ deg := m * m,\n num :=\n { val := \u2191(proj \ud835\udc9c m) f ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply m) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) f) ^ m \u2208\n \ud835\udc9c (m \u2022 m)) },\n den := { val := f ^ m, property := (_ : f ^ m \u2208 \ud835\udc9c (m * m)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ m, property := (_ : f ^ m \u2208 \ud835\udc9c (m * m)) }) }.num\n {\n val :=\n \u2191{ deg := m * m,\n num :=\n { val := \u2191(proj \ud835\udc9c m) f ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply m) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) f) ^ m \u2208\n \ud835\udc9c (m \u2022 m)) },\n den := { val := f ^ m, property := (_ : f ^ m \u2208 \ud835\udc9c (m * m)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ m, property := (_ : f ^ m \u2208 \ud835\udc9c (m * m)) }) }.den,\n property :=\n (_ :\n \u2191{ deg := m * m,\n num :=\n { val := \u2191(proj \ud835\udc9c m) f ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply m) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) f) ^ m \u2208\n \ud835\udc9c (m \u2022 m)) },\n den := { val := f ^ m, property := (_ : f ^ m \u2208 \ud835\udc9c (m * m)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ m, property := (_ : f ^ m \u2208 \ud835\udc9c (m * m)) }) }.den \u2208\n Submonoid.powers f) }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nrid : f \u2208 asIdeal f_deg hm q\ne_1\u271d :\n \u2191(CommRingCat.of (A\u2070_ f)) =\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f)))\n\u22a2 1 =\n Localization.mk (\u2191(\u2191(\u2191(decompose \ud835\udc9c) f) m) ^ m)\n { val := f ^ m,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ m, property := (_ : f ^ m \u2208 \ud835\udc9c (m * m)) }) }\n[PROOFSTEP]\nsimp_rw [decompose_of_mem_same _ f_deg]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nrid : f \u2208 asIdeal f_deg hm q\ne_1\u271d :\n \u2191(CommRingCat.of (A\u2070_ f)) =\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f)))\n\u22a2 1 =\n Localization.mk (f ^ m)\n { val := f ^ m,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ m, property := (_ : f ^ m \u2208 \ud835\udc9c (m * m)) }) }\n[PROOFSTEP]\nsimp only [mk_eq_monoidOf_mk', Submonoid.LocalizationMap.mk'_self]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\n\u22a2 (\u2200 (i : \u2115),\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) x ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) x) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal) \u2228\n \u2200 (i : \u2115),\n Quotient.mk''\n { deg := m * i,\n num :=\n { val := \u2191(proj \ud835\udc9c i) y ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply i) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) y) ^ m \u2208\n \ud835\udc9c (m \u2022 i)) },\n den := { val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ i, property := (_ : f ^ i \u2208 \ud835\udc9c (m * i)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nrw [\u2190 and_forall_ne nx, and_iff_left, \u2190 and_forall_ne ny, and_iff_left]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\n\u22a2 Quotient.mk''\n { deg := m * nx,\n num :=\n { val := \u2191(proj \ud835\udc9c nx) x ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply nx) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) x) ^ m \u2208\n \ud835\udc9c (m \u2022 nx)) },\n den := { val := f ^ nx, property := (_ : f ^ nx \u2208 \ud835\udc9c (m * nx)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ nx, property := (_ : f ^ nx \u2208 \ud835\udc9c (m * nx)) }) } \u2208\n q.asIdeal \u2228\n Quotient.mk''\n { deg := m * ny,\n num :=\n { val := \u2191(proj \ud835\udc9c ny) y ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply ny) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) y) ^ m \u2208\n \ud835\udc9c (m \u2022 ny)) },\n den := { val := f ^ ny, property := (_ : f ^ ny \u2208 \ud835\udc9c (m * ny)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ ny, property := (_ : f ^ ny \u2208 \ud835\udc9c (m * ny)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\napply q.2.mem_or_mem\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\n\u22a2 Quotient.mk''\n { deg := m * nx,\n num :=\n { val := \u2191(proj \ud835\udc9c nx) x ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply nx) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) x) ^ m \u2208\n \ud835\udc9c (m \u2022 nx)) },\n den := { val := f ^ nx, property := (_ : f ^ nx \u2208 \ud835\udc9c (m * nx)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ nx, property := (_ : f ^ nx \u2208 \ud835\udc9c (m * nx)) }) } *\n Quotient.mk''\n { deg := m * ny,\n num :=\n { val := \u2191(proj \ud835\udc9c ny) y ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply ny) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) y) ^ m \u2208\n \ud835\udc9c (m \u2022 ny)) },\n den := { val := f ^ ny, property := (_ : f ^ ny \u2208 \ud835\udc9c (m * ny)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ ny, property := (_ : f ^ ny \u2208 \ud835\udc9c (m * ny)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nconvert hxy (nx + ny) using 1\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\ne_1\u271d :\n \u2191(CommRingCat.of (A\u2070_ f)) =\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f)))\n\u22a2 Quotient.mk''\n { deg := m * nx,\n num :=\n { val := \u2191(proj \ud835\udc9c nx) x ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply nx) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) x) ^ m \u2208\n \ud835\udc9c (m \u2022 nx)) },\n den := { val := f ^ nx, property := (_ : f ^ nx \u2208 \ud835\udc9c (m * nx)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ nx, property := (_ : f ^ nx \u2208 \ud835\udc9c (m * nx)) }) } *\n Quotient.mk''\n { deg := m * ny,\n num :=\n { val := \u2191(proj \ud835\udc9c ny) y ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply ny) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) y) ^ m \u2208\n \ud835\udc9c (m \u2022 ny)) },\n den := { val := f ^ ny, property := (_ : f ^ ny \u2208 \ud835\udc9c (m * ny)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ ny, property := (_ : f ^ ny \u2208 \ud835\udc9c (m * ny)) }) } =\n Quotient.mk''\n { deg := m * (nx + ny),\n num :=\n { val := \u2191(proj \ud835\udc9c (nx + ny)) (x * y) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply (nx + ny)) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) (x * y)) ^\n m \u2208\n \ud835\udc9c (m \u2022 (nx + ny))) },\n den := { val := f ^ (nx + ny), property := (_ : f ^ (nx + ny) \u2208 \ud835\udc9c (m * (nx + ny))) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ (nx + ny), property := (_ : f ^ (nx + ny) \u2208 \ud835\udc9c (m * (nx + ny))) }) }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\ne_1\u271d :\n \u2191(CommRingCat.of (A\u2070_ f)) =\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f)))\n\u22a2 Quotient.mk''\n { deg := m * nx,\n num :=\n { val := \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) nx) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply nx) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) x) ^ m \u2208\n \ud835\udc9c (m \u2022 nx)) },\n den := { val := f ^ nx, property := (_ : f ^ nx \u2208 \ud835\udc9c (m * nx)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ nx, property := (_ : f ^ nx \u2208 \ud835\udc9c (m * nx)) }) } *\n Quotient.mk''\n { deg := m * ny,\n num :=\n { val := \u2191(\u2191(\u2191(decompose \ud835\udc9c) y) ny) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply ny) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) y) ^ m \u2208\n \ud835\udc9c (m \u2022 ny)) },\n den := { val := f ^ ny, property := (_ : f ^ ny \u2208 \ud835\udc9c (m * ny)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ ny, property := (_ : f ^ ny \u2208 \ud835\udc9c (m * ny)) }) } =\n Quotient.mk''\n { deg := m * (nx + ny),\n num :=\n { val := \u2191(\u2191(\u2191(decompose \ud835\udc9c) (x * y)) (nx + ny)) ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply (nx + ny)) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) (x * y)) ^\n m \u2208\n \ud835\udc9c (m \u2022 (nx + ny))) },\n den := { val := f ^ (nx + ny), property := (_ : f ^ (nx + ny) \u2208 \ud835\udc9c (m * (nx + ny))) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ (nx + ny), property := (_ : f ^ (nx + ny) \u2208 \ud835\udc9c (m * (nx + ny))) }) }\n[PROOFSTEP]\nsimp_rw [decompose_of_mem_same \ud835\udc9c hnx, decompose_of_mem_same \ud835\udc9c hny,\n decompose_of_mem_same \ud835\udc9c (SetLike.GradedMonoid.toGradedMul.mul_mem hnx hny), mul_pow, pow_add]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\ne_1\u271d :\n \u2191(CommRingCat.of (A\u2070_ f)) =\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f)))\n\u22a2 Quotient.mk''\n { deg := m * nx, num := { val := x ^ m, property := (_ : (fun x => x \u2208 \ud835\udc9c (m * nx)) (x ^ m)) },\n den := { val := f ^ nx, property := (_ : f ^ nx \u2208 \ud835\udc9c (m * nx)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ nx, property := (_ : f ^ nx \u2208 \ud835\udc9c (m * nx)) }) } *\n Quotient.mk''\n { deg := m * ny, num := { val := y ^ m, property := (_ : (fun x => x \u2208 \ud835\udc9c (m * ny)) (y ^ m)) },\n den := { val := f ^ ny, property := (_ : f ^ ny \u2208 \ud835\udc9c (m * ny)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ ny, property := (_ : f ^ ny \u2208 \ud835\udc9c (m * ny)) }) } =\n Quotient.mk''\n { deg := m * (nx + ny),\n num := { val := x ^ m * y ^ m, property := (_ : (fun x => x \u2208 \ud835\udc9c (m * (nx + ny))) (x ^ m * y ^ m)) },\n den := { val := f ^ nx * f ^ ny, property := (_ : (fun x => x \u2208 \ud835\udc9c (m * (nx + ny))) (f ^ nx * f ^ ny)) },\n den_mem :=\n (_ :\n \u2191{ val := f ^ nx * f ^ ny, property := (_ : (fun x => x \u2208 \ud835\udc9c (m * (nx + ny))) (f ^ nx * f ^ ny)) } \u2208\n Submonoid.powers f) }\n[PROOFSTEP]\nsimp only [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'', HomogeneousLocalization.mul_val,\n mk_mul]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\ne_1\u271d :\n \u2191(CommRingCat.of (A\u2070_ f)) =\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f)))\n\u22a2 Localization.mk (x ^ m * y ^ m)\n ({ val := f ^ nx,\n property :=\n (_ :\n \u2191{ deg := m * nx, num := { val := x ^ m, property := (_ : (fun x => x \u2208 \ud835\udc9c (m * nx)) (x ^ m)) },\n den := { val := f ^ nx, property := (_ : f ^ nx \u2208 \ud835\udc9c (m * nx)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ nx, property := (_ : f ^ nx \u2208 \ud835\udc9c (m * nx)) }) }.den \u2208\n Submonoid.powers f) } *\n { val := f ^ ny,\n property :=\n (_ :\n \u2191{ deg := m * ny, num := { val := y ^ m, property := (_ : (fun x => x \u2208 \ud835\udc9c (m * ny)) (y ^ m)) },\n den := { val := f ^ ny, property := (_ : f ^ ny \u2208 \ud835\udc9c (m * ny)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y =\n \u2191{ val := f ^ ny, property := (_ : f ^ ny \u2208 \ud835\udc9c (m * ny)) }) }.den \u2208\n Submonoid.powers f) }) =\n Localization.mk (x ^ m * y ^ m)\n { val := f ^ nx * f ^ ny,\n property :=\n (_ :\n \u2191{ deg := m * (nx + ny),\n num := { val := x ^ m * y ^ m, property := (_ : (fun x => x \u2208 \ud835\udc9c (m * (nx + ny))) (x ^ m * y ^ m)) },\n den :=\n { val := f ^ nx * f ^ ny, property := (_ : (fun x => x \u2208 \ud835\udc9c (m * (nx + ny))) (f ^ nx * f ^ ny)) },\n den_mem :=\n (_ :\n \u2191{ val := f ^ nx * f ^ ny,\n property := (_ : (fun x => x \u2208 \ud835\udc9c (m * (nx + ny))) (f ^ nx * f ^ ny)) } \u2208\n Submonoid.powers f) }.den \u2208\n Submonoid.powers f) }\n[PROOFSTEP]\nsimp only [Submonoid.mk_mul_mk, mk_eq_monoidOf_mk']\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\n\u22a2 \u2200 (b : \u2115),\n b \u2260 ny \u2192\n Quotient.mk''\n { deg := m * b,\n num :=\n { val := \u2191(proj \ud835\udc9c b) y ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply b) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) y) ^ m \u2208\n \ud835\udc9c (m \u2022 b)) },\n den := { val := f ^ b, property := (_ : f ^ b \u2208 \ud835\udc9c (m * b)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ b, property := (_ : f ^ b \u2208 \ud835\udc9c (m * b)) }) } \u2208\n q.asIdeal\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\n\u22a2 \u2200 (b : \u2115),\n b \u2260 nx \u2192\n Quotient.mk''\n { deg := m * b,\n num :=\n { val := \u2191(proj \ud835\udc9c b) x ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply b) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) x) ^ m \u2208\n \ud835\udc9c (m \u2022 b)) },\n den := { val := f ^ b, property := (_ : f ^ b \u2208 \ud835\udc9c (m * b)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ b, property := (_ : f ^ b \u2208 \ud835\udc9c (m * b)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nall_goals\n intro n hn; convert q.1.zero_mem using 1\n rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'', HomogeneousLocalization.zero_val];\n simp_rw [proj_apply]\n convert mk_zero (S := Submonoid.powers f) _\n rw [decompose_of_mem_ne \ud835\udc9c _ hn.symm, zero_pow hm]\n \u00b7\n first\n | exact hnx\n | exact hny\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\n\u22a2 \u2200 (b : \u2115),\n b \u2260 ny \u2192\n Quotient.mk''\n { deg := m * b,\n num :=\n { val := \u2191(proj \ud835\udc9c b) y ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply b) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) y) ^ m \u2208\n \ud835\udc9c (m \u2022 b)) },\n den := { val := f ^ b, property := (_ : f ^ b \u2208 \ud835\udc9c (m * b)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ b, property := (_ : f ^ b \u2208 \ud835\udc9c (m * b)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nintro n hn\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\nn : \u2115\nhn : n \u2260 ny\n\u22a2 Quotient.mk''\n { deg := m * n,\n num :=\n { val := \u2191(proj \ud835\udc9c n) y ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply n) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) y) ^ m \u2208\n \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nconvert q.1.zero_mem using 1\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\nn : \u2115\nhn : n \u2260 ny\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 Quotient.mk''\n { deg := m * n,\n num :=\n { val := \u2191(proj \ud835\udc9c n) y ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply n) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) y) ^ m \u2208\n \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) }) } =\n 0\n[PROOFSTEP]\nrw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'', HomogeneousLocalization.zero_val]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\nn : \u2115\nhn : n \u2260 ny\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 Localization.mk\n \u2191{ deg := m * n,\n num :=\n { val := \u2191(proj \ud835\udc9c n) y ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply n) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) y) ^ m \u2208\n \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) }) }.num\n {\n val :=\n \u2191{ deg := m * n,\n num :=\n { val := \u2191(proj \ud835\udc9c n) y ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply n) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) y) ^ m \u2208\n \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) }) }.den,\n property :=\n (_ :\n \u2191{ deg := m * n,\n num :=\n { val := \u2191(proj \ud835\udc9c n) y ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply n) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) y) ^ m \u2208\n \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) }) }.den \u2208\n Submonoid.powers f) } =\n 0\n[PROOFSTEP]\nsimp_rw [proj_apply]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\nn : \u2115\nhn : n \u2260 ny\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 Localization.mk (\u2191(\u2191(\u2191(decompose \ud835\udc9c) y) n) ^ m)\n { val := f ^ n,\n property :=\n (_ :\n \u2191{ deg := m * n,\n num :=\n { val := \u2191(proj \ud835\udc9c n) y ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply n) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) y) ^ m \u2208\n \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) }) }.den \u2208\n Submonoid.powers f) } =\n 0\n[PROOFSTEP]\nconvert mk_zero (S := Submonoid.powers f) _\n[GOAL]\ncase h.e'_2.h.e'_4\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\nn : \u2115\nhn : n \u2260 ny\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) y) n) ^ m = 0\n[PROOFSTEP]\nrw [decompose_of_mem_ne \ud835\udc9c _ hn.symm, zero_pow hm]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\nn : \u2115\nhn : n \u2260 ny\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 y \u2208 \ud835\udc9c ny\n[PROOFSTEP]\nfirst\n| exact hnx\n| exact hny\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\nn : \u2115\nhn : n \u2260 ny\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 y \u2208 \ud835\udc9c ny\n[PROOFSTEP]\nexact hnx\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\nn : \u2115\nhn : n \u2260 ny\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 y \u2208 \ud835\udc9c ny\n[PROOFSTEP]\nexact hny\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\n\u22a2 \u2200 (b : \u2115),\n b \u2260 nx \u2192\n Quotient.mk''\n { deg := m * b,\n num :=\n { val := \u2191(proj \ud835\udc9c b) x ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply b) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) x) ^ m \u2208\n \ud835\udc9c (m \u2022 b)) },\n den := { val := f ^ b, property := (_ : f ^ b \u2208 \ud835\udc9c (m * b)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ b, property := (_ : f ^ b \u2208 \ud835\udc9c (m * b)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nintro n hn\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\nn : \u2115\nhn : n \u2260 nx\n\u22a2 Quotient.mk''\n { deg := m * n,\n num :=\n { val := \u2191(proj \ud835\udc9c n) x ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply n) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) x) ^ m \u2208\n \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) }) } \u2208\n q.asIdeal\n[PROOFSTEP]\nconvert q.1.zero_mem using 1\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\nn : \u2115\nhn : n \u2260 nx\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 Quotient.mk''\n { deg := m * n,\n num :=\n { val := \u2191(proj \ud835\udc9c n) x ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply n) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) x) ^ m \u2208\n \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) },\n den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) }) } =\n 0\n[PROOFSTEP]\nrw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'', HomogeneousLocalization.zero_val]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\nn : \u2115\nhn : n \u2260 nx\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 Localization.mk\n \u2191{ deg := m * n,\n num :=\n { val := \u2191(proj \ud835\udc9c n) x ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply n) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) x) ^ m \u2208\n \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) }) }.num\n {\n val :=\n \u2191{ deg := m * n,\n num :=\n { val := \u2191(proj \ud835\udc9c n) x ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply n) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) x) ^ m \u2208\n \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) },\n den_mem :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) }) }.den,\n property :=\n (_ :\n \u2191{ deg := m * n,\n num :=\n { val := \u2191(proj \ud835\udc9c n) x ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply n) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) x) ^ m \u2208\n \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) }) }.den \u2208\n Submonoid.powers f) } =\n 0\n[PROOFSTEP]\nsimp_rw [proj_apply]\n[GOAL]\ncase h.e'_4.h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\nn : \u2115\nhn : n \u2260 nx\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 Localization.mk (\u2191(\u2191(\u2191(decompose \ud835\udc9c) x) n) ^ m)\n { val := f ^ n,\n property :=\n (_ :\n \u2191{ deg := m * n,\n num :=\n { val := \u2191(proj \ud835\udc9c n) x ^ m,\n property :=\n (_ :\n \u2191(\u2191(LinearMap.comp (DFinsupp.lapply n) (AlgHom.toLinearMap \u2191(decomposeAlgEquiv \ud835\udc9c))) x) ^ m \u2208\n \ud835\udc9c (m \u2022 n)) },\n den := { val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) },\n den_mem :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) f y = \u2191{ val := f ^ n, property := (_ : f ^ n \u2208 \ud835\udc9c (m * n)) }) }.den \u2208\n Submonoid.powers f) } =\n 0\n[PROOFSTEP]\nconvert mk_zero (S := Submonoid.powers f) _\n[GOAL]\ncase h.e'_2.h.e'_4\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\nn : \u2115\nhn : n \u2260 nx\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) n) ^ m = 0\n[PROOFSTEP]\nrw [decompose_of_mem_ne \ud835\udc9c _ hn.symm, zero_pow hm]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\nn : \u2115\nhn : n \u2260 nx\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 x \u2208 \ud835\udc9c nx\n[PROOFSTEP]\nfirst\n| exact hnx\n| exact hny\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nf : A\nm : \u2115\nf_deg : f \u2208 \ud835\udc9c m\nhm : 0 < m\nq : \u2191\u2191(Spec A\u2070_ f).toSheafedSpace.toPresheafedSpace\nx y : A\nx\u271d\u00b9 : Homogeneous \ud835\udc9c x\nx\u271d : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 asIdeal f_deg hm q\nnx : \u2115\nhnx : x \u2208 \ud835\udc9c nx\nny : \u2115\nhny : y \u2208 \ud835\udc9c ny\nn : \u2115\nhn : n \u2260 nx\ne_1\u271d :\n Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding \ud835\udc9c (Submonoid.powers f))) =\n \u2191(CommRingCat.of (A\u2070_ f))\n\u22a2 x \u2208 \ud835\udc9c nx\n[PROOFSTEP]\nexact hnx\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme", "llama_tokens": 221168, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833841649233, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.408450617205427}} {"text": "[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU : Set \u03b1\n\u03b5 : \u211d\u22650\u221e\nH : InnerRegular \u03bc p q\nhU : q U\n\u22a2 \u2191\u2191\u03bc U = \u2a06 (K : Set \u03b1) (_ : K \u2286 U) (_ : p K), \u2191\u2191\u03bc K\n[PROOFSTEP]\nrefine' le_antisymm (le_of_forall_lt fun r hr => _) (iSup\u2082_le fun K hK => iSup_le fun _ => \u03bc.mono hK)\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU : Set \u03b1\n\u03b5 : \u211d\u22650\u221e\nH : InnerRegular \u03bc p q\nhU : q U\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc U\n\u22a2 r < \u2a06 (K : Set \u03b1) (_ : K \u2286 U) (_ : p K), \u2191\u2191\u03bc K\n[PROOFSTEP]\nsimpa only [lt_iSup_iff, exists_prop] using H hU r hr\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU : Set \u03b1\n\u03b5 : \u211d\u22650\u221e\nH : InnerRegular \u03bc p q\nh0 : p \u2205\nhU : q U\nh\u03bcU : \u2191\u2191\u03bc U \u2260 \u22a4\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 K, K \u2286 U \u2227 p K \u2227 \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n[PROOFSTEP]\ncases' eq_or_ne (\u03bc U) 0 with h\u2080 h\u2080\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU : Set \u03b1\n\u03b5 : \u211d\u22650\u221e\nH : InnerRegular \u03bc p q\nh0 : p \u2205\nhU : q U\nh\u03bcU : \u2191\u2191\u03bc U \u2260 \u22a4\nh\u03b5 : \u03b5 \u2260 0\nh\u2080 : \u2191\u2191\u03bc U = 0\n\u22a2 \u2203 K, K \u2286 U \u2227 p K \u2227 \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n[PROOFSTEP]\nrefine' \u27e8\u2205, empty_subset _, h0, _\u27e9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU : Set \u03b1\n\u03b5 : \u211d\u22650\u221e\nH : InnerRegular \u03bc p q\nh0 : p \u2205\nhU : q U\nh\u03bcU : \u2191\u2191\u03bc U \u2260 \u22a4\nh\u03b5 : \u03b5 \u2260 0\nh\u2080 : \u2191\u2191\u03bc U = 0\n\u22a2 \u2191\u2191\u03bc U < \u2191\u2191\u03bc \u2205 + \u03b5\n[PROOFSTEP]\nrwa [measure_empty, h\u2080, zero_add, pos_iff_ne_zero]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU : Set \u03b1\n\u03b5 : \u211d\u22650\u221e\nH : InnerRegular \u03bc p q\nh0 : p \u2205\nhU : q U\nh\u03bcU : \u2191\u2191\u03bc U \u2260 \u22a4\nh\u03b5 : \u03b5 \u2260 0\nh\u2080 : \u2191\u2191\u03bc U \u2260 0\n\u22a2 \u2203 K, K \u2286 U \u2227 p K \u2227 \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n[PROOFSTEP]\nrcases H hU _ (ENNReal.sub_lt_self h\u03bcU h\u2080 h\u03b5) with \u27e8K, hKU, hKc, hrK\u27e9\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU : Set \u03b1\n\u03b5 : \u211d\u22650\u221e\nH : InnerRegular \u03bc p q\nh0 : p \u2205\nhU : q U\nh\u03bcU : \u2191\u2191\u03bc U \u2260 \u22a4\nh\u03b5 : \u03b5 \u2260 0\nh\u2080 : \u2191\u2191\u03bc U \u2260 0\nK : Set \u03b1\nhKU : K \u2286 U\nhKc : p K\nhrK : \u2191\u2191\u03bc U - \u03b5 < \u2191\u2191\u03bc K\n\u22a2 \u2203 K, K \u2286 U \u2227 p K \u2227 \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n[PROOFSTEP]\nexact \u27e8K, hKU, hKc, ENNReal.lt_add_of_sub_lt_right (Or.inl h\u03bcU) hrK\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\nm : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\np q : Set \u03b1\u271d \u2192 Prop\nU : Set \u03b1\u271d\n\u03b5 : \u211d\u22650\u221e\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\npa qa : Set \u03b1 \u2192 Prop\nH : InnerRegular \u03bc pa qa\nf : \u03b1 \u2243 \u03b2\nhf : AEMeasurable \u2191f\npb qb : Set \u03b2 \u2192 Prop\nhAB : \u2200 (U : Set \u03b2), qb U \u2192 qa (\u2191f \u207b\u00b9' U)\nhAB' : \u2200 (K : Set \u03b1), pa K \u2192 pb (\u2191f '' K)\nhB\u2081 : \u2200 (K : Set \u03b2), pb K \u2192 MeasurableSet K\nhB\u2082 : \u2200 (U : Set \u03b2), qb U \u2192 MeasurableSet U\n\u22a2 InnerRegular (Measure.map (\u2191f) \u03bc) pb qb\n[PROOFSTEP]\nintro U hU r hr\n[GOAL]\n\u03b1\u271d : Type u_1\nm : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\np q : Set \u03b1\u271d \u2192 Prop\nU\u271d : Set \u03b1\u271d\n\u03b5 : \u211d\u22650\u221e\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\npa qa : Set \u03b1 \u2192 Prop\nH : InnerRegular \u03bc pa qa\nf : \u03b1 \u2243 \u03b2\nhf : AEMeasurable \u2191f\npb qb : Set \u03b2 \u2192 Prop\nhAB : \u2200 (U : Set \u03b2), qb U \u2192 qa (\u2191f \u207b\u00b9' U)\nhAB' : \u2200 (K : Set \u03b1), pa K \u2192 pb (\u2191f '' K)\nhB\u2081 : \u2200 (K : Set \u03b2), pb K \u2192 MeasurableSet K\nhB\u2082 : \u2200 (U : Set \u03b2), qb U \u2192 MeasurableSet U\nU : Set \u03b2\nhU : qb U\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191(Measure.map (\u2191f) \u03bc) U\n\u22a2 \u2203 K, K \u2286 U \u2227 pb K \u2227 r < \u2191\u2191(Measure.map (\u2191f) \u03bc) K\n[PROOFSTEP]\nrw [map_apply_of_aemeasurable hf (hB\u2082 _ hU)] at hr \n[GOAL]\n\u03b1\u271d : Type u_1\nm : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\np q : Set \u03b1\u271d \u2192 Prop\nU\u271d : Set \u03b1\u271d\n\u03b5 : \u211d\u22650\u221e\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\npa qa : Set \u03b1 \u2192 Prop\nH : InnerRegular \u03bc pa qa\nf : \u03b1 \u2243 \u03b2\nhf : AEMeasurable \u2191f\npb qb : Set \u03b2 \u2192 Prop\nhAB : \u2200 (U : Set \u03b2), qb U \u2192 qa (\u2191f \u207b\u00b9' U)\nhAB' : \u2200 (K : Set \u03b1), pa K \u2192 pb (\u2191f '' K)\nhB\u2081 : \u2200 (K : Set \u03b2), pb K \u2192 MeasurableSet K\nhB\u2082 : \u2200 (U : Set \u03b2), qb U \u2192 MeasurableSet U\nU : Set \u03b2\nhU : qb U\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc (\u2191f \u207b\u00b9' U)\n\u22a2 \u2203 K, K \u2286 U \u2227 pb K \u2227 r < \u2191\u2191(Measure.map (\u2191f) \u03bc) K\n[PROOFSTEP]\nrcases H (hAB U hU) r hr with \u27e8K, hKU, hKc, hK\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1\u271d : Type u_1\nm : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\np q : Set \u03b1\u271d \u2192 Prop\nU\u271d : Set \u03b1\u271d\n\u03b5 : \u211d\u22650\u221e\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\npa qa : Set \u03b1 \u2192 Prop\nH : InnerRegular \u03bc pa qa\nf : \u03b1 \u2243 \u03b2\nhf : AEMeasurable \u2191f\npb qb : Set \u03b2 \u2192 Prop\nhAB : \u2200 (U : Set \u03b2), qb U \u2192 qa (\u2191f \u207b\u00b9' U)\nhAB' : \u2200 (K : Set \u03b1), pa K \u2192 pb (\u2191f '' K)\nhB\u2081 : \u2200 (K : Set \u03b2), pb K \u2192 MeasurableSet K\nhB\u2082 : \u2200 (U : Set \u03b2), qb U \u2192 MeasurableSet U\nU : Set \u03b2\nhU : qb U\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc (\u2191f \u207b\u00b9' U)\nK : Set \u03b1\nhKU : K \u2286 \u2191f \u207b\u00b9' U\nhKc : pa K\nhK : r < \u2191\u2191\u03bc K\n\u22a2 \u2203 K, K \u2286 U \u2227 pb K \u2227 r < \u2191\u2191(Measure.map (\u2191f) \u03bc) K\n[PROOFSTEP]\nrefine' \u27e8f '' K, image_subset_iff.2 hKU, hAB' _ hKc, _\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1\u271d : Type u_1\nm : MeasurableSpace \u03b1\u271d\n\u03bc\u271d : Measure \u03b1\u271d\np q : Set \u03b1\u271d \u2192 Prop\nU\u271d : Set \u03b1\u271d\n\u03b5 : \u211d\u22650\u221e\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\n\u03bc : Measure \u03b1\npa qa : Set \u03b1 \u2192 Prop\nH : InnerRegular \u03bc pa qa\nf : \u03b1 \u2243 \u03b2\nhf : AEMeasurable \u2191f\npb qb : Set \u03b2 \u2192 Prop\nhAB : \u2200 (U : Set \u03b2), qb U \u2192 qa (\u2191f \u207b\u00b9' U)\nhAB' : \u2200 (K : Set \u03b1), pa K \u2192 pb (\u2191f '' K)\nhB\u2081 : \u2200 (K : Set \u03b2), pb K \u2192 MeasurableSet K\nhB\u2082 : \u2200 (U : Set \u03b2), qb U \u2192 MeasurableSet U\nU : Set \u03b2\nhU : qb U\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc (\u2191f \u207b\u00b9' U)\nK : Set \u03b1\nhKU : K \u2286 \u2191f \u207b\u00b9' U\nhKc : pa K\nhK : r < \u2191\u2191\u03bc K\n\u22a2 r < \u2191\u2191(Measure.map (\u2191f) \u03bc) (\u2191f '' K)\n[PROOFSTEP]\nrwa [map_apply_of_aemeasurable hf (hB\u2081 _ <| hAB' _ hKc), f.preimage_image]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU : Set \u03b1\n\u03b5 : \u211d\u22650\u221e\nH : InnerRegular \u03bc p q\nc : \u211d\u22650\u221e\n\u22a2 InnerRegular (c \u2022 \u03bc) p q\n[PROOFSTEP]\nintro U hU r hr\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d : Set \u03b1\n\u03b5 : \u211d\u22650\u221e\nH : InnerRegular \u03bc p q\nc : \u211d\u22650\u221e\nU : Set \u03b1\nhU : q U\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191(c \u2022 \u03bc) U\n\u22a2 \u2203 K, K \u2286 U \u2227 p K \u2227 r < \u2191\u2191(c \u2022 \u03bc) K\n[PROOFSTEP]\nrw [smul_apply, H.measure_eq_iSup hU, smul_eq_mul] at hr \n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d : Set \u03b1\n\u03b5 : \u211d\u22650\u221e\nH : InnerRegular \u03bc p q\nc : \u211d\u22650\u221e\nU : Set \u03b1\nhU : q U\nr : \u211d\u22650\u221e\nhr : r < c * \u2a06 (K : Set \u03b1) (_ : K \u2286 U) (_ : p K), \u2191\u2191\u03bc K\n\u22a2 \u2203 K, K \u2286 U \u2227 p K \u2227 r < \u2191\u2191(c \u2022 \u03bc) K\n[PROOFSTEP]\nsimpa only [ENNReal.mul_iSup, lt_iSup_iff, exists_prop] using hr\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU : Set \u03b1\n\u03b5 : \u211d\u22650\u221e\nq' : Set \u03b1 \u2192 Prop\nH : InnerRegular \u03bc p q\nH' : InnerRegular \u03bc q q'\n\u22a2 InnerRegular \u03bc p q'\n[PROOFSTEP]\nintro U hU r hr\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d : Set \u03b1\n\u03b5 : \u211d\u22650\u221e\nq' : Set \u03b1 \u2192 Prop\nH : InnerRegular \u03bc p q\nH' : InnerRegular \u03bc q q'\nU : Set \u03b1\nhU : q' U\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc U\n\u22a2 \u2203 K, K \u2286 U \u2227 p K \u2227 r < \u2191\u2191\u03bc K\n[PROOFSTEP]\nrcases H' hU r hr with \u27e8F, hFU, hqF, hF\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d : Set \u03b1\n\u03b5 : \u211d\u22650\u221e\nq' : Set \u03b1 \u2192 Prop\nH : InnerRegular \u03bc p q\nH' : InnerRegular \u03bc q q'\nU : Set \u03b1\nhU : q' U\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc U\nF : Set \u03b1\nhFU : F \u2286 U\nhqF : q F\nhF : r < \u2191\u2191\u03bc F\n\u22a2 \u2203 K, K \u2286 U \u2227 p K \u2227 r < \u2191\u2191\u03bc K\n[PROOFSTEP]\nrcases H hqF _ hF with \u27e8K, hKF, hpK, hrK\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d : Set \u03b1\n\u03b5 : \u211d\u22650\u221e\nq' : Set \u03b1 \u2192 Prop\nH : InnerRegular \u03bc p q\nH' : InnerRegular \u03bc q q'\nU : Set \u03b1\nhU : q' U\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc U\nF : Set \u03b1\nhFU : F \u2286 U\nhqF : q F\nhF : r < \u2191\u2191\u03bc F\nK : Set \u03b1\nhKF : K \u2286 F\nhpK : p K\nhrK : r < \u2191\u2191\u03bc K\n\u22a2 \u2203 K, K \u2286 U \u2227 p K \u2227 r < \u2191\u2191\u03bc K\n[PROOFSTEP]\nexact \u27e8K, hKF.trans hFU, hpK, hrK\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nA : Set \u03b1\nr : \u211d\u22650\u221e\nhr : \u2191\u2191\u03bc A < r\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nrcases OuterRegular.outerRegular (measurableSet_toMeasurable \u03bc A) r (by rwa [measure_toMeasurable]) with\n \u27e8U, hAU, hUo, hU\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nA : Set \u03b1\nr : \u211d\u22650\u221e\nhr : \u2191\u2191\u03bc A < r\n\u22a2 r > \u2191\u2191?m.8589 (toMeasurable \u03bc A)\n[PROOFSTEP]\nrwa [measure_toMeasurable]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nA : Set \u03b1\nr : \u211d\u22650\u221e\nhr : \u2191\u2191\u03bc A < r\nU : Set \u03b1\nhAU : U \u2287 toMeasurable \u03bc A\nhUo : IsOpen U\nhU : \u2191\u2191\u03bc U < r\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nexact \u27e8U, (subset_toMeasurable _ _).trans hAU, hUo, hU\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nA : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\n\u22a2 \u2191\u2191\u03bc A = \u2a05 (U : Set \u03b1) (_ : A \u2286 U) (_ : IsOpen U), \u2191\u2191\u03bc U\n[PROOFSTEP]\nrefine' le_antisymm (le_iInf\u2082 fun s hs => le_iInf fun _ => \u03bc.mono hs) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nA : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\n\u22a2 \u2a05 (U : Set \u03b1) (_ : A \u2286 U) (_ : IsOpen U), \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc A\n[PROOFSTEP]\nrefine' le_of_forall_lt' fun r hr => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nA : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nr : \u211d\u22650\u221e\nhr : \u2191\u2191\u03bc A < r\n\u22a2 \u2a05 (U : Set \u03b1) (_ : A \u2286 U) (_ : IsOpen U), \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nsimpa only [iInf_lt_iff, exists_prop] using A.exists_isOpen_lt_of_lt r hr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nA : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc A + \u03b5\n[PROOFSTEP]\nrcases eq_or_ne (\u03bc A) \u221e with (H | H)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nA : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nH : \u2191\u2191\u03bc A = \u22a4\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc A + \u03b5\n[PROOFSTEP]\nexact \u27e8univ, subset_univ _, isOpen_univ, by simp only [H, _root_.top_add, le_top]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nA : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nH : \u2191\u2191\u03bc A = \u22a4\n\u22a2 \u2191\u2191\u03bc univ \u2264 \u2191\u2191\u03bc A + \u03b5\n[PROOFSTEP]\nsimp only [H, _root_.top_add, le_top]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nA : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nH : \u2191\u2191\u03bc A \u2260 \u22a4\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc A + \u03b5\n[PROOFSTEP]\nrcases A.exists_isOpen_lt_add H h\u03b5 with \u27e8U, AU, U_open, hU\u27e9\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nA : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nH : \u2191\u2191\u03bc A \u2260 \u22a4\nU : Set \u03b1\nAU : U \u2287 A\nU_open : IsOpen U\nhU : \u2191\u2191\u03bc U < \u2191\u2191\u03bc A + \u03b5\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc A + \u03b5\n[PROOFSTEP]\nexact \u27e8U, AU, U_open, hU.le\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nhA' : \u2191\u2191\u03bc A \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < \u22a4 \u2227 \u2191\u2191\u03bc (U \\ A) < \u03b5\n[PROOFSTEP]\nrcases A.exists_isOpen_lt_add hA' h\u03b5 with \u27e8U, hAU, hUo, hU\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nhA' : \u2191\u2191\u03bc A \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nU : Set \u03b1\nhAU : U \u2287 A\nhUo : IsOpen U\nhU : \u2191\u2191\u03bc U < \u2191\u2191\u03bc A + \u03b5\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < \u22a4 \u2227 \u2191\u2191\u03bc (U \\ A) < \u03b5\n[PROOFSTEP]\nuse U, hAU, hUo, hU.trans_le le_top\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nhA' : \u2191\u2191\u03bc A \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nU : Set \u03b1\nhAU : U \u2287 A\nhUo : IsOpen U\nhU : \u2191\u2191\u03bc U < \u2191\u2191\u03bc A + \u03b5\n\u22a2 \u2191\u2191\u03bc (U \\ A) < \u03b5\n[PROOFSTEP]\nexact measure_diff_lt_of_lt_add hA hAU hA' hU\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u2074 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\nf : \u03b1 \u2243\u209c \u03b2\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\n\u22a2 OuterRegular (map (\u2191f) \u03bc)\n[PROOFSTEP]\nrefine' \u27e8fun A hA r hr => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u2074 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\nf : \u03b1 \u2243\u209c \u03b2\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nA : Set \u03b2\nhA : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191(map (\u2191f) \u03bc) A\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191(map (\u2191f) \u03bc) U < r\n[PROOFSTEP]\nrw [map_apply f.measurable hA, \u2190 f.image_symm] at hr \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u2074 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\nf : \u03b1 \u2243\u209c \u03b2\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nA : Set \u03b2\nhA : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc (\u2191(Homeomorph.symm f) '' A)\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191(map (\u2191f) \u03bc) U < r\n[PROOFSTEP]\nrcases Set.exists_isOpen_lt_of_lt _ r hr with \u27e8U, hAU, hUo, hU\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u2074 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\nf : \u03b1 \u2243\u209c \u03b2\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nA : Set \u03b2\nhA : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc (\u2191(Homeomorph.symm f) '' A)\nU : Set \u03b1\nhAU : U \u2287 \u2191(Homeomorph.symm f) '' A\nhUo : IsOpen U\nhU : \u2191\u2191\u03bc U < r\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191(map (\u2191f) \u03bc) U < r\n[PROOFSTEP]\nhave : IsOpen (f.symm \u207b\u00b9' U) := hUo.preimage f.symm.continuous\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u2074 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\nf : \u03b1 \u2243\u209c \u03b2\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nA : Set \u03b2\nhA : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc (\u2191(Homeomorph.symm f) '' A)\nU : Set \u03b1\nhAU : U \u2287 \u2191(Homeomorph.symm f) '' A\nhUo : IsOpen U\nhU : \u2191\u2191\u03bc U < r\nthis : IsOpen (\u2191(Homeomorph.symm f) \u207b\u00b9' U)\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191(map (\u2191f) \u03bc) U < r\n[PROOFSTEP]\nrefine' \u27e8f.symm \u207b\u00b9' U, image_subset_iff.1 hAU, this, _\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d\u2074 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\nf : \u03b1 \u2243\u209c \u03b2\n\u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nA : Set \u03b2\nhA : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc (\u2191(Homeomorph.symm f) '' A)\nU : Set \u03b1\nhAU : U \u2287 \u2191(Homeomorph.symm f) '' A\nhUo : IsOpen U\nhU : \u2191\u2191\u03bc U < r\nthis : IsOpen (\u2191(Homeomorph.symm f) \u207b\u00b9' U)\n\u22a2 \u2191\u2191(map (\u2191f) \u03bc) (\u2191(Homeomorph.symm f) \u207b\u00b9' U) < r\n[PROOFSTEP]\nrwa [map_apply f.measurable this.measurableSet, f.preimage_symm, f.preimage_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\n\u22a2 OuterRegular (x \u2022 \u03bc)\n[PROOFSTEP]\nrcases eq_or_ne x 0 with (rfl | h0)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nhx : 0 \u2260 \u22a4\n\u22a2 OuterRegular (0 \u2022 \u03bc)\n[PROOFSTEP]\nrw [zero_smul]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nhx : 0 \u2260 \u22a4\n\u22a2 OuterRegular 0\n[PROOFSTEP]\nexact OuterRegular.zero\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\nh0 : x \u2260 0\n\u22a2 OuterRegular (x \u2022 \u03bc)\n[PROOFSTEP]\nrefine' \u27e8fun A _ r hr => _\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\nh0 : x \u2260 0\nA : Set \u03b1\nx\u271d : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191(x \u2022 \u03bc) A\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191(x \u2022 \u03bc) U < r\n[PROOFSTEP]\nrw [smul_apply, A.measure_eq_iInf_isOpen, smul_eq_mul] at hr \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\ninst\u271d : OuterRegular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\nh0 : x \u2260 0\nA : Set \u03b1\nx\u271d : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > x * \u2a05 (U : Set \u03b1) (_ : A \u2286 U) (_ : IsOpen U), \u2191\u2191\u03bc U\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191(x \u2022 \u03bc) U < r\n[PROOFSTEP]\nsimpa only [ENNReal.mul_iInf_of_ne h0 hx, gt_iff_lt, iInf_lt_iff, exists_prop] using hr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\n\u22a2 OuterRegular \u03bc\n[PROOFSTEP]\nrefine' \u27e8fun A hA r hr => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nA : Set \u03b1\nhA : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc A\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nhave hm : \u2200 n, MeasurableSet (s.set n) := fun n => (s.set_mem n).1.measurableSet\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nA : Set \u03b1\nhA : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc A\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nhaveI : \u2200 n, OuterRegular (\u03bc.restrict (s.set n)) := fun n =>\n (s.set_mem n).2\n -- Note that `A = \u22c3 n, A \u2229 disjointed s n`. We replace `A` with this sequence.\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nA : Set \u03b1\nhA : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc A\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\n\u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nobtain \u27e8A, hAm, hAs, hAd, rfl\u27e9 :\n \u2203 A' : \u2115 \u2192 Set \u03b1, (\u2200 n, MeasurableSet (A' n)) \u2227 (\u2200 n, A' n \u2286 s.set n) \u2227 Pairwise (Disjoint on A') \u2227 A = \u22c3 n, A' n :=\n by\n refine'\n \u27e8fun n => A \u2229 disjointed s.set n, fun n => hA.inter (MeasurableSet.disjointed hm _), fun n =>\n (inter_subset_right _ _).trans (disjointed_subset _ _),\n (disjoint_disjointed s.set).mono fun k l hkl => hkl.mono inf_le_right inf_le_right, _\u27e9\n rw [\u2190 inter_iUnion, iUnion_disjointed, s.spanning, inter_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nA : Set \u03b1\nhA : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc A\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\n\u22a2 \u2203 A',\n (\u2200 (n : \u2115), MeasurableSet (A' n)) \u2227\n (\u2200 (n : \u2115), A' n \u2286 FiniteSpanningSetsIn.set s n) \u2227 Pairwise (Disjoint on A') \u2227 A = \u22c3 (n : \u2115), A' n\n[PROOFSTEP]\nrefine'\n \u27e8fun n => A \u2229 disjointed s.set n, fun n => hA.inter (MeasurableSet.disjointed hm _), fun n =>\n (inter_subset_right _ _).trans (disjointed_subset _ _),\n (disjoint_disjointed s.set).mono fun k l hkl => hkl.mono inf_le_right inf_le_right, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nA : Set \u03b1\nhA : MeasurableSet A\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc A\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\n\u22a2 A = \u22c3 (n : \u2115), (fun n => A \u2229 disjointed s.set n) n\n[PROOFSTEP]\nrw [\u2190 inter_iUnion, iUnion_disjointed, s.spanning, inter_univ]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nr : \u211d\u22650\u221e\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\nA : \u2115 \u2192 Set \u03b1\nhAm : \u2200 (n : \u2115), MeasurableSet (A n)\nhAs : \u2200 (n : \u2115), A n \u2286 FiniteSpanningSetsIn.set s n\nhAd : Pairwise (Disjoint on A)\nhA : MeasurableSet (\u22c3 (n : \u2115), A n)\nhr : r > \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\n\u22a2 \u2203 U, U \u2287 \u22c3 (n : \u2115), A n \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nrcases ENNReal.exists_pos_sum_of_countable' (tsub_pos_iff_lt.2 hr).ne' \u2115 with \u27e8\u03b4, \u03b40, h\u03b4\u03b5\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nr : \u211d\u22650\u221e\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\nA : \u2115 \u2192 Set \u03b1\nhAm : \u2200 (n : \u2115), MeasurableSet (A n)\nhAs : \u2200 (n : \u2115), A n \u2286 FiniteSpanningSetsIn.set s n\nhAd : Pairwise (Disjoint on A)\nhA : MeasurableSet (\u22c3 (n : \u2115), A n)\nhr : r > \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < r - \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\n\u22a2 \u2203 U, U \u2287 \u22c3 (n : \u2115), A n \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nrw [lt_tsub_iff_right, add_comm] at h\u03b4\u03b5 \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nr : \u211d\u22650\u221e\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\nA : \u2115 \u2192 Set \u03b1\nhAm : \u2200 (n : \u2115), MeasurableSet (A n)\nhAs : \u2200 (n : \u2115), A n \u2286 FiniteSpanningSetsIn.set s n\nhAd : Pairwise (Disjoint on A)\nhA : MeasurableSet (\u22c3 (n : \u2115), A n)\nhr : r > \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5\u271d : \u2211' (i : \u2115), \u03b4 i < r - \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\nh\u03b4\u03b5 : \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n) + \u2211' (i : \u2115), \u03b4 i < r\n\u22a2 \u2203 U, U \u2287 \u22c3 (n : \u2115), A n \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nhave : \u2200 n, \u2203 (U : _) (_ : U \u2287 A n), IsOpen U \u2227 \u03bc U < \u03bc (A n) + \u03b4 n :=\n by\n intro n\n have H\u2081 : \u2200 t, \u03bc.restrict (s.set n) t = \u03bc (t \u2229 s.set n) := fun t => restrict_apply' (hm n)\n have Ht : \u03bc.restrict (s.set n) (A n) \u2260 \u22a4 := by\n rw [H\u2081]\n exact ((measure_mono <| inter_subset_right _ _).trans_lt (s.finite n)).ne\n rcases(A n).exists_isOpen_lt_add Ht (\u03b40 n).ne' with \u27e8U, hAU, hUo, hU\u27e9\n rw [H\u2081, H\u2081, inter_eq_self_of_subset_left (hAs _)] at hU \n exact \u27e8U \u2229 s.set n, subset_inter hAU (hAs _), hUo.inter (s.set_mem n).1, hU\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nr : \u211d\u22650\u221e\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\nA : \u2115 \u2192 Set \u03b1\nhAm : \u2200 (n : \u2115), MeasurableSet (A n)\nhAs : \u2200 (n : \u2115), A n \u2286 FiniteSpanningSetsIn.set s n\nhAd : Pairwise (Disjoint on A)\nhA : MeasurableSet (\u22c3 (n : \u2115), A n)\nhr : r > \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5\u271d : \u2211' (i : \u2115), \u03b4 i < r - \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\nh\u03b4\u03b5 : \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n) + \u2211' (i : \u2115), \u03b4 i < r\n\u22a2 \u2200 (n : \u2115), \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u2191\u2191\u03bc (A n) + \u03b4 n\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nr : \u211d\u22650\u221e\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\nA : \u2115 \u2192 Set \u03b1\nhAm : \u2200 (n : \u2115), MeasurableSet (A n)\nhAs : \u2200 (n : \u2115), A n \u2286 FiniteSpanningSetsIn.set s n\nhAd : Pairwise (Disjoint on A)\nhA : MeasurableSet (\u22c3 (n : \u2115), A n)\nhr : r > \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5\u271d : \u2211' (i : \u2115), \u03b4 i < r - \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\nh\u03b4\u03b5 : \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n) + \u2211' (i : \u2115), \u03b4 i < r\nn : \u2115\n\u22a2 \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u2191\u2191\u03bc (A n) + \u03b4 n\n[PROOFSTEP]\nhave H\u2081 : \u2200 t, \u03bc.restrict (s.set n) t = \u03bc (t \u2229 s.set n) := fun t => restrict_apply' (hm n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nr : \u211d\u22650\u221e\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\nA : \u2115 \u2192 Set \u03b1\nhAm : \u2200 (n : \u2115), MeasurableSet (A n)\nhAs : \u2200 (n : \u2115), A n \u2286 FiniteSpanningSetsIn.set s n\nhAd : Pairwise (Disjoint on A)\nhA : MeasurableSet (\u22c3 (n : \u2115), A n)\nhr : r > \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5\u271d : \u2211' (i : \u2115), \u03b4 i < r - \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\nh\u03b4\u03b5 : \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n) + \u2211' (i : \u2115), \u03b4 i < r\nn : \u2115\nH\u2081 : \u2200 (t : Set \u03b1), \u2191\u2191(restrict \u03bc (FiniteSpanningSetsIn.set s n)) t = \u2191\u2191\u03bc (t \u2229 FiniteSpanningSetsIn.set s n)\n\u22a2 \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u2191\u2191\u03bc (A n) + \u03b4 n\n[PROOFSTEP]\nhave Ht : \u03bc.restrict (s.set n) (A n) \u2260 \u22a4 := by\n rw [H\u2081]\n exact ((measure_mono <| inter_subset_right _ _).trans_lt (s.finite n)).ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nr : \u211d\u22650\u221e\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\nA : \u2115 \u2192 Set \u03b1\nhAm : \u2200 (n : \u2115), MeasurableSet (A n)\nhAs : \u2200 (n : \u2115), A n \u2286 FiniteSpanningSetsIn.set s n\nhAd : Pairwise (Disjoint on A)\nhA : MeasurableSet (\u22c3 (n : \u2115), A n)\nhr : r > \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5\u271d : \u2211' (i : \u2115), \u03b4 i < r - \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\nh\u03b4\u03b5 : \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n) + \u2211' (i : \u2115), \u03b4 i < r\nn : \u2115\nH\u2081 : \u2200 (t : Set \u03b1), \u2191\u2191(restrict \u03bc (FiniteSpanningSetsIn.set s n)) t = \u2191\u2191\u03bc (t \u2229 FiniteSpanningSetsIn.set s n)\n\u22a2 \u2191\u2191(restrict \u03bc (FiniteSpanningSetsIn.set s n)) (A n) \u2260 \u22a4\n[PROOFSTEP]\nrw [H\u2081]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nr : \u211d\u22650\u221e\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\nA : \u2115 \u2192 Set \u03b1\nhAm : \u2200 (n : \u2115), MeasurableSet (A n)\nhAs : \u2200 (n : \u2115), A n \u2286 FiniteSpanningSetsIn.set s n\nhAd : Pairwise (Disjoint on A)\nhA : MeasurableSet (\u22c3 (n : \u2115), A n)\nhr : r > \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5\u271d : \u2211' (i : \u2115), \u03b4 i < r - \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\nh\u03b4\u03b5 : \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n) + \u2211' (i : \u2115), \u03b4 i < r\nn : \u2115\nH\u2081 : \u2200 (t : Set \u03b1), \u2191\u2191(restrict \u03bc (FiniteSpanningSetsIn.set s n)) t = \u2191\u2191\u03bc (t \u2229 FiniteSpanningSetsIn.set s n)\n\u22a2 \u2191\u2191\u03bc (A n \u2229 FiniteSpanningSetsIn.set s n) \u2260 \u22a4\n[PROOFSTEP]\nexact ((measure_mono <| inter_subset_right _ _).trans_lt (s.finite n)).ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nr : \u211d\u22650\u221e\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\nA : \u2115 \u2192 Set \u03b1\nhAm : \u2200 (n : \u2115), MeasurableSet (A n)\nhAs : \u2200 (n : \u2115), A n \u2286 FiniteSpanningSetsIn.set s n\nhAd : Pairwise (Disjoint on A)\nhA : MeasurableSet (\u22c3 (n : \u2115), A n)\nhr : r > \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5\u271d : \u2211' (i : \u2115), \u03b4 i < r - \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\nh\u03b4\u03b5 : \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n) + \u2211' (i : \u2115), \u03b4 i < r\nn : \u2115\nH\u2081 : \u2200 (t : Set \u03b1), \u2191\u2191(restrict \u03bc (FiniteSpanningSetsIn.set s n)) t = \u2191\u2191\u03bc (t \u2229 FiniteSpanningSetsIn.set s n)\nHt : \u2191\u2191(restrict \u03bc (FiniteSpanningSetsIn.set s n)) (A n) \u2260 \u22a4\n\u22a2 \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u2191\u2191\u03bc (A n) + \u03b4 n\n[PROOFSTEP]\nrcases(A n).exists_isOpen_lt_add Ht (\u03b40 n).ne' with \u27e8U, hAU, hUo, hU\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nr : \u211d\u22650\u221e\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\nA : \u2115 \u2192 Set \u03b1\nhAm : \u2200 (n : \u2115), MeasurableSet (A n)\nhAs : \u2200 (n : \u2115), A n \u2286 FiniteSpanningSetsIn.set s n\nhAd : Pairwise (Disjoint on A)\nhA : MeasurableSet (\u22c3 (n : \u2115), A n)\nhr : r > \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5\u271d : \u2211' (i : \u2115), \u03b4 i < r - \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\nh\u03b4\u03b5 : \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n) + \u2211' (i : \u2115), \u03b4 i < r\nn : \u2115\nH\u2081 : \u2200 (t : Set \u03b1), \u2191\u2191(restrict \u03bc (FiniteSpanningSetsIn.set s n)) t = \u2191\u2191\u03bc (t \u2229 FiniteSpanningSetsIn.set s n)\nHt : \u2191\u2191(restrict \u03bc (FiniteSpanningSetsIn.set s n)) (A n) \u2260 \u22a4\nU : Set \u03b1\nhAU : U \u2287 A n\nhUo : IsOpen U\nhU : \u2191\u2191(restrict \u03bc (FiniteSpanningSetsIn.set s n)) U < \u2191\u2191(restrict \u03bc (FiniteSpanningSetsIn.set s n)) (A n) + \u03b4 n\n\u22a2 \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u2191\u2191\u03bc (A n) + \u03b4 n\n[PROOFSTEP]\nrw [H\u2081, H\u2081, inter_eq_self_of_subset_left (hAs _)] at hU \n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nr : \u211d\u22650\u221e\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\nA : \u2115 \u2192 Set \u03b1\nhAm : \u2200 (n : \u2115), MeasurableSet (A n)\nhAs : \u2200 (n : \u2115), A n \u2286 FiniteSpanningSetsIn.set s n\nhAd : Pairwise (Disjoint on A)\nhA : MeasurableSet (\u22c3 (n : \u2115), A n)\nhr : r > \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5\u271d : \u2211' (i : \u2115), \u03b4 i < r - \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\nh\u03b4\u03b5 : \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n) + \u2211' (i : \u2115), \u03b4 i < r\nn : \u2115\nH\u2081 : \u2200 (t : Set \u03b1), \u2191\u2191(restrict \u03bc (FiniteSpanningSetsIn.set s n)) t = \u2191\u2191\u03bc (t \u2229 FiniteSpanningSetsIn.set s n)\nHt : \u2191\u2191(restrict \u03bc (FiniteSpanningSetsIn.set s n)) (A n) \u2260 \u22a4\nU : Set \u03b1\nhAU : U \u2287 A n\nhUo : IsOpen U\nhU : \u2191\u2191\u03bc (U \u2229 FiniteSpanningSetsIn.set s n) < \u2191\u2191\u03bc (A n) + \u03b4 n\n\u22a2 \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u2191\u2191\u03bc (A n) + \u03b4 n\n[PROOFSTEP]\nexact \u27e8U \u2229 s.set n, subset_inter hAU (hAs _), hUo.inter (s.set_mem n).1, hU\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nr : \u211d\u22650\u221e\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis\u271d : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\nA : \u2115 \u2192 Set \u03b1\nhAm : \u2200 (n : \u2115), MeasurableSet (A n)\nhAs : \u2200 (n : \u2115), A n \u2286 FiniteSpanningSetsIn.set s n\nhAd : Pairwise (Disjoint on A)\nhA : MeasurableSet (\u22c3 (n : \u2115), A n)\nhr : r > \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5\u271d : \u2211' (i : \u2115), \u03b4 i < r - \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\nh\u03b4\u03b5 : \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n) + \u2211' (i : \u2115), \u03b4 i < r\nthis : \u2200 (n : \u2115), \u2203 U x, IsOpen U \u2227 \u2191\u2191\u03bc U < \u2191\u2191\u03bc (A n) + \u03b4 n\n\u22a2 \u2203 U, U \u2287 \u22c3 (n : \u2115), A n \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nchoose U hAU hUo hU using this\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nr : \u211d\u22650\u221e\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\nA : \u2115 \u2192 Set \u03b1\nhAm : \u2200 (n : \u2115), MeasurableSet (A n)\nhAs : \u2200 (n : \u2115), A n \u2286 FiniteSpanningSetsIn.set s n\nhAd : Pairwise (Disjoint on A)\nhA : MeasurableSet (\u22c3 (n : \u2115), A n)\nhr : r > \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5\u271d : \u2211' (i : \u2115), \u03b4 i < r - \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\nh\u03b4\u03b5 : \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n) + \u2211' (i : \u2115), \u03b4 i < r\nU : \u2115 \u2192 Set \u03b1\nhAU : \u2200 (n : \u2115), U n \u2287 A n\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) < \u2191\u2191\u03bc (A n) + \u03b4 n\n\u22a2 \u2203 U, U \u2287 \u22c3 (n : \u2115), A n \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nrefine' \u27e8\u22c3 n, U n, iUnion_mono hAU, isOpen_iUnion hUo, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : FiniteSpanningSetsIn \u03bc {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\nr : \u211d\u22650\u221e\nhm : \u2200 (n : \u2115), MeasurableSet (FiniteSpanningSetsIn.set s n)\nthis : \u2200 (n : \u2115), OuterRegular (restrict \u03bc (FiniteSpanningSetsIn.set s n))\nA : \u2115 \u2192 Set \u03b1\nhAm : \u2200 (n : \u2115), MeasurableSet (A n)\nhAs : \u2200 (n : \u2115), A n \u2286 FiniteSpanningSetsIn.set s n\nhAd : Pairwise (Disjoint on A)\nhA : MeasurableSet (\u22c3 (n : \u2115), A n)\nhr : r > \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5\u271d : \u2211' (i : \u2115), \u03b4 i < r - \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n)\nh\u03b4\u03b5 : \u2191\u2191\u03bc (\u22c3 (n : \u2115), A n) + \u2211' (i : \u2115), \u03b4 i < r\nU : \u2115 \u2192 Set \u03b1\nhAU : \u2200 (n : \u2115), U n \u2287 A n\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) < \u2191\u2191\u03bc (A n) + \u03b4 n\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), U n) < r\n[PROOFSTEP]\ncalc\n \u03bc (\u22c3 n, U n) \u2264 \u2211' n, \u03bc (U n) := measure_iUnion_le _\n _ \u2264 \u2211' n, (\u03bc (A n) + \u03b4 n) := (ENNReal.tsum_le_tsum fun n => (hU n).le)\n _ = \u2211' n, \u03bc (A n) + \u2211' n, \u03b4 n := ENNReal.tsum_add\n _ = \u03bc (\u22c3 n, A n) + \u2211' n, \u03b4 n := (congr_arg\u2082 (\u00b7 + \u00b7) (measure_iUnion hAd hAm).symm rfl)\n _ < r := h\u03b4\u03b5\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\n\u22a2 InnerRegular \u03bc p fun s => MeasurableSet s \u2227 \u2191\u2191\u03bc s \u2260 \u22a4\n[PROOFSTEP]\nrintro s \u27e8hs, h\u03bcs\u27e9 r hr\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc s\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 r < \u2191\u2191\u03bc K\n[PROOFSTEP]\nobtain \u27e8\u03b5, h\u03b5, h\u03b5s, rfl\u27e9 : \u2203 (\u03b5 : _) (_ : \u03b5 \u2260 0), \u03b5 + \u03b5 \u2264 \u03bc s \u2227 r = \u03bc s - (\u03b5 + \u03b5) :=\n by\n use(\u03bc s - r) / 2\n simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc s\n\u22a2 \u2203 \u03b5 x, \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s \u2227 r = \u2191\u2191\u03bc s - (\u03b5 + \u03b5)\n[PROOFSTEP]\nuse(\u03bc s - r) / 2\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc s\n\u22a2 \u2203 x, (\u2191\u2191\u03bc s - r) / 2 + (\u2191\u2191\u03bc s - r) / 2 \u2264 \u2191\u2191\u03bc s \u2227 r = \u2191\u2191\u03bc s - ((\u2191\u2191\u03bc s - r) / 2 + (\u2191\u2191\u03bc s - r) / 2)\n[PROOFSTEP]\nsimp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K\n[PROOFSTEP]\nrcases hs.exists_isOpen_diff_lt h\u03bcs h\u03b5 with \u27e8U, hsU, hUo, hUt, h\u03bcU\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K\n[PROOFSTEP]\nrcases(U \\ s).exists_isOpen_lt_of_lt _ h\u03bcU with \u27e8U', hsU', hU'o, h\u03bcU'\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhsU' : U' \u2287 U \\ s\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K\n[PROOFSTEP]\nreplace hsU' := diff_subset_comm.1 hsU'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K\n[PROOFSTEP]\nrcases H.exists_subset_lt_add h0 hUo hUt.ne h\u03b5 with \u27e8K, hKU, hKc, hKr\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\nK : Set \u03b1\nhKU : K \u2286 U\nhKc : p K\nhKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n\u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K\n[PROOFSTEP]\nrefine' \u27e8K \\ U', fun x hx => hsU' \u27e8hKU hx.1, hx.2\u27e9, hd hKc hU'o, ENNReal.sub_lt_of_lt_add h\u03b5s _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\nK : Set \u03b1\nhKU : K \u2286 U\nhKc : p K\nhKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n\u22a2 \u2191\u2191\u03bc s < \u2191\u2191\u03bc (K \\ U') + (\u03b5 + \u03b5)\n[PROOFSTEP]\ncalc\n \u03bc s \u2264 \u03bc U := \u03bc.mono hsU\n _ < \u03bc K + \u03b5 := hKr\n _ \u2264 \u03bc (K \\ U') + \u03bc U' + \u03b5 := (add_le_add_right (tsub_le_iff_right.1 le_measure_diff) _)\n _ \u2264 \u03bc (K \\ U') + \u03b5 + \u03b5 := by\n apply add_le_add_right; apply add_le_add_left\n exact h\u03bcU'.le\n _ = \u03bc (K \\ U') + (\u03b5 + \u03b5) := add_assoc _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\nK : Set \u03b1\nhKU : K \u2286 U\nhKc : p K\nhKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n\u22a2 \u2191\u2191\u03bc (K \\ U') + \u2191\u2191\u03bc U' + \u03b5 \u2264 \u2191\u2191\u03bc (K \\ U') + \u03b5 + \u03b5\n[PROOFSTEP]\napply add_le_add_right\n[GOAL]\ncase bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\nK : Set \u03b1\nhKU : K \u2286 U\nhKc : p K\nhKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n\u22a2 \u2191\u2191\u03bc (K \\ U') + \u2191\u2191\u03bc U' \u2264 \u2191\u2191\u03bc (K \\ U') + \u03b5\n[PROOFSTEP]\napply add_le_add_left\n[GOAL]\ncase bc.bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d : OuterRegular \u03bc\nH : InnerRegular \u03bc p IsOpen\nh0 : p \u2205\nhd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nh\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s\nhr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nhUt : \u2191\u2191\u03bc U < \u22a4\nh\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5\nU' : Set \u03b1\nhU'o : IsOpen U'\nh\u03bcU' : \u2191\u2191\u03bc U' < \u03b5\nhsU' : U \\ U' \u2286 s\nK : Set \u03b1\nhKU : K \u2286 U\nhKc : p K\nhKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5\n\u22a2 \u2191\u2191\u03bc U' \u2264 \u03b5\n[PROOFSTEP]\nexact h\u03bcU'.le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\n\u22a2 WeaklyRegular \u03bc\n[PROOFSTEP]\nhave hfin : \u2200 {s}, \u03bc s \u2260 \u22a4 := @(measure_ne_top \u03bc)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 WeaklyRegular \u03bc\n[PROOFSTEP]\nsuffices\n \u2200 s, MeasurableSet s \u2192 \u2200 \u03b5, \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u03bc s \u2264 \u03bc F + \u03b5 \u2227 \u03bc U \u2264 \u03bc s + \u03b5\n by\n refine'\n { outerRegular := fun s hs r hr => _\n innerRegular := H }\n rcases exists_between hr with \u27e8r', hsr', hr'r\u27e9\n rcases this s hs _ (tsub_pos_iff_lt.2 hsr').ne' with \u27e8-, -, U, hsU, -, hUo, -, H\u27e9\n refine' \u27e8U, hsU, hUo, _\u27e9\n rw [add_tsub_cancel_of_le hsr'.le] at H \n exact H.trans_lt hr'r\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n \u2200 (s : Set \u03b1),\n MeasurableSet s \u2192\n \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u22a2 WeaklyRegular \u03bc\n[PROOFSTEP]\nrefine'\n { outerRegular := fun s hs r hr => _\n innerRegular := H }\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n \u2200 (s : Set \u03b1),\n MeasurableSet s \u2192\n \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc s\n\u22a2 \u2203 U, U \u2287 s \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nrcases exists_between hr with \u27e8r', hsr', hr'r\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n \u2200 (s : Set \u03b1),\n MeasurableSet s \u2192\n \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc s\nr' : \u211d\u22650\u221e\nhsr' : \u2191\u2191\u03bc s < r'\nhr'r : r' < r\n\u22a2 \u2203 U, U \u2287 s \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nrcases this s hs _ (tsub_pos_iff_lt.2 hsr').ne' with \u27e8-, -, U, hsU, -, hUo, -, H\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n \u2200 (s : Set \u03b1),\n MeasurableSet s \u2192\n \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc s\nr' : \u211d\u22650\u221e\nhsr' : \u2191\u2191\u03bc s < r'\nhr'r : r' < r\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nH : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + (r' - \u2191\u2191\u03bc s)\n\u22a2 \u2203 U, U \u2287 s \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nrefine' \u27e8U, hsU, hUo, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n \u2200 (s : Set \u03b1),\n MeasurableSet s \u2192\n \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc s\nr' : \u211d\u22650\u221e\nhsr' : \u2191\u2191\u03bc s < r'\nhr'r : r' < r\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nH : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + (r' - \u2191\u2191\u03bc s)\n\u22a2 \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nrw [add_tsub_cancel_of_le hsr'.le] at H \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nthis :\n \u2200 (s : Set \u03b1),\n MeasurableSet s \u2192\n \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nr : \u211d\u22650\u221e\nhr : r > \u2191\u2191\u03bc s\nr' : \u211d\u22650\u221e\nhsr' : \u2191\u2191\u03bc s < r'\nhr'r : r' < r\nU : Set \u03b1\nhsU : U \u2287 s\nhUo : IsOpen U\nH : \u2191\u2191\u03bc U \u2264 r'\n\u22a2 \u2191\u2191\u03bc U < r\n[PROOFSTEP]\nexact H.trans_lt hr'r\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (s : Set \u03b1),\n MeasurableSet s \u2192\n \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nrefine'\n MeasurableSet.induction_on_open _ _\n _\n /- The proof is by measurable induction: we should check that the property is true for the empty\n set, for open sets, and is stable by taking the complement and by taking countable disjoint\n unions. The point of the property we are proving is that it is stable by taking complements\n (exchanging the roles of closed and open sets and thanks to the finiteness of the measure). -/\n -- check for open set\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (U : Set \u03b1),\n IsOpen U \u2192\n \u2200 (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 U \u2227 \u2203 U_1, U_1 \u2287 U \u2227 IsClosed F \u2227 IsOpen U_1 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U_1 \u2264 \u2191\u2191\u03bc U + \u03b5\n[PROOFSTEP]\nintro U hU \u03b5 h\u03b5\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nU : Set \u03b1\nhU : IsOpen U\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 F, F \u2286 U \u2227 \u2203 U_1, U_1 \u2287 U \u2227 IsClosed F \u2227 IsOpen U_1 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U_1 \u2264 \u2191\u2191\u03bc U + \u03b5\n[PROOFSTEP]\nrcases H.exists_subset_lt_add isClosed_empty hU hfin h\u03b5 with \u27e8F, hsF, hFc, hF\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\nU : Set \u03b1\nhU : IsOpen U\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nF : Set \u03b1\nhsF : F \u2286 U\nhFc : IsClosed F\nhF : \u2191\u2191\u03bc U < \u2191\u2191\u03bc F + \u03b5\n\u22a2 \u2203 F, F \u2286 U \u2227 \u2203 U_1, U_1 \u2287 U \u2227 IsClosed F \u2227 IsOpen U_1 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U_1 \u2264 \u2191\u2191\u03bc U + \u03b5\n[PROOFSTEP]\nexact\n \u27e8F, hsF, U, Subset.rfl, hFc, hU, hF.le, le_self_add\u27e9\n -- check for complements\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (t : Set \u03b1),\n MeasurableSet t \u2192\n (\u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 t \u2227 \u2203 U, U \u2287 t \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc t + \u03b5) \u2192\n \u2200 (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 t\u1d9c \u2227 \u2203 U, U \u2287 t\u1d9c \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc t\u1d9c \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc t\u1d9c + \u03b5\n[PROOFSTEP]\nrintro s hs H \u03b5 h\u03b5\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nH : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 F, F \u2286 s\u1d9c \u2227 \u2203 U, U \u2287 s\u1d9c \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s\u1d9c \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s\u1d9c + \u03b5\n[PROOFSTEP]\nrcases H \u03b5 h\u03b5 with \u27e8F, hFs, U, hsU, hFc, hUo, hF, hU\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nH : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nF : Set \u03b1\nhFs : F \u2286 s\nU : Set \u03b1\nhsU : U \u2287 s\nhFc : IsClosed F\nhUo : IsOpen U\nhF : \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5\nhU : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u22a2 \u2203 F, F \u2286 s\u1d9c \u2227 \u2203 U, U \u2287 s\u1d9c \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s\u1d9c \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s\u1d9c + \u03b5\n[PROOFSTEP]\nrefine' \u27e8U\u1d9c, compl_subset_compl.2 hsU, F\u1d9c, compl_subset_compl.2 hFs, hUo.isClosed_compl, hFc.isOpen_compl, _\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : Set \u03b1\nhs : MeasurableSet s\nH : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s \u2227 \u2203 U, U \u2287 s \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nF : Set \u03b1\nhFs : F \u2286 s\nU : Set \u03b1\nhsU : U \u2287 s\nhFc : IsClosed F\nhUo : IsOpen U\nhF : \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc F + \u03b5\nhU : \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc s + \u03b5\n\u22a2 \u2191\u2191\u03bc s\u1d9c \u2264 \u2191\u2191\u03bc U\u1d9c + \u03b5 \u2227 \u2191\u2191\u03bc F\u1d9c \u2264 \u2191\u2191\u03bc s\u1d9c + \u03b5\n[PROOFSTEP]\nsimp only [measure_compl_le_add_iff, *, hUo.measurableSet, hFc.measurableSet, true_and_iff]\n -- check for disjoint unions\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1),\n Pairwise (Disjoint on f) \u2192\n (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n (\u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192\n \u2203 F, F \u2286 f i \u2227 \u2203 U, U \u2287 f i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (f i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (f i) + \u03b5) \u2192\n \u2200 (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192\n \u2203 F,\n F \u2286 \u22c3 (i : \u2115), f i \u2227\n \u2203 U,\n U \u2287 \u22c3 (i : \u2115), f i \u2227\n IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 (i : \u2115), f i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 (i : \u2115), f i) + \u03b5\n[PROOFSTEP]\nintro s hsd hsm H \u03b5 \u03b50\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 F,\n F \u2286 \u22c3 (i : \u2115), s i \u2227\n \u2203 U,\n U \u2287 \u22c3 (i : \u2115), s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 (i : \u2115), s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 (i : \u2115), s i) + \u03b5\n[PROOFSTEP]\nhave \u03b50' : \u03b5 / 2 \u2260 0 := (ENNReal.half_pos \u03b50).ne'\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u22a2 \u2203 F,\n F \u2286 \u22c3 (i : \u2115), s i \u2227\n \u2203 U,\n U \u2287 \u22c3 (i : \u2115), s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 (i : \u2115), s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 (i : \u2115), s i) + \u03b5\n[PROOFSTEP]\nrcases ENNReal.exists_pos_sum_of_countable' \u03b50' \u2115 with \u27e8\u03b4, \u03b40, h\u03b4\u03b5\u27e9\n[GOAL]\ncase refine'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\n\u22a2 \u2203 F,\n F \u2286 \u22c3 (i : \u2115), s i \u2227\n \u2203 U,\n U \u2287 \u22c3 (i : \u2115), s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 (i : \u2115), s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 (i : \u2115), s i) + \u03b5\n[PROOFSTEP]\nchoose F hFs U hsU hFc hUo hF hU using fun n => H n (\u03b4 n) (\u03b40 n).ne'\n[GOAL]\ncase refine'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\n\u22a2 \u2203 F,\n F \u2286 \u22c3 (i : \u2115), s i \u2227\n \u2203 U,\n U \u2287 \u22c3 (i : \u2115), s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 (i : \u2115), s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 (i : \u2115), s i) + \u03b5\n[PROOFSTEP]\nhave : Tendsto (fun t => (\u2211 k in t, \u03bc (s k)) + \u03b5 / 2) atTop (\ud835\udcdd <| \u03bc (\u22c3 n, s n) + \u03b5 / 2) :=\n by\n rw [measure_iUnion hsd hsm]\n exact Tendsto.add ENNReal.summable.hasSum tendsto_const_nhds\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\n\u22a2 Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) + \u03b5 / 2))\n[PROOFSTEP]\nrw [measure_iUnion hsd hsm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\n\u22a2 Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2211' (i : \u2115), \u2191\u2191\u03bc (s i) + \u03b5 / 2))\n[PROOFSTEP]\nexact Tendsto.add ENNReal.summable.hasSum tendsto_const_nhds\n[GOAL]\ncase refine'_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) + \u03b5 / 2))\n\u22a2 \u2203 F,\n F \u2286 \u22c3 (i : \u2115), s i \u2227\n \u2203 U,\n U \u2287 \u22c3 (i : \u2115), s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 (i : \u2115), s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 (i : \u2115), s i) + \u03b5\n[PROOFSTEP]\nrcases(this.eventually <| lt_mem_nhds <| ENNReal.lt_add_right hfin \u03b50').exists with\n \u27e8t, ht\u27e9\n -- the approximating open set is constructed by taking for each `s n` an approximating open set\n -- `U n` with measure at most `\u03bc (s n) + \u03b4 n` for a summable `\u03b4`, and taking the union of these.\n[GOAL]\ncase refine'_3.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2203 F,\n F \u2286 \u22c3 (i : \u2115), s i \u2227\n \u2203 U,\n U \u2287 \u22c3 (i : \u2115), s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (\u22c3 (i : \u2115), s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (\u22c3 (i : \u2115), s i) + \u03b5\n[PROOFSTEP]\nrefine'\n \u27e8\u22c3 k \u2208 t, F k, iUnion_mono fun k => iUnion_subset fun _ => hFs _, \u22c3 n, U n, iUnion_mono hsU,\n isClosed_biUnion t.finite_toSet fun k _ => hFc k, isOpen_iUnion hUo, ht.le.trans _, _\u27e9\n[GOAL]\ncase refine'_3.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u2264 \u2191\u2191\u03bc (\u22c3 (k : \u2115) (_ : k \u2208 t), F k) + \u03b5\n[PROOFSTEP]\ncalc\n (\u2211 k in t, \u03bc (s k)) + \u03b5 / 2 \u2264 ((\u2211 k in t, \u03bc (F k)) + \u2211 k in t, \u03b4 k) + \u03b5 / 2 :=\n by\n rw [\u2190 sum_add_distrib]\n exact add_le_add_right (sum_le_sum fun k _ => hF k) _\n _ \u2264 (\u2211 k in t, \u03bc (F k)) + \u03b5 / 2 + \u03b5 / 2 :=\n (add_le_add_right (add_le_add_left ((ENNReal.sum_le_tsum _).trans h\u03b4\u03b5.le) _) _)\n _ = \u03bc (\u22c3 k \u2208 t, F k) + \u03b5 := by\n rw [measure_biUnion_finset, add_assoc, ENNReal.add_halves]\n exacts [fun k _ n _ hkn => (hsd hkn).mono (hFs k) (hFs n), fun k _ => (hFc k).measurableSet]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u2264 \u2211 k in t, \u2191\u2191\u03bc (F k) + \u2211 k in t, \u03b4 k + \u03b5 / 2\n[PROOFSTEP]\nrw [\u2190 sum_add_distrib]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2 \u2264 \u2211 x in t, (\u2191\u2191\u03bc (F x) + \u03b4 x) + \u03b5 / 2\n[PROOFSTEP]\nexact add_le_add_right (sum_le_sum fun k _ => hF k) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2211 k in t, \u2191\u2191\u03bc (F k) + \u03b5 / 2 + \u03b5 / 2 = \u2191\u2191\u03bc (\u22c3 (k : \u2115) (_ : k \u2208 t), F k) + \u03b5\n[PROOFSTEP]\nrw [measure_biUnion_finset, add_assoc, ENNReal.add_halves]\n[GOAL]\ncase hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 PairwiseDisjoint \u2191t fun k => F k\ncase hm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2200 (b : \u2115), b \u2208 t \u2192 MeasurableSet (F b)\n[PROOFSTEP]\nexacts [fun k _ n _ hkn => (hsd hkn).mono (hFs k) (hFs n), fun k _ => (hFc k).measurableSet]\n[GOAL]\ncase refine'_3.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), U n) \u2264 \u2191\u2191\u03bc (\u22c3 (i : \u2115), s i) + \u03b5\n[PROOFSTEP]\ncalc\n \u03bc (\u22c3 n, U n) \u2264 \u2211' n, \u03bc (U n) := measure_iUnion_le _\n _ \u2264 \u2211' n, (\u03bc (s n) + \u03b4 n) := (ENNReal.tsum_le_tsum hU)\n _ = \u03bc (\u22c3 n, s n) + \u2211' n, \u03b4 n := by rw [measure_iUnion hsd hsm, ENNReal.tsum_add]\n _ \u2264 \u03bc (\u22c3 n, s n) + \u03b5 := add_le_add_left (h\u03b4\u03b5.le.trans ENNReal.half_le_self) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s\u271d : Set \u03b1\n\u03b5\u271d r : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nH\u271d : InnerRegular \u03bc IsClosed IsOpen\nhfin : \u2200 {s : Set \u03b1}, \u2191\u2191\u03bc s \u2260 \u22a4\ns : \u2115 \u2192 Set \u03b1\nhsd : Pairwise (Disjoint on s)\nhsm : \u2200 (i : \u2115), MeasurableSet (s i)\nH :\n \u2200 (i : \u2115) (\u03b5 : \u211d\u22650\u221e),\n \u03b5 \u2260 0 \u2192 \u2203 F, F \u2286 s i \u2227 \u2203 U, U \u2287 s i \u2227 IsClosed F \u2227 IsOpen U \u2227 \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc F + \u03b5 \u2227 \u2191\u2191\u03bc U \u2264 \u2191\u2191\u03bc (s i) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u03b50' : \u03b5 / 2 \u2260 0\n\u03b4 : \u2115 \u2192 \u211d\u22650\u221e\n\u03b40 : \u2200 (i : \u2115), 0 < \u03b4 i\nh\u03b4\u03b5 : \u2211' (i : \u2115), \u03b4 i < \u03b5 / 2\nF : \u2115 \u2192 Set \u03b1\nhFs : \u2200 (n : \u2115), F n \u2286 s n\nU : \u2115 \u2192 Set \u03b1\nhsU : \u2200 (n : \u2115), U n \u2287 s n\nhFc : \u2200 (n : \u2115), IsClosed (F n)\nhUo : \u2200 (n : \u2115), IsOpen (U n)\nhF : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) \u2264 \u2191\u2191\u03bc (F n) + \u03b4 n\nhU : \u2200 (n : \u2115), \u2191\u2191\u03bc (U n) \u2264 \u2191\u2191\u03bc (s n) + \u03b4 n\nthis : Tendsto (fun t => \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) + \u03b5 / 2))\nt : Finset \u2115\nht : \u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) < \u2211 k in t, \u2191\u2191\u03bc (s k) + \u03b5 / 2\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (s n) + \u03b4 n) = \u2191\u2191\u03bc (\u22c3 (n : \u2115), s n) + \u2211' (n : \u2115), \u03b4 n\n[PROOFSTEP]\nrw [measure_iUnion hsd hsm, ENNReal.tsum_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b9 : PseudoEMetricSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\n\u22a2 InnerRegular \u03bc IsClosed IsOpen\n[PROOFSTEP]\nintro U hU r hr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU\u271d s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b9 : PseudoEMetricSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nU : Set X\nhU : IsOpen U\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc U\n\u22a2 \u2203 K, K \u2286 U \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc K\n[PROOFSTEP]\nrcases hU.exists_iUnion_isClosed with \u27e8F, F_closed, -, rfl, F_mono\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b9 : PseudoEMetricSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nr : \u211d\u22650\u221e\nF : \u2115 \u2192 Set X\nF_closed : \u2200 (n : \u2115), IsClosed (F n)\nF_mono : Monotone F\nhU : IsOpen (\u22c3 (n : \u2115), F n)\nhr : r < \u2191\u2191\u03bc (\u22c3 (n : \u2115), F n)\n\u22a2 \u2203 K, K \u2286 \u22c3 (n : \u2115), F n \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc K\n[PROOFSTEP]\nrw [measure_iUnion_eq_iSup F_mono.directed_le] at hr \n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b9 : PseudoEMetricSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nr : \u211d\u22650\u221e\nF : \u2115 \u2192 Set X\nF_closed : \u2200 (n : \u2115), IsClosed (F n)\nF_mono : Monotone F\nhU : IsOpen (\u22c3 (n : \u2115), F n)\nhr : r < \u2a06 (i : \u2115), \u2191\u2191\u03bc (F i)\n\u22a2 \u2203 K, K \u2286 \u22c3 (n : \u2115), F n \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc K\n[PROOFSTEP]\nrcases lt_iSup_iff.1 hr with \u27e8n, hn\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b9 : PseudoEMetricSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nr : \u211d\u22650\u221e\nF : \u2115 \u2192 Set X\nF_closed : \u2200 (n : \u2115), IsClosed (F n)\nF_mono : Monotone F\nhU : IsOpen (\u22c3 (n : \u2115), F n)\nhr : r < \u2a06 (i : \u2115), \u2191\u2191\u03bc (F i)\nn : \u2115\nhn : r < \u2191\u2191\u03bc (F n)\n\u22a2 \u2203 K, K \u2286 \u22c3 (n : \u2115), F n \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc K\n[PROOFSTEP]\nexact \u27e8F n, subset_iUnion _ _, F_closed n, hn\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\n\u22a2 InnerRegular \u03bc IsCompact IsClosed\n[PROOFSTEP]\nintro F hF r hr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K\n[PROOFSTEP]\nset B : \u2115 \u2192 Set X := compactCovering X\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K\n[PROOFSTEP]\nhave hBc : \u2200 n, IsCompact (F \u2229 B n) := fun n => (isCompact_compactCovering X n).inter_left hF\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K\n[PROOFSTEP]\nhave hBU : \u22c3 n, F \u2229 B n = F := by rw [\u2190 inter_iUnion, iUnion_compactCovering, Set.inter_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\n\u22a2 \u22c3 (n : \u2115), F \u2229 B n = F\n[PROOFSTEP]\nrw [\u2190 inter_iUnion, iUnion_compactCovering, Set.inter_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 (n : \u2115), F \u2229 B n = F\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K\n[PROOFSTEP]\nhave : \u03bc F = \u2a06 n, \u03bc (F \u2229 B n) := by\n rw [\u2190 measure_iUnion_eq_iSup, hBU]\n exact Monotone.directed_le fun m n h => inter_subset_inter_right _ (compactCovering_subset _ h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 (n : \u2115), F \u2229 B n = F\n\u22a2 \u2191\u2191\u03bc F = \u2a06 (n : \u2115), \u2191\u2191\u03bc (F \u2229 B n)\n[PROOFSTEP]\nrw [\u2190 measure_iUnion_eq_iSup, hBU]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 (n : \u2115), F \u2229 B n = F\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => F \u2229 B n\n[PROOFSTEP]\nexact Monotone.directed_le fun m n h => inter_subset_inter_right _ (compactCovering_subset _ h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc F\nB : \u2115 \u2192 Set X := compactCovering X\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 (n : \u2115), F \u2229 B n = F\nthis : \u2191\u2191\u03bc F = \u2a06 (n : \u2115), \u2191\u2191\u03bc (F \u2229 B n)\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K\n[PROOFSTEP]\nrw [this] at hr \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nB : \u2115 \u2192 Set X := compactCovering X\nhr : r < \u2a06 (n : \u2115), \u2191\u2191\u03bc (F \u2229 B n)\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 (n : \u2115), F \u2229 B n = F\nthis : \u2191\u2191\u03bc F = \u2a06 (n : \u2115), \u2191\u2191\u03bc (F \u2229 B n)\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K\n[PROOFSTEP]\nrcases lt_iSup_iff.1 hr with \u27e8n, hn\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\np q : Set \u03b1 \u2192 Prop\nU s : Set \u03b1\n\u03b5 r\u271d : \u211d\u22650\u221e\nX : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : SigmaCompactSpace X\ninst\u271d : MeasurableSpace X\n\u03bc : Measure X\nF : Set X\nhF : IsClosed F\nr : \u211d\u22650\u221e\nB : \u2115 \u2192 Set X := compactCovering X\nhr : r < \u2a06 (n : \u2115), \u2191\u2191\u03bc (F \u2229 B n)\nhBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n)\nhBU : \u22c3 (n : \u2115), F \u2229 B n = F\nthis : \u2191\u2191\u03bc F = \u2a06 (n : \u2115), \u2191\u2191\u03bc (F \u2229 B n)\nn : \u2115\nhn : r < \u2191\u2191\u03bc (F \u2229 B n)\n\u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K\n[PROOFSTEP]\nexact \u27e8_, inter_subset_left _ _, hBc n, hn\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : Regular \u03bc\n\u22a2 (\u2203 K, IsCompact K \u2227 \u2191\u2191\u03bc K \u2260 0) \u2194 \u03bc \u2260 0\n[PROOFSTEP]\nsimp_rw [Ne.def, \u2190 measure_univ_eq_zero, isOpen_univ.measure_eq_iSup_isCompact, ENNReal.iSup_eq_zero, not_forall,\n exists_prop, subset_univ, true_and_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : Regular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 K, K \u2286 A \u2227 IsCompact K \u2227 \u2191\u2191\u03bc (A \\ K) < \u03b5\n[PROOFSTEP]\nrcases hA.exists_isCompact_lt_add h'A h\u03b5 with \u27e8K, hKA, hKc, hK\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : Regular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nK : Set \u03b1\nhKA : K \u2286 A\nhKc : IsCompact K\nhK : \u2191\u2191\u03bc A < \u2191\u2191\u03bc K + \u03b5\n\u22a2 \u2203 K, K \u2286 A \u2227 IsCompact K \u2227 \u2191\u2191\u03bc (A \\ K) < \u03b5\n[PROOFSTEP]\nexact \u27e8K, hKA, hKc, measure_diff_lt_of_lt_add hKc.measurableSet hKA (ne_top_of_le_ne_top h'A <| measure_mono hKA) hK\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : OpensMeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\ninst\u271d : Regular \u03bc\nf : \u03b1 \u2243\u209c \u03b2\n\u22a2 Regular (map (\u2191f) \u03bc)\n[PROOFSTEP]\nhaveI := OuterRegular.map f \u03bc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : OpensMeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\ninst\u271d : Regular \u03bc\nf : \u03b1 \u2243\u209c \u03b2\nthis : OuterRegular (map (\u2191f) \u03bc)\n\u22a2 Regular (map (\u2191f) \u03bc)\n[PROOFSTEP]\nhaveI := IsFiniteMeasureOnCompacts.map \u03bc f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : OpensMeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\ninst\u271d : Regular \u03bc\nf : \u03b1 \u2243\u209c \u03b2\nthis\u271d : OuterRegular (map (\u2191f) \u03bc)\nthis : IsFiniteMeasureOnCompacts (map (\u2191f) \u03bc)\n\u22a2 Regular (map (\u2191f) \u03bc)\n[PROOFSTEP]\nexact\n \u27e8Regular.innerRegular.map f.toEquiv f.measurable.aemeasurable (fun U hU => hU.preimage f.continuous)\n (fun K hK => hK.image f.continuous) (fun K hK => hK.measurableSet) fun U hU => hU.measurableSet\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : Regular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\n\u22a2 Regular (x \u2022 \u03bc)\n[PROOFSTEP]\nhaveI := OuterRegular.smul \u03bc hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : Regular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\nthis : OuterRegular (x \u2022 \u03bc)\n\u22a2 Regular (x \u2022 \u03bc)\n[PROOFSTEP]\nhaveI := IsFiniteMeasureOnCompacts.smul \u03bc hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : Regular \u03bc\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\nthis\u271d : OuterRegular (x \u2022 \u03bc)\nthis : IsFiniteMeasureOnCompacts (x \u2022 \u03bc)\n\u22a2 Regular (x \u2022 \u03bc)\n[PROOFSTEP]\nexact \u27e8Regular.innerRegular.smul x\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 F, F \u2286 A \u2227 IsClosed F \u2227 \u2191\u2191\u03bc (A \\ F) < \u03b5\n[PROOFSTEP]\nrcases hA.exists_isClosed_lt_add h'A h\u03b5 with \u27e8F, hFA, hFc, hF\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nF : Set \u03b1\nhFA : F \u2286 A\nhFc : IsClosed F\nhF : \u2191\u2191\u03bc A < \u2191\u2191\u03bc F + \u03b5\n\u22a2 \u2203 F, F \u2286 A \u2227 IsClosed F \u2227 \u2191\u2191\u03bc (A \\ F) < \u03b5\n[PROOFSTEP]\nexact \u27e8F, hFA, hFc, measure_diff_lt_of_lt_add hFc.measurableSet hFA (ne_top_of_le_ne_top h'A <| measure_mono hFA) hF\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\n\u22a2 WeaklyRegular (restrict \u03bc A)\n[PROOFSTEP]\nhaveI : Fact (\u03bc A < \u221e) := \u27e8h'A.lt_top\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis : Fact (\u2191\u2191\u03bc A < \u22a4)\n\u22a2 WeaklyRegular (restrict \u03bc A)\n[PROOFSTEP]\nrefine' InnerRegular.weaklyRegular_of_finite (\u03bc.restrict A) fun V V_open => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\n\u22a2 \u2200 (r : \u211d\u22650\u221e), r < \u2191\u2191(restrict \u03bc A) V \u2192 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191(restrict \u03bc A) K\n[PROOFSTEP]\nsimp only [restrict_apply' hA]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\n\u22a2 \u2200 (r : \u211d\u22650\u221e), r < \u2191\u2191\u03bc (V \u2229 A) \u2192 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A)\n[PROOFSTEP]\nintro r hr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc (V \u2229 A)\n\u22a2 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A)\n[PROOFSTEP]\nhave : \u03bc (V \u2229 A) \u2260 \u221e := ne_top_of_le_ne_top h'A (measure_mono <| inter_subset_right _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis\u271d : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc (V \u2229 A)\nthis : \u2191\u2191\u03bc (V \u2229 A) \u2260 \u22a4\n\u22a2 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A)\n[PROOFSTEP]\nrcases(V_open.measurableSet.inter hA).exists_lt_isClosed_of_ne_top this hr with \u27e8F, hFVA, hFc, hF\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis\u271d : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc (V \u2229 A)\nthis : \u2191\u2191\u03bc (V \u2229 A) \u2260 \u22a4\nF : Set \u03b1\nhFVA : F \u2286 V \u2229 A\nhFc : IsClosed F\nhF : r < \u2191\u2191\u03bc F\n\u22a2 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A)\n[PROOFSTEP]\nrefine' \u27e8F, hFVA.trans (inter_subset_left _ _), hFc, _\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : WeaklyRegular \u03bc\nA : Set \u03b1\nhA : MeasurableSet A\nh'A : \u2191\u2191\u03bc A \u2260 \u22a4\nthis\u271d : Fact (\u2191\u2191\u03bc A < \u22a4)\nV : Set \u03b1\nV_open : IsOpen V\nr : \u211d\u22650\u221e\nhr : r < \u2191\u2191\u03bc (V \u2229 A)\nthis : \u2191\u2191\u03bc (V \u2229 A) \u2260 \u22a4\nF : Set \u03b1\nhFVA : F \u2286 V \u2229 A\nhFc : IsClosed F\nhF : r < \u2191\u2191\u03bc F\n\u22a2 r < \u2191\u2191\u03bc (F \u2229 A)\n[PROOFSTEP]\nrwa [inter_eq_self_of_subset_left (hFVA.trans <| inter_subset_right _ _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_3\ninst\u271d\u2074 : PseudoEMetricSpace X\ninst\u271d\u00b3 : TopologicalSpace.SecondCountableTopology X\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03bc : Measure X\ninst\u271d : IsLocallyFiniteMeasure \u03bc\n\u22a2 OuterRegular \u03bc\n[PROOFSTEP]\nrefine' (\u03bc.finiteSpanningSetsInOpen'.mono' fun U hU => _).outerRegular\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_3\ninst\u271d\u2074 : PseudoEMetricSpace X\ninst\u271d\u00b3 : TopologicalSpace.SecondCountableTopology X\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03bc : Measure X\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nU : Set X\nhU : U \u2208 {K | IsOpen K} \u2229 {s | \u2191\u2191\u03bc s < \u22a4}\n\u22a2 U \u2208 {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\n[PROOFSTEP]\nhave : Fact (\u03bc U < \u221e) := \u27e8hU.2\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03bc\u271d : Measure \u03b1\nX : Type u_3\ninst\u271d\u2074 : PseudoEMetricSpace X\ninst\u271d\u00b3 : TopologicalSpace.SecondCountableTopology X\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03bc : Measure X\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nU : Set X\nhU : U \u2208 {K | IsOpen K} \u2229 {s | \u2191\u2191\u03bc s < \u22a4}\nthis : Fact (\u2191\u2191\u03bc U < \u22a4)\n\u22a2 U \u2208 {U | IsOpen U \u2227 OuterRegular (restrict \u03bc U)}\n[PROOFSTEP]\nexact \u27e8hU.1, inferInstance\u27e9\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.Regular", "llama_tokens": 49458, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.4084384508486044}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 Monotone (\u2191toDual \u2218 f \u2218 \u2191ofDual) \u2194 Monotone f\n[PROOFSTEP]\nrw [monotone_toDual_comp_iff, antitone_comp_ofDual_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 Antitone (\u2191toDual \u2218 f \u2218 \u2191ofDual) \u2194 Antitone f\n[PROOFSTEP]\nrw [antitone_toDual_comp_iff, monotone_comp_ofDual_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 MonotoneOn (\u2191toDual \u2218 f \u2218 \u2191ofDual) s \u2194 MonotoneOn f s\n[PROOFSTEP]\nrw [monotoneOn_toDual_comp_iff, antitoneOn_comp_ofDual_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 AntitoneOn (\u2191toDual \u2218 f \u2218 \u2191ofDual) s \u2194 AntitoneOn f s\n[PROOFSTEP]\nrw [antitoneOn_toDual_comp_iff, monotoneOn_comp_ofDual_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 StrictMono (\u2191toDual \u2218 f \u2218 \u2191ofDual) \u2194 StrictMono f\n[PROOFSTEP]\nrw [strictMono_toDual_comp_iff, strictAnti_comp_ofDual_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 StrictAnti (\u2191toDual \u2218 f \u2218 \u2191ofDual) \u2194 StrictAnti f\n[PROOFSTEP]\nrw [strictAnti_toDual_comp_iff, strictMono_comp_ofDual_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 StrictMonoOn (\u2191toDual \u2218 f \u2218 \u2191ofDual) s \u2194 StrictMonoOn f s\n[PROOFSTEP]\nrw [strictMonoOn_toDual_comp_iff, strictAntiOn_comp_ofDual_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 StrictAntiOn (\u2191toDual \u2218 f \u2218 \u2191ofDual) s \u2194 StrictAntiOn f s\n[PROOFSTEP]\nrw [strictAntiOn_toDual_comp_iff, strictMonoOn_comp_ofDual_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (x y : \u03b1), x < y \u2192 f x \u2260 f y\n\u22a2 Injective f\n[PROOFSTEP]\nintro x y hf\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (x y : \u03b1), x < y \u2192 f x \u2260 f y\nx y : \u03b1\nhf : f x = f y\n\u22a2 x = y\n[PROOFSTEP]\nrcases lt_trichotomy x y with (hxy | rfl | hxy)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (x y : \u03b1), x < y \u2192 f x \u2260 f y\nx y : \u03b1\nhf : f x = f y\nhxy : x < y\n\u22a2 x = y\n[PROOFSTEP]\nexact absurd hf <| h _ _ hxy\n[GOAL]\ncase inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (x y : \u03b1), x < y \u2192 f x \u2260 f y\nx : \u03b1\nhf : f x = f x\n\u22a2 x = x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (x y : \u03b1), x < y \u2192 f x \u2260 f y\nx y : \u03b1\nhf : f x = f y\nhxy : y < x\n\u22a2 x = y\n[PROOFSTEP]\nexact absurd hf.symm <| h _ _ hxy\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : StrictMono f\nhg : StrictMono g\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u2200 \u2983x y : \u03b1\u2984, x < y \u2192 p y \u2192 p x\nhfg : \u2200 \u2983x y : \u03b1\u2984, p x \u2192 \u00acp y \u2192 x < y \u2192 f x < g y\n\u22a2 StrictMono fun x => if p x then f x else g x\n[PROOFSTEP]\nintro x y h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : StrictMono f\nhg : StrictMono g\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u2200 \u2983x y : \u03b1\u2984, x < y \u2192 p y \u2192 p x\nhfg : \u2200 \u2983x y : \u03b1\u2984, p x \u2192 \u00acp y \u2192 x < y \u2192 f x < g y\nx y : \u03b1\nh : x < y\n\u22a2 (fun x => if p x then f x else g x) x < (fun x => if p x then f x else g x) y\n[PROOFSTEP]\nby_cases hy : p y\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : StrictMono f\nhg : StrictMono g\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u2200 \u2983x y : \u03b1\u2984, x < y \u2192 p y \u2192 p x\nhfg : \u2200 \u2983x y : \u03b1\u2984, p x \u2192 \u00acp y \u2192 x < y \u2192 f x < g y\nx y : \u03b1\nh : x < y\nhy : p y\n\u22a2 (fun x => if p x then f x else g x) x < (fun x => if p x then f x else g x) y\n[PROOFSTEP]\nhave hx : p x := hp h hy\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : StrictMono f\nhg : StrictMono g\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u2200 \u2983x y : \u03b1\u2984, x < y \u2192 p y \u2192 p x\nhfg : \u2200 \u2983x y : \u03b1\u2984, p x \u2192 \u00acp y \u2192 x < y \u2192 f x < g y\nx y : \u03b1\nh : x < y\nhy : p y\nhx : p x\n\u22a2 (fun x => if p x then f x else g x) x < (fun x => if p x then f x else g x) y\n[PROOFSTEP]\nsimpa [hx, hy] using hf h\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : StrictMono f\nhg : StrictMono g\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u2200 \u2983x y : \u03b1\u2984, x < y \u2192 p y \u2192 p x\nhfg : \u2200 \u2983x y : \u03b1\u2984, p x \u2192 \u00acp y \u2192 x < y \u2192 f x < g y\nx y : \u03b1\nh : x < y\nhy : \u00acp y\n\u22a2 (fun x => if p x then f x else g x) x < (fun x => if p x then f x else g x) y\n[PROOFSTEP]\nby_cases hx : p x\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : StrictMono f\nhg : StrictMono g\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u2200 \u2983x y : \u03b1\u2984, x < y \u2192 p y \u2192 p x\nhfg : \u2200 \u2983x y : \u03b1\u2984, p x \u2192 \u00acp y \u2192 x < y \u2192 f x < g y\nx y : \u03b1\nh : x < y\nhy : \u00acp y\nhx : p x\n\u22a2 (fun x => if p x then f x else g x) x < (fun x => if p x then f x else g x) y\n[PROOFSTEP]\nsimpa [hx, hy] using hfg hx hy h\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : StrictMono f\nhg : StrictMono g\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u2200 \u2983x y : \u03b1\u2984, x < y \u2192 p y \u2192 p x\nhfg : \u2200 \u2983x y : \u03b1\u2984, p x \u2192 \u00acp y \u2192 x < y \u2192 f x < g y\nx y : \u03b1\nh : x < y\nhy : \u00acp y\nhx : \u00acp x\n\u22a2 (fun x => if p x then f x else g x) x < (fun x => if p x then f x else g x) y\n[PROOFSTEP]\nsimpa [hx, hy] using hg h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhf : StrictMonoOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 a = b \u2192 f a = f b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhf : StrictMonoOn f s\na : \u03b1\nha hb : a \u2208 s\n\u22a2 f a = f a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhf : StrictMonoOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 f a < f b \u2194 a < b\n[PROOFSTEP]\nrw [lt_iff_le_not_le, lt_iff_le_not_le, hf.le_iff_le ha hb, hf.le_iff_le hb ha]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y : \u03b1\n\u22a2 \u00acMonotone f \u2227 \u00acAntitone f \u2194 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nsimp_rw [Monotone, Antitone, not_forall, not_le]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y : \u03b1\n\u22a2 ((\u2203 x x_1 h, f x_1 < f x) \u2227 \u2203 x x_1 h, f x < f x_1) \u2194\n \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nrefine' Iff.symm \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y : \u03b1\n\u22a2 (\u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)) \u2192\n (\u2203 x x_1 h, f x_1 < f x) \u2227 \u2203 x x_1 h, f x < f x_1\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y : \u03b1\n\u22a2 ((\u2203 x x_1 h, f x_1 < f x) \u2227 \u2203 x x_1 h, f x < f x_1) \u2192\n \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\n{ rintro \u27e8a, b, c, hab, hbc, \u27e8hfab, hfcb\u27e9 | \u27e8hfba, hfbc\u27e9\u27e9\n exacts [\u27e8\u27e8_, _, hbc, hfcb\u27e9, _, _, hab, hfab\u27e9, \u27e8\u27e8_, _, hab, hfba\u27e9, _, _, hbc, hfbc\u27e9]\n}\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y : \u03b1\n\u22a2 (\u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)) \u2192\n (\u2203 x x_1 h, f x_1 < f x) \u2227 \u2203 x x_1 h, f x < f x_1\n[PROOFSTEP]\nrintro \u27e8a, b, c, hab, hbc, \u27e8hfab, hfcb\u27e9 | \u27e8hfba, hfbc\u27e9\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.inl.intro\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\nhfab : f a < f b\nhfcb : f c < f b\n\u22a2 (\u2203 x x_1 h, f x_1 < f x) \u2227 \u2203 x x_1 h, f x < f x_1\ncase refine'_1.intro.intro.intro.intro.intro.inr.intro\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\nhfba : f b < f a\nhfbc : f b < f c\n\u22a2 (\u2203 x x_1 h, f x_1 < f x) \u2227 \u2203 x x_1 h, f x < f x_1\n[PROOFSTEP]\nexacts [\u27e8\u27e8_, _, hbc, hfcb\u27e9, _, _, hab, hfab\u27e9, \u27e8\u27e8_, _, hab, hfba\u27e9, _, _, hbc, hfbc\u27e9]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y : \u03b1\n\u22a2 ((\u2203 x x_1 h, f x_1 < f x) \u2227 \u2203 x x_1 h, f x < f x_1) \u2192\n \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nrintro \u27e8\u27e8a, b, hab, hfba\u27e9, c, d, hcd, hfcd\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nobtain hda | had := le_total d a\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhda : d \u2264 a\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\n{ obtain hfad | hfda := le_total (f a) (f d)\n {exact \u27e8c, d, b, hcd, hda.trans hab, Or.inl \u27e8hfcd, hfba.trans_le hfad\u27e9\u27e9\n }\n {exact \u27e8c, a, b, hcd.trans hda, hab, Or.inl \u27e8hfcd.trans_le hfda, hfba\u27e9\u27e9\n }\n}\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhda : d \u2264 a\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nobtain hfad | hfda := le_total (f a) (f d)\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inl.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhda : d \u2264 a\nhfad : f a \u2264 f d\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inl.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhda : d \u2264 a\nhfda : f d \u2264 f a\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\n{exact \u27e8c, d, b, hcd, hda.trans hab, Or.inl \u27e8hfcd, hfba.trans_le hfad\u27e9\u27e9\n}\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inl.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhda : d \u2264 a\nhfad : f a \u2264 f d\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nexact \u27e8c, d, b, hcd, hda.trans hab, Or.inl \u27e8hfcd, hfba.trans_le hfad\u27e9\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inl.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhda : d \u2264 a\nhfda : f d \u2264 f a\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\n{exact \u27e8c, a, b, hcd.trans hda, hab, Or.inl \u27e8hfcd.trans_le hfda, hfba\u27e9\u27e9\n}\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inl.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhda : d \u2264 a\nhfda : f d \u2264 f a\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nexact \u27e8c, a, b, hcd.trans hda, hab, Or.inl \u27e8hfcd.trans_le hfda, hfba\u27e9\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nobtain hac | hca := le_total a c\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhac : a \u2264 c\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\n{ obtain hfdb | hfbd := le_or_lt (f d) (f b)\n {exact \u27e8a, c, d, hac, hcd, Or.inr \u27e8hfcd.trans $ hfdb.trans_lt hfba, hfcd\u27e9\u27e9\n }\n obtain hfca | hfac := lt_or_le (f c) (f a)\n {exact \u27e8a, c, d, hac, hcd, Or.inr \u27e8hfca, hfcd\u27e9\u27e9\n }\n obtain hbd | hdb := le_total b d\n {exact \u27e8a, b, d, hab, hbd, Or.inr \u27e8hfba, hfbd\u27e9\u27e9\n }\n {exact \u27e8a, d, b, had, hdb, Or.inl \u27e8hfac.trans_lt hfcd, hfbd\u27e9\u27e9\n }\n}\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhac : a \u2264 c\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nobtain hfdb | hfbd := le_or_lt (f d) (f b)\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inl.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhac : a \u2264 c\nhfdb : f d \u2264 f b\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inl.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhac : a \u2264 c\nhfbd : f b < f d\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\n{exact \u27e8a, c, d, hac, hcd, Or.inr \u27e8hfcd.trans $ hfdb.trans_lt hfba, hfcd\u27e9\u27e9\n}\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inl.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhac : a \u2264 c\nhfdb : f d \u2264 f b\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nexact \u27e8a, c, d, hac, hcd, Or.inr \u27e8hfcd.trans $ hfdb.trans_lt hfba, hfcd\u27e9\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inl.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhac : a \u2264 c\nhfbd : f b < f d\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nobtain hfca | hfac := lt_or_le (f c) (f a)\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inl.inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhac : a \u2264 c\nhfbd : f b < f d\nhfca : f c < f a\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inl.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhac : a \u2264 c\nhfbd : f b < f d\nhfac : f a \u2264 f c\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\n{exact \u27e8a, c, d, hac, hcd, Or.inr \u27e8hfca, hfcd\u27e9\u27e9\n}\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inl.inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhac : a \u2264 c\nhfbd : f b < f d\nhfca : f c < f a\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nexact \u27e8a, c, d, hac, hcd, Or.inr \u27e8hfca, hfcd\u27e9\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inl.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhac : a \u2264 c\nhfbd : f b < f d\nhfac : f a \u2264 f c\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nobtain hbd | hdb := le_total b d\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inl.inr.inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhac : a \u2264 c\nhfbd : f b < f d\nhfac : f a \u2264 f c\nhbd : b \u2264 d\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inl.inr.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhac : a \u2264 c\nhfbd : f b < f d\nhfac : f a \u2264 f c\nhdb : d \u2264 b\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\n{exact \u27e8a, b, d, hab, hbd, Or.inr \u27e8hfba, hfbd\u27e9\u27e9\n}\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inl.inr.inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhac : a \u2264 c\nhfbd : f b < f d\nhfac : f a \u2264 f c\nhbd : b \u2264 d\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nexact \u27e8a, b, d, hab, hbd, Or.inr \u27e8hfba, hfbd\u27e9\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inl.inr.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhac : a \u2264 c\nhfbd : f b < f d\nhfac : f a \u2264 f c\nhdb : d \u2264 b\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\n{exact \u27e8a, d, b, had, hdb, Or.inl \u27e8hfac.trans_lt hfcd, hfbd\u27e9\u27e9\n}\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inl.inr.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhac : a \u2264 c\nhfbd : f b < f d\nhfac : f a \u2264 f c\nhdb : d \u2264 b\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nexact \u27e8a, d, b, had, hdb, Or.inl \u27e8hfac.trans_lt hfcd, hfbd\u27e9\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\n{ obtain hfdb | hfbd := le_or_lt (f d) (f b)\n {exact \u27e8c, a, b, hca, hab, Or.inl \u27e8hfcd.trans $ hfdb.trans_lt hfba, hfba\u27e9\u27e9\n }\n obtain hfca | hfac := lt_or_le (f c) (f a)\n {exact \u27e8c, a, b, hca, hab, Or.inl \u27e8hfca, hfba\u27e9\u27e9\n }\n obtain hbd | hdb := le_total b d\n {exact \u27e8a, b, d, hab, hbd, Or.inr \u27e8hfba, hfbd\u27e9\u27e9\n }\n {exact \u27e8a, d, b, had, hdb, Or.inl \u27e8hfac.trans_lt hfcd, hfbd\u27e9\u27e9\n }\n}\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nobtain hfdb | hfbd := le_or_lt (f d) (f b)\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\nhfdb : f d \u2264 f b\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\nhfbd : f b < f d\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\n{exact \u27e8c, a, b, hca, hab, Or.inl \u27e8hfcd.trans $ hfdb.trans_lt hfba, hfba\u27e9\u27e9\n}\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\nhfdb : f d \u2264 f b\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nexact \u27e8c, a, b, hca, hab, Or.inl \u27e8hfcd.trans $ hfdb.trans_lt hfba, hfba\u27e9\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\nhfbd : f b < f d\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nobtain hfca | hfac := lt_or_le (f c) (f a)\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr.inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\nhfbd : f b < f d\nhfca : f c < f a\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\nhfbd : f b < f d\nhfac : f a \u2264 f c\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\n{exact \u27e8c, a, b, hca, hab, Or.inl \u27e8hfca, hfba\u27e9\u27e9\n}\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr.inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\nhfbd : f b < f d\nhfca : f c < f a\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nexact \u27e8c, a, b, hca, hab, Or.inl \u27e8hfca, hfba\u27e9\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\nhfbd : f b < f d\nhfac : f a \u2264 f c\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nobtain hbd | hdb := le_total b d\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr.inr.inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\nhfbd : f b < f d\nhfac : f a \u2264 f c\nhbd : b \u2264 d\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr.inr.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\nhfbd : f b < f d\nhfac : f a \u2264 f c\nhdb : d \u2264 b\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\n{exact \u27e8a, b, d, hab, hbd, Or.inr \u27e8hfba, hfbd\u27e9\u27e9\n}\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr.inr.inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\nhfbd : f b < f d\nhfac : f a \u2264 f c\nhbd : b \u2264 d\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nexact \u27e8a, b, d, hab, hbd, Or.inr \u27e8hfba, hfbd\u27e9\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr.inr.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\nhfbd : f b < f d\nhfac : f a \u2264 f c\nhdb : d \u2264 b\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\n{exact \u27e8a, d, b, had, hdb, Or.inl \u27e8hfac.trans_lt hfcd, hfbd\u27e9\u27e9\n}\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.inr.inr.inr.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nhab : a \u2264 b\nhfba : f b < f a\nc d : \u03b1\nhcd : c \u2264 d\nhfcd : f c < f d\nhad : a \u2264 d\nhca : c \u2264 a\nhfbd : f b < f d\nhfac : f a \u2264 f c\nhdb : d \u2264 b\n\u22a2 \u2203 a b c, a \u2264 b \u2227 b \u2264 c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nexact \u27e8a, d, b, had, hdb, Or.inl \u27e8hfac.trans_lt hfcd, hfbd\u27e9\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y : \u03b1\n\u22a2 \u00acMonotone f \u2227 \u00acAntitone f \u2194 \u2203 a b c, a < b \u2227 b < c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nsimp_rw [not_monotone_not_antitone_iff_exists_le_le, \u2190 and_assoc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y : \u03b1\n\u22a2 (\u2203 a b c, (a \u2264 b \u2227 b \u2264 c) \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)) \u2194\n \u2203 a b c, (a < b \u2227 b < c) \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c)\n[PROOFSTEP]\nrefine' exists\u2083_congr (fun a b c \u21a6 and_congr_left $ fun h \u21a6 (Ne.le_iff_lt _).and $ Ne.le_iff_lt _)\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b c : \u03b1\nh : f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c\n\u22a2 a \u2260 b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a c : \u03b1\nh : f a < f a \u2227 f c < f a \u2228 f a < f a \u2227 f a < f c\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b c : \u03b1\nh : f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c\n\u22a2 b \u2260 c\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nx y a b : \u03b1\nh : f a < f b \u2227 f b < f b \u2228 f b < f a \u2227 f b < f b\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 r\nf : \u2115 \u2192 \u03b2\na : \u2115\nh : \u2200 (n : \u2115), a \u2264 n \u2192 r (f n) (f (n + 1))\nb c : \u2115\nhab : a \u2264 b\nhbc : b < c\n\u22a2 r (f b) (f c)\n[PROOFSTEP]\ninduction' hbc with k b_lt_k r_b_k\n[GOAL]\ncase refl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 r\nf : \u2115 \u2192 \u03b2\na : \u2115\nh : \u2200 (n : \u2115), a \u2264 n \u2192 r (f n) (f (n + 1))\nb c : \u2115\nhab : a \u2264 b\n\u22a2 r (f b) (f (succ b))\ncase step\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 r\nf : \u2115 \u2192 \u03b2\na : \u2115\nh : \u2200 (n : \u2115), a \u2264 n \u2192 r (f n) (f (n + 1))\nb c : \u2115\nhab : a \u2264 b\nk : \u2115\nb_lt_k : Nat.le (succ b) k\nr_b_k : r (f b) (f k)\n\u22a2 r (f b) (f (succ k))\n[PROOFSTEP]\nexacts [h _ hab, _root_.trans r_b_k (h _ (hab.trans_lt b_lt_k).le)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMaxOrder \u03b1\na : \u03b1\n\u22a2 \u2203 f, StrictMono f \u2227 f 0 = a\n[PROOFSTEP]\nchoose g hg using fun x : \u03b1 \u21a6 exists_gt x\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMaxOrder \u03b1\na : \u03b1\ng : \u03b1 \u2192 \u03b1\nhg : \u2200 (x : \u03b1), x < g x\n\u22a2 \u2203 f, StrictMono f \u2227 f 0 = a\n[PROOFSTEP]\nexact \u27e8fun n \u21a6 Nat.recOn n a fun _ \u21a6 g, strictMono_nat_of_lt_succ fun n \u21a6 hg _, rfl\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 r\nf : \u2124 \u2192 \u03b2\nh : \u2200 (n : \u2124), r (f n) (f (n + 1))\na b : \u2124\nhab : a < b\n\u22a2 r (f a) (f b)\n[PROOFSTEP]\nrcases lt.dest hab with \u27e8n, rfl\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 r\nf : \u2124 \u2192 \u03b2\nh : \u2200 (n : \u2124), r (f n) (f (n + 1))\na : \u2124\nn : \u2115\nhab : a < a + \u2191(Nat.succ n)\n\u22a2 r (f a) (f (a + \u2191(Nat.succ n)))\n[PROOFSTEP]\nclear hab\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 r\nf : \u2124 \u2192 \u03b2\nh : \u2200 (n : \u2124), r (f n) (f (n + 1))\na : \u2124\nn : \u2115\n\u22a2 r (f a) (f (a + \u2191(Nat.succ n)))\n[PROOFSTEP]\ninduction' n with n ihn\n[GOAL]\ncase intro.zero\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 r\nf : \u2124 \u2192 \u03b2\nh : \u2200 (n : \u2124), r (f n) (f (n + 1))\na : \u2124\n\u22a2 r (f a) (f (a + \u2191(Nat.succ Nat.zero)))\n[PROOFSTEP]\nrw [Int.ofNat_one]\n[GOAL]\ncase intro.zero\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 r\nf : \u2124 \u2192 \u03b2\nh : \u2200 (n : \u2124), r (f n) (f (n + 1))\na : \u2124\n\u22a2 r (f a) (f (a + 1))\n[PROOFSTEP]\napply h\n[GOAL]\ncase intro.succ\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 r\nf : \u2124 \u2192 \u03b2\nh : \u2200 (n : \u2124), r (f n) (f (n + 1))\na : \u2124\nn : \u2115\nihn : r (f a) (f (a + \u2191(Nat.succ n)))\n\u22a2 r (f a) (f (a + \u2191(Nat.succ (Nat.succ n))))\n[PROOFSTEP]\nrw [Int.ofNat_succ, \u2190 Int.add_assoc]\n[GOAL]\ncase intro.succ\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 r\nf : \u2124 \u2192 \u03b2\nh : \u2200 (n : \u2124), r (f n) (f (n + 1))\na : \u2124\nn : \u2115\nihn : r (f a) (f (a + \u2191(Nat.succ n)))\n\u22a2 r (f a) (f (a + \u2191(n + 1) + 1))\n[PROOFSTEP]\nexact _root_.trans ihn (h _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u2074 inst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\n\u22a2 \u2203 f, StrictMono f\n[PROOFSTEP]\ninhabit \u03b1\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u2074 inst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\ninhabited_h : Inhabited \u03b1\n\u22a2 \u2203 f, StrictMono f\n[PROOFSTEP]\nrcases Nat.exists_strictMono' (default : \u03b1) with \u27e8f, hf, hf\u2080\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u2074 inst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\ninhabited_h : Inhabited \u03b1\nf : \u2115 \u2192 \u03b1\nhf : StrictMono f\nhf\u2080 : f 0 = default\n\u22a2 \u2203 f, StrictMono f\n[PROOFSTEP]\nrcases Nat.exists_strictAnti' (default : \u03b1) with \u27e8g, hg, hg\u2080\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u2074 inst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\ninhabited_h : Inhabited \u03b1\nf : \u2115 \u2192 \u03b1\nhf : StrictMono f\nhf\u2080 : f 0 = default\ng : \u2115 \u2192 \u03b1\nhg : StrictAnti g\nhg\u2080 : g 0 = default\n\u22a2 \u2203 f, StrictMono f\n[PROOFSTEP]\nrefine' \u27e8fun n \u21a6 Int.casesOn n f fun n \u21a6 g (n + 1), strictMono_int_of_lt_succ _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u2074 inst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\ninhabited_h : Inhabited \u03b1\nf : \u2115 \u2192 \u03b1\nhf : StrictMono f\nhf\u2080 : f 0 = default\ng : \u2115 \u2192 \u03b1\nhg : StrictAnti g\nhg\u2080 : g 0 = default\n\u22a2 \u2200 (n : \u2124), (Int.casesOn n f fun n => g (n + 1)) < Int.casesOn (n + 1) f fun n => g (n + 1)\n[PROOFSTEP]\nrintro (n | _ | n)\n[GOAL]\ncase intro.intro.intro.intro.ofNat\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u2074 inst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\ninhabited_h : Inhabited \u03b1\nf : \u2115 \u2192 \u03b1\nhf : StrictMono f\nhf\u2080 : f 0 = default\ng : \u2115 \u2192 \u03b1\nhg : StrictAnti g\nhg\u2080 : g 0 = default\nn : \u2115\n\u22a2 (Int.casesOn (ofNat n) f fun n => g (n + 1)) < Int.casesOn (ofNat n + 1) f fun n => g (n + 1)\n[PROOFSTEP]\nexact hf n.lt_succ_self\n[GOAL]\ncase intro.intro.intro.intro.negSucc.zero\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u2074 inst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\ninhabited_h : Inhabited \u03b1\nf : \u2115 \u2192 \u03b1\nhf : StrictMono f\nhf\u2080 : f 0 = default\ng : \u2115 \u2192 \u03b1\nhg : StrictAnti g\nhg\u2080 : g 0 = default\n\u22a2 (Int.casesOn -[Nat.zero+1] f fun n => g (n + 1)) < Int.casesOn (-[Nat.zero+1] + 1) f fun n => g (n + 1)\n[PROOFSTEP]\nshow g 1 < f 0\n[GOAL]\ncase intro.intro.intro.intro.negSucc.zero\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u2074 inst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\ninhabited_h : Inhabited \u03b1\nf : \u2115 \u2192 \u03b1\nhf : StrictMono f\nhf\u2080 : f 0 = default\ng : \u2115 \u2192 \u03b1\nhg : StrictAnti g\nhg\u2080 : g 0 = default\n\u22a2 g 1 < f 0\n[PROOFSTEP]\nrw [hf\u2080, \u2190 hg\u2080]\n[GOAL]\ncase intro.intro.intro.intro.negSucc.zero\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u2074 inst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\ninhabited_h : Inhabited \u03b1\nf : \u2115 \u2192 \u03b1\nhf : StrictMono f\nhf\u2080 : f 0 = default\ng : \u2115 \u2192 \u03b1\nhg : StrictAnti g\nhg\u2080 : g 0 = default\n\u22a2 g 1 < g 0\n[PROOFSTEP]\nexact hg Nat.zero_lt_one\n[GOAL]\ncase intro.intro.intro.intro.negSucc.succ\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u2074 inst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Nonempty \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\ninhabited_h : Inhabited \u03b1\nf : \u2115 \u2192 \u03b1\nhf : StrictMono f\nhf\u2080 : f 0 = default\ng : \u2115 \u2192 \u03b1\nhg : StrictAnti g\nhg\u2080 : g 0 = default\nn : \u2115\n\u22a2 (Int.casesOn -[Nat.succ n+1] f fun n => g (n + 1)) < Int.casesOn (-[Nat.succ n+1] + 1) f fun n => g (n + 1)\n[PROOFSTEP]\nexact hg (Nat.lt_succ_self _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Preorder \u03b1\nf : \u2115 \u2192 \u03b1\nhf : Monotone f\nn : \u2115\nx : \u03b1\nh1 : f n < x\nh2 : x < f (n + 1)\na : \u2115\n\u22a2 f a \u2260 x\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Preorder \u03b1\nf : \u2115 \u2192 \u03b1\nhf : Monotone f\nn a : \u2115\nh1 : f n < f a\nh2 : f a < f (n + 1)\n\u22a2 False\n[PROOFSTEP]\nexact (hf.reflect_lt h1).not_le (Nat.le_of_lt_succ <| hf.reflect_lt h2)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Preorder \u03b1\nf : \u2115 \u2192 \u03b1\nhf : Antitone f\nn : \u2115\nx : \u03b1\nh1 : f (n + 1) < x\nh2 : x < f n\na : \u2115\n\u22a2 f a \u2260 x\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Preorder \u03b1\nf : \u2115 \u2192 \u03b1\nhf : Antitone f\nn a : \u2115\nh1 : f (n + 1) < f a\nh2 : f a < f n\n\u22a2 False\n[PROOFSTEP]\nexact (hf.reflect_lt h2).not_le (Nat.le_of_lt_succ <| hf.reflect_lt h1)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Preorder \u03b1\nf : \u2124 \u2192 \u03b1\nhf : Monotone f\nn : \u2124\nx : \u03b1\nh1 : f n < x\nh2 : x < f (n + 1)\na : \u2124\n\u22a2 f a \u2260 x\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Preorder \u03b1\nf : \u2124 \u2192 \u03b1\nhf : Monotone f\nn a : \u2124\nh1 : f n < f a\nh2 : f a < f (n + 1)\n\u22a2 False\n[PROOFSTEP]\nexact (hf.reflect_lt h1).not_le (Int.le_of_lt_add_one <| hf.reflect_lt h2)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Preorder \u03b1\nf : \u2124 \u2192 \u03b1\nhf : Antitone f\nn : \u2124\nx : \u03b1\nh1 : f (n + 1) < x\nh2 : x < f n\na : \u2124\n\u22a2 f a \u2260 x\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Preorder \u03b1\nf : \u2124 \u2192 \u03b1\nhf : Antitone f\nn a : \u2124\nh1 : f (n + 1) < f a\nh2 : f a < f n\n\u22a2 False\n[PROOFSTEP]\nexact (hf.reflect_lt h2).not_le (Int.le_of_lt_add_one <| hf.reflect_lt h1)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : Preorder \u03b3\ninst\u271d : Preorder \u03b4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nhf : StrictMono f\nhg : StrictMono g\na b : \u03b1 \u00d7 \u03b2\n\u22a2 a < b \u2192 Prod.map f g a < Prod.map f g b\n[PROOFSTEP]\nsimp only [Prod.lt_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : Preorder \u03b3\ninst\u271d : Preorder \u03b4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nhf : StrictMono f\nhg : StrictMono g\na b : \u03b1 \u00d7 \u03b2\n\u22a2 a.fst < b.fst \u2227 a.snd \u2264 b.snd \u2228 a.fst \u2264 b.fst \u2227 a.snd < b.snd \u2192\n (Prod.map f g a).fst < (Prod.map f g b).fst \u2227 (Prod.map f g a).snd \u2264 (Prod.map f g b).snd \u2228\n (Prod.map f g a).fst \u2264 (Prod.map f g b).fst \u2227 (Prod.map f g a).snd < (Prod.map f g b).snd\n[PROOFSTEP]\nexact Or.imp (And.imp hf.imp hg.monotone.imp) (And.imp hf.monotone.imp hg.imp)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : Preorder \u03b3\ninst\u271d : Preorder \u03b4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nhf : StrictAnti f\nhg : StrictAnti g\na b : \u03b1 \u00d7 \u03b2\n\u22a2 a < b \u2192 Prod.map f g b < Prod.map f g a\n[PROOFSTEP]\nsimp only [Prod.lt_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : Preorder \u03b3\ninst\u271d : Preorder \u03b4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nhf : StrictAnti f\nhg : StrictAnti g\na b : \u03b1 \u00d7 \u03b2\n\u22a2 a.fst < b.fst \u2227 a.snd \u2264 b.snd \u2228 a.fst \u2264 b.fst \u2227 a.snd < b.snd \u2192\n (Prod.map f g b).fst < (Prod.map f g a).fst \u2227 (Prod.map f g b).snd \u2264 (Prod.map f g a).snd \u2228\n (Prod.map f g b).fst \u2264 (Prod.map f g a).fst \u2227 (Prod.map f g b).snd < (Prod.map f g a).snd\n[PROOFSTEP]\nexact Or.imp (And.imp hf.imp hg.antitone.imp) (And.imp hf.antitone.imp hg.imp)\n", "meta": {"mathlib_filename": "Mathlib.Order.Monotone.Basic", "llama_tokens": 24452, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300698514778, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.40838290087522944}} {"text": "[GOAL]\n\u03b1 : Type ?u.6639\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\n\u22a2 WithLowerTopology.instTopologicalSpaceWithLowerTopology = induced (\u2191WithLowerTopology.ofLower) inst\u271d\u00b9\n[PROOFSTEP]\nerw [topology_eq \u03b1, induced_id]\n[GOAL]\n\u03b1 : Type ?u.6639\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\n\u22a2 WithLowerTopology.instTopologicalSpaceWithLowerTopology = generateFrom {s | \u2203 a, (Ici a)\u1d9c = s}\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\n\u22a2 IsOpen s \u2194 GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} s\n[PROOFSTEP]\nrw [topology_eq \u03b1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\n\u22a2 IsOpen s \u2194 GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\nh : Set.Finite s\n\u22a2 IsClosed \u2191(upperClosure s)\n[PROOFSTEP]\nsimp only [\u2190 UpperSet.iInf_Ici, UpperSet.coe_iInf]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\nh : Set.Finite s\n\u22a2 IsClosed (\u22c3 (i : \u03b1) (_ : i \u2208 s), \u2191(UpperSet.Ici i))\n[PROOFSTEP]\nexact isClosed_biUnion h fun _ _ => isClosed_Ici\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\nh : IsOpen s\n\u22a2 IsLowerSet s\n[PROOFSTEP]\nreplace h := isOpen_iff_generate_Ici_compl.1 h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\nh : GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} s\n\u22a2 IsLowerSet s\n[PROOFSTEP]\ninduction h\n[GOAL]\ncase basic\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns s\u271d : Set \u03b1\na\u271d : s\u271d \u2208 {t | \u2203 a, (Ici a)\u1d9c = t}\n\u22a2 IsLowerSet s\u271d\ncase univ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\n\u22a2 IsLowerSet univ\ncase inter\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns s\u271d t\u271d : Set \u03b1\na\u271d\u00b9 : GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} s\u271d\na\u271d : GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} t\u271d\na_ih\u271d\u00b9 : IsLowerSet s\u271d\na_ih\u271d : IsLowerSet t\u271d\n\u22a2 IsLowerSet (s\u271d \u2229 t\u271d)\ncase sUnion\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s : Set \u03b1), s \u2208 S\u271d \u2192 GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} s\na_ih\u271d : \u2200 (s : Set \u03b1), s \u2208 S\u271d \u2192 IsLowerSet s\n\u22a2 IsLowerSet (\u22c3\u2080 S\u271d)\n[PROOFSTEP]\ncase basic u h' => obtain \u27e8a, rfl\u27e9 := h'; exact (isUpperSet_Ici a).compl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns u : Set \u03b1\nh' : u \u2208 {t | \u2203 a, (Ici a)\u1d9c = t}\n\u22a2 IsLowerSet u\n[PROOFSTEP]\ncase basic u h' => obtain \u27e8a, rfl\u27e9 := h'; exact (isUpperSet_Ici a).compl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns u : Set \u03b1\nh' : u \u2208 {t | \u2203 a, (Ici a)\u1d9c = t}\n\u22a2 IsLowerSet u\n[PROOFSTEP]\nobtain \u27e8a, rfl\u27e9 := h'\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 IsLowerSet (Ici a)\u1d9c\n[PROOFSTEP]\nexact (isUpperSet_Ici a).compl\n[GOAL]\ncase univ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\n\u22a2 IsLowerSet univ\ncase inter\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns s\u271d t\u271d : Set \u03b1\na\u271d\u00b9 : GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} s\u271d\na\u271d : GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} t\u271d\na_ih\u271d\u00b9 : IsLowerSet s\u271d\na_ih\u271d : IsLowerSet t\u271d\n\u22a2 IsLowerSet (s\u271d \u2229 t\u271d)\ncase sUnion\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s : Set \u03b1), s \u2208 S\u271d \u2192 GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} s\na_ih\u271d : \u2200 (s : Set \u03b1), s \u2208 S\u271d \u2192 IsLowerSet s\n\u22a2 IsLowerSet (\u22c3\u2080 S\u271d)\n[PROOFSTEP]\ncase univ => exact isLowerSet_univ\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\n\u22a2 IsLowerSet univ\n[PROOFSTEP]\ncase univ => exact isLowerSet_univ\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\n\u22a2 IsLowerSet univ\n[PROOFSTEP]\nexact isLowerSet_univ\n[GOAL]\ncase inter\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns s\u271d t\u271d : Set \u03b1\na\u271d\u00b9 : GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} s\u271d\na\u271d : GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} t\u271d\na_ih\u271d\u00b9 : IsLowerSet s\u271d\na_ih\u271d : IsLowerSet t\u271d\n\u22a2 IsLowerSet (s\u271d \u2229 t\u271d)\ncase sUnion\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s : Set \u03b1), s \u2208 S\u271d \u2192 GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} s\na_ih\u271d : \u2200 (s : Set \u03b1), s \u2208 S\u271d \u2192 IsLowerSet s\n\u22a2 IsLowerSet (\u22c3\u2080 S\u271d)\n[PROOFSTEP]\ncase inter u v _ _ hu2 hv2 => exact hu2.inter hv2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns u v : Set \u03b1\na\u271d\u00b9 : GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} u\na\u271d : GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} v\nhu2 : IsLowerSet u\nhv2 : IsLowerSet v\n\u22a2 IsLowerSet (u \u2229 v)\n[PROOFSTEP]\ncase inter u v _ _ hu2 hv2 => exact hu2.inter hv2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns u v : Set \u03b1\na\u271d\u00b9 : GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} u\na\u271d : GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} v\nhu2 : IsLowerSet u\nhv2 : IsLowerSet v\n\u22a2 IsLowerSet (u \u2229 v)\n[PROOFSTEP]\nexact hu2.inter hv2\n[GOAL]\ncase sUnion\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s : Set \u03b1), s \u2208 S\u271d \u2192 GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} s\na_ih\u271d : \u2200 (s : Set \u03b1), s \u2208 S\u271d \u2192 IsLowerSet s\n\u22a2 IsLowerSet (\u22c3\u2080 S\u271d)\n[PROOFSTEP]\ncase sUnion _ _ ih => exact isLowerSet_sUnion ih\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s : Set \u03b1), s \u2208 S\u271d \u2192 GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} s\nih : \u2200 (s : Set \u03b1), s \u2208 S\u271d \u2192 IsLowerSet s\n\u22a2 IsLowerSet (\u22c3\u2080 S\u271d)\n[PROOFSTEP]\ncase sUnion _ _ ih => exact isLowerSet_sUnion ih\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s : Set \u03b1), s \u2208 S\u271d \u2192 GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} s\nih : \u2200 (s : Set \u03b1), s \u2208 S\u271d \u2192 IsLowerSet s\n\u22a2 IsLowerSet (\u22c3\u2080 S\u271d)\n[PROOFSTEP]\nexact isLowerSet_sUnion ih\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\n\u22a2 IsTopologicalBasis (lowerBasis \u03b1)\n[PROOFSTEP]\nconvert isTopologicalBasis_of_subbasis (topology_eq \u03b1)\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\n\u22a2 lowerBasis \u03b1 = (fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}}\n[PROOFSTEP]\nsimp_rw [lowerBasis, coe_upperClosure, compl_iUnion]\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\n\u22a2 {s | \u2203 t, Set.Finite t \u2227 \u22c2 (i : \u03b1) (_ : i \u2208 t), (Ici i)\u1d9c = s} =\n (fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}}\n[PROOFSTEP]\next s\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns\u271d s : Set \u03b1\n\u22a2 s \u2208 {s | \u2203 t, Set.Finite t \u2227 \u22c2 (i : \u03b1) (_ : i \u2208 t), (Ici i)\u1d9c = s} \u2194\n s \u2208 (fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.e'_3.h.mp\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns\u271d s : Set \u03b1\n\u22a2 s \u2208 {s | \u2203 t, Set.Finite t \u2227 \u22c2 (i : \u03b1) (_ : i \u2208 t), (Ici i)\u1d9c = s} \u2192\n s \u2208 (fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}}\n[PROOFSTEP]\nrintro \u27e8F, hF, rfl\u27e9\n[GOAL]\ncase h.e'_3.h.mp.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns F : Set \u03b1\nhF : Set.Finite F\n\u22a2 \u22c2 (i : \u03b1) (_ : i \u2208 F), (Ici i)\u1d9c \u2208 (fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}}\n[PROOFSTEP]\nrefine' \u27e8(fun a => (Ici a)\u1d9c) '' F, \u27e8hF.image _, image_subset_iff.2 fun _ _ => \u27e8_, rfl\u27e9\u27e9, _\u27e9\n[GOAL]\ncase h.e'_3.h.mp.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns F : Set \u03b1\nhF : Set.Finite F\n\u22a2 (fun f => \u22c2\u2080 f) ((fun a => (Ici a)\u1d9c) '' F) = \u22c2 (i : \u03b1) (_ : i \u2208 F), (Ici i)\u1d9c\n[PROOFSTEP]\nsimp only [sInter_image]\n[GOAL]\ncase h.e'_3.h.mpr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns\u271d s : Set \u03b1\n\u22a2 s \u2208 (fun f => \u22c2\u2080 f) '' {f | Set.Finite f \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}} \u2192\n s \u2208 {s | \u2203 t, Set.Finite t \u2227 \u22c2 (i : \u03b1) (_ : i \u2208 t), (Ici i)\u1d9c = s}\n[PROOFSTEP]\nrintro \u27e8F, \u27e8hF, hs\u27e9, rfl\u27e9\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\nF : Set (Set \u03b1)\nhF : Set.Finite F\nhs : F \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}\n\u22a2 (fun f => \u22c2\u2080 f) F \u2208 {s | \u2203 t, Set.Finite t \u2227 \u22c2 (i : \u03b1) (_ : i \u2208 t), (Ici i)\u1d9c = s}\n[PROOFSTEP]\nhaveI := hF.to_subtype\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\nF : Set (Set \u03b1)\nhF : Set.Finite F\nhs : F \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}\nthis : Finite \u2191F\n\u22a2 (fun f => \u22c2\u2080 f) F \u2208 {s | \u2203 t, Set.Finite t \u2227 \u22c2 (i : \u03b1) (_ : i \u2208 t), (Ici i)\u1d9c = s}\n[PROOFSTEP]\nrw [subset_def, Subtype.forall'] at hs \n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\nF : Set (Set \u03b1)\nhF : Set.Finite F\nhs : \u2200 (x : { a // a \u2208 F }), \u2191x \u2208 {s | \u2203 a, (Ici a)\u1d9c = s}\nthis : Finite \u2191F\n\u22a2 (fun f => \u22c2\u2080 f) F \u2208 {s | \u2203 t, Set.Finite t \u2227 \u22c2 (i : \u03b1) (_ : i \u2208 t), (Ici i)\u1d9c = s}\n[PROOFSTEP]\nchoose f hf using hs\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\nF : Set (Set \u03b1)\nhF : Set.Finite F\nthis : Finite \u2191F\nf : { a // a \u2208 F } \u2192 \u03b1\nhf : \u2200 (x : { a // a \u2208 F }), (Ici (f x))\u1d9c = \u2191x\n\u22a2 (fun f => \u22c2\u2080 f) F \u2208 {s | \u2203 t, Set.Finite t \u2227 \u22c2 (i : \u03b1) (_ : i \u2208 t), (Ici i)\u1d9c = s}\n[PROOFSTEP]\nexact \u27e8_, finite_range f, by simp_rw [biInter_range, hf, sInter_eq_iInter]\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\ns : Set \u03b1\nF : Set (Set \u03b1)\nhF : Set.Finite F\nthis : Finite \u2191F\nf : { a // a \u2208 F } \u2192 \u03b1\nhf : \u2200 (x : { a // a \u2208 F }), (Ici (f x))\u1d9c = \u2191x\n\u22a2 \u22c2 (i : \u03b1) (_ : i \u2208 range f), (Ici i)\u1d9c = (fun f => \u22c2\u2080 f) F\n[PROOFSTEP]\nsimp_rw [biInter_range, hf, sInter_eq_iInter]\n[GOAL]\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LowerTopology \u03b1\ns : Set \u03b1\n\u03b2 : Type u_1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (a : \u03b1), IsClosed (f \u207b\u00b9' Ici a)\n\u22a2 Continuous f\n[PROOFSTEP]\nobtain rfl := LowerTopology.topology_eq \u03b1\n[GOAL]\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ns : Set \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (a : \u03b1), IsClosed (f \u207b\u00b9' Ici a)\ninst\u271d : LowerTopology \u03b1\n\u22a2 Continuous f\n[PROOFSTEP]\nrefine continuous_generateFrom ?_\n[GOAL]\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ns : Set \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (a : \u03b1), IsClosed (f \u207b\u00b9' Ici a)\ninst\u271d : LowerTopology \u03b1\n\u22a2 \u2200 (s : Set \u03b1), s \u2208 {s | \u2203 a, (Ici a)\u1d9c = s} \u2192 IsOpen (f \u207b\u00b9' s)\n[PROOFSTEP]\nrintro _ \u27e8a, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ns : Set \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (a : \u03b1), IsClosed (f \u207b\u00b9' Ici a)\ninst\u271d : LowerTopology \u03b1\na : \u03b1\n\u22a2 IsOpen (f \u207b\u00b9' (Ici a)\u1d9c)\n[PROOFSTEP]\nexact (h a).isOpen_compl\n[GOAL]\n\u03b1 : Type ?u.15319\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LowerTopology \u03b1\nx y : \u03b1\nh : Inseparable x y\n\u22a2 Ici x = Ici y\n[PROOFSTEP]\nsimpa only [inseparable_iff_closure_eq, closure_singleton] using h\n[GOAL]\n\u03b1 : Type ?u.16169\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UpperTopology \u03b1\ns : Set \u03b1\n\u22a2 WithUpperTopology.instTopologicalSpaceWithUpperTopology = induced (\u2191WithUpperTopology.ofUpper) inst\u271d\u00b9\n[PROOFSTEP]\nerw [topology_eq \u03b1, induced_id]\n[GOAL]\n\u03b1 : Type ?u.16169\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UpperTopology \u03b1\ns : Set \u03b1\n\u22a2 WithUpperTopology.instTopologicalSpaceWithUpperTopology = generateFrom {s | \u2203 a, (Iic a)\u1d9c = s}\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UpperTopology \u03b1\ns : Set \u03b1\n\u22a2 IsOpen s \u2194 GenerateOpen {t | \u2203 a, (Iic a)\u1d9c = t} s\n[PROOFSTEP]\nrw [topology_eq \u03b1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UpperTopology \u03b1\ns : Set \u03b1\n\u22a2 IsOpen s \u2194 GenerateOpen {t | \u2203 a, (Iic a)\u1d9c = t} s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type ?u.18877\n\u03b2 : Type ?u.18906\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : LowerTopology \u03b1\ninst\u271d\u2074 : OrderBot \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : LowerTopology \u03b2\ninst\u271d : OrderBot \u03b2\n\u22a2 instTopologicalSpaceProd = generateFrom {s | \u2203 a, (Ici a)\u1d9c = s}\n[PROOFSTEP]\nrefine' le_antisymm (le_generateFrom _) _\n[GOAL]\ncase refine'_1\n\u03b1 : Type ?u.18877\n\u03b2 : Type ?u.18906\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : LowerTopology \u03b1\ninst\u271d\u2074 : OrderBot \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : LowerTopology \u03b2\ninst\u271d : OrderBot \u03b2\n\u22a2 \u2200 (s : Set (\u03b1 \u00d7 \u03b2)), s \u2208 {s | \u2203 a, (Ici a)\u1d9c = s} \u2192 IsOpen s\n[PROOFSTEP]\nrintro _ \u27e8x, rfl\u27e9\n[GOAL]\ncase refine'_1.intro\n\u03b1 : Type ?u.18877\n\u03b2 : Type ?u.18906\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : LowerTopology \u03b1\ninst\u271d\u2074 : OrderBot \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : LowerTopology \u03b2\ninst\u271d : OrderBot \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 IsOpen (Ici x)\u1d9c\n[PROOFSTEP]\nexact (isClosed_Ici.prod isClosed_Ici).isOpen_compl\n[GOAL]\ncase refine'_2\n\u03b1 : Type ?u.18877\n\u03b2 : Type ?u.18906\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : LowerTopology \u03b1\ninst\u271d\u2074 : OrderBot \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : LowerTopology \u03b2\ninst\u271d : OrderBot \u03b2\n\u22a2 generateFrom {s | \u2203 a, (Ici a)\u1d9c = s} \u2264 instTopologicalSpaceProd\n[PROOFSTEP]\nrw [(LowerTopology.isTopologicalBasis.prod LowerTopology.isTopologicalBasis).eq_generateFrom,\n le_generateFrom_iff_subset_isOpen, image2_subset_iff]\n[GOAL]\ncase refine'_2\n\u03b1 : Type ?u.18877\n\u03b2 : Type ?u.18906\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : LowerTopology \u03b1\ninst\u271d\u2074 : OrderBot \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : LowerTopology \u03b2\ninst\u271d : OrderBot \u03b2\n\u22a2 \u2200 (x : Set \u03b1),\n x \u2208 LowerTopology.lowerBasis \u03b1 \u2192 \u2200 (y : Set \u03b2), y \u2208 LowerTopology.lowerBasis \u03b2 \u2192 x \u00d7\u02e2 y \u2208 {s | IsOpen s}\n[PROOFSTEP]\nrintro _ \u27e8s, hs, rfl\u27e9 _ \u27e8t, ht, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\n\u03b1 : Type ?u.18877\n\u03b2 : Type ?u.18906\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : LowerTopology \u03b1\ninst\u271d\u2074 : OrderBot \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : LowerTopology \u03b2\ninst\u271d : OrderBot \u03b2\ns : Set \u03b1\nhs : Set.Finite s\nt : Set \u03b2\nht : Set.Finite t\n\u22a2 (\u2191(upperClosure s))\u1d9c \u00d7\u02e2 (\u2191(upperClosure t))\u1d9c \u2208 {s | IsOpen s}\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\n\u03b1 : Type ?u.18877\n\u03b2 : Type ?u.18906\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : LowerTopology \u03b1\ninst\u271d\u2074 : OrderBot \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : LowerTopology \u03b2\ninst\u271d : OrderBot \u03b2\ns : Set \u03b1\nhs : Set.Finite s\nt : Set \u03b2\nht : Set.Finite t\n\u22a2 IsOpen ((\u2191(upperClosure s))\u1d9c \u00d7\u02e2 (\u2191(upperClosure t))\u1d9c)\n[PROOFSTEP]\nsimp_rw [coe_upperClosure, compl_iUnion, prod_eq, preimage_iInter, preimage_compl]\n -- without `let`, `refine` tries to use the product topology and fails\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\n\u03b1 : Type ?u.18877\n\u03b2 : Type ?u.18906\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : LowerTopology \u03b1\ninst\u271d\u2074 : OrderBot \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : LowerTopology \u03b2\ninst\u271d : OrderBot \u03b2\ns : Set \u03b1\nhs : Set.Finite s\nt : Set \u03b2\nht : Set.Finite t\n\u22a2 IsOpen ((\u22c2 (i : \u03b1) (_ : i \u2208 s), (Prod.fst \u207b\u00b9' Ici i)\u1d9c) \u2229 \u22c2 (i : \u03b2) (_ : i \u2208 t), (Prod.snd \u207b\u00b9' Ici i)\u1d9c)\n[PROOFSTEP]\nlet _ : TopologicalSpace (\u03b1 \u00d7 \u03b2) := generateFrom {s | \u2203 a, (Ici a)\u1d9c = s}\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\n\u03b1 : Type ?u.18877\n\u03b2 : Type ?u.18906\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : LowerTopology \u03b1\ninst\u271d\u2074 : OrderBot \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : LowerTopology \u03b2\ninst\u271d : OrderBot \u03b2\ns : Set \u03b1\nhs : Set.Finite s\nt : Set \u03b2\nht : Set.Finite t\nx\u271d : TopologicalSpace (\u03b1 \u00d7 \u03b2) := generateFrom {s | \u2203 a, (Ici a)\u1d9c = s}\n\u22a2 IsOpen ((\u22c2 (i : \u03b1) (_ : i \u2208 s), (Prod.fst \u207b\u00b9' Ici i)\u1d9c) \u2229 \u22c2 (i : \u03b2) (_ : i \u2208 t), (Prod.snd \u207b\u00b9' Ici i)\u1d9c)\n[PROOFSTEP]\nrefine (isOpen_biInter hs fun a _ => ?_).inter (isOpen_biInter ht fun b _ => ?_)\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.refine_1\n\u03b1 : Type ?u.18877\n\u03b2 : Type ?u.18906\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : LowerTopology \u03b1\ninst\u271d\u2074 : OrderBot \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : LowerTopology \u03b2\ninst\u271d : OrderBot \u03b2\ns : Set \u03b1\nhs : Set.Finite s\nt : Set \u03b2\nht : Set.Finite t\nx\u271d\u00b9 : TopologicalSpace (\u03b1 \u00d7 \u03b2) := generateFrom {s | \u2203 a, (Ici a)\u1d9c = s}\na : \u03b1\nx\u271d : a \u2208 s\n\u22a2 IsOpen (Prod.fst \u207b\u00b9' Ici a)\u1d9c\n[PROOFSTEP]\nexact GenerateOpen.basic _ \u27e8(a, \u22a5), by simp [Ici_prod_eq, prod_univ]\u27e9\n[GOAL]\n\u03b1 : Type ?u.18877\n\u03b2 : Type ?u.18906\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : LowerTopology \u03b1\ninst\u271d\u2074 : OrderBot \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : LowerTopology \u03b2\ninst\u271d : OrderBot \u03b2\ns : Set \u03b1\nhs : Set.Finite s\nt : Set \u03b2\nht : Set.Finite t\nx\u271d\u00b9 : TopologicalSpace (\u03b1 \u00d7 \u03b2) := generateFrom {s | \u2203 a, (Ici a)\u1d9c = s}\na : \u03b1\nx\u271d : a \u2208 s\n\u22a2 (Ici (a, \u22a5))\u1d9c = (Prod.fst \u207b\u00b9' Ici a)\u1d9c\n[PROOFSTEP]\nsimp [Ici_prod_eq, prod_univ]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.refine_2\n\u03b1 : Type ?u.18877\n\u03b2 : Type ?u.18906\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : LowerTopology \u03b1\ninst\u271d\u2074 : OrderBot \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : LowerTopology \u03b2\ninst\u271d : OrderBot \u03b2\ns : Set \u03b1\nhs : Set.Finite s\nt : Set \u03b2\nht : Set.Finite t\nx\u271d\u00b9 : TopologicalSpace (\u03b1 \u00d7 \u03b2) := generateFrom {s | \u2203 a, (Ici a)\u1d9c = s}\nb : \u03b2\nx\u271d : b \u2208 t\n\u22a2 IsOpen (Prod.snd \u207b\u00b9' Ici b)\u1d9c\n[PROOFSTEP]\nexact GenerateOpen.basic _ \u27e8(\u22a5, b), by simp [Ici_prod_eq, univ_prod]\u27e9\n[GOAL]\n\u03b1 : Type ?u.18877\n\u03b2 : Type ?u.18906\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : LowerTopology \u03b1\ninst\u271d\u2074 : OrderBot \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : LowerTopology \u03b2\ninst\u271d : OrderBot \u03b2\ns : Set \u03b1\nhs : Set.Finite s\nt : Set \u03b2\nht : Set.Finite t\nx\u271d\u00b9 : TopologicalSpace (\u03b1 \u00d7 \u03b2) := generateFrom {s | \u2203 a, (Ici a)\u1d9c = s}\nb : \u03b2\nx\u271d : b \u2208 t\n\u22a2 (Ici (\u22a5, b))\u1d9c = (Prod.snd \u207b\u00b9' Ici b)\u1d9c\n[PROOFSTEP]\nsimp [Ici_prod_eq, univ_prod]\n[GOAL]\n\u03b1 : Type ?u.24860\n\u03b2 : Type ?u.24889\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : UpperTopology \u03b1\ninst\u271d\u2074 : OrderTop \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : UpperTopology \u03b2\ninst\u271d : OrderTop \u03b2\n\u22a2 instTopologicalSpaceProd = generateFrom {s | \u2203 a, (Iic a)\u1d9c = s}\n[PROOFSTEP]\nsuffices : LowerTopology (\u03b1 \u00d7 \u03b2)\u1d52\u1d48\n[GOAL]\n\u03b1 : Type ?u.24860\n\u03b2 : Type ?u.24889\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : UpperTopology \u03b1\ninst\u271d\u2074 : OrderTop \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : UpperTopology \u03b2\ninst\u271d : OrderTop \u03b2\nthis : LowerTopology (\u03b1 \u00d7 \u03b2)\u1d52\u1d48\n\u22a2 instTopologicalSpaceProd = generateFrom {s | \u2203 a, (Iic a)\u1d9c = s}\n[PROOFSTEP]\nexact LowerTopology.topology_eq_lowerTopology (\u03b1 := (\u03b1 \u00d7 \u03b2)\u1d52\u1d48)\n[GOAL]\ncase this\n\u03b1 : Type ?u.24860\n\u03b2 : Type ?u.24889\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : UpperTopology \u03b1\ninst\u271d\u2074 : OrderTop \u03b1\ninst\u271d\u00b3 : Preorder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : UpperTopology \u03b2\ninst\u271d : OrderTop \u03b2\n\u22a2 LowerTopology (\u03b1 \u00d7 \u03b2)\u1d52\u1d48\n[PROOFSTEP]\nexact instLowerTopologyProd (\u03b1 := \u03b1\u1d52\u1d48) (\u03b2 := \u03b2\u1d52\u1d48)\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u2075 : CompleteLattice \u03b1\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LowerTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : LowerTopology \u03b2\nf : sInfHom \u03b1 \u03b2\n\u22a2 Continuous \u2191f\n[PROOFSTEP]\nrefine LowerTopology.continuous_of_Ici fun b => ?_\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u2075 : CompleteLattice \u03b1\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LowerTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : LowerTopology \u03b2\nf : sInfHom \u03b1 \u03b2\nb : \u03b2\n\u22a2 IsClosed (\u2191f \u207b\u00b9' Ici b)\n[PROOFSTEP]\nconvert isClosed_Ici (a := sInf <| f \u207b\u00b9' Ici b)\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u2075 : CompleteLattice \u03b1\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LowerTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : LowerTopology \u03b2\nf : sInfHom \u03b1 \u03b2\nb : \u03b2\n\u22a2 \u2191f \u207b\u00b9' Ici b = Ici (sInf (\u2191f \u207b\u00b9' Ici b))\n[PROOFSTEP]\nrefine' Subset.antisymm (fun a => sInf_le) fun a ha => le_trans _ <| OrderHomClass.mono (f : \u03b1 \u2192o \u03b2) ha\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u2075 : CompleteLattice \u03b1\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LowerTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : LowerTopology \u03b2\nf : sInfHom \u03b1 \u03b2\nb : \u03b2\na : \u03b1\nha : a \u2208 Ici (sInf (\u2191f \u207b\u00b9' Ici b))\n\u22a2 b \u2264 \u2191\u2191f (sInf (\u2191f \u207b\u00b9' Ici b))\n[PROOFSTEP]\nrefine' LE.le.trans _ (map_sInf f _).ge\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u2075 : CompleteLattice \u03b1\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LowerTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : LowerTopology \u03b2\nf : sInfHom \u03b1 \u03b2\nb : \u03b2\na : \u03b1\nha : a \u2208 Ici (sInf (\u2191f \u207b\u00b9' Ici b))\n\u22a2 b \u2264 sInf (\u2191f '' (\u2191f \u207b\u00b9' Ici b))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 UpperTopology \u03b1\u1d52\u1d48 \u2194 LowerTopology \u03b1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 UpperTopology \u03b1\u1d52\u1d48 \u2192 LowerTopology \u03b1\n[PROOFSTEP]\napply UpperTopology.instLowerTopologyDual\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 LowerTopology \u03b1 \u2192 UpperTopology \u03b1\u1d52\u1d48\n[PROOFSTEP]\napply LowerTopology.instUpperTopologyDual\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 LowerTopology \u03b1\u1d52\u1d48 \u2194 UpperTopology \u03b1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 LowerTopology \u03b1\u1d52\u1d48 \u2192 UpperTopology \u03b1\n[PROOFSTEP]\napply LowerTopology.instUpperTopologyDual\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 UpperTopology \u03b1 \u2192 LowerTopology \u03b1\u1d52\u1d48\n[PROOFSTEP]\napply UpperTopology.instLowerTopologyDual\n", "meta": {"mathlib_filename": "Mathlib.Topology.Order.LowerUpperTopology", "llama_tokens": 11321, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952052, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4083828936343596}} {"text": "[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.129, u_1} C\ninst\u271d\u2074 : Category.{?u.133, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\nX Y : C\neval : Y \u2297 X \u27f6 \ud835\udfd9_ C\ncoeval : \ud835\udfd9_ C \u27f6 X \u2297 Y\ninst\u271d : ExactPairing (F.obj X) (F.obj Y)\nmap_eval : F.map eval = inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X) \u226b \u03b5_ (F.obj X) (F.obj Y) \u226b F.\u03b5\nmap_coeval : F.map coeval = inv F.\u03b5 \u226b \u03b7_ (F.obj X) (F.obj Y) \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y\n\u22a2 F.map ((\ud835\udfd9 Y \u2297 coeval) \u226b (\u03b1_ Y X Y).inv \u226b (eval \u2297 \ud835\udfd9 Y)) = F.map ((\u03c1_ Y).hom \u226b (\u03bb_ Y).inv)\n[PROOFSTEP]\nsimp [map_eval, map_coeval, MonoidalFunctor.map_tensor]\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.129, u_1} C\ninst\u271d\u2074 : Category.{?u.133, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\nX Y : C\neval : Y \u2297 X \u27f6 \ud835\udfd9_ C\ncoeval : \ud835\udfd9_ C \u27f6 X \u2297 Y\ninst\u271d : ExactPairing (F.obj X) (F.obj Y)\nmap_eval : F.map eval = inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X) \u226b \u03b5_ (F.obj X) (F.obj Y) \u226b F.\u03b5\nmap_coeval : F.map coeval = inv F.\u03b5 \u226b \u03b7_ (F.obj X) (F.obj Y) \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y\n\u22a2 F.map ((coeval \u2297 \ud835\udfd9 X) \u226b (\u03b1_ X Y X).hom \u226b (\ud835\udfd9 X \u2297 eval)) = F.map ((\u03bb_ X).hom \u226b (\u03c1_ X).inv)\n[PROOFSTEP]\nsimp [map_eval, map_coeval, MonoidalFunctor.map_tensor]\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2076 : Category.{?u.23115, u_1} C\ninst\u271d\u2075 : Category.{?u.23119, u_2} D\ninst\u271d\u2074 : MonoidalCategory C\ninst\u271d\u00b3 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b2 : Full F.toFunctor\ninst\u271d\u00b9 : Faithful F.toFunctor\nX Y : C\ninst\u271d : ExactPairing (F.obj X) (F.obj Y)\n\u22a2 F.map (F.preimage (inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X) \u226b \u03b5_ (F.obj X) (F.obj Y) \u226b F.\u03b5)) =\n inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X) \u226b \u03b5_ (F.obj X) (F.obj Y) \u226b F.\u03b5\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2076 : Category.{?u.23115, u_1} C\ninst\u271d\u2075 : Category.{?u.23119, u_2} D\ninst\u271d\u2074 : MonoidalCategory C\ninst\u271d\u00b3 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b2 : Full F.toFunctor\ninst\u271d\u00b9 : Faithful F.toFunctor\nX Y : C\ninst\u271d : ExactPairing (F.obj X) (F.obj Y)\n\u22a2 F.map (F.preimage (inv F.\u03b5 \u226b \u03b7_ (F.obj X) (F.obj Y) \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y)) =\n inv F.\u03b5 \u226b \u03b7_ (F.obj X) (F.obj Y) \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.27280, u_1} C\ninst\u271d\u2074 : Category.{?u.27284, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : IsEquivalence F.toFunctor\nX : C\ninst\u271d : HasLeftDual (F.obj X)\n\u22a2 ExactPairing ((Functor.inv F.toFunctor).obj \u1601(F.obj X)) X\n[PROOFSTEP]\nrefine @exactPairingOfFullyFaithful _ _ _ _ _ _ F _ _ _ _ ?_\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.27280, u_1} C\ninst\u271d\u2074 : Category.{?u.27284, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : IsEquivalence F.toFunctor\nX : C\ninst\u271d : HasLeftDual (F.obj X)\n\u22a2 ExactPairing (F.obj ((Functor.inv F.toFunctor).obj \u1601(F.obj X))) (F.obj X)\n[PROOFSTEP]\nrefine @exactPairingCongrLeft _ _ _ _ _ _ ?_ (F.toFunctor.asEquivalence.counitIso.app _)\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.27280, u_1} C\ninst\u271d\u2074 : Category.{?u.27284, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : IsEquivalence F.toFunctor\nX : C\ninst\u271d : HasLeftDual (F.obj X)\n\u22a2 ExactPairing ((\ud835\udfed D).obj \u1601(F.obj X)) (F.obj X)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.27280, u_1} C\ninst\u271d\u2074 : Category.{?u.27284, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : IsEquivalence F.toFunctor\nX : C\ninst\u271d : HasLeftDual (F.obj X)\n\u22a2 ExactPairing (\u1601(F.obj X)) (F.obj X)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.30444, u_1} C\ninst\u271d\u2074 : Category.{?u.30448, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : IsEquivalence F.toFunctor\nX : C\ninst\u271d : HasRightDual (F.obj X)\n\u22a2 ExactPairing X ((Functor.inv F.toFunctor).obj (F.obj X)\u1601)\n[PROOFSTEP]\nrefine @exactPairingOfFullyFaithful _ _ _ _ _ _ F _ _ _ _ ?_\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.30444, u_1} C\ninst\u271d\u2074 : Category.{?u.30448, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : IsEquivalence F.toFunctor\nX : C\ninst\u271d : HasRightDual (F.obj X)\n\u22a2 ExactPairing (F.obj X) (F.obj ((Functor.inv F.toFunctor).obj (F.obj X)\u1601))\n[PROOFSTEP]\nrefine @exactPairingCongrRight _ _ _ _ _ _ ?_ (F.toFunctor.asEquivalence.counitIso.app _)\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.30444, u_1} C\ninst\u271d\u2074 : Category.{?u.30448, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : IsEquivalence F.toFunctor\nX : C\ninst\u271d : HasRightDual (F.obj X)\n\u22a2 ExactPairing (F.obj X) ((\ud835\udfed D).obj (F.obj X)\u1601)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.30444, u_1} C\ninst\u271d\u2074 : Category.{?u.30448, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : IsEquivalence F.toFunctor\nX : C\ninst\u271d : HasRightDual (F.obj X)\n\u22a2 ExactPairing (F.obj X) (F.obj X)\u1601\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence", "llama_tokens": 2823, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929104825006, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.40826416137260946}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b : \u03b1\na : \u03b1 \u00d7 \u03b1\nh : a \u2208 offDiag s\n\u22a2 \u00acSym2.IsDiag (Quotient.mk (Sym2.Rel.setoid \u03b1) a)\n[PROOFSTEP]\nrw [Sym2.isDiag_iff_proj_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b : \u03b1\na : \u03b1 \u00d7 \u03b1\nh : a \u2208 offDiag s\n\u22a2 \u00aca.fst = a.snd\n[PROOFSTEP]\nexact (mem_offDiag.1 h).2.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\n\u22a2 m \u2208 Finset.sym2 s \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\n[PROOFSTEP]\nrefine mem_image.trans \u27e8?_, fun h \u21a6 \u27e8m.out, mem_product.2 \u27e8h _ m.out_fst_mem, h _ m.out_snd_mem\u27e9, m.out_eq\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\n\u22a2 (\u2203 a, a \u2208 s \u00d7\u02e2 s \u2227 Quotient.mk' a = m) \u2192 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\n[PROOFSTEP]\nrintro \u27e8\u27e8a, b\u27e9, h, rfl\u27e9\n[GOAL]\ncase intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d a b : \u03b1\nh : (a, b) \u2208 s \u00d7\u02e2 s\n\u22a2 \u2200 (a_1 : \u03b1), a_1 \u2208 Quotient.mk' (a, b) \u2192 a_1 \u2208 s\n[PROOFSTEP]\nrw [Quotient.mk', @Sym2.ball _ (fun x \u21a6 x \u2208 s)]\n[GOAL]\ncase intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d a b : \u03b1\nh : (a, b) \u2208 s \u00d7\u02e2 s\n\u22a2 a \u2208 s \u2227 b \u2208 s\n[PROOFSTEP]\nrwa [mem_product] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\n\u22a2 Quotient.mk (Sym2.Rel.setoid \u03b1) (a, b) \u2208 Finset.sym2 s \u2194 a \u2208 s \u2227 b \u2208 s\n[PROOFSTEP]\nrw [mem_sym2_iff, Sym2.ball]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\n\u22a2 Finset.sym2 s = \u2205 \u2194 s = \u2205\n[PROOFSTEP]\nrw [Finset.sym2, image_eq_empty, product_eq_empty, or_self_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\n\u22a2 Finset.Nonempty (Finset.sym2 s) \u2194 Finset.Nonempty s\n[PROOFSTEP]\nrw [Finset.sym2, Nonempty.image_iff, nonempty_product, and_self_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\ninst\u271d : Fintype \u03b1\n\u22a2 Finset.sym2 univ = univ\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\ninst\u271d : Fintype \u03b1\na\u271d : Sym2 \u03b1\n\u22a2 a\u271d \u2208 Finset.sym2 univ \u2194 a\u271d \u2208 univ\n[PROOFSTEP]\nsimp only [mem_sym2_iff, mem_univ, implies_true]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b : \u03b1\nm : Sym2 \u03b1\na : \u03b1\n\u22a2 Finset.sym2 {a} = {Sym2.diag a}\n[PROOFSTEP]\nrw [Finset.sym2, singleton_product_singleton, image_singleton, Sym2.diag, Quotient.mk']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\n\u22a2 (\u2200 (b : \u03b1), b \u2208 Sym2.diag a \u2192 b \u2208 s) \u2194 a \u2208 s\n[PROOFSTEP]\nrw [\u2190 mem_sym2_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\n\u22a2 Sym2.diag a \u2208 Finset.sym2 s \u2194 a \u2208 s\n[PROOFSTEP]\nexact mk'_mem_sym2_iff.trans <| and_self_iff _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\n\u22a2 Sym2.diag a \u2208 Finset.sym2 s \u2194 a \u2208 s\n[PROOFSTEP]\nsimp [diag_mem_sym2_mem_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\n\u22a2 image Quotient.mk' (diag s) \u222a image Quotient.mk' (offDiag s) = Finset.sym2 s\n[PROOFSTEP]\nrw [\u2190 image_union, diag_union_offDiag]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\n\u22a2 image Quotient.mk' (s \u00d7\u02e2 s) = Finset.sym2 s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 n\n\u22a2 m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nm : Sym \u03b1 Nat.zero\n\u22a2 m \u2208 Finset.sym s Nat.zero \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\n[PROOFSTEP]\nrefine' mem_singleton.trans \u27e8_, fun _ \u21a6 Sym.eq_nil_of_card_zero _\u27e9\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nm : Sym \u03b1 Nat.zero\n\u22a2 m = \u2205 \u2192 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 n\n\u22a2 \u2200 (a : \u03b1), a \u2208 \u2205 \u2192 a \u2208 s\n[PROOFSTEP]\nexact fun a ha \u21a6 (Finset.not_mem_empty _ ha).elim\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nm : Sym \u03b1 (Nat.succ n)\n\u22a2 m \u2208 Finset.sym s (Nat.succ n) \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\n[PROOFSTEP]\nrefine' mem_sup.trans \u27e8_, fun h \u21a6 _\u27e9\n[GOAL]\ncase succ.refine'_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nm : Sym \u03b1 (Nat.succ n)\n\u22a2 (\u2203 v, v \u2208 s \u2227 m \u2208 image (Sym.cons v) (Finset.sym s n)) \u2192 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\n[PROOFSTEP]\nrintro \u27e8a, ha, he\u27e9 b hb\n[GOAL]\ncase succ.refine'_1.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nm : Sym \u03b1 (Nat.succ n)\na : \u03b1\nha : a \u2208 s\nhe : m \u2208 image (Sym.cons a) (Finset.sym s n)\nb : \u03b1\nhb : b \u2208 m\n\u22a2 b \u2208 s\n[PROOFSTEP]\nrw [mem_image] at he \n[GOAL]\ncase succ.refine'_1.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nm : Sym \u03b1 (Nat.succ n)\na : \u03b1\nha : a \u2208 s\nhe : \u2203 a_1, a_1 \u2208 Finset.sym s n \u2227 a ::\u209b a_1 = m\nb : \u03b1\nhb : b \u2208 m\n\u22a2 b \u2208 s\n[PROOFSTEP]\nobtain \u27e8m, he, rfl\u27e9 := he\n[GOAL]\ncase succ.refine'_1.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 Finset.sym s n\nhb : b \u2208 a ::\u209b m\n\u22a2 b \u2208 s\n[PROOFSTEP]\nrw [Sym.mem_cons] at hb \n[GOAL]\ncase succ.refine'_1.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 Finset.sym s n\nhb : b = a \u2228 b \u2208 m\n\u22a2 b \u2208 s\n[PROOFSTEP]\nobtain rfl | hb := hb\n[GOAL]\ncase succ.refine'_1.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 Finset.sym s n\nha : b \u2208 s\n\u22a2 b \u2208 s\n[PROOFSTEP]\nexact ha\n[GOAL]\ncase succ.refine'_1.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b\u271d : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nm : Sym \u03b1 n\nhe : m \u2208 Finset.sym s n\nhb : b \u2208 m\n\u22a2 b \u2208 s\n[PROOFSTEP]\nexact ih.1 he _ hb\n[GOAL]\ncase succ.refine'_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\nm : Sym \u03b1 (Nat.succ n)\nh : \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\n\u22a2 \u2203 v, v \u2208 s \u2227 m \u2208 image (Sym.cons v) (Finset.sym s n)\n[PROOFSTEP]\nobtain \u27e8a, m, rfl\u27e9 := m.exists_eq_cons_of_succ\n[GOAL]\ncase succ.refine'_2.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d b : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\nn : \u2115\nih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s\na : \u03b1\nm : Sym \u03b1 n\nh : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u209b m \u2192 a_1 \u2208 s\n\u22a2 \u2203 v, v \u2208 s \u2227 a ::\u209b m \u2208 image (Sym.cons v) (Finset.sym s n)\n[PROOFSTEP]\nexact \u27e8a, h _ <| Sym.mem_cons_self _ _, mem_image_of_mem _ <| ih.2 fun b hb \u21a6 h _ <| Sym.mem_cons_of_mem hb\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b\u271d : \u03b1\nn\u271d : \u2115\nm : Sym \u03b1 n\u271d\nha : a \u2208 s\nn : \u2115\nb : \u03b1\nhb : b \u2208 Sym.replicate n a\n\u22a2 b \u2208 s\n[PROOFSTEP]\nrwa [(Sym.mem_replicate.1 hb).2]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 n\nh : Finset.sym s n = \u2205\n\u22a2 s = \u2205\n[PROOFSTEP]\nrw [\u2190 not_nonempty_iff_eq_empty] at h \u22a2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 n\nh : \u00acFinset.Nonempty (Finset.sym s n)\n\u22a2 \u00acFinset.Nonempty s\n[PROOFSTEP]\nexact fun hs \u21a6 h (hs.sym _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 n\n\u22a2 Finset.sym s n = \u2205 \u2194 n \u2260 0 \u2227 s = \u2205\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym \u03b1 Nat.zero\n\u22a2 Finset.sym s Nat.zero = \u2205 \u2194 Nat.zero \u2260 0 \u2227 s = \u2205\n[PROOFSTEP]\nexact iff_of_false (singleton_ne_empty _) fun h \u21a6 (h.1 rfl).elim\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm : Sym \u03b1 (Nat.succ n\u271d)\n\u22a2 Finset.sym s (Nat.succ n\u271d) = \u2205 \u2194 Nat.succ n\u271d \u2260 0 \u2227 s = \u2205\n[PROOFSTEP]\nrefine \u27e8fun h \u21a6 \u27e8Nat.succ_ne_zero _, eq_empty_of_sym_eq_empty h\u27e9, ?_\u27e9\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm : Sym \u03b1 (Nat.succ n\u271d)\n\u22a2 Nat.succ n\u271d \u2260 0 \u2227 s = \u2205 \u2192 Finset.sym s (Nat.succ n\u271d) = \u2205\n[PROOFSTEP]\nrintro \u27e8_, rfl\u27e9\n[GOAL]\ncase succ.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nt : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm : Sym \u03b1 (Nat.succ n\u271d)\nleft\u271d : Nat.succ n\u271d \u2260 0\n\u22a2 Finset.sym \u2205 (Nat.succ n\u271d) = \u2205\n[PROOFSTEP]\nexact sym_empty _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 n\n\u22a2 Finset.Nonempty (Finset.sym s n) \u2194 n = 0 \u2228 Finset.Nonempty s\n[PROOFSTEP]\nsimp_rw [nonempty_iff_ne_empty, Ne.def]\n -- Porting note: using simp_rw does not work here, it does nothing...\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm : Sym \u03b1 n\n\u22a2 \u00acFinset.sym s n = \u2205 \u2194 n = 0 \u2228 \u00acs = \u2205\n[PROOFSTEP]\nrwa [sym_eq_empty, not_and_or, not_ne_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm : Sym \u03b1 n\u271d\ns t : Finset \u03b1\nn : \u2115\n\u22a2 Finset.sym (s \u2229 t) n = Finset.sym s n \u2229 Finset.sym t n\n[PROOFSTEP]\next m\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\na b : \u03b1\nn\u271d : \u2115\nm\u271d : Sym \u03b1 n\u271d\ns t : Finset \u03b1\nn : \u2115\nm : Sym \u03b1 n\n\u22a2 m \u2208 Finset.sym (s \u2229 t) n \u2194 m \u2208 Finset.sym s n \u2229 Finset.sym t n\n[PROOFSTEP]\nsimp only [mem_inter, mem_sym_iff, imp_and, forall_and]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nm : { x // x \u2208 Finset.sym (insert a s) n }\n\u22a2 (Sym.filterNe a \u2191m).snd \u2208 Finset.sym s (n - \u2191(Sym.filterNe a \u2191m).fst)\n[PROOFSTEP]\nconvert sym_filterNe_mem a m.2\n[GOAL]\ncase h.e'_5.h.e'_3\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nm : { x // x \u2208 Finset.sym (insert a s) n }\n\u22a2 s = erase (insert a s) a\n[PROOFSTEP]\nrw [erase_insert h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nx\u271d : (i : Fin (n + 1)) \u00d7 { x // x \u2208 Finset.sym s (n - \u2191i) }\ni : Fin (n + 1)\nm : Sym \u03b1 (n - \u2191i)\nhm : m \u2208 Finset.sym s (n - \u2191i)\n\u22a2 (fun m =>\n { fst := (Sym.filterNe a \u2191m).fst,\n snd :=\n { val := (Sym.filterNe a \u2191m).snd,\n property := (_ : (Sym.filterNe a \u2191m).snd \u2208 Finset.sym s (n - \u2191(Sym.filterNe a \u2191m).fst)) } })\n ((fun m =>\n { val := Sym.fill a m.fst \u2191m.snd, property := (_ : Sym.fill a m.fst \u2191m.snd \u2208 Finset.sym (insert a s) n) })\n { fst := i, snd := { val := m, property := hm } }) =\n { fst := i, snd := { val := m, property := hm } }\n[PROOFSTEP]\nrefine' Function.Injective.sigma_map (Function.injective_id) (fun i \u21a6 _) _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nx\u271d : (i : Fin (n + 1)) \u00d7 { x // x \u2208 Finset.sym s (n - \u2191i) }\ni : Fin (n + 1)\nm : Sym \u03b1 (n - \u2191i)\nhm : m \u2208 Finset.sym s (n - \u2191i)\n\u22a2 Fin (n + 1) \u2192 Type ?u.22728\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nx\u271d : (i : Fin (n + 1)) \u00d7 { x // x \u2208 Finset.sym s (n - \u2191i) }\ni : Fin (n + 1)\nm : Sym \u03b1 (n - \u2191i)\nhm : m \u2208 Finset.sym s (n - \u2191i)\n\u22a2 (a : Fin (n + 1)) \u2192 { x // x \u2208 Finset.sym s (n - \u2191a) } \u2192 ?refine'_1 (id a)\ncase refine'_3\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nx\u271d : (i : Fin (n + 1)) \u00d7 { x // x \u2208 Finset.sym s (n - \u2191i) }\ni\u271d : Fin (n + 1)\nm : Sym \u03b1 (n - \u2191i\u271d)\nhm : m \u2208 Finset.sym s (n - \u2191i\u271d)\ni : Fin (n + 1)\n\u22a2 Function.Injective (?refine'_2 i)\ncase refine'_4\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nx\u271d : (i : Fin (n + 1)) \u00d7 { x // x \u2208 Finset.sym s (n - \u2191i) }\ni : Fin (n + 1)\nm : Sym \u03b1 (n - \u2191i)\nhm : m \u2208 Finset.sym s (n - \u2191i)\n\u22a2 Sigma.map id ?refine'_2\n ((fun m =>\n { fst := (Sym.filterNe a \u2191m).fst,\n snd :=\n { val := (Sym.filterNe a \u2191m).snd,\n property := (_ : (Sym.filterNe a \u2191m).snd \u2208 Finset.sym s (n - \u2191(Sym.filterNe a \u2191m).fst)) } })\n ((fun m =>\n { val := Sym.fill a m.fst \u2191m.snd, property := (_ : Sym.fill a m.fst \u2191m.snd \u2208 Finset.sym (insert a s) n) })\n { fst := i, snd := { val := m, property := hm } })) =\n Sigma.map id ?refine'_2 { fst := i, snd := { val := m, property := hm } }\n[PROOFSTEP]\nexact fun i \u21a6 Sym \u03b1 (n - i)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nx\u271d : (i : Fin (n + 1)) \u00d7 { x // x \u2208 Finset.sym s (n - \u2191i) }\ni : Fin (n + 1)\nm : Sym \u03b1 (n - \u2191i)\nhm : m \u2208 Finset.sym s (n - \u2191i)\n\u22a2 (a : Fin (n + 1)) \u2192 { x // x \u2208 Finset.sym s (n - \u2191a) } \u2192 Sym \u03b1 (n - \u2191(id a))\ncase refine'_3\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nx\u271d : (i : Fin (n + 1)) \u00d7 { x // x \u2208 Finset.sym s (n - \u2191i) }\ni\u271d : Fin (n + 1)\nm : Sym \u03b1 (n - \u2191i\u271d)\nhm : m \u2208 Finset.sym s (n - \u2191i\u271d)\ni : Fin (n + 1)\n\u22a2 Function.Injective (?refine'_2 i)\ncase refine'_4\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nx\u271d : (i : Fin (n + 1)) \u00d7 { x // x \u2208 Finset.sym s (n - \u2191i) }\ni : Fin (n + 1)\nm : Sym \u03b1 (n - \u2191i)\nhm : m \u2208 Finset.sym s (n - \u2191i)\n\u22a2 Sigma.map id ?refine'_2\n ((fun m =>\n { fst := (Sym.filterNe a \u2191m).fst,\n snd :=\n { val := (Sym.filterNe a \u2191m).snd,\n property := (_ : (Sym.filterNe a \u2191m).snd \u2208 Finset.sym s (n - \u2191(Sym.filterNe a \u2191m).fst)) } })\n ((fun m =>\n { val := Sym.fill a m.fst \u2191m.snd, property := (_ : Sym.fill a m.fst \u2191m.snd \u2208 Finset.sym (insert a s) n) })\n { fst := i, snd := { val := m, property := hm } })) =\n Sigma.map id ?refine'_2 { fst := i, snd := { val := m, property := hm } }\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nx\u271d : (i : Fin (n + 1)) \u00d7 { x // x \u2208 Finset.sym s (n - \u2191i) }\ni\u271d : Fin (n + 1)\nm : Sym \u03b1 (n - \u2191i\u271d)\nhm : m \u2208 Finset.sym s (n - \u2191i\u271d)\ni : Fin (n + 1)\n\u22a2 Function.Injective (?refine'_2 i)\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nx\u271d : (i : Fin (n + 1)) \u00d7 { x // x \u2208 Finset.sym s (n - \u2191i) }\ni : Fin (n + 1)\nm : Sym \u03b1 (n - \u2191i)\nhm : m \u2208 Finset.sym s (n - \u2191i)\n\u22a2 (a : Fin (n + 1)) \u2192 { x // x \u2208 Finset.sym s (n - \u2191a) } \u2192 Sym \u03b1 (n - \u2191(id a))\ncase refine'_4\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nx\u271d : (i : Fin (n + 1)) \u00d7 { x // x \u2208 Finset.sym s (n - \u2191i) }\ni : Fin (n + 1)\nm : Sym \u03b1 (n - \u2191i)\nhm : m \u2208 Finset.sym s (n - \u2191i)\n\u22a2 Sigma.map id ?refine'_2\n ((fun m =>\n { fst := (Sym.filterNe a \u2191m).fst,\n snd :=\n { val := (Sym.filterNe a \u2191m).snd,\n property := (_ : (Sym.filterNe a \u2191m).snd \u2208 Finset.sym s (n - \u2191(Sym.filterNe a \u2191m).fst)) } })\n ((fun m =>\n { val := Sym.fill a m.fst \u2191m.snd, property := (_ : Sym.fill a m.fst \u2191m.snd \u2208 Finset.sym (insert a s) n) })\n { fst := i, snd := { val := m, property := hm } })) =\n Sigma.map id ?refine'_2 { fst := i, snd := { val := m, property := hm } }\n[PROOFSTEP]\nexact Subtype.coe_injective\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nx\u271d : (i : Fin (n + 1)) \u00d7 { x // x \u2208 Finset.sym s (n - \u2191i) }\ni : Fin (n + 1)\nm : Sym \u03b1 (n - \u2191i)\nhm : m \u2208 Finset.sym s (n - \u2191i)\n\u22a2 Sigma.map id (fun i a => \u2191a)\n ((fun m =>\n { fst := (Sym.filterNe a \u2191m).fst,\n snd :=\n { val := (Sym.filterNe a \u2191m).snd,\n property := (_ : (Sym.filterNe a \u2191m).snd \u2208 Finset.sym s (n - \u2191(Sym.filterNe a \u2191m).fst)) } })\n ((fun m =>\n { val := Sym.fill a m.fst \u2191m.snd, property := (_ : Sym.fill a m.fst \u2191m.snd \u2208 Finset.sym (insert a s) n) })\n { fst := i, snd := { val := m, property := hm } })) =\n Sigma.map id (fun i a => \u2191a) { fst := i, snd := { val := m, property := hm } }\n[PROOFSTEP]\nrefine Eq.trans ?_ (Sym.filter_ne_fill a _ ?_)\n[GOAL]\ncase refine'_4.refine_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nx\u271d : (i : Fin (n + 1)) \u00d7 { x // x \u2208 Finset.sym s (n - \u2191i) }\ni : Fin (n + 1)\nm : Sym \u03b1 (n - \u2191i)\nhm : m \u2208 Finset.sym s (n - \u2191i)\n\u22a2 Sigma.map id (fun i a => \u2191a)\n ((fun m =>\n { fst := (Sym.filterNe a \u2191m).fst,\n snd :=\n { val := (Sym.filterNe a \u2191m).snd,\n property := (_ : (Sym.filterNe a \u2191m).snd \u2208 Finset.sym s (n - \u2191(Sym.filterNe a \u2191m).fst)) } })\n ((fun m =>\n { val := Sym.fill a m.fst \u2191m.snd, property := (_ : Sym.fill a m.fst \u2191m.snd \u2208 Finset.sym (insert a s) n) })\n { fst := i, snd := { val := m, property := hm } })) =\n Sym.filterNe a\n (Sym.fill a (Sigma.map id (fun i a => \u2191a) { fst := i, snd := { val := m, property := hm } }).fst\n (Sigma.map id (fun i a => \u2191a) { fst := i, snd := { val := m, property := hm } }).snd)\ncase refine'_4.refine_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nn : \u2115\nm\u271d : Sym \u03b1 n\nh : \u00aca \u2208 s\nx\u271d : (i : Fin (n + 1)) \u00d7 { x // x \u2208 Finset.sym s (n - \u2191i) }\ni : Fin (n + 1)\nm : Sym \u03b1 (n - \u2191i)\nhm : m \u2208 Finset.sym s (n - \u2191i)\n\u22a2 \u00aca \u2208 (Sigma.map id (fun i a => \u2191a) { fst := i, snd := { val := m, property := hm } }).snd\n[PROOFSTEP]\nexacts [rfl, h \u2218 mem_sym_iff.1 hm a]\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.Sym", "llama_tokens": 9404, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.6297745935070806, "lm_q1q2_score": 0.4079668593854015}} {"text": "[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nc : \ud835\udd5c \u2192 \ud835\udd5c'\nc' : \ud835\udd5c'\nhc : HasDerivWithinAt c c' s x\nhf : HasDerivWithinAt f f' s x\n\u22a2 HasDerivWithinAt (fun y => c y \u2022 f y) (c x \u2022 f' + c' \u2022 f x) s x\n[PROOFSTEP]\nsimpa using (HasFDerivWithinAt.smul hc hf).hasDerivWithinAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nc : \ud835\udd5c \u2192 \ud835\udd5c'\nc' : \ud835\udd5c'\nhc : HasDerivAt c c' x\nhf : HasDerivAt f f' x\n\u22a2 HasDerivAt (fun y => c y \u2022 f y) (c x \u2022 f' + c' \u2022 f x) x\n[PROOFSTEP]\nrw [\u2190 hasDerivWithinAt_univ] at *\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nc : \ud835\udd5c \u2192 \ud835\udd5c'\nc' : \ud835\udd5c'\nhc : HasDerivWithinAt c c' univ x\nhf : HasDerivWithinAt f f' univ x\n\u22a2 HasDerivWithinAt (fun y => c y \u2022 f y) (c x \u2022 f' + c' \u2022 f x) univ x\n[PROOFSTEP]\nexact hc.smul hf\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nc : \ud835\udd5c \u2192 \ud835\udd5c'\nc' : \ud835\udd5c'\nhc : HasStrictDerivAt c c' x\nhf : HasStrictDerivAt f f' x\n\u22a2 HasStrictDerivAt (fun y => c y \u2022 f y) (c x \u2022 f' + c' \u2022 f x) x\n[PROOFSTEP]\nsimpa using (hc.smul hf).hasStrictDerivAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nc : \ud835\udd5c \u2192 \ud835\udd5c'\nc' : \ud835\udd5c'\nhc : HasStrictDerivAt c c' x\nf : F\n\u22a2 HasStrictDerivAt (fun y => c y \u2022 f) (c' \u2022 f) x\n[PROOFSTEP]\nhave := hc.smul (hasStrictDerivAt_const x f)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nc : \ud835\udd5c \u2192 \ud835\udd5c'\nc' : \ud835\udd5c'\nhc : HasStrictDerivAt c c' x\nf : F\nthis : HasStrictDerivAt (fun y => c y \u2022 f) (c x \u2022 0 + c' \u2022 f) x\n\u22a2 HasStrictDerivAt (fun y => c y \u2022 f) (c' \u2022 f) x\n[PROOFSTEP]\nrwa [smul_zero, zero_add] at this \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nc : \ud835\udd5c \u2192 \ud835\udd5c'\nc' : \ud835\udd5c'\nhc : HasDerivWithinAt c c' s x\nf : F\n\u22a2 HasDerivWithinAt (fun y => c y \u2022 f) (c' \u2022 f) s x\n[PROOFSTEP]\nhave := hc.smul (hasDerivWithinAt_const x s f)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nc : \ud835\udd5c \u2192 \ud835\udd5c'\nc' : \ud835\udd5c'\nhc : HasDerivWithinAt c c' s x\nf : F\nthis : HasDerivWithinAt (fun y => c y \u2022 f) (c x \u2022 0 + c' \u2022 f) s x\n\u22a2 HasDerivWithinAt (fun y => c y \u2022 f) (c' \u2022 f) s x\n[PROOFSTEP]\nrwa [smul_zero, zero_add] at this \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nc : \ud835\udd5c \u2192 \ud835\udd5c'\nc' : \ud835\udd5c'\nhc : HasDerivAt c c' x\nf : F\n\u22a2 HasDerivAt (fun y => c y \u2022 f) (c' \u2022 f) x\n[PROOFSTEP]\nrw [\u2190 hasDerivWithinAt_univ] at *\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\nc : \ud835\udd5c \u2192 \ud835\udd5c'\nc' : \ud835\udd5c'\nhc : HasDerivWithinAt c c' univ x\nf : F\n\u22a2 HasDerivWithinAt (fun y => c y \u2022 f) (c' \u2022 f) univ x\n[PROOFSTEP]\nexact hc.smul_const f\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR : Type u_1\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhf : HasStrictDerivAt f f' x\n\u22a2 HasStrictDerivAt (fun y => c \u2022 f y) (c \u2022 f') x\n[PROOFSTEP]\nsimpa using (hf.const_smul c).hasStrictDerivAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR : Type u_1\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhf : HasDerivAtFilter f f' x L\n\u22a2 HasDerivAtFilter (fun y => c \u2022 f y) (c \u2022 f') x L\n[PROOFSTEP]\nsimpa using (hf.const_smul c).hasDerivAtFilter\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhc : HasDerivWithinAt c c' s x\nhd : HasDerivWithinAt d d' s x\n\u22a2 HasDerivWithinAt (fun y => c y * d y) (c' * d x + c x * d') s x\n[PROOFSTEP]\nhave := (HasFDerivWithinAt.mul' hc hd).hasDerivWithinAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhc : HasDerivWithinAt c c' s x\nhd : HasDerivWithinAt d d' s x\nthis : HasDerivWithinAt (fun y => c y * d y) (\u2191(c x \u2022 smulRight 1 d' + smulRight (smulRight 1 c') (d x)) 1) s x\n\u22a2 HasDerivWithinAt (fun y => c y * d y) (c' * d x + c x * d') s x\n[PROOFSTEP]\nrwa [ContinuousLinearMap.add_apply, ContinuousLinearMap.smul_apply, ContinuousLinearMap.smulRight_apply,\n ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply, one_smul,\n one_smul, add_comm] at this \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhc : HasDerivAt c c' x\nhd : HasDerivAt d d' x\n\u22a2 HasDerivAt (fun y => c y * d y) (c' * d x + c x * d') x\n[PROOFSTEP]\nrw [\u2190 hasDerivWithinAt_univ] at *\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhc : HasDerivWithinAt c c' univ x\nhd : HasDerivWithinAt d d' univ x\n\u22a2 HasDerivWithinAt (fun y => c y * d y) (c' * d x + c x * d') univ x\n[PROOFSTEP]\nexact hc.mul hd\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhc : HasStrictDerivAt c c' x\nhd : HasStrictDerivAt d d' x\n\u22a2 HasStrictDerivAt (fun y => c y * d y) (c' * d x + c x * d') x\n[PROOFSTEP]\nhave := (HasStrictFDerivAt.mul' hc hd).hasStrictDerivAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhc : HasStrictDerivAt c c' x\nhd : HasStrictDerivAt d d' x\nthis : HasStrictDerivAt (fun y => c y * d y) (\u2191(c x \u2022 smulRight 1 d' + smulRight (smulRight 1 c') (d x)) 1) x\n\u22a2 HasStrictDerivAt (fun y => c y * d y) (c' * d x + c x * d') x\n[PROOFSTEP]\nrwa [ContinuousLinearMap.add_apply, ContinuousLinearMap.smul_apply, ContinuousLinearMap.smulRight_apply,\n ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply, one_smul,\n one_smul, add_comm] at this \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d\u271d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhc : HasDerivWithinAt c c' s x\nd : \ud835\udd38\n\u22a2 HasDerivWithinAt (fun y => c y * d) (c' * d) s x\n[PROOFSTEP]\nconvert hc.mul (hasDerivWithinAt_const x s d) using 1\n[GOAL]\ncase h.e'_7\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d\u271d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhc : HasDerivWithinAt c c' s x\nd : \ud835\udd38\n\u22a2 c' * d = c' * d + c x * 0\n[PROOFSTEP]\nrw [mul_zero, add_zero]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d\u271d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhc : HasDerivAt c c' x\nd : \ud835\udd38\n\u22a2 HasDerivAt (fun y => c y * d) (c' * d) x\n[PROOFSTEP]\nrw [\u2190 hasDerivWithinAt_univ] at *\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d\u271d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhc : HasDerivWithinAt c c' univ x\nd : \ud835\udd38\n\u22a2 HasDerivWithinAt (fun y => c y * d) (c' * d) univ x\n[PROOFSTEP]\nexact hc.mul_const d\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc\u271d d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nc : \ud835\udd5c\n\u22a2 HasDerivAt (fun x => x * c) c x\n[PROOFSTEP]\nsimpa only [one_mul] using (hasDerivAt_id' x).mul_const c\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d\u271d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhc : HasStrictDerivAt c c' x\nd : \ud835\udd38\n\u22a2 HasStrictDerivAt (fun y => c y * d) (c' * d) x\n[PROOFSTEP]\nconvert hc.mul (hasStrictDerivAt_const x d) using 1\n[GOAL]\ncase h.e'_7\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d\u271d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhc : HasStrictDerivAt c c' x\nd : \ud835\udd38\n\u22a2 c' * d = c' * d + c x * 0\n[PROOFSTEP]\nrw [mul_zero, add_zero]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\n\u22a2 deriv (fun y => u y * v) x = deriv u x * v\n[PROOFSTEP]\nby_cases hu : DifferentiableAt \ud835\udd5c u x\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\nhu : DifferentiableAt \ud835\udd5c u x\n\u22a2 deriv (fun y => u y * v) x = deriv u x * v\n[PROOFSTEP]\nexact deriv_mul_const hu v\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\nhu : \u00acDifferentiableAt \ud835\udd5c u x\n\u22a2 deriv (fun y => u y * v) x = deriv u x * v\n[PROOFSTEP]\nrw [deriv_zero_of_not_differentiableAt hu, zero_mul]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\nhu : \u00acDifferentiableAt \ud835\udd5c u x\n\u22a2 deriv (fun y => u y * v) x = 0\n[PROOFSTEP]\nrcases eq_or_ne v 0 with (rfl | hd)\n[GOAL]\ncase neg.inl\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhu : \u00acDifferentiableAt \ud835\udd5c u x\n\u22a2 deriv (fun y => u y * 0) x = 0\n[PROOFSTEP]\nsimp only [mul_zero, deriv_const]\n[GOAL]\ncase neg.inr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\nhu : \u00acDifferentiableAt \ud835\udd5c u x\nhd : v \u2260 0\n\u22a2 deriv (fun y => u y * v) x = 0\n[PROOFSTEP]\nrefine' deriv_zero_of_not_differentiableAt (mt (fun H => _) hu)\n[GOAL]\ncase neg.inr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\nhu : \u00acDifferentiableAt \ud835\udd5c u x\nhd : v \u2260 0\nH : DifferentiableAt \ud835\udd5c (fun y => u y * v) x\n\u22a2 DifferentiableAt \ud835\udd5c u x\n[PROOFSTEP]\nsimpa only [mul_inv_cancel_right\u2080 hd] using H.mul_const v\u207b\u00b9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc\u271d d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nc : \ud835\udd38\nhd : HasDerivWithinAt d d' s x\n\u22a2 HasDerivWithinAt (fun y => c * d y) (c * d') s x\n[PROOFSTEP]\nconvert (hasDerivWithinAt_const x s c).mul hd using 1\n[GOAL]\ncase h.e'_7\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc\u271d d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nc : \ud835\udd38\nhd : HasDerivWithinAt d d' s x\n\u22a2 c * d' = 0 * d x + c * d'\n[PROOFSTEP]\nrw [zero_mul, zero_add]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc\u271d d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nc : \ud835\udd38\nhd : HasDerivAt d d' x\n\u22a2 HasDerivAt (fun y => c * d y) (c * d') x\n[PROOFSTEP]\nrw [\u2190 hasDerivWithinAt_univ] at *\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc\u271d d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nc : \ud835\udd38\nhd : HasDerivWithinAt d d' univ x\n\u22a2 HasDerivWithinAt (fun y => c * d y) (c * d') univ x\n[PROOFSTEP]\nexact hd.const_mul c\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc\u271d d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nc : \ud835\udd38\nhd : HasStrictDerivAt d d' x\n\u22a2 HasStrictDerivAt (fun y => c * d y) (c * d') x\n[PROOFSTEP]\nconvert (hasStrictDerivAt_const _ _).mul hd using 1\n[GOAL]\ncase h.e'_7\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc\u271d d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nc : \ud835\udd38\nhd : HasStrictDerivAt d d' x\n\u22a2 c * d' = 0 * d x + c * d'\n[PROOFSTEP]\nrw [zero_mul, zero_add]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu\u271d v : \ud835\udd5c \u2192 \ud835\udd5c'\nu : \ud835\udd5c'\n\u22a2 deriv (fun y => u * v y) x = u * deriv v x\n[PROOFSTEP]\nsimp only [mul_comm u, deriv_mul_const_field]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : HasDerivAt c c' x\nd : \ud835\udd5c'\n\u22a2 HasDerivAt (fun x => c x / d) (c' / d) x\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using hc.mul_const d\u207b\u00b9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : HasDerivWithinAt c c' s x\nd : \ud835\udd5c'\n\u22a2 HasDerivWithinAt (fun x => c x / d) (c' / d) s x\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using hc.mul_const d\u207b\u00b9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : HasStrictDerivAt c c' x\nd : \ud835\udd5c'\n\u22a2 HasStrictDerivAt (fun x => c x / d) (c' / d) x\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using hc.mul_const d\u207b\u00b9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : DifferentiableWithinAt \ud835\udd5c c s x\nd : \ud835\udd5c'\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 derivWithin (fun x => c x / d) s x = derivWithin c s x / d\n[PROOFSTEP]\nsimp [div_eq_inv_mul, derivWithin_const_mul, hc, hxs]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' d : \ud835\udd5c'\n\u22a2 deriv (fun x => c x / d) x = deriv c x / d\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, deriv_mul_const_field]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nc : \ud835\udd5c \u2192 F \u2192L[\ud835\udd5c] G\nc' : F \u2192L[\ud835\udd5c] G\nd : \ud835\udd5c \u2192 E \u2192L[\ud835\udd5c] F\nd' : E \u2192L[\ud835\udd5c] F\nu : \ud835\udd5c \u2192 F\nu' : F\nhc : HasStrictDerivAt c c' x\nhd : HasStrictDerivAt d d' x\n\u22a2 HasStrictDerivAt (fun y => comp (c y) (d y)) (comp c' (d x) + comp (c x) d') x\n[PROOFSTEP]\nhave := (hc.hasStrictFDerivAt.clm_comp hd.hasStrictFDerivAt).hasStrictDerivAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nc : \ud835\udd5c \u2192 F \u2192L[\ud835\udd5c] G\nc' : F \u2192L[\ud835\udd5c] G\nd : \ud835\udd5c \u2192 E \u2192L[\ud835\udd5c] F\nd' : E \u2192L[\ud835\udd5c] F\nu : \ud835\udd5c \u2192 F\nu' : F\nhc : HasStrictDerivAt c c' x\nhd : HasStrictDerivAt d d' x\nthis :\n HasStrictDerivAt (fun y => comp (c y) (d y))\n (\u2191(comp (\u2191(compL \ud835\udd5c E F G) (c x)) (smulRight 1 d') +\n comp (\u2191(ContinuousLinearMap.flip (compL \ud835\udd5c E F G)) (d x)) (smulRight 1 c'))\n 1)\n x\n\u22a2 HasStrictDerivAt (fun y => comp (c y) (d y)) (comp c' (d x) + comp (c x) d') x\n[PROOFSTEP]\nrwa [add_apply, comp_apply, comp_apply, smulRight_apply, smulRight_apply, one_apply, one_smul, one_smul, add_comm] at\n this \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nc : \ud835\udd5c \u2192 F \u2192L[\ud835\udd5c] G\nc' : F \u2192L[\ud835\udd5c] G\nd : \ud835\udd5c \u2192 E \u2192L[\ud835\udd5c] F\nd' : E \u2192L[\ud835\udd5c] F\nu : \ud835\udd5c \u2192 F\nu' : F\nhc : HasDerivWithinAt c c' s x\nhd : HasDerivWithinAt d d' s x\n\u22a2 HasDerivWithinAt (fun y => comp (c y) (d y)) (comp c' (d x) + comp (c x) d') s x\n[PROOFSTEP]\nhave := (hc.hasFDerivWithinAt.clm_comp hd.hasFDerivWithinAt).hasDerivWithinAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nc : \ud835\udd5c \u2192 F \u2192L[\ud835\udd5c] G\nc' : F \u2192L[\ud835\udd5c] G\nd : \ud835\udd5c \u2192 E \u2192L[\ud835\udd5c] F\nd' : E \u2192L[\ud835\udd5c] F\nu : \ud835\udd5c \u2192 F\nu' : F\nhc : HasDerivWithinAt c c' s x\nhd : HasDerivWithinAt d d' s x\nthis :\n HasDerivWithinAt (fun y => comp (c y) (d y))\n (\u2191(comp (\u2191(compL \ud835\udd5c E F G) (c x)) (smulRight 1 d') +\n comp (\u2191(ContinuousLinearMap.flip (compL \ud835\udd5c E F G)) (d x)) (smulRight 1 c'))\n 1)\n s x\n\u22a2 HasDerivWithinAt (fun y => comp (c y) (d y)) (comp c' (d x) + comp (c x) d') s x\n[PROOFSTEP]\nrwa [add_apply, comp_apply, comp_apply, smulRight_apply, smulRight_apply, one_apply, one_smul, one_smul, add_comm] at\n this \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nc : \ud835\udd5c \u2192 F \u2192L[\ud835\udd5c] G\nc' : F \u2192L[\ud835\udd5c] G\nd : \ud835\udd5c \u2192 E \u2192L[\ud835\udd5c] F\nd' : E \u2192L[\ud835\udd5c] F\nu : \ud835\udd5c \u2192 F\nu' : F\nhc : HasDerivAt c c' x\nhd : HasDerivAt d d' x\n\u22a2 HasDerivAt (fun y => comp (c y) (d y)) (comp c' (d x) + comp (c x) d') x\n[PROOFSTEP]\nrw [\u2190 hasDerivWithinAt_univ] at *\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nc : \ud835\udd5c \u2192 F \u2192L[\ud835\udd5c] G\nc' : F \u2192L[\ud835\udd5c] G\nd : \ud835\udd5c \u2192 E \u2192L[\ud835\udd5c] F\nd' : E \u2192L[\ud835\udd5c] F\nu : \ud835\udd5c \u2192 F\nu' : F\nhc : HasDerivWithinAt c c' univ x\nhd : HasDerivWithinAt d d' univ x\n\u22a2 HasDerivWithinAt (fun y => comp (c y) (d y)) (comp c' (d x) + comp (c x) d') univ x\n[PROOFSTEP]\nexact hc.clm_comp hd\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nc : \ud835\udd5c \u2192 F \u2192L[\ud835\udd5c] G\nc' : F \u2192L[\ud835\udd5c] G\nd : \ud835\udd5c \u2192 E \u2192L[\ud835\udd5c] F\nd' : E \u2192L[\ud835\udd5c] F\nu : \ud835\udd5c \u2192 F\nu' : F\nhc : HasStrictDerivAt c c' x\nhu : HasStrictDerivAt u u' x\n\u22a2 HasStrictDerivAt (fun y => \u2191(c y) (u y)) (\u2191c' (u x) + \u2191(c x) u') x\n[PROOFSTEP]\nhave := (hc.hasStrictFDerivAt.clm_apply hu.hasStrictFDerivAt).hasStrictDerivAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nc : \ud835\udd5c \u2192 F \u2192L[\ud835\udd5c] G\nc' : F \u2192L[\ud835\udd5c] G\nd : \ud835\udd5c \u2192 E \u2192L[\ud835\udd5c] F\nd' : E \u2192L[\ud835\udd5c] F\nu : \ud835\udd5c \u2192 F\nu' : F\nhc : HasStrictDerivAt c c' x\nhu : HasStrictDerivAt u u' x\nthis :\n HasStrictDerivAt (fun y => \u2191(c y) (u y))\n (\u2191(comp (c x) (smulRight 1 u') + \u2191(ContinuousLinearMap.flip (smulRight 1 c')) (u x)) 1) x\n\u22a2 HasStrictDerivAt (fun y => \u2191(c y) (u y)) (\u2191c' (u x) + \u2191(c x) u') x\n[PROOFSTEP]\nrwa [add_apply, comp_apply, flip_apply, smulRight_apply, smulRight_apply, one_apply, one_smul, one_smul, add_comm] at\n this \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nc : \ud835\udd5c \u2192 F \u2192L[\ud835\udd5c] G\nc' : F \u2192L[\ud835\udd5c] G\nd : \ud835\udd5c \u2192 E \u2192L[\ud835\udd5c] F\nd' : E \u2192L[\ud835\udd5c] F\nu : \ud835\udd5c \u2192 F\nu' : F\nhc : HasDerivWithinAt c c' s x\nhu : HasDerivWithinAt u u' s x\n\u22a2 HasDerivWithinAt (fun y => \u2191(c y) (u y)) (\u2191c' (u x) + \u2191(c x) u') s x\n[PROOFSTEP]\nhave := (hc.hasFDerivWithinAt.clm_apply hu.hasFDerivWithinAt).hasDerivWithinAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nc : \ud835\udd5c \u2192 F \u2192L[\ud835\udd5c] G\nc' : F \u2192L[\ud835\udd5c] G\nd : \ud835\udd5c \u2192 E \u2192L[\ud835\udd5c] F\nd' : E \u2192L[\ud835\udd5c] F\nu : \ud835\udd5c \u2192 F\nu' : F\nhc : HasDerivWithinAt c c' s x\nhu : HasDerivWithinAt u u' s x\nthis :\n HasDerivWithinAt (fun y => \u2191(c y) (u y))\n (\u2191(comp (c x) (smulRight 1 u') + \u2191(ContinuousLinearMap.flip (smulRight 1 c')) (u x)) 1) s x\n\u22a2 HasDerivWithinAt (fun y => \u2191(c y) (u y)) (\u2191c' (u x) + \u2191(c x) u') s x\n[PROOFSTEP]\nrwa [add_apply, comp_apply, flip_apply, smulRight_apply, smulRight_apply, one_apply, one_smul, one_smul, add_comm] at\n this \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nc : \ud835\udd5c \u2192 F \u2192L[\ud835\udd5c] G\nc' : F \u2192L[\ud835\udd5c] G\nd : \ud835\udd5c \u2192 E \u2192L[\ud835\udd5c] F\nd' : E \u2192L[\ud835\udd5c] F\nu : \ud835\udd5c \u2192 F\nu' : F\nhc : HasDerivAt c c' x\nhu : HasDerivAt u u' x\n\u22a2 HasDerivAt (fun y => \u2191(c y) (u y)) (\u2191c' (u x) + \u2191(c x) u') x\n[PROOFSTEP]\nhave := (hc.hasFDerivAt.clm_apply hu.hasFDerivAt).hasDerivAt\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nc : \ud835\udd5c \u2192 F \u2192L[\ud835\udd5c] G\nc' : F \u2192L[\ud835\udd5c] G\nd : \ud835\udd5c \u2192 E \u2192L[\ud835\udd5c] F\nd' : E \u2192L[\ud835\udd5c] F\nu : \ud835\udd5c \u2192 F\nu' : F\nhc : HasDerivAt c c' x\nhu : HasDerivAt u u' x\nthis :\n HasDerivAt (fun y => \u2191(c y) (u y))\n (\u2191(comp (c x) (smulRight 1 u') + \u2191(ContinuousLinearMap.flip (smulRight 1 c')) (u x)) 1) x\n\u22a2 HasDerivAt (fun y => \u2191(c y) (u y)) (\u2191c' (u x) + \u2191(c x) u') x\n[PROOFSTEP]\nrwa [add_apply, comp_apply, flip_apply, smulRight_apply, smulRight_apply, one_apply, one_smul, one_smul, add_comm] at\n this \n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.Deriv.Mul", "llama_tokens": 18352, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.647798211152541, "lm_q1q2_score": 0.4079668551032055}} {"text": "[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : StrictOrderedSemiring R\ninst\u271d : Archimedean R\np : R \u2192 Prop\nh : \u2200\u1da0 (x : R) in atTop, p x\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, p \u2191n\n[PROOFSTEP]\nrw [\u2190 Nat.comap_cast_atTop (R := R)]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : StrictOrderedSemiring R\ninst\u271d : Archimedean R\np : R \u2192 Prop\nh : \u2200\u1da0 (x : R) in atTop, p x\n\u22a2 \u2200\u1da0 (n : \u2115) in Filter.comap Nat.cast atTop, p \u2191n\n[PROOFSTEP]\nexact h.comap _\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : StrictOrderedRing R\ninst\u271d : Archimedean R\nr : R\nn : \u2115\nhn : r \u2264 \u2191n\n\u22a2 r \u2264 \u2191\u2191n\n[PROOFSTEP]\nexact_mod_cast hn\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : StrictOrderedRing R\ninst\u271d : Archimedean R\nr : R\nn : \u2115\nhn : -r \u2264 \u2191n\n\u22a2 \u2191(-\u2191n) \u2264 r\n[PROOFSTEP]\nsimpa [neg_le] using hn\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : StrictOrderedRing R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2124\nl : Filter \u03b1\n\u22a2 Tendsto (fun n => \u2191(f n)) l atTop \u2194 Tendsto f l atTop\n[PROOFSTEP]\nrw [\u2190 @Int.comap_cast_atTop R, tendsto_comap_iff]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : StrictOrderedRing R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2124\nl : Filter \u03b1\n\u22a2 Tendsto (fun n => \u2191(f n)) l atTop \u2194 Tendsto (Int.cast \u2218 f) l atTop\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : StrictOrderedRing R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2124\nl : Filter \u03b1\n\u22a2 Tendsto (fun n => \u2191(f n)) l atBot \u2194 Tendsto f l atBot\n[PROOFSTEP]\nrw [\u2190 @Int.comap_cast_atBot R, tendsto_comap_iff]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : StrictOrderedRing R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2124\nl : Filter \u03b1\n\u22a2 Tendsto (fun n => \u2191(f n)) l atBot \u2194 Tendsto (Int.cast \u2218 f) l atBot\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : StrictOrderedRing R\ninst\u271d : Archimedean R\np : R \u2192 Prop\nh : \u2200\u1da0 (x : R) in atTop, p x\n\u22a2 \u2200\u1da0 (n : \u2124) in atTop, p \u2191n\n[PROOFSTEP]\nrw [\u2190 Int.comap_cast_atTop (R := R)]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : StrictOrderedRing R\ninst\u271d : Archimedean R\np : R \u2192 Prop\nh : \u2200\u1da0 (x : R) in atTop, p x\n\u22a2 \u2200\u1da0 (n : \u2124) in Filter.comap Int.cast atTop, p \u2191n\n[PROOFSTEP]\nexact h.comap _\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : StrictOrderedRing R\ninst\u271d : Archimedean R\np : R \u2192 Prop\nh : \u2200\u1da0 (x : R) in atBot, p x\n\u22a2 \u2200\u1da0 (n : \u2124) in atBot, p \u2191n\n[PROOFSTEP]\nrw [\u2190 Int.comap_cast_atBot (R := R)]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : StrictOrderedRing R\ninst\u271d : Archimedean R\np : R \u2192 Prop\nh : \u2200\u1da0 (x : R) in atBot, p x\n\u22a2 \u2200\u1da0 (n : \u2124) in Filter.comap Int.cast atBot, p \u2191n\n[PROOFSTEP]\nexact h.comap _\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : LinearOrderedField R\ninst\u271d : Archimedean R\nr : R\nn : \u2115\nhn : r \u2264 \u2191n\n\u22a2 r \u2264 \u2191\u2191n\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : LinearOrderedField R\ninst\u271d : Archimedean R\nr : R\nn : \u2115\nhn : -r \u2264 \u2191n\n\u22a2 \u2191(-\u2191n) \u2264 r\n[PROOFSTEP]\nsimpa [neg_le]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : LinearOrderedField R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u211a\nl : Filter \u03b1\n\u22a2 Tendsto (fun n => \u2191(f n)) l atTop \u2194 Tendsto f l atTop\n[PROOFSTEP]\nrw [\u2190 @Rat.comap_cast_atTop R, tendsto_comap_iff]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : LinearOrderedField R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u211a\nl : Filter \u03b1\n\u22a2 Tendsto (fun n => \u2191(f n)) l atTop \u2194 Tendsto (Rat.cast \u2218 f) l atTop\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : LinearOrderedField R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u211a\nl : Filter \u03b1\n\u22a2 Tendsto (fun n => \u2191(f n)) l atBot \u2194 Tendsto f l atBot\n[PROOFSTEP]\nrw [\u2190 @Rat.comap_cast_atBot R, tendsto_comap_iff]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : LinearOrderedField R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u211a\nl : Filter \u03b1\n\u22a2 Tendsto (fun n => \u2191(f n)) l atBot \u2194 Tendsto (Rat.cast \u2218 f) l atBot\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : LinearOrderedField R\ninst\u271d : Archimedean R\np : R \u2192 Prop\nh : \u2200\u1da0 (x : R) in atTop, p x\n\u22a2 \u2200\u1da0 (n : \u211a) in atTop, p \u2191n\n[PROOFSTEP]\nrw [\u2190 Rat.comap_cast_atTop (R := R)]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : LinearOrderedField R\ninst\u271d : Archimedean R\np : R \u2192 Prop\nh : \u2200\u1da0 (x : R) in atTop, p x\n\u22a2 \u2200\u1da0 (n : \u211a) in Filter.comap Rat.cast atTop, p \u2191n\n[PROOFSTEP]\nexact h.comap _\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : LinearOrderedField R\ninst\u271d : Archimedean R\np : R \u2192 Prop\nh : \u2200\u1da0 (x : R) in atBot, p x\n\u22a2 \u2200\u1da0 (n : \u211a) in atBot, p \u2191n\n[PROOFSTEP]\nrw [\u2190 Rat.comap_cast_atBot (R := R)]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : LinearOrderedField R\ninst\u271d : Archimedean R\np : R \u2192 Prop\nh : \u2200\u1da0 (x : R) in atBot, p x\n\u22a2 \u2200\u1da0 (n : \u211a) in Filter.comap Rat.cast atBot, p \u2191n\n[PROOFSTEP]\nexact\n h.comap\n _\n -- porting note: new lemma\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\n\u22a2 Tendsto (fun x => r * f x) l atTop\n[PROOFSTEP]\nrefine' tendsto_atTop.2 fun b => _\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\n\u22a2 \u2200\u1da0 (a : \u03b1) in l, b \u2264 r * f a\n[PROOFSTEP]\nobtain \u27e8n : \u2115, hn : 1 \u2264 n \u2022 r\u27e9 := Archimedean.arch 1 hr\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 n \u2022 r\n\u22a2 \u2200\u1da0 (a : \u03b1) in l, b \u2264 r * f a\n[PROOFSTEP]\nrw [nsmul_eq_mul'] at hn \n[GOAL]\ncase intro\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 r * \u2191n\n\u22a2 \u2200\u1da0 (a : \u03b1) in l, b \u2264 r * f a\n[PROOFSTEP]\nfilter_upwards [tendsto_atTop.1 hf (n * max b 0)] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 r * \u2191n\nx : \u03b1\nhx : \u2191n * max b 0 \u2264 f x\n\u22a2 b \u2264 r * f x\n[PROOFSTEP]\ncalc\n b \u2264 1 * max b 0 := by {\n rw [one_mul]\n exact le_max_left _ _\n }\n _ \u2264 r * n * max b 0 := by gcongr\n _ = r * (n * max b 0) := by rw [mul_assoc]\n _ \u2264 r * f x := by gcongr\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 r * \u2191n\nx : \u03b1\nhx : \u2191n * max b 0 \u2264 f x\n\u22a2 b \u2264 1 * max b 0\n[PROOFSTEP]\n{ rw [one_mul]\n exact le_max_left _ _\n}\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 r * \u2191n\nx : \u03b1\nhx : \u2191n * max b 0 \u2264 f x\n\u22a2 b \u2264 1 * max b 0\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 r * \u2191n\nx : \u03b1\nhx : \u2191n * max b 0 \u2264 f x\n\u22a2 b \u2264 max b 0\n[PROOFSTEP]\nexact le_max_left _ _\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 r * \u2191n\nx : \u03b1\nhx : \u2191n * max b 0 \u2264 f x\n\u22a2 1 * max b 0 \u2264 r * \u2191n * max b 0\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 r * \u2191n\nx : \u03b1\nhx : \u2191n * max b 0 \u2264 f x\n\u22a2 r * \u2191n * max b 0 = r * (\u2191n * max b 0)\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 r * \u2191n\nx : \u03b1\nhx : \u2191n * max b 0 \u2264 f x\n\u22a2 r * (\u2191n * max b 0) \u2264 r * f x\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\n\u22a2 Tendsto (fun x => f x * r) l atTop\n[PROOFSTEP]\nrefine' tendsto_atTop.2 fun b => _\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\n\u22a2 \u2200\u1da0 (a : \u03b1) in l, b \u2264 f a * r\n[PROOFSTEP]\nobtain \u27e8n : \u2115, hn : 1 \u2264 n \u2022 r\u27e9 := Archimedean.arch 1 hr\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 n \u2022 r\n\u22a2 \u2200\u1da0 (a : \u03b1) in l, b \u2264 f a * r\n[PROOFSTEP]\nhave hn' : 1 \u2264 (n : R) * r := by rwa [nsmul_eq_mul] at hn \n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 n \u2022 r\n\u22a2 1 \u2264 \u2191n * r\n[PROOFSTEP]\nrwa [nsmul_eq_mul] at hn \n[GOAL]\ncase intro\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 n \u2022 r\nhn' : 1 \u2264 \u2191n * r\n\u22a2 \u2200\u1da0 (a : \u03b1) in l, b \u2264 f a * r\n[PROOFSTEP]\nfilter_upwards [tendsto_atTop.1 hf (max b 0 * n)] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 n \u2022 r\nhn' : 1 \u2264 \u2191n * r\nx : \u03b1\nhx : max b 0 * \u2191n \u2264 f x\n\u22a2 b \u2264 f x * r\n[PROOFSTEP]\ncalc\n b \u2264 max b 0 * 1 := by {\n rw [mul_one]\n exact le_max_left _ _\n }\n _ \u2264 max b 0 * (n * r) := by gcongr\n _ = max b 0 * n * r := by rw [mul_assoc]\n _ \u2264 f x * r := by gcongr\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 n \u2022 r\nhn' : 1 \u2264 \u2191n * r\nx : \u03b1\nhx : max b 0 * \u2191n \u2264 f x\n\u22a2 b \u2264 max b 0 * 1\n[PROOFSTEP]\n{ rw [mul_one]\n exact le_max_left _ _\n}\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 n \u2022 r\nhn' : 1 \u2264 \u2191n * r\nx : \u03b1\nhx : max b 0 * \u2191n \u2264 f x\n\u22a2 b \u2264 max b 0 * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 n \u2022 r\nhn' : 1 \u2264 \u2191n * r\nx : \u03b1\nhx : max b 0 * \u2191n \u2264 f x\n\u22a2 b \u2264 max b 0\n[PROOFSTEP]\nexact le_max_left _ _\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 n \u2022 r\nhn' : 1 \u2264 \u2191n * r\nx : \u03b1\nhx : max b 0 * \u2191n \u2264 f x\n\u22a2 max b 0 * 1 \u2264 max b 0 * (\u2191n * r)\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 n \u2022 r\nhn' : 1 \u2264 \u2191n * r\nx : \u03b1\nhx : max b 0 * \u2191n \u2264 f x\n\u22a2 max b 0 * (\u2191n * r) = max b 0 * \u2191n * r\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedSemiring R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : \u2115\nhn : 1 \u2264 n \u2022 r\nhn' : 1 \u2264 \u2191n * r\nx : \u03b1\nhx : max b 0 * \u2191n \u2264 f x\n\u22a2 max b 0 * \u2191n * r \u2264 f x * r\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedRing R\ninst\u271d : Archimedean R\nhr : r < 0\nhf : Tendsto f l atTop\n\u22a2 Tendsto (fun x => f x * r) l atBot\n[PROOFSTEP]\nsimpa only [tendsto_neg_atTop_iff, mul_neg] using hf.atTop_mul_const' (neg_pos.mpr hr)\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedRing R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atBot\n\u22a2 Tendsto (fun x => f x * r) l atBot\n[PROOFSTEP]\nsimp only [\u2190 tendsto_neg_atTop_iff, \u2190 neg_mul] at hf \u22a2\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedRing R\ninst\u271d : Archimedean R\nhr : 0 < r\nhf : Tendsto (fun x => -f x) l atTop\n\u22a2 Tendsto (fun x => -f x * r) l atTop\n[PROOFSTEP]\nexact hf.atTop_mul_const' hr\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedRing R\ninst\u271d : Archimedean R\nhr : r < 0\nhf : Tendsto f l atBot\n\u22a2 Tendsto (fun x => f x * r) l atTop\n[PROOFSTEP]\nsimpa only [mul_neg, tendsto_neg_atBot_iff] using hf.atBot_mul_const' (neg_pos.2 hr)\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedCancelAddCommMonoid R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2115\nhr : 0 < r\nhf : Tendsto f l atTop\n\u22a2 Tendsto (fun x => f x \u2022 r) l atTop\n[PROOFSTEP]\nrefine' tendsto_atTop.mpr fun s => _\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedCancelAddCommMonoid R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2115\nhr : 0 < r\nhf : Tendsto f l atTop\ns : R\n\u22a2 \u2200\u1da0 (a : \u03b1) in l, s \u2264 f a \u2022 r\n[PROOFSTEP]\nobtain \u27e8n : \u2115, hn : s \u2264 n \u2022 r\u27e9 := Archimedean.arch s hr\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedCancelAddCommMonoid R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2115\nhr : 0 < r\nhf : Tendsto f l atTop\ns : R\nn : \u2115\nhn : s \u2264 n \u2022 r\n\u22a2 \u2200\u1da0 (a : \u03b1) in l, s \u2264 f a \u2022 r\n[PROOFSTEP]\nexact (tendsto_atTop.mp hf n).mono fun a ha => hn.trans (nsmul_le_nsmul hr.le ha)\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedAddCommGroup R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2115\nhr : r < 0\nhf : Tendsto f l atTop\n\u22a2 Tendsto (fun x => f x \u2022 r) l atBot\n[PROOFSTEP]\nsimpa using hf.atTop_nsmul_const (neg_pos.2 hr)\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedAddCommGroup R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2124\nhr : 0 < r\nhf : Tendsto f l atTop\n\u22a2 Tendsto (fun x => f x \u2022 r) l atTop\n[PROOFSTEP]\nrefine' tendsto_atTop.mpr fun s => _\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedAddCommGroup R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2124\nhr : 0 < r\nhf : Tendsto f l atTop\ns : R\n\u22a2 \u2200\u1da0 (a : \u03b1) in l, s \u2264 f a \u2022 r\n[PROOFSTEP]\nobtain \u27e8n : \u2115, hn : s \u2264 n \u2022 r\u27e9 := Archimedean.arch s hr\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedAddCommGroup R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2124\nhr : 0 < r\nhf : Tendsto f l atTop\ns : R\nn : \u2115\nhn : s \u2264 n \u2022 r\n\u22a2 \u2200\u1da0 (a : \u03b1) in l, s \u2264 f a \u2022 r\n[PROOFSTEP]\nreplace hn : s \u2264 (n : \u2124) \u2022 r\n[GOAL]\ncase hn\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedAddCommGroup R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2124\nhr : 0 < r\nhf : Tendsto f l atTop\ns : R\nn : \u2115\nhn : s \u2264 n \u2022 r\n\u22a2 s \u2264 \u2191n \u2022 r\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedAddCommGroup R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2124\nhr : 0 < r\nhf : Tendsto f l atTop\ns : R\nn : \u2115\nhn : s \u2264 \u2191n \u2022 r\n\u22a2 \u2200\u1da0 (a : \u03b1) in l, s \u2264 f a \u2022 r\n[PROOFSTEP]\nexact (tendsto_atTop.mp hf n).mono fun a ha => hn.trans (zsmul_le_zsmul hr.le ha)\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedAddCommGroup R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2124\nhr : r < 0\nhf : Tendsto f l atTop\n\u22a2 Tendsto (fun x => f x \u2022 r) l atBot\n[PROOFSTEP]\nsimpa using hf.atTop_zsmul_const (neg_pos.2 hr)\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedAddCommGroup R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2124\nhr : 0 < r\nhf : Tendsto f l atBot\n\u22a2 Tendsto (fun x => f x \u2022 r) l atBot\n[PROOFSTEP]\nsimp only [\u2190 tendsto_neg_atTop_iff, \u2190 neg_zsmul] at hf \u22a2\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedAddCommGroup R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2124\nhr : 0 < r\nhf : Tendsto (fun x => -f x) l atTop\n\u22a2 Tendsto (fun x => -f x \u2022 r) l atTop\n[PROOFSTEP]\nexact hf.atTop_zsmul_const hr\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 R\nr : R\ninst\u271d\u00b9 : LinearOrderedAddCommGroup R\ninst\u271d : Archimedean R\nf : \u03b1 \u2192 \u2124\nhr : r < 0\nhf : Tendsto f l atBot\n\u22a2 Tendsto (fun x => f x \u2022 r) l atTop\n[PROOFSTEP]\nsimpa using hf.atBot_zsmul_const (neg_pos.2 hr)\n", "meta": {"mathlib_filename": "Mathlib.Order.Filter.Archimedean", "llama_tokens": 8357, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.6513548511303338, "lm_q1q2_score": 0.4078290202476443}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nX Y : C\nf g : X \u27f6 Y\nw : \u2200 (x : (forget C).obj X), \u2191f x = \u2191g x\n\u22a2 f = g\n[PROOFSTEP]\napply @Faithful.map_injective C _ (Type w) _ (forget C) _ X Y\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nX Y : C\nf g : X \u27f6 Y\nw : \u2200 (x : (forget C).obj X), \u2191f x = \u2191g x\n\u22a2 (forget C).map f = (forget C).map g\n[PROOFSTEP]\ndsimp [forget]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nX Y : C\nf g : X \u27f6 Y\nw : \u2200 (x : (forget C).obj X), \u2191f x = \u2191g x\n\u22a2 ConcreteCategory.forget.map f = ConcreteCategory.forget.map g\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase a.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nX Y : C\nf g : X \u27f6 Y\nw : \u2200 (x : (forget C).obj X), \u2191f x = \u2191g x\nx : ConcreteCategory.forget.obj X\n\u22a2 ConcreteCategory.forget.map f x = ConcreteCategory.forget.map g x\n[PROOFSTEP]\nexact w x\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : ConcreteCategory C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsIso f\n\u22a2 Function.Bijective ((forget C).map f)\n[PROOFSTEP]\nrw [\u2190 isIso_iff_bijective]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : ConcreteCategory C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsIso f\n\u22a2 IsIso ((forget C).map f)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\nD : Type u'\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : Category.{v', u'} D\ninst\u271d\u00b2 : ConcreteCategory D\ninst\u271d\u00b9 : HasForget\u2082 C D\ninst\u271d : Functor.PreservesMonomorphisms (forget C)\n\u22a2 Functor.PreservesMonomorphisms (forget\u2082 C D \u22d9 forget D)\n[PROOFSTEP]\nsimp only [HasForget\u2082.forget_comp]\n[GOAL]\nC : Type u\nD : Type u'\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : Category.{v', u'} D\ninst\u271d\u00b2 : ConcreteCategory D\ninst\u271d\u00b9 : HasForget\u2082 C D\ninst\u271d : Functor.PreservesMonomorphisms (forget C)\n\u22a2 Functor.PreservesMonomorphisms (forget C)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\nD : Type u'\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : Category.{v', u'} D\ninst\u271d\u00b2 : ConcreteCategory D\ninst\u271d\u00b9 : HasForget\u2082 C D\ninst\u271d : Functor.PreservesEpimorphisms (forget C)\n\u22a2 Functor.PreservesEpimorphisms (forget\u2082 C D \u22d9 forget D)\n[PROOFSTEP]\nsimp only [HasForget\u2082.forget_comp]\n[GOAL]\nC : Type u\nD : Type u'\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : ConcreteCategory C\ninst\u271d\u00b3 : Category.{v', u'} D\ninst\u271d\u00b2 : ConcreteCategory D\ninst\u271d\u00b9 : HasForget\u2082 C D\ninst\u271d : Functor.PreservesEpimorphisms (forget C)\n\u22a2 Functor.PreservesEpimorphisms (forget C)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\nD : Type u'\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : ConcreteCategory D\nobj : C \u2192 D\nh_obj : \u2200 (X : C), (forget D).obj (obj X) = (forget C).obj X\nmap : {X Y : C} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nh_map : \u2200 {X Y : C} {f : X \u27f6 Y}, HEq ((forget D).map (map f)) ((forget C).map f)\n\u22a2 Faithful.div (forget C) (forget D) (fun X => obj X) h_obj (fun X Y f => map f) h_map \u22d9 forget D = forget C\n[PROOFSTEP]\napply Faithful.div_comp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.ConcreteCategory.Basic", "llama_tokens": 1413, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154240079185319, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4076862677977183}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : HasCokernels C\nX Y : C\nf : X \u27f6 Y\n\u22a2 (imageMonoFactorisation f).e = cokernel.\u03c0 (kernel.\u03b9 f) \u226b coimageImageComparison f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : HasCokernels C\nX Y : C\nf : X \u27f6 Y\n\u22a2 kernel.lift (cokernel.\u03c0 f) f (_ : f \u226b cokernel.\u03c0 f = 0) = cokernel.\u03c0 (kernel.\u03b9 f) \u226b coimageImageComparison f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : HasCokernels C\nX Y : C\nf : X \u27f6 Y\n\u22a2 kernel.lift (cokernel.\u03c0 f) f (_ : f \u226b cokernel.\u03c0 f = 0) \u226b equalizer.\u03b9 (cokernel.\u03c0 f) 0 =\n (cokernel.\u03c0 (kernel.\u03b9 f) \u226b coimageImageComparison f) \u226b equalizer.\u03b9 (cokernel.\u03c0 f) 0\n[PROOFSTEP]\nsimp only [Abelian.coimageImageComparison, imageMonoFactorisation_e, Category.assoc, cokernel.\u03c0_desc_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : HasKernels C\ninst\u271d\u00b9 : HasCokernels C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsIso (coimageImageComparison f)\nF : MonoFactorisation f\n\u22a2 (fun F => inv (coimageImageComparison f) \u226b cokernel.desc (kernel.\u03b9 f) F.e (_ : kernel.\u03b9 f \u226b F.e = 0)) F \u226b F.m =\n (imageMonoFactorisation f).m\n[PROOFSTEP]\nrw [imageMonoFactorisation_m]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : HasKernels C\ninst\u271d\u00b9 : HasCokernels C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsIso (coimageImageComparison f)\nF : MonoFactorisation f\n\u22a2 (fun F => inv (coimageImageComparison f) \u226b cokernel.desc (kernel.\u03b9 f) F.e (_ : kernel.\u03b9 f \u226b F.e = 0)) F \u226b F.m =\n kernel.\u03b9 (cokernel.\u03c0 f)\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : HasKernels C\ninst\u271d\u00b9 : HasCokernels C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsIso (coimageImageComparison f)\nF : MonoFactorisation f\n\u22a2 inv (coimageImageComparison f) \u226b cokernel.desc (kernel.\u03b9 f) F.e (_ : kernel.\u03b9 f \u226b F.e = 0) \u226b F.m =\n kernel.\u03b9 (cokernel.\u03c0 f)\n[PROOFSTEP]\nrw [IsIso.inv_comp_eq]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : HasKernels C\ninst\u271d\u00b9 : HasCokernels C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsIso (coimageImageComparison f)\nF : MonoFactorisation f\n\u22a2 cokernel.desc (kernel.\u03b9 f) F.e (_ : kernel.\u03b9 f \u226b F.e = 0) \u226b F.m = coimageImageComparison f \u226b kernel.\u03b9 (cokernel.\u03c0 f)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : HasKernels C\ninst\u271d\u00b9 : HasCokernels C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsIso (coimageImageComparison f)\nF : MonoFactorisation f\n\u22a2 coequalizer.\u03c0 (kernel.\u03b9 f) 0 \u226b cokernel.desc (kernel.\u03b9 f) F.e (_ : kernel.\u03b9 f \u226b F.e = 0) \u226b F.m =\n coequalizer.\u03c0 (kernel.\u03b9 f) 0 \u226b coimageImageComparison f \u226b kernel.\u03b9 (cokernel.\u03c0 f)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : Preadditive C\ninst\u271d\u2074 : HasKernels C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Mono f\ninst\u271d : IsIso (coimageImageComparison f)\n\u22a2 IsIso (imageMonoFactorisation f).e\n[PROOFSTEP]\nrw [imageMonoFactorisation_e']\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : Preadditive C\ninst\u271d\u2074 : HasKernels C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Mono f\ninst\u271d : IsIso (coimageImageComparison f)\n\u22a2 IsIso (cokernel.\u03c0 (kernel.\u03b9 f) \u226b coimageImageComparison f)\n[PROOFSTEP]\nexact IsIso.comp_isIso\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : Epi f\n\u22a2 IsIso (imageMonoFactorisation f).m\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : Epi f\n\u22a2 IsIso (kernel.\u03b9 (cokernel.\u03c0 f))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\n\u22a2 f \u226b cokernel.\u03c0 f = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\n\u22a2 IsLimit (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0))\n[PROOFSTEP]\nhaveI : Limits.HasImages C := hasImages\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\nthis : HasImages C\n\u22a2 IsLimit (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0))\n[PROOFSTEP]\nhaveI : HasEqualizers C := Preadditive.hasEqualizers_of_hasKernels\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\nthis\u271d : HasImages C\nthis : HasEqualizers C\n\u22a2 IsLimit (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0))\n[PROOFSTEP]\nhaveI : HasZeroObject C := Limits.hasZeroObject_of_hasFiniteBiproducts _\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\n\u22a2 IsLimit (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0))\n[PROOFSTEP]\nhave aux :\n \u2200 (s : KernelFork (cokernel.\u03c0 f)),\n (limit.lift (parallelPair (cokernel.\u03c0 f) 0) s \u226b inv (imageMonoFactorisation f).e) \u226b\n Fork.\u03b9 (KernelFork.of\u03b9 f (by simp)) =\n Fork.\u03b9 s :=\n ?_\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\ns : KernelFork (cokernel.\u03c0 f)\n\u22a2 f \u226b cokernel.\u03c0 f = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_2\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\naux :\n \u2200 (s : KernelFork (cokernel.\u03c0 f)),\n (limit.lift (parallelPair (cokernel.\u03c0 f) 0) s \u226b inv (imageMonoFactorisation f).e) \u226b\n Fork.\u03b9 (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0)) =\n Fork.\u03b9 s\n\u22a2 IsLimit (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0))\ncase refine_1\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\n\u22a2 \u2200 (s : KernelFork (cokernel.\u03c0 f)),\n (limit.lift (parallelPair (cokernel.\u03c0 f) 0) s \u226b inv (imageMonoFactorisation f).e) \u226b\n Fork.\u03b9 (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0)) =\n Fork.\u03b9 s\n[PROOFSTEP]\nrefine' isLimitAux _ (fun A => limit.lift _ _ \u226b inv (imageMonoFactorisation f).e) aux _\n[GOAL]\ncase refine_2\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\naux :\n \u2200 (s : KernelFork (cokernel.\u03c0 f)),\n (limit.lift (parallelPair (cokernel.\u03c0 f) 0) s \u226b inv (imageMonoFactorisation f).e) \u226b\n Fork.\u03b9 (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0)) =\n Fork.\u03b9 s\n\u22a2 \u2200 (s : KernelFork (cokernel.\u03c0 f)) (m_1 : s.pt \u27f6 (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0)).pt),\n m_1 \u226b Fork.\u03b9 (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0)) = Fork.\u03b9 s \u2192\n m_1 = (fun A => limit.lift (parallelPair (cokernel.\u03c0 f) 0) A \u226b inv (imageMonoFactorisation f).e) s\n[PROOFSTEP]\nintro A g hg\n[GOAL]\ncase refine_2\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\naux :\n \u2200 (s : KernelFork (cokernel.\u03c0 f)),\n (limit.lift (parallelPair (cokernel.\u03c0 f) 0) s \u226b inv (imageMonoFactorisation f).e) \u226b\n Fork.\u03b9 (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0)) =\n Fork.\u03b9 s\nA : KernelFork (cokernel.\u03c0 f)\ng : A.pt \u27f6 (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0)).pt\nhg : g \u226b Fork.\u03b9 (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0)) = Fork.\u03b9 A\n\u22a2 g = (fun A => limit.lift (parallelPair (cokernel.\u03c0 f) 0) A \u226b inv (imageMonoFactorisation f).e) A\n[PROOFSTEP]\nrw [KernelFork.\u03b9_of\u03b9] at hg \n[GOAL]\ncase refine_2\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\naux :\n \u2200 (s : KernelFork (cokernel.\u03c0 f)),\n (limit.lift (parallelPair (cokernel.\u03c0 f) 0) s \u226b inv (imageMonoFactorisation f).e) \u226b\n Fork.\u03b9 (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0)) =\n Fork.\u03b9 s\nA : KernelFork (cokernel.\u03c0 f)\ng : A.pt \u27f6 (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0)).pt\nhg : g \u226b f = Fork.\u03b9 A\n\u22a2 g = (fun A => limit.lift (parallelPair (cokernel.\u03c0 f) 0) A \u226b inv (imageMonoFactorisation f).e) A\n[PROOFSTEP]\nrw [\u2190 cancel_mono f, hg, \u2190 aux, KernelFork.\u03b9_of\u03b9]\n[GOAL]\ncase refine_1\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\n\u22a2 \u2200 (s : KernelFork (cokernel.\u03c0 f)),\n (limit.lift (parallelPair (cokernel.\u03c0 f) 0) s \u226b inv (imageMonoFactorisation f).e) \u226b\n Fork.\u03b9 (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0)) =\n Fork.\u03b9 s\n[PROOFSTEP]\nintro A\n[GOAL]\ncase refine_1\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\nA : KernelFork (cokernel.\u03c0 f)\n\u22a2 (limit.lift (parallelPair (cokernel.\u03c0 f) 0) A \u226b inv (imageMonoFactorisation f).e) \u226b\n Fork.\u03b9 (KernelFork.of\u03b9 f (_ : f \u226b cokernel.\u03c0 f = 0)) =\n Fork.\u03b9 A\n[PROOFSTEP]\nsimp only [KernelFork.\u03b9_of\u03b9, Category.assoc]\n[GOAL]\ncase refine_1\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\nA : KernelFork (cokernel.\u03c0 f)\n\u22a2 limit.lift (parallelPair (cokernel.\u03c0 f) 0) A \u226b inv (imageMonoFactorisation f).e \u226b f = Fork.\u03b9 A\n[PROOFSTEP]\nconvert limit.lift_\u03c0 A WalkingParallelPair.zero using 2\n[GOAL]\ncase h.e'_2.h.e'_7\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\nA : KernelFork (cokernel.\u03c0 f)\n\u22a2 inv (imageMonoFactorisation f).e \u226b f = limit.\u03c0 (parallelPair (cokernel.\u03c0 f) 0) WalkingParallelPair.zero\n[PROOFSTEP]\nrw [IsIso.inv_comp_eq, eq_comm]\n[GOAL]\ncase h.e'_2.h.e'_7\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Mono f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\nA : KernelFork (cokernel.\u03c0 f)\n\u22a2 (imageMonoFactorisation f).e \u226b limit.\u03c0 (parallelPair (cokernel.\u03c0 f) 0) WalkingParallelPair.zero = f\n[PROOFSTEP]\nexact (imageMonoFactorisation f).fac\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Epi f\n\u22a2 IsColimit (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0))\n[PROOFSTEP]\nhaveI : Limits.HasImages C := hasImages\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Epi f\nthis : HasImages C\n\u22a2 IsColimit (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0))\n[PROOFSTEP]\nhaveI : HasEqualizers C := Preadditive.hasEqualizers_of_hasKernels\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Epi f\nthis\u271d : HasImages C\nthis : HasEqualizers C\n\u22a2 IsColimit (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0))\n[PROOFSTEP]\nhaveI : HasZeroObject C := Limits.hasZeroObject_of_hasFiniteBiproducts _\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Epi f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\n\u22a2 IsColimit (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0))\n[PROOFSTEP]\nhave aux :\n \u2200 (s : CokernelCofork (kernel.\u03b9 f)),\n Cofork.\u03c0 (CokernelCofork.of\u03c0 f (by simp)) \u226b\n inv (imageMonoFactorisation f).m \u226b\n inv (Abelian.coimageImageComparison f) \u226b colimit.desc (parallelPair (kernel.\u03b9 f) 0) s =\n Cofork.\u03c0 s :=\n ?_\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Epi f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\ns : CokernelCofork (kernel.\u03b9 f)\n\u22a2 kernel.\u03b9 f \u226b f = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_2\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Epi f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\naux :\n \u2200 (s : CokernelCofork (kernel.\u03b9 f)),\n Cofork.\u03c0 (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0)) \u226b\n inv (imageMonoFactorisation f).m \u226b\n inv (coimageImageComparison f) \u226b colimit.desc (parallelPair (kernel.\u03b9 f) 0) s =\n Cofork.\u03c0 s\n\u22a2 IsColimit (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0))\ncase refine_1\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Epi f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\n\u22a2 \u2200 (s : CokernelCofork (kernel.\u03b9 f)),\n Cofork.\u03c0 (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0)) \u226b\n inv (imageMonoFactorisation f).m \u226b\n inv (coimageImageComparison f) \u226b colimit.desc (parallelPair (kernel.\u03b9 f) 0) s =\n Cofork.\u03c0 s\n[PROOFSTEP]\nrefine'\n isColimitAux _ (fun A => inv (imageMonoFactorisation f).m \u226b inv (Abelian.coimageImageComparison f) \u226b colimit.desc _ _)\n aux _\n[GOAL]\ncase refine_2\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Epi f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\naux :\n \u2200 (s : CokernelCofork (kernel.\u03b9 f)),\n Cofork.\u03c0 (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0)) \u226b\n inv (imageMonoFactorisation f).m \u226b\n inv (coimageImageComparison f) \u226b colimit.desc (parallelPair (kernel.\u03b9 f) 0) s =\n Cofork.\u03c0 s\n\u22a2 \u2200 (s : CokernelCofork (kernel.\u03b9 f)) (m_1 : (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0)).pt \u27f6 s.pt),\n Cofork.\u03c0 (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0)) \u226b m_1 = Cofork.\u03c0 s \u2192\n m_1 =\n (fun A =>\n inv (imageMonoFactorisation f).m \u226b\n inv (coimageImageComparison f) \u226b colimit.desc (parallelPair (kernel.\u03b9 f) 0) A)\n s\n[PROOFSTEP]\nintro A g hg\n[GOAL]\ncase refine_2\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Epi f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\naux :\n \u2200 (s : CokernelCofork (kernel.\u03b9 f)),\n Cofork.\u03c0 (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0)) \u226b\n inv (imageMonoFactorisation f).m \u226b\n inv (coimageImageComparison f) \u226b colimit.desc (parallelPair (kernel.\u03b9 f) 0) s =\n Cofork.\u03c0 s\nA : CokernelCofork (kernel.\u03b9 f)\ng : (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0)).pt \u27f6 A.pt\nhg : Cofork.\u03c0 (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0)) \u226b g = Cofork.\u03c0 A\n\u22a2 g =\n (fun A =>\n inv (imageMonoFactorisation f).m \u226b\n inv (coimageImageComparison f) \u226b colimit.desc (parallelPair (kernel.\u03b9 f) 0) A)\n A\n[PROOFSTEP]\nrw [CokernelCofork.\u03c0_of\u03c0] at hg \n[GOAL]\ncase refine_2\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Epi f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\naux :\n \u2200 (s : CokernelCofork (kernel.\u03b9 f)),\n Cofork.\u03c0 (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0)) \u226b\n inv (imageMonoFactorisation f).m \u226b\n inv (coimageImageComparison f) \u226b colimit.desc (parallelPair (kernel.\u03b9 f) 0) s =\n Cofork.\u03c0 s\nA : CokernelCofork (kernel.\u03b9 f)\ng : (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0)).pt \u27f6 A.pt\nhg : f \u226b g = Cofork.\u03c0 A\n\u22a2 g =\n (fun A =>\n inv (imageMonoFactorisation f).m \u226b\n inv (coimageImageComparison f) \u226b colimit.desc (parallelPair (kernel.\u03b9 f) 0) A)\n A\n[PROOFSTEP]\nrw [\u2190 cancel_epi f, hg, \u2190 aux, CokernelCofork.\u03c0_of\u03c0]\n[GOAL]\ncase refine_1\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Epi f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\n\u22a2 \u2200 (s : CokernelCofork (kernel.\u03b9 f)),\n Cofork.\u03c0 (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0)) \u226b\n inv (imageMonoFactorisation f).m \u226b\n inv (coimageImageComparison f) \u226b colimit.desc (parallelPair (kernel.\u03b9 f) 0) s =\n Cofork.\u03c0 s\n[PROOFSTEP]\nintro A\n[GOAL]\ncase refine_1\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Epi f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\nA : CokernelCofork (kernel.\u03b9 f)\n\u22a2 Cofork.\u03c0 (CokernelCofork.of\u03c0 f (_ : kernel.\u03b9 f \u226b f = 0)) \u226b\n inv (imageMonoFactorisation f).m \u226b inv (coimageImageComparison f) \u226b colimit.desc (parallelPair (kernel.\u03b9 f) 0) A =\n Cofork.\u03c0 A\n[PROOFSTEP]\nsimp only [CokernelCofork.\u03c0_of\u03c0, \u2190 Category.assoc]\n[GOAL]\ncase refine_1\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Epi f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\nA : CokernelCofork (kernel.\u03b9 f)\n\u22a2 ((f \u226b inv (imageMonoFactorisation f).m) \u226b inv (coimageImageComparison f)) \u226b\n colimit.desc (parallelPair (kernel.\u03b9 f) 0) A =\n Cofork.\u03c0 A\n[PROOFSTEP]\nconvert colimit.\u03b9_desc A WalkingParallelPair.one using 2\n[GOAL]\ncase h.e'_2.h.e'_6\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Epi f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\nA : CokernelCofork (kernel.\u03b9 f)\n\u22a2 (f \u226b inv (imageMonoFactorisation f).m) \u226b inv (coimageImageComparison f) =\n colimit.\u03b9 (parallelPair (kernel.\u03b9 f) 0) WalkingParallelPair.one\n[PROOFSTEP]\nrw [IsIso.comp_inv_eq, IsIso.comp_inv_eq, eq_comm, \u2190 imageMonoFactorisation_e']\n[GOAL]\ncase h.e'_2.h.e'_6\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (coimageImageComparison f)\ninst\u271d : HasFiniteProducts C\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nm : Epi f\nthis\u271d\u00b9 : HasImages C\nthis\u271d : HasEqualizers C\nthis : HasZeroObject C\nA : CokernelCofork (kernel.\u03b9 f)\n\u22a2 (imageMonoFactorisation f).e \u226b (imageMonoFactorisation f).m = f\n[PROOFSTEP]\nexact (imageMonoFactorisation f).fac\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nP Q : C\nf : P \u27f6 Q\n\u22a2 Epi (Abelian.factorThruImage f)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\nP Q : C\nf : P \u27f6 Q\ninst\u271d : Mono f\n\u22a2 IsIso (Abelian.factorThruImage f)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nP Q : C\nf : P \u27f6 Q\n\u22a2 Mono (Abelian.factorThruCoimage f)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\nP Q : C\nf : P \u27f6 Q\ninst\u271d : Epi f\n\u22a2 IsIso (Abelian.factorThruCoimage f)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nP Q : C\nf : P \u27f6 Q\nh : cokernel.\u03c0 f = 0\n\u22a2 Epi f\n[PROOFSTEP]\napply NormalMonoCategory.epi_of_zero_cokernel _ (cokernel f)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nP Q : C\nf : P \u27f6 Q\nh : cokernel.\u03c0 f = 0\n\u22a2 IsColimit (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0))\n[PROOFSTEP]\nsimp_rw [\u2190 h]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nP Q : C\nf : P \u27f6 Q\nh : cokernel.\u03c0 f = 0\n\u22a2 IsColimit (CokernelCofork.of\u03c0 (cokernel.\u03c0 f) (_ : f \u226b cokernel.\u03c0 f = 0))\n[PROOFSTEP]\nexact IsColimit.ofIsoColimit (colimit.isColimit (parallelPair f 0)) (isoOf\u03c0 _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nP Q : C\nf : P \u27f6 Q\nR : C\ng : Q \u27f6 R\nh : f \u226b g = 0\n\u22a2 Abelian.factorThruImage f \u226b image.\u03b9 f \u226b g = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nP Q : C\nf : P \u27f6 Q\nR : C\ng : Q \u27f6 R\nh : f \u226b g = 0\n\u22a2 (f \u226b coimage.\u03c0 g) \u226b Abelian.factorThruCoimage g = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nP Q : C\nf : P \u27f6 Q\n\u22a2 Mono (image.\u03b9 f)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nP Q : C\nf : P \u27f6 Q\n\u22a2 Mono (Abelian.factorThruCoimage f)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\n\u22a2 HasImages C\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\n\u22a2 HasImageMaps C\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nX Y : C\nf : X \u27f6 Y\n\u22a2 IsIso (coimageImageComparison f)\n[PROOFSTEP]\nconvert\n IsIso.of_iso\n (IsImage.isoExt (coimageStrongEpiMonoFactorisation f).toMonoIsImage\n (imageStrongEpiMonoFactorisation f).toMonoIsImage)\n[GOAL]\ncase h.e'_5.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nX Y : C\nf : X \u27f6 Y\ne_3\u271d : Abelian.coimage f = (coimageStrongEpiMonoFactorisation f).toMonoFactorisation.I\ne_4\u271d : Abelian.image f = (imageStrongEpiMonoFactorisation f).toMonoFactorisation.I\n\u22a2 coimageImageComparison f =\n (IsImage.isoExt (StrongEpiMonoFactorisation.toMonoIsImage (coimageStrongEpiMonoFactorisation f))\n (StrongEpiMonoFactorisation.toMonoIsImage (imageStrongEpiMonoFactorisation f))).hom\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_5.h.h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nX Y : C\nf : X \u27f6 Y\ne_3\u271d : Abelian.coimage f = (coimageStrongEpiMonoFactorisation f).toMonoFactorisation.I\ne_4\u271d : Abelian.image f = (imageStrongEpiMonoFactorisation f).toMonoFactorisation.I\n\u22a2 (coequalizer.\u03c0 (kernel.\u03b9 f) 0 \u226b coimageImageComparison f) \u226b equalizer.\u03b9 (cokernel.\u03c0 f) 0 =\n (coequalizer.\u03c0 (kernel.\u03b9 f) 0 \u226b\n (IsImage.isoExt (StrongEpiMonoFactorisation.toMonoIsImage (coimageStrongEpiMonoFactorisation f))\n (StrongEpiMonoFactorisation.toMonoIsImage (imageStrongEpiMonoFactorisation f))).hom) \u226b\n equalizer.\u03b9 (cokernel.\u03c0 f) 0\n[PROOFSTEP]\nchange _ = _ \u226b (imageStrongEpiMonoFactorisation f).m\n[GOAL]\ncase h.e'_5.h.h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nX Y : C\nf : X \u27f6 Y\ne_3\u271d : Abelian.coimage f = (coimageStrongEpiMonoFactorisation f).toMonoFactorisation.I\ne_4\u271d : Abelian.image f = (imageStrongEpiMonoFactorisation f).toMonoFactorisation.I\n\u22a2 (coequalizer.\u03c0 (kernel.\u03b9 f) 0 \u226b coimageImageComparison f) \u226b equalizer.\u03b9 (cokernel.\u03c0 f) 0 =\n (coequalizer.\u03c0 (kernel.\u03b9 f) 0 \u226b\n (IsImage.isoExt (StrongEpiMonoFactorisation.toMonoIsImage (coimageStrongEpiMonoFactorisation f))\n (StrongEpiMonoFactorisation.toMonoIsImage (imageStrongEpiMonoFactorisation f))).hom) \u226b\n (imageStrongEpiMonoFactorisation f).toMonoFactorisation.m\n[PROOFSTEP]\nsimp [-imageStrongEpiMonoFactorisation_m]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nX Y : C\nf : X \u27f6 Y\n\u22a2 kernel.\u03b9 f \u226b factorThruImage f = 0\n[PROOFSTEP]\nsimp [\u2190 cancel_mono (Limits.image.\u03b9 f)]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nX Y : C\nf : X \u27f6 Y\n\u22a2 (coimageIsoImage' f).hom = cokernel.desc (kernel.\u03b9 f) (factorThruImage f) (_ : kernel.\u03b9 f \u226b factorThruImage f = 0)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nX Y : C\nf : X \u27f6 Y\n\u22a2 coequalizer.\u03c0 (kernel.\u03b9 f) 0 \u226b (coimageIsoImage' f).hom =\n coequalizer.\u03c0 (kernel.\u03b9 f) 0 \u226b\n cokernel.desc (kernel.\u03b9 f) (factorThruImage f) (_ : kernel.\u03b9 f \u226b factorThruImage f = 0)\n[PROOFSTEP]\nsimp only [\u2190 cancel_mono (Limits.image.\u03b9 f), IsImage.isoExt_hom, cokernel.\u03c0_desc, Category.assoc, IsImage.lift_\u03b9,\n coimageStrongEpiMonoFactorisation_m, Limits.image.fac]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nX Y : C\nf : X \u27f6 Y\n\u22a2 factorThruImage f \u226b (coimageIsoImage' f).inv = cokernel.\u03c0 (kernel.\u03b9 f)\n[PROOFSTEP]\nsimp only [IsImage.isoExt_inv, image.isImage_lift, image.fac_lift, coimageStrongEpiMonoFactorisation_e]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nX Y : C\nf : X \u27f6 Y\n\u22a2 (imageIsoImage f).hom \u226b Limits.image.\u03b9 f = kernel.\u03b9 (cokernel.\u03c0 f)\n[PROOFSTEP]\nsimp only [IsImage.isoExt_hom, IsImage.lift_\u03b9, imageStrongEpiMonoFactorisation_m]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nX Y : C\nf : X \u27f6 Y\n\u22a2 Limits.image.\u03b9 f \u226b cokernel.\u03c0 f = 0\n[PROOFSTEP]\nsimp [\u2190 cancel_epi (factorThruImage f)]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nX Y : C\nf : X \u27f6 Y\n\u22a2 (imageIsoImage f).inv = kernel.lift (cokernel.\u03c0 f) (Limits.image.\u03b9 f) (_ : Limits.image.\u03b9 f \u226b cokernel.\u03c0 f = 0)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nX Y : C\nf : X \u27f6 Y\n\u22a2 (factorThruImage f \u226b (imageIsoImage f).inv) \u226b equalizer.\u03b9 (cokernel.\u03c0 f) 0 =\n (factorThruImage f \u226b kernel.lift (cokernel.\u03c0 f) (Limits.image.\u03b9 f) (_ : Limits.image.\u03b9 f \u226b cokernel.\u03c0 f = 0)) \u226b\n equalizer.\u03b9 (cokernel.\u03c0 f) 0\n[PROOFSTEP]\nrw [IsImage.isoExt_inv, image.isImage_lift, Limits.image.fac_lift, imageStrongEpiMonoFactorisation_e, Category.assoc,\n kernel.lift_\u03b9, equalizer_as_kernel, kernel.lift_\u03b9, Limits.image.fac]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPullbacks C\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 pullbackToBiproduct f g \u226b biprod.desc f (-g) = 0\n[PROOFSTEP]\nrw [biprod.lift_desc, comp_neg, pullback.condition, add_right_neg]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPullbacks C\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ns : Fork (biprod.desc f (-g)) 0\n\u22a2 (Fork.\u03b9 s \u226b biprod.fst) \u226b f - (Fork.\u03b9 s \u226b biprod.snd) \u226b g = 0\n[PROOFSTEP]\nrw [Category.assoc, Category.assoc, \u2190 comp_sub, sub_eq_add_neg, \u2190 comp_neg, \u2190 biprod.desc_eq, KernelFork.condition s]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPullbacks C\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ns : Fork (biprod.desc f (-g)) 0\n\u22a2 (fun s =>\n pullback.lift (Fork.\u03b9 s \u226b biprod.fst) (Fork.\u03b9 s \u226b biprod.snd)\n (_ : (Fork.\u03b9 s \u226b biprod.fst) \u226b f = (Fork.\u03b9 s \u226b biprod.snd) \u226b g))\n s \u226b\n Fork.\u03b9 (pullbackToBiproductFork f g) =\n Fork.\u03b9 s\n[PROOFSTEP]\napply biprod.hom_ext\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPullbacks C\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ns : Fork (biprod.desc f (-g)) 0\n\u22a2 ((fun s =>\n pullback.lift (Fork.\u03b9 s \u226b biprod.fst) (Fork.\u03b9 s \u226b biprod.snd)\n (_ : (Fork.\u03b9 s \u226b biprod.fst) \u226b f = (Fork.\u03b9 s \u226b biprod.snd) \u226b g))\n s \u226b\n Fork.\u03b9 (pullbackToBiproductFork f g)) \u226b\n biprod.fst =\n Fork.\u03b9 s \u226b biprod.fst\n[PROOFSTEP]\nrw [Fork.\u03b9_of\u03b9, Category.assoc]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPullbacks C\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ns : Fork (biprod.desc f (-g)) 0\n\u22a2 ((fun s =>\n pullback.lift (Fork.\u03b9 s \u226b biprod.fst) (Fork.\u03b9 s \u226b biprod.snd)\n (_ : (Fork.\u03b9 s \u226b biprod.fst) \u226b f = (Fork.\u03b9 s \u226b biprod.snd) \u226b g))\n s \u226b\n Fork.\u03b9 (pullbackToBiproductFork f g)) \u226b\n biprod.snd =\n Fork.\u03b9 s \u226b biprod.snd\n[PROOFSTEP]\nrw [Fork.\u03b9_of\u03b9, Category.assoc]\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPullbacks C\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ns : Fork (biprod.desc f (-g)) 0\n\u22a2 (fun s =>\n pullback.lift (Fork.\u03b9 s \u226b biprod.fst) (Fork.\u03b9 s \u226b biprod.snd)\n (_ : (Fork.\u03b9 s \u226b biprod.fst) \u226b f = (Fork.\u03b9 s \u226b biprod.snd) \u226b g))\n s \u226b\n pullbackToBiproduct f g \u226b biprod.fst =\n Fork.\u03b9 s \u226b biprod.fst\n[PROOFSTEP]\nrw [biprod.lift_fst, pullback.lift_fst]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPullbacks C\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ns : Fork (biprod.desc f (-g)) 0\n\u22a2 (fun s =>\n pullback.lift (Fork.\u03b9 s \u226b biprod.fst) (Fork.\u03b9 s \u226b biprod.snd)\n (_ : (Fork.\u03b9 s \u226b biprod.fst) \u226b f = (Fork.\u03b9 s \u226b biprod.snd) \u226b g))\n s \u226b\n pullbackToBiproduct f g \u226b biprod.snd =\n Fork.\u03b9 s \u226b biprod.snd\n[PROOFSTEP]\nrw [biprod.lift_snd, pullback.lift_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPullbacks C\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ns : Fork (biprod.desc f (-g)) 0\nm : s.pt \u27f6 (pullbackToBiproductFork f g).pt\nh : m \u226b Fork.\u03b9 (pullbackToBiproductFork f g) = Fork.\u03b9 s\n\u22a2 m =\n (fun s =>\n pullback.lift (Fork.\u03b9 s \u226b biprod.fst) (Fork.\u03b9 s \u226b biprod.snd)\n (_ : (Fork.\u03b9 s \u226b biprod.fst) \u226b f = (Fork.\u03b9 s \u226b biprod.snd) \u226b g))\n s\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPullbacks C\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ns : Fork (biprod.desc f (-g)) 0\nm : s.pt \u27f6 (pullbackToBiproductFork f g).pt\nh : m \u226b Fork.\u03b9 (pullbackToBiproductFork f g) = Fork.\u03b9 s\n\u22a2 m \u226b pullback.fst =\n (fun s =>\n pullback.lift (Fork.\u03b9 s \u226b biprod.fst) (Fork.\u03b9 s \u226b biprod.snd)\n (_ : (Fork.\u03b9 s \u226b biprod.fst) \u226b f = (Fork.\u03b9 s \u226b biprod.snd) \u226b g))\n s \u226b\n pullback.fst\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPullbacks C\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ns : Fork (biprod.desc f (-g)) 0\nm : s.pt \u27f6 (pullbackToBiproductFork f g).pt\nh : m \u226b Fork.\u03b9 (pullbackToBiproductFork f g) = Fork.\u03b9 s\n\u22a2 m \u226b pullback.snd =\n (fun s =>\n pullback.lift (Fork.\u03b9 s \u226b biprod.fst) (Fork.\u03b9 s \u226b biprod.snd)\n (_ : (Fork.\u03b9 s \u226b biprod.fst) \u226b f = (Fork.\u03b9 s \u226b biprod.snd) \u226b g))\n s \u226b\n pullback.snd\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\n\u22a2 biprod.lift f (-g) \u226b biproductToPushout f g = 0\n[PROOFSTEP]\nrw [biprod.lift_desc, neg_comp, pushout.condition, add_right_neg]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ns : Cofork (biprod.lift f (-g)) 0\n\u22a2 f \u226b biprod.inl \u226b Cofork.\u03c0 s - g \u226b biprod.inr \u226b Cofork.\u03c0 s = 0\n[PROOFSTEP]\nrw [\u2190 Category.assoc, \u2190 Category.assoc, \u2190 sub_comp, sub_eq_add_neg, \u2190 neg_comp, \u2190 biprod.lift_eq, Cofork.condition s,\n zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ns : Cofork (biprod.lift f (-g)) 0\n\u22a2 Cofork.\u03c0 (biproductToPushoutCofork f g) \u226b\n (fun s =>\n pushout.desc (biprod.inl \u226b Cofork.\u03c0 s) (biprod.inr \u226b Cofork.\u03c0 s)\n (_ : f \u226b biprod.inl \u226b Cofork.\u03c0 s = g \u226b biprod.inr \u226b Cofork.\u03c0 s))\n s =\n Cofork.\u03c0 s\n[PROOFSTEP]\napply biprod.hom_ext'\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ns : Cofork (biprod.lift f (-g)) 0\n\u22a2 biprod.inl \u226b\n Cofork.\u03c0 (biproductToPushoutCofork f g) \u226b\n (fun s =>\n pushout.desc (biprod.inl \u226b Cofork.\u03c0 s) (biprod.inr \u226b Cofork.\u03c0 s)\n (_ : f \u226b biprod.inl \u226b Cofork.\u03c0 s = g \u226b biprod.inr \u226b Cofork.\u03c0 s))\n s =\n biprod.inl \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ns : Cofork (biprod.lift f (-g)) 0\n\u22a2 biprod.inr \u226b\n Cofork.\u03c0 (biproductToPushoutCofork f g) \u226b\n (fun s =>\n pushout.desc (biprod.inl \u226b Cofork.\u03c0 s) (biprod.inr \u226b Cofork.\u03c0 s)\n (_ : f \u226b biprod.inl \u226b Cofork.\u03c0 s = g \u226b biprod.inr \u226b Cofork.\u03c0 s))\n s =\n biprod.inr \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ns : Cofork (biprod.lift f (-g)) 0\nm : (biproductToPushoutCofork f g).pt \u27f6 s.pt\nh : Cofork.\u03c0 (biproductToPushoutCofork f g) \u226b m = Cofork.\u03c0 s\n\u22a2 m =\n (fun s =>\n pushout.desc (biprod.inl \u226b Cofork.\u03c0 s) (biprod.inr \u226b Cofork.\u03c0 s)\n (_ : f \u226b biprod.inl \u226b Cofork.\u03c0 s = g \u226b biprod.inr \u226b Cofork.\u03c0 s))\n s\n[PROOFSTEP]\napply pushout.hom_ext\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ns : Cofork (biprod.lift f (-g)) 0\nm : (biproductToPushoutCofork f g).pt \u27f6 s.pt\nh : Cofork.\u03c0 (biproductToPushoutCofork f g) \u226b m = Cofork.\u03c0 s\n\u22a2 pushout.inl \u226b m =\n pushout.inl \u226b\n (fun s =>\n pushout.desc (biprod.inl \u226b Cofork.\u03c0 s) (biprod.inr \u226b Cofork.\u03c0 s)\n (_ : f \u226b biprod.inl \u226b Cofork.\u03c0 s = g \u226b biprod.inr \u226b Cofork.\u03c0 s))\n s\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Abelian C\ninst\u271d : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ns : Cofork (biprod.lift f (-g)) 0\nm : (biproductToPushoutCofork f g).pt \u27f6 s.pt\nh : Cofork.\u03c0 (biproductToPushoutCofork f g) \u226b m = Cofork.\u03c0 s\n\u22a2 pushout.inr \u226b m =\n pushout.inr \u226b\n (fun s =>\n pushout.desc (biprod.inl \u226b Cofork.\u03c0 s) (biprod.inr \u226b Cofork.\u03c0 s)\n (_ : f \u226b biprod.inl \u226b Cofork.\u03c0 s = g \u226b biprod.inr \u226b Cofork.\u03c0 s))\n s\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\n\u22a2 e = 0\n[PROOFSTEP]\nlet u := biprod.desc (0 : X \u27f6 R) e\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\n\u22a2 e = 0\n[PROOFSTEP]\nhave hu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0 := by\n simpa\n -- pullback_to_biproduct f g is a kernel of (f, -g), so (f, -g) is a\n -- cokernel of pullback_to_biproduct f g\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\n\u22a2 PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\n[PROOFSTEP]\nsimpa\n -- pullback_to_biproduct f g is a kernel of (f, -g), so (f, -g) is a\n -- cokernel of pullback_to_biproduct f g\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\n\u22a2 e = 0\n[PROOFSTEP]\nhave :=\n epiIsCokernelOfKernel _\n (PullbackToBiproductIsKernel.isLimitPullbackToBiproduct f g)\n -- We use this fact to obtain a factorization of u through (f, -g) via some d : Z \u27f6 R.\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\n\u22a2 e = 0\n[PROOFSTEP]\nobtain \u27e8d, hd\u27e9 := CokernelCofork.IsColimit.desc' this u hu\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd :\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0)).pt \u27f6\n R\nhd :\n Cofork.\u03c0\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0)) \u226b\n d =\n u\n\u22a2 e = 0\n[PROOFSTEP]\ndsimp at d \n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd :\n Cofork.\u03c0\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0)) \u226b\n d =\n u\n\u22a2 e = 0\n[PROOFSTEP]\ndsimp at hd \n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc 0 e\n\u22a2 e = 0\n[PROOFSTEP]\nhave : f \u226b d = 0\n[GOAL]\ncase this\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc 0 e\n\u22a2 f \u226b d = 0\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc 0 e\nthis : f \u226b d = 0\n\u22a2 e = 0\n[PROOFSTEP]\ncalc\n f \u226b d = (biprod.inl \u226b biprod.desc f (-g)) \u226b d := by rw [biprod.inl_desc]\n _ = biprod.inl \u226b u := by rw [Category.assoc, hd]\n _ = 0 :=\n biprod.inl_desc _\n _\n -- But f is an epimorphism, so d = 0...\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc 0 e\n\u22a2 f \u226b d = (biprod.inl \u226b biprod.desc f (-g)) \u226b d\n[PROOFSTEP]\nrw [biprod.inl_desc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc 0 e\n\u22a2 (biprod.inl \u226b biprod.desc f (-g)) \u226b d = biprod.inl \u226b u\n[PROOFSTEP]\nrw [Category.assoc, hd]\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc 0 e\nthis : f \u226b d = 0\n\u22a2 e = 0\n[PROOFSTEP]\nhave : d = 0 :=\n (cancel_epi f).1\n (by simpa)\n -- ...or, in other words, e = 0.\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc 0 e\nthis : f \u226b d = 0\n\u22a2 f \u226b d = f \u226b 0\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d\u00b9 :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc 0 e\nthis\u271d : f \u226b d = 0\nthis : d = 0\n\u22a2 e = 0\n[PROOFSTEP]\ncalc\n e = biprod.inr \u226b biprod.desc (0 : X \u27f6 R) e := by rw [biprod.inr_desc]\n _ = biprod.inr \u226b biprod.desc f (-g) \u226b d := by rw [\u2190 hd]\n _ = biprod.inr \u226b biprod.desc f (-g) \u226b 0 := by rw [this]\n _ = (biprod.inr \u226b biprod.desc f (-g)) \u226b 0 := by rw [\u2190 Category.assoc]\n _ = 0 := HasZeroMorphisms.comp_zero _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d\u00b9 :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc 0 e\nthis\u271d : f \u226b d = 0\nthis : d = 0\n\u22a2 e = biprod.inr \u226b biprod.desc 0 e\n[PROOFSTEP]\nrw [biprod.inr_desc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d\u00b9 :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc 0 e\nthis\u271d : f \u226b d = 0\nthis : d = 0\n\u22a2 biprod.inr \u226b biprod.desc 0 e = biprod.inr \u226b biprod.desc f (-g) \u226b d\n[PROOFSTEP]\nrw [\u2190 hd]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d\u00b9 :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc 0 e\nthis\u271d : f \u226b d = 0\nthis : d = 0\n\u22a2 biprod.inr \u226b biprod.desc f (-g) \u226b d = biprod.inr \u226b biprod.desc f (-g) \u226b 0\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\nR : C\ne : Y \u27f6 R\nh : pullback.snd \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d\u00b9 :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc 0 e\nthis\u271d : f \u226b d = 0\nthis : d = 0\n\u22a2 biprod.inr \u226b biprod.desc f (-g) \u226b 0 = (biprod.inr \u226b biprod.desc f (-g)) \u226b 0\n[PROOFSTEP]\nrw [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\n\u22a2 e = 0\n[PROOFSTEP]\nlet u :=\n biprod.desc e\n (0 : Y \u27f6 R)\n -- The composite pullback f g \u27f6 X \u229e Y \u27f6 R is zero by assumption.\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\n\u22a2 e = 0\n[PROOFSTEP]\nhave hu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0 := by\n simpa\n -- pullback_to_biproduct f g is a kernel of (f, -g), so (f, -g) is a\n -- cokernel of pullback_to_biproduct f g\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\n\u22a2 PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\n[PROOFSTEP]\nsimpa\n -- pullback_to_biproduct f g is a kernel of (f, -g), so (f, -g) is a\n -- cokernel of pullback_to_biproduct f g\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\n\u22a2 e = 0\n[PROOFSTEP]\nhave :=\n epiIsCokernelOfKernel _\n (PullbackToBiproductIsKernel.isLimitPullbackToBiproduct f g)\n -- We use this fact to obtain a factorization of u through (f, -g) via some d : Z \u27f6 R.\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\n\u22a2 e = 0\n[PROOFSTEP]\nobtain \u27e8d, hd\u27e9 := CokernelCofork.IsColimit.desc' this u hu\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd :\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0)).pt \u27f6\n R\nhd :\n Cofork.\u03c0\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0)) \u226b\n d =\n u\n\u22a2 e = 0\n[PROOFSTEP]\ndsimp at d \n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd :\n Cofork.\u03c0\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0)) \u226b\n d =\n u\n\u22a2 e = 0\n[PROOFSTEP]\ndsimp at hd \n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc e 0\n\u22a2 e = 0\n[PROOFSTEP]\nhave : (-g) \u226b d = 0\n[GOAL]\ncase this\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc e 0\n\u22a2 (-g) \u226b d = 0\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc e 0\nthis : (-g) \u226b d = 0\n\u22a2 e = 0\n[PROOFSTEP]\ncalc\n (-g) \u226b d = (biprod.inr \u226b biprod.desc f (-g)) \u226b d := by rw [biprod.inr_desc]\n _ = biprod.inr \u226b u := by rw [Category.assoc, hd]\n _ = 0 :=\n biprod.inr_desc _\n _\n -- But g is an epimorphism, thus so is -g, so d = 0...\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc e 0\n\u22a2 (-g) \u226b d = (biprod.inr \u226b biprod.desc f (-g)) \u226b d\n[PROOFSTEP]\nrw [biprod.inr_desc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc e 0\n\u22a2 (biprod.inr \u226b biprod.desc f (-g)) \u226b d = biprod.inr \u226b u\n[PROOFSTEP]\nrw [Category.assoc, hd]\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc e 0\nthis : (-g) \u226b d = 0\n\u22a2 e = 0\n[PROOFSTEP]\nhave : d = 0 :=\n (cancel_epi (-g)).1\n (by simpa)\n -- ...or, in other words, e = 0.\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc e 0\nthis : (-g) \u226b d = 0\n\u22a2 (-g) \u226b d = (-g) \u226b 0\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d\u00b9 :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc e 0\nthis\u271d : (-g) \u226b d = 0\nthis : d = 0\n\u22a2 e = 0\n[PROOFSTEP]\ncalc\n e = biprod.inl \u226b biprod.desc e (0 : Y \u27f6 R) := by rw [biprod.inl_desc]\n _ = biprod.inl \u226b biprod.desc f (-g) \u226b d := by rw [\u2190 hd]\n _ = biprod.inl \u226b biprod.desc f (-g) \u226b 0 := by rw [this]\n _ = (biprod.inl \u226b biprod.desc f (-g)) \u226b 0 := by rw [\u2190 Category.assoc]\n _ = 0 := HasZeroMorphisms.comp_zero _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d\u00b9 :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc e 0\nthis\u271d : (-g) \u226b d = 0\nthis : d = 0\n\u22a2 e = biprod.inl \u226b biprod.desc e 0\n[PROOFSTEP]\nrw [biprod.inl_desc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d\u00b9 :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc e 0\nthis\u271d : (-g) \u226b d = 0\nthis : d = 0\n\u22a2 biprod.inl \u226b biprod.desc e 0 = biprod.inl \u226b biprod.desc f (-g) \u226b d\n[PROOFSTEP]\nrw [\u2190 hd]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d\u00b9 :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc e 0\nthis\u271d : (-g) \u226b d = 0\nthis : d = 0\n\u22a2 biprod.inl \u226b biprod.desc f (-g) \u226b d = biprod.inl \u226b biprod.desc f (-g) \u226b 0\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\nR : C\ne : X \u27f6 R\nh : pullback.fst \u226b e = 0\nu : X \u229e Y \u27f6 R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g \u226b u = 0\nthis\u271d\u00b9 :\n IsColimit\n (CokernelCofork.of\u03c0 (biprod.desc f (-g))\n (_ : Fork.\u03b9 (PullbackToBiproductIsKernel.pullbackToBiproductFork f g) \u226b biprod.desc f (-g) = 0))\nd : Z \u27f6 R\nhd : biprod.desc f (-g) \u226b d = biprod.desc e 0\nthis\u271d : (-g) \u226b d = 0\nthis : d = 0\n\u22a2 biprod.inl \u226b biprod.desc f (-g) \u226b 0 = (biprod.inl \u226b biprod.desc f (-g)) \u226b 0\n[PROOFSTEP]\nrw [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\ns : PullbackCone f g\nhs : IsLimit s\n\u22a2 Epi (PullbackCone.snd s)\n[PROOFSTEP]\nhaveI : Epi (NatTrans.app (limit.cone (cospan f g)).\u03c0 WalkingCospan.right) := Abelian.epi_pullback_of_epi_f f g\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi f\ns : PullbackCone f g\nhs : IsLimit s\nthis : Epi (NatTrans.app (limit.cone (cospan f g)).\u03c0 WalkingCospan.right)\n\u22a2 Epi (PullbackCone.snd s)\n[PROOFSTEP]\napply epi_of_epi_fac (IsLimit.conePointUniqueUpToIso_hom_comp (limit.isLimit _) hs _)\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\ns : PullbackCone f g\nhs : IsLimit s\n\u22a2 Epi (PullbackCone.fst s)\n[PROOFSTEP]\nhaveI : Epi (NatTrans.app (limit.cone (cospan f g)).\u03c0 WalkingCospan.left) := Abelian.epi_pullback_of_epi_g f g\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : Epi g\ns : PullbackCone f g\nhs : IsLimit s\nthis : Epi (NatTrans.app (limit.cone (cospan f g)).\u03c0 WalkingCospan.left)\n\u22a2 Epi (PullbackCone.fst s)\n[PROOFSTEP]\napply epi_of_epi_fac (IsLimit.conePointUniqueUpToIso_hom_comp (limit.isLimit _) hs _)\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : Abelian C\ninst\u271d\u00b2 : HasPullbacks C\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\ng\u2081 : Y \u27f6 W\ninst\u271d\u00b9 : Epi g\u2081\ng\u2082 : W \u27f6 Z\ninst\u271d : Mono g\u2082\nhg : g\u2081 \u226b g\u2082 = g\nf' : X \u27f6 W\nhf : f' \u226b g\u2082 = f\nt : PullbackCone f g\nht : IsLimit t\n\u22a2 Epi (PullbackCone.fst t)\n[PROOFSTEP]\napply epi_fst_of_isLimit _ _ (PullbackCone.isLimitOfFactors f g g\u2082 f' g\u2081 hf hg t ht)\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\n\u22a2 e = 0\n[PROOFSTEP]\nlet u := biprod.lift (0 : R \u27f6 Y) e\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\n\u22a2 e = 0\n[PROOFSTEP]\nhave hu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0 := by simpa\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\n\u22a2 u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\n[PROOFSTEP]\nsimpa\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\n\u22a2 e = 0\n[PROOFSTEP]\nhave := monoIsKernelOfCokernel _ (BiproductToPushoutIsCokernel.isColimitBiproductToPushout f g)\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\n\u22a2 e = 0\n[PROOFSTEP]\nobtain \u27e8d, hd\u27e9 := KernelFork.IsLimit.lift' this u hu\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd :\n R \u27f6\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0)).pt\nhd :\n d \u226b\n Fork.\u03b9\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0)) =\n u\n\u22a2 e = 0\n[PROOFSTEP]\ndsimp at d \n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd :\n d \u226b\n Fork.\u03b9\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0)) =\n u\n\u22a2 e = 0\n[PROOFSTEP]\ndsimp at hd \n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift 0 e\n\u22a2 e = 0\n[PROOFSTEP]\nhave : d \u226b f = 0\n[GOAL]\ncase this\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift 0 e\n\u22a2 d \u226b f = 0\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift 0 e\nthis : d \u226b f = 0\n\u22a2 e = 0\n[PROOFSTEP]\ncalc\n d \u226b f = d \u226b biprod.lift f (-g) \u226b biprod.fst := by rw [biprod.lift_fst]\n _ = u \u226b biprod.fst := by rw [\u2190 Category.assoc, hd]\n _ = 0 := biprod.lift_fst _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift 0 e\n\u22a2 d \u226b f = d \u226b biprod.lift f (-g) \u226b biprod.fst\n[PROOFSTEP]\nrw [biprod.lift_fst]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift 0 e\n\u22a2 d \u226b biprod.lift f (-g) \u226b biprod.fst = u \u226b biprod.fst\n[PROOFSTEP]\nrw [\u2190 Category.assoc, hd]\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift 0 e\nthis : d \u226b f = 0\n\u22a2 e = 0\n[PROOFSTEP]\nhave : d = 0 := (cancel_mono f).1 (by simpa)\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift 0 e\nthis : d \u226b f = 0\n\u22a2 d \u226b f = 0 \u226b f\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d\u00b9 :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift 0 e\nthis\u271d : d \u226b f = 0\nthis : d = 0\n\u22a2 e = 0\n[PROOFSTEP]\ncalc\n e = biprod.lift (0 : R \u27f6 Y) e \u226b biprod.snd := by rw [biprod.lift_snd]\n _ = (d \u226b biprod.lift f (-g)) \u226b biprod.snd := by rw [\u2190 hd]\n _ = (0 \u226b biprod.lift f (-g)) \u226b biprod.snd := by rw [this]\n _ = 0 \u226b biprod.lift f (-g) \u226b biprod.snd := by rw [Category.assoc]\n _ = 0 := zero_comp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d\u00b9 :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift 0 e\nthis\u271d : d \u226b f = 0\nthis : d = 0\n\u22a2 e = biprod.lift 0 e \u226b biprod.snd\n[PROOFSTEP]\nrw [biprod.lift_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d\u00b9 :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift 0 e\nthis\u271d : d \u226b f = 0\nthis : d = 0\n\u22a2 biprod.lift 0 e \u226b biprod.snd = (d \u226b biprod.lift f (-g)) \u226b biprod.snd\n[PROOFSTEP]\nrw [\u2190 hd]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d\u00b9 :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift 0 e\nthis\u271d : d \u226b f = 0\nthis : d = 0\n\u22a2 (d \u226b biprod.lift f (-g)) \u226b biprod.snd = (0 \u226b biprod.lift f (-g)) \u226b biprod.snd\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\nR : C\ne : R \u27f6 Z\nh : e \u226b pushout.inr = 0\nu : R \u27f6 Y \u229e Z := biprod.lift 0 e\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d\u00b9 :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift 0 e\nthis\u271d : d \u226b f = 0\nthis : d = 0\n\u22a2 (0 \u226b biprod.lift f (-g)) \u226b biprod.snd = 0 \u226b biprod.lift f (-g) \u226b biprod.snd\n[PROOFSTEP]\nrw [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\n\u22a2 e = 0\n[PROOFSTEP]\nlet u := biprod.lift e (0 : R \u27f6 Z)\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\n\u22a2 e = 0\n[PROOFSTEP]\nhave hu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0 := by simpa\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\n\u22a2 u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\n[PROOFSTEP]\nsimpa\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\n\u22a2 e = 0\n[PROOFSTEP]\nhave := monoIsKernelOfCokernel _ (BiproductToPushoutIsCokernel.isColimitBiproductToPushout f g)\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\n\u22a2 e = 0\n[PROOFSTEP]\nobtain \u27e8d, hd\u27e9 := KernelFork.IsLimit.lift' this u hu\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd :\n R \u27f6\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0)).pt\nhd :\n d \u226b\n Fork.\u03b9\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0)) =\n u\n\u22a2 e = 0\n[PROOFSTEP]\ndsimp at d \n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd :\n d \u226b\n Fork.\u03b9\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0)) =\n u\n\u22a2 e = 0\n[PROOFSTEP]\ndsimp at hd \n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift e 0\n\u22a2 e = 0\n[PROOFSTEP]\nhave : d \u226b (-g) = 0\n[GOAL]\ncase this\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift e 0\n\u22a2 d \u226b (-g) = 0\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift e 0\nthis : d \u226b (-g) = 0\n\u22a2 e = 0\n[PROOFSTEP]\ncalc\n d \u226b (-g) = d \u226b biprod.lift f (-g) \u226b biprod.snd := by rw [biprod.lift_snd]\n _ = biprod.lift e (0 : R \u27f6 Z) \u226b biprod.snd := by rw [\u2190 Category.assoc, hd]\n _ = 0 := biprod.lift_snd _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift e 0\n\u22a2 d \u226b (-g) = d \u226b biprod.lift f (-g) \u226b biprod.snd\n[PROOFSTEP]\nrw [biprod.lift_snd]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift e 0\n\u22a2 d \u226b biprod.lift f (-g) \u226b biprod.snd = biprod.lift e 0 \u226b biprod.snd\n[PROOFSTEP]\nrw [\u2190 Category.assoc, hd]\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift e 0\nthis : d \u226b (-g) = 0\n\u22a2 e = 0\n[PROOFSTEP]\nhave : d = 0 := (cancel_mono (-g)).1 (by simpa)\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift e 0\nthis : d \u226b (-g) = 0\n\u22a2 d \u226b (-g) = 0 \u226b (-g)\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d\u00b9 :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift e 0\nthis\u271d : d \u226b (-g) = 0\nthis : d = 0\n\u22a2 e = 0\n[PROOFSTEP]\ncalc\n e = biprod.lift e (0 : R \u27f6 Z) \u226b biprod.fst := by rw [biprod.lift_fst]\n _ = (d \u226b biprod.lift f (-g)) \u226b biprod.fst := by rw [\u2190 hd]\n _ = (0 \u226b biprod.lift f (-g)) \u226b biprod.fst := by rw [this]\n _ = 0 \u226b biprod.lift f (-g) \u226b biprod.fst := by rw [Category.assoc]\n _ = 0 := zero_comp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d\u00b9 :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift e 0\nthis\u271d : d \u226b (-g) = 0\nthis : d = 0\n\u22a2 e = biprod.lift e 0 \u226b biprod.fst\n[PROOFSTEP]\nrw [biprod.lift_fst]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d\u00b9 :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift e 0\nthis\u271d : d \u226b (-g) = 0\nthis : d = 0\n\u22a2 biprod.lift e 0 \u226b biprod.fst = (d \u226b biprod.lift f (-g)) \u226b biprod.fst\n[PROOFSTEP]\nrw [\u2190 hd]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d\u00b9 :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift e 0\nthis\u271d : d \u226b (-g) = 0\nthis : d = 0\n\u22a2 (d \u226b biprod.lift f (-g)) \u226b biprod.fst = (0 \u226b biprod.lift f (-g)) \u226b biprod.fst\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\nR : C\ne : R \u27f6 Y\nh : e \u226b pushout.inl = 0\nu : R \u27f6 Y \u229e Z := biprod.lift e 0\nhu : u \u226b BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis\u271d\u00b9 :\n IsLimit\n (KernelFork.of\u03b9 (biprod.lift f (-g))\n (_ : biprod.lift f (-g) \u226b Cofork.\u03c0 (BiproductToPushoutIsCokernel.biproductToPushoutCofork f g) = 0))\nd : R \u27f6 X\nhd : d \u226b biprod.lift f (-g) = biprod.lift e 0\nthis\u271d : d \u226b (-g) = 0\nthis : d = 0\n\u22a2 (0 \u226b biprod.lift f (-g)) \u226b biprod.fst = 0 \u226b biprod.lift f (-g) \u226b biprod.fst\n[PROOFSTEP]\nrw [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\ns : PushoutCocone f g\nhs : IsColimit s\n\u22a2 Mono (PushoutCocone.inr s)\n[PROOFSTEP]\nhaveI : Mono (NatTrans.app (colimit.cocone (span f g)).\u03b9 WalkingCospan.right) := Abelian.mono_pushout_of_mono_f f g\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono f\ns : PushoutCocone f g\nhs : IsColimit s\nthis : Mono (NatTrans.app (colimit.cocone (span f g)).\u03b9 WalkingCospan.right)\n\u22a2 Mono (PushoutCocone.inr s)\n[PROOFSTEP]\napply mono_of_mono_fac (IsColimit.comp_coconePointUniqueUpToIso_hom hs (colimit.isColimit _) _)\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\ns : PushoutCocone f g\nhs : IsColimit s\n\u22a2 Mono (PushoutCocone.inl s)\n[PROOFSTEP]\nhaveI : Mono (NatTrans.app (colimit.cocone (span f g)).\u03b9 WalkingCospan.left) := Abelian.mono_pushout_of_mono_g f g\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Abelian C\ninst\u271d\u00b9 : HasPushouts C\nW X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : Mono g\ns : PushoutCocone f g\nhs : IsColimit s\nthis : Mono (NatTrans.app (colimit.cocone (span f g)).\u03b9 WalkingCospan.left)\n\u22a2 Mono (PushoutCocone.inl s)\n[PROOFSTEP]\napply mono_of_mono_fac (IsColimit.comp_coconePointUniqueUpToIso_hom hs (colimit.isColimit _) _)\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : Abelian C\ninst\u271d\u00b2 : HasPushouts C\nW X Y Z : C\nf\u271d : X \u27f6 Y\ng\u271d : X \u27f6 Z\nf : X \u27f6 Y\ng : X \u27f6 Z\ng\u2081 : X \u27f6 W\ninst\u271d\u00b9 : Epi g\u2081\ng\u2082 : W \u27f6 Z\ninst\u271d : Mono g\u2082\nhg : g\u2081 \u226b g\u2082 = g\nf' : W \u27f6 Y\nhf : g\u2081 \u226b f' = f\nt : PushoutCocone f g\nht : IsColimit t\n\u22a2 Mono (PushoutCocone.inl t)\n[PROOFSTEP]\napply mono_inl_of_isColimit _ _ (PushoutCocone.isColimitOfFactors _ _ _ _ _ hf hg t ht)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nsrc\u271d : Preadditive C := preadditive\n\u22a2 {X Y : C} \u2192 (f : X \u27f6 Y) \u2192 [inst : Mono f] \u2192 NormalMono f\n[PROOFSTEP]\nintro _ _ f _\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : NonPreadditiveAbelian C\nsrc\u271d : Preadditive C := preadditive\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\ninst\u271d : Mono f\n\u22a2 NormalMono f\n[PROOFSTEP]\nconvert normalMonoOfMono f\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nsrc\u271d : Preadditive C := preadditive\n\u22a2 {X Y : C} \u2192 (f : X \u27f6 Y) \u2192 [inst : Epi f] \u2192 NormalEpi f\n[PROOFSTEP]\nintro _ _ f _\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : NonPreadditiveAbelian C\nsrc\u271d : Preadditive C := preadditive\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\ninst\u271d : Epi f\n\u22a2 NormalEpi f\n[PROOFSTEP]\nconvert normalEpiOfEpi f\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nsrc\u271d : Preadditive C := preadditive\n\u22a2 HasFiniteProducts C\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nsrc\u271d : Preadditive C := preadditive\n\u22a2 HasKernels C\n[PROOFSTEP]\nconvert (by infer_instance : Limits.HasKernels C)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nsrc\u271d : Preadditive C := preadditive\n\u22a2 HasKernels C\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nsrc\u271d : Preadditive C := preadditive\n\u22a2 HasCokernels C\n[PROOFSTEP]\nconvert (by infer_instance : Limits.HasCokernels C)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nsrc\u271d : Preadditive C := preadditive\n\u22a2 HasCokernels C\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Abelian.Basic", "llama_tokens": 44036, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.40768626088252297}} {"text": "[GOAL]\n\u03b9\u271d : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\n\u03b9 : Type u_4\nS : Set \u03b9\nhS : Set.Countable S\ns : (i : \u03b9) \u2192 i \u2208 S \u2192 Set \u03b1\n\u22a2 \u22c2 (i : \u03b9) (hi : i \u2208 S), s i hi \u2208 l \u2194 \u2200 (i : \u03b9) (hi : i \u2208 S), s i hi \u2208 l\n[PROOFSTEP]\nrw [biInter_eq_iInter]\n[GOAL]\n\u03b9\u271d : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\n\u03b9 : Type u_4\nS : Set \u03b9\nhS : Set.Countable S\ns : (i : \u03b9) \u2192 i \u2208 S \u2192 Set \u03b1\n\u22a2 \u22c2 (x : \u2191S), s \u2191x (_ : \u2191x \u2208 S) \u2208 l \u2194 \u2200 (i : \u03b9) (hi : i \u2208 S), s i hi \u2208 l\n[PROOFSTEP]\nhaveI := hS.toEncodable\n[GOAL]\n\u03b9\u271d : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\n\u03b9 : Type u_4\nS : Set \u03b9\nhS : Set.Countable S\ns : (i : \u03b9) \u2192 i \u2208 S \u2192 Set \u03b1\nthis : Encodable \u2191S\n\u22a2 \u22c2 (x : \u2191S), s \u2191x (_ : \u2191x \u2208 S) \u2208 l \u2194 \u2200 (i : \u03b9) (hi : i \u2208 S), s i hi \u2208 l\n[PROOFSTEP]\nexact countable_iInter_mem.trans Subtype.forall\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\ninst\u271d : Countable \u03b9\np : \u03b1 \u2192 \u03b9 \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : \u03b1) in l, \u2200 (i : \u03b9), p x i) \u2194 \u2200 (i : \u03b9), \u2200\u1da0 (x : \u03b1) in l, p x i\n[PROOFSTEP]\nsimpa only [Filter.Eventually, setOf_forall] using @countable_iInter_mem _ _ l _ _ fun i => {x | p x i}\n[GOAL]\n\u03b9\u271d : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\n\u03b9 : Type u_4\nS : Set \u03b9\nhS : Set.Countable S\np : \u03b1 \u2192 (i : \u03b9) \u2192 i \u2208 S \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : \u03b1) in l, \u2200 (i : \u03b9) (hi : i \u2208 S), p x i hi) \u2194 \u2200 (i : \u03b9) (hi : i \u2208 S), \u2200\u1da0 (x : \u03b1) in l, p x i hi\n[PROOFSTEP]\nsimpa only [Filter.Eventually, setOf_forall] using @countable_bInter_mem _ l _ _ _ hS fun i hi => {x | p x i hi}\n[GOAL]\n\u03b9\u271d : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\n\u03b9 : Type u_4\nS : Set \u03b9\nhS : Set.Countable S\ns t : (i : \u03b9) \u2192 i \u2208 S \u2192 Set \u03b1\nh : \u2200 (i : \u03b9) (hi : i \u2208 S), s i hi \u2264\u1da0[l] t i hi\n\u22a2 \u22c3 (i : \u03b9) (h : i \u2208 S), s i h \u2264\u1da0[l] \u22c3 (i : \u03b9) (h : i \u2208 S), t i h\n[PROOFSTEP]\nsimp only [biUnion_eq_iUnion]\n[GOAL]\n\u03b9\u271d : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\n\u03b9 : Type u_4\nS : Set \u03b9\nhS : Set.Countable S\ns t : (i : \u03b9) \u2192 i \u2208 S \u2192 Set \u03b1\nh : \u2200 (i : \u03b9) (hi : i \u2208 S), s i hi \u2264\u1da0[l] t i hi\n\u22a2 \u22c3 (x : \u2191S), s \u2191x (_ : \u2191x \u2208 S) \u2264\u1da0[l] \u22c3 (x : \u2191S), t \u2191x (_ : \u2191x \u2208 S)\n[PROOFSTEP]\nhaveI := hS.toEncodable\n[GOAL]\n\u03b9\u271d : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\n\u03b9 : Type u_4\nS : Set \u03b9\nhS : Set.Countable S\ns t : (i : \u03b9) \u2192 i \u2208 S \u2192 Set \u03b1\nh : \u2200 (i : \u03b9) (hi : i \u2208 S), s i hi \u2264\u1da0[l] t i hi\nthis : Encodable \u2191S\n\u22a2 \u22c3 (x : \u2191S), s \u2191x (_ : \u2191x \u2208 S) \u2264\u1da0[l] \u22c3 (x : \u2191S), t \u2191x (_ : \u2191x \u2208 S)\n[PROOFSTEP]\nexact EventuallyLE.countable_iUnion fun i => h i i.2\n[GOAL]\n\u03b9\u271d : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\n\u03b9 : Type u_4\nS : Set \u03b9\nhS : Set.Countable S\ns t : (i : \u03b9) \u2192 i \u2208 S \u2192 Set \u03b1\nh : \u2200 (i : \u03b9) (hi : i \u2208 S), s i hi \u2264\u1da0[l] t i hi\n\u22a2 \u22c2 (i : \u03b9) (h : i \u2208 S), s i h \u2264\u1da0[l] \u22c2 (i : \u03b9) (h : i \u2208 S), t i h\n[PROOFSTEP]\nsimp only [biInter_eq_iInter]\n[GOAL]\n\u03b9\u271d : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\n\u03b9 : Type u_4\nS : Set \u03b9\nhS : Set.Countable S\ns t : (i : \u03b9) \u2192 i \u2208 S \u2192 Set \u03b1\nh : \u2200 (i : \u03b9) (hi : i \u2208 S), s i hi \u2264\u1da0[l] t i hi\n\u22a2 \u22c2 (x : \u2191S), s \u2191x (_ : \u2191x \u2208 S) \u2264\u1da0[l] \u22c2 (x : \u2191S), t \u2191x (_ : \u2191x \u2208 S)\n[PROOFSTEP]\nhaveI := hS.toEncodable\n[GOAL]\n\u03b9\u271d : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\n\u03b9 : Type u_4\nS : Set \u03b9\nhS : Set.Countable S\ns t : (i : \u03b9) \u2192 i \u2208 S \u2192 Set \u03b1\nh : \u2200 (i : \u03b9) (hi : i \u2208 S), s i hi \u2264\u1da0[l] t i hi\nthis : Encodable \u2191S\n\u22a2 \u22c2 (x : \u2191S), s \u2191x (_ : \u2191x \u2208 S) \u2264\u1da0[l] \u22c2 (x : \u2191S), t \u2191x (_ : \u2191x \u2208 S)\n[PROOFSTEP]\nexact EventuallyLE.countable_iInter fun i => h i i.2\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\n\u22a2 CountableInterFilter \u22a5\n[PROOFSTEP]\nrw [\u2190 principal_empty]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\n\u22a2 CountableInterFilter (\ud835\udcdf \u2205)\n[PROOFSTEP]\napply countableInterFilter_principal\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\n\u22a2 CountableInterFilter \u22a4\n[PROOFSTEP]\nrw [\u2190 principal_univ]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\n\u22a2 CountableInterFilter (\ud835\udcdf univ)\n[PROOFSTEP]\napply countableInterFilter_principal\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl\u271d : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u271d\nl : Filter \u03b2\ninst\u271d : CountableInterFilter l\nf : \u03b1 \u2192 \u03b2\n\u22a2 CountableInterFilter (comap f l)\n[PROOFSTEP]\nrefine' \u27e8fun S hSc hS => _\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl\u271d : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u271d\nl : Filter \u03b2\ninst\u271d : CountableInterFilter l\nf : \u03b1 \u2192 \u03b2\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhS : \u2200 (s : Set \u03b1), s \u2208 S \u2192 s \u2208 comap f l\n\u22a2 \u22c2\u2080 S \u2208 comap f l\n[PROOFSTEP]\nchoose! t htl ht using hS\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl\u271d : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u271d\nl : Filter \u03b2\ninst\u271d : CountableInterFilter l\nf : \u03b1 \u2192 \u03b2\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nt : Set \u03b1 \u2192 Set \u03b2\nhtl : \u2200 (s : Set \u03b1), s \u2208 S \u2192 t s \u2208 l\nht : \u2200 (s : Set \u03b1), s \u2208 S \u2192 f \u207b\u00b9' t s \u2286 s\n\u22a2 \u22c2\u2080 S \u2208 comap f l\n[PROOFSTEP]\nhave : (\u22c2 s \u2208 S, t s) \u2208 l := (countable_bInter_mem hSc).2 htl\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl\u271d : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u271d\nl : Filter \u03b2\ninst\u271d : CountableInterFilter l\nf : \u03b1 \u2192 \u03b2\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nt : Set \u03b1 \u2192 Set \u03b2\nhtl : \u2200 (s : Set \u03b1), s \u2208 S \u2192 t s \u2208 l\nht : \u2200 (s : Set \u03b1), s \u2208 S \u2192 f \u207b\u00b9' t s \u2286 s\nthis : \u22c2 (s : Set \u03b1) (_ : s \u2208 S), t s \u2208 l\n\u22a2 \u22c2\u2080 S \u2208 comap f l\n[PROOFSTEP]\nrefine' \u27e8_, this, _\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl\u271d : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u271d\nl : Filter \u03b2\ninst\u271d : CountableInterFilter l\nf : \u03b1 \u2192 \u03b2\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nt : Set \u03b1 \u2192 Set \u03b2\nhtl : \u2200 (s : Set \u03b1), s \u2208 S \u2192 t s \u2208 l\nht : \u2200 (s : Set \u03b1), s \u2208 S \u2192 f \u207b\u00b9' t s \u2286 s\nthis : \u22c2 (s : Set \u03b1) (_ : s \u2208 S), t s \u2208 l\n\u22a2 f \u207b\u00b9' \u22c2 (s : Set \u03b1) (_ : s \u2208 S), t s \u2286 \u22c2\u2080 S\n[PROOFSTEP]\nsimpa [preimage_iInter] using iInter\u2082_mono ht\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl\u271d : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u271d\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf : \u03b1 \u2192 \u03b2\n\u22a2 CountableInterFilter (map f l)\n[PROOFSTEP]\nrefine' \u27e8fun S hSc hS => _\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl\u271d : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u271d\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf : \u03b1 \u2192 \u03b2\nS : Set (Set \u03b2)\nhSc : Set.Countable S\nhS : \u2200 (s : Set \u03b2), s \u2208 S \u2192 s \u2208 map f l\n\u22a2 \u22c2\u2080 S \u2208 map f l\n[PROOFSTEP]\nsimp only [mem_map, sInter_eq_biInter, preimage_iInter\u2082] at hS \u22a2\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl\u271d : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u271d\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf : \u03b1 \u2192 \u03b2\nS : Set (Set \u03b2)\nhSc : Set.Countable S\nhS : \u2200 (s : Set \u03b2), s \u2208 S \u2192 f \u207b\u00b9' s \u2208 l\n\u22a2 \u22c2 (i : Set \u03b2) (_ : i \u2208 S), f \u207b\u00b9' i \u2208 l\n[PROOFSTEP]\nexact (countable_bInter_mem hSc).2 hS\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b2 : CountableInterFilter l\nl\u2081 l\u2082 : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u2081\ninst\u271d : CountableInterFilter l\u2082\n\u22a2 CountableInterFilter (l\u2081 \u2293 l\u2082)\n[PROOFSTEP]\nrefine' \u27e8fun S hSc hS => _\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b2 : CountableInterFilter l\nl\u2081 l\u2082 : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u2081\ninst\u271d : CountableInterFilter l\u2082\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhS : \u2200 (s : Set \u03b1), s \u2208 S \u2192 s \u2208 l\u2081 \u2293 l\u2082\n\u22a2 \u22c2\u2080 S \u2208 l\u2081 \u2293 l\u2082\n[PROOFSTEP]\nchoose s hs t ht hst using hS\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b2 : CountableInterFilter l\nl\u2081 l\u2082 : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u2081\ninst\u271d : CountableInterFilter l\u2082\nS : Set (Set \u03b1)\nhSc : Set.Countable S\ns : (s : Set \u03b1) \u2192 s \u2208 S \u2192 Set \u03b1\nhs : \u2200 (s_1 : Set \u03b1) (a : s_1 \u2208 S), s s_1 a \u2208 l\u2081\nt : (s : Set \u03b1) \u2192 s \u2208 S \u2192 Set \u03b1\nht : \u2200 (s : Set \u03b1) (a : s \u2208 S), t s a \u2208 l\u2082\nhst : \u2200 (s_1 : Set \u03b1) (a : s_1 \u2208 S), s_1 = s s_1 a \u2229 t s_1 a\n\u22a2 \u22c2\u2080 S \u2208 l\u2081 \u2293 l\u2082\n[PROOFSTEP]\nreplace hs : (\u22c2 i \u2208 S, s i \u2039_\u203a) \u2208 l\u2081 := (countable_bInter_mem hSc).2 hs\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b2 : CountableInterFilter l\nl\u2081 l\u2082 : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u2081\ninst\u271d : CountableInterFilter l\u2082\nS : Set (Set \u03b1)\nhSc : Set.Countable S\ns t : (s : Set \u03b1) \u2192 s \u2208 S \u2192 Set \u03b1\nht : \u2200 (s : Set \u03b1) (a : s \u2208 S), t s a \u2208 l\u2082\nhst : \u2200 (s_1 : Set \u03b1) (a : s_1 \u2208 S), s_1 = s s_1 a \u2229 t s_1 a\nhs : \u22c2 (i : Set \u03b1) (h : i \u2208 S), s i h \u2208 l\u2081\n\u22a2 \u22c2\u2080 S \u2208 l\u2081 \u2293 l\u2082\n[PROOFSTEP]\nreplace ht : (\u22c2 i \u2208 S, t i \u2039_\u203a) \u2208 l\u2082 := (countable_bInter_mem hSc).2 ht\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b2 : CountableInterFilter l\nl\u2081 l\u2082 : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u2081\ninst\u271d : CountableInterFilter l\u2082\nS : Set (Set \u03b1)\nhSc : Set.Countable S\ns t : (s : Set \u03b1) \u2192 s \u2208 S \u2192 Set \u03b1\nhst : \u2200 (s_1 : Set \u03b1) (a : s_1 \u2208 S), s_1 = s s_1 a \u2229 t s_1 a\nhs : \u22c2 (i : Set \u03b1) (h : i \u2208 S), s i h \u2208 l\u2081\nht : \u22c2 (i : Set \u03b1) (h : i \u2208 S), t i h \u2208 l\u2082\n\u22a2 \u22c2\u2080 S \u2208 l\u2081 \u2293 l\u2082\n[PROOFSTEP]\nrefine' mem_of_superset (inter_mem_inf hs ht) (subset_sInter fun i hi => _)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b2 : CountableInterFilter l\nl\u2081 l\u2082 : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u2081\ninst\u271d : CountableInterFilter l\u2082\nS : Set (Set \u03b1)\nhSc : Set.Countable S\ns t : (s : Set \u03b1) \u2192 s \u2208 S \u2192 Set \u03b1\nhst : \u2200 (s_1 : Set \u03b1) (a : s_1 \u2208 S), s_1 = s s_1 a \u2229 t s_1 a\nhs : \u22c2 (i : Set \u03b1) (h : i \u2208 S), s i h \u2208 l\u2081\nht : \u22c2 (i : Set \u03b1) (h : i \u2208 S), t i h \u2208 l\u2082\ni : Set \u03b1\nhi : i \u2208 S\n\u22a2 (\u22c2 (i : Set \u03b1) (h : i \u2208 S), s i h) \u2229 \u22c2 (i : Set \u03b1) (h : i \u2208 S), t i h \u2286 i\n[PROOFSTEP]\nrw [hst i hi]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b2 : CountableInterFilter l\nl\u2081 l\u2082 : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u2081\ninst\u271d : CountableInterFilter l\u2082\nS : Set (Set \u03b1)\nhSc : Set.Countable S\ns t : (s : Set \u03b1) \u2192 s \u2208 S \u2192 Set \u03b1\nhst : \u2200 (s_1 : Set \u03b1) (a : s_1 \u2208 S), s_1 = s s_1 a \u2229 t s_1 a\nhs : \u22c2 (i : Set \u03b1) (h : i \u2208 S), s i h \u2208 l\u2081\nht : \u22c2 (i : Set \u03b1) (h : i \u2208 S), t i h \u2208 l\u2082\ni : Set \u03b1\nhi : i \u2208 S\n\u22a2 (\u22c2 (i : Set \u03b1) (h : i \u2208 S), s i h) \u2229 \u22c2 (i : Set \u03b1) (h : i \u2208 S), t i h \u2286 s i hi \u2229 t i hi\n[PROOFSTEP]\napply inter_subset_inter\n[GOAL]\ncase h\u2081\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b2 : CountableInterFilter l\nl\u2081 l\u2082 : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u2081\ninst\u271d : CountableInterFilter l\u2082\nS : Set (Set \u03b1)\nhSc : Set.Countable S\ns t : (s : Set \u03b1) \u2192 s \u2208 S \u2192 Set \u03b1\nhst : \u2200 (s_1 : Set \u03b1) (a : s_1 \u2208 S), s_1 = s s_1 a \u2229 t s_1 a\nhs : \u22c2 (i : Set \u03b1) (h : i \u2208 S), s i h \u2208 l\u2081\nht : \u22c2 (i : Set \u03b1) (h : i \u2208 S), t i h \u2208 l\u2082\ni : Set \u03b1\nhi : i \u2208 S\n\u22a2 \u22c2 (i : Set \u03b1) (h : i \u2208 S), s i h \u2286 s i hi\n[PROOFSTEP]\nexact iInter_subset_of_subset i (iInter_subset _ _)\n[GOAL]\ncase h\u2082\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b2 : CountableInterFilter l\nl\u2081 l\u2082 : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u2081\ninst\u271d : CountableInterFilter l\u2082\nS : Set (Set \u03b1)\nhSc : Set.Countable S\ns t : (s : Set \u03b1) \u2192 s \u2208 S \u2192 Set \u03b1\nhst : \u2200 (s_1 : Set \u03b1) (a : s_1 \u2208 S), s_1 = s s_1 a \u2229 t s_1 a\nhs : \u22c2 (i : Set \u03b1) (h : i \u2208 S), s i h \u2208 l\u2081\nht : \u22c2 (i : Set \u03b1) (h : i \u2208 S), t i h \u2208 l\u2082\ni : Set \u03b1\nhi : i \u2208 S\n\u22a2 \u22c2 (i : Set \u03b1) (h : i \u2208 S), t i h \u2286 t i hi\n[PROOFSTEP]\nexact iInter_subset_of_subset i (iInter_subset _ _)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b2 : CountableInterFilter l\nl\u2081 l\u2082 : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u2081\ninst\u271d : CountableInterFilter l\u2082\n\u22a2 CountableInterFilter (l\u2081 \u2294 l\u2082)\n[PROOFSTEP]\nrefine' \u27e8fun S hSc hS => \u27e8_, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b2 : CountableInterFilter l\nl\u2081 l\u2082 : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u2081\ninst\u271d : CountableInterFilter l\u2082\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhS : \u2200 (s : Set \u03b1), s \u2208 S \u2192 s \u2208 l\u2081 \u2294 l\u2082\n\u22a2 \u22c2\u2080 S \u2208 l\u2081.sets\n[PROOFSTEP]\nrefine' (countable_sInter_mem hSc).2 fun s hs => _\n[GOAL]\ncase refine'_2\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b2 : CountableInterFilter l\nl\u2081 l\u2082 : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u2081\ninst\u271d : CountableInterFilter l\u2082\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhS : \u2200 (s : Set \u03b1), s \u2208 S \u2192 s \u2208 l\u2081 \u2294 l\u2082\n\u22a2 \u22c2\u2080 S \u2208 l\u2082.sets\n[PROOFSTEP]\nrefine' (countable_sInter_mem hSc).2 fun s hs => _\n[GOAL]\ncase refine'_1\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b2 : CountableInterFilter l\nl\u2081 l\u2082 : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u2081\ninst\u271d : CountableInterFilter l\u2082\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhS : \u2200 (s : Set \u03b1), s \u2208 S \u2192 s \u2208 l\u2081 \u2294 l\u2082\ns : Set \u03b1\nhs : s \u2208 S\n\u22a2 s \u2208 l\u2081\ncase refine'_2\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b2 : CountableInterFilter l\nl\u2081 l\u2082 : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\u2081\ninst\u271d : CountableInterFilter l\u2082\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhS : \u2200 (s : Set \u03b1), s \u2208 S \u2192 s \u2208 l\u2081 \u2294 l\u2082\ns : Set \u03b1\nhs : s \u2208 S\n\u22a2 s \u2208 l\u2082\n[PROOFSTEP]\nexacts [(hS s hs).1, (hS s hs).2]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\n\u22a2 CountableInterFilter (countableGenerate g)\n[PROOFSTEP]\ndelta countableGenerate\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\n\u22a2 CountableInterFilter\n (ofCountableInter (CountableGenerateSets g)\n (_ :\n \u2200 (x : Set (Set \u03b1)),\n Set.Countable x \u2192 (\u2200 (s : Set \u03b1), s \u2208 x \u2192 CountableGenerateSets g s) \u2192 CountableGenerateSets g (\u22c2\u2080 x))\n (_ : \u2200 (x x_1 : Set \u03b1), CountableGenerateSets g x \u2192 x \u2286 x_1 \u2192 CountableGenerateSets g x_1))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns : Set \u03b1\n\u22a2 s \u2208 countableGenerate g \u2194 \u2203 S, S \u2286 g \u2227 Set.Countable S \u2227 \u22c2\u2080 S \u2286 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns : Set \u03b1\n\u22a2 s \u2208 countableGenerate g \u2192 \u2203 S, S \u2286 g \u2227 Set.Countable S \u2227 \u22c2\u2080 S \u2286 s\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns : Set \u03b1\n\u22a2 (\u2203 S, S \u2286 g \u2227 Set.Countable S \u2227 \u22c2\u2080 S \u2286 s) \u2192 s \u2208 countableGenerate g\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns : Set \u03b1\nh : s \u2208 countableGenerate g\n\u22a2 \u2203 S, S \u2286 g \u2227 Set.Countable S \u2227 \u22c2\u2080 S \u2286 s\n[PROOFSTEP]\ninduction' h with s hs s t _ st ih S Sct _ ih\n[GOAL]\ncase mp.basic\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns\u271d s : Set \u03b1\nhs : s \u2208 g\n\u22a2 \u2203 S, S \u2286 g \u2227 Set.Countable S \u2227 \u22c2\u2080 S \u2286 s\n[PROOFSTEP]\nexact \u27e8{ s }, by simp [hs, subset_refl]\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns\u271d s : Set \u03b1\nhs : s \u2208 g\n\u22a2 {s} \u2286 g \u2227 Set.Countable {s} \u2227 \u22c2\u2080 {s} \u2286 s\n[PROOFSTEP]\nsimp [hs, subset_refl]\n[GOAL]\ncase mp.univ\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns : Set \u03b1\n\u22a2 \u2203 S, S \u2286 g \u2227 Set.Countable S \u2227 \u22c2\u2080 S \u2286 univ\n[PROOFSTEP]\nexact \u27e8\u2205, by simp\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns : Set \u03b1\n\u22a2 \u2205 \u2286 g \u2227 Set.Countable \u2205 \u2227 \u22c2\u2080 \u2205 \u2286 univ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mp.superset\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns\u271d s t : Set \u03b1\na\u271d : CountableGenerateSets g s\nst : s \u2286 t\nih : \u2203 S, S \u2286 g \u2227 Set.Countable S \u2227 \u22c2\u2080 S \u2286 s\n\u22a2 \u2203 S, S \u2286 g \u2227 Set.Countable S \u2227 \u22c2\u2080 S \u2286 t\n[PROOFSTEP]\nrefine' Exists.imp (fun S => _) ih\n[GOAL]\ncase mp.superset\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns\u271d s t : Set \u03b1\na\u271d : CountableGenerateSets g s\nst : s \u2286 t\nih : \u2203 S, S \u2286 g \u2227 Set.Countable S \u2227 \u22c2\u2080 S \u2286 s\nS : Set (Set \u03b1)\n\u22a2 S \u2286 g \u2227 Set.Countable S \u2227 \u22c2\u2080 S \u2286 s \u2192 S \u2286 g \u2227 Set.Countable S \u2227 \u22c2\u2080 S \u2286 t\n[PROOFSTEP]\ntauto\n[GOAL]\ncase mp.sInter\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns : Set \u03b1\nS : Set (Set \u03b1)\nSct : Set.Countable S\na\u271d : \u2200 (s : Set \u03b1), s \u2208 S \u2192 CountableGenerateSets g s\nih : \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2203 S, S \u2286 g \u2227 Set.Countable S \u2227 \u22c2\u2080 S \u2286 s\n\u22a2 \u2203 S_1, S_1 \u2286 g \u2227 Set.Countable S_1 \u2227 \u22c2\u2080 S_1 \u2286 \u22c2\u2080 S\n[PROOFSTEP]\nchoose T Tg Tct hT using ih\n[GOAL]\ncase mp.sInter\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns : Set \u03b1\nS : Set (Set \u03b1)\nSct : Set.Countable S\na\u271d : \u2200 (s : Set \u03b1), s \u2208 S \u2192 CountableGenerateSets g s\nT : (s : Set \u03b1) \u2192 s \u2208 S \u2192 Set (Set \u03b1)\nTg : \u2200 (s : Set \u03b1) (a : s \u2208 S), T s a \u2286 g\nTct : \u2200 (s : Set \u03b1) (a : s \u2208 S), Set.Countable (T s a)\nhT : \u2200 (s : Set \u03b1) (a : s \u2208 S), \u22c2\u2080 T s a \u2286 s\n\u22a2 \u2203 S_1, S_1 \u2286 g \u2227 Set.Countable S_1 \u2227 \u22c2\u2080 S_1 \u2286 \u22c2\u2080 S\n[PROOFSTEP]\nrefine' \u27e8\u22c3 (s) (H : s \u2208 S), T s H, by simpa, Sct.biUnion Tct, _\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns : Set \u03b1\nS : Set (Set \u03b1)\nSct : Set.Countable S\na\u271d : \u2200 (s : Set \u03b1), s \u2208 S \u2192 CountableGenerateSets g s\nT : (s : Set \u03b1) \u2192 s \u2208 S \u2192 Set (Set \u03b1)\nTg : \u2200 (s : Set \u03b1) (a : s \u2208 S), T s a \u2286 g\nTct : \u2200 (s : Set \u03b1) (a : s \u2208 S), Set.Countable (T s a)\nhT : \u2200 (s : Set \u03b1) (a : s \u2208 S), \u22c2\u2080 T s a \u2286 s\n\u22a2 \u22c3 (s : Set \u03b1) (H : s \u2208 S), T s H \u2286 g\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase mp.sInter\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns : Set \u03b1\nS : Set (Set \u03b1)\nSct : Set.Countable S\na\u271d : \u2200 (s : Set \u03b1), s \u2208 S \u2192 CountableGenerateSets g s\nT : (s : Set \u03b1) \u2192 s \u2208 S \u2192 Set (Set \u03b1)\nTg : \u2200 (s : Set \u03b1) (a : s \u2208 S), T s a \u2286 g\nTct : \u2200 (s : Set \u03b1) (a : s \u2208 S), Set.Countable (T s a)\nhT : \u2200 (s : Set \u03b1) (a : s \u2208 S), \u22c2\u2080 T s a \u2286 s\n\u22a2 \u22c2\u2080 \u22c3 (s : Set \u03b1) (H : s \u2208 S), T s H \u2286 \u22c2\u2080 S\n[PROOFSTEP]\napply subset_sInter\n[GOAL]\ncase mp.sInter.h\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns : Set \u03b1\nS : Set (Set \u03b1)\nSct : Set.Countable S\na\u271d : \u2200 (s : Set \u03b1), s \u2208 S \u2192 CountableGenerateSets g s\nT : (s : Set \u03b1) \u2192 s \u2208 S \u2192 Set (Set \u03b1)\nTg : \u2200 (s : Set \u03b1) (a : s \u2208 S), T s a \u2286 g\nTct : \u2200 (s : Set \u03b1) (a : s \u2208 S), Set.Countable (T s a)\nhT : \u2200 (s : Set \u03b1) (a : s \u2208 S), \u22c2\u2080 T s a \u2286 s\n\u22a2 \u2200 (t' : Set \u03b1), t' \u2208 S \u2192 \u22c2\u2080 \u22c3 (s : Set \u03b1) (H : s \u2208 S), T s H \u2286 t'\n[PROOFSTEP]\nintro s H\n[GOAL]\ncase mp.sInter.h\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns\u271d : Set \u03b1\nS : Set (Set \u03b1)\nSct : Set.Countable S\na\u271d : \u2200 (s : Set \u03b1), s \u2208 S \u2192 CountableGenerateSets g s\nT : (s : Set \u03b1) \u2192 s \u2208 S \u2192 Set (Set \u03b1)\nTg : \u2200 (s : Set \u03b1) (a : s \u2208 S), T s a \u2286 g\nTct : \u2200 (s : Set \u03b1) (a : s \u2208 S), Set.Countable (T s a)\nhT : \u2200 (s : Set \u03b1) (a : s \u2208 S), \u22c2\u2080 T s a \u2286 s\ns : Set \u03b1\nH : s \u2208 S\n\u22a2 \u22c2\u2080 \u22c3 (s : Set \u03b1) (H : s \u2208 S), T s H \u2286 s\n[PROOFSTEP]\nrefine' subset_trans (sInter_subset_sInter (subset_iUnion\u2082 s H)) (hT s H)\n[GOAL]\ncase mpr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns : Set \u03b1\nh : \u2203 S, S \u2286 g \u2227 Set.Countable S \u2227 \u22c2\u2080 S \u2286 s\n\u22a2 s \u2208 countableGenerate g\n[PROOFSTEP]\nrcases h with \u27e8S, Sg, Sct, hS\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns : Set \u03b1\nS : Set (Set \u03b1)\nSg : S \u2286 g\nSct : Set.Countable S\nhS : \u22c2\u2080 S \u2286 s\n\u22a2 s \u2208 countableGenerate g\n[PROOFSTEP]\nrefine' mem_of_superset ((countable_sInter_mem Sct).mpr _) hS\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns : Set \u03b1\nS : Set (Set \u03b1)\nSg : S \u2286 g\nSct : Set.Countable S\nhS : \u22c2\u2080 S \u2286 s\n\u22a2 \u2200 (s : Set \u03b1), s \u2208 S \u2192 s \u2208 countableGenerate g\n[PROOFSTEP]\nintro s H\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\ns\u271d : Set \u03b1\nS : Set (Set \u03b1)\nSg : S \u2286 g\nSct : Set.Countable S\nhS : \u22c2\u2080 S \u2286 s\u271d\ns : Set \u03b1\nH : s \u2208 S\n\u22a2 s \u2208 countableGenerate g\n[PROOFSTEP]\nexact CountableGenerateSets.basic (Sg H)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\ng : Set (Set \u03b1)\nf : Filter \u03b1\ninst\u271d : CountableInterFilter f\n\u22a2 f \u2264 countableGenerate g \u2194 g \u2286 f.sets\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\ng : Set (Set \u03b1)\nf : Filter \u03b1\ninst\u271d : CountableInterFilter f\n\u22a2 f \u2264 countableGenerate g \u2192 g \u2286 f.sets\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\ng : Set (Set \u03b1)\nf : Filter \u03b1\ninst\u271d : CountableInterFilter f\n\u22a2 g \u2286 f.sets \u2192 f \u2264 countableGenerate g\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\ng : Set (Set \u03b1)\nf : Filter \u03b1\ninst\u271d : CountableInterFilter f\nh : f \u2264 countableGenerate g\n\u22a2 g \u2286 f.sets\n[PROOFSTEP]\nexact subset_trans (fun s => CountableGenerateSets.basic) h\n[GOAL]\ncase mpr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\ng : Set (Set \u03b1)\nf : Filter \u03b1\ninst\u271d : CountableInterFilter f\nh : g \u2286 f.sets\n\u22a2 f \u2264 countableGenerate g\n[PROOFSTEP]\nintro s hs\n[GOAL]\ncase mpr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\ng : Set (Set \u03b1)\nf : Filter \u03b1\ninst\u271d : CountableInterFilter f\nh : g \u2286 f.sets\ns : Set \u03b1\nhs : s \u2208 countableGenerate g\n\u22a2 s \u2208 f\n[PROOFSTEP]\ninduction' hs with s hs s t _ st ih S Sct _ ih\n[GOAL]\ncase mpr.basic\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\ng : Set (Set \u03b1)\nf : Filter \u03b1\ninst\u271d : CountableInterFilter f\nh : g \u2286 f.sets\ns\u271d s : Set \u03b1\nhs : s \u2208 g\n\u22a2 s \u2208 f\n[PROOFSTEP]\nexact h hs\n[GOAL]\ncase mpr.univ\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\ng : Set (Set \u03b1)\nf : Filter \u03b1\ninst\u271d : CountableInterFilter f\nh : g \u2286 f.sets\ns : Set \u03b1\n\u22a2 univ \u2208 f\n[PROOFSTEP]\nexact univ_mem\n[GOAL]\ncase mpr.superset\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\ng : Set (Set \u03b1)\nf : Filter \u03b1\ninst\u271d : CountableInterFilter f\nh : g \u2286 f.sets\ns\u271d s t : Set \u03b1\na\u271d : CountableGenerateSets g s\nst : s \u2286 t\nih : s \u2208 f\n\u22a2 t \u2208 f\n[PROOFSTEP]\nexact mem_of_superset ih st\n[GOAL]\ncase mpr.sInter\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d\u00b9 : CountableInterFilter l\ng : Set (Set \u03b1)\nf : Filter \u03b1\ninst\u271d : CountableInterFilter f\nh : g \u2286 f.sets\ns : Set \u03b1\nS : Set (Set \u03b1)\nSct : Set.Countable S\na\u271d : \u2200 (s : Set \u03b1), s \u2208 S \u2192 CountableGenerateSets g s\nih : \u2200 (s : Set \u03b1), s \u2208 S \u2192 s \u2208 f\n\u22a2 \u22c2\u2080 S \u2208 f\n[PROOFSTEP]\nexact (countable_sInter_mem Sct).mpr ih\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\n\u22a2 IsGreatest {f | CountableInterFilter f \u2227 g \u2286 f.sets} (countableGenerate g)\n[PROOFSTEP]\nrefine' \u27e8\u27e8inferInstance, fun s => CountableGenerateSets.basic\u27e9, _\u27e9\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\n\u22a2 countableGenerate g \u2208 upperBounds {f | CountableInterFilter f \u2227 g \u2286 f.sets}\n[PROOFSTEP]\nrintro f \u27e8fct, hf\u27e9\n[GOAL]\ncase intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\ng : Set (Set \u03b1)\nf : Filter \u03b1\nfct : CountableInterFilter f\nhf : g \u2286 f.sets\n\u22a2 f \u2264 countableGenerate g\n[PROOFSTEP]\nrwa [@le_countableGenerate_iff_of_countableInterFilter _ _ _ fct]\n", "meta": {"mathlib_filename": "Mathlib.Order.Filter.CountableInter", "llama_tokens": 11770, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743505760728, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.4075454492074489}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\n\u22a2 MeasurableSet {x | Integrable (f x)}\n[PROOFSTEP]\nsimp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\n\u22a2 MeasurableSet {x | HasFiniteIntegral (f x)}\n[PROOFSTEP]\nexact measurableSet_lt (Measurable.lintegral_prod_right hf.ennnorm) measurable_const\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd\n[PROOFSTEP]\nborelize E\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd\n[PROOFSTEP]\nhaveI : SeparableSpace (range (uncurry f) \u222a {0} : Set E) := hf.separableSpace_range_union_singleton\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd\n[PROOFSTEP]\nlet s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E := SimpleFunc.approxOn _ hf.measurable (range (uncurry f) \u222a {0}) 0 (by simp)\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\n\u22a2 0 \u2208 range (uncurry f) \u222a {0}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd\n[PROOFSTEP]\nlet s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => (s n).comp (Prod.mk x) measurable_prod_mk_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd\n[PROOFSTEP]\nlet f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => {x | Integrable (f x) \u03bd}.indicator fun x => (s' n x).integral \u03bd\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd\n[PROOFSTEP]\nhave hf' : \u2200 n, StronglyMeasurable (f' n) := by\n intro n; refine' StronglyMeasurable.indicator _ (measurableSet_integrable hf)\n have : \u2200 x, ((s' n x).range.filter fun x => x \u2260 0) \u2286 (s n).range :=\n by\n intro x; refine' Finset.Subset.trans (Finset.filter_subset _ _) _; intro y\n simp_rw [SimpleFunc.mem_range]; rintro \u27e8z, rfl\u27e9; exact \u27e8(x, z), rfl\u27e9\n simp only [SimpleFunc.integral_eq_sum_of_subset (this _)]\n refine' Finset.stronglyMeasurable_sum _ fun x _ => _\n refine' (Measurable.ennreal_toReal _).stronglyMeasurable.smul_const _\n simp (config := { singlePass := true }) only [SimpleFunc.coe_comp, preimage_comp]\n apply measurable_measure_prod_mk_left\n exact (s n).measurableSet_fiber x\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\n\u22a2 \u2200 (n : \u2115), StronglyMeasurable (f' n)\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nn : \u2115\n\u22a2 StronglyMeasurable (f' n)\n[PROOFSTEP]\nrefine' StronglyMeasurable.indicator _ (measurableSet_integrable hf)\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nn : \u2115\n\u22a2 StronglyMeasurable fun x => SimpleFunc.integral \u03bd (s' n x)\n[PROOFSTEP]\nhave : \u2200 x, ((s' n x).range.filter fun x => x \u2260 0) \u2286 (s n).range :=\n by\n intro x; refine' Finset.Subset.trans (Finset.filter_subset _ _) _; intro y\n simp_rw [SimpleFunc.mem_range]; rintro \u27e8z, rfl\u27e9; exact \u27e8(x, z), rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nn : \u2115\n\u22a2 \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nn : \u2115\nx : \u03b1\n\u22a2 Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\n[PROOFSTEP]\nrefine' Finset.Subset.trans (Finset.filter_subset _ _) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nn : \u2115\nx : \u03b1\n\u22a2 SimpleFunc.range (s' n x) \u2286 SimpleFunc.range (s n)\n[PROOFSTEP]\nintro y\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nn : \u2115\nx : \u03b1\ny : E\n\u22a2 y \u2208 SimpleFunc.range (s' n x) \u2192 y \u2208 SimpleFunc.range (s n)\n[PROOFSTEP]\nsimp_rw [SimpleFunc.mem_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nn : \u2115\nx : \u03b1\ny : E\n\u22a2 y \u2208\n range\n \u2191(SimpleFunc.comp\n (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n (Prod.mk x) (_ : Measurable (Prod.mk x))) \u2192\n y \u2208\n range\n \u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n[PROOFSTEP]\nrintro \u27e8z, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nn : \u2115\nx : \u03b1\nz : \u03b2\n\u22a2 \u2191(SimpleFunc.comp\n (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n (Prod.mk x) (_ : Measurable (Prod.mk x)))\n z \u2208\n range\n \u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n[PROOFSTEP]\nexact \u27e8(x, z), rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\n\u22a2 StronglyMeasurable fun x => SimpleFunc.integral \u03bd (s' n x)\n[PROOFSTEP]\nsimp only [SimpleFunc.integral_eq_sum_of_subset (this _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\n\u22a2 StronglyMeasurable fun x =>\n Finset.sum\n (SimpleFunc.range\n (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n))\n fun x_1 =>\n ENNReal.toReal\n (\u2191\u2191\u03bd\n (\u2191(SimpleFunc.comp\n (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n (Prod.mk x) (_ : Measurable (Prod.mk x))) \u207b\u00b9'\n {x_1})) \u2022\n x_1\n[PROOFSTEP]\nrefine' Finset.stronglyMeasurable_sum _ fun x _ => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\nx : E\nx\u271d :\n x \u2208\n SimpleFunc.range\n (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n\u22a2 StronglyMeasurable fun x_1 =>\n ENNReal.toReal\n (\u2191\u2191\u03bd\n (\u2191(SimpleFunc.comp\n (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n (Prod.mk x_1) (_ : Measurable (Prod.mk x_1))) \u207b\u00b9'\n {x})) \u2022\n x\n[PROOFSTEP]\nrefine' (Measurable.ennreal_toReal _).stronglyMeasurable.smul_const _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\nx : E\nx\u271d :\n x \u2208\n SimpleFunc.range\n (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n\u22a2 Measurable fun x_1 =>\n \u2191\u2191\u03bd\n (\u2191(SimpleFunc.comp\n (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n (Prod.mk x_1) (_ : Measurable (Prod.mk x_1))) \u207b\u00b9'\n {x})\n[PROOFSTEP]\nsimp (config := { singlePass := true }) only [SimpleFunc.coe_comp, preimage_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\nx : E\nx\u271d :\n x \u2208\n SimpleFunc.range\n (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n\u22a2 Measurable fun x_1 =>\n \u2191\u2191\u03bd\n (Prod.mk x_1 \u207b\u00b9'\n (\u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u207b\u00b9'\n {x}))\n[PROOFSTEP]\napply measurable_measure_prod_mk_left\n[GOAL]\ncase hs\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nn : \u2115\nthis : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n)\nx : E\nx\u271d :\n x \u2208\n SimpleFunc.range\n (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n\u22a2 MeasurableSet\n (\u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u207b\u00b9'\n {x})\n[PROOFSTEP]\nexact (s n).measurableSet_fiber x\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd\n[PROOFSTEP]\nhave h2f' : Tendsto f' atTop (\ud835\udcdd fun x : \u03b1 => \u222b y : \u03b2, f x y \u2202\u03bd) :=\n by\n rw [tendsto_pi_nhds]; intro x\n by_cases hfx : Integrable (f x) \u03bd\n \u00b7 have : \u2200 n, Integrable (s' n x) \u03bd := by\n intro n; apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable\n apply eventually_of_forall; intro y\n simp_rw [SimpleFunc.coe_comp]; exact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n\n simp only [hfx, SimpleFunc.integral_eq_integral _ (this _), indicator_of_mem, mem_setOf_eq]\n refine'\n tendsto_integral_of_dominated_convergence (fun y => \u2016f x y\u2016 + \u2016f x y\u2016) (fun n => (s' n x).aestronglyMeasurable)\n (hfx.norm.add hfx.norm) _ _\n \u00b7 refine' fun n => eventually_of_forall fun y => SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n\n \u00b7 exact hf.measurable\n \u00b7 simp\n \u00b7 refine'\n eventually_of_forall fun y =>\n SimpleFunc.tendsto_approxOn _ _\n _\n -- Porting note: Lean 3 solved the following two subgoals on its own\n \u00b7 exact hf.measurable.of_uncurry_left\n \u00b7 simp\n apply subset_closure\n simp [-uncurry_apply_pair]\n \u00b7 simp [hfx, integral_undef]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\n\u22a2 Tendsto f' atTop (\ud835\udcdd fun x => \u222b (y : \u03b2), f x y \u2202\u03bd)\n[PROOFSTEP]\nrw [tendsto_pi_nhds]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u03bd))\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\n\u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u03bd))\n[PROOFSTEP]\nby_cases hfx : Integrable (f x) \u03bd\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\n\u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u03bd))\n[PROOFSTEP]\nhave : \u2200 n, Integrable (s' n x) \u03bd := by\n intro n; apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable\n apply eventually_of_forall; intro y\n simp_rw [SimpleFunc.coe_comp]; exact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\n\u22a2 \u2200 (n : \u2115), Integrable \u2191(s' n x)\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nn : \u2115\n\u22a2 Integrable \u2191(s' n x)\n[PROOFSTEP]\napply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nn : \u2115\n\u22a2 \u2200\u1d50 (a : \u03b2) \u2202\u03bd, \u2016\u2191(s' n x) a\u2016 \u2264 ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) a\n[PROOFSTEP]\napply eventually_of_forall\n[GOAL]\ncase hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nn : \u2115\n\u22a2 \u2200 (x_1 : \u03b2), \u2016\u2191(s' n x) x_1\u2016 \u2264 ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) x_1\n[PROOFSTEP]\nintro y\n[GOAL]\ncase hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nn : \u2115\ny : \u03b2\n\u22a2 \u2016\u2191(s' n x) y\u2016 \u2264 ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) y\n[PROOFSTEP]\nsimp_rw [SimpleFunc.coe_comp]\n[GOAL]\ncase hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nn : \u2115\ny : \u03b2\n\u22a2 \u2016(\u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u2218\n Prod.mk x)\n y\u2016 \u2264\n ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) y\n[PROOFSTEP]\nexact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\n\u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u03bd))\n[PROOFSTEP]\nsimp only [hfx, SimpleFunc.integral_eq_integral _ (this _), indicator_of_mem, mem_setOf_eq]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\n\u22a2 Tendsto\n (fun i =>\n \u222b (x_1 : \u03b2),\n \u2191(SimpleFunc.comp\n (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) i)\n (Prod.mk x) (_ : Measurable (Prod.mk x)))\n x_1 \u2202\u03bd)\n atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u03bd))\n[PROOFSTEP]\nrefine'\n tendsto_integral_of_dominated_convergence (fun y => \u2016f x y\u2016 + \u2016f x y\u2016) (fun n => (s' n x).aestronglyMeasurable)\n (hfx.norm.add hfx.norm) _ _\n[GOAL]\ncase pos.refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\n\u22a2 \u2200 (n : \u2115),\n \u2200\u1d50 (a : \u03b2) \u2202\u03bd,\n \u2016\u2191(SimpleFunc.comp\n (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n (Prod.mk x) (_ : Measurable (Prod.mk x)))\n a\u2016 \u2264\n (fun y => \u2016f x y\u2016 + \u2016f x y\u2016) a\n[PROOFSTEP]\nrefine' fun n => eventually_of_forall fun y => SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n\n[GOAL]\ncase pos.refine'_1.refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\nn : \u2115\ny : \u03b2\n\u22a2 Measurable (uncurry f)\n[PROOFSTEP]\nexact hf.measurable\n[GOAL]\ncase pos.refine'_1.refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\nn : \u2115\ny : \u03b2\n\u22a2 0 \u2208 range (uncurry f) \u222a {0}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos.refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\n\u22a2 \u2200\u1d50 (a : \u03b2) \u2202\u03bd,\n Tendsto\n (fun n =>\n \u2191(SimpleFunc.comp\n (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)\n (Prod.mk x) (_ : Measurable (Prod.mk x)))\n a)\n atTop (\ud835\udcdd (f x a))\n[PROOFSTEP]\nrefine'\n eventually_of_forall fun y =>\n SimpleFunc.tendsto_approxOn _ _\n _\n -- Porting note: Lean 3 solved the following two subgoals on its own\n[GOAL]\ncase pos.refine'_2.refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\ny : \u03b2\n\u22a2 Measurable fun x_1 => uncurry f (x, x_1)\n[PROOFSTEP]\nexact hf.measurable.of_uncurry_left\n[GOAL]\ncase pos.refine'_2.refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\ny : \u03b2\n\u22a2 0 \u2208 range (uncurry f) \u222a {0}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos.refine'_2.refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\ny : \u03b2\n\u22a2 uncurry f (x, y) \u2208 closure (range (uncurry f) \u222a {0})\n[PROOFSTEP]\napply subset_closure\n[GOAL]\ncase pos.refine'_2.refine'_3.a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b2 : MeasurableSpace E := borel E\nthis\u271d\u00b9 : BorelSpace E\nthis\u271d : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : Integrable (f x)\nthis : \u2200 (n : \u2115), Integrable \u2191(s' n x)\ny : \u03b2\n\u22a2 uncurry f (x, y) \u2208 range (uncurry f) \u222a {0}\n[PROOFSTEP]\nsimp [-uncurry_apply_pair]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nx : \u03b1\nhfx : \u00acIntegrable (f x)\n\u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u03bd))\n[PROOFSTEP]\nsimp [hfx, integral_undef]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u2192 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry f)\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nthis : SeparableSpace \u2191(range (uncurry f) \u222a {0})\ns : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0\n (_ : 0 \u2208 range (uncurry f) \u222a {0})\ns' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))\nf' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral \u03bd (s' n x)\nhf' : \u2200 (n : \u2115), StronglyMeasurable (f' n)\nh2f' : Tendsto f' atTop (\ud835\udcdd fun x => \u222b (y : \u03b2), f x y \u2202\u03bd)\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u03bd\n[PROOFSTEP]\nexact stronglyMeasurable_of_tendsto _ hf' h2f'\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : StronglyMeasurable f\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd\n[PROOFSTEP]\nrw [\u2190 uncurry_curry f] at hf \n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : StronglyMeasurable (uncurry (curry f))\n\u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd\n[PROOFSTEP]\nexact hf.integral_prod_right\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 Integrable fun x => ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s))\n[PROOFSTEP]\nrefine' \u27e8(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 HasFiniteIntegral fun x => ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s))\n[PROOFSTEP]\nsimp_rw [HasFiniteIntegral, ennnorm_eq_ofReal toReal_nonneg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s))) \u2202\u03bc < \u22a4\n[PROOFSTEP]\nconvert h2s.lt_top using 1\n -- Porting note: was `simp_rw`\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s))) \u2202\u03bc = \u2191\u2191(Measure.prod \u03bc \u03bd) s\n[PROOFSTEP]\nrw [prod_apply hs]\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s))) \u2202\u03bc = \u222b\u207b (x : \u03b1), \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) \u2202\u03bc\n[PROOFSTEP]\napply lintegral_congr_ae\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)))) =\u1da0[ae \u03bc] fun a => \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)\n[PROOFSTEP]\nrefine' (ae_measure_lt_top hs h2s).mp _\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) < \u22a4 \u2192\n (fun a => ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)))) x = (fun a => \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)) x\n[PROOFSTEP]\napply eventually_of_forall\n[GOAL]\ncase h.e'_3.h.hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 \u2200 (x : \u03b1),\n \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) < \u22a4 \u2192\n (fun a => ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)))) x = (fun a => \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h.e'_3.h.hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\nx : \u03b1\nhx : \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) < \u22a4\n\u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)))) x = (fun a => \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)) x\n[PROOFSTEP]\nrw [lt_top_iff_ne_top] at hx \n[GOAL]\ncase h.e'_3.h.hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\nx : \u03b1\nhx : \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) \u2260 \u22a4\n\u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)))) x = (fun a => \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)) x\n[PROOFSTEP]\nsimp [ofReal_toReal, hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\u271d\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b3 : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nf : \u03b2 \u00d7 \u03b1 \u2192 \u03b3\nhf : AEStronglyMeasurable f (Measure.prod \u03bd \u03bc)\n\u22a2 AEStronglyMeasurable (fun z => f (Prod.swap z)) (Measure.prod \u03bc \u03bd)\n[PROOFSTEP]\nrw [\u2190 prod_swap] at hf \n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\u271d\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03b3 : Type u_7\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nf : \u03b2 \u00d7 \u03b1 \u2192 \u03b3\nhf : AEStronglyMeasurable f (map Prod.swap (Measure.prod \u03bc \u03bd))\n\u22a2 AEStronglyMeasurable (fun z => f (Prod.swap z)) (Measure.prod \u03bc \u03bd)\n[PROOFSTEP]\nexact hf.comp_measurable measurable_swap\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 (fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd) =\u1da0[ae \u03bc] fun x => \u222b (y : \u03b2), AEStronglyMeasurable.mk f hf (x, y) \u2202\u03bd\n[PROOFSTEP]\nfilter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with _ hx using integral_congr_ae hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03b3 : Type u_7\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\nhf : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, AEStronglyMeasurable (fun y => f (x, y)) \u03bd\n[PROOFSTEP]\nfilter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3\u271d : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03b3 : Type u_7\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\nhf : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\nx : \u03b1\nhx : \u2200\u1d50 (y : \u03b2) \u2202\u03bd, f (x, y) = AEStronglyMeasurable.mk f hf (x, y)\n\u22a2 AEStronglyMeasurable (fun y => f (x, y)) \u03bd\n[PROOFSTEP]\nexact \u27e8fun y => hf.mk f (x, y), hf.stronglyMeasurable_mk.comp_measurable measurable_prod_mk_left, hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\n\u22a2 HasFiniteIntegral f \u2194\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => f (x, y)) \u2227 HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd\n[PROOFSTEP]\nsimp only [HasFiniteIntegral]\n -- Porting note: was `simp`\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\n\u22a2 \u222b\u207b (a : \u03b1 \u00d7 \u03b2), \u2191\u2016f a\u2016\u208a \u2202Measure.prod \u03bc \u03bd < \u22a4 \u2194\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4) \u2227 \u222b\u207b (a : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2016f (a, y)\u2016 \u2202\u03bd\u2016\u208a \u2202\u03bc < \u22a4\n[PROOFSTEP]\nrw [lintegral_prod_of_measurable _ h1f.ennnorm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\n\u22a2 \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016f (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc < \u22a4 \u2194\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4) \u2227 \u222b\u207b (a : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2016f (a, y)\u2016 \u2202\u03bd\u2016\u208a \u2202\u03bc < \u22a4\n[PROOFSTEP]\nhave : \u2200 x, \u2200\u1d50 y \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016 := fun x => eventually_of_forall fun y => norm_nonneg _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\n\u22a2 \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016f (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc < \u22a4 \u2194\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4) \u2227 \u222b\u207b (a : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2016f (a, y)\u2016 \u2202\u03bd\u2016\u208a \u2202\u03bc < \u22a4\n[PROOFSTEP]\nsimp_rw [integral_eq_lintegral_of_nonneg_ae (this _)\n (h1f.norm.comp_measurable measurable_prod_mk_left).aestronglyMeasurable,\n ennnorm_eq_ofReal toReal_nonneg, ofReal_norm_eq_coe_nnnorm]\n -- this fact is probably too specialized to be its own lemma\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\n\u22a2 \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016f (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc < \u22a4 \u2194\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4) \u2227\n \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2016f (a, a_1)\u2016\u208a \u2202\u03bd)) \u2202\u03bc < \u22a4\n[PROOFSTEP]\nhave : \u2200 {p q r : Prop} (_ : r \u2192 p), (r \u2194 p \u2227 q) \u2194 p \u2192 (r \u2194 q) := fun {p q r} h1 => by\n rw [\u2190 and_congr_right_iff, and_iff_right_of_imp h1]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\np q r : Prop\nh1 : r \u2192 p\n\u22a2 (r \u2194 p \u2227 q) \u2194 p \u2192 (r \u2194 q)\n[PROOFSTEP]\nrw [\u2190 and_congr_right_iff, and_iff_right_of_imp h1]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis\u271d : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\nthis : \u2200 {p q r : Prop}, (r \u2192 p) \u2192 ((r \u2194 p \u2227 q) \u2194 p \u2192 (r \u2194 q))\n\u22a2 \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016f (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc < \u22a4 \u2194\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4) \u2227\n \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2016f (a, a_1)\u2016\u208a \u2202\u03bd)) \u2202\u03bc < \u22a4\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis\u271d : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\nthis : \u2200 {p q r : Prop}, (r \u2192 p) \u2192 ((r \u2194 p \u2227 q) \u2194 p \u2192 (r \u2194 q))\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4) \u2192\n (\u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016f (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc < \u22a4 \u2194\n \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2016f (a, a_1)\u2016\u208a \u2202\u03bd)) \u2202\u03bc < \u22a4)\n[PROOFSTEP]\nintro h2f\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis\u271d : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\nthis : \u2200 {p q r : Prop}, (r \u2192 p) \u2192 ((r \u2194 p \u2227 q) \u2194 p \u2192 (r \u2194 q))\nh2f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016f (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc < \u22a4 \u2194\n \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2016f (a, a_1)\u2016\u208a \u2202\u03bd)) \u2202\u03bc < \u22a4\n[PROOFSTEP]\nrw [lintegral_congr_ae]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis\u271d : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\nthis : \u2200 {p q r : Prop}, (r \u2192 p) \u2192 ((r \u2194 p \u2227 q) \u2194 p \u2192 (r \u2194 q))\nh2f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4\n\u22a2 (fun x => \u222b\u207b (y : \u03b2), \u2191\u2016f (x, y)\u2016\u208a \u2202\u03bd) =\u1da0[ae \u03bc] fun a =>\n ENNReal.ofReal (ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2016f (a, a_1)\u2016\u208a \u2202\u03bd))\n[PROOFSTEP]\nrefine' h2f.mp _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis\u271d : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\nthis : \u2200 {p q r : Prop}, (r \u2192 p) \u2192 ((r \u2194 p \u2227 q) \u2194 p \u2192 (r \u2194 q))\nh2f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4 \u2192\n (fun x => \u222b\u207b (y : \u03b2), \u2191\u2016f (x, y)\u2016\u208a \u2202\u03bd) x =\n (fun a => ENNReal.ofReal (ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2016f (a, a_1)\u2016\u208a \u2202\u03bd))) x\n[PROOFSTEP]\napply eventually_of_forall\n[GOAL]\ncase hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis\u271d : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\nthis : \u2200 {p q r : Prop}, (r \u2192 p) \u2192 ((r \u2194 p \u2227 q) \u2194 p \u2192 (r \u2194 q))\nh2f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4\n\u22a2 \u2200 (x : \u03b1),\n \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4 \u2192\n (fun x => \u222b\u207b (y : \u03b2), \u2191\u2016f (x, y)\u2016\u208a \u2202\u03bd) x =\n (fun a => ENNReal.ofReal (ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2016f (a, a_1)\u2016\u208a \u2202\u03bd))) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis\u271d : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\nthis : \u2200 {p q r : Prop}, (r \u2192 p) \u2192 ((r \u2194 p \u2227 q) \u2194 p \u2192 (r \u2194 q))\nh2f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4\nx : \u03b1\nhx : \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4\n\u22a2 (fun x => \u222b\u207b (y : \u03b2), \u2191\u2016f (x, y)\u2016\u208a \u2202\u03bd) x =\n (fun a => ENNReal.ofReal (ENNReal.toReal (\u222b\u207b (a_1 : \u03b2), \u2191\u2016f (a, a_1)\u2016\u208a \u2202\u03bd))) x\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis\u271d : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\nthis : \u2200 {p q r : Prop}, (r \u2192 p) \u2192 ((r \u2194 p \u2227 q) \u2194 p \u2192 (r \u2194 q))\nh2f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4\nx : \u03b1\nhx : \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4\n\u22a2 \u222b\u207b (y : \u03b2), \u2191\u2016f (x, y)\u2016\u208a \u2202\u03bd = ENNReal.ofReal (ENNReal.toReal (\u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd))\n[PROOFSTEP]\nrw [ofReal_toReal]\n[GOAL]\ncase hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis\u271d : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\nthis : \u2200 {p q r : Prop}, (r \u2192 p) \u2192 ((r \u2194 p \u2227 q) \u2194 p \u2192 (r \u2194 q))\nh2f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4\nx : \u03b1\nhx : \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4\n\u22a2 \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd \u2260 \u22a4\n[PROOFSTEP]\nrw [\u2190 lt_top_iff_ne_top]\n[GOAL]\ncase hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis\u271d : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\nthis : \u2200 {p q r : Prop}, (r \u2192 p) \u2192 ((r \u2194 p \u2227 q) \u2194 p \u2192 (r \u2194 q))\nh2f : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4\nx : \u03b1\nhx : \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4\n\u22a2 \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4\n[PROOFSTEP]\nexact hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis\u271d : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\nthis : \u2200 {p q r : Prop}, (r \u2192 p) \u2192 ((r \u2194 p \u2227 q) \u2194 p \u2192 (r \u2194 q))\n\u22a2 \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016f (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc < \u22a4 \u2192 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4\n[PROOFSTEP]\nintro h2f\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis\u271d : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\nthis : \u2200 {p q r : Prop}, (r \u2192 p) \u2192 ((r \u2194 p \u2227 q) \u2194 p \u2192 (r \u2194 q))\nh2f : \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016f (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd < \u22a4\n[PROOFSTEP]\nrefine' ae_lt_top _ h2f.ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : StronglyMeasurable f\nthis\u271d : \u2200 (x : \u03b1), \u2200\u1d50 (y : \u03b2) \u2202\u03bd, 0 \u2264 \u2016f (x, y)\u2016\nthis : \u2200 {p q r : Prop}, (r \u2192 p) \u2192 ((r \u2194 p \u2227 q) \u2194 p \u2192 (r \u2194 q))\nh2f : \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016f (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc < \u22a4\n\u22a2 Measurable fun x => \u222b\u207b (a : \u03b2), \u2191\u2016f (x, a)\u2016\u208a \u2202\u03bd\n[PROOFSTEP]\nexact h1f.ennnorm.lintegral_prod_right'\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 HasFiniteIntegral f \u2194\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => f (x, y)) \u2227 HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd\n[PROOFSTEP]\nrw [hasFiniteIntegral_congr h1f.ae_eq_mk, hasFiniteIntegral_prod_iff h1f.stronglyMeasurable_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 ((\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2227\n HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03bd) \u2194\n (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => f (x, y)) \u2227 HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd\n[PROOFSTEP]\napply and_congr\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194\n \u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => f (x, y)\n[PROOFSTEP]\napply eventually_congr\n[GOAL]\ncase h\u2081.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194 HasFiniteIntegral fun y => f (x, y)\n[PROOFSTEP]\nfilter_upwards [ae_ae_of_ae_prod h1f.ae_eq_mk.symm]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 \u2200 (a : \u03b1),\n (\u2200\u1d50 (y : \u03b2) \u2202\u03bd, AEStronglyMeasurable.mk f h1f (a, y) = f (a, y)) \u2192\n ((HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (a, y)) \u2194 HasFiniteIntegral fun y => f (a, y))\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\nx : \u03b1\nhx : \u2200\u1d50 (y : \u03b2) \u2202\u03bd, AEStronglyMeasurable.mk f h1f (x, y) = f (x, y)\n\u22a2 (HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194 HasFiniteIntegral fun y => f (x, y)\n[PROOFSTEP]\nexact hasFiniteIntegral_congr hx\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 (HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03bd) \u2194\n HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd\n[PROOFSTEP]\napply hasFiniteIntegral_congr\n[GOAL]\ncase h\u2082.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 (fun x => \u222b (y : \u03b2), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03bd) =\u1da0[ae \u03bc] fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd\n[PROOFSTEP]\nfilter_upwards [ae_ae_of_ae_prod h1f.ae_eq_mk.symm] with _ hx using integral_congr_ae (EventuallyEq.fun_comp hx _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 Integrable f \u2194 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, Integrable fun y => f (x, y)) \u2227 Integrable fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd\n[PROOFSTEP]\nsimp [Integrable, h1f, hasFiniteIntegral_prod_iff', h1f.norm.integral_prod_right', h1f.prod_mk_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 Integrable f \u2194 (\u2200\u1d50 (y : \u03b2) \u2202\u03bd, Integrable fun x => f (x, y)) \u2227 Integrable fun y => \u222b (x : \u03b1), \u2016f (x, y)\u2016 \u2202\u03bc\n[PROOFSTEP]\nconvert integrable_prod_iff h1f.prod_swap using 1\n[GOAL]\ncase h.e'_1.a\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nh1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 Integrable f \u2194 Integrable fun z => f (Prod.swap z)\n[PROOFSTEP]\nrw [funext fun _ => Function.comp_apply.symm, integrable_swap_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\nL : Type u_7\ninst\u271d : IsROrC L\nf : \u03b1 \u2192 L\ng : \u03b2 \u2192 L\nhf : Integrable f\nhg : Integrable g\n\u22a2 Integrable fun z => f z.fst * g z.snd\n[PROOFSTEP]\nrefine' (integrable_prod_iff _).2 \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\nL : Type u_7\ninst\u271d : IsROrC L\nf : \u03b1 \u2192 L\ng : \u03b2 \u2192 L\nhf : Integrable f\nhg : Integrable g\n\u22a2 AEStronglyMeasurable (fun z => f z.fst * g z.snd) (Measure.prod \u03bc \u03bd)\n[PROOFSTEP]\nexact hf.1.fst.mul hg.1.snd\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\nL : Type u_7\ninst\u271d : IsROrC L\nf : \u03b1 \u2192 L\ng : \u03b2 \u2192 L\nhf : Integrable f\nhg : Integrable g\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Integrable fun y => f (x, y).fst * g (x, y).snd\n[PROOFSTEP]\nexact eventually_of_forall fun x => hg.const_mul (f x)\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SigmaFinite \u03bd\nL : Type u_7\ninst\u271d : IsROrC L\nf : \u03b1 \u2192 L\ng : \u03b2 \u2192 L\nhf : Integrable f\nhg : Integrable g\n\u22a2 Integrable fun x => \u222b (y : \u03b2), \u2016f (x, y).fst * g (x, y).snd\u2016 \u2202\u03bd\n[PROOFSTEP]\nsimpa only [norm_mul, integral_mul_left] using hf.norm.mul_const _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1'\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : MeasurableSpace \u03b2'\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd)\n\u22a2 \u222b (z : \u03b2 \u00d7 \u03b1), f (Prod.swap z) \u2202Measure.prod \u03bd \u03bc = \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd\n[PROOFSTEP]\nrw [\u2190 prod_swap] at hf \n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1'\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : MeasurableSpace \u03b2'\ninst\u271d\u2075 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : AEStronglyMeasurable f (map Prod.swap (Measure.prod \u03bd \u03bc))\n\u22a2 \u222b (z : \u03b2 \u00d7 \u03b1), f (Prod.swap z) \u2202Measure.prod \u03bd \u03bc = \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd\n[PROOFSTEP]\nrw [\u2190 integral_map measurable_swap.aemeasurable hf, prod_swap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nF : E \u2192 E'\nhf : Integrable f\nhg : Integrable g\n\u22a2 \u222b (x : \u03b1), F (\u222b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd) \u2202\u03bc =\n \u222b (x : \u03b1), F (\u222b (y : \u03b2), f (x, y) \u2202\u03bd + \u222b (y : \u03b2), g (x, y) \u2202\u03bd) \u2202\u03bc\n[PROOFSTEP]\nrefine' integral_congr_ae _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nF : E \u2192 E'\nhf : Integrable f\nhg : Integrable g\n\u22a2 (fun x => F (\u222b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd)) =\u1da0[ae \u03bc] fun x => F (\u222b (y : \u03b2), f (x, y) \u2202\u03bd + \u222b (y : \u03b2), g (x, y) \u2202\u03bd)\n[PROOFSTEP]\nfilter_upwards [hf.prod_right_ae, hg.prod_right_ae] with _ h2f h2g\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nF : E \u2192 E'\nhf : Integrable f\nhg : Integrable g\na\u271d : \u03b1\nh2f : Integrable fun y => f (a\u271d, y)\nh2g : Integrable fun y => g (a\u271d, y)\n\u22a2 F (\u222b (y : \u03b2), f (a\u271d, y) + g (a\u271d, y) \u2202\u03bd) = F (\u222b (y : \u03b2), f (a\u271d, y) \u2202\u03bd + \u222b (y : \u03b2), g (a\u271d, y) \u2202\u03bd)\n[PROOFSTEP]\nsimp [integral_add h2f h2g]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nF : E \u2192 E'\nhf : Integrable f\nhg : Integrable g\n\u22a2 \u222b (x : \u03b1), F (\u222b (y : \u03b2), f (x, y) - g (x, y) \u2202\u03bd) \u2202\u03bc =\n \u222b (x : \u03b1), F (\u222b (y : \u03b2), f (x, y) \u2202\u03bd - \u222b (y : \u03b2), g (x, y) \u2202\u03bd) \u2202\u03bc\n[PROOFSTEP]\nrefine' integral_congr_ae _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nF : E \u2192 E'\nhf : Integrable f\nhg : Integrable g\n\u22a2 (fun x => F (\u222b (y : \u03b2), f (x, y) - g (x, y) \u2202\u03bd)) =\u1da0[ae \u03bc] fun x => F (\u222b (y : \u03b2), f (x, y) \u2202\u03bd - \u222b (y : \u03b2), g (x, y) \u2202\u03bd)\n[PROOFSTEP]\nfilter_upwards [hf.prod_right_ae, hg.prod_right_ae] with _ h2f h2g\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nF : E \u2192 E'\nhf : Integrable f\nhg : Integrable g\na\u271d : \u03b1\nh2f : Integrable fun y => f (a\u271d, y)\nh2g : Integrable fun y => g (a\u271d, y)\n\u22a2 F (\u222b (y : \u03b2), f (a\u271d, y) - g (a\u271d, y) \u2202\u03bd) = F (\u222b (y : \u03b2), f (a\u271d, y) \u2202\u03bd - \u222b (y : \u03b2), g (a\u271d, y) \u2202\u03bd)\n[PROOFSTEP]\nsimp [integral_sub h2f h2g]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nF : E \u2192 \u211d\u22650\u221e\nhf : Integrable f\nhg : Integrable g\n\u22a2 \u222b\u207b (x : \u03b1), F (\u222b (y : \u03b2), f (x, y) - g (x, y) \u2202\u03bd) \u2202\u03bc =\n \u222b\u207b (x : \u03b1), F (\u222b (y : \u03b2), f (x, y) \u2202\u03bd - \u222b (y : \u03b2), g (x, y) \u2202\u03bd) \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_congr_ae _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nF : E \u2192 \u211d\u22650\u221e\nhf : Integrable f\nhg : Integrable g\n\u22a2 (fun x => F (\u222b (y : \u03b2), f (x, y) - g (x, y) \u2202\u03bd)) =\u1da0[ae \u03bc] fun x => F (\u222b (y : \u03b2), f (x, y) \u2202\u03bd - \u222b (y : \u03b2), g (x, y) \u2202\u03bd)\n[PROOFSTEP]\nfilter_upwards [hf.prod_right_ae, hg.prod_right_ae] with _ h2f h2g\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nF : E \u2192 \u211d\u22650\u221e\nhf : Integrable f\nhg : Integrable g\na\u271d : \u03b1\nh2f : Integrable fun y => f (a\u271d, y)\nh2g : Integrable fun y => g (a\u271d, y)\n\u22a2 F (\u222b (y : \u03b2), f (a\u271d, y) - g (a\u271d, y) \u2202\u03bd) = F (\u222b (y : \u03b2), f (a\u271d, y) \u2202\u03bd - \u222b (y : \u03b2), g (a\u271d, y) \u2202\u03bd)\n[PROOFSTEP]\nsimp [integral_sub h2f h2g]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\n\u22a2 Continuous fun f => \u222b (x : \u03b1), \u222b (y : \u03b2), \u2191\u2191f (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nrw [continuous_iff_continuousAt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\n\u22a2 \u2200 (x : { x // x \u2208 Lp E 1 }), ContinuousAt (fun f => \u222b (x : \u03b1), \u222b (y : \u03b2), \u2191\u2191f (x, y) \u2202\u03bd \u2202\u03bc) x\n[PROOFSTEP]\nintro g\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousAt (fun f => \u222b (x : \u03b1), \u222b (y : \u03b2), \u2191\u2191f (x, y) \u2202\u03bd \u2202\u03bc) g\n[PROOFSTEP]\nrefine'\n tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_prod_left\n (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_prod_left) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) \u2202\u03bd - \u222b (y : \u03b2), \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp_rw [\u2190 lintegral_fn_integral_sub (fun x => (\u2016x\u2016\u208a : \u211d\u22650\u221e)) (L1.integrable_coeFn _) (L1.integrable_coeFn g)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (fun i => zero_le _) _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 { x // x \u2208 Lp E 1 } \u2192 \u211d\u22650\u221e\n[PROOFSTEP]\nexact fun i => \u222b\u207b x, \u222b\u207b y, \u2016i (x, y) - g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) \u2264 fun i =>\n \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 (fun i => \u222b\u207b (x : \u03b1), \u2191\u2016\u222b (y : \u03b2), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u03bd\u2016\u208a \u2202\u03bc) \u2264 fun i =>\n \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nexact fun i => lintegral_mono fun x => ennnorm_integral_le_lintegral_ennnorm _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)\n[PROOFSTEP]\nshow Tendsto (fun i : \u03b1 \u00d7 \u03b2 \u2192\u2081[\u03bc.prod \u03bd] E => \u222b\u207b x, \u222b\u207b y : \u03b2, \u2016i (x, y) - g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : \u2200 i : \u03b1 \u00d7 \u03b2 \u2192\u2081[\u03bc.prod \u03bd] E, Measurable fun z => (\u2016i z - g z\u2016\u208a : \u211d\u22650\u221e) := fun i =>\n ((Lp.stronglyMeasurable i).sub (Lp.stronglyMeasurable g)).ennnorm\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)\n[PROOFSTEP]\nconv =>\n congr\n ext\n rw [\u2190 lintegral_prod_of_measurable _ (this _), \u2190 L1.ofReal_norm_sub_eq_lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n| Tendsto (fun i => \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)\n[PROOFSTEP]\n congr\n ext\n rw [\u2190 lintegral_prod_of_measurable _ (this _), \u2190 L1.ofReal_norm_sub_eq_lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n| Tendsto (fun i => \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)\n[PROOFSTEP]\n congr\n ext\n rw [\u2190 lintegral_prod_of_measurable _ (this _), \u2190 L1.ofReal_norm_sub_eq_lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n| Tendsto (fun i => \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc) (\ud835\udcdd g) (\ud835\udcdd 0)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase f\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n| fun i => \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc\ncase l\u2081\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n| \ud835\udcdd g\ncase l\u2082\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n| \ud835\udcdd 0\n[PROOFSTEP]\next\n[GOAL]\ncase f.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\nx\u271d : { x // x \u2208 Lp E 1 }\n| \u222b\u207b (x : \u03b1), \u222b\u207b (y : \u03b2), \u2191\u2016\u2191\u2191x\u271d (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u03bd \u2202\u03bc\ncase l\u2081\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n| \ud835\udcdd g\ncase l\u2082\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n| \ud835\udcdd 0\n[PROOFSTEP]\nrw [\u2190 lintegral_prod_of_measurable _ (this _), \u2190 L1.ofReal_norm_sub_eq_lintegral]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n\u22a2 Tendsto (fun x => ENNReal.ofReal \u2016x - g\u2016) (\ud835\udcdd g) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 ofReal_zero]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n\u22a2 Tendsto (fun x => ENNReal.ofReal \u2016x - g\u2016) (\ud835\udcdd g) (\ud835\udcdd (ENNReal.ofReal 0))\n[PROOFSTEP]\nrefine' (continuous_ofReal.tendsto 0).comp _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n\u22a2 Tendsto (fun x => \u2016x - g\u2016) (\ud835\udcdd g) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 tendsto_iff_norm_tendsto_zero]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\ng : { x // x \u2208 Lp E 1 }\nthis : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a\n\u22a2 Tendsto (fun x => x) (\ud835\udcdd g) (\ud835\udcdd g)\n[PROOFSTEP]\nexact tendsto_id\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\n\u22a2 \u2200 (f : \u03b1 \u00d7 \u03b2 \u2192 E), Integrable f \u2192 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\napply Integrable.induction\n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\n\u22a2 \u2200 (c : E) \u2983s : Set (\u03b1 \u00d7 \u03b2)\u2984,\n MeasurableSet s \u2192\n \u2191\u2191(Measure.prod \u03bc \u03bd) s < \u22a4 \u2192\n \u222b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =\n \u222b (x : \u03b1), \u222b (y : \u03b2), indicator s (fun x => c) (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nintro c s hs h2s\n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nc : E\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s < \u22a4\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =\n \u222b (x : \u03b1), \u222b (y : \u03b2), indicator s (fun x => c) (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [integral_indicator hs, \u2190 indicator_comp_right, Function.comp, integral_indicator (measurable_prod_mk_left hs),\n set_integral_const, integral_smul_const,\n integral_toReal (measurable_measure_prod_mk_left hs).aemeasurable (ae_measure_lt_top hs h2s.ne)]\n -- Porting note: was `simp_rw`\n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nc : E\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s < \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191(Measure.prod \u03bc \u03bd) s) \u2022 c = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s) \u2202\u03bc) \u2022 c\n[PROOFSTEP]\nrw [prod_apply hs]\n[GOAL]\ncase h_add\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\n\u22a2 \u2200 \u2983f g : \u03b1 \u00d7 \u03b2 \u2192 E\u2984,\n Disjoint (support f) (support g) \u2192\n Integrable f \u2192\n Integrable g \u2192\n \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc \u2192\n \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc \u2192\n \u222b (z : \u03b1 \u00d7 \u03b2), (f + g) z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), (f + g) (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nrintro f g - i_f i_g hf hg\n[GOAL]\ncase h_add\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\ni_f : Integrable f\ni_g : Integrable g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nhg : \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), (f + g) z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), (f + g) (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [integral_add' i_f i_g, integral_integral_add' i_f i_g, hf, hg]\n[GOAL]\ncase h_closed\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\n\u22a2 IsClosed {f | \u222b (z : \u03b1 \u00d7 \u03b2), \u2191\u2191f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), \u2191\u2191f (x, y) \u2202\u03bd \u2202\u03bc}\n[PROOFSTEP]\nexact isClosed_eq continuous_integral continuous_integral_integral\n[GOAL]\ncase h_ae\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\n\u22a2 \u2200 \u2983f g : \u03b1 \u00d7 \u03b2 \u2192 E\u2984,\n f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g \u2192\n Integrable f \u2192\n \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc \u2192\n \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nrintro f g hfg - hf\n[GOAL]\ncase h_ae\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nconvert hf using 1\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd\n[PROOFSTEP]\nexact integral_congr_ae hfg.symm\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nrefine' integral_congr_ae _\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) =\u1da0[ae \u03bc] fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd\n[PROOFSTEP]\nrefine' (ae_ae_of_ae_prod hfg).mp _\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n (\u2200\u1d50 (y : \u03b2) \u2202\u03bd, f (x, y) = g (x, y)) \u2192 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) x = (fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd) x\n[PROOFSTEP]\napply eventually_of_forall\n[GOAL]\ncase h.e'_3.hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\n\u22a2 \u2200 (x : \u03b1),\n (\u2200\u1d50 (y : \u03b2) \u2202\u03bd, f (x, y) = g (x, y)) \u2192 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) x = (fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd) x\n[PROOFSTEP]\nintro x hfgx\n[GOAL]\ncase h.e'_3.hp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nhfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g\nhf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc\nx : \u03b1\nhfgx : \u2200\u1d50 (y : \u03b2) \u2202\u03bd, f (x, y) = g (x, y)\n\u22a2 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) x = (fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd) x\n[PROOFSTEP]\nexact integral_congr_ae (ae_eq_symm hfgx)\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : Integrable f\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (y : \u03b2), \u222b (x : \u03b1), f (x, y) \u2202\u03bc \u2202\u03bd\n[PROOFSTEP]\nsimp_rw [\u2190 integral_prod_swap f hf.aestronglyMeasurable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf : \u03b1 \u00d7 \u03b2 \u2192 E\nhf : Integrable f\n\u22a2 \u222b (z : \u03b2 \u00d7 \u03b1), f (Prod.swap z) \u2202Measure.prod \u03bd \u03bc = \u222b (y : \u03b2), \u222b (x : \u03b1), f (x, y) \u2202\u03bc \u2202\u03bd\n[PROOFSTEP]\nexact integral_prod _ hf.swap\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf : \u03b1 \u00d7 \u03b2 \u2192 E\ns : Set \u03b1\nt : Set \u03b2\nhf : IntegrableOn f (s \u00d7\u02e2 t)\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2) in s \u00d7\u02e2 t, f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1) in s, \u222b (y : \u03b2) in t, f (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nsimp only [\u2190 Measure.prod_restrict s t, IntegrableOn] at hf \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : MeasurableSpace \u03b2'\ninst\u271d\u2078 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : SigmaFinite \u03bd\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : CompleteSpace E'\ninst\u271d : NormedSpace \u211d E'\nf : \u03b1 \u00d7 \u03b2 \u2192 E\ns : Set \u03b1\nt : Set \u03b2\nhf : Integrable f\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod (Measure.restrict \u03bc s) (Measure.restrict \u03bd t) =\n \u222b (x : \u03b1) in s, \u222b (y : \u03b2) in t, f (x, y) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nexact integral_prod f hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2'\ninst\u271d\u2079 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : CompleteSpace E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\nL : Type u_8\ninst\u271d : IsROrC L\nf : \u03b1 \u2192 L\ng : \u03b2 \u2192 L\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.fst * g z.snd \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) * \u222b (y : \u03b2), g y \u2202\u03bd\n[PROOFSTEP]\nby_cases h : Integrable (fun z : \u03b1 \u00d7 \u03b2 => f z.1 * g z.2) (\u03bc.prod \u03bd)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2'\ninst\u271d\u2079 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : CompleteSpace E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\nL : Type u_8\ninst\u271d : IsROrC L\nf : \u03b1 \u2192 L\ng : \u03b2 \u2192 L\nh : Integrable fun z => f z.fst * g z.snd\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.fst * g z.snd \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) * \u222b (y : \u03b2), g y \u2202\u03bd\n[PROOFSTEP]\nrw [integral_prod _ h]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2'\ninst\u271d\u2079 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : CompleteSpace E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\nL : Type u_8\ninst\u271d : IsROrC L\nf : \u03b1 \u2192 L\ng : \u03b2 \u2192 L\nh : Integrable fun z => f z.fst * g z.snd\n\u22a2 \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y).fst * g (x, y).snd \u2202\u03bd \u2202\u03bc = (\u222b (x : \u03b1), f x \u2202\u03bc) * \u222b (y : \u03b2), g y \u2202\u03bd\n[PROOFSTEP]\nsimp_rw [integral_mul_left, integral_mul_right]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2'\ninst\u271d\u2079 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : CompleteSpace E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\nL : Type u_8\ninst\u271d : IsROrC L\nf : \u03b1 \u2192 L\ng : \u03b2 \u2192 L\nh : \u00acIntegrable fun z => f z.fst * g z.snd\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.fst * g z.snd \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) * \u222b (y : \u03b2), g y \u2202\u03bd\n[PROOFSTEP]\nhave H : \u00acIntegrable f \u03bc \u2228 \u00acIntegrable g \u03bd := by\n contrapose! h\n exact integrable_prod_mul h.1 h.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2'\ninst\u271d\u2079 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : CompleteSpace E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\nL : Type u_8\ninst\u271d : IsROrC L\nf : \u03b1 \u2192 L\ng : \u03b2 \u2192 L\nh : \u00acIntegrable fun z => f z.fst * g z.snd\n\u22a2 \u00acIntegrable f \u2228 \u00acIntegrable g\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2'\ninst\u271d\u2079 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : CompleteSpace E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\nL : Type u_8\ninst\u271d : IsROrC L\nf : \u03b1 \u2192 L\ng : \u03b2 \u2192 L\nh : Integrable f \u2227 Integrable g\n\u22a2 Integrable fun z => f z.fst * g z.snd\n[PROOFSTEP]\nexact integrable_prod_mul h.1 h.2\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2'\ninst\u271d\u2079 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : CompleteSpace E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\nL : Type u_8\ninst\u271d : IsROrC L\nf : \u03b1 \u2192 L\ng : \u03b2 \u2192 L\nh : \u00acIntegrable fun z => f z.fst * g z.snd\nH : \u00acIntegrable f \u2228 \u00acIntegrable g\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.fst * g z.snd \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) * \u222b (y : \u03b2), g y \u2202\u03bd\n[PROOFSTEP]\ncases' H with H H\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2'\ninst\u271d\u2079 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : CompleteSpace E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\nL : Type u_8\ninst\u271d : IsROrC L\nf : \u03b1 \u2192 L\ng : \u03b2 \u2192 L\nh : \u00acIntegrable fun z => f z.fst * g z.snd\nH : \u00acIntegrable f\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.fst * g z.snd \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) * \u222b (y : \u03b2), g y \u2202\u03bd\n[PROOFSTEP]\nsimp [integral_undef h, integral_undef H]\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2'\ninst\u271d\u2079 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : CompleteSpace E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\nL : Type u_8\ninst\u271d : IsROrC L\nf : \u03b1 \u2192 L\ng : \u03b2 \u2192 L\nh : \u00acIntegrable fun z => f z.fst * g z.snd\nH : \u00acIntegrable g\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.fst * g z.snd \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1), f x \u2202\u03bc) * \u222b (y : \u03b2), g y \u2202\u03bd\n[PROOFSTEP]\nsimp [integral_undef h, integral_undef H]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2'\ninst\u271d\u2079 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : CompleteSpace E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\nL : Type u_8\ninst\u271d : IsROrC L\nf : \u03b1 \u2192 L\ng : \u03b2 \u2192 L\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2) in s \u00d7\u02e2 t, f z.fst * g z.snd \u2202Measure.prod \u03bc \u03bd = (\u222b (x : \u03b1) in s, f x \u2202\u03bc) * \u222b (y : \u03b2) in t, g y \u2202\u03bd\n[PROOFSTEP]\nrw [\u2190 Measure.prod_restrict s t]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u00b9\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1'\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2'\ninst\u271d\u2079 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : SigmaFinite \u03bc\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : CompleteSpace E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\nL : Type u_8\ninst\u271d : IsROrC L\nf : \u03b1 \u2192 L\ng : \u03b2 \u2192 L\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z.fst * g z.snd \u2202Measure.prod (Measure.restrict \u03bc s) (Measure.restrict \u03bd t) =\n (\u222b (x : \u03b1) in s, f x \u2202\u03bc) * \u222b (y : \u03b2) in t, g y \u2202\u03bd\n[PROOFSTEP]\nsimp only [\u2190 Measure.prod_restrict s t, IntegrableOn, integral_prod_mul]\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Constructions.Prod.Integral", "llama_tokens": 68293, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702761768248, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.4075050354572094}} {"text": "[GOAL]\n\u22a2 Dense {x | Irrational x}\n[PROOFSTEP]\nrefine' Real.isTopologicalBasis_Ioo_rat.dense_iff.2 _\n[GOAL]\n\u22a2 \u2200 (o : Set \u211d), o \u2208 \u22c3 (a : \u211a) (b : \u211a) (_ : a < b), {Ioo \u2191a \u2191b} \u2192 Set.Nonempty o \u2192 Set.Nonempty (o \u2229 {x | Irrational x})\n[PROOFSTEP]\nsimp only [gt_iff_lt, Rat.cast_lt, not_lt, ge_iff_le, Rat.cast_le, mem_iUnion, mem_singleton_iff, exists_prop,\n forall_exists_index, and_imp]\n[GOAL]\n\u22a2 \u2200 (o : Set \u211d) (x x_1 : \u211a), x < x_1 \u2192 o = Ioo \u2191x \u2191x_1 \u2192 Set.Nonempty o \u2192 Set.Nonempty (o \u2229 {x | Irrational x})\n[PROOFSTEP]\nrintro _ a b hlt rfl _\n[GOAL]\na b : \u211a\nhlt : a < b\na\u271d : Set.Nonempty (Ioo \u2191a \u2191b)\n\u22a2 Set.Nonempty (Ioo \u2191a \u2191b \u2229 {x | Irrational x})\n[PROOFSTEP]\nrw [inter_comm]\n[GOAL]\na b : \u211a\nhlt : a < b\na\u271d : Set.Nonempty (Ioo \u2191a \u2191b)\n\u22a2 Set.Nonempty ({x | Irrational x} \u2229 Ioo \u2191a \u2191b)\n[PROOFSTEP]\nexact exists_irrational_btwn (Rat.cast_lt.2 hlt)\n[GOAL]\nx\u271d\u00b9 : \u211d\nx\u271d : { x // Irrational x }\nx : \u211d\nhx : Irrational x\n\u22a2 { val := x, property := hx } < { val := x + \u21911, property := (_ : Irrational (x + \u21911)) }\n[PROOFSTEP]\nsimp\n[GOAL]\nx\u271d\u00b9 : \u211d\nx\u271d : { x // Irrational x }\nx : \u211d\nhx : Irrational x\n\u22a2 { val := x - \u21911, property := (_ : Irrational (x - \u21911)) } < { val := x, property := hx }\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u211d\nhx : Irrational x\nn : \u2115\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd 0, \u2200 (m : \u2124), \u03b5 \u2264 dist x (\u2191m / \u2191n)\n[PROOFSTEP]\nhave A : IsClosed (range (fun m => (n : \u211d)\u207b\u00b9 * m : \u2124 \u2192 \u211d)) :=\n ((isClosedMap_smul\u2080 (n\u207b\u00b9 : \u211d)).comp Int.closedEmbedding_coe_real.isClosedMap).closed_range\n[GOAL]\nx : \u211d\nhx : Irrational x\nn : \u2115\nA : IsClosed (range fun m => (\u2191n)\u207b\u00b9 * \u2191m)\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd 0, \u2200 (m : \u2124), \u03b5 \u2264 dist x (\u2191m / \u2191n)\n[PROOFSTEP]\nhave B : x \u2209 range (fun m => (n : \u211d)\u207b\u00b9 * m : \u2124 \u2192 \u211d) :=\n by\n rintro \u27e8m, rfl\u27e9\n simp at hx \n[GOAL]\nx : \u211d\nhx : Irrational x\nn : \u2115\nA : IsClosed (range fun m => (\u2191n)\u207b\u00b9 * \u2191m)\n\u22a2 \u00acx \u2208 range fun m => (\u2191n)\u207b\u00b9 * \u2191m\n[PROOFSTEP]\nrintro \u27e8m, rfl\u27e9\n[GOAL]\ncase intro\nn : \u2115\nA : IsClosed (range fun m => (\u2191n)\u207b\u00b9 * \u2191m)\nm : \u2124\nhx : Irrational ((fun m => (\u2191n)\u207b\u00b9 * \u2191m) m)\n\u22a2 False\n[PROOFSTEP]\nsimp at hx \n[GOAL]\nx : \u211d\nhx : Irrational x\nn : \u2115\nA : IsClosed (range fun m => (\u2191n)\u207b\u00b9 * \u2191m)\nB : \u00acx \u2208 range fun m => (\u2191n)\u207b\u00b9 * \u2191m\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd 0, \u2200 (m : \u2124), \u03b5 \u2264 dist x (\u2191m / \u2191n)\n[PROOFSTEP]\nrcases Metric.mem_nhds_iff.1 (A.isOpen_compl.mem_nhds B) with \u27e8\u03b5, \u03b50, h\u03b5\u27e9\n[GOAL]\ncase intro.intro\nx : \u211d\nhx : Irrational x\nn : \u2115\nA : IsClosed (range fun m => (\u2191n)\u207b\u00b9 * \u2191m)\nB : \u00acx \u2208 range fun m => (\u2191n)\u207b\u00b9 * \u2191m\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 (range fun m => (\u2191n)\u207b\u00b9 * \u2191m)\u1d9c\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd 0, \u2200 (m : \u2124), \u03b5 \u2264 dist x (\u2191m / \u2191n)\n[PROOFSTEP]\nrefine' (ge_mem_nhds \u03b50).mono fun \u03b4 h\u03b4 m => not_lt.1 fun hlt => _\n[GOAL]\ncase intro.intro\nx : \u211d\nhx : Irrational x\nn : \u2115\nA : IsClosed (range fun m => (\u2191n)\u207b\u00b9 * \u2191m)\nB : \u00acx \u2208 range fun m => (\u2191n)\u207b\u00b9 * \u2191m\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 (range fun m => (\u2191n)\u207b\u00b9 * \u2191m)\u1d9c\n\u03b4 : \u211d\nh\u03b4 : \u03b4 \u2264 \u03b5\nm : \u2124\nhlt : dist x (\u2191m / \u2191n) < \u03b4\n\u22a2 False\n[PROOFSTEP]\nrw [dist_comm] at hlt \n[GOAL]\ncase intro.intro\nx : \u211d\nhx : Irrational x\nn : \u2115\nA : IsClosed (range fun m => (\u2191n)\u207b\u00b9 * \u2191m)\nB : \u00acx \u2208 range fun m => (\u2191n)\u207b\u00b9 * \u2191m\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 (range fun m => (\u2191n)\u207b\u00b9 * \u2191m)\u1d9c\n\u03b4 : \u211d\nh\u03b4 : \u03b4 \u2264 \u03b5\nm : \u2124\nhlt : dist (\u2191m / \u2191n) x < \u03b4\n\u22a2 False\n[PROOFSTEP]\nrefine' h\u03b5 (ball_subset_ball h\u03b4 hlt) \u27e8m, _\u27e9\n[GOAL]\ncase intro.intro\nx : \u211d\nhx : Irrational x\nn : \u2115\nA : IsClosed (range fun m => (\u2191n)\u207b\u00b9 * \u2191m)\nB : \u00acx \u2208 range fun m => (\u2191n)\u207b\u00b9 * \u2191m\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 (range fun m => (\u2191n)\u207b\u00b9 * \u2191m)\u1d9c\n\u03b4 : \u211d\nh\u03b4 : \u03b4 \u2264 \u03b5\nm : \u2124\nhlt : dist (\u2191m / \u2191n) x < \u03b4\n\u22a2 (fun m => (\u2191n)\u207b\u00b9 * \u2191m) m = \u2191m / \u2191n\n[PROOFSTEP]\nsimp [div_eq_inv_mul]\n[GOAL]\nx : \u211d\nhx : Irrational x\nn : \u2115\n\u03b5 : \u211d\nH : \u2200 (k : \u2115), k \u2264 n \u2192 \u2200 (m : \u2124), \u03b5 \u2264 dist x (\u2191m / \u2191k)\nr : \u211a\nhr : r.den \u2264 n\n\u22a2 \u03b5 \u2264 dist x \u2191r\n[PROOFSTEP]\nsimpa only [Rat.cast_def] using H r.den hr r.num\n", "meta": {"mathlib_filename": "Mathlib.Topology.Instances.Irrational", "llama_tokens": 2095, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583376458152, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.4072060266339285}} {"text": "[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\nn : \u2115\nx : X\nhx : x \u2208 c.C\n\u22a2 approx n c x = 0\n[PROOFSTEP]\ninduction' n with n ihn generalizing c\n[GOAL]\ncase zero\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : x \u2208 c\u271d.C\nc : CU X\nhx : x \u2208 c.C\n\u22a2 approx Nat.zero c x = 0\n[PROOFSTEP]\nexact indicator_of_not_mem (fun (hU : x \u2208 c.U\u1d9c) => hU <| c.subset hx) _\n[GOAL]\ncase succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : x \u2208 c\u271d.C\nn : \u2115\nihn : \u2200 (c : CU X), x \u2208 c.C \u2192 approx n c x = 0\nc : CU X\nhx : x \u2208 c.C\n\u22a2 approx (Nat.succ n) c x = 0\n[PROOFSTEP]\nsimp only [approx]\n[GOAL]\ncase succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : x \u2208 c\u271d.C\nn : \u2115\nihn : \u2200 (c : CU X), x \u2208 c.C \u2192 approx n c x = 0\nc : CU X\nhx : x \u2208 c.C\n\u22a2 midpoint \u211d (approx n (left c) x) (approx n (right c) x) = 0\n[PROOFSTEP]\nrw [ihn, ihn, midpoint_self]\n[GOAL]\ncase succ.hx\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : x \u2208 c\u271d.C\nn : \u2115\nihn : \u2200 (c : CU X), x \u2208 c.C \u2192 approx n c x = 0\nc : CU X\nhx : x \u2208 c.C\n\u22a2 x \u2208 (right c).C\ncase succ.hx\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : x \u2208 c\u271d.C\nn : \u2115\nihn : \u2200 (c : CU X), x \u2208 c.C \u2192 approx n c x = 0\nc : CU X\nhx : x \u2208 c.C\n\u22a2 x \u2208 (left c).C\n[PROOFSTEP]\nexacts [c.subset_right_C hx, hx]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\nn : \u2115\nx : X\nhx : \u00acx \u2208 c.U\n\u22a2 approx n c x = 1\n[PROOFSTEP]\ninduction' n with n ihn generalizing c\n[GOAL]\ncase zero\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nc : CU X\nhx : \u00acx \u2208 c.U\n\u22a2 approx Nat.zero c x = 1\n[PROOFSTEP]\nrw [\u2190 mem_compl_iff] at hx \n[GOAL]\ncase zero\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nc : CU X\nhx : x \u2208 c.U\u1d9c\n\u22a2 approx Nat.zero c x = 1\n[PROOFSTEP]\nexact indicator_of_mem hx _\n[GOAL]\ncase succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nn : \u2115\nihn : \u2200 (c : CU X), \u00acx \u2208 c.U \u2192 approx n c x = 1\nc : CU X\nhx : \u00acx \u2208 c.U\n\u22a2 approx (Nat.succ n) c x = 1\n[PROOFSTEP]\nsimp only [approx]\n[GOAL]\ncase succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nn : \u2115\nihn : \u2200 (c : CU X), \u00acx \u2208 c.U \u2192 approx n c x = 1\nc : CU X\nhx : \u00acx \u2208 c.U\n\u22a2 midpoint \u211d (approx n (left c) x) (approx n (right c) x) = 1\n[PROOFSTEP]\nrw [ihn, ihn, midpoint_self]\n[GOAL]\ncase succ.hx\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nn : \u2115\nihn : \u2200 (c : CU X), \u00acx \u2208 c.U \u2192 approx n c x = 1\nc : CU X\nhx : \u00acx \u2208 c.U\n\u22a2 \u00acx \u2208 (right c).U\ncase succ.hx\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nn : \u2115\nihn : \u2200 (c : CU X), \u00acx \u2208 c.U \u2192 approx n c x = 1\nc : CU X\nhx : \u00acx \u2208 c.U\n\u22a2 \u00acx \u2208 (left c).U\n[PROOFSTEP]\nexacts [hx, fun hU => hx <| c.left_U_subset hU]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\nn : \u2115\nx : X\n\u22a2 0 \u2264 approx n c x\n[PROOFSTEP]\ninduction' n with n ihn generalizing c\n[GOAL]\ncase zero\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nc : CU X\n\u22a2 0 \u2264 approx Nat.zero c x\n[PROOFSTEP]\nexact indicator_nonneg (fun _ _ => zero_le_one) _\n[GOAL]\ncase succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), 0 \u2264 approx n c x\nc : CU X\n\u22a2 0 \u2264 approx (Nat.succ n) c x\n[PROOFSTEP]\nsimp only [approx, midpoint_eq_smul_add, invOf_eq_inv]\n[GOAL]\ncase succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), 0 \u2264 approx n c x\nc : CU X\n\u22a2 0 \u2264 2\u207b\u00b9 \u2022 (approx n (left c) x + approx n (right c) x)\n[PROOFSTEP]\nrefine' mul_nonneg (inv_nonneg.2 zero_le_two) (add_nonneg _ _)\n[GOAL]\ncase succ.refine'_1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), 0 \u2264 approx n c x\nc : CU X\n\u22a2 0 \u2264 approx n (left c) x\n[PROOFSTEP]\napply ihn\n[GOAL]\ncase succ.refine'_2\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), 0 \u2264 approx n c x\nc : CU X\n\u22a2 0 \u2264 approx n (right c) x\n[PROOFSTEP]\napply ihn\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\nn : \u2115\nx : X\n\u22a2 approx n c x \u2264 1\n[PROOFSTEP]\ninduction' n with n ihn generalizing c\n[GOAL]\ncase zero\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nc : CU X\n\u22a2 approx Nat.zero c x \u2264 1\n[PROOFSTEP]\nexact indicator_apply_le' (fun _ => le_rfl) fun _ => zero_le_one\n[GOAL]\ncase succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), approx n c x \u2264 1\nc : CU X\n\u22a2 approx (Nat.succ n) c x \u2264 1\n[PROOFSTEP]\nsimp only [approx, midpoint_eq_smul_add, invOf_eq_inv, smul_eq_mul, \u2190 div_eq_inv_mul]\n[GOAL]\ncase succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), approx n c x \u2264 1\nc : CU X\n\u22a2 (approx n (left c) x + approx n (right c) x) / 2 \u2264 1\n[PROOFSTEP]\nhave := add_le_add (ihn (left c)) (ihn (right c))\n[GOAL]\ncase succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), approx n c x \u2264 1\nc : CU X\nthis : approx n (left c) x + approx n (right c) x \u2264 1 + 1\n\u22a2 (approx n (left c) x + approx n (right c) x) / 2 \u2264 1\n[PROOFSTEP]\nnorm_num at this \n[GOAL]\ncase succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), approx n c x \u2264 1\nc : CU X\nthis : approx n (left c) x + approx n (right c) x \u2264 2\n\u22a2 (approx n (left c) x + approx n (right c) x) / 2 \u2264 1\n[PROOFSTEP]\nexact Iff.mpr (div_le_one zero_lt_two) this\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u2081 c\u2082 : CU X\nh : c\u2081.U \u2286 c\u2082.C\nn\u2081 n\u2082 : \u2115\nx : X\n\u22a2 approx n\u2082 c\u2082 x \u2264 approx n\u2081 c\u2081 x\n[PROOFSTEP]\nby_cases hx : x \u2208 c\u2081.U\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u2081 c\u2082 : CU X\nh : c\u2081.U \u2286 c\u2082.C\nn\u2081 n\u2082 : \u2115\nx : X\nhx : x \u2208 c\u2081.U\n\u22a2 approx n\u2082 c\u2082 x \u2264 approx n\u2081 c\u2081 x\n[PROOFSTEP]\ncalc\n approx n\u2082 c\u2082 x = 0 := approx_of_mem_C _ _ (h hx)\n _ \u2264 approx n\u2081 c\u2081 x := approx_nonneg _ _ _\n[GOAL]\ncase neg\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u2081 c\u2082 : CU X\nh : c\u2081.U \u2286 c\u2082.C\nn\u2081 n\u2082 : \u2115\nx : X\nhx : \u00acx \u2208 c\u2081.U\n\u22a2 approx n\u2082 c\u2082 x \u2264 approx n\u2081 c\u2081 x\n[PROOFSTEP]\ncalc\n approx n\u2082 c\u2082 x \u2264 1 := approx_le_one _ _ _\n _ = approx n\u2081 c\u2081 x := (approx_of_nmem_U _ _ hx).symm\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\nn : \u2115\nx : X\n\u22a2 approx n c x \u2208 Icc (approx n (right c) x) (approx n (left c) x)\n[PROOFSTEP]\ninduction' n with n ihn generalizing c\n[GOAL]\ncase zero\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nc : CU X\n\u22a2 approx Nat.zero c x \u2208 Icc (approx Nat.zero (right c) x) (approx Nat.zero (left c) x)\n[PROOFSTEP]\nexact \u27e8le_rfl, indicator_le_indicator_of_subset (compl_subset_compl.2 c.left_U_subset) (fun _ => zero_le_one) _\u27e9\n[GOAL]\ncase succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), approx n c x \u2208 Icc (approx n (right c) x) (approx n (left c) x)\nc : CU X\n\u22a2 approx (Nat.succ n) c x \u2208 Icc (approx (Nat.succ n) (right c) x) (approx (Nat.succ n) (left c) x)\n[PROOFSTEP]\nsimp only [approx, mem_Icc]\n[GOAL]\ncase succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), approx n c x \u2208 Icc (approx n (right c) x) (approx n (left c) x)\nc : CU X\n\u22a2 midpoint \u211d (approx n (left (right c)) x) (approx n (right (right c)) x) \u2264\n midpoint \u211d (approx n (left c) x) (approx n (right c) x) \u2227\n midpoint \u211d (approx n (left c) x) (approx n (right c) x) \u2264\n midpoint \u211d (approx n (left (left c)) x) (approx n (right (left c)) x)\n[PROOFSTEP]\nrefine' \u27e8midpoint_le_midpoint _ (ihn _).1, midpoint_le_midpoint (ihn _).2 _\u27e9\n[GOAL]\ncase succ.refine'_1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), approx n c x \u2208 Icc (approx n (right c) x) (approx n (left c) x)\nc : CU X\n\u22a2 approx n (left (right c)) x \u2264 approx n (left c) x\n[PROOFSTEP]\napply approx_le_approx_of_U_sub_C\n[GOAL]\ncase succ.refine'_2\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), approx n c x \u2208 Icc (approx n (right c) x) (approx n (left c) x)\nc : CU X\n\u22a2 approx n (right c) x \u2264 approx n (right (left c)) x\n[PROOFSTEP]\napply approx_le_approx_of_U_sub_C\n[GOAL]\ncase succ.refine'_1.h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), approx n c x \u2208 Icc (approx n (right c) x) (approx n (left c) x)\nc : CU X\n\u22a2 (left c).U \u2286 (left (right c)).C\ncase succ.refine'_2.h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), approx n c x \u2208 Icc (approx n (right c) x) (approx n (left c) x)\nc : CU X\n\u22a2 (right (left c)).U \u2286 (right c).C\n[PROOFSTEP]\nexacts [subset_closure, subset_closure]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\nn : \u2115\nx : X\n\u22a2 approx n c x \u2264 approx (n + 1) c x\n[PROOFSTEP]\ninduction' n with n ihn generalizing c\n[GOAL]\ncase zero\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nc : CU X\n\u22a2 approx Nat.zero c x \u2264 approx (Nat.zero + 1) c x\n[PROOFSTEP]\nsimp only [approx, right_U, right_le_midpoint]\n[GOAL]\ncase zero\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nc : CU X\n\u22a2 indicator c.U\u1d9c 1 x \u2264 indicator (left c).U\u1d9c 1 x\n[PROOFSTEP]\nexact (approx_mem_Icc_right_left c 0 x).2\n[GOAL]\ncase succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), approx n c x \u2264 approx (n + 1) c x\nc : CU X\n\u22a2 approx (Nat.succ n) c x \u2264 approx (Nat.succ n + 1) c x\n[PROOFSTEP]\nrw [approx, approx]\n[GOAL]\ncase succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nn : \u2115\nihn : \u2200 (c : CU X), approx n c x \u2264 approx (n + 1) c x\nc : CU X\n\u22a2 midpoint \u211d (approx n (left c) x) (approx n (right c) x) \u2264\n midpoint \u211d (approx (n + 1) (left c) x) (approx (n + 1) (right c) x)\n[PROOFSTEP]\nexact midpoint_le_midpoint (ihn _) (ihn _)\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\nx : X\nh : x \u2208 c.C\n\u22a2 CU.lim c x = 0\n[PROOFSTEP]\nsimp only [CU.lim, approx_of_mem_C, h, ciSup_const]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\nx : X\nh : \u00acx \u2208 c.U\n\u22a2 CU.lim c x = 1\n[PROOFSTEP]\nsimp only [CU.lim, approx_of_nmem_U c _ h, ciSup_const]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\nx : X\n\u22a2 CU.lim c x = midpoint \u211d (CU.lim (left c) x) (CU.lim (right c) x)\n[PROOFSTEP]\nrefine' tendsto_nhds_unique (c.tendsto_approx_atTop x) ((tendsto_add_atTop_iff_nat 1).1 _)\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\nx : X\n\u22a2 Tendsto (fun n => approx (n + 1) c x) atTop (\ud835\udcdd (midpoint \u211d (CU.lim (left c) x) (CU.lim (right c) x)))\n[PROOFSTEP]\nsimp only [approx]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\nx : X\n\u22a2 Tendsto (fun n => midpoint \u211d (approx (Nat.add n 0) (left c) x) (approx (Nat.add n 0) (right c) x)) atTop\n (\ud835\udcdd (midpoint \u211d (CU.lim (left c) x) (CU.lim (right c) x)))\n[PROOFSTEP]\nexact (c.left.tendsto_approx_atTop x).midpoint (c.right.tendsto_approx_atTop x)\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\n\u22a2 Continuous (CU.lim c)\n[PROOFSTEP]\nobtain \u27e8h0, h1234, h1\u27e9 : 0 < (2\u207b\u00b9 : \u211d) \u2227 (2\u207b\u00b9 : \u211d) < 3 / 4 \u2227 (3 / 4 : \u211d) < 1 := by norm_num\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\n\u22a2 0 < 2\u207b\u00b9 \u2227 2\u207b\u00b9 < 3 / 4 \u2227 3 / 4 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.intro\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\n\u22a2 Continuous (CU.lim c)\n[PROOFSTEP]\nrefine'\n continuous_iff_continuousAt.2 fun x =>\n (Metric.nhds_basis_closedBall_pow (h0.trans h1234) h1).tendsto_right_iff.2 fun n _ => _\n[GOAL]\ncase intro.intro\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nn : \u2115\nx\u271d : True\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, CU.lim c x_1 \u2208 Metric.closedBall (CU.lim c x) ((3 / 4) ^ n)\n[PROOFSTEP]\nsimp only [Metric.mem_closedBall]\n[GOAL]\ncase intro.intro\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nn : \u2115\nx\u271d : True\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\n[PROOFSTEP]\ninduction' n with n ihn generalizing c\n[GOAL]\ncase intro.intro.zero\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nc : CU X\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ Nat.zero\n[PROOFSTEP]\nrefine' eventually_of_forall fun y => _\n[GOAL]\ncase intro.intro.zero\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nc : CU X\ny : X\n\u22a2 dist (CU.lim c y) (CU.lim c x) \u2264 (3 / 4) ^ Nat.zero\n[PROOFSTEP]\nrw [pow_zero]\n[GOAL]\ncase intro.intro.zero\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nc : CU X\ny : X\n\u22a2 dist (CU.lim c y) (CU.lim c x) \u2264 1\n[PROOFSTEP]\nexact Real.dist_le_of_mem_Icc_01 (c.lim_mem_Icc _) (c.lim_mem_Icc _)\n[GOAL]\ncase intro.intro.succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ Nat.succ n\n[PROOFSTEP]\nby_cases hxl : x \u2208 c.left.U\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\nhxl : x \u2208 (left c).U\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ Nat.succ n\n[PROOFSTEP]\nfilter_upwards [IsOpen.mem_nhds c.left.open_U hxl, ihn c.left] with _ hyl hyd\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\nhxl : x \u2208 (left c).U\na\u271d : X\nhyl : a\u271d \u2208 (left c).U\nhyd : dist (CU.lim (left c) a\u271d) (CU.lim (left c) x) \u2264 (3 / 4) ^ n\n\u22a2 dist (CU.lim c a\u271d) (CU.lim c x) \u2264 (3 / 4) ^ Nat.succ n\n[PROOFSTEP]\nrw [pow_succ, c.lim_eq_midpoint, c.lim_eq_midpoint, c.right.lim_of_mem_C _ (c.left_U_subset_right_C hyl),\n c.right.lim_of_mem_C _ (c.left_U_subset_right_C hxl)]\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\nhxl : x \u2208 (left c).U\na\u271d : X\nhyl : a\u271d \u2208 (left c).U\nhyd : dist (CU.lim (left c) a\u271d) (CU.lim (left c) x) \u2264 (3 / 4) ^ n\n\u22a2 dist (midpoint \u211d (CU.lim (left c) a\u271d) 0) (midpoint \u211d (CU.lim (left c) x) 0) \u2264 3 / 4 * (3 / 4) ^ n\n[PROOFSTEP]\nrefine' (dist_midpoint_midpoint_le _ _ _ _).trans _\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\nhxl : x \u2208 (left c).U\na\u271d : X\nhyl : a\u271d \u2208 (left c).U\nhyd : dist (CU.lim (left c) a\u271d) (CU.lim (left c) x) \u2264 (3 / 4) ^ n\n\u22a2 (dist (CU.lim (left c) a\u271d) (CU.lim (left c) x) + dist 0 0) / 2 \u2264 3 / 4 * (3 / 4) ^ n\n[PROOFSTEP]\nrw [dist_self, add_zero, div_eq_inv_mul]\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\nhxl : x \u2208 (left c).U\na\u271d : X\nhyl : a\u271d \u2208 (left c).U\nhyd : dist (CU.lim (left c) a\u271d) (CU.lim (left c) x) \u2264 (3 / 4) ^ n\n\u22a2 2\u207b\u00b9 * dist (CU.lim (left c) a\u271d) (CU.lim (left c) x) \u2264 3 / 4 * (3 / 4) ^ n\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase neg\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\nhxl : \u00acx \u2208 (left c).U\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ Nat.succ n\n[PROOFSTEP]\nreplace hxl : x \u2208 c.left.right.C\u1d9c\n[GOAL]\ncase hxl\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\nhxl : \u00acx \u2208 (left c).U\n\u22a2 x \u2208 (right (left c)).C\u1d9c\ncase neg\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\nhxl : x \u2208 (right (left c)).C\u1d9c\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ Nat.succ n\n[PROOFSTEP]\nexact compl_subset_compl.2 c.left.right.subset hxl\n[GOAL]\ncase neg\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\nhxl : x \u2208 (right (left c)).C\u1d9c\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ Nat.succ n\n[PROOFSTEP]\nfilter_upwards [IsOpen.mem_nhds (isOpen_compl_iff.2 c.left.right.closed_C) hxl, ihn c.left.right, ihn c.right] with y\n hyl hydl hydr\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\nhxl : x \u2208 (right (left c)).C\u1d9c\ny : X\nhyl : y \u2208 (right (left c)).C\u1d9c\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) \u2264 (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) \u2264 (3 / 4) ^ n\n\u22a2 dist (CU.lim c y) (CU.lim c x) \u2264 (3 / 4) ^ Nat.succ n\n[PROOFSTEP]\nreplace hxl : x \u2209 c.left.left.U\n[GOAL]\ncase hxl\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\nhxl : x \u2208 (right (left c)).C\u1d9c\ny : X\nhyl : y \u2208 (right (left c)).C\u1d9c\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) \u2264 (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) \u2264 (3 / 4) ^ n\n\u22a2 \u00acx \u2208 (left (left c)).U\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\ny : X\nhyl : y \u2208 (right (left c)).C\u1d9c\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) \u2264 (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) \u2264 (3 / 4) ^ n\nhxl : \u00acx \u2208 (left (left c)).U\n\u22a2 dist (CU.lim c y) (CU.lim c x) \u2264 (3 / 4) ^ Nat.succ n\n[PROOFSTEP]\nexact compl_subset_compl.2 c.left.left_U_subset_right_C hxl\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\ny : X\nhyl : y \u2208 (right (left c)).C\u1d9c\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) \u2264 (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) \u2264 (3 / 4) ^ n\nhxl : \u00acx \u2208 (left (left c)).U\n\u22a2 dist (CU.lim c y) (CU.lim c x) \u2264 (3 / 4) ^ Nat.succ n\n[PROOFSTEP]\nreplace hyl : y \u2209 c.left.left.U\n[GOAL]\ncase hyl\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\ny : X\nhyl : y \u2208 (right (left c)).C\u1d9c\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) \u2264 (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) \u2264 (3 / 4) ^ n\nhxl : \u00acx \u2208 (left (left c)).U\n\u22a2 \u00acy \u2208 (left (left c)).U\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) \u2264 (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) \u2264 (3 / 4) ^ n\nhxl : \u00acx \u2208 (left (left c)).U\nhyl : \u00acy \u2208 (left (left c)).U\n\u22a2 dist (CU.lim c y) (CU.lim c x) \u2264 (3 / 4) ^ Nat.succ n\n[PROOFSTEP]\nexact compl_subset_compl.2 c.left.left_U_subset_right_C hyl\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) \u2264 (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) \u2264 (3 / 4) ^ n\nhxl : \u00acx \u2208 (left (left c)).U\nhyl : \u00acy \u2208 (left (left c)).U\n\u22a2 dist (CU.lim c y) (CU.lim c x) \u2264 (3 / 4) ^ Nat.succ n\n[PROOFSTEP]\nsimp only [pow_succ, c.lim_eq_midpoint, c.left.lim_eq_midpoint, c.left.left.lim_of_nmem_U _ hxl,\n c.left.left.lim_of_nmem_U _ hyl]\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) \u2264 (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) \u2264 (3 / 4) ^ n\nhxl : \u00acx \u2208 (left (left c)).U\nhyl : \u00acy \u2208 (left (left c)).U\n\u22a2 dist (midpoint \u211d (midpoint \u211d 1 (CU.lim (right (left c)) y)) (CU.lim (right c) y))\n (midpoint \u211d (midpoint \u211d 1 (CU.lim (right (left c)) x)) (CU.lim (right c) x)) \u2264\n 3 / 4 * (3 / 4) ^ n\n[PROOFSTEP]\nrefine' (dist_midpoint_midpoint_le _ _ _ _).trans _\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) \u2264 (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) \u2264 (3 / 4) ^ n\nhxl : \u00acx \u2208 (left (left c)).U\nhyl : \u00acy \u2208 (left (left c)).U\n\u22a2 (dist (midpoint \u211d 1 (CU.lim (right (left c)) y)) (midpoint \u211d 1 (CU.lim (right (left c)) x)) +\n dist (CU.lim (right c) y) (CU.lim (right c) x)) /\n 2 \u2264\n 3 / 4 * (3 / 4) ^ n\n[PROOFSTEP]\nrefine' (div_le_div_of_le_of_nonneg (add_le_add_right (dist_midpoint_midpoint_le _ _ _ _) _) zero_le_two).trans _\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) \u2264 (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) \u2264 (3 / 4) ^ n\nhxl : \u00acx \u2208 (left (left c)).U\nhyl : \u00acy \u2208 (left (left c)).U\n\u22a2 ((dist 1 1 + dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x)) / 2 +\n dist (CU.lim (right c) y) (CU.lim (right c) x)) /\n 2 \u2264\n 3 / 4 * (3 / 4) ^ n\n[PROOFSTEP]\nrw [dist_self, zero_add]\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 (3 / 4) ^ n\nc : CU X\ny : X\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) \u2264 (3 / 4) ^ n\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) \u2264 (3 / 4) ^ n\nhxl : \u00acx \u2208 (left (left c)).U\nhyl : \u00acy \u2208 (left (left c)).U\n\u22a2 (dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) / 2 + dist (CU.lim (right c) y) (CU.lim (right c) x)) /\n 2 \u2264\n 3 / 4 * (3 / 4) ^ n\n[PROOFSTEP]\nset r := (3 / 4 : \u211d) ^ n\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nc : CU X\ny : X\nhxl : \u00acx \u2208 (left (left c)).U\nhyl : \u00acy \u2208 (left (left c)).U\nr : \u211d := (3 / 4) ^ n\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 r\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) \u2264 r\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) \u2264 r\n\u22a2 (dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) / 2 + dist (CU.lim (right c) y) (CU.lim (right c) x)) /\n 2 \u2264\n 3 / 4 * r\n[PROOFSTEP]\ncalc\n _ \u2264 (r / 2 + r) / 2 := by gcongr\n _ = _ := by field_simp; ring\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nc : CU X\ny : X\nhxl : \u00acx \u2208 (left (left c)).U\nhyl : \u00acy \u2208 (left (left c)).U\nr : \u211d := (3 / 4) ^ n\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 r\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) \u2264 r\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) \u2264 r\n\u22a2 (dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) / 2 + dist (CU.lim (right c) y) (CU.lim (right c) x)) /\n 2 \u2264\n (r / 2 + r) / 2\n[PROOFSTEP]\ngcongr\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nc : CU X\ny : X\nhxl : \u00acx \u2208 (left (left c)).U\nhyl : \u00acy \u2208 (left (left c)).U\nr : \u211d := (3 / 4) ^ n\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 r\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) \u2264 r\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) \u2264 r\n\u22a2 (r / 2 + r) / 2 = 3 / 4 * r\n[PROOFSTEP]\nfield_simp\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nc : CU X\ny : X\nhxl : \u00acx \u2208 (left (left c)).U\nhyl : \u00acy \u2208 (left (left c)).U\nr : \u211d := (3 / 4) ^ n\nihn : \u2200 (c : CU X), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (CU.lim c x_1) (CU.lim c x) \u2264 r\nhydl : dist (CU.lim (right (left c)) y) (CU.lim (right (left c)) x) \u2264 r\nhydr : dist (CU.lim (right c) y) (CU.lim (right c) x) \u2264 r\n\u22a2 (3 ^ n * 4 ^ n + 3 ^ n * (4 ^ n * 2)) * (4 * 4 ^ n) = 3 * 3 ^ n * (4 ^ n * 2 * 4 ^ n * 2)\n[PROOFSTEP]\nring\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\ns t : Set X\nhs : IsClosed s\nht : IsClosed t\nhd : Disjoint s t\n\u22a2 \u2203 f, EqOn (\u2191f) 0 s \u2227 EqOn (\u2191f) 1 t \u2227 \u2200 (x : X), \u2191f x \u2208 Icc 0 1\n[PROOFSTEP]\nset c : Urysohns.CU X := \u27e8s, t\u1d9c, hs, ht.isOpen_compl, disjoint_left.1 hd\u27e9\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\ns t : Set X\nhs : IsClosed s\nht : IsClosed t\nhd : Disjoint s t\nc : Urysohns.CU X :=\n { C := s, U := t\u1d9c, closed_C := hs, open_U := (_ : IsOpen t\u1d9c), subset := (_ : \u2200 \u2983a : X\u2984, a \u2208 s \u2192 \u00aca \u2208 t) }\n\u22a2 \u2203 f, EqOn (\u2191f) 0 s \u2227 EqOn (\u2191f) 1 t \u2227 \u2200 (x : X), \u2191f x \u2208 Icc 0 1\n[PROOFSTEP]\nexact \u27e8\u27e8c.lim, c.continuous_lim\u27e9, c.lim_of_mem_C, fun x hx => c.lim_of_nmem_U _ fun h => h hx, c.lim_mem_Icc\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.UrysohnsLemma", "llama_tokens": 15770, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.4071407222236004}} {"text": "[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns\u271d S : Set G\ns : G\nhs : s \u2208 S\u207b\u00b9\n\u22a2 s \u2208 \u2191(closure S)\n[PROOFSTEP]\nrw [SetLike.mem_coe, \u2190 Subgroup.inv_mem_iff]\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns\u271d S : Set G\ns : G\nhs : s \u2208 S\u207b\u00b9\n\u22a2 s\u207b\u00b9 \u2208 closure S\n[PROOFSTEP]\nexact subset_closure (mem_inv.mp hs)\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns S : Set G\n\u22a2 (closure S).toSubmonoid = Submonoid.closure (S \u222a S\u207b\u00b9)\n[PROOFSTEP]\nrefine le_antisymm (fun x hx => ?_) (Submonoid.closure_le.2 ?_)\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns S : Set G\nx : G\nhx : x \u2208 (closure S).toSubmonoid\n\u22a2 x \u2208 Submonoid.closure (S \u222a S\u207b\u00b9)\n[PROOFSTEP]\nrefine'\n closure_induction hx (fun x hx => Submonoid.closure_mono (subset_union_left S S\u207b\u00b9) (Submonoid.subset_closure hx))\n (Submonoid.one_mem _) (fun x y hx hy => Submonoid.mul_mem _ hx hy) fun x hx => _\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns S : Set G\nx\u271d : G\nhx\u271d : x\u271d \u2208 (closure S).toSubmonoid\nx : G\nhx : x \u2208 Submonoid.closure (S \u222a S\u207b\u00b9)\n\u22a2 x\u207b\u00b9 \u2208 Submonoid.closure (S \u222a S\u207b\u00b9)\n[PROOFSTEP]\nrwa [\u2190 Submonoid.mem_closure_inv, Set.union_inv, inv_inv, Set.union_comm]\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns S : Set G\n\u22a2 S \u222a S\u207b\u00b9 \u2286 \u2191(closure S).toSubmonoid\n[PROOFSTEP]\nsimp only [true_and_iff, coe_toSubmonoid, union_subset_iff, subset_closure, inv_subset_closure]\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns\u271d s : Set G\n\u22a2 closure s\u207b\u00b9 = closure s\n[PROOFSTEP]\nsimp only [\u2190 toSubmonoid_eq, closure_toSubmonoid, inv_inv, union_comm]\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\n\u03b9 : Sort u_5\nS : \u03b9 \u2192 Subgroup G\nC : G \u2192 Prop\nx : G\nhx : x \u2208 \u2a06 (i : \u03b9), S i\nhp : \u2200 (i : \u03b9) (x : G), x \u2208 S i \u2192 C x\nh1 : C 1\nhmul : \u2200 (x y : G), C x \u2192 C y \u2192 C (x * y)\n\u22a2 C x\n[PROOFSTEP]\nrw [iSup_eq_closure] at hx \n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\n\u03b9 : Sort u_5\nS : \u03b9 \u2192 Subgroup G\nC : G \u2192 Prop\nx : G\nhx : x \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nhp : \u2200 (i : \u03b9) (x : G), x \u2208 S i \u2192 C x\nh1 : C 1\nhmul : \u2200 (x y : G), C x \u2192 C y \u2192 C (x * y)\n\u22a2 C x\n[PROOFSTEP]\nrefine' closure_induction'' hx (fun x hx => _) (fun x hx => _) h1 hmul\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\n\u03b9 : Sort u_5\nS : \u03b9 \u2192 Subgroup G\nC : G \u2192 Prop\nx\u271d : G\nhx\u271d : x\u271d \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nhp : \u2200 (i : \u03b9) (x : G), x \u2208 S i \u2192 C x\nh1 : C 1\nhmul : \u2200 (x y : G), C x \u2192 C y \u2192 C (x * y)\nx : G\nhx : x \u2208 \u22c3 (i : \u03b9), \u2191(S i)\n\u22a2 C x\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := Set.mem_iUnion.mp hx\n[GOAL]\ncase refine'_1.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\n\u03b9 : Sort u_5\nS : \u03b9 \u2192 Subgroup G\nC : G \u2192 Prop\nx\u271d : G\nhx\u271d : x\u271d \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nhp : \u2200 (i : \u03b9) (x : G), x \u2208 S i \u2192 C x\nh1 : C 1\nhmul : \u2200 (x y : G), C x \u2192 C y \u2192 C (x * y)\nx : G\nhx : x \u2208 \u22c3 (i : \u03b9), \u2191(S i)\ni : \u03b9\nhi : x \u2208 \u2191(S i)\n\u22a2 C x\n[PROOFSTEP]\nexact hp _ _ hi\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\n\u03b9 : Sort u_5\nS : \u03b9 \u2192 Subgroup G\nC : G \u2192 Prop\nx\u271d : G\nhx\u271d : x\u271d \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nhp : \u2200 (i : \u03b9) (x : G), x \u2208 S i \u2192 C x\nh1 : C 1\nhmul : \u2200 (x y : G), C x \u2192 C y \u2192 C (x * y)\nx : G\nhx : x \u2208 \u22c3 (i : \u03b9), \u2191(S i)\n\u22a2 C x\u207b\u00b9\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := Set.mem_iUnion.mp hx\n[GOAL]\ncase refine'_2.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\n\u03b9 : Sort u_5\nS : \u03b9 \u2192 Subgroup G\nC : G \u2192 Prop\nx\u271d : G\nhx\u271d : x\u271d \u2208 closure (\u22c3 (i : \u03b9), \u2191(S i))\nhp : \u2200 (i : \u03b9) (x : G), x \u2208 S i \u2192 C x\nh1 : C 1\nhmul : \u2200 (x y : G), C x \u2192 C y \u2192 C (x * y)\nx : G\nhx : x \u2208 \u22c3 (i : \u03b9), \u2191(S i)\ni : \u03b9\nhi : x \u2208 \u2191(S i)\n\u22a2 C x\u207b\u00b9\n[PROOFSTEP]\nexact hp _ _ (inv_mem hi)\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\n\u03b9 : Sort u_5\nS : \u03b9 \u2192 Subgroup G\nC : (x : G) \u2192 x \u2208 \u2a06 (i : \u03b9), S i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : G) (hx : x \u2208 S i), C x (_ : x \u2208 \u2a06 (i : \u03b9), S i)\nh1 : C 1 (_ : 1 \u2208 \u2a06 (i : \u03b9), S i)\nhmul :\n \u2200 (x y : G) (hx : x \u2208 \u2a06 (i : \u03b9), S i) (hy : y \u2208 \u2a06 (i : \u03b9), S i),\n C x hx \u2192 C y hy \u2192 C (x * y) (_ : x * y \u2208 \u2a06 (i : \u03b9), S i)\nx : G\nhx : x \u2208 \u2a06 (i : \u03b9), S i\n\u22a2 C x hx\n[PROOFSTEP]\nsuffices : \u2203 h, C x h\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\n\u03b9 : Sort u_5\nS : \u03b9 \u2192 Subgroup G\nC : (x : G) \u2192 x \u2208 \u2a06 (i : \u03b9), S i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : G) (hx : x \u2208 S i), C x (_ : x \u2208 \u2a06 (i : \u03b9), S i)\nh1 : C 1 (_ : 1 \u2208 \u2a06 (i : \u03b9), S i)\nhmul :\n \u2200 (x y : G) (hx : x \u2208 \u2a06 (i : \u03b9), S i) (hy : y \u2208 \u2a06 (i : \u03b9), S i),\n C x hx \u2192 C y hy \u2192 C (x * y) (_ : x * y \u2208 \u2a06 (i : \u03b9), S i)\nx : G\nhx : x \u2208 \u2a06 (i : \u03b9), S i\nthis : \u2203 h, C x h\n\u22a2 C x hx\ncase this\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\n\u03b9 : Sort u_5\nS : \u03b9 \u2192 Subgroup G\nC : (x : G) \u2192 x \u2208 \u2a06 (i : \u03b9), S i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : G) (hx : x \u2208 S i), C x (_ : x \u2208 \u2a06 (i : \u03b9), S i)\nh1 : C 1 (_ : 1 \u2208 \u2a06 (i : \u03b9), S i)\nhmul :\n \u2200 (x y : G) (hx : x \u2208 \u2a06 (i : \u03b9), S i) (hy : y \u2208 \u2a06 (i : \u03b9), S i),\n C x hx \u2192 C y hy \u2192 C (x * y) (_ : x * y \u2208 \u2a06 (i : \u03b9), S i)\nx : G\nhx : x \u2208 \u2a06 (i : \u03b9), S i\n\u22a2 \u2203 h, C x h\n[PROOFSTEP]\nexact this.snd\n[GOAL]\ncase this\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\n\u03b9 : Sort u_5\nS : \u03b9 \u2192 Subgroup G\nC : (x : G) \u2192 x \u2208 \u2a06 (i : \u03b9), S i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : G) (hx : x \u2208 S i), C x (_ : x \u2208 \u2a06 (i : \u03b9), S i)\nh1 : C 1 (_ : 1 \u2208 \u2a06 (i : \u03b9), S i)\nhmul :\n \u2200 (x y : G) (hx : x \u2208 \u2a06 (i : \u03b9), S i) (hy : y \u2208 \u2a06 (i : \u03b9), S i),\n C x hx \u2192 C y hy \u2192 C (x * y) (_ : x * y \u2208 \u2a06 (i : \u03b9), S i)\nx : G\nhx : x \u2208 \u2a06 (i : \u03b9), S i\n\u22a2 \u2203 h, C x h\n[PROOFSTEP]\nrefine' iSup_induction S (C := fun x => \u2203 h, C x h) hx (fun i x hx => _) _ fun x y => _\n[GOAL]\ncase this.refine'_1\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\n\u03b9 : Sort u_5\nS : \u03b9 \u2192 Subgroup G\nC : (x : G) \u2192 x \u2208 \u2a06 (i : \u03b9), S i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : G) (hx : x \u2208 S i), C x (_ : x \u2208 \u2a06 (i : \u03b9), S i)\nh1 : C 1 (_ : 1 \u2208 \u2a06 (i : \u03b9), S i)\nhmul :\n \u2200 (x y : G) (hx : x \u2208 \u2a06 (i : \u03b9), S i) (hy : y \u2208 \u2a06 (i : \u03b9), S i),\n C x hx \u2192 C y hy \u2192 C (x * y) (_ : x * y \u2208 \u2a06 (i : \u03b9), S i)\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2a06 (i : \u03b9), S i\ni : \u03b9\nx : G\nhx : x \u2208 S i\n\u22a2 (fun x => \u2203 h, C x h) x\n[PROOFSTEP]\nexact \u27e8_, hp i _ hx\u27e9\n[GOAL]\ncase this.refine'_2\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\n\u03b9 : Sort u_5\nS : \u03b9 \u2192 Subgroup G\nC : (x : G) \u2192 x \u2208 \u2a06 (i : \u03b9), S i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : G) (hx : x \u2208 S i), C x (_ : x \u2208 \u2a06 (i : \u03b9), S i)\nh1 : C 1 (_ : 1 \u2208 \u2a06 (i : \u03b9), S i)\nhmul :\n \u2200 (x y : G) (hx : x \u2208 \u2a06 (i : \u03b9), S i) (hy : y \u2208 \u2a06 (i : \u03b9), S i),\n C x hx \u2192 C y hy \u2192 C (x * y) (_ : x * y \u2208 \u2a06 (i : \u03b9), S i)\nx : G\nhx : x \u2208 \u2a06 (i : \u03b9), S i\n\u22a2 (fun x => \u2203 h, C x h) 1\n[PROOFSTEP]\nexact \u27e8_, h1\u27e9\n[GOAL]\ncase this.refine'_3\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\n\u03b9 : Sort u_5\nS : \u03b9 \u2192 Subgroup G\nC : (x : G) \u2192 x \u2208 \u2a06 (i : \u03b9), S i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : G) (hx : x \u2208 S i), C x (_ : x \u2208 \u2a06 (i : \u03b9), S i)\nh1 : C 1 (_ : 1 \u2208 \u2a06 (i : \u03b9), S i)\nhmul :\n \u2200 (x y : G) (hx : x \u2208 \u2a06 (i : \u03b9), S i) (hy : y \u2208 \u2a06 (i : \u03b9), S i),\n C x hx \u2192 C y hy \u2192 C (x * y) (_ : x * y \u2208 \u2a06 (i : \u03b9), S i)\nx\u271d : G\nhx : x\u271d \u2208 \u2a06 (i : \u03b9), S i\nx y : G\n\u22a2 (fun x => \u2203 h, C x h) x \u2192 (fun x => \u2203 h, C x h) y \u2192 (fun x => \u2203 h, C x h) (x * y)\n[PROOFSTEP]\nrintro \u27e8_, Cx\u27e9 \u27e8_, Cy\u27e9\n[GOAL]\ncase this.refine'_3.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\n\u03b9 : Sort u_5\nS : \u03b9 \u2192 Subgroup G\nC : (x : G) \u2192 x \u2208 \u2a06 (i : \u03b9), S i \u2192 Prop\nhp : \u2200 (i : \u03b9) (x : G) (hx : x \u2208 S i), C x (_ : x \u2208 \u2a06 (i : \u03b9), S i)\nh1 : C 1 (_ : 1 \u2208 \u2a06 (i : \u03b9), S i)\nhmul :\n \u2200 (x y : G) (hx : x \u2208 \u2a06 (i : \u03b9), S i) (hy : y \u2208 \u2a06 (i : \u03b9), S i),\n C x hx \u2192 C y hy \u2192 C (x * y) (_ : x * y \u2208 \u2a06 (i : \u03b9), S i)\nx\u271d : G\nhx : x\u271d \u2208 \u2a06 (i : \u03b9), S i\nx y : G\nw\u271d\u00b9 : x \u2208 \u2a06 (i : \u03b9), S i\nCx : C x w\u271d\u00b9\nw\u271d : y \u2208 \u2a06 (i : \u03b9), S i\nCy : C y w\u271d\n\u22a2 \u2203 h, C (x * y) h\n[PROOFSTEP]\nrefine' \u27e8_, hmul _ _ _ _ Cx Cy\u27e9\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH K : Subgroup G\n\u22a2 closure \u2191H \u2294 closure \u2191K \u2264 H \u2294 K\n[PROOFSTEP]\nrw [closure_eq, closure_eq]\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns\u271d s : Set G\nN : Subgroup G\nhN : Normal N\n\u22a2 s * \u2191N = \u2191N * s\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns\u271d s : Set G\nN : Subgroup G\nhN : Normal N\nx : G\n\u22a2 x \u2208 s * \u2191N \u2194 x \u2208 \u2191N * s\n[PROOFSTEP]\nrefine (exists_congr fun y => ?_).trans exists_swap\n[GOAL]\ncase h\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns\u271d s : Set G\nN : Subgroup G\nhN : Normal N\nx y : G\n\u22a2 (\u2203 b, y \u2208 s \u2227 b \u2208 \u2191N \u2227 (fun x x_1 => x * x_1) y b = x) \u2194 \u2203 y_1, y_1 \u2208 \u2191N \u2227 y \u2208 s \u2227 (fun x x_1 => x * x_1) y_1 y = x\n[PROOFSTEP]\nsimp only [exists_and_left, @and_left_comm _ (y \u2208 s), \u2190 eq_inv_mul_iff_mul_eq (b := y), \u2190\n eq_mul_inv_iff_mul_eq (c := y), exists_eq_right, SetLike.mem_coe, hN.mem_comm_iff]\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\n\u22a2 \u2191(H \u2294 N) = \u2191H * \u2191N\n[PROOFSTEP]\nrw [sup_eq_closure]\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\n\u22a2 \u2191(closure (\u2191H * \u2191N)) = \u2191H * \u2191N\n[PROOFSTEP]\nrefine Set.Subset.antisymm (fun x hx => ?_) subset_closure\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx : G\nhx : x \u2208 \u2191(closure (\u2191H * \u2191N))\n\u22a2 x \u2208 \u2191H * \u2191N\n[PROOFSTEP]\nrefine closure_induction'' (p := fun x => x \u2208 (H : Set G) * (N : Set G)) hx ?_ ?_ ?_ ?_\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx : G\nhx : x \u2208 \u2191(closure (\u2191H * \u2191N))\n\u22a2 \u2200 (x : G), x \u2208 \u2191H * \u2191N \u2192 (fun x => x \u2208 \u2191H * \u2191N) x\n[PROOFSTEP]\nrintro _ \u27e8x, y, hx, hy, rfl\u27e9\n[GOAL]\ncase refine_1.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N))\nx y : G\nhx : x \u2208 \u2191H\nhy : y \u2208 \u2191N\n\u22a2 (fun x x_1 => x * x_1) x y \u2208 \u2191H * \u2191N\n[PROOFSTEP]\nexact mul_mem_mul hx hy\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx : G\nhx : x \u2208 \u2191(closure (\u2191H * \u2191N))\n\u22a2 \u2200 (x : G), x \u2208 \u2191H * \u2191N \u2192 (fun x => x \u2208 \u2191H * \u2191N) x\u207b\u00b9\n[PROOFSTEP]\nrintro _ \u27e8x, y, hx, hy, rfl\u27e9\n[GOAL]\ncase refine_2.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N))\nx y : G\nhx : x \u2208 \u2191H\nhy : y \u2208 \u2191N\n\u22a2 ((fun x x_1 => x * x_1) x y)\u207b\u00b9 \u2208 \u2191H * \u2191N\n[PROOFSTEP]\nsimpa only [mul_inv_rev, mul_assoc, inv_inv, inv_mul_cancel_left] using\n mul_mem_mul (inv_mem hx) (hN.conj_mem _ (inv_mem hy) x)\n[GOAL]\ncase refine_3\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx : G\nhx : x \u2208 \u2191(closure (\u2191H * \u2191N))\n\u22a2 (fun x => x \u2208 \u2191H * \u2191N) 1\n[PROOFSTEP]\nexact \u27e81, 1, one_mem _, one_mem _, mul_one 1\u27e9\n[GOAL]\ncase refine_4\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx : G\nhx : x \u2208 \u2191(closure (\u2191H * \u2191N))\n\u22a2 \u2200 (x y : G), (fun x => x \u2208 \u2191H * \u2191N) x \u2192 (fun x => x \u2208 \u2191H * \u2191N) y \u2192 (fun x => x \u2208 \u2191H * \u2191N) (x * y)\n[PROOFSTEP]\nrintro _ _ \u27e8x, y, hx, hy, rfl\u27e9 \u27e8x', y', hx', hy', rfl\u27e9\n[GOAL]\ncase refine_4.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N))\nx y : G\nhx : x \u2208 \u2191H\nhy : y \u2208 \u2191N\nx' y' : G\nhx' : x' \u2208 \u2191H\nhy' : y' \u2208 \u2191N\n\u22a2 (fun x x_1 => x * x_1) x y * (fun x x_1 => x * x_1) x' y' \u2208 \u2191H * \u2191N\n[PROOFSTEP]\nrefine \u27e8x * x', x'\u207b\u00b9 * y * x' * y', mul_mem hx hx', mul_mem ?_ hy', ?_\u27e9\n[GOAL]\ncase refine_4.intro.intro.intro.intro.intro.intro.intro.intro.refine_1\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N))\nx y : G\nhx : x \u2208 \u2191H\nhy : y \u2208 \u2191N\nx' y' : G\nhx' : x' \u2208 \u2191H\nhy' : y' \u2208 \u2191N\n\u22a2 x'\u207b\u00b9 * y * x' \u2208 N\n[PROOFSTEP]\nsimpa using hN.conj_mem _ hy x'\u207b\u00b9\n[GOAL]\ncase refine_4.intro.intro.intro.intro.intro.intro.intro.intro.refine_2\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N))\nx y : G\nhx : x \u2208 \u2191H\nhy : y \u2208 \u2191N\nx' y' : G\nhx' : x' \u2208 \u2191H\nhy' : y' \u2208 \u2191N\n\u22a2 (fun x x_1 => x * x_1) (x * x') (x'\u207b\u00b9 * y * x' * y') = (fun x x_1 => x * x_1) x y * (fun x x_1 => x * x_1) x' y'\n[PROOFSTEP]\nsimp only [mul_assoc, mul_inv_cancel_left]\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN H : Subgroup G\ninst\u271d : Normal N\n\u22a2 \u2191(N \u2294 H) = \u2191N * \u2191H\n[PROOFSTEP]\nrw [\u2190 set_mul_normal_comm, sup_comm, mul_normal]\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : A \u2264 C\n\u22a2 \u2191A * \u2191(B \u2293 C) = \u2191A * \u2191B \u2293 \u2191C\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : A \u2264 C\nx\u271d : G\n\u22a2 x\u271d \u2208 \u2191A * \u2191(B \u2293 C) \u2194 x\u271d \u2208 \u2191A * \u2191B \u2293 \u2191C\n[PROOFSTEP]\nsimp only [coe_inf, Set.inf_eq_inter, Set.mem_mul, Set.mem_inter_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : A \u2264 C\nx\u271d : G\n\u22a2 (\u2203 x y, x \u2208 \u2191A \u2227 (y \u2208 \u2191B \u2227 y \u2208 \u2191C) \u2227 x * y = x\u271d) \u2194 (\u2203 x y, x \u2208 \u2191A \u2227 y \u2208 \u2191B \u2227 x * y = x\u271d) \u2227 x\u271d \u2208 \u2191C\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : A \u2264 C\nx\u271d : G\n\u22a2 (\u2203 x y, x \u2208 \u2191A \u2227 (y \u2208 \u2191B \u2227 y \u2208 \u2191C) \u2227 x * y = x\u271d) \u2192 (\u2203 x y, x \u2208 \u2191A \u2227 y \u2208 \u2191B \u2227 x * y = x\u271d) \u2227 x\u271d \u2208 \u2191C\n[PROOFSTEP]\nrintro \u27e8y, z, hy, \u27e8hzB, hzC\u27e9, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : A \u2264 C\ny z : G\nhy : y \u2208 \u2191A\nhzB : z \u2208 \u2191B\nhzC : z \u2208 \u2191C\n\u22a2 (\u2203 x y_1, x \u2208 \u2191A \u2227 y_1 \u2208 \u2191B \u2227 x * y_1 = y * z) \u2227 y * z \u2208 \u2191C\n[PROOFSTEP]\nrefine' \u27e8_, mul_mem (h hy) hzC\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : A \u2264 C\ny z : G\nhy : y \u2208 \u2191A\nhzB : z \u2208 \u2191B\nhzC : z \u2208 \u2191C\n\u22a2 \u2203 x y_1, x \u2208 \u2191A \u2227 y_1 \u2208 \u2191B \u2227 x * y_1 = y * z\n[PROOFSTEP]\nexact \u27e8y, z, hy, hzB, rfl\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : A \u2264 C\nx\u271d : G\n\u22a2 (\u2203 x y, x \u2208 \u2191A \u2227 y \u2208 \u2191B \u2227 x * y = x\u271d) \u2227 x\u271d \u2208 \u2191C \u2192 \u2203 x y, x \u2208 \u2191A \u2227 (y \u2208 \u2191B \u2227 y \u2208 \u2191C) \u2227 x * y = x\u271d\n[PROOFSTEP]\nrintro \u27e8\u27e8y, z, hy, hz, rfl\u27e9, hyz\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : A \u2264 C\ny z : G\nhy : y \u2208 \u2191A\nhz : z \u2208 \u2191B\nhyz : y * z \u2208 \u2191C\n\u22a2 \u2203 x y_1, x \u2208 \u2191A \u2227 (y_1 \u2208 \u2191B \u2227 y_1 \u2208 \u2191C) \u2227 x * y_1 = y * z\n[PROOFSTEP]\nrefine' \u27e8y, z, hy, \u27e8hz, _\u27e9, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : A \u2264 C\ny z : G\nhy : y \u2208 \u2191A\nhz : z \u2208 \u2191B\nhyz : y * z \u2208 \u2191C\n\u22a2 z \u2208 \u2191C\n[PROOFSTEP]\nsuffices y\u207b\u00b9 * (y * z) \u2208 C by simpa\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : A \u2264 C\ny z : G\nhy : y \u2208 \u2191A\nhz : z \u2208 \u2191B\nhyz : y * z \u2208 \u2191C\nthis : y\u207b\u00b9 * (y * z) \u2208 C\n\u22a2 z \u2208 \u2191C\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : A \u2264 C\ny z : G\nhy : y \u2208 \u2191A\nhz : z \u2208 \u2191B\nhyz : y * z \u2208 \u2191C\n\u22a2 y\u207b\u00b9 * (y * z) \u2208 C\n[PROOFSTEP]\nexact mul_mem (inv_mem (h hy)) hyz\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\n\u22a2 \u2191(A \u2293 B) * \u2191C = \u2191A \u2293 \u2191B * \u2191C\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\nx\u271d : G\n\u22a2 x\u271d \u2208 \u2191(A \u2293 B) * \u2191C \u2194 x\u271d \u2208 \u2191A \u2293 \u2191B * \u2191C\n[PROOFSTEP]\nsimp only [coe_inf, Set.inf_eq_inter, Set.mem_mul, Set.mem_inter_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\nx\u271d : G\n\u22a2 (\u2203 x y, (x \u2208 \u2191A \u2227 x \u2208 \u2191B) \u2227 y \u2208 \u2191C \u2227 x * y = x\u271d) \u2194 x\u271d \u2208 \u2191A \u2227 \u2203 x y, x \u2208 \u2191B \u2227 y \u2208 \u2191C \u2227 x * y = x\u271d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\nx\u271d : G\n\u22a2 (\u2203 x y, (x \u2208 \u2191A \u2227 x \u2208 \u2191B) \u2227 y \u2208 \u2191C \u2227 x * y = x\u271d) \u2192 x\u271d \u2208 \u2191A \u2227 \u2203 x y, x \u2208 \u2191B \u2227 y \u2208 \u2191C \u2227 x * y = x\u271d\n[PROOFSTEP]\nrintro \u27e8y, z, \u27e8hyA, hyB\u27e9, hz, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny z : G\nhyA : y \u2208 \u2191A\nhyB : y \u2208 \u2191B\nhz : z \u2208 \u2191C\n\u22a2 y * z \u2208 \u2191A \u2227 \u2203 x y_1, x \u2208 \u2191B \u2227 y_1 \u2208 \u2191C \u2227 x * y_1 = y * z\n[PROOFSTEP]\nrefine' \u27e8A.mul_mem hyA (h hz), _\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny z : G\nhyA : y \u2208 \u2191A\nhyB : y \u2208 \u2191B\nhz : z \u2208 \u2191C\n\u22a2 \u2203 x y_1, x \u2208 \u2191B \u2227 y_1 \u2208 \u2191C \u2227 x * y_1 = y * z\n[PROOFSTEP]\nexact \u27e8y, z, hyB, hz, rfl\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\nx\u271d : G\n\u22a2 (x\u271d \u2208 \u2191A \u2227 \u2203 x y, x \u2208 \u2191B \u2227 y \u2208 \u2191C \u2227 x * y = x\u271d) \u2192 \u2203 x y, (x \u2208 \u2191A \u2227 x \u2208 \u2191B) \u2227 y \u2208 \u2191C \u2227 x * y = x\u271d\n[PROOFSTEP]\nrintro \u27e8hyz, y, z, hy, hz, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny z : G\nhy : y \u2208 \u2191B\nhz : z \u2208 \u2191C\nhyz : y * z \u2208 \u2191A\n\u22a2 \u2203 x y_1, (x \u2208 \u2191A \u2227 x \u2208 \u2191B) \u2227 y_1 \u2208 \u2191C \u2227 x * y_1 = y * z\n[PROOFSTEP]\nrefine' \u27e8y, z, \u27e8_, hy\u27e9, hz, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny z : G\nhy : y \u2208 \u2191B\nhz : z \u2208 \u2191C\nhyz : y * z \u2208 \u2191A\n\u22a2 y \u2208 \u2191A\n[PROOFSTEP]\nsuffices y * z * z\u207b\u00b9 \u2208 A by simpa\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny z : G\nhy : y \u2208 \u2191B\nhz : z \u2208 \u2191C\nhyz : y * z \u2208 \u2191A\nthis : y * z * z\u207b\u00b9 \u2208 A\n\u22a2 y \u2208 \u2191A\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny z : G\nhy : y \u2208 \u2191B\nhz : z \u2208 \u2191C\nhyz : y * z \u2208 \u2191A\n\u22a2 y * z * z\u207b\u00b9 \u2208 A\n[PROOFSTEP]\nexact mul_mem hyz (inv_mem (h hz))\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH K : Subgroup G\nhH : Normal H\nhK : Normal K\nn : G\nhmem : n \u2208 H \u2294 K\ng : G\n\u22a2 g * n * g\u207b\u00b9 \u2208 H \u2294 K\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, normal_mul] at hmem \u22a2\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH K : Subgroup G\nhH : Normal H\nhK : Normal K\nn : G\nhmem : n \u2208 \u2191H * \u2191K\ng : G\n\u22a2 g * n * g\u207b\u00b9 \u2208 \u2191H * \u2191K\n[PROOFSTEP]\nrcases hmem with \u27e8h, k, hh, hk, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH K : Subgroup G\nhH : Normal H\nhK : Normal K\ng h k : G\nhh : h \u2208 \u2191H\nhk : k \u2208 \u2191K\n\u22a2 g * (fun x x_1 => x * x_1) h k * g\u207b\u00b9 \u2208 \u2191H * \u2191K\n[PROOFSTEP]\nrefine \u27e8g * h * g\u207b\u00b9, g * k * g\u207b\u00b9, hH.conj_mem h hh g, hK.conj_mem k hk g, ?_\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH K : Subgroup G\nhH : Normal H\nhK : Normal K\ng h k : G\nhh : h \u2208 \u2191H\nhk : k \u2208 \u2191K\n\u22a2 (fun x x_1 => x * x_1) (g * h * g\u207b\u00b9) (g * k * g\u207b\u00b9) = g * (fun x x_1 => x * x_1) h k * g\u207b\u00b9\n[PROOFSTEP]\nsimp only [mul_assoc, inv_mul_cancel_left]\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns\u271d : Set G\ng : G\nh : G\u1d50\u1d52\u1d56\ns : Set G\n\u22a2 (fun y => h \u2022 y) '' ((fun x => g * x) \u207b\u00b9' s) = (fun x => g * x) \u207b\u00b9' ((fun y => h \u2022 y) '' s)\n[PROOFSTEP]\nsimp [preimage_preimage, mul_assoc]\n -- porting note: deprecate?\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddGroup A\ns : Set G\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : MulDistribMulAction \u03b1 G\nS : Subgroup G\n\u22a2 1 \u2022 S = S\n[PROOFSTEP]\nchange S.map _ = S\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddGroup A\ns : Set G\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : MulDistribMulAction \u03b1 G\nS : Subgroup G\n\u22a2 map (\u2191(MulDistribMulAction.toMonoidEnd \u03b1 G) 1) S = S\n[PROOFSTEP]\nsimpa only [map_one] using S.map_id\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddGroup A\ns : Set G\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : MulDistribMulAction \u03b1 G\nP H : Subgroup G\nhP : P \u2264 H\nh : { x // x \u2208 H }\n\u22a2 \u2191MulAut.conj \u2191h \u2022 P \u2264 H\n[PROOFSTEP]\nrintro - \u27e8g, hg, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddGroup A\ns : Set G\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : MulDistribMulAction \u03b1 G\nP H : Subgroup G\nhP : P \u2264 H\nh : { x // x \u2208 H }\ng : G\nhg : g \u2208 \u2191P\n\u22a2 \u2191(\u2191(MulDistribMulAction.toMonoidEnd ((fun x => MulAut G) \u2191h) G) (\u2191MulAut.conj \u2191h)) g \u2208 H\n[PROOFSTEP]\nexact H.mul_mem (H.mul_mem h.2 (hP hg)) (H.inv_mem h.2)\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddGroup A\ns : Set G\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : MulDistribMulAction \u03b1 G\nP H : Subgroup G\nhP : P \u2264 H\nh : { x // x \u2208 H }\n\u22a2 \u2191MulAut.conj h \u2022 subgroupOf P H = subgroupOf (\u2191MulAut.conj \u2191h \u2022 P) H\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddGroup A\ns : Set G\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : MulDistribMulAction \u03b1 G\nP H : Subgroup G\nhP : P \u2264 H\nh : { x // x \u2208 H }\n\u22a2 \u2191MulAut.conj h \u2022 subgroupOf P H \u2264 subgroupOf (\u2191MulAut.conj \u2191h \u2022 P) H\n[PROOFSTEP]\nrintro - \u27e8g, hg, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddGroup A\ns : Set G\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : MulDistribMulAction \u03b1 G\nP H : Subgroup G\nhP : P \u2264 H\nh g : { x // x \u2208 H }\nhg : g \u2208 \u2191(subgroupOf P H)\n\u22a2 \u2191(\u2191(MulDistribMulAction.toMonoidEnd ((fun x => MulAut { x // x \u2208 H }) h) { x // x \u2208 H }) (\u2191MulAut.conj h)) g \u2208\n subgroupOf (\u2191MulAut.conj \u2191h \u2022 P) H\n[PROOFSTEP]\nexact \u27e8g, hg, rfl\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddGroup A\ns : Set G\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : MulDistribMulAction \u03b1 G\nP H : Subgroup G\nhP : P \u2264 H\nh : { x // x \u2208 H }\n\u22a2 subgroupOf (\u2191MulAut.conj \u2191h \u2022 P) H \u2264 \u2191MulAut.conj h \u2022 subgroupOf P H\n[PROOFSTEP]\nrintro p \u27e8g, hg, hp\u27e9\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddGroup A\ns : Set G\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : MulDistribMulAction \u03b1 G\nP H : Subgroup G\nhP : P \u2264 H\nh p : { x // x \u2208 H }\ng : G\nhg : g \u2208 \u2191P\nhp : \u2191(\u2191(MulDistribMulAction.toMonoidEnd ((fun x => MulAut G) \u2191h) G) (\u2191MulAut.conj \u2191h)) g = \u2191(Subgroup.subtype H) p\n\u22a2 p \u2208 \u2191MulAut.conj h \u2022 subgroupOf P H\n[PROOFSTEP]\nexact \u27e8\u27e8g, hP hg\u27e9, hg, Subtype.ext hp\u27e9\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddGroup A\ns : Set G\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulDistribMulAction \u03b1 G\na : \u03b1\nS T : Subgroup G\n\u22a2 a \u2022 (S \u2293 T) = a \u2022 S \u2293 a \u2022 T\n[PROOFSTEP]\nsimp [SetLike.ext_iff, mem_pointwise_smul_iff_inv_smul_mem]\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddGroup A\ns : Set G\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulDistribMulAction \u03b1 G\nH : Subgroup G\nh : G\nhh : h \u2208 H\nthis : {x | x \u2208 H} = \u2191H\n\u22a2 \u2191H * {h} = \u2191H\n[PROOFSTEP]\nsimpa [preimage, mul_mem_cancel_right (inv_mem hh)]\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddGroup A\ns : Set G\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulDistribMulAction \u03b1 G\nH : Subgroup G\nh : G\nhh : h \u2208 H\nthis : {x | x \u2208 H} = \u2191H\n\u22a2 {h} * \u2191H = \u2191H\n[PROOFSTEP]\nsimpa [preimage, mul_mem_cancel_left (inv_mem hh)]\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddGroup A\ns : Set G\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : DistribMulAction \u03b1 A\nS : AddSubgroup A\n\u22a2 1 \u2022 S = S\n[PROOFSTEP]\nchange S.map _ = S\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddGroup A\ns : Set G\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : DistribMulAction \u03b1 A\nS : AddSubgroup A\n\u22a2 map (\u2191(DistribMulAction.toAddMonoidEnd \u03b1 A) 1) S = S\n[PROOFSTEP]\nsimpa only [map_one] using S.map_id\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Subgroup.Pointwise", "llama_tokens": 14957, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.40710959997313734}} {"text": "[GOAL]\nz w : \u2102\nH : z = w\n\u22a2 z.re = w.re \u2227 z.im = w.im\n[PROOFSTEP]\nsimp [H]\n[GOAL]\nz w : \u211d\n\u22a2 z = w \u2192 \u2191z = \u2191w\n[PROOFSTEP]\napply congrArg\n[GOAL]\nr s : \u211d\n\u22a2 (\u2191(r + s)).re = (\u2191r + \u2191s).re \u2227 (\u2191(r + s)).im = (\u2191r + \u2191s).im\n[PROOFSTEP]\nsimp [ofReal']\n[GOAL]\nr : \u211d\n\u22a2 (\u2191(bit0 r)).re = (bit0 \u2191r).re \u2227 (\u2191(bit0 r)).im = (bit0 \u2191r).im\n[PROOFSTEP]\nsimp [bit0]\n[GOAL]\nr : \u211d\n\u22a2 (\u2191(bit1 r)).re = (bit1 \u2191r).re \u2227 (\u2191(bit1 r)).im = (bit1 \u2191r).im\n[PROOFSTEP]\nsimp [bit1]\n[GOAL]\nr : \u211d\n\u22a2 (\u2191(-r)).re = (-\u2191r).re \u2227 (\u2191(-r)).im = (-\u2191r).im\n[PROOFSTEP]\nsimp [ofReal']\n[GOAL]\nr s : \u211d\n\u22a2 (\u2191(r * s)).re = (\u2191r * \u2191s).re \u2227 (\u2191(r * s)).im = (\u2191r * \u2191s).im\n[PROOFSTEP]\nsimp [ofReal']\n[GOAL]\nr : \u211d\nz : \u2102\n\u22a2 (\u2191r * z).re = r * z.re\n[PROOFSTEP]\nsimp [ofReal']\n[GOAL]\nr : \u211d\nz : \u2102\n\u22a2 (\u2191r * z).im = r * z.im\n[PROOFSTEP]\nsimp [ofReal']\n[GOAL]\n\u22a2 (I * I).re = (-1).re \u2227 (I * I).im = (-1).im\n[PROOFSTEP]\nsimp\n[GOAL]\nz : \u2102\n\u22a2 (I * z).re = { re := -z.im, im := z.re }.re \u2227 (I * z).im = { re := -z.im, im := z.re }.im\n[PROOFSTEP]\nsimp\n[GOAL]\na b : \u211d\n\u22a2 { re := a, im := b }.re = (\u2191a + \u2191b * I).re \u2227 { re := a, im := b }.im = (\u2191a + \u2191b * I).im\n[PROOFSTEP]\nsimp [ofReal']\n[GOAL]\nz : \u2102\n\u22a2 (\u2191z.re + \u2191z.im * I).re = z.re \u2227 (\u2191z.re + \u2191z.im * I).im = z.im\n[PROOFSTEP]\nsimp [ofReal']\n[GOAL]\nz : \u2102\n\u22a2 (z * I).re = -z.im\n[PROOFSTEP]\nsimp\n[GOAL]\nz : \u2102\n\u22a2 (z * I).im = z.re\n[PROOFSTEP]\nsimp\n[GOAL]\nz : \u2102\n\u22a2 (I * z).re = -z.im\n[PROOFSTEP]\nsimp\n[GOAL]\nz : \u2102\n\u22a2 (I * z).im = z.re\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u211d \u00d7 \u211d\n\u22a2 \u2191equivRealProd.symm p = \u2191p.fst + \u2191p.snd * I\n[PROOFSTEP]\next\n[GOAL]\ncase a\np : \u211d \u00d7 \u211d\n\u22a2 (\u2191equivRealProd.symm p).re = (\u2191p.fst + \u2191p.snd * I).re\n[PROOFSTEP]\nsimp [Complex.equivRealProd, ofReal']\n[GOAL]\ncase a\np : \u211d \u00d7 \u211d\n\u22a2 (\u2191equivRealProd.symm p).im = (\u2191p.fst + \u2191p.snd * I).im\n[PROOFSTEP]\nsimp [Complex.equivRealProd, ofReal']\n[GOAL]\nR : Type u_1\ninst\u271d : SMul R \u211d\nr : R\nz : \u2102\n\u22a2 (r \u2022 z).re = r \u2022 z.re\n[PROOFSTEP]\nsimp [(\u00b7 \u2022 \u00b7), SMul.smul]\n[GOAL]\nR : Type u_1\ninst\u271d : SMul R \u211d\nr : R\nz : \u2102\n\u22a2 (r \u2022 z).im = r \u2022 z.im\n[PROOFSTEP]\nsimp [(\u00b7 \u2022 \u00b7), SMul.smul]\n[GOAL]\n\u22a2 \u2200 (a b c : \u2102), a + b + c = a + (b + c)\n[PROOFSTEP]\nintros\n[GOAL]\na\u271d b\u271d c\u271d : \u2102\n\u22a2 a\u271d + b\u271d + c\u271d = a\u271d + (b\u271d + c\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase a\na\u271d b\u271d c\u271d : \u2102\n\u22a2 (a\u271d + b\u271d + c\u271d).re = (a\u271d + (b\u271d + c\u271d)).re\n[PROOFSTEP]\nsimp [add_assoc]\n[GOAL]\ncase a\na\u271d b\u271d c\u271d : \u2102\n\u22a2 (a\u271d + b\u271d + c\u271d).im = (a\u271d + (b\u271d + c\u271d)).im\n[PROOFSTEP]\nsimp [add_assoc]\n[GOAL]\n\u22a2 \u2200 (a : \u2102), 0 + a = a\n[PROOFSTEP]\nintros\n[GOAL]\na\u271d : \u2102\n\u22a2 0 + a\u271d = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a\na\u271d : \u2102\n\u22a2 (0 + a\u271d).re = a\u271d.re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\na\u271d : \u2102\n\u22a2 (0 + a\u271d).im = a\u271d.im\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2200 (a : \u2102), a + 0 = a\n[PROOFSTEP]\nintros\n[GOAL]\na\u271d : \u2102\n\u22a2 a\u271d + 0 = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a\na\u271d : \u2102\n\u22a2 (a\u271d + 0).re = a\u271d.re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\na\u271d : \u2102\n\u22a2 (a\u271d + 0).im = a\u271d.im\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2200 (x : \u2102), (fun n z => n \u2022 z) 0 x = 0\n[PROOFSTEP]\nintros\n[GOAL]\nx\u271d : \u2102\n\u22a2 (fun n z => n \u2022 z) 0 x\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase a\nx\u271d : \u2102\n\u22a2 ((fun n z => n \u2022 z) 0 x\u271d).re = 0.re\n[PROOFSTEP]\nsimp [smul_re, smul_im]\n[GOAL]\ncase a\nx\u271d : \u2102\n\u22a2 ((fun n z => n \u2022 z) 0 x\u271d).im = 0.im\n[PROOFSTEP]\nsimp [smul_re, smul_im]\n[GOAL]\n\u22a2 \u2200 (n : \u2115) (x : \u2102), (fun n z => n \u2022 z) (n + 1) x = x + (fun n z => n \u2022 z) n x\n[PROOFSTEP]\nintros\n[GOAL]\nn\u271d : \u2115\nx\u271d : \u2102\n\u22a2 (fun n z => n \u2022 z) (n\u271d + 1) x\u271d = x\u271d + (fun n z => n \u2022 z) n\u271d x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a\nn\u271d : \u2115\nx\u271d : \u2102\n\u22a2 ((fun n z => n \u2022 z) (n\u271d + 1) x\u271d).re = (x\u271d + (fun n z => n \u2022 z) n\u271d x\u271d).re\n[PROOFSTEP]\nsimp [AddMonoid.nsmul_succ, add_mul, add_comm, smul_re, smul_im]\n[GOAL]\ncase a\nn\u271d : \u2115\nx\u271d : \u2102\n\u22a2 ((fun n z => n \u2022 z) (n\u271d + 1) x\u271d).im = (x\u271d + (fun n z => n \u2022 z) n\u271d x\u271d).im\n[PROOFSTEP]\nsimp [AddMonoid.nsmul_succ, add_mul, add_comm, smul_re, smul_im]\n[GOAL]\n\u22a2 \u2200 (a : \u2102), (fun n z => n \u2022 z) 0 a = 0\n[PROOFSTEP]\nintros\n[GOAL]\na\u271d : \u2102\n\u22a2 (fun n z => n \u2022 z) 0 a\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase a\na\u271d : \u2102\n\u22a2 ((fun n z => n \u2022 z) 0 a\u271d).re = 0.re\n[PROOFSTEP]\nsimp [smul_re, smul_im]\n[GOAL]\ncase a\na\u271d : \u2102\n\u22a2 ((fun n z => n \u2022 z) 0 a\u271d).im = 0.im\n[PROOFSTEP]\nsimp [smul_re, smul_im]\n[GOAL]\n\u22a2 \u2200 (n : \u2115) (a : \u2102), (fun n z => n \u2022 z) (Int.ofNat (Nat.succ n)) a = a + (fun n z => n \u2022 z) (Int.ofNat n) a\n[PROOFSTEP]\nintros\n[GOAL]\nn\u271d : \u2115\na\u271d : \u2102\n\u22a2 (fun n z => n \u2022 z) (Int.ofNat (Nat.succ n\u271d)) a\u271d = a\u271d + (fun n z => n \u2022 z) (Int.ofNat n\u271d) a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a\nn\u271d : \u2115\na\u271d : \u2102\n\u22a2 ((fun n z => n \u2022 z) (Int.ofNat (Nat.succ n\u271d)) a\u271d).re = (a\u271d + (fun n z => n \u2022 z) (Int.ofNat n\u271d) a\u271d).re\n[PROOFSTEP]\nsimp [SubNegMonoid.zsmul_succ', add_mul, add_comm, smul_re, smul_im]\n[GOAL]\ncase a\nn\u271d : \u2115\na\u271d : \u2102\n\u22a2 ((fun n z => n \u2022 z) (Int.ofNat (Nat.succ n\u271d)) a\u271d).im = (a\u271d + (fun n z => n \u2022 z) (Int.ofNat n\u271d) a\u271d).im\n[PROOFSTEP]\nsimp [SubNegMonoid.zsmul_succ', add_mul, add_comm, smul_re, smul_im]\n[GOAL]\n\u22a2 \u2200 (n : \u2115) (a : \u2102), (fun n z => n \u2022 z) (Int.negSucc n) a = -(fun n z => n \u2022 z) (\u2191(Nat.succ n)) a\n[PROOFSTEP]\nintros\n[GOAL]\nn\u271d : \u2115\na\u271d : \u2102\n\u22a2 (fun n z => n \u2022 z) (Int.negSucc n\u271d) a\u271d = -(fun n z => n \u2022 z) (\u2191(Nat.succ n\u271d)) a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a\nn\u271d : \u2115\na\u271d : \u2102\n\u22a2 ((fun n z => n \u2022 z) (Int.negSucc n\u271d) a\u271d).re = (-(fun n z => n \u2022 z) (\u2191(Nat.succ n\u271d)) a\u271d).re\n[PROOFSTEP]\nsimp [zsmul_neg', add_mul, smul_re, smul_im]\n[GOAL]\ncase a\nn\u271d : \u2115\na\u271d : \u2102\n\u22a2 ((fun n z => n \u2022 z) (Int.negSucc n\u271d) a\u271d).im = (-(fun n z => n \u2022 z) (\u2191(Nat.succ n\u271d)) a\u271d).im\n[PROOFSTEP]\nsimp [zsmul_neg', add_mul, smul_re, smul_im]\n[GOAL]\n\u22a2 \u2200 (a : \u2102), -a + a = 0\n[PROOFSTEP]\nintros\n[GOAL]\na\u271d : \u2102\n\u22a2 -a\u271d + a\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase a\na\u271d : \u2102\n\u22a2 (-a\u271d + a\u271d).re = 0.re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\na\u271d : \u2102\n\u22a2 (-a\u271d + a\u271d).im = 0.im\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2200 (a b : \u2102), a + b = b + a\n[PROOFSTEP]\nintros\n[GOAL]\na\u271d b\u271d : \u2102\n\u22a2 a\u271d + b\u271d = b\u271d + a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a\na\u271d b\u271d : \u2102\n\u22a2 (a\u271d + b\u271d).re = (b\u271d + a\u271d).re\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\ncase a\na\u271d b\u271d : \u2102\n\u22a2 (a\u271d + b\u271d).im = (b\u271d + a\u271d).im\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\n\u22a2 NatCast.natCast 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase a\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\n\u22a2 (NatCast.natCast 0).re = 0.re\n[PROOFSTEP]\nsimp [Nat.cast, AddMonoidWithOne.natCast_zero]\n[GOAL]\ncase a\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\n\u22a2 (NatCast.natCast 0).im = 0.im\n[PROOFSTEP]\nsimp [Nat.cast, AddMonoidWithOne.natCast_zero]\n[GOAL]\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\nx\u271d : \u2115\n\u22a2 NatCast.natCast (x\u271d + 1) = NatCast.natCast x\u271d + 1\n[PROOFSTEP]\next\n[GOAL]\ncase a\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\nx\u271d : \u2115\n\u22a2 (NatCast.natCast (x\u271d + 1)).re = (NatCast.natCast x\u271d + 1).re\n[PROOFSTEP]\nsimp [Nat.cast, AddMonoidWithOne.natCast_succ]\n[GOAL]\ncase a\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\nx\u271d : \u2115\n\u22a2 (NatCast.natCast (x\u271d + 1)).im = (NatCast.natCast x\u271d + 1).im\n[PROOFSTEP]\nsimp [Nat.cast, AddMonoidWithOne.natCast_succ]\n[GOAL]\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\nx\u271d : \u2115\n\u22a2 IntCast.intCast \u2191x\u271d = \u2191x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\nx\u271d : \u2115\n\u22a2 (IntCast.intCast \u2191x\u271d).re = (\u2191x\u271d).re\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\nx\u271d : \u2115\n\u22a2 (IntCast.intCast \u2191x\u271d).im = (\u2191x\u271d).im\n[PROOFSTEP]\nrfl\n[GOAL]\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\nn : \u2115\n\u22a2 IntCast.intCast (Int.negSucc n) = -\u2191(n + 1)\n[PROOFSTEP]\next\n[GOAL]\ncase a\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\nn : \u2115\n\u22a2 (IntCast.intCast (Int.negSucc n)).re = (-\u2191(n + 1)).re\n[PROOFSTEP]\nsimp [AddGroupWithOne.intCast_negSucc]\n[GOAL]\ncase a\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\nn : \u2115\n\u22a2 -1 + -\u2191n = -(\u2191(n + 1)).re\n[PROOFSTEP]\nshow -(1 : \u211d) + (-n) = -(\u2191(n + 1))\n[GOAL]\ncase a\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\nn : \u2115\n\u22a2 -1 + -\u2191n = -\u2191(n + 1)\n[PROOFSTEP]\nsimp [Nat.cast_add, add_comm]\n[GOAL]\ncase a\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\nn : \u2115\n\u22a2 (IntCast.intCast (Int.negSucc n)).im = (-\u2191(n + 1)).im\n[PROOFSTEP]\nsimp [AddGroupWithOne.intCast_negSucc]\n[GOAL]\ncase a\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\nn : \u2115\n\u22a2 (\u2191(n + 1)).im = 0\n[PROOFSTEP]\nshow im \u27e8n, 0\u27e9 = 0\n[GOAL]\ncase a\nsrc\u271d : AddCommGroup \u2102 := addCommGroup\nn : \u2115\n\u22a2 { re := \u2191n, im := 0 }.im = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\n\u22a2 \u2200 (a b : \u2102), a + b = b + a\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d : \u2102\n\u22a2 a\u271d + b\u271d = b\u271d + a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d : \u2102\n\u22a2 (a\u271d + b\u271d).re = (b\u271d + a\u271d).re\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d : \u2102\n\u22a2 (a\u271d + b\u271d).im = (b\u271d + a\u271d).im\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\n\u22a2 \u2200 (a b c : \u2102), a * (b + c) = a * b + a * c\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d c\u271d : \u2102\n\u22a2 a\u271d * (b\u271d + c\u271d) = a\u271d * b\u271d + a\u271d * c\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d c\u271d : \u2102\n\u22a2 (a\u271d * (b\u271d + c\u271d)).re = (a\u271d * b\u271d + a\u271d * c\u271d).re\n[PROOFSTEP]\nsimp [mul_re, mul_im]\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d c\u271d : \u2102\n\u22a2 (a\u271d * (b\u271d + c\u271d)).im = (a\u271d * b\u271d + a\u271d * c\u271d).im\n[PROOFSTEP]\nsimp [mul_re, mul_im]\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d c\u271d : \u2102\n\u22a2 a\u271d.re * (b\u271d.re + c\u271d.re) - a\u271d.im * (b\u271d.im + c\u271d.im) = a\u271d.re * b\u271d.re - a\u271d.im * b\u271d.im + (a\u271d.re * c\u271d.re - a\u271d.im * c\u271d.im)\n[PROOFSTEP]\nring\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d c\u271d : \u2102\n\u22a2 a\u271d.re * (b\u271d.im + c\u271d.im) + a\u271d.im * (b\u271d.re + c\u271d.re) = a\u271d.re * b\u271d.im + a\u271d.im * b\u271d.re + (a\u271d.re * c\u271d.im + a\u271d.im * c\u271d.re)\n[PROOFSTEP]\nring\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\n\u22a2 \u2200 (a b c : \u2102), (a + b) * c = a * c + b * c\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d c\u271d : \u2102\n\u22a2 (a\u271d + b\u271d) * c\u271d = a\u271d * c\u271d + b\u271d * c\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d c\u271d : \u2102\n\u22a2 ((a\u271d + b\u271d) * c\u271d).re = (a\u271d * c\u271d + b\u271d * c\u271d).re\n[PROOFSTEP]\nsimp [mul_re, mul_im]\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d c\u271d : \u2102\n\u22a2 ((a\u271d + b\u271d) * c\u271d).im = (a\u271d * c\u271d + b\u271d * c\u271d).im\n[PROOFSTEP]\nsimp [mul_re, mul_im]\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d c\u271d : \u2102\n\u22a2 (a\u271d.re + b\u271d.re) * c\u271d.re - (a\u271d.im + b\u271d.im) * c\u271d.im = a\u271d.re * c\u271d.re - a\u271d.im * c\u271d.im + (b\u271d.re * c\u271d.re - b\u271d.im * c\u271d.im)\n[PROOFSTEP]\nring\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d c\u271d : \u2102\n\u22a2 (a\u271d.re + b\u271d.re) * c\u271d.im + (a\u271d.im + b\u271d.im) * c\u271d.re = a\u271d.re * c\u271d.im + a\u271d.im * c\u271d.re + (b\u271d.re * c\u271d.im + b\u271d.im * c\u271d.re)\n[PROOFSTEP]\nring\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\n\u22a2 \u2200 (a : \u2102), 0 * a = 0\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d : \u2102\n\u22a2 0 * a\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d : \u2102\n\u22a2 (0 * a\u271d).re = 0.re\n[PROOFSTEP]\nsimp [zero_mul]\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d : \u2102\n\u22a2 (0 * a\u271d).im = 0.im\n[PROOFSTEP]\nsimp [zero_mul]\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\n\u22a2 \u2200 (a : \u2102), a * 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d : \u2102\n\u22a2 a\u271d * 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d : \u2102\n\u22a2 (a\u271d * 0).re = 0.re\n[PROOFSTEP]\nsimp [mul_zero]\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d : \u2102\n\u22a2 (a\u271d * 0).im = 0.im\n[PROOFSTEP]\nsimp [mul_zero]\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\n\u22a2 \u2200 (a b c : \u2102), a * b * c = a * (b * c)\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d c\u271d : \u2102\n\u22a2 a\u271d * b\u271d * c\u271d = a\u271d * (b\u271d * c\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d c\u271d : \u2102\n\u22a2 (a\u271d * b\u271d * c\u271d).re = (a\u271d * (b\u271d * c\u271d)).re\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d c\u271d : \u2102\n\u22a2 (a\u271d * b\u271d * c\u271d).im = (a\u271d * (b\u271d * c\u271d)).im\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d c\u271d : \u2102\n\u22a2 (a\u271d.re * b\u271d.re - a\u271d.im * b\u271d.im) * c\u271d.re - (a\u271d.re * b\u271d.im + a\u271d.im * b\u271d.re) * c\u271d.im =\n a\u271d.re * (b\u271d.re * c\u271d.re - b\u271d.im * c\u271d.im) - a\u271d.im * (b\u271d.re * c\u271d.im + b\u271d.im * c\u271d.re)\n[PROOFSTEP]\nring\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d c\u271d : \u2102\n\u22a2 (a\u271d.re * b\u271d.re - a\u271d.im * b\u271d.im) * c\u271d.im + (a\u271d.re * b\u271d.im + a\u271d.im * b\u271d.re) * c\u271d.re =\n a\u271d.re * (b\u271d.re * c\u271d.im + b\u271d.im * c\u271d.re) + a\u271d.im * (b\u271d.re * c\u271d.re - b\u271d.im * c\u271d.im)\n[PROOFSTEP]\nring\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\n\u22a2 \u2200 (a : \u2102), 1 * a = a\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d : \u2102\n\u22a2 1 * a\u271d = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d : \u2102\n\u22a2 (1 * a\u271d).re = a\u271d.re\n[PROOFSTEP]\nsimp [one_mul]\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d : \u2102\n\u22a2 (1 * a\u271d).im = a\u271d.im\n[PROOFSTEP]\nsimp [one_mul]\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\n\u22a2 \u2200 (a : \u2102), a * 1 = a\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d : \u2102\n\u22a2 a\u271d * 1 = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d : \u2102\n\u22a2 (a\u271d * 1).re = a\u271d.re\n[PROOFSTEP]\nsimp [mul_one]\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d : \u2102\n\u22a2 (a\u271d * 1).im = a\u271d.im\n[PROOFSTEP]\nsimp [mul_one]\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\n\u22a2 \u2200 (a b : \u2102), a * b = b * a\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d : \u2102\n\u22a2 a\u271d * b\u271d = b\u271d * a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d : \u2102\n\u22a2 (a\u271d * b\u271d).re = (b\u271d * a\u271d).re\n[PROOFSTEP]\nsimp [mul_comm]\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d : \u2102\n\u22a2 (a\u271d * b\u271d).im = (b\u271d * a\u271d).im\n[PROOFSTEP]\nsimp [mul_comm]\n[GOAL]\ncase a\nsrc\u271d : AddGroupWithOne \u2102 := Complex.addGroupWithOne\na\u271d b\u271d : \u2102\n\u22a2 a\u271d.re * b\u271d.im + b\u271d.re * a\u271d.im = b\u271d.re * a\u271d.im + a\u271d.re * b\u271d.im\n[PROOFSTEP]\nring\n[GOAL]\n\u22a2 Ring \u2102\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nn : \u2115\n\u22a2 I ^ bit0 n = (-1) ^ n\n[PROOFSTEP]\nrw [pow_bit0', Complex.I_mul_I]\n[GOAL]\nn : \u2115\n\u22a2 I ^ bit1 n = (-1) ^ n * I\n[PROOFSTEP]\nrw [pow_bit1', Complex.I_mul_I]\n[GOAL]\nx : \u2102\n\u22a2 star (star x) = x\n[PROOFSTEP]\nsimp only [eta, neg_neg]\n[GOAL]\na b : \u2102\n\u22a2 star (a * b) = star b * star a\n[PROOFSTEP]\next\n[GOAL]\ncase a\na b : \u2102\n\u22a2 (star (a * b)).re = (star b * star a).re\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\ncase a\na b : \u2102\n\u22a2 (star (a * b)).im = (star b * star a).im\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\ncase a\na b : \u2102\n\u22a2 a.re * b.re - a.im * b.im = b.re * a.re - b.im * a.im\n[PROOFSTEP]\nring\n[GOAL]\ncase a\na b : \u2102\n\u22a2 -(a.re * b.im) + -(a.im * b.re) = -(b.re * a.im) + -(b.im * a.re)\n[PROOFSTEP]\nring\n[GOAL]\na b : \u2102\n\u22a2 star (a + b) = star a + star b\n[PROOFSTEP]\next\n[GOAL]\ncase a\na b : \u2102\n\u22a2 (star (a + b)).re = (star a + star b).re\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\ncase a\na b : \u2102\n\u22a2 (star (a + b)).im = (star a + star b).im\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\nr : \u211d\n\u22a2 (\u2191(starRingEnd \u2102) \u2191r).re = (\u2191r).re \u2227 (\u2191(starRingEnd \u2102) \u2191r).im = (\u2191r).im\n[PROOFSTEP]\nsimp [star]\n[GOAL]\n\u22a2 (\u2191(starRingEnd \u2102) I).re = (-I).re \u2227 (\u2191(starRingEnd \u2102) I).im = (-I).im\n[PROOFSTEP]\nsimp\n[GOAL]\nz : \u2102\n\u22a2 (\u2191(starRingEnd \u2102) (bit0 z)).re = (bit0 (\u2191(starRingEnd \u2102) z)).re \u2227\n (\u2191(starRingEnd \u2102) (bit0 z)).im = (bit0 (\u2191(starRingEnd \u2102) z)).im\n[PROOFSTEP]\nsimp [bit0]\n[GOAL]\nz : \u2102\n\u22a2 (\u2191(starRingEnd \u2102) (bit1 z)).re = (bit1 (\u2191(starRingEnd \u2102) z)).re \u2227\n (\u2191(starRingEnd \u2102) (bit1 z)).im = (bit1 (\u2191(starRingEnd \u2102) z)).im\n[PROOFSTEP]\nsimp [bit0]\n[GOAL]\n\u22a2 (\u2191(starRingEnd \u2102) (-I)).re = I.re \u2227 (\u2191(starRingEnd \u2102) (-I)).im = I.im\n[PROOFSTEP]\nsimp\n[GOAL]\nz : \u2102\nx\u271d : \u2203 r, z = \u2191r\nh : \u211d\ne : z = \u2191h\n\u22a2 \u2191(starRingEnd \u2102) z = z\n[PROOFSTEP]\nrw [e, conj_ofReal]\n[GOAL]\nz : \u2102\n\u22a2 (\u2203 r, z = \u2191r) \u2192 \u2191z.re = z\n[PROOFSTEP]\nrintro \u27e8r, rfl\u27e9\n[GOAL]\ncase intro\nr : \u211d\n\u22a2 \u2191(\u2191r).re = \u2191r\n[PROOFSTEP]\nsimp [ofReal']\n[GOAL]\n\u22a2 (fun z => z.re * z.re + z.im * z.im) 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 ZeroHom.toFun { toFun := fun z => z.re * z.re + z.im * z.im, map_zero' := (_ : 0 * 0 + 0 * 0 = 0) } 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nz w : \u2102\n\u22a2 ZeroHom.toFun { toFun := fun z => z.re * z.re + z.im * z.im, map_zero' := (_ : 0 * 0 + 0 * 0 = 0) } (z * w) =\n ZeroHom.toFun { toFun := fun z => z.re * z.re + z.im * z.im, map_zero' := (_ : 0 * 0 + 0 * 0 = 0) } z *\n ZeroHom.toFun { toFun := fun z => z.re * z.re + z.im * z.im, map_zero' := (_ : 0 * 0 + 0 * 0 = 0) } w\n[PROOFSTEP]\ndsimp\n[GOAL]\nz w : \u2102\n\u22a2 (z.re * w.re - z.im * w.im) * (z.re * w.re - z.im * w.im) +\n (z.re * w.im + z.im * w.re) * (z.re * w.im + z.im * w.re) =\n (z.re * z.re + z.im * z.im) * (w.re * w.re + w.im * w.im)\n[PROOFSTEP]\nring\n[GOAL]\nr : \u211d\n\u22a2 \u2191normSq \u2191r = r * r\n[PROOFSTEP]\nsimp [normSq, ofReal']\n[GOAL]\nx y : \u211d\n\u22a2 \u2191normSq (\u2191x + \u2191y * I) = x ^ 2 + y ^ 2\n[PROOFSTEP]\nrw [\u2190 mk_eq_add_mul_I, normSq_mk, sq, sq]\n[GOAL]\nz : \u2102\n\u22a2 \u2191(\u2191normSq z) = \u2191(starRingEnd \u2102) z * z\n[PROOFSTEP]\next\n[GOAL]\ncase a\nz : \u2102\n\u22a2 (\u2191(\u2191normSq z)).re = (\u2191(starRingEnd \u2102) z * z).re\n[PROOFSTEP]\nsimp [normSq, mul_comm, ofReal']\n[GOAL]\ncase a\nz : \u2102\n\u22a2 (\u2191(\u2191normSq z)).im = (\u2191(starRingEnd \u2102) z * z).im\n[PROOFSTEP]\nsimp [normSq, mul_comm, ofReal']\n[GOAL]\n\u22a2 \u2191normSq I = 1\n[PROOFSTEP]\nsimp [normSq]\n[GOAL]\nx : \u211d\nhx : x \u2208 Ici 0\n\u22a2 \u2191normSq \u2191(Real.sqrt x) = x\n[PROOFSTEP]\nrw [normSq_ofReal, Real.mul_self_sqrt hx]\n[GOAL]\nz : \u2102\n\u22a2 \u2191normSq (-z) = \u2191normSq z\n[PROOFSTEP]\nsimp [normSq]\n[GOAL]\nz : \u2102\n\u22a2 \u2191normSq (\u2191(starRingEnd \u2102) z) = \u2191normSq z\n[PROOFSTEP]\nsimp [normSq]\n[GOAL]\nz w : \u2102\n\u22a2 \u2191normSq (z + w) = \u2191normSq z + \u2191normSq w + 2 * (z * \u2191(starRingEnd \u2102) w).re\n[PROOFSTEP]\ndsimp [normSq]\n[GOAL]\nz w : \u2102\n\u22a2 (z.re + w.re) * (z.re + w.re) + (z.im + w.im) * (z.im + w.im) =\n z.re * z.re + z.im * z.im + (w.re * w.re + w.im * w.im) + 2 * (z.re * w.re - z.im * -w.im)\n[PROOFSTEP]\nring\n[GOAL]\nz : \u2102\n\u22a2 (z * \u2191(starRingEnd \u2102) z).re = (\u2191(\u2191normSq z)).re \u2227 (z * \u2191(starRingEnd \u2102) z).im = (\u2191(\u2191normSq z)).im\n[PROOFSTEP]\nsimp [normSq, mul_comm, sub_eq_neg_add, add_comm, ofReal']\n[GOAL]\nz : \u2102\n\u22a2 (z + \u2191(starRingEnd \u2102) z).re = (\u2191(2 * z.re)).re \u2227 (z + \u2191(starRingEnd \u2102) z).im = (\u2191(2 * z.re)).im\n[PROOFSTEP]\nsimp [two_mul, ofReal']\n[GOAL]\n\u22a2 I ^ 2 = -1\n[PROOFSTEP]\nrw [sq, I_mul_I]\n[GOAL]\nr s : \u211d\n\u22a2 (\u2191(r - s)).re = (\u2191r - \u2191s).re \u2227 (\u2191(r - s)).im = (\u2191r - \u2191s).im\n[PROOFSTEP]\nsimp [ofReal']\n[GOAL]\nr : \u211d\nn : \u2115\n\u22a2 \u2191(r ^ n) = \u2191r ^ n\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\nr : \u211d\n\u22a2 \u2191(r ^ Nat.zero) = \u2191r ^ Nat.zero\n[PROOFSTEP]\nsimp [*, ofReal_mul, pow_succ]\n[GOAL]\ncase succ\nr : \u211d\nn\u271d : \u2115\nn_ih\u271d : \u2191(r ^ n\u271d) = \u2191r ^ n\u271d\n\u22a2 \u2191(r ^ Nat.succ n\u271d) = \u2191r ^ Nat.succ n\u271d\n[PROOFSTEP]\nsimp [*, ofReal_mul, pow_succ]\n[GOAL]\nz : \u2102\n\u22a2 (z - \u2191(starRingEnd \u2102) z).re = (\u2191(2 * z.im) * I).re \u2227 (z - \u2191(starRingEnd \u2102) z).im = (\u2191(2 * z.im) * I).im\n[PROOFSTEP]\nsimp [two_mul, sub_eq_add_neg, ofReal']\n[GOAL]\nz w : \u2102\n\u22a2 \u2191normSq (z - w) = \u2191normSq z + \u2191normSq w - 2 * (z * \u2191(starRingEnd \u2102) w).re\n[PROOFSTEP]\nrw [sub_eq_add_neg, normSq_add]\n[GOAL]\nz w : \u2102\n\u22a2 \u2191normSq z + \u2191normSq (-w) + 2 * (z * \u2191(starRingEnd \u2102) (-w)).re =\n \u2191normSq z + \u2191normSq w - 2 * (z * \u2191(starRingEnd \u2102) w).re\n[PROOFSTEP]\nsimp only [RingHom.map_neg, mul_neg, neg_re, normSq_neg]\n[GOAL]\nz w : \u2102\n\u22a2 \u2191normSq z + \u2191normSq w + -(2 * (z * \u2191(starRingEnd \u2102) w).re) = \u2191normSq z + \u2191normSq w - 2 * (z * \u2191(starRingEnd \u2102) w).re\n[PROOFSTEP]\nring\n[GOAL]\nz : \u2102\n\u22a2 z\u207b\u00b9.re = z.re / \u2191normSq z\n[PROOFSTEP]\nsimp [inv_def, division_def, ofReal']\n[GOAL]\nz : \u2102\n\u22a2 z\u207b\u00b9.im = -z.im / \u2191normSq z\n[PROOFSTEP]\nsimp [inv_def, division_def, ofReal']\n[GOAL]\nr : \u211d\n\u22a2 (\u2191r\u207b\u00b9).re = (\u2191r)\u207b\u00b9.re \u2227 (\u2191r\u207b\u00b9).im = (\u2191r)\u207b\u00b9.im\n[PROOFSTEP]\nsimp [ofReal']\n[GOAL]\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\nrw [\u2190 ofReal_zero, \u2190 ofReal_inv, inv_zero]\n[GOAL]\nz : \u2102\nh : z \u2260 0\n\u22a2 z * z\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [inv_def, \u2190 mul_assoc, mul_conj, \u2190 ofReal_mul, mul_inv_cancel (mt normSq_eq_zero.1 h), ofReal_one]\n[GOAL]\nn : \u2115\n\u22a2 (\u2191n).re = \u2191n\n[PROOFSTEP]\nrw [\u2190 ofReal_nat_cast, ofReal_re]\n[GOAL]\nn : \u2115\n\u22a2 (\u2191n).im = 0\n[PROOFSTEP]\nrw [\u2190 ofReal_nat_cast, ofReal_im]\n[GOAL]\nn : \u2124\n\u22a2 (\u2191n).re = \u2191n\n[PROOFSTEP]\nrw [\u2190 ofReal_int_cast, ofReal_re]\n[GOAL]\nn : \u2124\n\u22a2 (\u2191n).im = 0\n[PROOFSTEP]\nrw [\u2190 ofReal_int_cast, ofReal_im]\n[GOAL]\nq : \u211a\n\u22a2 (\u2191q).im = 0\n[PROOFSTEP]\nshow (Rat.castRec q : \u2102).im = 0\n[GOAL]\nq : \u211a\n\u22a2 (Rat.castRec q).im = 0\n[PROOFSTEP]\ncases q\n[GOAL]\ncase mk'\nnum\u271d : \u2124\nden\u271d : \u2115\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : Nat.coprime (Int.natAbs num\u271d) den\u271d\n\u22a2 (Rat.castRec (Rat.mk' num\u271d den\u271d)).im = 0\n[PROOFSTEP]\nsimp [Rat.castRec]\n[GOAL]\nq : \u211a\n\u22a2 (\u2191q).re = \u2191q\n[PROOFSTEP]\nshow (Rat.castRec q : \u2102).re = _\n[GOAL]\nq : \u211a\n\u22a2 (Rat.castRec q).re = \u2191q\n[PROOFSTEP]\ncases q\n[GOAL]\ncase mk'\nnum\u271d : \u2124\nden\u271d : \u2115\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : Nat.coprime (Int.natAbs num\u271d) den\u271d\n\u22a2 (Rat.castRec (Rat.mk' num\u271d den\u271d)).re = \u2191(Rat.mk' num\u271d den\u271d)\n[PROOFSTEP]\nsimp [Rat.castRec, normSq, Rat.mk_eq_divInt, Rat.mkRat_eq_div, div_eq_mul_inv, *]\n[GOAL]\nn : \u211a\nz : \u2102\n\u22a2 ((fun n z => n \u2022 z) n z).re = (\u2191n * z).re \u2227 ((fun n z => n \u2022 z) n z).im = (\u2191n * z).im\n[PROOFSTEP]\nsimp [Rat.smul_def, smul_re, smul_im]\n[GOAL]\nn : \u2124\n\u22a2 I ^ bit0 n = (-1) ^ n\n[PROOFSTEP]\nrw [zpow_bit0', I_mul_I]\n[GOAL]\nn : \u2124\n\u22a2 I ^ bit1 n = (-1) ^ n * I\n[PROOFSTEP]\nrw [zpow_bit1', I_mul_I]\n[GOAL]\nz w : \u2102\n\u22a2 (z / w).re = z.re * w.re / \u2191normSq w + z.im * w.im / \u2191normSq w\n[PROOFSTEP]\nsimp [div_eq_mul_inv, mul_assoc, sub_eq_add_neg]\n[GOAL]\nz w : \u2102\n\u22a2 (z / w).im = z.im * w.re / \u2191normSq w - z.re * w.im / \u2191normSq w\n[PROOFSTEP]\nsimp [div_eq_mul_inv, mul_assoc, sub_eq_add_neg, add_comm]\n[GOAL]\nz : \u2102\n\u22a2 -(z * I) * I = z\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\n\u22a2 I\u207b\u00b9 = -I\n[PROOFSTEP]\nrw [inv_eq_one_div, div_I, one_mul]\n[GOAL]\nn : \u2115\nh : \u2191n = 0\n\u22a2 n = 0\n[PROOFSTEP]\nrwa [\u2190 ofReal_nat_cast, ofReal_eq_zero, Nat.cast_eq_zero] at h \n[GOAL]\nz : \u2102\n\u22a2 \u2191z.re = (z + \u2191(starRingEnd \u2102) z) / 2\n[PROOFSTEP]\nhave : (\u2191(\u21912 : \u211d) : \u2102) = (2 : \u2102) := by rfl\n[GOAL]\nz : \u2102\n\u22a2 \u21912 = 2\n[PROOFSTEP]\nrfl\n[GOAL]\nz : \u2102\nthis : \u21912 = 2\n\u22a2 \u2191z.re = (z + \u2191(starRingEnd \u2102) z) / 2\n[PROOFSTEP]\nsimp only [add_conj, ofReal_mul, ofReal_one, ofReal_bit0, this, mul_div_cancel_left (z.re : \u2102) two_ne_zero]\n[GOAL]\nz : \u2102\n\u22a2 \u2191z.im = (z - \u2191(starRingEnd \u2102) z) / (2 * I)\n[PROOFSTEP]\nhave : (\u21912 : \u211d) * I = 2 * I := by rfl\n[GOAL]\nz : \u2102\n\u22a2 \u21912 * I = 2 * I\n[PROOFSTEP]\nrfl\n[GOAL]\nz : \u2102\nthis : \u21912 * I = 2 * I\n\u22a2 \u2191z.im = (z - \u2191(starRingEnd \u2102) z) / (2 * I)\n[PROOFSTEP]\nsimp only [sub_conj, ofReal_mul, ofReal_one, ofReal_bit0, mul_right_comm, this,\n mul_div_cancel_left _ (mul_ne_zero two_ne_zero I_ne_zero : 2 * I \u2260 0)]\n[GOAL]\nz : \u2102\n\u22a2 Real.sqrt (\u2191normSq (\u2191(starRingEnd \u2102) z)) = Real.sqrt (\u2191normSq z)\n[PROOFSTEP]\nsimp\n[GOAL]\nz : \u2102\n\u22a2 |z.re| \u2264 Real.sqrt (\u2191normSq z)\n[PROOFSTEP]\nrw [mul_self_le_mul_self_iff (abs_nonneg z.re) (abs_nonneg' _), abs_mul_abs_self, mul_self_abs]\n[GOAL]\nz : \u2102\n\u22a2 z.re * z.re \u2264 \u2191normSq z\n[PROOFSTEP]\napply re_sq_le_normSq\n[GOAL]\nz w : \u2102\n\u22a2 Real.sqrt (\u2191normSq (z * w)) = Real.sqrt (\u2191normSq z) * Real.sqrt (\u2191normSq w)\n[PROOFSTEP]\nrw [normSq_mul, Real.sqrt_mul (normSq_nonneg _)]\n[GOAL]\nz w : \u2102\n\u22a2 Real.sqrt (\u2191normSq (z + w)) * Real.sqrt (\u2191normSq (z + w)) \u2264\n (Real.sqrt (\u2191normSq z) + Real.sqrt (\u2191normSq w)) * (Real.sqrt (\u2191normSq z) + Real.sqrt (\u2191normSq w))\n[PROOFSTEP]\nrw [mul_self_abs, add_mul_self_eq, mul_self_abs, mul_self_abs, add_right_comm, normSq_add, add_le_add_iff_left,\n mul_assoc, mul_le_mul_left (zero_lt_two' \u211d), \u2190 Real.sqrt_mul <| normSq_nonneg z, \u2190 normSq_conj w, \u2190 map_mul]\n[GOAL]\nz w : \u2102\n\u22a2 (z * \u2191(starRingEnd \u2102) w).re \u2264 Real.sqrt (\u2191normSq (z * \u2191(starRingEnd \u2102) w))\n[PROOFSTEP]\nexact re_le_abs (z * conj w)\n[GOAL]\nr : \u211d\n\u22a2 \u2191abs \u2191r = |r|\n[PROOFSTEP]\nsimp [Complex.abs, normSq_ofReal, Real.sqrt_mul_self_eq_abs]\n[GOAL]\nn : \u2115\n\u22a2 \u2191abs \u2191n = \u2191abs \u2191\u2191n\n[PROOFSTEP]\nrw [ofReal_nat_cast]\n[GOAL]\nz : \u2102\n\u22a2 \u2191abs z ^ 2 - z.re ^ 2 = z.im ^ 2\n[PROOFSTEP]\nrw [sq_abs, normSq_apply, \u2190 sq, \u2190 sq, add_sub_cancel']\n[GOAL]\nz : \u2102\n\u22a2 \u2191abs z ^ 2 - z.im ^ 2 = z.re ^ 2\n[PROOFSTEP]\nrw [\u2190 sq_abs_sub_sq_re, sub_sub_cancel]\n[GOAL]\n\u22a2 \u2191abs I = 1\n[PROOFSTEP]\nsimp [Complex.abs]\n[GOAL]\n\u22a2 \u2191abs 2 = \u2191abs \u21912\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 0 \u2264 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 range \u2191abs \u2286 Ici 0\n[PROOFSTEP]\nsimp only [range_subset_iff, Ici, mem_setOf_eq, map_nonneg, forall_const]\n[GOAL]\nz : \u2102\n\u22a2 z.re ^ 2 \u2264 \u2191normSq z\n[PROOFSTEP]\nrw [normSq_apply, \u2190 sq]\n[GOAL]\nz : \u2102\n\u22a2 z.re ^ 2 \u2264 z.re ^ 2 + z.im * z.im\n[PROOFSTEP]\nexact le_add_of_nonneg_right (mul_self_nonneg _)\n[GOAL]\nz : \u2102\n\u22a2 z.im ^ 2 \u2264 \u2191normSq z\n[PROOFSTEP]\nrw [normSq_apply, \u2190 sq, \u2190 sq]\n[GOAL]\nz : \u2102\n\u22a2 z.im ^ 2 \u2264 z.re ^ 2 + z.im ^ 2\n[PROOFSTEP]\nexact le_add_of_nonneg_left (sq_nonneg _)\n[GOAL]\nz : \u2102\n\u22a2 |z.re| < \u2191abs z \u2194 z.im \u2260 0\n[PROOFSTEP]\nrw [Complex.abs, AbsoluteValue.coe_mk, MulHom.coe_mk, Real.lt_sqrt (abs_nonneg _), normSq_apply, _root_.sq_abs, \u2190 sq,\n lt_add_iff_pos_right, mul_self_pos]\n[GOAL]\nz : \u2102\n\u22a2 |z.im| < \u2191abs z \u2194 z.re \u2260 0\n[PROOFSTEP]\nsimpa using @abs_re_lt_abs (z * I)\n[GOAL]\nz : \u2102\n\u22a2 \u2191abs z \u2264 |z.re| + |z.im|\n[PROOFSTEP]\nsimpa [re_add_im] using Complex.abs.add_le z.re (z.im * I)\n[GOAL]\n\u22a2 1 \u2260 0\n[PROOFSTEP]\napply one_ne_zero\n[GOAL]\nz : \u2102\n\u22a2 \u2191abs z \u2264 Real.sqrt 2 * max |z.re| |z.im|\n[PROOFSTEP]\ncases' z with x y\n[GOAL]\ncase mk\nx y : \u211d\n\u22a2 \u2191abs { re := x, im := y } \u2264 Real.sqrt 2 * max |{ re := x, im := y }.re| |{ re := x, im := y }.im|\n[PROOFSTEP]\nsimp only [abs_apply, normSq_mk, \u2190 sq]\n[GOAL]\ncase mk\nx y : \u211d\n\u22a2 Real.sqrt (x ^ 2 + y ^ 2) \u2264 Real.sqrt 2 * max |x| |y|\n[PROOFSTEP]\nby_cases hle : |x| \u2264 |y|\n[GOAL]\ncase pos\nx y : \u211d\nhle : |x| \u2264 |y|\n\u22a2 Real.sqrt (x ^ 2 + y ^ 2) \u2264 Real.sqrt 2 * max |x| |y|\n[PROOFSTEP]\ncalc\n Real.sqrt (x ^ 2 + y ^ 2) \u2264 Real.sqrt (y ^ 2 + y ^ 2) := Real.sqrt_le_sqrt (add_le_add_right (sq_le_sq.2 hle) _)\n _ = Real.sqrt 2 * max |x| |y| := by rw [max_eq_right hle, \u2190 two_mul, Real.sqrt_mul two_pos.le, Real.sqrt_sq_eq_abs]\n[GOAL]\nx y : \u211d\nhle : |x| \u2264 |y|\n\u22a2 Real.sqrt (y ^ 2 + y ^ 2) = Real.sqrt 2 * max |x| |y|\n[PROOFSTEP]\nrw [max_eq_right hle, \u2190 two_mul, Real.sqrt_mul two_pos.le, Real.sqrt_sq_eq_abs]\n[GOAL]\ncase neg\nx y : \u211d\nhle : \u00ac|x| \u2264 |y|\n\u22a2 Real.sqrt (x ^ 2 + y ^ 2) \u2264 Real.sqrt 2 * max |x| |y|\n[PROOFSTEP]\nhave hle' := le_of_not_le hle\n[GOAL]\ncase neg\nx y : \u211d\nhle : \u00ac|x| \u2264 |y|\nhle' : |y| \u2264 |x|\n\u22a2 Real.sqrt (x ^ 2 + y ^ 2) \u2264 Real.sqrt 2 * max |x| |y|\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\ncase neg\nx y : \u211d\nhle : \u00ac|x| \u2264 |y|\nhle' : |y| \u2264 |x|\n\u22a2 Real.sqrt (y ^ 2 + x ^ 2) \u2264 Real.sqrt 2 * max |x| |y|\n[PROOFSTEP]\ncalc\n Real.sqrt (y ^ 2 + x ^ 2) \u2264 Real.sqrt (x ^ 2 + x ^ 2) := Real.sqrt_le_sqrt (add_le_add_right (sq_le_sq.2 hle') _)\n _ = Real.sqrt 2 * max |x| |y| := by rw [max_eq_left hle', \u2190 two_mul, Real.sqrt_mul two_pos.le, Real.sqrt_sq_eq_abs]\n[GOAL]\nx y : \u211d\nhle : \u00ac|x| \u2264 |y|\nhle' : |y| \u2264 |x|\n\u22a2 Real.sqrt (x ^ 2 + x ^ 2) = Real.sqrt 2 * max |x| |y|\n[PROOFSTEP]\nrw [max_eq_left hle', \u2190 two_mul, Real.sqrt_mul two_pos.le, Real.sqrt_sq_eq_abs]\n[GOAL]\nz : \u2102\nhz : z = 0\n\u22a2 |z.re / \u2191abs z| \u2264 1\n[PROOFSTEP]\nsimp [hz, zero_le_one]\n[GOAL]\nz : \u2102\nhz : \u00acz = 0\n\u22a2 |z.re / \u2191abs z| \u2264 1\n[PROOFSTEP]\nsimp_rw [_root_.abs_div, abs_abs, div_le_iff (AbsoluteValue.pos Complex.abs hz), one_mul, abs_re_le_abs]\n[GOAL]\nz : \u2102\nhz : z = 0\n\u22a2 |z.im / \u2191abs z| \u2264 1\n[PROOFSTEP]\nsimp [hz, zero_le_one]\n[GOAL]\nz : \u2102\nhz : \u00acz = 0\n\u22a2 |z.im / \u2191abs z| \u2264 1\n[PROOFSTEP]\nsimp_rw [_root_.abs_div, abs_abs, div_le_iff (AbsoluteValue.pos Complex.abs hz), one_mul, abs_im_le_abs]\n[GOAL]\nn : \u2115\n\u22a2 \u2191abs \u2191n = \u2191n\n[PROOFSTEP]\nrw [\u2190 ofReal_nat_cast, abs_of_nonneg (Nat.cast_nonneg n)]\n[GOAL]\nn : \u2124\n\u22a2 |\u2191n| = \u2191abs \u2191n\n[PROOFSTEP]\nrw [\u2190 ofReal_int_cast, abs_ofReal]\n[GOAL]\nx : \u2102\n\u22a2 \u2191normSq x = \u2191abs x ^ 2\n[PROOFSTEP]\nsimp [abs, sq, abs_def, Real.mul_self_sqrt (normSq_nonneg _)]\n[GOAL]\nf : CauSeq \u2102 \u2191abs\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 \u2191abs (\u2191f j - \u2191f i) < \u03b5\nj : \u2115\nij : j \u2265 i\n\u22a2 abs' ((fun n => (\u2191f n).re) j - (fun n => (\u2191f n).re) i) \u2264 \u2191abs (\u2191f j - \u2191f i)\n[PROOFSTEP]\nsimpa using abs_re_le_abs (f j - f i)\n[GOAL]\nf : CauSeq \u2102 \u2191abs\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nH : \u2200 (j : \u2115), j \u2265 i \u2192 \u2191abs (\u2191f j - \u2191f i) < \u03b5\nj : \u2115\nij : j \u2265 i\n\u22a2 abs' ((fun n => (\u2191f n).im) j - (fun n => (\u2191f n).im) i) \u2264 \u2191abs (\u2191f j - \u2191f i)\n[PROOFSTEP]\nsimpa using abs_im_le_abs (f j - f i)\n[GOAL]\nf : CauSeq \u2102 \u2191abs\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nH :\n \u2200 (j : \u2115),\n j \u2265 i \u2192\n abs'\n (\u2191({ val := fun n => (\u2191f n).re, property := (_ : IsCauSeq abs' fun n => (\u2191f n).re) } -\n CauSeq.const abs'\n (CauSeq.lim { val := fun n => (\u2191f n).re, property := (_ : IsCauSeq abs' fun n => (\u2191f n).re) }))\n j) <\n \u03b5 / 2 \u2227\n abs'\n (\u2191({ val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) } -\n CauSeq.const abs'\n (CauSeq.lim { val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) }))\n j) <\n \u03b5 / 2\nj : \u2115\nij : j \u2265 i\n\u22a2 \u2191abs (\u2191(f - CauSeq.const (\u2191abs) (limAux f)) j) < \u03b5\n[PROOFSTEP]\ncases' H _ ij with H\u2081 H\u2082\n[GOAL]\ncase intro\nf : CauSeq \u2102 \u2191abs\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nH :\n \u2200 (j : \u2115),\n j \u2265 i \u2192\n abs'\n (\u2191({ val := fun n => (\u2191f n).re, property := (_ : IsCauSeq abs' fun n => (\u2191f n).re) } -\n CauSeq.const abs'\n (CauSeq.lim { val := fun n => (\u2191f n).re, property := (_ : IsCauSeq abs' fun n => (\u2191f n).re) }))\n j) <\n \u03b5 / 2 \u2227\n abs'\n (\u2191({ val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) } -\n CauSeq.const abs'\n (CauSeq.lim { val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) }))\n j) <\n \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nH\u2081 :\n abs'\n (\u2191({ val := fun n => (\u2191f n).re, property := (_ : IsCauSeq abs' fun n => (\u2191f n).re) } -\n CauSeq.const abs'\n (CauSeq.lim { val := fun n => (\u2191f n).re, property := (_ : IsCauSeq abs' fun n => (\u2191f n).re) }))\n j) <\n \u03b5 / 2\nH\u2082 :\n abs'\n (\u2191({ val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) } -\n CauSeq.const abs'\n (CauSeq.lim { val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) }))\n j) <\n \u03b5 / 2\n\u22a2 \u2191abs (\u2191(f - CauSeq.const (\u2191abs) (limAux f)) j) < \u03b5\n[PROOFSTEP]\napply lt_of_le_of_lt (abs_le_abs_re_add_abs_im _)\n[GOAL]\ncase intro\nf : CauSeq \u2102 \u2191abs\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nH :\n \u2200 (j : \u2115),\n j \u2265 i \u2192\n abs'\n (\u2191({ val := fun n => (\u2191f n).re, property := (_ : IsCauSeq abs' fun n => (\u2191f n).re) } -\n CauSeq.const abs'\n (CauSeq.lim { val := fun n => (\u2191f n).re, property := (_ : IsCauSeq abs' fun n => (\u2191f n).re) }))\n j) <\n \u03b5 / 2 \u2227\n abs'\n (\u2191({ val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) } -\n CauSeq.const abs'\n (CauSeq.lim { val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) }))\n j) <\n \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nH\u2081 :\n abs'\n (\u2191({ val := fun n => (\u2191f n).re, property := (_ : IsCauSeq abs' fun n => (\u2191f n).re) } -\n CauSeq.const abs'\n (CauSeq.lim { val := fun n => (\u2191f n).re, property := (_ : IsCauSeq abs' fun n => (\u2191f n).re) }))\n j) <\n \u03b5 / 2\nH\u2082 :\n abs'\n (\u2191({ val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) } -\n CauSeq.const abs'\n (CauSeq.lim { val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) }))\n j) <\n \u03b5 / 2\n\u22a2 abs' (\u2191(f - CauSeq.const (\u2191abs) (limAux f)) j).re + abs' (\u2191(f - CauSeq.const (\u2191abs) (limAux f)) j).im < \u03b5\n[PROOFSTEP]\ndsimp [limAux] at *\n[GOAL]\ncase intro\nf : CauSeq \u2102 \u2191abs\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nH :\n \u2200 (j : \u2115),\n j \u2265 i \u2192\n abs' ((\u2191f j).re - CauSeq.lim { val := fun n => (\u2191f n).re, property := (_ : IsCauSeq abs' fun n => (\u2191f n).re) }) <\n \u03b5 / 2 \u2227\n abs'\n ((\u2191f j).im - CauSeq.lim { val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) }) <\n \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nH\u2081 :\n abs' ((\u2191f j).re - CauSeq.lim { val := fun n => (\u2191f n).re, property := (_ : IsCauSeq abs' fun n => (\u2191f n).re) }) <\n \u03b5 / 2\nH\u2082 :\n abs' ((\u2191f j).im - CauSeq.lim { val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) }) <\n \u03b5 / 2\n\u22a2 abs' ((\u2191f j).re - CauSeq.lim (cauSeqRe f)) + abs' ((\u2191f j).im - CauSeq.lim (cauSeqIm f)) < \u03b5\n[PROOFSTEP]\nhave := add_lt_add H\u2081 H\u2082\n[GOAL]\ncase intro\nf : CauSeq \u2102 \u2191abs\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nH :\n \u2200 (j : \u2115),\n j \u2265 i \u2192\n abs' ((\u2191f j).re - CauSeq.lim { val := fun n => (\u2191f n).re, property := (_ : IsCauSeq abs' fun n => (\u2191f n).re) }) <\n \u03b5 / 2 \u2227\n abs'\n ((\u2191f j).im - CauSeq.lim { val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) }) <\n \u03b5 / 2\nj : \u2115\nij : j \u2265 i\nH\u2081 :\n abs' ((\u2191f j).re - CauSeq.lim { val := fun n => (\u2191f n).re, property := (_ : IsCauSeq abs' fun n => (\u2191f n).re) }) <\n \u03b5 / 2\nH\u2082 :\n abs' ((\u2191f j).im - CauSeq.lim { val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) }) <\n \u03b5 / 2\nthis :\n abs' ((\u2191f j).re - CauSeq.lim { val := fun n => (\u2191f n).re, property := (_ : IsCauSeq abs' fun n => (\u2191f n).re) }) +\n abs' ((\u2191f j).im - CauSeq.lim { val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) }) <\n \u03b5 / 2 + \u03b5 / 2\n\u22a2 abs' ((\u2191f j).re - CauSeq.lim (cauSeqRe f)) + abs' ((\u2191f j).im - CauSeq.lim (cauSeqIm f)) < \u03b5\n[PROOFSTEP]\nrwa [add_halves] at this \n[GOAL]\nf : CauSeq \u2102 \u2191abs\nx\u271d : \u2115\n\u22a2 (\u2191(CauSeq.const (\u2191abs) (limAux f)) x\u271d).re =\n (\u2191(CauSeq.const (\u2191abs) (\u2191(CauSeq.lim (cauSeqRe f)) + \u2191(CauSeq.lim (cauSeqIm f)) * I)) x\u271d).re\n[PROOFSTEP]\nsimp [limAux, cauSeqRe, ofReal']\n[GOAL]\nf : CauSeq \u2102 \u2191abs\nx\u271d : \u2115\n\u22a2 (\u2191(CauSeq.const (\u2191abs) (limAux f)) x\u271d).im =\n (\u2191(CauSeq.const (\u2191abs) (\u2191(CauSeq.lim (cauSeqRe f)) + \u2191(CauSeq.lim (cauSeqIm f)) * I)) x\u271d).im\n[PROOFSTEP]\nsimp [limAux, cauSeqIm, ofReal']\n[GOAL]\nf : CauSeq \u2102 \u2191abs\n\u22a2 CauSeq.lim (cauSeqRe f) = (CauSeq.lim f).re\n[PROOFSTEP]\nrw [lim_eq_lim_im_add_lim_re]\n[GOAL]\nf : CauSeq \u2102 \u2191abs\n\u22a2 CauSeq.lim (cauSeqRe f) = (\u2191(CauSeq.lim (cauSeqRe f)) + \u2191(CauSeq.lim (cauSeqIm f)) * I).re\n[PROOFSTEP]\nsimp [ofReal']\n[GOAL]\nf : CauSeq \u2102 \u2191abs\n\u22a2 CauSeq.lim (cauSeqIm f) = (CauSeq.lim f).im\n[PROOFSTEP]\nrw [lim_eq_lim_im_add_lim_re]\n[GOAL]\nf : CauSeq \u2102 \u2191abs\n\u22a2 CauSeq.lim (cauSeqIm f) = (\u2191(CauSeq.lim (cauSeqRe f)) + \u2191(CauSeq.lim (cauSeqIm f)) * I).im\n[PROOFSTEP]\nsimp [ofReal']\n[GOAL]\nf : CauSeq \u2102 \u2191abs\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 \u2191abs (\u2191f j - \u2191f i) < \u03b5\nj : \u2115\nhj : j \u2265 i\n\u22a2 \u2191abs ((fun n => \u2191(starRingEnd \u2102) (\u2191f n)) j - (fun n => \u2191(starRingEnd \u2102) (\u2191f n)) i) < \u03b5\n[PROOFSTEP]\nrw [\u2190 RingHom.map_sub, abs_conj]\n[GOAL]\nf : CauSeq \u2102 \u2191abs\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 \u2191abs (\u2191f j - \u2191f i) < \u03b5\nj : \u2115\nhj : j \u2265 i\n\u22a2 \u2191abs (\u2191f j - \u2191f i) < \u03b5\n[PROOFSTEP]\nexact hi j hj\n[GOAL]\nf : CauSeq \u2102 \u2191abs\n\u22a2 (CauSeq.lim (cauSeqConj f)).re = (\u2191(starRingEnd \u2102) (CauSeq.lim f)).re\n[PROOFSTEP]\nsimp [cauSeqConj, (lim_re _).symm, cauSeqRe]\n[GOAL]\nf : CauSeq \u2102 \u2191abs\n\u22a2 (CauSeq.lim (cauSeqConj f)).im = (\u2191(starRingEnd \u2102) (CauSeq.lim f)).im\n[PROOFSTEP]\nsimp [cauSeqConj, (lim_im _).symm, cauSeqIm, (lim_neg _).symm]\n[GOAL]\nf : CauSeq \u2102 \u2191abs\n\u22a2 CauSeq.lim { val := fun n => -(\u2191f n).im, property := (_ : (fun f => IsCauSeq abs' f) fun n => -(\u2191f n).im) } =\n CauSeq.lim (-{ val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) })\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Data.Complex.Basic", "llama_tokens": 19774, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737214979745, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.40710959227059107}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nn : \u2115\nm : M\n\u22a2 \u2191(\u03b9 Q) m * \u2191(\u03b9 Q) m = \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191Q m)\n[PROOFSTEP]\nerw [\u2190 AlgHom.map_mul, RingQuot.mkAlgHom_rel R (Rel.of m), AlgHom.commutes]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nn : \u2115\nm : M\n\u22a2 \u2191(algebraMap R (RingQuot (Rel Q))) (\u2191Q m) = \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191Q m)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\ng : CliffordAlgebra Q \u2192\u2090[R] A\nm : M\n\u22a2 \u2191g (\u2191(\u03b9 Q) m) * \u2191g (\u2191(\u03b9 Q) m) = \u2191(algebraMap R A) (\u2191Q m)\n[PROOFSTEP]\nrw [\u2190 AlgHom.map_mul, \u03b9_sq_scalar, AlgHom.commutes]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nf : { f // \u2200 (m : M), \u2191f m * \u2191f m = \u2191(algebraMap R A) (\u2191Q m) }\nx y : TensorAlgebra R M\nh : Rel Q x y\n\u22a2 \u2191(\u2191(TensorAlgebra.lift R) \u2191f) x = \u2191(\u2191(TensorAlgebra.lift R) \u2191f) y\n[PROOFSTEP]\ninduction h\n[GOAL]\ncase of\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nf : { f // \u2200 (m : M), \u2191f m * \u2191f m = \u2191(algebraMap R A) (\u2191Q m) }\nx y : TensorAlgebra R M\nm\u271d : M\n\u22a2 \u2191(\u2191(TensorAlgebra.lift R) \u2191f) (\u2191(TensorAlgebra.\u03b9 R) m\u271d * \u2191(TensorAlgebra.\u03b9 R) m\u271d) =\n \u2191(\u2191(TensorAlgebra.lift R) \u2191f) (\u2191(algebraMap R (TensorAlgebra R M)) (\u2191Q m\u271d))\n[PROOFSTEP]\nrw [AlgHom.commutes, AlgHom.map_mul, TensorAlgebra.lift_\u03b9_apply, f.prop]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nF : CliffordAlgebra Q \u2192\u2090[R] A\nm : M\n\u22a2 \u2191(LinearMap.comp (AlgHom.toLinearMap F) (\u03b9 Q)) m * \u2191(LinearMap.comp (AlgHom.toLinearMap F) (\u03b9 Q)) m =\n \u2191(algebraMap R A) (\u2191Q m)\n[PROOFSTEP]\nrw [LinearMap.comp_apply, AlgHom.toLinearMap_apply, comp_\u03b9_sq_scalar]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nf : { f // \u2200 (m : M), \u2191f m * \u2191f m = \u2191(algebraMap R A) (\u2191Q m) }\n\u22a2 (fun F =>\n { val := LinearMap.comp (AlgHom.toLinearMap F) (\u03b9 Q),\n property :=\n (_ :\n \u2200 (m : M),\n \u2191(LinearMap.comp (AlgHom.toLinearMap F) (\u03b9 Q)) m * \u2191(LinearMap.comp (AlgHom.toLinearMap F) (\u03b9 Q)) m =\n \u2191(algebraMap R A) (\u2191Q m)) })\n ((fun f =>\n \u2191(RingQuot.liftAlgHom R)\n { val := \u2191(TensorAlgebra.lift R) \u2191f,\n property :=\n (_ :\n \u2200 (x y : TensorAlgebra R M),\n Rel Q x y \u2192 \u2191(\u2191(TensorAlgebra.lift R) \u2191f) x = \u2191(\u2191(TensorAlgebra.lift R) \u2191f) y) })\n f) =\n f\n[PROOFSTEP]\next x\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nf : { f // \u2200 (m : M), \u2191f m * \u2191f m = \u2191(algebraMap R A) (\u2191Q m) }\nx : M\n\u22a2 \u2191\u2191((fun F =>\n { val := LinearMap.comp (AlgHom.toLinearMap F) (\u03b9 Q),\n property :=\n (_ :\n \u2200 (m : M),\n \u2191(LinearMap.comp (AlgHom.toLinearMap F) (\u03b9 Q)) m *\n \u2191(LinearMap.comp (AlgHom.toLinearMap F) (\u03b9 Q)) m =\n \u2191(algebraMap R A) (\u2191Q m)) })\n ((fun f =>\n \u2191(RingQuot.liftAlgHom R)\n { val := \u2191(TensorAlgebra.lift R) \u2191f,\n property :=\n (_ :\n \u2200 (x y : TensorAlgebra R M),\n Rel Q x y \u2192 \u2191(\u2191(TensorAlgebra.lift R) \u2191f) x = \u2191(\u2191(TensorAlgebra.lift R) \u2191f) y) })\n f))\n x =\n \u2191\u2191f x\n[PROOFSTEP]\nexact (RingQuot.liftAlgHom_mkAlgHom_apply _ _ _ _).trans (TensorAlgebra.lift_\u03b9_apply _ x)\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nF : CliffordAlgebra Q \u2192\u2090[R] A\nx : M\n\u22a2 \u2191(LinearMap.comp\n (AlgHom.toLinearMap\n (AlgHom.comp\n ((fun f =>\n \u2191(RingQuot.liftAlgHom R)\n { val := \u2191(TensorAlgebra.lift R) \u2191f,\n property :=\n (_ :\n \u2200 (x y : TensorAlgebra R M),\n Rel Q x y \u2192 \u2191(\u2191(TensorAlgebra.lift R) \u2191f) x = \u2191(\u2191(TensorAlgebra.lift R) \u2191f) y) })\n ((fun F =>\n { val := LinearMap.comp (AlgHom.toLinearMap F) (\u03b9 Q),\n property :=\n (_ :\n \u2200 (m : M),\n \u2191(LinearMap.comp (AlgHom.toLinearMap F) (\u03b9 Q)) m *\n \u2191(LinearMap.comp (AlgHom.toLinearMap F) (\u03b9 Q)) m =\n \u2191(algebraMap R A) (\u2191Q m)) })\n F))\n (RingQuot.mkAlgHom R fun x y => Rel Q x y)))\n (TensorAlgebra.\u03b9 R))\n x =\n \u2191(LinearMap.comp (AlgHom.toLinearMap (AlgHom.comp F (RingQuot.mkAlgHom R fun x y => Rel Q x y)))\n (TensorAlgebra.\u03b9 R))\n x\n[PROOFSTEP]\nexact (RingQuot.liftAlgHom_mkAlgHom_apply _ _ _ _).trans (TensorAlgebra.lift_\u03b9_apply _ _)\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nf : M \u2192\u2097[R] A\ncond : \u2200 (m : M), \u2191f m * \u2191f m = \u2191(algebraMap R A) (\u2191Q m)\ng : CliffordAlgebra Q \u2192\u2090[R] A\n\u22a2 LinearMap.comp (AlgHom.toLinearMap g) (\u03b9 Q) = f \u2194 g = \u2191(lift Q) { val := f, property := cond }\n[PROOFSTEP]\nconvert (lift Q : _ \u2243 (CliffordAlgebra Q \u2192\u2090[R] A)).symm_apply_eq\n[GOAL]\ncase h.e'_1.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nf : M \u2192\u2097[R] A\ncond : \u2200 (m : M), \u2191f m * \u2191f m = \u2191(algebraMap R A) (\u2191Q m)\ng : CliffordAlgebra Q \u2192\u2090[R] A\n\u22a2 LinearMap.comp (AlgHom.toLinearMap g) (\u03b9 Q) = f \u2194 \u2191(lift Q).symm g = { val := f, property := cond }\n[PROOFSTEP]\nrw [lift_symm_apply, Subtype.mk_eq_mk]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\ng : CliffordAlgebra Q \u2192\u2090[R] A\n\u22a2 \u2191(lift Q)\n { val := LinearMap.comp (AlgHom.toLinearMap g) (\u03b9 Q),\n property := (_ : \u2200 (m : M), \u2191g (\u2191(\u03b9 Q) m) * \u2191g (\u2191(\u03b9 Q) m) = \u2191(algebraMap R A) (\u2191Q m)) } =\n g\n[PROOFSTEP]\nexact (lift Q : _ \u2243 (CliffordAlgebra Q \u2192\u2090[R] A)).apply_symm_apply g\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA\u271d : Type u_3\ninst\u271d\u00b3 : Semiring A\u271d\ninst\u271d\u00b2 : Algebra R A\u271d\nA : Type u_4\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nf g : CliffordAlgebra Q \u2192\u2090[R] A\n\u22a2 LinearMap.comp (AlgHom.toLinearMap f) (\u03b9 Q) = LinearMap.comp (AlgHom.toLinearMap g) (\u03b9 Q) \u2192 f = g\n[PROOFSTEP]\nintro h\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA\u271d : Type u_3\ninst\u271d\u00b3 : Semiring A\u271d\ninst\u271d\u00b2 : Algebra R A\u271d\nA : Type u_4\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nf g : CliffordAlgebra Q \u2192\u2090[R] A\nh : LinearMap.comp (AlgHom.toLinearMap f) (\u03b9 Q) = LinearMap.comp (AlgHom.toLinearMap g) (\u03b9 Q)\n\u22a2 f = g\n[PROOFSTEP]\napply (lift Q).symm.injective\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA\u271d : Type u_3\ninst\u271d\u00b3 : Semiring A\u271d\ninst\u271d\u00b2 : Algebra R A\u271d\nA : Type u_4\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nf g : CliffordAlgebra Q \u2192\u2090[R] A\nh : LinearMap.comp (AlgHom.toLinearMap f) (\u03b9 Q) = LinearMap.comp (AlgHom.toLinearMap g) (\u03b9 Q)\n\u22a2 \u2191(lift Q).symm f = \u2191(lift Q).symm g\n[PROOFSTEP]\nrw [lift_symm_apply, lift_symm_apply]\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA\u271d : Type u_3\ninst\u271d\u00b3 : Semiring A\u271d\ninst\u271d\u00b2 : Algebra R A\u271d\nA : Type u_4\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nf g : CliffordAlgebra Q \u2192\u2090[R] A\nh : LinearMap.comp (AlgHom.toLinearMap f) (\u03b9 Q) = LinearMap.comp (AlgHom.toLinearMap g) (\u03b9 Q)\n\u22a2 { val := LinearMap.comp (AlgHom.toLinearMap f) (\u03b9 Q),\n property :=\n (_ :\n \u2200 (m : M),\n \u2191(LinearMap.comp (AlgHom.toLinearMap f) (\u03b9 Q)) m * \u2191(LinearMap.comp (AlgHom.toLinearMap f) (\u03b9 Q)) m =\n \u2191(algebraMap R A) (\u2191Q m)) } =\n { val := LinearMap.comp (AlgHom.toLinearMap g) (\u03b9 Q),\n property :=\n (_ :\n \u2200 (m : M),\n \u2191(LinearMap.comp (AlgHom.toLinearMap g) (\u03b9 Q)) m * \u2191(LinearMap.comp (AlgHom.toLinearMap g) (\u03b9 Q)) m =\n \u2191(algebraMap R A) (\u2191Q m)) }\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nh_grade0 : \u2200 (r : R), C (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_grade1 : \u2200 (x : M), C (\u2191(\u03b9 Q) x)\nh_mul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nh_add : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\n\u22a2 C a\n[PROOFSTEP]\nlet s : Subalgebra R (CliffordAlgebra Q) :=\n { carrier := C\n mul_mem' := @h_mul\n add_mem' := @h_add\n algebraMap_mem' := h_grade0 }\n -- porting note: Added `h`. `h` is needed for `of`.\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nh_grade0 : \u2200 (r : R), C (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_grade1 : \u2200 (x : M), C (\u2191(\u03b9 Q) x)\nh_mul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nh_add : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n {\n toSubsemiring :=\n { toSubmonoid := { toSubsemigroup := { carrier := C, mul_mem' := h_mul }, one_mem' := (_ : 1 \u2208 C) },\n add_mem' := h_add, zero_mem' := (_ : 0 \u2208 C) },\n algebraMap_mem' := h_grade0 }\n\u22a2 C a\n[PROOFSTEP]\nletI h : AddCommMonoid s := inferInstanceAs (AddCommMonoid (Subalgebra.toSubmodule s))\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nh_grade0 : \u2200 (r : R), C (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_grade1 : \u2200 (x : M), C (\u2191(\u03b9 Q) x)\nh_mul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nh_add : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n {\n toSubsemiring :=\n { toSubmonoid := { toSubsemigroup := { carrier := C, mul_mem' := h_mul }, one_mem' := (_ : 1 \u2208 C) },\n add_mem' := h_add, zero_mem' := (_ : 0 \u2208 C) },\n algebraMap_mem' := h_grade0 }\nh : AddCommMonoid { x // x \u2208 s } := inferInstanceAs (AddCommMonoid { x // x \u2208 \u2191Subalgebra.toSubmodule s })\n\u22a2 C a\n[PROOFSTEP]\nlet of : { f : M \u2192\u2097[R] s // \u2200 m, f m * f m = algebraMap _ _ (Q m) } :=\n \u27e8(\u03b9 Q).codRestrict (Subalgebra.toSubmodule s) h_grade1, fun m => Subtype.eq <| \u03b9_sq_scalar Q m\u27e9\n -- the mapping through the subalgebra is the identity\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nh_grade0 : \u2200 (r : R), C (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_grade1 : \u2200 (x : M), C (\u2191(\u03b9 Q) x)\nh_mul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nh_add : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n {\n toSubsemiring :=\n { toSubmonoid := { toSubsemigroup := { carrier := C, mul_mem' := h_mul }, one_mem' := (_ : 1 \u2208 C) },\n add_mem' := h_add, zero_mem' := (_ : 0 \u2208 C) },\n algebraMap_mem' := h_grade0 }\nh : AddCommMonoid { x // x \u2208 s } := inferInstanceAs (AddCommMonoid { x // x \u2208 \u2191Subalgebra.toSubmodule s })\nof : { f // \u2200 (m : M), \u2191f m * \u2191f m = \u2191(algebraMap R { x // x \u2208 s }) (\u2191Q m) } :=\n { val := LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1,\n property :=\n (_ :\n \u2200 (m : M),\n \u2191(LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1) m *\n \u2191(LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1) m =\n \u2191(algebraMap R { x // x \u2208 s }) (\u2191Q m)) }\n\u22a2 C a\n[PROOFSTEP]\nhave of_id : AlgHom.id R (CliffordAlgebra Q) = s.val.comp (lift Q of) :=\n by\n ext\n simp\n -- porting note: `simp` can't apply this\n erw [LinearMap.codRestrict_apply]\n -- finding a proof is finding an element of the subalgebra\n -- porting note: was `convert Subtype.prop (lift Q of a); exact AlgHom.congr_fun of_id a`\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nh_grade0 : \u2200 (r : R), C (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_grade1 : \u2200 (x : M), C (\u2191(\u03b9 Q) x)\nh_mul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nh_add : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n {\n toSubsemiring :=\n { toSubmonoid := { toSubsemigroup := { carrier := C, mul_mem' := h_mul }, one_mem' := (_ : 1 \u2208 C) },\n add_mem' := h_add, zero_mem' := (_ : 0 \u2208 C) },\n algebraMap_mem' := h_grade0 }\nh : AddCommMonoid { x // x \u2208 s } := inferInstanceAs (AddCommMonoid { x // x \u2208 \u2191Subalgebra.toSubmodule s })\nof : { f // \u2200 (m : M), \u2191f m * \u2191f m = \u2191(algebraMap R { x // x \u2208 s }) (\u2191Q m) } :=\n { val := LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1,\n property :=\n (_ :\n \u2200 (m : M),\n \u2191(LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1) m *\n \u2191(LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1) m =\n \u2191(algebraMap R { x // x \u2208 s }) (\u2191Q m)) }\n\u22a2 AlgHom.id R (CliffordAlgebra Q) = AlgHom.comp (Subalgebra.val s) (\u2191(lift Q) of)\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nh_grade0 : \u2200 (r : R), C (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_grade1 : \u2200 (x : M), C (\u2191(\u03b9 Q) x)\nh_mul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nh_add : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n {\n toSubsemiring :=\n { toSubmonoid := { toSubsemigroup := { carrier := C, mul_mem' := h_mul }, one_mem' := (_ : 1 \u2208 C) },\n add_mem' := h_add, zero_mem' := (_ : 0 \u2208 C) },\n algebraMap_mem' := h_grade0 }\nh : AddCommMonoid { x // x \u2208 s } := inferInstanceAs (AddCommMonoid { x // x \u2208 \u2191Subalgebra.toSubmodule s })\nof : { f // \u2200 (m : M), \u2191f m * \u2191f m = \u2191(algebraMap R { x // x \u2208 s }) (\u2191Q m) } :=\n { val := LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1,\n property :=\n (_ :\n \u2200 (m : M),\n \u2191(LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1) m *\n \u2191(LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1) m =\n \u2191(algebraMap R { x // x \u2208 s }) (\u2191Q m)) }\nx\u271d : M\n\u22a2 \u2191(LinearMap.comp (AlgHom.toLinearMap (AlgHom.id R (CliffordAlgebra Q))) (\u03b9 Q)) x\u271d =\n \u2191(LinearMap.comp (AlgHom.toLinearMap (AlgHom.comp (Subalgebra.val s) (\u2191(lift Q) of))) (\u03b9 Q)) x\u271d\n[PROOFSTEP]\nsimp\n -- porting note: `simp` can't apply this\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nh_grade0 : \u2200 (r : R), C (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_grade1 : \u2200 (x : M), C (\u2191(\u03b9 Q) x)\nh_mul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nh_add : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n {\n toSubsemiring :=\n { toSubmonoid := { toSubsemigroup := { carrier := C, mul_mem' := h_mul }, one_mem' := (_ : 1 \u2208 C) },\n add_mem' := h_add, zero_mem' := (_ : 0 \u2208 C) },\n algebraMap_mem' := h_grade0 }\nh : AddCommMonoid { x // x \u2208 s } := inferInstanceAs (AddCommMonoid { x // x \u2208 \u2191Subalgebra.toSubmodule s })\nof : { f // \u2200 (m : M), \u2191f m * \u2191f m = \u2191(algebraMap R { x // x \u2208 s }) (\u2191Q m) } :=\n { val := LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1,\n property :=\n (_ :\n \u2200 (m : M),\n \u2191(LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1) m *\n \u2191(LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1) m =\n \u2191(algebraMap R { x // x \u2208 s }) (\u2191Q m)) }\nx\u271d : M\n\u22a2 \u2191(\u03b9 Q) x\u271d = \u2191(\u2191(LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1) x\u271d)\n[PROOFSTEP]\nerw [LinearMap.codRestrict_apply]\n -- finding a proof is finding an element of the subalgebra\n -- porting note: was `convert Subtype.prop (lift Q of a); exact AlgHom.congr_fun of_id a`\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nh_grade0 : \u2200 (r : R), C (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_grade1 : \u2200 (x : M), C (\u2191(\u03b9 Q) x)\nh_mul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nh_add : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n {\n toSubsemiring :=\n { toSubmonoid := { toSubsemigroup := { carrier := C, mul_mem' := h_mul }, one_mem' := (_ : 1 \u2208 C) },\n add_mem' := h_add, zero_mem' := (_ : 0 \u2208 C) },\n algebraMap_mem' := h_grade0 }\nh : AddCommMonoid { x // x \u2208 s } := inferInstanceAs (AddCommMonoid { x // x \u2208 \u2191Subalgebra.toSubmodule s })\nof : { f // \u2200 (m : M), \u2191f m * \u2191f m = \u2191(algebraMap R { x // x \u2208 s }) (\u2191Q m) } :=\n { val := LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1,\n property :=\n (_ :\n \u2200 (m : M),\n \u2191(LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1) m *\n \u2191(LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1) m =\n \u2191(algebraMap R { x // x \u2208 s }) (\u2191Q m)) }\nof_id : AlgHom.id R (CliffordAlgebra Q) = AlgHom.comp (Subalgebra.val s) (\u2191(lift Q) of)\n\u22a2 C a\n[PROOFSTEP]\nrw [\u2190 AlgHom.id_apply (R := R) a, of_id]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nC : CliffordAlgebra Q \u2192 Prop\nh_grade0 : \u2200 (r : R), C (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_grade1 : \u2200 (x : M), C (\u2191(\u03b9 Q) x)\nh_mul : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a * b)\nh_add : \u2200 (a b : CliffordAlgebra Q), C a \u2192 C b \u2192 C (a + b)\na : CliffordAlgebra Q\ns : Subalgebra R (CliffordAlgebra Q) :=\n {\n toSubsemiring :=\n { toSubmonoid := { toSubsemigroup := { carrier := C, mul_mem' := h_mul }, one_mem' := (_ : 1 \u2208 C) },\n add_mem' := h_add, zero_mem' := (_ : 0 \u2208 C) },\n algebraMap_mem' := h_grade0 }\nh : AddCommMonoid { x // x \u2208 s } := inferInstanceAs (AddCommMonoid { x // x \u2208 \u2191Subalgebra.toSubmodule s })\nof : { f // \u2200 (m : M), \u2191f m * \u2191f m = \u2191(algebraMap R { x // x \u2208 s }) (\u2191Q m) } :=\n { val := LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1,\n property :=\n (_ :\n \u2200 (m : M),\n \u2191(LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1) m *\n \u2191(LinearMap.codRestrict (\u2191Subalgebra.toSubmodule s) (\u03b9 Q) h_grade1) m =\n \u2191(algebraMap R { x // x \u2208 s }) (\u2191Q m)) }\nof_id : AlgHom.id R (CliffordAlgebra Q) = AlgHom.comp (Subalgebra.val s) (\u2191(lift Q) of)\n\u22a2 C (\u2191(AlgHom.comp (Subalgebra.val s) (\u2191(lift Q) of)) a)\n[PROOFSTEP]\nexact Subtype.prop (lift Q of a)\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na b : M\n\u22a2 \u2191(\u03b9 Q) a * \u2191(\u03b9 Q) b + \u2191(\u03b9 Q) b * \u2191(\u03b9 Q) a =\n \u2191(\u03b9 Q) (a + b) * \u2191(\u03b9 Q) (a + b) - \u2191(\u03b9 Q) a * \u2191(\u03b9 Q) a - \u2191(\u03b9 Q) b * \u2191(\u03b9 Q) b\n[PROOFSTEP]\nrw [(\u03b9 Q).map_add, mul_add, add_mul, add_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na b : M\n\u22a2 \u2191(\u03b9 Q) a * \u2191(\u03b9 Q) b + \u2191(\u03b9 Q) b * \u2191(\u03b9 Q) a =\n \u2191(\u03b9 Q) a * \u2191(\u03b9 Q) a + \u2191(\u03b9 Q) b * \u2191(\u03b9 Q) a + (\u2191(\u03b9 Q) a * \u2191(\u03b9 Q) b + \u2191(\u03b9 Q) b * \u2191(\u03b9 Q) b) - \u2191(\u03b9 Q) a * \u2191(\u03b9 Q) a -\n \u2191(\u03b9 Q) b * \u2191(\u03b9 Q) b\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na b : M\n\u22a2 \u2191(\u03b9 Q) a * \u2191(\u03b9 Q) b + \u2191(\u03b9 Q) b * \u2191(\u03b9 Q) a =\n \u2191(\u03b9 Q) a * \u2191(\u03b9 Q) a + \u2191(\u03b9 Q) b * \u2191(\u03b9 Q) a + (\u2191(\u03b9 Q) a * \u2191(\u03b9 Q) b + \u2191(\u03b9 Q) b * \u2191(\u03b9 Q) b) - \u2191(\u03b9 Q) a * \u2191(\u03b9 Q) a -\n \u2191(\u03b9 Q) b * \u2191(\u03b9 Q) b\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na b : M\n\u22a2 \u2191(\u03b9 Q) (a + b) * \u2191(\u03b9 Q) (a + b) - \u2191(\u03b9 Q) a * \u2191(\u03b9 Q) a - \u2191(\u03b9 Q) b * \u2191(\u03b9 Q) b =\n \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191Q (a + b)) - \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191Q a) -\n \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191Q b)\n[PROOFSTEP]\nrw [\u03b9_sq_scalar, \u03b9_sq_scalar, \u03b9_sq_scalar]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na b : M\n\u22a2 \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191Q (a + b)) - \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191Q a) -\n \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191Q b) =\n \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191Q (a + b) - \u2191Q a - \u2191Q b)\n[PROOFSTEP]\nrw [\u2190 RingHom.map_sub, \u2190 RingHom.map_sub]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na b : M\n\u22a2 \u2191(\u03b9 Q) a * \u2191(\u03b9 Q) b * \u2191(\u03b9 Q) a = \u2191(\u03b9 Q) (QuadraticForm.polar (\u2191Q) a b \u2022 a - \u2191Q a \u2022 b)\n[PROOFSTEP]\nrw [\u03b9_mul_comm, sub_mul, mul_assoc, \u03b9_sq_scalar, \u2190 Algebra.smul_def, \u2190 Algebra.commutes, \u2190 Algebra.smul_def, \u2190 map_smul,\n \u2190 map_smul, \u2190 map_sub]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nf : M \u2192\u2097[R] A\ncond : \u2200 (m : M), \u2191f m * \u2191f m = \u2191(algebraMap R A) (\u2191Q m)\n\u22a2 Submodule.map (AlgHom.toLinearMap (\u2191(lift Q) { val := f, property := cond })) (LinearMap.range (\u03b9 Q)) =\n LinearMap.range f\n[PROOFSTEP]\nrw [\u2190 LinearMap.range_comp, \u03b9_comp_lift]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\n\u22a2 map Q\u2081 Q\u2081 LinearMap.id (_ : \u2200 (m : M\u2081), \u2191Q\u2081 (\u2191LinearMap.id m) = \u2191Q\u2081 (\u2191LinearMap.id m)) =\n AlgHom.id R (CliffordAlgebra Q\u2081)\n[PROOFSTEP]\next m\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\nm : M\u2081\n\u22a2 \u2191(LinearMap.comp\n (AlgHom.toLinearMap (map Q\u2081 Q\u2081 LinearMap.id (_ : \u2200 (m : M\u2081), \u2191Q\u2081 (\u2191LinearMap.id m) = \u2191Q\u2081 (\u2191LinearMap.id m))))\n (\u03b9 Q\u2081))\n m =\n \u2191(LinearMap.comp (AlgHom.toLinearMap (AlgHom.id R (CliffordAlgebra Q\u2081))) (\u03b9 Q\u2081)) m\n[PROOFSTEP]\nexact map_apply_\u03b9 _ _ _ _ m\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\nf : M\u2082 \u2192\u2097[R] M\u2083\nhf : \u2200 (m : M\u2082), \u2191Q\u2083 (\u2191f m) = \u2191Q\u2082 m\ng : M\u2081 \u2192\u2097[R] M\u2082\nhg : \u2200 (m : M\u2081), \u2191Q\u2082 (\u2191g m) = \u2191Q\u2081 m\n\u22a2 AlgHom.comp (map Q\u2082 Q\u2083 f hf) (map Q\u2081 Q\u2082 g hg) =\n map Q\u2081 Q\u2083 (LinearMap.comp f g) (_ : \u2200 (m : M\u2081), \u2191Q\u2083 (\u2191f (\u2191g m)) = \u2191Q\u2081 m)\n[PROOFSTEP]\next m\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\nf : M\u2082 \u2192\u2097[R] M\u2083\nhf : \u2200 (m : M\u2082), \u2191Q\u2083 (\u2191f m) = \u2191Q\u2082 m\ng : M\u2081 \u2192\u2097[R] M\u2082\nhg : \u2200 (m : M\u2081), \u2191Q\u2082 (\u2191g m) = \u2191Q\u2081 m\nm : M\u2081\n\u22a2 \u2191(LinearMap.comp (AlgHom.toLinearMap (AlgHom.comp (map Q\u2082 Q\u2083 f hf) (map Q\u2081 Q\u2082 g hg))) (\u03b9 Q\u2081)) m =\n \u2191(LinearMap.comp (AlgHom.toLinearMap (map Q\u2081 Q\u2083 (LinearMap.comp f g) (_ : \u2200 (m : M\u2081), \u2191Q\u2083 (\u2191f (\u2191g m)) = \u2191Q\u2081 m)))\n (\u03b9 Q\u2081))\n m\n[PROOFSTEP]\ndsimp only [LinearMap.comp_apply, AlgHom.comp_apply, AlgHom.toLinearMap_apply, AlgHom.id_apply]\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\nf : M\u2082 \u2192\u2097[R] M\u2083\nhf : \u2200 (m : M\u2082), \u2191Q\u2083 (\u2191f m) = \u2191Q\u2082 m\ng : M\u2081 \u2192\u2097[R] M\u2082\nhg : \u2200 (m : M\u2081), \u2191Q\u2082 (\u2191g m) = \u2191Q\u2081 m\nm : M\u2081\n\u22a2 \u2191(map Q\u2082 Q\u2083 f hf) (\u2191(map Q\u2081 Q\u2082 g hg) (\u2191(\u03b9 Q\u2081) m)) =\n \u2191(map Q\u2081 Q\u2083 (LinearMap.comp f g) (_ : \u2200 (m : M\u2081), \u2191Q\u2083 (\u2191f (\u2191g m)) = \u2191Q\u2081 m)) (\u2191(\u03b9 Q\u2081) m)\n[PROOFSTEP]\nrw [map_apply_\u03b9, map_apply_\u03b9, map_apply_\u03b9, LinearMap.comp_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\ne : QuadraticForm.IsometryEquiv Q\u2081 Q\u2082\n\u22a2 map Q\u2082 Q\u2082 (LinearMap.comp \u2191\u2191e \u2191\u2191(QuadraticForm.IsometryEquiv.symm e))\n (_ : \u2200 (m : M\u2082), \u2191Q\u2082 (\u2191\u2191\u2191e (\u2191\u2191\u2191(QuadraticForm.IsometryEquiv.symm e) m)) = \u2191Q\u2082 m) =\n AlgHom.id R (CliffordAlgebra Q\u2082)\n[PROOFSTEP]\nconvert map_id Q\u2082 using 2\n -- porting note: replaced `_` with `Q\u2082`\n[GOAL]\ncase h.e'_2.h.e'_11\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\ne : QuadraticForm.IsometryEquiv Q\u2081 Q\u2082\n\u22a2 LinearMap.comp \u2191\u2191e \u2191\u2191(QuadraticForm.IsometryEquiv.symm e) = LinearMap.id\n[PROOFSTEP]\next m\n[GOAL]\ncase h.e'_2.h.e'_11.h\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\ne : QuadraticForm.IsometryEquiv Q\u2081 Q\u2082\nm : M\u2082\n\u22a2 \u2191(LinearMap.comp \u2191\u2191e \u2191\u2191(QuadraticForm.IsometryEquiv.symm e)) m = \u2191LinearMap.id m\n[PROOFSTEP]\nexact e.toLinearEquiv.apply_symm_apply m\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\ne : QuadraticForm.IsometryEquiv Q\u2081 Q\u2082\n\u22a2 map Q\u2081 Q\u2081 (LinearMap.comp \u2191\u2191(QuadraticForm.IsometryEquiv.symm e) \u2191\u2191e)\n (_ : \u2200 (m : M\u2081), \u2191Q\u2081 (\u2191\u2191\u2191(QuadraticForm.IsometryEquiv.symm e) (\u2191\u2191\u2191e m)) = \u2191Q\u2081 m) =\n AlgHom.id R (CliffordAlgebra Q\u2081)\n[PROOFSTEP]\nconvert map_id Q\u2081 using 2\n -- porting note: replaced `_` with `Q\u2081`\n[GOAL]\ncase h.e'_2.h.e'_11\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\ne : QuadraticForm.IsometryEquiv Q\u2081 Q\u2082\n\u22a2 LinearMap.comp \u2191\u2191(QuadraticForm.IsometryEquiv.symm e) \u2191\u2191e = LinearMap.id\n[PROOFSTEP]\next m\n[GOAL]\ncase h.e'_2.h.e'_11.h\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\ne : QuadraticForm.IsometryEquiv Q\u2081 Q\u2082\nm : M\u2081\n\u22a2 \u2191(LinearMap.comp \u2191\u2191(QuadraticForm.IsometryEquiv.symm e) \u2191\u2191e) m = \u2191LinearMap.id m\n[PROOFSTEP]\nexact e.toLinearEquiv.symm_apply_apply m\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\ne\u2081\u2082 : QuadraticForm.IsometryEquiv Q\u2081 Q\u2082\ne\u2082\u2083 : QuadraticForm.IsometryEquiv Q\u2082 Q\u2083\n\u22a2 AlgEquiv.trans (equivOfIsometry e\u2081\u2082) (equivOfIsometry e\u2082\u2083) =\n equivOfIsometry (QuadraticForm.IsometryEquiv.trans e\u2081\u2082 e\u2082\u2083)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\ne\u2081\u2082 : QuadraticForm.IsometryEquiv Q\u2081 Q\u2082\ne\u2082\u2083 : QuadraticForm.IsometryEquiv Q\u2082 Q\u2083\nx : CliffordAlgebra Q\u2081\n\u22a2 \u2191(AlgEquiv.trans (equivOfIsometry e\u2081\u2082) (equivOfIsometry e\u2082\u2083)) x =\n \u2191(equivOfIsometry (QuadraticForm.IsometryEquiv.trans e\u2081\u2082 e\u2082\u2083)) x\n[PROOFSTEP]\nexact AlgHom.congr_fun (map_comp_map Q\u2081 Q\u2082 Q\u2083 _ _ _ _) x\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\n\u22a2 equivOfIsometry (QuadraticForm.IsometryEquiv.refl Q\u2081) = AlgEquiv.refl\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : Type u_2\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u2077 : Semiring A\ninst\u271d\u2076 : Algebra R A\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2075 : AddCommGroup M\u2081\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nQ\u2081 : QuadraticForm R M\u2081\nQ\u2082 : QuadraticForm R M\u2082\nQ\u2083 : QuadraticForm R M\u2083\nx : CliffordAlgebra Q\u2081\n\u22a2 \u2191(equivOfIsometry (QuadraticForm.IsometryEquiv.refl Q\u2081)) x = \u2191AlgEquiv.refl x\n[PROOFSTEP]\nexact AlgHom.congr_fun (map_id Q\u2081) x\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra R A\nm : M\ninst\u271d : Invertible (\u2191Q m)\n\u22a2 \u2191(\u03b9 Q) (\u215f(\u2191Q m) \u2022 m) * \u2191(\u03b9 Q) m = 1\n[PROOFSTEP]\nrw [map_smul, smul_mul_assoc, \u03b9_sq_scalar, Algebra.smul_def, \u2190 map_mul, invOf_mul_self, map_one]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra R A\nm : M\ninst\u271d : Invertible (\u2191Q m)\n\u22a2 \u2191(\u03b9 Q) m * \u2191(\u03b9 Q) (\u215f(\u2191Q m) \u2022 m) = 1\n[PROOFSTEP]\nrw [map_smul, mul_smul_comm, \u03b9_sq_scalar, Algebra.smul_def, \u2190 map_mul, invOf_mul_self, map_one]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nm : M\ninst\u271d\u00b9 : Invertible (\u2191Q m)\ninst\u271d : Invertible (\u2191(\u03b9 Q) m)\n\u22a2 \u215f(\u2191(\u03b9 Q) m) = \u2191(\u03b9 Q) (\u215f(\u2191Q m) \u2022 m)\n[PROOFSTEP]\nletI := invertible\u03b9OfInvertible Q m\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nm : M\ninst\u271d\u00b9 : Invertible (\u2191Q m)\ninst\u271d : Invertible (\u2191(\u03b9 Q) m)\nthis : Invertible (\u2191(\u03b9 Q) m) := invertible\u03b9OfInvertible Q m\n\u22a2 \u215f(\u2191(\u03b9 Q) m) = \u2191(\u03b9 Q) (\u215f(\u2191Q m) \u2022 m)\n[PROOFSTEP]\nconvert (rfl : \u215f(\u03b9 Q m) = _)\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nm : M\nh : IsUnit (\u2191Q m)\n\u22a2 IsUnit (\u2191(\u03b9 Q) m)\n[PROOFSTEP]\ncases h.nonempty_invertible\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nm : M\nh : IsUnit (\u2191Q m)\nval\u271d : Invertible (\u2191Q m)\n\u22a2 IsUnit (\u2191(\u03b9 Q) m)\n[PROOFSTEP]\nletI := invertible\u03b9OfInvertible Q m\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nm : M\nh : IsUnit (\u2191Q m)\nval\u271d : Invertible (\u2191Q m)\nthis : Invertible (\u2191(\u03b9 Q) m) := invertible\u03b9OfInvertible Q m\n\u22a2 IsUnit (\u2191(\u03b9 Q) m)\n[PROOFSTEP]\nexact isUnit_of_invertible (\u03b9 Q m)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\na b : M\ninst\u271d\u00b9 : Invertible (\u2191(\u03b9 Q) a)\ninst\u271d : Invertible (\u2191Q a)\n\u22a2 \u2191(\u03b9 Q) a * \u2191(\u03b9 Q) b * \u215f(\u2191(\u03b9 Q) a) = \u2191(\u03b9 Q) ((\u215f(\u2191Q a) * QuadraticForm.polar (\u2191Q) a b) \u2022 a - b)\n[PROOFSTEP]\nrw [invOf_\u03b9, map_smul, mul_smul_comm, \u03b9_mul_\u03b9_mul_\u03b9, \u2190 map_smul, smul_sub, smul_smul, smul_smul, invOf_mul_self,\n one_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\na b : M\ninst\u271d\u00b9 : Invertible (\u2191(\u03b9 Q) a)\ninst\u271d : Invertible (\u2191Q a)\n\u22a2 \u215f(\u2191(\u03b9 Q) a) * \u2191(\u03b9 Q) b * \u2191(\u03b9 Q) a = \u2191(\u03b9 Q) ((\u215f(\u2191Q a) * QuadraticForm.polar (\u2191Q) a b) \u2022 a - b)\n[PROOFSTEP]\nrw [invOf_\u03b9, map_smul, smul_mul_assoc, smul_mul_assoc, \u03b9_mul_\u03b9_mul_\u03b9, \u2190 map_smul, smul_sub, smul_smul, smul_smul,\n invOf_mul_self, one_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nn : \u2115\nm : M\n\u22a2 \u2191toClifford (\u2191(\u03b9 R) m) = \u2191(CliffordAlgebra.\u03b9 Q) m\n[PROOFSTEP]\nsimp [toClifford]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.CliffordAlgebra.Basic", "llama_tokens": 17852, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936430859598, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.4070623659639856}} {"text": "[GOAL]\nM\u271d : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\u271d\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk\u271d l k : \u2115+\nM : Type u_7\ninst\u271d : CommMonoid M\na\u271d b\u271d : M\u02e3\nx\u271d\u00b9 : a\u271d \u2208 {\u03b6 | \u03b6 ^ \u2191k = 1}\nx\u271d : b\u271d \u2208 {\u03b6 | \u03b6 ^ \u2191k = 1}\n\u22a2 a\u271d * b\u271d \u2208 {\u03b6 | \u03b6 ^ \u2191k = 1}\n[PROOFSTEP]\nsimp_all only [Set.mem_setOf_eq, mul_pow, one_mul]\n[GOAL]\nM\u271d : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\u271d\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk\u271d l k : \u2115+\nM : Type u_7\ninst\u271d : CommMonoid M\nx\u271d\u00b9 : M\u02e3\nx\u271d :\n x\u271d\u00b9 \u2208\n {\n toSubsemigroup :=\n { carrier := {\u03b6 | \u03b6 ^ \u2191k = 1},\n mul_mem' := (_ : \u2200 {a b : M\u02e3}, a \u2208 {\u03b6 | \u03b6 ^ \u2191k = 1} \u2192 b \u2208 {\u03b6 | \u03b6 ^ \u2191k = 1} \u2192 (a * b) ^ \u2191k = 1) },\n one_mem' := (_ : 1 ^ \u2191k = 1) }.toSubsemigroup.carrier\n\u22a2 x\u271d\u00b9\u207b\u00b9 \u2208\n {\n toSubsemigroup :=\n { carrier := {\u03b6 | \u03b6 ^ \u2191k = 1},\n mul_mem' := (_ : \u2200 {a b : M\u02e3}, a \u2208 {\u03b6 | \u03b6 ^ \u2191k = 1} \u2192 b \u2208 {\u03b6 | \u03b6 ^ \u2191k = 1} \u2192 (a * b) ^ \u2191k = 1) },\n one_mem' := (_ : 1 ^ \u2191k = 1) }.toSubsemigroup.carrier\n[PROOFSTEP]\nsimp_all only [Set.mem_setOf_eq, inv_pow, inv_one]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk\u271d l k : \u2115+\n\u03b6 : M\u02e3\n\u22a2 \u03b6 \u2208 rootsOfUnity k M \u2194 \u2191\u03b6 ^ \u2191k = 1\n[PROOFSTEP]\nrw [mem_rootsOfUnity]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk\u271d l k : \u2115+\n\u03b6 : M\u02e3\n\u22a2 \u03b6 ^ \u2191k = 1 \u2194 \u2191\u03b6 ^ \u2191k = 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115+\nh : k \u2223 l\n\u22a2 rootsOfUnity k M \u2264 rootsOfUnity l M\n[PROOFSTEP]\nobtain \u27e8d, rfl\u27e9 := h\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk d : \u2115+\n\u22a2 rootsOfUnity k M \u2264 rootsOfUnity (k * d) M\n[PROOFSTEP]\nintro \u03b6 h\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk d : \u2115+\n\u03b6 : M\u02e3\nh : \u03b6 \u2208 rootsOfUnity k M\n\u22a2 \u03b6 \u2208 rootsOfUnity (k * d) M\n[PROOFSTEP]\nsimp_all only [mem_rootsOfUnity, PNat.mul_coe, pow_mul, one_pow]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk\u271d l : \u2115+\nf : M\u02e3 \u2192* N\u02e3\nk : \u2115+\n\u22a2 Subgroup.map f (rootsOfUnity k M) \u2264 rootsOfUnity k N\n[PROOFSTEP]\nrintro _ \u27e8\u03b6, h, rfl\u27e9\n[GOAL]\ncase intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk\u271d l : \u2115+\nf : M\u02e3 \u2192* N\u02e3\nk : \u2115+\n\u03b6 : M\u02e3\nh : \u03b6 \u2208 \u2191(rootsOfUnity k M)\n\u22a2 \u2191f \u03b6 \u2208 rootsOfUnity k N\n[PROOFSTEP]\nsimp_all only [\u2190 map_pow, mem_rootsOfUnity, SetLike.mem_coe, MonoidHom.map_one]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d : CommMonoid R\n\u03b6 : { x // x \u2208 rootsOfUnity k R }\nm : \u2115\n\u22a2 \u2191\u2191(\u03b6 ^ m) = \u2191\u2191\u03b6 ^ m\n[PROOFSTEP]\nrw [Subgroup.coe_pow, Units.val_pow_eq_pow_val]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\ninst\u271d\u2074 : CommMonoid N\ninst\u271d\u00b3 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : RingHomClass F R S\n\u03c3 : F\nn : \u2115+\n\u03be : { x // x \u2208 rootsOfUnity n R }\n\u22a2 \u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1\n[PROOFSTEP]\nrw [\u2190 map_pow, \u2190 Units.val_pow_eq_pow_val, show (\u03be : R\u02e3) ^ (n : \u2115) = 1 from \u03be.2, Units.val_one, map_one \u03c3]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\ninst\u271d\u2074 : CommMonoid N\ninst\u271d\u00b3 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : RingHomClass F R S\n\u03c3 : F\nn : \u2115+\nh : \u2200 (\u03be : { x // x \u2208 rootsOfUnity n R }), \u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1 :=\n fun \u03be =>\n Eq.mpr (id (Eq.symm (map_pow \u03c3 \u2191\u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1)))\n (Eq.mpr (id (Eq.symm (Units.val_pow_eq_pow_val \u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 (\u2191\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr\n (id\n ((let_fun this := \u03be.property;\n this) \u25b8\n Eq.refl (\u2191\u03c3 \u2191(\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr (id (Units.val_one \u25b8 Eq.refl (\u2191\u03c3 \u21911 = 1)))\n (Eq.mpr (id (map_one \u03c3 \u25b8 Eq.refl (\u2191\u03c3 1 = 1))) (Eq.refl 1)))))\n\u03be : { x // x \u2208 rootsOfUnity n R }\n\u22a2 unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) \u2208 rootsOfUnity n S\n[PROOFSTEP]\next\n[GOAL]\ncase a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\ninst\u271d\u2074 : CommMonoid N\ninst\u271d\u00b3 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : RingHomClass F R S\n\u03c3 : F\nn : \u2115+\nh : \u2200 (\u03be : { x // x \u2208 rootsOfUnity n R }), \u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1 :=\n fun \u03be =>\n Eq.mpr (id (Eq.symm (map_pow \u03c3 \u2191\u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1)))\n (Eq.mpr (id (Eq.symm (Units.val_pow_eq_pow_val \u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 (\u2191\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr\n (id\n ((let_fun this := \u03be.property;\n this) \u25b8\n Eq.refl (\u2191\u03c3 \u2191(\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr (id (Units.val_one \u25b8 Eq.refl (\u2191\u03c3 \u21911 = 1)))\n (Eq.mpr (id (map_one \u03c3 \u25b8 Eq.refl (\u2191\u03c3 1 = 1))) (Eq.refl 1)))))\n\u03be : { x // x \u2208 rootsOfUnity n R }\n\u22a2 \u2191(unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n) = \u21911\n[PROOFSTEP]\nrw [Units.val_pow_eq_pow_val]\n[GOAL]\ncase a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\ninst\u271d\u2074 : CommMonoid N\ninst\u271d\u00b3 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : RingHomClass F R S\n\u03c3 : F\nn : \u2115+\nh : \u2200 (\u03be : { x // x \u2208 rootsOfUnity n R }), \u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1 :=\n fun \u03be =>\n Eq.mpr (id (Eq.symm (map_pow \u03c3 \u2191\u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1)))\n (Eq.mpr (id (Eq.symm (Units.val_pow_eq_pow_val \u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 (\u2191\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr\n (id\n ((let_fun this := \u03be.property;\n this) \u25b8\n Eq.refl (\u2191\u03c3 \u2191(\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr (id (Units.val_one \u25b8 Eq.refl (\u2191\u03c3 \u21911 = 1)))\n (Eq.mpr (id (map_one \u03c3 \u25b8 Eq.refl (\u2191\u03c3 1 = 1))) (Eq.refl 1)))))\n\u03be : { x // x \u2208 rootsOfUnity n R }\n\u22a2 \u2191(unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be)) ^ \u2191n = \u21911\n[PROOFSTEP]\nexact h \u03be\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\ninst\u271d\u2074 : CommMonoid N\ninst\u271d\u00b3 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : RingHomClass F R S\n\u03c3 : F\nn : \u2115+\nh : \u2200 (\u03be : { x // x \u2208 rootsOfUnity n R }), \u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1 :=\n fun \u03be =>\n Eq.mpr (id (Eq.symm (map_pow \u03c3 \u2191\u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1)))\n (Eq.mpr (id (Eq.symm (Units.val_pow_eq_pow_val \u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 (\u2191\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr\n (id\n ((let_fun this := \u03be.property;\n this) \u25b8\n Eq.refl (\u2191\u03c3 \u2191(\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr (id (Units.val_one \u25b8 Eq.refl (\u2191\u03c3 \u21911 = 1)))\n (Eq.mpr (id (map_one \u03c3 \u25b8 Eq.refl (\u2191\u03c3 1 = 1))) (Eq.refl 1)))))\n\u22a2 (fun \u03be => { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) }) 1 = 1\n[PROOFSTEP]\next\n[GOAL]\ncase a.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\ninst\u271d\u2074 : CommMonoid N\ninst\u271d\u00b3 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : RingHomClass F R S\n\u03c3 : F\nn : \u2115+\nh : \u2200 (\u03be : { x // x \u2208 rootsOfUnity n R }), \u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1 :=\n fun \u03be =>\n Eq.mpr (id (Eq.symm (map_pow \u03c3 \u2191\u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1)))\n (Eq.mpr (id (Eq.symm (Units.val_pow_eq_pow_val \u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 (\u2191\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr\n (id\n ((let_fun this := \u03be.property;\n this) \u25b8\n Eq.refl (\u2191\u03c3 \u2191(\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr (id (Units.val_one \u25b8 Eq.refl (\u2191\u03c3 \u21911 = 1)))\n (Eq.mpr (id (map_one \u03c3 \u25b8 Eq.refl (\u2191\u03c3 1 = 1))) (Eq.refl 1)))))\n\u22a2 \u2191\u2191((fun \u03be => { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) }) 1) = \u2191\u21911\n[PROOFSTEP]\nexact map_one \u03c3\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\ninst\u271d\u2074 : CommMonoid N\ninst\u271d\u00b3 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : RingHomClass F R S\n\u03c3 : F\nn : \u2115+\nh : \u2200 (\u03be : { x // x \u2208 rootsOfUnity n R }), \u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1 :=\n fun \u03be =>\n Eq.mpr (id (Eq.symm (map_pow \u03c3 \u2191\u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1)))\n (Eq.mpr (id (Eq.symm (Units.val_pow_eq_pow_val \u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 (\u2191\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr\n (id\n ((let_fun this := \u03be.property;\n this) \u25b8\n Eq.refl (\u2191\u03c3 \u2191(\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr (id (Units.val_one \u25b8 Eq.refl (\u2191\u03c3 \u21911 = 1)))\n (Eq.mpr (id (map_one \u03c3 \u25b8 Eq.refl (\u2191\u03c3 1 = 1))) (Eq.refl 1)))))\n\u03be\u2081 \u03be\u2082 : { x // x \u2208 rootsOfUnity n R }\n\u22a2 OneHom.toFun\n { toFun := fun \u03be => { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) },\n map_one' :=\n (_ :\n (fun \u03be => { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) }) 1 =\n 1) }\n (\u03be\u2081 * \u03be\u2082) =\n OneHom.toFun\n { toFun := fun \u03be => { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) },\n map_one' :=\n (_ :\n (fun \u03be => { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) }) 1 =\n 1) }\n \u03be\u2081 *\n OneHom.toFun\n { toFun := fun \u03be => { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) },\n map_one' :=\n (_ :\n (fun \u03be => { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) }) 1 =\n 1) }\n \u03be\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase a.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\ninst\u271d\u2074 : CommMonoid N\ninst\u271d\u00b3 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : RingHomClass F R S\n\u03c3 : F\nn : \u2115+\nh : \u2200 (\u03be : { x // x \u2208 rootsOfUnity n R }), \u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1 :=\n fun \u03be =>\n Eq.mpr (id (Eq.symm (map_pow \u03c3 \u2191\u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1)))\n (Eq.mpr (id (Eq.symm (Units.val_pow_eq_pow_val \u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 (\u2191\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr\n (id\n ((let_fun this := \u03be.property;\n this) \u25b8\n Eq.refl (\u2191\u03c3 \u2191(\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr (id (Units.val_one \u25b8 Eq.refl (\u2191\u03c3 \u21911 = 1)))\n (Eq.mpr (id (map_one \u03c3 \u25b8 Eq.refl (\u2191\u03c3 1 = 1))) (Eq.refl 1)))))\n\u03be\u2081 \u03be\u2082 : { x // x \u2208 rootsOfUnity n R }\n\u22a2 \u2191\u2191(OneHom.toFun\n {\n toFun := fun \u03be =>\n { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) },\n map_one' :=\n (_ :\n (fun \u03be => { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) })\n 1 =\n 1) }\n (\u03be\u2081 * \u03be\u2082)) =\n \u2191\u2191(OneHom.toFun\n {\n toFun := fun \u03be =>\n { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) },\n map_one' :=\n (_ :\n (fun \u03be => { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) })\n 1 =\n 1) }\n \u03be\u2081 *\n OneHom.toFun\n {\n toFun := fun \u03be =>\n { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) },\n map_one' :=\n (_ :\n (fun \u03be => { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) })\n 1 =\n 1) }\n \u03be\u2082)\n[PROOFSTEP]\nrw [Subgroup.coe_mul, Units.val_mul]\n[GOAL]\ncase a.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\ninst\u271d\u2074 : CommMonoid N\ninst\u271d\u00b3 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : RingHomClass F R S\n\u03c3 : F\nn : \u2115+\nh : \u2200 (\u03be : { x // x \u2208 rootsOfUnity n R }), \u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1 :=\n fun \u03be =>\n Eq.mpr (id (Eq.symm (map_pow \u03c3 \u2191\u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 \u2191\u2191\u03be ^ \u2191n = 1)))\n (Eq.mpr (id (Eq.symm (Units.val_pow_eq_pow_val \u2191\u03be \u2191n) \u25b8 Eq.refl (\u2191\u03c3 (\u2191\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr\n (id\n ((let_fun this := \u03be.property;\n this) \u25b8\n Eq.refl (\u2191\u03c3 \u2191(\u2191\u03be ^ \u2191n) = 1)))\n (Eq.mpr (id (Units.val_one \u25b8 Eq.refl (\u2191\u03c3 \u21911 = 1)))\n (Eq.mpr (id (map_one \u03c3 \u25b8 Eq.refl (\u2191\u03c3 1 = 1))) (Eq.refl 1)))))\n\u03be\u2081 \u03be\u2082 : { x // x \u2208 rootsOfUnity n R }\n\u22a2 \u2191\u2191(OneHom.toFun\n {\n toFun := fun \u03be =>\n { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) },\n map_one' :=\n (_ :\n (fun \u03be => { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) })\n 1 =\n 1) }\n (\u03be\u2081 * \u03be\u2082)) =\n \u2191\u2191(OneHom.toFun\n {\n toFun := fun \u03be =>\n { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) },\n map_one' :=\n (_ :\n (fun \u03be => { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) })\n 1 =\n 1) }\n \u03be\u2081) *\n \u2191\u2191(OneHom.toFun\n {\n toFun := fun \u03be =>\n { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) },\n map_one' :=\n (_ :\n (fun \u03be => { val := unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be), property := (_ : unitOfInvertible (\u2191\u03c3 \u2191\u2191\u03be) ^ \u2191n = 1) })\n 1 =\n 1) }\n \u03be\u2082)\n[PROOFSTEP]\nexact map_mul \u03c3 _ _\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c3 : R \u2243+* S\nn : \u2115+\n\u03be : { x // x \u2208 rootsOfUnity n R }\n\u22a2 \u2191(_root_.restrictRootsOfUnity (toRingHom (RingEquiv.symm \u03c3)) n) (\u2191(_root_.restrictRootsOfUnity (toRingHom \u03c3) n) \u03be) = \u03be\n[PROOFSTEP]\next\n[GOAL]\ncase a.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c3 : R \u2243+* S\nn : \u2115+\n\u03be : { x // x \u2208 rootsOfUnity n R }\n\u22a2 \u2191\u2191(\u2191(_root_.restrictRootsOfUnity (toRingHom (RingEquiv.symm \u03c3)) n)\n (\u2191(_root_.restrictRootsOfUnity (toRingHom \u03c3) n) \u03be)) =\n \u2191\u2191\u03be\n[PROOFSTEP]\nexact \u03c3.symm_apply_apply (\u03be : R\u02e3)\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c3 : R \u2243+* S\nn : \u2115+\n\u03be : { x // x \u2208 rootsOfUnity n S }\n\u22a2 \u2191(_root_.restrictRootsOfUnity (toRingHom \u03c3) n) (\u2191(_root_.restrictRootsOfUnity (toRingHom (RingEquiv.symm \u03c3)) n) \u03be) = \u03be\n[PROOFSTEP]\next\n[GOAL]\ncase a.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c3 : R \u2243+* S\nn : \u2115+\n\u03be : { x // x \u2208 rootsOfUnity n S }\n\u22a2 \u2191\u2191(\u2191(_root_.restrictRootsOfUnity (toRingHom \u03c3) n)\n (\u2191(_root_.restrictRootsOfUnity (toRingHom (RingEquiv.symm \u03c3)) n) \u03be)) =\n \u2191\u2191\u03be\n[PROOFSTEP]\nexact \u03c3.apply_symm_apply (\u03be : S\u02e3)\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\u02e3\n\u22a2 \u03b6 \u2208 rootsOfUnity k R \u2194 \u2191\u03b6 \u2208 nthRoots (\u2191k) 1\n[PROOFSTEP]\nsimp only [mem_rootsOfUnity, mem_nthRoots k.pos, Units.ext_iff, Units.val_one, Units.val_pow_eq_pow_val]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 { x // x \u2208 rootsOfUnity k R } \u2243 { x // x \u2208 nthRoots (\u2191k) 1 }\n[PROOFSTEP]\nrefine'\n { toFun := fun x => \u27e8(x : R\u02e3), mem_rootsOfUnity_iff_mem_nthRoots.mp x.2\u27e9\n invFun := fun x => \u27e8\u27e8x, \u2191x ^ (k - 1 : \u2115), _, _\u27e9, _\u27e9\n left_inv := _\n right_inv := _ }\n[GOAL]\ncase refine'_1\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : { x // x \u2208 nthRoots (\u2191k) 1 }\n\u22a2 \u2191x * \u2191x ^ (\u2191k - 1) = 1\ncase refine'_2\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : { x // x \u2208 nthRoots (\u2191k) 1 }\n\u22a2 \u2191x ^ (\u2191k - 1) * \u2191x = 1\ncase refine'_3\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : { x // x \u2208 nthRoots (\u2191k) 1 }\n\u22a2 { val := \u2191x, inv := \u2191x ^ (\u2191k - 1), val_inv := ?refine'_1, inv_val := ?refine'_2 } \u2208 rootsOfUnity k R\ncase refine'_4\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 Function.LeftInverse\n (fun x =>\n {\n val :=\n { val := \u2191x, inv := \u2191x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200034 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200036 x ^ (\u2191k - 1) * \u2191x = 1) },\n property :=\n (_ :\n { val := \u2191x, inv := ?m.1200038 x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200039 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200041 x ^ (\u2191k - 1) * \u2191x = 1) } \u2208\n rootsOfUnity k R) })\n fun x => { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 nthRoots (\u2191k) 1) }\ncase refine'_5\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 Function.RightInverse\n (fun x =>\n {\n val :=\n { val := \u2191x, inv := \u2191x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200034 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200036 x ^ (\u2191k - 1) * \u2191x = 1) },\n property :=\n (_ :\n { val := \u2191x, inv := ?m.1200038 x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200039 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200041 x ^ (\u2191k - 1) * \u2191x = 1) } \u2208\n rootsOfUnity k R) })\n fun x => { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 nthRoots (\u2191k) 1) }\n[PROOFSTEP]\npick_goal 4\n[GOAL]\ncase refine'_4\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 Function.LeftInverse\n (fun x =>\n {\n val :=\n { val := \u2191x, inv := \u2191x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200034 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200036 x ^ (\u2191k - 1) * \u2191x = 1) },\n property :=\n (_ :\n { val := \u2191x, inv := ?m.1200038 x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200039 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200041 x ^ (\u2191k - 1) * \u2191x = 1) } \u2208\n rootsOfUnity k R) })\n fun x => { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 nthRoots (\u2191k) 1) }\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase refine'_4.mk\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : R\u02e3\nhx : x \u2208 rootsOfUnity k R\n\u22a2 (fun x =>\n {\n val :=\n { val := \u2191x, inv := \u2191x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200034 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200036 x ^ (\u2191k - 1) * \u2191x = 1) },\n property :=\n (_ :\n { val := \u2191x, inv := ?m.1200038 x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200039 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200041 x ^ (\u2191k - 1) * \u2191x = 1) } \u2208\n rootsOfUnity k R) })\n ((fun x => { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 nthRoots (\u2191k) 1) }) { val := x, property := hx }) =\n { val := x, property := hx }\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_4.mk.a.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : R\u02e3\nhx : x \u2208 rootsOfUnity k R\n\u22a2 \u2191\u2191((fun x =>\n {\n val :=\n { val := \u2191x, inv := \u2191x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200034 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200036 x ^ (\u2191k - 1) * \u2191x = 1) },\n property :=\n (_ :\n { val := \u2191x, inv := ?m.1200038 x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200039 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200041 x ^ (\u2191k - 1) * \u2191x = 1) } \u2208\n rootsOfUnity k R) })\n ((fun x => { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 nthRoots (\u2191k) 1) }) { val := x, property := hx })) =\n \u2191\u2191{ val := x, property := hx }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : { x // x \u2208 nthRoots (\u2191k) 1 }\n\u22a2 \u2191x * \u2191x ^ (\u2191k - 1) = 1\ncase refine'_2\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : { x // x \u2208 nthRoots (\u2191k) 1 }\n\u22a2 \u2191x ^ (\u2191k - 1) * \u2191x = 1\ncase refine'_3\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : { x // x \u2208 nthRoots (\u2191k) 1 }\n\u22a2 { val := \u2191x, inv := \u2191x ^ (\u2191k - 1), val_inv := ?refine'_1, inv_val := ?refine'_2 } \u2208 rootsOfUnity k R\ncase refine'_5\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 Function.RightInverse\n (fun x =>\n {\n val :=\n { val := \u2191x, inv := \u2191x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200034 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200036 x ^ (\u2191k - 1) * \u2191x = 1) },\n property :=\n (_ :\n { val := \u2191x, inv := ?m.1200038 x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200039 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200041 x ^ (\u2191k - 1) * \u2191x = 1) } \u2208\n rootsOfUnity k R) })\n fun x => { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 nthRoots (\u2191k) 1) }\n[PROOFSTEP]\npick_goal 4\n[GOAL]\ncase refine'_5\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 Function.RightInverse\n (fun x =>\n {\n val :=\n { val := \u2191x, inv := \u2191x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200034 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200036 x ^ (\u2191k - 1) * \u2191x = 1) },\n property :=\n (_ :\n { val := \u2191x, inv := ?m.1200038 x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200039 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200041 x ^ (\u2191k - 1) * \u2191x = 1) } \u2208\n rootsOfUnity k R) })\n fun x => { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 nthRoots (\u2191k) 1) }\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase refine'_5.mk\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : R\nhx : x \u2208 nthRoots (\u2191k) 1\n\u22a2 (fun x => { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 nthRoots (\u2191k) 1) })\n ((fun x =>\n {\n val :=\n { val := \u2191x, inv := \u2191x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200034 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200036 x ^ (\u2191k - 1) * \u2191x = 1) },\n property :=\n (_ :\n { val := \u2191x, inv := ?m.1200038 x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200039 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200041 x ^ (\u2191k - 1) * \u2191x = 1) } \u2208\n rootsOfUnity k R) })\n { val := x, property := hx }) =\n { val := x, property := hx }\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_5.mk.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : R\nhx : x \u2208 nthRoots (\u2191k) 1\n\u22a2 \u2191((fun x => { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 nthRoots (\u2191k) 1) })\n ((fun x =>\n {\n val :=\n { val := \u2191x, inv := \u2191x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200034 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200036 x ^ (\u2191k - 1) * \u2191x = 1) },\n property :=\n (_ :\n { val := \u2191x, inv := ?m.1200038 x ^ (\u2191k - 1), val_inv := (_ : \u2191x * ?m.1200039 x ^ (\u2191k - 1) = 1),\n inv_val := (_ : ?m.1200041 x ^ (\u2191k - 1) * \u2191x = 1) } \u2208\n rootsOfUnity k R) })\n { val := x, property := hx })) =\n \u2191{ val := x, property := hx }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : { x // x \u2208 nthRoots (\u2191k) 1 }\n\u22a2 \u2191x * \u2191x ^ (\u2191k - 1) = 1\ncase refine'_2\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : { x // x \u2208 nthRoots (\u2191k) 1 }\n\u22a2 \u2191x ^ (\u2191k - 1) * \u2191x = 1\ncase refine'_3\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : { x // x \u2208 nthRoots (\u2191k) 1 }\n\u22a2 { val := \u2191x, inv := \u2191x ^ (\u2191k - 1), val_inv := ?refine'_1, inv_val := ?refine'_2 } \u2208 rootsOfUnity k R\n[PROOFSTEP]\nall_goals\n rcases x with \u27e8x, hx\u27e9; rw [mem_nthRoots k.pos] at hx \n simp only [Subtype.coe_mk, \u2190 pow_succ, \u2190 pow_succ', hx, tsub_add_cancel_of_le (show 1 \u2264 (k : \u2115) from k.one_le)]\n[GOAL]\ncase refine'_1\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : { x // x \u2208 nthRoots (\u2191k) 1 }\n\u22a2 \u2191x * \u2191x ^ (\u2191k - 1) = 1\n[PROOFSTEP]\nrcases x with \u27e8x, hx\u27e9\n[GOAL]\ncase refine'_1.mk\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : R\nhx : x \u2208 nthRoots (\u2191k) 1\n\u22a2 \u2191{ val := x, property := hx } * \u2191{ val := x, property := hx } ^ (\u2191k - 1) = 1\n[PROOFSTEP]\nrw [mem_nthRoots k.pos] at hx \n[GOAL]\ncase refine'_1.mk\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : R\nhx\u271d : x \u2208 nthRoots (\u2191k) 1\nhx : x ^ \u2191k = 1\n\u22a2 \u2191{ val := x, property := hx\u271d } * \u2191{ val := x, property := hx\u271d } ^ (\u2191k - 1) = 1\n[PROOFSTEP]\nsimp only [Subtype.coe_mk, \u2190 pow_succ, \u2190 pow_succ', hx, tsub_add_cancel_of_le (show 1 \u2264 (k : \u2115) from k.one_le)]\n[GOAL]\ncase refine'_2\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : { x // x \u2208 nthRoots (\u2191k) 1 }\n\u22a2 \u2191x ^ (\u2191k - 1) * \u2191x = 1\n[PROOFSTEP]\nrcases x with \u27e8x, hx\u27e9\n[GOAL]\ncase refine'_2.mk\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : R\nhx : x \u2208 nthRoots (\u2191k) 1\n\u22a2 \u2191{ val := x, property := hx } ^ (\u2191k - 1) * \u2191{ val := x, property := hx } = 1\n[PROOFSTEP]\nrw [mem_nthRoots k.pos] at hx \n[GOAL]\ncase refine'_2.mk\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : R\nhx\u271d : x \u2208 nthRoots (\u2191k) 1\nhx : x ^ \u2191k = 1\n\u22a2 \u2191{ val := x, property := hx\u271d } ^ (\u2191k - 1) * \u2191{ val := x, property := hx\u271d } = 1\n[PROOFSTEP]\nsimp only [Subtype.coe_mk, \u2190 pow_succ, \u2190 pow_succ', hx, tsub_add_cancel_of_le (show 1 \u2264 (k : \u2115) from k.one_le)]\n[GOAL]\ncase refine'_3\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : { x // x \u2208 nthRoots (\u2191k) 1 }\n\u22a2 { val := \u2191x, inv := \u2191x ^ (\u2191k - 1), val_inv := (_ : \u2191x * \u2191x ^ (\u2191k - 1) = 1),\n inv_val := (_ : \u2191x ^ (\u2191k - 1) * \u2191x = 1) } \u2208\n rootsOfUnity k R\n[PROOFSTEP]\nrcases x with \u27e8x, hx\u27e9\n[GOAL]\ncase refine'_3.mk\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : R\nhx : x \u2208 nthRoots (\u2191k) 1\n\u22a2 { val := \u2191{ val := x, property := hx }, inv := \u2191{ val := x, property := hx } ^ (\u2191k - 1),\n val_inv := (_ : \u2191{ val := x, property := hx } * \u2191{ val := x, property := hx } ^ (\u2191k - 1) = 1),\n inv_val := (_ : \u2191{ val := x, property := hx } ^ (\u2191k - 1) * \u2191{ val := x, property := hx } = 1) } \u2208\n rootsOfUnity k R\n[PROOFSTEP]\nrw [mem_nthRoots k.pos] at hx \n[GOAL]\ncase refine'_3.mk\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : R\nhx\u271d : x \u2208 nthRoots (\u2191k) 1\nhx : x ^ \u2191k = 1\n\u22a2 { val := \u2191{ val := x, property := hx\u271d }, inv := \u2191{ val := x, property := hx\u271d } ^ (\u2191k - 1),\n val_inv := (_ : \u2191{ val := x, property := hx\u271d } * \u2191{ val := x, property := hx\u271d } ^ (\u2191k - 1) = 1),\n inv_val := (_ : \u2191{ val := x, property := hx\u271d } ^ (\u2191k - 1) * \u2191{ val := x, property := hx\u271d } = 1) } \u2208\n rootsOfUnity k R\n[PROOFSTEP]\nsimp only [Subtype.coe_mk, \u2190 pow_succ, \u2190 pow_succ', hx, tsub_add_cancel_of_le (show 1 \u2264 (k : \u2115) from k.one_le)]\n[GOAL]\ncase refine'_3.mk\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : R\nhx\u271d : x \u2208 nthRoots (\u2191k) 1\nhx : x ^ \u2191k = 1\n\u22a2 { val := x, inv := x ^ (\u2191k - 1),\n val_inv := (_ : \u2191{ val := x, property := hx\u271d } * \u2191{ val := x, property := hx\u271d } ^ (\u2191k - 1) = 1),\n inv_val := (_ : \u2191{ val := x, property := hx\u271d } ^ (\u2191k - 1) * \u2191{ val := x, property := hx\u271d } = 1) } \u2208\n rootsOfUnity k R\n[PROOFSTEP]\nshow (_ : R\u02e3) ^ (k : \u2115) = 1\n[GOAL]\ncase refine'_3.mk\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : R\nhx\u271d : x \u2208 nthRoots (\u2191k) 1\nhx : x ^ \u2191k = 1\n\u22a2 { val := x, inv := x ^ (\u2191k - 1),\n val_inv := (_ : \u2191{ val := x, property := hx\u271d } * \u2191{ val := x, property := hx\u271d } ^ (\u2191k - 1) = 1),\n inv_val := (_ : \u2191{ val := x, property := hx\u271d } ^ (\u2191k - 1) * \u2191{ val := x, property := hx\u271d } = 1) } ^\n \u2191k =\n 1\n[PROOFSTEP]\nsimp only [Units.ext_iff, hx, Units.val_mk, Units.val_one, Subtype.coe_mk, Units.val_pow_eq_pow_val]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\ninst\u271d\u2074 : CommMonoid N\ninst\u271d\u00b3 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : RingHomClass F R R\n\u03c3 : F\n\u03b6 : { x // x \u2208 rootsOfUnity k R }\n\u22a2 \u2203 m, \u2191\u03c3 \u2191\u2191\u03b6 = \u2191\u2191\u03b6 ^ m\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := MonoidHom.map_cyclic (restrictRootsOfUnity \u03c3 k)\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\ninst\u271d\u2074 : CommMonoid N\ninst\u271d\u00b3 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : RingHomClass F R R\n\u03c3 : F\n\u03b6 : { x // x \u2208 rootsOfUnity k R }\nm : \u2124\nhm : \u2200 (g : { x // x \u2208 rootsOfUnity k R }), \u2191(restrictRootsOfUnity \u03c3 k) g = g ^ m\n\u22a2 \u2203 m, \u2191\u03c3 \u2191\u2191\u03b6 = \u2191\u2191\u03b6 ^ m\n[PROOFSTEP]\nrw [\u2190 restrictRootsOfUnity_coe_apply, hm, zpow_eq_mod_orderOf, \u2190\n Int.toNat_of_nonneg (m.emod_nonneg (Int.coe_nat_ne_zero.mpr (pos_iff_ne_zero.mp (orderOf_pos \u03b6)))), zpow_ofNat,\n rootsOfUnity.coe_pow]\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\ninst\u271d\u2074 : CommMonoid N\ninst\u271d\u00b3 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : RingHomClass F R R\n\u03c3 : F\n\u03b6 : { x // x \u2208 rootsOfUnity k R }\nm : \u2124\nhm : \u2200 (g : { x // x \u2208 rootsOfUnity k R }), \u2191(restrictRootsOfUnity \u03c3 k) g = g ^ m\n\u22a2 \u2203 m_1, \u2191\u2191\u03b6 ^ Int.toNat (m % \u2191(orderOf \u03b6)) = \u2191\u2191\u03b6 ^ m_1\n[PROOFSTEP]\nexact \u27e8(m % orderOf \u03b6).toNat, rfl\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\ninst\u271d\u2074 : CommMonoid N\ninst\u271d\u00b3 : DivisionCommMonoid G\nk\u271d l : \u2115+\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsReduced R\np k : \u2115\nm : \u2115+\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R p\n\u03b6 : R\u02e3\n\u22a2 \u03b6 \u2208 rootsOfUnity ({ val := p, property := (_ : 0 < p) } ^ k * m) R \u2194 \u03b6 \u2208 rootsOfUnity m R\n[PROOFSTEP]\nsimp only [mem_rootsOfUnity', PNat.mul_coe, PNat.pow_coe, PNat.mk_coe, CharP.pow_prime_pow_mul_eq_one_iff]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\ninst\u271d\u2074 : CommMonoid N\ninst\u271d\u00b3 : DivisionCommMonoid G\nk\u271d l : \u2115+\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsReduced R\np k : \u2115\nm : \u2115+\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R p\n\u03b6 : R\u02e3\n\u22a2 \u03b6 ^ (p ^ k * \u2191m) = 1 \u2194 \u03b6 \u2208 rootsOfUnity m R\n[PROOFSTEP]\nrw [\u2190 PNat.mk_coe p hp.1.pos, \u2190 PNat.pow_coe, \u2190 PNat.mul_coe, \u2190 mem_rootsOfUnity, mem_rootsOfUnity_prime_pow_mul_iff]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nh0 : 0 < k\n\u22a2 \u03b6 \u2208 primitiveRoots k R \u2194 IsPrimitiveRoot \u03b6 k\n[PROOFSTEP]\nrw [primitiveRoots, mem_filter, Multiset.mem_toFinset, mem_nthRoots h0, and_iff_right_iff_imp]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nh0 : 0 < k\n\u22a2 IsPrimitiveRoot \u03b6 k \u2192 \u03b6 ^ k = 1\n[PROOFSTEP]\nexact IsPrimitiveRoot.pow_eq_one\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 primitiveRoots 0 R = \u2205\n[PROOFSTEP]\nrw [primitiveRoots, nthRoots_zero, Multiset.toFinset_zero, Finset.filter_empty]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nh : \u03b6 \u2208 primitiveRoots k R\nhk : k = 0\n\u22a2 IsPrimitiveRoot \u03b6 k\n[PROOFSTEP]\nsimp [hk] at h \n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nhk : 0 < k\nh1 : \u03b6 ^ k = 1\nh : \u2200 (l : \u2115), 0 < l \u2192 l < k \u2192 \u03b6 ^ l \u2260 1\n\u22a2 IsPrimitiveRoot \u03b6 k\n[PROOFSTEP]\nrefine' \u27e8h1, fun l hl => _\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nhk : 0 < k\nh1 : \u03b6 ^ k = 1\nh : \u2200 (l : \u2115), 0 < l \u2192 l < k \u2192 \u03b6 ^ l \u2260 1\nl : \u2115\nhl : \u03b6 ^ l = 1\n\u22a2 k \u2223 l\n[PROOFSTEP]\nsuffices k.gcd l = k by exact this \u25b8 k.gcd_dvd_right l\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nhk : 0 < k\nh1 : \u03b6 ^ k = 1\nh : \u2200 (l : \u2115), 0 < l \u2192 l < k \u2192 \u03b6 ^ l \u2260 1\nl : \u2115\nhl : \u03b6 ^ l = 1\nthis : Nat.gcd k l = k\n\u22a2 k \u2223 l\n[PROOFSTEP]\nexact this \u25b8 k.gcd_dvd_right l\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nhk : 0 < k\nh1 : \u03b6 ^ k = 1\nh : \u2200 (l : \u2115), 0 < l \u2192 l < k \u2192 \u03b6 ^ l \u2260 1\nl : \u2115\nhl : \u03b6 ^ l = 1\n\u22a2 Nat.gcd k l = k\n[PROOFSTEP]\nrw [eq_iff_le_not_lt]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nhk : 0 < k\nh1 : \u03b6 ^ k = 1\nh : \u2200 (l : \u2115), 0 < l \u2192 l < k \u2192 \u03b6 ^ l \u2260 1\nl : \u2115\nhl : \u03b6 ^ l = 1\n\u22a2 Nat.gcd k l \u2264 k \u2227 \u00acNat.gcd k l < k\n[PROOFSTEP]\nrefine' \u27e8Nat.le_of_dvd hk (k.gcd_dvd_left l), _\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nhk : 0 < k\nh1 : \u03b6 ^ k = 1\nh : \u2200 (l : \u2115), 0 < l \u2192 l < k \u2192 \u03b6 ^ l \u2260 1\nl : \u2115\nhl : \u03b6 ^ l = 1\n\u22a2 \u00acNat.gcd k l < k\n[PROOFSTEP]\nintro h'\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nhk : 0 < k\nh1 : \u03b6 ^ k = 1\nh : \u2200 (l : \u2115), 0 < l \u2192 l < k \u2192 \u03b6 ^ l \u2260 1\nl : \u2115\nhl : \u03b6 ^ l = 1\nh' : Nat.gcd k l < k\n\u22a2 False\n[PROOFSTEP]\napply h _ (Nat.gcd_pos_of_pos_left _ hk) h'\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nhk : 0 < k\nh1 : \u03b6 ^ k = 1\nh : \u2200 (l : \u2115), 0 < l \u2192 l < k \u2192 \u03b6 ^ l \u2260 1\nl : \u2115\nhl : \u03b6 ^ l = 1\nh' : Nat.gcd k l < k\n\u22a2 \u03b6 ^ Nat.gcd k l = 1\n[PROOFSTEP]\nexact pow_gcd_eq_one _ h1 hl\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nf : F\nh : IsPrimitiveRoot \u03b6 k\nl : \u2115\n\u22a2 k \u2223 l \u2192 \u03b6 ^ l = 1\n[PROOFSTEP]\nrintro \u27e8i, rfl\u27e9\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh : IsPrimitiveRoot \u03b6 k\ni : \u2115\n\u22a2 \u03b6 ^ (k * i) = 1\n[PROOFSTEP]\nsimp only [pow_mul, h.pow_eq_one, one_pow, PNat.mul_coe]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\n\u22a2 IsUnit \u03b6\n[PROOFSTEP]\napply isUnit_of_mul_eq_one \u03b6 (\u03b6 ^ (k - 1))\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\n\u22a2 \u03b6 * \u03b6 ^ (k - 1) = 1\n[PROOFSTEP]\nrw [\u2190 pow_succ, tsub_add_cancel_of_le h0.nat_succ_le, h.pow_eq_one]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni j : \u2115\nhi : i < k\nhj : j < k\nH : \u03b6 ^ i = \u03b6 ^ j\n\u22a2 i = j\n[PROOFSTEP]\nwlog hij : i \u2264 j generalizing i j\n[GOAL]\ncase inr\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni j : \u2115\nhi : i < k\nhj : j < k\nH : \u03b6 ^ i = \u03b6 ^ j\nthis : \u2200 \u2983i j : \u2115\u2984, i < k \u2192 j < k \u2192 \u03b6 ^ i = \u03b6 ^ j \u2192 i \u2264 j \u2192 i = j\nhij : \u00aci \u2264 j\n\u22a2 i = j\n[PROOFSTEP]\nexact (this hj hi H.symm (le_of_not_le hij)).symm\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni j : \u2115\nhi : i < k\nhj : j < k\nH : \u03b6 ^ i = \u03b6 ^ j\nhij : i \u2264 j\n\u22a2 i = j\n[PROOFSTEP]\napply le_antisymm hij\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni j : \u2115\nhi : i < k\nhj : j < k\nH : \u03b6 ^ i = \u03b6 ^ j\nhij : i \u2264 j\n\u22a2 j \u2264 i\n[PROOFSTEP]\nrw [\u2190 tsub_eq_zero_iff_le]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni j : \u2115\nhi : i < k\nhj : j < k\nH : \u03b6 ^ i = \u03b6 ^ j\nhij : i \u2264 j\n\u22a2 j - i = 0\n[PROOFSTEP]\napply Nat.eq_zero_of_dvd_of_lt _ (lt_of_le_of_lt tsub_le_self hj)\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni j : \u2115\nhi : i < k\nhj : j < k\nH : \u03b6 ^ i = \u03b6 ^ j\nhij : i \u2264 j\n\u22a2 k \u2223 j - i\n[PROOFSTEP]\napply h.dvd_of_pow_eq_one\n[GOAL]\ncase a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni j : \u2115\nhi : i < k\nhj : j < k\nH : \u03b6 ^ i = \u03b6 ^ j\nhij : i \u2264 j\n\u22a2 \u03b6 ^ (j - i) = 1\n[PROOFSTEP]\nrw [\u2190 ((h.isUnit (lt_of_le_of_lt (Nat.zero_le _) hi)).pow i).mul_left_inj, \u2190 pow_add, tsub_add_cancel_of_le hij, H,\n one_mul]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh : IsPrimitiveRoot \u03b6 k\n\u22a2 IsPrimitiveRoot \u03b6 1 \u2194 \u03b6 = 1\n[PROOFSTEP]\nclear h\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\n\u22a2 IsPrimitiveRoot \u03b6 1 \u2194 \u03b6 = 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\n\u22a2 IsPrimitiveRoot \u03b6 1 \u2192 \u03b6 = 1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh : IsPrimitiveRoot \u03b6 1\n\u22a2 \u03b6 = 1\n[PROOFSTEP]\nrw [\u2190 pow_one \u03b6, h.pow_eq_one]\n[GOAL]\ncase mpr\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\n\u22a2 \u03b6 = 1 \u2192 IsPrimitiveRoot \u03b6 1\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mpr\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\nf : F\n\u22a2 IsPrimitiveRoot 1 1\n[PROOFSTEP]\nexact one\n[GOAL]\nM\u271d : Type u_1\nN\u271d : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\u271d\ninst\u271d\u2074 : CommMonoid N\u271d\ninst\u271d\u00b3 : DivisionCommMonoid G\nk l : \u2115\n\u03b6\u271d : M\u271d\nf : F\nh : IsPrimitiveRoot \u03b6\u271d k\nM : Type u_7\nB : Type u_8\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nN : B\n\u03b6 : { x // x \u2208 N }\n\u22a2 IsPrimitiveRoot (\u2191\u03b6) k \u2194 IsPrimitiveRoot \u03b6 k\n[PROOFSTEP]\nsimp_rw [iff_def]\n[GOAL]\nM\u271d : Type u_1\nN\u271d : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2075 : CommMonoid M\u271d\ninst\u271d\u2074 : CommMonoid N\u271d\ninst\u271d\u00b3 : DivisionCommMonoid G\nk l : \u2115\n\u03b6\u271d : M\u271d\nf : F\nh : IsPrimitiveRoot \u03b6\u271d k\nM : Type u_7\nB : Type u_8\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : SetLike B M\ninst\u271d : SubmonoidClass B M\nN : B\n\u03b6 : { x // x \u2208 N }\n\u22a2 (\u2191\u03b6 ^ k = 1 \u2227 \u2200 (l : \u2115), \u2191\u03b6 ^ l = 1 \u2192 k \u2223 l) \u2194 \u03b6 ^ k = 1 \u2227 \u2200 (l : \u2115), \u03b6 ^ l = 1 \u2192 k \u2223 l\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6\u271d : M\nf : F\nh : IsPrimitiveRoot \u03b6\u271d k\n\u03b6 : M\u02e3\n\u22a2 IsPrimitiveRoot (\u2191\u03b6) k \u2194 IsPrimitiveRoot \u03b6 k\n[PROOFSTEP]\nsimp only [iff_def, Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh : IsPrimitiveRoot \u03b6 k\ni : \u2115\nhi : Nat.coprime i k\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) k\n[PROOFSTEP]\nby_cases h0 : k = 0\n[GOAL]\ncase pos\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh : IsPrimitiveRoot \u03b6 k\ni : \u2115\nhi : Nat.coprime i k\nh0 : k = 0\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) k\n[PROOFSTEP]\nsubst k\n[GOAL]\ncase pos\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nl : \u2115\n\u03b6 : M\nf : F\ni : \u2115\nh : IsPrimitiveRoot \u03b6 0\nhi : Nat.coprime i 0\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) 0\n[PROOFSTEP]\nsimp_all only [pow_one, Nat.coprime_zero_right]\n[GOAL]\ncase neg\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh : IsPrimitiveRoot \u03b6 k\ni : \u2115\nhi : Nat.coprime i k\nh0 : \u00ack = 0\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) k\n[PROOFSTEP]\nrcases h.isUnit (Nat.pos_of_ne_zero h0) with \u27e8\u03b6, rfl\u27e9\n[GOAL]\ncase neg.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\nf : F\ni : \u2115\nhi : Nat.coprime i k\nh0 : \u00ack = 0\n\u03b6 : M\u02e3\nh : IsPrimitiveRoot (\u2191\u03b6) k\n\u22a2 IsPrimitiveRoot (\u2191\u03b6 ^ i) k\n[PROOFSTEP]\nrw [\u2190 Units.val_pow_eq_pow_val]\n[GOAL]\ncase neg.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\nf : F\ni : \u2115\nhi : Nat.coprime i k\nh0 : \u00ack = 0\n\u03b6 : M\u02e3\nh : IsPrimitiveRoot (\u2191\u03b6) k\n\u22a2 IsPrimitiveRoot (\u2191(\u03b6 ^ i)) k\n[PROOFSTEP]\nrw [coe_units_iff] at h \u22a2\n[GOAL]\ncase neg.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\nf : F\ni : \u2115\nhi : Nat.coprime i k\nh0 : \u00ack = 0\n\u03b6 : M\u02e3\nh : IsPrimitiveRoot \u03b6 k\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) k\n[PROOFSTEP]\nrefine'\n { pow_eq_one := by rw [\u2190 pow_mul', pow_mul, h.pow_eq_one, one_pow]\n dvd_of_pow_eq_one := _ }\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\nf : F\ni : \u2115\nhi : Nat.coprime i k\nh0 : \u00ack = 0\n\u03b6 : M\u02e3\nh : IsPrimitiveRoot \u03b6 k\n\u22a2 (\u03b6 ^ i) ^ k = 1\n[PROOFSTEP]\nrw [\u2190 pow_mul', pow_mul, h.pow_eq_one, one_pow]\n[GOAL]\ncase neg.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\nf : F\ni : \u2115\nhi : Nat.coprime i k\nh0 : \u00ack = 0\n\u03b6 : M\u02e3\nh : IsPrimitiveRoot \u03b6 k\n\u22a2 \u2200 (l : \u2115), (\u03b6 ^ i) ^ l = 1 \u2192 k \u2223 l\n[PROOFSTEP]\nintro l hl\n[GOAL]\ncase neg.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\nf : F\ni : \u2115\nhi : Nat.coprime i k\nh0 : \u00ack = 0\n\u03b6 : M\u02e3\nh : IsPrimitiveRoot \u03b6 k\nl : \u2115\nhl : (\u03b6 ^ i) ^ l = 1\n\u22a2 k \u2223 l\n[PROOFSTEP]\napply h.dvd_of_pow_eq_one\n[GOAL]\ncase neg.intro.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\nf : F\ni : \u2115\nhi : Nat.coprime i k\nh0 : \u00ack = 0\n\u03b6 : M\u02e3\nh : IsPrimitiveRoot \u03b6 k\nl : \u2115\nhl : (\u03b6 ^ i) ^ l = 1\n\u22a2 \u03b6 ^ l = 1\n[PROOFSTEP]\nrw [\u2190 pow_one \u03b6, \u2190 zpow_ofNat \u03b6, \u2190 hi.gcd_eq_one, Nat.gcd_eq_gcd_ab, zpow_add, mul_pow, \u2190 zpow_ofNat, \u2190 zpow_mul,\n mul_right_comm]\n[GOAL]\ncase neg.intro.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\nf : F\ni : \u2115\nhi : Nat.coprime i k\nh0 : \u00ack = 0\n\u03b6 : M\u02e3\nh : IsPrimitiveRoot \u03b6 k\nl : \u2115\nhl : (\u03b6 ^ i) ^ l = 1\n\u22a2 \u03b6 ^ (\u2191i * \u2191l * Nat.gcdA i k) * (\u03b6 ^ (\u2191k * Nat.gcdB i k)) ^ l = 1\n[PROOFSTEP]\nsimp only [zpow_mul, hl, h.pow_eq_one, one_zpow, one_pow, one_mul, zpow_ofNat]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\ni : \u2115\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) k \u2194 Nat.coprime i k\n[PROOFSTEP]\nrefine' \u27e8_, h.pow_of_coprime i\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\ni : \u2115\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) k \u2192 Nat.coprime i k\n[PROOFSTEP]\nintro hi\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\ni : \u2115\nhi : IsPrimitiveRoot (\u03b6 ^ i) k\n\u22a2 Nat.coprime i k\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := i.gcd_dvd_left k\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\ni : \u2115\nhi : IsPrimitiveRoot (\u03b6 ^ i) k\na : \u2115\nha : i = Nat.gcd i k * a\n\u22a2 Nat.coprime i k\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 := i.gcd_dvd_right k\n[GOAL]\ncase intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\ni : \u2115\nhi : IsPrimitiveRoot (\u03b6 ^ i) k\na : \u2115\nha : i = Nat.gcd i k * a\nb : \u2115\nhb : k = Nat.gcd i k * b\n\u22a2 Nat.coprime i k\n[PROOFSTEP]\nsuffices b = k by\n -- Porting note: was `rwa [this, \u2190 one_mul k, mul_left_inj' h0.ne', eq_comm] at hb`\n rw [this, eq_comm, Nat.mul_left_eq_self_iff h0] at hb \n rwa [Nat.coprime]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\ni : \u2115\nhi : IsPrimitiveRoot (\u03b6 ^ i) k\na : \u2115\nha : i = Nat.gcd i k * a\nb : \u2115\nhb : k = Nat.gcd i k * b\nthis : b = k\n\u22a2 Nat.coprime i k\n[PROOFSTEP]\nrw [this, eq_comm, Nat.mul_left_eq_self_iff h0] at hb \n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\ni : \u2115\nhi : IsPrimitiveRoot (\u03b6 ^ i) k\na : \u2115\nha : i = Nat.gcd i k * a\nb : \u2115\nhb : Nat.gcd i k = 1\nthis : b = k\n\u22a2 Nat.coprime i k\n[PROOFSTEP]\nrwa [Nat.coprime]\n[GOAL]\ncase intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\ni : \u2115\nhi : IsPrimitiveRoot (\u03b6 ^ i) k\na : \u2115\nha : i = Nat.gcd i k * a\nb : \u2115\nhb : k = Nat.gcd i k * b\n\u22a2 b = k\n[PROOFSTEP]\nrw [ha] at hi \n[GOAL]\ncase intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\ni a : \u2115\nhi : IsPrimitiveRoot (\u03b6 ^ (Nat.gcd i k * a)) k\nha : i = Nat.gcd i k * a\nb : \u2115\nhb : k = Nat.gcd i k * b\n\u22a2 b = k\n[PROOFSTEP]\nrw [mul_comm] at hb \n[GOAL]\ncase intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\ni a : \u2115\nhi : IsPrimitiveRoot (\u03b6 ^ (Nat.gcd i k * a)) k\nha : i = Nat.gcd i k * a\nb : \u2115\nhb : k = b * Nat.gcd i k\n\u22a2 b = k\n[PROOFSTEP]\napply Nat.dvd_antisymm \u27e8i.gcd k, hb\u27e9 (hi.dvd_of_pow_eq_one b _)\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\ni a : \u2115\nhi : IsPrimitiveRoot (\u03b6 ^ (Nat.gcd i k * a)) k\nha : i = Nat.gcd i k * a\nb : \u2115\nhb : k = b * Nat.gcd i k\n\u22a2 (\u03b6 ^ (Nat.gcd i k * a)) ^ b = 1\n[PROOFSTEP]\nrw [\u2190 pow_mul', \u2190 mul_assoc, \u2190 hb, pow_mul, h.pow_eq_one, one_pow]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh : IsPrimitiveRoot \u03b6 k\nhk : 0 < k\n\u22a2 IsPrimitiveRoot \u03b6 k \u2194 \u03b6 ^ k = 1 \u2227 \u2200 (l : \u2115), 0 < l \u2192 l < k \u2192 \u03b6 ^ l \u2260 1\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8h.pow_eq_one, fun l hl' hl => _\u27e9, fun \u27e8h\u03b6, hl\u27e9 => IsPrimitiveRoot.mk_of_lt \u03b6 hk h\u03b6 hl\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\nhk : 0 < k\nh : IsPrimitiveRoot \u03b6 k\nl : \u2115\nhl' : 0 < l\nhl : l < k\n\u22a2 \u03b6 ^ l \u2260 1\n[PROOFSTEP]\nrw [h.eq_orderOf] at hl \n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\nhk : 0 < k\nh : IsPrimitiveRoot \u03b6 k\nl : \u2115\nhl' : 0 < l\nhl : l < orderOf \u03b6\n\u22a2 \u03b6 ^ l \u2260 1\n[PROOFSTEP]\nexact pow_ne_one_of_lt_orderOf' hl'.ne' hl\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\np : \u2115\nhp : p \u2260 0\nhdiv : p \u2223 k\n\u22a2 IsPrimitiveRoot (\u03b6 ^ p) (k / p)\n[PROOFSTEP]\nsuffices orderOf (\u03b6 ^ p) = k / p by exact this \u25b8 IsPrimitiveRoot.orderOf (\u03b6 ^ p)\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\np : \u2115\nhp : p \u2260 0\nhdiv : p \u2223 k\nthis : orderOf (\u03b6 ^ p) = k / p\n\u22a2 IsPrimitiveRoot (\u03b6 ^ p) (k / p)\n[PROOFSTEP]\nexact this \u25b8 IsPrimitiveRoot.orderOf (\u03b6 ^ p)\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d h : IsPrimitiveRoot \u03b6 k\np : \u2115\nhp : p \u2260 0\nhdiv : p \u2223 k\n\u22a2 orderOf (\u03b6 ^ p) = k / p\n[PROOFSTEP]\nrw [orderOf_pow' _ hp, \u2190 eq_orderOf h, Nat.gcd_eq_right hdiv]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\nn a b : \u2115\nhn : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nhprod : n = a * b\n\u22a2 IsPrimitiveRoot (\u03b6 ^ a) b\n[PROOFSTEP]\nsubst n\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\na b : \u2115\nhn : 0 < a * b\nh : IsPrimitiveRoot \u03b6 (a * b)\n\u22a2 IsPrimitiveRoot (\u03b6 ^ a) b\n[PROOFSTEP]\nsimp only [iff_def, \u2190 pow_mul, h.pow_eq_one, eq_self_iff_true, true_and_iff]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\na b : \u2115\nhn : 0 < a * b\nh : IsPrimitiveRoot \u03b6 (a * b)\n\u22a2 \u2200 (l : \u2115), \u03b6 ^ (a * l) = 1 \u2192 b \u2223 l\n[PROOFSTEP]\nintro l hl\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\na b : \u2115\nhn : 0 < a * b\nh : IsPrimitiveRoot \u03b6 (a * b)\nl : \u2115\nhl : \u03b6 ^ (a * l) = 1\n\u22a2 b \u2223 l\n[PROOFSTEP]\nhave ha0 : a \u2260 0 := left_ne_zero_of_mul hn.ne'\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\na b : \u2115\nhn : 0 < a * b\nh : IsPrimitiveRoot \u03b6 (a * b)\nl : \u2115\nhl : \u03b6 ^ (a * l) = 1\nha0 : a \u2260 0\n\u22a2 b \u2223 l\n[PROOFSTEP]\nrw [\u2190 mul_dvd_mul_iff_left ha0]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\na b : \u2115\nhn : 0 < a * b\nh : IsPrimitiveRoot \u03b6 (a * b)\nl : \u2115\nhl : \u03b6 ^ (a * l) = 1\nha0 : a \u2260 0\n\u22a2 a * b \u2223 a * l\n[PROOFSTEP]\nexact h.dvd_of_pow_eq_one _ hl\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : MonoidHomClass F M N\nh : IsPrimitiveRoot \u03b6 k\nhf : Injective \u2191f\n\u22a2 \u2191f \u03b6 ^ k = 1\n[PROOFSTEP]\nrw [\u2190 map_pow, h.pow_eq_one, _root_.map_one]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : MonoidHomClass F M N\nh : IsPrimitiveRoot \u03b6 k\nhf : Injective \u2191f\n\u22a2 \u2200 (l : \u2115), \u2191f \u03b6 ^ l = 1 \u2192 k \u2223 l\n[PROOFSTEP]\nrw [h.eq_orderOf]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : MonoidHomClass F M N\nh : IsPrimitiveRoot \u03b6 k\nhf : Injective \u2191f\n\u22a2 \u2200 (l : \u2115), \u2191f \u03b6 ^ l = 1 \u2192 orderOf \u03b6 \u2223 l\n[PROOFSTEP]\nintro l hl\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : MonoidHomClass F M N\nh : IsPrimitiveRoot \u03b6 k\nhf : Injective \u2191f\nl : \u2115\nhl : \u2191f \u03b6 ^ l = 1\n\u22a2 orderOf \u03b6 \u2223 l\n[PROOFSTEP]\nrw [\u2190 map_pow, \u2190 map_one f] at hl \n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : MonoidHomClass F M N\nh : IsPrimitiveRoot \u03b6 k\nhf : Injective \u2191f\nl : \u2115\nhl : \u2191f (\u03b6 ^ l) = \u2191f 1\n\u22a2 orderOf \u03b6 \u2223 l\n[PROOFSTEP]\nexact orderOf_dvd_of_pow_eq_one (hf hl)\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : MonoidHomClass F M N\nh : IsPrimitiveRoot (\u2191f \u03b6) k\nhf : Injective \u2191f\n\u22a2 \u03b6 ^ k = 1\n[PROOFSTEP]\napply_fun f\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : MonoidHomClass F M N\nh : IsPrimitiveRoot (\u2191f \u03b6) k\nhf : Injective \u2191f\n\u22a2 \u2191f (\u03b6 ^ k) = \u2191f 1\n[PROOFSTEP]\nrw [map_pow, _root_.map_one, h.pow_eq_one]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : MonoidHomClass F M N\nh : IsPrimitiveRoot (\u2191f \u03b6) k\nhf : Injective \u2191f\n\u22a2 \u2200 (l : \u2115), \u03b6 ^ l = 1 \u2192 k \u2223 l\n[PROOFSTEP]\nrw [h.eq_orderOf]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : MonoidHomClass F M N\nh : IsPrimitiveRoot (\u2191f \u03b6) k\nhf : Injective \u2191f\n\u22a2 \u2200 (l : \u2115), \u03b6 ^ l = 1 \u2192 orderOf (\u2191f \u03b6) \u2223 l\n[PROOFSTEP]\nintro l hl\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : MonoidHomClass F M N\nh : IsPrimitiveRoot (\u2191f \u03b6) k\nhf : Injective \u2191f\nl : \u2115\nhl : \u03b6 ^ l = 1\n\u22a2 orderOf (\u2191f \u03b6) \u2223 l\n[PROOFSTEP]\napply_fun f at hl \n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : MonoidHomClass F M N\nh : IsPrimitiveRoot (\u2191f \u03b6) k\nhf : Injective \u2191f\nl : \u2115\nhl : \u2191f (\u03b6 ^ l) = \u2191f 1\n\u22a2 orderOf (\u2191f \u03b6) \u2223 l\n[PROOFSTEP]\nrw [map_pow, _root_.map_one] at hl \n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nf : F\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : MonoidHomClass F M N\nh : IsPrimitiveRoot (\u2191f \u03b6) k\nhf : Injective \u2191f\nl : \u2115\nhl : \u2191f \u03b6 ^ l = 1\n\u22a2 orderOf (\u2191f \u03b6) \u2223 l\n[PROOFSTEP]\nexact orderOf_dvd_of_pow_eq_one hl\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l\u271d : \u2115\nM\u2080 : Type u_7\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\ninst\u271d : Nontrivial M\u2080\nl : \u2115\nhl : 0 ^ l = 1\n\u22a2 0 \u2223 l\n[PROOFSTEP]\nsimpa [zero_pow_eq, show \u2200 p, \u00acp \u2192 False \u2194 p from @Classical.not_not] using hl\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\n\u22a2 \u03b6 ^ \u2191k = 1\n[PROOFSTEP]\nrw [zpow_ofNat]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\n\u22a2 \u03b6 ^ k = 1\n[PROOFSTEP]\nexact h.pow_eq_one\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\nl : \u2124\n\u22a2 \u03b6 ^ l = 1 \u2194 \u2191k \u2223 l\n[PROOFSTEP]\nby_cases h0 : 0 \u2264 l\n[GOAL]\ncase pos\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\nl : \u2124\nh0 : 0 \u2264 l\n\u22a2 \u03b6 ^ l = 1 \u2194 \u2191k \u2223 l\n[PROOFSTEP]\nlift l to \u2115 using h0\n[GOAL]\ncase pos.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\nl : \u2115\n\u22a2 \u03b6 ^ \u2191l = 1 \u2194 \u2191k \u2223 \u2191l\n[PROOFSTEP]\nrw [zpow_ofNat]\n[GOAL]\ncase pos.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\nl : \u2115\n\u22a2 \u03b6 ^ l = 1 \u2194 \u2191k \u2223 \u2191l\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase pos.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\nl : \u2115\n\u22a2 \u03b6 ^ l = 1 \u2194 k \u2223 l\n[PROOFSTEP]\nexact h.pow_eq_one_iff_dvd l\n[GOAL]\ncase neg\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\nl : \u2124\nh0 : \u00ac0 \u2264 l\n\u22a2 \u03b6 ^ l = 1 \u2194 \u2191k \u2223 l\n[PROOFSTEP]\nhave : 0 \u2264 -l := by simp only [not_le, neg_nonneg] at h0 \u22a2; exact le_of_lt h0\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\nl : \u2124\nh0 : \u00ac0 \u2264 l\n\u22a2 0 \u2264 -l\n[PROOFSTEP]\nsimp only [not_le, neg_nonneg] at h0 \u22a2\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\nl : \u2124\nh0 : l < 0\n\u22a2 l \u2264 0\n[PROOFSTEP]\nexact le_of_lt h0\n[GOAL]\ncase neg\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\nl : \u2124\nh0 : \u00ac0 \u2264 l\nthis : 0 \u2264 -l\n\u22a2 \u03b6 ^ l = 1 \u2194 \u2191k \u2223 l\n[PROOFSTEP]\nlift -l to \u2115 using this with l' hl'\n[GOAL]\ncase neg.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\nl : \u2124\nh0 : \u00ac0 \u2264 l\nl' : \u2115\nhl' : \u2191l' = -l\n\u22a2 \u03b6 ^ l = 1 \u2194 \u2191k \u2223 l\n[PROOFSTEP]\nrw [\u2190 dvd_neg, \u2190 hl']\n[GOAL]\ncase neg.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\nl : \u2124\nh0 : \u00ac0 \u2264 l\nl' : \u2115\nhl' : \u2191l' = -l\n\u22a2 \u03b6 ^ l = 1 \u2194 \u2191k \u2223 \u2191l'\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase neg.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\nl : \u2124\nh0 : \u00ac0 \u2264 l\nl' : \u2115\nhl' : \u2191l' = -l\n\u22a2 \u03b6 ^ l = 1 \u2194 k \u2223 l'\n[PROOFSTEP]\nrw [\u2190 h.pow_eq_one_iff_dvd, \u2190 inv_inj, \u2190 zpow_neg, \u2190 hl', zpow_ofNat, inv_one]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\n\u22a2 \u03b6\u207b\u00b9 ^ k = 1\n[PROOFSTEP]\nsimp only [h.pow_eq_one, inv_one, eq_self_iff_true, inv_pow]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\n\u22a2 \u2200 (l : \u2115), \u03b6\u207b\u00b9 ^ l = 1 \u2192 k \u2223 l\n[PROOFSTEP]\nintro l hl\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\nl : \u2115\nhl : \u03b6\u207b\u00b9 ^ l = 1\n\u22a2 k \u2223 l\n[PROOFSTEP]\napply h.dvd_of_pow_eq_one l\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\nl : \u2115\nhl : \u03b6\u207b\u00b9 ^ l = 1\n\u22a2 \u03b6 ^ l = 1\n[PROOFSTEP]\nrw [\u2190 inv_inj, \u2190 inv_pow, hl, inv_one]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\n\u22a2 IsPrimitiveRoot \u03b6\u207b\u00b9 k \u2194 IsPrimitiveRoot \u03b6 k\n[PROOFSTEP]\nrefine' \u27e8_, fun h => inv h\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\n\u22a2 IsPrimitiveRoot \u03b6\u207b\u00b9 k \u2192 IsPrimitiveRoot \u03b6 k\n[PROOFSTEP]\nintro h\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6\u207b\u00b9 k\n\u22a2 IsPrimitiveRoot \u03b6 k\n[PROOFSTEP]\nrw [\u2190 inv_inv \u03b6]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6\u207b\u00b9 k\n\u22a2 IsPrimitiveRoot \u03b6\u207b\u00b9\u207b\u00b9 k\n[PROOFSTEP]\nexact inv h\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\ni : \u2124\nhi : Int.gcd i \u2191k = 1\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) k\n[PROOFSTEP]\nby_cases h0 : 0 \u2264 i\n[GOAL]\ncase pos\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\ni : \u2124\nhi : Int.gcd i \u2191k = 1\nh0 : 0 \u2264 i\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) k\n[PROOFSTEP]\nlift i to \u2115 using h0\n[GOAL]\ncase pos.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\ni : \u2115\nhi : Int.gcd \u2191i \u2191k = 1\n\u22a2 IsPrimitiveRoot (\u03b6 ^ \u2191i) k\n[PROOFSTEP]\nrw [zpow_ofNat]\n[GOAL]\ncase pos.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\ni : \u2115\nhi : Int.gcd \u2191i \u2191k = 1\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) k\n[PROOFSTEP]\nexact h.pow_of_coprime i hi\n[GOAL]\ncase neg\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\ni : \u2124\nhi : Int.gcd i \u2191k = 1\nh0 : \u00ac0 \u2264 i\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) k\n[PROOFSTEP]\nhave : 0 \u2264 -i := by simp only [not_le, neg_nonneg] at h0 \u22a2; exact le_of_lt h0\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\ni : \u2124\nhi : Int.gcd i \u2191k = 1\nh0 : \u00ac0 \u2264 i\n\u22a2 0 \u2264 -i\n[PROOFSTEP]\nsimp only [not_le, neg_nonneg] at h0 \u22a2\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\ni : \u2124\nhi : Int.gcd i \u2191k = 1\nh0 : i < 0\n\u22a2 i \u2264 0\n[PROOFSTEP]\nexact le_of_lt h0\n[GOAL]\ncase neg\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\ni : \u2124\nhi : Int.gcd i \u2191k = 1\nh0 : \u00ac0 \u2264 i\nthis : 0 \u2264 -i\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) k\n[PROOFSTEP]\nlift -i to \u2115 using this with i' hi'\n[GOAL]\ncase neg.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\ni : \u2124\nhi : Int.gcd i \u2191k = 1\nh0 : \u00ac0 \u2264 i\ni' : \u2115\nhi' : \u2191i' = -i\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) k\n[PROOFSTEP]\nrw [\u2190 inv_iff, \u2190 zpow_neg, \u2190 hi', zpow_ofNat]\n[GOAL]\ncase neg.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\ni : \u2124\nhi : Int.gcd i \u2191k = 1\nh0 : \u00ac0 \u2264 i\ni' : \u2115\nhi' : \u2191i' = -i\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i') k\n[PROOFSTEP]\napply h.pow_of_coprime\n[GOAL]\ncase neg.intro.hi\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\ni : \u2124\nhi : Int.gcd i \u2191k = 1\nh0 : \u00ac0 \u2264 i\ni' : \u2115\nhi' : \u2191i' = -i\n\u22a2 Nat.coprime i' k\n[PROOFSTEP]\nrw [Int.gcd, \u2190 Int.natAbs_neg, \u2190 hi'] at hi \n[GOAL]\ncase neg.intro.hi\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : G\nh : IsPrimitiveRoot \u03b6 k\ni : \u2124\nh0 : \u00ac0 \u2264 i\ni' : \u2115\nhi : Nat.gcd (Int.natAbs \u2191i') (Int.natAbs \u2191k) = 1\nhi' : \u2191i' = -i\n\u22a2 Nat.coprime i' k\n[PROOFSTEP]\nexact hi\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 primitiveRoots 1 R = {1}\n[PROOFSTEP]\napply Finset.eq_singleton_iff_unique_mem.2\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 1 \u2208 primitiveRoots 1 R \u2227 \u2200 (x : R), x \u2208 primitiveRoots 1 R \u2192 x = 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 1 \u2208 primitiveRoots 1 R\n[PROOFSTEP]\nsimp only [IsPrimitiveRoot.one_right_iff, mem_primitiveRoots zero_lt_one]\n[GOAL]\ncase right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u22a2 \u2200 (x : R), x \u2208 primitiveRoots 1 R \u2192 x = 1\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : R\nhx : x \u2208 primitiveRoots 1 R\n\u22a2 x = 1\n[PROOFSTEP]\nrw [mem_primitiveRoots zero_lt_one, IsPrimitiveRoot.one_right_iff] at hx \n[GOAL]\ncase right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : R\nhx : x = 1\n\u22a2 x = 1\n[PROOFSTEP]\nexact hx\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 NeZero \u2191\u2191n\n[PROOFSTEP]\nlet p := ringChar R\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\n\u22a2 NeZero \u2191\u2191n\n[PROOFSTEP]\nhave hfin := multiplicity.finite_nat_iff.2 \u27e8CharP.char_ne_one R p, n.pos\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\n\u22a2 NeZero \u2191\u2191n\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := multiplicity.exists_eq_pow_mul_and_not_dvd hfin\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\n\u22a2 NeZero \u2191\u2191n\n[PROOFSTEP]\nby_cases hp : p \u2223 n\n[GOAL]\ncase pos\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\nhp : p \u2223 \u2191n\n\u22a2 NeZero \u2191\u2191n\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := Nat.exists_eq_succ_of_ne_zero (multiplicity.pos_of_dvd hfin hp).ne'\n[GOAL]\ncase pos.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\nhp : p \u2223 \u2191n\nk : \u2115\nhk : Part.get (multiplicity p \u2191n) hfin = Nat.succ k\n\u22a2 NeZero \u2191\u2191n\n[PROOFSTEP]\nhaveI : NeZero p := NeZero.of_pos (Nat.pos_of_dvd_of_pos hp n.pos)\n[GOAL]\ncase pos.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\nhp : p \u2223 \u2191n\nk : \u2115\nhk : Part.get (multiplicity p \u2191n) hfin = Nat.succ k\nthis : NeZero p\n\u22a2 NeZero \u2191\u2191n\n[PROOFSTEP]\nhaveI hpri : Fact p.Prime := CharP.char_is_prime_of_pos R p\n[GOAL]\ncase pos.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\nhp : p \u2223 \u2191n\nk : \u2115\nhk : Part.get (multiplicity p \u2191n) hfin = Nat.succ k\nthis : NeZero p\nhpri : Fact (Nat.Prime p)\n\u22a2 NeZero \u2191\u2191n\n[PROOFSTEP]\nhave := h\u03b6.pow_eq_one\n[GOAL]\ncase pos.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\nhp : p \u2223 \u2191n\nk : \u2115\nhk : Part.get (multiplicity p \u2191n) hfin = Nat.succ k\nthis\u271d : NeZero p\nhpri : Fact (Nat.Prime p)\nthis : \u03b6 ^ \u2191n = 1\n\u22a2 NeZero \u2191\u2191n\n[PROOFSTEP]\nrw [hm.1, hk, pow_succ, mul_assoc, pow_mul', \u2190 frobenius_def, \u2190 frobenius_one p] at this \n[GOAL]\ncase pos.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\nhp : p \u2223 \u2191n\nk : \u2115\nhk : Part.get (multiplicity p \u2191n) hfin = Nat.succ k\nthis\u271d\u00b9 : NeZero p\nhpri : Fact (Nat.Prime p)\nthis\u271d : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = 1\nthis : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = \u2191(frobenius R p) 1\n\u22a2 NeZero \u2191\u2191n\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.intro.h\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\nhp : p \u2223 \u2191n\nk : \u2115\nhk : Part.get (multiplicity p \u2191n) hfin = Nat.succ k\nthis\u271d\u00b9 : NeZero p\nhpri : Fact (Nat.Prime p)\nthis\u271d : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = 1\nthis : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = \u2191(frobenius R p) 1\n\u22a2 False\n[PROOFSTEP]\nhave hpos : 0 < p ^ k * m :=\n by\n refine' mul_pos (pow_pos hpri.1.pos _) (Nat.pos_of_ne_zero fun h => _)\n have H := hm.1\n rw [h] at H \n simp at H \n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\nhp : p \u2223 \u2191n\nk : \u2115\nhk : Part.get (multiplicity p \u2191n) hfin = Nat.succ k\nthis\u271d\u00b9 : NeZero p\nhpri : Fact (Nat.Prime p)\nthis\u271d : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = 1\nthis : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = \u2191(frobenius R p) 1\n\u22a2 0 < p ^ k * m\n[PROOFSTEP]\nrefine' mul_pos (pow_pos hpri.1.pos _) (Nat.pos_of_ne_zero fun h => _)\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\nhp : p \u2223 \u2191n\nk : \u2115\nhk : Part.get (multiplicity p \u2191n) hfin = Nat.succ k\nthis\u271d\u00b9 : NeZero p\nhpri : Fact (Nat.Prime p)\nthis\u271d : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = 1\nthis : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = \u2191(frobenius R p) 1\nh : m = 0\n\u22a2 False\n[PROOFSTEP]\nhave H := hm.1\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\nhp : p \u2223 \u2191n\nk : \u2115\nhk : Part.get (multiplicity p \u2191n) hfin = Nat.succ k\nthis\u271d\u00b9 : NeZero p\nhpri : Fact (Nat.Prime p)\nthis\u271d : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = 1\nthis : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = \u2191(frobenius R p) 1\nh : m = 0\nH : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m\n\u22a2 False\n[PROOFSTEP]\nrw [h] at H \n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\nhp : p \u2223 \u2191n\nk : \u2115\nhk : Part.get (multiplicity p \u2191n) hfin = Nat.succ k\nthis\u271d\u00b9 : NeZero p\nhpri : Fact (Nat.Prime p)\nthis\u271d : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = 1\nthis : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = \u2191(frobenius R p) 1\nh : m = 0\nH : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * 0\n\u22a2 False\n[PROOFSTEP]\nsimp at H \n[GOAL]\ncase pos.intro.h\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\nhp : p \u2223 \u2191n\nk : \u2115\nhk : Part.get (multiplicity p \u2191n) hfin = Nat.succ k\nthis\u271d\u00b9 : NeZero p\nhpri : Fact (Nat.Prime p)\nthis\u271d : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = 1\nthis : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = \u2191(frobenius R p) 1\nhpos : 0 < p ^ k * m\n\u22a2 False\n[PROOFSTEP]\nrefine' h\u03b6.pow_ne_one_of_pos_of_lt hpos _ (frobenius_inj R p this)\n[GOAL]\ncase pos.intro.h\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\nhp : p \u2223 \u2191n\nk : \u2115\nhk : Part.get (multiplicity p \u2191n) hfin = Nat.succ k\nthis\u271d\u00b9 : NeZero p\nhpri : Fact (Nat.Prime p)\nthis\u271d : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = 1\nthis : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = \u2191(frobenius R p) 1\nhpos : 0 < p ^ k * m\n\u22a2 p ^ k * m < \u2191n\n[PROOFSTEP]\nrw [hm.1, hk, pow_succ, mul_assoc, mul_comm p]\n[GOAL]\ncase pos.intro.h\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\nhp : p \u2223 \u2191n\nk : \u2115\nhk : Part.get (multiplicity p \u2191n) hfin = Nat.succ k\nthis\u271d\u00b9 : NeZero p\nhpri : Fact (Nat.Prime p)\nthis\u271d : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = 1\nthis : \u2191(frobenius R p) (\u03b6 ^ (p ^ k * m)) = \u2191(frobenius R p) 1\nhpos : 0 < p ^ k * m\n\u22a2 p ^ k * m < p ^ k * m * p\n[PROOFSTEP]\nexact lt_mul_of_one_lt_right hpos hpri.1.one_lt\n[GOAL]\ncase neg\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : R\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115+\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\np : \u2115 := ringChar R\nhfin : multiplicity.Finite p \u2191n\nm : \u2115\nhm : \u2191n = p ^ Part.get (multiplicity p \u2191n) hfin * m \u2227 \u00acp \u2223 m\nhp : \u00acp \u2223 \u2191n\n\u22a2 NeZero \u2191\u2191n\n[PROOFSTEP]\nexact NeZero.of_not_dvd R hp\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : NoZeroDivisors R\n\u03b6 : R\nh : IsPrimitiveRoot \u03b6 2\n\u22a2 \u03b6 = -1\n[PROOFSTEP]\napply (eq_or_eq_neg_of_sq_eq_sq \u03b6 1 _).resolve_left\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : NoZeroDivisors R\n\u03b6 : R\nh : IsPrimitiveRoot \u03b6 2\n\u22a2 \u00ac\u03b6 = 1\n[PROOFSTEP]\nrw [\u2190 pow_one \u03b6]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : NoZeroDivisors R\n\u03b6 : R\nh : IsPrimitiveRoot \u03b6 2\n\u22a2 \u00ac\u03b6 ^ 1 = 1\n[PROOFSTEP]\napply h.pow_ne_one_of_pos_of_lt\n[GOAL]\ncase h0\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : NoZeroDivisors R\n\u03b6 : R\nh : IsPrimitiveRoot \u03b6 2\n\u22a2 0 < 1\n[PROOFSTEP]\ndecide\n[GOAL]\ncase hl\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : NoZeroDivisors R\n\u03b6 : R\nh : IsPrimitiveRoot \u03b6 2\n\u22a2 1 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : NoZeroDivisors R\n\u03b6 : R\nh : IsPrimitiveRoot \u03b6 2\n\u22a2 \u03b6 ^ 2 = 1 ^ 2\n[PROOFSTEP]\nsimp only [h.pow_eq_one, one_pow]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6 : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6 k\np : \u2115\ninst\u271d : Nontrivial R\nh : CharP R p\nhp : p \u2260 2\n\u22a2 IsPrimitiveRoot (-1) 2\n[PROOFSTEP]\nconvert IsPrimitiveRoot.orderOf (-1 : R)\n[GOAL]\ncase h.e'_4\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6 : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6 k\np : \u2115\ninst\u271d : Nontrivial R\nh : CharP R p\nhp : p \u2260 2\n\u22a2 2 = orderOf (-1)\n[PROOFSTEP]\nrw [orderOf_neg_one, if_neg]\n[GOAL]\ncase h.e'_4.hnc\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6 : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6 k\np : \u2115\ninst\u271d : Nontrivial R\nh : CharP R p\nhp : p \u2260 2\n\u22a2 \u00acringChar R = 2\n[PROOFSTEP]\nrwa [ringChar.eq_iff.mpr h]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nh\u03b6 : IsPrimitiveRoot \u03b6 k\nhk : 1 < k\n\u22a2 \u2211 i in range k, \u03b6 ^ i = 0\n[PROOFSTEP]\nrefine' eq_zero_of_ne_zero_of_mul_left_eq_zero (sub_ne_zero_of_ne (h\u03b6.ne_one hk).symm) _\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nh\u03b6 : IsPrimitiveRoot \u03b6 k\nhk : 1 < k\n\u22a2 (1 - \u03b6) * \u2211 i in range k, \u03b6 ^ i = 0\n[PROOFSTEP]\nrw [mul_neg_geom_sum, h\u03b6.pow_eq_one, sub_self]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nh\u03b6 : IsPrimitiveRoot \u03b6 k\nhk : 1 < k\n\u22a2 \u03b6 ^ Nat.pred k = -\u2211 i in range (Nat.pred k), \u03b6 ^ i\n[PROOFSTEP]\nrw [eq_neg_iff_add_eq_zero, add_comm, \u2190 sum_range_succ, \u2190 Nat.succ_eq_add_one, Nat.succ_pred_eq_of_pos (pos_of_gt hk),\n h\u03b6.geom_sum_eq_zero hk]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\n\u22a2 (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0\n[PROOFSTEP]\nsimp only [zpow_zero]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\n\u22a2 \u2191Additive.ofMul { val := 1, property := (_ : (fun x => x \u2208 Subgroup.zpowers \u03b6) 1) } = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\n\u22a2 \u2200 (x y : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (x + y) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n x +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n y\n[PROOFSTEP]\nintro i j\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni j : \u2124\n\u22a2 ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j\n[PROOFSTEP]\nsimp only [zpow_add]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni j : \u2124\n\u22a2 \u2191Additive.ofMul { val := \u03b6 ^ i * \u03b6 ^ j, property := (_ : (fun x => x \u2208 Subgroup.zpowers \u03b6) (\u03b6 ^ i * \u03b6 ^ j)) } =\n \u2191Additive.ofMul { val := \u03b6 ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) } +\n \u2191Additive.ofMul { val := \u03b6 ^ j, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 j) }\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni : \u2124\nhi : i \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod k))\n\u22a2 i \u2208\n AddMonoidHom.ker\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) }\n[PROOFSTEP]\nsimp only [AddMonoidHom.mem_ker, CharP.int_cast_eq_zero_iff (ZMod k) k, AddMonoidHom.coe_mk, Int.coe_castAddHom] at hi \u22a2\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni : \u2124\nhi : \u2191k \u2223 i\n\u22a2 \u2191{\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := \u03b6 ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i =\n 0\n[PROOFSTEP]\nobtain \u27e8i, rfl\u27e9 := hi\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni : \u2124\n\u22a2 \u2191{\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := \u03b6 ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (\u2191k * i) =\n 0\n[PROOFSTEP]\nsimp [zpow_mul, h.pow_eq_one, one_zpow, zpow_ofNat]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\n\u22a2 Function.Bijective\n \u2191(\u2191(AddMonoidHom.liftOfRightInverse (Int.castAddHom (ZMod k)) ZMod.cast\n (_ : Function.RightInverse ZMod.cast Int.cast))\n {\n val :=\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) },\n property :=\n (_ :\n \u2200 (i : \u2124),\n i \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod k)) \u2192\n i \u2208\n AddMonoidHom.ker\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) }) })\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\n\u22a2 Function.Injective\n \u2191(\u2191(AddMonoidHom.liftOfRightInverse (Int.castAddHom (ZMod k)) ZMod.cast\n (_ : Function.RightInverse ZMod.cast Int.cast))\n {\n val :=\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) },\n property :=\n (_ :\n \u2200 (i : \u2124),\n i \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod k)) \u2192\n i \u2208\n AddMonoidHom.ker\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) }) })\n[PROOFSTEP]\nrw [injective_iff_map_eq_zero]\n[GOAL]\ncase left\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\n\u22a2 \u2200 (a : ZMod k),\n \u2191(\u2191(AddMonoidHom.liftOfRightInverse (Int.castAddHom (ZMod k)) ZMod.cast\n (_ : Function.RightInverse ZMod.cast Int.cast))\n {\n val :=\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) },\n property :=\n (_ :\n \u2200 (i : \u2124),\n i \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod k)) \u2192\n i \u2208\n AddMonoidHom.ker\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y =\n (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y =\n (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y =\n (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) }) })\n a =\n 0 \u2192\n a = 0\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase left\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni : ZMod k\nhi :\n \u2191(\u2191(AddMonoidHom.liftOfRightInverse (Int.castAddHom (ZMod k)) ZMod.cast\n (_ : Function.RightInverse ZMod.cast Int.cast))\n {\n val :=\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) },\n property :=\n (_ :\n \u2200 (i : \u2124),\n i \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod k)) \u2192\n i \u2208\n AddMonoidHom.ker\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) }) })\n i =\n 0\n\u22a2 i = 0\n[PROOFSTEP]\nrw [Subtype.ext_iff] at hi \n[GOAL]\ncase left\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni : ZMod k\nhi :\n \u2191(\u2191(\u2191(AddMonoidHom.liftOfRightInverse (Int.castAddHom (ZMod k)) ZMod.cast\n (_ : Function.RightInverse ZMod.cast Int.cast))\n {\n val :=\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) },\n property :=\n (_ :\n \u2200 (i : \u2124),\n i \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod k)) \u2192\n i \u2208\n AddMonoidHom.ker\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y =\n (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y =\n (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) }) })\n i) =\n \u21910\n\u22a2 i = 0\n[PROOFSTEP]\nhave := (h.zpow_eq_one_iff_dvd _).mp hi\n[GOAL]\ncase left\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni : ZMod k\nhi :\n \u2191(\u2191(\u2191(AddMonoidHom.liftOfRightInverse (Int.castAddHom (ZMod k)) ZMod.cast\n (_ : Function.RightInverse ZMod.cast Int.cast))\n {\n val :=\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) },\n property :=\n (_ :\n \u2200 (i : \u2124),\n i \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod k)) \u2192\n i \u2208\n AddMonoidHom.ker\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y =\n (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y =\n (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) }) })\n i) =\n \u21910\nthis : \u2191k \u2223 \u2191i\n\u22a2 i = 0\n[PROOFSTEP]\nrw [\u2190 (CharP.int_cast_eq_zero_iff (ZMod k) k _).mpr this, eq_comm]\n[GOAL]\ncase left\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni : ZMod k\nhi :\n \u2191(\u2191(\u2191(AddMonoidHom.liftOfRightInverse (Int.castAddHom (ZMod k)) ZMod.cast\n (_ : Function.RightInverse ZMod.cast Int.cast))\n {\n val :=\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) },\n property :=\n (_ :\n \u2200 (i : \u2124),\n i \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod k)) \u2192\n i \u2208\n AddMonoidHom.ker\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y =\n (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y =\n (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) }) })\n i) =\n \u21910\nthis : \u2191k \u2223 \u2191i\n\u22a2 \u2191\u2191i = i\n[PROOFSTEP]\nexact ZMod.int_cast_rightInverse i\n[GOAL]\ncase right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\n\u22a2 Function.Surjective\n \u2191(\u2191(AddMonoidHom.liftOfRightInverse (Int.castAddHom (ZMod k)) ZMod.cast\n (_ : Function.RightInverse ZMod.cast Int.cast))\n {\n val :=\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) },\n property :=\n (_ :\n \u2200 (i : \u2124),\n i \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod k)) \u2192\n i \u2208\n AddMonoidHom.ker\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) }) })\n[PROOFSTEP]\nrintro \u27e8\u03be, i, rfl\u27e9\n[GOAL]\ncase right.mk.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni : \u2124\n\u22a2 \u2203 a,\n \u2191(\u2191(AddMonoidHom.liftOfRightInverse (Int.castAddHom (ZMod k)) ZMod.cast\n (_ : Function.RightInverse ZMod.cast Int.cast))\n {\n val :=\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) },\n property :=\n (_ :\n \u2200 (i : \u2124),\n i \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod k)) \u2192\n i \u2208\n AddMonoidHom.ker\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y =\n (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y =\n (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) }) })\n a =\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) }\n[PROOFSTEP]\nrefine' \u27e8Int.castAddHom (ZMod k) i, _\u27e9\n[GOAL]\ncase right.mk.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni : \u2124\n\u22a2 \u2191(\u2191(AddMonoidHom.liftOfRightInverse (Int.castAddHom (ZMod k)) ZMod.cast\n (_ : Function.RightInverse ZMod.cast Int.cast))\n {\n val :=\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) },\n property :=\n (_ :\n \u2200 (i : \u2124),\n i \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod k)) \u2192\n i \u2208\n AddMonoidHom.ker\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) }) })\n (\u2191(Int.castAddHom (ZMod k)) i) =\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) }\n[PROOFSTEP]\nrw [AddMonoidHom.liftOfRightInverse_comp_apply]\n[GOAL]\ncase right.mk.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh\u271d h : IsPrimitiveRoot \u03b6 k\ni : \u2124\n\u22a2 \u2191\u2191{\n val :=\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) },\n property :=\n (_ :\n \u2200 (i : \u2124),\n i \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod k)) \u2192\n i \u2208\n AddMonoidHom.ker\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) }) }\n i =\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) }\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 k\ni : \u2124\n\u22a2 \u2191(zmodEquivZpowers h) \u2191i = \u2191Additive.ofMul { val := \u03b6 ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = \u03b6 ^ i) }\n[PROOFSTEP]\nrw [zmodEquivZpowers, AddEquiv.ofBijective_apply]\n -- Porting note: Original proof didn't have `rw`\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 k\ni : \u2124\n\u22a2 \u2191(\u2191(AddMonoidHom.liftOfRightInverse (Int.castAddHom (ZMod k)) ZMod.cast\n (_ : Function.RightInverse ZMod.cast Int.cast))\n {\n val :=\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) },\n property :=\n (_ :\n \u2200 (i : \u2124),\n i \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod k)) \u2192\n i \u2208\n AddMonoidHom.ker\n {\n toZeroHom :=\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) },\n map_add' :=\n (_ :\n \u2200 (i j : \u2124),\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n (i + j) =\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n i +\n ZeroHom.toFun\n {\n toFun := fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) },\n map_zero' :=\n (_ :\n (fun i =>\n \u2191Additive.ofMul\n { val := (fun x x_1 => x ^ x_1) \u03b6 i,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) \u03b6 y = (fun x x_1 => x ^ x_1) \u03b6 i) })\n 0 =\n 0) }\n j) }) })\n \u2191i =\n \u2191Additive.ofMul { val := \u03b6 ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = \u03b6 ^ i) }\n[PROOFSTEP]\nexact AddMonoidHom.liftOfRightInverse_comp_apply _ _ ZMod.int_cast_rightInverse _ _\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 k\ni : \u2115\n\u22a2 \u2191(zmodEquivZpowers h) \u2191i = \u2191Additive.ofMul { val := \u03b6 ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = \u03b6 ^ i) }\n[PROOFSTEP]\nhave : (i : ZMod k) = (i : \u2124) := by norm_cast\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 k\ni : \u2115\n\u22a2 \u2191i = \u2191\u2191i\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 k\ni : \u2115\nthis : \u2191i = \u2191\u2191i\n\u22a2 \u2191(zmodEquivZpowers h) \u2191i = \u2191Additive.ofMul { val := \u03b6 ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = \u03b6 ^ i) }\n[PROOFSTEP]\nsimp only [this, zmodEquivZpowers_apply_coe_int, zpow_ofNat]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 k\ni : \u2124\n\u22a2 \u2191(AddEquiv.symm (zmodEquivZpowers h))\n (\u2191Additive.ofMul { val := \u03b6 ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = \u03b6 ^ i) }) =\n \u2191i\n[PROOFSTEP]\nrw [\u2190 h.zmodEquivZpowers.symm_apply_apply i, zmodEquivZpowers_apply_coe_int]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b3 : CommMonoid M\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d : CommRing R\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 k\ni : \u2115\n\u22a2 \u2191(AddEquiv.symm (zmodEquivZpowers h))\n (\u2191Additive.ofMul { val := \u03b6 ^ i, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) \u03b6 y = \u03b6 ^ i) }) =\n \u2191i\n[PROOFSTEP]\nrw [\u2190 h.zmodEquivZpowers.symm_apply_apply i, zmodEquivZpowers_apply_coe_nat]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\n\u22a2 Subgroup.zpowers \u03b6 = rootsOfUnity k R\n[PROOFSTEP]\napply SetLike.coe_injective\n[GOAL]\ncase a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\n\u22a2 \u2191(Subgroup.zpowers \u03b6) = \u2191(rootsOfUnity k R)\n[PROOFSTEP]\nhaveI F : Fintype (Subgroup.zpowers \u03b6) := Fintype.ofEquiv _ h.zmodEquivZpowers.toEquiv\n[GOAL]\ncase a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF\u271d : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\nF : Fintype { x // x \u2208 Subgroup.zpowers \u03b6 }\n\u22a2 \u2191(Subgroup.zpowers \u03b6) = \u2191(rootsOfUnity k R)\n[PROOFSTEP]\nrefine'\n @Set.eq_of_subset_of_card_le R\u02e3 (Subgroup.zpowers \u03b6) (rootsOfUnity k R) F (rootsOfUnity.fintype R k)\n (Subgroup.zpowers_le_of_mem <| show \u03b6 \u2208 rootsOfUnity k R from h.pow_eq_one) _\n[GOAL]\ncase a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF\u271d : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\nF : Fintype { x // x \u2208 Subgroup.zpowers \u03b6 }\n\u22a2 Fintype.card \u2191\u2191(rootsOfUnity k R) \u2264 Fintype.card \u2191\u2191(Subgroup.zpowers \u03b6)\n[PROOFSTEP]\ncalc\n Fintype.card (rootsOfUnity k R) \u2264 k := card_rootsOfUnity R k\n _ = Fintype.card (ZMod k) := (ZMod.card k).symm\n _ = Fintype.card (Subgroup.zpowers \u03b6) := Fintype.card_congr h.zmodEquivZpowers.toEquiv\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 \u03be : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\nh\u03be : \u03be \u2208 rootsOfUnity k R\n\u22a2 \u2203 i, i < \u2191k \u2227 \u03b6 ^ i = \u03be\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 : \u2203 n : \u2124, \u03b6 ^ n = \u03be := by rwa [\u2190 h.zpowers_eq] at h\u03be \n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 \u03be : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\nh\u03be : \u03be \u2208 rootsOfUnity k R\n\u22a2 \u2203 n, \u03b6 ^ n = \u03be\n[PROOFSTEP]\nrwa [\u2190 h.zpowers_eq] at h\u03be \n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\nn : \u2124\nh\u03be : \u03b6 ^ n \u2208 rootsOfUnity k R\n\u22a2 \u2203 i, i < \u2191k \u2227 \u03b6 ^ i = \u03b6 ^ n\n[PROOFSTEP]\nhave hk0 : (0 : \u2124) < k := by exact_mod_cast k.pos\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\nn : \u2124\nh\u03be : \u03b6 ^ n \u2208 rootsOfUnity k R\n\u22a2 0 < \u2191\u2191k\n[PROOFSTEP]\nexact_mod_cast k.pos\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\nn : \u2124\nh\u03be : \u03b6 ^ n \u2208 rootsOfUnity k R\nhk0 : 0 < \u2191\u2191k\n\u22a2 \u2203 i, i < \u2191k \u2227 \u03b6 ^ i = \u03b6 ^ n\n[PROOFSTEP]\nlet i := n % k\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\nn : \u2124\nh\u03be : \u03b6 ^ n \u2208 rootsOfUnity k R\nhk0 : 0 < \u2191\u2191k\ni : \u2124 := n % \u2191\u2191k\n\u22a2 \u2203 i, i < \u2191k \u2227 \u03b6 ^ i = \u03b6 ^ n\n[PROOFSTEP]\nhave hi0 : 0 \u2264 i := Int.emod_nonneg _ (ne_of_gt hk0)\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\nn : \u2124\nh\u03be : \u03b6 ^ n \u2208 rootsOfUnity k R\nhk0 : 0 < \u2191\u2191k\ni : \u2124 := n % \u2191\u2191k\nhi0 : 0 \u2264 i\n\u22a2 \u2203 i, i < \u2191k \u2227 \u03b6 ^ i = \u03b6 ^ n\n[PROOFSTEP]\nlift i to \u2115 using hi0 with i\u2080 hi\u2080\n[GOAL]\ncase intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\nn : \u2124\nh\u03be : \u03b6 ^ n \u2208 rootsOfUnity k R\nhk0 : 0 < \u2191\u2191k\ni : \u2124 := n % \u2191\u2191k\ni\u2080 : \u2115\nhi\u2080 : \u2191i\u2080 = i\n\u22a2 \u2203 i, i < \u2191k \u2227 \u03b6 ^ i = \u03b6 ^ n\n[PROOFSTEP]\nrefine' \u27e8i\u2080, _, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\nn : \u2124\nh\u03be : \u03b6 ^ n \u2208 rootsOfUnity k R\nhk0 : 0 < \u2191\u2191k\ni : \u2124 := n % \u2191\u2191k\ni\u2080 : \u2115\nhi\u2080 : \u2191i\u2080 = i\n\u22a2 i\u2080 < \u2191k\n[PROOFSTEP]\nzify\n[GOAL]\ncase intro.intro.refine'_1\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\nn : \u2124\nh\u03be : \u03b6 ^ n \u2208 rootsOfUnity k R\nhk0 : 0 < \u2191\u2191k\ni : \u2124 := n % \u2191\u2191k\ni\u2080 : \u2115\nhi\u2080 : \u2191i\u2080 = i\n\u22a2 \u2191i\u2080 < \u2191\u2191k\n[PROOFSTEP]\nrw [hi\u2080]\n[GOAL]\ncase intro.intro.refine'_1\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\nn : \u2124\nh\u03be : \u03b6 ^ n \u2208 rootsOfUnity k R\nhk0 : 0 < \u2191\u2191k\ni : \u2124 := n % \u2191\u2191k\ni\u2080 : \u2115\nhi\u2080 : \u2191i\u2080 = i\n\u22a2 i < \u2191\u2191k\n[PROOFSTEP]\nexact Int.emod_lt_of_pos _ hk0\n[GOAL]\ncase intro.intro.refine'_2\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\nn : \u2124\nh\u03be : \u03b6 ^ n \u2208 rootsOfUnity k R\nhk0 : 0 < \u2191\u2191k\ni : \u2124 := n % \u2191\u2191k\ni\u2080 : \u2115\nhi\u2080 : \u2191i\u2080 = i\n\u22a2 \u03b6 ^ i\u2080 = \u03b6 ^ n\n[PROOFSTEP]\nrw [\u2190 zpow_ofNat, hi\u2080, \u2190 Int.emod_add_ediv n k, zpow_add, zpow_mul, h.zpow_eq_one, one_zpow, mul_one]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115\n\u03b6 \u03be : R\nh : IsPrimitiveRoot \u03b6 k\nh\u03be : \u03be ^ k = 1\nh0 : 0 < k\n\u22a2 \u2203 i, i < k \u2227 \u03b6 ^ i = \u03be\n[PROOFSTEP]\nlift \u03b6 to R\u02e3 using h.isUnit h0\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115\n\u03be : R\nh\u03be : \u03be ^ k = 1\nh0 : 0 < k\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot (\u2191\u03b6) k\n\u22a2 \u2203 i, i < k \u2227 \u2191\u03b6 ^ i = \u03be\n[PROOFSTEP]\nlift \u03be to R\u02e3 using isUnit_ofPowEqOne h\u03be h0.ne'\n[GOAL]\ncase intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115\nh0 : 0 < k\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot (\u2191\u03b6) k\n\u03be : R\u02e3\nh\u03be : \u2191\u03be ^ k = 1\n\u22a2 \u2203 i, i < k \u2227 \u2191\u03b6 ^ i = \u2191\u03be\n[PROOFSTEP]\nlift k to \u2115+ using h0\n[GOAL]\ncase intro.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 \u03be : R\u02e3\nk : \u2115+\nh : IsPrimitiveRoot \u2191\u03b6 \u2191k\nh\u03be : \u2191\u03be ^ \u2191k = 1\n\u22a2 \u2203 i, i < \u2191k \u2227 \u2191\u03b6 ^ i = \u2191\u03be\n[PROOFSTEP]\nsimp only [\u2190 Units.val_pow_eq_pow_val, \u2190 Units.ext_iff]\n[GOAL]\ncase intro.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 \u03be : R\u02e3\nk : \u2115+\nh : IsPrimitiveRoot \u2191\u03b6 \u2191k\nh\u03be : \u2191\u03be ^ \u2191k = 1\n\u22a2 \u2203 i, i < \u2191k \u2227 \u03b6 ^ i = \u03be\n[PROOFSTEP]\nrw [coe_units_iff] at h \n[GOAL]\ncase intro.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 \u03be : R\u02e3\nk : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191k\nh\u03be : \u2191\u03be ^ \u2191k = 1\n\u22a2 \u2203 i, i < \u2191k \u2227 \u03b6 ^ i = \u03be\n[PROOFSTEP]\napply h.eq_pow_of_mem_rootsOfUnity\n[GOAL]\ncase intro.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 \u03be : R\u02e3\nk : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191k\nh\u03be : \u2191\u03be ^ \u2191k = 1\n\u22a2 \u03be \u2208 rootsOfUnity k R\n[PROOFSTEP]\nrw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, h\u03be, Units.val_one]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 \u03be : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\n\u22a2 IsPrimitiveRoot \u03be \u2191k \u2194 \u2203 i, i < \u2191k \u2227 Nat.coprime i \u2191k \u2227 \u03b6 ^ i = \u03be\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 \u03be : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\n\u22a2 IsPrimitiveRoot \u03be \u2191k \u2192 \u2203 i, i < \u2191k \u2227 Nat.coprime i \u2191k \u2227 \u03b6 ^ i = \u03be\n[PROOFSTEP]\nintro h\u03be\n[GOAL]\ncase mp\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 \u03be : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\nh\u03be : IsPrimitiveRoot \u03be \u2191k\n\u22a2 \u2203 i, i < \u2191k \u2227 Nat.coprime i \u2191k \u2227 \u03b6 ^ i = \u03be\n[PROOFSTEP]\nobtain \u27e8i, hik, rfl\u27e9 := h.eq_pow_of_mem_rootsOfUnity h\u03be.pow_eq_one\n[GOAL]\ncase mp.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\ni : \u2115\nhik : i < \u2191k\nh\u03be : IsPrimitiveRoot (\u03b6 ^ i) \u2191k\n\u22a2 \u2203 i_1, i_1 < \u2191k \u2227 Nat.coprime i_1 \u2191k \u2227 \u03b6 ^ i_1 = \u03b6 ^ i\n[PROOFSTEP]\nrw [h.pow_iff_coprime k.pos] at h\u03be \n[GOAL]\ncase mp.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\ni : \u2115\nhik : i < \u2191k\nh\u03be : Nat.coprime i \u2191k\n\u22a2 \u2203 i_1, i_1 < \u2191k \u2227 Nat.coprime i_1 \u2191k \u2227 \u03b6 ^ i_1 = \u03b6 ^ i\n[PROOFSTEP]\nexact \u27e8i, hik, h\u03be, rfl\u27e9\n[GOAL]\ncase mpr\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 \u03be : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\n\u22a2 (\u2203 i, i < \u2191k \u2227 Nat.coprime i \u2191k \u2227 \u03b6 ^ i = \u03be) \u2192 IsPrimitiveRoot \u03be \u2191k\n[PROOFSTEP]\nrintro \u27e8i, -, hi, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191k\ni : \u2115\nhi : Nat.coprime i \u2191k\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) \u2191k\n[PROOFSTEP]\nexact h.pow_of_coprime i hi\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115\n\u03b6 \u03be : R\nh : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\n\u22a2 IsPrimitiveRoot \u03be k \u2194 \u2203 i, i < k \u2227 Nat.coprime i k \u2227 \u03b6 ^ i = \u03be\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115\n\u03b6 \u03be : R\nh : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\n\u22a2 IsPrimitiveRoot \u03be k \u2192 \u2203 i, i < k \u2227 Nat.coprime i k \u2227 \u03b6 ^ i = \u03be\n[PROOFSTEP]\nintro h\u03be\n[GOAL]\ncase mp\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115\n\u03b6 \u03be : R\nh : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\nh\u03be : IsPrimitiveRoot \u03be k\n\u22a2 \u2203 i, i < k \u2227 Nat.coprime i k \u2227 \u03b6 ^ i = \u03be\n[PROOFSTEP]\nobtain \u27e8i, hik, rfl\u27e9 := h.eq_pow_of_pow_eq_one h\u03be.pow_eq_one h0\n[GOAL]\ncase mp.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115\n\u03b6 : R\nh : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\ni : \u2115\nhik : i < k\nh\u03be : IsPrimitiveRoot (\u03b6 ^ i) k\n\u22a2 \u2203 i_1, i_1 < k \u2227 Nat.coprime i_1 k \u2227 \u03b6 ^ i_1 = \u03b6 ^ i\n[PROOFSTEP]\nrw [h.pow_iff_coprime h0] at h\u03be \n[GOAL]\ncase mp.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115\n\u03b6 : R\nh : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\ni : \u2115\nhik : i < k\nh\u03be : Nat.coprime i k\n\u22a2 \u2203 i_1, i_1 < k \u2227 Nat.coprime i_1 k \u2227 \u03b6 ^ i_1 = \u03b6 ^ i\n[PROOFSTEP]\nexact \u27e8i, hik, h\u03be, rfl\u27e9\n[GOAL]\ncase mpr\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115\n\u03b6 \u03be : R\nh : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\n\u22a2 (\u2203 i, i < k \u2227 Nat.coprime i k \u2227 \u03b6 ^ i = \u03be) \u2192 IsPrimitiveRoot \u03be k\n[PROOFSTEP]\nrintro \u27e8i, -, hi, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\nk : \u2115\n\u03b6 : R\nh : IsPrimitiveRoot \u03b6 k\nh0 : 0 < k\ni : \u2115\nhi : Nat.coprime i k\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) k\n[PROOFSTEP]\nexact h.pow_of_coprime i hi\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6 : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : IsDomain R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 Fintype.card { x // x \u2208 rootsOfUnity n R } = \u2191n\n[PROOFSTEP]\nlet e := h.zmodEquivZpowers\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6 : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : IsDomain R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\ne : ZMod \u2191n \u2243+ Additive { x // x \u2208 Subgroup.zpowers \u03b6 } := zmodEquivZpowers h\n\u22a2 Fintype.card { x // x \u2208 rootsOfUnity n R } = \u2191n\n[PROOFSTEP]\nhaveI F : Fintype (Subgroup.zpowers \u03b6) := Fintype.ofEquiv _ e.toEquiv\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF\u271d : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6 : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : IsDomain R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\ne : ZMod \u2191n \u2243+ Additive { x // x \u2208 Subgroup.zpowers \u03b6 } := zmodEquivZpowers h\nF : Fintype { x // x \u2208 Subgroup.zpowers \u03b6 }\n\u22a2 Fintype.card { x // x \u2208 rootsOfUnity n R } = \u2191n\n[PROOFSTEP]\ncalc\n Fintype.card (rootsOfUnity n R) = Fintype.card (Subgroup.zpowers \u03b6) := Fintype.card_congr <| by rw [h.zpowers_eq]\n _ = Fintype.card (ZMod n) := (Fintype.card_congr e.toEquiv.symm)\n _ = n := ZMod.card n\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF\u271d : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6 : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : IsDomain R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\ne : ZMod \u2191n \u2243+ Additive { x // x \u2208 Subgroup.zpowers \u03b6 } := zmodEquivZpowers h\nF : Fintype { x // x \u2208 Subgroup.zpowers \u03b6 }\n\u22a2 { x // x \u2208 rootsOfUnity n R } \u2243 { x // x \u2208 Subgroup.zpowers \u03b6 }\n[PROOFSTEP]\nrw [h.zpowers_eq]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 Fintype.card { x // x \u2208 rootsOfUnity n R } = \u2191n\n[PROOFSTEP]\nobtain \u27e8\u03b6, h\u03b6\u27e9 := h.isUnit n.pos\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d\u00b9 : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d\u00b9 k\ninst\u271d : IsDomain R\n\u03b6\u271d : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6\u271d \u2191n\n\u03b6 : R\u02e3\nh\u03b6 : \u2191\u03b6 = \u03b6\u271d\n\u22a2 Fintype.card { x // x \u2208 rootsOfUnity n R } = \u2191n\n[PROOFSTEP]\nrw [\u2190 h\u03b6, IsPrimitiveRoot.coe_units_iff] at h \n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d\u00b9 : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d\u00b9 k\ninst\u271d : IsDomain R\n\u03b6\u271d : R\nn : \u2115+\n\u03b6 : R\u02e3\nh : IsPrimitiveRoot \u03b6 \u2191n\nh\u03b6 : \u2191\u03b6 = \u03b6\u271d\n\u22a2 Fintype.card { x // x \u2208 rootsOfUnity n R } = \u2191n\n[PROOFSTEP]\nexact h.card_rootsOfUnity'\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 \u2191Multiset.card (nthRoots n 1) = n\n[PROOFSTEP]\ncases' Nat.eq_zero_or_pos n with hzero hpos\n[GOAL]\ncase inl\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhzero : n = 0\n\u22a2 \u2191Multiset.card (nthRoots n 1) = n\n[PROOFSTEP]\nsimp only [hzero, Multiset.card_zero, nthRoots_zero]\n[GOAL]\ncase inr\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\n\u22a2 \u2191Multiset.card (nthRoots n 1) = n\n[PROOFSTEP]\nrw [eq_iff_le_not_lt]\n[GOAL]\ncase inr\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\n\u22a2 \u2191Multiset.card (nthRoots n 1) \u2264 n \u2227 \u00ac\u2191Multiset.card (nthRoots n 1) < n\n[PROOFSTEP]\nuse card_nthRoots n 1\n[GOAL]\ncase right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\n\u22a2 \u00ac\u2191Multiset.card (nthRoots n 1) < n\n[PROOFSTEP]\nrw [not_lt]\n[GOAL]\ncase right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\n\u22a2 n \u2264 \u2191Multiset.card (nthRoots n 1)\n[PROOFSTEP]\nhave hcard : Fintype.card { x // x \u2208 nthRoots n (1 : R) } \u2264 Multiset.card (nthRoots n (1 : R)).attach :=\n Multiset.card_le_of_le (Multiset.dedup_le _)\n[GOAL]\ncase right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\nhcard : Fintype.card { x // x \u2208 nthRoots n 1 } \u2264 \u2191Multiset.card (Multiset.attach (nthRoots n 1))\n\u22a2 n \u2264 \u2191Multiset.card (nthRoots n 1)\n[PROOFSTEP]\nrw [Multiset.card_attach] at hcard \n[GOAL]\ncase right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\nhcard : Fintype.card { x // x \u2208 nthRoots n 1 } \u2264 \u2191Multiset.card (nthRoots n 1)\n\u22a2 n \u2264 \u2191Multiset.card (nthRoots n 1)\n[PROOFSTEP]\nrw [\u2190 PNat.toPNat'_coe hpos] at hcard h \u22a2\n[GOAL]\ncase right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 \u2191(Nat.toPNat' n)\nhpos : n > 0\nhcard : Fintype.card { x // x \u2208 nthRoots (\u2191(Nat.toPNat' n)) 1 } \u2264 \u2191Multiset.card (nthRoots (\u2191(Nat.toPNat' n)) 1)\n\u22a2 \u2191(Nat.toPNat' n) \u2264 \u2191Multiset.card (nthRoots (\u2191(Nat.toPNat' n)) 1)\n[PROOFSTEP]\nset m := Nat.toPNat' n\n[GOAL]\ncase right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : n > 0\nm : \u2115+ := Nat.toPNat' n\nh : IsPrimitiveRoot \u03b6 \u2191m\nhcard : Fintype.card { x // x \u2208 nthRoots (\u2191m) 1 } \u2264 \u2191Multiset.card (nthRoots (\u2191m) 1)\n\u22a2 \u2191m \u2264 \u2191Multiset.card (nthRoots (\u2191m) 1)\n[PROOFSTEP]\nrw [\u2190 Fintype.card_congr (rootsOfUnityEquivNthRoots R m), card_rootsOfUnity h] at hcard \n[GOAL]\ncase right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : n > 0\nm : \u2115+ := Nat.toPNat' n\nh : IsPrimitiveRoot \u03b6 \u2191m\nhcard : \u2191m \u2264 \u2191Multiset.card (nthRoots (\u2191m) 1)\n\u22a2 \u2191m \u2264 \u2191Multiset.card (nthRoots (\u2191m) 1)\n[PROOFSTEP]\nexact hcard\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 Multiset.Nodup (nthRoots n 1)\n[PROOFSTEP]\ncases' Nat.eq_zero_or_pos n with hzero hpos\n[GOAL]\ncase inl\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhzero : n = 0\n\u22a2 Multiset.Nodup (nthRoots n 1)\n[PROOFSTEP]\nsimp only [hzero, Multiset.nodup_zero, nthRoots_zero]\n[GOAL]\ncase inr\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\n\u22a2 Multiset.Nodup (nthRoots n 1)\n[PROOFSTEP]\napply (Multiset.dedup_eq_self (\u03b1 := R)).1\n[GOAL]\ncase inr\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\n\u22a2 Multiset.dedup (nthRoots n 1) = nthRoots n 1\n[PROOFSTEP]\nrw [eq_iff_le_not_lt]\n[GOAL]\ncase inr\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\n\u22a2 Multiset.dedup (nthRoots n 1) \u2264 nthRoots n 1 \u2227 \u00acMultiset.dedup (nthRoots n 1) < nthRoots n 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase inr.left\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\n\u22a2 Multiset.dedup (nthRoots n 1) \u2264 nthRoots n 1\n[PROOFSTEP]\nexact Multiset.dedup_le (nthRoots n (1 : R))\n[GOAL]\ncase inr.right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\n\u22a2 \u00acMultiset.dedup (nthRoots n 1) < nthRoots n 1\n[PROOFSTEP]\nby_contra ha\n[GOAL]\ncase inr.right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\nha : Multiset.dedup (nthRoots n 1) < nthRoots n 1\n\u22a2 False\n[PROOFSTEP]\nreplace ha := Multiset.card_lt_of_lt ha\n[GOAL]\ncase inr.right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\nha : \u2191Multiset.card (Multiset.dedup (nthRoots n 1)) < \u2191Multiset.card (nthRoots n 1)\n\u22a2 False\n[PROOFSTEP]\nrw [card_nthRoots h] at ha \n[GOAL]\ncase inr.right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\nha : \u2191Multiset.card (Multiset.dedup (nthRoots n 1)) < n\n\u22a2 False\n[PROOFSTEP]\nhave hrw : Multiset.card (nthRoots n (1 : R)).dedup = Fintype.card { x // x \u2208 nthRoots n (1 : R) } :=\n by\n set fs := (\u27e8(nthRoots n (1 : R)).dedup, Multiset.nodup_dedup _\u27e9 : Finset R)\n rw [\u2190 Finset.card_mk, Fintype.card_of_subtype fs _]\n intro x\n simp only [Multiset.mem_dedup, Finset.mem_mk]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\nha : \u2191Multiset.card (Multiset.dedup (nthRoots n 1)) < n\n\u22a2 \u2191Multiset.card (Multiset.dedup (nthRoots n 1)) = Fintype.card { x // x \u2208 nthRoots n 1 }\n[PROOFSTEP]\nset fs := (\u27e8(nthRoots n (1 : R)).dedup, Multiset.nodup_dedup _\u27e9 : Finset R)\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\nha : \u2191Multiset.card (Multiset.dedup (nthRoots n 1)) < n\nfs : Finset R := { val := Multiset.dedup (nthRoots n 1), nodup := (_ : Multiset.Nodup (Multiset.dedup (nthRoots n 1))) }\n\u22a2 \u2191Multiset.card (Multiset.dedup (nthRoots n 1)) = Fintype.card { x // x \u2208 nthRoots n 1 }\n[PROOFSTEP]\nrw [\u2190 Finset.card_mk, Fintype.card_of_subtype fs _]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\nha : \u2191Multiset.card (Multiset.dedup (nthRoots n 1)) < n\nfs : Finset R := { val := Multiset.dedup (nthRoots n 1), nodup := (_ : Multiset.Nodup (Multiset.dedup (nthRoots n 1))) }\n\u22a2 \u2200 (x : R), x \u2208 fs \u2194 x \u2208 nthRoots n 1\n[PROOFSTEP]\nintro x\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\nha : \u2191Multiset.card (Multiset.dedup (nthRoots n 1)) < n\nfs : Finset R := { val := Multiset.dedup (nthRoots n 1), nodup := (_ : Multiset.Nodup (Multiset.dedup (nthRoots n 1))) }\nx : R\n\u22a2 x \u2208 fs \u2194 x \u2208 nthRoots n 1\n[PROOFSTEP]\nsimp only [Multiset.mem_dedup, Finset.mem_mk]\n[GOAL]\ncase inr.right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhpos : n > 0\nha : \u2191Multiset.card (Multiset.dedup (nthRoots n 1)) < n\nhrw : \u2191Multiset.card (Multiset.dedup (nthRoots n 1)) = Fintype.card { x // x \u2208 nthRoots n 1 }\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 PNat.toPNat'_coe hpos] at h hrw ha \n[GOAL]\ncase inr.right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 \u2191(Nat.toPNat' n)\nhpos : n > 0\nha : \u2191Multiset.card (Multiset.dedup (nthRoots (\u2191(Nat.toPNat' n)) 1)) < \u2191(Nat.toPNat' n)\nhrw :\n \u2191Multiset.card (Multiset.dedup (nthRoots (\u2191(Nat.toPNat' n)) 1)) =\n Fintype.card { x // x \u2208 nthRoots (\u2191(Nat.toPNat' n)) 1 }\n\u22a2 False\n[PROOFSTEP]\nset m := Nat.toPNat' n\n[GOAL]\ncase inr.right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : n > 0\nm : \u2115+ := Nat.toPNat' n\nh : IsPrimitiveRoot \u03b6 \u2191m\nha : \u2191Multiset.card (Multiset.dedup (nthRoots (\u2191m) 1)) < \u2191m\nhrw : \u2191Multiset.card (Multiset.dedup (nthRoots (\u2191m) 1)) = Fintype.card { x // x \u2208 nthRoots (\u2191m) 1 }\n\u22a2 False\n[PROOFSTEP]\nrw [hrw, \u2190 Fintype.card_congr (rootsOfUnityEquivNthRoots R m), card_rootsOfUnity h] at ha \n[GOAL]\ncase inr.right\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : n > 0\nm : \u2115+ := Nat.toPNat' n\nh : IsPrimitiveRoot \u03b6 \u2191m\nha : \u2191m < \u2191m\nhrw : \u2191Multiset.card (Multiset.dedup (nthRoots (\u2191m) 1)) = Fintype.card { x // x \u2208 nthRoots (\u2191m) 1 }\n\u22a2 False\n[PROOFSTEP]\nexact Nat.lt_asymm ha ha\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 card (nthRootsFinset n R) = n\n[PROOFSTEP]\nrw [nthRootsFinset, \u2190 Multiset.toFinset_eq (nthRoots_nodup h), card_mk, h.card_nthRoots]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\n\u22a2 card (primitiveRoots k R) = \u03c6 k\n[PROOFSTEP]\nby_cases h0 : k = 0\n[GOAL]\ncase pos\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\nh0 : k = 0\n\u22a2 card (primitiveRoots k R) = \u03c6 k\n[PROOFSTEP]\nsimp [h0]\n[GOAL]\ncase neg\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\nh0 : \u00ack = 0\n\u22a2 card (primitiveRoots k R) = \u03c6 k\n[PROOFSTEP]\nsymm\n[GOAL]\ncase neg\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\nh0 : \u00ack = 0\n\u22a2 \u03c6 k = card (primitiveRoots k R)\n[PROOFSTEP]\nrefine' Finset.card_congr (fun i _ => \u03b6 ^ i) _ _ _\n[GOAL]\ncase neg.refine'_1\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\nh0 : \u00ack = 0\n\u22a2 \u2200 (a : \u2115) (ha : a \u2208 filter (Nat.coprime k) (range k)), (fun i x => \u03b6 ^ i) a ha \u2208 primitiveRoots k R\n[PROOFSTEP]\nsimp only [true_and_iff, and_imp, mem_filter, mem_range, mem_univ]\n[GOAL]\ncase neg.refine'_1\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\nh0 : \u00ack = 0\n\u22a2 \u2200 (a : \u2115), a < k \u2192 Nat.coprime k a \u2192 \u03b6 ^ a \u2208 primitiveRoots k R\n[PROOFSTEP]\nrintro i - hi\n[GOAL]\ncase neg.refine'_1\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\nh0 : \u00ack = 0\ni : \u2115\nhi : Nat.coprime k i\n\u22a2 \u03b6 ^ i \u2208 primitiveRoots k R\n[PROOFSTEP]\nrw [mem_primitiveRoots (Nat.pos_of_ne_zero h0)]\n[GOAL]\ncase neg.refine'_1\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\nh0 : \u00ack = 0\ni : \u2115\nhi : Nat.coprime k i\n\u22a2 IsPrimitiveRoot (\u03b6 ^ i) k\n[PROOFSTEP]\nexact h.pow_of_coprime i hi.symm\n[GOAL]\ncase neg.refine'_2\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\nh0 : \u00ack = 0\n\u22a2 \u2200 (a b : \u2115) (ha : a \u2208 filter (Nat.coprime k) (range k)) (hb : b \u2208 filter (Nat.coprime k) (range k)),\n (fun i x => \u03b6 ^ i) a ha = (fun i x => \u03b6 ^ i) b hb \u2192 a = b\n[PROOFSTEP]\nsimp only [true_and_iff, and_imp, mem_filter, mem_range, mem_univ]\n[GOAL]\ncase neg.refine'_2\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\nh0 : \u00ack = 0\n\u22a2 \u2200 (a b : \u2115), a < k \u2192 Nat.coprime k a \u2192 b < k \u2192 Nat.coprime k b \u2192 \u03b6 ^ a = \u03b6 ^ b \u2192 a = b\n[PROOFSTEP]\nrintro i j hi - hj - H\n[GOAL]\ncase neg.refine'_2\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\nh0 : \u00ack = 0\ni j : \u2115\nhi : i < k\nhj : j < k\nH : \u03b6 ^ i = \u03b6 ^ j\n\u22a2 i = j\n[PROOFSTEP]\nexact h.pow_inj hi hj H\n[GOAL]\ncase neg.refine'_3\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\nh0 : \u00ack = 0\n\u22a2 \u2200 (b : R), b \u2208 primitiveRoots k R \u2192 \u2203 a ha, (fun i x => \u03b6 ^ i) a ha = b\n[PROOFSTEP]\nsimp only [exists_prop, true_and_iff, mem_filter, mem_range, mem_univ]\n[GOAL]\ncase neg.refine'_3\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\nh0 : \u00ack = 0\n\u22a2 \u2200 (b : R), b \u2208 primitiveRoots k R \u2192 \u2203 a, (a < k \u2227 Nat.coprime k a) \u2227 \u03b6 ^ a = b\n[PROOFSTEP]\nintro \u03be h\u03be\n[GOAL]\ncase neg.refine'_3\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\nh0 : \u00ack = 0\n\u03be : R\nh\u03be : \u03be \u2208 primitiveRoots k R\n\u22a2 \u2203 a, (a < k \u2227 Nat.coprime k a) \u2227 \u03b6 ^ a = \u03be\n[PROOFSTEP]\nrw [mem_primitiveRoots (Nat.pos_of_ne_zero h0), h.isPrimitiveRoot_iff (Nat.pos_of_ne_zero h0)] at h\u03be \n[GOAL]\ncase neg.refine'_3\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\nh0 : \u00ack = 0\n\u03be : R\nh\u03be : \u2203 i, i < k \u2227 Nat.coprime i k \u2227 \u03b6 ^ i = \u03be\n\u22a2 \u2203 a, (a < k \u2227 Nat.coprime k a) \u2227 \u03b6 ^ a = \u03be\n[PROOFSTEP]\nrcases h\u03be with \u27e8i, hin, hi, H\u27e9\n[GOAL]\ncase neg.refine'_3.intro.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nk : \u2115\nh : IsPrimitiveRoot \u03b6 k\nh0 : \u00ack = 0\n\u03be : R\ni : \u2115\nhin : i < k\nhi : Nat.coprime i k\nH : \u03b6 ^ i = \u03be\n\u22a2 \u2203 a, (a < k \u2227 Nat.coprime k a) \u2227 \u03b6 ^ a = \u03be\n[PROOFSTEP]\nexact \u27e8i, \u27e8hin, hi.symm\u27e9, H\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 nthRootsFinset (\u2191n) R = Finset.biUnion (Nat.divisors \u2191n) fun i => primitiveRoots i R\n[PROOFSTEP]\nsymm\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 (Finset.biUnion (Nat.divisors \u2191n) fun i => primitiveRoots i R) = nthRootsFinset (\u2191n) R\n[PROOFSTEP]\napply Finset.eq_of_subset_of_card_le\n[GOAL]\ncase h\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 (Finset.biUnion (Nat.divisors \u2191n) fun i => primitiveRoots i R) \u2286 nthRootsFinset (\u2191n) R\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx : R\n\u22a2 (x \u2208 Finset.biUnion (Nat.divisors \u2191n) fun i => primitiveRoots i R) \u2192 x \u2208 nthRootsFinset (\u2191n) R\n[PROOFSTEP]\nsimp only [nthRootsFinset, \u2190 Multiset.toFinset_eq (nthRoots_nodup h), exists_prop, Finset.mem_biUnion,\n Finset.mem_filter, Finset.mem_range, mem_nthRoots, Finset.mem_mk, Nat.mem_divisors, and_true_iff, Ne.def,\n PNat.ne_zero, PNat.pos, not_false_iff]\n[GOAL]\ncase h\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx : R\n\u22a2 (\u2203 a, a \u2223 \u2191n \u2227 x \u2208 primitiveRoots a R) \u2192 x ^ \u2191n = 1\n[PROOFSTEP]\nrintro \u27e8a, \u27e8d, hd\u27e9, ha\u27e9\n[GOAL]\ncase h.intro.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx : R\na : \u2115\nha : x \u2208 primitiveRoots a R\nd : \u2115\nhd : \u2191n = a * d\n\u22a2 x ^ \u2191n = 1\n[PROOFSTEP]\nhave hazero : 0 < a := by\n contrapose! hd with ha0\n simp_all only [nonpos_iff_eq_zero, zero_mul]\n exact n.ne_zero\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx : R\na : \u2115\nha : x \u2208 primitiveRoots a R\nd : \u2115\nhd : \u2191n = a * d\n\u22a2 0 < a\n[PROOFSTEP]\ncontrapose! hd with ha0\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx : R\na : \u2115\nha : x \u2208 primitiveRoots a R\nd : \u2115\nha0 : a \u2264 0\n\u22a2 \u2191n \u2260 a * d\n[PROOFSTEP]\nsimp_all only [nonpos_iff_eq_zero, zero_mul]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx : R\na d : \u2115\nha : x \u2208 primitiveRoots 0 R\nha0 : a = 0\n\u22a2 \u2191n \u2260 0\n[PROOFSTEP]\nexact n.ne_zero\n[GOAL]\ncase h.intro.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx : R\na : \u2115\nha : x \u2208 primitiveRoots a R\nd : \u2115\nhd : \u2191n = a * d\nhazero : 0 < a\n\u22a2 x ^ \u2191n = 1\n[PROOFSTEP]\nrw [mem_primitiveRoots hazero] at ha \n[GOAL]\ncase h.intro.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nx : R\na : \u2115\nha : IsPrimitiveRoot x a\nd : \u2115\nhd : \u2191n = a * d\nhazero : 0 < a\n\u22a2 x ^ \u2191n = 1\n[PROOFSTEP]\nrw [hd, pow_mul, ha.pow_eq_one, one_pow]\n[GOAL]\ncase h\u2082\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 card (nthRootsFinset (\u2191n) R) \u2264 card (Finset.biUnion (Nat.divisors \u2191n) fun i => primitiveRoots i R)\n[PROOFSTEP]\napply le_of_eq\n[GOAL]\ncase h\u2082.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 card (nthRootsFinset (\u2191n) R) = card (Finset.biUnion (Nat.divisors \u2191n) fun i => primitiveRoots i R)\n[PROOFSTEP]\nrw [h.card_nthRootsFinset, Finset.card_biUnion]\n[GOAL]\ncase h\u2082.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 \u2191n = \u2211 u in Nat.divisors \u2191n, card (primitiveRoots u R)\n[PROOFSTEP]\nnth_rw 1 [\u2190 Nat.sum_totient n]\n[GOAL]\ncase h\u2082.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 Finset.sum (Nat.divisors \u2191n) \u03c6 = \u2211 u in Nat.divisors \u2191n, card (primitiveRoots u R)\n[PROOFSTEP]\nrefine' sum_congr rfl _\n[GOAL]\ncase h\u2082.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 \u2200 (x : \u2115), x \u2208 Nat.divisors \u2191n \u2192 \u03c6 x = card (primitiveRoots x R)\n[PROOFSTEP]\nsimp only [Nat.mem_divisors]\n[GOAL]\ncase h\u2082.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 \u2200 (x : \u2115), x \u2223 \u2191n \u2227 \u2191n \u2260 0 \u2192 \u03c6 x = card (primitiveRoots x R)\n[PROOFSTEP]\nrintro k \u27e8\u27e8d, hd\u27e9, -\u27e9\n[GOAL]\ncase h\u2082.a.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nk d : \u2115\nhd : \u2191n = k * d\n\u22a2 \u03c6 k = card (primitiveRoots k R)\n[PROOFSTEP]\nrw [mul_comm] at hd \n[GOAL]\ncase h\u2082.a.intro.intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk\u271d l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\u271d\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nk d : \u2115\nhd : \u2191n = d * k\n\u22a2 \u03c6 k = card (primitiveRoots k R)\n[PROOFSTEP]\nrw [(h.pow n.pos hd).card_primitiveRoots]\n[GOAL]\ncase h\u2082.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 \u2200 (x : \u2115),\n x \u2208 Nat.divisors \u2191n \u2192 \u2200 (y : \u2115), y \u2208 Nat.divisors \u2191n \u2192 x \u2260 y \u2192 Disjoint (primitiveRoots x R) (primitiveRoots y R)\n[PROOFSTEP]\nintro i _ j _ hdiff\n[GOAL]\ncase h\u2082.a\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\ni : \u2115\na\u271d\u00b9 : i \u2208 Nat.divisors \u2191n\nj : \u2115\na\u271d : j \u2208 Nat.divisors \u2191n\nhdiff : i \u2260 j\n\u22a2 Disjoint (primitiveRoots i R) (primitiveRoots j R)\n[PROOFSTEP]\nexact disjoint hdiff\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 nthRootsFinset n R = Finset.biUnion (Nat.divisors n) fun i => primitiveRoots i R\n[PROOFSTEP]\nby_cases hn : n = 0\n[GOAL]\ncase pos\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhn : n = 0\n\u22a2 nthRootsFinset n R = Finset.biUnion (Nat.divisors n) fun i => primitiveRoots i R\n[PROOFSTEP]\nsimp [hn]\n[GOAL]\ncase neg\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nh : IsPrimitiveRoot \u03b6 n\nhn : \u00acn = 0\n\u22a2 nthRootsFinset n R = Finset.biUnion (Nat.divisors n) fun i => primitiveRoots i R\n[PROOFSTEP]\nexact nthRoots_one_eq_biUnion_primitiveRoots' (n := \u27e8n, Nat.pos_of_ne_zero hn\u27e9) h\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\n\u22a2 orderOf \u03bc' = \u2191n\n[PROOFSTEP]\nrw [h\u03bc.eq_orderOf, \u2190 h\u03bc.toRootsOfUnity_coe_val, orderOf_units, orderOf_subgroup]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf \u03bc' = \u2191n\n\u22a2 (fun \u03c3 => \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191\u03c3 \u2191\u2191\u03bc' = \u2191\u2191\u03bc' ^ m))) 1 = 1\n[PROOFSTEP]\ndsimp only\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf \u03bc' = \u2191n\n\u22a2 \u2191(Exists.choose (_ : \u2203 m, \u2191\u21911 \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m)) = 1\n[PROOFSTEP]\ngeneralize_proofs h1\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf \u03bc' = \u2191n\nh1 : \u2203 m, \u2191\u21911 \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\n\u22a2 \u2191(Exists.choose h1) = 1\n[PROOFSTEP]\nhave h := h1.choose_spec\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf \u03bc' = \u2191n\nh1 : \u2203 m, \u2191\u21911 \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nh : \u2191\u21911 \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose h1\n\u22a2 \u2191(Exists.choose h1) = 1\n[PROOFSTEP]\ndsimp only [AlgEquiv.one_apply, AlgEquiv.toRingEquiv_eq_coe, RingEquiv.toRingHom_eq_coe, RingEquiv.coe_toRingHom,\n AlgEquiv.coe_ringEquiv] at *\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf (toRootsOfUnity h\u03bc) = \u2191n\nh1 : \u2203 m, \u2191\u21911 \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nh : \u2191\u21911 \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose h1\n\u22a2 \u2191(Exists.choose h1) = 1\n[PROOFSTEP]\nreplace h : \u03bc' = \u03bc' ^ h1.choose := rootsOfUnity.coe_injective (by simpa only [rootsOfUnity.coe_pow] using h)\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf (toRootsOfUnity h\u03bc) = \u2191n\nh1 : \u2203 m, \u2191\u21911 \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nh : \u2191\u21911 \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose h1\n\u22a2 (fun x => \u2191\u2191x) \u03bc' = (fun x => \u2191\u2191x) (\u03bc' ^ Exists.choose h1)\n[PROOFSTEP]\nsimpa only [rootsOfUnity.coe_pow] using h\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf (toRootsOfUnity h\u03bc) = \u2191n\nh1 : \u2203 m, \u2191\u21911 \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nh : \u03bc' = \u03bc' ^ Exists.choose h1\n\u22a2 \u2191(Exists.choose h1) = 1\n[PROOFSTEP]\nnth_rw 1 [\u2190 pow_one \u03bc'] at h \n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf (toRootsOfUnity h\u03bc) = \u2191n\nh1 : \u2203 m, \u2191\u21911 \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nh : \u03bc' ^ 1 = \u03bc' ^ Exists.choose h1\n\u22a2 \u2191(Exists.choose h1) = 1\n[PROOFSTEP]\nrw [\u2190 Nat.cast_one, ZMod.nat_cast_eq_nat_cast_iff, \u2190 ho, \u2190 pow_eq_pow_iff_modEq \u03bc', h]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf \u03bc' = \u2191n\n\u22a2 \u2200 (x y : S \u2243\u2090[R] S),\n OneHom.toFun\n { toFun := fun \u03c3 => \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191\u03c3 \u2191\u2191\u03bc' = \u2191\u2191\u03bc' ^ m)),\n map_one' := (_ : (fun \u03c3 => \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191\u03c3 \u2191\u2191\u03bc' = \u2191\u2191\u03bc' ^ m))) 1 = 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun \u03c3 => \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191\u03c3 \u2191\u2191\u03bc' = \u2191\u2191\u03bc' ^ m)),\n map_one' := (_ : (fun \u03c3 => \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191\u03c3 \u2191\u2191\u03bc' = \u2191\u2191\u03bc' ^ m))) 1 = 1) }\n x *\n OneHom.toFun\n { toFun := fun \u03c3 => \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191\u03c3 \u2191\u2191\u03bc' = \u2191\u2191\u03bc' ^ m)),\n map_one' := (_ : (fun \u03c3 => \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191\u03c3 \u2191\u2191\u03bc' = \u2191\u2191\u03bc' ^ m))) 1 = 1) }\n y\n[PROOFSTEP]\nintro x y\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf \u03bc' = \u2191n\nx y : S \u2243\u2090[R] S\n\u22a2 OneHom.toFun\n { toFun := fun \u03c3 => \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191\u03c3 \u2191\u2191\u03bc' = \u2191\u2191\u03bc' ^ m)),\n map_one' := (_ : (fun \u03c3 => \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191\u03c3 \u2191\u2191\u03bc' = \u2191\u2191\u03bc' ^ m))) 1 = 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun \u03c3 => \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191\u03c3 \u2191\u2191\u03bc' = \u2191\u2191\u03bc' ^ m)),\n map_one' := (_ : (fun \u03c3 => \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191\u03c3 \u2191\u2191\u03bc' = \u2191\u2191\u03bc' ^ m))) 1 = 1) }\n x *\n OneHom.toFun\n { toFun := fun \u03c3 => \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191\u03c3 \u2191\u2191\u03bc' = \u2191\u2191\u03bc' ^ m)),\n map_one' := (_ : (fun \u03c3 => \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191\u03c3 \u2191\u2191\u03bc' = \u2191\u2191\u03bc' ^ m))) 1 = 1) }\n y\n[PROOFSTEP]\ndsimp only\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf \u03bc' = \u2191n\nx y : S \u2243\u2090[R] S\n\u22a2 \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m)) =\n \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m)) *\n \u2191(Exists.choose (_ : \u2203 m, \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m))\n[PROOFSTEP]\ngeneralize_proofs hxy' hx' hy'\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf \u03bc' = \u2191n\nx y : S \u2243\u2090[R] S\nhxy' : \u2203 m, \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx' : \u2203 m, \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhy' : \u2203 m, \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\n\u22a2 \u2191(Exists.choose hxy') = \u2191(Exists.choose hx') * \u2191(Exists.choose hy')\n[PROOFSTEP]\nhave hxy := hxy'.choose_spec\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf \u03bc' = \u2191n\nx y : S \u2243\u2090[R] S\nhxy' : \u2203 m, \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx' : \u2203 m, \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhy' : \u2203 m, \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhxy : \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hxy'\n\u22a2 \u2191(Exists.choose hxy') = \u2191(Exists.choose hx') * \u2191(Exists.choose hy')\n[PROOFSTEP]\nhave hx := hx'.choose_spec\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf \u03bc' = \u2191n\nx y : S \u2243\u2090[R] S\nhxy' : \u2203 m, \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx' : \u2203 m, \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhy' : \u2203 m, \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhxy : \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hxy'\nhx : \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hx'\n\u22a2 \u2191(Exists.choose hxy') = \u2191(Exists.choose hx') * \u2191(Exists.choose hy')\n[PROOFSTEP]\nhave hy := hy'.choose_spec\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf \u03bc' = \u2191n\nx y : S \u2243\u2090[R] S\nhxy' : \u2203 m, \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx' : \u2203 m, \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhy' : \u2203 m, \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhxy : \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hxy'\nhx : \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hx'\nhy : \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hy'\n\u22a2 \u2191(Exists.choose hxy') = \u2191(Exists.choose hx') * \u2191(Exists.choose hy')\n[PROOFSTEP]\ndsimp only [AlgEquiv.toRingEquiv_eq_coe, RingEquiv.toRingHom_eq_coe, RingEquiv.coe_toRingHom, AlgEquiv.coe_ringEquiv,\n AlgEquiv.mul_apply] at *\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf (toRootsOfUnity h\u03bc) = \u2191n\nx y : S \u2243\u2090[R] S\nhxy' : \u2203 m, \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx' : \u2203 m, \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhy' : \u2203 m, \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhxy : \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hxy'\nhx : \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hx'\nhy : \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hy'\n\u22a2 \u2191(Exists.choose hxy') = \u2191(Exists.choose hx') * \u2191(Exists.choose hy')\n[PROOFSTEP]\nreplace hxy : x (((\u03bc' : S\u02e3) : S) ^ hy'.choose) = ((\u03bc' : S\u02e3) : S) ^ hxy'.choose := hy \u25b8 hxy\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf (toRootsOfUnity h\u03bc) = \u2191n\nx y : S \u2243\u2090[R] S\nhxy' : \u2203 m, \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx' : \u2203 m, \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhy' : \u2203 m, \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx : \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hx'\nhy : \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hy'\nhxy : \u2191x (\u2191\u2191\u03bc' ^ Exists.choose hy') = \u2191\u2191\u03bc' ^ Exists.choose hxy'\n\u22a2 \u2191(Exists.choose hxy') = \u2191(Exists.choose hx') * \u2191(Exists.choose hy')\n[PROOFSTEP]\nrw [x.map_pow] at hxy \n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf (toRootsOfUnity h\u03bc) = \u2191n\nx y : S \u2243\u2090[R] S\nhxy' : \u2203 m, \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx' : \u2203 m, \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhy' : \u2203 m, \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx : \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hx'\nhy : \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hy'\nhxy : \u2191x \u2191\u2191\u03bc' ^ Exists.choose hy' = \u2191\u2191\u03bc' ^ Exists.choose hxy'\n\u22a2 \u2191(Exists.choose hxy') = \u2191(Exists.choose hx') * \u2191(Exists.choose hy')\n[PROOFSTEP]\nreplace hxy : (((\u03bc' : S\u02e3) : S) ^ hx'.choose) ^ hy'.choose = ((\u03bc' : S\u02e3) : S) ^ hxy'.choose := hx \u25b8 hxy\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf (toRootsOfUnity h\u03bc) = \u2191n\nx y : S \u2243\u2090[R] S\nhxy' : \u2203 m, \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx' : \u2203 m, \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhy' : \u2203 m, \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx : \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hx'\nhy : \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hy'\nhxy : (\u2191\u2191\u03bc' ^ Exists.choose hx') ^ Exists.choose hy' = \u2191\u2191\u03bc' ^ Exists.choose hxy'\n\u22a2 \u2191(Exists.choose hxy') = \u2191(Exists.choose hx') * \u2191(Exists.choose hy')\n[PROOFSTEP]\nrw [\u2190 pow_mul] at hxy \n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf (toRootsOfUnity h\u03bc) = \u2191n\nx y : S \u2243\u2090[R] S\nhxy' : \u2203 m, \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx' : \u2203 m, \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhy' : \u2203 m, \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx : \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hx'\nhy : \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hy'\nhxy : \u2191\u2191\u03bc' ^ (Exists.choose hx' * Exists.choose hy') = \u2191\u2191\u03bc' ^ Exists.choose hxy'\n\u22a2 \u2191(Exists.choose hxy') = \u2191(Exists.choose hx') * \u2191(Exists.choose hy')\n[PROOFSTEP]\nreplace hxy : \u03bc' ^ (hx'.choose * hy'.choose) = \u03bc' ^ hxy'.choose :=\n rootsOfUnity.coe_injective (by simpa only [rootsOfUnity.coe_pow] using hxy)\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf (toRootsOfUnity h\u03bc) = \u2191n\nx y : S \u2243\u2090[R] S\nhxy' : \u2203 m, \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx' : \u2203 m, \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhy' : \u2203 m, \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx : \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hx'\nhy : \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hy'\nhxy : \u2191\u2191\u03bc' ^ (Exists.choose hx' * Exists.choose hy') = \u2191\u2191\u03bc' ^ Exists.choose hxy'\n\u22a2 (fun x => \u2191\u2191x) (\u03bc' ^ (Exists.choose hx' * Exists.choose hy')) = (fun x => \u2191\u2191x) (\u03bc' ^ Exists.choose hxy')\n[PROOFSTEP]\nsimpa only [rootsOfUnity.coe_pow] using hxy\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\n\u03bc' : { x // x \u2208 rootsOfUnity n S } := toRootsOfUnity h\u03bc\nho : orderOf (toRootsOfUnity h\u03bc) = \u2191n\nx y : S \u2243\u2090[R] S\nhxy' : \u2203 m, \u2191\u2191(x * y) \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx' : \u2203 m, \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhy' : \u2203 m, \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhx : \u2191\u2191x \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hx'\nhy : \u2191\u2191y \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hy'\nhxy : \u03bc' ^ (Exists.choose hx' * Exists.choose hy') = \u03bc' ^ Exists.choose hxy'\n\u22a2 \u2191(Exists.choose hxy') = \u2191(Exists.choose hx') * \u2191(Exists.choose hy')\n[PROOFSTEP]\nrw [\u2190 Nat.cast_mul, ZMod.nat_cast_eq_nat_cast_iff, \u2190 ho, \u2190 pow_eq_pow_iff_modEq \u03bc', hxy]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\nf : S \u2243\u2090[R] S\n\u22a2 \u03bc ^ ZMod.val \u2191(\u2191(autToPow R h\u03bc) f) = \u2191f \u03bc\n[PROOFSTEP]\nrw [IsPrimitiveRoot.coe_autToPow_apply]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\nf : S \u2243\u2090[R] S\n\u22a2 \u03bc ^ ZMod.val \u2191(Exists.choose (_ : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m)) = \u2191f \u03bc\n[PROOFSTEP]\ngeneralize_proofs h\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\nf : S \u2243\u2090[R] S\nh : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\n\u22a2 \u03bc ^ ZMod.val \u2191(Exists.choose h) = \u2191f \u03bc\n[PROOFSTEP]\nhave := h.choose_spec\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\nf : S \u2243\u2090[R] S\nh : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nthis : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose h\n\u22a2 \u03bc ^ ZMod.val \u2191(Exists.choose h) = \u2191f \u03bc\n[PROOFSTEP]\ndsimp only [AlgEquiv.toAlgHom_eq_coe, AlgEquiv.coe_algHom] at this \n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\nf : S \u2243\u2090[R] S\nh : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nthis : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose h\n\u22a2 \u03bc ^ ZMod.val \u2191(Exists.choose h) = \u2191f \u03bc\n[PROOFSTEP]\nrefine' (_ : ((h\u03bc.toRootsOfUnity : S\u02e3) : S) ^ _ = _).trans this.symm\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\nf : S \u2243\u2090[R] S\nh : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nthis : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose h\n\u22a2 \u2191\u2191(toRootsOfUnity h\u03bc) ^ ZMod.val \u2191(Exists.choose h) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose h\n[PROOFSTEP]\nrw [\u2190 rootsOfUnity.coe_pow, \u2190 rootsOfUnity.coe_pow]\n[GOAL]\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\nf : S \u2243\u2090[R] S\nh : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nthis : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose h\n\u22a2 \u2191\u2191(toRootsOfUnity h\u03bc ^ ZMod.val \u2191(Exists.choose h)) = \u2191\u2191(toRootsOfUnity h\u03bc ^ Exists.choose h)\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_self.e_self\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\nf : S \u2243\u2090[R] S\nh : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nthis : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose h\n\u22a2 toRootsOfUnity h\u03bc ^ ZMod.val \u2191(Exists.choose h) = toRootsOfUnity h\u03bc ^ Exists.choose h\n[PROOFSTEP]\nrw [pow_eq_pow_iff_modEq, ZMod.val_nat_cast, h\u03bc.eq_orderOf, \u2190 orderOf_subgroup, \u2190 orderOf_units]\n[GOAL]\ncase e_self.e_self\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2076 : CommMonoid M\ninst\u271d\u2075 : CommMonoid N\ninst\u271d\u2074 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\n\u03bc : S\nn : \u2115+\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R S\nf : S \u2243\u2090[R] S\nh : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nthis : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose h\n\u22a2 Exists.choose h % orderOf \u03bc \u2261 Exists.choose h [MOD orderOf \u2191\u2191(toRootsOfUnity h\u03bc)]\n[PROOFSTEP]\nexact Nat.mod_modEq _ _\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.RootsOfUnity.Basic", "llama_tokens": 119656, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195152660687, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.4070475889661006}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CommSemigroup \u03b2\nf : \u03b1 \u2192 \u03b2\nn m : \u03b1\n\u22a2 f (min n m) * f (max n m) = f n * f m\n[PROOFSTEP]\ncases' le_total n m with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CommSemigroup \u03b2\nf : \u03b1 \u2192 \u03b2\nn m : \u03b1\nh : n \u2264 m\n\u22a2 f (min n m) * f (max n m) = f n * f m\n[PROOFSTEP]\nsimp [h, mul_comm]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CommSemigroup \u03b2\nf : \u03b1 \u2192 \u03b2\nn m : \u03b1\nh : m \u2264 n\n\u22a2 f (min n m) * f (max n m) = f n * f m\n[PROOFSTEP]\nsimp [h, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\n\u22a2 a\u2081 * b\u2081 < a\u2082 * b\u2082 \u2192 a\u2081 < a\u2082 \u2228 b\u2081 < b\u2082\n[PROOFSTEP]\ncontrapose!\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\n\u22a2 a\u2082 \u2264 a\u2081 \u2227 b\u2082 \u2264 b\u2081 \u2192 a\u2082 * b\u2082 \u2264 a\u2081 * b\u2081\n[PROOFSTEP]\nexact fun h => mul_le_mul' h.1 h.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\n\u22a2 a\u2081 * b\u2081 \u2264 a\u2082 * b\u2082 \u2192 a\u2081 \u2264 a\u2082 \u2228 b\u2081 < b\u2082\n[PROOFSTEP]\ncontrapose!\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\n\u22a2 a\u2082 < a\u2081 \u2227 b\u2082 \u2264 b\u2081 \u2192 a\u2082 * b\u2082 < a\u2081 * b\u2081\n[PROOFSTEP]\nexact fun h => mul_lt_mul_of_lt_of_le h.1 h.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\n\u22a2 a\u2081 * b\u2081 \u2264 a\u2082 * b\u2082 \u2192 a\u2081 < a\u2082 \u2228 b\u2081 \u2264 b\u2082\n[PROOFSTEP]\ncontrapose!\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\n\u22a2 a\u2082 \u2264 a\u2081 \u2227 b\u2082 < b\u2081 \u2192 a\u2082 * b\u2082 < a\u2081 * b\u2081\n[PROOFSTEP]\nexact fun h => mul_lt_mul_of_le_of_lt h.1 h.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\n\u22a2 a\u2081 * b\u2081 \u2264 a\u2082 * b\u2082 \u2192 a\u2081 \u2264 a\u2082 \u2228 b\u2081 \u2264 b\u2082\n[PROOFSTEP]\ncontrapose!\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\n\u22a2 a\u2082 < a\u2081 \u2227 b\u2082 < b\u2081 \u2192 a\u2082 * b\u2082 < a\u2081 * b\u2081\n[PROOFSTEP]\nexact fun h => mul_lt_mul_of_lt_of_lt h.1 h.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : Mul \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\nha : a\u2081 \u2264 a\u2082\nhb : b\u2081 \u2264 b\u2082\n\u22a2 a\u2081 * b\u2081 < a\u2082 * b\u2082 \u2194 a\u2081 < a\u2082 \u2228 b\u2081 < b\u2082\n[PROOFSTEP]\nrefine' \u27e8lt_or_lt_of_mul_lt_mul, fun h => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : Mul \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\nha : a\u2081 \u2264 a\u2082\nhb : b\u2081 \u2264 b\u2082\nh : a\u2081 < a\u2082 \u2228 b\u2081 < b\u2082\n\u22a2 a\u2081 * b\u2081 < a\u2082 * b\u2082\n[PROOFSTEP]\ncases' h with ha' hb'\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : Mul \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\nha : a\u2081 \u2264 a\u2082\nhb : b\u2081 \u2264 b\u2082\nha' : a\u2081 < a\u2082\n\u22a2 a\u2081 * b\u2081 < a\u2082 * b\u2082\n[PROOFSTEP]\nexact mul_lt_mul_of_lt_of_le ha' hb\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : Mul \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\nha : a\u2081 \u2264 a\u2082\nhb : b\u2081 \u2264 b\u2082\nhb' : b\u2081 < b\u2082\n\u22a2 a\u2081 * b\u2081 < a\u2082 * b\u2082\n[PROOFSTEP]\nexact mul_lt_mul_of_le_of_lt ha hb'\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Monoid.MinMax", "llama_tokens": 2762, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149758396752, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.4068865050234334}} {"text": "[GOAL]\n\u03b1 \u03b2 : BoolAlgCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\n\u22a2 ((let src :=\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b1),\n SupHom.toFun\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n a \u2294\n SupHom.toFun\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191e \u22a4 = \u22a4), map_bot' := (_ : \u2191e \u22a5 = \u22a5) }) \u226b\n let src :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191(OrderIso.symm e) \u22a4 = \u22a4), map_bot' := (_ : \u2191(OrderIso.symm e) \u22a5 = \u22a5) }) =\n \ud835\udfd9 \u03b1\n[PROOFSTEP]\next\n[GOAL]\ncase w\n\u03b1 \u03b2 : BoolAlgCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\nx\u271d : (forget BoolAlgCat).obj \u03b1\n\u22a2 \u2191((let src :=\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b1),\n SupHom.toFun\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n a \u2294\n SupHom.toFun\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191e \u22a4 = \u22a4), map_bot' := (_ : \u2191e \u22a5 = \u22a5) }) \u226b\n let src :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2294 b) =\n \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2294 b) =\n \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191(OrderIso.symm e) \u22a4 = \u22a4), map_bot' := (_ : \u2191(OrderIso.symm e) \u22a5 = \u22a5) })\n x\u271d =\n \u2191(\ud835\udfd9 \u03b1) x\u271d\n[PROOFSTEP]\nexact e.symm_apply_apply _\n[GOAL]\n\u03b1 \u03b2 : BoolAlgCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\n\u22a2 ((let src :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191(OrderIso.symm e) \u22a4 = \u22a4), map_bot' := (_ : \u2191(OrderIso.symm e) \u22a5 = \u22a5) }) \u226b\n let src :=\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b1),\n SupHom.toFun\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n a \u2294\n SupHom.toFun\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191e \u22a4 = \u22a4), map_bot' := (_ : \u2191e \u22a5 = \u22a5) }) =\n \ud835\udfd9 \u03b2\n[PROOFSTEP]\next\n[GOAL]\ncase w\n\u03b1 \u03b2 : BoolAlgCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\nx\u271d : (forget BoolAlgCat).obj \u03b2\n\u22a2 \u2191((let src :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2294 b) =\n \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2294 b) =\n \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n \u2191(OrderIso.symm e) (a \u2293 b) =\n \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b2),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191(OrderIso.symm e) \u22a4 = \u22a4), map_bot' := (_ : \u2191(OrderIso.symm e) \u22a5 = \u22a5) }) \u226b\n let src :=\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) };\n {\n toLatticeHom :=\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b1),\n SupHom.toFun\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n a \u2294\n SupHom.toFun\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1),\n SupHom.toFun src.toSupHom (a \u2293 b) = SupHom.toFun src.toSupHom a \u2293 SupHom.toFun src.toSupHom b) },\n map_top' := (_ : \u2191e \u22a4 = \u22a4), map_bot' := (_ : \u2191e \u22a5 = \u22a5) })\n x\u271d =\n \u2191(\ud835\udfd9 \u03b2) x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n", "meta": {"mathlib_filename": "Mathlib.Order.Category.BoolAlgCat", "llama_tokens": 6973, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8397339756938818, "lm_q2_score": 0.4843800842769843, "lm_q1q2_score": 0.4067504139168495}} {"text": "[GOAL]\nH : Type u_1\ninst\u271d\u2075 : TopologicalSpace H\nH' : Type u_2\ninst\u271d\u2074 : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\nM : Type u\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\nM' : Type u\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nhG : LocalInvariantProp G G' P\n\u22a2 \u2200 {U V : Opens \u2191(TopCat.of M)} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 M'),\n (fun {U} f => ChartedSpace.LiftProp P f) f \u2192\n (fun {U} f => ChartedSpace.LiftProp P f) fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)\n[PROOFSTEP]\nintro U V i f h x\n[GOAL]\nH : Type u_1\ninst\u271d\u2075 : TopologicalSpace H\nH' : Type u_2\ninst\u271d\u2074 : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\nM : Type u\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\nM' : Type u\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nhG : LocalInvariantProp G G' P\nU V : Opens \u2191(TopCat.of M)\ni : U \u27f6 V\nf : { x // x \u2208 V } \u2192 M'\nh : ChartedSpace.LiftProp P f\nx : { x // x \u2208 U }\n\u22a2 ChartedSpace.LiftPropAt P (fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) x\n[PROOFSTEP]\nhave hUV : U \u2264 V := CategoryTheory.leOfHom i\n[GOAL]\nH : Type u_1\ninst\u271d\u2075 : TopologicalSpace H\nH' : Type u_2\ninst\u271d\u2074 : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\nM : Type u\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\nM' : Type u\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nhG : LocalInvariantProp G G' P\nU V : Opens \u2191(TopCat.of M)\ni : U \u27f6 V\nf : { x // x \u2208 V } \u2192 M'\nh : ChartedSpace.LiftProp P f\nx : { x // x \u2208 U }\nhUV : U \u2264 V\n\u22a2 ChartedSpace.LiftPropAt P (fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) x\n[PROOFSTEP]\nshow ChartedSpace.LiftPropAt P (f \u2218 Set.inclusion hUV) x\n[GOAL]\nH : Type u_1\ninst\u271d\u2075 : TopologicalSpace H\nH' : Type u_2\ninst\u271d\u2074 : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\nM : Type u\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\nM' : Type u\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nhG : LocalInvariantProp G G' P\nU V : Opens \u2191(TopCat.of M)\ni : U \u27f6 V\nf : { x // x \u2208 V } \u2192 M'\nh : ChartedSpace.LiftProp P f\nx : { x // x \u2208 U }\nhUV : U \u2264 V\n\u22a2 ChartedSpace.LiftPropAt P (f \u2218 inclusion hUV) x\n[PROOFSTEP]\nrw [\u2190 hG.liftPropAt_iff_comp_inclusion hUV]\n[GOAL]\nH : Type u_1\ninst\u271d\u2075 : TopologicalSpace H\nH' : Type u_2\ninst\u271d\u2074 : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\nM : Type u\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\nM' : Type u\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nhG : LocalInvariantProp G G' P\nU V : Opens \u2191(TopCat.of M)\ni : U \u27f6 V\nf : { x // x \u2208 V } \u2192 M'\nh : ChartedSpace.LiftProp P f\nx : { x // x \u2208 U }\nhUV : U \u2264 V\n\u22a2 ChartedSpace.LiftPropAt P f (inclusion hUV x)\n[PROOFSTEP]\napply h\n[GOAL]\nH : Type u_1\ninst\u271d\u2075 : TopologicalSpace H\nH' : Type u_2\ninst\u271d\u2074 : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\nM : Type u\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\nM' : Type u\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nhG : LocalInvariantProp G G' P\n\u22a2 \u2200 {U : Opens \u2191(TopCat.of M)} (f : { x // x \u2208 U } \u2192 M'),\n (\u2200 (x : { x // x \u2208 U }),\n \u2203 V x i,\n TopCat.PrelocalPredicate.pred\n { pred := fun {U} f => ChartedSpace.LiftProp P f,\n res :=\n (_ :\n \u2200 {U V : Opens \u2191(TopCat.of M)} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 M'),\n ChartedSpace.LiftProp P f \u2192\n \u2200 (x : { x // x \u2208 U }), ChartedSpace.LiftPropAt P (f \u2218 inclusion (_ : U \u2264 V)) x) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)) \u2192\n TopCat.PrelocalPredicate.pred\n { pred := fun {U} f => ChartedSpace.LiftProp P f,\n res :=\n (_ :\n \u2200 {U V : Opens \u2191(TopCat.of M)} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 M'),\n ChartedSpace.LiftProp P f \u2192\n \u2200 (x : { x // x \u2208 U }), ChartedSpace.LiftPropAt P (f \u2218 inclusion (_ : U \u2264 V)) x) }\n f\n[PROOFSTEP]\nintro V f h x\n[GOAL]\nH : Type u_1\ninst\u271d\u2075 : TopologicalSpace H\nH' : Type u_2\ninst\u271d\u2074 : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\nM : Type u\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\nM' : Type u\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nhG : LocalInvariantProp G G' P\nV : Opens \u2191(TopCat.of M)\nf : { x // x \u2208 V } \u2192 M'\nh :\n \u2200 (x : { x // x \u2208 V }),\n \u2203 V_1 x i,\n TopCat.PrelocalPredicate.pred\n { pred := fun {U} f => ChartedSpace.LiftProp P f,\n res :=\n (_ :\n \u2200 {U V : Opens \u2191(TopCat.of M)} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 M'),\n ChartedSpace.LiftProp P f \u2192\n \u2200 (x : { x // x \u2208 U }), ChartedSpace.LiftPropAt P (f \u2218 inclusion (_ : U \u2264 V)) x) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)\nx : { x // x \u2208 V }\n\u22a2 ChartedSpace.LiftPropAt P f x\n[PROOFSTEP]\nobtain \u27e8U, hxU, i, hU : ChartedSpace.LiftProp P (f \u2218 i)\u27e9 := h x\n[GOAL]\ncase intro.intro.intro\nH : Type u_1\ninst\u271d\u2075 : TopologicalSpace H\nH' : Type u_2\ninst\u271d\u2074 : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\nM : Type u\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\nM' : Type u\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nhG : LocalInvariantProp G G' P\nV : Opens \u2191(TopCat.of M)\nf : { x // x \u2208 V } \u2192 M'\nh :\n \u2200 (x : { x // x \u2208 V }),\n \u2203 V_1 x i,\n TopCat.PrelocalPredicate.pred\n { pred := fun {U} f => ChartedSpace.LiftProp P f,\n res :=\n (_ :\n \u2200 {U V : Opens \u2191(TopCat.of M)} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 M'),\n ChartedSpace.LiftProp P f \u2192\n \u2200 (x : { x // x \u2208 U }), ChartedSpace.LiftPropAt P (f \u2218 inclusion (_ : U \u2264 V)) x) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)\nx : { x // x \u2208 V }\nU : Opens \u2191(TopCat.of M)\nhxU : \u2191x \u2208 U\ni : U \u27f6 V\nhU : ChartedSpace.LiftProp P (f \u2218 fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) })\n\u22a2 ChartedSpace.LiftPropAt P f x\n[PROOFSTEP]\nlet x' : U := \u27e8x, hxU\u27e9\n[GOAL]\ncase intro.intro.intro\nH : Type u_1\ninst\u271d\u2075 : TopologicalSpace H\nH' : Type u_2\ninst\u271d\u2074 : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\nM : Type u\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\nM' : Type u\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nhG : LocalInvariantProp G G' P\nV : Opens \u2191(TopCat.of M)\nf : { x // x \u2208 V } \u2192 M'\nh :\n \u2200 (x : { x // x \u2208 V }),\n \u2203 V_1 x i,\n TopCat.PrelocalPredicate.pred\n { pred := fun {U} f => ChartedSpace.LiftProp P f,\n res :=\n (_ :\n \u2200 {U V : Opens \u2191(TopCat.of M)} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 M'),\n ChartedSpace.LiftProp P f \u2192\n \u2200 (x : { x // x \u2208 U }), ChartedSpace.LiftPropAt P (f \u2218 inclusion (_ : U \u2264 V)) x) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)\nx : { x // x \u2208 V }\nU : Opens \u2191(TopCat.of M)\nhxU : \u2191x \u2208 U\ni : U \u27f6 V\nhU : ChartedSpace.LiftProp P (f \u2218 fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) })\nx' : { x // x \u2208 U } := { val := \u2191x, property := hxU }\n\u22a2 ChartedSpace.LiftPropAt P f x\n[PROOFSTEP]\nhave hUV : U \u2264 V := CategoryTheory.leOfHom i\n[GOAL]\ncase intro.intro.intro\nH : Type u_1\ninst\u271d\u2075 : TopologicalSpace H\nH' : Type u_2\ninst\u271d\u2074 : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\nM : Type u\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\nM' : Type u\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nhG : LocalInvariantProp G G' P\nV : Opens \u2191(TopCat.of M)\nf : { x // x \u2208 V } \u2192 M'\nh :\n \u2200 (x : { x // x \u2208 V }),\n \u2203 V_1 x i,\n TopCat.PrelocalPredicate.pred\n { pred := fun {U} f => ChartedSpace.LiftProp P f,\n res :=\n (_ :\n \u2200 {U V : Opens \u2191(TopCat.of M)} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 M'),\n ChartedSpace.LiftProp P f \u2192\n \u2200 (x : { x // x \u2208 U }), ChartedSpace.LiftPropAt P (f \u2218 inclusion (_ : U \u2264 V)) x) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)\nx : { x // x \u2208 V }\nU : Opens \u2191(TopCat.of M)\nhxU : \u2191x \u2208 U\ni : U \u27f6 V\nhU : ChartedSpace.LiftProp P (f \u2218 fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) })\nx' : { x // x \u2208 U } := { val := \u2191x, property := hxU }\nhUV : U \u2264 V\n\u22a2 ChartedSpace.LiftPropAt P f x\n[PROOFSTEP]\nhave : ChartedSpace.LiftPropAt P f (inclusion hUV x') :=\n by\n rw [hG.liftPropAt_iff_comp_inclusion hUV]\n exact hU x'\n[GOAL]\nH : Type u_1\ninst\u271d\u2075 : TopologicalSpace H\nH' : Type u_2\ninst\u271d\u2074 : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\nM : Type u\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\nM' : Type u\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nhG : LocalInvariantProp G G' P\nV : Opens \u2191(TopCat.of M)\nf : { x // x \u2208 V } \u2192 M'\nh :\n \u2200 (x : { x // x \u2208 V }),\n \u2203 V_1 x i,\n TopCat.PrelocalPredicate.pred\n { pred := fun {U} f => ChartedSpace.LiftProp P f,\n res :=\n (_ :\n \u2200 {U V : Opens \u2191(TopCat.of M)} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 M'),\n ChartedSpace.LiftProp P f \u2192\n \u2200 (x : { x // x \u2208 U }), ChartedSpace.LiftPropAt P (f \u2218 inclusion (_ : U \u2264 V)) x) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)\nx : { x // x \u2208 V }\nU : Opens \u2191(TopCat.of M)\nhxU : \u2191x \u2208 U\ni : U \u27f6 V\nhU : ChartedSpace.LiftProp P (f \u2218 fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) })\nx' : { x // x \u2208 U } := { val := \u2191x, property := hxU }\nhUV : U \u2264 V\n\u22a2 ChartedSpace.LiftPropAt P f (inclusion hUV x')\n[PROOFSTEP]\nrw [hG.liftPropAt_iff_comp_inclusion hUV]\n[GOAL]\nH : Type u_1\ninst\u271d\u2075 : TopologicalSpace H\nH' : Type u_2\ninst\u271d\u2074 : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\nM : Type u\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\nM' : Type u\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nhG : LocalInvariantProp G G' P\nV : Opens \u2191(TopCat.of M)\nf : { x // x \u2208 V } \u2192 M'\nh :\n \u2200 (x : { x // x \u2208 V }),\n \u2203 V_1 x i,\n TopCat.PrelocalPredicate.pred\n { pred := fun {U} f => ChartedSpace.LiftProp P f,\n res :=\n (_ :\n \u2200 {U V : Opens \u2191(TopCat.of M)} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 M'),\n ChartedSpace.LiftProp P f \u2192\n \u2200 (x : { x // x \u2208 U }), ChartedSpace.LiftPropAt P (f \u2218 inclusion (_ : U \u2264 V)) x) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)\nx : { x // x \u2208 V }\nU : Opens \u2191(TopCat.of M)\nhxU : \u2191x \u2208 U\ni : U \u27f6 V\nhU : ChartedSpace.LiftProp P (f \u2218 fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) })\nx' : { x // x \u2208 U } := { val := \u2191x, property := hxU }\nhUV : U \u2264 V\n\u22a2 ChartedSpace.LiftPropAt P (f \u2218 inclusion hUV) x'\n[PROOFSTEP]\nexact hU x'\n[GOAL]\ncase intro.intro.intro\nH : Type u_1\ninst\u271d\u2075 : TopologicalSpace H\nH' : Type u_2\ninst\u271d\u2074 : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\nM : Type u\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\nM' : Type u\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nhG : LocalInvariantProp G G' P\nV : Opens \u2191(TopCat.of M)\nf : { x // x \u2208 V } \u2192 M'\nh :\n \u2200 (x : { x // x \u2208 V }),\n \u2203 V_1 x i,\n TopCat.PrelocalPredicate.pred\n { pred := fun {U} f => ChartedSpace.LiftProp P f,\n res :=\n (_ :\n \u2200 {U V : Opens \u2191(TopCat.of M)} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 M'),\n ChartedSpace.LiftProp P f \u2192\n \u2200 (x : { x // x \u2208 U }), ChartedSpace.LiftPropAt P (f \u2218 inclusion (_ : U \u2264 V)) x) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)\nx : { x // x \u2208 V }\nU : Opens \u2191(TopCat.of M)\nhxU : \u2191x \u2208 U\ni : U \u27f6 V\nhU : ChartedSpace.LiftProp P (f \u2218 fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) })\nx' : { x // x \u2208 U } := { val := \u2191x, property := hxU }\nhUV : U \u2264 V\nthis : ChartedSpace.LiftPropAt P f (inclusion hUV x')\n\u22a2 ChartedSpace.LiftPropAt P f x\n[PROOFSTEP]\nconvert this\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Manifold.Sheaf.Basic", "llama_tokens": 5523, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.8198933315126791, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.4067440225877951}} {"text": "[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\nf : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\n\u03c0 : Prepartition I\n\u03c0i : (J : Box \u03b9) \u2192 TaggedPrepartition J\n\u22a2 integralSum f vol (Prepartition.biUnionTagged \u03c0 \u03c0i) = \u2211 J in \u03c0.boxes, integralSum f vol (\u03c0i J)\n[PROOFSTEP]\nrefine (\u03c0.sum_biUnion_boxes _ _).trans <| sum_congr rfl fun J hJ => sum_congr rfl fun J' hJ' => ?_\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\nf : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\n\u03c0 : Prepartition I\n\u03c0i : (J : Box \u03b9) \u2192 TaggedPrepartition J\nJ : Box \u03b9\nhJ : J \u2208 \u03c0.boxes\nJ' : Box \u03b9\nhJ' : J' \u2208 (\u03c0i J).boxes\n\u22a2 \u2191(\u2191vol J') (f (tag (Prepartition.biUnionTagged \u03c0 \u03c0i) J')) = \u2191(\u2191vol J') (f (tag (\u03c0i J) J'))\n[PROOFSTEP]\nrw [\u03c0.tag_biUnionTagged hJ hJ']\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\nf : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\n\u03c0 : TaggedPrepartition I\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nh\u03c0i : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 Prepartition.IsPartition (\u03c0i J)\n\u22a2 integralSum f vol (biUnionPrepartition \u03c0 \u03c0i) = integralSum f vol \u03c0\n[PROOFSTEP]\nrefine' (\u03c0.sum_biUnion_boxes _ _).trans (sum_congr rfl fun J hJ => _)\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\nf : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\n\u03c0 : TaggedPrepartition I\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nh\u03c0i : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 Prepartition.IsPartition (\u03c0i J)\nJ : Box \u03b9\nhJ : J \u2208 \u03c0.boxes\n\u22a2 \u2211 J' in (\u03c0i J).boxes, \u2191(\u2191vol J') (f (tag (biUnionPrepartition \u03c0 \u03c0i) J')) = \u2191(\u2191vol J) (f (tag \u03c0 J))\n[PROOFSTEP]\ncalc\n (\u2211 J' in (\u03c0i J).boxes, vol J' (f (\u03c0.tag <| \u03c0.toPrepartition.biUnionIndex \u03c0i J'))) =\n \u2211 J' in (\u03c0i J).boxes, vol J' (f (\u03c0.tag J)) :=\n sum_congr rfl fun J' hJ' => by rw [Prepartition.biUnionIndex_of_mem _ hJ hJ']\n _ = vol J (f (\u03c0.tag J)) :=\n (vol.map \u27e8\u27e8fun g : E \u2192L[\u211d] F => g (f (\u03c0.tag J)), rfl\u27e9, fun _ _ => rfl\u27e9).sum_partition_boxes le_top (h\u03c0i J hJ)\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\nf : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\n\u03c0 : TaggedPrepartition I\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nh\u03c0i : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 Prepartition.IsPartition (\u03c0i J)\nJ : Box \u03b9\nhJ : J \u2208 \u03c0.boxes\nJ' : Box \u03b9\nhJ' : J' \u2208 (\u03c0i J).boxes\n\u22a2 \u2191(\u2191vol J') (f (tag \u03c0 (Prepartition.biUnionIndex \u03c0.toPrepartition \u03c0i J'))) = \u2191(\u2191vol J') (f (tag \u03c0 J))\n[PROOFSTEP]\nrw [Prepartition.biUnionIndex_of_mem _ hJ hJ']\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\nf : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : IsPartition \u03c0\u2081\nh\u2082 : IsPartition \u03c0\u2082\n\u22a2 integralSum f vol \u03c0\u2081 - integralSum f vol \u03c0\u2082 =\n \u2211 J in (\u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition).boxes,\n (\u2191(\u2191vol J) (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J)) -\n \u2191(\u2191vol J) (f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J)))\n[PROOFSTEP]\nrw [\u2190 integralSum_inf_partition f vol \u03c0\u2081 h\u2082, \u2190 integralSum_inf_partition f vol \u03c0\u2082 h\u2081, integralSum, integralSum,\n Finset.sum_sub_distrib]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\nf : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : IsPartition \u03c0\u2081\nh\u2082 : IsPartition \u03c0\u2082\n\u22a2 \u2211 J in (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition).toPrepartition.boxes,\n \u2191(\u2191vol J) (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J)) -\n \u2211 J in (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition).toPrepartition.boxes,\n \u2191(\u2191vol J) (f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J)) =\n \u2211 x in (\u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition).boxes, \u2191(\u2191vol x) (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) x)) -\n \u2211 x in (\u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition).boxes, \u2191(\u2191vol x) (f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) x))\n[PROOFSTEP]\nsimp only [infPrepartition_toPrepartition, inf_comm]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\nf : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Disjoint (TaggedPrepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\n\u22a2 integralSum f vol (TaggedPrepartition.disjUnion \u03c0\u2081 \u03c0\u2082 h) = integralSum f vol \u03c0\u2081 + integralSum f vol \u03c0\u2082\n[PROOFSTEP]\nrefine'\n (Prepartition.sum_disj_union_boxes h _).trans\n (congr_arg\u2082 (\u00b7 + \u00b7) (sum_congr rfl fun J hJ => _) (sum_congr rfl fun J hJ => _))\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\nf : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Disjoint (TaggedPrepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.boxes\n\u22a2 \u2191(\u2191vol J) (f (tag (TaggedPrepartition.disjUnion \u03c0\u2081 \u03c0\u2082 h) J)) = \u2191(\u2191vol J) (f (tag \u03c0\u2081 J))\n[PROOFSTEP]\nrw [disjUnion_tag_of_mem_left _ hJ]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\nf : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Disjoint (TaggedPrepartition.iUnion \u03c0\u2081) (TaggedPrepartition.iUnion \u03c0\u2082)\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2082.boxes\n\u22a2 \u2191(\u2191vol J) (f (tag (TaggedPrepartition.disjUnion \u03c0\u2081 \u03c0\u2082 h) J)) = \u2191(\u2191vol J) (f (tag \u03c0\u2082 J))\n[PROOFSTEP]\nrw [disjUnion_tag_of_mem_right _ hJ]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\n\u03c0 : TaggedPrepartition I\n\u22a2 integralSum (f + g) vol \u03c0 = integralSum f vol \u03c0 + integralSum g vol \u03c0\n[PROOFSTEP]\nsimp only [integralSum, Pi.add_apply, (vol _).map_add, Finset.sum_add_distrib]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\nf : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\n\u03c0 : TaggedPrepartition I\n\u22a2 integralSum (-f) vol \u03c0 = -integralSum f vol \u03c0\n[PROOFSTEP]\nsimp only [integralSum, Pi.neg_apply, (vol _).map_neg, Finset.sum_neg_distrib]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\nc : \u211d\nf : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\n\u03c0 : TaggedPrepartition I\n\u22a2 integralSum (c \u2022 f) vol \u03c0 = c \u2022 integralSum f vol \u03c0\n[PROOFSTEP]\nsimp only [integralSum, Finset.smul_sum, Pi.smul_apply, ContinuousLinearMap.map_smul]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\n\u22a2 (\u2200 (ib : \u211d),\n 0 < ib \u2192\n \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0 \u2227 IsPartition \u03c0} \u2192 integralSum f vol x \u2208 closedBall y ib) \u2194\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 \u03b5\n[PROOFSTEP]\nsimp [@forall_swap \u211d\u22650 (TaggedPrepartition I)]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\na : \u211d\nh :\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 a * \u03b5\n\u22a2 HasIntegral I l f vol y\n[PROOFSTEP]\nrefine' hasIntegral_iff.2 fun \u03b5 h\u03b5 => _\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\na : \u211d\nh :\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 a * \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 \u03b5\n[PROOFSTEP]\nrcases exists_pos_mul_lt h\u03b5 a with \u27e8\u03b5', h\u03b5', ha\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\na : \u211d\nh :\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 a * \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211d\nh\u03b5' : 0 < \u03b5'\nha : a * \u03b5' < \u03b5\n\u22a2 \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 \u03b5\n[PROOFSTEP]\nrcases h \u03b5' h\u03b5' with \u27e8r, hr, H\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\na : \u211d\nh :\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 a * \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211d\nh\u03b5' : 0 < \u03b5'\nha : a * \u03b5' < \u03b5\nr : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\nhr : \u2200 (c : \u211d\u22650), RCond l (r c)\nH :\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 a * \u03b5'\n\u22a2 \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) y \u2264 \u03b5\n[PROOFSTEP]\nexact \u27e8r, hr, fun c \u03c0 h\u03c0 h\u03c0p => (H c \u03c0 h\u03c0 h\u03c0p).trans ha.le\u27e9\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d : CompleteSpace F\n\u22a2 Integrable I l f vol \u2194\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c\u2081 c\u2082 : \u211d\u22650) (\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I),\n MemBaseSet l I c\u2081 (r c\u2081) \u03c0\u2081 \u2192\n IsPartition \u03c0\u2081 \u2192\n MemBaseSet l I c\u2082 (r c\u2082) \u03c0\u2082 \u2192 IsPartition \u03c0\u2082 \u2192 dist (integralSum f vol \u03c0\u2081) (integralSum f vol \u03c0\u2082) \u2264 \u03b5\n[PROOFSTEP]\nrw [integrable_iff_cauchy, cauchy_map_iff',\n (l.hasBasis_toFilteriUnion_top _).prod_self.tendsto_iff uniformity_basis_dist_le]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d : CompleteSpace F\n\u22a2 (\u2200 (ib : \u211d),\n 0 < ib \u2192\n \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I \u00d7 TaggedPrepartition I),\n x \u2208\n {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0 \u2227 IsPartition \u03c0} \u00d7\u02e2\n {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0 \u2227 IsPartition \u03c0} \u2192\n (integralSum f vol x.fst, integralSum f vol x.snd) \u2208 {p | dist p.fst p.snd \u2264 ib}) \u2194\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c\u2081 c\u2082 : \u211d\u22650) (\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I),\n MemBaseSet l I c\u2081 (r c\u2081) \u03c0\u2081 \u2192\n IsPartition \u03c0\u2081 \u2192\n MemBaseSet l I c\u2082 (r c\u2082) \u03c0\u2082 \u2192 IsPartition \u03c0\u2082 \u2192 dist (integralSum f vol \u03c0\u2081) (integralSum f vol \u03c0\u2082) \u2264 \u03b5\n[PROOFSTEP]\nrefine' forall\u2082_congr fun \u03b5 _ => exists_congr fun r => _\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d : CompleteSpace F\n\u03b5 : \u211d\nx\u271d : 0 < \u03b5\nr : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u22a2 ((\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (x : TaggedPrepartition I \u00d7 TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (r c) \u03c0 \u2227 IsPartition \u03c0} \u00d7\u02e2 {\u03c0 | \u2203 c, MemBaseSet l I c (r c) \u03c0 \u2227 IsPartition \u03c0} \u2192\n (integralSum f vol x.fst, integralSum f vol x.snd) \u2208 {p | dist p.fst p.snd \u2264 \u03b5}) \u2194\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c\u2081 c\u2082 : \u211d\u22650) (\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I),\n MemBaseSet l I c\u2081 (r c\u2081) \u03c0\u2081 \u2192\n IsPartition \u03c0\u2081 \u2192\n MemBaseSet l I c\u2082 (r c\u2082) \u03c0\u2082 \u2192 IsPartition \u03c0\u2082 \u2192 dist (integralSum f vol \u03c0\u2081) (integralSum f vol \u03c0\u2082) \u2264 \u03b5\n[PROOFSTEP]\nsimp only [exists_prop, Prod.forall, Set.mem_iUnion, exists_imp, prod_mk_mem_set_prod_eq, and_imp, mem_inter_iff,\n mem_setOf_eq]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d : CompleteSpace F\n\u03b5 : \u211d\nx\u271d : 0 < \u03b5\nr : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u22a2 ((\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (a b : TaggedPrepartition I) (x : \u211d\u22650),\n MemBaseSet l I x (r x) a \u2192\n IsPartition a \u2192\n \u2200 (x : \u211d\u22650),\n MemBaseSet l I x (r x) b \u2192 IsPartition b \u2192 dist (integralSum f vol a) (integralSum f vol b) \u2264 \u03b5) \u2194\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c\u2081 c\u2082 : \u211d\u22650) (\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I),\n MemBaseSet l I c\u2081 (r c\u2081) \u03c0\u2081 \u2192\n IsPartition \u03c0\u2081 \u2192\n MemBaseSet l I c\u2082 (r c\u2082) \u03c0\u2082 \u2192 IsPartition \u03c0\u2082 \u2192 dist (integralSum f vol \u03c0\u2081) (integralSum f vol \u03c0\u2082) \u2264 \u03b5\n[PROOFSTEP]\nexact\n and_congr Iff.rfl\n \u27e8fun H c\u2081 c\u2082 \u03c0\u2081 \u03c0\u2082 h\u2081 hU\u2081 h\u2082 hU\u2082 => H \u03c0\u2081 \u03c0\u2082 c\u2081 h\u2081 hU\u2081 c\u2082 h\u2082 hU\u2082, fun H \u03c0\u2081 \u03c0\u2082 c\u2081 h\u2081 hU\u2081 c\u2082 h\u2082 hU\u2082 =>\n H c\u2081 c\u2082 \u03c0\u2081 \u03c0\u2082 h\u2081 hU\u2081 h\u2082 hU\u2082\u27e9\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nh : Integrable I l f vol\n\u22a2 HasIntegral I l f vol (integral I l f vol)\n[PROOFSTEP]\nrw [integral, dif_pos h]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nh : Integrable I l f vol\n\u22a2 HasIntegral I l f vol (Exists.choose h)\n[PROOFSTEP]\nexact Classical.choose_spec h\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nh : HasIntegral I l f vol y\nh' : HasIntegral I l g vol y'\n\u22a2 HasIntegral I l (f + g) vol (y + y')\n[PROOFSTEP]\nsimpa only [HasIntegral, \u2190 integralSum_add] using h.add h'\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhf : HasIntegral I l f vol y\n\u22a2 HasIntegral I l (-f) vol (-y)\n[PROOFSTEP]\nsimpa only [HasIntegral, \u2190 integralSum_neg] using hf.neg\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nh : \u00acIntegrable I l f vol\n\u22a2 integral I l (-f) vol = -integral I l f vol\n[PROOFSTEP]\nrw [integral, integral, dif_neg h, dif_neg (mt Integrable.of_neg h), neg_zero]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nh : HasIntegral I l f vol y\nh' : HasIntegral I l g vol y'\n\u22a2 HasIntegral I l (f - g) vol (y - y')\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using h.add h'.neg\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\n\u22a2 HasIntegral I l (fun x => 0) vol 0\n[PROOFSTEP]\nsimpa only [\u2190 (vol I).map_zero] using hasIntegral_const (0 : E)\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf\u271d g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 (\u03b9 \u2192 \u211d) \u2192 E\ng : \u03b1 \u2192 F\nh : \u2200 (i : \u03b1), i \u2208 s \u2192 HasIntegral I l (f i) vol (g i)\n\u22a2 HasIntegral I l (fun x => \u2211 i in s, f i x) vol (\u2211 i in s, g i)\n[PROOFSTEP]\ninduction' s using Finset.induction_on with a s ha ihs\n[GOAL]\ncase empty\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf\u271d g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 (\u03b9 \u2192 \u211d) \u2192 E\ng : \u03b1 \u2192 F\nh\u271d : \u2200 (i : \u03b1), i \u2208 s \u2192 HasIntegral I l (f i) vol (g i)\nh : \u2200 (i : \u03b1), i \u2208 \u2205 \u2192 HasIntegral I l (f i) vol (g i)\n\u22a2 HasIntegral I l (fun x => \u2211 i in \u2205, f i x) vol (\u2211 i in \u2205, g i)\n[PROOFSTEP]\nsimp [hasIntegral_zero]\n[GOAL]\ncase insert\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf\u271d g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\n\u03b1 : Type u_1\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 (\u03b9 \u2192 \u211d) \u2192 E\ng : \u03b1 \u2192 F\nh\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 HasIntegral I l (f i) vol (g i)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nihs :\n (\u2200 (i : \u03b1), i \u2208 s \u2192 HasIntegral I l (f i) vol (g i)) \u2192 HasIntegral I l (fun x => \u2211 i in s, f i x) vol (\u2211 i in s, g i)\nh : \u2200 (i : \u03b1), i \u2208 insert a s \u2192 HasIntegral I l (f i) vol (g i)\n\u22a2 HasIntegral I l (fun x => \u2211 i in insert a s, f i x) vol (\u2211 i in insert a s, g i)\n[PROOFSTEP]\nsimp only [Finset.sum_insert ha]\n[GOAL]\ncase insert\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf\u271d g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\n\u03b1 : Type u_1\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 (\u03b9 \u2192 \u211d) \u2192 E\ng : \u03b1 \u2192 F\nh\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 HasIntegral I l (f i) vol (g i)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nihs :\n (\u2200 (i : \u03b1), i \u2208 s \u2192 HasIntegral I l (f i) vol (g i)) \u2192 HasIntegral I l (fun x => \u2211 i in s, f i x) vol (\u2211 i in s, g i)\nh : \u2200 (i : \u03b1), i \u2208 insert a s \u2192 HasIntegral I l (f i) vol (g i)\n\u22a2 HasIntegral I l (fun x => f a x + \u2211 i in s, f i x) vol (g a + \u2211 i in s, g i)\n[PROOFSTEP]\nrw [Finset.forall_mem_insert] at h \n[GOAL]\ncase insert\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf\u271d g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\n\u03b1 : Type u_1\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 (\u03b9 \u2192 \u211d) \u2192 E\ng : \u03b1 \u2192 F\nh\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 HasIntegral I l (f i) vol (g i)\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nihs :\n (\u2200 (i : \u03b1), i \u2208 s \u2192 HasIntegral I l (f i) vol (g i)) \u2192 HasIntegral I l (fun x => \u2211 i in s, f i x) vol (\u2211 i in s, g i)\nh : HasIntegral I l (f a) vol (g a) \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 HasIntegral I l (f x) vol (g x)\n\u22a2 HasIntegral I l (fun x => f a x + \u2211 i in s, f i x) vol (g a + \u2211 i in s, g i)\n[PROOFSTEP]\nexact h.1.add (ihs h.2)\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhf : HasIntegral I l f vol y\nc : \u211d\n\u22a2 HasIntegral I l (c \u2022 f) vol (c \u2022 y)\n[PROOFSTEP]\nsimpa only [HasIntegral, \u2190 integralSum_smul] using (tendsto_const_nhds : Tendsto _ _ (\ud835\udcdd c)).smul hf\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc : \u211d\nhf : Integrable I l (c \u2022 f) vol\nhc : c \u2260 0\n\u22a2 Integrable I l f vol\n[PROOFSTEP]\nsimpa [inv_smul_smul\u2080 hc] using hf.smul c\u207b\u00b9\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc : \u211d\n\u22a2 integral I l (fun x => c \u2022 f x) vol = c \u2022 integral I l f vol\n[PROOFSTEP]\nrcases eq_or_ne c 0 with (rfl | hc)\n[GOAL]\ncase inl\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\n\u22a2 integral I l (fun x => 0 \u2022 f x) vol = 0 \u2022 integral I l f vol\n[PROOFSTEP]\nsimp only [zero_smul, integral_zero]\n[GOAL]\ncase inr\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc : \u211d\nhc : c \u2260 0\n\u22a2 integral I l (fun x => c \u2022 f x) vol = c \u2022 integral I l f vol\n[PROOFSTEP]\nby_cases hf : Integrable I l f vol\n[GOAL]\ncase pos\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc : \u211d\nhc : c \u2260 0\nhf : Integrable I l f vol\n\u22a2 integral I l (fun x => c \u2022 f x) vol = c \u2022 integral I l f vol\n[PROOFSTEP]\nexact (hf.hasIntegral.smul c).integral_eq\n[GOAL]\ncase neg\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc : \u211d\nhc : c \u2260 0\nhf : \u00acIntegrable I l f vol\n\u22a2 integral I l (fun x => c \u2022 f x) vol = c \u2022 integral I l f vol\n[PROOFSTEP]\nhave : \u00acIntegrable I l (fun x => c \u2022 f x) vol := mt (fun h => h.of_smul hc) hf\n[GOAL]\ncase neg\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc : \u211d\nhc : c \u2260 0\nhf : \u00acIntegrable I l f vol\nthis : \u00acIntegrable I l (fun x => c \u2022 f x) vol\n\u22a2 integral I l (fun x => c \u2022 f x) vol = c \u2022 integral I l f vol\n[PROOFSTEP]\nrw [integral, integral, dif_neg hf, dif_neg this, smul_zero]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ng : (\u03b9 \u2192 \u211d) \u2192 \u211d\nhg : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 0 \u2264 g x\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\n\u22a2 0 \u2264 integral I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n[PROOFSTEP]\nby_cases hgi : Integrable I l g \u03bc.toBoxAdditive.toSMul\n[GOAL]\ncase pos\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ng : (\u03b9 \u2192 \u211d) \u2192 \u211d\nhg : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 0 \u2264 g x\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhgi : Integrable I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n\u22a2 0 \u2264 integral I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n[PROOFSTEP]\nrefine ge_of_tendsto' hgi.hasIntegral fun \u03c0 => sum_nonneg fun J _ => ?_\n[GOAL]\ncase pos\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ng : (\u03b9 \u2192 \u211d) \u2192 \u211d\nhg : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 0 \u2264 g x\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhgi : Integrable I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n\u03c0 : TaggedPrepartition I\nJ : Box \u03b9\nx\u271d : J \u2208 \u03c0.boxes\n\u22a2 0 \u2264 \u2191(\u2191(BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) J) (g (tag \u03c0 J))\n[PROOFSTEP]\nexact mul_nonneg ENNReal.toReal_nonneg (hg _ <| \u03c0.tag_mem_Icc _)\n[GOAL]\ncase neg\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ng : (\u03b9 \u2192 \u211d) \u2192 \u211d\nhg : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 0 \u2264 g x\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhgi : \u00acIntegrable I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n\u22a2 0 \u2264 integral I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n[PROOFSTEP]\nrw [integral, dif_neg hgi]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ng : (\u03b9 \u2192 \u211d) \u2192 \u211d\nhle : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2016f x\u2016 \u2264 g x\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhg : Integrable I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n\u22a2 \u2016integral I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\u2016 \u2264\n integral I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n[PROOFSTEP]\nby_cases hfi : Integrable.{u, v, v} I l f \u03bc.toBoxAdditive.toSMul\n[GOAL]\ncase pos\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ng : (\u03b9 \u2192 \u211d) \u2192 \u211d\nhle : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2016f x\u2016 \u2264 g x\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhg : Integrable I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\nhfi : Integrable I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n\u22a2 \u2016integral I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\u2016 \u2264\n integral I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n[PROOFSTEP]\nrefine' le_of_tendsto_of_tendsto' hfi.hasIntegral.norm hg.hasIntegral fun \u03c0 => _\n[GOAL]\ncase pos\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ng : (\u03b9 \u2192 \u211d) \u2192 \u211d\nhle : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2016f x\u2016 \u2264 g x\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhg : Integrable I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\nhfi : Integrable I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n\u03c0 : TaggedPrepartition I\n\u22a2 \u2016integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2016 \u2264\n integralSum g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\n[PROOFSTEP]\nrefine' norm_sum_le_of_le _ fun J _ => _\n[GOAL]\ncase pos\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ng : (\u03b9 \u2192 \u211d) \u2192 \u211d\nhle : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2016f x\u2016 \u2264 g x\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhg : Integrable I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\nhfi : Integrable I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n\u03c0 : TaggedPrepartition I\nJ : Box \u03b9\nx\u271d : J \u2208 \u03c0.boxes\n\u22a2 \u2016\u2191(\u2191(BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) J) (f (tag \u03c0 J))\u2016 \u2264\n \u2191(\u2191(BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) J) (g (tag \u03c0 J))\n[PROOFSTEP]\nsimp only [BoxAdditiveMap.toSMul_apply, norm_smul, smul_eq_mul, Real.norm_eq_abs, \u03bc.toBoxAdditive_apply,\n abs_of_nonneg ENNReal.toReal_nonneg]\n[GOAL]\ncase pos\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ng : (\u03b9 \u2192 \u211d) \u2192 \u211d\nhle : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2016f x\u2016 \u2264 g x\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhg : Integrable I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\nhfi : Integrable I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n\u03c0 : TaggedPrepartition I\nJ : Box \u03b9\nx\u271d : J \u2208 \u03c0.boxes\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc \u2191J) * \u2016f (tag \u03c0 J)\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc \u2191J) * g (tag \u03c0 J)\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_left (hle _ <| \u03c0.tag_mem_Icc _) ENNReal.toReal_nonneg\n[GOAL]\ncase neg\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ng : (\u03b9 \u2192 \u211d) \u2192 \u211d\nhle : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2016f x\u2016 \u2264 g x\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhg : Integrable I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\nhfi : \u00acIntegrable I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n\u22a2 \u2016integral I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\u2016 \u2264\n integral I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n[PROOFSTEP]\nrw [integral, dif_neg hfi, norm_zero]\n[GOAL]\ncase neg\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ng : (\u03b9 \u2192 \u211d) \u2192 \u211d\nhle : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2016f x\u2016 \u2264 g x\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhg : Integrable I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\nhfi : \u00acIntegrable I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n\u22a2 0 \u2264 integral I l g (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n[PROOFSTEP]\nexact integral_nonneg (fun x hx => (norm_nonneg _).trans (hle x hx)) \u03bc\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc : \u211d\nhc : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2016f x\u2016 \u2264 c\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\n\u22a2 \u2016integral I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc \u2191I) * c\n[PROOFSTEP]\nsimpa only [integral_const] using norm_integral_le_of_norm_le hc \u03bc (integrable_const c)\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc\u271d c\u2081 c\u2082 : \u211d\u22650\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\n\u03b5 : \u211d\nc : \u211d\u22650\n\u22a2 RCond l (convergenceR h \u03b5 c)\n[PROOFSTEP]\nrw [convergenceR]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc\u271d c\u2081 c\u2082 : \u211d\u22650\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\n\u03b5 : \u211d\nc : \u211d\u22650\n\u22a2 RCond l\n (dite (0 < \u03b5)\n (fun h\u03b5 =>\n Exists.choose\n (_ :\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) (integral I l f vol) \u2264 \u03b5))\n (fun h\u03b5 x x => { val := 1, property := convergenceR.proof_3 }) c)\n[PROOFSTEP]\nsplit_ifs with h\u2080\n[GOAL]\ncase pos\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc\u271d c\u2081 c\u2082 : \u211d\u22650\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\n\u03b5 : \u211d\nc : \u211d\u22650\nh\u2080 : 0 < \u03b5\n\u22a2 RCond l\n (Exists.choose\n (_ :\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) (integral I l f vol) \u2264 \u03b5)\n c)\ncase neg\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc\u271d c\u2081 c\u2082 : \u211d\u22650\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\n\u03b5 : \u211d\nc : \u211d\u22650\nh\u2080 : \u00ac0 < \u03b5\n\u22a2 RCond l ((fun x x => { val := 1, property := convergenceR.proof_3 }) c)\n[PROOFSTEP]\nexacts [(hasIntegral_iff.1 h.hasIntegral \u03b5 h\u2080).choose_spec.1 _, fun _ x => rfl]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\nh\u2080 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\nh\u03c0p : IsPartition \u03c0\n\u22a2 dist (integralSum f vol \u03c0) (integral I l f vol) \u2264 \u03b5\n[PROOFSTEP]\nrw [convergenceR, dif_pos h\u2080] at h\u03c0 \n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\nh\u2080 : 0 < \u03b5\nh\u03c0 :\n MemBaseSet l I c\n (Exists.choose\n (_ :\n \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c : \u211d\u22650) (\u03c0 : TaggedPrepartition I),\n MemBaseSet l I c (r c) \u03c0 \u2192 IsPartition \u03c0 \u2192 dist (integralSum f vol \u03c0) (integral I l f vol) \u2264 \u03b5)\n c)\n \u03c0\nh\u03c0p : IsPartition \u03c0\n\u22a2 dist (integralSum f vol \u03c0) (integral I l f vol) \u2264 \u03b5\n[PROOFSTEP]\nexact (hasIntegral_iff.1 h.hasIntegral \u03b5 h\u2080).choose_spec.2 c _ h\u03c0 h\u03c0p\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\nhpos\u2081 : 0 < \u03b5\u2081\nhpos\u2082 : 0 < \u03b5\u2082\nh\u2081 : MemBaseSet l I c\u2081 (convergenceR h \u03b5\u2081 c\u2081) \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 (convergenceR h \u03b5\u2082 c\u2082) \u03c0\u2082\nHU : TaggedPrepartition.iUnion \u03c0\u2081 = TaggedPrepartition.iUnion \u03c0\u2082\n\u22a2 dist (integralSum f vol \u03c0\u2081) (integralSum f vol \u03c0\u2082) \u2264 \u03b5\u2081 + \u03b5\u2082\n[PROOFSTEP]\nrcases h\u2081.exists_common_compl h\u2082 HU with \u27e8\u03c0, h\u03c0U, h\u03c0c\u2081, h\u03c0c\u2082\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\nhpos\u2081 : 0 < \u03b5\u2081\nhpos\u2082 : 0 < \u03b5\u2082\nh\u2081 : MemBaseSet l I c\u2081 (convergenceR h \u03b5\u2081 c\u2081) \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 (convergenceR h \u03b5\u2082 c\u2082) \u03c0\u2082\nHU : TaggedPrepartition.iUnion \u03c0\u2081 = TaggedPrepartition.iUnion \u03c0\u2082\n\u03c0 : Prepartition I\nh\u03c0U : Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081\nh\u03c0c\u2081 : l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2081\nh\u03c0c\u2082 : l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2082\n\u22a2 dist (integralSum f vol \u03c0\u2081) (integralSum f vol \u03c0\u2082) \u2264 \u03b5\u2081 + \u03b5\u2082\n[PROOFSTEP]\nset r : \u211d\u207f \u2192 Ioi (0 : \u211d) := fun x => min (h.convergenceR \u03b5\u2081 c\u2081 x) (h.convergenceR \u03b5\u2082 c\u2082 x)\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\nhpos\u2081 : 0 < \u03b5\u2081\nhpos\u2082 : 0 < \u03b5\u2082\nh\u2081 : MemBaseSet l I c\u2081 (convergenceR h \u03b5\u2081 c\u2081) \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 (convergenceR h \u03b5\u2082 c\u2082) \u03c0\u2082\nHU : TaggedPrepartition.iUnion \u03c0\u2081 = TaggedPrepartition.iUnion \u03c0\u2082\n\u03c0 : Prepartition I\nh\u03c0U : Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081\nh\u03c0c\u2081 : l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2081\nh\u03c0c\u2082 : l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2082\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun x => min (convergenceR h \u03b5\u2081 c\u2081 x) (convergenceR h \u03b5\u2082 c\u2082 x)\n\u22a2 dist (integralSum f vol \u03c0\u2081) (integralSum f vol \u03c0\u2082) \u2264 \u03b5\u2081 + \u03b5\u2082\n[PROOFSTEP]\nhave hr : l.RCond r := (h.convergenceR_cond _ c\u2081).min (h.convergenceR_cond _ c\u2082)\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\nhpos\u2081 : 0 < \u03b5\u2081\nhpos\u2082 : 0 < \u03b5\u2082\nh\u2081 : MemBaseSet l I c\u2081 (convergenceR h \u03b5\u2081 c\u2081) \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 (convergenceR h \u03b5\u2082 c\u2082) \u03c0\u2082\nHU : TaggedPrepartition.iUnion \u03c0\u2081 = TaggedPrepartition.iUnion \u03c0\u2082\n\u03c0 : Prepartition I\nh\u03c0U : Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081\nh\u03c0c\u2081 : l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2081\nh\u03c0c\u2082 : l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2082\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun x => min (convergenceR h \u03b5\u2081 c\u2081 x) (convergenceR h \u03b5\u2082 c\u2082 x)\nhr : RCond l r\n\u22a2 dist (integralSum f vol \u03c0\u2081) (integralSum f vol \u03c0\u2082) \u2264 \u03b5\u2081 + \u03b5\u2082\n[PROOFSTEP]\nset \u03c0r := \u03c0.toSubordinate r\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\nhpos\u2081 : 0 < \u03b5\u2081\nhpos\u2082 : 0 < \u03b5\u2082\nh\u2081 : MemBaseSet l I c\u2081 (convergenceR h \u03b5\u2081 c\u2081) \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 (convergenceR h \u03b5\u2082 c\u2082) \u03c0\u2082\nHU : TaggedPrepartition.iUnion \u03c0\u2081 = TaggedPrepartition.iUnion \u03c0\u2082\n\u03c0 : Prepartition I\nh\u03c0U : Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081\nh\u03c0c\u2081 : l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2081\nh\u03c0c\u2082 : l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2082\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun x => min (convergenceR h \u03b5\u2081 c\u2081 x) (convergenceR h \u03b5\u2082 c\u2082 x)\nhr : RCond l r\n\u03c0r : TaggedPrepartition I := Prepartition.toSubordinate \u03c0 r\n\u22a2 dist (integralSum f vol \u03c0\u2081) (integralSum f vol \u03c0\u2082) \u2264 \u03b5\u2081 + \u03b5\u2082\n[PROOFSTEP]\nhave H\u2081 : dist (integralSum f vol (\u03c0\u2081.unionComplToSubordinate \u03c0 h\u03c0U r)) (integral I l f vol) \u2264 \u03b5\u2081 :=\n h.dist_integralSum_integral_le_of_memBaseSet hpos\u2081 (h\u2081.unionComplToSubordinate (fun _ _ => min_le_left _ _) h\u03c0U h\u03c0c\u2081)\n (isPartition_unionComplToSubordinate _ _ _ _)\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\nhpos\u2081 : 0 < \u03b5\u2081\nhpos\u2082 : 0 < \u03b5\u2082\nh\u2081 : MemBaseSet l I c\u2081 (convergenceR h \u03b5\u2081 c\u2081) \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 (convergenceR h \u03b5\u2082 c\u2082) \u03c0\u2082\nHU : TaggedPrepartition.iUnion \u03c0\u2081 = TaggedPrepartition.iUnion \u03c0\u2082\n\u03c0 : Prepartition I\nh\u03c0U : Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081\nh\u03c0c\u2081 : l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2081\nh\u03c0c\u2082 : l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2082\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun x => min (convergenceR h \u03b5\u2081 c\u2081 x) (convergenceR h \u03b5\u2082 c\u2082 x)\nhr : RCond l r\n\u03c0r : TaggedPrepartition I := Prepartition.toSubordinate \u03c0 r\nH\u2081 : dist (integralSum f vol (unionComplToSubordinate \u03c0\u2081 \u03c0 h\u03c0U r)) (integral I l f vol) \u2264 \u03b5\u2081\n\u22a2 dist (integralSum f vol \u03c0\u2081) (integralSum f vol \u03c0\u2082) \u2264 \u03b5\u2081 + \u03b5\u2082\n[PROOFSTEP]\nrw [HU] at h\u03c0U \n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\nhpos\u2081 : 0 < \u03b5\u2081\nhpos\u2082 : 0 < \u03b5\u2082\nh\u2081 : MemBaseSet l I c\u2081 (convergenceR h \u03b5\u2081 c\u2081) \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 (convergenceR h \u03b5\u2082 c\u2082) \u03c0\u2082\nHU : TaggedPrepartition.iUnion \u03c0\u2081 = TaggedPrepartition.iUnion \u03c0\u2082\n\u03c0 : Prepartition I\nh\u03c0U\u271d : Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081\nh\u03c0U : Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2082\nh\u03c0c\u2081 : l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2081\nh\u03c0c\u2082 : l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2082\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun x => min (convergenceR h \u03b5\u2081 c\u2081 x) (convergenceR h \u03b5\u2082 c\u2082 x)\nhr : RCond l r\n\u03c0r : TaggedPrepartition I := Prepartition.toSubordinate \u03c0 r\nH\u2081 : dist (integralSum f vol (unionComplToSubordinate \u03c0\u2081 \u03c0 h\u03c0U\u271d r)) (integral I l f vol) \u2264 \u03b5\u2081\n\u22a2 dist (integralSum f vol \u03c0\u2081) (integralSum f vol \u03c0\u2082) \u2264 \u03b5\u2081 + \u03b5\u2082\n[PROOFSTEP]\nhave H\u2082 : dist (integralSum f vol (\u03c0\u2082.unionComplToSubordinate \u03c0 h\u03c0U r)) (integral I l f vol) \u2264 \u03b5\u2082 :=\n h.dist_integralSum_integral_le_of_memBaseSet hpos\u2082 (h\u2082.unionComplToSubordinate (fun _ _ => min_le_right _ _) h\u03c0U h\u03c0c\u2082)\n (isPartition_unionComplToSubordinate _ _ _ _)\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\nhpos\u2081 : 0 < \u03b5\u2081\nhpos\u2082 : 0 < \u03b5\u2082\nh\u2081 : MemBaseSet l I c\u2081 (convergenceR h \u03b5\u2081 c\u2081) \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 (convergenceR h \u03b5\u2082 c\u2082) \u03c0\u2082\nHU : TaggedPrepartition.iUnion \u03c0\u2081 = TaggedPrepartition.iUnion \u03c0\u2082\n\u03c0 : Prepartition I\nh\u03c0U\u271d : Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2081\nh\u03c0U : Prepartition.iUnion \u03c0 = \u2191I \\ TaggedPrepartition.iUnion \u03c0\u2082\nh\u03c0c\u2081 : l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2081\nh\u03c0c\u2082 : l.bDistortion = true \u2192 Prepartition.distortion \u03c0 \u2264 c\u2082\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun x => min (convergenceR h \u03b5\u2081 c\u2081 x) (convergenceR h \u03b5\u2082 c\u2082 x)\nhr : RCond l r\n\u03c0r : TaggedPrepartition I := Prepartition.toSubordinate \u03c0 r\nH\u2081 : dist (integralSum f vol (unionComplToSubordinate \u03c0\u2081 \u03c0 h\u03c0U\u271d r)) (integral I l f vol) \u2264 \u03b5\u2081\nH\u2082 : dist (integralSum f vol (unionComplToSubordinate \u03c0\u2082 \u03c0 h\u03c0U r)) (integral I l f vol) \u2264 \u03b5\u2082\n\u22a2 dist (integralSum f vol \u03c0\u2081) (integralSum f vol \u03c0\u2082) \u2264 \u03b5\u2081 + \u03b5\u2082\n[PROOFSTEP]\nsimpa [unionComplToSubordinate] using (dist_triangle_right _ _ _).trans (add_le_add H\u2081 H\u2082)\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\n\u22a2 Tendsto (fun \u03c0 => (integralSum f vol \u03c0.fst, integralSum f vol \u03c0.snd))\n (toFilter l I \u00d7\u02e2 toFilter l I \u2293 \ud835\udcdf {\u03c0 | TaggedPrepartition.iUnion \u03c0.fst = TaggedPrepartition.iUnion \u03c0.snd}) (\ud835\udce4 F)\n[PROOFSTEP]\nrefine (((l.hasBasis_toFilter I).prod_self.inf_principal _).tendsto_iff uniformity_basis_dist_le).2 fun \u03b5 \u03b50 => ?_\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I \u00d7 TaggedPrepartition I),\n x \u2208\n {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0} \u00d7\u02e2 {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0} \u2229\n {\u03c0 | TaggedPrepartition.iUnion \u03c0.fst = TaggedPrepartition.iUnion \u03c0.snd} \u2192\n (integralSum f vol x.fst, integralSum f vol x.snd) \u2208 {p | dist p.fst p.snd \u2264 \u03b5}\n[PROOFSTEP]\nreplace \u03b50 := half_pos \u03b50\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5 / 2\n\u22a2 \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I \u00d7 TaggedPrepartition I),\n x \u2208\n {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0} \u00d7\u02e2 {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0} \u2229\n {\u03c0 | TaggedPrepartition.iUnion \u03c0.fst = TaggedPrepartition.iUnion \u03c0.snd} \u2192\n (integralSum f vol x.fst, integralSum f vol x.snd) \u2208 {p | dist p.fst p.snd \u2264 \u03b5}\n[PROOFSTEP]\nuse h.convergenceR (\u03b5 / 2), h.convergenceR_cond (\u03b5 / 2)\n[GOAL]\ncase right\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5 / 2\n\u22a2 \u2200 (x : TaggedPrepartition I \u00d7 TaggedPrepartition I),\n x \u2208\n {\u03c0 | \u2203 c, MemBaseSet l I c (convergenceR h (\u03b5 / 2) c) \u03c0} \u00d7\u02e2\n {\u03c0 | \u2203 c, MemBaseSet l I c (convergenceR h (\u03b5 / 2) c) \u03c0} \u2229\n {\u03c0 | TaggedPrepartition.iUnion \u03c0.fst = TaggedPrepartition.iUnion \u03c0.snd} \u2192\n (integralSum f vol x.fst, integralSum f vol x.snd) \u2208 {p | dist p.fst p.snd \u2264 \u03b5}\n[PROOFSTEP]\nrintro \u27e8\u03c0\u2081, \u03c0\u2082\u27e9 \u27e8\u27e8h\u2081, h\u2082\u27e9, hU\u27e9\n[GOAL]\ncase right.mk.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nh : Integrable I l f vol\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5 / 2\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nhU : (\u03c0\u2081, \u03c0\u2082) \u2208 {\u03c0 | TaggedPrepartition.iUnion \u03c0.fst = TaggedPrepartition.iUnion \u03c0.snd}\nh\u2081 : (\u03c0\u2081, \u03c0\u2082).fst \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (convergenceR h (\u03b5 / 2) c) \u03c0}\nh\u2082 : (\u03c0\u2081, \u03c0\u2082).snd \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (convergenceR h (\u03b5 / 2) c) \u03c0}\n\u22a2 (integralSum f vol (\u03c0\u2081, \u03c0\u2082).fst, integralSum f vol (\u03c0\u2081, \u03c0\u2082).snd) \u2208 {p | dist p.fst p.snd \u2264 \u03b5}\n[PROOFSTEP]\nrw [\u2190 add_halves \u03b5]\n[GOAL]\ncase right.mk.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\nh : Integrable I l f vol\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5 / 2\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nhU : (\u03c0\u2081, \u03c0\u2082) \u2208 {\u03c0 | TaggedPrepartition.iUnion \u03c0.fst = TaggedPrepartition.iUnion \u03c0.snd}\nh\u2081 : (\u03c0\u2081, \u03c0\u2082).fst \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (convergenceR h (\u03b5 / 2) c) \u03c0}\nh\u2082 : (\u03c0\u2081, \u03c0\u2082).snd \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (convergenceR h (\u03b5 / 2) c) \u03c0}\n\u22a2 (integralSum f vol (\u03c0\u2081, \u03c0\u2082).fst, integralSum f vol (\u03c0\u2081, \u03c0\u2082).snd) \u2208 {p | dist p.fst p.snd \u2264 \u03b5 / 2 + \u03b5 / 2}\n[PROOFSTEP]\nexact h.dist_integralSum_le_of_memBaseSet \u03b50 \u03b50 h\u2081.choose_spec h\u2082.choose_spec hU\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\n\u03c0\u2080 : Prepartition I\n\u22a2 Cauchy (Filter.map (integralSum f vol) (toFilteriUnion l I \u03c0\u2080))\n[PROOFSTEP]\nrefine' \u27e8inferInstance, _\u27e9\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\n\u03c0\u2080 : Prepartition I\n\u22a2 Filter.map (integralSum f vol) (toFilteriUnion l I \u03c0\u2080) \u00d7\u02e2 Filter.map (integralSum f vol) (toFilteriUnion l I \u03c0\u2080) \u2264 \ud835\udce4 F\n[PROOFSTEP]\nrw [prod_map_map_eq, \u2190 toFilter_inf_iUnion_eq, \u2190 prod_inf_prod, prod_principal_principal]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh : Integrable I l f vol\n\u03c0\u2080 : Prepartition I\n\u22a2 Filter.map (fun p => (integralSum f vol p.fst, integralSum f vol p.snd))\n (toFilter l I \u00d7\u02e2 toFilter l I \u2293\n \ud835\udcdf\n ({\u03c0 | TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080} \u00d7\u02e2\n {\u03c0 | TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080})) \u2264\n \ud835\udce4 F\n[PROOFSTEP]\nexact\n h.tendsto_integralSum_toFilter_prod_self_inf_iUnion_eq_uniformity.mono_left\n (inf_le_inf_left _ <| principal_mono.2 fun \u03c0 h => h.1.trans h.2.symm)\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nhJ : J \u2264 I\n\u22a2 \u2203 y, HasIntegral J l f vol y \u2227 Tendsto (integralSum f vol) (toFilteriUnion l I (Prepartition.single I J hJ)) (\ud835\udcdd y)\n[PROOFSTEP]\nrefine\n (cauchy_map_iff_exists_tendsto.1 (h.cauchy_map_integralSum_toFilteriUnion (.single I J hJ))).imp fun y hy \u21a6 \u27e8?_, hy\u27e9\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny\u271d y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nhJ : J \u2264 I\ny : F\nhy : Tendsto (integralSum f vol) (toFilteriUnion l I (Prepartition.single I J hJ)) (\ud835\udcdd y)\n\u22a2 HasIntegral J l f vol y\n[PROOFSTEP]\nconvert\n hy.comp\n (l.tendsto_embedBox_toFilteriUnion_top hJ)\n -- faster than `exact` here\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u22a2 dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u2264 \u03b5\n[PROOFSTEP]\nrefine'\n le_of_forall_pos_le_add fun \u03b4 \u03b40 =>\n _\n -- First we choose some constants.\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u22a2 dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u2264 \u03b5 + \u03b4\n[PROOFSTEP]\nset \u03b4' : \u211d := \u03b4 / (\u03c0\u2080.boxes.card + 1)\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\n\u22a2 dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u2264 \u03b5 + \u03b4\n[PROOFSTEP]\nhave H0 : 0 < (\u03c0\u2080.boxes.card + 1 : \u211d) := Nat.cast_add_one_pos _\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u22a2 dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u2264 \u03b5 + \u03b4\n[PROOFSTEP]\nhave \u03b4'0 : 0 < \u03b4' := div_pos \u03b40 H0\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\n\u22a2 dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u2264 \u03b5 + \u03b4\n[PROOFSTEP]\nset C := max \u03c0\u2080.distortion \u03c0\u2080.compl.distortion\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\n\u22a2 dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u2264 \u03b5 + \u03b4\n[PROOFSTEP]\nhave :\n \u2200 J \u2208 \u03c0\u2080,\n \u2203 \u03c0i : TaggedPrepartition J,\n \u03c0i.IsPartition \u2227\n dist (integralSum f vol \u03c0i) (integral J l f vol) \u2264 \u03b4' \u2227 l.MemBaseSet J C (h.convergenceR \u03b4' C) \u03c0i :=\n by\n intro J hJ\n have Hle : J \u2264 I := \u03c0\u2080.le_of_mem hJ\n have HJi : Integrable J l f vol := h.to_subbox Hle\n set r := fun x => min (h.convergenceR \u03b4' C x) (HJi.convergenceR \u03b4' C x)\n have hJd : J.distortion \u2264 C := le_trans (Finset.le_sup hJ) (le_max_left _ _)\n rcases l.exists_memBaseSet_isPartition J hJd r with \u27e8\u03c0J, hC, hp\u27e9\n have hC\u2081 : l.MemBaseSet J C (HJi.convergenceR \u03b4' C) \u03c0J := by refine' hC.mono J le_rfl le_rfl fun x _ => _;\n exact min_le_right _ _\n have hC\u2082 : l.MemBaseSet J C (h.convergenceR \u03b4' C) \u03c0J := by refine' hC.mono J le_rfl le_rfl fun x _ => _;\n exact min_le_left _ _\n exact\n \u27e8\u03c0J, hp, HJi.dist_integralSum_integral_le_of_memBaseSet \u03b4'0 hC\u2081 hp, hC\u2082\u27e9\n /- Now we combine these tagged partitions into a tagged prepartition of `I` that covers the\n same part of `I` as `\u03c0\u2080` and apply `BoxIntegral.dist_integralSum_le_of_memBaseSet` to\n `\u03c0` and this prepartition. -/\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\n\u22a2 \u2200 (J : Box \u03b9),\n J \u2208 \u03c0\u2080 \u2192\n \u2203 \u03c0i,\n IsPartition \u03c0i \u2227\n dist (integralSum f vol \u03c0i) (integral J l f vol) \u2264 \u03b4' \u2227 MemBaseSet l J C (convergenceR h \u03b4' C) \u03c0i\n[PROOFSTEP]\nintro J hJ\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2080\n\u22a2 \u2203 \u03c0i,\n IsPartition \u03c0i \u2227 dist (integralSum f vol \u03c0i) (integral J l f vol) \u2264 \u03b4' \u2227 MemBaseSet l J C (convergenceR h \u03b4' C) \u03c0i\n[PROOFSTEP]\nhave Hle : J \u2264 I := \u03c0\u2080.le_of_mem hJ\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2080\nHle : J \u2264 I\n\u22a2 \u2203 \u03c0i,\n IsPartition \u03c0i \u2227 dist (integralSum f vol \u03c0i) (integral J l f vol) \u2264 \u03b4' \u2227 MemBaseSet l J C (convergenceR h \u03b4' C) \u03c0i\n[PROOFSTEP]\nhave HJi : Integrable J l f vol := h.to_subbox Hle\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2080\nHle : J \u2264 I\nHJi : Integrable J l f vol\n\u22a2 \u2203 \u03c0i,\n IsPartition \u03c0i \u2227 dist (integralSum f vol \u03c0i) (integral J l f vol) \u2264 \u03b4' \u2227 MemBaseSet l J C (convergenceR h \u03b4' C) \u03c0i\n[PROOFSTEP]\nset r := fun x => min (h.convergenceR \u03b4' C x) (HJi.convergenceR \u03b4' C x)\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2080\nHle : J \u2264 I\nHJi : Integrable J l f vol\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun x => min (convergenceR h \u03b4' C x) (convergenceR HJi \u03b4' C x)\n\u22a2 \u2203 \u03c0i,\n IsPartition \u03c0i \u2227 dist (integralSum f vol \u03c0i) (integral J l f vol) \u2264 \u03b4' \u2227 MemBaseSet l J C (convergenceR h \u03b4' C) \u03c0i\n[PROOFSTEP]\nhave hJd : J.distortion \u2264 C := le_trans (Finset.le_sup hJ) (le_max_left _ _)\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2080\nHle : J \u2264 I\nHJi : Integrable J l f vol\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun x => min (convergenceR h \u03b4' C x) (convergenceR HJi \u03b4' C x)\nhJd : Box.distortion J \u2264 C\n\u22a2 \u2203 \u03c0i,\n IsPartition \u03c0i \u2227 dist (integralSum f vol \u03c0i) (integral J l f vol) \u2264 \u03b4' \u2227 MemBaseSet l J C (convergenceR h \u03b4' C) \u03c0i\n[PROOFSTEP]\nrcases l.exists_memBaseSet_isPartition J hJd r with \u27e8\u03c0J, hC, hp\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2080\nHle : J \u2264 I\nHJi : Integrable J l f vol\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun x => min (convergenceR h \u03b4' C x) (convergenceR HJi \u03b4' C x)\nhJd : Box.distortion J \u2264 C\n\u03c0J : TaggedPrepartition J\nhC : MemBaseSet l J C r \u03c0J\nhp : IsPartition \u03c0J\n\u22a2 \u2203 \u03c0i,\n IsPartition \u03c0i \u2227 dist (integralSum f vol \u03c0i) (integral J l f vol) \u2264 \u03b4' \u2227 MemBaseSet l J C (convergenceR h \u03b4' C) \u03c0i\n[PROOFSTEP]\nhave hC\u2081 : l.MemBaseSet J C (HJi.convergenceR \u03b4' C) \u03c0J := by refine' hC.mono J le_rfl le_rfl fun x _ => _;\n exact min_le_right _ _\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2080\nHle : J \u2264 I\nHJi : Integrable J l f vol\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun x => min (convergenceR h \u03b4' C x) (convergenceR HJi \u03b4' C x)\nhJd : Box.distortion J \u2264 C\n\u03c0J : TaggedPrepartition J\nhC : MemBaseSet l J C r \u03c0J\nhp : IsPartition \u03c0J\n\u22a2 MemBaseSet l J C (convergenceR HJi \u03b4' C) \u03c0J\n[PROOFSTEP]\nrefine' hC.mono J le_rfl le_rfl fun x _ => _\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2080\nHle : J \u2264 I\nHJi : Integrable J l f vol\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun x => min (convergenceR h \u03b4' C x) (convergenceR HJi \u03b4' C x)\nhJd : Box.distortion J \u2264 C\n\u03c0J : TaggedPrepartition J\nhC : MemBaseSet l J C r \u03c0J\nhp : IsPartition \u03c0J\nx : \u03b9 \u2192 \u211d\nx\u271d : x \u2208 \u2191Box.Icc J\n\u22a2 r x \u2264 convergenceR HJi \u03b4' C x\n[PROOFSTEP]\nexact min_le_right _ _\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2080\nHle : J \u2264 I\nHJi : Integrable J l f vol\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun x => min (convergenceR h \u03b4' C x) (convergenceR HJi \u03b4' C x)\nhJd : Box.distortion J \u2264 C\n\u03c0J : TaggedPrepartition J\nhC : MemBaseSet l J C r \u03c0J\nhp : IsPartition \u03c0J\nhC\u2081 : MemBaseSet l J C (convergenceR HJi \u03b4' C) \u03c0J\n\u22a2 \u2203 \u03c0i,\n IsPartition \u03c0i \u2227 dist (integralSum f vol \u03c0i) (integral J l f vol) \u2264 \u03b4' \u2227 MemBaseSet l J C (convergenceR h \u03b4' C) \u03c0i\n[PROOFSTEP]\nhave hC\u2082 : l.MemBaseSet J C (h.convergenceR \u03b4' C) \u03c0J := by refine' hC.mono J le_rfl le_rfl fun x _ => _;\n exact min_le_left _ _\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2080\nHle : J \u2264 I\nHJi : Integrable J l f vol\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun x => min (convergenceR h \u03b4' C x) (convergenceR HJi \u03b4' C x)\nhJd : Box.distortion J \u2264 C\n\u03c0J : TaggedPrepartition J\nhC : MemBaseSet l J C r \u03c0J\nhp : IsPartition \u03c0J\nhC\u2081 : MemBaseSet l J C (convergenceR HJi \u03b4' C) \u03c0J\n\u22a2 MemBaseSet l J C (convergenceR h \u03b4' C) \u03c0J\n[PROOFSTEP]\nrefine' hC.mono J le_rfl le_rfl fun x _ => _\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2080\nHle : J \u2264 I\nHJi : Integrable J l f vol\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun x => min (convergenceR h \u03b4' C x) (convergenceR HJi \u03b4' C x)\nhJd : Box.distortion J \u2264 C\n\u03c0J : TaggedPrepartition J\nhC : MemBaseSet l J C r \u03c0J\nhp : IsPartition \u03c0J\nhC\u2081 : MemBaseSet l J C (convergenceR HJi \u03b4' C) \u03c0J\nx : \u03b9 \u2192 \u211d\nx\u271d : x \u2208 \u2191Box.Icc J\n\u22a2 r x \u2264 convergenceR h \u03b4' C x\n[PROOFSTEP]\nexact min_le_left _ _\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2080\nHle : J \u2264 I\nHJi : Integrable J l f vol\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun x => min (convergenceR h \u03b4' C x) (convergenceR HJi \u03b4' C x)\nhJd : Box.distortion J \u2264 C\n\u03c0J : TaggedPrepartition J\nhC : MemBaseSet l J C r \u03c0J\nhp : IsPartition \u03c0J\nhC\u2081 : MemBaseSet l J C (convergenceR HJi \u03b4' C) \u03c0J\nhC\u2082 : MemBaseSet l J C (convergenceR h \u03b4' C) \u03c0J\n\u22a2 \u2203 \u03c0i,\n IsPartition \u03c0i \u2227 dist (integralSum f vol \u03c0i) (integral J l f vol) \u2264 \u03b4' \u2227 MemBaseSet l J C (convergenceR h \u03b4' C) \u03c0i\n[PROOFSTEP]\nexact\n \u27e8\u03c0J, hp, HJi.dist_integralSum_integral_le_of_memBaseSet \u03b4'0 hC\u2081 hp, hC\u2082\u27e9\n /- Now we combine these tagged partitions into a tagged prepartition of `I` that covers the\n same part of `I` as `\u03c0\u2080` and apply `BoxIntegral.dist_integralSum_le_of_memBaseSet` to\n `\u03c0` and this prepartition. -/\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\nthis :\n \u2200 (J : Box \u03b9),\n J \u2208 \u03c0\u2080 \u2192\n \u2203 \u03c0i,\n IsPartition \u03c0i \u2227\n dist (integralSum f vol \u03c0i) (integral J l f vol) \u2264 \u03b4' \u2227 MemBaseSet l J C (convergenceR h \u03b4' C) \u03c0i\n\u22a2 dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u2264 \u03b5 + \u03b4\n[PROOFSTEP]\nchoose! \u03c0i h\u03c0ip h\u03c0i\u03b4' h\u03c0iC using this\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\n\u03c0i : (J : Box \u03b9) \u2192 TaggedPrepartition J\nh\u03c0ip : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 IsPartition (\u03c0i J)\nh\u03c0i\u03b4' : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 dist (integralSum f vol (\u03c0i J)) (integral J l f vol) \u2264 \u03b4'\nh\u03c0iC : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 MemBaseSet l J C (convergenceR h \u03b4' C) (\u03c0i J)\n\u22a2 dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u2264 \u03b5 + \u03b4\n[PROOFSTEP]\nhave : l.MemBaseSet I C (h.convergenceR \u03b4' C) (\u03c0\u2080.biUnionTagged \u03c0i) :=\n biUnionTagged_memBaseSet h\u03c0iC h\u03c0ip fun _ => le_max_right _ _\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\n\u03c0i : (J : Box \u03b9) \u2192 TaggedPrepartition J\nh\u03c0ip : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 IsPartition (\u03c0i J)\nh\u03c0i\u03b4' : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 dist (integralSum f vol (\u03c0i J)) (integral J l f vol) \u2264 \u03b4'\nh\u03c0iC : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 MemBaseSet l J C (convergenceR h \u03b4' C) (\u03c0i J)\nthis : MemBaseSet l I C (convergenceR h \u03b4' C) (Prepartition.biUnionTagged \u03c0\u2080 \u03c0i)\n\u22a2 dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u2264 \u03b5 + \u03b4\n[PROOFSTEP]\nhave hU' : \u03c0.iUnion = (\u03c0\u2080.biUnionTagged \u03c0i).iUnion := hU.trans (Prepartition.iUnion_biUnion_partition _ h\u03c0ip).symm\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\n\u03c0i : (J : Box \u03b9) \u2192 TaggedPrepartition J\nh\u03c0ip : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 IsPartition (\u03c0i J)\nh\u03c0i\u03b4' : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 dist (integralSum f vol (\u03c0i J)) (integral J l f vol) \u2264 \u03b4'\nh\u03c0iC : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 MemBaseSet l J C (convergenceR h \u03b4' C) (\u03c0i J)\nthis : MemBaseSet l I C (convergenceR h \u03b4' C) (Prepartition.biUnionTagged \u03c0\u2080 \u03c0i)\nhU' : TaggedPrepartition.iUnion \u03c0 = TaggedPrepartition.iUnion (Prepartition.biUnionTagged \u03c0\u2080 \u03c0i)\n\u22a2 dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u2264 \u03b5 + \u03b4\n[PROOFSTEP]\nhave := h.dist_integralSum_le_of_memBaseSet h0 \u03b4'0 h\u03c0 this hU'\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\n\u03c0i : (J : Box \u03b9) \u2192 TaggedPrepartition J\nh\u03c0ip : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 IsPartition (\u03c0i J)\nh\u03c0i\u03b4' : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 dist (integralSum f vol (\u03c0i J)) (integral J l f vol) \u2264 \u03b4'\nh\u03c0iC : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 MemBaseSet l J C (convergenceR h \u03b4' C) (\u03c0i J)\nthis\u271d : MemBaseSet l I C (convergenceR h \u03b4' C) (Prepartition.biUnionTagged \u03c0\u2080 \u03c0i)\nhU' : TaggedPrepartition.iUnion \u03c0 = TaggedPrepartition.iUnion (Prepartition.biUnionTagged \u03c0\u2080 \u03c0i)\nthis : dist (integralSum f vol \u03c0) (integralSum f vol (Prepartition.biUnionTagged \u03c0\u2080 \u03c0i)) \u2264 \u03b5 + \u03b4'\n\u22a2 dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u2264 \u03b5 + \u03b4\n[PROOFSTEP]\nrw [integralSum_biUnionTagged] at this \n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\n\u03c0i : (J : Box \u03b9) \u2192 TaggedPrepartition J\nh\u03c0ip : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 IsPartition (\u03c0i J)\nh\u03c0i\u03b4' : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 dist (integralSum f vol (\u03c0i J)) (integral J l f vol) \u2264 \u03b4'\nh\u03c0iC : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 MemBaseSet l J C (convergenceR h \u03b4' C) (\u03c0i J)\nthis\u271d : MemBaseSet l I C (convergenceR h \u03b4' C) (Prepartition.biUnionTagged \u03c0\u2080 \u03c0i)\nhU' : TaggedPrepartition.iUnion \u03c0 = TaggedPrepartition.iUnion (Prepartition.biUnionTagged \u03c0\u2080 \u03c0i)\nthis : dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integralSum f vol (\u03c0i J)) \u2264 \u03b5 + \u03b4'\n\u22a2 dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u2264 \u03b5 + \u03b4\n[PROOFSTEP]\ncalc\n dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u2264\n dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integralSum f vol (\u03c0i J)) +\n dist (\u2211 J in \u03c0\u2080.boxes, integralSum f vol (\u03c0i J)) (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) :=\n dist_triangle _ _ _\n _ \u2264 \u03b5 + \u03b4' + \u2211 _J in \u03c0\u2080.boxes, \u03b4' := (add_le_add this (dist_sum_sum_le_of_le _ h\u03c0i\u03b4'))\n _ = \u03b5 + \u03b4 := by field_simp [H0.ne']; ring\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\n\u03c0i : (J : Box \u03b9) \u2192 TaggedPrepartition J\nh\u03c0ip : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 IsPartition (\u03c0i J)\nh\u03c0i\u03b4' : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 dist (integralSum f vol (\u03c0i J)) (integral J l f vol) \u2264 \u03b4'\nh\u03c0iC : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 MemBaseSet l J C (convergenceR h \u03b4' C) (\u03c0i J)\nthis\u271d : MemBaseSet l I C (convergenceR h \u03b4' C) (Prepartition.biUnionTagged \u03c0\u2080 \u03c0i)\nhU' : TaggedPrepartition.iUnion \u03c0 = TaggedPrepartition.iUnion (Prepartition.biUnionTagged \u03c0\u2080 \u03c0i)\nthis : dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integralSum f vol (\u03c0i J)) \u2264 \u03b5 + \u03b4'\n\u22a2 \u03b5 + \u03b4' + \u2211 _J in \u03c0\u2080.boxes, \u03b4' = \u03b5 + \u03b4\n[PROOFSTEP]\nfield_simp [H0.ne']\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nh0 : 0 < \u03b5\nh\u03c0 : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\n\u03c0\u2080 : Prepartition I\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u03b4' : \u211d := \u03b4 / (\u2191(card \u03c0\u2080.boxes) + 1)\nH0 : 0 < \u2191(card \u03c0\u2080.boxes) + 1\n\u03b4'0 : 0 < \u03b4'\nC : \u211d\u22650 := max (Prepartition.distortion \u03c0\u2080) (Prepartition.distortion (Prepartition.compl \u03c0\u2080))\n\u03c0i : (J : Box \u03b9) \u2192 TaggedPrepartition J\nh\u03c0ip : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 IsPartition (\u03c0i J)\nh\u03c0i\u03b4' : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 dist (integralSum f vol (\u03c0i J)) (integral J l f vol) \u2264 \u03b4'\nh\u03c0iC : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2080 \u2192 MemBaseSet l J C (convergenceR h \u03b4' C) (\u03c0i J)\nthis\u271d : MemBaseSet l I C (convergenceR h \u03b4' C) (Prepartition.biUnionTagged \u03c0\u2080 \u03c0i)\nhU' : TaggedPrepartition.iUnion \u03c0 = TaggedPrepartition.iUnion (Prepartition.biUnionTagged \u03c0\u2080 \u03c0i)\nthis : dist (integralSum f vol \u03c0) (\u2211 J in \u03c0\u2080.boxes, integralSum f vol (\u03c0i J)) \u2264 \u03b5 + \u03b4'\n\u22a2 \u03b5 * (\u2191(card \u03c0\u2080.boxes) + 1) + \u03b4 + \u2191(card \u03c0\u2080.boxes) * \u03b4 = (\u03b5 + \u03b4) * (\u2191(card \u03c0\u2080.boxes) + 1)\n[PROOFSTEP]\nring\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\n\u03c0\u2080 : Prepartition I\n\u22a2 Tendsto (integralSum f vol) (toFilteriUnion l I \u03c0\u2080) (\ud835\udcdd (\u2211 J in \u03c0\u2080.boxes, integral J l f vol))\n[PROOFSTEP]\nrefine' ((l.hasBasis_toFilteriUnion I \u03c0\u2080).tendsto_iff nhds_basis_closedBall).2 fun \u03b5 \u03b50 => _\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\n\u03c0\u2080 : Prepartition I\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0 \u2227 TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080} \u2192\n integralSum f vol x \u2208 closedBall (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u03b5\n[PROOFSTEP]\nrefine' \u27e8h.convergenceR \u03b5, h.convergenceR_cond \u03b5, _\u27e9\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\n\u03c0\u2080 : Prepartition I\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0 \u2227 TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080} \u2192\n integralSum f vol x \u2208 closedBall (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u03b5\n[PROOFSTEP]\nsimp only [mem_inter_iff, Set.mem_iUnion, mem_setOf_eq]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\n\u03c0\u2080 : Prepartition I\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 \u2200 (x : TaggedPrepartition I),\n (\u2203 c, MemBaseSet l I c (convergenceR h \u03b5 c) x \u2227 TaggedPrepartition.iUnion x = Prepartition.iUnion \u03c0\u2080) \u2192\n integralSum f vol x \u2208 closedBall (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u03b5\n[PROOFSTEP]\nrintro \u03c0 \u27e8c, hc, hU\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc\u271d c\u2081 c\u2082 : \u211d\u22650\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\n\u03c0\u2080 : Prepartition I\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nhc : MemBaseSet l I c (convergenceR h \u03b5 c) \u03c0\nhU : TaggedPrepartition.iUnion \u03c0 = Prepartition.iUnion \u03c0\u2080\n\u22a2 integralSum f vol \u03c0 \u2208 closedBall (\u2211 J in \u03c0\u2080.boxes, integral J l f vol) \u03b5\n[PROOFSTEP]\nexact h.dist_integralSum_sum_integral_le_of_memBaseSet_of_iUnion_eq \u03b50 hc hU\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nhU : Prepartition.iUnion \u03c0\u2081 = Prepartition.iUnion \u03c0\u2082\n\u22a2 \u2211 J in \u03c0\u2081.boxes, integral J l f vol = \u2211 J in \u03c0\u2082.boxes, integral J l f vol\n[PROOFSTEP]\nrefine' tendsto_nhds_unique (h.tendsto_integralSum_sum_integral \u03c0\u2081) _\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nhU : Prepartition.iUnion \u03c0\u2081 = Prepartition.iUnion \u03c0\u2082\n\u22a2 Tendsto (integralSum f vol) (toFilteriUnion l I \u03c0\u2081) (\ud835\udcdd (\u2211 J in \u03c0\u2082.boxes, integral J l f vol))\n[PROOFSTEP]\nrw [l.toFilteriUnion_congr _ hU]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nhU : Prepartition.iUnion \u03c0\u2081 = Prepartition.iUnion \u03c0\u2082\n\u22a2 Tendsto (integralSum f vol) (toFilteriUnion l I \u03c0\u2082) (\ud835\udcdd (\u2211 J in \u03c0\u2082.boxes, integral J l f vol))\n[PROOFSTEP]\nexact h.tendsto_integralSum_sum_integral \u03c0\u2082\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nJ : Box \u03b9\nhJ : \u2191J \u2264 \u2191I\n\u03c0 : Prepartition J\nh\u03c0 : Prepartition.IsPartition \u03c0\n\u22a2 \u2211 Ji in \u03c0.boxes, (fun J => integral J l f vol) Ji = (fun J => integral J l f vol) J\n[PROOFSTEP]\nreplace h\u03c0 := h\u03c0.iUnion_eq\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nJ : Box \u03b9\nhJ : \u2191J \u2264 \u2191I\n\u03c0 : Prepartition J\nh\u03c0 : Prepartition.iUnion \u03c0 = \u2191J\n\u22a2 \u2211 Ji in \u03c0.boxes, (fun J => integral J l f vol) Ji = (fun J => integral J l f vol) J\n[PROOFSTEP]\nrw [\u2190 Prepartition.iUnion_top] at h\u03c0 \n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nc c\u2081 c\u2082 : \u211d\u22650\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\ninst\u271d : CompleteSpace F\nh : Integrable I l f vol\nJ : Box \u03b9\nhJ : \u2191J \u2264 \u2191I\n\u03c0 : Prepartition J\nh\u03c0 : Prepartition.iUnion \u03c0 = Prepartition.iUnion \u22a4\n\u22a2 \u2211 Ji in \u03c0.boxes, (fun J => integral J l f vol) Ji = (fun J => integral J l f vol) J\n[PROOFSTEP]\nrw [(h.to_subbox (WithTop.coe_le_coe.1 hJ)).sum_integral_congr h\u03c0, Prepartition.top_boxes, sum_singleton]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\n\u22a2 Integrable I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n[PROOFSTEP]\nhave huc := I.isCompact_Icc.uniformContinuousOn_of_continuous hc\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc : UniformContinuousOn f (\u2191Box.Icc I)\n\u22a2 Integrable I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n[PROOFSTEP]\nrw [Metric.uniformContinuousOn_iff_le] at huc \n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u22a2 Integrable I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc))\n[PROOFSTEP]\nrefine' integrable_iff_cauchy_basis.2 fun \u03b5 \u03b50 => _\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c\u2081 c\u2082 : \u211d\u22650) (\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I),\n MemBaseSet l I c\u2081 (r c\u2081) \u03c0\u2081 \u2192\n IsPartition \u03c0\u2081 \u2192\n MemBaseSet l I c\u2082 (r c\u2082) \u03c0\u2082 \u2192\n IsPartition \u03c0\u2082 \u2192\n dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2081)\n (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2082) \u2264\n \u03b5\n[PROOFSTEP]\nrcases exists_pos_mul_lt \u03b50 (\u03bc.toBoxAdditive I) with \u27e8\u03b5', \u03b50', h\u03b5\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u22a2 \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c\u2081 c\u2082 : \u211d\u22650) (\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I),\n MemBaseSet l I c\u2081 (r c\u2081) \u03c0\u2081 \u2192\n IsPartition \u03c0\u2081 \u2192\n MemBaseSet l I c\u2082 (r c\u2082) \u03c0\u2082 \u2192\n IsPartition \u03c0\u2082 \u2192\n dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2081)\n (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2082) \u2264\n \u03b5\n[PROOFSTEP]\nrcases huc \u03b5' \u03b50' with \u27e8\u03b4, \u03b40 : 0 < \u03b4, H\u03b4\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\n\u22a2 \u2203 r,\n (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227\n \u2200 (c\u2081 c\u2082 : \u211d\u22650) (\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I),\n MemBaseSet l I c\u2081 (r c\u2081) \u03c0\u2081 \u2192\n IsPartition \u03c0\u2081 \u2192\n MemBaseSet l I c\u2082 (r c\u2082) \u03c0\u2082 \u2192\n IsPartition \u03c0\u2082 \u2192\n dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2081)\n (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2082) \u2264\n \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun _ _ => \u27e8\u03b4 / 2, half_pos \u03b40\u27e9, fun _ _ _ => rfl, fun c\u2081 c\u2082 \u03c0\u2081 \u03c0\u2082 h\u2081 h\u2081p h\u2082 h\u2082p => _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\n\u22a2 dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2081)\n (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2082) \u2264\n \u03b5\n[PROOFSTEP]\nsimp only [dist_eq_norm, integralSum_sub_partitions _ _ h\u2081p h\u2082p, BoxAdditiveMap.toSMul_apply, \u2190 smul_sub]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\n\u22a2 \u2016\u2211 x in (\u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition).boxes,\n \u2191(Measure.toBoxAdditive \u03bc) x \u2022\n (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) x) - f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) x))\u2016 \u2264\n \u03b5\n[PROOFSTEP]\nhave :\n \u2200 J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition,\n \u2016\u03bc.toBoxAdditive J \u2022\n (f ((\u03c0\u2081.infPrepartition \u03c0\u2082.toPrepartition).tag J) - f ((\u03c0\u2082.infPrepartition \u03c0\u2081.toPrepartition).tag J))\u2016 \u2264\n \u03bc.toBoxAdditive J * \u03b5' :=\n by\n intro J hJ\n have : 0 \u2264 \u03bc.toBoxAdditive J := ENNReal.toReal_nonneg\n rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg this, \u2190 dist_eq_norm]\n refine' mul_le_mul_of_nonneg_left _ this\n refine' H\u03b4 _ (TaggedPrepartition.tag_mem_Icc _ _) _ (TaggedPrepartition.tag_mem_Icc _ _) _\n rw [\u2190 add_halves \u03b4]\n refine' (dist_triangle_left _ _ J.upper).trans (add_le_add (h\u2081.1 _ _ _) (h\u2082.1 _ _ _))\n \u00b7 exact Prepartition.biUnionIndex_mem _ hJ\n \u00b7 exact Box.le_iff_Icc.1 (Prepartition.le_biUnionIndex _ hJ) J.upper_mem_Icc\n \u00b7 rw [_root_.inf_comm] at hJ \n exact Prepartition.biUnionIndex_mem _ hJ\n \u00b7 rw [_root_.inf_comm] at hJ \n exact Box.le_iff_Icc.1 (Prepartition.le_biUnionIndex _ hJ) J.upper_mem_Icc\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\n\u22a2 \u2200 (J : Box \u03b9),\n J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition \u2192\n \u2016\u2191(Measure.toBoxAdditive \u03bc) J \u2022\n (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) - f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J))\u2016 \u2264\n \u2191(Measure.toBoxAdditive \u03bc) J * \u03b5'\n[PROOFSTEP]\nintro J hJ\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition\n\u22a2 \u2016\u2191(Measure.toBoxAdditive \u03bc) J \u2022\n (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) - f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J))\u2016 \u2264\n \u2191(Measure.toBoxAdditive \u03bc) J * \u03b5'\n[PROOFSTEP]\nhave : 0 \u2264 \u03bc.toBoxAdditive J := ENNReal.toReal_nonneg\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition\nthis : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J\n\u22a2 \u2016\u2191(Measure.toBoxAdditive \u03bc) J \u2022\n (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) - f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J))\u2016 \u2264\n \u2191(Measure.toBoxAdditive \u03bc) J * \u03b5'\n[PROOFSTEP]\nrw [norm_smul, Real.norm_eq_abs, abs_of_nonneg this, \u2190 dist_eq_norm]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition\nthis : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J\n\u22a2 \u2191(Measure.toBoxAdditive \u03bc) J *\n dist (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J)) (f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J)) \u2264\n \u2191(Measure.toBoxAdditive \u03bc) J * \u03b5'\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left _ this\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition\nthis : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J\n\u22a2 dist (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J)) (f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J)) \u2264 \u03b5'\n[PROOFSTEP]\nrefine' H\u03b4 _ (TaggedPrepartition.tag_mem_Icc _ _) _ (TaggedPrepartition.tag_mem_Icc _ _) _\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition\nthis : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J\n\u22a2 dist (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J) \u2264 \u03b4\n[PROOFSTEP]\nrw [\u2190 add_halves \u03b4]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition\nthis : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J\n\u22a2 dist (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J) \u2264 \u03b4 / 2 + \u03b4 / 2\n[PROOFSTEP]\nrefine' (dist_triangle_left _ _ J.upper).trans (add_le_add (h\u2081.1 _ _ _) (h\u2082.1 _ _ _))\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition\nthis : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J\n\u22a2 Prepartition.biUnionIndex \u03c0\u2081.toPrepartition (fun J => Prepartition.restrict \u03c0\u2082.toPrepartition J) J \u2208 \u03c0\u2081\n[PROOFSTEP]\nexact Prepartition.biUnionIndex_mem _ hJ\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition\nthis : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J\n\u22a2 J.upper \u2208\n \u2191Box.Icc (Prepartition.biUnionIndex \u03c0\u2081.toPrepartition (fun J => Prepartition.restrict \u03c0\u2082.toPrepartition J) J)\n[PROOFSTEP]\nexact Box.le_iff_Icc.1 (Prepartition.le_biUnionIndex _ hJ) J.upper_mem_Icc\n[GOAL]\ncase refine'_3\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition\nthis : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J\n\u22a2 Prepartition.biUnionIndex \u03c0\u2082.toPrepartition (fun J => Prepartition.restrict \u03c0\u2081.toPrepartition J) J \u2208 \u03c0\u2082\n[PROOFSTEP]\nrw [_root_.inf_comm] at hJ \n[GOAL]\ncase refine'_3\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2082.toPrepartition \u2293 \u03c0\u2081.toPrepartition\nthis : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J\n\u22a2 Prepartition.biUnionIndex \u03c0\u2082.toPrepartition (fun J => Prepartition.restrict \u03c0\u2081.toPrepartition J) J \u2208 \u03c0\u2082\n[PROOFSTEP]\nexact Prepartition.biUnionIndex_mem _ hJ\n[GOAL]\ncase refine'_4\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition\nthis : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J\n\u22a2 J.upper \u2208\n \u2191Box.Icc (Prepartition.biUnionIndex \u03c0\u2082.toPrepartition (fun J => Prepartition.restrict \u03c0\u2081.toPrepartition J) J)\n[PROOFSTEP]\nrw [_root_.inf_comm] at hJ \n[GOAL]\ncase refine'_4\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J\u271d : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2082.toPrepartition \u2293 \u03c0\u2081.toPrepartition\nthis : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J\n\u22a2 J.upper \u2208\n \u2191Box.Icc (Prepartition.biUnionIndex \u03c0\u2082.toPrepartition (fun J => Prepartition.restrict \u03c0\u2081.toPrepartition J) J)\n[PROOFSTEP]\nexact Box.le_iff_Icc.1 (Prepartition.le_biUnionIndex _ hJ) J.upper_mem_Icc\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\nthis :\n \u2200 (J : Box \u03b9),\n J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition \u2192\n \u2016\u2191(Measure.toBoxAdditive \u03bc) J \u2022\n (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) - f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J))\u2016 \u2264\n \u2191(Measure.toBoxAdditive \u03bc) J * \u03b5'\n\u22a2 \u2016\u2211 x in (\u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition).boxes,\n \u2191(Measure.toBoxAdditive \u03bc) x \u2022\n (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) x) - f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) x))\u2016 \u2264\n \u03b5\n[PROOFSTEP]\nrefine' (norm_sum_le_of_le _ this).trans _\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\nthis :\n \u2200 (J : Box \u03b9),\n J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition \u2192\n \u2016\u2191(Measure.toBoxAdditive \u03bc) J \u2022\n (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) - f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J))\u2016 \u2264\n \u2191(Measure.toBoxAdditive \u03bc) J * \u03b5'\n\u22a2 \u2211 b in (\u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition).boxes, \u2191(Measure.toBoxAdditive \u03bc) b * \u03b5' \u2264 \u03b5\n[PROOFSTEP]\nrw [\u2190 Finset.sum_mul, \u03bc.toBoxAdditive.sum_partition_boxes le_top (h\u2081p.inf h\u2082p)]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nI\u271d J : Box \u03b9\n\u03c0 : TaggedPrepartition I\u271d\ninst\u271d\u00b2 : Fintype \u03b9\nl : IntegrationParams\nf\u271d g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d\u00b9 : CompleteSpace E\nI : Box \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192 E\nhc : ContinuousOn f (\u2191Box.Icc I)\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsLocallyFiniteMeasure \u03bc\nhuc :\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b5' : \u211d\n\u03b50' : 0 < \u03b5'\nh\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nH\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5'\nc\u2081 c\u2082 : \u211d\u22650\n\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nh\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081\nh\u2081p : IsPartition \u03c0\u2081\nh\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082\nh\u2082p : IsPartition \u03c0\u2082\nthis :\n \u2200 (J : Box \u03b9),\n J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition \u2192\n \u2016\u2191(Measure.toBoxAdditive \u03bc) J \u2022\n (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) - f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J))\u2016 \u2264\n \u2191(Measure.toBoxAdditive \u03bc) J * \u03b5'\n\u22a2 \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' \u2264 \u03b5\n[PROOFSTEP]\nexact h\u03b5.le\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\nH\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u03b4 \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\nH\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u03b4 \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u22a2 HasIntegral I l f vol (\u2191g I)\n[PROOFSTEP]\nrefine' ((l.hasBasis_toFilteriUnion_top _).tendsto_iff Metric.nhds_basis_closedBall).2 _\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\nH\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u03b4 \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\nH\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u03b4 \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u22a2 \u2200 (ib : \u211d),\n 0 < ib \u2192\n \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0 \u2227 IsPartition \u03c0} \u2192 integralSum f vol x \u2208 closedBall (\u2191g I) ib\n[PROOFSTEP]\nintro \u03b5 \u03b50\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\nH\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u03b4 \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\nH\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u03b4 \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0 \u2227 IsPartition \u03c0} \u2192 integralSum f vol x \u2208 closedBall (\u2191g I) \u03b5\n[PROOFSTEP]\nsimp only [\u2190 exists_prop, gt_iff_lt, Subtype.exists'] at H\u2081 H\u2082 \n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nH\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2203 x_1,\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191x_1 \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\nH\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2203 x_1,\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191x_1 \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u22a2 \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0 \u2227 IsPartition \u03c0} \u2192 integralSum f vol x \u2208 closedBall (\u2191g I) \u03b5\n[PROOFSTEP]\nchoose! \u03b4\u2081 H\u03b4\u2081 using H\u2081\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nH\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2203 x_1,\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191x_1 \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u22a2 \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0 \u2227 IsPartition \u03c0} \u2192 integralSum f vol x \u2208 closedBall (\u2191g I) \u03b5\n[PROOFSTEP]\nchoose! \u03b4\u2082 H\u03b4\u2082 using H\u2082\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u22a2 \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0 \u2227 IsPartition \u03c0} \u2192 integralSum f vol x \u2208 closedBall (\u2191g I) \u03b5\n[PROOFSTEP]\nhave \u03b50' := half_pos \u03b50\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\n\u22a2 \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0 \u2227 IsPartition \u03c0} \u2192 integralSum f vol x \u2208 closedBall (\u2191g I) \u03b5\n[PROOFSTEP]\nhave H0 : 0 < (2 : \u211d) ^ Fintype.card \u03b9 := pow_pos zero_lt_two _\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u22a2 \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0 \u2227 IsPartition \u03c0} \u2192 integralSum f vol x \u2208 closedBall (\u2191g I) \u03b5\n[PROOFSTEP]\nrcases hs.exists_pos_forall_sum_le (div_pos \u03b50' H0) with \u27e8\u03b5s, h\u03b5s0, h\u03b5s\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 i in t, \u03b5s i \u2264 \u03b5 / 2 / 2 ^ Fintype.card \u03b9\n\u22a2 \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0 \u2227 IsPartition \u03c0} \u2192 integralSum f vol x \u2208 closedBall (\u2191g I) \u03b5\n[PROOFSTEP]\nsimp only [le_div_iff' H0, mul_sum] at h\u03b5s \n[GOAL]\ncase intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u22a2 \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0 \u2227 IsPartition \u03c0} \u2192 integralSum f vol x \u2208 closedBall (\u2191g I) \u03b5\n[PROOFSTEP]\nrcases exists_pos_mul_lt \u03b50' (B I) with \u27e8\u03b5', \u03b5'0, h\u03b5I\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u22a2 \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0 \u2227 IsPartition \u03c0} \u2192 integralSum f vol x \u2208 closedBall (\u2191g I) \u03b5\n[PROOFSTEP]\nset \u03b4 : \u211d\u22650 \u2192 \u211d\u207f \u2192 Ioi (0 : \u211d) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else (\u03b4\u2082 c) x \u03b5'\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u22a2 \u2203 ia,\n (\u2200 (c : \u211d\u22650), RCond l (ia c)) \u2227\n \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (ia c) \u03c0 \u2227 IsPartition \u03c0} \u2192 integralSum f vol x \u2208 closedBall (\u2191g I) \u03b5\n[PROOFSTEP]\nrefine' \u27e8\u03b4, fun c => l.rCond_of_bRiemann_eq_false hl, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u22a2 \u2200 (x : TaggedPrepartition I),\n x \u2208 {\u03c0 | \u2203 c, MemBaseSet l I c (\u03b4 c) \u03c0 \u2227 IsPartition \u03c0} \u2192 integralSum f vol x \u2208 closedBall (\u2191g I) \u03b5\n[PROOFSTEP]\nsimp only [Set.mem_iUnion, mem_inter_iff, mem_setOf_eq]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u22a2 \u2200 (x : TaggedPrepartition I),\n (\u2203 c, MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') x \u2227 IsPartition x) \u2192\n integralSum f vol x \u2208 closedBall (\u2191g I) \u03b5\n[PROOFSTEP]\nrintro \u03c0\n \u27e8c, h\u03c0\u03b4, h\u03c0p\u27e9\n -- Now we split the sum into two parts based on whether `\u03c0.tag J` belongs to `s` or not.\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\n\u22a2 integralSum f vol \u03c0 \u2208 closedBall (\u2191g I) \u03b5\n[PROOFSTEP]\nrw [\u2190 g.sum_partition_boxes le_rfl h\u03c0p, mem_closedBall, integralSum, \u2190\n sum_filter_add_sum_filter_not \u03c0.boxes fun J => \u03c0.tag J \u2208 s, \u2190\n sum_filter_add_sum_filter_not \u03c0.boxes fun J => \u03c0.tag J \u2208 s, \u2190 add_halves \u03b5]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\n\u22a2 dist\n (\u2211 x in Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes, \u2191(\u2191vol x) (f (tag \u03c0 x)) +\n \u2211 x in Finset.filter (fun x => \u00actag \u03c0 x \u2208 s) \u03c0.boxes, \u2191(\u2191vol x) (f (tag \u03c0 x)))\n (\u2211 x in Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes, \u2191g x +\n \u2211 x in Finset.filter (fun x => \u00actag \u03c0 x \u2208 s) \u03c0.boxes, \u2191g x) \u2264\n \u03b5 / 2 + \u03b5 / 2\n[PROOFSTEP]\nrefine' dist_add_add_le_of_le _ _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\n\u22a2 dist (\u2211 x in Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes, \u2191(\u2191vol x) (f (tag \u03c0 x)))\n (\u2211 x in Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes, \u2191g x) \u2264\n \u03b5 / 2\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | hsne)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1.inl\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 \u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0p : IsPartition \u03c0\nhs : Set.Countable \u2205\nhlH : Set.Nonempty \u2205 \u2192 l.bHenstock = true\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 \u2205 \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ \u2205 \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 \u2205 \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 \u2205 then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 \u2205 then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\n\u22a2 dist (\u2211 x in Finset.filter (fun J => tag \u03c0 J \u2208 \u2205) \u03c0.boxes, \u2191(\u2191vol x) (f (tag \u03c0 x)))\n (\u2211 x in Finset.filter (fun J => tag \u03c0 J \u2208 \u2205) \u03c0.boxes, \u2191g x) \u2264\n \u03b5 / 2\n[PROOFSTEP]\nsimp [\u03b50'.le]\n /- For the boxes such that `\u03c0.tag J \u2208 s`, we use the fact that at most `2 ^ #\u03b9` boxes have the\n same tag. -/\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1.inr\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\n\u22a2 dist (\u2211 x in Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes, \u2191(\u2191vol x) (f (tag \u03c0 x)))\n (\u2211 x in Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes, \u2191g x) \u2264\n \u03b5 / 2\n[PROOFSTEP]\nspecialize hlH hsne\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1.inr\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\n\u22a2 dist (\u2211 x in Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes, \u2191(\u2191vol x) (f (tag \u03c0 x)))\n (\u2211 x in Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes, \u2191g x) \u2264\n \u03b5 / 2\n[PROOFSTEP]\nhave : \u2200 J \u2208 \u03c0.boxes.filter fun J => \u03c0.tag J \u2208 s, dist (vol J (f <| \u03c0.tag J)) (g J) \u2264 \u03b5s (\u03c0.tag J) := fun J hJ \u21a6\n by\n rw [Finset.mem_filter] at hJ ; cases' hJ with hJ hJs\n refine'\n H\u03b4\u2081 c _ \u27e8\u03c0.tag_mem_Icc _, hJs\u27e9 _ (h\u03b5s0 _) _ (\u03c0.le_of_mem' _ hJ) _ (h\u03c0\u03b4.2 hlH J hJ) fun hD =>\n (Finset.le_sup hJ).trans (h\u03c0\u03b4.3 hD)\n convert h\u03c0\u03b4.1 J hJ using 3; exact (if_pos hJs).symm\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\nJ : Box \u03b9\nhJ : J \u2208 Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes\n\u22a2 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5s (tag \u03c0 J)\n[PROOFSTEP]\nrw [Finset.mem_filter] at hJ \n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\nJ : Box \u03b9\nhJ : J \u2208 \u03c0.boxes \u2227 tag \u03c0 J \u2208 s\n\u22a2 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5s (tag \u03c0 J)\n[PROOFSTEP]\ncases' hJ with hJ hJs\n[GOAL]\ncase intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\nJ : Box \u03b9\nhJ : J \u2208 \u03c0.boxes\nhJs : tag \u03c0 J \u2208 s\n\u22a2 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5s (tag \u03c0 J)\n[PROOFSTEP]\nrefine'\n H\u03b4\u2081 c _ \u27e8\u03c0.tag_mem_Icc _, hJs\u27e9 _ (h\u03b5s0 _) _ (\u03c0.le_of_mem' _ hJ) _ (h\u03c0\u03b4.2 hlH J hJ) fun hD =>\n (Finset.le_sup hJ).trans (h\u03c0\u03b4.3 hD)\n[GOAL]\ncase intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\nJ : Box \u03b9\nhJ : J \u2208 \u03c0.boxes\nhJs : tag \u03c0 J \u2208 s\n\u22a2 \u2191Box.Icc J \u2286 closedBall (tag \u03c0 J) \u2191(\u03b4\u2081 c (tag \u03c0 J) (\u03b5s (tag \u03c0 J)))\n[PROOFSTEP]\nconvert h\u03c0\u03b4.1 J hJ using 3\n[GOAL]\ncase h.e'_4.h.e'_4.h.e'_3.h\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\nJ : Box \u03b9\nhJ : J \u2208 \u03c0.boxes\nhJs : tag \u03c0 J \u2208 s\ne_2\u271d : (fun a => 0 < a) = fun x => x \u2208 Set.Ioi 0\n\u22a2 \u03b4\u2081 c (tag \u03c0 J) (\u03b5s (tag \u03c0 J)) = (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') (tag \u03c0 J)\n[PROOFSTEP]\nexact (if_pos hJs).symm\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1.inr\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\nthis :\n \u2200 (J : Box \u03b9), J \u2208 Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes \u2192 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5s (tag \u03c0 J)\n\u22a2 dist (\u2211 x in Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes, \u2191(\u2191vol x) (f (tag \u03c0 x)))\n (\u2211 x in Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes, \u2191g x) \u2264\n \u03b5 / 2\n[PROOFSTEP]\nrefine' (dist_sum_sum_le_of_le _ this).trans _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1.inr\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\nthis :\n \u2200 (J : Box \u03b9), J \u2208 Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes \u2192 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5s (tag \u03c0 J)\n\u22a2 \u2211 b in Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes, \u03b5s (tag \u03c0 b) \u2264 \u03b5 / 2\n[PROOFSTEP]\nrw [sum_comp]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1.inr\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\nthis :\n \u2200 (J : Box \u03b9), J \u2208 Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes \u2192 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5s (tag \u03c0 J)\n\u22a2 \u2211 b in Finset.image (fun b => tag \u03c0 b) (Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes),\n card (Finset.filter (fun a => tag \u03c0 a = b) (Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes)) \u2022 \u03b5s b \u2264\n \u03b5 / 2\n[PROOFSTEP]\nrefine' (sum_le_sum _).trans (h\u03b5s _ _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1.inr.refine'_1\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\nthis :\n \u2200 (J : Box \u03b9), J \u2208 Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes \u2192 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5s (tag \u03c0 J)\n\u22a2 \u2200 (i : \u03b9 \u2192 \u211d),\n i \u2208 Finset.image (fun b => tag \u03c0 b) (Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes) \u2192\n card (Finset.filter (fun a => tag \u03c0 a = i) (Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes)) \u2022 \u03b5s i \u2264\n 2 ^ Fintype.card \u03b9 * \u03b5s i\n[PROOFSTEP]\nrintro b -\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1.inr.refine'_1\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\nthis :\n \u2200 (J : Box \u03b9), J \u2208 Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes \u2192 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5s (tag \u03c0 J)\nb : \u03b9 \u2192 \u211d\n\u22a2 card (Finset.filter (fun a => tag \u03c0 a = b) (Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes)) \u2022 \u03b5s b \u2264\n 2 ^ Fintype.card \u03b9 * \u03b5s b\n[PROOFSTEP]\nrw [\u2190 Nat.cast_two, \u2190 Nat.cast_pow, \u2190 nsmul_eq_mul]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1.inr.refine'_1\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\nthis :\n \u2200 (J : Box \u03b9), J \u2208 Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes \u2192 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5s (tag \u03c0 J)\nb : \u03b9 \u2192 \u211d\n\u22a2 card (Finset.filter (fun a => tag \u03c0 a = b) (Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes)) \u2022 \u03b5s b \u2264\n 2 ^ Fintype.card \u03b9 \u2022 \u03b5s b\n[PROOFSTEP]\nrefine' nsmul_le_nsmul (h\u03b5s0 _).le _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1.inr.refine'_1\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\nthis :\n \u2200 (J : Box \u03b9), J \u2208 Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes \u2192 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5s (tag \u03c0 J)\nb : \u03b9 \u2192 \u211d\n\u22a2 card (Finset.filter (fun a => tag \u03c0 a = b) (Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes)) \u2264 2 ^ Fintype.card \u03b9\n[PROOFSTEP]\nrefine' (Finset.card_le_of_subset _).trans ((h\u03c0\u03b4.isHenstock hlH).card_filter_tag_eq_le b)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1.inr.refine'_1\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\nthis :\n \u2200 (J : Box \u03b9), J \u2208 Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes \u2192 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5s (tag \u03c0 J)\nb : \u03b9 \u2192 \u211d\n\u22a2 Finset.filter (fun a => tag \u03c0 a = b) (Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes) \u2286\n Finset.filter (fun J => tag \u03c0 J = b) \u03c0.boxes\n[PROOFSTEP]\nexact filter_subset_filter _ (filter_subset _ _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1.inr.refine'_2\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\nthis :\n \u2200 (J : Box \u03b9), J \u2208 Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes \u2192 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5s (tag \u03c0 J)\n\u22a2 \u2191(Finset.image (fun b => tag \u03c0 b) (Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes)) \u2286 s\n[PROOFSTEP]\nrw [Finset.coe_image, Set.image_subset_iff]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1.inr.refine'_2\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nhsne : Set.Nonempty s\nhlH : l.bHenstock = true\nthis :\n \u2200 (J : Box \u03b9), J \u2208 Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes \u2192 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5s (tag \u03c0 J)\n\u22a2 \u2191(Finset.filter (fun J => tag \u03c0 J \u2208 s) \u03c0.boxes) \u2286 (fun b => tag \u03c0 b) \u207b\u00b9' s\n[PROOFSTEP]\nexact fun J hJ =>\n (Finset.mem_filter.1 hJ).2\n /- Now we deal with boxes such that `\u03c0.tag J \u2209 s`.\n In this case the estimate is straightforward. -/\n -- porting note: avoided strange elaboration issues by rewriting using `calc`\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\n\u22a2 dist (\u2211 x in Finset.filter (fun x => \u00actag \u03c0 x \u2208 s) \u03c0.boxes, \u2191(\u2191vol x) (f (tag \u03c0 x)))\n (\u2211 x in Finset.filter (fun x => \u00actag \u03c0 x \u2208 s) \u03c0.boxes, \u2191g x) \u2264\n \u03b5 / 2\n[PROOFSTEP]\ncalc\n dist (\u2211 J in \u03c0.boxes.filter (\u00actag \u03c0 \u00b7 \u2208 s), vol J (f (tag \u03c0 J))) (\u2211 J in \u03c0.boxes.filter (\u00actag \u03c0 \u00b7 \u2208 s), g J) \u2264\n \u2211 J in \u03c0.boxes.filter (\u00actag \u03c0 \u00b7 \u2208 s), \u03b5' * B J :=\n dist_sum_sum_le_of_le _ fun J hJ \u21a6 by\n rw [Finset.mem_filter] at hJ ; cases' hJ with hJ hJs\n refine'\n H\u03b4\u2082 c _ \u27e8\u03c0.tag_mem_Icc _, hJs\u27e9 _ \u03b5'0 _ (\u03c0.le_of_mem' _ hJ) _ (fun hH => h\u03c0\u03b4.2 hH J hJ) fun hD =>\n (Finset.le_sup hJ).trans (h\u03c0\u03b4.3 hD)\n convert h\u03c0\u03b4.1 J hJ using 3; exact (if_neg hJs).symm\n _ \u2264 \u2211 J in \u03c0.boxes, \u03b5' * B J :=\n (sum_le_sum_of_subset_of_nonneg (filter_subset _ _) fun _ _ _ \u21a6 mul_nonneg \u03b5'0.le (hB0 _))\n _ = B I * \u03b5' := by rw [\u2190 mul_sum, B.sum_partition_boxes le_rfl h\u03c0p, mul_comm]\n _ \u2264 \u03b5 / 2 := h\u03b5I.le\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nJ : Box \u03b9\nhJ : J \u2208 Finset.filter (fun x => \u00actag \u03c0 x \u2208 s) \u03c0.boxes\n\u22a2 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5' * \u2191B J\n[PROOFSTEP]\nrw [Finset.mem_filter] at hJ \n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nJ : Box \u03b9\nhJ : J \u2208 \u03c0.boxes \u2227 \u00actag \u03c0 J \u2208 s\n\u22a2 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5' * \u2191B J\n[PROOFSTEP]\ncases' hJ with hJ hJs\n[GOAL]\ncase intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nJ : Box \u03b9\nhJ : J \u2208 \u03c0.boxes\nhJs : \u00actag \u03c0 J \u2208 s\n\u22a2 dist (\u2191(\u2191vol J) (f (tag \u03c0 J))) (\u2191g J) \u2264 \u03b5' * \u2191B J\n[PROOFSTEP]\nrefine'\n H\u03b4\u2082 c _ \u27e8\u03c0.tag_mem_Icc _, hJs\u27e9 _ \u03b5'0 _ (\u03c0.le_of_mem' _ hJ) _ (fun hH => h\u03c0\u03b4.2 hH J hJ) fun hD =>\n (Finset.le_sup hJ).trans (h\u03c0\u03b4.3 hD)\n[GOAL]\ncase intro\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nJ : Box \u03b9\nhJ : J \u2208 \u03c0.boxes\nhJs : \u00actag \u03c0 J \u2208 s\n\u22a2 \u2191Box.Icc J \u2286 closedBall (tag \u03c0 J) \u2191(\u03b4\u2082 c (tag \u03c0 J) \u03b5')\n[PROOFSTEP]\nconvert h\u03c0\u03b4.1 J hJ using 3\n[GOAL]\ncase h.e'_4.h.e'_4.h.e'_3.h\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J\u271d : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\nJ : Box \u03b9\nhJ : J \u2208 \u03c0.boxes\nhJs : \u00actag \u03c0 J \u2208 s\ne_2\u271d : (fun a => 0 < a) = fun x => x \u2208 Set.Ioi 0\n\u22a2 \u03b4\u2082 c (tag \u03c0 J) \u03b5' = (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') (tag \u03c0 J)\n[PROOFSTEP]\nexact (if_neg hJs).symm\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0\u271d : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l.bRiemann = false\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nhlH : Set.Nonempty s \u2192 l.bHenstock = true\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4\u2081 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2081 c x \u03b5) \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\n\u03b4\u2082 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u211d \u2192 { a // 0 < a }\nH\u03b4\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n 0 < \u03b5 \u2192\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u2191(\u03b4\u2082 c x \u03b5) \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u03b50' : 0 < \u03b5 / 2\nH0 : 0 < 2 ^ Fintype.card \u03b9\n\u03b5s : (\u03b9 \u2192 \u211d) \u2192 \u211d\nh\u03b5s0 : \u2200 (i : \u03b9 \u2192 \u211d), 0 < \u03b5s i\nh\u03b5s : \u2200 (t : Finset (\u03b9 \u2192 \u211d)), \u2191t \u2286 s \u2192 \u2211 x in t, 2 ^ Fintype.card \u03b9 * \u03b5s x \u2264 \u03b5 / 2\n\u03b5' : \u211d\n\u03b5'0 : 0 < \u03b5'\nh\u03b5I : \u2191B I * \u03b5' < \u03b5 / 2\n\u03b4 : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0) := fun c x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5'\n\u03c0 : TaggedPrepartition I\nc : \u211d\u22650\nh\u03c0\u03b4 : MemBaseSet l I c (fun x => if x \u2208 s then \u03b4\u2081 c x (\u03b5s x) else \u03b4\u2082 c x \u03b5') \u03c0\nh\u03c0p : IsPartition \u03c0\n\u22a2 \u2211 J in \u03c0.boxes, \u03b5' * \u2191B J = \u2191B I * \u03b5'\n[PROOFSTEP]\nrw [\u2190 mul_sum, B.sum_partition_boxes le_rfl h\u03c0p, mul_comm]\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l \u2264 Henstock\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nH\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u03b4 \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\nH\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u03b4 \u2192\n x \u2208 \u2191Box.Icc J \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u22a2 \u00ac(\u2191equivProd Henstock).snd.fst = false\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l \u2264 Henstock\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nH\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u03b4 \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\nH\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u03b4 \u2192\n x \u2208 \u2191Box.Icc J \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\nA : l.bHenstock = true\n\u22a2 \u00ac(\u2191equivProd Henstock).fst = true\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nhl : l \u2264 Henstock\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\ns : Set (\u03b9 \u2192 \u211d)\nhs : Set.Countable s\nH\u2081 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2229 s \u2192\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u03b4 \u2192\n x \u2208 \u2191Box.Icc J \u2192 (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5\nH\u2082 :\n \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u03b4 \u2192\n x \u2208 \u2191Box.Icc J \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\nA : l.bHenstock = true\n\u22a2 \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ s \u2192\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u03b4 \u2192\n (l.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (l.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n[PROOFSTEP]\nsimpa only [A, true_imp_iff] using H\u2082\n[GOAL]\n\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d : Fintype \u03b9\nl : IntegrationParams\nf g\u271d : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\nB : \u03b9 \u2192\u1d47\u1d43[\u2191I] \u211d\nhB0 : \u2200 (J : Box \u03b9), 0 \u2264 \u2191B J\ng : \u03b9 \u2192\u1d47\u1d43[\u2191I] F\nH :\n \u211d\u22650 \u2192\n \u2200 (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \u2192\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (J : Box \u03b9), J \u2264 I \u2192 \u2191Box.Icc J \u2286 closedBall x \u03b4 \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n\u22a2 \u2200 (c : \u211d\u22650) (x : \u03b9 \u2192 \u211d),\n x \u2208 \u2191Box.Icc I \\ \u2205 \u2192\n \u2200 (\u03b5 : \u211d),\n \u03b5 > 0 \u2192\n \u2203 \u03b4,\n \u03b4 > 0 \u2227\n \u2200 (J : Box \u03b9),\n J \u2264 I \u2192\n \u2191Box.Icc J \u2286 closedBall x \u03b4 \u2192\n (McShane.bHenstock = true \u2192 x \u2208 \u2191Box.Icc J) \u2192\n (McShane.bDistortion = true \u2192 Box.distortion J \u2264 c) \u2192 dist (\u2191(\u2191vol J) (f x)) (\u2191g J) \u2264 \u03b5 * \u2191B J\n[PROOFSTEP]\nsimpa only [McShane, Bool.coe_sort_false, false_imp_iff, true_imp_iff, diff_empty] using H\n", "meta": {"mathlib_filename": "Mathlib.Analysis.BoxIntegral.Basic", "llama_tokens": 107440, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185944046238982, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.4067377523053335}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Preorder \u03b1\nh : \u2200 {a b : \u03b1}, a \u2264 b \u2192 IsCompact (Icc a b)\na b : \u03b1\nhab : \u00aca \u2264 b\n\u22a2 IsCompact (Icc a b)\n[PROOFSTEP]\nrw [Icc_eq_empty hab]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : Preorder \u03b1\nh : \u2200 {a b : \u03b1}, a \u2264 b \u2192 IsCompact (Icc a b)\na b : \u03b1\nhab : \u00aca \u2264 b\n\u22a2 IsCompact \u2205\n[PROOFSTEP]\nexact isCompact_empty\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PartialOrder \u03b1\nh : \u2200 {a b : \u03b1}, a < b \u2192 IsCompact (Icc a b)\na\u271d b\u271d : \u03b1\nhab : a\u271d \u2264 b\u271d\n\u22a2 a\u271d = b\u271d \u2192 IsCompact (Icc a\u271d b\u271d)\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PartialOrder \u03b1\nh : \u2200 {a b : \u03b1}, a < b \u2192 IsCompact (Icc a b)\na\u271d : \u03b1\nhab : a\u271d \u2264 a\u271d\n\u22a2 IsCompact (Icc a\u271d a\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\n\u22a2 CompactIccSpace \u03b1\n[PROOFSTEP]\nrefine' .mk'' fun {a b} hlt => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\n\u22a2 IsCompact (Icc a b)\n[PROOFSTEP]\ncases' le_or_lt a b with hab hab\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\n\u22a2 IsCompact (Icc a b)\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : b < a\n\u22a2 IsCompact (Icc a b)\n[PROOFSTEP]\nswap\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : b < a\n\u22a2 IsCompact (Icc a b)\n[PROOFSTEP]\nsimp [hab]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\n\u22a2 IsCompact (Icc a b)\n[PROOFSTEP]\nrefine' isCompact_iff_ultrafilter_le_nhds.2 fun f hf => _\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhf : \u2191f \u2264 \ud835\udcdf (Icc a b)\n\u22a2 \u2203 a_1, a_1 \u2208 Icc a b \u2227 \u2191f \u2264 \ud835\udcdd a_1\n[PROOFSTEP]\ncontrapose! hf\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 Icc a b \u2192 \u00ac\u2191f \u2264 \ud835\udcdd a_1\n\u22a2 \u00ac\u2191f \u2264 \ud835\udcdf (Icc a b)\n[PROOFSTEP]\nrw [le_principal_iff]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 Icc a b \u2192 \u00ac\u2191f \u2264 \ud835\udcdd a_1\n\u22a2 \u00acIcc a b \u2208 \u2191f\n[PROOFSTEP]\nhave hpt : \u2200 x \u2208 Icc a b, { x } \u2209 f := fun x hx hxf => hf x hx ((le_pure_iff.2 hxf).trans (pure_le_nhds x))\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 Icc a b \u2192 \u00ac\u2191f \u2264 \ud835\udcdd a_1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\n\u22a2 \u00acIcc a b \u2208 \u2191f\n[PROOFSTEP]\nset s := {x \u2208 Icc a b | Icc a x \u2209 f}\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 Icc a b \u2192 \u00ac\u2191f \u2264 \ud835\udcdd a_1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\n\u22a2 \u00acIcc a b \u2208 \u2191f\n[PROOFSTEP]\nhave hsb : b \u2208 upperBounds s := fun x hx => hx.1.2\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 Icc a b \u2192 \u00ac\u2191f \u2264 \ud835\udcdd a_1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\n\u22a2 \u00acIcc a b \u2208 \u2191f\n[PROOFSTEP]\nhave sbd : BddAbove s := \u27e8b, hsb\u27e9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 Icc a b \u2192 \u00ac\u2191f \u2264 \ud835\udcdd a_1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\n\u22a2 \u00acIcc a b \u2208 \u2191f\n[PROOFSTEP]\nhave ha : a \u2208 s := by simp [hpt, hab]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 Icc a b \u2192 \u00ac\u2191f \u2264 \ud835\udcdd a_1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\n\u22a2 a \u2208 s\n[PROOFSTEP]\nsimp [hpt, hab]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 Icc a b \u2192 \u00ac\u2191f \u2264 \ud835\udcdd a_1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n\u22a2 \u00acIcc a b \u2208 \u2191f\n[PROOFSTEP]\nrcases hab.eq_or_lt with (rfl | _hlt)\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na : \u03b1\nf : Ultrafilter \u03b1\nhlt : a < a\nhab : a \u2264 a\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 Icc a a \u2192 \u00ac\u2191f \u2264 \ud835\udcdd a_1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a a \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a a \u2227 \u00acIcc a x \u2208 f}\nhsb : a \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n\u22a2 \u00acIcc a a \u2208 \u2191f\n[PROOFSTEP]\nexact\n ha.2\n -- porting note: the `obtain` below was instead\n -- `set c := Sup s`\n -- `have hsc : IsLUB s c := isLUB_csSup \u27e8a, ha\u27e9 sbd`\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 Icc a b \u2192 \u00ac\u2191f \u2264 \ud835\udcdd a_1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\n\u22a2 \u00acIcc a b \u2208 \u2191f\n[PROOFSTEP]\nobtain \u27e8c, hsc\u27e9 : \u2203 c, IsLUB s c := \u27e8sSup s, isLUB_csSup \u27e8a, ha\u27e9 \u27e8b, hsb\u27e9\u27e9\n[GOAL]\ncase inl.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 Icc a b \u2192 \u00ac\u2191f \u2264 \ud835\udcdd a_1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\n\u22a2 \u00acIcc a b \u2208 \u2191f\n[PROOFSTEP]\nhave hc : c \u2208 Icc a b := \u27e8hsc.1 ha, hsc.2 hsb\u27e9\n[GOAL]\ncase inl.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhf : \u2200 (a_1 : \u03b1), a_1 \u2208 Icc a b \u2192 \u00ac\u2191f \u2264 \ud835\udcdd a_1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\n\u22a2 \u00acIcc a b \u2208 \u2191f\n[PROOFSTEP]\nspecialize hf c hc\n[GOAL]\ncase inl.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhf : \u00ac\u2191f \u2264 \ud835\udcdd c\n\u22a2 \u00acIcc a b \u2208 \u2191f\n[PROOFSTEP]\nhave hcs : c \u2208 s := by\n rcases hc.1.eq_or_lt with (rfl | hlt); \u00b7 assumption\n refine' \u27e8hc, fun hcf => hf fun U hU => _\u27e9\n rcases(mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hlt).1 (mem_nhdsWithin_of_mem_nhds hU) with \u27e8x, hxc, hxU\u27e9\n rcases((hsc.frequently_mem \u27e8a, ha\u27e9).and_eventually (Ioc_mem_nhdsWithin_Iic \u27e8hxc, le_rfl\u27e9)).exists with\n \u27e8y, \u27e8_hyab, hyf\u27e9, hy\u27e9\n refine' mem_of_superset (f.diff_mem_iff.2 \u27e8hcf, hyf\u27e9) (Subset.trans _ hxU)\n rw [diff_subset_iff]\n exact Subset.trans Icc_subset_Icc_union_Ioc <| union_subset_union Subset.rfl <| Ioc_subset_Ioc_left hy.1.le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhf : \u00ac\u2191f \u2264 \ud835\udcdd c\n\u22a2 c \u2208 s\n[PROOFSTEP]\nrcases hc.1.eq_or_lt with (rfl | hlt)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nhsc : IsLUB s a\nhc : a \u2208 Icc a b\nhf : \u00ac\u2191f \u2264 \ud835\udcdd a\n\u22a2 a \u2208 s\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhf : \u00ac\u2191f \u2264 \ud835\udcdd c\nhlt : a < c\n\u22a2 c \u2208 s\n[PROOFSTEP]\nrefine' \u27e8hc, fun hcf => hf fun U hU => _\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhf : \u00ac\u2191f \u2264 \ud835\udcdd c\nhlt : a < c\nhcf : Icc a c \u2208 f\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd c\n\u22a2 U \u2208 \u2191f\n[PROOFSTEP]\nrcases(mem_nhdsWithin_Iic_iff_exists_Ioc_subset' hlt).1 (mem_nhdsWithin_of_mem_nhds hU) with \u27e8x, hxc, hxU\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhf : \u00ac\u2191f \u2264 \ud835\udcdd c\nhlt : a < c\nhcf : Icc a c \u2208 f\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd c\nx : \u03b1\nhxc : x \u2208 Iio c\nhxU : Ioc x c \u2286 U\n\u22a2 U \u2208 \u2191f\n[PROOFSTEP]\nrcases((hsc.frequently_mem \u27e8a, ha\u27e9).and_eventually (Ioc_mem_nhdsWithin_Iic \u27e8hxc, le_rfl\u27e9)).exists with\n \u27e8y, \u27e8_hyab, hyf\u27e9, hy\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhf : \u00ac\u2191f \u2264 \ud835\udcdd c\nhlt : a < c\nhcf : Icc a c \u2208 f\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd c\nx : \u03b1\nhxc : x \u2208 Iio c\nhxU : Ioc x c \u2286 U\ny : \u03b1\nhy : x < y \u2227 y \u2264 c\n_hyab : y \u2208 Icc a b\nhyf : \u00acIcc a y \u2208 f\n\u22a2 U \u2208 \u2191f\n[PROOFSTEP]\nrefine' mem_of_superset (f.diff_mem_iff.2 \u27e8hcf, hyf\u27e9) (Subset.trans _ hxU)\n[GOAL]\ncase inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhf : \u00ac\u2191f \u2264 \ud835\udcdd c\nhlt : a < c\nhcf : Icc a c \u2208 f\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd c\nx : \u03b1\nhxc : x \u2208 Iio c\nhxU : Ioc x c \u2286 U\ny : \u03b1\nhy : x < y \u2227 y \u2264 c\n_hyab : y \u2208 Icc a b\nhyf : \u00acIcc a y \u2208 f\n\u22a2 Icc a c \\ Icc a y \u2286 Ioc x c\n[PROOFSTEP]\nrw [diff_subset_iff]\n[GOAL]\ncase inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhf : \u00ac\u2191f \u2264 \ud835\udcdd c\nhlt : a < c\nhcf : Icc a c \u2208 f\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd c\nx : \u03b1\nhxc : x \u2208 Iio c\nhxU : Ioc x c \u2286 U\ny : \u03b1\nhy : x < y \u2227 y \u2264 c\n_hyab : y \u2208 Icc a b\nhyf : \u00acIcc a y \u2208 f\n\u22a2 Icc a c \u2286 Icc a y \u222a Ioc x c\n[PROOFSTEP]\nexact Subset.trans Icc_subset_Icc_union_Ioc <| union_subset_union Subset.rfl <| Ioc_subset_Ioc_left hy.1.le\n[GOAL]\ncase inl.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhf : \u00ac\u2191f \u2264 \ud835\udcdd c\nhcs : c \u2208 s\n\u22a2 \u00acIcc a b \u2208 \u2191f\n[PROOFSTEP]\nrcases hc.2.eq_or_lt with (rfl | hlt)\n[GOAL]\ncase inl.inr.intro.inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na : \u03b1\nf : Ultrafilter \u03b1\nc : \u03b1\nhf : \u00ac\u2191f \u2264 \ud835\udcdd c\nhlt : a < c\nhab : a \u2264 c\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a c \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a c \u2227 \u00acIcc a x \u2208 f}\nhsb : c \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < c\nhsc : IsLUB s c\nhc : c \u2208 Icc a c\nhcs : c \u2208 s\n\u22a2 \u00acIcc a c \u2208 \u2191f\n[PROOFSTEP]\nexact hcs.2\n[GOAL]\ncase inl.inr.intro.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhf : \u00ac\u2191f \u2264 \ud835\udcdd c\nhcs : c \u2208 s\nhlt : c < b\n\u22a2 \u00acIcc a b \u2208 \u2191f\n[PROOFSTEP]\ncontrapose! hf\n[GOAL]\ncase inl.inr.intro.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhcs : c \u2208 s\nhlt : c < b\nhf : Icc a b \u2208 \u2191f\n\u22a2 \u2191f \u2264 \ud835\udcdd c\n[PROOFSTEP]\nintro U hU\n[GOAL]\ncase inl.inr.intro.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhcs : c \u2208 s\nhlt : c < b\nhf : Icc a b \u2208 \u2191f\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd c\n\u22a2 U \u2208 \u2191f\n[PROOFSTEP]\nrcases(mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset hlt).1 (mem_nhdsWithin_of_mem_nhds hU) with \u27e8y, hxy, hyU\u27e9\n[GOAL]\ncase inl.inr.intro.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhcs : c \u2208 s\nhlt : c < b\nhf : Icc a b \u2208 \u2191f\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd c\ny : \u03b1\nhxy : y \u2208 Ioc c b\nhyU : Ico c y \u2286 U\n\u22a2 U \u2208 \u2191f\n[PROOFSTEP]\nrefine' mem_of_superset _ hyU\n[GOAL]\ncase inl.inr.intro.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhcs : c \u2208 s\nhlt : c < b\nhf : Icc a b \u2208 \u2191f\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd c\ny : \u03b1\nhxy : y \u2208 Ioc c b\nhyU : Ico c y \u2286 U\n\u22a2 Ico c y \u2208 \u2191f\n[PROOFSTEP]\nclear! U\n[GOAL]\ncase inl.inr.intro.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhcs : c \u2208 s\nhlt : c < b\nhf : Icc a b \u2208 \u2191f\ny : \u03b1\nhxy : y \u2208 Ioc c b\n\u22a2 Ico c y \u2208 \u2191f\n[PROOFSTEP]\nhave hy : y \u2208 Icc a b := \u27e8hc.1.trans hxy.1.le, hxy.2\u27e9\n[GOAL]\ncase inl.inr.intro.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhcs : c \u2208 s\nhlt : c < b\nhf : Icc a b \u2208 \u2191f\ny : \u03b1\nhxy : y \u2208 Ioc c b\nhy : y \u2208 Icc a b\n\u22a2 Ico c y \u2208 \u2191f\n[PROOFSTEP]\nby_cases hay : Icc a y \u2208 f\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhcs : c \u2208 s\nhlt : c < b\nhf : Icc a b \u2208 \u2191f\ny : \u03b1\nhxy : y \u2208 Ioc c b\nhy : y \u2208 Icc a b\nhay : Icc a y \u2208 f\n\u22a2 Ico c y \u2208 \u2191f\n[PROOFSTEP]\nrefine' mem_of_superset (f.diff_mem_iff.2 \u27e8f.diff_mem_iff.2 \u27e8hay, hcs.2\u27e9, hpt y hy\u27e9) _\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhcs : c \u2208 s\nhlt : c < b\nhf : Icc a b \u2208 \u2191f\ny : \u03b1\nhxy : y \u2208 Ioc c b\nhy : y \u2208 Icc a b\nhay : Icc a y \u2208 f\n\u22a2 (Icc a y \\ Icc a c) \\ {y} \u2286 Ico c y\n[PROOFSTEP]\nrw [diff_subset_iff, union_comm, Ico_union_right hxy.1.le, diff_subset_iff]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhcs : c \u2208 s\nhlt : c < b\nhf : Icc a b \u2208 \u2191f\ny : \u03b1\nhxy : y \u2208 Ioc c b\nhy : y \u2208 Icc a b\nhay : Icc a y \u2208 f\n\u22a2 Icc a y \u2286 Icc a c \u222a Icc c y\n[PROOFSTEP]\nexact Icc_subset_Icc_union_Icc\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\na b : \u03b1\nhlt\u271d : a < b\nhab : a \u2264 b\nf : Ultrafilter \u03b1\nhpt : \u2200 (x : \u03b1), x \u2208 Icc a b \u2192 \u00ac{x} \u2208 f\ns : Set \u03b1 := {x | x \u2208 Icc a b \u2227 \u00acIcc a x \u2208 f}\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nha : a \u2208 s\n_hlt : a < b\nc : \u03b1\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhcs : c \u2208 s\nhlt : c < b\nhf : Icc a b \u2208 \u2191f\ny : \u03b1\nhxy : y \u2208 Ioc c b\nhy : y \u2208 Icc a b\nhay : \u00acIcc a y \u2208 f\n\u22a2 Ico c y \u2208 \u2191f\n[PROOFSTEP]\nexact ((hsc.1 \u27e8hy, hay\u27e9).not_lt hxy.1).elim\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CompleteLinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderTopology \u03b1\n\u22a2 IsCompact univ\n[PROOFSTEP]\nsimp only [\u2190 Icc_bot_top, isCompact_Icc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b1\nhs : IsCompact s\nne_s : Set.Nonempty s\n\u22a2 \u2203 x, IsLeast s x\n[PROOFSTEP]\nhaveI : Nonempty s := ne_s.to_subtype\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b1\nhs : IsCompact s\nne_s : Set.Nonempty s\nthis : Nonempty \u2191s\n\u22a2 \u2203 x, IsLeast s x\n[PROOFSTEP]\nsuffices : (s \u2229 \u22c2 x \u2208 s, Iic x).Nonempty\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b1\nhs : IsCompact s\nne_s : Set.Nonempty s\nthis\u271d : Nonempty \u2191s\nthis : Set.Nonempty (s \u2229 \u22c2 (x : \u03b1) (_ : x \u2208 s), Iic x)\n\u22a2 \u2203 x, IsLeast s x\n[PROOFSTEP]\nexact \u27e8this.choose, this.choose_spec.1, mem_iInter\u2082.mp this.choose_spec.2\u27e9\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b1\nhs : IsCompact s\nne_s : Set.Nonempty s\nthis : Nonempty \u2191s\n\u22a2 Set.Nonempty (s \u2229 \u22c2 (x : \u03b1) (_ : x \u2208 s), Iic x)\n[PROOFSTEP]\nrw [biInter_eq_iInter]\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b1\nhs : IsCompact s\nne_s : Set.Nonempty s\nthis : Nonempty \u2191s\n\u22a2 Set.Nonempty (s \u2229 \u22c2 (x : \u2191s), Iic \u2191x)\n[PROOFSTEP]\nby_contra H\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b1\nhs : IsCompact s\nne_s : Set.Nonempty s\nthis : Nonempty \u2191s\nH : \u00acSet.Nonempty (s \u2229 \u22c2 (x : \u2191s), Iic \u2191x)\n\u22a2 False\n[PROOFSTEP]\nrw [not_nonempty_iff_eq_empty] at H \n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b1\nhs : IsCompact s\nne_s : Set.Nonempty s\nthis : Nonempty \u2191s\nH : s \u2229 \u22c2 (x : \u2191s), Iic \u2191x = \u2205\n\u22a2 False\n[PROOFSTEP]\nrcases hs.elim_directed_family_closed (fun x : s => Iic \u2191x) (fun x => isClosed_Iic) H\n (directed_of_inf fun _ _ h => Iic_subset_Iic.mpr h) with\n \u27e8x, hx\u27e9\n[GOAL]\ncase this.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b1\nhs : IsCompact s\nne_s : Set.Nonempty s\nthis : Nonempty \u2191s\nH : s \u2229 \u22c2 (x : \u2191s), Iic \u2191x = \u2205\nx : \u2191s\nhx : s \u2229 Iic \u2191x = \u2205\n\u22a2 False\n[PROOFSTEP]\nexact not_nonempty_iff_eq_empty.mpr hx \u27e8x, x.2, le_rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nhs : IsCompact s\nne_s : Set.Nonempty s\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\n\u22a2 \u2203 x, x \u2208 s \u2227 IsMinOn f s x\n[PROOFSTEP]\nrcases(hs.image_of_continuousOn hf).exists_isLeast (ne_s.image f) with \u27e8_, \u27e8x, hxs, rfl\u27e9, hx\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nhs : IsCompact s\nne_s : Set.Nonempty s\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nx : \u03b2\nhxs : x \u2208 s\nhx : f x \u2208 lowerBounds (f '' s)\n\u22a2 \u2203 x, x \u2208 s \u2227 IsMinOn f s x\n[PROOFSTEP]\nexact \u27e8x, hxs, ball_image_iff.1 hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\n\u22a2 \u2203 x, x \u2208 s \u2227 IsMinOn f s x\n[PROOFSTEP]\nrcases(hasBasis_cocompact.inf_principal _).eventually_iff.1 hc with \u27e8K, hK, hKf\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\n\u22a2 \u2203 x, x \u2208 s \u2227 IsMinOn f s x\n[PROOFSTEP]\nhave hsub : insert x\u2080 (K \u2229 s) \u2286 s := insert_subset_iff.2 \u27e8h\u2080, inter_subset_right _ _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\nhsub : insert x\u2080 (K \u2229 s) \u2286 s\n\u22a2 \u2203 x, x \u2208 s \u2227 IsMinOn f s x\n[PROOFSTEP]\nobtain \u27e8x, hx, hxf\u27e9 : \u2203 x \u2208 insert x\u2080 (K \u2229 s), \u2200 y \u2208 insert x\u2080 (K \u2229 s), f x \u2264 f y :=\n ((hK.inter_right hsc).insert x\u2080).exists_forall_le (insert_nonempty _ _) (hf.mono hsub)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\nhsub : insert x\u2080 (K \u2229 s) \u2286 s\nx : \u03b2\nhx : x \u2208 insert x\u2080 (K \u2229 s)\nhxf : \u2200 (y : \u03b2), y \u2208 insert x\u2080 (K \u2229 s) \u2192 f x \u2264 f y\n\u22a2 \u2203 x, x \u2208 s \u2227 IsMinOn f s x\n[PROOFSTEP]\nrefine' \u27e8x, hsub hx, fun y hy => _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\nhsub : insert x\u2080 (K \u2229 s) \u2286 s\nx : \u03b2\nhx : x \u2208 insert x\u2080 (K \u2229 s)\nhxf : \u2200 (y : \u03b2), y \u2208 insert x\u2080 (K \u2229 s) \u2192 f x \u2264 f y\ny : \u03b2\nhy : y \u2208 s\n\u22a2 y \u2208 {x_1 | (fun x_2 => f x \u2264 f x_2) x_1}\n[PROOFSTEP]\nby_cases hyK : y \u2208 K\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\nhsub : insert x\u2080 (K \u2229 s) \u2286 s\nx : \u03b2\nhx : x \u2208 insert x\u2080 (K \u2229 s)\nhxf : \u2200 (y : \u03b2), y \u2208 insert x\u2080 (K \u2229 s) \u2192 f x \u2264 f y\ny : \u03b2\nhy : y \u2208 s\nhyK : y \u2208 K\n\u22a2 y \u2208 {x_1 | (fun x_2 => f x \u2264 f x_2) x_1}\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\nhsub : insert x\u2080 (K \u2229 s) \u2286 s\nx : \u03b2\nhx : x \u2208 insert x\u2080 (K \u2229 s)\nhxf : \u2200 (y : \u03b2), y \u2208 insert x\u2080 (K \u2229 s) \u2192 f x \u2264 f y\ny : \u03b2\nhy : y \u2208 s\nhyK : \u00acy \u2208 K\n\u22a2 y \u2208 {x_1 | (fun x_2 => f x \u2264 f x_2) x_1}\n[PROOFSTEP]\nexacts [hxf _ (Or.inr \u27e8hyK, hy\u27e9), (hxf _ (Or.inl rfl)).trans (hKf \u27e8hyK, hy\u27e9)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\nf : \u03b2 \u2192 \u03b1\nhf : Continuous f\nx\u2080 : \u03b2\nh : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2, f x\u2080 \u2264 f x\n\u22a2 \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf univ, f x\u2080 \u2264 f x\n[PROOFSTEP]\nrwa [principal_univ, inf_top_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : ClosedIicTopology \u03b1\ninst\u271d : Nonempty \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Continuous f\nhlim : Tendsto f (cocompact \u03b2) atTop\n\u22a2 \u2203 x, \u2200 (y : \u03b2), f x \u2264 f y\n[PROOFSTEP]\ninhabit \u03b2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : ClosedIicTopology \u03b1\ninst\u271d : Nonempty \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Continuous f\nhlim : Tendsto f (cocompact \u03b2) atTop\ninhabited_h : Inhabited \u03b2\n\u22a2 \u2203 x, \u2200 (y : \u03b2), f x \u2264 f y\n[PROOFSTEP]\nexact hf.exists_forall_le' default (hlim.eventually <| eventually_ge_atTop _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2076 : LinearOrder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : ClosedIicTopology \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : One \u03b1\nf : \u03b2 \u2192 \u03b1\nhf : Continuous f\nh : HasCompactMulSupport f\n\u22a2 \u2203 x, \u2200 (y : \u03b2), f x \u2264 f y\n[PROOFSTEP]\nobtain \u27e8_, \u27e8x, rfl\u27e9, hx\u27e9 := (h.isCompact_range hf).exists_isLeast (range_nonempty _)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2076 : LinearOrder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : ClosedIicTopology \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : One \u03b1\nf : \u03b2 \u2192 \u03b1\nhf : Continuous f\nh : HasCompactMulSupport f\nx : \u03b2\nhx : f x \u2208 lowerBounds (range f)\n\u22a2 \u2203 x, \u2200 (y : \u03b2), f x \u2264 f y\n[PROOFSTEP]\nrw [mem_lowerBounds, forall_range_iff] at hx \n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2076 : LinearOrder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : ClosedIicTopology \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : One \u03b1\nf : \u03b2 \u2192 \u03b1\nhf : Continuous f\nh : HasCompactMulSupport f\nx : \u03b2\nhx : \u2200 (i : \u03b2), f x \u2264 f i\n\u22a2 \u2203 x, \u2200 (y : \u03b2), f x \u2264 f y\n[PROOFSTEP]\nexact \u27e8x, hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : ClosedIicTopology \u03b1\ninst\u271d : Nonempty \u03b1\ns : Set \u03b1\nhs : IsCompact s\n\u22a2 BddBelow s\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with rfl | hne\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : ClosedIicTopology \u03b1\ninst\u271d : Nonempty \u03b1\nhs : IsCompact \u2205\n\u22a2 BddBelow \u2205\n[PROOFSTEP]\nexact bddBelow_empty\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : ClosedIicTopology \u03b1\ninst\u271d : Nonempty \u03b1\ns : Set \u03b1\nhs : IsCompact s\nhne : Set.Nonempty s\n\u22a2 BddBelow s\n[PROOFSTEP]\nobtain \u27e8a, -, has\u27e9 := hs.exists_isLeast hne\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : ClosedIicTopology \u03b1\ninst\u271d : Nonempty \u03b1\ns : Set \u03b1\nhs : IsCompact s\nhne : Set.Nonempty s\na : \u03b1\nhas : a \u2208 lowerBounds s\n\u22a2 BddBelow s\n[PROOFSTEP]\nexact \u27e8a, has\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIciTopology \u03b1\nf : \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nh0K : Set.Nonempty K\nhf : ContinuousOn f K\ny : \u03b1\n\u22a2 sSup (f '' K) < y \u2194 \u2200 (x : \u03b2), x \u2208 K \u2192 f x < y\n[PROOFSTEP]\nrefine' \u27e8fun h x hx => (le_csSup (hK.bddAbove_image hf) <| mem_image_of_mem f hx).trans_lt h, fun h => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIciTopology \u03b1\nf : \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nh0K : Set.Nonempty K\nhf : ContinuousOn f K\ny : \u03b1\nh : \u2200 (x : \u03b2), x \u2208 K \u2192 f x < y\n\u22a2 sSup (f '' K) < y\n[PROOFSTEP]\nobtain \u27e8x, hx, h2x\u27e9 := hK.exists_forall_ge h0K hf\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIciTopology \u03b1\nf : \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nh0K : Set.Nonempty K\nhf : ContinuousOn f K\ny : \u03b1\nh : \u2200 (x : \u03b2), x \u2208 K \u2192 f x < y\nx : \u03b2\nhx : x \u2208 K\nh2x : \u2200 (y : \u03b2), y \u2208 K \u2192 f y \u2264 f x\n\u22a2 sSup (f '' K) < y\n[PROOFSTEP]\nrefine' (csSup_le (h0K.image f) _).trans_lt (h x hx)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIciTopology \u03b1\nf : \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nh0K : Set.Nonempty K\nhf : ContinuousOn f K\ny : \u03b1\nh : \u2200 (x : \u03b2), x \u2208 K \u2192 f x < y\nx : \u03b2\nhx : x \u2208 K\nh2x : \u2200 (y : \u03b2), y \u2208 K \u2192 f y \u2264 f x\n\u22a2 \u2200 (b : \u03b1), b \u2208 f '' K \u2192 b \u2264 f x\n[PROOFSTEP]\nrintro _ \u27e8x', hx', rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIciTopology \u03b1\nf : \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nh0K : Set.Nonempty K\nhf : ContinuousOn f K\ny : \u03b1\nh : \u2200 (x : \u03b2), x \u2208 K \u2192 f x < y\nx : \u03b2\nhx : x \u2208 K\nh2x : \u2200 (y : \u03b2), y \u2208 K \u2192 f y \u2264 f x\nx' : \u03b2\nhx' : x' \u2208 K\n\u22a2 f x' \u2264 f x\n[PROOFSTEP]\nexact h2x x' hx'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\n\u22a2 Continuous fun x => sSup (f x '' K)\n[PROOFSTEP]\nrcases eq_empty_or_nonempty K with (rfl | h0K)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nhf : Continuous \u21bff\nhK : IsCompact \u2205\n\u22a2 Continuous fun x => sSup (f x '' \u2205)\n[PROOFSTEP]\nsimp_rw [image_empty]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nhf : Continuous \u21bff\nhK : IsCompact \u2205\n\u22a2 Continuous fun x => sSup \u2205\n[PROOFSTEP]\nexact continuous_const\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\nh0K : Set.Nonempty K\n\u22a2 Continuous fun x => sSup (f x '' K)\n[PROOFSTEP]\nrw [continuous_iff_continuousAt]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\nh0K : Set.Nonempty K\n\u22a2 \u2200 (x : \u03b3), ContinuousAt (fun x => sSup (f x '' K)) x\n[PROOFSTEP]\nintro x\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\nh0K : Set.Nonempty K\nx : \u03b3\n\u22a2 ContinuousAt (fun x => sSup (f x '' K)) x\n[PROOFSTEP]\nobtain \u27e8y, hyK, h2y, hy\u27e9 :=\n hK.exists_sSup_image_eq_and_ge h0K (show Continuous fun y => f x y from hf.comp <| Continuous.Prod.mk x).continuousOn\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\nh0K : Set.Nonempty K\nx : \u03b3\ny : \u03b2\nhyK : y \u2208 K\nh2y : sSup ((fun y => f x y) '' K) = f x y\nhy : \u2200 (y_1 : \u03b2), y_1 \u2208 K \u2192 f x y_1 \u2264 f x y\n\u22a2 ContinuousAt (fun x => sSup (f x '' K)) x\n[PROOFSTEP]\nrw [ContinuousAt, h2y, tendsto_order]\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\nh0K : Set.Nonempty K\nx : \u03b3\ny : \u03b2\nhyK : y \u2208 K\nh2y : sSup ((fun y => f x y) '' K) = f x y\nhy : \u2200 (y_1 : \u03b2), y_1 \u2208 K \u2192 f x y_1 \u2264 f x y\n\u22a2 (\u2200 (a' : \u03b1), a' < f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, a' < sSup (f b '' K)) \u2227\n \u2200 (a' : \u03b1), a' > f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, sSup (f b '' K) < a'\n[PROOFSTEP]\nhave :=\n tendsto_order.mp ((show Continuous fun x => f x y from hf.comp <| continuous_id.prod_mk continuous_const).tendsto x)\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\nh0K : Set.Nonempty K\nx : \u03b3\ny : \u03b2\nhyK : y \u2208 K\nh2y : sSup ((fun y => f x y) '' K) = f x y\nhy : \u2200 (y_1 : \u03b2), y_1 \u2208 K \u2192 f x y_1 \u2264 f x y\nthis : (\u2200 (a' : \u03b1), a' < f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, a' < f b y) \u2227 \u2200 (a' : \u03b1), a' > f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, f b y < a'\n\u22a2 (\u2200 (a' : \u03b1), a' < f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, a' < sSup (f b '' K)) \u2227\n \u2200 (a' : \u03b1), a' > f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, sSup (f b '' K) < a'\n[PROOFSTEP]\nrefine' \u27e8fun z hz => _, fun z hz => _\u27e9\n[GOAL]\ncase inr.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\nh0K : Set.Nonempty K\nx : \u03b3\ny : \u03b2\nhyK : y \u2208 K\nh2y : sSup ((fun y => f x y) '' K) = f x y\nhy : \u2200 (y_1 : \u03b2), y_1 \u2208 K \u2192 f x y_1 \u2264 f x y\nthis : (\u2200 (a' : \u03b1), a' < f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, a' < f b y) \u2227 \u2200 (a' : \u03b1), a' > f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, f b y < a'\nz : \u03b1\nhz : z < f x y\n\u22a2 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, z < sSup (f b '' K)\n[PROOFSTEP]\nrefine' (this.1 z hz).mono fun x' hx' => hx'.trans_le <| le_csSup _ <| mem_image_of_mem (f x') hyK\n[GOAL]\ncase inr.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\nh0K : Set.Nonempty K\nx : \u03b3\ny : \u03b2\nhyK : y \u2208 K\nh2y : sSup ((fun y => f x y) '' K) = f x y\nhy : \u2200 (y_1 : \u03b2), y_1 \u2208 K \u2192 f x y_1 \u2264 f x y\nthis : (\u2200 (a' : \u03b1), a' < f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, a' < f b y) \u2227 \u2200 (a' : \u03b1), a' > f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, f b y < a'\nz : \u03b1\nhz : z < f x y\nx' : \u03b3\nhx' : z < f x' y\n\u22a2 BddAbove (f x' '' K)\n[PROOFSTEP]\nexact hK.bddAbove_image (hf.comp <| Continuous.Prod.mk x').continuousOn\n[GOAL]\ncase inr.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\nh0K : Set.Nonempty K\nx : \u03b3\ny : \u03b2\nhyK : y \u2208 K\nh2y : sSup ((fun y => f x y) '' K) = f x y\nhy : \u2200 (y_1 : \u03b2), y_1 \u2208 K \u2192 f x y_1 \u2264 f x y\nthis : (\u2200 (a' : \u03b1), a' < f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, a' < f b y) \u2227 \u2200 (a' : \u03b1), a' > f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, f b y < a'\nz : \u03b1\nhz : z > f x y\n\u22a2 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, sSup (f b '' K) < z\n[PROOFSTEP]\nhave h : ({ x } : Set \u03b3) \u00d7\u02e2 K \u2286 \u21bff \u207b\u00b9' Iio z :=\n by\n rintro \u27e8x', y'\u27e9 \u27e8(rfl : x' = x), hy'\u27e9\n exact (hy y' hy').trans_lt hz\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\nh0K : Set.Nonempty K\nx : \u03b3\ny : \u03b2\nhyK : y \u2208 K\nh2y : sSup ((fun y => f x y) '' K) = f x y\nhy : \u2200 (y_1 : \u03b2), y_1 \u2208 K \u2192 f x y_1 \u2264 f x y\nthis : (\u2200 (a' : \u03b1), a' < f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, a' < f b y) \u2227 \u2200 (a' : \u03b1), a' > f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, f b y < a'\nz : \u03b1\nhz : z > f x y\n\u22a2 {x} \u00d7\u02e2 K \u2286 \u21bff \u207b\u00b9' Iio z\n[PROOFSTEP]\nrintro \u27e8x', y'\u27e9 \u27e8(rfl : x' = x), hy'\u27e9\n[GOAL]\ncase mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\nh0K : Set.Nonempty K\ny : \u03b2\nhyK : y \u2208 K\nz : \u03b1\nx' : \u03b3\ny' : \u03b2\nhy' : (x', y').snd \u2208 K\nh2y : sSup ((fun y => f x' y) '' K) = f x' y\nhy : \u2200 (y_1 : \u03b2), y_1 \u2208 K \u2192 f x' y_1 \u2264 f x' y\nthis :\n (\u2200 (a' : \u03b1), a' < f x' y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x', a' < f b y) \u2227 \u2200 (a' : \u03b1), a' > f x' y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x', f b y < a'\nhz : z > f x' y\n\u22a2 (x', y') \u2208 \u21bff \u207b\u00b9' Iio z\n[PROOFSTEP]\nexact (hy y' hy').trans_lt hz\n[GOAL]\ncase inr.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\nh0K : Set.Nonempty K\nx : \u03b3\ny : \u03b2\nhyK : y \u2208 K\nh2y : sSup ((fun y => f x y) '' K) = f x y\nhy : \u2200 (y_1 : \u03b2), y_1 \u2208 K \u2192 f x y_1 \u2264 f x y\nthis : (\u2200 (a' : \u03b1), a' < f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, a' < f b y) \u2227 \u2200 (a' : \u03b1), a' > f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, f b y < a'\nz : \u03b1\nhz : z > f x y\nh : {x} \u00d7\u02e2 K \u2286 \u21bff \u207b\u00b9' Iio z\n\u22a2 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, sSup (f b '' K) < z\n[PROOFSTEP]\nobtain \u27e8u, v, hu, _, hxu, hKv, huv\u27e9 := generalized_tube_lemma isCompact_singleton hK (isOpen_Iio.preimage hf) h\n[GOAL]\ncase inr.intro.intro.intro.refine'_2.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\nh0K : Set.Nonempty K\nx : \u03b3\ny : \u03b2\nhyK : y \u2208 K\nh2y : sSup ((fun y => f x y) '' K) = f x y\nhy : \u2200 (y_1 : \u03b2), y_1 \u2208 K \u2192 f x y_1 \u2264 f x y\nthis : (\u2200 (a' : \u03b1), a' < f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, a' < f b y) \u2227 \u2200 (a' : \u03b1), a' > f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, f b y < a'\nz : \u03b1\nhz : z > f x y\nh : {x} \u00d7\u02e2 K \u2286 \u21bff \u207b\u00b9' Iio z\nu : Set \u03b3\nv : Set \u03b2\nhu : IsOpen u\nleft\u271d : IsOpen v\nhxu : {x} \u2286 u\nhKv : K \u2286 v\nhuv : u \u00d7\u02e2 v \u2286 \u21bff \u207b\u00b9' Iio z\n\u22a2 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, sSup (f b '' K) < z\n[PROOFSTEP]\nrefine' eventually_of_mem (hu.mem_nhds (singleton_subset_iff.mp hxu)) fun x' hx' => _\n[GOAL]\ncase inr.intro.intro.intro.refine'_2.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\nh0K : Set.Nonempty K\nx : \u03b3\ny : \u03b2\nhyK : y \u2208 K\nh2y : sSup ((fun y => f x y) '' K) = f x y\nhy : \u2200 (y_1 : \u03b2), y_1 \u2208 K \u2192 f x y_1 \u2264 f x y\nthis : (\u2200 (a' : \u03b1), a' < f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, a' < f b y) \u2227 \u2200 (a' : \u03b1), a' > f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, f b y < a'\nz : \u03b1\nhz : z > f x y\nh : {x} \u00d7\u02e2 K \u2286 \u21bff \u207b\u00b9' Iio z\nu : Set \u03b3\nv : Set \u03b2\nhu : IsOpen u\nleft\u271d : IsOpen v\nhxu : {x} \u2286 u\nhKv : K \u2286 v\nhuv : u \u00d7\u02e2 v \u2286 \u21bff \u207b\u00b9' Iio z\nx' : \u03b3\nhx' : x' \u2208 u\n\u22a2 sSup (f x' '' K) < z\n[PROOFSTEP]\nrw [hK.sSup_lt_iff_of_continuous h0K (show Continuous (f x') from hf.comp <| Continuous.Prod.mk x').continuousOn]\n[GOAL]\ncase inr.intro.intro.intro.refine'_2.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b1\nK : Set \u03b2\nhK : IsCompact K\nhf : Continuous \u21bff\nh0K : Set.Nonempty K\nx : \u03b3\ny : \u03b2\nhyK : y \u2208 K\nh2y : sSup ((fun y => f x y) '' K) = f x y\nhy : \u2200 (y_1 : \u03b2), y_1 \u2208 K \u2192 f x y_1 \u2264 f x y\nthis : (\u2200 (a' : \u03b1), a' < f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, a' < f b y) \u2227 \u2200 (a' : \u03b1), a' > f x y \u2192 \u2200\u1da0 (b : \u03b3) in \ud835\udcdd x, f b y < a'\nz : \u03b1\nhz : z > f x y\nh : {x} \u00d7\u02e2 K \u2286 \u21bff \u207b\u00b9' Iio z\nu : Set \u03b3\nv : Set \u03b2\nhu : IsOpen u\nleft\u271d : IsOpen v\nhxu : {x} \u2286 u\nhKv : K \u2286 v\nhuv : u \u00d7\u02e2 v \u2286 \u21bff \u207b\u00b9' Iio z\nx' : \u03b3\nhx' : x' \u2208 u\n\u22a2 \u2200 (x : \u03b2), x \u2208 K \u2192 f x' x < z\n[PROOFSTEP]\nexact fun y' hy' => huv (mk_mem_prod hx' (hKv hy'))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : DenselyOrdered \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\na b c : \u03b1\nh : ContinuousOn f [[a, b]]\n\u22a2 f '' [[a, b]] = [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]\n[PROOFSTEP]\nrefine' h.image_uIcc_eq_Icc.trans (uIcc_of_le _).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : DenselyOrdered \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\na b c : \u03b1\nh : ContinuousOn f [[a, b]]\n\u22a2 sInf (f '' [[a, b]]) \u2264 sSup (f '' [[a, b]])\n[PROOFSTEP]\nrefine' csInf_le_csSup _ _ (nonempty_uIcc.image _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : DenselyOrdered \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\na b c : \u03b1\nh : ContinuousOn f [[a, b]]\n\u22a2 BddBelow (f '' [[a, b]])\n[PROOFSTEP]\nrw [h.image_uIcc_eq_Icc]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : DenselyOrdered \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\na b c : \u03b1\nh : ContinuousOn f [[a, b]]\n\u22a2 BddAbove (f '' [[a, b]])\n[PROOFSTEP]\nrw [h.image_uIcc_eq_Icc]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : DenselyOrdered \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\na b c : \u03b1\nh : ContinuousOn f [[a, b]]\n\u22a2 BddBelow (Icc (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])))\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : DenselyOrdered \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\na b c : \u03b1\nh : ContinuousOn f [[a, b]]\n\u22a2 BddAbove (Icc (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])))\n[PROOFSTEP]\nexacts [bddBelow_Icc, bddAbove_Icc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : DenselyOrdered \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\na b c : \u03b1\nh : ContinuousOn f (Icc a b)\nhc : c \u2208 Icc a b\n\u22a2 sInf (f '' Icc a b) \u2264 f c\n[PROOFSTEP]\nhave := mem_image_of_mem f hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : DenselyOrdered \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\na b c : \u03b1\nh : ContinuousOn f (Icc a b)\nhc : c \u2208 Icc a b\nthis : f c \u2208 f '' Icc a b\n\u22a2 sInf (f '' Icc a b) \u2264 f c\n[PROOFSTEP]\nrw [h.image_Icc (hc.1.trans hc.2)] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : DenselyOrdered \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\na b c : \u03b1\nh : ContinuousOn f (Icc a b)\nhc : c \u2208 Icc a b\nthis : f c \u2208 Icc (sInf (f '' Icc a b)) (sSup (f '' Icc a b))\n\u22a2 sInf (f '' Icc a b) \u2264 f c\n[PROOFSTEP]\nexact this.1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : DenselyOrdered \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\na b c : \u03b1\nh : ContinuousOn f (Icc a b)\nhc : c \u2208 Icc a b\n\u22a2 f c \u2264 sSup (f '' Icc a b)\n[PROOFSTEP]\nhave := mem_image_of_mem f hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : DenselyOrdered \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\na b c : \u03b1\nh : ContinuousOn f (Icc a b)\nhc : c \u2208 Icc a b\nthis : f c \u2208 f '' Icc a b\n\u22a2 f c \u2264 sSup (f '' Icc a b)\n[PROOFSTEP]\nrw [h.image_Icc (hc.1.trans hc.2)] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : DenselyOrdered \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\na b c : \u03b1\nh : ContinuousOn f (Icc a b)\nhc : c \u2208 Icc a b\nthis : f c \u2208 Icc (sInf (f '' Icc a b)) (sSup (f '' Icc a b))\n\u22a2 f c \u2264 sSup (f '' Icc a b)\n[PROOFSTEP]\nexact this.2\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Order.Compact", "llama_tokens": 25874, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943925708561, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.4067377454830883}} {"text": "[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\ni : \u2115\n\u22a2 coeff (scaleRoots p s) i = coeff p i * s ^ (natDegree p - i)\n[PROOFSTEP]\nsimp (config := { contextual := true }) [scaleRoots, coeff_monomial]\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\n\u22a2 coeff (scaleRoots p s) (natDegree p) = leadingCoeff p\n[PROOFSTEP]\nrw [leadingCoeff, coeff_scaleRoots, tsub_self, pow_zero, mul_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\ns : R\n\u22a2 scaleRoots 0 s = 0\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\ns : R\nn\u271d : \u2115\n\u22a2 coeff (scaleRoots 0 s) n\u271d = coeff 0 n\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nhp : p \u2260 0\ns : R\n\u22a2 scaleRoots p s \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nhp : p \u2260 0\ns : R\nh : scaleRoots p s = 0\n\u22a2 False\n[PROOFSTEP]\nhave : p.coeff p.natDegree \u2260 0 := mt leadingCoeff_eq_zero.mp hp\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nhp : p \u2260 0\ns : R\nh : scaleRoots p s = 0\nthis : coeff p (natDegree p) \u2260 0\n\u22a2 False\n[PROOFSTEP]\nhave : (scaleRoots p s).coeff p.natDegree = 0 := congr_fun (congr_arg (coeff : R[X] \u2192 \u2115 \u2192 R) h) p.natDegree\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nhp : p \u2260 0\ns : R\nh : scaleRoots p s = 0\nthis\u271d : coeff p (natDegree p) \u2260 0\nthis : coeff (scaleRoots p s) (natDegree p) = 0\n\u22a2 False\n[PROOFSTEP]\nrw [coeff_scaleRoots_natDegree] at this \n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nhp : p \u2260 0\ns : R\nh : scaleRoots p s = 0\nthis\u271d : coeff p (natDegree p) \u2260 0\nthis : leadingCoeff p = 0\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\n\u22a2 support (scaleRoots p s) \u2264 support p\n[PROOFSTEP]\nintro\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\na\u271d : \u2115\n\u22a2 a\u271d \u2208 support (scaleRoots p s) \u2192 a\u271d \u2208 support p\n[PROOFSTEP]\nsimpa using left_ne_zero_of_mul\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\nhs : s \u2208 nonZeroDivisors R\n\u22a2 support p \u2264 support (scaleRoots p s)\n[PROOFSTEP]\nintro i\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\nhs : s \u2208 nonZeroDivisors R\ni : \u2115\n\u22a2 i \u2208 support p \u2192 i \u2208 support (scaleRoots p s)\n[PROOFSTEP]\nsimp only [coeff_scaleRoots, Polynomial.mem_support_iff]\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\nhs : s \u2208 nonZeroDivisors R\ni : \u2115\n\u22a2 coeff p i \u2260 0 \u2192 coeff p i * s ^ (natDegree p - i) \u2260 0\n[PROOFSTEP]\nintro p_ne_zero ps_zero\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\nhs : s \u2208 nonZeroDivisors R\ni : \u2115\np_ne_zero : coeff p i \u2260 0\nps_zero : coeff p i * s ^ (natDegree p - i) = 0\n\u22a2 False\n[PROOFSTEP]\nhave := pow_mem hs (p.natDegree - i) _ ps_zero\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\nhs : s \u2208 nonZeroDivisors R\ni : \u2115\np_ne_zero : coeff p i \u2260 0\nps_zero : coeff p i * s ^ (natDegree p - i) = 0\nthis : coeff p i = 0\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\n\u22a2 degree (scaleRoots p s) = degree p\n[PROOFSTEP]\nhaveI := Classical.propDecidable\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\nthis : (a : Prop) \u2192 Decidable a\n\u22a2 degree (scaleRoots p s) = degree p\n[PROOFSTEP]\nby_cases hp : p = 0\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\nthis : (a : Prop) \u2192 Decidable a\nhp : p = 0\n\u22a2 degree (scaleRoots p s) = degree p\n[PROOFSTEP]\nrw [hp, zero_scaleRoots]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\nthis : (a : Prop) \u2192 Decidable a\nhp : \u00acp = 0\n\u22a2 degree (scaleRoots p s) = degree p\n[PROOFSTEP]\nrefine' le_antisymm (Finset.sup_mono (support_scaleRoots_le p s)) (degree_le_degree _)\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\nthis : (a : Prop) \u2192 Decidable a\nhp : \u00acp = 0\n\u22a2 coeff (scaleRoots p s) (natDegree p) \u2260 0\n[PROOFSTEP]\nrw [coeff_scaleRoots_natDegree]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\nthis : (a : Prop) \u2192 Decidable a\nhp : \u00acp = 0\n\u22a2 leadingCoeff p \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\nthis : (a : Prop) \u2192 Decidable a\nhp : \u00acp = 0\nh : leadingCoeff p = 0\n\u22a2 False\n[PROOFSTEP]\nhave := leadingCoeff_eq_zero.mp h\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\nthis\u271d : (a : Prop) \u2192 Decidable a\nhp : \u00acp = 0\nh : leadingCoeff p = 0\nthis : p = 0\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\n\u22a2 natDegree (scaleRoots p s) = natDegree p\n[PROOFSTEP]\nsimp only [natDegree, degree_scaleRoots]\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\ns : R\n\u22a2 Monic (scaleRoots p s) \u2194 Monic p\n[PROOFSTEP]\nsimp only [Monic, leadingCoeff, natDegree_scaleRoots, coeff_scaleRoots_natDegree]\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nx : R\nf : R \u2192+* S\nh : \u2191f (leadingCoeff p) \u2260 0\n\u22a2 map f (scaleRoots p x) = scaleRoots (map f p) (\u2191f x)\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nx : R\nf : R \u2192+* S\nh : \u2191f (leadingCoeff p) \u2260 0\nn\u271d : \u2115\n\u22a2 coeff (map f (scaleRoots p x)) n\u271d = coeff (scaleRoots (map f p) (\u2191f x)) n\u271d\n[PROOFSTEP]\nsimp [Polynomial.natDegree_map_of_leadingCoeff_ne_zero _ h]\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Field K\np : S[X]\nf : S \u2192+* R\nr : R\ns : S\n\u22a2 eval\u2082 f (\u2191f s * r) (scaleRoots p s) =\n \u2211 i in support (scaleRoots p s), \u2191f (coeff p i * s ^ (natDegree p - i)) * (\u2191f s * r) ^ i\n[PROOFSTEP]\nsimp [eval\u2082_eq_sum, sum_def]\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Field K\np : S[X]\nf : S \u2192+* R\nr : R\ns : S\ni : \u2115\n_hi : i \u2208 support p\nhi' : \u00aci \u2208 support (scaleRoots p s)\n\u22a2 \u2191f (coeff p i * s ^ (natDegree p - i)) * (\u2191f s * r) ^ i = 0\n[PROOFSTEP]\nlet this : coeff p i * s ^ (p.natDegree - i) = 0 := by simpa using hi'\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Field K\np : S[X]\nf : S \u2192+* R\nr : R\ns : S\ni : \u2115\n_hi : i \u2208 support p\nhi' : \u00aci \u2208 support (scaleRoots p s)\n\u22a2 coeff p i * s ^ (natDegree p - i) = 0\n[PROOFSTEP]\nsimpa using hi'\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Field K\np : S[X]\nf : S \u2192+* R\nr : R\ns : S\ni : \u2115\n_hi : i \u2208 support p\nhi' : \u00aci \u2208 support (scaleRoots p s)\nthis : coeff p i * s ^ (natDegree p - i) = 0 :=\n Eq.mp\n (Eq.trans\n (congrArg Not\n (Eq.trans Mathlib.Data.Polynomial.Basic._auxLemma.17 (congrFun (congrArg Ne (coeff_scaleRoots p s i)) 0)))\n Mathlib.Logic.Basic._auxLemma.9)\n hi'\n\u22a2 \u2191f (coeff p i * s ^ (natDegree p - i)) * (\u2191f s * r) ^ i = 0\n[PROOFSTEP]\nsimp [this]\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Field K\np : S[X]\nf : S \u2192+* R\nr : R\ns : S\ni : \u2115\n_hi : i \u2208 support p\n\u22a2 \u2191f (coeff p i * s ^ (natDegree p - i)) * (\u2191f s * r) ^ i = \u2191f (coeff p i) * \u2191f s ^ (natDegree p - i + i) * r ^ i\n[PROOFSTEP]\nsimp_rw [f.map_mul, f.map_pow, pow_add, mul_pow, mul_assoc]\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Field K\np : S[X]\nf : S \u2192+* R\nr : R\ns : S\ni : \u2115\nhi : i \u2208 support p\n\u22a2 \u2191f (coeff p i) * \u2191f s ^ (natDegree p - i + i) * r ^ i = \u2191f s ^ natDegree p * (\u2191f (coeff p i) * r ^ i)\n[PROOFSTEP]\nrw [mul_assoc, mul_left_comm, tsub_add_cancel_of_le]\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Field K\np : S[X]\nf : S \u2192+* R\nr : R\ns : S\ni : \u2115\nhi : i \u2208 support p\n\u22a2 i \u2264 natDegree p\n[PROOFSTEP]\nexact le_natDegree_of_ne_zero (Polynomial.mem_support_iff.mp hi)\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Field K\np : S[X]\nf : S \u2192+* R\nr : R\ns : S\n\u22a2 \u2191f s ^ natDegree p * \u2211 i in support p, \u2191f (coeff p i) * r ^ i = \u2191f s ^ natDegree p * eval\u2082 f r p\n[PROOFSTEP]\nsimp [eval\u2082_eq_sum, sum_def]\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Field K\np : S[X]\nf : S \u2192+* R\nr : R\ns : S\nhr : eval\u2082 f r p = 0\n\u22a2 eval\u2082 f (\u2191f s * r) (scaleRoots p s) = 0\n[PROOFSTEP]\nrw [scaleRoots_eval\u2082_mul, hr, mul_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Field K\np : S[X]\nf : S \u2192+* K\nhf : Function.Injective \u2191f\nr s : S\nhr : eval\u2082 f (\u2191f r / \u2191f s) p = 0\nhs : s \u2208 nonZeroDivisors S\n\u22a2 eval\u2082 f (\u2191f r) (scaleRoots p s) = 0\n[PROOFSTEP]\nnontriviality S using Subsingleton.eq_zero\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Field K\np : S[X]\nf : S \u2192+* K\nhf : Function.Injective \u2191f\nr s : S\nhr : eval\u2082 f (\u2191f r / \u2191f s) p = 0\nhs : s \u2208 nonZeroDivisors S\n\u271d : Nontrivial S\n\u22a2 eval\u2082 f (\u2191f r) (scaleRoots p s) = 0\n[PROOFSTEP]\nconvert @scaleRoots_eval\u2082_eq_zero _ _ _ _ p f _ s hr\n[GOAL]\ncase h.e'_2.h.e'_6\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Field K\np : S[X]\nf : S \u2192+* K\nhf : Function.Injective \u2191f\nr s : S\nhr : eval\u2082 f (\u2191f r / \u2191f s) p = 0\nhs : s \u2208 nonZeroDivisors S\n\u271d : Nontrivial S\n\u22a2 \u2191f r = \u2191f s * (\u2191f r / \u2191f s)\n[PROOFSTEP]\nrw [\u2190 mul_div_assoc, mul_comm, mul_div_cancel]\n[GOAL]\ncase h.e'_2.h.e'_6.h\nR : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Field K\np : S[X]\nf : S \u2192+* K\nhf : Function.Injective \u2191f\nr s : S\nhr : eval\u2082 f (\u2191f r / \u2191f s) p = 0\nhs : s \u2208 nonZeroDivisors S\n\u271d : Nontrivial S\n\u22a2 \u2191f s \u2260 0\n[PROOFSTEP]\nexact map_ne_zero_of_mem_nonZeroDivisors _ hf hs\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Polynomial.ScaleRoots", "llama_tokens": 5881, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494421679929, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.4065393236300952}} {"text": "[GOAL]\nR\u271d : Type ?u.74090\ninst\u271d\u00b9 : CommSemiring R\u271d\nR : Type u_1\ninst\u271d : Ring R\na b : R\n\u22a2 a * -b = -(a * b)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d : CommSemiring R\n\u22a2 Nat.rawCast 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d : CommSemiring R\n\u22a2 Nat.rawCast 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type ?u.76702\ninst\u271d\u00b9 : CommSemiring R\u271d\nR : Type u_1\ninst\u271d : Ring R\n\u22a2 Int.rawCast (Int.negOfNat 1) = -1\n[PROOFSTEP]\nsimp [Int.negOfNat_eq]\n[GOAL]\nR\u271d : Type ?u.78211\ninst\u271d\u00b2 : CommSemiring R\u271d\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nat.AtLeastTwo n\n\u22a2 Int.rawCast (Int.negOfNat n) = -OfNat.ofNat n\n[PROOFSTEP]\nsimp [Int.negOfNat_eq, OfNat.ofNat]\n[GOAL]\nR\u271d : Type ?u.80333\ninst\u271d\u00b9 : CommSemiring R\u271d\nn : \u2124\nd : \u2115\nR : Type u_1\ninst\u271d : DivisionRing R\n\u22a2 Rat.rawCast n d = \u2191n / \u2191d\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Tactic.Ring.RingNF", "llama_tokens": 452, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544085240401, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4064787488216858}} {"text": "[GOAL]\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nlet F' : J \u2964 TopCat := F \u22d9 TopCat.discrete\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\nF' : J \u2964 TopCat := F \u22d9 TopCat.discrete\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nhaveI : \u2200 j, DiscreteTopology (F'.obj j) := fun _ => \u27e8rfl\u27e9\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\nF' : J \u2964 TopCat := F \u22d9 TopCat.discrete\nthis : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j)\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nhaveI : \u2200 j, Finite (F'.obj j) := hf\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\nF' : J \u2964 TopCat := F \u22d9 TopCat.discrete\nthis\u271d : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j)\nthis : \u2200 (j : J), Finite \u2191(F'.obj j)\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nhaveI : \u2200 j, Nonempty (F'.obj j) := hne\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\nF' : J \u2964 TopCat := F \u22d9 TopCat.discrete\nthis\u271d\u00b9 : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j)\nthis\u271d : \u2200 (j : J), Finite \u2191(F'.obj j)\nthis : \u2200 (j : J), _root_.Nonempty \u2191(F'.obj j)\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nobtain \u27e8\u27e8u, hu\u27e9\u27e9 := TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system.{u} F'\n[GOAL]\ncase intro.mk\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\nF' : J \u2964 TopCat := F \u22d9 TopCat.discrete\nthis\u271d\u00b9 : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j)\nthis\u271d : \u2200 (j : J), Finite \u2191(F'.obj j)\nthis : \u2200 (j : J), _root_.Nonempty \u2191(F'.obj j)\nu : (j : J) \u2192 \u2191(F'.obj j)\nhu : u \u2208 {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F'.map f) (u i) = u j}\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nexact \u27e8u, hu\u27e9\n[GOAL]\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nlet J' : Type max w v u := AsSmall.{max w v} J\n[GOAL]\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nlet down : J' \u2964 J := AsSmall.down\n[GOAL]\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nlet F' : J' \u2964 Type max u v w := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\n[GOAL]\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nhaveI : \u2200 i, Nonempty (F'.obj i) := fun i => \u27e8\u27e8Classical.arbitrary (F.obj (down.obj i))\u27e9\u27e9\n[GOAL]\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis : \u2200 (i : J'), _root_.Nonempty (F'.obj i)\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nhaveI : \u2200 i, Finite (F'.obj i) := fun i => Finite.of_equiv (F.obj (down.obj i)) Equiv.ulift.symm\n[GOAL]\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d : \u2200 (i : J'), _root_.Nonempty (F'.obj i)\nthis : \u2200 (i : J'), Finite (F'.obj i)\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\ncases isEmpty_or_nonempty J\n[GOAL]\ncase inl\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d : \u2200 (i : J'), _root_.Nonempty (F'.obj i)\nthis : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : IsEmpty J\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase inl.w\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d : \u2200 (i : J'), _root_.Nonempty (F'.obj i)\nthis : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : IsEmpty J\n\u22a2 (j : J) \u2192 F.obj j\n[PROOFSTEP]\napply isEmptyElim\n[GOAL]\ncase inl.h\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d : \u2200 (i : J'), _root_.Nonempty (F'.obj i)\nthis : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : IsEmpty J\n\u22a2 isEmptyElim \u2208 Functor.sections F\n[PROOFSTEP]\napply isEmptyElim\n[GOAL]\ncase inr\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d : \u2200 (i : J'), _root_.Nonempty (F'.obj i)\nthis : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : _root_.Nonempty J\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nhaveI : IsCofiltered J := \u27e8\u27e9\n[GOAL]\ncase inr\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), _root_.Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : _root_.Nonempty J\nthis : IsCofiltered J\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nobtain \u27e8u, hu\u27e9 := nonempty_sections_of_finite_cofiltered_system.init F'\n[GOAL]\ncase inr.intro\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), _root_.Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : _root_.Nonempty J\nthis : IsCofiltered J\nu : (j : J') \u2192 F'.obj j\nhu : u \u2208 Functor.sections F'\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nuse fun j => (u \u27e8j\u27e9).down\n[GOAL]\ncase h\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), _root_.Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : _root_.Nonempty J\nthis : IsCofiltered J\nu : (j : J') \u2192 F'.obj j\nhu : u \u2208 Functor.sections F'\n\u22a2 (fun j => (u { down := j }).down) \u2208 Functor.sections F\n[PROOFSTEP]\nintro j j' f\n[GOAL]\ncase h\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), _root_.Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : _root_.Nonempty J\nthis : IsCofiltered J\nu : (j : J') \u2192 F'.obj j\nhu : u \u2208 Functor.sections F'\nj j' : J\nf : j \u27f6 j'\n\u22a2 F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'\n[PROOFSTEP]\nhave h := @hu (\u27e8j\u27e9 : J') (\u27e8j'\u27e9 : J') (ULift.up f)\n[GOAL]\ncase h\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), _root_.Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : _root_.Nonempty J\nthis : IsCofiltered J\nu : (j : J') \u2192 F'.obj j\nhu : u \u2208 Functor.sections F'\nj j' : J\nf : j \u27f6 j'\nh : F'.map { down := f } (u { down := j }) = u { down := j' }\n\u22a2 F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'\n[PROOFSTEP]\nsimp only [AsSmall.down, Functor.comp_map, uliftFunctor_map, Functor.op_map] at h \n[GOAL]\ncase h\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), _root_.Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : _root_.Nonempty J\nthis : IsCofiltered J\nu : (j : J') \u2192 F'.obj j\nhu : u \u2208 Functor.sections F'\nj j' : J\nf : j \u27f6 j'\nh : { down := F.map f (u { down := j }).down } = u { down := j' }\n\u22a2 F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'\n[PROOFSTEP]\nsimp_rw [\u2190 h]\n[GOAL]\nJ : Type u\ninst\u271d\u00b3 : Preorder J\ninst\u271d\u00b2 : IsDirected J fun x x_1 => x \u2264 x_1\nF : J\u1d52\u1d56 \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J\u1d52\u1d56), Finite (F.obj j)\ninst\u271d : \u2200 (j : J\u1d52\u1d56), _root_.Nonempty (F.obj j)\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\ncases isEmpty_or_nonempty J\n[GOAL]\ncase inl\nJ : Type u\ninst\u271d\u00b3 : Preorder J\ninst\u271d\u00b2 : IsDirected J fun x x_1 => x \u2264 x_1\nF : J\u1d52\u1d56 \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J\u1d52\u1d56), Finite (F.obj j)\ninst\u271d : \u2200 (j : J\u1d52\u1d56), _root_.Nonempty (F.obj j)\nh\u271d : IsEmpty J\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nhaveI : IsEmpty J\u1d52\u1d56 :=\n \u27e8fun j => isEmptyElim j.unop\u27e9\n -- TODO: this should be a global instance\n[GOAL]\ncase inl\nJ : Type u\ninst\u271d\u00b3 : Preorder J\ninst\u271d\u00b2 : IsDirected J fun x x_1 => x \u2264 x_1\nF : J\u1d52\u1d56 \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J\u1d52\u1d56), Finite (F.obj j)\ninst\u271d : \u2200 (j : J\u1d52\u1d56), _root_.Nonempty (F.obj j)\nh\u271d : IsEmpty J\nthis : IsEmpty J\u1d52\u1d56\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nexact \u27e8isEmptyElim, by apply isEmptyElim\u27e9\n[GOAL]\nJ : Type u\ninst\u271d\u00b3 : Preorder J\ninst\u271d\u00b2 : IsDirected J fun x x_1 => x \u2264 x_1\nF : J\u1d52\u1d56 \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J\u1d52\u1d56), Finite (F.obj j)\ninst\u271d : \u2200 (j : J\u1d52\u1d56), _root_.Nonempty (F.obj j)\nh\u271d : IsEmpty J\nthis : IsEmpty J\u1d52\u1d56\n\u22a2 (fun a => isEmptyElim a) \u2208 Functor.sections F\n[PROOFSTEP]\napply isEmptyElim\n[GOAL]\ncase inr\nJ : Type u\ninst\u271d\u00b3 : Preorder J\ninst\u271d\u00b2 : IsDirected J fun x x_1 => x \u2264 x_1\nF : J\u1d52\u1d56 \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J\u1d52\u1d56), Finite (F.obj j)\ninst\u271d : \u2200 (j : J\u1d52\u1d56), _root_.Nonempty (F.obj j)\nh\u271d : _root_.Nonempty J\n\u22a2 Set.Nonempty (Functor.sections F)\n[PROOFSTEP]\nexact nonempty_sections_of_finite_cofiltered_system _\n[GOAL]\nJ : Type u\ninst\u271d : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\nh : IsMittagLeffler F\nf : j \u27f6 i\n\u22a2 eventualRange F i \u2286 F.map f '' eventualRange F j\n[PROOFSTEP]\nobtain \u27e8k, g, hg\u27e9 := F.isMittagLeffler_iff_eventualRange.1 h j\n[GOAL]\ncase intro.intro\nJ : Type u\ninst\u271d : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\nh : IsMittagLeffler F\nf : j \u27f6 i\nk : J\ng : k \u27f6 j\nhg : eventualRange F j = range (F.map g)\n\u22a2 eventualRange F i \u2286 F.map f '' eventualRange F j\n[PROOFSTEP]\nrw [hg]\n[GOAL]\ncase intro.intro\nJ : Type u\ninst\u271d : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\nh : IsMittagLeffler F\nf : j \u27f6 i\nk : J\ng : k \u27f6 j\nhg : eventualRange F j = range (F.map g)\n\u22a2 eventualRange F i \u2286 F.map f '' range (F.map g)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro\nJ : Type u\ninst\u271d : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\nh : IsMittagLeffler F\nf : j \u27f6 i\nk : J\ng : k \u27f6 j\nhg : eventualRange F j = range (F.map g)\nx : F.obj i\nhx : x \u2208 eventualRange F i\n\u22a2 x \u2208 F.map f '' range (F.map g)\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 := F.mem_eventualRange_iff.1 hx (g \u226b f)\n[GOAL]\ncase intro.intro.intro\nJ : Type u\ninst\u271d : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\nh : IsMittagLeffler F\nf : j \u27f6 i\nk : J\ng : k \u27f6 j\nhg : eventualRange F j = range (F.map g)\nx : F.obj k\nhx : F.map (g \u226b f) x \u2208 eventualRange F i\n\u22a2 F.map (g \u226b f) x \u2208 F.map f '' range (F.map g)\n[PROOFSTEP]\nrefine' \u27e8_, \u27e8x, rfl\u27e9, by rw [map_comp_apply]\u27e9\n[GOAL]\nJ : Type u\ninst\u271d : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\nh : IsMittagLeffler F\nf : j \u27f6 i\nk : J\ng : k \u27f6 j\nhg : eventualRange F j = range (F.map g)\nx : F.obj k\nhx : F.map (g \u226b f) x \u2208 eventualRange F i\n\u22a2 F.map f (F.map g x) = F.map (g \u226b f) x\n[PROOFSTEP]\nrw [map_comp_apply]\n[GOAL]\nJ : Type u\ninst\u271d : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\nf : i \u27f6 j\ng : j \u27f6 k\nh : eventualRange F k = range (F.map g)\n\u22a2 eventualRange F k = range (F.map (f \u226b g))\n[PROOFSTEP]\napply subset_antisymm\n[GOAL]\ncase a\nJ : Type u\ninst\u271d : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\nf : i \u27f6 j\ng : j \u27f6 k\nh : eventualRange F k = range (F.map g)\n\u22a2 eventualRange F k \u2286 range (F.map (f \u226b g))\n[PROOFSTEP]\napply iInter\u2082_subset\n[GOAL]\ncase a\nJ : Type u\ninst\u271d : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\nf : i \u27f6 j\ng : j \u27f6 k\nh : eventualRange F k = range (F.map g)\n\u22a2 range (F.map (f \u226b g)) \u2286 eventualRange F k\n[PROOFSTEP]\nrw [h, F.map_comp]\n[GOAL]\ncase a\nJ : Type u\ninst\u271d : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\nf : i \u27f6 j\ng : j \u27f6 k\nh : eventualRange F k = range (F.map g)\n\u22a2 range (F.map f \u226b F.map g) \u2286 range (F.map g)\n[PROOFSTEP]\napply range_comp_subset_range\n[GOAL]\nJ : Type u\ninst\u271d : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k\u271d : J\ns : Set (F.obj i)\nh : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\nj k : J\ng : k \u27f6 j\n\u22a2 range (F.map (\ud835\udfd9 j)) \u2286 range (F.map g)\n[PROOFSTEP]\nrw [map_id, types_id, range_id, (h g).range_eq]\n[GOAL]\nJ : Type u\ninst\u271d : Category.{?u.11956, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\nX\u271d Y\u271d : J\ng : X\u271d \u27f6 Y\u271d\nx : F.obj X\u271d\nh : x \u2208 \u22c2 (f : X\u271d \u27f6 i), F.map f \u207b\u00b9' s\n\u22a2 F.map g x \u2208 \u22c2 (f : Y\u271d \u27f6 i), F.map f \u207b\u00b9' s\n[PROOFSTEP]\nrw [mem_iInter] at h \u22a2\n[GOAL]\nJ : Type u\ninst\u271d : Category.{?u.11956, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\nX\u271d Y\u271d : J\ng : X\u271d \u27f6 Y\u271d\nx : F.obj X\u271d\nh : \u2200 (i_1 : X\u271d \u27f6 i), x \u2208 F.map i_1 \u207b\u00b9' s\n\u22a2 \u2200 (i_1 : Y\u271d \u27f6 i), F.map g x \u2208 F.map i_1 \u207b\u00b9' s\n[PROOFSTEP]\nintro f\n[GOAL]\nJ : Type u\ninst\u271d : Category.{?u.11956, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\nX\u271d Y\u271d : J\ng : X\u271d \u27f6 Y\u271d\nx : F.obj X\u271d\nh : \u2200 (i_1 : X\u271d \u27f6 i), x \u2208 F.map i_1 \u207b\u00b9' s\nf : Y\u271d \u27f6 i\n\u22a2 F.map g x \u2208 F.map f \u207b\u00b9' s\n[PROOFSTEP]\nrw [\u2190 mem_preimage, preimage_preimage, mem_preimage]\n[GOAL]\nJ : Type u\ninst\u271d : Category.{?u.11956, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\nX\u271d Y\u271d : J\ng : X\u271d \u27f6 Y\u271d\nx : F.obj X\u271d\nh : \u2200 (i_1 : X\u271d \u27f6 i), x \u2208 F.map i_1 \u207b\u00b9' s\nf : Y\u271d \u27f6 i\n\u22a2 F.map f (F.map g x) \u2208 s\n[PROOFSTEP]\nconvert h (g \u226b f)\n[GOAL]\ncase a\nJ : Type u\ninst\u271d : Category.{?u.11956, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\nX\u271d Y\u271d : J\ng : X\u271d \u27f6 Y\u271d\nx : F.obj X\u271d\nh : \u2200 (i_1 : X\u271d \u27f6 i), x \u2208 F.map i_1 \u207b\u00b9' s\nf : Y\u271d \u27f6 i\n\u22a2 F.map f (F.map g x) \u2208 s \u2194 x \u2208 F.map (g \u226b f) \u207b\u00b9' s\n[PROOFSTEP]\nrw [F.map_comp]\n[GOAL]\ncase a\nJ : Type u\ninst\u271d : Category.{?u.11956, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\nX\u271d Y\u271d : J\ng : X\u271d \u27f6 Y\u271d\nx : F.obj X\u271d\nh : \u2200 (i_1 : X\u271d \u27f6 i), x \u2208 F.map i_1 \u207b\u00b9' s\nf : Y\u271d \u27f6 i\n\u22a2 F.map f (F.map g x) \u2208 s \u2194 x \u2208 (F.map g \u226b F.map f) \u207b\u00b9' s\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type u\ninst\u271d : Category.{?u.11956, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\nj : J\n\u22a2 { obj := fun j => \u2191(\u22c2 (f : j \u27f6 i), F.map f \u207b\u00b9' s),\n map := fun {X Y} g =>\n MapsTo.restrict (F.map g) (\u22c2 (f : X \u27f6 i), F.map f \u207b\u00b9' s) (\u22c2 (f : Y \u27f6 i), F.map f \u207b\u00b9' s)\n (_ : \u2200 (x : F.obj X), x \u2208 \u22c2 (f : X \u27f6 i), F.map f \u207b\u00b9' s \u2192 F.map g x \u2208 \u22c2 (f : Y \u27f6 i), F.map f \u207b\u00b9' s) }.map\n (\ud835\udfd9 j) =\n \ud835\udfd9\n ({ obj := fun j => \u2191(\u22c2 (f : j \u27f6 i), F.map f \u207b\u00b9' s),\n map := fun {X Y} g =>\n MapsTo.restrict (F.map g) (\u22c2 (f : X \u27f6 i), F.map f \u207b\u00b9' s) (\u22c2 (f : Y \u27f6 i), F.map f \u207b\u00b9' s)\n (_ : \u2200 (x : F.obj X), x \u2208 \u22c2 (f : X \u27f6 i), F.map f \u207b\u00b9' s \u2192 F.map g x \u2208 \u22c2 (f : Y \u27f6 i), F.map f \u207b\u00b9' s) }.obj\n j)\n[PROOFSTEP]\nsimp_rw [MapsTo.restrict, Subtype.map, F.map_id]\n[GOAL]\nJ : Type u\ninst\u271d : Category.{?u.11956, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\nj : J\n\u22a2 (fun x => { val := \ud835\udfd9 (F.obj j) \u2191x, property := (_ : (fun x => x \u2208 \u22c2 (f : j \u27f6 i), F.map f \u207b\u00b9' s) (\ud835\udfd9 (F.obj j) \u2191x)) }) =\n \ud835\udfd9 \u2191(\u22c2 (f : j \u27f6 i), F.map f \u207b\u00b9' s)\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\nJ : Type u\ninst\u271d : Category.{?u.11956, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\nj : J\na\u271d : { x // x \u2208 \u22c2 (f : j \u27f6 i), F.map f \u207b\u00b9' s }\n\u22a2 \u2191{ val := \ud835\udfd9 (F.obj j) \u2191a\u271d, property := (_ : (fun x => x \u2208 \u22c2 (f : j \u27f6 i), F.map f \u207b\u00b9' s) (\ud835\udfd9 (F.obj j) \u2191a\u271d)) } =\n \u2191(\ud835\udfd9 (\u2191(\u22c2 (f : j \u27f6 i), F.map f \u207b\u00b9' s)) a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type u\ninst\u271d : Category.{?u.11956, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\nX\u271d Y\u271d Z\u271d : J\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun j => \u2191(\u22c2 (f : j \u27f6 i), F.map f \u207b\u00b9' s),\n map := fun {X Y} g =>\n MapsTo.restrict (F.map g) (\u22c2 (f : X \u27f6 i), F.map f \u207b\u00b9' s) (\u22c2 (f : Y \u27f6 i), F.map f \u207b\u00b9' s)\n (_ : \u2200 (x : F.obj X), x \u2208 \u22c2 (f : X \u27f6 i), F.map f \u207b\u00b9' s \u2192 F.map g x \u2208 \u22c2 (f : Y \u27f6 i), F.map f \u207b\u00b9' s) }.map\n (f \u226b g) =\n { obj := fun j => \u2191(\u22c2 (f : j \u27f6 i), F.map f \u207b\u00b9' s),\n map := fun {X Y} g =>\n MapsTo.restrict (F.map g) (\u22c2 (f : X \u27f6 i), F.map f \u207b\u00b9' s) (\u22c2 (f : Y \u27f6 i), F.map f \u207b\u00b9' s)\n (_ : \u2200 (x : F.obj X), x \u2208 \u22c2 (f : X \u27f6 i), F.map f \u207b\u00b9' s \u2192 F.map g x \u2208 \u22c2 (f : Y \u27f6 i), F.map f \u207b\u00b9' s) }.map\n f \u226b\n { obj := fun j => \u2191(\u22c2 (f : j \u27f6 i), F.map f \u207b\u00b9' s),\n map := fun {X Y} g =>\n MapsTo.restrict (F.map g) (\u22c2 (f : X \u27f6 i), F.map f \u207b\u00b9' s) (\u22c2 (f : Y \u27f6 i), F.map f \u207b\u00b9' s)\n (_ : \u2200 (x : F.obj X), x \u2208 \u22c2 (f : X \u27f6 i), F.map f \u207b\u00b9' s \u2192 F.map g x \u2208 \u22c2 (f : Y \u27f6 i), F.map f \u207b\u00b9' s) }.map\n g\n[PROOFSTEP]\nsimp_rw [MapsTo.restrict, Subtype.map, F.map_comp]\n[GOAL]\nJ : Type u\ninst\u271d : Category.{?u.11956, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\nX\u271d Y\u271d Z\u271d : J\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 (fun x =>\n { val := (F.map f \u226b F.map g) \u2191x,\n property := (_ : (fun x => x \u2208 \u22c2 (f : Z\u271d \u27f6 i), F.map f \u207b\u00b9' s) ((F.map f \u226b F.map g) \u2191x)) }) =\n (fun x => { val := F.map f \u2191x, property := (_ : F.map f \u2191x \u2208 \u22c2 (f : Y\u271d \u27f6 i), F.map f \u207b\u00b9' s) }) \u226b fun x =>\n { val := F.map g \u2191x, property := (_ : F.map g \u2191x \u2208 \u22c2 (f : Z\u271d \u27f6 i), F.map f \u207b\u00b9' s) }\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nx : F.obj j\nhx : x \u2208 eventualRange F j\n\u22a2 F.map f x \u2208 eventualRange F i\n[PROOFSTEP]\nrw [mem_eventualRange_iff] at hx \u22a2\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nx : F.obj j\nhx : \u2200 \u2983i : J\u2984 (f : i \u27f6 j), x \u2208 range (F.map f)\n\u22a2 \u2200 \u2983i_1 : J\u2984 (f_1 : i_1 \u27f6 i), F.map f x \u2208 range (F.map f_1)\n[PROOFSTEP]\nintro k f'\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nx : F.obj j\nhx : \u2200 \u2983i : J\u2984 (f : i \u27f6 j), x \u2208 range (F.map f)\nk : J\nf' : k \u27f6 i\n\u22a2 F.map f x \u2208 range (F.map f')\n[PROOFSTEP]\nobtain \u27e8l, g, g', he\u27e9 := cospan f f'\n[GOAL]\ncase intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nx : F.obj j\nhx : \u2200 \u2983i : J\u2984 (f : i \u27f6 j), x \u2208 range (F.map f)\nk : J\nf' : k \u27f6 i\nl : J\ng : l \u27f6 j\ng' : l \u27f6 k\nhe : g \u226b f = g' \u226b f'\n\u22a2 F.map f x \u2208 range (F.map f')\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 := hx g\n[GOAL]\ncase intro.intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nk : J\nf' : k \u27f6 i\nl : J\ng : l \u27f6 j\ng' : l \u27f6 k\nhe : g \u226b f = g' \u226b f'\nx : F.obj l\nhx : \u2200 \u2983i : J\u2984 (f : i \u27f6 j), F.map g x \u2208 range (F.map f)\n\u22a2 F.map f (F.map g x) \u2208 range (F.map f')\n[PROOFSTEP]\nrw [\u2190 map_comp_apply, he, F.map_comp]\n[GOAL]\ncase intro.intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nk : J\nf' : k \u27f6 i\nl : J\ng : l \u27f6 j\ng' : l \u27f6 k\nhe : g \u226b f = g' \u226b f'\nx : F.obj l\nhx : \u2200 \u2983i : J\u2984 (f : i \u27f6 j), F.map g x \u2208 range (F.map f)\n\u22a2 (F.map g' \u226b F.map f') x \u2208 range (F.map f')\n[PROOFSTEP]\nexact \u27e8_, rfl\u27e9\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : i \u27f6 j\n\u22a2 eventualRange F j = range (F.map f) \u2194 \u2200 \u2983k : J\u2984 (g : k \u27f6 i), range (F.map f) \u2286 range (F.map (g \u226b f))\n[PROOFSTEP]\nrw [subset_antisymm_iff, eventualRange, and_iff_right (iInter\u2082_subset _ _), subset_iInter\u2082_iff]\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : i \u27f6 j\n\u22a2 (\u2200 (i_1 : J) (j_1 : i_1 \u27f6 j), range (F.map f) \u2286 range (F.map j_1)) \u2194\n \u2200 \u2983k : J\u2984 (g : k \u27f6 i), range (F.map f) \u2286 range (F.map (g \u226b f))\n[PROOFSTEP]\nrefine' \u27e8fun h k g => h _ _, fun h j' f' => _\u27e9\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : i \u27f6 j\nh : \u2200 \u2983k : J\u2984 (g : k \u27f6 i), range (F.map f) \u2286 range (F.map (g \u226b f))\nj' : J\nf' : j' \u27f6 j\n\u22a2 range (F.map f) \u2286 range (F.map f')\n[PROOFSTEP]\nobtain \u27e8k, g, g', he\u27e9 := cospan f f'\n[GOAL]\ncase intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : i \u27f6 j\nh : \u2200 \u2983k : J\u2984 (g : k \u27f6 i), range (F.map f) \u2286 range (F.map (g \u226b f))\nj' : J\nf' : j' \u27f6 j\nk : J\ng : k \u27f6 i\ng' : k \u27f6 j'\nhe : g \u226b f = g' \u226b f'\n\u22a2 range (F.map f) \u2286 range (F.map f')\n[PROOFSTEP]\nrefine' (h g).trans _\n[GOAL]\ncase intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : i \u27f6 j\nh : \u2200 \u2983k : J\u2984 (g : k \u27f6 i), range (F.map f) \u2286 range (F.map (g \u226b f))\nj' : J\nf' : j' \u27f6 j\nk : J\ng : k \u27f6 i\ng' : k \u27f6 j'\nhe : g \u226b f = g' \u226b f'\n\u22a2 range (F.map (g \u226b f)) \u2286 range (F.map f')\n[PROOFSTEP]\nrw [he, F.map_comp]\n[GOAL]\ncase intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : i \u27f6 j\nh : \u2200 \u2983k : J\u2984 (g : k \u27f6 i), range (F.map f) \u2286 range (F.map (g \u226b f))\nj' : J\nf' : j' \u27f6 j\nk : J\ng : k \u27f6 i\ng' : k \u27f6 j'\nhe : g \u226b f = g' \u226b f'\n\u22a2 range (F.map g' \u226b F.map f') \u2286 range (F.map f')\n[PROOFSTEP]\napply range_comp_subset_range\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\n\u22a2 IsMittagLeffler F \u2194 \u2200 (j : J), \u2203 i f, \u2200 \u2983k : J\u2984 (g : k \u27f6 i), range (F.map f) \u2286 range (F.map (g \u226b f))\n[PROOFSTEP]\nsimp_rw [isMittagLeffler_iff_eventualRange, eventualRange_eq_iff]\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj : J\n\u22a2 \u2203 i_1 f,\n \u2200 \u2983k : J\u2984 (g : k \u27f6 i_1), range ((Functor.toPreimages F s).map f) \u2286 range ((Functor.toPreimages F s).map (g \u226b f))\n[PROOFSTEP]\nobtain \u27e8j\u2081, g\u2081, f\u2081, -\u27e9 := IsCofilteredOrEmpty.cone_objs i j\n[GOAL]\ncase intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j\u2081 : J\ng\u2081 : j\u2081 \u27f6 i\nf\u2081 : j\u2081 \u27f6 j\n\u22a2 \u2203 i_1 f,\n \u2200 \u2983k : J\u2984 (g : k \u27f6 i_1), range ((Functor.toPreimages F s).map f) \u2286 range ((Functor.toPreimages F s).map (g \u226b f))\n[PROOFSTEP]\nobtain \u27e8j\u2082, f\u2082, h\u2082\u27e9 := F.isMittagLeffler_iff_eventualRange.1 h j\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j\u2081 : J\ng\u2081 : j\u2081 \u27f6 i\nf\u2081 : j\u2081 \u27f6 j\nj\u2082 : J\nf\u2082 : j\u2082 \u27f6 j\u2081\nh\u2082 : eventualRange F j\u2081 = range (F.map f\u2082)\n\u22a2 \u2203 i_1 f,\n \u2200 \u2983k : J\u2984 (g : k \u27f6 i_1), range ((Functor.toPreimages F s).map f) \u2286 range ((Functor.toPreimages F s).map (g \u226b f))\n[PROOFSTEP]\nrefine' \u27e8j\u2082, f\u2082 \u226b f\u2081, fun j\u2083 f\u2083 => _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j\u2081 : J\ng\u2081 : j\u2081 \u27f6 i\nf\u2081 : j\u2081 \u27f6 j\nj\u2082 : J\nf\u2082 : j\u2082 \u27f6 j\u2081\nh\u2082 : eventualRange F j\u2081 = range (F.map f\u2082)\nj\u2083 : J\nf\u2083 : j\u2083 \u27f6 j\u2082\n\u22a2 range ((Functor.toPreimages F s).map (f\u2082 \u226b f\u2081)) \u2286 range ((Functor.toPreimages F s).map (f\u2083 \u226b f\u2082 \u226b f\u2081))\n[PROOFSTEP]\nrintro _ \u27e8\u27e8x, hx\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j\u2081 : J\ng\u2081 : j\u2081 \u27f6 i\nf\u2081 : j\u2081 \u27f6 j\nj\u2082 : J\nf\u2082 : j\u2082 \u27f6 j\u2081\nh\u2082 : eventualRange F j\u2081 = range (F.map f\u2082)\nj\u2083 : J\nf\u2083 : j\u2083 \u27f6 j\u2082\nx : F.obj j\u2082\nhx : x \u2208 \u22c2 (f : j\u2082 \u27f6 i), F.map f \u207b\u00b9' s\n\u22a2 (Functor.toPreimages F s).map (f\u2082 \u226b f\u2081) { val := x, property := hx } \u2208\n range ((Functor.toPreimages F s).map (f\u2083 \u226b f\u2082 \u226b f\u2081))\n[PROOFSTEP]\nhave : F.map f\u2082 x \u2208 F.eventualRange j\u2081 := by\n rw [h\u2082]\n exact \u27e8_, rfl\u27e9\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j\u2081 : J\ng\u2081 : j\u2081 \u27f6 i\nf\u2081 : j\u2081 \u27f6 j\nj\u2082 : J\nf\u2082 : j\u2082 \u27f6 j\u2081\nh\u2082 : eventualRange F j\u2081 = range (F.map f\u2082)\nj\u2083 : J\nf\u2083 : j\u2083 \u27f6 j\u2082\nx : F.obj j\u2082\nhx : x \u2208 \u22c2 (f : j\u2082 \u27f6 i), F.map f \u207b\u00b9' s\n\u22a2 F.map f\u2082 x \u2208 eventualRange F j\u2081\n[PROOFSTEP]\nrw [h\u2082]\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j\u2081 : J\ng\u2081 : j\u2081 \u27f6 i\nf\u2081 : j\u2081 \u27f6 j\nj\u2082 : J\nf\u2082 : j\u2082 \u27f6 j\u2081\nh\u2082 : eventualRange F j\u2081 = range (F.map f\u2082)\nj\u2083 : J\nf\u2083 : j\u2083 \u27f6 j\u2082\nx : F.obj j\u2082\nhx : x \u2208 \u22c2 (f : j\u2082 \u27f6 i), F.map f \u207b\u00b9' s\n\u22a2 F.map f\u2082 x \u2208 range (F.map f\u2082)\n[PROOFSTEP]\nexact \u27e8_, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j\u2081 : J\ng\u2081 : j\u2081 \u27f6 i\nf\u2081 : j\u2081 \u27f6 j\nj\u2082 : J\nf\u2082 : j\u2082 \u27f6 j\u2081\nh\u2082 : eventualRange F j\u2081 = range (F.map f\u2082)\nj\u2083 : J\nf\u2083 : j\u2083 \u27f6 j\u2082\nx : F.obj j\u2082\nhx : x \u2208 \u22c2 (f : j\u2082 \u27f6 i), F.map f \u207b\u00b9' s\nthis : F.map f\u2082 x \u2208 eventualRange F j\u2081\n\u22a2 (Functor.toPreimages F s).map (f\u2082 \u226b f\u2081) { val := x, property := hx } \u2208\n range ((Functor.toPreimages F s).map (f\u2083 \u226b f\u2082 \u226b f\u2081))\n[PROOFSTEP]\nobtain \u27e8y, hy, h\u2083\u27e9 := h.subset_image_eventualRange F (f\u2083 \u226b f\u2082) this\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j\u2081 : J\ng\u2081 : j\u2081 \u27f6 i\nf\u2081 : j\u2081 \u27f6 j\nj\u2082 : J\nf\u2082 : j\u2082 \u27f6 j\u2081\nh\u2082 : eventualRange F j\u2081 = range (F.map f\u2082)\nj\u2083 : J\nf\u2083 : j\u2083 \u27f6 j\u2082\nx : F.obj j\u2082\nhx : x \u2208 \u22c2 (f : j\u2082 \u27f6 i), F.map f \u207b\u00b9' s\nthis : F.map f\u2082 x \u2208 eventualRange F j\u2081\ny : F.obj j\u2083\nhy : y \u2208 eventualRange F j\u2083\nh\u2083 : F.map (f\u2083 \u226b f\u2082) y = F.map f\u2082 x\n\u22a2 (Functor.toPreimages F s).map (f\u2082 \u226b f\u2081) { val := x, property := hx } \u2208\n range ((Functor.toPreimages F s).map (f\u2083 \u226b f\u2082 \u226b f\u2081))\n[PROOFSTEP]\nrefine' \u27e8\u27e8y, mem_iInter.2 fun g\u2082 => _\u27e9, Subtype.ext _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_1\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j\u2081 : J\ng\u2081 : j\u2081 \u27f6 i\nf\u2081 : j\u2081 \u27f6 j\nj\u2082 : J\nf\u2082 : j\u2082 \u27f6 j\u2081\nh\u2082 : eventualRange F j\u2081 = range (F.map f\u2082)\nj\u2083 : J\nf\u2083 : j\u2083 \u27f6 j\u2082\nx : F.obj j\u2082\nhx : x \u2208 \u22c2 (f : j\u2082 \u27f6 i), F.map f \u207b\u00b9' s\nthis : F.map f\u2082 x \u2208 eventualRange F j\u2081\ny : F.obj j\u2083\nhy : y \u2208 eventualRange F j\u2083\nh\u2083 : F.map (f\u2083 \u226b f\u2082) y = F.map f\u2082 x\ng\u2082 : j\u2083 \u27f6 i\n\u22a2 y \u2208 F.map g\u2082 \u207b\u00b9' s\n[PROOFSTEP]\nobtain \u27e8j\u2084, f\u2084, h\u2084\u27e9 := IsCofilteredOrEmpty.cone_maps g\u2082 ((f\u2083 \u226b f\u2082) \u226b g\u2081)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_1.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j\u2081 : J\ng\u2081 : j\u2081 \u27f6 i\nf\u2081 : j\u2081 \u27f6 j\nj\u2082 : J\nf\u2082 : j\u2082 \u27f6 j\u2081\nh\u2082 : eventualRange F j\u2081 = range (F.map f\u2082)\nj\u2083 : J\nf\u2083 : j\u2083 \u27f6 j\u2082\nx : F.obj j\u2082\nhx : x \u2208 \u22c2 (f : j\u2082 \u27f6 i), F.map f \u207b\u00b9' s\nthis : F.map f\u2082 x \u2208 eventualRange F j\u2081\ny : F.obj j\u2083\nhy : y \u2208 eventualRange F j\u2083\nh\u2083 : F.map (f\u2083 \u226b f\u2082) y = F.map f\u2082 x\ng\u2082 : j\u2083 \u27f6 i\nj\u2084 : J\nf\u2084 : j\u2084 \u27f6 j\u2083\nh\u2084 : f\u2084 \u226b g\u2082 = f\u2084 \u226b (f\u2083 \u226b f\u2082) \u226b g\u2081\n\u22a2 y \u2208 F.map g\u2082 \u207b\u00b9' s\n[PROOFSTEP]\nobtain \u27e8y, rfl\u27e9 := F.mem_eventualRange_iff.1 hy f\u2084\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_1.intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j\u2081 : J\ng\u2081 : j\u2081 \u27f6 i\nf\u2081 : j\u2081 \u27f6 j\nj\u2082 : J\nf\u2082 : j\u2082 \u27f6 j\u2081\nh\u2082 : eventualRange F j\u2081 = range (F.map f\u2082)\nj\u2083 : J\nf\u2083 : j\u2083 \u27f6 j\u2082\nx : F.obj j\u2082\nhx : x \u2208 \u22c2 (f : j\u2082 \u27f6 i), F.map f \u207b\u00b9' s\nthis : F.map f\u2082 x \u2208 eventualRange F j\u2081\ng\u2082 : j\u2083 \u27f6 i\nj\u2084 : J\nf\u2084 : j\u2084 \u27f6 j\u2083\nh\u2084 : f\u2084 \u226b g\u2082 = f\u2084 \u226b (f\u2083 \u226b f\u2082) \u226b g\u2081\ny : F.obj j\u2084\nhy : F.map f\u2084 y \u2208 eventualRange F j\u2083\nh\u2083 : F.map (f\u2083 \u226b f\u2082) (F.map f\u2084 y) = F.map f\u2082 x\n\u22a2 F.map f\u2084 y \u2208 F.map g\u2082 \u207b\u00b9' s\n[PROOFSTEP]\nrw [\u2190 map_comp_apply] at h\u2083 \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_1.intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j\u2081 : J\ng\u2081 : j\u2081 \u27f6 i\nf\u2081 : j\u2081 \u27f6 j\nj\u2082 : J\nf\u2082 : j\u2082 \u27f6 j\u2081\nh\u2082 : eventualRange F j\u2081 = range (F.map f\u2082)\nj\u2083 : J\nf\u2083 : j\u2083 \u27f6 j\u2082\nx : F.obj j\u2082\nhx : x \u2208 \u22c2 (f : j\u2082 \u27f6 i), F.map f \u207b\u00b9' s\nthis : F.map f\u2082 x \u2208 eventualRange F j\u2081\ng\u2082 : j\u2083 \u27f6 i\nj\u2084 : J\nf\u2084 : j\u2084 \u27f6 j\u2083\nh\u2084 : f\u2084 \u226b g\u2082 = f\u2084 \u226b (f\u2083 \u226b f\u2082) \u226b g\u2081\ny : F.obj j\u2084\nhy : F.map f\u2084 y \u2208 eventualRange F j\u2083\nh\u2083 : F.map (f\u2084 \u226b f\u2083 \u226b f\u2082) y = F.map f\u2082 x\n\u22a2 F.map f\u2084 y \u2208 F.map g\u2082 \u207b\u00b9' s\n[PROOFSTEP]\nrw [mem_preimage, \u2190 map_comp_apply, h\u2084, \u2190 Category.assoc, map_comp_apply, h\u2083, \u2190 map_comp_apply]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_1.intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j\u2081 : J\ng\u2081 : j\u2081 \u27f6 i\nf\u2081 : j\u2081 \u27f6 j\nj\u2082 : J\nf\u2082 : j\u2082 \u27f6 j\u2081\nh\u2082 : eventualRange F j\u2081 = range (F.map f\u2082)\nj\u2083 : J\nf\u2083 : j\u2083 \u27f6 j\u2082\nx : F.obj j\u2082\nhx : x \u2208 \u22c2 (f : j\u2082 \u27f6 i), F.map f \u207b\u00b9' s\nthis : F.map f\u2082 x \u2208 eventualRange F j\u2081\ng\u2082 : j\u2083 \u27f6 i\nj\u2084 : J\nf\u2084 : j\u2084 \u27f6 j\u2083\nh\u2084 : f\u2084 \u226b g\u2082 = f\u2084 \u226b (f\u2083 \u226b f\u2082) \u226b g\u2081\ny : F.obj j\u2084\nhy : F.map f\u2084 y \u2208 eventualRange F j\u2083\nh\u2083 : F.map (f\u2084 \u226b f\u2083 \u226b f\u2082) y = F.map f\u2082 x\n\u22a2 F.map (f\u2082 \u226b g\u2081) x \u2208 s\n[PROOFSTEP]\napply mem_iInter.1 hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_2\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j\u2081 : J\ng\u2081 : j\u2081 \u27f6 i\nf\u2081 : j\u2081 \u27f6 j\nj\u2082 : J\nf\u2082 : j\u2082 \u27f6 j\u2081\nh\u2082 : eventualRange F j\u2081 = range (F.map f\u2082)\nj\u2083 : J\nf\u2083 : j\u2083 \u27f6 j\u2082\nx : F.obj j\u2082\nhx : x \u2208 \u22c2 (f : j\u2082 \u27f6 i), F.map f \u207b\u00b9' s\nthis : F.map f\u2082 x \u2208 eventualRange F j\u2081\ny : F.obj j\u2083\nhy : y \u2208 eventualRange F j\u2083\nh\u2083 : F.map (f\u2083 \u226b f\u2082) y = F.map f\u2082 x\n\u22a2 \u2191((Functor.toPreimages F s).map (f\u2083 \u226b f\u2082 \u226b f\u2081)\n { val := y, property := (_ : y \u2208 \u22c2 (i_1 : j\u2083 \u27f6 i), F.map i_1 \u207b\u00b9' s) }) =\n \u2191((Functor.toPreimages F s).map (f\u2082 \u226b f\u2081) { val := x, property := hx })\n[PROOFSTEP]\nsimp_rw [toPreimages_map, MapsTo.val_restrict_apply]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk.intro.intro.refine'_2\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\nj j\u2081 : J\ng\u2081 : j\u2081 \u27f6 i\nf\u2081 : j\u2081 \u27f6 j\nj\u2082 : J\nf\u2082 : j\u2082 \u27f6 j\u2081\nh\u2082 : eventualRange F j\u2081 = range (F.map f\u2082)\nj\u2083 : J\nf\u2083 : j\u2083 \u27f6 j\u2082\nx : F.obj j\u2082\nhx : x \u2208 \u22c2 (f : j\u2082 \u27f6 i), F.map f \u207b\u00b9' s\nthis : F.map f\u2082 x \u2208 eventualRange F j\u2081\ny : F.obj j\u2083\nhy : y \u2208 eventualRange F j\u2083\nh\u2083 : F.map (f\u2083 \u226b f\u2082) y = F.map f\u2082 x\n\u22a2 F.map (f\u2083 \u226b f\u2082 \u226b f\u2081) y = F.map (f\u2082 \u226b f\u2081) x\n[PROOFSTEP]\nrw [\u2190 Category.assoc, map_comp_apply, h\u2083, map_comp_apply]\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : \u2200 (j : J), \u2203 i f, Set.Finite (range (F.map f))\n\u22a2 IsMittagLeffler F\n[PROOFSTEP]\nintro j\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nh : \u2200 (j : J), \u2203 i f, Set.Finite (range (F.map f))\nj : J\n\u22a2 \u2203 i f, \u2200 \u2983k : J\u2984 (g : k \u27f6 j), range (F.map f) \u2286 range (F.map g)\n[PROOFSTEP]\nobtain \u27e8i, hi, hf\u27e9 := h j\n[GOAL]\ncase intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j\u271d k : J\ns : Set (F.obj i\u271d)\ninst\u271d : IsCofilteredOrEmpty J\nh : \u2200 (j : J), \u2203 i f, Set.Finite (range (F.map f))\nj i : J\nhi : i \u27f6 j\nhf : Set.Finite (range (F.map hi))\n\u22a2 \u2203 i f, \u2200 \u2983k : J\u2984 (g : k \u27f6 j), range (F.map f) \u2286 range (F.map g)\n[PROOFSTEP]\nobtain \u27e8m, \u27e8i, f, hm\u27e9, hmin\u27e9 :=\n Finset.wellFoundedLT.wf.has_min {s : Finset (F.obj j) | \u2203 (i : _) (f : i \u27f6 j), \u2191s = range (F.map f)}\n \u27e8_, i, hi, hf.coe_toFinset\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d\u00b9 j\u271d k : J\ns : Set (F.obj i\u271d\u00b9)\ninst\u271d : IsCofilteredOrEmpty J\nh : \u2200 (j : J), \u2203 i f, Set.Finite (range (F.map f))\nj i\u271d : J\nhi : i\u271d \u27f6 j\nhf : Set.Finite (range (F.map hi))\nm : Finset (F.obj j)\nhmin : \u2200 (x : Finset (F.obj j)), x \u2208 {s | \u2203 i f, \u2191s = range (F.map f)} \u2192 \u00acx < m\ni : J\nf : i \u27f6 j\nhm : \u2191m = range (F.map f)\n\u22a2 \u2203 i f, \u2200 \u2983k : J\u2984 (g : k \u27f6 j), range (F.map f) \u2286 range (F.map g)\n[PROOFSTEP]\nrefine' \u27e8i, f, fun k g => (directedOn_range.mp <| F.ranges_directed j).is_bot_of_is_min \u27e8\u27e8i, f\u27e9, rfl\u27e9 _ _ \u27e8\u27e8k, g\u27e9, rfl\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d\u00b9 j\u271d k\u271d : J\ns : Set (F.obj i\u271d\u00b9)\ninst\u271d : IsCofilteredOrEmpty J\nh : \u2200 (j : J), \u2203 i f, Set.Finite (range (F.map f))\nj i\u271d : J\nhi : i\u271d \u27f6 j\nhf : Set.Finite (range (F.map hi))\nm : Finset (F.obj j)\nhmin : \u2200 (x : Finset (F.obj j)), x \u2208 {s | \u2203 i f, \u2191s = range (F.map f)} \u2192 \u00acx < m\ni : J\nf : i \u27f6 j\nhm : \u2191m = range (F.map f)\nk : J\ng : k \u27f6 j\n\u22a2 \u2200 (a : Set (F.obj j)),\n (a \u2208 range fun f => range (F.map f.snd)) \u2192\n a \u2264 (fun f => range (F.map f.snd)) { fst := i, snd := f } \u2192\n (fun f => range (F.map f.snd)) { fst := i, snd := f } \u2264 a\n[PROOFSTEP]\nrintro _ \u27e8\u27e8k', g'\u27e9, rfl\u27e9 hl\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.mk\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d\u00b9 j\u271d k\u271d : J\ns : Set (F.obj i\u271d\u00b9)\ninst\u271d : IsCofilteredOrEmpty J\nh : \u2200 (j : J), \u2203 i f, Set.Finite (range (F.map f))\nj i\u271d : J\nhi : i\u271d \u27f6 j\nhf : Set.Finite (range (F.map hi))\nm : Finset (F.obj j)\nhmin : \u2200 (x : Finset (F.obj j)), x \u2208 {s | \u2203 i f, \u2191s = range (F.map f)} \u2192 \u00acx < m\ni : J\nf : i \u27f6 j\nhm : \u2191m = range (F.map f)\nk : J\ng : k \u27f6 j\nk' : J\ng' : k' \u27f6 j\nhl : (fun f => range (F.map f.snd)) { fst := k', snd := g' } \u2264 (fun f => range (F.map f.snd)) { fst := i, snd := f }\n\u22a2 (fun f => range (F.map f.snd)) { fst := i, snd := f } \u2264 (fun f => range (F.map f.snd)) { fst := k', snd := g' }\n[PROOFSTEP]\nrefine' (eq_of_le_of_not_lt hl _).ge\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.mk\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d\u00b9 j\u271d k\u271d : J\ns : Set (F.obj i\u271d\u00b9)\ninst\u271d : IsCofilteredOrEmpty J\nh : \u2200 (j : J), \u2203 i f, Set.Finite (range (F.map f))\nj i\u271d : J\nhi : i\u271d \u27f6 j\nhf : Set.Finite (range (F.map hi))\nm : Finset (F.obj j)\nhmin : \u2200 (x : Finset (F.obj j)), x \u2208 {s | \u2203 i f, \u2191s = range (F.map f)} \u2192 \u00acx < m\ni : J\nf : i \u27f6 j\nhm : \u2191m = range (F.map f)\nk : J\ng : k \u27f6 j\nk' : J\ng' : k' \u27f6 j\nhl : (fun f => range (F.map f.snd)) { fst := k', snd := g' } \u2264 (fun f => range (F.map f.snd)) { fst := i, snd := f }\n\u22a2 \u00ac(fun f => range (F.map f.snd)) { fst := k', snd := g' } < (fun f => range (F.map f.snd)) { fst := i, snd := f }\n[PROOFSTEP]\nhave := hmin _ \u27e8k', g', (m.finite_toSet.subset <| hm.substr hl).coe_toFinset\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.mk\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d\u00b9 j\u271d k\u271d : J\ns : Set (F.obj i\u271d\u00b9)\ninst\u271d : IsCofilteredOrEmpty J\nh : \u2200 (j : J), \u2203 i f, Set.Finite (range (F.map f))\nj i\u271d : J\nhi : i\u271d \u27f6 j\nhf : Set.Finite (range (F.map hi))\nm : Finset (F.obj j)\nhmin : \u2200 (x : Finset (F.obj j)), x \u2208 {s | \u2203 i f, \u2191s = range (F.map f)} \u2192 \u00acx < m\ni : J\nf : i \u27f6 j\nhm : \u2191m = range (F.map f)\nk : J\ng : k \u27f6 j\nk' : J\ng' : k' \u27f6 j\nhl : (fun f => range (F.map f.snd)) { fst := k', snd := g' } \u2264 (fun f => range (F.map f.snd)) { fst := i, snd := f }\nthis : \u00acFinite.toFinset (_ : Set.Finite (range (F.map g'))) < m\n\u22a2 \u00ac(fun f => range (F.map f.snd)) { fst := k', snd := g' } < (fun f => range (F.map f.snd)) { fst := i, snd := f }\n[PROOFSTEP]\nrwa [Finset.lt_iff_ssubset, \u2190 Finset.coe_ssubset, Set.Finite.coe_toFinset, hm] at this \n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{?u.27549, u} J\nF : J \u2964 Type v\ni\u271d j k : J\ns : Set (F.obj i\u271d)\ninst\u271d : IsCofilteredOrEmpty J\ni : J\n\u22a2 { obj := fun j => \u2191(eventualRange F j),\n map := fun {X Y} f =>\n MapsTo.restrict (F.map f) (eventualRange F X) (eventualRange F Y)\n (_ : MapsTo (F.map f) (eventualRange F X) (eventualRange F Y)) }.map\n (\ud835\udfd9 i) =\n \ud835\udfd9\n ({ obj := fun j => \u2191(eventualRange F j),\n map := fun {X Y} f =>\n MapsTo.restrict (F.map f) (eventualRange F X) (eventualRange F Y)\n (_ : MapsTo (F.map f) (eventualRange F X) (eventualRange F Y)) }.obj\n i)\n[PROOFSTEP]\nsimp_rw [MapsTo.restrict, Subtype.map, F.map_id]\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{?u.27549, u} J\nF : J \u2964 Type v\ni\u271d j k : J\ns : Set (F.obj i\u271d)\ninst\u271d : IsCofilteredOrEmpty J\ni : J\n\u22a2 (fun x => { val := \ud835\udfd9 (F.obj i) \u2191x, property := (_ : (fun x => x \u2208 eventualRange F i) (\ud835\udfd9 (F.obj i) \u2191x)) }) =\n \ud835\udfd9 \u2191(eventualRange F i)\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\nJ : Type u\ninst\u271d\u00b9 : Category.{?u.27549, u} J\nF : J \u2964 Type v\ni\u271d j k : J\ns : Set (F.obj i\u271d)\ninst\u271d : IsCofilteredOrEmpty J\ni : J\na\u271d : { x // x \u2208 eventualRange F i }\n\u22a2 \u2191{ val := \ud835\udfd9 (F.obj i) \u2191a\u271d, property := (_ : (fun x => x \u2208 eventualRange F i) (\ud835\udfd9 (F.obj i) \u2191a\u271d)) } =\n \u2191(\ud835\udfd9 (\u2191(eventualRange F i)) a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{?u.27549, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nX\u271d Y\u271d Z\u271d : J\nx\u271d\u00b9 : X\u271d \u27f6 Y\u271d\nx\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun j => \u2191(eventualRange F j),\n map := fun {X Y} f =>\n MapsTo.restrict (F.map f) (eventualRange F X) (eventualRange F Y)\n (_ : MapsTo (F.map f) (eventualRange F X) (eventualRange F Y)) }.map\n (x\u271d\u00b9 \u226b x\u271d) =\n { obj := fun j => \u2191(eventualRange F j),\n map := fun {X Y} f =>\n MapsTo.restrict (F.map f) (eventualRange F X) (eventualRange F Y)\n (_ : MapsTo (F.map f) (eventualRange F X) (eventualRange F Y)) }.map\n x\u271d\u00b9 \u226b\n { obj := fun j => \u2191(eventualRange F j),\n map := fun {X Y} f =>\n MapsTo.restrict (F.map f) (eventualRange F X) (eventualRange F Y)\n (_ : MapsTo (F.map f) (eventualRange F X) (eventualRange F Y)) }.map\n x\u271d\n[PROOFSTEP]\nsimp_rw [MapsTo.restrict, Subtype.map, F.map_comp]\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{?u.27549, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nX\u271d Y\u271d Z\u271d : J\nx\u271d\u00b9 : X\u271d \u27f6 Y\u271d\nx\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 (fun x =>\n { val := (F.map x\u271d\u00b9 \u226b F.map x\u271d) \u2191x,\n property := (_ : (fun x => x \u2208 eventualRange F Z\u271d) ((F.map x\u271d\u00b9 \u226b F.map x\u271d) \u2191x)) }) =\n (fun x => { val := F.map x\u271d\u00b9 \u2191x, property := (_ : F.map x\u271d\u00b9 \u2191x \u2208 eventualRange F Y\u271d) }) \u226b fun x =>\n { val := F.map x\u271d \u2191x, property := (_ : F.map x\u271d \u2191x \u2208 eventualRange F Z\u271d) }\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{?u.30110, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nx\u271d : \u2191(sections (toEventualRanges F))\n\u22a2 (fun s =>\n { val := fun j => { val := \u2191s j, property := (_ : \u2191s j \u2208 \u22c2 (i : J) (j_1 : i \u27f6 j), range (F.map j_1)) },\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n (toEventualRanges F).map f\n ((fun j => { val := \u2191s j, property := (_ : \u2191s j \u2208 \u22c2 (i : J) (j_1 : i \u27f6 j), range (F.map j_1)) })\n j) =\n (fun j => { val := \u2191s j, property := (_ : \u2191s j \u2208 \u22c2 (i : J) (j_1 : i \u27f6 j), range (F.map j_1)) }) j') })\n ((fun s =>\n { val := fun {j} => \u2191(\u2191s j),\n property := (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191((toEventualRanges F).map f (\u2191s j)) = \u2191(\u2191s j')) })\n x\u271d) =\n x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nJ : Type u\ninst\u271d\u00b9 : Category.{?u.30110, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nx\u271d\u00b9 : \u2191(sections (toEventualRanges F))\nx\u271d : J\n\u22a2 \u2191((fun s =>\n { val := fun j => { val := \u2191s j, property := (_ : \u2191s j \u2208 \u22c2 (i : J) (j_1 : i \u27f6 j), range (F.map j_1)) },\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n (toEventualRanges F).map f\n ((fun j => { val := \u2191s j, property := (_ : \u2191s j \u2208 \u22c2 (i : J) (j_1 : i \u27f6 j), range (F.map j_1)) })\n j) =\n (fun j => { val := \u2191s j, property := (_ : \u2191s j \u2208 \u22c2 (i : J) (j_1 : i \u27f6 j), range (F.map j_1)) })\n j') })\n ((fun s =>\n { val := fun {j} => \u2191(\u2191s j),\n property := (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191((toEventualRanges F).map f (\u2191s j)) = \u2191(\u2191s j')) })\n x\u271d\u00b9))\n x\u271d =\n \u2191x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{?u.30110, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nx\u271d : \u2191(sections F)\n\u22a2 (fun s =>\n { val := fun {j} => \u2191(\u2191s j),\n property := (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191((toEventualRanges F).map f (\u2191s j)) = \u2191(\u2191s j')) })\n ((fun s =>\n { val := fun j => { val := \u2191s j, property := (_ : \u2191s j \u2208 \u22c2 (i : J) (j_1 : i \u27f6 j), range (F.map j_1)) },\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n (toEventualRanges F).map f\n ((fun j => { val := \u2191s j, property := (_ : \u2191s j \u2208 \u22c2 (i : J) (j_1 : i \u27f6 j), range (F.map j_1)) })\n j) =\n (fun j => { val := \u2191s j, property := (_ : \u2191s j \u2208 \u22c2 (i : J) (j_1 : i \u27f6 j), range (F.map j_1)) })\n j') })\n x\u271d) =\n x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nJ : Type u\ninst\u271d\u00b9 : Category.{?u.30110, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nx\u271d\u00b9 : \u2191(sections F)\nx\u271d : J\n\u22a2 \u2191((fun s =>\n { val := fun {j} => \u2191(\u2191s j),\n property := (_ : \u2200 {j j' : J} (f : j \u27f6 j'), \u2191((toEventualRanges F).map f (\u2191s j)) = \u2191(\u2191s j')) })\n ((fun s =>\n { val := fun j => { val := \u2191s j, property := (_ : \u2191s j \u2208 \u22c2 (i : J) (j_1 : i \u27f6 j), range (F.map j_1)) },\n property :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n (toEventualRanges F).map f\n ((fun j =>\n { val := \u2191s j, property := (_ : \u2191s j \u2208 \u22c2 (i : J) (j_1 : i \u27f6 j), range (F.map j_1)) })\n j) =\n (fun j => { val := \u2191s j, property := (_ : \u2191s j \u2208 \u22c2 (i : J) (j_1 : i \u27f6 j), range (F.map j_1)) })\n j') })\n x\u271d\u00b9))\n x\u271d =\n \u2191x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j\u271d k : J\ns : Set (F.obj i\u271d)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\ni j : J\nf : i \u27f6 j\nx\u271d : (toEventualRanges F).obj j\nx : F.obj j\nhx : x \u2208 eventualRange F j\n\u22a2 \u2203 a, (toEventualRanges F).map f a = { val := x, property := hx }\n[PROOFSTEP]\nobtain \u27e8y, hy, rfl\u27e9 := h.subset_image_eventualRange F f hx\n[GOAL]\ncase intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j\u271d k : J\ns : Set (F.obj i\u271d)\ninst\u271d : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\ni j : J\nf : i \u27f6 j\nx\u271d : (toEventualRanges F).obj j\ny : F.obj i\nhy : y \u2208 eventualRange F i\nhx : F.map f y \u2208 eventualRange F j\n\u22a2 \u2203 a, (toEventualRanges F).map f a = { val := F.map f y, property := hx }\n[PROOFSTEP]\nexact \u27e8\u27e8y, hy\u27e9, rfl\u27e9\n[GOAL]\nJ : Type u\ninst\u271d\u00b2 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d\u00b9 : IsCofilteredOrEmpty J\nh : IsMittagLeffler F\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nj : J\n\u22a2 _root_.Nonempty ((toEventualRanges F).obj j)\n[PROOFSTEP]\nlet \u27e8i, f, h\u27e9 := F.isMittagLeffler_iff_eventualRange.1 h j\n[GOAL]\nJ : Type u\ninst\u271d\u00b2 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j\u271d k : J\ns : Set (F.obj i\u271d)\ninst\u271d\u00b9 : IsCofilteredOrEmpty J\nh\u271d : IsMittagLeffler F\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nj i : J\nf : i \u27f6 j\nh : eventualRange F j = range (F.map f)\n\u22a2 _root_.Nonempty ((toEventualRanges F).obj j)\n[PROOFSTEP]\nrw [toEventualRanges_obj, h]\n[GOAL]\nJ : Type u\ninst\u271d\u00b2 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j\u271d k : J\ns : Set (F.obj i\u271d)\ninst\u271d\u00b9 : IsCofilteredOrEmpty J\nh\u271d : IsMittagLeffler F\ninst\u271d : \u2200 (j : J), _root_.Nonempty (F.obj j)\nj i : J\nf : i \u27f6 j\nh : eventualRange F j = range (F.map f)\n\u22a2 _root_.Nonempty \u2191(range (F.map f))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j\u271d k\u271d : J\ns : Set (F.obj i\u271d)\ninst\u271d : IsCofilteredOrEmpty J\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ni j : J\nf g : i \u27f6 j\nk : J\n\u03c6 : k \u27f6 i\nh\u03c6 : \u03c6 \u226b f = \u03c6 \u226b g\n\u22a2 (fun g => g \u2218 F.map \u03c6) (F.map f) = (fun g => g \u2218 F.map \u03c6) (F.map g)\n[PROOFSTEP]\nsimp_rw [\u2190 types_comp, \u2190 F.map_comp, h\u03c6]\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nhFn : \u2200 (j : J), _root_.Nonempty (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\nhs : Set.Nonempty s\nj : J\n\u22a2 _root_.Nonempty ((toPreimages F s).obj j)\n[PROOFSTEP]\nsimp only [toPreimages_obj, nonempty_coe_sort, nonempty_iInter, mem_preimage]\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nhFn : \u2200 (j : J), _root_.Nonempty (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\nhs : Set.Nonempty s\nj : J\n\u22a2 \u2203 x, \u2200 (i_1 : j \u27f6 i), F.map i_1 x \u2208 s\n[PROOFSTEP]\nobtain h | \u27e8\u27e8ji\u27e9\u27e9 := isEmpty_or_nonempty (j \u27f6 i)\n[GOAL]\ncase inl\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nhFn : \u2200 (j : J), _root_.Nonempty (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\nhs : Set.Nonempty s\nj : J\nh : IsEmpty (j \u27f6 i)\n\u22a2 \u2203 x, \u2200 (i_1 : j \u27f6 i), F.map i_1 x \u2208 s\n[PROOFSTEP]\nexact \u27e8(hFn j).some, fun ji => h.elim ji\u27e9\n[GOAL]\ncase inr.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nhFn : \u2200 (j : J), _root_.Nonempty (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\nhs : Set.Nonempty s\nj : J\nji : j \u27f6 i\n\u22a2 \u2203 x, \u2200 (i_1 : j \u27f6 i), F.map i_1 x \u2208 s\n[PROOFSTEP]\nobtain \u27e8y, ys\u27e9 := hs\n[GOAL]\ncase inr.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nhFn : \u2200 (j : J), _root_.Nonempty (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\nj : J\nji : j \u27f6 i\ny : F.obj i\nys : y \u2208 s\n\u22a2 \u2203 x, \u2200 (i_1 : j \u27f6 i), F.map i_1 x \u2208 s\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 := Fsur ji y\n[GOAL]\ncase inr.intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j\u271d k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nhFn : \u2200 (j : J), _root_.Nonempty (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\nj : J\nji : j \u27f6 i\nx : F.obj j\nys : F.map ji x \u2208 s\n\u22a2 \u2203 x, \u2200 (i_1 : j \u27f6 i), F.map i_1 x \u2208 s\n[PROOFSTEP]\nexact \u27e8x, fun ji' => (F.thin_diagram_of_surjective Fsur ji' ji).symm \u25b8 ys\u27e9\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j\u271d k : J\ns : Set (F.obj i\u271d)\ninst\u271d : IsCofilteredOrEmpty J\nj : J\nFinj : \u2200 (i : J) (f : i \u27f6 j), Function.Injective (F.map f)\ni : J\nf : i \u27f6 j\n\u22a2 Function.Injective fun s => \u2191s j\n[PROOFSTEP]\nrefine' fun s\u2080 s\u2081 h => Subtype.ext <| funext fun k => _\n[GOAL]\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j\u271d k\u271d : J\ns : Set (F.obj i\u271d)\ninst\u271d : IsCofilteredOrEmpty J\nj : J\nFinj : \u2200 (i : J) (f : i \u27f6 j), Function.Injective (F.map f)\ni : J\nf : i \u27f6 j\ns\u2080 s\u2081 : \u2191(sections F)\nh : (fun s => \u2191s j) s\u2080 = (fun s => \u2191s j) s\u2081\nk : J\n\u22a2 \u2191s\u2080 k = \u2191s\u2081 k\n[PROOFSTEP]\nobtain \u27e8m, mi, mk, _\u27e9 := IsCofilteredOrEmpty.cone_objs i k\n[GOAL]\ncase intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j\u271d k\u271d : J\ns : Set (F.obj i\u271d)\ninst\u271d : IsCofilteredOrEmpty J\nj : J\nFinj : \u2200 (i : J) (f : i \u27f6 j), Function.Injective (F.map f)\ni : J\nf : i \u27f6 j\ns\u2080 s\u2081 : \u2191(sections F)\nh : (fun s => \u2191s j) s\u2080 = (fun s => \u2191s j) s\u2081\nk m : J\nmi : m \u27f6 i\nmk : m \u27f6 k\nh\u271d : True\n\u22a2 \u2191s\u2080 k = \u2191s\u2081 k\n[PROOFSTEP]\ndsimp at h \n[GOAL]\ncase intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j\u271d k\u271d : J\ns : Set (F.obj i\u271d)\ninst\u271d : IsCofilteredOrEmpty J\nj : J\nFinj : \u2200 (i : J) (f : i \u27f6 j), Function.Injective (F.map f)\ni : J\nf : i \u27f6 j\ns\u2080 s\u2081 : \u2191(sections F)\nh : \u2191s\u2080 j = \u2191s\u2081 j\nk m : J\nmi : m \u27f6 i\nmk : m \u27f6 k\nh\u271d : True\n\u22a2 \u2191s\u2080 k = \u2191s\u2081 k\n[PROOFSTEP]\nrw [\u2190 s\u2080.prop (mi \u226b f), \u2190 s\u2081.prop (mi \u226b f)] at h \n[GOAL]\ncase intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j\u271d k\u271d : J\ns : Set (F.obj i\u271d)\ninst\u271d : IsCofilteredOrEmpty J\nj : J\nFinj : \u2200 (i : J) (f : i \u27f6 j), Function.Injective (F.map f)\ni : J\nf : i \u27f6 j\ns\u2080 s\u2081 : \u2191(sections F)\nk m : J\nmi : m \u27f6 i\nh : F.map (mi \u226b f) (\u2191s\u2080 m) = F.map (mi \u226b f) (\u2191s\u2081 m)\nmk : m \u27f6 k\nh\u271d : True\n\u22a2 \u2191s\u2080 k = \u2191s\u2081 k\n[PROOFSTEP]\nrw [\u2190 s\u2080.prop mk, \u2190 s\u2081.prop mk]\n[GOAL]\ncase intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j\u271d k\u271d : J\ns : Set (F.obj i\u271d)\ninst\u271d : IsCofilteredOrEmpty J\nj : J\nFinj : \u2200 (i : J) (f : i \u27f6 j), Function.Injective (F.map f)\ni : J\nf : i \u27f6 j\ns\u2080 s\u2081 : \u2191(sections F)\nk m : J\nmi : m \u27f6 i\nh : F.map (mi \u226b f) (\u2191s\u2080 m) = F.map (mi \u226b f) (\u2191s\u2081 m)\nmk : m \u27f6 k\nh\u271d : True\n\u22a2 F.map mk (\u2191s\u2080 m) = F.map mk (\u2191s\u2081 m)\n[PROOFSTEP]\nrefine' congr_arg _ (Finj m (mi \u226b f) h)\n[GOAL]\nJ : Type u\ninst\u271d\u00b3 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j k : J\ns : Set (F.obj i\u271d)\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\ninst\u271d\u00b9 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\n\u22a2 \u2203 a, (fun s => \u2191s i) a = x\n[PROOFSTEP]\nlet s : Set (F.obj i) := { x }\n[GOAL]\nJ : Type u\ninst\u271d\u00b3 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j k : J\ns\u271d : Set (F.obj i\u271d)\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\ninst\u271d\u00b9 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\n\u22a2 \u2203 a, (fun s => \u2191s i) a = x\n[PROOFSTEP]\nhaveI := F.toPreimages_nonempty_of_surjective s Fsur (singleton_nonempty x)\n[GOAL]\nJ : Type u\ninst\u271d\u00b3 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j k : J\ns\u271d : Set (F.obj i\u271d)\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\ninst\u271d\u00b9 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis : \u2200 (j : J), _root_.Nonempty ((toPreimages F s).obj j)\n\u22a2 \u2203 a, (fun s => \u2191s i) a = x\n[PROOFSTEP]\nobtain \u27e8sec, h\u27e9 := nonempty_sections_of_finite_cofiltered_system (F.toPreimages s)\n[GOAL]\ncase intro\nJ : Type u\ninst\u271d\u00b3 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j k : J\ns\u271d : Set (F.obj i\u271d)\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\ninst\u271d\u00b9 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis : \u2200 (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) \u2192 (toPreimages F s).obj j\nh : sec \u2208 sections (toPreimages F s)\n\u22a2 \u2203 a, (fun s => \u2191s i) a = x\n[PROOFSTEP]\nrefine' \u27e8\u27e8fun j => (sec j).val, fun jk => by simpa [Subtype.ext_iff] using h jk\u27e9, _\u27e9\n[GOAL]\nJ : Type u\ninst\u271d\u00b3 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j k : J\ns\u271d : Set (F.obj i\u271d)\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\ninst\u271d\u00b9 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis : \u2200 (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) \u2192 (toPreimages F s).obj j\nh : sec \u2208 sections (toPreimages F s)\nj\u271d j'\u271d : J\njk : j\u271d \u27f6 j'\u271d\n\u22a2 F.map jk ((fun j => \u2191(sec j)) j\u271d) = (fun j => \u2191(sec j)) j'\u271d\n[PROOFSTEP]\nsimpa [Subtype.ext_iff] using h jk\n[GOAL]\ncase intro\nJ : Type u\ninst\u271d\u00b3 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j k : J\ns\u271d : Set (F.obj i\u271d)\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\ninst\u271d\u00b9 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis : \u2200 (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) \u2192 (toPreimages F s).obj j\nh : sec \u2208 sections (toPreimages F s)\n\u22a2 (fun s => \u2191s i)\n { val := fun j => \u2191(sec j),\n property := (_ : \u2200 {j j' : J} (jk : j \u27f6 j'), F.map jk ((fun j => \u2191(sec j)) j) = (fun j => \u2191(sec j)) j') } =\n x\n[PROOFSTEP]\nhave := (sec i).prop\n[GOAL]\ncase intro\nJ : Type u\ninst\u271d\u00b3 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j k : J\ns\u271d : Set (F.obj i\u271d)\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\ninst\u271d\u00b9 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis\u271d : \u2200 (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) \u2192 (toPreimages F s).obj j\nh : sec \u2208 sections (toPreimages F s)\nthis : \u2191(sec i) \u2208 \u22c2 (f : i \u27f6 i), F.map f \u207b\u00b9' s\n\u22a2 (fun s => \u2191s i)\n { val := fun j => \u2191(sec j),\n property := (_ : \u2200 {j j' : J} (jk : j \u27f6 j'), F.map jk ((fun j => \u2191(sec j)) j) = (fun j => \u2191(sec j)) j') } =\n x\n[PROOFSTEP]\nsimp only [mem_iInter, mem_preimage, mem_singleton_iff] at this \n[GOAL]\ncase intro\nJ : Type u\ninst\u271d\u00b3 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j k : J\ns\u271d : Set (F.obj i\u271d)\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\ninst\u271d\u00b9 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis\u271d : \u2200 (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) \u2192 (toPreimages F s).obj j\nh : sec \u2208 sections (toPreimages F s)\nthis : \u2200 (i_1 : i \u27f6 i), F.map i_1 \u2191(sec i) = x\n\u22a2 (fun s => \u2191s i)\n { val := fun j => \u2191(sec j),\n property := (_ : \u2200 {j j' : J} (jk : j \u27f6 j'), F.map jk ((fun j => \u2191(sec j)) j) = (fun j => \u2191(sec j)) j') } =\n x\n[PROOFSTEP]\nhave := this (\ud835\udfd9 i)\n[GOAL]\ncase intro\nJ : Type u\ninst\u271d\u00b3 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j k : J\ns\u271d : Set (F.obj i\u271d)\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\ninst\u271d\u00b9 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ni : J\nx : F.obj i\ns : Set (F.obj i) := {x}\nthis\u271d\u00b9 : \u2200 (j : J), _root_.Nonempty ((toPreimages F s).obj j)\nsec : (j : J) \u2192 (toPreimages F s).obj j\nh : sec \u2208 sections (toPreimages F s)\nthis\u271d : \u2200 (i_1 : i \u27f6 i), F.map i_1 \u2191(sec i) = x\nthis : F.map (\ud835\udfd9 i) \u2191(sec i) = x\n\u22a2 (fun s => \u2191s i)\n { val := fun j => \u2191(sec j),\n property := (_ : \u2200 {j j' : J} (jk : j \u27f6 j'), F.map jk ((fun j => \u2191(sec j)) j) = (fun j => \u2191(sec j)) j') } =\n x\n[PROOFSTEP]\nrwa [map_id_apply] at this \n[GOAL]\nJ : Type u\ninst\u271d\u2075 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d\u2074 : IsCofilteredOrEmpty J\ninst\u271d\u00b3 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d\u00b2 : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ninst\u271d\u00b9 : _root_.Nonempty J\ninst\u271d : Finite \u2191(sections F)\n\u22a2 \u2203 j, \u2200 (i : J) (f : i \u27f6 j), Function.Injective (F.map f)\n[PROOFSTEP]\nhaveI : \u2200 j, Fintype (F.obj j) := fun j => Fintype.ofFinite (F.obj j)\n[GOAL]\nJ : Type u\ninst\u271d\u2075 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d\u2074 : IsCofilteredOrEmpty J\ninst\u271d\u00b3 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d\u00b2 : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ninst\u271d\u00b9 : _root_.Nonempty J\ninst\u271d : Finite \u2191(sections F)\nthis : (j : J) \u2192 Fintype (F.obj j)\n\u22a2 \u2203 j, \u2200 (i : J) (f : i \u27f6 j), Function.Injective (F.map f)\n[PROOFSTEP]\nhaveI : Fintype F.sections := Fintype.ofFinite F.sections\n[GOAL]\nJ : Type u\ninst\u271d\u2075 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d\u2074 : IsCofilteredOrEmpty J\ninst\u271d\u00b3 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d\u00b2 : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ninst\u271d\u00b9 : _root_.Nonempty J\ninst\u271d : Finite \u2191(sections F)\nthis\u271d : (j : J) \u2192 Fintype (F.obj j)\nthis : Fintype \u2191(sections F)\n\u22a2 \u2203 j, \u2200 (i : J) (f : i \u27f6 j), Function.Injective (F.map f)\n[PROOFSTEP]\nhave card_le : \u2200 j, Fintype.card (F.obj j) \u2264 Fintype.card F.sections := fun j =>\n Fintype.card_le_of_surjective _ (F.eval_section_surjective_of_surjective Fsur j)\n[GOAL]\nJ : Type u\ninst\u271d\u2075 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d\u2074 : IsCofilteredOrEmpty J\ninst\u271d\u00b3 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d\u00b2 : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ninst\u271d\u00b9 : _root_.Nonempty J\ninst\u271d : Finite \u2191(sections F)\nthis\u271d : (j : J) \u2192 Fintype (F.obj j)\nthis : Fintype \u2191(sections F)\ncard_le : \u2200 (j : J), Fintype.card (F.obj j) \u2264 Fintype.card \u2191(sections F)\n\u22a2 \u2203 j, \u2200 (i : J) (f : i \u27f6 j), Function.Injective (F.map f)\n[PROOFSTEP]\nlet fn j := Fintype.card F.sections - Fintype.card (F.obj j)\n[GOAL]\nJ : Type u\ninst\u271d\u2075 : Category.{u_1, u} J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d\u2074 : IsCofilteredOrEmpty J\ninst\u271d\u00b3 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d\u00b2 : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ninst\u271d\u00b9 : _root_.Nonempty J\ninst\u271d : Finite \u2191(sections F)\nthis\u271d : (j : J) \u2192 Fintype (F.obj j)\nthis : Fintype \u2191(sections F)\ncard_le : \u2200 (j : J), Fintype.card (F.obj j) \u2264 Fintype.card \u2191(sections F)\nfn : J \u2192 \u2115 := fun j => Fintype.card \u2191(sections F) - Fintype.card (F.obj j)\n\u22a2 \u2203 j, \u2200 (i : J) (f : i \u27f6 j), Function.Injective (F.map f)\n[PROOFSTEP]\nrefine'\n \u27e8fn.argmin Nat.lt_wfRel.wf, fun i f =>\n ((Fintype.bijective_iff_surjective_and_card _).2\n \u27e8Fsur f, le_antisymm _ (Fintype.card_le_of_surjective _ <| Fsur f)\u27e9).1\u27e9\n[GOAL]\nJ : Type u\ninst\u271d\u2075 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j k : J\ns : Set (F.obj i\u271d)\ninst\u271d\u2074 : IsCofilteredOrEmpty J\ninst\u271d\u00b3 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d\u00b2 : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ninst\u271d\u00b9 : _root_.Nonempty J\ninst\u271d : Finite \u2191(sections F)\nthis\u271d : (j : J) \u2192 Fintype (F.obj j)\nthis : Fintype \u2191(sections F)\ncard_le : \u2200 (j : J), Fintype.card (F.obj j) \u2264 Fintype.card \u2191(sections F)\nfn : J \u2192 \u2115 := fun j => Fintype.card \u2191(sections F) - Fintype.card (F.obj j)\ni : J\nf : i \u27f6 Function.argmin fn (_ : WellFounded WellFoundedRelation.rel)\n\u22a2 Fintype.card (F.obj i) \u2264 Fintype.card (F.obj (Function.argmin fn (_ : WellFounded WellFoundedRelation.rel)))\n[PROOFSTEP]\nrw [\u2190 Nat.sub_sub_self (card_le i), tsub_le_iff_tsub_le]\n[GOAL]\nJ : Type u\ninst\u271d\u2075 : Category.{u_1, u} J\nF : J \u2964 Type v\ni\u271d j k : J\ns : Set (F.obj i\u271d)\ninst\u271d\u2074 : IsCofilteredOrEmpty J\ninst\u271d\u00b3 : \u2200 (j : J), _root_.Nonempty (F.obj j)\ninst\u271d\u00b2 : \u2200 (j : J), Finite (F.obj j)\nFsur : \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), Function.Surjective (F.map f)\ninst\u271d\u00b9 : _root_.Nonempty J\ninst\u271d : Finite \u2191(sections F)\nthis\u271d : (j : J) \u2192 Fintype (F.obj j)\nthis : Fintype \u2191(sections F)\ncard_le : \u2200 (j : J), Fintype.card (F.obj j) \u2264 Fintype.card \u2191(sections F)\nfn : J \u2192 \u2115 := fun j => Fintype.card \u2191(sections F) - Fintype.card (F.obj j)\ni : J\nf : i \u27f6 Function.argmin fn (_ : WellFounded WellFoundedRelation.rel)\n\u22a2 Fintype.card \u2191(sections F) - Fintype.card (F.obj (Function.argmin fn (_ : WellFounded WellFoundedRelation.rel))) \u2264\n Fintype.card \u2191(sections F) - Fintype.card (F.obj i)\n[PROOFSTEP]\napply fn.argmin_le\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.CofilteredSystem", "llama_tokens": 32271, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544085240401, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4064787488216858}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 IsBigOWith c l f g \u2194 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nrw [IsBigOWith_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 f =O[l] g \u2194 \u2203 c, IsBigOWith c l f g\n[PROOFSTEP]\nrw [IsBigO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 f =O[l] g \u2194 \u2203 c, \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nsimp only [IsBigO_def, IsBigOWith_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\n\u22a2 f =O[l] g \u2194 \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nrefine \u27e8fun h => ?mp, fun h => ?mpr\u27e9\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : f =O[l] g\n\u22a2 \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 f =O[l] g\n[PROOFSTEP]\ncase mp =>\n rw [isBigO_iff] at h \n obtain \u27e8c, hc\u27e9 := h\n refine' \u27e8max c 1, zero_lt_one.trans_le (le_max_right _ _), _\u27e9\n filter_upwards [hc] with x hx\n apply hx.trans\n gcongr\n exact le_max_left _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : f =O[l] g\n\u22a2 \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\ncase mp =>\n rw [isBigO_iff] at h \n obtain \u27e8c, hc\u27e9 := h\n refine' \u27e8max c 1, zero_lt_one.trans_le (le_max_right _ _), _\u27e9\n filter_upwards [hc] with x hx\n apply hx.trans\n gcongr\n exact le_max_left _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : f =O[l] g\n\u22a2 \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nrw [isBigO_iff] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : \u2203 c, \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 := h\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nhc : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nrefine' \u27e8max c 1, zero_lt_one.trans_le (le_max_right _ _), _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nhc : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 max c 1 * \u2016g x\u2016\n[PROOFSTEP]\nfilter_upwards [hc] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nhc : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nx : \u03b1\nhx : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 \u2016f x\u2016 \u2264 max c 1 * \u2016g x\u2016\n[PROOFSTEP]\napply hx.trans\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nhc : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nx : \u03b1\nhx : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 c * \u2016g x\u2016 \u2264 max c 1 * \u2016g x\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nhc : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nx : \u03b1\nhx : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 c \u2264 max c 1\n[PROOFSTEP]\nexact le_max_left _ _\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 f =O[l] g\n[PROOFSTEP]\ncase mpr =>\n rw [isBigO_iff]\n obtain \u27e8c, \u27e8_, hc\u27e9\u27e9 := h\n exact \u27e8c, hc\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 f =O[l] g\n[PROOFSTEP]\ncase mpr =>\n rw [isBigO_iff]\n obtain \u27e8c, \u27e8_, hc\u27e9\u27e9 := h\n exact \u27e8c, hc\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 f =O[l] g\n[PROOFSTEP]\nrw [isBigO_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 \u2203 c, \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nobtain \u27e8c, \u27e8_, hc\u27e9\u27e9 := h\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nleft\u271d : c > 0\nhc : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 \u2203 c, \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nexact \u27e8c, hc\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\n\u22a2 f =O[l] g \u2194 \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nrefine \u27e8fun h => ?mp, fun h => ?mpr\u27e9\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : f =O[l] g\n\u22a2 \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u22a2 f =O[l] g\n[PROOFSTEP]\ncase mp =>\n rw [isBigO_iff'] at h \n obtain \u27e8c, \u27e8hc_pos, hc\u27e9\u27e9 := h\n refine \u27e8c\u207b\u00b9, \u27e8by positivity, ?_\u27e9\u27e9\n filter_upwards [hc] with x hx\n rwa [inv_mul_le_iff (by positivity)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : f =O[l] g\n\u22a2 \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\ncase mp =>\n rw [isBigO_iff'] at h \n obtain \u27e8c, \u27e8hc_pos, hc\u27e9\u27e9 := h\n refine \u27e8c\u207b\u00b9, \u27e8by positivity, ?_\u27e9\u27e9\n filter_upwards [hc] with x hx\n rwa [inv_mul_le_iff (by positivity)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : f =O[l] g\n\u22a2 \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nrw [isBigO_iff'] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nobtain \u27e8c, \u27e8hc_pos, hc\u27e9\u27e9 := h\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nhc_pos : c > 0\nhc : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nrefine \u27e8c\u207b\u00b9, \u27e8by positivity, ?_\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nhc_pos : c > 0\nhc : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 c\u207b\u00b9 > 0\n[PROOFSTEP]\npositivity\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nhc_pos : c > 0\nhc : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, c\u207b\u00b9 * \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nfilter_upwards [hc] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nhc_pos : c > 0\nhc : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nx : \u03b1\nhx : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 c\u207b\u00b9 * \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nrwa [inv_mul_le_iff (by positivity)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nhc_pos : c > 0\nhc : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nx : \u03b1\nhx : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 0 < c\n[PROOFSTEP]\npositivity\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u22a2 f =O[l] g\n[PROOFSTEP]\ncase mpr =>\n rw [isBigO_iff']\n obtain \u27e8c, \u27e8hc_pos, hc\u27e9\u27e9 := h\n refine \u27e8c\u207b\u00b9, \u27e8by positivity, ?_\u27e9\u27e9\n filter_upwards [hc] with x hx\n rwa [\u2190 inv_inv c, inv_mul_le_iff (by positivity)] at hx \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u22a2 f =O[l] g\n[PROOFSTEP]\ncase mpr =>\n rw [isBigO_iff']\n obtain \u27e8c, \u27e8hc_pos, hc\u27e9\u27e9 := h\n refine \u27e8c\u207b\u00b9, \u27e8by positivity, ?_\u27e9\u27e9\n filter_upwards [hc] with x hx\n rwa [\u2190 inv_inv c, inv_mul_le_iff (by positivity)] at hx \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u22a2 f =O[l] g\n[PROOFSTEP]\nrw [isBigO_iff']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nh : \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u22a2 \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nobtain \u27e8c, \u27e8hc_pos, hc\u27e9\u27e9 := h\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nhc_pos : c > 0\nhc : \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u22a2 \u2203 c, c > 0 \u2227 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nrefine \u27e8c\u207b\u00b9, \u27e8by positivity, ?_\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nhc_pos : c > 0\nhc : \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u22a2 c\u207b\u00b9 > 0\n[PROOFSTEP]\npositivity\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nhc_pos : c > 0\nhc : \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c\u207b\u00b9 * \u2016g x\u2016\n[PROOFSTEP]\nfilter_upwards [hc] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nhc_pos : c > 0\nhc : \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\nx : \u03b1\nhx : c * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u22a2 \u2016f x\u2016 \u2264 c\u207b\u00b9 * \u2016g x\u2016\n[PROOFSTEP]\nrwa [\u2190 inv_inv c, inv_mul_le_iff (by positivity)] at hx \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ng : \u03b1 \u2192 E'''\nc : \u211d\nhc_pos : c > 0\nhc : \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\nx : \u03b1\nhx : c\u207b\u00b9\u207b\u00b9 * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u22a2 0 < c\u207b\u00b9\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 1 * \u2016g x\u2016\n[PROOFSTEP]\nsimp_rw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 f =o[l] g \u2194 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f g\n[PROOFSTEP]\nrw [IsLittleO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 f =o[l] g \u2194 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nsimp only [IsLittleO_def, IsBigOWith_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : f =o[l] g\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nsimpa using h.def zero_lt_one\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : IsBigOWith c l f g\n\u22a2 f =O[l] g\n[PROOFSTEP]\nrw [IsBigO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : IsBigOWith c l f g\n\u22a2 \u2203 c, IsBigOWith c l f g\n[PROOFSTEP]\nexact \u27e8c, h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : IsBigOWith c l f g'\nhc : c \u2264 c'\nx : \u03b1\nhx : x \u2208 {x | (fun x => \u2016f x\u2016 \u2264 c * \u2016g' x\u2016) x}\n\u22a2 c * \u2016g' x\u2016 \u2264 c' * \u2016g' x\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 (\u2200\u1da0 (c : \u211d) in atTop, IsBigOWith c l f g') \u2194 \u2200\u1da0 (c : \u211d) in atTop, \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g' x\u2016\n[PROOFSTEP]\nsimp only [IsBigOWith_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Sort u_17\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nh\u271d : f =O[l] g'\nhb : HasBasis l p s\nc : \u211d\n_hc : 0 < c\nh : IsBigOWith c l f g'\n\u22a2 \u2203 i _hi, \u2200 (x : \u03b1), x \u2208 s i \u2192 \u2016f x\u2016 \u2264 c * \u2016g' x\u2016\n[PROOFSTEP]\nsimpa only [isBigOWith_iff, hb.eventually_iff, exists_prop] using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhc : 0 < c\n\u22a2 IsBigOWith c\u207b\u00b9 l f g \u2194 \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nsimp only [IsBigOWith_def, \u2190 div_eq_inv_mul, le_div_iff' hc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh\u2080 : (\u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016) \u2228 \u2200 (x : \u03b1), 0 \u2264 \u2016g x\u2016\n\u22a2 f =o[l] g \u2194 \u2200 (n : \u2115), \u2200\u1da0 (x : \u03b1) in l, \u2191n * \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh\u2080 : (\u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016) \u2228 \u2200 (x : \u03b1), 0 \u2264 \u2016g x\u2016\n\u22a2 f =o[l] g \u2192 \u2200 (n : \u2115), \u2200\u1da0 (x : \u03b1) in l, \u2191n * \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nrintro H (_ | n)\n[GOAL]\ncase mp.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh\u2080 : (\u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016) \u2228 \u2200 (x : \u03b1), 0 \u2264 \u2016g x\u2016\nH : f =o[l] g\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2191Nat.zero * \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nrefine' (H.def one_pos).mono fun x h\u2080' => _\n[GOAL]\ncase mp.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh\u2080 : (\u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016) \u2228 \u2200 (x : \u03b1), 0 \u2264 \u2016g x\u2016\nH : f =o[l] g\nx : \u03b1\nh\u2080' : \u2016f x\u2016 \u2264 1 * \u2016g x\u2016\n\u22a2 \u2191Nat.zero * \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nrw [Nat.cast_zero, zero_mul]\n[GOAL]\ncase mp.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh\u2080 : (\u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016) \u2228 \u2200 (x : \u03b1), 0 \u2264 \u2016g x\u2016\nH : f =o[l] g\nx : \u03b1\nh\u2080' : \u2016f x\u2016 \u2264 1 * \u2016g x\u2016\n\u22a2 0 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nrefine' h\u2080.elim (fun hf => (hf x).trans _) fun hg => hg x\n[GOAL]\ncase mp.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh\u2080 : (\u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016) \u2228 \u2200 (x : \u03b1), 0 \u2264 \u2016g x\u2016\nH : f =o[l] g\nx : \u03b1\nh\u2080' : \u2016f x\u2016 \u2264 1 * \u2016g x\u2016\nhf : \u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016\n\u22a2 \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nrwa [one_mul] at h\u2080' \n[GOAL]\ncase mp.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh\u2080 : (\u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016) \u2228 \u2200 (x : \u03b1), 0 \u2264 \u2016g x\u2016\nH : f =o[l] g\nn : \u2115\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2191(Nat.succ n) * \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nhave : (0 : \u211d) < n.succ := Nat.cast_pos.2 n.succ_pos\n[GOAL]\ncase mp.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh\u2080 : (\u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016) \u2228 \u2200 (x : \u03b1), 0 \u2264 \u2016g x\u2016\nH : f =o[l] g\nn : \u2115\nthis : 0 < \u2191(Nat.succ n)\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2191(Nat.succ n) * \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nexact (isBigOWith_inv this).1 (H.def' <| inv_pos.2 this)\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh\u2080 : (\u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016) \u2228 \u2200 (x : \u03b1), 0 \u2264 \u2016g x\u2016\n\u22a2 (\u2200 (n : \u2115), \u2200\u1da0 (x : \u03b1) in l, \u2191n * \u2016f x\u2016 \u2264 \u2016g x\u2016) \u2192 f =o[l] g\n[PROOFSTEP]\nrefine' fun H => isLittleO_iff.2 fun \u03b5 \u03b50 => _\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh\u2080 : (\u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016) \u2228 \u2200 (x : \u03b1), 0 \u2264 \u2016g x\u2016\nH : \u2200 (n : \u2115), \u2200\u1da0 (x : \u03b1) in l, \u2191n * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 \u03b5 * \u2016g x\u2016\n[PROOFSTEP]\nrcases exists_nat_gt \u03b5\u207b\u00b9 with \u27e8n, hn\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh\u2080 : (\u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016) \u2228 \u2200 (x : \u03b1), 0 \u2264 \u2016g x\u2016\nH : \u2200 (n : \u2115), \u2200\u1da0 (x : \u03b1) in l, \u2191n * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nn : \u2115\nhn : \u03b5\u207b\u00b9 < \u2191n\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 \u03b5 * \u2016g x\u2016\n[PROOFSTEP]\nhave hn\u2080 : (0 : \u211d) < n := (inv_pos.2 \u03b50).trans hn\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh\u2080 : (\u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016) \u2228 \u2200 (x : \u03b1), 0 \u2264 \u2016g x\u2016\nH : \u2200 (n : \u2115), \u2200\u1da0 (x : \u03b1) in l, \u2191n * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nn : \u2115\nhn : \u03b5\u207b\u00b9 < \u2191n\nhn\u2080 : 0 < \u2191n\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 \u03b5 * \u2016g x\u2016\n[PROOFSTEP]\nrefine' ((isBigOWith_inv hn\u2080).2 (H n)).bound.mono fun x hfg => _\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh\u2080 : (\u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016) \u2228 \u2200 (x : \u03b1), 0 \u2264 \u2016g x\u2016\nH : \u2200 (n : \u2115), \u2200\u1da0 (x : \u03b1) in l, \u2191n * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nn : \u2115\nhn : \u03b5\u207b\u00b9 < \u2191n\nhn\u2080 : 0 < \u2191n\nx : \u03b1\nhfg : \u2016f x\u2016 \u2264 (\u2191n)\u207b\u00b9 * \u2016g x\u2016\n\u22a2 \u2016f x\u2016 \u2264 \u03b5 * \u2016g x\u2016\n[PROOFSTEP]\nrefine' hfg.trans (mul_le_mul_of_nonneg_right (inv_le_of_inv_le \u03b50 hn.le) _)\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh\u2080 : (\u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016) \u2228 \u2200 (x : \u03b1), 0 \u2264 \u2016g x\u2016\nH : \u2200 (n : \u2115), \u2200\u1da0 (x : \u03b1) in l, \u2191n * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nn : \u2115\nhn : \u03b5\u207b\u00b9 < \u2191n\nhn\u2080 : 0 < \u2191n\nx : \u03b1\nhfg : \u2016f x\u2016 \u2264 (\u2191n)\u207b\u00b9 * \u2016g x\u2016\n\u22a2 0 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nrefine' h\u2080.elim (fun hf => nonneg_of_mul_nonneg_right ((hf x).trans hfg) _) fun h => h x\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh\u2080 : (\u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016) \u2228 \u2200 (x : \u03b1), 0 \u2264 \u2016g x\u2016\nH : \u2200 (n : \u2115), \u2200\u1da0 (x : \u03b1) in l, \u2191n * \u2016f x\u2016 \u2264 \u2016g x\u2016\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\nn : \u2115\nhn : \u03b5\u207b\u00b9 < \u2191n\nhn\u2080 : 0 < \u2191n\nx : \u03b1\nhfg : \u2016f x\u2016 \u2264 (\u2191n)\u207b\u00b9 * \u2016g x\u2016\nhf : \u2200 (x : \u03b1), 0 \u2264 \u2016f x\u2016\n\u22a2 0 < (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nexact inv_pos.2 hn\u2080\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : Subsingleton E'\nc : \u211d\nhc : 0 < c\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016f' x\u2016 \u2264 c * \u2016g' x\u2016\n[PROOFSTEP]\nsimp [Subsingleton.elim (f' _) 0, mul_nonneg hc.le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E\ng\u2081 g\u2082 : \u03b1 \u2192 F\nhc : c\u2081 = c\u2082\nhf : f\u2081 =\u1da0[l] f\u2082\nhg : g\u2081 =\u1da0[l] g\u2082\n\u22a2 IsBigOWith c\u2081 l f\u2081 g\u2081 \u2194 IsBigOWith c\u2082 l f\u2082 g\u2082\n[PROOFSTEP]\nsimp only [IsBigOWith_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E\ng\u2081 g\u2082 : \u03b1 \u2192 F\nhc : c\u2081 = c\u2082\nhf : f\u2081 =\u1da0[l] f\u2082\nhg : g\u2081 =\u1da0[l] g\u2082\n\u22a2 (\u2200\u1da0 (x : \u03b1) in l, \u2016f\u2081 x\u2016 \u2264 c\u2081 * \u2016g\u2081 x\u2016) \u2194 \u2200\u1da0 (x : \u03b1) in l, \u2016f\u2082 x\u2016 \u2264 c\u2082 * \u2016g\u2082 x\u2016\n[PROOFSTEP]\nsubst c\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E\ng\u2081 g\u2082 : \u03b1 \u2192 F\nhf : f\u2081 =\u1da0[l] f\u2082\nhg : g\u2081 =\u1da0[l] g\u2082\n\u22a2 (\u2200\u1da0 (x : \u03b1) in l, \u2016f\u2081 x\u2016 \u2264 c\u2081 * \u2016g\u2081 x\u2016) \u2194 \u2200\u1da0 (x : \u03b1) in l, \u2016f\u2082 x\u2016 \u2264 c\u2081 * \u2016g\u2082 x\u2016\n[PROOFSTEP]\napply Filter.eventually_congr\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E\ng\u2081 g\u2082 : \u03b1 \u2192 F\nhf : f\u2081 =\u1da0[l] f\u2082\nhg : g\u2081 =\u1da0[l] g\u2082\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016f\u2081 x\u2016 \u2264 c\u2081 * \u2016g\u2081 x\u2016 \u2194 \u2016f\u2082 x\u2016 \u2264 c\u2081 * \u2016g\u2082 x\u2016\n[PROOFSTEP]\nfilter_upwards [hf, hg] with _ e\u2081 e\u2082\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E\ng\u2081 g\u2082 : \u03b1 \u2192 F\nhf : f\u2081 =\u1da0[l] f\u2082\nhg : g\u2081 =\u1da0[l] g\u2082\na\u271d : \u03b1\ne\u2081 : f\u2081 a\u271d = f\u2082 a\u271d\ne\u2082 : g\u2081 a\u271d = g\u2082 a\u271d\n\u22a2 \u2016f\u2081 a\u271d\u2016 \u2264 c\u2081 * \u2016g\u2081 a\u271d\u2016 \u2194 \u2016f\u2082 a\u271d\u2016 \u2264 c\u2081 * \u2016g\u2082 a\u271d\u2016\n[PROOFSTEP]\nrw [e\u2081, e\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E\ng\u2081 g\u2082 : \u03b1 \u2192 F\nhf : f\u2081 =\u1da0[l] f\u2082\nhg : g\u2081 =\u1da0[l] g\u2082\n\u22a2 f\u2081 =O[l] g\u2081 \u2194 f\u2082 =O[l] g\u2082\n[PROOFSTEP]\nsimp only [IsBigO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E\ng\u2081 g\u2082 : \u03b1 \u2192 F\nhf : f\u2081 =\u1da0[l] f\u2082\nhg : g\u2081 =\u1da0[l] g\u2082\n\u22a2 (\u2203 c, IsBigOWith c l f\u2081 g\u2081) \u2194 \u2203 c, IsBigOWith c l f\u2082 g\u2082\n[PROOFSTEP]\nexact exists_congr fun c => isBigOWith_congr rfl hf hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E\ng\u2081 g\u2082 : \u03b1 \u2192 F\nhf : f\u2081 =\u1da0[l] f\u2082\nhg : g\u2081 =\u1da0[l] g\u2082\n\u22a2 f\u2081 =o[l] g\u2081 \u2194 f\u2082 =o[l] g\u2082\n[PROOFSTEP]\nsimp only [IsLittleO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E\ng\u2081 g\u2082 : \u03b1 \u2192 F\nhf : f\u2081 =\u1da0[l] f\u2082\nhg : g\u2081 =\u1da0[l] g\u2082\n\u22a2 (\u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f\u2081 g\u2081) \u2194 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f\u2082 g\u2082\n[PROOFSTEP]\nexact forall\u2082_congr fun c _hc => isBigOWith_congr (Eq.refl c) hf hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk\u271d : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\nk : \u03b2 \u2192 \u03b1\nl : Filter \u03b2\n\u22a2 IsBigOWith c (map k l) f g \u2194 IsBigOWith c l (f \u2218 k) (g \u2218 k)\n[PROOFSTEP]\nsimp only [IsBigOWith_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk\u271d : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\nk : \u03b2 \u2192 \u03b1\nl : Filter \u03b2\n\u22a2 (\u2200\u1da0 (x : \u03b1) in map k l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016) \u2194 \u2200\u1da0 (x : \u03b2) in l, \u2016(f \u2218 k) x\u2016 \u2264 c * \u2016(g \u2218 k) x\u2016\n[PROOFSTEP]\nexact eventually_map\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk\u271d : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\nk : \u03b2 \u2192 \u03b1\nl : Filter \u03b2\n\u22a2 f =O[map k l] g \u2194 (f \u2218 k) =O[l] (g \u2218 k)\n[PROOFSTEP]\nsimp only [IsBigO_def, isBigOWith_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk\u271d : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\nk : \u03b2 \u2192 \u03b1\nl : Filter \u03b2\n\u22a2 f =o[map k l] g \u2194 (f \u2218 k) =o[l] (g \u2218 k)\n[PROOFSTEP]\nsimp only [IsLittleO_def, isBigOWith_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhfg : IsBigOWith c l f g\nhgk : IsBigOWith c' l g k\nhc : 0 \u2264 c\n\u22a2 IsBigOWith (c * c') l f k\n[PROOFSTEP]\nsimp only [IsBigOWith_def] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhc : 0 \u2264 c\nhfg : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nhgk : \u2200\u1da0 (x : \u03b1) in l, \u2016g x\u2016 \u2264 c' * \u2016k x\u2016\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * c' * \u2016k x\u2016\n[PROOFSTEP]\nfilter_upwards [hfg, hgk] with x hx hx'\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhc : 0 \u2264 c\nhfg : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nhgk : \u2200\u1da0 (x : \u03b1) in l, \u2016g x\u2016 \u2264 c' * \u2016k x\u2016\nx : \u03b1\nhx : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nhx' : \u2016g x\u2016 \u2264 c' * \u2016k x\u2016\n\u22a2 \u2016f x\u2016 \u2264 c * c' * \u2016k x\u2016\n[PROOFSTEP]\ncalc\n \u2016f x\u2016 \u2264 c * \u2016g x\u2016 := hx\n _ \u2264 c * (c' * \u2016k x\u2016) := by gcongr\n _ = c * c' * \u2016k x\u2016 := (mul_assoc _ _ _).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhc : 0 \u2264 c\nhfg : \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nhgk : \u2200\u1da0 (x : \u03b1) in l, \u2016g x\u2016 \u2264 c' * \u2016k x\u2016\nx : \u03b1\nhx : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nhx' : \u2016g x\u2016 \u2264 c' * \u2016k x\u2016\n\u22a2 c * \u2016g x\u2016 \u2264 c * (c' * \u2016k x\u2016)\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhfg : f =o[l] g\nhgk : IsBigOWith c l g k\nhc : 0 < c\n\u22a2 f =o[l] k\n[PROOFSTEP]\nsimp only [IsLittleO_def] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhgk : IsBigOWith c l g k\nhc : 0 < c\nhfg : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f g\n\u22a2 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f k\n[PROOFSTEP]\nintro c' c'pos\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c'\u271d c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhgk : IsBigOWith c l g k\nhc : 0 < c\nhfg : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f g\nc' : \u211d\nc'pos : 0 < c'\n\u22a2 IsBigOWith c' l f k\n[PROOFSTEP]\nhave : 0 < c' / c := div_pos c'pos hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c'\u271d c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhgk : IsBigOWith c l g k\nhc : 0 < c\nhfg : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f g\nc' : \u211d\nc'pos : 0 < c'\nthis : 0 < c' / c\n\u22a2 IsBigOWith c' l f k\n[PROOFSTEP]\nexact ((hfg this).trans hgk this.le).congr_const (div_mul_cancel _ hc.ne')\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhfg : IsBigOWith c l f g\nhgk : g =o[l] k\nhc : 0 < c\n\u22a2 f =o[l] k\n[PROOFSTEP]\nsimp only [IsLittleO_def] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhfg : IsBigOWith c l f g\nhc : 0 < c\nhgk : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l g k\n\u22a2 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f k\n[PROOFSTEP]\nintro c' c'pos\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c'\u271d c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhfg : IsBigOWith c l f g\nhc : 0 < c\nhgk : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l g k\nc' : \u211d\nc'pos : 0 < c'\n\u22a2 IsBigOWith c' l f k\n[PROOFSTEP]\nhave : 0 < c' / c := div_pos c'pos hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c'\u271d c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhfg : IsBigOWith c l f g\nhc : 0 < c\nhgk : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l g k\nc' : \u211d\nc'pos : 0 < c'\nthis : 0 < c' / c\n\u22a2 IsBigOWith c' l f k\n[PROOFSTEP]\nexact (hfg.trans (hgk this) hc.le).congr_const (mul_div_cancel' _ hc.ne')\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhfg : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016g x\u2016\nx : \u03b1\n\u22a2 \u2016f x\u2016 \u2264 1 * \u2016g x\u2016\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhfg : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016g x\u2016\nx : \u03b1\n\u22a2 \u2016f x\u2016 \u2264 \u2016g x\u2016\n[PROOFSTEP]\nexact hfg x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : \u2203\u1da0 (x : \u03b1) in l, \u2016f' x\u2016 \u2260 0\n\u22a2 \u00acf' =o[l] f'\n[PROOFSTEP]\nintro ho\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : \u2203\u1da0 (x : \u03b1) in l, \u2016f' x\u2016 \u2260 0\nho : f' =o[l] f'\n\u22a2 False\n[PROOFSTEP]\nrcases((ho.bound one_half_pos).and_frequently h).exists with \u27e8x, hle, hne\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : \u2203\u1da0 (x : \u03b1) in l, \u2016f' x\u2016 \u2260 0\nho : f' =o[l] f'\nx : \u03b1\nhle : \u2016f' x\u2016 \u2264 1 / 2 * \u2016f' x\u2016\nhne : \u2016f' x\u2016 \u2260 0\n\u22a2 False\n[PROOFSTEP]\nrw [one_div, \u2190 div_eq_inv_mul] at hle \n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : \u2203\u1da0 (x : \u03b1) in l, \u2016f' x\u2016 \u2260 0\nho : f' =o[l] f'\nx : \u03b1\nhle : \u2016f' x\u2016 \u2264 \u2016f' x\u2016 / 2\nhne : \u2016f' x\u2016 \u2260 0\n\u22a2 False\n[PROOFSTEP]\nexact (half_lt_self (lt_of_le_of_ne (norm_nonneg _) hne.symm)).not_le hle\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng'\u271d : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : TopologicalSpace \u03b1\nx : \u03b1\ns : Set \u03b1\nC : \u211d\ng : \u03b1 \u2192 E\ng' : \u03b1 \u2192 F\nh : \u2016g x\u2016 \u2264 C * \u2016g' x\u2016\n\u22a2 IsBigOWith C (\ud835\udcdd[insert x s] x) g g' \u2194 IsBigOWith C (\ud835\udcdd[s] x) g g'\n[PROOFSTEP]\nsimp_rw [IsBigOWith_def, nhdsWithin_insert, eventually_sup, eventually_pure, h, true_and_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng'\u271d : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : TopologicalSpace \u03b1\nx : \u03b1\ns : Set \u03b1\ng : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nh : g x = 0\n\u22a2 g =o[\ud835\udcdd[insert x s] x] g' \u2194 g =o[\ud835\udcdd[s] x] g'\n[PROOFSTEP]\nsimp_rw [IsLittleO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng'\u271d : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : TopologicalSpace \u03b1\nx : \u03b1\ns : Set \u03b1\ng : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nh : g x = 0\n\u22a2 (\u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c (\ud835\udcdd[insert x s] x) g g') \u2194 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c (\ud835\udcdd[s] x) g g'\n[PROOFSTEP]\nrefine' forall_congr' fun c => forall_congr' fun hc => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng'\u271d : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : TopologicalSpace \u03b1\nx : \u03b1\ns : Set \u03b1\ng : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nh : g x = 0\nc : \u211d\nhc : 0 < c\n\u22a2 IsBigOWith c (\ud835\udcdd[insert x s] x) g g' \u2194 IsBigOWith c (\ud835\udcdd[s] x) g g'\n[PROOFSTEP]\nrw [isBigOWith_insert]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng'\u271d : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : TopologicalSpace \u03b1\nx : \u03b1\ns : Set \u03b1\ng : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nh : g x = 0\nc : \u211d\nhc : 0 < c\n\u22a2 \u2016g x\u2016 \u2264 c * \u2016g' x\u2016\n[PROOFSTEP]\nrw [h, norm_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng'\u271d : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : TopologicalSpace \u03b1\nx : \u03b1\ns : Set \u03b1\ng : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nh : g x = 0\nc : \u211d\nhc : 0 < c\n\u22a2 0 \u2264 c * \u2016g' x\u2016\n[PROOFSTEP]\nexact mul_nonneg hc.le (norm_nonneg _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \u211d\n\u22a2 (IsBigOWith c l f fun x => \u2016g' x\u2016) \u2194 IsBigOWith c l f g'\n[PROOFSTEP]\nsimp only [IsBigOWith_def, norm_norm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \u211d\n\u22a2 (f =O[l] fun x => \u2016g' x\u2016) \u2194 f =O[l] g'\n[PROOFSTEP]\nsimp only [IsBigO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \u211d\n\u22a2 (\u2203 c, IsBigOWith c l f fun x => \u2016g' x\u2016) \u2194 \u2203 c, IsBigOWith c l f g'\n[PROOFSTEP]\nexact exists_congr fun _ => isBigOWith_norm_right\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \u211d\n\u22a2 (f =o[l] fun x => \u2016g' x\u2016) \u2194 f =o[l] g'\n[PROOFSTEP]\nsimp only [IsLittleO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \u211d\n\u22a2 (\u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f fun x => \u2016g' x\u2016) \u2194 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f g'\n[PROOFSTEP]\nexact forall\u2082_congr fun _ _ => isBigOWith_norm_right\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \u211d\n\u22a2 IsBigOWith c l (fun x => \u2016f' x\u2016) g \u2194 IsBigOWith c l f' g\n[PROOFSTEP]\nsimp only [IsBigOWith_def, norm_norm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \u211d\n\u22a2 (fun x => \u2016f' x\u2016) =O[l] g \u2194 f' =O[l] g\n[PROOFSTEP]\nsimp only [IsBigO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \u211d\n\u22a2 (\u2203 c, IsBigOWith c l (fun x => \u2016f' x\u2016) g) \u2194 \u2203 c, IsBigOWith c l f' g\n[PROOFSTEP]\nexact exists_congr fun _ => isBigOWith_norm_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \u211d\n\u22a2 (fun x => \u2016f' x\u2016) =o[l] g \u2194 f' =o[l] g\n[PROOFSTEP]\nsimp only [IsLittleO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \u211d\n\u22a2 (\u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l (fun x => \u2016f' x\u2016) g) \u2194 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f' g\n[PROOFSTEP]\nexact forall\u2082_congr fun _ _ => isBigOWith_norm_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 (IsBigOWith c l f fun x => -g' x) \u2194 IsBigOWith c l f g'\n[PROOFSTEP]\nsimp only [IsBigOWith_def, norm_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 (f =O[l] fun x => -g' x) \u2194 f =O[l] g'\n[PROOFSTEP]\nsimp only [IsBigO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 (\u2203 c, IsBigOWith c l f fun x => -g' x) \u2194 \u2203 c, IsBigOWith c l f g'\n[PROOFSTEP]\nexact exists_congr fun _ => isBigOWith_neg_right\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 (f =o[l] fun x => -g' x) \u2194 f =o[l] g'\n[PROOFSTEP]\nsimp only [IsLittleO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 (\u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f fun x => -g' x) \u2194 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f g'\n[PROOFSTEP]\nexact forall\u2082_congr fun _ _ => isBigOWith_neg_right\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 IsBigOWith c l (fun x => -f' x) g \u2194 IsBigOWith c l f' g\n[PROOFSTEP]\nsimp only [IsBigOWith_def, norm_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 (fun x => -f' x) =O[l] g \u2194 f' =O[l] g\n[PROOFSTEP]\nsimp only [IsBigO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 (\u2203 c, IsBigOWith c l (fun x => -f' x) g) \u2194 \u2203 c, IsBigOWith c l f' g\n[PROOFSTEP]\nexact exists_congr fun _ => isBigOWith_neg_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 (fun x => -f' x) =o[l] g \u2194 f' =o[l] g\n[PROOFSTEP]\nsimp only [IsLittleO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 (\u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l (fun x => -f' x) g) \u2194 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f' g\n[PROOFSTEP]\nexact forall\u2082_congr fun _ _ => isBigOWith_neg_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf'\u271d : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf' : \u03b1 \u2192 E' \u00d7 F'\n\u22a2 (fun x => (f' x).fst) =O[l] f'\n[PROOFSTEP]\nsimpa [IsBigO_def, IsBigOWith_def] using isBigO_fst_prod (E' := E') (F' := F')\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf'\u271d : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf' : \u03b1 \u2192 E' \u00d7 F'\n\u22a2 (fun x => (f' x).snd) =O[l] f'\n[PROOFSTEP]\nsimpa [IsBigO_def, IsBigOWith_def] using isBigO_snd_prod (E' := E') (F' := F')\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhf : IsBigOWith c l f' k'\nhg : IsBigOWith c l g' k'\n\u22a2 IsBigOWith c l (fun x => (f' x, g' x)) k'\n[PROOFSTEP]\nrw [isBigOWith_iff] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhf : \u2200\u1da0 (x : \u03b1) in l, \u2016f' x\u2016 \u2264 c * \u2016k' x\u2016\nhg : \u2200\u1da0 (x : \u03b1) in l, \u2016g' x\u2016 \u2264 c * \u2016k' x\u2016\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016(f' x, g' x)\u2016 \u2264 c * \u2016k' x\u2016\n[PROOFSTEP]\nfilter_upwards [hf, hg] with x using max_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : IsBigOWith c l f'' g''\nx : \u03b1\nhx : \u2016f'' x\u2016 \u2264 c * \u2016g'' x\u2016\nhg : g'' x = 0\n\u22a2 \u2016f'' x\u2016 \u2264 0\n[PROOFSTEP]\nsimpa [hg] using hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E'\ng\u2081 g\u2082 : \u03b1 \u2192 F'\nh\u2081 : IsBigOWith c\u2081 l f\u2081 g\nh\u2082 : IsBigOWith c\u2082 l f\u2082 g\n\u22a2 IsBigOWith (c\u2081 + c\u2082) l (fun x => f\u2081 x + f\u2082 x) g\n[PROOFSTEP]\nrw [IsBigOWith_def] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E'\ng\u2081 g\u2082 : \u03b1 \u2192 F'\nh\u2081 : \u2200\u1da0 (x : \u03b1) in l, \u2016f\u2081 x\u2016 \u2264 c\u2081 * \u2016g x\u2016\nh\u2082 : \u2200\u1da0 (x : \u03b1) in l, \u2016f\u2082 x\u2016 \u2264 c\u2082 * \u2016g x\u2016\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016f\u2081 x + f\u2082 x\u2016 \u2264 (c\u2081 + c\u2082) * \u2016g x\u2016\n[PROOFSTEP]\nfilter_upwards [h\u2081, h\u2082] with x hx\u2081 hx\u2082 using\n calc\n \u2016f\u2081 x + f\u2082 x\u2016 \u2264 c\u2081 * \u2016g x\u2016 + c\u2082 * \u2016g x\u2016 := norm_add_le_of_le hx\u2081 hx\u2082\n _ = (c\u2081 + c\u2082) * \u2016g x\u2016 := (add_mul _ _ _).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E'\ng\u2081 g\u2082 : \u03b1 \u2192 F'\nh\u2081 : f\u2081 =o[l] g\u2081\nh\u2082 : f\u2082 =o[l] g\u2082\n\u22a2 (fun x => f\u2081 x + f\u2082 x) =o[l] fun x => \u2016g\u2081 x\u2016 + \u2016g\u2082 x\u2016\n[PROOFSTEP]\nrefine' (h\u2081.trans_le fun x => _).add (h\u2082.trans_le _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E'\ng\u2081 g\u2082 : \u03b1 \u2192 F'\nh\u2081 : f\u2081 =o[l] g\u2081\nh\u2082 : f\u2082 =o[l] g\u2082\nx : \u03b1\n\u22a2 \u2016g\u2081 x\u2016 \u2264 \u2016\u2016g\u2081 x\u2016 + \u2016g\u2082 x\u2016\u2016\n[PROOFSTEP]\nsimp [abs_of_nonneg, add_nonneg]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E'\ng\u2081 g\u2082 : \u03b1 \u2192 F'\nh\u2081 : f\u2081 =o[l] g\u2081\nh\u2082 : f\u2082 =o[l] g\u2082\n\u22a2 \u2200 (x : \u03b1), \u2016g\u2082 x\u2016 \u2264 \u2016\u2016g\u2081 x\u2016 + \u2016g\u2082 x\u2016\u2016\n[PROOFSTEP]\nsimp [abs_of_nonneg, add_nonneg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E'\ng\u2081 g\u2082 : \u03b1 \u2192 F'\nh\u2081 : IsBigOWith c\u2081 l f\u2081 g\nh\u2082 : IsBigOWith c\u2082 l f\u2082 g\n\u22a2 IsBigOWith (c\u2081 + c\u2082) l (fun x => f\u2081 x - f\u2082 x) g\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using h\u2081.add h\u2082.neg_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E'\ng\u2081 g\u2082 : \u03b1 \u2192 F'\nh\u2081 : IsBigOWith c\u2081 l f\u2081 g\nh\u2082 : f\u2082 =o[l] g\nhc : c\u2081 < c\u2082\n\u22a2 IsBigOWith c\u2082 l (fun x => f\u2081 x - f\u2082 x) g\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using h\u2081.add_isLittleO h\u2082.neg_left hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E'\ng\u2081 g\u2082 : \u03b1 \u2192 F'\nh\u2081 : f\u2081 =O[l] g\nh\u2082 : f\u2082 =O[l] g\n\u22a2 (fun x => f\u2081 x - f\u2082 x) =O[l] g\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using h\u2081.add h\u2082.neg_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E'\ng\u2081 g\u2082 : \u03b1 \u2192 F'\nh\u2081 : f\u2081 =o[l] g\nh\u2082 : f\u2082 =o[l] g\n\u22a2 (fun x => f\u2081 x - f\u2082 x) =o[l] g\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using h\u2081.add h\u2082.neg_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 f\u2083 : \u03b1 \u2192 E'\nh : (fun x => f\u2081 x - f\u2082 x) =o[l] g\n\u22a2 (fun x => f\u2082 x - f\u2081 x) =o[l] g\n[PROOFSTEP]\nsimpa only [neg_sub] using h.neg_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : \u211d\nhc : 0 < c\nx : \u03b1\n\u22a2 x \u2208 {x | (fun x => \u20160\u2016 \u2264 c * \u2016g' x\u2016) x}\n[PROOFSTEP]\nsimpa using mul_nonneg hc.le (norm_nonneg <| g' x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhc : 0 \u2264 c\nx : \u03b1\n\u22a2 x \u2208 {x | (fun x => \u20160\u2016 \u2264 c * \u2016g' x\u2016) x}\n[PROOFSTEP]\nsimpa using mul_nonneg hc (norm_nonneg <| g' x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nx : \u03b1\n\u22a2 x \u2208 {x | (fun x => \u20160\u2016 \u2264 0 * \u2016g x\u2016) x}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 (IsBigOWith c l f'' fun _x => 0) \u2194 f'' =\u1da0[l] 0\n[PROOFSTEP]\nsimp only [IsBigOWith_def, exists_prop, true_and_iff, norm_zero, mul_zero, norm_le_zero_iff, EventuallyEq,\n Pi.zero_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c'\u271d c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\nc : E\nc' : F''\nhc' : c' \u2260 0\nl : Filter \u03b1\n\u22a2 IsBigOWith (\u2016c\u2016 / \u2016c'\u2016) l (fun _x => c) fun _x => c'\n[PROOFSTEP]\nsimp only [IsBigOWith_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c'\u271d c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\nc : E\nc' : F''\nhc' : c' \u2260 0\nl : Filter \u03b1\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016c\u2016 \u2264 \u2016c\u2016 / \u2016c'\u2016 * \u2016c'\u2016\n[PROOFSTEP]\napply univ_mem'\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c'\u271d c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\nc : E\nc' : F''\nhc' : c' \u2260 0\nl : Filter \u03b1\n\u22a2 \u2200 (a : \u03b1), a \u2208 {x | (fun x => \u2016c\u2016 \u2264 \u2016c\u2016 / \u2016c'\u2016 * \u2016c'\u2016) x}\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c'\u271d c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\nc : E\nc' : F''\nhc' : c' \u2260 0\nl : Filter \u03b1\nx : \u03b1\n\u22a2 x \u2208 {x | (fun x => \u2016c\u2016 \u2264 \u2016c\u2016 / \u2016c'\u2016 * \u2016c'\u2016) x}\n[PROOFSTEP]\nsimp only [mem_setOf_eq]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c'\u271d c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\nc : E\nc' : F''\nhc' : c' \u2260 0\nl : Filter \u03b1\nx : \u03b1\n\u22a2 \u2016c\u2016 \u2264 \u2016c\u2016 / \u2016c'\u2016 * \u2016c'\u2016\n[PROOFSTEP]\nrw [div_mul_cancel]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c'\u271d c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\nc : E\nc' : F''\nhc' : c' \u2260 0\nl : Filter \u03b1\nx : \u03b1\n\u22a2 \u2016c'\u2016 \u2260 0\n[PROOFSTEP]\nrwa [Ne.def, norm_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c'\u271d c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\nc : E''\nc' : F''\nl : Filter \u03b1\ninst\u271d : NeBot l\n\u22a2 ((fun _x => c) =O[l] fun _x => c') \u2194 c' = 0 \u2192 c = 0\n[PROOFSTEP]\nrcases eq_or_ne c' 0 with (rfl | hc')\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\nc : E''\nl : Filter \u03b1\ninst\u271d : NeBot l\n\u22a2 ((fun _x => c) =O[l] fun _x => 0) \u2194 0 = 0 \u2192 c = 0\n[PROOFSTEP]\nsimp [EventuallyEq]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c'\u271d c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\nc : E''\nc' : F''\nl : Filter \u03b1\ninst\u271d : NeBot l\nhc' : c' \u2260 0\n\u22a2 ((fun _x => c) =O[l] fun _x => c') \u2194 c' = 0 \u2192 c = 0\n[PROOFSTEP]\nsimp [hc', isBigO_const_const _ hc']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 IsBigOWith c \u22a4 f g \u2194 \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nrw [IsBigOWith_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 (\u2200\u1da0 (x : \u03b1) in \u22a4, \u2016f x\u2016 \u2264 c * \u2016g x\u2016) \u2194 \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 f =O[\u22a4] g \u2194 \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C * \u2016g x\u2016\n[PROOFSTEP]\nrw [isBigO_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 (\u2203 c, \u2200\u1da0 (x : \u03b1) in \u22a4, \u2016f x\u2016 \u2264 c * \u2016g x\u2016) \u2194 \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C * \u2016g x\u2016\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 f'' =o[\u22a4] g'' \u2194 \u2200 (x : \u03b1), f'' x = 0\n[PROOFSTEP]\nrefine' \u27e8_, fun h => (isLittleO_zero g'' \u22a4).congr (fun x => (h x).symm) fun x => rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 f'' =o[\u22a4] g'' \u2192 \u2200 (x : \u03b1), f'' x = 0\n[PROOFSTEP]\nsimp only [isLittleO_iff, eventually_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 (\u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200 (x : \u03b1), \u2016f'' x\u2016 \u2264 c * \u2016g'' x\u2016) \u2192 \u2200 (x : \u03b1), f'' x = 0\n[PROOFSTEP]\nrefine' fun h x => norm_le_zero_iff.1 _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200 (x : \u03b1), \u2016f'' x\u2016 \u2264 c * \u2016g'' x\u2016\nx : \u03b1\n\u22a2 \u2016f'' x\u2016 \u2264 0\n[PROOFSTEP]\nhave : Tendsto (fun c : \u211d => c * \u2016g'' x\u2016) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) :=\n ((continuous_id.mul continuous_const).tendsto' _ _ (zero_mul _)).mono_left inf_le_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200 (x : \u03b1), \u2016f'' x\u2016 \u2264 c * \u2016g'' x\u2016\nx : \u03b1\nthis : Tendsto (fun c => c * \u2016g'' x\u2016) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2016f'' x\u2016 \u2264 0\n[PROOFSTEP]\nexact\n le_of_tendsto_of_tendsto tendsto_const_nhds this\n (eventually_nhdsWithin_iff.2 <| eventually_of_forall fun c hc => h hc x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ns : Set \u03b1\n\u22a2 IsBigOWith c (\ud835\udcdf s) f g \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nrw [IsBigOWith_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ns : Set \u03b1\n\u22a2 (\u2200\u1da0 (x : \u03b1) in \ud835\udcdf s, \u2016f x\u2016 \u2264 c * \u2016g x\u2016) \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ns : Set \u03b1\n\u22a2 f =O[\ud835\udcdf s] g \u2194 \u2203 c, \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nrw [isBigO_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ns : Set \u03b1\n\u22a2 (\u2203 c, \u2200\u1da0 (x : \u03b1) in \ud835\udcdf s, \u2016f x\u2016 \u2264 c * \u2016g x\u2016) \u2194 \u2203 c, \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\ninst\u271d\u00b9 : One F\ninst\u271d : NormOneClass F\nc : E\nl : Filter \u03b1\n\u22a2 IsBigOWith \u2016c\u2016 l (fun _x => c) fun _x => 1\n[PROOFSTEP]\nsimp [isBigOWith_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : One F\ninst\u271d : NormOneClass F\n\u22a2 (f' =o[l] fun _x => 1) \u2194 Tendsto f' l (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp only [isLittleO_iff, norm_one, mul_one, Metric.nhds_basis_closedBall.tendsto_right_iff, Metric.mem_closedBall,\n dist_zero_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : One F\ninst\u271d : NormOneClass F\n\u22a2 (f =O[l] fun _x => 1) \u2194 IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016f x\u2016\n[PROOFSTEP]\nsimp only [isBigO_iff, norm_one, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : One F\ninst\u271d : NormOneClass F\n\u22a2 (\u2203 c, \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c) \u2194 IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016f x\u2016\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : One F\ninst\u271d : NormOneClass F\n_x : \u03b1\n\u22a2 0 \u2264 \u20161\u2016\n[PROOFSTEP]\nsimp only [norm_one, zero_le_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : One F\ninst\u271d : NormOneClass F\n\u22a2 (\u2200 (n : \u2115), \u2200\u1da0 (x : \u03b1) in l, \u2191n * \u20161\u2016 \u2264 \u2016f x\u2016) \u2194 \u2200 (n : \u2115), True \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2208 Ici \u2191n\n[PROOFSTEP]\nsimp only [norm_one, mul_one, true_imp_iff, mem_Ici]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : F''\n\u22a2 (f'' =O[l] fun _x => c) \u2194 (c = 0 \u2192 f'' =\u1da0[l] 0) \u2227 IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016f'' x\u2016\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8fun hc => isBigO_zero_right_iff.1 (by rwa [\u2190 hc]), h.isBoundedUnder_le\u27e9, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : F''\nh : f'' =O[l] fun _x => c\nhc : c = 0\n\u22a2 f'' =O[l] fun _x => 0\n[PROOFSTEP]\nrwa [\u2190 hc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : F''\n\u22a2 ((c = 0 \u2192 f'' =\u1da0[l] 0) \u2227 IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016f'' x\u2016) \u2192 f'' =O[l] fun _x => c\n[PROOFSTEP]\nrintro \u27e8hcf, hf\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : F''\nhcf : c = 0 \u2192 f'' =\u1da0[l] 0\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016f'' x\u2016\n\u22a2 f'' =O[l] fun _x => c\n[PROOFSTEP]\nrcases eq_or_ne c 0 with (hc | hc)\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : F''\nhcf : c = 0 \u2192 f'' =\u1da0[l] 0\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016f'' x\u2016\nhc : c = 0\n\u22a2 f'' =O[l] fun _x => c\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : F''\nhcf : c = 0 \u2192 f'' =\u1da0[l] 0\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016f'' x\u2016\nhc : c \u2260 0\n\u22a2 f'' =O[l] fun _x => c\n[PROOFSTEP]\nexacts [(hcf hc).trans_isBigO (isBigO_zero _ _), hf.isBigO_const hc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : \u2200\u1da0 (x : \u03b1) in l, g'' x \u2260 0\n\u22a2 f =O[l] g'' \u2194 IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016f x\u2016 / \u2016g'' x\u2016\n[PROOFSTEP]\nsimp only [isBigO_iff, IsBoundedUnder, IsBounded, eventually_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : \u2200\u1da0 (x : \u03b1) in l, g'' x \u2260 0\n\u22a2 (\u2203 c, \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 \u2264 c * \u2016g'' x\u2016) \u2194 \u2203 b, \u2200\u1da0 (a : \u03b1) in l, \u2016f a\u2016 / \u2016g'' a\u2016 \u2264 b\n[PROOFSTEP]\nexact exists_congr fun c => eventually_congr <| h.mono fun x hx => (div_le_iff <| norm_pos_iff.2 hx).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : E''\nhc : c \u2260 0\n\u22a2 (fun _x => c) =O[l] f' \u2194 \u2203 b, 0 < b \u2227 \u2200\u1da0 (x : \u03b1) in l, b \u2264 \u2016f' x\u2016\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : E''\nhc : c \u2260 0\n\u22a2 (fun _x => c) =O[l] f' \u2192 \u2203 b, 0 < b \u2227 \u2200\u1da0 (x : \u03b1) in l, b \u2264 \u2016f' x\u2016\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : E''\nhc : c \u2260 0\nh : (fun _x => c) =O[l] f'\n\u22a2 \u2203 b, 0 < b \u2227 \u2200\u1da0 (x : \u03b1) in l, b \u2264 \u2016f' x\u2016\n[PROOFSTEP]\nrcases h.exists_pos with \u27e8C, hC\u2080, hC\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : E''\nhc : c \u2260 0\nh : (fun _x => c) =O[l] f'\nC : \u211d\nhC\u2080 : 0 < C\nhC : IsBigOWith C l (fun _x => c) f'\n\u22a2 \u2203 b, 0 < b \u2227 \u2200\u1da0 (x : \u03b1) in l, b \u2264 \u2016f' x\u2016\n[PROOFSTEP]\nrefine' \u27e8\u2016c\u2016 / C, div_pos (norm_pos_iff.2 hc) hC\u2080, _\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : E''\nhc : c \u2260 0\nh : (fun _x => c) =O[l] f'\nC : \u211d\nhC\u2080 : 0 < C\nhC : IsBigOWith C l (fun _x => c) f'\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016c\u2016 / C \u2264 \u2016f' x\u2016\n[PROOFSTEP]\nexact hC.bound.mono fun x => (div_le_iff' hC\u2080).2\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : E''\nhc : c \u2260 0\n\u22a2 (\u2203 b, 0 < b \u2227 \u2200\u1da0 (x : \u03b1) in l, b \u2264 \u2016f' x\u2016) \u2192 (fun _x => c) =O[l] f'\n[PROOFSTEP]\nrintro \u27e8b, hb\u2080, hb\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : E''\nhc : c \u2260 0\nb : \u211d\nhb\u2080 : 0 < b\nhb : \u2200\u1da0 (x : \u03b1) in l, b \u2264 \u2016f' x\u2016\n\u22a2 (fun _x => c) =O[l] f'\n[PROOFSTEP]\nrefine' IsBigO.of_bound (\u2016c\u2016 / b) (hb.mono fun x hx => _)\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : E''\nhc : c \u2260 0\nb : \u211d\nhb\u2080 : 0 < b\nhb : \u2200\u1da0 (x : \u03b1) in l, b \u2264 \u2016f' x\u2016\nx : \u03b1\nhx : b \u2264 \u2016f' x\u2016\n\u22a2 \u2016c\u2016 \u2264 \u2016c\u2016 / b * \u2016f' x\u2016\n[PROOFSTEP]\nrw [div_mul_eq_mul_div, mul_div_assoc]\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : E''\nhc : c \u2260 0\nb : \u211d\nhb\u2080 : 0 < b\nhb : \u2200\u1da0 (x : \u03b1) in l, b \u2264 \u2016f' x\u2016\nx : \u03b1\nhx : b \u2264 \u2016f' x\u2016\n\u22a2 \u2016c\u2016 \u2264 \u2016c\u2016 * (\u2016f' x\u2016 / b)\n[PROOFSTEP]\nexact le_mul_of_one_le_right (norm_nonneg _) ((one_le_div hb\u2080).2 hx)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\nu : R\u02e3\nf : \u03b1 \u2192 R\nl : Filter \u03b1\n\u22a2 IsBigOWith \u2016\u2191u\u207b\u00b9\u2016 l f fun x => \u2191u * f x\n[PROOFSTEP]\nrefine' (isBigOWith_const_mul_self \u2191u\u207b\u00b9 _ l).congr_left _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\nu : R\u02e3\nf : \u03b1 \u2192 R\nl : Filter \u03b1\n\u22a2 \u2200 (x : \u03b1), \u2191u\u207b\u00b9 * (\u2191u * f x) = f x\n[PROOFSTEP]\nexact fun x =>\n u.inv_mul_cancel_left\n (f x)\n -- porting note: Lean just had trouble elaborating correctly, but this fixes it.\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081\u271d c\u2082\u271d : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 R\ng\u2081 g\u2082 : \u03b1 \u2192 \ud835\udd5c\nc\u2081 c\u2082 : \u211d\nh\u2081 : IsBigOWith c\u2081 l f\u2081 g\u2081\nh\u2082 : IsBigOWith c\u2082 l f\u2082 g\u2082\n\u22a2 IsBigOWith (c\u2081 * c\u2082) l (fun x => f\u2081 x * f\u2082 x) fun x => g\u2081 x * g\u2082 x\n[PROOFSTEP]\nsimp only [IsBigOWith_def] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081\u271d c\u2082\u271d : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 R\ng\u2081 g\u2082 : \u03b1 \u2192 \ud835\udd5c\nc\u2081 c\u2082 : \u211d\nh\u2081 : \u2200\u1da0 (x : \u03b1) in l, \u2016f\u2081 x\u2016 \u2264 c\u2081 * \u2016g\u2081 x\u2016\nh\u2082 : \u2200\u1da0 (x : \u03b1) in l, \u2016f\u2082 x\u2016 \u2264 c\u2082 * \u2016g\u2082 x\u2016\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016f\u2081 x * f\u2082 x\u2016 \u2264 c\u2081 * c\u2082 * \u2016g\u2081 x * g\u2082 x\u2016\n[PROOFSTEP]\nfilter_upwards [h\u2081, h\u2082] with _ hx\u2081 hx\u2082\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081\u271d c\u2082\u271d : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 R\ng\u2081 g\u2082 : \u03b1 \u2192 \ud835\udd5c\nc\u2081 c\u2082 : \u211d\nh\u2081 : \u2200\u1da0 (x : \u03b1) in l, \u2016f\u2081 x\u2016 \u2264 c\u2081 * \u2016g\u2081 x\u2016\nh\u2082 : \u2200\u1da0 (x : \u03b1) in l, \u2016f\u2082 x\u2016 \u2264 c\u2082 * \u2016g\u2082 x\u2016\na\u271d : \u03b1\nhx\u2081 : \u2016f\u2081 a\u271d\u2016 \u2264 c\u2081 * \u2016g\u2081 a\u271d\u2016\nhx\u2082 : \u2016f\u2082 a\u271d\u2016 \u2264 c\u2082 * \u2016g\u2082 a\u271d\u2016\n\u22a2 \u2016f\u2081 a\u271d * f\u2082 a\u271d\u2016 \u2264 c\u2081 * c\u2082 * \u2016g\u2081 a\u271d * g\u2082 a\u271d\u2016\n[PROOFSTEP]\napply le_trans (norm_mul_le _ _)\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081\u271d c\u2082\u271d : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 R\ng\u2081 g\u2082 : \u03b1 \u2192 \ud835\udd5c\nc\u2081 c\u2082 : \u211d\nh\u2081 : \u2200\u1da0 (x : \u03b1) in l, \u2016f\u2081 x\u2016 \u2264 c\u2081 * \u2016g\u2081 x\u2016\nh\u2082 : \u2200\u1da0 (x : \u03b1) in l, \u2016f\u2082 x\u2016 \u2264 c\u2082 * \u2016g\u2082 x\u2016\na\u271d : \u03b1\nhx\u2081 : \u2016f\u2081 a\u271d\u2016 \u2264 c\u2081 * \u2016g\u2081 a\u271d\u2016\nhx\u2082 : \u2016f\u2082 a\u271d\u2016 \u2264 c\u2082 * \u2016g\u2082 a\u271d\u2016\n\u22a2 \u2016f\u2081 a\u271d\u2016 * \u2016f\u2082 a\u271d\u2016 \u2264 c\u2081 * c\u2082 * \u2016g\u2081 a\u271d * g\u2082 a\u271d\u2016\n[PROOFSTEP]\nconvert mul_le_mul hx\u2081 hx\u2082 (norm_nonneg _) (le_trans (norm_nonneg _) hx\u2081) using 1\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081\u271d c\u2082\u271d : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 R\ng\u2081 g\u2082 : \u03b1 \u2192 \ud835\udd5c\nc\u2081 c\u2082 : \u211d\nh\u2081 : \u2200\u1da0 (x : \u03b1) in l, \u2016f\u2081 x\u2016 \u2264 c\u2081 * \u2016g\u2081 x\u2016\nh\u2082 : \u2200\u1da0 (x : \u03b1) in l, \u2016f\u2082 x\u2016 \u2264 c\u2082 * \u2016g\u2082 x\u2016\na\u271d : \u03b1\nhx\u2081 : \u2016f\u2081 a\u271d\u2016 \u2264 c\u2081 * \u2016g\u2081 a\u271d\u2016\nhx\u2082 : \u2016f\u2082 a\u271d\u2016 \u2264 c\u2082 * \u2016g\u2082 a\u271d\u2016\n\u22a2 c\u2081 * c\u2082 * \u2016g\u2081 a\u271d * g\u2082 a\u271d\u2016 = c\u2081 * \u2016g\u2081 a\u271d\u2016 * (c\u2082 * \u2016g\u2082 a\u271d\u2016)\n[PROOFSTEP]\nrw [norm_mul, mul_mul_mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 R\ng\u2081 g\u2082 : \u03b1 \u2192 \ud835\udd5c\nh\u2081 : f\u2081 =O[l] g\u2081\nh\u2082 : f\u2082 =o[l] g\u2082\n\u22a2 (fun x => f\u2081 x * f\u2082 x) =o[l] fun x => g\u2081 x * g\u2082 x\n[PROOFSTEP]\nsimp only [IsLittleO_def] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 R\ng\u2081 g\u2082 : \u03b1 \u2192 \ud835\udd5c\nh\u2081 : f\u2081 =O[l] g\u2081\nh\u2082 : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f\u2082 g\u2082\n\u22a2 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l (fun x => f\u2081 x * f\u2082 x) fun x => g\u2081 x * g\u2082 x\n[PROOFSTEP]\nintro c cpos\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 R\ng\u2081 g\u2082 : \u03b1 \u2192 \ud835\udd5c\nh\u2081 : f\u2081 =O[l] g\u2081\nh\u2082 : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f\u2082 g\u2082\nc : \u211d\ncpos : 0 < c\n\u22a2 IsBigOWith c l (fun x => f\u2081 x * f\u2082 x) fun x => g\u2081 x * g\u2082 x\n[PROOFSTEP]\nrcases h\u2081.exists_pos with \u27e8c', c'pos, hc'\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c'\u271d c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 R\ng\u2081 g\u2082 : \u03b1 \u2192 \ud835\udd5c\nh\u2081 : f\u2081 =O[l] g\u2081\nh\u2082 : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f\u2082 g\u2082\nc : \u211d\ncpos : 0 < c\nc' : \u211d\nc'pos : 0 < c'\nhc' : IsBigOWith c' l f\u2081 g\u2081\n\u22a2 IsBigOWith c l (fun x => f\u2081 x * f\u2082 x) fun x => g\u2081 x * g\u2082 x\n[PROOFSTEP]\nexact (hc'.mul (h\u2082 (div_pos cpos c'pos))).congr_const (mul_div_cancel' _ (ne_of_gt c'pos))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 R\ng\u2081 g\u2082 : \u03b1 \u2192 \ud835\udd5c\nh\u2081 : f\u2081 =o[l] g\u2081\nh\u2082 : f\u2082 =O[l] g\u2082\n\u22a2 (fun x => f\u2081 x * f\u2082 x) =o[l] fun x => g\u2081 x * g\u2082 x\n[PROOFSTEP]\nsimp only [IsLittleO_def] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 R\ng\u2081 g\u2082 : \u03b1 \u2192 \ud835\udd5c\nh\u2082 : f\u2082 =O[l] g\u2082\nh\u2081 : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f\u2081 g\u2081\n\u22a2 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l (fun x => f\u2081 x * f\u2082 x) fun x => g\u2081 x * g\u2082 x\n[PROOFSTEP]\nintro c cpos\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 R\ng\u2081 g\u2082 : \u03b1 \u2192 \ud835\udd5c\nh\u2082 : f\u2082 =O[l] g\u2082\nh\u2081 : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f\u2081 g\u2081\nc : \u211d\ncpos : 0 < c\n\u22a2 IsBigOWith c l (fun x => f\u2081 x * f\u2082 x) fun x => g\u2081 x * g\u2082 x\n[PROOFSTEP]\nrcases h\u2082.exists_pos with \u27e8c', c'pos, hc'\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c'\u271d c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 R\ng\u2081 g\u2082 : \u03b1 \u2192 \ud835\udd5c\nh\u2082 : f\u2082 =O[l] g\u2082\nh\u2081 : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f\u2081 g\u2081\nc : \u211d\ncpos : 0 < c\nc' : \u211d\nc'pos : 0 < c'\nhc' : IsBigOWith c' l f\u2082 g\u2082\n\u22a2 IsBigOWith c l (fun x => f\u2081 x * f\u2082 x) fun x => g\u2081 x * g\u2082 x\n[PROOFSTEP]\nexact ((h\u2081 (div_pos cpos c'pos)).mul hc').congr_const (div_mul_cancel _ (ne_of_gt c'pos))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 R\ng : \u03b1 \u2192 \ud835\udd5c\nh : IsBigOWith c l f g\n\u22a2 IsBigOWith (Nat.casesOn 0 \u20161\u2016 fun n => c ^ (n + 1)) l (fun x => f x ^ 0) fun x => g x ^ 0\n[PROOFSTEP]\nsimpa using isBigOWith_const_const (1 : R) (one_ne_zero' \ud835\udd5c) l\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 R\ng : \u03b1 \u2192 \ud835\udd5c\nh : IsBigOWith c l f g\n\u22a2 IsBigOWith (Nat.casesOn 1 \u20161\u2016 fun n => c ^ (n + 1)) l (fun x => f x ^ 1) fun x => g x ^ 1\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 R\ng : \u03b1 \u2192 \ud835\udd5c\nh : IsBigOWith c l f g\nn : \u2115\n\u22a2 IsBigOWith (Nat.casesOn (n + 2) \u20161\u2016 fun n => c ^ (n + 1)) l (fun x => f x ^ (n + 2)) fun x => g x ^ (n + 2)\n[PROOFSTEP]\nsimpa [pow_succ] using h.mul (IsBigOWith.pow' h (n + 1))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormOneClass R\nf : \u03b1 \u2192 R\ng : \u03b1 \u2192 \ud835\udd5c\nh : IsBigOWith c l f g\n\u22a2 IsBigOWith (c ^ 0) l (fun x => f x ^ 0) fun x => g x ^ 0\n[PROOFSTEP]\nsimpa using h.pow' 0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nn : \u2115\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b1 \u2192 R\nh : IsBigOWith c l (f ^ n) (g ^ n)\nhn : n \u2260 0\nhc : c \u2264 c' ^ n\nhc' : 0 \u2264 c'\nx : \u03b1\nhx : \u2016(f ^ n) x\u2016 \u2264 c' ^ n * \u2016(g ^ n) x\u2016\n\u22a2 c' ^ n * \u2016g x ^ n\u2016 \u2264 c' ^ n * \u2016g x\u2016 ^ n\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nn : \u2115\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b1 \u2192 R\nh : IsBigOWith c l (f ^ n) (g ^ n)\nhn : n \u2260 0\nhc : c \u2264 c' ^ n\nhc' : 0 \u2264 c'\nx : \u03b1\nhx : \u2016(f ^ n) x\u2016 \u2264 c' ^ n * \u2016(g ^ n) x\u2016\n\u22a2 \u2016g x ^ n\u2016 \u2264 \u2016g x\u2016 ^ n\n[PROOFSTEP]\nexact norm_pow_le' _ hn.bot_lt\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b1 \u2192 R\nn : \u2115\nhn : n \u2260 0\nh : (f ^ n) =O[l] (g ^ n)\n\u22a2 f =O[l] g\n[PROOFSTEP]\nrcases h.exists_pos with \u27e8C, _hC\u2080, hC\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b1 \u2192 R\nn : \u2115\nhn : n \u2260 0\nh : (f ^ n) =O[l] (g ^ n)\nC : \u211d\n_hC\u2080 : 0 < C\nhC : IsBigOWith C l (f ^ n) (g ^ n)\n\u22a2 f =O[l] g\n[PROOFSTEP]\nobtain \u27e8c, hc\u2080, hc\u27e9 : \u2203 c : \u211d, 0 \u2264 c \u2227 C \u2264 c ^ n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b1 \u2192 R\nn : \u2115\nhn : n \u2260 0\nh : (f ^ n) =O[l] (g ^ n)\nC : \u211d\n_hC\u2080 : 0 < C\nhC : IsBigOWith C l (f ^ n) (g ^ n)\n\u22a2 \u2203 c, 0 \u2264 c \u2227 C \u2264 c ^ n\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b1 \u2192 R\nn : \u2115\nhn : n \u2260 0\nh : (f ^ n) =O[l] (g ^ n)\nC : \u211d\n_hC\u2080 : 0 < C\nhC : IsBigOWith C l (f ^ n) (g ^ n)\nc : \u211d\nhc\u2080 : 0 \u2264 c\nhc : C \u2264 c ^ n\n\u22a2 f =O[l] g\n[PROOFSTEP]\nexact ((eventually_ge_atTop _).and <| (tendsto_pow_atTop hn).eventually_ge_atTop C).exists\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b1 \u2192 R\nn : \u2115\nhn : n \u2260 0\nh : (f ^ n) =O[l] (g ^ n)\nC : \u211d\n_hC\u2080 : 0 < C\nhC : IsBigOWith C l (f ^ n) (g ^ n)\nc : \u211d\nhc\u2080 : 0 \u2264 c\nhc : C \u2264 c ^ n\n\u22a2 f =O[l] g\n[PROOFSTEP]\nexact (hC.of_pow hn hc hc\u2080).isBigO\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 R\ng : \u03b1 \u2192 \ud835\udd5c\nh : f =o[l] g\nn : \u2115\nhn : 0 < n\n\u22a2 (fun x => f x ^ n) =o[l] fun x => g x ^ n\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 R\ng : \u03b1 \u2192 \ud835\udd5c\nh : f =o[l] g\nhn : 0 < Nat.zero\n\u22a2 (fun x => f x ^ Nat.zero) =o[l] fun x => g x ^ Nat.zero\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 R\ng : \u03b1 \u2192 \ud835\udd5c\nh : f =o[l] g\nn : \u2115\nhn : 0 < Nat.succ n\n\u22a2 (fun x => f x ^ Nat.succ n) =o[l] fun x => g x ^ Nat.succ n\n[PROOFSTEP]\nexact hn.false.elim\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 R\ng : \u03b1 \u2192 \ud835\udd5c\nh : f =o[l] g\nn : \u2115\nhn : 0 < Nat.succ n\n\u22a2 (fun x => f x ^ Nat.succ n) =o[l] fun x => g x ^ Nat.succ n\n[PROOFSTEP]\nclear hn\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 R\ng : \u03b1 \u2192 \ud835\udd5c\nh : f =o[l] g\nn : \u2115\n\u22a2 (fun x => f x ^ Nat.succ n) =o[l] fun x => g x ^ Nat.succ n\n[PROOFSTEP]\ninduction' n with n ihn\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 R\ng : \u03b1 \u2192 \ud835\udd5c\nh : f =o[l] g\n\u22a2 (fun x => f x ^ Nat.succ Nat.zero) =o[l] fun x => g x ^ Nat.succ Nat.zero\n[PROOFSTEP]\nsimpa only [Nat.zero_eq, \u2190 Nat.one_eq_succ_zero, pow_one]\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 R\ng : \u03b1 \u2192 \ud835\udd5c\nh : f =o[l] g\nn : \u2115\nihn : (fun x => f x ^ Nat.succ n) =o[l] fun x => g x ^ Nat.succ n\n\u22a2 (fun x => f x ^ Nat.succ (Nat.succ n)) =o[l] fun x => g x ^ Nat.succ (Nat.succ n)\n[PROOFSTEP]\nconvert h.mul ihn\n[GOAL]\ncase h.e'_7.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 R\ng : \u03b1 \u2192 \ud835\udd5c\nh : f =o[l] g\nn : \u2115\nihn : (fun x => f x ^ Nat.succ n) =o[l] fun x => g x ^ Nat.succ n\nx\u271d : \u03b1\n\u22a2 f x\u271d ^ Nat.succ (Nat.succ n) = f x\u271d * f x\u271d ^ Nat.succ n\n[PROOFSTEP]\nsimp [pow_succ]\n[GOAL]\ncase h.e'_8.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 R\ng : \u03b1 \u2192 \ud835\udd5c\nh : f =o[l] g\nn : \u2115\nihn : (fun x => f x ^ Nat.succ n) =o[l] fun x => g x ^ Nat.succ n\nx\u271d : \u03b1\n\u22a2 g x\u271d ^ Nat.succ (Nat.succ n) = g x\u271d * g x\u271d ^ Nat.succ n\n[PROOFSTEP]\nsimp [pow_succ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b1 \u2192 \ud835\udd5c'\nh : IsBigOWith c l f g\nh\u2080 : \u2200\u1da0 (x : \u03b1) in l, f x = 0 \u2192 g x = 0\n\u22a2 IsBigOWith c l (fun x => (g x)\u207b\u00b9) fun x => (f x)\u207b\u00b9\n[PROOFSTEP]\nrefine' IsBigOWith.of_bound (h.bound.mp (h\u2080.mono fun x h\u2080 hle => _))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b1 \u2192 \ud835\udd5c'\nh : IsBigOWith c l f g\nh\u2080\u271d : \u2200\u1da0 (x : \u03b1) in l, f x = 0 \u2192 g x = 0\nx : \u03b1\nh\u2080 : f x = 0 \u2192 g x = 0\nhle : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 \u2016(g x)\u207b\u00b9\u2016 \u2264 c * \u2016(f x)\u207b\u00b9\u2016\n[PROOFSTEP]\ncases' eq_or_ne (f x) 0 with hx hx\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b1 \u2192 \ud835\udd5c'\nh : IsBigOWith c l f g\nh\u2080\u271d : \u2200\u1da0 (x : \u03b1) in l, f x = 0 \u2192 g x = 0\nx : \u03b1\nh\u2080 : f x = 0 \u2192 g x = 0\nhle : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nhx : f x = 0\n\u22a2 \u2016(g x)\u207b\u00b9\u2016 \u2264 c * \u2016(f x)\u207b\u00b9\u2016\n[PROOFSTEP]\nsimp only [hx, h\u2080 hx, inv_zero, norm_zero, mul_zero]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b1 \u2192 \ud835\udd5c'\nh : IsBigOWith c l f g\nh\u2080\u271d : \u2200\u1da0 (x : \u03b1) in l, f x = 0 \u2192 g x = 0\nx : \u03b1\nh\u2080 : f x = 0 \u2192 g x = 0\nhle : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nhx : f x = 0\n\u22a2 0 \u2264 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b1 \u2192 \ud835\udd5c'\nh : IsBigOWith c l f g\nh\u2080\u271d : \u2200\u1da0 (x : \u03b1) in l, f x = 0 \u2192 g x = 0\nx : \u03b1\nh\u2080 : f x = 0 \u2192 g x = 0\nhle : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nhx : f x \u2260 0\n\u22a2 \u2016(g x)\u207b\u00b9\u2016 \u2264 c * \u2016(f x)\u207b\u00b9\u2016\n[PROOFSTEP]\nhave hc : 0 < c := pos_of_mul_pos_left ((norm_pos_iff.2 hx).trans_le hle) (norm_nonneg _)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b1 \u2192 \ud835\udd5c'\nh : IsBigOWith c l f g\nh\u2080\u271d : \u2200\u1da0 (x : \u03b1) in l, f x = 0 \u2192 g x = 0\nx : \u03b1\nh\u2080 : f x = 0 \u2192 g x = 0\nhle : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\nhx : f x \u2260 0\nhc : 0 < c\n\u22a2 \u2016(g x)\u207b\u00b9\u2016 \u2264 c * \u2016(f x)\u207b\u00b9\u2016\n[PROOFSTEP]\nreplace hle := inv_le_inv_of_le (norm_pos_iff.2 hx) hle\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\ng : \u03b1 \u2192 \ud835\udd5c'\nh : IsBigOWith c l f g\nh\u2080\u271d : \u2200\u1da0 (x : \u03b1) in l, f x = 0 \u2192 g x = 0\nx : \u03b1\nh\u2080 : f x = 0 \u2192 g x = 0\nhx : f x \u2260 0\nhc : 0 < c\nhle : (c * \u2016g x\u2016)\u207b\u00b9 \u2264 \u2016f x\u2016\u207b\u00b9\n\u22a2 \u2016(g x)\u207b\u00b9\u2016 \u2264 c * \u2016(f x)\u207b\u00b9\u2016\n[PROOFSTEP]\nsimpa only [norm_inv, mul_inv, \u2190 div_eq_inv_mul, div_le_iff hc] using hle\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc c'\u271d c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nh : IsBigOWith c l f' g\nc' : \ud835\udd5c\n\u22a2 IsBigOWith (\u2016c'\u2016 * c) l (fun x => \u2016c' \u2022 f' x\u2016) g\n[PROOFSTEP]\nsimpa only [norm_smul, _root_.norm_norm] using\n h.norm_left.const_mul_left\n \u2016c'\u2016\n -- porting note: probably `Asymptotics.IsBigO.norm_norm` and `Asymptotics.IsLittleO.norm_norm`\n -- should be protected.\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nh : f' =o[l] g\nc : \ud835\udd5c\n\u22a2 (fun x => \u2016(c \u2022 f') x\u2016) =o[l] g\n[PROOFSTEP]\nsimpa only [\u2190 norm_smul] using h.norm_left.const_mul_left \u2016c\u2016\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\n\u22a2 (fun x => c \u2022 f' x) =O[l] g \u2194 f' =O[l] g\n[PROOFSTEP]\nhave cne0 : \u2016c\u2016 \u2260 0 := mt norm_eq_zero.mp hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\ncne0 : \u2016c\u2016 \u2260 0\n\u22a2 (fun x => c \u2022 f' x) =O[l] g \u2194 f' =O[l] g\n[PROOFSTEP]\nrw [\u2190 isBigO_norm_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\ncne0 : \u2016c\u2016 \u2260 0\n\u22a2 (fun x => \u2016c \u2022 f' x\u2016) =O[l] g \u2194 f' =O[l] g\n[PROOFSTEP]\nsimp only [norm_smul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\ncne0 : \u2016c\u2016 \u2260 0\n\u22a2 (fun x => \u2016c\u2016 * \u2016f' x\u2016) =O[l] g \u2194 f' =O[l] g\n[PROOFSTEP]\nrw [isBigO_const_mul_left_iff cne0, isBigO_norm_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\n\u22a2 (fun x => c \u2022 f' x) =o[l] g \u2194 f' =o[l] g\n[PROOFSTEP]\nhave cne0 : \u2016c\u2016 \u2260 0 := mt norm_eq_zero.mp hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\ncne0 : \u2016c\u2016 \u2260 0\n\u22a2 (fun x => c \u2022 f' x) =o[l] g \u2194 f' =o[l] g\n[PROOFSTEP]\nrw [\u2190 isLittleO_norm_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\ncne0 : \u2016c\u2016 \u2260 0\n\u22a2 (fun x => \u2016c \u2022 f' x\u2016) =o[l] g \u2194 f' =o[l] g\n[PROOFSTEP]\nsimp only [norm_smul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\ncne0 : \u2016c\u2016 \u2260 0\n\u22a2 (fun x => \u2016c\u2016 * \u2016f' x\u2016) =o[l] g \u2194 f' =o[l] g\n[PROOFSTEP]\nrw [isLittleO_const_mul_left_iff cne0, isLittleO_norm_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\n\u22a2 (f =O[l] fun x => c \u2022 f' x) \u2194 f =O[l] f'\n[PROOFSTEP]\nhave cne0 : \u2016c\u2016 \u2260 0 := mt norm_eq_zero.mp hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\ncne0 : \u2016c\u2016 \u2260 0\n\u22a2 (f =O[l] fun x => c \u2022 f' x) \u2194 f =O[l] f'\n[PROOFSTEP]\nrw [\u2190 isBigO_norm_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\ncne0 : \u2016c\u2016 \u2260 0\n\u22a2 (f =O[l] fun x => \u2016c \u2022 f' x\u2016) \u2194 f =O[l] f'\n[PROOFSTEP]\nsimp only [norm_smul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\ncne0 : \u2016c\u2016 \u2260 0\n\u22a2 (f =O[l] fun x => \u2016c\u2016 * \u2016f' x\u2016) \u2194 f =O[l] f'\n[PROOFSTEP]\nrw [isBigO_const_mul_right_iff cne0, isBigO_norm_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\n\u22a2 (f =o[l] fun x => c \u2022 f' x) \u2194 f =o[l] f'\n[PROOFSTEP]\nhave cne0 : \u2016c\u2016 \u2260 0 := mt norm_eq_zero.mp hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\ncne0 : \u2016c\u2016 \u2260 0\n\u22a2 (f =o[l] fun x => c \u2022 f' x) \u2194 f =o[l] f'\n[PROOFSTEP]\nrw [\u2190 isLittleO_norm_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\ncne0 : \u2016c\u2016 \u2260 0\n\u22a2 (f =o[l] fun x => \u2016c \u2022 f' x\u2016) \u2194 f =o[l] f'\n[PROOFSTEP]\nsimp only [norm_smul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nc : \ud835\udd5c\nhc : c \u2260 0\ncne0 : \u2016c\u2016 \u2260 0\n\u22a2 (f =o[l] fun x => \u2016c\u2016 * \u2016f' x\u2016) \u2194 f =o[l] f'\n[PROOFSTEP]\nrw [isLittleO_const_mul_right_iff cne0, isLittleO_norm_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : IsBigOWith c l k\u2081 k\u2082\nh\u2082 : IsBigOWith c' l f' g'\n\u22a2 IsBigOWith (c * c') l (fun x => k\u2081 x \u2022 f' x) fun x => k\u2082 x \u2022 g' x\n[PROOFSTEP]\nrefine' ((h\u2081.norm_norm.mul h\u2082.norm_norm).congr rfl _ _).of_norm_norm\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : IsBigOWith c l k\u2081 k\u2082\nh\u2082 : IsBigOWith c' l f' g'\n\u22a2 \u2200 (x : \u03b1), \u2016k\u2081 x\u2016 * \u2016f' x\u2016 = \u2016k\u2081 x \u2022 f' x\u2016\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : IsBigOWith c l k\u2081 k\u2082\nh\u2082 : IsBigOWith c' l f' g'\nx\u271d : \u03b1\n\u22a2 \u2016k\u2081 x\u271d\u2016 * \u2016f' x\u271d\u2016 = \u2016k\u2081 x\u271d \u2022 f' x\u271d\u2016\n[PROOFSTEP]\nsimp only [norm_smul]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : IsBigOWith c l k\u2081 k\u2082\nh\u2082 : IsBigOWith c' l f' g'\n\u22a2 \u2200 (x : \u03b1), \u2016k\u2082 x\u2016 * \u2016g' x\u2016 = \u2016k\u2082 x \u2022 g' x\u2016\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : IsBigOWith c l k\u2081 k\u2082\nh\u2082 : IsBigOWith c' l f' g'\nx\u271d : \u03b1\n\u22a2 \u2016k\u2082 x\u271d\u2016 * \u2016g' x\u271d\u2016 = \u2016k\u2082 x\u271d \u2022 g' x\u271d\u2016\n[PROOFSTEP]\nsimp only [norm_smul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =O[l] k\u2082\nh\u2082 : f' =O[l] g'\n\u22a2 (fun x => k\u2081 x \u2022 f' x) =O[l] fun x => k\u2082 x \u2022 g' x\n[PROOFSTEP]\nrefine' ((h\u2081.norm_norm.mul h\u2082.norm_norm).congr _ _).of_norm_norm\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =O[l] k\u2082\nh\u2082 : f' =O[l] g'\n\u22a2 \u2200 (x : \u03b1), \u2016k\u2081 x\u2016 * \u2016f' x\u2016 = \u2016k\u2081 x \u2022 f' x\u2016\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =O[l] k\u2082\nh\u2082 : f' =O[l] g'\nx\u271d : \u03b1\n\u22a2 \u2016k\u2081 x\u271d\u2016 * \u2016f' x\u271d\u2016 = \u2016k\u2081 x\u271d \u2022 f' x\u271d\u2016\n[PROOFSTEP]\nsimp only [norm_smul]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =O[l] k\u2082\nh\u2082 : f' =O[l] g'\n\u22a2 \u2200 (x : \u03b1), \u2016k\u2082 x\u2016 * \u2016g' x\u2016 = \u2016k\u2082 x \u2022 g' x\u2016\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =O[l] k\u2082\nh\u2082 : f' =O[l] g'\nx\u271d : \u03b1\n\u22a2 \u2016k\u2082 x\u271d\u2016 * \u2016g' x\u271d\u2016 = \u2016k\u2082 x\u271d \u2022 g' x\u271d\u2016\n[PROOFSTEP]\nsimp only [norm_smul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =O[l] k\u2082\nh\u2082 : f' =o[l] g'\n\u22a2 (fun x => k\u2081 x \u2022 f' x) =o[l] fun x => k\u2082 x \u2022 g' x\n[PROOFSTEP]\nrefine' ((h\u2081.norm_norm.mul_isLittleO h\u2082.norm_norm).congr _ _).of_norm_norm\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =O[l] k\u2082\nh\u2082 : f' =o[l] g'\n\u22a2 \u2200 (x : \u03b1), \u2016k\u2081 x\u2016 * \u2016f' x\u2016 = \u2016k\u2081 x \u2022 f' x\u2016\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =O[l] k\u2082\nh\u2082 : f' =o[l] g'\nx\u271d : \u03b1\n\u22a2 \u2016k\u2081 x\u271d\u2016 * \u2016f' x\u271d\u2016 = \u2016k\u2081 x\u271d \u2022 f' x\u271d\u2016\n[PROOFSTEP]\nsimp only [norm_smul]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =O[l] k\u2082\nh\u2082 : f' =o[l] g'\n\u22a2 \u2200 (x : \u03b1), \u2016k\u2082 x\u2016 * \u2016g' x\u2016 = \u2016k\u2082 x \u2022 g' x\u2016\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =O[l] k\u2082\nh\u2082 : f' =o[l] g'\nx\u271d : \u03b1\n\u22a2 \u2016k\u2082 x\u271d\u2016 * \u2016g' x\u271d\u2016 = \u2016k\u2082 x\u271d \u2022 g' x\u271d\u2016\n[PROOFSTEP]\nsimp only [norm_smul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =o[l] k\u2082\nh\u2082 : f' =O[l] g'\n\u22a2 (fun x => k\u2081 x \u2022 f' x) =o[l] fun x => k\u2082 x \u2022 g' x\n[PROOFSTEP]\nrefine' ((h\u2081.norm_norm.mul_isBigO h\u2082.norm_norm).congr _ _).of_norm_norm\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =o[l] k\u2082\nh\u2082 : f' =O[l] g'\n\u22a2 \u2200 (x : \u03b1), \u2016k\u2081 x\u2016 * \u2016f' x\u2016 = \u2016k\u2081 x \u2022 f' x\u2016\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =o[l] k\u2082\nh\u2082 : f' =O[l] g'\nx\u271d : \u03b1\n\u22a2 \u2016k\u2081 x\u271d\u2016 * \u2016f' x\u271d\u2016 = \u2016k\u2081 x\u271d \u2022 f' x\u271d\u2016\n[PROOFSTEP]\nsimp only [norm_smul]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =o[l] k\u2082\nh\u2082 : f' =O[l] g'\n\u22a2 \u2200 (x : \u03b1), \u2016k\u2082 x\u2016 * \u2016g' x\u2016 = \u2016k\u2082 x \u2022 g' x\u2016\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =o[l] k\u2082\nh\u2082 : f' =O[l] g'\nx\u271d : \u03b1\n\u22a2 \u2016k\u2082 x\u271d\u2016 * \u2016g' x\u271d\u2016 = \u2016k\u2082 x\u271d \u2022 g' x\u271d\u2016\n[PROOFSTEP]\nsimp only [norm_smul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =o[l] k\u2082\nh\u2082 : f' =o[l] g'\n\u22a2 (fun x => k\u2081 x \u2022 f' x) =o[l] fun x => k\u2082 x \u2022 g' x\n[PROOFSTEP]\nrefine' ((h\u2081.norm_norm.mul h\u2082.norm_norm).congr _ _).of_norm_norm\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =o[l] k\u2082\nh\u2082 : f' =o[l] g'\n\u22a2 \u2200 (x : \u03b1), \u2016k\u2081 x\u2016 * \u2016f' x\u2016 = \u2016k\u2081 x \u2022 f' x\u2016\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =o[l] k\u2082\nh\u2082 : f' =o[l] g'\nx\u271d : \u03b1\n\u22a2 \u2016k\u2081 x\u271d\u2016 * \u2016f' x\u271d\u2016 = \u2016k\u2081 x\u271d \u2022 f' x\u271d\u2016\n[PROOFSTEP]\nsimp only [norm_smul]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =o[l] k\u2082\nh\u2082 : f' =o[l] g'\n\u22a2 \u2200 (x : \u03b1), \u2016k\u2082 x\u2016 * \u2016g' x\u2016 = \u2016k\u2082 x \u2022 g' x\u2016\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : NormedSpace \ud835\udd5c' F'\nk\u2081 : \u03b1 \u2192 \ud835\udd5c\nk\u2082 : \u03b1 \u2192 \ud835\udd5c'\nh\u2081 : k\u2081 =o[l] k\u2082\nh\u2082 : f' =o[l] g'\nx\u271d : \u03b1\n\u22a2 \u2016k\u2082 x\u271d\u2016 * \u2016g' x\u271d\u2016 = \u2016k\u2082 x\u271d \u2022 g' x\u271d\u2016\n[PROOFSTEP]\nsimp only [norm_smul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\nA : \u03b9 \u2192 \u03b1 \u2192 E'\nC : \u03b9 \u2192 \u211d\ns : Finset \u03b9\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 IsBigOWith (C i) l (A i) g\n\u22a2 IsBigOWith (\u2211 i in s, C i) l (fun x => \u2211 i in s, A i x) g\n[PROOFSTEP]\ninduction' s using Finset.induction_on with i s is IH\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\nA : \u03b9 \u2192 \u03b1 \u2192 E'\nC : \u03b9 \u2192 \u211d\ns : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsBigOWith (C i) l (A i) g\nh : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 IsBigOWith (C i) l (A i) g\n\u22a2 IsBigOWith (\u2211 i in \u2205, C i) l (fun x => \u2211 i in \u2205, A i x) g\n[PROOFSTEP]\nsimp only [isBigOWith_zero', Finset.sum_empty, forall_true_iff]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\nA : \u03b9 \u2192 \u03b1 \u2192 E'\nC : \u03b9 \u2192 \u211d\ns\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsBigOWith (C i) l (A i) g\ni : \u03b9\ns : Finset \u03b9\nis : \u00aci \u2208 s\nIH : (\u2200 (i : \u03b9), i \u2208 s \u2192 IsBigOWith (C i) l (A i) g) \u2192 IsBigOWith (\u2211 i in s, C i) l (fun x => \u2211 i in s, A i x) g\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 IsBigOWith (C i_1) l (A i_1) g\n\u22a2 IsBigOWith (\u2211 i in insert i s, C i) l (fun x => \u2211 i in insert i s, A i x) g\n[PROOFSTEP]\nsimp only [is, Finset.sum_insert, not_false_iff]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\nA : \u03b9 \u2192 \u03b1 \u2192 E'\nC : \u03b9 \u2192 \u211d\ns\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsBigOWith (C i) l (A i) g\ni : \u03b9\ns : Finset \u03b9\nis : \u00aci \u2208 s\nIH : (\u2200 (i : \u03b9), i \u2208 s \u2192 IsBigOWith (C i) l (A i) g) \u2192 IsBigOWith (\u2211 i in s, C i) l (fun x => \u2211 i in s, A i x) g\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 IsBigOWith (C i_1) l (A i_1) g\n\u22a2 IsBigOWith (C i + \u2211 i in s, C i) l (fun x => A i x + \u2211 i in s, A i x) g\n[PROOFSTEP]\nexact (h _ (Finset.mem_insert_self i s)).add (IH fun j hj => h _ (Finset.mem_insert_of_mem hj))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\nA : \u03b9 \u2192 \u03b1 \u2192 E'\nC : \u03b9 \u2192 \u211d\ns : Finset \u03b9\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 A i =O[l] g\n\u22a2 (fun x => \u2211 i in s, A i x) =O[l] g\n[PROOFSTEP]\nsimp only [IsBigO_def] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\nA : \u03b9 \u2192 \u03b1 \u2192 E'\nC : \u03b9 \u2192 \u211d\ns : Finset \u03b9\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2203 c, IsBigOWith c l (A i) g\n\u22a2 \u2203 c, IsBigOWith c l (fun x => \u2211 i in s, A i x) g\n[PROOFSTEP]\nchoose! C hC using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\nA : \u03b9 \u2192 \u03b1 \u2192 E'\nC\u271d : \u03b9 \u2192 \u211d\ns : Finset \u03b9\nC : \u03b9 \u2192 \u211d\nhC : \u2200 (i : \u03b9), i \u2208 s \u2192 IsBigOWith (C i) l (A i) g\n\u22a2 \u2203 c, IsBigOWith c l (fun x => \u2211 i in s, A i x) g\n[PROOFSTEP]\nexact \u27e8_, IsBigOWith.sum hC\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\nA : \u03b9 \u2192 \u03b1 \u2192 E'\nC : \u03b9 \u2192 \u211d\ns : Finset \u03b9\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 A i =o[l] g'\n\u22a2 (fun x => \u2211 i in s, A i x) =o[l] g'\n[PROOFSTEP]\ninduction' s using Finset.induction_on with i s is IH\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\nA : \u03b9 \u2192 \u03b1 \u2192 E'\nC : \u03b9 \u2192 \u211d\ns : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 A i =o[l] g'\nh : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 A i =o[l] g'\n\u22a2 (fun x => \u2211 i in \u2205, A i x) =o[l] g'\n[PROOFSTEP]\nsimp only [isLittleO_zero, Finset.sum_empty, forall_true_iff]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\nA : \u03b9 \u2192 \u03b1 \u2192 E'\nC : \u03b9 \u2192 \u211d\ns\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 A i =o[l] g'\ni : \u03b9\ns : Finset \u03b9\nis : \u00aci \u2208 s\nIH : (\u2200 (i : \u03b9), i \u2208 s \u2192 A i =o[l] g') \u2192 (fun x => \u2211 i in s, A i x) =o[l] g'\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 A i_1 =o[l] g'\n\u22a2 (fun x => \u2211 i in insert i s, A i x) =o[l] g'\n[PROOFSTEP]\nsimp only [is, Finset.sum_insert, not_false_iff]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\nA : \u03b9 \u2192 \u03b1 \u2192 E'\nC : \u03b9 \u2192 \u211d\ns\u271d : Finset \u03b9\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 A i =o[l] g'\ni : \u03b9\ns : Finset \u03b9\nis : \u00aci \u2208 s\nIH : (\u2200 (i : \u03b9), i \u2208 s \u2192 A i =o[l] g') \u2192 (fun x => \u2211 i in s, A i x) =o[l] g'\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 A i_1 =o[l] g'\n\u22a2 (fun x => A i x + \u2211 i in s, A i x) =o[l] g'\n[PROOFSTEP]\nexact (h _ (Finset.mem_insert_self i s)).add (IH fun j hj => h _ (Finset.mem_insert_of_mem hj))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nh : f =o[l] g\n\u22a2 (fun x => f x / g x) =o[l] fun _x => 1\n[PROOFSTEP]\ncalc\n (fun x => f x / g x) =o[l] fun x => g x / g x := by simpa only [div_eq_mul_inv] using h.mul_isBigO (isBigO_refl _ _)\n _ =O[l] fun _x => (1 : \ud835\udd5c) := isBigO_of_le _ fun x => by simp [div_self_le_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nh : f =o[l] g\n\u22a2 (fun x => f x / g x) =o[l] fun x => g x / g x\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using h.mul_isBigO (isBigO_refl _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nh : f =o[l] g\nx : \u03b1\n\u22a2 \u2016g x / g x\u2016 \u2264 \u20161\u2016\n[PROOFSTEP]\nsimp [div_self_le_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl\u271d l' : Filter \u03b1\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : \u03b1 \u2192 E'\ng : \u03b1 \u2192 \ud835\udd5c\nl : Filter \u03b1\nh : f =o[l] g\n\u22a2 Tendsto (fun x => (g x)\u207b\u00b9 \u2022 f x) l (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa only [div_eq_inv_mul, \u2190 norm_inv, \u2190 norm_smul, \u2190 tendsto_zero_iff_norm_tendsto_zero] using\n h.norm_norm.tendsto_div_nhds_zero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : E''\nhc : c \u2260 0\n\u22a2 (fun _x => c) =o[l] g \u2194 Tendsto (fun x => \u2016g x\u2016) l atTop\n[PROOFSTEP]\nsimp only [\u2190 isLittleO_one_left_iff \u211d]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : E''\nhc : c \u2260 0\n\u22a2 (fun _x => c) =o[l] g \u2194 (fun _x => 1) =o[l] fun x => g x\n[PROOFSTEP]\nexact \u27e8(isBigO_const_const (1 : \u211d) hc l).trans_isLittleO, (isBigO_const_one \u211d c l).trans_isLittleO\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : E''\n\u22a2 (fun _x => c) =o[l] g'' \u2194 c = 0 \u2228 Tendsto (norm \u2218 g'') l atTop\n[PROOFSTEP]\nrcases eq_or_ne c 0 with (rfl | hc)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u22a2 (fun _x => 0) =o[l] g'' \u2194 0 = 0 \u2228 Tendsto (norm \u2218 g'') l atTop\n[PROOFSTEP]\nsimp only [isLittleO_zero, eq_self_iff_true, true_or_iff]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : E''\nhc : c \u2260 0\n\u22a2 (fun _x => c) =o[l] g'' \u2194 c = 0 \u2228 Tendsto (norm \u2218 g'') l atTop\n[PROOFSTEP]\nsimp only [hc, false_or_iff, isLittleO_const_left_of_ne hc]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nc : E''\nhc : c \u2260 0\n\u22a2 Tendsto (fun x => \u2016g'' x\u2016) l atTop \u2194 Tendsto (norm \u2218 g'') l atTop\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NeBot l\nd : E''\nc : F''\n\u22a2 ((fun _x => d) =o[l] fun _x => c) \u2194 d = 0\n[PROOFSTEP]\nhave : \u00acTendsto (Function.const \u03b1 \u2016c\u2016) l atTop := not_tendsto_atTop_of_tendsto_nhds tendsto_const_nhds\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NeBot l\nd : E''\nc : F''\nthis : \u00acTendsto (Function.const \u03b1 \u2016c\u2016) l atTop\n\u22a2 ((fun _x => d) =o[l] fun _x => c) \u2194 d = 0\n[PROOFSTEP]\nsimp only [isLittleO_const_left, or_iff_left_iff_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : NeBot l\nd : E''\nc : F''\nthis : \u00acTendsto (Function.const \u03b1 \u2016c\u2016) l atTop\n\u22a2 Tendsto (norm \u2218 fun _x => c) l atTop \u2192 d = 0\n[PROOFSTEP]\nexact fun h => (this h).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\nh : IsBigOWith c l u v\ny : \u03b1\nhy : \u2016u y\u2016 \u2264 c * \u2016v y\u2016\nhv : v y = 0\n\u22a2 u y = 0\n[PROOFSTEP]\nsimpa [hv] using hy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v \u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x\u2016 \u2264 c\nh : u =\u1da0[l] \u03c6 * v\n\u22a2 IsBigOWith c l u v\n[PROOFSTEP]\nsimp only [IsBigOWith_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v \u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x\u2016 \u2264 c\nh : u =\u1da0[l] \u03c6 * v\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 c * \u2016v x\u2016\n[PROOFSTEP]\nrefine' h.symm.rw (fun x a => \u2016a\u2016 \u2264 c * \u2016v x\u2016) (h\u03c6.mono fun x hx => _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v \u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x\u2016 \u2264 c\nh : u =\u1da0[l] \u03c6 * v\nx : \u03b1\nhx : \u2016\u03c6 x\u2016 \u2264 c\n\u22a2 (fun x a => \u2016a\u2016 \u2264 c * \u2016v x\u2016) x ((\u03c6 * v) x)\n[PROOFSTEP]\nsimp only [norm_mul, Pi.mul_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v \u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x\u2016 \u2264 c\nh : u =\u1da0[l] \u03c6 * v\nx : \u03b1\nhx : \u2016\u03c6 x\u2016 \u2264 c\n\u22a2 \u2016\u03c6 x\u2016 * \u2016v x\u2016 \u2264 c * \u2016v x\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\nhc : 0 \u2264 c\n\u22a2 IsBigOWith c l u v \u2194 \u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\nhc : 0 \u2264 c\n\u22a2 IsBigOWith c l u v \u2192 \u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\nhc : 0 \u2264 c\nh : IsBigOWith c l u v\n\u22a2 \u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nuse fun x => u x / v x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\nhc : 0 \u2264 c\nh : IsBigOWith c l u v\n\u22a2 \u2203 _h\u03c6, u =\u1da0[l] (fun x => u x / v x) * v\n[PROOFSTEP]\nrefine' \u27e8Eventually.mono h.bound fun y hy => _, h.eventually_mul_div_cancel.symm\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\nhc : 0 \u2264 c\nh : IsBigOWith c l u v\ny : \u03b1\nhy : \u2016u y\u2016 \u2264 c * \u2016v y\u2016\n\u22a2 \u2016u y / v y\u2016 \u2264 c\n[PROOFSTEP]\nsimpa using div_le_of_nonneg_of_le_mul (norm_nonneg _) hc hy\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\nhc : 0 \u2264 c\n\u22a2 (\u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v) \u2192 IsBigOWith c l u v\n[PROOFSTEP]\nrintro \u27e8\u03c6, h\u03c6, h\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\nhc : 0 \u2264 c\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x\u2016 \u2264 c\nh : u =\u1da0[l] \u03c6 * v\n\u22a2 IsBigOWith c l u v\n[PROOFSTEP]\nexact isBigOWith_of_eq_mul \u03c6 h\u03c6 h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\n\u22a2 u =O[l] v \u2194 \u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\n\u22a2 u =O[l] v \u2192 \u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nrintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\nh : u =O[l] v\n\u22a2 \u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nrcases h.exists_nonneg with \u27e8c, hnnc, hc\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\nh : u =O[l] v\nc : \u211d\nhnnc : 0 \u2264 c\nhc : IsBigOWith c l u v\n\u22a2 \u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nrcases hc.exists_eq_mul hnnc with \u27e8\u03c6, h\u03c6, huv\u03c6\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\nh : u =O[l] v\nc : \u211d\nhnnc : 0 \u2264 c\nhc : IsBigOWith c l u v\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x\u2016 \u2264 c\nhuv\u03c6 : u =\u1da0[l] \u03c6 * v\n\u22a2 \u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nexact \u27e8\u03c6, \u27e8c, h\u03c6\u27e9, huv\u03c6\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\n\u22a2 (\u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v) \u2192 u =O[l] v\n[PROOFSTEP]\nrintro \u27e8\u03c6, \u27e8c, h\u03c6\u27e9, huv\u03c6\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v \u03c6 : \u03b1 \u2192 \ud835\udd5c\nhuv\u03c6 : u =\u1da0[l] \u03c6 * v\nc : \u211d\nh\u03c6 : \u2200\u1da0 (x : \u211d) in map (norm \u2218 \u03c6) l, (fun x x_1 => x \u2264 x_1) x c\n\u22a2 u =O[l] v\n[PROOFSTEP]\nexact isBigO_iff_isBigOWith.2 \u27e8c, isBigOWith_of_eq_mul \u03c6 h\u03c6 huv\u03c6\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\n\u22a2 u =o[l] v \u2194 \u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\n\u22a2 u =o[l] v \u2192 \u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nexact fun h => \u27e8fun x => u x / v x, h.tendsto_div_nhds_zero, h.eventually_mul_div_cancel.symm\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\n\u22a2 (\u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v) \u2192 u =o[l] v\n[PROOFSTEP]\nsimp only [IsLittleO_def]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\n\u22a2 (\u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v) \u2192 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l u v\n[PROOFSTEP]\nrintro \u27e8\u03c6, h\u03c6, huv\u03c6\u27e9 c hpos\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v \u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 0)\nhuv\u03c6 : u =\u1da0[l] \u03c6 * v\nc : \u211d\nhpos : 0 < c\n\u22a2 IsBigOWith c l u v\n[PROOFSTEP]\nrw [NormedAddCommGroup.tendsto_nhds_zero] at h\u03c6 \n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v \u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x\u2016 < \u03b5\nhuv\u03c6 : u =\u1da0[l] \u03c6 * v\nc : \u211d\nhpos : 0 < c\n\u22a2 IsBigOWith c l u v\n[PROOFSTEP]\nexact isBigOWith_of_eq_mul _ ((h\u03c6 c hpos).mono fun x => le_of_lt) huv\u03c6\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nh : f =O[l] g\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016f x / g x\u2016\n[PROOFSTEP]\nobtain \u27e8c, h\u2080, hc\u27e9 := h.exists_nonneg\n[GOAL]\ncase intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nh : f =O[l] g\nc : \u211d\nh\u2080 : 0 \u2264 c\nhc : IsBigOWith c l f g\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016f x / g x\u2016\n[PROOFSTEP]\nrefine' \u27e8c, eventually_map.2 (hc.bound.mono fun x hx => _)\u27e9\n[GOAL]\ncase intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nh : f =O[l] g\nc : \u211d\nh\u2080 : 0 \u2264 c\nhc : IsBigOWith c l f g\nx : \u03b1\nhx : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 (fun x x_1 => x \u2264 x_1) \u2016f x / g x\u2016 c\n[PROOFSTEP]\nrw [norm_div]\n[GOAL]\ncase intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nh : f =O[l] g\nc : \u211d\nh\u2080 : 0 \u2264 c\nhc : IsBigOWith c l f g\nx : \u03b1\nhx : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 (fun x x_1 => x \u2264 x_1) (\u2016f x\u2016 / \u2016g x\u2016) c\n[PROOFSTEP]\nexact div_le_of_nonneg_of_le_mul (norm_nonneg _) h\u2080 hx\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nhgf : \u2200\u1da0 (x : \u03b1) in l, g x = 0 \u2192 f x = 0\n\u22a2 f =O[l] g \u2194 IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016f x / g x\u2016\n[PROOFSTEP]\nrefine' \u27e8div_isBoundedUnder_of_isBigO, fun h => _\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nhgf : \u2200\u1da0 (x : \u03b1) in l, g x = 0 \u2192 f x = 0\nh : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016f x / g x\u2016\n\u22a2 f =O[l] g\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 := h\n[GOAL]\ncase intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nhgf : \u2200\u1da0 (x : \u03b1) in l, g x = 0 \u2192 f x = 0\nc : \u211d\nhc : \u2200\u1da0 (x : \u211d) in map (fun x => \u2016f x / g x\u2016) l, (fun x x_1 => x \u2264 x_1) x c\n\u22a2 f =O[l] g\n[PROOFSTEP]\nsimp only [eventually_map, norm_div] at hc \n[GOAL]\ncase intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nhgf : \u2200\u1da0 (x : \u03b1) in l, g x = 0 \u2192 f x = 0\nc : \u211d\nhc : \u2200\u1da0 (a : \u03b1) in l, \u2016f a\u2016 / \u2016g a\u2016 \u2264 c\n\u22a2 f =O[l] g\n[PROOFSTEP]\nrefine' IsBigO.of_bound c (hc.mp <| hgf.mono fun x hx\u2081 hx\u2082 => _)\n[GOAL]\ncase intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nhgf : \u2200\u1da0 (x : \u03b1) in l, g x = 0 \u2192 f x = 0\nc : \u211d\nhc : \u2200\u1da0 (a : \u03b1) in l, \u2016f a\u2016 / \u2016g a\u2016 \u2264 c\nx : \u03b1\nhx\u2081 : g x = 0 \u2192 f x = 0\nhx\u2082 : \u2016f x\u2016 / \u2016g x\u2016 \u2264 c\n\u22a2 \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nby_cases hgx : g x = 0\n[GOAL]\ncase pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nhgf : \u2200\u1da0 (x : \u03b1) in l, g x = 0 \u2192 f x = 0\nc : \u211d\nhc : \u2200\u1da0 (a : \u03b1) in l, \u2016f a\u2016 / \u2016g a\u2016 \u2264 c\nx : \u03b1\nhx\u2081 : g x = 0 \u2192 f x = 0\nhx\u2082 : \u2016f x\u2016 / \u2016g x\u2016 \u2264 c\nhgx : g x = 0\n\u22a2 \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nsimp [hx\u2081 hgx, hgx]\n[GOAL]\ncase neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nhgf : \u2200\u1da0 (x : \u03b1) in l, g x = 0 \u2192 f x = 0\nc : \u211d\nhc : \u2200\u1da0 (a : \u03b1) in l, \u2016f a\u2016 / \u2016g a\u2016 \u2264 c\nx : \u03b1\nhx\u2081 : g x = 0 \u2192 f x = 0\nhx\u2082 : \u2016f x\u2016 / \u2016g x\u2016 \u2264 c\nhgx : \u00acg x = 0\n\u22a2 \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n[PROOFSTEP]\nexact (div_le_iff (norm_pos_iff.2 hgx)).mp hx\u2082\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c\u271d : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\u271d\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\u271d\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1\u271d \u2192 E\u271d\ng : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\nE : Type u_18\n\ud835\udd5c : Type u_19\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedField \ud835\udd5c\nu : \u03b1 \u2192 E\nv : \u03b1 \u2192 \ud835\udd5c\nl : Filter \u03b1\ny : \ud835\udd5c\nhuv : u =o[l] v\nhv : Tendsto v l (\ud835\udcdd y)\n\u22a2 Tendsto u l (\ud835\udcdd 0)\n[PROOFSTEP]\nsuffices h : u =o[l] fun _x => (1 : \ud835\udd5c)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c\u271d : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\u271d\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\u271d\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1\u271d \u2192 E\u271d\ng : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\nE : Type u_18\n\ud835\udd5c : Type u_19\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedField \ud835\udd5c\nu : \u03b1 \u2192 E\nv : \u03b1 \u2192 \ud835\udd5c\nl : Filter \u03b1\ny : \ud835\udd5c\nhuv : u =o[l] v\nhv : Tendsto v l (\ud835\udcdd y)\nh : u =o[l] fun _x => 1\n\u22a2 Tendsto u l (\ud835\udcdd 0)\n[PROOFSTEP]\nrwa [isLittleO_one_iff] at h \n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE\u271d : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c\u271d : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\u271d\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\u271d\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1\u271d \u2192 E\u271d\ng : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\nE : Type u_18\n\ud835\udd5c : Type u_19\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedField \ud835\udd5c\nu : \u03b1 \u2192 E\nv : \u03b1 \u2192 \ud835\udd5c\nl : Filter \u03b1\ny : \ud835\udd5c\nhuv : u =o[l] v\nhv : Tendsto v l (\ud835\udcdd y)\n\u22a2 u =o[l] fun _x => 1\n[PROOFSTEP]\nexact huv.trans_isBigO (hv.isBigO_one \ud835\udd5c)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nm n : \u2115\nh : m < n\n\u22a2 (fun x => x ^ n) =o[\ud835\udcdd 0] fun x => x ^ m\n[PROOFSTEP]\nrcases lt_iff_exists_add.1 h with \u27e8p, hp0 : 0 < p, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nm p : \u2115\nhp0 : 0 < p\nh : m < m + p\n\u22a2 (fun x => x ^ (m + p)) =o[\ud835\udcdd 0] fun x => x ^ m\n[PROOFSTEP]\nsuffices (fun x : \ud835\udd5c => x ^ m * x ^ p) =o[\ud835\udcdd 0] fun x => x ^ m * 1 ^ p by simpa only [pow_add, one_pow, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nm p : \u2115\nhp0 : 0 < p\nh : m < m + p\nthis : (fun x => x ^ m * x ^ p) =o[\ud835\udcdd 0] fun x => x ^ m * 1 ^ p\n\u22a2 (fun x => x ^ (m + p)) =o[\ud835\udcdd 0] fun x => x ^ m\n[PROOFSTEP]\nsimpa only [pow_add, one_pow, mul_one]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nm p : \u2115\nhp0 : 0 < p\nh : m < m + p\n\u22a2 (fun x => x ^ m * x ^ p) =o[\ud835\udcdd 0] fun x => x ^ m * 1 ^ p\n[PROOFSTEP]\nexact IsBigO.mul_isLittleO (isBigO_refl _ _) (IsLittleO.pow ((isLittleO_one_iff _).2 tendsto_id) hp0)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nn : \u2115\nh : 1 < n\n\u22a2 (fun x => x ^ n) =o[\ud835\udcdd 0] fun x => x\n[PROOFSTEP]\nconvert isLittleO_pow_pow h (\ud835\udd5c := \ud835\udd5c)\n[GOAL]\ncase h.e'_8.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nn : \u2115\nh : 1 < n\nx\u271d : \ud835\udd5c\n\u22a2 x\u271d = x\u271d ^ 1\n[PROOFSTEP]\nsimp only [pow_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nn : \u2115\nh : 1 < n\n\u22a2 (fun x => \u2016x\u2016 ^ n) =o[\ud835\udcdd 0] fun x => x\n[PROOFSTEP]\nhave := @isLittleO_norm_pow_norm_pow E' _ _ _ h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nn : \u2115\nh : 1 < n\nthis : (fun x => \u2016x\u2016 ^ n) =o[\ud835\udcdd 0] fun x => \u2016x\u2016 ^ 1\n\u22a2 (fun x => \u2016x\u2016 ^ n) =o[\ud835\udcdd 0] fun x => x\n[PROOFSTEP]\nsimp only [pow_one] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nn : \u2115\nh : 1 < n\nthis : (fun x => \u2016x\u2016 ^ n) =o[\ud835\udcdd 0] fun x => \u2016x\u2016\n\u22a2 (fun x => \u2016x\u2016 ^ n) =o[\ud835\udcdd 0] fun x => x\n[PROOFSTEP]\nexact isLittleO_norm_right.mp this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : E'' \u2192 F''\ns : Set E''\nx\u2080 : E''\nn : \u2115\nh : f =O[\ud835\udcdd[s] x\u2080] fun x => \u2016x - x\u2080\u2016 ^ n\nhx\u2080 : x\u2080 \u2208 s\nhn : 0 < n\n\u22a2 \u2016x\u2080 - x\u2080\u2016 ^ n = 0\n[PROOFSTEP]\nsimp_rw [sub_self, norm_zero, zero_pow hn]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : E'' \u2192 F''\nx\u2080 : E''\nn : \u2115\nh : f =O[\ud835\udcdd x\u2080] fun x => \u2016x - x\u2080\u2016 ^ n\nhn : 0 < n\n\u22a2 f x\u2080 = 0\n[PROOFSTEP]\nrw [\u2190 nhdsWithin_univ] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : E'' \u2192 F''\nx\u2080 : E''\nn : \u2115\nh : f =O[\ud835\udcdd[univ] x\u2080] fun x => \u2016x - x\u2080\u2016 ^ n\nhn : 0 < n\n\u22a2 f x\u2080 = 0\n[PROOFSTEP]\nexact h.eq_zero_of_norm_pow_within (mem_univ _) hn\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nx\u2080 : E'\nn m : \u2115\nh : n < m\n\u22a2 Tendsto (fun x => \u2016x - x\u2080\u2016) (\ud835\udcdd x\u2080) (\ud835\udcdd 0)\n[PROOFSTEP]\napply tendsto_norm_zero.comp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nx\u2080 : E'\nn m : \u2115\nh : n < m\n\u22a2 Tendsto (fun x => x - x\u2080) (\ud835\udcdd x\u2080) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 sub_self x\u2080]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nx\u2080 : E'\nn m : \u2115\nh : n < m\n\u22a2 Tendsto (fun x => x - x\u2080) (\ud835\udcdd x\u2080) (\ud835\udcdd (x\u2080 - x\u2080))\n[PROOFSTEP]\nexact tendsto_id.sub tendsto_const_nhds\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nx\u2080 : E'\nm : \u2115\nh : 1 < m\n\u22a2 (fun x => \u2016x - x\u2080\u2016 ^ m) =o[\ud835\udcdd x\u2080] fun x => x - x\u2080\n[PROOFSTEP]\nsimpa only [isLittleO_norm_right, pow_one] using isLittleO_pow_sub_pow_sub x\u2080 h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E'\nh : IsBigOWith c l f\u2081 f\u2082\nhc : c < 1\nx : \u03b1\nhx : x \u2208 {x | (fun x => \u2016f\u2081 x\u2016 \u2264 c * \u2016f\u2082 x\u2016) x}\n\u22a2 x \u2208 {x | (fun x => \u2016f\u2082 x\u2016 \u2264 1 / (1 - c) * \u2016f\u2082 x - f\u2081 x\u2016) x}\n[PROOFSTEP]\nsimp only [mem_setOf_eq] at hx \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E'\nh : IsBigOWith c l f\u2081 f\u2082\nhc : c < 1\nx : \u03b1\nhx : \u2016f\u2081 x\u2016 \u2264 c * \u2016f\u2082 x\u2016\n\u22a2 \u2016f\u2082 x\u2016 \u2264 1 / (1 - c) * \u2016f\u2082 x - f\u2081 x\u2016\n[PROOFSTEP]\nrw [mul_comm, one_div, \u2190 div_eq_mul_inv, _root_.le_div_iff, mul_sub, mul_one, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E'\nh : IsBigOWith c l f\u2081 f\u2082\nhc : c < 1\nx : \u03b1\nhx : \u2016f\u2081 x\u2016 \u2264 c * \u2016f\u2082 x\u2016\n\u22a2 \u2016f\u2082 x\u2016 - c * \u2016f\u2082 x\u2016 \u2264 \u2016f\u2082 x - f\u2081 x\u2016\n[PROOFSTEP]\nexact le_trans (sub_le_sub_left hx _) (norm_sub_norm_le _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E'\nh : IsBigOWith c l f\u2081 f\u2082\nhc : c < 1\nx : \u03b1\nhx : \u2016f\u2081 x\u2016 \u2264 c * \u2016f\u2082 x\u2016\n\u22a2 0 < 1 - c\n[PROOFSTEP]\nexact sub_pos.2 hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf\u2081 f\u2082 : \u03b1 \u2192 E'\nh : IsBigOWith c l f\u2081 f\u2082\nhc : c < 1\nx : \u03b1\n\u22a2 -(-f\u2082 x - f\u2081 x) = f\u2081 x + f\u2082 x\n[PROOFSTEP]\nrw [neg_sub, sub_neg_eq_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : f =O[cofinite] g''\n\u22a2 \u2203 C, C > 0 \u2227 \u2200 \u2983x : \u03b1\u2984, g'' x \u2260 0 \u2192 \u2016f x\u2016 \u2264 C * \u2016g'' x\u2016\n[PROOFSTEP]\nrcases h.exists_pos with \u27e8C, C\u2080, hC\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : f =O[cofinite] g''\nC : \u211d\nC\u2080 : 0 < C\nhC : IsBigOWith C cofinite f g''\n\u22a2 \u2203 C, C > 0 \u2227 \u2200 \u2983x : \u03b1\u2984, g'' x \u2260 0 \u2192 \u2016f x\u2016 \u2264 C * \u2016g'' x\u2016\n[PROOFSTEP]\nrw [IsBigOWith_def, eventually_cofinite] at hC \n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : f =O[cofinite] g''\nC : \u211d\nC\u2080 : 0 < C\nhC : Set.Finite {x | \u00ac\u2016f x\u2016 \u2264 C * \u2016g'' x\u2016}\n\u22a2 \u2203 C, C > 0 \u2227 \u2200 \u2983x : \u03b1\u2984, g'' x \u2260 0 \u2192 \u2016f x\u2016 \u2264 C * \u2016g'' x\u2016\n[PROOFSTEP]\nrcases(hC.toFinset.image fun x => \u2016f x\u2016 / \u2016g'' x\u2016).exists_le with \u27e8C', hC'\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : f =O[cofinite] g''\nC : \u211d\nC\u2080 : 0 < C\nhC : Set.Finite {x | \u00ac\u2016f x\u2016 \u2264 C * \u2016g'' x\u2016}\nC' : \u211d\nhC' : \u2200 (i : \u211d), i \u2208 Finset.image (fun x => \u2016f x\u2016 / \u2016g'' x\u2016) (Finite.toFinset hC) \u2192 i \u2264 C'\n\u22a2 \u2203 C, C > 0 \u2227 \u2200 \u2983x : \u03b1\u2984, g'' x \u2260 0 \u2192 \u2016f x\u2016 \u2264 C * \u2016g'' x\u2016\n[PROOFSTEP]\nhave : \u2200 x, C * \u2016g'' x\u2016 < \u2016f x\u2016 \u2192 \u2016f x\u2016 / \u2016g'' x\u2016 \u2264 C' := by simpa using hC'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : f =O[cofinite] g''\nC : \u211d\nC\u2080 : 0 < C\nhC : Set.Finite {x | \u00ac\u2016f x\u2016 \u2264 C * \u2016g'' x\u2016}\nC' : \u211d\nhC' : \u2200 (i : \u211d), i \u2208 Finset.image (fun x => \u2016f x\u2016 / \u2016g'' x\u2016) (Finite.toFinset hC) \u2192 i \u2264 C'\n\u22a2 \u2200 (x : \u03b1), C * \u2016g'' x\u2016 < \u2016f x\u2016 \u2192 \u2016f x\u2016 / \u2016g'' x\u2016 \u2264 C'\n[PROOFSTEP]\nsimpa using hC'\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : f =O[cofinite] g''\nC : \u211d\nC\u2080 : 0 < C\nhC : Set.Finite {x | \u00ac\u2016f x\u2016 \u2264 C * \u2016g'' x\u2016}\nC' : \u211d\nhC' : \u2200 (i : \u211d), i \u2208 Finset.image (fun x => \u2016f x\u2016 / \u2016g'' x\u2016) (Finite.toFinset hC) \u2192 i \u2264 C'\nthis : \u2200 (x : \u03b1), C * \u2016g'' x\u2016 < \u2016f x\u2016 \u2192 \u2016f x\u2016 / \u2016g'' x\u2016 \u2264 C'\n\u22a2 \u2203 C, C > 0 \u2227 \u2200 \u2983x : \u03b1\u2984, g'' x \u2260 0 \u2192 \u2016f x\u2016 \u2264 C * \u2016g'' x\u2016\n[PROOFSTEP]\nrefine' \u27e8max C C', lt_max_iff.2 (Or.inl C\u2080), fun x h\u2080 => _\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : f =O[cofinite] g''\nC : \u211d\nC\u2080 : 0 < C\nhC : Set.Finite {x | \u00ac\u2016f x\u2016 \u2264 C * \u2016g'' x\u2016}\nC' : \u211d\nhC' : \u2200 (i : \u211d), i \u2208 Finset.image (fun x => \u2016f x\u2016 / \u2016g'' x\u2016) (Finite.toFinset hC) \u2192 i \u2264 C'\nthis : \u2200 (x : \u03b1), C * \u2016g'' x\u2016 < \u2016f x\u2016 \u2192 \u2016f x\u2016 / \u2016g'' x\u2016 \u2264 C'\nx : \u03b1\nh\u2080 : g'' x \u2260 0\n\u22a2 \u2016f x\u2016 \u2264 max C C' * \u2016g'' x\u2016\n[PROOFSTEP]\nrw [max_mul_of_nonneg _ _ (norm_nonneg _), le_max_iff, or_iff_not_imp_left, not_le]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : f =O[cofinite] g''\nC : \u211d\nC\u2080 : 0 < C\nhC : Set.Finite {x | \u00ac\u2016f x\u2016 \u2264 C * \u2016g'' x\u2016}\nC' : \u211d\nhC' : \u2200 (i : \u211d), i \u2208 Finset.image (fun x => \u2016f x\u2016 / \u2016g'' x\u2016) (Finite.toFinset hC) \u2192 i \u2264 C'\nthis : \u2200 (x : \u03b1), C * \u2016g'' x\u2016 < \u2016f x\u2016 \u2192 \u2016f x\u2016 / \u2016g'' x\u2016 \u2264 C'\nx : \u03b1\nh\u2080 : g'' x \u2260 0\n\u22a2 C * \u2016g'' x\u2016 < \u2016f x\u2016 \u2192 \u2016f x\u2016 \u2264 C' * \u2016g'' x\u2016\n[PROOFSTEP]\nexact fun hx => (div_le_iff (norm_pos_iff.2 h\u2080)).1 (this _ hx)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nh : \u2200 (x : \u03b1), g'' x = 0 \u2192 f'' x = 0\nh' : f'' =O[cofinite] g''\nC : \u211d\n_C\u2080 : C > 0\nhC : \u2200 \u2983x : \u03b1\u2984, g'' x \u2260 0 \u2192 \u2016f'' x\u2016 \u2264 C * \u2016g'' x\u2016\nx : \u03b1\nhx : g'' x = 0\n\u22a2 \u2016f'' x\u2016 \u2264 C * \u2016g'' x\u2016\n[PROOFSTEP]\nsimp [h _ hx, hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng''\u271d : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u2115 \u2192 E\ng'' : \u2115 \u2192 E''\nh : f =O[atTop] g''\n\u22a2 (fun x => f x) =O[cofinite] fun x => g'' x\n[PROOFSTEP]\nrwa [Nat.cofinite_eq_atTop]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng\u271d : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u2115 \u2192 E''\ng : \u2115 \u2192 F''\nh : \u2200 (x : \u2115), g x = 0 \u2192 f x = 0\n\u22a2 f =O[atTop] g \u2194 \u2203 C, \u2200 (x : \u2115), \u2016f x\u2016 \u2264 C * \u2016g x\u2016\n[PROOFSTEP]\nrw [\u2190 Nat.cofinite_eq_atTop, isBigO_cofinite_iff h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nf : \u2115 \u2192 E''\n\u22a2 (\u2203 C, \u2200 (x : \u2115), \u2016f x\u2016 \u2264 C * \u20161\u2016) \u2194 \u2203 C, \u2200 (n : \u2115), \u2016f n\u2016 \u2264 C\n[PROOFSTEP]\nsimp only [norm_one, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE'\u271d : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\u271d\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\ninst\u271d\u00b9 : Fintype \u03b9\nE' : \u03b9 \u2192 Type u_18\ninst\u271d : (i : \u03b9) \u2192 NormedAddCommGroup (E' i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 E' i\nC : \u211d\nhC : 0 \u2264 C\n\u22a2 IsBigOWith C l f g' \u2194 \u2200 (i : \u03b9), IsBigOWith C l (fun x => f x i) g'\n[PROOFSTEP]\nhave : \u2200 x, 0 \u2264 C * \u2016g' x\u2016 := fun x => mul_nonneg hC (norm_nonneg _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE'\u271d : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\u271d\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\ninst\u271d\u00b9 : Fintype \u03b9\nE' : \u03b9 \u2192 Type u_18\ninst\u271d : (i : \u03b9) \u2192 NormedAddCommGroup (E' i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 E' i\nC : \u211d\nhC : 0 \u2264 C\nthis : \u2200 (x : \u03b1), 0 \u2264 C * \u2016g' x\u2016\n\u22a2 IsBigOWith C l f g' \u2194 \u2200 (i : \u03b9), IsBigOWith C l (fun x => f x i) g'\n[PROOFSTEP]\nsimp only [isBigOWith_iff, pi_norm_le_iff_of_nonneg (this _), eventually_all]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE'\u271d : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\u271d\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\ninst\u271d\u00b9 : Fintype \u03b9\nE' : \u03b9 \u2192 Type u_18\ninst\u271d : (i : \u03b9) \u2192 NormedAddCommGroup (E' i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 E' i\n\u22a2 f =O[l] g' \u2194 \u2200 (i : \u03b9), (fun x => f x i) =O[l] g'\n[PROOFSTEP]\nsimp only [isBigO_iff_eventually_isBigOWith, \u2190 eventually_all]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE'\u271d : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\u271d\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\ninst\u271d\u00b9 : Fintype \u03b9\nE' : \u03b9 \u2192 Type u_18\ninst\u271d : (i : \u03b9) \u2192 NormedAddCommGroup (E' i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 E' i\n\u22a2 (\u2200\u1da0 (c : \u211d) in atTop, IsBigOWith c l f g') \u2194 \u2200\u1da0 (x : \u211d) in atTop, \u2200 (i : \u03b9), IsBigOWith x l (fun x => f x i) g'\n[PROOFSTEP]\nexact eventually_congr (eventually_atTop.2 \u27e80, fun c => isBigOWith_pi\u27e9)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE'\u271d : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\u271d\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\ninst\u271d\u00b9 : Fintype \u03b9\nE' : \u03b9 \u2192 Type u_18\ninst\u271d : (i : \u03b9) \u2192 NormedAddCommGroup (E' i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 E' i\n\u22a2 f =o[l] g' \u2194 \u2200 (i : \u03b9), (fun x => f x i) =o[l] g'\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [IsLittleO_def, isBigOWith_pi, le_of_lt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE'\u271d : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\u271d\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\n\u03b9 : Type u_17\ninst\u271d\u00b9 : Fintype \u03b9\nE' : \u03b9 \u2192 Type u_18\ninst\u271d : (i : \u03b9) \u2192 NormedAddCommGroup (E' i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 E' i\n\u22a2 (\u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200 (i : \u03b9), IsBigOWith c l (fun x => f x i) g') \u2194\n \u2200 (i : \u03b9) \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l (fun x => f x i) g'\n[PROOFSTEP]\nexact \u27e8fun h i c hc => h hc i, fun h c hc i => h i hc\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Nonempty \u03b1\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\n\u22a2 f =O[atTop] g \u2194 \u2200\u1da0 (n\u2080 : \u03b1) in atTop, \u2203 c, \u2200 (n : \u03b1), n \u2265 n\u2080 \u2192 \u2016f n\u2016 \u2264 c * \u2016g n\u2016\n[PROOFSTEP]\nrw [isBigO_iff, exists_eventually_atTop]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2075 : Norm E\ninst\u271d\u00b9\u2074 : Norm F\ninst\u271d\u00b9\u00b3 : Norm G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup G'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F''\ninst\u271d\u2077 : NormedAddCommGroup G''\ninst\u271d\u2076 : SeminormedRing R\ninst\u271d\u2075 : SeminormedAddGroup E'''\ninst\u271d\u2074 : SeminormedRing R'\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl l' : Filter \u03b1\u271d\n\u03b1 : Type u_17\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Nonempty \u03b1\nf : \u03b1 \u2192 G\ng : \u03b1 \u2192 G'\n\u22a2 f =O[atTop] g \u2194 \u2200\u1da0 (n\u2080 : \u03b1) in atTop, \u2203 c, c > 0 \u2227 \u2200 (n : \u03b1), n \u2265 n\u2080 \u2192 c * \u2016f n\u2016 \u2264 \u2016g n\u2016\n[PROOFSTEP]\nsimp_rw [isBigO_iff'', \u2190 exists_prop, Subtype.exists', exists_eventually_atTop]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nE : Type u_3\ninst\u271d\u00b9 : Norm E\nF : Type u_4\ninst\u271d : Norm F\ne : LocalHomeomorph \u03b1 \u03b2\nb : \u03b2\nhb : b \u2208 e.target\nf : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nC : \u211d\nh : IsBigOWith C (\ud835\udcdd b) f g\n\u22a2 Tendsto (\u2191e) (\ud835\udcdd (\u2191(LocalHomeomorph.symm e) b)) (\ud835\udcdd b)\n[PROOFSTEP]\nhave := e.continuousAt (e.map_target hb)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nE : Type u_3\ninst\u271d\u00b9 : Norm E\nF : Type u_4\ninst\u271d : Norm F\ne : LocalHomeomorph \u03b1 \u03b2\nb : \u03b2\nhb : b \u2208 e.target\nf : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nC : \u211d\nh : IsBigOWith C (\ud835\udcdd b) f g\nthis : ContinuousAt (\u2191e) (\u2191(LocalHomeomorph.symm e) b)\n\u22a2 Tendsto (\u2191e) (\ud835\udcdd (\u2191(LocalHomeomorph.symm e) b)) (\ud835\udcdd b)\n[PROOFSTEP]\nrwa [ContinuousAt, e.rightInvOn hb] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nE : Type u_3\ninst\u271d\u00b9 : Norm E\nF : Type u_4\ninst\u271d : Norm F\ne : LocalHomeomorph \u03b1 \u03b2\nb : \u03b2\nhb : b \u2208 e.target\nf : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\n\u22a2 f =O[\ud835\udcdd b] g \u2194 (f \u2218 \u2191e) =O[\ud835\udcdd (\u2191(LocalHomeomorph.symm e) b)] (g \u2218 \u2191e)\n[PROOFSTEP]\nsimp only [IsBigO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nE : Type u_3\ninst\u271d\u00b9 : Norm E\nF : Type u_4\ninst\u271d : Norm F\ne : LocalHomeomorph \u03b1 \u03b2\nb : \u03b2\nhb : b \u2208 e.target\nf : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\n\u22a2 (\u2203 c, IsBigOWith c (\ud835\udcdd b) f g) \u2194 \u2203 c, IsBigOWith c (\ud835\udcdd (\u2191(LocalHomeomorph.symm e) b)) (f \u2218 \u2191e) (g \u2218 \u2191e)\n[PROOFSTEP]\nexact exists_congr fun C => e.isBigOWith_congr hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nE : Type u_3\ninst\u271d\u00b9 : Norm E\nF : Type u_4\ninst\u271d : Norm F\ne : LocalHomeomorph \u03b1 \u03b2\nb : \u03b2\nhb : b \u2208 e.target\nf : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\n\u22a2 f =o[\ud835\udcdd b] g \u2194 (f \u2218 \u2191e) =o[\ud835\udcdd (\u2191(LocalHomeomorph.symm e) b)] (g \u2218 \u2191e)\n[PROOFSTEP]\nsimp only [IsLittleO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nE : Type u_3\ninst\u271d\u00b9 : Norm E\nF : Type u_4\ninst\u271d : Norm F\ne : LocalHomeomorph \u03b1 \u03b2\nb : \u03b2\nhb : b \u2208 e.target\nf : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\n\u22a2 (\u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c (\ud835\udcdd b) f g) \u2194\n \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c (\ud835\udcdd (\u2191(LocalHomeomorph.symm e) b)) (f \u2218 \u2191e) (g \u2218 \u2191e)\n[PROOFSTEP]\nexact forall\u2082_congr fun c _hc => e.isBigOWith_congr hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nE : Type u_3\ninst\u271d\u00b9 : Norm E\nF : Type u_4\ninst\u271d : Norm F\ne : \u03b1 \u2243\u209c \u03b2\nb : \u03b2\nf : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\n\u22a2 f =O[\ud835\udcdd b] g \u2194 (f \u2218 \u2191e) =O[\ud835\udcdd (\u2191(Homeomorph.symm e) b)] (g \u2218 \u2191e)\n[PROOFSTEP]\nsimp only [IsBigO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nE : Type u_3\ninst\u271d\u00b9 : Norm E\nF : Type u_4\ninst\u271d : Norm F\ne : \u03b1 \u2243\u209c \u03b2\nb : \u03b2\nf : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\n\u22a2 (\u2203 c, IsBigOWith c (\ud835\udcdd b) f g) \u2194 \u2203 c, IsBigOWith c (\ud835\udcdd (\u2191(Homeomorph.symm e) b)) (f \u2218 \u2191e) (g \u2218 \u2191e)\n[PROOFSTEP]\nexact exists_congr fun C => e.isBigOWith_congr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nE : Type u_3\ninst\u271d\u00b9 : Norm E\nF : Type u_4\ninst\u271d : Norm F\ne : \u03b1 \u2243\u209c \u03b2\nb : \u03b2\nf : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\n\u22a2 f =o[\ud835\udcdd b] g \u2194 (f \u2218 \u2191e) =o[\ud835\udcdd (\u2191(Homeomorph.symm e) b)] (g \u2218 \u2191e)\n[PROOFSTEP]\nsimp only [IsLittleO_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nE : Type u_3\ninst\u271d\u00b9 : Norm E\nF : Type u_4\ninst\u271d : Norm F\ne : \u03b1 \u2243\u209c \u03b2\nb : \u03b2\nf : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\n\u22a2 (\u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c (\ud835\udcdd b) f g) \u2194\n \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c (\ud835\udcdd (\u2191(Homeomorph.symm e) b)) (f \u2218 \u2191e) (g \u2218 \u2191e)\n[PROOFSTEP]\nexact forall\u2082_congr fun c _hc => e.isBigOWith_congr\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Asymptotics.Asymptotics", "llama_tokens": 197360, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124812, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.4063183036428736}} {"text": "[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nP Q : Presheaf C X\nf g : P \u27f6 Q\nw : \u2200 (U : Opens \u2191X), NatTrans.app f (op U) = NatTrans.app g (op U)\n\u22a2 f = g\n[PROOFSTEP]\napply NatTrans.ext\n[GOAL]\ncase app\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nP Q : Presheaf C X\nf g : P \u27f6 Q\nw : \u2200 (U : Opens \u2191X), NatTrans.app f (op U) = NatTrans.app g (op U)\n\u22a2 f.app = g.app\n[PROOFSTEP]\next U\n[GOAL]\ncase app.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nP Q : Presheaf C X\nf g : P \u27f6 Q\nw : \u2200 (U : Opens \u2191X), NatTrans.app f (op U) = NatTrans.app g (op U)\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 NatTrans.app f U = NatTrans.app g U\n[PROOFSTEP]\ninduction U with\n| _ U => ?_\n[GOAL]\ncase app.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nP Q : Presheaf C X\nf g : P \u27f6 Q\nw : \u2200 (U : Opens \u2191X), NatTrans.app f (op U) = NatTrans.app g (op U)\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 NatTrans.app f U = NatTrans.app g U\n[PROOFSTEP]\ninduction U with\n| _ U => ?_\n[GOAL]\ncase app.h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nP Q : Presheaf C X\nf g : P \u27f6 Q\nw : \u2200 (U : Opens \u2191X), NatTrans.app f (op U) = NatTrans.app g (op U)\nU : Opens \u2191X\n\u22a2 NatTrans.app f (op U) = NatTrans.app g (op U)\n[PROOFSTEP]\napply w\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nv w x y z : Opens \u2191X\nh\u2080 : v \u2264 x\nh\u2081 : x \u2264 z \u2293 w\nh\u2082 : x \u2264 y \u2293 z\n\u22a2 v \u2264 y\n[PROOFSTEP]\nrestrict_tac\n[GOAL]\nC\u271d : Type u\ninst\u271d\u00b2 : Category.{v, u} C\u271d\nX : TopCat\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_3, u_1} C\ninst\u271d : ConcreteCategory C\nF : Presheaf C X\nU V W : Opens \u2191X\ne\u2081 : U \u2264 V\ne\u2082 : V \u2264 W\nx : (forget C).obj (F.obj (op W))\n\u22a2 x |_ V |_ U = x |_ U\n[PROOFSTEP]\ndelta restrictOpen restrict\n[GOAL]\nC\u271d : Type u\ninst\u271d\u00b2 : Category.{v, u} C\u271d\nX : TopCat\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_3, u_1} C\ninst\u271d : ConcreteCategory C\nF : Presheaf C X\nU V W : Opens \u2191X\ne\u2081 : U \u2264 V\ne\u2082 : V \u2264 W\nx : (forget C).obj (F.obj (op W))\n\u22a2 \u2191(F.map (homOfLE (_ : \u2200 \u2983a : \u2191X\u2984, a \u2208 \u2191U \u2192 a \u2208 \u2191V)).op) (\u2191(F.map (homOfLE (_ : \u2200 \u2983a : \u2191X\u2984, a \u2208 \u2191V \u2192 a \u2208 \u2191W)).op) x) =\n \u2191(F.map (homOfLE (_ : \u2200 \u2983a : \u2191X\u2984, a \u2208 \u2191U \u2192 a \u2208 \u2191W)).op) x\n[PROOFSTEP]\nrw [\u2190 comp_apply, \u2190 Functor.map_comp]\n[GOAL]\nC\u271d : Type u\ninst\u271d\u00b2 : Category.{v, u} C\u271d\nX : TopCat\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_3, u_1} C\ninst\u271d : ConcreteCategory C\nF : Presheaf C X\nU V W : Opens \u2191X\ne\u2081 : U \u2264 V\ne\u2082 : V \u2264 W\nx : (forget C).obj (F.obj (op W))\n\u22a2 \u2191(F.map ((homOfLE (_ : \u2200 \u2983a : \u2191X\u2984, a \u2208 \u2191V \u2192 a \u2208 \u2191W)).op \u226b (homOfLE (_ : \u2200 \u2983a : \u2191X\u2984, a \u2208 \u2191U \u2192 a \u2208 \u2191V)).op)) x =\n \u2191(F.map (homOfLE (_ : \u2200 \u2983a : \u2191X\u2984, a \u2208 \u2191U \u2192 a \u2208 \u2191W)).op) x\n[PROOFSTEP]\nrfl\n[GOAL]\nC\u271d : Type u\ninst\u271d\u00b2 : Category.{v, u} C\u271d\nX : TopCat\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_3, u_1} C\ninst\u271d : ConcreteCategory C\nF G : Presheaf C X\ne : F \u27f6 G\nU V : Opens \u2191X\nh : U \u2264 V\nx : (forget C).obj (F.obj (op V))\n\u22a2 \u2191(NatTrans.app e (op U)) (x |_ U) = \u2191(NatTrans.app e (op V)) x |_ U\n[PROOFSTEP]\ndelta restrictOpen restrict\n[GOAL]\nC\u271d : Type u\ninst\u271d\u00b2 : Category.{v, u} C\u271d\nX : TopCat\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_3, u_1} C\ninst\u271d : ConcreteCategory C\nF G : Presheaf C X\ne : F \u27f6 G\nU V : Opens \u2191X\nh : U \u2264 V\nx : (forget C).obj (F.obj (op V))\n\u22a2 \u2191(NatTrans.app e (op U)) (\u2191(F.map (homOfLE (_ : \u2200 \u2983a : \u2191X\u2984, a \u2208 \u2191U \u2192 a \u2208 \u2191V)).op) x) =\n \u2191(G.map (homOfLE (_ : \u2200 \u2983a : \u2191X\u2984, a \u2208 \u2191U \u2192 a \u2208 \u2191V)).op) (\u2191(NatTrans.app e (op V)) x)\n[PROOFSTEP]\nrw [\u2190 comp_apply, NatTrans.naturality, comp_apply]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf g : X \u27f6 Y\nh : f = g\n\u2131 : Presheaf C X\n\u22a2 f _* \u2131 = g _* \u2131\n[PROOFSTEP]\nrw [h]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf g : X \u27f6 Y\nh : f = g\n\u2131 : Presheaf C X\nU : (Opens \u2191Y)\u1d52\u1d56\n\u22a2 (Opens.map f).op.obj U \u27f6 (Opens.map g).op.obj U\n[PROOFSTEP]\ndsimp [Functor.op]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf g : X \u27f6 Y\nh : f = g\n\u2131 : Presheaf C X\nU : (Opens \u2191Y)\u1d52\u1d56\n\u22a2 op ((Opens.map f).obj U.unop) \u27f6 op ((Opens.map g).obj U.unop)\n[PROOFSTEP]\napply Quiver.Hom.op\n[GOAL]\ncase f\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf g : X \u27f6 Y\nh : f = g\n\u2131 : Presheaf C X\nU : (Opens \u2191Y)\u1d52\u1d56\n\u22a2 (Opens.map g).obj U.unop \u27f6 (Opens.map f).obj U.unop\n[PROOFSTEP]\napply eqToHom\n[GOAL]\ncase f.p\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf g : X \u27f6 Y\nh : f = g\n\u2131 : Presheaf C X\nU : (Opens \u2191Y)\u1d52\u1d56\n\u22a2 (Opens.map g).obj U.unop = (Opens.map f).obj U.unop\n[PROOFSTEP]\nrw [h]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf g : X \u27f6 Y\nh : f = g\n\u2131 : Presheaf C X\nU : (Opens \u2191Y)\u1d52\u1d56\n\u22a2 NatTrans.app (pushforwardEq h \u2131).hom U =\n \u2131.map (id (eqToHom (_ : (Opens.map g).obj U.unop = (Opens.map f).obj U.unop)).op)\n[PROOFSTEP]\nsimp [pushforwardEq]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf g : X \u27f6 Y\nh : f = g\n\u2131 : Presheaf C X\nU : (Opens \u2191Y)\u1d52\u1d56\n\u22a2 (Opens.map f).op.obj U = (Opens.map g).op.obj U\n[PROOFSTEP]\nrw [h]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf g : X \u27f6 Y\nh : f = g\n\u2131 : Presheaf C X\nU : (Opens \u2191Y)\u1d52\u1d56\n\u22a2 NatTrans.app (eqToHom (_ : f _* \u2131 = g _* \u2131)) U = \u2131.map (eqToHom (_ : (Opens.map f).op.obj U = (Opens.map g).op.obj U))\n[PROOFSTEP]\nrw [eqToHom_app, eqToHom_map]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C X\nU : Opens \u2191Y\n\u22a2 NatTrans.app (pushforwardEq (_ : f = f) \u2131).hom (op U) = \ud835\udfd9 ((f _* \u2131).obj (op U))\n[PROOFSTEP]\ndsimp [pushforwardEq]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C X\nU : Opens \u2191Y\n\u22a2 \u2131.map (\ud835\udfd9 (op ((Opens.map f).obj U))) = \ud835\udfd9 (\u2131.obj (op ((Opens.map f).obj U)))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u2131 : Presheaf C X\n\u22a2 \ud835\udfd9 X _* \u2131 = \u2131\n[PROOFSTEP]\nunfold pushforwardObj\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u2131 : Presheaf C X\n\u22a2 (Opens.map (\ud835\udfd9 X)).op \u22d9 \u2131 = \u2131\n[PROOFSTEP]\nrw [Opens.map_id_eq]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u2131 : Presheaf C X\n\u22a2 (\ud835\udfed (Opens \u2191X)).op \u22d9 \u2131 = \u2131\n[PROOFSTEP]\nerw [Functor.id_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u2131 : Presheaf C X\nU : Set \u2191X\np : IsOpen U\n\u22a2 NatTrans.app (id \u2131).hom (op { carrier := U, is_open' := p }) = \u2131.map (\ud835\udfd9 (op { carrier := U, is_open' := p }))\n[PROOFSTEP]\ndsimp [id]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u2131 : Presheaf C X\nU : Set \u2191X\np : IsOpen U\n\u22a2 NatTrans.app (whiskerRight (NatTrans.op (Opens.mapId X).inv) \u2131 \u226b (Functor.leftUnitor \u2131).hom)\n (op { carrier := U, is_open' := p }) =\n \u2131.map (\ud835\udfd9 (op { carrier := U, is_open' := p }))\n[PROOFSTEP]\nsimp [CategoryStruct.comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u2131 : Presheaf C X\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 NatTrans.app (id \u2131).hom U = \u2131.map (eqToHom (_ : (Opens.map (\ud835\udfd9 X)).op.obj U = U))\n[PROOFSTEP]\ninduction U\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u2131 : Presheaf C X\nX\u271d : Opens \u2191X\n\u22a2 NatTrans.app (id \u2131).hom (op X\u271d) = \u2131.map (eqToHom (_ : (Opens.map (\ud835\udfd9 X)).op.obj (op X\u271d) = op X\u271d))\n[PROOFSTEP]\napply id_hom_app'\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u2131 : Presheaf C X\nU : Set \u2191X\np : IsOpen U\n\u22a2 NatTrans.app (id \u2131).inv (op { carrier := U, is_open' := p }) = \u2131.map (\ud835\udfd9 (op { carrier := U, is_open' := p }))\n[PROOFSTEP]\ndsimp [id]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u2131 : Presheaf C X\nU : Set \u2191X\np : IsOpen U\n\u22a2 NatTrans.app ((Functor.leftUnitor \u2131).inv \u226b whiskerRight (NatTrans.op (Opens.mapId X).hom) \u2131)\n (op { carrier := U, is_open' := p }) =\n \u2131.map (\ud835\udfd9 (op { carrier := U, is_open' := p }))\n[PROOFSTEP]\nsimp [CategoryStruct.comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u2131 : Presheaf C X\nY Z : TopCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nU : (Opens \u2191Z)\u1d52\u1d56\n\u22a2 NatTrans.app (comp \u2131 f g).hom U = \ud835\udfd9 (((f \u226b g) _* \u2131).obj U)\n[PROOFSTEP]\nsimp [comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u2131 : Presheaf C X\nY Z : TopCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nU : (Opens \u2191Z)\u1d52\u1d56\n\u22a2 NatTrans.app (comp \u2131 f g).inv U = \ud835\udfd9 ((g _* (f _* \u2131)).obj U)\n[PROOFSTEP]\nsimp [comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 \ud835\udca2 : Presheaf C X\n\u03b1 : \u2131 \u27f6 \ud835\udca2\nx\u271d\u00b9 x\u271d : (Opens \u2191Y)\u1d52\u1d56\ni : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 (f _* \u2131).map i \u226b (fun U => NatTrans.app \u03b1 ((Opens.map f).op.obj U)) x\u271d =\n (fun U => NatTrans.app \u03b1 ((Opens.map f).op.obj U)) x\u271d\u00b9 \u226b (f _* \ud835\udca2).map i\n[PROOFSTEP]\nerw [\u03b1.naturality]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 \ud835\udca2 : Presheaf C X\n\u03b1 : \u2131 \u27f6 \ud835\udca2\nx\u271d\u00b9 x\u271d : (Opens \u2191Y)\u1d52\u1d56\ni : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 NatTrans.app \u03b1 ((Opens.map f).op.obj x\u271d\u00b9) \u226b \ud835\udca2.map ((Opens.map f).op.map i) =\n (fun U => NatTrans.app \u03b1 ((Opens.map f).op.obj U)) x\u271d\u00b9 \u226b (f _* \ud835\udca2).map i\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C Y\nU : Opens \u2191X\nH : IsOpen (\u2191f '' \u2191U)\n\u22a2 (pullbackObj f \u2131).obj (op U) \u2245 \u2131.obj (op { carrier := \u2191f '' \u2191U, is_open' := H })\n[PROOFSTEP]\nlet x : CostructuredArrow (Opens.map f).op (op U) :=\n CostructuredArrow.mk (@homOfLE _ _ _ ((Opens.map f).obj \u27e8_, H\u27e9) (Set.image_preimage.le_u_l _)).op\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C Y\nU : Opens \u2191X\nH : IsOpen (\u2191f '' \u2191U)\nx : CostructuredArrow (Opens.map f).op (op U) :=\n CostructuredArrow.mk (homOfLE (_ : \u2191U \u2264 (fun a => \u2191f a) \u207b\u00b9' ((fun a => \u2191f a) '' \u2191U))).op\n\u22a2 (pullbackObj f \u2131).obj (op U) \u2245 \u2131.obj (op { carrier := \u2191f '' \u2191U, is_open' := H })\n[PROOFSTEP]\nhave hx : IsTerminal x :=\n { lift := fun s \u21a6 by\n fapply CostructuredArrow.homMk\n change op (unop _) \u27f6 op (\u27e8_, H\u27e9 : Opens _)\n \u00b7 refine' (homOfLE _).op\n apply (Set.image_subset f s.pt.hom.unop.le).trans\n exact Set.image_preimage.l_u_le (SetLike.coe s.pt.left.unop)\n \u00b7\n simp\n -- porting note : add `fac`, `uniq` manually\n fac := fun _ _ => by ext; simp\n uniq := fun _ _ _ => by ext; simp }\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C Y\nU : Opens \u2191X\nH : IsOpen (\u2191f '' \u2191U)\nx : CostructuredArrow (Opens.map f).op (op U) :=\n CostructuredArrow.mk (homOfLE (_ : \u2191U \u2264 (fun a => \u2191f a) \u207b\u00b9' ((fun a => \u2191f a) '' \u2191U))).op\ns : Cone (Functor.empty (CostructuredArrow (Opens.map f).op (op U)))\n\u22a2 s.pt \u27f6 (asEmptyCone x).pt\n[PROOFSTEP]\nfapply CostructuredArrow.homMk\n[GOAL]\ncase g\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C Y\nU : Opens \u2191X\nH : IsOpen (\u2191f '' \u2191U)\nx : CostructuredArrow (Opens.map f).op (op U) :=\n CostructuredArrow.mk (homOfLE (_ : \u2191U \u2264 (fun a => \u2191f a) \u207b\u00b9' ((fun a => \u2191f a) '' \u2191U))).op\ns : Cone (Functor.empty (CostructuredArrow (Opens.map f).op (op U)))\n\u22a2 s.pt.left \u27f6 (asEmptyCone x).pt.left\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C Y\nU : Opens \u2191X\nH : IsOpen (\u2191f '' \u2191U)\nx : CostructuredArrow (Opens.map f).op (op U) :=\n CostructuredArrow.mk (homOfLE (_ : \u2191U \u2264 (fun a => \u2191f a) \u207b\u00b9' ((fun a => \u2191f a) '' \u2191U))).op\ns : Cone (Functor.empty (CostructuredArrow (Opens.map f).op (op U)))\n\u22a2 autoParam ((Opens.map f).op.map ?g \u226b (asEmptyCone x).pt.hom = s.pt.hom) _auto\u271d\n[PROOFSTEP]\nchange op (unop _) \u27f6 op (\u27e8_, H\u27e9 : Opens _)\n[GOAL]\ncase g\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C Y\nU : Opens \u2191X\nH : IsOpen (\u2191f '' \u2191U)\nx : CostructuredArrow (Opens.map f).op (op U) :=\n CostructuredArrow.mk (homOfLE (_ : \u2191U \u2264 (fun a => \u2191f a) \u207b\u00b9' ((fun a => \u2191f a) '' \u2191U))).op\ns : Cone (Functor.empty (CostructuredArrow (Opens.map f).op (op U)))\n\u22a2 op s.pt.1.unop \u27f6 op { carrier := \u2191f '' \u2191U, is_open' := H }\n[PROOFSTEP]\nrefine' (homOfLE _).op\n[GOAL]\ncase g\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C Y\nU : Opens \u2191X\nH : IsOpen (\u2191f '' \u2191U)\nx : CostructuredArrow (Opens.map f).op (op U) :=\n CostructuredArrow.mk (homOfLE (_ : \u2191U \u2264 (fun a => \u2191f a) \u207b\u00b9' ((fun a => \u2191f a) '' \u2191U))).op\ns : Cone (Functor.empty (CostructuredArrow (Opens.map f).op (op U)))\n\u22a2 { carrier := \u2191f '' \u2191U, is_open' := H } \u2264 s.pt.1.unop\n[PROOFSTEP]\napply (Set.image_subset f s.pt.hom.unop.le).trans\n[GOAL]\ncase g\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C Y\nU : Opens \u2191X\nH : IsOpen (\u2191f '' \u2191U)\nx : CostructuredArrow (Opens.map f).op (op U) :=\n CostructuredArrow.mk (homOfLE (_ : \u2191U \u2264 (fun a => \u2191f a) \u207b\u00b9' ((fun a => \u2191f a) '' \u2191U))).op\ns : Cone (Functor.empty (CostructuredArrow (Opens.map f).op (op U)))\n\u22a2 \u2191f '' \u2191((Opens.map f).op.obj s.pt.left).unop \u2286 \u2191s.pt.1.unop\n[PROOFSTEP]\nexact Set.image_preimage.l_u_le (SetLike.coe s.pt.left.unop)\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C Y\nU : Opens \u2191X\nH : IsOpen (\u2191f '' \u2191U)\nx : CostructuredArrow (Opens.map f).op (op U) :=\n CostructuredArrow.mk (homOfLE (_ : \u2191U \u2264 (fun a => \u2191f a) \u207b\u00b9' ((fun a => \u2191f a) '' \u2191U))).op\ns : Cone (Functor.empty (CostructuredArrow (Opens.map f).op (op U)))\n\u22a2 autoParam\n ((Opens.map f).op.map\n (let_fun this := (homOfLE (_ : \u2191f '' \u2191((Functor.fromPUnit (op U)).obj s.pt.right).unop \u2286 \u2191s.pt.1.unop)).op;\n this) \u226b\n (asEmptyCone x).pt.hom =\n s.pt.hom)\n _auto\u271d\n[PROOFSTEP]\nsimp\n -- porting note : add `fac`, `uniq` manually\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C Y\nU : Opens \u2191X\nH : IsOpen (\u2191f '' \u2191U)\nx : CostructuredArrow (Opens.map f).op (op U) :=\n CostructuredArrow.mk (homOfLE (_ : \u2191U \u2264 (fun a => \u2191f a) \u207b\u00b9' ((fun a => \u2191f a) '' \u2191U))).op\nx\u271d\u00b9 : Cone (Functor.empty (CostructuredArrow (Opens.map f).op (op U)))\nx\u271d : Discrete PEmpty\n\u22a2 (fun s =>\n CostructuredArrow.homMk\n (let_fun this := (homOfLE (_ : \u2191f '' \u2191((Functor.fromPUnit (op U)).obj s.pt.right).unop \u2286 \u2191s.pt.1.unop)).op;\n this))\n x\u271d\u00b9 \u226b\n NatTrans.app (asEmptyCone x).\u03c0 x\u271d =\n NatTrans.app x\u271d\u00b9.\u03c0 x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C Y\nU : Opens \u2191X\nH : IsOpen (\u2191f '' \u2191U)\nx : CostructuredArrow (Opens.map f).op (op U) :=\n CostructuredArrow.mk (homOfLE (_ : \u2191U \u2264 (fun a => \u2191f a) \u207b\u00b9' ((fun a => \u2191f a) '' \u2191U))).op\nx\u271d\u00b9 : Cone (Functor.empty (CostructuredArrow (Opens.map f).op (op U)))\nx\u271d : Discrete PEmpty\n\u22a2 ((fun s =>\n CostructuredArrow.homMk\n (let_fun this :=\n (homOfLE (_ : \u2191f '' \u2191((Functor.fromPUnit (op U)).obj s.pt.right).unop \u2286 \u2191s.pt.1.unop)).op;\n this))\n x\u271d\u00b9 \u226b\n NatTrans.app (asEmptyCone x).\u03c0 x\u271d).left =\n (NatTrans.app x\u271d\u00b9.\u03c0 x\u271d).left\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C Y\nU : Opens \u2191X\nH : IsOpen (\u2191f '' \u2191U)\nx : CostructuredArrow (Opens.map f).op (op U) :=\n CostructuredArrow.mk (homOfLE (_ : \u2191U \u2264 (fun a => \u2191f a) \u207b\u00b9' ((fun a => \u2191f a) '' \u2191U))).op\nx\u271d\u00b2 : Cone (Functor.empty (CostructuredArrow (Opens.map f).op (op U)))\nx\u271d\u00b9 : x\u271d\u00b2.pt \u27f6 (asEmptyCone x).pt\nx\u271d : \u2200 (j : Discrete PEmpty), x\u271d\u00b9 \u226b NatTrans.app (asEmptyCone x).\u03c0 j = NatTrans.app x\u271d\u00b2.\u03c0 j\n\u22a2 x\u271d\u00b9 =\n (fun s =>\n CostructuredArrow.homMk\n (let_fun this := (homOfLE (_ : \u2191f '' \u2191((Functor.fromPUnit (op U)).obj s.pt.right).unop \u2286 \u2191s.pt.1.unop)).op;\n this))\n x\u271d\u00b2\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C Y\nU : Opens \u2191X\nH : IsOpen (\u2191f '' \u2191U)\nx : CostructuredArrow (Opens.map f).op (op U) :=\n CostructuredArrow.mk (homOfLE (_ : \u2191U \u2264 (fun a => \u2191f a) \u207b\u00b9' ((fun a => \u2191f a) '' \u2191U))).op\nx\u271d\u00b2 : Cone (Functor.empty (CostructuredArrow (Opens.map f).op (op U)))\nx\u271d\u00b9 : x\u271d\u00b2.pt \u27f6 (asEmptyCone x).pt\nx\u271d : \u2200 (j : Discrete PEmpty), x\u271d\u00b9 \u226b NatTrans.app (asEmptyCone x).\u03c0 j = NatTrans.app x\u271d\u00b2.\u03c0 j\n\u22a2 x\u271d\u00b9.left =\n ((fun s =>\n CostructuredArrow.homMk\n (let_fun this := (homOfLE (_ : \u2191f '' \u2191((Functor.fromPUnit (op U)).obj s.pt.right).unop \u2286 \u2191s.pt.1.unop)).op;\n this))\n x\u271d\u00b2).left\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\nf : X \u27f6 Y\n\u2131 : Presheaf C Y\nU : Opens \u2191X\nH : IsOpen (\u2191f '' \u2191U)\nx : CostructuredArrow (Opens.map f).op (op U) :=\n CostructuredArrow.mk (homOfLE (_ : \u2191U \u2264 (fun a => \u2191f a) \u207b\u00b9' ((fun a => \u2191f a) '' \u2191U))).op\nhx : IsTerminal x\n\u22a2 (pullbackObj f \u2131).obj (op U) \u2245 \u2131.obj (op { carrier := \u2191f '' \u2191U, is_open' := H })\n[PROOFSTEP]\nexact IsColimit.coconePointUniqueUpToIso (colimit.isColimit _) (colimitOfDiagramTerminal hx _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU : (Opens \u2191Y)\u1d52\u1d56\n\u22a2 IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191U.unop)\n[PROOFSTEP]\nsimpa using U.unop.2\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU : (Opens \u2191Y)\u1d52\u1d56\n\u22a2 op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191U.unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191U.unop)) } = U\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU V : (Opens \u2191Y)\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 (pullbackObj (\ud835\udfd9 Y) \u2131).map i \u226b\n ((fun U =>\n pullbackObjObjOfImageOpen (\ud835\udfd9 Y) \u2131 U.unop (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191U.unop)) \u226a\u226b\n \u2131.mapIso\n (eqToIso (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191U.unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191U.unop)) } = U)))\n V).hom =\n ((fun U =>\n pullbackObjObjOfImageOpen (\ud835\udfd9 Y) \u2131 U.unop (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191U.unop)) \u226a\u226b\n \u2131.mapIso\n (eqToIso (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191U.unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191U.unop)) } = U)))\n U).hom \u226b\n \u2131.map i\n[PROOFSTEP]\nsimp only [pullbackObj_obj]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU V : (Opens \u2191Y)\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 (pullbackObj (\ud835\udfd9 Y) \u2131).map i \u226b\n (pullbackObjObjOfImageOpen (\ud835\udfd9 Y) \u2131 V.unop (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191V.unop)) \u226a\u226b\n \u2131.mapIso\n (eqToIso (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191V.unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191V.unop)) } = V))).hom =\n (pullbackObjObjOfImageOpen (\ud835\udfd9 Y) \u2131 U.unop (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191U.unop)) \u226a\u226b\n \u2131.mapIso\n (eqToIso (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191U.unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191U.unop)) } = U))).hom \u226b\n \u2131.map i\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU V : (Opens \u2191Y)\u1d52\u1d56\ni : U \u27f6 V\nj\u271d : CostructuredArrow (Opens.map (\ud835\udfd9 Y)).op U\n\u22a2 colimit.\u03b9 (Lan.diagram (Opens.map (\ud835\udfd9 Y)).op \u2131 U) j\u271d \u226b\n (pullbackObj (\ud835\udfd9 Y) \u2131).map i \u226b\n (pullbackObjObjOfImageOpen (\ud835\udfd9 Y) \u2131 V.unop (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191V.unop)) \u226a\u226b\n \u2131.mapIso\n (eqToIso\n (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191V.unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191V.unop)) } = V))).hom =\n colimit.\u03b9 (Lan.diagram (Opens.map (\ud835\udfd9 Y)).op \u2131 U) j\u271d \u226b\n (pullbackObjObjOfImageOpen (\ud835\udfd9 Y) \u2131 U.unop (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191U.unop)) \u226a\u226b\n \u2131.mapIso\n (eqToIso\n (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191U.unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191U.unop)) } = U))).hom \u226b\n \u2131.map i\n[PROOFSTEP]\nsimp only [Functor.comp_obj, CostructuredArrow.proj_obj, pullbackObj_map, Iso.trans_hom, Functor.mapIso_hom,\n eqToIso.hom, Category.assoc]\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU V : (Opens \u2191Y)\u1d52\u1d56\ni : U \u27f6 V\nj\u271d : CostructuredArrow (Opens.map (\ud835\udfd9 Y)).op U\n\u22a2 colimit.\u03b9 (Lan.diagram (Opens.map (\ud835\udfd9 Y)).op \u2131 U) j\u271d \u226b\n colimit.pre (Lan.diagram (Opens.map (\ud835\udfd9 Y)).op \u2131 V) (CostructuredArrow.map i) \u226b\n (pullbackObjObjOfImageOpen (\ud835\udfd9 Y) \u2131 V.unop (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191V.unop))).hom \u226b\n \u2131.map (eqToHom (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191V.unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191V.unop)) } = V)) =\n colimit.\u03b9 (Lan.diagram (Opens.map (\ud835\udfd9 Y)).op \u2131 U) j\u271d \u226b\n (pullbackObjObjOfImageOpen (\ud835\udfd9 Y) \u2131 U.unop (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191U.unop))).hom \u226b\n \u2131.map (eqToHom (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191U.unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191U.unop)) } = U)) \u226b\n \u2131.map i\n[PROOFSTEP]\nerw [colimit.pre_desc_assoc, colimit.\u03b9_desc_assoc, colimit.\u03b9_desc_assoc]\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU V : (Opens \u2191Y)\u1d52\u1d56\ni : U \u27f6 V\nj\u271d : CostructuredArrow (Opens.map (\ud835\udfd9 Y)).op U\n\u22a2 NatTrans.app\n (Cocone.whisker (CostructuredArrow.map i)\n (coconeOfDiagramTerminal\n (IsLimit.mk fun s =>\n CostructuredArrow.homMk\n (let_fun this :=\n (homOfLE (_ : \u2191(\ud835\udfd9 Y) '' \u2191((Functor.fromPUnit (op V.unop)).obj s.pt.right).unop \u2286 \u2191s.pt.1.unop)).op;\n this))\n (Lan.diagram (Opens.map (\ud835\udfd9 Y)).op \u2131 (op V.unop)))).\u03b9\n j\u271d \u226b\n \u2131.map (eqToHom (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191V.unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191V.unop)) } = V)) =\n NatTrans.app\n (coconeOfDiagramTerminal\n (IsLimit.mk fun s =>\n CostructuredArrow.homMk\n (let_fun this :=\n (homOfLE (_ : \u2191(\ud835\udfd9 Y) '' \u2191((Functor.fromPUnit (op U.unop)).obj s.pt.right).unop \u2286 \u2191s.pt.1.unop)).op;\n this))\n (Lan.diagram (Opens.map (\ud835\udfd9 Y)).op \u2131 (op U.unop))).\u03b9\n j\u271d \u226b\n \u2131.map (eqToHom (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191U.unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191U.unop)) } = U)) \u226b\n \u2131.map i\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU V : (Opens \u2191Y)\u1d52\u1d56\ni : U \u27f6 V\nj\u271d : CostructuredArrow (Opens.map (\ud835\udfd9 Y)).op U\n\u22a2 \u2131.map\n (IsTerminal.from\n (IsLimit.mk fun s =>\n CostructuredArrow.homMk\n (homOfLE (_ : \u2191(\ud835\udfd9 Y) '' \u2191((Functor.fromPUnit (op V.unop)).obj s.pt.right).unop \u2286 \u2191s.pt.1.unop)).op)\n ((CostructuredArrow.map i).obj j\u271d)).left \u226b\n \u2131.map (eqToHom (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191V.unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191V.unop)) } = V)) =\n \u2131.map\n (IsTerminal.from\n (IsLimit.mk fun s =>\n CostructuredArrow.homMk\n (homOfLE (_ : \u2191(\ud835\udfd9 Y) '' \u2191((Functor.fromPUnit (op U.unop)).obj s.pt.right).unop \u2286 \u2191s.pt.1.unop)).op)\n j\u271d).left \u226b\n \u2131.map (eqToHom (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191U.unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191U.unop)) } = U)) \u226b\n \u2131.map i\n[PROOFSTEP]\nsimp only [\u2190 \u2131.map_comp]\n -- Porting note : `congr` does not work, but `congr 1` does\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU V : (Opens \u2191Y)\u1d52\u1d56\ni : U \u27f6 V\nj\u271d : CostructuredArrow (Opens.map (\ud835\udfd9 Y)).op U\n\u22a2 \u2131.map\n ((IsTerminal.from\n (IsLimit.mk fun s =>\n CostructuredArrow.homMk\n (homOfLE (_ : \u2191(\ud835\udfd9 Y) '' \u2191((Functor.fromPUnit (op V.unop)).obj s.pt.right).unop \u2286 \u2191s.pt.1.unop)).op)\n ((CostructuredArrow.map i).obj j\u271d)).left \u226b\n eqToHom (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191V.unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191V.unop)) } = V)) =\n \u2131.map\n ((IsTerminal.from\n (IsLimit.mk fun s =>\n CostructuredArrow.homMk\n (homOfLE (_ : \u2191(\ud835\udfd9 Y) '' \u2191((Functor.fromPUnit (op U.unop)).obj s.pt.right).unop \u2286 \u2191s.pt.1.unop)).op)\n j\u271d).left \u226b\n eqToHom (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191U.unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191U.unop)) } = U) \u226b i)\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU : Opens \u2191Y\n\u22a2 (Opens.map (\ud835\udfd9 Y)).op.obj (op U) = op U\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU : Opens \u2191Y\n\u22a2 NatTrans.app (id \u2131).inv (op U) =\n colimit.\u03b9 (Lan.diagram (Opens.map (\ud835\udfd9 Y)).op \u2131 (op U))\n (CostructuredArrow.mk (eqToHom (_ : (Opens.map (\ud835\udfd9 Y)).op.obj (op U) = op U)))\n[PROOFSTEP]\nrw [\u2190 Category.id_comp ((id \u2131).inv.app (op U)), \u2190 NatIso.app_inv, Iso.comp_inv_eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU : Opens \u2191Y\n\u22a2 \ud835\udfd9 (\u2131.obj (op U)) =\n colimit.\u03b9 (Lan.diagram (Opens.map (\ud835\udfd9 Y)).op \u2131 (op U))\n (CostructuredArrow.mk (eqToHom (_ : (Opens.map (\ud835\udfd9 Y)).op.obj (op U) = op U))) \u226b\n ((id \u2131).app (op U)).hom\n[PROOFSTEP]\ndsimp [id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU : Opens \u2191Y\n\u22a2 \ud835\udfd9 (\u2131.obj (op U)) =\n colimit.\u03b9 (Lan.diagram (Opens.map (\ud835\udfd9 Y)).op \u2131 (op U)) (CostructuredArrow.mk (\ud835\udfd9 (op ((Opens.map (\ud835\udfd9 Y)).obj U)))) \u226b\n colimit.desc (Lan.diagram (Opens.map (\ud835\udfd9 Y)).op \u2131 (op U))\n (coconeOfDiagramTerminal\n (IsLimit.mk fun s =>\n CostructuredArrow.homMk\n (homOfLE (_ : \u2191(\ud835\udfd9 Y) '' \u2191((Functor.fromPUnit (op U)).obj s.pt.right).unop \u2286 \u2191s.pt.1.unop)).op)\n (Lan.diagram (Opens.map (\ud835\udfd9 Y)).op \u2131 (op U))) \u226b\n \u2131.map\n (eqToHom\n (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191(op U).unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191(op U).unop)) } = op U))\n[PROOFSTEP]\nerw [colimit.\u03b9_desc_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU : Opens \u2191Y\n\u22a2 \ud835\udfd9 (\u2131.obj (op U)) =\n NatTrans.app\n (coconeOfDiagramTerminal\n (IsLimit.mk fun s =>\n CostructuredArrow.homMk\n (homOfLE (_ : \u2191(\ud835\udfd9 Y) '' \u2191((Functor.fromPUnit (op U)).obj s.pt.right).unop \u2286 \u2191s.pt.1.unop)).op)\n (Lan.diagram (Opens.map (\ud835\udfd9 Y)).op \u2131 (op U))).\u03b9\n (CostructuredArrow.mk (\ud835\udfd9 (op ((Opens.map (\ud835\udfd9 Y)).obj U)))) \u226b\n \u2131.map\n (eqToHom\n (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191(op U).unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191(op U).unop)) } = op U))\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU : Opens \u2191Y\n\u22a2 \ud835\udfd9 (\u2131.obj (op U)) =\n \u2131.map\n (IsTerminal.from\n (IsLimit.mk fun s =>\n CostructuredArrow.homMk\n (homOfLE (_ : \u2191(\ud835\udfd9 Y) '' \u2191((Functor.fromPUnit (op U)).obj s.pt.right).unop \u2286 \u2191s.pt.1.unop)).op)\n (CostructuredArrow.mk (\ud835\udfd9 (op ((Opens.map (\ud835\udfd9 Y)).obj U))))).left \u226b\n \u2131.map\n (eqToHom\n (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191(op U).unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191(op U).unop)) } = op U))\n[PROOFSTEP]\nrw [\u2190 \u2131.map_comp, \u2190 \u2131.map_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y : TopCat\n\u2131 : Presheaf C Y\nU : Opens \u2191Y\n\u22a2 \u2131.map (\ud835\udfd9 (op U)) =\n \u2131.map\n ((IsTerminal.from\n (IsLimit.mk fun s =>\n CostructuredArrow.homMk\n (homOfLE (_ : \u2191(\ud835\udfd9 Y) '' \u2191((Functor.fromPUnit (op U)).obj s.pt.right).unop \u2286 \u2191s.pt.1.unop)).op)\n (CostructuredArrow.mk (\ud835\udfd9 (op ((Opens.map (\ud835\udfd9 Y)).obj U))))).left \u226b\n eqToHom\n (_ : op { carrier := \u2191(\ud835\udfd9 Y) '' \u2191(op U).unop, is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 Y) '' \u2191(op U).unop)) } = op U))\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u22a2 pushforward C (\ud835\udfd9 X) = \ud835\udfed (Presheaf C X)\n[PROOFSTEP]\napply CategoryTheory.Functor.ext\n[GOAL]\ncase h_map\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u22a2 autoParam\n (\u2200 (X_1 Y : Presheaf C X) (f : X_1 \u27f6 Y),\n (pushforward C (\ud835\udfd9 X)).map f =\n eqToHom (_ : ?F.obj X_1 = ?G.obj X_1) \u226b\n (\ud835\udfed (Presheaf C X)).map f \u226b eqToHom (_ : (\ud835\udfed (Presheaf C X)).obj Y = (pushforward C (\ud835\udfd9 X)).obj Y))\n _auto\u271d\n[PROOFSTEP]\nintros a b f\n[GOAL]\ncase h_map\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\na b : Presheaf C X\nf : a \u27f6 b\n\u22a2 (pushforward C (\ud835\udfd9 X)).map f =\n eqToHom (_ : ?F.obj a = ?G.obj a) \u226b\n (\ud835\udfed (Presheaf C X)).map f \u226b eqToHom (_ : (\ud835\udfed (Presheaf C X)).obj b = (pushforward C (\ud835\udfd9 X)).obj b)\n[PROOFSTEP]\next U\n[GOAL]\ncase h_map.w\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\na b : Presheaf C X\nf : a \u27f6 b\nU : Opens \u2191X\n\u22a2 NatTrans.app ((pushforward C (\ud835\udfd9 X)).map f) (op U) =\n NatTrans.app\n (eqToHom (_ : ?F.obj a = ?G.obj a) \u226b\n (\ud835\udfed (Presheaf C X)).map f \u226b eqToHom (_ : (\ud835\udfed (Presheaf C X)).obj b = (pushforward C (\ud835\udfd9 X)).obj b))\n (op U)\n[PROOFSTEP]\nerw [NatTrans.congr f (Opens.op_map_id_obj (op U))]\n[GOAL]\ncase h_map.w\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\na b : Presheaf C X\nf : a \u27f6 b\nU : Opens \u2191X\n\u22a2 a.map (eqToHom (_ : (Opens.map (\ud835\udfd9 X)).op.obj (op U) = op U)) \u226b\n NatTrans.app f (op U) \u226b b.map (eqToHom (_ : op U = (Opens.map (\ud835\udfd9 X)).op.obj (op U))) =\n NatTrans.app\n (eqToHom (_ : ?F.obj a = ?G.obj a) \u226b\n (\ud835\udfed (Presheaf C X)).map f \u226b eqToHom (_ : (\ud835\udfed (Presheaf C X)).obj b = (pushforward C (\ud835\udfd9 X)).obj b))\n (op U)\ncase h_obj\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u22a2 \u2200 (X_1 : Presheaf C X), (pushforward C (\ud835\udfd9 X)).obj X_1 = (\ud835\udfed (Presheaf C X)).obj X_1\n[PROOFSTEP]\nsimp only [Functor.op_obj, eqToHom_refl, CategoryTheory.Functor.map_id, Category.comp_id, Category.id_comp,\n Functor.id_obj, Functor.id_map]\n[GOAL]\ncase h_obj\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u22a2 \u2200 (X_1 : Presheaf C X), (pushforward C (\ud835\udfd9 X)).obj X_1 = (\ud835\udfed (Presheaf C X)).obj X_1\n[PROOFSTEP]\napply Pushforward.id_eq\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nH\u2081 : X \u2245 Y\n\u2131 : Presheaf C X\n\ud835\udca2 : Presheaf C Y\nH\u2082 : H\u2081.hom _* \u2131 \u27f6 \ud835\udca2\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 U = (Opens.map H\u2081.hom).op.obj (op ((Opens.map H\u2081.inv).obj U.unop))\n[PROOFSTEP]\nsimp [Opens.map, Set.preimage_preimage]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nH\u2081 : X \u2245 Y\n\u2131 : Presheaf C X\n\ud835\udca2 : Presheaf C Y\nH\u2082 : H\u2081.hom _* \u2131 \u27f6 \ud835\udca2\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 NatTrans.app (toPushforwardOfIso H\u2081 H\u2082) U =\n \u2131.map\n (eqToHom\n (_ :\n U =\n op\n {\n carrier :=\n \u2191H\u2081.hom \u207b\u00b9'\n \u2191(op { carrier := \u2191H\u2081.inv \u207b\u00b9' \u2191U.unop, is_open' := (_ : IsOpen (\u2191H\u2081.inv \u207b\u00b9' \u2191U.unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191H\u2081.hom \u207b\u00b9'\n \u2191(op\n { carrier := \u2191H\u2081.inv \u207b\u00b9' \u2191U.unop,\n is_open' := (_ : IsOpen (\u2191H\u2081.inv \u207b\u00b9' \u2191U.unop)) }).unop)) })) \u226b\n NatTrans.app H\u2082 (op ((Opens.map H\u2081.inv).obj U.unop))\n[PROOFSTEP]\ndelta toPushforwardOfIso\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nH\u2081 : X \u2245 Y\n\u2131 : Presheaf C X\n\ud835\udca2 : Presheaf C Y\nH\u2082 : H\u2081.hom _* \u2131 \u27f6 \ud835\udca2\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 NatTrans.app (\u2191(Adjunction.homEquiv (Equivalence.toAdjunction (presheafEquivOfIso C H\u2081)) \u2131 \ud835\udca2) H\u2082) U =\n \u2131.map\n (eqToHom\n (_ :\n U =\n op\n {\n carrier :=\n \u2191H\u2081.hom \u207b\u00b9'\n \u2191(op { carrier := \u2191H\u2081.inv \u207b\u00b9' \u2191U.unop, is_open' := (_ : IsOpen (\u2191H\u2081.inv \u207b\u00b9' \u2191U.unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191H\u2081.hom \u207b\u00b9'\n \u2191(op\n { carrier := \u2191H\u2081.inv \u207b\u00b9' \u2191U.unop,\n is_open' := (_ : IsOpen (\u2191H\u2081.inv \u207b\u00b9' \u2191U.unop)) }).unop)) })) \u226b\n NatTrans.app H\u2082 (op ((Opens.map H\u2081.inv).obj U.unop))\n[PROOFSTEP]\nsimp only [pushforwardObj_obj, Functor.op_obj, Equivalence.toAdjunction, Adjunction.homEquiv_unit, Functor.id_obj,\n Functor.comp_obj, Adjunction.mkOfUnitCounit_unit, unop_op, eqToHom_map]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nH\u2081 : X \u2245 Y\n\u2131 : Presheaf C X\n\ud835\udca2 : Presheaf C Y\nH\u2082 : H\u2081.hom _* \u2131 \u27f6 \ud835\udca2\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 NatTrans.app (NatTrans.app (Equivalence.unit (presheafEquivOfIso C H\u2081)) \u2131 \u226b (presheafEquivOfIso C H\u2081).inverse.map H\u2082)\n U =\n eqToHom (_ : \u2131.obj U = \u2131.obj (op ((Opens.map H\u2081.hom).obj ((Opens.map H\u2081.inv).obj U.unop)))) \u226b\n NatTrans.app H\u2082 (op ((Opens.map H\u2081.inv).obj U.unop))\n[PROOFSTEP]\nrw [NatTrans.comp_app, presheafEquivOfIso_inverse_map_app, Equivalence.Equivalence_mk'_unit]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nH\u2081 : X \u2245 Y\n\u2131 : Presheaf C X\n\ud835\udca2 : Presheaf C Y\nH\u2082 : H\u2081.hom _* \u2131 \u27f6 \ud835\udca2\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 NatTrans.app (NatTrans.app (presheafEquivOfIso C H\u2081).unitIso.hom \u2131) U \u226b\n NatTrans.app H\u2082 (op ((Opens.map H\u2081.inv).obj U.unop)) =\n eqToHom (_ : \u2131.obj U = \u2131.obj (op ((Opens.map H\u2081.hom).obj ((Opens.map H\u2081.inv).obj U.unop)))) \u226b\n NatTrans.app H\u2082 (op ((Opens.map H\u2081.inv).obj U.unop))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nH\u2081 : X \u2245 Y\n\u2131 : Presheaf C X\n\ud835\udca2 : Presheaf C Y\nH\u2082 : H\u2081.hom _* \u2131 \u27f6 \ud835\udca2\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 NatTrans.app (NatTrans.app (presheafEquivOfIso C H\u2081).unitIso.hom \u2131) U =\n eqToHom (_ : \u2131.obj U = \u2131.obj (op ((Opens.map H\u2081.hom).obj ((Opens.map H\u2081.inv).obj U.unop))))\n[PROOFSTEP]\nsimp only [Equivalence.unit, Equivalence.op, CategoryTheory.Equivalence.symm, Opens.mapMapIso, Functor.id_obj,\n Functor.comp_obj, Iso.symm_hom, NatIso.op_inv, Iso.symm_inv, NatTrans.op_app, NatIso.ofComponents_hom_app,\n eqToIso.hom, eqToHom_op, Equivalence.Equivalence_mk'_unitInv, Equivalence.Equivalence_mk'_counitInv, NatIso.op_hom,\n unop_op, op_unop, eqToIso.inv, NatIso.ofComponents_inv_app, eqToHom_unop, \u2190 \u2131.map_comp, eqToHom_trans, eqToHom_map,\n presheafEquivOfIso_unitIso_hom_app_app]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nH\u2081 : X \u2245 Y\n\u2131 : Presheaf C Y\n\ud835\udca2 : Presheaf C X\nH\u2082 : \u2131 \u27f6 H\u2081.hom _* \ud835\udca2\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 (Opens.map H\u2081.hom).op.obj (op ((Opens.map H\u2081.inv).obj U.unop)) = U\n[PROOFSTEP]\nsimp [Opens.map, Set.preimage_preimage]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : TopCat\nH\u2081 : X \u2245 Y\n\u2131 : Presheaf C Y\n\ud835\udca2 : Presheaf C X\nH\u2082 : \u2131 \u27f6 H\u2081.hom _* \ud835\udca2\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 NatTrans.app (pushforwardToOfIso H\u2081 H\u2082) U =\n NatTrans.app H\u2082 (op ((Opens.map H\u2081.inv).obj U.unop)) \u226b\n \ud835\udca2.map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191H\u2081.hom \u207b\u00b9'\n \u2191(op { carrier := \u2191H\u2081.inv \u207b\u00b9' \u2191U.unop, is_open' := (_ : IsOpen (\u2191H\u2081.inv \u207b\u00b9' \u2191U.unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191H\u2081.hom \u207b\u00b9'\n \u2191(op\n { carrier := \u2191H\u2081.inv \u207b\u00b9' \u2191U.unop,\n is_open' := (_ : IsOpen (\u2191H\u2081.inv \u207b\u00b9' \u2191U.unop)) }).unop)) } =\n U))\n[PROOFSTEP]\nsimp [pushforwardToOfIso, Equivalence.toAdjunction, CategoryStruct.comp]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Sheaves.Presheaf", "llama_tokens": 17455, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124812, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.4063183036428736}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\n\u22a2 MeasurableSet univ\u1d9c\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\ns t : Set \u03b1\nhs : MeasurableSet s\nh : s = t\n\u22a2 MeasurableSet t\n[PROOFSTEP]\nrwa [\u2190 h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u2200 (b : \u03b9), MeasurableSet (f b)\n\u22a2 MeasurableSet (\u22c3 (b : \u03b9), f b)\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u2200 (b : \u03b9), MeasurableSet (f b)\nh\u271d : IsEmpty \u03b9\n\u22a2 MeasurableSet (\u22c3 (b : \u03b9), f b)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u2200 (b : \u03b9), MeasurableSet (f b)\nh\u271d : Nonempty \u03b9\n\u22a2 MeasurableSet (\u22c3 (b : \u03b9), f b)\n[PROOFSTEP]\nrcases exists_surjective_nat \u03b9 with \u27e8e, he\u27e9\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u2200 (b : \u03b9), MeasurableSet (f b)\nh\u271d : Nonempty \u03b9\ne : \u2115 \u2192 \u03b9\nhe : Surjective e\n\u22a2 MeasurableSet (\u22c3 (b : \u03b9), f b)\n[PROOFSTEP]\nrw [\u2190 iUnion_congr_of_surjective _ he (fun _ => rfl)]\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u2200 (b : \u03b9), MeasurableSet (f b)\nh\u271d : Nonempty \u03b9\ne : \u2115 \u2192 \u03b9\nhe : Surjective e\n\u22a2 MeasurableSet (\u22c3 (x : \u2115), f (e x))\n[PROOFSTEP]\nexact m.measurableSet_iUnion _ fun _ => h _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b2 \u2192 Set \u03b1\ns : Set \u03b2\nhs : Set.Countable s\nh : \u2200 (b : \u03b2), b \u2208 s \u2192 MeasurableSet (f b)\n\u22a2 MeasurableSet (\u22c3 (b : \u03b2) (_ : b \u2208 s), f b)\n[PROOFSTEP]\nrw [biUnion_eq_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b2 \u2192 Set \u03b1\ns : Set \u03b2\nhs : Set.Countable s\nh : \u2200 (b : \u03b2), b \u2208 s \u2192 MeasurableSet (f b)\n\u22a2 MeasurableSet (\u22c3 (x : \u2191s), f \u2191x)\n[PROOFSTEP]\nhave := hs.to_subtype\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b2 \u2192 Set \u03b1\ns : Set \u03b2\nhs : Set.Countable s\nh : \u2200 (b : \u03b2), b \u2208 s \u2192 MeasurableSet (f b)\nthis : Countable \u2191s\n\u22a2 MeasurableSet (\u22c3 (x : \u2191s), f \u2191x)\n[PROOFSTEP]\nexact MeasurableSet.iUnion (by simpa using h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b2 \u2192 Set \u03b1\ns : Set \u03b2\nhs : Set.Countable s\nh : \u2200 (b : \u03b2), b \u2208 s \u2192 MeasurableSet (f b)\nthis : Countable \u2191s\n\u22a2 \u2200 (b : \u2191s), MeasurableSet (f \u2191b)\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns : Set (Set \u03b1)\nhs : Set.Countable s\nh : \u2200 (t : Set \u03b1), t \u2208 s \u2192 MeasurableSet t\n\u22a2 MeasurableSet (\u22c3\u2080 s)\n[PROOFSTEP]\nrw [sUnion_eq_biUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns : Set (Set \u03b1)\nhs : Set.Countable s\nh : \u2200 (t : Set \u03b1), t \u2208 s \u2192 MeasurableSet t\n\u22a2 MeasurableSet (\u22c3 (i : Set \u03b1) (_ : i \u2208 s), i)\n[PROOFSTEP]\nexact .biUnion hs h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u2200 (b : \u03b9), MeasurableSet (f b)\n\u22a2 MeasurableSet (\u22c2 (b : \u03b9), f b)\u1d9c\n[PROOFSTEP]\nrw [compl_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u2200 (b : \u03b9), MeasurableSet (f b)\n\u22a2 MeasurableSet (\u22c3 (i : \u03b9), (f i)\u1d9c)\n[PROOFSTEP]\nexact .iUnion fun b => (h b).compl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b2 \u2192 Set \u03b1\ns : Set \u03b2\nhs : Set.Countable s\nh : \u2200 (b : \u03b2), b \u2208 s \u2192 MeasurableSet (f b)\n\u22a2 MeasurableSet (\u22c2 (b : \u03b2) (_ : b \u2208 s), f b)\u1d9c\n[PROOFSTEP]\nrw [compl_iInter\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : MeasurableSpace \u03b1\nf : \u03b2 \u2192 Set \u03b1\ns : Set \u03b2\nhs : Set.Countable s\nh : \u2200 (b : \u03b2), b \u2208 s \u2192 MeasurableSet (f b)\n\u22a2 MeasurableSet (\u22c3 (i : \u03b2) (_ : i \u2208 s), (f i)\u1d9c)\n[PROOFSTEP]\nexact .biUnion hs fun b hb => (h b hb).compl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns : Set (Set \u03b1)\nhs : Set.Countable s\nh : \u2200 (t : Set \u03b1), t \u2208 s \u2192 MeasurableSet t\n\u22a2 MeasurableSet (\u22c2\u2080 s)\n[PROOFSTEP]\nrw [sInter_eq_biInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns : Set (Set \u03b1)\nhs : Set.Countable s\nh : \u2200 (t : Set \u03b1), t \u2208 s \u2192 MeasurableSet t\n\u22a2 MeasurableSet (\u22c2 (i : Set \u03b1) (_ : i \u2208 s), i)\n[PROOFSTEP]\nexact MeasurableSet.biInter hs h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nh\u2082 : MeasurableSet s\u2082\n\u22a2 MeasurableSet (s\u2081 \u222a s\u2082)\n[PROOFSTEP]\nrw [union_eq_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nh\u2082 : MeasurableSet s\u2082\n\u22a2 MeasurableSet (\u22c3 (b : Bool), bif b then s\u2081 else s\u2082)\n[PROOFSTEP]\nexact .iUnion (Bool.forall_bool.2 \u27e8h\u2082, h\u2081\u27e9)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nh\u2082 : MeasurableSet s\u2082\n\u22a2 MeasurableSet (s\u2081 \u2229 s\u2082)\n[PROOFSTEP]\nrw [inter_eq_compl_compl_union_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nh\u2082 : MeasurableSet s\u2082\n\u22a2 MeasurableSet (s\u2081\u1d9c \u222a s\u2082\u1d9c)\u1d9c\n[PROOFSTEP]\nexact (h\u2081.compl.union h\u2082.compl).compl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\ns t : Set \u03b1\np : Prop\nhs : p \u2192 MeasurableSet s\nht : \u00acp \u2192 MeasurableSet t\n\u22a2 MeasurableSet (if p then s else t)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\ns t : Set \u03b1\np : Prop\nhs : p \u2192 MeasurableSet s\nht : \u00acp \u2192 MeasurableSet t\nh : p\n\u22a2 MeasurableSet s\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t\u271d u : Set \u03b1\nm : MeasurableSpace \u03b1\ns t : Set \u03b1\np : Prop\nhs : p \u2192 MeasurableSet s\nht : \u00acp \u2192 MeasurableSet t\nh : \u00acp\n\u22a2 MeasurableSet t\n[PROOFSTEP]\nexacts [hs h, ht h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nh\u2082 : MeasurableSet s\u2082\ni : Bool\n\u22a2 MeasurableSet (bif i then s\u2081 else s\u2082)\n[PROOFSTEP]\ncases i\n[GOAL]\ncase false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nh\u2082 : MeasurableSet s\u2082\n\u22a2 MeasurableSet (bif false then s\u2081 else s\u2082)\ncase true\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ns\u2081 s\u2082 : Set \u03b1\nh\u2081 : MeasurableSet s\u2081\nh\u2082 : MeasurableSet s\u2082\n\u22a2 MeasurableSet (bif true then s\u2081 else s\u2082)\n[PROOFSTEP]\nexacts [h\u2082, h\u2081]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\np : Prop\n\u22a2 MeasurableSet {_a | p}\n[PROOFSTEP]\nby_cases p\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\np : Prop\n\u22a2 MeasurableSet {_a | p}\n[PROOFSTEP]\nby_cases p\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\np : Prop\nh : p\n\u22a2 MeasurableSet {_a | p}\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\np : Prop\nh : \u00acp\n\u22a2 MeasurableSet {_a | p}\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nMeasurableSet'\u271d\u00b9 : Set \u03b1 \u2192 Prop\nmeasurableSet_empty\u271d\u00b9 : MeasurableSet'\u271d\u00b9 \u2205\nmeasurableSet_compl\u271d\u00b9 : \u2200 (s : Set \u03b1), MeasurableSet'\u271d\u00b9 s \u2192 MeasurableSet'\u271d\u00b9 s\u1d9c\nmeasurableSet_iUnion\u271d\u00b9 : \u2200 (f : \u2115 \u2192 Set \u03b1), (\u2200 (i : \u2115), MeasurableSet'\u271d\u00b9 (f i)) \u2192 MeasurableSet'\u271d\u00b9 (\u22c3 (i : \u2115), f i)\nMeasurableSet'\u271d : Set \u03b1 \u2192 Prop\nmeasurableSet_empty\u271d : MeasurableSet'\u271d \u2205\nmeasurableSet_compl\u271d : \u2200 (s : Set \u03b1), MeasurableSet'\u271d s \u2192 MeasurableSet'\u271d s\u1d9c\nmeasurableSet_iUnion\u271d : \u2200 (f : \u2115 \u2192 Set \u03b1), (\u2200 (i : \u2115), MeasurableSet'\u271d (f i)) \u2192 MeasurableSet'\u271d (\u22c3 (i : \u2115), f i)\nx\u271d : MeasurableSet = MeasurableSet\n\u22a2 { MeasurableSet' := MeasurableSet'\u271d\u00b9, measurableSet_empty := measurableSet_empty\u271d\u00b9,\n measurableSet_compl := measurableSet_compl\u271d\u00b9, measurableSet_iUnion := measurableSet_iUnion\u271d\u00b9 } =\n { MeasurableSet' := MeasurableSet'\u271d, measurableSet_empty := measurableSet_empty\u271d,\n measurableSet_compl := measurableSet_compl\u271d, measurableSet_iUnion := measurableSet_iUnion\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\na : \u03b1\ns : Set \u03b1\nh : MeasurableSet (insert a s)\nha : a \u2208 s\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nrwa [\u2190 insert_eq_of_mem ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\ns : Set \u03b1\nhs : Set.Countable s\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nrw [\u2190 biUnion_of_singleton s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\ns : Set \u03b1\nhs : Set.Countable s\n\u22a2 MeasurableSet (\u22c3 (x : \u03b1) (_ : x \u2208 s), {x})\n[PROOFSTEP]\nexact .biUnion hs fun b _ => .singleton b\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\np : Set \u03b1 \u2192 Prop\nh : \u2200 (s : Set \u03b1), p s \u2194 MeasurableSet s\n\u22a2 p \u2205\n[PROOFSTEP]\nsimpa only [h] using m.measurableSet_empty\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\np : Set \u03b1 \u2192 Prop\nh : \u2200 (s : Set \u03b1), p s \u2194 MeasurableSet s\n\u22a2 \u2200 (s : Set \u03b1), p s \u2192 p s\u1d9c\n[PROOFSTEP]\nsimpa only [h] using m.measurableSet_compl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\np : Set \u03b1 \u2192 Prop\nh : \u2200 (s : Set \u03b1), p s \u2194 MeasurableSet s\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1), (\u2200 (i : \u2115), p (f i)) \u2192 p (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nsimpa only [h] using m.measurableSet_iUnion\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\np : Set \u03b1 \u2192 Prop\nC : Set (Set \u03b1)\nhC : \u2200 (t : Set \u03b1), t \u2208 C \u2192 p t\nh_empty : p \u2205\nh_compl : \u2200 (t : Set \u03b1), p t \u2192 p t\u1d9c\nh_Union : \u2200 (f : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), p (f n)) \u2192 p (\u22c3 (i : \u2115), f i)\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 p s\n[PROOFSTEP]\ninduction hs\n[GOAL]\ncase basic\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\np : Set \u03b1 \u2192 Prop\nC : Set (Set \u03b1)\nhC : \u2200 (t : Set \u03b1), t \u2208 C \u2192 p t\nh_empty : p \u2205\nh_compl : \u2200 (t : Set \u03b1), p t \u2192 p t\u1d9c\nh_Union : \u2200 (f : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), p (f n)) \u2192 p (\u22c3 (i : \u2115), f i)\ns u\u271d : Set \u03b1\na\u271d : u\u271d \u2208 C\n\u22a2 p u\u271d\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\np : Set \u03b1 \u2192 Prop\nC : Set (Set \u03b1)\nhC : \u2200 (t : Set \u03b1), t \u2208 C \u2192 p t\nh_empty : p \u2205\nh_compl : \u2200 (t : Set \u03b1), p t \u2192 p t\u1d9c\nh_Union : \u2200 (f : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), p (f n)) \u2192 p (\u22c3 (i : \u2115), f i)\ns : Set \u03b1\n\u22a2 p \u2205\ncase compl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\np : Set \u03b1 \u2192 Prop\nC : Set (Set \u03b1)\nhC : \u2200 (t : Set \u03b1), t \u2208 C \u2192 p t\nh_empty : p \u2205\nh_compl : \u2200 (t : Set \u03b1), p t \u2192 p t\u1d9c\nh_Union : \u2200 (f : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), p (f n)) \u2192 p (\u22c3 (i : \u2115), f i)\ns t\u271d : Set \u03b1\na\u271d : GenerateMeasurable C t\u271d\na_ih\u271d : p t\u271d\n\u22a2 p t\u271d\u1d9c\ncase iUnion\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\np : Set \u03b1 \u2192 Prop\nC : Set (Set \u03b1)\nhC : \u2200 (t : Set \u03b1), t \u2208 C \u2192 p t\nh_empty : p \u2205\nh_compl : \u2200 (t : Set \u03b1), p t \u2192 p t\u1d9c\nh_Union : \u2200 (f : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), p (f n)) \u2192 p (\u22c3 (i : \u2115), f i)\ns : Set \u03b1\nf\u271d : \u2115 \u2192 Set \u03b1\na\u271d : \u2200 (n : \u2115), GenerateMeasurable C (f\u271d n)\na_ih\u271d : \u2200 (n : \u2115), p (f\u271d n)\n\u22a2 p (\u22c3 (i : \u2115), f\u271d i)\n[PROOFSTEP]\nexacts [hC _ \u2039_\u203a, h_empty, h_compl _ \u2039_\u203a, h_Union \u2039_\u203a \u2039_\u203a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nS : Set (Set \u03b1)\nx y : \u03b1\n\u22a2 (\u2200 (s : Set \u03b1), MeasurableSet s \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2194 \u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)\n[PROOFSTEP]\nrefine \u27e8fun H s hs \u21a6 H s (.basic s hs), fun H s \u21a6 ?_\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nS : Set (Set \u03b1)\nx y : \u03b1\nH : \u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)\ns : Set \u03b1\n\u22a2 MeasurableSet s \u2192 (x \u2208 s \u2194 y \u2208 s)\n[PROOFSTEP]\napply generateFrom_induction\n[GOAL]\ncase hC\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nS : Set (Set \u03b1)\nx y : \u03b1\nH : \u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)\ns : Set \u03b1\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 S \u2192 (x \u2208 t \u2194 y \u2208 t)\n[PROOFSTEP]\nexact H\n[GOAL]\ncase h_empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nS : Set (Set \u03b1)\nx y : \u03b1\nH : \u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)\ns : Set \u03b1\n\u22a2 x \u2208 \u2205 \u2194 y \u2208 \u2205\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_compl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nS : Set (Set \u03b1)\nx y : \u03b1\nH : \u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)\ns : Set \u03b1\n\u22a2 \u2200 (t : Set \u03b1), (x \u2208 t \u2194 y \u2208 t) \u2192 (x \u2208 t\u1d9c \u2194 y \u2208 t\u1d9c)\n[PROOFSTEP]\nexact fun _ \u21a6 Iff.not\n[GOAL]\ncase h_Union\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nS : Set (Set \u03b1)\nx y : \u03b1\nH : \u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)\ns : Set \u03b1\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), x \u2208 f n \u2194 y \u2208 f n) \u2192 (x \u2208 \u22c3 (i : \u2115), f i \u2194 y \u2208 \u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nintro f hf\n[GOAL]\ncase h_Union\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nS : Set (Set \u03b1)\nx y : \u03b1\nH : \u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)\ns : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : \u2200 (n : \u2115), x \u2208 f n \u2194 y \u2208 f n\n\u22a2 x \u2208 \u22c3 (i : \u2115), f i \u2194 y \u2208 \u22c3 (i : \u2115), f i\n[PROOFSTEP]\nsimp only [mem_iUnion, hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 {\u2205} \u2192 MeasurableSet t\n[PROOFSTEP]\nsimp [@MeasurableSet.empty \u03b1 \u22a5]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 {univ} \u2192 MeasurableSet t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nS : Set (Set \u03b1)\n\u22a2 generateFrom (insert univ S) = generateFrom S\n[PROOFSTEP]\nrw [insert_eq, \u2190 generateFrom_sup_generateFrom, generateFrom_singleton_univ, bot_sup_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nS : Set (Set \u03b1)\n\u22a2 generateFrom (insert \u2205 S) = generateFrom S\n[PROOFSTEP]\nrw [insert_eq, \u2190 generateFrom_sup_generateFrom, generateFrom_singleton_empty, bot_sup_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u s : Set \u03b1\n\u22a2 \u2200 (s : Set \u03b1), (fun s => s = \u2205 \u2228 s = univ) s \u2192 (fun s => s = \u2205 \u2228 s = univ) s\u1d9c\n[PROOFSTEP]\nsimp (config := { contextual := true }) [or_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nms : Set (MeasurableSpace \u03b1)\ns : Set \u03b1\n\u22a2 s \u2208 \u22c2\u2080 ((fun m => {t | MeasurableSet t}) '' ms) \u2194 \u2200 (m : MeasurableSpace \u03b1), m \u2208 ms \u2192 MeasurableSet s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9\u271d : Sort u_6\ns\u271d t u : Set \u03b1\n\u03b9 : Sort u_7\nm : \u03b9 \u2192 MeasurableSpace \u03b1\ns : Set \u03b1\n\u22a2 MeasurableSet s \u2194 \u2200 (i : \u03b9), MeasurableSet s\n[PROOFSTEP]\nrw [iInf, measurableSet_sInf, forall_range_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nms : Set (MeasurableSpace \u03b1)\ns : Set \u03b1\n\u22a2 MeasurableSet s \u2194 GenerateMeasurable {s | \u2203 m, m \u2208 ms \u2227 MeasurableSet s} s\n[PROOFSTEP]\nchange GenerateMeasurable (\u22c3\u2080 _) _ \u2194 _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nms : Set (MeasurableSpace \u03b1)\ns : Set \u03b1\n\u22a2 GenerateMeasurable (\u22c3\u2080 ((fun m => {t | MeasurableSet t}) '' ms)) s \u2194\n GenerateMeasurable {s | \u2203 m, m \u2208 ms \u2227 MeasurableSet s} s\n[PROOFSTEP]\nsimp [\u2190 setOf_exists]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9\u271d : Sort u_6\ns\u271d t u : Set \u03b1\n\u03b9 : Sort u_7\nm : \u03b9 \u2192 MeasurableSpace \u03b1\ns : Set \u03b1\n\u22a2 MeasurableSet s \u2194 GenerateMeasurable {s | \u2203 i, MeasurableSet s} s\n[PROOFSTEP]\nsimp only [iSup, measurableSet_sSup, exists_range_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns t u : Set \u03b1\nm : \u03b9 \u2192 MeasurableSpace \u03b1\n\u22a2 \u2a06 (n : \u03b9), m n = generateFrom {s | \u2203 n, MeasurableSet s}\n[PROOFSTEP]\next s\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : \u03b9 \u2192 MeasurableSpace \u03b1\ns : Set \u03b1\n\u22a2 MeasurableSet s \u2194 MeasurableSet s\n[PROOFSTEP]\nrw [measurableSet_iSup]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u_6\ns\u271d t u : Set \u03b1\nm : \u03b9 \u2192 MeasurableSpace \u03b1\ns : Set \u03b1\n\u22a2 GenerateMeasurable {s | \u2203 i, MeasurableSet s} s \u2194 MeasurableSet s\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.MeasurableSpaceDef", "llama_tokens": 9530, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.689305616785446, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4059245855021656}} {"text": "[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nV : OpensLeCover U\nA B : StructuredArrow V (pairwiseToOpensLeCover U)\n\u22a2 \u2203 l, List.Chain Zag A l \u2227 List.getLast (A :: l) (_ : A :: l \u2260 []) = B\n[PROOFSTEP]\nrcases A with \u27e8\u27e8\u27e8\u27e9\u27e9, \u27e8i\u27e9 | \u27e8i, j\u27e9, a\u27e9\n[GOAL]\ncase mk.mk.unit.single\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nV : OpensLeCover U\nB : StructuredArrow V (pairwiseToOpensLeCover U)\ni : \u03b9\na : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (single i)\n\u22a2 \u2203 l,\n List.Chain Zag { left := { as := PUnit.unit }, right := single i, hom := a } l \u2227\n List.getLast ({ left := { as := PUnit.unit }, right := single i, hom := a } :: l)\n (_ : { left := { as := PUnit.unit }, right := single i, hom := a } :: l \u2260 []) =\n B\n[PROOFSTEP]\nrcases B with \u27e8\u27e8\u27e8\u27e9\u27e9, \u27e8i'\u27e9 | \u27e8i', j'\u27e9, b\u27e9\n[GOAL]\ncase mk.mk.unit.pair\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nV : OpensLeCover U\nB : StructuredArrow V (pairwiseToOpensLeCover U)\ni j : \u03b9\na : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (Pairwise.pair i j)\n\u22a2 \u2203 l,\n List.Chain Zag { left := { as := PUnit.unit }, right := Pairwise.pair i j, hom := a } l \u2227\n List.getLast ({ left := { as := PUnit.unit }, right := Pairwise.pair i j, hom := a } :: l)\n (_ : { left := { as := PUnit.unit }, right := Pairwise.pair i j, hom := a } :: l \u2260 []) =\n B\n[PROOFSTEP]\nrcases B with \u27e8\u27e8\u27e8\u27e9\u27e9, \u27e8i'\u27e9 | \u27e8i', j'\u27e9, b\u27e9\n[GOAL]\ncase mk.mk.unit.single.mk.mk.unit.single\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nV : OpensLeCover U\ni : \u03b9\na : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (single i)\ni' : \u03b9\nb : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (single i')\n\u22a2 \u2203 l,\n List.Chain Zag { left := { as := PUnit.unit }, right := single i, hom := a } l \u2227\n List.getLast ({ left := { as := PUnit.unit }, right := single i, hom := a } :: l)\n (_ : { left := { as := PUnit.unit }, right := single i, hom := a } :: l \u2260 []) =\n { left := { as := PUnit.unit }, right := single i', hom := b }\n[PROOFSTEP]\nrefine'\n \u27e8[{ left := \u27e8\u27e8\u27e9\u27e9\n right := pair i i'\n hom := (le_inf a.le b.le).hom }, _], _, rfl\u27e9\n[GOAL]\ncase mk.mk.unit.single.mk.mk.unit.single\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nV : OpensLeCover U\ni : \u03b9\na : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (single i)\ni' : \u03b9\nb : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (single i')\n\u22a2 List.Chain Zag { left := { as := PUnit.unit }, right := single i, hom := a }\n [{ left := { as := PUnit.unit }, right := Pairwise.pair i i',\n hom :=\n LE.le.hom\n (_ :\n ((Functor.fromPUnit V).obj { as := PUnit.unit }).obj \u2264\n ((pairwiseToOpensLeCover U).obj (single i)).obj \u2293 ((pairwiseToOpensLeCover U).obj (single i')).obj) },\n { left := { as := PUnit.unit }, right := single i', hom := b }]\n[PROOFSTEP]\nexact\n List.Chain.cons\n (Or.inr\n \u27e8{ left := \ud835\udfd9 _\n right := left i i' }\u27e9)\n (List.Chain.cons\n (Or.inl\n \u27e8{ left := \ud835\udfd9 _\n right := right i i' }\u27e9)\n List.Chain.nil)\n[GOAL]\ncase mk.mk.unit.single.mk.mk.unit.pair\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nV : OpensLeCover U\ni : \u03b9\na : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (single i)\ni' j' : \u03b9\nb : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (Pairwise.pair i' j')\n\u22a2 \u2203 l,\n List.Chain Zag { left := { as := PUnit.unit }, right := single i, hom := a } l \u2227\n List.getLast ({ left := { as := PUnit.unit }, right := single i, hom := a } :: l)\n (_ : { left := { as := PUnit.unit }, right := single i, hom := a } :: l \u2260 []) =\n { left := { as := PUnit.unit }, right := Pairwise.pair i' j', hom := b }\n[PROOFSTEP]\nrefine'\n \u27e8[{ left := \u27e8\u27e8\u27e9\u27e9\n right := pair i' i\n hom := (le_inf (b.le.trans inf_le_left) a.le).hom },\n { left := \u27e8\u27e8\u27e9\u27e9\n right := single i'\n hom := (b.le.trans inf_le_left).hom }, _], _, rfl\u27e9\n[GOAL]\ncase mk.mk.unit.single.mk.mk.unit.pair\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nV : OpensLeCover U\ni : \u03b9\na : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (single i)\ni' j' : \u03b9\nb : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (Pairwise.pair i' j')\n\u22a2 List.Chain Zag { left := { as := PUnit.unit }, right := single i, hom := a }\n [{ left := { as := PUnit.unit }, right := Pairwise.pair i' i,\n hom :=\n LE.le.hom\n (_ :\n ((Functor.fromPUnit V).obj { as := PUnit.unit }).obj \u2264\n U i' \u2293 ((pairwiseToOpensLeCover U).obj (single i)).obj) },\n { left := { as := PUnit.unit }, right := single i',\n hom := LE.le.hom (_ : ((Functor.fromPUnit V).obj { as := PUnit.unit }).obj \u2264 U i') },\n { left := { as := PUnit.unit }, right := Pairwise.pair i' j', hom := b }]\n[PROOFSTEP]\nexact\n List.Chain.cons\n (Or.inr\n \u27e8{ left := \ud835\udfd9 _\n right := right i' i }\u27e9)\n (List.Chain.cons\n (Or.inl\n \u27e8{ left := \ud835\udfd9 _\n right := left i' i }\u27e9)\n (List.Chain.cons\n (Or.inr\n \u27e8{ left := \ud835\udfd9 _\n right := left i' j' }\u27e9)\n List.Chain.nil))\n[GOAL]\ncase mk.mk.unit.pair.mk.mk.unit.single\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nV : OpensLeCover U\ni j : \u03b9\na : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (Pairwise.pair i j)\ni' : \u03b9\nb : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (single i')\n\u22a2 \u2203 l,\n List.Chain Zag { left := { as := PUnit.unit }, right := Pairwise.pair i j, hom := a } l \u2227\n List.getLast ({ left := { as := PUnit.unit }, right := Pairwise.pair i j, hom := a } :: l)\n (_ : { left := { as := PUnit.unit }, right := Pairwise.pair i j, hom := a } :: l \u2260 []) =\n { left := { as := PUnit.unit }, right := single i', hom := b }\n[PROOFSTEP]\nrefine'\n \u27e8[{ left := \u27e8\u27e8\u27e9\u27e9\n right := single i\n hom := (a.le.trans inf_le_left).hom },\n { left := \u27e8\u27e8\u27e9\u27e9\n right := pair i i'\n hom := (le_inf (a.le.trans inf_le_left) b.le).hom }, _], _, rfl\u27e9\n[GOAL]\ncase mk.mk.unit.pair.mk.mk.unit.single\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nV : OpensLeCover U\ni j : \u03b9\na : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (Pairwise.pair i j)\ni' : \u03b9\nb : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (single i')\n\u22a2 List.Chain Zag { left := { as := PUnit.unit }, right := Pairwise.pair i j, hom := a }\n [{ left := { as := PUnit.unit }, right := single i,\n hom := LE.le.hom (_ : ((Functor.fromPUnit V).obj { as := PUnit.unit }).obj \u2264 U i) },\n { left := { as := PUnit.unit }, right := Pairwise.pair i i',\n hom :=\n LE.le.hom\n (_ :\n ((Functor.fromPUnit V).obj { as := PUnit.unit }).obj \u2264\n U i \u2293 ((pairwiseToOpensLeCover U).obj (single i')).obj) },\n { left := { as := PUnit.unit }, right := single i', hom := b }]\n[PROOFSTEP]\nexact\n List.Chain.cons\n (Or.inl\n \u27e8{ left := \ud835\udfd9 _\n right := left i j }\u27e9)\n (List.Chain.cons\n (Or.inr\n \u27e8{ left := \ud835\udfd9 _\n right := left i i' }\u27e9)\n (List.Chain.cons\n (Or.inl\n \u27e8{ left := \ud835\udfd9 _\n right := right i i' }\u27e9)\n List.Chain.nil))\n[GOAL]\ncase mk.mk.unit.pair.mk.mk.unit.pair\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nV : OpensLeCover U\ni j : \u03b9\na : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (Pairwise.pair i j)\ni' j' : \u03b9\nb : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (Pairwise.pair i' j')\n\u22a2 \u2203 l,\n List.Chain Zag { left := { as := PUnit.unit }, right := Pairwise.pair i j, hom := a } l \u2227\n List.getLast ({ left := { as := PUnit.unit }, right := Pairwise.pair i j, hom := a } :: l)\n (_ : { left := { as := PUnit.unit }, right := Pairwise.pair i j, hom := a } :: l \u2260 []) =\n { left := { as := PUnit.unit }, right := Pairwise.pair i' j', hom := b }\n[PROOFSTEP]\nrefine'\n \u27e8[{ left := \u27e8\u27e8\u27e9\u27e9\n right := single i\n hom := (a.le.trans inf_le_left).hom },\n { left := \u27e8\u27e8\u27e9\u27e9\n right := pair i i'\n hom := (le_inf (a.le.trans inf_le_left) (b.le.trans inf_le_left)).hom },\n { left := \u27e8\u27e8\u27e9\u27e9\n right := single i'\n hom := (b.le.trans inf_le_left).hom }, _], _, rfl\u27e9\n[GOAL]\ncase mk.mk.unit.pair.mk.mk.unit.pair\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nV : OpensLeCover U\ni j : \u03b9\na : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (Pairwise.pair i j)\ni' j' : \u03b9\nb : (Functor.fromPUnit V).obj { as := PUnit.unit } \u27f6 (pairwiseToOpensLeCover U).obj (Pairwise.pair i' j')\n\u22a2 List.Chain Zag { left := { as := PUnit.unit }, right := Pairwise.pair i j, hom := a }\n [{ left := { as := PUnit.unit }, right := single i,\n hom := LE.le.hom (_ : ((Functor.fromPUnit V).obj { as := PUnit.unit }).obj \u2264 U i) },\n { left := { as := PUnit.unit }, right := Pairwise.pair i i',\n hom := LE.le.hom (_ : ((Functor.fromPUnit V).obj { as := PUnit.unit }).obj \u2264 U i \u2293 U i') },\n { left := { as := PUnit.unit }, right := single i',\n hom := LE.le.hom (_ : ((Functor.fromPUnit V).obj { as := PUnit.unit }).obj \u2264 U i') },\n { left := { as := PUnit.unit }, right := Pairwise.pair i' j', hom := b }]\n[PROOFSTEP]\nexact\n List.Chain.cons\n (Or.inl\n \u27e8{ left := \ud835\udfd9 _\n right := left i j }\u27e9)\n (List.Chain.cons\n (Or.inr\n \u27e8{ left := \ud835\udfd9 _\n right := left i i' }\u27e9)\n (List.Chain.cons\n (Or.inl\n \u27e8{ left := \ud835\udfd9 _\n right := right i i' }\u27e9)\n (List.Chain.cons\n (Or.inr\n \u27e8{ left := \ud835\udfd9 _\n right := left i' j' }\u27e9)\n List.Chain.nil)))\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 (X_1 : CategoryTheory.Pairwise \u03b9) \u2192\n (diagram U).obj X_1 \u27f6 (pairwiseToOpensLeCover U \u22d9 fullSubcategoryInclusion fun V => \u2203 i, V \u2264 U i).obj X_1\n[PROOFSTEP]\nrintro (i | \u27e8i, j\u27e9)\n[GOAL]\ncase single\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\ni : \u03b9\n\u22a2 (diagram U).obj (single i) \u27f6\n (pairwiseToOpensLeCover U \u22d9 fullSubcategoryInclusion fun V => \u2203 i, V \u2264 U i).obj (single i)\n[PROOFSTEP]\nexact \ud835\udfd9 _\n[GOAL]\ncase pair\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\ni j : \u03b9\n\u22a2 (diagram U).obj (Pairwise.pair i j) \u27f6\n (pairwiseToOpensLeCover U \u22d9 fullSubcategoryInclusion fun V => \u2203 i, V \u2264 U i).obj (Pairwise.pair i j)\n[PROOFSTEP]\nexact \ud835\udfd9 _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 (X_1 : CategoryTheory.Pairwise \u03b9) \u2192\n (pairwiseToOpensLeCover U \u22d9 fullSubcategoryInclusion fun V => \u2203 i, V \u2264 U i).obj X_1 \u27f6 (diagram U).obj X_1\n[PROOFSTEP]\nrintro (i | \u27e8i, j\u27e9)\n[GOAL]\ncase single\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\ni : \u03b9\n\u22a2 (pairwiseToOpensLeCover U \u22d9 fullSubcategoryInclusion fun V => \u2203 i, V \u2264 U i).obj (single i) \u27f6\n (diagram U).obj (single i)\n[PROOFSTEP]\nexact \ud835\udfd9 _\n[GOAL]\ncase pair\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\ni j : \u03b9\n\u22a2 (pairwiseToOpensLeCover U \u22d9 fullSubcategoryInclusion fun V => \u2203 i, V \u2264 U i).obj (Pairwise.pair i j) \u27f6\n (diagram U).obj (Pairwise.pair i j)\n[PROOFSTEP]\nexact \ud835\udfd9 _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 \u2200 (j : (CategoryTheory.Pairwise \u03b9)\u1d52\u1d56),\n NatTrans.app (Cocone.op (cocone U)).\u03c0 j =\n (Iso.refl (Cocone.op (cocone U)).pt).hom \u226b\n NatTrans.app\n ((Cones.postcomposeEquivalence (NatIso.op (pairwiseDiagramIso U))).functor.obj\n (Cone.whisker (pairwiseToOpensLeCover U).op (Cocone.op (opensLeCoverCocone U)))).\u03c0\n j\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u22a2 IsSheaf F \u2194 IsSheafPairwiseIntersections F\n[PROOFSTEP]\nrw [isSheaf_iff_isSheafOpensLeCover, isSheafOpensLeCover_iff_isSheafPairwiseIntersections]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u22a2 IsSheaf F \u2194 IsSheafPreservesLimitPairwiseIntersections F\n[PROOFSTEP]\nrw [isSheaf_iff_isSheafPairwiseIntersections]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u22a2 IsSheafPairwiseIntersections F \u2194 IsSheafPreservesLimitPairwiseIntersections F\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u22a2 IsSheafPairwiseIntersections F \u2192 IsSheafPreservesLimitPairwiseIntersections F\n[PROOFSTEP]\nintro h \u03b9 U\n[GOAL]\ncase mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Nonempty (PreservesLimit (Pairwise.diagram U).op F)\n[PROOFSTEP]\nexact \u27e8preservesLimitOfPreservesLimitCone (Pairwise.coconeIsColimit U).op (h U).some\u27e9\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u22a2 IsSheafPreservesLimitPairwiseIntersections F \u2192 IsSheafPairwiseIntersections F\n[PROOFSTEP]\nintro h \u03b9 U\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\nh : IsSheafPreservesLimitPairwiseIntersections F\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Nonempty (IsLimit (F.mapCone (Cocone.op (Pairwise.cocone U))))\n[PROOFSTEP]\nhaveI := (h U).some\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\nh : IsSheafPreservesLimitPairwiseIntersections F\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nthis : PreservesLimit (Pairwise.diagram U).op F\n\u22a2 Nonempty (IsLimit (F.mapCone (Cocone.op (Pairwise.cocone U))))\n[PROOFSTEP]\nexact \u27e8PreservesLimit.preserves (Pairwise.coconeIsColimit U).op\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\n\u22a2 F.val.map (homOfLE (_ : U \u2264 U \u2294 V)).op \u226b F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op =\n F.val.map (homOfLE (_ : V \u2264 U \u2294 V)).op \u226b F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op\n[PROOFSTEP]\nrw [\u2190 F.1.map_comp, \u2190 F.1.map_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\n\u22a2 F.val.map ((homOfLE (_ : U \u2264 U \u2294 V)).op \u226b (homOfLE (_ : U \u2293 V \u2264 U)).op) =\n F.val.map ((homOfLE (_ : V \u2264 U \u2294 V)).op \u226b (homOfLE (_ : U \u2293 V \u2264 V)).op)\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u22a2 s.pt \u27f6 F.val.obj (op (U \u2294 V))\n[PROOFSTEP]\nlet \u03b9 : ULift.{w} WalkingPair \u2192 Opens X := fun j => WalkingPair.casesOn j.down U V\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\n\u22a2 s.pt \u27f6 F.val.obj (op (U \u2294 V))\n[PROOFSTEP]\nhave h\u03b9 : U \u2294 V = iSup \u03b9 := by\n ext\n rw [Opens.coe_iSup, Set.mem_iUnion]\n constructor\n \u00b7 rintro (h | h)\n exacts [\u27e8\u27e8WalkingPair.left\u27e9, h\u27e9, \u27e8\u27e8WalkingPair.right\u27e9, h\u27e9]\n \u00b7 rintro \u27e8\u27e8_ | _\u27e9, h\u27e9\n exacts [Or.inl h, Or.inr h]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\n\u22a2 U \u2294 V = iSup \u03b9\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nx\u271d : \u2191X\n\u22a2 x\u271d \u2208 \u2191(U \u2294 V) \u2194 x\u271d \u2208 \u2191(iSup \u03b9)\n[PROOFSTEP]\nrw [Opens.coe_iSup, Set.mem_iUnion]\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nx\u271d : \u2191X\n\u22a2 x\u271d \u2208 \u2191(U \u2294 V) \u2194 \u2203 i, x\u271d \u2208 \u2191(WalkingPair.casesOn i.down U V)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h.mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nx\u271d : \u2191X\n\u22a2 x\u271d \u2208 \u2191(U \u2294 V) \u2192 \u2203 i, x\u271d \u2208 \u2191(WalkingPair.casesOn i.down U V)\n[PROOFSTEP]\nrintro (h | h)\n[GOAL]\ncase h.h.mp.inl\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nx\u271d : \u2191X\nh : x\u271d \u2208 \u2191U\n\u22a2 \u2203 i, x\u271d \u2208 \u2191(WalkingPair.casesOn i.down U V)\ncase h.h.mp.inr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nx\u271d : \u2191X\nh : x\u271d \u2208 \u2191V\n\u22a2 \u2203 i, x\u271d \u2208 \u2191(WalkingPair.casesOn i.down U V)\n[PROOFSTEP]\nexacts [\u27e8\u27e8WalkingPair.left\u27e9, h\u27e9, \u27e8\u27e8WalkingPair.right\u27e9, h\u27e9]\n[GOAL]\ncase h.h.mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nx\u271d : \u2191X\n\u22a2 (\u2203 i, x\u271d \u2208 \u2191(WalkingPair.casesOn i.down U V)) \u2192 x\u271d \u2208 \u2191(U \u2294 V)\n[PROOFSTEP]\nrintro \u27e8\u27e8_ | _\u27e9, h\u27e9\n[GOAL]\ncase h.h.mpr.intro.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nx\u271d : \u2191X\nh : x\u271d \u2208 \u2191(WalkingPair.casesOn { down := WalkingPair.left }.down U V)\n\u22a2 x\u271d \u2208 \u2191(U \u2294 V)\ncase h.h.mpr.intro.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nx\u271d : \u2191X\nh : x\u271d \u2208 \u2191(WalkingPair.casesOn { down := WalkingPair.right }.down U V)\n\u22a2 x\u271d \u2208 \u2191(U \u2294 V)\n[PROOFSTEP]\nexacts [Or.inl h, Or.inr h]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 s.pt \u27f6 F.val.obj (op (U \u2294 V))\n[PROOFSTEP]\nrefine'\n (F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 \u03b9).some.lift\n \u27e8s.pt,\n { app := _\n naturality := _ }\u27e9 \u226b\n F.1.map (eqToHom h\u03b9).op\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 (X_1 : (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56) \u2192\n ((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).obj X_1 \u27f6\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).obj X_1\n[PROOFSTEP]\nrintro ((_ | _) | (_ | _))\n[GOAL]\ncase refine'_1.mk.single.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 ((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).obj\n { unop := Pairwise.single { down := WalkingPair.left } } \u27f6\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).obj { unop := Pairwise.single { down := WalkingPair.left } }\ncase refine'_1.mk.single.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 ((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).obj\n { unop := Pairwise.single { down := WalkingPair.right } } \u27f6\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).obj { unop := Pairwise.single { down := WalkingPair.right } }\ncase refine'_1.mk.pair.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\na\u271d : ULift WalkingPair\n\u22a2 ((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).obj\n { unop := Pairwise.pair { down := WalkingPair.left } a\u271d } \u27f6\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).obj { unop := Pairwise.pair { down := WalkingPair.left } a\u271d }\ncase refine'_1.mk.pair.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\na\u271d : ULift WalkingPair\n\u22a2 ((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).obj\n { unop := Pairwise.pair { down := WalkingPair.right } a\u271d } \u27f6\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).obj { unop := Pairwise.pair { down := WalkingPair.right } a\u271d }\n[PROOFSTEP]\nexacts [s.fst, s.snd, s.fst \u226b F.1.map (homOfLE inf_le_left).op, s.snd \u226b F.1.map (homOfLE inf_le_left).op]\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 \u2200 \u2983X_1 Y : (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56\u2984 (f : X_1 \u27f6 Y),\n (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map f \u226b\n Opposite.casesOn Y fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn X_1 fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map f\n[PROOFSTEP]\nrintro \u27e8i\u27e9 \u27e8j\u27e9 f\n[GOAL]\ncase refine'_2.mk.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ni j : CategoryTheory.Pairwise (ULift WalkingPair)\nf : { unop := i } \u27f6 { unop := j }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map f \u226b\n Opposite.casesOn { unop := j } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := i } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map f\n[PROOFSTEP]\nlet g : j \u27f6 i := f.unop\n[GOAL]\ncase refine'_2.mk.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ni j : CategoryTheory.Pairwise (ULift WalkingPair)\nf : { unop := i } \u27f6 { unop := j }\ng : j \u27f6 i := f.unop\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map f \u226b\n Opposite.casesOn { unop := j } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := i } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map f\n[PROOFSTEP]\nhave : f = g.op := rfl\n[GOAL]\ncase refine'_2.mk.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ni j : CategoryTheory.Pairwise (ULift WalkingPair)\nf : { unop := i } \u27f6 { unop := j }\ng : j \u27f6 i := f.unop\nthis : f = g.op\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map f \u226b\n Opposite.casesOn { unop := j } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := i } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map f\n[PROOFSTEP]\nclear_value g\n[GOAL]\ncase refine'_2.mk.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ni j : CategoryTheory.Pairwise (ULift WalkingPair)\nf : { unop := i } \u27f6 { unop := j }\ng : j \u27f6 i\nthis : f = g.op\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map f \u226b\n Opposite.casesOn { unop := j } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := i } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map f\n[PROOFSTEP]\nsubst this\n[GOAL]\ncase refine'_2.mk.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ni j : CategoryTheory.Pairwise (ULift WalkingPair)\ng : j \u27f6 i\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := j } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := i } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases i with (\u27e8\u27e8_ | _\u27e9\u27e9 | \u27e8\u27e8_ | _\u27e9, \u27e8_\u27e9\u27e9)\n[GOAL]\ncase refine'_2.mk.mk.single.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\nj : CategoryTheory.Pairwise (ULift WalkingPair)\ng : j \u27f6 Pairwise.single { down := WalkingPair.left }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := j } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases j with (\u27e8\u27e8_ | _\u27e9\u27e9 | \u27e8\u27e8_ | _\u27e9, \u27e8_\u27e9\u27e9)\n[GOAL]\ncase refine'_2.mk.mk.single.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\nj : CategoryTheory.Pairwise (ULift WalkingPair)\ng : j \u27f6 Pairwise.single { down := WalkingPair.right }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := j } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases j with (\u27e8\u27e8_ | _\u27e9\u27e9 | \u27e8\u27e8_ | _\u27e9, \u27e8_\u27e9\u27e9)\n[GOAL]\ncase refine'_2.mk.mk.pair.up.left.up\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\nj : CategoryTheory.Pairwise (ULift WalkingPair)\ndown\u271d : WalkingPair\ng : j \u27f6 Pairwise.pair { down := WalkingPair.left } { down := down\u271d }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := j } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.left } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases j with (\u27e8\u27e8_ | _\u27e9\u27e9 | \u27e8\u27e8_ | _\u27e9, \u27e8_\u27e9\u27e9)\n[GOAL]\ncase refine'_2.mk.mk.pair.up.right.up\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\nj : CategoryTheory.Pairwise (ULift WalkingPair)\ndown\u271d : WalkingPair\ng : j \u27f6 Pairwise.pair { down := WalkingPair.right } { down := down\u271d }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := j } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.right } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases j with (\u27e8\u27e8_ | _\u27e9\u27e9 | \u27e8\u27e8_ | _\u27e9, \u27e8_\u27e9\u27e9)\n[GOAL]\ncase refine'_2.mk.mk.single.up.left.single.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ng : Pairwise.single { down := WalkingPair.left } \u27f6 Pairwise.single { down := WalkingPair.left }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.single.up.left.single.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ng : Pairwise.single { down := WalkingPair.right } \u27f6 Pairwise.single { down := WalkingPair.left }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.single.up.left.pair.up.left.up\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\ng : Pairwise.pair { down := WalkingPair.left } { down := down\u271d } \u27f6 Pairwise.single { down := WalkingPair.left }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.left } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.single.up.left.pair.up.right.up\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\ng : Pairwise.pair { down := WalkingPair.right } { down := down\u271d } \u27f6 Pairwise.single { down := WalkingPair.left }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.right } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.single.up.right.single.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ng : Pairwise.single { down := WalkingPair.left } \u27f6 Pairwise.single { down := WalkingPair.right }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.single.up.right.single.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ng : Pairwise.single { down := WalkingPair.right } \u27f6 Pairwise.single { down := WalkingPair.right }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.single.up.right.pair.up.left.up\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\ng : Pairwise.pair { down := WalkingPair.left } { down := down\u271d } \u27f6 Pairwise.single { down := WalkingPair.right }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.left } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.single.up.right.pair.up.right.up\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\ng : Pairwise.pair { down := WalkingPair.right } { down := down\u271d } \u27f6 Pairwise.single { down := WalkingPair.right }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.right } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.pair.up.left.up.single.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\ng : Pairwise.single { down := WalkingPair.left } \u27f6 Pairwise.pair { down := WalkingPair.left } { down := down\u271d }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.left } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.pair.up.left.up.single.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\ng : Pairwise.single { down := WalkingPair.right } \u27f6 Pairwise.pair { down := WalkingPair.left } { down := down\u271d }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.left } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.pair.up.left.up.pair.up.left.up\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d\u00b9 down\u271d : WalkingPair\ng :\n Pairwise.pair { down := WalkingPair.left } { down := down\u271d } \u27f6\n Pairwise.pair { down := WalkingPair.left } { down := down\u271d\u00b9 }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.left } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.left } { down := down\u271d\u00b9 } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.pair.up.left.up.pair.up.right.up\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d\u00b9 down\u271d : WalkingPair\ng :\n Pairwise.pair { down := WalkingPair.right } { down := down\u271d } \u27f6\n Pairwise.pair { down := WalkingPair.left } { down := down\u271d\u00b9 }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.right } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.left } { down := down\u271d\u00b9 } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.pair.up.right.up.single.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\ng : Pairwise.single { down := WalkingPair.left } \u27f6 Pairwise.pair { down := WalkingPair.right } { down := down\u271d }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.right } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.pair.up.right.up.single.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\ng : Pairwise.single { down := WalkingPair.right } \u27f6 Pairwise.pair { down := WalkingPair.right } { down := down\u271d }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.right } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.pair.up.right.up.pair.up.left.up\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d\u00b9 down\u271d : WalkingPair\ng :\n Pairwise.pair { down := WalkingPair.left } { down := down\u271d } \u27f6\n Pairwise.pair { down := WalkingPair.right } { down := down\u271d\u00b9 }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.left } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.right } { down := down\u271d\u00b9 } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.pair.up.right.up.pair.up.right.up\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d\u00b9 down\u271d : WalkingPair\ng :\n Pairwise.pair { down := WalkingPair.right } { down := down\u271d } \u27f6\n Pairwise.pair { down := WalkingPair.right } { down := down\u271d\u00b9 }\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map g.op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.right } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.right } { down := down\u271d\u00b9 } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map g.op\n[PROOFSTEP]\nrcases g with \u27e8\u27e9\n[GOAL]\ncase refine'_2.mk.mk.single.up.left.single.up.left.id_single\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map\n (Pairwise.Hom.id_single { down := WalkingPair.left }).op \u226b\n Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map (Pairwise.Hom.id_single { down := WalkingPair.left }).op\n[PROOFSTEP]\ndsimp [Pairwise.diagram]\n[GOAL]\ncase refine'_2.mk.mk.single.up.left.pair.up.left.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map\n (Pairwise.Hom.left { down := WalkingPair.left } { down := down\u271d }).op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.left } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map (Pairwise.Hom.left { down := WalkingPair.left } { down := down\u271d }).op\n[PROOFSTEP]\ndsimp [Pairwise.diagram]\n[GOAL]\ncase refine'_2.mk.mk.single.up.left.pair.up.left.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map\n (Pairwise.Hom.right { down := WalkingPair.left } { down := WalkingPair.left }).op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.left } { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map\n (Pairwise.Hom.right { down := WalkingPair.left } { down := WalkingPair.left }).op\n[PROOFSTEP]\ndsimp [Pairwise.diagram]\n[GOAL]\ncase refine'_2.mk.mk.single.up.left.pair.up.right.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map\n (Pairwise.Hom.right { down := WalkingPair.right } { down := WalkingPair.left }).op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.right } { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.left } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map\n (Pairwise.Hom.right { down := WalkingPair.right } { down := WalkingPair.left }).op\n[PROOFSTEP]\ndsimp [Pairwise.diagram]\n[GOAL]\ncase refine'_2.mk.mk.single.up.right.single.up.right.id_single\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map\n (Pairwise.Hom.id_single { down := WalkingPair.right }).op \u226b\n Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map (Pairwise.Hom.id_single { down := WalkingPair.right }).op\n[PROOFSTEP]\ndsimp [Pairwise.diagram]\n[GOAL]\ncase refine'_2.mk.mk.single.up.right.pair.up.left.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map\n (Pairwise.Hom.right { down := WalkingPair.left } { down := WalkingPair.right }).op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.left } { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map\n (Pairwise.Hom.right { down := WalkingPair.left } { down := WalkingPair.right }).op\n[PROOFSTEP]\ndsimp [Pairwise.diagram]\n[GOAL]\ncase refine'_2.mk.mk.single.up.right.pair.up.right.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map\n (Pairwise.Hom.left { down := WalkingPair.right } { down := down\u271d }).op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.right } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map (Pairwise.Hom.left { down := WalkingPair.right } { down := down\u271d }).op\n[PROOFSTEP]\ndsimp [Pairwise.diagram]\n[GOAL]\ncase refine'_2.mk.mk.single.up.right.pair.up.right.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map\n (Pairwise.Hom.right { down := WalkingPair.right } { down := WalkingPair.right }).op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.right } { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.single { down := WalkingPair.right } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map\n (Pairwise.Hom.right { down := WalkingPair.right } { down := WalkingPair.right }).op\n[PROOFSTEP]\ndsimp [Pairwise.diagram]\n[GOAL]\ncase refine'_2.mk.mk.pair.up.left.up.pair.up.left.up.id_pair\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map\n (Pairwise.Hom.id_pair { down := WalkingPair.left } { down := down\u271d }).op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.left } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.left } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map\n (Pairwise.Hom.id_pair { down := WalkingPair.left } { down := down\u271d }).op\n[PROOFSTEP]\ndsimp [Pairwise.diagram]\n[GOAL]\ncase refine'_2.mk.mk.pair.up.right.up.pair.up.right.up.id_pair\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\n\u22a2 (((Functor.const (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56).obj s.pt).map\n (Pairwise.Hom.id_pair { down := WalkingPair.right } { down := down\u271d }).op \u226b\n Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.right } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) =\n (Opposite.casesOn { unop := Pairwise.pair { down := WalkingPair.right } { down := down\u271d } } fun unop =>\n Pairwise.casesOn unop\n (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 \u03b9 a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 \u03b9 a_1 \u2264 V)).op)) \u226b\n ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).map\n (Pairwise.Hom.id_pair { down := WalkingPair.right } { down := down\u271d }).op\n[PROOFSTEP]\ndsimp [Pairwise.diagram]\n[GOAL]\ncase refine'_2.mk.mk.single.up.left.single.up.left.id_single\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 \ud835\udfd9 s.pt \u226b PullbackCone.fst s = PullbackCone.fst s \u226b (presheaf F).map (\ud835\udfd9 (op U))\n[PROOFSTEP]\nsimp only [Category.id_comp, s.condition, CategoryTheory.Functor.map_id, Category.comp_id]\n[GOAL]\ncase refine'_2.mk.mk.single.up.left.pair.up.left.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\n\u22a2 \ud835\udfd9 s.pt \u226b PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 WalkingPair.rec U V down\u271d \u2264 U)).op =\n PullbackCone.fst s \u226b\n (presheaf F).map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2293 WalkingPair.rec U V { down := down\u271d }.down \u2264\n WalkingPair.rec U V { down := WalkingPair.left }.down)).op\n[PROOFSTEP]\nsimp only [Category.id_comp, s.condition, CategoryTheory.Functor.map_id, Category.comp_id]\n[GOAL]\ncase refine'_2.mk.mk.single.up.left.pair.up.left.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 \ud835\udfd9 s.pt \u226b PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 U \u2264 U)).op =\n PullbackCone.fst s \u226b\n (presheaf F).map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2293\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2264\n WalkingPair.rec U V { down := WalkingPair.left }.down)).op\n[PROOFSTEP]\nsimp only [Category.id_comp, s.condition, CategoryTheory.Functor.map_id, Category.comp_id]\n[GOAL]\ncase refine'_2.mk.mk.single.up.left.pair.up.right.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 \ud835\udfd9 s.pt \u226b PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 U \u2264 V)).op =\n PullbackCone.fst s \u226b\n (presheaf F).map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2293\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2264\n WalkingPair.rec U V { down := WalkingPair.left }.down)).op\n[PROOFSTEP]\nsimp only [Category.id_comp, s.condition, CategoryTheory.Functor.map_id, Category.comp_id]\n[GOAL]\ncase refine'_2.mk.mk.single.up.right.single.up.right.id_single\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 \ud835\udfd9 s.pt \u226b PullbackCone.snd s = PullbackCone.snd s \u226b (presheaf F).map (\ud835\udfd9 (op V))\n[PROOFSTEP]\nsimp only [Category.id_comp, s.condition, CategoryTheory.Functor.map_id, Category.comp_id]\n[GOAL]\ncase refine'_2.mk.mk.single.up.right.pair.up.left.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 \ud835\udfd9 s.pt \u226b PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op =\n PullbackCone.snd s \u226b\n (presheaf F).map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2293\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2264\n WalkingPair.rec U V { down := WalkingPair.right }.down)).op\n[PROOFSTEP]\nsimp only [Category.id_comp, s.condition, CategoryTheory.Functor.map_id, Category.comp_id]\n[GOAL]\ncase refine'_2.mk.mk.single.up.right.pair.up.right.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\n\u22a2 \ud835\udfd9 s.pt \u226b PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 WalkingPair.rec U V down\u271d \u2264 V)).op =\n PullbackCone.snd s \u226b\n (presheaf F).map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2293 WalkingPair.rec U V { down := down\u271d }.down \u2264\n WalkingPair.rec U V { down := WalkingPair.right }.down)).op\n[PROOFSTEP]\nsimp only [Category.id_comp, s.condition, CategoryTheory.Functor.map_id, Category.comp_id]\n[GOAL]\ncase refine'_2.mk.mk.single.up.right.pair.up.right.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 \ud835\udfd9 s.pt \u226b PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 V \u2264 V)).op =\n PullbackCone.snd s \u226b\n (presheaf F).map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2293\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2264\n WalkingPair.rec U V { down := WalkingPair.right }.down)).op\n[PROOFSTEP]\nsimp only [Category.id_comp, s.condition, CategoryTheory.Functor.map_id, Category.comp_id]\n[GOAL]\ncase refine'_2.mk.mk.pair.up.left.up.pair.up.left.up.id_pair\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\n\u22a2 \ud835\udfd9 s.pt \u226b PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 WalkingPair.rec U V down\u271d \u2264 U)).op =\n (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 WalkingPair.rec U V down\u271d \u2264 U)).op) \u226b\n (presheaf F).map (\ud835\udfd9 (op (U \u2293 WalkingPair.rec U V down\u271d)))\n[PROOFSTEP]\nsimp only [Category.id_comp, s.condition, CategoryTheory.Functor.map_id, Category.comp_id]\n[GOAL]\ncase refine'_2.mk.mk.pair.up.right.up.pair.up.right.up.id_pair\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ndown\u271d : WalkingPair\n\u22a2 \ud835\udfd9 s.pt \u226b PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 WalkingPair.rec U V down\u271d \u2264 V)).op =\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 WalkingPair.rec U V down\u271d \u2264 V)).op) \u226b\n (presheaf F).map (\ud835\udfd9 (op (V \u2293 WalkingPair.rec U V down\u271d)))\n[PROOFSTEP]\nsimp only [Category.id_comp, s.condition, CategoryTheory.Functor.map_id, Category.comp_id]\n[GOAL]\ncase refine'_2.mk.mk.single.up.left.pair.up.right.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 U \u2264 V)).op =\n PullbackCone.fst s \u226b\n (presheaf F).map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2293\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2264\n WalkingPair.rec U V { down := WalkingPair.left }.down)).op\n[PROOFSTEP]\nrw [\u2190 cancel_mono (F.1.map (eqToHom <| inf_comm : U \u2293 V \u27f6 _).op), Category.assoc, Category.assoc, \u2190 F.1.map_comp, \u2190\n F.1.map_comp]\n[GOAL]\ncase refine'_2.mk.mk.single.up.left.pair.up.right.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X := fun j => WalkingPair.casesOn j.down U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 PullbackCone.snd s \u226b F.val.map ((homOfLE (_ : V \u2293 U \u2264 V)).op \u226b (eqToHom (_ : U \u2293 V = V \u2293 U)).op) =\n PullbackCone.fst s \u226b\n F.val.map\n ((homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2293\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2264\n WalkingPair.rec U V { down := WalkingPair.left }.down)).op \u226b\n (eqToHom (_ : U \u2293 V = V \u2293 U)).op)\n[PROOFSTEP]\nexact s.condition.symm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u22a2 interUnionPullbackConeLift F U V s \u226b F.val.map (homOfLE (_ : U \u2264 U \u2294 V)).op = PullbackCone.fst s\n[PROOFSTEP]\nerw [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u22a2 IsLimit.lift\n (Nonempty.some\n (_ :\n Nonempty\n (IsLimit ((presheaf F).mapCone (Cocone.op (Pairwise.cocone fun j => WalkingPair.casesOn j.down U V))))))\n { pt := s.pt,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Opposite.casesOn X_1 fun unop =>\n Pairwise.casesOn unop\n (fun a =>\n ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down\n (PullbackCone.fst s \u226b\n F.val.map (homOfLE (_ : U \u2293 (fun j => WalkingPair.casesOn j.down U V) a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b\n F.val.map (homOfLE (_ : V \u2293 (fun j => WalkingPair.casesOn j.down U V) a_1 \u2264 V)).op) } \u226b\n F.val.map (eqToHom (_ : U \u2294 V = \u2a06 (j : ULift WalkingPair), WalkingPair.casesOn j.down U V)).op \u226b\n F.val.map (homOfLE (_ : U \u2264 U \u2294 V)).op =\n PullbackCone.fst s\n[PROOFSTEP]\nsimp_rw [\u2190 F.1.map_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u22a2 IsLimit.lift\n (Nonempty.some\n (_ :\n Nonempty\n (IsLimit ((presheaf F).mapCone (Cocone.op (Pairwise.cocone fun j => WalkingPair.casesOn j.down U V))))))\n { pt := s.pt,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Pairwise.rec (fun a => WalkingPair.rec (PullbackCone.fst s) (PullbackCone.snd s) a.down)\n (fun a a_1 =>\n WalkingPair.rec\n (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 WalkingPair.rec U V a_1.down \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 WalkingPair.rec U V a_1.down \u2264 V)).op) a.down)\n X_1.unop } \u226b\n F.val.map\n ((eqToHom (_ : U \u2294 V = \u2a06 (j : ULift WalkingPair), WalkingPair.casesOn j.down U V)).op \u226b\n (homOfLE (_ : U \u2264 U \u2294 V)).op) =\n PullbackCone.fst s\n[PROOFSTEP]\nexact\n (F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 _).some.fac _ <|\n op <| Pairwise.single <| ULift.up WalkingPair.left\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u22a2 interUnionPullbackConeLift F U V s \u226b F.val.map (homOfLE (_ : V \u2264 U \u2294 V)).op = PullbackCone.snd s\n[PROOFSTEP]\nerw [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u22a2 IsLimit.lift\n (Nonempty.some\n (_ :\n Nonempty\n (IsLimit ((presheaf F).mapCone (Cocone.op (Pairwise.cocone fun j => WalkingPair.casesOn j.down U V))))))\n { pt := s.pt,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Opposite.casesOn X_1 fun unop =>\n Pairwise.casesOn unop\n (fun a =>\n ULift.casesOn a fun down => WalkingPair.casesOn down (PullbackCone.fst s) (PullbackCone.snd s))\n fun a a_1 =>\n ULift.casesOn a fun down =>\n WalkingPair.casesOn down\n (PullbackCone.fst s \u226b\n F.val.map (homOfLE (_ : U \u2293 (fun j => WalkingPair.casesOn j.down U V) a_1 \u2264 U)).op)\n (PullbackCone.snd s \u226b\n F.val.map (homOfLE (_ : V \u2293 (fun j => WalkingPair.casesOn j.down U V) a_1 \u2264 V)).op) } \u226b\n F.val.map (eqToHom (_ : U \u2294 V = \u2a06 (j : ULift WalkingPair), WalkingPair.casesOn j.down U V)).op \u226b\n F.val.map (homOfLE (_ : V \u2264 U \u2294 V)).op =\n PullbackCone.snd s\n[PROOFSTEP]\nsimp_rw [\u2190 F.1.map_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u22a2 IsLimit.lift\n (Nonempty.some\n (_ :\n Nonempty\n (IsLimit ((presheaf F).mapCone (Cocone.op (Pairwise.cocone fun j => WalkingPair.casesOn j.down U V))))))\n { pt := s.pt,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Pairwise.rec (fun a => WalkingPair.rec (PullbackCone.fst s) (PullbackCone.snd s) a.down)\n (fun a a_1 =>\n WalkingPair.rec\n (PullbackCone.fst s \u226b F.val.map (homOfLE (_ : U \u2293 WalkingPair.rec U V a_1.down \u2264 U)).op)\n (PullbackCone.snd s \u226b F.val.map (homOfLE (_ : V \u2293 WalkingPair.rec U V a_1.down \u2264 V)).op) a.down)\n X_1.unop } \u226b\n F.val.map\n ((eqToHom (_ : U \u2294 V = \u2a06 (j : ULift WalkingPair), WalkingPair.casesOn j.down U V)).op \u226b\n (homOfLE (_ : V \u2264 U \u2294 V)).op) =\n PullbackCone.snd s\n[PROOFSTEP]\nexact\n (F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 _).some.fac _ <|\n op <| Pairwise.single <| ULift.up WalkingPair.right\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u22a2 IsLimit (interUnionPullbackCone F U V)\n[PROOFSTEP]\nlet \u03b9 : ULift.{w} WalkingPair \u2192 Opens X := fun \u27e8j\u27e9 => WalkingPair.casesOn j U V\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\n\u22a2 IsLimit (interUnionPullbackCone F U V)\n[PROOFSTEP]\nhave h\u03b9 : U \u2294 V = iSup \u03b9 := by\n ext\n rw [Opens.coe_iSup, Set.mem_iUnion]\n constructor\n \u00b7 rintro (h | h)\n exacts [\u27e8\u27e8WalkingPair.left\u27e9, h\u27e9, \u27e8\u27e8WalkingPair.right\u27e9, h\u27e9]\n \u00b7 rintro \u27e8\u27e8_ | _\u27e9, h\u27e9\n exacts [Or.inl h, Or.inr h]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\n\u22a2 U \u2294 V = iSup \u03b9\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nx\u271d : \u2191X\n\u22a2 x\u271d \u2208 \u2191(U \u2294 V) \u2194 x\u271d \u2208 \u2191(iSup \u03b9)\n[PROOFSTEP]\nrw [Opens.coe_iSup, Set.mem_iUnion]\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nx\u271d : \u2191X\n\u22a2 x\u271d \u2208 \u2191(U \u2294 V) \u2194\n \u2203 i,\n x\u271d \u2208\n \u2191(match i with\n | { down := j } => WalkingPair.casesOn j U V)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h.mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nx\u271d : \u2191X\n\u22a2 x\u271d \u2208 \u2191(U \u2294 V) \u2192\n \u2203 i,\n x\u271d \u2208\n \u2191(match i with\n | { down := j } => WalkingPair.casesOn j U V)\n[PROOFSTEP]\nrintro (h | h)\n[GOAL]\ncase h.h.mp.inl\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nx\u271d : \u2191X\nh : x\u271d \u2208 \u2191U\n\u22a2 \u2203 i,\n x\u271d \u2208\n \u2191(match i with\n | { down := j } => WalkingPair.casesOn j U V)\ncase h.h.mp.inr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nx\u271d : \u2191X\nh : x\u271d \u2208 \u2191V\n\u22a2 \u2203 i,\n x\u271d \u2208\n \u2191(match i with\n | { down := j } => WalkingPair.casesOn j U V)\n[PROOFSTEP]\nexacts [\u27e8\u27e8WalkingPair.left\u27e9, h\u27e9, \u27e8\u27e8WalkingPair.right\u27e9, h\u27e9]\n[GOAL]\ncase h.h.mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nx\u271d : \u2191X\n\u22a2 (\u2203 i,\n x\u271d \u2208\n \u2191(match i with\n | { down := j } => WalkingPair.casesOn j U V)) \u2192\n x\u271d \u2208 \u2191(U \u2294 V)\n[PROOFSTEP]\nrintro \u27e8\u27e8_ | _\u27e9, h\u27e9\n[GOAL]\ncase h.h.mpr.intro.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nx\u271d : \u2191X\nh :\n x\u271d \u2208\n \u2191(match { down := WalkingPair.left } with\n | { down := j } => WalkingPair.casesOn j U V)\n\u22a2 x\u271d \u2208 \u2191(U \u2294 V)\ncase h.h.mpr.intro.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nx\u271d : \u2191X\nh :\n x\u271d \u2208\n \u2191(match { down := WalkingPair.right } with\n | { down := j } => WalkingPair.casesOn j U V)\n\u22a2 x\u271d \u2208 \u2191(U \u2294 V)\n[PROOFSTEP]\nexacts [Or.inl h, Or.inr h]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 IsLimit (interUnionPullbackCone F U V)\n[PROOFSTEP]\napply PullbackCone.isLimitAux'\n[GOAL]\ncase create\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\n\u22a2 (s : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)) \u2192\n { l //\n l \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (interUnionPullbackCone F U V).pt},\n m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s \u2192 m = l }\n[PROOFSTEP]\nintro s\n[GOAL]\ncase create\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u22a2 { l //\n l \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (interUnionPullbackCone F U V).pt},\n m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s \u2192 m = l }\n[PROOFSTEP]\nuse interUnionPullbackConeLift F U V s\n[GOAL]\ncase property\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u22a2 interUnionPullbackConeLift F U V s \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s \u2227\n interUnionPullbackConeLift F U V s \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (interUnionPullbackCone F U V).pt},\n m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s \u2192\n m = interUnionPullbackConeLift F U V s\n[PROOFSTEP]\nrefine' \u27e8_, _, _\u27e9\n[GOAL]\ncase property.refine'_1\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u22a2 interUnionPullbackConeLift F U V s \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\n[PROOFSTEP]\napply interUnionPullbackConeLift_left\n[GOAL]\ncase property.refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u22a2 interUnionPullbackConeLift F U V s \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\n[PROOFSTEP]\napply interUnionPullbackConeLift_right\n[GOAL]\ncase property.refine'_3\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u22a2 \u2200 {m : s.pt \u27f6 (interUnionPullbackCone F U V).pt},\n m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s \u2192 m = interUnionPullbackConeLift F U V s\n[PROOFSTEP]\nintro m h\u2081 h\u2082\n[GOAL]\ncase property.refine'_3\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\n\u22a2 m = interUnionPullbackConeLift F U V s\n[PROOFSTEP]\nrw [\u2190 cancel_mono (F.1.map (eqToHom h\u03b9.symm).op)]\n[GOAL]\ncase property.refine'_3\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\n\u22a2 m \u226b F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op =\n interUnionPullbackConeLift F U V s \u226b F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op\n[PROOFSTEP]\napply (F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 \u03b9).some.hom_ext\n[GOAL]\ncase property.refine'_3\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\n\u22a2 \u2200 (j : (CategoryTheory.Pairwise (ULift WalkingPair))\u1d52\u1d56),\n (m \u226b F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op) \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0 j =\n (interUnionPullbackConeLift F U V s \u226b F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op) \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0 j\n[PROOFSTEP]\nrintro ((_ | _) | (_ | _))\n[GOAL]\ncase property.refine'_3.mk.single.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\n\u22a2 (m \u226b F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op) \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.single { down := WalkingPair.left } } =\n (interUnionPullbackConeLift F U V s \u226b F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op) \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.single { down := WalkingPair.left } }\n[PROOFSTEP]\nrw [Category.assoc, Category.assoc]\n[GOAL]\ncase property.refine'_3.mk.single.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\n\u22a2 (m \u226b F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op) \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.single { down := WalkingPair.right } } =\n (interUnionPullbackConeLift F U V s \u226b F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op) \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.single { down := WalkingPair.right } }\n[PROOFSTEP]\nrw [Category.assoc, Category.assoc]\n[GOAL]\ncase property.refine'_3.mk.pair.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 (m \u226b F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op) \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.pair { down := WalkingPair.left } a\u271d } =\n (interUnionPullbackConeLift F U V s \u226b F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op) \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.pair { down := WalkingPair.left } a\u271d }\n[PROOFSTEP]\nrw [Category.assoc, Category.assoc]\n[GOAL]\ncase property.refine'_3.mk.pair.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 (m \u226b F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op) \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.pair { down := WalkingPair.right } a\u271d } =\n (interUnionPullbackConeLift F U V s \u226b F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op) \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.pair { down := WalkingPair.right } a\u271d }\n[PROOFSTEP]\nrw [Category.assoc, Category.assoc]\n[GOAL]\ncase property.refine'_3.mk.single.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\n\u22a2 m \u226b\n F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.single { down := WalkingPair.left } } =\n interUnionPullbackConeLift F U V s \u226b\n F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.single { down := WalkingPair.left } }\n[PROOFSTEP]\nerw [\u2190 F.1.map_comp]\n[GOAL]\ncase property.refine'_3.mk.single.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\n\u22a2 m \u226b\n F.val.map\n ((eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app (Cocone.op (Pairwise.cocone \u03b9)).\u03c0 { unop := Pairwise.single { down := WalkingPair.left } }) =\n interUnionPullbackConeLift F U V s \u226b\n F.val.map\n ((eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app (Cocone.op (Pairwise.cocone \u03b9)).\u03c0 { unop := Pairwise.single { down := WalkingPair.left } })\n[PROOFSTEP]\nconvert h\u2081\n[GOAL]\ncase h.e'_3.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\ne_1\u271d :\n (s.pt \u27f6 ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).obj { unop := Pairwise.single { down := WalkingPair.left } }) =\n (s.pt \u27f6 F.val.obj (op U))\n\u22a2 interUnionPullbackConeLift F U V s \u226b\n F.val.map\n ((eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app (Cocone.op (Pairwise.cocone \u03b9)).\u03c0 { unop := Pairwise.single { down := WalkingPair.left } }) =\n PullbackCone.fst s\n[PROOFSTEP]\napply interUnionPullbackConeLift_left\n[GOAL]\ncase property.refine'_3.mk.single.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\n\u22a2 m \u226b\n F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.single { down := WalkingPair.right } } =\n interUnionPullbackConeLift F U V s \u226b\n F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.single { down := WalkingPair.right } }\n[PROOFSTEP]\nerw [\u2190 F.1.map_comp]\n[GOAL]\ncase property.refine'_3.mk.single.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\n\u22a2 m \u226b\n F.val.map\n ((eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app (Cocone.op (Pairwise.cocone \u03b9)).\u03c0 { unop := Pairwise.single { down := WalkingPair.right } }) =\n interUnionPullbackConeLift F U V s \u226b\n F.val.map\n ((eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app (Cocone.op (Pairwise.cocone \u03b9)).\u03c0 { unop := Pairwise.single { down := WalkingPair.right } })\n[PROOFSTEP]\nconvert h\u2082\n[GOAL]\ncase h.e'_3.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\ne_1\u271d :\n (s.pt \u27f6 ((Pairwise.diagram \u03b9).op \u22d9 presheaf F).obj { unop := Pairwise.single { down := WalkingPair.right } }) =\n (s.pt \u27f6 F.val.obj (op V))\n\u22a2 interUnionPullbackConeLift F U V s \u226b\n F.val.map\n ((eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app (Cocone.op (Pairwise.cocone \u03b9)).\u03c0 { unop := Pairwise.single { down := WalkingPair.right } }) =\n PullbackCone.snd s\n[PROOFSTEP]\napply interUnionPullbackConeLift_right\n[GOAL]\ncase property.refine'_3.mk.pair.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 m \u226b\n F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.pair { down := WalkingPair.left } a\u271d } =\n interUnionPullbackConeLift F U V s \u226b\n F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.pair { down := WalkingPair.left } a\u271d }\ncase property.refine'_3.mk.pair.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 m \u226b\n F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.pair { down := WalkingPair.right } a\u271d } =\n interUnionPullbackConeLift F U V s \u226b\n F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.pair { down := WalkingPair.right } a\u271d }\n[PROOFSTEP]\nall_goals\n dsimp only [Functor.op, Pairwise.cocone_\u03b9_app, Functor.mapCone_\u03c0_app, Cocone.op, Pairwise.cocone\u03b9App, unop_op,\n op_comp, NatTrans.op]\n simp_rw [F.1.map_comp, \u2190 Category.assoc]\n congr 1\n simp_rw [Category.assoc, \u2190 F.1.map_comp]\n[GOAL]\ncase property.refine'_3.mk.pair.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 m \u226b\n F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.pair { down := WalkingPair.left } a\u271d } =\n interUnionPullbackConeLift F U V s \u226b\n F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.pair { down := WalkingPair.left } a\u271d }\n[PROOFSTEP]\ndsimp only [Functor.op, Pairwise.cocone_\u03b9_app, Functor.mapCone_\u03c0_app, Cocone.op, Pairwise.cocone\u03b9App, unop_op, op_comp,\n NatTrans.op]\n[GOAL]\ncase property.refine'_3.mk.pair.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 m \u226b\n F.val.map (eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op \u226b\n (presheaf F).map\n ((homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op \u226b\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2293 WalkingPair.rec U V a\u271d.down \u2264\n WalkingPair.rec U V { down := WalkingPair.left }.down)).op) =\n interUnionPullbackConeLift F U V s \u226b\n F.val.map (eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op \u226b\n (presheaf F).map\n ((homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op \u226b\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2293 WalkingPair.rec U V a\u271d.down \u2264\n WalkingPair.rec U V { down := WalkingPair.left }.down)).op)\n[PROOFSTEP]\nsimp_rw [F.1.map_comp, \u2190 Category.assoc]\n[GOAL]\ncase property.refine'_3.mk.pair.up.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 ((m \u226b F.val.map (eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op) \u226b\n F.val.map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op) \u226b\n F.val.map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2293 WalkingPair.rec U V a\u271d.down \u2264\n WalkingPair.rec U V { down := WalkingPair.left }.down)).op =\n ((interUnionPullbackConeLift F U V s \u226b\n F.val.map (eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op) \u226b\n F.val.map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op) \u226b\n F.val.map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2293 WalkingPair.rec U V a\u271d.down \u2264\n WalkingPair.rec U V { down := WalkingPair.left }.down)).op\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase property.refine'_3.mk.pair.up.left.e_a\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 (m \u226b F.val.map (eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op) \u226b\n F.val.map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op =\n (interUnionPullbackConeLift F U V s \u226b\n F.val.map (eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op) \u226b\n F.val.map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op\n[PROOFSTEP]\nsimp_rw [Category.assoc, \u2190 F.1.map_comp]\n[GOAL]\ncase property.refine'_3.mk.pair.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 m \u226b\n F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.pair { down := WalkingPair.right } a\u271d } =\n interUnionPullbackConeLift F U V s \u226b\n F.val.map (eqToHom (_ : iSup \u03b9 = U \u2294 V)).op \u226b\n NatTrans.app ((presheaf F).mapCone (Cocone.op (Pairwise.cocone \u03b9))).\u03c0\n { unop := Pairwise.pair { down := WalkingPair.right } a\u271d }\n[PROOFSTEP]\ndsimp only [Functor.op, Pairwise.cocone_\u03b9_app, Functor.mapCone_\u03c0_app, Cocone.op, Pairwise.cocone\u03b9App, unop_op, op_comp,\n NatTrans.op]\n[GOAL]\ncase property.refine'_3.mk.pair.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 m \u226b\n F.val.map (eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op \u226b\n (presheaf F).map\n ((homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op \u226b\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2293 WalkingPair.rec U V a\u271d.down \u2264\n WalkingPair.rec U V { down := WalkingPair.right }.down)).op) =\n interUnionPullbackConeLift F U V s \u226b\n F.val.map (eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op \u226b\n (presheaf F).map\n ((homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op \u226b\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2293 WalkingPair.rec U V a\u271d.down \u2264\n WalkingPair.rec U V { down := WalkingPair.right }.down)).op)\n[PROOFSTEP]\nsimp_rw [F.1.map_comp, \u2190 Category.assoc]\n[GOAL]\ncase property.refine'_3.mk.pair.up.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 ((m \u226b F.val.map (eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op) \u226b\n F.val.map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op) \u226b\n F.val.map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2293 WalkingPair.rec U V a\u271d.down \u2264\n WalkingPair.rec U V { down := WalkingPair.right }.down)).op =\n ((interUnionPullbackConeLift F U V s \u226b\n F.val.map (eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op) \u226b\n F.val.map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op) \u226b\n F.val.map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2293 WalkingPair.rec U V a\u271d.down \u2264\n WalkingPair.rec U V { down := WalkingPair.right }.down)).op\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase property.refine'_3.mk.pair.up.right.e_a\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 (m \u226b F.val.map (eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op) \u226b\n F.val.map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op =\n (interUnionPullbackConeLift F U V s \u226b\n F.val.map (eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op) \u226b\n F.val.map\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op\n[PROOFSTEP]\nsimp_rw [Category.assoc, \u2190 F.1.map_comp]\n[GOAL]\ncase property.refine'_3.mk.pair.up.left.e_a\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 m \u226b\n F.val.map\n ((eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op \u226b\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op) =\n interUnionPullbackConeLift F U V s \u226b\n F.val.map\n ((eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op \u226b\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op)\n[PROOFSTEP]\nconvert h\u2081\n[GOAL]\ncase h.e'_3\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 interUnionPullbackConeLift F U V s \u226b\n F.val.map\n ((eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op \u226b\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.left }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op) =\n PullbackCone.fst s\n[PROOFSTEP]\napply interUnionPullbackConeLift_left\n[GOAL]\ncase property.refine'_3.mk.pair.up.right.e_a\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 m \u226b\n F.val.map\n ((eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op \u226b\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op) =\n interUnionPullbackConeLift F U V s \u226b\n F.val.map\n ((eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op \u226b\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op)\n[PROOFSTEP]\nconvert h\u2082\n[GOAL]\ncase h.e'_3\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns\u271d : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\n\u03b9 : ULift WalkingPair \u2192 Opens \u2191X :=\n fun x =>\n match x with\n | { down := j } => WalkingPair.casesOn j U V\nh\u03b9 : U \u2294 V = iSup \u03b9\ns : PullbackCone (F.val.map (homOfLE (_ : U \u2293 V \u2264 U)).op) (F.val.map (homOfLE (_ : U \u2293 V \u2264 V)).op)\nm : s.pt \u27f6 (interUnionPullbackCone F U V).pt\nh\u2081 : m \u226b PullbackCone.fst (interUnionPullbackCone F U V) = PullbackCone.fst s\nh\u2082 : m \u226b PullbackCone.snd (interUnionPullbackCone F U V) = PullbackCone.snd s\na\u271d : ULift WalkingPair\n\u22a2 interUnionPullbackConeLift F U V s \u226b\n F.val.map\n ((eqToHom (_ : \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down = U \u2294 V)).op \u226b\n (homOfLE\n (_ :\n WalkingPair.rec U V { down := WalkingPair.right }.down \u2264\n \u2a06 (x : ULift WalkingPair), WalkingPair.rec U V x.down)).op) =\n PullbackCone.snd s\n[PROOFSTEP]\napply interUnionPullbackConeLift_right\n", "meta": {"mathlib_filename": "Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections", "llama_tokens": 62412, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581626286834, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.40588471390777764}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u22a2 LowerSemicontinuousWithinAt f univ x \u2194 LowerSemicontinuousAt f x\n[PROOFSTEP]\nsimp [LowerSemicontinuousWithinAt, LowerSemicontinuousAt, nhdsWithin_univ]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u22a2 LowerSemicontinuousOn f univ \u2194 LowerSemicontinuous f\n[PROOFSTEP]\nsimp [LowerSemicontinuousOn, LowerSemicontinuous, lowerSemicontinuousWithinAt_univ_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsOpen s\nhy : 0 \u2264 y\n\u22a2 LowerSemicontinuous (indicator s fun _x => y)\n[PROOFSTEP]\nintro x z hz\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsOpen s\nhy : 0 \u2264 y\nx : \u03b1\nz : \u03b2\nhz : z < indicator s (fun _x => y) x\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd x, z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nby_cases h : x \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsOpen s\nhy : 0 \u2264 y\nx : \u03b1\nz : \u03b2\nhz : z < indicator s (fun _x => y) x\nh : x \u2208 s\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd x, z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nsimp [h] at hz \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsOpen s\nhy : 0 \u2264 y\nx : \u03b1\nz : \u03b2\nhz : z < indicator s (fun _x => y) x\nh : \u00acx \u2208 s\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd x, z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nsimp [h] at hz \n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsOpen s\nhy : 0 \u2264 y\nx : \u03b1\nz : \u03b2\nh : x \u2208 s\nhz : z < y\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd x, z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nfilter_upwards [hs.mem_nhds h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsOpen s\nhy : 0 \u2264 y\nx : \u03b1\nz : \u03b2\nh : x \u2208 s\nhz : z < y\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2192 z < indicator s (fun _x => y) a\n[PROOFSTEP]\nsimp (config := { contextual := true }) [hz]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsOpen s\nhy : 0 \u2264 y\nx : \u03b1\nz : \u03b2\nh : \u00acx \u2208 s\nhz : z < 0\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd x, z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nrefine Filter.eventually_of_forall fun x' => ?_\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsOpen s\nhy : 0 \u2264 y\nx : \u03b1\nz : \u03b2\nh : \u00acx \u2208 s\nhz : z < 0\nx' : \u03b1\n\u22a2 z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nby_cases h' : x' \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsOpen s\nhy : 0 \u2264 y\nx : \u03b1\nz : \u03b2\nh : \u00acx \u2208 s\nhz : z < 0\nx' : \u03b1\nh' : x' \u2208 s\n\u22a2 z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nsimp [h', hz.trans_le hy, hz]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsOpen s\nhy : 0 \u2264 y\nx : \u03b1\nz : \u03b2\nh : \u00acx \u2208 s\nhz : z < 0\nx' : \u03b1\nh' : \u00acx' \u2208 s\n\u22a2 z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nsimp [h', hz.trans_le hy, hz]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsClosed s\nhy : y \u2264 0\n\u22a2 LowerSemicontinuous (indicator s fun _x => y)\n[PROOFSTEP]\nintro x z hz\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsClosed s\nhy : y \u2264 0\nx : \u03b1\nz : \u03b2\nhz : z < indicator s (fun _x => y) x\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd x, z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nby_cases h : x \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsClosed s\nhy : y \u2264 0\nx : \u03b1\nz : \u03b2\nhz : z < indicator s (fun _x => y) x\nh : x \u2208 s\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd x, z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nsimp [h] at hz \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsClosed s\nhy : y \u2264 0\nx : \u03b1\nz : \u03b2\nhz : z < indicator s (fun _x => y) x\nh : \u00acx \u2208 s\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd x, z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nsimp [h] at hz \n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsClosed s\nhy : y \u2264 0\nx : \u03b1\nz : \u03b2\nh : x \u2208 s\nhz : z < y\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd x, z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nrefine Filter.eventually_of_forall fun x' => ?_\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsClosed s\nhy : y \u2264 0\nx : \u03b1\nz : \u03b2\nh : x \u2208 s\nhz : z < y\nx' : \u03b1\n\u22a2 z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nby_cases h' : x' \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsClosed s\nhy : y \u2264 0\nx : \u03b1\nz : \u03b2\nh : x \u2208 s\nhz : z < y\nx' : \u03b1\nh' : x' \u2208 s\n\u22a2 z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nsimp [h', hz, hz.trans_le hy]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsClosed s\nhy : y \u2264 0\nx : \u03b1\nz : \u03b2\nh : x \u2208 s\nhz : z < y\nx' : \u03b1\nh' : \u00acx' \u2208 s\n\u22a2 z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nsimp [h', hz, hz.trans_le hy]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsClosed s\nhy : y \u2264 0\nx : \u03b1\nz : \u03b2\nh : \u00acx \u2208 s\nhz : z < 0\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd x, z < indicator s (fun _x => y) x'\n[PROOFSTEP]\nfilter_upwards [hs.isOpen_compl.mem_nhds h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx\u271d : \u03b1\ns t : Set \u03b1\ny z\u271d : \u03b2\ninst\u271d : Zero \u03b2\nhs : IsClosed s\nhy : y \u2264 0\nx : \u03b1\nz : \u03b2\nh : \u00acx \u2208 s\nhz : z < 0\n\u22a2 \u2200 (a : \u03b1), a \u2208 s\u1d9c \u2192 z < indicator s (fun _x => y) a\n[PROOFSTEP]\nsimp (config := { contextual := true }) [hz]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d : LinearOrder \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 LowerSemicontinuous f \u2194 \u2200 (y : \u03b3), IsClosed (f \u207b\u00b9' Iic y)\n[PROOFSTEP]\nrw [lowerSemicontinuous_iff_isOpen_preimage]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d : LinearOrder \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 (\u2200 (y : \u03b3), IsOpen (f \u207b\u00b9' Ioi y)) \u2194 \u2200 (y : \u03b3), IsClosed (f \u207b\u00b9' Iic y)\n[PROOFSTEP]\nsimp only [\u2190 isOpen_compl_iff, \u2190 preimage_compl, compl_Iic]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b3\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : OrderTopology \u03b3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LinearOrder \u03b4\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : OrderTopology \u03b4\ng : \u03b3 \u2192 \u03b4\nf : \u03b1 \u2192 \u03b3\nhg : ContinuousAt g (f x)\nhf : LowerSemicontinuousWithinAt f s x\ngmon : Monotone g\n\u22a2 LowerSemicontinuousWithinAt (g \u2218 f) s x\n[PROOFSTEP]\nintro y hy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b3\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : OrderTopology \u03b3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LinearOrder \u03b4\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : OrderTopology \u03b4\ng : \u03b3 \u2192 \u03b4\nf : \u03b1 \u2192 \u03b3\nhg : ContinuousAt g (f x)\nhf : LowerSemicontinuousWithinAt f s x\ngmon : Monotone g\ny : \u03b4\nhy : y < (g \u2218 f) x\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (g \u2218 f) x'\n[PROOFSTEP]\nby_cases h : \u2203 l, l < f x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b3\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : OrderTopology \u03b3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LinearOrder \u03b4\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : OrderTopology \u03b4\ng : \u03b3 \u2192 \u03b4\nf : \u03b1 \u2192 \u03b3\nhg : ContinuousAt g (f x)\nhf : LowerSemicontinuousWithinAt f s x\ngmon : Monotone g\ny : \u03b4\nhy : y < (g \u2218 f) x\nh : \u2203 l, l < f x\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (g \u2218 f) x'\n[PROOFSTEP]\nobtain \u27e8z, zlt, hz\u27e9 : \u2203 z < f x, Ioc z (f x) \u2286 g \u207b\u00b9' Ioi y := exists_Ioc_subset_of_mem_nhds (hg (Ioi_mem_nhds hy)) h\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b3\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : OrderTopology \u03b3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LinearOrder \u03b4\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : OrderTopology \u03b4\ng : \u03b3 \u2192 \u03b4\nf : \u03b1 \u2192 \u03b3\nhg : ContinuousAt g (f x)\nhf : LowerSemicontinuousWithinAt f s x\ngmon : Monotone g\ny : \u03b4\nhy : y < (g \u2218 f) x\nh : \u2203 l, l < f x\nz : \u03b3\nzlt : z < f x\nhz : Ioc z (f x) \u2286 g \u207b\u00b9' Ioi y\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (g \u2218 f) x'\n[PROOFSTEP]\nfilter_upwards [hf z zlt] with a ha\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b3\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : OrderTopology \u03b3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LinearOrder \u03b4\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : OrderTopology \u03b4\ng : \u03b3 \u2192 \u03b4\nf : \u03b1 \u2192 \u03b3\nhg : ContinuousAt g (f x)\nhf : LowerSemicontinuousWithinAt f s x\ngmon : Monotone g\ny : \u03b4\nhy : y < (g \u2218 f) x\nh : \u2203 l, l < f x\nz : \u03b3\nzlt : z < f x\nhz : Ioc z (f x) \u2286 g \u207b\u00b9' Ioi y\na : \u03b1\nha : z < f a\n\u22a2 y < (g \u2218 f) a\n[PROOFSTEP]\ncalc\n y < g (min (f x) (f a)) := hz (by simp [zlt, ha, le_refl])\n _ \u2264 g (f a) := gmon (min_le_right _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b3\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : OrderTopology \u03b3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LinearOrder \u03b4\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : OrderTopology \u03b4\ng : \u03b3 \u2192 \u03b4\nf : \u03b1 \u2192 \u03b3\nhg : ContinuousAt g (f x)\nhf : LowerSemicontinuousWithinAt f s x\ngmon : Monotone g\ny : \u03b4\nhy : y < (g \u2218 f) x\nh : \u2203 l, l < f x\nz : \u03b3\nzlt : z < f x\nhz : Ioc z (f x) \u2286 g \u207b\u00b9' Ioi y\na : \u03b1\nha : z < f a\n\u22a2 min (f x) (f a) \u2208 Ioc z (f x)\n[PROOFSTEP]\nsimp [zlt, ha, le_refl]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b3\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : OrderTopology \u03b3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LinearOrder \u03b4\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : OrderTopology \u03b4\ng : \u03b3 \u2192 \u03b4\nf : \u03b1 \u2192 \u03b3\nhg : ContinuousAt g (f x)\nhf : LowerSemicontinuousWithinAt f s x\ngmon : Monotone g\ny : \u03b4\nhy : y < (g \u2218 f) x\nh : \u00ac\u2203 l, l < f x\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (g \u2218 f) x'\n[PROOFSTEP]\nsimp only [not_exists, not_lt] at h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b3\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : OrderTopology \u03b3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LinearOrder \u03b4\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : OrderTopology \u03b4\ng : \u03b3 \u2192 \u03b4\nf : \u03b1 \u2192 \u03b3\nhg : ContinuousAt g (f x)\nhf : LowerSemicontinuousWithinAt f s x\ngmon : Monotone g\ny : \u03b4\nhy : y < (g \u2218 f) x\nh : \u2200 (x_1 : \u03b3), f x \u2264 x_1\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (g \u2218 f) x'\n[PROOFSTEP]\nexact Filter.eventually_of_forall fun a => hy.trans_le (gmon (h (f a)))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b3\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : OrderTopology \u03b3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LinearOrder \u03b4\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : OrderTopology \u03b4\ng : \u03b3 \u2192 \u03b4\nf : \u03b1 \u2192 \u03b3\nhg : ContinuousAt g (f x)\nhf : LowerSemicontinuousAt f x\ngmon : Monotone g\n\u22a2 LowerSemicontinuousAt (g \u2218 f) x\n[PROOFSTEP]\nsimp only [\u2190 lowerSemicontinuousWithinAt_univ_iff] at hf \u22a2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrder \u03b3\ninst\u271d\u2074 : TopologicalSpace \u03b3\ninst\u271d\u00b3 : OrderTopology \u03b3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : LinearOrder \u03b4\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : OrderTopology \u03b4\ng : \u03b3 \u2192 \u03b4\nf : \u03b1 \u2192 \u03b3\nhg : ContinuousAt g (f x)\ngmon : Monotone g\nhf : LowerSemicontinuousWithinAt f univ x\n\u22a2 LowerSemicontinuousWithinAt (g \u2218 f) univ x\n[PROOFSTEP]\nexact hg.comp_lowerSemicontinuousWithinAt hf gmon\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\n\u22a2 LowerSemicontinuousWithinAt (fun z => f z + g z) s x\n[PROOFSTEP]\nintro y hy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun z => f z + g z) x'\n[PROOFSTEP]\nobtain \u27e8u, v, u_open, xu, v_open, xv, h\u27e9 :\n \u2203 u v : Set \u03b3, IsOpen u \u2227 f x \u2208 u \u2227 IsOpen v \u2227 g x \u2208 v \u2227 u \u00d7\u02e2 v \u2286 {p : \u03b3 \u00d7 \u03b3 | y < p.fst + p.snd} :=\n mem_nhds_prod_iff'.1 (hcont (isOpen_Ioi.mem_nhds hy))\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun z => f z + g z) x'\n[PROOFSTEP]\nby_cases hx\u2081 : \u2203 l, l < f x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun z => f z + g z) x'\n[PROOFSTEP]\nobtain \u27e8z\u2081, z\u2081lt, h\u2081\u27e9 : \u2203 z\u2081 < f x, Ioc z\u2081 (f x) \u2286 u := exists_Ioc_subset_of_mem_nhds (u_open.mem_nhds xu) hx\u2081\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun z => f z + g z) x'\n[PROOFSTEP]\nby_cases hx\u2082 : \u2203 l, l < g x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun z => f z + g z) x'\n[PROOFSTEP]\nobtain \u27e8z\u2082, z\u2082lt, h\u2082\u27e9 : \u2203 z\u2082 < g x, Ioc z\u2082 (g x) \u2286 v := exists_Ioc_subset_of_mem_nhds (v_open.mem_nhds xv) hx\u2082\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun z => f z + g z) x'\n[PROOFSTEP]\nfilter_upwards [hf z\u2081 z\u2081lt, hg z\u2082 z\u2082lt] with z h\u2081z h\u2082z\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2081z : z\u2081 < f z\nh\u2082z : z\u2082 < g z\n\u22a2 y < f z + g z\n[PROOFSTEP]\nhave A1 : min (f z) (f x) \u2208 u := by\n by_cases H : f z \u2264 f x\n \u00b7 simp [H]\n exact h\u2081 \u27e8h\u2081z, H\u27e9\n \u00b7 simp [le_of_not_le H]\n exact h\u2081 \u27e8z\u2081lt, le_rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2081z : z\u2081 < f z\nh\u2082z : z\u2082 < g z\n\u22a2 min (f z) (f x) \u2208 u\n[PROOFSTEP]\nby_cases H : f z \u2264 f x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2081z : z\u2081 < f z\nh\u2082z : z\u2082 < g z\nH : f z \u2264 f x\n\u22a2 min (f z) (f x) \u2208 u\n[PROOFSTEP]\nsimp [H]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2081z : z\u2081 < f z\nh\u2082z : z\u2082 < g z\nH : f z \u2264 f x\n\u22a2 f z \u2208 u\n[PROOFSTEP]\nexact h\u2081 \u27e8h\u2081z, H\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2081z : z\u2081 < f z\nh\u2082z : z\u2082 < g z\nH : \u00acf z \u2264 f x\n\u22a2 min (f z) (f x) \u2208 u\n[PROOFSTEP]\nsimp [le_of_not_le H]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2081z : z\u2081 < f z\nh\u2082z : z\u2082 < g z\nH : \u00acf z \u2264 f x\n\u22a2 f x \u2208 u\n[PROOFSTEP]\nexact h\u2081 \u27e8z\u2081lt, le_rfl\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2081z : z\u2081 < f z\nh\u2082z : z\u2082 < g z\nA1 : min (f z) (f x) \u2208 u\n\u22a2 y < f z + g z\n[PROOFSTEP]\nhave A2 : min (g z) (g x) \u2208 v := by\n by_cases H : g z \u2264 g x\n \u00b7 simp [H]\n exact h\u2082 \u27e8h\u2082z, H\u27e9\n \u00b7 simp [le_of_not_le H]\n exact h\u2082 \u27e8z\u2082lt, le_rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2081z : z\u2081 < f z\nh\u2082z : z\u2082 < g z\nA1 : min (f z) (f x) \u2208 u\n\u22a2 min (g z) (g x) \u2208 v\n[PROOFSTEP]\nby_cases H : g z \u2264 g x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2081z : z\u2081 < f z\nh\u2082z : z\u2082 < g z\nA1 : min (f z) (f x) \u2208 u\nH : g z \u2264 g x\n\u22a2 min (g z) (g x) \u2208 v\n[PROOFSTEP]\nsimp [H]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2081z : z\u2081 < f z\nh\u2082z : z\u2082 < g z\nA1 : min (f z) (f x) \u2208 u\nH : g z \u2264 g x\n\u22a2 g z \u2208 v\n[PROOFSTEP]\nexact h\u2082 \u27e8h\u2082z, H\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2081z : z\u2081 < f z\nh\u2082z : z\u2082 < g z\nA1 : min (f z) (f x) \u2208 u\nH : \u00acg z \u2264 g x\n\u22a2 min (g z) (g x) \u2208 v\n[PROOFSTEP]\nsimp [le_of_not_le H]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2081z : z\u2081 < f z\nh\u2082z : z\u2082 < g z\nA1 : min (f z) (f x) \u2208 u\nH : \u00acg z \u2264 g x\n\u22a2 g x \u2208 v\n[PROOFSTEP]\nexact h\u2082 \u27e8z\u2082lt, le_rfl\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2081z : z\u2081 < f z\nh\u2082z : z\u2082 < g z\nA1 : min (f z) (f x) \u2208 u\nA2 : min (g z) (g x) \u2208 v\n\u22a2 y < f z + g z\n[PROOFSTEP]\nhave : (min (f z) (f x), min (g z) (g x)) \u2208 u \u00d7\u02e2 v := \u27e8A1, A2\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2081z : z\u2081 < f z\nh\u2082z : z\u2082 < g z\nA1 : min (f z) (f x) \u2208 u\nA2 : min (g z) (g x) \u2208 v\nthis : (min (f z) (f x), min (g z) (g x)) \u2208 u \u00d7\u02e2 v\n\u22a2 y < f z + g z\n[PROOFSTEP]\ncalc\n y < min (f z) (f x) + min (g z) (g x) := h this\n _ \u2264 f z + g z := add_le_add (min_le_left _ _) (min_le_left _ _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u00ac\u2203 l, l < g x\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun z => f z + g z) x'\n[PROOFSTEP]\nsimp only [not_exists, not_lt] at hx\u2082 \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2200 (x_1 : \u03b3), g x \u2264 x_1\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun z => f z + g z) x'\n[PROOFSTEP]\nfilter_upwards [hf z\u2081 z\u2081lt] with z h\u2081z\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2200 (x_1 : \u03b3), g x \u2264 x_1\nz : \u03b1\nh\u2081z : z\u2081 < f z\n\u22a2 y < f z + g z\n[PROOFSTEP]\nhave A1 : min (f z) (f x) \u2208 u := by\n by_cases H : f z \u2264 f x\n \u00b7 simp [H]\n exact h\u2081 \u27e8h\u2081z, H\u27e9\n \u00b7 simp [le_of_not_le H]\n exact h\u2081 \u27e8z\u2081lt, le_rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2200 (x_1 : \u03b3), g x \u2264 x_1\nz : \u03b1\nh\u2081z : z\u2081 < f z\n\u22a2 min (f z) (f x) \u2208 u\n[PROOFSTEP]\nby_cases H : f z \u2264 f x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2200 (x_1 : \u03b3), g x \u2264 x_1\nz : \u03b1\nh\u2081z : z\u2081 < f z\nH : f z \u2264 f x\n\u22a2 min (f z) (f x) \u2208 u\n[PROOFSTEP]\nsimp [H]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2200 (x_1 : \u03b3), g x \u2264 x_1\nz : \u03b1\nh\u2081z : z\u2081 < f z\nH : f z \u2264 f x\n\u22a2 f z \u2208 u\n[PROOFSTEP]\nexact h\u2081 \u27e8h\u2081z, H\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2200 (x_1 : \u03b3), g x \u2264 x_1\nz : \u03b1\nh\u2081z : z\u2081 < f z\nH : \u00acf z \u2264 f x\n\u22a2 min (f z) (f x) \u2208 u\n[PROOFSTEP]\nsimp [le_of_not_le H]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2200 (x_1 : \u03b3), g x \u2264 x_1\nz : \u03b1\nh\u2081z : z\u2081 < f z\nH : \u00acf z \u2264 f x\n\u22a2 f x \u2208 u\n[PROOFSTEP]\nexact h\u2081 \u27e8z\u2081lt, le_rfl\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2200 (x_1 : \u03b3), g x \u2264 x_1\nz : \u03b1\nh\u2081z : z\u2081 < f z\nA1 : min (f z) (f x) \u2208 u\n\u22a2 y < f z + g z\n[PROOFSTEP]\nhave : (min (f z) (f x), g x) \u2208 u \u00d7\u02e2 v := \u27e8A1, xv\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2203 l, l < f x\nz\u2081 : \u03b3\nz\u2081lt : z\u2081 < f x\nh\u2081 : Ioc z\u2081 (f x) \u2286 u\nhx\u2082 : \u2200 (x_1 : \u03b3), g x \u2264 x_1\nz : \u03b1\nh\u2081z : z\u2081 < f z\nA1 : min (f z) (f x) \u2208 u\nthis : (min (f z) (f x), g x) \u2208 u \u00d7\u02e2 v\n\u22a2 y < f z + g z\n[PROOFSTEP]\ncalc\n y < min (f z) (f x) + g x := h this\n _ \u2264 f z + g z := add_le_add (min_le_left _ _) (hx\u2082 (g z))\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u00ac\u2203 l, l < f x\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun z => f z + g z) x'\n[PROOFSTEP]\nsimp only [not_exists, not_lt] at hx\u2081 \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun z => f z + g z) x'\n[PROOFSTEP]\nby_cases hx\u2082 : \u2203 l, l < g x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nhx\u2082 : \u2203 l, l < g x\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun z => f z + g z) x'\n[PROOFSTEP]\nobtain \u27e8z\u2082, z\u2082lt, h\u2082\u27e9 : \u2203 z\u2082 < g x, Ioc z\u2082 (g x) \u2286 v := exists_Ioc_subset_of_mem_nhds (v_open.mem_nhds xv) hx\u2082\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun z => f z + g z) x'\n[PROOFSTEP]\nfilter_upwards [hg z\u2082 z\u2082lt] with z h\u2082z\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2082z : z\u2082 < g z\n\u22a2 y < f z + g z\n[PROOFSTEP]\nhave A2 : min (g z) (g x) \u2208 v := by\n by_cases H : g z \u2264 g x\n \u00b7 simp [H]\n exact h\u2082 \u27e8h\u2082z, H\u27e9\n \u00b7 simp [le_of_not_le H]\n exact h\u2082 \u27e8z\u2082lt, le_rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2082z : z\u2082 < g z\n\u22a2 min (g z) (g x) \u2208 v\n[PROOFSTEP]\nby_cases H : g z \u2264 g x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2082z : z\u2082 < g z\nH : g z \u2264 g x\n\u22a2 min (g z) (g x) \u2208 v\n[PROOFSTEP]\nsimp [H]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2082z : z\u2082 < g z\nH : g z \u2264 g x\n\u22a2 g z \u2208 v\n[PROOFSTEP]\nexact h\u2082 \u27e8h\u2082z, H\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2082z : z\u2082 < g z\nH : \u00acg z \u2264 g x\n\u22a2 min (g z) (g x) \u2208 v\n[PROOFSTEP]\nsimp [le_of_not_le H]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2082z : z\u2082 < g z\nH : \u00acg z \u2264 g x\n\u22a2 g x \u2208 v\n[PROOFSTEP]\nexact h\u2082 \u27e8z\u2082lt, le_rfl\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2082z : z\u2082 < g z\nA2 : min (g z) (g x) \u2208 v\n\u22a2 y < f z + g z\n[PROOFSTEP]\nhave : (f x, min (g z) (g x)) \u2208 u \u00d7\u02e2 v := \u27e8xu, A2\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nhx\u2082 : \u2203 l, l < g x\nz\u2082 : \u03b3\nz\u2082lt : z\u2082 < g x\nh\u2082 : Ioc z\u2082 (g x) \u2286 v\nz : \u03b1\nh\u2082z : z\u2082 < g z\nA2 : min (g z) (g x) \u2208 v\nthis : (f x, min (g z) (g x)) \u2208 u \u00d7\u02e2 v\n\u22a2 y < f z + g z\n[PROOFSTEP]\ncalc\n y < f x + min (g z) (g x) := h this\n _ \u2264 f z + g z := add_le_add (hx\u2081 (f z)) (min_le_left _ _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nhx\u2082 : \u00ac\u2203 l, l < g x\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun z => f z + g z) x'\n[PROOFSTEP]\nsimp only [not_exists, not_lt] at hx\u2081 hx\u2082 \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nhx\u2082 : \u2200 (x_1 : \u03b3), g x \u2264 x_1\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun z => f z + g z) x'\n[PROOFSTEP]\napply Filter.eventually_of_forall\n[GOAL]\ncase neg.hp\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nhx\u2082 : \u2200 (x_1 : \u03b3), g x \u2264 x_1\n\u22a2 \u2200 (x : \u03b1), y < (fun z => f z + g z) x\n[PROOFSTEP]\nintro z\n[GOAL]\ncase neg.hp\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nhx\u2082 : \u2200 (x_1 : \u03b3), g x \u2264 x_1\nz : \u03b1\n\u22a2 y < (fun z => f z + g z) z\n[PROOFSTEP]\nhave : (f x, g x) \u2208 u \u00d7\u02e2 v := \u27e8xu, xv\u27e9\n[GOAL]\ncase neg.hp\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z\u271d : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\ny : \u03b3\nhy : y < (fun z => f z + g z) x\nu v : Set \u03b3\nu_open : IsOpen u\nxu : f x \u2208 u\nv_open : IsOpen v\nxv : g x \u2208 v\nh : u \u00d7\u02e2 v \u2286 {p | y < p.fst + p.snd}\nhx\u2081 : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nhx\u2082 : \u2200 (x_1 : \u03b3), g x \u2264 x_1\nz : \u03b1\nthis : (f x, g x) \u2208 u \u00d7\u02e2 v\n\u22a2 y < (fun z => f z + g z) z\n[PROOFSTEP]\ncalc\n y < f x + g x := h this\n _ \u2264 f z + g z := add_le_add (hx\u2081 (f z)) (hx\u2082 (g z))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousAt f x\nhg : LowerSemicontinuousAt g x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\n\u22a2 LowerSemicontinuousAt (fun z => f z + g z) x\n[PROOFSTEP]\nsimp_rw [\u2190 lowerSemicontinuousWithinAt_univ_iff] at *\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\nhf : LowerSemicontinuousWithinAt f univ x\nhg : LowerSemicontinuousWithinAt g univ x\n\u22a2 LowerSemicontinuousWithinAt (fun z => f z + g z) univ x\n[PROOFSTEP]\nexact hf.add' hg hcont\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : OrderTopology \u03b3\ninst\u271d : ContinuousAdd \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b3\na : Finset \u03b9\nha : \u2200 (i : \u03b9), i \u2208 a \u2192 LowerSemicontinuousWithinAt (f i) s x\n\u22a2 LowerSemicontinuousWithinAt (fun z => \u2211 i in a, f i z) s x\n[PROOFSTEP]\nclassical\ninduction' a using Finset.induction_on with i a ia IH\n\u00b7 exact lowerSemicontinuousWithinAt_const\n\u00b7 simp only [ia, Finset.sum_insert, not_false_iff]\n exact\n LowerSemicontinuousWithinAt.add (ha _ (Finset.mem_insert_self i a))\n (IH fun j ja => ha j (Finset.mem_insert_of_mem ja))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : OrderTopology \u03b3\ninst\u271d : ContinuousAdd \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b3\na : Finset \u03b9\nha : \u2200 (i : \u03b9), i \u2208 a \u2192 LowerSemicontinuousWithinAt (f i) s x\n\u22a2 LowerSemicontinuousWithinAt (fun z => \u2211 i in a, f i z) s x\n[PROOFSTEP]\ninduction' a using Finset.induction_on with i a ia IH\n[GOAL]\ncase empty\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : OrderTopology \u03b3\ninst\u271d : ContinuousAdd \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b3\na : Finset \u03b9\nha\u271d : \u2200 (i : \u03b9), i \u2208 a \u2192 LowerSemicontinuousWithinAt (f i) s x\nha : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 LowerSemicontinuousWithinAt (f i) s x\n\u22a2 LowerSemicontinuousWithinAt (fun z => \u2211 i in \u2205, f i z) s x\n[PROOFSTEP]\nexact lowerSemicontinuousWithinAt_const\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : OrderTopology \u03b3\ninst\u271d : ContinuousAdd \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b3\na\u271d : Finset \u03b9\nha\u271d : \u2200 (i : \u03b9), i \u2208 a\u271d \u2192 LowerSemicontinuousWithinAt (f i) s x\ni : \u03b9\na : Finset \u03b9\nia : \u00aci \u2208 a\nIH :\n (\u2200 (i : \u03b9), i \u2208 a \u2192 LowerSemicontinuousWithinAt (f i) s x) \u2192\n LowerSemicontinuousWithinAt (fun z => \u2211 i in a, f i z) s x\nha : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i a \u2192 LowerSemicontinuousWithinAt (f i_1) s x\n\u22a2 LowerSemicontinuousWithinAt (fun z => \u2211 i in insert i a, f i z) s x\n[PROOFSTEP]\nsimp only [ia, Finset.sum_insert, not_false_iff]\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : OrderTopology \u03b3\ninst\u271d : ContinuousAdd \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b3\na\u271d : Finset \u03b9\nha\u271d : \u2200 (i : \u03b9), i \u2208 a\u271d \u2192 LowerSemicontinuousWithinAt (f i) s x\ni : \u03b9\na : Finset \u03b9\nia : \u00aci \u2208 a\nIH :\n (\u2200 (i : \u03b9), i \u2208 a \u2192 LowerSemicontinuousWithinAt (f i) s x) \u2192\n LowerSemicontinuousWithinAt (fun z => \u2211 i in a, f i z) s x\nha : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i a \u2192 LowerSemicontinuousWithinAt (f i_1) s x\n\u22a2 LowerSemicontinuousWithinAt (fun z => f i z + \u2211 i in a, f i z) s x\n[PROOFSTEP]\nexact\n LowerSemicontinuousWithinAt.add (ha _ (Finset.mem_insert_self i a))\n (IH fun j ja => ha j (Finset.mem_insert_of_mem ja))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : OrderTopology \u03b3\ninst\u271d : ContinuousAdd \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b3\na : Finset \u03b9\nha : \u2200 (i : \u03b9), i \u2208 a \u2192 LowerSemicontinuousAt (f i) x\n\u22a2 LowerSemicontinuousAt (fun z => \u2211 i in a, f i z) x\n[PROOFSTEP]\nsimp_rw [\u2190 lowerSemicontinuousWithinAt_univ_iff] at *\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : OrderTopology \u03b3\ninst\u271d : ContinuousAdd \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b3\na : Finset \u03b9\nha : \u2200 (i : \u03b9), i \u2208 a \u2192 LowerSemicontinuousWithinAt (f i) univ x\n\u22a2 LowerSemicontinuousWithinAt (fun z => \u2211 i in a, f i z) univ x\n[PROOFSTEP]\nexact lowerSemicontinuousWithinAt_sum ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Sort u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\ninst\u271d\u00b9 : CompleteLinearOrder \u03b4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b4'\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b4'\nbdd : \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, BddAbove (range fun i => f i y)\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) s x\n\u22a2 LowerSemicontinuousWithinAt (fun x' => \u2a06 (i : \u03b9), f i x') s x\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Sort u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\ninst\u271d\u00b9 : CompleteLinearOrder \u03b4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b4'\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b4'\nbdd : \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, BddAbove (range fun i => f i y)\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) s x\nh\u271d : IsEmpty \u03b9\n\u22a2 LowerSemicontinuousWithinAt (fun x' => \u2a06 (i : \u03b9), f i x') s x\n[PROOFSTEP]\nsimpa only [iSup_of_empty'] using lowerSemicontinuousWithinAt_const\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Sort u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\ninst\u271d\u00b9 : CompleteLinearOrder \u03b4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b4'\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b4'\nbdd : \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, BddAbove (range fun i => f i y)\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) s x\nh\u271d : Nonempty \u03b9\n\u22a2 LowerSemicontinuousWithinAt (fun x' => \u2a06 (i : \u03b9), f i x') s x\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Sort u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\ninst\u271d\u00b9 : CompleteLinearOrder \u03b4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b4'\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b4'\nbdd : \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, BddAbove (range fun i => f i y)\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) s x\nh\u271d : Nonempty \u03b9\ny : \u03b4'\nhy : y < (fun x' => \u2a06 (i : \u03b9), f i x') x\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun x' => \u2a06 (i : \u03b9), f i x') x'\n[PROOFSTEP]\nrcases exists_lt_of_lt_ciSup hy with \u27e8i, hi\u27e9\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b9 : Sort u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\ninst\u271d\u00b9 : CompleteLinearOrder \u03b4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b4'\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b4'\nbdd : \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, BddAbove (range fun i => f i y)\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) s x\nh\u271d : Nonempty \u03b9\ny : \u03b4'\nhy : y < (fun x' => \u2a06 (i : \u03b9), f i x') x\ni : \u03b9\nhi : y < f i x\n\u22a2 \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < (fun x' => \u2a06 (i : \u03b9), f i x') x'\n[PROOFSTEP]\nfilter_upwards [h i y hi, bdd] with y hy hy' using hy.trans_le (le_ciSup hy' i)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Sort u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\ninst\u271d\u00b9 : CompleteLinearOrder \u03b4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b4'\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b4\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) s x\n\u22a2 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, BddAbove (range fun i => f i y)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Sort u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\ninst\u271d\u00b9 : CompleteLinearOrder \u03b4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b4'\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b4'\nbdd : \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, BddAbove (range fun i => f i y)\nh : \u2200 (i : \u03b9), LowerSemicontinuousAt (f i) x\n\u22a2 LowerSemicontinuousAt (fun x' => \u2a06 (i : \u03b9), f i x') x\n[PROOFSTEP]\nsimp_rw [\u2190 lowerSemicontinuousWithinAt_univ_iff] at *\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Sort u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\ninst\u271d\u00b9 : CompleteLinearOrder \u03b4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b4'\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b4'\nbdd : \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, BddAbove (range fun i => f i y)\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) univ x\n\u22a2 LowerSemicontinuousWithinAt (fun x' => \u2a06 (i : \u03b9), f i x') univ x\n[PROOFSTEP]\nrw [\u2190 nhdsWithin_univ] at bdd \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Sort u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\ninst\u271d\u00b9 : CompleteLinearOrder \u03b4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b4'\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b4'\nbdd : \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[univ] x, BddAbove (range fun i => f i y)\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) univ x\n\u22a2 LowerSemicontinuousWithinAt (fun x' => \u2a06 (i : \u03b9), f i x') univ x\n[PROOFSTEP]\nexact lowerSemicontinuousWithinAt_ciSup bdd h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Sort u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\ninst\u271d\u00b9 : CompleteLinearOrder \u03b4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b4'\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b4\nh : \u2200 (i : \u03b9), LowerSemicontinuousAt (f i) x\n\u22a2 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, BddAbove (range fun i => f i y)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Sort u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\ninst\u271d\u00b9 : CompleteLinearOrder \u03b4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b4'\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b4\nh : \u2200 (i : \u03b9), LowerSemicontinuousOn (f i) s\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 BddAbove (range fun i => f i x)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Sort u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\ninst\u271d\u00b9 : CompleteLinearOrder \u03b4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b4'\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b4\nh : \u2200 (i : \u03b9), LowerSemicontinuous (f i)\n\u22a2 \u2200 (x : \u03b1), BddAbove (range fun i => f i x)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) s x\n\u22a2 LowerSemicontinuousWithinAt (fun x' => \u2211' (i : \u03b9), f i x') s x\n[PROOFSTEP]\nsimp_rw [ENNReal.tsum_eq_iSup_sum]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) s x\n\u22a2 LowerSemicontinuousWithinAt (fun x' => \u2a06 (s : Finset \u03b9), \u2211 i in s, f i x') s x\n[PROOFSTEP]\nrefine lowerSemicontinuousWithinAt_iSup fun b => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) s x\nb : Finset \u03b9\n\u22a2 LowerSemicontinuousWithinAt (fun x' => \u2211 i in b, f i x') s x\n[PROOFSTEP]\nexact lowerSemicontinuousWithinAt_sum fun i _hi => h i\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (i : \u03b9), LowerSemicontinuousAt (f i) x\n\u22a2 LowerSemicontinuousAt (fun x' => \u2211' (i : \u03b9), f i x') x\n[PROOFSTEP]\nsimp_rw [\u2190 lowerSemicontinuousWithinAt_univ_iff] at *\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) univ x\n\u22a2 LowerSemicontinuousWithinAt (fun x' => \u2211' (i : \u03b9), f i x') univ x\n[PROOFSTEP]\nexact lowerSemicontinuousWithinAt_tsum h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u22a2 UpperSemicontinuousWithinAt f univ x \u2194 UpperSemicontinuousAt f x\n[PROOFSTEP]\nsimp [UpperSemicontinuousWithinAt, UpperSemicontinuousAt, nhdsWithin_univ]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u22a2 UpperSemicontinuousOn f univ \u2194 UpperSemicontinuous f\n[PROOFSTEP]\nsimp [UpperSemicontinuousOn, UpperSemicontinuous, upperSemicontinuousWithinAt_univ_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d : LinearOrder \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 UpperSemicontinuous f \u2194 \u2200 (y : \u03b3), IsClosed (f \u207b\u00b9' Ici y)\n[PROOFSTEP]\nrw [upperSemicontinuous_iff_isOpen_preimage]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d : LinearOrder \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 (\u2200 (y : \u03b3), IsOpen (f \u207b\u00b9' Iio y)) \u2194 \u2200 (y : \u03b3), IsClosed (f \u207b\u00b9' Ici y)\n[PROOFSTEP]\nsimp only [\u2190 isOpen_compl_iff, \u2190 preimage_compl, compl_Ici]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhf : UpperSemicontinuousAt f x\nhg : UpperSemicontinuousAt g x\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\n\u22a2 UpperSemicontinuousAt (fun z => f z + g z) x\n[PROOFSTEP]\nsimp_rw [\u2190 upperSemicontinuousWithinAt_univ_iff] at *\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf g : \u03b1 \u2192 \u03b3\nhcont : ContinuousAt (fun p => p.fst + p.snd) (f x, g x)\nhf : UpperSemicontinuousWithinAt f univ x\nhg : UpperSemicontinuousWithinAt g univ x\n\u22a2 UpperSemicontinuousWithinAt (fun z => f z + g z) univ x\n[PROOFSTEP]\nexact hf.add' hg hcont\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : OrderTopology \u03b3\ninst\u271d : ContinuousAdd \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b3\na : Finset \u03b9\nha : \u2200 (i : \u03b9), i \u2208 a \u2192 UpperSemicontinuousAt (f i) x\n\u22a2 UpperSemicontinuousAt (fun z => \u2211 i in a, f i z) x\n[PROOFSTEP]\nsimp_rw [\u2190 upperSemicontinuousWithinAt_univ_iff] at *\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : LinearOrderedAddCommMonoid \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : OrderTopology \u03b3\ninst\u271d : ContinuousAdd \u03b3\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b3\na : Finset \u03b9\nha : \u2200 (i : \u03b9), i \u2208 a \u2192 UpperSemicontinuousWithinAt (f i) univ x\n\u22a2 UpperSemicontinuousWithinAt (fun z => \u2211 i in a, f i z) univ x\n[PROOFSTEP]\nexact upperSemicontinuousWithinAt_sum ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 ContinuousWithinAt f s x \u2194 LowerSemicontinuousWithinAt f s x \u2227 UpperSemicontinuousWithinAt f s x\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8h.lowerSemicontinuousWithinAt, h.upperSemicontinuousWithinAt\u27e9, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 LowerSemicontinuousWithinAt f s x \u2227 UpperSemicontinuousWithinAt f s x \u2192 ContinuousWithinAt f s x\n[PROOFSTEP]\nrintro \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\n\u22a2 ContinuousWithinAt f s x\n[PROOFSTEP]\nintro v hv\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\n\u22a2 v \u2208 map f (\ud835\udcdd[s] x)\n[PROOFSTEP]\nsimp only [Filter.mem_map]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\nby_cases Hl : \u2203 l, l < f x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2203 l, l < f x\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\nrcases exists_Ioc_subset_of_mem_nhds hv Hl with \u27e8l, lfx, hl\u27e9\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2203 l, l < f x\nl : \u03b3\nlfx : l < f x\nhl : Ioc l (f x) \u2286 v\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\nby_cases Hu : \u2203 u, f x < u\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2203 l, l < f x\nl : \u03b3\nlfx : l < f x\nhl : Ioc l (f x) \u2286 v\nHu : \u2203 u, f x < u\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\nrcases exists_Ico_subset_of_mem_nhds hv Hu with \u27e8u, fxu, hu\u27e9\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2203 l, l < f x\nl : \u03b3\nlfx : l < f x\nhl : Ioc l (f x) \u2286 v\nHu : \u2203 u, f x < u\nu : \u03b3\nfxu : f x < u\nhu : Ico (f x) u \u2286 v\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\nfilter_upwards [h\u2081 l lfx, h\u2082 u fxu] with a lfa fau\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2203 l, l < f x\nl : \u03b3\nlfx : l < f x\nhl : Ioc l (f x) \u2286 v\nHu : \u2203 u, f x < u\nu : \u03b3\nfxu : f x < u\nhu : Ico (f x) u \u2286 v\na : \u03b1\nlfa : l < f a\nfau : f a < u\n\u22a2 a \u2208 f \u207b\u00b9' v\n[PROOFSTEP]\ncases' le_or_gt (f a) (f x) with h h\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2203 l, l < f x\nl : \u03b3\nlfx : l < f x\nhl : Ioc l (f x) \u2286 v\nHu : \u2203 u, f x < u\nu : \u03b3\nfxu : f x < u\nhu : Ico (f x) u \u2286 v\na : \u03b1\nlfa : l < f a\nfau : f a < u\nh : f a \u2264 f x\n\u22a2 a \u2208 f \u207b\u00b9' v\n[PROOFSTEP]\nexact hl \u27e8lfa, h\u27e9\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2203 l, l < f x\nl : \u03b3\nlfx : l < f x\nhl : Ioc l (f x) \u2286 v\nHu : \u2203 u, f x < u\nu : \u03b3\nfxu : f x < u\nhu : Ico (f x) u \u2286 v\na : \u03b1\nlfa : l < f a\nfau : f a < u\nh : f a > f x\n\u22a2 a \u2208 f \u207b\u00b9' v\n[PROOFSTEP]\nexact hu \u27e8le_of_lt h, fau\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2203 l, l < f x\nl : \u03b3\nlfx : l < f x\nhl : Ioc l (f x) \u2286 v\nHu : \u00ac\u2203 u, f x < u\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\nsimp only [not_exists, not_lt] at Hu \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2203 l, l < f x\nl : \u03b3\nlfx : l < f x\nhl : Ioc l (f x) \u2286 v\nHu : \u2200 (x_1 : \u03b3), x_1 \u2264 f x\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\nfilter_upwards [h\u2081 l lfx] with a lfa using hl \u27e8lfa, Hu (f a)\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u00ac\u2203 l, l < f x\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\nsimp only [not_exists, not_lt] at Hl \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2200 (x_1 : \u03b3), f x \u2264 x_1\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\nby_cases Hu : \u2203 u, f x < u\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nHu : \u2203 u, f x < u\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\nrcases exists_Ico_subset_of_mem_nhds hv Hu with \u27e8u, fxu, hu\u27e9\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nHu : \u2203 u, f x < u\nu : \u03b3\nfxu : f x < u\nhu : Ico (f x) u \u2286 v\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\nfilter_upwards [h\u2082 u fxu] with a lfa\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nHu : \u2203 u, f x < u\nu : \u03b3\nfxu : f x < u\nhu : Ico (f x) u \u2286 v\na : \u03b1\nlfa : f a < u\n\u22a2 a \u2208 f \u207b\u00b9' v\n[PROOFSTEP]\napply hu\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nHu : \u2203 u, f x < u\nu : \u03b3\nfxu : f x < u\nhu : Ico (f x) u \u2286 v\na : \u03b1\nlfa : f a < u\n\u22a2 f a \u2208 Ico (f x) u\n[PROOFSTEP]\nexact \u27e8Hl (f a), lfa\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nHu : \u00ac\u2203 u, f x < u\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\nsimp only [not_exists, not_lt] at Hu \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nHu : \u2200 (x_1 : \u03b3), x_1 \u2264 f x\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\napply Filter.eventually_of_forall\n[GOAL]\ncase neg.hp\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nHu : \u2200 (x_1 : \u03b3), x_1 \u2264 f x\n\u22a2 \u2200 (x : \u03b1), f x \u2208 v\n[PROOFSTEP]\nintro a\n[GOAL]\ncase neg.hp\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nHu : \u2200 (x_1 : \u03b3), x_1 \u2264 f x\na : \u03b1\n\u22a2 f a \u2208 v\n[PROOFSTEP]\nhave : f a = f x := le_antisymm (Hu _) (Hl _)\n[GOAL]\ncase neg.hp\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nHu : \u2200 (x_1 : \u03b3), x_1 \u2264 f x\na : \u03b1\nthis : f a = f x\n\u22a2 f a \u2208 v\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase neg.hp\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\nh\u2081 : LowerSemicontinuousWithinAt f s x\nh\u2082 : UpperSemicontinuousWithinAt f s x\nv : Set \u03b3\nhv : v \u2208 \ud835\udcdd (f x)\nHl : \u2200 (x_1 : \u03b3), f x \u2264 x_1\nHu : \u2200 (x_1 : \u03b3), x_1 \u2264 f x\na : \u03b1\nthis : f a = f x\n\u22a2 f x \u2208 v\n[PROOFSTEP]\nexact mem_of_mem_nhds hv\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 ContinuousAt f x \u2194 LowerSemicontinuousAt f x \u2227 UpperSemicontinuousAt f x\n[PROOFSTEP]\nsimp_rw [\u2190 continuousWithinAt_univ, \u2190 lowerSemicontinuousWithinAt_univ_iff, \u2190 upperSemicontinuousWithinAt_univ_iff,\n continuousWithinAt_iff_lower_upperSemicontinuousWithinAt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 ContinuousOn f s \u2194 LowerSemicontinuousOn f s \u2227 UpperSemicontinuousOn f s\n[PROOFSTEP]\nsimp only [ContinuousOn, continuousWithinAt_iff_lower_upperSemicontinuousWithinAt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 LowerSemicontinuousWithinAt f s x \u2227 UpperSemicontinuousWithinAt f s x) \u2194\n LowerSemicontinuousOn f s \u2227 UpperSemicontinuousOn f s\n[PROOFSTEP]\nexact \u27e8fun H => \u27e8fun x hx => (H x hx).1, fun x hx => (H x hx).2\u27e9, fun H x hx => \u27e8H.1 x hx, H.2 x hx\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : OrderTopology \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 Continuous f \u2194 LowerSemicontinuous f \u2227 UpperSemicontinuous f\n[PROOFSTEP]\nsimp_rw [continuous_iff_continuousOn_univ, continuousOn_iff_lower_upperSemicontinuousOn, lowerSemicontinuousOn_univ_iff,\n upperSemicontinuousOn_univ_iff]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Semicontinuous", "llama_tokens": 42428, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.712232184238947, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.40586739861106036}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : Ring R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsSimpleModule R M\n\u22a2 \u2203 y, 0 \u2260 y\n[PROOFSTEP]\nhave h : (\u22a5 : Submodule R M) \u2260 \u22a4 := bot_ne_top\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : Ring R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsSimpleModule R M\nh : \u22a5 \u2260 \u22a4\n\u22a2 \u2203 y, 0 \u2260 y\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : Ring R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsSimpleModule R M\nh : \u2200 (y : M), 0 = y\n\u22a2 \u22a5 = \u22a4\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b3 : Ring R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsSimpleModule R M\nh : \u2200 (y : M), 0 = y\nx : M\n\u22a2 x \u2208 \u22a5 \u2194 x \u2208 \u22a4\n[PROOFSTEP]\nsimp [Submodule.mem_bot, Submodule.mem_top, h x]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\n\u22a2 IsSimpleModule R { x // x \u2208 m } \u2194 IsAtom m\n[PROOFSTEP]\nrw [\u2190 Set.isSimpleOrder_Iic_iff_isAtom]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\n\u22a2 IsSimpleModule R { x // x \u2208 m } \u2194 IsSimpleOrder \u2191(Set.Iic m)\n[PROOFSTEP]\napply OrderIso.isSimpleOrder_iff\n[GOAL]\ncase f\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\n\u22a2 Submodule R { x // x \u2208 m } \u2243o \u2191(Set.Iic m)\n[PROOFSTEP]\nexact Submodule.MapSubtype.relIso m\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\n\u22a2 IsSimpleModule R (M \u29f8 m) \u2194 IsCoatom m\n[PROOFSTEP]\nrw [\u2190 Set.isSimpleOrder_Ici_iff_isCoatom]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\n\u22a2 IsSimpleModule R (M \u29f8 m) \u2194 IsSimpleOrder \u2191(Set.Ici m)\n[PROOFSTEP]\napply OrderIso.isSimpleOrder_iff\n[GOAL]\ncase f\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\n\u22a2 Submodule R (M \u29f8 m) \u2243o \u2191(Set.Ici m)\n[PROOFSTEP]\nexact Submodule.comapMkQRelIso m\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nA B : Submodule R M\nhAB : A \u2264 B\n\u22a2 A \u22d6 B \u2194 IsSimpleModule R ({ x // x \u2208 B } \u29f8 Submodule.comap (Submodule.subtype B) A)\n[PROOFSTEP]\nset f : Submodule R B \u2243o Set.Iic B := Submodule.MapSubtype.relIso B with hf\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nA B : Submodule R M\nhAB : A \u2264 B\nf : Submodule R { x // x \u2208 B } \u2243o \u2191(Set.Iic B) := Submodule.MapSubtype.relIso B\nhf : f = Submodule.MapSubtype.relIso B\n\u22a2 A \u22d6 B \u2194 IsSimpleModule R ({ x // x \u2208 B } \u29f8 Submodule.comap (Submodule.subtype B) A)\n[PROOFSTEP]\nrw [covby_iff_coatom_Iic hAB, isSimpleModule_iff_isCoatom, \u2190 OrderIso.isCoatom_iff f, hf]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nA B : Submodule R M\nhAB : A \u2264 B\nf : Submodule R { x // x \u2208 B } \u2243o \u2191(Set.Iic B) := Submodule.MapSubtype.relIso B\nhf : f = Submodule.MapSubtype.relIso B\n\u22a2 IsCoatom { val := A, property := hAB } \u2194\n IsCoatom (\u2191(Submodule.MapSubtype.relIso B) (Submodule.comap (Submodule.subtype B) A))\n[PROOFSTEP]\nsimp [-OrderIso.isCoatom_iff, Submodule.MapSubtype.relIso, Submodule.map_comap_subtype, inf_eq_right.2 hAB]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nh : sSup {m | IsSimpleModule R { x // x \u2208 m }} = \u22a4\n\u22a2 sSup {a | IsAtom a} = \u22a4\n[PROOFSTEP]\nsimp_rw [\u2190 h, isSimpleModule_iff_isAtom]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : Ring R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : IsSemisimpleModule R M\n\u22a2 sSup {m | IsSimpleModule R { x // x \u2208 m }} = \u22a4\n[PROOFSTEP]\nsimp_rw [isSimpleModule_iff_isAtom]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : Ring R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : IsSemisimpleModule R M\n\u22a2 sSup {m | IsAtom m} = \u22a4\n[PROOFSTEP]\nexact sSup_atoms_eq_top\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\n\u22a2 IsSemisimpleModule R M \u2192 sSup {m | IsSimpleModule R { x // x \u2208 m }} = \u22a4\n[PROOFSTEP]\nintro\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\na\u271d : IsSemisimpleModule R M\n\u22a2 sSup {m | IsSimpleModule R { x // x \u2208 m }} = \u22a4\n[PROOFSTEP]\nexact IsSemisimpleModule.sSup_simples_eq_top\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : Ring R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : IsSimpleModule R M\nf : M \u2192\u2097[R] N\n\u22a2 Function.Injective \u2191f \u2228 f = 0\n[PROOFSTEP]\nrw [\u2190 ker_eq_bot, \u2190 ker_eq_top]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : Ring R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : IsSimpleModule R M\nf : M \u2192\u2097[R] N\n\u22a2 ker f = \u22a5 \u2228 ker f = \u22a4\n[PROOFSTEP]\napply eq_bot_or_eq_top\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : Ring R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : IsSimpleModule R N\nf : M \u2192\u2097[R] N\n\u22a2 Function.Surjective \u2191f \u2228 f = 0\n[PROOFSTEP]\nrw [\u2190 range_eq_top, \u2190 range_eq_bot, or_comm]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : Ring R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : IsSimpleModule R N\nf : M \u2192\u2097[R] N\n\u22a2 range f = \u22a5 \u2228 range f = \u22a4\n[PROOFSTEP]\napply eq_bot_or_eq_top\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : IsSimpleModule R M\ninst\u271d : IsSimpleModule R N\nf : M \u2192\u2097[R] N\n\u22a2 Function.Bijective \u2191f \u2228 f = 0\n[PROOFSTEP]\nby_cases h : f = 0\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2076 : Ring R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : IsSimpleModule R M\ninst\u271d : IsSimpleModule R N\nf : M \u2192\u2097[R] N\nh : f = 0\n\u22a2 Function.Bijective \u2191f \u2228 f = 0\n[PROOFSTEP]\nright\n[GOAL]\ncase pos.h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : IsSimpleModule R M\ninst\u271d : IsSimpleModule R N\nf : M \u2192\u2097[R] N\nh : f = 0\n\u22a2 f = 0\n[PROOFSTEP]\nexact h\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2076 : Ring R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : IsSimpleModule R M\ninst\u271d : IsSimpleModule R N\nf : M \u2192\u2097[R] N\nh : \u00acf = 0\n\u22a2 Function.Bijective \u2191f \u2228 f = 0\n[PROOFSTEP]\nexact Or.intro_left _ \u27e8injective_of_ne_zero h, surjective_of_ne_zero h\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : Ring R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : IsSimpleModule R N\nf : M \u2192\u2097[R] N\nhf : Function.Surjective \u2191f\n\u22a2 IsCoatom (ker f)\n[PROOFSTEP]\nrw [\u2190 isSimpleModule_iff_isCoatom]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : Ring R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R N\ninst\u271d : IsSimpleModule R N\nf : M \u2192\u2097[R] N\nhf : Function.Surjective \u2191f\n\u22a2 IsSimpleModule R (M \u29f8 ker f)\n[PROOFSTEP]\nexact IsSimpleModule.congr (f.quotKerEquivOfSurjective hf)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : DecidableEq (Module.End R M)\ninst\u271d : IsSimpleModule R M\nsrc\u271d : Ring (Module.End R M) := Module.End.ring\n\u22a2 0 \u2260 1\n[PROOFSTEP]\nhaveI := IsSimpleModule.nontrivial R M\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : DecidableEq (Module.End R M)\ninst\u271d : IsSimpleModule R M\nsrc\u271d : Ring (Module.End R M) := Module.End.ring\nthis : Nontrivial M\n\u22a2 0 \u2260 1\n[PROOFSTEP]\nhave h := exists_pair_ne M\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : DecidableEq (Module.End R M)\ninst\u271d : IsSimpleModule R M\nsrc\u271d : Ring (Module.End R M) := Module.End.ring\nthis : Nontrivial M\nh : \u2203 x y, x \u2260 y\n\u22a2 0 \u2260 1\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : DecidableEq (Module.End R M)\ninst\u271d : IsSimpleModule R M\nsrc\u271d : Ring (Module.End R M) := Module.End.ring\nthis : Nontrivial M\nh : 0 = 1\n\u22a2 \u2200 (x y : M), x = y\n[PROOFSTEP]\nintro x y\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : DecidableEq (Module.End R M)\ninst\u271d : IsSimpleModule R M\nsrc\u271d : Ring (Module.End R M) := Module.End.ring\nthis : Nontrivial M\nh : 0 = 1\nx y : M\n\u22a2 x = y\n[PROOFSTEP]\nsimp_rw [ext_iff, one_apply, zero_apply] at h \n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : DecidableEq (Module.End R M)\ninst\u271d : IsSimpleModule R M\nsrc\u271d : Ring (Module.End R M) := Module.End.ring\nthis : Nontrivial M\nx y : M\nh : \u2200 (x : M), 0 = x\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 h x, h y]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : DecidableEq (Module.End R M)\ninst\u271d : IsSimpleModule R M\nsrc\u271d : Ring (Module.End R M) := Module.End.ring\n\u22a2 \u2200 (a : Module.End R M), a \u2260 0 \u2192 a * a\u207b\u00b9 = 1\n[PROOFSTEP]\nintro a a0\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : DecidableEq (Module.End R M)\ninst\u271d : IsSimpleModule R M\nsrc\u271d : Ring (Module.End R M) := Module.End.ring\na : Module.End R M\na0 : a \u2260 0\n\u22a2 a * a\u207b\u00b9 = 1\n[PROOFSTEP]\nchange a * dite _ _ _ = 1\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : Ring R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : DecidableEq (Module.End R M)\ninst\u271d : IsSimpleModule R M\nsrc\u271d : Ring (Module.End R M) := Module.End.ring\na : Module.End R M\na0 : a \u2260 0\n\u22a2 (a *\n if h : a = 0 then 0\n else\n inverse a (Equiv.ofBijective \u2191a (_ : Function.Bijective \u2191a)).invFun\n (_ :\n Function.LeftInverse (Equiv.ofBijective \u2191a (_ : Function.Bijective \u2191a)).invFun\n (Equiv.ofBijective \u2191a (_ : Function.Bijective \u2191a)).toFun)\n (_ :\n Function.RightInverse (Equiv.ofBijective \u2191a (_ : Function.Bijective \u2191a)).invFun\n (Equiv.ofBijective \u2191a (_ : Function.Bijective \u2191a)).toFun)) =\n 1\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : DecidableEq (Module.End R M)\ninst\u271d : IsSimpleModule R M\nsrc\u271d : Ring (Module.End R M) := Module.End.ring\na : Module.End R M\na0 : a \u2260 0\nx : M\n\u22a2 \u2191(a *\n if h : a = 0 then 0\n else\n inverse a (Equiv.ofBijective \u2191a (_ : Function.Bijective \u2191a)).invFun\n (_ :\n Function.LeftInverse (Equiv.ofBijective \u2191a (_ : Function.Bijective \u2191a)).invFun\n (Equiv.ofBijective \u2191a (_ : Function.Bijective \u2191a)).toFun)\n (_ :\n Function.RightInverse (Equiv.ofBijective \u2191a (_ : Function.Bijective \u2191a)).invFun\n (Equiv.ofBijective \u2191a (_ : Function.Bijective \u2191a)).toFun))\n x =\n \u21911 x\n[PROOFSTEP]\nrw [dif_neg a0, mul_eq_comp, one_apply, comp_apply]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : DecidableEq (Module.End R M)\ninst\u271d : IsSimpleModule R M\nsrc\u271d : Ring (Module.End R M) := Module.End.ring\na : Module.End R M\na0 : a \u2260 0\nx : M\n\u22a2 \u2191a\n (\u2191(inverse a (Equiv.ofBijective \u2191a (_ : Function.Bijective \u2191a)).invFun\n (_ :\n Function.LeftInverse (Equiv.ofBijective \u2191a (_ : Function.Bijective \u2191a)).invFun\n (Equiv.ofBijective \u2191a (_ : Function.Bijective \u2191a)).toFun)\n (_ :\n Function.RightInverse (Equiv.ofBijective \u2191a (_ : Function.Bijective \u2191a)).invFun\n (Equiv.ofBijective \u2191a (_ : Function.Bijective \u2191a)).toFun))\n x) =\n x\n[PROOFSTEP]\nexact (Equiv.ofBijective _ (bijective_of_ne_zero a0)).right_inv x\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nX Y : Submodule R M\nx\u271d : X \u22d6 X \u2294 Y\n\u22a2 Iso (X, X \u2294 Y) (X \u2293 Y, Y)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase val\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nX Y : Submodule R M\nx\u271d : X \u22d6 X \u2294 Y\n\u22a2 ({ x // x \u2208 (X, X \u2294 Y).snd } \u29f8 Submodule.comap (Submodule.subtype (X, X \u2294 Y).snd) (X, X \u2294 Y).fst) \u2243\u2097[R]\n { x // x \u2208 (X \u2293 Y, Y).snd } \u29f8 Submodule.comap (Submodule.subtype (X \u2293 Y, Y).snd) (X \u2293 Y, Y).fst\n[PROOFSTEP]\nrw [sup_comm, inf_comm]\n[GOAL]\ncase val\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nX Y : Submodule R M\nx\u271d : X \u22d6 X \u2294 Y\n\u22a2 ({ x // x \u2208 (X, Y \u2294 X).snd } \u29f8 Submodule.comap (Submodule.subtype (X, Y \u2294 X).snd) (X, Y \u2294 X).fst) \u2243\u2097[R]\n { x // x \u2208 (Y \u2293 X, Y).snd } \u29f8 Submodule.comap (Submodule.subtype (Y \u2293 X, Y).snd) (Y \u2293 X, Y).fst\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase val\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type u_3\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nX Y : Submodule R M\nx\u271d : X \u22d6 X \u2294 Y\n\u22a2 ({ x // x \u2208 Y \u2294 X } \u29f8 Submodule.comap (Submodule.subtype (Y \u2294 X)) X) \u2243\u2097[R]\n { x // x \u2208 Y } \u29f8 Submodule.comap (Submodule.subtype Y) Y \u2293 Submodule.comap (Submodule.subtype Y) X\n[PROOFSTEP]\nexact (LinearMap.quotientInfEquivSupQuotient Y X).symm\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.SimpleModule", "llama_tokens": 7375, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984444, "lm_q2_score": 0.6370308013713525, "lm_q1q2_score": 0.40580822872654837}} {"text": "[GOAL]\nn : Type u\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nR : Type v\ninst\u271d : CommRing R\nA : SpecialLinearGroup n R\n\u22a2 det (adjugate \u2191A) = 1\n[PROOFSTEP]\nrw [det_adjugate, A.prop, one_pow]\n[GOAL]\nn : Type u\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nR : Type v\ninst\u271d : CommRing R\nA B : SpecialLinearGroup n R\n\u22a2 det (\u2191A * \u2191B) = 1\n[PROOFSTEP]\nrw [det_mul, A.prop, B.prop, one_mul]\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nA B : SpecialLinearGroup n R\ninst\u271d : Nontrivial R\ng : SpecialLinearGroup n R\n\u22a2 det \u2191g \u2260 0\n[PROOFSTEP]\nrw [g.det_coe]\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nA B : SpecialLinearGroup n R\ninst\u271d : Nontrivial R\ng : SpecialLinearGroup n R\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nA B : SpecialLinearGroup n R\ninst\u271d : Nontrivial R\ng : SpecialLinearGroup n R\ni : n\nh : \u2191g i = 0\n\u22a2 \u2200 (j : n), \u2191g i j = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nn : Type u\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nR : Type v\ninst\u271d : CommRing R\nsrc\u271d\u00b9 : Monoid (SpecialLinearGroup n R) := monoid\nsrc\u271d : Inv (SpecialLinearGroup n R) := hasInv\nA : SpecialLinearGroup n R\n\u22a2 A\u207b\u00b9 * A = 1\n[PROOFSTEP]\next1\n[GOAL]\ncase a\nn : Type u\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nR : Type v\ninst\u271d : CommRing R\nsrc\u271d\u00b9 : Monoid (SpecialLinearGroup n R) := monoid\nsrc\u271d : Inv (SpecialLinearGroup n R) := hasInv\nA : SpecialLinearGroup n R\ni\u271d j\u271d : n\n\u22a2 \u2191(A\u207b\u00b9 * A) i\u271d j\u271d = \u21911 i\u271d j\u271d\n[PROOFSTEP]\nsimp [adjugate_mul]\n[GOAL]\nn : Type u\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nR : Type v\ninst\u271d : CommRing R\nA : SpecialLinearGroup n R\n\u22a2 comp (\u2191Matrix.toLin' \u2191A) (\u2191Matrix.toLin' \u2191A\u207b\u00b9) = LinearMap.id\n[PROOFSTEP]\nrw [\u2190 toLin'_mul, \u2190 coe_mul, mul_right_inv, coe_one, toLin'_one]\n[GOAL]\nn : Type u\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nR : Type v\ninst\u271d : CommRing R\nA : SpecialLinearGroup n R\n\u22a2 comp (\u2191Matrix.toLin' \u2191A\u207b\u00b9) (\u2191Matrix.toLin' \u2191A) = LinearMap.id\n[PROOFSTEP]\nrw [\u2190 toLin'_mul, \u2190 coe_mul, mul_left_inv, coe_one, toLin'_one]\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nf : R \u2192+* S\ng : SpecialLinearGroup n R\n\u22a2 det (\u2191(RingHom.mapMatrix f) \u2191g) = 1\n[PROOFSTEP]\nrw [\u2190 f.map_det]\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nf : R \u2192+* S\ng : SpecialLinearGroup n R\n\u22a2 \u2191f (det \u2191g) = 1\n[PROOFSTEP]\nsimp [g.prop]\n[GOAL]\nn : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\nR : Type v\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Fact (Even (Fintype.card n))\ng : SpecialLinearGroup n R\n\u22a2 det (-\u2191g) = 1\n[PROOFSTEP]\nsimpa [(@Fact.out <| Even <| Fintype.card n).neg_one_pow, g.det_coe] using det_smul (\u2191\u2098g) (-1)\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nA : SL(2, R)\n\u22a2 det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1\n[PROOFSTEP]\nrw [Matrix.det_fin_two, mul_comm]\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nA : SL(2, R)\n\u22a2 vecCons ![\u2191A 1 1, -\u2191A 0 1] ![![-\u2191A 1 0, \u2191A 0 0]] 1 1 * vecCons ![\u2191A 1 1, -\u2191A 0 1] ![![-\u2191A 1 0, \u2191A 0 0]] 0 0 -\n vecCons ![\u2191A 1 1, -\u2191A 0 1] ![![-\u2191A 1 0, \u2191A 0 0]] 0 1 * vecCons ![\u2191A 1 1, -\u2191A 0 1] ![![-\u2191A 1 0, \u2191A 0 0]] 1 0 =\n 1\n[PROOFSTEP]\nsimp only [cons_val_zero, cons_val_one, head_cons, mul_neg, neg_mul, neg_neg]\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nA : SL(2, R)\n\u22a2 \u2191A 0 0 * \u2191A 1 1 - \u2191A 0 1 * \u2191A 1 0 = 1\n[PROOFSTEP]\nhave := A.2\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nA : SL(2, R)\nthis : det \u2191A = 1\n\u22a2 \u2191A 0 0 * \u2191A 1 1 - \u2191A 0 1 * \u2191A 1 0 = 1\n[PROOFSTEP]\nrw [Matrix.det_fin_two] at this \n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nA : SL(2, R)\nthis : \u2191A 0 0 * \u2191A 1 1 - \u2191A 0 1 * \u2191A 1 0 = 1\n\u22a2 \u2191A 0 0 * \u2191A 1 1 - \u2191A 0 1 * \u2191A 1 0 = 1\n[PROOFSTEP]\nconvert this\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nA : SL(2, R)\n\u22a2 A\u207b\u00b9 =\n { val := ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]],\n property := (_ : det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1) }\n[PROOFSTEP]\next\n[GOAL]\ncase a\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nA : SL(2, R)\ni\u271d j\u271d : Fin 2\n\u22a2 \u2191A\u207b\u00b9 i\u271d j\u271d =\n \u2191{ val := ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]],\n property := (_ : det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1) }\n i\u271d j\u271d\n[PROOFSTEP]\nhave := Matrix.adjugate_fin_two A.1\n[GOAL]\ncase a\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nA : SL(2, R)\ni\u271d j\u271d : Fin 2\nthis : adjugate \u2191A = \u2191of ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]]\n\u22a2 \u2191A\u207b\u00b9 i\u271d j\u271d =\n \u2191{ val := ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]],\n property := (_ : det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1) }\n i\u271d j\u271d\n[PROOFSTEP]\nrw [coe_inv, this]\n[GOAL]\ncase a\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nA : SL(2, R)\ni\u271d j\u271d : Fin 2\nthis : adjugate \u2191A = \u2191of ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]]\n\u22a2 \u2191of ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] i\u271d j\u271d =\n \u2191{ val := ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]],\n property := (_ : det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1) }\n i\u271d j\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nP : SL(2, R) \u2192 Prop\na b c d : R\nhdet : a * d - b * c = 1\n\u22a2 det (\u2191of ![![a, b], ![c, d]]) = 1\n[PROOFSTEP]\nrwa [det_fin_two_of]\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nP : SL(2, R) \u2192 Prop\nh :\n \u2200 (a b c d : R) (hdet : a * d - b * c = 1),\n P { val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) }\ng : SL(2, R)\n\u22a2 P g\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := g\n[GOAL]\ncase mk\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nP : SL(2, R) \u2192 Prop\nh :\n \u2200 (a b c d : R) (hdet : a * d - b * c = 1),\n P { val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) }\nm : Matrix (Fin 2) (Fin 2) R\nhm : det m = 1\n\u22a2 P { val := m, property := hm }\n[PROOFSTEP]\nconvert h (m 0 0) (m 0 1) (m 1 0) (m 1 1) (by rwa [det_fin_two] at hm )\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nP : SL(2, R) \u2192 Prop\nh :\n \u2200 (a b c d : R) (hdet : a * d - b * c = 1),\n P { val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) }\nm : Matrix (Fin 2) (Fin 2) R\nhm : det m = 1\n\u22a2 m 0 0 * m 1 1 - m 0 1 * m 1 0 = 1\n[PROOFSTEP]\nrwa [det_fin_two] at hm \n[GOAL]\ncase h.e'_1.h.e'_3\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nP : SL(2, R) \u2192 Prop\nh :\n \u2200 (a b c d : R) (hdet : a * d - b * c = 1),\n P { val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) }\nm : Matrix (Fin 2) (Fin 2) R\nhm : det m = 1\n\u22a2 m = \u2191of ![![m 0 0, m 0 1], ![m 1 0, m 1 1]]\n[PROOFSTEP]\next i j\n[GOAL]\ncase h.e'_1.h.e'_3.a.h\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nP : SL(2, R) \u2192 Prop\nh :\n \u2200 (a b c d : R) (hdet : a * d - b * c = 1),\n P { val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) }\nm : Matrix (Fin 2) (Fin 2) R\nhm : det m = 1\ni j : Fin 2\n\u22a2 m i j = \u2191of ![![m 0 0, m 0 1], ![m 1 0, m 1 1]] i j\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase h.e'_1.h.e'_3.a.h.head\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nP : SL(2, R) \u2192 Prop\nh :\n \u2200 (a b c d : R) (hdet : a * d - b * c = 1),\n P { val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) }\nm : Matrix (Fin 2) (Fin 2) R\nhm : det m = 1\nj : Fin 2\n\u22a2 m { val := 0, isLt := (_ : 0 < 2) } j = \u2191of ![![m 0 0, m 0 1], ![m 1 0, m 1 1]] { val := 0, isLt := (_ : 0 < 2) } j\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase h.e'_1.h.e'_3.a.h.tail.head\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nP : SL(2, R) \u2192 Prop\nh :\n \u2200 (a b c d : R) (hdet : a * d - b * c = 1),\n P { val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) }\nm : Matrix (Fin 2) (Fin 2) R\nhm : det m = 1\nj : Fin 2\n\u22a2 m { val := 1, isLt := (_ : (fun a => a < 2) 1) } j =\n \u2191of ![![m 0 0, m 0 1], ![m 1 0, m 1 1]] { val := 1, isLt := (_ : (fun a => a < 2) 1) } j\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase h.e'_1.h.e'_3.a.h.head.head\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nP : SL(2, R) \u2192 Prop\nh :\n \u2200 (a b c d : R) (hdet : a * d - b * c = 1),\n P { val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) }\nm : Matrix (Fin 2) (Fin 2) R\nhm : det m = 1\n\u22a2 m { val := 0, isLt := (_ : 0 < 2) } { val := 0, isLt := (_ : 0 < 2) } =\n \u2191of ![![m 0 0, m 0 1], ![m 1 0, m 1 1]] { val := 0, isLt := (_ : 0 < 2) } { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_1.h.e'_3.a.h.head.tail.head\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nP : SL(2, R) \u2192 Prop\nh :\n \u2200 (a b c d : R) (hdet : a * d - b * c = 1),\n P { val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) }\nm : Matrix (Fin 2) (Fin 2) R\nhm : det m = 1\n\u22a2 m { val := 0, isLt := (_ : 0 < 2) } { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n \u2191of ![![m 0 0, m 0 1], ![m 1 0, m 1 1]] { val := 0, isLt := (_ : 0 < 2) }\n { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_1.h.e'_3.a.h.tail.head.head\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nP : SL(2, R) \u2192 Prop\nh :\n \u2200 (a b c d : R) (hdet : a * d - b * c = 1),\n P { val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) }\nm : Matrix (Fin 2) (Fin 2) R\nhm : det m = 1\n\u22a2 m { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 0, isLt := (_ : 0 < 2) } =\n \u2191of ![![m 0 0, m 0 1], ![m 1 0, m 1 1]] { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_1.h.e'_3.a.h.tail.head.tail.head\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type u_1\ninst\u271d : CommRing S\nP : SL(2, R) \u2192 Prop\nh :\n \u2200 (a b c d : R) (hdet : a * d - b * c = 1),\n P { val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) }\nm : Matrix (Fin 2) (Fin 2) R\nhm : det m = 1\n\u22a2 m { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n \u2191of ![![m 0 0, m 0 1], ![m 1 0, m 1 1]] { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nrfl\n[GOAL]\nn : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\nR\u271d : Type v\ninst\u271d\u00b2 : CommRing R\u271d\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nR : Type u_2\ninst\u271d : Field R\ng : SL(2, R)\nhg : \u2191g 1 0 = 0\na b : R\nh : a \u2260 0\n\u22a2 det (\u2191of ![![a, b], ![0, a\u207b\u00b9]]) = 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nn : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\nR\u271d : Type v\ninst\u271d\u00b2 : CommRing R\u271d\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nR : Type u_2\ninst\u271d : Field R\ng : SL(2, R)\nhg : \u2191g 1 0 = 0\n\u22a2 \u2203 a b h, g = { val := \u2191of ![![a, b], ![0, a\u207b\u00b9]], property := (_ : det (\u2191of ![![a, b], ![0, a\u207b\u00b9]]) = 1) }\n[PROOFSTEP]\ninduction' g using Matrix.SpecialLinearGroup.fin_two_induction with a b c d h_det\n[GOAL]\ncase h\nn : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\nR\u271d : Type v\ninst\u271d\u00b2 : CommRing R\u271d\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nR : Type u_2\ninst\u271d : Field R\ng : SL(2, R)\nhg\u271d : \u2191g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : \u2191{ val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) } 1 0 = 0\n\u22a2 \u2203 a_1 b_1 h,\n { val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) } =\n { val := \u2191of ![![a_1, b_1], ![0, a_1\u207b\u00b9]], property := (_ : det (\u2191of ![![a_1, b_1], ![0, a_1\u207b\u00b9]]) = 1) }\n[PROOFSTEP]\nreplace hg : c = 0 := by simpa using hg\n[GOAL]\nn : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\nR\u271d : Type v\ninst\u271d\u00b2 : CommRing R\u271d\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nR : Type u_2\ninst\u271d : Field R\ng : SL(2, R)\nhg\u271d : \u2191g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : \u2191{ val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) } 1 0 = 0\n\u22a2 c = 0\n[PROOFSTEP]\nsimpa using hg\n[GOAL]\ncase h\nn : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\nR\u271d : Type v\ninst\u271d\u00b2 : CommRing R\u271d\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nR : Type u_2\ninst\u271d : Field R\ng : SL(2, R)\nhg\u271d : \u2191g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\n\u22a2 \u2203 a_1 b_1 h,\n { val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) } =\n { val := \u2191of ![![a_1, b_1], ![0, a_1\u207b\u00b9]], property := (_ : det (\u2191of ![![a_1, b_1], ![0, a_1\u207b\u00b9]]) = 1) }\n[PROOFSTEP]\nhave had : a * d = 1 := by rwa [hg, mul_zero, sub_zero] at h_det \n[GOAL]\nn : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\nR\u271d : Type v\ninst\u271d\u00b2 : CommRing R\u271d\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nR : Type u_2\ninst\u271d : Field R\ng : SL(2, R)\nhg\u271d : \u2191g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\n\u22a2 a * d = 1\n[PROOFSTEP]\nrwa [hg, mul_zero, sub_zero] at h_det \n[GOAL]\ncase h\nn : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\nR\u271d : Type v\ninst\u271d\u00b2 : CommRing R\u271d\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nR : Type u_2\ninst\u271d : Field R\ng : SL(2, R)\nhg\u271d : \u2191g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\nhad : a * d = 1\n\u22a2 \u2203 a_1 b_1 h,\n { val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) } =\n { val := \u2191of ![![a_1, b_1], ![0, a_1\u207b\u00b9]], property := (_ : det (\u2191of ![![a_1, b_1], ![0, a_1\u207b\u00b9]]) = 1) }\n[PROOFSTEP]\nrefine' \u27e8a, b, left_ne_zero_of_mul_eq_one had, _\u27e9\n[GOAL]\ncase h\nn : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\nR\u271d : Type v\ninst\u271d\u00b2 : CommRing R\u271d\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nR : Type u_2\ninst\u271d : Field R\ng : SL(2, R)\nhg\u271d : \u2191g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\nhad : a * d = 1\n\u22a2 { val := \u2191of ![![a, b], ![c, d]], property := (_ : det (\u2191of ![![a, b], ![c, d]]) = 1) } =\n { val := \u2191of ![![a, b], ![0, a\u207b\u00b9]], property := (_ : det (\u2191of ![![a, b], ![0, a\u207b\u00b9]]) = 1) }\n[PROOFSTEP]\nsimp_rw [eq_inv_of_mul_eq_one_right had, hg]\n[GOAL]\n\u22a2 det (\u2191of ![![0, -1], ![1, 0]]) = 1\n[PROOFSTEP]\nnorm_num [Matrix.det_fin_two_of]\n[GOAL]\n\u22a2 det (\u2191of ![![1, 1], ![0, 1]]) = 1\n[PROOFSTEP]\nnorm_num [Matrix.det_fin_two_of]\n[GOAL]\n\u22a2 \u2191T\u207b\u00b9 = \u2191of ![![1, -1], ![0, 1]]\n[PROOFSTEP]\nsimp [coe_inv, coe_T, adjugate_fin_two]\n[GOAL]\nn : \u2124\n\u22a2 \u2191(T ^ n) = \u2191of ![![1, n], ![0, 1]]\n[PROOFSTEP]\ninduction' n using Int.induction_on with n h n h\n[GOAL]\ncase hz\n\u22a2 \u2191(T ^ 0) = \u2191of ![![1, 0], ![0, 1]]\n[PROOFSTEP]\nrw [zpow_zero, coe_one, Matrix.one_fin_two]\n[GOAL]\ncase hp\nn : \u2115\nh : \u2191(T ^ \u2191n) = \u2191of ![![1, \u2191n], ![0, 1]]\n\u22a2 \u2191(T ^ (\u2191n + 1)) = \u2191of ![![1, \u2191n + 1], ![0, 1]]\n[PROOFSTEP]\nsimp_rw [zpow_add, zpow_one, coe_mul, h, coe_T, Matrix.mul_fin_two]\n -- Porting note: was congrm !![_, _; _, _]\n[GOAL]\ncase hp\nn : \u2115\nh : \u2191(T ^ \u2191n) = \u2191of ![![1, \u2191n], ![0, 1]]\n\u22a2 \u2191of ![![1 * 1 + \u2191n * 0, 1 * 1 + \u2191n * 1], ![0 * 1 + 1 * 0, 0 * 1 + 1 * 1]] = \u2191of ![![1, \u2191n + 1], ![0, 1]]\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase hn\nn : \u2115\nh : \u2191(T ^ (-\u2191n)) = \u2191of ![![1, -\u2191n], ![0, 1]]\n\u22a2 \u2191(T ^ (-\u2191n - 1)) = \u2191of ![![1, -\u2191n - 1], ![0, 1]]\n[PROOFSTEP]\nsimp_rw [zpow_sub, zpow_one, coe_mul, h, coe_T_inv, Matrix.mul_fin_two]\n -- Porting note: was congrm !![_, _; _, _]\n[GOAL]\ncase hn\nn : \u2115\nh : \u2191(T ^ (-\u2191n)) = \u2191of ![![1, -\u2191n], ![0, 1]]\n\u22a2 \u2191of ![![1 * 1 + -\u2191n * 0, 1 * -1 + -\u2191n * 1], ![0 * 1 + 1 * 0, 0 * -1 + 1 * 1]] = \u2191of ![![1, -\u2191n - 1], ![0, 1]]\n[PROOFSTEP]\nring_nf\n[GOAL]\nn : \u2124\ng : SL(2, \u2124)\n\u22a2 \u2191(T ^ n * g) 1 = \u2191g 1\n[PROOFSTEP]\next j\n[GOAL]\ncase h\nn : \u2124\ng : SL(2, \u2124)\nj : Fin 2\n\u22a2 \u2191(T ^ n * g) 1 j = \u2191g 1 j\n[PROOFSTEP]\nsimp [coe_T_zpow, Matrix.vecMul, Matrix.dotProduct, Fin.sum_univ_succ, vecTail]\n[GOAL]\ng : SL(2, \u2124)\n\u22a2 \u2191(T * g) 1 = \u2191g 1\n[PROOFSTEP]\nsimpa using T_pow_mul_apply_one 1 g\n[GOAL]\ng : SL(2, \u2124)\n\u22a2 \u2191(T\u207b\u00b9 * g) 1 = \u2191g 1\n[PROOFSTEP]\nsimpa using T_pow_mul_apply_one (-1) g\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "llama_tokens": 9583, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.63341027059799, "lm_q1q2_score": 0.4057853301689145}} {"text": "[GOAL]\nX : Type u_1\ns : Set X\n\u22a2 (some '' s)\u1d9c = some '' s\u1d9c \u222a {\u221e}\n[PROOFSTEP]\nrw [coe_injective.compl_image_eq, compl_range_coe]\n[GOAL]\nX : Type u_1\nx : OnePoint X\n\u22a2 x \u2260 \u221e \u2194 \u2203 y, \u2191y = x\n[PROOFSTEP]\ninduction x using OnePoint.rec\n[GOAL]\ncase h\u2081\nX : Type u_1\n\u22a2 \u221e \u2260 \u221e \u2194 \u2203 y, \u2191y = \u221e\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nX : Type u_1\nx\u271d : X\n\u22a2 \u2191x\u271d \u2260 \u221e \u2194 \u2203 y, \u2191y = \u2191x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\nx : OnePoint X\n\u22a2 \u00acx \u2208 range some \u2194 x = \u221e\n[PROOFSTEP]\nrw [\u2190 mem_compl_iff, compl_range_coe, mem_singleton_iff]\n[GOAL]\nX : Type u_1\n\u22a2 some \u207b\u00b9' {\u221e} = \u2205\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX : Type u_1\nx\u271d : X\n\u22a2 x\u271d \u2208 some \u207b\u00b9' {\u221e} \u2194 x\u271d \u2208 \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u22a2 (fun s => (\u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c) \u2227 IsOpen (some \u207b\u00b9' s)) univ\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns t : Set (OnePoint X)\n\u22a2 (fun s => (\u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c) \u2227 IsOpen (some \u207b\u00b9' s)) s \u2192\n (fun s => (\u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c) \u2227 IsOpen (some \u207b\u00b9' s)) t \u2192\n (fun s => (\u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c) \u2227 IsOpen (some \u207b\u00b9' s)) (s \u2229 t)\n[PROOFSTEP]\nrintro \u27e8hms, hs\u27e9 \u27e8hmt, ht\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns t : Set (OnePoint X)\nhms : \u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c\nhs : IsOpen (some \u207b\u00b9' s)\nhmt : \u221e \u2208 t \u2192 IsCompact (some \u207b\u00b9' t)\u1d9c\nht : IsOpen (some \u207b\u00b9' t)\n\u22a2 (\u221e \u2208 s \u2229 t \u2192 IsCompact (some \u207b\u00b9' (s \u2229 t))\u1d9c) \u2227 IsOpen (some \u207b\u00b9' (s \u2229 t))\n[PROOFSTEP]\nrefine' \u27e8_, hs.inter ht\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns t : Set (OnePoint X)\nhms : \u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c\nhs : IsOpen (some \u207b\u00b9' s)\nhmt : \u221e \u2208 t \u2192 IsCompact (some \u207b\u00b9' t)\u1d9c\nht : IsOpen (some \u207b\u00b9' t)\n\u22a2 \u221e \u2208 s \u2229 t \u2192 IsCompact (some \u207b\u00b9' (s \u2229 t))\u1d9c\n[PROOFSTEP]\nrintro \u27e8hms', hmt'\u27e9\n[GOAL]\ncase intro.intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns t : Set (OnePoint X)\nhms : \u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c\nhs : IsOpen (some \u207b\u00b9' s)\nhmt : \u221e \u2208 t \u2192 IsCompact (some \u207b\u00b9' t)\u1d9c\nht : IsOpen (some \u207b\u00b9' t)\nhms' : \u221e \u2208 s\nhmt' : \u221e \u2208 t\n\u22a2 IsCompact (some \u207b\u00b9' (s \u2229 t))\u1d9c\n[PROOFSTEP]\nsimpa [compl_inter] using (hms hms').union (hmt hmt')\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\nS : Set (Set (OnePoint X))\nho : \u2200 (t : Set (OnePoint X)), t \u2208 S \u2192 (fun s => (\u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c) \u2227 IsOpen (some \u207b\u00b9' s)) t\n\u22a2 (fun s => (\u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c) \u2227 IsOpen (some \u207b\u00b9' s)) (\u22c3\u2080 S)\n[PROOFSTEP]\nsuffices IsOpen ((\u2191) \u207b\u00b9' \u22c3\u2080 S : Set X) by\n refine' \u27e8_, this\u27e9\n rintro \u27e8s, hsS : s \u2208 S, hs : \u221e \u2208 s\u27e9\n refine' isCompact_of_isClosed_subset ((ho s hsS).1 hs) this.isClosed_compl _\n exact compl_subset_compl.mpr (preimage_mono <| subset_sUnion_of_mem hsS)\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\nS : Set (Set (OnePoint X))\nho : \u2200 (t : Set (OnePoint X)), t \u2208 S \u2192 (fun s => (\u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c) \u2227 IsOpen (some \u207b\u00b9' s)) t\nthis : IsOpen (some \u207b\u00b9' \u22c3\u2080 S)\n\u22a2 (fun s => (\u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c) \u2227 IsOpen (some \u207b\u00b9' s)) (\u22c3\u2080 S)\n[PROOFSTEP]\nrefine' \u27e8_, this\u27e9\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\nS : Set (Set (OnePoint X))\nho : \u2200 (t : Set (OnePoint X)), t \u2208 S \u2192 (fun s => (\u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c) \u2227 IsOpen (some \u207b\u00b9' s)) t\nthis : IsOpen (some \u207b\u00b9' \u22c3\u2080 S)\n\u22a2 \u221e \u2208 \u22c3\u2080 S \u2192 IsCompact (some \u207b\u00b9' \u22c3\u2080 S)\u1d9c\n[PROOFSTEP]\nrintro \u27e8s, hsS : s \u2208 S, hs : \u221e \u2208 s\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\nS : Set (Set (OnePoint X))\nho : \u2200 (t : Set (OnePoint X)), t \u2208 S \u2192 (fun s => (\u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c) \u2227 IsOpen (some \u207b\u00b9' s)) t\nthis : IsOpen (some \u207b\u00b9' \u22c3\u2080 S)\ns : Set (OnePoint X)\nhsS : s \u2208 S\nhs : \u221e \u2208 s\n\u22a2 IsCompact (some \u207b\u00b9' \u22c3\u2080 S)\u1d9c\n[PROOFSTEP]\nrefine' isCompact_of_isClosed_subset ((ho s hsS).1 hs) this.isClosed_compl _\n[GOAL]\ncase intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\nS : Set (Set (OnePoint X))\nho : \u2200 (t : Set (OnePoint X)), t \u2208 S \u2192 (fun s => (\u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c) \u2227 IsOpen (some \u207b\u00b9' s)) t\nthis : IsOpen (some \u207b\u00b9' \u22c3\u2080 S)\ns : Set (OnePoint X)\nhsS : s \u2208 S\nhs : \u221e \u2208 s\n\u22a2 (some \u207b\u00b9' \u22c3\u2080 S)\u1d9c \u2286 (some \u207b\u00b9' s)\u1d9c\n[PROOFSTEP]\nexact compl_subset_compl.mpr (preimage_mono <| subset_sUnion_of_mem hsS)\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\nS : Set (Set (OnePoint X))\nho : \u2200 (t : Set (OnePoint X)), t \u2208 S \u2192 (fun s => (\u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c) \u2227 IsOpen (some \u207b\u00b9' s)) t\n\u22a2 IsOpen (some \u207b\u00b9' \u22c3\u2080 S)\n[PROOFSTEP]\nrw [preimage_sUnion]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\nS : Set (Set (OnePoint X))\nho : \u2200 (t : Set (OnePoint X)), t \u2208 S \u2192 (fun s => (\u221e \u2208 s \u2192 IsCompact (some \u207b\u00b9' s)\u1d9c) \u2227 IsOpen (some \u207b\u00b9' s)) t\n\u22a2 IsOpen (\u22c3 (t : Set (OnePoint X)) (_ : t \u2208 S), some \u207b\u00b9' t)\n[PROOFSTEP]\nexact isOpen_biUnion fun s hs => (ho s hs).2\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nh : \u221e \u2208 s\n\u22a2 IsOpen s \u2194 IsCompact (some \u207b\u00b9' s)\u1d9c \u2227 IsOpen (some \u207b\u00b9' s)\n[PROOFSTEP]\nsimp [isOpen_def, h]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nh : \u221e \u2208 s\n\u22a2 IsOpen s \u2194 IsClosed (some \u207b\u00b9' s)\u1d9c \u2227 IsCompact (some \u207b\u00b9' s)\u1d9c\n[PROOFSTEP]\nsimp only [isOpen_iff_of_mem' h, isClosed_compl_iff, and_comm]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nh : \u00ac\u221e \u2208 s\n\u22a2 IsOpen s \u2194 IsOpen (some \u207b\u00b9' s)\n[PROOFSTEP]\nsimp [isOpen_def, h]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nh : \u221e \u2208 s\n\u22a2 IsClosed s \u2194 IsClosed (some \u207b\u00b9' s)\n[PROOFSTEP]\nhave : \u221e \u2209 s\u1d9c := fun H => H h\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nh : \u221e \u2208 s\nthis : \u00ac\u221e \u2208 s\u1d9c\n\u22a2 IsClosed s \u2194 IsClosed (some \u207b\u00b9' s)\n[PROOFSTEP]\nrw [\u2190 isOpen_compl_iff, isOpen_iff_of_not_mem this, \u2190 isOpen_compl_iff, preimage_compl]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nh : \u00ac\u221e \u2208 s\n\u22a2 IsClosed s \u2194 IsClosed (some \u207b\u00b9' s) \u2227 IsCompact (some \u207b\u00b9' s)\n[PROOFSTEP]\nrw [\u2190 isOpen_compl_iff, isOpen_iff_of_mem (mem_compl h), \u2190 preimage_compl, compl_compl]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns\u271d : Set (OnePoint X)\nt s : Set X\n\u22a2 IsOpen (some '' s) \u2194 IsOpen s\n[PROOFSTEP]\nrw [isOpen_iff_of_not_mem infty_not_mem_image_coe, preimage_image_eq _ coe_injective]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns\u271d : Set (OnePoint X)\nt s : Set X\n\u22a2 IsOpen (some '' s)\u1d9c \u2194 IsClosed s \u2227 IsCompact s\n[PROOFSTEP]\nrw [isOpen_iff_of_mem, \u2190 preimage_compl, compl_compl, preimage_image_eq _ coe_injective]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns\u271d : Set (OnePoint X)\nt s : Set X\n\u22a2 \u221e \u2208 (some '' s)\u1d9c\n[PROOFSTEP]\nexact infty_not_mem_image_coe\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns\u271d : Set (OnePoint X)\nt s : Set X\n\u22a2 IsClosed (some '' s) \u2194 IsClosed s \u2227 IsCompact s\n[PROOFSTEP]\nrw [\u2190 isOpen_compl_iff, isOpen_compl_image_coe]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n\u22a2 IsClosed {\u221e}\n[PROOFSTEP]\nrw [\u2190 compl_range_coe, isClosed_compl_iff]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n\u22a2 IsOpen (range some)\n[PROOFSTEP]\nexact isOpen_range_coe\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nx : X\nh : NeBot (\ud835\udcdd[{x}\u1d9c] x)\n\u22a2 NeBot (\ud835\udcdd[{\u2191x}\u1d9c] \u2191x)\n[PROOFSTEP]\nsimpa [nhdsWithin_coe, preimage, coe_eq_coe] using h.map some\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n\u22a2 \ud835\udcdd[{\u221e}\u1d9c] \u221e = map some (coclosedCompact X)\n[PROOFSTEP]\nrefine' (nhdsWithin_basis_open \u221e _).ext (hasBasis_coclosedCompact.map _) _ _\n[GOAL]\ncase refine'_1\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n\u22a2 \u2200 (i : Set (OnePoint X)), \u221e \u2208 i \u2227 IsOpen i \u2192 \u2203 i', (IsClosed i' \u2227 IsCompact i') \u2227 some '' i'\u1d9c \u2286 i \u2229 {\u221e}\u1d9c\n[PROOFSTEP]\nrintro s \u27e8hs, hso\u27e9\n[GOAL]\ncase refine'_1.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns\u271d : Set (OnePoint X)\nt : Set X\ns : Set (OnePoint X)\nhs : \u221e \u2208 s\nhso : IsOpen s\n\u22a2 \u2203 i', (IsClosed i' \u2227 IsCompact i') \u2227 some '' i'\u1d9c \u2286 s \u2229 {\u221e}\u1d9c\n[PROOFSTEP]\nrefine' \u27e8_, (isOpen_iff_of_mem hs).mp hso, _\u27e9\n[GOAL]\ncase refine'_1.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns\u271d : Set (OnePoint X)\nt : Set X\ns : Set (OnePoint X)\nhs : \u221e \u2208 s\nhso : IsOpen s\n\u22a2 some '' (some \u207b\u00b9' s)\u1d9c\u1d9c \u2286 s \u2229 {\u221e}\u1d9c\n[PROOFSTEP]\nsimp [Subset.rfl]\n[GOAL]\ncase refine'_2\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n\u22a2 \u2200 (i' : Set X), IsClosed i' \u2227 IsCompact i' \u2192 \u2203 i, (\u221e \u2208 i \u2227 IsOpen i) \u2227 i \u2229 {\u221e}\u1d9c \u2286 some '' i'\u1d9c\n[PROOFSTEP]\nrintro s \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase refine'_2.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns\u271d : Set (OnePoint X)\nt s : Set X\nh\u2081 : IsClosed s\nh\u2082 : IsCompact s\n\u22a2 \u2203 i, (\u221e \u2208 i \u2227 IsOpen i) \u2227 i \u2229 {\u221e}\u1d9c \u2286 some '' s\u1d9c\n[PROOFSTEP]\nrefine' \u27e8_, \u27e8mem_compl infty_not_mem_image_coe, isOpen_compl_image_coe.2 \u27e8h\u2081, h\u2082\u27e9\u27e9, _\u27e9\n[GOAL]\ncase refine'_2.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns\u271d : Set (OnePoint X)\nt s : Set X\nh\u2081 : IsClosed s\nh\u2082 : IsCompact s\n\u22a2 (some '' s)\u1d9c \u2229 {\u221e}\u1d9c \u2286 some '' s\u1d9c\n[PROOFSTEP]\nsimp [compl_image_coe, \u2190 diff_eq, subset_preimage_image]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d : NoncompactSpace X\n\u22a2 NeBot (\ud835\udcdd[{\u221e}\u1d9c] \u221e)\n[PROOFSTEP]\nrw [nhdsWithin_compl_infty_eq]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d : NoncompactSpace X\n\u22a2 NeBot (map some (coclosedCompact X))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n\u22a2 \ud835\udcdd \u221e = map some (coclosedCompact X) \u2294 pure \u221e\n[PROOFSTEP]\nrw [\u2190 nhdsWithin_compl_infty_eq, nhdsWithin_compl_singleton_sup_pure]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n\u22a2 HasBasis (\ud835\udcdd \u221e) (fun s => IsClosed s \u2227 IsCompact s) fun s => some '' s\u1d9c \u222a {\u221e}\n[PROOFSTEP]\nrw [nhds_infty_eq]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n\u22a2 HasBasis (map some (coclosedCompact X) \u2294 pure \u221e) (fun s => IsClosed s \u2227 IsCompact s) fun s => some '' s\u1d9c \u222a {\u221e}\n[PROOFSTEP]\nexact (hasBasis_coclosedCompact.map _).sup_pure _\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n\u22a2 comap some (\ud835\udcdd \u221e) = coclosedCompact X\n[PROOFSTEP]\nsimp [nhds_infty_eq, comap_sup, comap_map coe_injective]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nf : Filter (OnePoint X)\n\u22a2 f \u2264 \ud835\udcdd \u221e \u2194 \u2200 (s : Set X), IsClosed s \u2192 IsCompact s \u2192 some '' s\u1d9c \u222a {\u221e} \u2208 f\n[PROOFSTEP]\nsimp only [hasBasis_nhds_infty.ge_iff, and_imp]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nf : Ultrafilter (OnePoint X)\n\u22a2 \u2191f \u2264 \ud835\udcdd \u221e \u2194 \u2200 (s : Set X), IsClosed s \u2192 IsCompact s \u2192 \u00acsome '' s \u2208 f\n[PROOFSTEP]\nsimp only [le_nhds_infty, \u2190 compl_image_coe, Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n\u03b1 : Type u_2\nf : OnePoint X \u2192 \u03b1\nl : Filter \u03b1\n\u22a2 Tendsto f (\ud835\udcdd \u221e) l \u2194 Tendsto f (pure \u221e) l \u2227 Tendsto (f \u2218 some) (coclosedCompact X) l\n[PROOFSTEP]\nsimp [nhds_infty_eq, and_comm]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n\u03b1 : Type u_2\nf : OnePoint X \u2192 \u03b1\nl : Filter \u03b1\n\u22a2 Tendsto f (pure \u221e) l \u2227 Tendsto (f \u2218 some) (coclosedCompact X) l \u2194\n \u2200 (s : Set \u03b1), s \u2208 l \u2192 f \u221e \u2208 s \u2227 \u2203 t, IsClosed t \u2227 IsCompact t \u2227 MapsTo (f \u2218 some) t\u1d9c s\n[PROOFSTEP]\nsimp only [tendsto_pure_left, hasBasis_coclosedCompact.tendsto_left_iff, forall_and, and_assoc, exists_prop]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nY : Type u_2\ninst\u271d : TopologicalSpace Y\nf : OnePoint X \u2192 Y\n\u22a2 Tendsto (f \u2218 some) (coclosedCompact X) (\ud835\udcdd (f \u221e)) \u2194\n \u2200 (s : Set Y), s \u2208 \ud835\udcdd (f \u221e) \u2192 \u2203 t, IsClosed t \u2227 IsCompact t \u2227 MapsTo (f \u2218 some) t\u1d9c s\n[PROOFSTEP]\nsimp only [hasBasis_coclosedCompact.tendsto_left_iff, and_assoc]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nY : Type u_2\ninst\u271d : TopologicalSpace Y\nf : OnePoint X \u2192 Y\nx : X\n\u22a2 ContinuousAt f \u2191x \u2194 ContinuousAt (f \u2218 some) x\n[PROOFSTEP]\nrw [ContinuousAt, nhds_coe_eq, tendsto_map'_iff, ContinuousAt]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nY : Type u_2\ninst\u271d : TopologicalSpace Y\nf : OnePoint X \u2192 Y\nx : X\n\u22a2 Tendsto (f \u2218 some) (\ud835\udcdd x) (\ud835\udcdd (f \u2191x)) \u2194 Tendsto (f \u2218 some) (\ud835\udcdd x) (\ud835\udcdd ((f \u2218 some) x))\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d : NoncompactSpace X\n\u22a2 DenseRange some\n[PROOFSTEP]\nrw [DenseRange, \u2190 compl_infty]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d : NoncompactSpace X\n\u22a2 Dense {\u221e}\u1d9c\n[PROOFSTEP]\nexact dense_compl_singleton _\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nx y : OnePoint X\n\u22a2 Inseparable x y \u2194 x = \u221e \u2227 y = \u221e \u2228 \u2203 x', x = \u2191x' \u2227 \u2203 y', y = \u2191y' \u2227 Inseparable x' y'\n[PROOFSTEP]\ninduction x using OnePoint.rec\n[GOAL]\ncase h\u2081\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ny : OnePoint X\n\u22a2 Inseparable \u221e y \u2194 \u221e = \u221e \u2227 y = \u221e \u2228 \u2203 x', \u221e = \u2191x' \u2227 \u2203 y', y = \u2191y' \u2227 Inseparable x' y'\n[PROOFSTEP]\ninduction y using OnePoint.rec\n[GOAL]\ncase h\u2082\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ny : OnePoint X\nx\u271d : X\n\u22a2 Inseparable (\u2191x\u271d) y \u2194 \u2191x\u271d = \u221e \u2227 y = \u221e \u2228 \u2203 x', \u2191x\u271d = \u2191x' \u2227 \u2203 y', y = \u2191y' \u2227 Inseparable x' y'\n[PROOFSTEP]\ninduction y using OnePoint.rec\n[GOAL]\ncase h\u2081.h\u2081\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n\u22a2 Inseparable \u221e \u221e \u2194 \u221e = \u221e \u2227 \u221e = \u221e \u2228 \u2203 x', \u221e = \u2191x' \u2227 \u2203 y', \u221e = \u2191y' \u2227 Inseparable x' y'\n[PROOFSTEP]\nsimp [not_inseparable_infty_coe, not_inseparable_coe_infty, coe_eq_coe, Inseparable.refl]\n[GOAL]\ncase h\u2081.h\u2082\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nx\u271d : X\n\u22a2 Inseparable \u221e \u2191x\u271d \u2194 \u221e = \u221e \u2227 \u2191x\u271d = \u221e \u2228 \u2203 x', \u221e = \u2191x' \u2227 \u2203 y', \u2191x\u271d = \u2191y' \u2227 Inseparable x' y'\n[PROOFSTEP]\nsimp [not_inseparable_infty_coe, not_inseparable_coe_infty, coe_eq_coe, Inseparable.refl]\n[GOAL]\ncase h\u2082.h\u2081\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nx\u271d : X\n\u22a2 Inseparable \u2191x\u271d \u221e \u2194 \u2191x\u271d = \u221e \u2227 \u221e = \u221e \u2228 \u2203 x', \u2191x\u271d = \u2191x' \u2227 \u2203 y', \u221e = \u2191y' \u2227 Inseparable x' y'\n[PROOFSTEP]\nsimp [not_inseparable_infty_coe, not_inseparable_coe_infty, coe_eq_coe, Inseparable.refl]\n[GOAL]\ncase h\u2082.h\u2082\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nx\u271d\u00b9 x\u271d : X\n\u22a2 Inseparable \u2191x\u271d\u00b9 \u2191x\u271d \u2194 \u2191x\u271d\u00b9 = \u221e \u2227 \u2191x\u271d = \u221e \u2228 \u2203 x', \u2191x\u271d\u00b9 = \u2191x' \u2227 \u2203 y', \u2191x\u271d = \u2191y' \u2227 Inseparable x' y'\n[PROOFSTEP]\nsimp [not_inseparable_infty_coe, not_inseparable_coe_infty, coe_eq_coe, Inseparable.refl]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n\u22a2 IsCompact univ\n[PROOFSTEP]\nhave : Tendsto ((\u2191) : X \u2192 OnePoint X) (cocompact X) (\ud835\udcdd \u221e) :=\n by\n rw [nhds_infty_eq]\n exact (tendsto_map.mono_left cocompact_le_coclosedCompact).mono_right le_sup_left\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n\u22a2 Tendsto some (cocompact X) (\ud835\udcdd \u221e)\n[PROOFSTEP]\nrw [nhds_infty_eq]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n\u22a2 Tendsto some (cocompact X) (map some (coclosedCompact X) \u2294 pure \u221e)\n[PROOFSTEP]\nexact (tendsto_map.mono_left cocompact_le_coclosedCompact).mono_right le_sup_left\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nthis : Tendsto some (cocompact X) (\ud835\udcdd \u221e)\n\u22a2 IsCompact univ\n[PROOFSTEP]\nrw [\u2190 insert_none_range_some X]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nthis : Tendsto some (cocompact X) (\ud835\udcdd \u221e)\n\u22a2 IsCompact (insert none (range Option.some))\n[PROOFSTEP]\nexact this.isCompact_insert_range_of_cocompact continuous_coe\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d : T0Space X\n\u22a2 T0Space (OnePoint X)\n[PROOFSTEP]\nrefine' \u27e8fun x y hxy => _\u27e9\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d : T0Space X\nx y : OnePoint X\nhxy : Inseparable x y\n\u22a2 x = y\n[PROOFSTEP]\nrcases inseparable_iff.1 hxy with (\u27e8rfl, rfl\u27e9 | \u27e8x, rfl, y, rfl, h\u27e9)\n[GOAL]\ncase inl.intro\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d : T0Space X\nhxy : Inseparable \u221e \u221e\n\u22a2 \u221e = \u221e\ncase inr.intro.intro.intro.intro\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d : T0Space X\nx y : X\nh : Inseparable x y\nhxy : Inseparable \u2191x \u2191y\n\u22a2 \u2191x = \u2191y\n[PROOFSTEP]\nexacts [rfl, congr_arg some h.eq]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d : T1Space X\nz : OnePoint X\n\u22a2 IsClosed {z}\n[PROOFSTEP]\ninduction z using OnePoint.rec\n[GOAL]\ncase h\u2081\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d : T1Space X\n\u22a2 IsClosed {\u221e}\n[PROOFSTEP]\nexact isClosed_infty\n[GOAL]\ncase h\u2082\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d : T1Space X\nx\u271d : X\n\u22a2 IsClosed {\u2191x\u271d}\n[PROOFSTEP]\nrw [\u2190 image_singleton, isClosed_image_coe]\n[GOAL]\ncase h\u2082\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d : T1Space X\nx\u271d : X\n\u22a2 IsClosed {x\u271d} \u2227 IsCompact {x\u271d}\n[PROOFSTEP]\nexact \u27e8isClosed_singleton, isCompact_singleton\u27e9\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\n\u22a2 NormalSpace (OnePoint X)\n[PROOFSTEP]\nhave key : \u2200 z : X, Disjoint (\ud835\udcdd (some z)) (\ud835\udcdd \u221e) := fun z =>\n by\n rw [nhds_infty_eq, disjoint_sup_right, nhds_coe_eq, coclosedCompact_eq_cocompact, disjoint_map coe_injective, \u2190\n principal_singleton, disjoint_principal_right, compl_infty]\n exact \u27e8disjoint_nhds_cocompact z, range_mem_map\u27e9\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nz : X\n\u22a2 Disjoint (\ud835\udcdd \u2191z) (\ud835\udcdd \u221e)\n[PROOFSTEP]\nrw [nhds_infty_eq, disjoint_sup_right, nhds_coe_eq, coclosedCompact_eq_cocompact, disjoint_map coe_injective, \u2190\n principal_singleton, disjoint_principal_right, compl_infty]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nz : X\n\u22a2 Disjoint (\ud835\udcdd z) (cocompact X) \u2227 range some \u2208 map some (\ud835\udcdd z)\n[PROOFSTEP]\nexact \u27e8disjoint_nhds_cocompact z, range_mem_map\u27e9\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nkey : \u2200 (z : X), Disjoint (\ud835\udcdd \u2191z) (\ud835\udcdd \u221e)\n\u22a2 NormalSpace (OnePoint X)\n[PROOFSTEP]\nsuffices : T2Space (OnePoint X)\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nkey : \u2200 (z : X), Disjoint (\ud835\udcdd \u2191z) (\ud835\udcdd \u221e)\nthis : T2Space (OnePoint X)\n\u22a2 NormalSpace (OnePoint X)\ncase this\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nkey : \u2200 (z : X), Disjoint (\ud835\udcdd \u2191z) (\ud835\udcdd \u221e)\n\u22a2 T2Space (OnePoint X)\n[PROOFSTEP]\nexact normalOfCompactT2\n[GOAL]\ncase this\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nkey : \u2200 (z : X), Disjoint (\ud835\udcdd \u2191z) (\ud835\udcdd \u221e)\n\u22a2 T2Space (OnePoint X)\n[PROOFSTEP]\nrefine t2Space_iff_disjoint_nhds.2 fun x y hxy => ?_\n[GOAL]\ncase this\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nkey : \u2200 (z : X), Disjoint (\ud835\udcdd \u2191z) (\ud835\udcdd \u221e)\nx y : OnePoint X\nhxy : x \u2260 y\n\u22a2 Disjoint (\ud835\udcdd x) (\ud835\udcdd y)\n[PROOFSTEP]\ninduction x using OnePoint.rec\n[GOAL]\ncase this.h\u2081\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nkey : \u2200 (z : X), Disjoint (\ud835\udcdd \u2191z) (\ud835\udcdd \u221e)\ny : OnePoint X\nhxy : \u221e \u2260 y\n\u22a2 Disjoint (\ud835\udcdd \u221e) (\ud835\udcdd y)\n[PROOFSTEP]\ninduction y using OnePoint.rec\n[GOAL]\ncase this.h\u2082\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nkey : \u2200 (z : X), Disjoint (\ud835\udcdd \u2191z) (\ud835\udcdd \u221e)\ny : OnePoint X\nx\u271d : X\nhxy : \u2191x\u271d \u2260 y\n\u22a2 Disjoint (\ud835\udcdd \u2191x\u271d) (\ud835\udcdd y)\n[PROOFSTEP]\ninduction y using OnePoint.rec\n[GOAL]\ncase this.h\u2081.h\u2081\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nkey : \u2200 (z : X), Disjoint (\ud835\udcdd \u2191z) (\ud835\udcdd \u221e)\nhxy : \u221e \u2260 \u221e\n\u22a2 Disjoint (\ud835\udcdd \u221e) (\ud835\udcdd \u221e)\n[PROOFSTEP]\nexact (hxy rfl).elim\n[GOAL]\ncase this.h\u2081.h\u2082\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nkey : \u2200 (z : X), Disjoint (\ud835\udcdd \u2191z) (\ud835\udcdd \u221e)\nx\u271d : X\nhxy : \u221e \u2260 \u2191x\u271d\n\u22a2 Disjoint (\ud835\udcdd \u221e) (\ud835\udcdd \u2191x\u271d)\n[PROOFSTEP]\nexact (key _).symm\n[GOAL]\ncase this.h\u2082.h\u2081\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nkey : \u2200 (z : X), Disjoint (\ud835\udcdd \u2191z) (\ud835\udcdd \u221e)\nx\u271d : X\nhxy : \u2191x\u271d \u2260 \u221e\n\u22a2 Disjoint (\ud835\udcdd \u2191x\u271d) (\ud835\udcdd \u221e)\n[PROOFSTEP]\nexact key _\n[GOAL]\ncase this.h\u2082.h\u2082\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : LocallyCompactSpace X\ninst\u271d : T2Space X\nkey : \u2200 (z : X), Disjoint (\ud835\udcdd \u2191z) (\ud835\udcdd \u221e)\nx\u271d\u00b9 x\u271d : X\nhxy : \u2191x\u271d\u00b9 \u2260 \u2191x\u271d\n\u22a2 Disjoint (\ud835\udcdd \u2191x\u271d\u00b9) (\ud835\udcdd \u2191x\u271d)\n[PROOFSTEP]\nrwa [nhds_coe_eq, nhds_coe_eq, disjoint_map coe_injective, disjoint_nhds_nhds, \u2190 coe_injective.ne_iff]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : Infinite X\ninst\u271d : DiscreteTopology X\n\u22a2 \u00acContinuous \u2191CofiniteTopology.of.symm\n[PROOFSTEP]\ninhabit X\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : Infinite X\ninst\u271d : DiscreteTopology X\ninhabited_h : Inhabited X\n\u22a2 \u00acContinuous \u2191CofiniteTopology.of.symm\n[PROOFSTEP]\nsimp only [continuous_iff_continuousAt, ContinuousAt, not_forall]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : Infinite X\ninst\u271d : DiscreteTopology X\ninhabited_h : Inhabited X\n\u22a2 \u2203 x, \u00acTendsto (\u2191CofiniteTopology.of.symm) (\ud835\udcdd x) (\ud835\udcdd (\u2191CofiniteTopology.of.symm x))\n[PROOFSTEP]\nuse CofiniteTopology.of \u2191(default : X)\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d\u00b9 : Infinite X\ninst\u271d : DiscreteTopology X\ninhabited_h : Inhabited X\n\u22a2 \u00acTendsto (\u2191CofiniteTopology.of.symm) (\ud835\udcdd (\u2191CofiniteTopology.of \u2191default))\n (\ud835\udcdd (\u2191CofiniteTopology.of.symm (\u2191CofiniteTopology.of \u2191default)))\n[PROOFSTEP]\nsimpa [nhds_coe_eq, nhds_discrete, CofiniteTopology.nhds_eq] using\n (finite_singleton ((default : X) : OnePoint X)).infinite_compl\n", "meta": {"mathlib_filename": "Mathlib.Topology.Compactification.OnePoint", "llama_tokens": 11057, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.4057470702544803}} {"text": "[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous reflTransSymmAux\n[PROOFSTEP]\nrefine' continuous_if_le _ _ (Continuous.continuousOn _) (Continuous.continuousOn _) _\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous fun x => \u2191x.snd\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_2\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous fun x => 1 / 2\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_3\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous fun x => \u2191x.fst * 2 * \u2191x.snd\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_4\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous fun x => \u2191x.fst * (2 - 2 * \u2191x.snd)\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_5\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 \u2200 (x : \u2191I \u00d7 \u2191I), \u2191x.snd = 1 / 2 \u2192 \u2191x.fst * 2 * \u2191x.snd = \u2191x.fst * (2 - 2 * \u2191x.snd)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_5\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nhx : \u2191x.snd = 1 / 2\n\u22a2 \u2191x.fst * 2 * \u2191x.snd = \u2191x.fst * (2 - 2 * \u2191x.snd)\n[PROOFSTEP]\nrw [hx, mul_assoc]\n[GOAL]\ncase refine'_5\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nhx : \u2191x.snd = 1 / 2\n\u22a2 \u2191x.fst * (2 * (1 / 2)) = \u2191x.fst * (2 - 2 * (1 / 2))\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\n\u22a2 reflTransSymmAux x \u2208 I\n[PROOFSTEP]\ndsimp only [reflTransSymmAux]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\n\u22a2 (if \u2191x.snd \u2264 1 / 2 then \u2191x.fst * 2 * \u2191x.snd else \u2191x.fst * (2 - 2 * \u2191x.snd)) \u2208 I\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u2191x.snd \u2264 1 / 2\n\u22a2 \u2191x.fst * 2 * \u2191x.snd \u2208 I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase pos.left\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u2191x.snd \u2264 1 / 2\n\u22a2 0 \u2264 \u2191x.fst * 2 * \u2191x.snd\n[PROOFSTEP]\napply mul_nonneg\n[GOAL]\ncase pos.left.ha\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u2191x.snd \u2264 1 / 2\n\u22a2 0 \u2264 \u2191x.fst * 2\n[PROOFSTEP]\napply mul_nonneg\n[GOAL]\ncase pos.left.ha.ha\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u2191x.snd \u2264 1 / 2\n\u22a2 0 \u2264 \u2191x.fst\n[PROOFSTEP]\nunit_interval\n[GOAL]\ncase pos.left.ha.hb\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u2191x.snd \u2264 1 / 2\n\u22a2 0 \u2264 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase pos.left.hb\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u2191x.snd \u2264 1 / 2\n\u22a2 0 \u2264 \u2191x.snd\n[PROOFSTEP]\nunit_interval\n[GOAL]\ncase pos.right\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u2191x.snd \u2264 1 / 2\n\u22a2 \u2191x.fst * 2 * \u2191x.snd \u2264 1\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\ncase pos.right\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u2191x.snd \u2264 1 / 2\n\u22a2 \u2191x.fst * (2 * \u2191x.snd) \u2264 1\n[PROOFSTEP]\napply mul_le_one\n[GOAL]\ncase pos.right.ha\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u2191x.snd \u2264 1 / 2\n\u22a2 \u2191x.fst \u2264 1\n[PROOFSTEP]\nunit_interval\n[GOAL]\ncase pos.right.hb'\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u2191x.snd \u2264 1 / 2\n\u22a2 0 \u2264 2 * \u2191x.snd\n[PROOFSTEP]\napply mul_nonneg\n[GOAL]\ncase pos.right.hb'.ha\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u2191x.snd \u2264 1 / 2\n\u22a2 0 \u2264 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase pos.right.hb'.hb\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u2191x.snd \u2264 1 / 2\n\u22a2 0 \u2264 \u2191x.snd\n[PROOFSTEP]\nunit_interval\n[GOAL]\ncase pos.right.hb\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u2191x.snd \u2264 1 / 2\n\u22a2 2 * \u2191x.snd \u2264 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u00ac\u2191x.snd \u2264 1 / 2\n\u22a2 \u2191x.fst * (2 - 2 * \u2191x.snd) \u2208 I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase neg.left\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u00ac\u2191x.snd \u2264 1 / 2\n\u22a2 0 \u2264 \u2191x.fst * (2 - 2 * \u2191x.snd)\n[PROOFSTEP]\napply mul_nonneg\n[GOAL]\ncase neg.left.ha\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u00ac\u2191x.snd \u2264 1 / 2\n\u22a2 0 \u2264 \u2191x.fst\n[PROOFSTEP]\nunit_interval\n[GOAL]\ncase neg.left.hb\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u00ac\u2191x.snd \u2264 1 / 2\n\u22a2 0 \u2264 2 - 2 * \u2191x.snd\n[PROOFSTEP]\nlinarith [unitInterval.nonneg x.2, unitInterval.le_one x.2]\n[GOAL]\ncase neg.right\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u00ac\u2191x.snd \u2264 1 / 2\n\u22a2 \u2191x.fst * (2 - 2 * \u2191x.snd) \u2264 1\n[PROOFSTEP]\napply mul_le_one\n[GOAL]\ncase neg.right.ha\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u00ac\u2191x.snd \u2264 1 / 2\n\u22a2 \u2191x.fst \u2264 1\n[PROOFSTEP]\nunit_interval\n[GOAL]\ncase neg.right.hb'\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u00ac\u2191x.snd \u2264 1 / 2\n\u22a2 0 \u2264 2 - 2 * \u2191x.snd\n[PROOFSTEP]\nlinarith [unitInterval.nonneg x.2, unitInterval.le_one x.2]\n[GOAL]\ncase neg.right.hb\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I \u00d7 \u2191I\nh\u271d : \u00ac\u2191x.snd \u2264 1 / 2\n\u22a2 2 - 2 * \u2191x.snd \u2264 1\n[PROOFSTEP]\nlinarith [unitInterval.nonneg x.2, unitInterval.le_one x.2]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\n\u22a2 Continuous fun x => \u2191p { val := reflTransSymmAux x, property := (_ : reflTransSymmAux x \u2208 I) }\n[PROOFSTEP]\ncontinuity\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\n\u22a2 \u2200 (x : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun x => \u2191p { val := reflTransSymmAux x, property := (_ : reflTransSymmAux x \u2208 I) }) (0, x) =\n \u2191(Path.refl x\u2080).toContinuousMap x\n[PROOFSTEP]\nsimp [reflTransSymmAux]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx : \u2191I\n\u22a2 ContinuousMap.toFun\n (ContinuousMap.mk fun x => \u2191p { val := reflTransSymmAux x, property := (_ : reflTransSymmAux x \u2208 I) }) (1, x) =\n \u2191(Path.trans p (Path.symm p)).toContinuousMap x\n[PROOFSTEP]\ndsimp only [reflTransSymmAux, Path.coe_toContinuousMap, Path.trans]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx : \u2191I\n\u22a2 \u2191p { val := if \u2191x \u2264 1 / 2 then \u21911 * 2 * \u2191x else \u21911 * (2 - 2 * \u2191x), property := (_ : reflTransSymmAux (1, x) \u2208 I) } =\n \u2191(ContinuousMap.mk\n ((fun t => if t \u2264 1 / 2 then extend p (2 * t) else extend (Path.symm p) (2 * t - 1)) \u2218 Subtype.val))\n x\n[PROOFSTEP]\nchange _ = ite _ _ _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx : \u2191I\n\u22a2 \u2191p { val := if \u2191x \u2264 1 / 2 then \u21911 * 2 * \u2191x else \u21911 * (2 - 2 * \u2191x), property := (_ : reflTransSymmAux (1, x) \u2208 I) } =\n if \u2191x \u2264 1 / 2 then extend p (2 * \u2191x) else extend (Path.symm p) (2 * \u2191x - 1)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx : \u2191I\nh : \u2191x \u2264 1 / 2\n\u22a2 \u2191p { val := \u21911 * 2 * \u2191x, property := (_ : (fun x => x \u2208 I) (\u21911 * 2 * \u2191x)) } = extend p (2 * \u2191x)\n[PROOFSTEP]\nrw [Path.extend, Set.IccExtend_of_mem]\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx : \u2191I\nh : \u2191x \u2264 1 / 2\n\u22a2 \u2191p { val := \u21911 * 2 * \u2191x, property := (_ : (fun x => x \u2208 I) (\u21911 * 2 * \u2191x)) } =\n \u2191p { val := 2 * \u2191x, property := ?pos.hx\u271d }\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase pos.hx\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx : \u2191I\nh : \u2191x \u2264 1 / 2\n\u22a2 2 * \u2191x \u2208 Set.Icc 0 1\n[PROOFSTEP]\nrw [unitInterval.mul_pos_mem_iff zero_lt_two]\n[GOAL]\ncase pos.hx\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx : \u2191I\nh : \u2191x \u2264 1 / 2\n\u22a2 \u2191x \u2208 Set.Icc 0 (1 / 2)\n[PROOFSTEP]\nexact \u27e8unitInterval.nonneg x, h\u27e9\n[GOAL]\ncase neg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx : \u2191I\nh : \u00ac\u2191x \u2264 1 / 2\n\u22a2 \u2191p { val := \u21911 * (2 - 2 * \u2191x), property := (_ : (fun x => x \u2208 I) (\u21911 * (2 - 2 * \u2191x))) } =\n extend (Path.symm p) (2 * \u2191x - 1)\n[PROOFSTEP]\nrw [Path.symm, Path.extend, Set.IccExtend_of_mem]\n[GOAL]\ncase neg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx : \u2191I\nh : \u00ac\u2191x \u2264 1 / 2\n\u22a2 \u2191p { val := \u21911 * (2 - 2 * \u2191x), property := (_ : (fun x => x \u2208 I) (\u21911 * (2 - 2 * \u2191x))) } =\n \u2191{ toContinuousMap := ContinuousMap.mk (\u2191p \u2218 \u03c3),\n source' := (_ : ContinuousMap.toFun (ContinuousMap.mk (\u2191p \u2218 \u03c3)) 0 = x\u2081),\n target' := (_ : ContinuousMap.toFun (ContinuousMap.mk (\u2191p \u2218 \u03c3)) 1 = x\u2080) }\n { val := 2 * \u2191x - 1, property := ?neg.hx\u271d }\n[PROOFSTEP]\nsimp only [Set.Icc.coe_one, one_mul, coe_mk_mk, Function.comp_apply]\n[GOAL]\ncase neg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx : \u2191I\nh : \u00ac\u2191x \u2264 1 / 2\n\u22a2 \u2191p { val := 2 - 2 * \u2191x, property := (_ : (fun x => x \u2208 I) (2 - 2 * \u2191x)) } =\n \u2191p (\u03c3 { val := 2 * \u2191x - 1, property := ?neg.hx\u271d })\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase neg.h.e_6.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx : \u2191I\nh : \u00ac\u2191x \u2264 1 / 2\n\u22a2 { val := 2 - 2 * \u2191x, property := (_ : (fun x => x \u2208 I) (2 - 2 * \u2191x)) } = \u03c3 { val := 2 * \u2191x - 1, property := ?neg.hx\u271d }\n[PROOFSTEP]\next\n -- Porting note: norm_num ignores arguments.\n[GOAL]\ncase neg.h.e_6.h.a\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx : \u2191I\nh : \u00ac\u2191x \u2264 1 / 2\n\u22a2 \u2191{ val := 2 - 2 * \u2191x, property := (_ : (fun x => x \u2208 I) (2 - 2 * \u2191x)) } =\n \u2191(\u03c3 { val := 2 * \u2191x - 1, property := ?neg.hx\u271d })\n[PROOFSTEP]\nsimp [sub_sub_eq_add_sub]\n[GOAL]\ncase neg.h.e_6.h.a\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx : \u2191I\nh : \u00ac\u2191x \u2264 1 / 2\n\u22a2 2 = 1 + 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase neg.hx\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx : \u2191I\nh : \u00ac\u2191x \u2264 1 / 2\n\u22a2 2 * \u2191x - 1 \u2208 Set.Icc 0 1\n[PROOFSTEP]\nrw [unitInterval.two_mul_sub_one_mem_iff]\n[GOAL]\ncase neg.hx\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx : \u2191I\nh : \u00ac\u2191x \u2264 1 / 2\n\u22a2 \u2191x \u2208 Set.Icc (1 / 2) 1\n[PROOFSTEP]\nexact \u27e8(not_le.1 h).le, unitInterval.le_one x\u27e9\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nt x : \u2191I\nhx : x \u2208 {0, 1}\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p { val := reflTransSymmAux x, property := (_ : reflTransSymmAux x \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (a : \u2191I),\n \u2191p { val := reflTransSymmAux (0, a), property := (_ : reflTransSymmAux (0, a) \u2208 I) } = x\u2080),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun x =>\n \u2191p { val := reflTransSymmAux x, property := (_ : reflTransSymmAux x \u2208 I) })\n (1, x) =\n \u2191(Path.trans p (Path.symm p)).toContinuousMap x) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.refl x\u2080).toContinuousMap x \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p { val := reflTransSymmAux x, property := (_ : reflTransSymmAux x \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (a : \u2191I),\n \u2191p { val := reflTransSymmAux (0, a), property := (_ : reflTransSymmAux (0, a) \u2208 I) } = x\u2080),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun x =>\n \u2191p { val := reflTransSymmAux x, property := (_ : reflTransSymmAux x \u2208 I) })\n (1, x) =\n \u2191(Path.trans p (Path.symm p)).toContinuousMap x) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.trans p (Path.symm p)).toContinuousMap x\n[PROOFSTEP]\nsimp only [Set.mem_singleton_iff, Set.mem_insert_iff] at hx \n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nt x : \u2191I\nhx : x = 0 \u2228 x = 1\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p { val := reflTransSymmAux x, property := (_ : reflTransSymmAux x \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (a : \u2191I),\n \u2191p { val := reflTransSymmAux (0, a), property := (_ : reflTransSymmAux (0, a) \u2208 I) } = x\u2080),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun x =>\n \u2191p { val := reflTransSymmAux x, property := (_ : reflTransSymmAux x \u2208 I) })\n (1, x) =\n \u2191(Path.trans p (Path.symm p)).toContinuousMap x) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.refl x\u2080).toContinuousMap x \u2227\n \u2191(ContinuousMap.mk fun x =>\n ContinuousMap.toFun\n {\n toContinuousMap :=\n ContinuousMap.mk fun x =>\n \u2191p { val := reflTransSymmAux x, property := (_ : reflTransSymmAux x \u2208 I) },\n map_zero_left :=\n (_ :\n \u2200 (a : \u2191I),\n \u2191p { val := reflTransSymmAux (0, a), property := (_ : reflTransSymmAux (0, a) \u2208 I) } = x\u2080),\n map_one_left :=\n (_ :\n \u2200 (x : \u2191I),\n ContinuousMap.toFun\n (ContinuousMap.mk fun x =>\n \u2191p { val := reflTransSymmAux x, property := (_ : reflTransSymmAux x \u2208 I) })\n (1, x) =\n \u2191(Path.trans p (Path.symm p)).toContinuousMap x) }.toContinuousMap\n (t, x))\n x =\n \u2191(Path.trans p (Path.symm p)).toContinuousMap x\n[PROOFSTEP]\nsimp only [ContinuousMap.coe_mk, coe_toContinuousMap, Path.refl_apply]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nt x : \u2191I\nhx : x = 0 \u2228 x = 1\n\u22a2 \u2191p { val := reflTransSymmAux (t, x), property := (_ : reflTransSymmAux (t, x) \u2208 I) } = x\u2080 \u2227\n \u2191p { val := reflTransSymmAux (t, x), property := (_ : reflTransSymmAux (t, x) \u2208 I) } =\n \u2191(Path.trans p (Path.symm p)) x\n[PROOFSTEP]\ncases hx with\n| inl hx| inr hx =>\n rw [hx]\n simp only [reflTransSymmAux, Set.Icc.coe_zero, Set.Icc.coe_one, one_div, mul_one, inv_nonneg, mul_zero, sub_zero,\n sub_self, Path.source, Path.target, and_self]\n norm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nt x : \u2191I\nhx : x = 0 \u2228 x = 1\n\u22a2 \u2191p { val := reflTransSymmAux (t, x), property := (_ : reflTransSymmAux (t, x) \u2208 I) } = x\u2080 \u2227\n \u2191p { val := reflTransSymmAux (t, x), property := (_ : reflTransSymmAux (t, x) \u2208 I) } =\n \u2191(Path.trans p (Path.symm p)) x\n[PROOFSTEP]\ncases hx with\n| inl hx =>\n rw [hx]\n simp only [reflTransSymmAux, Set.Icc.coe_zero, Set.Icc.coe_one, one_div, mul_one, inv_nonneg, mul_zero, sub_zero,\n sub_self, Path.source, Path.target, and_self]\n norm_num\n| inr hx =>\n rw [hx]\n simp only [reflTransSymmAux, Set.Icc.coe_zero, Set.Icc.coe_one, one_div, mul_one, inv_nonneg, mul_zero, sub_zero,\n sub_self, Path.source, Path.target, and_self]\n norm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nt x : \u2191I\nhx : x = 0 \u2228 x = 1\n\u22a2 \u2191p { val := reflTransSymmAux (t, x), property := (_ : reflTransSymmAux (t, x) \u2208 I) } = x\u2080 \u2227\n \u2191p { val := reflTransSymmAux (t, x), property := (_ : reflTransSymmAux (t, x) \u2208 I) } =\n \u2191(Path.trans p (Path.symm p)) x\n[PROOFSTEP]\ncases hx with\n| inl hx =>\n rw [hx]\n simp only [reflTransSymmAux, Set.Icc.coe_zero, Set.Icc.coe_one, one_div, mul_one, inv_nonneg, mul_zero, sub_zero,\n sub_self, Path.source, Path.target, and_self]\n norm_num\n| inr hx =>\n rw [hx]\n simp only [reflTransSymmAux, Set.Icc.coe_zero, Set.Icc.coe_one, one_div, mul_one, inv_nonneg, mul_zero, sub_zero,\n sub_self, Path.source, Path.target, and_self]\n norm_num\n[GOAL]\ncase inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nt x : \u2191I\nhx : x = 0\n\u22a2 \u2191p { val := reflTransSymmAux (t, x), property := (_ : reflTransSymmAux (t, x) \u2208 I) } = x\u2080 \u2227\n \u2191p { val := reflTransSymmAux (t, x), property := (_ : reflTransSymmAux (t, x) \u2208 I) } =\n \u2191(Path.trans p (Path.symm p)) x\n[PROOFSTEP]\n\n| inl hx =>\n rw [hx]\n simp only [reflTransSymmAux, Set.Icc.coe_zero, Set.Icc.coe_one, one_div, mul_one, inv_nonneg, mul_zero, sub_zero,\n sub_self, Path.source, Path.target, and_self]\n norm_num\n[GOAL]\ncase inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nt x : \u2191I\nhx : x = 0\n\u22a2 \u2191p { val := reflTransSymmAux (t, x), property := (_ : reflTransSymmAux (t, x) \u2208 I) } = x\u2080 \u2227\n \u2191p { val := reflTransSymmAux (t, x), property := (_ : reflTransSymmAux (t, x) \u2208 I) } =\n \u2191(Path.trans p (Path.symm p)) x\n[PROOFSTEP]\nrw [hx]\n[GOAL]\ncase inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nt x : \u2191I\nhx : x = 0\n\u22a2 \u2191p { val := reflTransSymmAux (t, 0), property := (_ : reflTransSymmAux (t, 0) \u2208 I) } = x\u2080 \u2227\n \u2191p { val := reflTransSymmAux (t, 0), property := (_ : reflTransSymmAux (t, 0) \u2208 I) } =\n \u2191(Path.trans p (Path.symm p)) 0\n[PROOFSTEP]\nsimp only [reflTransSymmAux, Set.Icc.coe_zero, Set.Icc.coe_one, one_div, mul_one, inv_nonneg, mul_zero, sub_zero,\n sub_self, Path.source, Path.target, and_self]\n[GOAL]\ncase inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nt x : \u2191I\nhx : x = 0\n\u22a2 \u2191p { val := if 0 \u2264 2 then 0 else \u2191t * 2, property := (_ : (fun x => x \u2208 I) (if 0 \u2264 2 then 0 else \u2191t * 2)) } = x\u2080\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nt x : \u2191I\nhx : x = 1\n\u22a2 \u2191p { val := reflTransSymmAux (t, x), property := (_ : reflTransSymmAux (t, x) \u2208 I) } = x\u2080 \u2227\n \u2191p { val := reflTransSymmAux (t, x), property := (_ : reflTransSymmAux (t, x) \u2208 I) } =\n \u2191(Path.trans p (Path.symm p)) x\n[PROOFSTEP]\n\n| inr hx =>\n rw [hx]\n simp only [reflTransSymmAux, Set.Icc.coe_zero, Set.Icc.coe_one, one_div, mul_one, inv_nonneg, mul_zero, sub_zero,\n sub_self, Path.source, Path.target, and_self]\n norm_num\n[GOAL]\ncase inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nt x : \u2191I\nhx : x = 1\n\u22a2 \u2191p { val := reflTransSymmAux (t, x), property := (_ : reflTransSymmAux (t, x) \u2208 I) } = x\u2080 \u2227\n \u2191p { val := reflTransSymmAux (t, x), property := (_ : reflTransSymmAux (t, x) \u2208 I) } =\n \u2191(Path.trans p (Path.symm p)) x\n[PROOFSTEP]\nrw [hx]\n[GOAL]\ncase inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nt x : \u2191I\nhx : x = 1\n\u22a2 \u2191p { val := reflTransSymmAux (t, 1), property := (_ : reflTransSymmAux (t, 1) \u2208 I) } = x\u2080 \u2227\n \u2191p { val := reflTransSymmAux (t, 1), property := (_ : reflTransSymmAux (t, 1) \u2208 I) } =\n \u2191(Path.trans p (Path.symm p)) 1\n[PROOFSTEP]\nsimp only [reflTransSymmAux, Set.Icc.coe_zero, Set.Icc.coe_one, one_div, mul_one, inv_nonneg, mul_zero, sub_zero,\n sub_self, Path.source, Path.target, and_self]\n[GOAL]\ncase inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nt x : \u2191I\nhx : x = 1\n\u22a2 \u2191p { val := if 1 \u2264 2\u207b\u00b9 then \u2191t * 2 else 0, property := (_ : (fun x => x \u2208 I) (if 1 \u2264 2\u207b\u00b9 then \u2191t * 2 else 0)) } = x\u2080\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous transReflReparamAux\n[PROOFSTEP]\nrefine' continuous_if_le _ _ (Continuous.continuousOn _) (Continuous.continuousOn _) _ <;> [continuity; continuity;\n continuity; continuity; skip]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous transReflReparamAux\n[PROOFSTEP]\nrefine' continuous_if_le _ _ (Continuous.continuousOn _) (Continuous.continuousOn _) _\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous fun x => \u2191x\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_2\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous fun x => 1 / 2\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_3\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous fun x => 2 * \u2191x\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_4\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous fun x => 1\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_5\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 \u2200 (x : \u2191I), \u2191x = 1 / 2 \u2192 2 * \u2191x = 1\n[PROOFSTEP]\nskip\n[GOAL]\ncase refine'_5\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 \u2200 (x : \u2191I), \u2191x = 1 / 2 \u2192 2 * \u2191x = 1\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_5\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I\nhx : \u2191x = 1 / 2\n\u22a2 2 * \u2191x = 1\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\n\u22a2 transReflReparamAux t \u2208 I\n[PROOFSTEP]\nunfold transReflReparamAux\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\n\u22a2 (if \u2191t \u2264 1 / 2 then 2 * \u2191t else 1) \u2208 I\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\nh\u271d : \u2191t \u2264 1 / 2\n\u22a2 2 * \u2191t \u2208 I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase neg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\nh\u271d : \u00ac\u2191t \u2264 1 / 2\n\u22a2 1 \u2208 I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase pos.left\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\nh\u271d : \u2191t \u2264 1 / 2\n\u22a2 0 \u2264 2 * \u2191t\n[PROOFSTEP]\nlinarith [unitInterval.le_one t, unitInterval.nonneg t]\n[GOAL]\ncase pos.right\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\nh\u271d : \u2191t \u2264 1 / 2\n\u22a2 2 * \u2191t \u2264 1\n[PROOFSTEP]\nlinarith [unitInterval.le_one t, unitInterval.nonneg t]\n[GOAL]\ncase neg.left\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\nh\u271d : \u00ac\u2191t \u2264 1 / 2\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nlinarith [unitInterval.le_one t, unitInterval.nonneg t]\n[GOAL]\ncase neg.right\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\nh\u271d : \u00ac\u2191t \u2264 1 / 2\n\u22a2 1 \u2264 1\n[PROOFSTEP]\nlinarith [unitInterval.le_one t, unitInterval.nonneg t]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 transReflReparamAux 0 = 0\n[PROOFSTEP]\nsimp [transReflReparamAux]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 2 < 0 \u2192 False\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 transReflReparamAux 1 = 1\n[PROOFSTEP]\nsimp [transReflReparamAux]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 1 \u2264 2\u207b\u00b9 \u2192 False\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\n\u22a2 Continuous fun t => { val := transReflReparamAux t, property := (_ : transReflReparamAux t \u2208 I) }\n[PROOFSTEP]\ncontinuity\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\n\u22a2 Path.trans p (Path.refl x\u2081) =\n Path.reparam p (fun t => { val := transReflReparamAux t, property := (_ : transReflReparamAux t \u2208 I) })\n (_ : Continuous fun x => { val := transReflReparamAux x, property := (_ : transReflReparamAux x \u2208 I) })\n (_ : (fun t => { val := transReflReparamAux t, property := (_ : transReflReparamAux t \u2208 I) }) 0 = 0)\n (_ : (fun t => { val := transReflReparamAux t, property := (_ : transReflReparamAux t \u2208 I) }) 1 = 1)\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx\u271d : \u2191I\n\u22a2 \u2191(Path.trans p (Path.refl x\u2081)) x\u271d =\n \u2191(Path.reparam p (fun t => { val := transReflReparamAux t, property := (_ : transReflReparamAux t \u2208 I) })\n (_ : Continuous fun x => { val := transReflReparamAux x, property := (_ : transReflReparamAux x \u2208 I) })\n (_ : (fun t => { val := transReflReparamAux t, property := (_ : transReflReparamAux t \u2208 I) }) 0 = 0)\n (_ : (fun t => { val := transReflReparamAux t, property := (_ : transReflReparamAux t \u2208 I) }) 1 = 1))\n x\u271d\n[PROOFSTEP]\nunfold transReflReparamAux\n[GOAL]\ncase a.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx\u271d : \u2191I\n\u22a2 \u2191(Path.trans p (Path.refl x\u2081)) x\u271d =\n \u2191(Path.reparam p (fun t => { val := if \u2191t \u2264 1 / 2 then 2 * \u2191t else 1, property := (_ : transReflReparamAux t \u2208 I) })\n (_ :\n Continuous fun x =>\n { val := if \u2191x \u2264 1 / 2 then 2 * \u2191x else 1, property := (_ : transReflReparamAux x \u2208 I) })\n (_ :\n (fun t => { val := if \u2191t \u2264 1 / 2 then 2 * \u2191t else 1, property := (_ : transReflReparamAux t \u2208 I) }) 0 = 0)\n (_ :\n (fun t => { val := if \u2191t \u2264 1 / 2 then 2 * \u2191t else 1, property := (_ : transReflReparamAux t \u2208 I) }) 1 = 1))\n x\u271d\n[PROOFSTEP]\nsimp only [Path.trans_apply, not_le, coe_reparam, Function.comp_apply, one_div, Path.refl_apply]\n[GOAL]\ncase a.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx\u271d : \u2191I\n\u22a2 (if h : \u2191x\u271d \u2264 2\u207b\u00b9 then \u2191p { val := 2 * \u2191x\u271d, property := (_ : 2 * \u2191x\u271d \u2208 I) } else x\u2081) =\n \u2191p\n { val := if \u2191x\u271d \u2264 2\u207b\u00b9 then 2 * \u2191x\u271d else 1, property := (_ : (fun x => x \u2208 I) (if \u2191x\u271d \u2264 2\u207b\u00b9 then 2 * \u2191x\u271d else 1)) }\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx\u271d : \u2191I\nh\u271d\u00b9 : \u2191x\u271d \u2264 2\u207b\u00b9\nh\u271d : \u2191x\u271d \u2264 2\u207b\u00b9\n\u22a2 \u2191p { val := 2 * \u2191x\u271d, property := (_ : 2 * \u2191x\u271d \u2208 I) } =\n \u2191p { val := 2 * \u2191x\u271d, property := (_ : (fun x => x \u2208 I) (2 * \u2191x\u271d)) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx\u271d : \u2191I\nh\u271d\u00b9 : \u2191x\u271d \u2264 2\u207b\u00b9\nh\u271d : \u00ac\u2191x\u271d \u2264 2\u207b\u00b9\n\u22a2 \u2191p { val := 2 * \u2191x\u271d, property := (_ : 2 * \u2191x\u271d \u2208 I) } =\n \u2191p { val := 2 * \u2191x\u271d, property := (_ : (fun x => x \u2208 I) (2 * \u2191x\u271d)) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx\u271d : \u2191I\nh\u271d\u00b9 : \u00ac\u2191x\u271d \u2264 2\u207b\u00b9\nh\u271d : \u2191x\u271d \u2264 2\u207b\u00b9\n\u22a2 x\u2081 = \u2191p { val := 1, property := (_ : (fun x => x \u2208 I) 1) }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\nx\u271d : \u2191I\nh\u271d\u00b9 : \u00ac\u2191x\u271d \u2264 2\u207b\u00b9\nh\u271d : \u00ac\u2191x\u271d \u2264 2\u207b\u00b9\n\u22a2 x\u2081 = \u2191p { val := 1, property := (_ : (fun x => x \u2208 I) 1) }\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\n\u22a2 Continuous fun t => { val := transReflReparamAux t, property := (_ : transReflReparamAux t \u2208 I) }\n[PROOFSTEP]\ncontinuity\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\n\u22a2 Path.symm (Path.trans (Path.symm p) (Path.refl x\u2080)) = Path.trans (Path.refl x\u2080) p\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\np : Path x\u2080 x\u2081\n\u22a2 Path.symm (Path.symm p) = p\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous transAssocReparamAux\n[PROOFSTEP]\nrefine'\n continuous_if_le _ _ (Continuous.continuousOn _)\n (continuous_if_le _ _ (Continuous.continuousOn _) (Continuous.continuousOn _) _).continuousOn _ <;>\n [continuity; continuity; continuity; continuity; continuity; continuity; continuity; skip; skip]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous transAssocReparamAux\n[PROOFSTEP]\nrefine'\n continuous_if_le _ _ (Continuous.continuousOn _)\n (continuous_if_le _ _ (Continuous.continuousOn _) (Continuous.continuousOn _) _).continuousOn _\n[GOAL]\ncase refine'_1\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous fun x => \u2191x\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_2\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous fun x => 1 / 4\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_3\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous fun x => 2 * \u2191x\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_4\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous fun x => \u2191x\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_5\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous fun x => 1 / 2\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_6\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous fun x => \u2191x + 1 / 4\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_7\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 Continuous fun x => 1 / 2 * (\u2191x + 1)\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_8\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 \u2200 (x : \u2191I), \u2191x = 1 / 2 \u2192 \u2191x + 1 / 4 = 1 / 2 * (\u2191x + 1)\n[PROOFSTEP]\nskip\n[GOAL]\ncase refine'_9\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 \u2200 (x : \u2191I), \u2191x = 1 / 4 \u2192 2 * \u2191x = if \u2191x \u2264 1 / 2 then \u2191x + 1 / 4 else 1 / 2 * (\u2191x + 1)\n[PROOFSTEP]\nskip\n[GOAL]\ncase refine'_8\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 \u2200 (x : \u2191I), \u2191x = 1 / 2 \u2192 \u2191x + 1 / 4 = 1 / 2 * (\u2191x + 1)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_8\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I\nhx : \u2191x = 1 / 2\n\u22a2 \u2191x + 1 / 4 = 1 / 2 * (\u2191x + 1)\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase refine'_8\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I\nhx : \u2191x = 1 / 2\n\u22a2 2\u207b\u00b9 + 4\u207b\u00b9 = 2\u207b\u00b9 * (2\u207b\u00b9 + 1)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine'_9\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 \u2200 (x : \u2191I), \u2191x = 1 / 4 \u2192 2 * \u2191x = if \u2191x \u2264 1 / 2 then \u2191x + 1 / 4 else 1 / 2 * (\u2191x + 1)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_9\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I\nhx : \u2191x = 1 / 4\n\u22a2 2 * \u2191x = if \u2191x \u2264 1 / 2 then \u2191x + 1 / 4 else 1 / 2 * (\u2191x + 1)\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase refine'_9\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx : \u2191I\nhx : \u2191x = 1 / 4\n\u22a2 2 * 4\u207b\u00b9 = if 4\u207b\u00b9 \u2264 2\u207b\u00b9 then 4\u207b\u00b9 + 4\u207b\u00b9 else 2\u207b\u00b9 * (4\u207b\u00b9 + 1)\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\n\u22a2 transAssocReparamAux t \u2208 I\n[PROOFSTEP]\nunfold transAssocReparamAux\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\n\u22a2 (if \u2191t \u2264 1 / 4 then 2 * \u2191t else if \u2191t \u2264 1 / 2 then \u2191t + 1 / 4 else 1 / 2 * (\u2191t + 1)) \u2208 I\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\nh\u271d : \u2191t \u2264 1 / 4\n\u22a2 2 * \u2191t \u2208 I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\nh\u271d\u00b9 : \u00ac\u2191t \u2264 1 / 4\nh\u271d : \u2191t \u2264 1 / 2\n\u22a2 \u2191t + 1 / 4 \u2208 I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase neg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\nh\u271d\u00b9 : \u00ac\u2191t \u2264 1 / 4\nh\u271d : \u00ac\u2191t \u2264 1 / 2\n\u22a2 1 / 2 * (\u2191t + 1) \u2208 I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase pos.left\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\nh\u271d : \u2191t \u2264 1 / 4\n\u22a2 0 \u2264 2 * \u2191t\n[PROOFSTEP]\nlinarith [unitInterval.le_one t, unitInterval.nonneg t]\n[GOAL]\ncase pos.right\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\nh\u271d : \u2191t \u2264 1 / 4\n\u22a2 2 * \u2191t \u2264 1\n[PROOFSTEP]\nlinarith [unitInterval.le_one t, unitInterval.nonneg t]\n[GOAL]\ncase pos.left\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\nh\u271d\u00b9 : \u00ac\u2191t \u2264 1 / 4\nh\u271d : \u2191t \u2264 1 / 2\n\u22a2 0 \u2264 \u2191t + 1 / 4\n[PROOFSTEP]\nlinarith [unitInterval.le_one t, unitInterval.nonneg t]\n[GOAL]\ncase pos.right\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\nh\u271d\u00b9 : \u00ac\u2191t \u2264 1 / 4\nh\u271d : \u2191t \u2264 1 / 2\n\u22a2 \u2191t + 1 / 4 \u2264 1\n[PROOFSTEP]\nlinarith [unitInterval.le_one t, unitInterval.nonneg t]\n[GOAL]\ncase neg.left\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\nh\u271d\u00b9 : \u00ac\u2191t \u2264 1 / 4\nh\u271d : \u00ac\u2191t \u2264 1 / 2\n\u22a2 0 \u2264 1 / 2 * (\u2191t + 1)\n[PROOFSTEP]\nlinarith [unitInterval.le_one t, unitInterval.nonneg t]\n[GOAL]\ncase neg.right\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nt : \u2191I\nh\u271d\u00b9 : \u00ac\u2191t \u2264 1 / 4\nh\u271d : \u00ac\u2191t \u2264 1 / 2\n\u22a2 1 / 2 * (\u2191t + 1) \u2264 1\n[PROOFSTEP]\nlinarith [unitInterval.le_one t, unitInterval.nonneg t]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 transAssocReparamAux 0 = 0\n[PROOFSTEP]\nsimp [transAssocReparamAux]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 4 < 0 \u2192 (if 0 \u2264 2 then 4\u207b\u00b9 else 2\u207b\u00b9) = 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 transAssocReparamAux 1 = 1\n[PROOFSTEP]\nsimp [transAssocReparamAux]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 (if 1 \u2264 4\u207b\u00b9 then 2 else if 1 \u2264 2\u207b\u00b9 then 1 + 4\u207b\u00b9 else 2\u207b\u00b9 * (1 + 1)) = 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\n\u22a2 Continuous fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) }\n[PROOFSTEP]\ncontinuity\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\n\u22a2 Path.trans (Path.trans p q) r =\n Path.reparam (Path.trans p (Path.trans q r))\n (fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) })\n (_ : Continuous fun x => { val := transAssocReparamAux x, property := (_ : transAssocReparamAux x \u2208 I) })\n (_ : (fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) }) 0 = 0)\n (_ : (fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) }) 1 = 1)\n[PROOFSTEP]\next x\n[GOAL]\ncase a.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\n\u22a2 \u2191(Path.trans (Path.trans p q) r) x =\n \u2191(Path.reparam (Path.trans p (Path.trans q r))\n (fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) })\n (_ : Continuous fun x => { val := transAssocReparamAux x, property := (_ : transAssocReparamAux x \u2208 I) })\n (_ : (fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) }) 0 = 0)\n (_ : (fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) }) 1 = 1))\n x\n[PROOFSTEP]\nsimp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left\u2080, not_le, Function.comp_apply, Ne.def,\n not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk, Path.coe_reparam]\n -- TODO: why does split_ifs not reduce the ifs??????\n[GOAL]\ncase a.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\n\u22a2 (if h : \u2191x \u2264 1 / 2 then\n if h_1 : 2 * \u2191x \u2264 1 / 2 then\n \u2191p { val := 2 * (2 * \u2191x), property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } \u2208 I) }\n else\n \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) }\n else \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) }) =\n if h : (if \u2191x \u2264 1 / 4 then 2 * \u2191x else if \u2191x \u2264 1 / 2 then \u2191x + 1 / 4 else 1 / 2 * (\u2191x + 1)) \u2264 1 / 2 then\n \u2191p\n { val := if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x) else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1)),\n property :=\n (_ :\n (fun x => x \u2208 I)\n (if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1)))) }\n else\n if h_1 :\n (if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x) else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) - 1 \u2264\n 1 / 2 then\n \u2191q\n {\n val :=\n 2 *\n ((if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n 1),\n property :=\n (_ :\n 2 *\n \u2191{\n val :=\n (if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n 1,\n property :=\n (_ :\n (fun x => x \u2208 I)\n ((if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n 1)) } \u2208\n I) }\n else\n \u2191r\n {\n val :=\n 2 *\n ((if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n 1) -\n 1,\n property :=\n (_ :\n 2 *\n \u2191{\n val :=\n (if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n 1,\n property :=\n (_ :\n (fun x => x \u2208 I)\n ((if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n 1)) } -\n 1 \u2208\n I) }\n[PROOFSTEP]\nsplit_ifs with h\u2081 h\u2082 h\u2083 h\u2084 h\u2085\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u2191x \u2264 1 / 4\n\u22a2 \u2191p { val := 2 * (2 * \u2191x), property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } \u2208 I) } =\n \u2191p { val := 2 * (2 * \u2191x), property := (_ : (fun x => x \u2208 I) (2 * (2 * \u2191x))) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u2191x + 1 / 4 \u2264 1 / 2\n\u22a2 \u2191p { val := 2 * (2 * \u2191x), property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } \u2208 I) } =\n \u2191p { val := 2 * (\u2191x + 1 / 4), property := (_ : (fun x => x \u2208 I) (2 * (\u2191x + 1 / 4))) }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u2191x + 1 / 4 \u2264 1 / 2\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u2085 : 2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 \u2191p { val := 2 * (2 * \u2191x), property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } \u2208 I) } =\n \u2191q { val := 2 * (2 * (\u2191x + 1 / 4) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (\u2191x + 1 / 4) - 1))) }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u2085 : 2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u2085 : \u00ac2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 \u2191p { val := 2 * (2 * \u2191x), property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } \u2208 I) } =\n \u2191r { val := 2 * (2 * (\u2191x + 1 / 4) - 1) - 1, property := (_ : (fun x => x \u2208 I) (2 * (2 * (\u2191x + 1 / 4) - 1) - 1)) }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u2085 : \u00ac2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : 2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n \u2191q { val := 2 * (2 * (2 * \u2191x) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (2 * \u2191x) - 1))) }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : 2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : \u00ac2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n \u2191r { val := 2 * (2 * (2 * \u2191x) - 1) - 1, property := (_ : (fun x => x \u2208 I) (2 * (2 * (2 * \u2191x) - 1) - 1)) }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : \u00ac2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u00ac\u2191x \u2264 1 / 4\nh\u271d : \u2191x + 1 / 4 \u2264 1 / 2\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n \u2191p { val := 2 * (\u2191x + 1 / 4), property := (_ : (fun x => x \u2208 I) (2 * (\u2191x + 1 / 4))) }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u00ac\u2191x \u2264 1 / 4\nh\u271d : \u2191x + 1 / 4 \u2264 1 / 2\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u271d : 2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n \u2191q { val := 2 * (2 * (\u2191x + 1 / 4) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (\u2191x + 1 / 4) - 1))) }\n[PROOFSTEP]\nhave h : 2 * (2 * (x : \u211d)) - 1 = 2 * (2 * (\u2191x + 1 / 4) - 1) := by linarith\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u271d : 2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 2 * (2 * \u2191x) - 1 = 2 * (2 * (\u2191x + 1 / 4) - 1)\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u271d : 2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\nh : 2 * (2 * \u2191x) - 1 = 2 * (2 * (\u2191x + 1 / 4) - 1)\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n \u2191q { val := 2 * (2 * (\u2191x + 1 / 4) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (\u2191x + 1 / 4) - 1))) }\n[PROOFSTEP]\nsimp [h\u2082, h\u2081, h, dif_neg (show \u00acFalse from id), dif_pos True.intro, if_false, if_true]\n[GOAL]\ncase neg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u271d : \u00ac2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n \u2191r { val := 2 * (2 * (\u2191x + 1 / 4) - 1) - 1, property := (_ : (fun x => x \u2208 I) (2 * (2 * (\u2191x + 1 / 4) - 1) - 1)) }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u271d : \u00ac2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : 2 * \u2191x \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n \u2191p { val := 2 * (2 * \u2191x), property := (_ : (fun x => x \u2208 I) (2 * (2 * \u2191x))) }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : 2 * \u2191x \u2264 1 / 2\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d : 2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n \u2191q { val := 2 * (2 * (2 * \u2191x) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (2 * \u2191x) - 1))) }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d : 2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d : \u00ac2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n \u2191r { val := 2 * (2 * (2 * \u2191x) - 1) - 1, property := (_ : (fun x => x \u2208 I) (2 * (2 * (2 * \u2191x) - 1) - 1)) }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d : \u00ac2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u00ac\u2191x \u2264 1 / 4\nh\u271d : 1 / 2 * (\u2191x + 1) \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n \u2191p { val := 2 * (1 / 2 * (\u2191x + 1)), property := (_ : (fun x => x \u2208 I) (2 * (1 / 2 * (\u2191x + 1)))) }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u00ac\u2191x \u2264 1 / 4\nh\u271d : 1 / 2 * (\u2191x + 1) \u2264 1 / 2\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase pos\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac1 / 2 * (\u2191x + 1) \u2264 1 / 2\nh\u271d : 2 * (1 / 2 * (\u2191x + 1)) - 1 \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n \u2191q\n { val := 2 * (2 * (1 / 2 * (\u2191x + 1)) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (1 / 2 * (\u2191x + 1)) - 1))) }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac1 / 2 * (\u2191x + 1) \u2264 1 / 2\nh\u271d : 2 * (1 / 2 * (\u2191x + 1)) - 1 \u2264 1 / 2\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac1 / 2 * (\u2191x + 1) \u2264 1 / 2\nh\u271d : \u00ac2 * (1 / 2 * (\u2191x + 1)) - 1 \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n \u2191r\n { val := 2 * (2 * (1 / 2 * (\u2191x + 1)) - 1) - 1,\n property := (_ : (fun x => x \u2208 I) (2 * (2 * (1 / 2 * (\u2191x + 1)) - 1) - 1)) }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase neg.h.e_6.h.e_val.e_a.e_a\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac1 / 2 * (\u2191x + 1) \u2264 1 / 2\nh\u271d : \u00ac2 * (1 / 2 * (\u2191x + 1)) - 1 \u2264 1 / 2\n\u22a2 \u2191x = 2 * (1 / 2 * (\u2191x + 1)) - 1\n[PROOFSTEP]\nring\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\n\u22a2 Continuous fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) }\n[PROOFSTEP]\ncontinuity\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx y : FundamentalGroupoid X\np : x \u27f6 y\n\u22a2 \u2200 (a b : Path x y),\n a \u2248 b \u2192\n (fun l => Quotient.mk (Path.Homotopic.setoid y x) (Path.symm l)) a =\n (fun l => Quotient.mk (Path.Homotopic.setoid y x) (Path.symm l)) b\n[PROOFSTEP]\nrintro a b \u27e8h\u27e9\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx y : FundamentalGroupoid X\np : x \u27f6 y\na b : Path x y\nh : Path.Homotopy a b\n\u22a2 (fun l => Quotient.mk (Path.Homotopic.setoid y x) (Path.symm l)) a =\n (fun l => Quotient.mk (Path.Homotopic.setoid y x) (Path.symm l)) b\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx y : FundamentalGroupoid X\np : x \u27f6 y\na b : Path x y\nh : Path.Homotopy a b\n\u22a2 Quotient.mk (Path.Homotopic.setoid y x) (Path.symm a) = Quotient.mk (Path.Homotopic.setoid y x) (Path.symm b)\n[PROOFSTEP]\nrw [Quotient.eq]\n[GOAL]\ncase intro\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nx y : FundamentalGroupoid X\np : x \u27f6 y\na b : Path x y\nh : Path.Homotopy a b\n\u22a2 Path.symm a \u2248 Path.symm b\n[PROOFSTEP]\nexact \u27e8h.symm\u2082\u27e9\n[GOAL]\nX\u271d\u00b9 : Type u\nY\u271d\u00b9 : Type v\ninst\u271d\u00b9 : TopologicalSpace X\u271d\u00b9\ninst\u271d : TopologicalSpace Y\u271d\u00b9\nx\u2080 x\u2081 : X\u271d\u00b9\nX\u271d Y\u271d : TopCat\nf : X\u271d \u27f6 Y\u271d\nX Y : \u2191((fun X => Bundled.mk (FundamentalGroupoid \u2191X)) X\u271d)\np : X \u27f6 Y\n\u22a2 \u2191f X \u27f6 \u2191f Y\n[PROOFSTEP]\nexact Path.Homotopic.Quotient.mapFn p f\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nX\u271d Y\u271d : TopCat\nf : X\u271d \u27f6 Y\u271d\nx y z : \u2191((fun X => Bundled.mk (FundamentalGroupoid \u2191X)) X\u271d)\np : x \u27f6 y\nq : y \u27f6 z\n\u22a2 { obj := \u2191f, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f }.map (p \u226b q) =\n { obj := \u2191f, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f }.map p \u226b\n { obj := \u2191f, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f }.map q\n[PROOFSTEP]\nrefine' Quotient.inductionOn\u2082 p q fun a b => _\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nX\u271d Y\u271d : TopCat\nf : X\u271d \u27f6 Y\u271d\nx y z : \u2191((fun X => Bundled.mk (FundamentalGroupoid \u2191X)) X\u271d)\np : x \u27f6 y\nq : y \u27f6 z\na : Path x y\nb : Path y z\n\u22a2 { obj := \u2191f, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f }.map\n (Quotient.mk (Path.Homotopic.setoid x y) a \u226b Quotient.mk (Path.Homotopic.setoid y z) b) =\n { obj := \u2191f, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f }.map\n (Quotient.mk (Path.Homotopic.setoid x y) a) \u226b\n { obj := \u2191f, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f }.map\n (Quotient.mk (Path.Homotopic.setoid y z) b)\n[PROOFSTEP]\nsimp only [comp_eq, \u2190 Path.Homotopic.map_lift, \u2190 Path.Homotopic.comp_lift, Path.map_trans]\n[GOAL]\nX\u271d : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\u271d\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\u271d\nX : TopCat\n\u22a2 { obj := fun X => Bundled.mk (FundamentalGroupoid \u2191X),\n map := fun {X Y} f =>\n CategoryTheory.Functor.mk { obj := \u2191f, map := fun {X_1 Y_1} p => Path.Homotopic.Quotient.mapFn p f } }.map\n (\ud835\udfd9 X) =\n \ud835\udfd9\n ({ obj := fun X => Bundled.mk (FundamentalGroupoid \u2191X),\n map := fun {X Y} f =>\n CategoryTheory.Functor.mk { obj := \u2191f, map := fun {X_1 Y_1} p => Path.Homotopic.Quotient.mapFn p f } }.obj\n X)\n[PROOFSTEP]\nsimp only\n[GOAL]\nX\u271d : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\u271d\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\u271d\nX : TopCat\n\u22a2 CategoryTheory.Functor.mk { obj := \u2191(\ud835\udfd9 X), map := fun {X_1 Y} p => Path.Homotopic.Quotient.mapFn p (\ud835\udfd9 X) } =\n \ud835\udfd9 (Bundled.mk (FundamentalGroupoid \u2191X))\n[PROOFSTEP]\nchange _ = (\u27e8_, _, _\u27e9 : FundamentalGroupoid X \u2964 FundamentalGroupoid X)\n[GOAL]\nX\u271d : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\u271d\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\u271d\nX : TopCat\n\u22a2 CategoryTheory.Functor.mk { obj := \u2191(\ud835\udfd9 X), map := fun {X_1 Y} p => Path.Homotopic.Quotient.mapFn p (\ud835\udfd9 X) } =\n CategoryTheory.Functor.mk { obj := fun X => X, map := fun {X_1 Y} f => f }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_toPrefunctor.e_map\nX\u271d : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\u271d\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\u271d\nX : TopCat\n\u22a2 (fun {X_1 Y} p => Path.Homotopic.Quotient.mapFn p (\ud835\udfd9 X)) = fun {X_1 Y} f => f\n[PROOFSTEP]\next x y p\n[GOAL]\ncase e_toPrefunctor.e_map.h.h.h\nX\u271d : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\u271d\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\u271d\nX : TopCat\nx y : FundamentalGroupoid \u2191X\np : x \u27f6 y\n\u22a2 Path.Homotopic.Quotient.mapFn p (\ud835\udfd9 X) = p\n[PROOFSTEP]\nrefine' Quotient.inductionOn p fun q => _\n[GOAL]\ncase e_toPrefunctor.e_map.h.h.h\nX\u271d : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\u271d\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\u271d\nX : TopCat\nx y : FundamentalGroupoid \u2191X\np : x \u27f6 y\nq : Path x y\n\u22a2 Path.Homotopic.Quotient.mapFn (Quotient.mk (Path.Homotopic.setoid x y) q) (\ud835\udfd9 X) =\n Quotient.mk (Path.Homotopic.setoid x y) q\n[PROOFSTEP]\nrw [\u2190 Path.Homotopic.map_lift]\n[GOAL]\ncase e_toPrefunctor.e_map.h.h.h\nX\u271d : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\u271d\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\u271d\nX : TopCat\nx y : FundamentalGroupoid \u2191X\np : x \u27f6 y\nq : Path x y\n\u22a2 Quotient.mk (Path.Homotopic.setoid (\u2191(\ud835\udfd9 X) x) (\u2191(\ud835\udfd9 X) y)) (Path.map q (_ : Continuous \u2191(\ud835\udfd9 X))) =\n Quotient.mk (Path.Homotopic.setoid x y) q\n[PROOFSTEP]\nconv_rhs => rw [\u2190 q.map_id]\n[GOAL]\nX\u271d : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\u271d\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\u271d\nX : TopCat\nx y : FundamentalGroupoid \u2191X\np : x \u27f6 y\nq : Path x y\n| Quotient.mk (Path.Homotopic.setoid x y) q\n[PROOFSTEP]\nrw [\u2190 q.map_id]\n[GOAL]\nX\u271d : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\u271d\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\u271d\nX : TopCat\nx y : FundamentalGroupoid \u2191X\np : x \u27f6 y\nq : Path x y\n| Quotient.mk (Path.Homotopic.setoid x y) q\n[PROOFSTEP]\nrw [\u2190 q.map_id]\n[GOAL]\nX\u271d : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\u271d\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\u271d\nX : TopCat\nx y : FundamentalGroupoid \u2191X\np : x \u27f6 y\nq : Path x y\n| Quotient.mk (Path.Homotopic.setoid x y) q\n[PROOFSTEP]\nrw [\u2190 q.map_id]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nX\u271d Y\u271d Z\u271d : TopCat\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun X => Bundled.mk (FundamentalGroupoid \u2191X),\n map := fun {X Y} f =>\n CategoryTheory.Functor.mk { obj := \u2191f, map := fun {X_1 Y_1} p => Path.Homotopic.Quotient.mapFn p f } }.map\n (f \u226b g) =\n { obj := fun X => Bundled.mk (FundamentalGroupoid \u2191X),\n map := fun {X Y} f =>\n CategoryTheory.Functor.mk { obj := \u2191f, map := fun {X_1 Y_1} p => Path.Homotopic.Quotient.mapFn p f } }.map\n f \u226b\n { obj := fun X => Bundled.mk (FundamentalGroupoid \u2191X),\n map := fun {X Y} f =>\n CategoryTheory.Functor.mk { obj := \u2191f, map := fun {X_1 Y_1} p => Path.Homotopic.Quotient.mapFn p f } }.map\n g\n[PROOFSTEP]\nsimp only\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nX\u271d Y\u271d Z\u271d : TopCat\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 CategoryTheory.Functor.mk { obj := \u2191(f \u226b g), map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p (f \u226b g) } =\n CategoryTheory.Functor.mk { obj := \u2191f, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f } \u226b\n CategoryTheory.Functor.mk { obj := \u2191g, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p g }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_toPrefunctor.e_map\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nX\u271d Y\u271d Z\u271d : TopCat\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 (fun {X Y} p => Path.Homotopic.Quotient.mapFn p (f \u226b g)) = fun {X Y} f_1 =>\n (CategoryTheory.Functor.mk { obj := \u2191g, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p g }).map\n ((CategoryTheory.Functor.mk { obj := \u2191f, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f }).map f_1)\n[PROOFSTEP]\next x y p\n[GOAL]\ncase e_toPrefunctor.e_map.h.h.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nX\u271d Y\u271d Z\u271d : TopCat\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\nx y : FundamentalGroupoid \u2191X\u271d\np : x \u27f6 y\n\u22a2 Path.Homotopic.Quotient.mapFn p (f \u226b g) =\n (CategoryTheory.Functor.mk { obj := \u2191g, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p g }).map\n ((CategoryTheory.Functor.mk { obj := \u2191f, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f }).map p)\n[PROOFSTEP]\nrefine' Quotient.inductionOn p fun q => _\n[GOAL]\ncase e_toPrefunctor.e_map.h.h.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nX\u271d Y\u271d Z\u271d : TopCat\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\nx y : FundamentalGroupoid \u2191X\u271d\np : x \u27f6 y\nq : Path x y\n\u22a2 Path.Homotopic.Quotient.mapFn (Quotient.mk (Path.Homotopic.setoid x y) q) (f \u226b g) =\n (CategoryTheory.Functor.mk { obj := \u2191g, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p g }).map\n ((CategoryTheory.Functor.mk { obj := \u2191f, map := fun {X Y} p => Path.Homotopic.Quotient.mapFn p f }).map\n (Quotient.mk (Path.Homotopic.setoid x y) q))\n[PROOFSTEP]\nsimp only [Quotient.map_mk, Path.map_map, Quotient.eq']\n[GOAL]\ncase e_toPrefunctor.e_map.h.h.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\nX\u271d Y\u271d Z\u271d : TopCat\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\nx y : FundamentalGroupoid \u2191X\u271d\np : x \u27f6 y\nq : Path x y\n\u22a2 Path.Homotopic.Quotient.mapFn (Quotient.mk (Path.Homotopic.setoid x y) q) (f \u226b g) =\n Path.Homotopic.Quotient.mapFn (Path.Homotopic.Quotient.mapFn (Quotient.mk (Path.Homotopic.setoid x y) q) f) g\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic", "llama_tokens": 31694, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.405586054970079}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nh : u ~[l] v\n\u22a2 v =O[l] u\n[PROOFSTEP]\nconvert h.isLittleO.right_isBigO_add\n[GOAL]\ncase h.e'_8.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nh : u ~[l] v\nx\u271d : \u03b1\n\u22a2 u x\u271d = (u - v) x\u271d + v x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u22a2 u ~[l] u\n[PROOFSTEP]\nrw [IsEquivalent, sub_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u22a2 0 =o[l] u\n[PROOFSTEP]\nexact isLittleO_zero _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u22a2 u ~[l] 0 \u2194 u =\u1da0[l] 0\n[PROOFSTEP]\nrw [IsEquivalent, sub_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u22a2 u =o[l] 0 \u2194 u =\u1da0[l] 0\n[PROOFSTEP]\nexact isLittleO_zero_right_iff\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u22a2 u ~[l] 0 \u2194 u =O[l] 0\n[PROOFSTEP]\nrefine' \u27e8IsEquivalent.isBigO, fun h \u21a6 _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nh : u =O[l] 0\n\u22a2 u ~[l] 0\n[PROOFSTEP]\nrw [isEquivalent_zero_iff_eventually_zero, eventuallyEq_iff_exists_mem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nh : u =O[l] 0\n\u22a2 \u2203 s, s \u2208 l \u2227 Set.EqOn u 0 s\n[PROOFSTEP]\nexact \u27e8{x : \u03b1 | u x = 0}, isBigO_zero_right_iff.mp h, fun x hx \u21a6 hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nc : \u03b2\nh : c \u2260 0\n\u22a2 u ~[l] const \u03b1 c \u2194 Tendsto u l (\ud835\udcdd c)\n[PROOFSTEP]\nsimp_rw [IsEquivalent, const, isLittleO_const_iff h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nc : \u03b2\nh : c \u2260 0\n\u22a2 Tendsto (u - fun x => c) l (\ud835\udcdd 0) \u2194 Tendsto u l (\ud835\udcdd c)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nc : \u03b2\nh : c \u2260 0\n\u22a2 Tendsto (u - fun x => c) l (\ud835\udcdd 0) \u2192 Tendsto u l (\ud835\udcdd c)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nc : \u03b2\nh : c \u2260 0\n\u22a2 Tendsto u l (\ud835\udcdd c) \u2192 Tendsto (u - fun x => c) l (\ud835\udcdd 0)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nc : \u03b2\nh\u271d : c \u2260 0\nh : Tendsto (u - fun x => c) l (\ud835\udcdd 0)\n\u22a2 Tendsto u l (\ud835\udcdd c)\n[PROOFSTEP]\nhave := h.sub (tendsto_const_nhds (a := -c))\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nc : \u03b2\nh\u271d : c \u2260 0\nh : Tendsto (u - fun x => c) l (\ud835\udcdd 0)\nthis : Tendsto (fun x => (u - fun x => c) x - -c) l (\ud835\udcdd (0 - -c))\n\u22a2 Tendsto u l (\ud835\udcdd c)\n[PROOFSTEP]\nsimp only [Pi.sub_apply, sub_neg_eq_add, sub_add_cancel, zero_add] at this \n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nc : \u03b2\nh\u271d : c \u2260 0\nh : Tendsto (u - fun x => c) l (\ud835\udcdd 0)\nthis : Tendsto (fun x => u x) l (\ud835\udcdd c)\n\u22a2 Tendsto u l (\ud835\udcdd c)\n[PROOFSTEP]\nexact this\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nc : \u03b2\nh\u271d : c \u2260 0\nh : Tendsto u l (\ud835\udcdd c)\n\u22a2 Tendsto (u - fun x => c) l (\ud835\udcdd 0)\n[PROOFSTEP]\nhave := h.sub (tendsto_const_nhds (a := c))\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nc : \u03b2\nh\u271d : c \u2260 0\nh : Tendsto u l (\ud835\udcdd c)\nthis : Tendsto (fun x => u x - c) l (\ud835\udcdd (c - c))\n\u22a2 Tendsto (u - fun x => c) l (\ud835\udcdd 0)\n[PROOFSTEP]\nrwa [sub_self] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nc : \u03b2\nhu : u ~[l] const \u03b1 c\n\u22a2 Tendsto u l (\ud835\udcdd c)\n[PROOFSTEP]\nrcases em <| c = 0 with rfl | h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhu : u ~[l] const \u03b1 0\n\u22a2 Tendsto u l (\ud835\udcdd 0)\n[PROOFSTEP]\nexact (tendsto_congr' <| isEquivalent_zero_iff_eventually_zero.mp hu).mpr tendsto_const_nhds\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nc : \u03b2\nhu : u ~[l] const \u03b1 c\nh : \u00acc = 0\n\u22a2 Tendsto u l (\ud835\udcdd c)\n[PROOFSTEP]\nexact (isEquivalent_const_iff_tendsto h).mp hu\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nc : \u03b2\nhuv : u ~[l] v\nhu : Tendsto u l (\ud835\udcdd c)\n\u22a2 Tendsto v l (\ud835\udcdd c)\n[PROOFSTEP]\nby_cases h : c = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nc : \u03b2\nhuv : u ~[l] v\nhu : Tendsto u l (\ud835\udcdd c)\nh : c = 0\n\u22a2 Tendsto v l (\ud835\udcdd c)\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhuv : u ~[l] v\nhu : Tendsto u l (\ud835\udcdd 0)\n\u22a2 Tendsto v l (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 isLittleO_one_iff \u211d] at hu \u22a2\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhuv : u ~[l] v\nhu : u =o[l] fun _x => 1\n\u22a2 v =o[l] fun _x => 1\n[PROOFSTEP]\nsimpa using (huv.symm.isLittleO.trans hu).add hu\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nc : \u03b2\nhuv : u ~[l] v\nhu : Tendsto u l (\ud835\udcdd c)\nh : \u00acc = 0\n\u22a2 Tendsto v l (\ud835\udcdd c)\n[PROOFSTEP]\nrw [\u2190 isEquivalent_const_iff_tendsto h] at hu \u22a2\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nc : \u03b2\nhuv : u ~[l] v\nhu : u ~[l] const \u03b1 c\nh : \u00acc = 0\n\u22a2 v ~[l] const \u03b1 c\n[PROOFSTEP]\nexact huv.symm.trans hu\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhuv : u ~[l] v\nhwv : w =o[l] v\n\u22a2 u + w ~[l] v\n[PROOFSTEP]\nsimpa only [IsEquivalent, add_sub_right_comm] using huv.add hwv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhuv : u ~[l] v\nhwv : w =o[l] v\n\u22a2 u - w ~[l] v\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using huv.add_isLittleO hwv.neg_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhuv : u ~[l] v\n\u22a2 (fun x => -u x) ~[l] fun x => -v x\n[PROOFSTEP]\nrw [IsEquivalent]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhuv : u ~[l] v\n\u22a2 ((fun x => -u x) - fun x => -v x) =o[l] fun x => -v x\n[PROOFSTEP]\nconvert huv.isLittleO.neg_left.neg_right\n[GOAL]\ncase h.e'_7.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nu v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhuv : u ~[l] v\nx\u271d : \u03b1\n\u22a2 ((fun x => -u x) - fun x => -v x) x\u271d = -(u - v) x\u271d\n[PROOFSTEP]\nsimp [neg_add_eq_sub]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u22a2 u ~[l] v \u2194 \u2203 \u03c6 x, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nrw [IsEquivalent, isLittleO_iff_exists_eq_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u22a2 (\u2203 \u03c6 _h\u03c6, u - v =\u1da0[l] \u03c6 * v) \u2194 \u2203 \u03c6 x, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nconstructor <;> rintro \u27e8\u03c6, h\u03c6, h\u27e9 <;> [refine' \u27e8\u03c6 + 1, _, _\u27e9; refine' \u27e8\u03c6 - 1, _, _\u27e9]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u22a2 (\u2203 \u03c6 _h\u03c6, u - v =\u1da0[l] \u03c6 * v) \u2194 \u2203 \u03c6 x, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u22a2 (\u2203 \u03c6 _h\u03c6, u - v =\u1da0[l] \u03c6 * v) \u2192 \u2203 \u03c6 x, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nrintro \u27e8\u03c6, h\u03c6, h\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u22a2 (\u2203 \u03c6 x, u =\u1da0[l] \u03c6 * v) \u2192 \u2203 \u03c6 _h\u03c6, u - v =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nrintro \u27e8\u03c6, h\u03c6, h\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 0)\nh : u - v =\u1da0[l] \u03c6 * v\n\u22a2 \u2203 \u03c6 x, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nrefine' \u27e8\u03c6 + 1, _, _\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nh : u =\u1da0[l] \u03c6 * v\n\u22a2 \u2203 \u03c6 _h\u03c6, u - v =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nrefine' \u27e8\u03c6 - 1, _, _\u27e9\n[GOAL]\ncase mp.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 0)\nh : u - v =\u1da0[l] \u03c6 * v\n\u22a2 Tendsto (\u03c6 + 1) l (\ud835\udcdd 1)\n[PROOFSTEP]\nconv in \ud835\udcdd _ => rw [\u2190 zero_add (1 : \u03b2)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 0)\nh : u - v =\u1da0[l] \u03c6 * v\n| \ud835\udcdd 1\n[PROOFSTEP]\nrw [\u2190 zero_add (1 : \u03b2)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 0)\nh : u - v =\u1da0[l] \u03c6 * v\n| \ud835\udcdd 1\n[PROOFSTEP]\nrw [\u2190 zero_add (1 : \u03b2)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 0)\nh : u - v =\u1da0[l] \u03c6 * v\n| \ud835\udcdd 1\n[PROOFSTEP]\nrw [\u2190 zero_add (1 : \u03b2)]\n[GOAL]\ncase mp.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 0)\nh : u - v =\u1da0[l] \u03c6 * v\n\u22a2 Tendsto (\u03c6 + 1) l (\ud835\udcdd (0 + 1))\n[PROOFSTEP]\nexact h\u03c6.add tendsto_const_nhds\n[GOAL]\ncase mp.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 0)\nh : u - v =\u1da0[l] \u03c6 * v\n\u22a2 u =\u1da0[l] (\u03c6 + 1) * v\n[PROOFSTEP]\nconvert h.add (EventuallyEq.refl l v)\n[GOAL]\ncase h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 0)\nh : u - v =\u1da0[l] \u03c6 * v\nx\u271d : \u03b1\n\u22a2 u x\u271d = (u - v) x\u271d + v x\u271d\n[PROOFSTEP]\nsimp [add_mul]\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 0)\nh : u - v =\u1da0[l] \u03c6 * v\nx\u271d : \u03b1\n\u22a2 ((\u03c6 + 1) * v) x\u271d = (\u03c6 * v) x\u271d + v x\u271d\n[PROOFSTEP]\nsimp [add_mul]\n[GOAL]\ncase mpr.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nh : u =\u1da0[l] \u03c6 * v\n\u22a2 Tendsto (\u03c6 - 1) l (\ud835\udcdd 0)\n[PROOFSTEP]\nconv in \ud835\udcdd _ => rw [\u2190 sub_self (1 : \u03b2)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nh : u =\u1da0[l] \u03c6 * v\n| \ud835\udcdd 0\n[PROOFSTEP]\nrw [\u2190 sub_self (1 : \u03b2)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nh : u =\u1da0[l] \u03c6 * v\n| \ud835\udcdd 0\n[PROOFSTEP]\nrw [\u2190 sub_self (1 : \u03b2)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nh : u =\u1da0[l] \u03c6 * v\n| \ud835\udcdd 0\n[PROOFSTEP]\nrw [\u2190 sub_self (1 : \u03b2)]\n[GOAL]\ncase mpr.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nh : u =\u1da0[l] \u03c6 * v\n\u22a2 Tendsto (\u03c6 - 1) l (\ud835\udcdd (1 - 1))\n[PROOFSTEP]\nexact h\u03c6.sub tendsto_const_nhds\n[GOAL]\ncase mpr.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nh : u =\u1da0[l] \u03c6 * v\n\u22a2 u - v =\u1da0[l] (\u03c6 - 1) * v\n[PROOFSTEP]\nconvert h.sub (EventuallyEq.refl l v)\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nh : u =\u1da0[l] \u03c6 * v\nx\u271d : \u03b1\n\u22a2 ((\u03c6 - 1) * v) x\u271d = (\u03c6 * v) x\u271d - v x\u271d\n[PROOFSTEP]\nsimp [sub_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhz : \u2200\u1da0 (x : \u03b1) in l, v x = 0 \u2192 u x = 0\nhuv : Tendsto (u / v) l (\ud835\udcdd 1)\n\u22a2 u ~[l] v\n[PROOFSTEP]\nrw [isEquivalent_iff_exists_eq_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhz : \u2200\u1da0 (x : \u03b1) in l, v x = 0 \u2192 u x = 0\nhuv : Tendsto (u / v) l (\ud835\udcdd 1)\n\u22a2 \u2203 \u03c6 x, u =\u1da0[l] \u03c6 * v\n[PROOFSTEP]\nrefine' \u27e8u / v, huv, hz.mono fun x hz' \u21a6 (div_mul_cancel_of_imp hz').symm\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhz : \u2200\u1da0 (x : \u03b1) in l, v x \u2260 0\n\u22a2 u ~[l] v \u2194 Tendsto (u / v) l (\ud835\udcdd 1)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhz : \u2200\u1da0 (x : \u03b1) in l, v x \u2260 0\n\u22a2 u ~[l] v \u2192 Tendsto (u / v) l (\ud835\udcdd 1)\n[PROOFSTEP]\nintro hequiv\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhz : \u2200\u1da0 (x : \u03b1) in l, v x \u2260 0\nhequiv : u ~[l] v\n\u22a2 Tendsto (u / v) l (\ud835\udcdd 1)\n[PROOFSTEP]\nhave := hequiv.isLittleO.tendsto_div_nhds_zero\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhz : \u2200\u1da0 (x : \u03b1) in l, v x \u2260 0\nhequiv : u ~[l] v\nthis : Tendsto (fun x => (u - v) x / v x) l (\ud835\udcdd 0)\n\u22a2 Tendsto (u / v) l (\ud835\udcdd 1)\n[PROOFSTEP]\nsimp only [Pi.sub_apply, sub_div] at this \n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhz : \u2200\u1da0 (x : \u03b1) in l, v x \u2260 0\nhequiv : u ~[l] v\nthis : Tendsto (fun x => u x / v x - v x / v x) l (\ud835\udcdd 0)\n\u22a2 Tendsto (u / v) l (\ud835\udcdd 1)\n[PROOFSTEP]\nhave key : Tendsto (fun x \u21a6 v x / v x) l (\ud835\udcdd 1) :=\n (tendsto_congr' <| hz.mono fun x hnz \u21a6 @div_self _ _ (v x) hnz).mpr tendsto_const_nhds\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhz : \u2200\u1da0 (x : \u03b1) in l, v x \u2260 0\nhequiv : u ~[l] v\nthis : Tendsto (fun x => u x / v x - v x / v x) l (\ud835\udcdd 0)\nkey : Tendsto (fun x => v x / v x) l (\ud835\udcdd 1)\n\u22a2 Tendsto (u / v) l (\ud835\udcdd 1)\n[PROOFSTEP]\nconvert this.add key\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhz : \u2200\u1da0 (x : \u03b1) in l, v x \u2260 0\nhequiv : u ~[l] v\nthis : Tendsto (fun x => u x / v x - v x / v x) l (\ud835\udcdd 0)\nkey : Tendsto (fun x => v x / v x) l (\ud835\udcdd 1)\nx\u271d : \u03b1\n\u22a2 (u / v) x\u271d = u x\u271d / v x\u271d - v x\u271d / v x\u271d + v x\u271d / v x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhz : \u2200\u1da0 (x : \u03b1) in l, v x \u2260 0\nhequiv : u ~[l] v\nthis : Tendsto (fun x => u x / v x - v x / v x) l (\ud835\udcdd 0)\nkey : Tendsto (fun x => v x / v x) l (\ud835\udcdd 1)\n\u22a2 1 = 0 + 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhz : \u2200\u1da0 (x : \u03b1) in l, v x \u2260 0\n\u22a2 Tendsto (u / v) l (\ud835\udcdd 1) \u2192 u ~[l] v\n[PROOFSTEP]\nexact isEquivalent_of_tendsto_one (hz.mono fun x hnvz hz \u21a6 (hnvz hz).elim)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : a ~[l] b\nhuv : u ~[l] v\n\u22a2 (fun x => a x \u2022 u x) ~[l] fun x => b x \u2022 v x\n[PROOFSTEP]\nrcases hab.exists_eq_mul with \u27e8\u03c6, h\u03c6, hab\u03c6\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : a ~[l] b\nhuv : u ~[l] v\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\n\u22a2 (fun x => a x \u2022 u x) ~[l] fun x => b x \u2022 v x\n[PROOFSTEP]\nhave : ((fun x \u21a6 a x \u2022 u x) - (fun x \u21a6 b x \u2022 v x)) =\u1da0[l] fun x \u21a6 b x \u2022 (\u03c6 x \u2022 u x - v x) := by\n -- Porting note: `convert` has become too strong, so we need to specify `using 1`.\n convert (hab\u03c6.comp\u2082 (\u00b7 \u2022 \u00b7) <| EventuallyEq.refl _ u).sub (EventuallyEq.refl _ fun x \u21a6 b x \u2022 v x) using 1\n ext\n rw [Pi.mul_apply, mul_comm, mul_smul, \u2190 smul_sub]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : a ~[l] b\nhuv : u ~[l] v\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\n\u22a2 ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\n[PROOFSTEP]\nconvert (hab\u03c6.comp\u2082 (\u00b7 \u2022 \u00b7) <| EventuallyEq.refl _ u).sub (EventuallyEq.refl _ fun x \u21a6 b x \u2022 v x) using 1\n[GOAL]\ncase h.e'_5\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : a ~[l] b\nhuv : u ~[l] v\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\n\u22a2 (fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)) = fun x => (\u03c6 * b) x \u2022 u x - b x \u2022 v x\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : a ~[l] b\nhuv : u ~[l] v\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nx\u271d : \u03b1\n\u22a2 b x\u271d \u2022 (\u03c6 x\u271d \u2022 u x\u271d - v x\u271d) = (\u03c6 * b) x\u271d \u2022 u x\u271d - b x\u271d \u2022 v x\u271d\n[PROOFSTEP]\nrw [Pi.mul_apply, mul_comm, mul_smul, \u2190 smul_sub]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : a ~[l] b\nhuv : u ~[l] v\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\n\u22a2 (fun x => a x \u2022 u x) ~[l] fun x => b x \u2022 v x\n[PROOFSTEP]\nrefine' (isLittleO_congr this.symm <| EventuallyEq.rfl).mp ((isBigO_refl b l).smul_isLittleO _)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : a ~[l] b\nhuv : u ~[l] v\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\n\u22a2 (fun x => \u03c6 x \u2022 u x - v x) =o[l] fun x => v x\n[PROOFSTEP]\nrcases huv.isBigO.exists_pos with \u27e8C, hC, hCuv\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : a ~[l] b\nhuv : u ~[l] v\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : IsBigOWith C l u v\n\u22a2 (fun x => \u03c6 x \u2022 u x - v x) =o[l] fun x => v x\n[PROOFSTEP]\nrw [IsEquivalent] at *\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : (a - b) =o[l] b\nhuv : (u - v) =o[l] v\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : IsBigOWith C l u v\n\u22a2 (fun x => \u03c6 x \u2022 u x - v x) =o[l] fun x => v x\n[PROOFSTEP]\nrw [isLittleO_iff] at *\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\nhuv : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c * \u2016v x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : IsBigOWith C l u v\n\u22a2 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x \u2022 u x - v x\u2016 \u2264 c * \u2016v x\u2016\n[PROOFSTEP]\nrw [IsBigOWith] at hCuv \n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\nhuv : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c * \u2016v x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\n\u22a2 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x \u2022 u x - v x\u2016 \u2264 c * \u2016v x\u2016\n[PROOFSTEP]\nsimp only [Metric.tendsto_nhds, dist_eq_norm] at h\u03c6 \n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\nhuv : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c * \u2016v x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nh\u03c6 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < \u03b5\n\u22a2 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x \u2022 u x - v x\u2016 \u2264 c * \u2016v x\u2016\n[PROOFSTEP]\nintro c hc\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\nhuv : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c * \u2016v x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nh\u03c6 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < \u03b5\nc : \u211d\nhc : 0 < c\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x \u2022 u x - v x\u2016 \u2264 c * \u2016v x\u2016\n[PROOFSTEP]\nspecialize h\u03c6 (c / 2 / C) (div_pos (div_pos hc zero_lt_two) hC)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\nhuv : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c * \u2016v x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nc : \u211d\nhc : 0 < c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < c / 2 / C\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x \u2022 u x - v x\u2016 \u2264 c * \u2016v x\u2016\n[PROOFSTEP]\nspecialize huv (div_pos hc zero_lt_two)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nc : \u211d\nhc : 0 < c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < c / 2 / C\nhuv : \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x \u2022 u x - v x\u2016 \u2264 c * \u2016v x\u2016\n[PROOFSTEP]\nrefine' h\u03c6.mp (huv.mp <| hCuv.mono fun x hCuvx huvx h\u03c6x \u21a6 _)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nc : \u211d\nhc : 0 < c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < c / 2 / C\nhuv : \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nx : \u03b1\nhCuvx : \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nhuvx : \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nh\u03c6x : \u2016\u03c6 x - 1\u2016 < c / 2 / C\n\u22a2 \u2016\u03c6 x \u2022 u x - v x\u2016 \u2264 c * \u2016v x\u2016\n[PROOFSTEP]\nhave key :=\n calc\n \u2016\u03c6 x - 1\u2016 * \u2016u x\u2016 \u2264 c / 2 / C * \u2016u x\u2016 := by gcongr\n _ \u2264 c / 2 / C * (C * \u2016v x\u2016) := by gcongr\n _ = c / 2 * \u2016v x\u2016 := by\n field_simp [hC.ne.symm]\n ring\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nc : \u211d\nhc : 0 < c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < c / 2 / C\nhuv : \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nx : \u03b1\nhCuvx : \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nhuvx : \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nh\u03c6x : \u2016\u03c6 x - 1\u2016 < c / 2 / C\n\u22a2 \u2016\u03c6 x - 1\u2016 * \u2016u x\u2016 \u2264 c / 2 / C * \u2016u x\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nc : \u211d\nhc : 0 < c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < c / 2 / C\nhuv : \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nx : \u03b1\nhCuvx : \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nhuvx : \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nh\u03c6x : \u2016\u03c6 x - 1\u2016 < c / 2 / C\n\u22a2 c / 2 / C * \u2016u x\u2016 \u2264 c / 2 / C * (C * \u2016v x\u2016)\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nc : \u211d\nhc : 0 < c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < c / 2 / C\nhuv : \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nx : \u03b1\nhCuvx : \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nhuvx : \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nh\u03c6x : \u2016\u03c6 x - 1\u2016 < c / 2 / C\n\u22a2 c / 2 / C * (C * \u2016v x\u2016) = c / 2 * \u2016v x\u2016\n[PROOFSTEP]\nfield_simp [hC.ne.symm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nc : \u211d\nhc : 0 < c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < c / 2 / C\nhuv : \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nx : \u03b1\nhCuvx : \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nhuvx : \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nh\u03c6x : \u2016\u03c6 x - 1\u2016 < c / 2 / C\n\u22a2 c * (C * \u2016v x\u2016) * 2 = c * \u2016v x\u2016 * (2 * C)\n[PROOFSTEP]\nring\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nc : \u211d\nhc : 0 < c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < c / 2 / C\nhuv : \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nx : \u03b1\nhCuvx : \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nhuvx : \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nh\u03c6x : \u2016\u03c6 x - 1\u2016 < c / 2 / C\nkey : \u2016\u03c6 x - 1\u2016 * \u2016u x\u2016 \u2264 c / 2 * \u2016v x\u2016\n\u22a2 \u2016\u03c6 x \u2022 u x - v x\u2016 \u2264 c * \u2016v x\u2016\n[PROOFSTEP]\ncalc\n \u2016((fun x : \u03b1 \u21a6 \u03c6 x \u2022 u x) - v) x\u2016 = \u2016(\u03c6 x - 1) \u2022 u x + (u x - v x)\u2016 := by simp [sub_smul, sub_add]\n _ \u2264 \u2016(\u03c6 x - 1) \u2022 u x\u2016 + \u2016u x - v x\u2016 := (norm_add_le _ _)\n _ = \u2016\u03c6 x - 1\u2016 * \u2016u x\u2016 + \u2016u x - v x\u2016 := by rw [norm_smul]\n _ \u2264 c / 2 * \u2016v x\u2016 + \u2016u x - v x\u2016 := by gcongr\n _ \u2264 c / 2 * \u2016v x\u2016 + c / 2 * \u2016v x\u2016 := by gcongr; exact huvx\n _ = c * \u2016v x\u2016 := by ring\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nc : \u211d\nhc : 0 < c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < c / 2 / C\nhuv : \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nx : \u03b1\nhCuvx : \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nhuvx : \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nh\u03c6x : \u2016\u03c6 x - 1\u2016 < c / 2 / C\nkey : \u2016\u03c6 x - 1\u2016 * \u2016u x\u2016 \u2264 c / 2 * \u2016v x\u2016\n\u22a2 \u2016((fun x => \u03c6 x \u2022 u x) - v) x\u2016 = \u2016(\u03c6 x - 1) \u2022 u x + (u x - v x)\u2016\n[PROOFSTEP]\nsimp [sub_smul, sub_add]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nc : \u211d\nhc : 0 < c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < c / 2 / C\nhuv : \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nx : \u03b1\nhCuvx : \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nhuvx : \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nh\u03c6x : \u2016\u03c6 x - 1\u2016 < c / 2 / C\nkey : \u2016\u03c6 x - 1\u2016 * \u2016u x\u2016 \u2264 c / 2 * \u2016v x\u2016\n\u22a2 \u2016(\u03c6 x - 1) \u2022 u x\u2016 + \u2016u x - v x\u2016 = \u2016\u03c6 x - 1\u2016 * \u2016u x\u2016 + \u2016u x - v x\u2016\n[PROOFSTEP]\nrw [norm_smul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nc : \u211d\nhc : 0 < c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < c / 2 / C\nhuv : \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nx : \u03b1\nhCuvx : \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nhuvx : \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nh\u03c6x : \u2016\u03c6 x - 1\u2016 < c / 2 / C\nkey : \u2016\u03c6 x - 1\u2016 * \u2016u x\u2016 \u2264 c / 2 * \u2016v x\u2016\n\u22a2 \u2016\u03c6 x - 1\u2016 * \u2016u x\u2016 + \u2016u x - v x\u2016 \u2264 c / 2 * \u2016v x\u2016 + \u2016u x - v x\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nc : \u211d\nhc : 0 < c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < c / 2 / C\nhuv : \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nx : \u03b1\nhCuvx : \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nhuvx : \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nh\u03c6x : \u2016\u03c6 x - 1\u2016 < c / 2 / C\nkey : \u2016\u03c6 x - 1\u2016 * \u2016u x\u2016 \u2264 c / 2 * \u2016v x\u2016\n\u22a2 c / 2 * \u2016v x\u2016 + \u2016u x - v x\u2016 \u2264 c / 2 * \u2016v x\u2016 + c / 2 * \u2016v x\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase bc\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nc : \u211d\nhc : 0 < c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < c / 2 / C\nhuv : \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nx : \u03b1\nhCuvx : \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nhuvx : \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nh\u03c6x : \u2016\u03c6 x - 1\u2016 < c / 2 / C\nkey : \u2016\u03c6 x - 1\u2016 * \u2016u x\u2016 \u2264 c / 2 * \u2016v x\u2016\n\u22a2 \u2016u x - v x\u2016 \u2264 c / 2 * \u2016v x\u2016\n[PROOFSTEP]\nexact huvx\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\na b : \u03b1 \u2192 \ud835\udd5c\nu v : \u03b1 \u2192 E\nl : Filter \u03b1\nhab : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016(a - b) x\u2016 \u2264 c * \u2016b x\u2016\n\u03c6 : \u03b1 \u2192 \ud835\udd5c\nhab\u03c6 : a =\u1da0[l] \u03c6 * b\nthis : ((fun x => a x \u2022 u x) - fun x => b x \u2022 v x) =\u1da0[l] fun x => b x \u2022 (\u03c6 x \u2022 u x - v x)\nC : \u211d\nhC : 0 < C\nhCuv : \u2200\u1da0 (x : \u03b1) in l, \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nc : \u211d\nhc : 0 < c\nh\u03c6 : \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x - 1\u2016 < c / 2 / C\nhuv : \u2200\u1da0 (x : \u03b1) in l, \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nx : \u03b1\nhCuvx : \u2016u x\u2016 \u2264 C * \u2016v x\u2016\nhuvx : \u2016(u - v) x\u2016 \u2264 c / 2 * \u2016v x\u2016\nh\u03c6x : \u2016\u03c6 x - 1\u2016 < c / 2 / C\nkey : \u2016\u03c6 x - 1\u2016 * \u2016u x\u2016 \u2264 c / 2 * \u2016v x\u2016\n\u22a2 c / 2 * \u2016v x\u2016 + c / 2 * \u2016v x\u2016 = c * \u2016v x\u2016\n[PROOFSTEP]\nring\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhuv : u ~[l] v\n\u22a2 (fun x => (u x)\u207b\u00b9) ~[l] fun x => (v x)\u207b\u00b9\n[PROOFSTEP]\nrw [isEquivalent_iff_exists_eq_mul] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhuv : \u2203 \u03c6 x, u =\u1da0[l] \u03c6 * v\n\u22a2 \u2203 \u03c6 x, (fun x => (u x)\u207b\u00b9) =\u1da0[l] \u03c6 * fun x => (v x)\u207b\u00b9\n[PROOFSTEP]\nrcases huv with \u27e8\u03c6, h\u03c6, h\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nh : u =\u1da0[l] \u03c6 * v\n\u22a2 \u2203 \u03c6 x, (fun x => (u x)\u207b\u00b9) =\u1da0[l] \u03c6 * fun x => (v x)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 inv_one]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nh : u =\u1da0[l] \u03c6 * v\n\u22a2 \u2203 \u03c6 x, (fun x => (u x)\u207b\u00b9) =\u1da0[l] \u03c6 * fun x => (v x)\u207b\u00b9\n[PROOFSTEP]\nrefine' \u27e8fun x \u21a6 (\u03c6 x)\u207b\u00b9, Tendsto.inv\u2080 h\u03c6 (by norm_num), _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nh : u =\u1da0[l] \u03c6 * v\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nh : u =\u1da0[l] \u03c6 * v\n\u22a2 (fun x => (u x)\u207b\u00b9) =\u1da0[l] (fun x => (\u03c6 x)\u207b\u00b9) * fun x => (v x)\u207b\u00b9\n[PROOFSTEP]\nconvert h.inv\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\n\u03c6 : \u03b1 \u2192 \u03b2\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 1)\nh : u =\u1da0[l] \u03c6 * v\nx\u271d : \u03b1\n\u22a2 ((fun x => (\u03c6 x)\u207b\u00b9) * fun x => (v x)\u207b\u00b9) x\u271d = ((\u03c6 * v) x\u271d)\u207b\u00b9\n[PROOFSTEP]\nsimp [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nt u v w : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\nhtu : t ~[l] u\nhvw : v ~[l] w\n\u22a2 (fun x => t x / v x) ~[l] fun x => u x / w x\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using htu.mul hvw.inv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\nu v : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\ninst\u271d : OrderTopology \u03b2\nhuv : u ~[l] v\nhu : Tendsto u l atBot\n\u22a2 Tendsto v l atBot\n[PROOFSTEP]\nconvert tendsto_neg_atTop_atBot.comp (huv.neg.tendsto_atTop <| tendsto_neg_atBot_atTop.comp hu)\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\nu v : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\ninst\u271d : OrderTopology \u03b2\nhuv : u ~[l] v\nhu : Tendsto u l atBot\n\u22a2 v = Neg.neg \u2218 fun x => -v x\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\nu v : \u03b1 \u2192 \u03b2\nl : Filter \u03b1\ninst\u271d : OrderTopology \u03b2\nhuv : u ~[l] v\nhu : Tendsto u l atBot\nx\u271d : \u03b1\n\u22a2 v x\u271d = (Neg.neg \u2218 fun x => -v x) x\u271d\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "llama_tokens": 19841, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.40558604713174917}} {"text": "[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v : V\np : Walk G u v\nht : IsTrail p\nx : V\n\u22a2 Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 u \u2260 v \u2192 x \u2260 u \u2227 x \u2260 v\n[PROOFSTEP]\ninduction' p with u u v w huv p ih\n[GOAL]\ncase nil\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v : V\np : Walk G u\u271d v\nht\u271d : IsTrail p\nx u : V\nht : IsTrail Walk.nil\n\u22a2 Even (List.countp (fun e => decide (x \u2208 e)) (edges Walk.nil)) \u2194 u \u2260 u \u2192 x \u2260 u \u2227 x \u2260 u\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v w : V\nhuv : Adj G u v\np : Walk G v w\nih : IsTrail p \u2192 (Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 v \u2260 w \u2192 x \u2260 v \u2227 x \u2260 w)\nht : IsTrail (cons huv p)\n\u22a2 Even (List.countp (fun e => decide (x \u2208 e)) (edges (cons huv p))) \u2194 u \u2260 w \u2192 x \u2260 u \u2227 x \u2260 w\n[PROOFSTEP]\nrw [cons_isTrail_iff] at ht \n[GOAL]\ncase cons\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v w : V\nhuv : Adj G u v\np : Walk G v w\nih : IsTrail p \u2192 (Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 v \u2260 w \u2192 x \u2260 v \u2227 x \u2260 w)\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, v) \u2208 edges p\n\u22a2 Even (List.countp (fun e => decide (x \u2208 e)) (edges (cons huv p))) \u2194 u \u2260 w \u2192 x \u2260 u \u2227 x \u2260 w\n[PROOFSTEP]\nspecialize ih ht.1\n[GOAL]\ncase cons\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v w : V\nhuv : Adj G u v\np : Walk G v w\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, v) \u2208 edges p\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 v \u2260 w \u2192 x \u2260 v \u2227 x \u2260 w\n\u22a2 Even (List.countp (fun e => decide (x \u2208 e)) (edges (cons huv p))) \u2194 u \u2260 w \u2192 x \u2260 u \u2227 x \u2260 w\n[PROOFSTEP]\nsimp only [List.countp_cons, Ne.def, edges_cons, Sym2.mem_iff]\n[GOAL]\ncase cons\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v w : V\nhuv : Adj G u v\np : Walk G v w\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, v) \u2208 edges p\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 v \u2260 w \u2192 x \u2260 v \u2227 x \u2260 w\n\u22a2 Even (List.countp (fun e => decide (x \u2208 e)) (edges p) + if decide (x = u \u2228 x = v) = true then 1 else 0) \u2194\n \u00acu = w \u2192 \u00acx = u \u2227 \u00acx = w\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v w : V\nhuv : Adj G u v\np : Walk G v w\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, v) \u2208 edges p\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 v \u2260 w \u2192 x \u2260 v \u2227 x \u2260 w\nh : decide (x = u \u2228 x = v) = true\n\u22a2 Even (List.countp (fun e => decide (x \u2208 e)) (edges p) + 1) \u2194 \u00acu = w \u2192 \u00acx = u \u2227 \u00acx = w\n[PROOFSTEP]\nrw [decide_eq_true_eq] at h \n[GOAL]\ncase pos\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v w : V\nhuv : Adj G u v\np : Walk G v w\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, v) \u2208 edges p\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 v \u2260 w \u2192 x \u2260 v \u2227 x \u2260 w\nh : x = u \u2228 x = v\n\u22a2 Even (List.countp (fun e => decide (x \u2208 e)) (edges p) + 1) \u2194 \u00acu = w \u2192 \u00acx = u \u2227 \u00acx = w\n[PROOFSTEP]\nobtain (rfl | rfl) := h\n[GOAL]\ncase pos.inl\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v\u271d : V\np\u271d : Walk G u v\u271d\nht\u271d : IsTrail p\u271d\nx v w : V\np : Walk G v w\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 v \u2260 w \u2192 x \u2260 v \u2227 x \u2260 w\nhuv : Adj G x v\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (x, v) \u2208 edges p\n\u22a2 Even (List.countp (fun e => decide (x \u2208 e)) (edges p) + 1) \u2194 \u00acx = w \u2192 \u00acx = x \u2227 \u00acx = w\n[PROOFSTEP]\nrw [Nat.even_add_one, ih]\n[GOAL]\ncase pos.inl\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v\u271d : V\np\u271d : Walk G u v\u271d\nht\u271d : IsTrail p\u271d\nx v w : V\np : Walk G v w\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 v \u2260 w \u2192 x \u2260 v \u2227 x \u2260 w\nhuv : Adj G x v\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (x, v) \u2208 edges p\n\u22a2 \u00ac(v \u2260 w \u2192 x \u2260 v \u2227 x \u2260 w) \u2194 \u00acx = w \u2192 \u00acx = x \u2227 \u00acx = w\n[PROOFSTEP]\nsimp only [huv.ne, imp_false, Ne.def, not_false_iff, true_and_iff, not_forall, Classical.not_not, exists_prop,\n eq_self_iff_true, not_true, false_and_iff, and_iff_right_iff_imp]\n[GOAL]\ncase pos.inl\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v\u271d : V\np\u271d : Walk G u v\u271d\nht\u271d : IsTrail p\u271d\nx v w : V\np : Walk G v w\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 v \u2260 w \u2192 x \u2260 v \u2227 x \u2260 w\nhuv : Adj G x v\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (x, v) \u2208 edges p\n\u22a2 x = w \u2192 \u00acv = w\n[PROOFSTEP]\nrintro rfl rfl\n[GOAL]\ncase pos.inl\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v\u271d : V\np\u271d : Walk G u v\u271d\nht\u271d : IsTrail p\u271d\nv : V\nhuv : Adj G v v\np : Walk G v v\nih : Even (List.countp (fun e => decide (v \u2208 e)) (edges p)) \u2194 v \u2260 v \u2192 v \u2260 v \u2227 v \u2260 v\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (v, v) \u2208 edges p\n\u22a2 False\n[PROOFSTEP]\nexact G.loopless _ huv\n[GOAL]\ncase pos.inr\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v : V\np\u271d : Walk G u\u271d v\nht\u271d : IsTrail p\u271d\nx u w : V\nhuv : Adj G u x\np : Walk G x w\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, x) \u2208 edges p\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 x \u2260 w \u2192 x \u2260 x \u2227 x \u2260 w\n\u22a2 Even (List.countp (fun e => decide (x \u2208 e)) (edges p) + 1) \u2194 \u00acu = w \u2192 \u00acx = u \u2227 \u00acx = w\n[PROOFSTEP]\nrw [Nat.even_add_one, ih, \u2190 not_iff_not]\n[GOAL]\ncase pos.inr\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v : V\np\u271d : Walk G u\u271d v\nht\u271d : IsTrail p\u271d\nx u w : V\nhuv : Adj G u x\np : Walk G x w\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, x) \u2208 edges p\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 x \u2260 w \u2192 x \u2260 x \u2227 x \u2260 w\n\u22a2 \u00ac\u00ac(x \u2260 w \u2192 x \u2260 x \u2227 x \u2260 w) \u2194 \u00ac(\u00acu = w \u2192 \u00acx = u \u2227 \u00acx = w)\n[PROOFSTEP]\nsimp only [huv.ne.symm, Ne.def, eq_self_iff_true, not_true, false_and_iff, not_forall, not_false_iff, exists_prop,\n and_true_iff, Classical.not_not, true_and_iff, iff_and_self]\n[GOAL]\ncase pos.inr\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v : V\np\u271d : Walk G u\u271d v\nht\u271d : IsTrail p\u271d\nx u w : V\nhuv : Adj G u x\np : Walk G x w\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, x) \u2208 edges p\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 x \u2260 w \u2192 x \u2260 x \u2227 x \u2260 w\n\u22a2 x = w \u2192 \u00acu = w\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase pos.inr\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v : V\np\u271d : Walk G u\u271d v\nht\u271d : IsTrail p\u271d\nx u : V\nhuv : Adj G u x\np : Walk G x x\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, x) \u2208 edges p\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 x \u2260 x \u2192 x \u2260 x \u2227 x \u2260 x\n\u22a2 \u00acu = x\n[PROOFSTEP]\nexact huv.ne\n[GOAL]\ncase neg\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v w : V\nhuv : Adj G u v\np : Walk G v w\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, v) \u2208 edges p\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 v \u2260 w \u2192 x \u2260 v \u2227 x \u2260 w\nh : \u00acdecide (x = u \u2228 x = v) = true\n\u22a2 Even (List.countp (fun e => decide (x \u2208 e)) (edges p) + 0) \u2194 \u00acu = w \u2192 \u00acx = u \u2227 \u00acx = w\n[PROOFSTEP]\nrw [decide_eq_true_eq, not_or] at h \n[GOAL]\ncase neg\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v w : V\nhuv : Adj G u v\np : Walk G v w\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, v) \u2208 edges p\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 v \u2260 w \u2192 x \u2260 v \u2227 x \u2260 w\nh : \u00acx = u \u2227 \u00acx = v\n\u22a2 Even (List.countp (fun e => decide (x \u2208 e)) (edges p) + 0) \u2194 \u00acu = w \u2192 \u00acx = u \u2227 \u00acx = w\n[PROOFSTEP]\nsimp only [h.1, h.2, not_false_iff, true_and_iff, add_zero, Ne.def] at ih \u22a2\n[GOAL]\ncase neg\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v w : V\nhuv : Adj G u v\np : Walk G v w\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, v) \u2208 edges p\nh : \u00acx = u \u2227 \u00acx = v\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 \u00acv = w \u2192 \u00acx = w\n\u22a2 Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 \u00acu = w \u2192 \u00acx = w\n[PROOFSTEP]\nrw [ih]\n[GOAL]\ncase neg\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v w : V\nhuv : Adj G u v\np : Walk G v w\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, v) \u2208 edges p\nh : \u00acx = u \u2227 \u00acx = v\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 \u00acv = w \u2192 \u00acx = w\n\u22a2 \u00acv = w \u2192 \u00acx = w \u2194 \u00acu = w \u2192 \u00acx = w\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase neg.mp\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v w : V\nhuv : Adj G u v\np : Walk G v w\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, v) \u2208 edges p\nh : \u00acx = u \u2227 \u00acx = v\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 \u00acv = w \u2192 \u00acx = w\n\u22a2 (\u00acv = w \u2192 \u00acx = w) \u2192 \u00acu = w \u2192 \u00acx = w\n[PROOFSTEP]\nrintro h' h'' rfl\n[GOAL]\ncase neg.mp\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v : V\nhuv : Adj G u v\nh : \u00acx = u \u2227 \u00acx = v\np : Walk G v x\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, v) \u2208 edges p\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 \u00acv = x \u2192 \u00acx = x\nh' : \u00acv = x \u2192 \u00acx = x\nh'' : \u00acu = x\n\u22a2 False\n[PROOFSTEP]\nsimp only [imp_false, eq_self_iff_true, not_true, Classical.not_not] at h' \n[GOAL]\ncase neg.mp\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v : V\nhuv : Adj G u v\nh : \u00acx = u \u2227 \u00acx = v\np : Walk G v x\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, v) \u2208 edges p\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 \u00acv = x \u2192 \u00acx = x\nh'' : \u00acu = x\nh' : v = x\n\u22a2 False\n[PROOFSTEP]\ncases h'\n[GOAL]\ncase neg.mp.refl\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v : V\np\u271d : Walk G u\u271d v\nht\u271d : IsTrail p\u271d\nx u : V\nh'' : \u00acu = x\nhuv : Adj G u x\nh : \u00acx = u \u2227 \u00acx = x\np : Walk G x x\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, x) \u2208 edges p\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 \u00acx = x \u2192 \u00acx = x\n\u22a2 False\n[PROOFSTEP]\nsimp only [not_true, and_false, false_and] at h \n[GOAL]\ncase neg.mpr\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v w : V\nhuv : Adj G u v\np : Walk G v w\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, v) \u2208 edges p\nh : \u00acx = u \u2227 \u00acx = v\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 \u00acv = w \u2192 \u00acx = w\n\u22a2 (\u00acu = w \u2192 \u00acx = w) \u2192 \u00acv = w \u2192 \u00acx = w\n[PROOFSTEP]\nrintro h' h'' rfl\n[GOAL]\ncase neg.mpr\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v : V\nhuv : Adj G u v\nh : \u00acx = u \u2227 \u00acx = v\np : Walk G v x\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, v) \u2208 edges p\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 \u00acv = x \u2192 \u00acx = x\nh' : \u00acu = x \u2192 \u00acx = x\nh'' : \u00acv = x\n\u22a2 False\n[PROOFSTEP]\nsimp only [imp_false, eq_self_iff_true, not_true, Classical.not_not] at h' \n[GOAL]\ncase neg.mpr\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu\u271d v\u271d : V\np\u271d : Walk G u\u271d v\u271d\nht\u271d : IsTrail p\u271d\nx u v : V\nhuv : Adj G u v\nh : \u00acx = u \u2227 \u00acx = v\np : Walk G v x\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (u, v) \u2208 edges p\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 \u00acv = x \u2192 \u00acx = x\nh'' : \u00acv = x\nh' : u = x\n\u22a2 False\n[PROOFSTEP]\ncases h'\n[GOAL]\ncase neg.mpr.refl\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v\u271d : V\np\u271d : Walk G u v\u271d\nht\u271d : IsTrail p\u271d\nx v : V\np : Walk G v x\nih : Even (List.countp (fun e => decide (x \u2208 e)) (edges p)) \u2194 \u00acv = x \u2192 \u00acx = x\nh'' : \u00acv = x\nhuv : Adj G x v\nh : \u00acx = x \u2227 \u00acx = v\nht : IsTrail p \u2227 \u00acQuotient.mk (Sym2.Rel.setoid V) (x, v) \u2208 edges p\n\u22a2 False\n[PROOFSTEP]\nsimp only [not_true, and_false, false_and] at h \n[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\n\u22a2 IsTrail p\n[PROOFSTEP]\nrw [isTrail_def, List.nodup_iff_count_le_one]\n[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\n\u22a2 \u2200 (a : Sym2 V), List.count a (edges p) \u2264 1\n[PROOFSTEP]\nintro e\n[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\ne : Sym2 V\n\u22a2 List.count e (edges p) \u2264 1\n[PROOFSTEP]\nby_cases he : e \u2208 p.edges\n[GOAL]\ncase pos\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\ne : Sym2 V\nhe : e \u2208 edges p\n\u22a2 List.count e (edges p) \u2264 1\n[PROOFSTEP]\nexact (h e (edges_subset_edgeSet _ he)).le\n[GOAL]\ncase neg\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\ne : Sym2 V\nhe : \u00ace \u2208 edges p\n\u22a2 List.count e (edges p) \u2264 1\n[PROOFSTEP]\nsimp [he]\n[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\ne : Sym2 V\nhe : e \u2208 edgeSet G\n\u22a2 e \u2208 edges p\n[PROOFSTEP]\nsimpa using (h e he).ge\n[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\ne : Sym2 V\n\u22a2 e \u2208 IsTrail.edgesFinset (_ : IsTrail p) \u2194 e \u2208 edgeSet G\n[PROOFSTEP]\nsimp only [Finset.mem_mk, Multiset.mem_coe, h.mem_edges_iff]\n[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v : V\np : Walk G u v\n\u22a2 IsEulerian p \u2194 IsTrail p \u2227 \u2200 (e : Sym2 V), e \u2208 edgeSet G \u2192 e \u2208 edges p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v : V\np : Walk G u v\n\u22a2 IsEulerian p \u2192 IsTrail p \u2227 \u2200 (e : Sym2 V), e \u2208 edgeSet G \u2192 e \u2208 edges p\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v : V\np : Walk G u v\nh : IsEulerian p\n\u22a2 IsTrail p \u2227 \u2200 (e : Sym2 V), e \u2208 edgeSet G \u2192 e \u2208 edges p\n[PROOFSTEP]\nexact \u27e8h.isTrail, fun _ => h.mem_edges_iff.mpr\u27e9\n[GOAL]\ncase mpr\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v : V\np : Walk G u v\n\u22a2 (IsTrail p \u2227 \u2200 (e : Sym2 V), e \u2208 edgeSet G \u2192 e \u2208 edges p) \u2192 IsEulerian p\n[PROOFSTEP]\nrintro \u27e8h, hl\u27e9\n[GOAL]\ncase mpr.intro\nV : Type u_1\nG : SimpleGraph V\ninst\u271d : DecidableEq V\nu v : V\np : Walk G u v\nh : IsTrail p\nhl : \u2200 (e : Sym2 V), e \u2208 edgeSet G \u2192 e \u2208 edges p\n\u22a2 IsEulerian p\n[PROOFSTEP]\nexact h.isEulerian_of_forall_mem hl\n[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : Fintype \u2191(edgeSet G)\nu v : V\np : Walk G u v\nh : IsEulerian p\n\u22a2 IsTrail.edgesFinset (_ : IsTrail p) = edgeFinset G\n[PROOFSTEP]\next e\n[GOAL]\ncase a\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b9 : DecidableEq V\ninst\u271d : Fintype \u2191(edgeSet G)\nu v : V\np : Walk G u v\nh : IsEulerian p\ne : Sym2 V\n\u22a2 e \u2208 IsTrail.edgesFinset (_ : IsTrail p) \u2194 e \u2208 edgeFinset G\n[PROOFSTEP]\nsimp [h.mem_edges_iff]\n[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\nx u v : V\np : Walk G u v\nht : IsEulerian p\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\n\u22a2 Even (degree G x) \u2194 u \u2260 v \u2192 x \u2260 u \u2227 x \u2260 v\n[PROOFSTEP]\nconvert ht.isTrail.even_countp_edges_iff x\n[GOAL]\ncase h.e'_1.h.e'_3\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\nx u v : V\np : Walk G u v\nht : IsEulerian p\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\n\u22a2 degree G x = List.countp (fun e => decide (x \u2208 e)) (edges p)\n[PROOFSTEP]\nrw [\u2190 Multiset.coe_countp, Multiset.countp_eq_card_filter, \u2190 card_incidenceFinset_eq_degree]\n[GOAL]\ncase h.e'_1.h.e'_3\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\nx u v : V\np : Walk G u v\nht : IsEulerian p\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\n\u22a2 Finset.card (incidenceFinset G x) = \u2191Multiset.card (Multiset.filter (fun e => x \u2208 e) \u2191(edges p))\n[PROOFSTEP]\nchange Multiset.card _ = _\n[GOAL]\ncase h.e'_1.h.e'_3\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\nx u v : V\np : Walk G u v\nht : IsEulerian p\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\n\u22a2 \u2191Multiset.card (incidenceFinset G x).val = \u2191Multiset.card (Multiset.filter (fun e => x \u2208 e) \u2191(edges p))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e'_1.h.e'_3.h.e_6.h\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\nx u v : V\np : Walk G u v\nht : IsEulerian p\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\n\u22a2 (incidenceFinset G x).val = Multiset.filter (fun e => x \u2208 e) \u2191(edges p)\n[PROOFSTEP]\nconvert_to _ = (ht.isTrail.edgesFinset.filter (Membership.mem x)).val\n[GOAL]\ncase h.e'_1.h.e'_3.h.e_6.h.convert_2\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\nx u v : V\np : Walk G u v\nht : IsEulerian p\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\n\u22a2 (incidenceFinset G x).val = (Finset.filter (Membership.mem x) (IsTrail.edgesFinset (_ : IsTrail p))).val\n[PROOFSTEP]\nhave : Fintype G.edgeSet := fintypeEdgeSet ht\n[GOAL]\ncase h.e'_1.h.e'_3.h.e_6.h.convert_2\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\nx u v : V\np : Walk G u v\nht : IsEulerian p\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nthis : Fintype \u2191(edgeSet G)\n\u22a2 (incidenceFinset G x).val = (Finset.filter (Membership.mem x) (IsTrail.edgesFinset (_ : IsTrail p))).val\n[PROOFSTEP]\nrw [ht.edgesFinset_eq, G.incidenceFinset_eq_filter x]\n[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\ns : Finset V\nh : s = Finset.filter (fun v => Odd (degree G v)) Finset.univ\n\u22a2 Finset.card s = 0 \u2228 Finset.card s = 2\n[PROOFSTEP]\nsubst s\n[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\n\u22a2 Finset.card (Finset.filter (fun v => Odd (degree G v)) Finset.univ) = 0 \u2228\n Finset.card (Finset.filter (fun v => Odd (degree G v)) Finset.univ) = 2\n[PROOFSTEP]\nsimp only [Nat.odd_iff_not_even, Finset.card_eq_zero]\n[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\n\u22a2 Finset.filter (fun v => \u00acEven (degree G v)) Finset.univ = \u2205 \u2228\n Finset.card (Finset.filter (fun v => \u00acEven (degree G v)) Finset.univ) = 2\n[PROOFSTEP]\nsimp only [ht.even_degree_iff, Ne.def, not_forall, not_and, Classical.not_not, exists_prop]\n[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\n\u22a2 Finset.filter (fun v_1 => \u00acu = v \u2227 (\u00acv_1 = u \u2192 v_1 = v)) Finset.univ = \u2205 \u2228\n Finset.card (Finset.filter (fun v_1 => \u00acu = v \u2227 (\u00acv_1 = u \u2192 v_1 = v)) Finset.univ) = 2\n[PROOFSTEP]\nobtain rfl | hn := eq_or_ne u v\n[GOAL]\ncase inl\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu : V\np : Walk G u u\nht : IsEulerian p\n\u22a2 Finset.filter (fun v => \u00acu = u \u2227 (\u00acv = u \u2192 v = u)) Finset.univ = \u2205 \u2228\n Finset.card (Finset.filter (fun v => \u00acu = u \u2227 (\u00acv = u \u2192 v = u)) Finset.univ) = 2\n[PROOFSTEP]\nleft\n[GOAL]\ncase inl.h\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu : V\np : Walk G u u\nht : IsEulerian p\n\u22a2 Finset.filter (fun v => \u00acu = u \u2227 (\u00acv = u \u2192 v = u)) Finset.univ = \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u \u2260 v\n\u22a2 Finset.filter (fun v_1 => \u00acu = v \u2227 (\u00acv_1 = u \u2192 v_1 = v)) Finset.univ = \u2205 \u2228\n Finset.card (Finset.filter (fun v_1 => \u00acu = v \u2227 (\u00acv_1 = u \u2192 v_1 = v)) Finset.univ) = 2\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.h\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u \u2260 v\n\u22a2 Finset.card (Finset.filter (fun v_1 => \u00acu = v \u2227 (\u00acv_1 = u \u2192 v_1 = v)) Finset.univ) = 2\n[PROOFSTEP]\nconvert_to _ = ({ u, v } : Finset V).card\n[GOAL]\ncase h.e'_3\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u \u2260 v\n\u22a2 2 = Finset.card {u, v}\n[PROOFSTEP]\nsimp [hn]\n[GOAL]\ncase inr.h.convert_2\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u \u2260 v\n\u22a2 Finset.card (Finset.filter (fun v_1 => \u00acu = v \u2227 (\u00acv_1 = u \u2192 v_1 = v)) Finset.univ) = Finset.card {u, v}\n[PROOFSTEP]\ncongr\n[GOAL]\ncase inr.h.convert_2.e_s\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u \u2260 v\n\u22a2 Finset.filter (fun v_1 => \u00acu = v \u2227 (\u00acv_1 = u \u2192 v_1 = v)) Finset.univ = {u, v}\n[PROOFSTEP]\next x\n[GOAL]\ncase inr.h.convert_2.e_s.a\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u \u2260 v\nx : V\n\u22a2 x \u2208 Finset.filter (fun v_1 => \u00acu = v \u2227 (\u00acv_1 = u \u2192 v_1 = v)) Finset.univ \u2194 x \u2208 {u, v}\n[PROOFSTEP]\nsimp [hn, imp_iff_not_or]\n[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\n\u22a2 Fintype.card \u2191{v | Odd (degree G v)} = 0 \u2228 Fintype.card \u2191{v | Odd (degree G v)} = 2\n[PROOFSTEP]\nrw [\u2190 Set.toFinset_card]\n[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\n\u22a2 Finset.card (Set.toFinset {v | Odd (degree G v)}) = 0 \u2228 Finset.card (Set.toFinset {v | Odd (degree G v)}) = 2\n[PROOFSTEP]\napply IsEulerian.card_filter_odd_degree ht\n[GOAL]\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\n\u22a2 Set.toFinset {v | Odd (degree G v)} = Finset.filter (fun v => Odd (degree G v)) Finset.univ\n[PROOFSTEP]\next v\n[GOAL]\ncase a\nV : Type u_1\nG : SimpleGraph V\ninst\u271d\u00b2 : DecidableEq V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nu v\u271d : V\np : Walk G u v\u271d\nht : IsEulerian p\nv : V\n\u22a2 v \u2208 Set.toFinset {v | Odd (degree G v)} \u2194 v \u2208 Finset.filter (fun v => Odd (degree G v)) Finset.univ\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SimpleGraph.Trails", "llama_tokens": 11315, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585786300049, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.4053503900062879}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E x)\nHB : Type u_7\ninst\u271d\u00b3 : TopologicalSpace HB\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nx : TotalSpace F E\n\u22a2 chartAt (ModelProd HB F) x =\n (trivializationAt F E x.proj).toLocalHomeomorph \u226b\u2095 LocalHomeomorph.prod (chartAt HB x.proj) (LocalHomeomorph.refl F)\n[PROOFSTEP]\ndsimp only [chartAt_comp, prodChartedSpace_chartAt, FiberBundle.chartedSpace'_chartAt, chartAt_self_eq]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E x)\nHB : Type u_7\ninst\u271d\u00b3 : TopologicalSpace HB\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nx : TotalSpace F E\n\u22a2 (trivializationAt F E x.proj).toLocalHomeomorph \u226b\u2095\n LocalHomeomorph.prod (chartAt HB (\u2191(trivializationAt F E x.proj).toLocalHomeomorph x).fst)\n (LocalHomeomorph.refl F) =\n (trivializationAt F E x.proj).toLocalHomeomorph \u226b\u2095 LocalHomeomorph.prod (chartAt HB x.proj) (LocalHomeomorph.refl F)\n[PROOFSTEP]\nrw [Trivialization.coe_coe, Trivialization.coe_fst' _ (mem_baseSet_trivializationAt F E x.proj)]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E x)\nHB : Type u_7\ninst\u271d\u00b3 : TopologicalSpace HB\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nx : TotalSpace F E\ny : ModelProd HB F\nhy : y \u2208 (chartAt (ModelProd HB F) x).toLocalEquiv.target\n\u22a2 (\u2191(LocalHomeomorph.symm (chartAt (ModelProd HB F) x)) y).proj = \u2191(LocalHomeomorph.symm (chartAt HB x.proj)) y.fst\n[PROOFSTEP]\nsimp only [FiberBundle.chartedSpace_chartAt, mfld_simps] at hy \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E x)\nHB : Type u_7\ninst\u271d\u00b3 : TopologicalSpace HB\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nx : TotalSpace F E\ny : ModelProd HB F\nhy :\n y \u2208 (chartAt HB x.proj).toLocalEquiv.target \u00d7\u02e2 univ \u2227\n \u2191(LocalEquiv.symm (LocalEquiv.prod (chartAt HB x.proj).toLocalEquiv (LocalEquiv.refl F))) y \u2208\n (trivializationAt F E x.proj).toLocalHomeomorph.toLocalEquiv.target\n\u22a2 (\u2191(LocalHomeomorph.symm (trivializationAt F E x.proj).toLocalHomeomorph)\n (\u2191(LocalHomeomorph.symm (LocalHomeomorph.prod (chartAt HB x.proj) (LocalHomeomorph.refl F))) y)).proj =\n \u2191(LocalHomeomorph.symm (chartAt HB x.proj)) y.fst\n[PROOFSTEP]\nexact (trivializationAt F E x.proj).proj_symm_apply hy.2\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nx : TotalSpace F E\n\u22a2 extChartAt (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) x =\n (trivializationAt F E x.proj).toLocalHomeomorph.toLocalEquiv \u226b\n LocalEquiv.prod (extChartAt IB x.proj) (LocalEquiv.refl F)\n[PROOFSTEP]\nsimp_rw [extChartAt, FiberBundle.chartedSpace_chartAt, extend]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nx : TotalSpace F E\n\u22a2 ((trivializationAt F E x.proj).toLocalHomeomorph \u226b\u2095\n LocalHomeomorph.prod (chartAt HB x.proj) (LocalHomeomorph.refl F)).toLocalEquiv \u226b\n (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)).toLocalEquiv =\n (trivializationAt F E x.proj).toLocalHomeomorph.toLocalEquiv \u226b\n LocalEquiv.prod ((chartAt HB x.proj).toLocalEquiv \u226b IB.toLocalEquiv) (LocalEquiv.refl F)\n[PROOFSTEP]\nsimp only [LocalEquiv.trans_assoc, mfld_simps]\n -- porting note: should not be needed\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nx : TotalSpace F E\n\u22a2 (trivializationAt F E x.proj).toLocalHomeomorph.toLocalEquiv \u226b\n LocalEquiv.prod (chartAt HB x.proj).toLocalEquiv (LocalEquiv.refl F) \u226b\n LocalEquiv.prod IB.toLocalEquiv (LocalEquiv.refl F) =\n (trivializationAt F E x.proj).toLocalHomeomorph.toLocalEquiv \u226b\n LocalEquiv.prod ((chartAt HB x.proj).toLocalEquiv \u226b IB.toLocalEquiv) (LocalEquiv.refl F)\n[PROOFSTEP]\nrw [LocalEquiv.prod_trans, LocalEquiv.refl_trans]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nx : TotalSpace F E\n\u22a2 (extChartAt (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) x).target =\n ((extChartAt IB x.proj).target \u2229\n \u2191(LocalEquiv.symm (extChartAt IB x.proj)) \u207b\u00b9' (trivializationAt F E x.proj).baseSet) \u00d7\u02e2\n univ\n[PROOFSTEP]\nrw [FiberBundle.extChartAt, LocalEquiv.trans_target, Trivialization.target_eq, inter_prod]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nx : TotalSpace F E\n\u22a2 (LocalEquiv.prod (extChartAt IB x.proj) (LocalEquiv.refl F)).target \u2229\n \u2191(LocalEquiv.symm (LocalEquiv.prod (extChartAt IB x.proj) (LocalEquiv.refl F))) \u207b\u00b9'\n (trivializationAt F E x.proj).baseSet \u00d7\u02e2 univ =\n (extChartAt IB x.proj).target \u00d7\u02e2 univ \u2229\n (\u2191(LocalEquiv.symm (extChartAt IB x.proj)) \u207b\u00b9' (trivializationAt F E x.proj).baseSet) \u00d7\u02e2 univ\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nf : M \u2192 TotalSpace F E\ns : Set M\nx\u2080 : M\n\u22a2 ContMDiffWithinAt IM (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) n f s x\u2080 \u2194\n ContMDiffWithinAt IM IB n (fun x => (f x).proj) s x\u2080 \u2227\n ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n (fun x => (\u2191(trivializationAt F E (f x\u2080).proj) (f x)).snd) s x\u2080\n[PROOFSTEP]\nsimp (config := { singlePass := true }) only [contMDiffWithinAt_iff_target]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nf : M \u2192 TotalSpace F E\ns : Set M\nx\u2080 : M\n\u22a2 ContinuousWithinAt f s x\u2080 \u2227\n ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, EB \u00d7 F) n (\u2191(extChartAt (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (f x\u2080)) \u2218 f) s x\u2080 \u2194\n (ContinuousWithinAt (fun x => (f x).proj) s x\u2080 \u2227\n ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, EB) n (\u2191(extChartAt IB (f x\u2080).proj) \u2218 fun x => (f x).proj) s x\u2080) \u2227\n ContinuousWithinAt (fun x => (\u2191(trivializationAt F E (f x\u2080).proj) (f x)).snd) s x\u2080 \u2227\n ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n\n (\u2191(extChartAt \ud835\udcd8(\ud835\udd5c, F) (\u2191(trivializationAt F E (f x\u2080).proj) (f x\u2080)).snd) \u2218 fun x =>\n (\u2191(trivializationAt F E (f x\u2080).proj) (f x)).snd)\n s x\u2080\n[PROOFSTEP]\nrw [and_and_and_comm, \u2190 FiberBundle.continuousWithinAt_totalSpace, and_congr_right_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nf : M \u2192 TotalSpace F E\ns : Set M\nx\u2080 : M\n\u22a2 ContinuousWithinAt f s x\u2080 \u2192\n (ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, EB \u00d7 F) n (\u2191(extChartAt (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (f x\u2080)) \u2218 f) s x\u2080 \u2194\n ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, EB) n (\u2191(extChartAt IB (f x\u2080).proj) \u2218 fun x => (f x).proj) s x\u2080 \u2227\n ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n\n (\u2191(extChartAt \ud835\udcd8(\ud835\udd5c, F) (\u2191(trivializationAt F E (f x\u2080).proj) (f x\u2080)).snd) \u2218 fun x =>\n (\u2191(trivializationAt F E (f x\u2080).proj) (f x)).snd)\n s x\u2080)\n[PROOFSTEP]\nintro hf\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nf : M \u2192 TotalSpace F E\ns : Set M\nx\u2080 : M\nhf : ContinuousWithinAt f s x\u2080\n\u22a2 ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, EB \u00d7 F) n (\u2191(extChartAt (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (f x\u2080)) \u2218 f) s x\u2080 \u2194\n ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, EB) n (\u2191(extChartAt IB (f x\u2080).proj) \u2218 fun x => (f x).proj) s x\u2080 \u2227\n ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n\n (\u2191(extChartAt \ud835\udcd8(\ud835\udd5c, F) (\u2191(trivializationAt F E (f x\u2080).proj) (f x\u2080)).snd) \u2218 fun x =>\n (\u2191(trivializationAt F E (f x\u2080).proj) (f x)).snd)\n s x\u2080\n[PROOFSTEP]\nsimp_rw [modelWithCornersSelf_prod, FiberBundle.extChartAt, Function.comp, LocalEquiv.trans_apply, LocalEquiv.prod_coe,\n LocalEquiv.refl_coe, extChartAt_self_apply, modelWithCornersSelf_coe, id_def]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nf : M \u2192 TotalSpace F E\ns : Set M\nx\u2080 : M\nhf : ContinuousWithinAt f s x\u2080\n\u22a2 ContMDiffWithinAt IM (ModelWithCorners.prod \ud835\udcd8(\ud835\udd5c, EB) \ud835\udcd8(\ud835\udd5c, F)) n\n (fun x =>\n (\u2191(extChartAt IB (f x\u2080).proj) (\u2191(trivializationAt F E (f x\u2080).proj).toLocalHomeomorph.toLocalEquiv (f x)).fst,\n (\u2191(trivializationAt F E (f x\u2080).proj).toLocalHomeomorph.toLocalEquiv (f x)).snd))\n s x\u2080 \u2194\n ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, EB) n (fun x => \u2191(extChartAt IB (f x\u2080).proj) (f x).proj) s x\u2080 \u2227\n ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n (fun x => (\u2191(trivializationAt F E (f x\u2080).proj) (f x)).snd) s x\u2080\n[PROOFSTEP]\nrefine (contMDiffWithinAt_prod_iff _).trans (and_congr ?_ Iff.rfl)\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nf : M \u2192 TotalSpace F E\ns : Set M\nx\u2080 : M\nhf : ContinuousWithinAt f s x\u2080\n\u22a2 ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, EB) n\n (Prod.fst \u2218 fun x =>\n (\u2191(extChartAt IB (f x\u2080).proj) (\u2191(trivializationAt F E (f x\u2080).proj).toLocalHomeomorph.toLocalEquiv (f x)).fst,\n (\u2191(trivializationAt F E (f x\u2080).proj).toLocalHomeomorph.toLocalEquiv (f x)).snd))\n s x\u2080 \u2194\n ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, EB) n (fun x => \u2191(extChartAt IB (f x\u2080).proj) (f x).proj) s x\u2080\n[PROOFSTEP]\nhave h1 : (fun x => (f x).proj) \u207b\u00b9' (trivializationAt F E (f x\u2080).proj).baseSet \u2208 \ud835\udcdd[s] x\u2080 :=\n ((FiberBundle.continuous_proj F E).continuousWithinAt.comp hf (mapsTo_image f s))\n ((Trivialization.open_baseSet _).mem_nhds (mem_baseSet_trivializationAt F E _))\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nf : M \u2192 TotalSpace F E\ns : Set M\nx\u2080 : M\nhf : ContinuousWithinAt f s x\u2080\nh1 : (fun x => (f x).proj) \u207b\u00b9' (trivializationAt F E (f x\u2080).proj).baseSet \u2208 \ud835\udcdd[s] x\u2080\n\u22a2 ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, EB) n\n (Prod.fst \u2218 fun x =>\n (\u2191(extChartAt IB (f x\u2080).proj) (\u2191(trivializationAt F E (f x\u2080).proj).toLocalHomeomorph.toLocalEquiv (f x)).fst,\n (\u2191(trivializationAt F E (f x\u2080).proj).toLocalHomeomorph.toLocalEquiv (f x)).snd))\n s x\u2080 \u2194\n ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, EB) n (fun x => \u2191(extChartAt IB (f x\u2080).proj) (f x).proj) s x\u2080\n[PROOFSTEP]\nrefine EventuallyEq.contMDiffWithinAt_iff (eventually_of_mem h1 fun x hx => ?_) ?_\n[GOAL]\ncase refine_1\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nf : M \u2192 TotalSpace F E\ns : Set M\nx\u2080 : M\nhf : ContinuousWithinAt f s x\u2080\nh1 : (fun x => (f x).proj) \u207b\u00b9' (trivializationAt F E (f x\u2080).proj).baseSet \u2208 \ud835\udcdd[s] x\u2080\nx : M\nhx : x \u2208 (fun x => (f x).proj) \u207b\u00b9' (trivializationAt F E (f x\u2080).proj).baseSet\n\u22a2 (Prod.fst \u2218 fun x =>\n (\u2191(extChartAt IB (f x\u2080).proj) (\u2191(trivializationAt F E (f x\u2080).proj).toLocalHomeomorph.toLocalEquiv (f x)).fst,\n (\u2191(trivializationAt F E (f x\u2080).proj).toLocalHomeomorph.toLocalEquiv (f x)).snd))\n x =\n (fun x => \u2191(extChartAt IB (f x\u2080).proj) (f x).proj) x\n[PROOFSTEP]\nsimp_rw [Function.comp, LocalHomeomorph.coe_coe, Trivialization.coe_coe]\n[GOAL]\ncase refine_1\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nf : M \u2192 TotalSpace F E\ns : Set M\nx\u2080 : M\nhf : ContinuousWithinAt f s x\u2080\nh1 : (fun x => (f x).proj) \u207b\u00b9' (trivializationAt F E (f x\u2080).proj).baseSet \u2208 \ud835\udcdd[s] x\u2080\nx : M\nhx : x \u2208 (fun x => (f x).proj) \u207b\u00b9' (trivializationAt F E (f x\u2080).proj).baseSet\n\u22a2 \u2191(extChartAt IB (f x\u2080).proj) (\u2191(trivializationAt F E (f x\u2080).proj) (f x)).fst = \u2191(extChartAt IB (f x\u2080).proj) (f x).proj\n[PROOFSTEP]\nrw [Trivialization.coe_fst']\n[GOAL]\ncase refine_1.ex\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nf : M \u2192 TotalSpace F E\ns : Set M\nx\u2080 : M\nhf : ContinuousWithinAt f s x\u2080\nh1 : (fun x => (f x).proj) \u207b\u00b9' (trivializationAt F E (f x\u2080).proj).baseSet \u2208 \ud835\udcdd[s] x\u2080\nx : M\nhx : x \u2208 (fun x => (f x).proj) \u207b\u00b9' (trivializationAt F E (f x\u2080).proj).baseSet\n\u22a2 (f x).proj \u2208 (trivializationAt F E (f x\u2080).proj).baseSet\n[PROOFSTEP]\nexact hx\n[GOAL]\ncase refine_2\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nf : M \u2192 TotalSpace F E\ns : Set M\nx\u2080 : M\nhf : ContinuousWithinAt f s x\u2080\nh1 : (fun x => (f x).proj) \u207b\u00b9' (trivializationAt F E (f x\u2080).proj).baseSet \u2208 \ud835\udcdd[s] x\u2080\n\u22a2 (Prod.fst \u2218 fun x =>\n (\u2191(extChartAt IB (f x\u2080).proj) (\u2191(trivializationAt F E (f x\u2080).proj).toLocalHomeomorph.toLocalEquiv (f x)).fst,\n (\u2191(trivializationAt F E (f x\u2080).proj).toLocalHomeomorph.toLocalEquiv (f x)).snd))\n x\u2080 =\n \u2191(extChartAt IB (f x\u2080).proj) (f x\u2080).proj\n[PROOFSTEP]\nsimp only [mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nf : M \u2192 TotalSpace F E\nx\u2080 : M\n\u22a2 ContMDiffAt IM (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) n f x\u2080 \u2194\n ContMDiffAt IM IB n (fun x => (f x).proj) x\u2080 \u2227\n ContMDiffAt IM \ud835\udcd8(\ud835\udd5c, F) n (fun x => (\u2191(trivializationAt F E (f x\u2080).proj) (f x)).snd) x\u2080\n[PROOFSTEP]\nsimp_rw [\u2190 contMDiffWithinAt_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nf : M \u2192 TotalSpace F E\nx\u2080 : M\n\u22a2 ContMDiffWithinAt IM (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) n f univ x\u2080 \u2194\n ContMDiffWithinAt IM IB n (fun x => (f x).proj) univ x\u2080 \u2227\n ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n (fun x => (\u2191(trivializationAt F E (f x\u2080).proj) (f x)).snd) univ x\u2080\n[PROOFSTEP]\nexact contMDiffWithinAt_totalSpace f\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\ns : (x : B) \u2192 E x\nx\u2080 : B\n\u22a2 ContMDiffAt IB (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) n (fun x => TotalSpace.mk' F x (s x)) x\u2080 \u2194\n ContMDiffAt IB \ud835\udcd8(\ud835\udd5c, F) n (fun x => (\u2191(trivializationAt F E x\u2080) { proj := x, snd := s x }).snd) x\u2080\n[PROOFSTEP]\nsimp_rw [contMDiffAt_totalSpace, and_iff_right_iff_imp]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\ns : (x : B) \u2192 E x\nx\u2080 : B\n\u22a2 ContMDiffAt IB \ud835\udcd8(\ud835\udd5c, F) n (fun x => (\u2191(trivializationAt F E x\u2080) (TotalSpace.mk' F x (s x))).snd) x\u2080 \u2192\n ContMDiffAt IB IB n (fun x => x) x\u2080\n[PROOFSTEP]\nintro\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\ns : (x : B) \u2192 E x\nx\u2080 : B\na\u271d : ContMDiffAt IB \ud835\udcd8(\ud835\udd5c, F) n (fun x => (\u2191(trivializationAt F E x\u2080) (TotalSpace.mk' F x (s x))).snd) x\u2080\n\u22a2 ContMDiffAt IB IB n (fun x => x) x\u2080\n[PROOFSTEP]\nexact contMDiffAt_id\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nx : TotalSpace F E\n\u22a2 ContMDiffAt (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) IB n TotalSpace.proj x\n[PROOFSTEP]\nhave : ContMDiffAt (IB.prod \ud835\udcd8(\ud835\udd5c, F)) (IB.prod \ud835\udcd8(\ud835\udd5c, F)) n id x := contMDiffAt_id\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nx : TotalSpace F E\nthis : ContMDiffAt (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) n id x\n\u22a2 ContMDiffAt (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) IB n TotalSpace.proj x\n[PROOFSTEP]\nrw [contMDiffAt_totalSpace] at this \n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\nx : TotalSpace F E\nthis :\n ContMDiffAt (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) IB n (fun x => (id x).proj) x \u2227\n ContMDiffAt (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) \ud835\udcd8(\ud835\udd5c, F) n\n (fun x_1 => (\u2191(trivializationAt F E (id x).proj) (id x_1)).snd) x\n\u22a2 ContMDiffAt (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) IB n TotalSpace.proj x\n[PROOFSTEP]\nexact this.1\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EB\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b2 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u00b9\u2070 : NormedAddCommGroup EM\ninst\u271d\u2079 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2078 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : ChartedSpace HB B\ninst\u271d\u00b3 : FiberBundle F E\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d : VectorBundle \ud835\udd5c F E\nx : B\n\u22a2 ContMDiffAt IB (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) \u22a4 (zeroSection F E) x\n[PROOFSTEP]\nunfold zeroSection\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EB\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b2 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u00b9\u2070 : NormedAddCommGroup EM\ninst\u271d\u2079 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2078 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : ChartedSpace HB B\ninst\u271d\u00b3 : FiberBundle F E\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d : VectorBundle \ud835\udd5c F E\nx : B\n\u22a2 ContMDiffAt IB (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) \u22a4 (fun x => { proj := x, snd := 0 }) x\n[PROOFSTEP]\nrw [Bundle.contMDiffAt_section]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EB\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b2 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u00b9\u2070 : NormedAddCommGroup EM\ninst\u271d\u2079 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2078 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : ChartedSpace HB B\ninst\u271d\u00b3 : FiberBundle F E\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d : VectorBundle \ud835\udd5c F E\nx : B\n\u22a2 ContMDiffAt IB \ud835\udcd8(\ud835\udd5c, F) \u22a4 (fun x_1 => (\u2191(trivializationAt F E x) { proj := x_1, snd := 0 }).snd) x\n[PROOFSTEP]\napply (contMDiffAt_const (c := 0)).congr_of_eventuallyEq\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EB\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b2 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u00b9\u2070 : NormedAddCommGroup EM\ninst\u271d\u2079 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2078 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : ChartedSpace HB B\ninst\u271d\u00b3 : FiberBundle F E\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d : VectorBundle \ud835\udd5c F E\nx : B\n\u22a2 (fun x_1 => (\u2191(trivializationAt F E x) { proj := x_1, snd := 0 }).snd) =\u1da0[\ud835\udcdd x] fun x => 0\n[PROOFSTEP]\nfilter_upwards [(trivializationAt F E x).open_baseSet.mem_nhds (mem_baseSet_trivializationAt F E x)] with y hy using\n congr_arg Prod.snd <| (trivializationAt F E x).zeroSection \ud835\udd5c hy\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\n\u22a2 SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun b => \u2191(ContinuousLinearEquiv.symm (Trivialization.coordChangeL \ud835\udd5c e e' b)))\n (e.baseSet \u2229 e'.baseSet)\n[PROOFSTEP]\nrw [inter_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\n\u22a2 SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun b => \u2191(ContinuousLinearEquiv.symm (Trivialization.coordChangeL \ud835\udd5c e e' b)))\n (e'.baseSet \u2229 e.baseSet)\n[PROOFSTEP]\nrefine (SmoothVectorBundle.smoothOn_coordChangeL e' e).congr fun b hb \u21a6 ?_\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\nb : B\nhb : b \u2208 e'.baseSet \u2229 e.baseSet\n\u22a2 \u2191(ContinuousLinearEquiv.symm (Trivialization.coordChangeL \ud835\udd5c e e' b)) = \u2191(Trivialization.coordChangeL \ud835\udd5c e' e b)\n[PROOFSTEP]\nrw [e.symm_coordChangeL e' hb]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\ns : Set M\nx : M\nf : M \u2192 B\ng : M \u2192 F\nhf : ContMDiffWithinAt IM IB n f s x\nhg : ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n g s x\nhe : f x \u2208 e.baseSet\nhe' : f x \u2208 e'.baseSet\n\u22a2 ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n (fun y => Trivialization.coordChange e e' (f y) (g y)) s x\n[PROOFSTEP]\nrefine ((hf.coordChangeL he he').clm_apply hg).congr_of_eventuallyEq ?_ ?_\n[GOAL]\ncase refine_1\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\ns : Set M\nx : M\nf : M \u2192 B\ng : M \u2192 F\nhf : ContMDiffWithinAt IM IB n f s x\nhg : ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n g s x\nhe : f x \u2208 e.baseSet\nhe' : f x \u2208 e'.baseSet\n\u22a2 (fun y => Trivialization.coordChange e e' (f y) (g y)) =\u1da0[\ud835\udcdd[s] x] fun x =>\n \u2191\u2191(Trivialization.coordChangeL \ud835\udd5c e e' (f x)) (g x)\n[PROOFSTEP]\nhave : e.baseSet \u2229 e'.baseSet \u2208 \ud835\udcdd (f x) := (e.open_baseSet.inter e'.open_baseSet).mem_nhds \u27e8he, he'\u27e9\n[GOAL]\ncase refine_1\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\ns : Set M\nx : M\nf : M \u2192 B\ng : M \u2192 F\nhf : ContMDiffWithinAt IM IB n f s x\nhg : ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n g s x\nhe : f x \u2208 e.baseSet\nhe' : f x \u2208 e'.baseSet\nthis : e.baseSet \u2229 e'.baseSet \u2208 \ud835\udcdd (f x)\n\u22a2 (fun y => Trivialization.coordChange e e' (f y) (g y)) =\u1da0[\ud835\udcdd[s] x] fun x =>\n \u2191\u2191(Trivialization.coordChangeL \ud835\udd5c e e' (f x)) (g x)\n[PROOFSTEP]\nfilter_upwards [hf.continuousWithinAt this] with y hy\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\ns : Set M\nx : M\nf : M \u2192 B\ng : M \u2192 F\nhf : ContMDiffWithinAt IM IB n f s x\nhg : ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n g s x\nhe : f x \u2208 e.baseSet\nhe' : f x \u2208 e'.baseSet\nthis : e.baseSet \u2229 e'.baseSet \u2208 \ud835\udcdd (f x)\ny : M\nhy : y \u2208 f \u207b\u00b9' (e.baseSet \u2229 e'.baseSet)\n\u22a2 Trivialization.coordChange e e' (f y) (g y) = \u2191\u2191(Trivialization.coordChangeL \ud835\udd5c e e' (f y)) (g y)\n[PROOFSTEP]\nexact (Trivialization.coordChangeL_apply' e e' hy (g y)).symm\n[GOAL]\ncase refine_2\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\ns : Set M\nx : M\nf : M \u2192 B\ng : M \u2192 F\nhf : ContMDiffWithinAt IM IB n f s x\nhg : ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n g s x\nhe : f x \u2208 e.baseSet\nhe' : f x \u2208 e'.baseSet\n\u22a2 Trivialization.coordChange e e' (f x) (g x) = \u2191\u2191(Trivialization.coordChangeL \ud835\udd5c e e' (f x)) (g x)\n[PROOFSTEP]\nexact (Trivialization.coordChangeL_apply' e e' \u27e8he, he'\u27e9 (g x)).symm\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\n\u22a2 ContMDiffOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) n\n (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph \u226b\u2095 e'.toLocalHomeomorph)) (e.target \u2229 e'.target)\n[PROOFSTEP]\nhave Hmaps : MapsTo Prod.fst (e.target \u2229 e'.target) (e.baseSet \u2229 e'.baseSet) := fun x hx \u21a6\n \u27e8e.mem_target.1 hx.1, e'.mem_target.1 hx.2\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\nHmaps : MapsTo Prod.fst (e.target \u2229 e'.target) (e.baseSet \u2229 e'.baseSet)\n\u22a2 ContMDiffOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) n\n (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph \u226b\u2095 e'.toLocalHomeomorph)) (e.target \u2229 e'.target)\n[PROOFSTEP]\nrw [mapsTo_inter] at Hmaps \n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\nHmaps : MapsTo Prod.fst (e.target \u2229 e'.target) e.baseSet \u2227 MapsTo Prod.fst (e.target \u2229 e'.target) e'.baseSet\n\u22a2 ContMDiffOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) n\n (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph \u226b\u2095 e'.toLocalHomeomorph)) (e.target \u2229 e'.target)\n[PROOFSTEP]\nrefine (contMDiffOn_fst.prod_mk (contMDiffOn_fst.coordChange contMDiffOn_snd Hmaps.1 Hmaps.2)).congr ?_\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\nHmaps : MapsTo Prod.fst (e.target \u2229 e'.target) e.baseSet \u2227 MapsTo Prod.fst (e.target \u2229 e'.target) e'.baseSet\n\u22a2 \u2200 (y : B \u00d7 F),\n y \u2208 e.target \u2229 e'.target \u2192\n \u2191(LocalHomeomorph.symm e.toLocalHomeomorph \u226b\u2095 e'.toLocalHomeomorph) y = (y.fst, coordChange e e' y.fst y.snd)\n[PROOFSTEP]\nrintro \u27e8b, x\u27e9 hb\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\nHmaps : MapsTo Prod.fst (e.target \u2229 e'.target) e.baseSet \u2227 MapsTo Prod.fst (e.target \u2229 e'.target) e'.baseSet\nb : B\nx : F\nhb : (b, x) \u2208 e.target \u2229 e'.target\n\u22a2 \u2191(LocalHomeomorph.symm e.toLocalHomeomorph \u226b\u2095 e'.toLocalHomeomorph) (b, x) =\n ((b, x).fst, coordChange e e' (b, x).fst (b, x).snd)\n[PROOFSTEP]\nrefine Prod.ext ?_ rfl\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\nHmaps : MapsTo Prod.fst (e.target \u2229 e'.target) e.baseSet \u2227 MapsTo Prod.fst (e.target \u2229 e'.target) e'.baseSet\nb : B\nx : F\nhb : (b, x) \u2208 e.target \u2229 e'.target\n\u22a2 (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph \u226b\u2095 e'.toLocalHomeomorph) (b, x)).fst =\n ((b, x).fst, coordChange e e' (b, x).fst (b, x).snd).fst\n[PROOFSTEP]\nhave : (e.toLocalHomeomorph.symm (b, x)).1 \u2208 e'.baseSet\n[GOAL]\ncase this\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\nHmaps : MapsTo Prod.fst (e.target \u2229 e'.target) e.baseSet \u2227 MapsTo Prod.fst (e.target \u2229 e'.target) e'.baseSet\nb : B\nx : F\nhb : (b, x) \u2208 e.target \u2229 e'.target\n\u22a2 (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph) (b, x)).proj \u2208 e'.baseSet\n[PROOFSTEP]\nsimp_all only [Trivialization.mem_target, mfld_simps]\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\nHmaps : MapsTo Prod.fst (e.target \u2229 e'.target) e.baseSet \u2227 MapsTo Prod.fst (e.target \u2229 e'.target) e'.baseSet\nb : B\nx : F\nhb : (b, x) \u2208 e.target \u2229 e'.target\nthis : (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph) (b, x)).proj \u2208 e'.baseSet\n\u22a2 (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph \u226b\u2095 e'.toLocalHomeomorph) (b, x)).fst =\n ((b, x).fst, coordChange e e' (b, x).fst (b, x).snd).fst\n[PROOFSTEP]\nexact (e'.coe_fst' this).trans (e.proj_symm_apply hb.1)\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\ns : Set M\nx : M\nf : M \u2192 TotalSpace F E\nhp : ContMDiffWithinAt IM IB n (TotalSpace.proj \u2218 f) s x\nhf : ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n (fun y => (\u2191e (f y)).snd) s x\nhe : f x \u2208 e.source\nhe' : f x \u2208 e'.source\n\u22a2 ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n (fun y => (\u2191e' (f y)).snd) s x\n[PROOFSTEP]\nrw [Trivialization.mem_source] at he he' \n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\ns : Set M\nx : M\nf : M \u2192 TotalSpace F E\nhp : ContMDiffWithinAt IM IB n (TotalSpace.proj \u2218 f) s x\nhf : ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n (fun y => (\u2191e (f y)).snd) s x\nhe : (f x).proj \u2208 e.baseSet\nhe' : (f x).proj \u2208 e'.baseSet\n\u22a2 ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n (fun y => (\u2191e' (f y)).snd) s x\n[PROOFSTEP]\nrefine (hp.coordChange hf he he').congr_of_eventuallyEq ?_ ?_\n[GOAL]\ncase refine_1\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\ns : Set M\nx : M\nf : M \u2192 TotalSpace F E\nhp : ContMDiffWithinAt IM IB n (TotalSpace.proj \u2218 f) s x\nhf : ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n (fun y => (\u2191e (f y)).snd) s x\nhe : (f x).proj \u2208 e.baseSet\nhe' : (f x).proj \u2208 e'.baseSet\n\u22a2 (fun y => (\u2191e' (f y)).snd) =\u1da0[\ud835\udcdd[s] x] fun y =>\n Trivialization.coordChange e e' ((TotalSpace.proj \u2218 f) y) (\u2191e (f y)).snd\n[PROOFSTEP]\nfilter_upwards [hp.continuousWithinAt (e.open_baseSet.mem_nhds he)] with y hy\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\ns : Set M\nx : M\nf : M \u2192 TotalSpace F E\nhp : ContMDiffWithinAt IM IB n (TotalSpace.proj \u2218 f) s x\nhf : ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n (fun y => (\u2191e (f y)).snd) s x\nhe : (f x).proj \u2208 e.baseSet\nhe' : (f x).proj \u2208 e'.baseSet\ny : M\nhy : y \u2208 TotalSpace.proj \u2218 f \u207b\u00b9' e.baseSet\n\u22a2 (\u2191e' (f y)).snd = Trivialization.coordChange e e' ((TotalSpace.proj \u2218 f) y) (\u2191e (f y)).snd\n[PROOFSTEP]\nrw [Function.comp_apply, e.coordChange_apply_snd _ hy]\n[GOAL]\ncase refine_2\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\ns : Set M\nx : M\nf : M \u2192 TotalSpace F E\nhp : ContMDiffWithinAt IM IB n (TotalSpace.proj \u2218 f) s x\nhf : ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n (fun y => (\u2191e (f y)).snd) s x\nhe : (f x).proj \u2208 e.baseSet\nhe' : (f x).proj \u2208 e'.baseSet\n\u22a2 (\u2191e' (f x)).snd = Trivialization.coordChange e e' ((TotalSpace.proj \u2218 f) x) (\u2191e (f x)).snd\n[PROOFSTEP]\nrw [Function.comp_apply, e.coordChange_apply_snd _ he]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\n\u22a2 \u2200 {e e' : LocalHomeomorph (TotalSpace F E) (B \u00d7 F)},\n e \u2208 atlas (B \u00d7 F) (TotalSpace F E) \u2192\n e' \u2208 atlas (B \u00d7 F) (TotalSpace F E) \u2192 LocalHomeomorph.symm e \u226b\u2095 e' \u2208 smoothFiberwiseLinear B F IB\n[PROOFSTEP]\nrintro _ _ \u27e8e, he, rfl\u27e9 \u27e8e', he', rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : Trivialization F TotalSpace.proj\nhe : e \u2208 trivializationAtlas F E\ne' : Trivialization F TotalSpace.proj\nhe' : e' \u2208 trivializationAtlas F E\n\u22a2 LocalHomeomorph.symm ((fun e => e.toLocalHomeomorph) e) \u226b\u2095 (fun e => e.toLocalHomeomorph) e' \u2208\n smoothFiberwiseLinear B F IB\n[PROOFSTEP]\nhaveI : MemTrivializationAtlas e := \u27e8he\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : Trivialization F TotalSpace.proj\nhe : e \u2208 trivializationAtlas F E\ne' : Trivialization F TotalSpace.proj\nhe' : e' \u2208 trivializationAtlas F E\nthis : MemTrivializationAtlas e\n\u22a2 LocalHomeomorph.symm ((fun e => e.toLocalHomeomorph) e) \u226b\u2095 (fun e => e.toLocalHomeomorph) e' \u2208\n smoothFiberwiseLinear B F IB\n[PROOFSTEP]\nhaveI : MemTrivializationAtlas e' := \u27e8he'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : Trivialization F TotalSpace.proj\nhe : e \u2208 trivializationAtlas F E\ne' : Trivialization F TotalSpace.proj\nhe' : e' \u2208 trivializationAtlas F E\nthis\u271d : MemTrivializationAtlas e\nthis : MemTrivializationAtlas e'\n\u22a2 LocalHomeomorph.symm ((fun e => e.toLocalHomeomorph) e) \u226b\u2095 (fun e => e.toLocalHomeomorph) e' \u2208\n smoothFiberwiseLinear B F IB\n[PROOFSTEP]\nrw [mem_smoothFiberwiseLinear_iff]\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : Trivialization F TotalSpace.proj\nhe : e \u2208 trivializationAtlas F E\ne' : Trivialization F TotalSpace.proj\nhe' : e' \u2208 trivializationAtlas F E\nthis\u271d : MemTrivializationAtlas e\nthis : MemTrivializationAtlas e'\n\u22a2 \u2203 \u03c6 U hU h\u03c6 h2\u03c6,\n EqOnSource (LocalHomeomorph.symm ((fun e => e.toLocalHomeomorph) e) \u226b\u2095 (fun e => e.toLocalHomeomorph) e')\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U))\n[PROOFSTEP]\nrefine'\n \u27e8_, _, e.open_baseSet.inter e'.open_baseSet, smoothOn_coordChangeL IB e e', smoothOn_symm_coordChangeL IB e e', _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : Trivialization F TotalSpace.proj\nhe : e \u2208 trivializationAtlas F E\ne' : Trivialization F TotalSpace.proj\nhe' : e' \u2208 trivializationAtlas F E\nthis\u271d : MemTrivializationAtlas e\nthis : MemTrivializationAtlas e'\n\u22a2 EqOnSource (LocalHomeomorph.symm ((fun e => e.toLocalHomeomorph) e) \u226b\u2095 (fun e => e.toLocalHomeomorph) e')\n (FiberwiseLinear.localHomeomorph (fun x => Trivialization.coordChangeL \ud835\udd5c e e' x)\n (_ : IsOpen (e.baseSet \u2229 e'.baseSet))\n (_ : ContinuousOn (fun x => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' x)) (e.baseSet \u2229 e'.baseSet))\n (_ :\n ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (Trivialization.coordChangeL \ud835\udd5c e e' x)))\n (e.baseSet \u2229 e'.baseSet)))\n[PROOFSTEP]\nrefine LocalHomeomorph.eqOnSourceSetoid.symm \u27e8?_, ?_\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine_1\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : Trivialization F TotalSpace.proj\nhe : e \u2208 trivializationAtlas F E\ne' : Trivialization F TotalSpace.proj\nhe' : e' \u2208 trivializationAtlas F E\nthis\u271d : MemTrivializationAtlas e\nthis : MemTrivializationAtlas e'\n\u22a2 (FiberwiseLinear.localHomeomorph (fun x => Trivialization.coordChangeL \ud835\udd5c e e' x) (_ : IsOpen (e.baseSet \u2229 e'.baseSet))\n (_ : ContinuousOn (fun x => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' x)) (e.baseSet \u2229 e'.baseSet))\n (_ :\n ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (Trivialization.coordChangeL \ud835\udd5c e e' x)))\n (e.baseSet \u2229 e'.baseSet))).toLocalEquiv.source =\n (LocalHomeomorph.symm ((fun e => e.toLocalHomeomorph) e) \u226b\u2095 (fun e => e.toLocalHomeomorph) e').toLocalEquiv.source\n[PROOFSTEP]\nsimp only [e.symm_trans_source_eq e', FiberwiseLinear.localHomeomorph, trans_toLocalEquiv, symm_toLocalEquiv]\n[GOAL]\ncase intro.intro.intro.intro.refine_2\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : Trivialization F TotalSpace.proj\nhe : e \u2208 trivializationAtlas F E\ne' : Trivialization F TotalSpace.proj\nhe' : e' \u2208 trivializationAtlas F E\nthis\u271d : MemTrivializationAtlas e\nthis : MemTrivializationAtlas e'\n\u22a2 EqOn\n (\u2191(FiberwiseLinear.localHomeomorph (fun x => Trivialization.coordChangeL \ud835\udd5c e e' x)\n (_ : IsOpen (e.baseSet \u2229 e'.baseSet))\n (_ : ContinuousOn (fun x => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' x)) (e.baseSet \u2229 e'.baseSet))\n (_ :\n ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (Trivialization.coordChangeL \ud835\udd5c e e' x)))\n (e.baseSet \u2229 e'.baseSet))))\n (\u2191(LocalHomeomorph.symm ((fun e => e.toLocalHomeomorph) e) \u226b\u2095 (fun e => e.toLocalHomeomorph) e'))\n (FiberwiseLinear.localHomeomorph (fun x => Trivialization.coordChangeL \ud835\udd5c e e' x)\n (_ : IsOpen (e.baseSet \u2229 e'.baseSet))\n (_ : ContinuousOn (fun x => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' x)) (e.baseSet \u2229 e'.baseSet))\n (_ :\n ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (Trivialization.coordChangeL \ud835\udd5c e e' x)))\n (e.baseSet \u2229 e'.baseSet))).toLocalEquiv.source\n[PROOFSTEP]\nrintro \u27e8b, v\u27e9 hb\n[GOAL]\ncase intro.intro.intro.intro.refine_2.mk\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : Trivialization F TotalSpace.proj\nhe : e \u2208 trivializationAtlas F E\ne' : Trivialization F TotalSpace.proj\nhe' : e' \u2208 trivializationAtlas F E\nthis\u271d : MemTrivializationAtlas e\nthis : MemTrivializationAtlas e'\nb : B\nv : F\nhb :\n (b, v) \u2208\n (FiberwiseLinear.localHomeomorph (fun x => Trivialization.coordChangeL \ud835\udd5c e e' x)\n (_ : IsOpen (e.baseSet \u2229 e'.baseSet))\n (_ : ContinuousOn (fun x => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' x)) (e.baseSet \u2229 e'.baseSet))\n (_ :\n ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (Trivialization.coordChangeL \ud835\udd5c e e' x)))\n (e.baseSet \u2229 e'.baseSet))).toLocalEquiv.source\n\u22a2 \u2191(FiberwiseLinear.localHomeomorph (fun x => Trivialization.coordChangeL \ud835\udd5c e e' x)\n (_ : IsOpen (e.baseSet \u2229 e'.baseSet))\n (_ : ContinuousOn (fun x => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' x)) (e.baseSet \u2229 e'.baseSet))\n (_ :\n ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (Trivialization.coordChangeL \ud835\udd5c e e' x)))\n (e.baseSet \u2229 e'.baseSet)))\n (b, v) =\n \u2191(LocalHomeomorph.symm ((fun e => e.toLocalHomeomorph) e) \u226b\u2095 (fun e => e.toLocalHomeomorph) e') (b, v)\n[PROOFSTEP]\nexact (e.apply_symm_apply_eq_coordChangeL e' hb.1 v).symm\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\n\u22a2 SmoothManifoldWithCorners (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (TotalSpace F E)\n[PROOFSTEP]\nrefine' { StructureGroupoid.HasGroupoid.comp (smoothFiberwiseLinear B F IB) _ with }\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\n\u22a2 \u2200 (e : LocalHomeomorph (B \u00d7 F) (B \u00d7 F)),\n e \u2208 smoothFiberwiseLinear B F IB \u2192\n ChartedSpace.LiftPropOn\n (StructureGroupoid.IsLocalStructomorphWithinAt (contDiffGroupoid \u22a4 (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)))) (\u2191e)\n e.source\n[PROOFSTEP]\nintro e he\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : LocalHomeomorph (B \u00d7 F) (B \u00d7 F)\nhe : e \u2208 smoothFiberwiseLinear B F IB\n\u22a2 ChartedSpace.LiftPropOn\n (StructureGroupoid.IsLocalStructomorphWithinAt (contDiffGroupoid \u22a4 (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)))) (\u2191e)\n e.source\n[PROOFSTEP]\nrw [mem_smoothFiberwiseLinear_iff] at he \n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : LocalHomeomorph (B \u00d7 F) (B \u00d7 F)\nhe :\n \u2203 \u03c6 U hU h\u03c6 h2\u03c6,\n EqOnSource e\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U))\n\u22a2 ChartedSpace.LiftPropOn\n (StructureGroupoid.IsLocalStructomorphWithinAt (contDiffGroupoid \u22a4 (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)))) (\u2191e)\n e.source\n[PROOFSTEP]\nobtain \u27e8\u03c6, U, hU, h\u03c6, h2\u03c6, he\u03c6\u27e9 := he\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : LocalHomeomorph (B \u00d7 F) (B \u00d7 F)\n\u03c6 : B \u2192 F \u2243L[\ud835\udd5c] F\nU : Set B\nhU : IsOpen U\nh\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(\u03c6 x)) U\nh2\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U\nhe\u03c6 :\n EqOnSource e\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U))\n\u22a2 ChartedSpace.LiftPropOn\n (StructureGroupoid.IsLocalStructomorphWithinAt (contDiffGroupoid \u22a4 (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)))) (\u2191e)\n e.source\n[PROOFSTEP]\nrw [isLocalStructomorphOn_contDiffGroupoid_iff]\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : LocalHomeomorph (B \u00d7 F) (B \u00d7 F)\n\u03c6 : B \u2192 F \u2243L[\ud835\udd5c] F\nU : Set B\nhU : IsOpen U\nh\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(\u03c6 x)) U\nh2\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U\nhe\u03c6 :\n EqOnSource e\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U))\n\u22a2 SmoothOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (\u2191e) e.source \u2227\n SmoothOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (\u2191(LocalHomeomorph.symm e)) e.target\n[PROOFSTEP]\nrefine' \u27e8ContMDiffOn.congr _ (EqOnSource.eqOn he\u03c6), ContMDiffOn.congr _ (EqOnSource.eqOn (EqOnSource.symm' he\u03c6))\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : LocalHomeomorph (B \u00d7 F) (B \u00d7 F)\n\u03c6 : B \u2192 F \u2243L[\ud835\udd5c] F\nU : Set B\nhU : IsOpen U\nh\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(\u03c6 x)) U\nh2\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U\nhe\u03c6 :\n EqOnSource e\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U))\n\u22a2 ContMDiffOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) \u22a4\n (fun y =>\n \u2191(FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U))\n y)\n e.source\n[PROOFSTEP]\nrw [EqOnSource.source_eq he\u03c6]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : LocalHomeomorph (B \u00d7 F) (B \u00d7 F)\n\u03c6 : B \u2192 F \u2243L[\ud835\udd5c] F\nU : Set B\nhU : IsOpen U\nh\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(\u03c6 x)) U\nh2\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U\nhe\u03c6 :\n EqOnSource e\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U))\n\u22a2 ContMDiffOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) \u22a4\n (fun y =>\n \u2191(FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U))\n y)\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U)).toLocalEquiv.source\n[PROOFSTEP]\napply smoothOn_fst.prod_mk\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : LocalHomeomorph (B \u00d7 F) (B \u00d7 F)\n\u03c6 : B \u2192 F \u2243L[\ud835\udd5c] F\nU : Set B\nhU : IsOpen U\nh\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(\u03c6 x)) U\nh2\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U\nhe\u03c6 :\n EqOnSource e\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U))\n\u22a2 SmoothOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) \ud835\udcd8(\ud835\udd5c, F) (fun x => \u2191(\u03c6 x.fst) x.snd)\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U)).toLocalEquiv.source\n[PROOFSTEP]\nexact (h\u03c6.comp contMDiffOn_fst <| prod_subset_preimage_fst _ _).clm_apply contMDiffOn_snd\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : LocalHomeomorph (B \u00d7 F) (B \u00d7 F)\n\u03c6 : B \u2192 F \u2243L[\ud835\udd5c] F\nU : Set B\nhU : IsOpen U\nh\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(\u03c6 x)) U\nh2\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U\nhe\u03c6 :\n EqOnSource e\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U))\n\u22a2 ContMDiffOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) \u22a4\n (fun y =>\n \u2191(LocalHomeomorph.symm\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U)))\n y)\n e.target\n[PROOFSTEP]\nrw [EqOnSource.target_eq he\u03c6]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : LocalHomeomorph (B \u00d7 F) (B \u00d7 F)\n\u03c6 : B \u2192 F \u2243L[\ud835\udd5c] F\nU : Set B\nhU : IsOpen U\nh\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(\u03c6 x)) U\nh2\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U\nhe\u03c6 :\n EqOnSource e\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U))\n\u22a2 ContMDiffOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) \u22a4\n (fun y =>\n \u2191(LocalHomeomorph.symm\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U)))\n y)\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U)).toLocalEquiv.target\n[PROOFSTEP]\napply smoothOn_fst.prod_mk\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne : LocalHomeomorph (B \u00d7 F) (B \u00d7 F)\n\u03c6 : B \u2192 F \u2243L[\ud835\udd5c] F\nU : Set B\nhU : IsOpen U\nh\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(\u03c6 x)) U\nh2\u03c6 : SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U\nhe\u03c6 :\n EqOnSource e\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U))\n\u22a2 SmoothOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) \ud835\udcd8(\ud835\udd5c, F) (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x.fst)) x.snd)\n (FiberwiseLinear.localHomeomorph \u03c6 hU (_ : ContinuousOn (fun x => \u2191(\u03c6 x)) U)\n (_ : ContinuousOn (fun x => \u2191(ContinuousLinearEquiv.symm (\u03c6 x))) U)).toLocalEquiv.target\n[PROOFSTEP]\nexact (h2\u03c6.comp contMDiffOn_fst <| prod_subset_preimage_fst _ _).clm_apply contMDiffOn_snd\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\nf : M \u2192 TotalSpace F E\ns : Set M\nhe : MapsTo f s e.source\n\u22a2 ContMDiffOn IM (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) n f s \u2194\n ContMDiffOn IM IB n (fun x => (f x).proj) s \u2227 ContMDiffOn IM \ud835\udcd8(\ud835\udd5c, F) n (fun x => (\u2191e (f x)).snd) s\n[PROOFSTEP]\nsimp only [ContMDiffOn, \u2190 forall_and]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2078 : TopologicalSpace B\ninst\u271d\u00b9\u2077 : ChartedSpace HB B\ninst\u271d\u00b9\u2076 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b3 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u2070 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2079 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b2 : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : MemTrivializationAtlas e\ninst\u271d : MemTrivializationAtlas e'\nf : M \u2192 TotalSpace F E\ns : Set M\nhe : MapsTo f s e.source\n\u22a2 (\u2200 (x : M), x \u2208 s \u2192 ContMDiffWithinAt IM (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) n f s x) \u2194\n \u2200 (x : M),\n x \u2208 s \u2192\n ContMDiffWithinAt IM IB n (fun x => (f x).proj) s x \u2227\n ContMDiffWithinAt IM \ud835\udcd8(\ud835\udd5c, F) n (fun x => (\u2191e (f x)).snd) s x\n[PROOFSTEP]\nexact forall\u2082_congr fun x hx \u21a6 e.contMDiffWithinAt_iff IB (he hx)\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b3 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup EB\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b2\u2070 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2079 : TopologicalSpace B\ninst\u271d\u00b9\u2078 : ChartedSpace HB B\ninst\u271d\u00b9\u2077 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2076 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u2074 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2075 : FiberBundle F E\ninst\u271d\u2074 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b3 : SmoothVectorBundle F E IB\ne\u271d e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b2 : MemTrivializationAtlas e\u271d\ninst\u271d\u00b9 : MemTrivializationAtlas e'\ne : Trivialization F TotalSpace.proj\ninst\u271d : MemTrivializationAtlas e\n\u22a2 SmoothOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (\u2191e) e.source\n[PROOFSTEP]\nhave : SmoothOn (IB.prod \ud835\udcd8(\ud835\udd5c, F)) (IB.prod \ud835\udcd8(\ud835\udd5c, F)) id e.source := smoothOn_id\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b3 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup EB\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b2\u2070 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2079 : TopologicalSpace B\ninst\u271d\u00b9\u2078 : ChartedSpace HB B\ninst\u271d\u00b9\u2077 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2076 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u2074 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2075 : FiberBundle F E\ninst\u271d\u2074 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b3 : SmoothVectorBundle F E IB\ne\u271d e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b2 : MemTrivializationAtlas e\u271d\ninst\u271d\u00b9 : MemTrivializationAtlas e'\ne : Trivialization F TotalSpace.proj\ninst\u271d : MemTrivializationAtlas e\nthis : SmoothOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) id e.source\n\u22a2 SmoothOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (\u2191e) e.source\n[PROOFSTEP]\nrw [e.smoothOn_iff IB (mapsTo_id _)] at this \n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b3 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup EB\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b2\u2070 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2079 : TopologicalSpace B\ninst\u271d\u00b9\u2078 : ChartedSpace HB B\ninst\u271d\u00b9\u2077 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2076 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u2074 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2075 : FiberBundle F E\ninst\u271d\u2074 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b3 : SmoothVectorBundle F E IB\ne\u271d e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b2 : MemTrivializationAtlas e\u271d\ninst\u271d\u00b9 : MemTrivializationAtlas e'\ne : Trivialization F TotalSpace.proj\ninst\u271d : MemTrivializationAtlas e\nthis :\n SmoothOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) IB (fun x => (id x).proj) e.source \u2227\n SmoothOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) \ud835\udcd8(\ud835\udd5c, F) (fun x => (\u2191e (id x)).snd) e.source\n\u22a2 SmoothOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (\u2191e) e.source\n[PROOFSTEP]\nexact (this.1.prod_mk this.2).congr fun x hx \u21a6 (e.mk_proj_snd hx).symm\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b3 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup EB\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b2\u2070 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2079 : TopologicalSpace B\ninst\u271d\u00b9\u2078 : ChartedSpace HB B\ninst\u271d\u00b9\u2077 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2076 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u2074 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2075 : FiberBundle F E\ninst\u271d\u2074 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b3 : SmoothVectorBundle F E IB\ne\u271d e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b2 : MemTrivializationAtlas e\u271d\ninst\u271d\u00b9 : MemTrivializationAtlas e'\ne : Trivialization F TotalSpace.proj\ninst\u271d : MemTrivializationAtlas e\n\u22a2 SmoothOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F))\n (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph)) e.target\n[PROOFSTEP]\nrw [e.smoothOn_iff IB e.toLocalHomeomorph.symm_mapsTo]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b3 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup EB\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b2\u2070 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2079 : TopologicalSpace B\ninst\u271d\u00b9\u2078 : ChartedSpace HB B\ninst\u271d\u00b9\u2077 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2076 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u2074 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2075 : FiberBundle F E\ninst\u271d\u2074 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b3 : SmoothVectorBundle F E IB\ne\u271d e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b2 : MemTrivializationAtlas e\u271d\ninst\u271d\u00b9 : MemTrivializationAtlas e'\ne : Trivialization F TotalSpace.proj\ninst\u271d : MemTrivializationAtlas e\n\u22a2 SmoothOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) IB (fun x => (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph) x).proj)\n e.target \u2227\n SmoothOn (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) \ud835\udcd8(\ud835\udd5c, F)\n (fun x => (\u2191e (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph) x)).snd) e.target\n[PROOFSTEP]\nrefine \u27e8smoothOn_fst.congr fun x hx \u21a6 e.proj_symm_apply hx, smoothOn_snd.congr fun x hx \u21a6 ?_\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b3 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup EB\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b2\u2070 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2079 : TopologicalSpace B\ninst\u271d\u00b9\u2078 : ChartedSpace HB B\ninst\u271d\u00b9\u2077 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2076 : NormedAddCommGroup EM\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u2074 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u2075 : FiberBundle F E\ninst\u271d\u2074 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b3 : SmoothVectorBundle F E IB\ne\u271d e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b2 : MemTrivializationAtlas e\u271d\ninst\u271d\u00b9 : MemTrivializationAtlas e'\ne : Trivialization F TotalSpace.proj\ninst\u271d : MemTrivializationAtlas e\nx : B \u00d7 F\nhx : x \u2208 e.target\n\u22a2 (\u2191e (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph) x)).snd = x.snd\n[PROOFSTEP]\nrw [e.apply_symm_apply hx]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u2070 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2078 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2077 : TopologicalSpace B\ninst\u271d\u00b9\u2076 : ChartedSpace HB B\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b2 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b3 : FiberBundle F E\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b9 : SmoothVectorBundle F E IB\n\u03b9 : Type u_11\nZ : VectorBundleCore \ud835\udd5c B F \u03b9\ninst\u271d : IsSmooth Z IB\n\u22a2 \u2200 (e e' : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e] [inst_1 : MemTrivializationAtlas e'],\n SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' b)) (e.baseSet \u2229 e'.baseSet)\n[PROOFSTEP]\nrintro - - \u27e8i, rfl\u27e9\n \u27e8i', rfl\u27e9\n -- Porting note: Originally `Z.smoothOn_coordChange IB i i'`\n[GOAL]\ncase mk.intro.mk.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u2070 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2078 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2077 : TopologicalSpace B\ninst\u271d\u00b9\u2076 : ChartedSpace HB B\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b2 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b3 : FiberBundle F E\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b9 : SmoothVectorBundle F E IB\n\u03b9 : Type u_11\nZ : VectorBundleCore \ud835\udd5c B F \u03b9\ninst\u271d : IsSmooth Z IB\ni i' : \u03b9\n\u22a2 SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F)\n (fun b =>\n \u2191(Trivialization.coordChangeL \ud835\udd5c (FiberBundleCore.localTriv (toFiberBundleCore Z) i)\n (FiberBundleCore.localTriv (toFiberBundleCore Z) i') b))\n ((FiberBundleCore.localTriv (toFiberBundleCore Z) i).baseSet \u2229\n (FiberBundleCore.localTriv (toFiberBundleCore Z) i').baseSet)\n[PROOFSTEP]\nrefine' (VectorBundleCore.IsSmooth.smoothOn_coordChange (Z := Z) (IB := IB) i i').congr fun b hb => _\n[GOAL]\ncase mk.intro.mk.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u2070 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2078 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2077 : TopologicalSpace B\ninst\u271d\u00b9\u2076 : ChartedSpace HB B\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b2 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b3 : FiberBundle F E\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b9 : SmoothVectorBundle F E IB\n\u03b9 : Type u_11\nZ : VectorBundleCore \ud835\udd5c B F \u03b9\ninst\u271d : IsSmooth Z IB\ni i' : \u03b9\nb : B\nhb : b \u2208 baseSet Z i \u2229 baseSet Z i'\n\u22a2 \u2191(Trivialization.coordChangeL \ud835\udd5c (FiberBundleCore.localTriv (toFiberBundleCore Z) i)\n (FiberBundleCore.localTriv (toFiberBundleCore Z) i') b) =\n coordChange Z i i' b\n[PROOFSTEP]\next v\n[GOAL]\ncase mk.intro.mk.intro.h\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b2\u2070 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2078 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2077 : TopologicalSpace B\ninst\u271d\u00b9\u2076 : ChartedSpace HB B\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u2074 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b2 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b3 : FiberBundle F E\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b9 : SmoothVectorBundle F E IB\n\u03b9 : Type u_11\nZ : VectorBundleCore \ud835\udd5c B F \u03b9\ninst\u271d : IsSmooth Z IB\ni i' : \u03b9\nb : B\nhb : b \u2208 baseSet Z i \u2229 baseSet Z i'\nv : F\n\u22a2 \u2191\u2191(Trivialization.coordChangeL \ud835\udd5c (FiberBundleCore.localTriv (toFiberBundleCore Z) i)\n (FiberBundleCore.localTriv (toFiberBundleCore Z) i') b)\n v =\n \u2191(coordChange Z i i' b) v\n[PROOFSTEP]\nexact Z.localTriv_coordChange_eq i i' hb v\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\n\u22a2 \u2200 (e e' : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e] [inst_1 : MemTrivializationAtlas e'],\n SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' b)) (e.baseSet \u2229 e'.baseSet)\n[PROOFSTEP]\nintro e e' he he'\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne e' : Trivialization F TotalSpace.proj\nhe : MemTrivializationAtlas e\nhe' : MemTrivializationAtlas e'\n\u22a2 SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' b)) (e.baseSet \u2229 e'.baseSet)\n[PROOFSTEP]\nobtain rfl := Bundle.Trivial.eq_trivialization B F e\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\ne' : Trivialization F TotalSpace.proj\nhe' : MemTrivializationAtlas e'\nhe : MemTrivializationAtlas (trivialization B F)\n\u22a2 SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c (trivialization B F) e' b))\n ((trivialization B F).baseSet \u2229 e'.baseSet)\n[PROOFSTEP]\nobtain rfl := Bundle.Trivial.eq_trivialization B F e'\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\nhe he' : MemTrivializationAtlas (trivialization B F)\n\u22a2 SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c (trivialization B F) (trivialization B F) b))\n ((trivialization B F).baseSet \u2229 (trivialization B F).baseSet)\n[PROOFSTEP]\nsimp_rw [Bundle.Trivial.trivialization.coordChangeL]\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2079 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u2077 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u2076 : TopologicalSpace B\ninst\u271d\u00b9\u2075 : ChartedSpace HB B\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup EM\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b9\u00b9 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b9\u2070 : TopologicalSpace M\ninst\u271d\u2079 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F E\ninst\u271d : SmoothVectorBundle F E IB\nhe he' : MemTrivializationAtlas (trivialization B F)\n\u22a2 SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun b => \u2191(ContinuousLinearEquiv.refl \ud835\udd5c F))\n ((trivialization B F).baseSet \u2229 (trivialization B F).baseSet)\n[PROOFSTEP]\nexact smooth_const.smoothOn\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b3\u2078 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b3\u2077 : NormedAddCommGroup EB\ninst\u271d\u00b3\u2076 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b3\u2075 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b3\u2074 : TopologicalSpace B\ninst\u271d\u00b3\u00b3 : ChartedSpace HB B\ninst\u271d\u00b3\u00b2 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup EM\ninst\u271d\u00b3\u2070 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b2\u2079 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b2\u2078 : TopologicalSpace M\ninst\u271d\u00b2\u2077 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2\u2076 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2\u2075 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2\u2074 : NormedAddCommGroup F\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2\u2070 : FiberBundle F E\ninst\u271d\u00b9\u2079 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b9\u2078 : SmoothVectorBundle F E IB\nF\u2081 : Type u_11\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_12\ninst\u271d\u00b9\u2075 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\ninst\u271d\u00b9\u2074 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\nF\u2082 : Type u_13\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_14\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u2075 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c F\u2082 E\u2082\ninst\u271d\u00b9 : SmoothVectorBundle F\u2081 E\u2081 IB\ninst\u271d : SmoothVectorBundle F\u2082 E\u2082 IB\n\u22a2 \u2200 (e e' : Trivialization (F\u2081 \u00d7 F\u2082) TotalSpace.proj) [inst : MemTrivializationAtlas e]\n [inst_1 : MemTrivializationAtlas e'],\n SmoothOn IB \ud835\udcd8(\ud835\udd5c, F\u2081 \u00d7 F\u2082 \u2192L[\ud835\udd5c] F\u2081 \u00d7 F\u2082) (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' b)) (e.baseSet \u2229 e'.baseSet)\n[PROOFSTEP]\nrintro _ _ \u27e8e\u2081, e\u2082, i\u2081, i\u2082, rfl\u27e9 \u27e8e\u2081', e\u2082', i\u2081', i\u2082', rfl\u27e9\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b3\u2078 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b3\u2077 : NormedAddCommGroup EB\ninst\u271d\u00b3\u2076 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b3\u2075 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b3\u2074 : TopologicalSpace B\ninst\u271d\u00b3\u00b3 : ChartedSpace HB B\ninst\u271d\u00b3\u00b2 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup EM\ninst\u271d\u00b3\u2070 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b2\u2079 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b2\u2078 : TopologicalSpace M\ninst\u271d\u00b2\u2077 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2\u2076 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2\u2075 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2\u2074 : NormedAddCommGroup F\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2\u2070 : FiberBundle F E\ninst\u271d\u00b9\u2079 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b9\u2078 : SmoothVectorBundle F E IB\nF\u2081 : Type u_11\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_12\ninst\u271d\u00b9\u2075 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\ninst\u271d\u00b9\u2074 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\nF\u2082 : Type u_13\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_14\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u2075 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c F\u2082 E\u2082\ninst\u271d\u00b9 : SmoothVectorBundle F\u2081 E\u2081 IB\ninst\u271d : SmoothVectorBundle F\u2082 E\u2082 IB\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\ni\u2081 : MemTrivializationAtlas e\u2081\ni\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\ni\u2081' : MemTrivializationAtlas e\u2081'\ni\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 SmoothOn IB \ud835\udcd8(\ud835\udd5c, F\u2081 \u00d7 F\u2082 \u2192L[\ud835\udd5c] F\u2081 \u00d7 F\u2082)\n (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c (Trivialization.prod e\u2081 e\u2082) (Trivialization.prod e\u2081' e\u2082') b))\n ((Trivialization.prod e\u2081 e\u2082).baseSet \u2229 (Trivialization.prod e\u2081' e\u2082').baseSet)\n[PROOFSTEP]\nrw [SmoothOn]\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b3\u2078 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b3\u2077 : NormedAddCommGroup EB\ninst\u271d\u00b3\u2076 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b3\u2075 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b3\u2074 : TopologicalSpace B\ninst\u271d\u00b3\u00b3 : ChartedSpace HB B\ninst\u271d\u00b3\u00b2 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup EM\ninst\u271d\u00b3\u2070 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b2\u2079 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b2\u2078 : TopologicalSpace M\ninst\u271d\u00b2\u2077 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2\u2076 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2\u2075 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2\u2074 : NormedAddCommGroup F\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2\u2070 : FiberBundle F E\ninst\u271d\u00b9\u2079 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b9\u2078 : SmoothVectorBundle F E IB\nF\u2081 : Type u_11\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_12\ninst\u271d\u00b9\u2075 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\ninst\u271d\u00b9\u2074 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\nF\u2082 : Type u_13\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_14\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u2075 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c F\u2082 E\u2082\ninst\u271d\u00b9 : SmoothVectorBundle F\u2081 E\u2081 IB\ninst\u271d : SmoothVectorBundle F\u2082 E\u2082 IB\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\ni\u2081 : MemTrivializationAtlas e\u2081\ni\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\ni\u2081' : MemTrivializationAtlas e\u2081'\ni\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 ContMDiffOn IB \ud835\udcd8(\ud835\udd5c, F\u2081 \u00d7 F\u2082 \u2192L[\ud835\udd5c] F\u2081 \u00d7 F\u2082) \u22a4\n (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c (Trivialization.prod e\u2081 e\u2082) (Trivialization.prod e\u2081' e\u2082') b))\n ((Trivialization.prod e\u2081 e\u2082).baseSet \u2229 (Trivialization.prod e\u2081' e\u2082').baseSet)\n[PROOFSTEP]\nrefine' ContMDiffOn.congr _ (e\u2081.coordChangeL_prod \ud835\udd5c e\u2081' e\u2082 e\u2082')\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b3\u2078 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b3\u2077 : NormedAddCommGroup EB\ninst\u271d\u00b3\u2076 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b3\u2075 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b3\u2074 : TopologicalSpace B\ninst\u271d\u00b3\u00b3 : ChartedSpace HB B\ninst\u271d\u00b3\u00b2 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup EM\ninst\u271d\u00b3\u2070 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b2\u2079 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b2\u2078 : TopologicalSpace M\ninst\u271d\u00b2\u2077 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2\u2076 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2\u2075 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2\u2074 : NormedAddCommGroup F\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2\u2070 : FiberBundle F E\ninst\u271d\u00b9\u2079 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b9\u2078 : SmoothVectorBundle F E IB\nF\u2081 : Type u_11\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_12\ninst\u271d\u00b9\u2075 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\ninst\u271d\u00b9\u2074 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\nF\u2082 : Type u_13\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_14\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u2075 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c F\u2082 E\u2082\ninst\u271d\u00b9 : SmoothVectorBundle F\u2081 E\u2081 IB\ninst\u271d : SmoothVectorBundle F\u2082 E\u2082 IB\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\ni\u2081 : MemTrivializationAtlas e\u2081\ni\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\ni\u2081' : MemTrivializationAtlas e\u2081'\ni\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 ContMDiffOn IB \ud835\udcd8(\ud835\udd5c, F\u2081 \u00d7 F\u2082 \u2192L[\ud835\udd5c] F\u2081 \u00d7 F\u2082) \u22a4\n (fun y =>\n ContinuousLinearMap.prodMap \u2191(Trivialization.coordChangeL \ud835\udd5c e\u2081 e\u2081' y) \u2191(Trivialization.coordChangeL \ud835\udd5c e\u2082 e\u2082' y))\n ((Trivialization.prod e\u2081 e\u2082).baseSet \u2229 (Trivialization.prod e\u2081' e\u2082').baseSet)\n[PROOFSTEP]\nrefine' ContMDiffOn.clm_prodMap _ _\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b3\u2078 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b3\u2077 : NormedAddCommGroup EB\ninst\u271d\u00b3\u2076 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b3\u2075 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b3\u2074 : TopologicalSpace B\ninst\u271d\u00b3\u00b3 : ChartedSpace HB B\ninst\u271d\u00b3\u00b2 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup EM\ninst\u271d\u00b3\u2070 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b2\u2079 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b2\u2078 : TopologicalSpace M\ninst\u271d\u00b2\u2077 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2\u2076 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2\u2075 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2\u2074 : NormedAddCommGroup F\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2\u2070 : FiberBundle F E\ninst\u271d\u00b9\u2079 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b9\u2078 : SmoothVectorBundle F E IB\nF\u2081 : Type u_11\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_12\ninst\u271d\u00b9\u2075 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\ninst\u271d\u00b9\u2074 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\nF\u2082 : Type u_13\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_14\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u2075 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c F\u2082 E\u2082\ninst\u271d\u00b9 : SmoothVectorBundle F\u2081 E\u2081 IB\ninst\u271d : SmoothVectorBundle F\u2082 E\u2082 IB\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\ni\u2081 : MemTrivializationAtlas e\u2081\ni\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\ni\u2081' : MemTrivializationAtlas e\u2081'\ni\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 ContMDiffOn IB \ud835\udcd8(\ud835\udd5c, F\u2081 \u2192L[\ud835\udd5c] F\u2081) \u22a4 (fun y => \u2191(Trivialization.coordChangeL \ud835\udd5c e\u2081 e\u2081' y))\n ((Trivialization.prod e\u2081 e\u2082).baseSet \u2229 (Trivialization.prod e\u2081' e\u2082').baseSet)\n[PROOFSTEP]\nrefine' (smoothOn_coordChangeL IB e\u2081 e\u2081').mono _\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b3\u2078 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b3\u2077 : NormedAddCommGroup EB\ninst\u271d\u00b3\u2076 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b3\u2075 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b3\u2074 : TopologicalSpace B\ninst\u271d\u00b3\u00b3 : ChartedSpace HB B\ninst\u271d\u00b3\u00b2 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup EM\ninst\u271d\u00b3\u2070 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b2\u2079 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b2\u2078 : TopologicalSpace M\ninst\u271d\u00b2\u2077 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2\u2076 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2\u2075 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2\u2074 : NormedAddCommGroup F\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2\u2070 : FiberBundle F E\ninst\u271d\u00b9\u2079 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b9\u2078 : SmoothVectorBundle F E IB\nF\u2081 : Type u_11\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_12\ninst\u271d\u00b9\u2075 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\ninst\u271d\u00b9\u2074 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\nF\u2082 : Type u_13\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_14\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u2075 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c F\u2082 E\u2082\ninst\u271d\u00b9 : SmoothVectorBundle F\u2081 E\u2081 IB\ninst\u271d : SmoothVectorBundle F\u2082 E\u2082 IB\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\ni\u2081 : MemTrivializationAtlas e\u2081\ni\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\ni\u2081' : MemTrivializationAtlas e\u2081'\ni\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 (Trivialization.prod e\u2081 e\u2082).baseSet \u2229 (Trivialization.prod e\u2081' e\u2082').baseSet \u2286 e\u2081.baseSet \u2229 e\u2081'.baseSet\n[PROOFSTEP]\nsimp only [Trivialization.baseSet_prod, mfld_simps]\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b3\u2078 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b3\u2077 : NormedAddCommGroup EB\ninst\u271d\u00b3\u2076 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b3\u2075 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b3\u2074 : TopologicalSpace B\ninst\u271d\u00b3\u00b3 : ChartedSpace HB B\ninst\u271d\u00b3\u00b2 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup EM\ninst\u271d\u00b3\u2070 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b2\u2079 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b2\u2078 : TopologicalSpace M\ninst\u271d\u00b2\u2077 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2\u2076 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2\u2075 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2\u2074 : NormedAddCommGroup F\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2\u2070 : FiberBundle F E\ninst\u271d\u00b9\u2079 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b9\u2078 : SmoothVectorBundle F E IB\nF\u2081 : Type u_11\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_12\ninst\u271d\u00b9\u2075 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\ninst\u271d\u00b9\u2074 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\nF\u2082 : Type u_13\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_14\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u2075 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c F\u2082 E\u2082\ninst\u271d\u00b9 : SmoothVectorBundle F\u2081 E\u2081 IB\ninst\u271d : SmoothVectorBundle F\u2082 E\u2082 IB\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\ni\u2081 : MemTrivializationAtlas e\u2081\ni\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\ni\u2081' : MemTrivializationAtlas e\u2081'\ni\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet) \u2286 e\u2081.baseSet \u2229 e\u2081'.baseSet\n[PROOFSTEP]\nmfld_set_tac\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b3\u2078 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b3\u2077 : NormedAddCommGroup EB\ninst\u271d\u00b3\u2076 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b3\u2075 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b3\u2074 : TopologicalSpace B\ninst\u271d\u00b3\u00b3 : ChartedSpace HB B\ninst\u271d\u00b3\u00b2 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup EM\ninst\u271d\u00b3\u2070 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b2\u2079 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b2\u2078 : TopologicalSpace M\ninst\u271d\u00b2\u2077 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2\u2076 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2\u2075 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2\u2074 : NormedAddCommGroup F\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2\u2070 : FiberBundle F E\ninst\u271d\u00b9\u2079 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b9\u2078 : SmoothVectorBundle F E IB\nF\u2081 : Type u_11\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_12\ninst\u271d\u00b9\u2075 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\ninst\u271d\u00b9\u2074 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\nF\u2082 : Type u_13\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_14\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u2075 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c F\u2082 E\u2082\ninst\u271d\u00b9 : SmoothVectorBundle F\u2081 E\u2081 IB\ninst\u271d : SmoothVectorBundle F\u2082 E\u2082 IB\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\ni\u2081 : MemTrivializationAtlas e\u2081\ni\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\ni\u2081' : MemTrivializationAtlas e\u2081'\ni\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 ContMDiffOn IB \ud835\udcd8(\ud835\udd5c, F\u2082 \u2192L[\ud835\udd5c] F\u2082) \u22a4 (fun y => \u2191(Trivialization.coordChangeL \ud835\udd5c e\u2082 e\u2082' y))\n ((Trivialization.prod e\u2081 e\u2082).baseSet \u2229 (Trivialization.prod e\u2081' e\u2082').baseSet)\n[PROOFSTEP]\nrefine' (smoothOn_coordChangeL IB e\u2082 e\u2082').mono _\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b3\u2078 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b3\u2077 : NormedAddCommGroup EB\ninst\u271d\u00b3\u2076 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b3\u2075 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b3\u2074 : TopologicalSpace B\ninst\u271d\u00b3\u00b3 : ChartedSpace HB B\ninst\u271d\u00b3\u00b2 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup EM\ninst\u271d\u00b3\u2070 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b2\u2079 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b2\u2078 : TopologicalSpace M\ninst\u271d\u00b2\u2077 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2\u2076 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2\u2075 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2\u2074 : NormedAddCommGroup F\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2\u2070 : FiberBundle F E\ninst\u271d\u00b9\u2079 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b9\u2078 : SmoothVectorBundle F E IB\nF\u2081 : Type u_11\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_12\ninst\u271d\u00b9\u2075 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\ninst\u271d\u00b9\u2074 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\nF\u2082 : Type u_13\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_14\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u2075 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c F\u2082 E\u2082\ninst\u271d\u00b9 : SmoothVectorBundle F\u2081 E\u2081 IB\ninst\u271d : SmoothVectorBundle F\u2082 E\u2082 IB\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\ni\u2081 : MemTrivializationAtlas e\u2081\ni\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\ni\u2081' : MemTrivializationAtlas e\u2081'\ni\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 (Trivialization.prod e\u2081 e\u2082).baseSet \u2229 (Trivialization.prod e\u2081' e\u2082').baseSet \u2286 e\u2082.baseSet \u2229 e\u2082'.baseSet\n[PROOFSTEP]\nsimp only [Trivialization.baseSet_prod, mfld_simps]\n[GOAL]\ncase mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b3\u2078 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b3\u2077 : NormedAddCommGroup EB\ninst\u271d\u00b3\u2076 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b3\u2075 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b3\u2074 : TopologicalSpace B\ninst\u271d\u00b3\u00b3 : ChartedSpace HB B\ninst\u271d\u00b3\u00b2 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup EM\ninst\u271d\u00b3\u2070 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u00b2\u2079 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u00b2\u2078 : TopologicalSpace M\ninst\u271d\u00b2\u2077 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2\u2076 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2\u2075 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2\u2074 : NormedAddCommGroup F\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2\u2070 : FiberBundle F E\ninst\u271d\u00b9\u2079 : VectorBundle \ud835\udd5c F E\ninst\u271d\u00b9\u2078 : SmoothVectorBundle F E IB\nF\u2081 : Type u_11\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c F\u2081\nE\u2081 : B \u2192 Type u_12\ninst\u271d\u00b9\u2075 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\ninst\u271d\u00b9\u2074 : (x : B) \u2192 AddCommMonoid (E\u2081 x)\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E\u2081 x)\nF\u2082 : Type u_13\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2082\nE\u2082 : B \u2192 Type u_14\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E\u2082 x)\ninst\u271d\u2078 : (x : B) \u2192 Module \ud835\udd5c (E\u2082 x)\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u2075 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c F\u2081 E\u2081\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c F\u2082 E\u2082\ninst\u271d\u00b9 : SmoothVectorBundle F\u2081 E\u2081 IB\ninst\u271d : SmoothVectorBundle F\u2082 E\u2082 IB\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\ni\u2081 : MemTrivializationAtlas e\u2081\ni\u2082 : MemTrivializationAtlas e\u2082\ne\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082' : Trivialization F\u2082 TotalSpace.proj\ni\u2081' : MemTrivializationAtlas e\u2081'\ni\u2082' : MemTrivializationAtlas e\u2082'\n\u22a2 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet) \u2286 e\u2082.baseSet \u2229 e\u2082'.baseSet\n[PROOFSTEP]\nmfld_set_tac\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b3 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b2 : TopologicalSpace B\ninst\u271d\u00b9\u00b9 : ChartedSpace HB B\ninst\u271d\u00b9\u2070 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u2079 : NormedAddCommGroup EM\ninst\u271d\u2078 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u2077 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2074 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle \ud835\udd5c F E\nha : IsSmooth IB a\ne e' : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nhe' : e' \u2208 a.pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 (b, \u2191(smoothCoordChange IB a he he' b) v) = \u2191e' { proj := b, snd := Pretrivialization.symm e b v }\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b3 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b2 : TopologicalSpace B\ninst\u271d\u00b9\u00b9 : ChartedSpace HB B\ninst\u271d\u00b9\u2070 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u2079 : NormedAddCommGroup EM\ninst\u271d\u2078 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u2077 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2074 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle \ud835\udd5c F E\nha : IsSmooth IB a\ne e' : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nhe' : e' \u2208 a.pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 (b, \u2191(smoothCoordChange IB a he he' b) v).fst = (\u2191e' { proj := b, snd := Pretrivialization.symm e b v }).fst\n[PROOFSTEP]\nrw [e.mk_symm hb.1 v, e'.coe_fst', e.proj_symm_apply' hb.1]\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b3 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b2 : TopologicalSpace B\ninst\u271d\u00b9\u00b9 : ChartedSpace HB B\ninst\u271d\u00b9\u2070 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u2079 : NormedAddCommGroup EM\ninst\u271d\u2078 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u2077 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2074 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle \ud835\udd5c F E\nha : IsSmooth IB a\ne e' : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nhe' : e' \u2208 a.pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 (\u2191(LocalEquiv.symm e.toLocalEquiv) (b, v)).proj \u2208 e'.baseSet\n[PROOFSTEP]\nrw [e.proj_symm_apply' hb.1]\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b3 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b2 : TopologicalSpace B\ninst\u271d\u00b9\u00b9 : ChartedSpace HB B\ninst\u271d\u00b9\u2070 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u2079 : NormedAddCommGroup EM\ninst\u271d\u2078 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u2077 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2074 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle \ud835\udd5c F E\nha : IsSmooth IB a\ne e' : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nhe' : e' \u2208 a.pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 b \u2208 e'.baseSet\n[PROOFSTEP]\nexact hb.2\n[GOAL]\ncase h\u2082\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b3 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b2 : TopologicalSpace B\ninst\u271d\u00b9\u00b9 : ChartedSpace HB B\ninst\u271d\u00b9\u2070 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u2079 : NormedAddCommGroup EM\ninst\u271d\u2078 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u2077 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2074 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle \ud835\udd5c F E\nha : IsSmooth IB a\ne e' : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nhe' : e' \u2208 a.pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 (b, \u2191(smoothCoordChange IB a he he' b) v).snd = (\u2191e' { proj := b, snd := Pretrivialization.symm e b v }).snd\n[PROOFSTEP]\nexact a.smoothCoordChange_apply he he' hb v\n[GOAL]\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b3 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b2 : TopologicalSpace B\ninst\u271d\u00b9\u00b9 : ChartedSpace HB B\ninst\u271d\u00b9\u2070 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u2079 : NormedAddCommGroup EM\ninst\u271d\u2078 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u2077 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2074 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle \ud835\udd5c F E\nha : IsSmooth IB a\ne e' : Pretrivialization F TotalSpace.proj\nthis\u271d\u00b9 : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis\u271d : FiberBundle F E := toFiberBundle a\nthis : VectorBundle \ud835\udd5c F E := toVectorBundle a\n\u22a2 \u2200 (e e' : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e] [inst_1 : MemTrivializationAtlas e'],\n SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c e e' b)) (e.baseSet \u2229 e'.baseSet)\n[PROOFSTEP]\nrintro _ _ \u27e8e, he, rfl\u27e9 \u27e8e', he', rfl\u27e9\n[GOAL]\ncase mk.intro.intro.mk.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b3 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b2 : TopologicalSpace B\ninst\u271d\u00b9\u00b9 : ChartedSpace HB B\ninst\u271d\u00b9\u2070 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u2079 : NormedAddCommGroup EM\ninst\u271d\u2078 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u2077 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2074 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle \ud835\udd5c F E\nha : IsSmooth IB a\ne\u271d e'\u271d : Pretrivialization F TotalSpace.proj\nthis\u271d\u00b9 : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis\u271d : FiberBundle F E := toFiberBundle a\nthis : VectorBundle \ud835\udd5c F E := toVectorBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 (toFiberPrebundle a).pretrivializationAtlas\n\u22a2 SmoothOn IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F)\n (fun b =>\n \u2191(Trivialization.coordChangeL \ud835\udd5c (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b))\n ((FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he).baseSet \u2229\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he').baseSet)\n[PROOFSTEP]\nrefine' (a.smoothOn_smoothCoordChange he he').congr _\n[GOAL]\ncase mk.intro.intro.mk.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b3 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b2 : TopologicalSpace B\ninst\u271d\u00b9\u00b9 : ChartedSpace HB B\ninst\u271d\u00b9\u2070 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u2079 : NormedAddCommGroup EM\ninst\u271d\u2078 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u2077 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2074 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle \ud835\udd5c F E\nha : IsSmooth IB a\ne\u271d e'\u271d : Pretrivialization F TotalSpace.proj\nthis\u271d\u00b9 : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis\u271d : FiberBundle F E := toFiberBundle a\nthis : VectorBundle \ud835\udd5c F E := toVectorBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 (toFiberPrebundle a).pretrivializationAtlas\n\u22a2 \u2200 (y : B),\n y \u2208 e.baseSet \u2229 e'.baseSet \u2192\n \u2191(Trivialization.coordChangeL \ud835\udd5c (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') y) =\n smoothCoordChange IB a he he' y\n[PROOFSTEP]\nintro b hb\n[GOAL]\ncase mk.intro.intro.mk.intro.intro\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b3 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b2 : TopologicalSpace B\ninst\u271d\u00b9\u00b9 : ChartedSpace HB B\ninst\u271d\u00b9\u2070 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u2079 : NormedAddCommGroup EM\ninst\u271d\u2078 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u2077 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2074 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle \ud835\udd5c F E\nha : IsSmooth IB a\ne\u271d e'\u271d : Pretrivialization F TotalSpace.proj\nthis\u271d\u00b9 : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis\u271d : FiberBundle F E := toFiberBundle a\nthis : VectorBundle \ud835\udd5c F E := toVectorBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 \u2191(Trivialization.coordChangeL \ud835\udd5c (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b) =\n smoothCoordChange IB a he he' b\n[PROOFSTEP]\next v\n[GOAL]\ncase mk.intro.intro.mk.intro.intro.h\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b3 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b2 : TopologicalSpace B\ninst\u271d\u00b9\u00b9 : ChartedSpace HB B\ninst\u271d\u00b9\u2070 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u2079 : NormedAddCommGroup EM\ninst\u271d\u2078 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u2077 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2074 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle \ud835\udd5c F E\nha : IsSmooth IB a\ne\u271d e'\u271d : Pretrivialization F TotalSpace.proj\nthis\u271d\u00b9 : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis\u271d : FiberBundle F E := toFiberBundle a\nthis : VectorBundle \ud835\udd5c F E := toVectorBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 \u2191\u2191(Trivialization.coordChangeL \ud835\udd5c (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b)\n v =\n \u2191(smoothCoordChange IB a he he' b) v\n[PROOFSTEP]\nrw [a.smoothCoordChange_apply he he' hb v, ContinuousLinearEquiv.coe_coe, Trivialization.coordChangeL_apply]\n[GOAL]\ncase mk.intro.intro.mk.intro.intro.h\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b3 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b2 : TopologicalSpace B\ninst\u271d\u00b9\u00b9 : ChartedSpace HB B\ninst\u271d\u00b9\u2070 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u2079 : NormedAddCommGroup EM\ninst\u271d\u2078 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u2077 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2074 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle \ud835\udd5c F E\nha : IsSmooth IB a\ne\u271d e'\u271d : Pretrivialization F TotalSpace.proj\nthis\u271d\u00b9 : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis\u271d : FiberBundle F E := toFiberBundle a\nthis : VectorBundle \ud835\udd5c F E := toVectorBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 (\u2191(FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he')\n { proj := b,\n snd :=\n Trivialization.symm (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) b\n v }).snd =\n (\u2191e' { proj := b, snd := Pretrivialization.symm e b v }).snd\ncase mk.intro.intro.mk.intro.intro.h.hb\n\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nEB : Type u_7\ninst\u271d\u00b9\u2075 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u00b9\u00b3 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\ninst\u271d\u00b9\u00b2 : TopologicalSpace B\ninst\u271d\u00b9\u00b9 : ChartedSpace HB B\ninst\u271d\u00b9\u2070 : SmoothManifoldWithCorners IB B\nEM : Type u_9\ninst\u271d\u2079 : NormedAddCommGroup EM\ninst\u271d\u2078 : NormedSpace \ud835\udd5c EM\nHM : Type u_10\ninst\u271d\u2077 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u2074 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b3 : (x : B) \u2192 Module \ud835\udd5c (E x)\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle \ud835\udd5c F E\nha : IsSmooth IB a\ne\u271d e'\u271d : Pretrivialization F TotalSpace.proj\nthis\u271d\u00b9 : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis\u271d : FiberBundle F E := toFiberBundle a\nthis : VectorBundle \ud835\udd5c F E := toVectorBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 b \u2208\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he).baseSet \u2229\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he').baseSet\n[PROOFSTEP]\nexacts [rfl, hb]\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Manifold.VectorBundle.Basic", "llama_tokens": 74084, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125848754472, "lm_q2_score": 0.538983220687684, "lm_q1q2_score": 0.40532216499383883}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nx : \u03b1\nU : Set \u03b1\nh_acc : AccPt x (\ud835\udcdf C)\nhU : U \u2208 \ud835\udcdd x\n\u22a2 AccPt x (\ud835\udcdf (U \u2229 C))\n[PROOFSTEP]\nhave : \ud835\udcdd[\u2260] x \u2264 \ud835\udcdf U := by\n rw [le_principal_iff]\n exact mem_nhdsWithin_of_mem_nhds hU\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nx : \u03b1\nU : Set \u03b1\nh_acc : AccPt x (\ud835\udcdf C)\nhU : U \u2208 \ud835\udcdd x\n\u22a2 \ud835\udcdd[{x}\u1d9c] x \u2264 \ud835\udcdf U\n[PROOFSTEP]\nrw [le_principal_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nx : \u03b1\nU : Set \u03b1\nh_acc : AccPt x (\ud835\udcdf C)\nhU : U \u2208 \ud835\udcdd x\n\u22a2 U \u2208 \ud835\udcdd[{x}\u1d9c] x\n[PROOFSTEP]\nexact mem_nhdsWithin_of_mem_nhds hU\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nx : \u03b1\nU : Set \u03b1\nh_acc : AccPt x (\ud835\udcdf C)\nhU : U \u2208 \ud835\udcdd x\nthis : \ud835\udcdd[{x}\u1d9c] x \u2264 \ud835\udcdf U\n\u22a2 AccPt x (\ud835\udcdf (U \u2229 C))\n[PROOFSTEP]\nrw [AccPt, \u2190 inf_principal, \u2190 inf_assoc, inf_of_le_left this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nx : \u03b1\nU : Set \u03b1\nh_acc : AccPt x (\ud835\udcdf C)\nhU : U \u2208 \ud835\udcdd x\nthis : \ud835\udcdd[{x}\u1d9c] x \u2264 \ud835\udcdf U\n\u22a2 NeBot (\ud835\udcdd[{x}\u1d9c] x \u2293 \ud835\udcdf C)\n[PROOFSTEP]\nexact h_acc\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\n\u22a2 Preperfect C \u2194 \u2200 (x : \u03b1), x \u2208 C \u2192 \u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 y, y \u2208 U \u2229 C \u2227 y \u2260 x\n[PROOFSTEP]\nsimp only [Preperfect, accPt_iff_nhds]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC U : Set \u03b1\nhC : Preperfect C\nhU : IsOpen U\n\u22a2 Preperfect (U \u2229 C)\n[PROOFSTEP]\nrintro x \u27e8xU, xC\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC U : Set \u03b1\nhC : Preperfect C\nhU : IsOpen U\nx : \u03b1\nxU : x \u2208 U\nxC : x \u2208 C\n\u22a2 AccPt x (\ud835\udcdf (U \u2229 C))\n[PROOFSTEP]\napply (hC _ xC).nhds_inter\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC U : Set \u03b1\nhC : Preperfect C\nhU : IsOpen U\nx : \u03b1\nxU : x \u2208 U\nxC : x \u2208 C\n\u22a2 U \u2208 \ud835\udcdd x\n[PROOFSTEP]\nexact hU.mem_nhds xU\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\n\u22a2 Perfect (closure C)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase closed\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\n\u22a2 IsClosed (closure C)\n[PROOFSTEP]\nexact isClosed_closure\n[GOAL]\ncase acc\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\n\u22a2 Preperfect (closure C)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase acc\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\n\u22a2 AccPt x (\ud835\udcdf (closure C))\n[PROOFSTEP]\nby_cases h : x \u2208 C\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : x \u2208 C\n\u22a2 AccPt x (\ud835\udcdf (closure C))\n[PROOFSTEP]\napply AccPt.mono _ (principal_mono.mpr subset_closure)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : \u00acx \u2208 C\n\u22a2 AccPt x (\ud835\udcdf (closure C))\n[PROOFSTEP]\napply AccPt.mono _ (principal_mono.mpr subset_closure)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : x \u2208 C\n\u22a2 AccPt x (\ud835\udcdf C)\n[PROOFSTEP]\nexact hC _ h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : \u00acx \u2208 C\n\u22a2 AccPt x (\ud835\udcdf C)\n[PROOFSTEP]\nhave : { x }\u1d9c \u2229 C = C := by simp [h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : \u00acx \u2208 C\n\u22a2 {x}\u1d9c \u2229 C = C\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : \u00acx \u2208 C\nthis : {x}\u1d9c \u2229 C = C\n\u22a2 AccPt x (\ud835\udcdf C)\n[PROOFSTEP]\nrw [AccPt, nhdsWithin, inf_assoc, inf_principal, this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : \u00acx \u2208 C\nthis : {x}\u1d9c \u2229 C = C\n\u22a2 NeBot (\ud835\udcdd x \u2293 \ud835\udcdf C)\n[PROOFSTEP]\nrw [closure_eq_cluster_pts] at hx \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 {a | ClusterPt a (\ud835\udcdf C)}\nh : \u00acx \u2208 C\nthis : {x}\u1d9c \u2229 C = C\n\u22a2 NeBot (\ud835\udcdd x \u2293 \ud835\udcdf C)\n[PROOFSTEP]\nexact hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T1Space \u03b1\n\u22a2 Preperfect C \u2194 Perfect (closure C)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T1Space \u03b1\n\u22a2 Preperfect C \u2192 Perfect (closure C)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T1Space \u03b1\n\u22a2 Perfect (closure C) \u2192 Preperfect C\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T1Space \u03b1\nh : Preperfect C\n\u22a2 Perfect (closure C)\n[PROOFSTEP]\nexact h.perfect_closure\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T1Space \u03b1\nh : Perfect (closure C)\n\u22a2 Preperfect C\n[PROOFSTEP]\nintro x xC\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T1Space \u03b1\nh : Perfect (closure C)\nx : \u03b1\nxC : x \u2208 C\n\u22a2 AccPt x (\ud835\udcdf C)\n[PROOFSTEP]\nhave H : AccPt x (\ud835\udcdf (closure C)) := h.acc _ (subset_closure xC)\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T1Space \u03b1\nh : Perfect (closure C)\nx : \u03b1\nxC : x \u2208 C\nH : AccPt x (\ud835\udcdf (closure C))\n\u22a2 AccPt x (\ud835\udcdf C)\n[PROOFSTEP]\nrw [accPt_iff_frequently] at *\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T1Space \u03b1\nh : Perfect (closure C)\nx : \u03b1\nxC : x \u2208 C\nH : \u2203\u1da0 (y : \u03b1) in \ud835\udcdd x, y \u2260 x \u2227 y \u2208 closure C\n\u22a2 \u2203\u1da0 (y : \u03b1) in \ud835\udcdd x, y \u2260 x \u2227 y \u2208 C\n[PROOFSTEP]\nhave : \u2200 y, y \u2260 x \u2227 y \u2208 closure C \u2192 \u2203\u1da0 z in \ud835\udcdd y, z \u2260 x \u2227 z \u2208 C :=\n by\n rintro y \u27e8hyx, yC\u27e9\n simp only [\u2190 mem_compl_singleton_iff, and_comm, \u2190 frequently_nhdsWithin_iff, hyx.nhdsWithin_compl_singleton, \u2190\n mem_closure_iff_frequently]\n exact yC\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T1Space \u03b1\nh : Perfect (closure C)\nx : \u03b1\nxC : x \u2208 C\nH : \u2203\u1da0 (y : \u03b1) in \ud835\udcdd x, y \u2260 x \u2227 y \u2208 closure C\n\u22a2 \u2200 (y : \u03b1), y \u2260 x \u2227 y \u2208 closure C \u2192 \u2203\u1da0 (z : \u03b1) in \ud835\udcdd y, z \u2260 x \u2227 z \u2208 C\n[PROOFSTEP]\nrintro y \u27e8hyx, yC\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T1Space \u03b1\nh : Perfect (closure C)\nx : \u03b1\nxC : x \u2208 C\nH : \u2203\u1da0 (y : \u03b1) in \ud835\udcdd x, y \u2260 x \u2227 y \u2208 closure C\ny : \u03b1\nhyx : y \u2260 x\nyC : y \u2208 closure C\n\u22a2 \u2203\u1da0 (z : \u03b1) in \ud835\udcdd y, z \u2260 x \u2227 z \u2208 C\n[PROOFSTEP]\nsimp only [\u2190 mem_compl_singleton_iff, and_comm, \u2190 frequently_nhdsWithin_iff, hyx.nhdsWithin_compl_singleton, \u2190\n mem_closure_iff_frequently]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T1Space \u03b1\nh : Perfect (closure C)\nx : \u03b1\nxC : x \u2208 C\nH : \u2203\u1da0 (y : \u03b1) in \ud835\udcdd x, y \u2260 x \u2227 y \u2208 closure C\ny : \u03b1\nhyx : y \u2260 x\nyC : y \u2208 closure C\n\u22a2 y \u2208 closure C\n[PROOFSTEP]\nexact yC\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T1Space \u03b1\nh : Perfect (closure C)\nx : \u03b1\nxC : x \u2208 C\nH : \u2203\u1da0 (y : \u03b1) in \ud835\udcdd x, y \u2260 x \u2227 y \u2208 closure C\nthis : \u2200 (y : \u03b1), y \u2260 x \u2227 y \u2208 closure C \u2192 \u2203\u1da0 (z : \u03b1) in \ud835\udcdd y, z \u2260 x \u2227 z \u2208 C\n\u22a2 \u2203\u1da0 (y : \u03b1) in \ud835\udcdd x, y \u2260 x \u2227 y \u2208 C\n[PROOFSTEP]\nrw [\u2190 frequently_frequently_nhds]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T1Space \u03b1\nh : Perfect (closure C)\nx : \u03b1\nxC : x \u2208 C\nH : \u2203\u1da0 (y : \u03b1) in \ud835\udcdd x, y \u2260 x \u2227 y \u2208 closure C\nthis : \u2200 (y : \u03b1), y \u2260 x \u2227 y \u2208 closure C \u2192 \u2203\u1da0 (z : \u03b1) in \ud835\udcdd y, z \u2260 x \u2227 z \u2208 C\n\u22a2 \u2203\u1da0 (y : \u03b1) in \ud835\udcdd x, \u2203\u1da0 (x_1 : \u03b1) in \ud835\udcdd y, x_1 \u2260 x \u2227 x_1 \u2208 C\n[PROOFSTEP]\nexact H.mono this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC U : Set \u03b1\nhC : Perfect C\nx : \u03b1\nxC : x \u2208 C\nxU : x \u2208 U\nUop : IsOpen U\n\u22a2 Perfect (closure (U \u2229 C)) \u2227 Set.Nonempty (closure (U \u2229 C))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC U : Set \u03b1\nhC : Perfect C\nx : \u03b1\nxC : x \u2208 C\nxU : x \u2208 U\nUop : IsOpen U\n\u22a2 Perfect (closure (U \u2229 C))\n[PROOFSTEP]\napply Preperfect.perfect_closure\n[GOAL]\ncase left.hC\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC U : Set \u03b1\nhC : Perfect C\nx : \u03b1\nxC : x \u2208 C\nxU : x \u2208 U\nUop : IsOpen U\n\u22a2 Preperfect (U \u2229 C)\n[PROOFSTEP]\nexact hC.acc.open_inter Uop\n[GOAL]\ncase right\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC U : Set \u03b1\nhC : Perfect C\nx : \u03b1\nxC : x \u2208 C\nxU : x \u2208 U\nUop : IsOpen U\n\u22a2 Set.Nonempty (closure (U \u2229 C))\n[PROOFSTEP]\napply Nonempty.closure\n[GOAL]\ncase right.a\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC U : Set \u03b1\nhC : Perfect C\nx : \u03b1\nxC : x \u2208 C\nxU : x \u2208 U\nUop : IsOpen U\n\u22a2 Set.Nonempty (U \u2229 C)\n[PROOFSTEP]\nexact \u27e8x, \u27e8xU, xC\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\nhnonempty : Set.Nonempty C\n\u22a2 \u2203 C\u2080 C\u2081, (Perfect C\u2080 \u2227 Set.Nonempty C\u2080 \u2227 C\u2080 \u2286 C) \u2227 (Perfect C\u2081 \u2227 Set.Nonempty C\u2081 \u2227 C\u2081 \u2286 C) \u2227 Disjoint C\u2080 C\u2081\n[PROOFSTEP]\ncases' hnonempty with y yC\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\n\u22a2 \u2203 C\u2080 C\u2081, (Perfect C\u2080 \u2227 Set.Nonempty C\u2080 \u2227 C\u2080 \u2286 C) \u2227 (Perfect C\u2081 \u2227 Set.Nonempty C\u2081 \u2227 C\u2081 \u2286 C) \u2227 Disjoint C\u2080 C\u2081\n[PROOFSTEP]\nobtain \u27e8x, xC, hxy\u27e9 : \u2203 x \u2208 C, x \u2260 y := by\n have := hC.acc _ yC\n rw [accPt_iff_nhds] at this \n rcases this univ univ_mem with \u27e8x, xC, hxy\u27e9\n exact \u27e8x, xC.2, hxy\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\n\u22a2 \u2203 x, x \u2208 C \u2227 x \u2260 y\n[PROOFSTEP]\nhave := hC.acc _ yC\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nthis : AccPt y (\ud835\udcdf C)\n\u22a2 \u2203 x, x \u2208 C \u2227 x \u2260 y\n[PROOFSTEP]\nrw [accPt_iff_nhds] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nthis : \u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd y \u2192 \u2203 y_1, y_1 \u2208 U \u2229 C \u2227 y_1 \u2260 y\n\u22a2 \u2203 x, x \u2208 C \u2227 x \u2260 y\n[PROOFSTEP]\nrcases this univ univ_mem with \u27e8x, xC, hxy\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nthis : \u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd y \u2192 \u2203 y_1, y_1 \u2208 U \u2229 C \u2227 y_1 \u2260 y\nx : \u03b1\nxC : x \u2208 univ \u2229 C\nhxy : x \u2260 y\n\u22a2 \u2203 x, x \u2208 C \u2227 x \u2260 y\n[PROOFSTEP]\nexact \u27e8x, xC.2, hxy\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\n\u22a2 \u2203 C\u2080 C\u2081, (Perfect C\u2080 \u2227 Set.Nonempty C\u2080 \u2227 C\u2080 \u2286 C) \u2227 (Perfect C\u2081 \u2227 Set.Nonempty C\u2081 \u2227 C\u2081 \u2286 C) \u2227 Disjoint C\u2080 C\u2081\n[PROOFSTEP]\nobtain \u27e8U, xU, Uop, V, yV, Vop, hUV\u27e9 := exists_open_nhds_disjoint_closure hxy\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 \u2203 C\u2080 C\u2081, (Perfect C\u2080 \u2227 Set.Nonempty C\u2080 \u2227 C\u2080 \u2286 C) \u2227 (Perfect C\u2081 \u2227 Set.Nonempty C\u2081 \u2227 C\u2081 \u2286 C) \u2227 Disjoint C\u2080 C\u2081\n[PROOFSTEP]\nuse closure (U \u2229 C), closure (V \u2229 C)\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 (Perfect (closure (U \u2229 C)) \u2227 Set.Nonempty (closure (U \u2229 C)) \u2227 closure (U \u2229 C) \u2286 C) \u2227\n (Perfect (closure (V \u2229 C)) \u2227 Set.Nonempty (closure (V \u2229 C)) \u2227 closure (V \u2229 C) \u2286 C) \u2227\n Disjoint (closure (U \u2229 C)) (closure (V \u2229 C))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 Perfect (closure (U \u2229 C)) \u2227 Set.Nonempty (closure (U \u2229 C)) \u2227 closure (U \u2229 C) \u2286 C\n[PROOFSTEP]\nrw [\u2190 and_assoc]\n[GOAL]\ncase h.right\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 (Perfect (closure (V \u2229 C)) \u2227 Set.Nonempty (closure (V \u2229 C)) \u2227 closure (V \u2229 C) \u2286 C) \u2227\n Disjoint (closure (U \u2229 C)) (closure (V \u2229 C))\n[PROOFSTEP]\nrw [\u2190 and_assoc]\n[GOAL]\ncase h.left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 (Perfect (closure (U \u2229 C)) \u2227 Set.Nonempty (closure (U \u2229 C))) \u2227 closure (U \u2229 C) \u2286 C\n[PROOFSTEP]\nrefine' \u27e8hC.closure_nhds_inter x xC xU Uop, _\u27e9\n[GOAL]\ncase h.left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 closure (U \u2229 C) \u2286 C\n[PROOFSTEP]\nrw [hC.closed.closure_subset_iff]\n[GOAL]\ncase h.left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 U \u2229 C \u2286 C\n[PROOFSTEP]\nexact inter_subset_right _ _\n[GOAL]\ncase h.right\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 ((Perfect (closure (V \u2229 C)) \u2227 Set.Nonempty (closure (V \u2229 C))) \u2227 closure (V \u2229 C) \u2286 C) \u2227\n Disjoint (closure (U \u2229 C)) (closure (V \u2229 C))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.right.left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 (Perfect (closure (V \u2229 C)) \u2227 Set.Nonempty (closure (V \u2229 C))) \u2227 closure (V \u2229 C) \u2286 C\n[PROOFSTEP]\nrefine' \u27e8hC.closure_nhds_inter y yC yV Vop, _\u27e9\n[GOAL]\ncase h.right.left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 closure (V \u2229 C) \u2286 C\n[PROOFSTEP]\nrw [hC.closed.closure_subset_iff]\n[GOAL]\ncase h.right.left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 V \u2229 C \u2286 C\n[PROOFSTEP]\nexact inter_subset_right _ _\n[GOAL]\ncase h.right.right\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 Disjoint (closure (U \u2229 C)) (closure (V \u2229 C))\n[PROOFSTEP]\napply Disjoint.mono _ _ hUV\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 closure (U \u2229 C) \u2264 closure U\n[PROOFSTEP]\napply closure_mono\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 closure (V \u2229 C) \u2264 closure V\n[PROOFSTEP]\napply closure_mono\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 U \u2229 C \u2286 U\n[PROOFSTEP]\nexact inter_subset_left _ _\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : T25Space \u03b1\nhC : Perfect C\ny : \u03b1\nyC : y \u2208 C\nx : \u03b1\nxC : x \u2208 C\nhxy : x \u2260 y\nU : Set \u03b1\nxU : x \u2208 U\nUop : IsOpen U\nV : Set \u03b1\nyV : y \u2208 V\nVop : IsOpen V\nhUV : Disjoint (closure U) (closure V)\n\u22a2 V \u2229 C \u2286 V\n[PROOFSTEP]\nexact inter_subset_left _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\n\u22a2 \u2203 V D, Set.Countable V \u2227 Perfect D \u2227 C = V \u222a D\n[PROOFSTEP]\nobtain \u27e8b, bct, _, bbasis\u27e9 := TopologicalSpace.exists_countable_basis \u03b1\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\n\u22a2 \u2203 V D, Set.Countable V \u2227 Perfect D \u2227 C = V \u222a D\n[PROOFSTEP]\nlet v := {U \u2208 b | (U \u2229 C).Countable}\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\n\u22a2 \u2203 V D, Set.Countable V \u2227 Perfect D \u2227 C = V \u222a D\n[PROOFSTEP]\nlet V := \u22c3 U \u2208 v, U\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\n\u22a2 \u2203 V D, Set.Countable V \u2227 Perfect D \u2227 C = V \u222a D\n[PROOFSTEP]\nlet D := C \\ V\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\n\u22a2 \u2203 V D, Set.Countable V \u2227 Perfect D \u2227 C = V \u222a D\n[PROOFSTEP]\nhave Vct : (V \u2229 C).Countable := by\n simp only [iUnion_inter, mem_sep_iff]\n apply Countable.biUnion\n \u00b7 exact Countable.mono (inter_subset_left _ _) bct\n \u00b7 exact inter_subset_right _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\n\u22a2 Set.Countable (V \u2229 C)\n[PROOFSTEP]\nsimp only [iUnion_inter, mem_sep_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\n\u22a2 Set.Countable (\u22c3 (i : Set \u03b1) (_ : i \u2208 b \u2227 Set.Countable (i \u2229 C)), i \u2229 C)\n[PROOFSTEP]\napply Countable.biUnion\n[GOAL]\ncase hs\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\n\u22a2 Set.Countable fun a => a \u2208 b \u2227 Set.Countable (a \u2229 C)\n[PROOFSTEP]\nexact Countable.mono (inter_subset_left _ _) bct\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\n\u22a2 \u2200 (a : Set \u03b1), (a \u2208 fun a => a \u2208 b \u2227 Set.Countable (a \u2229 C)) \u2192 Set.Countable (a \u2229 C)\n[PROOFSTEP]\nexact inter_subset_right _ _\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\n\u22a2 \u2203 V D, Set.Countable V \u2227 Perfect D \u2227 C = V \u222a D\n[PROOFSTEP]\nrefine' \u27e8V \u2229 C, D, Vct, \u27e8_, _\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\n\u22a2 IsClosed D\n[PROOFSTEP]\nrefine' hclosed.sdiff (isOpen_biUnion fun _ \u21a6 _)\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx\u271d : Set \u03b1\n\u22a2 x\u271d \u2208 v \u2192 IsOpen x\u271d\n[PROOFSTEP]\nexact fun \u27e8Ub, _\u27e9 \u21a6 IsTopologicalBasis.isOpen bbasis Ub\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\n\u22a2 Preperfect D\n[PROOFSTEP]\nrw [preperfect_iff_nhds]\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\n\u22a2 \u2200 (x : \u03b1), x \u2208 D \u2192 \u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 y, y \u2208 U \u2229 D \u2227 y \u2260 x\n[PROOFSTEP]\nintro x xD E xE\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx : \u03b1\nxD : x \u2208 D\nE : Set \u03b1\nxE : E \u2208 \ud835\udcdd x\n\u22a2 \u2203 y, y \u2208 E \u2229 D \u2227 y \u2260 x\n[PROOFSTEP]\nhave : \u00ac(E \u2229 D).Countable := by\n intro h\n obtain \u27e8U, hUb, xU, hU\u27e9 : \u2203 U \u2208 b, x \u2208 U \u2227 U \u2286 E := (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE\n have hU_cnt : (U \u2229 C).Countable := by\n apply @Countable.mono _ _ (E \u2229 D \u222a V \u2229 C)\n \u00b7 rintro y \u27e8yU, yC\u27e9\n by_cases h : y \u2208 V\n \u00b7 exact mem_union_right _ (mem_inter h yC)\n \u00b7 exact mem_union_left _ (mem_inter (hU yU) \u27e8yC, h\u27e9)\n exact Countable.union h Vct\n have : U \u2208 v := \u27e8hUb, hU_cnt\u27e9\n apply xD.2\n exact mem_biUnion this xU\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx : \u03b1\nxD : x \u2208 D\nE : Set \u03b1\nxE : E \u2208 \ud835\udcdd x\n\u22a2 \u00acSet.Countable (E \u2229 D)\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx : \u03b1\nxD : x \u2208 D\nE : Set \u03b1\nxE : E \u2208 \ud835\udcdd x\nh : Set.Countable (E \u2229 D)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8U, hUb, xU, hU\u27e9 : \u2203 U \u2208 b, x \u2208 U \u2227 U \u2286 E := (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx : \u03b1\nxD : x \u2208 D\nE : Set \u03b1\nxE : E \u2208 \ud835\udcdd x\nh : Set.Countable (E \u2229 D)\nU : Set \u03b1\nhUb : U \u2208 b\nxU : x \u2208 U\nhU : U \u2286 E\n\u22a2 False\n[PROOFSTEP]\nhave hU_cnt : (U \u2229 C).Countable := by\n apply @Countable.mono _ _ (E \u2229 D \u222a V \u2229 C)\n \u00b7 rintro y \u27e8yU, yC\u27e9\n by_cases h : y \u2208 V\n \u00b7 exact mem_union_right _ (mem_inter h yC)\n \u00b7 exact mem_union_left _ (mem_inter (hU yU) \u27e8yC, h\u27e9)\n exact Countable.union h Vct\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx : \u03b1\nxD : x \u2208 D\nE : Set \u03b1\nxE : E \u2208 \ud835\udcdd x\nh : Set.Countable (E \u2229 D)\nU : Set \u03b1\nhUb : U \u2208 b\nxU : x \u2208 U\nhU : U \u2286 E\n\u22a2 Set.Countable (U \u2229 C)\n[PROOFSTEP]\napply @Countable.mono _ _ (E \u2229 D \u222a V \u2229 C)\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx : \u03b1\nxD : x \u2208 D\nE : Set \u03b1\nxE : E \u2208 \ud835\udcdd x\nh : Set.Countable (E \u2229 D)\nU : Set \u03b1\nhUb : U \u2208 b\nxU : x \u2208 U\nhU : U \u2286 E\n\u22a2 U \u2229 C \u2286 E \u2229 D \u222a V \u2229 C\n[PROOFSTEP]\nrintro y \u27e8yU, yC\u27e9\n[GOAL]\ncase h.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx : \u03b1\nxD : x \u2208 D\nE : Set \u03b1\nxE : E \u2208 \ud835\udcdd x\nh : Set.Countable (E \u2229 D)\nU : Set \u03b1\nhUb : U \u2208 b\nxU : x \u2208 U\nhU : U \u2286 E\ny : \u03b1\nyU : y \u2208 U\nyC : y \u2208 C\n\u22a2 y \u2208 E \u2229 D \u222a V \u2229 C\n[PROOFSTEP]\nby_cases h : y \u2208 V\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx : \u03b1\nxD : x \u2208 D\nE : Set \u03b1\nxE : E \u2208 \ud835\udcdd x\nh\u271d : Set.Countable (E \u2229 D)\nU : Set \u03b1\nhUb : U \u2208 b\nxU : x \u2208 U\nhU : U \u2286 E\ny : \u03b1\nyU : y \u2208 U\nyC : y \u2208 C\nh : y \u2208 V\n\u22a2 y \u2208 E \u2229 D \u222a V \u2229 C\n[PROOFSTEP]\nexact mem_union_right _ (mem_inter h yC)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx : \u03b1\nxD : x \u2208 D\nE : Set \u03b1\nxE : E \u2208 \ud835\udcdd x\nh\u271d : Set.Countable (E \u2229 D)\nU : Set \u03b1\nhUb : U \u2208 b\nxU : x \u2208 U\nhU : U \u2286 E\ny : \u03b1\nyU : y \u2208 U\nyC : y \u2208 C\nh : \u00acy \u2208 V\n\u22a2 y \u2208 E \u2229 D \u222a V \u2229 C\n[PROOFSTEP]\nexact mem_union_left _ (mem_inter (hU yU) \u27e8yC, h\u27e9)\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx : \u03b1\nxD : x \u2208 D\nE : Set \u03b1\nxE : E \u2208 \ud835\udcdd x\nh : Set.Countable (E \u2229 D)\nU : Set \u03b1\nhUb : U \u2208 b\nxU : x \u2208 U\nhU : U \u2286 E\n\u22a2 Set.Countable (E \u2229 D \u222a V \u2229 C)\n[PROOFSTEP]\nexact Countable.union h Vct\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx : \u03b1\nxD : x \u2208 D\nE : Set \u03b1\nxE : E \u2208 \ud835\udcdd x\nh : Set.Countable (E \u2229 D)\nU : Set \u03b1\nhUb : U \u2208 b\nxU : x \u2208 U\nhU : U \u2286 E\nhU_cnt : Set.Countable (U \u2229 C)\n\u22a2 False\n[PROOFSTEP]\nhave : U \u2208 v := \u27e8hUb, hU_cnt\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx : \u03b1\nxD : x \u2208 D\nE : Set \u03b1\nxE : E \u2208 \ud835\udcdd x\nh : Set.Countable (E \u2229 D)\nU : Set \u03b1\nhUb : U \u2208 b\nxU : x \u2208 U\nhU : U \u2286 E\nhU_cnt : Set.Countable (U \u2229 C)\nthis : U \u2208 v\n\u22a2 False\n[PROOFSTEP]\napply xD.2\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx : \u03b1\nxD : x \u2208 D\nE : Set \u03b1\nxE : E \u2208 \ud835\udcdd x\nh : Set.Countable (E \u2229 D)\nU : Set \u03b1\nhUb : U \u2208 b\nxU : x \u2208 U\nhU : U \u2286 E\nhU_cnt : Set.Countable (U \u2229 C)\nthis : U \u2208 v\n\u22a2 x \u2208 V\n[PROOFSTEP]\nexact mem_biUnion this xU\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx : \u03b1\nxD : x \u2208 D\nE : Set \u03b1\nxE : E \u2208 \ud835\udcdd x\nthis : \u00acSet.Countable (E \u2229 D)\n\u22a2 \u2203 y, y \u2208 E \u2229 D \u2227 y \u2260 x\n[PROOFSTEP]\nby_contra' h\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\nx : \u03b1\nxD : x \u2208 D\nE : Set \u03b1\nxE : E \u2208 \ud835\udcdd x\nthis : \u00acSet.Countable (E \u2229 D)\nh : \u2200 (y : \u03b1), y \u2208 E \u2229 (C \\ V) \u2192 y = x\n\u22a2 False\n[PROOFSTEP]\nexact absurd (Countable.mono h (Set.countable_singleton _)) this\n[GOAL]\ncase intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nb : Set (Set \u03b1)\nbct : Set.Countable b\nleft\u271d : \u00ac\u2205 \u2208 b\nbbasis : IsTopologicalBasis b\nv : Set (Set \u03b1) := {U | U \u2208 b \u2227 Set.Countable (U \u2229 C)}\nV : Set \u03b1 := \u22c3 (U : Set \u03b1) (_ : U \u2208 v), U\nD : Set \u03b1 := C \\ V\nVct : Set.Countable (V \u2229 C)\n\u22a2 C = V \u2229 C \u222a D\n[PROOFSTEP]\nrw [inter_comm, inter_union_diff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nhunc : \u00acSet.Countable C\n\u22a2 \u2203 D, Perfect D \u2227 Set.Nonempty D \u2227 D \u2286 C\n[PROOFSTEP]\nrcases exists_countable_union_perfect_of_isClosed hclosed with \u27e8V, D, Vct, Dperf, VD\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nhunc : \u00acSet.Countable C\nV D : Set \u03b1\nVct : Set.Countable V\nDperf : Perfect D\nVD : C = V \u222a D\n\u22a2 \u2203 D, Perfect D \u2227 Set.Nonempty D \u2227 D \u2286 C\n[PROOFSTEP]\nrefine' \u27e8D, \u27e8Dperf, _\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nhunc : \u00acSet.Countable C\nV D : Set \u03b1\nVct : Set.Countable V\nDperf : Perfect D\nVD : C = V \u222a D\n\u22a2 Set.Nonempty D \u2227 D \u2286 C\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.intro.intro.left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nhunc : \u00acSet.Countable C\nV D : Set \u03b1\nVct : Set.Countable V\nDperf : Perfect D\nVD : C = V \u222a D\n\u22a2 Set.Nonempty D\n[PROOFSTEP]\nrw [nonempty_iff_ne_empty]\n[GOAL]\ncase intro.intro.intro.intro.left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nhunc : \u00acSet.Countable C\nV D : Set \u03b1\nVct : Set.Countable V\nDperf : Perfect D\nVD : C = V \u222a D\n\u22a2 D \u2260 \u2205\n[PROOFSTEP]\nby_contra h\n[GOAL]\ncase intro.intro.intro.intro.left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nhunc : \u00acSet.Countable C\nV D : Set \u03b1\nVct : Set.Countable V\nDperf : Perfect D\nVD : C = V \u222a D\nh : D = \u2205\n\u22a2 False\n[PROOFSTEP]\nrw [h, union_empty] at VD \n[GOAL]\ncase intro.intro.intro.intro.left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nhunc : \u00acSet.Countable C\nV D : Set \u03b1\nVct : Set.Countable V\nDperf : Perfect D\nVD : C = V\nh : D = \u2205\n\u22a2 False\n[PROOFSTEP]\nrw [VD] at hunc \n[GOAL]\ncase intro.intro.intro.intro.left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nV : Set \u03b1\nhunc : \u00acSet.Countable V\nD : Set \u03b1\nVct : Set.Countable V\nDperf : Perfect D\nVD : C = V\nh : D = \u2205\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase intro.intro.intro.intro.right\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nhunc : \u00acSet.Countable C\nV D : Set \u03b1\nVct : Set.Countable V\nDperf : Perfect D\nVD : C = V \u222a D\n\u22a2 D \u2286 C\n[PROOFSTEP]\nrw [VD]\n[GOAL]\ncase intro.intro.intro.intro.right\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nC : Set \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhclosed : IsClosed C\nhunc : \u00acSet.Countable C\nV D : Set \u03b1\nVct : Set.Countable V\nDperf : Perfect D\nVD : C = V \u222a D\n\u22a2 D \u2286 V \u222a D\n[PROOFSTEP]\nexact subset_union_right _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nx : \u03b1\nxC : x \u2208 C\n\u22a2 let D := closure (EMetric.ball x (\u03b5 / 2) \u2229 C);\n Perfect D \u2227 Set.Nonempty D \u2227 D \u2286 C \u2227 EMetric.diam D \u2264 \u03b5\n[PROOFSTEP]\nhave : x \u2208 EMetric.ball x (\u03b5 / 2) := by\n apply EMetric.mem_ball_self\n rw [ENNReal.div_pos_iff]\n exact \u27e8ne_of_gt \u03b5_pos, by norm_num\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nx : \u03b1\nxC : x \u2208 C\n\u22a2 x \u2208 EMetric.ball x (\u03b5 / 2)\n[PROOFSTEP]\napply EMetric.mem_ball_self\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nx : \u03b1\nxC : x \u2208 C\n\u22a2 0 < \u03b5 / 2\n[PROOFSTEP]\nrw [ENNReal.div_pos_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nx : \u03b1\nxC : x \u2208 C\n\u22a2 \u03b5 \u2260 0 \u2227 2 \u2260 \u22a4\n[PROOFSTEP]\nexact \u27e8ne_of_gt \u03b5_pos, by norm_num\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nx : \u03b1\nxC : x \u2208 C\n\u22a2 2 \u2260 \u22a4\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nx : \u03b1\nxC : x \u2208 C\nthis : x \u2208 EMetric.ball x (\u03b5 / 2)\n\u22a2 let D := closure (EMetric.ball x (\u03b5 / 2) \u2229 C);\n Perfect D \u2227 Set.Nonempty D \u2227 D \u2286 C \u2227 EMetric.diam D \u2264 \u03b5\n[PROOFSTEP]\nhave := hC.closure_nhds_inter x xC this EMetric.isOpen_ball\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nx : \u03b1\nxC : x \u2208 C\nthis\u271d : x \u2208 EMetric.ball x (\u03b5 / 2)\nthis : Perfect (closure (EMetric.ball x (\u03b5 / 2) \u2229 C)) \u2227 Set.Nonempty (closure (EMetric.ball x (\u03b5 / 2) \u2229 C))\n\u22a2 let D := closure (EMetric.ball x (\u03b5 / 2) \u2229 C);\n Perfect D \u2227 Set.Nonempty D \u2227 D \u2286 C \u2227 EMetric.diam D \u2264 \u03b5\n[PROOFSTEP]\nrefine' \u27e8this.1, this.2, _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nx : \u03b1\nxC : x \u2208 C\nthis\u271d : x \u2208 EMetric.ball x (\u03b5 / 2)\nthis : Perfect (closure (EMetric.ball x (\u03b5 / 2) \u2229 C)) \u2227 Set.Nonempty (closure (EMetric.ball x (\u03b5 / 2) \u2229 C))\n\u22a2 closure (EMetric.ball x (\u03b5 / 2) \u2229 C) \u2286 C\n[PROOFSTEP]\nrw [IsClosed.closure_subset_iff hC.closed]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nx : \u03b1\nxC : x \u2208 C\nthis\u271d : x \u2208 EMetric.ball x (\u03b5 / 2)\nthis : Perfect (closure (EMetric.ball x (\u03b5 / 2) \u2229 C)) \u2227 Set.Nonempty (closure (EMetric.ball x (\u03b5 / 2) \u2229 C))\n\u22a2 EMetric.ball x (\u03b5 / 2) \u2229 C \u2286 C\n[PROOFSTEP]\napply inter_subset_right\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nx : \u03b1\nxC : x \u2208 C\nthis\u271d : x \u2208 EMetric.ball x (\u03b5 / 2)\nthis : Perfect (closure (EMetric.ball x (\u03b5 / 2) \u2229 C)) \u2227 Set.Nonempty (closure (EMetric.ball x (\u03b5 / 2) \u2229 C))\n\u22a2 EMetric.diam (closure (EMetric.ball x (\u03b5 / 2) \u2229 C)) \u2264 \u03b5\n[PROOFSTEP]\nrw [EMetric.diam_closure]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nx : \u03b1\nxC : x \u2208 C\nthis\u271d : x \u2208 EMetric.ball x (\u03b5 / 2)\nthis : Perfect (closure (EMetric.ball x (\u03b5 / 2) \u2229 C)) \u2227 Set.Nonempty (closure (EMetric.ball x (\u03b5 / 2) \u2229 C))\n\u22a2 EMetric.diam (EMetric.ball x (\u03b5 / 2) \u2229 C) \u2264 \u03b5\n[PROOFSTEP]\napply le_trans (EMetric.diam_mono (inter_subset_left _ _))\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nx : \u03b1\nxC : x \u2208 C\nthis\u271d : x \u2208 EMetric.ball x (\u03b5 / 2)\nthis : Perfect (closure (EMetric.ball x (\u03b5 / 2) \u2229 C)) \u2227 Set.Nonempty (closure (EMetric.ball x (\u03b5 / 2) \u2229 C))\n\u22a2 EMetric.diam (EMetric.ball x (\u03b5 / 2)) \u2264 \u03b5\n[PROOFSTEP]\nconvert EMetric.diam_ball (x := x)\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nx : \u03b1\nxC : x \u2208 C\nthis\u271d : x \u2208 EMetric.ball x (\u03b5 / 2)\nthis : Perfect (closure (EMetric.ball x (\u03b5 / 2) \u2229 C)) \u2227 Set.Nonempty (closure (EMetric.ball x (\u03b5 / 2) \u2229 C))\n\u22a2 \u03b5 = 2 * (\u03b5 / 2)\n[PROOFSTEP]\nrw [mul_comm, ENNReal.div_mul_cancel]\n[GOAL]\ncase h.e'_4.h0\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nx : \u03b1\nxC : x \u2208 C\nthis\u271d : x \u2208 EMetric.ball x (\u03b5 / 2)\nthis : Perfect (closure (EMetric.ball x (\u03b5 / 2) \u2229 C)) \u2227 Set.Nonempty (closure (EMetric.ball x (\u03b5 / 2) \u2229 C))\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_4.hI\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\n\u03b5_pos : 0 < \u03b5\nx : \u03b1\nxC : x \u2208 C\nthis\u271d : x \u2208 EMetric.ball x (\u03b5 / 2)\nthis : Perfect (closure (EMetric.ball x (\u03b5 / 2) \u2229 C)) \u2227 Set.Nonempty (closure (EMetric.ball x (\u03b5 / 2) \u2229 C))\n\u22a2 2 \u2260 \u22a4\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\n\u03b5_pos : 0 < \u03b5\n\u22a2 \u2203 C\u2080 C\u2081,\n (Perfect C\u2080 \u2227 Set.Nonempty C\u2080 \u2227 C\u2080 \u2286 C \u2227 EMetric.diam C\u2080 \u2264 \u03b5) \u2227\n (Perfect C\u2081 \u2227 Set.Nonempty C\u2081 \u2227 C\u2081 \u2286 C \u2227 EMetric.diam C\u2081 \u2264 \u03b5) \u2227 Disjoint C\u2080 C\u2081\n[PROOFSTEP]\nrcases hC.splitting hnonempty with \u27e8D\u2080, D\u2081, \u27e8perf0, non0, sub0\u27e9, \u27e8perf1, non1, sub1\u27e9, hdisj\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\n\u03b5_pos : 0 < \u03b5\nD\u2080 D\u2081 : Set \u03b1\nperf0 : Perfect D\u2080\nnon0 : Set.Nonempty D\u2080\nsub0 : D\u2080 \u2286 C\nhdisj : Disjoint D\u2080 D\u2081\nperf1 : Perfect D\u2081\nnon1 : Set.Nonempty D\u2081\nsub1 : D\u2081 \u2286 C\n\u22a2 \u2203 C\u2080 C\u2081,\n (Perfect C\u2080 \u2227 Set.Nonempty C\u2080 \u2227 C\u2080 \u2286 C \u2227 EMetric.diam C\u2080 \u2264 \u03b5) \u2227\n (Perfect C\u2081 \u2227 Set.Nonempty C\u2081 \u2227 C\u2081 \u2286 C \u2227 EMetric.diam C\u2081 \u2264 \u03b5) \u2227 Disjoint C\u2080 C\u2081\n[PROOFSTEP]\ncases' non0 with x\u2080 hx\u2080\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\n\u03b5_pos : 0 < \u03b5\nD\u2080 D\u2081 : Set \u03b1\nperf0 : Perfect D\u2080\nsub0 : D\u2080 \u2286 C\nhdisj : Disjoint D\u2080 D\u2081\nperf1 : Perfect D\u2081\nnon1 : Set.Nonempty D\u2081\nsub1 : D\u2081 \u2286 C\nx\u2080 : \u03b1\nhx\u2080 : x\u2080 \u2208 D\u2080\n\u22a2 \u2203 C\u2080 C\u2081,\n (Perfect C\u2080 \u2227 Set.Nonempty C\u2080 \u2227 C\u2080 \u2286 C \u2227 EMetric.diam C\u2080 \u2264 \u03b5) \u2227\n (Perfect C\u2081 \u2227 Set.Nonempty C\u2081 \u2227 C\u2081 \u2286 C \u2227 EMetric.diam C\u2081 \u2264 \u03b5) \u2227 Disjoint C\u2080 C\u2081\n[PROOFSTEP]\ncases' non1 with x\u2081 hx\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\n\u03b5_pos : 0 < \u03b5\nD\u2080 D\u2081 : Set \u03b1\nperf0 : Perfect D\u2080\nsub0 : D\u2080 \u2286 C\nhdisj : Disjoint D\u2080 D\u2081\nperf1 : Perfect D\u2081\nsub1 : D\u2081 \u2286 C\nx\u2080 : \u03b1\nhx\u2080 : x\u2080 \u2208 D\u2080\nx\u2081 : \u03b1\nhx\u2081 : x\u2081 \u2208 D\u2081\n\u22a2 \u2203 C\u2080 C\u2081,\n (Perfect C\u2080 \u2227 Set.Nonempty C\u2080 \u2227 C\u2080 \u2286 C \u2227 EMetric.diam C\u2080 \u2264 \u03b5) \u2227\n (Perfect C\u2081 \u2227 Set.Nonempty C\u2081 \u2227 C\u2081 \u2286 C \u2227 EMetric.diam C\u2081 \u2264 \u03b5) \u2227 Disjoint C\u2080 C\u2081\n[PROOFSTEP]\nrcases perf0.small_diam_aux \u03b5_pos hx\u2080 with \u27e8perf0', non0', sub0', diam0\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\n\u03b5_pos : 0 < \u03b5\nD\u2080 D\u2081 : Set \u03b1\nperf0 : Perfect D\u2080\nsub0 : D\u2080 \u2286 C\nhdisj : Disjoint D\u2080 D\u2081\nperf1 : Perfect D\u2081\nsub1 : D\u2081 \u2286 C\nx\u2080 : \u03b1\nhx\u2080 : x\u2080 \u2208 D\u2080\nx\u2081 : \u03b1\nhx\u2081 : x\u2081 \u2208 D\u2081\nperf0' : Perfect (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080))\nnon0' : Set.Nonempty (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080))\nsub0' : closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080) \u2286 D\u2080\ndiam0 : EMetric.diam (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080)) \u2264 \u03b5\n\u22a2 \u2203 C\u2080 C\u2081,\n (Perfect C\u2080 \u2227 Set.Nonempty C\u2080 \u2227 C\u2080 \u2286 C \u2227 EMetric.diam C\u2080 \u2264 \u03b5) \u2227\n (Perfect C\u2081 \u2227 Set.Nonempty C\u2081 \u2227 C\u2081 \u2286 C \u2227 EMetric.diam C\u2081 \u2264 \u03b5) \u2227 Disjoint C\u2080 C\u2081\n[PROOFSTEP]\nrcases perf1.small_diam_aux \u03b5_pos hx\u2081 with \u27e8perf1', non1', sub1', diam1\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\n\u03b5_pos : 0 < \u03b5\nD\u2080 D\u2081 : Set \u03b1\nperf0 : Perfect D\u2080\nsub0 : D\u2080 \u2286 C\nhdisj : Disjoint D\u2080 D\u2081\nperf1 : Perfect D\u2081\nsub1 : D\u2081 \u2286 C\nx\u2080 : \u03b1\nhx\u2080 : x\u2080 \u2208 D\u2080\nx\u2081 : \u03b1\nhx\u2081 : x\u2081 \u2208 D\u2081\nperf0' : Perfect (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080))\nnon0' : Set.Nonempty (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080))\nsub0' : closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080) \u2286 D\u2080\ndiam0 : EMetric.diam (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080)) \u2264 \u03b5\nperf1' : Perfect (closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081))\nnon1' : Set.Nonempty (closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081))\nsub1' : closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081) \u2286 D\u2081\ndiam1 : EMetric.diam (closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081)) \u2264 \u03b5\n\u22a2 \u2203 C\u2080 C\u2081,\n (Perfect C\u2080 \u2227 Set.Nonempty C\u2080 \u2227 C\u2080 \u2286 C \u2227 EMetric.diam C\u2080 \u2264 \u03b5) \u2227\n (Perfect C\u2081 \u2227 Set.Nonempty C\u2081 \u2227 C\u2081 \u2286 C \u2227 EMetric.diam C\u2081 \u2264 \u03b5) \u2227 Disjoint C\u2080 C\u2081\n[PROOFSTEP]\nrefine'\n \u27e8closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080), closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081),\n \u27e8perf0', non0', sub0'.trans sub0, diam0\u27e9, \u27e8perf1', non1', sub1'.trans sub1, diam1\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\n\u03b5_pos : 0 < \u03b5\nD\u2080 D\u2081 : Set \u03b1\nperf0 : Perfect D\u2080\nsub0 : D\u2080 \u2286 C\nhdisj : Disjoint D\u2080 D\u2081\nperf1 : Perfect D\u2081\nsub1 : D\u2081 \u2286 C\nx\u2080 : \u03b1\nhx\u2080 : x\u2080 \u2208 D\u2080\nx\u2081 : \u03b1\nhx\u2081 : x\u2081 \u2208 D\u2081\nperf0' : Perfect (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080))\nnon0' : Set.Nonempty (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080))\nsub0' : closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080) \u2286 D\u2080\ndiam0 : EMetric.diam (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080)) \u2264 \u03b5\nperf1' : Perfect (closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081))\nnon1' : Set.Nonempty (closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081))\nsub1' : closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081) \u2286 D\u2081\ndiam1 : EMetric.diam (closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081)) \u2264 \u03b5\n\u22a2 Disjoint (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080)) (closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081))\n[PROOFSTEP]\napply Disjoint.mono _ _ hdisj\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\n\u03b5_pos : 0 < \u03b5\nD\u2080 D\u2081 : Set \u03b1\nperf0 : Perfect D\u2080\nsub0 : D\u2080 \u2286 C\nhdisj : Disjoint D\u2080 D\u2081\nperf1 : Perfect D\u2081\nsub1 : D\u2081 \u2286 C\nx\u2080 : \u03b1\nhx\u2080 : x\u2080 \u2208 D\u2080\nx\u2081 : \u03b1\nhx\u2081 : x\u2081 \u2208 D\u2081\nperf0' : Perfect (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080))\nnon0' : Set.Nonempty (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080))\nsub0' : closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080) \u2286 D\u2080\ndiam0 : EMetric.diam (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080)) \u2264 \u03b5\nperf1' : Perfect (closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081))\nnon1' : Set.Nonempty (closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081))\nsub1' : closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081) \u2286 D\u2081\ndiam1 : EMetric.diam (closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081)) \u2264 \u03b5\n\u22a2 closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080) \u2264 D\u2080\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\n\u03b5_pos : 0 < \u03b5\nD\u2080 D\u2081 : Set \u03b1\nperf0 : Perfect D\u2080\nsub0 : D\u2080 \u2286 C\nhdisj : Disjoint D\u2080 D\u2081\nperf1 : Perfect D\u2081\nsub1 : D\u2081 \u2286 C\nx\u2080 : \u03b1\nhx\u2080 : x\u2080 \u2208 D\u2080\nx\u2081 : \u03b1\nhx\u2081 : x\u2081 \u2208 D\u2081\nperf0' : Perfect (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080))\nnon0' : Set.Nonempty (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080))\nsub0' : closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080) \u2286 D\u2080\ndiam0 : EMetric.diam (closure (EMetric.ball x\u2080 (\u03b5 / 2) \u2229 D\u2080)) \u2264 \u03b5\nperf1' : Perfect (closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081))\nnon1' : Set.Nonempty (closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081))\nsub1' : closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081) \u2286 D\u2081\ndiam1 : EMetric.diam (closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081)) \u2264 \u03b5\n\u22a2 closure (EMetric.ball x\u2081 (\u03b5 / 2) \u2229 D\u2081) \u2264 D\u2081\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\n\u22a2 \u2203 f, range f \u2286 C \u2227 Continuous f \u2227 Injective f\n[PROOFSTEP]\nobtain \u27e8u, -, upos', hu\u27e9 := exists_seq_strictAnti_tendsto' (zero_lt_one' \u211d\u22650\u221e)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\n\u22a2 \u2203 f, range f \u2286 C \u2227 Continuous f \u2227 Injective f\n[PROOFSTEP]\nhave upos := fun n => (upos' n).1\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\n\u22a2 \u2203 f, range f \u2286 C \u2227 Continuous f \u2227 Injective f\n[PROOFSTEP]\nlet P := Subtype fun E : Set \u03b1 => Perfect E \u2227 E.Nonempty\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\n\u22a2 \u2203 f, range f \u2286 C \u2227 Continuous f \u2227 Injective f\n[PROOFSTEP]\nchoose C0 C1 h0 h1 hdisj using fun {C : Set \u03b1} (hC : Perfect C) (hnonempty : C.Nonempty) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5) =>\n hC.small_diam_splitting hnonempty h\u03b5\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\n\u22a2 \u2203 f, range f \u2286 C \u2227 Continuous f \u2227 Injective f\n[PROOFSTEP]\nlet DP : List Bool \u2192 P := fun l => by\n induction' l with a l ih; \u00b7 exact \u27e8C, \u27e8hC, hnonempty\u27e9\u27e9\n cases a\n \u00b7 use C0 ih.property.1 ih.property.2 (upos l.length.succ)\n exact \u27e8(h0 _ _ _).1, (h0 _ _ _).2.1\u27e9\n use C1 ih.property.1 ih.property.2 (upos l.length.succ)\n exact \u27e8(h1 _ _ _).1, (h1 _ _ _).2.1\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nl : List Bool\n\u22a2 P\n[PROOFSTEP]\ninduction' l with a l ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\n\u22a2 P\n[PROOFSTEP]\nexact \u27e8C, \u27e8hC, hnonempty\u27e9\u27e9\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\na : Bool\nl : List Bool\nih : P\n\u22a2 P\n[PROOFSTEP]\ncases a\n[GOAL]\ncase cons.false\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nl : List Bool\nih : P\n\u22a2 P\n[PROOFSTEP]\nuse C0 ih.property.1 ih.property.2 (upos l.length.succ)\n[GOAL]\ncase property\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nl : List Bool\nih : P\n\u22a2 Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))\n[PROOFSTEP]\nexact \u27e8(h0 _ _ _).1, (h0 _ _ _).2.1\u27e9\n[GOAL]\ncase cons.true\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nl : List Bool\nih : P\n\u22a2 P\n[PROOFSTEP]\nuse C1 ih.property.1 ih.property.2 (upos l.length.succ)\n[GOAL]\ncase property\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nl : List Bool\nih : P\n\u22a2 Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))\n[PROOFSTEP]\nexact \u27e8(h1 _ _ _).1, (h1 _ _ _).2.1\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\n\u22a2 \u2203 f, range f \u2286 C \u2227 Continuous f \u2227 Injective f\n[PROOFSTEP]\nlet D : List Bool \u2192 Set \u03b1 := fun l => (DP l).val\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\n\u22a2 \u2203 f, range f \u2286 C \u2227 Continuous f \u2227 Injective f\n[PROOFSTEP]\nhave hanti : ClosureAntitone D :=\n by\n refine' Antitone.closureAntitone _ fun l => (DP l).property.1.closed\n intro l a\n cases a\n \u00b7 exact (h0 _ _ _).2.2.1\n exact (h1 _ _ _).2.2.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\n\u22a2 ClosureAntitone D\n[PROOFSTEP]\nrefine' Antitone.closureAntitone _ fun l => (DP l).property.1.closed\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\n\u22a2 CantorScheme.Antitone D\n[PROOFSTEP]\nintro l a\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nl : List Bool\na : Bool\n\u22a2 D (a :: l) \u2286 D l\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nl : List Bool\n\u22a2 D (false :: l) \u2286 D l\n[PROOFSTEP]\nexact (h0 _ _ _).2.2.1\n[GOAL]\ncase true\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nl : List Bool\n\u22a2 D (true :: l) \u2286 D l\n[PROOFSTEP]\nexact (h1 _ _ _).2.2.1\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\n\u22a2 \u2203 f, range f \u2286 C \u2227 Continuous f \u2227 Injective f\n[PROOFSTEP]\nhave hdiam : VanishingDiam D := by\n intro x\n apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hu\n \u00b7 simp\n rw [eventually_atTop]\n refine' \u27e81, fun m (hm : 1 \u2264 m) => _\u27e9\n rw [Nat.one_le_iff_ne_zero] at hm \n rcases Nat.exists_eq_succ_of_ne_zero hm with \u27e8n, rfl\u27e9\n dsimp\n cases x n\n \u00b7 convert (h0 _ _ _).2.2.2\n rw [PiNat.res_length]\n convert (h1 _ _ _).2.2.2\n rw [PiNat.res_length]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\n\u22a2 VanishingDiam D\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nx : \u2115 \u2192 Bool\n\u22a2 Tendsto (fun n => EMetric.diam (D (PiNat.res x n))) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\napply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hu\n[GOAL]\ncase hgf\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nx : \u2115 \u2192 Bool\n\u22a2 \u2200\u1da0 (b : \u2115) in atTop, 0 \u2264 EMetric.diam (D (PiNat.res x b))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hfh\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nx : \u2115 \u2192 Bool\n\u22a2 \u2200\u1da0 (b : \u2115) in atTop, EMetric.diam (D (PiNat.res x b)) \u2264 u b\n[PROOFSTEP]\nrw [eventually_atTop]\n[GOAL]\ncase hfh\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nx : \u2115 \u2192 Bool\n\u22a2 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 EMetric.diam (D (PiNat.res x b)) \u2264 u b\n[PROOFSTEP]\nrefine' \u27e81, fun m (hm : 1 \u2264 m) => _\u27e9\n[GOAL]\ncase hfh\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nx : \u2115 \u2192 Bool\nm : \u2115\nhm : 1 \u2264 m\n\u22a2 EMetric.diam (D (PiNat.res x m)) \u2264 u m\n[PROOFSTEP]\nrw [Nat.one_le_iff_ne_zero] at hm \n[GOAL]\ncase hfh\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nx : \u2115 \u2192 Bool\nm : \u2115\nhm : m \u2260 0\n\u22a2 EMetric.diam (D (PiNat.res x m)) \u2264 u m\n[PROOFSTEP]\nrcases Nat.exists_eq_succ_of_ne_zero hm with \u27e8n, rfl\u27e9\n[GOAL]\ncase hfh.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nx : \u2115 \u2192 Bool\nn : \u2115\nhm : Nat.succ n \u2260 0\n\u22a2 EMetric.diam (D (PiNat.res x (Nat.succ n))) \u2264 u (Nat.succ n)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase hfh.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nx : \u2115 \u2192 Bool\nn : \u2115\nhm : Nat.succ n \u2260 0\n\u22a2 EMetric.diam\n \u2191(Bool.rec (motive := fun t => x n = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = x n) \u25b8\n {\n val :=\n C0\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C0\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) \u2227\n Set.Nonempty\n (C0\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n (fun h =>\n (_ : true = x n) \u25b8\n {\n val :=\n C1\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C1\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) \u2227\n Set.Nonempty\n (C1\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n (x n) (_ : x n = x n)) \u2264\n u (Nat.succ n)\n[PROOFSTEP]\ncases x n\n[GOAL]\ncase hfh.intro.false\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nx : \u2115 \u2192 Bool\nn : \u2115\nhm : Nat.succ n \u2260 0\n\u22a2 EMetric.diam\n \u2191(Bool.rec (motive := fun t => false = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = false) \u25b8\n {\n val :=\n C0\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C0\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) \u2227\n Set.Nonempty\n (C0\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n (fun h =>\n (_ : true = false) \u25b8\n {\n val :=\n C1\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C1\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) \u2227\n Set.Nonempty\n (C1\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n false (_ : false = false)) \u2264\n u (Nat.succ n)\n[PROOFSTEP]\nconvert (h0 _ _ _).2.2.2\n[GOAL]\ncase h.e'_4.h.e'_1.h.e'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nx : \u2115 \u2192 Bool\nn : \u2115\nhm : Nat.succ n \u2260 0\n\u22a2 n = List.length (PiNat.res x n)\n[PROOFSTEP]\nrw [PiNat.res_length]\n[GOAL]\ncase hfh.intro.true\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nx : \u2115 \u2192 Bool\nn : \u2115\nhm : Nat.succ n \u2260 0\n\u22a2 EMetric.diam\n \u2191(Bool.rec (motive := fun t => true = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = true) \u25b8\n {\n val :=\n C0\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C0\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) \u2227\n Set.Nonempty\n (C0\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n (fun h =>\n (_ : true = true) \u25b8\n {\n val :=\n C1\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C1\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) \u2227\n Set.Nonempty\n (C1\n (_ :\n Perfect\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n \u2191(List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t \u2192 { E // Perfect E \u2227 Set.Nonempty E })\n (fun h =>\n (_ : false = a) \u25b8\n {\n val :=\n C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n {\n val :=\n C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n true (_ : true = true)) \u2264\n u (Nat.succ n)\n[PROOFSTEP]\nconvert (h1 _ _ _).2.2.2\n[GOAL]\ncase h.e'_4.h.e'_1.h.e'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nx : \u2115 \u2192 Bool\nn : \u2115\nhm : Nat.succ n \u2260 0\n\u22a2 n = List.length (PiNat.res x n)\n[PROOFSTEP]\nrw [PiNat.res_length]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\n\u22a2 \u2203 f, range f \u2286 C \u2227 Continuous f \u2227 Injective f\n[PROOFSTEP]\nhave hdisj' : CantorScheme.Disjoint D :=\n by\n rintro l (a | a) (b | b) hab <;> try contradiction\n \u00b7 exact hdisj _ _ _\n exact (hdisj _ _ _).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\n\u22a2 CantorScheme.Disjoint D\n[PROOFSTEP]\nrintro l (a | a) (b | b) hab\n[GOAL]\ncase false.false\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nl : List Bool\nhab : false \u2260 false\n\u22a2 Disjoint (D (false :: l)) (D (false :: l))\n[PROOFSTEP]\ntry contradiction\n[GOAL]\ncase false.false\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nl : List Bool\nhab : false \u2260 false\n\u22a2 Disjoint (D (false :: l)) (D (false :: l))\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase false.true\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nl : List Bool\nhab : false \u2260 true\n\u22a2 Disjoint (D (false :: l)) (D (true :: l))\n[PROOFSTEP]\ntry contradiction\n[GOAL]\ncase false.true\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nl : List Bool\nhab : false \u2260 true\n\u22a2 Disjoint (D (false :: l)) (D (true :: l))\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase true.false\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nl : List Bool\nhab : true \u2260 false\n\u22a2 Disjoint (D (true :: l)) (D (false :: l))\n[PROOFSTEP]\ntry contradiction\n[GOAL]\ncase true.false\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nl : List Bool\nhab : true \u2260 false\n\u22a2 Disjoint (D (true :: l)) (D (false :: l))\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase true.true\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nl : List Bool\nhab : true \u2260 true\n\u22a2 Disjoint (D (true :: l)) (D (true :: l))\n[PROOFSTEP]\ntry contradiction\n[GOAL]\ncase true.true\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nl : List Bool\nhab : true \u2260 true\n\u22a2 Disjoint (D (true :: l)) (D (true :: l))\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase false.true\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nl : List Bool\nhab : false \u2260 true\n\u22a2 Disjoint (D (false :: l)) (D (true :: l))\n[PROOFSTEP]\nexact hdisj _ _ _\n[GOAL]\ncase true.false\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nl : List Bool\nhab : true \u2260 false\n\u22a2 Disjoint (D (true :: l)) (D (false :: l))\n[PROOFSTEP]\nexact (hdisj _ _ _).symm\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\n\u22a2 \u2203 f, range f \u2286 C \u2227 Continuous f \u2227 Injective f\n[PROOFSTEP]\nhave hdom : \u2200 {x : \u2115 \u2192 Bool}, x \u2208 (inducedMap D).1 := fun {x} =>\n by\n rw [hanti.map_of_vanishingDiam hdiam fun l => (DP l).property.2]\n apply mem_univ\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nx : \u2115 \u2192 Bool\n\u22a2 x \u2208 (inducedMap D).fst\n[PROOFSTEP]\nrw [hanti.map_of_vanishingDiam hdiam fun l => (DP l).property.2]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nx : \u2115 \u2192 Bool\n\u22a2 x \u2208 univ\n[PROOFSTEP]\napply mem_univ\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : \u2200 {x : \u2115 \u2192 Bool}, x \u2208 (inducedMap D).fst\n\u22a2 \u2203 f, range f \u2286 C \u2227 Continuous f \u2227 Injective f\n[PROOFSTEP]\nrefine' \u27e8fun x => (inducedMap D).2 \u27e8x, hdom\u27e9, _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : \u2200 {x : \u2115 \u2192 Bool}, x \u2208 (inducedMap D).fst\n\u22a2 (range fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x \u2208 (inducedMap D).fst) }) \u2286 C\n[PROOFSTEP]\nrintro y \u27e8x, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : \u2200 {x : \u2115 \u2192 Bool}, x \u2208 (inducedMap D).fst\nx : \u2115 \u2192 Bool\n\u22a2 (fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x \u2208 (inducedMap D).fst) }) x \u2208 C\n[PROOFSTEP]\nexact map_mem \u27e8_, hdom\u27e9 0\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : \u2200 {x : \u2115 \u2192 Bool}, x \u2208 (inducedMap D).fst\n\u22a2 Continuous fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x \u2208 (inducedMap D).fst) }\n[PROOFSTEP]\napply hdiam.map_continuous.comp\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : \u2200 {x : \u2115 \u2192 Bool}, x \u2208 (inducedMap D).fst\n\u22a2 Continuous fun x => { val := x, property := (_ : x \u2208 (inducedMap D).fst) }\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : \u2200 {x : \u2115 \u2192 Bool}, x \u2208 (inducedMap D).fst\n\u22a2 Injective fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x \u2208 (inducedMap D).fst) }\n[PROOFSTEP]\nintro x y hxy\n[GOAL]\ncase intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\nC : Set \u03b1\nhC : Perfect C\n\u03b5 : \u211d\u22650\u221e\nhnonempty : Set.Nonempty C\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 \u211d\u22650\u221e\nupos' : \u2200 (n : \u2115), u n \u2208 Ioo 0 1\nhu : Tendsto u atTop (\ud835\udcdd 0)\nupos : \u2200 (n : \u2115), 0 < u n\nP : Type u_1 := { E // Perfect E \u2227 Set.Nonempty E }\nC0 C1 : {C : Set \u03b1} \u2192 Perfect C \u2192 Set.Nonempty C \u2192 {\u03b5 : \u211d\u22650\u221e} \u2192 0 < \u03b5 \u2192 Set \u03b1\nh0 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C0 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C0 hC hnonempty h\u03b5) \u2227 C0 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C0 hC hnonempty h\u03b5) \u2264 \u03b5\nh1 :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Perfect (C1 hC hnonempty h\u03b5) \u2227\n Set.Nonempty (C1 hC hnonempty h\u03b5) \u2227 C1 hC hnonempty h\u03b5 \u2286 C \u2227 EMetric.diam (C1 hC hnonempty h\u03b5) \u2264 \u03b5\nhdisj :\n \u2200 {C : Set \u03b1} (hC : Perfect C) (hnonempty : Set.Nonempty C) {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : 0 < \u03b5),\n Disjoint (C0 hC hnonempty h\u03b5) (C1 hC hnonempty h\u03b5)\nDP : List Bool \u2192 P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C \u2227 Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t \u2192 P) a\n (fun h =>\n (_ : false = a) \u25b8\n { val := C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C0 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) \u25b8\n { val := C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l)))) \u2227\n Set.Nonempty\n (C1 (_ : Perfect \u2191ih) (_ : Set.Nonempty \u2191ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool \u2192 Set \u03b1 := fun l => \u2191(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : \u2200 {x : \u2115 \u2192 Bool}, x \u2208 (inducedMap D).fst\nx y : \u2115 \u2192 Bool\nhxy :\n (fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x \u2208 (inducedMap D).fst) }) x =\n (fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x \u2208 (inducedMap D).fst) }) y\n\u22a2 x = y\n[PROOFSTEP]\nsimpa only [\u2190 Subtype.val_inj] using hdisj'.map_injective hxy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\nC : Set \u03b1\nhC : IsClosed C\nhunc : \u00acSet.Countable C\n\u22a2 \u2203 f, range f \u2286 C \u2227 Continuous f \u2227 Function.Injective f\n[PROOFSTEP]\nletI := upgradePolishSpace \u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\nC : Set \u03b1\nhC : IsClosed C\nhunc : \u00acSet.Countable C\nthis : UpgradedPolishSpace \u03b1 := upgradePolishSpace \u03b1\n\u22a2 \u2203 f, range f \u2286 C \u2227 Continuous f \u2227 Function.Injective f\n[PROOFSTEP]\nobtain \u27e8D, hD, Dnonempty, hDC\u27e9 := exists_perfect_nonempty_of_isClosed_of_not_countable hC hunc\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\nC : Set \u03b1\nhC : IsClosed C\nhunc : \u00acSet.Countable C\nthis : UpgradedPolishSpace \u03b1 := upgradePolishSpace \u03b1\nD : Set \u03b1\nhD : Perfect D\nDnonempty : Set.Nonempty D\nhDC : D \u2286 C\n\u22a2 \u2203 f, range f \u2286 C \u2227 Continuous f \u2227 Function.Injective f\n[PROOFSTEP]\nobtain \u27e8f, hfD, hf\u27e9 := hD.exists_nat_bool_injection Dnonempty\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\nC : Set \u03b1\nhC : IsClosed C\nhunc : \u00acSet.Countable C\nthis : UpgradedPolishSpace \u03b1 := upgradePolishSpace \u03b1\nD : Set \u03b1\nhD : Perfect D\nDnonempty : Set.Nonempty D\nhDC : D \u2286 C\nf : (\u2115 \u2192 Bool) \u2192 \u03b1\nhfD : range f \u2286 D\nhf : Continuous f \u2227 Function.Injective f\n\u22a2 \u2203 f, range f \u2286 C \u2227 Continuous f \u2227 Function.Injective f\n[PROOFSTEP]\nexact \u27e8f, hfD.trans hDC, hf\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.Perfect", "llama_tokens": 92588, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389930307512, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.40505146345970056}} {"text": "[GOAL]\nR : Type u\ninst\u271d : Ring R\nM N : ModuleCat R\nf : M \u27f6 N\nhf : Mono f\n\u22a2 (LinearEquiv.toModuleIso'\n (LinearEquiv.trans (LinearEquiv.symm (Submodule.quotEquivOfEqBot (LinearMap.ker f) (_ : LinearMap.ker f = \u22a5)))\n (LinearEquiv.trans (LinearMap.quotKerEquivRange f)\n (LinearEquiv.ofEq (LinearMap.range f) (LinearMap.ker (Submodule.mkQ (LinearMap.range f)))\n (_ : LinearMap.range f = LinearMap.ker (Submodule.mkQ (LinearMap.range f))))))).hom \u226b\n Fork.\u03b9 (kernelCone (Submodule.mkQ (LinearMap.range f))) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\ninst\u271d : Ring R\nM N : ModuleCat R\nf : M \u27f6 N\nhf : Mono f\nx\u271d : \u2191M\n\u22a2 \u2191((LinearEquiv.toModuleIso'\n (LinearEquiv.trans\n (LinearEquiv.symm (Submodule.quotEquivOfEqBot (LinearMap.ker f) (_ : LinearMap.ker f = \u22a5)))\n (LinearEquiv.trans (LinearMap.quotKerEquivRange f)\n (LinearEquiv.ofEq (LinearMap.range f) (LinearMap.ker (Submodule.mkQ (LinearMap.range f)))\n (_ : LinearMap.range f = LinearMap.ker (Submodule.mkQ (LinearMap.range f))))))).hom \u226b\n Fork.\u03b9 (kernelCone (Submodule.mkQ (LinearMap.range f))))\n x\u271d =\n \u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nM N : ModuleCat R\nf : M \u27f6 N\nhf : Epi f\n\u22a2 Cofork.\u03c0 (cokernelCocone (Submodule.subtype (LinearMap.ker f))) \u226b\n (LinearEquiv.toModuleIso'\n (LinearEquiv.trans\n (LinearEquiv.trans\n (Submodule.quotEquivOfEq (LinearMap.range (Submodule.subtype (LinearMap.ker f))) (LinearMap.ker f)\n (_ : LinearMap.range (Submodule.subtype (LinearMap.ker f)) = LinearMap.ker f))\n (LinearMap.quotKerEquivRange f))\n (LinearEquiv.ofTop (LinearMap.range f) (_ : LinearMap.range f = \u22a4)))).hom =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\ninst\u271d : Ring R\nM N : ModuleCat R\nf : M \u27f6 N\nhf : Epi f\nx\u271d : \u2191((parallelPair (Submodule.subtype (LinearMap.ker f)) 0).obj WalkingParallelPair.one)\n\u22a2 \u2191(Cofork.\u03c0 (cokernelCocone (Submodule.subtype (LinearMap.ker f))) \u226b\n (LinearEquiv.toModuleIso'\n (LinearEquiv.trans\n (LinearEquiv.trans\n (Submodule.quotEquivOfEq (LinearMap.range (Submodule.subtype (LinearMap.ker f))) (LinearMap.ker f)\n (_ : LinearMap.range (Submodule.subtype (LinearMap.ker f)) = LinearMap.ker f))\n (LinearMap.quotKerEquivRange f))\n (LinearEquiv.ofTop (LinearMap.range f) (_ : LinearMap.range f = \u22a4)))).hom)\n x\u271d =\n \u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nM N : ModuleCat R\nf : M \u27f6 N\nO : ModuleCat R\ng : N \u27f6 O\n\u22a2 Exact f g \u2194 range f = ker g\n[PROOFSTEP]\nrw [abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nM N : ModuleCat R\nf : M \u27f6 N\nO : ModuleCat R\ng : N \u27f6 O\n\u22a2 f \u226b g = 0 \u2227 Fork.\u03b9 (kernelCone g) \u226b Cofork.\u03c0 (cokernelCocone f) = 0 \u2194 range f = ker g\n[PROOFSTEP]\nexact\n \u27e8fun h => le_antisymm (range_le_ker_iff.2 h.1) (ker_le_range_iff.2 h.2), fun h =>\n \u27e8range_le_ker_iff.1 <| le_of_eq h, ker_le_range_iff.1 <| le_of_eq h.symm\u27e9\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.ModuleCat.Abelian", "llama_tokens": 1234, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389817407016, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.40505145733074716}} {"text": "[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2076 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2075 : CommMonoid \u03b1\ninst\u271d\u2074 : CommMonoid \u03b2\ninst\u271d\u00b3 : CommMonoid \u03b3\ninst\u271d\u00b2 : CommMonoid \u03b4\ninst\u271d\u00b9 : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\ninst\u271d : FreimanHomClass F A \u03b2 2\nf : F\nha : a \u2208 A\nhb : b \u2208 A\nhc : c \u2208 A\nhd : d \u2208 A\nh : a * b = c * d\n\u22a2 \u2191f a * \u2191f b = \u2191f c * \u2191f d\n[PROOFSTEP]\nsimp_rw [\u2190 prod_pair] at h \u22a2\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2076 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2075 : CommMonoid \u03b1\ninst\u271d\u2074 : CommMonoid \u03b2\ninst\u271d\u00b3 : CommMonoid \u03b3\ninst\u271d\u00b2 : CommMonoid \u03b4\ninst\u271d\u00b9 : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\ninst\u271d : FreimanHomClass F A \u03b2 2\nf : F\nha : a \u2208 A\nhb : b \u2208 A\nhc : c \u2208 A\nhd : d \u2208 A\nh : prod {a, b} = prod {c, d}\n\u22a2 prod {\u2191f a, \u2191f b} = prod {\u2191f c, \u2191f d}\n[PROOFSTEP]\nrefine' map_prod_eq_map_prod f _ _ (card_pair _ _) (card_pair _ _) h\n[GOAL]\ncase refine'_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2076 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2075 : CommMonoid \u03b1\ninst\u271d\u2074 : CommMonoid \u03b2\ninst\u271d\u00b3 : CommMonoid \u03b3\ninst\u271d\u00b2 : CommMonoid \u03b4\ninst\u271d\u00b9 : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\ninst\u271d : FreimanHomClass F A \u03b2 2\nf : F\nha : a \u2208 A\nhb : b \u2208 A\nhc : c \u2208 A\nhd : d \u2208 A\nh : prod {a, b} = prod {c, d}\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 {a, b} \u2192 x \u2208 A\n[PROOFSTEP]\nsimp [ha, hb, hc, hd]\n[GOAL]\ncase refine'_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2076 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2075 : CommMonoid \u03b1\ninst\u271d\u2074 : CommMonoid \u03b2\ninst\u271d\u00b3 : CommMonoid \u03b3\ninst\u271d\u00b2 : CommMonoid \u03b4\ninst\u271d\u00b9 : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\ninst\u271d : FreimanHomClass F A \u03b2 2\nf : F\nha : a \u2208 A\nhb : b \u2208 A\nhc : c \u2208 A\nhd : d \u2208 A\nh : prod {a, b} = prod {c, d}\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 {c, d} \u2192 x \u2208 A\n[PROOFSTEP]\nsimp [ha, hb, hc, hd]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\nf g : A \u2192*[n] \u03b2\nh : f.toFun = g.toFun\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\ng : A \u2192*[n] \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_prod_eq_map_prod'\u271d :\n \u2200 {s t : Multiset \u03b1},\n (\u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 x \u2208 A) \u2192\n (\u2200 \u2983x : \u03b1\u2984, x \u2208 t \u2192 x \u2208 A) \u2192\n \u2191card s = n \u2192 \u2191card t = n \u2192 prod s = prod t \u2192 prod (map toFun\u271d s) = prod (map toFun\u271d t)\nh : { toFun := toFun\u271d, map_prod_eq_map_prod' := map_prod_eq_map_prod'\u271d }.toFun = g.toFun\n\u22a2 { toFun := toFun\u271d, map_prod_eq_map_prod' := map_prod_eq_map_prod'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmap_prod_eq_map_prod'\u271d\u00b9 :\n \u2200 {s t : Multiset \u03b1},\n (\u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 x \u2208 A) \u2192\n (\u2200 \u2983x : \u03b1\u2984, x \u2208 t \u2192 x \u2208 A) \u2192\n \u2191card s = n \u2192 \u2191card t = n \u2192 prod s = prod t \u2192 prod (map toFun\u271d\u00b9 s) = prod (map toFun\u271d\u00b9 t)\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_prod_eq_map_prod'\u271d :\n \u2200 {s t : Multiset \u03b1},\n (\u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 x \u2208 A) \u2192\n (\u2200 \u2983x : \u03b1\u2984, x \u2208 t \u2192 x \u2208 A) \u2192\n \u2191card s = n \u2192 \u2191card t = n \u2192 prod s = prod t \u2192 prod (map toFun\u271d s) = prod (map toFun\u271d t)\nh :\n { toFun := toFun\u271d\u00b9, map_prod_eq_map_prod' := map_prod_eq_map_prod'\u271d\u00b9 }.toFun =\n { toFun := toFun\u271d, map_prod_eq_map_prod' := map_prod_eq_map_prod'\u271d }.toFun\n\u22a2 { toFun := toFun\u271d\u00b9, map_prod_eq_map_prod' := map_prod_eq_map_prod'\u271d\u00b9 } =\n { toFun := toFun\u271d, map_prod_eq_map_prod' := map_prod_eq_map_prod'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA\u271d : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn\u271d : \u2115\na b c d : \u03b1\nA : Set \u03b1\nn : \u2115\ns\u271d t\u271d : Multiset \u03b1\nx\u271d\u00b3 : \u2200 \u2983x : \u03b1\u2984, x \u2208 s\u271d \u2192 x \u2208 A\nx\u271d\u00b2 : \u2200 \u2983x : \u03b1\u2984, x \u2208 t\u271d \u2192 x \u2208 A\nx\u271d\u00b9 : \u2191card s\u271d = n\nx\u271d : \u2191card t\u271d = n\nh : prod s\u271d = prod t\u271d\n\u22a2 prod (map (fun x => x) s\u271d) = prod (map (fun x => x) t\u271d)\n[PROOFSTEP]\nrw [map_id', map_id', h]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\nf : B \u2192*[n] \u03b3\ng : A \u2192*[n] \u03b2\nhAB : Set.MapsTo (\u2191g) A B\ns\u271d t\u271d : Multiset \u03b1\nhsA : \u2200 \u2983x : \u03b1\u2984, x \u2208 s\u271d \u2192 x \u2208 A\nhtA : \u2200 \u2983x : \u03b1\u2984, x \u2208 t\u271d \u2192 x \u2208 A\nhs : \u2191card s\u271d = n\nht : \u2191card t\u271d = n\nh : prod s\u271d = prod t\u271d\n\u22a2 prod (map (\u2191f \u2218 \u2191g) s\u271d) = prod (map (\u2191f \u2218 \u2191g) t\u271d)\n[PROOFSTEP]\nrw [\u2190 map_map, \u2190 map_map]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\nf : B \u2192*[n] \u03b3\ng : A \u2192*[n] \u03b2\nhAB : Set.MapsTo (\u2191g) A B\ns\u271d t\u271d : Multiset \u03b1\nhsA : \u2200 \u2983x : \u03b1\u2984, x \u2208 s\u271d \u2192 x \u2208 A\nhtA : \u2200 \u2983x : \u03b1\u2984, x \u2208 t\u271d \u2192 x \u2208 A\nhs : \u2191card s\u271d = n\nht : \u2191card t\u271d = n\nh : prod s\u271d = prod t\u271d\n\u22a2 prod (map (\u2191f) (map (\u2191g) s\u271d)) = prod (map (\u2191f) (map (\u2191g) t\u271d))\n[PROOFSTEP]\napply map_prod_eq_map_prod f _ _ ((card_map _ _).trans hs)\n[GOAL]\ncase ht\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\nf : B \u2192*[n] \u03b3\ng : A \u2192*[n] \u03b2\nhAB : Set.MapsTo (\u2191g) A B\ns\u271d t\u271d : Multiset \u03b1\nhsA : \u2200 \u2983x : \u03b1\u2984, x \u2208 s\u271d \u2192 x \u2208 A\nhtA : \u2200 \u2983x : \u03b1\u2984, x \u2208 t\u271d \u2192 x \u2208 A\nhs : \u2191card s\u271d = n\nht : \u2191card t\u271d = n\nh : prod s\u271d = prod t\u271d\n\u22a2 \u2191card (map (\u2191g) t\u271d) = n\n[PROOFSTEP]\nrwa [card_map]\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\nf : B \u2192*[n] \u03b3\ng : A \u2192*[n] \u03b2\nhAB : Set.MapsTo (\u2191g) A B\ns\u271d t\u271d : Multiset \u03b1\nhsA : \u2200 \u2983x : \u03b1\u2984, x \u2208 s\u271d \u2192 x \u2208 A\nhtA : \u2200 \u2983x : \u03b1\u2984, x \u2208 t\u271d \u2192 x \u2208 A\nhs : \u2191card s\u271d = n\nht : \u2191card t\u271d = n\nh : prod s\u271d = prod t\u271d\n\u22a2 prod (map (\u2191g) s\u271d) = prod (map (\u2191g) t\u271d)\n[PROOFSTEP]\napply (map_prod_eq_map_prod g hsA htA hs ht h)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\nf : B \u2192*[n] \u03b3\ng : A \u2192*[n] \u03b2\nhAB : Set.MapsTo (\u2191g) A B\ns\u271d t\u271d : Multiset \u03b1\nhsA : \u2200 \u2983x : \u03b1\u2984, x \u2208 s\u271d \u2192 x \u2208 A\nhtA : \u2200 \u2983x : \u03b1\u2984, x \u2208 t\u271d \u2192 x \u2208 A\nhs : \u2191card s\u271d = n\nht : \u2191card t\u271d = n\nh : prod s\u271d = prod t\u271d\n\u22a2 \u2200 \u2983x : \u03b2\u2984, x \u2208 map (\u2191g) s\u271d \u2192 x \u2208 B\n[PROOFSTEP]\nsimpa using fun a h => hAB (hsA h)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\nf : B \u2192*[n] \u03b3\ng : A \u2192*[n] \u03b2\nhAB : Set.MapsTo (\u2191g) A B\ns\u271d t\u271d : Multiset \u03b1\nhsA : \u2200 \u2983x : \u03b1\u2984, x \u2208 s\u271d \u2192 x \u2208 A\nhtA : \u2200 \u2983x : \u03b1\u2984, x \u2208 t\u271d \u2192 x \u2208 A\nhs : \u2191card s\u271d = n\nht : \u2191card t\u271d = n\nh : prod s\u271d = prod t\u271d\n\u22a2 \u2200 \u2983x : \u03b2\u2984, x \u2208 map (\u2191g) t\u271d \u2192 x \u2208 B\n[PROOFSTEP]\nsimpa using fun a h => hAB (htA h)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\ng\u2081 g\u2082 : B \u2192*[n] \u03b3\nf : A \u2192*[n] \u03b2\nhf : Set.SurjOn (\u2191f) A B\nhf' : Set.MapsTo (\u2191f) A B\n\u22a2 Set.EqOn (\u2191(FreimanHom.comp g\u2081 f hf')) (\u2191(FreimanHom.comp g\u2082 f hf')) A \u2194 Set.EqOn (\u2191g\u2081) (\u2191g\u2082) B\n[PROOFSTEP]\nsimp [hf.cancel_right hf']\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\ng : B \u2192*[n] \u03b3\nf\u2081 f\u2082 : A \u2192*[n] \u03b2\nhg : Set.InjOn (\u2191g) B\nhf\u2081 : Set.MapsTo (\u2191f\u2081) A B\nhf\u2082 : Set.MapsTo (\u2191f\u2082) A B\n\u22a2 Set.EqOn (\u2191(FreimanHom.comp g f\u2081 hf\u2081)) (\u2191(FreimanHom.comp g f\u2082 hf\u2082)) A \u2194 Set.EqOn (\u2191f\u2081) (\u2191f\u2082) A\n[PROOFSTEP]\nsimp [hg.cancel_left hf\u2081 hf\u2082]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA\u271d : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn\u271d : \u2115\na b\u271d c d : \u03b1\nA : Set \u03b1\nn : \u2115\nb : \u03b2\ns\u271d t\u271d : Multiset \u03b1\nx\u271d\u00b2 : \u2200 \u2983x : \u03b1\u2984, x \u2208 s\u271d \u2192 x \u2208 A\nx\u271d\u00b9 : \u2200 \u2983x : \u03b1\u2984, x \u2208 t\u271d \u2192 x \u2208 A\nhs : \u2191card s\u271d = n\nht : \u2191card t\u271d = n\nx\u271d : prod s\u271d = prod t\u271d\n\u22a2 prod (map (fun x => b) s\u271d) = prod (map (fun x => b) t\u271d)\n[PROOFSTEP]\nsimp only [map_const', hs, prod_replicate, ht]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\nf g : A \u2192*[n] \u03b2\ns\u271d t\u271d : Multiset \u03b1\nhsA : \u2200 \u2983x : \u03b1\u2984, x \u2208 s\u271d \u2192 x \u2208 A\nhtA : \u2200 \u2983x : \u03b1\u2984, x \u2208 t\u271d \u2192 x \u2208 A\nhs : \u2191card s\u271d = n\nht : \u2191card t\u271d = n\nh : prod s\u271d = prod t\u271d\n\u22a2 prod (map (fun x => \u2191f x * \u2191g x) s\u271d) = prod (map (fun x => \u2191f x * \u2191g x) t\u271d)\n[PROOFSTEP]\nrw [prod_map_mul, prod_map_mul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\nf g : A \u2192*[n] \u03b2\ns\u271d t\u271d : Multiset \u03b1\nhsA : \u2200 \u2983x : \u03b1\u2984, x \u2208 s\u271d \u2192 x \u2208 A\nhtA : \u2200 \u2983x : \u03b1\u2984, x \u2208 t\u271d \u2192 x \u2208 A\nhs : \u2191card s\u271d = n\nht : \u2191card t\u271d = n\nh : prod s\u271d = prod t\u271d\n\u22a2 prod (map (fun x => \u2191f x) s\u271d) * prod (map (fun x => \u2191g x) s\u271d) =\n prod (map (fun x => \u2191f x) t\u271d) * prod (map (fun x => \u2191g x) t\u271d)\n[PROOFSTEP]\nrw [map_prod_eq_map_prod f hsA htA hs ht h]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\nf g : A \u2192*[n] \u03b2\ns\u271d t\u271d : Multiset \u03b1\nhsA : \u2200 \u2983x : \u03b1\u2984, x \u2208 s\u271d \u2192 x \u2208 A\nhtA : \u2200 \u2983x : \u03b1\u2984, x \u2208 t\u271d \u2192 x \u2208 A\nhs : \u2191card s\u271d = n\nht : \u2191card t\u271d = n\nh : prod s\u271d = prod t\u271d\n\u22a2 prod (map (\u2191f) t\u271d) * prod (map (fun x => \u2191g x) s\u271d) = prod (map (fun x => \u2191f x) t\u271d) * prod (map (fun x => \u2191g x) t\u271d)\n[PROOFSTEP]\nrw [map_prod_eq_map_prod g hsA htA hs ht h]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\nf : A \u2192*[n] G\ns\u271d t\u271d : Multiset \u03b1\nhsA : \u2200 \u2983x : \u03b1\u2984, x \u2208 s\u271d \u2192 x \u2208 A\nhtA : \u2200 \u2983x : \u03b1\u2984, x \u2208 t\u271d \u2192 x \u2208 A\nhs : \u2191card s\u271d = n\nht : \u2191card t\u271d = n\nh : prod s\u271d = prod t\u271d\n\u22a2 prod (map (fun x => (\u2191f x)\u207b\u00b9) s\u271d) = prod (map (fun x => (\u2191f x)\u207b\u00b9) t\u271d)\n[PROOFSTEP]\nrw [prod_map_inv, prod_map_inv, map_prod_eq_map_prod f hsA htA hs ht h]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\nf g : A \u2192*[n] G\ns\u271d t\u271d : Multiset \u03b1\nhsA : \u2200 \u2983x : \u03b1\u2984, x \u2208 s\u271d \u2192 x \u2208 A\nhtA : \u2200 \u2983x : \u03b1\u2984, x \u2208 t\u271d \u2192 x \u2208 A\nhs : \u2191card s\u271d = n\nht : \u2191card t\u271d = n\nh : prod s\u271d = prod t\u271d\n\u22a2 prod (map (fun x => \u2191f x / \u2191g x) s\u271d) = prod (map (fun x => \u2191f x / \u2191g x) t\u271d)\n[PROOFSTEP]\nrw [prod_map_div, prod_map_div, map_prod_eq_map_prod f hsA htA hs ht h, map_prod_eq_map_prod g hsA htA hs ht h]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na\u271d b\u271d c\u271d d : \u03b1\na b c : A \u2192*[n] \u03b2\n\u22a2 a * b * c = a * (b * c)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na\u271d b\u271d c\u271d d : \u03b1\na b c : A \u2192*[n] \u03b2\nx\u271d : \u03b1\n\u22a2 \u2191(a * b * c) x\u271d = \u2191(a * (b * c)) x\u271d\n[PROOFSTEP]\napply mul_assoc\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na\u271d b c d : \u03b1\na : A \u2192*[n] \u03b2\n\u22a2 1 * a = a\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na\u271d b c d : \u03b1\na : A \u2192*[n] \u03b2\nx\u271d : \u03b1\n\u22a2 \u2191(1 * a) x\u271d = \u2191a x\u271d\n[PROOFSTEP]\napply one_mul\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na\u271d b c d : \u03b1\na : A \u2192*[n] \u03b2\n\u22a2 a * 1 = a\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na\u271d b c d : \u03b1\na : A \u2192*[n] \u03b2\nx\u271d : \u03b1\n\u22a2 \u2191(a * 1) x\u271d = \u2191a x\u271d\n[PROOFSTEP]\napply mul_one\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na\u271d b\u271d c d : \u03b1\na b : A \u2192*[n] \u03b2\n\u22a2 a * b = b * a\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na\u271d b\u271d c d : \u03b1\na b : A \u2192*[n] \u03b2\nx\u271d : \u03b1\n\u22a2 \u2191(a * b) x\u271d = \u2191(b * a) x\u271d\n[PROOFSTEP]\napply mul_comm\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2\u271d : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2076 : FunLike F \u03b1 fun x => \u03b2\u271d\ninst\u271d\u2075 : CommMonoid \u03b1\ninst\u271d\u2074 : CommMonoid \u03b2\u271d\ninst\u271d\u00b3 : CommMonoid \u03b3\ninst\u271d\u00b2 : CommMonoid \u03b4\ninst\u271d\u00b9 : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\u271d\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\n\u03b2 : Type ?u.68728\ninst\u271d : CommGroup \u03b2\nsrc\u271d : CommMonoid (A \u2192*[n] \u03b2) := commMonoid\n\u22a2 \u2200 (a b : A \u2192*[n] \u03b2), a / b = a * b\u207b\u00b9\n[PROOFSTEP]\nintros\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2\u271d : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2076 : FunLike F \u03b1 fun x => \u03b2\u271d\ninst\u271d\u2075 : CommMonoid \u03b1\ninst\u271d\u2074 : CommMonoid \u03b2\u271d\ninst\u271d\u00b3 : CommMonoid \u03b3\ninst\u271d\u00b2 : CommMonoid \u03b4\ninst\u271d\u00b9 : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\u271d\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\n\u03b2 : Type ?u.68728\ninst\u271d : CommGroup \u03b2\nsrc\u271d : CommMonoid (A \u2192*[n] \u03b2) := commMonoid\na\u271d b\u271d : A \u2192*[n] \u03b2\n\u22a2 a\u271d / b\u271d = a\u271d * b\u271d\u207b\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2\u271d : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2076 : FunLike F \u03b1 fun x => \u03b2\u271d\ninst\u271d\u2075 : CommMonoid \u03b1\ninst\u271d\u2074 : CommMonoid \u03b2\u271d\ninst\u271d\u00b3 : CommMonoid \u03b3\ninst\u271d\u00b2 : CommMonoid \u03b4\ninst\u271d\u00b9 : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\u271d\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\n\u03b2 : Type ?u.68728\ninst\u271d : CommGroup \u03b2\nsrc\u271d : CommMonoid (A \u2192*[n] \u03b2) := commMonoid\na\u271d b\u271d : A \u2192*[n] \u03b2\nx\u271d : \u03b1\n\u22a2 \u2191(a\u271d / b\u271d) x\u271d = \u2191(a\u271d * b\u271d\u207b\u00b9) x\u271d\n[PROOFSTEP]\napply div_eq_mul_inv\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2\u271d : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2076 : FunLike F \u03b1 fun x => \u03b2\u271d\ninst\u271d\u2075 : CommMonoid \u03b1\ninst\u271d\u2074 : CommMonoid \u03b2\u271d\ninst\u271d\u00b3 : CommMonoid \u03b3\ninst\u271d\u00b2 : CommMonoid \u03b4\ninst\u271d\u00b9 : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\u271d\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\n\u03b2 : Type ?u.68728\ninst\u271d : CommGroup \u03b2\nsrc\u271d : CommMonoid (A \u2192*[n] \u03b2) := commMonoid\n\u22a2 \u2200 (a : A \u2192*[n] \u03b2), a\u207b\u00b9 * a = 1\n[PROOFSTEP]\nintros\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2\u271d : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2076 : FunLike F \u03b1 fun x => \u03b2\u271d\ninst\u271d\u2075 : CommMonoid \u03b1\ninst\u271d\u2074 : CommMonoid \u03b2\u271d\ninst\u271d\u00b3 : CommMonoid \u03b3\ninst\u271d\u00b2 : CommMonoid \u03b4\ninst\u271d\u00b9 : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\u271d\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\n\u03b2 : Type ?u.68728\ninst\u271d : CommGroup \u03b2\nsrc\u271d : CommMonoid (A \u2192*[n] \u03b2) := commMonoid\na\u271d : A \u2192*[n] \u03b2\n\u22a2 a\u271d\u207b\u00b9 * a\u271d = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2\u271d : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2076 : FunLike F \u03b1 fun x => \u03b2\u271d\ninst\u271d\u2075 : CommMonoid \u03b1\ninst\u271d\u2074 : CommMonoid \u03b2\u271d\ninst\u271d\u00b3 : CommMonoid \u03b3\ninst\u271d\u00b2 : CommMonoid \u03b4\ninst\u271d\u00b9 : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\u271d\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\n\u03b2 : Type ?u.68728\ninst\u271d : CommGroup \u03b2\nsrc\u271d : CommMonoid (A \u2192*[n] \u03b2) := commMonoid\na\u271d : A \u2192*[n] \u03b2\nx\u271d : \u03b1\n\u22a2 \u2191(a\u271d\u207b\u00b9 * a\u271d) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\napply mul_left_inv\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\nf : \u03b1 \u2192* \u03b2\ns t : Multiset \u03b1\nx\u271d\u00b3 : \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 x \u2208 Set.univ\nx\u271d\u00b2 : \u2200 \u2983x : \u03b1\u2984, x \u2208 t \u2192 x \u2208 Set.univ\nx\u271d\u00b9 : \u2191card s = n\nx\u271d : \u2191card t = n\nh : prod s = prod t\n\u22a2 prod (map (\u2191f) s) = prod (map (\u2191f) t)\n[PROOFSTEP]\nrw [\u2190 f.map_multiset_prod, h, f.map_multiset_prod]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u2075 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u2074 : CommMonoid \u03b1\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b3\ninst\u271d\u00b9 : CommMonoid \u03b4\ninst\u271d : CommGroup G\nA : Set \u03b1\nB : Set \u03b2\nC : Set \u03b3\nn : \u2115\na b c d : \u03b1\nf g : \u03b1 \u2192* \u03b2\nh : toFreimanHom A n f = toFreimanHom A n g\n\u22a2 f = g\n[PROOFSTEP]\nrwa [toFreimanHom, toFreimanHom, FreimanHom.mk.injEq, FunLike.coe_fn_eq] at h \n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\n\u22a2 prod (map (\u2191f) s) = prod (map (\u2191f) t)\n[PROOFSTEP]\nobtain rfl | hm := m.eq_zero_or_pos\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nn : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhst : prod s = prod t\nhs : \u2191card s = 0\nht : \u2191card t = 0\nh : 0 \u2264 n\n\u22a2 prod (map (\u2191f) s) = prod (map (\u2191f) t)\n[PROOFSTEP]\nrw [card_eq_zero] at hs ht \n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nn : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhst : prod s = prod t\nhs : s = 0\nht : t = 0\nh : 0 \u2264 n\n\u22a2 prod (map (\u2191f) s) = prod (map (\u2191f) t)\n[PROOFSTEP]\nrw [hs, ht]\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\nhm : m > 0\n\u22a2 prod (map (\u2191f) s) = prod (map (\u2191f) t)\n[PROOFSTEP]\nsimp [\u2190 hs, card_pos_iff_exists_mem] at hm \n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\nhm : \u2203 a, a \u2208 s\n\u22a2 prod (map (\u2191f) s) = prod (map (\u2191f) t)\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := hm\n[GOAL]\ncase inr.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 s\n\u22a2 prod (map (\u2191f) s) = prod (map (\u2191f) t)\n[PROOFSTEP]\nsuffices ((s + Multiset.replicate (n - m) a).map f).prod = ((t + Multiset.replicate (n - m) a).map f).prod\n by\n simp_rw [Multiset.map_add, prod_add] at this \n exact mul_right_cancel this\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 s\nthis : prod (map (\u2191f) (s + replicate (n - m) a)) = prod (map (\u2191f) (t + replicate (n - m) a))\n\u22a2 prod (map (\u2191f) s) = prod (map (\u2191f) t)\n[PROOFSTEP]\nsimp_rw [Multiset.map_add, prod_add] at this \n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 s\nthis :\n prod (map (fun x => \u2191f x) s) * prod (map (fun x => \u2191f x) (replicate (n - m) a)) =\n prod (map (fun x => \u2191f x) t) * prod (map (fun x => \u2191f x) (replicate (n - m) a))\n\u22a2 prod (map (\u2191f) s) = prod (map (\u2191f) t)\n[PROOFSTEP]\nexact mul_right_cancel this\n[GOAL]\ncase inr.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 s\n\u22a2 prod (map (\u2191f) (s + replicate (n - m) a)) = prod (map (\u2191f) (t + replicate (n - m) a))\n[PROOFSTEP]\nreplace ha := hsA _ ha\n[GOAL]\ncase inr.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\n\u22a2 prod (map (\u2191f) (s + replicate (n - m) a)) = prod (map (\u2191f) (t + replicate (n - m) a))\n[PROOFSTEP]\napply\n map_prod_eq_map_prod f (A := A) (\u03b2 := \u03b2) (n := n) (fun x hx => _) (fun x hx => _) _ _\n _\n -- porting note: below could be golfed when wlog is available\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\n\u22a2 \u2200 (x : \u03b1), x \u2208 s + replicate (n - m) a \u2192 x \u2208 A\n[PROOFSTEP]\nintro x hx\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\nx : \u03b1\nhx : x \u2208 s + replicate (n - m) a\n\u22a2 x \u2208 A\n[PROOFSTEP]\nrw [mem_add] at hx \n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\nx : \u03b1\nhx : x \u2208 s \u2228 x \u2208 replicate (n - m) a\n\u22a2 x \u2208 A\n[PROOFSTEP]\ncases' hx with hx hx\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 A\n[PROOFSTEP]\nexact hsA x hx\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\nx : \u03b1\nhx : x \u2208 replicate (n - m) a\n\u22a2 x \u2208 A\n[PROOFSTEP]\nrwa [eq_of_mem_replicate hx]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\n\u22a2 \u2200 (x : \u03b1), x \u2208 t + replicate (n - m) a \u2192 x \u2208 A\n[PROOFSTEP]\nintro x hx\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\nx : \u03b1\nhx : x \u2208 t + replicate (n - m) a\n\u22a2 x \u2208 A\n[PROOFSTEP]\nrw [mem_add] at hx \n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\nx : \u03b1\nhx : x \u2208 t \u2228 x \u2208 replicate (n - m) a\n\u22a2 x \u2208 A\n[PROOFSTEP]\ncases' hx with hx hx\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 A\n[PROOFSTEP]\nexact htA x hx\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\nx : \u03b1\nhx : x \u2208 replicate (n - m) a\n\u22a2 x \u2208 A\n[PROOFSTEP]\nrwa [eq_of_mem_replicate hx]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\n\u22a2 \u2191card (s + replicate (n - m) a) = n\n[PROOFSTEP]\nrw [_root_.map_add, card_replicate, hs]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\n\u22a2 m + (n - m) = n\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\n\u22a2 \u2191card (t + replicate (n - m) a) = n\n[PROOFSTEP]\nrw [_root_.map_add, card_replicate, ht]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\n\u22a2 m + (n - m) = n\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\ninst\u271d : FreimanHomClass F A \u03b2 n\nf : F\ns t : Multiset \u03b1\nhsA : \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 A\nhtA : \u2200 (x : \u03b1), x \u2208 t \u2192 x \u2208 A\nhs : \u2191card s = m\nht : \u2191card t = m\nhst : prod s = prod t\nh : m \u2264 n\na : \u03b1\nha : a \u2208 A\n\u22a2 prod (s + replicate (n - m) a) = prod (t + replicate (n - m) a)\n[PROOFSTEP]\nrw [prod_add, prod_add, hst]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\nG : Type u_6\ninst\u271d\u00b2 : FunLike F \u03b1 fun x => \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : CancelCommMonoid \u03b2\nA : Set \u03b1\nm n : \u2115\nh : m \u2264 n\nf g : A \u2192*[n] \u03b2\nhfg : toFreimanHom h f = toFreimanHom h g\n\u22a2 \u2200 (x : \u03b1), \u2191f x = \u2191g x\n[PROOFSTEP]\nconvert FunLike.ext_iff.1 hfg using 0\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Hom.Freiman", "llama_tokens": 18077, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389817407016, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.40505145733074716}} {"text": "[GOAL]\nA : Type u_1\ninst\u271d : SeminormedGroup A\nn : \u2115\n\u03b4 : \u211d\na : A\n\u22a2 a \u2208 approxOrderOf A n \u03b4 \u2194 \u2203 b, orderOf b = n \u2227 a \u2208 ball b \u03b4\n[PROOFSTEP]\nsimp only [approxOrderOf, thickening_eq_biUnion_ball, mem_iUnion\u2082, mem_setOf_eq, exists_prop]\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhm : 0 < m\nhmn : Nat.coprime n m\n\u22a2 (fun y => y ^ m) '' approxOrderOf A n \u03b4 \u2286 approxOrderOf A n (\u2191m * \u03b4)\n[PROOFSTEP]\nrintro - \u27e8a, ha, rfl\u27e9\n[GOAL]\ncase intro.intro\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na\u271d : A\nm n : \u2115\n\u03b4 : \u211d\nhm : 0 < m\nhmn : Nat.coprime n m\na : A\nha : a \u2208 approxOrderOf A n \u03b4\n\u22a2 (fun y => y ^ m) a \u2208 approxOrderOf A n (\u2191m * \u03b4)\n[PROOFSTEP]\nobtain \u27e8b, hb, hab\u27e9 := mem_approxOrderOf_iff.mp ha\n[GOAL]\ncase intro.intro.intro.intro\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na\u271d : A\nm n : \u2115\n\u03b4 : \u211d\nhm : 0 < m\nhmn : Nat.coprime n m\na : A\nha : a \u2208 approxOrderOf A n \u03b4\nb : A\nhb : orderOf b = n\nhab : a \u2208 ball b \u03b4\n\u22a2 (fun y => y ^ m) a \u2208 approxOrderOf A n (\u2191m * \u03b4)\n[PROOFSTEP]\nreplace hb : b ^ m \u2208 {u : A | orderOf u = n} := by rw [\u2190 hb] at hmn \u22a2; exact orderOf_pow_coprime hmn\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na\u271d : A\nm n : \u2115\n\u03b4 : \u211d\nhm : 0 < m\nhmn : Nat.coprime n m\na : A\nha : a \u2208 approxOrderOf A n \u03b4\nb : A\nhb : orderOf b = n\nhab : a \u2208 ball b \u03b4\n\u22a2 b ^ m \u2208 {u | orderOf u = n}\n[PROOFSTEP]\nrw [\u2190 hb] at hmn \u22a2\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na\u271d : A\nm n : \u2115\n\u03b4 : \u211d\nhm : 0 < m\na : A\nha : a \u2208 approxOrderOf A n \u03b4\nb : A\nhmn : Nat.coprime (orderOf b) m\nhb : orderOf b = n\nhab : a \u2208 ball b \u03b4\n\u22a2 b ^ m \u2208 {u | orderOf u = orderOf b}\n[PROOFSTEP]\nexact orderOf_pow_coprime hmn\n[GOAL]\ncase intro.intro.intro.intro\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na\u271d : A\nm n : \u2115\n\u03b4 : \u211d\nhm : 0 < m\nhmn : Nat.coprime n m\na : A\nha : a \u2208 approxOrderOf A n \u03b4\nb : A\nhab : a \u2208 ball b \u03b4\nhb : b ^ m \u2208 {u | orderOf u = n}\n\u22a2 (fun y => y ^ m) a \u2208 approxOrderOf A n (\u2191m * \u03b4)\n[PROOFSTEP]\napply ball_subset_thickening hb ((m : \u211d) \u2022 \u03b4)\n[GOAL]\ncase intro.intro.intro.intro.a\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na\u271d : A\nm n : \u2115\n\u03b4 : \u211d\nhm : 0 < m\nhmn : Nat.coprime n m\na : A\nha : a \u2208 approxOrderOf A n \u03b4\nb : A\nhab : a \u2208 ball b \u03b4\nhb : b ^ m \u2208 {u | orderOf u = n}\n\u22a2 (fun y => y ^ m) a \u2208 ball (b ^ m) (\u2191m \u2022 \u03b4)\n[PROOFSTEP]\nconvert pow_mem_ball hm hab using 1\n[GOAL]\ncase h.e'_5\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na\u271d : A\nm n : \u2115\n\u03b4 : \u211d\nhm : 0 < m\nhmn : Nat.coprime n m\na : A\nha : a \u2208 approxOrderOf A n \u03b4\nb : A\nhab : a \u2208 ball b \u03b4\nhb : b ^ m \u2208 {u | orderOf u = n}\n\u22a2 ball (b ^ m) (\u2191m \u2022 \u03b4) = ball (b ^ m) (m \u2022 \u03b4)\n[PROOFSTEP]\nsimp only [nsmul_eq_mul, Algebra.id.smul_eq_mul]\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n\u271d : \u2115\n\u03b4 : \u211d\nn : \u2115\nhm : 0 < m\n\u22a2 (fun y => y ^ m) '' approxOrderOf A (n * m) \u03b4 \u2286 approxOrderOf A n (\u2191m * \u03b4)\n[PROOFSTEP]\nrintro - \u27e8a, ha, rfl\u27e9\n[GOAL]\ncase intro.intro\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na\u271d : A\nm n\u271d : \u2115\n\u03b4 : \u211d\nn : \u2115\nhm : 0 < m\na : A\nha : a \u2208 approxOrderOf A (n * m) \u03b4\n\u22a2 (fun y => y ^ m) a \u2208 approxOrderOf A n (\u2191m * \u03b4)\n[PROOFSTEP]\nobtain \u27e8b, hb : orderOf b = n * m, hab : a \u2208 ball b \u03b4\u27e9 := mem_approxOrderOf_iff.mp ha\n[GOAL]\ncase intro.intro.intro.intro\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na\u271d : A\nm n\u271d : \u2115\n\u03b4 : \u211d\nn : \u2115\nhm : 0 < m\na : A\nha : a \u2208 approxOrderOf A (n * m) \u03b4\nb : A\nhb : orderOf b = n * m\nhab : a \u2208 ball b \u03b4\n\u22a2 (fun y => y ^ m) a \u2208 approxOrderOf A n (\u2191m * \u03b4)\n[PROOFSTEP]\nreplace hb : b ^ m \u2208 {y : A | orderOf y = n}\n[GOAL]\ncase hb\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na\u271d : A\nm n\u271d : \u2115\n\u03b4 : \u211d\nn : \u2115\nhm : 0 < m\na : A\nha : a \u2208 approxOrderOf A (n * m) \u03b4\nb : A\nhb : orderOf b = n * m\nhab : a \u2208 ball b \u03b4\n\u22a2 b ^ m \u2208 {y | orderOf y = n}\n[PROOFSTEP]\nrw [mem_setOf_eq, orderOf_pow' b hm.ne', hb, Nat.gcd_mul_left_left, n.mul_div_cancel hm]\n[GOAL]\ncase intro.intro.intro.intro\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na\u271d : A\nm n\u271d : \u2115\n\u03b4 : \u211d\nn : \u2115\nhm : 0 < m\na : A\nha : a \u2208 approxOrderOf A (n * m) \u03b4\nb : A\nhab : a \u2208 ball b \u03b4\nhb : b ^ m \u2208 {y | orderOf y = n}\n\u22a2 (fun y => y ^ m) a \u2208 approxOrderOf A n (\u2191m * \u03b4)\n[PROOFSTEP]\napply ball_subset_thickening hb (m * \u03b4)\n[GOAL]\ncase intro.intro.intro.intro.a\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na\u271d : A\nm n\u271d : \u2115\n\u03b4 : \u211d\nn : \u2115\nhm : 0 < m\na : A\nha : a \u2208 approxOrderOf A (n * m) \u03b4\nb : A\nhab : a \u2208 ball b \u03b4\nhb : b ^ m \u2208 {y | orderOf y = n}\n\u22a2 (fun y => y ^ m) a \u2208 ball (b ^ m) (\u2191m * \u03b4)\n[PROOFSTEP]\nconvert pow_mem_ball hm hab using 1\n[GOAL]\ncase h.e'_5\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na\u271d : A\nm n\u271d : \u2115\n\u03b4 : \u211d\nn : \u2115\nhm : 0 < m\na : A\nha : a \u2208 approxOrderOf A (n * m) \u03b4\nb : A\nhab : a \u2208 ball b \u03b4\nhb : b ^ m \u2208 {y | orderOf y = n}\n\u22a2 ball (b ^ m) (\u2191m * \u03b4) = ball (b ^ m) (m \u2022 \u03b4)\n[PROOFSTEP]\nsimp only [nsmul_eq_mul]\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhan : Nat.coprime (orderOf a) n\n\u22a2 a \u2022 approxOrderOf A n \u03b4 \u2286 approxOrderOf A (orderOf a * n) \u03b4\n[PROOFSTEP]\nsimp_rw [approxOrderOf, thickening_eq_biUnion_ball, \u2190 image_smul, image_iUnion\u2082, image_smul, smul_ball'', smul_eq_mul,\n mem_setOf_eq]\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhan : Nat.coprime (orderOf a) n\n\u22a2 \u22c3 (i : A) (_ : orderOf i = n), ball (a * i) \u03b4 \u2286 \u22c3 (x : A) (_ : orderOf x = orderOf a * n), ball x \u03b4\n[PROOFSTEP]\nrefine' iUnion\u2082_subset_iff.mpr fun b hb c hc => _\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhan : Nat.coprime (orderOf a) n\nb : A\nhb : orderOf b = n\nc : A\nhc : c \u2208 ball (a * b) \u03b4\n\u22a2 c \u2208 \u22c3 (x : A) (_ : orderOf x = orderOf a * n), ball x \u03b4\n[PROOFSTEP]\nsimp only [mem_iUnion, exists_prop]\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhan : Nat.coprime (orderOf a) n\nb : A\nhb : orderOf b = n\nc : A\nhc : c \u2208 ball (a * b) \u03b4\n\u22a2 \u2203 i, orderOf i = orderOf a * n \u2227 c \u2208 ball i \u03b4\n[PROOFSTEP]\nrefine' \u27e8a * b, _, hc\u27e9\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhan : Nat.coprime (orderOf a) n\nb : A\nhb : orderOf b = n\nc : A\nhc : c \u2208 ball (a * b) \u03b4\n\u22a2 orderOf (a * b) = orderOf a * n\n[PROOFSTEP]\nrw [\u2190 hb] at han \u22a2\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nb : A\nhan : Nat.coprime (orderOf a) (orderOf b)\nhb : orderOf b = n\nc : A\nhc : c \u2208 ball (a * b) \u03b4\n\u22a2 orderOf (a * b) = orderOf a * orderOf b\n[PROOFSTEP]\nexact (Commute.all a b).orderOf_mul_eq_mul_orderOf_of_coprime han\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhn : 0 < n\nhan : orderOf a ^ 2 \u2223 n\n\u22a2 a \u2022 approxOrderOf A n \u03b4 = approxOrderOf A n \u03b4\n[PROOFSTEP]\nsimp_rw [approxOrderOf, thickening_eq_biUnion_ball, \u2190 image_smul, image_iUnion\u2082, image_smul, smul_ball'', smul_eq_mul,\n mem_setOf_eq]\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhn : 0 < n\nhan : orderOf a ^ 2 \u2223 n\n\u22a2 \u22c3 (i : A) (_ : orderOf i = n), ball (a * i) \u03b4 = \u22c3 (x : A) (_ : orderOf x = n), ball x \u03b4\n[PROOFSTEP]\nreplace han : \u2200 {b : A}, orderOf b = n \u2192 orderOf (a * b) = n\n[GOAL]\ncase han\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhn : 0 < n\nhan : orderOf a ^ 2 \u2223 n\n\u22a2 \u2200 {b : A}, orderOf b = n \u2192 orderOf (a * b) = n\n[PROOFSTEP]\nintro b hb\n[GOAL]\ncase han\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhn : 0 < n\nhan : orderOf a ^ 2 \u2223 n\nb : A\nhb : orderOf b = n\n\u22a2 orderOf (a * b) = n\n[PROOFSTEP]\nrw [\u2190 hb] at han hn \n[GOAL]\ncase han\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nb : A\nhn : 0 < orderOf b\nhan : orderOf a ^ 2 \u2223 orderOf b\nhb : orderOf b = n\n\u22a2 orderOf (a * b) = n\n[PROOFSTEP]\nrw [sq] at han \n[GOAL]\ncase han\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nb : A\nhn : 0 < orderOf b\nhan : orderOf a * orderOf a \u2223 orderOf b\nhb : orderOf b = n\n\u22a2 orderOf (a * b) = n\n[PROOFSTEP]\nrwa [(Commute.all a b).orderOf_mul_eq_right_of_forall_prime_mul_dvd (orderOf_pos_iff.mp hn) fun p _ hp' =>\n dvd_trans (mul_dvd_mul_right hp' <| orderOf a) han]\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhn : 0 < n\nhan : \u2200 {b : A}, orderOf b = n \u2192 orderOf (a * b) = n\n\u22a2 \u22c3 (i : A) (_ : orderOf i = n), ball (a * i) \u03b4 = \u22c3 (x : A) (_ : orderOf x = n), ball x \u03b4\n[PROOFSTEP]\nlet f : {b : A | orderOf b = n} \u2192 {b : A | orderOf b = n} := fun b => \u27e8a * b, han b.property\u27e9\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhn : 0 < n\nhan : \u2200 {b : A}, orderOf b = n \u2192 orderOf (a * b) = n\nf : \u2191{b | orderOf b = n} \u2192 \u2191{b | orderOf b = n} := fun b => { val := a * \u2191b, property := (_ : orderOf (a * \u2191b) = n) }\n\u22a2 \u22c3 (i : A) (_ : orderOf i = n), ball (a * i) \u03b4 = \u22c3 (x : A) (_ : orderOf x = n), ball x \u03b4\n[PROOFSTEP]\nhave hf : Surjective f := by\n rintro \u27e8b, hb\u27e9\n refine' \u27e8\u27e8a\u207b\u00b9 * b, _\u27e9, _\u27e9\n \u00b7 rw [mem_setOf_eq, \u2190 orderOf_inv, mul_inv_rev, inv_inv, mul_comm]\n apply han\n simpa\n \u00b7 simp only [Subtype.mk_eq_mk, Subtype.coe_mk, mul_inv_cancel_left]\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhn : 0 < n\nhan : \u2200 {b : A}, orderOf b = n \u2192 orderOf (a * b) = n\nf : \u2191{b | orderOf b = n} \u2192 \u2191{b | orderOf b = n} := fun b => { val := a * \u2191b, property := (_ : orderOf (a * \u2191b) = n) }\n\u22a2 Surjective f\n[PROOFSTEP]\nrintro \u27e8b, hb\u27e9\n[GOAL]\ncase mk\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhn : 0 < n\nhan : \u2200 {b : A}, orderOf b = n \u2192 orderOf (a * b) = n\nf : \u2191{b | orderOf b = n} \u2192 \u2191{b | orderOf b = n} := fun b => { val := a * \u2191b, property := (_ : orderOf (a * \u2191b) = n) }\nb : A\nhb : b \u2208 {b | orderOf b = n}\n\u22a2 \u2203 a, f a = { val := b, property := hb }\n[PROOFSTEP]\nrefine' \u27e8\u27e8a\u207b\u00b9 * b, _\u27e9, _\u27e9\n[GOAL]\ncase mk.refine'_1\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhn : 0 < n\nhan : \u2200 {b : A}, orderOf b = n \u2192 orderOf (a * b) = n\nf : \u2191{b | orderOf b = n} \u2192 \u2191{b | orderOf b = n} := fun b => { val := a * \u2191b, property := (_ : orderOf (a * \u2191b) = n) }\nb : A\nhb : b \u2208 {b | orderOf b = n}\n\u22a2 a\u207b\u00b9 * b \u2208 {b | orderOf b = n}\n[PROOFSTEP]\nrw [mem_setOf_eq, \u2190 orderOf_inv, mul_inv_rev, inv_inv, mul_comm]\n[GOAL]\ncase mk.refine'_1\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhn : 0 < n\nhan : \u2200 {b : A}, orderOf b = n \u2192 orderOf (a * b) = n\nf : \u2191{b | orderOf b = n} \u2192 \u2191{b | orderOf b = n} := fun b => { val := a * \u2191b, property := (_ : orderOf (a * \u2191b) = n) }\nb : A\nhb : b \u2208 {b | orderOf b = n}\n\u22a2 orderOf (a * b\u207b\u00b9) = n\n[PROOFSTEP]\napply han\n[GOAL]\ncase mk.refine'_1.a\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhn : 0 < n\nhan : \u2200 {b : A}, orderOf b = n \u2192 orderOf (a * b) = n\nf : \u2191{b | orderOf b = n} \u2192 \u2191{b | orderOf b = n} := fun b => { val := a * \u2191b, property := (_ : orderOf (a * \u2191b) = n) }\nb : A\nhb : b \u2208 {b | orderOf b = n}\n\u22a2 orderOf b\u207b\u00b9 = n\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase mk.refine'_2\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhn : 0 < n\nhan : \u2200 {b : A}, orderOf b = n \u2192 orderOf (a * b) = n\nf : \u2191{b | orderOf b = n} \u2192 \u2191{b | orderOf b = n} := fun b => { val := a * \u2191b, property := (_ : orderOf (a * \u2191b) = n) }\nb : A\nhb : b \u2208 {b | orderOf b = n}\n\u22a2 f { val := a\u207b\u00b9 * b, property := (_ : a\u207b\u00b9 * b \u2208 {b | orderOf b = n}) } = { val := b, property := hb }\n[PROOFSTEP]\nsimp only [Subtype.mk_eq_mk, Subtype.coe_mk, mul_inv_cancel_left]\n[GOAL]\nA : Type u_1\ninst\u271d : SeminormedCommGroup A\na : A\nm n : \u2115\n\u03b4 : \u211d\nhn : 0 < n\nhan : \u2200 {b : A}, orderOf b = n \u2192 orderOf (a * b) = n\nf : \u2191{b | orderOf b = n} \u2192 \u2191{b | orderOf b = n} := fun b => { val := a * \u2191b, property := (_ : orderOf (a * \u2191b) = n) }\nhf : Surjective f\n\u22a2 \u22c3 (i : A) (_ : orderOf i = n), ball (a * i) \u03b4 = \u22c3 (x : A) (_ : orderOf x = n), ball x \u03b4\n[PROOFSTEP]\nsimpa only [mem_setOf_eq, Subtype.coe_mk, iUnion_coe_set] using hf.iUnion_comp fun b => ball (b : A) \u03b4\n[GOAL]\n\u03b4 : \u211d\nx : UnitAddCircle\nn : \u2115\nhn : 0 < n\n\u22a2 x \u2208 approxAddOrderOf UnitAddCircle n \u03b4 \u2194 \u2203 m, m < n \u2227 gcd m n = 1 \u2227 \u2016x - \u2191(\u2191m / \u2191n)\u2016 < \u03b4\n[PROOFSTEP]\nhaveI := Real.fact_zero_lt_one\n[GOAL]\n\u03b4 : \u211d\nx : UnitAddCircle\nn : \u2115\nhn : 0 < n\nthis : Fact (0 < 1)\n\u22a2 x \u2208 approxAddOrderOf UnitAddCircle n \u03b4 \u2194 \u2203 m, m < n \u2227 gcd m n = 1 \u2227 \u2016x - \u2191(\u2191m / \u2191n)\u2016 < \u03b4\n[PROOFSTEP]\nsimp only [mem_approx_add_orderOf_iff, mem_setOf_eq, ball, exists_prop, dist_eq_norm,\n AddCircle.addOrderOf_eq_pos_iff hn, mul_one]\n[GOAL]\n\u03b4 : \u211d\nx : UnitAddCircle\nn : \u2115\nhn : 0 < n\nthis : Fact (0 < 1)\n\u22a2 (\u2203 b, (\u2203 m, m < n \u2227 Nat.gcd m n = 1 \u2227 \u2191(\u2191m / \u2191n) = b) \u2227 \u2016x - b\u2016 < \u03b4) \u2194 \u2203 m, m < n \u2227 gcd m n = 1 \u2227 \u2016x - \u2191(\u2191m / \u2191n)\u2016 < \u03b4\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b4 : \u211d\nx : UnitAddCircle\nn : \u2115\nhn : 0 < n\nthis : Fact (0 < 1)\n\u22a2 (\u2203 b, (\u2203 m, m < n \u2227 Nat.gcd m n = 1 \u2227 \u2191(\u2191m / \u2191n) = b) \u2227 \u2016x - b\u2016 < \u03b4) \u2192 \u2203 m, m < n \u2227 gcd m n = 1 \u2227 \u2016x - \u2191(\u2191m / \u2191n)\u2016 < \u03b4\n[PROOFSTEP]\nrintro \u27e8y, \u27e8m, hm\u2081, hm\u2082, rfl\u27e9, hx\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b4 : \u211d\nx : UnitAddCircle\nn : \u2115\nhn : 0 < n\nthis : Fact (0 < 1)\nm : \u2115\nhm\u2081 : m < n\nhm\u2082 : Nat.gcd m n = 1\nhx : \u2016x - \u2191(\u2191m / \u2191n)\u2016 < \u03b4\n\u22a2 \u2203 m, m < n \u2227 gcd m n = 1 \u2227 \u2016x - \u2191(\u2191m / \u2191n)\u2016 < \u03b4\n[PROOFSTEP]\nexact \u27e8m, hm\u2081, hm\u2082, hx\u27e9\n[GOAL]\ncase mpr\n\u03b4 : \u211d\nx : UnitAddCircle\nn : \u2115\nhn : 0 < n\nthis : Fact (0 < 1)\n\u22a2 (\u2203 m, m < n \u2227 gcd m n = 1 \u2227 \u2016x - \u2191(\u2191m / \u2191n)\u2016 < \u03b4) \u2192 \u2203 b, (\u2203 m, m < n \u2227 Nat.gcd m n = 1 \u2227 \u2191(\u2191m / \u2191n) = b) \u2227 \u2016x - b\u2016 < \u03b4\n[PROOFSTEP]\nrintro \u27e8m, hm\u2081, hm\u2082, hx\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b4 : \u211d\nx : UnitAddCircle\nn : \u2115\nhn : 0 < n\nthis : Fact (0 < 1)\nm : \u2115\nhm\u2081 : m < n\nhm\u2082 : gcd m n = 1\nhx : \u2016x - \u2191(\u2191m / \u2191n)\u2016 < \u03b4\n\u22a2 \u2203 b, (\u2203 m, m < n \u2227 Nat.gcd m n = 1 \u2227 \u2191(\u2191m / \u2191n) = b) \u2227 \u2016x - b\u2016 < \u03b4\n[PROOFSTEP]\nexact \u27e8\u2191((m : \u211d) / n), \u27e8m, hm\u2081, hm\u2082, rfl\u27e9, hx\u27e9\n[GOAL]\n\u03b4 : \u2115 \u2192 \u211d\nx : UnitAddCircle\n\u22a2 x \u2208 addWellApproximable UnitAddCircle \u03b4 \u2194 Set.Infinite {n | \u2203 m, m < n \u2227 gcd m n = 1 \u2227 \u2016x - \u2191(\u2191m / \u2191n)\u2016 < \u03b4 n}\n[PROOFSTEP]\nsimp only [mem_add_wellApproximable_iff, \u2190 Nat.cofinite_eq_atTop, cofinite.blimsup_set_eq, mem_setOf_eq]\n[GOAL]\n\u03b4 : \u2115 \u2192 \u211d\nx : UnitAddCircle\n\u22a2 Set.Infinite {n | 0 < n \u2227 x \u2208 approxAddOrderOf UnitAddCircle n (\u03b4 n)} \u2194\n Set.Infinite {n | \u2203 m, m < n \u2227 gcd m n = 1 \u2227 \u2016x - \u2191(\u2191m / \u2191n)\u2016 < \u03b4 n}\n[PROOFSTEP]\nrefine' iff_of_eq (congr_arg Set.Infinite <| ext fun n => \u27e8fun hn => _, fun hn => _\u27e9)\n[GOAL]\ncase refine'_1\n\u03b4 : \u2115 \u2192 \u211d\nx : UnitAddCircle\nn : \u2115\nhn : n \u2208 {n | 0 < n \u2227 x \u2208 approxAddOrderOf UnitAddCircle n (\u03b4 n)}\n\u22a2 n \u2208 {n | \u2203 m, m < n \u2227 gcd m n = 1 \u2227 \u2016x - \u2191(\u2191m / \u2191n)\u2016 < \u03b4 n}\n[PROOFSTEP]\nexact (mem_approxAddOrderOf_iff hn.1).mp hn.2\n[GOAL]\ncase refine'_2\n\u03b4 : \u2115 \u2192 \u211d\nx : UnitAddCircle\nn : \u2115\nhn : n \u2208 {n | \u2203 m, m < n \u2227 gcd m n = 1 \u2227 \u2016x - \u2191(\u2191m / \u2191n)\u2016 < \u03b4 n}\n\u22a2 n \u2208 {n | 0 < n \u2227 x \u2208 approxAddOrderOf UnitAddCircle n (\u03b4 n)}\n[PROOFSTEP]\nhave h : 0 < n := by obtain \u27e8m, hm\u2081, _, _\u27e9 := hn; exact pos_of_gt hm\u2081\n[GOAL]\n\u03b4 : \u2115 \u2192 \u211d\nx : UnitAddCircle\nn : \u2115\nhn : n \u2208 {n | \u2203 m, m < n \u2227 gcd m n = 1 \u2227 \u2016x - \u2191(\u2191m / \u2191n)\u2016 < \u03b4 n}\n\u22a2 0 < n\n[PROOFSTEP]\nobtain \u27e8m, hm\u2081, _, _\u27e9 := hn\n[GOAL]\ncase intro.intro.intro\n\u03b4 : \u2115 \u2192 \u211d\nx : UnitAddCircle\nn m : \u2115\nhm\u2081 : m < n\nleft\u271d : gcd m n = 1\nright\u271d : \u2016x - \u2191(\u2191m / \u2191n)\u2016 < \u03b4 n\n\u22a2 0 < n\n[PROOFSTEP]\nexact pos_of_gt hm\u2081\n[GOAL]\ncase refine'_2\n\u03b4 : \u2115 \u2192 \u211d\nx : UnitAddCircle\nn : \u2115\nhn : n \u2208 {n | \u2203 m, m < n \u2227 gcd m n = 1 \u2227 \u2016x - \u2191(\u2191m / \u2191n)\u2016 < \u03b4 n}\nh : 0 < n\n\u22a2 n \u2208 {n | 0 < n \u2227 x \u2208 approxAddOrderOf UnitAddCircle n (\u03b4 n)}\n[PROOFSTEP]\nexact \u27e8h, (mem_approxAddOrderOf_iff h).mpr hn\u27e9\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a), \u00acaddWellApproximable \ud835\udd4a \u03b4 x) \u2228 \u2200\u1d50 (x : \ud835\udd4a), addWellApproximable \ud835\udd4a \u03b4 x\n[PROOFSTEP]\nletI : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup _\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a), \u00acaddWellApproximable \ud835\udd4a \u03b4 x) \u2228 \u2200\u1d50 (x : \ud835\udd4a), addWellApproximable \ud835\udd4a \u03b4 x\n[PROOFSTEP]\nset \u03bc : Measure \ud835\udd4a := volume\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acaddWellApproximable \ud835\udd4a \u03b4 x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, addWellApproximable \ud835\udd4a \u03b4 x\n[PROOFSTEP]\nset u : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191((\u2191(1 : \u2115) : \u211d) / ((p : \u2115) : \u211d) * T)\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acaddWellApproximable \ud835\udd4a \u03b4 x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, addWellApproximable \ud835\udd4a \u03b4 x\n[PROOFSTEP]\nhave hu\u2080 : \u2200 p : Nat.Primes, addOrderOf (u p) = (p : \u2115) := by rintro \u27e8p, hp\u27e9;\n exact addOrderOf_div_of_gcd_eq_one hp.pos (gcd_one_left p)\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\n\u22a2 \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\n[PROOFSTEP]\nrintro \u27e8p, hp\u27e9\n[GOAL]\ncase mk\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\np : \u2115\nhp : Nat.Prime p\n\u22a2 addOrderOf (u { val := p, property := hp }) = \u2191{ val := p, property := hp }\n[PROOFSTEP]\nexact addOrderOf_div_of_gcd_eq_one hp.pos (gcd_one_left p)\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acaddWellApproximable \ud835\udd4a \u03b4 x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, addWellApproximable \ud835\udd4a \u03b4 x\n[PROOFSTEP]\nhave hu : Tendsto (addOrderOf \u2218 u) atTop atTop :=\n by\n rw [(funext hu\u2080 : addOrderOf \u2218 u = (\u2191))]\n have h_mono : Monotone ((\u2191) : Nat.Primes \u2192 \u2115) := fun p q hpq => hpq\n refine' h_mono.tendsto_atTop_atTop fun n => _\n obtain \u27e8p, hp, hp'\u27e9 := n.exists_infinite_primes\n exact \u27e8\u27e8p, hp'\u27e9, hp\u27e9\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\n\u22a2 Tendsto (addOrderOf \u2218 u) atTop atTop\n[PROOFSTEP]\nrw [(funext hu\u2080 : addOrderOf \u2218 u = (\u2191))]\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\n\u22a2 Tendsto Subtype.val atTop atTop\n[PROOFSTEP]\nhave h_mono : Monotone ((\u2191) : Nat.Primes \u2192 \u2115) := fun p q hpq => hpq\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nh_mono : Monotone Subtype.val\n\u22a2 Tendsto Subtype.val atTop atTop\n[PROOFSTEP]\nrefine' h_mono.tendsto_atTop_atTop fun n => _\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nh_mono : Monotone Subtype.val\nn : \u2115\n\u22a2 \u2203 a, n \u2264 \u2191a\n[PROOFSTEP]\nobtain \u27e8p, hp, hp'\u27e9 := n.exists_infinite_primes\n[GOAL]\ncase intro.intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nh_mono : Monotone Subtype.val\nn p : \u2115\nhp : n \u2264 p\nhp' : Nat.Prime p\n\u22a2 \u2203 a, n \u2264 \u2191a\n[PROOFSTEP]\nexact \u27e8\u27e8p, hp'\u27e9, hp\u27e9\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acaddWellApproximable \ud835\udd4a \u03b4 x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, addWellApproximable \ud835\udd4a \u03b4 x\n[PROOFSTEP]\nset E := addWellApproximable \ud835\udd4a \u03b4\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acE x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, E x\n[PROOFSTEP]\nset X : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acE x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, E x\n[PROOFSTEP]\nset A : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p\u2224n\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acE x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, E x\n[PROOFSTEP]\nset B : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p\u2223\u2223n\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acE x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, E x\n[PROOFSTEP]\nset C : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acE x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, E x\n[PROOFSTEP]\nhave hA\u2080 : \u2200 p, MeasurableSet (A p) := fun p =>\n MeasurableSet.measurableSet_blimsup fun n _ => isOpen_thickening.measurableSet\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acE x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, E x\n[PROOFSTEP]\nhave hB\u2080 : \u2200 p, MeasurableSet (B p) := fun p =>\n MeasurableSet.measurableSet_blimsup fun n _ => isOpen_thickening.measurableSet\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acE x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, E x\n[PROOFSTEP]\nhave hE\u2080 : NullMeasurableSet E \u03bc :=\n by\n refine' (MeasurableSet.measurableSet_blimsup fun n hn => IsOpen.measurableSet _).nullMeasurableSet\n exact isOpen_thickening\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\n\u22a2 NullMeasurableSet E\n[PROOFSTEP]\nrefine' (MeasurableSet.measurableSet_blimsup fun n hn => IsOpen.measurableSet _).nullMeasurableSet\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nn : \u2115\nhn : 0 < n\n\u22a2 IsOpen (approxAddOrderOf \ud835\udd4a n (\u03b4 n))\n[PROOFSTEP]\nexact isOpen_thickening\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acE x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, E x\n[PROOFSTEP]\nhave hE\u2081 : \u2200 p, E = A p \u222a B p \u222a C p := by\n intro p\n simp only [addWellApproximable, \u2190 blimsup_or_eq_sup, \u2190 and_or_left, \u2190 sup_eq_union, sq]\n congr\n ext n\n tauto\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\n\u22a2 \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\n[PROOFSTEP]\nintro p\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\np : \u2115\n\u22a2 E = A p \u222a B p \u222a C p\n[PROOFSTEP]\nsimp only [addWellApproximable, \u2190 blimsup_or_eq_sup, \u2190 and_or_left, \u2190 sup_eq_union, sq]\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\np : \u2115\n\u22a2 (blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun n => 0 < n) =\n blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun x => 0 < x \u2227 ((\u00acp \u2223 x \u2228 p \u2223 x \u2227 \u00acp * p \u2223 x) \u2228 p * p \u2223 x)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_p\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\np : \u2115\n\u22a2 (fun n => 0 < n) = fun x => 0 < x \u2227 ((\u00acp \u2223 x \u2228 p \u2223 x \u2227 \u00acp * p \u2223 x) \u2228 p * p \u2223 x)\n[PROOFSTEP]\next n\n[GOAL]\ncase e_p.h.a\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\np n : \u2115\n\u22a2 0 < n \u2194 0 < n \u2227 ((\u00acp \u2223 n \u2228 p \u2223 n \u2227 \u00acp * p \u2223 n) \u2228 p * p \u2223 n)\n[PROOFSTEP]\ntauto\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acE x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, E x\n[PROOFSTEP]\nhave hE\u2082 : \u2200 p : Nat.Primes, A p =\u1d50[\u03bc] (\u2205 : Set \ud835\udd4a) \u2227 B p =\u1d50[\u03bc] (\u2205 : Set \ud835\udd4a) \u2192 E =\u1d50[\u03bc] C p :=\n by\n rintro p \u27e8hA, hB\u27e9\n rw [hE\u2081 p]\n exact union_ae_eq_right_of_ae_eq_empty ((union_ae_eq_right_of_ae_eq_empty hA).trans hB)\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\n\u22a2 \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\n[PROOFSTEP]\nrintro p \u27e8hA, hB\u27e9\n[GOAL]\ncase intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\np : Nat.Primes\nhA : A \u2191p =\u1d50[\u03bc] \u2205\nhB : B \u2191p =\u1d50[\u03bc] \u2205\n\u22a2 E =\u1d50[\u03bc] C \u2191p\n[PROOFSTEP]\nrw [hE\u2081 p]\n[GOAL]\ncase intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\np : Nat.Primes\nhA : A \u2191p =\u1d50[\u03bc] \u2205\nhB : B \u2191p =\u1d50[\u03bc] \u2205\n\u22a2 A \u2191p \u222a B \u2191p \u222a C \u2191p =\u1d50[\u03bc] C \u2191p\n[PROOFSTEP]\nexact union_ae_eq_right_of_ae_eq_empty ((union_ae_eq_right_of_ae_eq_empty hA).trans hB)\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acE x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, E x\n[PROOFSTEP]\nhave hA : \u2200 p : Nat.Primes, A p =\u1d50[\u03bc] (\u2205 : Set \ud835\udd4a) \u2228 A p =\u1d50[\u03bc] univ :=\n by\n rintro \u27e8p, hp\u27e9\n let f : \ud835\udd4a \u2192 \ud835\udd4a := fun y => (p : \u2115) \u2022 y\n suffices f '' A p \u2286 blimsup (fun n => approxAddOrderOf \ud835\udd4a n (p * \u03b4 n)) atTop fun n => 0 < n \u2227 p\u2224n\n by\n apply (ergodic_nsmul hp.one_lt).ae_empty_or_univ_of_image_ae_le (hA\u2080 p)\n apply (HasSubset.Subset.eventuallyLE this).congr EventuallyEq.rfl\n exact\n blimsup_thickening_mul_ae_eq \u03bc (fun n => 0 < n \u2227 p\u2224n) (fun n => {y | addOrderOf y = n}) (Nat.cast_pos.mpr hp.pos)\n _ h\u03b4\n refine' (SupHom.apply_blimsup_le (sSupHom.setImage f)).trans (mono_blimsup fun n hn => _)\n replace hn := Nat.coprime_comm.mp (hp.coprime_iff_not_dvd.2 hn.2)\n exact approxAddOrderOf.image_nsmul_subset_of_coprime (\u03b4 n) hp.pos hn\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\n\u22a2 \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\n[PROOFSTEP]\nrintro \u27e8p, hp\u27e9\n[GOAL]\ncase mk\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\np : \u2115\nhp : Nat.Prime p\n\u22a2 A \u2191{ val := p, property := hp } =\u1d50[\u03bc] \u2205 \u2228 A \u2191{ val := p, property := hp } =\u1d50[\u03bc] univ\n[PROOFSTEP]\nlet f : \ud835\udd4a \u2192 \ud835\udd4a := fun y => (p : \u2115) \u2022 y\n[GOAL]\ncase mk\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\np : \u2115\nhp : Nat.Prime p\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y\n\u22a2 A \u2191{ val := p, property := hp } =\u1d50[\u03bc] \u2205 \u2228 A \u2191{ val := p, property := hp } =\u1d50[\u03bc] univ\n[PROOFSTEP]\nsuffices f '' A p \u2286 blimsup (fun n => approxAddOrderOf \ud835\udd4a n (p * \u03b4 n)) atTop fun n => 0 < n \u2227 p\u2224n\n by\n apply (ergodic_nsmul hp.one_lt).ae_empty_or_univ_of_image_ae_le (hA\u2080 p)\n apply (HasSubset.Subset.eventuallyLE this).congr EventuallyEq.rfl\n exact\n blimsup_thickening_mul_ae_eq \u03bc (fun n => 0 < n \u2227 p\u2224n) (fun n => {y | addOrderOf y = n}) (Nat.cast_pos.mpr hp.pos) _\n h\u03b4\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis\u271d : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\np : \u2115\nhp : Nat.Prime p\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y\nthis : f '' A p \u2286 blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)) atTop fun n => 0 < n \u2227 \u00acp \u2223 n\n\u22a2 A \u2191{ val := p, property := hp } =\u1d50[\u03bc] \u2205 \u2228 A \u2191{ val := p, property := hp } =\u1d50[\u03bc] univ\n[PROOFSTEP]\napply (ergodic_nsmul hp.one_lt).ae_empty_or_univ_of_image_ae_le (hA\u2080 p)\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis\u271d : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\np : \u2115\nhp : Nat.Prime p\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y\nthis : f '' A p \u2286 blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)) atTop fun n => 0 < n \u2227 \u00acp \u2223 n\n\u22a2 (fun y => p \u2022 y) '' A p \u2264\u1d50[volume] A p\n[PROOFSTEP]\napply (HasSubset.Subset.eventuallyLE this).congr EventuallyEq.rfl\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis\u271d : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\np : \u2115\nhp : Nat.Prime p\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y\nthis : f '' A p \u2286 blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)) atTop fun n => 0 < n \u2227 \u00acp \u2223 n\n\u22a2 (blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)) atTop fun n => 0 < n \u2227 \u00acp \u2223 n) =\u1d50[volume] A p\n[PROOFSTEP]\nexact\n blimsup_thickening_mul_ae_eq \u03bc (fun n => 0 < n \u2227 p\u2224n) (fun n => {y | addOrderOf y = n}) (Nat.cast_pos.mpr hp.pos) _ h\u03b4\n[GOAL]\ncase mk\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\np : \u2115\nhp : Nat.Prime p\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y\n\u22a2 f '' A p \u2286 blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)) atTop fun n => 0 < n \u2227 \u00acp \u2223 n\n[PROOFSTEP]\nrefine' (SupHom.apply_blimsup_le (sSupHom.setImage f)).trans (mono_blimsup fun n hn => _)\n[GOAL]\ncase mk\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\np : \u2115\nhp : Nat.Prime p\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y\nn : \u2115\nhn : 0 < n \u2227 \u00acp \u2223 n\n\u22a2 (\u2191(sSupHom.setImage f) \u2218 X) n \u2264 approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)\n[PROOFSTEP]\nreplace hn := Nat.coprime_comm.mp (hp.coprime_iff_not_dvd.2 hn.2)\n[GOAL]\ncase mk\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\np : \u2115\nhp : Nat.Prime p\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y\nn : \u2115\nhn : Nat.coprime n p\n\u22a2 (\u2191(sSupHom.setImage f) \u2218 X) n \u2264 approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)\n[PROOFSTEP]\nexact approxAddOrderOf.image_nsmul_subset_of_coprime (\u03b4 n) hp.pos hn\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acE x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, E x\n[PROOFSTEP]\nhave hB : \u2200 p : Nat.Primes, B p =\u1d50[\u03bc] (\u2205 : Set \ud835\udd4a) \u2228 B p =\u1d50[\u03bc] univ :=\n by\n rintro \u27e8p, hp\u27e9\n let x := u \u27e8p, hp\u27e9\n let f : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\n suffices f '' B p \u2286 blimsup (fun n => approxAddOrderOf \ud835\udd4a n (p * \u03b4 n)) atTop fun n => 0 < n \u2227 p\u2223\u2223n\n by\n apply (ergodic_nsmul_add x hp.one_lt).ae_empty_or_univ_of_image_ae_le (hB\u2080 p)\n apply (HasSubset.Subset.eventuallyLE this).congr EventuallyEq.rfl\n exact\n blimsup_thickening_mul_ae_eq \u03bc (fun n => 0 < n \u2227 p\u2223\u2223n) (fun n => {y | addOrderOf y = n}) (Nat.cast_pos.mpr hp.pos)\n _ h\u03b4\n refine' (SupHom.apply_blimsup_le (sSupHom.setImage f)).trans (mono_blimsup _)\n rintro n \u27e8hn, h_div, h_ndiv\u27e9\n have h_cop : (addOrderOf x).coprime (n / p) :=\n by\n obtain \u27e8q, rfl\u27e9 := h_div\n rw [hu\u2080, Subtype.coe_mk, hp.coprime_iff_not_dvd, q.mul_div_cancel_left hp.pos]\n exact fun contra => h_ndiv (mul_dvd_mul_left p contra)\n replace h_div : n / p * p = n := Nat.div_mul_cancel h_div\n have hf : f = (fun y => x + y) \u2218 fun y => p \u2022 y := by ext; simp [add_comm x]; ac_rfl\n simp only at hf \n simp_rw [Function.comp_apply, le_eq_subset]\n rw [sSupHom.setImage_toFun, hf, image_comp]\n have := @monotone_image \ud835\udd4a \ud835\udd4a fun y => x + y\n specialize this (approxAddOrderOf.image_nsmul_subset (\u03b4 n) (n / p) hp.pos)\n simp only [h_div] at this \u22a2\n refine' this.trans _\n convert approxAddOrderOf.vadd_subset_of_coprime (p * \u03b4 n) h_cop\n rw [hu\u2080, Subtype.coe_mk, mul_comm p, h_div]\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\n\u22a2 \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\n[PROOFSTEP]\nrintro \u27e8p, hp\u27e9\n[GOAL]\ncase mk\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\n\u22a2 B \u2191{ val := p, property := hp } =\u1d50[\u03bc] \u2205 \u2228 B \u2191{ val := p, property := hp } =\u1d50[\u03bc] univ\n[PROOFSTEP]\nlet x := u \u27e8p, hp\u27e9\n[GOAL]\ncase mk\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\n\u22a2 B \u2191{ val := p, property := hp } =\u1d50[\u03bc] \u2205 \u2228 B \u2191{ val := p, property := hp } =\u1d50[\u03bc] univ\n[PROOFSTEP]\nlet f : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\n[GOAL]\ncase mk\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\n\u22a2 B \u2191{ val := p, property := hp } =\u1d50[\u03bc] \u2205 \u2228 B \u2191{ val := p, property := hp } =\u1d50[\u03bc] univ\n[PROOFSTEP]\nsuffices f '' B p \u2286 blimsup (fun n => approxAddOrderOf \ud835\udd4a n (p * \u03b4 n)) atTop fun n => 0 < n \u2227 p\u2223\u2223n\n by\n apply (ergodic_nsmul_add x hp.one_lt).ae_empty_or_univ_of_image_ae_le (hB\u2080 p)\n apply (HasSubset.Subset.eventuallyLE this).congr EventuallyEq.rfl\n exact\n blimsup_thickening_mul_ae_eq \u03bc (fun n => 0 < n \u2227 p\u2223\u2223n) (fun n => {y | addOrderOf y = n}) (Nat.cast_pos.mpr hp.pos) _\n h\u03b4\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis\u271d : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nthis : f '' B p \u2286 blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)) atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\n\u22a2 B \u2191{ val := p, property := hp } =\u1d50[\u03bc] \u2205 \u2228 B \u2191{ val := p, property := hp } =\u1d50[\u03bc] univ\n[PROOFSTEP]\napply (ergodic_nsmul_add x hp.one_lt).ae_empty_or_univ_of_image_ae_le (hB\u2080 p)\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis\u271d : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nthis : f '' B p \u2286 blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)) atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\n\u22a2 (fun y => p \u2022 y + x) '' B p \u2264\u1d50[volume] B p\n[PROOFSTEP]\napply (HasSubset.Subset.eventuallyLE this).congr EventuallyEq.rfl\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis\u271d : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nthis : f '' B p \u2286 blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)) atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\n\u22a2 (blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)) atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n) =\u1d50[volume] B p\n[PROOFSTEP]\nexact\n blimsup_thickening_mul_ae_eq \u03bc (fun n => 0 < n \u2227 p\u2223\u2223n) (fun n => {y | addOrderOf y = n}) (Nat.cast_pos.mpr hp.pos) _\n h\u03b4\n[GOAL]\ncase mk\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\n\u22a2 f '' B p \u2286 blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)) atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\n[PROOFSTEP]\nrefine' (SupHom.apply_blimsup_le (sSupHom.setImage f)).trans (mono_blimsup _)\n[GOAL]\ncase mk\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\n\u22a2 \u2200 (x : \u2115), 0 < x \u2227 p \u2223 x \u2227 \u00acp * p \u2223 x \u2192 (\u2191(sSupHom.setImage f) \u2218 X) x \u2264 approxAddOrderOf \ud835\udd4a x (\u2191p * \u03b4 x)\n[PROOFSTEP]\nrintro n \u27e8hn, h_div, h_ndiv\u27e9\n[GOAL]\ncase mk.intro.intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_div : p \u2223 n\nh_ndiv : \u00acp * p \u2223 n\n\u22a2 (\u2191(sSupHom.setImage f) \u2218 X) n \u2264 approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)\n[PROOFSTEP]\nhave h_cop : (addOrderOf x).coprime (n / p) := by\n obtain \u27e8q, rfl\u27e9 := h_div\n rw [hu\u2080, Subtype.coe_mk, hp.coprime_iff_not_dvd, q.mul_div_cancel_left hp.pos]\n exact fun contra => h_ndiv (mul_dvd_mul_left p contra)\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_div : p \u2223 n\nh_ndiv : \u00acp * p \u2223 n\n\u22a2 Nat.coprime (addOrderOf x) (n / p)\n[PROOFSTEP]\nobtain \u27e8q, rfl\u27e9 := h_div\n[GOAL]\ncase intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nq : \u2115\nhn : 0 < p * q\nh_ndiv : \u00acp * p \u2223 p * q\n\u22a2 Nat.coprime (addOrderOf x) (p * q / p)\n[PROOFSTEP]\nrw [hu\u2080, Subtype.coe_mk, hp.coprime_iff_not_dvd, q.mul_div_cancel_left hp.pos]\n[GOAL]\ncase intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nq : \u2115\nhn : 0 < p * q\nh_ndiv : \u00acp * p \u2223 p * q\n\u22a2 \u00acp \u2223 q\n[PROOFSTEP]\nexact fun contra => h_ndiv (mul_dvd_mul_left p contra)\n[GOAL]\ncase mk.intro.intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_div : p \u2223 n\nh_ndiv : \u00acp * p \u2223 n\nh_cop : Nat.coprime (addOrderOf x) (n / p)\n\u22a2 (\u2191(sSupHom.setImage f) \u2218 X) n \u2264 approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)\n[PROOFSTEP]\nreplace h_div : n / p * p = n := Nat.div_mul_cancel h_div\n[GOAL]\ncase mk.intro.intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_ndiv : \u00acp * p \u2223 n\nh_cop : Nat.coprime (addOrderOf x) (n / p)\nh_div : n / p * p = n\n\u22a2 (\u2191(sSupHom.setImage f) \u2218 X) n \u2264 approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)\n[PROOFSTEP]\nhave hf : f = (fun y => x + y) \u2218 fun y => p \u2022 y := by ext; simp [add_comm x]; ac_rfl\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_ndiv : \u00acp * p \u2223 n\nh_cop : Nat.coprime (addOrderOf x) (n / p)\nh_div : n / p * p = n\n\u22a2 f = (fun y => x + y) \u2218 fun y => p \u2022 y\n[PROOFSTEP]\next\n[GOAL]\ncase h\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_ndiv : \u00acp * p \u2223 n\nh_cop : Nat.coprime (addOrderOf x) (n / p)\nh_div : n / p * p = n\nx\u271d : \ud835\udd4a\n\u22a2 f x\u271d = ((fun y => x + y) \u2218 fun y => p \u2022 y) x\u271d\n[PROOFSTEP]\nsimp [add_comm x]\n[GOAL]\ncase h\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_ndiv : \u00acp * p \u2223 n\nh_cop : Nat.coprime (addOrderOf x) (n / p)\nh_div : n / p * p = n\nx\u271d : \ud835\udd4a\n\u22a2 p \u2022 x\u271d + \u2191((\u2191p)\u207b\u00b9 * T) = \u2191((\u2191p)\u207b\u00b9 * T) + p \u2022 x\u271d\n[PROOFSTEP]\nac_rfl\n[GOAL]\ncase mk.intro.intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_ndiv : \u00acp * p \u2223 n\nh_cop : Nat.coprime (addOrderOf x) (n / p)\nh_div : n / p * p = n\nhf : f = (fun y => x + y) \u2218 fun y => p \u2022 y\n\u22a2 (\u2191(sSupHom.setImage f) \u2218 X) n \u2264 approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)\n[PROOFSTEP]\nsimp only at hf \n[GOAL]\ncase mk.intro.intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_ndiv : \u00acp * p \u2223 n\nh_cop : Nat.coprime (addOrderOf x) (n / p)\nh_div : n / p * p = n\nhf : (fun y => p \u2022 y + \u2191(\u21911 / \u2191p * T)) = (fun y => \u2191(\u21911 / \u2191p * T) + y) \u2218 fun y => p \u2022 y\n\u22a2 (\u2191(sSupHom.setImage f) \u2218 X) n \u2264 approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)\n[PROOFSTEP]\nsimp_rw [Function.comp_apply, le_eq_subset]\n[GOAL]\ncase mk.intro.intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_ndiv : \u00acp * p \u2223 n\nh_cop : Nat.coprime (addOrderOf x) (n / p)\nh_div : n / p * p = n\nhf : (fun y => p \u2022 y + \u2191(\u21911 / \u2191p * T)) = (fun y => \u2191(\u21911 / \u2191p * T) + y) \u2218 fun y => p \u2022 y\n\u22a2 \u2191(sSupHom.setImage fun y => p \u2022 y + \u2191(\u21911 / \u2191p * T)) (approxAddOrderOf \ud835\udd4a n (\u03b4 n)) \u2286 approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)\n[PROOFSTEP]\nrw [sSupHom.setImage_toFun, hf, image_comp]\n[GOAL]\ncase mk.intro.intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_ndiv : \u00acp * p \u2223 n\nh_cop : Nat.coprime (addOrderOf x) (n / p)\nh_div : n / p * p = n\nhf : (fun y => p \u2022 y + \u2191(\u21911 / \u2191p * T)) = (fun y => \u2191(\u21911 / \u2191p * T) + y) \u2218 fun y => p \u2022 y\n\u22a2 (fun y => \u2191(\u21911 / \u2191p * T) + y) '' ((fun y => p \u2022 y) '' approxAddOrderOf \ud835\udd4a n (\u03b4 n)) \u2286 approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)\n[PROOFSTEP]\nhave := @monotone_image \ud835\udd4a \ud835\udd4a fun y => x + y\n[GOAL]\ncase mk.intro.intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis\u271d : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_ndiv : \u00acp * p \u2223 n\nh_cop : Nat.coprime (addOrderOf x) (n / p)\nh_div : n / p * p = n\nhf : (fun y => p \u2022 y + \u2191(\u21911 / \u2191p * T)) = (fun y => \u2191(\u21911 / \u2191p * T) + y) \u2218 fun y => p \u2022 y\nthis : Monotone (image fun y => x + y)\n\u22a2 (fun y => \u2191(\u21911 / \u2191p * T) + y) '' ((fun y => p \u2022 y) '' approxAddOrderOf \ud835\udd4a n (\u03b4 n)) \u2286 approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)\n[PROOFSTEP]\nspecialize this (approxAddOrderOf.image_nsmul_subset (\u03b4 n) (n / p) hp.pos)\n[GOAL]\ncase mk.intro.intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis\u271d : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_ndiv : \u00acp * p \u2223 n\nh_cop : Nat.coprime (addOrderOf x) (n / p)\nh_div : n / p * p = n\nhf : (fun y => p \u2022 y + \u2191(\u21911 / \u2191p * T)) = (fun y => \u2191(\u21911 / \u2191p * T) + y) \u2218 fun y => p \u2022 y\nthis :\n (fun y => x + y) '' ((fun y => p \u2022 y) '' approxAddOrderOf \ud835\udd4a (n / p * p) (\u03b4 n)) \u2264\n (fun y => x + y) '' approxAddOrderOf \ud835\udd4a (n / p) (\u2191p * \u03b4 n)\n\u22a2 (fun y => \u2191(\u21911 / \u2191p * T) + y) '' ((fun y => p \u2022 y) '' approxAddOrderOf \ud835\udd4a n (\u03b4 n)) \u2286 approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)\n[PROOFSTEP]\nsimp only [h_div] at this \u22a2\n[GOAL]\ncase mk.intro.intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis\u271d : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_ndiv : \u00acp * p \u2223 n\nh_cop : Nat.coprime (addOrderOf x) (n / p)\nh_div : n / p * p = n\nhf : (fun y => p \u2022 y + \u2191(\u21911 / \u2191p * T)) = (fun y => \u2191(\u21911 / \u2191p * T) + y) \u2218 fun y => p \u2022 y\nthis :\n (fun a => \u2191(\u21911 / \u2191p * T) + a) '' ((fun y => p \u2022 y) '' approxAddOrderOf \ud835\udd4a n (\u03b4 n)) \u2264\n (fun a => \u2191(\u21911 / \u2191p * T) + a) '' approxAddOrderOf \ud835\udd4a (n / p) (\u2191p * \u03b4 n)\n\u22a2 (fun y => \u2191(\u21911 / \u2191p * T) + y) '' ((fun y => p \u2022 y) '' approxAddOrderOf \ud835\udd4a n (\u03b4 n)) \u2286 approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)\n[PROOFSTEP]\nrefine' this.trans _\n[GOAL]\ncase mk.intro.intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis\u271d : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_ndiv : \u00acp * p \u2223 n\nh_cop : Nat.coprime (addOrderOf x) (n / p)\nh_div : n / p * p = n\nhf : (fun y => p \u2022 y + \u2191(\u21911 / \u2191p * T)) = (fun y => \u2191(\u21911 / \u2191p * T) + y) \u2218 fun y => p \u2022 y\nthis :\n (fun a => \u2191(\u21911 / \u2191p * T) + a) '' ((fun y => p \u2022 y) '' approxAddOrderOf \ud835\udd4a n (\u03b4 n)) \u2264\n (fun a => \u2191(\u21911 / \u2191p * T) + a) '' approxAddOrderOf \ud835\udd4a (n / p) (\u2191p * \u03b4 n)\n\u22a2 (fun a => \u2191(\u21911 / \u2191p * T) + a) '' approxAddOrderOf \ud835\udd4a (n / p) (\u2191p * \u03b4 n) \u2264 approxAddOrderOf \ud835\udd4a n (\u2191p * \u03b4 n)\n[PROOFSTEP]\nconvert approxAddOrderOf.vadd_subset_of_coprime (p * \u03b4 n) h_cop\n[GOAL]\ncase h.e'_2.h.e'_3\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis\u271d : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\np : \u2115\nhp : Nat.Prime p\nx : \ud835\udd4a := u { val := p, property := hp }\nf : \ud835\udd4a \u2192 \ud835\udd4a := fun y => p \u2022 y + x\nn : \u2115\nhn : 0 < n\nh_ndiv : \u00acp * p \u2223 n\nh_cop : Nat.coprime (addOrderOf x) (n / p)\nh_div : n / p * p = n\nhf : (fun y => p \u2022 y + \u2191(\u21911 / \u2191p * T)) = (fun y => \u2191(\u21911 / \u2191p * T) + y) \u2218 fun y => p \u2022 y\nthis :\n (fun a => \u2191(\u21911 / \u2191p * T) + a) '' ((fun y => p \u2022 y) '' approxAddOrderOf \ud835\udd4a n (\u03b4 n)) \u2264\n (fun a => \u2191(\u21911 / \u2191p * T) + a) '' approxAddOrderOf \ud835\udd4a (n / p) (\u2191p * \u03b4 n)\n\u22a2 n = addOrderOf x * (n / p)\n[PROOFSTEP]\nrw [hu\u2080, Subtype.coe_mk, mul_comm p, h_div]\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, \u00acE x) \u2228 \u2200\u1d50 (x : \ud835\udd4a) \u2202\u03bc, E x\n[PROOFSTEP]\nchange (\u2200\u1d50 x, x \u2209 E) \u2228 E \u2208 volume.ae\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\n\u22a2 (\u2200\u1d50 (x : \ud835\udd4a), \u00acx \u2208 E) \u2228 E \u2208 Measure.ae volume\n[PROOFSTEP]\nrw [\u2190 eventuallyEq_empty, \u2190 eventuallyEq_univ]\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\n\u22a2 E =\u1d50[volume] \u2205 \u2228 E =\u1d50[volume] univ\n[PROOFSTEP]\nhave hC : \u2200 p : Nat.Primes, u p +\u1d65 C p = C p := by\n intro p\n let e := (AddAction.toPerm (u p) : Equiv.Perm \ud835\udd4a).toOrderIsoSet\n change e (C p) = C p\n rw [OrderIso.apply_blimsup e, \u2190 hu\u2080 p]\n exact blimsup_congr (eventually_of_forall fun n hn => approxAddOrderOf.vadd_eq_of_mul_dvd (\u03b4 n) hn.1 hn.2)\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\n\u22a2 \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\n[PROOFSTEP]\nintro p\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\np : Nat.Primes\n\u22a2 u p +\u1d65 C \u2191p = C \u2191p\n[PROOFSTEP]\nlet e := (AddAction.toPerm (u p) : Equiv.Perm \ud835\udd4a).toOrderIsoSet\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\np : Nat.Primes\ne : Set \ud835\udd4a \u2243o Set \ud835\udd4a := Equiv.toOrderIsoSet (AddAction.toPerm (u p))\n\u22a2 u p +\u1d65 C \u2191p = C \u2191p\n[PROOFSTEP]\nchange e (C p) = C p\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\np : Nat.Primes\ne : Set \ud835\udd4a \u2243o Set \ud835\udd4a := Equiv.toOrderIsoSet (AddAction.toPerm (u p))\n\u22a2 \u2191e (C \u2191p) = C \u2191p\n[PROOFSTEP]\nrw [OrderIso.apply_blimsup e, \u2190 hu\u2080 p]\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\np : Nat.Primes\ne : Set \ud835\udd4a \u2243o Set \ud835\udd4a := Equiv.toOrderIsoSet (AddAction.toPerm (u p))\n\u22a2 (blimsup (\u2191e \u2218 X) atTop fun n => 0 < n \u2227 addOrderOf (u p) ^ 2 \u2223 n) = C (addOrderOf (u p))\n[PROOFSTEP]\nexact blimsup_congr (eventually_of_forall fun n hn => approxAddOrderOf.vadd_eq_of_mul_dvd (\u03b4 n) hn.1 hn.2)\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\n\u22a2 E =\u1d50[volume] \u2205 \u2228 E =\u1d50[volume] univ\n[PROOFSTEP]\nby_cases h : \u2200 p : Nat.Primes, A p =\u1d50[\u03bc] (\u2205 : Set \ud835\udd4a) \u2227 B p =\u1d50[\u03bc] (\u2205 : Set \ud835\udd4a)\n[GOAL]\ncase pos\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\nh : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205\n\u22a2 E =\u1d50[volume] \u2205 \u2228 E =\u1d50[volume] univ\n[PROOFSTEP]\nreplace h : \u2200 p : Nat.Primes, (u p +\u1d65 E : Set _) =\u1d50[\u03bc] E\n[GOAL]\ncase h\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\nh : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205\n\u22a2 \u2200 (p : Nat.Primes), u p +\u1d65 E =\u1d50[\u03bc] E\n[PROOFSTEP]\nintro p\n[GOAL]\ncase h\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\nh : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205\np : Nat.Primes\n\u22a2 u p +\u1d65 E =\u1d50[\u03bc] E\n[PROOFSTEP]\nreplace hE\u2082 : E =\u1d50[\u03bc] C p := hE\u2082 p (h p)\n[GOAL]\ncase h\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\nh : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205\np : Nat.Primes\nhE\u2082 : E =\u1d50[\u03bc] C \u2191p\n\u22a2 u p +\u1d65 E =\u1d50[\u03bc] E\n[PROOFSTEP]\nhave h_qmp : MeasureTheory.Measure.QuasiMeasurePreserving ((\u00b7 +\u1d65 \u00b7) (-u p)) \u03bc \u03bc :=\n (measurePreserving_vadd _ \u03bc).quasiMeasurePreserving\n[GOAL]\ncase h\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\nh : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205\np : Nat.Primes\nhE\u2082 : E =\u1d50[\u03bc] C \u2191p\nh_qmp : Measure.QuasiMeasurePreserving ((fun x x_1 => x +\u1d65 x_1) (-u p))\n\u22a2 u p +\u1d65 E =\u1d50[\u03bc] E\n[PROOFSTEP]\nrefine' (h_qmp.vadd_ae_eq_of_ae_eq (u p) hE\u2082).trans (ae_eq_trans _ hE\u2082.symm)\n[GOAL]\ncase h\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\nh : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205\np : Nat.Primes\nhE\u2082 : E =\u1d50[\u03bc] C \u2191p\nh_qmp : Measure.QuasiMeasurePreserving ((fun x x_1 => x +\u1d65 x_1) (-u p))\n\u22a2 u p +\u1d65 C \u2191p =\u1d50[\u03bc] C \u2191p\n[PROOFSTEP]\nrw [hC]\n[GOAL]\ncase pos\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\nh : \u2200 (p : Nat.Primes), u p +\u1d65 E =\u1d50[\u03bc] E\n\u22a2 E =\u1d50[volume] \u2205 \u2228 E =\u1d50[volume] univ\n[PROOFSTEP]\nexact ae_empty_or_univ_of_forall_vadd_ae_eq_self hE\u2080 h hu\n[GOAL]\ncase neg\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\nh : \u00ac\u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205\n\u22a2 E =\u1d50[volume] \u2205 \u2228 E =\u1d50[volume] univ\n[PROOFSTEP]\nright\n[GOAL]\ncase neg.h\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\nh : \u00ac\u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205\n\u22a2 E =\u1d50[volume] univ\n[PROOFSTEP]\nsimp only [not_forall, not_and_or] at h \n[GOAL]\ncase neg.h\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\nh :\n \u2203 x,\n \u00ac(blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun n => 0 < n \u2227 \u00ac\u2191x \u2223 n) =\u1d50[volume] \u2205 \u2228\n \u00ac(blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun n => 0 < n \u2227 \u2191x \u2223 n \u2227 \u00ac\u2191x * \u2191x \u2223 n) =\u1d50[volume] \u2205\n\u22a2 E =\u1d50[volume] univ\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := h\n[GOAL]\ncase neg.h.intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\np : Nat.Primes\nhp :\n \u00ac(blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun n => 0 < n \u2227 \u00ac\u2191p \u2223 n) =\u1d50[volume] \u2205 \u2228\n \u00ac(blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun n => 0 < n \u2227 \u2191p \u2223 n \u2227 \u00ac\u2191p * \u2191p \u2223 n) =\u1d50[volume] \u2205\n\u22a2 E =\u1d50[volume] univ\n[PROOFSTEP]\nrw [hE\u2081 p]\n[GOAL]\ncase neg.h.intro\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\np : Nat.Primes\nhp :\n \u00ac(blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun n => 0 < n \u2227 \u00ac\u2191p \u2223 n) =\u1d50[volume] \u2205 \u2228\n \u00ac(blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun n => 0 < n \u2227 \u2191p \u2223 n \u2227 \u00ac\u2191p * \u2191p \u2223 n) =\u1d50[volume] \u2205\n\u22a2 A \u2191p \u222a B \u2191p \u222a C \u2191p =\u1d50[volume] univ\n[PROOFSTEP]\ncases hp\n[GOAL]\ncase neg.h.intro.inl\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\np : Nat.Primes\nh\u271d : \u00ac(blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun n => 0 < n \u2227 \u00ac\u2191p \u2223 n) =\u1d50[volume] \u2205\n\u22a2 A \u2191p \u222a B \u2191p \u222a C \u2191p =\u1d50[volume] univ\n[PROOFSTEP]\ncases' hA p with _ h\n[GOAL]\ncase neg.h.intro.inl.inl\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\np : Nat.Primes\nh\u271d\u00b9 : \u00ac(blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun n => 0 < n \u2227 \u00ac\u2191p \u2223 n) =\u1d50[volume] \u2205\nh\u271d : A \u2191p =\u1d50[\u03bc] \u2205\n\u22a2 A \u2191p \u222a B \u2191p \u222a C \u2191p =\u1d50[volume] univ\n[PROOFSTEP]\ncontradiction\n -- Porting note: was `simp only [h, union_ae_eq_univ_of_ae_eq_univ_left]`\n[GOAL]\ncase neg.h.intro.inl.inr\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\np : Nat.Primes\nh\u271d : \u00ac(blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun n => 0 < n \u2227 \u00ac\u2191p \u2223 n) =\u1d50[volume] \u2205\nh : A \u2191p =\u1d50[\u03bc] univ\n\u22a2 A \u2191p \u222a B \u2191p \u222a C \u2191p =\u1d50[volume] univ\n[PROOFSTEP]\nhave := union_ae_eq_univ_of_ae_eq_univ_left (t := B \u2191p) h\n[GOAL]\ncase neg.h.intro.inl.inr\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis\u271d : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\np : Nat.Primes\nh\u271d : \u00ac(blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun n => 0 < n \u2227 \u00ac\u2191p \u2223 n) =\u1d50[volume] \u2205\nh : A \u2191p =\u1d50[\u03bc] univ\nthis : A \u2191p \u222a B \u2191p =\u1d50[\u03bc] univ\n\u22a2 A \u2191p \u222a B \u2191p \u222a C \u2191p =\u1d50[volume] univ\n[PROOFSTEP]\nexact union_ae_eq_univ_of_ae_eq_univ_left (t := C \u2191p) this\n[GOAL]\ncase neg.h.intro.inr\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\np : Nat.Primes\nh\u271d : \u00ac(blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun n => 0 < n \u2227 \u2191p \u2223 n \u2227 \u00ac\u2191p * \u2191p \u2223 n) =\u1d50[volume] \u2205\n\u22a2 A \u2191p \u222a B \u2191p \u222a C \u2191p =\u1d50[volume] univ\n[PROOFSTEP]\ncases' hB p with _ h\n[GOAL]\ncase neg.h.intro.inr.inl\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\np : Nat.Primes\nh\u271d\u00b9 : \u00ac(blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun n => 0 < n \u2227 \u2191p \u2223 n \u2227 \u00ac\u2191p * \u2191p \u2223 n) =\u1d50[volume] \u2205\nh\u271d : B \u2191p =\u1d50[\u03bc] \u2205\n\u22a2 A \u2191p \u222a B \u2191p \u222a C \u2191p =\u1d50[volume] univ\n[PROOFSTEP]\ncontradiction\n -- Porting note: was\n -- `simp only [h, union_ae_eq_univ_of_ae_eq_univ_left, union_ae_eq_univ_of_ae_eq_univ_right]`\n[GOAL]\ncase neg.h.intro.inr.inr\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\np : Nat.Primes\nh\u271d : \u00ac(blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun n => 0 < n \u2227 \u2191p \u2223 n \u2227 \u00ac\u2191p * \u2191p \u2223 n) =\u1d50[volume] \u2205\nh : B \u2191p =\u1d50[\u03bc] univ\n\u22a2 A \u2191p \u222a B \u2191p \u222a C \u2191p =\u1d50[volume] univ\n[PROOFSTEP]\nhave := union_ae_eq_univ_of_ae_eq_univ_right (s := A \u2191p) h\n[GOAL]\ncase neg.h.intro.inr.inr\nT : \u211d\nhT : Fact (0 < T)\n\u03b4 : \u2115 \u2192 \u211d\nh\u03b4 : Tendsto \u03b4 atTop (\ud835\udcdd 0)\nthis\u271d : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p => Irreducible p\n\u03bc : Measure \ud835\udd4a := volume\nu : Nat.Primes \u2192 \ud835\udd4a := fun p => \u2191(\u21911 / \u2191\u2191p * T)\nhu\u2080 : \u2200 (p : Nat.Primes), addOrderOf (u p) = \u2191p\nhu : Tendsto (addOrderOf \u2218 u) atTop atTop\nE : Set \ud835\udd4a := addWellApproximable \ud835\udd4a \u03b4\nX : \u2115 \u2192 Set \ud835\udd4a := fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)\nA : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 \u00acp \u2223 n\nB : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p \u2223 n \u2227 \u00acp * p \u2223 n\nC : \u2115 \u2192 Set \ud835\udd4a := fun p => blimsup X atTop fun n => 0 < n \u2227 p ^ 2 \u2223 n\nhA\u2080 : \u2200 (p : \u2115), MeasurableSet (A p)\nhB\u2080 : \u2200 (p : \u2115), MeasurableSet (B p)\nhE\u2080 : NullMeasurableSet E\nhE\u2081 : \u2200 (p : \u2115), E = A p \u222a B p \u222a C p\nhE\u2082 : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2227 B \u2191p =\u1d50[\u03bc] \u2205 \u2192 E =\u1d50[\u03bc] C \u2191p\nhA : \u2200 (p : Nat.Primes), A \u2191p =\u1d50[\u03bc] \u2205 \u2228 A \u2191p =\u1d50[\u03bc] univ\nhB : \u2200 (p : Nat.Primes), B \u2191p =\u1d50[\u03bc] \u2205 \u2228 B \u2191p =\u1d50[\u03bc] univ\nhC : \u2200 (p : Nat.Primes), u p +\u1d65 C \u2191p = C \u2191p\np : Nat.Primes\nh\u271d : \u00ac(blimsup (fun n => approxAddOrderOf \ud835\udd4a n (\u03b4 n)) atTop fun n => 0 < n \u2227 \u2191p \u2223 n \u2227 \u00ac\u2191p * \u2191p \u2223 n) =\u1d50[volume] \u2205\nh : B \u2191p =\u1d50[\u03bc] univ\nthis : A \u2191p \u222a B \u2191p =\u1d50[\u03bc] univ\n\u22a2 A \u2191p \u222a B \u2191p \u222a C \u2191p =\u1d50[volume] univ\n[PROOFSTEP]\nexact union_ae_eq_univ_of_ae_eq_univ_left (t := C \u2191p) this\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\n\u22a2 \u2203 j, j \u2208 Icc 1 n \u2227 \u2016j \u2022 \u03be\u2016 \u2264 \u03b4\n[PROOFSTEP]\nhave : IsFiniteMeasure \u03bc := CompactSpace.isFiniteMeasure\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis : IsFiniteMeasure \u03bc\n\u22a2 \u2203 j, j \u2208 Icc 1 n \u2227 \u2016j \u2022 \u03be\u2016 \u2264 \u03b4\n[PROOFSTEP]\nlet B : Icc 0 n \u2192 Set A := fun j \u21a6 closedBall ((j : \u2115) \u2022 \u03be) (\u03b4 / 2)\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\n\u22a2 \u2203 j, j \u2208 Icc 1 n \u2227 \u2016j \u2022 \u03be\u2016 \u2264 \u03b4\n[PROOFSTEP]\nhave hB : \u2200 j, IsClosed (B j) := fun j \u21a6 isClosed_ball\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\n\u22a2 \u2203 j, j \u2208 Icc 1 n \u2227 \u2016j \u2022 \u03be\u2016 \u2264 \u03b4\n[PROOFSTEP]\nsuffices : \u00acPairwise (Disjoint on B)\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis\u271d : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nthis : \u00acPairwise (Disjoint on B)\n\u22a2 \u2203 j, j \u2208 Icc 1 n \u2227 \u2016j \u2022 \u03be\u2016 \u2264 \u03b4\n[PROOFSTEP]\nobtain \u27e8i, j, hij, x, hx\u27e9 := exists_lt_mem_inter_of_not_pairwise_disjoint this\n[GOAL]\ncase intro.intro.intro.intro\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis\u271d : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nthis : \u00acPairwise (Disjoint on B)\ni j : \u2191(Icc 0 n)\nhij : i < j\nx : A\nhx : x \u2208 B i \u2229 B j\n\u22a2 \u2203 j, j \u2208 Icc 1 n \u2227 \u2016j \u2022 \u03be\u2016 \u2264 \u03b4\n[PROOFSTEP]\nrefine' \u27e8j - i, \u27e8le_tsub_of_add_le_left hij, _\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis\u271d : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nthis : \u00acPairwise (Disjoint on B)\ni j : \u2191(Icc 0 n)\nhij : i < j\nx : A\nhx : x \u2208 B i \u2229 B j\n\u22a2 \u2191j - \u2191i \u2264 n\n[PROOFSTEP]\nsimpa only [tsub_le_iff_right] using j.property.2.trans le_self_add\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis\u271d : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nthis : \u00acPairwise (Disjoint on B)\ni j : \u2191(Icc 0 n)\nhij : i < j\nx : A\nhx : x \u2208 B i \u2229 B j\n\u22a2 \u2016(\u2191j - \u2191i) \u2022 \u03be\u2016 \u2264 \u03b4\n[PROOFSTEP]\nrw [sub_nsmul _ (Subtype.coe_le_coe.mpr hij.le), \u2190 sub_eq_add_neg, \u2190 dist_eq_norm]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis\u271d : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nthis : \u00acPairwise (Disjoint on B)\ni j : \u2191(Icc 0 n)\nhij : i < j\nx : A\nhx : x \u2208 B i \u2229 B j\n\u22a2 dist (\u2191j \u2022 \u03be) (\u2191i \u2022 \u03be) \u2264 \u03b4\n[PROOFSTEP]\nrefine' (dist_triangle (\u2191j \u2022 \u03be) x (\u2191i \u2022 \u03be)).trans _\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis\u271d : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nthis : \u00acPairwise (Disjoint on B)\ni j : \u2191(Icc 0 n)\nhij : i < j\nx : A\nhx : x \u2208 B i \u2229 B j\n\u22a2 dist (\u2191j \u2022 \u03be) x + dist x (\u2191i \u2022 \u03be) \u2264 \u03b4\n[PROOFSTEP]\nlinarith [mem_closedBall.mp hx.1, mem_closedBall'.mp hx.2]\n[GOAL]\ncase this\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\n\u22a2 \u00acPairwise (Disjoint on B)\n[PROOFSTEP]\nby_contra h\n[GOAL]\ncase this\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\n\u22a2 False\n[PROOFSTEP]\napply hn.ne'\n[GOAL]\ncase this\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\n\u22a2 n = 0\n[PROOFSTEP]\nhave h' : \u22c3 j, B j = univ := by\n rw [\u2190 (isClosed_iUnion hB).measure_eq_univ_iff_eq (\u03bc := \u03bc)]\n refine' le_antisymm (\u03bc.mono (subset_univ _)) _\n simp_rw [measure_iUnion h (fun _ \u21a6 measurableSet_closedBall), tsum_fintype, \u03bc.addHaar_closedBall_center,\n Finset.sum_const, Finset.card_univ, Nat.card_fintypeIcc, tsub_zero]\n exact h\u03b4\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\n\u22a2 \u22c3 (j : \u2191(Icc 0 n)), B j = univ\n[PROOFSTEP]\nrw [\u2190 (isClosed_iUnion hB).measure_eq_univ_iff_eq (\u03bc := \u03bc)]\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u2191(Icc 0 n)), B i) = \u2191\u2191\u03bc univ\n[PROOFSTEP]\nrefine' le_antisymm (\u03bc.mono (subset_univ _)) _\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\n\u22a2 \u2191\u2191\u03bc univ \u2264 \u2191\u2191\u03bc (\u22c3 (i : \u2191(Icc 0 n)), B i)\n[PROOFSTEP]\nsimp_rw [measure_iUnion h (fun _ \u21a6 measurableSet_closedBall), tsum_fintype, \u03bc.addHaar_closedBall_center,\n Finset.sum_const, Finset.card_univ, Nat.card_fintypeIcc, tsub_zero]\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\n\u22a2 \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\n[PROOFSTEP]\nexact h\u03b4\n[GOAL]\ncase this\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\nh' : \u22c3 (j : \u2191(Icc 0 n)), B j = univ\n\u22a2 n = 0\n[PROOFSTEP]\nreplace h\u03b4 : 0 \u2264 \u03b4 / 2 := by\n by_contra contra\n suffices : \u03bc (closedBall 0 (\u03b4 / 2)) = 0\n \u00b7 apply isOpen_univ.measure_ne_zero \u03bc univ_nonempty $ le_zero_iff.mp $ le_trans h\u03b4 _\n simp [this]\n rw [not_le, \u2190 closedBall_eq_empty (x := (0 : A))] at contra \n simp [contra]\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\nh' : \u22c3 (j : \u2191(Icc 0 n)), B j = univ\n\u22a2 0 \u2264 \u03b4 / 2\n[PROOFSTEP]\nby_contra contra\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\nh' : \u22c3 (j : \u2191(Icc 0 n)), B j = univ\ncontra : \u00ac0 \u2264 \u03b4 / 2\n\u22a2 False\n[PROOFSTEP]\nsuffices : \u03bc (closedBall 0 (\u03b4 / 2)) = 0\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis\u271d : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\nh' : \u22c3 (j : \u2191(Icc 0 n)), B j = univ\ncontra : \u00ac0 \u2264 \u03b4 / 2\nthis : \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2)) = 0\n\u22a2 False\n[PROOFSTEP]\napply isOpen_univ.measure_ne_zero \u03bc univ_nonempty $ le_zero_iff.mp $ le_trans h\u03b4 _\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis\u271d : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\nh' : \u22c3 (j : \u2191(Icc 0 n)), B j = univ\ncontra : \u00ac0 \u2264 \u03b4 / 2\nthis : \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2)) = 0\n\u22a2 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2)) \u2264 0\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase this\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\nh' : \u22c3 (j : \u2191(Icc 0 n)), B j = univ\ncontra : \u00ac0 \u2264 \u03b4 / 2\n\u22a2 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2)) = 0\n[PROOFSTEP]\nrw [not_le, \u2190 closedBall_eq_empty (x := (0 : A))] at contra \n[GOAL]\ncase this\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nh\u03b4 : \u2191\u2191\u03bc univ \u2264 (n + 1) \u2022 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2))\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\nh' : \u22c3 (j : \u2191(Icc 0 n)), B j = univ\ncontra : closedBall 0 (\u03b4 / 2) = \u2205\n\u22a2 \u2191\u2191\u03bc (closedBall 0 (\u03b4 / 2)) = 0\n[PROOFSTEP]\nsimp [contra]\n[GOAL]\ncase this\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\nh' : \u22c3 (j : \u2191(Icc 0 n)), B j = univ\nh\u03b4 : 0 \u2264 \u03b4 / 2\n\u22a2 n = 0\n[PROOFSTEP]\nhave h'' : \u2200 j, (B j).Nonempty := by intro j; rwa [nonempty_closedBall]\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\nh' : \u22c3 (j : \u2191(Icc 0 n)), B j = univ\nh\u03b4 : 0 \u2264 \u03b4 / 2\n\u22a2 \u2200 (j : \u2191(Icc 0 n)), Set.Nonempty (B j)\n[PROOFSTEP]\nintro j\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\nh' : \u22c3 (j : \u2191(Icc 0 n)), B j = univ\nh\u03b4 : 0 \u2264 \u03b4 / 2\nj : \u2191(Icc 0 n)\n\u22a2 Set.Nonempty (B j)\n[PROOFSTEP]\nrwa [nonempty_closedBall]\n[GOAL]\ncase this\nT : \u211d\nhT : Fact (0 < T)\nA : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup A\ninst\u271d\u2074 : CompactSpace A\ninst\u271d\u00b3 : ConnectedSpace A\ninst\u271d\u00b2 : MeasurableSpace A\ninst\u271d\u00b9 : BorelSpace A\n\u03bc : Measure A\ninst\u271d : Measure.IsAddHaarMeasure \u03bc\n\u03be : A\nn : \u2115\nhn : 0 < n\n\u03b4 : \u211d\nthis : IsFiniteMeasure \u03bc\nB : \u2191(Icc 0 n) \u2192 Set A := fun j => closedBall (\u2191j \u2022 \u03be) (\u03b4 / 2)\nhB : \u2200 (j : \u2191(Icc 0 n)), IsClosed (B j)\nh : Pairwise (Disjoint on B)\nh' : \u22c3 (j : \u2191(Icc 0 n)), B j = univ\nh\u03b4 : 0 \u2264 \u03b4 / 2\nh'' : \u2200 (j : \u2191(Icc 0 n)), Set.Nonempty (B j)\n\u22a2 n = 0\n[PROOFSTEP]\nsimpa using subsingleton_of_disjoint_isClosed_iUnion_eq_univ h'' h hB h'\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03be : \ud835\udd4a\nn : \u2115\nhn : 0 < n\n\u22a2 \u2203 j, j \u2208 Icc 1 n \u2227 \u2016j \u2022 \u03be\u2016 \u2264 T / \u2191(n + 1)\n[PROOFSTEP]\napply NormedAddCommGroup.exists_norm_nsmul_le (\u03bc := volume) \u03be hn\n[GOAL]\ncase h\u03b4\nT : \u211d\nhT : Fact (0 < T)\n\u03be : \ud835\udd4a\nn : \u2115\nhn : 0 < n\n\u22a2 \u2191\u2191volume univ \u2264 (n + 1) \u2022 \u2191\u2191volume (closedBall 0 (T / \u2191(n + 1) / 2))\n[PROOFSTEP]\nrw [AddCircle.measure_univ, volume_closedBall, \u2190 ENNReal.ofReal_nsmul, mul_div_cancel' _ two_ne_zero,\n min_eq_right (div_le_self hT.out.le $ by simp), nsmul_eq_mul, mul_div_cancel' _ (Nat.cast_ne_zero.mpr n.succ_ne_zero)]\n[GOAL]\nT : \u211d\nhT : Fact (0 < T)\n\u03be : \ud835\udd4a\nn : \u2115\nhn : 0 < n\n\u22a2 1 \u2264 \u2191(n + 1)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.WellApproximable", "llama_tokens": 75600, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.4049846084547341}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.11751, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nk : A \u27f6 S.X\u2082\nx : A \u27f6 S.X\u2081\nhx : k = x \u226b S.f\n\u22a2 k \u226b S.g = 0\n[PROOFSTEP]\nrw [hx, assoc, S.zero, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nk : A \u27f6 S.X\u2082\nx : A \u27f6 S.X\u2081\nhx : k = x \u226b S.f\n\u22a2 liftK h k (_ : k \u226b S.g = 0) \u226b h.\u03c0 = 0\n[PROOFSTEP]\nrw [show 0 = (x \u226b h.f') \u226b h.\u03c0 by simp]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nk : A \u27f6 S.X\u2082\nx : A \u27f6 S.X\u2081\nhx : k = x \u226b S.f\n\u22a2 0 = (x \u226b f' h) \u226b h.\u03c0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nk : A \u27f6 S.X\u2082\nx : A \u27f6 S.X\u2081\nhx : k = x \u226b S.f\n\u22a2 liftK h k (_ : k \u226b S.g = 0) \u226b h.\u03c0 = (x \u226b f' h) \u226b h.\u03c0\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nk : A \u27f6 S.X\u2082\nx : A \u27f6 S.X\u2081\nhx : k = x \u226b S.f\n\u22a2 liftK h k (_ : k \u226b S.g = 0) = x \u226b f' h\n[PROOFSTEP]\nsimp only [\u2190 cancel_mono h.i, hx, liftK_i, assoc, f'_i]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhg : S.g = 0\n\u22a2 \ud835\udfd9 S.X\u2082 \u226b S.g = 0\n[PROOFSTEP]\nrw [hg, id_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhg : S.g = 0\n\u22a2 h.i \u226b liftK h (\ud835\udfd9 S.X\u2082) (_ : \ud835\udfd9 S.X\u2082 \u226b S.g = 0) = \ud835\udfd9 h.K\n[PROOFSTEP]\nsimp only [\u2190 cancel_mono h.i, id_comp, assoc, liftK_i, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhf : S.f = 0\n\u22a2 IsIso h.\u03c0\n[PROOFSTEP]\nhave \u27e8\u03c6, h\u03c6\u27e9 := CokernelCofork.IsColimit.desc' h.h\u03c0' (\ud835\udfd9 _) (by rw [\u2190 cancel_mono h.i, comp_id, f'_i, zero_comp, hf])\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhf : S.f = 0\n\u22a2 f' h \u226b \ud835\udfd9 h.K = 0\n[PROOFSTEP]\nrw [\u2190 cancel_mono h.i, comp_id, f'_i, zero_comp, hf]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhf : S.f = 0\n\u03c6 : (CokernelCofork.of\u03c0 h.\u03c0 (_ : f' h \u226b h.\u03c0 = 0)).pt \u27f6 h.K\nh\u03c6 : Cofork.\u03c0 (CokernelCofork.of\u03c0 h.\u03c0 (_ : f' h \u226b h.\u03c0 = 0)) \u226b \u03c6 = \ud835\udfd9 h.K\n\u22a2 IsIso h.\u03c0\n[PROOFSTEP]\ndsimp at h\u03c6 \n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhf : S.f = 0\n\u03c6 : (CokernelCofork.of\u03c0 h.\u03c0 (_ : f' h \u226b h.\u03c0 = 0)).pt \u27f6 h.K\nh\u03c6 : h.\u03c0 \u226b \u03c6 = \ud835\udfd9 h.K\n\u22a2 IsIso h.\u03c0\n[PROOFSTEP]\nexact \u27e8\u03c6, h\u03c6, by rw [\u2190 cancel_epi h.\u03c0, reassoc_of% h\u03c6, comp_id]\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhf : S.f = 0\n\u03c6 : (CokernelCofork.of\u03c0 h.\u03c0 (_ : f' h \u226b h.\u03c0 = 0)).pt \u27f6 h.K\nh\u03c6 : h.\u03c0 \u226b \u03c6 = \ud835\udfd9 h.K\n\u22a2 \u03c6 \u226b h.\u03c0 = \ud835\udfd9 h.H\n[PROOFSTEP]\nrw [\u2190 cancel_epi h.\u03c0, reassoc_of% h\u03c6, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.20415, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhg : S.g = 0\nc : CokernelCofork S.f\nhc : IsColimit c\n\u22a2 \ud835\udfd9 S.X\u2082 \u226b S.g = 0\n[PROOFSTEP]\nrw [id_comp, hg]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhg : S.g = 0\nc : CokernelCofork S.f\nhc : IsColimit c\n\u22a2 f' (ofIsColimitCokernelCofork S hg c hc) = S.f\n[PROOFSTEP]\nrw [\u2190 cancel_mono (ofIsColimitCokernelCofork S hg c hc).i, f'_i, ofIsColimitCokernelCofork_i]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhg : S.g = 0\nc : CokernelCofork S.f\nhc : IsColimit c\n\u22a2 S.f = S.f \u226b \ud835\udfd9 S.X\u2082\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhg : S.g = 0\nc : CokernelCofork S.f\nhc : IsColimit c\n\u22a2 S.f = S.f \u226b \ud835\udfd9 S.X\u2082\n[PROOFSTEP]\nrw [comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.25743, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhf : S.f = 0\nc : KernelFork S.g\nhc : IsLimit c\n\u22a2 (IsLimit.lift (IsLimit.ofIsoLimit hc (Fork.ext (Iso.refl c.pt))) (KernelFork.of\u03b9 S.f (_ : S.f \u226b S.g = 0)) \u226b \ud835\udfd9 c.pt) \u226b\n Fork.\u03b9 c =\n 0 \u226b Fork.\u03b9 c\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.25743, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhf : S.f = 0\nc : KernelFork S.g\nhc : IsLimit c\n\u22a2 ((IsLimit.lift hc (KernelFork.of\u03b9 S.f (_ : S.f \u226b S.g = 0)) \u226b \ud835\udfd9 c.pt) \u226b \ud835\udfd9 c.pt) \u226b Fork.\u03b9 c = 0 \u226b Fork.\u03b9 c\n[PROOFSTEP]\nsimp only [comp_id, zero_comp, Fork.IsLimit.lift_\u03b9, Fork.\u03b9_of\u03b9, hf]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.25743, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhf : S.f = 0\nc : KernelFork S.g\nhc : IsLimit c\n\u22a2 IsLimit.lift (IsLimit.ofIsoLimit hc (Fork.ext (Iso.refl c.pt))) (KernelFork.of\u03b9 S.f (_ : S.f \u226b S.g = 0)) \u226b Fork.\u03b9 c =\n 0 \u226b Fork.\u03b9 c\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.25743, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhf : S.f = 0\nc : KernelFork S.g\nhc : IsLimit c\n\u22a2 (IsLimit.lift hc (KernelFork.of\u03b9 S.f (_ : S.f \u226b S.g = 0)) \u226b \ud835\udfd9 c.pt) \u226b Fork.\u03b9 c = 0 \u226b Fork.\u03b9 c\n[PROOFSTEP]\nsimp only [comp_id, zero_comp, Fork.IsLimit.lift_\u03b9, Fork.\u03b9_of\u03b9, hf]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhf : S.f = 0\nc : KernelFork S.g\nhc : IsLimit c\n\u22a2 f' (ofIsLimitKernelFork S hf c hc) = 0\n[PROOFSTEP]\nrw [\u2190 cancel_mono (ofIsLimitKernelFork S hf c hc).i, f'_i, hf, zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.30745, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhf : S.f = 0\nhg : S.g = 0\n\u22a2 \ud835\udfd9 S.X\u2082 \u226b S.g = 0\n[PROOFSTEP]\nrw [id_comp, hg]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.30745, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhf : S.f = 0\nhg : S.g = 0\n\u22a2 IsLimit.lift (KernelFork.IsLimit.ofId S.g hg) (KernelFork.of\u03b9 S.f (_ : S.f \u226b S.g = 0)) \u226b \ud835\udfd9 S.X\u2082 = 0\n[PROOFSTEP]\nchange S.f \u226b \ud835\udfd9 _ = 0\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.30745, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhf : S.f = 0\nhg : S.g = 0\n\u22a2 S.f \u226b \ud835\udfd9 S.X\u2082 = 0\n[PROOFSTEP]\nsimp only [hf, zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nhf : S.f = 0\nhg : S.g = 0\n\u22a2 f' (ofZeros S hf hg) = 0\n[PROOFSTEP]\nrw [\u2190 cancel_mono ((ofZeros S hf hg).i), zero_comp, f'_i, hf]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.51232, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03c8\u2081 \u03c8\u2082 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\n\u22a2 \u03c8\u2081 = \u03c8\u2082\n[PROOFSTEP]\nhave hK : \u03c8\u2081.\u03c6K = \u03c8\u2082.\u03c6K := by rw [\u2190 cancel_mono h\u2082.i, commi, commi]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.51232, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03c8\u2081 \u03c8\u2082 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\n\u22a2 \u03c8\u2081.\u03c6K = \u03c8\u2082.\u03c6K\n[PROOFSTEP]\nrw [\u2190 cancel_mono h\u2082.i, commi, commi]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.51232, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03c8\u2081 \u03c8\u2082 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\nhK : \u03c8\u2081.\u03c6K = \u03c8\u2082.\u03c6K\n\u22a2 \u03c8\u2081 = \u03c8\u2082\n[PROOFSTEP]\nhave hH : \u03c8\u2081.\u03c6H = \u03c8\u2082.\u03c6H := by rw [\u2190 cancel_epi h\u2081.\u03c0, comm\u03c0, comm\u03c0, hK]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.51232, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03c8\u2081 \u03c8\u2082 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\nhK : \u03c8\u2081.\u03c6K = \u03c8\u2082.\u03c6K\n\u22a2 \u03c8\u2081.\u03c6H = \u03c8\u2082.\u03c6H\n[PROOFSTEP]\nrw [\u2190 cancel_epi h\u2081.\u03c0, comm\u03c0, comm\u03c0, hK]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.51232, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03c8\u2081 \u03c8\u2082 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\nhK : \u03c8\u2081.\u03c6K = \u03c8\u2082.\u03c6K\nhH : \u03c8\u2081.\u03c6H = \u03c8\u2082.\u03c6H\n\u22a2 \u03c8\u2081 = \u03c8\u2082\n[PROOFSTEP]\ncases \u03c8\u2081\n[GOAL]\ncase mk\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.51232, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03c8\u2082 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\n\u03c6K\u271d : h\u2081.K \u27f6 h\u2082.K\n\u03c6H\u271d : h\u2081.H \u27f6 h\u2082.H\ncommi\u271d : \u03c6K\u271d \u226b h\u2082.i = h\u2081.i \u226b \u03c6.\u03c4\u2082\ncommf'\u271d : LeftHomologyData.f' h\u2081 \u226b \u03c6K\u271d = \u03c6.\u03c4\u2081 \u226b LeftHomologyData.f' h\u2082\ncomm\u03c0\u271d : h\u2081.\u03c0 \u226b \u03c6H\u271d = \u03c6K\u271d \u226b h\u2082.\u03c0\nhK : (mk \u03c6K\u271d \u03c6H\u271d).\u03c6K = \u03c8\u2082.\u03c6K\nhH : (mk \u03c6K\u271d \u03c6H\u271d).\u03c6H = \u03c8\u2082.\u03c6H\n\u22a2 mk \u03c6K\u271d \u03c6H\u271d = \u03c8\u2082\n[PROOFSTEP]\ncases \u03c8\u2082\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.51232, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03c6K\u271d\u00b9 : h\u2081.K \u27f6 h\u2082.K\n\u03c6H\u271d\u00b9 : h\u2081.H \u27f6 h\u2082.H\ncommi\u271d\u00b9 : \u03c6K\u271d\u00b9 \u226b h\u2082.i = h\u2081.i \u226b \u03c6.\u03c4\u2082\ncommf'\u271d\u00b9 : LeftHomologyData.f' h\u2081 \u226b \u03c6K\u271d\u00b9 = \u03c6.\u03c4\u2081 \u226b LeftHomologyData.f' h\u2082\ncomm\u03c0\u271d\u00b9 : h\u2081.\u03c0 \u226b \u03c6H\u271d\u00b9 = \u03c6K\u271d\u00b9 \u226b h\u2082.\u03c0\n\u03c6K\u271d : h\u2081.K \u27f6 h\u2082.K\n\u03c6H\u271d : h\u2081.H \u27f6 h\u2082.H\ncommi\u271d : \u03c6K\u271d \u226b h\u2082.i = h\u2081.i \u226b \u03c6.\u03c4\u2082\ncommf'\u271d : LeftHomologyData.f' h\u2081 \u226b \u03c6K\u271d = \u03c6.\u03c4\u2081 \u226b LeftHomologyData.f' h\u2082\ncomm\u03c0\u271d : h\u2081.\u03c0 \u226b \u03c6H\u271d = \u03c6K\u271d \u226b h\u2082.\u03c0\nhK : (mk \u03c6K\u271d\u00b9 \u03c6H\u271d\u00b9).\u03c6K = (mk \u03c6K\u271d \u03c6H\u271d).\u03c6K\nhH : (mk \u03c6K\u271d\u00b9 \u03c6H\u271d\u00b9).\u03c6H = (mk \u03c6K\u271d \u03c6H\u271d).\u03c6H\n\u22a2 mk \u03c6K\u271d\u00b9 \u03c6H\u271d\u00b9 = mk \u03c6K\u271d \u03c6H\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.53678, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u22a2 LeftHomologyMapData \u03c6 h\u2081 h\u2082\n[PROOFSTEP]\nlet \u03c6K : h\u2081.K \u27f6 h\u2082.K := h\u2082.liftK (h\u2081.i \u226b \u03c6.\u03c4\u2082) (by rw [assoc, \u03c6.comm\u2082\u2083, h\u2081.wi_assoc, zero_comp])\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.53678, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u22a2 (h\u2081.i \u226b \u03c6.\u03c4\u2082) \u226b S\u2082.g = 0\n[PROOFSTEP]\nrw [assoc, \u03c6.comm\u2082\u2083, h\u2081.wi_assoc, zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.53678, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03c6K : h\u2081.K \u27f6 h\u2082.K := LeftHomologyData.liftK h\u2082 (h\u2081.i \u226b \u03c6.\u03c4\u2082) (_ : (h\u2081.i \u226b \u03c6.\u03c4\u2082) \u226b S\u2082.g = 0)\n\u22a2 LeftHomologyMapData \u03c6 h\u2081 h\u2082\n[PROOFSTEP]\nhave commf' : h\u2081.f' \u226b \u03c6K = \u03c6.\u03c4\u2081 \u226b h\u2082.f' := by\n rw [\u2190 cancel_mono h\u2082.i, assoc, assoc, LeftHomologyData.liftK_i, LeftHomologyData.f'_i_assoc, LeftHomologyData.f'_i,\n \u03c6.comm\u2081\u2082]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.53678, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03c6K : h\u2081.K \u27f6 h\u2082.K := LeftHomologyData.liftK h\u2082 (h\u2081.i \u226b \u03c6.\u03c4\u2082) (_ : (h\u2081.i \u226b \u03c6.\u03c4\u2082) \u226b S\u2082.g = 0)\n\u22a2 LeftHomologyData.f' h\u2081 \u226b \u03c6K = \u03c6.\u03c4\u2081 \u226b LeftHomologyData.f' h\u2082\n[PROOFSTEP]\nrw [\u2190 cancel_mono h\u2082.i, assoc, assoc, LeftHomologyData.liftK_i, LeftHomologyData.f'_i_assoc, LeftHomologyData.f'_i,\n \u03c6.comm\u2081\u2082]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.53678, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03c6K : h\u2081.K \u27f6 h\u2082.K := LeftHomologyData.liftK h\u2082 (h\u2081.i \u226b \u03c6.\u03c4\u2082) (_ : (h\u2081.i \u226b \u03c6.\u03c4\u2082) \u226b S\u2082.g = 0)\ncommf' : LeftHomologyData.f' h\u2081 \u226b \u03c6K = \u03c6.\u03c4\u2081 \u226b LeftHomologyData.f' h\u2082\n\u22a2 LeftHomologyMapData \u03c6 h\u2081 h\u2082\n[PROOFSTEP]\nlet \u03c6H : h\u2081.H \u27f6 h\u2082.H := h\u2081.descH (\u03c6K \u226b h\u2082.\u03c0) (by rw [reassoc_of% commf', h\u2082.f'_\u03c0, comp_zero])\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.53678, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03c6K : h\u2081.K \u27f6 h\u2082.K := LeftHomologyData.liftK h\u2082 (h\u2081.i \u226b \u03c6.\u03c4\u2082) (_ : (h\u2081.i \u226b \u03c6.\u03c4\u2082) \u226b S\u2082.g = 0)\ncommf' : LeftHomologyData.f' h\u2081 \u226b \u03c6K = \u03c6.\u03c4\u2081 \u226b LeftHomologyData.f' h\u2082\n\u22a2 LeftHomologyData.f' h\u2081 \u226b \u03c6K \u226b h\u2082.\u03c0 = 0\n[PROOFSTEP]\nrw [reassoc_of% commf', h\u2082.f'_\u03c0, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.53678, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03c6K : h\u2081.K \u27f6 h\u2082.K := LeftHomologyData.liftK h\u2082 (h\u2081.i \u226b \u03c6.\u03c4\u2082) (_ : (h\u2081.i \u226b \u03c6.\u03c4\u2082) \u226b S\u2082.g = 0)\ncommf' : LeftHomologyData.f' h\u2081 \u226b \u03c6K = \u03c6.\u03c4\u2081 \u226b LeftHomologyData.f' h\u2082\n\u03c6H : h\u2081.H \u27f6 h\u2082.H := LeftHomologyData.descH h\u2081 (\u03c6K \u226b h\u2082.\u03c0) (_ : LeftHomologyData.f' h\u2081 \u226b \u03c6K \u226b h\u2082.\u03c0 = 0)\n\u22a2 LeftHomologyMapData \u03c6 h\u2081 h\u2082\n[PROOFSTEP]\nexact \u27e8\u03c6K, \u03c6H, by simp, commf', by simp\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.53678, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03c6K : h\u2081.K \u27f6 h\u2082.K := LeftHomologyData.liftK h\u2082 (h\u2081.i \u226b \u03c6.\u03c4\u2082) (_ : (h\u2081.i \u226b \u03c6.\u03c4\u2082) \u226b S\u2082.g = 0)\ncommf' : LeftHomologyData.f' h\u2081 \u226b \u03c6K = \u03c6.\u03c4\u2081 \u226b LeftHomologyData.f' h\u2082\n\u03c6H : h\u2081.H \u27f6 h\u2082.H := LeftHomologyData.descH h\u2081 (\u03c6K \u226b h\u2082.\u03c0) (_ : LeftHomologyData.f' h\u2081 \u226b \u03c6K \u226b h\u2082.\u03c0 = 0)\n\u22a2 \u03c6K \u226b h\u2082.i = h\u2081.i \u226b \u03c6.\u03c4\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.53678, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03c6K : h\u2081.K \u27f6 h\u2082.K := LeftHomologyData.liftK h\u2082 (h\u2081.i \u226b \u03c6.\u03c4\u2082) (_ : (h\u2081.i \u226b \u03c6.\u03c4\u2082) \u226b S\u2082.g = 0)\ncommf' : LeftHomologyData.f' h\u2081 \u226b \u03c6K = \u03c6.\u03c4\u2081 \u226b LeftHomologyData.f' h\u2082\n\u03c6H : h\u2081.H \u27f6 h\u2082.H := LeftHomologyData.descH h\u2081 (\u03c6K \u226b h\u2082.\u03c0) (_ : LeftHomologyData.f' h\u2081 \u226b \u03c6K \u226b h\u2082.\u03c0 = 0)\n\u22a2 h\u2081.\u03c0 \u226b \u03c6H = \u03c6K \u226b h\u2082.\u03c0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03b3\u2081 \u03b3\u2082 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\neq : \u03b3\u2081 = \u03b3\u2082\n\u22a2 \u03b3\u2081.\u03c6H = \u03b3\u2082.\u03c6H\n[PROOFSTEP]\nrw [eq]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03b3\u2081 \u03b3\u2082 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\neq : \u03b3\u2081 = \u03b3\u2082\n\u22a2 \u03b3\u2081.\u03c6K = \u03b3\u2082.\u03c6K\n[PROOFSTEP]\nrw [eq]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.62763, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6\u271d : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03c6 : S\u2081 \u27f6 S\u2082\nhg\u2081 : S\u2081.g = 0\nc\u2081 : CokernelCofork S\u2081.f\nhc\u2081 : IsColimit c\u2081\nhg\u2082 : S\u2082.g = 0\nc\u2082 : CokernelCofork S\u2082.f\nhc\u2082 : IsColimit c\u2082\nf : c\u2081.pt \u27f6 c\u2082.pt\ncomm : \u03c6.\u03c4\u2082 \u226b Cofork.\u03c0 c\u2082 = Cofork.\u03c0 c\u2081 \u226b f\n\u22a2 LeftHomologyData.f' (LeftHomologyData.ofIsColimitCokernelCofork S\u2081 hg\u2081 c\u2081 hc\u2081) \u226b \u03c6.\u03c4\u2082 =\n \u03c6.\u03c4\u2081 \u226b LeftHomologyData.f' (LeftHomologyData.ofIsColimitCokernelCofork S\u2082 hg\u2082 c\u2082 hc\u2082)\n[PROOFSTEP]\nsimp only [LeftHomologyData.ofIsColimitCokernelCofork_f', \u03c6.comm\u2081\u2082]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.82184, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d : HasLeftHomology S\n\u22a2 Mono (iCycles S)\n[PROOFSTEP]\ndsimp only [iCycles]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.82184, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d : HasLeftHomology S\n\u22a2 Mono (leftHomologyData S).i\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.82564, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d : HasLeftHomology S\n\u22a2 Epi (leftHomology\u03c0 S)\n[PROOFSTEP]\ndsimp only [leftHomology\u03c0]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.82564, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d : HasLeftHomology S\n\u22a2 Epi (leftHomologyData S).\u03c0\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d : HasLeftHomology S\nA : C\nf\u2081 f\u2082 : leftHomology S \u27f6 A\n\u22a2 f\u2081 = f\u2082 \u2194 leftHomology\u03c0 S \u226b f\u2081 = leftHomology\u03c0 S \u226b f\u2082\n[PROOFSTEP]\nrw [cancel_epi]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d : HasLeftHomology S\nA : C\nf\u2081 f\u2082 : leftHomology S \u27f6 A\nh : leftHomology\u03c0 S \u226b f\u2081 = leftHomology\u03c0 S \u226b f\u2082\n\u22a2 f\u2081 = f\u2082\n[PROOFSTEP]\nsimpa only [leftHomology_ext_iff] using h\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d : HasLeftHomology S\nA : C\nf\u2081 f\u2082 : A \u27f6 cycles S\n\u22a2 f\u2081 = f\u2082 \u2194 f\u2081 \u226b iCycles S = f\u2082 \u226b iCycles S\n[PROOFSTEP]\nrw [cancel_mono]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d : HasLeftHomology S\nA : C\nf\u2081 f\u2082 : A \u27f6 cycles S\nh : f\u2081 \u226b iCycles S = f\u2082 \u226b iCycles S\n\u22a2 f\u2081 = f\u2082\n[PROOFSTEP]\nsimpa only [cycles_ext_iff] using h\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.86307, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d : HasLeftHomology S\nhg : S.g = 0\n\u22a2 cycles S \u2245 S.X\u2082\n[PROOFSTEP]\nhave := S.isIso_iCycles hg\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.86307, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d : HasLeftHomology S\nhg : S.g = 0\nthis : IsIso (iCycles S)\n\u22a2 cycles S \u2245 S.X\u2082\n[PROOFSTEP]\nexact asIso S.iCycles\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.89230, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d : HasLeftHomology S\nhf : S.f = 0\n\u22a2 cycles S \u2245 leftHomology S\n[PROOFSTEP]\nhave := S.isIso_leftHomology\u03c0 hf\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.89230, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d : HasLeftHomology S\nhf : S.f = 0\nthis : IsIso (leftHomology\u03c0 S)\n\u22a2 cycles S \u2245 leftHomology S\n[PROOFSTEP]\nexact asIso S.leftHomology\u03c0\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u22a2 LeftHomologyData.f' h\u2081 \u226b cyclesMap' \u03c6 h\u2081 h\u2082 = \u03c6.\u03c4\u2081 \u226b LeftHomologyData.f' h\u2082\n[PROOFSTEP]\nsimp only [\u2190 cancel_mono h\u2082.i, assoc, \u03c6.comm\u2081\u2082, cyclesMap'_i, LeftHomologyData.f'_i_assoc, LeftHomologyData.f'_i]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6\u2081 : S\u2081 \u27f6 S\u2082\n\u03c6\u2082 : S\u2082 \u27f6 S\u2083\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\nh\u2083 : LeftHomologyData S\u2083\n\u22a2 leftHomologyMap' (\u03c6\u2081 \u226b \u03c6\u2082) h\u2081 h\u2083 = leftHomologyMap' \u03c6\u2081 h\u2081 h\u2082 \u226b leftHomologyMap' \u03c6\u2082 h\u2082 h\u2083\n[PROOFSTEP]\nlet \u03b3\u2081 := leftHomologyMapData \u03c6\u2081 h\u2081 h\u2082\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6\u2081 : S\u2081 \u27f6 S\u2082\n\u03c6\u2082 : S\u2082 \u27f6 S\u2083\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\nh\u2083 : LeftHomologyData S\u2083\n\u03b3\u2081 : LeftHomologyMapData \u03c6\u2081 h\u2081 h\u2082 := leftHomologyMapData \u03c6\u2081 h\u2081 h\u2082\n\u22a2 leftHomologyMap' (\u03c6\u2081 \u226b \u03c6\u2082) h\u2081 h\u2083 = leftHomologyMap' \u03c6\u2081 h\u2081 h\u2082 \u226b leftHomologyMap' \u03c6\u2082 h\u2082 h\u2083\n[PROOFSTEP]\nlet \u03b3\u2082 := leftHomologyMapData \u03c6\u2082 h\u2082 h\u2083\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6\u2081 : S\u2081 \u27f6 S\u2082\n\u03c6\u2082 : S\u2082 \u27f6 S\u2083\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\nh\u2083 : LeftHomologyData S\u2083\n\u03b3\u2081 : LeftHomologyMapData \u03c6\u2081 h\u2081 h\u2082 := leftHomologyMapData \u03c6\u2081 h\u2081 h\u2082\n\u03b3\u2082 : LeftHomologyMapData \u03c6\u2082 h\u2082 h\u2083 := leftHomologyMapData \u03c6\u2082 h\u2082 h\u2083\n\u22a2 leftHomologyMap' (\u03c6\u2081 \u226b \u03c6\u2082) h\u2081 h\u2083 = leftHomologyMap' \u03c6\u2081 h\u2081 h\u2082 \u226b leftHomologyMap' \u03c6\u2082 h\u2082 h\u2083\n[PROOFSTEP]\nrw [\u03b3\u2081.leftHomologyMap'_eq, \u03b3\u2082.leftHomologyMap'_eq, (\u03b3\u2081.comp \u03b3\u2082).leftHomologyMap'_eq, LeftHomologyMapData.comp_\u03c6H]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6\u2081 : S\u2081 \u27f6 S\u2082\n\u03c6\u2082 : S\u2082 \u27f6 S\u2083\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\nh\u2083 : LeftHomologyData S\u2083\n\u22a2 cyclesMap' (\u03c6\u2081 \u226b \u03c6\u2082) h\u2081 h\u2083 = cyclesMap' \u03c6\u2081 h\u2081 h\u2082 \u226b cyclesMap' \u03c6\u2082 h\u2082 h\u2083\n[PROOFSTEP]\nlet \u03b3\u2081 := leftHomologyMapData \u03c6\u2081 h\u2081 h\u2082\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6\u2081 : S\u2081 \u27f6 S\u2082\n\u03c6\u2082 : S\u2082 \u27f6 S\u2083\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\nh\u2083 : LeftHomologyData S\u2083\n\u03b3\u2081 : LeftHomologyMapData \u03c6\u2081 h\u2081 h\u2082 := leftHomologyMapData \u03c6\u2081 h\u2081 h\u2082\n\u22a2 cyclesMap' (\u03c6\u2081 \u226b \u03c6\u2082) h\u2081 h\u2083 = cyclesMap' \u03c6\u2081 h\u2081 h\u2082 \u226b cyclesMap' \u03c6\u2082 h\u2082 h\u2083\n[PROOFSTEP]\nlet \u03b3\u2082 := leftHomologyMapData \u03c6\u2082 h\u2082 h\u2083\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6\u2081 : S\u2081 \u27f6 S\u2082\n\u03c6\u2082 : S\u2082 \u27f6 S\u2083\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\nh\u2083 : LeftHomologyData S\u2083\n\u03b3\u2081 : LeftHomologyMapData \u03c6\u2081 h\u2081 h\u2082 := leftHomologyMapData \u03c6\u2081 h\u2081 h\u2082\n\u03b3\u2082 : LeftHomologyMapData \u03c6\u2082 h\u2082 h\u2083 := leftHomologyMapData \u03c6\u2082 h\u2082 h\u2083\n\u22a2 cyclesMap' (\u03c6\u2081 \u226b \u03c6\u2082) h\u2081 h\u2083 = cyclesMap' \u03c6\u2081 h\u2081 h\u2082 \u226b cyclesMap' \u03c6\u2082 h\u2082 h\u2083\n[PROOFSTEP]\nrw [\u03b3\u2081.cyclesMap'_eq, \u03b3\u2082.cyclesMap'_eq, (\u03b3\u2081.comp \u03b3\u2082).cyclesMap'_eq, LeftHomologyMapData.comp_\u03c6K]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.116872, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ne : S\u2081 \u2245 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u22a2 leftHomologyMap' e.hom h\u2081 h\u2082 \u226b leftHomologyMap' e.inv h\u2082 h\u2081 = \ud835\udfd9 h\u2081.H\n[PROOFSTEP]\nrw [\u2190 leftHomologyMap'_comp, e.hom_inv_id, leftHomologyMap'_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.116872, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ne : S\u2081 \u2245 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u22a2 leftHomologyMap' e.inv h\u2082 h\u2081 \u226b leftHomologyMap' e.hom h\u2081 h\u2082 = \ud835\udfd9 h\u2082.H\n[PROOFSTEP]\nrw [\u2190 leftHomologyMap'_comp, e.inv_hom_id, leftHomologyMap'_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.119499, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ne : S\u2081 \u2245 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u22a2 cyclesMap' e.hom h\u2081 h\u2082 \u226b cyclesMap' e.inv h\u2082 h\u2081 = \ud835\udfd9 h\u2081.K\n[PROOFSTEP]\nrw [\u2190 cyclesMap'_comp, e.hom_inv_id, cyclesMap'_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.119499, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ne : S\u2081 \u2245 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u22a2 cyclesMap' e.inv h\u2082 h\u2081 \u226b cyclesMap' e.hom h\u2081 h\u2082 = \ud835\udfd9 h\u2082.K\n[PROOFSTEP]\nrw [\u2190 cyclesMap'_comp, e.inv_hom_id, cyclesMap'_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{?u.122110, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ne : S\u2081 \u2245 S\u2082\ninst\u271d\u00b9 : HasLeftHomology S\u2081\ninst\u271d : HasLeftHomology S\u2082\n\u22a2 leftHomologyMap e.hom \u226b leftHomologyMap e.inv = \ud835\udfd9 (leftHomology S\u2081)\n[PROOFSTEP]\nrw [\u2190 leftHomologyMap_comp, e.hom_inv_id, leftHomologyMap_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{?u.122110, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ne : S\u2081 \u2245 S\u2082\ninst\u271d\u00b9 : HasLeftHomology S\u2081\ninst\u271d : HasLeftHomology S\u2082\n\u22a2 leftHomologyMap e.inv \u226b leftHomologyMap e.hom = \ud835\udfd9 (leftHomology S\u2082)\n[PROOFSTEP]\nrw [\u2190 leftHomologyMap_comp, e.inv_hom_id, leftHomologyMap_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{?u.125711, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ne : S\u2081 \u2245 S\u2082\ninst\u271d\u00b9 : HasLeftHomology S\u2081\ninst\u271d : HasLeftHomology S\u2082\n\u22a2 cyclesMap e.hom \u226b cyclesMap e.inv = \ud835\udfd9 (cycles S\u2081)\n[PROOFSTEP]\nrw [\u2190 cyclesMap_comp, e.hom_inv_id, cyclesMap_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{?u.125711, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ne : S\u2081 \u2245 S\u2082\ninst\u271d\u00b9 : HasLeftHomology S\u2081\ninst\u271d : HasLeftHomology S\u2082\n\u22a2 cyclesMap e.inv \u226b cyclesMap e.hom = \ud835\udfd9 (cycles S\u2082)\n[PROOFSTEP]\nrw [\u2190 cyclesMap_comp, e.inv_hom_id, cyclesMap_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\ninst\u271d : HasLeftHomology S\n\u22a2 (cyclesIso h).hom \u226b h.i = iCycles S\n[PROOFSTEP]\ndsimp [iCycles, LeftHomologyData.cyclesIso]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\ninst\u271d : HasLeftHomology S\n\u22a2 cyclesMap' (\ud835\udfd9 S) (leftHomologyData S) h \u226b h.i = (leftHomologyData S).i\n[PROOFSTEP]\nsimp only [cyclesMap'_i, id_\u03c4\u2082, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\ninst\u271d : HasLeftHomology S\n\u22a2 (cyclesIso h).inv \u226b iCycles S = h.i\n[PROOFSTEP]\nsimp only [\u2190 h.cyclesIso_hom_comp_i, Iso.inv_hom_id_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\ninst\u271d : HasLeftHomology S\n\u22a2 leftHomology\u03c0 S \u226b (leftHomologyIso h).hom = (cyclesIso h).hom \u226b h.\u03c0\n[PROOFSTEP]\ndsimp only [leftHomology\u03c0, leftHomologyIso, cyclesIso, leftHomologyMapIso', cyclesMapIso', Iso.refl]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\ninst\u271d : HasLeftHomology S\n\u22a2 (leftHomologyData S).\u03c0 \u226b leftHomologyMap' (\ud835\udfd9 S) (leftHomologyData S) h = cyclesMap' (\ud835\udfd9 S) (leftHomologyData S) h \u226b h.\u03c0\n[PROOFSTEP]\nrw [\u2190 leftHomology\u03c0_naturality']\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\ninst\u271d : HasLeftHomology S\n\u22a2 h.\u03c0 \u226b (leftHomologyIso h).inv = (cyclesIso h).inv \u226b leftHomology\u03c0 S\n[PROOFSTEP]\nsimp only [\u2190 cancel_epi h.cyclesIso.hom, \u2190 cancel_mono h.leftHomologyIso.hom, assoc, Iso.inv_hom_id, comp_id,\n Iso.hom_inv_id_assoc, LeftHomologyData.leftHomology\u03c0_comp_leftHomologyIso_hom]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\ninst\u271d\u00b9 : HasLeftHomology S\u2081\ninst\u271d : HasLeftHomology S\u2082\n\u22a2 leftHomologyMap \u03c6 = (LeftHomologyData.leftHomologyIso h\u2081).hom \u226b \u03b3.\u03c6H \u226b (LeftHomologyData.leftHomologyIso h\u2082).inv\n[PROOFSTEP]\ndsimp [LeftHomologyData.leftHomologyIso, leftHomologyMapIso']\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\ninst\u271d\u00b9 : HasLeftHomology S\u2081\ninst\u271d : HasLeftHomology S\u2082\n\u22a2 leftHomologyMap \u03c6 =\n leftHomologyMap' (\ud835\udfd9 S\u2081) (leftHomologyData S\u2081) h\u2081 \u226b \u03b3.\u03c6H \u226b leftHomologyMap' (\ud835\udfd9 S\u2082) h\u2082 (leftHomologyData S\u2082)\n[PROOFSTEP]\nrw [\u2190 \u03b3.leftHomologyMap'_eq, \u2190 leftHomologyMap'_comp, \u2190 leftHomologyMap'_comp, id_comp, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\ninst\u271d\u00b9 : HasLeftHomology S\u2081\ninst\u271d : HasLeftHomology S\u2082\n\u22a2 leftHomologyMap \u03c6 = leftHomologyMap' \u03c6 (leftHomologyData S\u2081) (leftHomologyData S\u2082)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\ninst\u271d\u00b9 : HasLeftHomology S\u2081\ninst\u271d : HasLeftHomology S\u2082\n\u22a2 cyclesMap \u03c6 = (LeftHomologyData.cyclesIso h\u2081).hom \u226b \u03b3.\u03c6K \u226b (LeftHomologyData.cyclesIso h\u2082).inv\n[PROOFSTEP]\ndsimp [LeftHomologyData.cyclesIso, cyclesMapIso']\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\ninst\u271d\u00b9 : HasLeftHomology S\u2081\ninst\u271d : HasLeftHomology S\u2082\n\u22a2 cyclesMap \u03c6 = cyclesMap' (\ud835\udfd9 S\u2081) (leftHomologyData S\u2081) h\u2081 \u226b \u03b3.\u03c6K \u226b cyclesMap' (\ud835\udfd9 S\u2082) h\u2082 (leftHomologyData S\u2082)\n[PROOFSTEP]\nrw [\u2190 \u03b3.cyclesMap'_eq, \u2190 cyclesMap'_comp, \u2190 cyclesMap'_comp, id_comp, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\ninst\u271d\u00b9 : HasLeftHomology S\u2081\ninst\u271d : HasLeftHomology S\u2082\n\u22a2 cyclesMap \u03c6 = cyclesMap' \u03c6 (leftHomologyData S\u2081) (leftHomologyData S\u2082)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\ninst\u271d\u00b9 : HasLeftHomology S\u2081\ninst\u271d : HasLeftHomology S\u2082\n\u22a2 leftHomologyMap \u03c6 \u226b (LeftHomologyData.leftHomologyIso h\u2082).hom = (LeftHomologyData.leftHomologyIso h\u2081).hom \u226b \u03b3.\u03c6H\n[PROOFSTEP]\nsimp only [\u03b3.leftHomologyMap_eq, assoc, Iso.inv_hom_id, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\ninst\u271d\u00b9 : HasLeftHomology S\u2081\ninst\u271d : HasLeftHomology S\u2082\n\u22a2 cyclesMap \u03c6 \u226b (LeftHomologyData.cyclesIso h\u2082).hom = (LeftHomologyData.cyclesIso h\u2081).hom \u226b \u03b3.\u03c6K\n[PROOFSTEP]\nsimp only [\u03b3.cyclesMap_eq, assoc, Iso.inv_hom_id, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\n\u22a2 LeftHomologyData S\u2082\n[PROOFSTEP]\nlet i : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\n\u22a2 LeftHomologyData S\u2082\n[PROOFSTEP]\nhave wi : i \u226b S\u2082.g = 0 := by simp only [assoc, \u03c6.comm\u2082\u2083, h.wi_assoc, zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\n\u22a2 i \u226b S\u2082.g = 0\n[PROOFSTEP]\nsimp only [assoc, \u03c6.comm\u2082\u2083, h.wi_assoc, zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\n\u22a2 LeftHomologyData S\u2082\n[PROOFSTEP]\nhave hi : IsLimit (KernelFork.of\u03b9 i wi) :=\n KernelFork.IsLimit.of\u03b9 _ _\n (fun x hx =>\n h.liftK (x \u226b inv \u03c6.\u03c4\u2082)\n (by rw [assoc, \u2190 cancel_mono \u03c6.\u03c4\u2083, assoc, assoc, \u2190 \u03c6.comm\u2082\u2083, IsIso.inv_hom_id_assoc, hx, zero_comp]))\n (fun x hx => by simp)\n (fun x hx b hb => by\n dsimp\n rw [\u2190 cancel_mono h.i, \u2190 cancel_mono \u03c6.\u03c4\u2082, assoc, assoc, liftK_i_assoc, assoc, IsIso.inv_hom_id, comp_id, hb])\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nW'\u271d : C\nx : W'\u271d \u27f6 S\u2082.X\u2082\nhx : x \u226b S\u2082.g = 0\n\u22a2 (x \u226b inv \u03c6.\u03c4\u2082) \u226b S\u2081.g = 0\n[PROOFSTEP]\nrw [assoc, \u2190 cancel_mono \u03c6.\u03c4\u2083, assoc, assoc, \u2190 \u03c6.comm\u2082\u2083, IsIso.inv_hom_id_assoc, hx, zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nW'\u271d : C\nx : W'\u271d \u27f6 S\u2082.X\u2082\nhx : x \u226b S\u2082.g = 0\n\u22a2 (fun {W'} x hx => liftK h (x \u226b inv \u03c6.\u03c4\u2082) (_ : (x \u226b inv \u03c6.\u03c4\u2082) \u226b S\u2081.g = 0)) x hx \u226b i = x\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nW'\u271d : C\nx : W'\u271d \u27f6 S\u2082.X\u2082\nhx : x \u226b S\u2082.g = 0\nb : W'\u271d \u27f6 h.K\nhb : b \u226b i = x\n\u22a2 b = (fun {W'} x hx => liftK h (x \u226b inv \u03c6.\u03c4\u2082) (_ : (x \u226b inv \u03c6.\u03c4\u2082) \u226b S\u2081.g = 0)) x hx\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nW'\u271d : C\nx : W'\u271d \u27f6 S\u2082.X\u2082\nhx : x \u226b S\u2082.g = 0\nb : W'\u271d \u27f6 h.K\nhb : b \u226b i = x\n\u22a2 b = liftK h (x \u226b inv \u03c6.\u03c4\u2082) (_ : (x \u226b inv \u03c6.\u03c4\u2082) \u226b S\u2081.g = 0)\n[PROOFSTEP]\nrw [\u2190 cancel_mono h.i, \u2190 cancel_mono \u03c6.\u03c4\u2082, assoc, assoc, liftK_i_assoc, assoc, IsIso.inv_hom_id, comp_id, hb]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\n\u22a2 LeftHomologyData S\u2082\n[PROOFSTEP]\nlet f' := hi.lift (KernelFork.of\u03b9 S\u2082.f S\u2082.zero)\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0))\n\u22a2 LeftHomologyData S\u2082\n[PROOFSTEP]\nhave hf' : \u03c6.\u03c4\u2081 \u226b f' = h.f' :=\n by\n have eq := @Fork.IsLimit.lift_\u03b9 _ _ _ _ _ _ _ ((KernelFork.of\u03b9 S\u2082.f S\u2082.zero)) hi\n simp only [Fork.\u03b9_of\u03b9] at eq \n rw [\u2190 cancel_mono h.i, \u2190 cancel_mono \u03c6.\u03c4\u2082, assoc, assoc, eq, f'_i, \u03c6.comm\u2081\u2082]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0))\n\u22a2 \u03c6.\u03c4\u2081 \u226b f' = CategoryTheory.ShortComplex.LeftHomologyData.f' h\n[PROOFSTEP]\nhave eq := @Fork.IsLimit.lift_\u03b9 _ _ _ _ _ _ _ ((KernelFork.of\u03b9 S\u2082.f S\u2082.zero)) hi\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0))\neq :\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0)) \u226b Fork.\u03b9 (KernelFork.of\u03b9 i wi) =\n Fork.\u03b9 (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0))\n\u22a2 \u03c6.\u03c4\u2081 \u226b f' = CategoryTheory.ShortComplex.LeftHomologyData.f' h\n[PROOFSTEP]\nsimp only [Fork.\u03b9_of\u03b9] at eq \n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0))\neq : IsLimit.lift hi (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0)) \u226b h.i \u226b \u03c6.\u03c4\u2082 = S\u2082.f\n\u22a2 \u03c6.\u03c4\u2081 \u226b f' = CategoryTheory.ShortComplex.LeftHomologyData.f' h\n[PROOFSTEP]\nrw [\u2190 cancel_mono h.i, \u2190 cancel_mono \u03c6.\u03c4\u2082, assoc, assoc, eq, f'_i, \u03c6.comm\u2081\u2082]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0))\nhf' : \u03c6.\u03c4\u2081 \u226b f' = CategoryTheory.ShortComplex.LeftHomologyData.f' h\n\u22a2 LeftHomologyData S\u2082\n[PROOFSTEP]\nhave w\u03c0 : f' \u226b h.\u03c0 = 0 := by rw [\u2190 cancel_epi \u03c6.\u03c4\u2081, comp_zero, reassoc_of% hf', h.f'_\u03c0]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0))\nhf' : \u03c6.\u03c4\u2081 \u226b f' = CategoryTheory.ShortComplex.LeftHomologyData.f' h\n\u22a2 f' \u226b h.\u03c0 = 0\n[PROOFSTEP]\nrw [\u2190 cancel_epi \u03c6.\u03c4\u2081, comp_zero, reassoc_of% hf', h.f'_\u03c0]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0))\nhf' : \u03c6.\u03c4\u2081 \u226b f' = CategoryTheory.ShortComplex.LeftHomologyData.f' h\nw\u03c0 : f' \u226b h.\u03c0 = 0\n\u22a2 LeftHomologyData S\u2082\n[PROOFSTEP]\nhave h\u03c0 : IsColimit (CokernelCofork.of\u03c0 h.\u03c0 w\u03c0) :=\n CokernelCofork.IsColimit.of\u03c0 _ _ (fun x hx => h.descH x (by rw [\u2190 hf', assoc, hx, comp_zero])) (fun x hx => by simp)\n (fun x hx b hb => by rw [\u2190 cancel_epi h.\u03c0, \u03c0_descH, hb])\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0))\nhf' : \u03c6.\u03c4\u2081 \u226b f' = CategoryTheory.ShortComplex.LeftHomologyData.f' h\nw\u03c0 : f' \u226b h.\u03c0 = 0\nZ'\u271d : C\nx : h.K \u27f6 Z'\u271d\nhx : f' \u226b x = 0\n\u22a2 CategoryTheory.ShortComplex.LeftHomologyData.f' h \u226b x = 0\n[PROOFSTEP]\nrw [\u2190 hf', assoc, hx, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0))\nhf' : \u03c6.\u03c4\u2081 \u226b f' = CategoryTheory.ShortComplex.LeftHomologyData.f' h\nw\u03c0 : f' \u226b h.\u03c0 = 0\nZ'\u271d : C\nx : h.K \u27f6 Z'\u271d\nhx : f' \u226b x = 0\n\u22a2 h.\u03c0 \u226b (fun {Z'} x hx => descH h x (_ : CategoryTheory.ShortComplex.LeftHomologyData.f' h \u226b x = 0)) x hx = x\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0))\nhf' : \u03c6.\u03c4\u2081 \u226b f' = CategoryTheory.ShortComplex.LeftHomologyData.f' h\nw\u03c0 : f' \u226b h.\u03c0 = 0\nZ'\u271d : C\nx : h.K \u27f6 Z'\u271d\nhx : f' \u226b x = 0\nb : h.H \u27f6 Z'\u271d\nhb : h.\u03c0 \u226b b = x\n\u22a2 b = (fun {Z'} x hx => descH h x (_ : CategoryTheory.ShortComplex.LeftHomologyData.f' h \u226b x = 0)) x hx\n[PROOFSTEP]\nrw [\u2190 cancel_epi h.\u03c0, \u03c0_descH, hb]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.150309, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2082.X\u2082 := h.i \u226b \u03c6.\u03c4\u2082\nwi : i \u226b S\u2082.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2082.f (_ : S\u2082.f \u226b S\u2082.g = 0))\nhf' : \u03c6.\u03c4\u2081 \u226b f' = CategoryTheory.ShortComplex.LeftHomologyData.f' h\nw\u03c0 : f' \u226b h.\u03c0 = 0\nh\u03c0 : IsColimit (CokernelCofork.of\u03c0 h.\u03c0 w\u03c0)\n\u22a2 LeftHomologyData S\u2082\n[PROOFSTEP]\nexact \u27e8h.K, h.H, i, h.\u03c0, wi, hi, w\u03c0, h\u03c0\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2081\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\n\u22a2 \u03c6.\u03c4\u2081 \u226b f' (ofEpiOfIsIsoOfMono \u03c6 h) = f' h\n[PROOFSTEP]\nrw [\u2190 cancel_mono (ofEpiOfIsIsoOfMono \u03c6 h).i, assoc, f'_i, ofEpiOfIsIsoOfMono_i, f'_i_assoc, \u03c6.comm\u2081\u2082]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\n\u22a2 LeftHomologyData S\u2081\n[PROOFSTEP]\nlet i : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\n\u22a2 LeftHomologyData S\u2081\n[PROOFSTEP]\nhave wi : i \u226b S\u2081.g = 0 := by\n rw [assoc, \u2190 cancel_mono \u03c6.\u03c4\u2083, zero_comp, assoc, assoc, \u2190 \u03c6.comm\u2082\u2083, IsIso.inv_hom_id_assoc, h.wi]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\n\u22a2 i \u226b S\u2081.g = 0\n[PROOFSTEP]\nrw [assoc, \u2190 cancel_mono \u03c6.\u03c4\u2083, zero_comp, assoc, assoc, \u2190 \u03c6.comm\u2082\u2083, IsIso.inv_hom_id_assoc, h.wi]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\nwi : i \u226b S\u2081.g = 0\n\u22a2 LeftHomologyData S\u2081\n[PROOFSTEP]\nhave hi : IsLimit (KernelFork.of\u03b9 i wi) :=\n KernelFork.IsLimit.of\u03b9 _ _ (fun x hx => h.liftK (x \u226b \u03c6.\u03c4\u2082) (by rw [assoc, \u03c6.comm\u2082\u2083, reassoc_of% hx, zero_comp]))\n (fun x hx => by simp)\n (fun x hx b hb => by\n rw [\u2190 cancel_mono h.i, \u2190 cancel_mono (inv \u03c6.\u03c4\u2082), assoc, assoc, hb, liftK_i_assoc, assoc, IsIso.hom_inv_id,\n comp_id])\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\nwi : i \u226b S\u2081.g = 0\nW'\u271d : C\nx : W'\u271d \u27f6 S\u2081.X\u2082\nhx : x \u226b S\u2081.g = 0\n\u22a2 (x \u226b \u03c6.\u03c4\u2082) \u226b S\u2082.g = 0\n[PROOFSTEP]\nrw [assoc, \u03c6.comm\u2082\u2083, reassoc_of% hx, zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\nwi : i \u226b S\u2081.g = 0\nW'\u271d : C\nx : W'\u271d \u27f6 S\u2081.X\u2082\nhx : x \u226b S\u2081.g = 0\n\u22a2 (fun {W'} x hx => liftK h (x \u226b \u03c6.\u03c4\u2082) (_ : (x \u226b \u03c6.\u03c4\u2082) \u226b S\u2082.g = 0)) x hx \u226b i = x\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\nwi : i \u226b S\u2081.g = 0\nW'\u271d : C\nx : W'\u271d \u27f6 S\u2081.X\u2082\nhx : x \u226b S\u2081.g = 0\nb : W'\u271d \u27f6 h.K\nhb : b \u226b i = x\n\u22a2 b = (fun {W'} x hx => liftK h (x \u226b \u03c6.\u03c4\u2082) (_ : (x \u226b \u03c6.\u03c4\u2082) \u226b S\u2082.g = 0)) x hx\n[PROOFSTEP]\nrw [\u2190 cancel_mono h.i, \u2190 cancel_mono (inv \u03c6.\u03c4\u2082), assoc, assoc, hb, liftK_i_assoc, assoc, IsIso.hom_inv_id, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\nwi : i \u226b S\u2081.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\n\u22a2 LeftHomologyData S\u2081\n[PROOFSTEP]\nlet f' := hi.lift (KernelFork.of\u03b9 S\u2081.f S\u2081.zero)\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\nwi : i \u226b S\u2081.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0))\n\u22a2 LeftHomologyData S\u2081\n[PROOFSTEP]\nhave hf' : f' \u226b i = S\u2081.f := Fork.IsLimit.lift_\u03b9 _\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\nwi : i \u226b S\u2081.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0))\nhf' : f' \u226b i = S\u2081.f\n\u22a2 LeftHomologyData S\u2081\n[PROOFSTEP]\nhave hf'' : f' = \u03c6.\u03c4\u2081 \u226b h.f' := by\n rw [\u2190 cancel_mono h.i, \u2190 cancel_mono (inv \u03c6.\u03c4\u2082), assoc, assoc, assoc, hf', f'_i_assoc, \u03c6.comm\u2081\u2082_assoc,\n IsIso.hom_inv_id, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\nwi : i \u226b S\u2081.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0))\nhf' : f' \u226b i = S\u2081.f\n\u22a2 f' = \u03c6.\u03c4\u2081 \u226b CategoryTheory.ShortComplex.LeftHomologyData.f' h\n[PROOFSTEP]\nrw [\u2190 cancel_mono h.i, \u2190 cancel_mono (inv \u03c6.\u03c4\u2082), assoc, assoc, assoc, hf', f'_i_assoc, \u03c6.comm\u2081\u2082_assoc, IsIso.hom_inv_id,\n comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\nwi : i \u226b S\u2081.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0))\nhf' : f' \u226b i = S\u2081.f\nhf'' : f' = \u03c6.\u03c4\u2081 \u226b CategoryTheory.ShortComplex.LeftHomologyData.f' h\n\u22a2 LeftHomologyData S\u2081\n[PROOFSTEP]\nhave w\u03c0 : f' \u226b h.\u03c0 = 0 := by simp only [hf'', assoc, f'_\u03c0, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\nwi : i \u226b S\u2081.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0))\nhf' : f' \u226b i = S\u2081.f\nhf'' : f' = \u03c6.\u03c4\u2081 \u226b CategoryTheory.ShortComplex.LeftHomologyData.f' h\n\u22a2 f' \u226b h.\u03c0 = 0\n[PROOFSTEP]\nsimp only [hf'', assoc, f'_\u03c0, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\nwi : i \u226b S\u2081.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0))\nhf' : f' \u226b i = S\u2081.f\nhf'' : f' = \u03c6.\u03c4\u2081 \u226b CategoryTheory.ShortComplex.LeftHomologyData.f' h\nw\u03c0 : f' \u226b h.\u03c0 = 0\n\u22a2 LeftHomologyData S\u2081\n[PROOFSTEP]\nhave h\u03c0 : IsColimit (CokernelCofork.of\u03c0 h.\u03c0 w\u03c0) :=\n CokernelCofork.IsColimit.of\u03c0 _ _\n (fun x hx => h.descH x (by rw [\u2190 cancel_epi \u03c6.\u03c4\u2081, \u2190 reassoc_of% hf'', hx, comp_zero])) (fun x hx => \u03c0_descH _ _ _)\n (fun x hx b hx => by rw [\u2190 cancel_epi h.\u03c0, \u03c0_descH, hx])\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\nwi : i \u226b S\u2081.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0))\nhf' : f' \u226b i = S\u2081.f\nhf'' : f' = \u03c6.\u03c4\u2081 \u226b CategoryTheory.ShortComplex.LeftHomologyData.f' h\nw\u03c0 : f' \u226b h.\u03c0 = 0\nZ'\u271d : C\nx : h.K \u27f6 Z'\u271d\nhx : f' \u226b x = 0\n\u22a2 CategoryTheory.ShortComplex.LeftHomologyData.f' h \u226b x = 0\n[PROOFSTEP]\nrw [\u2190 cancel_epi \u03c6.\u03c4\u2081, \u2190 reassoc_of% hf'', hx, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\nwi : i \u226b S\u2081.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0))\nhf' : f' \u226b i = S\u2081.f\nhf'' : f' = \u03c6.\u03c4\u2081 \u226b CategoryTheory.ShortComplex.LeftHomologyData.f' h\nw\u03c0 : f' \u226b h.\u03c0 = 0\nZ'\u271d : C\nx : h.K \u27f6 Z'\u271d\nhx\u271d : f' \u226b x = 0\nb : h.H \u27f6 Z'\u271d\nhx : h.\u03c0 \u226b b = x\n\u22a2 b = (fun {Z'} x hx => descH h x (_ : CategoryTheory.ShortComplex.LeftHomologyData.f' h \u226b x = 0)) x hx\u271d\n[PROOFSTEP]\nrw [\u2190 cancel_epi h.\u03c0, \u03c0_descH, hx]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.166978, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\ni : h.K \u27f6 S\u2081.X\u2082 := h.i \u226b inv \u03c6.\u03c4\u2082\nwi : i \u226b S\u2081.g = 0\nhi : IsLimit (KernelFork.of\u03b9 i wi)\nf' : (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0)).pt \u27f6 (KernelFork.of\u03b9 i wi).pt :=\n IsLimit.lift hi (KernelFork.of\u03b9 S\u2081.f (_ : S\u2081.f \u226b S\u2081.g = 0))\nhf' : f' \u226b i = S\u2081.f\nhf'' : f' = \u03c6.\u03c4\u2081 \u226b CategoryTheory.ShortComplex.LeftHomologyData.f' h\nw\u03c0 : f' \u226b h.\u03c0 = 0\nh\u03c0 : IsColimit (CokernelCofork.of\u03c0 h.\u03c0 w\u03c0)\n\u22a2 LeftHomologyData S\u2081\n[PROOFSTEP]\nexact \u27e8h.K, h.H, i, h.\u03c0, wi, hi, w\u03c0, h\u03c0\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\n\u22a2 f' (ofEpiOfIsIsoOfMono' \u03c6 h) = \u03c6.\u03c4\u2081 \u226b f' h\n[PROOFSTEP]\nrw [\u2190 cancel_mono (ofEpiOfIsIsoOfMono' \u03c6 h).i, f'_i, ofEpiOfIsIsoOfMono'_i, assoc, f'_i_assoc, \u03c6.comm\u2081\u2082_assoc,\n IsIso.hom_inv_id, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.195682, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\n\u22a2 IsIso (leftHomologyMap' \u03c6 h\u2081 h\u2082)\n[PROOFSTEP]\nlet h\u2082' := LeftHomologyData.ofEpiOfIsIsoOfMono \u03c6 h\u2081\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.195682, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\nh\u2082' : LeftHomologyData S\u2082 := LeftHomologyData.ofEpiOfIsIsoOfMono \u03c6 h\u2081\n\u22a2 IsIso (leftHomologyMap' \u03c6 h\u2081 h\u2082)\n[PROOFSTEP]\nhave : IsIso (leftHomologyMap' \u03c6 h\u2081 h\u2082') :=\n by\n rw [(LeftHomologyMapData.ofEpiOfIsIsoOfMono \u03c6 h\u2081).leftHomologyMap'_eq]\n dsimp\n infer_instance\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.195682, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\nh\u2082' : LeftHomologyData S\u2082 := LeftHomologyData.ofEpiOfIsIsoOfMono \u03c6 h\u2081\n\u22a2 IsIso (leftHomologyMap' \u03c6 h\u2081 h\u2082')\n[PROOFSTEP]\nrw [(LeftHomologyMapData.ofEpiOfIsIsoOfMono \u03c6 h\u2081).leftHomologyMap'_eq]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.195682, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\nh\u2082' : LeftHomologyData S\u2082 := LeftHomologyData.ofEpiOfIsIsoOfMono \u03c6 h\u2081\n\u22a2 IsIso (LeftHomologyMapData.ofEpiOfIsIsoOfMono \u03c6 h\u2081).\u03c6H\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.195682, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\nh\u2082' : LeftHomologyData S\u2082 := LeftHomologyData.ofEpiOfIsIsoOfMono \u03c6 h\u2081\n\u22a2 IsIso (\ud835\udfd9 h\u2081.H)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.195682, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\nh\u2082' : LeftHomologyData S\u2082 := LeftHomologyData.ofEpiOfIsIsoOfMono \u03c6 h\u2081\nthis : IsIso (leftHomologyMap' \u03c6 h\u2081 h\u2082')\n\u22a2 IsIso (leftHomologyMap' \u03c6 h\u2081 h\u2082)\n[PROOFSTEP]\nhave eq := leftHomologyMap'_comp \u03c6 (\ud835\udfd9 S\u2082) h\u2081 h\u2082' h\u2082\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.195682, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\nh\u2082' : LeftHomologyData S\u2082 := LeftHomologyData.ofEpiOfIsIsoOfMono \u03c6 h\u2081\nthis : IsIso (leftHomologyMap' \u03c6 h\u2081 h\u2082')\neq : leftHomologyMap' (\u03c6 \u226b \ud835\udfd9 S\u2082) h\u2081 h\u2082 = leftHomologyMap' \u03c6 h\u2081 h\u2082' \u226b leftHomologyMap' (\ud835\udfd9 S\u2082) h\u2082' h\u2082\n\u22a2 IsIso (leftHomologyMap' \u03c6 h\u2081 h\u2082)\n[PROOFSTEP]\nrw [comp_id] at eq \n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.195682, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\nh\u2082' : LeftHomologyData S\u2082 := LeftHomologyData.ofEpiOfIsIsoOfMono \u03c6 h\u2081\nthis : IsIso (leftHomologyMap' \u03c6 h\u2081 h\u2082')\neq : leftHomologyMap' \u03c6 h\u2081 h\u2082 = leftHomologyMap' \u03c6 h\u2081 h\u2082' \u226b leftHomologyMap' (\ud835\udfd9 S\u2082) h\u2082' h\u2082\n\u22a2 IsIso (leftHomologyMap' \u03c6 h\u2081 h\u2082)\n[PROOFSTEP]\nrw [eq]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.195682, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\nh\u2082' : LeftHomologyData S\u2082 := LeftHomologyData.ofEpiOfIsIsoOfMono \u03c6 h\u2081\nthis : IsIso (leftHomologyMap' \u03c6 h\u2081 h\u2082')\neq : leftHomologyMap' \u03c6 h\u2081 h\u2082 = leftHomologyMap' \u03c6 h\u2081 h\u2082' \u226b leftHomologyMap' (\ud835\udfd9 S\u2082) h\u2082' h\u2082\n\u22a2 IsIso (leftHomologyMap' \u03c6 h\u2081 h\u2082' \u226b leftHomologyMap' (\ud835\udfd9 S\u2082) h\u2082' h\u2082)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.197546, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\ninst\u271d\u2074 : HasLeftHomology S\u2081\ninst\u271d\u00b3 : HasLeftHomology S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\n\u22a2 IsIso (leftHomologyMap \u03c6)\n[PROOFSTEP]\ndsimp only [leftHomologyMap]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.197546, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\ninst\u271d\u2074 : HasLeftHomology S\u2081\ninst\u271d\u00b3 : HasLeftHomology S\u2082\ninst\u271d\u00b2 : Epi \u03c6.\u03c4\u2081\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2082\ninst\u271d : Mono \u03c6.\u03c4\u2083\n\u22a2 IsIso (leftHomologyMap' \u03c6 (leftHomologyData S\u2081) (leftHomologyData S\u2082))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.201841, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nk : A \u27f6 S.X\u2082\nhk : k \u226b S.g = 0\ninst\u271d : HasLeftHomology S\nA' : C\n\u03b1 : A' \u27f6 A\n\u22a2 (\u03b1 \u226b k) \u226b S.g = 0\n[PROOFSTEP]\nrw [assoc, hk, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nk : A \u27f6 S.X\u2082\nhk : k \u226b S.g = 0\ninst\u271d : HasLeftHomology S\nA' : C\n\u03b1 : A' \u27f6 A\n\u22a2 \u03b1 \u226b liftCycles S k hk = liftCycles S (\u03b1 \u226b k) (_ : (\u03b1 \u226b k) \u226b S.g = 0)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{?u.205152, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nk : A \u27f6 S.X\u2082\nhk : k \u226b S.g = 0\ninst\u271d\u00b9 : HasLeftHomology S\ninst\u271d : HasKernel S.g\n\u22a2 iCycles S \u226b S.g = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{?u.205152, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nk : A \u27f6 S.X\u2082\nhk : k \u226b S.g = 0\ninst\u271d\u00b9 : HasLeftHomology S\ninst\u271d : HasKernel S.g\n\u22a2 kernel.\u03b9 S.g \u226b S.g = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.209151, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nk : A \u27f6 S.X\u2082\nhk : k \u226b S.g = 0\ninst\u271d : HasLeftHomology S\nx : A \u27f6 S.X\u2081\nhx : k = x \u226b S.f\n\u22a2 k \u226b S.g = 0\n[PROOFSTEP]\nrw [hx, assoc, S.zero, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nk : A \u27f6 S.X\u2082\nhk : k \u226b S.g = 0\ninst\u271d : HasLeftHomology S\n\u22a2 S.f = \ud835\udfd9 S.X\u2081 \u226b S.f\n[PROOFSTEP]\nrw [id_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{?u.213429, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nk : A \u27f6 S.X\u2082\nhk : k \u226b S.g = 0\ninst\u271d\u00b9 : HasLeftHomology S\n\u03c6 : S \u27f6 S\u2081\ninst\u271d : HasLeftHomology S\u2081\n\u22a2 (k \u226b \u03c6.\u03c4\u2082) \u226b S\u2081.g = 0\n[PROOFSTEP]\nrw [assoc, \u03c6.comm\u2082\u2083, reassoc_of% hk, zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nk : A \u27f6 S.X\u2082\nhk : k \u226b S.g = 0\ninst\u271d\u00b9 : HasLeftHomology S\n\u03c6 : S \u27f6 S\u2081\ninst\u271d : HasLeftHomology S\u2081\n\u22a2 liftCycles S k hk \u226b cyclesMap \u03c6 = liftCycles S\u2081 (k \u226b \u03c6.\u03c4\u2082) (_ : (k \u226b \u03c6.\u03c4\u2082) \u226b S\u2081.g = 0)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nk : A \u27f6 S.X\u2082\nhk : k \u226b S.g = 0\ninst\u271d : HasLeftHomology S\n\u22a2 liftCycles S k hk \u226b (cyclesIso h).hom = liftK h k hk\n[PROOFSTEP]\nsimp only [\u2190 cancel_mono h.i, assoc, LeftHomologyData.cyclesIso_hom_comp_i, liftCycles_i, LeftHomologyData.liftK_i]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\nh : LeftHomologyData S\nA : C\nk : A \u27f6 S.X\u2082\nhk : k \u226b S.g = 0\ninst\u271d : HasLeftHomology S\n\u22a2 liftK h k hk \u226b (cyclesIso h).inv = liftCycles S k hk\n[PROOFSTEP]\nrw [\u2190 h.liftCycles_comp_cyclesIso_hom, assoc, Iso.hom_inv_id, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d\u00b9 : HasLeftHomology S\ninst\u271d : HasKernel S.g\n\u22a2 HasCokernel (kernel.lift S.g S.f (_ : S.f \u226b S.g = 0))\n[PROOFSTEP]\nlet h := S.leftHomologyData\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d\u00b9 : HasLeftHomology S\ninst\u271d : HasKernel S.g\nh : LeftHomologyData S := leftHomologyData S\n\u22a2 HasCokernel (kernel.lift S.g S.f (_ : S.f \u226b S.g = 0))\n[PROOFSTEP]\nhaveI : HasColimit (parallelPair h.f' 0) := \u27e8\u27e8\u27e8_, h.h\u03c0'\u27e9\u27e9\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d\u00b9 : HasLeftHomology S\ninst\u271d : HasKernel S.g\nh : LeftHomologyData S := leftHomologyData S\nthis : HasColimit (parallelPair (LeftHomologyData.f' h) 0)\n\u22a2 HasCokernel (kernel.lift S.g S.f (_ : S.f \u226b S.g = 0))\n[PROOFSTEP]\nlet e : parallelPair (kernel.lift S.g S.f S.zero) 0 \u2245 parallelPair h.f' 0 :=\n parallelPair.ext (Iso.refl _) (IsLimit.conePointUniqueUpToIso (kernelIsKernel S.g) h.hi) (by aesop_cat) (by aesop_cat)\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d\u00b9 : HasLeftHomology S\ninst\u271d : HasKernel S.g\nh : LeftHomologyData S := leftHomologyData S\nthis : HasColimit (parallelPair (LeftHomologyData.f' h) 0)\n\u22a2 (parallelPair (kernel.lift S.g S.f (_ : S.f \u226b S.g = 0)) 0).map WalkingParallelPairHom.left \u226b\n (IsLimit.conePointUniqueUpToIso (kernelIsKernel S.g) h.hi).hom =\n (Iso.refl ((parallelPair (kernel.lift S.g S.f (_ : S.f \u226b S.g = 0)) 0).obj WalkingParallelPair.zero)).hom \u226b\n (parallelPair (LeftHomologyData.f' h) 0).map WalkingParallelPairHom.left\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d\u00b9 : HasLeftHomology S\ninst\u271d : HasKernel S.g\nh : LeftHomologyData S := leftHomologyData S\nthis : HasColimit (parallelPair (LeftHomologyData.f' h) 0)\n\u22a2 (parallelPair (kernel.lift S.g S.f (_ : S.f \u226b S.g = 0)) 0).map WalkingParallelPairHom.right \u226b\n (IsLimit.conePointUniqueUpToIso (kernelIsKernel S.g) h.hi).hom =\n (Iso.refl ((parallelPair (kernel.lift S.g S.f (_ : S.f \u226b S.g = 0)) 0).obj WalkingParallelPair.zero)).hom \u226b\n (parallelPair (LeftHomologyData.f' h) 0).map WalkingParallelPairHom.right\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\ninst\u271d\u00b9 : HasLeftHomology S\ninst\u271d : HasKernel S.g\nh : LeftHomologyData S := leftHomologyData S\nthis : HasColimit (parallelPair (LeftHomologyData.f' h) 0)\ne : parallelPair (kernel.lift S.g S.f (_ : S.f \u226b S.g = 0)) 0 \u2245 parallelPair (LeftHomologyData.f' h) 0 :=\n parallelPair.ext (Iso.refl ((parallelPair (kernel.lift S.g S.f (_ : S.f \u226b S.g = 0)) 0).obj WalkingParallelPair.zero))\n (IsLimit.conePointUniqueUpToIso (kernelIsKernel S.g) h.hi)\n (_ :\n (parallelPair (kernel.lift S.g S.f (_ : S.f \u226b S.g = 0)) 0).map WalkingParallelPairHom.left \u226b\n (IsLimit.conePointUniqueUpToIso (kernelIsKernel S.g) h.hi).hom =\n (Iso.refl ((parallelPair (kernel.lift S.g S.f (_ : S.f \u226b S.g = 0)) 0).obj WalkingParallelPair.zero)).hom \u226b\n (parallelPair (LeftHomologyData.f' h) 0).map WalkingParallelPairHom.left)\n (_ : 0 \u226b (IsLimit.conePointUniqueUpToIso (kernelIsKernel S.g) (leftHomologyData S).hi).hom = \ud835\udfd9 S.X\u2081 \u226b 0)\n\u22a2 HasCokernel (kernel.lift S.g S.f (_ : S.f \u226b S.g = 0))\n[PROOFSTEP]\nexact hasColimitOfIso e\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2082\u271d : IsIso \u03c6.\u03c4\u2082\nh\u2083 : Mono \u03c6.\u03c4\u2083\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u22a2 IsIso (cyclesMap' \u03c6 h\u2081 h\u2082)\n[PROOFSTEP]\nrefine' \u27e8h\u2081.liftK (h\u2082.i \u226b inv \u03c6.\u03c4\u2082) _, _, _\u27e9\n[GOAL]\ncase refine'_1\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2082\u271d : IsIso \u03c6.\u03c4\u2082\nh\u2083 : Mono \u03c6.\u03c4\u2083\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u22a2 (h\u2082.i \u226b inv \u03c6.\u03c4\u2082) \u226b S\u2081.g = 0\n[PROOFSTEP]\nsimp only [assoc, \u2190 cancel_mono \u03c6.\u03c4\u2083, zero_comp, \u2190 \u03c6.comm\u2082\u2083, IsIso.inv_hom_id_assoc, h\u2082.wi]\n[GOAL]\ncase refine'_2\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2082\u271d : IsIso \u03c6.\u03c4\u2082\nh\u2083 : Mono \u03c6.\u03c4\u2083\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u22a2 cyclesMap' \u03c6 h\u2081 h\u2082 \u226b LeftHomologyData.liftK h\u2081 (h\u2082.i \u226b inv \u03c6.\u03c4\u2082) (_ : (h\u2082.i \u226b inv \u03c6.\u03c4\u2082) \u226b S\u2081.g = 0) = \ud835\udfd9 h\u2081.K\n[PROOFSTEP]\nsimp only [\u2190 cancel_mono h\u2081.i, assoc, h\u2081.liftK_i, cyclesMap'_i_assoc, IsIso.hom_inv_id, comp_id, id_comp]\n[GOAL]\ncase refine'_3\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasZeroMorphisms C\nS S\u2081 S\u2082 S\u2083 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2082\u271d : IsIso \u03c6.\u03c4\u2082\nh\u2083 : Mono \u03c6.\u03c4\u2083\nh\u2081 : LeftHomologyData S\u2081\nh\u2082 : LeftHomologyData S\u2082\n\u22a2 LeftHomologyData.liftK h\u2081 (h\u2082.i \u226b inv \u03c6.\u03c4\u2082) (_ : (h\u2082.i \u226b inv \u03c6.\u03c4\u2082) \u226b S\u2081.g = 0) \u226b cyclesMap' \u03c6 h\u2081 h\u2082 = \ud835\udfd9 h\u2082.K\n[PROOFSTEP]\nsimp only [\u2190 cancel_mono h\u2082.i, assoc, cyclesMap'_i, h\u2081.liftK_i_assoc, IsIso.inv_hom_id, comp_id, id_comp]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Homology.ShortComplex.LeftHomology", "llama_tokens": 34217, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.4049432421375034}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : MonoidalCategory C\nX Y : C\n\u22a2 (\u03b1_ (\ud835\udfd9_ C) X Y).hom \u226b (\u03bb_ (X \u2297 Y)).hom = (\u03bb_ X).hom \u2297 \ud835\udfd9 Y\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : MonoidalCategory C\nX Y : C\n\u22a2 (\u03bb_ (X \u2297 Y)).hom = (\u03b1_ (\ud835\udfd9_ C) X Y).inv \u226b ((\u03bb_ X).hom \u2297 \ud835\udfd9 Y)\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : MonoidalCategory C\nX Y : C\n\u22a2 (\u03bb_ (X \u2297 Y)).inv = ((\u03bb_ X).inv \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ (\ud835\udfd9_ C) X Y).hom\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : MonoidalCategory C\nX Y : C\n\u22a2 \ud835\udfd9 X \u2297 (\u03c1_ Y).inv = (\u03c1_ (X \u2297 Y)).inv \u226b (\u03b1_ X Y tensorUnit').hom\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : MonoidalCategory C\nX Y : C\n\u22a2 (\u03bb_ X).inv \u2297 \ud835\udfd9 Y = (\u03bb_ (X \u2297 Y)).inv \u226b (\u03b1_ tensorUnit' X Y).inv\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : MonoidalCategory C\nW X Y Z : C\n\u22a2 (\u03b1_ W (X \u2297 Y) Z).inv \u226b ((\u03b1_ W X Y).inv \u2297 \ud835\udfd9 Z) \u226b (\u03b1_ (W \u2297 X) Y Z).hom = (\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).hom) \u226b (\u03b1_ W X (Y \u2297 Z)).inv\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : MonoidalCategory C\nX Y : C\n\u22a2 ((\u03c1_ X).inv \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ X (\ud835\udfd9_ C) Y).hom = \ud835\udfd9 X \u2297 (\u03bb_ Y).inv\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : MonoidalCategory C\n\u22a2 (\u03bb_ (\ud835\udfd9_ C)).hom = (\u03c1_ (\ud835\udfd9_ C)).hom\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : MonoidalCategory C\n\u22a2 (\u03bb_ (\ud835\udfd9_ C)).inv = (\u03c1_ (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : MonoidalCategory C\nW X Y Z : C\n\u22a2 (\u03b1_ W X (Y \u2297 Z)).hom \u226b (\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv) = (\u03b1_ (W \u2297 X) Y Z).inv \u226b ((\u03b1_ W X Y).hom \u2297 \ud835\udfd9 Z) \u226b (\u03b1_ W (X \u2297 Y) Z).hom\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : MonoidalCategory C\nW X Y Z : C\n\u22a2 (\u03b1_ (W \u2297 X) Y Z).inv \u226b ((\u03b1_ W X Y).hom \u2297 \ud835\udfd9 Z) = (\u03b1_ W X (Y \u2297 Z)).hom \u226b (\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv) \u226b (\u03b1_ W (X \u2297 Y) Z).inv\n[PROOFSTEP]\ncoherence\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monoidal.CoherenceLemmas", "llama_tokens": 1191, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850402140659, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4048936378738108}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\n\u22a2 zipWith f l [] = []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\n\u22a2 zipWith f l l' = [] \u2194 l = [] \u2228 l' = []\n[PROOFSTEP]\ncases l\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl' : List \u03b2\n\u22a2 zipWith f [] l' = [] \u2194 [] = [] \u2228 l' = []\n[PROOFSTEP]\ncases l'\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl' : List \u03b2\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\n\u22a2 zipWith f (head\u271d :: tail\u271d) l' = [] \u2194 head\u271d :: tail\u271d = [] \u2228 l' = []\n[PROOFSTEP]\ncases l'\n[GOAL]\ncase nil.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u22a2 zipWith f [] [] = [] \u2194 [] = [] \u2228 [] = []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase nil.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\n\u22a2 zipWith f [] (head\u271d :: tail\u271d) = [] \u2194 [] = [] \u2228 head\u271d :: tail\u271d = []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\n\u22a2 zipWith f (head\u271d :: tail\u271d) [] = [] \u2194 head\u271d :: tail\u271d = [] \u2228 [] = []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nhead\u271d\u00b9 : \u03b1\ntail\u271d\u00b9 : List \u03b1\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\n\u22a2 zipWith f (head\u271d\u00b9 :: tail\u271d\u00b9) (head\u271d :: tail\u271d) = [] \u2194 head\u271d\u00b9 :: tail\u271d\u00b9 = [] \u2228 head\u271d :: tail\u271d = []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 : List \u03b1\n\u22a2 map Prod.swap (zip l\u2081 []) = zip [] l\u2081\n[PROOFSTEP]\nrw [zip_nil_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 : List \u03b1\n\u22a2 map Prod.swap [] = zip [] l\u2081\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b2\nl\u2082 : List \u03b2\n\u22a2 map Prod.swap (zip (a :: l\u2081) (b :: l\u2082)) = zip (b :: l\u2082) (a :: l\u2081)\n[PROOFSTEP]\nsimp only [zip_cons_cons, map_cons, zip_swap l\u2081 l\u2082, Prod.swap_prod_mk]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\np : \u03b3 \u2192 Prop\nx\u271d : length [] = length []\n\u22a2 All\u2082 p (zipWith f [] []) \u2194 Forall\u2082 (fun x y => p (f x y)) [] []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\np : \u03b3 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b2\nl\u2082 : List \u03b2\nh : length (a :: l\u2081) = length (b :: l\u2082)\n\u22a2 All\u2082 p (zipWith f (a :: l\u2081) (b :: l\u2082)) \u2194 Forall\u2082 (fun x y => p (f x y)) (a :: l\u2081) (b :: l\u2082)\n[PROOFSTEP]\nsimp only [length_cons, succ_inj'] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\np : \u03b3 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b2\nl\u2082 : List \u03b2\nh : length l\u2081 = length l\u2082\n\u22a2 All\u2082 p (zipWith f (a :: l\u2081) (b :: l\u2082)) \u2194 Forall\u2082 (fun x y => p (f x y)) (a :: l\u2081) (b :: l\u2082)\n[PROOFSTEP]\nsimp [all\u2082_zipWith h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ni : \u2115\nl : List \u03b1\nl' : List \u03b2\nh : i < length (zipWith f l l')\n\u22a2 i < length l\n[PROOFSTEP]\nrw [length_zipWith, lt_min_iff] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ni : \u2115\nl : List \u03b1\nl' : List \u03b2\nh : i < length l \u2227 i < length l'\n\u22a2 i < length l\n[PROOFSTEP]\nexact h.left\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ni : \u2115\nl : List \u03b1\nl' : List \u03b2\nh : i < length (zipWith f l l')\n\u22a2 i < length l'\n[PROOFSTEP]\nrw [length_zipWith, lt_min_iff] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ni : \u2115\nl : List \u03b1\nl' : List \u03b2\nh : i < length l \u2227 i < length l'\n\u22a2 i < length l'\n[PROOFSTEP]\nexact h.right\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nr\u2081 : List \u03b1\nl\u2082 r\u2082 : List \u03b2\nh : length [] = length l\u2082\n\u22a2 zip ([] ++ r\u2081) (l\u2082 ++ r\u2082) = zip [] l\u2082 ++ zip r\u2081 r\u2082\n[PROOFSTEP]\nsimp only [eq_nil_of_length_eq_zero h.symm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nr\u2081 : List \u03b1\nl\u2082 r\u2082 : List \u03b2\nh : length [] = length l\u2082\n\u22a2 zip ([] ++ r\u2081) ([] ++ r\u2082) = zip [] [] ++ zip r\u2081 r\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 r\u2081 : List \u03b1\nr\u2082 : List \u03b2\nh : length l\u2081 = length []\n\u22a2 zip (l\u2081 ++ r\u2081) ([] ++ r\u2082) = zip l\u2081 [] ++ zip r\u2081 r\u2082\n[PROOFSTEP]\nsimp only [eq_nil_of_length_eq_zero h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 r\u2081 : List \u03b1\nr\u2082 : List \u03b2\nh : length l\u2081 = length []\n\u22a2 zip ([] ++ r\u2081) ([] ++ r\u2082) = zip [] [] ++ zip r\u2081 r\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\na : \u03b1\nl\u2081 r\u2081 : List \u03b1\nb : \u03b2\nl\u2082 r\u2082 : List \u03b2\nh : length (a :: l\u2081) = length (b :: l\u2082)\n\u22a2 zip (a :: l\u2081 ++ r\u2081) (b :: l\u2082 ++ r\u2082) = zip (a :: l\u2081) (b :: l\u2082) ++ zip r\u2081 r\u2082\n[PROOFSTEP]\nsimp only [cons_append, zip_cons_cons, zip_append (succ.inj h)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nl\u2081 : List \u03b1\n\u22a2 zip (map f l\u2081) (map g []) = map (Prod.map f g) (zip l\u2081 [])\n[PROOFSTEP]\nsimp only [map, zip_nil_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b2\nl\u2082 : List \u03b2\n\u22a2 zip (map f (a :: l\u2081)) (map g (b :: l\u2082)) = map (Prod.map f g) (zip (a :: l\u2081) (b :: l\u2082))\n[PROOFSTEP]\nsimp only [map, zip_cons_cons, zip_map, Prod.map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b2\nl\u2082 : List \u03b2\n\u22a2 (f a, g b) :: map (fun x => (f x.fst, g x.snd)) (zip l\u2081 l\u2082) =\n (f a, g b) :: map (fun x => (f x.fst, g x.snd)) (zipWith Prod.mk l\u2081 l\u2082)\n[PROOFSTEP]\nconstructor\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b3\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\n\u22a2 zip (map f l\u2081) l\u2082 = map (Prod.map f id) (zip l\u2081 l\u2082)\n[PROOFSTEP]\nrw [\u2190 zip_map, map_id]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b2 \u2192 \u03b3\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\n\u22a2 zip l\u2081 (map f l\u2082) = map (Prod.map id f) (zip l\u2081 l\u2082)\n[PROOFSTEP]\nrw [\u2190 zip_map, map_id]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03bc : Type u_5\nf : \u03b3 \u2192 \u03b4 \u2192 \u03bc\ng : \u03b1 \u2192 \u03b3\nh : \u03b2 \u2192 \u03b4\nas : List \u03b1\nbs : List \u03b2\n\u22a2 zipWith f (map g as) (map h bs) = zipWith (fun a b => f (g a) (h b)) as bs\n[PROOFSTEP]\ninduction as generalizing bs\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03bc : Type u_5\nf : \u03b3 \u2192 \u03b4 \u2192 \u03bc\ng : \u03b1 \u2192 \u03b3\nh : \u03b2 \u2192 \u03b4\nbs : List \u03b2\n\u22a2 zipWith f (map g []) (map h bs) = zipWith (fun a b => f (g a) (h b)) [] bs\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03bc : Type u_5\nf : \u03b3 \u2192 \u03b4 \u2192 \u03bc\ng : \u03b1 \u2192 \u03b3\nh : \u03b2 \u2192 \u03b4\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (bs : List \u03b2), zipWith f (map g tail\u271d) (map h bs) = zipWith (fun a b => f (g a) (h b)) tail\u271d bs\nbs : List \u03b2\n\u22a2 zipWith f (map g (head\u271d :: tail\u271d)) (map h bs) = zipWith (fun a b => f (g a) (h b)) (head\u271d :: tail\u271d) bs\n[PROOFSTEP]\ncases bs\n[GOAL]\ncase cons.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03bc : Type u_5\nf : \u03b3 \u2192 \u03b4 \u2192 \u03bc\ng : \u03b1 \u2192 \u03b3\nh : \u03b2 \u2192 \u03b4\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (bs : List \u03b2), zipWith f (map g tail\u271d) (map h bs) = zipWith (fun a b => f (g a) (h b)) tail\u271d bs\n\u22a2 zipWith f (map g (head\u271d :: tail\u271d)) (map h []) = zipWith (fun a b => f (g a) (h b)) (head\u271d :: tail\u271d) []\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03bc : Type u_5\nf : \u03b3 \u2192 \u03b4 \u2192 \u03bc\ng : \u03b1 \u2192 \u03b3\nh : \u03b2 \u2192 \u03b4\nhead\u271d\u00b9 : \u03b1\ntail\u271d\u00b9 : List \u03b1\ntail_ih\u271d : \u2200 (bs : List \u03b2), zipWith f (map g tail\u271d\u00b9) (map h bs) = zipWith (fun a b => f (g a) (h b)) tail\u271d\u00b9 bs\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\n\u22a2 zipWith f (map g (head\u271d\u00b9 :: tail\u271d\u00b9)) (map h (head\u271d :: tail\u271d)) =\n zipWith (fun a b => f (g a) (h b)) (head\u271d\u00b9 :: tail\u271d\u00b9) (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03b4 \u2192 \u03b1\nl : List \u03b4\nl' : List \u03b2\n\u22a2 zipWith f (map g l) l' = zipWith (f \u2218 g) l l'\n[PROOFSTEP]\nconvert zipWith_map f g id l l'\n[GOAL]\ncase h.e'_2.h.e'_6\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03b4 \u2192 \u03b1\nl : List \u03b4\nl' : List \u03b2\n\u22a2 l' = map id l'\n[PROOFSTEP]\nexact Eq.symm (List.map_id _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\ng : \u03b4 \u2192 \u03b2\nl' : List \u03b4\n\u22a2 zipWith f l (map g l') = zipWith (fun x => f x \u2218 g) l l'\n[PROOFSTEP]\nconvert List.zipWith_map f id g l l'\n[GOAL]\ncase h.e'_2.h.e'_5\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\ng : \u03b4 \u2192 \u03b2\nl' : List \u03b4\n\u22a2 l = map id l\n[PROOFSTEP]\nexact Eq.symm (List.map_id _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\na : \u03b1\nl : List \u03b1\n\u22a2 zip (map f (a :: l)) (map g (a :: l)) = map (fun a => (f a, g a)) (a :: l)\n[PROOFSTEP]\nsimp only [map, zip_cons_cons, zip_map']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4\u271d : Type u_3\n\u03b5 : Type u_4\n\u03b4 : Type u_5\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4 \u2192 \u03b1\nl : List \u03b3\nl' : List \u03b4\n\u22a2 map f (zipWith g l l') = zipWith (fun x y => f (g x y)) l l'\n[PROOFSTEP]\ninduction' l with hd tl hl generalizing l'\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4\u271d : Type u_3\n\u03b5 : Type u_4\n\u03b4 : Type u_5\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4 \u2192 \u03b1\nl'\u271d l' : List \u03b4\n\u22a2 map f (zipWith g [] l') = zipWith (fun x y => f (g x y)) [] l'\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4\u271d : Type u_3\n\u03b5 : Type u_4\n\u03b4 : Type u_5\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4 \u2192 \u03b1\nl'\u271d : List \u03b4\nhd : \u03b3\ntl : List \u03b3\nhl : \u2200 (l' : List \u03b4), map f (zipWith g tl l') = zipWith (fun x y => f (g x y)) tl l'\nl' : List \u03b4\n\u22a2 map f (zipWith g (hd :: tl) l') = zipWith (fun x y => f (g x y)) (hd :: tl) l'\n[PROOFSTEP]\ncases l'\n[GOAL]\ncase cons.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4\u271d : Type u_3\n\u03b5 : Type u_4\n\u03b4 : Type u_5\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4 \u2192 \u03b1\nl' : List \u03b4\nhd : \u03b3\ntl : List \u03b3\nhl : \u2200 (l' : List \u03b4), map f (zipWith g tl l') = zipWith (fun x y => f (g x y)) tl l'\n\u22a2 map f (zipWith g (hd :: tl) []) = zipWith (fun x y => f (g x y)) (hd :: tl) []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4\u271d : Type u_3\n\u03b5 : Type u_4\n\u03b4 : Type u_5\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4 \u2192 \u03b1\nl' : List \u03b4\nhd : \u03b3\ntl : List \u03b3\nhl : \u2200 (l' : List \u03b4), map f (zipWith g tl l') = zipWith (fun x y => f (g x y)) tl l'\nhead\u271d : \u03b4\ntail\u271d : List \u03b4\n\u22a2 map f (zipWith g (hd :: tl) (head\u271d :: tail\u271d)) = zipWith (fun x y => f (g x y)) (hd :: tl) (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp [hl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\na : \u03b1\nb : \u03b2\nhead\u271d\u00b9 : \u03b1\nl\u2081 : List \u03b1\nhead\u271d : \u03b2\nl\u2082 : List \u03b2\nh : (a, b) \u2208 zip (head\u271d\u00b9 :: l\u2081) (head\u271d :: l\u2082)\n\u22a2 a \u2208 head\u271d\u00b9 :: l\u2081 \u2227 b \u2208 head\u271d :: l\u2082\n[PROOFSTEP]\ncases' h with _ _ _ h\n[GOAL]\ncase head\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\na : \u03b1\nb : \u03b2\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\n\u22a2 a \u2208 a :: l\u2081 \u2227 b \u2208 b :: l\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase tail\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\na : \u03b1\nb : \u03b2\nhead\u271d\u00b9 : \u03b1\nl\u2081 : List \u03b1\nhead\u271d : \u03b2\nl\u2082 : List \u03b2\nh : Mem (a, b) (zipWith Prod.mk l\u2081 l\u2082)\n\u22a2 a \u2208 head\u271d\u00b9 :: l\u2081 \u2227 b \u2208 head\u271d :: l\u2082\n[PROOFSTEP]\nhave := mem_zip h\n[GOAL]\ncase tail\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\na : \u03b1\nb : \u03b2\nhead\u271d\u00b9 : \u03b1\nl\u2081 : List \u03b1\nhead\u271d : \u03b2\nl\u2082 : List \u03b2\nh : Mem (a, b) (zipWith Prod.mk l\u2081 l\u2082)\nthis : a \u2208 l\u2081 \u2227 b \u2208 l\u2082\n\u22a2 a \u2208 head\u271d\u00b9 :: l\u2081 \u2227 b \u2208 head\u271d :: l\u2082\n[PROOFSTEP]\nexact \u27e8Mem.tail _ this.1, Mem.tail _ this.2\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhead\u271d\u00b9 : \u03b1\nas : List \u03b1\nhead\u271d : \u03b2\nbs : List \u03b2\nh : length (head\u271d\u00b9 :: as) \u2264 length (head\u271d :: bs)\n\u22a2 map Prod.fst (zip (head\u271d\u00b9 :: as) (head\u271d :: bs)) = head\u271d\u00b9 :: as\n[PROOFSTEP]\nsimp [succ_le_succ_iff] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhead\u271d\u00b9 : \u03b1\nas : List \u03b1\nhead\u271d : \u03b2\nbs : List \u03b2\nh : length as \u2264 length bs\n\u22a2 map Prod.fst (zip (head\u271d\u00b9 :: as) (head\u271d :: bs)) = head\u271d\u00b9 :: as\n[PROOFSTEP]\nchange _ :: map Prod.fst (zip as bs) = _ :: as\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhead\u271d\u00b9 : \u03b1\nas : List \u03b1\nhead\u271d : \u03b2\nbs : List \u03b2\nh : length as \u2264 length bs\n\u22a2 (head\u271d\u00b9, head\u271d).fst :: map Prod.fst (zip as bs) = head\u271d\u00b9 :: as\n[PROOFSTEP]\nrw [map_fst_zip as bs h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\na : \u03b1\nas : List \u03b1\nh : length (a :: as) \u2264 length []\n\u22a2 map Prod.fst (zip (a :: as) []) = a :: as\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nx\u271d\u00b9 : List \u03b1\nx\u271d : length [] \u2264 length x\u271d\u00b9\n\u22a2 map Prod.snd (zip x\u271d\u00b9 []) = []\n[PROOFSTEP]\nrw [zip_nil_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nx\u271d\u00b9 : List \u03b1\nx\u271d : length [] \u2264 length x\u271d\u00b9\n\u22a2 map Prod.snd [] = []\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nb : \u03b2\nbs : List \u03b2\nh : length (b :: bs) \u2264 length []\n\u22a2 map Prod.snd (zip [] (b :: bs)) = b :: bs\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\na : \u03b1\nas : List \u03b1\nb : \u03b2\nbs : List \u03b2\nh : length (b :: bs) \u2264 length (a :: as)\n\u22a2 map Prod.snd (zip (a :: as) (b :: bs)) = b :: bs\n[PROOFSTEP]\nsimp [succ_le_succ_iff] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\na : \u03b1\nas : List \u03b1\nb : \u03b2\nbs : List \u03b2\nh : length bs \u2264 length as\n\u22a2 map Prod.snd (zip (a :: as) (b :: bs)) = b :: bs\n[PROOFSTEP]\nchange _ :: map Prod.snd (zip as bs) = _ :: bs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\na : \u03b1\nas : List \u03b1\nb : \u03b2\nbs : List \u03b2\nh : length bs \u2264 length as\n\u22a2 (a, b).snd :: map Prod.snd (zip as bs) = b :: bs\n[PROOFSTEP]\nrw [map_snd_zip as bs h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\na : \u03b1\nb : \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\n\u22a2 unzip ((a, b) :: l) = (map Prod.fst ((a, b) :: l), map Prod.snd ((a, b) :: l))\n[PROOFSTEP]\nsimp only [unzip_cons, map_cons, unzip_eq_map l]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List (\u03b1 \u00d7 \u03b2)\n\u22a2 (unzip l).fst = map Prod.fst l\n[PROOFSTEP]\nsimp only [unzip_eq_map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List (\u03b1 \u00d7 \u03b2)\n\u22a2 (unzip l).snd = map Prod.snd l\n[PROOFSTEP]\nsimp only [unzip_eq_map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List (\u03b1 \u00d7 \u03b2)\n\u22a2 unzip (map Prod.swap l) = Prod.swap (unzip l)\n[PROOFSTEP]\nsimp only [unzip_eq_map, map_map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List (\u03b1 \u00d7 \u03b2)\n\u22a2 (map (Prod.fst \u2218 Prod.swap) l, map (Prod.snd \u2218 Prod.swap) l) = Prod.swap (map Prod.fst l, map Prod.snd l)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\na : \u03b1\nb : \u03b2\nl : List (\u03b1 \u00d7 \u03b2)\n\u22a2 zip (unzip ((a, b) :: l)).fst (unzip ((a, b) :: l)).snd = (a, b) :: l\n[PROOFSTEP]\nsimp only [unzip_cons, zip_cons_cons, zip_unzip l]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 : List \u03b1\nh : length l\u2081 \u2264 length []\n\u22a2 (unzip (zip l\u2081 [])).fst = l\u2081\n[PROOFSTEP]\nrw [eq_nil_of_length_eq_zero (Nat.eq_zero_of_le_zero h)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 : List \u03b1\nh : length l\u2081 \u2264 length []\n\u22a2 (unzip (zip [] [])).fst = []\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b2\nl\u2082 : List \u03b2\nh : length (a :: l\u2081) \u2264 length (b :: l\u2082)\n\u22a2 (unzip (zip (a :: l\u2081) (b :: l\u2082))).fst = a :: l\u2081\n[PROOFSTEP]\nsimp only [zip_cons_cons, unzip_cons, unzip_zip_left (le_of_succ_le_succ h)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\nh : length l\u2082 \u2264 length l\u2081\n\u22a2 (unzip (zip l\u2081 l\u2082)).snd = l\u2082\n[PROOFSTEP]\nrw [\u2190 zip_swap, unzip_swap]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\nh : length l\u2082 \u2264 length l\u2081\n\u22a2 (Prod.swap (unzip (zip l\u2082 l\u2081))).snd = l\u2082\n[PROOFSTEP]\nexact unzip_zip_left h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\nh : length l\u2081 = length l\u2082\n\u22a2 unzip (zip l\u2081 l\u2082) = (l\u2081, l\u2082)\n[PROOFSTEP]\nrw [\u2190 Prod.mk.eta (p := unzip (zip l\u2081 l\u2082)), unzip_zip_left (le_of_eq h), unzip_zip_right (ge_of_eq h)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List \u03b1\nl' : List \u03b2\nlp : List (\u03b1 \u00d7 \u03b2)\nhl : map Prod.fst lp = l\nhr : map Prod.snd lp = l'\n\u22a2 lp = zip l l'\n[PROOFSTEP]\nrw [\u2190 hl, \u2190 hr, \u2190 zip_unzip lp, \u2190 unzip_left, \u2190 unzip_right, zip_unzip, zip_unzip]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 map (fun x => (x, f x)) l = zip l (map f l)\n[PROOFSTEP]\nrw [\u2190 zip_map']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 zip (map (fun x => x) l) (map (fun x => f x) l) = zip l (map f l)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 map (fun x => x) l = l\n[PROOFSTEP]\nexact map_id _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 map (fun x => (f x, x)) l = zip (map f l) l\n[PROOFSTEP]\nrw [\u2190 zip_map']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 zip (map (fun x => f x) l) (map (fun x => x) l) = zip (map f l) l\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 map (fun x => x) l = l\n[PROOFSTEP]\nexact map_id _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf g : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nla : List \u03b1\nlb : List \u03b2\nh : Forall\u2082 (fun a b => f a b = g a b) la lb\n\u22a2 zipWith f la lb = zipWith g la lb\n[PROOFSTEP]\ninduction' h with a b as bs hfg _ ih\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf g : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nla : List \u03b1\nlb : List \u03b2\n\u22a2 zipWith f [] [] = zipWith g [] []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf g : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nla : List \u03b1\nlb : List \u03b2\na : \u03b1\nb : \u03b2\nas : List \u03b1\nbs : List \u03b2\nhfg : f a b = g a b\na\u271d : Forall\u2082 (fun a b => f a b = g a b) as bs\nih : zipWith f as bs = zipWith g as bs\n\u22a2 zipWith f (a :: as) (b :: bs) = zipWith g (a :: as) (b :: bs)\n[PROOFSTEP]\nexact congr_arg\u2082 _ hfg ih\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b2\ncomm : \u2200 (x y : \u03b1), f x y = f y x\nl l' : List \u03b1\n\u22a2 zipWith f l l' = zipWith f l' l\n[PROOFSTEP]\nrw [zipWith_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b2\ncomm : \u2200 (x y : \u03b1), f x y = f y x\nl l' : List \u03b1\n\u22a2 zipWith (fun b a => f a b) l' l = zipWith f l' l\n[PROOFSTEP]\nsimp only [comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List \u03b1\n\u22a2 length (revzip l) = length l\n[PROOFSTEP]\nsimp only [revzip, length_zip, length_reverse, min_self]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List \u03b1\n\u22a2 map Prod.fst (revzip l) = l\n[PROOFSTEP]\nrw [\u2190 unzip_left, unzip_revzip]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List \u03b1\n\u22a2 map Prod.snd (revzip l) = reverse l\n[PROOFSTEP]\nrw [\u2190 unzip_right, unzip_revzip]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List \u03b1\n\u22a2 reverse (revzip l) = revzip (reverse l)\n[PROOFSTEP]\nrw [\u2190 zip_unzip (revzip l).reverse]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List \u03b1\n\u22a2 zip (unzip (reverse (revzip l))).fst (unzip (reverse (revzip l))).snd = revzip (reverse l)\n[PROOFSTEP]\nsimp [unzip_eq_map, revzip, map_reverse, map_fst_zip, map_snd_zip]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl : List \u03b1\n\u22a2 map Prod.swap (revzip l) = revzip (reverse l)\n[PROOFSTEP]\nsimp [revzip]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\ni : \u2115\n\u22a2 get? (zipWith f l\u2081 l\u2082) i = Option.bind (Option.map f (get? l\u2081 i)) fun g => Option.map g (get? l\u2082 i)\n[PROOFSTEP]\ninduction' l\u2081 with head tail generalizing l\u2082 i\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2082\u271d : List \u03b2\ni\u271d : \u2115\nl\u2082 : List \u03b2\ni : \u2115\n\u22a2 get? (zipWith f [] l\u2082) i = Option.bind (Option.map f (get? [] i)) fun g => Option.map g (get? l\u2082 i)\n[PROOFSTEP]\nrw [zipWith]\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2082\u271d : List \u03b2\ni\u271d : \u2115\nl\u2082 : List \u03b2\ni : \u2115\n\u22a2 get? [] i = Option.bind (Option.map f (get? [] i)) fun g => Option.map g (get? l\u2082 i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase nil.x_4\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2082\u271d : List \u03b2\ni\u271d : \u2115\nl\u2082 : List \u03b2\ni : \u2115\n\u22a2 \u2200 (x : \u03b1) (xs : List \u03b1) (y : \u03b2) (ys : List \u03b2), [] = x :: xs \u2192 l\u2082 = y :: ys \u2192 False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2082\u271d : List \u03b2\ni\u271d : \u2115\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d :\n \u2200 (l\u2082 : List \u03b2) (i : \u2115),\n get? (zipWith f tail l\u2082) i = Option.bind (Option.map f (get? tail i)) fun g => Option.map g (get? l\u2082 i)\nl\u2082 : List \u03b2\ni : \u2115\n\u22a2 get? (zipWith f (head :: tail) l\u2082) i =\n Option.bind (Option.map f (get? (head :: tail) i)) fun g => Option.map g (get? l\u2082 i)\n[PROOFSTEP]\ncases l\u2082\n[GOAL]\ncase cons.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2082 : List \u03b2\ni\u271d : \u2115\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d :\n \u2200 (l\u2082 : List \u03b2) (i : \u2115),\n get? (zipWith f tail l\u2082) i = Option.bind (Option.map f (get? tail i)) fun g => Option.map g (get? l\u2082 i)\ni : \u2115\n\u22a2 get? (zipWith f (head :: tail) []) i =\n Option.bind (Option.map f (get? (head :: tail) i)) fun g => Option.map g (get? [] i)\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2082 : List \u03b2\ni\u271d : \u2115\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d :\n \u2200 (l\u2082 : List \u03b2) (i : \u2115),\n get? (zipWith f tail l\u2082) i = Option.bind (Option.map f (get? tail i)) fun g => Option.map g (get? l\u2082 i)\ni : \u2115\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\n\u22a2 get? (zipWith f (head :: tail) (head\u271d :: tail\u271d)) i =\n Option.bind (Option.map f (get? (head :: tail) i)) fun g => Option.map g (get? (head\u271d :: tail\u271d) i)\n[PROOFSTEP]\nsimp only [zipWith, Seq.seq, Functor.map, get?, Option.map_none']\n[GOAL]\ncase cons.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2082 : List \u03b2\ni\u271d : \u2115\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d :\n \u2200 (l\u2082 : List \u03b2) (i : \u2115),\n get? (zipWith f tail l\u2082) i = Option.bind (Option.map f (get? tail i)) fun g => Option.map g (get? l\u2082 i)\ni : \u2115\n\u22a2 none = Option.bind (Option.map f (get? (head :: tail) i)) fun g => none\n[PROOFSTEP]\ncases (head :: tail).get? i\n[GOAL]\ncase cons.nil.none\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2082 : List \u03b2\ni\u271d : \u2115\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d :\n \u2200 (l\u2082 : List \u03b2) (i : \u2115),\n get? (zipWith f tail l\u2082) i = Option.bind (Option.map f (get? tail i)) fun g => Option.map g (get? l\u2082 i)\ni : \u2115\n\u22a2 none = Option.bind (Option.map f none) fun g => none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons.nil.some\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2082 : List \u03b2\ni\u271d : \u2115\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d :\n \u2200 (l\u2082 : List \u03b2) (i : \u2115),\n get? (zipWith f tail l\u2082) i = Option.bind (Option.map f (get? tail i)) fun g => Option.map g (get? l\u2082 i)\ni : \u2115\nval\u271d : \u03b1\n\u22a2 none = Option.bind (Option.map f (some val\u271d)) fun g => none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2082 : List \u03b2\ni\u271d : \u2115\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d :\n \u2200 (l\u2082 : List \u03b2) (i : \u2115),\n get? (zipWith f tail l\u2082) i = Option.bind (Option.map f (get? tail i)) fun g => Option.map g (get? l\u2082 i)\ni : \u2115\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\n\u22a2 get? (zipWith f (head :: tail) (head\u271d :: tail\u271d)) i =\n Option.bind (Option.map f (get? (head :: tail) i)) fun g => Option.map g (get? (head\u271d :: tail\u271d) i)\n[PROOFSTEP]\ncases i\n[GOAL]\ncase cons.cons.zero\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2082 : List \u03b2\ni : \u2115\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d :\n \u2200 (l\u2082 : List \u03b2) (i : \u2115),\n get? (zipWith f tail l\u2082) i = Option.bind (Option.map f (get? tail i)) fun g => Option.map g (get? l\u2082 i)\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\n\u22a2 get? (zipWith f (head :: tail) (head\u271d :: tail\u271d)) zero =\n Option.bind (Option.map f (get? (head :: tail) zero)) fun g => Option.map g (get? (head\u271d :: tail\u271d) zero)\n[PROOFSTEP]\nsimp only [Option.map_some', get?, Option.some_bind', *]\n[GOAL]\ncase cons.cons.succ\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2082 : List \u03b2\ni : \u2115\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d :\n \u2200 (l\u2082 : List \u03b2) (i : \u2115),\n get? (zipWith f tail l\u2082) i = Option.bind (Option.map f (get? tail i)) fun g => Option.map g (get? l\u2082 i)\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\nn\u271d : \u2115\n\u22a2 get? (zipWith f (head :: tail) (head\u271d :: tail\u271d)) (succ n\u271d) =\n Option.bind (Option.map f (get? (head :: tail) (succ n\u271d))) fun g => Option.map g (get? (head\u271d :: tail\u271d) (succ n\u271d))\n[PROOFSTEP]\nsimp only [Option.map_some', get?, Option.some_bind', *]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\nz : \u03b3\ni : \u2115\n\u22a2 get? (zipWith f l\u2081 l\u2082) i = some z \u2194 \u2203 x y, get? l\u2081 i = some x \u2227 get? l\u2082 i = some y \u2227 f x y = z\n[PROOFSTEP]\ninduction l\u2081 generalizing l\u2082 i\n[GOAL]\ncase nil\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nz : \u03b3\nl\u2082 : List \u03b2\ni : \u2115\n\u22a2 get? (zipWith f [] l\u2082) i = some z \u2194 \u2203 x y, get? [] i = some x \u2227 get? l\u2082 i = some y \u2227 f x y = z\n[PROOFSTEP]\nsimp [zipWith]\n[GOAL]\ncase cons\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nz : \u03b3\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d :\n \u2200 (l\u2082 : List \u03b2) (i : \u2115),\n get? (zipWith f tail\u271d l\u2082) i = some z \u2194 \u2203 x y, get? tail\u271d i = some x \u2227 get? l\u2082 i = some y \u2227 f x y = z\nl\u2082 : List \u03b2\ni : \u2115\n\u22a2 get? (zipWith f (head\u271d :: tail\u271d) l\u2082) i = some z \u2194\n \u2203 x y, get? (head\u271d :: tail\u271d) i = some x \u2227 get? l\u2082 i = some y \u2227 f x y = z\n[PROOFSTEP]\ncases l\u2082\n[GOAL]\ncase cons.nil\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nz : \u03b3\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d :\n \u2200 (l\u2082 : List \u03b2) (i : \u2115),\n get? (zipWith f tail\u271d l\u2082) i = some z \u2194 \u2203 x y, get? tail\u271d i = some x \u2227 get? l\u2082 i = some y \u2227 f x y = z\ni : \u2115\n\u22a2 get? (zipWith f (head\u271d :: tail\u271d) []) i = some z \u2194\n \u2203 x y, get? (head\u271d :: tail\u271d) i = some x \u2227 get? [] i = some y \u2227 f x y = z\n[PROOFSTEP]\nsimp only [zipWith, get?, exists_false, and_false_iff, false_and_iff]\n[GOAL]\ncase cons.cons\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nz : \u03b3\nhead\u271d\u00b9 : \u03b1\ntail\u271d\u00b9 : List \u03b1\ntail_ih\u271d :\n \u2200 (l\u2082 : List \u03b2) (i : \u2115),\n get? (zipWith f tail\u271d\u00b9 l\u2082) i = some z \u2194 \u2203 x y, get? tail\u271d\u00b9 i = some x \u2227 get? l\u2082 i = some y \u2227 f x y = z\ni : \u2115\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\n\u22a2 get? (zipWith f (head\u271d\u00b9 :: tail\u271d\u00b9) (head\u271d :: tail\u271d)) i = some z \u2194\n \u2203 x y, get? (head\u271d\u00b9 :: tail\u271d\u00b9) i = some x \u2227 get? (head\u271d :: tail\u271d) i = some y \u2227 f x y = z\n[PROOFSTEP]\nsimp only [zipWith, get?, exists_false, and_false_iff, false_and_iff]\n[GOAL]\ncase cons.cons\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nz : \u03b3\nhead\u271d\u00b9 : \u03b1\ntail\u271d\u00b9 : List \u03b1\ntail_ih\u271d :\n \u2200 (l\u2082 : List \u03b2) (i : \u2115),\n get? (zipWith f tail\u271d\u00b9 l\u2082) i = some z \u2194 \u2203 x y, get? tail\u271d\u00b9 i = some x \u2227 get? l\u2082 i = some y \u2227 f x y = z\ni : \u2115\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\n\u22a2 get? (f head\u271d\u00b9 head\u271d :: zipWith f tail\u271d\u00b9 tail\u271d) i = some z \u2194\n \u2203 x y, get? (head\u271d\u00b9 :: tail\u271d\u00b9) i = some x \u2227 get? (head\u271d :: tail\u271d) i = some y \u2227 f x y = z\n[PROOFSTEP]\ncases i\n[GOAL]\ncase cons.cons.zero\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nz : \u03b3\nhead\u271d\u00b9 : \u03b1\ntail\u271d\u00b9 : List \u03b1\ntail_ih\u271d :\n \u2200 (l\u2082 : List \u03b2) (i : \u2115),\n get? (zipWith f tail\u271d\u00b9 l\u2082) i = some z \u2194 \u2203 x y, get? tail\u271d\u00b9 i = some x \u2227 get? l\u2082 i = some y \u2227 f x y = z\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\n\u22a2 get? (f head\u271d\u00b9 head\u271d :: zipWith f tail\u271d\u00b9 tail\u271d) zero = some z \u2194\n \u2203 x y, get? (head\u271d\u00b9 :: tail\u271d\u00b9) zero = some x \u2227 get? (head\u271d :: tail\u271d) zero = some y \u2227 f x y = z\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase cons.cons.succ\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_7\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nz : \u03b3\nhead\u271d\u00b9 : \u03b1\ntail\u271d\u00b9 : List \u03b1\ntail_ih\u271d :\n \u2200 (l\u2082 : List \u03b2) (i : \u2115),\n get? (zipWith f tail\u271d\u00b9 l\u2082) i = some z \u2194 \u2203 x y, get? tail\u271d\u00b9 i = some x \u2227 get? l\u2082 i = some y \u2227 f x y = z\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\nn\u271d : \u2115\n\u22a2 get? (f head\u271d\u00b9 head\u271d :: zipWith f tail\u271d\u00b9 tail\u271d) (succ n\u271d) = some z \u2194\n \u2203 x y, get? (head\u271d\u00b9 :: tail\u271d\u00b9) (succ n\u271d) = some x \u2227 get? (head\u271d :: tail\u271d) (succ n\u271d) = some y \u2227 f x y = z\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\nz : \u03b1 \u00d7 \u03b2\ni : \u2115\n\u22a2 get? (zip l\u2081 l\u2082) i = some z \u2194 get? l\u2081 i = some z.fst \u2227 get? l\u2082 i = some z.snd\n[PROOFSTEP]\ncases z\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\ni : \u2115\nfst\u271d : \u03b1\nsnd\u271d : \u03b2\n\u22a2 get? (zip l\u2081 l\u2082) i = some (fst\u271d, snd\u271d) \u2194 get? l\u2081 i = some (fst\u271d, snd\u271d).fst \u2227 get? l\u2082 i = some (fst\u271d, snd\u271d).snd\n[PROOFSTEP]\nrw [zip, get?_zip_with_eq_some]\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\ni : \u2115\nfst\u271d : \u03b1\nsnd\u271d : \u03b2\n\u22a2 (\u2203 x y, get? l\u2081 i = some x \u2227 get? l\u2082 i = some y \u2227 (x, y) = (fst\u271d, snd\u271d)) \u2194\n get? l\u2081 i = some (fst\u271d, snd\u271d).fst \u2227 get? l\u2082 i = some (fst\u271d, snd\u271d).snd\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.mp\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\ni : \u2115\nfst\u271d : \u03b1\nsnd\u271d : \u03b2\n\u22a2 (\u2203 x y, get? l\u2081 i = some x \u2227 get? l\u2082 i = some y \u2227 (x, y) = (fst\u271d, snd\u271d)) \u2192\n get? l\u2081 i = some (fst\u271d, snd\u271d).fst \u2227 get? l\u2082 i = some (fst\u271d, snd\u271d).snd\n[PROOFSTEP]\nrintro \u27e8x, y, h\u2080, h\u2081, h\u2082\u27e9\n[GOAL]\ncase mk.mp.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\ni : \u2115\nfst\u271d : \u03b1\nsnd\u271d : \u03b2\nx : \u03b1\ny : \u03b2\nh\u2080 : get? l\u2081 i = some x\nh\u2081 : get? l\u2082 i = some y\nh\u2082 : (x, y) = (fst\u271d, snd\u271d)\n\u22a2 get? l\u2081 i = some (fst\u271d, snd\u271d).fst \u2227 get? l\u2082 i = some (fst\u271d, snd\u271d).snd\n[PROOFSTEP]\nsimpa [h\u2080, h\u2081] using h\u2082\n[GOAL]\ncase mk.mpr\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\ni : \u2115\nfst\u271d : \u03b1\nsnd\u271d : \u03b2\n\u22a2 get? l\u2081 i = some (fst\u271d, snd\u271d).fst \u2227 get? l\u2082 i = some (fst\u271d, snd\u271d).snd \u2192\n \u2203 x y, get? l\u2081 i = some x \u2227 get? l\u2082 i = some y \u2227 (x, y) = (fst\u271d, snd\u271d)\n[PROOFSTEP]\nrintro \u27e8h\u2080, h\u2081\u27e9\n[GOAL]\ncase mk.mpr.intro\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\ni : \u2115\nfst\u271d : \u03b1\nsnd\u271d : \u03b2\nh\u2080 : get? l\u2081 i = some (fst\u271d, snd\u271d).fst\nh\u2081 : get? l\u2082 i = some (fst\u271d, snd\u271d).snd\n\u22a2 \u2203 x y, get? l\u2081 i = some x \u2227 get? l\u2082 i = some y \u2227 (x, y) = (fst\u271d, snd\u271d)\n[PROOFSTEP]\nexact \u27e8_, _, h\u2080, h\u2081, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\ni : Fin (length (zipWith f l l'))\n\u22a2 get (zipWith f l l') i =\n f (get l { val := \u2191i, isLt := (_ : \u2191i < length l) }) (get l' { val := \u2191i, isLt := (_ : \u2191i < length l') })\n[PROOFSTEP]\nrw [\u2190 Option.some_inj, \u2190 get?_eq_get, get?_zip_with_eq_some]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\ni : Fin (length (zipWith f l l'))\n\u22a2 \u2203 x y,\n get? l \u2191i = some x \u2227\n get? l' \u2191i = some y \u2227\n f x y =\n f (get l { val := \u2191i, isLt := (_ : \u2191i < length l) }) (get l' { val := \u2191i, isLt := (_ : \u2191i < length l') })\n[PROOFSTEP]\nexact\n \u27e8l.get \u27e8i, lt_length_left_of_zipWith i.isLt\u27e9, l'.get \u27e8i, lt_length_right_of_zipWith i.isLt\u27e9, by rw [get?_eq_get], by\n rw [get?_eq_get]; exact \u27e8rfl, rfl\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\ni : Fin (length (zipWith f l l'))\n\u22a2 get? l \u2191i = some (get l { val := \u2191i, isLt := (_ : \u2191i < length l) })\n[PROOFSTEP]\nrw [get?_eq_get]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\ni : Fin (length (zipWith f l l'))\n\u22a2 get? l' \u2191i = some (get l' { val := \u2191i, isLt := (_ : \u2191i < length l') }) \u2227\n f (get l { val := \u2191i, isLt := (_ : \u2191i < length l) }) (get l' { val := \u2191i, isLt := (_ : \u2191i < length l') }) =\n f (get l { val := \u2191i, isLt := (_ : \u2191i < length l) }) (get l' { val := \u2191i, isLt := (_ : \u2191i < length l') })\n[PROOFSTEP]\nrw [get?_eq_get]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\ni : Fin (length (zipWith f l l'))\n\u22a2 some (get l' { val := \u2191i, isLt := ?m.129801 }) = some (get l' { val := \u2191i, isLt := (_ : \u2191i < length l') }) \u2227\n f (get l { val := \u2191i, isLt := (_ : \u2191i < length l) }) (get l' { val := \u2191i, isLt := (_ : \u2191i < length l') }) =\n f (get l { val := \u2191i, isLt := (_ : \u2191i < length l) }) (get l' { val := \u2191i, isLt := (_ : \u2191i < length l') })\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\ni : Fin (length (zipWith f l l'))\n\u22a2 \u2191i < length l'\n[PROOFSTEP]\nexact \u27e8rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nl init tail : List \u03b1\n\u22a2 (init, tail) \u2208 zip (inits l) (tails l) \u2194 init ++ tail = l\n[PROOFSTEP]\ninduction' l with hd tl ih generalizing init tail\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit\u271d tail\u271d init tail : List \u03b1\n\u22a2 (init, tail) \u2208 zip (inits []) (tails []) \u2194 init ++ tail = []\n[PROOFSTEP]\nsimp_rw [tails, inits, zip_cons_cons]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit\u271d tail\u271d : List \u03b1\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 zip (inits tl) (tails tl) \u2194 init ++ tail = tl\ninit tail : List \u03b1\n\u22a2 (init, tail) \u2208 zip (inits (hd :: tl)) (tails (hd :: tl)) \u2194 init ++ tail = hd :: tl\n[PROOFSTEP]\nsimp_rw [tails, inits, zip_cons_cons]\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit\u271d tail\u271d init tail : List \u03b1\n\u22a2 (init, tail) \u2208 ([], []) :: zip [] [] \u2194 init ++ tail = []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit\u271d tail\u271d : List \u03b1\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 zip (inits tl) (tails tl) \u2194 init ++ tail = tl\ninit tail : List \u03b1\n\u22a2 (init, tail) \u2208 ([], hd :: tl) :: zip (map (fun t => hd :: t) (inits tl)) (tails tl) \u2194 init ++ tail = hd :: tl\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons.mp\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit\u271d tail\u271d : List \u03b1\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 zip (inits tl) (tails tl) \u2194 init ++ tail = tl\ninit tail : List \u03b1\n\u22a2 (init, tail) \u2208 ([], hd :: tl) :: zip (map (fun t => hd :: t) (inits tl)) (tails tl) \u2192 init ++ tail = hd :: tl\n[PROOFSTEP]\nrw [mem_cons, zip_map_left, mem_map, Prod.exists]\n[GOAL]\ncase cons.mpr\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit\u271d tail\u271d : List \u03b1\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 zip (inits tl) (tails tl) \u2194 init ++ tail = tl\ninit tail : List \u03b1\n\u22a2 init ++ tail = hd :: tl \u2192 (init, tail) \u2208 ([], hd :: tl) :: zip (map (fun t => hd :: t) (inits tl)) (tails tl)\n[PROOFSTEP]\nrw [mem_cons, zip_map_left, mem_map, Prod.exists]\n[GOAL]\ncase cons.mp\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit\u271d tail\u271d : List \u03b1\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 zip (inits tl) (tails tl) \u2194 init ++ tail = tl\ninit tail : List \u03b1\n\u22a2 ((init, tail) = ([], hd :: tl) \u2228\n \u2203 a b, (a, b) \u2208 zip (inits tl) (tails tl) \u2227 Prod.map (fun t => hd :: t) id (a, b) = (init, tail)) \u2192\n init ++ tail = hd :: tl\n[PROOFSTEP]\nrintro (\u27e8rfl, rfl\u27e9 | \u27e8_, _, h, rfl, rfl\u27e9)\n[GOAL]\ncase cons.mp.inl.refl\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit tail : List \u03b1\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 zip (inits tl) (tails tl) \u2194 init ++ tail = tl\n\u22a2 [] ++ hd :: tl = hd :: tl\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.mp.inr.intro.intro.intro.refl\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit tail : List \u03b1\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 zip (inits tl) (tails tl) \u2194 init ++ tail = tl\nw\u271d\u00b9 w\u271d : List \u03b1\nh : (w\u271d\u00b9, w\u271d) \u2208 zip (inits tl) (tails tl)\n\u22a2 (fun t => hd :: t) w\u271d\u00b9 ++ id w\u271d = hd :: tl\n[PROOFSTEP]\nsimp [ih.mp h]\n[GOAL]\ncase cons.mpr\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit\u271d tail\u271d : List \u03b1\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 zip (inits tl) (tails tl) \u2194 init ++ tail = tl\ninit tail : List \u03b1\n\u22a2 init ++ tail = hd :: tl \u2192\n (init, tail) = ([], hd :: tl) \u2228\n \u2203 a b, (a, b) \u2208 zip (inits tl) (tails tl) \u2227 Prod.map (fun t => hd :: t) id (a, b) = (init, tail)\n[PROOFSTEP]\ncases' init with hd' tl'\n[GOAL]\ncase cons.mpr.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit tail\u271d : List \u03b1\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 zip (inits tl) (tails tl) \u2194 init ++ tail = tl\ntail : List \u03b1\n\u22a2 [] ++ tail = hd :: tl \u2192\n ([], tail) = ([], hd :: tl) \u2228\n \u2203 a b, (a, b) \u2208 zip (inits tl) (tails tl) \u2227 Prod.map (fun t => hd :: t) id (a, b) = ([], tail)\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase cons.mpr.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit tail : List \u03b1\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 zip (inits tl) (tails tl) \u2194 init ++ tail = tl\n\u22a2 ([], hd :: tl) = ([], hd :: tl) \u2228\n \u2203 a b, (a, b) \u2208 zip (inits tl) (tails tl) \u2227 Prod.map (fun t => hd :: t) id (a, b) = ([], hd :: tl)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.mpr.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit tail\u271d : List \u03b1\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 zip (inits tl) (tails tl) \u2194 init ++ tail = tl\ntail : List \u03b1\nhd' : \u03b1\ntl' : List \u03b1\n\u22a2 hd' :: tl' ++ tail = hd :: tl \u2192\n (hd' :: tl', tail) = ([], hd :: tl) \u2228\n \u2203 a b, (a, b) \u2208 zip (inits tl) (tails tl) \u2227 Prod.map (fun t => hd :: t) id (a, b) = (hd' :: tl', tail)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase cons.mpr.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit tail\u271d : List \u03b1\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 zip (inits tl) (tails tl) \u2194 init ++ tail = tl\ntail : List \u03b1\nhd' : \u03b1\ntl' : List \u03b1\nh : hd' :: tl' ++ tail = hd :: tl\n\u22a2 (hd' :: tl', tail) = ([], hd :: tl) \u2228\n \u2203 a b, (a, b) \u2208 zip (inits tl) (tails tl) \u2227 Prod.map (fun t => hd :: t) id (a, b) = (hd' :: tl', tail)\n[PROOFSTEP]\nright\n[GOAL]\ncase cons.mpr.cons.h\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit tail\u271d : List \u03b1\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 zip (inits tl) (tails tl) \u2194 init ++ tail = tl\ntail : List \u03b1\nhd' : \u03b1\ntl' : List \u03b1\nh : hd' :: tl' ++ tail = hd :: tl\n\u22a2 \u2203 a b, (a, b) \u2208 zip (inits tl) (tails tl) \u2227 Prod.map (fun t => hd :: t) id (a, b) = (hd' :: tl', tail)\n[PROOFSTEP]\nuse tl', tail\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninit tail\u271d : List \u03b1\nhd : \u03b1\ntl : List \u03b1\nih : \u2200 {init tail : List \u03b1}, (init, tail) \u2208 zip (inits tl) (tails tl) \u2194 init ++ tail = tl\ntail : List \u03b1\nhd' : \u03b1\ntl' : List \u03b1\nh : hd' :: tl' ++ tail = hd :: tl\n\u22a2 (tl', tail) \u2208 zip (inits tl) (tails tl) \u2227 Prod.map (fun t => hd :: t) id (tl', tail) = (hd' :: tl', tail)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\n\u22a2 map (Function.uncurry f) (zip l l') = zipWith f l l'\n[PROOFSTEP]\nrw [zip]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\n\u22a2 map (Function.uncurry f) (zipWith Prod.mk l l') = zipWith f l l'\n[PROOFSTEP]\ninduction' l with hd tl hl generalizing l'\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl'\u271d l' : List \u03b2\n\u22a2 map (Function.uncurry f) (zipWith Prod.mk [] l') = zipWith f [] l'\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl'\u271d : List \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), map (Function.uncurry f) (zipWith Prod.mk tl l') = zipWith f tl l'\nl' : List \u03b2\n\u22a2 map (Function.uncurry f) (zipWith Prod.mk (hd :: tl) l') = zipWith f (hd :: tl) l'\n[PROOFSTEP]\ncases' l' with hd' tl'\n[GOAL]\ncase cons.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl' : List \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), map (Function.uncurry f) (zipWith Prod.mk tl l') = zipWith f tl l'\n\u22a2 map (Function.uncurry f) (zipWith Prod.mk (hd :: tl) []) = zipWith f (hd :: tl) []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl' : List \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), map (Function.uncurry f) (zipWith Prod.mk tl l') = zipWith f tl l'\nhd' : \u03b2\ntl' : List \u03b2\n\u22a2 map (Function.uncurry f) (zipWith Prod.mk (hd :: tl) (hd' :: tl')) = zipWith f (hd :: tl) (hd' :: tl')\n[PROOFSTEP]\nsimp [hl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3\u271d : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Semiring \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn : \u03b3\nl : List \u03b1\nl' : List \u03b2\n\u22a2 sum (zipWith (fun x y => n * f x y) l l') = n * sum (zipWith f l l')\n[PROOFSTEP]\ninduction' l with hd tl hl generalizing f n l'\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3\u271d : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Semiring \u03b3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u03b3\nl'\u271d : List \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn : \u03b3\nl' : List \u03b2\n\u22a2 sum (zipWith (fun x y => n * f x y) [] l') = n * sum (zipWith f [] l')\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3\u271d : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Semiring \u03b3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u03b3\nl'\u271d : List \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (n : \u03b3) (l' : List \u03b2), sum (zipWith (fun x y => n * f x y) tl l') = n * sum (zipWith f tl l')\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn : \u03b3\nl' : List \u03b2\n\u22a2 sum (zipWith (fun x y => n * f x y) (hd :: tl) l') = n * sum (zipWith f (hd :: tl) l')\n[PROOFSTEP]\ncases' l' with hd' tl'\n[GOAL]\ncase cons.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3\u271d : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Semiring \u03b3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u03b3\nl' : List \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (n : \u03b3) (l' : List \u03b2), sum (zipWith (fun x y => n * f x y) tl l') = n * sum (zipWith f tl l')\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn : \u03b3\n\u22a2 sum (zipWith (fun x y => n * f x y) (hd :: tl) []) = n * sum (zipWith f (hd :: tl) [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3\u271d : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Semiring \u03b3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn\u271d : \u03b3\nl' : List \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (n : \u03b3) (l' : List \u03b2), sum (zipWith (fun x y => n * f x y) tl l') = n * sum (zipWith f tl l')\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nn : \u03b3\nhd' : \u03b2\ntl' : List \u03b2\n\u22a2 sum (zipWith (fun x y => n * f x y) (hd :: tl) (hd' :: tl')) = n * sum (zipWith f (hd :: tl) (hd' :: tl'))\n[PROOFSTEP]\nsimp [hl, mul_add]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn : \u2115\n\u22a2 take n (zipWith f l l') = zipWith f (take n l) (take n l')\n[PROOFSTEP]\ninduction' l with hd tl hl generalizing l' n\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl'\u271d : List \u03b2\nn\u271d : \u2115\nl' : List \u03b2\nn : \u2115\n\u22a2 take n (zipWith f [] l') = zipWith f (take n []) (take n l')\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl'\u271d : List \u03b2\nn\u271d : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2) (n : \u2115), take n (zipWith f tl l') = zipWith f (take n tl) (take n l')\nl' : List \u03b2\nn : \u2115\n\u22a2 take n (zipWith f (hd :: tl) l') = zipWith f (take n (hd :: tl)) (take n l')\n[PROOFSTEP]\ncases l'\n[GOAL]\ncase cons.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn\u271d : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2) (n : \u2115), take n (zipWith f tl l') = zipWith f (take n tl) (take n l')\nn : \u2115\n\u22a2 take n (zipWith f (hd :: tl) []) = zipWith f (take n (hd :: tl)) (take n [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn\u271d : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2) (n : \u2115), take n (zipWith f tl l') = zipWith f (take n tl) (take n l')\nn : \u2115\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\n\u22a2 take n (zipWith f (hd :: tl) (head\u271d :: tail\u271d)) = zipWith f (take n (hd :: tl)) (take n (head\u271d :: tail\u271d))\n[PROOFSTEP]\ncases n\n[GOAL]\ncase cons.cons.zero\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2) (n : \u2115), take n (zipWith f tl l') = zipWith f (take n tl) (take n l')\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\n\u22a2 take zero (zipWith f (hd :: tl) (head\u271d :: tail\u271d)) = zipWith f (take zero (hd :: tl)) (take zero (head\u271d :: tail\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.cons.succ\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2) (n : \u2115), take n (zipWith f tl l') = zipWith f (take n tl) (take n l')\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\nn\u271d : \u2115\n\u22a2 take (succ n\u271d) (zipWith f (hd :: tl) (head\u271d :: tail\u271d)) =\n zipWith f (take (succ n\u271d) (hd :: tl)) (take (succ n\u271d) (head\u271d :: tail\u271d))\n[PROOFSTEP]\nsimp [hl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn : \u2115\n\u22a2 drop n (zipWith f l l') = zipWith f (drop n l) (drop n l')\n[PROOFSTEP]\ninduction' l with hd tl hl generalizing l' n\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl'\u271d : List \u03b2\nn\u271d : \u2115\nl' : List \u03b2\nn : \u2115\n\u22a2 drop n (zipWith f [] l') = zipWith f (drop n []) (drop n l')\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl'\u271d : List \u03b2\nn\u271d : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2) (n : \u2115), drop n (zipWith f tl l') = zipWith f (drop n tl) (drop n l')\nl' : List \u03b2\nn : \u2115\n\u22a2 drop n (zipWith f (hd :: tl) l') = zipWith f (drop n (hd :: tl)) (drop n l')\n[PROOFSTEP]\ncases l'\n[GOAL]\ncase cons.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn\u271d : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2) (n : \u2115), drop n (zipWith f tl l') = zipWith f (drop n tl) (drop n l')\nn : \u2115\n\u22a2 drop n (zipWith f (hd :: tl) []) = zipWith f (drop n (hd :: tl)) (drop n [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn\u271d : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2) (n : \u2115), drop n (zipWith f tl l') = zipWith f (drop n tl) (drop n l')\nn : \u2115\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\n\u22a2 drop n (zipWith f (hd :: tl) (head\u271d :: tail\u271d)) = zipWith f (drop n (hd :: tl)) (drop n (head\u271d :: tail\u271d))\n[PROOFSTEP]\ncases n\n[GOAL]\ncase cons.cons.zero\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2) (n : \u2115), drop n (zipWith f tl l') = zipWith f (drop n tl) (drop n l')\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\n\u22a2 drop zero (zipWith f (hd :: tl) (head\u271d :: tail\u271d)) = zipWith f (drop zero (hd :: tl)) (drop zero (head\u271d :: tail\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.cons.succ\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2) (n : \u2115), drop n (zipWith f tl l') = zipWith f (drop n tl) (drop n l')\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\nn\u271d : \u2115\n\u22a2 drop (succ n\u271d) (zipWith f (hd :: tl) (head\u271d :: tail\u271d)) =\n zipWith f (drop (succ n\u271d) (hd :: tl)) (drop (succ n\u271d) (head\u271d :: tail\u271d))\n[PROOFSTEP]\nsimp [hl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn : \u2115\n\u22a2 tail (zipWith f l l') = zipWith f (tail l) (tail l')\n[PROOFSTEP]\nsimp_rw [\u2190 drop_one, zipWith_distrib_drop]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl' lb : List \u03b2\nh : length l = length l'\n\u22a2 zipWith f (l ++ la) (l' ++ lb) = zipWith f l l' ++ zipWith f la lb\n[PROOFSTEP]\ninduction' l with hd tl hl generalizing l'\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d\u00b9 : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl'\u271d lb : List \u03b2\nh\u271d : length l = length l'\u271d\nl' : List \u03b2\nh : length [] = length l'\n\u22a2 zipWith f ([] ++ la) (l' ++ lb) = zipWith f [] l' ++ zipWith f la lb\n[PROOFSTEP]\nhave : l' = [] := eq_nil_of_length_eq_zero (by simpa using h.symm)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d\u00b9 : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl'\u271d lb : List \u03b2\nh\u271d : length l = length l'\u271d\nl' : List \u03b2\nh : length [] = length l'\n\u22a2 length l' = 0\n[PROOFSTEP]\nsimpa using h.symm\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d\u00b9 : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl'\u271d lb : List \u03b2\nh\u271d : length l = length l'\u271d\nl' : List \u03b2\nh : length [] = length l'\nthis : l' = []\n\u22a2 zipWith f ([] ++ la) (l' ++ lb) = zipWith f [] l' ++ zipWith f la lb\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d\u00b9 : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl'\u271d lb : List \u03b2\nh\u271d : length l = length l'\u271d\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 zipWith f (tl ++ la) (l' ++ lb) = zipWith f tl l' ++ zipWith f la lb\nl' : List \u03b2\nh : length (hd :: tl) = length l'\n\u22a2 zipWith f (hd :: tl ++ la) (l' ++ lb) = zipWith f (hd :: tl) l' ++ zipWith f la lb\n[PROOFSTEP]\ncases l'\n[GOAL]\ncase cons.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl' lb : List \u03b2\nh\u271d : length l = length l'\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 zipWith f (tl ++ la) (l' ++ lb) = zipWith f tl l' ++ zipWith f la lb\nh : length (hd :: tl) = length []\n\u22a2 zipWith f (hd :: tl ++ la) ([] ++ lb) = zipWith f (hd :: tl) [] ++ zipWith f la lb\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl' lb : List \u03b2\nh\u271d : length l = length l'\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 zipWith f (tl ++ la) (l' ++ lb) = zipWith f tl l' ++ zipWith f la lb\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\nh : length (hd :: tl) = length (head\u271d :: tail\u271d)\n\u22a2 zipWith f (hd :: tl ++ la) (head\u271d :: tail\u271d ++ lb) = zipWith f (hd :: tl) (head\u271d :: tail\u271d) ++ zipWith f la lb\n[PROOFSTEP]\nsimp only [add_left_inj, length] at h \n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl' lb : List \u03b2\nh\u271d : length l = length l'\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 zipWith f (tl ++ la) (l' ++ lb) = zipWith f tl l' ++ zipWith f la lb\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\nh : length tl = length tail\u271d\n\u22a2 zipWith f (hd :: tl ++ la) (head\u271d :: tail\u271d ++ lb) = zipWith f (hd :: tl) (head\u271d :: tail\u271d) ++ zipWith f la lb\n[PROOFSTEP]\nsimp [hl _ h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn : \u2115\nh : length l = length l'\n\u22a2 reverse (zipWith f l l') = zipWith f (reverse l) (reverse l')\n[PROOFSTEP]\ninduction' l with hd tl hl generalizing l'\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl'\u271d : List \u03b2\nn : \u2115\nh\u271d : length l = length l'\u271d\nl' : List \u03b2\nh : length [] = length l'\n\u22a2 reverse (zipWith f [] l') = zipWith f (reverse []) (reverse l')\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl'\u271d : List \u03b2\nn : \u2115\nh\u271d : length l = length l'\u271d\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 reverse (zipWith f tl l') = zipWith f (reverse tl) (reverse l')\nl' : List \u03b2\nh : length (hd :: tl) = length l'\n\u22a2 reverse (zipWith f (hd :: tl) l') = zipWith f (reverse (hd :: tl)) (reverse l')\n[PROOFSTEP]\ncases' l' with hd' tl'\n[GOAL]\ncase cons.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn : \u2115\nh\u271d : length l = length l'\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 reverse (zipWith f tl l') = zipWith f (reverse tl) (reverse l')\nh : length (hd :: tl) = length []\n\u22a2 reverse (zipWith f (hd :: tl) []) = zipWith f (reverse (hd :: tl)) (reverse [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn : \u2115\nh\u271d : length l = length l'\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 reverse (zipWith f tl l') = zipWith f (reverse tl) (reverse l')\nhd' : \u03b2\ntl' : List \u03b2\nh : length (hd :: tl) = length (hd' :: tl')\n\u22a2 reverse (zipWith f (hd :: tl) (hd' :: tl')) = zipWith f (reverse (hd :: tl)) (reverse (hd' :: tl'))\n[PROOFSTEP]\nsimp only [add_left_inj, length] at h \n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn : \u2115\nh\u271d : length l = length l'\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 reverse (zipWith f tl l') = zipWith f (reverse tl) (reverse l')\nhd' : \u03b2\ntl' : List \u03b2\nh : length tl = length tl'\n\u22a2 reverse (zipWith f (hd :: tl) (hd' :: tl')) = zipWith f (reverse (hd :: tl)) (reverse (hd' :: tl'))\n[PROOFSTEP]\nhave : tl.reverse.length = tl'.reverse.length := by simp [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn : \u2115\nh\u271d : length l = length l'\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 reverse (zipWith f tl l') = zipWith f (reverse tl) (reverse l')\nhd' : \u03b2\ntl' : List \u03b2\nh : length tl = length tl'\n\u22a2 length (reverse tl) = length (reverse tl')\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\nn : \u2115\nh\u271d : length l = length l'\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 reverse (zipWith f tl l') = zipWith f (reverse tl) (reverse l')\nhd' : \u03b2\ntl' : List \u03b2\nh : length tl = length tl'\nthis : length (reverse tl) = length (reverse tl')\n\u22a2 reverse (zipWith f (hd :: tl) (hd' :: tl')) = zipWith f (reverse (hd :: tl)) (reverse (hd' :: tl'))\n[PROOFSTEP]\nsimp [hl _ h, zipWith_append _ _ _ _ _ this]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nys : List \u03b1\n\u22a2 prod [] * prod ys =\n prod (zipWith (fun x x_1 => x * x_1) [] ys) * prod (drop (length ys) []) * prod (drop (length []) ys)\n[PROOFSTEP]\nsimp [Nat.zero_le]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nxs : List \u03b1\n\u22a2 prod xs * prod [] =\n prod (zipWith (fun x x_1 => x * x_1) xs []) * prod (drop (length []) xs) * prod (drop (length xs) [])\n[PROOFSTEP]\nsimp [Nat.zero_le]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nx : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\n\u22a2 prod (x :: xs) * prod (y :: ys) =\n prod (zipWith (fun x x_1 => x * x_1) (x :: xs) (y :: ys)) * prod (drop (length (y :: ys)) (x :: xs)) *\n prod (drop (length (x :: xs)) (y :: ys))\n[PROOFSTEP]\nsimp only [drop, length, zipWith_cons_cons, prod_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nx : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\n\u22a2 x * prod xs * (y * prod ys) =\n x * y * prod (zipWith (fun x x_1 => x * x_1) xs ys) * prod (drop (Nat.add (length ys) 0) xs) *\n prod (drop (Nat.add (length xs) 0) ys)\n[PROOFSTEP]\nconv =>\n lhs; rw [mul_assoc]; right; rw [mul_comm, mul_assoc]; right\n rw [mul_comm, prod_mul_prod_eq_prod_zipWith_mul_prod_drop xs ys]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nx : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\n| x * prod xs * (y * prod ys) =\n x * y * prod (zipWith (fun x x_1 => x * x_1) xs ys) * prod (drop (Nat.add (length ys) 0) xs) *\n prod (drop (Nat.add (length xs) 0) ys)\n[PROOFSTEP]\n lhs; rw [mul_assoc]; right; rw [mul_comm, mul_assoc]; right\n rw [mul_comm, prod_mul_prod_eq_prod_zipWith_mul_prod_drop xs ys]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nx : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\n| x * prod xs * (y * prod ys) =\n x * y * prod (zipWith (fun x x_1 => x * x_1) xs ys) * prod (drop (Nat.add (length ys) 0) xs) *\n prod (drop (Nat.add (length xs) 0) ys)\n[PROOFSTEP]\n lhs; rw [mul_assoc]; right; rw [mul_comm, mul_assoc]; right\n rw [mul_comm, prod_mul_prod_eq_prod_zipWith_mul_prod_drop xs ys]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nx : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\n| x * prod xs * (y * prod ys) =\n x * y * prod (zipWith (fun x x_1 => x * x_1) xs ys) * prod (drop (Nat.add (length ys) 0) xs) *\n prod (drop (Nat.add (length xs) 0) ys)\n[PROOFSTEP]\nlhs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nx : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\n| x * prod xs * (y * prod ys)\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nx : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\n| x * (prod xs * (y * prod ys))\n[PROOFSTEP]\nright\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nx : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\n| prod xs * (y * prod ys)\n[PROOFSTEP]\nrw [mul_comm, mul_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nx : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\n| y * (prod ys * prod xs)\n[PROOFSTEP]\nright\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nx : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\n| prod ys * prod xs\n[PROOFSTEP]\nrw [mul_comm, prod_mul_prod_eq_prod_zipWith_mul_prod_drop xs ys]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nx : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\n\u22a2 x * (y * (prod (zipWith (fun x x_1 => x * x_1) xs ys) * prod (drop (length ys) xs) * prod (drop (length xs) ys))) =\n x * y * prod (zipWith (fun x x_1 => x * x_1) xs ys) * prod (drop (Nat.add (length ys) 0) xs) *\n prod (drop (Nat.add (length xs) 0) ys)\n[PROOFSTEP]\nsimp only [add_eq, add_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nx : \u03b1\nxs : List \u03b1\ny : \u03b1\nys : List \u03b1\n\u22a2 x * (y * (prod (zipWith (fun x x_1 => x * x_1) xs ys) * prod (drop (length ys) xs) * prod (drop (length xs) ys))) =\n x * y * prod (zipWith (fun x x_1 => x * x_1) xs ys) * prod (drop (length ys) xs) * prod (drop (length xs) ys)\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nL L' : List \u03b1\nh : length L = length L'\n\u22a2 prod L * prod L' = prod (zipWith (fun x x_1 => x * x_1) L L')\n[PROOFSTEP]\napply (prod_mul_prod_eq_prod_zipWith_mul_prod_drop L L').trans\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\ninst\u271d : CommMonoid \u03b1\nL L' : List \u03b1\nh : length L = length L'\n\u22a2 prod (zipWith (fun x x_1 => x * x_1) L L') * prod (drop (length L') L) * prod (drop (length L) L') =\n prod (zipWith (fun x x_1 => x * x_1) L L')\n[PROOFSTEP]\nrw [\u2190 h, drop_length, h, drop_length, prod_nil, mul_one, mul_one]\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Zip", "llama_tokens": 30430, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4048936307733618}} {"text": "[GOAL]\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\n\u22a2 \u2191\u2191(AlternatingMap.measure (volumeForm o)) (parallelepiped \u2191b) = 1\n[PROOFSTEP]\nhave e : \u03b9 \u2243 Fin n := by\n refine' Fintype.equivFinOfCardEq _\n rw [\u2190 _i.out, finrank_eq_card_basis b.toBasis]\n[GOAL]\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\n\u22a2 \u03b9 \u2243 Fin n\n[PROOFSTEP]\nrefine' Fintype.equivFinOfCardEq _\n[GOAL]\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\n\u22a2 Fintype.card \u03b9 = n\n[PROOFSTEP]\nrw [\u2190 _i.out, finrank_eq_card_basis b.toBasis]\n[GOAL]\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\ne : \u03b9 \u2243 Fin n\n\u22a2 \u2191\u2191(AlternatingMap.measure (volumeForm o)) (parallelepiped \u2191b) = 1\n[PROOFSTEP]\nhave A : \u21d1b = b.reindex e \u2218 e := by\n ext x\n simp only [OrthonormalBasis.coe_reindex, Function.comp_apply, Equiv.symm_apply_apply]\n[GOAL]\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\ne : \u03b9 \u2243 Fin n\n\u22a2 \u2191b = \u2191(OrthonormalBasis.reindex b e) \u2218 \u2191e\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\ne : \u03b9 \u2243 Fin n\nx : \u03b9\n\u22a2 \u2191b x = (\u2191(OrthonormalBasis.reindex b e) \u2218 \u2191e) x\n[PROOFSTEP]\nsimp only [OrthonormalBasis.coe_reindex, Function.comp_apply, Equiv.symm_apply_apply]\n[GOAL]\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nb : OrthonormalBasis \u03b9 \u211d F\ne : \u03b9 \u2243 Fin n\nA : \u2191b = \u2191(OrthonormalBasis.reindex b e) \u2218 \u2191e\n\u22a2 \u2191\u2191(AlternatingMap.measure (volumeForm o)) (parallelepiped \u2191b) = 1\n[PROOFSTEP]\nrw [A, parallelepiped_comp_equiv, AlternatingMap.measure_parallelepiped, o.abs_volumeForm_apply_of_orthonormal,\n ENNReal.ofReal_one]\n[GOAL]\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\n\u22a2 AlternatingMap.measure (volumeForm o) = volume\n[PROOFSTEP]\nhave A : o.volumeForm.measure (stdOrthonormalBasis \u211d F).toBasis.parallelepiped = 1 :=\n Orientation.measure_orthonormalBasis o (stdOrthonormalBasis \u211d F)\n[GOAL]\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nA :\n \u2191\u2191(AlternatingMap.measure (volumeForm o))\n \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d F))) =\n 1\n\u22a2 AlternatingMap.measure (volumeForm o) = volume\n[PROOFSTEP]\nrw [addHaarMeasure_unique o.volumeForm.measure (stdOrthonormalBasis \u211d F).toBasis.parallelepiped, A, one_smul]\n[GOAL]\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nm n : \u2115\n_i : Fact (finrank \u211d F = n)\no : Orientation \u211d F (Fin n)\nA :\n \u2191\u2191(AlternatingMap.measure (volumeForm o))\n \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d F))) =\n 1\n\u22a2 addHaarMeasure (Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d F))) = volume\n[PROOFSTEP]\nsimp only [volume, Basis.addHaar]\n[GOAL]\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nb : OrthonormalBasis \u03b9 \u211d F\n\u22a2 \u2191\u2191volume (parallelepiped \u2191b) = 1\n[PROOFSTEP]\nhaveI : Fact (finrank \u211d F = finrank \u211d F) := \u27e8rfl\u27e9\n[GOAL]\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nb : OrthonormalBasis \u03b9 \u211d F\nthis : Fact (finrank \u211d F = finrank \u211d F)\n\u22a2 \u2191\u2191volume (parallelepiped \u2191b) = 1\n[PROOFSTEP]\nlet o := (stdOrthonormalBasis \u211d F).toBasis.orientation\n[GOAL]\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nb : OrthonormalBasis \u03b9 \u211d F\nthis : Fact (finrank \u211d F = finrank \u211d F)\no : Orientation \u211d F (Fin (finrank \u211d F)) := Basis.orientation (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d F))\n\u22a2 \u2191\u2191volume (parallelepiped \u2191b) = 1\n[PROOFSTEP]\nrw [\u2190 o.measure_eq_volume]\n[GOAL]\n\u03b9 : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : FiniteDimensional \u211d F\ninst\u271d\u00b9 : MeasurableSpace F\ninst\u271d : BorelSpace F\nb : OrthonormalBasis \u03b9 \u211d F\nthis : Fact (finrank \u211d F = finrank \u211d F)\no : Orientation \u211d F (Fin (finrank \u211d F)) := Basis.orientation (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d F))\n\u22a2 \u2191\u2191(AlternatingMap.measure (Orientation.volumeForm o)) (parallelepiped \u2191b) = 1\n[PROOFSTEP]\nexact o.measure_orthonormalBasis b\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "llama_tokens": 2939, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583376458152, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.40456273263319087}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nN : Submonoid R\nh\u2081 : M \u2264 N\nh\u2082 : \u2200 (r : R), r \u2208 N \u2192 IsUnit (\u2191(algebraMap R S) r)\ns : S\n\u22a2 \u2203 x, s * \u2191(algebraMap R S) \u2191x.snd = \u2191(algebraMap R S) x.fst\n[PROOFSTEP]\nobtain \u27e8\u27e8x, y, hy\u27e9, H\u27e9 := IsLocalization.surj M s\n[GOAL]\ncase intro.mk.mk\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nN : Submonoid R\nh\u2081 : M \u2264 N\nh\u2082 : \u2200 (r : R), r \u2208 N \u2192 IsUnit (\u2191(algebraMap R S) r)\ns : S\nx y : R\nhy : y \u2208 M\nH :\n s * \u2191(algebraMap R S) \u2191(x, { val := y, property := hy }).snd = \u2191(algebraMap R S) (x, { val := y, property := hy }).fst\n\u22a2 \u2203 x, s * \u2191(algebraMap R S) \u2191x.snd = \u2191(algebraMap R S) x.fst\n[PROOFSTEP]\nexact \u27e8\u27e8x, y, h\u2081 hy\u27e9, H\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nN : Submonoid R\nh\u2081 : M \u2264 N\nh\u2082 : \u2200 (r : R), r \u2208 N \u2192 IsUnit (\u2191(algebraMap R S) r)\nx y : R\n\u22a2 \u2191(algebraMap R S) x = \u2191(algebraMap R S) y \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nN : Submonoid R\nh\u2081 : M \u2264 N\nh\u2082 : \u2200 (r : R), r \u2208 N \u2192 IsUnit (\u2191(algebraMap R S) r)\nx y : R\n\u22a2 \u2191(algebraMap R S) x = \u2191(algebraMap R S) y \u2192 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nrw [IsLocalization.eq_iff_exists M]\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nN : Submonoid R\nh\u2081 : M \u2264 N\nh\u2082 : \u2200 (r : R), r \u2208 N \u2192 IsUnit (\u2191(algebraMap R S) r)\nx y : R\n\u22a2 (\u2203 c, \u2191c * x = \u2191c * y) \u2192 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nrintro \u27e8c, hc\u27e9\n[GOAL]\ncase mp.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nN : Submonoid R\nh\u2081 : M \u2264 N\nh\u2082 : \u2200 (r : R), r \u2208 N \u2192 IsUnit (\u2191(algebraMap R S) r)\nx y : R\nc : { x // x \u2208 M }\nhc : \u2191c * x = \u2191c * y\n\u22a2 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nexact \u27e8\u27e8c, h\u2081 c.2\u27e9, hc\u27e9\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nN : Submonoid R\nh\u2081 : M \u2264 N\nh\u2082 : \u2200 (r : R), r \u2208 N \u2192 IsUnit (\u2191(algebraMap R S) r)\nx y : R\n\u22a2 (\u2203 c, \u2191c * x = \u2191c * y) \u2192 \u2191(algebraMap R S) x = \u2191(algebraMap R S) y\n[PROOFSTEP]\nrintro \u27e8c, h\u27e9\n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nN : Submonoid R\nh\u2081 : M \u2264 N\nh\u2082 : \u2200 (r : R), r \u2208 N \u2192 IsUnit (\u2191(algebraMap R S) r)\nx y : R\nc : { x // x \u2208 N }\nh : \u2191c * x = \u2191c * y\n\u22a2 \u2191(algebraMap R S) x = \u2191(algebraMap R S) y\n[PROOFSTEP]\nsimpa only [map_mul, (h\u2082 c c.2).mul_right_inj] using congr_arg (algebraMap R S) h\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nz : S\n\u22a2 \u2191(algebraMap R S) (sec M z).fst = \u2191(algebraMap R S) \u2191(sec M z).snd * z\n[PROOFSTEP]\nrw [mul_comm, sec_spec]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nz : S\nx : R\ny : { x // x \u2208 M }\nh : z * \u2191(algebraMap R S) \u2191y = \u2191(algebraMap R S) x\nhx : x = 0\n\u22a2 z = 0\n[PROOFSTEP]\nrw [hx, (algebraMap R S).map_zero] at h \n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nz : S\nx : R\ny : { x // x \u2208 M }\nh : z * \u2191(algebraMap R S) \u2191y = 0\nhx : x = 0\n\u22a2 z = 0\n[PROOFSTEP]\nexact (IsUnit.mul_left_eq_zero (IsLocalization.map_units S y)).1 h\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nr : R\n\u22a2 \u2191(algebraMap R S) r = 0 \u2194 \u2203 m, \u2191m * r = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nr : R\n\u22a2 \u2191(algebraMap R S) r = 0 \u2192 \u2203 m, \u2191m * r = 0\ncase mpr\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nr : R\n\u22a2 (\u2203 m, \u2191m * r = 0) \u2192 \u2191(algebraMap R S) r = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nr : R\nh : \u2191(algebraMap R S) r = 0\n\u22a2 \u2203 m, \u2191m * r = 0\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := (IsLocalization.eq_iff_exists M S).mp ((algebraMap R S).map_zero.trans h.symm)\n[GOAL]\ncase mp.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nr : R\nh : \u2191(algebraMap R S) r = 0\nm : { x // x \u2208 M }\nhm : \u2191m * 0 = \u2191m * r\n\u22a2 \u2203 m, \u2191m * r = 0\n[PROOFSTEP]\nexact \u27e8m, by simpa using hm.symm\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nr : R\nh : \u2191(algebraMap R S) r = 0\nm : { x // x \u2208 M }\nhm : \u2191m * 0 = \u2191m * r\n\u22a2 \u2191m * r = 0\n[PROOFSTEP]\nsimpa using hm.symm\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nr : R\n\u22a2 (\u2203 m, \u2191m * r = 0) \u2192 \u2191(algebraMap R S) r = 0\n[PROOFSTEP]\nrintro \u27e8m, hm\u27e9\n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nr : R\nm : { x // x \u2208 M }\nhm : \u2191m * r = 0\n\u22a2 \u2191(algebraMap R S) r = 0\n[PROOFSTEP]\nrw [\u2190 (IsLocalization.map_units S m).mul_right_inj, mul_zero, \u2190 RingHom.map_mul, hm, RingHom.map_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx : R\ny : { x // x \u2208 M }\nz\u2081 z\u2082 : S\n\u22a2 mk' S x y + z\u2081 = z\u2082 \u2194 \u2191(algebraMap R S) x + z\u2081 * \u2191(algebraMap R S) \u2191y = z\u2082 * \u2191(algebraMap R S) \u2191y\n[PROOFSTEP]\nrw [\u2190 mk'_spec S x y, \u2190 IsUnit.mul_left_inj (IsLocalization.map_units S y), right_distrib]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : 0 \u2208 M\n\u22a2 0 = 1\n[PROOFSTEP]\nsimpa using IsLocalization.map_units S \u27e80, h\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx : R\ny : { x // x \u2208 M }\nI : Ideal S\n\u22a2 mk' S x y \u2208 I \u2194 \u2191(algebraMap R S) x \u2208 I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx : R\ny : { x // x \u2208 M }\nI : Ideal S\n\u22a2 mk' S x y \u2208 I \u2192 \u2191(algebraMap R S) x \u2208 I\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx : R\ny : { x // x \u2208 M }\nI : Ideal S\n\u22a2 \u2191(algebraMap R S) x \u2208 I \u2192 mk' S x y \u2208 I\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx : R\ny : { x // x \u2208 M }\nI : Ideal S\nh : mk' S x y \u2208 I\n\u22a2 \u2191(algebraMap R S) x \u2208 I\n[PROOFSTEP]\nrw [\u2190 mk'_spec S x y, mul_comm]\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx : R\ny : { x // x \u2208 M }\nI : Ideal S\nh : mk' S x y \u2208 I\n\u22a2 \u2191(algebraMap R S) \u2191y * mk' S x y \u2208 I\n[PROOFSTEP]\nexact I.mul_mem_left ((algebraMap R S) y) h\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx : R\ny : { x // x \u2208 M }\nI : Ideal S\nh : \u2191(algebraMap R S) x \u2208 I\n\u22a2 mk' S x y \u2208 I\n[PROOFSTEP]\nrw [\u2190 mk'_spec S x y] at h \n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx : R\ny : { x // x \u2208 M }\nI : Ideal S\nh : mk' S x y * \u2191(algebraMap R S) \u2191y \u2208 I\n\u22a2 mk' S x y \u2208 I\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 := isUnit_iff_exists_inv.1 (map_units S y)\n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx : R\ny : { x // x \u2208 M }\nI : Ideal S\nh : mk' S x y * \u2191(algebraMap R S) \u2191y \u2208 I\nb : S\nhb : \u2191(algebraMap R S) \u2191y * b = 1\n\u22a2 mk' S x y \u2208 I\n[PROOFSTEP]\nhave := I.mul_mem_left b h\n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx : R\ny : { x // x \u2208 M }\nI : Ideal S\nh : mk' S x y * \u2191(algebraMap R S) \u2191y \u2208 I\nb : S\nhb : \u2191(algebraMap R S) \u2191y * b = 1\nthis : b * (mk' S x y * \u2191(algebraMap R S) \u2191y) \u2208 I\n\u22a2 mk' S x y \u2208 I\n[PROOFSTEP]\nrwa [mul_comm, mul_assoc, hb, mul_one] at this \n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx : R\ns : { x // x \u2208 M }\n\u22a2 mk' S x s = 0 \u2194 \u2203 m, \u2191m * x = 0\n[PROOFSTEP]\nrw [\u2190 (map_units S s).mul_left_inj, mk'_spec, zero_mul, map_eq_zero_iff M]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\ns : { x // x \u2208 M }\n\u22a2 mk' S 0 s = 0\n[PROOFSTEP]\nrw [eq_comm, IsLocalization.eq_mk'_iff_mul_eq, zero_mul, map_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx : R\ny : { x // x \u2208 M }\nhxy : mk' S x y \u2260 0\n\u22a2 x \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\ny : { x // x \u2208 M }\nhxy : mk' S 0 y \u2260 0\n\u22a2 False\n[PROOFSTEP]\nexact hxy (IsLocalization.mk'_zero _)\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx y : { x // x \u2208 M }\n\u22a2 mk' S (\u2191x) y * mk' S (\u2191y) x = 1\n[PROOFSTEP]\nrw [\u2190 mk'_mul, mul_comm]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx y : { x // x \u2208 M }\n\u22a2 mk' S (\u2191y * \u2191x) (y * x) = 1\n[PROOFSTEP]\nexact mk'_self _ _\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx y : R\nm : { x // x \u2208 M }\n\u22a2 x \u2022 mk' S y m = mk' S (x * y) m\n[PROOFSTEP]\nnth_rw 2 [\u2190 one_mul m]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx y : R\nm : { x // x \u2208 M }\n\u22a2 x \u2022 mk' S y m = mk' S (x * y) (1 * m)\n[PROOFSTEP]\nrw [mk'_mul, mk'_one, Algebra.smul_def]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx : R\nm : { x // x \u2208 M }\n\u22a2 x \u2022 mk' S 1 m = mk' S x m\n[PROOFSTEP]\nrw [smul_mk', mul_one]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nm : { x // x \u2208 M }\nr : R\n\u22a2 \u2191m \u2022 mk' S r m = \u2191(algebraMap R S) r\n[PROOFSTEP]\nrw [smul_mk', mk'_mul_cancel_left]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx\u2081 x\u2082 : R\ny\u2081 y\u2082 : { x // x \u2208 M }\n\u22a2 (mk' S x\u2081 y\u2081 + mk' S x\u2082 y\u2082) * \u2191(algebraMap R S) \u2191(y\u2081 * y\u2082) = \u2191(algebraMap R S) (x\u2081 * \u2191y\u2082 + x\u2082 * \u2191y\u2081)\n[PROOFSTEP]\nrw [mul_comm (_ + _), mul_add, mul_mk'_eq_mk'_of_mul, mk'_add_eq_iff_add_mul_eq_mul, mul_comm (_ * _), \u2190 mul_assoc,\n add_comm, \u2190 map_mul, mul_mk'_eq_mk'_of_mul, add_comm _ (mk' _ _ _), mk'_add_eq_iff_add_mul_eq_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx\u2081 x\u2082 : R\ny\u2081 y\u2082 : { x // x \u2208 M }\n\u22a2 \u2191(algebraMap R ((fun x => (fun x => S) (x\u2082 * \u2191y\u2081 + x\u2081 * \u2191y\u2082)) (\u2191(y\u2081 * y\u2082) * x\u2081))) (\u2191y\u2081 * \u2191(y\u2081 * y\u2082) * x\u2082) +\n \u2191(algebraMap R ((fun x => S) (x\u2082 * \u2191y\u2081 + x\u2081 * \u2191y\u2082))) (\u2191(y\u2081 * y\u2082) * x\u2081) *\n \u2191(algebraMap R ((fun x => (fun x => S) (x\u2082 * \u2191y\u2081 + x\u2081 * \u2191y\u2082)) (\u2191(y\u2081 * y\u2082) * x\u2081))) \u2191y\u2082 =\n \u2191(algebraMap R S) (x\u2082 * \u2191y\u2081 + x\u2081 * \u2191y\u2082) * \u2191(algebraMap R ((fun x => S) (x\u2082 * \u2191y\u2081 + x\u2081 * \u2191y\u2082))) \u2191y\u2081 *\n \u2191(algebraMap R ((fun x => (fun x => S) (x\u2082 * \u2191y\u2081 + x\u2081 * \u2191y\u2082)) (\u2191(y\u2081 * y\u2082) * x\u2081))) \u2191y\u2082\n[PROOFSTEP]\nsimp only [map_add, Submonoid.coe_mul, map_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx\u2081 x\u2082 : R\ny\u2081 y\u2082 : { x // x \u2208 M }\n\u22a2 \u2191(algebraMap R S) \u2191y\u2081 * (\u2191(algebraMap R S) \u2191y\u2081 * \u2191(algebraMap R S) \u2191y\u2082) * \u2191(algebraMap R S) x\u2082 +\n \u2191(algebraMap R S) \u2191y\u2081 * \u2191(algebraMap R S) \u2191y\u2082 * \u2191(algebraMap R S) x\u2081 * \u2191(algebraMap R S) \u2191y\u2082 =\n (\u2191(algebraMap R S) x\u2082 * \u2191(algebraMap R S) \u2191y\u2081 + \u2191(algebraMap R S) x\u2081 * \u2191(algebraMap R S) \u2191y\u2082) *\n \u2191(algebraMap R S) \u2191y\u2081 *\n \u2191(algebraMap R S) \u2191y\u2082\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\ng : R \u2192+* P\nh : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\ny : { x // x \u2208 M }\nw z\u2081 z\u2082 : P\n\u22a2 w * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict (\u2191g) M) h) y)\u207b\u00b9 + z\u2081 = z\u2082 \u2194 w + \u2191g \u2191y * z\u2081 = \u2191g \u2191y * z\u2082\n[PROOFSTEP]\nrw [mul_comm, \u2190 one_mul z\u2081, \u2190 Units.inv_mul (IsUnit.liftRight (g.toMonoidHom.restrict M) h y), mul_assoc, \u2190 mul_add,\n Units.inv_mul_eq_iff_eq_mul, Units.inv_mul_cancel_left, IsUnit.coe_liftRight]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\ng : R \u2192+* P\nh : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\ny : { x // x \u2208 M }\nw z\u2081 z\u2082 : P\n\u22a2 w + \u2191(MonoidHom.restrict (\u2191g) M) y * z\u2081 = \u2191(MonoidHom.restrict (\u2191g) M) y * z\u2082 \u2194 w + \u2191g \u2191y * z\u2081 = \u2191g \u2191y * z\u2082\n[PROOFSTEP]\nsimp [RingHom.toMonoidHom_eq_coe, MonoidHom.restrict_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nz : S\nw w' v : (fun x => P) z\n\u22a2 \u2191(Submonoid.LocalizationWithZeroMap.lift (toLocalizationWithZeroMap M S) (RingHom.toMonoidWithZeroHom g) hg) z * w +\n w' =\n v \u2194\n \u2191g (Submonoid.LocalizationMap.sec (toLocalizationMap M S) z).fst * w +\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) z).snd * w' =\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) z).snd * v\n[PROOFSTEP]\nerw [mul_comm, \u2190 mul_assoc, mul_add_inv_left hg, mul_comm]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nz : S\nw w' v : (fun x => P) z\n\u22a2 \u2191\u2191(RingHom.toMonoidWithZeroHom g)\n (Submonoid.LocalizationMap.sec (toLocalizationWithZeroMap M S).toLocalizationMap z).fst *\n w +\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationWithZeroMap M S).toLocalizationMap z).snd * w' =\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationWithZeroMap M S).toLocalizationMap z).snd * v \u2194\n \u2191g (Submonoid.LocalizationMap.sec (toLocalizationMap M S) z).fst * w +\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) z).snd * w' =\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) z).snd * v\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nsrc\u271d : S \u2192*\u2080 P :=\n Submonoid.LocalizationWithZeroMap.lift (toLocalizationWithZeroMap M S) (RingHom.toMonoidWithZeroHom g) hg\n\u22a2 \u2200 (x y : S),\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : ZeroHom.toFun (\u2191src\u271d) 1 = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), ZeroHom.toFun (\u2191src\u271d) (x * y) = ZeroHom.toFun (\u2191src\u271d) x * ZeroHom.toFun (\u2191src\u271d) y) })\n (x + y) =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : ZeroHom.toFun (\u2191src\u271d) 1 = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), ZeroHom.toFun (\u2191src\u271d) (x * y) = ZeroHom.toFun (\u2191src\u271d) x * ZeroHom.toFun (\u2191src\u271d) y) })\n x +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : ZeroHom.toFun (\u2191src\u271d) 1 = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), ZeroHom.toFun (\u2191src\u271d) (x * y) = ZeroHom.toFun (\u2191src\u271d) x * ZeroHom.toFun (\u2191src\u271d) y) })\n y\n[PROOFSTEP]\nintro x y\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nsrc\u271d : S \u2192*\u2080 P :=\n Submonoid.LocalizationWithZeroMap.lift (toLocalizationWithZeroMap M S) (RingHom.toMonoidWithZeroHom g) hg\nx y : S\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : ZeroHom.toFun (\u2191src\u271d) 1 = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), ZeroHom.toFun (\u2191src\u271d) (x * y) = ZeroHom.toFun (\u2191src\u271d) x * ZeroHom.toFun (\u2191src\u271d) y) })\n (x + y) =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : ZeroHom.toFun (\u2191src\u271d) 1 = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), ZeroHom.toFun (\u2191src\u271d) (x * y) = ZeroHom.toFun (\u2191src\u271d) x * ZeroHom.toFun (\u2191src\u271d) y) })\n x +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : ZeroHom.toFun (\u2191src\u271d) 1 = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), ZeroHom.toFun (\u2191src\u271d) (x * y) = ZeroHom.toFun (\u2191src\u271d) x * ZeroHom.toFun (\u2191src\u271d) y) })\n y\n[PROOFSTEP]\nerw [(toLocalizationMap M S).lift_spec, mul_add, mul_comm, eq_comm, lift_spec_mul_add, add_comm, mul_comm, mul_assoc,\n mul_comm, mul_assoc, lift_spec_mul_add]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nsrc\u271d : S \u2192*\u2080 P :=\n Submonoid.LocalizationWithZeroMap.lift (toLocalizationWithZeroMap M S) (RingHom.toMonoidWithZeroHom g) hg\nx y : S\n\u22a2 \u2191g (Submonoid.LocalizationMap.sec (toLocalizationMap M S) y).fst *\n (\u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) x).snd *\n \u2191\u2191(RingHom.toMonoidWithZeroHom g) \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) (x + y)).snd) +\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) y).snd *\n (\u2191g (Submonoid.LocalizationMap.sec (toLocalizationMap M S) x).fst *\n \u2191\u2191(RingHom.toMonoidWithZeroHom g) \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) (x + y)).snd) =\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) y).snd *\n (\u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) x).snd *\n \u2191\u2191(RingHom.toMonoidWithZeroHom g) (Submonoid.LocalizationMap.sec (toLocalizationMap M S) (x + y)).fst)\n[PROOFSTEP]\nsimp_rw [\u2190 mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nsrc\u271d : S \u2192*\u2080 P :=\n Submonoid.LocalizationWithZeroMap.lift (toLocalizationWithZeroMap M S) (RingHom.toMonoidWithZeroHom g) hg\nx y : S\n\u22a2 \u2191g (Submonoid.LocalizationMap.sec (toLocalizationMap M S) y).fst *\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) x).snd *\n \u2191\u2191(RingHom.toMonoidWithZeroHom g) \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) (x + y)).snd +\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) y).snd *\n \u2191g (Submonoid.LocalizationMap.sec (toLocalizationMap M S) x).fst *\n \u2191\u2191(RingHom.toMonoidWithZeroHom g) \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) (x + y)).snd =\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) y).snd *\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) x).snd *\n \u2191\u2191(RingHom.toMonoidWithZeroHom g) (Submonoid.LocalizationMap.sec (toLocalizationMap M S) (x + y)).fst\n[PROOFSTEP]\nshow g _ * g _ * g _ + g _ * g _ * g _ = g _ * g _ * g _\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nsrc\u271d : S \u2192*\u2080 P :=\n Submonoid.LocalizationWithZeroMap.lift (toLocalizationWithZeroMap M S) (RingHom.toMonoidWithZeroHom g) hg\nx y : S\n\u22a2 \u2191g (Submonoid.LocalizationMap.sec (toLocalizationMap M S) y).fst *\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) x).snd *\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) (x + y)).snd +\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) y).snd *\n \u2191g (Submonoid.LocalizationMap.sec (toLocalizationMap M S) x).fst *\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) (x + y)).snd =\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) y).snd *\n \u2191g \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) x).snd *\n \u2191g (Submonoid.LocalizationMap.sec (toLocalizationMap M S) (x + y)).fst\n[PROOFSTEP]\nsimp_rw [\u2190 map_mul g, \u2190 map_add g]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nsrc\u271d : S \u2192*\u2080 P :=\n Submonoid.LocalizationWithZeroMap.lift (toLocalizationWithZeroMap M S) (RingHom.toMonoidWithZeroHom g) hg\nx y : S\n\u22a2 \u2191g\n ((Submonoid.LocalizationMap.sec (toLocalizationMap M S) y).fst *\n \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) x).snd *\n \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) (x + y)).snd +\n \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) y).snd *\n (Submonoid.LocalizationMap.sec (toLocalizationMap M S) x).fst *\n \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) (x + y)).snd) =\n \u2191g\n (\u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) y).snd *\n \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) x).snd *\n (Submonoid.LocalizationMap.sec (toLocalizationMap M S) (x + y)).fst)\n[PROOFSTEP]\napply @eq_of_eq _ _ _ S _ _ _ _ _ g hg\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nsrc\u271d : S \u2192*\u2080 P :=\n Submonoid.LocalizationWithZeroMap.lift (toLocalizationWithZeroMap M S) (RingHom.toMonoidWithZeroHom g) hg\nx y : S\n\u22a2 \u2191(algebraMap R S)\n ((Submonoid.LocalizationMap.sec (toLocalizationMap M S) y).fst *\n \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) x).snd *\n \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) (x + y)).snd +\n \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) y).snd *\n (Submonoid.LocalizationMap.sec (toLocalizationMap M S) x).fst *\n \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) (x + y)).snd) =\n \u2191(algebraMap R S)\n (\u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) y).snd *\n \u2191(Submonoid.LocalizationMap.sec (toLocalizationMap M S) x).snd *\n (Submonoid.LocalizationMap.sec (toLocalizationMap M S) (x + y)).fst)\n[PROOFSTEP]\nsimp only [sec_spec', toLocalizationMap_sec, map_add, map_mul]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nsrc\u271d : S \u2192*\u2080 P :=\n Submonoid.LocalizationWithZeroMap.lift (toLocalizationWithZeroMap M S) (RingHom.toMonoidWithZeroHom g) hg\nx y : S\n\u22a2 \u2191(algebraMap R S) \u2191(sec M y).snd * y * \u2191(algebraMap R S) \u2191(sec M x).snd * \u2191(algebraMap R S) \u2191(sec M (x + y)).snd +\n \u2191(algebraMap R S) \u2191(sec M y).snd * (\u2191(algebraMap R S) \u2191(sec M x).snd * x) *\n \u2191(algebraMap R S) \u2191(sec M (x + y)).snd =\n \u2191(algebraMap R S) \u2191(sec M y).snd * \u2191(algebraMap R S) \u2191(sec M x).snd *\n (\u2191(algebraMap R S) \u2191(sec M (x + y)).snd * (x + y))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\ninst\u271d\u00b9 : IsLocalization M S\ng\u271d : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g\u271d \u2191y)\ninst\u271d : Algebra R P\nf g : S \u2192\u2090[R] P\n\u22a2 RingHom.comp (\u2191f) (algebraMap R S) = RingHom.comp (\u2191g) (algebraMap R S)\n[PROOFSTEP]\nrw [f.comp_algebraMap, g.comp_algebraMap]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nj k : S \u2192 P\nhj1 : j 1 = 1\nhk1 : k 1 = 1\nhjm : \u2200 (a b : S), j (a * b) = j a * j b\nhkm : \u2200 (a b : S), k (a * b) = k a * k b\nh : \u2200 (a : R), j (\u2191(algebraMap R S) a) = k (\u2191(algebraMap R S) a)\nj' : S \u2192* P := { toOneHom := { toFun := j, map_one' := hj1 }, map_mul' := hjm }\nk' : S \u2192* P := { toOneHom := { toFun := k, map_one' := hk1 }, map_mul' := hkm }\nthis : j' = k'\n\u22a2 j'.toFun = k'.toFun\n[PROOFSTEP]\nrw [this]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommSemiring S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommSemiring P\ninst\u271d\u2077 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d\u2076 : CommSemiring Q\nhy : M \u2264 Submonoid.comap g T\ninst\u271d\u2075 : Algebra P Q\ninst\u271d\u2074 : IsLocalization T Q\nA : Type u_5\ninst\u271d\u00b3 : CommSemiring A\nU : Submonoid A\nW : Type u_6\ninst\u271d\u00b2 : CommSemiring W\ninst\u271d\u00b9 : Algebra A W\ninst\u271d : IsLocalization U W\nl : P \u2192+* A\nhl : T \u2264 Submonoid.comap l U\nx : S\n\u22a2 \u2191(map W l hl) (\u2191(map Q g hy) x) = \u2191(map W (RingHom.comp l g) (_ : \u2200 (x : R), x \u2208 M \u2192 \u2191g x \u2208 Submonoid.comap l U)) x\n[PROOFSTEP]\nrw [\u2190 map_comp_map (Q := Q) hy hl]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommSemiring S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommSemiring P\ninst\u271d\u2077 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d\u2076 : CommSemiring Q\nhy : M \u2264 Submonoid.comap g T\ninst\u271d\u2075 : Algebra P Q\ninst\u271d\u2074 : IsLocalization T Q\nA : Type u_5\ninst\u271d\u00b3 : CommSemiring A\nU : Submonoid A\nW : Type u_6\ninst\u271d\u00b2 : CommSemiring W\ninst\u271d\u00b9 : Algebra A W\ninst\u271d : IsLocalization U W\nl : P \u2192+* A\nhl : T \u2264 Submonoid.comap l U\nx : S\n\u22a2 \u2191(map W l hl) (\u2191(map Q g hy) x) = \u2191(RingHom.comp (map W l hl) (map Q g hy)) x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommSemiring S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommSemiring P\ninst\u271d\u00b3 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d\u00b2 : CommSemiring Q\nhy : M \u2264 Submonoid.comap g T\ninst\u271d\u00b9 : Algebra P Q\ninst\u271d : IsLocalization T Q\nx : S\nz : R\n\u22a2 \u2191(map Q g hy) (z \u2022 x) = \u2191g z \u2022 \u2191(map Q g hy) x\n[PROOFSTEP]\nrw [Algebra.smul_def, Algebra.smul_def, RingHom.map_mul, map_eq]\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommSemiring S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommSemiring P\ninst\u271d\u00b3 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d\u00b2 : CommSemiring Q\nhy : M \u2264 Submonoid.comap g T\ninst\u271d\u00b9 : Algebra P Q\ninst\u271d : IsLocalization T Q\nh : R \u2243+* P\nH : Submonoid.map (RingEquiv.toMonoidHom h) M = T\n\u22a2 Submonoid.map (RingEquiv.toMonoidHom (RingEquiv.symm h)) T = M\n[PROOFSTEP]\nrw [\u2190 M.map_id, \u2190 H, Submonoid.map_map]\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommSemiring S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommSemiring P\ninst\u271d\u00b3 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d\u00b2 : CommSemiring Q\nhy : M \u2264 Submonoid.comap g T\ninst\u271d\u00b9 : Algebra P Q\ninst\u271d : IsLocalization T Q\nh : R \u2243+* P\nH : Submonoid.map (RingEquiv.toMonoidHom h) M = T\n\u22a2 Submonoid.map (MonoidHom.comp (RingEquiv.toMonoidHom (RingEquiv.symm h)) (RingEquiv.toMonoidHom h)) M =\n Submonoid.map (MonoidHom.id R) M\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommSemiring S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommSemiring P\ninst\u271d\u00b3 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d\u00b2 : CommSemiring Q\nhy : M \u2264 Submonoid.comap g T\ninst\u271d\u00b9 : Algebra P Q\ninst\u271d : IsLocalization T Q\nh : R \u2243+* P\nH : Submonoid.map (RingEquiv.toMonoidHom h) M = T\n\u22a2 MonoidHom.comp (RingEquiv.toMonoidHom (RingEquiv.symm h)) (RingEquiv.toMonoidHom h) = MonoidHom.id R\n[PROOFSTEP]\next\n[GOAL]\ncase e_f.h\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommSemiring S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommSemiring P\ninst\u271d\u00b3 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d\u00b2 : CommSemiring Q\nhy : M \u2264 Submonoid.comap g T\ninst\u271d\u00b9 : Algebra P Q\ninst\u271d : IsLocalization T Q\nh : R \u2243+* P\nH : Submonoid.map (RingEquiv.toMonoidHom h) M = T\nx\u271d : R\n\u22a2 \u2191(MonoidHom.comp (RingEquiv.toMonoidHom (RingEquiv.symm h)) (RingEquiv.toMonoidHom h)) x\u271d = \u2191(MonoidHom.id R) x\u271d\n[PROOFSTEP]\napply h.symm_apply_apply\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommSemiring S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommSemiring P\ninst\u271d\u00b3 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d\u00b2 : CommSemiring Q\nhy : M \u2264 Submonoid.comap g T\ninst\u271d\u00b9 : Algebra P Q\ninst\u271d : IsLocalization T Q\nh : R \u2243+* P\nH : Submonoid.map (RingEquiv.toMonoidHom h) M = T\nH' : Submonoid.map (RingEquiv.toMonoidHom (RingEquiv.symm h)) T = M\nsrc\u271d : S \u2192+* Q := map Q \u2191h (_ : M \u2264 Submonoid.comap (\u2191h) T)\nx : S\n\u22a2 \u2191(map S \u2191(RingEquiv.symm h) (_ : T \u2264 Submonoid.comap (\u2191(RingEquiv.symm h)) M))\n (\u2191(map Q \u2191h (_ : M \u2264 Submonoid.comap (\u2191h) T)) x) =\n x\n[PROOFSTEP]\nrw [map_map, map_unique _ (RingHom.id _), RingHom.id_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommSemiring S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommSemiring P\ninst\u271d\u00b3 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d\u00b2 : CommSemiring Q\nhy : M \u2264 Submonoid.comap g T\ninst\u271d\u00b9 : Algebra P Q\ninst\u271d : IsLocalization T Q\nh : R \u2243+* P\nH : Submonoid.map (RingEquiv.toMonoidHom h) M = T\nH' : Submonoid.map (RingEquiv.toMonoidHom (RingEquiv.symm h)) T = M\nsrc\u271d : S \u2192+* Q := map Q \u2191h (_ : M \u2264 Submonoid.comap (\u2191h) T)\nx : S\n\u22a2 \u2200 (x : R), \u2191(RingHom.id S) (\u2191(algebraMap R S) x) = \u2191(algebraMap R S) (\u2191(RingHom.comp \u2191(RingEquiv.symm h) \u2191h) x)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommSemiring S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommSemiring P\ninst\u271d\u00b3 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d\u00b2 : CommSemiring Q\nhy : M \u2264 Submonoid.comap g T\ninst\u271d\u00b9 : Algebra P Q\ninst\u271d : IsLocalization T Q\nh : R \u2243+* P\nH : Submonoid.map (RingEquiv.toMonoidHom h) M = T\nH' : Submonoid.map (RingEquiv.toMonoidHom (RingEquiv.symm h)) T = M\nsrc\u271d : S \u2192+* Q := map Q \u2191h (_ : M \u2264 Submonoid.comap (\u2191h) T)\nx : Q\n\u22a2 \u2191(map Q \u2191h (_ : M \u2264 Submonoid.comap (\u2191h) T))\n (\u2191(map S \u2191(RingEquiv.symm h) (_ : T \u2264 Submonoid.comap (\u2191(RingEquiv.symm h)) M)) x) =\n x\n[PROOFSTEP]\nrw [map_map, map_unique _ (RingHom.id _), RingHom.id_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommSemiring S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommSemiring P\ninst\u271d\u00b3 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d\u00b2 : CommSemiring Q\nhy : M \u2264 Submonoid.comap g T\ninst\u271d\u00b9 : Algebra P Q\ninst\u271d : IsLocalization T Q\nh : R \u2243+* P\nH : Submonoid.map (RingEquiv.toMonoidHom h) M = T\nH' : Submonoid.map (RingEquiv.toMonoidHom (RingEquiv.symm h)) T = M\nsrc\u271d : S \u2192+* Q := map Q \u2191h (_ : M \u2264 Submonoid.comap (\u2191h) T)\nx : Q\n\u22a2 \u2200 (x : P), \u2191(RingHom.id Q) (\u2191(algebraMap P Q) x) = \u2191(algebraMap P Q) (\u2191(RingHom.comp \u2191h \u2191(RingEquiv.symm h)) x)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommSemiring S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommSemiring P\ninst\u271d\u00b3 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d\u00b2 : CommSemiring Q\nhy : M \u2264 Submonoid.comap g T\ninst\u271d\u00b9 : Algebra P Q\ninst\u271d : IsLocalization T Q\nj : R \u2243+* P\nH : Submonoid.map (RingEquiv.toMonoidHom j) M = T\nx : R\n\u22a2 \u2191(ringEquivOfRingEquiv S Q j H) (\u2191(algebraMap R S) x) = \u2191(algebraMap P Q) (\u2191j x)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommSemiring S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommSemiring P\ninst\u271d\u00b3 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d\u00b2 : CommSemiring Q\nhy : M \u2264 Submonoid.comap g T\ninst\u271d\u00b9 : Algebra P Q\ninst\u271d : IsLocalization T Q\nj : R \u2243+* P\nH : Submonoid.map (RingEquiv.toMonoidHom j) M = T\nx : R\ny : { x // x \u2208 M }\n\u22a2 \u2191(ringEquivOfRingEquiv S Q j H) (mk' S x y) = mk' Q (\u2191j x) { val := \u2191j \u2191y, property := (_ : \u2191j \u2191y \u2208 T) }\n[PROOFSTEP]\nsimp [map_mk']\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommSemiring S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommSemiring P\ninst\u271d\u00b3 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nQ : Type u_4\ninst\u271d\u00b2 : CommSemiring Q\ninst\u271d\u00b9 : Algebra R Q\ninst\u271d : IsLocalization M Q\nx : R\ny : { x // x \u2208 M }\n\u22a2 \u2191(algEquiv M S Q) (mk' S x y) = mk' Q x y\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommSemiring S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommSemiring P\ninst\u271d\u00b3 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nQ : Type u_4\ninst\u271d\u00b2 : CommSemiring Q\ninst\u271d\u00b9 : Algebra R Q\ninst\u271d : IsLocalization M Q\nx : R\ny : { x // x \u2208 M }\n\u22a2 \u2191(AlgEquiv.symm (algEquiv M S Q)) (mk' Q x y) = mk' S x y\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\ninst\u271d\u00b9 : Algebra R P\ninst\u271d : IsLocalization M S\nh : S \u2243\u2090[R] P\n\u22a2 IsLocalization M P\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase map_units'\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\ninst\u271d\u00b9 : Algebra R P\ninst\u271d : IsLocalization M S\nh : S \u2243\u2090[R] P\n\u22a2 \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191(algebraMap R P) \u2191y)\n[PROOFSTEP]\nintro y\n[GOAL]\ncase map_units'\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\ninst\u271d\u00b9 : Algebra R P\ninst\u271d : IsLocalization M S\nh : S \u2243\u2090[R] P\ny : { x // x \u2208 M }\n\u22a2 IsUnit (\u2191(algebraMap R P) \u2191y)\n[PROOFSTEP]\nconvert (IsLocalization.map_units S y).map h.toAlgHom.toRingHom.toMonoidHom\n[GOAL]\ncase h.e'_3\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\ninst\u271d\u00b9 : Algebra R P\ninst\u271d : IsLocalization M S\nh : S \u2243\u2090[R] P\ny : { x // x \u2208 M }\n\u22a2 \u2191(algebraMap R P) \u2191y = \u2191\u2191\u2191\u2191h (\u2191(algebraMap R S) \u2191y)\n[PROOFSTEP]\nexact (h.commutes y).symm\n[GOAL]\ncase surj'\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\ninst\u271d\u00b9 : Algebra R P\ninst\u271d : IsLocalization M S\nh : S \u2243\u2090[R] P\n\u22a2 \u2200 (z : P), \u2203 x, z * \u2191(algebraMap R P) \u2191x.snd = \u2191(algebraMap R P) x.fst\n[PROOFSTEP]\nintro y\n[GOAL]\ncase surj'\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\ninst\u271d\u00b9 : Algebra R P\ninst\u271d : IsLocalization M S\nh : S \u2243\u2090[R] P\ny : P\n\u22a2 \u2203 x, y * \u2191(algebraMap R P) \u2191x.snd = \u2191(algebraMap R P) x.fst\n[PROOFSTEP]\nobtain \u27e8\u27e8x, s\u27e9, e\u27e9 := IsLocalization.surj M (h.symm y)\n[GOAL]\ncase surj'.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\ninst\u271d\u00b9 : Algebra R P\ninst\u271d : IsLocalization M S\nh : S \u2243\u2090[R] P\ny : P\nx : R\ns : { x // x \u2208 M }\ne : \u2191(AlgEquiv.symm h) y * \u2191(algebraMap R ((fun x => S) y)) \u2191(x, s).snd = \u2191(algebraMap R ((fun x => S) y)) (x, s).fst\n\u22a2 \u2203 x, y * \u2191(algebraMap R P) \u2191x.snd = \u2191(algebraMap R P) x.fst\n[PROOFSTEP]\napply_fun (show S \u2192 P from h) at e \n[GOAL]\ncase surj'.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\ninst\u271d\u00b9 : Algebra R P\ninst\u271d : IsLocalization M S\nh : S \u2243\u2090[R] P\ny : P\nx : R\ns : { x // x \u2208 M }\ne :\n (let_fun this := \u2191h;\n this)\n (\u2191(AlgEquiv.symm h) y * \u2191(algebraMap R ((fun x => S) y)) \u2191(x, s).snd) =\n (let_fun this := \u2191h;\n this)\n (\u2191(algebraMap R ((fun x => S) y)) (x, s).fst)\n\u22a2 \u2203 x, y * \u2191(algebraMap R P) \u2191x.snd = \u2191(algebraMap R P) x.fst\n[PROOFSTEP]\nsimp only [h.map_mul, h.apply_symm_apply, h.commutes] at e \n[GOAL]\ncase surj'.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\ninst\u271d\u00b9 : Algebra R P\ninst\u271d : IsLocalization M S\nh : S \u2243\u2090[R] P\ny : P\nx : R\ns : { x // x \u2208 M }\ne : y * \u2191(algebraMap R P) \u2191s = \u2191(algebraMap R P) x\n\u22a2 \u2203 x, y * \u2191(algebraMap R P) \u2191x.snd = \u2191(algebraMap R P) x.fst\n[PROOFSTEP]\nexact \u27e8\u27e8x, s\u27e9, e\u27e9\n[GOAL]\ncase eq_iff_exists'\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\ninst\u271d\u00b9 : Algebra R P\ninst\u271d : IsLocalization M S\nh : S \u2243\u2090[R] P\n\u22a2 \u2200 {x y : R}, \u2191(algebraMap R P) x = \u2191(algebraMap R P) y \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase eq_iff_exists'\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\ninst\u271d\u00b9 : Algebra R P\ninst\u271d : IsLocalization M S\nh : S \u2243\u2090[R] P\nx y : R\n\u22a2 \u2191(algebraMap R P) x = \u2191(algebraMap R P) y \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nrw [\u2190 h.symm.toEquiv.injective.eq_iff, \u2190 IsLocalization.eq_iff_exists M S, \u2190 h.symm.commutes, \u2190 h.symm.commutes]\n[GOAL]\ncase eq_iff_exists'\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\ninst\u271d\u00b9 : Algebra R P\ninst\u271d : IsLocalization M S\nh : S \u2243\u2090[R] P\nx y : R\n\u22a2 \u2191(AlgEquiv.symm h).toEquiv (\u2191(algebraMap R P) x) = \u2191(AlgEquiv.symm h).toEquiv (\u2191(algebraMap R P) y) \u2194\n \u2191(AlgEquiv.symm h) (\u2191(algebraMap R P) x) = \u2191(AlgEquiv.symm h) (\u2191(algebraMap R P) y)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\n\u22a2 IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n[PROOFSTEP]\nletI : Algebra P S := ((algebraMap R S).comp h.symm.toRingHom).toAlgebra\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n\u22a2 IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase map_units'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n\u22a2 \u2200 (y : { x // x \u2208 Submonoid.map (RingEquiv.toMonoidHom h) M }), IsUnit (\u2191(algebraMap P S) \u2191y)\n[PROOFSTEP]\nrintro \u27e8_, \u27e8y, hy, rfl\u27e9\u27e9\n[GOAL]\ncase map_units'.mk.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : R\nhy : y \u2208 \u2191M\n\u22a2 IsUnit\n (\u2191(algebraMap P S)\n \u2191{ val := \u2191(RingEquiv.toMonoidHom h) y,\n property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) y) })\n[PROOFSTEP]\nconvert IsLocalization.map_units S \u27e8y, hy\u27e9\n[GOAL]\ncase h.e'_3\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : R\nhy : y \u2208 \u2191M\n\u22a2 \u2191(algebraMap P S)\n \u2191{ val := \u2191(RingEquiv.toMonoidHom h) y,\n property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) y) } =\n \u2191(algebraMap R S) \u2191{ val := y, property := hy }\n[PROOFSTEP]\ndsimp only [RingHom.algebraMap_toAlgebra, RingHom.comp_apply]\n[GOAL]\ncase h.e'_3\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : R\nhy : y \u2208 \u2191M\n\u22a2 \u2191(algebraMap R S) (\u2191(RingEquiv.toRingHom (RingEquiv.symm h)) (\u2191(RingEquiv.toMonoidHom h) y)) = \u2191(algebraMap R S) y\n[PROOFSTEP]\nexact congr_arg _ (h.symm_apply_apply _)\n[GOAL]\ncase surj'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n\u22a2 \u2200 (z : S), \u2203 x, z * \u2191(algebraMap P S) \u2191x.snd = \u2191(algebraMap P S) x.fst\n[PROOFSTEP]\nintro y\n[GOAL]\ncase surj'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\n\u22a2 \u2203 x, y * \u2191(algebraMap P S) \u2191x.snd = \u2191(algebraMap P S) x.fst\n[PROOFSTEP]\nobtain \u27e8\u27e8x, s\u27e9, e\u27e9 := IsLocalization.surj M y\n[GOAL]\ncase surj'.intro.mk\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x \u2208 M }\ne : y * \u2191(algebraMap R S) \u2191(x, s).snd = \u2191(algebraMap R S) (x, s).fst\n\u22a2 \u2203 x, y * \u2191(algebraMap P S) \u2191x.snd = \u2191(algebraMap P S) x.fst\n[PROOFSTEP]\nrefine' \u27e8\u27e8h x, _, _, s.prop, rfl\u27e9, _\u27e9\n[GOAL]\ncase surj'.intro.mk\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x \u2208 M }\ne : y * \u2191(algebraMap R S) \u2191(x, s).snd = \u2191(algebraMap R S) (x, s).fst\n\u22a2 y *\n \u2191(algebraMap P S)\n \u2191(\u2191h x,\n { val := \u2191(RingEquiv.toMonoidHom h) \u2191s,\n property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) \u2191s) }).snd =\n \u2191(algebraMap P S)\n (\u2191h x,\n { val := \u2191(RingEquiv.toMonoidHom h) \u2191s,\n property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) \u2191s) }).fst\n[PROOFSTEP]\ndsimp only [RingHom.algebraMap_toAlgebra, RingHom.comp_apply] at e \u22a2\n[GOAL]\ncase surj'.intro.mk\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x \u2208 M }\ne : y * \u2191(algebraMap R S) \u2191s = \u2191(algebraMap R S) x\n\u22a2 y * \u2191(algebraMap R S) (\u2191(RingEquiv.toRingHom (RingEquiv.symm h)) (\u2191(RingEquiv.toMonoidHom h) \u2191s)) =\n \u2191(algebraMap R S) (\u2191(RingEquiv.toRingHom (RingEquiv.symm h)) (\u2191h x))\n[PROOFSTEP]\nconvert e\n[GOAL]\ncase h.e'_2.h.e'_6.h.e'_6\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x \u2208 M }\ne : y * \u2191(algebraMap R S) \u2191s = \u2191(algebraMap R S) x\n\u22a2 \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) (\u2191(RingEquiv.toMonoidHom h) \u2191s) = \u2191s\n[PROOFSTEP]\nexact h.symm_apply_apply _\n[GOAL]\ncase h.e'_3.h.e'_6\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x \u2208 M }\ne : y * \u2191(algebraMap R S) \u2191s = \u2191(algebraMap R S) x\n\u22a2 \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) (\u2191h x) = x\n[PROOFSTEP]\nexact h.symm_apply_apply _\n[GOAL]\ncase eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n\u22a2 \u2200 {x y : P}, \u2191(algebraMap P S) x = \u2191(algebraMap P S) y \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n\u22a2 \u2191(algebraMap P S) x = \u2191(algebraMap P S) y \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nrw [RingHom.algebraMap_toAlgebra, RingHom.comp_apply, RingHom.comp_apply, IsLocalization.eq_iff_exists M S]\n[GOAL]\ncase eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n\u22a2 (\u2203 c, \u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) x = \u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) y) \u2194\n \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nsimp_rw [\u2190 h.toEquiv.apply_eq_iff_eq]\n[GOAL]\ncase eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n\u22a2 (\u2203 c,\n \u2191h.toEquiv (\u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) x) =\n \u2191h.toEquiv (\u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) y)) \u2194\n \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nchange (\u2203 c : M, h (c * h.symm x) = h (c * h.symm y)) \u2194 _\n[GOAL]\ncase eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n\u22a2 (\u2203 c, \u2191h (\u2191c * \u2191(RingEquiv.symm h) x) = \u2191h (\u2191c * \u2191(RingEquiv.symm h) y)) \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nsimp only [RingEquiv.apply_symm_apply, RingEquiv.map_mul]\n[GOAL]\ncase eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n\u22a2 (\u2203 c, \u2191h \u2191c * x = \u2191h \u2191c * y) \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nexact \u27e8fun \u27e8c, e\u27e9 => \u27e8\u27e8_, _, c.prop, rfl\u27e9, e\u27e9, fun \u27e8\u27e8_, c, h, e\u2081\u27e9, e\u2082\u27e9 => \u27e8\u27e8_, h\u27e9, e\u2081.symm \u25b8 e\u2082\u27e9\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nh : R \u2243+* P\n\u22a2 IsLocalization M S \u2194 IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n[PROOFSTEP]\nletI : Algebra P S := ((algebraMap R S).comp h.symm.toRingHom).toAlgebra\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n\u22a2 IsLocalization M S \u2194 IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n[PROOFSTEP]\nrefine' \u27e8fun _ => isLocalization_of_base_ringEquiv M S h, _\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n\u22a2 IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S \u2192 IsLocalization M S\n[PROOFSTEP]\nintro H\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n\u22a2 IsLocalization M S\n[PROOFSTEP]\nconvert isLocalization_of_base_ringEquiv (Submonoid.map (RingEquiv.toMonoidHom h) M) S h.symm\n[GOAL]\ncase h.e'_3\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n\u22a2 M = Submonoid.map (RingEquiv.toMonoidHom (RingEquiv.symm h)) (Submonoid.map (RingEquiv.toMonoidHom h) M)\n[PROOFSTEP]\nerw [Submonoid.map_equiv_eq_comap_symm, Submonoid.comap_map_eq_of_injective]\n[GOAL]\ncase h.e'_3.hf\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n\u22a2 Injective \u2191(MulEquiv.toMonoidHom (MulEquiv.symm (RingEquiv.toMulEquiv (RingEquiv.symm h))))\n[PROOFSTEP]\nexact h.toEquiv.injective\n[GOAL]\ncase h.e'_6\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n\u22a2 inst\u271d\u00b9 = RingHom.toAlgebra (RingHom.comp (algebraMap P S) (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.symm h))))\n[PROOFSTEP]\nrw [RingHom.algebraMap_toAlgebra, RingHom.comp_assoc]\n[GOAL]\ncase h.e'_6\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n\u22a2 inst\u271d\u00b9 =\n RingHom.toAlgebra\n (RingHom.comp (algebraMap R S)\n (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm h))\n (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.symm h)))))\n[PROOFSTEP]\nsimp only [RingHom.comp_id, RingEquiv.symm_symm, RingEquiv.symm_toRingHom_comp_toRingHom]\n[GOAL]\ncase h.e'_6\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n\u22a2 inst\u271d\u00b9 = RingHom.toAlgebra (algebraMap R S)\n[PROOFSTEP]\napply Algebra.algebra_ext\n[GOAL]\ncase h.e'_6.h\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n\u22a2 \u2200 (r : R), \u2191(algebraMap R S) r = \u2191(algebraMap R S) r\n[PROOFSTEP]\nintro r\n[GOAL]\ncase h.e'_6.h\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\nr : R\n\u22a2 \u2191(algebraMap R S) r = \u2191(algebraMap R S) r\n[PROOFSTEP]\nrw [RingHom.algebraMap_toAlgebra]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\n\u22a2 nonZeroDivisors R \u2264 Submonoid.comap (algebraMap R S) (nonZeroDivisors S)\n[PROOFSTEP]\nrintro a ha b (e : b * algebraMap R S a = 0)\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\na : R\nha : a \u2208 nonZeroDivisors R\nb : S\ne : b * \u2191(algebraMap R S) a = 0\n\u22a2 b = 0\n[PROOFSTEP]\nobtain \u27e8x, s, rfl\u27e9 := mk'_surjective M b\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\na : R\nha : a \u2208 nonZeroDivisors R\nx : R\ns : { x // x \u2208 M }\ne : mk' S x s * \u2191(algebraMap R S) a = 0\n\u22a2 mk' S x s = 0\n[PROOFSTEP]\nrw [\u2190 @mk'_one R _ M, \u2190 mk'_mul, \u2190 (algebraMap R S).map_zero, \u2190 @mk'_one R _ M, IsLocalization.eq] at e \n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\na : R\nha : a \u2208 nonZeroDivisors R\nx : R\ns : { x // x \u2208 M }\ne : \u2203 c, \u2191c * (\u21911 * (x * a)) = \u2191c * (\u2191(s * 1) * 0)\n\u22a2 mk' S x s = 0\n[PROOFSTEP]\nobtain \u27e8c, e\u27e9 := e\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\na : R\nha : a \u2208 nonZeroDivisors R\nx : R\ns c : { x // x \u2208 M }\ne : \u2191c * (\u21911 * (x * a)) = \u2191c * (\u2191(s * 1) * 0)\n\u22a2 mk' S x s = 0\n[PROOFSTEP]\nrw [mul_zero, mul_zero, Submonoid.coe_one, one_mul, \u2190 mul_assoc] at e \n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\na : R\nha : a \u2208 nonZeroDivisors R\nx : R\ns c : { x // x \u2208 M }\ne : \u2191c * x * a = 0\n\u22a2 mk' S x s = 0\n[PROOFSTEP]\nrw [mk'_eq_zero_iff]\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\na : R\nha : a \u2208 nonZeroDivisors R\nx : R\ns c : { x // x \u2208 M }\ne : \u2191c * x * a = 0\n\u22a2 \u2203 m, \u2191m * x = 0\n[PROOFSTEP]\nexact \u27e8c, ha _ e\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : Subsingleton R\na : Localization M\nx\u271d : R \u00d7 { x // x \u2208 M }\n\u22a2 mk x\u271d.fst x\u271d.snd = mk 1 1\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_x\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : Subsingleton R\na : Localization M\nx\u271d : R \u00d7 { x // x \u2208 M }\n\u22a2 x\u271d.fst = 1\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\ncase e_y\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : Subsingleton R\na : Localization M\nx\u271d : R \u00d7 { x // x \u2208 M }\n\u22a2 x\u271d.snd = 1\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nz w : Localization M\na a' : R\nb b' : { x // x \u2208 M }\nc c' : R\nd d' : { x // x \u2208 M }\nh1 : \u2191(r M) (a, b) (a', b')\nh2 : \u2191(r M) (c, d) (c', d')\n\u22a2 \u2191(r M) (\u2191b * c + \u2191d * a, b * d) (\u2191b' * c' + \u2191d' * a', b' * d')\n[PROOFSTEP]\nrw [r_eq_r'] at h1 h2 \u22a2\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nz w : Localization M\na a' : R\nb b' : { x // x \u2208 M }\nc c' : R\nd d' : { x // x \u2208 M }\nh1 : \u2191(r' M) (a, b) (a', b')\nh2 : \u2191(r' M) (c, d) (c', d')\n\u22a2 \u2191(r' M) (\u2191b * c + \u2191d * a, b * d) (\u2191b' * c' + \u2191d' * a', b' * d')\n[PROOFSTEP]\ncases' h1 with t\u2085 ht\u2085\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nz w : Localization M\na a' : R\nb b' : { x // x \u2208 M }\nc c' : R\nd d' : { x // x \u2208 M }\nh2 : \u2191(r' M) (c, d) (c', d')\nt\u2085 : { x // x \u2208 M }\nht\u2085 : \u2191t\u2085 * (\u2191(a', b').snd * (a, b).fst) = \u2191t\u2085 * (\u2191(a, b).snd * (a', b').fst)\n\u22a2 \u2191(r' M) (\u2191b * c + \u2191d * a, b * d) (\u2191b' * c' + \u2191d' * a', b' * d')\n[PROOFSTEP]\ncases' h2 with t\u2086 ht\u2086\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nz w : Localization M\na a' : R\nb b' : { x // x \u2208 M }\nc c' : R\nd d' t\u2085 : { x // x \u2208 M }\nht\u2085 : \u2191t\u2085 * (\u2191(a', b').snd * (a, b).fst) = \u2191t\u2085 * (\u2191(a, b).snd * (a', b').fst)\nt\u2086 : { x // x \u2208 M }\nht\u2086 : \u2191t\u2086 * (\u2191(c', d').snd * (c, d).fst) = \u2191t\u2086 * (\u2191(c, d).snd * (c', d').fst)\n\u22a2 \u2191(r' M) (\u2191b * c + \u2191d * a, b * d) (\u2191b' * c' + \u2191d' * a', b' * d')\n[PROOFSTEP]\nuse t\u2085 * t\u2086\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nz w : Localization M\na a' : R\nb b' : { x // x \u2208 M }\nc c' : R\nd d' t\u2085 : { x // x \u2208 M }\nht\u2085 : \u2191t\u2085 * (\u2191(a', b').snd * (a, b).fst) = \u2191t\u2085 * (\u2191(a, b).snd * (a', b').fst)\nt\u2086 : { x // x \u2208 M }\nht\u2086 : \u2191t\u2086 * (\u2191(c', d').snd * (c, d).fst) = \u2191t\u2086 * (\u2191(c, d).snd * (c', d').fst)\n\u22a2 \u2191(t\u2085 * t\u2086) * (\u2191(\u2191b' * c' + \u2191d' * a', b' * d').snd * (\u2191b * c + \u2191d * a, b * d).fst) =\n \u2191(t\u2085 * t\u2086) * (\u2191(\u2191b * c + \u2191d * a, b * d).snd * (\u2191b' * c' + \u2191d' * a', b' * d').fst)\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nz w : Localization M\na a' : R\nb b' : { x // x \u2208 M }\nc c' : R\nd d' t\u2085 : { x // x \u2208 M }\nht\u2085 : \u2191t\u2085 * (\u2191(a', b').snd * (a, b).fst) = \u2191t\u2085 * (\u2191(a, b).snd * (a', b').fst)\nt\u2086 : { x // x \u2208 M }\nht\u2086 : \u2191t\u2086 * (\u2191(c', d').snd * (c, d).fst) = \u2191t\u2086 * (\u2191(c, d).snd * (c', d').fst)\n\u22a2 \u2191(t\u2085 * t\u2086) * (\u2191(b' * d') * (\u2191b * c + \u2191d * a)) = \u2191(t\u2085 * t\u2086) * (\u2191(b * d) * (\u2191b' * c' + \u2191d' * a'))\n[PROOFSTEP]\ncalc\n \u2191t\u2085 * \u2191t\u2086 * (\u2191b' * \u2191d' * ((b : R) * c + d * a)) = t\u2086 * (d' * c) * (t\u2085 * (b' * b)) + t\u2085 * (b' * a) * (t\u2086 * (d' * d)) :=\n by ring\n _ = t\u2085 * t\u2086 * (b * d * (b' * c' + d' * a')) := by rw [ht\u2086, ht\u2085]; ring\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nz w : Localization M\na a' : R\nb b' : { x // x \u2208 M }\nc c' : R\nd d' t\u2085 : { x // x \u2208 M }\nht\u2085 : \u2191t\u2085 * (\u2191(a', b').snd * (a, b).fst) = \u2191t\u2085 * (\u2191(a, b).snd * (a', b').fst)\nt\u2086 : { x // x \u2208 M }\nht\u2086 : \u2191t\u2086 * (\u2191(c', d').snd * (c, d).fst) = \u2191t\u2086 * (\u2191(c, d).snd * (c', d').fst)\n\u22a2 \u2191t\u2085 * \u2191t\u2086 * (\u2191b' * \u2191d' * (\u2191b * c + \u2191d * a)) =\n \u2191t\u2086 * (\u2191d' * c) * (\u2191t\u2085 * (\u2191b' * \u2191b)) + \u2191t\u2085 * (\u2191b' * a) * (\u2191t\u2086 * (\u2191d' * \u2191d))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nz w : Localization M\na a' : R\nb b' : { x // x \u2208 M }\nc c' : R\nd d' t\u2085 : { x // x \u2208 M }\nht\u2085 : \u2191t\u2085 * (\u2191(a', b').snd * (a, b).fst) = \u2191t\u2085 * (\u2191(a, b).snd * (a', b').fst)\nt\u2086 : { x // x \u2208 M }\nht\u2086 : \u2191t\u2086 * (\u2191(c', d').snd * (c, d).fst) = \u2191t\u2086 * (\u2191(c, d).snd * (c', d').fst)\n\u22a2 \u2191t\u2086 * (\u2191d' * c) * (\u2191t\u2085 * (\u2191b' * \u2191b)) + \u2191t\u2085 * (\u2191b' * a) * (\u2191t\u2086 * (\u2191d' * \u2191d)) =\n \u2191t\u2085 * \u2191t\u2086 * (\u2191b * \u2191d * (\u2191b' * c' + \u2191d' * a'))\n[PROOFSTEP]\nrw [ht\u2086, ht\u2085]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nz w : Localization M\na a' : R\nb b' : { x // x \u2208 M }\nc c' : R\nd d' t\u2085 : { x // x \u2208 M }\nht\u2085 : \u2191t\u2085 * (\u2191(a', b').snd * (a, b).fst) = \u2191t\u2085 * (\u2191(a, b).snd * (a', b').fst)\nt\u2086 : { x // x \u2208 M }\nht\u2086 : \u2191t\u2086 * (\u2191(c', d').snd * (c, d).fst) = \u2191t\u2086 * (\u2191(c, d).snd * (c', d').fst)\n\u22a2 \u2191t\u2086 * (\u2191(c, d).snd * (c', d').fst) * (\u2191t\u2085 * (\u2191b' * \u2191b)) + \u2191t\u2085 * (\u2191(a, b).snd * (a', b').fst) * (\u2191t\u2086 * (\u2191d' * \u2191d)) =\n \u2191t\u2085 * \u2191t\u2086 * (\u2191b * \u2191d * (\u2191b' * c' + \u2191d' * a'))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\na : R\nb : { x // x \u2208 M }\nc : R\nd : { x // x \u2208 M }\n\u22a2 mk a b + mk c d = mk (\u2191b * c + \u2191d * a) (b * d)\n[PROOFSTEP]\nshow Localization.add (mk a b) (mk c d) = mk _ _\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\na : R\nb : { x // x \u2208 M }\nc : R\nd : { x // x \u2208 M }\n\u22a2 Localization.add (mk a b) (mk c d) = mk (\u2191b * c + \u2191d * a) (b * d)\n[PROOFSTEP]\nsimp [Localization.add_def]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\na : R\nb : { x // x \u2208 M }\nc : R\n\u22a2 mk a b + mk c b = mk (a + c) b\n[PROOFSTEP]\nrw [add_mk, mk_eq_mk_iff, r_eq_r']\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\na : R\nb : { x // x \u2208 M }\nc : R\n\u22a2 \u2191(r' M) (\u2191b * c + \u2191b * a, b * b) (a + c, b)\n[PROOFSTEP]\nrefine' (r' M).symm \u27e81, _\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\na : R\nb : { x // x \u2208 M }\nc : R\n\u22a2 \u21911 * (\u2191(\u2191b * c + \u2191b * a, b * b).snd * (a + c, b).fst) = \u21911 * (\u2191(a + c, b).snd * (\u2191b * c + \u2191b * a, b * b).fst)\n[PROOFSTEP]\nsimp only [Submonoid.coe_one, Submonoid.coe_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\na : R\nb : { x // x \u2208 M }\nc : R\n\u22a2 1 * (\u2191b * \u2191b * (a + c)) = 1 * (\u2191b * (\u2191b * c + \u2191b * a))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\n\u22a2 CommMonoidWithZero (Localization M)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nsrc\u271d : CommMonoidWithZero (Localization M) :=\n let_fun this := inferInstance;\n this\nm n k : Localization M\n\u22a2 \u2200 (x y z : R \u00d7 { x // x \u2208 M }),\n mk x.fst x.snd + mk y.fst y.snd + mk z.fst z.snd = mk x.fst x.snd + (mk y.fst y.snd + mk z.fst z.snd)\n[PROOFSTEP]\nlocalization_tac\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nsrc\u271d : CommMonoidWithZero (Localization M) :=\n let_fun this := inferInstance;\n this\ny : Localization M\n\u22a2 \u2200 (y : R \u00d7 { x // x \u2208 M }), 0 + mk y.fst y.snd = mk y.fst y.snd\n[PROOFSTEP]\nlocalization_tac\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nsrc\u271d : CommMonoidWithZero (Localization M) :=\n let_fun this := inferInstance;\n this\ny : Localization M\n\u22a2 \u2200 (y : R \u00d7 { x // x \u2208 M }), mk y.fst y.snd + 0 = mk y.fst y.snd\n[PROOFSTEP]\nlocalization_tac\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nsrc\u271d : CommMonoidWithZero (Localization M) :=\n let_fun this := inferInstance;\n this\nx\u271d : Localization M\nx : R \u00d7 { x // x \u2208 M }\n\u22a2 (fun x x_1 => x \u2022 x_1) 0 (mk x.fst x.snd) = 0\n[PROOFSTEP]\nsimp only [smul_mk, zero_nsmul, mk_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nsrc\u271d : CommMonoidWithZero (Localization M) :=\n let_fun this := inferInstance;\n this\nn : \u2115\nx\u271d : Localization M\nx : R \u00d7 { x // x \u2208 M }\n\u22a2 (fun x x_1 => x \u2022 x_1) (n + 1) (mk x.fst x.snd) = mk x.fst x.snd + (fun x x_1 => x \u2022 x_1) n (mk x.fst x.snd)\n[PROOFSTEP]\nsimp only [smul_mk, succ_nsmul, add_mk_self]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nsrc\u271d : CommMonoidWithZero (Localization M) :=\n let_fun this := inferInstance;\n this\ny z : Localization M\n\u22a2 \u2200 (x y : R \u00d7 { x // x \u2208 M }), mk y.fst y.snd + mk x.fst x.snd = mk x.fst x.snd + mk y.fst y.snd\n[PROOFSTEP]\nlocalization_tac\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nsrc\u271d : CommMonoidWithZero (Localization M) :=\n let_fun this := inferInstance;\n this\nm n k : Localization M\n\u22a2 \u2200 (x y z : R \u00d7 { x // x \u2208 M }),\n mk x.fst x.snd * (mk y.fst y.snd + mk z.fst z.snd) =\n mk x.fst x.snd * mk y.fst y.snd + mk x.fst x.snd * mk z.fst z.snd\n[PROOFSTEP]\nlocalization_tac\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nsrc\u271d : CommMonoidWithZero (Localization M) :=\n let_fun this := inferInstance;\n this\nm n k : Localization M\n\u22a2 \u2200 (x y z : R \u00d7 { x // x \u2208 M }),\n (mk x.fst x.snd + mk y.fst y.snd) * mk z.fst z.snd =\n mk x.fst x.snd * mk z.fst z.snd + mk y.fst y.snd * mk z.fst z.snd\n[PROOFSTEP]\nlocalization_tac\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring S\u271d\ninst\u271d\u2074 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b3 : CommSemiring P\nS : Type u_4\ninst\u271d\u00b2 : Monoid S\ninst\u271d\u00b9 : DistribMulAction S R\ninst\u271d : IsScalarTower S R R\ns : S\n\u22a2 s \u2022 0 = 0\n[PROOFSTEP]\nsimp only [\u2190 Localization.mk_zero 1, Localization.smul_mk, smul_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring S\u271d\ninst\u271d\u2074 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b3 : CommSemiring P\nS : Type u_4\ninst\u271d\u00b2 : Monoid S\ninst\u271d\u00b9 : DistribMulAction S R\ninst\u271d : IsScalarTower S R R\ns : S\nx y : Localization M\nr\u2081 : R\nx\u2081 : { x // x \u2208 M }\nr\u2082 : R\nx\u2082 : { x // x \u2208 M }\n\u22a2 s \u2022 (mk (r\u2081, x\u2081).fst (r\u2081, x\u2081).snd + mk (r\u2082, x\u2082).fst (r\u2082, x\u2082).snd) =\n s \u2022 mk (r\u2081, x\u2081).fst (r\u2081, x\u2081).snd + s \u2022 mk (r\u2082, x\u2082).fst (r\u2082, x\u2082).snd\n[PROOFSTEP]\nsimp only [Localization.smul_mk, Localization.add_mk, smul_add, mul_comm _ (s \u2022 _), mul_comm _ r\u2081, mul_comm _ r\u2082,\n smul_mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring S\u271d\ninst\u271d\u2074 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b3 : CommSemiring P\nS : Type u_4\ninst\u271d\u00b2 : Semiring S\ninst\u271d\u00b9 : Module S R\ninst\u271d : IsScalarTower S R R\nsrc\u271d : DistribMulAction S (Localization M) := inferInstanceAs (DistribMulAction S (Localization M))\ns\u2081 s\u2082 : S\n\u22a2 \u2200 (fst : R) (snd : { x // x \u2208 M }),\n (s\u2081 + s\u2082) \u2022 mk (fst, snd).fst (fst, snd).snd =\n s\u2081 \u2022 mk (fst, snd).fst (fst, snd).snd + s\u2082 \u2022 mk (fst, snd).fst (fst, snd).snd\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring S\u271d\ninst\u271d\u2074 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b3 : CommSemiring P\nS : Type u_4\ninst\u271d\u00b2 : Semiring S\ninst\u271d\u00b9 : Module S R\ninst\u271d : IsScalarTower S R R\nsrc\u271d : DistribMulAction S (Localization M) := inferInstanceAs (DistribMulAction S (Localization M))\ns\u2081 s\u2082 : S\nfst\u271d : R\nsnd\u271d : { x // x \u2208 M }\n\u22a2 (s\u2081 + s\u2082) \u2022 mk (fst\u271d, snd\u271d).fst (fst\u271d, snd\u271d).snd =\n s\u2081 \u2022 mk (fst\u271d, snd\u271d).fst (fst\u271d, snd\u271d).snd + s\u2082 \u2022 mk (fst\u271d, snd\u271d).fst (fst\u271d, snd\u271d).snd\n[PROOFSTEP]\nsimp only [Localization.smul_mk, add_smul, add_mk_self]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring S\u271d\ninst\u271d\u2074 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b3 : CommSemiring P\nS : Type u_4\ninst\u271d\u00b2 : Semiring S\ninst\u271d\u00b9 : Module S R\ninst\u271d : IsScalarTower S R R\nsrc\u271d : DistribMulAction S (Localization M) := inferInstanceAs (DistribMulAction S (Localization M))\n\u22a2 \u2200 (fst : R) (snd : { x // x \u2208 M }), 0 \u2022 mk (fst, snd).fst (fst, snd).snd = 0\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring S\u271d\ninst\u271d\u2074 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b3 : CommSemiring P\nS : Type u_4\ninst\u271d\u00b2 : Semiring S\ninst\u271d\u00b9 : Module S R\ninst\u271d : IsScalarTower S R R\nsrc\u271d : DistribMulAction S (Localization M) := inferInstanceAs (DistribMulAction S (Localization M))\nfst\u271d : R\nsnd\u271d : { x // x \u2208 M }\n\u22a2 0 \u2022 mk (fst\u271d, snd\u271d).fst (fst\u271d, snd\u271d).snd = 0\n[PROOFSTEP]\nsimp only [Localization.smul_mk, zero_smul, mk_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring S\u271d\ninst\u271d\u00b3 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\nS : Type u_4\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\nsrc\u271d : Submonoid.LocalizationMap M (Localization M) := monoidOf M\n\u22a2 OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src\u271d.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src\u271d.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src\u271d.toMonoidHom) x * OneHom.toFun (\u2191src\u271d.toMonoidHom) y) })\n 0 =\n 0\n[PROOFSTEP]\nrw [\u2190 mk_zero (1 : M), mk_one_eq_monoidOf_mk]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring S\u271d\ninst\u271d\u00b3 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\nS : Type u_4\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\nsrc\u271d : Submonoid.LocalizationMap M (Localization M) := monoidOf M\nx y : R\n\u22a2 OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src\u271d.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src\u271d.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src\u271d.toMonoidHom) x * OneHom.toFun (\u2191src\u271d.toMonoidHom) y) })\n (x + y) =\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src\u271d.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src\u271d.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src\u271d.toMonoidHom) x * OneHom.toFun (\u2191src\u271d.toMonoidHom) y) })\n x +\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src\u271d.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src\u271d.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src\u271d.toMonoidHom) x * OneHom.toFun (\u2191src\u271d.toMonoidHom) y) })\n y\n[PROOFSTEP]\nsimp only [\u2190 mk_one_eq_monoidOf_mk, add_mk, Submonoid.coe_one, one_mul, add_comm]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring S\u271d\ninst\u271d\u00b3 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\nS : Type u_4\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\n\u22a2 \u2200 (fst : R) (snd : { x // x \u2208 M }),\n \u2191(RingHom.comp\n (let src := monoidOf M;\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src.toMonoidHom) x * OneHom.toFun (\u2191src.toMonoidHom) y) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src.toMonoidHom) x * OneHom.toFun (\u2191src.toMonoidHom) y) })\n 0 =\n 0),\n map_add' := (_ : \u2200 (x y : R), mk (x + y) 1 = mk x 1 + mk y 1) })\n (algebraMap S R))\n s *\n mk (fst, snd).fst (fst, snd).snd =\n mk (fst, snd).fst (fst, snd).snd *\n \u2191(RingHom.comp\n (let src := monoidOf M;\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src.toMonoidHom) x * OneHom.toFun (\u2191src.toMonoidHom) y) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src.toMonoidHom) x * OneHom.toFun (\u2191src.toMonoidHom) y) })\n 0 =\n 0),\n map_add' := (_ : \u2200 (x y : R), mk (x + y) 1 = mk x 1 + mk y 1) })\n (algebraMap S R))\n s\n[PROOFSTEP]\nintro r x\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring S\u271d\ninst\u271d\u00b3 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\nS : Type u_4\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\nr : R\nx : { x // x \u2208 M }\n\u22a2 \u2191(RingHom.comp\n (let src := monoidOf M;\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src.toMonoidHom) x * OneHom.toFun (\u2191src.toMonoidHom) y) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src.toMonoidHom) x * OneHom.toFun (\u2191src.toMonoidHom) y) })\n 0 =\n 0),\n map_add' := (_ : \u2200 (x y : R), mk (x + y) 1 = mk x 1 + mk y 1) })\n (algebraMap S R))\n s *\n mk (r, x).fst (r, x).snd =\n mk (r, x).fst (r, x).snd *\n \u2191(RingHom.comp\n (let src := monoidOf M;\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src.toMonoidHom) x * OneHom.toFun (\u2191src.toMonoidHom) y) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src.toMonoidHom) x * OneHom.toFun (\u2191src.toMonoidHom) y) })\n 0 =\n 0),\n map_add' := (_ : \u2200 (x y : R), mk (x + y) 1 = mk x 1 + mk y 1) })\n (algebraMap S R))\n s\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring S\u271d\ninst\u271d\u00b3 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\nS : Type u_4\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\nr : R\nx : { x // x \u2208 M }\n\u22a2 \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)) (\u2191(algebraMap S R) s) * mk r x =\n mk r x * \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)) (\u2191(algebraMap S R) s)\n[PROOFSTEP]\nsimp only [\u2190 mk_one_eq_monoidOf_mk, mk_mul, Localization.smul_mk, one_mul, mul_one, Algebra.commutes]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring S\u271d\ninst\u271d\u00b3 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\nS : Type u_4\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\n\u22a2 \u2200 (fst : R) (snd : { x // x \u2208 M }),\n s \u2022 mk (fst, snd).fst (fst, snd).snd =\n \u2191(RingHom.comp\n (let src := monoidOf M;\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src.toMonoidHom) x * OneHom.toFun (\u2191src.toMonoidHom) y) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src.toMonoidHom) x * OneHom.toFun (\u2191src.toMonoidHom) y) })\n 0 =\n 0),\n map_add' := (_ : \u2200 (x y : R), mk (x + y) 1 = mk x 1 + mk y 1) })\n (algebraMap S R))\n s *\n mk (fst, snd).fst (fst, snd).snd\n[PROOFSTEP]\nintro r x\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring S\u271d\ninst\u271d\u00b3 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\nS : Type u_4\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\nr : R\nx : { x // x \u2208 M }\n\u22a2 s \u2022 mk (r, x).fst (r, x).snd =\n \u2191(RingHom.comp\n (let src := monoidOf M;\n {\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src.toMonoidHom) x * OneHom.toFun (\u2191src.toMonoidHom) y) },\n map_zero' :=\n (_ :\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)),\n map_one' := (_ : OneHom.toFun (\u2191src.toMonoidHom) 1 = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun (\u2191src.toMonoidHom) (x * y) =\n OneHom.toFun (\u2191src.toMonoidHom) x * OneHom.toFun (\u2191src.toMonoidHom) y) })\n 0 =\n 0),\n map_add' := (_ : \u2200 (x y : R), mk (x + y) 1 = mk x 1 + mk y 1) })\n (algebraMap S R))\n s *\n mk (r, x).fst (r, x).snd\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u2074 : CommSemiring S\u271d\ninst\u271d\u00b3 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\nS : Type u_4\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\nr : R\nx : { x // x \u2208 M }\n\u22a2 s \u2022 mk r x = \u2191(Submonoid.LocalizationMap.toMap (monoidOf M)) (\u2191(algebraMap S R) s) * mk r x\n[PROOFSTEP]\nsimp only [\u2190 mk_one_eq_monoidOf_mk, mk_mul, Localization.smul_mk, one_mul, Algebra.smul_def]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nx : R\ny : { x // x \u2208 M }\n\u22a2 mk x y = mk' (Localization M) x y\n[PROOFSTEP]\nrw [mk_eq_monoidOf_mk'_apply, mk', toLocalizationMap_eq_monoidOf]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\nA : Type u_4\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra A R\nm : A\n\u22a2 mk (\u2191(algebraMap A R) m) 1 = \u2191(algebraMap A (Localization M)) m\n[PROOFSTEP]\nrw [mk_eq_mk', mk'_eq_iff_eq_mul, Submonoid.coe_one, map_one, mul_one]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommSemiring P\nA : Type u_4\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra A R\nm : A\n\u22a2 \u2191(algebraMap ((fun x => R) m) (Localization M)) (\u2191(algebraMap A R) m) = \u2191(algebraMap A (Localization M)) m\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nm : \u2115\n\u22a2 mk (\u2191m) 1 = \u2191m\n[PROOFSTEP]\nsimpa using @mk_algebraMap R _ M \u2115 _ _ m\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx : R\ny : { x // x \u2208 M }\n\u22a2 \u2191(algEquiv M S) (mk x y) = mk' S x y\n[PROOFSTEP]\nrw [mk_eq_mk', algEquiv_mk']\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nx : R\ny : { x // x \u2208 M }\n\u22a2 \u2191(AlgEquiv.symm (algEquiv M S)) (mk' S x y) = mk x y\n[PROOFSTEP]\nrw [mk_eq_mk', algEquiv_symm_mk']\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nz : Localization M\na b : R\nc d : { x // x \u2208 M }\nh : \u2191(r M) (a, c) (b, d)\n\u22a2 \u2191(r M) (-a, c) (-b, d)\n[PROOFSTEP]\nrw [r_eq_r'] at h \u22a2\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nz : Localization M\na b : R\nc d : { x // x \u2208 M }\nh : \u2191(r' M) (a, c) (b, d)\n\u22a2 \u2191(r' M) (-a, c) (-b, d)\n[PROOFSTEP]\ncases' h with t ht\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nz : Localization M\na b : R\nc d t : { x // x \u2208 M }\nht : \u2191t * (\u2191(b, d).snd * (a, c).fst) = \u2191t * (\u2191(a, c).snd * (b, d).fst)\n\u22a2 \u2191(r' M) (-a, c) (-b, d)\n[PROOFSTEP]\nuse t\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nz : Localization M\na b : R\nc d t : { x // x \u2208 M }\nht : \u2191t * (\u2191(b, d).snd * (a, c).fst) = \u2191t * (\u2191(a, c).snd * (b, d).fst)\n\u22a2 \u2191t * (\u2191(-b, d).snd * (-a, c).fst) = \u2191t * (\u2191(-a, c).snd * (-b, d).fst)\n[PROOFSTEP]\nrw [mul_neg, mul_neg, ht]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nz : Localization M\na b : R\nc d t : { x // x \u2208 M }\nht : \u2191t * (\u2191(b, d).snd * (a, c).fst) = \u2191t * (\u2191(a, c).snd * (b, d).fst)\n\u22a2 -(\u2191t * (\u2191(a, c).snd * (b, d).fst)) = \u2191t * (\u2191(-a, c).snd * (-b, d).fst)\n[PROOFSTEP]\nring_nf\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\na : R\nb : { x // x \u2208 M }\n\u22a2 -mk a b = mk (-a) b\n[PROOFSTEP]\nshow Localization.neg (mk a b) = mk (-a) b\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\na : R\nb : { x // x \u2208 M }\n\u22a2 Localization.neg (mk a b) = mk (-a) b\n[PROOFSTEP]\nrw [Localization.neg_def]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\na : R\nb : { x // x \u2208 M }\n\u22a2 liftOn (mk a b) (fun a b => mk (-a) b)\n (_ : \u2200 (a b : R) (c d : { x // x \u2208 M }), \u2191(r M) (a, c) (b, d) \u2192 mk (-a) c = mk (-b) d) =\n mk (-a) b\n[PROOFSTEP]\napply liftOn_mk\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nsrc\u271d : CommSemiring (Localization M) := inferInstanceAs (CommSemiring (Localization M))\nx\u271d : Localization M\nx : R \u00d7 { x // x \u2208 M }\n\u22a2 (fun x x_1 => x \u2022 x_1) 0 (mk x.fst x.snd) = 0\n[PROOFSTEP]\nsimp only [smul_mk, zero_zsmul, mk_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nsrc\u271d : CommSemiring (Localization M) := inferInstanceAs (CommSemiring (Localization M))\nn : \u2115\nx\u271d : Localization M\nx : R \u00d7 { x // x \u2208 M }\n\u22a2 (fun x x_1 => x \u2022 x_1) (Int.ofNat (Nat.succ n)) (mk x.fst x.snd) =\n mk x.fst x.snd + (fun x x_1 => x \u2022 x_1) (Int.ofNat n) (mk x.fst x.snd)\n[PROOFSTEP]\nsimp [smul_mk, add_mk_self, -mk_eq_monoidOf_mk', add_comm (n : \u2124) 1, add_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nsrc\u271d : CommSemiring (Localization M) := inferInstanceAs (CommSemiring (Localization M))\nn : \u2115\nx\u271d : Localization M\nx : R \u00d7 { x // x \u2208 M }\n\u22a2 (fun x x_1 => x \u2022 x_1) (Int.negSucc n) (mk x.fst x.snd) = -(fun x x_1 => x \u2022 x_1) (\u2191(Nat.succ n)) (mk x.fst x.snd)\n[PROOFSTEP]\ndsimp only\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nsrc\u271d : CommSemiring (Localization M) := inferInstanceAs (CommSemiring (Localization M))\nn : \u2115\nx\u271d : Localization M\nx : R \u00d7 { x // x \u2208 M }\n\u22a2 Int.negSucc n \u2022 mk x.fst x.snd = -(\u2191(Nat.succ n) \u2022 mk x.fst x.snd)\n[PROOFSTEP]\nrw [smul_mk, smul_mk, neg_mk, \u2190 neg_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nsrc\u271d : CommSemiring (Localization M) := inferInstanceAs (CommSemiring (Localization M))\nn : \u2115\nx\u271d : Localization M\nx : R \u00d7 { x // x \u2208 M }\n\u22a2 mk (Int.negSucc n \u2022 x.fst) x.snd = mk (-\u2191(Nat.succ n) \u2022 x.fst) x.snd\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nsrc\u271d : CommSemiring (Localization M) := inferInstanceAs (CommSemiring (Localization M))\ny : Localization M\n\u22a2 \u2200 (y : R \u00d7 { x // x \u2208 M }), -mk y.fst y.snd + mk y.fst y.snd = 0\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nsrc\u271d : CommSemiring (Localization M) := inferInstanceAs (CommSemiring (Localization M))\ny : Localization M\ny\u271d : R \u00d7 { x // x \u2208 M }\n\u22a2 -mk y\u271d.fst y\u271d.snd + mk y\u271d.fst y\u271d.snd = 0\n[PROOFSTEP]\nsimp only [add_mk, Localization.mk_mul, neg_mk, \u2190 mk_zero 1]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nsrc\u271d : CommSemiring (Localization M) := inferInstanceAs (CommSemiring (Localization M))\ny : Localization M\ny\u271d : R \u00d7 { x // x \u2208 M }\n\u22a2 mk (\u2191y\u271d.snd * y\u271d.fst + \u2191y\u271d.snd * -y\u271d.fst) (y\u271d.snd * y\u271d.snd) = mk 0 1\n[PROOFSTEP]\nrefine' mk_eq_mk_iff.mpr (r_of_eq _)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nsrc\u271d : CommSemiring (Localization M) := inferInstanceAs (CommSemiring (Localization M))\ny : Localization M\ny\u271d : R \u00d7 { x // x \u2208 M }\n\u22a2 \u2191(0, 1).snd * (\u2191y\u271d.snd * y\u271d.fst + \u2191y\u271d.snd * -y\u271d.fst, y\u271d.snd * y\u271d.snd).fst =\n \u2191(\u2191y\u271d.snd * y\u271d.fst + \u2191y\u271d.snd * -y\u271d.fst, y\u271d.snd * y\u271d.snd).snd * (0, 1).fst\n[PROOFSTEP]\nsimp only [Submonoid.coe_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nsrc\u271d : CommSemiring (Localization M) := inferInstanceAs (CommSemiring (Localization M))\ny : Localization M\ny\u271d : R \u00d7 { x // x \u2208 M }\n\u22a2 \u21911 * (\u2191y\u271d.snd * y\u271d.fst + \u2191y\u271d.snd * -y\u271d.fst) = \u2191y\u271d.snd * \u2191y\u271d.snd * 0\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\na c : R\nb d : { x // x \u2208 M }\n\u22a2 mk a b + -mk c d = mk a b + mk (-c) d\n[PROOFSTEP]\nrw [neg_mk]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\na c : R\nb d : { x // x \u2208 M }\n\u22a2 mk (\u2191b * -c + \u2191d * a) (b * d) = mk (\u2191d * a - \u2191b * c) (b * d)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_x\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\na c : R\nb d : { x // x \u2208 M }\n\u22a2 \u2191b * -c + \u2191d * a = \u2191d * a - \u2191b * c\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommRing P\nm : \u2124\n\u22a2 mk (\u2191m) 1 = \u2191m\n[PROOFSTEP]\nsimpa using @mk_algebraMap R _ M \u2124 _ _ m\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommRing P\nK : Type u_4\ninst\u271d : IsLocalization M S\nx : R\nhM : M \u2264 nonZeroDivisors R\n\u22a2 \u2191(algebraMap R S) x = 0 \u2194 x = 0\n[PROOFSTEP]\nrw [\u2190 (algebraMap R S).map_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommRing P\nK : Type u_4\ninst\u271d : IsLocalization M S\nx : R\nhM : M \u2264 nonZeroDivisors R\n\u22a2 \u2191(algebraMap R S) x = \u2191(algebraMap R S) 0 \u2194 x = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommRing P\nK : Type u_4\ninst\u271d : IsLocalization M S\nx : R\nhM : M \u2264 nonZeroDivisors R\n\u22a2 \u2191(algebraMap R S) x = \u2191(algebraMap R S) 0 \u2192 x = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommRing P\nK : Type u_4\ninst\u271d : IsLocalization M S\nx : R\nhM : M \u2264 nonZeroDivisors R\n\u22a2 x = 0 \u2192 \u2191(algebraMap R S) x = \u2191(algebraMap R S) 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommRing P\nK : Type u_4\ninst\u271d : IsLocalization M S\nx : R\nhM : M \u2264 nonZeroDivisors R\nh : \u2191(algebraMap R S) x = \u2191(algebraMap R S) 0\n\u22a2 x = 0\n[PROOFSTEP]\ncases' (eq_iff_exists M S).mp h with c hc\n[GOAL]\ncase mp.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommRing P\nK : Type u_4\ninst\u271d : IsLocalization M S\nx : R\nhM : M \u2264 nonZeroDivisors R\nh : \u2191(algebraMap R S) x = \u2191(algebraMap R S) 0\nc : { x // x \u2208 M }\nhc : \u2191c * x = \u2191c * 0\n\u22a2 x = 0\n[PROOFSTEP]\nrw [mul_zero, mul_comm] at hc \n[GOAL]\ncase mp.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommRing P\nK : Type u_4\ninst\u271d : IsLocalization M S\nx : R\nhM : M \u2264 nonZeroDivisors R\nh : \u2191(algebraMap R S) x = \u2191(algebraMap R S) 0\nc : { x // x \u2208 M }\nhc : x * \u2191c = 0\n\u22a2 x = 0\n[PROOFSTEP]\nexact hM c.2 x hc\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommRing P\nK : Type u_4\ninst\u271d : IsLocalization M S\nx : R\nhM : M \u2264 nonZeroDivisors R\nh : x = 0\n\u22a2 \u2191(algebraMap R S) x = \u2191(algebraMap R S) 0\n[PROOFSTEP]\nrw [h]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommRing P\nK : Type u_4\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors R\n\u22a2 Injective \u2191(algebraMap R S)\n[PROOFSTEP]\nrw [injective_iff_map_eq_zero (algebraMap R S)]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommRing P\nK : Type u_4\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors R\n\u22a2 \u2200 (a : R), \u2191(algebraMap R S) a = 0 \u2192 a = 0\n[PROOFSTEP]\nintro a ha\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommRing P\nK : Type u_4\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors R\na : R\nha : \u2191(algebraMap R S) a = 0\n\u22a2 a = 0\n[PROOFSTEP]\nrwa [to_map_eq_zero_iff S hM] at ha \n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nP : Type u_3\ninst\u271d\u00b3 : CommRing P\nK : Type u_4\ninst\u271d\u00b2 : IsLocalization M S\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors S\nhM : M \u2264 nonZeroDivisors R\nx : S\nhx : x \u2260 0\n\u22a2 (sec M x).fst \u2260 0\n[PROOFSTEP]\nhave hsec := sec_spec M x\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nP : Type u_3\ninst\u271d\u00b3 : CommRing P\nK : Type u_4\ninst\u271d\u00b2 : IsLocalization M S\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors S\nhM : M \u2264 nonZeroDivisors R\nx : S\nhx : x \u2260 0\nhsec : x * \u2191(algebraMap R S) \u2191(sec M x).snd = \u2191(algebraMap R S) (sec M x).fst\n\u22a2 (sec M x).fst \u2260 0\n[PROOFSTEP]\nintro hfst\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nP : Type u_3\ninst\u271d\u00b3 : CommRing P\nK : Type u_4\ninst\u271d\u00b2 : IsLocalization M S\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors S\nhM : M \u2264 nonZeroDivisors R\nx : S\nhx : x \u2260 0\nhsec : x * \u2191(algebraMap R S) \u2191(sec M x).snd = \u2191(algebraMap R S) (sec M x).fst\nhfst : (sec M x).fst = 0\n\u22a2 False\n[PROOFSTEP]\nrw [hfst, map_zero, mul_eq_zero, _root_.map_eq_zero_iff] at hsec \n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nP : Type u_3\ninst\u271d\u00b3 : CommRing P\nK : Type u_4\ninst\u271d\u00b2 : IsLocalization M S\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors S\nhM : M \u2264 nonZeroDivisors R\nx : S\nhx : x \u2260 0\nhsec : x = 0 \u2228 \u2191(sec M x).snd = 0\nhfst : (sec M x).fst = 0\n\u22a2 False\n[PROOFSTEP]\nexact Or.elim hsec hx (sec_snd_ne_zero hM x)\n[GOAL]\ncase hf\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nP : Type u_3\ninst\u271d\u00b3 : CommRing P\nK : Type u_4\ninst\u271d\u00b2 : IsLocalization M S\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors S\nhM : M \u2264 nonZeroDivisors R\nx : S\nhx : x \u2260 0\nhsec : x = 0 \u2228 \u2191(algebraMap R S) \u2191(sec M x).snd = 0\nhfst : (sec M x).fst = 0\n\u22a2 Injective \u2191(algebraMap R S)\n[PROOFSTEP]\nexact IsLocalization.injective S hM\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nP : Type u_3\ninst\u271d\u00b3 : CommRing P\nK : Type u_4\ninst\u271d\u00b2 : IsLocalization M S\nQ : Type u_5\ninst\u271d\u00b9 : CommRing Q\ng : R \u2192+* P\ninst\u271d : Algebra P Q\nhg : Injective \u2191g\ni : IsLocalization (Submonoid.map g M) Q\n\u22a2 Injective \u2191(map Q g (_ : M \u2264 Submonoid.comap g (Submonoid.map g M)))\n[PROOFSTEP]\nrw [injective_iff_map_eq_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nP : Type u_3\ninst\u271d\u00b3 : CommRing P\nK : Type u_4\ninst\u271d\u00b2 : IsLocalization M S\nQ : Type u_5\ninst\u271d\u00b9 : CommRing Q\ng : R \u2192+* P\ninst\u271d : Algebra P Q\nhg : Injective \u2191g\ni : IsLocalization (Submonoid.map g M) Q\n\u22a2 \u2200 (a : S), \u2191(map Q g (_ : M \u2264 Submonoid.comap g (Submonoid.map g M))) a = 0 \u2192 a = 0\n[PROOFSTEP]\nintro z hz\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nP : Type u_3\ninst\u271d\u00b3 : CommRing P\nK : Type u_4\ninst\u271d\u00b2 : IsLocalization M S\nQ : Type u_5\ninst\u271d\u00b9 : CommRing Q\ng : R \u2192+* P\ninst\u271d : Algebra P Q\nhg : Injective \u2191g\ni : IsLocalization (Submonoid.map g M) Q\nz : S\nhz : \u2191(map Q g (_ : M \u2264 Submonoid.comap g (Submonoid.map g M))) z = 0\n\u22a2 z = 0\n[PROOFSTEP]\nobtain \u27e8a, b, rfl\u27e9 := mk'_surjective M z\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nP : Type u_3\ninst\u271d\u00b3 : CommRing P\nK : Type u_4\ninst\u271d\u00b2 : IsLocalization M S\nQ : Type u_5\ninst\u271d\u00b9 : CommRing Q\ng : R \u2192+* P\ninst\u271d : Algebra P Q\nhg : Injective \u2191g\ni : IsLocalization (Submonoid.map g M) Q\na : R\nb : { x // x \u2208 M }\nhz : \u2191(map Q g (_ : M \u2264 Submonoid.comap g (Submonoid.map g M))) (mk' S a b) = 0\n\u22a2 mk' S a b = 0\n[PROOFSTEP]\nrw [map_mk', mk'_eq_zero_iff] at hz \n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nP : Type u_3\ninst\u271d\u00b3 : CommRing P\nK : Type u_4\ninst\u271d\u00b2 : IsLocalization M S\nQ : Type u_5\ninst\u271d\u00b9 : CommRing Q\ng : R \u2192+* P\ninst\u271d : Algebra P Q\nhg : Injective \u2191g\ni : IsLocalization (Submonoid.map g M) Q\na : R\nb : { x // x \u2208 M }\nhz : \u2203 m, \u2191m * \u2191g a = 0\n\u22a2 mk' S a b = 0\n[PROOFSTEP]\nobtain \u27e8\u27e8m', hm'\u27e9, hm\u27e9 := hz\n[GOAL]\ncase intro.intro.intro.mk\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nP : Type u_3\ninst\u271d\u00b3 : CommRing P\nK : Type u_4\ninst\u271d\u00b2 : IsLocalization M S\nQ : Type u_5\ninst\u271d\u00b9 : CommRing Q\ng : R \u2192+* P\ninst\u271d : Algebra P Q\nhg : Injective \u2191g\ni : IsLocalization (Submonoid.map g M) Q\na : R\nb : { x // x \u2208 M }\nm' : P\nhm' : m' \u2208 Submonoid.map g M\nhm : \u2191{ val := m', property := hm' } * \u2191g a = 0\n\u22a2 mk' S a b = 0\n[PROOFSTEP]\nrw [Submonoid.mem_map] at hm' \n[GOAL]\ncase intro.intro.intro.mk\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nP : Type u_3\ninst\u271d\u00b3 : CommRing P\nK : Type u_4\ninst\u271d\u00b2 : IsLocalization M S\nQ : Type u_5\ninst\u271d\u00b9 : CommRing Q\ng : R \u2192+* P\ninst\u271d : Algebra P Q\nhg : Injective \u2191g\ni : IsLocalization (Submonoid.map g M) Q\na : R\nb : { x // x \u2208 M }\nm' : P\nhm'\u271d : m' \u2208 Submonoid.map g M\nhm' : \u2203 x, x \u2208 M \u2227 \u2191g x = m'\nhm : \u2191{ val := m', property := hm'\u271d } * \u2191g a = 0\n\u22a2 mk' S a b = 0\n[PROOFSTEP]\nobtain \u27e8n, hn, hnm\u27e9 := hm'\n[GOAL]\ncase intro.intro.intro.mk.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nP : Type u_3\ninst\u271d\u00b3 : CommRing P\nK : Type u_4\ninst\u271d\u00b2 : IsLocalization M S\nQ : Type u_5\ninst\u271d\u00b9 : CommRing Q\ng : R \u2192+* P\ninst\u271d : Algebra P Q\nhg : Injective \u2191g\ni : IsLocalization (Submonoid.map g M) Q\na : R\nb : { x // x \u2208 M }\nm' : P\nhm' : m' \u2208 Submonoid.map g M\nhm : \u2191{ val := m', property := hm' } * \u2191g a = 0\nn : R\nhn : n \u2208 M\nhnm : \u2191g n = m'\n\u22a2 mk' S a b = 0\n[PROOFSTEP]\nrw [Subtype.coe_mk, \u2190 hnm, \u2190 map_mul, \u2190 map_zero g] at hm \n[GOAL]\ncase intro.intro.intro.mk.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nP : Type u_3\ninst\u271d\u00b3 : CommRing P\nK : Type u_4\ninst\u271d\u00b2 : IsLocalization M S\nQ : Type u_5\ninst\u271d\u00b9 : CommRing Q\ng : R \u2192+* P\ninst\u271d : Algebra P Q\nhg : Injective \u2191g\ni : IsLocalization (Submonoid.map g M) Q\na : R\nb : { x // x \u2208 M }\nm' : P\nhm' : m' \u2208 Submonoid.map g M\nn : R\nhm : \u2191g (n * a) = \u2191g 0\nhn : n \u2208 M\nhnm : \u2191g n = m'\n\u22a2 mk' S a b = 0\n[PROOFSTEP]\nrw [mk'_eq_zero_iff]\n[GOAL]\ncase intro.intro.intro.mk.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nP : Type u_3\ninst\u271d\u00b3 : CommRing P\nK : Type u_4\ninst\u271d\u00b2 : IsLocalization M S\nQ : Type u_5\ninst\u271d\u00b9 : CommRing Q\ng : R \u2192+* P\ninst\u271d : Algebra P Q\nhg : Injective \u2191g\ni : IsLocalization (Submonoid.map g M) Q\na : R\nb : { x // x \u2208 M }\nm' : P\nhm' : m' \u2208 Submonoid.map g M\nn : R\nhm : \u2191g (n * a) = \u2191g 0\nhn : n \u2208 M\nhnm : \u2191g n = m'\n\u22a2 \u2203 m, \u2191m * a = 0\n[PROOFSTEP]\nexact \u27e8\u27e8n, hn\u27e9, hg hm\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type u_3\ninst\u271d\u2077 : CommRing P\nK : Type u_4\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type u_5\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_6\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\n\u22a2 \u2200 {a b : S}, a * b = 0 \u2192 a = 0 \u2228 b = 0\n[PROOFSTEP]\nintro z w h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type u_3\ninst\u271d\u2077 : CommRing P\nK : Type u_4\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type u_5\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_6\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\nz w : S\nh : z * w = 0\n\u22a2 z = 0 \u2228 w = 0\n[PROOFSTEP]\ncases' surj M z with x hx\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type u_3\ninst\u271d\u2077 : CommRing P\nK : Type u_4\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type u_5\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_6\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\nz w : S\nh : z * w = 0\nx : A \u00d7 { x // x \u2208 M }\nhx : z * \u2191(algebraMap A S) \u2191x.snd = \u2191(algebraMap A S) x.fst\n\u22a2 z = 0 \u2228 w = 0\n[PROOFSTEP]\ncases' surj M w with y hy\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type u_3\ninst\u271d\u2077 : CommRing P\nK : Type u_4\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type u_5\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_6\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\nz w : S\nh : z * w = 0\nx : A \u00d7 { x // x \u2208 M }\nhx : z * \u2191(algebraMap A S) \u2191x.snd = \u2191(algebraMap A S) x.fst\ny : A \u00d7 { x // x \u2208 M }\nhy : w * \u2191(algebraMap A S) \u2191y.snd = \u2191(algebraMap A S) y.fst\n\u22a2 z = 0 \u2228 w = 0\n[PROOFSTEP]\nhave : z * w * algebraMap A S y.2 * algebraMap A S x.2 = algebraMap A S x.1 * algebraMap A S y.1 := by\n rw [mul_assoc z, hy, \u2190 hx]; ring\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type u_3\ninst\u271d\u2077 : CommRing P\nK : Type u_4\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type u_5\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_6\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\nz w : S\nh : z * w = 0\nx : A \u00d7 { x // x \u2208 M }\nhx : z * \u2191(algebraMap A S) \u2191x.snd = \u2191(algebraMap A S) x.fst\ny : A \u00d7 { x // x \u2208 M }\nhy : w * \u2191(algebraMap A S) \u2191y.snd = \u2191(algebraMap A S) y.fst\n\u22a2 z * w * \u2191(algebraMap A S) \u2191y.snd * \u2191(algebraMap A S) \u2191x.snd = \u2191(algebraMap A S) x.fst * \u2191(algebraMap A S) y.fst\n[PROOFSTEP]\nrw [mul_assoc z, hy, \u2190 hx]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type u_3\ninst\u271d\u2077 : CommRing P\nK : Type u_4\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type u_5\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_6\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\nz w : S\nh : z * w = 0\nx : A \u00d7 { x // x \u2208 M }\nhx : z * \u2191(algebraMap A S) \u2191x.snd = \u2191(algebraMap A S) x.fst\ny : A \u00d7 { x // x \u2208 M }\nhy : w * \u2191(algebraMap A S) \u2191y.snd = \u2191(algebraMap A S) y.fst\n\u22a2 z * \u2191(algebraMap A S) y.fst * \u2191(algebraMap A S) \u2191x.snd = z * \u2191(algebraMap A S) \u2191x.snd * \u2191(algebraMap A S) y.fst\n[PROOFSTEP]\nring\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type u_3\ninst\u271d\u2077 : CommRing P\nK : Type u_4\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type u_5\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_6\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\nz w : S\nh : z * w = 0\nx : A \u00d7 { x // x \u2208 M }\nhx : z * \u2191(algebraMap A S) \u2191x.snd = \u2191(algebraMap A S) x.fst\ny : A \u00d7 { x // x \u2208 M }\nhy : w * \u2191(algebraMap A S) \u2191y.snd = \u2191(algebraMap A S) y.fst\nthis : z * w * \u2191(algebraMap A S) \u2191y.snd * \u2191(algebraMap A S) \u2191x.snd = \u2191(algebraMap A S) x.fst * \u2191(algebraMap A S) y.fst\n\u22a2 z = 0 \u2228 w = 0\n[PROOFSTEP]\nrw [h, zero_mul, zero_mul, \u2190 (algebraMap A S).map_mul] at this \n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type u_3\ninst\u271d\u2077 : CommRing P\nK : Type u_4\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type u_5\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_6\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\nz w : S\nh : z * w = 0\nx : A \u00d7 { x // x \u2208 M }\nhx : z * \u2191(algebraMap A S) \u2191x.snd = \u2191(algebraMap A S) x.fst\ny : A \u00d7 { x // x \u2208 M }\nhy : w * \u2191(algebraMap A S) \u2191y.snd = \u2191(algebraMap A S) y.fst\nthis : 0 = \u2191(algebraMap A S) (x.fst * y.fst)\n\u22a2 z = 0 \u2228 w = 0\n[PROOFSTEP]\ncases' eq_zero_or_eq_zero_of_mul_eq_zero ((to_map_eq_zero_iff S hM).mp this.symm) with H H\n[GOAL]\ncase intro.intro.inl\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type u_3\ninst\u271d\u2077 : CommRing P\nK : Type u_4\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type u_5\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_6\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\nz w : S\nh : z * w = 0\nx : A \u00d7 { x // x \u2208 M }\nhx : z * \u2191(algebraMap A S) \u2191x.snd = \u2191(algebraMap A S) x.fst\ny : A \u00d7 { x // x \u2208 M }\nhy : w * \u2191(algebraMap A S) \u2191y.snd = \u2191(algebraMap A S) y.fst\nthis : 0 = \u2191(algebraMap A S) (x.fst * y.fst)\nH : x.fst = 0\n\u22a2 z = 0 \u2228 w = 0\n[PROOFSTEP]\nexact Or.inl (eq_zero_of_fst_eq_zero hx H)\n[GOAL]\ncase intro.intro.inr\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type u_3\ninst\u271d\u2077 : CommRing P\nK : Type u_4\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type u_5\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_6\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\nz w : S\nh : z * w = 0\nx : A \u00d7 { x // x \u2208 M }\nhx : z * \u2191(algebraMap A S) \u2191x.snd = \u2191(algebraMap A S) x.fst\ny : A \u00d7 { x // x \u2208 M }\nhy : w * \u2191(algebraMap A S) \u2191y.snd = \u2191(algebraMap A S) y.fst\nthis : 0 = \u2191(algebraMap A S) (x.fst * y.fst)\nH : y.fst = 0\n\u22a2 z = 0 \u2228 w = 0\n[PROOFSTEP]\nexact Or.inr (eq_zero_of_fst_eq_zero hy H)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type u_3\ninst\u271d\u2077 : CommRing P\nK : Type u_4\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type u_5\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_6\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\n\u22a2 IsDomain S\n[PROOFSTEP]\napply @NoZeroDivisors.to_isDomain _ _ (id _) (id _)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type u_3\ninst\u271d\u2077 : CommRing P\nK : Type u_4\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type u_5\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_6\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\n\u22a2 Nontrivial S\n[PROOFSTEP]\nexact \u27e8\u27e8(algebraMap A S) 0, (algebraMap A S) 1, fun h => zero_ne_one (IsLocalization.injective S hM h)\u27e9\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type u_3\ninst\u271d\u2077 : CommRing P\nK : Type u_4\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type u_5\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_6\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\n\u22a2 NoZeroDivisors S\n[PROOFSTEP]\nexact noZeroDivisors_of_le_nonZeroDivisors _ hM\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u2077 : CommRing R\u271d\nM\u271d : Submonoid R\u271d\nS : Type u_2\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R\u271d S\nP : Type u_3\ninst\u271d\u2074 : CommRing P\nR : Type u_4\nR\u2098 : Type u_5\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing R\u2098\nM : Submonoid R\nhM : \u00ac0 \u2208 M\nhR : IsField R\ninst\u271d\u00b9 : Algebra R R\u2098\ninst\u271d : IsLocalization M R\u2098\n\u22a2 Bijective \u2191(algebraMap R R\u2098)\n[PROOFSTEP]\nletI := hR.toField\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u2077 : CommRing R\u271d\nM\u271d : Submonoid R\u271d\nS : Type u_2\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R\u271d S\nP : Type u_3\ninst\u271d\u2074 : CommRing P\nR : Type u_4\nR\u2098 : Type u_5\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing R\u2098\nM : Submonoid R\nhM : \u00ac0 \u2208 M\nhR : IsField R\ninst\u271d\u00b9 : Algebra R R\u2098\ninst\u271d : IsLocalization M R\u2098\nthis : Field R := toField hR\n\u22a2 Bijective \u2191(algebraMap R R\u2098)\n[PROOFSTEP]\nreplace hM := le_nonZeroDivisors_of_noZeroDivisors hM\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u2077 : CommRing R\u271d\nM\u271d : Submonoid R\u271d\nS : Type u_2\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R\u271d S\nP : Type u_3\ninst\u271d\u2074 : CommRing P\nR : Type u_4\nR\u2098 : Type u_5\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing R\u2098\nM : Submonoid R\nhR : IsField R\ninst\u271d\u00b9 : Algebra R R\u2098\ninst\u271d : IsLocalization M R\u2098\nthis : Field R := toField hR\nhM : M \u2264 nonZeroDivisors R\n\u22a2 Bijective \u2191(algebraMap R R\u2098)\n[PROOFSTEP]\nrefine' \u27e8IsLocalization.injective _ hM, fun x => _\u27e9\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u2077 : CommRing R\u271d\nM\u271d : Submonoid R\u271d\nS : Type u_2\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R\u271d S\nP : Type u_3\ninst\u271d\u2074 : CommRing P\nR : Type u_4\nR\u2098 : Type u_5\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing R\u2098\nM : Submonoid R\nhR : IsField R\ninst\u271d\u00b9 : Algebra R R\u2098\ninst\u271d : IsLocalization M R\u2098\nthis : Field R := toField hR\nhM : M \u2264 nonZeroDivisors R\nx : R\u2098\n\u22a2 \u2203 a, \u2191(algebraMap R R\u2098) a = x\n[PROOFSTEP]\nobtain \u27e8r, \u27e8m, hm\u27e9, rfl\u27e9 := mk'_surjective M x\n[GOAL]\ncase intro.intro.mk\nR\u271d : Type u_1\ninst\u271d\u2077 : CommRing R\u271d\nM\u271d : Submonoid R\u271d\nS : Type u_2\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R\u271d S\nP : Type u_3\ninst\u271d\u2074 : CommRing P\nR : Type u_4\nR\u2098 : Type u_5\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing R\u2098\nM : Submonoid R\nhR : IsField R\ninst\u271d\u00b9 : Algebra R R\u2098\ninst\u271d : IsLocalization M R\u2098\nthis : Field R := toField hR\nhM : M \u2264 nonZeroDivisors R\nr m : R\nhm : m \u2208 M\n\u22a2 \u2203 a, \u2191(algebraMap R R\u2098) a = mk' R\u2098 r { val := m, property := hm }\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := hR.mul_inv_cancel (nonZeroDivisors.ne_zero <| hM hm)\n[GOAL]\ncase intro.intro.mk.intro\nR\u271d : Type u_1\ninst\u271d\u2077 : CommRing R\u271d\nM\u271d : Submonoid R\u271d\nS : Type u_2\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R\u271d S\nP : Type u_3\ninst\u271d\u2074 : CommRing P\nR : Type u_4\nR\u2098 : Type u_5\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing R\u2098\nM : Submonoid R\nhR : IsField R\ninst\u271d\u00b9 : Algebra R R\u2098\ninst\u271d : IsLocalization M R\u2098\nthis : Field R := toField hR\nhM : M \u2264 nonZeroDivisors R\nr m : R\nhm : m \u2208 M\nn : R\nhn : m * n = 1\n\u22a2 \u2203 a, \u2191(algebraMap R R\u2098) a = mk' R\u2098 r { val := m, property := hm }\n[PROOFSTEP]\nexact \u27e8r * n, by erw [eq_mk'_iff_mul_eq, \u2190 map_mul, mul_assoc, _root_.mul_comm n, hn, mul_one]\u27e9\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u2077 : CommRing R\u271d\nM\u271d : Submonoid R\u271d\nS : Type u_2\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R\u271d S\nP : Type u_3\ninst\u271d\u2074 : CommRing P\nR : Type u_4\nR\u2098 : Type u_5\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing R\u2098\nM : Submonoid R\nhR : IsField R\ninst\u271d\u00b9 : Algebra R R\u2098\ninst\u271d : IsLocalization M R\u2098\nthis : Field R := toField hR\nhM : M \u2264 nonZeroDivisors R\nr m : R\nhm : m \u2208 M\nn : R\nhn : m * n = 1\n\u22a2 \u2191(algebraMap R R\u2098) (r * n) = mk' R\u2098 r { val := m, property := hm }\n[PROOFSTEP]\nerw [eq_mk'_iff_mul_eq, \u2190 map_mul, mul_assoc, _root_.mul_comm n, hn, mul_one]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nR\u2098 : Type u_4\nS\u2098 : Type u_5\ninst\u271d\u2078 : CommRing R\u2098\ninst\u271d\u2077 : CommRing S\u2098\ninst\u271d\u2076 : Algebra R R\u2098\ninst\u271d\u2075 : IsLocalization M R\u2098\ninst\u271d\u2074 : Algebra S S\u2098\ni : IsLocalization (Algebra.algebraMapSubmonoid S M) S\u2098\ninst\u271d\u00b3 : Algebra R\u2098 S\u2098\ninst\u271d\u00b2 : Algebra R S\u2098\ninst\u271d\u00b9 : IsScalarTower R R\u2098 S\u2098\ninst\u271d : IsScalarTower R S S\u2098\ny : { x // x \u2208 M }\n\u22a2 IsUnit (\u2191(algebraMap R S\u2098) \u2191y)\n[PROOFSTEP]\nrw [IsScalarTower.algebraMap_apply _ S]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nR\u2098 : Type u_4\nS\u2098 : Type u_5\ninst\u271d\u2078 : CommRing R\u2098\ninst\u271d\u2077 : CommRing S\u2098\ninst\u271d\u2076 : Algebra R R\u2098\ninst\u271d\u2075 : IsLocalization M R\u2098\ninst\u271d\u2074 : Algebra S S\u2098\ni : IsLocalization (Algebra.algebraMapSubmonoid S M) S\u2098\ninst\u271d\u00b3 : Algebra R\u2098 S\u2098\ninst\u271d\u00b2 : Algebra R S\u2098\ninst\u271d\u00b9 : IsScalarTower R R\u2098 S\u2098\ninst\u271d : IsScalarTower R S S\u2098\ny : { x // x \u2208 M }\n\u22a2 IsUnit (\u2191(algebraMap S S\u2098) (\u2191(algebraMap R S) \u2191y))\n[PROOFSTEP]\nexact IsLocalization.map_units S\u2098 \u27e8algebraMap R S y, Algebra.mem_algebraMapSubmonoid_of_mem y\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nR\u2098 : Type u_4\nS\u2098 : Type u_5\ninst\u271d\u2078 : CommRing R\u2098\ninst\u271d\u2077 : CommRing S\u2098\ninst\u271d\u2076 : Algebra R R\u2098\ninst\u271d\u2075 : IsLocalization M R\u2098\ninst\u271d\u2074 : Algebra S S\u2098\ni : IsLocalization (Algebra.algebraMapSubmonoid S M) S\u2098\ninst\u271d\u00b3 : Algebra R\u2098 S\u2098\ninst\u271d\u00b2 : Algebra R S\u2098\ninst\u271d\u00b9 : IsScalarTower R R\u2098 S\u2098\ninst\u271d : IsScalarTower R S S\u2098\nx : R\ny : { x // x \u2208 M }\n\u22a2 \u2191(algebraMap R\u2098 S\u2098) (mk' R\u2098 x y) =\n mk' S\u2098 (\u2191(algebraMap R S) x)\n { val := \u2191(algebraMap R S) \u2191y, property := (_ : \u2191(algebraMap R S) \u2191y \u2208 Algebra.algebraMapSubmonoid S M) }\n[PROOFSTEP]\nrw [IsLocalization.eq_mk'_iff_mul_eq, Subtype.coe_mk, \u2190 IsScalarTower.algebraMap_apply, \u2190\n IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply R R\u2098 S\u2098, IsScalarTower.algebraMap_apply R R\u2098 S\u2098, \u2190\n _root_.map_mul, mul_comm, IsLocalization.mul_mk'_eq_mk'_of_mul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nR\u2098 : Type u_4\nS\u2098 : Type u_5\ninst\u271d\u2078 : CommRing R\u2098\ninst\u271d\u2077 : CommRing S\u2098\ninst\u271d\u2076 : Algebra R R\u2098\ninst\u271d\u2075 : IsLocalization M R\u2098\ninst\u271d\u2074 : Algebra S S\u2098\ni : IsLocalization (Algebra.algebraMapSubmonoid S M) S\u2098\ninst\u271d\u00b3 : Algebra R\u2098 S\u2098\ninst\u271d\u00b2 : Algebra R S\u2098\ninst\u271d\u00b9 : IsScalarTower R R\u2098 S\u2098\ninst\u271d : IsScalarTower R S S\u2098\nx : R\ny : { x // x \u2208 M }\n\u22a2 \u2191(algebraMap R\u2098 S\u2098) (mk' R\u2098 (\u2191y * x) y) = \u2191(algebraMap R\u2098 S\u2098) (\u2191(algebraMap R R\u2098) x)\n[PROOFSTEP]\nexact congr_arg (algebraMap R\u2098 S\u2098) (IsLocalization.mk'_mul_cancel_left x y)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nR\u2098 : Type u_4\nS\u2098 : Type u_5\ninst\u271d\u2078 : CommRing R\u2098\ninst\u271d\u2077 : CommRing S\u2098\ninst\u271d\u2076 : Algebra R R\u2098\ninst\u271d\u2075 : IsLocalization M R\u2098\ninst\u271d\u2074 : Algebra S S\u2098\ni : IsLocalization (Algebra.algebraMapSubmonoid S M) S\u2098\ninst\u271d\u00b3 : Algebra R\u2098 S\u2098\ninst\u271d\u00b2 : Algebra R S\u2098\ninst\u271d\u00b9 : IsScalarTower R R\u2098 S\u2098\ninst\u271d : IsScalarTower R S S\u2098\nx : R\n\u22a2 \u2191(algebraMap R\u2098 S\u2098) (\u2191(algebraMap R R\u2098) x) = \u2191(algebraMap S S\u2098) (\u2191(algebraMap R S) x)\n[PROOFSTEP]\nrw [\u2190 IsScalarTower.algebraMap_apply R S S\u2098, \u2190 IsScalarTower.algebraMap_apply R R\u2098 S\u2098]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Localization.Basic", "llama_tokens": 60309, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195385342972, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.4042116251003158}} {"text": "[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7\u2080 \u03c7\u2081 : MulChar R R'\nh : (fun \u03c7 => \u03c7.toFun) \u03c7\u2080 = (fun \u03c7 => \u03c7.toFun) \u03c7\u2081\n\u22a2 \u03c7\u2080 = \u03c7\u2081\n[PROOFSTEP]\ncases \u03c7\u2080\n[GOAL]\ncase mk\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7\u2081 : MulChar R R'\ntoMonoidHom\u271d : R \u2192* R'\nmap_nonunit'\u271d : \u2200 (a : R), \u00acIsUnit a \u2192 OneHom.toFun (\u2191toMonoidHom\u271d) a = 0\nh : (fun \u03c7 => \u03c7.toFun) { toMonoidHom := toMonoidHom\u271d, map_nonunit' := map_nonunit'\u271d } = (fun \u03c7 => \u03c7.toFun) \u03c7\u2081\n\u22a2 { toMonoidHom := toMonoidHom\u271d, map_nonunit' := map_nonunit'\u271d } = \u03c7\u2081\n[PROOFSTEP]\ncases \u03c7\u2081\n[GOAL]\ncase mk.mk\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\ntoMonoidHom\u271d\u00b9 : R \u2192* R'\nmap_nonunit'\u271d\u00b9 : \u2200 (a : R), \u00acIsUnit a \u2192 OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) a = 0\ntoMonoidHom\u271d : R \u2192* R'\nmap_nonunit'\u271d : \u2200 (a : R), \u00acIsUnit a \u2192 OneHom.toFun (\u2191toMonoidHom\u271d) a = 0\nh :\n (fun \u03c7 => \u03c7.toFun) { toMonoidHom := toMonoidHom\u271d\u00b9, map_nonunit' := map_nonunit'\u271d\u00b9 } =\n (fun \u03c7 => \u03c7.toFun) { toMonoidHom := toMonoidHom\u271d, map_nonunit' := map_nonunit'\u271d }\n\u22a2 { toMonoidHom := toMonoidHom\u271d\u00b9, map_nonunit' := map_nonunit'\u271d\u00b9 } =\n { toMonoidHom := toMonoidHom\u271d, map_nonunit' := map_nonunit'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toMonoidHom\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\ntoMonoidHom\u271d\u00b9 : R \u2192* R'\nmap_nonunit'\u271d\u00b9 : \u2200 (a : R), \u00acIsUnit a \u2192 OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) a = 0\ntoMonoidHom\u271d : R \u2192* R'\nmap_nonunit'\u271d : \u2200 (a : R), \u00acIsUnit a \u2192 OneHom.toFun (\u2191toMonoidHom\u271d) a = 0\nh :\n (fun \u03c7 => \u03c7.toFun) { toMonoidHom := toMonoidHom\u271d\u00b9, map_nonunit' := map_nonunit'\u271d\u00b9 } =\n (fun \u03c7 => \u03c7.toFun) { toMonoidHom := toMonoidHom\u271d, map_nonunit' := map_nonunit'\u271d }\n\u22a2 toMonoidHom\u271d\u00b9 = toMonoidHom\u271d\n[PROOFSTEP]\napply MonoidHom.ext (fun _ => congr_fun h _)\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u22a2 R \u2192 R'\n[PROOFSTEP]\nclassical exact fun x => if IsUnit x then 1 else 0\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u22a2 R \u2192 R'\n[PROOFSTEP]\nexact fun x => if IsUnit x then 1 else 0\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u22a2 (if IsUnit 1 then 1 else 0) = 1\n[PROOFSTEP]\nsimp only [isUnit_one, if_true]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u22a2 \u2200 (x y : R),\n OneHom.toFun { toFun := fun x => if IsUnit x then 1 else 0, map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) }\n (x * y) =\n OneHom.toFun { toFun := fun x => if IsUnit x then 1 else 0, map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) }\n x *\n OneHom.toFun { toFun := fun x => if IsUnit x then 1 else 0, map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) }\n y\n[PROOFSTEP]\nintro x y\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nx y : R\n\u22a2 OneHom.toFun { toFun := fun x => if IsUnit x then 1 else 0, map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) }\n (x * y) =\n OneHom.toFun { toFun := fun x => if IsUnit x then 1 else 0, map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) } x *\n OneHom.toFun { toFun := fun x => if IsUnit x then 1 else 0, map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) } y\n[PROOFSTEP]\nclassical\nsimp only [IsUnit.mul_iff, boole_mul]\nsplit_ifs <;> tauto\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nx y : R\n\u22a2 OneHom.toFun { toFun := fun x => if IsUnit x then 1 else 0, map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) }\n (x * y) =\n OneHom.toFun { toFun := fun x => if IsUnit x then 1 else 0, map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) } x *\n OneHom.toFun { toFun := fun x => if IsUnit x then 1 else 0, map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) } y\n[PROOFSTEP]\nsimp only [IsUnit.mul_iff, boole_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nx y : R\n\u22a2 (if IsUnit x \u2227 IsUnit y then 1 else 0) = if IsUnit x then if IsUnit y then 1 else 0 else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nx y : R\nh\u271d\u00b2 : IsUnit x \u2227 IsUnit y\nh\u271d\u00b9 : IsUnit x\nh\u271d : IsUnit y\n\u22a2 1 = 1\n[PROOFSTEP]\ntauto\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nx y : R\nh\u271d\u00b2 : IsUnit x \u2227 IsUnit y\nh\u271d\u00b9 : IsUnit x\nh\u271d : \u00acIsUnit y\n\u22a2 1 = 0\n[PROOFSTEP]\ntauto\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nx y : R\nh\u271d\u00b9 : IsUnit x \u2227 IsUnit y\nh\u271d : \u00acIsUnit x\n\u22a2 1 = 0\n[PROOFSTEP]\ntauto\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nx y : R\nh\u271d\u00b2 : \u00ac(IsUnit x \u2227 IsUnit y)\nh\u271d\u00b9 : IsUnit x\nh\u271d : IsUnit y\n\u22a2 0 = 1\n[PROOFSTEP]\ntauto\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nx y : R\nh\u271d\u00b2 : \u00ac(IsUnit x \u2227 IsUnit y)\nh\u271d\u00b9 : IsUnit x\nh\u271d : \u00acIsUnit y\n\u22a2 0 = 0\n[PROOFSTEP]\ntauto\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nx y : R\nh\u271d\u00b9 : \u00ac(IsUnit x \u2227 IsUnit y)\nh\u271d : \u00acIsUnit x\n\u22a2 0 = 0\n[PROOFSTEP]\ntauto\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u22a2 \u2200 (a : R),\n \u00acIsUnit a \u2192\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun x => if IsUnit x then 1 else 0, map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun\n { toFun := fun x => if IsUnit x then 1 else 0,\n map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun x => if IsUnit x then 1 else 0,\n map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) }\n x *\n OneHom.toFun\n { toFun := fun x => if IsUnit x then 1 else 0,\n map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) }\n y) })\n a =\n 0\n[PROOFSTEP]\nintro a ha\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\na : R\nha : \u00acIsUnit a\n\u22a2 OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun x => if IsUnit x then 1 else 0, map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun\n { toFun := fun x => if IsUnit x then 1 else 0, map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun x => if IsUnit x then 1 else 0, map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) }\n x *\n OneHom.toFun\n { toFun := fun x => if IsUnit x then 1 else 0, map_one' := (_ : (if IsUnit 1 then 1 else 0) = 1) }\n y) })\n a =\n 0\n[PROOFSTEP]\nsimp only [ha, if_false]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 \u03c7' : MulChar R R'\nh : \u2200 (a : R), \u2191\u03c7 a = \u2191\u03c7' a\n\u22a2 \u03c7 = \u03c7'\n[PROOFSTEP]\ncases \u03c7\n[GOAL]\ncase mk\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7' : MulChar R R'\ntoMonoidHom\u271d : R \u2192* R'\nmap_nonunit'\u271d : \u2200 (a : R), \u00acIsUnit a \u2192 OneHom.toFun (\u2191toMonoidHom\u271d) a = 0\nh : \u2200 (a : R), \u2191{ toMonoidHom := toMonoidHom\u271d, map_nonunit' := map_nonunit'\u271d } a = \u2191\u03c7' a\n\u22a2 { toMonoidHom := toMonoidHom\u271d, map_nonunit' := map_nonunit'\u271d } = \u03c7'\n[PROOFSTEP]\ncases \u03c7'\n[GOAL]\ncase mk.mk\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\ntoMonoidHom\u271d\u00b9 : R \u2192* R'\nmap_nonunit'\u271d\u00b9 : \u2200 (a : R), \u00acIsUnit a \u2192 OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) a = 0\ntoMonoidHom\u271d : R \u2192* R'\nmap_nonunit'\u271d : \u2200 (a : R), \u00acIsUnit a \u2192 OneHom.toFun (\u2191toMonoidHom\u271d) a = 0\nh :\n \u2200 (a : R),\n \u2191{ toMonoidHom := toMonoidHom\u271d\u00b9, map_nonunit' := map_nonunit'\u271d\u00b9 } a =\n \u2191{ toMonoidHom := toMonoidHom\u271d, map_nonunit' := map_nonunit'\u271d } a\n\u22a2 { toMonoidHom := toMonoidHom\u271d\u00b9, map_nonunit' := map_nonunit'\u271d\u00b9 } =\n { toMonoidHom := toMonoidHom\u271d, map_nonunit' := map_nonunit'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toMonoidHom\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\ntoMonoidHom\u271d\u00b9 : R \u2192* R'\nmap_nonunit'\u271d\u00b9 : \u2200 (a : R), \u00acIsUnit a \u2192 OneHom.toFun (\u2191toMonoidHom\u271d\u00b9) a = 0\ntoMonoidHom\u271d : R \u2192* R'\nmap_nonunit'\u271d : \u2200 (a : R), \u00acIsUnit a \u2192 OneHom.toFun (\u2191toMonoidHom\u271d) a = 0\nh :\n \u2200 (a : R),\n \u2191{ toMonoidHom := toMonoidHom\u271d\u00b9, map_nonunit' := map_nonunit'\u271d\u00b9 } a =\n \u2191{ toMonoidHom := toMonoidHom\u271d, map_nonunit' := map_nonunit'\u271d } a\n\u22a2 toMonoidHom\u271d\u00b9 = toMonoidHom\u271d\n[PROOFSTEP]\nexact MonoidHom.ext h\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 \u03c7' : MulChar R R'\nh : \u2200 (a : R\u02e3), \u2191\u03c7 \u2191a = \u2191\u03c7' \u2191a\n\u22a2 \u03c7 = \u03c7'\n[PROOFSTEP]\napply ext'\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 \u03c7' : MulChar R R'\nh : \u2200 (a : R\u02e3), \u2191\u03c7 \u2191a = \u2191\u03c7' \u2191a\n\u22a2 \u2200 (a : R), \u2191\u03c7 a = \u2191\u03c7' a\n[PROOFSTEP]\nintro a\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 \u03c7' : MulChar R R'\nh : \u2200 (a : R\u02e3), \u2191\u03c7 \u2191a = \u2191\u03c7' \u2191a\na : R\n\u22a2 \u2191\u03c7 a = \u2191\u03c7' a\n[PROOFSTEP]\nby_cases ha : IsUnit a\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 \u03c7' : MulChar R R'\nh : \u2200 (a : R\u02e3), \u2191\u03c7 \u2191a = \u2191\u03c7' \u2191a\na : R\nha : IsUnit a\n\u22a2 \u2191\u03c7 a = \u2191\u03c7' a\n[PROOFSTEP]\nexact h ha.unit\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 \u03c7' : MulChar R R'\nh : \u2200 (a : R\u02e3), \u2191\u03c7 \u2191a = \u2191\u03c7' \u2191a\na : R\nha : \u00acIsUnit a\n\u22a2 \u2191\u03c7 a = \u2191\u03c7' a\n[PROOFSTEP]\nrw [map_nonunit \u03c7 ha, map_nonunit \u03c7' ha]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 \u03c7' : MulChar R R'\n\u22a2 \u03c7 = \u03c7' \u2192 \u2200 (a : R\u02e3), \u2191\u03c7 \u2191a = \u2191\u03c7' \u2191a\n[PROOFSTEP]\nrintro rfl a\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\na : R\u02e3\n\u22a2 \u2191\u03c7 \u2191a = \u2191\u03c7 \u2191a\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\n\u22a2 R \u2192 R'\n[PROOFSTEP]\nclassical exact fun x => if hx : IsUnit x then f hx.unit else 0\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\n\u22a2 R \u2192 R'\n[PROOFSTEP]\nexact fun x => if hx : IsUnit x then f hx.unit else 0\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\n\u22a2 (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1\n[PROOFSTEP]\nhave h1 : (isUnit_one.unit : R\u02e3) = 1 := Units.eq_iff.mp rfl\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\nh1 : IsUnit.unit (_ : IsUnit 1) = 1\n\u22a2 (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1\n[PROOFSTEP]\nsimp only [h1, dif_pos, Units.val_eq_one, map_one, isUnit_one]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\n\u22a2 \u2200 (x y : R),\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n x *\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n y\n[PROOFSTEP]\nclassical\nintro x y\nby_cases hx : IsUnit x\n\u00b7 simp only [hx, IsUnit.mul_iff, true_and_iff, dif_pos]\n by_cases hy : IsUnit y\n \u00b7 simp only [hy, dif_pos]\n have hm : (IsUnit.mul_iff.mpr \u27e8hx, hy\u27e9).unit = hx.unit * hy.unit := Units.eq_iff.mp rfl\n rw [hm, map_mul]\n norm_cast\n \u00b7 simp only [hy, not_false_iff, dif_neg, mul_zero]\n\u00b7 simp only [hx, IsUnit.mul_iff, false_and_iff, not_false_iff, dif_neg, zero_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\n\u22a2 \u2200 (x y : R),\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n x *\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n y\n[PROOFSTEP]\nintro x y\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\nx y : R\n\u22a2 OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n x *\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n y\n[PROOFSTEP]\nby_cases hx : IsUnit x\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\nx y : R\nhx : IsUnit x\n\u22a2 OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n x *\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n y\n[PROOFSTEP]\nsimp only [hx, IsUnit.mul_iff, true_and_iff, dif_pos]\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\nx y : R\nhx : IsUnit x\n\u22a2 (if h : IsUnit y then \u2191(\u2191f (IsUnit.unit (_ : IsUnit (x * y)))) else 0) =\n \u2191(\u2191f (IsUnit.unit (_ : IsUnit x))) * if h : IsUnit y then \u2191(\u2191f (IsUnit.unit (_ : IsUnit y))) else 0\n[PROOFSTEP]\nby_cases hy : IsUnit y\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\nx y : R\nhx : IsUnit x\nhy : IsUnit y\n\u22a2 (if h : IsUnit y then \u2191(\u2191f (IsUnit.unit (_ : IsUnit (x * y)))) else 0) =\n \u2191(\u2191f (IsUnit.unit (_ : IsUnit x))) * if h : IsUnit y then \u2191(\u2191f (IsUnit.unit (_ : IsUnit y))) else 0\n[PROOFSTEP]\nsimp only [hy, dif_pos]\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\nx y : R\nhx : IsUnit x\nhy : IsUnit y\n\u22a2 \u2191(\u2191f (IsUnit.unit (_ : IsUnit (x * y)))) = \u2191(\u2191f (IsUnit.unit (_ : IsUnit x))) * \u2191(\u2191f (IsUnit.unit (_ : IsUnit y)))\n[PROOFSTEP]\nhave hm : (IsUnit.mul_iff.mpr \u27e8hx, hy\u27e9).unit = hx.unit * hy.unit := Units.eq_iff.mp rfl\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\nx y : R\nhx : IsUnit x\nhy : IsUnit y\nhm : IsUnit.unit (_ : IsUnit (x * y)) = IsUnit.unit hx * IsUnit.unit hy\n\u22a2 \u2191(\u2191f (IsUnit.unit (_ : IsUnit (x * y)))) = \u2191(\u2191f (IsUnit.unit (_ : IsUnit x))) * \u2191(\u2191f (IsUnit.unit (_ : IsUnit y)))\n[PROOFSTEP]\nrw [hm, map_mul]\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\nx y : R\nhx : IsUnit x\nhy : IsUnit y\nhm : IsUnit.unit (_ : IsUnit (x * y)) = IsUnit.unit hx * IsUnit.unit hy\n\u22a2 \u2191(\u2191f (IsUnit.unit hx) * \u2191f (IsUnit.unit hy)) = \u2191(\u2191f (IsUnit.unit (_ : IsUnit x))) * \u2191(\u2191f (IsUnit.unit (_ : IsUnit y)))\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\nx y : R\nhx : IsUnit x\nhy : \u00acIsUnit y\n\u22a2 (if h : IsUnit y then \u2191(\u2191f (IsUnit.unit (_ : IsUnit (x * y)))) else 0) =\n \u2191(\u2191f (IsUnit.unit (_ : IsUnit x))) * if h : IsUnit y then \u2191(\u2191f (IsUnit.unit (_ : IsUnit y))) else 0\n[PROOFSTEP]\nsimp only [hy, not_false_iff, dif_neg, mul_zero]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\nx y : R\nhx : \u00acIsUnit x\n\u22a2 OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n x *\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n y\n[PROOFSTEP]\nsimp only [hx, IsUnit.mul_iff, false_and_iff, not_false_iff, dif_neg, zero_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\n\u22a2 \u2200 (a : R),\n \u00acIsUnit a \u2192\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n x *\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n y) })\n a =\n 0\n[PROOFSTEP]\nintro a ha\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\na : R\nha : \u00acIsUnit a\n\u22a2 OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n x *\n OneHom.toFun\n { toFun := fun x => if hx : IsUnit x then \u2191(\u2191f (IsUnit.unit hx)) else 0,\n map_one' := (_ : (if hx : IsUnit 1 then \u2191(\u2191f (IsUnit.unit hx)) else 0) = 1) }\n y) })\n a =\n 0\n[PROOFSTEP]\nsimp only [ha, not_false_iff, dif_neg]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\na : R\u02e3\n\u22a2 \u2191(ofUnitHom f) \u2191a = \u2191(\u2191f a)\n[PROOFSTEP]\nsimp [ofUnitHom]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u22a2 Function.LeftInverse ofUnitHom toUnitHom\n[PROOFSTEP]\nintro \u03c7\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\n\u22a2 ofUnitHom (toUnitHom \u03c7) = \u03c7\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\nx : R\u02e3\n\u22a2 \u2191(ofUnitHom (toUnitHom \u03c7)) \u2191x = \u2191\u03c7 \u2191x\n[PROOFSTEP]\nrw [ofUnitHom_coe, coe_toUnitHom]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u22a2 Function.RightInverse ofUnitHom toUnitHom\n[PROOFSTEP]\nintro f\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\n\u22a2 toUnitHom (ofUnitHom f) = f\n[PROOFSTEP]\next x\n[GOAL]\ncase h.a\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\nf : R\u02e3 \u2192* R'\u02e3\nx : R\u02e3\n\u22a2 \u2191(\u2191(toUnitHom (ofUnitHom f)) x) = \u2191(\u2191f x)\n[PROOFSTEP]\nsimp only [coe_toUnitHom, ofUnitHom_coe]\n[GOAL]\nR\u271d : Type u\ninst\u271d\u00b3 : CommMonoid R\u271d\nR' : Type v\ninst\u271d\u00b2 : CommMonoidWithZero R'\nR : Type u\ninst\u271d\u00b9 : CommMonoidWithZero R\ninst\u271d : Nontrivial R\n\u03c7 : MulChar R R'\n\u22a2 \u2191\u03c7 0 = 0\n[PROOFSTEP]\nrw [map_nonunit \u03c7 not_isUnit_zero]\n[GOAL]\nR\u271d : Type u\ninst\u271d\u00b3 : CommMonoid R\u271d\nR' : Type v\ninst\u271d\u00b2 : CommMonoidWithZero R'\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\n\u03c7 : MulChar R R'\n\u22a2 \u2191\u03c7 \u2191(ringChar R) = 0\n[PROOFSTEP]\nrw [ringChar.Nat.cast_ringChar, \u03c7.map_zero]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\na : R\u02e3\n\u22a2 \u21911 \u2191a = 1\n[PROOFSTEP]\nclassical exact dif_pos a.isUnit\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\na : R\u02e3\n\u22a2 \u21911 \u2191a = 1\n[PROOFSTEP]\nexact dif_pos a.isUnit\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 \u03c7' : MulChar R R'\nsrc\u271d : R \u2192* R' := \u03c7.toMonoidHom * \u03c7'.toMonoidHom\na : R\nha : \u00acIsUnit a\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191\u03c7 * \u2191\u03c7', map_one' := (_ : OneHom.toFun (\u2191src\u271d) 1 = 1) },\n map_mul' :=\n (_ : \u2200 (x y : R), OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n a =\n 0\n[PROOFSTEP]\nsimp only [map_nonunit \u03c7 ha, zero_mul, Pi.mul_apply]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\n\u22a2 1 * \u03c7 = \u03c7\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\na\u271d : R\u02e3\n\u22a2 \u2191(1 * \u03c7) \u2191a\u271d = \u2191\u03c7 \u2191a\u271d\n[PROOFSTEP]\nsimp only [one_mul, Pi.mul_apply, MulChar.coeToFun_mul, MulChar.one_apply_coe]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\n\u22a2 \u03c7 * 1 = \u03c7\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\na\u271d : R\u02e3\n\u22a2 \u2191(\u03c7 * 1) \u2191a\u271d = \u2191\u03c7 \u2191a\u271d\n[PROOFSTEP]\nsimp only [mul_one, Pi.mul_apply, MulChar.coeToFun_mul, MulChar.one_apply_coe]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\nsrc\u271d : R \u2192* R' := MonoidHom.comp (\u2191MonoidWithZero.inverse) \u03c7.toMonoidHom\na : R\nha : \u00acIsUnit a\n\u22a2 OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun a => \u2191MonoidWithZero.inverse (\u2191\u03c7 a), map_one' := (_ : OneHom.toFun (\u2191src\u271d) 1 = 1) },\n map_mul' :=\n (_ : \u2200 (x y : R), OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n a =\n 0\n[PROOFSTEP]\nsimp [map_nonunit _ ha]\n[GOAL]\nR\u271d : Type u\ninst\u271d\u00b2 : CommMonoid R\u271d\nR' : Type v\ninst\u271d\u00b9 : CommMonoidWithZero R'\nR : Type u\ninst\u271d : CommMonoidWithZero R\n\u03c7 : MulChar R R'\na : R\n\u22a2 \u2191\u03c7\u207b\u00b9 a = \u2191\u03c7 (Ring.inverse a)\n[PROOFSTEP]\nby_cases ha : IsUnit a\n[GOAL]\ncase pos\nR\u271d : Type u\ninst\u271d\u00b2 : CommMonoid R\u271d\nR' : Type v\ninst\u271d\u00b9 : CommMonoidWithZero R'\nR : Type u\ninst\u271d : CommMonoidWithZero R\n\u03c7 : MulChar R R'\na : R\nha : IsUnit a\n\u22a2 \u2191\u03c7\u207b\u00b9 a = \u2191\u03c7 (Ring.inverse a)\n[PROOFSTEP]\nrw [inv_apply_eq_inv]\n[GOAL]\ncase pos\nR\u271d : Type u\ninst\u271d\u00b2 : CommMonoid R\u271d\nR' : Type v\ninst\u271d\u00b9 : CommMonoidWithZero R'\nR : Type u\ninst\u271d : CommMonoidWithZero R\n\u03c7 : MulChar R R'\na : R\nha : IsUnit a\n\u22a2 Ring.inverse (\u2191\u03c7 a) = \u2191\u03c7 (Ring.inverse a)\n[PROOFSTEP]\nhave h := IsUnit.map \u03c7 ha\n[GOAL]\ncase pos\nR\u271d : Type u\ninst\u271d\u00b2 : CommMonoid R\u271d\nR' : Type v\ninst\u271d\u00b9 : CommMonoidWithZero R'\nR : Type u\ninst\u271d : CommMonoidWithZero R\n\u03c7 : MulChar R R'\na : R\nha : IsUnit a\nh : IsUnit (\u2191\u03c7 a)\n\u22a2 Ring.inverse (\u2191\u03c7 a) = \u2191\u03c7 (Ring.inverse a)\n[PROOFSTEP]\napply IsUnit.mul_right_injective h\n[GOAL]\ncase pos.a\nR\u271d : Type u\ninst\u271d\u00b2 : CommMonoid R\u271d\nR' : Type v\ninst\u271d\u00b9 : CommMonoidWithZero R'\nR : Type u\ninst\u271d : CommMonoidWithZero R\n\u03c7 : MulChar R R'\na : R\nha : IsUnit a\nh : IsUnit (\u2191\u03c7 a)\n\u22a2 (fun x x_1 => x * x_1) (\u2191\u03c7 a) (Ring.inverse (\u2191\u03c7 a)) = (fun x x_1 => x * x_1) (\u2191\u03c7 a) (\u2191\u03c7 (Ring.inverse a))\n[PROOFSTEP]\ndsimp only\n -- Porting note: was\n -- rw [Ring.mul_inverse_cancel _ h, \u2190 map_mul, Ring.mul_inverse_cancel _ ha, MulChar.map_one]\n[GOAL]\ncase pos.a\nR\u271d : Type u\ninst\u271d\u00b2 : CommMonoid R\u271d\nR' : Type v\ninst\u271d\u00b9 : CommMonoidWithZero R'\nR : Type u\ninst\u271d : CommMonoidWithZero R\n\u03c7 : MulChar R R'\na : R\nha : IsUnit a\nh : IsUnit (\u2191\u03c7 a)\n\u22a2 \u2191\u03c7 a * Ring.inverse (\u2191\u03c7 a) = \u2191\u03c7 a * \u2191\u03c7 (Ring.inverse a)\n[PROOFSTEP]\nerw [Ring.mul_inverse_cancel _ h, \u2190 map_mul, Ring.mul_inverse_cancel _ ha]\n[GOAL]\ncase pos.a\nR\u271d : Type u\ninst\u271d\u00b2 : CommMonoid R\u271d\nR' : Type v\ninst\u271d\u00b9 : CommMonoidWithZero R'\nR : Type u\ninst\u271d : CommMonoidWithZero R\n\u03c7 : MulChar R R'\na : R\nha : IsUnit a\nh : IsUnit (\u2191\u03c7 a)\n\u22a2 1 = \u2191\u03c7 1\n[PROOFSTEP]\nexact (MulChar.map_one \u03c7).symm\n[GOAL]\ncase neg\nR\u271d : Type u\ninst\u271d\u00b2 : CommMonoid R\u271d\nR' : Type v\ninst\u271d\u00b9 : CommMonoidWithZero R'\nR : Type u\ninst\u271d : CommMonoidWithZero R\n\u03c7 : MulChar R R'\na : R\nha : \u00acIsUnit a\n\u22a2 \u2191\u03c7\u207b\u00b9 a = \u2191\u03c7 (Ring.inverse a)\n[PROOFSTEP]\nrevert ha\n[GOAL]\ncase neg\nR\u271d : Type u\ninst\u271d\u00b2 : CommMonoid R\u271d\nR' : Type v\ninst\u271d\u00b9 : CommMonoidWithZero R'\nR : Type u\ninst\u271d : CommMonoidWithZero R\n\u03c7 : MulChar R R'\na : R\n\u22a2 \u00acIsUnit a \u2192 \u2191\u03c7\u207b\u00b9 a = \u2191\u03c7 (Ring.inverse a)\n[PROOFSTEP]\nnontriviality R\n[GOAL]\nR\u271d : Type u\ninst\u271d\u00b2 : CommMonoid R\u271d\nR' : Type v\ninst\u271d\u00b9 : CommMonoidWithZero R'\nR : Type u\ninst\u271d : CommMonoidWithZero R\n\u03c7 : MulChar R R'\na : R\n\u271d : Nontrivial R\n\u22a2 \u00acIsUnit a \u2192 \u2191\u03c7\u207b\u00b9 a = \u2191\u03c7 (Ring.inverse a)\n[PROOFSTEP]\nintro ha\n[GOAL]\nR\u271d : Type u\ninst\u271d\u00b2 : CommMonoid R\u271d\nR' : Type v\ninst\u271d\u00b9 : CommMonoidWithZero R'\nR : Type u\ninst\u271d : CommMonoidWithZero R\n\u03c7 : MulChar R R'\na : R\n\u271d : Nontrivial R\nha : \u00acIsUnit a\n\u22a2 \u2191\u03c7\u207b\u00b9 a = \u2191\u03c7 (Ring.inverse a)\n[PROOFSTEP]\nrw [map_nonunit _ ha, Ring.inverse_non_unit a ha, MulChar.map_zero \u03c7]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\n\u22a2 \u03c7\u207b\u00b9 * \u03c7 = 1\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\nx : R\u02e3\n\u22a2 \u2191(\u03c7\u207b\u00b9 * \u03c7) \u2191x = \u21911 \u2191x\n[PROOFSTEP]\nrw [coeToFun_mul, Pi.mul_apply, inv_apply_eq_inv]\n -- Porting note: was\n -- simp only [Ring.inverse_mul_cancel _ (IsUnit.map _ x.isUnit)]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\nx : R\u02e3\n\u22a2 Ring.inverse (\u2191\u03c7 \u2191x) * \u2191\u03c7 \u2191x = \u21911 \u2191x\n[PROOFSTEP]\nerw [Ring.inverse_mul_cancel _ (IsUnit.map \u03c7 x.isUnit)]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\nx : R\u02e3\n\u22a2 1 = \u21911 \u2191x\n[PROOFSTEP]\nrw [one_apply_coe]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u22a2 \u2200 (a b c : MulChar R R'), a * b * c = a * (b * c)\n[PROOFSTEP]\nintro \u03c7\u2081 \u03c7\u2082 \u03c7\u2083\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : MulChar R R'\n\u22a2 \u03c7\u2081 * \u03c7\u2082 * \u03c7\u2083 = \u03c7\u2081 * (\u03c7\u2082 * \u03c7\u2083)\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : MulChar R R'\na : R\u02e3\n\u22a2 \u2191(\u03c7\u2081 * \u03c7\u2082 * \u03c7\u2083) \u2191a = \u2191(\u03c7\u2081 * (\u03c7\u2082 * \u03c7\u2083)) \u2191a\n[PROOFSTEP]\nsimp only [mul_assoc, Pi.mul_apply, MulChar.coeToFun_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u22a2 \u2200 (a b : MulChar R R'), a * b = b * a\n[PROOFSTEP]\nintro \u03c7\u2081 \u03c7\u2082\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7\u2081 \u03c7\u2082 : MulChar R R'\n\u22a2 \u03c7\u2081 * \u03c7\u2082 = \u03c7\u2082 * \u03c7\u2081\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7\u2081 \u03c7\u2082 : MulChar R R'\na : R\u02e3\n\u22a2 \u2191(\u03c7\u2081 * \u03c7\u2082) \u2191a = \u2191(\u03c7\u2082 * \u03c7\u2081) \u2191a\n[PROOFSTEP]\nsimp only [mul_comm, Pi.mul_apply, MulChar.coeToFun_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\nn : \u2115\na : R\u02e3\n\u22a2 \u2191(\u03c7 ^ n) \u2191a = \u2191\u03c7 \u2191a ^ n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\na : R\u02e3\n\u22a2 \u2191(\u03c7 ^ Nat.zero) \u2191a = \u2191\u03c7 \u2191a ^ Nat.zero\n[PROOFSTEP]\nrw [pow_zero, pow_zero, one_apply_coe]\n[GOAL]\ncase succ\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\na : R\u02e3\nn : \u2115\nih : \u2191(\u03c7 ^ n) \u2191a = \u2191\u03c7 \u2191a ^ n\n\u22a2 \u2191(\u03c7 ^ Nat.succ n) \u2191a = \u2191\u03c7 \u2191a ^ Nat.succ n\n[PROOFSTEP]\nrw [pow_succ, pow_succ, mul_apply, ih]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\nn : \u2115\nhn : 0 < n\na : R\n\u22a2 \u2191(\u03c7 ^ n) a = \u2191\u03c7 a ^ n\n[PROOFSTEP]\nby_cases ha : IsUnit a\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\nn : \u2115\nhn : 0 < n\na : R\nha : IsUnit a\n\u22a2 \u2191(\u03c7 ^ n) a = \u2191\u03c7 a ^ n\n[PROOFSTEP]\nexact pow_apply_coe \u03c7 n ha.unit\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\nn : \u2115\nhn : 0 < n\na : R\nha : \u00acIsUnit a\n\u22a2 \u2191(\u03c7 ^ n) a = \u2191\u03c7 a ^ n\n[PROOFSTEP]\nrw [map_nonunit (\u03c7 ^ n) ha, map_nonunit \u03c7 ha, zero_pow hn]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\n\u22a2 IsNontrivial \u03c7 \u2194 \u03c7 \u2260 1\n[PROOFSTEP]\nsimp only [IsNontrivial, Ne.def, ext_iff, not_forall, one_apply_coe]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nf : R' \u2192+* R''\nsrc\u271d : R \u2192* R'' := MonoidHom.comp (\u2191f) \u03c7.toMonoidHom\na : R\nha : \u00acIsUnit a\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := fun a => \u2191f (\u2191\u03c7 a), map_one' := (_ : OneHom.toFun (\u2191src\u271d) 1 = 1) },\n map_mul' :=\n (_ : \u2200 (x y : R), OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n a =\n 0\n[PROOFSTEP]\nsimp only [map_nonunit \u03c7 ha, map_zero]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsNontrivial \u03c7\nf : R' \u2192+* R''\nhf : Function.Injective \u2191f\n\u22a2 IsNontrivial (ringHomComp \u03c7 f)\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := h\u03c7\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nf : R' \u2192+* R''\nhf : Function.Injective \u2191f\na : R\u02e3\nha : \u2191\u03c7 \u2191a \u2260 1\n\u22a2 IsNontrivial (ringHomComp \u03c7 f)\n[PROOFSTEP]\nuse a\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nf : R' \u2192+* R''\nhf : Function.Injective \u2191f\na : R\u02e3\nha : \u2191\u03c7 \u2191a \u2260 1\n\u22a2 \u2191(ringHomComp \u03c7 f) \u2191a \u2260 1\n[PROOFSTEP]\nsimp_rw [ringHomComp_apply, \u2190 RingHom.map_one f]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nf : R' \u2192+* R''\nhf : Function.Injective \u2191f\na : R\u02e3\nha : \u2191\u03c7 \u2191a \u2260 1\n\u22a2 \u2191f (\u2191\u03c7 \u2191a) \u2260 \u2191f 1\n[PROOFSTEP]\nexact fun h => ha (hf h)\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nf : R' \u2192+* R''\n\u22a2 IsQuadratic (ringHomComp \u03c7 f)\n[PROOFSTEP]\nintro a\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nf : R' \u2192+* R''\na : R\n\u22a2 \u2191(ringHomComp \u03c7 f) a = 0 \u2228 \u2191(ringHomComp \u03c7 f) a = 1 \u2228 \u2191(ringHomComp \u03c7 f) a = -1\n[PROOFSTEP]\nrcases h\u03c7 a with (ha | ha | ha)\n[GOAL]\ncase inl\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nf : R' \u2192+* R''\na : R\nha : \u2191\u03c7 a = 0\n\u22a2 \u2191(ringHomComp \u03c7 f) a = 0 \u2228 \u2191(ringHomComp \u03c7 f) a = 1 \u2228 \u2191(ringHomComp \u03c7 f) a = -1\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\ncase inr.inl\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nf : R' \u2192+* R''\na : R\nha : \u2191\u03c7 a = 1\n\u22a2 \u2191(ringHomComp \u03c7 f) a = 0 \u2228 \u2191(ringHomComp \u03c7 f) a = 1 \u2228 \u2191(ringHomComp \u03c7 f) a = -1\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\ncase inr.inr\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nf : R' \u2192+* R''\na : R\nha : \u2191\u03c7 a = -1\n\u22a2 \u2191(ringHomComp \u03c7 f) a = 0 \u2228 \u2191(ringHomComp \u03c7 f) a = 1 \u2228 \u2191(ringHomComp \u03c7 f) a = -1\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\n\u22a2 \u03c7\u207b\u00b9 = \u03c7\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nx : R\u02e3\n\u22a2 \u2191\u03c7\u207b\u00b9 \u2191x = \u2191\u03c7 \u2191x\n[PROOFSTEP]\nrw [inv_apply_eq_inv]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nx : R\u02e3\n\u22a2 Ring.inverse (\u2191\u03c7 \u2191x) = \u2191\u03c7 \u2191x\n[PROOFSTEP]\nrcases h\u03c7 x with (h\u2080 | h\u2081 | h\u2082)\n[GOAL]\ncase h.inl\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nx : R\u02e3\nh\u2080 : \u2191\u03c7 \u2191x = 0\n\u22a2 Ring.inverse (\u2191\u03c7 \u2191x) = \u2191\u03c7 \u2191x\n[PROOFSTEP]\nrw [h\u2080, Ring.inverse_zero]\n[GOAL]\ncase h.inr.inl\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nx : R\u02e3\nh\u2081 : \u2191\u03c7 \u2191x = 1\n\u22a2 Ring.inverse (\u2191\u03c7 \u2191x) = \u2191\u03c7 \u2191x\n[PROOFSTEP]\nrw [h\u2081, Ring.inverse_one]\n[GOAL]\ncase h.inr.inr\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nx : R\u02e3\nh\u2082 : \u2191\u03c7 \u2191x = -1\n\u22a2 Ring.inverse (\u2191\u03c7 \u2191x) = \u2191\u03c7 \u2191x\n[PROOFSTEP]\nhave : (-1 : R') = (-1 : R'\u02e3) := by rw [Units.val_neg, Units.val_one]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nx : R\u02e3\nh\u2082 : \u2191\u03c7 \u2191x = -1\n\u22a2 -1 = \u2191(-1)\n[PROOFSTEP]\nrw [Units.val_neg, Units.val_one]\n[GOAL]\ncase h.inr.inr\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nx : R\u02e3\nh\u2082 : \u2191\u03c7 \u2191x = -1\nthis : -1 = \u2191(-1)\n\u22a2 Ring.inverse (\u2191\u03c7 \u2191x) = \u2191\u03c7 \u2191x\n[PROOFSTEP]\nrw [h\u2082, this, Ring.inverse_unit (-1 : R'\u02e3)]\n[GOAL]\ncase h.inr.inr\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nx : R\u02e3\nh\u2082 : \u2191\u03c7 \u2191x = -1\nthis : -1 = \u2191(-1)\n\u22a2 \u2191(-1)\u207b\u00b9 = \u2191(-1)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\n\u22a2 \u03c7 ^ 2 = 1\n[PROOFSTEP]\nrw [\u2190 mul_left_inv \u03c7, pow_two, h\u03c7.inv]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nR' : Type v\ninst\u271d\u00b2 : CommRing R'\nR'' : Type w\ninst\u271d\u00b9 : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\np : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R' p\n\u22a2 \u03c7 ^ p = \u03c7\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b3 : CommRing R\nR' : Type v\ninst\u271d\u00b2 : CommRing R'\nR'' : Type w\ninst\u271d\u00b9 : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\np : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R' p\nx : R\u02e3\n\u22a2 \u2191(\u03c7 ^ p) \u2191x = \u2191\u03c7 \u2191x\n[PROOFSTEP]\nrw [pow_apply_coe]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b3 : CommRing R\nR' : Type v\ninst\u271d\u00b2 : CommRing R'\nR'' : Type w\ninst\u271d\u00b9 : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\np : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R' p\nx : R\u02e3\n\u22a2 \u2191\u03c7 \u2191x ^ p = \u2191\u03c7 \u2191x\n[PROOFSTEP]\nrcases h\u03c7 x with (hx | hx | hx)\n[GOAL]\ncase h.inl\nR : Type u\ninst\u271d\u00b3 : CommRing R\nR' : Type v\ninst\u271d\u00b2 : CommRing R'\nR'' : Type w\ninst\u271d\u00b9 : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\np : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R' p\nx : R\u02e3\nhx : \u2191\u03c7 \u2191x = 0\n\u22a2 \u2191\u03c7 \u2191x ^ p = \u2191\u03c7 \u2191x\n[PROOFSTEP]\nrw [hx]\n[GOAL]\ncase h.inr.inl\nR : Type u\ninst\u271d\u00b3 : CommRing R\nR' : Type v\ninst\u271d\u00b2 : CommRing R'\nR'' : Type w\ninst\u271d\u00b9 : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\np : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R' p\nx : R\u02e3\nhx : \u2191\u03c7 \u2191x = 1\n\u22a2 \u2191\u03c7 \u2191x ^ p = \u2191\u03c7 \u2191x\n[PROOFSTEP]\nrw [hx]\n[GOAL]\ncase h.inr.inr\nR : Type u\ninst\u271d\u00b3 : CommRing R\nR' : Type v\ninst\u271d\u00b2 : CommRing R'\nR'' : Type w\ninst\u271d\u00b9 : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\np : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R' p\nx : R\u02e3\nhx : \u2191\u03c7 \u2191x = -1\n\u22a2 \u2191\u03c7 \u2191x ^ p = \u2191\u03c7 \u2191x\n[PROOFSTEP]\nrw [hx]\n[GOAL]\ncase h.inl\nR : Type u\ninst\u271d\u00b3 : CommRing R\nR' : Type v\ninst\u271d\u00b2 : CommRing R'\nR'' : Type w\ninst\u271d\u00b9 : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\np : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R' p\nx : R\u02e3\nhx : \u2191\u03c7 \u2191x = 0\n\u22a2 0 ^ p = 0\n[PROOFSTEP]\nrw [zero_pow (@Fact.out p.Prime).pos]\n[GOAL]\ncase h.inr.inl\nR : Type u\ninst\u271d\u00b3 : CommRing R\nR' : Type v\ninst\u271d\u00b2 : CommRing R'\nR'' : Type w\ninst\u271d\u00b9 : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\np : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R' p\nx : R\u02e3\nhx : \u2191\u03c7 \u2191x = 1\n\u22a2 1 ^ p = 1\n[PROOFSTEP]\nrw [one_pow]\n[GOAL]\ncase h.inr.inr\nR : Type u\ninst\u271d\u00b3 : CommRing R\nR' : Type v\ninst\u271d\u00b2 : CommRing R'\nR'' : Type w\ninst\u271d\u00b9 : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\np : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R' p\nx : R\u02e3\nhx : \u2191\u03c7 \u2191x = -1\n\u22a2 (-1) ^ p = -1\n[PROOFSTEP]\nexact CharP.neg_one_pow_char R' p\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nn : \u2115\nhn : Even n\n\u22a2 \u03c7 ^ n = 1\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := even_iff_two_dvd.mp hn\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nn : \u2115\nhn : Even (2 * n)\n\u22a2 \u03c7 ^ (2 * n) = 1\n[PROOFSTEP]\nrw [pow_mul, h\u03c7.sq_eq_one, one_pow]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nn : \u2115\nhn : Odd n\n\u22a2 \u03c7 ^ n = \u03c7\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := hn\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u00b2 : CommRing R\nR' : Type v\ninst\u271d\u00b9 : CommRing R'\nR'' : Type w\ninst\u271d : CommRing R''\n\u03c7 : MulChar R R'\nh\u03c7 : IsQuadratic \u03c7\nn : \u2115\n\u22a2 \u03c7 ^ (2 * n + 1) = \u03c7\n[PROOFSTEP]\nrw [pow_add, pow_one, h\u03c7.pow_even (even_two_mul _), one_mul]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nR' : Type v\ninst\u271d\u00b3 : CommRing R'\nR'' : Type w\ninst\u271d\u00b2 : CommRing R''\ninst\u271d\u00b9 : Fintype R\ninst\u271d : IsDomain R'\n\u03c7 : MulChar R R'\nh\u03c7 : IsNontrivial \u03c7\n\u22a2 \u2211 a : R, \u2191\u03c7 a = 0\n[PROOFSTEP]\nrcases h\u03c7 with \u27e8b, hb\u27e9\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u2074 : CommRing R\nR' : Type v\ninst\u271d\u00b3 : CommRing R'\nR'' : Type w\ninst\u271d\u00b2 : CommRing R''\ninst\u271d\u00b9 : Fintype R\ninst\u271d : IsDomain R'\n\u03c7 : MulChar R R'\nb : R\u02e3\nhb : \u2191\u03c7 \u2191b \u2260 1\n\u22a2 \u2211 a : R, \u2191\u03c7 a = 0\n[PROOFSTEP]\nrefine'\n eq_zero_of_mul_eq_self_left hb\n _\n -- POrting note: `map_mul` isn't applied\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u2074 : CommRing R\nR' : Type v\ninst\u271d\u00b3 : CommRing R'\nR'' : Type w\ninst\u271d\u00b2 : CommRing R''\ninst\u271d\u00b9 : Fintype R\ninst\u271d : IsDomain R'\n\u03c7 : MulChar R R'\nb : R\u02e3\nhb : \u2191\u03c7 \u2191b \u2260 1\n\u22a2 \u2191\u03c7 \u2191b * \u2211 a : R, \u2191\u03c7 a = \u2211 a : R, \u2191\u03c7 a\n[PROOFSTEP]\nsimp only [Finset.mul_sum, \u2190 map_mul]\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u2074 : CommRing R\nR' : Type v\ninst\u271d\u00b3 : CommRing R'\nR'' : Type w\ninst\u271d\u00b2 : CommRing R''\ninst\u271d\u00b9 : Fintype R\ninst\u271d : IsDomain R'\n\u03c7 : MulChar R R'\nb : R\u02e3\nhb : \u2191\u03c7 \u2191b \u2260 1\n\u22a2 \u2211 x : R, \u2191\u03c7 (\u2191b * x) = \u2211 x : R, \u2191\u03c7 x\n[PROOFSTEP]\nrefine Fintype.sum_bijective _ (Units.mulLeft_bijective b) _ _ fun x => rfl\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nR' : Type v\ninst\u271d\u00b3 : CommRing R'\nR'' : Type w\ninst\u271d\u00b2 : CommRing R''\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\n\u22a2 \u2211 a : R, \u21911 a = \u2191(Fintype.card R\u02e3)\n[PROOFSTEP]\ncalc\n (\u2211 a, (1 : MulChar R R') a) = \u2211 a : R, if IsUnit a then 1 else 0 := Finset.sum_congr rfl fun a _ => ?_\n _ = ((Finset.univ : Finset R).filter IsUnit).card := Finset.sum_boole\n _ = (Finset.univ.map \u27e8((\u2191) : R\u02e3 \u2192 R), Units.ext\u27e9).card := ?_\n _ = Fintype.card R\u02e3 := congr_arg _ (Finset.card_map _)\n[GOAL]\ncase calc_1\nR : Type u\ninst\u271d\u2074 : CommRing R\nR' : Type v\ninst\u271d\u00b3 : CommRing R'\nR'' : Type w\ninst\u271d\u00b2 : CommRing R''\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\na : R\nx\u271d : a \u2208 Finset.univ\n\u22a2 \u21911 a = if IsUnit a then 1 else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2074 : CommRing R\nR' : Type v\ninst\u271d\u00b3 : CommRing R'\nR'' : Type w\ninst\u271d\u00b2 : CommRing R''\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\na : R\nx\u271d : a \u2208 Finset.univ\nh : IsUnit a\n\u22a2 \u21911 a = 1\n[PROOFSTEP]\nexact one_apply_coe h.unit\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\nR' : Type v\ninst\u271d\u00b3 : CommRing R'\nR'' : Type w\ninst\u271d\u00b2 : CommRing R''\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\na : R\nx\u271d : a \u2208 Finset.univ\nh : \u00acIsUnit a\n\u22a2 \u21911 a = 0\n[PROOFSTEP]\nexact map_nonunit _ h\n[GOAL]\ncase calc_2\nR : Type u\ninst\u271d\u2074 : CommRing R\nR' : Type v\ninst\u271d\u00b3 : CommRing R'\nR'' : Type w\ninst\u271d\u00b2 : CommRing R''\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\n\u22a2 \u2191(Finset.card (Finset.filter IsUnit Finset.univ)) =\n \u2191(Finset.card (Finset.map { toFun := Units.val, inj' := (_ : Function.Injective Units.val) } Finset.univ))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase calc_2.e_a.e_s\nR : Type u\ninst\u271d\u2074 : CommRing R\nR' : Type v\ninst\u271d\u00b3 : CommRing R'\nR'' : Type w\ninst\u271d\u00b2 : CommRing R''\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\n\u22a2 Finset.filter IsUnit Finset.univ =\n Finset.map { toFun := Units.val, inj' := (_ : Function.Injective Units.val) } Finset.univ\n[PROOFSTEP]\next a\n[GOAL]\ncase calc_2.e_a.e_s.a\nR : Type u\ninst\u271d\u2074 : CommRing R\nR' : Type v\ninst\u271d\u00b3 : CommRing R'\nR'' : Type w\ninst\u271d\u00b2 : CommRing R''\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\na : R\n\u22a2 a \u2208 Finset.filter IsUnit Finset.univ \u2194\n a \u2208 Finset.map { toFun := Units.val, inj' := (_ : Function.Injective Units.val) } Finset.univ\n[PROOFSTEP]\nsimp only [Finset.mem_filter, Finset.mem_univ, true_and_iff, Finset.mem_map, Function.Embedding.coeFn_mk,\n exists_true_left, IsUnit]\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "llama_tokens": 21225, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4040364332908043}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b a : \u03b1\nl : List \u03b1\np : Chain R a l\n\u22a2 Chain (fun x y => x \u2208 a :: l \u2227 y \u2208 l \u2227 R x y) a l\n[PROOFSTEP]\ninduction' p with _ a b l r _ IH <;> constructor <;> [exact \u27e8mem_cons_self _ _, mem_cons_self _ _, r\u27e9;\n exact IH.imp fun a b \u27e8am, bm, h\u27e9 => \u27e8mem_cons_of_mem _ am, mem_cons_of_mem _ bm, h\u27e9]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b a : \u03b1\nl : List \u03b1\np : Chain R a l\n\u22a2 Chain (fun x y => x \u2208 a :: l \u2227 y \u2208 l \u2227 R x y) a l\n[PROOFSTEP]\ninduction' p with _ a b l r _ IH\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d\u00b9 b a : \u03b1\nl : List \u03b1\na\u271d : \u03b1\n\u22a2 Chain (fun x y => x \u2208 [a\u271d] \u2227 y \u2208 [] \u2227 R x y) a\u271d []\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\nR r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d\u00b9 l\u2081 l\u2082 : List \u03b1\na\u271d\u00b2 b\u271d a\u271d\u00b9 : \u03b1\nl\u271d : List \u03b1\na b : \u03b1\nl : List \u03b1\nr : R a b\na\u271d : Chain R b l\nIH : Chain (fun x y => x \u2208 b :: l \u2227 y \u2208 l \u2227 R x y) b l\n\u22a2 Chain (fun x y => x \u2208 a :: b :: l \u2227 y \u2208 b :: l \u2227 R x y) a (b :: l)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons.a\n\u03b1 : Type u\n\u03b2 : Type v\nR r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d\u00b9 l\u2081 l\u2082 : List \u03b1\na\u271d\u00b2 b\u271d a\u271d\u00b9 : \u03b1\nl\u271d : List \u03b1\na b : \u03b1\nl : List \u03b1\nr : R a b\na\u271d : Chain R b l\nIH : Chain (fun x y => x \u2208 b :: l \u2227 y \u2208 l \u2227 R x y) b l\n\u22a2 a \u2208 a :: b :: l \u2227 b \u2208 b :: l \u2227 R a b\n[PROOFSTEP]\nexact \u27e8mem_cons_self _ _, mem_cons_self _ _, r\u27e9\n[GOAL]\ncase cons.a\n\u03b1 : Type u\n\u03b2 : Type v\nR r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d\u00b9 l\u2081 l\u2082 : List \u03b1\na\u271d\u00b2 b\u271d a\u271d\u00b9 : \u03b1\nl\u271d : List \u03b1\na b : \u03b1\nl : List \u03b1\nr : R a b\na\u271d : Chain R b l\nIH : Chain (fun x y => x \u2208 b :: l \u2227 y \u2208 l \u2227 R x y) b l\n\u22a2 Chain (fun x y => x \u2208 a :: b :: l \u2227 y \u2208 b :: l \u2227 R x y) b l\n[PROOFSTEP]\nexact IH.imp fun a b \u27e8am, bm, h\u27e9 => \u27e8mem_cons_of_mem _ am, mem_cons_of_mem _ bm, h\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d a b : \u03b1\n\u22a2 Chain R a [b] \u2194 R a b\n[PROOFSTEP]\nsimp only [chain_cons, Chain.nil, and_true_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d b\u271d a b : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 Chain R a (l\u2081 ++ b :: l\u2082) \u2194 Chain R a (l\u2081 ++ [b]) \u2227 Chain R b l\u2082\n[PROOFSTEP]\ninduction' l\u2081 with x l\u2081 IH generalizing a\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082\u271d : List \u03b1\na\u271d\u00b9 b\u271d a\u271d b : \u03b1\nl\u2082 : List \u03b1\na : \u03b1\n\u22a2 Chain R a ([] ++ b :: l\u2082) \u2194 Chain R a ([] ++ [b]) \u2227 Chain R b l\u2082\n[PROOFSTEP]\nsimp only [*, nil_append, cons_append, Chain.nil, chain_cons, and_true_iff, and_assoc]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d\u00b9 b\u271d a\u271d b : \u03b1\nl\u2082 : List \u03b1\nx : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {a : \u03b1}, Chain R a (l\u2081 ++ b :: l\u2082) \u2194 Chain R a (l\u2081 ++ [b]) \u2227 Chain R b l\u2082\na : \u03b1\n\u22a2 Chain R a (x :: l\u2081 ++ b :: l\u2082) \u2194 Chain R a (x :: l\u2081 ++ [b]) \u2227 Chain R b l\u2082\n[PROOFSTEP]\nsimp only [*, nil_append, cons_append, Chain.nil, chain_cons, and_true_iff, and_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d b\u271d a b c : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 Chain R a (l\u2081 ++ b :: c :: l\u2082) \u2194 Chain R a (l\u2081 ++ [b]) \u2227 R b c \u2227 Chain R c l\u2082\n[PROOFSTEP]\nrw [chain_split, chain_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b a : \u03b1\n\u22a2 Chain R a [] \u2194 [] = [] \u2228 Forall\u2082 R (a :: dropLast []) []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d a b : \u03b1\nl : List \u03b1\n\u22a2 Chain R a (b :: l) \u2194 b :: l = [] \u2228 Forall\u2082 R (a :: dropLast (b :: l)) (b :: l)\n[PROOFSTEP]\nby_cases h : l = []\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d a b : \u03b1\nl : List \u03b1\nh : l = []\n\u22a2 Chain R a (b :: l) \u2194 b :: l = [] \u2228 Forall\u2082 R (a :: dropLast (b :: l)) (b :: l)\n[PROOFSTEP]\nsimp [@chain_iff_forall\u2082 b l, *]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d a b : \u03b1\nl : List \u03b1\nh : \u00acl = []\n\u22a2 Chain R a (b :: l) \u2194 b :: l = [] \u2228 Forall\u2082 R (a :: dropLast (b :: l)) (b :: l)\n[PROOFSTEP]\nsimp [@chain_iff_forall\u2082 b l, *]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\n\u22a2 Chain R a (l ++ [b]) \u2194 Forall\u2082 R (a :: l) (l ++ [b])\n[PROOFSTEP]\nsimp [chain_iff_forall\u2082]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na b\u271d : \u03b1\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nl : List \u03b2\n\u22a2 Chain R (f b) (map f l) \u2194 Chain (fun a b => R (f a) (f b)) b l\n[PROOFSTEP]\ninduction l generalizing b\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b\u271d : \u03b1\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\n\u22a2 Chain R (f b) (map f []) \u2194 Chain (fun a b => R (f a) (f b)) b []\n[PROOFSTEP]\nsimp only [map, Chain.nil, chain_cons, *]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b\u271d : \u03b1\nf : \u03b2 \u2192 \u03b1\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\ntail_ih\u271d : \u2200 {b : \u03b2}, Chain R (f b) (map f tail\u271d) \u2194 Chain (fun a b => R (f a) (f b)) b tail\u271d\nb : \u03b2\n\u22a2 Chain R (f b) (map f (head\u271d :: tail\u271d)) \u2194 Chain (fun a b => R (f a) (f b)) b (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp only [map, Chain.nil, chain_cons, *]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b : \u03b1\nS : \u03b2 \u2192 \u03b2 \u2192 Prop\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nH : \u2200 (a b : \u03b1) (ha : p a) (hb : p b), R a b \u2192 S (f a ha) (f b hb)\na : \u03b1\nl : List \u03b1\nhl\u2081 : Chain R a l\nha : p a\nhl\u2082 : \u2200 (a : \u03b1), a \u2208 l \u2192 p a\n\u22a2 Chain S (f a ha) (pmap f l hl\u2082)\n[PROOFSTEP]\ninduction' l with lh lt l_ih generalizing a\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d\u00b9 b : \u03b1\nS : \u03b2 \u2192 \u03b2 \u2192 Prop\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nH : \u2200 (a b : \u03b1) (ha : p a) (hb : p b), R a b \u2192 S (f a ha) (f b hb)\na\u271d : \u03b1\nl : List \u03b1\nhl\u2081\u271d : Chain R a\u271d l\nha\u271d : p a\u271d\nhl\u2082\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 p a\na : \u03b1\nhl\u2081 : Chain R a []\nha : p a\nhl\u2082 : \u2200 (a : \u03b1), a \u2208 [] \u2192 p a\n\u22a2 Chain S (f a ha) (pmap f [] hl\u2082)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d\u00b9 b : \u03b1\nS : \u03b2 \u2192 \u03b2 \u2192 Prop\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nH : \u2200 (a b : \u03b1) (ha : p a) (hb : p b), R a b \u2192 S (f a ha) (f b hb)\na\u271d : \u03b1\nl : List \u03b1\nhl\u2081\u271d : Chain R a\u271d l\nha\u271d : p a\u271d\nhl\u2082\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 p a\nlh : \u03b1\nlt : List \u03b1\nl_ih : \u2200 {a : \u03b1}, Chain R a lt \u2192 \u2200 (ha : p a) (hl\u2082 : \u2200 (a : \u03b1), a \u2208 lt \u2192 p a), Chain S (f a ha) (pmap f lt hl\u2082)\na : \u03b1\nhl\u2081 : Chain R a (lh :: lt)\nha : p a\nhl\u2082 : \u2200 (a : \u03b1), a \u2208 lh :: lt \u2192 p a\n\u22a2 Chain S (f a ha) (pmap f (lh :: lt) hl\u2082)\n[PROOFSTEP]\nsimp [H _ _ _ _ (rel_of_chain_cons hl\u2081), l_ih (chain_of_chain_cons hl\u2081)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b : \u03b1\nS : \u03b2 \u2192 \u03b2 \u2192 Prop\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nl : List \u03b1\nhl\u2081 : \u2200 (a : \u03b1), a \u2208 l \u2192 p a\na : \u03b1\nha : p a\nhl\u2082 : Chain S (f a ha) (pmap f l hl\u2081)\nH : \u2200 (a b : \u03b1) (ha : p a) (hb : p b), S (f a ha) (f b hb) \u2192 R a b\n\u22a2 Chain R a l\n[PROOFSTEP]\ninduction' l with lh lt l_ih generalizing a\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d\u00b9 b : \u03b1\nS : \u03b2 \u2192 \u03b2 \u2192 Prop\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nl : List \u03b1\nhl\u2081\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 p a\na\u271d : \u03b1\nha\u271d : p a\u271d\nhl\u2082\u271d : Chain S (f a\u271d ha\u271d) (pmap f l hl\u2081\u271d)\nH : \u2200 (a b : \u03b1) (ha : p a) (hb : p b), S (f a ha) (f b hb) \u2192 R a b\nhl\u2081 : \u2200 (a : \u03b1), a \u2208 [] \u2192 p a\na : \u03b1\nha : p a\nhl\u2082 : Chain S (f a ha) (pmap f [] hl\u2081)\n\u22a2 Chain R a []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d\u00b9 b : \u03b1\nS : \u03b2 \u2192 \u03b2 \u2192 Prop\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nl : List \u03b1\nhl\u2081\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 p a\na\u271d : \u03b1\nha\u271d : p a\u271d\nhl\u2082\u271d : Chain S (f a\u271d ha\u271d) (pmap f l hl\u2081\u271d)\nH : \u2200 (a b : \u03b1) (ha : p a) (hb : p b), S (f a ha) (f b hb) \u2192 R a b\nlh : \u03b1\nlt : List \u03b1\nl_ih : \u2200 (hl\u2081 : \u2200 (a : \u03b1), a \u2208 lt \u2192 p a) {a : \u03b1} (ha : p a), Chain S (f a ha) (pmap f lt hl\u2081) \u2192 Chain R a lt\nhl\u2081 : \u2200 (a : \u03b1), a \u2208 lh :: lt \u2192 p a\na : \u03b1\nha : p a\nhl\u2082 : Chain S (f a ha) (pmap f (lh :: lt) hl\u2081)\n\u22a2 Chain R a (lh :: lt)\n[PROOFSTEP]\nsimp [H _ _ _ _ (rel_of_chain_cons hl\u2082), l_ih _ _ (chain_of_chain_cons hl\u2082)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : IsTrans \u03b1 R\na b : \u03b1\nl : List \u03b1\nh : R a b\nhb : Chain R b l\n\u22a2 \u2200 (a' : \u03b1), a' \u2208 b :: l \u2192 R a a'\n[PROOFSTEP]\nsimp only [mem_cons, forall_eq_or_imp, h, true_and_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d : IsTrans \u03b1 R\na b : \u03b1\nl : List \u03b1\nh : R a b\nhb : Chain R b l\n\u22a2 \u2200 (a_1 : \u03b1), a_1 \u2208 l \u2192 R a a_1\n[PROOFSTEP]\nexact fun c hc => _root_.trans h (rel_of_pairwise_cons hb.pairwise hc)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\ninst\u271d : IsTrans \u03b1 R\nhl : Chain R a l\u2082\nh : l\u2081 <+ l\u2082\n\u22a2 Chain R a l\u2081\n[PROOFSTEP]\nrw [chain_iff_pairwise] at hl \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\ninst\u271d : IsTrans \u03b1 R\nhl : Pairwise R (a :: l\u2082)\nh : l\u2081 <+ l\u2082\n\u22a2 Pairwise R (a :: l\u2081)\n[PROOFSTEP]\nexact hl.sublist (h.cons_cons a)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\ninst\u271d : IsTrans \u03b1 R\nhl : Chain R a l\nhb : b \u2208 l\n\u22a2 R a b\n[PROOFSTEP]\nrw [chain_iff_pairwise] at hl \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\ninst\u271d : IsTrans \u03b1 R\nhl : Pairwise R (a :: l)\nhb : b \u2208 l\n\u22a2 R a b\n[PROOFSTEP]\nexact rel_of_pairwise_cons hl hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\n\u22a2 Chain R a []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nh : 0 < length []\n\u22a2 R a (get [] { val := 0, isLt := h })\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nx\u271d : \u2115\nh : x\u271d < length [] - 1\n\u22a2 R (get [] { val := x\u271d, isLt := (_ : x\u271d < length []) }) (get [] { val := x\u271d + 1, isLt := (_ : succ x\u271d < length []) })\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\n\u22a2 Chain R a (b :: t) \u2194\n (\u2200 (h : 0 < length (b :: t)), R a (get (b :: t) { val := 0, isLt := h })) \u2227\n \u2200 (i : \u2115) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })\n[PROOFSTEP]\nrw [chain_cons, @chain_iff_get _ _ t]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\n\u22a2 (R a b \u2227\n (\u2200 (h : 0 < length t), R b (get t { val := 0, isLt := h })) \u2227\n \u2200 (i : \u2115) (h : i < length t - 1),\n R (get t { val := i, isLt := (_ : i < length t) })\n (get t { val := i + 1, isLt := (_ : succ i < length t) })) \u2194\n (\u2200 (h : 0 < length (b :: t)), R a (get (b :: t) { val := 0, isLt := h })) \u2227\n \u2200 (i : \u2115) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\n\u22a2 (R a b \u2227\n (\u2200 (h : 0 < length t), R b (get t { val := 0, isLt := h })) \u2227\n \u2200 (i : \u2115) (h : i < length t - 1),\n R (get t { val := i, isLt := (_ : i < length t) })\n (get t { val := i + 1, isLt := (_ : succ i < length t) })) \u2192\n (\u2200 (h : 0 < length (b :: t)), R a (get (b :: t) { val := 0, isLt := h })) \u2227\n \u2200 (i : \u2115) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })\n[PROOFSTEP]\nrintro \u27e8R, \u27e8h0, h\u27e9\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d\u00b9 r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\nR : R\u271d a b\nh0 : \u2200 (h : 0 < length t), R\u271d b (get t { val := 0, isLt := h })\nh :\n \u2200 (i : \u2115) (h : i < length t - 1),\n R\u271d (get t { val := i, isLt := (_ : i < length t) }) (get t { val := i + 1, isLt := (_ : succ i < length t) })\n\u22a2 (\u2200 (h : 0 < length (b :: t)), R\u271d a (get (b :: t) { val := 0, isLt := h })) \u2227\n \u2200 (i : \u2115) (h : i < length (b :: t) - 1),\n R\u271d (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.intro.intro.left\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d\u00b9 r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\nR : R\u271d a b\nh0 : \u2200 (h : 0 < length t), R\u271d b (get t { val := 0, isLt := h })\nh :\n \u2200 (i : \u2115) (h : i < length t - 1),\n R\u271d (get t { val := i, isLt := (_ : i < length t) }) (get t { val := i + 1, isLt := (_ : succ i < length t) })\n\u22a2 \u2200 (h : 0 < length (b :: t)), R\u271d a (get (b :: t) { val := 0, isLt := h })\n[PROOFSTEP]\nintro _\n[GOAL]\ncase mp.intro.intro.left\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d\u00b9 r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\nR : R\u271d a b\nh0 : \u2200 (h : 0 < length t), R\u271d b (get t { val := 0, isLt := h })\nh :\n \u2200 (i : \u2115) (h : i < length t - 1),\n R\u271d (get t { val := i, isLt := (_ : i < length t) }) (get t { val := i + 1, isLt := (_ : succ i < length t) })\nh\u271d : 0 < length (b :: t)\n\u22a2 R\u271d a (get (b :: t) { val := 0, isLt := h\u271d })\n[PROOFSTEP]\nexact R\n[GOAL]\ncase mp.intro.intro.right\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d\u00b9 r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\nR : R\u271d a b\nh0 : \u2200 (h : 0 < length t), R\u271d b (get t { val := 0, isLt := h })\nh :\n \u2200 (i : \u2115) (h : i < length t - 1),\n R\u271d (get t { val := i, isLt := (_ : i < length t) }) (get t { val := i + 1, isLt := (_ : succ i < length t) })\n\u22a2 \u2200 (i : \u2115) (h : i < length (b :: t) - 1),\n R\u271d (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })\n[PROOFSTEP]\nintro i w\n[GOAL]\ncase mp.intro.intro.right\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d\u00b9 r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\nR : R\u271d a b\nh0 : \u2200 (h : 0 < length t), R\u271d b (get t { val := 0, isLt := h })\nh :\n \u2200 (i : \u2115) (h : i < length t - 1),\n R\u271d (get t { val := i, isLt := (_ : i < length t) }) (get t { val := i + 1, isLt := (_ : succ i < length t) })\ni : \u2115\nw : i < length (b :: t) - 1\n\u22a2 R\u271d (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })\n[PROOFSTEP]\ncases' i with i\n[GOAL]\ncase mp.intro.intro.right.zero\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d\u00b9 r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\nR : R\u271d a b\nh0 : \u2200 (h : 0 < length t), R\u271d b (get t { val := 0, isLt := h })\nh :\n \u2200 (i : \u2115) (h : i < length t - 1),\n R\u271d (get t { val := i, isLt := (_ : i < length t) }) (get t { val := i + 1, isLt := (_ : succ i < length t) })\nw : zero < length (b :: t) - 1\n\u22a2 R\u271d (get (b :: t) { val := zero, isLt := (_ : zero < length (b :: t)) })\n (get (b :: t) { val := zero + 1, isLt := (_ : succ zero < length (b :: t)) })\n[PROOFSTEP]\napply h0\n[GOAL]\ncase mp.intro.intro.right.succ\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d\u00b9 r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\nR : R\u271d a b\nh0 : \u2200 (h : 0 < length t), R\u271d b (get t { val := 0, isLt := h })\nh :\n \u2200 (i : \u2115) (h : i < length t - 1),\n R\u271d (get t { val := i, isLt := (_ : i < length t) }) (get t { val := i + 1, isLt := (_ : succ i < length t) })\ni : \u2115\nw : succ i < length (b :: t) - 1\n\u22a2 R\u271d (get (b :: t) { val := succ i, isLt := (_ : succ i < length (b :: t)) })\n (get (b :: t) { val := succ i + 1, isLt := (_ : succ (succ i) < length (b :: t)) })\n[PROOFSTEP]\nexact h i (lt_pred_iff.2 <| by simpa using w)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d\u00b9 r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\nR : R\u271d a b\nh0 : \u2200 (h : 0 < length t), R\u271d b (get t { val := 0, isLt := h })\nh :\n \u2200 (i : \u2115) (h : i < length t - 1),\n R\u271d (get t { val := i, isLt := (_ : i < length t) }) (get t { val := i + 1, isLt := (_ : succ i < length t) })\ni : \u2115\nw : succ i < length (b :: t) - 1\n\u22a2 succ i < Nat.sub (length t) 0\n[PROOFSTEP]\nsimpa using w\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\n\u22a2 ((\u2200 (h : 0 < length (b :: t)), R a (get (b :: t) { val := 0, isLt := h })) \u2227\n \u2200 (i : \u2115) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })) \u2192\n R a b \u2227\n (\u2200 (h : 0 < length t), R b (get t { val := 0, isLt := h })) \u2227\n \u2200 (i : \u2115) (h : i < length t - 1),\n R (get t { val := i, isLt := (_ : i < length t) }) (get t { val := i + 1, isLt := (_ : succ i < length t) })\n[PROOFSTEP]\nrintro \u27e8h0, h\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\nh0 : \u2200 (h : 0 < length (b :: t)), R a (get (b :: t) { val := 0, isLt := h })\nh :\n \u2200 (i : \u2115) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })\n\u22a2 R a b \u2227\n (\u2200 (h : 0 < length t), R b (get t { val := 0, isLt := h })) \u2227\n \u2200 (i : \u2115) (h : i < length t - 1),\n R (get t { val := i, isLt := (_ : i < length t) }) (get t { val := i + 1, isLt := (_ : succ i < length t) })\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.intro.left\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\nh0 : \u2200 (h : 0 < length (b :: t)), R a (get (b :: t) { val := 0, isLt := h })\nh :\n \u2200 (i : \u2115) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })\n\u22a2 R a b\n[PROOFSTEP]\napply h0\n[GOAL]\ncase mpr.intro.left\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\nh0 : \u2200 (h : 0 < length (b :: t)), R a (get (b :: t) { val := 0, isLt := h })\nh :\n \u2200 (i : \u2115) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })\n\u22a2 0 < length (b :: t)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr.intro.right\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\nh0 : \u2200 (h : 0 < length (b :: t)), R a (get (b :: t) { val := 0, isLt := h })\nh :\n \u2200 (i : \u2115) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })\n\u22a2 (\u2200 (h : 0 < length t), R b (get t { val := 0, isLt := h })) \u2227\n \u2200 (i : \u2115) (h : i < length t - 1),\n R (get t { val := i, isLt := (_ : i < length t) }) (get t { val := i + 1, isLt := (_ : succ i < length t) })\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.intro.right.left\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\nh0 : \u2200 (h : 0 < length (b :: t)), R a (get (b :: t) { val := 0, isLt := h })\nh :\n \u2200 (i : \u2115) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })\n\u22a2 \u2200 (h : 0 < length t), R b (get t { val := 0, isLt := h })\n[PROOFSTEP]\napply h 0\n[GOAL]\ncase mpr.intro.right.right\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\nh0 : \u2200 (h : 0 < length (b :: t)), R a (get (b :: t) { val := 0, isLt := h })\nh :\n \u2200 (i : \u2115) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })\n\u22a2 \u2200 (i : \u2115) (h : i < length t - 1),\n R (get t { val := i, isLt := (_ : i < length t) }) (get t { val := i + 1, isLt := (_ : succ i < length t) })\n[PROOFSTEP]\nintro i w\n[GOAL]\ncase mpr.intro.right.right\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\nh0 : \u2200 (h : 0 < length (b :: t)), R a (get (b :: t) { val := 0, isLt := h })\nh :\n \u2200 (i : \u2115) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })\ni : \u2115\nw : i < length t - 1\n\u22a2 R (get t { val := i, isLt := (_ : i < length t) }) (get t { val := i + 1, isLt := (_ : succ i < length t) })\n[PROOFSTEP]\nexact h (i + 1) (lt_pred_iff.mp w)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\n\u22a2 Chain R a l \u2194\n (\u2200 (h : 0 < length l), R a (nthLe l 0 h)) \u2227\n \u2200 (i : \u2115) (h : i < length l - 1), R (nthLe l i (_ : i < length l)) (nthLe l (i + 1) (_ : succ i < length l))\n[PROOFSTEP]\nrw [chain_iff_get]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\n\u22a2 ((\u2200 (h : 0 < length l), R a (get l { val := 0, isLt := h })) \u2227\n \u2200 (i : \u2115) (h : i < length l - 1),\n R (get l { val := i, isLt := (_ : i < length l) }) (get l { val := i + 1, isLt := (_ : succ i < length l) })) \u2194\n (\u2200 (h : 0 < length l), R a (nthLe l 0 h)) \u2227\n \u2200 (i : \u2115) (h : i < length l - 1), R (nthLe l i (_ : i < length l)) (nthLe l (i + 1) (_ : succ i < length l))\n[PROOFSTEP]\nsimp [nthLe]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nS : \u03b1 \u2192 \u03b1 \u2192 Prop\nH : \u2200 (a b : \u03b1), R a b \u2192 S a b\nl : List \u03b1\np : Chain' R l\n\u22a2 Chain' S l\n[PROOFSTEP]\ncases l <;> [trivial; exact Chain.imp H p]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nS : \u03b1 \u2192 \u03b1 \u2192 Prop\nH : \u2200 (a b : \u03b1), R a b \u2192 S a b\nl : List \u03b1\np : Chain' R l\n\u22a2 Chain' S l\n[PROOFSTEP]\ncases l\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nS : \u03b1 \u2192 \u03b1 \u2192 Prop\nH : \u2200 (a b : \u03b1), R a b \u2192 S a b\np : Chain' R []\n\u22a2 Chain' S []\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nS : \u03b1 \u2192 \u03b1 \u2192 Prop\nH : \u2200 (a b : \u03b1), R a b \u2192 S a b\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\np : Chain' R (head\u271d :: tail\u271d)\n\u22a2 Chain' S (head\u271d :: tail\u271d)\n[PROOFSTEP]\nexact Chain.imp H p\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d a b : \u03b1\nl : List \u03b1\n\u22a2 [] ++ [a, b] ++ l = a :: b :: l\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d a b : \u03b1\nl : List \u03b1\nx y : \u03b1\nh : [x, y] <:+: b :: l\n\u22a2 [a] ++ b :: l ++ [] = a :: b :: l\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081\u271d l\u2082\u271d : List \u03b1\na b\u271d b c : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 Chain' R (l\u2081 ++ b :: c :: l\u2082) \u2194 Chain' R (l\u2081 ++ [b]) \u2227 R b c \u2227 Chain' R (c :: l\u2082)\n[PROOFSTEP]\nrw [chain'_split, chain'_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\nl : List \u03b2\n\u22a2 Chain' R (map f l) \u2194 Chain' (fun a b => R (f a) (f b)) l\n[PROOFSTEP]\ncases l <;> [rfl; exact chain_map _]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\nl : List \u03b2\n\u22a2 Chain' R (map f l) \u2194 Chain' (fun a b => R (f a) (f b)) l\n[PROOFSTEP]\ncases l\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\n\u22a2 Chain' R (map f []) \u2194 Chain' (fun a b => R (f a) (f b)) []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\n\u22a2 Chain' R (map f (head\u271d :: tail\u271d)) \u2194 Chain' (fun a b => R (f a) (f b)) (head\u271d :: tail\u271d)\n[PROOFSTEP]\nexact chain_map _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\ninst\u271d : IsTrans \u03b1 R\nhl : Chain' R l\u2082\nh : l\u2081 <+ l\u2082\n\u22a2 Chain' R l\u2081\n[PROOFSTEP]\nrw [chain'_iff_pairwise] at hl \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\ninst\u271d : IsTrans \u03b1 R\nhl : Pairwise R l\u2082\nh : l\u2081 <+ l\u2082\n\u22a2 Pairwise R l\u2081\n[PROOFSTEP]\nexact hl.sublist h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na b x : \u03b1\nl : List \u03b1\nh : Chain' R (x :: l)\ny : \u03b1\nhy : y \u2208 head? l\n\u22a2 R x y\n[PROOFSTEP]\nrw [\u2190 cons_head?_tail hy] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na b x : \u03b1\nl : List \u03b1\ny : \u03b1\nh : Chain' R (x :: y :: List.tail l)\nhy : y \u2208 head? l\n\u22a2 R x y\n[PROOFSTEP]\nexact h.rel_head\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 Chain' R ([] ++ l) \u2194 Chain' R [] \u2227 Chain' R l \u2227 \u2200 (x : \u03b1), x \u2208 getLast? [] \u2192 \u2200 (y : \u03b1), y \u2208 head? l \u2192 R x y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b a : \u03b1\nl : List \u03b1\n\u22a2 Chain' R ([a] ++ l) \u2194 Chain' R [a] \u2227 Chain' R l \u2227 \u2200 (x : \u03b1), x \u2208 getLast? [a] \u2192 \u2200 (y : \u03b1), y \u2208 head? l \u2192 R x y\n[PROOFSTEP]\nsimp [chain'_cons', and_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d b\u271d a b : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 Chain' R (a :: b :: l\u2081 ++ l\u2082) \u2194\n Chain' R (a :: b :: l\u2081) \u2227 Chain' R l\u2082 \u2227 \u2200 (x : \u03b1), x \u2208 getLast? (a :: b :: l\u2081) \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y\n[PROOFSTEP]\nrw [cons_append, cons_append, chain'_cons, chain'_cons, \u2190 cons_append, chain'_append, and_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d b\u271d a b : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 (R a b \u2227 Chain' R (b :: l\u2081) \u2227 Chain' R l\u2082 \u2227 \u2200 (x : \u03b1), x \u2208 getLast? (b :: l\u2081) \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y) \u2194\n R a b \u2227 Chain' R (b :: l\u2081) \u2227 Chain' R l\u2082 \u2227 \u2200 (x : \u03b1), x \u2208 getLast? (a :: b :: l\u2081) \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nh : Chain' R l\nh' : l\u2081 <:+: l\n\u22a2 Chain' R l\u2081\n[PROOFSTEP]\nrcases h' with \u27e8l\u2082, l\u2083, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u2081 l\u2082\u271d : List \u03b1\na b : \u03b1\nl\u2082 l\u2083 : List \u03b1\nh : Chain' R (l\u2082 ++ l\u2081 ++ l\u2083)\n\u22a2 Chain' R l\u2081\n[PROOFSTEP]\nexact h.left_of_append.right_of_append\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b x y : \u03b1\n\u22a2 Chain' R [x, y] \u2194 R x y\n[PROOFSTEP]\nsimp only [chain'_singleton, chain'_cons, and_true_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b a : \u03b1\n\u22a2 Chain' R (reverse [a]) \u2194 Chain' (flip R) [a]\n[PROOFSTEP]\nsimp only [chain'_singleton, reverse_singleton]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d a b : \u03b1\nl : List \u03b1\n\u22a2 Chain' R (reverse (a :: b :: l)) \u2194 Chain' (flip R) (a :: b :: l)\n[PROOFSTEP]\nrw [chain'_cons, reverse_cons, reverse_cons, append_assoc, cons_append, nil_append, chain'_split, \u2190 reverse_cons,\n @chain'_reverse (b :: l), and_comm, chain'_pair, flip]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 Chain' R []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nx\u271d : \u2115\nh : x\u271d < length [] - 1\n\u22a2 R (get [] { val := x\u271d, isLt := (_ : x\u271d < length []) }) (get [] { val := x\u271d + 1, isLt := (_ : succ x\u271d < length []) })\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\n\u22a2 Chain' R [a]\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nx\u271d : \u2115\nh : x\u271d < length [a] - 1\n\u22a2 R (get [a] { val := x\u271d, isLt := (_ : x\u271d < length [a]) })\n (get [a] { val := x\u271d + 1, isLt := (_ : succ x\u271d < length [a]) })\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\n\u22a2 Chain' R (a :: b :: t) \u2194\n \u2200 (i : \u2115) (h : i < length (a :: b :: t) - 1),\n R (get (a :: b :: t) { val := i, isLt := (_ : i < length (a :: b :: t)) })\n (get (a :: b :: t) { val := i + 1, isLt := (_ : succ i < length (a :: b :: t)) })\n[PROOFSTEP]\nrw [\u2190 and_forall_succ, chain'_cons, chain'_iff_get]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\n\u22a2 (R a b \u2227\n \u2200 (i : \u2115) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })) \u2194\n (\u2200 (h : 0 < length (a :: b :: t) - 1),\n R (get (a :: b :: t) { val := 0, isLt := (_ : 0 < length (a :: b :: t)) })\n (get (a :: b :: t) { val := 0 + 1, isLt := (_ : succ 0 < length (a :: b :: t)) })) \u2227\n \u2200 (n : \u2115) (h : n + 1 < length (a :: b :: t) - 1),\n R (get (a :: b :: t) { val := n + 1, isLt := (_ : n + 1 < length (a :: b :: t)) })\n (get (a :: b :: t) { val := n + 1 + 1, isLt := (_ : succ (n + 1) < length (a :: b :: t)) })\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\n\u22a2 R a b \u2192\n ((\u2200 (i : \u2115) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get t { val := i, isLt := (_ : i < length t) })) \u2194\n \u2200 (n : \u2115) (h : n + 1 < length (a :: b :: t) - 1),\n R (get (b :: t) { val := n, isLt := (_ : n < length (b :: t)) })\n (get t { val := n, isLt := (_ : n < length t) }))\n[PROOFSTEP]\ndsimp [succ_sub_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na\u271d b\u271d : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nt : List \u03b1\n\u22a2 R a b \u2192\n ((\u2200 (i : \u2115) (h : i < length t),\n R (get (b :: t) { val := i, isLt := (_ : i < succ (length t)) })\n (get t { val := i, isLt := (_ : succ i \u2264 length t) })) \u2194\n \u2200 (n : \u2115) (h : n + 1 < length t + 1),\n R (get (b :: t) { val := n, isLt := (_ : succ n \u2264 length (b :: t)) })\n (get t { val := n, isLt := (_ : succ n \u2264 length t) }))\n[PROOFSTEP]\nexact fun _ => \u27e8fun h i hi => h i (Nat.lt_of_succ_lt_succ hi), fun h i hi => h i (Nat.succ_lt_succ hi)\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR\u271d r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nl : List \u03b1\n\u22a2 (\u2200 (i : \u2115) (h : i < length l - 1),\n R (get l { val := i, isLt := (_ : i < length l) }) (get l { val := i + 1, isLt := (_ : succ i < length l) })) \u2194\n \u2200 (i : \u2115) (h : i < length l - 1), R (nthLe l i (_ : i < length l)) (nthLe l (i + 1) (_ : succ i < length l))\n[PROOFSTEP]\nsimp [nthLe]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081\u271d l\u2082\u271d : List \u03b1\na b : \u03b1\nl\u2081 l\u2082 l\u2083 : List \u03b1\nh\u2081 : Chain' R (l\u2081 ++ l\u2082)\nh\u2082 : Chain' R (l\u2082 ++ l\u2083)\nhn : l\u2082 \u2260 []\n\u22a2 \u2200 (x : \u03b1), x \u2208 getLast? (l\u2081 ++ l\u2082) \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2083 \u2192 R x y\n[PROOFSTEP]\nsimpa only [getLast?_append_of_ne_nil _ hn] using (chain'_append.1 h\u2082).2.2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nx\u271d : \u00ac[] \u2208 []\n\u22a2 Chain' R (join []) \u2194\n (\u2200 (l : List \u03b1), l \u2208 [] \u2192 Chain' R l) \u2227\n Chain' (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y) []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\nx\u271d : \u00ac[] \u2208 [l]\n\u22a2 Chain' R (join [l]) \u2194\n (\u2200 (l_1 : List \u03b1), l_1 \u2208 [l] \u2192 Chain' R l_1) \u2227\n Chain' (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y) [l]\n[PROOFSTEP]\nsimp [join]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081\u271d l\u2082\u271d : List \u03b1\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\nL : List (List \u03b1)\nhL : \u00ac[] \u2208 l\u2081 :: l\u2082 :: L\n\u22a2 Chain' R (join (l\u2081 :: l\u2082 :: L)) \u2194\n (\u2200 (l : List \u03b1), l \u2208 l\u2081 :: l\u2082 :: L \u2192 Chain' R l) \u2227\n Chain' (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y) (l\u2081 :: l\u2082 :: L)\n[PROOFSTEP]\nrw [mem_cons, not_or, \u2190 Ne.def] at hL \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081\u271d l\u2082\u271d : List \u03b1\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\nL : List (List \u03b1)\nhL : [] \u2260 l\u2081 \u2227 \u00ac[] \u2208 l\u2082 :: L\n\u22a2 Chain' R (join (l\u2081 :: l\u2082 :: L)) \u2194\n (\u2200 (l : List \u03b1), l \u2208 l\u2081 :: l\u2082 :: L \u2192 Chain' R l) \u2227\n Chain' (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y) (l\u2081 :: l\u2082 :: L)\n[PROOFSTEP]\nrw [join, chain'_append, chain'_join hL.2, forall_mem_cons, chain'_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081\u271d l\u2082\u271d : List \u03b1\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\nL : List (List \u03b1)\nhL : [] \u2260 l\u2081 \u2227 \u00ac[] \u2208 l\u2082 :: L\n\u22a2 (Chain' R l\u2081 \u2227\n ((Chain' R l\u2082 \u2227 \u2200 (x : List \u03b1), x \u2208 L \u2192 Chain' R x) \u2227\n Chain' (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y) (l\u2082 :: L)) \u2227\n \u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? (join (l\u2082 :: L)) \u2192 R x y) \u2194\n (\u2200 (l : List \u03b1), l \u2208 l\u2081 :: l\u2082 :: L \u2192 Chain' R l) \u2227\n (\u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y) \u2227\n Chain' (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y) (l\u2082 :: L)\n[PROOFSTEP]\nrw [mem_cons, not_or, \u2190 Ne.def] at hL \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081\u271d l\u2082\u271d : List \u03b1\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\nL : List (List \u03b1)\nhL : [] \u2260 l\u2081 \u2227 [] \u2260 l\u2082 \u2227 \u00ac[] \u2208 L\n\u22a2 (Chain' R l\u2081 \u2227\n ((Chain' R l\u2082 \u2227 \u2200 (x : List \u03b1), x \u2208 L \u2192 Chain' R x) \u2227\n Chain' (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y) (l\u2082 :: L)) \u2227\n \u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? (join (l\u2082 :: L)) \u2192 R x y) \u2194\n (\u2200 (l : List \u03b1), l \u2208 l\u2081 :: l\u2082 :: L \u2192 Chain' R l) \u2227\n (\u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y) \u2227\n Chain' (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y) (l\u2082 :: L)\n[PROOFSTEP]\nsimp only [forall_mem_cons, and_assoc, join, head?_append_of_ne_nil _ hL.2.1.symm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081\u271d l\u2082\u271d : List \u03b1\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\nL : List (List \u03b1)\nhL : [] \u2260 l\u2081 \u2227 [] \u2260 l\u2082 \u2227 \u00ac[] \u2208 L\n\u22a2 (Chain' R l\u2081 \u2227\n Chain' R l\u2082 \u2227\n (\u2200 (x : List \u03b1), x \u2208 L \u2192 Chain' R x) \u2227\n Chain' (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y) (l\u2082 :: L) \u2227\n \u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y) \u2194\n Chain' R l\u2081 \u2227\n Chain' R l\u2082 \u2227\n (\u2200 (x : List \u03b1), x \u2208 L \u2192 Chain' R x) \u2227\n (\u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y) \u2227\n Chain' (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 getLast? l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 head? l\u2082 \u2192 R x y) (l\u2082 :: L)\n[PROOFSTEP]\nexact Iff.rfl.and (Iff.rfl.and $ Iff.rfl.and and_comm)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nh : Relation.ReflTransGen r a b\n\u22a2 \u2203 l, Chain r a l \u2227 getLast (a :: l) (_ : a :: l \u2260 []) = b\n[PROOFSTEP]\nrefine' Relation.ReflTransGen.head_induction_on h _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nh : Relation.ReflTransGen r a b\n\u22a2 \u2203 l, Chain r b l \u2227 getLast (b :: l) (_ : b :: l \u2260 []) = b\n[PROOFSTEP]\nexact \u27e8[], Chain.nil, rfl\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nh : Relation.ReflTransGen r a b\n\u22a2 \u2200 {a c : \u03b1},\n r a c \u2192\n Relation.ReflTransGen r c b \u2192\n (\u2203 l, Chain r c l \u2227 getLast (c :: l) (_ : c :: l \u2260 []) = b) \u2192\n \u2203 l, Chain r a l \u2227 getLast (a :: l) (_ : a :: l \u2260 []) = b\n[PROOFSTEP]\nintro c d e _ ih\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nh : Relation.ReflTransGen r a b\nc d : \u03b1\ne : r c d\nh\u271d : Relation.ReflTransGen r d b\nih : \u2203 l, Chain r d l \u2227 getLast (d :: l) (_ : d :: l \u2260 []) = b\n\u22a2 \u2203 l, Chain r c l \u2227 getLast (c :: l) (_ : c :: l \u2260 []) = b\n[PROOFSTEP]\nobtain \u27e8l, hl\u2081, hl\u2082\u27e9 := ih\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nh : Relation.ReflTransGen r a b\nc d : \u03b1\ne : r c d\nh\u271d : Relation.ReflTransGen r d b\nl : List \u03b1\nhl\u2081 : Chain r d l\nhl\u2082 : getLast (d :: l) (_ : d :: l \u2260 []) = b\n\u22a2 \u2203 l, Chain r c l \u2227 getLast (c :: l) (_ : c :: l \u2260 []) = b\n[PROOFSTEP]\nrefine' \u27e8d :: l, Chain.cons e hl\u2081, _\u27e9\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nh : Relation.ReflTransGen r a b\nc d : \u03b1\ne : r c d\nh\u271d : Relation.ReflTransGen r d b\nl : List \u03b1\nhl\u2081 : Chain r d l\nhl\u2082 : getLast (d :: l) (_ : d :: l \u2260 []) = b\n\u22a2 getLast (c :: d :: l) (_ : c :: d :: l \u2260 []) = b\n[PROOFSTEP]\nrwa [getLast_cons_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na b : \u03b1\np : \u03b1 \u2192 Prop\nl : List \u03b1\nh : Chain r a l\nhb : getLast (a :: l) (_ : a :: l \u2260 []) = b\ncarries : \u2200 \u2983x y : \u03b1\u2984, r x y \u2192 p y \u2192 p x\nfinal : p b\n\u22a2 \u2200 (i : \u03b1), i \u2208 a :: l \u2192 p i\n[PROOFSTEP]\ninduction' l with _ _ l_ih generalizing a\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b : \u03b1\np : \u03b1 \u2192 Prop\nl : List \u03b1\nh\u271d : Chain r a\u271d l\nhb\u271d : getLast (a\u271d :: l) (_ : a\u271d :: l \u2260 []) = b\ncarries : \u2200 \u2983x y : \u03b1\u2984, r x y \u2192 p y \u2192 p x\nfinal : p b\na : \u03b1\nh : Chain r a []\nhb : getLast [a] (_ : [a] \u2260 []) = b\n\u22a2 \u2200 (i : \u03b1), i \u2208 [a] \u2192 p i\n[PROOFSTEP]\ncases hb\n[GOAL]\ncase nil.refl\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d : \u03b1\np : \u03b1 \u2192 Prop\nl : List \u03b1\nh\u271d : Chain r a\u271d l\ncarries : \u2200 \u2983x y : \u03b1\u2984, r x y \u2192 p y \u2192 p x\na : \u03b1\nh : Chain r a []\nhb : getLast (a\u271d :: l) (_ : a\u271d :: l \u2260 []) = getLast [a] (_ : [a] \u2260 [])\nfinal : p (getLast [a] (_ : [a] \u2260 []))\n\u22a2 \u2200 (i : \u03b1), i \u2208 [a] \u2192 p i\n[PROOFSTEP]\nsimpa using final\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b : \u03b1\np : \u03b1 \u2192 Prop\nl : List \u03b1\nh\u271d : Chain r a\u271d l\nhb\u271d : getLast (a\u271d :: l) (_ : a\u271d :: l \u2260 []) = b\ncarries : \u2200 \u2983x y : \u03b1\u2984, r x y \u2192 p y \u2192 p x\nfinal : p b\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nl_ih : \u2200 {a : \u03b1}, Chain r a tail\u271d \u2192 getLast (a :: tail\u271d) (_ : a :: tail\u271d \u2260 []) = b \u2192 \u2200 (i : \u03b1), i \u2208 a :: tail\u271d \u2192 p i\na : \u03b1\nh : Chain r a (head\u271d :: tail\u271d)\nhb : getLast (a :: head\u271d :: tail\u271d) (_ : a :: head\u271d :: tail\u271d \u2260 []) = b\n\u22a2 \u2200 (i : \u03b1), i \u2208 a :: head\u271d :: tail\u271d \u2192 p i\n[PROOFSTEP]\nrw [chain_cons] at h \n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b : \u03b1\np : \u03b1 \u2192 Prop\nl : List \u03b1\nh\u271d : Chain r a\u271d l\nhb\u271d : getLast (a\u271d :: l) (_ : a\u271d :: l \u2260 []) = b\ncarries : \u2200 \u2983x y : \u03b1\u2984, r x y \u2192 p y \u2192 p x\nfinal : p b\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nl_ih : \u2200 {a : \u03b1}, Chain r a tail\u271d \u2192 getLast (a :: tail\u271d) (_ : a :: tail\u271d \u2260 []) = b \u2192 \u2200 (i : \u03b1), i \u2208 a :: tail\u271d \u2192 p i\na : \u03b1\nh : r a head\u271d \u2227 Chain r head\u271d tail\u271d\nhb : getLast (a :: head\u271d :: tail\u271d) (_ : a :: head\u271d :: tail\u271d \u2260 []) = b\n\u22a2 \u2200 (i : \u03b1), i \u2208 a :: head\u271d :: tail\u271d \u2192 p i\n[PROOFSTEP]\nsimp only [mem_cons]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b : \u03b1\np : \u03b1 \u2192 Prop\nl : List \u03b1\nh\u271d : Chain r a\u271d l\nhb\u271d : getLast (a\u271d :: l) (_ : a\u271d :: l \u2260 []) = b\ncarries : \u2200 \u2983x y : \u03b1\u2984, r x y \u2192 p y \u2192 p x\nfinal : p b\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nl_ih : \u2200 {a : \u03b1}, Chain r a tail\u271d \u2192 getLast (a :: tail\u271d) (_ : a :: tail\u271d \u2260 []) = b \u2192 \u2200 (i : \u03b1), i \u2208 a :: tail\u271d \u2192 p i\na : \u03b1\nh : r a head\u271d \u2227 Chain r head\u271d tail\u271d\nhb : getLast (a :: head\u271d :: tail\u271d) (_ : a :: head\u271d :: tail\u271d \u2260 []) = b\n\u22a2 \u2200 (i : \u03b1), i = a \u2228 i = head\u271d \u2228 i \u2208 tail\u271d \u2192 p i\n[PROOFSTEP]\nrintro _ (rfl | H)\n[GOAL]\ncase cons.inl\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na b : \u03b1\np : \u03b1 \u2192 Prop\nl : List \u03b1\nh\u271d : Chain r a l\nhb\u271d : getLast (a :: l) (_ : a :: l \u2260 []) = b\ncarries : \u2200 \u2983x y : \u03b1\u2984, r x y \u2192 p y \u2192 p x\nfinal : p b\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nl_ih : \u2200 {a : \u03b1}, Chain r a tail\u271d \u2192 getLast (a :: tail\u271d) (_ : a :: tail\u271d \u2260 []) = b \u2192 \u2200 (i : \u03b1), i \u2208 a :: tail\u271d \u2192 p i\ni\u271d : \u03b1\nh : r i\u271d head\u271d \u2227 Chain r head\u271d tail\u271d\nhb : getLast (i\u271d :: head\u271d :: tail\u271d) (_ : i\u271d :: head\u271d :: tail\u271d \u2260 []) = b\n\u22a2 p i\u271d\ncase cons.inr\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b : \u03b1\np : \u03b1 \u2192 Prop\nl : List \u03b1\nh\u271d : Chain r a\u271d l\nhb\u271d : getLast (a\u271d :: l) (_ : a\u271d :: l \u2260 []) = b\ncarries : \u2200 \u2983x y : \u03b1\u2984, r x y \u2192 p y \u2192 p x\nfinal : p b\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nl_ih : \u2200 {a : \u03b1}, Chain r a tail\u271d \u2192 getLast (a :: tail\u271d) (_ : a :: tail\u271d \u2260 []) = b \u2192 \u2200 (i : \u03b1), i \u2208 a :: tail\u271d \u2192 p i\na : \u03b1\nh : r a head\u271d \u2227 Chain r head\u271d tail\u271d\nhb : getLast (a :: head\u271d :: tail\u271d) (_ : a :: head\u271d :: tail\u271d \u2260 []) = b\ni\u271d : \u03b1\nH : i\u271d = head\u271d \u2228 i\u271d \u2208 tail\u271d\n\u22a2 p i\u271d\n[PROOFSTEP]\napply carries h.1 (l_ih h.2 hb _ (mem_cons.2 (Or.inl rfl)))\n[GOAL]\ncase cons.inr\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d b : \u03b1\np : \u03b1 \u2192 Prop\nl : List \u03b1\nh\u271d : Chain r a\u271d l\nhb\u271d : getLast (a\u271d :: l) (_ : a\u271d :: l \u2260 []) = b\ncarries : \u2200 \u2983x y : \u03b1\u2984, r x y \u2192 p y \u2192 p x\nfinal : p b\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nl_ih : \u2200 {a : \u03b1}, Chain r a tail\u271d \u2192 getLast (a :: tail\u271d) (_ : a :: tail\u271d \u2260 []) = b \u2192 \u2200 (i : \u03b1), i \u2208 a :: tail\u271d \u2192 p i\na : \u03b1\nh : r a head\u271d \u2227 Chain r head\u271d tail\u271d\nhb : getLast (a :: head\u271d :: tail\u271d) (_ : a :: head\u271d :: tail\u271d \u2260 []) = b\ni\u271d : \u03b1\nH : i\u271d = head\u271d \u2228 i\u271d \u2208 tail\u271d\n\u22a2 p i\u271d\n[PROOFSTEP]\napply l_ih h.2 hb _ (mem_cons.2 H)\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nl : List.chains r\nacc : \u2200 (a : \u03b1), a \u2208 List.head? \u2191l \u2192 Acc r a\n\u22a2 Acc (List.lex_chains r) l\n[PROOFSTEP]\nobtain \u27e8_ | \u27e8a, l\u27e9, hl\u27e9 := l\n[GOAL]\ncase mk.nil\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhl : List.Chain' (flip r) []\nacc : \u2200 (a : \u03b1), a \u2208 List.head? \u2191{ val := [], property := hl } \u2192 Acc r a\n\u22a2 Acc (List.lex_chains r) { val := [], property := hl }\n[PROOFSTEP]\napply Acc.intro\n[GOAL]\ncase mk.nil.h\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhl : List.Chain' (flip r) []\nacc : \u2200 (a : \u03b1), a \u2208 List.head? \u2191{ val := [], property := hl } \u2192 Acc r a\n\u22a2 \u2200 (y : List.chains r), List.lex_chains r y { val := [], property := hl } \u2192 Acc (List.lex_chains r) y\n[PROOFSTEP]\nrintro \u27e8_\u27e9 \u27e8_\u27e9\n[GOAL]\ncase mk.cons\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nhl : List.Chain' (flip r) (a :: l)\nacc : \u2200 (a_1 : \u03b1), a_1 \u2208 List.head? \u2191{ val := a :: l, property := hl } \u2192 Acc r a_1\n\u22a2 Acc (List.lex_chains r) { val := a :: l, property := hl }\n[PROOFSTEP]\nspecialize acc a _\n[GOAL]\ncase mk.cons\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nhl : List.Chain' (flip r) (a :: l)\nacc : \u2200 (a_1 : \u03b1), a_1 \u2208 List.head? \u2191{ val := a :: l, property := hl } \u2192 Acc r a_1\n\u22a2 a \u2208 List.head? \u2191{ val := a :: l, property := hl }\n[PROOFSTEP]\nrw [List.head?_cons, Option.mem_some_iff]\n /- For an r-decreasing chain of the form a :: l, apply induction on a -/\n[GOAL]\ncase mk.cons\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nhl : List.Chain' (flip r) (a :: l)\nacc : Acc r a\n\u22a2 Acc (List.lex_chains r) { val := a :: l, property := hl }\n[PROOFSTEP]\ninduction acc generalizing l with\n| intro a _ ih =>\n /- Bundle l with a proof that it is r-decreasing to form l' -/\n have hl' := (List.chain'_cons'.1 hl).2\n let l' : List.chains r := \u27e8l, hl'\u27e9\n have : Acc (List.lex_chains r) l'\n \u00b7 cases' l with b l\n \u00b7 apply Acc.intro;\n rintro \u27e8_\u27e9\n \u27e8_\u27e9\n /- l' is accessible by induction hypothesis -/\n \u00b7\n apply\n ih b\n (List.chain'_cons.1 hl).1\n /- make l' a free variable and induct on l' -/\n revert hl\n rw [(by rfl : l = l'.1)]\n clear_value l'\n induction this with\n | intro l _ ihl =>\n intro hl\n apply Acc.intro\n rintro \u27e8_ | \u27e8b, m\u27e9, hm\u27e9 (_ | hr | hr)\n \u00b7 apply Acc.intro; rintro \u27e8_\u27e9 \u27e8_\u27e9\n \u00b7 apply ihl \u27e8m, (List.chain'_cons'.1 hm).2\u27e9 hr\n \u00b7 apply ih b hr\n[GOAL]\ncase mk.cons\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nhl : List.Chain' (flip r) (a :: l)\nacc : Acc r a\n\u22a2 Acc (List.lex_chains r) { val := a :: l, property := hl }\n[PROOFSTEP]\ninduction acc generalizing l with\n| intro a _ ih =>\n /- Bundle l with a proof that it is r-decreasing to form l' -/\n have hl' := (List.chain'_cons'.1 hl).2\n let l' : List.chains r := \u27e8l, hl'\u27e9\n have : Acc (List.lex_chains r) l'\n \u00b7 cases' l with b l\n \u00b7 apply Acc.intro;\n rintro \u27e8_\u27e9\n \u27e8_\u27e9\n /- l' is accessible by induction hypothesis -/\n \u00b7\n apply\n ih b\n (List.chain'_cons.1 hl).1\n /- make l' a free variable and induct on l' -/\n revert hl\n rw [(by rfl : l = l'.1)]\n clear_value l'\n induction this with\n | intro l _ ihl =>\n intro hl\n apply Acc.intro\n rintro \u27e8_ | \u27e8b, m\u27e9, hm\u27e9 (_ | hr | hr)\n \u00b7 apply Acc.intro; rintro \u27e8_\u27e9 \u27e8_\u27e9\n \u00b7 apply ihl \u27e8m, (List.chain'_cons'.1 hm).2\u27e9 hr\n \u00b7 apply ih b hr\n[GOAL]\ncase mk.cons.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl : List \u03b1\nhl : List.Chain' (flip r) (a :: l)\n\u22a2 Acc (List.lex_chains r) { val := a :: l, property := hl }\n[PROOFSTEP]\n\n| intro a _ ih =>\n /- Bundle l with a proof that it is r-decreasing to form l' -/\n have hl' := (List.chain'_cons'.1 hl).2\n let l' : List.chains r := \u27e8l, hl'\u27e9\n have : Acc (List.lex_chains r) l'\n \u00b7 cases' l with b l\n \u00b7 apply Acc.intro;\n rintro \u27e8_\u27e9\n \u27e8_\u27e9\n /- l' is accessible by induction hypothesis -/\n \u00b7\n apply\n ih b\n (List.chain'_cons.1 hl).1\n /- make l' a free variable and induct on l' -/\n revert hl\n rw [(by rfl : l = l'.1)]\n clear_value l'\n induction this with\n | intro l _ ihl =>\n intro hl\n apply Acc.intro\n rintro \u27e8_ | \u27e8b, m\u27e9, hm\u27e9 (_ | hr | hr)\n \u00b7 apply Acc.intro; rintro \u27e8_\u27e9 \u27e8_\u27e9\n \u00b7 apply ihl \u27e8m, (List.chain'_cons'.1 hm).2\u27e9 hr\n \u00b7 apply ih b hr\n[GOAL]\ncase mk.cons.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl : List \u03b1\nhl : List.Chain' (flip r) (a :: l)\n\u22a2 Acc (List.lex_chains r) { val := a :: l, property := hl }\n[PROOFSTEP]\nhave hl' := (List.chain'_cons'.1 hl).2\n[GOAL]\ncase mk.cons.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl : List \u03b1\nhl : List.Chain' (flip r) (a :: l)\nhl' : List.Chain' (flip r) l\n\u22a2 Acc (List.lex_chains r) { val := a :: l, property := hl }\n[PROOFSTEP]\nlet l' : List.chains r := \u27e8l, hl'\u27e9\n[GOAL]\ncase mk.cons.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl : List \u03b1\nhl : List.Chain' (flip r) (a :: l)\nhl' : List.Chain' (flip r) l\nl' : List.chains r := { val := l, property := hl' }\n\u22a2 Acc (List.lex_chains r) { val := a :: l, property := hl }\n[PROOFSTEP]\nhave : Acc (List.lex_chains r) l'\n[GOAL]\ncase this\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl : List \u03b1\nhl : List.Chain' (flip r) (a :: l)\nhl' : List.Chain' (flip r) l\nl' : List.chains r := { val := l, property := hl' }\n\u22a2 Acc (List.lex_chains r) l'\n[PROOFSTEP]\ncases' l with b l\n[GOAL]\ncase this.nil\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nhl : List.Chain' (flip r) [a]\nhl' : List.Chain' (flip r) []\nl' : List.chains r := { val := [], property := hl' }\n\u22a2 Acc (List.lex_chains r) l'\n[PROOFSTEP]\napply Acc.intro\n[GOAL]\ncase this.nil.h\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nhl : List.Chain' (flip r) [a]\nhl' : List.Chain' (flip r) []\nl' : List.chains r := { val := [], property := hl' }\n\u22a2 \u2200 (y : List.chains r), List.lex_chains r y l' \u2192 Acc (List.lex_chains r) y\n[PROOFSTEP]\nrintro \u27e8_\u27e9\n \u27e8_\u27e9\n /- l' is accessible by induction hypothesis -/\n[GOAL]\ncase this.cons\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nb : \u03b1\nl : List \u03b1\nhl : List.Chain' (flip r) (a :: b :: l)\nhl' : List.Chain' (flip r) (b :: l)\nl' : List.chains r := { val := b :: l, property := hl' }\n\u22a2 Acc (List.lex_chains r) l'\n[PROOFSTEP]\napply\n ih b\n (List.chain'_cons.1 hl).1\n /- make l' a free variable and induct on l' -/\n[GOAL]\ncase mk.cons.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl : List \u03b1\nhl : List.Chain' (flip r) (a :: l)\nhl' : List.Chain' (flip r) l\nl' : List.chains r := { val := l, property := hl' }\nthis : Acc (List.lex_chains r) l'\n\u22a2 Acc (List.lex_chains r) { val := a :: l, property := hl }\n[PROOFSTEP]\nrevert hl\n[GOAL]\ncase mk.cons.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl : List \u03b1\nhl' : List.Chain' (flip r) l\nl' : List.chains r := { val := l, property := hl' }\nthis : Acc (List.lex_chains r) l'\n\u22a2 \u2200 (hl : List.Chain' (flip r) (a :: l)), Acc (List.lex_chains r) { val := a :: l, property := hl }\n[PROOFSTEP]\nrw [(by rfl : l = l'.1)]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl : List \u03b1\nhl' : List.Chain' (flip r) l\nl' : List.chains r := { val := l, property := hl' }\nthis : Acc (List.lex_chains r) l'\n\u22a2 l = \u2191l'\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.cons.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl : List \u03b1\nhl' : List.Chain' (flip r) l\nl' : List.chains r := { val := l, property := hl' }\nthis : Acc (List.lex_chains r) l'\n\u22a2 \u2200 (hl : List.Chain' (flip r) (a :: \u2191l')), Acc (List.lex_chains r) { val := a :: \u2191l', property := hl }\n[PROOFSTEP]\nclear_value l'\n[GOAL]\ncase mk.cons.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl : List \u03b1\nhl' : List.Chain' (flip r) l\nl' : List.chains r\nthis : Acc (List.lex_chains r) l'\n\u22a2 \u2200 (hl : List.Chain' (flip r) (a :: \u2191l')), Acc (List.lex_chains r) { val := a :: \u2191l', property := hl }\n[PROOFSTEP]\ninduction this with\n| intro l _ ihl =>\n intro hl\n apply Acc.intro\n rintro \u27e8_ | \u27e8b, m\u27e9, hm\u27e9 (_ | hr | hr)\n \u00b7 apply Acc.intro; rintro \u27e8_\u27e9 \u27e8_\u27e9\n \u00b7 apply ihl \u27e8m, (List.chain'_cons'.1 hm).2\u27e9 hr\n \u00b7 apply ih b hr\n[GOAL]\ncase mk.cons.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl : List \u03b1\nhl' : List.Chain' (flip r) l\nl' : List.chains r\nthis : Acc (List.lex_chains r) l'\n\u22a2 \u2200 (hl : List.Chain' (flip r) (a :: \u2191l')), Acc (List.lex_chains r) { val := a :: \u2191l', property := hl }\n[PROOFSTEP]\ninduction this with\n| intro l _ ihl =>\n intro hl\n apply Acc.intro\n rintro \u27e8_ | \u27e8b, m\u27e9, hm\u27e9 (_ | hr | hr)\n \u00b7 apply Acc.intro; rintro \u27e8_\u27e9 \u27e8_\u27e9\n \u00b7 apply ihl \u27e8m, (List.chain'_cons'.1 hm).2\u27e9 hr\n \u00b7 apply ih b hr\n[GOAL]\ncase mk.cons.intro.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d\u00b9 : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl\u271d : List \u03b1\nhl' : List.Chain' (flip r) l\u271d\nl' l : List.chains r\nh\u271d : \u2200 (y : List.chains r), List.lex_chains r y l \u2192 Acc (List.lex_chains r) y\nihl :\n \u2200 (y : List.chains r),\n List.lex_chains r y l \u2192\n \u2200 (hl : List.Chain' (flip r) (a :: \u2191y)), Acc (List.lex_chains r) { val := a :: \u2191y, property := hl }\n\u22a2 \u2200 (hl : List.Chain' (flip r) (a :: \u2191l)), Acc (List.lex_chains r) { val := a :: \u2191l, property := hl }\n[PROOFSTEP]\n\n| intro l _ ihl =>\n intro hl\n apply Acc.intro\n rintro \u27e8_ | \u27e8b, m\u27e9, hm\u27e9 (_ | hr | hr)\n \u00b7 apply Acc.intro; rintro \u27e8_\u27e9 \u27e8_\u27e9\n \u00b7 apply ihl \u27e8m, (List.chain'_cons'.1 hm).2\u27e9 hr\n \u00b7 apply ih b hr\n[GOAL]\ncase mk.cons.intro.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d\u00b9 : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl\u271d : List \u03b1\nhl' : List.Chain' (flip r) l\u271d\nl' l : List.chains r\nh\u271d : \u2200 (y : List.chains r), List.lex_chains r y l \u2192 Acc (List.lex_chains r) y\nihl :\n \u2200 (y : List.chains r),\n List.lex_chains r y l \u2192\n \u2200 (hl : List.Chain' (flip r) (a :: \u2191y)), Acc (List.lex_chains r) { val := a :: \u2191y, property := hl }\n\u22a2 \u2200 (hl : List.Chain' (flip r) (a :: \u2191l)), Acc (List.lex_chains r) { val := a :: \u2191l, property := hl }\n[PROOFSTEP]\nintro hl\n[GOAL]\ncase mk.cons.intro.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d\u00b9 : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl\u271d : List \u03b1\nhl' : List.Chain' (flip r) l\u271d\nl' l : List.chains r\nh\u271d : \u2200 (y : List.chains r), List.lex_chains r y l \u2192 Acc (List.lex_chains r) y\nihl :\n \u2200 (y : List.chains r),\n List.lex_chains r y l \u2192\n \u2200 (hl : List.Chain' (flip r) (a :: \u2191y)), Acc (List.lex_chains r) { val := a :: \u2191y, property := hl }\nhl : List.Chain' (flip r) (a :: \u2191l)\n\u22a2 Acc (List.lex_chains r) { val := a :: \u2191l, property := hl }\n[PROOFSTEP]\napply Acc.intro\n[GOAL]\ncase mk.cons.intro.intro.h\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d\u00b9 : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl\u271d : List \u03b1\nhl' : List.Chain' (flip r) l\u271d\nl' l : List.chains r\nh\u271d : \u2200 (y : List.chains r), List.lex_chains r y l \u2192 Acc (List.lex_chains r) y\nihl :\n \u2200 (y : List.chains r),\n List.lex_chains r y l \u2192\n \u2200 (hl : List.Chain' (flip r) (a :: \u2191y)), Acc (List.lex_chains r) { val := a :: \u2191y, property := hl }\nhl : List.Chain' (flip r) (a :: \u2191l)\n\u22a2 \u2200 (y : List.chains r), List.lex_chains r y { val := a :: \u2191l, property := hl } \u2192 Acc (List.lex_chains r) y\n[PROOFSTEP]\nrintro \u27e8_ | \u27e8b, m\u27e9, hm\u27e9 (_ | hr | hr)\n[GOAL]\ncase mk.cons.intro.intro.h.mk.nil.nil\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d\u00b9 : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl\u271d : List \u03b1\nhl' : List.Chain' (flip r) l\u271d\nl' l : List.chains r\nh\u271d : \u2200 (y : List.chains r), List.lex_chains r y l \u2192 Acc (List.lex_chains r) y\nihl :\n \u2200 (y : List.chains r),\n List.lex_chains r y l \u2192\n \u2200 (hl : List.Chain' (flip r) (a :: \u2191y)), Acc (List.lex_chains r) { val := a :: \u2191y, property := hl }\nhl : List.Chain' (flip r) (a :: \u2191l)\nhm : List.Chain' (flip r) []\n\u22a2 Acc (List.lex_chains r) { val := [], property := hm }\n[PROOFSTEP]\napply Acc.intro\n[GOAL]\ncase mk.cons.intro.intro.h.mk.nil.nil.h\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d\u00b9 : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl\u271d : List \u03b1\nhl' : List.Chain' (flip r) l\u271d\nl' l : List.chains r\nh\u271d : \u2200 (y : List.chains r), List.lex_chains r y l \u2192 Acc (List.lex_chains r) y\nihl :\n \u2200 (y : List.chains r),\n List.lex_chains r y l \u2192\n \u2200 (hl : List.Chain' (flip r) (a :: \u2191y)), Acc (List.lex_chains r) { val := a :: \u2191y, property := hl }\nhl : List.Chain' (flip r) (a :: \u2191l)\nhm : List.Chain' (flip r) []\n\u22a2 \u2200 (y : List.chains r), List.lex_chains r y { val := [], property := hm } \u2192 Acc (List.lex_chains r) y\n[PROOFSTEP]\nrintro \u27e8_\u27e9 \u27e8_\u27e9\n[GOAL]\ncase mk.cons.intro.intro.h.mk.cons.cons\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d\u00b9 : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl\u271d : List \u03b1\nhl' : List.Chain' (flip r) l\u271d\nl' l : List.chains r\nh\u271d : \u2200 (y : List.chains r), List.lex_chains r y l \u2192 Acc (List.lex_chains r) y\nihl :\n \u2200 (y : List.chains r),\n List.lex_chains r y l \u2192\n \u2200 (hl : List.Chain' (flip r) (a :: \u2191y)), Acc (List.lex_chains r) { val := a :: \u2191y, property := hl }\nhl : List.Chain' (flip r) (a :: \u2191l)\nm : List \u03b1\nhm : List.Chain' (flip r) (a :: m)\nhr : List.Lex r m \u2191l\n\u22a2 Acc (List.lex_chains r) { val := a :: m, property := hm }\n[PROOFSTEP]\napply ihl \u27e8m, (List.chain'_cons'.1 hm).2\u27e9 hr\n[GOAL]\ncase mk.cons.intro.intro.h.mk.cons.rel\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d a : \u03b1\nh\u271d\u00b9 : \u2200 (y : \u03b1), r y a \u2192 Acc r y\nih :\n \u2200 (y : \u03b1),\n r y a \u2192\n \u2200 (l : List \u03b1) (hl : List.Chain' (flip r) (y :: l)), Acc (List.lex_chains r) { val := y :: l, property := hl }\nl\u271d : List \u03b1\nhl' : List.Chain' (flip r) l\u271d\nl' l : List.chains r\nh\u271d : \u2200 (y : List.chains r), List.lex_chains r y l \u2192 Acc (List.lex_chains r) y\nihl :\n \u2200 (y : List.chains r),\n List.lex_chains r y l \u2192\n \u2200 (hl : List.Chain' (flip r) (a :: \u2191y)), Acc (List.lex_chains r) { val := a :: \u2191y, property := hl }\nhl : List.Chain' (flip r) (a :: \u2191l)\nb : \u03b1\nm : List \u03b1\nhm : List.Chain' (flip r) (b :: m)\nhr : r b a\n\u22a2 Acc (List.lex_chains r) { val := b :: m, property := hm }\n[PROOFSTEP]\napply ih b hr\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Chain", "llama_tokens": 28657, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.5888891307678319, "lm_q1q2_score": 0.40394817831044255}} {"text": "[GOAL]\nP : Type u_1\ninst\u271d : LE P\nI J s\u271d t\u271d : Ideal P\nx y : P\ns t : Ideal P\nx\u271d : s.toLowerSet = t.toLowerSet\n\u22a2 s = t\n[PROOFSTEP]\ncases s\n[GOAL]\ncase mk\nP : Type u_1\ninst\u271d : LE P\nI J s t\u271d : Ideal P\nx y : P\nt : Ideal P\ntoLowerSet\u271d : LowerSet P\nnonempty'\u271d : Set.Nonempty toLowerSet\u271d.carrier\ndirected'\u271d : DirectedOn (fun x x_1 => x \u2264 x_1) toLowerSet\u271d.carrier\nx\u271d : { toLowerSet := toLowerSet\u271d, nonempty' := nonempty'\u271d, directed' := directed'\u271d }.toLowerSet = t.toLowerSet\n\u22a2 { toLowerSet := toLowerSet\u271d, nonempty' := nonempty'\u271d, directed' := directed'\u271d } = t\n[PROOFSTEP]\ncases t\n[GOAL]\ncase mk.mk\nP : Type u_1\ninst\u271d : LE P\nI J s t : Ideal P\nx y : P\ntoLowerSet\u271d\u00b9 : LowerSet P\nnonempty'\u271d\u00b9 : Set.Nonempty toLowerSet\u271d\u00b9.carrier\ndirected'\u271d\u00b9 : DirectedOn (fun x x_1 => x \u2264 x_1) toLowerSet\u271d\u00b9.carrier\ntoLowerSet\u271d : LowerSet P\nnonempty'\u271d : Set.Nonempty toLowerSet\u271d.carrier\ndirected'\u271d : DirectedOn (fun x x_1 => x \u2264 x_1) toLowerSet\u271d.carrier\nx\u271d :\n { toLowerSet := toLowerSet\u271d\u00b9, nonempty' := nonempty'\u271d\u00b9, directed' := directed'\u271d\u00b9 }.toLowerSet =\n { toLowerSet := toLowerSet\u271d, nonempty' := nonempty'\u271d, directed' := directed'\u271d }.toLowerSet\n\u22a2 { toLowerSet := toLowerSet\u271d\u00b9, nonempty' := nonempty'\u271d\u00b9, directed' := directed'\u271d\u00b9 } =\n { toLowerSet := toLowerSet\u271d, nonempty' := nonempty'\u271d, directed' := directed'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nP : Type u_1\ninst\u271d : LE P\nI\u271d J s t : Ideal P\nx y : P\nI : Ideal P\np : P\nnmem : \u00acp \u2208 I\nhp : \u2191I = univ\n\u22a2 False\n[PROOFSTEP]\nhave := mem_univ p\n[GOAL]\nP : Type u_1\ninst\u271d : LE P\nI\u271d J s t : Ideal P\nx y : P\nI : Ideal P\np : P\nnmem : \u00acp \u2208 I\nhp : \u2191I = univ\nthis : p \u2208 univ\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 hp] at this \n[GOAL]\nP : Type u_1\ninst\u271d : LE P\nI\u271d J s t : Ideal P\nx y : P\nI : Ideal P\np : P\nnmem : \u00acp \u2208 I\nhp : \u2191I = univ\nthis : p \u2208 \u2191I\n\u22a2 False\n[PROOFSTEP]\nexact nmem this\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : LE P\nI\u271d J\u271d s t : Ideal P\nx y : P\ninst\u271d : IsDirected P fun x x_1 => x \u2265 x_1\nI J : Ideal P\n\u22a2 Set.Nonempty (\u2191I \u2229 \u2191J)\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := I.nonempty\n[GOAL]\ncase intro\nP : Type u_1\ninst\u271d\u00b9 : LE P\nI\u271d J\u271d s t : Ideal P\nx y : P\ninst\u271d : IsDirected P fun x x_1 => x \u2265 x_1\nI J : Ideal P\na : P\nha : a \u2208 \u2191I\n\u22a2 Set.Nonempty (\u2191I \u2229 \u2191J)\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 := J.nonempty\n[GOAL]\ncase intro.intro\nP : Type u_1\ninst\u271d\u00b9 : LE P\nI\u271d J\u271d s t : Ideal P\nx y : P\ninst\u271d : IsDirected P fun x x_1 => x \u2265 x_1\nI J : Ideal P\na : P\nha : a \u2208 \u2191I\nb : P\nhb : b \u2208 \u2191J\n\u22a2 Set.Nonempty (\u2191I \u2229 \u2191J)\n[PROOFSTEP]\nobtain \u27e8c, hac, hbc\u27e9 := exists_le_le a b\n[GOAL]\ncase intro.intro.intro.intro\nP : Type u_1\ninst\u271d\u00b9 : LE P\nI\u271d J\u271d s t : Ideal P\nx y : P\ninst\u271d : IsDirected P fun x x_1 => x \u2265 x_1\nI J : Ideal P\na : P\nha : a \u2208 \u2191I\nb : P\nhb : b \u2208 \u2191J\nc : P\nhac : c \u2264 a\nhbc : c \u2264 b\n\u22a2 Set.Nonempty (\u2191I \u2229 \u2191J)\n[PROOFSTEP]\nexact \u27e8c, I.lower hac ha, J.lower hbc hb\u27e9\n[GOAL]\nP : Type u_1\ninst\u271d\u00b2 : LE P\ninst\u271d\u00b9 : IsDirected P fun x x_1 => x \u2264 x_1\ninst\u271d : Nonempty P\nI : Ideal P\nhI : IsCoatom I\nsrc\u271d : IsProper I := IsCoatom.isProper hI\nx\u271d : Ideal P\nhJ : I < x\u271d\n\u22a2 \u2191x\u271d = univ\n[PROOFSTEP]\nsimp [hI.2 _ hJ]\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : LE P\ninst\u271d : OrderTop P\nI : Ideal P\nh : \u22a4 \u2208 I\n\u22a2 I = \u22a4\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nP : Type u_1\ninst\u271d\u00b9 : LE P\ninst\u271d : OrderTop P\nI : Ideal P\nh : \u22a4 \u2208 I\nx\u271d : P\n\u22a2 x\u271d \u2208 \u2191I \u2194 x\u271d \u2208 \u2191\u22a4\n[PROOFSTEP]\nexact iff_of_true (I.lower le_top h) trivial\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : Preorder P\ninst\u271d : OrderBot P\n\u22a2 \u2200 (a : Ideal P), \u22a5 \u2264 a\n[PROOFSTEP]\nsimp\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeSup P\ninst\u271d : IsDirected P fun x x_1 => x \u2265 x_1\nx\u271d : P\nI\u271d J\u271d K s t I J : Ideal P\nx : P\nhx : x \u2208 (I.toLowerSet \u2293 J.toLowerSet).carrier\ny : P\nhy : y \u2208 (I.toLowerSet \u2293 J.toLowerSet).carrier\n\u22a2 (fun x x_1 => x \u2264 x_1) x (x \u2294 y) \u2227 (fun x x_1 => x \u2264 x_1) y (x \u2294 y)\n[PROOFSTEP]\nsimp\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeSup P\ninst\u271d : IsDirected P fun x x_1 => x \u2265 x_1\nx : P\nI\u271d J\u271d K s t I J : Ideal P\n\u22a2 Set.Nonempty\n { carrier := {x | \u2203 i, i \u2208 I \u2227 \u2203 j, j \u2208 J \u2227 x \u2264 i \u2294 j},\n lower' :=\n (_ :\n \u2200 (x y : P),\n y \u2264 x \u2192\n x \u2208 {x | \u2203 i, i \u2208 I \u2227 \u2203 j, j \u2208 J \u2227 x \u2264 i \u2294 j} \u2192 y \u2208 {x | \u2203 i, i \u2208 I \u2227 \u2203 j, j \u2208 J \u2227 x \u2264 i \u2294 j}) }.carrier\n[PROOFSTEP]\ncases' inter_nonempty I J with w h\n[GOAL]\ncase intro\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeSup P\ninst\u271d : IsDirected P fun x x_1 => x \u2265 x_1\nx : P\nI\u271d J\u271d K s t I J : Ideal P\nw : P\nh : w \u2208 \u2191I \u2229 \u2191J\n\u22a2 Set.Nonempty\n { carrier := {x | \u2203 i, i \u2208 I \u2227 \u2203 j, j \u2208 J \u2227 x \u2264 i \u2294 j},\n lower' :=\n (_ :\n \u2200 (x y : P),\n y \u2264 x \u2192\n x \u2208 {x | \u2203 i, i \u2208 I \u2227 \u2203 j, j \u2208 J \u2227 x \u2264 i \u2294 j} \u2192 y \u2208 {x | \u2203 i, i \u2208 I \u2227 \u2203 j, j \u2208 J \u2227 x \u2264 i \u2294 j}) }.carrier\n[PROOFSTEP]\nexact \u27e8w, w, h.1, w, h.2, le_sup_left\u27e9\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeSup P\ninst\u271d : IsDirected P fun x x_1 => x \u2265 x_1\nx : P\nI J K s t : Ideal P\nhx : \u00acx \u2208 I\nh : I = I \u2294 principal x\n\u22a2 x \u2208 I\n[PROOFSTEP]\nsimpa only [left_eq_sup, principal_le_iff] using h\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeSup P\ninst\u271d : OrderBot P\nx : P\nI J K : Ideal P\nS : Set (Ideal P)\n\u22a2 \u22a5 \u2208 (\u2a05 (s : Ideal P) (_ : s \u2208 S), s.toLowerSet).carrier\n[PROOFSTEP]\nrw [LowerSet.carrier_eq_coe, LowerSet.coe_iInf\u2082, Set.mem_iInter\u2082]\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeSup P\ninst\u271d : OrderBot P\nx : P\nI J K : Ideal P\nS : Set (Ideal P)\n\u22a2 \u2200 (i : Ideal P), i \u2208 S \u2192 \u22a5 \u2208 \u2191i.toLowerSet\n[PROOFSTEP]\nexact fun s _ \u21a6 s.bot_mem\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeSup P\ninst\u271d : OrderBot P\nx : P\nI J K : Ideal P\nS : Set (Ideal P)\na : P\nha : a \u2208 (\u2a05 (s : Ideal P) (_ : s \u2208 S), s.toLowerSet).carrier\nb : P\nhb : b \u2208 (\u2a05 (s : Ideal P) (_ : s \u2208 S), s.toLowerSet).carrier\n\u22a2 a \u2294 b \u2208 (\u2a05 (s : Ideal P) (_ : s \u2208 S), s.toLowerSet).carrier\n[PROOFSTEP]\nrw [LowerSet.carrier_eq_coe, LowerSet.coe_iInf\u2082, Set.mem_iInter\u2082] at ha hb \u22a2\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeSup P\ninst\u271d : OrderBot P\nx : P\nI J K : Ideal P\nS : Set (Ideal P)\na : P\nha : \u2200 (i : Ideal P), i \u2208 S \u2192 a \u2208 \u2191i.toLowerSet\nb : P\nhb : \u2200 (i : Ideal P), i \u2208 S \u2192 b \u2208 \u2191i.toLowerSet\n\u22a2 \u2200 (i : Ideal P), i \u2208 S \u2192 a \u2294 b \u2208 \u2191i.toLowerSet\n[PROOFSTEP]\nexact fun s hs \u21a6 sup_mem (ha _ hs) (hb _ hs)\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeSup P\ninst\u271d : OrderBot P\nx : P\nI J K : Ideal P\nS : Set (Ideal P)\n\u22a2 x \u2208 sInf S \u2194 \u2200 (s : Ideal P), s \u2208 S \u2192 x \u2208 s\n[PROOFSTEP]\nsimp_rw [\u2190 SetLike.mem_coe, coe_sInf, mem_iInter\u2082]\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeSup P\ninst\u271d : OrderBot P\nx : P\nI J K : Ideal P\nS\u271d : Set (Ideal P)\nsrc\u271d : Lattice (Ideal P) := inferInstance\nS : Set (Ideal P)\n\u22a2 IsGLB S (sInf S)\n[PROOFSTEP]\nrefine' \u27e8fun s hs \u21a6 _, fun s hs \u21a6 by rwa [\u2190 coe_subset_coe, coe_sInf, subset_iInter\u2082_iff]\u27e9\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeSup P\ninst\u271d : OrderBot P\nx : P\nI J K : Ideal P\nS\u271d : Set (Ideal P)\nsrc\u271d : Lattice (Ideal P) := inferInstance\nS : Set (Ideal P)\ns : Ideal P\nhs : s \u2208 lowerBounds S\n\u22a2 s \u2264 sInf S\n[PROOFSTEP]\nrwa [\u2190 coe_subset_coe, coe_sInf, subset_iInter\u2082_iff]\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeSup P\ninst\u271d : OrderBot P\nx : P\nI J K : Ideal P\nS\u271d : Set (Ideal P)\nsrc\u271d : Lattice (Ideal P) := inferInstance\nS : Set (Ideal P)\ns : Ideal P\nhs : s \u2208 S\n\u22a2 sInf S \u2264 s\n[PROOFSTEP]\nrw [\u2190 coe_subset_coe, coe_sInf]\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeSup P\ninst\u271d : OrderBot P\nx : P\nI J K : Ideal P\nS\u271d : Set (Ideal P)\nsrc\u271d : Lattice (Ideal P) := inferInstance\nS : Set (Ideal P)\ns : Ideal P\nhs : s \u2208 S\n\u22a2 \u22c2 (s : Ideal P) (_ : s \u2208 S), \u2191s \u2286 \u2191s\n[PROOFSTEP]\nexact biInter_subset_of_mem hs\n[GOAL]\nP : Type u_1\ninst\u271d : DistribLattice P\nI J : Ideal P\nx i j : P\nhi : i \u2208 I\nhj : j \u2208 J\nhx : x \u2264 i \u2294 j\n\u22a2 \u2203 i', i' \u2208 I \u2227 \u2203 j', j' \u2208 J \u2227 x = i' \u2294 j'\n[PROOFSTEP]\nrefine' \u27e8x \u2293 i, I.lower inf_le_right hi, x \u2293 j, J.lower inf_le_right hj, _\u27e9\n[GOAL]\nP : Type u_1\ninst\u271d : DistribLattice P\nI J : Ideal P\nx i j : P\nhi : i \u2208 I\nhj : j \u2208 J\nhx : x \u2264 i \u2294 j\n\u22a2 x = x \u2293 i \u2294 x \u2293 j\n[PROOFSTEP]\ncalc\n x = x \u2293 (i \u2294 j) := left_eq_inf.mpr hx\n _ = x \u2293 i \u2294 x \u2293 j := inf_sup_left\n[GOAL]\nP : Type u_1\ninst\u271d : BooleanAlgebra P\nx : P\nI : Ideal P\nhI : IsProper I\nhxc : x\u1d9c \u2208 I\n\u22a2 \u00acx \u2208 I\n[PROOFSTEP]\nintro hx\n[GOAL]\nP : Type u_1\ninst\u271d : BooleanAlgebra P\nx : P\nI : Ideal P\nhI : IsProper I\nhxc : x\u1d9c \u2208 I\nhx : x \u2208 I\n\u22a2 False\n[PROOFSTEP]\napply hI.top_not_mem\n[GOAL]\nP : Type u_1\ninst\u271d : BooleanAlgebra P\nx : P\nI : Ideal P\nhI : IsProper I\nhxc : x\u1d9c \u2208 I\nhx : x \u2208 I\n\u22a2 \u22a4 \u2208 I\n[PROOFSTEP]\nhave ht : x \u2294 x\u1d9c \u2208 I := sup_mem \u2039_\u203a \u2039_\u203a\n[GOAL]\nP : Type u_1\ninst\u271d : BooleanAlgebra P\nx : P\nI : Ideal P\nhI : IsProper I\nhxc : x\u1d9c \u2208 I\nhx : x \u2208 I\nht : x \u2294 x\u1d9c \u2208 I\n\u22a2 \u22a4 \u2208 I\n[PROOFSTEP]\nrwa [sup_compl_eq_top] at ht \n[GOAL]\nP : Type u_1\ninst\u271d : BooleanAlgebra P\nx : P\nI : Ideal P\nhI : IsProper I\n\u22a2 \u00acx \u2208 I \u2228 \u00acx\u1d9c \u2208 I\n[PROOFSTEP]\nhave h : x\u1d9c \u2208 I \u2192 x \u2209 I := hI.not_mem_of_compl_mem\n[GOAL]\nP : Type u_1\ninst\u271d : BooleanAlgebra P\nx : P\nI : Ideal P\nhI : IsProper I\nh : x\u1d9c \u2208 I \u2192 \u00acx \u2208 I\n\u22a2 \u00acx \u2208 I \u2228 \u00acx\u1d9c \u2208 I\n[PROOFSTEP]\ntauto\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : Preorder P\np : P\n\u03b9 : Type u_2\ninst\u271d : Encodable \u03b9\n\ud835\udc9f : \u03b9 \u2192 Cofinal P\n\u22a2 Monotone (sequenceOfCofinals p \ud835\udc9f)\n[PROOFSTEP]\napply monotone_nat_of_le_succ\n[GOAL]\ncase hf\nP : Type u_1\ninst\u271d\u00b9 : Preorder P\np : P\n\u03b9 : Type u_2\ninst\u271d : Encodable \u03b9\n\ud835\udc9f : \u03b9 \u2192 Cofinal P\n\u22a2 \u2200 (n : \u2115), sequenceOfCofinals p \ud835\udc9f n \u2264 sequenceOfCofinals p \ud835\udc9f (n + 1)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase hf\nP : Type u_1\ninst\u271d\u00b9 : Preorder P\np : P\n\u03b9 : Type u_2\ninst\u271d : Encodable \u03b9\n\ud835\udc9f : \u03b9 \u2192 Cofinal P\nn : \u2115\n\u22a2 sequenceOfCofinals p \ud835\udc9f n \u2264 sequenceOfCofinals p \ud835\udc9f (n + 1)\n[PROOFSTEP]\ndsimp only [sequenceOfCofinals, Nat.add]\n[GOAL]\ncase hf\nP : Type u_1\ninst\u271d\u00b9 : Preorder P\np : P\n\u03b9 : Type u_2\ninst\u271d : Encodable \u03b9\n\ud835\udc9f : \u03b9 \u2192 Cofinal P\nn : \u2115\n\u22a2 sequenceOfCofinals p \ud835\udc9f n \u2264\n match Encodable.decode n with\n | none => sequenceOfCofinals p \ud835\udc9f n\n | some i => Cofinal.above (\ud835\udc9f i) (sequenceOfCofinals p \ud835\udc9f n)\n[PROOFSTEP]\ncases (Encodable.decode n : Option \u03b9)\n[GOAL]\ncase hf.none\nP : Type u_1\ninst\u271d\u00b9 : Preorder P\np : P\n\u03b9 : Type u_2\ninst\u271d : Encodable \u03b9\n\ud835\udc9f : \u03b9 \u2192 Cofinal P\nn : \u2115\n\u22a2 sequenceOfCofinals p \ud835\udc9f n \u2264\n match none with\n | none => sequenceOfCofinals p \ud835\udc9f n\n | some i => Cofinal.above (\ud835\udc9f i) (sequenceOfCofinals p \ud835\udc9f n)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase hf.some\nP : Type u_1\ninst\u271d\u00b9 : Preorder P\np : P\n\u03b9 : Type u_2\ninst\u271d : Encodable \u03b9\n\ud835\udc9f : \u03b9 \u2192 Cofinal P\nn : \u2115\nval\u271d : \u03b9\n\u22a2 sequenceOfCofinals p \ud835\udc9f n \u2264\n match some val\u271d with\n | none => sequenceOfCofinals p \ud835\udc9f n\n | some i => Cofinal.above (\ud835\udc9f i) (sequenceOfCofinals p \ud835\udc9f n)\n[PROOFSTEP]\napply Cofinal.le_above\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : Preorder P\np : P\n\u03b9 : Type u_2\ninst\u271d : Encodable \u03b9\n\ud835\udc9f : \u03b9 \u2192 Cofinal P\ni : \u03b9\n\u22a2 sequenceOfCofinals p \ud835\udc9f (Encodable.encode i + 1) \u2208 \ud835\udc9f i\n[PROOFSTEP]\ndsimp only [sequenceOfCofinals, Nat.add]\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : Preorder P\np : P\n\u03b9 : Type u_2\ninst\u271d : Encodable \u03b9\n\ud835\udc9f : \u03b9 \u2192 Cofinal P\ni : \u03b9\n\u22a2 (match Encodable.decode (Encodable.encode i) with\n | none => sequenceOfCofinals p \ud835\udc9f (Encodable.encode i)\n | some i_1 => Cofinal.above (\ud835\udc9f i_1) (sequenceOfCofinals p \ud835\udc9f (Encodable.encode i))) \u2208\n \ud835\udc9f i\n[PROOFSTEP]\nrw [Encodable.encodek]\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : Preorder P\np : P\n\u03b9 : Type u_2\ninst\u271d : Encodable \u03b9\n\ud835\udc9f : \u03b9 \u2192 Cofinal P\ni : \u03b9\n\u22a2 (match some i with\n | none => sequenceOfCofinals p \ud835\udc9f (Encodable.encode i)\n | some i_1 => Cofinal.above (\ud835\udc9f i_1) (sequenceOfCofinals p \ud835\udc9f (Encodable.encode i))) \u2208\n \ud835\udc9f i\n[PROOFSTEP]\napply Cofinal.above_mem\n", "meta": {"mathlib_filename": "Mathlib.Order.Ideal", "llama_tokens": 6022, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.40394817831044255}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nr : S\nhr : r \u2208 I\np : R[X]\nhp : eval\u2082 f r p \u2208 I\n\u22a2 coeff p 0 \u2208 comap f I\n[PROOFSTEP]\nrw [\u2190 p.divX_mul_X_add, eval\u2082_add, eval\u2082_C, eval\u2082_mul, eval\u2082_X] at hp \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nr : S\nhr : r \u2208 I\np : R[X]\nhp : eval\u2082 f r (divX p) * r + \u2191f (coeff p 0) \u2208 I\n\u22a2 coeff p 0 \u2208 comap f I\n[PROOFSTEP]\nrefine' mem_comap.mpr ((I.add_mem_iff_right _).mp hp)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nr : S\nhr : r \u2208 I\np : R[X]\nhp : eval\u2082 f r (divX p) * r + \u2191f (coeff p 0) \u2208 I\n\u22a2 eval\u2082 f r (divX p) * r \u2208 I\n[PROOFSTEP]\nexact I.mul_mem_left _ hr\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nr : S\nr_non_zero_divisor : \u2200 {x : S}, x * r = 0 \u2192 x = 0\nhr : r \u2208 I\np : R[X]\n\u22a2 p \u2260 0 \u2192 eval\u2082 f r p = 0 \u2192 \u2203 i, coeff p i \u2260 0 \u2227 coeff p i \u2208 comap f I\n[PROOFSTEP]\nrefine' p.recOnHorner _ _ _\n[GOAL]\ncase refine'_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nr : S\nr_non_zero_divisor : \u2200 {x : S}, x * r = 0 \u2192 x = 0\nhr : r \u2208 I\np : R[X]\n\u22a2 0 \u2260 0 \u2192 eval\u2082 f r 0 = 0 \u2192 \u2203 i, coeff 0 i \u2260 0 \u2227 coeff 0 i \u2208 comap f I\n[PROOFSTEP]\nintro h\n[GOAL]\ncase refine'_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nr : S\nr_non_zero_divisor : \u2200 {x : S}, x * r = 0 \u2192 x = 0\nhr : r \u2208 I\np : R[X]\nh : 0 \u2260 0\n\u22a2 eval\u2082 f r 0 = 0 \u2192 \u2203 i, coeff 0 i \u2260 0 \u2227 coeff 0 i \u2208 comap f I\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nr : S\nr_non_zero_divisor : \u2200 {x : S}, x * r = 0 \u2192 x = 0\nhr : r \u2208 I\np : R[X]\n\u22a2 \u2200 (p : R[X]) (a : R),\n coeff p 0 = 0 \u2192\n a \u2260 0 \u2192\n (p \u2260 0 \u2192 eval\u2082 f r p = 0 \u2192 \u2203 i, coeff p i \u2260 0 \u2227 coeff p i \u2208 comap f I) \u2192\n p + \u2191C a \u2260 0 \u2192 eval\u2082 f r (p + \u2191C a) = 0 \u2192 \u2203 i, coeff (p + \u2191C a) i \u2260 0 \u2227 coeff (p + \u2191C a) i \u2208 comap f I\n[PROOFSTEP]\nintro p a coeff_eq_zero a_ne_zero _ _ hp\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nr : S\nr_non_zero_divisor : \u2200 {x : S}, x * r = 0 \u2192 x = 0\nhr : r \u2208 I\np\u271d p : R[X]\na : R\ncoeff_eq_zero : coeff p 0 = 0\na_ne_zero : a \u2260 0\na\u271d : p \u2260 0 \u2192 eval\u2082 f r p = 0 \u2192 \u2203 i, coeff p i \u2260 0 \u2227 coeff p i \u2208 comap f I\nx\u271d : p + \u2191C a \u2260 0\nhp : eval\u2082 f r (p + \u2191C a) = 0\n\u22a2 \u2203 i, coeff (p + \u2191C a) i \u2260 0 \u2227 coeff (p + \u2191C a) i \u2208 comap f I\n[PROOFSTEP]\nrefine' \u27e80, _, coeff_zero_mem_comap_of_root_mem hr hp\u27e9\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nr : S\nr_non_zero_divisor : \u2200 {x : S}, x * r = 0 \u2192 x = 0\nhr : r \u2208 I\np\u271d p : R[X]\na : R\ncoeff_eq_zero : coeff p 0 = 0\na_ne_zero : a \u2260 0\na\u271d : p \u2260 0 \u2192 eval\u2082 f r p = 0 \u2192 \u2203 i, coeff p i \u2260 0 \u2227 coeff p i \u2208 comap f I\nx\u271d : p + \u2191C a \u2260 0\nhp : eval\u2082 f r (p + \u2191C a) = 0\n\u22a2 coeff (p + \u2191C a) 0 \u2260 0\n[PROOFSTEP]\nsimp [coeff_eq_zero, a_ne_zero]\n[GOAL]\ncase refine'_3\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nr : S\nr_non_zero_divisor : \u2200 {x : S}, x * r = 0 \u2192 x = 0\nhr : r \u2208 I\np : R[X]\n\u22a2 \u2200 (p : R[X]),\n p \u2260 0 \u2192\n (p \u2260 0 \u2192 eval\u2082 f r p = 0 \u2192 \u2203 i, coeff p i \u2260 0 \u2227 coeff p i \u2208 comap f I) \u2192\n p * X \u2260 0 \u2192 eval\u2082 f r (p * X) = 0 \u2192 \u2203 i, coeff (p * X) i \u2260 0 \u2227 coeff (p * X) i \u2208 comap f I\n[PROOFSTEP]\nintro p p_nonzero ih _ hp\n[GOAL]\ncase refine'_3\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nr : S\nr_non_zero_divisor : \u2200 {x : S}, x * r = 0 \u2192 x = 0\nhr : r \u2208 I\np\u271d p : R[X]\np_nonzero : p \u2260 0\nih : p \u2260 0 \u2192 eval\u2082 f r p = 0 \u2192 \u2203 i, coeff p i \u2260 0 \u2227 coeff p i \u2208 comap f I\nx\u271d : p * X \u2260 0\nhp : eval\u2082 f r (p * X) = 0\n\u22a2 \u2203 i, coeff (p * X) i \u2260 0 \u2227 coeff (p * X) i \u2208 comap f I\n[PROOFSTEP]\nrw [eval\u2082_mul, eval\u2082_X] at hp \n[GOAL]\ncase refine'_3\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nr : S\nr_non_zero_divisor : \u2200 {x : S}, x * r = 0 \u2192 x = 0\nhr : r \u2208 I\np\u271d p : R[X]\np_nonzero : p \u2260 0\nih : p \u2260 0 \u2192 eval\u2082 f r p = 0 \u2192 \u2203 i, coeff p i \u2260 0 \u2227 coeff p i \u2208 comap f I\nx\u271d : p * X \u2260 0\nhp : eval\u2082 f r p * r = 0\n\u22a2 \u2203 i, coeff (p * X) i \u2260 0 \u2227 coeff (p * X) i \u2208 comap f I\n[PROOFSTEP]\nobtain \u27e8i, hi, mem\u27e9 := ih p_nonzero (r_non_zero_divisor hp)\n[GOAL]\ncase refine'_3.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nr : S\nr_non_zero_divisor : \u2200 {x : S}, x * r = 0 \u2192 x = 0\nhr : r \u2208 I\np\u271d p : R[X]\np_nonzero : p \u2260 0\nih : p \u2260 0 \u2192 eval\u2082 f r p = 0 \u2192 \u2203 i, coeff p i \u2260 0 \u2227 coeff p i \u2208 comap f I\nx\u271d : p * X \u2260 0\nhp : eval\u2082 f r p * r = 0\ni : \u2115\nhi : coeff p i \u2260 0\nmem : coeff p i \u2208 comap f I\n\u22a2 \u2203 i, coeff (p * X) i \u2260 0 \u2227 coeff (p * X) i \u2208 comap f I\n[PROOFSTEP]\nrefine' \u27e8i + 1, _, _\u27e9\n[GOAL]\ncase refine'_3.intro.intro.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nr : S\nr_non_zero_divisor : \u2200 {x : S}, x * r = 0 \u2192 x = 0\nhr : r \u2208 I\np\u271d p : R[X]\np_nonzero : p \u2260 0\nih : p \u2260 0 \u2192 eval\u2082 f r p = 0 \u2192 \u2203 i, coeff p i \u2260 0 \u2227 coeff p i \u2208 comap f I\nx\u271d : p * X \u2260 0\nhp : eval\u2082 f r p * r = 0\ni : \u2115\nhi : coeff p i \u2260 0\nmem : coeff p i \u2208 comap f I\n\u22a2 coeff (p * X) (i + 1) \u2260 0\n[PROOFSTEP]\nsimp [hi, mem]\n[GOAL]\ncase refine'_3.intro.intro.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nr : S\nr_non_zero_divisor : \u2200 {x : S}, x * r = 0 \u2192 x = 0\nhr : r \u2208 I\np\u271d p : R[X]\np_nonzero : p \u2260 0\nih : p \u2260 0 \u2192 eval\u2082 f r p = 0 \u2192 \u2203 i, coeff p i \u2260 0 \u2227 coeff p i \u2208 comap f I\nx\u271d : p * X \u2260 0\nhp : eval\u2082 f r p * r = 0\ni : \u2115\nhi : coeff p i \u2260 0\nmem : coeff p i \u2208 comap f I\n\u22a2 coeff (p * X) (i + 1) \u2208 comap f I\n[PROOFSTEP]\nsimpa [hi] using mem\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\n\u22a2 Function.Injective\n \u2191(quotientMap (map (mapRingHom (Quotient.mk (comap C P))) P) (mapRingHom (Quotient.mk (comap C P)))\n (_ : P \u2264 comap (mapRingHom (Quotient.mk (comap C P))) (map (mapRingHom (Quotient.mk (comap C P))) P)))\n[PROOFSTEP]\nrefine' quotientMap_injective' (le_of_eq _)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\n\u22a2 comap (mapRingHom (Quotient.mk (comap C P))) (map (mapRingHom (Quotient.mk (comap C P))) P) = P\n[PROOFSTEP]\nrw [comap_map_of_surjective (mapRingHom (Ideal.Quotient.mk (P.comap (C : R \u2192+* R[X]))))\n (map_surjective (Ideal.Quotient.mk (P.comap (C : R \u2192+* R[X]))) Ideal.Quotient.mk_surjective)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\n\u22a2 P \u2294 comap (mapRingHom (Quotient.mk (comap C P))) \u22a5 = P\n[PROOFSTEP]\nrefine' le_antisymm (sup_le le_rfl _) (le_sup_of_le_left le_rfl)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\n\u22a2 comap (mapRingHom (Quotient.mk (comap C P))) \u22a5 \u2264 P\n[PROOFSTEP]\nrefine' fun p hp => polynomial_mem_ideal_of_coeff_mem_ideal P p fun n => Ideal.Quotient.eq_zero_iff_mem.mp _\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\np : R[X]\nhp : p \u2208 comap (mapRingHom (Quotient.mk (comap C P))) \u22a5\nn : \u2115\n\u22a2 \u2191(Quotient.mk (comap C P)) (coeff p n) = 0\n[PROOFSTEP]\nsimpa only [coeff_map, coe_mapRingHom] using ext_iff.mp (Ideal.mem_bot.mp (mem_comap.mp hp)) n\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\n\u22a2 RingHom.comp (RingHom.comp (Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)) C) (Quotient.mk (comap C P)) =\n RingHom.comp\n (quotientMap (map (mapRingHom (Quotient.mk (comap C P))) P) (mapRingHom (Quotient.mk (comap C P)))\n (_ : P \u2264 comap (mapRingHom (Quotient.mk (comap C P))) (map (mapRingHom (Quotient.mk (comap C P))) P)))\n (RingHom.comp (Quotient.mk P) C)\n[PROOFSTEP]\nrefine' RingHom.ext fun x => _\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\nx : R\n\u22a2 \u2191(RingHom.comp (RingHom.comp (Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)) C)\n (Quotient.mk (comap C P)))\n x =\n \u2191(RingHom.comp\n (quotientMap (map (mapRingHom (Quotient.mk (comap C P))) P) (mapRingHom (Quotient.mk (comap C P)))\n (_ : P \u2264 comap (mapRingHom (Quotient.mk (comap C P))) (map (mapRingHom (Quotient.mk (comap C P))) P)))\n (RingHom.comp (Quotient.mk P) C))\n x\n[PROOFSTEP]\nrepeat' rw [RingHom.coe_comp, Function.comp_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\nx : R\n\u22a2 \u2191(RingHom.comp (RingHom.comp (Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)) C)\n (Quotient.mk (comap C P)))\n x =\n \u2191(RingHom.comp\n (quotientMap (map (mapRingHom (Quotient.mk (comap C P))) P) (mapRingHom (Quotient.mk (comap C P)))\n (_ : P \u2264 comap (mapRingHom (Quotient.mk (comap C P))) (map (mapRingHom (Quotient.mk (comap C P))) P)))\n (RingHom.comp (Quotient.mk P) C))\n x\n[PROOFSTEP]\nrw [RingHom.coe_comp, Function.comp_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\nx : R\n\u22a2 \u2191(RingHom.comp (Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)) C) (\u2191(Quotient.mk (comap C P)) x) =\n \u2191(RingHom.comp\n (quotientMap (map (mapRingHom (Quotient.mk (comap C P))) P) (mapRingHom (Quotient.mk (comap C P)))\n (_ : P \u2264 comap (mapRingHom (Quotient.mk (comap C P))) (map (mapRingHom (Quotient.mk (comap C P))) P)))\n (RingHom.comp (Quotient.mk P) C))\n x\n[PROOFSTEP]\nrw [RingHom.coe_comp, Function.comp_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\nx : R\n\u22a2 \u2191(Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)) (\u2191C (\u2191(Quotient.mk (comap C P)) x)) =\n \u2191(RingHom.comp\n (quotientMap (map (mapRingHom (Quotient.mk (comap C P))) P) (mapRingHom (Quotient.mk (comap C P)))\n (_ : P \u2264 comap (mapRingHom (Quotient.mk (comap C P))) (map (mapRingHom (Quotient.mk (comap C P))) P)))\n (RingHom.comp (Quotient.mk P) C))\n x\n[PROOFSTEP]\nrw [RingHom.coe_comp, Function.comp_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\nx : R\n\u22a2 \u2191(Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)) (\u2191C (\u2191(Quotient.mk (comap C P)) x)) =\n \u2191(quotientMap (map (mapRingHom (Quotient.mk (comap C P))) P) (mapRingHom (Quotient.mk (comap C P)))\n (_ : P \u2264 comap (mapRingHom (Quotient.mk (comap C P))) (map (mapRingHom (Quotient.mk (comap C P))) P)))\n (\u2191(RingHom.comp (Quotient.mk P) C) x)\n[PROOFSTEP]\nrw [RingHom.coe_comp, Function.comp_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\nx : R\n\u22a2 \u2191(Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)) (\u2191C (\u2191(Quotient.mk (comap C P)) x)) =\n \u2191(quotientMap (map (mapRingHom (Quotient.mk (comap C P))) P) (mapRingHom (Quotient.mk (comap C P)))\n (_ : P \u2264 comap (mapRingHom (Quotient.mk (comap C P))) (map (mapRingHom (Quotient.mk (comap C P))) P)))\n (\u2191(Quotient.mk P) (\u2191C x))\n[PROOFSTEP]\nrw [RingHom.coe_comp, Function.comp_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\nx : R\n\u22a2 \u2191(Quotient.mk (map (mapRingHom (Quotient.mk (comap C P))) P)) (\u2191C (\u2191(Quotient.mk (comap C P)) x)) =\n \u2191(quotientMap (map (mapRingHom (Quotient.mk (comap C P))) P) (mapRingHom (Quotient.mk (comap C P)))\n (_ : P \u2264 comap (mapRingHom (Quotient.mk (comap C P))) (map (mapRingHom (Quotient.mk (comap C P))) P)))\n (\u2191(Quotient.mk P) (\u2191C x))\n[PROOFSTEP]\nrw [quotientMap_mk, coe_mapRingHom, map_C]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P \u2260 \u22a5\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\n\u22a2 \u2203 p, p \u2208 P \u2227 Polynomial.map (Quotient.mk (comap C P)) p \u2260 0\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := Submodule.nonzero_mem_of_bot_lt (bot_lt_iff_ne_bot.mpr Pb)\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P \u2260 \u22a5\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nm : { x // x \u2208 P }\nhm : m \u2260 0\n\u22a2 \u2203 p, p \u2208 P \u2227 Polynomial.map (Quotient.mk (comap C P)) p \u2260 0\n[PROOFSTEP]\nrefine' \u27e8m, Submodule.coe_mem m, fun pp0 => hm (Submodule.coe_eq_zero.mp _)\u27e9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P \u2260 \u22a5\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nm : { x // x \u2208 P }\nhm : m \u2260 0\npp0 : Polynomial.map (Quotient.mk (comap C P)) \u2191m = 0\n\u22a2 \u2191m = 0\n[PROOFSTEP]\nrefine' (injective_iff_map_eq_zero (Polynomial.mapRingHom (Ideal.Quotient.mk (P.comap (C : R \u2192+* R[X]))))).mp _ _ pp0\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P \u2260 \u22a5\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nm : { x // x \u2208 P }\nhm : m \u2260 0\npp0 : Polynomial.map (Quotient.mk (comap C P)) \u2191m = 0\n\u22a2 Function.Injective \u2191(mapRingHom (Quotient.mk (comap C P)))\n[PROOFSTEP]\nrefine' map_injective _ ((Ideal.Quotient.mk (P.comap C)).injective_iff_ker_eq_bot.mpr _)\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P \u2260 \u22a5\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nm : { x // x \u2208 P }\nhm : m \u2260 0\npp0 : Polynomial.map (Quotient.mk (comap C P)) \u2191m = 0\n\u22a2 RingHom.ker (Quotient.mk (comap C P)) = \u22a5\n[PROOFSTEP]\nrw [mk_ker]\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P \u2260 \u22a5\nhP : \u2200 (x : R), \u2191C x \u2208 P \u2192 x = 0\nm : { x // x \u2208 P }\nhm : m \u2260 0\npp0 : Polynomial.map (Quotient.mk (comap C P)) \u2191m = 0\n\u22a2 comap C P = \u22a5\n[PROOFSTEP]\nexact (Submodule.eq_bot_iff _).mpr fun x hx => hP x (mem_comap.mp hx)\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : Algebra (R \u29f8 p) (S \u29f8 P)\ninst\u271d : IsScalarTower R (R \u29f8 p) (S \u29f8 P)\nh : Function.Injective \u2191(algebraMap (R \u29f8 p) (S \u29f8 P))\n\u22a2 comap (algebraMap R S) P = p\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : Algebra (R \u29f8 p) (S \u29f8 P)\ninst\u271d : IsScalarTower R (R \u29f8 p) (S \u29f8 P)\nh : Function.Injective \u2191(algebraMap (R \u29f8 p) (S \u29f8 P))\nx : R\n\u22a2 x \u2208 comap (algebraMap R S) P \u2194 x \u2208 p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : Algebra (R \u29f8 p) (S \u29f8 P)\ninst\u271d : IsScalarTower R (R \u29f8 p) (S \u29f8 P)\nh : Function.Injective \u2191(algebraMap (R \u29f8 p) (S \u29f8 P))\nx : R\n\u22a2 x \u2208 comap (algebraMap R S) P \u2192 x \u2208 p\n[PROOFSTEP]\nrw [mem_comap, \u2190 Quotient.eq_zero_iff_mem, \u2190 Quotient.eq_zero_iff_mem, Quotient.mk_algebraMap,\n IsScalarTower.algebraMap_apply R (R \u29f8 p) (S \u29f8 P), Quotient.algebraMap_eq]\n[GOAL]\ncase h.mpr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : Algebra (R \u29f8 p) (S \u29f8 P)\ninst\u271d : IsScalarTower R (R \u29f8 p) (S \u29f8 P)\nh : Function.Injective \u2191(algebraMap (R \u29f8 p) (S \u29f8 P))\nx : R\n\u22a2 x \u2208 p \u2192 x \u2208 comap (algebraMap R S) P\n[PROOFSTEP]\nrw [mem_comap, \u2190 Quotient.eq_zero_iff_mem, \u2190 Quotient.eq_zero_iff_mem, Quotient.mk_algebraMap,\n IsScalarTower.algebraMap_apply R (R \u29f8 p) (S \u29f8 P), Quotient.algebraMap_eq]\n[GOAL]\ncase h.mp\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : Algebra (R \u29f8 p) (S \u29f8 P)\ninst\u271d : IsScalarTower R (R \u29f8 p) (S \u29f8 P)\nh : Function.Injective \u2191(algebraMap (R \u29f8 p) (S \u29f8 P))\nx : R\n\u22a2 \u2191(algebraMap (R \u29f8 p) (S \u29f8 P)) (\u2191(Quotient.mk p) x) = 0 \u2192 \u2191(Quotient.mk p) x = 0\n[PROOFSTEP]\nintro hx\n[GOAL]\ncase h.mp\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : Algebra (R \u29f8 p) (S \u29f8 P)\ninst\u271d : IsScalarTower R (R \u29f8 p) (S \u29f8 P)\nh : Function.Injective \u2191(algebraMap (R \u29f8 p) (S \u29f8 P))\nx : R\nhx : \u2191(algebraMap (R \u29f8 p) (S \u29f8 P)) (\u2191(Quotient.mk p) x) = 0\n\u22a2 \u2191(Quotient.mk p) x = 0\n[PROOFSTEP]\nexact (injective_iff_map_eq_zero (algebraMap (R \u29f8 p) (S \u29f8 P))).mp h _ hx\n[GOAL]\ncase h.mpr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : Algebra (R \u29f8 p) (S \u29f8 P)\ninst\u271d : IsScalarTower R (R \u29f8 p) (S \u29f8 P)\nh : Function.Injective \u2191(algebraMap (R \u29f8 p) (S \u29f8 P))\nx : R\n\u22a2 \u2191(Quotient.mk p) x = 0 \u2192 \u2191(algebraMap (R \u29f8 p) (S \u29f8 P)) (\u2191(Quotient.mk p) x) = 0\n[PROOFSTEP]\nintro hx\n[GOAL]\ncase h.mpr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : Algebra (R \u29f8 p) (S \u29f8 P)\ninst\u271d : IsScalarTower R (R \u29f8 p) (S \u29f8 P)\nh : Function.Injective \u2191(algebraMap (R \u29f8 p) (S \u29f8 P))\nx : R\nhx : \u2191(Quotient.mk p) x = 0\n\u22a2 \u2191(algebraMap (R \u29f8 p) (S \u29f8 P)) (\u2191(Quotient.mk p) x) = 0\n[PROOFSTEP]\nrw [hx, RingHom.map_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\np : Ideal R\nP : Ideal S\ninst\u271d : Algebra R S\nx : R\n\u22a2 \u2191(algebraMap R (S \u29f8 map (algebraMap R S) p)) x =\n \u2191(algebraMap (R \u29f8 p) (S \u29f8 map (algebraMap R S) p)) (\u2191(algebraMap R (R \u29f8 p)) x)\n[PROOFSTEP]\nrw [Quotient.algebraMap_eq, Quotient.algebraMap_quotient_map_quotient, Quotient.mk_algebraMap]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\nhr : r \u2208 \u2191J \\ \u2191I\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)\n[PROOFSTEP]\nobtain \u27e8hrJ, hrI\u27e9 := hr\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)\n[PROOFSTEP]\nhave rbar_ne_zero : Ideal.Quotient.mk I r \u2260 0 := mt (Quotient.mk_eq_zero I).mp hrI\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)\n[PROOFSTEP]\nhave rbar_mem_J : Ideal.Quotient.mk I r \u2208 J.map (Ideal.Quotient.mk I) := mem_map_of_mem _ hrJ\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)\n[PROOFSTEP]\nhave quotient_f : \u2200 x \u2208 I.comap f, (Ideal.Quotient.mk I).comp f x = 0 := by simp [Quotient.eq_zero_iff_mem]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\n\u22a2 \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\n[PROOFSTEP]\nsimp [Quotient.eq_zero_iff_mem]\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)\n[PROOFSTEP]\nhave rbar_root :\n (p.map (Ideal.Quotient.mk (I.comap f))).eval\u2082 (Quotient.lift (I.comap f) _ quotient_f) (Ideal.Quotient.mk I r) = 0 :=\n by\n convert Quotient.eq_zero_iff_mem.mpr hpI\n exact _root_.trans (eval\u2082_map _ _ _) (hom_eval\u2082 p f (Ideal.Quotient.mk I) r).symm\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\n\u22a2 eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n (Polynomial.map (Quotient.mk (comap f I)) p) =\n 0\n[PROOFSTEP]\nconvert Quotient.eq_zero_iff_mem.mpr hpI\n[GOAL]\ncase h.e'_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\n\u22a2 eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n (Polynomial.map (Quotient.mk (comap f I)) p) =\n \u2191(Quotient.mk I) (eval\u2082 f r p)\n[PROOFSTEP]\nexact _root_.trans (eval\u2082_map _ _ _) (hom_eval\u2082 p f (Ideal.Quotient.mk I) r).symm\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\nrbar_root :\n eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n (Polynomial.map (Quotient.mk (comap f I)) p) =\n 0\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)\n[PROOFSTEP]\nobtain \u27e8i, ne_zero, mem\u27e9 := exists_coeff_ne_zero_mem_comap_of_root_mem rbar_ne_zero rbar_mem_J p_ne_zero rbar_root\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\nrbar_root :\n eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n (Polynomial.map (Quotient.mk (comap f I)) p) =\n 0\ni : \u2115\nne_zero : coeff (Polynomial.map (Quotient.mk (comap f I)) p) i \u2260 0\nmem :\n coeff (Polynomial.map (Quotient.mk (comap f I)) p) i \u2208\n comap (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (map (Quotient.mk I) J)\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)\n[PROOFSTEP]\nrw [coeff_map] at ne_zero mem \n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\nrbar_root :\n eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n (Polynomial.map (Quotient.mk (comap f I)) p) =\n 0\ni : \u2115\nne_zero : \u2191(Quotient.mk (comap f I)) (coeff p i) \u2260 0\nmem :\n \u2191(Quotient.mk (comap f I)) (coeff p i) \u2208\n comap (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (map (Quotient.mk I) J)\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)\n[PROOFSTEP]\nrefine' \u27e8i, (mem_quotient_iff_mem hIJ).mp _, mt _ ne_zero\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\nrbar_root :\n eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n (Polynomial.map (Quotient.mk (comap f I)) p) =\n 0\ni : \u2115\nne_zero : \u2191(Quotient.mk (comap f I)) (coeff p i) \u2260 0\nmem :\n \u2191(Quotient.mk (comap f I)) (coeff p i) \u2208\n comap (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (map (Quotient.mk I) J)\n\u22a2 \u2191(Quotient.mk I) (\u2191f (coeff p i)) \u2208 map (Quotient.mk I) J\n[PROOFSTEP]\nsimpa using mem\n[GOAL]\ncase intro.intro.intro.refine'_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\nrbar_root :\n eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n (Polynomial.map (Quotient.mk (comap f I)) p) =\n 0\ni : \u2115\nne_zero : \u2191(Quotient.mk (comap f I)) (coeff p i) \u2260 0\nmem :\n \u2191(Quotient.mk (comap f I)) (coeff p i) \u2208\n comap (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (map (Quotient.mk I) J)\n\u22a2 coeff p i \u2208 \u2191(comap f I) \u2192 \u2191(Quotient.mk (comap f I)) (coeff p i) = 0\n[PROOFSTEP]\nsimp [Quotient.eq_zero_iff_mem]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : Algebra R S\nhI : IsPrime I\nhIJ : I \u2264 J\nx : S\nmem : x \u2208 \u2191J \\ \u2191I\nintegral : IsIntegral R x\n\u22a2 comap (algebraMap R S) I < comap (algebraMap R S) J\n[PROOFSTEP]\nobtain \u27e8p, p_monic, hpx\u27e9 := integral\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : Algebra R S\nhI : IsPrime I\nhIJ : I \u2264 J\nx : S\nmem : x \u2208 \u2191J \\ \u2191I\np : R[X]\np_monic : Monic p\nhpx : eval\u2082 (algebraMap R S) x p = 0\n\u22a2 comap (algebraMap R S) I < comap (algebraMap R S) J\n[PROOFSTEP]\nrefine' comap_lt_comap_of_root_mem_sdiff hIJ mem _ _\n[GOAL]\ncase intro.intro.refine'_1\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : Algebra R S\nhI : IsPrime I\nhIJ : I \u2264 J\nx : S\nmem : x \u2208 \u2191J \\ \u2191I\np : R[X]\np_monic : Monic p\nhpx : eval\u2082 (algebraMap R S) x p = 0\n\u22a2 R[X]\ncase intro.intro.refine'_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : Algebra R S\nhI : IsPrime I\nhIJ : I \u2264 J\nx : S\nmem : x \u2208 \u2191J \\ \u2191I\np : R[X]\np_monic : Monic p\nhpx : eval\u2082 (algebraMap R S) x p = 0\n\u22a2 Polynomial.map (Quotient.mk (comap (algebraMap R S) I)) ?intro.intro.refine'_1 \u2260 0\ncase intro.intro.refine'_3\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : Algebra R S\nhI : IsPrime I\nhIJ : I \u2264 J\nx : S\nmem : x \u2208 \u2191J \\ \u2191I\np : R[X]\np_monic : Monic p\nhpx : eval\u2082 (algebraMap R S) x p = 0\n\u22a2 eval\u2082 (algebraMap R S) x ?intro.intro.refine'_1 \u2208 I\n[PROOFSTEP]\nswap\n[GOAL]\ncase intro.intro.refine'_2\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : Algebra R S\nhI : IsPrime I\nhIJ : I \u2264 J\nx : S\nmem : x \u2208 \u2191J \\ \u2191I\np : R[X]\np_monic : Monic p\nhpx : eval\u2082 (algebraMap R S) x p = 0\n\u22a2 Polynomial.map (Quotient.mk (comap (algebraMap R S) I)) ?intro.intro.refine'_1 \u2260 0\n[PROOFSTEP]\napply map_monic_ne_zero p_monic\n[GOAL]\ncase intro.intro.refine'_3\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : Algebra R S\nhI : IsPrime I\nhIJ : I \u2264 J\nx : S\nmem : x \u2208 \u2191J \\ \u2191I\np : R[X]\np_monic : Monic p\nhpx : eval\u2082 (algebraMap R S) x p = 0\n\u22a2 eval\u2082 (algebraMap R S) x p \u2208 I\n[PROOFSTEP]\nconvert I.zero_mem\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsDomain S\nhRS : Algebra.IsIntegral R S\nhI : comap (algebraMap R S) I = \u22a5\n\u22a2 I = \u22a5\n[PROOFSTEP]\nrefine' eq_bot_iff.2 fun x hx => _\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsDomain S\nhRS : Algebra.IsIntegral R S\nhI : comap (algebraMap R S) I = \u22a5\nx : S\nhx : x \u2208 I\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nby_cases hx0 : x = 0\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsDomain S\nhRS : Algebra.IsIntegral R S\nhI : comap (algebraMap R S) I = \u22a5\nx : S\nhx : x \u2208 I\nhx0 : x = 0\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nexact hx0.symm \u25b8 Ideal.zero_mem \u22a5\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : IsDomain S\nhRS : Algebra.IsIntegral R S\nhI : comap (algebraMap R S) I = \u22a5\nx : S\nhx : x \u2208 I\nhx0 : \u00acx = 0\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nexact absurd hI (comap_ne_bot_of_integral_mem hx0 hx (hRS x))\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nI : Ideal S\nhI : IsMaximal I\n\u22a2 IsMaximal (comap (algebraMap R S) I)\n[PROOFSTEP]\nrefine' Ideal.Quotient.maximal_of_isField _ _\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nI : Ideal S\nhI : IsMaximal I\n\u22a2 IsField (R \u29f8 comap (algebraMap R S) I)\n[PROOFSTEP]\nhaveI : IsPrime (I.comap (algebraMap R S)) := comap_isPrime _ _\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nI : Ideal S\nhI : IsMaximal I\nthis : IsPrime (comap (algebraMap R S) I)\n\u22a2 IsField (R \u29f8 comap (algebraMap R S) I)\n[PROOFSTEP]\nexact\n isField_of_isIntegral_of_isField (isIntegral_quotient_of_isIntegral hRS) algebraMap_quotient_injective\n (by rwa [\u2190 Quotient.maximal_ideal_iff_isField_quotient])\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nI : Ideal S\nhI : IsMaximal I\nthis : IsPrime (comap (algebraMap R S) I)\n\u22a2 IsField (S \u29f8 I)\n[PROOFSTEP]\nrwa [\u2190 Quotient.maximal_ideal_iff_isField_quotient]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d\u2077 : Algebra R S\nA : Type u_3\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : Algebra A S\ninst\u271d\u00b3 : IsScalarTower R A S\ninst\u271d\u00b2 : IsIntegralClosure A R S\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : Nontrivial R\nI : Ideal A\n\u22a2 comap (algebraMap R A) I = \u22a5 \u2192 I = \u22a5\n[PROOFSTEP]\ncontrapose\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d\u2077 : Algebra R S\nA : Type u_3\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : Algebra A S\ninst\u271d\u00b3 : IsScalarTower R A S\ninst\u271d\u00b2 : IsIntegralClosure A R S\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : Nontrivial R\nI : Ideal A\n\u22a2 \u00acI = \u22a5 \u2192 \u00accomap (algebraMap R A) I = \u22a5\n[PROOFSTEP]\nexact (IsIntegralClosure.comap_ne_bot S)\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsDomain S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d : IsPrime P\nhP : RingHom.ker (algebraMap R S) \u2264 P\n\u22a2 \u2203 Q, IsPrime Q \u2227 comap (algebraMap R S) Q = P\n[PROOFSTEP]\nhave hP0 : (0 : S) \u2209 Algebra.algebraMapSubmonoid S P.primeCompl :=\n by\n rintro \u27e8x, \u27e8hx, x0\u27e9\u27e9\n exact absurd (hP x0) hx\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsDomain S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d : IsPrime P\nhP : RingHom.ker (algebraMap R S) \u2264 P\n\u22a2 \u00ac0 \u2208 Algebra.algebraMapSubmonoid S (primeCompl P)\n[PROOFSTEP]\nrintro \u27e8x, \u27e8hx, x0\u27e9\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsDomain S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d : IsPrime P\nhP : RingHom.ker (algebraMap R S) \u2264 P\nx : R\nhx : x \u2208 \u2191(primeCompl P)\nx0 : \u2191(algebraMap R S) x = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd (hP x0) hx\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsDomain S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d : IsPrime P\nhP : RingHom.ker (algebraMap R S) \u2264 P\nhP0 : \u00ac0 \u2208 Algebra.algebraMapSubmonoid S (primeCompl P)\n\u22a2 \u2203 Q, IsPrime Q \u2227 comap (algebraMap R S) Q = P\n[PROOFSTEP]\nlet R\u209a := Localization P.primeCompl\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsDomain S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d : IsPrime P\nhP : RingHom.ker (algebraMap R S) \u2264 P\nhP0 : \u00ac0 \u2208 Algebra.algebraMapSubmonoid S (primeCompl P)\nR\u209a : Type u_1 := Localization (primeCompl P)\n\u22a2 \u2203 Q, IsPrime Q \u2227 comap (algebraMap R S) Q = P\n[PROOFSTEP]\nlet S\u209a := Localization (Algebra.algebraMapSubmonoid S P.primeCompl)\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsDomain S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d : IsPrime P\nhP : RingHom.ker (algebraMap R S) \u2264 P\nhP0 : \u00ac0 \u2208 Algebra.algebraMapSubmonoid S (primeCompl P)\nR\u209a : Type u_1 := Localization (primeCompl P)\nS\u209a : Type u_2 := Localization (Algebra.algebraMapSubmonoid S (primeCompl P))\n\u22a2 \u2203 Q, IsPrime Q \u2227 comap (algebraMap R S) Q = P\n[PROOFSTEP]\nletI : IsDomain (Localization (Algebra.algebraMapSubmonoid S P.primeCompl)) :=\n IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsDomain S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d : IsPrime P\nhP : RingHom.ker (algebraMap R S) \u2264 P\nhP0 : \u00ac0 \u2208 Algebra.algebraMapSubmonoid S (primeCompl P)\nR\u209a : Type u_1 := Localization (primeCompl P)\nS\u209a : Type u_2 := Localization (Algebra.algebraMapSubmonoid S (primeCompl P))\nthis : IsDomain (Localization (Algebra.algebraMapSubmonoid S (primeCompl P))) :=\n IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)\n\u22a2 \u2203 Q, IsPrime Q \u2227 comap (algebraMap R S) Q = P\n[PROOFSTEP]\nobtain \u27e8Q\u209a : Ideal S\u209a, Q\u209a_maximal\u27e9 := exists_maximal S\u209a\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsDomain S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d : IsPrime P\nhP : RingHom.ker (algebraMap R S) \u2264 P\nhP0 : \u00ac0 \u2208 Algebra.algebraMapSubmonoid S (primeCompl P)\nR\u209a : Type u_1 := Localization (primeCompl P)\nS\u209a : Type u_2 := Localization (Algebra.algebraMapSubmonoid S (primeCompl P))\nthis : IsDomain (Localization (Algebra.algebraMapSubmonoid S (primeCompl P))) :=\n IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)\nQ\u209a : Ideal S\u209a\nQ\u209a_maximal : IsMaximal Q\u209a\n\u22a2 \u2203 Q, IsPrime Q \u2227 comap (algebraMap R S) Q = P\n[PROOFSTEP]\nhaveI Q\u209a_max : IsMaximal (comap _ Q\u209a) :=\n @isMaximal_comap_of_isIntegral_of_isMaximal R\u209a _ S\u209a _ (localizationAlgebra P.primeCompl S) (isIntegral_localization H)\n _ Q\u209a_maximal\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsDomain S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d : IsPrime P\nhP : RingHom.ker (algebraMap R S) \u2264 P\nhP0 : \u00ac0 \u2208 Algebra.algebraMapSubmonoid S (primeCompl P)\nR\u209a : Type u_1 := Localization (primeCompl P)\nS\u209a : Type u_2 := Localization (Algebra.algebraMapSubmonoid S (primeCompl P))\nthis : IsDomain (Localization (Algebra.algebraMapSubmonoid S (primeCompl P))) :=\n IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)\nQ\u209a : Ideal S\u209a\nQ\u209a_maximal : IsMaximal Q\u209a\nQ\u209a_max : IsMaximal (comap (algebraMap R\u209a S\u209a) Q\u209a)\n\u22a2 \u2203 Q, IsPrime Q \u2227 comap (algebraMap R S) Q = P\n[PROOFSTEP]\nrefine' \u27e8comap (algebraMap S S\u209a) Q\u209a, \u27e8comap_isPrime _ Q\u209a, _\u27e9\u27e9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsDomain S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d : IsPrime P\nhP : RingHom.ker (algebraMap R S) \u2264 P\nhP0 : \u00ac0 \u2208 Algebra.algebraMapSubmonoid S (primeCompl P)\nR\u209a : Type u_1 := Localization (primeCompl P)\nS\u209a : Type u_2 := Localization (Algebra.algebraMapSubmonoid S (primeCompl P))\nthis : IsDomain (Localization (Algebra.algebraMapSubmonoid S (primeCompl P))) :=\n IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)\nQ\u209a : Ideal S\u209a\nQ\u209a_maximal : IsMaximal Q\u209a\nQ\u209a_max : IsMaximal (comap (algebraMap R\u209a S\u209a) Q\u209a)\n\u22a2 comap (algebraMap R S) (comap (algebraMap S S\u209a) Q\u209a) = P\n[PROOFSTEP]\nconvert Localization.AtPrime.comap_maximalIdeal (I := P)\n[GOAL]\ncase h.e'_2\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsDomain S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d : IsPrime P\nhP : RingHom.ker (algebraMap R S) \u2264 P\nhP0 : \u00ac0 \u2208 Algebra.algebraMapSubmonoid S (primeCompl P)\nR\u209a : Type u_1 := Localization (primeCompl P)\nS\u209a : Type u_2 := Localization (Algebra.algebraMapSubmonoid S (primeCompl P))\nthis : IsDomain (Localization (Algebra.algebraMapSubmonoid S (primeCompl P))) :=\n IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)\nQ\u209a : Ideal S\u209a\nQ\u209a_maximal : IsMaximal Q\u209a\nQ\u209a_max : IsMaximal (comap (algebraMap R\u209a S\u209a) Q\u209a)\n\u22a2 comap (algebraMap R S) (comap (algebraMap S S\u209a) Q\u209a) =\n comap (algebraMap R (Localization.AtPrime P)) (LocalRing.maximalIdeal (Localization (primeCompl P)))\n[PROOFSTEP]\nrw [comap_comap, \u2190 LocalRing.eq_maximalIdeal Q\u209a_max, \u2190\n @IsLocalization.map_comp (P := S) (Q := S\u209a) (g := algebraMap R S) (M := P.primeCompl) (T :=\n Algebra.algebraMapSubmonoid S P.primeCompl) (S := R\u209a) _ _ _ _ _ _\n (fun p hp => Algebra.mem_algebraMapSubmonoid_of_mem \u27e8p, hp\u27e9) _ _]\n[GOAL]\ncase h.e'_2\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsDomain S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d : IsPrime P\nhP : RingHom.ker (algebraMap R S) \u2264 P\nhP0 : \u00ac0 \u2208 Algebra.algebraMapSubmonoid S (primeCompl P)\nR\u209a : Type u_1 := Localization (primeCompl P)\nS\u209a : Type u_2 := Localization (Algebra.algebraMapSubmonoid S (primeCompl P))\nthis : IsDomain (Localization (Algebra.algebraMapSubmonoid S (primeCompl P))) :=\n IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)\nQ\u209a : Ideal S\u209a\nQ\u209a_maximal : IsMaximal Q\u209a\nQ\u209a_max : IsMaximal (comap (algebraMap R\u209a S\u209a) Q\u209a)\n\u22a2 comap\n (RingHom.comp\n (IsLocalization.map S\u209a (algebraMap R S)\n (_ :\n \u2200 (p : R) (hp : p \u2208 primeCompl P),\n \u2191(algebraMap R S) \u2191{ val := p, property := hp } \u2208 Algebra.algebraMapSubmonoid S (primeCompl P)))\n (algebraMap R R\u209a))\n Q\u209a =\n comap (algebraMap R (Localization.AtPrime P)) (comap (algebraMap R\u209a S\u209a) Q\u209a)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d\u00b2 : Algebra R S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d\u00b9 : IsPrime P\nI : Ideal S\ninst\u271d : IsPrime I\nhIP : comap (algebraMap R S) I \u2264 P\n\u22a2 \u2203 Q, Q \u2265 I \u2227 IsPrime Q \u2227 comap (algebraMap R S) Q = P\n[PROOFSTEP]\nobtain \u27e8Q' : Ideal (S \u29f8 I), \u27e8Q'_prime, hQ'\u27e9\u27e9 :=\n @exists_ideal_over_prime_of_is_integral' (R \u29f8 I.comap (algebraMap R S)) _ (S \u29f8 I) _ Ideal.quotientAlgebra _\n (isIntegral_quotient_of_isIntegral H) (map (Ideal.Quotient.mk (I.comap (algebraMap R S))) P)\n (map_isPrime_of_surjective Quotient.mk_surjective (by simp [hIP]))\n (le_trans (le_of_eq ((RingHom.injective_iff_ker_eq_bot _).1 algebraMap_quotient_injective)) bot_le)\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d\u00b2 : Algebra R S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d\u00b9 : IsPrime P\nI : Ideal S\ninst\u271d : IsPrime I\nhIP : comap (algebraMap R S) I \u2264 P\n\u22a2 RingHom.ker (Quotient.mk (comap (algebraMap R S) I)) \u2264 P\n[PROOFSTEP]\nsimp [hIP]\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d\u00b2 : Algebra R S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d\u00b9 : IsPrime P\nI : Ideal S\ninst\u271d : IsPrime I\nhIP : comap (algebraMap R S) I \u2264 P\nQ' : Ideal (S \u29f8 I)\nQ'_prime : IsPrime Q'\nhQ' : comap (algebraMap (R \u29f8 comap (algebraMap R S) I) (S \u29f8 I)) Q' = map (Quotient.mk (comap (algebraMap R S) I)) P\n\u22a2 \u2203 Q, Q \u2265 I \u2227 IsPrime Q \u2227 comap (algebraMap R S) Q = P\n[PROOFSTEP]\nhaveI := Q'_prime\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d\u00b2 : Algebra R S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d\u00b9 : IsPrime P\nI : Ideal S\ninst\u271d : IsPrime I\nhIP : comap (algebraMap R S) I \u2264 P\nQ' : Ideal (S \u29f8 I)\nQ'_prime : IsPrime Q'\nhQ' : comap (algebraMap (R \u29f8 comap (algebraMap R S) I) (S \u29f8 I)) Q' = map (Quotient.mk (comap (algebraMap R S) I)) P\nthis : IsPrime Q'\n\u22a2 \u2203 Q, Q \u2265 I \u2227 IsPrime Q \u2227 comap (algebraMap R S) Q = P\n[PROOFSTEP]\nrefine' \u27e8Q'.comap _, le_trans (le_of_eq mk_ker.symm) (ker_le_comap _), \u27e8comap_isPrime _ Q', _\u27e9\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d\u00b2 : Algebra R S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d\u00b9 : IsPrime P\nI : Ideal S\ninst\u271d : IsPrime I\nhIP : comap (algebraMap R S) I \u2264 P\nQ' : Ideal (S \u29f8 I)\nQ'_prime : IsPrime Q'\nhQ' : comap (algebraMap (R \u29f8 comap (algebraMap R S) I) (S \u29f8 I)) Q' = map (Quotient.mk (comap (algebraMap R S) I)) P\nthis : IsPrime Q'\n\u22a2 comap (algebraMap R S) (comap (Quotient.mk I) Q') = P\n[PROOFSTEP]\nrw [comap_comap]\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d\u00b2 : Algebra R S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d\u00b9 : IsPrime P\nI : Ideal S\ninst\u271d : IsPrime I\nhIP : comap (algebraMap R S) I \u2264 P\nQ' : Ideal (S \u29f8 I)\nQ'_prime : IsPrime Q'\nhQ' : comap (algebraMap (R \u29f8 comap (algebraMap R S) I) (S \u29f8 I)) Q' = map (Quotient.mk (comap (algebraMap R S) I)) P\nthis : IsPrime Q'\n\u22a2 comap (RingHom.comp (Quotient.mk I) (algebraMap R S)) Q' = P\n[PROOFSTEP]\nrefine' _root_.trans _ (_root_.trans (congr_arg (comap (Ideal.Quotient.mk (comap (algebraMap R S) I))) hQ') _)\n[GOAL]\ncase intro.intro.refine'_1\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d\u00b2 : Algebra R S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d\u00b9 : IsPrime P\nI : Ideal S\ninst\u271d : IsPrime I\nhIP : comap (algebraMap R S) I \u2264 P\nQ' : Ideal (S \u29f8 I)\nQ'_prime : IsPrime Q'\nhQ' : comap (algebraMap (R \u29f8 comap (algebraMap R S) I) (S \u29f8 I)) Q' = map (Quotient.mk (comap (algebraMap R S) I)) P\nthis : IsPrime Q'\n\u22a2 comap (RingHom.comp (Quotient.mk I) (algebraMap R S)) Q' =\n comap (Quotient.mk (comap (algebraMap R S) I)) (comap (algebraMap (R \u29f8 comap (algebraMap R S) I) (S \u29f8 I)) Q')\n[PROOFSTEP]\nrw [comap_comap]\n[GOAL]\ncase intro.intro.refine'_1\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d\u00b2 : Algebra R S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d\u00b9 : IsPrime P\nI : Ideal S\ninst\u271d : IsPrime I\nhIP : comap (algebraMap R S) I \u2264 P\nQ' : Ideal (S \u29f8 I)\nQ'_prime : IsPrime Q'\nhQ' : comap (algebraMap (R \u29f8 comap (algebraMap R S) I) (S \u29f8 I)) Q' = map (Quotient.mk (comap (algebraMap R S) I)) P\nthis : IsPrime Q'\n\u22a2 comap (RingHom.comp (Quotient.mk I) (algebraMap R S)) Q' =\n comap (RingHom.comp (algebraMap (R \u29f8 comap (algebraMap R S) I) (S \u29f8 I)) (Quotient.mk (comap (algebraMap R S) I))) Q'\n[PROOFSTEP]\nexact congr_arg (comap . Q') (RingHom.ext fun r => rfl)\n[GOAL]\ncase intro.intro.refine'_2\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d\u00b2 : Algebra R S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d\u00b9 : IsPrime P\nI : Ideal S\ninst\u271d : IsPrime I\nhIP : comap (algebraMap R S) I \u2264 P\nQ' : Ideal (S \u29f8 I)\nQ'_prime : IsPrime Q'\nhQ' : comap (algebraMap (R \u29f8 comap (algebraMap R S) I) (S \u29f8 I)) Q' = map (Quotient.mk (comap (algebraMap R S) I)) P\nthis : IsPrime Q'\n\u22a2 comap (Quotient.mk (comap (algebraMap R S) I)) (map (Quotient.mk (comap (algebraMap R S) I)) P) = P\n[PROOFSTEP]\nrefine' _root_.trans (comap_map_of_surjective _ Quotient.mk_surjective _) (sup_eq_left.2 _)\n[GOAL]\ncase intro.intro.refine'_2\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nS : Type u_2\ninst\u271d\u00b3 : CommRing S\nf : R \u2192+* S\nI\u271d J : Ideal S\ninst\u271d\u00b2 : Algebra R S\nH : Algebra.IsIntegral R S\nP : Ideal R\ninst\u271d\u00b9 : IsPrime P\nI : Ideal S\ninst\u271d : IsPrime I\nhIP : comap (algebraMap R S) I \u2264 P\nQ' : Ideal (S \u29f8 I)\nQ'_prime : IsPrime Q'\nhQ' : comap (algebraMap (R \u29f8 comap (algebraMap R S) I) (S \u29f8 I)) Q' = map (Quotient.mk (comap (algebraMap R S) I)) P\nthis : IsPrime Q'\n\u22a2 comap (Quotient.mk (comap (algebraMap R S) I)) \u22a5 \u2264 P\n[PROOFSTEP]\nsimpa [\u2190 RingHom.ker_eq_comap_bot] using hIP\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsDomain S\nH : Algebra.IsIntegral R S\nP : Ideal R\nP_max : IsMaximal P\nhP : RingHom.ker (algebraMap R S) \u2264 P\n\u22a2 \u2203 Q, IsMaximal Q \u2227 comap (algebraMap R S) Q = P\n[PROOFSTEP]\nobtain \u27e8Q, \u27e8Q_prime, hQ\u27e9\u27e9 := exists_ideal_over_prime_of_is_integral' H P hP\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsDomain S\nH : Algebra.IsIntegral R S\nP : Ideal R\nP_max : IsMaximal P\nhP : RingHom.ker (algebraMap R S) \u2264 P\nQ : Ideal S\nQ_prime : IsPrime Q\nhQ : comap (algebraMap R S) Q = P\n\u22a2 \u2203 Q, IsMaximal Q \u2227 comap (algebraMap R S) Q = P\n[PROOFSTEP]\nhaveI : Q.IsPrime := Q_prime\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsDomain S\nH : Algebra.IsIntegral R S\nP : Ideal R\nP_max : IsMaximal P\nhP : RingHom.ker (algebraMap R S) \u2264 P\nQ : Ideal S\nQ_prime : IsPrime Q\nhQ : comap (algebraMap R S) Q = P\nthis : IsPrime Q\n\u22a2 \u2203 Q, IsMaximal Q \u2227 comap (algebraMap R S) Q = P\n[PROOFSTEP]\nexact \u27e8Q, isMaximal_of_isIntegral_of_isMaximal_comap H _ (hQ.symm \u25b8 P_max), hQ\u27e9\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Ideal.Over", "llama_tokens": 24597, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.4036408856346731}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\n\u22a2 Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id) \u2264\n 2 ^ (Finset.card P.parts - 1) * (Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\nhave hX : G.nonuniformWitness \u03b5 U V \u2208 P.nonuniformWitnesses G \u03b5 U := nonuniformWitness_mem_nonuniformWitnesses h\u2082 hV hUV\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\n\u22a2 Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id) \u2264\n 2 ^ (Finset.card P.parts - 1) * (Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\nhave q :\n G.nonuniformWitness \u03b5 U V \\ (star hP G \u03b5 hU V).biUnion id \u2286\n ((atomise U <| P.nonuniformWitnesses G \u03b5 U).parts.filter fun B =>\n B \u2286 G.nonuniformWitness \u03b5 U V \u2227 B.Nonempty).biUnion\n fun B => B \\ ((chunk hP G \u03b5 hU).parts.filter (\u00b7 \u2286 B)).biUnion id :=\n by\n intro x hx\n rw [\u2190 biUnion_filter_atomise hX (G.nonuniformWitness_subset h\u2082), star, mem_sdiff, mem_biUnion] at hx \n simp only [not_exists, mem_biUnion, and_imp, exists_prop, mem_filter, not_and, mem_sdiff, id.def, mem_sdiff] at hx \u22a2\n obtain \u27e8\u27e8B, hB\u2081, hB\u2082\u27e9, hx\u27e9 := hx\n exact \u27e8B, hB\u2081, hB\u2082, fun A hA AB => hx A hA <| AB.trans hB\u2081.2.1\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\n\u22a2 SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nx : \u03b1\nhx : x \u2208 SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id\n\u22a2 x \u2208\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\n[PROOFSTEP]\nrw [\u2190 biUnion_filter_atomise hX (G.nonuniformWitness_subset h\u2082), star, mem_sdiff, mem_biUnion] at hx \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nx : \u03b1\nhx :\n (\u2203 a,\n a \u2208\n filter (fun u => u \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty u)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts \u2227\n x \u2208 id a) \u2227\n \u00acx \u2208 Finset.biUnion (filter (fun x => x \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V) (chunk hP G \u03b5 hU).parts) id\n\u22a2 x \u2208\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\n[PROOFSTEP]\nsimp only [not_exists, mem_biUnion, and_imp, exists_prop, mem_filter, not_and, mem_sdiff, id.def, mem_sdiff] at hx \u22a2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nx : \u03b1\nhx :\n (\u2203 a,\n (a \u2208 (atomise U (nonuniformWitnesses P G \u03b5 U)).parts \u2227\n a \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty a) \u2227\n x \u2208 a) \u2227\n \u2200 (x_1 : Finset \u03b1), x_1 \u2208 (chunk hP G \u03b5 hU).parts \u2192 x_1 \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2192 \u00acx \u2208 x_1\n\u22a2 \u2203 a,\n (a \u2208 (atomise U (nonuniformWitnesses P G \u03b5 U)).parts \u2227\n a \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty a) \u2227\n x \u2208 a \u2227 \u2200 (x_1 : Finset \u03b1), x_1 \u2208 (chunk hP G \u03b5 hU).parts \u2192 x_1 \u2286 a \u2192 \u00acx \u2208 x_1\n[PROOFSTEP]\nobtain \u27e8\u27e8B, hB\u2081, hB\u2082\u27e9, hx\u27e9 := hx\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nx : \u03b1\nhx : \u2200 (x_1 : Finset \u03b1), x_1 \u2208 (chunk hP G \u03b5 hU).parts \u2192 x_1 \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2192 \u00acx \u2208 x_1\nB : Finset \u03b1\nhB\u2081 :\n B \u2208 (atomise U (nonuniformWitnesses P G \u03b5 U)).parts \u2227 B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B\nhB\u2082 : x \u2208 B\n\u22a2 \u2203 a,\n (a \u2208 (atomise U (nonuniformWitnesses P G \u03b5 U)).parts \u2227\n a \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty a) \u2227\n x \u2208 a \u2227 \u2200 (x_1 : Finset \u03b1), x_1 \u2208 (chunk hP G \u03b5 hU).parts \u2192 x_1 \u2286 a \u2192 \u00acx \u2208 x_1\n[PROOFSTEP]\nexact \u27e8B, hB\u2081, hB\u2082, fun A hA AB => hx A hA <| AB.trans hB\u2081.2.1\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nq :\n SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\n\u22a2 Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id) \u2264\n 2 ^ (Finset.card P.parts - 1) * (Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\napply (card_le_of_subset q).trans (card_biUnion_le.trans _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nq :\n SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\n\u22a2 \u2211 a in\n filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts,\n Finset.card (a \\ Finset.biUnion (filter (fun x => x \u2286 a) (chunk hP G \u03b5 hU).parts) id) \u2264\n 2 ^ (Finset.card P.parts - 1) * (Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\ntrans\n \u2211 _i in (atomise U <| P.nonuniformWitnesses G \u03b5 U).parts.filter fun B => B \u2286 G.nonuniformWitness \u03b5 U V \u2227 B.Nonempty, m\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nq :\n SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\n\u22a2 \u2211 a in\n filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts,\n Finset.card (a \\ Finset.biUnion (filter (fun x => x \u2286 a) (chunk hP G \u03b5 hU).parts) id) \u2264\n \u2211 _i in\n filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts,\n Fintype.card \u03b1 / stepBound (Finset.card P.parts)\n[PROOFSTEP]\nsuffices\n \u2200 B \u2208 (atomise U <| P.nonuniformWitnesses G \u03b5 U).parts,\n (B \\ ((chunk hP G \u03b5 hU).parts.filter (\u00b7 \u2286 B)).biUnion id).card \u2264 m\n by exact sum_le_sum fun B hB => this B <| filter_subset _ _ hB\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nq :\n SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\nthis :\n \u2200 (B : Finset \u03b1),\n B \u2208 (atomise U (nonuniformWitnesses P G \u03b5 U)).parts \u2192\n Finset.card (B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id) \u2264\n Fintype.card \u03b1 / stepBound (Finset.card P.parts)\n\u22a2 \u2211 a in\n filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts,\n Finset.card (a \\ Finset.biUnion (filter (fun x => x \u2286 a) (chunk hP G \u03b5 hU).parts) id) \u2264\n \u2211 _i in\n filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts,\n Fintype.card \u03b1 / stepBound (Finset.card P.parts)\n[PROOFSTEP]\nexact sum_le_sum fun B hB => this B <| filter_subset _ _ hB\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nq :\n SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\n\u22a2 \u2200 (B : Finset \u03b1),\n B \u2208 (atomise U (nonuniformWitnesses P G \u03b5 U)).parts \u2192\n Finset.card (B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id) \u2264\n Fintype.card \u03b1 / stepBound (Finset.card P.parts)\n[PROOFSTEP]\nintro B hB\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nq :\n SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\nB : Finset \u03b1\nhB : B \u2208 (atomise U (nonuniformWitnesses P G \u03b5 U)).parts\n\u22a2 Finset.card (B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id) \u2264\n Fintype.card \u03b1 / stepBound (Finset.card P.parts)\n[PROOFSTEP]\nunfold chunk\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nq :\n SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\nB : Finset \u03b1\nhB : B \u2208 (atomise U (nonuniformWitnesses P G \u03b5 U)).parts\n\u22a2 Finset.card\n (B \\\n Finset.biUnion\n (filter (fun x => x \u2286 B)\n (if hUcard :\n Finset.card U =\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts) then\n equitabilise\n (_ :\n (4 ^ Finset.card P.parts -\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts)) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts)) +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1) =\n Finset.card U)\n else\n equitabilise\n (_ :\n (4 ^ Finset.card P.parts -\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n 1)) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts)) +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n 1) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1) =\n Finset.card U)).parts)\n id) \u2264\n Fintype.card \u03b1 / stepBound (Finset.card P.parts)\n[PROOFSTEP]\nsplit_ifs with h\u2081\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nq :\n SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\nB : Finset \u03b1\nhB : B \u2208 (atomise U (nonuniformWitnesses P G \u03b5 U)).parts\nh\u2081 :\n Finset.card U =\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts)\n\u22a2 Finset.card\n (B \\\n Finset.biUnion\n (filter (fun x => x \u2286 B)\n (equitabilise\n (_ :\n (4 ^ Finset.card P.parts -\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts)) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts)) +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1) =\n Finset.card U)).parts)\n id) \u2264\n Fintype.card \u03b1 / stepBound (Finset.card P.parts)\n[PROOFSTEP]\nconvert card_parts_equitabilise_subset_le _ (card_aux\u2081 h\u2081) hB\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nq :\n SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\nB : Finset \u03b1\nhB : B \u2208 (atomise U (nonuniformWitnesses P G \u03b5 U)).parts\nh\u2081 :\n \u00acFinset.card U =\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts)\n\u22a2 Finset.card\n (B \\\n Finset.biUnion\n (filter (fun x => x \u2286 B)\n (equitabilise\n (_ :\n (4 ^ Finset.card P.parts -\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n 1)) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts)) +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n 1) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1) =\n Finset.card U)).parts)\n id) \u2264\n Fintype.card \u03b1 / stepBound (Finset.card P.parts)\n[PROOFSTEP]\nconvert card_parts_equitabilise_subset_le _ (card_aux\u2082 hP hU h\u2081) hB\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nq :\n SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\n\u22a2 \u2211 _i in\n filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts,\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) \u2264\n 2 ^ (Finset.card P.parts - 1) * (Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\nrw [sum_const]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nq :\n SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\n\u22a2 Finset.card\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts) \u2022\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264\n 2 ^ (Finset.card P.parts - 1) * (Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\nrefine' mul_le_mul_right' _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nq :\n SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\n\u22a2 Finset.card\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts) \u2264\n 2 ^ (Finset.card P.parts - 1)\n[PROOFSTEP]\nhave t := card_filter_atomise_le_two_pow (s := U) hX\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nq :\n SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\nt :\n Finset.card\n (filter (fun u => u \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty u)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts) \u2264\n 2 ^ (Finset.card (nonuniformWitnesses P G \u03b5 U) - 1)\n\u22a2 Finset.card\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts) \u2264\n 2 ^ (Finset.card P.parts - 1)\n[PROOFSTEP]\nrefine' t.trans (pow_le_pow (by norm_num) <| tsub_le_tsub_right _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nq :\n SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\nt :\n Finset.card\n (filter (fun u => u \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty u)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts) \u2264\n 2 ^ (Finset.card (nonuniformWitnesses P G \u03b5 U) - 1)\n\u22a2 1 \u2264 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nh\u2082 : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhX : SimpleGraph.nonuniformWitness G \u03b5 U V \u2208 nonuniformWitnesses P G \u03b5 U\nq :\n SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id \u2286\n Finset.biUnion\n (filter (fun B => B \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty B)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts)\n fun B => B \\ Finset.biUnion (filter (fun x => x \u2286 B) (chunk hP G \u03b5 hU).parts) id\nt :\n Finset.card\n (filter (fun u => u \u2286 SimpleGraph.nonuniformWitness G \u03b5 U V \u2227 Finset.Nonempty u)\n (atomise U (nonuniformWitnesses P G \u03b5 U)).parts) \u2264\n 2 ^ (Finset.card (nonuniformWitnesses P G \u03b5 U) - 1)\n\u22a2 Finset.card (nonuniformWitnesses P G \u03b5 U) \u2264 Finset.card P.parts\n[PROOFSTEP]\nexact card_image_le.trans (card_le_of_subset <| filter_subset _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 (1 - \u03b5 / 10) * \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V)) \u2264\n \u2191(Finset.card (Finset.biUnion (star hP G \u03b5 hU V) id))\n[PROOFSTEP]\nhave hP\u2081 : 0 < P.parts.card := Finset.card_pos.2 \u27e8_, hU\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 (1 - \u03b5 / 10) * \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V)) \u2264\n \u2191(Finset.card (Finset.biUnion (star hP G \u03b5 hU V) id))\n[PROOFSTEP]\nhave : (\u21912 ^ P.parts.card : \u211d) * m / (U.card * \u03b5) \u2264 \u03b5 / 10 :=\n by\n rw [\u2190 div_div, div_le_iff']\n swap\n sz_positivity\n refine' le_of_mul_le_mul_left _ (pow_pos zero_lt_two P.parts.card)\n calc\n \u21912 ^ P.parts.card * ((\u21912 ^ P.parts.card * m : \u211d) / U.card) = ((2 : \u211d) * 2) ^ P.parts.card * m / U.card := by\n rw [mul_pow, \u2190 mul_div_assoc, mul_assoc]\n _ = \u21914 ^ P.parts.card * m / U.card := by norm_num\n _ \u2264 1 := (div_le_one_of_le (pow_mul_m_le_card_part hP hU) (cast_nonneg _))\n _ \u2264 \u21912 ^ P.parts.card * \u03b5 ^ 2 / 10 :=\n by\n refine' (one_le_sq_iff <| by positivity).1 _\n rw [div_pow, mul_pow, pow_right_comm, \u2190 pow_mul \u03b5, one_le_div (sq_pos_of_ne_zero (10 : \u211d) <| by norm_num)]\n calc\n (\u219110 ^ 2) = 100 := by norm_num\n _ \u2264 \u21914 ^ P.parts.card * \u03b5 ^ 5 := hP\u03b5\n _ \u2264 \u21914 ^ P.parts.card * \u03b5 ^ 4 :=\n (mul_le_mul_of_nonneg_left (pow_le_pow_of_le_one (by sz_positivity) h\u03b5\u2081 <| le_succ _) (by positivity))\n _ = (\u21912 ^ 2) ^ P.parts.card * \u03b5 ^ (2 * 2) := by norm_num\n _ = \u21912 ^ P.parts.card * (\u03b5 * (\u03b5 / 10)) := by rw [mul_div_assoc, sq, mul_div_assoc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\n[PROOFSTEP]\nrw [\u2190 div_div, div_le_iff']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / \u2191(Finset.card U) \u2264 \u03b5 * (\u03b5 / 10)\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 0 < \u03b5\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 0 < \u03b5\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / \u2191(Finset.card U) \u2264 \u03b5 * (\u03b5 / 10)\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / \u2191(Finset.card U) \u2264 \u03b5 * (\u03b5 / 10)\n[PROOFSTEP]\nrefine' le_of_mul_le_mul_left _ (pow_pos zero_lt_two P.parts.card)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 2 ^ Finset.card P.parts *\n (2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / \u2191(Finset.card U)) \u2264\n 2 ^ Finset.card P.parts * (\u03b5 * (\u03b5 / 10))\n[PROOFSTEP]\ncalc\n \u21912 ^ P.parts.card * ((\u21912 ^ P.parts.card * m : \u211d) / U.card) = ((2 : \u211d) * 2) ^ P.parts.card * m / U.card := by\n rw [mul_pow, \u2190 mul_div_assoc, mul_assoc]\n _ = \u21914 ^ P.parts.card * m / U.card := by norm_num\n _ \u2264 1 := (div_le_one_of_le (pow_mul_m_le_card_part hP hU) (cast_nonneg _))\n _ \u2264 \u21912 ^ P.parts.card * \u03b5 ^ 2 / 10 :=\n by\n refine' (one_le_sq_iff <| by positivity).1 _\n rw [div_pow, mul_pow, pow_right_comm, \u2190 pow_mul \u03b5, one_le_div (sq_pos_of_ne_zero (10 : \u211d) <| by norm_num)]\n calc\n (\u219110 ^ 2) = 100 := by norm_num\n _ \u2264 \u21914 ^ P.parts.card * \u03b5 ^ 5 := hP\u03b5\n _ \u2264 \u21914 ^ P.parts.card * \u03b5 ^ 4 :=\n (mul_le_mul_of_nonneg_left (pow_le_pow_of_le_one (by sz_positivity) h\u03b5\u2081 <| le_succ _) (by positivity))\n _ = (\u21912 ^ 2) ^ P.parts.card * \u03b5 ^ (2 * 2) := by norm_num\n _ = \u21912 ^ P.parts.card * (\u03b5 * (\u03b5 / 10)) := by rw [mul_div_assoc, sq, mul_div_assoc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 2 ^ Finset.card P.parts *\n (2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / \u2191(Finset.card U)) =\n (2 * 2) ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / \u2191(Finset.card U)\n[PROOFSTEP]\nrw [mul_pow, \u2190 mul_div_assoc, mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 (2 * 2) ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / \u2191(Finset.card U) =\n 4 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / \u2191(Finset.card U)\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 1 \u2264 2 ^ Finset.card P.parts * \u03b5 ^ 2 / 10\n[PROOFSTEP]\nrefine' (one_le_sq_iff <| by positivity).1 _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 0 \u2264 2 ^ Finset.card P.parts * \u03b5 ^ 2 / 10\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 1 \u2264 (2 ^ Finset.card P.parts * \u03b5 ^ 2 / 10) ^ 2\n[PROOFSTEP]\nrw [div_pow, mul_pow, pow_right_comm, \u2190 pow_mul \u03b5, one_le_div (sq_pos_of_ne_zero (10 : \u211d) <| by norm_num)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 10 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 10 ^ 2 \u2264 (2 ^ 2) ^ Finset.card P.parts * \u03b5 ^ (2 * 2)\n[PROOFSTEP]\ncalc\n (\u219110 ^ 2) = 100 := by norm_num\n _ \u2264 \u21914 ^ P.parts.card * \u03b5 ^ 5 := hP\u03b5\n _ \u2264 \u21914 ^ P.parts.card * \u03b5 ^ 4 :=\n (mul_le_mul_of_nonneg_left (pow_le_pow_of_le_one (by sz_positivity) h\u03b5\u2081 <| le_succ _) (by positivity))\n _ = (\u21912 ^ 2) ^ P.parts.card * \u03b5 ^ (2 * 2) := by norm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 10 ^ 2 = 100\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 0 \u2264 \u03b5\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 0 \u2264 4 ^ Finset.card P.parts\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 4 ^ Finset.card P.parts * \u03b5 ^ 4 = (2 ^ 2) ^ Finset.card P.parts * \u03b5 ^ (2 * 2)\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\n\u22a2 2 ^ Finset.card P.parts * \u03b5 ^ 2 / 10 = 2 ^ Finset.card P.parts * (\u03b5 * (\u03b5 / 10))\n[PROOFSTEP]\nrw [mul_div_assoc, sq, mul_div_assoc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\nthis : 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\n\u22a2 (1 - \u03b5 / 10) * \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V)) \u2264\n \u2191(Finset.card (Finset.biUnion (star hP G \u03b5 hU V) id))\n[PROOFSTEP]\ncalc\n (\u21911 - \u03b5 / 10) * (G.nonuniformWitness \u03b5 U V).card \u2264\n (\u21911 - \u21912 ^ P.parts.card * m / (U.card * \u03b5)) * (G.nonuniformWitness \u03b5 U V).card :=\n mul_le_mul_of_nonneg_right (sub_le_sub_left this _) (cast_nonneg _)\n _ = (G.nonuniformWitness \u03b5 U V).card - \u21912 ^ P.parts.card * m / (U.card * \u03b5) * (G.nonuniformWitness \u03b5 U V).card := by\n rw [sub_mul, one_mul]\n _ \u2264 (G.nonuniformWitness \u03b5 U V).card - \u21912 ^ (P.parts.card - 1) * m :=\n by\n refine' sub_le_sub_left _ _\n have : (2 : \u211d) ^ P.parts.card = \u21912 ^ (P.parts.card - 1) * 2 := by\n rw [\u2190 _root_.pow_succ', tsub_add_cancel_of_le (succ_le_iff.2 hP\u2081)]\n rw [\u2190 mul_div_right_comm, this, mul_right_comm _ (2 : \u211d), mul_assoc, le_div_iff]\n refine' mul_le_mul_of_nonneg_left _ (by positivity)\n exact (G.le_card_nonuniformWitness hunif).trans (le_mul_of_one_le_left (cast_nonneg _) one_le_two)\n have := Finset.card_pos.mpr (P.nonempty_of_mem_parts hU)\n sz_positivity\n _ \u2264 ((star hP G \u03b5 hU V).biUnion id).card :=\n by\n rw [sub_le_comm, \u2190 cast_sub (card_le_of_subset <| biUnion_star_subset_nonuniformWitness hP G \u03b5 hU V), \u2190\n card_sdiff (biUnion_star_subset_nonuniformWitness hP G \u03b5 hU V)]\n exact_mod_cast card_nonuniformWitness_sdiff_biUnion_star hV hUV hunif\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\nthis : 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\n\u22a2 (1 - 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5)) *\n \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V)) =\n \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V)) -\n 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) *\n \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V))\n[PROOFSTEP]\nrw [sub_mul, one_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\nthis : 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\n\u22a2 \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V)) -\n 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) *\n \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V)) \u2264\n \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V)) -\n 2 ^ (Finset.card P.parts - 1) * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\nrefine' sub_le_sub_left _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\nthis : 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\n\u22a2 2 ^ (Finset.card P.parts - 1) * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264\n 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) *\n \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V))\n[PROOFSTEP]\nhave : (2 : \u211d) ^ P.parts.card = \u21912 ^ (P.parts.card - 1) * 2 := by\n rw [\u2190 _root_.pow_succ', tsub_add_cancel_of_le (succ_le_iff.2 hP\u2081)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\nthis : 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\n\u22a2 2 ^ Finset.card P.parts = 2 ^ (Finset.card P.parts - 1) * 2\n[PROOFSTEP]\nrw [\u2190 _root_.pow_succ', tsub_add_cancel_of_le (succ_le_iff.2 hP\u2081)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\nthis\u271d : 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\nthis : 2 ^ Finset.card P.parts = 2 ^ (Finset.card P.parts - 1) * 2\n\u22a2 2 ^ (Finset.card P.parts - 1) * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264\n 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) *\n \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V))\n[PROOFSTEP]\nrw [\u2190 mul_div_right_comm, this, mul_right_comm _ (2 : \u211d), mul_assoc, le_div_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\nthis\u271d : 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\nthis : 2 ^ Finset.card P.parts = 2 ^ (Finset.card P.parts - 1) * 2\n\u22a2 2 ^ (Finset.card P.parts - 1) * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) * (\u2191(Finset.card U) * \u03b5) \u2264\n 2 ^ (Finset.card P.parts - 1) * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) *\n (2 * \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V)))\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\nthis\u271d : 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\nthis : 2 ^ Finset.card P.parts = 2 ^ (Finset.card P.parts - 1) * 2\n\u22a2 0 < \u2191(Finset.card U) * \u03b5\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left _ (by positivity)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\nthis\u271d : 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\nthis : 2 ^ Finset.card P.parts = 2 ^ (Finset.card P.parts - 1) * 2\n\u22a2 0 \u2264 2 ^ (Finset.card P.parts - 1) * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\nthis\u271d : 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\nthis : 2 ^ Finset.card P.parts = 2 ^ (Finset.card P.parts - 1) * 2\n\u22a2 \u2191(Finset.card U) * \u03b5 \u2264 2 * \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V))\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\nthis\u271d : 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\nthis : 2 ^ Finset.card P.parts = 2 ^ (Finset.card P.parts - 1) * 2\n\u22a2 0 < \u2191(Finset.card U) * \u03b5\n[PROOFSTEP]\nexact (G.le_card_nonuniformWitness hunif).trans (le_mul_of_one_le_left (cast_nonneg _) one_le_two)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\nthis\u271d : 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\nthis : 2 ^ Finset.card P.parts = 2 ^ (Finset.card P.parts - 1) * 2\n\u22a2 0 < \u2191(Finset.card U) * \u03b5\n[PROOFSTEP]\nhave := Finset.card_pos.mpr (P.nonempty_of_mem_parts hU)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\nthis\u271d\u00b9 : 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\nthis\u271d : 2 ^ Finset.card P.parts = 2 ^ (Finset.card P.parts - 1) * 2\nthis : 0 < Finset.card U\n\u22a2 0 < \u2191(Finset.card U) * \u03b5\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\nthis : 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\n\u22a2 \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V)) -\n 2 ^ (Finset.card P.parts - 1) * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264\n \u2191(Finset.card (Finset.biUnion (star hP G \u03b5 hU V) id))\n[PROOFSTEP]\nrw [sub_le_comm, \u2190 cast_sub (card_le_of_subset <| biUnion_star_subset_nonuniformWitness hP G \u03b5 hU V), \u2190\n card_sdiff (biUnion_star_subset_nonuniformWitness hP G \u03b5 hU V)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2081 : 0 < Finset.card P.parts\nthis : 2 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Finset.card U) * \u03b5) \u2264 \u03b5 / 10\n\u22a2 \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V \\ Finset.biUnion (star hP G \u03b5 hU V) id)) \u2264\n 2 ^ (Finset.card P.parts - 1) * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\nexact_mod_cast card_nonuniformWitness_sdiff_biUnion_star hV hUV hunif\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhm : Fintype.card \u03b1 / stepBound (Finset.card P.parts) \u2260 0\n\u22a2 Finset.card (chunk hP G \u03b5 hU).parts = 4 ^ Finset.card P.parts\n[PROOFSTEP]\nunfold chunk\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhm : Fintype.card \u03b1 / stepBound (Finset.card P.parts) \u2260 0\n\u22a2 Finset.card\n (if hUcard :\n Finset.card U =\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts) then\n equitabilise\n (_ :\n (4 ^ Finset.card P.parts -\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts)) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts)) +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1) =\n Finset.card U)\n else\n equitabilise\n (_ :\n (4 ^ Finset.card P.parts -\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n 1)) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts)) +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n 1) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1) =\n Finset.card U)).parts =\n 4 ^ Finset.card P.parts\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhm : Fintype.card \u03b1 / stepBound (Finset.card P.parts) \u2260 0\nh\u271d :\n Finset.card U =\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts)\n\u22a2 Finset.card\n (equitabilise\n (_ :\n (4 ^ Finset.card P.parts -\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts)) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts)) +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1) =\n Finset.card U)).parts =\n 4 ^ Finset.card P.parts\n[PROOFSTEP]\nrw [card_parts_equitabilise _ _ hm, tsub_add_cancel_of_le]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhm : Fintype.card \u03b1 / stepBound (Finset.card P.parts) \u2260 0\nh\u271d :\n Finset.card U =\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts)\n\u22a2 Fintype.card \u03b1 / Finset.card P.parts - Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts \u2264\n 4 ^ Finset.card P.parts\n[PROOFSTEP]\nexact le_of_lt a_add_one_le_four_pow_parts_card\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhm : Fintype.card \u03b1 / stepBound (Finset.card P.parts) \u2260 0\nh\u271d :\n \u00acFinset.card U =\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts)\n\u22a2 Finset.card\n (equitabilise\n (_ :\n (4 ^ Finset.card P.parts -\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n 1)) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts)) +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n 1) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1) =\n Finset.card U)).parts =\n 4 ^ Finset.card P.parts\n[PROOFSTEP]\nrw [card_parts_equitabilise _ _ hm, tsub_add_cancel_of_le a_add_one_le_four_pow_parts_card]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhs : s \u2208 (chunk hP G \u03b5 hU).parts\n\u22a2 Finset.card s = Fintype.card \u03b1 / stepBound (Finset.card P.parts) \u2228\n Finset.card s = Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1\n[PROOFSTEP]\nunfold chunk at hs \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhs :\n s \u2208\n (if hUcard :\n Finset.card U =\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts) then\n equitabilise\n (_ :\n (4 ^ Finset.card P.parts -\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts)) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts)) +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1) =\n Finset.card U)\n else\n equitabilise\n (_ :\n (4 ^ Finset.card P.parts -\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n 1)) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts)) +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n 1) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1) =\n Finset.card U)).parts\n\u22a2 Finset.card s = Fintype.card \u03b1 / stepBound (Finset.card P.parts) \u2228\n Finset.card s = Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1\n[PROOFSTEP]\nsplit_ifs at hs \n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nh\u271d :\n Finset.card U =\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts)\nhs :\n s \u2208\n (equitabilise\n (_ :\n (4 ^ Finset.card P.parts -\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts)) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts)) +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1) =\n Finset.card U)).parts\n\u22a2 Finset.card s = Fintype.card \u03b1 / stepBound (Finset.card P.parts) \u2228\n Finset.card s = Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1\n[PROOFSTEP]\nexact card_eq_of_mem_parts_equitabilise hs\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nh\u271d :\n \u00acFinset.card U =\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts)\nhs :\n s \u2208\n (equitabilise\n (_ :\n (4 ^ Finset.card P.parts -\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n 1)) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts)) +\n (Fintype.card \u03b1 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n 1) *\n (Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1) =\n Finset.card U)).parts\n\u22a2 Finset.card s = Fintype.card \u03b1 / stepBound (Finset.card P.parts) \u2228\n Finset.card s = Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1\n[PROOFSTEP]\nexact card_eq_of_mem_parts_equitabilise hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhs : s \u2208 (chunk hP G \u03b5 hU).parts\ni : Finset.card s = Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1\n\u22a2 Fintype.card \u03b1 / stepBound (Finset.card P.parts) \u2264 Finset.card s\n[PROOFSTEP]\nsimp [i]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhs : s \u2208 (chunk hP G \u03b5 hU).parts\ni : Finset.card s = Fintype.card \u03b1 / stepBound (Finset.card P.parts)\n\u22a2 Finset.card s \u2264 Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1\n[PROOFSTEP]\nsimp [i]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\n\u22a2 \u2191(Finset.card (Finset.biUnion (star hP G \u03b5 hU V) id)) \u2264\n \u2191(Finset.card (star hP G \u03b5 hU V)) * (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n[PROOFSTEP]\nexact_mod_cast card_biUnion_le_card_mul _ _ _ fun s hs => card_le_m_add_one_of_mem_chunk_parts <| star_subset_chunk hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nh\ud835\udc9c : \ud835\udc9c \u2286 (chunk hP G \u03b5 hU).parts\nhs : s \u2208 \ud835\udc9c\n\u22a2 \u2191(Finset.card \ud835\udc9c) * \u2191(Finset.card s) *\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) /\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)) \u2264\n \u2191(Finset.card (sup \ud835\udc9c id))\n[PROOFSTEP]\nrw [mul_div_assoc', div_le_iff coe_m_add_one_pos, mul_right_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nh\ud835\udc9c : \ud835\udc9c \u2286 (chunk hP G \u03b5 hU).parts\nhs : s \u2208 \ud835\udc9c\n\u22a2 \u2191(Finset.card \ud835\udc9c) * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) * \u2191(Finset.card s) \u2264\n \u2191(Finset.card (sup \ud835\udc9c id)) * (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n[PROOFSTEP]\nrefine' mul_le_mul _ _ (cast_nonneg _) (cast_nonneg _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nh\ud835\udc9c : \ud835\udc9c \u2286 (chunk hP G \u03b5 hU).parts\nhs : s \u2208 \ud835\udc9c\n\u22a2 \u2191(Finset.card \ud835\udc9c) * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264 \u2191(Finset.card (sup \ud835\udc9c id))\n[PROOFSTEP]\nrw [\u2190 (ofSubset _ h\ud835\udc9c rfl).sum_card_parts, ofSubset_parts, \u2190 cast_mul, cast_le]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nh\ud835\udc9c : \ud835\udc9c \u2286 (chunk hP G \u03b5 hU).parts\nhs : s \u2208 \ud835\udc9c\n\u22a2 Finset.card \ud835\udc9c * (Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264 \u2211 i in \ud835\udc9c, Finset.card i\n[PROOFSTEP]\nexact card_nsmul_le_sum _ _ _ fun x hx => m_le_card_of_mem_chunk_parts <| h\ud835\udc9c hx\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nh\ud835\udc9c : \ud835\udc9c \u2286 (chunk hP G \u03b5 hU).parts\nhs : s \u2208 \ud835\udc9c\n\u22a2 \u2191(Finset.card s) \u2264 \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1\n[PROOFSTEP]\nexact_mod_cast card_le_m_add_one_of_mem_chunk_parts (h\ud835\udc9c hs)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nm_pos : 0 < \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\nh\ud835\udc9c : \ud835\udc9c \u2286 (chunk hP G \u03b5 hU).parts\nhs : s \u2208 \ud835\udc9c\n\u22a2 \u2191(Finset.card (sup \ud835\udc9c id)) \u2264\n \u2191(Finset.card \ud835\udc9c) * \u2191(Finset.card s) *\n ((\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\n[PROOFSTEP]\nrw [sup_eq_biUnion, mul_div_assoc', le_div_iff m_pos, mul_right_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nm_pos : 0 < \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\nh\ud835\udc9c : \ud835\udc9c \u2286 (chunk hP G \u03b5 hU).parts\nhs : s \u2208 \ud835\udc9c\n\u22a2 \u2191(Finset.card (Finset.biUnion \ud835\udc9c id)) * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264\n \u2191(Finset.card \ud835\udc9c) * (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) * \u2191(Finset.card s)\n[PROOFSTEP]\nrefine' mul_le_mul _ _ (cast_nonneg _) (by positivity)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nm_pos : 0 < \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\nh\ud835\udc9c : \ud835\udc9c \u2286 (chunk hP G \u03b5 hU).parts\nhs : s \u2208 \ud835\udc9c\n\u22a2 0 \u2264 \u2191(Finset.card \ud835\udc9c) * (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n[PROOFSTEP]\npositivity\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nm_pos : 0 < \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\nh\ud835\udc9c : \ud835\udc9c \u2286 (chunk hP G \u03b5 hU).parts\nhs : s \u2208 \ud835\udc9c\n\u22a2 \u2191(Finset.card (Finset.biUnion \ud835\udc9c id)) \u2264 \u2191(Finset.card \ud835\udc9c) * (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nm_pos : 0 < \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\nh\ud835\udc9c : \ud835\udc9c \u2286 (chunk hP G \u03b5 hU).parts\nhs : s \u2208 \ud835\udc9c\n\u22a2 Finset.card (Finset.biUnion \ud835\udc9c id) \u2264 Finset.card \ud835\udc9c * (Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1)\n[PROOFSTEP]\nrefine' card_biUnion_le_card_mul _ _ _ fun x hx => _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nm_pos : 0 < \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\nh\ud835\udc9c : \ud835\udc9c \u2286 (chunk hP G \u03b5 hU).parts\nhs : s \u2208 \ud835\udc9c\nx : Finset \u03b1\nhx : x \u2208 \ud835\udc9c\n\u22a2 Finset.card (id x) \u2264 Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1\n[PROOFSTEP]\napply card_le_m_add_one_of_mem_chunk_parts (h\ud835\udc9c hx)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nm_pos : 0 < \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\nh\ud835\udc9c : \ud835\udc9c \u2286 (chunk hP G \u03b5 hU).parts\nhs : s \u2208 \ud835\udc9c\n\u22a2 \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264 \u2191(Finset.card s)\n[PROOFSTEP]\nexact_mod_cast m_le_card_of_mem_chunk_parts (h\ud835\udc9c hs)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\n\u22a2 1 - \u03b5 ^ 5 / 50 \u2264\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)) ^\n 2\n[PROOFSTEP]\nhave : (m : \u211d) / (m + 1) = 1 - 1 / (m + 1) := by rw [one_sub_div coe_m_add_one_pos.ne', add_sub_cancel]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\n\u22a2 \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) =\n 1 - 1 / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n[PROOFSTEP]\nrw [one_sub_div coe_m_add_one_pos.ne', add_sub_cancel]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nthis :\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) =\n 1 - 1 / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n\u22a2 1 - \u03b5 ^ 5 / 50 \u2264\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)) ^\n 2\n[PROOFSTEP]\nrw [this, sub_sq, one_pow, mul_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nthis :\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) =\n 1 - 1 / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n\u22a2 1 - \u03b5 ^ 5 / 50 \u2264\n 1 - 2 * (1 / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)) +\n (1 / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)) ^ 2\n[PROOFSTEP]\nrefine' le_trans _ (le_add_of_nonneg_right <| sq_nonneg _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nthis :\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) =\n 1 - 1 / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n\u22a2 1 - \u03b5 ^ 5 / 50 \u2264 1 - 2 * (1 / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1))\n[PROOFSTEP]\nrw [sub_le_sub_iff_left, \u2190 le_div_iff' (show (0 : \u211d) < 2 by norm_num), div_div, one_div_le coe_m_add_one_pos,\n one_div_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nthis :\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) =\n 1 - 1 / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nthis :\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) =\n 1 - 1 / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n\u22a2 50 * 2 / \u03b5 ^ 5 \u2264 \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nthis :\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) =\n 1 - 1 / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n\u22a2 0 < \u03b5 ^ 5 / (50 * 2)\n[PROOFSTEP]\nrefine' le_trans _ (le_add_of_nonneg_right zero_le_one)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nthis :\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) =\n 1 - 1 / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n\u22a2 50 * 2 / \u03b5 ^ 5 \u2264 \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nthis :\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) =\n 1 - 1 / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n\u22a2 0 < \u03b5 ^ 5 / (50 * 2)\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nthis :\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) =\n 1 - 1 / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n\u22a2 100 / \u03b5 ^ 5 \u2264 \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nthis :\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) =\n 1 - 1 / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n\u22a2 0 < \u03b5 ^ 5 / (50 * 2)\n[PROOFSTEP]\napply hundred_div_\u03b5_pow_five_le_m hP\u03b1 hP\u03b5\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nthis :\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) =\n 1 - 1 / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n\u22a2 0 < \u03b5 ^ 5 / (50 * 2)\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 ((\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))) ^\n 2 \u2264\n 1 + \u03b5 ^ 5 / 49\n[PROOFSTEP]\nrw [same_add_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 (1 + 1 / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))) ^ 2 \u2264 1 + \u03b5 ^ 5 / 49\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2260 0\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2260 0\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 (1 + 1 / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))) ^ 2 \u2264 1 + \u03b5 ^ 5 / 49\n[PROOFSTEP]\nhave : \u21911 + \u21911 / (m : \u211d) \u2264 \u21911 + \u03b5 ^ 5 / 100 :=\n by\n rw [add_le_add_iff_left, \u2190 one_div_div (100 : \u211d)]\n exact one_div_le_one_div_of_le (by sz_positivity) (hundred_div_\u03b5_pow_five_le_m hP\u03b1 hP\u03b5)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 1 + 1 / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264 1 + \u03b5 ^ 5 / 100\n[PROOFSTEP]\nrw [add_le_add_iff_left, \u2190 one_div_div (100 : \u211d)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 1 / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264 1 / (100 / \u03b5 ^ 5)\n[PROOFSTEP]\nexact one_div_le_one_div_of_le (by sz_positivity) (hundred_div_\u03b5_pow_five_le_m hP\u03b1 hP\u03b5)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 0 < 100 / \u03b5 ^ 5\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : 1 + 1 / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264 1 + \u03b5 ^ 5 / 100\n\u22a2 (1 + 1 / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))) ^ 2 \u2264 1 + \u03b5 ^ 5 / 49\n[PROOFSTEP]\nrefine' (pow_le_pow_of_le_left _ this 2).trans _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : 1 + 1 / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264 1 + \u03b5 ^ 5 / 100\n\u22a2 0 \u2264 1 + 1 / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\npositivity\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : 1 + 1 / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264 1 + \u03b5 ^ 5 / 100\n\u22a2 (1 + \u03b5 ^ 5 / 100) ^ 2 \u2264 1 + \u03b5 ^ 5 / 49\n[PROOFSTEP]\nrw [add_sq, one_pow, add_assoc, add_le_add_iff_left, mul_one, \u2190 le_sub_iff_add_le', div_eq_mul_one_div _ (49 : \u211d),\n mul_div_left_comm (2 : \u211d), \u2190 mul_sub_left_distrib, div_pow, div_le_iff (show (0 : \u211d) < \u2191100 ^ 2 by norm_num),\n mul_assoc, sq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : 1 + 1 / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264 1 + \u03b5 ^ 5 / 100\n\u22a2 0 < 100 ^ 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : 1 + 1 / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264 1 + \u03b5 ^ 5 / 100\n\u22a2 \u03b5 ^ 5 * \u03b5 ^ 5 \u2264 \u03b5 ^ 5 * ((1 / 49 - 2 / 100) * 100 ^ 2)\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left _ (by sz_positivity)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : 1 + 1 / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264 1 + \u03b5 ^ 5 / 100\n\u22a2 0 \u2264 \u03b5 ^ 5\n[PROOFSTEP]\nsz_positivity\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : 1 + 1 / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264 1 + \u03b5 ^ 5 / 100\n\u22a2 \u03b5 ^ 5 \u2264 (1 / 49 - 2 / 100) * 100 ^ 2\n[PROOFSTEP]\nexact (pow_le_one 5 (by sz_positivity) h\u03b5\u2081).trans (by norm_num)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : 1 + 1 / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264 1 + \u03b5 ^ 5 / 100\n\u22a2 0 \u2264 \u03b5\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : 1 + 1 / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) \u2264 1 + \u03b5 ^ 5 / 100\n\u22a2 1 \u2264 (1 / 49 - 2 / 100) * 100 ^ 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B))\n[PROOFSTEP]\nhave :\n \u2191(G.edgeDensity (A.biUnion id) (B.biUnion id)) - \u03b5 ^ 5 / \u219150 \u2264\n (\u21911 - \u03b5 ^ 5 / 50) * G.edgeDensity (A.biUnion id) (B.biUnion id) :=\n by\n rw [sub_mul, one_mul, sub_le_sub_iff_left]\n refine' mul_le_of_le_one_right (by sz_positivity) _\n exact_mod_cast G.edgeDensity_le_one _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n[PROOFSTEP]\nrw [sub_mul, one_mul, sub_le_sub_iff_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 \u03b5 ^ 5 / 50 * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264 \u03b5 ^ 5 / 50\n[PROOFSTEP]\nrefine' mul_le_of_le_one_right (by sz_positivity) _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 0 \u2264 \u03b5 ^ 5 / 50\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264 1\n[PROOFSTEP]\nexact_mod_cast G.edgeDensity_le_one _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n\u22a2 \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B))\n[PROOFSTEP]\nrefine' this.trans _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n\u22a2 (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B))\n[PROOFSTEP]\nconv_rhs =>\n -- Porting note: LHS and RHS need separate treatment to get the desired formsimp only [SimpleGraph.edgeDensity_def,\n sum_div, Rat.cast_div, div_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n| (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B))\n[PROOFSTEP]\nsimp only [SimpleGraph.edgeDensity_def, sum_div, Rat.cast_div, div_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n| (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B))\n[PROOFSTEP]\nsimp only [SimpleGraph.edgeDensity_def, sum_div, Rat.cast_div, div_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n| (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B))\n[PROOFSTEP]\nsimp only [SimpleGraph.edgeDensity_def, sum_div, Rat.cast_div, div_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n\u22a2 (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2211 x in Finset.product A B,\n \u2191\u2191(Finset.card (SimpleGraph.interedges G x.fst x.snd)) /\n (\u2191(\u2191(Finset.card x.fst) * \u2191(Finset.card x.snd)) * (\u2191(Finset.card A) * \u2191(Finset.card B)))\n[PROOFSTEP]\nconv_lhs =>\n rw [SimpleGraph.edgeDensity_def, SimpleGraph.interedges, \u2190 sup_eq_biUnion, \u2190 sup_eq_biUnion,\n Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl), ofSubset_parts, ofSubset_parts]\n simp only [cast_sum, sum_div, mul_sum, Rat.cast_sum, Rat.cast_div, mul_div_left_comm ((1 : \u211d) - _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n| (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n[PROOFSTEP]\n rw [SimpleGraph.edgeDensity_def, SimpleGraph.interedges, \u2190 sup_eq_biUnion, \u2190 sup_eq_biUnion,\n Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl), ofSubset_parts, ofSubset_parts]\n simp only [cast_sum, sum_div, mul_sum, Rat.cast_sum, Rat.cast_div, mul_div_left_comm ((1 : \u211d) - _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n| (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n[PROOFSTEP]\n rw [SimpleGraph.edgeDensity_def, SimpleGraph.interedges, \u2190 sup_eq_biUnion, \u2190 sup_eq_biUnion,\n Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl), ofSubset_parts, ofSubset_parts]\n simp only [cast_sum, sum_div, mul_sum, Rat.cast_sum, Rat.cast_div, mul_div_left_comm ((1 : \u211d) - _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n| (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n[PROOFSTEP]\nrw [SimpleGraph.edgeDensity_def, SimpleGraph.interedges, \u2190 sup_eq_biUnion, \u2190 sup_eq_biUnion,\n Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl), ofSubset_parts, ofSubset_parts]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n| (1 - \u03b5 ^ 5 / 50) *\n \u2191(\u2191(\u2211 ab in A \u00d7\u02e2 B, Finset.card (interedges G.Adj ab.fst ab.snd)) /\n (\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))))\n[PROOFSTEP]\nsimp only [cast_sum, sum_div, mul_sum, Rat.cast_sum, Rat.cast_div, mul_div_left_comm ((1 : \u211d) - _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n\u22a2 \u2211 x in A \u00d7\u02e2 B,\n \u2191\u2191(Finset.card (interedges G.Adj x.fst x.snd)) *\n ((1 - \u03b5 ^ 5 / 50) / \u2191(\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id)))) \u2264\n \u2211 x in Finset.product A B,\n \u2191\u2191(Finset.card (SimpleGraph.interedges G x.fst x.snd)) /\n (\u2191(\u2191(Finset.card x.fst) * \u2191(Finset.card x.snd)) * (\u2191(Finset.card A) * \u2191(Finset.card B)))\n[PROOFSTEP]\npush_cast\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n\u22a2 \u2211 x in A \u00d7\u02e2 B,\n \u2191(Finset.card (interedges G.Adj x.fst x.snd)) *\n ((1 - \u03b5 ^ 5 / 50) / (\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id)))) \u2264\n \u2211 x in Finset.product A B,\n \u2191(Finset.card (SimpleGraph.interedges G x.fst x.snd)) /\n (\u2191(Finset.card x.fst) * \u2191(Finset.card x.snd) * (\u2191(Finset.card A) * \u2191(Finset.card B)))\n[PROOFSTEP]\napply sum_le_sum\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n\u22a2 \u2200 (i : Finset \u03b1 \u00d7 Finset \u03b1),\n i \u2208 A \u00d7\u02e2 B \u2192\n \u2191(Finset.card (interedges G.Adj i.fst i.snd)) *\n ((1 - \u03b5 ^ 5 / 50) / (\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id)))) \u2264\n \u2191(Finset.card (SimpleGraph.interedges G i.fst i.snd)) /\n (\u2191(Finset.card i.fst) * \u2191(Finset.card i.snd) * (\u2191(Finset.card A) * \u2191(Finset.card B)))\n[PROOFSTEP]\nsimp only [and_imp, Prod.forall, mem_product]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n\u22a2 \u2200 (a b : Finset \u03b1),\n a \u2208 A \u2192\n b \u2208 B \u2192\n \u2191(Finset.card (interedges G.Adj a b)) *\n ((1 - \u03b5 ^ 5 / 50) / (\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id)))) \u2264\n \u2191(Finset.card (SimpleGraph.interedges G a b)) /\n (\u2191(Finset.card a) * \u2191(Finset.card b) * (\u2191(Finset.card A) * \u2191(Finset.card B)))\n[PROOFSTEP]\nrintro x y hx hy\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 \u2191(Finset.card (interedges G.Adj x y)) * ((1 - \u03b5 ^ 5 / 50) / (\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id)))) \u2264\n \u2191(Finset.card (SimpleGraph.interedges G x y)) /\n (\u2191(Finset.card x) * \u2191(Finset.card y) * (\u2191(Finset.card A) * \u2191(Finset.card B)))\n[PROOFSTEP]\nrw [mul_mul_mul_comm, mul_comm (x.card : \u211d), mul_comm (y.card : \u211d), le_div_iff, mul_assoc]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 \u2191(Finset.card (interedges G.Adj x y)) *\n ((1 - \u03b5 ^ 5 / 50) / (\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))) *\n (\u2191(Finset.card A) * \u2191(Finset.card x) * (\u2191(Finset.card B) * \u2191(Finset.card y)))) \u2264\n \u2191(Finset.card (SimpleGraph.interedges G x y))\n[PROOFSTEP]\nrefine' mul_le_of_le_one_right (cast_nonneg _) _\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 (1 - \u03b5 ^ 5 / 50) / (\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))) *\n (\u2191(Finset.card A) * \u2191(Finset.card x) * (\u2191(Finset.card B) * \u2191(Finset.card y))) \u2264\n 1\n[PROOFSTEP]\nrw [div_mul_eq_mul_div, \u2190 mul_assoc, mul_assoc]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 (1 - \u03b5 ^ 5 / 50) * (\u2191(Finset.card A) * \u2191(Finset.card x) * (\u2191(Finset.card B) * \u2191(Finset.card y))) /\n (\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))) \u2264\n 1\n[PROOFSTEP]\nrefine' div_le_one_of_le _ (by positivity)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 \u2264 \u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))\n[PROOFSTEP]\npositivity\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 (1 - \u03b5 ^ 5 / 50) * (\u2191(Finset.card A) * \u2191(Finset.card x) * (\u2191(Finset.card B) * \u2191(Finset.card y))) \u2264\n \u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))\n[PROOFSTEP]\nrefine' (mul_le_mul_of_nonneg_right (one_sub_le_m_div_m_add_one_sq hP\u03b1 hP\u03b5) _).trans _\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 \u2264 \u2191(Finset.card A) * \u2191(Finset.card x) * (\u2191(Finset.card B) * \u2191(Finset.card y))\n[PROOFSTEP]\nexact_mod_cast _root_.zero_le _\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)) ^\n 2 *\n (\u2191(Finset.card A) * \u2191(Finset.card x) * (\u2191(Finset.card B) * \u2191(Finset.card y))) \u2264\n \u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))\n[PROOFSTEP]\nrw [sq, mul_mul_mul_comm, mul_comm ((m : \u211d) / _), mul_comm ((m : \u211d) / _)]\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 \u2191(Finset.card A) * \u2191(Finset.card x) *\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) /\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)) *\n (\u2191(Finset.card B) * \u2191(Finset.card y) *\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) /\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1))) \u2264\n \u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))\n[PROOFSTEP]\nrefine' mul_le_mul _ _ _ (cast_nonneg _)\n[GOAL]\ncase h.refine'_2.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 \u2191(Finset.card A) * \u2191(Finset.card x) *\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) /\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)) \u2264\n \u2191(Finset.card (sup A id))\ncase h.refine'_2.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 \u2191(Finset.card B) * \u2191(Finset.card y) *\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) /\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)) \u2264\n \u2191(Finset.card (sup B id))\ncase h.refine'_2.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 \u2264\n \u2191(Finset.card B) * \u2191(Finset.card y) *\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1))\n[PROOFSTEP]\napply le_sum_card_subset_chunk_parts hA hx\n[GOAL]\ncase h.refine'_2.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 \u2191(Finset.card B) * \u2191(Finset.card y) *\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) /\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)) \u2264\n \u2191(Finset.card (sup B id))\ncase h.refine'_2.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 \u2264\n \u2191(Finset.card B) * \u2191(Finset.card y) *\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1))\n[PROOFSTEP]\napply le_sum_card_subset_chunk_parts hB hy\n[GOAL]\ncase h.refine'_2.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 \u2264\n \u2191(Finset.card B) * \u2191(Finset.card y) *\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) / (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1))\n[PROOFSTEP]\npositivity\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < \u2191(Finset.card A) * \u2191(Finset.card x) * (\u2191(Finset.card B) * \u2191(Finset.card y))\n[PROOFSTEP]\nrefine' mul_pos (mul_pos _ _) (mul_pos _ _)\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < \u2191(Finset.card A)\n[PROOFSTEP]\nrw [cast_pos, Finset.card_pos]\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < \u2191(Finset.card x)\n[PROOFSTEP]\nrw [cast_pos, Finset.card_pos]\n[GOAL]\ncase h.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < \u2191(Finset.card B)\n[PROOFSTEP]\nrw [cast_pos, Finset.card_pos]\n[GOAL]\ncase h.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < \u2191(Finset.card y)\n[PROOFSTEP]\nrw [cast_pos, Finset.card_pos]\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 Finset.Nonempty A\ncase h.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 Finset.Nonempty x\ncase h.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 Finset.Nonempty B\ncase h.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (1 - \u03b5 ^ 5 / 50) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 Finset.Nonempty y\n[PROOFSTEP]\nexacts [\u27e8_, hx\u27e9, nonempty_of_mem_parts _ (hA hx), \u27e8_, hy\u27e9, nonempty_of_mem_parts _ (hB hy)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n[PROOFSTEP]\nhave :\n (\u21911 + \u03b5 ^ 5 / \u219149) * G.edgeDensity (A.biUnion id) (B.biUnion id) \u2264\n G.edgeDensity (A.biUnion id) (B.biUnion id) + \u03b5 ^ 5 / 49 :=\n by\n rw [add_mul, one_mul, add_le_add_iff_left]\n refine' mul_le_of_le_one_right (by sz_positivity) _\n exact_mod_cast G.edgeDensity_le_one _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n[PROOFSTEP]\nrw [add_mul, one_mul, add_le_add_iff_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 \u03b5 ^ 5 / 49 * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264 \u03b5 ^ 5 / 49\n[PROOFSTEP]\nrefine' mul_le_of_le_one_right (by sz_positivity) _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 0 \u2264 \u03b5 ^ 5 / 49\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264 1\n[PROOFSTEP]\nexact_mod_cast G.edgeDensity_le_one _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n\u22a2 (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n[PROOFSTEP]\nrefine' le_trans _ this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n\u22a2 (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B)) \u2264\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n[PROOFSTEP]\nconv_lhs =>\n -- Porting note: LHS and RHS need separate treatment to get the desired formsimp only [SimpleGraph.edgeDensity,\n edgeDensity, sum_div, Rat.cast_div, div_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n| (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B))\n[PROOFSTEP]\nsimp only [SimpleGraph.edgeDensity, edgeDensity, sum_div, Rat.cast_div, div_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n| (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B))\n[PROOFSTEP]\nsimp only [SimpleGraph.edgeDensity, edgeDensity, sum_div, Rat.cast_div, div_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n| (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B))\n[PROOFSTEP]\nsimp only [SimpleGraph.edgeDensity, edgeDensity, sum_div, Rat.cast_div, div_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n\u22a2 \u2211 x in Finset.product A B,\n \u2191\u2191(Finset.card (interedges G.Adj x.fst x.snd)) /\n (\u2191(\u2191(Finset.card x.fst) * \u2191(Finset.card x.snd)) * (\u2191(Finset.card A) * \u2191(Finset.card B))) \u2264\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n[PROOFSTEP]\nconv_rhs =>\n rw [SimpleGraph.edgeDensity, edgeDensity, \u2190 sup_eq_biUnion, \u2190 sup_eq_biUnion,\n Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl)]\n simp only [cast_sum, mul_sum, sum_div, Rat.cast_sum, Rat.cast_div, mul_div_left_comm ((1 : \u211d) + _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n| (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n[PROOFSTEP]\n rw [SimpleGraph.edgeDensity, edgeDensity, \u2190 sup_eq_biUnion, \u2190 sup_eq_biUnion,\n Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl)]\n simp only [cast_sum, mul_sum, sum_div, Rat.cast_sum, Rat.cast_div, mul_div_left_comm ((1 : \u211d) + _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n| (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n[PROOFSTEP]\n rw [SimpleGraph.edgeDensity, edgeDensity, \u2190 sup_eq_biUnion, \u2190 sup_eq_biUnion,\n Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl)]\n simp only [cast_sum, mul_sum, sum_div, Rat.cast_sum, Rat.cast_div, mul_div_left_comm ((1 : \u211d) + _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n| (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))\n[PROOFSTEP]\nrw [SimpleGraph.edgeDensity, edgeDensity, \u2190 sup_eq_biUnion, \u2190 sup_eq_biUnion,\n Rel.card_interedges_finpartition _ (ofSubset _ hA rfl) (ofSubset _ hB rfl)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n| (1 + \u03b5 ^ 5 / 49) *\n \u2191(\u2191(\u2211 ab in\n (ofSubset (chunk hP G \u03b5 hU) hA (_ : sup A id = sup A id)).parts \u00d7\u02e2\n (ofSubset (chunk hP G \u03b5 hV) hB (_ : sup B id = sup B id)).parts,\n Finset.card (interedges G.Adj ab.fst ab.snd)) /\n (\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))))\n[PROOFSTEP]\nsimp only [cast_sum, mul_sum, sum_div, Rat.cast_sum, Rat.cast_div, mul_div_left_comm ((1 : \u211d) + _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n\u22a2 \u2211 x in Finset.product A B,\n \u2191\u2191(Finset.card (interedges G.Adj x.fst x.snd)) /\n (\u2191(\u2191(Finset.card x.fst) * \u2191(Finset.card x.snd)) * (\u2191(Finset.card A) * \u2191(Finset.card B))) \u2264\n \u2211 x in\n (ofSubset (chunk hP G \u03b5 hU) hA (_ : sup A id = sup A id)).parts \u00d7\u02e2\n (ofSubset (chunk hP G \u03b5 hV) hB (_ : sup B id = sup B id)).parts,\n \u2191\u2191(Finset.card (interedges G.Adj x.fst x.snd)) *\n ((1 + \u03b5 ^ 5 / 49) / \u2191(\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))))\n[PROOFSTEP]\npush_cast\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n\u22a2 \u2211 x in Finset.product A B,\n \u2191(Finset.card (interedges G.Adj x.fst x.snd)) /\n (\u2191(Finset.card x.fst) * \u2191(Finset.card x.snd) * (\u2191(Finset.card A) * \u2191(Finset.card B))) \u2264\n \u2211 x in\n (ofSubset (chunk hP G \u03b5 hU) hA (_ : sup A id = sup A id)).parts \u00d7\u02e2\n (ofSubset (chunk hP G \u03b5 hV) hB (_ : sup B id = sup B id)).parts,\n \u2191(Finset.card (interedges G.Adj x.fst x.snd)) *\n ((1 + \u03b5 ^ 5 / 49) / (\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))))\n[PROOFSTEP]\napply sum_le_sum\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n\u22a2 \u2200 (i : Finset \u03b1 \u00d7 Finset \u03b1),\n i \u2208 Finset.product A B \u2192\n \u2191(Finset.card (interedges G.Adj i.fst i.snd)) /\n (\u2191(Finset.card i.fst) * \u2191(Finset.card i.snd) * (\u2191(Finset.card A) * \u2191(Finset.card B))) \u2264\n \u2191(Finset.card (interedges G.Adj i.fst i.snd)) *\n ((1 + \u03b5 ^ 5 / 49) / (\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))))\n[PROOFSTEP]\nsimp only [and_imp, Prod.forall, mem_product, show A.product B = A \u00d7\u02e2 B by rfl]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n\u22a2 Finset.product A B = A \u00d7\u02e2 B\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n\u22a2 \u2200 (a b : Finset \u03b1),\n a \u2208 A \u2192\n b \u2208 B \u2192\n \u2191(Finset.card (interedges G.Adj a b)) /\n (\u2191(Finset.card a) * \u2191(Finset.card b) * (\u2191(Finset.card A) * \u2191(Finset.card B))) \u2264\n \u2191(Finset.card (interedges G.Adj a b)) *\n ((1 + \u03b5 ^ 5 / 49) / (\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))))\n[PROOFSTEP]\nintro x y hx hy\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 \u2191(Finset.card (interedges G.Adj x y)) /\n (\u2191(Finset.card x) * \u2191(Finset.card y) * (\u2191(Finset.card A) * \u2191(Finset.card B))) \u2264\n \u2191(Finset.card (interedges G.Adj x y)) * ((1 + \u03b5 ^ 5 / 49) / (\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))))\n[PROOFSTEP]\nrw [mul_mul_mul_comm, mul_comm (x.card : \u211d), mul_comm (y.card : \u211d), div_le_iff, mul_assoc]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 \u2191(Finset.card (interedges G.Adj x y)) \u2264\n \u2191(Finset.card (interedges G.Adj x y)) *\n ((1 + \u03b5 ^ 5 / 49) / (\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))) *\n (\u2191(Finset.card A) * \u2191(Finset.card x) * (\u2191(Finset.card B) * \u2191(Finset.card y))))\n[PROOFSTEP]\nrefine' le_mul_of_one_le_right (cast_nonneg _) _\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 1 \u2264\n (1 + \u03b5 ^ 5 / 49) / (\u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))) *\n (\u2191(Finset.card A) * \u2191(Finset.card x) * (\u2191(Finset.card B) * \u2191(Finset.card y)))\n[PROOFSTEP]\nrw [div_mul_eq_mul_div, one_le_div]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 \u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id)) \u2264\n (1 + \u03b5 ^ 5 / 49) * (\u2191(Finset.card A) * \u2191(Finset.card x) * (\u2191(Finset.card B) * \u2191(Finset.card y)))\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < \u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))\n[PROOFSTEP]\nrefine' le_trans _ (mul_le_mul_of_nonneg_right (m_add_one_div_m_le_one_add hP\u03b1 hP\u03b5 h\u03b5\u2081) _)\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 \u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id)) \u2264\n ((\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))) ^\n 2 *\n (\u2191(Finset.card A) * \u2191(Finset.card x) * (\u2191(Finset.card B) * \u2191(Finset.card y)))\n[PROOFSTEP]\nrw [sq, mul_mul_mul_comm, mul_comm (_ / (m : \u211d)), mul_comm (_ / (m : \u211d))]\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 \u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id)) \u2264\n \u2191(Finset.card A) * \u2191(Finset.card x) *\n ((\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) /\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))) *\n (\u2191(Finset.card B) * \u2191(Finset.card y) *\n ((\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) /\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))))\n[PROOFSTEP]\nexact\n mul_le_mul (sum_card_subset_chunk_parts_le (by sz_positivity) hA hx)\n (sum_card_subset_chunk_parts_le (by sz_positivity) hB hy) (by positivity) (by positivity)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 \u2264 \u2191(Finset.card (sup B id))\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 \u2264\n \u2191(Finset.card A) * \u2191(Finset.card x) *\n ((\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) / \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\n[PROOFSTEP]\npositivity\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 \u2264 \u2191(Finset.card A) * \u2191(Finset.card x) * (\u2191(Finset.card B) * \u2191(Finset.card y))\n[PROOFSTEP]\nexact_mod_cast _root_.zero_le _\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < \u2191(Finset.card (sup A id)) * \u2191(Finset.card (sup B id))\n[PROOFSTEP]\nrw [\u2190 cast_mul, cast_pos]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < Finset.card (sup A id) * Finset.card (sup B id)\n[PROOFSTEP]\napply mul_pos\n[GOAL]\ncase h.ha\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < Finset.card (sup A id)\n[PROOFSTEP]\nrw [Finset.card_pos, sup_eq_biUnion, biUnion_nonempty]\n[GOAL]\ncase h.hb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < Finset.card (sup B id)\n[PROOFSTEP]\nrw [Finset.card_pos, sup_eq_biUnion, biUnion_nonempty]\n[GOAL]\ncase h.ha\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 \u2203 x, x \u2208 A \u2227 Finset.Nonempty (id x)\n[PROOFSTEP]\nexact \u27e8_, hx, nonempty_of_mem_parts _ (hA hx)\u27e9\n[GOAL]\ncase h.hb\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 \u2203 x, x \u2208 B \u2227 Finset.Nonempty (id x)\n[PROOFSTEP]\nexact \u27e8_, hy, nonempty_of_mem_parts _ (hB hy)\u27e9\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < \u2191(Finset.card A) * \u2191(Finset.card x) * (\u2191(Finset.card B) * \u2191(Finset.card y))\n[PROOFSTEP]\nrefine' mul_pos (mul_pos _ _) (mul_pos _ _)\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < \u2191(Finset.card A)\n[PROOFSTEP]\nrw [cast_pos, Finset.card_pos]\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < \u2191(Finset.card x)\n[PROOFSTEP]\nrw [cast_pos, Finset.card_pos]\n[GOAL]\ncase h.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < \u2191(Finset.card B)\n[PROOFSTEP]\nrw [cast_pos, Finset.card_pos]\n[GOAL]\ncase h.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 0 < \u2191(Finset.card y)\n[PROOFSTEP]\nrw [cast_pos, Finset.card_pos]\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 Finset.Nonempty A\ncase h.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 Finset.Nonempty x\ncase h.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 Finset.Nonempty B\ncase h.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n (1 + \u03b5 ^ 5 / 49) * \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\nx y : Finset \u03b1\nhx : x \u2208 A\nhy : y \u2208 B\n\u22a2 Finset.Nonempty y\n[PROOFSTEP]\nexacts [\u27e8_, hx\u27e9, nonempty_of_mem_parts _ (hA hx), \u27e8_, hy\u27e9, nonempty_of_mem_parts _ (hB hy)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 |(\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B)) -\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id))| \u2264\n \u03b5 ^ 5 / 49\n[PROOFSTEP]\nrw [abs_sub_le_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B)) -\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u03b5 ^ 5 / 49 \u2227\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) -\n (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card A) * \u2191(Finset.card B)) \u2264\n \u03b5 ^ 5 / 49\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B)) -\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) \u2264\n \u03b5 ^ 5 / 49\n[PROOFSTEP]\nrw [sub_le_iff_le_add']\n[GOAL]\ncase left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B)) \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) + \u03b5 ^ 5 / 49\n[PROOFSTEP]\nexact sum_density_div_card_le_density_add_eps hP\u03b1 hP\u03b5 h\u03b5\u2081 hA hB\n[GOAL]\ncase right\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) -\n (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B)) \u2264\n \u03b5 ^ 5 / 49\n[PROOFSTEP]\nsuffices\n (G.edgeDensity (A.biUnion id) (B.biUnion id) : \u211d) -\n (\u2211 ab in A.product B, (G.edgeDensity ab.1 ab.2 : \u211d)) / (A.card * B.card) \u2264\n \u03b5 ^ 5 / 50\n by\n apply this.trans\n exact div_le_div_of_le_left (by sz_positivity) (by norm_num) (by norm_num)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) -\n (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B)) \u2264\n \u03b5 ^ 5 / 50\n\u22a2 \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) -\n (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B)) \u2264\n \u03b5 ^ 5 / 49\n[PROOFSTEP]\napply this.trans\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) -\n (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B)) \u2264\n \u03b5 ^ 5 / 50\n\u22a2 \u03b5 ^ 5 / 50 \u2264 \u03b5 ^ 5 / 49\n[PROOFSTEP]\nexact div_le_div_of_le_left (by sz_positivity) (by norm_num) (by norm_num)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) -\n (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B)) \u2264\n \u03b5 ^ 5 / 50\n\u22a2 0 \u2264 \u03b5 ^ 5\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) -\n (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B)) \u2264\n \u03b5 ^ 5 / 50\n\u22a2 0 < 49\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\nthis :\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) -\n (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B)) \u2264\n \u03b5 ^ 5 / 50\n\u22a2 49 \u2264 50\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase right\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) -\n (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B)) \u2264\n \u03b5 ^ 5 / 50\n[PROOFSTEP]\nrw [sub_le_iff_le_add, \u2190 sub_le_iff_le_add']\n[GOAL]\ncase right\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nA B : Finset (Finset \u03b1)\nhA : A \u2286 (chunk hP G \u03b5 hU).parts\nhB : B \u2286 (chunk hP G \u03b5 hV).parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G (Finset.biUnion A id) (Finset.biUnion B id)) - \u03b5 ^ 5 / 50 \u2264\n (\u2211 ab in Finset.product A B, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) / (\u2191(Finset.card A) * \u2191(Finset.card B))\n[PROOFSTEP]\napply density_sub_eps_le_sum_density_div_card hP\u03b1 hP\u03b5 hA hB\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 \u2264\n ((\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts) ^\n 2\n[PROOFSTEP]\nobtain hG\u03b5 | hG\u03b5 := le_total (G.edgeDensity U V : \u211d) (\u03b5 ^ 5 / 50)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5 : \u2191(SimpleGraph.edgeDensity G U V) \u2264 \u03b5 ^ 5 / 50\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 \u2264\n ((\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts) ^\n 2\n[PROOFSTEP]\nrefine' (sub_nonpos_of_le <| (sq_le _ _).trans <| hG\u03b5.trans _).trans (sq_nonneg _)\n[GOAL]\ncase inl.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5 : \u2191(SimpleGraph.edgeDensity G U V) \u2264 \u03b5 ^ 5 / 50\n\u22a2 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\n[PROOFSTEP]\nexact_mod_cast G.edgeDensity_nonneg _ _\n[GOAL]\ncase inl.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5 : \u2191(SimpleGraph.edgeDensity G U V) \u2264 \u03b5 ^ 5 / 50\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) \u2264 1\n[PROOFSTEP]\nexact_mod_cast G.edgeDensity_le_one _ _\n[GOAL]\ncase inl.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5 : \u2191(SimpleGraph.edgeDensity G U V) \u2264 \u03b5 ^ 5 / 50\n\u22a2 \u03b5 ^ 5 / 50 \u2264 \u03b5 ^ 5 / 25\n[PROOFSTEP]\nexact div_le_div_of_le_left (by sz_positivity) (by norm_num) (by norm_num)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5 : \u2191(SimpleGraph.edgeDensity G U V) \u2264 \u03b5 ^ 5 / 50\n\u22a2 0 \u2264 \u03b5 ^ 5\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5 : \u2191(SimpleGraph.edgeDensity G U V) \u2264 \u03b5 ^ 5 / 50\n\u22a2 0 < 25\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5 : \u2191(SimpleGraph.edgeDensity G U V) \u2264 \u03b5 ^ 5 / 50\n\u22a2 25 \u2264 50\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5 : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 \u2264\n ((\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts) ^\n 2\n[PROOFSTEP]\nrw [\u2190 sub_nonneg] at hG\u03b5 \n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5\u271d : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\nhG\u03b5 : 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 \u2264\n ((\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts) ^\n 2\n[PROOFSTEP]\nhave :\n \u2191(G.edgeDensity U V) - \u03b5 ^ 5 / \u219150 \u2264\n (\u2211 ab in (chunk hP G \u03b5 hU).parts.product (chunk hP G \u03b5 hV).parts, (G.edgeDensity ab.1 ab.2 : \u211d)) /\n \u219116 ^ P.parts.card :=\n by\n have rflU := Set.Subset.refl (chunk hP G \u03b5 hU).parts.toSet\n have rflV := Set.Subset.refl (chunk hP G \u03b5 hV).parts.toSet\n refine' (le_trans _ <| density_sub_eps_le_sum_density_div_card hP\u03b1 hP\u03b5 rflU rflV).trans _\n \u00b7 rw [biUnion_parts, biUnion_parts]\n \u00b7 rw [card_chunk (m_pos hP\u03b1).ne', card_chunk (m_pos hP\u03b1).ne', \u2190 cast_mul, \u2190 mul_pow, cast_pow]\n norm_cast\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5\u271d : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\nhG\u03b5 : 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nhave rflU := Set.Subset.refl (chunk hP G \u03b5 hU).parts.toSet\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5\u271d : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\nhG\u03b5 : 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50\nrflU : \u2191(chunk hP G \u03b5 hU).parts \u2286 \u2191(chunk hP G \u03b5 hU).parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nhave rflV := Set.Subset.refl (chunk hP G \u03b5 hV).parts.toSet\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5\u271d : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\nhG\u03b5 : 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50\nrflU : \u2191(chunk hP G \u03b5 hU).parts \u2286 \u2191(chunk hP G \u03b5 hU).parts\nrflV : \u2191(chunk hP G \u03b5 hV).parts \u2286 \u2191(chunk hP G \u03b5 hV).parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nrefine' (le_trans _ <| density_sub_eps_le_sum_density_div_card hP\u03b1 hP\u03b5 rflU rflV).trans _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5\u271d : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\nhG\u03b5 : 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50\nrflU : \u2191(chunk hP G \u03b5 hU).parts \u2286 \u2191(chunk hP G \u03b5 hU).parts\nrflV : \u2191(chunk hP G \u03b5 hV).parts \u2286 \u2191(chunk hP G \u03b5 hV).parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50 \u2264\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (chunk hP G \u03b5 hU).parts id)\n (Finset.biUnion (chunk hP G \u03b5 hV).parts id)) -\n \u03b5 ^ 5 / 50\n[PROOFSTEP]\nrw [biUnion_parts, biUnion_parts]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5\u271d : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\nhG\u03b5 : 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50\nrflU : \u2191(chunk hP G \u03b5 hU).parts \u2286 \u2191(chunk hP G \u03b5 hU).parts\nrflV : \u2191(chunk hP G \u03b5 hV).parts \u2286 \u2191(chunk hP G \u03b5 hV).parts\n\u22a2 (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (chunk hP G \u03b5 hU).parts) * \u2191(Finset.card (chunk hP G \u03b5 hV).parts)) \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nrw [card_chunk (m_pos hP\u03b1).ne', card_chunk (m_pos hP\u03b1).ne', \u2190 cast_mul, \u2190 mul_pow, cast_pow]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5\u271d : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\nhG\u03b5 : 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50\nrflU : \u2191(chunk hP G \u03b5 hU).parts \u2286 \u2191(chunk hP G \u03b5 hU).parts\nrflV : \u2191(chunk hP G \u03b5 hV).parts \u2286 \u2191(chunk hP G \u03b5 hV).parts\n\u22a2 (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n \u2191(4 * 4) ^ Finset.card P.parts \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5\u271d : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\nhG\u03b5 : 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50\nthis :\n \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 \u2264\n ((\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts) ^\n 2\n[PROOFSTEP]\nrefine' le_trans _ (pow_le_pow_of_le_left hG\u03b5 this 2)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5\u271d : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\nhG\u03b5 : 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50\nthis :\n \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 \u2264 (\u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50) ^ 2\n[PROOFSTEP]\nrw [sub_sq, sub_add, sub_le_sub_iff_left]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5\u271d : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\nhG\u03b5 : 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50\nthis :\n \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts\n\u22a2 2 * \u2191(SimpleGraph.edgeDensity G U V) * (\u03b5 ^ 5 / 50) - (\u03b5 ^ 5 / 50) ^ 2 \u2264 \u03b5 ^ 5 / 25\n[PROOFSTEP]\nrefine' (sub_le_self _ <| sq_nonneg <| \u03b5 ^ 5 / 50).trans _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5\u271d : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\nhG\u03b5 : 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50\nthis :\n \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts\n\u22a2 2 * \u2191(SimpleGraph.edgeDensity G U V) * (\u03b5 ^ 5 / 50) \u2264 \u03b5 ^ 5 / 25\n[PROOFSTEP]\nrw [mul_right_comm, mul_div_left_comm, div_eq_mul_inv (\u03b5 ^ 5), show (2 : \u211d) / 50 = 25\u207b\u00b9 by norm_num]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5\u271d : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\nhG\u03b5 : 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50\nthis :\n \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts\n\u22a2 2 / 50 = 25\u207b\u00b9\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5\u271d : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\nhG\u03b5 : 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50\nthis :\n \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts\n\u22a2 \u03b5 ^ 5 * 25\u207b\u00b9 * \u2191(SimpleGraph.edgeDensity G U V) \u2264 \u03b5 ^ 5 * 25\u207b\u00b9\n[PROOFSTEP]\nexact mul_le_of_le_one_right (by sz_positivity) (by exact_mod_cast G.edgeDensity_le_one _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5\u271d : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\nhG\u03b5 : 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50\nthis :\n \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts\n\u22a2 0 \u2264 \u03b5 ^ 5 * 25\u207b\u00b9\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhG\u03b5\u271d : \u03b5 ^ 5 / 50 \u2264 \u2191(SimpleGraph.edgeDensity G U V)\nhG\u03b5 : 0 \u2264 \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50\nthis :\n \u2191(SimpleGraph.edgeDensity G U V) - \u03b5 ^ 5 / 50 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) \u2264 1\n[PROOFSTEP]\nexact_mod_cast G.edgeDensity_le_one _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV' : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\n\u22a2 |\u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id)) -\n \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))| \u2264\n \u03b5 / 5\n[PROOFSTEP]\nconvert\n abs_edgeDensity_sub_edgeDensity_le_two_mul G.Adj (biUnion_star_subset_nonuniformWitness hP G \u03b5 hU V)\n (biUnion_star_subset_nonuniformWitness hP G \u03b5 hV U) (by sz_positivity)\n (one_sub_eps_mul_card_nonuniformWitness_le_card_star hV hUV' hUV hP\u03b5 h\u03b5\u2081)\n (one_sub_eps_mul_card_nonuniformWitness_le_card_star hU hUV'.symm (fun hVU => hUV hVU.symm) hP\u03b5 h\u03b5\u2081) using\n 1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV' : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\n\u22a2 0 \u2264 \u03b5 / 10\n[PROOFSTEP]\nsz_positivity\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV' : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\n\u22a2 \u03b5 / 5 = 2 * (\u03b5 / 10)\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\n\u22a2 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\n[PROOFSTEP]\nhave hm : (0 : \u211d) \u2264 1 - (\u2191m)\u207b\u00b9 := sub_nonneg_of_le (inv_le_one <| one_le_m_coe hP\u03b1)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\n\u22a2 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\n[PROOFSTEP]\nhave h\u03b5 : 0 \u2264 1 - \u03b5 / 10 := sub_nonneg_of_le (div_le_one_of_le (h\u03b5\u2081.trans <| by norm_num) <| by norm_num)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\n\u22a2 1 \u2264 10\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\n\u22a2 0 \u2264 10\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\n[PROOFSTEP]\ncalc\n 4 / 5 * \u03b5 = (1 - 1 / 10) * (1 - 9\u207b\u00b9) * \u03b5 := by norm_num\n _ \u2264 (1 - \u03b5 / 10) * (1 - (\u2191m)\u207b\u00b9) * ((G.nonuniformWitness \u03b5 U V).card / U.card) :=\n (mul_le_mul\n (mul_le_mul (sub_le_sub_left (div_le_div_of_le_of_nonneg h\u03b5\u2081 <| by norm_num) _)\n (sub_le_sub_left\n (inv_le_inv_of_le (by norm_num) <| by\n exact_mod_cast (show 9 \u2264 100 by norm_num).trans (hundred_le_m hP\u03b1 hP\u03b5 h\u03b5\u2081))\n _)\n (by norm_num) h\u03b5)\n ((le_div_iff' <| (@cast_pos \u211d _ _ _).2 (P.nonempty_of_mem_parts hU).card_pos).2 <|\n G.le_card_nonuniformWitness hunif)\n (by sz_positivity) (by positivity))\n _ = (1 - \u03b5 / 10) * (G.nonuniformWitness \u03b5 U V).card * ((1 - (\u2191m)\u207b\u00b9) / U.card) := by\n rw [mul_assoc, mul_assoc, mul_div_left_comm]\n _ \u2264 ((star hP G \u03b5 hU V).biUnion id).card * ((1 - (\u2191m)\u207b\u00b9) / U.card) :=\n (mul_le_mul_of_nonneg_right (one_sub_eps_mul_card_nonuniformWitness_le_card_star hV hUV hunif hP\u03b5 h\u03b5\u2081)\n (by positivity))\n _ \u2264 (star hP G \u03b5 hU V).card * (m + 1) * ((1 - (\u2191m)\u207b\u00b9) / U.card) :=\n (mul_le_mul_of_nonneg_right card_biUnion_star_le_m_add_one_card_star_mul (by positivity))\n _ \u2264 (star hP G \u03b5 hU V).card * (m + \u21911) * ((\u21911 - (\u2191m)\u207b\u00b9) / (\u21914 ^ P.parts.card * m)) :=\n (mul_le_mul_of_nonneg_left (div_le_div_of_le_left hm (by sz_positivity) <| pow_mul_m_le_card_part hP hU)\n (by positivity))\n _ \u2264 (star hP G \u03b5 hU V).card / \u21914 ^ P.parts.card :=\n by\n rw [mul_assoc, mul_comm ((4 : \u211d) ^ P.parts.card), \u2190 div_div, \u2190 mul_div_assoc, \u2190 mul_comm_div]\n refine' mul_le_of_le_one_right (by positivity) _\n have hm : (0 : \u211d) < m := by sz_positivity\n rw [mul_div_assoc', div_le_one hm, \u2190 one_div, one_sub_div hm.ne', mul_div_assoc', div_le_iff hm]\n linarith\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 4 / 5 * \u03b5 = (1 - 1 / 10) * (1 - 9\u207b\u00b9) * \u03b5\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 0 \u2264 10\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 0 < 9\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 9 \u2264 \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\nexact_mod_cast (show 9 \u2264 100 by norm_num).trans (hundred_le_m hP\u03b1 hP\u03b5 h\u03b5\u2081)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 9 \u2264 100\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 0 \u2264 1 - 9\u207b\u00b9\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 0 \u2264 \u03b5\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 0 \u2264 (1 - \u03b5 / 10) * (1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9)\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 (1 - \u03b5 / 10) * (1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9) *\n (\u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V)) / \u2191(Finset.card U)) =\n (1 - \u03b5 / 10) * \u2191(Finset.card (SimpleGraph.nonuniformWitness G \u03b5 U V)) *\n ((1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9) / \u2191(Finset.card U))\n[PROOFSTEP]\nrw [mul_assoc, mul_assoc, mul_div_left_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 0 \u2264 (1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9) / \u2191(Finset.card U)\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 0 \u2264 (1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9) / \u2191(Finset.card U)\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 0 < 4 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 0 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) * (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1)\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 \u2191(Finset.card (star hP G \u03b5 hU V)) * (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) *\n ((1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9) /\n (4 ^ Finset.card P.parts * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))) \u2264\n \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\n[PROOFSTEP]\nrw [mul_assoc, mul_comm ((4 : \u211d) ^ P.parts.card), \u2190 div_div, \u2190 mul_div_assoc, \u2190 mul_comm_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts *\n ((\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) *\n ((1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9) /\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))) \u2264\n \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\n[PROOFSTEP]\nrefine' mul_le_of_le_one_right (by positivity) _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 0 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) *\n ((1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9) /\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))) \u2264\n 1\n[PROOFSTEP]\nhave hm : (0 : \u211d) < m := by sz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\n\u22a2 0 < \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm\u271d : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\nhm : 0 < \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n\u22a2 (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) *\n ((1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9) /\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))) \u2264\n 1\n[PROOFSTEP]\nrw [mul_div_assoc', div_le_one hm, \u2190 one_div, one_sub_div hm.ne', mul_div_assoc', div_le_iff hm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUV : U \u2260 V\nhunif : \u00acSimpleGraph.IsUniform G \u03b5 U V\nhm\u271d : 0 \u2264 1 - (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)))\u207b\u00b9\nh\u03b5 : 0 \u2264 1 - \u03b5 / 10\nhm : 0 < \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n\u22a2 (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) + 1) *\n (\u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) - 1) \u2264\n \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts)) * \u2191(Fintype.card \u03b1 / stepBound (Finset.card P.parts))\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\n\u22a2 3 / 4 * \u03b5 \u2264\n |(\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U))) -\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts|\n[PROOFSTEP]\nrw [show (16 : \u211d) = \u21914 ^ 2 by norm_num, pow_right_comm, sq ((4 : \u211d) ^ _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\n\u22a2 16 = 4 ^ 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\n\u22a2 3 / 4 * \u03b5 \u2264\n |(\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U))) -\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)|\n[PROOFSTEP]\nset p : \u211d :=\n (\u2211 ab in (star hP G \u03b5 hU V).product (star hP G \u03b5 hV U), (G.edgeDensity ab.1 ab.2 : \u211d)) /\n ((star hP G \u03b5 hU V).card * (star hP G \u03b5 hV U).card)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\n\u22a2 3 / 4 * \u03b5 \u2264\n |p -\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)|\n[PROOFSTEP]\nset q : \u211d :=\n (\u2211 ab in (chunk hP G \u03b5 hU).parts.product (chunk hP G \u03b5 hV).parts, (G.edgeDensity ab.1 ab.2 : \u211d)) /\n (\u21914 ^ P.parts.card * \u21914 ^ P.parts.card)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\n\u22a2 3 / 4 * \u03b5 \u2264 |p - q|\n[PROOFSTEP]\nchange _ \u2264 |p - q|\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\n\u22a2 3 / 4 * \u03b5 \u2264 |p - q|\n[PROOFSTEP]\nset r : \u211d := \u2191(G.edgeDensity ((star hP G \u03b5 hU V).biUnion id) ((star hP G \u03b5 hV U).biUnion id))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\n\u22a2 3 / 4 * \u03b5 \u2264 |p - q|\n[PROOFSTEP]\nset s : \u211d := \u2191(G.edgeDensity (G.nonuniformWitness \u03b5 U V) (G.nonuniformWitness \u03b5 V U))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\n\u22a2 3 / 4 * \u03b5 \u2264 |p - q|\n[PROOFSTEP]\nset t : \u211d := \u2191(G.edgeDensity U V)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\n\u22a2 3 / 4 * \u03b5 \u2264 |p - q|\n[PROOFSTEP]\nhave hrs : |r - s| \u2264 \u03b5 / 5 := abs_density_star_sub_density_le_eps hP\u03b5 h\u03b5\u2081 hUVne hUV\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\n\u22a2 3 / 4 * \u03b5 \u2264 |p - q|\n[PROOFSTEP]\nhave hst : \u03b5 \u2264 |s - t| := by exact_mod_cast G.nonuniformWitness_spec hUVne hUV\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\n\u22a2 \u03b5 \u2264 |s - t|\n[PROOFSTEP]\nexact_mod_cast G.nonuniformWitness_spec hUVne hUV\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\n\u22a2 3 / 4 * \u03b5 \u2264 |p - q|\n[PROOFSTEP]\nhave hpr : |p - r| \u2264 \u03b5 ^ 5 / 49 := average_density_near_total_density hP\u03b1 hP\u03b5 h\u03b5\u2081 star_subset_chunk star_subset_chunk\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\n\u22a2 3 / 4 * \u03b5 \u2264 |p - q|\n[PROOFSTEP]\nhave hqt : |q - t| \u2264 \u03b5 ^ 5 / 49 :=\n by\n have :=\n average_density_near_total_density hP\u03b1 hP\u03b5 h\u03b5\u2081 (Subset.refl (chunk hP G \u03b5 hU).parts)\n (Subset.refl (chunk hP G \u03b5 hV).parts)\n simp_rw [\u2190 sup_eq_biUnion, supParts, card_chunk (m_pos hP\u03b1).ne', cast_pow] at this \n norm_num at this \n exact this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\n\u22a2 |q - t| \u2264 \u03b5 ^ 5 / 49\n[PROOFSTEP]\nhave :=\n average_density_near_total_density hP\u03b1 hP\u03b5 h\u03b5\u2081 (Subset.refl (chunk hP G \u03b5 hU).parts)\n (Subset.refl (chunk hP G \u03b5 hV).parts)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nthis :\n |(\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (chunk hP G \u03b5 hU).parts) * \u2191(Finset.card (chunk hP G \u03b5 hV).parts)) -\n \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (chunk hP G \u03b5 hU).parts id)\n (Finset.biUnion (chunk hP G \u03b5 hV).parts id))| \u2264\n \u03b5 ^ 5 / 49\n\u22a2 |q - t| \u2264 \u03b5 ^ 5 / 49\n[PROOFSTEP]\nsimp_rw [\u2190 sup_eq_biUnion, supParts, card_chunk (m_pos hP\u03b1).ne', cast_pow] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nthis :\n |(\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u21914 ^ Finset.card P.parts * \u21914 ^ Finset.card P.parts) -\n \u2191(SimpleGraph.edgeDensity G U V)| \u2264\n \u03b5 ^ 5 / 49\n\u22a2 |q - t| \u2264 \u03b5 ^ 5 / 49\n[PROOFSTEP]\nnorm_num at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nthis :\n |(\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts) -\n \u2191(SimpleGraph.edgeDensity G U V)| \u2264\n \u03b5 ^ 5 / 49\n\u22a2 |q - t| \u2264 \u03b5 ^ 5 / 49\n[PROOFSTEP]\nexact this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\n\u22a2 3 / 4 * \u03b5 \u2264 |p - q|\n[PROOFSTEP]\nhave h\u03b5' : \u03b5 ^ 5 \u2264 \u03b5 := by simpa using pow_le_pow_of_le_one (by sz_positivity) h\u03b5\u2081 (show 1 \u2264 5 by norm_num)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\n\u22a2 \u03b5 ^ 5 \u2264 \u03b5\n[PROOFSTEP]\nsimpa using pow_le_pow_of_le_one (by sz_positivity) h\u03b5\u2081 (show 1 \u2264 5 by norm_num)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\n\u22a2 0 \u2264 \u03b5\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\n\u22a2 1 \u2264 5\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\nh\u03b5' : \u03b5 ^ 5 \u2264 \u03b5\n\u22a2 3 / 4 * \u03b5 \u2264 |p - q|\n[PROOFSTEP]\nhave hpr' : |p - r| \u2264 \u03b5 / 49 := hpr.trans (div_le_div_of_le_of_nonneg h\u03b5' <| by norm_num)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\nh\u03b5' : \u03b5 ^ 5 \u2264 \u03b5\n\u22a2 0 \u2264 49\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\nh\u03b5' : \u03b5 ^ 5 \u2264 \u03b5\nhpr' : |p - r| \u2264 \u03b5 / 49\n\u22a2 3 / 4 * \u03b5 \u2264 |p - q|\n[PROOFSTEP]\nhave hqt' : |q - t| \u2264 \u03b5 / 49 := hqt.trans (div_le_div_of_le_of_nonneg h\u03b5' <| by norm_num)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\nh\u03b5' : \u03b5 ^ 5 \u2264 \u03b5\nhpr' : |p - r| \u2264 \u03b5 / 49\n\u22a2 0 \u2264 49\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : |r - s| \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\nh\u03b5' : \u03b5 ^ 5 \u2264 \u03b5\nhpr' : |p - r| \u2264 \u03b5 / 49\nhqt' : |q - t| \u2264 \u03b5 / 49\n\u22a2 3 / 4 * \u03b5 \u2264 |p - q|\n[PROOFSTEP]\nrw [abs_sub_le_iff] at hrs hpr' hqt' \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : r - s \u2264 \u03b5 / 5 \u2227 s - r \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 |s - t|\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\nh\u03b5' : \u03b5 ^ 5 \u2264 \u03b5\nhpr' : p - r \u2264 \u03b5 / 49 \u2227 r - p \u2264 \u03b5 / 49\nhqt' : q - t \u2264 \u03b5 / 49 \u2227 t - q \u2264 \u03b5 / 49\n\u22a2 3 / 4 * \u03b5 \u2264 |p - q|\n[PROOFSTEP]\nrw [le_abs] at hst \u22a2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : r - s \u2264 \u03b5 / 5 \u2227 s - r \u2264 \u03b5 / 5\nhst : \u03b5 \u2264 s - t \u2228 \u03b5 \u2264 -(s - t)\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\nh\u03b5' : \u03b5 ^ 5 \u2264 \u03b5\nhpr' : p - r \u2264 \u03b5 / 49 \u2227 r - p \u2264 \u03b5 / 49\nhqt' : q - t \u2264 \u03b5 / 49 \u2227 t - q \u2264 \u03b5 / 49\n\u22a2 3 / 4 * \u03b5 \u2264 p - q \u2228 3 / 4 * \u03b5 \u2264 -(p - q)\n[PROOFSTEP]\ncases hst\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : r - s \u2264 \u03b5 / 5 \u2227 s - r \u2264 \u03b5 / 5\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\nh\u03b5' : \u03b5 ^ 5 \u2264 \u03b5\nhpr' : p - r \u2264 \u03b5 / 49 \u2227 r - p \u2264 \u03b5 / 49\nhqt' : q - t \u2264 \u03b5 / 49 \u2227 t - q \u2264 \u03b5 / 49\nh\u271d : \u03b5 \u2264 s - t\n\u22a2 3 / 4 * \u03b5 \u2264 p - q \u2228 3 / 4 * \u03b5 \u2264 -(p - q)\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : r - s \u2264 \u03b5 / 5 \u2227 s - r \u2264 \u03b5 / 5\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\nh\u03b5' : \u03b5 ^ 5 \u2264 \u03b5\nhpr' : p - r \u2264 \u03b5 / 49 \u2227 r - p \u2264 \u03b5 / 49\nhqt' : q - t \u2264 \u03b5 / 49 \u2227 t - q \u2264 \u03b5 / 49\nh\u271d : \u03b5 \u2264 -(s - t)\n\u22a2 3 / 4 * \u03b5 \u2264 p - q \u2228 3 / 4 * \u03b5 \u2264 -(p - q)\n[PROOFSTEP]\nleft\n[GOAL]\ncase inl.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : r - s \u2264 \u03b5 / 5 \u2227 s - r \u2264 \u03b5 / 5\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\nh\u03b5' : \u03b5 ^ 5 \u2264 \u03b5\nhpr' : p - r \u2264 \u03b5 / 49 \u2227 r - p \u2264 \u03b5 / 49\nhqt' : q - t \u2264 \u03b5 / 49 \u2227 t - q \u2264 \u03b5 / 49\nh\u271d : \u03b5 \u2264 s - t\n\u22a2 3 / 4 * \u03b5 \u2264 p - q\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : r - s \u2264 \u03b5 / 5 \u2227 s - r \u2264 \u03b5 / 5\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\nh\u03b5' : \u03b5 ^ 5 \u2264 \u03b5\nhpr' : p - r \u2264 \u03b5 / 49 \u2227 r - p \u2264 \u03b5 / 49\nhqt' : q - t \u2264 \u03b5 / 49 \u2227 t - q \u2264 \u03b5 / 49\nh\u271d : \u03b5 \u2264 -(s - t)\n\u22a2 3 / 4 * \u03b5 \u2264 p - q \u2228 3 / 4 * \u03b5 \u2264 -(p - q)\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : r - s \u2264 \u03b5 / 5 \u2227 s - r \u2264 \u03b5 / 5\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\nh\u03b5' : \u03b5 ^ 5 \u2264 \u03b5\nhpr' : p - r \u2264 \u03b5 / 49 \u2227 r - p \u2264 \u03b5 / 49\nhqt' : q - t \u2264 \u03b5 / 49 \u2227 t - q \u2264 \u03b5 / 49\nh\u271d : \u03b5 \u2264 -(s - t)\n\u22a2 3 / 4 * \u03b5 \u2264 p - q \u2228 3 / 4 * \u03b5 \u2264 -(p - q)\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\np : \u211d :=\n (\u2211 ab in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U), \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)))\nq : \u211d :=\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n (4 ^ Finset.card P.parts * 4 ^ Finset.card P.parts)\nr : \u211d := \u2191(SimpleGraph.edgeDensity G (Finset.biUnion (star hP G \u03b5 hU V) id) (Finset.biUnion (star hP G \u03b5 hV U) id))\ns : \u211d := \u2191(SimpleGraph.edgeDensity G (SimpleGraph.nonuniformWitness G \u03b5 U V) (SimpleGraph.nonuniformWitness G \u03b5 V U))\nt : \u211d := \u2191(SimpleGraph.edgeDensity G U V)\nhrs : r - s \u2264 \u03b5 / 5 \u2227 s - r \u2264 \u03b5 / 5\nhpr : |p - r| \u2264 \u03b5 ^ 5 / 49\nhqt : |q - t| \u2264 \u03b5 ^ 5 / 49\nh\u03b5' : \u03b5 ^ 5 \u2264 \u03b5\nhpr' : p - r \u2264 \u03b5 / 49 \u2227 r - p \u2264 \u03b5 / 49\nhqt' : q - t \u2264 \u03b5 / 49 \u2227 t - q \u2264 \u03b5 / 49\nh\u271d : \u03b5 \u2264 -(s - t)\n\u22a2 3 / 4 * \u03b5 \u2264 -(p - q)\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / \u219125 + \u03b5 ^ 4 / 3 \u2264\n \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 +\n \u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)) / 16 ^ Finset.card P.parts * (9 / 16) *\n \u03b5 ^ 2\n[PROOFSTEP]\napply add_le_add_left\n[GOAL]\ncase bc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\n\u22a2 \u03b5 ^ 4 / 3 \u2264\n \u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)) / 16 ^ Finset.card P.parts * (9 / 16) * \u03b5 ^ 2\n[PROOFSTEP]\nhave Ul : 4 / 5 * \u03b5 \u2264 (star hP G \u03b5 hU V).card / _ := eps_le_card_star_div hP\u03b1 hP\u03b5 h\u03b5\u2081 hU hV hUVne hUV\n[GOAL]\ncase bc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nUl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\n\u22a2 \u03b5 ^ 4 / 3 \u2264\n \u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)) / 16 ^ Finset.card P.parts * (9 / 16) * \u03b5 ^ 2\n[PROOFSTEP]\nhave Vl : 4 / 5 * \u03b5 \u2264 (star hP G \u03b5 hV U).card / _ :=\n eps_le_card_star_div hP\u03b1 hP\u03b5 h\u03b5\u2081 hV hU hUVne.symm fun h => hUV h.symm\n[GOAL]\ncase bc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nUl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\nVl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts\n\u22a2 \u03b5 ^ 4 / 3 \u2264\n \u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)) / 16 ^ Finset.card P.parts * (9 / 16) * \u03b5 ^ 2\n[PROOFSTEP]\nrw [show (16 : \u211d) = \u21914 ^ 2 by norm_num, pow_right_comm, sq ((4 : \u211d) ^ _), \u2190 _root_.div_mul_div_comm, mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nUl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\nVl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts\n\u22a2 16 = 4 ^ 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase bc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nUl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\nVl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts\n\u22a2 \u03b5 ^ 4 / 3 \u2264\n \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts *\n (\u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts) *\n (9 / 4 ^ 2 * \u03b5 ^ 2)\n[PROOFSTEP]\nhave : 0 < \u03b5 := by sz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nUl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\nVl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts\n\u22a2 0 < \u03b5\n[PROOFSTEP]\nsz_positivity\n[GOAL]\ncase bc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nUl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\nVl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts\nthis : 0 < \u03b5\n\u22a2 \u03b5 ^ 4 / 3 \u2264\n \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts *\n (\u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts) *\n (9 / 4 ^ 2 * \u03b5 ^ 2)\n[PROOFSTEP]\nhave UVl := mul_le_mul Ul Vl (by positivity) (by positivity)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nUl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\nVl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts\nthis : 0 < \u03b5\n\u22a2 0 \u2264 4 / 5 * \u03b5\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nUl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\nVl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts\nthis : 0 < \u03b5\n\u22a2 0 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\n[PROOFSTEP]\npositivity\n[GOAL]\ncase bc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nUl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\nVl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts\nthis : 0 < \u03b5\nUVl :\n 4 / 5 * \u03b5 * (4 / 5 * \u03b5) \u2264\n \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts *\n (\u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts)\n\u22a2 \u03b5 ^ 4 / 3 \u2264\n \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts *\n (\u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts) *\n (9 / 4 ^ 2 * \u03b5 ^ 2)\n[PROOFSTEP]\nrefine' le_trans _ (mul_le_mul_of_nonneg_right UVl _)\n[GOAL]\ncase bc.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nUl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\nVl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts\nthis : 0 < \u03b5\nUVl :\n 4 / 5 * \u03b5 * (4 / 5 * \u03b5) \u2264\n \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts *\n (\u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts)\n\u22a2 \u03b5 ^ 4 / 3 \u2264 4 / 5 * \u03b5 * (4 / 5 * \u03b5) * (9 / 4 ^ 2 * \u03b5 ^ 2)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase bc.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nUl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\nVl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts\nthis : 0 < \u03b5\nUVl :\n 4 / 5 * \u03b5 * (4 / 5 * \u03b5) \u2264\n \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts *\n (\u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts)\n\u22a2 \u03b5 ^ 4 / 3 \u2264 4 / 5 * \u03b5 * (4 / 5 * \u03b5) * (9 / 16 * \u03b5 ^ 2)\n[PROOFSTEP]\nnlinarith\n[GOAL]\ncase bc.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nUl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\nVl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts\nthis : 0 < \u03b5\nUVl :\n 4 / 5 * \u03b5 * (4 / 5 * \u03b5) \u2264\n \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts *\n (\u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts)\n\u22a2 0 \u2264 9 / 4 ^ 2 * \u03b5 ^ 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase bc.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nUl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts\nVl : 4 / 5 * \u03b5 \u2264 \u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts\nthis : 0 < \u03b5\nUVl :\n 4 / 5 * \u03b5 * (4 / 5 * \u03b5) \u2264\n \u2191(Finset.card (star hP G \u03b5 hU V)) / 4 ^ Finset.card P.parts *\n (\u2191(Finset.card (star hP G \u03b5 hV U)) / 4 ^ Finset.card P.parts)\n\u22a2 0 \u2264 \u03b5 ^ 2\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 +\n \u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)) / 16 ^ Finset.card P.parts * (9 / 16) *\n \u03b5 ^ 2 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd) ^ 2) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nhave t : (star hP G \u03b5 hU V).product (star hP G \u03b5 hV U) \u2286 (chunk hP G \u03b5 hU).parts.product (chunk hP G \u03b5 hV).parts :=\n product_subset_product star_subset_chunk star_subset_chunk\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 +\n \u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)) / 16 ^ Finset.card P.parts * (9 / 16) *\n \u03b5 ^ 2 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd) ^ 2) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nhave h\u03b5 : 0 \u2264 \u03b5 := by sz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\n\u22a2 0 \u2264 \u03b5\n[PROOFSTEP]\nsz_positivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 +\n \u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)) / 16 ^ Finset.card P.parts * (9 / 16) *\n \u03b5 ^ 2 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd) ^ 2) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nhave sp : \u2200 (a b : Finset (Finset \u03b1)), a.product b = a \u00d7\u02e2 b := fun a b => rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 +\n \u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)) / 16 ^ Finset.card P.parts * (9 / 16) *\n \u03b5 ^ 2 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd) ^ 2) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nhave :=\n add_div_le_sum_sq_div_card t (fun x => (G.edgeDensity x.1 x.2 : \u211d)) ((G.edgeDensity U V : \u211d) ^ 2 - \u03b5 ^ 5 / \u219125)\n (show 0 \u2264 3 / 4 * \u03b5 by linarith) ?_ ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 0 \u2264 3 / 4 * \u03b5\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\nthis :\n \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 +\n \u2191(Finset.card (Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U))) /\n \u2191(Finset.card (Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts)) *\n (3 / 4 * \u03b5) ^ 2 \u2264\n (\u2211 i in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n (fun x => \u2191(SimpleGraph.edgeDensity G x.fst x.snd)) i ^ 2) /\n \u2191(Finset.card (Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts))\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 +\n \u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)) / 16 ^ Finset.card P.parts * (9 / 16) *\n \u03b5 ^ 2 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd) ^ 2) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nsimp_rw [sp, card_product, card_chunk (m_pos hP\u03b1).ne', \u2190 mul_pow, cast_pow, mul_pow, div_pow, \u2190 mul_assoc] at this \n[GOAL]\ncase refine_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\nthis :\n \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 +\n \u2191(Finset.card (star hP G \u03b5 hU V) * Finset.card (star hP G \u03b5 hV U)) / \u2191(4 * 4) ^ Finset.card P.parts *\n (3 ^ 2 / 4 ^ 2) *\n \u03b5 ^ 2 \u2264\n (\u2211 x in (chunk hP G \u03b5 hU).parts \u00d7\u02e2 (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G x.fst x.snd) ^ 2) /\n \u2191(4 * 4) ^ Finset.card P.parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 +\n \u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)) / 16 ^ Finset.card P.parts * (9 / 16) *\n \u03b5 ^ 2 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd) ^ 2) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nnorm_num at this \n[GOAL]\ncase refine_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\nthis :\n \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 +\n \u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)) / 16 ^ Finset.card P.parts * (9 / 16) *\n \u03b5 ^ 2 \u2264\n (\u2211 x in (chunk hP G \u03b5 hU).parts \u00d7\u02e2 (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G x.fst x.snd) ^ 2) /\n 16 ^ Finset.card P.parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 +\n \u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U)) / 16 ^ Finset.card P.parts * (9 / 16) *\n \u03b5 ^ 2 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd) ^ 2) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nexact this\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 3 / 4 * \u03b5 \u2264\n |(\u2211 i in Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U),\n (fun x => \u2191(SimpleGraph.edgeDensity G x.fst x.snd)) i) /\n \u2191(Finset.card (Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U))) -\n (\u2211 i in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n (fun x => \u2191(SimpleGraph.edgeDensity G x.fst x.snd)) i) /\n \u2191(Finset.card (Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts))|\n[PROOFSTEP]\nsimp_rw [sp, card_product, card_chunk (m_pos hP\u03b1).ne', \u2190 mul_pow]\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 3 / 4 * \u03b5 \u2264\n |(\u2211 x in star hP G \u03b5 hU V \u00d7\u02e2 star hP G \u03b5 hV U, \u2191(SimpleGraph.edgeDensity G x.fst x.snd)) /\n \u2191(Finset.card (star hP G \u03b5 hU V) * Finset.card (star hP G \u03b5 hV U)) -\n (\u2211 x in (chunk hP G \u03b5 hU).parts \u00d7\u02e2 (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G x.fst x.snd)) /\n \u2191((4 * 4) ^ Finset.card P.parts)|\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 3 / 4 * \u03b5 \u2264\n |(\u2211 x in star hP G \u03b5 hU V \u00d7\u02e2 star hP G \u03b5 hV U, \u2191(SimpleGraph.edgeDensity G x.fst x.snd)) /\n (\u2191(Finset.card (star hP G \u03b5 hU V)) * \u2191(Finset.card (star hP G \u03b5 hV U))) -\n (\u2211 x in (chunk hP G \u03b5 hU).parts \u00d7\u02e2 (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G x.fst x.snd)) /\n 16 ^ Finset.card P.parts|\n[PROOFSTEP]\nexact edgeDensity_star_not_uniform hP\u03b1 hP\u03b5 h\u03b5\u2081 hUVne hUV\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 \u2264\n ((\u2211 i in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n (fun x => \u2191(SimpleGraph.edgeDensity G x.fst x.snd)) i) /\n \u2191(Finset.card (Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts))) ^\n 2\n[PROOFSTEP]\nrw [sp, card_product]\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 \u2264\n ((\u2211 i in (chunk hP G \u03b5 hU).parts \u00d7\u02e2 (chunk hP G \u03b5 hV).parts,\n (fun x => \u2191(SimpleGraph.edgeDensity G x.fst x.snd)) i) /\n \u2191(Finset.card (chunk hP G \u03b5 hU).parts * Finset.card (chunk hP G \u03b5 hV).parts)) ^\n 2\n[PROOFSTEP]\napply (edgeDensity_chunk_aux hP\u03b1 hP\u03b5 hU hV).trans\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 ((\u2211 ab in Finset.product (chunk ?m.2764430 G \u03b5 hU).parts (chunk ?m.2764430 G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts) ^\n 2 \u2264\n ((\u2211 i in (chunk hP G \u03b5 hU).parts \u00d7\u02e2 (chunk hP G \u03b5 hV).parts,\n (fun x => \u2191(SimpleGraph.edgeDensity G x.fst x.snd)) i) /\n \u2191(Finset.card (chunk hP G \u03b5 hU).parts * Finset.card (chunk hP G \u03b5 hV).parts)) ^\n 2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 IsEquipartition P\n[PROOFSTEP]\nrw [card_chunk (m_pos hP\u03b1).ne', card_chunk (m_pos hP\u03b1).ne', \u2190 mul_pow]\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 ((\u2211 ab in Finset.product (chunk ?m.2764430 G \u03b5 hU).parts (chunk ?m.2764430 G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts) ^\n 2 \u2264\n ((\u2211 i in (chunk hP G \u03b5 hU).parts \u00d7\u02e2 (chunk hP G \u03b5 hV).parts,\n (fun x => \u2191(SimpleGraph.edgeDensity G x.fst x.snd)) i) /\n \u2191((4 * 4) ^ Finset.card P.parts)) ^\n 2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 IsEquipartition P\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 IsEquipartition P\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 IsEquipartition P\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 IsEquipartition P\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 (\u2211 ab in Finset.product (chunk ?m.2764430 G \u03b5 hU).parts (chunk ?m.2764430 G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) ^\n 2 /\n (16 ^ Finset.card P.parts) ^ 2 \u2264\n (\u2211 ab in (chunk hP G \u03b5 hU).parts \u00d7\u02e2 (chunk hP G \u03b5 hV).parts, \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) ^ 2 /\n (16 ^ Finset.card P.parts) ^ 2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 IsEquipartition P\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 IsEquipartition P\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 IsEquipartition P\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nh\u03b5\u2081 : \u03b5 \u2264 1\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nhUVne : U \u2260 V\nhUV : \u00acSimpleGraph.IsUniform G \u03b5 U V\nt :\n Finset.product (star hP G \u03b5 hU V) (star hP G \u03b5 hV U) \u2286 Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts\nh\u03b5 : 0 \u2264 \u03b5\nsp : \u2200 (a b : Finset (Finset \u03b1)), Finset.product a b = a \u00d7\u02e2 b\n\u22a2 IsEquipartition P\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\n\u22a2 \u2191(SimpleGraph.edgeDensity G U V) ^ 2 - \u03b5 ^ 5 / 25 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd) ^ 2) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\napply (edgeDensity_chunk_aux (hP := hP) hP\u03b1 hP\u03b5 hU hV).trans\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\n\u22a2 ((\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts) ^\n 2 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd) ^ 2) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nhave key : \u219116 ^ P.parts.card = (((chunk hP G \u03b5 hU).parts \u00d7\u02e2 (chunk hP G \u03b5 hV).parts).card : \u211d) := by\n rw [card_product, cast_mul, card_chunk (m_pos hP\u03b1).ne', card_chunk (m_pos hP\u03b1).ne', \u2190 cast_mul, \u2190 mul_pow]; norm_cast\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\n\u22a2 16 ^ Finset.card P.parts = \u2191(Finset.card ((chunk hP G \u03b5 hU).parts \u00d7\u02e2 (chunk hP G \u03b5 hV).parts))\n[PROOFSTEP]\nrw [card_product, cast_mul, card_chunk (m_pos hP\u03b1).ne', card_chunk (m_pos hP\u03b1).ne', \u2190 cast_mul, \u2190 mul_pow]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\n\u22a2 16 ^ Finset.card P.parts = \u2191((4 * 4) ^ Finset.card P.parts)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nkey : 16 ^ Finset.card P.parts = \u2191(Finset.card ((chunk hP G \u03b5 hU).parts \u00d7\u02e2 (chunk hP G \u03b5 hV).parts))\n\u22a2 ((\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n 16 ^ Finset.card P.parts) ^\n 2 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd) ^ 2) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nsimp_rw [key]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nU : Finset \u03b1\nhU\u271d : U \u2208 P.parts\nV : Finset \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhU : U \u2208 P.parts\nhV : V \u2208 P.parts\nkey : 16 ^ Finset.card P.parts = \u2191(Finset.card ((chunk hP G \u03b5 hU).parts \u00d7\u02e2 (chunk hP G \u03b5 hV).parts))\n\u22a2 ((\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd)) /\n \u2191(Finset.card ((chunk hP G \u03b5 hU).parts \u00d7\u02e2 (chunk hP G \u03b5 hV).parts))) ^\n 2 \u2264\n (\u2211 ab in Finset.product (chunk hP G \u03b5 hU).parts (chunk hP G \u03b5 hV).parts,\n \u2191(SimpleGraph.edgeDensity G ab.fst ab.snd) ^ 2) /\n \u2191(Finset.card ((chunk hP G \u03b5 hU).parts \u00d7\u02e2 (chunk hP G \u03b5 hV).parts))\n[PROOFSTEP]\nconvert sum_div_card_sq_le_sum_sq_div_card (\u03b1 := \u211d)\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk", "llama_tokens": 142937, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754371026367, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.40364087909006924}} {"text": "[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : CommRing A\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : IsDomain A\ninst\u271d\u00b9\u00b2 : Algebra A K\ninst\u271d\u00b9\u00b9 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u2070 : Field L\nC : Type u_5\ninst\u271d\u2079 : CommRing C\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra A L\ninst\u271d\u2076 : IsScalarTower A K L\ninst\u271d\u2075 : Algebra C L\ninst\u271d\u2074 : IsIntegralClosure C A L\ninst\u271d\u00b3 : Algebra A C\ninst\u271d\u00b2 : IsScalarTower A C L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : NoZeroSMulDivisors A L\n\u22a2 IsLocalization (algebraMapSubmonoid C A\u2070) L\n[PROOFSTEP]\nhaveI : IsDomain C := (IsIntegralClosure.equiv A C L (integralClosure A L)).toMulEquiv.isDomain (integralClosure A L)\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : CommRing A\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : IsDomain A\ninst\u271d\u00b9\u00b2 : Algebra A K\ninst\u271d\u00b9\u00b9 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u2070 : Field L\nC : Type u_5\ninst\u271d\u2079 : CommRing C\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra A L\ninst\u271d\u2076 : IsScalarTower A K L\ninst\u271d\u2075 : Algebra C L\ninst\u271d\u2074 : IsIntegralClosure C A L\ninst\u271d\u00b3 : Algebra A C\ninst\u271d\u00b2 : IsScalarTower A C L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : NoZeroSMulDivisors A L\nthis : IsDomain C\n\u22a2 IsLocalization (algebraMapSubmonoid C A\u2070) L\n[PROOFSTEP]\nhaveI : NoZeroSMulDivisors A C := IsIntegralClosure.noZeroSMulDivisors A L\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : CommRing A\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : IsDomain A\ninst\u271d\u00b9\u00b2 : Algebra A K\ninst\u271d\u00b9\u00b9 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u2070 : Field L\nC : Type u_5\ninst\u271d\u2079 : CommRing C\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra A L\ninst\u271d\u2076 : IsScalarTower A K L\ninst\u271d\u2075 : Algebra C L\ninst\u271d\u2074 : IsIntegralClosure C A L\ninst\u271d\u00b3 : Algebra A C\ninst\u271d\u00b2 : IsScalarTower A C L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d : IsDomain C\nthis : NoZeroSMulDivisors A C\n\u22a2 IsLocalization (algebraMapSubmonoid C A\u2070) L\n[PROOFSTEP]\nrefine' \u27e8_, fun z => _, fun {x y} => \u27e8fun h => \u27e81, _\u27e9, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : CommRing A\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : IsDomain A\ninst\u271d\u00b9\u00b2 : Algebra A K\ninst\u271d\u00b9\u00b9 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u2070 : Field L\nC : Type u_5\ninst\u271d\u2079 : CommRing C\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra A L\ninst\u271d\u2076 : IsScalarTower A K L\ninst\u271d\u2075 : Algebra C L\ninst\u271d\u2074 : IsIntegralClosure C A L\ninst\u271d\u00b3 : Algebra A C\ninst\u271d\u00b2 : IsScalarTower A C L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d : IsDomain C\nthis : NoZeroSMulDivisors A C\n\u22a2 \u2200 (y : { x // x \u2208 algebraMapSubmonoid C A\u2070 }), IsUnit (\u2191(algebraMap C L) \u2191y)\n[PROOFSTEP]\nrintro \u27e8_, x, hx, rfl\u27e9\n[GOAL]\ncase refine'_1.mk.intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : CommRing A\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : IsDomain A\ninst\u271d\u00b9\u00b2 : Algebra A K\ninst\u271d\u00b9\u00b9 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u2070 : Field L\nC : Type u_5\ninst\u271d\u2079 : CommRing C\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra A L\ninst\u271d\u2076 : IsScalarTower A K L\ninst\u271d\u2075 : Algebra C L\ninst\u271d\u2074 : IsIntegralClosure C A L\ninst\u271d\u00b3 : Algebra A C\ninst\u271d\u00b2 : IsScalarTower A C L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d : IsDomain C\nthis : NoZeroSMulDivisors A C\nx : A\nhx : x \u2208 \u2191A\u2070\n\u22a2 IsUnit\n (\u2191(algebraMap C L)\n \u2191{ val := \u2191(algebraMap A C) x, property := (_ : \u2203 a, a \u2208 \u2191A\u2070 \u2227 \u2191(algebraMap A C) a = \u2191(algebraMap A C) x) })\n[PROOFSTEP]\nrw [isUnit_iff_ne_zero, map_ne_zero_iff _ (IsIntegralClosure.algebraMap_injective C A L), Subtype.coe_mk,\n map_ne_zero_iff _ (NoZeroSMulDivisors.algebraMap_injective A C)]\n[GOAL]\ncase refine'_1.mk.intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : CommRing A\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : IsDomain A\ninst\u271d\u00b9\u00b2 : Algebra A K\ninst\u271d\u00b9\u00b9 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u2070 : Field L\nC : Type u_5\ninst\u271d\u2079 : CommRing C\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra A L\ninst\u271d\u2076 : IsScalarTower A K L\ninst\u271d\u2075 : Algebra C L\ninst\u271d\u2074 : IsIntegralClosure C A L\ninst\u271d\u00b3 : Algebra A C\ninst\u271d\u00b2 : IsScalarTower A C L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d : IsDomain C\nthis : NoZeroSMulDivisors A C\nx : A\nhx : x \u2208 \u2191A\u2070\n\u22a2 x \u2260 0\n[PROOFSTEP]\nexact mem_nonZeroDivisors_iff_ne_zero.mp hx\n[GOAL]\ncase refine'_2\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : CommRing A\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : IsDomain A\ninst\u271d\u00b9\u00b2 : Algebra A K\ninst\u271d\u00b9\u00b9 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u2070 : Field L\nC : Type u_5\ninst\u271d\u2079 : CommRing C\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra A L\ninst\u271d\u2076 : IsScalarTower A K L\ninst\u271d\u2075 : Algebra C L\ninst\u271d\u2074 : IsIntegralClosure C A L\ninst\u271d\u00b3 : Algebra A C\ninst\u271d\u00b2 : IsScalarTower A C L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d : IsDomain C\nthis : NoZeroSMulDivisors A C\nz : L\n\u22a2 \u2203 x, z * \u2191(algebraMap C L) \u2191x.snd = \u2191(algebraMap C L) x.fst\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := IsIntegral.exists_multiple_integral_of_isLocalization A\u2070 z (IsSeparable.isIntegral K z)\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : CommRing A\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : IsDomain A\ninst\u271d\u00b9\u00b2 : Algebra A K\ninst\u271d\u00b9\u00b9 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u2070 : Field L\nC : Type u_5\ninst\u271d\u2079 : CommRing C\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra A L\ninst\u271d\u2076 : IsScalarTower A K L\ninst\u271d\u2075 : Algebra C L\ninst\u271d\u2074 : IsIntegralClosure C A L\ninst\u271d\u00b3 : Algebra A C\ninst\u271d\u00b2 : IsScalarTower A C L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d : IsDomain C\nthis : NoZeroSMulDivisors A C\nz : L\nm : { x // x \u2208 A\u2070 }\nhm : IsIntegral A (m \u2022 z)\n\u22a2 \u2203 x, z * \u2191(algebraMap C L) \u2191x.snd = \u2191(algebraMap C L) x.fst\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 : \u2203 x, algebraMap C L x = m \u2022 z := IsIntegralClosure.isIntegral_iff.mp hm\n[GOAL]\ncase refine'_2.intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : CommRing A\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : IsDomain A\ninst\u271d\u00b9\u00b2 : Algebra A K\ninst\u271d\u00b9\u00b9 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u2070 : Field L\nC : Type u_5\ninst\u271d\u2079 : CommRing C\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra A L\ninst\u271d\u2076 : IsScalarTower A K L\ninst\u271d\u2075 : Algebra C L\ninst\u271d\u2074 : IsIntegralClosure C A L\ninst\u271d\u00b3 : Algebra A C\ninst\u271d\u00b2 : IsScalarTower A C L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d : IsDomain C\nthis : NoZeroSMulDivisors A C\nz : L\nm : { x // x \u2208 A\u2070 }\nhm : IsIntegral A (m \u2022 z)\nx : C\nhx : \u2191(algebraMap C L) x = m \u2022 z\n\u22a2 \u2203 x, z * \u2191(algebraMap C L) \u2191x.snd = \u2191(algebraMap C L) x.fst\n[PROOFSTEP]\nrefine' \u27e8\u27e8x, algebraMap A C m, m, SetLike.coe_mem m, rfl\u27e9, _\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : CommRing A\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : IsDomain A\ninst\u271d\u00b9\u00b2 : Algebra A K\ninst\u271d\u00b9\u00b9 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u2070 : Field L\nC : Type u_5\ninst\u271d\u2079 : CommRing C\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra A L\ninst\u271d\u2076 : IsScalarTower A K L\ninst\u271d\u2075 : Algebra C L\ninst\u271d\u2074 : IsIntegralClosure C A L\ninst\u271d\u00b3 : Algebra A C\ninst\u271d\u00b2 : IsScalarTower A C L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d : IsDomain C\nthis : NoZeroSMulDivisors A C\nz : L\nm : { x // x \u2208 A\u2070 }\nhm : IsIntegral A (m \u2022 z)\nx : C\nhx : \u2191(algebraMap C L) x = m \u2022 z\n\u22a2 z *\n \u2191(algebraMap C L)\n \u2191(x,\n { val := \u2191(algebraMap A C) \u2191m,\n property := (_ : \u2203 a, a \u2208 \u2191A\u2070 \u2227 \u2191(algebraMap A C) a = \u2191(algebraMap A C) \u2191m) }).snd =\n \u2191(algebraMap C L)\n (x,\n { val := \u2191(algebraMap A C) \u2191m,\n property := (_ : \u2203 a, a \u2208 \u2191A\u2070 \u2227 \u2191(algebraMap A C) a = \u2191(algebraMap A C) \u2191m) }).fst\n[PROOFSTEP]\nrw [Subtype.coe_mk, \u2190 IsScalarTower.algebraMap_apply, hx, mul_comm, Submonoid.smul_def, smul_def]\n[GOAL]\ncase refine'_3\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : CommRing A\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : IsDomain A\ninst\u271d\u00b9\u00b2 : Algebra A K\ninst\u271d\u00b9\u00b9 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u2070 : Field L\nC : Type u_5\ninst\u271d\u2079 : CommRing C\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra A L\ninst\u271d\u2076 : IsScalarTower A K L\ninst\u271d\u2075 : Algebra C L\ninst\u271d\u2074 : IsIntegralClosure C A L\ninst\u271d\u00b3 : Algebra A C\ninst\u271d\u00b2 : IsScalarTower A C L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d : IsDomain C\nthis : NoZeroSMulDivisors A C\nx y : C\nh : \u2191(algebraMap C L) x = \u2191(algebraMap C L) y\n\u22a2 \u21911 * x = \u21911 * y\n[PROOFSTEP]\nsimp only [IsIntegralClosure.algebraMap_injective C A L h]\n[GOAL]\ncase refine'_4\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : CommRing A\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : IsDomain A\ninst\u271d\u00b9\u00b2 : Algebra A K\ninst\u271d\u00b9\u00b9 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u2070 : Field L\nC : Type u_5\ninst\u271d\u2079 : CommRing C\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra A L\ninst\u271d\u2076 : IsScalarTower A K L\ninst\u271d\u2075 : Algebra C L\ninst\u271d\u2074 : IsIntegralClosure C A L\ninst\u271d\u00b3 : Algebra A C\ninst\u271d\u00b2 : IsScalarTower A C L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d : IsDomain C\nthis : NoZeroSMulDivisors A C\nx y : C\n\u22a2 (\u2203 c, \u2191c * x = \u2191c * y) \u2192 \u2191(algebraMap C L) x = \u2191(algebraMap C L) y\n[PROOFSTEP]\nrintro \u27e8\u27e8_, m, hm, rfl\u27e9, h\u27e9\n[GOAL]\ncase refine'_4.intro.mk.intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : CommRing A\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : IsDomain A\ninst\u271d\u00b9\u00b2 : Algebra A K\ninst\u271d\u00b9\u00b9 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u2070 : Field L\nC : Type u_5\ninst\u271d\u2079 : CommRing C\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra A L\ninst\u271d\u2076 : IsScalarTower A K L\ninst\u271d\u2075 : Algebra C L\ninst\u271d\u2074 : IsIntegralClosure C A L\ninst\u271d\u00b3 : Algebra A C\ninst\u271d\u00b2 : IsScalarTower A C L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d : IsDomain C\nthis : NoZeroSMulDivisors A C\nx y : C\nm : A\nhm : m \u2208 \u2191A\u2070\nh :\n \u2191{ val := \u2191(algebraMap A C) m, property := (_ : \u2203 a, a \u2208 \u2191A\u2070 \u2227 \u2191(algebraMap A C) a = \u2191(algebraMap A C) m) } * x =\n \u2191{ val := \u2191(algebraMap A C) m, property := (_ : \u2203 a, a \u2208 \u2191A\u2070 \u2227 \u2191(algebraMap A C) a = \u2191(algebraMap A C) m) } * y\n\u22a2 \u2191(algebraMap C L) x = \u2191(algebraMap C L) y\n[PROOFSTEP]\nrefine' congr_arg (algebraMap C L) ((mul_right_inj' _).mp h)\n[GOAL]\ncase refine'_4.intro.mk.intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : CommRing A\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : IsDomain A\ninst\u271d\u00b9\u00b2 : Algebra A K\ninst\u271d\u00b9\u00b9 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u2070 : Field L\nC : Type u_5\ninst\u271d\u2079 : CommRing C\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra A L\ninst\u271d\u2076 : IsScalarTower A K L\ninst\u271d\u2075 : Algebra C L\ninst\u271d\u2074 : IsIntegralClosure C A L\ninst\u271d\u00b3 : Algebra A C\ninst\u271d\u00b2 : IsScalarTower A C L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d : IsDomain C\nthis : NoZeroSMulDivisors A C\nx y : C\nm : A\nhm : m \u2208 \u2191A\u2070\nh :\n \u2191{ val := \u2191(algebraMap A C) m, property := (_ : \u2203 a, a \u2208 \u2191A\u2070 \u2227 \u2191(algebraMap A C) a = \u2191(algebraMap A C) m) } * x =\n \u2191{ val := \u2191(algebraMap A C) m, property := (_ : \u2203 a, a \u2208 \u2191A\u2070 \u2227 \u2191(algebraMap A C) a = \u2191(algebraMap A C) m) } * y\n\u22a2 \u2191{ val := \u2191(algebraMap A C) m, property := (_ : \u2203 a, a \u2208 \u2191A\u2070 \u2227 \u2191(algebraMap A C) a = \u2191(algebraMap A C) m) } \u2260 0\n[PROOFSTEP]\nrw [Subtype.coe_mk, map_ne_zero_iff _ (NoZeroSMulDivisors.algebraMap_injective A C)]\n[GOAL]\ncase refine'_4.intro.mk.intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : CommRing A\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : IsDomain A\ninst\u271d\u00b9\u00b2 : Algebra A K\ninst\u271d\u00b9\u00b9 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u2070 : Field L\nC : Type u_5\ninst\u271d\u2079 : CommRing C\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra A L\ninst\u271d\u2076 : IsScalarTower A K L\ninst\u271d\u2075 : Algebra C L\ninst\u271d\u2074 : IsIntegralClosure C A L\ninst\u271d\u00b3 : Algebra A C\ninst\u271d\u00b2 : IsScalarTower A C L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d : IsDomain C\nthis : NoZeroSMulDivisors A C\nx y : C\nm : A\nhm : m \u2208 \u2191A\u2070\nh :\n \u2191{ val := \u2191(algebraMap A C) m, property := (_ : \u2203 a, a \u2208 \u2191A\u2070 \u2227 \u2191(algebraMap A C) a = \u2191(algebraMap A C) m) } * x =\n \u2191{ val := \u2191(algebraMap A C) m, property := (_ : \u2203 a, a \u2208 \u2191A\u2070 \u2227 \u2191(algebraMap A C) a = \u2191(algebraMap A C) m) } * y\n\u22a2 m \u2260 0\n[PROOFSTEP]\nexact mem_nonZeroDivisors_iff_ne_zero.mp hm\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\n\u22a2 LinearMap.range (\u2191A (Algebra.linearMap C L)) \u2264\n Submodule.span A (Set.range \u2191(BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b))\n[PROOFSTEP]\nlet db := (traceForm K L).dualBasis (traceForm_nondegenerate K L) b\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\n\u22a2 LinearMap.range (\u2191A (Algebra.linearMap C L)) \u2264\n Submodule.span A (Set.range \u2191(BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b))\n[PROOFSTEP]\nrintro _ \u27e8x, rfl\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\n\u22a2 \u2191(\u2191A (Algebra.linearMap C L)) x \u2208\n Submodule.span A (Set.range \u2191(BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b))\n[PROOFSTEP]\nsimp only [LinearMap.coe_restrictScalars, Algebra.linearMap_apply]\n[GOAL]\ncase intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\n\u22a2 \u2191(algebraMap C L) x \u2208\n Submodule.span A (Set.range \u2191(BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b))\n[PROOFSTEP]\nhave hx : IsIntegral A (algebraMap C L x) := (IsIntegralClosure.isIntegral A L x).algebraMap\n[GOAL]\ncase intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx : IsIntegral A (\u2191(algebraMap C L) x)\n\u22a2 \u2191(algebraMap C L) x \u2208\n Submodule.span A (Set.range \u2191(BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b))\n[PROOFSTEP]\nrsuffices \u27e8c, x_eq\u27e9 : \u2203 c : \u03b9 \u2192 A, algebraMap C L x = \u2211 i, c i \u2022 db i\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx : IsIntegral A (\u2191(algebraMap C L) x)\nc : \u03b9 \u2192 A\nx_eq : \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\n\u22a2 \u2191(algebraMap C L) x \u2208\n Submodule.span A (Set.range \u2191(BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b))\n[PROOFSTEP]\nrw [x_eq]\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx : IsIntegral A (\u2191(algebraMap C L) x)\nc : \u03b9 \u2192 A\nx_eq : \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\n\u22a2 \u2211 i : \u03b9, c i \u2022 \u2191db i \u2208\n Submodule.span A (Set.range \u2191(BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b))\n[PROOFSTEP]\nrefine' Submodule.sum_mem _ fun i _ => Submodule.smul_mem _ _ (Submodule.subset_span _)\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx : IsIntegral A (\u2191(algebraMap C L) x)\nc : \u03b9 \u2192 A\nx_eq : \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2191db i \u2208 Set.range \u2191(BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b)\n[PROOFSTEP]\nrw [Set.mem_range]\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx : IsIntegral A (\u2191(algebraMap C L) x)\nc : \u03b9 \u2192 A\nx_eq : \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2203 y, \u2191(BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b) y = \u2191db i\n[PROOFSTEP]\nexact \u27e8i, rfl\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx : IsIntegral A (\u2191(algebraMap C L) x)\n\u22a2 \u2203 c, \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\n[PROOFSTEP]\nsuffices \u2203 c : \u03b9 \u2192 K, (\u2200 i, IsIntegral A (c i)) \u2227 algebraMap C L x = \u2211 i, c i \u2022 db i\n by\n obtain \u27e8c, hc, hx\u27e9 := this\n have hc' : \u2200 i, IsLocalization.IsInteger A (c i) := fun i => IsIntegrallyClosed.isIntegral_iff.mp (hc i)\n use fun i => Classical.choose (hc' i)\n refine' hx.trans (Finset.sum_congr rfl fun i _ => _)\n conv_lhs => rw [\u2190 Classical.choose_spec (hc' i)]\n rw [\u2190 IsScalarTower.algebraMap_smul K (Classical.choose (hc' i)) (db i)]\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx : IsIntegral A (\u2191(algebraMap C L) x)\nthis : \u2203 c, (\u2200 (i : \u03b9), IsIntegral A (c i)) \u2227 \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\n\u22a2 \u2203 c, \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\n[PROOFSTEP]\nobtain \u27e8c, hc, hx\u27e9 := this\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx\u271d : IsIntegral A (\u2191(algebraMap C L) x)\nc : \u03b9 \u2192 K\nhc : \u2200 (i : \u03b9), IsIntegral A (c i)\nhx : \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\n\u22a2 \u2203 c, \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\n[PROOFSTEP]\nhave hc' : \u2200 i, IsLocalization.IsInteger A (c i) := fun i => IsIntegrallyClosed.isIntegral_iff.mp (hc i)\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx\u271d : IsIntegral A (\u2191(algebraMap C L) x)\nc : \u03b9 \u2192 K\nhc : \u2200 (i : \u03b9), IsIntegral A (c i)\nhx : \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\nhc' : \u2200 (i : \u03b9), IsLocalization.IsInteger A (c i)\n\u22a2 \u2203 c, \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\n[PROOFSTEP]\nuse fun i => Classical.choose (hc' i)\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx\u271d : IsIntegral A (\u2191(algebraMap C L) x)\nc : \u03b9 \u2192 K\nhc : \u2200 (i : \u03b9), IsIntegral A (c i)\nhx : \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\nhc' : \u2200 (i : \u03b9), IsLocalization.IsInteger A (c i)\n\u22a2 \u2191(algebraMap C L) x = \u2211 i : \u03b9, Classical.choose (_ : IsLocalization.IsInteger A (c i)) \u2022 \u2191db i\n[PROOFSTEP]\nrefine' hx.trans (Finset.sum_congr rfl fun i _ => _)\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx\u271d : IsIntegral A (\u2191(algebraMap C L) x)\nc : \u03b9 \u2192 K\nhc : \u2200 (i : \u03b9), IsIntegral A (c i)\nhx : \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\nhc' : \u2200 (i : \u03b9), IsLocalization.IsInteger A (c i)\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 c i \u2022 \u2191db i = Classical.choose (_ : IsLocalization.IsInteger A (c i)) \u2022 \u2191db i\n[PROOFSTEP]\nconv_lhs => rw [\u2190 Classical.choose_spec (hc' i)]\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx\u271d : IsIntegral A (\u2191(algebraMap C L) x)\nc : \u03b9 \u2192 K\nhc : \u2200 (i : \u03b9), IsIntegral A (c i)\nhx : \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\nhc' : \u2200 (i : \u03b9), IsLocalization.IsInteger A (c i)\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n| c i \u2022 \u2191db i\n[PROOFSTEP]\nrw [\u2190 Classical.choose_spec (hc' i)]\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx\u271d : IsIntegral A (\u2191(algebraMap C L) x)\nc : \u03b9 \u2192 K\nhc : \u2200 (i : \u03b9), IsIntegral A (c i)\nhx : \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\nhc' : \u2200 (i : \u03b9), IsLocalization.IsInteger A (c i)\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n| c i \u2022 \u2191db i\n[PROOFSTEP]\nrw [\u2190 Classical.choose_spec (hc' i)]\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx\u271d : IsIntegral A (\u2191(algebraMap C L) x)\nc : \u03b9 \u2192 K\nhc : \u2200 (i : \u03b9), IsIntegral A (c i)\nhx : \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\nhc' : \u2200 (i : \u03b9), IsLocalization.IsInteger A (c i)\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n| c i \u2022 \u2191db i\n[PROOFSTEP]\nrw [\u2190 Classical.choose_spec (hc' i)]\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx\u271d : IsIntegral A (\u2191(algebraMap C L) x)\nc : \u03b9 \u2192 K\nhc : \u2200 (i : \u03b9), IsIntegral A (c i)\nhx : \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\nhc' : \u2200 (i : \u03b9), IsLocalization.IsInteger A (c i)\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2191(algebraMap A K) (Classical.choose (_ : IsLocalization.IsInteger A (c i))) \u2022 \u2191db i =\n Classical.choose (_ : IsLocalization.IsInteger A (c i)) \u2022 \u2191db i\n[PROOFSTEP]\nrw [\u2190 IsScalarTower.algebraMap_smul K (Classical.choose (hc' i)) (db i)]\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx : IsIntegral A (\u2191(algebraMap C L) x)\n\u22a2 \u2203 c, (\u2200 (i : \u03b9), IsIntegral A (c i)) \u2227 \u2191(algebraMap C L) x = \u2211 i : \u03b9, c i \u2022 \u2191db i\n[PROOFSTEP]\nrefine' \u27e8fun i => db.repr (algebraMap C L x) i, fun i => _, (db.sum_repr _).symm\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx : IsIntegral A (\u2191(algebraMap C L) x)\ni : \u03b9\n\u22a2 IsIntegral A ((fun i => \u2191(\u2191db.repr (\u2191(algebraMap C L) x)) i) i)\n[PROOFSTEP]\nsimp_rw [BilinForm.dualBasis_repr_apply]\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\ndb : Basis \u03b9 K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nx : C\nhx : IsIntegral A (\u2191(algebraMap C L) x)\ni : \u03b9\n\u22a2 IsIntegral A (BilinForm.bilin (traceForm K L) (\u2191(algebraMap C L) x) (\u2191b i))\n[PROOFSTEP]\nexact isIntegral_trace (isIntegral_mul hx (hb_int i))\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\n\u22a2 \u2191Subalgebra.toSubmodule (integralClosure A L) \u2264\n Submodule.span A (Set.range \u2191(BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b))\n[PROOFSTEP]\nrefine' le_trans _ (IsIntegralClosure.range_le_span_dualBasis (integralClosure A L) b hb_int)\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\n\u22a2 \u2191Subalgebra.toSubmodule (integralClosure A L) \u2264\n LinearMap.range (\u2191A (Algebra.linearMap { x // x \u2208 integralClosure A L } L))\n[PROOFSTEP]\nintro x hx\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing A\ninst\u271d\u00b9\u2077 : Field K\ninst\u271d\u00b9\u2076 : IsDomain A\ninst\u271d\u00b9\u2075 : Algebra A K\ninst\u271d\u00b9\u2074 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b3 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing C\ninst\u271d\u00b9\u00b9 : Algebra K L\ninst\u271d\u00b9\u2070 : Algebra A L\ninst\u271d\u2079 : IsScalarTower A K L\ninst\u271d\u2078 : Algebra C L\ninst\u271d\u2077 : IsIntegralClosure C A L\ninst\u271d\u2076 : Algebra A C\ninst\u271d\u2075 : IsScalarTower A C L\ninst\u271d\u2074 : FiniteDimensional K L\ninst\u271d\u00b3 : IsSeparable K L\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nb : Basis \u03b9 K L\nhb_int : \u2200 (i : \u03b9), IsIntegral A (\u2191b i)\ninst\u271d : IsIntegrallyClosed A\nx : L\nhx : x \u2208 \u2191Subalgebra.toSubmodule (integralClosure A L)\n\u22a2 x \u2208 LinearMap.range (\u2191A (Algebra.linearMap { x // x \u2208 integralClosure A L } L))\n[PROOFSTEP]\nexact \u27e8\u27e8x, hx\u27e9, rfl\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns : Finset L\n\u22a2 \u2203 y x, \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\n[PROOFSTEP]\nhaveI := Classical.decEq L\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns : Finset L\nthis : DecidableEq L\n\u22a2 \u2203 y x, \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\n[PROOFSTEP]\nrefine' s.induction _ _\n[GOAL]\ncase refine'_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns : Finset L\nthis : DecidableEq L\n\u22a2 \u2203 y x, \u2200 (x : L), x \u2208 \u2205 \u2192 IsIntegral A (y \u2022 x)\n[PROOFSTEP]\nuse 1, one_ne_zero\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns : Finset L\nthis : DecidableEq L\n\u22a2 \u2200 (x : L), x \u2208 \u2205 \u2192 IsIntegral A (1 \u2022 x)\n[PROOFSTEP]\nrintro x \u27e8\u27e9\n[GOAL]\ncase refine'_2\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns : Finset L\nthis : DecidableEq L\n\u22a2 \u2200 \u2983a : L\u2984 {s : Finset L},\n \u00aca \u2208 s \u2192 (\u2203 y x, \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)) \u2192 \u2203 y x, \u2200 (x : L), x \u2208 insert a s \u2192 IsIntegral A (y \u2022 x)\n[PROOFSTEP]\nrintro x s hx \u27e8y, hy, hs\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns\u271d : Finset L\nthis : DecidableEq L\nx : L\ns : Finset L\nhx : \u00acx \u2208 s\ny : A\nhy : y \u2260 0\nhs : \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\n\u22a2 \u2203 y x_1, \u2200 (x_2 : L), x_2 \u2208 insert x s \u2192 IsIntegral A (y \u2022 x_2)\n[PROOFSTEP]\nhave :=\n exists_integral_multiple ((IsFractionRing.isAlgebraic_iff A K L).mpr (isAlgebraic_of_finite _ _ x))\n ((injective_iff_map_eq_zero (algebraMap A L)).mp ?_)\n[GOAL]\ncase refine'_2.intro.intro.refine_2\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns\u271d : Finset L\nthis\u271d : DecidableEq L\nx : L\ns : Finset L\nhx : \u00acx \u2208 s\ny : A\nhy : y \u2260 0\nhs : \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\nthis : \u2203 x_1 y x_2, x * \u2191(algebraMap A L) y = \u2191x_1\n\u22a2 \u2203 y x_1, \u2200 (x_2 : L), x_2 \u2208 insert x s \u2192 IsIntegral A (y \u2022 x_2)\ncase refine'_2.intro.intro.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns\u271d : Finset L\nthis : DecidableEq L\nx : L\ns : Finset L\nhx : \u00acx \u2208 s\ny : A\nhy : y \u2260 0\nhs : \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\n\u22a2 Function.Injective \u2191(algebraMap A L)\n[PROOFSTEP]\nrcases this with \u27e8x', y', hy', hx'\u27e9\n[GOAL]\ncase refine'_2.intro.intro.refine_2.intro.intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns\u271d : Finset L\nthis : DecidableEq L\nx : L\ns : Finset L\nhx : \u00acx \u2208 s\ny : A\nhy : y \u2260 0\nhs : \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\nx' : { x // x \u2208 integralClosure A L }\ny' : A\nhy' : y' \u2260 0\nhx' : x * \u2191(algebraMap A L) y' = \u2191x'\n\u22a2 \u2203 y x_1, \u2200 (x_2 : L), x_2 \u2208 insert x s \u2192 IsIntegral A (y \u2022 x_2)\ncase refine'_2.intro.intro.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns\u271d : Finset L\nthis : DecidableEq L\nx : L\ns : Finset L\nhx : \u00acx \u2208 s\ny : A\nhy : y \u2260 0\nhs : \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\n\u22a2 Function.Injective \u2191(algebraMap A L)\n[PROOFSTEP]\nrefine' \u27e8y * y', mul_ne_zero hy hy', fun x'' hx'' => _\u27e9\n[GOAL]\ncase refine'_2.intro.intro.refine_2.intro.intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns\u271d : Finset L\nthis : DecidableEq L\nx : L\ns : Finset L\nhx : \u00acx \u2208 s\ny : A\nhy : y \u2260 0\nhs : \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\nx' : { x // x \u2208 integralClosure A L }\ny' : A\nhy' : y' \u2260 0\nhx' : x * \u2191(algebraMap A L) y' = \u2191x'\nx'' : L\nhx'' : x'' \u2208 insert x s\n\u22a2 IsIntegral A ((y * y') \u2022 x'')\ncase refine'_2.intro.intro.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns\u271d : Finset L\nthis : DecidableEq L\nx : L\ns : Finset L\nhx : \u00acx \u2208 s\ny : A\nhy : y \u2260 0\nhs : \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\n\u22a2 Function.Injective \u2191(algebraMap A L)\n[PROOFSTEP]\nrcases Finset.mem_insert.mp hx'' with (rfl | hx'')\n[GOAL]\ncase refine'_2.intro.intro.refine_2.intro.intro.intro.inl\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns\u271d : Finset L\nthis : DecidableEq L\ns : Finset L\ny : A\nhy : y \u2260 0\nhs : \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\nx' : { x // x \u2208 integralClosure A L }\ny' : A\nhy' : y' \u2260 0\nx'' : L\nhx : \u00acx'' \u2208 s\nhx' : x'' * \u2191(algebraMap A L) y' = \u2191x'\nhx'' : x'' \u2208 insert x'' s\n\u22a2 IsIntegral A ((y * y') \u2022 x'')\n[PROOFSTEP]\nrw [mul_smul, Algebra.smul_def, Algebra.smul_def, mul_comm _ x'', hx']\n[GOAL]\ncase refine'_2.intro.intro.refine_2.intro.intro.intro.inl\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns\u271d : Finset L\nthis : DecidableEq L\ns : Finset L\ny : A\nhy : y \u2260 0\nhs : \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\nx' : { x // x \u2208 integralClosure A L }\ny' : A\nhy' : y' \u2260 0\nx'' : L\nhx : \u00acx'' \u2208 s\nhx' : x'' * \u2191(algebraMap A L) y' = \u2191x'\nhx'' : x'' \u2208 insert x'' s\n\u22a2 IsIntegral A (\u2191(algebraMap A L) y * \u2191x')\n[PROOFSTEP]\nexact isIntegral_mul isIntegral_algebraMap x'.2\n[GOAL]\ncase refine'_2.intro.intro.refine_2.intro.intro.intro.inr\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns\u271d : Finset L\nthis : DecidableEq L\nx : L\ns : Finset L\nhx : \u00acx \u2208 s\ny : A\nhy : y \u2260 0\nhs : \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\nx' : { x // x \u2208 integralClosure A L }\ny' : A\nhy' : y' \u2260 0\nhx' : x * \u2191(algebraMap A L) y' = \u2191x'\nx'' : L\nhx''\u271d : x'' \u2208 insert x s\nhx'' : x'' \u2208 s\n\u22a2 IsIntegral A ((y * y') \u2022 x'')\n[PROOFSTEP]\nrw [mul_comm, mul_smul, Algebra.smul_def]\n[GOAL]\ncase refine'_2.intro.intro.refine_2.intro.intro.intro.inr\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns\u271d : Finset L\nthis : DecidableEq L\nx : L\ns : Finset L\nhx : \u00acx \u2208 s\ny : A\nhy : y \u2260 0\nhs : \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\nx' : { x // x \u2208 integralClosure A L }\ny' : A\nhy' : y' \u2260 0\nhx' : x * \u2191(algebraMap A L) y' = \u2191x'\nx'' : L\nhx''\u271d : x'' \u2208 insert x s\nhx'' : x'' \u2208 s\n\u22a2 IsIntegral A (\u2191(algebraMap A L) y' * y \u2022 x'')\n[PROOFSTEP]\nexact isIntegral_mul isIntegral_algebraMap (hs _ hx'')\n[GOAL]\ncase refine'_2.intro.intro.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns\u271d : Finset L\nthis : DecidableEq L\nx : L\ns : Finset L\nhx : \u00acx \u2208 s\ny : A\nhy : y \u2260 0\nhs : \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\n\u22a2 Function.Injective \u2191(algebraMap A L)\n[PROOFSTEP]\nrw [IsScalarTower.algebraMap_eq A K L]\n[GOAL]\ncase refine'_2.intro.intro.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns\u271d : Finset L\nthis : DecidableEq L\nx : L\ns : Finset L\nhx : \u00acx \u2208 s\ny : A\nhy : y \u2260 0\nhs : \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\n\u22a2 Function.Injective \u2191(RingHom.comp (algebraMap K L) (algebraMap A K))\n[PROOFSTEP]\napply (algebraMap K L).injective.comp\n[GOAL]\ncase refine'_2.intro.intro.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\ns\u271d : Finset L\nthis : DecidableEq L\nx : L\ns : Finset L\nhx : \u00acx \u2208 s\ny : A\nhy : y \u2260 0\nhs : \u2200 (x : L), x \u2208 s \u2192 IsIntegral A (y \u2022 x)\n\u22a2 Function.Injective fun x => \u2191(algebraMap A K) x\n[PROOFSTEP]\nexact IsFractionRing.injective _ _\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\n\u22a2 \u2203 s b, \u2200 (x : { x // x \u2208 s }), IsIntegral A (\u2191b x)\n[PROOFSTEP]\nletI := Classical.decEq L\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis : DecidableEq L := Classical.decEq L\n\u22a2 \u2203 s b, \u2200 (x : { x // x \u2208 s }), IsIntegral A (\u2191b x)\n[PROOFSTEP]\nletI : IsNoetherian K L := IsNoetherian.iff_fg.2 inferInstance\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\n\u22a2 \u2203 s b, \u2200 (x : { x // x \u2208 s }), IsIntegral A (\u2191b x)\n[PROOFSTEP]\nlet s' := IsNoetherian.finsetBasisIndex K L\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\ns' : Finset L := IsNoetherian.finsetBasisIndex K L\n\u22a2 \u2203 s b, \u2200 (x : { x // x \u2208 s }), IsIntegral A (\u2191b x)\n[PROOFSTEP]\nlet bs' := IsNoetherian.finsetBasis K L\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\ns' : Finset L := IsNoetherian.finsetBasisIndex K L\nbs' : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex K L } K L := IsNoetherian.finsetBasis K L\n\u22a2 \u2203 s b, \u2200 (x : { x // x \u2208 s }), IsIntegral A (\u2191b x)\n[PROOFSTEP]\nobtain \u27e8y, hy, his'\u27e9 := exists_integral_multiples A K (Finset.univ.image bs')\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\ns' : Finset L := IsNoetherian.finsetBasisIndex K L\nbs' : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex K L } K L := IsNoetherian.finsetBasis K L\ny : A\nhy : y \u2260 0\nhis' : \u2200 (x : L), x \u2208 Finset.image (\u2191bs') Finset.univ \u2192 IsIntegral A (y \u2022 x)\n\u22a2 \u2203 s b, \u2200 (x : { x // x \u2208 s }), IsIntegral A (\u2191b x)\n[PROOFSTEP]\nhave hy' : algebraMap A L y \u2260 0 :=\n by\n refine' mt ((injective_iff_map_eq_zero (algebraMap A L)).mp _ _) hy\n rw [IsScalarTower.algebraMap_eq A K L]\n exact (algebraMap K L).injective.comp (IsFractionRing.injective A K)\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\ns' : Finset L := IsNoetherian.finsetBasisIndex K L\nbs' : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex K L } K L := IsNoetherian.finsetBasis K L\ny : A\nhy : y \u2260 0\nhis' : \u2200 (x : L), x \u2208 Finset.image (\u2191bs') Finset.univ \u2192 IsIntegral A (y \u2022 x)\n\u22a2 \u2191(algebraMap A L) y \u2260 0\n[PROOFSTEP]\nrefine' mt ((injective_iff_map_eq_zero (algebraMap A L)).mp _ _) hy\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\ns' : Finset L := IsNoetherian.finsetBasisIndex K L\nbs' : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex K L } K L := IsNoetherian.finsetBasis K L\ny : A\nhy : y \u2260 0\nhis' : \u2200 (x : L), x \u2208 Finset.image (\u2191bs') Finset.univ \u2192 IsIntegral A (y \u2022 x)\n\u22a2 Function.Injective \u2191(algebraMap A L)\n[PROOFSTEP]\nrw [IsScalarTower.algebraMap_eq A K L]\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\ns' : Finset L := IsNoetherian.finsetBasisIndex K L\nbs' : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex K L } K L := IsNoetherian.finsetBasis K L\ny : A\nhy : y \u2260 0\nhis' : \u2200 (x : L), x \u2208 Finset.image (\u2191bs') Finset.univ \u2192 IsIntegral A (y \u2022 x)\n\u22a2 Function.Injective \u2191(RingHom.comp (algebraMap K L) (algebraMap A K))\n[PROOFSTEP]\nexact (algebraMap K L).injective.comp (IsFractionRing.injective A K)\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\ns' : Finset L := IsNoetherian.finsetBasisIndex K L\nbs' : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex K L } K L := IsNoetherian.finsetBasis K L\ny : A\nhy : y \u2260 0\nhis' : \u2200 (x : L), x \u2208 Finset.image (\u2191bs') Finset.univ \u2192 IsIntegral A (y \u2022 x)\nhy' : \u2191(algebraMap A L) y \u2260 0\n\u22a2 \u2203 s b, \u2200 (x : { x // x \u2208 s }), IsIntegral A (\u2191b x)\n[PROOFSTEP]\nrefine\n \u27e8s',\n bs'.map\n {\n Algebra.lmul _ _ (algebraMap A L\n y) with\n toFun := fun x => algebraMap A L y * x\n invFun := fun x => (algebraMap A L y)\u207b\u00b9 * x\n left_inv := ?_\n right_inv := ?_ },\n ?_\u27e9\n[GOAL]\ncase intro.intro.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\ns' : Finset L := IsNoetherian.finsetBasisIndex K L\nbs' : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex K L } K L := IsNoetherian.finsetBasis K L\ny : A\nhy : y \u2260 0\nhis' : \u2200 (x : L), x \u2208 Finset.image (\u2191bs') Finset.univ \u2192 IsIntegral A (y \u2022 x)\nhy' : \u2191(algebraMap A L) y \u2260 0\nsrc\u271d : (fun x => Module.End K ((fun x => L) y)) (\u2191(algebraMap A L) y) :=\n \u2191(lmul K ((fun x => L) y)) (\u2191(algebraMap A L) y)\n\u22a2 Function.LeftInverse (fun x => (\u2191(algebraMap A L) y)\u207b\u00b9 * x)\n {\n toAddHom :=\n { toFun := fun x => \u2191(algebraMap A L) y * x,\n map_add' :=\n (_ :\n \u2200 (x y_1 : (fun x => L) y),\n AddHom.toFun src\u271d.toAddHom (x + y_1) =\n AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y_1) },\n map_smul' :=\n (_ :\n \u2200 (r : K) (x : (fun x => L) y),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id K) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom.toFun\n[PROOFSTEP]\nintro x\n[GOAL]\ncase intro.intro.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\ns' : Finset L := IsNoetherian.finsetBasisIndex K L\nbs' : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex K L } K L := IsNoetherian.finsetBasis K L\ny : A\nhy : y \u2260 0\nhis' : \u2200 (x : L), x \u2208 Finset.image (\u2191bs') Finset.univ \u2192 IsIntegral A (y \u2022 x)\nhy' : \u2191(algebraMap A L) y \u2260 0\nsrc\u271d : (fun x => Module.End K ((fun x => L) y)) (\u2191(algebraMap A L) y) :=\n \u2191(lmul K ((fun x => L) y)) (\u2191(algebraMap A L) y)\nx : L\n\u22a2 (fun x => (\u2191(algebraMap A L) y)\u207b\u00b9 * x)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun x => \u2191(algebraMap A L) y * x,\n map_add' :=\n (_ :\n \u2200 (x y_1 : (fun x => L) y),\n AddHom.toFun src\u271d.toAddHom (x + y_1) =\n AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y_1) },\n map_smul' :=\n (_ :\n \u2200 (r : K) (x : (fun x => L) y),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id K) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n x) =\n x\n[PROOFSTEP]\nsimp only [inv_mul_cancel_left\u2080 hy']\n[GOAL]\ncase intro.intro.refine_2\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\ns' : Finset L := IsNoetherian.finsetBasisIndex K L\nbs' : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex K L } K L := IsNoetherian.finsetBasis K L\ny : A\nhy : y \u2260 0\nhis' : \u2200 (x : L), x \u2208 Finset.image (\u2191bs') Finset.univ \u2192 IsIntegral A (y \u2022 x)\nhy' : \u2191(algebraMap A L) y \u2260 0\nsrc\u271d : (fun x => Module.End K ((fun x => L) y)) (\u2191(algebraMap A L) y) :=\n \u2191(lmul K ((fun x => L) y)) (\u2191(algebraMap A L) y)\n\u22a2 Function.RightInverse (fun x => (\u2191(algebraMap A L) y)\u207b\u00b9 * x)\n {\n toAddHom :=\n { toFun := fun x => \u2191(algebraMap A L) y * x,\n map_add' :=\n (_ :\n \u2200 (x y_1 : (fun x => L) y),\n AddHom.toFun src\u271d.toAddHom (x + y_1) =\n AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y_1) },\n map_smul' :=\n (_ :\n \u2200 (r : K) (x : (fun x => L) y),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id K) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom.toFun\n[PROOFSTEP]\nintro x\n[GOAL]\ncase intro.intro.refine_2\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\ns' : Finset L := IsNoetherian.finsetBasisIndex K L\nbs' : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex K L } K L := IsNoetherian.finsetBasis K L\ny : A\nhy : y \u2260 0\nhis' : \u2200 (x : L), x \u2208 Finset.image (\u2191bs') Finset.univ \u2192 IsIntegral A (y \u2022 x)\nhy' : \u2191(algebraMap A L) y \u2260 0\nsrc\u271d : (fun x => Module.End K ((fun x => L) y)) (\u2191(algebraMap A L) y) :=\n \u2191(lmul K ((fun x => L) y)) (\u2191(algebraMap A L) y)\nx : L\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun x => \u2191(algebraMap A L) y * x,\n map_add' :=\n (_ :\n \u2200 (x y_1 : (fun x => L) y),\n AddHom.toFun src\u271d.toAddHom (x + y_1) =\n AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y_1) },\n map_smul' :=\n (_ :\n \u2200 (r : K) (x : (fun x => L) y),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id K) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n ((fun x => (\u2191(algebraMap A L) y)\u207b\u00b9 * x) x) =\n x\n[PROOFSTEP]\nsimp only [mul_inv_cancel_left\u2080 hy']\n[GOAL]\ncase intro.intro.refine_3\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\ns' : Finset L := IsNoetherian.finsetBasisIndex K L\nbs' : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex K L } K L := IsNoetherian.finsetBasis K L\ny : A\nhy : y \u2260 0\nhis' : \u2200 (x : L), x \u2208 Finset.image (\u2191bs') Finset.univ \u2192 IsIntegral A (y \u2022 x)\nhy' : \u2191(algebraMap A L) y \u2260 0\n\u22a2 \u2200 (x : { x // x \u2208 s' }),\n IsIntegral A\n (\u2191(Basis.map bs'\n (let src := \u2191(lmul K ((fun x => L) y)) (\u2191(algebraMap A L) y);\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := fun x => \u2191(algebraMap A L) y * x,\n map_add' :=\n (_ :\n \u2200 (x y_1 : (fun x => L) y),\n AddHom.toFun src.toAddHom (x + y_1) =\n AddHom.toFun src.toAddHom x + AddHom.toFun src.toAddHom y_1) },\n map_smul' :=\n (_ :\n \u2200 (r : K) (x : (fun x => L) y),\n AddHom.toFun src.toAddHom (r \u2022 x) = \u2191(RingHom.id K) r \u2022 AddHom.toFun src.toAddHom x) },\n invFun := fun x => (\u2191(algebraMap A L) y)\u207b\u00b9 * x,\n left_inv := (_ : \u2200 (x : L), (\u2191(algebraMap A L) y)\u207b\u00b9 * (\u2191(algebraMap A L) y * x) = x),\n right_inv := (_ : \u2200 (x : L), \u2191(algebraMap A L) y * ((\u2191(algebraMap A L) y)\u207b\u00b9 * x) = x) }))\n x)\n[PROOFSTEP]\nrintro \u27e8x', hx'\u27e9\n[GOAL]\ncase intro.intro.refine_3.mk\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\ns' : Finset L := IsNoetherian.finsetBasisIndex K L\nbs' : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex K L } K L := IsNoetherian.finsetBasis K L\ny : A\nhy : y \u2260 0\nhis' : \u2200 (x : L), x \u2208 Finset.image (\u2191bs') Finset.univ \u2192 IsIntegral A (y \u2022 x)\nhy' : \u2191(algebraMap A L) y \u2260 0\nx' : L\nhx' : x' \u2208 s'\n\u22a2 IsIntegral A\n (\u2191(Basis.map bs'\n (let src := \u2191(lmul K ((fun x => L) y)) (\u2191(algebraMap A L) y);\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := fun x => \u2191(algebraMap A L) y * x,\n map_add' :=\n (_ :\n \u2200 (x y_1 : (fun x => L) y),\n AddHom.toFun src.toAddHom (x + y_1) =\n AddHom.toFun src.toAddHom x + AddHom.toFun src.toAddHom y_1) },\n map_smul' :=\n (_ :\n \u2200 (r : K) (x : (fun x => L) y),\n AddHom.toFun src.toAddHom (r \u2022 x) = \u2191(RingHom.id K) r \u2022 AddHom.toFun src.toAddHom x) },\n invFun := fun x => (\u2191(algebraMap A L) y)\u207b\u00b9 * x,\n left_inv := (_ : \u2200 (x : L), (\u2191(algebraMap A L) y)\u207b\u00b9 * (\u2191(algebraMap A L) y * x) = x),\n right_inv := (_ : \u2200 (x : L), \u2191(algebraMap A L) y * ((\u2191(algebraMap A L) y)\u207b\u00b9 * x) = x) }))\n { val := x', property := hx' })\n[PROOFSTEP]\nsimp only [Algebra.smul_def, Finset.mem_image, exists_prop, Finset.mem_univ, true_and_iff] at his' \n[GOAL]\ncase intro.intro.refine_3.mk\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\ns' : Finset L := IsNoetherian.finsetBasisIndex K L\nbs' : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex K L } K L := IsNoetherian.finsetBasis K L\ny : A\nhy : y \u2260 0\nhy' : \u2191(algebraMap A L) y \u2260 0\nx' : L\nhx' : x' \u2208 s'\nhis' : \u2200 (x : L), (\u2203 a, \u2191(IsNoetherian.finsetBasis K L) a = x) \u2192 IsIntegral A (\u2191(algebraMap A L) y * x)\n\u22a2 IsIntegral A\n (\u2191(Basis.map bs'\n (let src := \u2191(lmul K ((fun x => L) y)) (\u2191(algebraMap A L) y);\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := fun x => \u2191(algebraMap A L) y * x,\n map_add' :=\n (_ :\n \u2200 (x y_1 : (fun x => L) y),\n AddHom.toFun src.toAddHom (x + y_1) =\n AddHom.toFun src.toAddHom x + AddHom.toFun src.toAddHom y_1) },\n map_smul' :=\n (_ :\n \u2200 (r : K) (x : (fun x => L) y),\n AddHom.toFun src.toAddHom (r \u2022 x) = \u2191(RingHom.id K) r \u2022 AddHom.toFun src.toAddHom x) },\n invFun := fun x => (\u2191(algebraMap A L) y)\u207b\u00b9 * x,\n left_inv := (_ : \u2200 (x : L), (\u2191(algebraMap A L) y)\u207b\u00b9 * (\u2191(algebraMap A L) y * x) = x),\n right_inv := (_ : \u2200 (x : L), \u2191(algebraMap A L) y * ((\u2191(algebraMap A L) y)\u207b\u00b9 * x) = x) }))\n { val := x', property := hx' })\n[PROOFSTEP]\nsimp only [Basis.map_apply, LinearEquiv.coe_mk]\n[GOAL]\ncase intro.intro.refine_3.mk\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing A\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : IsDomain A\ninst\u271d\u00b9\u00b9 : Algebra A K\ninst\u271d\u00b9\u2070 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u2079 : Field L\nC : Type u_5\ninst\u271d\u2078 : CommRing C\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra A L\ninst\u271d\u2075 : IsScalarTower A K L\ninst\u271d\u2074 : Algebra C L\ninst\u271d\u00b3 : IsIntegralClosure C A L\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower A C L\ninst\u271d : FiniteDimensional K L\nthis\u271d : DecidableEq L := Classical.decEq L\nthis : IsNoetherian K L := Iff.mpr IsNoetherian.iff_fg inferInstance\ns' : Finset L := IsNoetherian.finsetBasisIndex K L\nbs' : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex K L } K L := IsNoetherian.finsetBasis K L\ny : A\nhy : y \u2260 0\nhy' : \u2191(algebraMap A L) y \u2260 0\nx' : L\nhx' : x' \u2208 s'\nhis' : \u2200 (x : L), (\u2203 a, \u2191(IsNoetherian.finsetBasis K L) a = x) \u2192 IsIntegral A (\u2191(algebraMap A L) y * x)\n\u22a2 IsIntegral A (\u2191(algebraMap A L) y * \u2191(IsNoetherian.finsetBasis K L) { val := x', property := hx' })\n[PROOFSTEP]\nexact his' _ \u27e8_, rfl\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : IsIntegrallyClosed A\ninst\u271d : IsNoetherianRing A\n\u22a2 IsNoetherian A C\n[PROOFSTEP]\nhaveI := Classical.decEq L\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : IsIntegrallyClosed A\ninst\u271d : IsNoetherianRing A\nthis : DecidableEq L\n\u22a2 IsNoetherian A C\n[PROOFSTEP]\nobtain \u27e8s, b, hb_int\u27e9 := FiniteDimensional.exists_is_basis_integral A K L\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : IsIntegrallyClosed A\ninst\u271d : IsNoetherianRing A\nthis : DecidableEq L\ns : Finset L\nb : Basis { x // x \u2208 s } K L\nhb_int : \u2200 (x : { x // x \u2208 s }), IsIntegral A (\u2191b x)\n\u22a2 IsNoetherian A C\n[PROOFSTEP]\nlet b' := (traceForm K L).dualBasis (traceForm_nondegenerate K L) b\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : IsIntegrallyClosed A\ninst\u271d : IsNoetherianRing A\nthis : DecidableEq L\ns : Finset L\nb : Basis { x // x \u2208 s } K L\nhb_int : \u2200 (x : { x // x \u2208 s }), IsIntegral A (\u2191b x)\nb' : Basis { x // x \u2208 s } K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\n\u22a2 IsNoetherian A C\n[PROOFSTEP]\nletI := isNoetherian_span_of_finite A (Set.finite_range b')\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : IsIntegrallyClosed A\ninst\u271d : IsNoetherianRing A\nthis\u271d : DecidableEq L\ns : Finset L\nb : Basis { x // x \u2208 s } K L\nhb_int : \u2200 (x : { x // x \u2208 s }), IsIntegral A (\u2191b x)\nb' : Basis { x // x \u2208 s } K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nthis : IsNoetherian A { x // x \u2208 Submodule.span A (Set.range \u2191b') } :=\n isNoetherian_span_of_finite A (Set.finite_range \u2191b')\n\u22a2 IsNoetherian A C\n[PROOFSTEP]\nlet f : C \u2192\u2097[A] Submodule.span A (Set.range b') :=\n (Submodule.ofLe (IsIntegralClosure.range_le_span_dualBasis C b hb_int)).comp\n ((Algebra.linearMap C L).restrictScalars A).rangeRestrict\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : IsIntegrallyClosed A\ninst\u271d : IsNoetherianRing A\nthis\u271d : DecidableEq L\ns : Finset L\nb : Basis { x // x \u2208 s } K L\nhb_int : \u2200 (x : { x // x \u2208 s }), IsIntegral A (\u2191b x)\nb' : Basis { x // x \u2208 s } K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nthis : IsNoetherian A { x // x \u2208 Submodule.span A (Set.range \u2191b') } :=\n isNoetherian_span_of_finite A (Set.finite_range \u2191b')\nf : C \u2192\u2097[A] { x // x \u2208 Submodule.span A (Set.range \u2191b') } :=\n LinearMap.comp\n (Submodule.ofLe\n (_ :\n LinearMap.range (\u2191A (Algebra.linearMap C L)) \u2264\n Submodule.span A\n (Set.range \u2191(BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b))))\n (LinearMap.rangeRestrict (\u2191A (Algebra.linearMap C L)))\n\u22a2 IsNoetherian A C\n[PROOFSTEP]\nrefine' isNoetherian_of_ker_bot f _\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : IsIntegrallyClosed A\ninst\u271d : IsNoetherianRing A\nthis\u271d : DecidableEq L\ns : Finset L\nb : Basis { x // x \u2208 s } K L\nhb_int : \u2200 (x : { x // x \u2208 s }), IsIntegral A (\u2191b x)\nb' : Basis { x // x \u2208 s } K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nthis : IsNoetherian A { x // x \u2208 Submodule.span A (Set.range \u2191b') } :=\n isNoetherian_span_of_finite A (Set.finite_range \u2191b')\nf : C \u2192\u2097[A] { x // x \u2208 Submodule.span A (Set.range \u2191b') } :=\n LinearMap.comp\n (Submodule.ofLe\n (_ :\n LinearMap.range (\u2191A (Algebra.linearMap C L)) \u2264\n Submodule.span A\n (Set.range \u2191(BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b))))\n (LinearMap.rangeRestrict (\u2191A (Algebra.linearMap C L)))\n\u22a2 LinearMap.ker f = \u22a5\n[PROOFSTEP]\nrw [LinearMap.ker_comp, Submodule.ker_ofLe, Submodule.comap_bot, LinearMap.ker_codRestrict]\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : IsIntegrallyClosed A\ninst\u271d : IsNoetherianRing A\nthis\u271d : DecidableEq L\ns : Finset L\nb : Basis { x // x \u2208 s } K L\nhb_int : \u2200 (x : { x // x \u2208 s }), IsIntegral A (\u2191b x)\nb' : Basis { x // x \u2208 s } K L := BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b\nthis : IsNoetherian A { x // x \u2208 Submodule.span A (Set.range \u2191b') } :=\n isNoetherian_span_of_finite A (Set.finite_range \u2191b')\nf : C \u2192\u2097[A] { x // x \u2208 Submodule.span A (Set.range \u2191b') } :=\n LinearMap.comp\n (Submodule.ofLe\n (_ :\n LinearMap.range (\u2191A (Algebra.linearMap C L)) \u2264\n Submodule.span A\n (Set.range \u2191(BilinForm.dualBasis (traceForm K L) (_ : BilinForm.Nondegenerate (traceForm K L)) b))))\n (LinearMap.rangeRestrict (\u2191A (Algebra.linearMap C L)))\n\u22a2 LinearMap.ker (\u2191A (Algebra.linearMap C L)) = \u22a5\n[PROOFSTEP]\nexact LinearMap.ker_eq_bot_of_injective (IsIntegralClosure.algebraMap_injective C A L)\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : NoZeroSMulDivisors A L\ninst\u271d : IsPrincipalIdealRing A\n\u22a2 Module.Free A C\n[PROOFSTEP]\nhaveI : NoZeroSMulDivisors A C := IsIntegralClosure.noZeroSMulDivisors A L\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : NoZeroSMulDivisors A L\ninst\u271d : IsPrincipalIdealRing A\nthis : NoZeroSMulDivisors A C\n\u22a2 Module.Free A C\n[PROOFSTEP]\nhaveI : IsNoetherian A C := IsIntegralClosure.isNoetherian A K L _\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : NoZeroSMulDivisors A L\ninst\u271d : IsPrincipalIdealRing A\nthis\u271d : NoZeroSMulDivisors A C\nthis : IsNoetherian A C\n\u22a2 Module.Free A C\n[PROOFSTEP]\nexact Module.free_of_finite_type_torsion_free'\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : IsPrincipalIdealRing A\ninst\u271d : NoZeroSMulDivisors A L\n\u22a2 FiniteDimensional.finrank A C = FiniteDimensional.finrank K L\n[PROOFSTEP]\nhaveI : Module.Free A C := IsIntegralClosure.module_free A K L C\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : IsPrincipalIdealRing A\ninst\u271d : NoZeroSMulDivisors A L\nthis : Module.Free A C\n\u22a2 FiniteDimensional.finrank A C = FiniteDimensional.finrank K L\n[PROOFSTEP]\nhaveI : IsNoetherian A C := IsIntegralClosure.isNoetherian A K L C\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : IsPrincipalIdealRing A\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d : Module.Free A C\nthis : IsNoetherian A C\n\u22a2 FiniteDimensional.finrank A C = FiniteDimensional.finrank K L\n[PROOFSTEP]\nhaveI : IsLocalization (Algebra.algebraMapSubmonoid C A\u2070) L := IsIntegralClosure.isLocalization A K L C\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : IsPrincipalIdealRing A\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d\u00b9 : Module.Free A C\nthis\u271d : IsNoetherian A C\nthis : IsLocalization (algebraMapSubmonoid C A\u2070) L\n\u22a2 FiniteDimensional.finrank A C = FiniteDimensional.finrank K L\n[PROOFSTEP]\nlet b := Basis.localizationLocalization K A\u2070 L (Module.Free.chooseBasis A C)\n[GOAL]\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : IsDomain A\ninst\u271d\u00b9\u2074 : Algebra A K\ninst\u271d\u00b9\u00b3 : IsFractionRing A K\nL : Type u_4\ninst\u271d\u00b9\u00b2 : Field L\nC : Type u_5\ninst\u271d\u00b9\u00b9 : CommRing C\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : Algebra C L\ninst\u271d\u2076 : IsIntegralClosure C A L\ninst\u271d\u2075 : Algebra A C\ninst\u271d\u2074 : IsScalarTower A C L\ninst\u271d\u00b3 : FiniteDimensional K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : IsPrincipalIdealRing A\ninst\u271d : NoZeroSMulDivisors A L\nthis\u271d\u00b9 : Module.Free A C\nthis\u271d : IsNoetherian A C\nthis : IsLocalization (algebraMapSubmonoid C A\u2070) L\nb : Basis (Module.Free.ChooseBasisIndex A C) K L := Basis.localizationLocalization K A\u2070 L (Module.Free.chooseBasis A C)\n\u22a2 FiniteDimensional.finrank A C = FiniteDimensional.finrank K L\n[PROOFSTEP]\nrw [FiniteDimensional.finrank_eq_card_chooseBasisIndex, FiniteDimensional.finrank_eq_card_basis b]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "llama_tokens": 42196, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056040203136, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.4033135045110409}} {"text": "[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : Finite \u03b1\n\u22a2 \u03b1 \u2243 Fin (Nat.card \u03b1)\n[PROOFSTEP]\nhave := (Finite.exists_equiv_fin \u03b1).choose_spec.some\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : Finite \u03b1\nthis : \u03b1 \u2243 Fin (Exists.choose (_ : \u2203 n, Nonempty (\u03b1 \u2243 Fin n)))\n\u22a2 \u03b1 \u2243 Fin (Nat.card \u03b1)\n[PROOFSTEP]\nrwa [Nat.card_eq_of_equiv_fin this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\nn : \u2115\nh : Nat.card \u03b1 = n\n\u22a2 \u03b1 \u2243 Fin n\n[PROOFSTEP]\nsubst h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\n\u22a2 \u03b1 \u2243 Fin (Nat.card \u03b1)\n[PROOFSTEP]\napply Finite.equivFin\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u22a2 Nat.card \u03b1 = if h : Finite \u03b1 then Fintype.card \u03b1 else 0\n[PROOFSTEP]\ncases finite_or_infinite \u03b1\n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\nh\u271d : Finite \u03b1\n\u22a2 Nat.card \u03b1 = if h : Finite \u03b1 then Fintype.card \u03b1 else 0\n[PROOFSTEP]\nletI := Fintype.ofFinite \u03b1\n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\nh\u271d : Finite \u03b1\nthis : Fintype \u03b1 := Fintype.ofFinite \u03b1\n\u22a2 Nat.card \u03b1 = if h : Finite \u03b1 then Fintype.card \u03b1 else 0\n[PROOFSTEP]\nsimp only [*, Nat.card_eq_fintype_card, dif_pos]\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\nh\u271d : Infinite \u03b1\n\u22a2 Nat.card \u03b1 = if h : Finite \u03b1 then Fintype.card \u03b1 else 0\n[PROOFSTEP]\nsimp only [*, card_eq_zero_of_infinite, not_finite_iff_infinite.mpr, dite_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\n\u22a2 0 < Nat.card \u03b1 \u2194 Nonempty \u03b1\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\nthis : Fintype \u03b1\n\u22a2 0 < Nat.card \u03b1 \u2194 Nonempty \u03b1\n[PROOFSTEP]\nrw [Nat.card_eq_fintype_card, Fintype.card_pos_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : Finite \u03b2\n\u22a2 Nat.card \u03b1 = Nat.card \u03b2 \u2194 Nonempty (\u03b1 \u2243 \u03b2)\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : Finite \u03b2\nthis : Fintype \u03b1\n\u22a2 Nat.card \u03b1 = Nat.card \u03b2 \u2194 Nonempty (\u03b1 \u2243 \u03b2)\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : Finite \u03b2\nthis\u271d : Fintype \u03b1\nthis : Fintype \u03b2\n\u22a2 Nat.card \u03b1 = Nat.card \u03b2 \u2194 Nonempty (\u03b1 \u2243 \u03b2)\n[PROOFSTEP]\nsimp only [Nat.card_eq_fintype_card, Fintype.card_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\n\u22a2 Nat.card \u03b1 \u2264 1 \u2194 Subsingleton \u03b1\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\nthis : Fintype \u03b1\n\u22a2 Nat.card \u03b1 \u2264 1 \u2194 Subsingleton \u03b1\n[PROOFSTEP]\nsimp only [Nat.card_eq_fintype_card, Fintype.card_le_one_iff_subsingleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\n\u22a2 1 < Nat.card \u03b1 \u2194 Nontrivial \u03b1\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\nthis : Fintype \u03b1\n\u22a2 1 < Nat.card \u03b1 \u2194 Nontrivial \u03b1\n[PROOFSTEP]\nsimp only [Nat.card_eq_fintype_card, Fintype.one_lt_card_iff_nontrivial]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\n\u22a2 Nat.card (Option \u03b1) = Nat.card \u03b1 + 1\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\nthis : Fintype \u03b1\n\u22a2 Nat.card (Option \u03b1) = Nat.card \u03b1 + 1\n[PROOFSTEP]\nsimp only [Nat.card_eq_fintype_card, Fintype.card_option]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Function.Injective f\n\u22a2 Nat.card \u03b1 \u2264 Nat.card \u03b2\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Function.Injective f\nthis : Fintype \u03b2\n\u22a2 Nat.card \u03b1 \u2264 Nat.card \u03b2\n[PROOFSTEP]\nhaveI := Fintype.ofInjective f hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Function.Injective f\nthis\u271d : Fintype \u03b2\nthis : Fintype \u03b1\n\u22a2 Nat.card \u03b1 \u2264 Nat.card \u03b2\n[PROOFSTEP]\nsimpa only [Nat.card_eq_fintype_card, ge_iff_le] using Fintype.card_le_of_injective f hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Function.Surjective f\n\u22a2 Nat.card \u03b2 \u2264 Nat.card \u03b1\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Function.Surjective f\nthis : Fintype \u03b1\n\u22a2 Nat.card \u03b2 \u2264 Nat.card \u03b1\n[PROOFSTEP]\nhaveI := Fintype.ofSurjective f hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Function.Surjective f\nthis\u271d : Fintype \u03b1\nthis : Fintype \u03b2\n\u22a2 Nat.card \u03b2 \u2264 Nat.card \u03b1\n[PROOFSTEP]\nsimpa only [Nat.card_eq_fintype_card, ge_iff_le] using Fintype.card_le_of_surjective f hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\n\u22a2 Nat.card \u03b1 = 0 \u2194 IsEmpty \u03b1\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\nthis : Fintype \u03b1\n\u22a2 Nat.card \u03b1 = 0 \u2194 IsEmpty \u03b1\n[PROOFSTEP]\nsimp only [Nat.card_eq_fintype_card, Fintype.card_eq_zero_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Function.Surjective f\nh : Nat.card \u03b2 = 0\n\u22a2 Nat.card \u03b1 = 0\n[PROOFSTEP]\ncases finite_or_infinite \u03b2\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Function.Surjective f\nh : Nat.card \u03b2 = 0\nh\u271d : Finite \u03b2\n\u22a2 Nat.card \u03b1 = 0\n[PROOFSTEP]\nhaveI := card_eq_zero_iff.mp h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Function.Surjective f\nh : Nat.card \u03b2 = 0\nh\u271d : Finite \u03b2\nthis : IsEmpty \u03b2\n\u22a2 Nat.card \u03b1 = 0\n[PROOFSTEP]\nhaveI := Function.isEmpty f\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Function.Surjective f\nh : Nat.card \u03b2 = 0\nh\u271d : Finite \u03b2\nthis\u271d : IsEmpty \u03b2\nthis : IsEmpty \u03b1\n\u22a2 Nat.card \u03b1 = 0\n[PROOFSTEP]\nexact Nat.card_of_isEmpty\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Function.Surjective f\nh : Nat.card \u03b2 = 0\nh\u271d : Infinite \u03b2\n\u22a2 Nat.card \u03b1 = 0\n[PROOFSTEP]\nhaveI := Infinite.of_surjective f hf\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhf : Function.Surjective f\nh : Nat.card \u03b2 = 0\nh\u271d : Infinite \u03b2\nthis : Infinite \u03b1\n\u22a2 Nat.card \u03b1 = 0\n[PROOFSTEP]\nexact Nat.card_eq_zero_of_infinite\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : Finite \u03b2\n\u22a2 Nat.card (\u03b1 \u2295 \u03b2) = Nat.card \u03b1 + Nat.card \u03b2\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : Finite \u03b2\nthis : Fintype \u03b1\n\u22a2 Nat.card (\u03b1 \u2295 \u03b2) = Nat.card \u03b1 + Nat.card \u03b2\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : Finite \u03b2\nthis\u271d : Fintype \u03b1\nthis : Fintype \u03b2\n\u22a2 Nat.card (\u03b1 \u2295 \u03b2) = Nat.card \u03b1 + Nat.card \u03b2\n[PROOFSTEP]\nsimp only [Nat.card_eq_fintype_card, Fintype.card_sum]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 Nat.card { x // p x } \u2264 Nat.card \u03b1\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\np : \u03b1 \u2192 Prop\nthis : Fintype \u03b1\n\u22a2 Nat.card { x // p x } \u2264 Nat.card \u03b1\n[PROOFSTEP]\nsimpa only [Nat.card_eq_fintype_card, ge_iff_le] using Fintype.card_subtype_le p\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\np : \u03b1 \u2192 Prop\nx : \u03b1\nhx : \u00acp x\n\u22a2 Nat.card { x // p x } < Nat.card \u03b1\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\np : \u03b1 \u2192 Prop\nx : \u03b1\nhx : \u00acp x\nthis : Fintype \u03b1\n\u22a2 Nat.card { x // p x } < Nat.card \u03b1\n[PROOFSTEP]\nsimpa only [Nat.card_eq_fintype_card, gt_iff_lt] using Fintype.card_subtype_lt hx\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : Finite \u03b1\n\u22a2 card \u03b1 = \u2191(Nat.card \u03b1)\n[PROOFSTEP]\nunfold PartENat.card\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : Finite \u03b1\n\u22a2 \u2191Cardinal.toPartENat (Cardinal.mk \u03b1) = \u2191(Nat.card \u03b1)\n[PROOFSTEP]\napply symm\n[GOAL]\ncase a\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : Finite \u03b1\n\u22a2 \u2191(Nat.card \u03b1) = \u2191Cardinal.toPartENat (Cardinal.mk \u03b1)\n[PROOFSTEP]\nrw [Cardinal.natCast_eq_toPartENat_iff]\n[GOAL]\ncase a\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : Finite \u03b1\n\u22a2 \u2191(Nat.card \u03b1) = Cardinal.mk \u03b1\n[PROOFSTEP]\nexact Finite.cast_card_eq_mk\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Set \u03b1\n\u22a2 Nat.card \u2191(s \u222a t) \u2264 Nat.card \u2191s + Nat.card \u2191t\n[PROOFSTEP]\ncases' _root_.finite_or_infinite (\u21a5(s \u222a t)) with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Set \u03b1\nh : Finite \u2191(s \u222a t)\n\u22a2 Nat.card \u2191(s \u222a t) \u2264 Nat.card \u2191s + Nat.card \u2191t\n[PROOFSTEP]\nrw [finite_coe_iff, finite_union, \u2190 finite_coe_iff, \u2190 finite_coe_iff] at h \n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Set \u03b1\nh : Finite \u2191s \u2227 Finite \u2191t\n\u22a2 Nat.card \u2191(s \u222a t) \u2264 Nat.card \u2191s + Nat.card \u2191t\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Set \u03b1\nleft\u271d : Finite \u2191s\nright\u271d : Finite \u2191t\n\u22a2 Nat.card \u2191(s \u222a t) \u2264 Nat.card \u2191s + Nat.card \u2191t\n[PROOFSTEP]\nrw [\u2190 Cardinal.natCast_le, Nat.cast_add, Finite.cast_card_eq_mk, Finite.cast_card_eq_mk, Finite.cast_card_eq_mk]\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Set \u03b1\nleft\u271d : Finite \u2191s\nright\u271d : Finite \u2191t\n\u22a2 Cardinal.mk \u2191(s \u222a t) \u2264 Cardinal.mk \u2191s + Cardinal.mk \u2191t\n[PROOFSTEP]\nexact Cardinal.mk_union_le s t\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Set \u03b1\nh : Infinite \u2191(s \u222a t)\n\u22a2 Nat.card \u2191(s \u222a t) \u2264 Nat.card \u2191s + Nat.card \u2191t\n[PROOFSTEP]\nexact Nat.card_eq_zero_of_infinite.trans_le (zero_le _)\n", "meta": {"mathlib_filename": "Mathlib.Data.Finite.Card", "llama_tokens": 4734, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300698514777, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.40302633165989615}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\ns t : Set \u03b1\nh : IsUpperSet s\u1d9c\n\u22a2 IsLowerSet s\n[PROOFSTEP]\nconvert h.compl\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\ns t : Set \u03b1\nh : IsUpperSet s\u1d9c\n\u22a2 s = s\u1d9c\u1d9c\n[PROOFSTEP]\nrw [compl_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\ns t : Set \u03b1\nh : IsLowerSet s\u1d9c\n\u22a2 IsUpperSet s\n[PROOFSTEP]\nconvert h.compl\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : LE \u03b2\ns t : Set \u03b1\nh : IsLowerSet s\u1d9c\n\u22a2 s = s\u1d9c\u1d9c\n[PROOFSTEP]\nrw [compl_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set \u03b1\np : \u03b1 \u2192 Prop\na : \u03b1\n\u22a2 IsUpperSet s \u2194 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 Ici a \u2286 s\n[PROOFSTEP]\nsimp [IsUpperSet, subset_def, @forall_swap (_ \u2208 s)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set \u03b1\np : \u03b1 \u2192 Prop\na : \u03b1\n\u22a2 IsLowerSet s \u2194 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 Iic a \u2286 s\n[PROOFSTEP]\nsimp [IsLowerSet, subset_def, @forall_swap (_ \u2208 s)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set \u03b1\np : \u03b1 \u2192 Prop\na : \u03b1\nhs : IsUpperSet s\nf : \u03b1 \u2243o \u03b2\n\u22a2 IsUpperSet (\u2191f '' s)\n[PROOFSTEP]\nchange IsUpperSet ((f : \u03b1 \u2243 \u03b2) '' s)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set \u03b1\np : \u03b1 \u2192 Prop\na : \u03b1\nhs : IsUpperSet s\nf : \u03b1 \u2243o \u03b2\n\u22a2 IsUpperSet (\u2191\u2191f '' s)\n[PROOFSTEP]\nrw [Set.image_equiv_eq_preimage_symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set \u03b1\np : \u03b1 \u2192 Prop\na : \u03b1\nhs : IsUpperSet s\nf : \u03b1 \u2243o \u03b2\n\u22a2 IsUpperSet (\u2191(\u2191f).symm \u207b\u00b9' s)\n[PROOFSTEP]\nexact hs.preimage f.symm.monotone\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set \u03b1\np : \u03b1 \u2192 Prop\na : \u03b1\nhs : IsLowerSet s\nf : \u03b1 \u2243o \u03b2\n\u22a2 IsLowerSet (\u2191f '' s)\n[PROOFSTEP]\nchange IsLowerSet ((f : \u03b1 \u2243 \u03b2) '' s)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set \u03b1\np : \u03b1 \u2192 Prop\na : \u03b1\nhs : IsLowerSet s\nf : \u03b1 \u2243o \u03b2\n\u22a2 IsLowerSet (\u2191\u2191f '' s)\n[PROOFSTEP]\nrw [Set.image_equiv_eq_preimage_symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set \u03b1\np : \u03b1 \u2192 Prop\na : \u03b1\nhs : IsLowerSet s\nf : \u03b1 \u2243o \u03b2\n\u22a2 IsLowerSet (\u2191(\u2191f).symm \u207b\u00b9' s)\n[PROOFSTEP]\nexact hs.preimage f.symm.monotone\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns : Set \u03b1\np : \u03b1 \u2192 Prop\na : \u03b1\ninst\u271d : NoMaxOrder \u03b1\nhs : IsUpperSet s\n\u22a2 Set.Nonempty s \u2192 \u00acBddAbove s\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns : Set \u03b1\np : \u03b1 \u2192 Prop\na\u271d : \u03b1\ninst\u271d : NoMaxOrder \u03b1\nhs : IsUpperSet s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 upperBounds s\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 := exists_gt b\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns : Set \u03b1\np : \u03b1 \u2192 Prop\na\u271d : \u03b1\ninst\u271d : NoMaxOrder \u03b1\nhs : IsUpperSet s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 upperBounds s\nc : \u03b1\nhc : b < c\n\u22a2 False\n[PROOFSTEP]\nexact hc.not_le (hb <| hs ((hb ha).trans hc.le) ha)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns : Set \u03b1\np : \u03b1 \u2192 Prop\na : \u03b1\ninst\u271d : NoMinOrder \u03b1\nhs : IsLowerSet s\n\u22a2 Set.Nonempty s \u2192 \u00acBddBelow s\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns : Set \u03b1\np : \u03b1 \u2192 Prop\na\u271d : \u03b1\ninst\u271d : NoMinOrder \u03b1\nhs : IsLowerSet s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 lowerBounds s\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 := exists_lt b\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ns : Set \u03b1\np : \u03b1 \u2192 Prop\na\u271d : \u03b1\ninst\u271d : NoMinOrder \u03b1\nhs : IsLowerSet s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 lowerBounds s\nc : \u03b1\nhc : c < b\n\u22a2 False\n[PROOFSTEP]\nexact hc.not_le (hb <| hs (hc.le.trans <| hb ha) ha)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : PartialOrder \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 (\u2200 \u2983b : \u03b1\u2984, a \u2264 b \u2192 a \u2208 s \u2192 b \u2208 s) \u2194 \u2200 \u2983b : \u03b1\u2984, a < b \u2192 a \u2208 s \u2192 b \u2208 s\n[PROOFSTEP]\nsimp [le_iff_eq_or_lt, or_imp, forall_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : PartialOrder \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 (\u2200 \u2983b : \u03b1\u2984, b \u2264 a \u2192 a \u2208 s \u2192 b \u2208 s) \u2194 \u2200 \u2983b : \u03b1\u2984, b < a \u2192 a \u2208 s \u2192 b \u2208 s\n[PROOFSTEP]\nsimp [le_iff_eq_or_lt, or_imp, forall_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : PartialOrder \u03b1\ns : Set \u03b1\n\u22a2 IsUpperSet s \u2194 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 Ioi a \u2286 s\n[PROOFSTEP]\nsimp [isUpperSet_iff_forall_lt, subset_def, @forall_swap (_ \u2208 s)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : PartialOrder \u03b1\ns : Set \u03b1\n\u22a2 IsLowerSet s \u2194 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 Iio a \u2286 s\n[PROOFSTEP]\nsimp [isLowerSet_iff_forall_lt, subset_def, @forall_swap (_ \u2208 s)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ns t : UpperSet \u03b1\nh : s.carrier = t.carrier\n\u22a2 s = t\n[PROOFSTEP]\ncases s\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nt : UpperSet \u03b1\ncarrier\u271d : Set \u03b1\nupper'\u271d : IsUpperSet carrier\u271d\nh : { carrier := carrier\u271d, upper' := upper'\u271d }.carrier = t.carrier\n\u22a2 { carrier := carrier\u271d, upper' := upper'\u271d } = t\n[PROOFSTEP]\ncases t\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ncarrier\u271d\u00b9 : Set \u03b1\nupper'\u271d\u00b9 : IsUpperSet carrier\u271d\u00b9\ncarrier\u271d : Set \u03b1\nupper'\u271d : IsUpperSet carrier\u271d\nh : { carrier := carrier\u271d\u00b9, upper' := upper'\u271d\u00b9 }.carrier = { carrier := carrier\u271d, upper' := upper'\u271d }.carrier\n\u22a2 { carrier := carrier\u271d\u00b9, upper' := upper'\u271d\u00b9 } = { carrier := carrier\u271d, upper' := upper'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ns t : LowerSet \u03b1\nh : s.carrier = t.carrier\n\u22a2 s = t\n[PROOFSTEP]\ncases s\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nt : LowerSet \u03b1\ncarrier\u271d : Set \u03b1\nlower'\u271d : IsLowerSet carrier\u271d\nh : { carrier := carrier\u271d, lower' := lower'\u271d }.carrier = t.carrier\n\u22a2 { carrier := carrier\u271d, lower' := lower'\u271d } = t\n[PROOFSTEP]\ncases t\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ncarrier\u271d\u00b9 : Set \u03b1\nlower'\u271d\u00b9 : IsLowerSet carrier\u271d\u00b9\ncarrier\u271d : Set \u03b1\nlower'\u271d : IsLowerSet carrier\u271d\nh : { carrier := carrier\u271d\u00b9, lower' := lower'\u271d\u00b9 }.carrier = { carrier := carrier\u271d, lower' := lower'\u271d }.carrier\n\u22a2 { carrier := carrier\u271d\u00b9, lower' := lower'\u271d\u00b9 } = { carrier := carrier\u271d, lower' := lower'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (UpperSet \u03b1)\ns t : UpperSet \u03b1\na : \u03b1\n\u22a2 \u2191s = univ \u2194 s = \u22a5\n[PROOFSTEP]\nsimp [SetLike.ext'_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (UpperSet \u03b1)\ns t : UpperSet \u03b1\na : \u03b1\n\u22a2 \u2191s = \u2205 \u2194 s = \u22a4\n[PROOFSTEP]\nsimp [SetLike.ext'_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (UpperSet \u03b1)\ns t : UpperSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 UpperSet \u03b1\n\u22a2 \u2191(\u2a06 (i : \u03b9), f i) = \u22c2 (i : \u03b9), \u2191(f i)\n[PROOFSTEP]\nsimp [iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (UpperSet \u03b1)\ns t : UpperSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 UpperSet \u03b1\n\u22a2 \u2191(\u2a05 (i : \u03b9), f i) = \u22c3 (i : \u03b9), \u2191(f i)\n[PROOFSTEP]\nsimp [iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (UpperSet \u03b1)\ns t : UpperSet \u03b1\na : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 UpperSet \u03b1\n\u22a2 \u2191(\u2a06 (i : \u03b9) (j : \u03ba i), f i j) = \u22c2 (i : \u03b9) (j : \u03ba i), \u2191(f i j)\n[PROOFSTEP]\nsimp_rw [coe_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (UpperSet \u03b1)\ns t : UpperSet \u03b1\na : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 UpperSet \u03b1\n\u22a2 \u2191(\u2a05 (i : \u03b9) (j : \u03ba i), f i j) = \u22c3 (i : \u03b9) (j : \u03ba i), \u2191(f i j)\n[PROOFSTEP]\nsimp_rw [coe_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (UpperSet \u03b1)\ns t : UpperSet \u03b1\na : \u03b1\n\u22a2 (\u2203 i j, a \u2208 \u2191i) \u2194 \u2203 s, s \u2208 S \u2227 a \u2208 s\n[PROOFSTEP]\nsimp only [exists_prop, SetLike.mem_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (UpperSet \u03b1)\ns t : UpperSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 UpperSet \u03b1\n\u22a2 a \u2208 \u2a06 (i : \u03b9), f i \u2194 \u2200 (i : \u03b9), a \u2208 f i\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, coe_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (UpperSet \u03b1)\ns t : UpperSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 UpperSet \u03b1\n\u22a2 a \u2208 \u22c2 (i : \u03b9), \u2191(f i) \u2194 \u2200 (i : \u03b9), a \u2208 f i\n[PROOFSTEP]\nexact mem_iInter\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (UpperSet \u03b1)\ns t : UpperSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 UpperSet \u03b1\n\u22a2 a \u2208 \u2a05 (i : \u03b9), f i \u2194 \u2203 i, a \u2208 f i\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, coe_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (UpperSet \u03b1)\ns t : UpperSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 UpperSet \u03b1\n\u22a2 a \u2208 \u22c3 (i : \u03b9), \u2191(f i) \u2194 \u2203 i, a \u2208 f i\n[PROOFSTEP]\nexact mem_iUnion\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (UpperSet \u03b1)\ns t : UpperSet \u03b1\na : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 UpperSet \u03b1\n\u22a2 a \u2208 \u2a06 (i : \u03b9) (j : \u03ba i), f i j \u2194 \u2200 (i : \u03b9) (j : \u03ba i), a \u2208 f i j\n[PROOFSTEP]\nsimp_rw [mem_iSup_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (UpperSet \u03b1)\ns t : UpperSet \u03b1\na : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 UpperSet \u03b1\n\u22a2 a \u2208 \u2a05 (i : \u03b9) (j : \u03ba i), f i j \u2194 \u2203 i j, a \u2208 f i j\n[PROOFSTEP]\nsimp_rw [mem_iInf_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (UpperSet \u03b1)\ns t : UpperSet \u03b1\na : \u03b1\n\u22a2 Codisjoint \u2191s \u2191t \u2194 Disjoint s t\n[PROOFSTEP]\nsimp [disjoint_iff, codisjoint_iff, SetLike.ext'_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (LowerSet \u03b1)\ns t : LowerSet \u03b1\na : \u03b1\n\u22a2 \u2191s = univ \u2194 s = \u22a4\n[PROOFSTEP]\nsimp [SetLike.ext'_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (LowerSet \u03b1)\ns t : LowerSet \u03b1\na : \u03b1\n\u22a2 \u2191s = \u2205 \u2194 s = \u22a5\n[PROOFSTEP]\nsimp [SetLike.ext'_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (LowerSet \u03b1)\ns t : LowerSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 LowerSet \u03b1\n\u22a2 \u2191(\u2a06 (i : \u03b9), f i) = \u22c3 (i : \u03b9), \u2191(f i)\n[PROOFSTEP]\nsimp_rw [iSup, coe_sSup, mem_range, iUnion_exists, iUnion_iUnion_eq']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (LowerSet \u03b1)\ns t : LowerSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 LowerSet \u03b1\n\u22a2 \u2191(\u2a05 (i : \u03b9), f i) = \u22c2 (i : \u03b9), \u2191(f i)\n[PROOFSTEP]\nsimp_rw [iInf, coe_sInf, mem_range, iInter_exists, iInter_iInter_eq']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (LowerSet \u03b1)\ns t : LowerSet \u03b1\na : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 LowerSet \u03b1\n\u22a2 \u2191(\u2a06 (i : \u03b9) (j : \u03ba i), f i j) = \u22c3 (i : \u03b9) (j : \u03ba i), \u2191(f i j)\n[PROOFSTEP]\nsimp_rw [coe_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (LowerSet \u03b1)\ns t : LowerSet \u03b1\na : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 LowerSet \u03b1\n\u22a2 \u2191(\u2a05 (i : \u03b9) (j : \u03ba i), f i j) = \u22c2 (i : \u03b9) (j : \u03ba i), \u2191(f i j)\n[PROOFSTEP]\nsimp_rw [coe_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (LowerSet \u03b1)\ns t : LowerSet \u03b1\na : \u03b1\n\u22a2 (\u2203 i j, a \u2208 \u2191i) \u2194 \u2203 s, s \u2208 S \u2227 a \u2208 s\n[PROOFSTEP]\nsimp only [exists_prop, SetLike.mem_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (LowerSet \u03b1)\ns t : LowerSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 LowerSet \u03b1\n\u22a2 a \u2208 \u2a06 (i : \u03b9), f i \u2194 \u2203 i, a \u2208 f i\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, coe_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (LowerSet \u03b1)\ns t : LowerSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 LowerSet \u03b1\n\u22a2 a \u2208 \u22c3 (i : \u03b9), \u2191(f i) \u2194 \u2203 i, a \u2208 f i\n[PROOFSTEP]\nexact mem_iUnion\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (LowerSet \u03b1)\ns t : LowerSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 LowerSet \u03b1\n\u22a2 a \u2208 \u2a05 (i : \u03b9), f i \u2194 \u2200 (i : \u03b9), a \u2208 f i\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, coe_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (LowerSet \u03b1)\ns t : LowerSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 LowerSet \u03b1\n\u22a2 a \u2208 \u22c2 (i : \u03b9), \u2191(f i) \u2194 \u2200 (i : \u03b9), a \u2208 f i\n[PROOFSTEP]\nexact mem_iInter\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (LowerSet \u03b1)\ns t : LowerSet \u03b1\na : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 LowerSet \u03b1\n\u22a2 a \u2208 \u2a06 (i : \u03b9) (j : \u03ba i), f i j \u2194 \u2203 i j, a \u2208 f i j\n[PROOFSTEP]\nsimp_rw [mem_iSup_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (LowerSet \u03b1)\ns t : LowerSet \u03b1\na : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 LowerSet \u03b1\n\u22a2 a \u2208 \u2a05 (i : \u03b9) (j : \u03ba i), f i j \u2194 \u2200 (i : \u03b9) (j : \u03ba i), a \u2208 f i j\n[PROOFSTEP]\nsimp_rw [mem_iInf_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\nS : Set (LowerSet \u03b1)\ns t : LowerSet \u03b1\na : \u03b1\n\u22a2 Disjoint \u2191s \u2191t \u2194 Disjoint s t\n[PROOFSTEP]\nsimp [disjoint_iff, SetLike.ext'_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ns t : UpperSet \u03b1\na : \u03b1\nS : Set (UpperSet \u03b1)\n\u22a2 \u2191(compl (sSup S)) = \u2191(\u2a06 (s : UpperSet \u03b1) (_ : s \u2208 S), compl s)\n[PROOFSTEP]\nsimp only [coe_compl, coe_sSup, compl_iInter\u2082, LowerSet.coe_iSup\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ns t : UpperSet \u03b1\na : \u03b1\nS : Set (UpperSet \u03b1)\n\u22a2 \u2191(compl (sInf S)) = \u2191(\u2a05 (s : UpperSet \u03b1) (_ : s \u2208 S), compl s)\n[PROOFSTEP]\nsimp only [coe_compl, coe_sInf, compl_iUnion\u2082, LowerSet.coe_iInf\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ns t : UpperSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 UpperSet \u03b1\n\u22a2 \u2191(compl (\u2a06 (i : \u03b9), f i)) = \u2191(\u2a06 (i : \u03b9), compl (f i))\n[PROOFSTEP]\nsimp only [coe_compl, coe_iSup, compl_iInter, LowerSet.coe_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ns t : UpperSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 UpperSet \u03b1\n\u22a2 \u2191(compl (\u2a05 (i : \u03b9), f i)) = \u2191(\u2a05 (i : \u03b9), compl (f i))\n[PROOFSTEP]\nsimp only [coe_compl, coe_iInf, compl_iUnion, LowerSet.coe_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ns t : UpperSet \u03b1\na : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 UpperSet \u03b1\n\u22a2 compl (\u2a06 (i : \u03b9) (j : \u03ba i), f i j) = \u2a06 (i : \u03b9) (j : \u03ba i), compl (f i j)\n[PROOFSTEP]\nsimp_rw [UpperSet.compl_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ns t : UpperSet \u03b1\na : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 UpperSet \u03b1\n\u22a2 compl (\u2a05 (i : \u03b9) (j : \u03ba i), f i j) = \u2a05 (i : \u03b9) (j : \u03ba i), compl (f i j)\n[PROOFSTEP]\nsimp_rw [UpperSet.compl_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ns t : LowerSet \u03b1\na : \u03b1\nS : Set (LowerSet \u03b1)\n\u22a2 \u2191(compl (sSup S)) = \u2191(\u2a06 (s : LowerSet \u03b1) (_ : s \u2208 S), compl s)\n[PROOFSTEP]\nsimp only [coe_compl, coe_sSup, compl_iUnion\u2082, UpperSet.coe_iSup\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ns t : LowerSet \u03b1\na : \u03b1\nS : Set (LowerSet \u03b1)\n\u22a2 \u2191(compl (sInf S)) = \u2191(\u2a05 (s : LowerSet \u03b1) (_ : s \u2208 S), compl s)\n[PROOFSTEP]\nsimp only [coe_compl, coe_sInf, compl_iInter\u2082, UpperSet.coe_iInf\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ns t : LowerSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 LowerSet \u03b1\n\u22a2 \u2191(compl (\u2a06 (i : \u03b9), f i)) = \u2191(\u2a06 (i : \u03b9), compl (f i))\n[PROOFSTEP]\nsimp only [coe_compl, coe_iSup, compl_iUnion, UpperSet.coe_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ns t : LowerSet \u03b1\na : \u03b1\nf : \u03b9 \u2192 LowerSet \u03b1\n\u22a2 \u2191(compl (\u2a05 (i : \u03b9), f i)) = \u2191(\u2a05 (i : \u03b9), compl (f i))\n[PROOFSTEP]\nsimp only [coe_compl, coe_iInf, compl_iInter, UpperSet.coe_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ns t : LowerSet \u03b1\na : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 LowerSet \u03b1\n\u22a2 compl (\u2a06 (i : \u03b9) (j : \u03ba i), f i j) = \u2a06 (i : \u03b9) (j : \u03ba i), compl (f i j)\n[PROOFSTEP]\nsimp_rw [LowerSet.compl_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : LE \u03b1\ns t : LowerSet \u03b1\na : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 LowerSet \u03b1\n\u22a2 compl (\u2a05 (i : \u03b9) (j : \u03ba i), f i j) = \u2a05 (i : \u03b9) (j : \u03ba i), compl (f i j)\n[PROOFSTEP]\nsimp_rw [LowerSet.compl_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf\u271d : \u03b1 \u2243o \u03b2\ns\u271d t : UpperSet \u03b1\na : \u03b1\nb : \u03b2\nf : \u03b1 \u2243o \u03b2\ns : UpperSet \u03b2\n\u22a2 \u2191(\u2191(OrderIso.symm (map f)) s) = \u2191(\u2191(map (OrderIso.symm f)) s)\n[PROOFSTEP]\nconvert Set.preimage_equiv_eq_image_symm s f.toEquiv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2243o \u03b2\ns t : UpperSet \u03b1\na : \u03b1\nb : \u03b2\n\u22a2 b \u2208 \u2191(map f) s \u2194 \u2191(OrderIso.symm f) b \u2208 s\n[PROOFSTEP]\nrw [\u2190 f.symm_symm, \u2190 symm_map, f.symm_symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2243o \u03b2\ns t : UpperSet \u03b1\na : \u03b1\nb : \u03b2\n\u22a2 b \u2208 \u2191(OrderIso.symm (map (OrderIso.symm f))) s \u2194 \u2191(OrderIso.symm f) b \u2208 s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2243o \u03b2\ns t : UpperSet \u03b1\na : \u03b1\nb : \u03b2\n\u22a2 map (OrderIso.refl \u03b1) = OrderIso.refl (UpperSet \u03b1)\n[PROOFSTEP]\next\n[GOAL]\ncase h.h.a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2243o \u03b2\ns t : UpperSet \u03b1\na : \u03b1\nb : \u03b2\nx\u271d\u00b9 : UpperSet \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(\u2191(map (OrderIso.refl \u03b1)) x\u271d\u00b9) \u2194 x\u271d \u2208 \u2191(\u2191(OrderIso.refl (UpperSet \u03b1)) x\u271d\u00b9)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf\u271d : \u03b1 \u2243o \u03b2\ns t : UpperSet \u03b1\na : \u03b1\nb : \u03b2\ng : \u03b2 \u2243o \u03b3\nf : \u03b1 \u2243o \u03b2\n\u22a2 \u2191(map g) (\u2191(map f) s) = \u2191(map (OrderIso.trans f g)) s\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf\u271d : \u03b1 \u2243o \u03b2\ns t : UpperSet \u03b1\na : \u03b1\nb : \u03b2\ng : \u03b2 \u2243o \u03b3\nf : \u03b1 \u2243o \u03b2\nx\u271d : \u03b3\n\u22a2 x\u271d \u2208 \u2191(\u2191(map g) (\u2191(map f) s)) \u2194 x\u271d \u2208 \u2191(\u2191(map (OrderIso.trans f g)) s)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf\u271d : \u03b1 \u2243o \u03b2\ns\u271d t : LowerSet \u03b1\na : \u03b1\nb : \u03b2\nf : \u03b1 \u2243o \u03b2\ns : LowerSet \u03b2\n\u22a2 \u2191(\u2191(OrderIso.symm (map f)) s) = \u2191(\u2191(map (OrderIso.symm f)) s)\n[PROOFSTEP]\nconvert Set.preimage_equiv_eq_image_symm s f.toEquiv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf\u271d : \u03b1 \u2243o \u03b2\ns t : LowerSet \u03b1\na : \u03b1\nb\u271d : \u03b2\nf : \u03b1 \u2243o \u03b2\nb : \u03b2\n\u22a2 b \u2208 \u2191(map f) s \u2194 \u2191(OrderIso.symm f) b \u2208 s\n[PROOFSTEP]\nrw [\u2190 f.symm_symm, \u2190 symm_map, f.symm_symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf\u271d : \u03b1 \u2243o \u03b2\ns t : LowerSet \u03b1\na : \u03b1\nb\u271d : \u03b2\nf : \u03b1 \u2243o \u03b2\nb : \u03b2\n\u22a2 b \u2208 \u2191(OrderIso.symm (map (OrderIso.symm f))) s \u2194 \u2191(OrderIso.symm f) b \u2208 s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2243o \u03b2\ns t : LowerSet \u03b1\na : \u03b1\nb : \u03b2\n\u22a2 map (OrderIso.refl \u03b1) = OrderIso.refl (LowerSet \u03b1)\n[PROOFSTEP]\next\n[GOAL]\ncase h.h.a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2243o \u03b2\ns t : LowerSet \u03b1\na : \u03b1\nb : \u03b2\nx\u271d\u00b9 : LowerSet \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(\u2191(map (OrderIso.refl \u03b1)) x\u271d\u00b9) \u2194 x\u271d \u2208 \u2191(\u2191(OrderIso.refl (LowerSet \u03b1)) x\u271d\u00b9)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf\u271d : \u03b1 \u2243o \u03b2\ns t : LowerSet \u03b1\na : \u03b1\nb : \u03b2\ng : \u03b2 \u2243o \u03b3\nf : \u03b1 \u2243o \u03b2\n\u22a2 \u2191(map g) (\u2191(map f) s) = \u2191(map (OrderIso.trans f g)) s\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf\u271d : \u03b1 \u2243o \u03b2\ns t : LowerSet \u03b1\na : \u03b1\nb : \u03b2\ng : \u03b2 \u2243o \u03b3\nf : \u03b1 \u2243o \u03b2\nx\u271d : \u03b3\n\u22a2 x\u271d \u2208 \u2191(\u2191(map g) (\u2191(map f) s)) \u2194 x\u271d \u2208 \u2191(\u2191(map (OrderIso.trans f g)) s)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : UpperSet \u03b1\na\u271d b : \u03b1\nf : \u03b1 \u2243o \u03b2\na : \u03b1\n\u22a2 \u2191(map f) (Ici a) = Ici (\u2191f a)\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : UpperSet \u03b1\na\u271d b : \u03b1\nf : \u03b1 \u2243o \u03b2\na : \u03b1\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 \u2191(\u2191(map f) (Ici a)) \u2194 x\u271d \u2208 \u2191(Ici (\u2191f a))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : UpperSet \u03b1\na\u271d b : \u03b1\nf : \u03b1 \u2243o \u03b2\na : \u03b1\n\u22a2 \u2191(map f) (Ioi a) = Ioi (\u2191f a)\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : UpperSet \u03b1\na\u271d b : \u03b1\nf : \u03b1 \u2243o \u03b2\na : \u03b1\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 \u2191(\u2191(map f) (Ioi a)) \u2194 x\u271d \u2208 \u2191(Ioi (\u2191f a))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : CompleteLattice \u03b1\nS : Set \u03b1\nc : \u03b1\n\u22a2 c \u2208 Ici (sSup S) \u2194 c \u2208 \u2a06 (a : \u03b1) (_ : a \u2208 S), Ici a\n[PROOFSTEP]\nsimp only [mem_Ici_iff, mem_iSup_iff, sSup_le_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\nc : \u03b1\n\u22a2 c \u2208 Ici (\u2a06 (i : \u03b9), f i) \u2194 c \u2208 \u2a06 (i : \u03b9), Ici (f i)\n[PROOFSTEP]\nsimp only [mem_Ici_iff, mem_iSup_iff, iSup_le_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : CompleteLattice \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 \u03b1\n\u22a2 Ici (\u2a06 (i : \u03b9) (j : \u03ba i), f i j) = \u2a06 (i : \u03b9) (j : \u03ba i), Ici (f i j)\n[PROOFSTEP]\nsimp_rw [Ici_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : LowerSet \u03b1\na\u271d b : \u03b1\nf : \u03b1 \u2243o \u03b2\na : \u03b1\n\u22a2 \u2191(map f) (Iic a) = Iic (\u2191f a)\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : LowerSet \u03b1\na\u271d b : \u03b1\nf : \u03b1 \u2243o \u03b2\na : \u03b1\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 \u2191(\u2191(map f) (Iic a)) \u2194 x\u271d \u2208 \u2191(Iic (\u2191f a))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : LowerSet \u03b1\na\u271d b : \u03b1\nf : \u03b1 \u2243o \u03b2\na : \u03b1\n\u22a2 \u2191(map f) (Iio a) = Iio (\u2191f a)\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : LowerSet \u03b1\na\u271d b : \u03b1\nf : \u03b1 \u2243o \u03b2\na : \u03b1\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 \u2191(\u2191(map f) (Iio a)) \u2194 x\u271d \u2208 \u2191(Iio (\u2191f a))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : CompleteLattice \u03b1\nS : Set \u03b1\nc : \u03b1\n\u22a2 c \u2208 Iic (sInf S) \u2194 c \u2208 \u2a05 (a : \u03b1) (_ : a \u2208 S), Iic a\n[PROOFSTEP]\nsimp only [mem_Iic_iff, mem_iInf\u2082_iff, le_sInf_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\nc : \u03b1\n\u22a2 c \u2208 Iic (\u2a05 (i : \u03b9), f i) \u2194 c \u2208 \u2a05 (i : \u03b9), Iic (f i)\n[PROOFSTEP]\nsimp only [mem_Iic_iff, mem_iInf_iff, le_iInf_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d : CompleteLattice \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 \u03b1\n\u22a2 Iic (\u2a05 (i : \u03b9) (j : \u03ba i), f i j) = \u2a05 (i : \u03b9) (j : \u03ba i), Iic (f i j)\n[PROOFSTEP]\nsimp_rw [Iic_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns\u271d t : Set \u03b1\nx : \u03b1\ns : Set \u03b1\n\u22a2 \u2191(upperClosure s) = \u22c3 (a : \u03b1) (_ : a \u2208 s), Ici a\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns\u271d t : Set \u03b1\nx : \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(upperClosure s) \u2194 x\u271d \u2208 \u22c3 (a : \u03b1) (_ : a \u2208 s), Ici a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns\u271d t : Set \u03b1\nx : \u03b1\ns : Set \u03b1\n\u22a2 \u2191(lowerClosure s) = \u22c3 (a : \u03b1) (_ : a \u2208 s), Iic a\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns\u271d t : Set \u03b1\nx : \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(lowerClosure s) \u2194 x\u271d \u2208 \u22c3 (a : \u03b1) (_ : a \u2208 s), Iic a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx : \u03b1\nf : \u03b1 \u2243o \u03b2\n\u22a2 upperClosure (\u2191f '' s) = \u2191(UpperSet.map f) (upperClosure s)\n[PROOFSTEP]\nrw [\u2190 f.symm_symm, \u2190 UpperSet.symm_map, f.symm_symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx : \u03b1\nf : \u03b1 \u2243o \u03b2\n\u22a2 upperClosure (\u2191f '' s) = \u2191(OrderIso.symm (UpperSet.map (OrderIso.symm f))) (upperClosure s)\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx : \u03b1\nf : \u03b1 \u2243o \u03b2\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 \u2191(upperClosure (\u2191f '' s)) \u2194 x\u271d \u2208 \u2191(\u2191(OrderIso.symm (UpperSet.map (OrderIso.symm f))) (upperClosure s))\n[PROOFSTEP]\nsimp [-UpperSet.symm_map, UpperSet.map, OrderIso.symm, \u2190 f.le_symm_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx : \u03b1\nf : \u03b1 \u2243o \u03b2\n\u22a2 lowerClosure (\u2191f '' s) = \u2191(LowerSet.map f) (lowerClosure s)\n[PROOFSTEP]\nrw [\u2190 f.symm_symm, \u2190 LowerSet.symm_map, f.symm_symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx : \u03b1\nf : \u03b1 \u2243o \u03b2\n\u22a2 lowerClosure (\u2191f '' s) = \u2191(OrderIso.symm (LowerSet.map (OrderIso.symm f))) (lowerClosure s)\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx : \u03b1\nf : \u03b1 \u2243o \u03b2\nx\u271d : \u03b2\n\u22a2 x\u271d \u2208 \u2191(lowerClosure (\u2191f '' s)) \u2194 x\u271d \u2208 \u2191(\u2191(OrderIso.symm (LowerSet.map (OrderIso.symm f))) (lowerClosure s))\n[PROOFSTEP]\nsimp [-LowerSet.symm_map, LowerSet.map, OrderIso.symm, \u2190 f.symm_apply_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns\u271d t : Set \u03b1\nx : \u03b1\ns : Set \u03b1\n\u22a2 \u2a05 (a : \u03b1) (_ : a \u2208 s), Ici a = upperClosure s\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns\u271d t : Set \u03b1\nx : \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(\u2a05 (a : \u03b1) (_ : a \u2208 s), Ici a) \u2194 x\u271d \u2208 \u2191(upperClosure s)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns\u271d t : Set \u03b1\nx : \u03b1\ns : Set \u03b1\n\u22a2 \u2a06 (a : \u03b1) (_ : a \u2208 s), Iic a = lowerClosure s\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns\u271d t : Set \u03b1\nx : \u03b1\ns : Set \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(\u2a06 (a : \u03b1) (_ : a \u2208 s), Iic a) \u2194 x\u271d \u2208 \u2191(lowerClosure s)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx a : \u03b1\n\u22a2 upperClosure {a} = UpperSet.Ici a\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx a x\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(upperClosure {a}) \u2194 x\u271d \u2208 \u2191(UpperSet.Ici a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx a : \u03b1\n\u22a2 lowerClosure {a} = LowerSet.Iic a\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx a x\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(lowerClosure {a}) \u2194 x\u271d \u2208 \u2191(LowerSet.Iic a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx : \u03b1\nS : Set (Set \u03b1)\n\u22a2 upperClosure (\u22c3\u2080 S) = \u2a05 (s : Set \u03b1) (_ : s \u2208 S), upperClosure s\n[PROOFSTEP]\nsimp_rw [sUnion_eq_biUnion, upperClosure_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx : \u03b1\nS : Set (Set \u03b1)\n\u22a2 lowerClosure (\u22c3\u2080 S) = \u2a06 (s : Set \u03b1) (_ : s \u2208 S), lowerClosure s\n[PROOFSTEP]\nsimp_rw [sUnion_eq_biUnion, lowerClosure_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx : \u03b1\n\u22a2 OrdConnected s \u2194 \u2191(upperClosure s) \u2229 \u2191(lowerClosure s) = s\n[PROOFSTEP]\nrefine' \u27e8Set.OrdConnected.upperClosure_inter_lowerClosure, fun h => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx : \u03b1\nh : \u2191(upperClosure s) \u2229 \u2191(lowerClosure s) = s\n\u22a2 OrdConnected s\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx : \u03b1\nh : \u2191(upperClosure s) \u2229 \u2191(lowerClosure s) = s\n\u22a2 OrdConnected (\u2191(upperClosure s) \u2229 \u2191(lowerClosure s))\n[PROOFSTEP]\nexact (UpperSet.upper _).ordConnected.inter (LowerSet.lower _).ordConnected\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx : \u03b1\n\u22a2 BddAbove \u2191(lowerClosure s) \u2194 BddAbove s\n[PROOFSTEP]\nsimp_rw [BddAbove, upperBounds_lowerClosure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns t : Set \u03b1\nx : \u03b1\n\u22a2 BddBelow \u2191(upperClosure s) \u2194 BddBelow s\n[PROOFSTEP]\nsimp_rw [BddBelow, lowerBounds_upperClosure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns s\u2081 s\u2082 : UpperSet \u03b1\nt t\u2081 t\u2082 : UpperSet \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 \u2191s\u2082 \u2286 \u2191s\u2081 \u2227 \u2191t\u2082 \u2286 \u2191t\u2081 \u2228 \u2191s\u2082 = \u2205 \u2228 \u2191t\u2082 = \u2205 \u2194 s\u2081 \u2264 s\u2082 \u2227 t\u2081 \u2264 t\u2082 \u2228 s\u2082 = \u22a4 \u2228 t\u2082 = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns s\u2081 s\u2082 : UpperSet \u03b1\nt t\u2081 t\u2082 : UpperSet \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 s \u00d7\u02e2 t = \u22a4 \u2194 s = \u22a4 \u2228 t = \u22a4\n[PROOFSTEP]\nsimp_rw [SetLike.ext'_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns s\u2081 s\u2082 : UpperSet \u03b1\nt t\u2081 t\u2082 : UpperSet \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 \u2191(s \u00d7\u02e2 t) = \u2191\u22a4 \u2194 \u2191s = \u2191\u22a4 \u2228 \u2191t = \u2191\u22a4\n[PROOFSTEP]\nexact prod_eq_empty_iff\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns s\u2081 s\u2082 : UpperSet \u03b1\nt t\u2081 t\u2082 : UpperSet \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 Codisjoint (s\u2081 \u00d7\u02e2 t\u2081) (s\u2082 \u00d7\u02e2 t\u2082) \u2194 Codisjoint s\u2081 s\u2082 \u2228 Codisjoint t\u2081 t\u2082\n[PROOFSTEP]\nsimp_rw [codisjoint_iff, prod_sup_prod, prod_eq_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns s\u2081 s\u2082 : LowerSet \u03b1\nt t\u2081 t\u2082 : LowerSet \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 \u2191s\u2081 \u2286 \u2191s\u2082 \u2227 \u2191t\u2081 \u2286 \u2191t\u2082 \u2228 \u2191s\u2081 = \u2205 \u2228 \u2191t\u2081 = \u2205 \u2194 s\u2081 \u2264 s\u2082 \u2227 t\u2081 \u2264 t\u2082 \u2228 s\u2081 = \u22a5 \u2228 t\u2081 = \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns s\u2081 s\u2082 : LowerSet \u03b1\nt t\u2081 t\u2082 : LowerSet \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 s \u00d7\u02e2 t = \u22a5 \u2194 s = \u22a5 \u2228 t = \u22a5\n[PROOFSTEP]\nsimp_rw [SetLike.ext'_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns s\u2081 s\u2082 : LowerSet \u03b1\nt t\u2081 t\u2082 : LowerSet \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 \u2191(s \u00d7\u02e2 t) = \u2191\u22a5 \u2194 \u2191s = \u2191\u22a5 \u2228 \u2191t = \u2191\u22a5\n[PROOFSTEP]\nexact prod_eq_empty_iff\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns s\u2081 s\u2082 : LowerSet \u03b1\nt t\u2081 t\u2082 : LowerSet \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 Disjoint (s\u2081 \u00d7\u02e2 t\u2081) (s\u2082 \u00d7\u02e2 t\u2082) \u2194 Disjoint s\u2081 s\u2082 \u2228 Disjoint t\u2081 t\u2082\n[PROOFSTEP]\nsimp_rw [disjoint_iff, prod_inf_prod, prod_eq_bot]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 upperClosure (s \u00d7\u02e2 t) = upperClosure s \u00d7\u02e2 upperClosure t\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set \u03b1\nt : Set \u03b2\nx\u271d : \u03b1 \u00d7 \u03b2\n\u22a2 x\u271d \u2208 \u2191(upperClosure (s \u00d7\u02e2 t)) \u2194 x\u271d \u2208 \u2191(upperClosure s \u00d7\u02e2 upperClosure t)\n[PROOFSTEP]\nsimp [Prod.le_def, @and_and_and_comm _ (_ \u2208 t)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 lowerClosure (s \u00d7\u02e2 t) = lowerClosure s \u00d7\u02e2 lowerClosure t\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set \u03b1\nt : Set \u03b2\nx\u271d : \u03b1 \u00d7 \u03b2\n\u22a2 x\u271d \u2208 \u2191(lowerClosure (s \u00d7\u02e2 t)) \u2194 x\u271d \u2208 \u2191(lowerClosure s \u00d7\u02e2 lowerClosure t)\n[PROOFSTEP]\nsimp [Prod.le_def, @and_and_and_comm _ (_ \u2208 t)]\n", "meta": {"mathlib_filename": "Mathlib.Order.UpperLower.Basic", "llama_tokens": 18816, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.40281779447529764}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 x \u2208 box n d \u2194 \u2200 (i : Fin n), x i < d\n[PROOFSTEP]\nsimp only [box, Fintype.mem_piFinset, mem_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 card (box n d) = d ^ n\n[PROOFSTEP]\nsimp [box]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 box (n + 1) 0 = \u2205\n[PROOFSTEP]\nsimp [box]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d x : Fin n \u2192 \u2115\n\u22a2 x \u2208 sphere n d 0 \u2192 x \u2208 0\n[PROOFSTEP]\nsimp [sphere, Function.funext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn k : \u2115\n\u22a2 sphere (n + 1) 0 k = \u2205\n[PROOFSTEP]\nsimp [sphere]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d x : Fin n \u2192 \u2115\nhx : x \u2208 sphere n d k\n\u22a2 \u2016\u2191(PiLp.equiv 2 fun a => \u211d).symm (Nat.cast \u2218 x)\u2016 = Real.sqrt \u2191k\n[PROOFSTEP]\nrw [EuclideanSpace.norm_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d x : Fin n \u2192 \u2115\nhx : x \u2208 sphere n d k\n\u22a2 Real.sqrt (\u2211 i : Fin n, \u2016\u2191(PiLp.equiv 2 fun a => \u211d).symm (Nat.cast \u2218 x) i\u2016 ^ 2) = Real.sqrt \u2191k\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d x : Fin n \u2192 \u2115\nhx : x \u2208 sphere n d k\n\u22a2 Real.sqrt (\u2211 i : Fin n, |\u2191(x i)| ^ 2) = Real.sqrt \u2191k\n[PROOFSTEP]\nsimp_rw [abs_cast, \u2190 cast_pow, \u2190 cast_sum, (mem_filter.1 hx).2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d x : Fin n \u2192 \u2115\nhx : x \u2208 \u2191(sphere n d k)\n\u22a2 x \u2208 (fun x => \u2191(PiLp.equiv 2 fun a => \u211d).symm (Nat.cast \u2218 x)) \u207b\u00b9' Metric.sphere 0 (Real.sqrt \u2191k)\n[PROOFSTEP]\nrw [Set.mem_preimage, mem_sphere_zero_iff_norm, norm_of_mem_sphere hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d\u271d k N : \u2115\nx : Fin n \u2192 \u2115\nd : \u2115\n\u22a2 (fun a => \u2211 i : Fin n, a i * d ^ \u2191i) 0 = 0\n[PROOFSTEP]\nsimp_rw [Pi.zero_apply, zero_mul, sum_const_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d\u271d k N : \u2115\nx : Fin n \u2192 \u2115\nd : \u2115\na b : Fin n \u2192 \u2115\n\u22a2 ZeroHom.toFun\n { toFun := fun a => \u2211 i : Fin n, a i * d ^ \u2191i, map_zero' := (_ : (fun a => \u2211 i : Fin n, a i * d ^ \u2191i) 0 = 0) }\n (a + b) =\n ZeroHom.toFun\n { toFun := fun a => \u2211 i : Fin n, a i * d ^ \u2191i, map_zero' := (_ : (fun a => \u2211 i : Fin n, a i * d ^ \u2191i) 0 = 0) }\n a +\n ZeroHom.toFun\n { toFun := fun a => \u2211 i : Fin n, a i * d ^ \u2191i, map_zero' := (_ : (fun a => \u2211 i : Fin n, a i * d ^ \u2191i) 0 = 0) } b\n[PROOFSTEP]\nsimp_rw [Pi.add_apply, add_mul, sum_add_distrib]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d\u271d k N : \u2115\nx : Fin n \u2192 \u2115\nd : \u2115\na : Fin 0 \u2192 \u2115\n\u22a2 \u2191(map d) a = 0\n[PROOFSTEP]\nsimp [map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\na : Fin (n + 1) \u2192 \u2115\n\u22a2 \u2191(map d) a = a 0 + (\u2211 x : Fin n, a (Fin.succ x) * d ^ \u2191x) * d\n[PROOFSTEP]\nsimp [map, Fin.sum_univ_succ, _root_.pow_succ', \u2190 mul_assoc, \u2190 sum_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d\u271d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nd : \u2115\nx y : Fin n \u2192 \u2115\nh : x \u2264 y\n\u22a2 \u2191(map d) x \u2264 \u2191(map d) y\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d\u271d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nd : \u2115\nx y : Fin n \u2192 \u2115\nh : x \u2264 y\n\u22a2 \u2211 i : Fin n, x i * d ^ \u2191i \u2264 \u2211 i : Fin n, y i * d ^ \u2191i\n[PROOFSTEP]\nexact sum_le_sum fun i _ => Nat.mul_le_mul_right _ <| h i\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\na : Fin (succ n) \u2192 \u2115\n\u22a2 \u2191(map d) a % d = a 0 % d\n[PROOFSTEP]\nrw [map_succ, Nat.add_mul_mod_self_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nx\u2081 x\u2082 : Fin (succ n) \u2192 \u2115\nhx\u2081 : \u2200 (i : Fin (succ n)), x\u2081 i < d\nhx\u2082 : \u2200 (i : Fin (succ n)), x\u2082 i < d\n\u22a2 \u2191(map d) x\u2081 = \u2191(map d) x\u2082 \u2194 x\u2081 0 = x\u2082 0 \u2227 \u2191(map d) (x\u2081 \u2218 Fin.succ) = \u2191(map d) (x\u2082 \u2218 Fin.succ)\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => by rw [map_succ', map_succ', h.1, h.2]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nx\u2081 x\u2082 : Fin (succ n) \u2192 \u2115\nhx\u2081 : \u2200 (i : Fin (succ n)), x\u2081 i < d\nhx\u2082 : \u2200 (i : Fin (succ n)), x\u2082 i < d\nh : x\u2081 0 = x\u2082 0 \u2227 \u2191(map d) (x\u2081 \u2218 Fin.succ) = \u2191(map d) (x\u2082 \u2218 Fin.succ)\n\u22a2 \u2191(map d) x\u2081 = \u2191(map d) x\u2082\n[PROOFSTEP]\nrw [map_succ', map_succ', h.1, h.2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nx\u2081 x\u2082 : Fin (succ n) \u2192 \u2115\nhx\u2081 : \u2200 (i : Fin (succ n)), x\u2081 i < d\nhx\u2082 : \u2200 (i : Fin (succ n)), x\u2082 i < d\nh : \u2191(map d) x\u2081 = \u2191(map d) x\u2082\n\u22a2 x\u2081 0 = x\u2082 0 \u2227 \u2191(map d) (x\u2081 \u2218 Fin.succ) = \u2191(map d) (x\u2082 \u2218 Fin.succ)\n[PROOFSTEP]\nhave : x\u2081 0 = x\u2082 0 := by rw [\u2190 mod_eq_of_lt (hx\u2081 _), \u2190 map_mod, \u2190 mod_eq_of_lt (hx\u2082 _), \u2190 map_mod, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nx\u2081 x\u2082 : Fin (succ n) \u2192 \u2115\nhx\u2081 : \u2200 (i : Fin (succ n)), x\u2081 i < d\nhx\u2082 : \u2200 (i : Fin (succ n)), x\u2082 i < d\nh : \u2191(map d) x\u2081 = \u2191(map d) x\u2082\n\u22a2 x\u2081 0 = x\u2082 0\n[PROOFSTEP]\nrw [\u2190 mod_eq_of_lt (hx\u2081 _), \u2190 map_mod, \u2190 mod_eq_of_lt (hx\u2082 _), \u2190 map_mod, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nx\u2081 x\u2082 : Fin (succ n) \u2192 \u2115\nhx\u2081 : \u2200 (i : Fin (succ n)), x\u2081 i < d\nhx\u2082 : \u2200 (i : Fin (succ n)), x\u2082 i < d\nh : \u2191(map d) x\u2081 = \u2191(map d) x\u2082\nthis : x\u2081 0 = x\u2082 0\n\u22a2 x\u2081 0 = x\u2082 0 \u2227 \u2191(map d) (x\u2081 \u2218 Fin.succ) = \u2191(map d) (x\u2082 \u2218 Fin.succ)\n[PROOFSTEP]\nrw [map_succ, map_succ, this, add_right_inj, mul_eq_mul_right_iff] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nx\u2081 x\u2082 : Fin (succ n) \u2192 \u2115\nhx\u2081 : \u2200 (i : Fin (succ n)), x\u2081 i < d\nhx\u2082 : \u2200 (i : Fin (succ n)), x\u2082 i < d\nh : \u2211 x : Fin n, x\u2081 (Fin.succ x) * d ^ \u2191x = \u2211 x : Fin n, x\u2082 (Fin.succ x) * d ^ \u2191x \u2228 d = 0\nthis : x\u2081 0 = x\u2082 0\n\u22a2 x\u2081 0 = x\u2082 0 \u2227 \u2191(map d) (x\u2081 \u2218 Fin.succ) = \u2191(map d) (x\u2082 \u2218 Fin.succ)\n[PROOFSTEP]\nexact \u27e8this, h.resolve_right (pos_of_gt (hx\u2081 0)).ne'\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 Set.InjOn \u2191(map d) {x | \u2200 (i : Fin n), x i < d}\n[PROOFSTEP]\nintro x\u2081 hx\u2081 x\u2082 hx\u2082 h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx x\u2081 : Fin n \u2192 \u2115\nhx\u2081 : x\u2081 \u2208 {x | \u2200 (i : Fin n), x i < d}\nx\u2082 : Fin n \u2192 \u2115\nhx\u2082 : x\u2082 \u2208 {x | \u2200 (i : Fin n), x i < d}\nh : \u2191(map d) x\u2081 = \u2191(map d) x\u2082\n\u22a2 x\u2081 = x\u2082\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d x\u2081\u271d : Fin n \u2192 \u2115\nhx\u2081\u271d : x\u2081\u271d \u2208 {x | \u2200 (i : Fin n), x i < d}\nx\u2082\u271d : Fin n \u2192 \u2115\nhx\u2082\u271d : x\u2082\u271d \u2208 {x | \u2200 (i : Fin n), x i < d}\nh\u271d : \u2191(map d) x\u2081\u271d = \u2191(map d) x\u2082\u271d\nx x\u2081 : Fin zero \u2192 \u2115\nhx\u2081 : x\u2081 \u2208 {x | \u2200 (i : Fin zero), x i < d}\nx\u2082 : Fin zero \u2192 \u2115\nhx\u2082 : x\u2082 \u2208 {x | \u2200 (i : Fin zero), x i < d}\nh : \u2191(map d) x\u2081 = \u2191(map d) x\u2082\n\u22a2 x\u2081 = x\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx\u271d x\u2081\u271d : Fin n\u271d \u2192 \u2115\nhx\u2081\u271d : x\u2081\u271d \u2208 {x | \u2200 (i : Fin n\u271d), x i < d}\nx\u2082\u271d : Fin n\u271d \u2192 \u2115\nhx\u2082\u271d : x\u2082\u271d \u2208 {x | \u2200 (i : Fin n\u271d), x i < d}\nh\u271d : \u2191(map d) x\u2081\u271d = \u2191(map d) x\u2082\u271d\nn : \u2115\nih :\n \u2200 {x : Fin n \u2192 \u2115} \u2983x\u2081 : Fin n \u2192 \u2115\u2984,\n x\u2081 \u2208 {x | \u2200 (i : Fin n), x i < d} \u2192\n \u2200 \u2983x\u2082 : Fin n \u2192 \u2115\u2984, x\u2082 \u2208 {x | \u2200 (i : Fin n), x i < d} \u2192 \u2191(map d) x\u2081 = \u2191(map d) x\u2082 \u2192 x\u2081 = x\u2082\nx x\u2081 : Fin (succ n) \u2192 \u2115\nhx\u2081 : x\u2081 \u2208 {x | \u2200 (i : Fin (succ n)), x i < d}\nx\u2082 : Fin (succ n) \u2192 \u2115\nhx\u2082 : x\u2082 \u2208 {x | \u2200 (i : Fin (succ n)), x i < d}\nh : \u2191(map d) x\u2081 = \u2191(map d) x\u2082\n\u22a2 x\u2081 = x\u2082\n[PROOFSTEP]\nrw [forall_const] at ih \n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx\u271d x\u2081\u271d : Fin n\u271d \u2192 \u2115\nhx\u2081\u271d : x\u2081\u271d \u2208 {x | \u2200 (i : Fin n\u271d), x i < d}\nx\u2082\u271d : Fin n\u271d \u2192 \u2115\nhx\u2082\u271d : x\u2082\u271d \u2208 {x | \u2200 (i : Fin n\u271d), x i < d}\nh\u271d : \u2191(map d) x\u2081\u271d = \u2191(map d) x\u2082\u271d\nn : \u2115\nih :\n \u2200 \u2983x\u2081 : Fin n \u2192 \u2115\u2984,\n x\u2081 \u2208 {x | \u2200 (i : Fin n), x i < d} \u2192\n \u2200 \u2983x\u2082 : Fin n \u2192 \u2115\u2984, x\u2082 \u2208 {x | \u2200 (i : Fin n), x i < d} \u2192 \u2191(map d) x\u2081 = \u2191(map d) x\u2082 \u2192 x\u2081 = x\u2082\nx x\u2081 : Fin (succ n) \u2192 \u2115\nhx\u2081 : x\u2081 \u2208 {x | \u2200 (i : Fin (succ n)), x i < d}\nx\u2082 : Fin (succ n) \u2192 \u2115\nhx\u2082 : x\u2082 \u2208 {x | \u2200 (i : Fin (succ n)), x i < d}\nh : \u2191(map d) x\u2081 = \u2191(map d) x\u2082\n\u22a2 x\u2081 = x\u2082\n[PROOFSTEP]\next i\n[GOAL]\ncase succ.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx\u271d x\u2081\u271d : Fin n\u271d \u2192 \u2115\nhx\u2081\u271d : x\u2081\u271d \u2208 {x | \u2200 (i : Fin n\u271d), x i < d}\nx\u2082\u271d : Fin n\u271d \u2192 \u2115\nhx\u2082\u271d : x\u2082\u271d \u2208 {x | \u2200 (i : Fin n\u271d), x i < d}\nh\u271d : \u2191(map d) x\u2081\u271d = \u2191(map d) x\u2082\u271d\nn : \u2115\nih :\n \u2200 \u2983x\u2081 : Fin n \u2192 \u2115\u2984,\n x\u2081 \u2208 {x | \u2200 (i : Fin n), x i < d} \u2192\n \u2200 \u2983x\u2082 : Fin n \u2192 \u2115\u2984, x\u2082 \u2208 {x | \u2200 (i : Fin n), x i < d} \u2192 \u2191(map d) x\u2081 = \u2191(map d) x\u2082 \u2192 x\u2081 = x\u2082\nx x\u2081 : Fin (succ n) \u2192 \u2115\nhx\u2081 : x\u2081 \u2208 {x | \u2200 (i : Fin (succ n)), x i < d}\nx\u2082 : Fin (succ n) \u2192 \u2115\nhx\u2082 : x\u2082 \u2208 {x | \u2200 (i : Fin (succ n)), x i < d}\nh : \u2191(map d) x\u2081 = \u2191(map d) x\u2082\ni : Fin (succ n)\n\u22a2 x\u2081 i = x\u2082 i\n[PROOFSTEP]\nhave x := (map_eq_iff hx\u2081 hx\u2082).1 h\n[GOAL]\ncase succ.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx\u271d\u00b9 x\u2081\u271d : Fin n\u271d \u2192 \u2115\nhx\u2081\u271d : x\u2081\u271d \u2208 {x | \u2200 (i : Fin n\u271d), x i < d}\nx\u2082\u271d : Fin n\u271d \u2192 \u2115\nhx\u2082\u271d : x\u2082\u271d \u2208 {x | \u2200 (i : Fin n\u271d), x i < d}\nh\u271d : \u2191(map d) x\u2081\u271d = \u2191(map d) x\u2082\u271d\nn : \u2115\nih :\n \u2200 \u2983x\u2081 : Fin n \u2192 \u2115\u2984,\n x\u2081 \u2208 {x | \u2200 (i : Fin n), x i < d} \u2192\n \u2200 \u2983x\u2082 : Fin n \u2192 \u2115\u2984, x\u2082 \u2208 {x | \u2200 (i : Fin n), x i < d} \u2192 \u2191(map d) x\u2081 = \u2191(map d) x\u2082 \u2192 x\u2081 = x\u2082\nx\u271d x\u2081 : Fin (succ n) \u2192 \u2115\nhx\u2081 : x\u2081 \u2208 {x | \u2200 (i : Fin (succ n)), x i < d}\nx\u2082 : Fin (succ n) \u2192 \u2115\nhx\u2082 : x\u2082 \u2208 {x | \u2200 (i : Fin (succ n)), x i < d}\nh : \u2191(map d) x\u2081 = \u2191(map d) x\u2082\ni : Fin (succ n)\nx : x\u2081 0 = x\u2082 0 \u2227 \u2191(map d) ((fun i => x\u2081 i) \u2218 Fin.succ) = \u2191(map d) ((fun i => x\u2082 i) \u2218 Fin.succ)\n\u22a2 x\u2081 i = x\u2082 i\n[PROOFSTEP]\nrefine' Fin.cases x.1 (congr_fun <| ih (fun _ => _) (fun _ => _) x.2) i\n[GOAL]\ncase succ.h.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx\u271d\u00b2 x\u2081\u271d : Fin n\u271d \u2192 \u2115\nhx\u2081\u271d : x\u2081\u271d \u2208 {x | \u2200 (i : Fin n\u271d), x i < d}\nx\u2082\u271d : Fin n\u271d \u2192 \u2115\nhx\u2082\u271d : x\u2082\u271d \u2208 {x | \u2200 (i : Fin n\u271d), x i < d}\nh\u271d : \u2191(map d) x\u2081\u271d = \u2191(map d) x\u2082\u271d\nn : \u2115\nih :\n \u2200 \u2983x\u2081 : Fin n \u2192 \u2115\u2984,\n x\u2081 \u2208 {x | \u2200 (i : Fin n), x i < d} \u2192\n \u2200 \u2983x\u2082 : Fin n \u2192 \u2115\u2984, x\u2082 \u2208 {x | \u2200 (i : Fin n), x i < d} \u2192 \u2191(map d) x\u2081 = \u2191(map d) x\u2082 \u2192 x\u2081 = x\u2082\nx\u271d\u00b9 x\u2081 : Fin (succ n) \u2192 \u2115\nhx\u2081 : x\u2081 \u2208 {x | \u2200 (i : Fin (succ n)), x i < d}\nx\u2082 : Fin (succ n) \u2192 \u2115\nhx\u2082 : x\u2082 \u2208 {x | \u2200 (i : Fin (succ n)), x i < d}\nh : \u2191(map d) x\u2081 = \u2191(map d) x\u2082\ni : Fin (succ n)\nx : x\u2081 0 = x\u2082 0 \u2227 \u2191(map d) ((fun i => x\u2081 i) \u2218 Fin.succ) = \u2191(map d) ((fun i => x\u2082 i) \u2218 Fin.succ)\nx\u271d : Fin n\n\u22a2 x\u2081 (Fin.succ x\u271d) < d\n[PROOFSTEP]\nexact hx\u2081 _\n[GOAL]\ncase succ.h.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx\u271d\u00b2 x\u2081\u271d : Fin n\u271d \u2192 \u2115\nhx\u2081\u271d : x\u2081\u271d \u2208 {x | \u2200 (i : Fin n\u271d), x i < d}\nx\u2082\u271d : Fin n\u271d \u2192 \u2115\nhx\u2082\u271d : x\u2082\u271d \u2208 {x | \u2200 (i : Fin n\u271d), x i < d}\nh\u271d : \u2191(map d) x\u2081\u271d = \u2191(map d) x\u2082\u271d\nn : \u2115\nih :\n \u2200 \u2983x\u2081 : Fin n \u2192 \u2115\u2984,\n x\u2081 \u2208 {x | \u2200 (i : Fin n), x i < d} \u2192\n \u2200 \u2983x\u2082 : Fin n \u2192 \u2115\u2984, x\u2082 \u2208 {x | \u2200 (i : Fin n), x i < d} \u2192 \u2191(map d) x\u2081 = \u2191(map d) x\u2082 \u2192 x\u2081 = x\u2082\nx\u271d\u00b9 x\u2081 : Fin (succ n) \u2192 \u2115\nhx\u2081 : x\u2081 \u2208 {x | \u2200 (i : Fin (succ n)), x i < d}\nx\u2082 : Fin (succ n) \u2192 \u2115\nhx\u2082 : x\u2082 \u2208 {x | \u2200 (i : Fin (succ n)), x i < d}\nh : \u2191(map d) x\u2081 = \u2191(map d) x\u2082\ni : Fin (succ n)\nx : x\u2081 0 = x\u2082 0 \u2227 \u2191(map d) ((fun i => x\u2081 i) \u2218 Fin.succ) = \u2191(map d) ((fun i => x\u2082 i) \u2218 Fin.succ)\nx\u271d : Fin n\n\u22a2 x\u2082 (Fin.succ x\u271d) < d\n[PROOFSTEP]\nexact hx\u2082 _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 AddSalemSpencer \u2191(sphere n d k)\n[PROOFSTEP]\nset f : (Fin n \u2192 \u2115) \u2192+ EuclideanSpace \u211d (Fin n) :=\n { toFun := fun f => ((\u2191) : \u2115 \u2192 \u211d) \u2218 f\n map_zero' := funext fun _ => cast_zero\n map_add' := fun _ _ => funext fun _ => cast_add _ _ }\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nf : (Fin n \u2192 \u2115) \u2192+ EuclideanSpace \u211d (Fin n) :=\n { toZeroHom := { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) },\n map_add' :=\n (_ :\n \u2200 (x x_1 : Fin n \u2192 \u2115),\n ZeroHom.toFun { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) } (x + x_1) =\n ZeroHom.toFun { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) } x +\n ZeroHom.toFun { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) } x_1) }\n\u22a2 AddSalemSpencer \u2191(sphere n d k)\n[PROOFSTEP]\nrefine' AddSalemSpencer.of_image (f.toAddFreimanHom (sphere n d k : Set (Fin n \u2192 \u2115)) 2) _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nf : (Fin n \u2192 \u2115) \u2192+ EuclideanSpace \u211d (Fin n) :=\n { toZeroHom := { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) },\n map_add' :=\n (_ :\n \u2200 (x x_1 : Fin n \u2192 \u2115),\n ZeroHom.toFun { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) } (x + x_1) =\n ZeroHom.toFun { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) } x +\n ZeroHom.toFun { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) } x_1) }\n\u22a2 Set.InjOn \u2191(AddMonoidHom.toAddFreimanHom (\u2191(sphere n d k)) 2 f) \u2191(sphere n d k)\n[PROOFSTEP]\nexact cast_injective.comp_left.injOn _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nf : (Fin n \u2192 \u2115) \u2192+ EuclideanSpace \u211d (Fin n) :=\n { toZeroHom := { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) },\n map_add' :=\n (_ :\n \u2200 (x x_1 : Fin n \u2192 \u2115),\n ZeroHom.toFun { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) } (x + x_1) =\n ZeroHom.toFun { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) } x +\n ZeroHom.toFun { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) } x_1) }\n\u22a2 AddSalemSpencer (\u2191(AddMonoidHom.toAddFreimanHom (\u2191(sphere n d k)) 2 f) '' \u2191(sphere n d k))\n[PROOFSTEP]\nrefine' (addSalemSpencer_sphere 0 <| Real.sqrt k).mono (Set.image_subset_iff.2 fun x => _)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nf : (Fin n \u2192 \u2115) \u2192+ EuclideanSpace \u211d (Fin n) :=\n { toZeroHom := { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) },\n map_add' :=\n (_ :\n \u2200 (x x_1 : Fin n \u2192 \u2115),\n ZeroHom.toFun { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) } (x + x_1) =\n ZeroHom.toFun { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) } x +\n ZeroHom.toFun { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) } x_1) }\nx : Fin n \u2192 \u2115\n\u22a2 x \u2208 \u2191(sphere n d k) \u2192 x \u2208 \u2191(AddMonoidHom.toAddFreimanHom (\u2191(sphere n d k)) 2 f) \u207b\u00b9' Metric.sphere 0 (Real.sqrt \u2191k)\n[PROOFSTEP]\nrw [Set.mem_preimage, mem_sphere_zero_iff_norm]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nf : (Fin n \u2192 \u2115) \u2192+ EuclideanSpace \u211d (Fin n) :=\n { toZeroHom := { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) },\n map_add' :=\n (_ :\n \u2200 (x x_1 : Fin n \u2192 \u2115),\n ZeroHom.toFun { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) } (x + x_1) =\n ZeroHom.toFun { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) } x +\n ZeroHom.toFun { toFun := fun f => Nat.cast \u2218 f, map_zero' := (_ : (fun f => Nat.cast \u2218 f) 0 = 0) } x_1) }\nx : Fin n \u2192 \u2115\n\u22a2 x \u2208 \u2191(sphere n d k) \u2192 \u2016\u2191(AddMonoidHom.toAddFreimanHom (\u2191(sphere n d k)) 2 f) x\u2016 = Real.sqrt \u2191k\n[PROOFSTEP]\nexact norm_of_mem_sphere\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 AddSalemSpencer \u2191(image (\u2191(map (2 * d - 1))) (sphere n d k))\n[PROOFSTEP]\nrw [coe_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 AddSalemSpencer (\u2191(map (2 * d - 1)) '' \u2191(sphere n d k))\n[PROOFSTEP]\nrefine'\n @AddSalemSpencer.image _ (Fin n \u2192 \u2115) \u2115 _ _ (sphere n d k) _ (map (2 * d - 1)) (map_injOn.mono _)\n addSalemSpencer_sphere\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 Fin n \u2192 \u2115\n[PROOFSTEP]\nexact x\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 \u2191(sphere n d k) + \u2191(sphere n d k) \u2286 {x | \u2200 (i : Fin n), x i < 2 * d - 1}\n[PROOFSTEP]\nrw [Set.add_subset_iff]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 \u2200 (x : Fin n \u2192 \u2115),\n x \u2208 \u2191(sphere n d k) \u2192 \u2200 (y : Fin n \u2192 \u2115), y \u2208 \u2191(sphere n d k) \u2192 x + y \u2208 {x | \u2200 (i : Fin n), x i < 2 * d - 1}\n[PROOFSTEP]\nrintro a ha b hb i\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx a : Fin n \u2192 \u2115\nha : a \u2208 \u2191(sphere n d k)\nb : Fin n \u2192 \u2115\nhb : b \u2208 \u2191(sphere n d k)\ni : Fin n\n\u22a2 (a + b) i < 2 * d - 1\n[PROOFSTEP]\nhave hai := mem_box.1 (sphere_subset_box ha) i\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx a : Fin n \u2192 \u2115\nha : a \u2208 \u2191(sphere n d k)\nb : Fin n \u2192 \u2115\nhb : b \u2208 \u2191(sphere n d k)\ni : Fin n\nhai : a i < d\n\u22a2 (a + b) i < 2 * d - 1\n[PROOFSTEP]\nhave hbi := mem_box.1 (sphere_subset_box hb) i\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx a : Fin n \u2192 \u2115\nha : a \u2208 \u2191(sphere n d k)\nb : Fin n \u2192 \u2115\nhb : b \u2208 \u2191(sphere n d k)\ni : Fin n\nhai : a i < d\nhbi : b i < d\n\u22a2 (a + b) i < 2 * d - 1\n[PROOFSTEP]\nrw [lt_tsub_iff_right, \u2190 succ_le_iff, two_mul]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx a : Fin n \u2192 \u2115\nha : a \u2208 \u2191(sphere n d k)\nb : Fin n \u2192 \u2115\nhb : b \u2208 \u2191(sphere n d k)\ni : Fin n\nhai : a i < d\nhbi : b i < d\n\u22a2 succ ((a + b) i + 1) \u2264 d + d\n[PROOFSTEP]\nexact (add_add_add_comm _ _ 1 1).trans_le (_root_.add_le_add hai hbi)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhx : x \u2208 box n d\n\u22a2 \u2211 i : Fin n, x i ^ 2 \u2264 n * (d - 1) ^ 2\n[PROOFSTEP]\nrw [mem_box] at hx \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhx : \u2200 (i : Fin n), x i < d\n\u22a2 \u2211 i : Fin n, x i ^ 2 \u2264 n * (d - 1) ^ 2\n[PROOFSTEP]\nhave : \u2200 i, x i ^ 2 \u2264 (d - 1) ^ 2 := fun i => Nat.pow_le_pow_of_le_left (Nat.le_pred_of_lt (hx i)) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhx : \u2200 (i : Fin n), x i < d\nthis : \u2200 (i : Fin n), x i ^ 2 \u2264 (d - 1) ^ 2\n\u22a2 \u2211 i : Fin n, x i ^ 2 \u2264 n * (d - 1) ^ 2\n[PROOFSTEP]\nexact (sum_le_card_nsmul univ _ _ fun i _ => this i).trans (by rw [card_fin, smul_eq_mul])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhx : \u2200 (i : Fin n), x i < d\nthis : \u2200 (i : Fin n), x i ^ 2 \u2264 (d - 1) ^ 2\n\u22a2 card univ \u2022 (d - 1) ^ 2 \u2264 n * (d - 1) ^ 2\n[PROOFSTEP]\nrw [card_fin, smul_eq_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 \u2211 i : Fin n, d * (2 * d + 1) ^ \u2191i = ((2 * d + 1) ^ n - 1) / 2\n[PROOFSTEP]\nrefine' (Nat.div_eq_of_eq_mul_left zero_lt_two _).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 (2 * d + 1) ^ n - 1 = (\u2211 i : Fin n, d * (2 * d + 1) ^ \u2191i) * 2\n[PROOFSTEP]\nrw [\u2190 sum_range fun i => d * (2 * d + 1) ^ (i : \u2115), \u2190 mul_sum, mul_right_comm, mul_comm d, \u2190 geom_sum_mul_add,\n add_tsub_cancel_right, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d k : \u2115\n\u22a2 card (sphere n d k) \u2264 \u2191rothNumberNat ((2 * d - 1) ^ n)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d\u271d k\u271d N : \u2115\nx : Fin n \u2192 \u2115\nd k : \u2115\n\u22a2 card (sphere zero d k) \u2264 \u2191rothNumberNat ((2 * d - 1) ^ zero)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d\u271d k\u271d N : \u2115\nx : Fin n \u2192 \u2115\nd k : \u2115\n\u22a2 card (sphere 0 d k) \u2264 \u2191rothNumberNat 1\n[PROOFSTEP]\nrefine' (card_le_univ _).trans_eq _\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d\u271d k\u271d N : \u2115\nx : Fin n \u2192 \u2115\nd k : \u2115\n\u22a2 Fintype.card (Fin 0 \u2192 \u2115) = \u2191rothNumberNat 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d\u271d k\u271d N : \u2115\nx : Fin n \u2192 \u2115\nd k n\u271d : \u2115\n\u22a2 card (sphere (succ n\u271d) d k) \u2264 \u2191rothNumberNat ((2 * d - 1) ^ succ n\u271d)\n[PROOFSTEP]\ncases d\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k\u271d N : \u2115\nx : Fin n \u2192 \u2115\nk n\u271d : \u2115\n\u22a2 card (sphere (succ n\u271d) zero k) \u2264 \u2191rothNumberNat ((2 * zero - 1) ^ succ n\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k\u271d N : \u2115\nx : Fin n \u2192 \u2115\nk n\u271d\u00b9 n\u271d : \u2115\n\u22a2 card (sphere (succ n\u271d\u00b9) (succ n\u271d) k) \u2264 \u2191rothNumberNat ((2 * succ n\u271d - 1) ^ succ n\u271d\u00b9)\n[PROOFSTEP]\nrefine' addSalemSpencer_image_sphere.le_rothNumberNat _ _ (card_image_of_injOn _)\n[GOAL]\ncase succ.succ.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k\u271d N : \u2115\nx : Fin n \u2192 \u2115\nk n\u271d\u00b9 n\u271d : \u2115\n\u22a2 Fin (succ n\u271d\u00b9) \u2192 \u2115\n[PROOFSTEP]\nintro\n[GOAL]\ncase succ.succ.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k\u271d N : \u2115\nx : Fin n \u2192 \u2115\nk n\u271d\u00b9 n\u271d : \u2115\na\u271d : Fin (succ n\u271d\u00b9)\n\u22a2 \u2115\n[PROOFSTEP]\nassumption\n[GOAL]\ncase succ.succ.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k\u271d N : \u2115\nx : Fin n \u2192 \u2115\nk n\u271d\u00b9 n\u271d : \u2115\n\u22a2 \u2200 (x : \u2115), x \u2208 image (\u2191(map (2 * succ n\u271d - 1))) (sphere (succ n\u271d\u00b9) (succ n\u271d) k) \u2192 x < (2 * succ n\u271d - 1) ^ succ n\u271d\u00b9\n[PROOFSTEP]\nsimp only [subset_iff, mem_image, and_imp, forall_exists_index, mem_range, forall_apply_eq_imp_iff\u2082, sphere, mem_filter]\n[GOAL]\ncase succ.succ.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k\u271d N : \u2115\nx : Fin n \u2192 \u2115\nk n\u271d\u00b9 n\u271d : \u2115\n\u22a2 \u2200 (x : \u2115) (x_1 : Fin (succ n\u271d\u00b9) \u2192 \u2115),\n x_1 \u2208 box (succ n\u271d\u00b9) (succ n\u271d) \u2192\n \u2211 i : Fin (succ n\u271d\u00b9), x_1 i ^ 2 = k \u2192 \u2191(map (2 * succ n\u271d - 1)) x_1 = x \u2192 x < (2 * succ n\u271d - 1) ^ succ n\u271d\u00b9\n[PROOFSTEP]\nrintro _ x hx _ rfl\n[GOAL]\ncase succ.succ.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k\u271d N : \u2115\nx\u271d : Fin n \u2192 \u2115\nk n\u271d\u00b9 n\u271d : \u2115\nx : Fin (succ n\u271d\u00b9) \u2192 \u2115\nhx : x \u2208 box (succ n\u271d\u00b9) (succ n\u271d)\na\u271d : \u2211 i : Fin (succ n\u271d\u00b9), x i ^ 2 = k\n\u22a2 \u2191(map (2 * succ n\u271d - 1)) x < (2 * succ n\u271d - 1) ^ succ n\u271d\u00b9\n[PROOFSTEP]\nexact (map_le_of_mem_box hx).trans_lt sum_lt\n[GOAL]\ncase succ.succ.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k\u271d N : \u2115\nx : Fin n \u2192 \u2115\nk n\u271d\u00b9 n\u271d : \u2115\n\u22a2 Set.InjOn \u2191(map (2 * succ n\u271d - 1)) \u2191(sphere (succ n\u271d\u00b9) (succ n\u271d) k)\n[PROOFSTEP]\nrefine' map_injOn.mono fun x => _\n[GOAL]\ncase succ.succ.refine'_3.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k\u271d N : \u2115\nx : Fin n \u2192 \u2115\nk n\u271d\u00b9 n\u271d : \u2115\n\u22a2 Fin (succ n\u271d\u00b9) \u2192 \u2115\n[PROOFSTEP]\nintro\n[GOAL]\ncase succ.succ.refine'_3.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k\u271d N : \u2115\nx : Fin n \u2192 \u2115\nk n\u271d\u00b9 n\u271d : \u2115\na\u271d : Fin (succ n\u271d\u00b9)\n\u22a2 \u2115\n[PROOFSTEP]\nassumption\n[GOAL]\ncase succ.succ.refine'_3.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k\u271d N : \u2115\nx\u271d : Fin n \u2192 \u2115\nk n\u271d\u00b9 n\u271d : \u2115\nx : Fin (succ n\u271d\u00b9) \u2192 \u2115\n\u22a2 x \u2208 \u2191(sphere (succ n\u271d\u00b9) (succ n\u271d) k) \u2192 x \u2208 {x | \u2200 (i : Fin (succ n\u271d\u00b9)), x i < 2 * succ n\u271d - 1}\n[PROOFSTEP]\nsimp only [mem_coe, sphere, mem_filter, mem_box, and_imp, two_mul]\n[GOAL]\ncase succ.succ.refine'_3.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k\u271d N : \u2115\nx\u271d : Fin n \u2192 \u2115\nk n\u271d\u00b9 n\u271d : \u2115\nx : Fin (succ n\u271d\u00b9) \u2192 \u2115\n\u22a2 (\u2200 (i : Fin (succ n\u271d\u00b9)), x i < succ n\u271d) \u2192\n \u2211 i : Fin (succ n\u271d\u00b9), x i ^ 2 = k \u2192 x \u2208 {x | \u2200 (i : Fin (succ n\u271d\u00b9)), x i < succ n\u271d + succ n\u271d - 1}\n[PROOFSTEP]\nexact fun h _ i => (h i).trans_le le_self_add\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d : \u2115\n\u22a2 \u2203 k, k \u2208 range (n * (d - 1) ^ 2 + 1) \u2227 \u2191(d ^ n) / (\u2191(n * (d - 1) ^ 2) + 1) \u2264 \u2191(card (sphere n d k))\n[PROOFSTEP]\nrefine' exists_le_card_fiber_of_nsmul_le_card_of_maps_to (fun x hx => _) nonempty_range_succ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k N : \u2115\nx\u271d : Fin n\u271d \u2192 \u2115\nn d : \u2115\nx : Fin n \u2192 \u2115\nhx : x \u2208 box n d\n\u22a2 \u2211 i : Fin n, x i ^ 2 \u2208 range (n * (d - 1) ^ 2 + 1)\n[PROOFSTEP]\nrw [mem_range, lt_succ_iff]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k N : \u2115\nx\u271d : Fin n\u271d \u2192 \u2115\nn d : \u2115\nx : Fin n \u2192 \u2115\nhx : x \u2208 box n d\n\u22a2 \u2211 i : Fin n, x i ^ 2 \u2264 n * (d - 1) ^ 2\n[PROOFSTEP]\nexact sum_sq_le_of_mem_box hx\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d : \u2115\n\u22a2 card (range (n * (d - 1) ^ 2 + 1)) \u2022 (\u2191(d ^ n) / (\u2191(n * (d - 1) ^ 2) + 1)) \u2264 \u2191(card (box n d))\n[PROOFSTEP]\nrw [card_range, _root_.nsmul_eq_mul, mul_div_assoc', cast_add_one, mul_div_cancel_left, card_box]\n[GOAL]\ncase refine'_2.ha\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d : \u2115\n\u22a2 \u2191(n * (d - 1) ^ 2) + 1 \u2260 0\n[PROOFSTEP]\nexact (cast_add_one_pos _).ne'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d : \u2115\n\u22a2 \u2203 k, \u2191d ^ \u2191n / \u2191(n * d ^ 2) \u2264 \u2191(card (sphere n d k))\n[PROOFSTEP]\nobtain \u27e8k, -, hk\u27e9 := exists_large_sphere_aux n d\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d k : \u2115\nhk : \u2191(d ^ n) / (\u2191(n * (d - 1) ^ 2) + 1) \u2264 \u2191(card (sphere n d k))\n\u22a2 \u2203 k, \u2191d ^ \u2191n / \u2191(n * d ^ 2) \u2264 \u2191(card (sphere n d k))\n[PROOFSTEP]\nrefine' \u27e8k, _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d k : \u2115\nhk : \u2191(d ^ n) / (\u2191(n * (d - 1) ^ 2) + 1) \u2264 \u2191(card (sphere n d k))\n\u22a2 \u2191d ^ \u2191n / \u2191(n * d ^ 2) \u2264 \u2191(card (sphere n d k))\n[PROOFSTEP]\nobtain rfl | hn := n.eq_zero_or_pos\n[GOAL]\ncase intro.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d\u271d k\u271d N : \u2115\nx : Fin n \u2192 \u2115\nd k : \u2115\nhk : \u2191(d ^ 0) / (\u2191(0 * (d - 1) ^ 2) + 1) \u2264 \u2191(card (sphere 0 d k))\n\u22a2 \u2191d ^ \u21910 / \u2191(0 * d ^ 2) \u2264 \u2191(card (sphere 0 d k))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d k : \u2115\nhk : \u2191(d ^ n) / (\u2191(n * (d - 1) ^ 2) + 1) \u2264 \u2191(card (sphere n d k))\nhn : n > 0\n\u22a2 \u2191d ^ \u2191n / \u2191(n * d ^ 2) \u2264 \u2191(card (sphere n d k))\n[PROOFSTEP]\nobtain rfl | hd := d.eq_zero_or_pos\n[GOAL]\ncase intro.intro.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn k : \u2115\nhn : n > 0\nhk : \u2191(0 ^ n) / (\u2191(n * (0 - 1) ^ 2) + 1) \u2264 \u2191(card (sphere n 0 k))\n\u22a2 \u21910 ^ \u2191n / \u2191(n * 0 ^ 2) \u2264 \u2191(card (sphere n 0 k))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d k : \u2115\nhk : \u2191(d ^ n) / (\u2191(n * (d - 1) ^ 2) + 1) \u2264 \u2191(card (sphere n d k))\nhn : n > 0\nhd : d > 0\n\u22a2 \u2191d ^ \u2191n / \u2191(n * d ^ 2) \u2264 \u2191(card (sphere n d k))\n[PROOFSTEP]\nrw [rpow_nat_cast, \u2190 cast_pow]\n[GOAL]\ncase intro.intro.inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d k : \u2115\nhk : \u2191(d ^ n) / (\u2191(n * (d - 1) ^ 2) + 1) \u2264 \u2191(card (sphere n d k))\nhn : n > 0\nhd : d > 0\n\u22a2 \u2191(d ^ n) / \u2191(n * d ^ 2) \u2264 \u2191(card (sphere n d k))\n[PROOFSTEP]\nrefine' (div_le_div_of_le_left _ _ _).trans hk\n[GOAL]\ncase intro.intro.inr.inr.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d k : \u2115\nhk : \u2191(d ^ n) / (\u2191(n * (d - 1) ^ 2) + 1) \u2264 \u2191(card (sphere n d k))\nhn : n > 0\nhd : d > 0\n\u22a2 0 \u2264 \u2191(d ^ n)\n[PROOFSTEP]\nexact cast_nonneg _\n[GOAL]\ncase intro.intro.inr.inr.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d k : \u2115\nhk : \u2191(d ^ n) / (\u2191(n * (d - 1) ^ 2) + 1) \u2264 \u2191(card (sphere n d k))\nhn : n > 0\nhd : d > 0\n\u22a2 0 < \u2191(n * (d - 1) ^ 2) + 1\n[PROOFSTEP]\nexact cast_add_one_pos _\n[GOAL]\ncase intro.intro.inr.inr.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d k : \u2115\nhk : \u2191(d ^ n) / (\u2191(n * (d - 1) ^ 2) + 1) \u2264 \u2191(card (sphere n d k))\nhn : n > 0\nhd : d > 0\n\u22a2 \u2191(n * (d - 1) ^ 2) + 1 \u2264 \u2191(n * d ^ 2)\n[PROOFSTEP]\nsimp only [\u2190 le_sub_iff_add_le', cast_mul, \u2190 mul_sub, cast_pow, cast_sub hd, sub_sq, one_pow, cast_one, mul_one,\n sub_add, sub_sub_self]\n[GOAL]\ncase intro.intro.inr.inr.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d k : \u2115\nhk : \u2191(d ^ n) / (\u2191(n * (d - 1) ^ 2) + 1) \u2264 \u2191(card (sphere n d k))\nhn : n > 0\nhd : d > 0\n\u22a2 1 \u2264 \u2191n * (2 * \u2191d - 1)\n[PROOFSTEP]\napply one_le_mul_of_one_le_of_one_le\n[GOAL]\ncase intro.intro.inr.inr.refine'_3.ha\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d k : \u2115\nhk : \u2191(d ^ n) / (\u2191(n * (d - 1) ^ 2) + 1) \u2264 \u2191(card (sphere n d k))\nhn : n > 0\nhd : d > 0\n\u22a2 1 \u2264 \u2191n\n[PROOFSTEP]\nrwa [one_le_cast]\n[GOAL]\ncase intro.intro.inr.inr.refine'_3.hb\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d k : \u2115\nhk : \u2191(d ^ n) / (\u2191(n * (d - 1) ^ 2) + 1) \u2264 \u2191(card (sphere n d k))\nhn : n > 0\nhd : d > 0\n\u22a2 1 \u2264 2 * \u2191d - 1\n[PROOFSTEP]\nrw [_root_.le_sub_iff_add_le]\n[GOAL]\ncase intro.intro.inr.inr.refine'_3.hb\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d k : \u2115\nhk : \u2191(d ^ n) / (\u2191(n * (d - 1) ^ 2) + 1) \u2264 \u2191(card (sphere n d k))\nhn : n > 0\nhd : d > 0\n\u22a2 1 + 1 \u2264 2 * \u2191d\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.intro.inr.inr.refine'_3.hb\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d\u271d k\u271d N : \u2115\nx : Fin n\u271d \u2192 \u2115\nn d k : \u2115\nhk : \u2191(d ^ n) / (\u2191(n * (d - 1) ^ 2) + 1) \u2264 \u2191(card (sphere n d k))\nhn : n > 0\nhd : d > 0\n\u22a2 1 \u2264 d\n[PROOFSTEP]\nexact one_le_cast.2 hd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhd : d \u2260 0\nhn : 2 \u2264 n\n\u22a2 \u2191d ^ (\u2191n - 2) / \u2191n \u2264 \u2191(\u2191rothNumberNat ((2 * d - 1) ^ n))\n[PROOFSTEP]\nconvert bound_aux' n d using 1\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhd : d \u2260 0\nhn : 2 \u2264 n\n\u22a2 \u2191d ^ (\u2191n - 2) / \u2191n = \u2191d ^ \u2191n / \u2191(n * d ^ 2)\n[PROOFSTEP]\nrw [cast_mul, cast_pow, mul_comm, \u2190 div_div, \u2190 cast_two, \u2190 cast_sub hn, rpow_nat_cast, rpow_nat_cast,\n pow_sub\u2080 _ (cast_ne_zero.2 hd) hn, \u2190 div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 log 2 * 2 \u2264 Real.sqrt (log 8)\n[PROOFSTEP]\nhave : (8 : \u211d) = 2 ^ ((3 : \u2115) : \u211d) := by rw [rpow_nat_cast]; norm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 8 = 2 ^ \u21913\n[PROOFSTEP]\nrw [rpow_nat_cast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 8 = 2 ^ 3\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nthis : 8 = 2 ^ \u21913\n\u22a2 log 2 * 2 \u2264 Real.sqrt (log 8)\n[PROOFSTEP]\nrw [this, log_rpow zero_lt_two (3 : \u2115)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nthis : 8 = 2 ^ \u21913\n\u22a2 log 2 * 2 \u2264 Real.sqrt (\u21913 * log 2)\n[PROOFSTEP]\napply le_sqrt_of_sq_le\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nthis : 8 = 2 ^ \u21913\n\u22a2 (log 2 * 2) ^ 2 \u2264 \u21913 * log 2\n[PROOFSTEP]\nrw [mul_pow, sq (log 2), mul_assoc, mul_comm]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nthis : 8 = 2 ^ \u21913\n\u22a2 log 2 * 2 ^ 2 * log 2 \u2264 \u21913 * log 2\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_right _ (log_nonneg one_le_two)\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nthis : 8 = 2 ^ \u21913\n\u22a2 log 2 * 2 ^ 2 \u2264 \u21913\n[PROOFSTEP]\nrw [\u2190 le_div_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nthis : 8 = 2 ^ \u21913\n\u22a2 log 2 \u2264 \u21913 / 2 ^ 2\ncase h \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 this : 8 = 2 ^ \u21913 \u22a2 0 < 2 ^ 2\n[PROOFSTEP]\napply log_two_lt_d9.le.trans\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nthis : 8 = 2 ^ \u21913\n\u22a2 0.6931471808 \u2264 \u21913 / 2 ^ 2\ncase h \u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 this : 8 = 2 ^ \u21913 \u22a2 0 < 2 ^ 2\n[PROOFSTEP]\nall_goals norm_num1\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nthis : 8 = 2 ^ \u21913\n\u22a2 0.6931471808 \u2264 \u21913 / 2 ^ 2\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nthis : 8 = 2 ^ \u21913\n\u22a2 0 < 2 ^ 2\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 2 / (1 - 2 / exp 1) \u2264 8\n[PROOFSTEP]\nrw [div_le_iff, mul_sub, mul_one, mul_div_assoc', le_sub_comm, div_le_iff (exp_pos _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 8 * 2 \u2264 (8 - 2) * exp 1\n[PROOFSTEP]\nhave : 16 < 6 * (2.7182818283 : \u211d) := by norm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 16 < 6 * 2.7182818283\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nthis : 16 < 6 * 2.7182818283\n\u22a2 8 * 2 \u2264 (8 - 2) * exp 1\n[PROOFSTEP]\nlinarith [exp_one_gt_d9]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 0 < 1 - 2 / exp 1\n[PROOFSTEP]\nrw [sub_pos, div_lt_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 2 < exp 1\n[PROOFSTEP]\nexact exp_one_gt_d9.trans' (by norm_num)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 2 < 2.7182818283\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 0 < exp 1\n[PROOFSTEP]\nexact exp_one_gt_d9.trans' (by norm_num)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 0 < 2.7182818283\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 log (2 / (1 - 2 / exp 1)) * (69 / 50) \u2264 Real.sqrt (log \u2191N)\n[PROOFSTEP]\nhave : ((12 : \u2115) : \u211d) * log 2 \u2264 log N :=\n by\n rw [\u2190 log_rpow zero_lt_two, log_le_log, rpow_nat_cast]\n \u00b7 norm_num1\n exact_mod_cast hN\n \u00b7 exact rpow_pos_of_pos zero_lt_two _\n rw [cast_pos]\n exact hN.trans_lt' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 \u219112 * log 2 \u2264 log \u2191N\n[PROOFSTEP]\nrw [\u2190 log_rpow zero_lt_two, log_le_log, rpow_nat_cast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 2 ^ 12 \u2264 \u2191N\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 4096 \u2264 \u2191N\n[PROOFSTEP]\nexact_mod_cast hN\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < 2 ^ \u219112\n[PROOFSTEP]\nexact rpow_pos_of_pos zero_lt_two _\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < \u2191N\n[PROOFSTEP]\nrw [cast_pos]\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < N\n[PROOFSTEP]\nexact hN.trans_lt' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < 4096\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 log (2 / (1 - 2 / exp 1)) * (69 / 50) \u2264 Real.sqrt (log \u2191N)\n[PROOFSTEP]\nrefine'\n (mul_le_mul_of_nonneg_right ((log_le_log _ <| by norm_num1).2 two_div_one_sub_two_div_e_le_eight) <| by\n norm_num1).trans\n _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 0 < 8\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 0 \u2264 69 / 50\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 0 < 2 / (1 - 2 / exp 1)\n[PROOFSTEP]\nrefine' div_pos zero_lt_two _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 0 < 1 - 2 / exp 1\n[PROOFSTEP]\nrw [sub_pos, div_lt_one (exp_pos _)]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 2 < exp 1\n[PROOFSTEP]\nexact exp_one_gt_d9.trans_le' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 2 \u2264 2.7182818283\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 log 8 * (69 / 50) \u2264 Real.sqrt (log \u2191N)\n[PROOFSTEP]\nhave l8 : log 8 = (3 : \u2115) * log 2 := by\n rw [\u2190 log_rpow zero_lt_two, rpow_nat_cast]\n norm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 log 8 = \u21913 * log 2\n[PROOFSTEP]\nrw [\u2190 log_rpow zero_lt_two, rpow_nat_cast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 log 8 = log (2 ^ 3)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 log 8 * (69 / 50) \u2264 Real.sqrt (log \u2191N)\n[PROOFSTEP]\nrw [l8]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 \u21913 * log 2 * (69 / 50) \u2264 Real.sqrt (log \u2191N)\n[PROOFSTEP]\napply le_sqrt_of_sq_le (le_trans _ this)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 (\u21913 * log 2 * (69 / 50)) ^ 2 \u2264 \u219112 * log 2\n[PROOFSTEP]\nrw [mul_right_comm, mul_pow, sq (log 2), \u2190 mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 (\u21913 * (69 / 50)) ^ 2 * log 2 * log 2 \u2264 \u219112 * log 2\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_right _ (log_nonneg one_le_two)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 (\u21913 * (69 / 50)) ^ 2 * log 2 \u2264 \u219112\n[PROOFSTEP]\nrw [\u2190 le_div_iff']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 log 2 \u2264 \u219112 / (\u21913 * (69 / 50)) ^ 2\n[PROOFSTEP]\nexact log_two_lt_d9.le.trans (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 0.6931471808 \u2264 \u219112 / (\u21913 * (69 / 50)) ^ 2\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 0 < (\u21913 * (69 / 50)) ^ 2\n[PROOFSTEP]\nexact sq_pos_of_ne_zero _ (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 \u21913 * (69 / 50) \u2260 0\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 0 < x\n\u22a2 exp (-2 * x) < exp (2 - \u2191\u2308x\u2309\u208a) / \u2191\u2308x\u2309\u208a\n[PROOFSTEP]\nhave h\u2081 := ceil_lt_add_one hx.le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 0 < x\nh\u2081 : \u2191\u2308x\u2309\u208a < x + 1\n\u22a2 exp (-2 * x) < exp (2 - \u2191\u2308x\u2309\u208a) / \u2191\u2308x\u2309\u208a\n[PROOFSTEP]\nhave h\u2082 : 1 - x \u2264 2 - \u2308x\u2309\u208a := by\n rw [_root_.le_sub_iff_add_le]\n apply (add_le_add_left h\u2081.le _).trans_eq\n rw [\u2190 add_assoc, sub_add_cancel]\n linarith\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 0 < x\nh\u2081 : \u2191\u2308x\u2309\u208a < x + 1\n\u22a2 1 - x \u2264 2 - \u2191\u2308x\u2309\u208a\n[PROOFSTEP]\nrw [_root_.le_sub_iff_add_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 0 < x\nh\u2081 : \u2191\u2308x\u2309\u208a < x + 1\n\u22a2 1 - x + \u2191\u2308x\u2309\u208a \u2264 2\n[PROOFSTEP]\napply (add_le_add_left h\u2081.le _).trans_eq\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 0 < x\nh\u2081 : \u2191\u2308x\u2309\u208a < x + 1\n\u22a2 1 - x + (x + 1) = 2\n[PROOFSTEP]\nrw [\u2190 add_assoc, sub_add_cancel]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 0 < x\nh\u2081 : \u2191\u2308x\u2309\u208a < x + 1\n\u22a2 1 + 1 = 2\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 0 < x\nh\u2081 : \u2191\u2308x\u2309\u208a < x + 1\nh\u2082 : 1 - x \u2264 2 - \u2191\u2308x\u2309\u208a\n\u22a2 exp (-2 * x) < exp (2 - \u2191\u2308x\u2309\u208a) / \u2191\u2308x\u2309\u208a\n[PROOFSTEP]\nrefine' lt_of_le_of_lt _ (div_lt_div_of_lt_left (exp_pos _) (cast_pos.2 <| ceil_pos.2 hx) h\u2081)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 0 < x\nh\u2081 : \u2191\u2308x\u2309\u208a < x + 1\nh\u2082 : 1 - x \u2264 2 - \u2191\u2308x\u2309\u208a\n\u22a2 exp (-2 * x) \u2264 exp (2 - \u2191\u2308x\u2309\u208a) / (x + 1)\n[PROOFSTEP]\nrefine' le_trans _ (div_le_div_of_le_of_nonneg (exp_le_exp.2 h\u2082) <| add_nonneg hx.le zero_le_one)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 0 < x\nh\u2081 : \u2191\u2308x\u2309\u208a < x + 1\nh\u2082 : 1 - x \u2264 2 - \u2191\u2308x\u2309\u208a\n\u22a2 exp (-2 * x) \u2264 exp (1 - x) / (x + 1)\n[PROOFSTEP]\nrw [le_div_iff (add_pos hx zero_lt_one), \u2190 le_div_iff' (exp_pos _), \u2190 exp_sub, neg_mul, sub_neg_eq_add, two_mul,\n sub_add_add_cancel, add_comm _ x]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 0 < x\nh\u2081 : \u2191\u2308x\u2309\u208a < x + 1\nh\u2082 : 1 - x \u2264 2 - \u2191\u2308x\u2309\u208a\n\u22a2 x + 1 \u2264 exp (x + 1)\n[PROOFSTEP]\nrefine' le_trans _ (add_one_le_exp_of_nonneg <| add_nonneg hx.le zero_le_one)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 0 < x\nh\u2081 : \u2191\u2308x\u2309\u208a < x + 1\nh\u2082 : 1 - x \u2264 2 - \u2191\u2308x\u2309\u208a\n\u22a2 x + 1 \u2264 x + 1 + 1\n[PROOFSTEP]\nexact le_add_of_nonneg_right zero_le_one\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 2 / (1 - 2 / exp 1) \u2264 x\n\u22a2 x / exp 1 < \u2191\u230ax / 2\u230b\u208a\n[PROOFSTEP]\napply lt_of_le_of_lt _ (sub_one_lt_floor _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 2 / (1 - 2 / exp 1) \u2264 x\n\u22a2 x / exp 1 \u2264 x / 2 - 1\n[PROOFSTEP]\nhave : 0 < 1 - 2 / exp 1 := by\n rw [sub_pos, div_lt_one (exp_pos _)]\n exact lt_of_le_of_lt (by norm_num) exp_one_gt_d9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 2 / (1 - 2 / exp 1) \u2264 x\n\u22a2 0 < 1 - 2 / exp 1\n[PROOFSTEP]\nrw [sub_pos, div_lt_one (exp_pos _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 2 / (1 - 2 / exp 1) \u2264 x\n\u22a2 2 < exp 1\n[PROOFSTEP]\nexact lt_of_le_of_lt (by norm_num) exp_one_gt_d9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 2 / (1 - 2 / exp 1) \u2264 x\n\u22a2 2 \u2264 2.7182818283\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 2 / (1 - 2 / exp 1) \u2264 x\nthis : 0 < 1 - 2 / exp 1\n\u22a2 x / exp 1 \u2264 x / 2 - 1\n[PROOFSTEP]\nrwa [le_sub_comm, div_eq_mul_one_div x, div_eq_mul_one_div x, \u2190 mul_sub, div_sub', \u2190 div_eq_mul_one_div, mul_div_assoc',\n one_le_div, \u2190 div_le_iff this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 2 / (1 - 2 / exp 1) \u2264 x\nthis : 0 < 1 - 2 / exp 1\n\u22a2 0 < 2\n[PROOFSTEP]\nexact zero_lt_two\n[GOAL]\ncase hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 2 / (1 - 2 / exp 1) \u2264 x\nthis : 0 < 1 - 2 / exp 1\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nexact two_ne_zero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 50 / 19 \u2264 x\n\u22a2 \u2191\u2308x\u2309\u208a < 1.38 * x\n[PROOFSTEP]\nrefine' (ceil_lt_add_one <| hx.trans' <| by norm_num).trans_le _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 50 / 19 \u2264 x\n\u22a2 0 \u2264 50 / 19\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 50 / 19 \u2264 x\n\u22a2 x + 1 \u2264 1.38 * x\n[PROOFSTEP]\nrw [\u2190 le_sub_iff_add_le', \u2190 sub_one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 50 / 19 \u2264 x\n\u22a2 1 \u2264 (1.38 - 1) * x\n[PROOFSTEP]\nhave : (1.38 : \u211d) = 69 / 50 := by norm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 50 / 19 \u2264 x\n\u22a2 1.38 = 69 / 50\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 50 / 19 \u2264 x\nthis : 1.38 = 69 / 50\n\u22a2 1 \u2264 (1.38 - 1) * x\n[PROOFSTEP]\nrwa [this, show (69 / 50 - 1 : \u211d) = (50 / 19)\u207b\u00b9 by norm_num1, \u2190 div_eq_inv_mul, one_le_div]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 50 / 19 \u2264 x\nthis : 1.38 = 69 / 50\n\u22a2 69 / 50 - 1 = (50 / 19)\u207b\u00b9\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx\u271d : Fin n \u2192 \u2115\nx : \u211d\nhx : 50 / 19 \u2264 x\nthis : 1.38 = 69 / 50\n\u22a2 0 < 50 / 19\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 3 \u2264 N\n\u22a2 2 \u2264 nValue N\n[PROOFSTEP]\nrefine' succ_le_of_lt (lt_ceil.2 <| lt_sqrt_of_sq_lt _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 3 \u2264 N\n\u22a2 \u21911 ^ 2 < log \u2191N\n[PROOFSTEP]\nrw [cast_one, one_pow, lt_log_iff_exp_lt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 3 \u2264 N\n\u22a2 exp 1 < \u2191N\n\u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 3 \u2264 N \u22a2 0 < \u2191N\n[PROOFSTEP]\nrefine' lt_of_lt_of_le _ (cast_le.2 hN)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 3 \u2264 N\n\u22a2 exp 1 < \u21913\n[PROOFSTEP]\nexact exp_one_lt_d9.trans_le (by norm_num)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 3 \u2264 N\n\u22a2 2.7182818286 \u2264 \u21913\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 3 \u2264 N\n\u22a2 0 < \u2191N\n[PROOFSTEP]\nrw [cast_pos]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 3 \u2264 N\n\u22a2 0 < N\n[PROOFSTEP]\nexact (zero_lt_succ _).trans_le hN\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 64 \u2264 N\n\u22a2 3 \u2264 nValue N\n[PROOFSTEP]\nrw [nValue, \u2190 lt_iff_add_one_le, lt_ceil, cast_two]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 64 \u2264 N\n\u22a2 2 < Real.sqrt (log \u2191N)\n[PROOFSTEP]\napply lt_sqrt_of_sq_lt\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 64 \u2264 N\n\u22a2 2 ^ 2 < log \u2191N\n[PROOFSTEP]\nhave : (2 : \u211d) ^ ((6 : \u2115) : \u211d) \u2264 N := by\n rw [rpow_nat_cast]\n exact (cast_le.2 hN).trans' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 64 \u2264 N\n\u22a2 2 ^ \u21916 \u2264 \u2191N\n[PROOFSTEP]\nrw [rpow_nat_cast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 64 \u2264 N\n\u22a2 2 ^ 6 \u2264 \u2191N\n[PROOFSTEP]\nexact (cast_le.2 hN).trans' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 64 \u2264 N\n\u22a2 2 ^ 6 \u2264 \u219164\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 64 \u2264 N\nthis : 2 ^ \u21916 \u2264 \u2191N\n\u22a2 2 ^ 2 < log \u2191N\n[PROOFSTEP]\napply lt_of_lt_of_le _ ((log_le_log (rpow_pos_of_pos zero_lt_two _) _).2 this)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 64 \u2264 N\nthis : 2 ^ \u21916 \u2264 \u2191N\n\u22a2 2 ^ 2 < log (2 ^ \u21916)\n\u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 64 \u2264 N this : 2 ^ \u21916 \u2264 \u2191N \u22a2 0 < \u2191N\n[PROOFSTEP]\nrw [log_rpow zero_lt_two, \u2190 div_lt_iff']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 64 \u2264 N\nthis : 2 ^ \u21916 \u2264 \u2191N\n\u22a2 2 ^ 2 / \u21916 < log 2\n[PROOFSTEP]\nexact log_two_gt_d9.trans_le' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 64 \u2264 N\nthis : 2 ^ \u21916 \u2264 \u2191N\n\u22a2 2 ^ 2 / \u21916 \u2264 0.6931471803\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 64 \u2264 N\nthis : 2 ^ \u21916 \u2264 \u2191N\n\u22a2 0 < \u21916\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 64 \u2264 N\nthis : 2 ^ \u21916 \u2264 \u2191N\n\u22a2 0 < \u2191N\n[PROOFSTEP]\nrw [cast_pos]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 64 \u2264 N\nthis : 2 ^ \u21916 \u2264 \u2191N\n\u22a2 0 < N\n[PROOFSTEP]\nexact hN.trans_lt' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 64 \u2264 N\nthis : 2 ^ \u21916 \u2264 \u2191N\n\u22a2 0 < 64\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\n\u22a2 0 < dValue N\n[PROOFSTEP]\nhave hN\u2080 : 0 < (N : \u211d) := cast_pos.2 (succ_pos'.trans_le hN\u2083)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 0 < dValue N\n[PROOFSTEP]\nrw [dValue, floor_pos, \u2190 log_le_log zero_lt_one, log_one, log_div _ two_ne_zero, log_rpow hN\u2080, div_mul_eq_mul_div,\n one_mul, sub_nonneg, le_div_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 log 2 * \u2191(nValue N) \u2264 log \u2191N\n[PROOFSTEP]\nhave : (nValue N : \u211d) \u2264 2 * sqrt (log N) :=\n by\n apply (ceil_lt_add_one <| sqrt_nonneg _).le.trans\n rw [two_mul, add_le_add_iff_left]\n apply le_sqrt_of_sq_le\n rw [one_pow, le_log_iff_exp_le hN\u2080]\n exact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN\u2083)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n[PROOFSTEP]\napply (ceil_lt_add_one <| sqrt_nonneg _).le.trans\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 Real.sqrt (log \u2191N) + 1 \u2264 2 * Real.sqrt (log \u2191N)\n[PROOFSTEP]\nrw [two_mul, add_le_add_iff_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 1 \u2264 Real.sqrt (log \u2191N)\n[PROOFSTEP]\napply le_sqrt_of_sq_le\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 1 ^ 2 \u2264 log \u2191N\n[PROOFSTEP]\nrw [one_pow, le_log_iff_exp_le hN\u2080]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 exp 1 \u2264 \u2191N\n[PROOFSTEP]\nexact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN\u2083)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 2.7182818286 \u2264 \u21918\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 log 2 * \u2191(nValue N) \u2264 log \u2191N\n[PROOFSTEP]\napply (mul_le_mul_of_nonneg_left this <| log_nonneg one_le_two).trans _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 log 2 * (2 * Real.sqrt (log \u2191N)) \u2264 log \u2191N\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 le_div_iff (Real.sqrt_pos.2 <| log_pos <| one_lt_cast.2 _), div_sqrt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 log 2 * 2 \u2264 Real.sqrt (log \u2191N)\n[PROOFSTEP]\napply log_two_mul_two_le_sqrt_log_eight.trans\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 Real.sqrt (log 8) \u2264 Real.sqrt (log \u2191N)\n[PROOFSTEP]\napply Real.sqrt_le_sqrt\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 log 8 \u2264 log \u2191N\n[PROOFSTEP]\nrw [log_le_log _ hN\u2080]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 8 \u2264 \u2191N\n[PROOFSTEP]\nexact_mod_cast hN\u2083\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 0 < 8\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 1 < N\n[PROOFSTEP]\nexact hN\u2083.trans_lt' (by norm_num)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 1 < 8\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 0 < \u2191(nValue N)\n[PROOFSTEP]\nexact cast_pos.2 (nValue_pos <| hN\u2083.trans' <| by norm_num)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 2 \u2264 8\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 \u2191N ^ (1 / \u2191(nValue N)) \u2260 0\n[PROOFSTEP]\nexact (rpow_pos_of_pos hN\u2080 _).ne'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 0 < \u2191N ^ (1 / \u2191(nValue N)) / 2\n[PROOFSTEP]\nexact div_pos (rpow_pos_of_pos hN\u2080 _) zero_lt_two\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\n\u22a2 (2 * dValue N - 1) ^ nValue N \u2264 N\n[PROOFSTEP]\nhave : (2 * dValue N - 1) ^ nValue N \u2264 (2 * dValue N) ^ nValue N := Nat.pow_le_pow_of_le_left (Nat.sub_le _ _) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\nthis : (2 * dValue N - 1) ^ nValue N \u2264 (2 * dValue N) ^ nValue N\n\u22a2 (2 * dValue N - 1) ^ nValue N \u2264 N\n[PROOFSTEP]\napply this.trans\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\nthis : (2 * dValue N - 1) ^ nValue N \u2264 (2 * dValue N) ^ nValue N\n\u22a2 (2 * dValue N) ^ nValue N \u2264 N\n[PROOFSTEP]\nsuffices ((2 * dValue N) ^ nValue N : \u211d) \u2264 N by exact_mod_cast this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\nthis\u271d : (2 * dValue N - 1) ^ nValue N \u2264 (2 * dValue N) ^ nValue N\nthis : (2 * \u2191(dValue N)) ^ \u2191(nValue N) \u2264 \u2191N\n\u22a2 (2 * dValue N) ^ nValue N \u2264 N\n[PROOFSTEP]\nexact_mod_cast this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\nthis : (2 * dValue N - 1) ^ nValue N \u2264 (2 * dValue N) ^ nValue N\n\u22a2 (2 * \u2191(dValue N)) ^ \u2191(nValue N) \u2264 \u2191N\n[PROOFSTEP]\nsuffices i : (2 * dValue N : \u211d) \u2264 (N : \u211d) ^ (1 / nValue N : \u211d)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\nthis : (2 * dValue N - 1) ^ nValue N \u2264 (2 * dValue N) ^ nValue N\ni : 2 * \u2191(dValue N) \u2264 \u2191N ^ (1 / \u2191(nValue N))\n\u22a2 (2 * \u2191(dValue N)) ^ \u2191(nValue N) \u2264 \u2191N\n[PROOFSTEP]\napply (rpow_le_rpow (mul_nonneg zero_le_two (cast_nonneg _)) i (cast_nonneg _)).trans\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\nthis : (2 * dValue N - 1) ^ nValue N \u2264 (2 * dValue N) ^ nValue N\ni : 2 * \u2191(dValue N) \u2264 \u2191N ^ (1 / \u2191(nValue N))\n\u22a2 (\u2191N ^ (1 / \u2191(nValue N))) ^ \u2191(nValue N) \u2264 \u2191N\n[PROOFSTEP]\nrw [\u2190 rpow_mul (cast_nonneg _), one_div_mul_cancel, rpow_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\nthis : (2 * dValue N - 1) ^ nValue N \u2264 (2 * dValue N) ^ nValue N\ni : 2 * \u2191(dValue N) \u2264 \u2191N ^ (1 / \u2191(nValue N))\n\u22a2 \u2191(nValue N) \u2260 0\n[PROOFSTEP]\nrw [cast_ne_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\nthis : (2 * dValue N - 1) ^ nValue N \u2264 (2 * dValue N) ^ nValue N\ni : 2 * \u2191(dValue N) \u2264 \u2191N ^ (1 / \u2191(nValue N))\n\u22a2 nValue N \u2260 0\n[PROOFSTEP]\napply (nValue_pos hN).ne'\n[GOAL]\ncase i\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\nthis : (2 * dValue N - 1) ^ nValue N \u2264 (2 * dValue N) ^ nValue N\n\u22a2 2 * \u2191(dValue N) \u2264 \u2191N ^ (1 / \u2191(nValue N))\n[PROOFSTEP]\nrw [\u2190 le_div_iff']\n[GOAL]\ncase i\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\nthis : (2 * dValue N - 1) ^ nValue N \u2264 (2 * dValue N) ^ nValue N\n\u22a2 \u2191(dValue N) \u2264 \u2191N ^ (1 / \u2191(nValue N)) / 2\n[PROOFSTEP]\nexact floor_le (div_nonneg (rpow_nonneg_of_nonneg (cast_nonneg _) _) zero_le_two)\n[GOAL]\ncase i\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\nthis : (2 * dValue N - 1) ^ nValue N \u2264 (2 * dValue N) ^ nValue N\n\u22a2 0 < 2\n[PROOFSTEP]\napply zero_lt_two\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 \u2191N ^ (1 / \u2191(nValue N)) / exp 1 < \u2191(dValue N)\n[PROOFSTEP]\napply div_lt_floor _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 2 / (1 - 2 / exp 1) \u2264 \u2191N ^ (1 / \u2191(nValue N))\n[PROOFSTEP]\nrw [\u2190 log_le_log, log_rpow, mul_comm, \u2190 div_eq_mul_one_div]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 log (2 / (1 - 2 / exp 1)) \u2264 log \u2191N / \u2191(nValue N)\n[PROOFSTEP]\napply le_trans _ (div_le_div_of_le_left _ _ (ceil_lt_mul _).le)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 log (2 / (1 - 2 / exp 1)) \u2264 log \u2191N / (1.38 * Real.sqrt (log \u2191N))\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 \u2264 log \u2191N\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < \u2191\u2308Real.sqrt (log \u2191N)\u2309\u208a\n\u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 hN : 4096 \u2264 N \u22a2 50 / 19 \u2264 Real.sqrt (log \u2191N)\n[PROOFSTEP]\nrw [mul_comm, \u2190 div_div, div_sqrt, le_div_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 log (2 / (1 - 2 / exp 1)) * 1.38 \u2264 Real.sqrt (log \u2191N)\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 log (2 / (1 - 2 / exp 1)) * (69 / 50) \u2264 Real.sqrt (log \u2191N)\n[PROOFSTEP]\nexact le_sqrt_log hN\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < 1.38\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 \u2264 log \u2191N\n[PROOFSTEP]\napply log_nonneg\n[GOAL]\ncase hx\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 1 \u2264 \u2191N\n[PROOFSTEP]\nrw [one_le_cast]\n[GOAL]\ncase hx\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 1 \u2264 N\n[PROOFSTEP]\nexact hN.trans' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 1 \u2264 4096\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < \u2191\u2308Real.sqrt (log \u2191N)\u2309\u208a\n[PROOFSTEP]\nrw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < log \u2191N\n[PROOFSTEP]\nrefine' log_pos _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 1 < \u2191N\n[PROOFSTEP]\nrw [one_lt_cast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 1 < N\n[PROOFSTEP]\nexact hN.trans_lt' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 1 < 4096\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 50 / 19 \u2264 Real.sqrt (log \u2191N)\n[PROOFSTEP]\napply le_sqrt_of_sq_le\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 (50 / 19) ^ 2 \u2264 log \u2191N\n[PROOFSTEP]\nhave : ((12 : \u2115) : \u211d) * log 2 \u2264 log N :=\n by\n rw [\u2190 log_rpow zero_lt_two, log_le_log, rpow_nat_cast]\n \u00b7 norm_num1\n exact_mod_cast hN\n \u00b7 exact rpow_pos_of_pos zero_lt_two _\n rw [cast_pos]\n exact hN.trans_lt' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 \u219112 * log 2 \u2264 log \u2191N\n[PROOFSTEP]\nrw [\u2190 log_rpow zero_lt_two, log_le_log, rpow_nat_cast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 2 ^ 12 \u2264 \u2191N\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 4096 \u2264 \u2191N\n[PROOFSTEP]\nexact_mod_cast hN\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < 2 ^ \u219112\n[PROOFSTEP]\nexact rpow_pos_of_pos zero_lt_two _\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < \u2191N\n[PROOFSTEP]\nrw [cast_pos]\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < N\n[PROOFSTEP]\nexact hN.trans_lt' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < 4096\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 (50 / 19) ^ 2 \u2264 log \u2191N\n[PROOFSTEP]\nrefine' le_trans _ this\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 (50 / 19) ^ 2 \u2264 \u219112 * log 2\n[PROOFSTEP]\nrw [\u2190 div_le_iff']\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 (50 / 19) ^ 2 / \u219112 \u2264 log 2\n[PROOFSTEP]\nexact log_two_gt_d9.le.trans' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 (50 / 19) ^ 2 / \u219112 \u2264 0.6931471803\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 0 < \u219112\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase hx\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < \u2191N\n[PROOFSTEP]\nrw [cast_pos]\n[GOAL]\ncase hx\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < N\n[PROOFSTEP]\nexact hN.trans_lt' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < 4096\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < 2 / (1 - 2 / exp 1)\n[PROOFSTEP]\nrefine' div_pos zero_lt_two _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < 1 - 2 / exp 1\n[PROOFSTEP]\nrw [sub_pos, div_lt_one (exp_pos _)]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 2 < exp 1\n[PROOFSTEP]\nexact lt_of_le_of_lt (by norm_num1) exp_one_gt_d9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 2 \u2264 2.7182818283\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < \u2191N ^ (1 / \u2191(nValue N))\n[PROOFSTEP]\napply rpow_pos_of_pos\n[GOAL]\ncase h\u2081.hx\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < \u2191N\n[PROOFSTEP]\nrw [cast_pos]\n[GOAL]\ncase h\u2081.hx\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < N\n[PROOFSTEP]\nexact hN.trans_lt' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < 4096\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) < \u2191(\u2191rothNumberNat N)\n[PROOFSTEP]\nlet n := nValue N\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) < \u2191(\u2191rothNumberNat N)\n[PROOFSTEP]\nhave hn : 0 < (n : \u211d) := cast_pos.2 (nValue_pos <| hN.trans' <| by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\n\u22a2 2 \u2264 4096\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) < \u2191(\u2191rothNumberNat N)\n[PROOFSTEP]\nhave hd : 0 < dValue N := dValue_pos (hN.trans' <| by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\n\u22a2 8 \u2264 4096\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) < \u2191(\u2191rothNumberNat N)\n[PROOFSTEP]\nhave hN\u2080 : 0 < (N : \u211d) := cast_pos.2 (hN.trans' <| by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\n\u22a2 succ 0 \u2264 4096\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) < \u2191(\u2191rothNumberNat N)\n[PROOFSTEP]\nhave hn\u2082 : 2 \u2264 n := two_le_nValue (hN.trans' <| by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\n\u22a2 3 \u2264 4096\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) < \u2191(\u2191rothNumberNat N)\n[PROOFSTEP]\nhave : (2 * dValue N - 1) ^ n \u2264 N := le_N (hN.trans' <| by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\n\u22a2 2 \u2264 4096\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis : (2 * dValue N - 1) ^ n \u2264 N\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) < \u2191(\u2191rothNumberNat N)\n[PROOFSTEP]\nrefine' ((bound_aux hd.ne' hn\u2082).trans <| cast_le.2 <| rothNumberNat.mono this).trans_lt' _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis : (2 * dValue N - 1) ^ n \u2264 N\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) < \u2191(dValue N) ^ (\u2191n - 2) / \u2191n\n[PROOFSTEP]\nconv_rhs => rw [\u2190 cast_two, \u2190 cast_sub hn\u2082, rpow_nat_cast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis : (2 * dValue N - 1) ^ n \u2264 N\n| \u2191(dValue N) ^ (\u2191n - 2) / \u2191n\n[PROOFSTEP]\nrw [\u2190 cast_two, \u2190 cast_sub hn\u2082, rpow_nat_cast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis : (2 * dValue N - 1) ^ n \u2264 N\n| \u2191(dValue N) ^ (\u2191n - 2) / \u2191n\n[PROOFSTEP]\nrw [\u2190 cast_two, \u2190 cast_sub hn\u2082, rpow_nat_cast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis : (2 * dValue N - 1) ^ n \u2264 N\n| \u2191(dValue N) ^ (\u2191n - 2) / \u2191n\n[PROOFSTEP]\nrw [\u2190 cast_two, \u2190 cast_sub hn\u2082, rpow_nat_cast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis : (2 * dValue N - 1) ^ n \u2264 N\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) < \u2191(dValue N) ^ (n - 2) / \u2191n\n[PROOFSTEP]\nrefine' (div_lt_div_of_lt hn <| pow_lt_pow_of_lt_left (bound hN) _ _).trans_le' _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis : (2 * dValue N - 1) ^ n \u2264 N\n\u22a2 0 \u2264 \u2191N ^ (1 / \u2191(nValue N)) / exp 1\n[PROOFSTEP]\nexact div_nonneg (rpow_nonneg_of_nonneg (cast_nonneg _) _) (exp_pos _).le\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis : (2 * dValue N - 1) ^ n \u2264 N\n\u22a2 0 < n - 2\n[PROOFSTEP]\nexact tsub_pos_of_lt (three_le_nValue <| hN.trans' <| by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis : (2 * dValue N - 1) ^ n \u2264 N\n\u22a2 64 \u2264 4096\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis : (2 * dValue N - 1) ^ n \u2264 N\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) \u2264 (\u2191N ^ (1 / \u2191(nValue N)) / exp 1) ^ (n - 2) / \u2191n\n[PROOFSTEP]\nrw [\u2190 rpow_nat_cast, div_rpow (rpow_nonneg_of_nonneg hN\u2080.le _) (exp_pos _).le, \u2190 rpow_mul hN\u2080.le, mul_comm (_ / _),\n mul_one_div, cast_sub hn\u2082, cast_two, same_sub_div hn.ne', exp_one_rpow, div_div, rpow_sub hN\u2080, rpow_one, div_div,\n div_eq_mul_inv]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis : (2 * dValue N - 1) ^ n \u2264 N\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) \u2264 \u2191N * (\u2191N ^ (2 / \u2191n) * (exp (\u2191n - 2) * \u2191n))\u207b\u00b9\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left _ (cast_nonneg _)\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis : (2 * dValue N - 1) ^ n \u2264 N\n\u22a2 exp (-4 * Real.sqrt (log \u2191N)) \u2264 (\u2191N ^ (2 / \u2191n) * (exp (\u2191n - 2) * \u2191n))\u207b\u00b9\n[PROOFSTEP]\nrw [mul_inv, mul_inv, \u2190 exp_neg, \u2190 rpow_neg (cast_nonneg _), neg_sub, \u2190 div_eq_mul_inv]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis : (2 * dValue N - 1) ^ n \u2264 N\n\u22a2 exp (-4 * Real.sqrt (log \u2191N)) \u2264 \u2191N ^ (-(2 / \u2191n)) * (exp (2 - \u2191n) / \u2191n)\n[PROOFSTEP]\nhave : exp (-4 * sqrt (log N)) = exp (-2 * sqrt (log N)) * exp (-2 * sqrt (log N)) :=\n by\n rw [\u2190 exp_add, \u2190 add_mul]\n norm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis : (2 * dValue N - 1) ^ n \u2264 N\n\u22a2 exp (-4 * Real.sqrt (log \u2191N)) = exp (-2 * Real.sqrt (log \u2191N)) * exp (-2 * Real.sqrt (log \u2191N))\n[PROOFSTEP]\nrw [\u2190 exp_add, \u2190 add_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis : (2 * dValue N - 1) ^ n \u2264 N\n\u22a2 exp (-4 * Real.sqrt (log \u2191N)) = exp ((-2 + -2) * Real.sqrt (log \u2191N))\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis\u271d : (2 * dValue N - 1) ^ n \u2264 N\nthis : exp (-4 * Real.sqrt (log \u2191N)) = exp (-2 * Real.sqrt (log \u2191N)) * exp (-2 * Real.sqrt (log \u2191N))\n\u22a2 exp (-4 * Real.sqrt (log \u2191N)) \u2264 \u2191N ^ (-(2 / \u2191n)) * (exp (2 - \u2191n) / \u2191n)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis\u271d : (2 * dValue N - 1) ^ n \u2264 N\nthis : exp (-4 * Real.sqrt (log \u2191N)) = exp (-2 * Real.sqrt (log \u2191N)) * exp (-2 * Real.sqrt (log \u2191N))\n\u22a2 exp (-2 * Real.sqrt (log \u2191N)) * exp (-2 * Real.sqrt (log \u2191N)) \u2264 \u2191N ^ (-(2 / \u2191n)) * (exp (2 - \u2191n) / \u2191n)\n[PROOFSTEP]\nrefine'\n mul_le_mul _ (exp_neg_two_mul_le <| Real.sqrt_pos.2 <| log_pos _).le (exp_pos _).le <|\n rpow_nonneg_of_nonneg (cast_nonneg _) _\n[GOAL]\ncase refine'_3.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis\u271d : (2 * dValue N - 1) ^ n \u2264 N\nthis : exp (-4 * Real.sqrt (log \u2191N)) = exp (-2 * Real.sqrt (log \u2191N)) * exp (-2 * Real.sqrt (log \u2191N))\n\u22a2 exp (-2 * Real.sqrt (log \u2191N)) \u2264 \u2191N ^ (-(2 / \u2191n))\n[PROOFSTEP]\nrw [\u2190 le_log_iff_exp_le (rpow_pos_of_pos hN\u2080 _), log_rpow hN\u2080, \u2190 le_div_iff, mul_div_assoc, div_sqrt, neg_mul,\n neg_le_neg_iff, div_mul_eq_mul_div, div_le_iff hn]\n[GOAL]\ncase refine'_3.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis\u271d : (2 * dValue N - 1) ^ n \u2264 N\nthis : exp (-4 * Real.sqrt (log \u2191N)) = exp (-2 * Real.sqrt (log \u2191N)) * exp (-2 * Real.sqrt (log \u2191N))\n\u22a2 2 * Real.sqrt (log \u2191N) \u2264 2 * \u2191n\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_left (le_ceil _) zero_le_two\n[GOAL]\ncase refine'_3.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis\u271d : (2 * dValue N - 1) ^ n \u2264 N\nthis : exp (-4 * Real.sqrt (log \u2191N)) = exp (-2 * Real.sqrt (log \u2191N)) * exp (-2 * Real.sqrt (log \u2191N))\n\u22a2 0 < Real.sqrt (log \u2191N)\n[PROOFSTEP]\nrefine' Real.sqrt_pos.2 (log_pos _)\n[GOAL]\ncase refine'_3.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis\u271d : (2 * dValue N - 1) ^ n \u2264 N\nthis : exp (-4 * Real.sqrt (log \u2191N)) = exp (-2 * Real.sqrt (log \u2191N)) * exp (-2 * Real.sqrt (log \u2191N))\n\u22a2 1 < \u2191N\n[PROOFSTEP]\nrw [one_lt_cast]\n[GOAL]\ncase refine'_3.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis\u271d : (2 * dValue N - 1) ^ n \u2264 N\nthis : exp (-4 * Real.sqrt (log \u2191N)) = exp (-2 * Real.sqrt (log \u2191N)) * exp (-2 * Real.sqrt (log \u2191N))\n\u22a2 1 < N\n[PROOFSTEP]\nexact hN.trans_lt' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis\u271d : (2 * dValue N - 1) ^ n \u2264 N\nthis : exp (-4 * Real.sqrt (log \u2191N)) = exp (-2 * Real.sqrt (log \u2191N)) * exp (-2 * Real.sqrt (log \u2191N))\n\u22a2 1 < 4096\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_3.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis\u271d : (2 * dValue N - 1) ^ n \u2264 N\nthis : exp (-4 * Real.sqrt (log \u2191N)) = exp (-2 * Real.sqrt (log \u2191N)) * exp (-2 * Real.sqrt (log \u2191N))\n\u22a2 1 < \u2191N\n[PROOFSTEP]\nrw [one_lt_cast]\n[GOAL]\ncase refine'_3.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis\u271d : (2 * dValue N - 1) ^ n \u2264 N\nthis : exp (-4 * Real.sqrt (log \u2191N)) = exp (-2 * Real.sqrt (log \u2191N)) * exp (-2 * Real.sqrt (log \u2191N))\n\u22a2 1 < N\n[PROOFSTEP]\nexact hN.trans_lt' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d d k N : \u2115\nx : Fin n\u271d \u2192 \u2115\nhN : 4096 \u2264 N\nn : \u2115 := nValue N\nhn : 0 < \u2191n\nhd : 0 < dValue N\nhN\u2080 : 0 < \u2191N\nhn\u2082 : 2 \u2264 n\nthis\u271d : (2 * dValue N - 1) ^ n \u2264 N\nthis : exp (-4 * Real.sqrt (log \u2191N)) = exp (-2 * Real.sqrt (log \u2191N)) * exp (-2 * Real.sqrt (log \u2191N))\n\u22a2 1 < 4096\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 exp 4 < 64\n[PROOFSTEP]\nrw [show (64 : \u211d) = 2 ^ ((6 : \u2115) : \u211d) by rw [rpow_nat_cast]; norm_num1, \u2190\n lt_log_iff_exp_lt (rpow_pos_of_pos zero_lt_two _), log_rpow zero_lt_two, \u2190 div_lt_iff']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 64 = 2 ^ \u21916\n[PROOFSTEP]\nrw [rpow_nat_cast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 64 = 2 ^ 6\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 4 / \u21916 < log 2\n\u03b1 : Type u_1 \u03b2 : Type u_2 n d k N : \u2115 x : Fin n \u2192 \u2115 \u22a2 0 < \u21916\n[PROOFSTEP]\nexact log_two_gt_d9.trans_le' (by norm_num1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 4 / \u21916 \u2264 0.6931471803\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 0 < \u21916\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 4096 < exp 16\n[PROOFSTEP]\nrw [\u2190 log_lt_iff_lt_exp (show (0 : \u211d) < 4096 by norm_num), show (4096 : \u211d) = 2 ^ 12 by norm_cast, log_rpow zero_lt_two]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 0 < 4096\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 4096 = 2 ^ 12\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 12 * log 2 < 16\n[PROOFSTEP]\nhave : 12 * (0.6931471808 : \u211d) < 16 := by norm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 12 * 0.6931471808 < 16\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nthis : 12 * 0.6931471808 < 16\n\u22a2 12 * log 2 < 16\n[PROOFSTEP]\nlinarith [log_two_lt_d9]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\nhN' : N \u2264 4096\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) \u2264 1\n[PROOFSTEP]\nrw [\u2190 log_le_log (mul_pos (cast_pos.2 (zero_lt_two.trans_le hN)) (exp_pos _)) zero_lt_one, log_one,\n log_mul (cast_pos.2 (zero_lt_two.trans_le hN)).ne' (exp_pos _).ne', log_exp, neg_mul, \u2190 sub_eq_add_neg, sub_nonpos, \u2190\n div_le_iff (Real.sqrt_pos.2 <| log_pos <| one_lt_cast.2 <| one_lt_two.trans_le hN), div_sqrt,\n sqrt_le_left zero_le_four, log_le_iff_le_exp (cast_pos.2 (zero_lt_two.trans_le hN))]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\nhN' : N \u2264 4096\n\u22a2 \u2191N \u2264 exp (4 ^ 2)\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\nhN' : N \u2264 4096\n\u22a2 \u2191N \u2264 exp 16\n[PROOFSTEP]\napply le_trans _ four_zero_nine_six_lt_exp_sixteen.le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 2 \u2264 N\nhN' : N \u2264 4096\n\u22a2 \u2191N \u2264 4096\n[PROOFSTEP]\nexact_mod_cast hN'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 1 \u2264 N\nhN' : N \u2264 4096\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) \u2264 1\n[PROOFSTEP]\nobtain rfl | hN := hN.eq_or_lt\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k : \u2115\nx : Fin n \u2192 \u2115\nhN : 1 \u2264 1\nhN' : 1 \u2264 4096\n\u22a2 \u21911 * exp (-4 * Real.sqrt (log \u21911)) \u2264 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u271d : 1 \u2264 N\nhN' : N \u2264 4096\nhN : 1 < N\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) \u2264 1\n[PROOFSTEP]\nexact lower_bound_le_one' hN hN'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) \u2264 \u2191(\u2191rothNumberNat N)\n[PROOFSTEP]\nobtain rfl | hN := Nat.eq_zero_or_pos N\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k : \u2115\nx : Fin n \u2192 \u2115\n\u22a2 \u21910 * exp (-4 * Real.sqrt (log \u21910)) \u2264 \u2191(\u2191rothNumberNat 0)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : N > 0\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) \u2264 \u2191(\u2191rothNumberNat N)\n[PROOFSTEP]\nobtain h\u2081 | h\u2081 := le_or_lt 4096 N\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : N > 0\nh\u2081 : 4096 \u2264 N\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) \u2264 \u2191(\u2191rothNumberNat N)\n[PROOFSTEP]\nexact (roth_lower_bound_explicit h\u2081).le\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : N > 0\nh\u2081 : N < 4096\n\u22a2 \u2191N * exp (-4 * Real.sqrt (log \u2191N)) \u2264 \u2191(\u2191rothNumberNat N)\n[PROOFSTEP]\napply (lower_bound_le_one hN h\u2081.le).trans\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : N > 0\nh\u2081 : N < 4096\n\u22a2 1 \u2264 \u2191(\u2191rothNumberNat N)\n[PROOFSTEP]\nsimpa using rothNumberNat.monotone hN\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.Additive.Behrend", "llama_tokens": 43258, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.4026349255062284}} {"text": "[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\nhf : Function.Surjective \u2191f\n\u22a2 index (comap f H) = index H\n[PROOFSTEP]\nletI := QuotientGroup.leftRel H\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\nhf : Function.Surjective \u2191f\nthis : Setoid G := QuotientGroup.leftRel H\n\u22a2 index (comap f H) = index H\n[PROOFSTEP]\nletI := QuotientGroup.leftRel (H.comap f)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\nhf : Function.Surjective \u2191f\nthis\u271d : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\n\u22a2 index (comap f H) = index H\n[PROOFSTEP]\nhave key : \u2200 x y : G', Setoid.r x y \u2194 Setoid.r (f x) (f y) :=\n by\n simp only [QuotientGroup.leftRel_apply]\n exact fun x y => iff_of_eq (congr_arg (\u00b7 \u2208 H) (by rw [f.map_mul, f.map_inv]))\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\nhf : Function.Surjective \u2191f\nthis\u271d : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\n\u22a2 \u2200 (x y : G'), Setoid.r x y \u2194 Setoid.r (\u2191f x) (\u2191f y)\n[PROOFSTEP]\nsimp only [QuotientGroup.leftRel_apply]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\nhf : Function.Surjective \u2191f\nthis\u271d : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\n\u22a2 \u2200 (x y : G'), x\u207b\u00b9 * y \u2208 comap f H \u2194 (\u2191f x)\u207b\u00b9 * \u2191f y \u2208 H\n[PROOFSTEP]\nexact fun x y => iff_of_eq (congr_arg (\u00b7 \u2208 H) (by rw [f.map_mul, f.map_inv]))\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\nhf : Function.Surjective \u2191f\nthis\u271d : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nx y : G'\n\u22a2 \u2191f (x\u207b\u00b9 * y) = (\u2191f x)\u207b\u00b9 * \u2191f y\n[PROOFSTEP]\nrw [f.map_mul, f.map_inv]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\nhf : Function.Surjective \u2191f\nthis\u271d : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey : \u2200 (x y : G'), Setoid.r x y \u2194 Setoid.r (\u2191f x) (\u2191f y)\n\u22a2 index (comap f H) = index H\n[PROOFSTEP]\nrefine' Cardinal.toNat_congr (Equiv.ofBijective (Quotient.map' f fun x y => (key x y).mp) \u27e8_, _\u27e9)\n[GOAL]\ncase refine'_1\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\nhf : Function.Surjective \u2191f\nthis\u271d : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey : \u2200 (x y : G'), Setoid.r x y \u2194 Setoid.r (\u2191f x) (\u2191f y)\n\u22a2 Function.Injective (Quotient.map' \u2191f (_ : \u2200 (x y : G'), Setoid.r x y \u2192 Setoid.r (\u2191f x) (\u2191f y)))\n[PROOFSTEP]\nsimp_rw [\u2190 Quotient.eq''] at key \n[GOAL]\ncase refine'_1\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\nhf : Function.Surjective \u2191f\nthis\u271d : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey\u271d : \u2200 (x y : G'), Setoid.r x y \u2194 Setoid.r (\u2191f x) (\u2191f y)\nkey : \u2200 (x y : G'), Quotient.mk'' x = Quotient.mk'' y \u2194 Quotient.mk'' (\u2191f x) = Quotient.mk'' (\u2191f y)\n\u22a2 Function.Injective (Quotient.map' \u2191f (_ : \u2200 (x y : G'), Setoid.r x y \u2192 Setoid.r (\u2191f x) (\u2191f y)))\n[PROOFSTEP]\nrefine' Quotient.ind' fun x => _\n[GOAL]\ncase refine'_1\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\nhf : Function.Surjective \u2191f\nthis\u271d : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey\u271d : \u2200 (x y : G'), Setoid.r x y \u2194 Setoid.r (\u2191f x) (\u2191f y)\nkey : \u2200 (x y : G'), Quotient.mk'' x = Quotient.mk'' y \u2194 Quotient.mk'' (\u2191f x) = Quotient.mk'' (\u2191f y)\nx : G'\n\u22a2 \u2200 \u2983a\u2082 : G' \u29f8 comap f H\u2984,\n Quotient.map' \u2191f (_ : \u2200 (x y : G'), Setoid.r x y \u2192 Setoid.r (\u2191f x) (\u2191f y)) (Quotient.mk'' x) =\n Quotient.map' \u2191f (_ : \u2200 (x y : G'), Setoid.r x y \u2192 Setoid.r (\u2191f x) (\u2191f y)) a\u2082 \u2192\n Quotient.mk'' x = a\u2082\n[PROOFSTEP]\nrefine' Quotient.ind' fun y => _\n[GOAL]\ncase refine'_1\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\nhf : Function.Surjective \u2191f\nthis\u271d : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey\u271d : \u2200 (x y : G'), Setoid.r x y \u2194 Setoid.r (\u2191f x) (\u2191f y)\nkey : \u2200 (x y : G'), Quotient.mk'' x = Quotient.mk'' y \u2194 Quotient.mk'' (\u2191f x) = Quotient.mk'' (\u2191f y)\nx y : G'\n\u22a2 Quotient.map' \u2191f (_ : \u2200 (x y : G'), Setoid.r x y \u2192 Setoid.r (\u2191f x) (\u2191f y)) (Quotient.mk'' x) =\n Quotient.map' \u2191f (_ : \u2200 (x y : G'), Setoid.r x y \u2192 Setoid.r (\u2191f x) (\u2191f y)) (Quotient.mk'' y) \u2192\n Quotient.mk'' x = Quotient.mk'' y\n[PROOFSTEP]\nexact (key x y).mpr\n[GOAL]\ncase refine'_2\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\nhf : Function.Surjective \u2191f\nthis\u271d : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey : \u2200 (x y : G'), Setoid.r x y \u2194 Setoid.r (\u2191f x) (\u2191f y)\n\u22a2 Function.Surjective (Quotient.map' \u2191f (_ : \u2200 (x y : G'), Setoid.r x y \u2192 Setoid.r (\u2191f x) (\u2191f y)))\n[PROOFSTEP]\nrefine' Quotient.ind' fun x => _\n[GOAL]\ncase refine'_2\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\nhf : Function.Surjective \u2191f\nthis\u271d : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey : \u2200 (x y : G'), Setoid.r x y \u2194 Setoid.r (\u2191f x) (\u2191f y)\nx : G\n\u22a2 \u2203 a, Quotient.map' \u2191f (_ : \u2200 (x y : G'), Setoid.r x y \u2192 Setoid.r (\u2191f x) (\u2191f y)) a = Quotient.mk'' x\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := hf x\n[GOAL]\ncase refine'_2.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\nhf : Function.Surjective \u2191f\nthis\u271d : Setoid G := QuotientGroup.leftRel H\nthis : Setoid G' := QuotientGroup.leftRel (comap f H)\nkey : \u2200 (x y : G'), Setoid.r x y \u2194 Setoid.r (\u2191f x) (\u2191f y)\nx : G\ny : G'\nhy : \u2191f y = x\n\u22a2 \u2203 a, Quotient.map' \u2191f (_ : \u2200 (x y : G'), Setoid.r x y \u2192 Setoid.r (\u2191f x) (\u2191f y)) a = Quotient.mk'' x\n[PROOFSTEP]\nexact \u27e8y, (Quotient.map'_mk'' f _ y).trans (congr_arg Quotient.mk'' hy)\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\n\u22a2 comap f H = comap (MonoidHom.rangeRestrict f) (subgroupOf H (MonoidHom.range f))\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K\u271d L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G' \u2192* G\nK : Subgroup G'\n\u22a2 relindex (comap f H) K = relindex H (map f K)\n[PROOFSTEP]\nrw [relindex, subgroupOf, comap_comap, index_comap, \u2190 f.map_range, K.subtype_range]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nhHK : H \u2264 K\nhKL : K \u2264 L\n\u22a2 relindex H K * relindex K L = relindex H L\n[PROOFSTEP]\nrw [\u2190 relindex_subgroupOf hKL]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nhHK : H \u2264 K\nhKL : K \u2264 L\n\u22a2 relindex (subgroupOf H L) (subgroupOf K L) * relindex K L = relindex H L\n[PROOFSTEP]\nexact relindex_mul_index fun x hx => hHK hx\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\n\u22a2 relindex (H \u2293 K) K = relindex H K\n[PROOFSTEP]\nrw [relindex, relindex, inf_subgroupOf_right]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\n\u22a2 relindex (H \u2293 K) H = relindex K H\n[PROOFSTEP]\nrw [inf_comm, inf_relindex_right]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\n\u22a2 relindex H (K \u2293 L) * relindex K L = relindex (H \u2293 K) L\n[PROOFSTEP]\nrw [\u2190 inf_relindex_right H (K \u2293 L), \u2190 inf_relindex_right K L, \u2190 inf_relindex_right (H \u2293 K) L, inf_assoc,\n relindex_mul_relindex (H \u2293 (K \u2293 L)) (K \u2293 L) L inf_le_right inf_le_right]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\ninst\u271d : Normal K\n\u22a2 relindex K (K \u2294 H) = relindex K H\n[PROOFSTEP]\nrw [sup_comm, relindex_sup_right]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\n\u22a2 index H = 2 \u2194 \u2203 a, \u2200 (b : G), Xor' (b * a \u2208 H) (b \u2208 H)\n[PROOFSTEP]\nsimp only [index, Nat.card_eq_two_iff' ((1 : G) : G \u29f8 H), ExistsUnique, inv_mem_iff, QuotientGroup.exists_mk,\n QuotientGroup.forall_mk, Ne.def, QuotientGroup.eq, mul_one, xor_iff_iff_not]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\n\u22a2 (\u2203 x, \u00acx \u2208 H \u2227 \u2200 (x_1 : G), \u00acx_1 \u2208 H \u2192 x_1\u207b\u00b9 * x \u2208 H) \u2194 \u2203 a, \u2200 (b : G), b * a \u2208 H \u2194 \u00acb \u2208 H\n[PROOFSTEP]\nrefine' exists_congr fun a => \u27e8fun ha b => \u27e8fun hba hb => _, fun hb => _\u27e9, fun ha => \u27e8_, fun b hb => _\u27e9\u27e9\n[GOAL]\ncase refine'_1\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\na : G\nha : \u00aca \u2208 H \u2227 \u2200 (x : G), \u00acx \u2208 H \u2192 x\u207b\u00b9 * a \u2208 H\nb : G\nhba : b * a \u2208 H\nhb : b \u2208 H\n\u22a2 False\n[PROOFSTEP]\nexact ha.1 ((mul_mem_cancel_left hb).1 hba)\n[GOAL]\ncase refine'_2\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\na : G\nha : \u00aca \u2208 H \u2227 \u2200 (x : G), \u00acx \u2208 H \u2192 x\u207b\u00b9 * a \u2208 H\nb : G\nhb : \u00acb \u2208 H\n\u22a2 b * a \u2208 H\n[PROOFSTEP]\nexact inv_inv b \u25b8 ha.2 _ (mt (inv_mem_iff (x := b)).1 hb)\n[GOAL]\ncase refine'_3\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\na : G\nha : \u2200 (b : G), b * a \u2208 H \u2194 \u00acb \u2208 H\n\u22a2 \u00aca \u2208 H\n[PROOFSTEP]\nrw [\u2190 inv_mem_iff (x := a), \u2190 ha, inv_mul_self]\n[GOAL]\ncase refine'_3\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\na : G\nha : \u2200 (b : G), b * a \u2208 H \u2194 \u00acb \u2208 H\n\u22a2 1 \u2208 H\n[PROOFSTEP]\nexact one_mem _\n[GOAL]\ncase refine'_4\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\na : G\nha : \u2200 (b : G), b * a \u2208 H \u2194 \u00acb \u2208 H\nb : G\nhb : \u00acb \u2208 H\n\u22a2 b\u207b\u00b9 * a \u2208 H\n[PROOFSTEP]\nrwa [ha, inv_mem_iff (x := b)]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nh : index H = 2\na b : G\n\u22a2 a * b \u2208 H \u2194 (a \u2208 H \u2194 b \u2208 H)\n[PROOFSTEP]\nby_cases ha : a \u2208 H\n[GOAL]\ncase pos\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nh : index H = 2\na b : G\nha : a \u2208 H\n\u22a2 a * b \u2208 H \u2194 (a \u2208 H \u2194 b \u2208 H)\n[PROOFSTEP]\nsimp only [ha, true_iff_iff, mul_mem_cancel_left ha]\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nh : index H = 2\na b : G\nha : \u00aca \u2208 H\n\u22a2 a * b \u2208 H \u2194 (a \u2208 H \u2194 b \u2208 H)\n[PROOFSTEP]\nby_cases hb : b \u2208 H\n[GOAL]\ncase pos\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nh : index H = 2\na b : G\nha : \u00aca \u2208 H\nhb : b \u2208 H\n\u22a2 a * b \u2208 H \u2194 (a \u2208 H \u2194 b \u2208 H)\n[PROOFSTEP]\nsimp only [hb, iff_true_iff, mul_mem_cancel_right hb]\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nh : index H = 2\na b : G\nha : \u00aca \u2208 H\nhb : \u00acb \u2208 H\n\u22a2 a * b \u2208 H \u2194 (a \u2208 H \u2194 b \u2208 H)\n[PROOFSTEP]\nsimp only [ha, hb, iff_self_iff, iff_true_iff]\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nh : index H = 2\na b : G\nha : \u00aca \u2208 H\nhb : \u00acb \u2208 H\n\u22a2 a * b \u2208 H\n[PROOFSTEP]\nrcases index_eq_two_iff.1 h with \u27e8c, hc\u27e9\n[GOAL]\ncase neg.intro\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nh : index H = 2\na b : G\nha : \u00aca \u2208 H\nhb : \u00acb \u2208 H\nc : G\nhc : \u2200 (b : G), Xor' (b * c \u2208 H) (b \u2208 H)\n\u22a2 a * b \u2208 H\n[PROOFSTEP]\nrefine' (hc _).or.resolve_left _\n[GOAL]\ncase neg.intro\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nh : index H = 2\na b : G\nha : \u00aca \u2208 H\nhb : \u00acb \u2208 H\nc : G\nhc : \u2200 (b : G), Xor' (b * c \u2208 H) (b \u2208 H)\n\u22a2 \u00aca * b * c \u2208 H\n[PROOFSTEP]\nrwa [mul_assoc, mul_mem_cancel_right ((hc _).or.resolve_right hb)]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nh : index H = 2\na : G\n\u22a2 a * a \u2208 H\n[PROOFSTEP]\nrw [mul_mem_iff_of_index_two h]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\n\u22a2 relindex H \u22a4 = index H\n[PROOFSTEP]\nrw [\u2190 relindex_mul_index (show H \u2264 \u22a4 from le_top), index_top, mul_one]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\n\u22a2 relindex \u22a5 H = Nat.card { x // x \u2208 H }\n[PROOFSTEP]\nrw [relindex, bot_subgroupOf, index_bot]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\n\u22a2 relindex H \u22a5 = 1\n[PROOFSTEP]\nrw [relindex, subgroupOf_bot_eq_top, index_top]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\n\u22a2 relindex H H = 1\n[PROOFSTEP]\nrw [relindex, subgroupOf_self, index_top]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d K L : Subgroup G\nH : Type u_2\ninst\u271d : Group H\nf : G \u2192* H\n\u22a2 index (MonoidHom.ker f) = Nat.card \u2191(Set.range \u2191f)\n[PROOFSTEP]\nrw [\u2190 MonoidHom.comap_bot, index_comap, relindex_bot_left]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d K L : Subgroup G\nH : Type u_2\ninst\u271d : Group H\nf : G \u2192* H\n\u22a2 Nat.card { x // x \u2208 MonoidHom.range f } = Nat.card \u2191(Set.range \u2191f)\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d K\u271d L : Subgroup G\nH : Type u_2\ninst\u271d : Group H\nf : G \u2192* H\nK : Subgroup G\n\u22a2 relindex (MonoidHom.ker f) K = Nat.card \u2191(\u2191f '' \u2191K)\n[PROOFSTEP]\nrw [\u2190 MonoidHom.comap_bot, relindex_comap, relindex_bot_left]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d K\u271d L : Subgroup G\nH : Type u_2\ninst\u271d : Group H\nf : G \u2192* H\nK : Subgroup G\n\u22a2 Nat.card { x // x \u2208 map f K } = Nat.card \u2191(\u2191f '' \u2191K)\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\n\u22a2 Nat.card { x // x \u2208 H } * index H = Nat.card G\n[PROOFSTEP]\nrw [\u2190 relindex_bot_left, \u2190 index_bot]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\n\u22a2 relindex \u22a5 H * index H = index \u22a5\n[PROOFSTEP]\nexact relindex_mul_index bot_le\n[GOAL]\nG\u271d : Type u_1\ninst\u271d\u00b2 : Group G\u271d\nH\u271d K L : Subgroup G\u271d\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192* H\nhf : Function.Injective \u2191f\n\u22a2 Nat.card G \u2223 Nat.card H\n[PROOFSTEP]\nrw [Nat.card_congr (MonoidHom.ofInjective hf).toEquiv]\n[GOAL]\nG\u271d : Type u_1\ninst\u271d\u00b2 : Group G\u271d\nH\u271d K L : Subgroup G\u271d\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192* H\nhf : Function.Injective \u2191f\n\u22a2 Nat.card { x // x \u2208 MonoidHom.range f } \u2223 Nat.card H\n[PROOFSTEP]\nexact Dvd.intro f.range.index f.range.card_mul_index\n[GOAL]\nG\u271d : Type u_1\ninst\u271d\u00b2 : Group G\u271d\nH\u271d K L : Subgroup G\u271d\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192* H\nhf : Function.Surjective \u2191f\n\u22a2 Nat.card H \u2223 Nat.card G\n[PROOFSTEP]\nrw [\u2190 Nat.card_congr (QuotientGroup.quotientKerEquivOfSurjective f hf).toEquiv]\n[GOAL]\nG\u271d : Type u_1\ninst\u271d\u00b2 : Group G\u271d\nH\u271d K L : Subgroup G\u271d\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192* H\nhf : Function.Surjective \u2191f\n\u22a2 Nat.card (G \u29f8 MonoidHom.ker f) \u2223 Nat.card G\n[PROOFSTEP]\nexact Dvd.intro_left (Nat.card f.ker) f.ker.card_mul_index\n[GOAL]\nG\u271d : Type u_1\ninst\u271d\u2074 : Group G\u271d\nH\u271d K L : Subgroup G\u271d\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\ninst\u271d\u00b9 : Fintype G\ninst\u271d : Fintype H\nf : G \u2192* H\nhf : Function.Surjective \u2191f\n\u22a2 Fintype.card H \u2223 Fintype.card G\n[PROOFSTEP]\nsimp only [\u2190 Nat.card_eq_fintype_card, nat_card_dvd_of_surjective f hf]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G \u2192* G'\n\u22a2 index (map f H) = index (H \u2294 MonoidHom.ker f) * index (MonoidHom.range f)\n[PROOFSTEP]\nrw [\u2190 comap_map_eq, index_comap, relindex_mul_index (H.map_le_range f)]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G \u2192* G'\nhf : Function.Surjective \u2191f\n\u22a2 index (map f H) \u2223 index H\n[PROOFSTEP]\nrw [index_map, f.range_top_of_surjective hf, index_top, mul_one]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G \u2192* G'\nhf : Function.Surjective \u2191f\n\u22a2 index (H \u2294 MonoidHom.ker f) \u2223 index H\n[PROOFSTEP]\nexact index_dvd_of_le le_sup_left\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G \u2192* G'\nhf : MonoidHom.ker f \u2264 H\n\u22a2 index H \u2223 index (map f H)\n[PROOFSTEP]\nrw [index_map, sup_of_le_left hf]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\nG' : Type u_2\ninst\u271d : Group G'\nf : G \u2192* G'\nhf : MonoidHom.ker f \u2264 H\n\u22a2 index H \u2223 index H * index (MonoidHom.range f)\n[PROOFSTEP]\napply dvd_mul_right\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\ninst\u271d : Fintype G\nhH : Fintype { x // x \u2208 H }\n\u22a2 index H * Fintype.card { x // x \u2208 H } = Fintype.card G\n[PROOFSTEP]\nrw [\u2190 relindex_bot_left_eq_card, \u2190 index_bot_eq_card, mul_comm]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\ninst\u271d : Fintype G\nhH : Fintype { x // x \u2208 H }\n\u22a2 relindex \u22a5 H * index H = index \u22a5\n[PROOFSTEP]\nexact relindex_mul_index bot_le\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\ninst\u271d : Fintype G\n\u22a2 index H \u2223 Fintype.card G\n[PROOFSTEP]\nclassical exact \u27e8Fintype.card H, H.index_mul_card.symm\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\ninst\u271d : Fintype G\n\u22a2 index H \u2223 Fintype.card G\n[PROOFSTEP]\nexact \u27e8Fintype.card H, H.index_mul_card.symm\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nhH : relindex H L \u2260 0\nhK : relindex K L \u2260 0\n\u22a2 relindex (H \u2293 K) L \u2260 0\n[PROOFSTEP]\nreplace hH : H.relindex (K \u2293 L) \u2260 0 := mt (relindex_eq_zero_of_le_right inf_le_right) hH\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nhK : relindex K L \u2260 0\nhH : relindex H (K \u2293 L) \u2260 0\n\u22a2 relindex (H \u2293 K) L \u2260 0\n[PROOFSTEP]\nrw [\u2190 inf_relindex_right] at hH hK \u22a2\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nhK : relindex (K \u2293 L) L \u2260 0\nhH : relindex (H \u2293 (K \u2293 L)) (K \u2293 L) \u2260 0\n\u22a2 relindex (H \u2293 K \u2293 L) L \u2260 0\n[PROOFSTEP]\nrw [inf_assoc]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nhK : relindex (K \u2293 L) L \u2260 0\nhH : relindex (H \u2293 (K \u2293 L)) (K \u2293 L) \u2260 0\n\u22a2 relindex (H \u2293 (K \u2293 L)) L \u2260 0\n[PROOFSTEP]\nexact relindex_ne_zero_trans hH hK\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nhH : index H \u2260 0\nhK : index K \u2260 0\n\u22a2 index (H \u2293 K) \u2260 0\n[PROOFSTEP]\nrw [\u2190 relindex_top_right] at hH hK \u22a2\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nhH : relindex H \u22a4 \u2260 0\nhK : relindex K \u22a4 \u2260 0\n\u22a2 relindex (H \u2293 K) \u22a4 \u2260 0\n[PROOFSTEP]\nexact relindex_inf_ne_zero hH hK\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\n\u22a2 relindex (H \u2293 K) L \u2264 relindex H L * relindex K L\n[PROOFSTEP]\nby_cases h : H.relindex L = 0\n[GOAL]\ncase pos\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nh : relindex H L = 0\n\u22a2 relindex (H \u2293 K) L \u2264 relindex H L * relindex K L\n[PROOFSTEP]\nexact (le_of_eq (relindex_eq_zero_of_le_left inf_le_left h)).trans (zero_le _)\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nh : \u00acrelindex H L = 0\n\u22a2 relindex (H \u2293 K) L \u2264 relindex H L * relindex K L\n[PROOFSTEP]\nrw [\u2190 inf_relindex_right, inf_assoc, \u2190 relindex_mul_relindex _ _ L inf_le_right inf_le_right, inf_relindex_right,\n inf_relindex_right]\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nh : \u00acrelindex H L = 0\n\u22a2 relindex H (K \u2293 L) * relindex K L \u2264 relindex H L * relindex K L\n[PROOFSTEP]\nexact mul_le_mul_right' (relindex_le_of_le_right inf_le_right h) (K.relindex L)\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\n\u22a2 index (H \u2293 K) \u2264 index H * index K\n[PROOFSTEP]\nsimp_rw [\u2190 relindex_top_right, relindex_inf_le]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 Subgroup G\nhf : \u2200 (i : \u03b9), index (f i) \u2260 0\n\u22a2 index (\u2a05 (i : \u03b9), f i) \u2260 0\n[PROOFSTEP]\nsimp_rw [\u2190 relindex_top_right] at hf \u22a2\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 Subgroup G\nhf : \u2200 (i : \u03b9), relindex (f i) \u22a4 \u2260 0\n\u22a2 relindex (\u2a05 (i : \u03b9), f i) \u22a4 \u2260 0\n[PROOFSTEP]\nexact relindex_iInf_ne_zero hf\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 Subgroup G\n\u22a2 index (\u2a05 (i : \u03b9), f i) \u2264 \u220f i : \u03b9, index (f i)\n[PROOFSTEP]\nsimp_rw [\u2190 relindex_top_right, relindex_iInf_le]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nhH : Finite (G \u29f8 H)\n\u22a2 index H \u2260 0\n[PROOFSTEP]\ncases nonempty_fintype (G \u29f8 H)\n[GOAL]\ncase intro\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nhH : Finite (G \u29f8 H)\nval\u271d : Fintype (G \u29f8 H)\n\u22a2 index H \u2260 0\n[PROOFSTEP]\nrw [index_eq_card]\n[GOAL]\ncase intro\nG : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nhH : Finite (G \u29f8 H)\nval\u271d : Fintype (G \u29f8 H)\n\u22a2 Fintype.card (G \u29f8 H) \u2260 0\n[PROOFSTEP]\nexact Fintype.card_ne_zero\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\ninst\u271d : FiniteIndex H\n\u22a2 FiniteIndex (normalCore H)\n[PROOFSTEP]\nrw [normalCore_eq_ker]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\ninst\u271d : FiniteIndex H\n\u22a2 FiniteIndex (MonoidHom.ker (MulAction.toPermHom G (G \u29f8 H)))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\nH K L : Subgroup G\ninst\u271d\u00b9 : Finite \u2191(commutatorSet G)\ninst\u271d : Group.FG G\n\u22a2 FiniteIndex (center G)\n[PROOFSTEP]\nobtain \u27e8S, -, hS\u27e9 := Group.rank_spec G\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\nH K L : Subgroup G\ninst\u271d\u00b9 : Finite \u2191(commutatorSet G)\ninst\u271d : Group.FG G\nS : Finset G\nhS : closure \u2191S = \u22a4\n\u22a2 FiniteIndex (center G)\n[PROOFSTEP]\nexact \u27e8mt (Finite.card_eq_zero_of_embedding (quotientCenterEmbedding hS)) Finite.card_pos.ne'\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\nH K L : Subgroup G\ninst\u271d\u00b9 : Finite \u2191(commutatorSet G)\ninst\u271d : Group.FG G\n\u22a2 index (center G) \u2264 Nat.card \u2191(commutatorSet G) ^ Group.rank G\n[PROOFSTEP]\nobtain \u27e8S, hS1, hS2\u27e9 := Group.rank_spec G\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\nH K L : Subgroup G\ninst\u271d\u00b9 : Finite \u2191(commutatorSet G)\ninst\u271d : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure \u2191S = \u22a4\n\u22a2 index (center G) \u2264 Nat.card \u2191(commutatorSet G) ^ Group.rank G\n[PROOFSTEP]\nrw [\u2190 hS1, \u2190 Fintype.card_coe, \u2190 Nat.card_eq_fintype_card, \u2190 Finset.coe_sort_coe, \u2190 Nat.card_fun]\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\nH K L : Subgroup G\ninst\u271d\u00b9 : Finite \u2191(commutatorSet G)\ninst\u271d : Group.FG G\nS : Finset G\nhS1 : Finset.card S = Group.rank G\nhS2 : closure \u2191S = \u22a4\n\u22a2 index (center G) \u2264 Nat.card (\u2191\u2191S \u2192 \u2191(commutatorSet G))\n[PROOFSTEP]\nexact Finite.card_le_of_embedding (quotientCenterEmbedding hS2)\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Index", "llama_tokens": 10672, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175005616829, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.40254459074480453}} {"text": "[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2115\ninst\u271d : StrictOrderedSemiring A\nh : NormNum.IsNat e n\nw : Nat.ble 1 n = true\n\u22a2 0 < e\n[PROOFSTEP]\nrw [NormNum.IsNat.to_eq h rfl]\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2115\ninst\u271d : StrictOrderedSemiring A\nh : NormNum.IsNat e n\nw : Nat.ble 1 n = true\n\u22a2 0 < \u2191n\n[PROOFSTEP]\napply Nat.cast_pos.2\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2115\ninst\u271d : StrictOrderedSemiring A\nh : NormNum.IsNat e n\nw : Nat.ble 1 n = true\n\u22a2 0 < n\n[PROOFSTEP]\nsimpa using w\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2115\ninst\u271d : OrderedSemiring A\nh : NormNum.IsNat e n\n\u22a2 0 \u2264 e\n[PROOFSTEP]\nrw [NormNum.IsNat.to_eq h rfl]\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2115\ninst\u271d : OrderedSemiring A\nh : NormNum.IsNat e n\n\u22a2 0 \u2264 \u2191n\n[PROOFSTEP]\nexact Nat.cast_nonneg n\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2115\ninst\u271d : StrictOrderedRing A\nh : NormNum.IsInt e (Int.negOfNat n)\nw : Nat.ble 1 n = true\n\u22a2 e \u2260 0\n[PROOFSTEP]\nrw [NormNum.IsInt.neg_to_eq h rfl]\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2115\ninst\u271d : StrictOrderedRing A\nh : NormNum.IsInt e (Int.negOfNat n)\nw : Nat.ble 1 n = true\n\u22a2 -\u2191n \u2260 0\n[PROOFSTEP]\nsimp only [ne_eq, neg_eq_zero]\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2115\ninst\u271d : StrictOrderedRing A\nh : NormNum.IsInt e (Int.negOfNat n)\nw : Nat.ble 1 n = true\n\u22a2 \u00ac\u2191n = 0\n[PROOFSTEP]\napply ne_of_gt\n[GOAL]\ncase h\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2115\ninst\u271d : StrictOrderedRing A\nh : NormNum.IsInt e (Int.negOfNat n)\nw : Nat.ble 1 n = true\n\u22a2 0 < \u2191n\n[PROOFSTEP]\nsimpa using w\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2124\nd : \u2115\ninst\u271d : LinearOrderedRing A\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (0 < n) = true\n\u22a2 0 < e\n[PROOFSTEP]\nhave pos_invOf_d : (0 < \u215f(d : A)) := pos_invOf_of_invertible_cast d\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2124\nd : \u2115\ninst\u271d : LinearOrderedRing A\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (0 < n) = true\npos_invOf_d : 0 < \u215f\u2191d\n\u22a2 0 < e\n[PROOFSTEP]\nhave pos_n : (0 < (n : A)) := Int.cast_pos (n := n) |>.2 (of_decide_eq_true h)\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2124\nd : \u2115\ninst\u271d : LinearOrderedRing A\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (0 < n) = true\npos_invOf_d : 0 < \u215f\u2191d\npos_n : 0 < \u2191n\n\u22a2 0 < e\n[PROOFSTEP]\nrw [eq]\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2124\nd : \u2115\ninst\u271d : LinearOrderedRing A\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (0 < n) = true\npos_invOf_d : 0 < \u215f\u2191d\npos_n : 0 < \u2191n\n\u22a2 0 < \u2191n * \u215f\u2191d\n[PROOFSTEP]\nexact mul_pos pos_n pos_invOf_d\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2124\nd : \u2115\ninst\u271d : LinearOrderedRing A\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (n = 0) = true\n\u22a2 0 \u2264 e\n[PROOFSTEP]\nrw [eq, of_decide_eq_true h]\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2124\nd : \u2115\ninst\u271d : LinearOrderedRing A\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (n = 0) = true\n\u22a2 0 \u2264 \u21910 * \u215f\u2191d\n[PROOFSTEP]\nsimp\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2124\nd : \u2115\ninst\u271d : LinearOrderedRing A\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (n < 0) = true\n\u22a2 e \u2260 0\n[PROOFSTEP]\nhave pos_invOf_d : (0 < \u215f(d : A)) := pos_invOf_of_invertible_cast d\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2124\nd : \u2115\ninst\u271d : LinearOrderedRing A\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (n < 0) = true\npos_invOf_d : 0 < \u215f\u2191d\n\u22a2 e \u2260 0\n[PROOFSTEP]\nhave neg_n : ((n : A) < 0) := Int.cast_lt_zero (n := n) |>.2 (of_decide_eq_true h)\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2124\nd : \u2115\ninst\u271d : LinearOrderedRing A\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (n < 0) = true\npos_invOf_d : 0 < \u215f\u2191d\nneg_n : \u2191n < 0\n\u22a2 e \u2260 0\n[PROOFSTEP]\nhave neg := mul_neg_of_neg_of_pos neg_n pos_invOf_d\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2124\nd : \u2115\ninst\u271d : LinearOrderedRing A\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (n < 0) = true\npos_invOf_d : 0 < \u215f\u2191d\nneg_n : \u2191n < 0\nneg : \u2191n * \u215f\u2191d < 0\n\u22a2 e \u2260 0\n[PROOFSTEP]\nrw [eq]\n[GOAL]\nu : Level\n\u03b1 : Q(Type u)\nz\u03b1 : Q(Zero \u00ab$\u03b1\u00bb)\np\u03b1 : Q(PartialOrder \u00ab$\u03b1\u00bb)\nA : Type u_1\ne : A\nn : \u2124\nd : \u2115\ninst\u271d : LinearOrderedRing A\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (n < 0) = true\npos_invOf_d : 0 < \u215f\u2191d\nneg_n : \u2191n < 0\nneg : \u2191n * \u215f\u2191d < 0\n\u22a2 \u2191n * \u215f\u2191d \u2260 0\n[PROOFSTEP]\nexact ne_iff_lt_or_gt.2 (Or.inl neg)\n", "meta": {"mathlib_filename": "Mathlib.Tactic.Positivity.Core", "llama_tokens": 2898, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.40252653040150815}} {"text": "[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nX Y : J\nf : X \u27f6 Y\n\u22a2 ((Functor.const J).obj (of \u2191{u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (u i) = u j})).map f \u226b\n (fun j => ContinuousMap.mk fun u => \u2191u j) Y =\n (fun j => ContinuousMap.mk fun u => \u2191u j) X \u226b F.map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nX Y : J\nf : X \u27f6 Y\n\u22a2 (\ud835\udfd9 (of { x // \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (x i) = x j }) \u226b ContinuousMap.mk fun u => \u2191u Y) =\n (ContinuousMap.mk fun u => \u2191u X) \u226b F.map f\n[PROOFSTEP]\nrw [Category.id_comp]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nX Y : J\nf : X \u27f6 Y\n\u22a2 (ContinuousMap.mk fun u => \u2191u Y) = (ContinuousMap.mk fun u => \u2191u X) \u226b F.map f\n[PROOFSTEP]\napply ContinuousMap.ext\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nX Y : J\nf : X \u27f6 Y\n\u22a2 \u2200 (a : { x // \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (x i) = x j }),\n \u2191(ContinuousMap.mk fun u => \u2191u Y) a = \u2191((ContinuousMap.mk fun u => \u2191u X) \u226b F.map f) a\n[PROOFSTEP]\nintro a\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nX Y : J\nf : X \u27f6 Y\na : { x // \u2200 {i j : J} (f : i \u27f6 j), \u2191(F.map f) (x i) = x j }\n\u22a2 \u2191(ContinuousMap.mk fun u => \u2191u Y) a = \u2191((ContinuousMap.mk fun u => \u2191u X) \u226b F.map f) a\n[PROOFSTEP]\nexact (a.2 f).symm\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nS : Cone F\nx : \u2191S.pt\ni\u271d j\u271d : J\nf : i\u271d \u27f6 j\u271d\n\u22a2 \u2191(F.map f) ((fun j => \u2191(NatTrans.app S.\u03c0 j) x) i\u271d) = (fun j => \u2191(NatTrans.app S.\u03c0 j) x) j\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nS : Cone F\nx : \u2191S.pt\ni\u271d j\u271d : J\nf : i\u271d \u27f6 j\u271d\n\u22a2 \u2191(F.map f) (\u2191(NatTrans.app S.\u03c0 i\u271d) x) = \u2191(NatTrans.app S.\u03c0 j\u271d) x\n[PROOFSTEP]\nerw [\u2190 S.w f]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nS : Cone F\nx : \u2191S.pt\ni\u271d j\u271d : J\nf : i\u271d \u27f6 j\u271d\n\u22a2 \u2191(F.map f) (\u2191(NatTrans.app S.\u03c0 i\u271d) x) = \u2191(NatTrans.app S.\u03c0 i\u271d \u226b F.map f) x\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nS : Cone F\nx : \u2191S.pt\ni j : J\nf : i \u27f6 j\n\u22a2 \u2191(F.map f) ((fun j => \u2191(NatTrans.app S.\u03c0 j) x) i) = (fun j => \u2191(NatTrans.app S.\u03c0 j) x) j\n[PROOFSTEP]\ndsimp\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nS : Cone F\nx : \u2191S.pt\ni j : J\nf : i \u27f6 j\n\u22a2 \u2191(F.map f) (\u2191(NatTrans.app S.\u03c0 i) x) = \u2191(NatTrans.app S.\u03c0 j) x\n[PROOFSTEP]\nrw [\u2190 S.w f]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nS : Cone F\nx : \u2191S.pt\ni j : J\nf : i \u27f6 j\n\u22a2 \u2191(F.map f) (\u2191(NatTrans.app S.\u03c0 i) x) = \u2191(NatTrans.app S.\u03c0 i \u226b F.map f) x\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nS : Cone F\nm : S.pt \u27f6 (limitCone F).pt\nh : \u2200 (j : J), m \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app S.\u03c0 j\n\u22a2 m =\n (fun S =>\n ContinuousMap.mk fun x =>\n { val := fun j => \u2191(NatTrans.app S.\u03c0 j) x,\n property :=\n (_ :\n \u2200 {i j : J} (f : i \u27f6 j),\n \u2191(F.map f) ((fun j => \u2191(NatTrans.app S.\u03c0 j) x) i) = (fun j => \u2191(NatTrans.app S.\u03c0 j) x) j) })\n S\n[PROOFSTEP]\napply ContinuousMap.ext\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nS : Cone F\nm : S.pt \u27f6 (limitCone F).pt\nh : \u2200 (j : J), m \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app S.\u03c0 j\n\u22a2 \u2200 (a : \u2191S.pt),\n \u2191m a =\n \u2191((fun S =>\n ContinuousMap.mk fun x =>\n { val := fun j => \u2191(NatTrans.app S.\u03c0 j) x,\n property :=\n (_ :\n \u2200 {i j : J} (f : i \u27f6 j),\n \u2191(F.map f) ((fun j => \u2191(NatTrans.app S.\u03c0 j) x) i) = (fun j => \u2191(NatTrans.app S.\u03c0 j) x) j) })\n S)\n a\n[PROOFSTEP]\nintros a\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nS : Cone F\nm : S.pt \u27f6 (limitCone F).pt\nh : \u2200 (j : J), m \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app S.\u03c0 j\na : \u2191S.pt\n\u22a2 \u2191m a =\n \u2191((fun S =>\n ContinuousMap.mk fun x =>\n { val := fun j => \u2191(NatTrans.app S.\u03c0 j) x,\n property :=\n (_ :\n \u2200 {i j : J} (f : i \u27f6 j),\n \u2191(F.map f) ((fun j => \u2191(NatTrans.app S.\u03c0 j) x) i) = (fun j => \u2191(NatTrans.app S.\u03c0 j) x) j) })\n S)\n a\n[PROOFSTEP]\napply Subtype.ext\n[GOAL]\ncase h.a\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nS : Cone F\nm : S.pt \u27f6 (limitCone F).pt\nh : \u2200 (j : J), m \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app S.\u03c0 j\na : \u2191S.pt\n\u22a2 \u2191(\u2191m a) =\n \u2191(\u2191((fun S =>\n ContinuousMap.mk fun x =>\n { val := fun j => \u2191(NatTrans.app S.\u03c0 j) x,\n property :=\n (_ :\n \u2200 {i j : J} (f : i \u27f6 j),\n \u2191(F.map f) ((fun j => \u2191(NatTrans.app S.\u03c0 j) x) i) = (fun j => \u2191(NatTrans.app S.\u03c0 j) x) j) })\n S)\n a)\n[PROOFSTEP]\nfunext j\n[GOAL]\ncase h.a.h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nS : Cone F\nm : S.pt \u27f6 (limitCone F).pt\nh : \u2200 (j : J), m \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app S.\u03c0 j\na : \u2191S.pt\nj : J\n\u22a2 \u2191(\u2191m a) j =\n \u2191(\u2191((fun S =>\n ContinuousMap.mk fun x =>\n { val := fun j => \u2191(NatTrans.app S.\u03c0 j) x,\n property :=\n (_ :\n \u2200 {i j : J} (f : i \u27f6 j),\n \u2191(F.map f) ((fun j => \u2191(NatTrans.app S.\u03c0 j) x) i) = (fun j => \u2191(NatTrans.app S.\u03c0 j) x) j) })\n S)\n a)\n j\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.a.h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nS : Cone F\nm : S.pt \u27f6 (limitCone F).pt\nh : \u2200 (j : J), m \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app S.\u03c0 j\na : \u2191S.pt\nj : J\n\u22a2 \u2191(\u2191m a) j = \u2191(NatTrans.app S.\u03c0 j) a\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase h.a.h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\nS : Cone F\nm : S.pt \u27f6 (limitCone F).pt\nh : \u2200 (j : J), m \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app S.\u03c0 j\na : \u2191S.pt\nj : J\n\u22a2 \u2191(\u2191m a) j = \u2191(m \u226b NatTrans.app (limitCone F).\u03c0 j) a\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\n\u22a2 IsLimit (limitConeInfi F)\n[PROOFSTEP]\nrefine\n IsLimit.ofFaithful forget\n (Types.limitConeIsLimit.{v, u} (F \u22d9 forget))\n -- Porting note: previously could infer all ?_ except continuity\n (fun s => \u27e8fun v => \u27e8fun j => (Functor.mapCone forget s).\u03c0.app j v, ?_\u27e9, ?_\u27e9) fun s => ?_\n[GOAL]\ncase refine_1\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\ns : Cone F\nv : \u2191s.pt\n\u22a2 (fun j => NatTrans.app (forget.mapCone s).\u03c0 j v) \u2208 Functor.sections (F \u22d9 forget)\n[PROOFSTEP]\ndsimp [Functor.sections]\n[GOAL]\ncase refine_1\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\ns : Cone F\nv : \u2191s.pt\n\u22a2 \u2200 {j j' : J} (f : j \u27f6 j'),\n forget.map (F.map f) (forget.map (NatTrans.app s.\u03c0 j) v) = forget.map (NatTrans.app s.\u03c0 j') v\n[PROOFSTEP]\nintro _ _ _\n[GOAL]\ncase refine_1\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\ns : Cone F\nv : \u2191s.pt\nj\u271d j'\u271d : J\nf\u271d : j\u271d \u27f6 j'\u271d\n\u22a2 forget.map (F.map f\u271d) (forget.map (NatTrans.app s.\u03c0 j\u271d) v) = forget.map (NatTrans.app s.\u03c0 j'\u271d) v\n[PROOFSTEP]\nrw [\u2190 comp_apply', forget_map_eq_coe, \u2190 s.\u03c0.naturality, forget_map_eq_coe]\n[GOAL]\ncase refine_1\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\ns : Cone F\nv : \u2191s.pt\nj\u271d j'\u271d : J\nf\u271d : j\u271d \u27f6 j'\u271d\n\u22a2 \u2191(((Functor.const J).obj s.pt).map f\u271d \u226b NatTrans.app s.\u03c0 j'\u271d) v = \u2191(NatTrans.app s.\u03c0 j'\u271d) v\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine_1\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\ns : Cone F\nv : \u2191s.pt\nj\u271d j'\u271d : J\nf\u271d : j\u271d \u27f6 j'\u271d\n\u22a2 \u2191(\ud835\udfd9 s.pt \u226b NatTrans.app s.\u03c0 j'\u271d) v = \u2191(NatTrans.app s.\u03c0 j'\u271d) v\n[PROOFSTEP]\nrw [Category.id_comp]\n[GOAL]\ncase refine_2\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\ns : Cone F\n\u22a2 Continuous fun v =>\n { val := fun j => NatTrans.app (forget.mapCone s).\u03c0 j v,\n property := (_ : (fun j => NatTrans.app (forget.mapCone s).\u03c0 j v) \u2208 Functor.sections (F \u22d9 forget)) }\n[PROOFSTEP]\nexact\n continuous_iff_coinduced_le.mpr\n (le_iInf fun j => coinduced_le_iff_le_induced.mp <| (continuous_iff_coinduced_le.mp (s.\u03c0.app j).continuous : _))\n[GOAL]\ncase refine_3\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\ns : Cone F\n\u22a2 forget.map\n ((fun s =>\n ContinuousMap.mk fun v =>\n { val := fun j => NatTrans.app (forget.mapCone s).\u03c0 j v,\n property := (_ : (fun j => NatTrans.app (forget.mapCone s).\u03c0 j v) \u2208 Functor.sections (F \u22d9 forget)) })\n s) =\n IsLimit.lift (Types.limitConeIsLimit (F \u22d9 forget)) (forget.mapCone s)\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\n\u22a2 IsColimit (colimitCocone F)\n[PROOFSTEP]\nrefine\n IsColimit.ofFaithful forget (Types.colimitCoconeIsColimit _)\n (fun s =>\n -- Porting note: it appears notation for forget breaks dot notation (also above)\n -- Porting note: previously function was inferred\n \u27e8Quot.lift (fun p => (Functor.mapCocone forget s).\u03b9.app p.fst p.snd) ?_, ?_\u27e9)\n fun s => ?_\n[GOAL]\ncase refine_1\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\ns : Cocone F\n\u22a2 \u2200 (a b : (j : J) \u00d7 (F \u22d9 forget).obj j),\n Types.Quot.Rel (F \u22d9 forget) a b \u2192\n (fun p => NatTrans.app (forget.mapCocone s).\u03b9 p.fst p.snd) a =\n (fun p => NatTrans.app (forget.mapCocone s).\u03b9 p.fst p.snd) b\n[PROOFSTEP]\nintro _ _ \u27e8_, h\u27e9\n[GOAL]\ncase refine_1\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\ns : Cocone F\na\u271d b\u271d : (j : J) \u00d7 (F \u22d9 forget).obj j\nw\u271d : a\u271d.fst \u27f6 b\u271d.fst\nh : b\u271d.snd = (F \u22d9 forget).map w\u271d a\u271d.snd\n\u22a2 (fun p => NatTrans.app (forget.mapCocone s).\u03b9 p.fst p.snd) a\u271d =\n (fun p => NatTrans.app (forget.mapCocone s).\u03b9 p.fst p.snd) b\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine_1\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\ns : Cocone F\na\u271d b\u271d : (j : J) \u00d7 (F \u22d9 forget).obj j\nw\u271d : a\u271d.fst \u27f6 b\u271d.fst\nh : b\u271d.snd = (F \u22d9 forget).map w\u271d a\u271d.snd\n\u22a2 forget.map (NatTrans.app s.\u03b9 a\u271d.fst) a\u271d.snd = forget.map (NatTrans.app s.\u03b9 b\u271d.fst) b\u271d.snd\n[PROOFSTEP]\nrw [h, Functor.comp_map, \u2190 comp_apply', s.\u03b9.naturality]\n[GOAL]\ncase refine_1\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\ns : Cocone F\na\u271d b\u271d : (j : J) \u00d7 (F \u22d9 forget).obj j\nw\u271d : a\u271d.fst \u27f6 b\u271d.fst\nh : b\u271d.snd = (F \u22d9 forget).map w\u271d a\u271d.snd\n\u22a2 forget.map (NatTrans.app s.\u03b9 a\u271d.fst) a\u271d.snd =\n forget.map (NatTrans.app s.\u03b9 a\u271d.fst \u226b ((Functor.const J).obj s.pt).map w\u271d) a\u271d.snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine_1\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\ns : Cocone F\na\u271d b\u271d : (j : J) \u00d7 (F \u22d9 forget).obj j\nw\u271d : a\u271d.fst \u27f6 b\u271d.fst\nh : b\u271d.snd = (F \u22d9 forget).map w\u271d a\u271d.snd\n\u22a2 forget.map (NatTrans.app s.\u03b9 a\u271d.fst) a\u271d.snd = forget.map (NatTrans.app s.\u03b9 a\u271d.fst \u226b \ud835\udfd9 s.pt) a\u271d.snd\n[PROOFSTEP]\nrw [Category.comp_id]\n[GOAL]\ncase refine_2\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\ns : Cocone F\n\u22a2 Continuous\n (Quot.lift (fun p => NatTrans.app (forget.mapCocone s).\u03b9 p.fst p.snd)\n (_ :\n \u2200 (a b : (j : J) \u00d7 (F \u22d9 forget).obj j),\n Types.Quot.Rel (F \u22d9 forget) a b \u2192\n (fun p => NatTrans.app (forget.mapCocone s).\u03b9 p.fst p.snd) a =\n (fun p => NatTrans.app (forget.mapCocone s).\u03b9 p.fst p.snd) b))\n[PROOFSTEP]\nexact\n continuous_iff_le_induced.mpr\n (iSup_le fun j => coinduced_le_iff_le_induced.mp <| (continuous_iff_coinduced_le.mp (s.\u03b9.app j).continuous : _))\n[GOAL]\ncase refine_3\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TopCatMax\ns : Cocone F\n\u22a2 forget.map\n ((fun s =>\n ContinuousMap.mk\n (Quot.lift (fun p => NatTrans.app (forget.mapCocone s).\u03b9 p.fst p.snd)\n (_ :\n \u2200 (a b : (j : J) \u00d7 (F \u22d9 forget).obj j),\n Types.Quot.Rel (F \u22d9 forget) a b \u2192\n (fun p => NatTrans.app (forget.mapCocone s).\u03b9 p.fst p.snd) a =\n (fun p => NatTrans.app (forget.mapCocone s).\u03b9 p.fst p.snd) b)))\n s) =\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget)) (forget.mapCocone s)\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nX : TopCat\n\u22a2 Continuous fun x => PUnit.unit\n[PROOFSTEP]\ncontinuity\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nX : TopCat\nf : X \u27f6 of PUnit\n\u22a2 f = default\n[PROOFSTEP]\next\n[GOAL]\ncase w\nJ : Type v\ninst\u271d : SmallCategory J\nX : TopCat\nf : X \u27f6 of PUnit\nx\u271d : forget.obj X\n\u22a2 \u2191f x\u271d = \u2191default x\u271d\n[PROOFSTEP]\naesop\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nX : TopCat\n\u22a2 Continuous fun x => PEmpty.elim x\n[PROOFSTEP]\ncontinuity\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nX : TopCat\nf : of PEmpty \u27f6 X\n\u22a2 f = default\n[PROOFSTEP]\next \u27e8\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.Category.TopCat.Limits.Basic", "llama_tokens": 6039, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.40252653040150815}} {"text": "[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\n\u22a2 \u2191(aeval f) (charpoly f) = 0\n[PROOFSTEP]\napply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\n\u22a2 \u2191(AlgEquiv.toLinearEquiv (algEquivMatrix (chooseBasis R M))) (\u2191(aeval f) (charpoly f)) = 0\n[PROOFSTEP]\nrw [AlgEquiv.toLinearEquiv_apply, \u2190 AlgEquiv.coe_algHom, \u2190 Polynomial.aeval_algHom_apply _ _ _, charpoly_def]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\n\u22a2 \u2191(aeval (\u2191\u2191(algEquivMatrix (chooseBasis R M)) f))\n (Matrix.charpoly (\u2191(toMatrix (chooseBasis R M) (chooseBasis R M)) f)) =\n 0\n[PROOFSTEP]\nexact Matrix.aeval_self_charpoly _\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nk : \u2115\n\u22a2 f ^ k = \u2191(aeval f) (X ^ k %\u2098 charpoly f)\n[PROOFSTEP]\nrw [\u2190 aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u2191f\n\u22a2 coeff (minpoly R f) 0 \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u2191f\nh : coeff (minpoly R f) 0 = 0\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8P, hP\u27e9 := X_dvd_iff.2 h\n[GOAL]\ncase intro\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u2191f\nh : coeff (minpoly R f) 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\n\u22a2 False\n[PROOFSTEP]\nhave hdegP : P.degree < (minpoly R f).degree := by\n rw [hP, mul_comm]\n refine' degree_lt_degree_mul_X fun h => _\n rw [h, mul_zero] at hP \n exact minpoly.ne_zero (isIntegral f) hP\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u2191f\nh : coeff (minpoly R f) 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\n\u22a2 degree P < degree (minpoly R f)\n[PROOFSTEP]\nrw [hP, mul_comm]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u2191f\nh : coeff (minpoly R f) 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\n\u22a2 degree P < degree (P * X)\n[PROOFSTEP]\nrefine' degree_lt_degree_mul_X fun h => _\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u2191f\nh\u271d : coeff (minpoly R f) 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nh : P = 0\n\u22a2 False\n[PROOFSTEP]\nrw [h, mul_zero] at hP \n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u2191f\nh\u271d : coeff (minpoly R f) 0 = 0\nP : R[X]\nhP : minpoly R f = 0\nh : P = 0\n\u22a2 False\n[PROOFSTEP]\nexact minpoly.ne_zero (isIntegral f) hP\n[GOAL]\ncase intro\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u2191f\nh : coeff (minpoly R f) 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : degree P < degree (minpoly R f)\n\u22a2 False\n[PROOFSTEP]\nhave hPmonic : P.Monic :=\n by\n suffices (minpoly R f).Monic by rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, \u2190 Monic.def] at this \n exact minpoly.monic (isIntegral f)\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u2191f\nh : coeff (minpoly R f) 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : degree P < degree (minpoly R f)\n\u22a2 Monic P\n[PROOFSTEP]\nsuffices (minpoly R f).Monic by rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, \u2190 Monic.def] at this \n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u2191f\nh : coeff (minpoly R f) 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : degree P < degree (minpoly R f)\nthis : Monic (minpoly R f)\n\u22a2 Monic P\n[PROOFSTEP]\nrwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, \u2190 Monic.def] at this \n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u2191f\nh : coeff (minpoly R f) 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : degree P < degree (minpoly R f)\n\u22a2 Monic (minpoly R f)\n[PROOFSTEP]\nexact minpoly.monic (isIntegral f)\n[GOAL]\ncase intro\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u2191f\nh : coeff (minpoly R f) 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : degree P < degree (minpoly R f)\nhPmonic : Monic P\n\u22a2 False\n[PROOFSTEP]\nhave hzero : aeval f (minpoly R f) = 0 := minpoly.aeval _ _\n[GOAL]\ncase intro\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u2191f\nh : coeff (minpoly R f) 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : degree P < degree (minpoly R f)\nhPmonic : Monic P\nhzero : \u2191(aeval f) (minpoly R f) = 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [hP, mul_eq_comp, ext_iff, hf, aeval_X, map_eq_zero_iff, coe_comp, AlgHom.map_mul, zero_apply,\n Function.comp_apply] at hzero \n[GOAL]\ncase intro\nR : Type u\nM : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Nontrivial R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module.Free R M\ninst\u271d : Module.Finite R M\nf : M \u2192\u2097[R] M\nhf : Function.Injective \u2191f\nh : coeff (minpoly R f) 0 = 0\nP : R[X]\nhP : minpoly R f = X * P\nhdegP : degree P < degree (minpoly R f)\nhPmonic : Monic P\nhzero : \u2200 (x : M), \u2191(\u2191(aeval f) P) x = 0\n\u22a2 False\n[PROOFSTEP]\nexact not_le.2 hdegP (minpoly.min _ _ hPmonic (ext hzero))\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Charpoly.Basic", "llama_tokens": 3388, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154240079185319, "lm_q2_score": 0.5621765008857982, "lm_q1q2_score": 0.4021945654213338}} {"text": "[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nx : X\n\u22a2 sum (zipWith d [x] ([] ++ [x])) = 0\n[PROOFSTEP]\nsimp [dist_self]\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nx y : X\n\u22a2 \u2a05 (l : List X), sum (zipWith d (x :: l) (l ++ [y])) = \u2a05 (l : List X), sum (zipWith d (y :: l) (l ++ [x]))\n[PROOFSTEP]\nrefine' reverse_surjective.iInf_congr _ fun l => _\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nx y : X\nl : List X\n\u22a2 sum (zipWith d (y :: reverse l) (reverse l ++ [x])) = sum (zipWith d (x :: l) (l ++ [y]))\n[PROOFSTEP]\nrw [\u2190 sum_reverse, zipWith_distrib_reverse, reverse_append, reverse_reverse, reverse_singleton, singleton_append,\n reverse_cons, reverse_reverse, zipWith_comm_of_comm _ dist_comm]\n[GOAL]\ncase h\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nx y : X\nl : List X\n\u22a2 length (y :: reverse l) = length (reverse l ++ [x])\n[PROOFSTEP]\nsimp only [length, length_append]\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nx y z : X\n\u22a2 dist x z \u2264 dist x y + dist y z\n[PROOFSTEP]\nunfold dist\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nx y z : X\n\u22a2 { dist := fun x y => \u2191(\u2a05 (l : List X), sum (zipWith d (x :: l) (l ++ [y]))) }.1 x z \u2264\n { dist := fun x y => \u2191(\u2a05 (l : List X), sum (zipWith d (x :: l) (l ++ [y]))) }.1 x y +\n { dist := fun x y => \u2191(\u2a05 (l : List X), sum (zipWith d (x :: l) (l ++ [y]))) }.1 y z\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_add, NNReal.coe_le_coe]\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nx y z : X\n\u22a2 \u2a05 (l : List X), sum (zipWith d (x :: l) (l ++ [z])) \u2264\n (\u2a05 (l : List X), sum (zipWith d (x :: l) (l ++ [y]))) + \u2a05 (l : List X), sum (zipWith d (y :: l) (l ++ [z]))\n[PROOFSTEP]\nrefine' NNReal.le_iInf_add_iInf fun lxy lyz => _\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nx y z : X\nlxy lyz : List X\n\u22a2 \u2a05 (l : List X), sum (zipWith d (x :: l) (l ++ [z])) \u2264\n sum (zipWith d (x :: lxy) (lxy ++ [y])) + sum (zipWith d (y :: lyz) (lyz ++ [z]))\n[PROOFSTEP]\ncalc\n \u2a05 l, (zipWith d (x :: l) (l ++ [z])).sum \u2264 (zipWith d (x :: lxy ++ y :: lyz) ((lxy ++ y :: lyz) ++ [z])).sum :=\n ciInf_le (OrderBot.bddBelow _) (lxy ++ y :: lyz)\n _ = (zipWith d (x :: lxy) (lxy ++ [y])).sum + (zipWith d (y :: lyz) (lyz ++ [z])).sum :=\n by\n rw [\u2190 sum_append, \u2190 zipWith_append, cons_append, \u2190 @singleton_append _ y, append_assoc, append_assoc, append_assoc]\n rw [length_cons, length_append, length_singleton]\n -- Porting note: `edist_dist` is no longer inferred\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nx y z : X\nlxy lyz : List X\n\u22a2 sum (zipWith d (x :: lxy ++ y :: lyz) (lxy ++ y :: lyz ++ [z])) =\n sum (zipWith d (x :: lxy) (lxy ++ [y])) + sum (zipWith d (y :: lyz) (lyz ++ [z]))\n[PROOFSTEP]\nrw [\u2190 sum_append, \u2190 zipWith_append, cons_append, \u2190 @singleton_append _ y, append_assoc, append_assoc, append_assoc]\n[GOAL]\ncase h\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nx y z : X\nlxy lyz : List X\n\u22a2 length (x :: lxy) = length (lxy ++ [y])\n[PROOFSTEP]\nrw [length_cons, length_append, length_singleton]\n -- Porting note: `edist_dist` is no longer inferred\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nx y : X\n\u22a2 sum (zipWith d [x] ([] ++ [y])) = d x y\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx y : X\n\u22a2 \u2191(d x y) \u2264 2 * dist x y\n[PROOFSTEP]\nrw [dist_ofPreNNDist, \u2190 NNReal.coe_two, \u2190 NNReal.coe_mul, NNReal.mul_iInf, NNReal.coe_le_coe]\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx y : X\n\u22a2 d x y \u2264 \u2a05 (i : List X), 2 * sum (zipWith d (x :: i) (i ++ [y]))\n[PROOFSTEP]\nrefine' le_ciInf fun l => _\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx y : X\nl : List X\n\u22a2 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\n[PROOFSTEP]\nhave hd\u2080_trans : Transitive fun x y => d x y = 0 :=\n by\n intro a b c hab hbc\n rw [\u2190 nonpos_iff_eq_zero]\n simpa only [nonpos_iff_eq_zero, hab, hbc, dist_self c, max_self, mul_zero] using hd a b c c\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx y : X\nl : List X\n\u22a2 Transitive fun x y => d x y = 0\n[PROOFSTEP]\nintro a b c hab hbc\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx y : X\nl : List X\na b c : X\nhab : d a b = 0\nhbc : d b c = 0\n\u22a2 d a c = 0\n[PROOFSTEP]\nrw [\u2190 nonpos_iff_eq_zero]\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx y : X\nl : List X\na b c : X\nhab : d a b = 0\nhbc : d b c = 0\n\u22a2 d a c \u2264 0\n[PROOFSTEP]\nsimpa only [nonpos_iff_eq_zero, hab, hbc, dist_self c, max_self, mul_zero] using hd a b c c\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx y : X\nl : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\n\u22a2 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\n[PROOFSTEP]\nhaveI : IsTrans X fun x y => d x y = 0 := \u27e8hd\u2080_trans\u27e9\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx y : X\nl : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\n\u22a2 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\n[PROOFSTEP]\ninduction' hn : length l using Nat.strong_induction_on with n ihn generalizing x y l\n[GOAL]\ncase h\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn\u271d : length l\u271d = x\u271d\nn : \u2115\nihn : \u2200 (m : \u2115), m < n \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nx y : X\nl : List X\nhn : length l = n\n\u22a2 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\n[PROOFSTEP]\nsimp only at ihn \n[GOAL]\ncase h\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn\u271d : length l\u271d = x\u271d\nn : \u2115\nihn : \u2200 (m : \u2115), m < n \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nx y : X\nl : List X\nhn : length l = n\n\u22a2 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\n[PROOFSTEP]\nsubst n\n[GOAL]\ncase h\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\n\u22a2 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\n[PROOFSTEP]\nset L := zipWith d (x :: l) (l ++ [y])\n[GOAL]\ncase h\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\n\u22a2 d x y \u2264 2 * sum L\n[PROOFSTEP]\nhave hL_len : length L = length l + 1 := by simp\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\n\u22a2 length L = length l + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\n\u22a2 d x y \u2264 2 * sum L\n[PROOFSTEP]\ncases' eq_or_ne (d x y) 0 with hd\u2080 hd\u2080\n[GOAL]\ncase h.inl\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y = 0\n\u22a2 d x y \u2264 2 * sum L\n[PROOFSTEP]\nsimp only [hd\u2080, zero_le]\n[GOAL]\ncase h.inr\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\n\u22a2 d x y \u2264 2 * sum L\n[PROOFSTEP]\nrsuffices \u27e8z, z', hxz, hzz', hz'y\u27e9 : \u2203 z z' : X, d x z \u2264 L.sum \u2227 d z z' \u2264 L.sum \u2227 d z' y \u2264 L.sum\n[GOAL]\ncase h.inr.intro.intro.intro.intro\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\nz z' : X\nhxz : d x z \u2264 sum L\nhzz' : d z z' \u2264 sum L\nhz'y : d z' y \u2264 sum L\n\u22a2 d x y \u2264 2 * sum L\n[PROOFSTEP]\nexact (hd x z z' y).trans (mul_le_mul_left' (max_le hxz (max_le hzz' hz'y)) _)\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\n\u22a2 \u2203 z z', d x z \u2264 sum L \u2227 d z z' \u2264 sum L \u2227 d z' y \u2264 sum L\n[PROOFSTEP]\nset s : Set \u2115 := {m : \u2115 | 2 * (take m L).sum \u2264 L.sum}\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\n\u22a2 \u2203 z z', d x z \u2264 sum L \u2227 d z z' \u2264 sum L \u2227 d z' y \u2264 sum L\n[PROOFSTEP]\nhave hs\u2080 : 0 \u2208 s := by simp\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\n\u22a2 0 \u2208 s\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\n\u22a2 \u2203 z z', d x z \u2264 sum L \u2227 d z z' \u2264 sum L \u2227 d z' y \u2264 sum L\n[PROOFSTEP]\nhave hsne : s.Nonempty := \u27e80, hs\u2080\u27e9\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\n\u22a2 \u2203 z z', d x z \u2264 sum L \u2227 d z z' \u2264 sum L \u2227 d z' y \u2264 sum L\n[PROOFSTEP]\nobtain \u27e8M, hMl, hMs\u27e9 : \u2203 M \u2264 length l, IsGreatest s M :=\n by\n have hs_ub : length l \u2208 upperBounds s := by\n intro m hm\n rw [\u2190 not_lt, Nat.lt_iff_add_one_le, \u2190 hL_len]\n intro hLm\n rw [mem_setOf_eq, take_all_of_le hLm, two_mul, add_le_iff_nonpos_left, nonpos_iff_eq_zero, sum_eq_zero_iff, \u2190\n all\u2082_iff_forall, all\u2082_zipWith, \u2190 chain_append_singleton_iff_forall\u2082] at hm <;>\n [skip; simp]\n exact hd\u2080 (hm.rel (mem_append.2 <| Or.inr <| mem_singleton_self _))\n have hs_bdd : BddAbove s := \u27e8length l, hs_ub\u27e9\n exact \u27e8sSup s, csSup_le hsne hs_ub, \u27e8Nat.sSup_mem hsne hs_bdd, fun k => le_csSup hs_bdd\u27e9\u27e9\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\n\u22a2 \u2203 M, M \u2264 length l \u2227 IsGreatest s M\n[PROOFSTEP]\nhave hs_ub : length l \u2208 upperBounds s := by\n intro m hm\n rw [\u2190 not_lt, Nat.lt_iff_add_one_le, \u2190 hL_len]\n intro hLm\n rw [mem_setOf_eq, take_all_of_le hLm, two_mul, add_le_iff_nonpos_left, nonpos_iff_eq_zero, sum_eq_zero_iff, \u2190\n all\u2082_iff_forall, all\u2082_zipWith, \u2190 chain_append_singleton_iff_forall\u2082] at hm <;>\n [skip; simp]\n exact hd\u2080 (hm.rel (mem_append.2 <| Or.inr <| mem_singleton_self _))\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\n\u22a2 length l \u2208 upperBounds s\n[PROOFSTEP]\nintro m hm\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nm : \u2115\nhm : m \u2208 s\n\u22a2 m \u2264 length l\n[PROOFSTEP]\nrw [\u2190 not_lt, Nat.lt_iff_add_one_le, \u2190 hL_len]\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nm : \u2115\nhm : m \u2208 s\n\u22a2 \u00aclength L \u2264 m\n[PROOFSTEP]\nintro hLm\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nm : \u2115\nhm : m \u2208 s\nhLm : length L \u2264 m\n\u22a2 False\n[PROOFSTEP]\nrw [mem_setOf_eq, take_all_of_le hLm, two_mul, add_le_iff_nonpos_left, nonpos_iff_eq_zero, sum_eq_zero_iff, \u2190\n all\u2082_iff_forall, all\u2082_zipWith, \u2190 chain_append_singleton_iff_forall\u2082] at hm <;>\n [skip; simp]\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nm : \u2115\nhm : m \u2208 s\nhLm : length L \u2264 m\n\u22a2 False\n[PROOFSTEP]\nrw [mem_setOf_eq, take_all_of_le hLm, two_mul, add_le_iff_nonpos_left, nonpos_iff_eq_zero, sum_eq_zero_iff, \u2190\n all\u2082_iff_forall, all\u2082_zipWith, \u2190 chain_append_singleton_iff_forall\u2082] at hm \n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nm : \u2115\nhm : Chain (fun x y => d x y = 0) x (l ++ [y])\nhLm : length L \u2264 m\n\u22a2 False\n[PROOFSTEP]\nskip\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nm : \u2115\nhm : All\u2082 (fun x => x = 0) L\nhLm : length L \u2264 m\n\u22a2 length (x :: l) = length (l ++ [y])\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nm : \u2115\nhm : Chain (fun x y => d x y = 0) x (l ++ [y])\nhLm : length L \u2264 m\n\u22a2 False\n[PROOFSTEP]\nexact hd\u2080 (hm.rel (mem_append.2 <| Or.inr <| mem_singleton_self _))\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nhs_ub : length l \u2208 upperBounds s\n\u22a2 \u2203 M, M \u2264 length l \u2227 IsGreatest s M\n[PROOFSTEP]\nhave hs_bdd : BddAbove s := \u27e8length l, hs_ub\u27e9\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nhs_ub : length l \u2208 upperBounds s\nhs_bdd : BddAbove s\n\u22a2 \u2203 M, M \u2264 length l \u2227 IsGreatest s M\n[PROOFSTEP]\nexact \u27e8sSup s, csSup_le hsne hs_ub, \u27e8Nat.sSup_mem hsne hs_bdd, fun k => le_csSup hs_bdd\u27e9\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : M \u2264 length l\nhMs : IsGreatest s M\n\u22a2 \u2203 z z', d x z \u2264 sum L \u2227 d z z' \u2264 sum L \u2227 d z' y \u2264 sum L\n[PROOFSTEP]\nhave hM_lt : M < length L := by rwa [hL_len, Nat.lt_succ_iff]\n[GOAL]\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : M \u2264 length l\nhMs : IsGreatest s M\n\u22a2 M < length L\n[PROOFSTEP]\nrwa [hL_len, Nat.lt_succ_iff]\n[GOAL]\ncase intro.intro\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\n\u22a2 \u2203 z z', d x z \u2264 sum L \u2227 d z z' \u2264 sum L \u2227 d z' y \u2264 sum L\n[PROOFSTEP]\nhave hM_ltx : M < length (x :: l) := lt_length_left_of_zipWith hM_lt\n[GOAL]\ncase intro.intro\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\nhM_ltx : M < length (x :: l)\n\u22a2 \u2203 z z', d x z \u2264 sum L \u2227 d z z' \u2264 sum L \u2227 d z' y \u2264 sum L\n[PROOFSTEP]\nhave hM_lty : M < length (l ++ [y]) := lt_length_right_of_zipWith hM_lt\n[GOAL]\ncase intro.intro\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\nhM_ltx : M < length (x :: l)\nhM_lty : M < length (l ++ [y])\n\u22a2 \u2203 z z', d x z \u2264 sum L \u2227 d z z' \u2264 sum L \u2227 d z' y \u2264 sum L\n[PROOFSTEP]\nrefine' \u27e8(x :: l).get \u27e8M, hM_ltx\u27e9, (l ++ [y]).get \u27e8M, hM_lty\u27e9, _, _, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\nhM_ltx : M < length (x :: l)\nhM_lty : M < length (l ++ [y])\n\u22a2 d x (List.get (x :: l) { val := M, isLt := hM_ltx }) \u2264 sum L\n[PROOFSTEP]\ncases M with\n| zero => simp [dist_self, List.get]\n| succ M =>\n rw [Nat.succ_le_iff] at hMl \n have hMl' : length (take M l) = M := (length_take _ _).trans (min_eq_left hMl.le)\n simp only [List.get]\n refine' (ihn _ hMl _ _ _ hMl').trans _\n convert hMs.1.out\n rw [zipWith_distrib_take, take, take_succ, get?_append hMl, get?_eq_get hMl, \u2190 Option.coe_def, Option.to_list_some,\n take_append_of_le_length hMl.le]\n[GOAL]\ncase intro.intro.refine'_1\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\nhM_ltx : M < length (x :: l)\nhM_lty : M < length (l ++ [y])\n\u22a2 d x (List.get (x :: l) { val := M, isLt := hM_ltx }) \u2264 sum L\n[PROOFSTEP]\ncases M with\n| zero => simp [dist_self, List.get]\n| succ M =>\n rw [Nat.succ_le_iff] at hMl \n have hMl' : length (take M l) = M := (length_take _ _).trans (min_eq_left hMl.le)\n simp only [List.get]\n refine' (ihn _ hMl _ _ _ hMl').trans _\n convert hMs.1.out\n rw [zipWith_distrib_take, take, take_succ, get?_append hMl, get?_eq_get hMl, \u2190 Option.coe_def, Option.to_list_some,\n take_append_of_le_length hMl.le]\n[GOAL]\ncase intro.intro.refine'_1.zero\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nhMl : Nat.zero \u2264 length l\nhMs : IsGreatest s Nat.zero\nhM_lt : Nat.zero < length L\nhM_ltx : Nat.zero < length (x :: l)\nhM_lty : Nat.zero < length (l ++ [y])\n\u22a2 d x (List.get (x :: l) { val := Nat.zero, isLt := hM_ltx }) \u2264 sum L\n[PROOFSTEP]\n\n| zero => simp [dist_self, List.get]\n[GOAL]\ncase intro.intro.refine'_1.zero\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nhMl : Nat.zero \u2264 length l\nhMs : IsGreatest s Nat.zero\nhM_lt : Nat.zero < length L\nhM_ltx : Nat.zero < length (x :: l)\nhM_lty : Nat.zero < length (l ++ [y])\n\u22a2 d x (List.get (x :: l) { val := Nat.zero, isLt := hM_ltx }) \u2264 sum L\n[PROOFSTEP]\nsimp [dist_self, List.get]\n[GOAL]\ncase intro.intro.refine'_1.succ\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : Nat.succ M \u2264 length l\nhMs : IsGreatest s (Nat.succ M)\nhM_lt : Nat.succ M < length L\nhM_ltx : Nat.succ M < length (x :: l)\nhM_lty : Nat.succ M < length (l ++ [y])\n\u22a2 d x (List.get (x :: l) { val := Nat.succ M, isLt := hM_ltx }) \u2264 sum L\n[PROOFSTEP]\n\n| succ M =>\n rw [Nat.succ_le_iff] at hMl \n have hMl' : length (take M l) = M := (length_take _ _).trans (min_eq_left hMl.le)\n simp only [List.get]\n refine' (ihn _ hMl _ _ _ hMl').trans _\n convert hMs.1.out\n rw [zipWith_distrib_take, take, take_succ, get?_append hMl, get?_eq_get hMl, \u2190 Option.coe_def, Option.to_list_some,\n take_append_of_le_length hMl.le]\n[GOAL]\ncase intro.intro.refine'_1.succ\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : Nat.succ M \u2264 length l\nhMs : IsGreatest s (Nat.succ M)\nhM_lt : Nat.succ M < length L\nhM_ltx : Nat.succ M < length (x :: l)\nhM_lty : Nat.succ M < length (l ++ [y])\n\u22a2 d x (List.get (x :: l) { val := Nat.succ M, isLt := hM_ltx }) \u2264 sum L\n[PROOFSTEP]\nrw [Nat.succ_le_iff] at hMl \n[GOAL]\ncase intro.intro.refine'_1.succ\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : M < length l\nhMs : IsGreatest s (Nat.succ M)\nhM_lt : Nat.succ M < length L\nhM_ltx : Nat.succ M < length (x :: l)\nhM_lty : Nat.succ M < length (l ++ [y])\n\u22a2 d x (List.get (x :: l) { val := Nat.succ M, isLt := hM_ltx }) \u2264 sum L\n[PROOFSTEP]\nhave hMl' : length (take M l) = M := (length_take _ _).trans (min_eq_left hMl.le)\n[GOAL]\ncase intro.intro.refine'_1.succ\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : M < length l\nhMs : IsGreatest s (Nat.succ M)\nhM_lt : Nat.succ M < length L\nhM_ltx : Nat.succ M < length (x :: l)\nhM_lty : Nat.succ M < length (l ++ [y])\nhMl' : length (take M l) = M\n\u22a2 d x (List.get (x :: l) { val := Nat.succ M, isLt := hM_ltx }) \u2264 sum L\n[PROOFSTEP]\nsimp only [List.get]\n[GOAL]\ncase intro.intro.refine'_1.succ\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : M < length l\nhMs : IsGreatest s (Nat.succ M)\nhM_lt : Nat.succ M < length L\nhM_ltx : Nat.succ M < length (x :: l)\nhM_lty : Nat.succ M < length (l ++ [y])\nhMl' : length (take M l) = M\n\u22a2 d x (List.get l { val := M, isLt := (_ : Nat.succ M \u2264 length l) }) \u2264 sum (zipWith d (x :: l) (l ++ [y]))\n[PROOFSTEP]\nrefine' (ihn _ hMl _ _ _ hMl').trans _\n[GOAL]\ncase intro.intro.refine'_1.succ\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : M < length l\nhMs : IsGreatest s (Nat.succ M)\nhM_lt : Nat.succ M < length L\nhM_ltx : Nat.succ M < length (x :: l)\nhM_lty : Nat.succ M < length (l ++ [y])\nhMl' : length (take M l) = M\n\u22a2 2 * sum (zipWith d (x :: take M l) (take M l ++ [List.get l { val := M, isLt := (_ : Nat.succ M \u2264 length l) }])) \u2264\n sum (zipWith d (x :: l) (l ++ [y]))\n[PROOFSTEP]\nconvert hMs.1.out\n[GOAL]\ncase h.e'_3.h.e'_6.h.e'_4\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : M < length l\nhMs : IsGreatest s (Nat.succ M)\nhM_lt : Nat.succ M < length L\nhM_ltx : Nat.succ M < length (x :: l)\nhM_lty : Nat.succ M < length (l ++ [y])\nhMl' : length (take M l) = M\n\u22a2 zipWith d (x :: take M l) (take M l ++ [List.get l { val := M, isLt := (_ : Nat.succ M \u2264 length l) }]) =\n take (Nat.succ M) L\n[PROOFSTEP]\nrw [zipWith_distrib_take, take, take_succ, get?_append hMl, get?_eq_get hMl, \u2190 Option.coe_def, Option.to_list_some,\n take_append_of_le_length hMl.le]\n[GOAL]\ncase intro.intro.refine'_2\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\nhM_ltx : M < length (x :: l)\nhM_lty : M < length (l ++ [y])\n\u22a2 d (List.get (x :: l) { val := M, isLt := hM_ltx }) (List.get (l ++ [y]) { val := M, isLt := hM_lty }) \u2264 sum L\n[PROOFSTEP]\nexact single_le_sum (fun x _ => zero_le x) _ (mem_iff_get.2 \u27e8\u27e8M, hM_lt\u27e9, get_zipWith\u27e9)\n[GOAL]\ncase intro.intro.refine'_3\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\nhM_ltx : M < length (x :: l)\nhM_lty : M < length (l ++ [y])\n\u22a2 d (List.get (l ++ [y]) { val := M, isLt := hM_lty }) y \u2264 sum L\n[PROOFSTEP]\nrcases hMl.eq_or_lt with (rfl | hMl)\n[GOAL]\ncase intro.intro.refine'_3.inl\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nhMl : length l \u2264 length l\nhMs : IsGreatest s (length l)\nhM_lt : length l < length L\nhM_ltx : length l < length (x :: l)\nhM_lty : length l < length (l ++ [y])\n\u22a2 d (List.get (l ++ [y]) { val := length l, isLt := hM_lty }) y \u2264 sum L\n[PROOFSTEP]\nsimp only [get_append_right' le_rfl, sub_self, get_singleton, dist_self, zero_le]\n[GOAL]\ncase intro.intro.refine'_3.inr\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl\u271d : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\nhM_ltx : M < length (x :: l)\nhM_lty : M < length (l ++ [y])\nhMl : M < length l\n\u22a2 d (List.get (l ++ [y]) { val := M, isLt := hM_lty }) y \u2264 sum L\n[PROOFSTEP]\nrw [get_append _ hMl]\n[GOAL]\ncase intro.intro.refine'_3.inr\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl\u271d : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\nhM_ltx : M < length (x :: l)\nhM_lty : M < length (l ++ [y])\nhMl : M < length l\n\u22a2 d (List.get l { val := M, isLt := hMl }) y \u2264 sum L\n[PROOFSTEP]\nhave hlen : length (drop (M + 1) l) = length l - (M + 1) := length_drop _ _\n[GOAL]\ncase intro.intro.refine'_3.inr\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl\u271d : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\nhM_ltx : M < length (x :: l)\nhM_lty : M < length (l ++ [y])\nhMl : M < length l\nhlen : length (drop (M + 1) l) = length l - (M + 1)\n\u22a2 d (List.get l { val := M, isLt := hMl }) y \u2264 sum L\n[PROOFSTEP]\nhave hlen_lt : length l - (M + 1) < length l := Nat.sub_lt_of_pos_le M.succ_pos hMl\n[GOAL]\ncase intro.intro.refine'_3.inr\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl\u271d : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\nhM_ltx : M < length (x :: l)\nhM_lty : M < length (l ++ [y])\nhMl : M < length l\nhlen : length (drop (M + 1) l) = length l - (M + 1)\nhlen_lt : length l - (M + 1) < length l\n\u22a2 d (List.get l { val := M, isLt := hMl }) y \u2264 sum L\n[PROOFSTEP]\nrefine' (ihn _ hlen_lt _ y _ hlen).trans _\n[GOAL]\ncase intro.intro.refine'_3.inr\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl\u271d : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\nhM_ltx : M < length (x :: l)\nhM_lty : M < length (l ++ [y])\nhMl : M < length l\nhlen : length (drop (M + 1) l) = length l - (M + 1)\nhlen_lt : length l - (M + 1) < length l\n\u22a2 2 * sum (zipWith d (List.get l { val := M, isLt := hMl } :: drop (M + 1) l) (drop (M + 1) l ++ [y])) \u2264 sum L\n[PROOFSTEP]\nrw [cons_get_drop_succ]\n[GOAL]\ncase intro.intro.refine'_3.inr\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl\u271d : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\nhM_ltx : M < length (x :: l)\nhM_lty : M < length (l ++ [y])\nhMl : M < length l\nhlen : length (drop (M + 1) l) = length l - (M + 1)\nhlen_lt : length l - (M + 1) < length l\n\u22a2 2 * sum (zipWith d (drop (\u2191{ val := M, isLt := hMl }) l) (drop (M + 1) l ++ [y])) \u2264 sum L\n[PROOFSTEP]\nhave hMs' : L.sum \u2264 2 * (L.take (M + 1)).sum := not_lt.1 fun h => (hMs.2 h.le).not_lt M.lt_succ_self\n[GOAL]\ncase intro.intro.refine'_3.inr\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl\u271d : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\nhM_ltx : M < length (x :: l)\nhM_lty : M < length (l ++ [y])\nhMl : M < length l\nhlen : length (drop (M + 1) l) = length l - (M + 1)\nhlen_lt : length l - (M + 1) < length l\nhMs' : sum L \u2264 2 * sum (take (M + 1) L)\n\u22a2 2 * sum (zipWith d (drop (\u2191{ val := M, isLt := hMl }) l) (drop (M + 1) l ++ [y])) \u2264 sum L\n[PROOFSTEP]\nrw [\u2190 sum_take_add_sum_drop L (M + 1), two_mul, add_le_add_iff_left, \u2190 add_le_add_iff_right, sum_take_add_sum_drop, \u2190\n two_mul] at hMs' \n[GOAL]\ncase intro.intro.refine'_3.inr\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl\u271d : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\nhM_ltx : M < length (x :: l)\nhM_lty : M < length (l ++ [y])\nhMl : M < length l\nhlen : length (drop (M + 1) l) = length l - (M + 1)\nhlen_lt : length l - (M + 1) < length l\nhMs'\u271d : sum (drop (M + 1) L) \u2264 sum (take (M + 1) L)\nhMs' : 2 * sum (drop (M + 1) L) \u2264 sum L\n\u22a2 2 * sum (zipWith d (drop (\u2191{ val := M, isLt := hMl }) l) (drop (M + 1) l ++ [y])) \u2264 sum L\n[PROOFSTEP]\nconvert hMs'\n[GOAL]\ncase h.e'_3.h.e'_6.h.e'_4\nX : Type u_1\nd : X \u2192 X \u2192 \u211d\u22650\ndist_self : \u2200 (x : X), d x x = 0\ndist_comm : \u2200 (x y : X), d x y = d y x\nhd : \u2200 (x\u2081 x\u2082 x\u2083 x\u2084 : X), d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\nx\u271d\u00b9 y\u271d : X\nl\u271d : List X\nhd\u2080_trans : Transitive fun x y => d x y = 0\nthis : IsTrans X fun x y => d x y = 0\nx\u271d : \u2115\nhn : length l\u271d = x\u271d\nx y : X\nl : List X\nihn : \u2200 (m : \u2115), m < length l \u2192 \u2200 (x y : X) (l : List X), length l = m \u2192 d x y \u2264 2 * sum (zipWith d (x :: l) (l ++ [y]))\nL : List \u211d\u22650 := zipWith d (x :: l) (l ++ [y])\nhL_len : length L = length l + 1\nhd\u2080 : d x y \u2260 0\ns : Set \u2115 := {m | 2 * sum (take m L) \u2264 sum L}\nhs\u2080 : 0 \u2208 s\nhsne : Set.Nonempty s\nM : \u2115\nhMl\u271d : M \u2264 length l\nhMs : IsGreatest s M\nhM_lt : M < length L\nhM_ltx : M < length (x :: l)\nhM_lty : M < length (l ++ [y])\nhMl : M < length l\nhlen : length (drop (M + 1) l) = length l - (M + 1)\nhlen_lt : length l - (M + 1) < length l\nhMs'\u271d : sum (drop (M + 1) L) \u2264 sum (take (M + 1) L)\nhMs' : 2 * sum (drop (M + 1) L) \u2264 sum L\n\u22a2 zipWith d (drop (\u2191{ val := M, isLt := hMl }) l) (drop (M + 1) l ++ [y]) = drop (M + 1) L\n[PROOFSTEP]\nrwa [zipWith_distrib_drop, drop, drop_append_of_le_length]\n[GOAL]\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\n\u22a2 \u2203 I, PseudoMetricSpace.toUniformSpace = inst\u271d\u00b9\n[PROOFSTEP]\nclassical\n /- Choose a fast decreasing antitone basis `U : \u2115 \u2192 set (X \u00d7 X)` of the uniformity filter `\ud835\udce4 X`.\n Define `d x y : \u211d\u22650` to be `(1 / 2) ^ n`, where `n` is the minimal index of `U n` that\n separates `x` and `y`: `(x, y) \u2209 U n`, or `0` if `x` is not separated from `y`. This function\n satisfies the assumptions of `PseudoMetricSpace.ofPreNNDist` and\n `PseudoMetricSpace.le_two_mul_dist_ofPreNNDist`, hence the distance given by the former pseudo\n metric space structure is Lipschitz equivalent to the `d`. Thus the uniformities generated by\n `d` and `dist` are equal. Since the former uniformity is equal to `\ud835\udce4 X`, the latter is equal to\n `\ud835\udce4 X` as well. -/\nobtain \u27e8U, hU_symm, hU_comp, hB\u27e9 :\n \u2203 U : \u2115 \u2192 Set (X \u00d7 X),\n (\u2200 n, SymmetricRel (U n)) \u2227 (\u2200 \u2983m n\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m) \u2227 (\ud835\udce4 X).HasAntitoneBasis U :=\n by\n rcases UniformSpace.has_seq_basis X with \u27e8V, hB, hV_symm\u27e9\n rcases hB.subbasis_with_rel fun m =>\n hB.tendsto_smallSets.eventually (eventually_uniformity_iterate_comp_subset (hB.mem m) 2) with\n \u27e8\u03c6, -, h\u03c6_comp, h\u03c6B\u27e9\n exact \u27e8V \u2218 \u03c6, fun n => hV_symm _, h\u03c6_comp, h\u03c6B\u27e9\nletI := UniformSpace.separationSetoid X\nset d : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, (x, y) \u2209 U n then (1 / 2) ^ Nat.find h else 0\nhave hd\u2080 : \u2200 {x y}, d x y = 0 \u2194 x \u2248 y := by\n intro x y\n refine' Iff.trans _ hB.mem_separationRel.symm\n simp only [true_imp_iff]\n split_ifs with h\n \u00b7 rw [\u2190 not_forall] at h \n simp [h, pow_eq_zero_iff']\n \u00b7 simpa only [not_exists, Classical.not_not, eq_self_iff_true, true_iff_iff] using h\nhave hd_symm : \u2200 x y, d x y = d y x := by\n intro x y\n simp only [@SymmetricRel.mk_mem_comm _ _ (hU_symm _) x y]\nhave hr : (1 / 2 : \u211d\u22650) \u2208 Ioo (0 : \u211d\u22650) 1 := \u27e8half_pos one_pos, NNReal.half_lt_self one_ne_zero\u27e9\nletI I := PseudoMetricSpace.ofPreNNDist d (fun x => hd\u2080.2 (Setoid.refl _)) hd_symm\nhave hdist_le : \u2200 x y, dist x y \u2264 d x y := PseudoMetricSpace.dist_ofPreNNDist_le _ _ _\nhave hle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 (x, y) \u2209 U n :=\n by\n intro x y n\n dsimp only []\n split_ifs with h\n \u00b7 rw [(strictAnti_pow hr.1 hr.2).le_iff_le, Nat.find_le_iff]\n exact \u27e8fun \u27e8m, hmn, hm\u27e9 hn => hm (hB.antitone hmn hn), fun h => \u27e8n, le_rfl, h\u27e9\u27e9\n \u00b7 push_neg at h \n simp only [h, not_true, (pow_pos hr.1 _).not_le]\nhave hd_le : \u2200 x y, \u2191(d x y) \u2264 2 * dist x y :=\n by\n refine' PseudoMetricSpace.le_two_mul_dist_ofPreNNDist _ _ _ fun x\u2081 x\u2082 x\u2083 x\u2084 => _\n by_cases H : \u2203 n, (x\u2081, x\u2084) \u2209 U n\n \u00b7 refine' (dif_pos H).trans_le _\n rw [\u2190 NNReal.div_le_iff' two_ne_zero, \u2190 mul_one_div (_ ^ _), \u2190 pow_succ']\n simp only [le_max_iff, hle_d, \u2190 not_and_or]\n rintro \u27e8h\u2081\u2082, h\u2082\u2083, h\u2083\u2084\u27e9\n refine' Nat.find_spec H (hU_comp (lt_add_one <| Nat.find H) _)\n exact \u27e8x\u2082, h\u2081\u2082, x\u2083, h\u2082\u2083, h\u2083\u2084\u27e9\n \u00b7\n exact\n (dif_neg H).trans_le\n (zero_le _)\n -- Porting note: without the next line, `uniformity_basis_dist_pow` ends up introducing some\n -- `Subtype.val` applications instead of `NNReal.toReal`.\nrw [mem_Ioo, \u2190 NNReal.coe_lt_coe, \u2190 NNReal.coe_lt_coe] at hr \nrefine' \u27e8I, uniformSpace_eq <| (uniformity_basis_dist_pow hr.1 hr.2).ext hB.toHasBasis _ _\u27e9\n\u00b7 refine' fun n hn => \u27e8n, hn, fun x hx => (hdist_le _ _).trans_lt _\u27e9\n rwa [\u2190 NNReal.coe_pow, NNReal.coe_lt_coe, \u2190 not_le, hle_d, Classical.not_not, Prod.mk.eta]\n\u00b7 refine' fun n _ => \u27e8n + 1, trivial, fun x hx => _\u27e9\n rw [mem_setOf_eq] at hx \n contrapose! hx\n refine' le_trans _ ((div_le_iff' (zero_lt_two' \u211d)).2 (hd_le x.1 x.2))\n rwa [\u2190 NNReal.coe_two, \u2190 NNReal.coe_div, \u2190 NNReal.coe_pow, NNReal.coe_le_coe, pow_succ', mul_one_div,\n NNReal.div_le_iff two_ne_zero, div_mul_cancel _ (two_ne_zero' \u211d\u22650), hle_d, Prod.mk.eta]\n[GOAL]\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\n\u22a2 \u2203 I, PseudoMetricSpace.toUniformSpace = inst\u271d\u00b9\n[PROOFSTEP]\nobtain \u27e8U, hU_symm, hU_comp, hB\u27e9 :\n \u2203 U : \u2115 \u2192 Set (X \u00d7 X),\n (\u2200 n, SymmetricRel (U n)) \u2227 (\u2200 \u2983m n\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m) \u2227 (\ud835\udce4 X).HasAntitoneBasis U :=\n by\n rcases UniformSpace.has_seq_basis X with \u27e8V, hB, hV_symm\u27e9\n rcases hB.subbasis_with_rel fun m =>\n hB.tendsto_smallSets.eventually (eventually_uniformity_iterate_comp_subset (hB.mem m) 2) with\n \u27e8\u03c6, -, h\u03c6_comp, h\u03c6B\u27e9\n exact \u27e8V \u2218 \u03c6, fun n => hV_symm _, h\u03c6_comp, h\u03c6B\u27e9\n[GOAL]\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\n\u22a2 \u2203 U, (\u2200 (n : \u2115), SymmetricRel (U n)) \u2227 (\u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m) \u2227 HasAntitoneBasis (\ud835\udce4 X) U\n[PROOFSTEP]\nrcases UniformSpace.has_seq_basis X with \u27e8V, hB, hV_symm\u27e9\n[GOAL]\ncase intro.intro\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nV : \u2115 \u2192 Set (X \u00d7 X)\nhB : HasAntitoneBasis (\ud835\udce4 X) V\nhV_symm : \u2200 (n : \u2115), SymmetricRel (V n)\n\u22a2 \u2203 U, (\u2200 (n : \u2115), SymmetricRel (U n)) \u2227 (\u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m) \u2227 HasAntitoneBasis (\ud835\udce4 X) U\n[PROOFSTEP]\nrcases hB.subbasis_with_rel fun m =>\n hB.tendsto_smallSets.eventually (eventually_uniformity_iterate_comp_subset (hB.mem m) 2) with\n \u27e8\u03c6, -, h\u03c6_comp, h\u03c6B\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nV : \u2115 \u2192 Set (X \u00d7 X)\nhB : HasAntitoneBasis (\ud835\udce4 X) V\nhV_symm : \u2200 (n : \u2115), SymmetricRel (V n)\n\u03c6 : \u2115 \u2192 \u2115\nh\u03c6_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 (fun x => V (\u03c6 n) \u25cb x)^[2] (V (\u03c6 n)) \u2286 V (\u03c6 m)\nh\u03c6B : HasAntitoneBasis (\ud835\udce4 X) (V \u2218 \u03c6)\n\u22a2 \u2203 U, (\u2200 (n : \u2115), SymmetricRel (U n)) \u2227 (\u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m) \u2227 HasAntitoneBasis (\ud835\udce4 X) U\n[PROOFSTEP]\nexact \u27e8V \u2218 \u03c6, fun n => hV_symm _, h\u03c6_comp, h\u03c6B\u27e9\n[GOAL]\ncase intro.intro.intro\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\n\u22a2 \u2203 I, PseudoMetricSpace.toUniformSpace = inst\u271d\u00b9\n[PROOFSTEP]\nletI := UniformSpace.separationSetoid X\n[GOAL]\ncase intro.intro.intro\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\n\u22a2 \u2203 I, PseudoMetricSpace.toUniformSpace = inst\u271d\u00b9\n[PROOFSTEP]\nset d : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, (x, y) \u2209 U n then (1 / 2) ^ Nat.find h else 0\n[GOAL]\ncase intro.intro.intro\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\n\u22a2 \u2203 I, PseudoMetricSpace.toUniformSpace = inst\u271d\u00b9\n[PROOFSTEP]\nhave hd\u2080 : \u2200 {x y}, d x y = 0 \u2194 x \u2248 y := by\n intro x y\n refine' Iff.trans _ hB.mem_separationRel.symm\n simp only [true_imp_iff]\n split_ifs with h\n \u00b7 rw [\u2190 not_forall] at h \n simp [h, pow_eq_zero_iff']\n \u00b7 simpa only [not_exists, Classical.not_not, eq_self_iff_true, true_iff_iff] using h\n[GOAL]\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\n\u22a2 \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\n[PROOFSTEP]\nintro x y\n[GOAL]\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nx y : X\n\u22a2 d x y = 0 \u2194 x \u2248 y\n[PROOFSTEP]\nrefine' Iff.trans _ hB.mem_separationRel.symm\n[GOAL]\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nx y : X\n\u22a2 d x y = 0 \u2194 \u2200 (i : \u2115), True \u2192 (x, y) \u2208 U i\n[PROOFSTEP]\nsimp only [true_imp_iff]\n[GOAL]\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nx y : X\n\u22a2 (if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0) = 0 \u2194 \u2200 (i : \u2115), (x, y) \u2208 U i\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nx y : X\nh : \u2203 n, \u00ac(x, y) \u2208 U n\n\u22a2 (1 / 2) ^ Nat.find h = 0 \u2194 \u2200 (i : \u2115), (x, y) \u2208 U i\n[PROOFSTEP]\nrw [\u2190 not_forall] at h \n[GOAL]\ncase pos\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nx y : X\nh\u271d : \u2203 n, \u00ac(x, y) \u2208 U n\nh : \u00ac\u2200 (x_1 : \u2115), (x, y) \u2208 U x_1\n\u22a2 (1 / 2) ^ Nat.find h\u271d = 0 \u2194 \u2200 (i : \u2115), (x, y) \u2208 U i\n[PROOFSTEP]\nsimp [h, pow_eq_zero_iff']\n[GOAL]\ncase neg\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nx y : X\nh : \u00ac\u2203 n, \u00ac(x, y) \u2208 U n\n\u22a2 0 = 0 \u2194 \u2200 (i : \u2115), (x, y) \u2208 U i\n[PROOFSTEP]\nsimpa only [not_exists, Classical.not_not, eq_self_iff_true, true_iff_iff] using h\n[GOAL]\ncase intro.intro.intro\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\n\u22a2 \u2203 I, PseudoMetricSpace.toUniformSpace = inst\u271d\u00b9\n[PROOFSTEP]\nhave hd_symm : \u2200 x y, d x y = d y x := by\n intro x y\n simp only [@SymmetricRel.mk_mem_comm _ _ (hU_symm _) x y]\n[GOAL]\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\n\u22a2 \u2200 (x y : X), d x y = d y x\n[PROOFSTEP]\nintro x y\n[GOAL]\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nx y : X\n\u22a2 d x y = d y x\n[PROOFSTEP]\nsimp only [@SymmetricRel.mk_mem_comm _ _ (hU_symm _) x y]\n[GOAL]\ncase intro.intro.intro\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\n\u22a2 \u2203 I, PseudoMetricSpace.toUniformSpace = inst\u271d\u00b9\n[PROOFSTEP]\nhave hr : (1 / 2 : \u211d\u22650) \u2208 Ioo (0 : \u211d\u22650) 1 := \u27e8half_pos one_pos, NNReal.half_lt_self one_ne_zero\u27e9\n[GOAL]\ncase intro.intro.intro\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\n\u22a2 \u2203 I, PseudoMetricSpace.toUniformSpace = inst\u271d\u00b9\n[PROOFSTEP]\nletI I := PseudoMetricSpace.ofPreNNDist d (fun x => hd\u2080.2 (Setoid.refl _)) hd_symm\n[GOAL]\ncase intro.intro.intro\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\n\u22a2 \u2203 I, PseudoMetricSpace.toUniformSpace = inst\u271d\u00b9\n[PROOFSTEP]\nhave hdist_le : \u2200 x y, dist x y \u2264 d x y := PseudoMetricSpace.dist_ofPreNNDist_le _ _ _\n[GOAL]\ncase intro.intro.intro\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\n\u22a2 \u2203 I, PseudoMetricSpace.toUniformSpace = inst\u271d\u00b9\n[PROOFSTEP]\nhave hle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 (x, y) \u2209 U n :=\n by\n intro x y n\n dsimp only []\n split_ifs with h\n \u00b7 rw [(strictAnti_pow hr.1 hr.2).le_iff_le, Nat.find_le_iff]\n exact \u27e8fun \u27e8m, hmn, hm\u27e9 hn => hm (hB.antitone hmn hn), fun h => \u27e8n, le_rfl, h\u27e9\u27e9\n \u00b7 push_neg at h \n simp only [h, not_true, (pow_pos hr.1 _).not_le]\n[GOAL]\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\n\u22a2 \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\n[PROOFSTEP]\nintro x y n\n[GOAL]\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nx y : X\nn : \u2115\n\u22a2 (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\n[PROOFSTEP]\ndsimp only []\n[GOAL]\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nx y : X\nn : \u2115\n\u22a2 ((1 / 2) ^ n \u2264 if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0) \u2194 \u00ac(x, y) \u2208 U n\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nx y : X\nn : \u2115\nh : \u2203 n, \u00ac(x, y) \u2208 U n\n\u22a2 (1 / 2) ^ n \u2264 (1 / 2) ^ Nat.find h \u2194 \u00ac(x, y) \u2208 U n\n[PROOFSTEP]\nrw [(strictAnti_pow hr.1 hr.2).le_iff_le, Nat.find_le_iff]\n[GOAL]\ncase pos\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nx y : X\nn : \u2115\nh : \u2203 n, \u00ac(x, y) \u2208 U n\n\u22a2 (\u2203 m, m \u2264 n \u2227 \u00ac(x, y) \u2208 U m) \u2194 \u00ac(x, y) \u2208 U n\n[PROOFSTEP]\nexact \u27e8fun \u27e8m, hmn, hm\u27e9 hn => hm (hB.antitone hmn hn), fun h => \u27e8n, le_rfl, h\u27e9\u27e9\n[GOAL]\ncase neg\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nx y : X\nn : \u2115\nh : \u00ac\u2203 n, \u00ac(x, y) \u2208 U n\n\u22a2 (1 / 2) ^ n \u2264 0 \u2194 \u00ac(x, y) \u2208 U n\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nx y : X\nn : \u2115\nh : \u2200 (n : \u2115), (x, y) \u2208 U n\n\u22a2 (1 / 2) ^ n \u2264 0 \u2194 \u00ac(x, y) \u2208 U n\n[PROOFSTEP]\nsimp only [h, not_true, (pow_pos hr.1 _).not_le]\n[GOAL]\ncase intro.intro.intro\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\n\u22a2 \u2203 I, PseudoMetricSpace.toUniformSpace = inst\u271d\u00b9\n[PROOFSTEP]\nhave hd_le : \u2200 x y, \u2191(d x y) \u2264 2 * dist x y :=\n by\n refine' PseudoMetricSpace.le_two_mul_dist_ofPreNNDist _ _ _ fun x\u2081 x\u2082 x\u2083 x\u2084 => _\n by_cases H : \u2203 n, (x\u2081, x\u2084) \u2209 U n\n \u00b7 refine' (dif_pos H).trans_le _\n rw [\u2190 NNReal.div_le_iff' two_ne_zero, \u2190 mul_one_div (_ ^ _), \u2190 pow_succ']\n simp only [le_max_iff, hle_d, \u2190 not_and_or]\n rintro \u27e8h\u2081\u2082, h\u2082\u2083, h\u2083\u2084\u27e9\n refine' Nat.find_spec H (hU_comp (lt_add_one <| Nat.find H) _)\n exact \u27e8x\u2082, h\u2081\u2082, x\u2083, h\u2082\u2083, h\u2083\u2084\u27e9\n \u00b7\n exact\n (dif_neg H).trans_le\n (zero_le _)\n -- Porting note: without the next line, `uniformity_basis_dist_pow` ends up introducing some\n -- `Subtype.val` applications instead of `NNReal.toReal`.\n[GOAL]\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\n\u22a2 \u2200 (x y : X), \u2191(d x y) \u2264 2 * dist x y\n[PROOFSTEP]\nrefine' PseudoMetricSpace.le_two_mul_dist_ofPreNNDist _ _ _ fun x\u2081 x\u2082 x\u2083 x\u2084 => _\n[GOAL]\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nx\u2081 x\u2082 x\u2083 x\u2084 : X\n\u22a2 d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\n[PROOFSTEP]\nby_cases H : \u2203 n, (x\u2081, x\u2084) \u2209 U n\n[GOAL]\ncase pos\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nx\u2081 x\u2082 x\u2083 x\u2084 : X\nH : \u2203 n, \u00ac(x\u2081, x\u2084) \u2208 U n\n\u22a2 d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\n[PROOFSTEP]\nrefine' (dif_pos H).trans_le _\n[GOAL]\ncase pos\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nx\u2081 x\u2082 x\u2083 x\u2084 : X\nH : \u2203 n, \u00ac(x\u2081, x\u2084) \u2208 U n\n\u22a2 (1 / 2) ^ Nat.find H \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\n[PROOFSTEP]\nrw [\u2190 NNReal.div_le_iff' two_ne_zero, \u2190 mul_one_div (_ ^ _), \u2190 pow_succ']\n[GOAL]\ncase pos\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nx\u2081 x\u2082 x\u2083 x\u2084 : X\nH : \u2203 n, \u00ac(x\u2081, x\u2084) \u2208 U n\n\u22a2 (1 / 2) ^ (Nat.find H + 1) \u2264 max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\n[PROOFSTEP]\nsimp only [le_max_iff, hle_d, \u2190 not_and_or]\n[GOAL]\ncase pos\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nx\u2081 x\u2082 x\u2083 x\u2084 : X\nH : \u2203 n, \u00ac(x\u2081, x\u2084) \u2208 U n\n\u22a2 \u00ac((x\u2081, x\u2082) \u2208 U (Nat.find H + 1) \u2227 (x\u2082, x\u2083) \u2208 U (Nat.find H + 1) \u2227 (x\u2083, x\u2084) \u2208 U (Nat.find H + 1))\n[PROOFSTEP]\nrintro \u27e8h\u2081\u2082, h\u2082\u2083, h\u2083\u2084\u27e9\n[GOAL]\ncase pos.intro.intro\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nx\u2081 x\u2082 x\u2083 x\u2084 : X\nH : \u2203 n, \u00ac(x\u2081, x\u2084) \u2208 U n\nh\u2081\u2082 : (x\u2081, x\u2082) \u2208 U (Nat.find H + 1)\nh\u2082\u2083 : (x\u2082, x\u2083) \u2208 U (Nat.find H + 1)\nh\u2083\u2084 : (x\u2083, x\u2084) \u2208 U (Nat.find H + 1)\n\u22a2 False\n[PROOFSTEP]\nrefine' Nat.find_spec H (hU_comp (lt_add_one <| Nat.find H) _)\n[GOAL]\ncase pos.intro.intro\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nx\u2081 x\u2082 x\u2083 x\u2084 : X\nH : \u2203 n, \u00ac(x\u2081, x\u2084) \u2208 U n\nh\u2081\u2082 : (x\u2081, x\u2082) \u2208 U (Nat.find H + 1)\nh\u2082\u2083 : (x\u2082, x\u2083) \u2208 U (Nat.find H + 1)\nh\u2083\u2084 : (x\u2083, x\u2084) \u2208 U (Nat.find H + 1)\n\u22a2 (x\u2081, x\u2084) \u2208 U (Nat.find H + 1) \u25cb (U (Nat.find H + 1) \u25cb U (Nat.find H + 1))\n[PROOFSTEP]\nexact \u27e8x\u2082, h\u2081\u2082, x\u2083, h\u2082\u2083, h\u2083\u2084\u27e9\n[GOAL]\ncase neg\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nx\u2081 x\u2082 x\u2083 x\u2084 : X\nH : \u00ac\u2203 n, \u00ac(x\u2081, x\u2084) \u2208 U n\n\u22a2 d x\u2081 x\u2084 \u2264 2 * max (d x\u2081 x\u2082) (max (d x\u2082 x\u2083) (d x\u2083 x\u2084))\n[PROOFSTEP]\nexact\n (dif_neg H).trans_le\n (zero_le _)\n -- Porting note: without the next line, `uniformity_basis_dist_pow` ends up introducing some\n -- `Subtype.val` applications instead of `NNReal.toReal`.\n[GOAL]\ncase intro.intro.intro\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : 1 / 2 \u2208 Ioo 0 1\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nhd_le : \u2200 (x y : X), \u2191(d x y) \u2264 2 * dist x y\n\u22a2 \u2203 I, PseudoMetricSpace.toUniformSpace = inst\u271d\u00b9\n[PROOFSTEP]\nrw [mem_Ioo, \u2190 NNReal.coe_lt_coe, \u2190 NNReal.coe_lt_coe] at hr \n[GOAL]\ncase intro.intro.intro\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : \u21910 < \u2191(1 / 2) \u2227 \u2191(1 / 2) < \u21911\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nhd_le : \u2200 (x y : X), \u2191(d x y) \u2264 2 * dist x y\n\u22a2 \u2203 I, PseudoMetricSpace.toUniformSpace = inst\u271d\u00b9\n[PROOFSTEP]\nrefine' \u27e8I, uniformSpace_eq <| (uniformity_basis_dist_pow hr.1 hr.2).ext hB.toHasBasis _ _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : \u21910 < \u2191(1 / 2) \u2227 \u2191(1 / 2) < \u21911\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nhd_le : \u2200 (x y : X), \u2191(d x y) \u2264 2 * dist x y\n\u22a2 \u2200 (i : \u2115), True \u2192 \u2203 i', True \u2227 U i' \u2286 {p | dist p.fst p.snd < \u2191(1 / 2) ^ i}\n[PROOFSTEP]\nrefine' fun n hn => \u27e8n, hn, fun x hx => (hdist_le _ _).trans_lt _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : \u21910 < \u2191(1 / 2) \u2227 \u2191(1 / 2) < \u21911\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nhd_le : \u2200 (x y : X), \u2191(d x y) \u2264 2 * dist x y\nn : \u2115\nhn : True\nx : X \u00d7 X\nhx : x \u2208 U n\n\u22a2 \u2191(d x.fst x.snd) < \u2191(1 / 2) ^ n\n[PROOFSTEP]\nrwa [\u2190 NNReal.coe_pow, NNReal.coe_lt_coe, \u2190 not_le, hle_d, Classical.not_not, Prod.mk.eta]\n[GOAL]\ncase intro.intro.intro.refine'_2\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : \u21910 < \u2191(1 / 2) \u2227 \u2191(1 / 2) < \u21911\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nhd_le : \u2200 (x y : X), \u2191(d x y) \u2264 2 * dist x y\n\u22a2 \u2200 (i' : \u2115), True \u2192 \u2203 i, True \u2227 {p | dist p.fst p.snd < \u2191(1 / 2) ^ i} \u2286 U i'\n[PROOFSTEP]\nrefine' fun n _ => \u27e8n + 1, trivial, fun x hx => _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_2\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : \u21910 < \u2191(1 / 2) \u2227 \u2191(1 / 2) < \u21911\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nhd_le : \u2200 (x y : X), \u2191(d x y) \u2264 2 * dist x y\nn : \u2115\nx\u271d : True\nx : X \u00d7 X\nhx : x \u2208 {p | dist p.fst p.snd < \u2191(1 / 2) ^ (n + 1)}\n\u22a2 x \u2208 U n\n[PROOFSTEP]\nrw [mem_setOf_eq] at hx \n[GOAL]\ncase intro.intro.intro.refine'_2\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : \u21910 < \u2191(1 / 2) \u2227 \u2191(1 / 2) < \u21911\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nhd_le : \u2200 (x y : X), \u2191(d x y) \u2264 2 * dist x y\nn : \u2115\nx\u271d : True\nx : X \u00d7 X\nhx : dist x.fst x.snd < \u2191(1 / 2) ^ (n + 1)\n\u22a2 x \u2208 U n\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\ncase intro.intro.intro.refine'_2\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : \u21910 < \u2191(1 / 2) \u2227 \u2191(1 / 2) < \u21911\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nhd_le : \u2200 (x y : X), \u2191(d x y) \u2264 2 * dist x y\nn : \u2115\nx\u271d : True\nx : X \u00d7 X\nhx : \u00acx \u2208 U n\n\u22a2 \u2191(1 / 2) ^ (n + 1) \u2264 dist x.fst x.snd\n[PROOFSTEP]\nrefine' le_trans _ ((div_le_iff' (zero_lt_two' \u211d)).2 (hd_le x.1 x.2))\n[GOAL]\ncase intro.intro.intro.refine'_2\nX\u271d : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nU : \u2115 \u2192 Set (X \u00d7 X)\nhU_symm : \u2200 (n : \u2115), SymmetricRel (U n)\nhU_comp : \u2200 \u2983m n : \u2115\u2984, m < n \u2192 U n \u25cb (U n \u25cb U n) \u2286 U m\nhB : HasAntitoneBasis (\ud835\udce4 X) U\nthis : Setoid X := separationSetoid X\nd : X \u2192 X \u2192 \u211d\u22650 := fun x y => if h : \u2203 n, \u00ac(x, y) \u2208 U n then (1 / 2) ^ Nat.find h else 0\nhd\u2080 : \u2200 {x y : X}, d x y = 0 \u2194 x \u2248 y\nhd_symm : \u2200 (x y : X), d x y = d y x\nhr : \u21910 < \u2191(1 / 2) \u2227 \u2191(1 / 2) < \u21911\nI : PseudoMetricSpace X := PseudoMetricSpace.ofPreNNDist d (_ : \u2200 (x : X), d x x = 0) hd_symm\nhdist_le : \u2200 (x y : X), dist x y \u2264 \u2191(d x y)\nhle_d : \u2200 {x y : X} {n : \u2115}, (1 / 2) ^ n \u2264 d x y \u2194 \u00ac(x, y) \u2208 U n\nhd_le : \u2200 (x y : X), \u2191(d x y) \u2264 2 * dist x y\nn : \u2115\nx\u271d : True\nx : X \u00d7 X\nhx : \u00acx \u2208 U n\n\u22a2 \u2191(1 / 2) ^ (n + 1) \u2264 \u2191(d x.fst x.snd) / 2\n[PROOFSTEP]\nrwa [\u2190 NNReal.coe_two, \u2190 NNReal.coe_div, \u2190 NNReal.coe_pow, NNReal.coe_le_coe, pow_succ', mul_one_div,\n NNReal.div_le_iff two_ne_zero, div_mul_cancel _ (two_ne_zero' \u211d\u22650), hle_d, Prod.mk.eta]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\n\u22a2 TopologicalSpace.PseudoMetrizableSpace X\n[PROOFSTEP]\nletI := UniformSpace.pseudoMetricSpace X\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : UniformSpace X\ninst\u271d : IsCountablyGenerated (\ud835\udce4 X)\nthis : PseudoMetricSpace X := UniformSpace.pseudoMetricSpace X\n\u22a2 TopologicalSpace.PseudoMetrizableSpace X\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : UniformSpace X\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 X)\ninst\u271d : T0Space X\n\u22a2 TopologicalSpace.MetrizableSpace X\n[PROOFSTEP]\nletI := UniformSpace.metricSpace X\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : UniformSpace X\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 X)\ninst\u271d : T0Space X\nthis : MetricSpace X := UniformSpace.metricSpace X\n\u22a2 TopologicalSpace.MetrizableSpace X\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.MetrizableUniformity", "llama_tokens": 49562, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154240079185319, "lm_q2_score": 0.5621765008857982, "lm_q1q2_score": 0.4021945654213338}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nh : Inducing f\n\u22a2 Inducing (restrictPreimage s f)\n[PROOFSTEP]\nsimp_rw [inducing_subtype_val.inducing_iff, inducing_iff_nhds, restrictPreimage, MapsTo.coe_restrict, restrict_eq, \u2190\n @Filter.comap_comap _ _ _ _ _ f, Function.comp_apply] at h \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nh : \u2200 (a : \u03b1), \ud835\udcdd a = comap f (\ud835\udcdd (f a))\n\u22a2 \u2200 (a : \u2191(f \u207b\u00b9' s)), \ud835\udcdd a = comap Subtype.val (comap f (\ud835\udcdd (f \u2191a)))\n[PROOFSTEP]\nintro a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nh : \u2200 (a : \u03b1), \ud835\udcdd a = comap f (\ud835\udcdd (f a))\na : \u2191(f \u207b\u00b9' s)\n\u22a2 \ud835\udcdd a = comap Subtype.val (comap f (\ud835\udcdd (f \u2191a)))\n[PROOFSTEP]\nrw [\u2190 h, \u2190 inducing_subtype_val.nhds_eq_comap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\n\u22a2 IsClosedMap (restrictPreimage s f)\n[PROOFSTEP]\nrintro t \u27e8u, hu, e\u27e9\n[GOAL]\ncase mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\n\u22a2 IsClosed (restrictPreimage s f '' t)\n[PROOFSTEP]\nrefine' \u27e8\u27e8_, (H _ (IsOpen.isClosed_compl hu)).1, _\u27e9\u27e9\n[GOAL]\ncase mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\n\u22a2 Subtype.val \u207b\u00b9' (f '' u\u1d9c)\u1d9c = (restrictPreimage s f '' t)\u1d9c\n[PROOFSTEP]\nrw [\u2190 (congr_arg HasCompl.compl e).trans (compl_compl t)]\n[GOAL]\ncase mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\n\u22a2 Subtype.val \u207b\u00b9' (f '' u\u1d9c)\u1d9c = (restrictPreimage s f '' (Subtype.val \u207b\u00b9' u)\u1d9c)\u1d9c\n[PROOFSTEP]\nsimp only [Set.preimage_compl, compl_inj_iff]\n[GOAL]\ncase mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\n\u22a2 Subtype.val \u207b\u00b9' (f '' u\u1d9c) = restrictPreimage s f '' (Subtype.val \u207b\u00b9' u)\u1d9c\n[PROOFSTEP]\next \u27e8x, hx\u27e9\n[GOAL]\ncase mk.intro.intro.h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\nx : \u03b2\nhx : x \u2208 s\n\u22a2 { val := x, property := hx } \u2208 Subtype.val \u207b\u00b9' (f '' u\u1d9c) \u2194\n { val := x, property := hx } \u2208 restrictPreimage s f '' (Subtype.val \u207b\u00b9' u)\u1d9c\n[PROOFSTEP]\nsuffices (\u2203 y, y \u2209 u \u2227 f y = x) \u2194 \u2203 y, y \u2209 u \u2227 f y \u2208 s \u2227 f y = x by simpa [Set.restrictPreimage, \u2190 Subtype.coe_inj]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\nx : \u03b2\nhx : x \u2208 s\nthis : (\u2203 y, \u00acy \u2208 u \u2227 f y = x) \u2194 \u2203 y, \u00acy \u2208 u \u2227 f y \u2208 s \u2227 f y = x\n\u22a2 { val := x, property := hx } \u2208 Subtype.val \u207b\u00b9' (f '' u\u1d9c) \u2194\n { val := x, property := hx } \u2208 restrictPreimage s f '' (Subtype.val \u207b\u00b9' u)\u1d9c\n[PROOFSTEP]\nsimpa [Set.restrictPreimage, \u2190 Subtype.coe_inj]\n[GOAL]\ncase mk.intro.intro.h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\nx : \u03b2\nhx : x \u2208 s\n\u22a2 (\u2203 y, \u00acy \u2208 u \u2227 f y = x) \u2194 \u2203 y, \u00acy \u2208 u \u2227 f y \u2208 s \u2227 f y = x\n[PROOFSTEP]\nexact \u27e8fun \u27e8a, b, c\u27e9 => \u27e8a, b, c.symm \u25b8 hx, c\u27e9, fun \u27e8a, b, _, c\u27e9 => \u27e8a, b, c\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\n\u22a2 IsOpen s \u2194 \u2200 (i : \u03b9), IsOpen (s \u2229 \u2191(U i))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\n\u22a2 IsOpen s \u2192 \u2200 (i : \u03b9), IsOpen (s \u2229 \u2191(U i))\n[PROOFSTEP]\nexact fun H i => H.inter (U i).2\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\n\u22a2 (\u2200 (i : \u03b9), IsOpen (s \u2229 \u2191(U i))) \u2192 IsOpen s\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : \u2200 (i : \u03b9), IsOpen (s \u2229 \u2191(U i))\n\u22a2 IsOpen s\n[PROOFSTEP]\nhave : \u22c3 i, (U i : Set \u03b2) = Set.univ := by\n convert congr_arg (SetLike.coe) hU\n simp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : \u2200 (i : \u03b9), IsOpen (s \u2229 \u2191(U i))\n\u22a2 \u22c3 (i : \u03b9), \u2191(U i) = univ\n[PROOFSTEP]\nconvert congr_arg (SetLike.coe) hU\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : \u2200 (i : \u03b9), IsOpen (s \u2229 \u2191(U i))\n\u22a2 \u22c3 (i : \u03b9), \u2191(U i) = \u2191(iSup U)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : \u2200 (i : \u03b9), IsOpen (s \u2229 \u2191(U i))\nthis : \u22c3 (i : \u03b9), \u2191(U i) = univ\n\u22a2 IsOpen s\n[PROOFSTEP]\nrw [\u2190 s.inter_univ, \u2190 this, Set.inter_iUnion]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : \u2200 (i : \u03b9), IsOpen (s \u2229 \u2191(U i))\nthis : \u22c3 (i : \u03b9), \u2191(U i) = univ\n\u22a2 IsOpen (\u22c3 (i : \u03b9), s \u2229 \u2191(U i))\n[PROOFSTEP]\nexact isOpen_iUnion H\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\n\u22a2 IsOpen s \u2194 \u2200 (i : \u03b9), IsOpen (Subtype.val \u207b\u00b9' s)\n[PROOFSTEP]\nrw [isOpen_iff_inter_of_iSup_eq_top hU s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\n\u22a2 (\u2200 (i : \u03b9), IsOpen (s \u2229 \u2191(U i))) \u2194 \u2200 (i : \u03b9), IsOpen (Subtype.val \u207b\u00b9' s)\n[PROOFSTEP]\nrefine forall_congr' fun i => ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\ni : \u03b9\n\u22a2 IsOpen (s \u2229 \u2191(U i)) \u2194 IsOpen (Subtype.val \u207b\u00b9' s)\n[PROOFSTEP]\nrw [(U _).2.openEmbedding_subtype_val.open_iff_image_open]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\ni : \u03b9\n\u22a2 IsOpen (s \u2229 \u2191(U i)) \u2194 IsOpen (Subtype.val '' (Subtype.val \u207b\u00b9' s))\n[PROOFSTEP]\nerw [Set.image_preimage_eq_inter_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\ni : \u03b9\n\u22a2 IsOpen (s \u2229 \u2191(U i)) \u2194 IsOpen (s \u2229 range Subtype.val)\n[PROOFSTEP]\nrw [Subtype.range_coe, Opens.carrier_eq_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\n\u22a2 IsClosed s \u2194 \u2200 (i : \u03b9), IsClosed (Subtype.val \u207b\u00b9' s)\n[PROOFSTEP]\nsimpa using isOpen_iff_coe_preimage_of_iSup_eq_top hU s\u1d9c\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\n\u22a2 IsClosedMap f \u2194 \u2200 (i : \u03b9), IsClosedMap (restrictPreimage (U i).carrier f)\n[PROOFSTEP]\nrefine' \u27e8fun h i => Set.restrictPreimage_isClosedMap _ h, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\n\u22a2 (\u2200 (i : \u03b9), IsClosedMap (restrictPreimage (U i).carrier f)) \u2192 IsClosedMap f\n[PROOFSTEP]\nrintro H s hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nH : \u2200 (i : \u03b9), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set \u03b1\nhs : IsClosed s\n\u22a2 IsClosed (f '' s)\n[PROOFSTEP]\nrw [isClosed_iff_coe_preimage_of_iSup_eq_top hU]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nH : \u2200 (i : \u03b9), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set \u03b1\nhs : IsClosed s\n\u22a2 \u2200 (i : \u03b9), IsClosed (Subtype.val \u207b\u00b9' (f '' s))\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nH : \u2200 (i : \u03b9), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set \u03b1\nhs : IsClosed s\ni : \u03b9\n\u22a2 IsClosed (Subtype.val \u207b\u00b9' (f '' s))\n[PROOFSTEP]\nconvert H i _ \u27e8\u27e8_, hs.1, eq_compl_comm.mpr rfl\u27e9\u27e9\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nH : \u2200 (i : \u03b9), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set \u03b1\nhs : IsClosed s\ni : \u03b9\ne_1\u271d : { x // x \u2208 U i } = \u2191(U i).carrier\n\u22a2 Subtype.val \u207b\u00b9' (f '' s) = restrictPreimage (U i).carrier f '' (Subtype.val \u207b\u00b9' s\u1d9c)\u1d9c\n[PROOFSTEP]\next \u27e8x, hx\u27e9\n[GOAL]\ncase h.e'_3.h.h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nH : \u2200 (i : \u03b9), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set \u03b1\nhs : IsClosed s\ni : \u03b9\ne_1\u271d : { x // x \u2208 U i } = \u2191(U i).carrier\nx : \u03b2\nhx : x \u2208 U i\n\u22a2 { val := x, property := hx } \u2208 Subtype.val \u207b\u00b9' (f '' s) \u2194\n { val := x, property := hx } \u2208 restrictPreimage (U i).carrier f '' (Subtype.val \u207b\u00b9' s\u1d9c)\u1d9c\n[PROOFSTEP]\nsuffices (\u2203 y, y \u2208 s \u2227 f y = x) \u2194 \u2203 y, y \u2208 s \u2227 f y \u2208 U i \u2227 f y = x by simpa [Set.restrictPreimage, \u2190 Subtype.coe_inj]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nH : \u2200 (i : \u03b9), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set \u03b1\nhs : IsClosed s\ni : \u03b9\ne_1\u271d : { x // x \u2208 U i } = \u2191(U i).carrier\nx : \u03b2\nhx : x \u2208 U i\nthis : (\u2203 y, y \u2208 s \u2227 f y = x) \u2194 \u2203 y, y \u2208 s \u2227 f y \u2208 U i \u2227 f y = x\n\u22a2 { val := x, property := hx } \u2208 Subtype.val \u207b\u00b9' (f '' s) \u2194\n { val := x, property := hx } \u2208 restrictPreimage (U i).carrier f '' (Subtype.val \u207b\u00b9' s\u1d9c)\u1d9c\n[PROOFSTEP]\nsimpa [Set.restrictPreimage, \u2190 Subtype.coe_inj]\n[GOAL]\ncase h.e'_3.h.h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nH : \u2200 (i : \u03b9), IsClosedMap (restrictPreimage (U i).carrier f)\ns : Set \u03b1\nhs : IsClosed s\ni : \u03b9\ne_1\u271d : { x // x \u2208 U i } = \u2191(U i).carrier\nx : \u03b2\nhx : x \u2208 U i\n\u22a2 (\u2203 y, y \u2208 s \u2227 f y = x) \u2194 \u2203 y, y \u2208 s \u2227 f y \u2208 U i \u2227 f y = x\n[PROOFSTEP]\nexact \u27e8fun \u27e8a, b, c\u27e9 => \u27e8a, b, c.symm \u25b8 hx, c\u27e9, fun \u27e8a, b, _, c\u27e9 => \u27e8a, b, c\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 Inducing f \u2194 \u2200 (i : \u03b9), Inducing (restrictPreimage (U i).carrier f)\n[PROOFSTEP]\nsimp_rw [inducing_subtype_val.inducing_iff, inducing_iff_nhds, restrictPreimage, MapsTo.coe_restrict, restrict_eq, \u2190\n @Filter.comap_comap _ _ _ _ _ f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 (\u2200 (a : \u03b1), \ud835\udcdd a = comap f (\ud835\udcdd (f a))) \u2194\n \u2200 (i : \u03b9) (a : \u2191(f \u207b\u00b9' (U i).carrier)), \ud835\udcdd a = comap Subtype.val (comap f (\ud835\udcdd ((f \u2218 Subtype.val) a)))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 (\u2200 (a : \u03b1), \ud835\udcdd a = comap f (\ud835\udcdd (f a))) \u2192\n \u2200 (i : \u03b9) (a : \u2191(f \u207b\u00b9' (U i).carrier)), \ud835\udcdd a = comap Subtype.val (comap f (\ud835\udcdd ((f \u2218 Subtype.val) a)))\n[PROOFSTEP]\nintro H i x\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\nH : \u2200 (a : \u03b1), \ud835\udcdd a = comap f (\ud835\udcdd (f a))\ni : \u03b9\nx : \u2191(f \u207b\u00b9' (U i).carrier)\n\u22a2 \ud835\udcdd x = comap Subtype.val (comap f (\ud835\udcdd ((f \u2218 Subtype.val) x)))\n[PROOFSTEP]\nrw [Function.comp_apply, \u2190 H, \u2190 inducing_subtype_val.nhds_eq_comap]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 (\u2200 (i : \u03b9) (a : \u2191(f \u207b\u00b9' (U i).carrier)), \ud835\udcdd a = comap Subtype.val (comap f (\ud835\udcdd ((f \u2218 Subtype.val) a)))) \u2192\n \u2200 (a : \u03b1), \ud835\udcdd a = comap f (\ud835\udcdd (f a))\n[PROOFSTEP]\nintro H x\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\nH : \u2200 (i : \u03b9) (a : \u2191(f \u207b\u00b9' (U i).carrier)), \ud835\udcdd a = comap Subtype.val (comap f (\ud835\udcdd ((f \u2218 Subtype.val) a)))\nx : \u03b1\n\u22a2 \ud835\udcdd x = comap f (\ud835\udcdd (f x))\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 :=\n Opens.mem_iSup.mp\n (show f x \u2208 iSup U by\n rw [hU]\n triv)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\nH : \u2200 (i : \u03b9) (a : \u2191(f \u207b\u00b9' (U i).carrier)), \ud835\udcdd a = comap Subtype.val (comap f (\ud835\udcdd ((f \u2218 Subtype.val) a)))\nx : \u03b1\n\u22a2 f x \u2208 iSup U\n[PROOFSTEP]\nrw [hU]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\nH : \u2200 (i : \u03b9) (a : \u2191(f \u207b\u00b9' (U i).carrier)), \ud835\udcdd a = comap Subtype.val (comap f (\ud835\udcdd ((f \u2218 Subtype.val) a)))\nx : \u03b1\n\u22a2 f x \u2208 \u22a4\n[PROOFSTEP]\ntriv\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\nH : \u2200 (i : \u03b9) (a : \u2191(f \u207b\u00b9' (U i).carrier)), \ud835\udcdd a = comap Subtype.val (comap f (\ud835\udcdd ((f \u2218 Subtype.val) a)))\nx : \u03b1\ni : \u03b9\nhi : f x \u2208 U i\n\u22a2 \ud835\udcdd x = comap f (\ud835\udcdd (f x))\n[PROOFSTEP]\nerw [\u2190 OpenEmbedding.map_nhds_eq (h.1 _ (U i).2).openEmbedding_subtype_val \u27e8x, hi\u27e9]\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\nH : \u2200 (i : \u03b9) (a : \u2191(f \u207b\u00b9' (U i).carrier)), \ud835\udcdd a = comap Subtype.val (comap f (\ud835\udcdd ((f \u2218 Subtype.val) a)))\nx : \u03b1\ni : \u03b9\nhi : f x \u2208 U i\n\u22a2 map Subtype.val (\ud835\udcdd { val := x, property := hi }) = comap f (\ud835\udcdd (f x))\n[PROOFSTEP]\nrw [(H i) \u27e8x, hi\u27e9, Filter.subtype_coe_map_comap, Function.comp_apply, Subtype.coe_mk, inf_eq_left,\n Filter.le_principal_iff]\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\nH : \u2200 (i : \u03b9) (a : \u2191(f \u207b\u00b9' (U i).carrier)), \ud835\udcdd a = comap Subtype.val (comap f (\ud835\udcdd ((f \u2218 Subtype.val) a)))\nx : \u03b1\ni : \u03b9\nhi : f x \u2208 U i\n\u22a2 f \u207b\u00b9' (U i).carrier \u2208 comap f (\ud835\udcdd (f x))\n[PROOFSTEP]\nexact Filter.preimage_mem_comap ((U i).2.mem_nhds hi)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 Embedding f \u2194 \u2200 (i : \u03b9), Embedding (restrictPreimage (U i).carrier f)\n[PROOFSTEP]\nsimp_rw [embedding_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 Inducing f \u2227 Function.Injective f \u2194\n \u2200 (i : \u03b9), Inducing (restrictPreimage (U i).carrier f) \u2227 Function.Injective (restrictPreimage (U i).carrier f)\n[PROOFSTEP]\nrw [forall_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 Inducing f \u2227 Function.Injective f \u2194\n (\u2200 (x : \u03b9), Inducing (restrictPreimage (U x).carrier f)) \u2227\n \u2200 (x : \u03b9), Function.Injective (restrictPreimage (U x).carrier f)\n[PROOFSTEP]\napply and_congr\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 Inducing f \u2194 \u2200 (x : \u03b9), Inducing (restrictPreimage (U x).carrier f)\n[PROOFSTEP]\napply inducing_iff_inducing_of_iSup_eq_top\n[GOAL]\ncase h\u2081.hU\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 \u2a06 (i : \u03b9), U i = \u22a4\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h\u2081.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 Continuous f\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 Function.Injective f \u2194 \u2200 (x : \u03b9), Function.Injective (restrictPreimage (U x).carrier f)\n[PROOFSTEP]\napply Set.injective_iff_injective_of_iUnion_eq_univ\n[GOAL]\ncase h\u2082.hU\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 \u22c3 (i : \u03b9), (U i).carrier = univ\n[PROOFSTEP]\nconvert congr_arg SetLike.coe hU\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 \u22c3 (i : \u03b9), (U i).carrier = \u2191(iSup U)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 OpenEmbedding f \u2194 \u2200 (i : \u03b9), OpenEmbedding (restrictPreimage (U i).carrier f)\n[PROOFSTEP]\nsimp_rw [openEmbedding_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 Embedding f \u2227 IsOpen (range f) \u2194\n \u2200 (i : \u03b9), Embedding (restrictPreimage (U i).carrier f) \u2227 IsOpen (range (restrictPreimage (U i).carrier f))\n[PROOFSTEP]\nrw [forall_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 Embedding f \u2227 IsOpen (range f) \u2194\n (\u2200 (x : \u03b9), Embedding (restrictPreimage (U x).carrier f)) \u2227\n \u2200 (x : \u03b9), IsOpen (range (restrictPreimage (U x).carrier f))\n[PROOFSTEP]\napply and_congr\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 Embedding f \u2194 \u2200 (x : \u03b9), Embedding (restrictPreimage (U x).carrier f)\n[PROOFSTEP]\napply embedding_iff_embedding_of_iSup_eq_top\n[GOAL]\ncase h\u2081.hU\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 \u2a06 (i : \u03b9), U i = \u22a4\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h\u2081.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 Continuous f\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 IsOpen (range f) \u2194 \u2200 (x : \u03b9), IsOpen (range (restrictPreimage (U x).carrier f))\n[PROOFSTEP]\nsimp_rw [Set.range_restrictPreimage]\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 IsOpen (range f) \u2194 \u2200 (x : \u03b9), IsOpen (Subtype.val \u207b\u00b9' range f)\n[PROOFSTEP]\napply isOpen_iff_coe_preimage_of_iSup_eq_top hU\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 ClosedEmbedding f \u2194 \u2200 (i : \u03b9), ClosedEmbedding (restrictPreimage (U i).carrier f)\n[PROOFSTEP]\nsimp_rw [closedEmbedding_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 Embedding f \u2227 IsClosed (range f) \u2194\n \u2200 (i : \u03b9), Embedding (restrictPreimage (U i).carrier f) \u2227 IsClosed (range (restrictPreimage (U i).carrier f))\n[PROOFSTEP]\nrw [forall_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 Embedding f \u2227 IsClosed (range f) \u2194\n (\u2200 (x : \u03b9), Embedding (restrictPreimage (U x).carrier f)) \u2227\n \u2200 (x : \u03b9), IsClosed (range (restrictPreimage (U x).carrier f))\n[PROOFSTEP]\napply and_congr\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 Embedding f \u2194 \u2200 (x : \u03b9), Embedding (restrictPreimage (U x).carrier f)\n[PROOFSTEP]\napply embedding_iff_embedding_of_iSup_eq_top\n[GOAL]\ncase h\u2081.hU\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 \u2a06 (i : \u03b9), U i = \u22a4\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h\u2081.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 Continuous f\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 IsClosed (range f) \u2194 \u2200 (x : \u03b9), IsClosed (range (restrictPreimage (U x).carrier f))\n[PROOFSTEP]\nsimp_rw [Set.range_restrictPreimage]\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u03b9 : Type u_3\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\nh : Continuous f\n\u22a2 IsClosed (range f) \u2194 \u2200 (x : \u03b9), IsClosed (Subtype.val \u207b\u00b9' range f)\n[PROOFSTEP]\napply isClosed_iff_coe_preimage_of_iSup_eq_top hU\n", "meta": {"mathlib_filename": "Mathlib.Topology.LocalAtTarget", "llama_tokens": 12327, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.40219346944133555}} {"text": "[GOAL]\nF : Type u_1\ninst\u271d : CommSemiring F\nq : \u2115\nf : F[X]\nhf\u271d : HasSeparableContraction q f\ng : F[X]\nhf : IsSeparableContraction q f g\n\u22a2 \u2203 m, natDegree g * q ^ m = natDegree f\n[PROOFSTEP]\nobtain \u27e8m, rfl\u27e9 := hf.2\n[GOAL]\ncase intro\nF : Type u_1\ninst\u271d : CommSemiring F\nq : \u2115\ng : F[X]\nm : \u2115\nhf\u271d : HasSeparableContraction q (\u2191(expand F (q ^ m)) g)\nhf : IsSeparableContraction q (\u2191(expand F (q ^ m)) g) g\n\u22a2 \u2203 m_1, natDegree g * q ^ m_1 = natDegree (\u2191(expand F (q ^ m)) g)\n[PROOFSTEP]\nuse m\n[GOAL]\ncase h\nF : Type u_1\ninst\u271d : CommSemiring F\nq : \u2115\ng : F[X]\nm : \u2115\nhf\u271d : HasSeparableContraction q (\u2191(expand F (q ^ m)) g)\nhf : IsSeparableContraction q (\u2191(expand F (q ^ m)) g) g\n\u22a2 natDegree g * q ^ m = natDegree (\u2191(expand F (q ^ m)) g)\n[PROOFSTEP]\nrw [natDegree_expand]\n[GOAL]\nF : Type u_1\ninst\u271d : CommSemiring F\nq : \u2115\nf\u271d : F[X]\nhf\u271d : HasSeparableContraction q f\u271d\nf : F[X]\nhf : HasSeparableContraction 1 f\n\u22a2 degree hf = natDegree f\n[PROOFSTEP]\nlet \u27e8a, ha\u27e9 := hf.dvd_degree'\n[GOAL]\nF : Type u_1\ninst\u271d : CommSemiring F\nq : \u2115\nf\u271d : F[X]\nhf\u271d : HasSeparableContraction q f\u271d\nf : F[X]\nhf : HasSeparableContraction 1 f\na : \u2115\nha : degree hf * 1 ^ a = natDegree f\n\u22a2 degree hf = natDegree f\n[PROOFSTEP]\nrw [\u2190 ha, one_pow a, mul_one]\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nq\u271d : \u2115\nf\u271d : F[X]\nhf : HasSeparableContraction q\u271d f\u271d\nq : \u2115\nhF : ExpChar F q\nf : F[X]\nirred : Irreducible f\n\u22a2 HasSeparableContraction q f\n[PROOFSTEP]\ncases hF\n[GOAL]\ncase zero\nF : Type u_1\ninst\u271d\u00b9 : Field F\nq : \u2115\nf\u271d : F[X]\nhf : HasSeparableContraction q f\u271d\nf : F[X]\nirred : Irreducible f\ninst\u271d : CharZero F\n\u22a2 HasSeparableContraction 1 f\n[PROOFSTEP]\nexact \u27e8f, irred.separable, \u27e80, by rw [pow_zero, expand_one]\u27e9\u27e9\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\nq : \u2115\nf\u271d : F[X]\nhf : HasSeparableContraction q f\u271d\nf : F[X]\nirred : Irreducible f\ninst\u271d : CharZero F\n\u22a2 \u2191(expand F (1 ^ 0)) f = f\n[PROOFSTEP]\nrw [pow_zero, expand_one]\n[GOAL]\ncase prime\nF : Type u_1\ninst\u271d : Field F\nq\u271d : \u2115\nf\u271d : F[X]\nhf : HasSeparableContraction q\u271d f\u271d\nq : \u2115\nf : F[X]\nirred : Irreducible f\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP F q\n\u22a2 HasSeparableContraction q f\n[PROOFSTEP]\nrcases exists_separable_of_irreducible q irred \u2039q.Prime\u203a.ne_zero with \u27e8n, g, hgs, hge\u27e9\n[GOAL]\ncase prime.intro.intro.intro\nF : Type u_1\ninst\u271d : Field F\nq\u271d : \u2115\nf\u271d : F[X]\nhf : HasSeparableContraction q\u271d f\u271d\nq : \u2115\nf : F[X]\nirred : Irreducible f\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP F q\nn : \u2115\ng : F[X]\nhgs : Separable g\nhge : \u2191(expand F (q ^ n)) g = f\n\u22a2 HasSeparableContraction q f\n[PROOFSTEP]\nexact \u27e8g, hgs, n, hge\u27e9\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\nhq : NeZero q\ninst\u271d : CharP F q\ng g' : F[X]\nm m' : \u2115\nh_expand : \u2191(expand F (q ^ m)) g = \u2191(expand F (q ^ m')) g'\nhg : Separable g\nhg' : Separable g'\n\u22a2 natDegree g = natDegree g'\n[PROOFSTEP]\nwlog hm : m \u2264 m'\n[GOAL]\ncase inr\nF : Type u_1\ninst\u271d\u00b9 : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\nhq : NeZero q\ninst\u271d : CharP F q\ng g' : F[X]\nm m' : \u2115\nh_expand : \u2191(expand F (q ^ m)) g = \u2191(expand F (q ^ m')) g'\nhg : Separable g\nhg' : Separable g'\nthis :\n \u2200 {F : Type u_1} [inst : Field F] (q : \u2115) {f : F[X]},\n HasSeparableContraction q f \u2192\n \u2200 [hq : NeZero q] [inst_1 : CharP F q] (g g' : F[X]) (m m' : \u2115),\n \u2191(expand F (q ^ m)) g = \u2191(expand F (q ^ m')) g' \u2192\n Separable g \u2192 Separable g' \u2192 m \u2264 m' \u2192 natDegree g = natDegree g'\nhm : \u00acm \u2264 m'\n\u22a2 natDegree g = natDegree g'\n[PROOFSTEP]\nexact (this q hf g' g m' m h_expand.symm hg' hg (le_of_not_le hm)).symm\n[GOAL]\nF\u271d : Type u_1\ninst\u271d\u00b2 : Field F\u271d\nq\u271d : \u2115\nF : Type u_1\ninst\u271d\u00b9 : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\nhq : NeZero q\ninst\u271d : CharP F q\ng g' : F[X]\nm m' : \u2115\nh_expand : \u2191(expand F (q ^ m)) g = \u2191(expand F (q ^ m')) g'\nhg : Separable g\nhg' : Separable g'\nhm : m \u2264 m'\n\u22a2 natDegree g = natDegree g'\n[PROOFSTEP]\nobtain \u27e8s, rfl\u27e9 := exists_add_of_le hm\n[GOAL]\ncase intro\nF\u271d : Type u_1\ninst\u271d\u00b2 : Field F\u271d\nq\u271d : \u2115\nF : Type u_1\ninst\u271d\u00b9 : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\nhq : NeZero q\ninst\u271d : CharP F q\ng g' : F[X]\nm : \u2115\nhg : Separable g\nhg' : Separable g'\ns : \u2115\nh_expand : \u2191(expand F (q ^ m)) g = \u2191(expand F (q ^ (m + s))) g'\nhm : m \u2264 m + s\n\u22a2 natDegree g = natDegree g'\n[PROOFSTEP]\nrw [pow_add, expand_mul, expand_inj (pow_pos (NeZero.pos q) m)] at h_expand \n[GOAL]\ncase intro\nF\u271d : Type u_1\ninst\u271d\u00b2 : Field F\u271d\nq\u271d : \u2115\nF : Type u_1\ninst\u271d\u00b9 : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\nhq : NeZero q\ninst\u271d : CharP F q\ng g' : F[X]\nm : \u2115\nhg : Separable g\nhg' : Separable g'\ns : \u2115\nh_expand : g = \u2191(expand F (q ^ s)) g'\nhm : m \u2264 m + s\n\u22a2 natDegree g = natDegree g'\n[PROOFSTEP]\nsubst h_expand\n[GOAL]\ncase intro\nF\u271d : Type u_1\ninst\u271d\u00b2 : Field F\u271d\nq\u271d : \u2115\nF : Type u_1\ninst\u271d\u00b9 : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\nhq : NeZero q\ninst\u271d : CharP F q\ng' : F[X]\nm : \u2115\nhg' : Separable g'\ns : \u2115\nhm : m \u2264 m + s\nhg : Separable (\u2191(expand F (q ^ s)) g')\n\u22a2 natDegree (\u2191(expand F (q ^ s)) g') = natDegree g'\n[PROOFSTEP]\nrcases isUnit_or_eq_zero_of_separable_expand q s (NeZero.pos q) hg with (h | rfl)\n[GOAL]\ncase intro.inl\nF\u271d : Type u_1\ninst\u271d\u00b2 : Field F\u271d\nq\u271d : \u2115\nF : Type u_1\ninst\u271d\u00b9 : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\nhq : NeZero q\ninst\u271d : CharP F q\ng' : F[X]\nm : \u2115\nhg' : Separable g'\ns : \u2115\nhm : m \u2264 m + s\nhg : Separable (\u2191(expand F (q ^ s)) g')\nh : IsUnit g'\n\u22a2 natDegree (\u2191(expand F (q ^ s)) g') = natDegree g'\n[PROOFSTEP]\nrw [natDegree_expand, natDegree_eq_zero_of_isUnit h, zero_mul]\n[GOAL]\ncase intro.inr\nF\u271d : Type u_1\ninst\u271d\u00b2 : Field F\u271d\nq\u271d : \u2115\nF : Type u_1\ninst\u271d\u00b9 : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\nhq : NeZero q\ninst\u271d : CharP F q\ng' : F[X]\nm : \u2115\nhg' : Separable g'\nhm : m \u2264 m + 0\nhg : Separable (\u2191(expand F (q ^ 0)) g')\n\u22a2 natDegree (\u2191(expand F (q ^ 0)) g') = natDegree g'\n[PROOFSTEP]\nrw [natDegree_expand, pow_zero, mul_one]\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\nhF : ExpChar F q\ng : F[X]\nhg : IsSeparableContraction q f g\n\u22a2 natDegree g = HasSeparableContraction.degree hf\n[PROOFSTEP]\ncases hF\n[GOAL]\ncase zero\nF : Type u_1\ninst\u271d\u00b9 : Field F\nf g : F[X]\ninst\u271d : CharZero F\nhf : HasSeparableContraction 1 f\nhg : IsSeparableContraction 1 f g\n\u22a2 natDegree g = HasSeparableContraction.degree hf\n[PROOFSTEP]\nrcases hg with \u27e8_, m, hm\u27e9\n[GOAL]\ncase zero.intro.intro\nF : Type u_1\ninst\u271d\u00b9 : Field F\nf g : F[X]\ninst\u271d : CharZero F\nhf : HasSeparableContraction 1 f\nleft\u271d : Separable g\nm : \u2115\nhm : \u2191(expand F (1 ^ m)) g = f\n\u22a2 natDegree g = HasSeparableContraction.degree hf\n[PROOFSTEP]\nrw [one_pow, expand_one] at hm \n[GOAL]\ncase zero.intro.intro\nF : Type u_1\ninst\u271d\u00b9 : Field F\nf g : F[X]\ninst\u271d : CharZero F\nhf : HasSeparableContraction 1 f\nleft\u271d : Separable g\nm : \u2115\nhm : g = f\n\u22a2 natDegree g = HasSeparableContraction.degree hf\n[PROOFSTEP]\nrw [hf.eq_degree, hm]\n[GOAL]\ncase prime\nF : Type u_1\ninst\u271d : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\ng : F[X]\nhg : IsSeparableContraction q f g\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP F q\n\u22a2 natDegree g = HasSeparableContraction.degree hf\n[PROOFSTEP]\nrcases hg with \u27e8hg, m, hm\u27e9\n[GOAL]\ncase prime.intro.intro\nF : Type u_1\ninst\u271d : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\ng : F[X]\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP F q\nhg : Separable g\nm : \u2115\nhm : \u2191(expand F (q ^ m)) g = f\n\u22a2 natDegree g = HasSeparableContraction.degree hf\n[PROOFSTEP]\nlet g' := Classical.choose hf\n[GOAL]\ncase prime.intro.intro\nF : Type u_1\ninst\u271d : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\ng : F[X]\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP F q\nhg : Separable g\nm : \u2115\nhm : \u2191(expand F (q ^ m)) g = f\ng' : F[X] := choose hf\n\u22a2 natDegree g = HasSeparableContraction.degree hf\n[PROOFSTEP]\ncases' (Classical.choose_spec hf).2 with m' hm'\n[GOAL]\ncase prime.intro.intro.intro\nF : Type u_1\ninst\u271d : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\ng : F[X]\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP F q\nhg : Separable g\nm : \u2115\nhm : \u2191(expand F (q ^ m)) g = f\ng' : F[X] := choose hf\nm' : \u2115\nhm' : \u2191(expand F (q ^ m')) (choose hf) = f\n\u22a2 natDegree g = HasSeparableContraction.degree hf\n[PROOFSTEP]\nhaveI : Fact q.Prime := \u27e8by assumption\u27e9\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\ng : F[X]\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP F q\nhg : Separable g\nm : \u2115\nhm : \u2191(expand F (q ^ m)) g = f\ng' : F[X] := choose hf\nm' : \u2115\nhm' : \u2191(expand F (q ^ m')) (choose hf) = f\n\u22a2 Nat.Prime q\n[PROOFSTEP]\nassumption\n[GOAL]\ncase prime.intro.intro.intro\nF : Type u_1\ninst\u271d : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\ng : F[X]\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP F q\nhg : Separable g\nm : \u2115\nhm : \u2191(expand F (q ^ m)) g = f\ng' : F[X] := choose hf\nm' : \u2115\nhm' : \u2191(expand F (q ^ m')) (choose hf) = f\nthis : Fact (Nat.Prime q)\n\u22a2 natDegree g = HasSeparableContraction.degree hf\n[PROOFSTEP]\napply contraction_degree_eq_or_insep q g g' m m'\n[GOAL]\ncase prime.intro.intro.intro.h_expand\nF : Type u_1\ninst\u271d : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\ng : F[X]\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP F q\nhg : Separable g\nm : \u2115\nhm : \u2191(expand F (q ^ m)) g = f\ng' : F[X] := choose hf\nm' : \u2115\nhm' : \u2191(expand F (q ^ m')) (choose hf) = f\nthis : Fact (Nat.Prime q)\n\u22a2 \u2191(expand F (q ^ m)) g = \u2191(expand F (q ^ m')) g'\ncase prime.intro.intro.intro.hg\nF : Type u_1\ninst\u271d : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\ng : F[X]\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP F q\nhg : Separable g\nm : \u2115\nhm : \u2191(expand F (q ^ m)) g = f\ng' : F[X] := choose hf\nm' : \u2115\nhm' : \u2191(expand F (q ^ m')) (choose hf) = f\nthis : Fact (Nat.Prime q)\n\u22a2 Separable g\ncase prime.intro.intro.intro.hg'\nF : Type u_1\ninst\u271d : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\ng : F[X]\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP F q\nhg : Separable g\nm : \u2115\nhm : \u2191(expand F (q ^ m)) g = f\ng' : F[X] := choose hf\nm' : \u2115\nhm' : \u2191(expand F (q ^ m')) (choose hf) = f\nthis : Fact (Nat.Prime q)\n\u22a2 Separable g'\n[PROOFSTEP]\nrw [hm, hm']\n[GOAL]\ncase prime.intro.intro.intro.hg\nF : Type u_1\ninst\u271d : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\ng : F[X]\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP F q\nhg : Separable g\nm : \u2115\nhm : \u2191(expand F (q ^ m)) g = f\ng' : F[X] := choose hf\nm' : \u2115\nhm' : \u2191(expand F (q ^ m')) (choose hf) = f\nthis : Fact (Nat.Prime q)\n\u22a2 Separable g\ncase prime.intro.intro.intro.hg'\nF : Type u_1\ninst\u271d : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\ng : F[X]\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP F q\nhg : Separable g\nm : \u2115\nhm : \u2191(expand F (q ^ m)) g = f\ng' : F[X] := choose hf\nm' : \u2115\nhm' : \u2191(expand F (q ^ m')) (choose hf) = f\nthis : Fact (Nat.Prime q)\n\u22a2 Separable g'\n[PROOFSTEP]\nexact hg\n[GOAL]\ncase prime.intro.intro.intro.hg'\nF : Type u_1\ninst\u271d : Field F\nq : \u2115\nf : F[X]\nhf : HasSeparableContraction q f\ng : F[X]\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP F q\nhg : Separable g\nm : \u2115\nhm : \u2191(expand F (q ^ m)) g = f\ng' : F[X] := choose hf\nm' : \u2115\nhm' : \u2191(expand F (q ^ m')) (choose hf) = f\nthis : Fact (Nat.Prime q)\n\u22a2 Separable g'\n[PROOFSTEP]\nexact (Classical.choose_spec hf).1\n", "meta": {"mathlib_filename": "Mathlib.FieldTheory.SeparableDegree", "llama_tokens": 5613, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850154599562, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4021934623882379}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\n\u22a2 Monotone card\n[PROOFSTEP]\napply card_le_of_subset\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 card ({a} \u2229 s) \u2264 1\n[PROOFSTEP]\ncases' Finset.decidableMem a s with h h\n[GOAL]\ncase isFalse\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : \u00aca \u2208 s\n\u22a2 card ({a} \u2229 s) \u2264 1\n[PROOFSTEP]\nsimp [Finset.singleton_inter_of_not_mem h]\n[GOAL]\ncase isTrue\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a \u2208 s\n\u22a2 card ({a} \u2229 s) \u2264 1\n[PROOFSTEP]\nsimp [Finset.singleton_inter_of_mem h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : \u00aca \u2208 s\n\u22a2 card (insert a s) = card s + 1\n[PROOFSTEP]\nrw [\u2190 cons_eq_insert _ _ h, card_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a \u2208 s\n\u22a2 card (insert a s) = card s\n[PROOFSTEP]\nrw [insert_eq_of_mem h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\n\u22a2 card (insert a s) \u2264 card s + 1\n[PROOFSTEP]\nby_cases h : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\nh : a \u2208 s\n\u22a2 card (insert a s) \u2264 card s + 1\n[PROOFSTEP]\nrw [insert_eq_of_mem h]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\nh : a \u2208 s\n\u22a2 card s \u2264 card s + 1\n[PROOFSTEP]\nexact Nat.le_succ _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\nh : \u00aca \u2208 s\n\u22a2 card (insert a s) \u2264 card s + 1\n[PROOFSTEP]\nrw [card_insert_of_not_mem h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 card (insert a s) = if a \u2208 s then card s else card s + 1\n[PROOFSTEP]\nby_cases h : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a \u2208 s\n\u22a2 card (insert a s) = if a \u2208 s then card s else card s + 1\n[PROOFSTEP]\nrw [card_insert_of_mem h, if_pos h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : \u00aca \u2208 s\n\u22a2 card (insert a s) = if a \u2208 s then card s else card s + 1\n[PROOFSTEP]\nrw [card_insert_of_not_mem h, if_neg h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a \u2260 b\n\u22a2 card {a, b} = 2\n[PROOFSTEP]\nrw [card_insert_of_not_mem (not_mem_singleton.2 h), card_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 card s - 1 \u2264 card (erase s a)\n[PROOFSTEP]\nby_cases h : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a \u2208 s\n\u22a2 card s - 1 \u2264 card (erase s a)\n[PROOFSTEP]\nexact (card_erase_of_mem h).ge\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : \u00aca \u2208 s\n\u22a2 card s - 1 \u2264 card (erase s a)\n[PROOFSTEP]\nrw [erase_eq_of_not_mem h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : \u00aca \u2208 s\n\u22a2 card s - 1 \u2264 card s\n[PROOFSTEP]\nexact Nat.sub_le _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ns : Finset \u03b1\n\u22a2 List.length (toList s) = card s\n[PROOFSTEP]\nrw [toList, \u2190 Multiset.coe_card, Multiset.coe_toList, card_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b2\n\u22a2 card (image f s) \u2264 card s\n[PROOFSTEP]\nsimpa only [card_map] using (s.1.map f).toFinset_card_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b2\nH : Set.InjOn f \u2191s\n\u22a2 card (image f s) = card s\n[PROOFSTEP]\nsimp only [card, image_val_of_injOn H, card_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b2\nH : card (image f s) = card s\n\u22a2 Set.InjOn f \u2191s\n[PROOFSTEP]\nrw [card_def, card_def, image, toFinset] at H \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b2\nH :\n \u2191Multiset.card { val := dedup (Multiset.map f s.val), nodup := (_ : Nodup (dedup (Multiset.map f s.val))) }.val =\n \u2191Multiset.card s.val\n\u22a2 Set.InjOn f \u2191s\n[PROOFSTEP]\ndsimp only at H \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b2\nH : \u2191Multiset.card (dedup (Multiset.map f s.val)) = \u2191Multiset.card s.val\n\u22a2 Set.InjOn f \u2191s\n[PROOFSTEP]\nhave : (s.1.map f).dedup = s.1.map f :=\n by\n refine Multiset.eq_of_le_of_card_le (Multiset.dedup_le _) ?_\n simp only [H, Multiset.card_map, le_rfl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b2\nH : \u2191Multiset.card (dedup (Multiset.map f s.val)) = \u2191Multiset.card s.val\n\u22a2 dedup (Multiset.map f s.val) = Multiset.map f s.val\n[PROOFSTEP]\nrefine Multiset.eq_of_le_of_card_le (Multiset.dedup_le _) ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b2\nH : \u2191Multiset.card (dedup (Multiset.map f s.val)) = \u2191Multiset.card s.val\n\u22a2 \u2191Multiset.card (Multiset.map f s.val) \u2264 \u2191Multiset.card (dedup (Multiset.map f s.val))\n[PROOFSTEP]\nsimp only [H, Multiset.card_map, le_rfl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b2\nH : \u2191Multiset.card (dedup (Multiset.map f s.val)) = \u2191Multiset.card s.val\nthis : dedup (Multiset.map f s.val) = Multiset.map f s.val\n\u22a2 Set.InjOn f \u2191s\n[PROOFSTEP]\nrw [Multiset.dedup_eq_self] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b2\nH : \u2191Multiset.card (dedup (Multiset.map f s.val)) = \u2191Multiset.card s.val\nthis : Nodup (Multiset.map f s.val)\n\u22a2 Set.InjOn f \u2191s\n[PROOFSTEP]\nexact inj_on_of_nodup_map this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : DecidableEq \u03b2\ny : \u03b2\n\u22a2 card (filter (fun x => f x = y) s) \u2260 0 \u2194 y \u2208 image f s\n[PROOFSTEP]\nrw [\u2190 pos_iff_ne_zero, card_pos, fiber_nonempty_iff_mem_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Finset \u03b1\n\u22a2 card (Finset.subtype p s) = card (filter p s)\n[PROOFSTEP]\nsimp [Finset.subtype]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : card (filter p s) = card s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 p x\n[PROOFSTEP]\nrw [\u2190 eq_of_subset_of_card_le (s.filter_subset p) h.ge, mem_filter] at hx \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : card (filter p s) = card s\nx : \u03b1\nhx : x \u2208 s \u2227 p x\n\u22a2 p x\n[PROOFSTEP]\nexact hx.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\n\u22a2 card s = n\n[PROOFSTEP]\nclassical\nhave : \u2200 a : \u03b1, a \u2208 s \u2194 \u2203 (i : _) (hi : i \u2208 range n), f i (mem_range.1 hi) = a := fun a =>\n \u27e8fun ha =>\n let \u27e8i, hi, eq\u27e9 := hf a ha\n \u27e8i, mem_range.2 hi, eq\u27e9,\n fun \u27e8i, hi, eq\u27e9 => eq \u25b8 hf' i (mem_range.1 hi)\u27e9\nhave : s = (range n).attach.image fun i => f i.1 (mem_range.1 i.2) := by\n simpa only [ext_iff, mem_image, exists_prop, Subtype.exists, mem_attach, true_and_iff]\ncalc\n s.card = card ((range n).attach.image fun i => f i.1 (mem_range.1 i.2)) := by rw [this]\n _ = card (range n).attach :=\n ((card_image_of_injective _) fun \u27e8i, hi\u27e9 \u27e8j, hj\u27e9 eq => Subtype.eq <| f_inj i j (mem_range.1 hi) (mem_range.1 hj) eq)\n _ = card (range n) := card_attach\n _ = n := card_range n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\n\u22a2 card s = n\n[PROOFSTEP]\nhave : \u2200 a : \u03b1, a \u2208 s \u2194 \u2203 (i : _) (hi : i \u2208 range n), f i (mem_range.1 hi) = a := fun a =>\n \u27e8fun ha =>\n let \u27e8i, hi, eq\u27e9 := hf a ha\n \u27e8i, mem_range.2 hi, eq\u27e9,\n fun \u27e8i, hi, eq\u27e9 => eq \u25b8 hf' i (mem_range.1 hi)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nthis : \u2200 (a : \u03b1), a \u2208 s \u2194 \u2203 i hi, f i (_ : i < n) = a\n\u22a2 card s = n\n[PROOFSTEP]\nhave : s = (range n).attach.image fun i => f i.1 (mem_range.1 i.2) := by\n simpa only [ext_iff, mem_image, exists_prop, Subtype.exists, mem_attach, true_and_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nthis : \u2200 (a : \u03b1), a \u2208 s \u2194 \u2203 i hi, f i (_ : i < n) = a\n\u22a2 s = image (fun i => f \u2191i (_ : \u2191i < n)) (attach (range n))\n[PROOFSTEP]\nsimpa only [ext_iff, mem_image, exists_prop, Subtype.exists, mem_attach, true_and_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nthis\u271d : \u2200 (a : \u03b1), a \u2208 s \u2194 \u2203 i hi, f i (_ : i < n) = a\nthis : s = image (fun i => f \u2191i (_ : \u2191i < n)) (attach (range n))\n\u22a2 card s = n\n[PROOFSTEP]\ncalc\n s.card = card ((range n).attach.image fun i => f i.1 (mem_range.1 i.2)) := by rw [this]\n _ = card (range n).attach :=\n ((card_image_of_injective _) fun \u27e8i, hi\u27e9 \u27e8j, hj\u27e9 eq => Subtype.eq <| f_inj i j (mem_range.1 hi) (mem_range.1 hj) eq)\n _ = card (range n) := card_attach\n _ = n := card_range n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nf : (i : \u2115) \u2192 i < n \u2192 \u03b1\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a\nhf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s\nf_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j\nthis\u271d : \u2200 (a : \u03b1), a \u2208 s \u2194 \u2203 i hi, f i (_ : i < n) = a\nthis : s = image (fun i => f \u2191i (_ : \u2191i < n)) (attach (range n))\n\u22a2 card s = card (image (fun i => f \u2191i (_ : \u2191i < n)) (attach (range n)))\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nh\u2081 : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nh\u2082 : \u2200 (a b : \u03b1) (ha : a \u2208 s) (hb : b \u2208 s), f a ha = f b hb \u2192 a = b\nh\u2083 : \u2200 (b : \u03b2), b \u2208 t \u2192 \u2203 a ha, f a ha = b\n\u22a2 card s = card t\n[PROOFSTEP]\nclassical calc\n s.card = s.attach.card := card_attach.symm\n _ = (s.attach.image fun a : { a // a \u2208 s } => f a.1 a.2).card :=\n (Eq.symm ((card_image_of_injective _) fun a b h => Subtype.eq <| h\u2082 _ _ _ _ h))\n _ = t.card :=\n congr_arg card\n (Finset.ext fun b =>\n \u27e8fun h =>\n let \u27e8a, _, ha\u2082\u27e9 := mem_image.1 h\n ha\u2082 \u25b8 h\u2081 _ _,\n fun h =>\n let \u27e8a, ha\u2081, ha\u2082\u27e9 := h\u2083 b h\n mem_image.2 \u27e8\u27e8a, ha\u2081\u27e9, by simp [ha\u2082]\u27e9\u27e9)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nh\u2081 : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nh\u2082 : \u2200 (a b : \u03b1) (ha : a \u2208 s) (hb : b \u2208 s), f a ha = f b hb \u2192 a = b\nh\u2083 : \u2200 (b : \u03b2), b \u2208 t \u2192 \u2203 a ha, f a ha = b\n\u22a2 card s = card t\n[PROOFSTEP]\ncalc\n s.card = s.attach.card := card_attach.symm\n _ = (s.attach.image fun a : { a // a \u2208 s } => f a.1 a.2).card :=\n (Eq.symm ((card_image_of_injective _) fun a b h => Subtype.eq <| h\u2082 _ _ _ _ h))\n _ = t.card :=\n congr_arg card\n (Finset.ext fun b =>\n \u27e8fun h =>\n let \u27e8a, _, ha\u2082\u27e9 := mem_image.1 h\n ha\u2082 \u25b8 h\u2081 _ _,\n fun h =>\n let \u27e8a, ha\u2081, ha\u2082\u27e9 := h\u2083 b h\n mem_image.2 \u27e8\u27e8a, ha\u2081\u27e9, by simp [ha\u2082]\u27e9\u27e9)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nh\u2081 : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nh\u2082 : \u2200 (a b : \u03b1) (ha : a \u2208 s) (hb : b \u2208 s), f a ha = f b hb \u2192 a = b\nh\u2083 : \u2200 (b : \u03b2), b \u2208 t \u2192 \u2203 a ha, f a ha = b\nb : \u03b2\nh : b \u2208 t\na : \u03b1\nha\u2081 : a \u2208 s\nha\u2082 : f a ha\u2081 = b\n\u22a2 { val := a, property := ha\u2081 } \u2208 attach s \u2227\n f \u2191{ val := a, property := ha\u2081 } (_ : \u2191{ val := a, property := ha\u2081 } \u2208 s) = b\n[PROOFSTEP]\nsimp [ha\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nf_inj : \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 f a\u2081 = f a\u2082 \u2192 a\u2081 = a\u2082\n\u22a2 card s \u2264 card t\n[PROOFSTEP]\nclassical calc\n s.card = (s.image f).card := (card_image_of_injOn f_inj).symm\n _ \u2264 t.card := card_le_of_subset <| image_subset_iff.2 hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nf_inj : \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 f a\u2081 = f a\u2082 \u2192 a\u2081 = a\u2082\n\u22a2 card s \u2264 card t\n[PROOFSTEP]\ncalc\n s.card = (s.image f).card := (card_image_of_injOn f_inj).symm\n _ \u2264 t.card := card_le_of_subset <| image_subset_iff.2 hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nhc : card t < card s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\n\u22a2 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y \u2227 f x = f y\n[PROOFSTEP]\nclassical\nby_contra' hz\nrefine' hc.not_le (card_le_card_of_inj_on f hf _)\nintro x hx y hy\ncontrapose\nexact hz x hx y hy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nhc : card t < card s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\n\u22a2 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y \u2227 f x = f y\n[PROOFSTEP]\nby_contra' hz\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nhc : card t < card s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nhz : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x \u2260 y \u2192 f x \u2260 f y\n\u22a2 False\n[PROOFSTEP]\nrefine' hc.not_le (card_le_card_of_inj_on f hf _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nhc : card t < card s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nhz : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x \u2260 y \u2192 f x \u2260 f y\n\u22a2 \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 f a\u2081 = f a\u2082 \u2192 a\u2081 = a\u2082\n[PROOFSTEP]\nintro x hx y hy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nhc : card t < card s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nhz : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x \u2260 y \u2192 f x \u2260 f y\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\n\u22a2 f x = f y \u2192 x = y\n[PROOFSTEP]\ncontrapose\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nhc : card t < card s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nhz : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x \u2260 y \u2192 f x \u2260 f y\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\n\u22a2 \u00acx = y \u2192 \u00acf x = f y\n[PROOFSTEP]\nexact hz x hx y hy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nf : \u2115 \u2192 \u03b1\nhf : \u2200 (i : \u2115), i < n \u2192 f i \u2208 s\nf_inj : \u2200 (i : \u2115), i < n \u2192 \u2200 (j : \u2115), j < n \u2192 f i = f j \u2192 i = j\n\u22a2 \u2200 (a : \u2115), a \u2208 range n \u2192 f a \u2208 s\n[PROOFSTEP]\nsimpa only [mem_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nf : \u2115 \u2192 \u03b1\nhf : \u2200 (i : \u2115), i < n \u2192 f i \u2208 s\nf_inj : \u2200 (i : \u2115), i < n \u2192 \u2200 (j : \u2115), j < n \u2192 f i = f j \u2192 i = j\n\u22a2 \u2200 (a\u2081 : \u2115), a\u2081 \u2208 range n \u2192 \u2200 (a\u2082 : \u2115), a\u2082 \u2208 range n \u2192 f a\u2081 = f a\u2082 \u2192 a\u2081 = a\u2082\n[PROOFSTEP]\nsimpa only [mem_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : card t \u2264 card s\n\u22a2 \u2200 (b : \u03b2), b \u2208 t \u2192 \u2203 a ha, b = f a ha\n[PROOFSTEP]\nclassical\nintro b hb\nhave h : (s.attach.image fun a : { a // a \u2208 s } => f a a.prop).card = s.card :=\n @card_attach _ s \u25b8 card_image_of_injective _ fun \u27e8a\u2081, ha\u2081\u27e9 \u27e8a\u2082, ha\u2082\u27e9 h => Subtype.eq <| hinj _ _ _ _ h\nhave h' : image (fun a : { a // a \u2208 s } => f a a.prop) s.attach = t :=\n eq_of_subset_of_card_le\n (fun b h =>\n let \u27e8a, _, ha\u2082\u27e9 := mem_image.1 h\n ha\u2082 \u25b8 hf _ _)\n (by simp [hst, h])\nrw [\u2190 h'] at hb \nobtain \u27e8a, _, ha\u2082\u27e9 := mem_image.1 hb\nexact \u27e8a, a.2, ha\u2082.symm\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : card t \u2264 card s\n\u22a2 \u2200 (b : \u03b2), b \u2208 t \u2192 \u2203 a ha, b = f a ha\n[PROOFSTEP]\nintro b hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : card t \u2264 card s\nb : \u03b2\nhb : b \u2208 t\n\u22a2 \u2203 a ha, b = f a ha\n[PROOFSTEP]\nhave h : (s.attach.image fun a : { a // a \u2208 s } => f a a.prop).card = s.card :=\n @card_attach _ s \u25b8 card_image_of_injective _ fun \u27e8a\u2081, ha\u2081\u27e9 \u27e8a\u2082, ha\u2082\u27e9 h => Subtype.eq <| hinj _ _ _ _ h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : card t \u2264 card s\nb : \u03b2\nhb : b \u2208 t\nh : card (image (fun a => f \u2191a (_ : \u2191a \u2208 s)) (attach s)) = card s\n\u22a2 \u2203 a ha, b = f a ha\n[PROOFSTEP]\nhave h' : image (fun a : { a // a \u2208 s } => f a a.prop) s.attach = t :=\n eq_of_subset_of_card_le\n (fun b h =>\n let \u27e8a, _, ha\u2082\u27e9 := mem_image.1 h\n ha\u2082 \u25b8 hf _ _)\n (by simp [hst, h])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : card t \u2264 card s\nb : \u03b2\nhb : b \u2208 t\nh : card (image (fun a => f \u2191a (_ : \u2191a \u2208 s)) (attach s)) = card s\n\u22a2 card t \u2264 card (image (fun a => f \u2191a (_ : \u2191a \u2208 s)) (attach s))\n[PROOFSTEP]\nsimp [hst, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : card t \u2264 card s\nb : \u03b2\nhb : b \u2208 t\nh : card (image (fun a => f \u2191a (_ : \u2191a \u2208 s)) (attach s)) = card s\nh' : image (fun a => f \u2191a (_ : \u2191a \u2208 s)) (attach s) = t\n\u22a2 \u2203 a ha, b = f a ha\n[PROOFSTEP]\nrw [\u2190 h'] at hb \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : card t \u2264 card s\nb : \u03b2\nhb : b \u2208 image (fun a => f \u2191a (_ : \u2191a \u2208 s)) (attach s)\nh : card (image (fun a => f \u2191a (_ : \u2191a \u2208 s)) (attach s)) = card s\nh' : image (fun a => f \u2191a (_ : \u2191a \u2208 s)) (attach s) = t\n\u22a2 \u2203 a ha, b = f a ha\n[PROOFSTEP]\nobtain \u27e8a, _, ha\u2082\u27e9 := mem_image.1 hb\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhinj : \u2200 (a\u2081 a\u2082 : \u03b1) (ha\u2081 : a\u2081 \u2208 s) (ha\u2082 : a\u2082 \u2208 s), f a\u2081 ha\u2081 = f a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\nhst : card t \u2264 card s\nb : \u03b2\nhb : b \u2208 image (fun a => f \u2191a (_ : \u2191a \u2208 s)) (attach s)\nh : card (image (fun a => f \u2191a (_ : \u2191a \u2208 s)) (attach s)) = card s\nh' : image (fun a => f \u2191a (_ : \u2191a \u2208 s)) (attach s) = t\na : { a // a \u2208 s }\nleft\u271d : a \u2208 attach s\nha\u2082 : f \u2191a (_ : \u2191a \u2208 s) = b\n\u22a2 \u2203 a ha, b = f a ha\n[PROOFSTEP]\nexact \u27e8a, a.2, ha\u2082.symm\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nhf : \u2200 (a : \u03b1) (ha : a \u2208 s), f a ha \u2208 t\nhsurj : \u2200 (b : \u03b2), b \u2208 t \u2192 \u2203 a ha, b = f a ha\nhst : card s \u2264 card t\na\u2081 a\u2082 : \u03b1\nha\u2081 : a\u2081 \u2208 s\nha\u2082 : a\u2082 \u2208 s\nha\u2081a\u2082 : f a\u2081 ha\u2081 = f a\u2082 ha\u2082\nthis : Inhabited { x // x \u2208 s }\nf' : { x // x \u2208 s } \u2192 { x // x \u2208 t } := fun x => { val := f \u2191x (_ : \u2191x \u2208 s), property := (_ : f \u2191x (_ : \u2191x \u2208 s) \u2208 t) }\ng : { x // x \u2208 t } \u2192 { x // x \u2208 s } := surjInv (_ : \u2200 (x : { x // x \u2208 t }), \u2203 a, f' a = x)\nhg : Injective g\nx : { x // x \u2208 s }\n\u22a2 card (attach s) \u2264 card (attach t)\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 card (s \u222a \u2205) + card (s \u2229 \u2205) = card s + card \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na : \u03b1\nr : Finset \u03b1\nhar : \u00aca \u2208 r\nh : card (s \u222a r) + card (s \u2229 r) = card s + card r\n\u22a2 card (s \u222a insert a r) + card (s \u2229 insert a r) = card s + card (insert a r)\n[PROOFSTEP]\nby_cases a \u2208 s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na : \u03b1\nr : Finset \u03b1\nhar : \u00aca \u2208 r\nh : card (s \u222a r) + card (s \u2229 r) = card s + card r\n\u22a2 card (s \u222a insert a r) + card (s \u2229 insert a r) = card s + card (insert a r)\n[PROOFSTEP]\nby_cases a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na : \u03b1\nr : Finset \u03b1\nhar : \u00aca \u2208 r\nh\u271d : card (s \u222a r) + card (s \u2229 r) = card s + card r\nh : a \u2208 s\n\u22a2 card (s \u222a insert a r) + card (s \u2229 insert a r) = card s + card (insert a r)\n[PROOFSTEP]\nsimp [*, \u2190 add_assoc, add_right_comm _ 1]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na : \u03b1\nr : Finset \u03b1\nhar : \u00aca \u2208 r\nh\u271d : card (s \u222a r) + card (s \u2229 r) = card s + card r\nh : \u00aca \u2208 s\n\u22a2 card (s \u222a insert a r) + card (s \u2229 insert a r) = card s + card (insert a r)\n[PROOFSTEP]\nsimp [*, \u2190 add_assoc, add_right_comm _ 1]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 card (s \u2229 t) + card (s \u222a t) = card s + card t\n[PROOFSTEP]\nrw [add_comm, card_union_add_card_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : _root_.Disjoint s t\n\u22a2 card (s \u222a t) = card s + card t\n[PROOFSTEP]\nrw [\u2190 disjUnion_eq_union s t h, card_disjUnion _ _ _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : s \u2286 t\n\u22a2 card (t \\ s) = card t - card s\n[PROOFSTEP]\nsuffices card (t \\ s) = card (t \\ s \u222a s) - s.card by rwa [sdiff_union_of_subset h] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : s \u2286 t\nthis : card (t \\ s) = card (t \\ s \u222a s) - card s\n\u22a2 card (t \\ s) = card t - card s\n[PROOFSTEP]\nrwa [sdiff_union_of_subset h] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : s \u2286 t\n\u22a2 card (t \\ s) = card (t \\ s \u222a s) - card s\n[PROOFSTEP]\nrw [card_disjoint_union sdiff_disjoint, add_tsub_cancel_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 card (t \\ (s \u2229 t)) \u2264 card (t \\ s)\n[PROOFSTEP]\nrw [sdiff_inter_self_right t s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\n\u22a2 card (s \\ t) + card t = card (s \u222a t)\n[PROOFSTEP]\nrw [\u2190 card_disjoint_union sdiff_disjoint, sdiff_union_self_eq_union]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : (x : \u03b1) \u2192 Decidable \u00acp x\n\u22a2 card (filter p s) + card (filter (fun a => \u00acp a) s) = card s\n[PROOFSTEP]\nclassical rw [\u2190 card_union_eq (disjoint_filter_filter_neg _ _ _), filter_union_filter_neg_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : (x : \u03b1) \u2192 Decidable \u00acp x\n\u22a2 card (filter p s) + card (filter (fun a => \u00acp a) s) = card s\n[PROOFSTEP]\nrw [\u2190 card_union_eq (disjoint_filter_filter_neg _ _ _), filter_union_filter_neg_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA B : Finset \u03b1\ni : \u2115\nh\u2081 : i + card B \u2264 card A\nh\u2082 : B \u2286 A\n\u22a2 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\n[PROOFSTEP]\nclassical\nrcases Nat.le.dest h\u2081 with \u27e8k, h\u27e9\nclear h\u2081\ninduction' k with k ih generalizing A\n\u00b7 exact \u27e8A, h\u2082, Subset.refl _, h.symm\u27e9\nobtain \u27e8a, ha\u27e9 : (A \\ B).Nonempty :=\n by\n rw [\u2190 card_pos, card_sdiff h\u2082, \u2190 h, Nat.add_right_comm, add_tsub_cancel_right, Nat.add_succ]\n apply Nat.succ_pos\nhave z : i + card B + k = card (erase A a) := by\n rw [card_erase_of_mem (mem_sdiff.1 ha).1, \u2190 h, Nat.add_sub_assoc (Nat.one_le_iff_ne_zero.mpr k.succ_ne_zero), \u2190\n pred_eq_sub_one, k.pred_succ]\nhave : B \u2286 A.erase a := by\n rintro t th\n apply mem_erase_of_ne_of_mem _ (h\u2082 th)\n rintro rfl\n exact not_mem_sdiff_of_mem_right th ha\nrcases ih this z with \u27e8B', hB', B'subA', cards\u27e9\nexact \u27e8B', hB', B'subA'.trans (erase_subset _ _), cards\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA B : Finset \u03b1\ni : \u2115\nh\u2081 : i + card B \u2264 card A\nh\u2082 : B \u2286 A\n\u22a2 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\n[PROOFSTEP]\nrcases Nat.le.dest h\u2081 with \u27e8k, h\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA B : Finset \u03b1\ni : \u2115\nh\u2081 : i + card B \u2264 card A\nh\u2082 : B \u2286 A\nk : \u2115\nh : i + card B + k = card A\n\u22a2 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\n[PROOFSTEP]\nclear h\u2081\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA B : Finset \u03b1\ni : \u2115\nh\u2082 : B \u2286 A\nk : \u2115\nh : i + card B + k = card A\n\u22a2 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\n[PROOFSTEP]\ninduction' k with k ih generalizing A\n[GOAL]\ncase intro.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA\u271d B : Finset \u03b1\ni : \u2115\nh\u2082\u271d : B \u2286 A\u271d\nk : \u2115\nh\u271d : i + card B + k = card A\u271d\nA : Finset \u03b1\nh\u2082 : B \u2286 A\nh : i + card B + zero = card A\n\u22a2 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\n[PROOFSTEP]\nexact \u27e8A, h\u2082, Subset.refl _, h.symm\u27e9\n[GOAL]\ncase intro.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA\u271d B : Finset \u03b1\ni : \u2115\nh\u2082\u271d : B \u2286 A\u271d\nk\u271d : \u2115\nh\u271d : i + card B + k\u271d = card A\u271d\nk : \u2115\nih : \u2200 {A : Finset \u03b1}, B \u2286 A \u2192 i + card B + k = card A \u2192 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\nA : Finset \u03b1\nh\u2082 : B \u2286 A\nh : i + card B + succ k = card A\n\u22a2 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 : (A \\ B).Nonempty :=\n by\n rw [\u2190 card_pos, card_sdiff h\u2082, \u2190 h, Nat.add_right_comm, add_tsub_cancel_right, Nat.add_succ]\n apply Nat.succ_pos\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA\u271d B : Finset \u03b1\ni : \u2115\nh\u2082\u271d : B \u2286 A\u271d\nk\u271d : \u2115\nh\u271d : i + card B + k\u271d = card A\u271d\nk : \u2115\nih : \u2200 {A : Finset \u03b1}, B \u2286 A \u2192 i + card B + k = card A \u2192 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\nA : Finset \u03b1\nh\u2082 : B \u2286 A\nh : i + card B + succ k = card A\n\u22a2 Finset.Nonempty (A \\ B)\n[PROOFSTEP]\nrw [\u2190 card_pos, card_sdiff h\u2082, \u2190 h, Nat.add_right_comm, add_tsub_cancel_right, Nat.add_succ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA\u271d B : Finset \u03b1\ni : \u2115\nh\u2082\u271d : B \u2286 A\u271d\nk\u271d : \u2115\nh\u271d : i + card B + k\u271d = card A\u271d\nk : \u2115\nih : \u2200 {A : Finset \u03b1}, B \u2286 A \u2192 i + card B + k = card A \u2192 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\nA : Finset \u03b1\nh\u2082 : B \u2286 A\nh : i + card B + succ k = card A\n\u22a2 0 < succ (i + k)\n[PROOFSTEP]\napply Nat.succ_pos\n[GOAL]\ncase intro.succ.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA\u271d B : Finset \u03b1\ni : \u2115\nh\u2082\u271d : B \u2286 A\u271d\nk\u271d : \u2115\nh\u271d : i + card B + k\u271d = card A\u271d\nk : \u2115\nih : \u2200 {A : Finset \u03b1}, B \u2286 A \u2192 i + card B + k = card A \u2192 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\nA : Finset \u03b1\nh\u2082 : B \u2286 A\nh : i + card B + succ k = card A\na : \u03b1\nha : a \u2208 A \\ B\n\u22a2 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\n[PROOFSTEP]\nhave z : i + card B + k = card (erase A a) := by\n rw [card_erase_of_mem (mem_sdiff.1 ha).1, \u2190 h, Nat.add_sub_assoc (Nat.one_le_iff_ne_zero.mpr k.succ_ne_zero), \u2190\n pred_eq_sub_one, k.pred_succ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA\u271d B : Finset \u03b1\ni : \u2115\nh\u2082\u271d : B \u2286 A\u271d\nk\u271d : \u2115\nh\u271d : i + card B + k\u271d = card A\u271d\nk : \u2115\nih : \u2200 {A : Finset \u03b1}, B \u2286 A \u2192 i + card B + k = card A \u2192 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\nA : Finset \u03b1\nh\u2082 : B \u2286 A\nh : i + card B + succ k = card A\na : \u03b1\nha : a \u2208 A \\ B\n\u22a2 i + card B + k = card (erase A a)\n[PROOFSTEP]\nrw [card_erase_of_mem (mem_sdiff.1 ha).1, \u2190 h, Nat.add_sub_assoc (Nat.one_le_iff_ne_zero.mpr k.succ_ne_zero), \u2190\n pred_eq_sub_one, k.pred_succ]\n[GOAL]\ncase intro.succ.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA\u271d B : Finset \u03b1\ni : \u2115\nh\u2082\u271d : B \u2286 A\u271d\nk\u271d : \u2115\nh\u271d : i + card B + k\u271d = card A\u271d\nk : \u2115\nih : \u2200 {A : Finset \u03b1}, B \u2286 A \u2192 i + card B + k = card A \u2192 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\nA : Finset \u03b1\nh\u2082 : B \u2286 A\nh : i + card B + succ k = card A\na : \u03b1\nha : a \u2208 A \\ B\nz : i + card B + k = card (erase A a)\n\u22a2 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\n[PROOFSTEP]\nhave : B \u2286 A.erase a := by\n rintro t th\n apply mem_erase_of_ne_of_mem _ (h\u2082 th)\n rintro rfl\n exact not_mem_sdiff_of_mem_right th ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA\u271d B : Finset \u03b1\ni : \u2115\nh\u2082\u271d : B \u2286 A\u271d\nk\u271d : \u2115\nh\u271d : i + card B + k\u271d = card A\u271d\nk : \u2115\nih : \u2200 {A : Finset \u03b1}, B \u2286 A \u2192 i + card B + k = card A \u2192 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\nA : Finset \u03b1\nh\u2082 : B \u2286 A\nh : i + card B + succ k = card A\na : \u03b1\nha : a \u2208 A \\ B\nz : i + card B + k = card (erase A a)\n\u22a2 B \u2286 erase A a\n[PROOFSTEP]\nrintro t th\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA\u271d B : Finset \u03b1\ni : \u2115\nh\u2082\u271d : B \u2286 A\u271d\nk\u271d : \u2115\nh\u271d : i + card B + k\u271d = card A\u271d\nk : \u2115\nih : \u2200 {A : Finset \u03b1}, B \u2286 A \u2192 i + card B + k = card A \u2192 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\nA : Finset \u03b1\nh\u2082 : B \u2286 A\nh : i + card B + succ k = card A\na : \u03b1\nha : a \u2208 A \\ B\nz : i + card B + k = card (erase A a)\nt : \u03b1\nth : t \u2208 B\n\u22a2 t \u2208 erase A a\n[PROOFSTEP]\napply mem_erase_of_ne_of_mem _ (h\u2082 th)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA\u271d B : Finset \u03b1\ni : \u2115\nh\u2082\u271d : B \u2286 A\u271d\nk\u271d : \u2115\nh\u271d : i + card B + k\u271d = card A\u271d\nk : \u2115\nih : \u2200 {A : Finset \u03b1}, B \u2286 A \u2192 i + card B + k = card A \u2192 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\nA : Finset \u03b1\nh\u2082 : B \u2286 A\nh : i + card B + succ k = card A\na : \u03b1\nha : a \u2208 A \\ B\nz : i + card B + k = card (erase A a)\nt : \u03b1\nth : t \u2208 B\n\u22a2 t \u2260 a\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA\u271d B : Finset \u03b1\ni : \u2115\nh\u2082\u271d : B \u2286 A\u271d\nk\u271d : \u2115\nh\u271d : i + card B + k\u271d = card A\u271d\nk : \u2115\nih : \u2200 {A : Finset \u03b1}, B \u2286 A \u2192 i + card B + k = card A \u2192 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\nA : Finset \u03b1\nh\u2082 : B \u2286 A\nh : i + card B + succ k = card A\nt : \u03b1\nth : t \u2208 B\nha : t \u2208 A \\ B\nz : i + card B + k = card (erase A t)\n\u22a2 False\n[PROOFSTEP]\nexact not_mem_sdiff_of_mem_right th ha\n[GOAL]\ncase intro.succ.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA\u271d B : Finset \u03b1\ni : \u2115\nh\u2082\u271d : B \u2286 A\u271d\nk\u271d : \u2115\nh\u271d : i + card B + k\u271d = card A\u271d\nk : \u2115\nih : \u2200 {A : Finset \u03b1}, B \u2286 A \u2192 i + card B + k = card A \u2192 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\nA : Finset \u03b1\nh\u2082 : B \u2286 A\nh : i + card B + succ k = card A\na : \u03b1\nha : a \u2208 A \\ B\nz : i + card B + k = card (erase A a)\nthis : B \u2286 erase A a\n\u22a2 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\n[PROOFSTEP]\nrcases ih this z with \u27e8B', hB', B'subA', cards\u27e9\n[GOAL]\ncase intro.succ.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA\u271d B : Finset \u03b1\ni : \u2115\nh\u2082\u271d : B \u2286 A\u271d\nk\u271d : \u2115\nh\u271d : i + card B + k\u271d = card A\u271d\nk : \u2115\nih : \u2200 {A : Finset \u03b1}, B \u2286 A \u2192 i + card B + k = card A \u2192 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\nA : Finset \u03b1\nh\u2082 : B \u2286 A\nh : i + card B + succ k = card A\na : \u03b1\nha : a \u2208 A \\ B\nz : i + card B + k = card (erase A a)\nthis : B \u2286 erase A a\nB' : Finset \u03b1\nhB' : B \u2286 B'\nB'subA' : B' \u2286 erase A a\ncards : card B' = i + card B\n\u22a2 \u2203 C, B \u2286 C \u2227 C \u2286 A \u2227 card C = i + card B\n[PROOFSTEP]\nexact \u27e8B', hB', B'subA'.trans (erase_subset _ _), cards\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nA : Finset \u03b1\ni : \u2115\nh\u2081 : i \u2264 card A\n\u22a2 i + card \u2205 \u2264 card A\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : 2 * n < card (X \u222a Y)\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)\n[PROOFSTEP]\nhave h\u2081 : (X \u2229 (Y \\ X)).card = 0 := Finset.card_eq_zero.mpr (Finset.inter_sdiff_self X Y)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : 2 * n < card (X \u222a Y)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)\n[PROOFSTEP]\nhave h\u2082 : (X \u222a Y).card = X.card + (Y \\ X).card := by\n rw [\u2190 card_union_add_card_inter X (Y \\ X), Finset.union_sdiff_self_eq_union, h\u2081, add_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : 2 * n < card (X \u222a Y)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\n\u22a2 card (X \u222a Y) = card X + card (Y \\ X)\n[PROOFSTEP]\nrw [\u2190 card_union_add_card_inter X (Y \\ X), Finset.union_sdiff_self_eq_union, h\u2081, add_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : 2 * n < card (X \u222a Y)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\nh\u2082 : card (X \u222a Y) = card X + card (Y \\ X)\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)\n[PROOFSTEP]\nrw [h\u2082, two_mul] at hXY \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : n + n < card X + card (Y \\ X)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\nh\u2082 : card (X \u222a Y) = card X + card (Y \\ X)\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)\n[PROOFSTEP]\nrcases lt_or_lt_of_add_lt_add hXY with (h | h)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : n + n < card X + card (Y \\ X)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\nh\u2082 : card (X \u222a Y) = card X + card (Y \\ X)\nh : n < card X\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)\n[PROOFSTEP]\nexact \u27e8X, h, Or.inl (Finset.Subset.refl X)\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\ninst\u271d : DecidableEq \u03b1\nX Y : Finset \u03b1\nn : \u2115\nhXY : n + n < card X + card (Y \\ X)\nh\u2081 : card (X \u2229 (Y \\ X)) = 0\nh\u2082 : card (X \u222a Y) = card X + card (Y \\ X)\nh : n < card (Y \\ X)\n\u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y)\n[PROOFSTEP]\nexact \u27e8Y \\ X, h, Or.inr (Finset.sdiff_subset Y X)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 card s = 1 \u2194 \u2203 a, s = {a}\n[PROOFSTEP]\ncases s\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\n\u22a2 card { val := val\u271d, nodup := nodup\u271d } = 1 \u2194 \u2203 a, { val := val\u271d, nodup := nodup\u271d } = {a}\n[PROOFSTEP]\nsimp only [Multiset.card_eq_one, Finset.card, \u2190 val_inj, singleton_val]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 (\u2203 a x, insert a s = t) \u2194 s \u2286 t \u2227 card s + 1 = card t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 (\u2203 a x, insert a s = t) \u2192 s \u2286 t \u2227 card s + 1 = card t\n[PROOFSTEP]\nrintro \u27e8a, ha, rfl\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 s \u2286 insert a s \u2227 card s + 1 = card (insert a s)\n[PROOFSTEP]\nexact \u27e8subset_insert _ _, (card_insert_of_not_mem ha).symm\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 s \u2286 t \u2227 card s + 1 = card t \u2192 \u2203 a x, insert a s = t\n[PROOFSTEP]\nrintro \u27e8hst, h\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nhst : s \u2286 t\nh : card s + 1 = card t\n\u22a2 \u2203 a x, insert a s = t\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 : \u2203 a, t \\ s = { a } := card_eq_one.1 (by rw [card_sdiff hst, \u2190 h, add_tsub_cancel_left])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nhst : s \u2286 t\nh : card s + 1 = card t\n\u22a2 card (t \\ s) = 1\n[PROOFSTEP]\nrw [card_sdiff hst, \u2190 h, add_tsub_cancel_left]\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nhst : s \u2286 t\nh : card s + 1 = card t\na : \u03b1\nha : t \\ s = {a}\n\u22a2 \u2203 a x, insert a s = t\n[PROOFSTEP]\nrefine' \u27e8a, fun hs => (_ : a \u2209 { a }) <| mem_singleton_self _, by rw [insert_eq, \u2190 ha, sdiff_union_of_subset hst]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nhst : s \u2286 t\nh : card s + 1 = card t\na : \u03b1\nha : t \\ s = {a}\n\u22a2 insert a s = t\n[PROOFSTEP]\nrw [insert_eq, \u2190 ha, sdiff_union_of_subset hst]\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nhst : s \u2286 t\nh : card s + 1 = card t\na : \u03b1\nha : t \\ s = {a}\nhs : a \u2208 s\n\u22a2 \u00aca \u2208 {a}\n[PROOFSTEP]\nrw [\u2190 ha]\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nhst : s \u2286 t\nh : card s + 1 = card t\na : \u03b1\nha : t \\ s = {a}\nhs : a \u2208 s\n\u22a2 \u00aca \u2208 t \\ s\n[PROOFSTEP]\nexact not_mem_sdiff_of_mem_right hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 card s \u2264 1 \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a = b\n[PROOFSTEP]\nobtain rfl | \u27e8x, hx\u27e9 := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 card \u2205 \u2264 1 \u2194 \u2200 (a : \u03b1), a \u2208 \u2205 \u2192 \u2200 (b : \u03b1), b \u2208 \u2205 \u2192 a = b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nx : \u03b1\nhx : x \u2208 s\n\u22a2 card s \u2264 1 \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a = b\n[PROOFSTEP]\nrefine' (Nat.succ_le_of_lt (card_pos.2 \u27e8x, hx\u27e9)).le_iff_eq.trans (card_eq_one.trans \u27e8_, _\u27e9)\n[GOAL]\ncase inr.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nx : \u03b1\nhx : x \u2208 s\n\u22a2 (\u2203 a, s = {a}) \u2192 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a = b\n[PROOFSTEP]\nrintro \u27e8y, rfl\u27e9\n[GOAL]\ncase inr.intro.refine'_1.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nx y : \u03b1\nhx : x \u2208 {y}\n\u22a2 \u2200 (a : \u03b1), a \u2208 {y} \u2192 \u2200 (b : \u03b1), b \u2208 {y} \u2192 a = b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nx : \u03b1\nhx : x \u2208 s\n\u22a2 (\u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a = b) \u2192 \u2203 a, s = {a}\n[PROOFSTEP]\nexact fun h => \u27e8x, eq_singleton_iff_unique_mem.2 \u27e8hx, fun y hy => h _ hy _ hx\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 card s \u2264 1 \u2194 \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 a = b\n[PROOFSTEP]\nrw [card_le_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 (\u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a = b) \u2194 \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 a = b\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : Nonempty \u03b1\n\u22a2 card s \u2264 1 \u2194 \u2203 x, s \u2286 {x}\n[PROOFSTEP]\nrefine' \u27e8fun H => _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : Nonempty \u03b1\nH : card s \u2264 1\n\u22a2 \u2203 x, s \u2286 {x}\n[PROOFSTEP]\nobtain rfl | \u27e8x, hx\u27e9 := s.eq_empty_or_nonempty\n[GOAL]\ncase refine'_1.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : Nonempty \u03b1\nH : card \u2205 \u2264 1\n\u22a2 \u2203 x, \u2205 \u2286 {x}\n[PROOFSTEP]\nexact \u27e8Classical.arbitrary \u03b1, empty_subset _\u27e9\n[GOAL]\ncase refine'_1.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : Nonempty \u03b1\nH : card s \u2264 1\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2203 x, s \u2286 {x}\n[PROOFSTEP]\nexact \u27e8x, fun y hy => by rw [card_le_one.1 H y hy x hx, mem_singleton]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : Nonempty \u03b1\nH : card s \u2264 1\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\n\u22a2 y \u2208 {x}\n[PROOFSTEP]\nrw [card_le_one.1 H y hy x hx, mem_singleton]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : Nonempty \u03b1\n\u22a2 (\u2203 x, s \u2286 {x}) \u2192 card s \u2264 1\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : Nonempty \u03b1\nx : \u03b1\nhx : s \u2286 {x}\n\u22a2 card s \u2264 1\n[PROOFSTEP]\nrw [\u2190 card_singleton x]\n[GOAL]\ncase refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : Nonempty \u03b1\nx : \u03b1\nhx : s \u2286 {x}\n\u22a2 card s \u2264 card {x}\n[PROOFSTEP]\nexact card_le_of_subset hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 1 < card s \u2194 \u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 s \u2227 a \u2260 b\n[PROOFSTEP]\nrw [\u2190 not_iff_not]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 \u00ac1 < card s \u2194 \u00ac\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 s \u2227 a \u2260 b\n[PROOFSTEP]\npush_neg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 card s \u2264 1 \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a = b\n[PROOFSTEP]\nexact card_le_one\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 1 < card s \u2194 \u2203 a b, a \u2208 s \u2227 b \u2208 s \u2227 a \u2260 b\n[PROOFSTEP]\nrw [one_lt_card]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 s \u2227 a \u2260 b) \u2194 \u2203 a b, a \u2208 s \u2227 b \u2208 s \u2227 a \u2260 b\n[PROOFSTEP]\nsimp only [exists_prop, exists_and_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 2 < card s \u2194 \u2203 a b c, a \u2208 s \u2227 b \u2208 s \u2227 c \u2208 s \u2227 a \u2260 b \u2227 a \u2260 c \u2227 b \u2260 c\n[PROOFSTEP]\nclassical\nrefine' \u27e8fun h => _, _\u27e9\n\u00b7 obtain \u27e8c, hc\u27e9 := card_pos.mp (pos_of_gt h)\n have : 1 < (s.erase c).card := by rwa [\u2190 add_lt_add_iff_right 1, card_erase_add_one hc]\n obtain \u27e8a, b, ha, hb, hab\u27e9 := one_lt_card_iff.mp this\n exact \u27e8a, b, c, mem_of_mem_erase ha, mem_of_mem_erase hb, hc, hab, ne_of_mem_erase ha, ne_of_mem_erase hb\u27e9\n\u00b7 rintro \u27e8a, b, c, ha, hb, hc, hab, hac, hbc\u27e9\n rw [\u2190 card_erase_add_one hc, \u2190 card_erase_add_one (mem_erase_of_ne_of_mem hbc hb), \u2190\n card_erase_add_one (mem_erase_of_ne_of_mem hab (mem_erase_of_ne_of_mem hac ha))]\n apply Nat.le_add_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 2 < card s \u2194 \u2203 a b c, a \u2208 s \u2227 b \u2208 s \u2227 c \u2208 s \u2227 a \u2260 b \u2227 a \u2260 c \u2227 b \u2260 c\n[PROOFSTEP]\nrefine' \u27e8fun h => _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nh : 2 < card s\n\u22a2 \u2203 a b c, a \u2208 s \u2227 b \u2208 s \u2227 c \u2208 s \u2227 a \u2260 b \u2227 a \u2260 c \u2227 b \u2260 c\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 := card_pos.mp (pos_of_gt h)\n[GOAL]\ncase refine'_1.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nh : 2 < card s\nc : \u03b1\nhc : c \u2208 s\n\u22a2 \u2203 a b c, a \u2208 s \u2227 b \u2208 s \u2227 c \u2208 s \u2227 a \u2260 b \u2227 a \u2260 c \u2227 b \u2260 c\n[PROOFSTEP]\nhave : 1 < (s.erase c).card := by rwa [\u2190 add_lt_add_iff_right 1, card_erase_add_one hc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nh : 2 < card s\nc : \u03b1\nhc : c \u2208 s\n\u22a2 1 < card (erase s c)\n[PROOFSTEP]\nrwa [\u2190 add_lt_add_iff_right 1, card_erase_add_one hc]\n[GOAL]\ncase refine'_1.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nh : 2 < card s\nc : \u03b1\nhc : c \u2208 s\nthis : 1 < card (erase s c)\n\u22a2 \u2203 a b c, a \u2208 s \u2227 b \u2208 s \u2227 c \u2208 s \u2227 a \u2260 b \u2227 a \u2260 c \u2227 b \u2260 c\n[PROOFSTEP]\nobtain \u27e8a, b, ha, hb, hab\u27e9 := one_lt_card_iff.mp this\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nh : 2 < card s\nc : \u03b1\nhc : c \u2208 s\nthis : 1 < card (erase s c)\na b : \u03b1\nha : a \u2208 erase s c\nhb : b \u2208 erase s c\nhab : a \u2260 b\n\u22a2 \u2203 a b c, a \u2208 s \u2227 b \u2208 s \u2227 c \u2208 s \u2227 a \u2260 b \u2227 a \u2260 c \u2227 b \u2260 c\n[PROOFSTEP]\nexact \u27e8a, b, c, mem_of_mem_erase ha, mem_of_mem_erase hb, hc, hab, ne_of_mem_erase ha, ne_of_mem_erase hb\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 (\u2203 a b c, a \u2208 s \u2227 b \u2208 s \u2227 c \u2208 s \u2227 a \u2260 b \u2227 a \u2260 c \u2227 b \u2260 c) \u2192 2 < card s\n[PROOFSTEP]\nrintro \u27e8a, b, c, ha, hb, hc, hab, hac, hbc\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\na b c : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nhc : c \u2208 s\nhab : a \u2260 b\nhac : a \u2260 c\nhbc : b \u2260 c\n\u22a2 2 < card s\n[PROOFSTEP]\nrw [\u2190 card_erase_add_one hc, \u2190 card_erase_add_one (mem_erase_of_ne_of_mem hbc hb), \u2190\n card_erase_add_one (mem_erase_of_ne_of_mem hab (mem_erase_of_ne_of_mem hac ha))]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\na b c : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nhc : c \u2208 s\nhab : a \u2260 b\nhac : a \u2260 c\nhbc : b \u2260 c\n\u22a2 2 < card (erase (erase (erase s c) b) a) + 1 + 1 + 1\n[PROOFSTEP]\napply Nat.le_add_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 2 < card s \u2194 \u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 s \u2227 \u2203 c, c \u2208 s \u2227 a \u2260 b \u2227 a \u2260 c \u2227 b \u2260 c\n[PROOFSTEP]\nsimp_rw [two_lt_card_iff, exists_and_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nhs : 1 < card s\na : \u03b1\n\u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a\n[PROOFSTEP]\nobtain \u27e8x, hx, y, hy, hxy\u27e9 := Finset.one_lt_card.mp hs\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nhs : 1 < card s\na x : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\n\u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a\n[PROOFSTEP]\nby_cases ha : y = a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nhs : 1 < card s\na x : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\nha : y = a\n\u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a\n[PROOFSTEP]\nexact \u27e8x, hx, ne_of_ne_of_eq hxy ha\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nhs : 1 < card s\na x : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nhxy : x \u2260 y\nha : \u00acy = a\n\u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a\n[PROOFSTEP]\nexact \u27e8y, hy, ha\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : card s = n + 1\na : \u03b1\nhas : a \u2208 s\n\u22a2 card (erase s a) = n\n[PROOFSTEP]\nsimp only [h, card_erase_of_mem has, add_tsub_cancel_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\n\u22a2 card s = 2 \u2194 \u2203 x y, x \u2260 y \u2227 s = {x, y}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\n\u22a2 card s = 2 \u2192 \u2203 x y, x \u2260 y \u2227 s = {x, y}\n[PROOFSTEP]\nrw [card_eq_succ]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\n\u22a2 (\u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 card t = 1) \u2192 \u2203 x y, x \u2260 y \u2227 s = {x, y}\n[PROOFSTEP]\nsimp_rw [card_eq_one]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\n\u22a2 (\u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 \u2203 a, t = {a}) \u2192 \u2203 x y, x \u2260 y \u2227 s = {x, y}\n[PROOFSTEP]\nrintro \u27e8a, _, hab, rfl, b, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nhab : \u00aca \u2208 {b}\n\u22a2 \u2203 x y, x \u2260 y \u2227 {a, b} = {x, y}\n[PROOFSTEP]\nexact \u27e8a, b, not_mem_singleton.1 hab, rfl\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\n\u22a2 (\u2203 x y, x \u2260 y \u2227 s = {x, y}) \u2192 card s = 2\n[PROOFSTEP]\nrintro \u27e8x, y, h, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nh : x \u2260 y\n\u22a2 card {x, y} = 2\n[PROOFSTEP]\nexact card_doubleton h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\n\u22a2 card s = 3 \u2194 \u2203 x y z, x \u2260 y \u2227 x \u2260 z \u2227 y \u2260 z \u2227 s = {x, y, z}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\n\u22a2 card s = 3 \u2192 \u2203 x y z, x \u2260 y \u2227 x \u2260 z \u2227 y \u2260 z \u2227 s = {x, y, z}\n[PROOFSTEP]\nrw [card_eq_succ]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\n\u22a2 (\u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 card t = 2) \u2192 \u2203 x y z, x \u2260 y \u2227 x \u2260 z \u2227 y \u2260 z \u2227 s = {x, y, z}\n[PROOFSTEP]\nsimp_rw [card_eq_two]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\n\u22a2 (\u2203 a t, \u00aca \u2208 t \u2227 insert a t = s \u2227 \u2203 x y, x \u2260 y \u2227 t = {x, y}) \u2192 \u2203 x y z, x \u2260 y \u2227 x \u2260 z \u2227 y \u2260 z \u2227 s = {x, y, z}\n[PROOFSTEP]\nrintro \u27e8a, _, abc, rfl, b, c, bc, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nbc : b \u2260 c\nabc : \u00aca \u2208 {b, c}\n\u22a2 \u2203 x y z, x \u2260 y \u2227 x \u2260 z \u2227 y \u2260 z \u2227 {a, b, c} = {x, y, z}\n[PROOFSTEP]\nrw [mem_insert, mem_singleton, not_or] at abc \n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nbc : b \u2260 c\nabc : \u00aca = b \u2227 \u00aca = c\n\u22a2 \u2203 x y z, x \u2260 y \u2227 x \u2260 z \u2227 y \u2260 z \u2227 {a, b, c} = {x, y, z}\n[PROOFSTEP]\nexact \u27e8a, b, c, abc.1, abc.2, bc, rfl\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\n\u22a2 (\u2203 x y z, x \u2260 y \u2227 x \u2260 z \u2227 y \u2260 z \u2227 s = {x, y, z}) \u2192 card s = 3\n[PROOFSTEP]\nrintro \u27e8x, y, z, xy, xz, yz, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nxy : x \u2260 y\nxz : x \u2260 z\nyz : y \u2260 z\n\u22a2 card {x, y, z} = 3\n[PROOFSTEP]\nsimp only [xy, xz, yz, mem_insert, card_insert_of_not_mem, not_false_iff, mem_singleton, or_self_iff, card_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : Finset \u03b1 \u2192 Sort u_3\nH : (s : Finset \u03b1) \u2192 ((t : Finset \u03b1) \u2192 t \u2282 s \u2192 p t) \u2192 p s\ns : Finset \u03b1\n\u22a2 strongInduction H s = H s fun t x => strongInduction H t\n[PROOFSTEP]\nrw [strongInduction]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : Finset \u03b1 \u2192 Sort u_3\ns : Finset \u03b1\nH : (s : Finset \u03b1) \u2192 ((t : Finset \u03b1) \u2192 t \u2282 s \u2192 p t) \u2192 p s\n\u22a2 strongInductionOn s H = H s fun t x => strongInductionOn t H\n[PROOFSTEP]\ndsimp only [strongInductionOn]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : Finset \u03b1 \u2192 Sort u_3\ns : Finset \u03b1\nH : (s : Finset \u03b1) \u2192 ((t : Finset \u03b1) \u2192 t \u2282 s \u2192 p t) \u2192 p s\n\u22a2 strongInduction H s = H s fun t x => strongInduction H t\n[PROOFSTEP]\nrw [strongInduction]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : Finset \u03b1 \u2192 Sort u_3\nH : (t\u2081 : Finset \u03b1) \u2192 ({t\u2082 : Finset \u03b1} \u2192 card t\u2082 \u2264 n \u2192 t\u2081 \u2282 t\u2082 \u2192 p t\u2082) \u2192 card t\u2081 \u2264 n \u2192 p t\u2081\ns : Finset \u03b1\n\u22a2 strongDownwardInduction H s = H s fun {t} ht x => strongDownwardInduction H t ht\n[PROOFSTEP]\nrw [strongDownwardInduction]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : Finset \u03b1 \u2192 Sort u_3\ns : Finset \u03b1\nH : (t\u2081 : Finset \u03b1) \u2192 ({t\u2082 : Finset \u03b1} \u2192 card t\u2082 \u2264 n \u2192 t\u2081 \u2282 t\u2082 \u2192 p t\u2082) \u2192 card t\u2081 \u2264 n \u2192 p t\u2081\n\u22a2 (fun a => strongDownwardInductionOn s H a) = H s fun {t} ht x => strongDownwardInductionOn t H ht\n[PROOFSTEP]\ndsimp only [strongDownwardInductionOn]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : Finset \u03b1 \u2192 Sort u_3\ns : Finset \u03b1\nH : (t\u2081 : Finset \u03b1) \u2192 ({t\u2082 : Finset \u03b1} \u2192 card t\u2082 \u2264 n \u2192 t\u2081 \u2282 t\u2082 \u2192 p t\u2082) \u2192 card t\u2081 \u2264 n \u2192 p t\u2081\n\u22a2 (fun a => strongDownwardInduction H s a) = H s fun {t} ht x => strongDownwardInduction H t ht\n[PROOFSTEP]\nrw [strongDownwardInduction]\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.Card", "llama_tokens": 26826, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737214979746, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.4019602455379065}} {"text": "[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 GroupCatMax\nj : J\n\u22a2 Group ((F \u22d9 forget GroupCat).obj j)\n[PROOFSTEP]\nchange Group (F.obj j)\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 GroupCatMax\nj : J\n\u22a2 Group \u2191(F.obj j)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 GroupCat\nsrc\u271d : Submonoid ((j : J) \u2192 \u2191((F \u22d9 forget\u2082 GroupCat MonCat).obj j)) :=\n MonCat.sectionsSubmonoid (F \u22d9 forget\u2082 GroupCat MonCat)\na : (j : J) \u2192 \u2191(F.obj j)\nah :\n a \u2208\n {\n toSubsemigroup :=\n { carrier := Functor.sections (F \u22d9 forget GroupCat),\n mul_mem' :=\n (_ : \u2200 {a b : (j : J) \u2192 \u2191(F.obj j)}, a \u2208 src\u271d.carrier \u2192 b \u2208 src\u271d.carrier \u2192 a * b \u2208 src\u271d.carrier) },\n one_mem' :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n ((F \u22d9 forget\u2082 GroupCat MonCat) \u22d9 forget MonCat).map f (OfNat.ofNat 1 j) =\n OfNat.ofNat 1 j') }.toSubsemigroup.carrier\nj j' : J\nf : j \u27f6 j'\n\u22a2 (F \u22d9 forget GroupCat).map f (a\u207b\u00b9 j) = a\u207b\u00b9 j'\n[PROOFSTEP]\nsimp only [Functor.comp_map, Pi.inv_apply, MonoidHom.map_inv, inv_inj]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 GroupCat\nsrc\u271d : Submonoid ((j : J) \u2192 \u2191((F \u22d9 forget\u2082 GroupCat MonCat).obj j)) :=\n MonCat.sectionsSubmonoid (F \u22d9 forget\u2082 GroupCat MonCat)\na : (j : J) \u2192 \u2191(F.obj j)\nah :\n a \u2208\n {\n toSubsemigroup :=\n { carrier := Functor.sections (F \u22d9 forget GroupCat),\n mul_mem' :=\n (_ : \u2200 {a b : (j : J) \u2192 \u2191(F.obj j)}, a \u2208 src\u271d.carrier \u2192 b \u2208 src\u271d.carrier \u2192 a * b \u2208 src\u271d.carrier) },\n one_mem' :=\n (_ :\n \u2200 {j j' : J} (f : j \u27f6 j'),\n ((F \u22d9 forget\u2082 GroupCat MonCat) \u22d9 forget MonCat).map f (OfNat.ofNat 1 j) =\n OfNat.ofNat 1 j') }.toSubsemigroup.carrier\nj j' : J\nf : j \u27f6 j'\n\u22a2 (forget GroupCat).map (F.map f) (a j)\u207b\u00b9 = (a j')\u207b\u00b9\n[PROOFSTEP]\ndsimp [Functor.sections] at ah \u22a2\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 GroupCat\nsrc\u271d : Submonoid ((j : J) \u2192 \u2191((F \u22d9 forget\u2082 GroupCat MonCat).obj j)) :=\n MonCat.sectionsSubmonoid (F \u22d9 forget\u2082 GroupCat MonCat)\na : (j : J) \u2192 \u2191(F.obj j)\nah : a \u2208 {u | \u2200 {j j' : J} (f : j \u27f6 j'), \u2191(F.map f) (u j) = u j'}\nj j' : J\nf : j \u27f6 j'\n\u22a2 \u2191(F.map f) (a j)\u207b\u00b9 = (a j')\u207b\u00b9\n[PROOFSTEP]\nrw [(F.map f).map_inv (a j), ah f]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 GroupCatMax\n\u22a2 Group (Types.limitCone (F \u22d9 forget GroupCat)).pt\n[PROOFSTEP]\nchange Group (sectionsSubgroup.{v, u} F)\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 GroupCatMax\n\u22a2 Group { x // x \u2208 sectionsSubgroup F }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 GroupCatMax\nthis : ReflectsIsomorphisms (forget\u2082 GroupCatMax MonCatMax) := reflectsIsomorphisms_forget\u2082 GroupCatMax MonCatMax\nc' : Cone (F \u22d9 forget\u2082 GroupCat MonCat)\nt : IsLimit c'\n\u22a2 (forget\u2082 GroupCat MonCat).mapCone\n { pt := of (Types.limitCone (F \u22d9 forget GroupCatMax)).pt,\n \u03c0 := NatTrans.mk (MonCat.limit\u03c0MonoidHom (F \u22d9 forget\u2082 GroupCatMax MonCatMax)) } \u2245\n c'\n[PROOFSTEP]\napply IsLimit.uniqueUpToIso (MonCat.HasLimits.limitConeIsLimit.{v, u} _) t\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CommGroupCatMax\nj : J\n\u22a2 CommGroup ((F \u22d9 forget CommGroupCatMax).obj j)\n[PROOFSTEP]\nchange CommGroup (F.obj j)\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CommGroupCatMax\nj : J\n\u22a2 CommGroup \u2191(F.obj j)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CommGroupCatMax\nthis : ReflectsIsomorphisms (forget\u2082 CommGroupCatMax GroupCatMax) :=\n reflectsIsomorphisms_forget\u2082 CommGroupCatMax GroupCatMax\nc' : Cone (F \u22d9 forget\u2082 CommGroupCat GroupCatMax)\nt : IsLimit c'\n\u22a2 (forget\u2082 CommGroupCat GroupCatMax).mapCone\n { pt := of (Types.limitCone (F \u22d9 forget CommGroupCat)).pt,\n \u03c0 := NatTrans.mk (MonCat.limit\u03c0MonoidHom (F \u22d9 forget\u2082 CommGroupCat GroupCat \u22d9 forget\u2082 GroupCat MonCat)) } \u2245\n c'\n[PROOFSTEP]\napply IsLimit.uniqueUpToIso (GroupCat.limitConeIsLimit _) t\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 CommGroupCatMax\nthis : ReflectsIsomorphisms (forget\u2082 CommGroupCatMax GroupCatMax) :=\n reflectsIsomorphisms_forget\u2082 CommGroupCatMax GroupCatMax\nc' : Cone (F \u22d9 forget\u2082 CommGroupCat GroupCatMax)\nt : IsLimit c'\n\u22a2 IsLimit\n ((forget\u2082 CommGroupCatMax GroupCat \u22d9 forget\u2082 GroupCat MonCat).mapCone\n {\n liftedCone :=\n { pt := of (Types.limitCone (F \u22d9 forget CommGroupCat)).pt,\n \u03c0 := NatTrans.mk (MonCat.limit\u03c0MonoidHom (F \u22d9 forget\u2082 CommGroupCat GroupCat \u22d9 forget\u2082 GroupCat MonCat)) },\n validLift :=\n IsLimit.uniqueUpToIso (GroupCat.limitConeIsLimit (F \u22d9 forget\u2082 CommGroupCat GroupCatMax)) t }.liftedCone)\n[PROOFSTEP]\napply MonCat.HasLimits.limitConeIsLimit _\n[GOAL]\nJ\u271d : Type v\ninst\u271d : SmallCategory J\u271d\nJ : Type v\n\ud835\udca5 : Category.{v, v} J\nF : J \u2964 CommGroupCatMax\n\u22a2 PreservesLimit F (forget\u2082 CommGroupCatMax GroupCatMax)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nf : \u2115 \u2192 AddCommGroupCat\n\u22a2 HasProduct f\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\n\u22a2 (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0\n[PROOFSTEP]\nrefine Subtype.ext ?_\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\n\u22a2 \u2191((fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0) = \u21910\n[PROOFSTEP]\nsimp [(AddSubgroup.coe_zero _).symm]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\ng g' : \u2191(kernel f)\n\u22a2 ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' := (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n (g + g') =\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' := (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n g +\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' := (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n g'\n[PROOFSTEP]\nrefine Subtype.ext ?_\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\ng g' : \u2191(kernel f)\n\u22a2 \u2191(ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' := (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n (g + g')) =\n \u2191(ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n g +\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n g')\n[PROOFSTEP]\nchange _ = _ + _\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\ng g' : \u2191(kernel f)\n\u22a2 \u2191(ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' := (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n (g + g')) =\n \u2191(ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n g) +\n \u2191(ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n g')\n[PROOFSTEP]\ndsimp\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\ng g' : \u2191(kernel f)\n\u22a2 \u2191(kernel.\u03b9 f) (g + g') = \u2191(kernel.\u03b9 f) g + \u2191(kernel.\u03b9 f) g'\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\n\u22a2 AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f = 0\n[PROOFSTEP]\nrefine FunLike.ext _ _ ?_\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\n\u22a2 \u2200 (x : \u2191(of { x // x \u2208 AddMonoidHom.ker f })), \u2191(AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f) x = \u21910 x\n[PROOFSTEP]\nrintro \u27e8x, (hx : f _ = 0)\u27e9\n[GOAL]\ncase mk\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\nx : \u2191G\nhx : \u2191f x = 0\n\u22a2 \u2191(AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f) { val := x, property := hx } = \u21910 { val := x, property := hx }\n[PROOFSTEP]\nexact hx\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\n\u22a2 {\n toZeroHom :=\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) },\n map_add' :=\n (_ :\n \u2200 (g g' : \u2191(kernel f)),\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n (g + g') =\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ :\n (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n g +\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ :\n (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n g') } \u226b\n kernel.lift f (AddSubgroup.subtype (AddMonoidHom.ker f)) (_ : AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f = 0) =\n \ud835\udfd9 (kernel f)\n[PROOFSTEP]\nrefine equalizer.hom_ext ?_\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\n\u22a2 ({\n toZeroHom :=\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) },\n map_add' :=\n (_ :\n \u2200 (g g' : \u2191(kernel f)),\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ :\n (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n (g + g') =\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ :\n (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n g +\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ :\n (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n g') } \u226b\n kernel.lift f (AddSubgroup.subtype (AddMonoidHom.ker f))\n (_ : AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f = 0)) \u226b\n equalizer.\u03b9 f 0 =\n \ud835\udfd9 (kernel f) \u226b equalizer.\u03b9 f 0\n[PROOFSTEP]\next x\n[GOAL]\ncase w\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\nx : \u2191(kernel f)\n\u22a2 \u2191(({\n toZeroHom :=\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) },\n map_add' :=\n (_ :\n \u2200 (g g' : \u2191(kernel f)),\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ :\n (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 =\n 0) }\n (g + g') =\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ :\n (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 =\n 0) }\n g +\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ :\n (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 =\n 0) }\n g') } \u226b\n kernel.lift f (AddSubgroup.subtype (AddMonoidHom.ker f))\n (_ : AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f = 0)) \u226b\n equalizer.\u03b9 f 0)\n x =\n \u2191(\ud835\udfd9 (kernel f) \u226b equalizer.\u03b9 f 0) x\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\nx : \u2191(kernel f)\n\u22a2 \u2191(kernel.\u03b9 f)\n (\u2191(kernel.lift f (AddSubgroup.subtype (AddMonoidHom.ker f))\n (_ : AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f = 0))\n { val := \u2191(kernel.\u03b9 f) x, property := (_ : \u2191(kernel.\u03b9 f \u226b f) x = \u21910 x) }) =\n \u2191(kernel.\u03b9 f) x\n[PROOFSTEP]\ngeneralize_proofs _ h1 h2\n[GOAL]\ncase w\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\nx : \u2191(kernel f)\nh\u271d : HasKernel f\nh1 : AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f = 0\nh2 : \u2191(kernel.\u03b9 f \u226b f) x = \u21910 x\n\u22a2 \u2191(kernel.\u03b9 f)\n (\u2191(kernel.lift f (AddSubgroup.subtype (AddMonoidHom.ker f)) h1) { val := \u2191(kernel.\u03b9 f) x, property := h2 }) =\n \u2191(kernel.\u03b9 f) x\n[PROOFSTEP]\nerw [FunLike.congr_fun (kernel.lift_\u03b9 f _ h1) \u27e8_, h2\u27e9]\n[GOAL]\ncase w\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\nx : \u2191(kernel f)\nh\u271d : HasKernel f\nh1 : AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f = 0\nh2 : \u2191(kernel.\u03b9 f \u226b f) x = \u21910 x\n\u22a2 \u2191(AddSubgroup.subtype (AddMonoidHom.ker f)) { val := \u2191(kernel.\u03b9 f) x, property := h2 } = \u2191(kernel.\u03b9 f) x\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\n\u22a2 kernel.lift f (AddSubgroup.subtype (AddMonoidHom.ker f)) (_ : AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f = 0) \u226b\n {\n toZeroHom :=\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) },\n map_add' :=\n (_ :\n \u2200 (g g' : \u2191(kernel f)),\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n (g + g') =\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ :\n (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n g +\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ :\n (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) }\n g') } =\n \ud835\udfd9 (of { x // x \u2208 AddMonoidHom.ker f })\n[PROOFSTEP]\napply AddCommGroupCat.ext\n[GOAL]\ncase w\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\n\u22a2 \u2200 (x : \u2191(of { x // x \u2208 AddMonoidHom.ker f })),\n \u2191(kernel.lift f (AddSubgroup.subtype (AddMonoidHom.ker f)) (_ : AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f = 0) \u226b\n {\n toZeroHom :=\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) },\n map_add' :=\n (_ :\n \u2200 (g g' : \u2191(kernel f)),\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ :\n (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 =\n 0) }\n (g + g') =\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ :\n (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 =\n 0) }\n g +\n ZeroHom.toFun\n { toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ :\n (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 =\n 0) }\n g') })\n x =\n \u2191(\ud835\udfd9 (of { x // x \u2208 AddMonoidHom.ker f })) x\n[PROOFSTEP]\nsimp only [AddMonoidHom.coe_mk, coe_id, coe_comp]\n[GOAL]\ncase w\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\n\u22a2 \u2200 (x : \u2191(of { x // x \u2208 AddMonoidHom.ker f })),\n (\u2191{ toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) } \u2218\n \u2191(kernel.lift f (AddSubgroup.subtype (AddMonoidHom.ker f))\n (_ : AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f = 0)))\n x =\n id x\n[PROOFSTEP]\nrintro \u27e8x, mem\u27e9\n[GOAL]\ncase w.mk\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\nx : \u2191G\nmem : x \u2208 AddMonoidHom.ker f\n\u22a2 (\u2191{ toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) } \u2218\n \u2191(kernel.lift f (AddSubgroup.subtype (AddMonoidHom.ker f))\n (_ : AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f = 0)))\n { val := x, property := mem } =\n id { val := x, property := mem }\n[PROOFSTEP]\nrefine Subtype.ext ?_\n[GOAL]\ncase w.mk\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\nx : \u2191G\nmem : x \u2208 AddMonoidHom.ker f\n\u22a2 \u2191((\u2191{ toFun := fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) },\n map_zero' :=\n (_ : (fun g => { val := \u2191(kernel.\u03b9 f) g, property := (_ : \u2191(kernel.\u03b9 f \u226b f) g = \u21910 g) }) 0 = 0) } \u2218\n \u2191(kernel.lift f (AddSubgroup.subtype (AddMonoidHom.ker f))\n (_ : AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f = 0)))\n { val := x, property := mem }) =\n \u2191(id { val := x, property := mem })\n[PROOFSTEP]\nsimp only [ZeroHom.coe_mk, Function.comp_apply, id_eq]\n[GOAL]\ncase w.mk\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\nx : \u2191G\nmem : x \u2208 AddMonoidHom.ker f\n\u22a2 \u2191(kernel.\u03b9 f)\n (\u2191(kernel.lift f (AddSubgroup.subtype (AddMonoidHom.ker f))\n (_ : AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f = 0))\n { val := x, property := mem }) =\n x\n[PROOFSTEP]\ngeneralize_proofs _ h1 h2\n[GOAL]\ncase w.mk\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\nx : \u2191G\nmem : x \u2208 AddMonoidHom.ker f\nh\u271d : HasKernel f\nh1 : AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f = 0\n\u22a2 \u2191(kernel.\u03b9 f) (\u2191(kernel.lift f (AddSubgroup.subtype (AddMonoidHom.ker f)) h1) { val := x, property := mem }) = x\n[PROOFSTEP]\nerw [FunLike.congr_fun (kernel.lift_\u03b9 f _ h1) \u27e8_, mem\u27e9]\n[GOAL]\ncase w.mk\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\nx : \u2191G\nmem : x \u2208 AddMonoidHom.ker f\nh\u271d : HasKernel f\nh1 : AddSubgroup.subtype (AddMonoidHom.ker f) \u226b f = 0\n\u22a2 \u2191(AddSubgroup.subtype (AddMonoidHom.ker f)) { val := x, property := mem } = x\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\n\u22a2 (kernelIsoKer f).hom \u226b AddSubgroup.subtype (AddMonoidHom.ker f) = kernel.\u03b9 f\n[PROOFSTEP]\next\n[GOAL]\ncase w\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\nx\u271d : \u2191(kernel f)\n\u22a2 \u2191((kernelIsoKer f).hom \u226b AddSubgroup.subtype (AddMonoidHom.ker f)) x\u271d = \u2191(kernel.\u03b9 f) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\n\u22a2 (kernelIsoKer f).inv \u226b kernel.\u03b9 f = AddSubgroup.subtype (AddMonoidHom.ker f)\n[PROOFSTEP]\next\n[GOAL]\ncase w\nJ : Type v\ninst\u271d : SmallCategory J\nG H : AddCommGroupCat\nf : G \u27f6 H\nx\u271d : \u2191(of { x // x \u2208 AddMonoidHom.ker f })\n\u22a2 \u2191((kernelIsoKer f).inv \u226b kernel.\u03b9 f) x\u271d = \u2191(AddSubgroup.subtype (AddMonoidHom.ker f)) x\u271d\n[PROOFSTEP]\nsimp [kernelIsoKer]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.GroupCat.Limits", "llama_tokens": 9401, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702880639791, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.4019191656588334}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\nh : \u2200 (c : Set \u03b1), IsChain r c \u2192 \u2203 ub, \u2200 (a : \u03b1), a \u2208 c \u2192 r a ub\ntrans : \u2200 {a b c : \u03b1}, r a b \u2192 r b c \u2192 r a c\nthis\u271d : \u2203 ub, \u2200 (a : \u03b1), a \u2208 maxChain r \u2192 r a ub\nub : \u03b1\nhub : \u2200 (a : \u03b1), a \u2208 maxChain r \u2192 r a ub\na : \u03b1\nha : r ub a\nthis : IsChain r (insert a (maxChain r))\n\u22a2 a \u2208 maxChain r\n[PROOFSTEP]\nrw [maxChain_spec.right this (subset_insert _ _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\nh : \u2200 (c : Set \u03b1), IsChain r c \u2192 \u2203 ub, \u2200 (a : \u03b1), a \u2208 c \u2192 r a ub\ntrans : \u2200 {a b c : \u03b1}, r a b \u2192 r b c \u2192 r a c\nthis\u271d : \u2203 ub, \u2200 (a : \u03b1), a \u2208 maxChain r \u2192 r a ub\nub : \u03b1\nhub : \u2200 (a : \u03b1), a \u2208 maxChain r \u2192 r a ub\na : \u03b1\nha : r ub a\nthis : IsChain r (insert a (maxChain r))\n\u22a2 a \u2208 insert a (maxChain r)\n[PROOFSTEP]\nexact mem_insert _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nih : \u2200 (c : Set \u03b1), c \u2286 s \u2192 IsChain (fun x x_1 => x \u2264 x_1) c \u2192 \u2203 ub, ub \u2208 s \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\nc : Set \u2191s\nhc : IsChain (fun x x_1 => x \u2264 x_1) c\n\u22a2 IsChain (fun x x_1 => x \u2264 x_1) (Subtype.val '' c)\n[PROOFSTEP]\nrintro _ \u27e8p, hpc, rfl\u27e9 _ \u27e8q, hqc, rfl\u27e9 hpq\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nih : \u2200 (c : Set \u03b1), c \u2286 s \u2192 IsChain (fun x x_1 => x \u2264 x_1) c \u2192 \u2203 ub, ub \u2208 s \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\nc : Set \u2191s\nhc : IsChain (fun x x_1 => x \u2264 x_1) c\np : { x // x \u2208 s }\nhpc : p \u2208 c\nq : { x // x \u2208 s }\nhqc : q \u2208 c\nhpq : \u2191p \u2260 \u2191q\n\u22a2 (fun x x_1 => x \u2264 x_1) \u2191p \u2191q \u2228 (fun x x_1 => x \u2264 x_1) \u2191q \u2191p\n[PROOFSTEP]\nrefine' hc hpc hqc fun t => hpq (Subtype.ext_iff.1 t)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nih :\n \u2200 (c : Set \u03b1), c \u2286 s \u2192 IsChain (fun x x_1 => x \u2264 x_1) c \u2192 \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 ub, ub \u2208 s \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\nx : \u03b1\nhxs : x \u2208 s\n\u22a2 \u2203 m, m \u2208 s \u2227 x \u2264 m \u2227 \u2200 (z : \u03b1), z \u2208 s \u2192 m \u2264 z \u2192 z \u2264 m\n[PROOFSTEP]\nhave H := zorn_preorder\u2080 ({y \u2208 s | x \u2264 y}) fun c hcs hc => ?_\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nih :\n \u2200 (c : Set \u03b1), c \u2286 s \u2192 IsChain (fun x x_1 => x \u2264 x_1) c \u2192 \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 ub, ub \u2208 s \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\nx : \u03b1\nhxs : x \u2208 s\nH : \u2203 m, m \u2208 {y | y \u2208 s \u2227 x \u2264 y} \u2227 \u2200 (z : \u03b1), z \u2208 {y | y \u2208 s \u2227 x \u2264 y} \u2192 m \u2264 z \u2192 z \u2264 m\n\u22a2 \u2203 m, m \u2208 s \u2227 x \u2264 m \u2227 \u2200 (z : \u03b1), z \u2208 s \u2192 m \u2264 z \u2192 z \u2264 m\n[PROOFSTEP]\nrcases H with \u27e8m, \u27e8hms, hxm\u27e9, hm\u27e9\n[GOAL]\ncase refine_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nih :\n \u2200 (c : Set \u03b1), c \u2286 s \u2192 IsChain (fun x x_1 => x \u2264 x_1) c \u2192 \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 ub, ub \u2208 s \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\nx : \u03b1\nhxs : x \u2208 s\nm : \u03b1\nhm : \u2200 (z : \u03b1), z \u2208 {y | y \u2208 s \u2227 x \u2264 y} \u2192 m \u2264 z \u2192 z \u2264 m\nhms : m \u2208 s\nhxm : x \u2264 m\n\u22a2 \u2203 m, m \u2208 s \u2227 x \u2264 m \u2227 \u2200 (z : \u03b1), z \u2208 s \u2192 m \u2264 z \u2192 z \u2264 m\n[PROOFSTEP]\nexact \u27e8m, hms, hxm, fun z hzs hmz => hm _ \u27e8hzs, hxm.trans hmz\u27e9 hmz\u27e9\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nih :\n \u2200 (c : Set \u03b1), c \u2286 s \u2192 IsChain (fun x x_1 => x \u2264 x_1) c \u2192 \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 ub, ub \u2208 s \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\nx : \u03b1\nhxs : x \u2208 s\nc : Set \u03b1\nhcs : c \u2286 {y | y \u2208 s \u2227 x \u2264 y}\nhc : IsChain (fun x x_1 => x \u2264 x_1) c\n\u22a2 \u2203 ub, ub \u2208 {y | y \u2208 s \u2227 x \u2264 y} \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\n[PROOFSTEP]\nrcases c.eq_empty_or_nonempty with (rfl | \u27e8y, hy\u27e9)\n[GOAL]\ncase refine_1.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nih :\n \u2200 (c : Set \u03b1), c \u2286 s \u2192 IsChain (fun x x_1 => x \u2264 x_1) c \u2192 \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 ub, ub \u2208 s \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\nx : \u03b1\nhxs : x \u2208 s\nhcs : \u2205 \u2286 {y | y \u2208 s \u2227 x \u2264 y}\nhc : IsChain (fun x x_1 => x \u2264 x_1) \u2205\n\u22a2 \u2203 ub, ub \u2208 {y | y \u2208 s \u2227 x \u2264 y} \u2227 \u2200 (z : \u03b1), z \u2208 \u2205 \u2192 z \u2264 ub\n[PROOFSTEP]\nexact \u27e8x, \u27e8hxs, le_rfl\u27e9, fun z => False.elim\u27e9\n[GOAL]\ncase refine_1.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nih :\n \u2200 (c : Set \u03b1), c \u2286 s \u2192 IsChain (fun x x_1 => x \u2264 x_1) c \u2192 \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 ub, ub \u2208 s \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\nx : \u03b1\nhxs : x \u2208 s\nc : Set \u03b1\nhcs : c \u2286 {y | y \u2208 s \u2227 x \u2264 y}\nhc : IsChain (fun x x_1 => x \u2264 x_1) c\ny : \u03b1\nhy : y \u2208 c\n\u22a2 \u2203 ub, ub \u2208 {y | y \u2208 s \u2227 x \u2264 y} \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\n[PROOFSTEP]\nrcases ih c (fun z hz => (hcs hz).1) hc y hy with \u27e8z, hzs, hz\u27e9\n[GOAL]\ncase refine_1.inr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nih :\n \u2200 (c : Set \u03b1), c \u2286 s \u2192 IsChain (fun x x_1 => x \u2264 x_1) c \u2192 \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 ub, ub \u2208 s \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\nx : \u03b1\nhxs : x \u2208 s\nc : Set \u03b1\nhcs : c \u2286 {y | y \u2208 s \u2227 x \u2264 y}\nhc : IsChain (fun x x_1 => x \u2264 x_1) c\ny : \u03b1\nhy : y \u2208 c\nz : \u03b1\nhzs : z \u2208 s\nhz : \u2200 (z_1 : \u03b1), z_1 \u2208 c \u2192 z_1 \u2264 z\n\u22a2 \u2203 ub, ub \u2208 {y | y \u2208 s \u2227 x \u2264 y} \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\n[PROOFSTEP]\nexact \u27e8z, \u27e8hzs, (hcs hy).2.trans <| hz _ hy\u27e9, hz\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nih :\n \u2200 (c : Set \u03b1),\n c \u2286 Ici a \u2192 IsChain (fun x x_1 => x \u2264 x_1) c \u2192 \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 ub, a \u2264 ub \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\nx : \u03b1\nhax : a \u2264 x\n\u22a2 \u2200 (c : Set \u03b1),\n c \u2286 Ici a \u2192 IsChain (fun x x_1 => x \u2264 x_1) c \u2192 \u2200 (y : \u03b1), y \u2208 c \u2192 \u2203 ub, ub \u2208 Ici a \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\n[PROOFSTEP]\nsimpa using ih\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\nhc : IsChain r c\n\u22a2 \u2203 M, IsMaxChain r M \u2227 c \u2286 M\n[PROOFSTEP]\nhave H := zorn_subset_nonempty {s | c \u2286 s \u2227 IsChain r s} ?_ c \u27e8Subset.rfl, hc\u27e9\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\nhc : IsChain r c\nH : \u2203 m, m \u2208 {s | c \u2286 s \u2227 IsChain r s} \u2227 c \u2286 m \u2227 \u2200 (a : Set \u03b1), a \u2208 {s | c \u2286 s \u2227 IsChain r s} \u2192 m \u2286 a \u2192 a = m\n\u22a2 \u2203 M, IsMaxChain r M \u2227 c \u2286 M\n[PROOFSTEP]\nobtain \u27e8M, \u27e8_, hM\u2080\u27e9, hM\u2081, hM\u2082\u27e9 := H\n[GOAL]\ncase refine_2.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\nhc : IsChain r c\nM : Set \u03b1\nleft\u271d : c \u2286 M\nhM\u2080 : IsChain r M\nhM\u2081 : c \u2286 M\nhM\u2082 : \u2200 (a : Set \u03b1), a \u2208 {s | c \u2286 s \u2227 IsChain r s} \u2192 M \u2286 a \u2192 a = M\n\u22a2 \u2203 M, IsMaxChain r M \u2227 c \u2286 M\n[PROOFSTEP]\nexact \u27e8M, \u27e8hM\u2080, fun d hd hMd => (hM\u2082 _ \u27e8hM\u2081.trans hMd, hd\u27e9 hMd).symm\u27e9, hM\u2081\u27e9\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\nhc : IsChain r c\n\u22a2 \u2200 (c_1 : Set (Set \u03b1)),\n c_1 \u2286 {s | c \u2286 s \u2227 IsChain r s} \u2192\n IsChain (fun x x_1 => x \u2286 x_1) c_1 \u2192\n Set.Nonempty c_1 \u2192 \u2203 ub, ub \u2208 {s | c \u2286 s \u2227 IsChain r s} \u2227 \u2200 (s : Set \u03b1), s \u2208 c_1 \u2192 s \u2286 ub\n[PROOFSTEP]\nrintro cs hcs\u2080 hcs\u2081 \u27e8s, hs\u27e9\n[GOAL]\ncase refine_1.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\nhc : IsChain r c\ncs : Set (Set \u03b1)\nhcs\u2080 : cs \u2286 {s | c \u2286 s \u2227 IsChain r s}\nhcs\u2081 : IsChain (fun x x_1 => x \u2286 x_1) cs\ns : Set \u03b1\nhs : s \u2208 cs\n\u22a2 \u2203 ub, ub \u2208 {s | c \u2286 s \u2227 IsChain r s} \u2227 \u2200 (s : Set \u03b1), s \u2208 cs \u2192 s \u2286 ub\n[PROOFSTEP]\nrefine' \u27e8\u22c3\u2080 cs, \u27e8fun _ ha => Set.mem_sUnion_of_mem ((hcs\u2080 hs).left ha) hs, _\u27e9, fun _ => Set.subset_sUnion_of_mem\u27e9\n[GOAL]\ncase refine_1.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\nhc : IsChain r c\ncs : Set (Set \u03b1)\nhcs\u2080 : cs \u2286 {s | c \u2286 s \u2227 IsChain r s}\nhcs\u2081 : IsChain (fun x x_1 => x \u2286 x_1) cs\ns : Set \u03b1\nhs : s \u2208 cs\n\u22a2 IsChain r (\u22c3\u2080 cs)\n[PROOFSTEP]\nrintro y \u27e8sy, hsy, hysy\u27e9 z \u27e8sz, hsz, hzsz\u27e9 hyz\n[GOAL]\ncase refine_1.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\nhc : IsChain r c\ncs : Set (Set \u03b1)\nhcs\u2080 : cs \u2286 {s | c \u2286 s \u2227 IsChain r s}\nhcs\u2081 : IsChain (fun x x_1 => x \u2286 x_1) cs\ns : Set \u03b1\nhs : s \u2208 cs\ny : \u03b1\nsy : Set \u03b1\nhsy : sy \u2208 cs\nhysy : y \u2208 sy\nz : \u03b1\nsz : Set \u03b1\nhsz : sz \u2208 cs\nhzsz : z \u2208 sz\nhyz : y \u2260 z\n\u22a2 r y z \u2228 r z y\n[PROOFSTEP]\nobtain rfl | hsseq := eq_or_ne sy sz\n[GOAL]\ncase refine_1.intro.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\nhc : IsChain r c\ncs : Set (Set \u03b1)\nhcs\u2080 : cs \u2286 {s | c \u2286 s \u2227 IsChain r s}\nhcs\u2081 : IsChain (fun x x_1 => x \u2286 x_1) cs\ns : Set \u03b1\nhs : s \u2208 cs\ny : \u03b1\nsy : Set \u03b1\nhsy : sy \u2208 cs\nhysy : y \u2208 sy\nz : \u03b1\nhyz : y \u2260 z\nhsz : sy \u2208 cs\nhzsz : z \u2208 sy\n\u22a2 r y z \u2228 r z y\n[PROOFSTEP]\nexact (hcs\u2080 hsy).right hysy hzsz hyz\n[GOAL]\ncase refine_1.intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\nhc : IsChain r c\ncs : Set (Set \u03b1)\nhcs\u2080 : cs \u2286 {s | c \u2286 s \u2227 IsChain r s}\nhcs\u2081 : IsChain (fun x x_1 => x \u2286 x_1) cs\ns : Set \u03b1\nhs : s \u2208 cs\ny : \u03b1\nsy : Set \u03b1\nhsy : sy \u2208 cs\nhysy : y \u2208 sy\nz : \u03b1\nsz : Set \u03b1\nhsz : sz \u2208 cs\nhzsz : z \u2208 sz\nhyz : y \u2260 z\nhsseq : sy \u2260 sz\n\u22a2 r y z \u2228 r z y\n[PROOFSTEP]\ncases' hcs\u2081 hsy hsz hsseq with h h\n[GOAL]\ncase refine_1.intro.intro.intro.intro.intro.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\nhc : IsChain r c\ncs : Set (Set \u03b1)\nhcs\u2080 : cs \u2286 {s | c \u2286 s \u2227 IsChain r s}\nhcs\u2081 : IsChain (fun x x_1 => x \u2286 x_1) cs\ns : Set \u03b1\nhs : s \u2208 cs\ny : \u03b1\nsy : Set \u03b1\nhsy : sy \u2208 cs\nhysy : y \u2208 sy\nz : \u03b1\nsz : Set \u03b1\nhsz : sz \u2208 cs\nhzsz : z \u2208 sz\nhyz : y \u2260 z\nhsseq : sy \u2260 sz\nh : (fun x x_1 => x \u2286 x_1) sy sz\n\u22a2 r y z \u2228 r z y\n[PROOFSTEP]\nexact (hcs\u2080 hsz).right (h hysy) hzsz hyz\n[GOAL]\ncase refine_1.intro.intro.intro.intro.intro.inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc : Set \u03b1\nhc : IsChain r c\ncs : Set (Set \u03b1)\nhcs\u2080 : cs \u2286 {s | c \u2286 s \u2227 IsChain r s}\nhcs\u2081 : IsChain (fun x x_1 => x \u2286 x_1) cs\ns : Set \u03b1\nhs : s \u2208 cs\ny : \u03b1\nsy : Set \u03b1\nhsy : sy \u2208 cs\nhysy : y \u2208 sy\nz : \u03b1\nsz : Set \u03b1\nhsz : sz \u2208 cs\nhzsz : z \u2208 sz\nhyz : y \u2260 z\nhsseq : sy \u2260 sz\nh : (fun x x_1 => x \u2286 x_1) sz sy\n\u22a2 r y z \u2228 r z y\n[PROOFSTEP]\nexact (hcs\u2080 hsy).right hysy (h hzsz) hyz\n", "meta": {"mathlib_filename": "Mathlib.Order.Zorn", "llama_tokens": 5170, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.4019127070692996}} {"text": "[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : Mul A\ninst\u271d : IsCommJordan A\na b : A\n\u22a2 a * b * a = a * (b * a)\n[PROOFSTEP]\nrw [IsCommJordan.mul_comm, IsCommJordan.mul_comm a b]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : Mul A\ninst\u271d : IsCommJordan A\na b : A\n\u22a2 a * a * (a * b) = a * (a * a * b)\n[PROOFSTEP]\nrw [IsCommJordan.mul_comm (a * a) (a * b), IsCommJordan.lmul_comm_rmul_rmul, IsCommJordan.mul_comm b (a * a)]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : Mul A\ninst\u271d : IsCommJordan A\na b : A\n\u22a2 a * a * (b * a) = a * a * b * a\n[PROOFSTEP]\nrw [IsCommJordan.mul_comm (a * a) (b * a), IsCommJordan.mul_comm b a, IsCommJordan.lmul_comm_rmul_rmul,\n IsCommJordan.mul_comm, IsCommJordan.mul_comm b (a * a)]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : Mul A\ninst\u271d : IsCommJordan A\na b : A\n\u22a2 b * a * (a * a) = b * (a * a) * a\n[PROOFSTEP]\nrw [IsCommJordan.mul_comm b a, IsCommJordan.lmul_comm_rmul_rmul, IsCommJordan.mul_comm]\n[GOAL]\nA : Type u_1\ninst\u271d : Semigroup A\na b : A\n\u22a2 a * b * a = a * (b * a)\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\nA : Type u_1\ninst\u271d : Semigroup A\na b : A\n\u22a2 a * a * (a * b) = a * (a * a * b)\n[PROOFSTEP]\nrw [mul_assoc, mul_assoc]\n[GOAL]\nA : Type u_1\ninst\u271d : Semigroup A\na b : A\n\u22a2 a * a * (b * a) = a * a * b * a\n[PROOFSTEP]\nrw [\u2190 mul_assoc]\n[GOAL]\nA : Type u_1\ninst\u271d : Semigroup A\na b : A\n\u22a2 a * b * (a * a) = a * (b * (a * a))\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\nA : Type u_1\ninst\u271d : Semigroup A\na b : A\n\u22a2 b * a * (a * a) = b * (a * a) * a\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 mul_assoc]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b : A\n\u22a2 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (b * a)\u2046) = \u2045\u2191L (a * a), \u2191L b\u2046 + \u2045\u2191L (b * b), \u2191L a\u2046\n[PROOFSTEP]\nsuffices 2 \u2022 \u2045L a, L (a * b)\u2046 + 2 \u2022 \u2045L b, L (b * a)\u2046 + \u2045L b, L (a * a)\u2046 + \u2045L a, L (b * b)\u2046 = 0 by\n rwa [\u2190 sub_eq_zero, \u2190 sub_sub, sub_eq_add_neg, sub_eq_add_neg, lie_skew, lie_skew, nsmul_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b : A\nthis : 2 \u2022 \u2045\u2191L a, \u2191L (a * b)\u2046 + 2 \u2022 \u2045\u2191L b, \u2191L (b * a)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 = 0\n\u22a2 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (b * a)\u2046) = \u2045\u2191L (a * a), \u2191L b\u2046 + \u2045\u2191L (b * b), \u2191L a\u2046\n[PROOFSTEP]\nrwa [\u2190 sub_eq_zero, \u2190 sub_sub, sub_eq_add_neg, sub_eq_add_neg, lie_skew, lie_skew, nsmul_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b : A\n\u22a2 2 \u2022 \u2045\u2191L a, \u2191L (a * b)\u2046 + 2 \u2022 \u2045\u2191L b, \u2191L (b * a)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 = 0\n[PROOFSTEP]\nconvert (commute_lmul_lmul_sq (a + b)).lie_eq using 1\n[GOAL]\ncase h.e'_2\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b : A\n\u22a2 2 \u2022 \u2045\u2191L a, \u2191L (a * b)\u2046 + 2 \u2022 \u2045\u2191L b, \u2191L (b * a)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 =\n \u2045\u2191L (a + b), \u2191L ((a + b) * (a + b))\u2046\n[PROOFSTEP]\nsimp only [add_mul, mul_add, map_add, lie_add, add_lie, IsCommJordan.mul_comm b a, (commute_lmul_lmul_sq a).lie_eq,\n (commute_lmul_lmul_sq b).lie_eq, zero_add, add_zero, two_smul]\n[GOAL]\ncase h.e'_2\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b : A\n\u22a2 \u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L a, \u2191L (a * b)\u2046 + (\u2045\u2191L b, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046) + \u2045\u2191L b, \u2191L (a * a)\u2046 +\n \u2045\u2191L a, \u2191L (b * b)\u2046 =\n \u2045\u2191L b, \u2191L (a * a)\u2046 + (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046) +\n (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_2\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b : A\n\u22a2 \u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L a, \u2191L (a * b)\u2046 + (\u2045\u2191L b, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046) + \u2045\u2191L b, \u2191L (a * a)\u2046 +\n \u2045\u2191L a, \u2191L (b * b)\u2046 =\n \u2045\u2191L b, \u2191L (a * a)\u2046 + (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046) +\n (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046)\n[PROOFSTEP]\nabel\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L (a + b + c), \u2191L ((a + b + c) * (a + b + c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [add_mul, add_mul]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L (a + b + c), \u2191L (a * (a + b + c) + b * (a + b + c) + c * (a + b + c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\niterate 6 rw [mul_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L (a + b + c), \u2191L (a * (a + b + c) + b * (a + b + c) + c * (a + b + c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [mul_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L (a + b + c), \u2191L (a * (a + b) + a * c + b * (a + b + c) + c * (a + b + c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [mul_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L (a + b + c), \u2191L (a * a + a * b + a * c + b * (a + b + c) + c * (a + b + c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [mul_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L (a + b + c), \u2191L (a * a + a * b + a * c + (b * (a + b) + b * c) + c * (a + b + c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [mul_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L (a + b + c), \u2191L (a * a + a * b + a * c + (b * a + b * b + b * c) + c * (a + b + c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [mul_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L (a + b + c), \u2191L (a * a + a * b + a * c + (b * a + b * b + b * c) + (c * (a + b) + c * c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [mul_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L (a + b + c), \u2191L (a * a + a * b + a * c + (b * a + b * b + b * c) + (c * a + c * b + c * c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\niterate 10 rw [map_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L (a + b + c), \u2191L (a * a + a * b + a * c + (b * a + b * b + b * c) + (c * a + c * b + c * c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [map_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L (a + b) + \u2191L c, \u2191L (a * a + a * b + a * c + (b * a + b * b + b * c) + (c * a + c * b + c * c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [map_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a + a * b + a * c + (b * a + b * b + b * c) + (c * a + c * b + c * c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [map_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a + a * b + a * c + (b * a + b * b + b * c)) + \u2191L (c * a + c * b + c * c)\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [map_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a + a * b + a * c) + \u2191L (b * a + b * b + b * c) + \u2191L (c * a + c * b + c * c)\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [map_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a + a * b) + \u2191L (a * c) + \u2191L (b * a + b * b + b * c) + \u2191L (c * a + c * b + c * c)\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [map_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (a * b) + \u2191L (a * c) + \u2191L (b * a + b * b + b * c) + \u2191L (c * a + c * b + c * c)\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [map_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c,\n \u2191L (a * a) + \u2191L (a * b) + \u2191L (a * c) + (\u2191L (b * a + b * b) + \u2191L (b * c)) + \u2191L (c * a + c * b + c * c)\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [map_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c,\n \u2191L (a * a) + \u2191L (a * b) + \u2191L (a * c) + (\u2191L (b * a) + \u2191L (b * b) + \u2191L (b * c)) + \u2191L (c * a + c * b + c * c)\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [map_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c,\n \u2191L (a * a) + \u2191L (a * b) + \u2191L (a * c) + (\u2191L (b * a) + \u2191L (b * b) + \u2191L (b * c)) +\n (\u2191L (c * a + c * b) + \u2191L (c * c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [map_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c,\n \u2191L (a * a) + \u2191L (a * b) + \u2191L (a * c) + (\u2191L (b * a) + \u2191L (b * b) + \u2191L (b * c)) +\n (\u2191L (c * a) + \u2191L (c * b) + \u2191L (c * c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [IsCommJordan.mul_comm b a, IsCommJordan.mul_comm c a, IsCommJordan.mul_comm c b]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c,\n \u2191L (a * a) + \u2191L (a * b) + \u2191L (a * c) + (\u2191L (a * b) + \u2191L (b * b) + \u2191L (b * c)) +\n (\u2191L (a * c) + \u2191L (b * c) + \u2191L (c * c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (a * c) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\niterate 3 rw [two_smul]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c,\n \u2191L (a * a) + \u2191L (a * b) + \u2191L (a * c) + (\u2191L (a * b) + \u2191L (b * b) + \u2191L (b * c)) +\n (\u2191L (a * c) + \u2191L (b * c) + \u2191L (c * c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (a * c) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [two_smul]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c,\n \u2191L (a * a) + \u2191L (a * b) + \u2191L (a * c) + (\u2191L (a * b) + \u2191L (b * b) + \u2191L (b * c)) +\n (\u2191L (a * c) + \u2191L (b * c) + \u2191L (c * c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c,\n \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + (\u2191L (a * b) + \u2191L (a * b)) + 2 \u2022 \u2191L (a * c) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [two_smul]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c,\n \u2191L (a * a) + \u2191L (a * b) + \u2191L (a * c) + (\u2191L (a * b) + \u2191L (b * b) + \u2191L (b * c)) +\n (\u2191L (a * c) + \u2191L (b * c) + \u2191L (c * c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c,\n \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + (\u2191L (a * b) + \u2191L (a * b)) + (\u2191L (a * c) + \u2191L (a * c)) + 2 \u2022 \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [two_smul]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c,\n \u2191L (a * a) + \u2191L (a * b) + \u2191L (a * c) + (\u2191L (a * b) + \u2191L (b * b) + \u2191L (b * c)) +\n (\u2191L (a * c) + \u2191L (b * c) + \u2191L (c * c))\u2046 =\n \u2045\u2191L a + \u2191L b + \u2191L c,\n \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + (\u2191L (a * b) + \u2191L (a * b)) + (\u2191L (a * c) + \u2191L (a * c)) +\n (\u2191L (b * c) + \u2191L (b * c))\u2046\n[PROOFSTEP]\nsimp only [lie_add, add_lie, commute_lmul_lmul_sq, zero_add, add_zero]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 +\n (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046 + \u2045\u2191L c, \u2191L (a * b)\u2046) +\n (\u2045\u2191L a, \u2191L (a * c)\u2046 + \u2045\u2191L b, \u2191L (a * c)\u2046 + \u2045\u2191L c, \u2191L (a * c)\u2046) +\n (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046 + \u2045\u2191L c, \u2191L (a * b)\u2046 +\n (\u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046) +\n (\u2045\u2191L a, \u2191L (b * c)\u2046 + \u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046)) +\n (\u2045\u2191L a, \u2191L (a * c)\u2046 + \u2045\u2191L b, \u2191L (a * c)\u2046 + \u2045\u2191L c, \u2191L (a * c)\u2046 +\n (\u2045\u2191L a, \u2191L (b * c)\u2046 + \u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046)) =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 +\n (\u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046) +\n (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046 + \u2045\u2191L c, \u2191L (a * b)\u2046 +\n (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046 + \u2045\u2191L c, \u2191L (a * b)\u2046)) +\n (\u2045\u2191L a, \u2191L (a * c)\u2046 + \u2045\u2191L b, \u2191L (a * c)\u2046 + \u2045\u2191L c, \u2191L (a * c)\u2046 +\n (\u2045\u2191L a, \u2191L (a * c)\u2046 + \u2045\u2191L b, \u2191L (a * c)\u2046 + \u2045\u2191L c, \u2191L (a * c)\u2046)) +\n (\u2045\u2191L a, \u2191L (b * c)\u2046 + \u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046 +\n (\u2045\u2191L a, \u2191L (b * c)\u2046 + \u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046))\n[PROOFSTEP]\nabel\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 +\n (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046 + \u2045\u2191L c, \u2191L (a * b)\u2046) +\n (\u2045\u2191L a, \u2191L (a * c)\u2046 + \u2045\u2191L b, \u2191L (a * c)\u2046 + \u2045\u2191L c, \u2191L (a * c)\u2046) +\n (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046 + \u2045\u2191L c, \u2191L (a * b)\u2046 +\n (\u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046) +\n (\u2045\u2191L a, \u2191L (b * c)\u2046 + \u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046)) +\n (\u2045\u2191L a, \u2191L (a * c)\u2046 + \u2045\u2191L b, \u2191L (a * c)\u2046 + \u2045\u2191L c, \u2191L (a * c)\u2046 +\n (\u2045\u2191L a, \u2191L (b * c)\u2046 + \u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046)) =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 +\n (\u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046) +\n (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046 + \u2045\u2191L c, \u2191L (a * b)\u2046 +\n (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046 + \u2045\u2191L c, \u2191L (a * b)\u2046)) +\n (\u2045\u2191L a, \u2191L (a * c)\u2046 + \u2045\u2191L b, \u2191L (a * c)\u2046 + \u2045\u2191L c, \u2191L (a * c)\u2046 +\n (\u2045\u2191L a, \u2191L (a * c)\u2046 + \u2045\u2191L b, \u2191L (a * c)\u2046 + \u2045\u2191L c, \u2191L (a * c)\u2046)) +\n (\u2045\u2191L a, \u2191L (b * c)\u2046 + \u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046 +\n (\u2045\u2191L a, \u2191L (b * c)\u2046 + \u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046))\n[PROOFSTEP]\nabel\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [add_lie, add_lie]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 +\n \u2045\u2191L b, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 +\n \u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\niterate 15 rw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 +\n \u2045\u2191L b, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 +\n \u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n \u2045\u2191L b, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 +\n \u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n \u2045\u2191L b, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 +\n \u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n \u2045\u2191L b, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 +\n \u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a) + \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n \u2045\u2191L b, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 +\n \u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n \u2045\u2191L b, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 +\n \u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n \u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n \u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n \u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a) + \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n \u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n \u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046) =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046) =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046) =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046) =\n \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046) =\n \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046)) +\n (2 \u2022 \u2045\u2191L a, \u2191L (b * c)\u2046 + 2 \u2022 \u2045\u2191L b, \u2191L (c * a)\u2046 + 2 \u2022 \u2045\u2191L c, \u2191L (a * b)\u2046)\n[PROOFSTEP]\nrw [(commute_lmul_lmul_sq a).lie_eq, (commute_lmul_lmul_sq b).lie_eq, (commute_lmul_lmul_sq c).lie_eq, zero_add,\n add_zero, add_zero]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046) =\n \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046)) +\n (2 \u2022 \u2045\u2191L a, \u2191L (b * c)\u2046 + 2 \u2022 \u2045\u2191L b, \u2191L (c * a)\u2046 + 2 \u2022 \u2045\u2191L c, \u2191L (a * b)\u2046)\n[PROOFSTEP]\nsimp only [lie_nsmul]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + 2 \u2022 \u2045\u2191L a, \u2191L (a * b)\u2046 + 2 \u2022 \u2045\u2191L a, \u2191L (c * a)\u2046 + 2 \u2022 \u2045\u2191L a, \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + 2 \u2022 \u2045\u2191L b, \u2191L (a * b)\u2046 + 2 \u2022 \u2045\u2191L b, \u2191L (c * a)\u2046 +\n 2 \u2022 \u2045\u2191L b, \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 \u2045\u2191L c, \u2191L (a * b)\u2046 + 2 \u2022 \u2045\u2191L c, \u2191L (c * a)\u2046 +\n 2 \u2022 \u2045\u2191L c, \u2191L (b * c)\u2046) =\n \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046)) +\n (2 \u2022 \u2045\u2191L a, \u2191L (b * c)\u2046 + 2 \u2022 \u2045\u2191L b, \u2191L (c * a)\u2046 + 2 \u2022 \u2045\u2191L c, \u2191L (a * b)\u2046)\n[PROOFSTEP]\nabel\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + 2 \u2022 \u2045\u2191L a, \u2191L (a * b)\u2046 + 2 \u2022 \u2045\u2191L a, \u2191L (c * a)\u2046 + 2 \u2022 \u2045\u2191L a, \u2191L (b * c)\u2046 +\n (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + 2 \u2022 \u2045\u2191L b, \u2191L (a * b)\u2046 + 2 \u2022 \u2045\u2191L b, \u2191L (c * a)\u2046 +\n 2 \u2022 \u2045\u2191L b, \u2191L (b * c)\u2046) +\n (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 \u2045\u2191L c, \u2191L (a * b)\u2046 + 2 \u2022 \u2045\u2191L c, \u2191L (c * a)\u2046 +\n 2 \u2022 \u2045\u2191L c, \u2191L (b * c)\u2046) =\n \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046)) +\n (2 \u2022 \u2045\u2191L a, \u2191L (b * c)\u2046 + 2 \u2022 \u2045\u2191L b, \u2191L (c * a)\u2046 + 2 \u2022 \u2045\u2191L c, \u2191L (a * b)\u2046)\n[PROOFSTEP]\nabel\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046)) +\n (2 \u2022 \u2045\u2191L a, \u2191L (b * c)\u2046 + 2 \u2022 \u2045\u2191L b, \u2191L (c * a)\u2046 + 2 \u2022 \u2045\u2191L c, \u2191L (a * b)\u2046) =\n 2 \u2022 \u2045\u2191L a, \u2191L (b * c)\u2046 + 2 \u2022 \u2045\u2191L b, \u2191L (c * a)\u2046 + 2 \u2022 \u2045\u2191L c, \u2191L (a * b)\u2046\n[PROOFSTEP]\nrw [add_left_eq_self]\n -- Porting note: was `nth_rw` instead of `conv_lhs`\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046)) =\n 0\n[PROOFSTEP]\nconv_lhs => enter [1, 1, 2, 2, 2]; rw [IsCommJordan.mul_comm a b]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n| \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046))\n[PROOFSTEP]\nenter [1, 1, 2, 2, 2]; rw [IsCommJordan.mul_comm a b]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n| \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046))\n[PROOFSTEP]\nenter [1, 1, 2, 2, 2]; rw [IsCommJordan.mul_comm a b]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n| \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (a * b)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046))\n[PROOFSTEP]\nenter [1, 1, 2, 2, 2]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n| \u2045\u2191L b, \u2191L (a * b)\u2046\n[PROOFSTEP]\nrw [IsCommJordan.mul_comm a b]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (b * a)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046)) =\n 0\n[PROOFSTEP]\nconv_lhs => enter [1, 2, 2, 2, 1]; rw [IsCommJordan.mul_comm c a]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n| \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (b * a)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046))\n[PROOFSTEP]\nenter [1, 2, 2, 2, 1]; rw [IsCommJordan.mul_comm c a]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n| \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (b * a)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046))\n[PROOFSTEP]\nenter [1, 2, 2, 2, 1]; rw [IsCommJordan.mul_comm c a]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n| \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (b * a)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046))\n[PROOFSTEP]\nenter [1, 2, 2, 2, 1]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n| \u2045\u2191L a, \u2191L (c * a)\u2046\n[PROOFSTEP]\nrw [IsCommJordan.mul_comm c a]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (b * a)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * c)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046)) =\n 0\n[PROOFSTEP]\nconv_lhs => enter [2, 2, 2, 2]; rw [IsCommJordan.mul_comm b c]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n| \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (b * a)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * c)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046))\n[PROOFSTEP]\nenter [2, 2, 2, 2]; rw [IsCommJordan.mul_comm b c]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n| \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (b * a)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * c)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046))\n[PROOFSTEP]\nenter [2, 2, 2, 2]; rw [IsCommJordan.mul_comm b c]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n| \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (b * a)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * c)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (b * c)\u2046))\n[PROOFSTEP]\nenter [2, 2, 2, 2]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n| \u2045\u2191L c, \u2191L (b * c)\u2046\n[PROOFSTEP]\nrw [IsCommJordan.mul_comm b c]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (b * a)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * c)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (c * b)\u2046)) =\n 0\n[PROOFSTEP]\niterate 3 rw [two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * b)\u2046 + \u2045\u2191L b, \u2191L (b * a)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * c)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (c * b)\u2046)) =\n 0\n[PROOFSTEP]\nrw [two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + (\u2045\u2191L (a * a), \u2191L b\u2046 + \u2045\u2191L (b * b), \u2191L a\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + 2 \u2022 (\u2045\u2191L a, \u2191L (a * c)\u2046 + \u2045\u2191L c, \u2191L (c * a)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (c * b)\u2046)) =\n 0\n[PROOFSTEP]\nrw [two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + (\u2045\u2191L (a * a), \u2191L b\u2046 + \u2045\u2191L (b * b), \u2191L a\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + (\u2045\u2191L (a * a), \u2191L c\u2046 + \u2045\u2191L (c * c), \u2191L a\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + 2 \u2022 (\u2045\u2191L b, \u2191L (b * c)\u2046 + \u2045\u2191L c, \u2191L (c * b)\u2046)) =\n 0\n[PROOFSTEP]\nrw [two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + (\u2045\u2191L (a * a), \u2191L b\u2046 + \u2045\u2191L (b * b), \u2191L a\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + (\u2045\u2191L (a * a), \u2191L c\u2046 + \u2045\u2191L (c * c), \u2191L a\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + (\u2045\u2191L (b * b), \u2191L c\u2046 + \u2045\u2191L (c * c), \u2191L b\u2046)) =\n 0\n[PROOFSTEP]\niterate 2 rw [\u2190 lie_skew (L (a * a)), \u2190 lie_skew (L (b * b)), \u2190 lie_skew (L (c * c))]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + (\u2045\u2191L (a * a), \u2191L b\u2046 + \u2045\u2191L (b * b), \u2191L a\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + (\u2045\u2191L (a * a), \u2191L c\u2046 + \u2045\u2191L (c * c), \u2191L a\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + (\u2045\u2191L (b * b), \u2191L c\u2046 + \u2045\u2191L (c * c), \u2191L b\u2046)) =\n 0\n[PROOFSTEP]\nrw [\u2190 lie_skew (L (a * a)), \u2190 lie_skew (L (b * b)), \u2190 lie_skew (L (c * c))]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + (-\u2045\u2191L b, \u2191L (a * a)\u2046 + -\u2045\u2191L a, \u2191L (b * b)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + (\u2045\u2191L (a * a), \u2191L c\u2046 + -\u2045\u2191L a, \u2191L (c * c)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + (\u2045\u2191L (b * b), \u2191L c\u2046 + \u2045\u2191L (c * c), \u2191L b\u2046)) =\n 0\n[PROOFSTEP]\nrw [\u2190 lie_skew (L (a * a)), \u2190 lie_skew (L (b * b)), \u2190 lie_skew (L (c * c))]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + (-\u2045\u2191L b, \u2191L (a * a)\u2046 + -\u2045\u2191L a, \u2191L (b * b)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + (-\u2045\u2191L c, \u2191L (a * a)\u2046 + -\u2045\u2191L a, \u2191L (c * c)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + (-\u2045\u2191L c, \u2191L (b * b)\u2046 + -\u2045\u2191L b, \u2191L (c * c)\u2046)) =\n 0\n[PROOFSTEP]\nabel\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (a * a)\u2046 + (-\u2045\u2191L b, \u2191L (a * a)\u2046 + -\u2045\u2191L a, \u2191L (b * b)\u2046) +\n (\u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (a * a)\u2046 + (-\u2045\u2191L c, \u2191L (a * a)\u2046 + -\u2045\u2191L a, \u2191L (c * c)\u2046)) +\n (\u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + (-\u2045\u2191L c, \u2191L (b * b)\u2046 + -\u2045\u2191L b, \u2191L (c * c)\u2046)) =\n 0\n[PROOFSTEP]\nabel\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 2 \u2022 (\u2045\u2191L a, \u2191L (b * c)\u2046 + \u2045\u2191L b, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (a * b)\u2046) = 0\n[PROOFSTEP]\nsymm\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 0 = 2 \u2022 (\u2045\u2191L a, \u2191L (b * c)\u2046 + \u2045\u2191L b, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (a * b)\u2046)\n[PROOFSTEP]\ncalc\n 0 = \u2045L (a + b + c), L ((a + b + c) * (a + b + c))\u2046 := by rw [(commute_lmul_lmul_sq (a + b + c)).lie_eq]\n _ = _ := by rw [aux0, aux1, aux2, aux3, nsmul_add, nsmul_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 0 = \u2045\u2191L (a + b + c), \u2191L ((a + b + c) * (a + b + c))\u2046\n[PROOFSTEP]\nrw [(commute_lmul_lmul_sq (a + b + c)).lie_eq]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L (a + b + c), \u2191L ((a + b + c) * (a + b + c))\u2046 = 2 \u2022 (\u2045\u2191L a, \u2191L (b * c)\u2046 + \u2045\u2191L b, \u2191L (c * a)\u2046 + \u2045\u2191L c, \u2191L (a * b)\u2046)\n[PROOFSTEP]\nrw [aux0, aux1, aux2, aux3, nsmul_add, nsmul_add]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Jordan.Basic", "llama_tokens": 34532, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.4019127070692996}} {"text": "[GOAL]\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : AddGroup A\ninst\u271d\u00b2 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : AddGroup \u03b1\na : \u03b1\ninst\u271d : AddAction \u03b1 \u03b2\nb : \u03b2\n\u22a2 zmultiples \u2191(minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b) \u2264\n comap (\u2191(zmultiplesHom { x // x \u2208 zmultiples a }) { val := a, property := (_ : a \u2208 zmultiples a) })\n (stabilizer { x // x \u2208 zmultiples a } b)\n[PROOFSTEP]\nrw [zmultiples_le, mem_comap, mem_stabilizer_iff, zmultiplesHom_apply, coe_nat_zsmul]\n[GOAL]\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : AddGroup A\ninst\u271d\u00b2 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : AddGroup \u03b1\na : \u03b1\ninst\u271d : AddAction \u03b1 \u03b2\nb : \u03b2\n\u22a2 minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b \u2022 { val := a, property := (_ : a \u2208 zmultiples a) } +\u1d65 b = b\n[PROOFSTEP]\nsimp_rw [\u2190 vadd_iterate]\n[GOAL]\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : AddGroup A\ninst\u271d\u00b2 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : AddGroup \u03b1\na : \u03b1\ninst\u271d : AddAction \u03b1 \u03b2\nb : \u03b2\n\u22a2 (fun x => { val := a, property := (_ : a \u2208 zmultiples a) } +\u1d65 x)^[minimalPeriod (fun x => a +\u1d65 x) b] b = b\n[PROOFSTEP]\nexact isPeriodicPt_minimalPeriod ((\u00b7 +\u1d65 \u00b7) a) b\n[GOAL]\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : AddGroup A\ninst\u271d\u00b2 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : AddGroup \u03b1\na : \u03b1\ninst\u271d : AddAction \u03b1 \u03b2\nb : \u03b2\n\u22a2 Injective\n \u2191(QuotientAddGroup.map (zmultiples \u2191(minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b))\n (stabilizer { x // x \u2208 zmultiples a } b)\n (\u2191(zmultiplesHom { x // x \u2208 zmultiples a }) { val := a, property := (_ : a \u2208 zmultiples a) })\n (_ :\n zmultiples \u2191(minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b) \u2264\n comap (\u2191(zmultiplesHom { x // x \u2208 zmultiples a }) { val := a, property := (_ : a \u2208 zmultiples a) })\n (stabilizer { x // x \u2208 zmultiples a } b)))\n[PROOFSTEP]\nrw [\u2190 ker_eq_bot_iff, eq_bot_iff]\n[GOAL]\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : AddGroup A\ninst\u271d\u00b2 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : AddGroup \u03b1\na : \u03b1\ninst\u271d : AddAction \u03b1 \u03b2\nb : \u03b2\n\u22a2 ker\n (QuotientAddGroup.map (zmultiples \u2191(minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b))\n (stabilizer { x // x \u2208 zmultiples a } b)\n (\u2191(zmultiplesHom { x // x \u2208 zmultiples a }) { val := a, property := (_ : a \u2208 zmultiples a) })\n (_ :\n zmultiples \u2191(minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b) \u2264\n comap (\u2191(zmultiplesHom { x // x \u2208 zmultiples a }) { val := a, property := (_ : a \u2208 zmultiples a) })\n (stabilizer { x // x \u2208 zmultiples a } b))) \u2264\n \u22a5\n[PROOFSTEP]\nrefine' fun q => induction_on' q fun n hn => _\n[GOAL]\nn\u271d : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : AddGroup A\ninst\u271d\u00b2 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : AddGroup \u03b1\na : \u03b1\ninst\u271d : AddAction \u03b1 \u03b2\nb : \u03b2\nq : \u2124 \u29f8 zmultiples \u2191(minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b)\nn : \u2124\nhn :\n \u2191n \u2208\n ker\n (QuotientAddGroup.map (zmultiples \u2191(minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b))\n (stabilizer { x // x \u2208 zmultiples a } b)\n (\u2191(zmultiplesHom { x // x \u2208 zmultiples a }) { val := a, property := (_ : a \u2208 zmultiples a) })\n (_ :\n zmultiples \u2191(minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b) \u2264\n comap (\u2191(zmultiplesHom { x // x \u2208 zmultiples a }) { val := a, property := (_ : a \u2208 zmultiples a) })\n (stabilizer { x // x \u2208 zmultiples a } b)))\n\u22a2 \u2191n \u2208 \u22a5\n[PROOFSTEP]\nrw [mem_bot, eq_zero_iff, Int.mem_zmultiples_iff, \u2190 zsmul_vadd_eq_iff_minimalPeriod_dvd]\n[GOAL]\nn\u271d : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : AddGroup A\ninst\u271d\u00b2 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : AddGroup \u03b1\na : \u03b1\ninst\u271d : AddAction \u03b1 \u03b2\nb : \u03b2\nq : \u2124 \u29f8 zmultiples \u2191(minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b)\nn : \u2124\nhn :\n \u2191n \u2208\n ker\n (QuotientAddGroup.map (zmultiples \u2191(minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b))\n (stabilizer { x // x \u2208 zmultiples a } b)\n (\u2191(zmultiplesHom { x // x \u2208 zmultiples a }) { val := a, property := (_ : a \u2208 zmultiples a) })\n (_ :\n zmultiples \u2191(minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b) \u2264\n comap (\u2191(zmultiplesHom { x // x \u2208 zmultiples a }) { val := a, property := (_ : a \u2208 zmultiples a) })\n (stabilizer { x // x \u2208 zmultiples a } b)))\n\u22a2 n \u2022 a +\u1d65 b = b\n[PROOFSTEP]\nexact (eq_zero_iff _).mp hn\n[GOAL]\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : AddGroup A\ninst\u271d\u00b2 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\na : \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\nb : \u03b2\nk : \u2124\n\u22a2 \u2191(orbitZpowersEquiv a b).symm \u2191k =\n { val := a, property := (_ : a \u2208 zpowers a) } ^ k \u2022\n { val := b, property := (_ : b \u2208 orbit { x // x \u2208 zpowers a } b) }\n[PROOFSTEP]\nrw [orbitZpowersEquiv_symm_apply, ZMod.coe_int_cast]\n[GOAL]\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : AddGroup A\ninst\u271d\u00b2 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\na : \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\nb : \u03b2\nk : \u2124\n\u22a2 { val := a, property := (_ : a \u2208 zpowers a) } ^ (k % \u2191(minimalPeriod ((fun x x_1 => x \u2022 x_1) a) b)) \u2022\n { val := b, property := (_ : b \u2208 orbit { x // x \u2208 zpowers a } b) } =\n { val := a, property := (_ : a \u2208 zpowers a) } ^ k \u2022\n { val := b, property := (_ : b \u2208 orbit { x // x \u2208 zpowers a } b) }\n[PROOFSTEP]\nexact Subtype.ext (zpow_smul_mod_minimalPeriod _ _ k)\n[GOAL]\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u2075 : AddGroup A\ninst\u271d\u2074 : Ring R\n\u03b1\u271d : Type u_3\n\u03b2\u271d : Type u_4\ninst\u271d\u00b3 : Group \u03b1\u271d\na\u271d : \u03b1\u271d\ninst\u271d\u00b2 : MulAction \u03b1\u271d \u03b2\u271d\nb\u271d : \u03b2\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : AddGroup \u03b1\na : \u03b1\ninst\u271d : AddAction \u03b1 \u03b2\nb : \u03b2\nk : \u2124\n\u22a2 \u2191(orbitZmultiplesEquiv a b).symm \u2191k =\n k \u2022 { val := a, property := (_ : a \u2208 zmultiples a) } +\u1d65\n { val := b, property := (_ : b \u2208 AddAction.orbit { x // x \u2208 zmultiples a } b) }\n[PROOFSTEP]\nrw [AddAction.orbit_zmultiples_equiv_symm_apply, ZMod.coe_int_cast]\n -- porting note: times out without `a b` explicit\n[GOAL]\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u2075 : AddGroup A\ninst\u271d\u2074 : Ring R\n\u03b1\u271d : Type u_3\n\u03b2\u271d : Type u_4\ninst\u271d\u00b3 : Group \u03b1\u271d\na\u271d : \u03b1\u271d\ninst\u271d\u00b2 : MulAction \u03b1\u271d \u03b2\u271d\nb\u271d : \u03b2\u271d\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b9 : AddGroup \u03b1\na : \u03b1\ninst\u271d : AddAction \u03b1 \u03b2\nb : \u03b2\nk : \u2124\n\u22a2 (k % \u2191(minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b)) \u2022 { val := a, property := (_ : a \u2208 zmultiples a) } +\u1d65\n { val := b, property := (_ : b \u2208 AddAction.orbit { x // x \u2208 zmultiples a } b) } =\n k \u2022 { val := a, property := (_ : a \u2208 zmultiples a) } +\u1d65\n { val := b, property := (_ : b \u2208 AddAction.orbit { x // x \u2208 zmultiples a } b) }\n[PROOFSTEP]\nexact Subtype.ext (zsmul_vadd_mod_minimalPeriod a b k)\n[GOAL]\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u2074 : AddGroup A\ninst\u271d\u00b3 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b2 : Group \u03b1\na : \u03b1\ninst\u271d\u00b9 : MulAction \u03b1 \u03b2\nb : \u03b2\ninst\u271d : Fintype \u2191(orbit { x // x \u2208 zpowers a } b)\n\u22a2 minimalPeriod ((fun x x_1 => x \u2022 x_1) a) b = Fintype.card \u2191(orbit { x // x \u2208 zpowers a } b)\n[PROOFSTEP]\nrw [\u2190 Fintype.ofEquiv_card (orbitZpowersEquiv a b), @ZMod.card _ (_)]\n[GOAL]\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u2074 : AddGroup A\ninst\u271d\u00b3 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b2 : Group \u03b1\na : \u03b1\ninst\u271d\u00b9 : MulAction \u03b1 \u03b2\nb : \u03b2\ninst\u271d : Finite \u2191(orbit { x // x \u2208 zpowers a } b)\n\u22a2 minimalPeriod ((fun x x_1 => x \u2022 x_1) a) b \u2260 0\n[PROOFSTEP]\ncases nonempty_fintype (orbit (zpowers a) b)\n[GOAL]\ncase intro\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u2074 : AddGroup A\ninst\u271d\u00b3 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b2 : Group \u03b1\na : \u03b1\ninst\u271d\u00b9 : MulAction \u03b1 \u03b2\nb : \u03b2\ninst\u271d : Finite \u2191(orbit { x // x \u2208 zpowers a } b)\nval\u271d : Fintype \u2191(orbit { x // x \u2208 zpowers a } b)\n\u22a2 minimalPeriod ((fun x x_1 => x \u2022 x_1) a) b \u2260 0\n[PROOFSTEP]\nhaveI : Nonempty (orbit (zpowers a) b) := (orbit_nonempty b).to_subtype\n[GOAL]\ncase intro\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u2074 : AddGroup A\ninst\u271d\u00b3 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b2 : Group \u03b1\na : \u03b1\ninst\u271d\u00b9 : MulAction \u03b1 \u03b2\nb : \u03b2\ninst\u271d : Finite \u2191(orbit { x // x \u2208 zpowers a } b)\nval\u271d : Fintype \u2191(orbit { x // x \u2208 zpowers a } b)\nthis : Nonempty \u2191(orbit { x // x \u2208 zpowers a } b)\n\u22a2 minimalPeriod ((fun x x_1 => x \u2022 x_1) a) b \u2260 0\n[PROOFSTEP]\nrw [minimalPeriod_eq_card]\n[GOAL]\ncase intro\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u2074 : AddGroup A\ninst\u271d\u00b3 : Ring R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b2 : Group \u03b1\na : \u03b1\ninst\u271d\u00b9 : MulAction \u03b1 \u03b2\nb : \u03b2\ninst\u271d : Finite \u2191(orbit { x // x \u2208 zpowers a } b)\nval\u271d : Fintype \u2191(orbit { x // x \u2208 zpowers a } b)\nthis : Nonempty \u2191(orbit { x // x \u2208 zpowers a } b)\n\u22a2 Fintype.card \u2191(orbit { x // x \u2208 zpowers a } b) \u2260 0\n[PROOFSTEP]\nexact Fintype.card_ne_zero\n[GOAL]\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : AddGroup A\ninst\u271d\u00b9 : Ring R\n\u03b1 : Type u_3\ninst\u271d : Group \u03b1\na : \u03b1\n\u22a2 orderOf a = Nat.card { x // x \u2208 zpowers a }\n[PROOFSTEP]\nhave := Nat.card_congr (MulAction.orbitZpowersEquiv a (1 : \u03b1))\n[GOAL]\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : AddGroup A\ninst\u271d\u00b9 : Ring R\n\u03b1 : Type u_3\ninst\u271d : Group \u03b1\na : \u03b1\nthis :\n Nat.card \u2191(MulAction.orbit { x // x \u2208 zpowers a } 1) =\n Nat.card (ZMod (Function.minimalPeriod ((fun x x_1 => x \u2022 x_1) a) 1))\n\u22a2 orderOf a = Nat.card { x // x \u2208 zpowers a }\n[PROOFSTEP]\nrwa [Nat.card_zmod, orbit_subgroup_one_eq_self, eq_comm] at this \n[GOAL]\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : AddGroup A\ninst\u271d\u00b9 : Ring R\n\u03b1 : Type u_3\ninst\u271d : Group \u03b1\na : \u03b1\nh : IsOfFinOrder a\n\u22a2 Finite { x // x \u2208 zpowers a }\n[PROOFSTEP]\nrw [\u2190 orderOf_pos_iff, order_eq_card_zpowers'] at h \n[GOAL]\nn : \u2115\nA : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : AddGroup A\ninst\u271d\u00b9 : Ring R\n\u03b1 : Type u_3\ninst\u271d : Group \u03b1\na : \u03b1\nh : 0 < Nat.card { x // x \u2208 zpowers a }\n\u22a2 Finite { x // x \u2208 zpowers a }\n[PROOFSTEP]\nexact Nat.finite_of_card_ne_zero h.ne.symm\n", "meta": {"mathlib_filename": "Mathlib.Data.ZMod.Quotient", "llama_tokens": 4546, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085909370422, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.40185544579931554}} {"text": "[GOAL]\nS\u271d : Type u'\nT : Type u''\nR\u271d : Type u\nM\u271d : Type v\ninst\u271d\u2075 : SMul R\u271d M\u271d\ninst\u271d\u2074 : SetLike S\u271d M\u271d\nhS : SMulMemClass S\u271d R\u271d M\u271d\ns : S\u271d\nR : Type u_1\nM : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : Monoid R\ninst\u271d\u00b2 : MulAction R M\ninst\u271d\u00b9 : SetLike S M\ninst\u271d : SMulMemClass S R M\nN : S\nx : M\nh : \u2200 (a : R), a \u2022 x \u2208 N\n\u22a2 x \u2208 N\n[PROOFSTEP]\nsimpa using h 1\n[GOAL]\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d : SMul R M\np q : SubMulAction R M\nh : p.carrier = q.carrier\n\u22a2 p = q\n[PROOFSTEP]\ncases p\n[GOAL]\ncase mk\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d : SMul R M\nq : SubMulAction R M\ncarrier\u271d : Set M\nsmul_mem'\u271d : \u2200 (c : R) {x : M}, x \u2208 carrier\u271d \u2192 c \u2022 x \u2208 carrier\u271d\nh : { carrier := carrier\u271d, smul_mem' := smul_mem'\u271d }.carrier = q.carrier\n\u22a2 { carrier := carrier\u271d, smul_mem' := smul_mem'\u271d } = q\n[PROOFSTEP]\ncases q\n[GOAL]\ncase mk.mk\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d : SMul R M\ncarrier\u271d\u00b9 : Set M\nsmul_mem'\u271d\u00b9 : \u2200 (c : R) {x : M}, x \u2208 carrier\u271d\u00b9 \u2192 c \u2022 x \u2208 carrier\u271d\u00b9\ncarrier\u271d : Set M\nsmul_mem'\u271d : \u2200 (c : R) {x : M}, x \u2208 carrier\u271d \u2192 c \u2022 x \u2208 carrier\u271d\nh :\n { carrier := carrier\u271d\u00b9, smul_mem' := smul_mem'\u271d\u00b9 }.carrier = { carrier := carrier\u271d, smul_mem' := smul_mem'\u271d }.carrier\n\u22a2 { carrier := carrier\u271d\u00b9, smul_mem' := smul_mem'\u271d\u00b9 } = { carrier := carrier\u271d, smul_mem' := smul_mem'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d : SMul R M\np : SubMulAction R M\nr : R\nx : M\n\u22a2 { x // x \u2208 p } \u2192[R] M\n[PROOFSTEP]\nrefine' { toFun := Subtype.val .. }\n[GOAL]\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d : SMul R M\np : SubMulAction R M\nr : R\nx : M\n\u22a2 \u2200 (m : R) (x : { x // x \u2208 p }), \u2191(m \u2022 x) = m \u2022 \u2191x\n[PROOFSTEP]\nsimp [val_smul]\n[GOAL]\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d\u2074 : Monoid R\ninst\u271d\u00b3 : MulAction R M\ninst\u271d\u00b2 : SMul S R\ninst\u271d\u00b9 : SMul S M\ninst\u271d : IsScalarTower S R M\np : SubMulAction R M\ns : S\nx : M\nh : x \u2208 p\n\u22a2 s \u2022 x \u2208 p\n[PROOFSTEP]\nrw [\u2190 one_smul R x, \u2190 smul_assoc]\n[GOAL]\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d\u2074 : Monoid R\ninst\u271d\u00b3 : MulAction R M\ninst\u271d\u00b2 : SMul S R\ninst\u271d\u00b9 : SMul S M\ninst\u271d : IsScalarTower S R M\np : SubMulAction R M\ns : S\nx : M\nh : x \u2208 p\n\u22a2 (s \u2022 1) \u2022 x \u2208 p\n[PROOFSTEP]\nexact p.smul_mem _ h\n[GOAL]\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d\u00b9 : Monoid R\ninst\u271d : MulAction R M\np : SubMulAction R M\nm : { x // x \u2208 p }\n\u22a2 MulAction.Stabilizer.submonoid R m = MulAction.Stabilizer.submonoid R \u2191m\n[PROOFSTEP]\next\n[GOAL]\ncase h\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d\u00b9 : Monoid R\ninst\u271d : MulAction R M\np : SubMulAction R M\nm : { x // x \u2208 p }\nx\u271d : R\n\u22a2 x\u271d \u2208 MulAction.Stabilizer.submonoid R m \u2194 x\u271d \u2208 MulAction.Stabilizer.submonoid R \u2191m\n[PROOFSTEP]\nsimp only [MulAction.mem_stabilizer_submonoid_iff, \u2190 SubMulAction.val_smul, SetLike.coe_eq_coe]\n[GOAL]\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d\u00b9 : Group R\ninst\u271d : MulAction R M\np : SubMulAction R M\nm : { x // x \u2208 p }\n\u22a2 MulAction.stabilizer R m = MulAction.stabilizer R \u2191m\n[PROOFSTEP]\nrw [\u2190 Subgroup.toSubmonoid_eq]\n[GOAL]\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d\u00b9 : Group R\ninst\u271d : MulAction R M\np : SubMulAction R M\nm : { x // x \u2208 p }\n\u22a2 (MulAction.stabilizer R m).toSubmonoid = (MulAction.stabilizer R \u2191m).toSubmonoid\n[PROOFSTEP]\nexact stabilizer_of_subMul.submonoid m\n[GOAL]\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\np p' : SubMulAction R M\nr : R\nx y : M\nhx : x \u2208 p\n\u22a2 -x \u2208 p\n[PROOFSTEP]\nrw [\u2190 neg_one_smul R]\n[GOAL]\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\np p' : SubMulAction R M\nr : R\nx y : M\nhx : x \u2208 p\n\u22a2 -1 \u2022 x \u2208 p\n[PROOFSTEP]\nexact p.smul_mem _ hx\n[GOAL]\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\np p' : SubMulAction R M\nr : R\nx y : M\nh : -x \u2208 p\n\u22a2 x \u2208 p\n[PROOFSTEP]\nrw [\u2190 neg_neg x]\n[GOAL]\nS : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\np p' : SubMulAction R M\nr : R\nx y : M\nh : -x \u2208 p\n\u22a2 - -x \u2208 p\n[PROOFSTEP]\nexact neg_mem _ h\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.GroupAction.SubMulAction", "llama_tokens": 2153, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.668880247169804, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.40145413809040503}} {"text": "[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderTop \u03b1\n\u22a2 \u2200 (J : Type) [\ud835\udca5 : SmallCategory J] [inst : FinCategory J], HasLimitsOfShape J \u03b1\n[PROOFSTEP]\nintro J \ud835\udca5\u2081 \ud835\udca5\u2082\n[GOAL]\n\u03b1 : Type u\nJ\u271d : Type w\ninst\u271d\u00b3 : SmallCategory J\u271d\ninst\u271d\u00b2 : FinCategory J\u271d\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderTop \u03b1\nJ : Type\n\ud835\udca5\u2081 : SmallCategory J\n\ud835\udca5\u2082 : FinCategory J\n\u22a2 HasLimitsOfShape J \u03b1\n[PROOFSTEP]\nexact { has_limit := fun F => HasLimit.mk (finiteLimitCone F) }\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : OrderBot \u03b1\n\u22a2 \u2200 (J : Type) [\ud835\udca5 : SmallCategory J] [inst : FinCategory J], HasColimitsOfShape J \u03b1\n[PROOFSTEP]\nintro J \ud835\udca5\u2081 \ud835\udca5\u2082\n[GOAL]\n\u03b1 : Type u\nJ\u271d : Type w\ninst\u271d\u00b3 : SmallCategory J\u271d\ninst\u271d\u00b2 : FinCategory J\u271d\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : OrderBot \u03b1\nJ : Type\n\ud835\udca5\u2081 : SmallCategory J\n\ud835\udca5\u2082 : FinCategory J\n\u22a2 HasColimitsOfShape J \u03b1\n[PROOFSTEP]\nexact { has_colimit := fun F => HasColimit.mk (finiteColimitCocone F) }\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u2074 : SmallCategory J\ninst\u271d\u00b3 : FinCategory J\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\n\u03b9 : Type u\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 \u220f f = Finset.inf Fintype.elems f\n[PROOFSTEP]\ntrans\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u2074 : SmallCategory J\ninst\u271d\u00b3 : FinCategory J\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\n\u03b9 : Type u\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 \u220f f = ?m.27522\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u2074 : SmallCategory J\ninst\u271d\u00b3 : FinCategory J\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\n\u03b9 : Type u\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 ?m.27522 = Finset.inf Fintype.elems f\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u2074 : SmallCategory J\ninst\u271d\u00b3 : FinCategory J\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\n\u03b9 : Type u\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 \u03b1\n[PROOFSTEP]\nexact (IsLimit.conePointUniqueUpToIso (limit.isLimit _) (finiteLimitCone (Discrete.functor f)).isLimit).to_eq\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u2074 : SmallCategory J\ninst\u271d\u00b3 : FinCategory J\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\n\u03b9 : Type u\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 (finiteLimitCone (Discrete.functor f)).cone.pt = Finset.inf Fintype.elems f\n[PROOFSTEP]\nchange Finset.univ.inf (f \u2218 discreteEquiv.toEmbedding) = Fintype.elems.inf f\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u2074 : SmallCategory J\ninst\u271d\u00b3 : FinCategory J\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\n\u03b9 : Type u\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 Finset.inf Finset.univ (f \u2218 \u2191(Equiv.toEmbedding discreteEquiv)) = Finset.inf Fintype.elems f\n[PROOFSTEP]\nsimp only [\u2190 Finset.inf_map, Finset.univ_map_equiv_to_embedding]\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u2074 : SmallCategory J\ninst\u271d\u00b3 : FinCategory J\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\n\u03b9 : Type u\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 Finset.inf Finset.univ f = Finset.inf Fintype.elems f\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u2074 : SmallCategory J\ninst\u271d\u00b3 : FinCategory J\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\n\u03b9 : Type u\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 \u2210 f = Finset.sup Fintype.elems f\n[PROOFSTEP]\ntrans\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u2074 : SmallCategory J\ninst\u271d\u00b3 : FinCategory J\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\n\u03b9 : Type u\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 \u2210 f = ?m.38444\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u2074 : SmallCategory J\ninst\u271d\u00b3 : FinCategory J\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\n\u03b9 : Type u\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 ?m.38444 = Finset.sup Fintype.elems f\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u2074 : SmallCategory J\ninst\u271d\u00b3 : FinCategory J\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\n\u03b9 : Type u\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 \u03b1\n[PROOFSTEP]\nexact\n (IsColimit.coconePointUniqueUpToIso (colimit.isColimit _) (finiteColimitCocone (Discrete.functor f)).isColimit).to_eq\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u2074 : SmallCategory J\ninst\u271d\u00b3 : FinCategory J\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\n\u03b9 : Type u\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 (finiteColimitCocone (Discrete.functor f)).cocone.pt = Finset.sup Fintype.elems f\n[PROOFSTEP]\nchange Finset.univ.sup (f \u2218 discreteEquiv.toEmbedding) = Fintype.elems.sup f\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u2074 : SmallCategory J\ninst\u271d\u00b3 : FinCategory J\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\n\u03b9 : Type u\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 Finset.sup Finset.univ (f \u2218 \u2191(Equiv.toEmbedding discreteEquiv)) = Finset.sup Fintype.elems f\n[PROOFSTEP]\nsimp only [\u2190 Finset.sup_map, Finset.univ_map_equiv_to_embedding]\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u2074 : SmallCategory J\ninst\u271d\u00b3 : FinCategory J\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\n\u03b9 : Type u\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 Finset.sup Finset.univ f = Finset.sup Fintype.elems f\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderTop \u03b1\n\u22a2 HasBinaryProducts \u03b1\n[PROOFSTEP]\nhave : \u2200 x y : \u03b1, HasLimit (pair x y) :=\n by\n letI := hasFiniteLimits_of_hasFiniteLimits_of_size.{u} \u03b1\n infer_instance\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderTop \u03b1\n\u22a2 \u2200 (x y : \u03b1), HasLimit (pair x y)\n[PROOFSTEP]\nletI := hasFiniteLimits_of_hasFiniteLimits_of_size.{u} \u03b1\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderTop \u03b1\nthis : (\u2200 (J : Type u) {\ud835\udca5 : SmallCategory J}, FinCategory J \u2192 HasLimitsOfShape J \u03b1) \u2192 HasFiniteLimits \u03b1 :=\n hasFiniteLimits_of_hasFiniteLimits_of_size \u03b1\n\u22a2 \u2200 (x y : \u03b1), HasLimit (pair x y)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderTop \u03b1\nthis : \u2200 (x y : \u03b1), HasLimit (pair x y)\n\u22a2 HasBinaryProducts \u03b1\n[PROOFSTEP]\napply hasBinaryProducts_of_hasLimit_pair\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderTop \u03b1\nx y : \u03b1\n\u22a2 limit (pair x y) = Finset.inf Finset.univ (pair x y).toPrefunctor.obj\n[PROOFSTEP]\nrw [finite_limit_eq_finset_univ_inf (pair.{u} x y)]\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderTop \u03b1\nx y : \u03b1\n\u22a2 x \u2293 (y \u2293 \u22a4) = x \u2293 y\n[PROOFSTEP]\nrw [inf_top_eq]\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : OrderBot \u03b1\n\u22a2 HasBinaryCoproducts \u03b1\n[PROOFSTEP]\nhave : \u2200 x y : \u03b1, HasColimit (pair x y) :=\n by\n letI := hasFiniteColimits_of_hasFiniteColimits_of_size.{u} \u03b1\n infer_instance\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : OrderBot \u03b1\n\u22a2 \u2200 (x y : \u03b1), HasColimit (pair x y)\n[PROOFSTEP]\nletI := hasFiniteColimits_of_hasFiniteColimits_of_size.{u} \u03b1\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : OrderBot \u03b1\nthis : (\u2200 (J : Type u) {\ud835\udca5 : SmallCategory J}, FinCategory J \u2192 HasColimitsOfShape J \u03b1) \u2192 HasFiniteColimits \u03b1 :=\n hasFiniteColimits_of_hasFiniteColimits_of_size \u03b1\n\u22a2 \u2200 (x y : \u03b1), HasColimit (pair x y)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : OrderBot \u03b1\nthis : \u2200 (x y : \u03b1), HasColimit (pair x y)\n\u22a2 HasBinaryCoproducts \u03b1\n[PROOFSTEP]\napply hasBinaryCoproducts_of_hasColimit_pair\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : OrderBot \u03b1\nx y : \u03b1\n\u22a2 colimit (pair x y) = Finset.sup Finset.univ (pair x y).toPrefunctor.obj\n[PROOFSTEP]\nrw [finite_colimit_eq_finset_univ_sup (pair x y)]\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : OrderBot \u03b1\nx y : \u03b1\n\u22a2 x \u2294 (y \u2294 \u22a5) = x \u2294 y\n[PROOFSTEP]\nrw [sup_bot_eq]\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderTop \u03b1\nx y z : \u03b1\nf : x \u27f6 z\ng : y \u27f6 z\n\u22a2 limit (cospan f g) = Finset.inf Finset.univ (cospan f g).toPrefunctor.obj\n[PROOFSTEP]\nrw [finite_limit_eq_finset_univ_inf]\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderTop \u03b1\nx y z : \u03b1\nf : x \u27f6 z\ng : y \u27f6 z\n\u22a2 z \u2293 (x \u2293 (y \u2293 \u22a4)) = z \u2293 (x \u2293 y)\n[PROOFSTEP]\nrw [inf_top_eq]\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : OrderBot \u03b1\nx y z : \u03b1\nf : z \u27f6 x\ng : z \u27f6 y\n\u22a2 colimit (span f g) = Finset.sup Finset.univ (span f g).toPrefunctor.obj\n[PROOFSTEP]\nrw [finite_colimit_eq_finset_univ_sup]\n[GOAL]\n\u03b1 : Type u\nJ : Type w\ninst\u271d\u00b3 : SmallCategory J\ninst\u271d\u00b2 : FinCategory J\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : OrderBot \u03b1\nx y z : \u03b1\nf : z \u27f6 x\ng : z \u27f6 y\n\u22a2 z \u2294 (x \u2294 (y \u2294 \u22a5)) = z \u2294 (x \u2294 y)\n[PROOFSTEP]\nrw [sup_bot_eq]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : CompleteLattice \u03b1\nJ : Type u\ninst\u271d : SmallCategory J\nF : J \u2964 \u03b1\ns : Cone F\n\u22a2 \u2200 (b : \u03b1), b \u2208 Set.range F.obj \u2192 s.pt \u2264 b\n[PROOFSTEP]\nrintro _ \u27e8j, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\ninst\u271d\u00b9 : CompleteLattice \u03b1\nJ : Type u\ninst\u271d : SmallCategory J\nF : J \u2964 \u03b1\ns : Cone F\nj : J\n\u22a2 s.pt \u2264 F.obj j\n[PROOFSTEP]\nexact (s.\u03c0.app j).le\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : CompleteLattice \u03b1\nJ : Type u\ninst\u271d : SmallCategory J\nF : J \u2964 \u03b1\ns : Cocone F\n\u22a2 \u2200 (b : \u03b1), b \u2208 Set.range F.obj \u2192 b \u2264 s.pt\n[PROOFSTEP]\nrintro _ \u27e8j, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\ninst\u271d\u00b9 : CompleteLattice \u03b1\nJ : Type u\ninst\u271d : SmallCategory J\nF : J \u2964 \u03b1\ns : Cocone F\nj : J\n\u22a2 F.obj j \u2264 s.pt\n[PROOFSTEP]\nexact (s.\u03b9.app j).le\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Lattice", "llama_tokens": 4518, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494678483918, "lm_q2_score": 0.585101154203231, "lm_q1q2_score": 0.4013498253631862}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nh : IsAssociatedPrime I M\nhf : Function.Injective \u2191f\n\u22a2 IsAssociatedPrime I M'\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 := h.2\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nJ : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhf : Function.Injective \u2191f\nx : M\nh : IsAssociatedPrime (Submodule.annihilator (Submodule.span R {x})) M\n\u22a2 IsAssociatedPrime (Submodule.annihilator (Submodule.span R {x})) M'\n[PROOFSTEP]\nrefine' \u27e8h.1, \u27e8f x, _\u27e9\u27e9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nJ : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhf : Function.Injective \u2191f\nx : M\nh : IsAssociatedPrime (Submodule.annihilator (Submodule.span R {x})) M\n\u22a2 Submodule.annihilator (Submodule.span R {x}) = Submodule.annihilator (Submodule.span R {\u2191f x})\n[PROOFSTEP]\next r\n[GOAL]\ncase intro.h\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nJ : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhf : Function.Injective \u2191f\nx : M\nh : IsAssociatedPrime (Submodule.annihilator (Submodule.span R {x})) M\nr : R\n\u22a2 r \u2208 Submodule.annihilator (Submodule.span R {x}) \u2194 r \u2208 Submodule.annihilator (Submodule.span R {\u2191f x})\n[PROOFSTEP]\nrw [Submodule.mem_annihilator_span_singleton, Submodule.mem_annihilator_span_singleton, \u2190 map_smul, \u2190 f.map_zero,\n hf.eq_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : Subsingleton M\n\u22a2 \u00acIsAssociatedPrime I M\n[PROOFSTEP]\nrintro \u27e8hI, x, hx\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : Subsingleton M\nhI : Ideal.IsPrime I\nx : M\nhx : I = Submodule.annihilator (Submodule.span R {x})\n\u22a2 False\n[PROOFSTEP]\napply hI.ne_top\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : Subsingleton M\nhI : Ideal.IsPrime I\nx : M\nhx : I = Submodule.annihilator (Submodule.span R {x})\n\u22a2 I = \u22a4\n[PROOFSTEP]\nrwa [Subsingleton.elim x 0, Submodule.span_singleton_eq_bot.mpr rfl, Submodule.annihilator_bot] at hx \n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nH : IsNoetherianRing R\nx : M\nhx : x \u2260 0\n\u22a2 \u2203 P, IsAssociatedPrime P M \u2227 Submodule.annihilator (Submodule.span R {x}) \u2264 P\n[PROOFSTEP]\nhave : (R \u2219 x).annihilator \u2260 \u22a4 := by\n rwa [Ne.def, Ideal.eq_top_iff_one, Submodule.mem_annihilator_span_singleton, one_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nH : IsNoetherianRing R\nx : M\nhx : x \u2260 0\n\u22a2 Submodule.annihilator (Submodule.span R {x}) \u2260 \u22a4\n[PROOFSTEP]\nrwa [Ne.def, Ideal.eq_top_iff_one, Submodule.mem_annihilator_span_singleton, one_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nH : IsNoetherianRing R\nx : M\nhx : x \u2260 0\nthis : Submodule.annihilator (Submodule.span R {x}) \u2260 \u22a4\n\u22a2 \u2203 P, IsAssociatedPrime P M \u2227 Submodule.annihilator (Submodule.span R {x}) \u2264 P\n[PROOFSTEP]\nobtain \u27e8P, \u27e8l, h\u2081, y, rfl\u27e9, h\u2083\u27e9 :=\n set_has_maximal_iff_noetherian.mpr H {P | (R \u2219 x).annihilator \u2264 P \u2227 P \u2260 \u22a4 \u2227 \u2203 y : M, P = (R \u2219 y).annihilator}\n \u27e8(R \u2219 x).annihilator, rfl.le, this, x, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nH : IsNoetherianRing R\nx : M\nhx : x \u2260 0\nthis : Submodule.annihilator (Submodule.span R {x}) \u2260 \u22a4\ny : M\nh\u2083 :\n \u2200 (I : Submodule R R),\n I \u2208\n {P |\n Submodule.annihilator (Submodule.span R {x}) \u2264 P \u2227\n P \u2260 \u22a4 \u2227 \u2203 y, P = Submodule.annihilator (Submodule.span R {y})} \u2192\n \u00acSubmodule.annihilator (Submodule.span R {y}) < I\nl : Submodule.annihilator (Submodule.span R {x}) \u2264 Submodule.annihilator (Submodule.span R {y})\nh\u2081 : Submodule.annihilator (Submodule.span R {y}) \u2260 \u22a4\n\u22a2 \u2203 P, IsAssociatedPrime P M \u2227 Submodule.annihilator (Submodule.span R {x}) \u2264 P\n[PROOFSTEP]\nrefine' \u27e8_, \u27e8\u27e8h\u2081, _\u27e9, y, rfl\u27e9, l\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nH : IsNoetherianRing R\nx : M\nhx : x \u2260 0\nthis : Submodule.annihilator (Submodule.span R {x}) \u2260 \u22a4\ny : M\nh\u2083 :\n \u2200 (I : Submodule R R),\n I \u2208\n {P |\n Submodule.annihilator (Submodule.span R {x}) \u2264 P \u2227\n P \u2260 \u22a4 \u2227 \u2203 y, P = Submodule.annihilator (Submodule.span R {y})} \u2192\n \u00acSubmodule.annihilator (Submodule.span R {y}) < I\nl : Submodule.annihilator (Submodule.span R {x}) \u2264 Submodule.annihilator (Submodule.span R {y})\nh\u2081 : Submodule.annihilator (Submodule.span R {y}) \u2260 \u22a4\n\u22a2 \u2200 {x y_1 : R},\n x * y_1 \u2208 Submodule.annihilator (Submodule.span R {y}) \u2192\n x \u2208 Submodule.annihilator (Submodule.span R {y}) \u2228 y_1 \u2208 Submodule.annihilator (Submodule.span R {y})\n[PROOFSTEP]\nintro a b hab\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nH : IsNoetherianRing R\nx : M\nhx : x \u2260 0\nthis : Submodule.annihilator (Submodule.span R {x}) \u2260 \u22a4\ny : M\nh\u2083 :\n \u2200 (I : Submodule R R),\n I \u2208\n {P |\n Submodule.annihilator (Submodule.span R {x}) \u2264 P \u2227\n P \u2260 \u22a4 \u2227 \u2203 y, P = Submodule.annihilator (Submodule.span R {y})} \u2192\n \u00acSubmodule.annihilator (Submodule.span R {y}) < I\nl : Submodule.annihilator (Submodule.span R {x}) \u2264 Submodule.annihilator (Submodule.span R {y})\nh\u2081 : Submodule.annihilator (Submodule.span R {y}) \u2260 \u22a4\na b : R\nhab : a * b \u2208 Submodule.annihilator (Submodule.span R {y})\n\u22a2 a \u2208 Submodule.annihilator (Submodule.span R {y}) \u2228 b \u2208 Submodule.annihilator (Submodule.span R {y})\n[PROOFSTEP]\nrw [or_iff_not_imp_left]\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nH : IsNoetherianRing R\nx : M\nhx : x \u2260 0\nthis : Submodule.annihilator (Submodule.span R {x}) \u2260 \u22a4\ny : M\nh\u2083 :\n \u2200 (I : Submodule R R),\n I \u2208\n {P |\n Submodule.annihilator (Submodule.span R {x}) \u2264 P \u2227\n P \u2260 \u22a4 \u2227 \u2203 y, P = Submodule.annihilator (Submodule.span R {y})} \u2192\n \u00acSubmodule.annihilator (Submodule.span R {y}) < I\nl : Submodule.annihilator (Submodule.span R {x}) \u2264 Submodule.annihilator (Submodule.span R {y})\nh\u2081 : Submodule.annihilator (Submodule.span R {y}) \u2260 \u22a4\na b : R\nhab : a * b \u2208 Submodule.annihilator (Submodule.span R {y})\n\u22a2 \u00aca \u2208 Submodule.annihilator (Submodule.span R {y}) \u2192 b \u2208 Submodule.annihilator (Submodule.span R {y})\n[PROOFSTEP]\nintro ha\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nH : IsNoetherianRing R\nx : M\nhx : x \u2260 0\nthis : Submodule.annihilator (Submodule.span R {x}) \u2260 \u22a4\ny : M\nh\u2083 :\n \u2200 (I : Submodule R R),\n I \u2208\n {P |\n Submodule.annihilator (Submodule.span R {x}) \u2264 P \u2227\n P \u2260 \u22a4 \u2227 \u2203 y, P = Submodule.annihilator (Submodule.span R {y})} \u2192\n \u00acSubmodule.annihilator (Submodule.span R {y}) < I\nl : Submodule.annihilator (Submodule.span R {x}) \u2264 Submodule.annihilator (Submodule.span R {y})\nh\u2081 : Submodule.annihilator (Submodule.span R {y}) \u2260 \u22a4\na b : R\nhab : a * b \u2208 Submodule.annihilator (Submodule.span R {y})\nha : \u00aca \u2208 Submodule.annihilator (Submodule.span R {y})\n\u22a2 b \u2208 Submodule.annihilator (Submodule.span R {y})\n[PROOFSTEP]\nrw [Submodule.mem_annihilator_span_singleton] at ha hab \n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nH : IsNoetherianRing R\nx : M\nhx : x \u2260 0\nthis : Submodule.annihilator (Submodule.span R {x}) \u2260 \u22a4\ny : M\nh\u2083 :\n \u2200 (I : Submodule R R),\n I \u2208\n {P |\n Submodule.annihilator (Submodule.span R {x}) \u2264 P \u2227\n P \u2260 \u22a4 \u2227 \u2203 y, P = Submodule.annihilator (Submodule.span R {y})} \u2192\n \u00acSubmodule.annihilator (Submodule.span R {y}) < I\nl : Submodule.annihilator (Submodule.span R {x}) \u2264 Submodule.annihilator (Submodule.span R {y})\nh\u2081 : Submodule.annihilator (Submodule.span R {y}) \u2260 \u22a4\na b : R\nhab : (a * b) \u2022 y = 0\nha : \u00aca \u2022 y = 0\n\u22a2 b \u2208 Submodule.annihilator (Submodule.span R {y})\n[PROOFSTEP]\nhave H\u2081 : (R \u2219 y).annihilator \u2264 (R \u2219 a \u2022 y).annihilator :=\n by\n intro c hc\n rw [Submodule.mem_annihilator_span_singleton] at hc \u22a2\n rw [smul_comm, hc, smul_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nH : IsNoetherianRing R\nx : M\nhx : x \u2260 0\nthis : Submodule.annihilator (Submodule.span R {x}) \u2260 \u22a4\ny : M\nh\u2083 :\n \u2200 (I : Submodule R R),\n I \u2208\n {P |\n Submodule.annihilator (Submodule.span R {x}) \u2264 P \u2227\n P \u2260 \u22a4 \u2227 \u2203 y, P = Submodule.annihilator (Submodule.span R {y})} \u2192\n \u00acSubmodule.annihilator (Submodule.span R {y}) < I\nl : Submodule.annihilator (Submodule.span R {x}) \u2264 Submodule.annihilator (Submodule.span R {y})\nh\u2081 : Submodule.annihilator (Submodule.span R {y}) \u2260 \u22a4\na b : R\nhab : (a * b) \u2022 y = 0\nha : \u00aca \u2022 y = 0\n\u22a2 Submodule.annihilator (Submodule.span R {y}) \u2264 Submodule.annihilator (Submodule.span R {a \u2022 y})\n[PROOFSTEP]\nintro c hc\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nH : IsNoetherianRing R\nx : M\nhx : x \u2260 0\nthis : Submodule.annihilator (Submodule.span R {x}) \u2260 \u22a4\ny : M\nh\u2083 :\n \u2200 (I : Submodule R R),\n I \u2208\n {P |\n Submodule.annihilator (Submodule.span R {x}) \u2264 P \u2227\n P \u2260 \u22a4 \u2227 \u2203 y, P = Submodule.annihilator (Submodule.span R {y})} \u2192\n \u00acSubmodule.annihilator (Submodule.span R {y}) < I\nl : Submodule.annihilator (Submodule.span R {x}) \u2264 Submodule.annihilator (Submodule.span R {y})\nh\u2081 : Submodule.annihilator (Submodule.span R {y}) \u2260 \u22a4\na b : R\nhab : (a * b) \u2022 y = 0\nha : \u00aca \u2022 y = 0\nc : R\nhc : c \u2208 Submodule.annihilator (Submodule.span R {y})\n\u22a2 c \u2208 Submodule.annihilator (Submodule.span R {a \u2022 y})\n[PROOFSTEP]\nrw [Submodule.mem_annihilator_span_singleton] at hc \u22a2\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nH : IsNoetherianRing R\nx : M\nhx : x \u2260 0\nthis : Submodule.annihilator (Submodule.span R {x}) \u2260 \u22a4\ny : M\nh\u2083 :\n \u2200 (I : Submodule R R),\n I \u2208\n {P |\n Submodule.annihilator (Submodule.span R {x}) \u2264 P \u2227\n P \u2260 \u22a4 \u2227 \u2203 y, P = Submodule.annihilator (Submodule.span R {y})} \u2192\n \u00acSubmodule.annihilator (Submodule.span R {y}) < I\nl : Submodule.annihilator (Submodule.span R {x}) \u2264 Submodule.annihilator (Submodule.span R {y})\nh\u2081 : Submodule.annihilator (Submodule.span R {y}) \u2260 \u22a4\na b : R\nhab : (a * b) \u2022 y = 0\nha : \u00aca \u2022 y = 0\nc : R\nhc : c \u2022 y = 0\n\u22a2 c \u2022 a \u2022 y = 0\n[PROOFSTEP]\nrw [smul_comm, hc, smul_zero]\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nH : IsNoetherianRing R\nx : M\nhx : x \u2260 0\nthis : Submodule.annihilator (Submodule.span R {x}) \u2260 \u22a4\ny : M\nh\u2083 :\n \u2200 (I : Submodule R R),\n I \u2208\n {P |\n Submodule.annihilator (Submodule.span R {x}) \u2264 P \u2227\n P \u2260 \u22a4 \u2227 \u2203 y, P = Submodule.annihilator (Submodule.span R {y})} \u2192\n \u00acSubmodule.annihilator (Submodule.span R {y}) < I\nl : Submodule.annihilator (Submodule.span R {x}) \u2264 Submodule.annihilator (Submodule.span R {y})\nh\u2081 : Submodule.annihilator (Submodule.span R {y}) \u2260 \u22a4\na b : R\nhab : (a * b) \u2022 y = 0\nha : \u00aca \u2022 y = 0\nH\u2081 : Submodule.annihilator (Submodule.span R {y}) \u2264 Submodule.annihilator (Submodule.span R {a \u2022 y})\n\u22a2 b \u2208 Submodule.annihilator (Submodule.span R {y})\n[PROOFSTEP]\nhave H\u2082 : (Submodule.span R {a \u2022 y}).annihilator \u2260 \u22a4 := by\n rwa [Ne.def, Submodule.annihilator_eq_top_iff, Submodule.span_singleton_eq_bot]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nH : IsNoetherianRing R\nx : M\nhx : x \u2260 0\nthis : Submodule.annihilator (Submodule.span R {x}) \u2260 \u22a4\ny : M\nh\u2083 :\n \u2200 (I : Submodule R R),\n I \u2208\n {P |\n Submodule.annihilator (Submodule.span R {x}) \u2264 P \u2227\n P \u2260 \u22a4 \u2227 \u2203 y, P = Submodule.annihilator (Submodule.span R {y})} \u2192\n \u00acSubmodule.annihilator (Submodule.span R {y}) < I\nl : Submodule.annihilator (Submodule.span R {x}) \u2264 Submodule.annihilator (Submodule.span R {y})\nh\u2081 : Submodule.annihilator (Submodule.span R {y}) \u2260 \u22a4\na b : R\nhab : (a * b) \u2022 y = 0\nha : \u00aca \u2022 y = 0\nH\u2081 : Submodule.annihilator (Submodule.span R {y}) \u2264 Submodule.annihilator (Submodule.span R {a \u2022 y})\n\u22a2 Submodule.annihilator (Submodule.span R {a \u2022 y}) \u2260 \u22a4\n[PROOFSTEP]\nrwa [Ne.def, Submodule.annihilator_eq_top_iff, Submodule.span_singleton_eq_bot]\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nH : IsNoetherianRing R\nx : M\nhx : x \u2260 0\nthis : Submodule.annihilator (Submodule.span R {x}) \u2260 \u22a4\ny : M\nh\u2083 :\n \u2200 (I : Submodule R R),\n I \u2208\n {P |\n Submodule.annihilator (Submodule.span R {x}) \u2264 P \u2227\n P \u2260 \u22a4 \u2227 \u2203 y, P = Submodule.annihilator (Submodule.span R {y})} \u2192\n \u00acSubmodule.annihilator (Submodule.span R {y}) < I\nl : Submodule.annihilator (Submodule.span R {x}) \u2264 Submodule.annihilator (Submodule.span R {y})\nh\u2081 : Submodule.annihilator (Submodule.span R {y}) \u2260 \u22a4\na b : R\nhab : (a * b) \u2022 y = 0\nha : \u00aca \u2022 y = 0\nH\u2081 : Submodule.annihilator (Submodule.span R {y}) \u2264 Submodule.annihilator (Submodule.span R {a \u2022 y})\nH\u2082 : Submodule.annihilator (Submodule.span R {a \u2022 y}) \u2260 \u22a4\n\u22a2 b \u2208 Submodule.annihilator (Submodule.span R {y})\n[PROOFSTEP]\nrwa [H\u2081.eq_of_not_lt (h\u2083 (R \u2219 a \u2022 y).annihilator \u27e8l.trans H\u2081, H\u2082, _, rfl\u27e9), Submodule.mem_annihilator_span_singleton,\n smul_comm, smul_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : Subsingleton M\n\u22a2 associatedPrimes R M = \u2205\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : Subsingleton M\nx\u271d : Ideal R\n\u22a2 x\u271d \u2208 associatedPrimes R M \u2194 x\u271d \u2208 \u2205\n[PROOFSTEP]\nsimp only [Set.mem_empty_iff_false, iff_false_iff]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : Subsingleton M\nx\u271d : Ideal R\n\u22a2 \u00acx\u271d \u2208 associatedPrimes R M\n[PROOFSTEP]\napply not_isAssociatedPrime_of_subsingleton\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Nontrivial M\n\u22a2 Set.Nonempty (associatedPrimes R M)\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := exists_ne (0 : M)\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Nontrivial M\nx : M\nhx : x \u2260 0\n\u22a2 Set.Nonempty (associatedPrimes R M)\n[PROOFSTEP]\nobtain \u27e8P, hP, _\u27e9 := exists_le_isAssociatedPrime_of_isNoetherianRing R x hx\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Nontrivial M\nx : M\nhx : x \u2260 0\nP : Ideal R\nhP : IsAssociatedPrime P M\nright\u271d : Submodule.annihilator (Submodule.span R {x}) \u2264 P\n\u22a2 Set.Nonempty (associatedPrimes R M)\n[PROOFSTEP]\nexact \u27e8P, hP\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nh : IsAssociatedPrime I M\n\u22a2 Submodule.annihilator \u22a4 \u2264 I\n[PROOFSTEP]\nobtain \u27e8hI, x, rfl\u27e9 := h\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nJ : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nx : M\nhI : Ideal.IsPrime (Submodule.annihilator (Submodule.span R {x}))\n\u22a2 Submodule.annihilator \u22a4 \u2264 Submodule.annihilator (Submodule.span R {x})\n[PROOFSTEP]\nexact Submodule.annihilator_mono le_top\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : Ideal.IsPrimary I\nh : IsAssociatedPrime J (R \u29f8 I)\n\u22a2 J = Ideal.radical I\n[PROOFSTEP]\nobtain \u27e8hJ, x, e\u27e9 := h\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : Ideal.IsPrimary I\nhJ : Ideal.IsPrime J\nx : R \u29f8 I\ne : J = Submodule.annihilator (Submodule.span R {x})\n\u22a2 J = Ideal.radical I\n[PROOFSTEP]\nhave : x \u2260 0 := by\n rintro rfl\n apply hJ.1\n rwa [Submodule.span_singleton_eq_bot.mpr rfl, Submodule.annihilator_bot] at e \n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : Ideal.IsPrimary I\nhJ : Ideal.IsPrime J\nx : R \u29f8 I\ne : J = Submodule.annihilator (Submodule.span R {x})\n\u22a2 x \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : Ideal.IsPrimary I\nhJ : Ideal.IsPrime J\ne : J = Submodule.annihilator (Submodule.span R {0})\n\u22a2 False\n[PROOFSTEP]\napply hJ.1\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : Ideal.IsPrimary I\nhJ : Ideal.IsPrime J\ne : J = Submodule.annihilator (Submodule.span R {0})\n\u22a2 J = \u22a4\n[PROOFSTEP]\nrwa [Submodule.span_singleton_eq_bot.mpr rfl, Submodule.annihilator_bot] at e \n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : Ideal.IsPrimary I\nhJ : Ideal.IsPrime J\nx : R \u29f8 I\ne : J = Submodule.annihilator (Submodule.span R {x})\nthis : x \u2260 0\n\u22a2 J = Ideal.radical I\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 := Ideal.Quotient.mk\u2090_surjective R _ x\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : Ideal.IsPrimary I\nhJ : Ideal.IsPrime J\nx : R\ne : J = Submodule.annihilator (Submodule.span R {\u2191(Ideal.Quotient.mk\u2090 R I) x})\nthis : \u2191(Ideal.Quotient.mk\u2090 R I) x \u2260 0\n\u22a2 J = Ideal.radical I\n[PROOFSTEP]\nreplace e : \u2200 {y}, y \u2208 J \u2194 x * y \u2208 I\n[GOAL]\ncase e\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : Ideal.IsPrimary I\nhJ : Ideal.IsPrime J\nx : R\ne : J = Submodule.annihilator (Submodule.span R {\u2191(Ideal.Quotient.mk\u2090 R I) x})\nthis : \u2191(Ideal.Quotient.mk\u2090 R I) x \u2260 0\n\u22a2 \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\n[PROOFSTEP]\nintro y\n[GOAL]\ncase e\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : Ideal.IsPrimary I\nhJ : Ideal.IsPrime J\nx : R\ne : J = Submodule.annihilator (Submodule.span R {\u2191(Ideal.Quotient.mk\u2090 R I) x})\nthis : \u2191(Ideal.Quotient.mk\u2090 R I) x \u2260 0\ny : R\n\u22a2 y \u2208 J \u2194 x * y \u2208 I\n[PROOFSTEP]\nrw [e, Submodule.mem_annihilator_span_singleton, \u2190 map_smul, smul_eq_mul, mul_comm, Ideal.Quotient.mk\u2090_eq_mk, \u2190\n Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero]\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : Ideal.IsPrimary I\nhJ : Ideal.IsPrime J\nx : R\nthis : \u2191(Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\n\u22a2 J = Ideal.radical I\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase intro.intro.intro.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : Ideal.IsPrimary I\nhJ : Ideal.IsPrime J\nx : R\nthis : \u2191(Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\n\u22a2 J \u2264 Ideal.radical I\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase intro.intro.intro.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : Ideal.IsPrimary I\nhJ : Ideal.IsPrime J\nx : R\nthis : \u2191(Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\ny : R\nhy : y \u2208 J\n\u22a2 y \u2208 Ideal.radical I\n[PROOFSTEP]\nexact (hI.2 <| e.mp hy).resolve_left ((Submodule.Quotient.mk_eq_zero I).not.mp this)\n[GOAL]\ncase intro.intro.intro.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : Ideal.IsPrimary I\nhJ : Ideal.IsPrime J\nx : R\nthis : \u2191(Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\n\u22a2 Ideal.radical I \u2264 J\n[PROOFSTEP]\nrw [hJ.radical_le_iff]\n[GOAL]\ncase intro.intro.intro.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : Ideal.IsPrimary I\nhJ : Ideal.IsPrime J\nx : R\nthis : \u2191(Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\n\u22a2 I \u2264 J\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase intro.intro.intro.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : Ideal.IsPrimary I\nhJ : Ideal.IsPrime J\nx : R\nthis : \u2191(Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\ny : R\nhy : y \u2208 I\n\u22a2 y \u2208 J\n[PROOFSTEP]\nexact e.mpr (I.mul_mem_left x hy)\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : IsNoetherianRing R\nhI : Ideal.IsPrimary I\n\u22a2 associatedPrimes R (R \u29f8 I) = {Ideal.radical I}\n[PROOFSTEP]\next J\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J\u271d : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : IsNoetherianRing R\nhI : Ideal.IsPrimary I\nJ : Ideal R\n\u22a2 J \u2208 associatedPrimes R (R \u29f8 I) \u2194 J \u2208 {Ideal.radical I}\n[PROOFSTEP]\nrw [Set.mem_singleton_iff]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J\u271d : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : IsNoetherianRing R\nhI : Ideal.IsPrimary I\nJ : Ideal R\n\u22a2 J \u2208 associatedPrimes R (R \u29f8 I) \u2194 J = Ideal.radical I\n[PROOFSTEP]\nrefine' \u27e8IsAssociatedPrime.eq_radical hI, _\u27e9\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J\u271d : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : IsNoetherianRing R\nhI : Ideal.IsPrimary I\nJ : Ideal R\n\u22a2 J = Ideal.radical I \u2192 J \u2208 associatedPrimes R (R \u29f8 I)\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : IsNoetherianRing R\nhI : Ideal.IsPrimary I\n\u22a2 Ideal.radical I \u2208 associatedPrimes R (R \u29f8 I)\n[PROOFSTEP]\nhaveI : Nontrivial (R \u29f8 I) :=\n by\n refine \u27e8(Ideal.Quotient.mk I : _) 1, (Ideal.Quotient.mk I : _) 0, ?_\u27e9\n rw [Ne.def, Ideal.Quotient.eq, sub_zero, \u2190 Ideal.eq_top_iff_one]\n exact hI.1\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : IsNoetherianRing R\nhI : Ideal.IsPrimary I\n\u22a2 Nontrivial (R \u29f8 I)\n[PROOFSTEP]\nrefine \u27e8(Ideal.Quotient.mk I : _) 1, (Ideal.Quotient.mk I : _) 0, ?_\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : IsNoetherianRing R\nhI : Ideal.IsPrimary I\n\u22a2 \u2191(Ideal.Quotient.mk I) 1 \u2260 \u2191(Ideal.Quotient.mk I) 0\n[PROOFSTEP]\nrw [Ne.def, Ideal.Quotient.eq, sub_zero, \u2190 Ideal.eq_top_iff_one]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : IsNoetherianRing R\nhI : Ideal.IsPrimary I\n\u22a2 \u00acI = \u22a4\n[PROOFSTEP]\nexact hI.1\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : IsNoetherianRing R\nhI : Ideal.IsPrimary I\nthis : Nontrivial (R \u29f8 I)\n\u22a2 Ideal.radical I \u2208 associatedPrimes R (R \u29f8 I)\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := associatedPrimes.nonempty R (R \u29f8 I)\n[GOAL]\ncase h.intro\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nM' : Type u_3\ninst\u271d\u00b2 : AddCommGroup M'\ninst\u271d\u00b9 : Module R M'\nf : M \u2192\u2097[R] M'\ninst\u271d : IsNoetherianRing R\nhI : Ideal.IsPrimary I\nthis : Nontrivial (R \u29f8 I)\na : Ideal R\nha : a \u2208 associatedPrimes R (R \u29f8 I)\n\u22a2 Ideal.radical I \u2208 associatedPrimes R (R \u29f8 I)\n[PROOFSTEP]\nexact ha.eq_radical hI \u25b8 ha\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Ideal.AssociatedPrime", "llama_tokens": 13591, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.4013498178503709}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\n\u22a2 classes (ker f) \u2286 Set.range fun y => {x | f x = y}\n[PROOFSTEP]\nrintro s \u27e8x, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 {x_1 | Rel (ker f) x_1 x} \u2208 Set.range fun y => {x | f x = y}\n[PROOFSTEP]\nrw [Set.mem_range]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 \u2203 y, {x | f x = y} = {x_1 | Rel (ker f) x_1 x}\n[PROOFSTEP]\nexact \u27e8f x, rfl\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u2191(classes (ker f))\n\u22a2 Fintype.card \u2191(classes (ker f)) \u2264 Fintype.card \u03b2\n[PROOFSTEP]\nclassical exact le_trans (Set.card_le_of_subset (classes_ker_subset_fiber_set f)) (Fintype.card_range_le _)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Fintype \u2191(classes (ker f))\n\u22a2 Fintype.card \u2191(classes (ker f)) \u2264 Fintype.card \u03b2\n[PROOFSTEP]\nexact le_trans (Set.card_le_of_subset (classes_ker_subset_fiber_set f)) (Fintype.card_range_le _)\n[GOAL]\n\u03b1 : Type u_1\nr : Setoid \u03b1\nx y : \u03b1\nx\u271d : \u2203 c, c \u2208 classes r \u2227 x \u2208 c \u2227 y \u2208 c\nc : Set \u03b1\nz : \u03b1\nhz : c = {x | Rel r x z}\nhx : x \u2208 c\nhy : y \u2208 c\n\u22a2 Rel r x y\n[PROOFSTEP]\nsubst c\n[GOAL]\n\u03b1 : Type u_1\nr : Setoid \u03b1\nx y : \u03b1\nx\u271d : \u2203 c, c \u2208 classes r \u2227 x \u2208 c \u2227 y \u2208 c\nz : \u03b1\nhx : x \u2208 {x | Rel r x z}\nhy : y \u2208 {x | Rel r x z}\n\u22a2 Rel r x y\n[PROOFSTEP]\nexact r.trans' hx (r.symm' hy)\n[GOAL]\n\u03b1 : Type u_1\nr\u2081 r\u2082 : Setoid \u03b1\nh : classes r\u2081 = classes r\u2082\na b : \u03b1\n\u22a2 Rel r\u2081 a b \u2194 Rel r\u2082 a b\n[PROOFSTEP]\nsimp only [rel_iff_exists_classes, exists_prop, h]\n[GOAL]\n\u03b1 : Type u_1\nr : Setoid \u03b1\na : \u03b1\n\u22a2 \u2200 (y : Set \u03b1), y \u2208 classes r \u2192 a \u2208 y \u2192 y = {x | Rel r x a}\n[PROOFSTEP]\nrintro _ \u27e8y, rfl\u27e9 ha\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nr : Setoid \u03b1\na y : \u03b1\nha : a \u2208 {x | Rel r x y}\n\u22a2 {x | Rel r x y} = {x | Rel r x a}\n[PROOFSTEP]\next x\n[GOAL]\ncase intro.h\n\u03b1 : Type u_1\nr : Setoid \u03b1\na y : \u03b1\nha : a \u2208 {x | Rel r x y}\nx : \u03b1\n\u22a2 x \u2208 {x | Rel r x y} \u2194 x \u2208 {x | Rel r x a}\n[PROOFSTEP]\nexact \u27e8fun hx => r.trans' hx (r.symm' ha), fun hx => r.trans' hx ha\u27e9\n[GOAL]\n\u03b1 : Type u_1\nc : Set (Set \u03b1)\nH : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b\nx : \u03b1\n\u22a2 {y | Rel (mkClasses c H) x y} \u2208 c\n[PROOFSTEP]\nconvert @Setoid.eqv_class_mem _ _ H x using 3\n[GOAL]\ncase h.e'_4.h.e'_2.h.a\n\u03b1 : Type u_1\nc : Set (Set \u03b1)\nH : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b\nx x\u271d : \u03b1\n\u22a2 Rel (mkClasses c H) x x\u271d \u2194 Rel (mkClasses c H) x\u271d x\n[PROOFSTEP]\nrw [Setoid.comm']\n[GOAL]\n\u03b1 : Type u_1\nc : Set (Set \u03b1)\nhu : \u22c3\u2080 c = Set.univ\nH : Set.PairwiseDisjoint c id\na : \u03b1\n\u22a2 a \u2208 \u22c3\u2080 c\n[PROOFSTEP]\nrw [hu]\n[GOAL]\n\u03b1 : Type u_1\nc : Set (Set \u03b1)\nhu : \u22c3\u2080 c = Set.univ\nH : Set.PairwiseDisjoint c id\na : \u03b1\n\u22a2 a \u2208 Set.univ\n[PROOFSTEP]\nexact Set.mem_univ a\n[GOAL]\n\u03b1 : Type u_1\nc : Set (Set \u03b1)\nhc : IsPartition c\nx : \u03b1\nt : Set \u03b1\nht : (fun b => \u2203! x_1, x \u2208 b) t \u2227 \u2200 (y : Set \u03b1), (fun b => \u2203! x_1, x \u2208 b) y \u2192 y = t\n\u22a2 t \u2208 c \u2227 x \u2208 t\n[PROOFSTEP]\nsimp only [exists_unique_iff_exists] at ht \n[GOAL]\n\u03b1 : Type u_1\nc : Set (Set \u03b1)\nhc : IsPartition c\nx : \u03b1\nt : Set \u03b1\nht : (\u2203 x_1, x \u2208 t) \u2227 \u2200 (y : Set \u03b1), (\u2203 x_1, x \u2208 y) \u2192 y = t\n\u22a2 t \u2208 c \u2227 x \u2208 t\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\nc : Set (Set \u03b1)\nhc : IsPartition c\ns : Set \u03b1\nx\u271d : s \u2208 classes (mkClasses c (_ : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b))\ny : \u03b1\nhs : s = {x | Rel (mkClasses c (_ : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b)) x y}\nb : Set \u03b1\nhm : b \u2208 c\nhb : y \u2208 b\n_hy : \u2200 (y_1 : Set \u03b1), y_1 \u2208 c \u2192 y \u2208 y_1 \u2192 y_1 = b\n\u22a2 s \u2208 c\n[PROOFSTEP]\nrwa [show s = b from\n hs.symm \u25b8\n Set.ext fun x =>\n \u27e8fun hx => symm' (mkClasses c hc.2) hx b hm hb, fun hx b' hc' hx' =>\n eq_of_mem_eqv_class hc.2 hm hx hc' hx' \u25b8 hb\u27e9]\n[GOAL]\n\u03b1 : Type u_1\nx y : Subtype IsPartition\nhx : x \u2264 y\nhy : y \u2264 x\n\u22a2 x = y\n[PROOFSTEP]\nlet h := @le_antisymm (Setoid \u03b1) _ _ _ hx hy\n[GOAL]\n\u03b1 : Type u_1\nx y : Subtype IsPartition\nhx : x \u2264 y\nhy : y \u2264 x\nh : mkClasses \u2191x (_ : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b) = mkClasses \u2191y (_ : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b) := le_antisymm hx hy\n\u22a2 x = y\n[PROOFSTEP]\nrw [Subtype.ext_iff_val, \u2190 classes_mkClasses x.1 x.2, \u2190 classes_mkClasses y.1 y.2, h]\n[GOAL]\n\u03b1 : Type u_1\nC : { C // IsPartition C }\n\u22a2 (fun r => { val := classes r, property := (_ : \u00ac\u2205 \u2208 classes r \u2227 \u2200 (a : \u03b1), \u2203! b x, a \u2208 b) })\n ((fun C => mkClasses \u2191C (_ : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b)) C) =\n C\n[PROOFSTEP]\nrw [Subtype.ext_iff_val, \u2190 classes_mkClasses C.1 C.2]\n[GOAL]\n\u03b1 : Type u_1\nr s : Setoid \u03b1\n\u22a2 \u2191{ toFun := fun r => { val := classes r, property := (_ : \u00ac\u2205 \u2208 classes r \u2227 \u2200 (a : \u03b1), \u2203! b x, a \u2208 b) },\n invFun := fun C => mkClasses \u2191C (_ : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b),\n left_inv := (_ : \u2200 (r : Setoid \u03b1), mkClasses (classes r) (_ : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b) = r),\n right_inv :=\n (_ :\n \u2200 (C : { C // IsPartition C }),\n (fun r => { val := classes r, property := (_ : \u00ac\u2205 \u2208 classes r \u2227 \u2200 (a : \u03b1), \u2203! b x, a \u2208 b) })\n ((fun C => mkClasses \u2191C (_ : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b)) C) =\n C) }\n r \u2264\n \u2191{ toFun := fun r => { val := classes r, property := (_ : \u00ac\u2205 \u2208 classes r \u2227 \u2200 (a : \u03b1), \u2203! b x, a \u2208 b) },\n invFun := fun C => mkClasses \u2191C (_ : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b),\n left_inv := (_ : \u2200 (r : Setoid \u03b1), mkClasses (classes r) (_ : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b) = r),\n right_inv :=\n (_ :\n \u2200 (C : { C // IsPartition C }),\n (fun r => { val := classes r, property := (_ : \u00ac\u2205 \u2208 classes r \u2227 \u2200 (a : \u03b1), \u2203! b x, a \u2208 b) })\n ((fun C => mkClasses \u2191C (_ : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b)) C) =\n C) }\n s \u2194\n r \u2264 s\n[PROOFSTEP]\nconv_rhs => rw [\u2190 mkClasses_classes r, \u2190 mkClasses_classes s]\n[GOAL]\n\u03b1 : Type u_1\nr s : Setoid \u03b1\n| r \u2264 s\n[PROOFSTEP]\nrw [\u2190 mkClasses_classes r, \u2190 mkClasses_classes s]\n[GOAL]\n\u03b1 : Type u_1\nr s : Setoid \u03b1\n| r \u2264 s\n[PROOFSTEP]\nrw [\u2190 mkClasses_classes r, \u2190 mkClasses_classes s]\n[GOAL]\n\u03b1 : Type u_1\nr s : Setoid \u03b1\n| r \u2264 s\n[PROOFSTEP]\nrw [\u2190 mkClasses_classes r, \u2190 mkClasses_classes s]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ns : \u03b9 \u2192 Set \u03b1\nhs : IndexedPartition s\n\u22a2 \u22c3 (i : \u03b9), s i = univ\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ns : \u03b9 \u2192 Set \u03b1\nhs : IndexedPartition s\nx : \u03b1\n\u22a2 x \u2208 \u22c3 (i : \u03b9), s i \u2194 x \u2208 univ\n[PROOFSTEP]\nsimp [hs.exists_mem x]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ns : \u03b9 \u2192 Set \u03b1\nhs : IndexedPartition s\nx\u271d : IndexedPartition.Quotient hs\nx : \u03b1\n\u22a2 proj hs \u207b\u00b9' {Quotient.mk'' x} = s (\u2191(equivQuotient hs).symm (Quotient.mk'' x))\n[PROOFSTEP]\next y\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ns : \u03b9 \u2192 Set \u03b1\nhs : IndexedPartition s\nx\u271d : IndexedPartition.Quotient hs\nx y : \u03b1\n\u22a2 y \u2208 proj hs \u207b\u00b9' {Quotient.mk'' x} \u2194 y \u2208 s (\u2191(equivQuotient hs).symm (Quotient.mk'' x))\n[PROOFSTEP]\nsimp only [Set.mem_preimage, Set.mem_singleton_iff, hs.mem_iff_index_eq]\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ns : \u03b9 \u2192 Set \u03b1\nhs : IndexedPartition s\nx\u271d : IndexedPartition.Quotient hs\nx y : \u03b1\n\u22a2 proj hs y = Quotient.mk'' x \u2194 index hs y = \u2191(equivQuotient hs).symm (Quotient.mk'' x)\n[PROOFSTEP]\nexact Quotient.eq''\n", "meta": {"mathlib_filename": "Mathlib.Data.Setoid.Partition", "llama_tokens": 3451, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.4013498178503708}} {"text": "[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : Field A\ninst\u271d\u00b9 : CharZero A\ninst\u271d : IsAlgClosed A\n\u22a2 Fintype.card (K \u2192+* A) = finrank \u211a K\n[PROOFSTEP]\nrw [Fintype.ofEquiv_card RingHom.equivRatAlgHom.symm, AlgHom.card]\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : Field A\ninst\u271d\u00b9 : CharZero A\ninst\u271d : IsAlgClosed A\n\u22a2 Nonempty (K \u2192+* A)\n[PROOFSTEP]\nrw [\u2190 Fintype.card_pos_iff, NumberField.Embeddings.card K A]\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : Field A\ninst\u271d\u00b9 : CharZero A\ninst\u271d : IsAlgClosed A\n\u22a2 0 < finrank \u211a K\n[PROOFSTEP]\nexact FiniteDimensional.finrank_pos\n[GOAL]\nK : Type u_1\nA : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\ninst\u271d\u00b2 : Field A\ninst\u271d\u00b9 : Algebra \u211a A\ninst\u271d : IsAlgClosed A\nx : K\n\u22a2 (range fun \u03c6 => \u2191\u03c6 x) = rootSet (minpoly \u211a x) A\n[PROOFSTEP]\nconvert (NumberField.isAlgebraic K).range_eval_eq_rootSet_minpoly A x using 1\n[GOAL]\ncase h.e'_2\nK : Type u_1\nA : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\ninst\u271d\u00b2 : Field A\ninst\u271d\u00b9 : Algebra \u211a A\ninst\u271d : IsAlgClosed A\nx : K\n\u22a2 (range fun \u03c6 => \u2191\u03c6 x) = range fun \u03c8 => \u2191\u03c8 x\n[PROOFSTEP]\next a\n[GOAL]\ncase h.e'_2.h\nK : Type u_1\nA : Type u_2\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\ninst\u271d\u00b2 : Field A\ninst\u271d\u00b9 : Algebra \u211a A\ninst\u271d : IsAlgClosed A\nx : K\na : (fun x => A) x\n\u22a2 (a \u2208 range fun \u03c6 => \u2191\u03c6 x) \u2194 a \u2208 range fun \u03c8 => \u2191\u03c8 x\n[PROOFSTEP]\nexact \u27e8fun \u27e8\u03c6, h\u03c6\u27e9 => \u27e8\u03c6.toRatAlgHom, h\u03c6\u27e9, fun \u27e8\u03c6, h\u03c6\u27e9 => \u27e8\u03c6.toRingHom, h\u03c6\u27e9\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nx : K\nh : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B\ni : \u2115\n\u22a2 \u2016coeff (minpoly \u211a x) i\u2016 \u2264 max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\n[PROOFSTEP]\nhave hx := IsSeparable.isIntegral \u211a x\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nx : K\nh : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B\ni : \u2115\nhx : IsIntegral \u211a x\n\u22a2 \u2016coeff (minpoly \u211a x) i\u2016 \u2264 max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\n[PROOFSTEP]\nrw [\u2190 norm_algebraMap' A, \u2190 coeff_map (algebraMap \u211a A)]\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nx : K\nh : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B\ni : \u2115\nhx : IsIntegral \u211a x\n\u22a2 \u2016coeff (map (algebraMap \u211a A) (minpoly \u211a x)) i\u2016 \u2264 max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\n[PROOFSTEP]\nrefine\n coeff_bdd_of_roots_le _ (minpoly.monic hx) (IsAlgClosed.splits_codomain _) (minpoly.natDegree_le x) (fun z hz => ?_) i\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nx : K\nh : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B\ni : \u2115\nhx : IsIntegral \u211a x\nz : (fun x => A) (coeff (minpoly \u211a x) i)\nhz : z \u2208 roots (map (algebraMap \u211a A) (minpoly \u211a x))\n\u22a2 \u2016z\u2016 \u2264 B\n[PROOFSTEP]\nclassical\nrw [\u2190 Multiset.mem_toFinset] at hz \nobtain \u27e8\u03c6, rfl\u27e9 := (range_eval_eq_rootSet_minpoly K A x).symm.subset hz\nexact h \u03c6\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nx : K\nh : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B\ni : \u2115\nhx : IsIntegral \u211a x\nz : (fun x => A) (coeff (minpoly \u211a x) i)\nhz : z \u2208 roots (map (algebraMap \u211a A) (minpoly \u211a x))\n\u22a2 \u2016z\u2016 \u2264 B\n[PROOFSTEP]\nrw [\u2190 Multiset.mem_toFinset] at hz \n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nx : K\nh : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B\ni : \u2115\nhx : IsIntegral \u211a x\nz : (fun x => A) (coeff (minpoly \u211a x) i)\nhz\u271d : z \u2208 roots (map (algebraMap \u211a A) (minpoly \u211a x))\nhz : z \u2208 Multiset.toFinset (roots (map (algebraMap \u211a A) (minpoly \u211a x)))\n\u22a2 \u2016z\u2016 \u2264 B\n[PROOFSTEP]\nobtain \u27e8\u03c6, rfl\u27e9 := (range_eval_eq_rootSet_minpoly K A x).symm.subset hz\n[GOAL]\ncase intro\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nx : K\nh : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B\ni : \u2115\nhx : IsIntegral \u211a x\n\u03c6 : K \u2192+* A\nhz\u271d : (fun \u03c6 => \u2191\u03c6 x) \u03c6 \u2208 roots (map (algebraMap \u211a A) (minpoly \u211a x))\nhz : (fun \u03c6 => \u2191\u03c6 x) \u03c6 \u2208 Multiset.toFinset (roots (map (algebraMap \u211a A) (minpoly \u211a x)))\n\u22a2 \u2016(fun \u03c6 => \u2191\u03c6 x) \u03c6\u2016 \u2264 B\n[PROOFSTEP]\nexact h \u03c6\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\n\u22a2 Set.Finite {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B}\n[PROOFSTEP]\nlet C := Nat.ceil (max B 1 ^ finrank \u211a K * (finrank \u211a K).choose (finrank \u211a K / 2))\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nC : \u2115 := \u2308max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\u2309\u208a\n\u22a2 Set.Finite {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B}\n[PROOFSTEP]\nhave := bUnion_roots_finite (algebraMap \u2124 K) (finrank \u211a K) (finite_Icc (-C : \u2124) C)\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nC : \u2115 := \u2308max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\u2309\u208a\nthis :\n Set.Finite\n (\u22c3 (f : \u2124[X]) (_ : natDegree f \u2264 finrank \u211a K \u2227 \u2200 (i : \u2115), coeff f i \u2208 Icc (-\u2191C) \u2191C),\n \u2191(Multiset.toFinset (roots (map (algebraMap \u2124 K) f))))\n\u22a2 Set.Finite {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B}\n[PROOFSTEP]\nrefine this.subset fun x hx => ?_\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nC : \u2115 := \u2308max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\u2309\u208a\nthis :\n Set.Finite\n (\u22c3 (f : \u2124[X]) (_ : natDegree f \u2264 finrank \u211a K \u2227 \u2200 (i : \u2115), coeff f i \u2208 Icc (-\u2191C) \u2191C),\n \u2191(Multiset.toFinset (roots (map (algebraMap \u2124 K) f))))\nx : K\nhx : x \u2208 {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B}\n\u22a2 x \u2208\n \u22c3 (f : \u2124[X]) (_ : natDegree f \u2264 finrank \u211a K \u2227 \u2200 (i : \u2115), coeff f i \u2208 Icc (-\u2191C) \u2191C),\n \u2191(Multiset.toFinset (roots (map (algebraMap \u2124 K) f)))\n[PROOFSTEP]\nsimp_rw [mem_iUnion]\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nC : \u2115 := \u2308max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\u2309\u208a\nthis :\n Set.Finite\n (\u22c3 (f : \u2124[X]) (_ : natDegree f \u2264 finrank \u211a K \u2227 \u2200 (i : \u2115), coeff f i \u2208 Icc (-\u2191C) \u2191C),\n \u2191(Multiset.toFinset (roots (map (algebraMap \u2124 K) f))))\nx : K\nhx : x \u2208 {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B}\n\u22a2 \u2203 i i_1, x \u2208 \u2191(Multiset.toFinset (roots (map (algebraMap \u2124 K) i)))\n[PROOFSTEP]\nhave h_map_\u211a_minpoly := minpoly.isIntegrallyClosed_eq_field_fractions' \u211a hx.1\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nC : \u2115 := \u2308max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\u2309\u208a\nthis :\n Set.Finite\n (\u22c3 (f : \u2124[X]) (_ : natDegree f \u2264 finrank \u211a K \u2227 \u2200 (i : \u2115), coeff f i \u2208 Icc (-\u2191C) \u2191C),\n \u2191(Multiset.toFinset (roots (map (algebraMap \u2124 K) f))))\nx : K\nhx : x \u2208 {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B}\nh_map_\u211a_minpoly : minpoly \u211a x = map (algebraMap \u2124 \u211a) (minpoly \u2124 x)\n\u22a2 \u2203 i i_1, x \u2208 \u2191(Multiset.toFinset (roots (map (algebraMap \u2124 K) i)))\n[PROOFSTEP]\nrefine \u27e8_, \u27e8?_, fun i => ?_\u27e9, mem_rootSet.2 \u27e8minpoly.ne_zero hx.1, minpoly.aeval \u2124 x\u27e9\u27e9\n[GOAL]\ncase refine_1\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nC : \u2115 := \u2308max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\u2309\u208a\nthis :\n Set.Finite\n (\u22c3 (f : \u2124[X]) (_ : natDegree f \u2264 finrank \u211a K \u2227 \u2200 (i : \u2115), coeff f i \u2208 Icc (-\u2191C) \u2191C),\n \u2191(Multiset.toFinset (roots (map (algebraMap \u2124 K) f))))\nx : K\nhx : x \u2208 {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B}\nh_map_\u211a_minpoly : minpoly \u211a x = map (algebraMap \u2124 \u211a) (minpoly \u2124 x)\n\u22a2 natDegree (minpoly \u2124 x) \u2264 finrank \u211a K\n[PROOFSTEP]\nrw [\u2190 (minpoly.monic hx.1).natDegree_map (algebraMap \u2124 \u211a), \u2190 h_map_\u211a_minpoly]\n[GOAL]\ncase refine_1\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nC : \u2115 := \u2308max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\u2309\u208a\nthis :\n Set.Finite\n (\u22c3 (f : \u2124[X]) (_ : natDegree f \u2264 finrank \u211a K \u2227 \u2200 (i : \u2115), coeff f i \u2208 Icc (-\u2191C) \u2191C),\n \u2191(Multiset.toFinset (roots (map (algebraMap \u2124 K) f))))\nx : K\nhx : x \u2208 {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B}\nh_map_\u211a_minpoly : minpoly \u211a x = map (algebraMap \u2124 \u211a) (minpoly \u2124 x)\n\u22a2 natDegree (minpoly \u211a x) \u2264 finrank \u211a K\n[PROOFSTEP]\nexact minpoly.natDegree_le x\n[GOAL]\ncase refine_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nC : \u2115 := \u2308max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\u2309\u208a\nthis :\n Set.Finite\n (\u22c3 (f : \u2124[X]) (_ : natDegree f \u2264 finrank \u211a K \u2227 \u2200 (i : \u2115), coeff f i \u2208 Icc (-\u2191C) \u2191C),\n \u2191(Multiset.toFinset (roots (map (algebraMap \u2124 K) f))))\nx : K\nhx : x \u2208 {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B}\nh_map_\u211a_minpoly : minpoly \u211a x = map (algebraMap \u2124 \u211a) (minpoly \u2124 x)\ni : \u2115\n\u22a2 coeff (minpoly \u2124 x) i \u2208\n Icc (-\u2191\u2308max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\u2309\u208a)\n \u2191\u2308max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\u2309\u208a\n[PROOFSTEP]\nrw [mem_Icc, \u2190 abs_le, \u2190 @Int.cast_le \u211d]\n[GOAL]\ncase refine_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nC : \u2115 := \u2308max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\u2309\u208a\nthis :\n Set.Finite\n (\u22c3 (f : \u2124[X]) (_ : natDegree f \u2264 finrank \u211a K \u2227 \u2200 (i : \u2115), coeff f i \u2208 Icc (-\u2191C) \u2191C),\n \u2191(Multiset.toFinset (roots (map (algebraMap \u2124 K) f))))\nx : K\nhx : x \u2208 {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B}\nh_map_\u211a_minpoly : minpoly \u211a x = map (algebraMap \u2124 \u211a) (minpoly \u2124 x)\ni : \u2115\n\u22a2 \u2191|coeff (minpoly \u2124 x) i| \u2264 \u2191\u2191\u2308max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\u2309\u208a\n[PROOFSTEP]\nrefine (Eq.trans_le ?_ <| coeff_bdd_of_norm_le hx.2 i).trans (Nat.le_ceil _)\n[GOAL]\ncase refine_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nB : \u211d\nC : \u2115 := \u2308max B 1 ^ finrank \u211a K * \u2191(Nat.choose (finrank \u211a K) (finrank \u211a K / 2))\u2309\u208a\nthis :\n Set.Finite\n (\u22c3 (f : \u2124[X]) (_ : natDegree f \u2264 finrank \u211a K \u2227 \u2200 (i : \u2115), coeff f i \u2208 Icc (-\u2191C) \u2191C),\n \u2191(Multiset.toFinset (roots (map (algebraMap \u2124 K) f))))\nx : K\nhx : x \u2208 {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 B}\nh_map_\u211a_minpoly : minpoly \u211a x = map (algebraMap \u2124 \u211a) (minpoly \u2124 x)\ni : \u2115\n\u22a2 \u2191|coeff (minpoly \u2124 x) i| = \u2016coeff (minpoly \u211a x) i\u2016\n[PROOFSTEP]\nrw [h_map_\u211a_minpoly, coeff_map, eq_intCast, Int.norm_cast_rat, Int.norm_eq_abs, Int.cast_abs]\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nx : K\nhxi : IsIntegral \u2124 x\nhx : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 = 1\n\u22a2 \u2203 n x_1, x ^ n = 1\n[PROOFSTEP]\nobtain \u27e8a, -, b, -, habne, h\u27e9 :=\n @Set.Infinite.exists_ne_map_eq_of_mapsTo _ _ _ _ ((\u00b7 ^ \u00b7) x : \u2115 \u2192 K) Set.infinite_univ\n (by exact fun a _ => \u27e8hxi.pow a, fun \u03c6 => by simp [hx \u03c6]\u27e9) (finite_of_norm_le K A (1 : \u211d))\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nx : K\nhxi : IsIntegral \u2124 x\nhx : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 = 1\n\u22a2 MapsTo ((fun x x_1 => x ^ x_1) x) univ {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 \u2264 1}\n[PROOFSTEP]\nexact fun a _ => \u27e8hxi.pow a, fun \u03c6 => by simp [hx \u03c6]\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nx : K\nhxi : IsIntegral \u2124 x\nhx : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 = 1\na : \u2115\nx\u271d : a \u2208 univ\n\u03c6 : K \u2192+* A\n\u22a2 \u2016\u2191\u03c6 ((fun x x_1 => x ^ x_1) x a)\u2016 \u2264 1\n[PROOFSTEP]\nsimp [hx \u03c6]\n[GOAL]\ncase intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nx : K\nhxi : IsIntegral \u2124 x\nhx : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 = 1\na b : \u2115\nhabne : a \u2260 b\nh : (fun x x_1 => x ^ x_1) x a = (fun x x_1 => x ^ x_1) x b\n\u22a2 \u2203 n x_1, x ^ n = 1\n[PROOFSTEP]\nwlog hlt : b < a\n[GOAL]\ncase intro.intro.intro.intro.intro.inr\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nx : K\nhxi : IsIntegral \u2124 x\nhx : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 = 1\na b : \u2115\nhabne : a \u2260 b\nh : (fun x x_1 => x ^ x_1) x a = (fun x x_1 => x ^ x_1) x b\nthis :\n \u2200 (K : Type u_1) [inst : Field K] [inst_1 : NumberField K] (A : Type u_2) [inst_2 : NormedField A]\n [inst_3 : IsAlgClosed A] [inst_4 : NormedAlgebra \u211a A] {x : K},\n IsIntegral \u2124 x \u2192\n (\u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 = 1) \u2192\n \u2200 (a b : \u2115), a \u2260 b \u2192 (fun x x_1 => x ^ x_1) x a = (fun x x_1 => x ^ x_1) x b \u2192 b < a \u2192 \u2203 n x_1, x ^ n = 1\nhlt : \u00acb < a\n\u22a2 \u2203 n x_1, x ^ n = 1\n[PROOFSTEP]\nexact this K A hxi hx b a habne.symm h.symm (habne.lt_or_lt.resolve_right hlt)\n[GOAL]\nK\u271d : Type u_1\ninst\u271d\u2076 : Field K\u271d\nA\u271d : Type u_2\ninst\u271d\u2075 : NormedField A\u271d\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nx : K\nhxi : IsIntegral \u2124 x\nhx : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 = 1\na b : \u2115\nhabne : a \u2260 b\nh : (fun x x_1 => x ^ x_1) x a = (fun x x_1 => x ^ x_1) x b\nhlt : b < a\n\u22a2 \u2203 n x_1, x ^ n = 1\n[PROOFSTEP]\nrefine \u27e8a - b, tsub_pos_of_lt hlt, ?_\u27e9\n[GOAL]\nK\u271d : Type u_1\ninst\u271d\u2076 : Field K\u271d\nA\u271d : Type u_2\ninst\u271d\u2075 : NormedField A\u271d\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nx : K\nhxi : IsIntegral \u2124 x\nhx : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 = 1\na b : \u2115\nhabne : a \u2260 b\nh : (fun x x_1 => x ^ x_1) x a = (fun x x_1 => x ^ x_1) x b\nhlt : b < a\n\u22a2 x ^ (a - b) = 1\n[PROOFSTEP]\ndsimp at h \n[GOAL]\nK\u271d : Type u_1\ninst\u271d\u2076 : Field K\u271d\nA\u271d : Type u_2\ninst\u271d\u2075 : NormedField A\u271d\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nx : K\nhxi : IsIntegral \u2124 x\nhx : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 = 1\na b : \u2115\nhabne : a \u2260 b\nh : x ^ a = x ^ b\nhlt : b < a\n\u22a2 x ^ (a - b) = 1\n[PROOFSTEP]\nrw [\u2190 Nat.sub_add_cancel hlt.le, pow_add, mul_left_eq_self\u2080] at h \n[GOAL]\nK\u271d : Type u_1\ninst\u271d\u2076 : Field K\u271d\nA\u271d : Type u_2\ninst\u271d\u2075 : NormedField A\u271d\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nx : K\nhxi : IsIntegral \u2124 x\nhx : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 = 1\na b : \u2115\nhabne : a \u2260 b\nh : x ^ (a - b) = 1 \u2228 x ^ b = 0\nhlt : b < a\n\u22a2 x ^ (a - b) = 1\n[PROOFSTEP]\nrefine h.resolve_right fun hp => ?_\n[GOAL]\nK\u271d : Type u_1\ninst\u271d\u2076 : Field K\u271d\nA\u271d : Type u_2\ninst\u271d\u2075 : NormedField A\u271d\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nx : K\nhxi : IsIntegral \u2124 x\nhx : \u2200 (\u03c6 : K \u2192+* A), \u2016\u2191\u03c6 x\u2016 = 1\na b : \u2115\nhabne : a \u2260 b\nh : x ^ (a - b) = 1 \u2228 x ^ b = 0\nhlt : b < a\nhp : x ^ b = 0\n\u22a2 False\n[PROOFSTEP]\nspecialize hx (IsAlgClosed.lift (NumberField.isAlgebraic K)).toRingHom\n[GOAL]\nK\u271d : Type u_1\ninst\u271d\u2076 : Field K\u271d\nA\u271d : Type u_2\ninst\u271d\u2075 : NormedField A\u271d\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nx : K\nhxi : IsIntegral \u2124 x\na b : \u2115\nhabne : a \u2260 b\nh : x ^ (a - b) = 1 \u2228 x ^ b = 0\nhlt : b < a\nhp : x ^ b = 0\nhx : \u2016\u2191\u2191(IsAlgClosed.lift (_ : Algebra.IsAlgebraic \u211a K)) x\u2016 = 1\n\u22a2 False\n[PROOFSTEP]\nrw [pow_eq_zero hp, map_zero, norm_zero] at hx \n[GOAL]\nK\u271d : Type u_1\ninst\u271d\u2076 : Field K\u271d\nA\u271d : Type u_2\ninst\u271d\u2075 : NormedField A\u271d\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : NumberField K\nA : Type u_2\ninst\u271d\u00b2 : NormedField A\ninst\u271d\u00b9 : IsAlgClosed A\ninst\u271d : NormedAlgebra \u211a A\nx : K\nhxi : IsIntegral \u2124 x\na b : \u2115\nhabne : a \u2260 b\nh : x ^ (a - b) = 1 \u2228 x ^ b = 0\nhlt : b < a\nhp : x ^ b = 0\nhx : 0 = 1\n\u22a2 False\n[PROOFSTEP]\nnorm_num at hx \n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\n\u22a2 place (conjugate \u03c6) = place \u03c6\n[PROOFSTEP]\next\n[GOAL]\ncase a\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nx\u271d : K\n\u22a2 \u2191(place (conjugate \u03c6)) x\u271d = \u2191(place \u03c6) x\u271d\n[PROOFSTEP]\nsimp only [place_apply, norm_eq_abs, abs_conj, conjugate_coe_eq]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : IsReal \u03c6\n\u22a2 (fun x => (\u2191\u03c6 x).re) 1 = 1\n[PROOFSTEP]\nsimp only [map_one, one_re]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : IsReal \u03c6\n\u22a2 \u2200 (x y : K),\n OneHom.toFun { toFun := fun x => (\u2191\u03c6 x).re, map_one' := (_ : (\u2191\u03c6 1).re = 1) } (x * y) =\n OneHom.toFun { toFun := fun x => (\u2191\u03c6 x).re, map_one' := (_ : (\u2191\u03c6 1).re = 1) } x *\n OneHom.toFun { toFun := fun x => (\u2191\u03c6 x).re, map_one' := (_ : (\u2191\u03c6 1).re = 1) } y\n[PROOFSTEP]\nsimp only [Complex.conj_eq_iff_im.mp (RingHom.congr_fun h\u03c6 _), map_mul, mul_re, mul_zero, tsub_zero, eq_self_iff_true,\n forall_const]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : IsReal \u03c6\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := fun x => (\u2191\u03c6 x).re, map_one' := (_ : (\u2191\u03c6 1).re = 1) },\n map_mul' := (_ : \u2200 (a a_1 : K), (\u2191\u03c6 (a * a_1)).re = (\u2191\u03c6 a).re * (\u2191\u03c6 a_1).re) })\n 0 =\n 0\n[PROOFSTEP]\nsimp only [map_zero, zero_re]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : IsReal \u03c6\n\u22a2 \u2200 (x y : K),\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := fun x => (\u2191\u03c6 x).re, map_one' := (_ : (\u2191\u03c6 1).re = 1) },\n map_mul' := (_ : \u2200 (a a_1 : K), (\u2191\u03c6 (a * a_1)).re = (\u2191\u03c6 a).re * (\u2191\u03c6 a_1).re) })\n (x + y) =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := fun x => (\u2191\u03c6 x).re, map_one' := (_ : (\u2191\u03c6 1).re = 1) },\n map_mul' := (_ : \u2200 (a a_1 : K), (\u2191\u03c6 (a * a_1)).re = (\u2191\u03c6 a).re * (\u2191\u03c6 a_1).re) })\n x +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := fun x => (\u2191\u03c6 x).re, map_one' := (_ : (\u2191\u03c6 1).re = 1) },\n map_mul' := (_ : \u2200 (a a_1 : K), (\u2191\u03c6 (a * a_1)).re = (\u2191\u03c6 a).re * (\u2191\u03c6 a_1).re) })\n y\n[PROOFSTEP]\nsimp only [map_add, add_re, eq_self_iff_true, forall_const]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : IsReal \u03c6\nx : K\n\u22a2 \u2191(\u2191(embedding h\u03c6) x) = \u2191\u03c6 x\n[PROOFSTEP]\next\n[GOAL]\ncase a\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : IsReal \u03c6\nx : K\n\u22a2 (\u2191(\u2191(embedding h\u03c6) x)).re = (\u2191\u03c6 x).re\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : IsReal \u03c6\nx : K\n\u22a2 (\u2191(\u2191(embedding h\u03c6) x)).im = (\u2191\u03c6 x).im\n[PROOFSTEP]\nrw [ofReal_im, eq_comm, \u2190 Complex.conj_eq_iff_im]\n[GOAL]\ncase a\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : IsReal \u03c6\nx : K\n\u22a2 \u2191(starRingEnd \u2102) (\u2191\u03c6 x) = \u2191\u03c6 x\n[PROOFSTEP]\nexact RingHom.congr_fun h\u03c6 x\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\n\u22a2 mk (ComplexEmbedding.conjugate \u03c6) = mk \u03c6\n[PROOFSTEP]\nrefine FunLike.ext _ _ (fun x => ?_)\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nx : K\n\u22a2 \u2191(mk (ComplexEmbedding.conjugate \u03c6)) x = \u2191(mk \u03c6) x\n[PROOFSTEP]\nrw [apply, apply, ComplexEmbedding.conjugate_coe_eq, Complex.abs_conj]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nw : InfinitePlace K\nx : K\n\u22a2 \u2016\u2191(embedding w) x\u2016 = \u2191w x\n[PROOFSTEP]\nnth_rewrite 2 [\u2190 mk_embedding w]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nw : InfinitePlace K\nx : K\n\u22a2 \u2016\u2191(embedding w) x\u2016 = \u2191(mk (embedding w)) x\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nx : K\nr : \u211d\n\u22a2 (\u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 = r) \u2192 \u2200 (w : InfinitePlace K), \u2191w x = r\n[PROOFSTEP]\nrintro h\u03c6 \u27e8w, \u27e8\u03c6, rfl\u27e9\u27e9\n[GOAL]\ncase mk.intro\nK : Type u_1\ninst\u271d : Field K\nx : K\nr : \u211d\nh\u03c6 : \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 = r\n\u03c6 : K \u2192+* \u2102\n\u22a2 \u2191{ val := place \u03c6, property := (_ : \u2203 \u03c6_1, place \u03c6_1 = place \u03c6) } x = r\n[PROOFSTEP]\nexact h\u03c6 \u03c6\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nx : K\nr : \u211d\n\u22a2 (\u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r) \u2192 \u2200 (w : InfinitePlace K), \u2191w x \u2264 r\n[PROOFSTEP]\nrintro h\u03c6 \u27e8w, \u27e8\u03c6, rfl\u27e9\u27e9\n[GOAL]\ncase mk.intro\nK : Type u_1\ninst\u271d : Field K\nx : K\nr : \u211d\nh\u03c6 : \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u2191\u03c6 x\u2016 \u2264 r\n\u03c6 : K \u2192+* \u2102\n\u22a2 \u2191{ val := place \u03c6, property := (_ : \u2203 \u03c6_1, place \u03c6_1 = place \u03c6) } x \u2264 r\n[PROOFSTEP]\nexact h\u03c6 \u03c6\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\n\u22a2 mk \u03c6 = mk \u03c8 \u2194 \u03c6 = \u03c8 \u2228 ComplexEmbedding.conjugate \u03c6 = \u03c8\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\n\u22a2 mk \u03c6 = mk \u03c8 \u2192 \u03c6 = \u03c8 \u2228 ComplexEmbedding.conjugate \u03c6 = \u03c8\n[PROOFSTEP]\nintro h\u2080\n[GOAL]\ncase mp\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\n\u22a2 \u03c6 = \u03c8 \u2228 ComplexEmbedding.conjugate \u03c6 = \u03c8\n[PROOFSTEP]\nobtain \u27e8j, hi\u03c6\u27e9 := (\u03c6.injective).hasLeftInverse\n[GOAL]\ncase mp.intro\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u22a2 \u03c6 = \u03c8 \u2228 ComplexEmbedding.conjugate \u03c6 = \u03c8\n[PROOFSTEP]\nlet \u03b9 := RingEquiv.ofLeftInverse hi\u03c6\n[GOAL]\ncase mp.intro\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\n\u22a2 \u03c6 = \u03c8 \u2228 ComplexEmbedding.conjugate \u03c6 = \u03c8\n[PROOFSTEP]\nhave hlip : LipschitzWith 1 (RingHom.comp \u03c8 \u03b9.symm.toRingHom) :=\n by\n change LipschitzWith 1 (\u03c8 \u2218 \u03b9.symm)\n apply LipschitzWith.of_dist_le_mul\n intro x y\n rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply, \u2190 map_sub, \u2190 map_sub]\n apply le_of_eq\n suffices \u2016\u03c6 (\u03b9.symm (x - y))\u2016 = \u2016\u03c8 (\u03b9.symm (x - y))\u2016\n by\n rw [\u2190 this, \u2190 RingEquiv.ofLeftInverse_apply hi\u03c6 _, RingEquiv.apply_symm_apply \u03b9 _]\n rfl\n exact congrFun (congrArg (\u2191) h\u2080) _\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\n\u22a2 LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\n[PROOFSTEP]\nchange LipschitzWith 1 (\u03c8 \u2218 \u03b9.symm)\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\n\u22a2 LipschitzWith 1 (\u2191\u03c8 \u2218 \u2191(RingEquiv.symm \u03b9))\n[PROOFSTEP]\napply LipschitzWith.of_dist_le_mul\n[GOAL]\ncase a\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\n\u22a2 \u2200 (x y : { x // x \u2208 RingHom.range \u03c6 }),\n dist ((\u2191\u03c8 \u2218 \u2191(RingEquiv.symm \u03b9)) x) ((\u2191\u03c8 \u2218 \u2191(RingEquiv.symm \u03b9)) y) \u2264 \u21911 * dist x y\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase a\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nx y : { x // x \u2208 RingHom.range \u03c6 }\n\u22a2 dist ((\u2191\u03c8 \u2218 \u2191(RingEquiv.symm \u03b9)) x) ((\u2191\u03c8 \u2218 \u2191(RingEquiv.symm \u03b9)) y) \u2264 \u21911 * dist x y\n[PROOFSTEP]\nrw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply, \u2190 map_sub, \u2190 map_sub]\n[GOAL]\ncase a\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nx y : { x // x \u2208 RingHom.range \u03c6 }\n\u22a2 \u2016\u2191\u03c8 (\u2191(RingEquiv.symm \u03b9) (x - y))\u2016 \u2264 dist x y\n[PROOFSTEP]\napply le_of_eq\n[GOAL]\ncase a.a\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nx y : { x // x \u2208 RingHom.range \u03c6 }\n\u22a2 \u2016\u2191\u03c8 (\u2191(RingEquiv.symm \u03b9) (x - y))\u2016 = dist x y\n[PROOFSTEP]\nsuffices \u2016\u03c6 (\u03b9.symm (x - y))\u2016 = \u2016\u03c8 (\u03b9.symm (x - y))\u2016\n by\n rw [\u2190 this, \u2190 RingEquiv.ofLeftInverse_apply hi\u03c6 _, RingEquiv.apply_symm_apply \u03b9 _]\n rfl\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nx y : { x // x \u2208 RingHom.range \u03c6 }\nthis : \u2016\u2191\u03c6 (\u2191(RingEquiv.symm \u03b9) (x - y))\u2016 = \u2016\u2191\u03c8 (\u2191(RingEquiv.symm \u03b9) (x - y))\u2016\n\u22a2 \u2016\u2191\u03c8 (\u2191(RingEquiv.symm \u03b9) (x - y))\u2016 = dist x y\n[PROOFSTEP]\nrw [\u2190 this, \u2190 RingEquiv.ofLeftInverse_apply hi\u03c6 _, RingEquiv.apply_symm_apply \u03b9 _]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nx y : { x // x \u2208 RingHom.range \u03c6 }\nthis : \u2016\u2191\u03c6 (\u2191(RingEquiv.symm \u03b9) (x - y))\u2016 = \u2016\u2191\u03c8 (\u2191(RingEquiv.symm \u03b9) (x - y))\u2016\n\u22a2 \u2016\u2191(x - y)\u2016 = dist x y\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.a\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nx y : { x // x \u2208 RingHom.range \u03c6 }\n\u22a2 \u2016\u2191\u03c6 (\u2191(RingEquiv.symm \u03b9) (x - y))\u2016 = \u2016\u2191\u03c8 (\u2191(RingEquiv.symm \u03b9) (x - y))\u2016\n[PROOFSTEP]\nexact congrFun (congrArg (\u2191) h\u2080) _\n[GOAL]\ncase mp.intro\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\n\u22a2 \u03c6 = \u03c8 \u2228 ComplexEmbedding.conjugate \u03c6 = \u03c8\n[PROOFSTEP]\ncases Complex.uniformContinuous_ringHom_eq_id_or_conj \u03c6.fieldRange hlip.uniformContinuous with\n| inl h =>\n left; ext1 x\n conv_rhs => rw [\u2190 hi\u03c6 x]\n exact (congrFun h (\u03b9 x)).symm\n| inr h =>\n right; ext1 x\n conv_rhs => rw [\u2190 hi\u03c6 x]\n exact (congrFun h (\u03b9 x)).symm\n[GOAL]\ncase mp.intro\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nx\u271d :\n (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun = \u2191(Subfield.subtype (RingHom.fieldRange \u03c6)) \u2228\n (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun =\n \u2191(starRingEnd \u2102) \u2218 \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\n\u22a2 \u03c6 = \u03c8 \u2228 ComplexEmbedding.conjugate \u03c6 = \u03c8\n[PROOFSTEP]\ncases Complex.uniformContinuous_ringHom_eq_id_or_conj \u03c6.fieldRange hlip.uniformContinuous with\n| inl h =>\n left; ext1 x\n conv_rhs => rw [\u2190 hi\u03c6 x]\n exact (congrFun h (\u03b9 x)).symm\n| inr h =>\n right; ext1 x\n conv_rhs => rw [\u2190 hi\u03c6 x]\n exact (congrFun h (\u03b9 x)).symm\n[GOAL]\ncase mp.intro.inl\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh : (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun = \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\n\u22a2 \u03c6 = \u03c8 \u2228 ComplexEmbedding.conjugate \u03c6 = \u03c8\n[PROOFSTEP]\n\n| inl h =>\n left; ext1 x\n conv_rhs => rw [\u2190 hi\u03c6 x]\n exact (congrFun h (\u03b9 x)).symm\n[GOAL]\ncase mp.intro.inl\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh : (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun = \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\n\u22a2 \u03c6 = \u03c8 \u2228 ComplexEmbedding.conjugate \u03c6 = \u03c8\n[PROOFSTEP]\nleft\n[GOAL]\ncase mp.intro.inl.h\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh : (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun = \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\n\u22a2 \u03c6 = \u03c8\n[PROOFSTEP]\next1 x\n[GOAL]\ncase mp.intro.inl.h.a\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh : (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun = \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\nx : K\n\u22a2 \u2191\u03c6 x = \u2191\u03c8 x\n[PROOFSTEP]\nconv_rhs => rw [\u2190 hi\u03c6 x]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh : (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun = \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\nx : K\n| \u2191\u03c8 x\n[PROOFSTEP]\nrw [\u2190 hi\u03c6 x]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh : (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun = \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\nx : K\n| \u2191\u03c8 x\n[PROOFSTEP]\nrw [\u2190 hi\u03c6 x]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh : (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun = \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\nx : K\n| \u2191\u03c8 x\n[PROOFSTEP]\nrw [\u2190 hi\u03c6 x]\n[GOAL]\ncase mp.intro.inl.h.a\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh : (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun = \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\nx : K\n\u22a2 \u2191\u03c6 x = \u2191\u03c8 (j (\u2191\u03c6 x))\n[PROOFSTEP]\nexact (congrFun h (\u03b9 x)).symm\n[GOAL]\ncase mp.intro.inr\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh :\n (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun =\n \u2191(starRingEnd \u2102) \u2218 \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\n\u22a2 \u03c6 = \u03c8 \u2228 ComplexEmbedding.conjugate \u03c6 = \u03c8\n[PROOFSTEP]\n\n| inr h =>\n right; ext1 x\n conv_rhs => rw [\u2190 hi\u03c6 x]\n exact (congrFun h (\u03b9 x)).symm\n[GOAL]\ncase mp.intro.inr\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh :\n (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun =\n \u2191(starRingEnd \u2102) \u2218 \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\n\u22a2 \u03c6 = \u03c8 \u2228 ComplexEmbedding.conjugate \u03c6 = \u03c8\n[PROOFSTEP]\nright\n[GOAL]\ncase mp.intro.inr.h\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh :\n (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun =\n \u2191(starRingEnd \u2102) \u2218 \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\n\u22a2 ComplexEmbedding.conjugate \u03c6 = \u03c8\n[PROOFSTEP]\next1 x\n[GOAL]\ncase mp.intro.inr.h.a\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh :\n (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun =\n \u2191(starRingEnd \u2102) \u2218 \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\nx : K\n\u22a2 \u2191(ComplexEmbedding.conjugate \u03c6) x = \u2191\u03c8 x\n[PROOFSTEP]\nconv_rhs => rw [\u2190 hi\u03c6 x]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh :\n (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun =\n \u2191(starRingEnd \u2102) \u2218 \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\nx : K\n| \u2191\u03c8 x\n[PROOFSTEP]\nrw [\u2190 hi\u03c6 x]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh :\n (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun =\n \u2191(starRingEnd \u2102) \u2218 \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\nx : K\n| \u2191\u03c8 x\n[PROOFSTEP]\nrw [\u2190 hi\u03c6 x]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh :\n (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun =\n \u2191(starRingEnd \u2102) \u2218 \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\nx : K\n| \u2191\u03c8 x\n[PROOFSTEP]\nrw [\u2190 hi\u03c6 x]\n[GOAL]\ncase mp.intro.inr.h.a\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh\u2080 : mk \u03c6 = mk \u03c8\nj : \u2102 \u2192 K\nhi\u03c6 : Function.LeftInverse j \u2191\u03c6\n\u03b9 : K \u2243+* { x // x \u2208 RingHom.range \u03c6 } := RingEquiv.ofLeftInverse hi\u03c6\nhlip : LipschitzWith 1 \u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))\nh :\n (\u2191\u2191(RingHom.comp \u03c8 (RingEquiv.toRingHom (RingEquiv.symm \u03b9)))).toFun =\n \u2191(starRingEnd \u2102) \u2218 \u2191(Subfield.subtype (RingHom.fieldRange \u03c6))\nx : K\n\u22a2 \u2191(ComplexEmbedding.conjugate \u03c6) x = \u2191\u03c8 (j (\u2191\u03c6 x))\n[PROOFSTEP]\nexact (congrFun h (\u03b9 x)).symm\n[GOAL]\ncase mpr\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\n\u22a2 \u03c6 = \u03c8 \u2228 ComplexEmbedding.conjugate \u03c6 = \u03c8 \u2192 mk \u03c6 = mk \u03c8\n[PROOFSTEP]\nrintro (\u27e8h\u27e9 | \u27e8h\u27e9)\n[GOAL]\ncase mpr.inl\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh : \u03c6 = \u03c8\n\u22a2 mk \u03c6 = mk \u03c8\n[PROOFSTEP]\nexact congr_arg mk h\n[GOAL]\ncase mpr.inr\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh : ComplexEmbedding.conjugate \u03c6 = \u03c8\n\u22a2 mk \u03c6 = mk \u03c8\n[PROOFSTEP]\nrw [\u2190 mk_conjugate_eq]\n[GOAL]\ncase mpr.inr\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : K \u2192+* \u2102\nh : ComplexEmbedding.conjugate \u03c6 = \u03c8\n\u22a2 mk (ComplexEmbedding.conjugate \u03c6) = mk \u03c8\n[PROOFSTEP]\nexact congr_arg mk h\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\n\u22a2 embedding (mk \u03c6) = \u03c6 \u2228 embedding (mk \u03c6) = ComplexEmbedding.conjugate \u03c6\n[PROOFSTEP]\nrw [@eq_comm _ _ \u03c6, @eq_comm _ _ (ComplexEmbedding.conjugate \u03c6), \u2190 mk_eq_iff, mk_embedding]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh : ComplexEmbedding.IsReal \u03c6\n\u22a2 embedding (mk \u03c6) = \u03c6\n[PROOFSTEP]\nhave := embedding_mk_eq \u03c6\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh : ComplexEmbedding.IsReal \u03c6\nthis : embedding (mk \u03c6) = \u03c6 \u2228 embedding (mk \u03c6) = ComplexEmbedding.conjugate \u03c6\n\u22a2 embedding (mk \u03c6) = \u03c6\n[PROOFSTEP]\nrwa [ComplexEmbedding.isReal_iff.mp h, or_self] at this \n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nw : InfinitePlace K\n\u22a2 IsReal w \u2194 ComplexEmbedding.IsReal (embedding w)\n[PROOFSTEP]\nrefine \u27e8?_, fun h => \u27e8embedding w, h, mk_embedding w\u27e9\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nw : InfinitePlace K\n\u22a2 IsReal w \u2192 ComplexEmbedding.IsReal (embedding w)\n[PROOFSTEP]\nrintro \u27e8\u03c6, \u27e8h\u03c6, rfl\u27e9\u27e9\n[GOAL]\ncase intro.intro\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : ComplexEmbedding.IsReal \u03c6\n\u22a2 ComplexEmbedding.IsReal (embedding (mk \u03c6))\n[PROOFSTEP]\nrwa [embedding_mk_eq_of_isReal h\u03c6]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nw : InfinitePlace K\n\u22a2 IsComplex w \u2194 \u00acComplexEmbedding.IsReal (embedding w)\n[PROOFSTEP]\nrefine \u27e8?_, fun h => \u27e8embedding w, h, mk_embedding w\u27e9\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nw : InfinitePlace K\n\u22a2 IsComplex w \u2192 \u00acComplexEmbedding.IsReal (embedding w)\n[PROOFSTEP]\nrintro \u27e8\u03c6, \u27e8h\u03c6, rfl\u27e9\u27e9\n[GOAL]\ncase intro.intro\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : \u00acComplexEmbedding.IsReal \u03c6\n\u22a2 \u00acComplexEmbedding.IsReal (embedding (mk \u03c6))\n[PROOFSTEP]\ncontrapose! h\u03c6\n[GOAL]\ncase intro.intro\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : ComplexEmbedding.IsReal (embedding (mk \u03c6))\n\u22a2 ComplexEmbedding.IsReal \u03c6\n[PROOFSTEP]\ncases mk_eq_iff.mp (mk_embedding (mk \u03c6)) with\n| inl h => rwa [h] at h\u03c6 \n| inr h => rwa [\u2190 ComplexEmbedding.isReal_conjugate_iff, h] at h\u03c6 \n[GOAL]\ncase intro.intro\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : ComplexEmbedding.IsReal (embedding (mk \u03c6))\nx\u271d : embedding (mk \u03c6) = \u03c6 \u2228 ComplexEmbedding.conjugate (embedding (mk \u03c6)) = \u03c6\n\u22a2 ComplexEmbedding.IsReal \u03c6\n[PROOFSTEP]\ncases mk_eq_iff.mp (mk_embedding (mk \u03c6)) with\n| inl h => rwa [h] at h\u03c6 \n| inr h => rwa [\u2190 ComplexEmbedding.isReal_conjugate_iff, h] at h\u03c6 \n[GOAL]\ncase intro.intro.inl\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : ComplexEmbedding.IsReal (embedding (mk \u03c6))\nh : embedding (mk \u03c6) = \u03c6\n\u22a2 ComplexEmbedding.IsReal \u03c6\n[PROOFSTEP]\n\n| inl h => rwa [h] at h\u03c6 \n[GOAL]\ncase intro.intro.inl\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : ComplexEmbedding.IsReal (embedding (mk \u03c6))\nh : embedding (mk \u03c6) = \u03c6\n\u22a2 ComplexEmbedding.IsReal \u03c6\n[PROOFSTEP]\nrwa [h] at h\u03c6 \n[GOAL]\ncase intro.intro.inr\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : ComplexEmbedding.IsReal (embedding (mk \u03c6))\nh : ComplexEmbedding.conjugate (embedding (mk \u03c6)) = \u03c6\n\u22a2 ComplexEmbedding.IsReal \u03c6\n[PROOFSTEP]\n\n| inr h => rwa [\u2190 ComplexEmbedding.isReal_conjugate_iff, h] at h\u03c6 \n[GOAL]\ncase intro.intro.inr\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : ComplexEmbedding.IsReal (embedding (mk \u03c6))\nh : ComplexEmbedding.conjugate (embedding (mk \u03c6)) = \u03c6\n\u22a2 ComplexEmbedding.IsReal \u03c6\n[PROOFSTEP]\nrwa [\u2190 ComplexEmbedding.isReal_conjugate_iff, h] at h\u03c6 \n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nw : InfinitePlace K\n\u22a2 \u00acIsReal w \u2194 IsComplex w\n[PROOFSTEP]\nrw [isComplex_iff, isReal_iff]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nw : InfinitePlace K\n\u22a2 IsReal w \u2228 IsComplex w\n[PROOFSTEP]\nrw [\u2190 not_isReal_iff_isComplex]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nw : InfinitePlace K\n\u22a2 IsReal w \u2228 \u00acIsReal w\n[PROOFSTEP]\nexact em _\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh : IsReal (mk \u03c6)\n\u22a2 ComplexEmbedding.IsReal \u03c6\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh : \u00acComplexEmbedding.IsReal \u03c6\n\u22a2 \u00acIsReal (mk \u03c6)\n[PROOFSTEP]\nrw [not_isReal_iff_isComplex]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh : \u00acComplexEmbedding.IsReal \u03c6\n\u22a2 IsComplex (mk \u03c6)\n[PROOFSTEP]\nexact \u27e8\u03c6, h, rfl\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh : IsComplex (mk \u03c6)\n\u22a2 \u00acComplexEmbedding.IsReal \u03c6\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh : ComplexEmbedding.IsReal \u03c6\n\u22a2 \u00acIsComplex (mk \u03c6)\n[PROOFSTEP]\nrw [\u2190 not_isReal_iff_isComplex.not, not_not]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : K \u2192+* \u2102\nh : ComplexEmbedding.IsReal \u03c6\n\u22a2 IsReal (mk \u03c6)\n[PROOFSTEP]\nexact \u27e8\u03c6, h, rfl\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\n\u22a2 Finset.card (Finset.filter (fun \u03c6 => mk \u03c6 = w) Finset.univ) = mult w\n[PROOFSTEP]\nconv_lhs =>\n congr; congr; ext\n rw [\u2190 mk_embedding w, mk_eq_iff, ComplexEmbedding.conjugate, star_involutive.eq_iff]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\n| Finset.card (Finset.filter (fun \u03c6 => mk \u03c6 = w) Finset.univ)\n[PROOFSTEP]\n congr; congr; ext\n rw [\u2190 mk_embedding w, mk_eq_iff, ComplexEmbedding.conjugate, star_involutive.eq_iff]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\n| Finset.card (Finset.filter (fun \u03c6 => mk \u03c6 = w) Finset.univ)\n[PROOFSTEP]\n congr; congr; ext\n rw [\u2190 mk_embedding w, mk_eq_iff, ComplexEmbedding.conjugate, star_involutive.eq_iff]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\n| Finset.card (Finset.filter (fun \u03c6 => mk \u03c6 = w) Finset.univ)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase s\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\n| Finset.filter (fun \u03c6 => mk \u03c6 = w) Finset.univ\n[PROOFSTEP]\ncongr\n[GOAL]\ncase s.p\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\n| fun \u03c6 => mk \u03c6 = w\ncase s.s\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\n| Finset.univ\nK : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : NumberField K w : InfinitePlace K \u22a2 DecidablePred ?m.1138470\n[PROOFSTEP]\next\n[GOAL]\ncase s.p.h\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\nx\u271d : K \u2192+* \u2102\n| mk x\u271d = w\ncase s.s\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\n| Finset.univ\nK : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : NumberField K w : InfinitePlace K \u22a2 DecidablePred ?m.1138470\n[PROOFSTEP]\nrw [\u2190 mk_embedding w, mk_eq_iff, ComplexEmbedding.conjugate, star_involutive.eq_iff]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\n\u22a2 Finset.card (Finset.filter (fun x => x = embedding w \u2228 x = star (embedding w)) Finset.univ) = mult w\n[PROOFSTEP]\nsimp_rw [Finset.filter_or, Finset.filter_eq' _ (embedding w),\n Finset.filter_eq' _ (ComplexEmbedding.conjugate (embedding w)), Finset.mem_univ, ite_true, mult]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\n\u22a2 Finset.card ({embedding w} \u222a {ComplexEmbedding.conjugate (embedding w)}) = if IsReal w then 1 else 2\n[PROOFSTEP]\nsplit_ifs with hw\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\nhw : IsReal w\n\u22a2 Finset.card ({embedding w} \u222a {ComplexEmbedding.conjugate (embedding w)}) = 1\n[PROOFSTEP]\nrw [ComplexEmbedding.isReal_iff.mp (isReal_iff.mp hw), Finset.union_idempotent, Finset.card_singleton]\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\nhw : \u00acIsReal w\n\u22a2 Finset.card ({embedding w} \u222a {ComplexEmbedding.conjugate (embedding w)}) = 2\n[PROOFSTEP]\nrefine Finset.card_doubleton ?_\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\nhw : \u00acIsReal w\n\u22a2 embedding w \u2260 ComplexEmbedding.conjugate (embedding w)\n[PROOFSTEP]\nrwa [Ne.def, eq_comm, \u2190 ComplexEmbedding.isReal_iff, \u2190 isReal_iff]\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\n\u22a2 { \u03c6 // ComplexEmbedding.IsReal \u03c6 } \u2243 { w // IsReal w }\n[PROOFSTEP]\nrefine (Equiv.ofBijective (fun \u03c6 => \u27e8mk \u03c6, ?_\u27e9) \u27e8fun \u03c6 \u03c8 h => ?_, fun w => ?_\u27e9)\n[GOAL]\ncase refine_1\nK : Type u_1\ninst\u271d : Field K\n\u03c6 : { \u03c6 // ComplexEmbedding.IsReal \u03c6 }\n\u22a2 IsReal (mk \u2191\u03c6)\n[PROOFSTEP]\nexact \u27e8\u03c6, \u03c6.prop, rfl\u27e9\n[GOAL]\ncase refine_2\nK : Type u_1\ninst\u271d : Field K\n\u03c6 \u03c8 : { \u03c6 // ComplexEmbedding.IsReal \u03c6 }\nh :\n (fun \u03c6 => { val := mk \u2191\u03c6, property := (_ : \u2203 \u03c6_1, ComplexEmbedding.IsReal \u03c6_1 \u2227 mk \u03c6_1 = mk \u2191\u03c6) }) \u03c6 =\n (fun \u03c6 => { val := mk \u2191\u03c6, property := (_ : \u2203 \u03c6_1, ComplexEmbedding.IsReal \u03c6_1 \u2227 mk \u03c6_1 = mk \u2191\u03c6) }) \u03c8\n\u22a2 \u03c6 = \u03c8\n[PROOFSTEP]\nrwa [Subtype.mk.injEq, mk_eq_iff, ComplexEmbedding.isReal_iff.mp \u03c6.prop, or_self, \u2190 Subtype.ext_iff] at h \n[GOAL]\ncase refine_3\nK : Type u_1\ninst\u271d : Field K\nw : { w // IsReal w }\n\u22a2 \u2203 a, (fun \u03c6 => { val := mk \u2191\u03c6, property := (_ : \u2203 \u03c6_1, ComplexEmbedding.IsReal \u03c6_1 \u2227 mk \u03c6_1 = mk \u2191\u03c6) }) a = w\n[PROOFSTEP]\nexact \u27e8\u27e8embedding w, isReal_iff.mp w.prop\u27e9, by simp\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\nw : { w // IsReal w }\n\u22a2 (fun \u03c6 => { val := mk \u2191\u03c6, property := (_ : \u2203 \u03c6_1, ComplexEmbedding.IsReal \u03c6_1 \u2227 mk \u03c6_1 = mk \u2191\u03c6) })\n { val := embedding \u2191w, property := (_ : ComplexEmbedding.IsReal (embedding \u2191w)) } =\n w\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\n\u22a2 \u220f w : InfinitePlace K, \u2191w x ^ mult w = \u2191|\u2191(Algebra.norm \u211a) x|\n[PROOFSTEP]\nconvert (congr_arg Complex.abs (@Algebra.norm_eq_prod_embeddings \u211a _ _ _ _ \u2102 _ _ _ _ _ x)).symm\n[GOAL]\ncase h.e'_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\n\u22a2 \u220f w : InfinitePlace K, \u2191w x ^ mult w = \u2191Complex.abs (\u220f \u03c3 : K \u2192\u2090[\u211a] \u2102, \u2191\u03c3 x)\n[PROOFSTEP]\nrw [map_prod, \u2190\n Equiv.prod_comp' RingHom.equivRatAlgHom (fun f => Complex.abs (f x)) (fun \u03c6 => Complex.abs (\u03c6 x)) fun _ => by\n simp [RingHom.equivRatAlgHom_apply]; rfl]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\nx\u271d : K \u2192+* \u2102\n\u22a2 (fun f => \u2191Complex.abs (\u2191f x)) x\u271d = (fun \u03c6 => \u2191Complex.abs (\u2191\u03c6 x)) (\u2191RingHom.equivRatAlgHom x\u271d)\n[PROOFSTEP]\nsimp [RingHom.equivRatAlgHom_apply]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\nx\u271d : K \u2192+* \u2102\n\u22a2 \u2191Complex.abs (\u2191x\u271d x) = \u2191Complex.abs (\u2191(RingHom.toRatAlgHom x\u271d) x)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\n\u22a2 \u220f w : InfinitePlace K, \u2191w x ^ mult w = \u220f i : K \u2192+* \u2102, \u2191Complex.abs (\u2191i x)\n[PROOFSTEP]\nrw [\u2190 Finset.prod_fiberwise Finset.univ (fun \u03c6 => mk \u03c6) (fun \u03c6 => Complex.abs (\u03c6 x))]\n[GOAL]\ncase h.e'_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\n\u22a2 \u220f w : InfinitePlace K, \u2191w x ^ mult w =\n \u220f b : InfinitePlace K, \u220f a in Finset.filter (fun a => mk a = b) Finset.univ, \u2191Complex.abs (\u2191a x)\n[PROOFSTEP]\nhave : \u2200 w : InfinitePlace K, \u2200 \u03c6 \u2208 Finset.filter (fun a \u21a6 mk a = w) Finset.univ, Complex.abs (\u03c6 x) = w x :=\n by\n intro _ _ h\u03c6\n rw [\u2190 (Finset.mem_filter.mp h\u03c6).2]\n rfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\n\u22a2 \u2200 (w : InfinitePlace K) (\u03c6 : K \u2192+* \u2102), \u03c6 \u2208 Finset.filter (fun a => mk a = w) Finset.univ \u2192 \u2191Complex.abs (\u2191\u03c6 x) = \u2191w x\n[PROOFSTEP]\nintro _ _ h\u03c6\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\nw\u271d : InfinitePlace K\n\u03c6\u271d : K \u2192+* \u2102\nh\u03c6 : \u03c6\u271d \u2208 Finset.filter (fun a => mk a = w\u271d) Finset.univ\n\u22a2 \u2191Complex.abs (\u2191\u03c6\u271d x) = \u2191w\u271d x\n[PROOFSTEP]\nrw [\u2190 (Finset.mem_filter.mp h\u03c6).2]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\nw\u271d : InfinitePlace K\n\u03c6\u271d : K \u2192+* \u2102\nh\u03c6 : \u03c6\u271d \u2208 Finset.filter (fun a => mk a = w\u271d) Finset.univ\n\u22a2 \u2191Complex.abs (\u2191\u03c6\u271d x) = \u2191(mk \u03c6\u271d) x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\nthis :\n \u2200 (w : InfinitePlace K) (\u03c6 : K \u2192+* \u2102), \u03c6 \u2208 Finset.filter (fun a => mk a = w) Finset.univ \u2192 \u2191Complex.abs (\u2191\u03c6 x) = \u2191w x\n\u22a2 \u220f w : InfinitePlace K, \u2191w x ^ mult w =\n \u220f b : InfinitePlace K, \u220f a in Finset.filter (fun a => mk a = b) Finset.univ, \u2191Complex.abs (\u2191a x)\n[PROOFSTEP]\nsimp_rw [Finset.prod_congr rfl (this _), Finset.prod_const, card_filter_mk_eq]\n[GOAL]\ncase h.e'_3\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nx : K\n\u22a2 \u2191|\u2191(Algebra.norm \u211a) x| = \u2191Complex.abs (\u2191(algebraMap \u211a \u2102) (\u2191(Algebra.norm \u211a) x))\n[PROOFSTEP]\nrw [eq_ratCast, Rat.cast_abs, \u2190 Complex.abs_ofReal, Complex.ofReal_rat_cast]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u22a2 card { \u03c6 // \u00acComplexEmbedding.IsReal \u03c6 } = 2 * card { w // IsComplex w }\n[PROOFSTEP]\nsuffices\n \u2200 w : { w : InfinitePlace K // IsComplex w },\n (Finset.univ.filter fun \u03c6 : { \u03c6 // \u00acComplexEmbedding.IsReal \u03c6 } => mkComplex \u03c6 = w).card = 2\n by\n rw [Fintype.card, Finset.card_eq_sum_ones, \u2190 Finset.sum_fiberwise _ (fun \u03c6 => mkComplex \u03c6)]\n simp_rw [Finset.sum_const, this, smul_eq_mul, mul_one, Fintype.card, Finset.card_eq_sum_ones, Finset.mul_sum]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nthis : \u2200 (w : { w // IsComplex w }), Finset.card (Finset.filter (fun \u03c6 => mkComplex \u03c6 = w) Finset.univ) = 2\n\u22a2 card { \u03c6 // \u00acComplexEmbedding.IsReal \u03c6 } = 2 * card { w // IsComplex w }\n[PROOFSTEP]\nrw [Fintype.card, Finset.card_eq_sum_ones, \u2190 Finset.sum_fiberwise _ (fun \u03c6 => mkComplex \u03c6)]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nthis : \u2200 (w : { w // IsComplex w }), Finset.card (Finset.filter (fun \u03c6 => mkComplex \u03c6 = w) Finset.univ) = 2\n\u22a2 \u2211 b : { w // IsComplex w }, \u2211 a in Finset.filter (fun a => mkComplex a = b) Finset.univ, 1 =\n 2 * card { w // IsComplex w }\n[PROOFSTEP]\nsimp_rw [Finset.sum_const, this, smul_eq_mul, mul_one, Fintype.card, Finset.card_eq_sum_ones, Finset.mul_sum]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u22a2 \u2200 (w : { w // IsComplex w }), Finset.card (Finset.filter (fun \u03c6 => mkComplex \u03c6 = w) Finset.univ) = 2\n[PROOFSTEP]\nrintro \u27e8w, hw\u27e9\n[GOAL]\ncase mk\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\nhw : IsComplex w\n\u22a2 Finset.card (Finset.filter (fun \u03c6 => mkComplex \u03c6 = { val := w, property := hw }) Finset.univ) = 2\n[PROOFSTEP]\nconvert card_filter_mk_eq w\n[GOAL]\ncase h.e'_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\nhw : IsComplex w\n\u22a2 Finset.card (Finset.filter (fun \u03c6 => mkComplex \u03c6 = { val := w, property := hw }) Finset.univ) =\n Finset.card (Finset.filter (fun \u03c6 => mk \u03c6 = w) Finset.univ)\n[PROOFSTEP]\nrw [\u2190 Fintype.card_subtype, \u2190 Fintype.card_subtype]\n[GOAL]\ncase h.e'_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\nhw : IsComplex w\n\u22a2 card { x // mkComplex x = { val := w, property := hw } } = card { x // mk x = w }\n[PROOFSTEP]\nrefine Fintype.card_congr (Equiv.ofBijective ?_ \u27e8fun _ _ h => ?_, fun \u27e8\u03c6, h\u03c6\u27e9 => ?_\u27e9)\n[GOAL]\ncase h.e'_2.refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\nhw : IsComplex w\n\u22a2 { x // mkComplex x = { val := w, property := hw } } \u2192 { x // mk x = w }\n[PROOFSTEP]\nexact fun \u27e8\u03c6, h\u03c6\u27e9 => \u27e8\u03c6.val, by rwa [Subtype.ext_iff] at h\u03c6 \u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\nhw : IsComplex w\nx\u271d : { x // mkComplex x = { val := w, property := hw } }\n\u03c6 : { \u03c6 // \u00acComplexEmbedding.IsReal \u03c6 }\nh\u03c6 : mkComplex \u03c6 = { val := w, property := hw }\n\u22a2 mk \u2191\u03c6 = w\n[PROOFSTEP]\nrwa [Subtype.ext_iff] at h\u03c6 \n[GOAL]\ncase h.e'_2.refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\nhw : IsComplex w\nx\u271d\u00b9 x\u271d : { x // mkComplex x = { val := w, property := hw } }\nh :\n (match x\u271d\u00b9 with\n | { val := \u03c6, property := h\u03c6 } => { val := \u2191\u03c6, property := (_ : \u2191(mkComplex \u03c6) = \u2191{ val := w, property := hw }) }) =\n match x\u271d with\n | { val := \u03c6, property := h\u03c6 } => { val := \u2191\u03c6, property := (_ : \u2191(mkComplex \u03c6) = \u2191{ val := w, property := hw }) }\n\u22a2 x\u271d\u00b9 = x\u271d\n[PROOFSTEP]\nrwa [Subtype.mk_eq_mk, \u2190 Subtype.ext_iff, \u2190 Subtype.ext_iff] at h \n[GOAL]\ncase h.e'_2.refine_3\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\nhw : IsComplex w\nx\u271d : { x // mk x = w }\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : mk \u03c6 = w\n\u22a2 \u2203 a,\n (match a with\n | { val := \u03c6, property := h\u03c6 } =>\n { val := \u2191\u03c6, property := (_ : \u2191(mkComplex \u03c6) = \u2191{ val := w, property := hw }) }) =\n { val := \u03c6, property := h\u03c6 }\n[PROOFSTEP]\nrefine \u27e8\u27e8\u27e8\u03c6, not_isReal_of_mk_isComplex (h\u03c6.symm \u25b8 hw)\u27e9, ?_\u27e9, rfl\u27e9\n[GOAL]\ncase h.e'_2.refine_3\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\nhw : IsComplex w\nx\u271d : { x // mk x = w }\n\u03c6 : K \u2192+* \u2102\nh\u03c6 : mk \u03c6 = w\n\u22a2 mkComplex { val := \u03c6, property := (_ : \u00acComplexEmbedding.IsReal \u03c6) } = { val := w, property := hw }\n[PROOFSTEP]\nrwa [Subtype.ext_iff, mkComplex_coe]\n[GOAL]\ncase h.e'_3\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw : InfinitePlace K\nhw : IsComplex w\n\u22a2 2 = mult w\n[PROOFSTEP]\nsimp_rw [mult, not_isReal_iff_isComplex.mpr hw]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u22a2 card { w // IsReal w } + 2 * card { w // IsComplex w } = finrank \u211a K\n[PROOFSTEP]\nrw [\u2190 card_real_embeddings, \u2190 card_complex_embeddings]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u22a2 card { \u03c6 // ComplexEmbedding.IsReal \u03c6 } + card { \u03c6 // \u00acComplexEmbedding.IsReal \u03c6 } = finrank \u211a K\n[PROOFSTEP]\nrw [Fintype.card_subtype_compl, \u2190 Embeddings.card K \u2102, Nat.add_sub_of_le]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\n\u22a2 card { \u03c6 // ComplexEmbedding.IsReal \u03c6 } \u2264 card (K \u2192+* \u2102)\n[PROOFSTEP]\nexact Fintype.card_subtype_le _\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.NumberField.Embeddings", "llama_tokens": 26440, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494421679929, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.40134981033755524}} {"text": "[GOAL]\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : AddCommMonoid N'\ninst\u271d\u00b9 : Module R N'\ninst\u271d : TopologicalSpace N'\nn : \u2115\nf\u271d g\u271d f g : M[\u039b^\u03b9]\u2192L[R]N\nh : toAlternatingMap f = toAlternatingMap g\n\u22a2 \u2191f = \u2191g\n[PROOFSTEP]\nconvert FunLike.ext'_iff.1 h\n[GOAL]\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : AddCommMonoid N'\ninst\u271d\u00b9 : Module R N'\ninst\u271d : TopologicalSpace N'\nn : \u2115\nf\u271d g : M[\u039b^\u03b9]\u2192L[R]N\ne : M \u2243L[R] M'\nf : M[\u039b^\u03b9]\u2192L[R]N\n\u22a2 (fun f => compContinuousLinearMap f \u2191e) ((fun f => compContinuousLinearMap f \u2191(ContinuousLinearEquiv.symm e)) f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : AddCommMonoid N'\ninst\u271d\u00b9 : Module R N'\ninst\u271d : TopologicalSpace N'\nn : \u2115\nf\u271d g : M[\u039b^\u03b9]\u2192L[R]N\ne : M \u2243L[R] M'\nf : M[\u039b^\u03b9]\u2192L[R]N\nx\u271d : \u03b9 \u2192 M\n\u22a2 \u2191((fun f => compContinuousLinearMap f \u2191e) ((fun f => compContinuousLinearMap f \u2191(ContinuousLinearEquiv.symm e)) f))\n x\u271d =\n \u2191f x\u271d\n[PROOFSTEP]\nsimp [(\u00b7 \u2218 \u00b7)]\n[GOAL]\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : AddCommMonoid N'\ninst\u271d\u00b9 : Module R N'\ninst\u271d : TopologicalSpace N'\nn : \u2115\nf\u271d g : M[\u039b^\u03b9]\u2192L[R]N\ne : M \u2243L[R] M'\nf : M'[\u039b^\u03b9]\u2192L[R]N\n\u22a2 (fun f => compContinuousLinearMap f \u2191(ContinuousLinearEquiv.symm e)) ((fun f => compContinuousLinearMap f \u2191e) f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : AddCommMonoid N'\ninst\u271d\u00b9 : Module R N'\ninst\u271d : TopologicalSpace N'\nn : \u2115\nf\u271d g : M[\u039b^\u03b9]\u2192L[R]N\ne : M \u2243L[R] M'\nf : M'[\u039b^\u03b9]\u2192L[R]N\nx\u271d : \u03b9 \u2192 M'\n\u22a2 \u2191((fun f => compContinuousLinearMap f \u2191(ContinuousLinearEquiv.symm e)) ((fun f => compContinuousLinearMap f \u2191e) f))\n x\u271d =\n \u2191f x\u271d\n[PROOFSTEP]\nsimp [(\u00b7 \u2218 \u00b7)]\n[GOAL]\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : AddCommMonoid N'\ninst\u271d\u00b9 : Module R N'\ninst\u271d : TopologicalSpace N'\nn : \u2115\nf\u271d g : M[\u039b^\u03b9]\u2192L[R]N\ne : N \u2243L[R] N'\nf : M[\u039b^\u03b9]\u2192L[R]N\n\u22a2 ContinuousLinearMap.compContinuousAlternatingMap (\u2191(ContinuousLinearEquiv.symm e))\n (ContinuousLinearMap.compContinuousAlternatingMap (\u2191e) f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : AddCommMonoid N'\ninst\u271d\u00b9 : Module R N'\ninst\u271d : TopologicalSpace N'\nn : \u2115\nf\u271d g : M[\u039b^\u03b9]\u2192L[R]N\ne : N \u2243L[R] N'\nf : M[\u039b^\u03b9]\u2192L[R]N\nx\u271d : \u03b9 \u2192 M\n\u22a2 \u2191(ContinuousLinearMap.compContinuousAlternatingMap (\u2191(ContinuousLinearEquiv.symm e))\n (ContinuousLinearMap.compContinuousAlternatingMap (\u2191e) f))\n x\u271d =\n \u2191f x\u271d\n[PROOFSTEP]\nsimp [(\u00b7 \u2218 \u00b7)]\n[GOAL]\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : AddCommMonoid N'\ninst\u271d\u00b9 : Module R N'\ninst\u271d : TopologicalSpace N'\nn : \u2115\nf\u271d g : M[\u039b^\u03b9]\u2192L[R]N\ne : N \u2243L[R] N'\nf : M[\u039b^\u03b9]\u2192L[R]N'\n\u22a2 ContinuousLinearMap.compContinuousAlternatingMap (\u2191e)\n (ContinuousLinearMap.compContinuousAlternatingMap (\u2191(ContinuousLinearEquiv.symm e)) f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : AddCommMonoid N'\ninst\u271d\u00b9 : Module R N'\ninst\u271d : TopologicalSpace N'\nn : \u2115\nf\u271d g : M[\u039b^\u03b9]\u2192L[R]N\ne : N \u2243L[R] N'\nf : M[\u039b^\u03b9]\u2192L[R]N'\nx\u271d : \u03b9 \u2192 M\n\u22a2 \u2191(ContinuousLinearMap.compContinuousAlternatingMap (\u2191e)\n (ContinuousLinearMap.compContinuousAlternatingMap (\u2191(ContinuousLinearEquiv.symm e)) f))\n x\u271d =\n \u2191f x\u271d\n[PROOFSTEP]\nsimp [(\u00b7 \u2218 \u00b7)]\n[GOAL]\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN\u271d : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M'\ninst\u271d\u00b9\u2070 : Module R M'\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : AddCommMonoid N\u271d\ninst\u271d\u2077 : Module R N\u271d\ninst\u271d\u2076 : TopologicalSpace N\u271d\ninst\u271d\u2075 : AddCommMonoid N'\ninst\u271d\u2074 : Module R N'\ninst\u271d\u00b3 : TopologicalSpace N'\nn : \u2115\nf\u271d g : M[\u039b^\u03b9]\u2192L[R]N\u271d\n\u03b9' : Type u_7\nN : \u03b9' \u2192 Type u_8\ninst\u271d\u00b2 : (i : \u03b9') \u2192 AddCommMonoid (N i)\ninst\u271d\u00b9 : (i : \u03b9') \u2192 TopologicalSpace (N i)\ninst\u271d : (i : \u03b9') \u2192 Module R (N i)\nf : (i : \u03b9') \u2192 M[\u039b^\u03b9]\u2192L[R]N i\n\u22a2 (fun f i => ContinuousLinearMap.compContinuousAlternatingMap (ContinuousLinearMap.proj i) f) (pi f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase h.H\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN\u271d : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M'\ninst\u271d\u00b9\u2070 : Module R M'\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : AddCommMonoid N\u271d\ninst\u271d\u2077 : Module R N\u271d\ninst\u271d\u2076 : TopologicalSpace N\u271d\ninst\u271d\u2075 : AddCommMonoid N'\ninst\u271d\u2074 : Module R N'\ninst\u271d\u00b3 : TopologicalSpace N'\nn : \u2115\nf\u271d g : M[\u039b^\u03b9]\u2192L[R]N\u271d\n\u03b9' : Type u_7\nN : \u03b9' \u2192 Type u_8\ninst\u271d\u00b2 : (i : \u03b9') \u2192 AddCommMonoid (N i)\ninst\u271d\u00b9 : (i : \u03b9') \u2192 TopologicalSpace (N i)\ninst\u271d : (i : \u03b9') \u2192 Module R (N i)\nf : (i : \u03b9') \u2192 M[\u039b^\u03b9]\u2192L[R]N i\nx\u271d\u00b9 : \u03b9'\nx\u271d : \u03b9 \u2192 M\n\u22a2 \u2191((fun f i => ContinuousLinearMap.compContinuousAlternatingMap (ContinuousLinearMap.proj i) f) (pi f) x\u271d\u00b9) x\u271d =\n \u2191(f x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN\u271d : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M'\ninst\u271d\u00b9\u2070 : Module R M'\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : AddCommMonoid N\u271d\ninst\u271d\u2077 : Module R N\u271d\ninst\u271d\u2076 : TopologicalSpace N\u271d\ninst\u271d\u2075 : AddCommMonoid N'\ninst\u271d\u2074 : Module R N'\ninst\u271d\u00b3 : TopologicalSpace N'\nn : \u2115\nf\u271d g : M[\u039b^\u03b9]\u2192L[R]N\u271d\n\u03b9' : Type u_7\nN : \u03b9' \u2192 Type u_8\ninst\u271d\u00b2 : (i : \u03b9') \u2192 AddCommMonoid (N i)\ninst\u271d\u00b9 : (i : \u03b9') \u2192 TopologicalSpace (N i)\ninst\u271d : (i : \u03b9') \u2192 Module R (N i)\nf : M[\u039b^\u03b9]\u2192L[R]((i : \u03b9') \u2192 N i)\n\u22a2 pi ((fun f i => ContinuousLinearMap.compContinuousAlternatingMap (ContinuousLinearMap.proj i) f) f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase H.h\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN\u271d : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : AddCommMonoid M'\ninst\u271d\u00b9\u2070 : Module R M'\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : AddCommMonoid N\u271d\ninst\u271d\u2077 : Module R N\u271d\ninst\u271d\u2076 : TopologicalSpace N\u271d\ninst\u271d\u2075 : AddCommMonoid N'\ninst\u271d\u2074 : Module R N'\ninst\u271d\u00b3 : TopologicalSpace N'\nn : \u2115\nf\u271d g : M[\u039b^\u03b9]\u2192L[R]N\u271d\n\u03b9' : Type u_7\nN : \u03b9' \u2192 Type u_8\ninst\u271d\u00b2 : (i : \u03b9') \u2192 AddCommMonoid (N i)\ninst\u271d\u00b9 : (i : \u03b9') \u2192 TopologicalSpace (N i)\ninst\u271d : (i : \u03b9') \u2192 Module R (N i)\nf : M[\u039b^\u03b9]\u2192L[R]((i : \u03b9') \u2192 N i)\nx\u271d\u00b9 : \u03b9 \u2192 M\nx\u271d : \u03b9'\n\u22a2 \u2191(pi ((fun f i => ContinuousLinearMap.compContinuousAlternatingMap (ContinuousLinearMap.proj i) f) f)) x\u271d\u00b9 x\u271d =\n \u2191f x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M'\ninst\u271d\u00b9\u00b9 : Module R M'\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : AddCommMonoid N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : TopologicalSpace N\ninst\u271d\u2076 : ContinuousAdd N\ninst\u271d\u2075 : ContinuousConstSMul R N\ninst\u271d\u2074 : AddCommMonoid N'\ninst\u271d\u00b3 : Module R N'\ninst\u271d\u00b2 : TopologicalSpace N'\ninst\u271d\u00b9 : ContinuousAdd N'\ninst\u271d : ContinuousConstSMul R N'\nf : M \u2192L[R] M'\ng g' : M'[\u039b^\u03b9]\u2192L[R]N\n\u22a2 (fun g => compContinuousLinearMap g f) (g + g') =\n (fun g => compContinuousLinearMap g f) g + (fun g => compContinuousLinearMap g f) g'\n[PROOFSTEP]\next\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M'\ninst\u271d\u00b9\u00b9 : Module R M'\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : AddCommMonoid N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : TopologicalSpace N\ninst\u271d\u2076 : ContinuousAdd N\ninst\u271d\u2075 : ContinuousConstSMul R N\ninst\u271d\u2074 : AddCommMonoid N'\ninst\u271d\u00b3 : Module R N'\ninst\u271d\u00b2 : TopologicalSpace N'\ninst\u271d\u00b9 : ContinuousAdd N'\ninst\u271d : ContinuousConstSMul R N'\nf : M \u2192L[R] M'\ng g' : M'[\u039b^\u03b9]\u2192L[R]N\nx\u271d : \u03b9 \u2192 M\n\u22a2 \u2191((fun g => compContinuousLinearMap g f) (g + g')) x\u271d =\n \u2191((fun g => compContinuousLinearMap g f) g + (fun g => compContinuousLinearMap g f) g') x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M'\ninst\u271d\u00b9\u00b9 : Module R M'\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : AddCommMonoid N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : TopologicalSpace N\ninst\u271d\u2076 : ContinuousAdd N\ninst\u271d\u2075 : ContinuousConstSMul R N\ninst\u271d\u2074 : AddCommMonoid N'\ninst\u271d\u00b3 : Module R N'\ninst\u271d\u00b2 : TopologicalSpace N'\ninst\u271d\u00b9 : ContinuousAdd N'\ninst\u271d : ContinuousConstSMul R N'\nf : M \u2192L[R] M'\nc : R\ng : M'[\u039b^\u03b9]\u2192L[R]N\n\u22a2 AddHom.toFun\n { toFun := fun g => compContinuousLinearMap g f,\n map_add' :=\n (_ :\n \u2200 (g g' : M'[\u039b^\u03b9]\u2192L[R]N),\n (fun g => compContinuousLinearMap g f) (g + g') =\n (fun g => compContinuousLinearMap g f) g + (fun g => compContinuousLinearMap g f) g') }\n (c \u2022 g) =\n \u2191(RingHom.id R) c \u2022\n AddHom.toFun\n { toFun := fun g => compContinuousLinearMap g f,\n map_add' :=\n (_ :\n \u2200 (g g' : M'[\u039b^\u03b9]\u2192L[R]N),\n (fun g => compContinuousLinearMap g f) (g + g') =\n (fun g => compContinuousLinearMap g f) g + (fun g => compContinuousLinearMap g f) g') }\n g\n[PROOFSTEP]\next\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M'\ninst\u271d\u00b9\u00b9 : Module R M'\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : AddCommMonoid N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : TopologicalSpace N\ninst\u271d\u2076 : ContinuousAdd N\ninst\u271d\u2075 : ContinuousConstSMul R N\ninst\u271d\u2074 : AddCommMonoid N'\ninst\u271d\u00b3 : Module R N'\ninst\u271d\u00b2 : TopologicalSpace N'\ninst\u271d\u00b9 : ContinuousAdd N'\ninst\u271d : ContinuousConstSMul R N'\nf : M \u2192L[R] M'\nc : R\ng : M'[\u039b^\u03b9]\u2192L[R]N\nx\u271d : \u03b9 \u2192 M\n\u22a2 \u2191(AddHom.toFun\n { toFun := fun g => compContinuousLinearMap g f,\n map_add' :=\n (_ :\n \u2200 (g g' : M'[\u039b^\u03b9]\u2192L[R]N),\n (fun g => compContinuousLinearMap g f) (g + g') =\n (fun g => compContinuousLinearMap g f) g + (fun g => compContinuousLinearMap g f) g') }\n (c \u2022 g))\n x\u271d =\n \u2191(\u2191(RingHom.id R) c \u2022\n AddHom.toFun\n { toFun := fun g => compContinuousLinearMap g f,\n map_add' :=\n (_ :\n \u2200 (g g' : M'[\u039b^\u03b9]\u2192L[R]N),\n (fun g => compContinuousLinearMap g f) (g + g') =\n (fun g => compContinuousLinearMap g f) g + (fun g => compContinuousLinearMap g f) g') }\n g)\n x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M'\ninst\u271d\u00b9\u00b9 : Module R M'\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : AddCommMonoid N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : TopologicalSpace N\ninst\u271d\u2076 : ContinuousAdd N\ninst\u271d\u2075 : ContinuousConstSMul R N\ninst\u271d\u2074 : AddCommMonoid N'\ninst\u271d\u00b3 : Module R N'\ninst\u271d\u00b2 : TopologicalSpace N'\ninst\u271d\u00b9 : ContinuousAdd N'\ninst\u271d : ContinuousConstSMul R N'\nf : N \u2192L[R] N'\ng\u2081 g\u2082 : M[\u039b^\u03b9]\u2192L[R]N\n\u22a2 ContinuousLinearMap.compContinuousAlternatingMap f (g\u2081 + g\u2082) =\n ContinuousLinearMap.compContinuousAlternatingMap f g\u2081 + ContinuousLinearMap.compContinuousAlternatingMap f g\u2082\n[PROOFSTEP]\next1\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M'\ninst\u271d\u00b9\u00b9 : Module R M'\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : AddCommMonoid N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : TopologicalSpace N\ninst\u271d\u2076 : ContinuousAdd N\ninst\u271d\u2075 : ContinuousConstSMul R N\ninst\u271d\u2074 : AddCommMonoid N'\ninst\u271d\u00b3 : Module R N'\ninst\u271d\u00b2 : TopologicalSpace N'\ninst\u271d\u00b9 : ContinuousAdd N'\ninst\u271d : ContinuousConstSMul R N'\nf : N \u2192L[R] N'\ng\u2081 g\u2082 : M[\u039b^\u03b9]\u2192L[R]N\nx\u271d : \u03b9 \u2192 M\n\u22a2 \u2191(ContinuousLinearMap.compContinuousAlternatingMap f (g\u2081 + g\u2082)) x\u271d =\n \u2191(ContinuousLinearMap.compContinuousAlternatingMap f g\u2081 + ContinuousLinearMap.compContinuousAlternatingMap f g\u2082) x\u271d\n[PROOFSTEP]\napply f.map_add\n[GOAL]\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M'\ninst\u271d\u00b9\u00b9 : Module R M'\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : AddCommMonoid N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : TopologicalSpace N\ninst\u271d\u2076 : ContinuousAdd N\ninst\u271d\u2075 : ContinuousConstSMul R N\ninst\u271d\u2074 : AddCommMonoid N'\ninst\u271d\u00b3 : Module R N'\ninst\u271d\u00b2 : TopologicalSpace N'\ninst\u271d\u00b9 : ContinuousAdd N'\ninst\u271d : ContinuousConstSMul R N'\nc : R\nf : N \u2192L[R] N'\ng : M[\u039b^\u03b9]\u2192L[R]N\n\u22a2 ContinuousLinearMap.compContinuousAlternatingMap f (c \u2022 g) = c \u2022 ContinuousLinearMap.compContinuousAlternatingMap f g\n[PROOFSTEP]\next1\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M'\ninst\u271d\u00b9\u00b9 : Module R M'\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : AddCommMonoid N\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : TopologicalSpace N\ninst\u271d\u2076 : ContinuousAdd N\ninst\u271d\u2075 : ContinuousConstSMul R N\ninst\u271d\u2074 : AddCommMonoid N'\ninst\u271d\u00b3 : Module R N'\ninst\u271d\u00b2 : TopologicalSpace N'\ninst\u271d\u00b9 : ContinuousAdd N'\ninst\u271d : ContinuousConstSMul R N'\nc : R\nf : N \u2192L[R] N'\ng : M[\u039b^\u03b9]\u2192L[R]N\nx\u271d : \u03b9 \u2192 M\n\u22a2 \u2191(ContinuousLinearMap.compContinuousAlternatingMap f (c \u2022 g)) x\u271d =\n \u2191(c \u2022 ContinuousLinearMap.compContinuousAlternatingMap f g) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : TopologicalAddGroup N\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf\u271d g f : ContinuousMultilinearMap R (fun x => M) N\nv : \u03b9 \u2192 M\ni j : \u03b9\nhv : v i = v j\nhne : i \u2260 j\n\u22a2 MultilinearMap.toFun (\u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f).toMultilinearMap v = 0\n[PROOFSTEP]\nsimpa [MultilinearMap.alternatization_apply] using f.1.alternatization.map_eq_zero_of_eq' v i j hv hne\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : TopologicalAddGroup N\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ContinuousMultilinearMap R (fun x => M) N\n\u22a2 (fun f =>\n { toContinuousMultilinearMap := \u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f,\n map_eq_zero_of_eq' :=\n (_ :\n \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9),\n v i = v j \u2192\n i \u2260 j \u2192\n MultilinearMap.toFun (\u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f).toMultilinearMap v =\n 0) })\n 0 =\n 0\n[PROOFSTEP]\next\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : TopologicalAddGroup N\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ContinuousMultilinearMap R (fun x => M) N\nx\u271d : \u03b9 \u2192 M\n\u22a2 \u2191((fun f =>\n { toContinuousMultilinearMap := \u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f,\n map_eq_zero_of_eq' :=\n (_ :\n \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9),\n v i = v j \u2192\n i \u2260 j \u2192\n MultilinearMap.toFun (\u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f).toMultilinearMap\n v =\n 0) })\n 0)\n x\u271d =\n \u21910 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : TopologicalAddGroup N\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf g x\u271d\u00b9 x\u271d : ContinuousMultilinearMap R (fun x => M) N\n\u22a2 ZeroHom.toFun\n {\n toFun := fun f =>\n { toContinuousMultilinearMap := \u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f,\n map_eq_zero_of_eq' :=\n (_ :\n \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9),\n v i = v j \u2192\n i \u2260 j \u2192\n MultilinearMap.toFun (\u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f).toMultilinearMap\n v =\n 0) },\n map_zero' :=\n (_ :\n (fun f =>\n { toContinuousMultilinearMap := \u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f,\n map_eq_zero_of_eq' :=\n (_ :\n \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9),\n v i = v j \u2192\n i \u2260 j \u2192\n MultilinearMap.toFun\n (\u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f).toMultilinearMap v =\n 0) })\n 0 =\n 0) }\n (x\u271d\u00b9 + x\u271d) =\n ZeroHom.toFun\n {\n toFun := fun f =>\n { toContinuousMultilinearMap := \u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f,\n map_eq_zero_of_eq' :=\n (_ :\n \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9),\n v i = v j \u2192\n i \u2260 j \u2192\n MultilinearMap.toFun (\u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f).toMultilinearMap\n v =\n 0) },\n map_zero' :=\n (_ :\n (fun f =>\n { toContinuousMultilinearMap := \u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f,\n map_eq_zero_of_eq' :=\n (_ :\n \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9),\n v i = v j \u2192\n i \u2260 j \u2192\n MultilinearMap.toFun\n (\u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f).toMultilinearMap v =\n 0) })\n 0 =\n 0) }\n x\u271d\u00b9 +\n ZeroHom.toFun\n {\n toFun := fun f =>\n { toContinuousMultilinearMap := \u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f,\n map_eq_zero_of_eq' :=\n (_ :\n \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9),\n v i = v j \u2192\n i \u2260 j \u2192\n MultilinearMap.toFun (\u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f).toMultilinearMap\n v =\n 0) },\n map_zero' :=\n (_ :\n (fun f =>\n { toContinuousMultilinearMap := \u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f,\n map_eq_zero_of_eq' :=\n (_ :\n \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9),\n v i = v j \u2192\n i \u2260 j \u2192\n MultilinearMap.toFun\n (\u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f).toMultilinearMap v =\n 0) })\n 0 =\n 0) }\n x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : TopologicalAddGroup N\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf g x\u271d\u00b2 x\u271d\u00b9 : ContinuousMultilinearMap R (fun x => M) N\nx\u271d : \u03b9 \u2192 M\n\u22a2 \u2191(ZeroHom.toFun\n {\n toFun := fun f =>\n { toContinuousMultilinearMap := \u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f,\n map_eq_zero_of_eq' :=\n (_ :\n \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9),\n v i = v j \u2192\n i \u2260 j \u2192\n MultilinearMap.toFun\n (\u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f).toMultilinearMap v =\n 0) },\n map_zero' :=\n (_ :\n (fun f =>\n { toContinuousMultilinearMap := \u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f,\n map_eq_zero_of_eq' :=\n (_ :\n \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9),\n v i = v j \u2192\n i \u2260 j \u2192\n MultilinearMap.toFun\n (\u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f).toMultilinearMap v =\n 0) })\n 0 =\n 0) }\n (x\u271d\u00b2 + x\u271d\u00b9))\n x\u271d =\n \u2191(ZeroHom.toFun\n {\n toFun := fun f =>\n { toContinuousMultilinearMap := \u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f,\n map_eq_zero_of_eq' :=\n (_ :\n \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9),\n v i = v j \u2192\n i \u2260 j \u2192\n MultilinearMap.toFun\n (\u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f).toMultilinearMap v =\n 0) },\n map_zero' :=\n (_ :\n (fun f =>\n { toContinuousMultilinearMap := \u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f,\n map_eq_zero_of_eq' :=\n (_ :\n \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9),\n v i = v j \u2192\n i \u2260 j \u2192\n MultilinearMap.toFun\n (\u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f).toMultilinearMap v =\n 0) })\n 0 =\n 0) }\n x\u271d\u00b2 +\n ZeroHom.toFun\n {\n toFun := fun f =>\n { toContinuousMultilinearMap := \u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f,\n map_eq_zero_of_eq' :=\n (_ :\n \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9),\n v i = v j \u2192\n i \u2260 j \u2192\n MultilinearMap.toFun\n (\u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f).toMultilinearMap v =\n 0) },\n map_zero' :=\n (_ :\n (fun f =>\n { toContinuousMultilinearMap := \u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f,\n map_eq_zero_of_eq' :=\n (_ :\n \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9),\n v i = v j \u2192\n i \u2260 j \u2192\n MultilinearMap.toFun\n (\u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 domDomCongr \u03c3 f).toMultilinearMap v =\n 0) })\n 0 =\n 0) }\n x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\nsimp [Finset.sum_add_distrib]\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : TopologicalAddGroup N\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ContinuousMultilinearMap R (fun x => M) N\nv : \u03b9 \u2192 M\n\u22a2 \u2191(\u2191alternatization f) v = \u2211 \u03c3 : Equiv.Perm \u03b9, \u2191Equiv.Perm.sign \u03c3 \u2022 \u2191f (v \u2218 \u2191\u03c3)\n[PROOFSTEP]\nsimp [alternatization, (\u00b7 \u2218 \u00b7)]\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : TopologicalAddGroup N\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ContinuousMultilinearMap R (fun x => M) N\n\u22a2 ContinuousAlternatingMap.toAlternatingMap (\u2191alternatization f) = \u2191MultilinearMap.alternatization f.toMultilinearMap\n[PROOFSTEP]\next v\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : TopologicalAddGroup N\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ContinuousMultilinearMap R (fun x => M) N\nv : \u03b9 \u2192 M\n\u22a2 \u2191(ContinuousAlternatingMap.toAlternatingMap (\u2191alternatization f)) v =\n \u2191(\u2191MultilinearMap.alternatization f.toMultilinearMap) v\n[PROOFSTEP]\nsimp [alternatization_apply_apply, MultilinearMap.alternatization_apply, (\u00b7 \u2218 \u00b7)]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Module.Alternating", "llama_tokens": 12389, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.40131797153104715}} {"text": "[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\np : \u2115\ninst\u271d : CharP R p\nS : Subsemiring R\nx : \u2115\nh : \u2191x = 0\n\u22a2 \u2191(Subsemiring.subtype S) \u2191x = 0\n[PROOFSTEP]\nrw [map_natCast, h]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Semiring R\np : \u2115\ninst\u271d : CharP R p\nS : Subsemiring R\nx : \u2115\nh : \u2191x = 0\n\u22a2 \u2191(Subsemiring.subtype S) \u2191x = 0\n[PROOFSTEP]\nrw [h, RingHom.map_zero]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\np : \u2115\ninst\u271d : CharP R p\nS : Subring R\nx : \u2115\nh : \u2191x = 0\n\u22a2 \u2191(Subring.subtype S) \u2191x = 0\n[PROOFSTEP]\nrw [map_natCast, h]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\np : \u2115\ninst\u271d : CharP R p\nS : Subring R\nx : \u2115\nh : \u2191x = 0\n\u22a2 \u2191(Subring.subtype S) \u2191x = 0\n[PROOFSTEP]\nrw [h, RingHom.map_zero]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.CharP.Subring", "llama_tokens": 366, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.6297746213017459, "lm_q1q2_score": 0.40118583603094876}} {"text": "[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns t s\u2081 s\u2082 t\u2081 t\u2082 u : Set E\nx y : E\n\u22a2 x \u2208 convexJoin \ud835\udd5c s t \u2194 \u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 x \u2208 segment \ud835\udd5c a b\n[PROOFSTEP]\nsimp [convexJoin]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d s\u2081 s\u2082 t\u2081 t\u2082 u : Set E\nx y : E\ns t : Set E\n\u22a2 \u22c3 (i\u2082 : E) (_ : i\u2082 \u2208 t) (i\u2081 : E) (_ : i\u2081 \u2208 s), segment \ud835\udd5c i\u2081 i\u2082 = convexJoin \ud835\udd5c t s\n[PROOFSTEP]\nsimp_rw [convexJoin, segment_symm]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns t\u271d s\u2081 s\u2082 t\u2081 t\u2082 u : Set E\nx y : E\nt : Set E\n\u22a2 convexJoin \ud835\udd5c \u2205 t = \u2205\n[PROOFSTEP]\nsimp [convexJoin]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t s\u2081 s\u2082 t\u2081 t\u2082 u : Set E\nx y : E\ns : Set E\n\u22a2 convexJoin \ud835\udd5c s \u2205 = \u2205\n[PROOFSTEP]\nsimp [convexJoin]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns t\u271d s\u2081 s\u2082 t\u2081 t\u2082 u : Set E\nx\u271d y : E\nt : Set E\nx : E\n\u22a2 convexJoin \ud835\udd5c {x} t = \u22c3 (y : E) (_ : y \u2208 t), segment \ud835\udd5c x y\n[PROOFSTEP]\nsimp [convexJoin]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t s\u2081 s\u2082 t\u2081 t\u2082 u : Set E\nx y\u271d : E\ns : Set E\ny : E\n\u22a2 convexJoin \ud835\udd5c s {y} = \u22c3 (x : E) (_ : x \u2208 s), segment \ud835\udd5c x y\n[PROOFSTEP]\nsimp [convexJoin]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns t s\u2081 s\u2082 t\u2081 t\u2082 u : Set E\nx\u271d y x : E\n\u22a2 convexJoin \ud835\udd5c {x} {y} = segment \ud835\udd5c x y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns t\u271d s\u2081\u271d s\u2082\u271d t\u2081 t\u2082 u : Set E\nx y : E\ns\u2081 s\u2082 t : Set E\n\u22a2 convexJoin \ud835\udd5c (s\u2081 \u222a s\u2082) t = convexJoin \ud835\udd5c s\u2081 t \u222a convexJoin \ud835\udd5c s\u2082 t\n[PROOFSTEP]\nsimp_rw [convexJoin, mem_union, iUnion_or, iUnion_union_distrib]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t s\u2081 s\u2082 t\u2081\u271d t\u2082\u271d u : Set E\nx y : E\ns t\u2081 t\u2082 : Set E\n\u22a2 convexJoin \ud835\udd5c s (t\u2081 \u222a t\u2082) = convexJoin \ud835\udd5c s t\u2081 \u222a convexJoin \ud835\udd5c s t\u2082\n[PROOFSTEP]\nsimp_rw [convexJoin_comm s, convexJoin_union_left]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d s\u2081 s\u2082 t\u2081 t\u2082 u : Set E\nx y : E\ns : \u03b9 \u2192 Set E\nt : Set E\n\u22a2 convexJoin \ud835\udd5c (\u22c3 (i : \u03b9), s i) t = \u22c3 (i : \u03b9), convexJoin \ud835\udd5c (s i) t\n[PROOFSTEP]\nsimp_rw [convexJoin, mem_iUnion, iUnion_exists]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d s\u2081 s\u2082 t\u2081 t\u2082 u : Set E\nx y : E\ns : \u03b9 \u2192 Set E\nt : Set E\n\u22a2 \u22c3 (x : E) (i : \u03b9) (_ : x \u2208 s i) (y : E) (_ : y \u2208 t), segment \ud835\udd5c x y =\n \u22c3 (i : \u03b9) (x : E) (_ : x \u2208 s i) (y : E) (_ : y \u2208 t), segment \ud835\udd5c x y\n[PROOFSTEP]\nexact iUnion_comm _\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d s\u2081 s\u2082 t\u2081 t\u2082 u : Set E\nx y : E\ns : Set E\nt : \u03b9 \u2192 Set E\n\u22a2 convexJoin \ud835\udd5c s (\u22c3 (i : \u03b9), t i) = \u22c3 (i : \u03b9), convexJoin \ud835\udd5c s (t i)\n[PROOFSTEP]\nsimp_rw [convexJoin_comm s, convexJoin_iUnion_left]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx y : E\ns t u : Set E\n\u22a2 convexJoin \ud835\udd5c (convexJoin \ud835\udd5c s t) u \u2286 convexJoin \ud835\udd5c s (convexJoin \ud835\udd5c t u)\n[PROOFSTEP]\nsimp_rw [subset_def, mem_convexJoin]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx y : E\ns t u : Set E\n\u22a2 \u2200 (x : E),\n (\u2203 a, (\u2203 a_1, a_1 \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2208 segment \ud835\udd5c a_1 b) \u2227 \u2203 b, b \u2208 u \u2227 x \u2208 segment \ud835\udd5c a b) \u2192\n \u2203 a, a \u2208 s \u2227 \u2203 b, (\u2203 a, a \u2208 t \u2227 \u2203 b_1, b_1 \u2208 u \u2227 b \u2208 segment \ud835\udd5c a b_1) \u2227 x \u2208 segment \ud835\udd5c a b\n[PROOFSTEP]\nrintro _ \u27e8z, \u27e8x, hx, y, hy, a\u2081, b\u2081, ha\u2081, hb\u2081, hab\u2081, rfl\u27e9, z, hz, a\u2082, b\u2082, ha\u2082, hb\u2082, hab\u2082, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx\u271d y\u271d : E\ns t u : Set E\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nz : E\nhz : z \u2208 u\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\n\u22a2 \u2203 a,\n a \u2208 s \u2227 \u2203 b, (\u2203 a, a \u2208 t \u2227 \u2203 b_1, b_1 \u2208 u \u2227 b \u2208 segment \ud835\udd5c a b_1) \u2227 a\u2082 \u2022 (a\u2081 \u2022 x + b\u2081 \u2022 y) + b\u2082 \u2022 z \u2208 segment \ud835\udd5c a b\n[PROOFSTEP]\nobtain rfl | hb\u2082 := hb\u2082.eq_or_lt\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inl\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx\u271d y\u271d : E\ns t u : Set E\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nz : E\nhz : z \u2208 u\na\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 0\nhab\u2082 : a\u2082 + 0 = 1\n\u22a2 \u2203 a, a \u2208 s \u2227 \u2203 b, (\u2203 a, a \u2208 t \u2227 \u2203 b_1, b_1 \u2208 u \u2227 b \u2208 segment \ud835\udd5c a b_1) \u2227 a\u2082 \u2022 (a\u2081 \u2022 x + b\u2081 \u2022 y) + 0 \u2022 z \u2208 segment \ud835\udd5c a b\n[PROOFSTEP]\nrefine' \u27e8x, hx, y, \u27e8y, hy, z, hz, left_mem_segment \ud835\udd5c _ _\u27e9, a\u2081, b\u2081, ha\u2081, hb\u2081, hab\u2081, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inl\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx\u271d y\u271d : E\ns t u : Set E\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nz : E\nhz : z \u2208 u\na\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 0\nhab\u2082 : a\u2082 + 0 = 1\n\u22a2 a\u2081 \u2022 x + b\u2081 \u2022 y = a\u2082 \u2022 (a\u2081 \u2022 x + b\u2081 \u2022 y) + 0 \u2022 z\n[PROOFSTEP]\nrw [add_zero] at hab\u2082 \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inl\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx\u271d y\u271d : E\ns t u : Set E\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nz : E\nhz : z \u2208 u\na\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 0\nhab\u2082 : a\u2082 = 1\n\u22a2 a\u2081 \u2022 x + b\u2081 \u2022 y = a\u2082 \u2022 (a\u2081 \u2022 x + b\u2081 \u2022 y) + 0 \u2022 z\n[PROOFSTEP]\nrw [hab\u2082, one_smul, zero_smul, add_zero]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx\u271d y\u271d : E\ns t u : Set E\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nz : E\nhz : z \u2208 u\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082\u271d : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\nhb\u2082 : 0 < b\u2082\n\u22a2 \u2203 a,\n a \u2208 s \u2227 \u2203 b, (\u2203 a, a \u2208 t \u2227 \u2203 b_1, b_1 \u2208 u \u2227 b \u2208 segment \ud835\udd5c a b_1) \u2227 a\u2082 \u2022 (a\u2081 \u2022 x + b\u2081 \u2022 y) + b\u2082 \u2022 z \u2208 segment \ud835\udd5c a b\n[PROOFSTEP]\nhave ha\u2082b\u2081 : 0 \u2264 a\u2082 * b\u2081 := mul_nonneg ha\u2082 hb\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx\u271d y\u271d : E\ns t u : Set E\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nz : E\nhz : z \u2208 u\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082\u271d : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\nhb\u2082 : 0 < b\u2082\nha\u2082b\u2081 : 0 \u2264 a\u2082 * b\u2081\n\u22a2 \u2203 a,\n a \u2208 s \u2227 \u2203 b, (\u2203 a, a \u2208 t \u2227 \u2203 b_1, b_1 \u2208 u \u2227 b \u2208 segment \ud835\udd5c a b_1) \u2227 a\u2082 \u2022 (a\u2081 \u2022 x + b\u2081 \u2022 y) + b\u2082 \u2022 z \u2208 segment \ud835\udd5c a b\n[PROOFSTEP]\nhave hab : 0 < a\u2082 * b\u2081 + b\u2082 := add_pos_of_nonneg_of_pos ha\u2082b\u2081 hb\u2082\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx\u271d y\u271d : E\ns t u : Set E\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nz : E\nhz : z \u2208 u\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082\u271d : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\nhb\u2082 : 0 < b\u2082\nha\u2082b\u2081 : 0 \u2264 a\u2082 * b\u2081\nhab : 0 < a\u2082 * b\u2081 + b\u2082\n\u22a2 \u2203 a,\n a \u2208 s \u2227 \u2203 b, (\u2203 a, a \u2208 t \u2227 \u2203 b_1, b_1 \u2208 u \u2227 b \u2208 segment \ud835\udd5c a b_1) \u2227 a\u2082 \u2022 (a\u2081 \u2022 x + b\u2081 \u2022 y) + b\u2082 \u2022 z \u2208 segment \ud835\udd5c a b\n[PROOFSTEP]\nrefine'\n \u27e8x, hx, (a\u2082 * b\u2081 / (a\u2082 * b\u2081 + b\u2082)) \u2022 y + (b\u2082 / (a\u2082 * b\u2081 + b\u2082)) \u2022 z, \u27e8y, hy, z, hz, _, _, _, _, _, rfl\u27e9, a\u2082 * a\u2081,\n a\u2082 * b\u2081 + b\u2082, mul_nonneg ha\u2082 ha\u2081, hab.le, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.refine'_1\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx\u271d y\u271d : E\ns t u : Set E\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nz : E\nhz : z \u2208 u\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082\u271d : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\nhb\u2082 : 0 < b\u2082\nha\u2082b\u2081 : 0 \u2264 a\u2082 * b\u2081\nhab : 0 < a\u2082 * b\u2081 + b\u2082\n\u22a2 0 \u2264 a\u2082 * b\u2081 / (a\u2082 * b\u2081 + b\u2082)\n[PROOFSTEP]\nexact div_nonneg ha\u2082b\u2081 hab.le\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.refine'_2\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx\u271d y\u271d : E\ns t u : Set E\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nz : E\nhz : z \u2208 u\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082\u271d : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\nhb\u2082 : 0 < b\u2082\nha\u2082b\u2081 : 0 \u2264 a\u2082 * b\u2081\nhab : 0 < a\u2082 * b\u2081 + b\u2082\n\u22a2 0 \u2264 b\u2082 / (a\u2082 * b\u2081 + b\u2082)\n[PROOFSTEP]\nexact div_nonneg hb\u2082.le hab.le\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.refine'_3\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx\u271d y\u271d : E\ns t u : Set E\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nz : E\nhz : z \u2208 u\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082\u271d : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\nhb\u2082 : 0 < b\u2082\nha\u2082b\u2081 : 0 \u2264 a\u2082 * b\u2081\nhab : 0 < a\u2082 * b\u2081 + b\u2082\n\u22a2 a\u2082 * b\u2081 / (a\u2082 * b\u2081 + b\u2082) + b\u2082 / (a\u2082 * b\u2081 + b\u2082) = 1\n[PROOFSTEP]\nrw [\u2190 add_div, div_self hab.ne']\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.refine'_4\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx\u271d y\u271d : E\ns t u : Set E\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nz : E\nhz : z \u2208 u\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082\u271d : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\nhb\u2082 : 0 < b\u2082\nha\u2082b\u2081 : 0 \u2264 a\u2082 * b\u2081\nhab : 0 < a\u2082 * b\u2081 + b\u2082\n\u22a2 a\u2082 * a\u2081 + (a\u2082 * b\u2081 + b\u2082) = 1\n[PROOFSTEP]\nrw [\u2190 add_assoc, \u2190 mul_add, hab\u2081, mul_one, hab\u2082]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.refine'_5\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx\u271d y\u271d : E\ns t u : Set E\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nz : E\nhz : z \u2208 u\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082\u271d : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\nhb\u2082 : 0 < b\u2082\nha\u2082b\u2081 : 0 \u2264 a\u2082 * b\u2081\nhab : 0 < a\u2082 * b\u2081 + b\u2082\n\u22a2 (a\u2082 * a\u2081) \u2022 x + (a\u2082 * b\u2081 + b\u2082) \u2022 ((a\u2082 * b\u2081 / (a\u2082 * b\u2081 + b\u2082)) \u2022 y + (b\u2082 / (a\u2082 * b\u2081 + b\u2082)) \u2022 z) =\n a\u2082 \u2022 (a\u2081 \u2022 x + b\u2081 \u2022 y) + b\u2082 \u2022 z\n[PROOFSTEP]\nsimp_rw [smul_add, \u2190 mul_smul, mul_div_cancel' _ hab.ne', add_assoc]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx y : E\ns t u : Set E\n\u22a2 convexJoin \ud835\udd5c (convexJoin \ud835\udd5c s t) u = convexJoin \ud835\udd5c s (convexJoin \ud835\udd5c t u)\n[PROOFSTEP]\nrefine' (convexJoin_assoc_aux _ _ _).antisymm _\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx y : E\ns t u : Set E\n\u22a2 convexJoin \ud835\udd5c s (convexJoin \ud835\udd5c t u) \u2286 convexJoin \ud835\udd5c (convexJoin \ud835\udd5c s t) u\n[PROOFSTEP]\nsimp_rw [convexJoin_comm s, convexJoin_comm _ u]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx y : E\ns t u : Set E\n\u22a2 convexJoin \ud835\udd5c (convexJoin \ud835\udd5c u t) s \u2286 convexJoin \ud835\udd5c u (convexJoin \ud835\udd5c t s)\n[PROOFSTEP]\nexact convexJoin_assoc_aux _ _ _\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx y : E\ns t u : Set E\n\u22a2 convexJoin \ud835\udd5c s (convexJoin \ud835\udd5c t u) = convexJoin \ud835\udd5c t (convexJoin \ud835\udd5c s u)\n[PROOFSTEP]\nsimp_rw [\u2190 convexJoin_assoc, convexJoin_comm]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx y : E\ns t u : Set E\n\u22a2 convexJoin \ud835\udd5c (convexJoin \ud835\udd5c s t) u = convexJoin \ud835\udd5c (convexJoin \ud835\udd5c s u) t\n[PROOFSTEP]\nsimp_rw [convexJoin_assoc, convexJoin_comm]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d t\u271d u\u271d : Set E\nx y : E\ns t u v : Set E\n\u22a2 convexJoin \ud835\udd5c (convexJoin \ud835\udd5c s t) (convexJoin \ud835\udd5c u v) = convexJoin \ud835\udd5c (convexJoin \ud835\udd5c s u) (convexJoin \ud835\udd5c t v)\n[PROOFSTEP]\nsimp_rw [\u2190 convexJoin_assoc, convexJoin_right_comm]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx y : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\n\u22a2 Convex \ud835\udd5c (convexJoin \ud835\udd5c s t)\n[PROOFSTEP]\nsimp only [Convex, StarConvex, convexJoin, mem_iUnion]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx y : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\n\u22a2 \u2200 \u2983x : E\u2984,\n (\u2203 i h i_1 i_2, x \u2208 [i-[\ud835\udd5c]i_1]) \u2192\n \u2200 \u2983y : E\u2984,\n (\u2203 i h i_1 i_2, y \u2208 [i-[\ud835\udd5c]i_1]) \u2192\n \u2200 \u2983a b : \ud835\udd5c\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 \u2203 i h i_1 i_2, a \u2022 x + b \u2022 y \u2208 [i-[\ud835\udd5c]i_1]\n[PROOFSTEP]\nrintro _ \u27e8x\u2081, hx\u2081, y\u2081, hy\u2081, a\u2081, b\u2081, ha\u2081, hb\u2081, hab\u2081, rfl\u27e9 _ \u27e8x\u2082, hx\u2082, y\u2082, hy\u2082, a\u2082, b\u2082, ha\u2082, hb\u2082, hab\u2082, rfl\u27e9 p q hp hq hpq\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx y : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\ny\u2081 : E\nhy\u2081 : y\u2081 \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 t\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\np q : \ud835\udd5c\nhp : 0 \u2264 p\nhq : 0 \u2264 q\nhpq : p + q = 1\n\u22a2 \u2203 i h i_1 i_2, p \u2022 (a\u2081 \u2022 x\u2081 + b\u2081 \u2022 y\u2081) + q \u2022 (a\u2082 \u2022 x\u2082 + b\u2082 \u2022 y\u2082) \u2208 [i-[\ud835\udd5c]i_1]\n[PROOFSTEP]\nrcases hs.exists_mem_add_smul_eq hx\u2081 hx\u2082 (mul_nonneg hp ha\u2081) (mul_nonneg hq ha\u2082) with \u27e8x, hxs, hx\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx\u271d y : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\ny\u2081 : E\nhy\u2081 : y\u2081 \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 t\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\np q : \ud835\udd5c\nhp : 0 \u2264 p\nhq : 0 \u2264 q\nhpq : p + q = 1\nx : E\nhxs : x \u2208 s\nhx : (p * a\u2081 + q * a\u2082) \u2022 x = (p * a\u2081) \u2022 x\u2081 + (q * a\u2082) \u2022 x\u2082\n\u22a2 \u2203 i h i_1 i_2, p \u2022 (a\u2081 \u2022 x\u2081 + b\u2081 \u2022 y\u2081) + q \u2022 (a\u2082 \u2022 x\u2082 + b\u2082 \u2022 y\u2082) \u2208 [i-[\ud835\udd5c]i_1]\n[PROOFSTEP]\nrcases ht.exists_mem_add_smul_eq hy\u2081 hy\u2082 (mul_nonneg hp hb\u2081) (mul_nonneg hq hb\u2082) with \u27e8y, hyt, hy\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx\u271d y\u271d : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\ny\u2081 : E\nhy\u2081 : y\u2081 \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 t\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\np q : \ud835\udd5c\nhp : 0 \u2264 p\nhq : 0 \u2264 q\nhpq : p + q = 1\nx : E\nhxs : x \u2208 s\nhx : (p * a\u2081 + q * a\u2082) \u2022 x = (p * a\u2081) \u2022 x\u2081 + (q * a\u2082) \u2022 x\u2082\ny : E\nhyt : y \u2208 t\nhy : (p * b\u2081 + q * b\u2082) \u2022 y = (p * b\u2081) \u2022 y\u2081 + (q * b\u2082) \u2022 y\u2082\n\u22a2 \u2203 i h i_1 i_2, p \u2022 (a\u2081 \u2022 x\u2081 + b\u2081 \u2022 y\u2081) + q \u2022 (a\u2082 \u2022 x\u2082 + b\u2082 \u2022 y\u2082) \u2208 [i-[\ud835\udd5c]i_1]\n[PROOFSTEP]\nrefine \u27e8_, hxs, _, hyt, p * a\u2081 + q * a\u2082, p * b\u2081 + q * b\u2082, ?_, ?_, ?_, ?_\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx\u271d y\u271d : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\ny\u2081 : E\nhy\u2081 : y\u2081 \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 t\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\np q : \ud835\udd5c\nhp : 0 \u2264 p\nhq : 0 \u2264 q\nhpq : p + q = 1\nx : E\nhxs : x \u2208 s\nhx : (p * a\u2081 + q * a\u2082) \u2022 x = (p * a\u2081) \u2022 x\u2081 + (q * a\u2082) \u2022 x\u2082\ny : E\nhyt : y \u2208 t\nhy : (p * b\u2081 + q * b\u2082) \u2022 y = (p * b\u2081) \u2022 y\u2081 + (q * b\u2082) \u2022 y\u2082\n\u22a2 0 \u2264 p * a\u2081 + q * a\u2082\n[PROOFSTEP]\ntry positivity\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx\u271d y\u271d : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\ny\u2081 : E\nhy\u2081 : y\u2081 \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 t\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\np q : \ud835\udd5c\nhp : 0 \u2264 p\nhq : 0 \u2264 q\nhpq : p + q = 1\nx : E\nhxs : x \u2208 s\nhx : (p * a\u2081 + q * a\u2082) \u2022 x = (p * a\u2081) \u2022 x\u2081 + (q * a\u2082) \u2022 x\u2082\ny : E\nhyt : y \u2208 t\nhy : (p * b\u2081 + q * b\u2082) \u2022 y = (p * b\u2081) \u2022 y\u2081 + (q * b\u2082) \u2022 y\u2082\n\u22a2 0 \u2264 p * a\u2081 + q * a\u2082\n[PROOFSTEP]\npositivity\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx\u271d y\u271d : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\ny\u2081 : E\nhy\u2081 : y\u2081 \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 t\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\np q : \ud835\udd5c\nhp : 0 \u2264 p\nhq : 0 \u2264 q\nhpq : p + q = 1\nx : E\nhxs : x \u2208 s\nhx : (p * a\u2081 + q * a\u2082) \u2022 x = (p * a\u2081) \u2022 x\u2081 + (q * a\u2082) \u2022 x\u2082\ny : E\nhyt : y \u2208 t\nhy : (p * b\u2081 + q * b\u2082) \u2022 y = (p * b\u2081) \u2022 y\u2081 + (q * b\u2082) \u2022 y\u2082\n\u22a2 0 \u2264 p * b\u2081 + q * b\u2082\n[PROOFSTEP]\ntry positivity\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx\u271d y\u271d : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\ny\u2081 : E\nhy\u2081 : y\u2081 \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 t\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\np q : \ud835\udd5c\nhp : 0 \u2264 p\nhq : 0 \u2264 q\nhpq : p + q = 1\nx : E\nhxs : x \u2208 s\nhx : (p * a\u2081 + q * a\u2082) \u2022 x = (p * a\u2081) \u2022 x\u2081 + (q * a\u2082) \u2022 x\u2082\ny : E\nhyt : y \u2208 t\nhy : (p * b\u2081 + q * b\u2082) \u2022 y = (p * b\u2081) \u2022 y\u2081 + (q * b\u2082) \u2022 y\u2082\n\u22a2 0 \u2264 p * b\u2081 + q * b\u2082\n[PROOFSTEP]\npositivity\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_3\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx\u271d y\u271d : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\ny\u2081 : E\nhy\u2081 : y\u2081 \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 t\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\np q : \ud835\udd5c\nhp : 0 \u2264 p\nhq : 0 \u2264 q\nhpq : p + q = 1\nx : E\nhxs : x \u2208 s\nhx : (p * a\u2081 + q * a\u2082) \u2022 x = (p * a\u2081) \u2022 x\u2081 + (q * a\u2082) \u2022 x\u2082\ny : E\nhyt : y \u2208 t\nhy : (p * b\u2081 + q * b\u2082) \u2022 y = (p * b\u2081) \u2022 y\u2081 + (q * b\u2082) \u2022 y\u2082\n\u22a2 p * a\u2081 + q * a\u2082 + (p * b\u2081 + q * b\u2082) = 1\n[PROOFSTEP]\ntry positivity\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_3\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx\u271d y\u271d : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\ny\u2081 : E\nhy\u2081 : y\u2081 \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 t\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\np q : \ud835\udd5c\nhp : 0 \u2264 p\nhq : 0 \u2264 q\nhpq : p + q = 1\nx : E\nhxs : x \u2208 s\nhx : (p * a\u2081 + q * a\u2082) \u2022 x = (p * a\u2081) \u2022 x\u2081 + (q * a\u2082) \u2022 x\u2082\ny : E\nhyt : y \u2208 t\nhy : (p * b\u2081 + q * b\u2082) \u2022 y = (p * b\u2081) \u2022 y\u2081 + (q * b\u2082) \u2022 y\u2082\n\u22a2 p * a\u2081 + q * a\u2082 + (p * b\u2081 + q * b\u2082) = 1\n[PROOFSTEP]\npositivity\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_4\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx\u271d y\u271d : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\ny\u2081 : E\nhy\u2081 : y\u2081 \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 t\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\np q : \ud835\udd5c\nhp : 0 \u2264 p\nhq : 0 \u2264 q\nhpq : p + q = 1\nx : E\nhxs : x \u2208 s\nhx : (p * a\u2081 + q * a\u2082) \u2022 x = (p * a\u2081) \u2022 x\u2081 + (q * a\u2082) \u2022 x\u2082\ny : E\nhyt : y \u2208 t\nhy : (p * b\u2081 + q * b\u2082) \u2022 y = (p * b\u2081) \u2022 y\u2081 + (q * b\u2082) \u2022 y\u2082\n\u22a2 (p * a\u2081 + q * a\u2082) \u2022 x + (p * b\u2081 + q * b\u2082) \u2022 y = p \u2022 (a\u2081 \u2022 x\u2081 + b\u2081 \u2022 y\u2081) + q \u2022 (a\u2082 \u2022 x\u2082 + b\u2082 \u2022 y\u2082)\n[PROOFSTEP]\ntry positivity\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_4\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx\u271d y\u271d : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\ny\u2081 : E\nhy\u2081 : y\u2081 \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 t\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\np q : \ud835\udd5c\nhp : 0 \u2264 p\nhq : 0 \u2264 q\nhpq : p + q = 1\nx : E\nhxs : x \u2208 s\nhx : (p * a\u2081 + q * a\u2082) \u2022 x = (p * a\u2081) \u2022 x\u2081 + (q * a\u2082) \u2022 x\u2082\ny : E\nhyt : y \u2208 t\nhy : (p * b\u2081 + q * b\u2082) \u2022 y = (p * b\u2081) \u2022 y\u2081 + (q * b\u2082) \u2022 y\u2082\n\u22a2 (p * a\u2081 + q * a\u2082) \u2022 x + (p * b\u2081 + q * b\u2082) \u2022 y = p \u2022 (a\u2081 \u2022 x\u2081 + b\u2081 \u2022 y\u2081) + q \u2022 (a\u2082 \u2022 x\u2082 + b\u2082 \u2022 y\u2082)\n[PROOFSTEP]\npositivity\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_3\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx\u271d y\u271d : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\ny\u2081 : E\nhy\u2081 : y\u2081 \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 t\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\np q : \ud835\udd5c\nhp : 0 \u2264 p\nhq : 0 \u2264 q\nhpq : p + q = 1\nx : E\nhxs : x \u2208 s\nhx : (p * a\u2081 + q * a\u2082) \u2022 x = (p * a\u2081) \u2022 x\u2081 + (q * a\u2082) \u2022 x\u2082\ny : E\nhyt : y \u2208 t\nhy : (p * b\u2081 + q * b\u2082) \u2022 y = (p * b\u2081) \u2022 y\u2081 + (q * b\u2082) \u2022 y\u2082\n\u22a2 p * a\u2081 + q * a\u2082 + (p * b\u2081 + q * b\u2082) = 1\n[PROOFSTEP]\nrwa [add_add_add_comm, \u2190 mul_add, \u2190 mul_add, hab\u2081, hab\u2082, mul_one, mul_one]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_4\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx\u271d y\u271d : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\ny\u2081 : E\nhy\u2081 : y\u2081 \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 t\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\np q : \ud835\udd5c\nhp : 0 \u2264 p\nhq : 0 \u2264 q\nhpq : p + q = 1\nx : E\nhxs : x \u2208 s\nhx : (p * a\u2081 + q * a\u2082) \u2022 x = (p * a\u2081) \u2022 x\u2081 + (q * a\u2082) \u2022 x\u2082\ny : E\nhyt : y \u2208 t\nhy : (p * b\u2081 + q * b\u2082) \u2022 y = (p * b\u2081) \u2022 y\u2081 + (q * b\u2082) \u2022 y\u2082\n\u22a2 (p * a\u2081 + q * a\u2082) \u2022 x + (p * b\u2081 + q * b\u2082) \u2022 y = p \u2022 (a\u2081 \u2022 x\u2081 + b\u2081 \u2022 y\u2081) + q \u2022 (a\u2082 \u2022 x\u2082 + b\u2082 \u2022 y\u2082)\n[PROOFSTEP]\nrw [hx, hy, add_add_add_comm]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_4\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx\u271d y\u271d : E\nhs : Convex \ud835\udd5c s\nht : Convex \ud835\udd5c t\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 s\ny\u2081 : E\nhy\u2081 : y\u2081 \u2208 t\na\u2081 b\u2081 : \ud835\udd5c\nha\u2081 : 0 \u2264 a\u2081\nhb\u2081 : 0 \u2264 b\u2081\nhab\u2081 : a\u2081 + b\u2081 = 1\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 s\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 t\na\u2082 b\u2082 : \ud835\udd5c\nha\u2082 : 0 \u2264 a\u2082\nhb\u2082 : 0 \u2264 b\u2082\nhab\u2082 : a\u2082 + b\u2082 = 1\np q : \ud835\udd5c\nhp : 0 \u2264 p\nhq : 0 \u2264 q\nhpq : p + q = 1\nx : E\nhxs : x \u2208 s\nhx : (p * a\u2081 + q * a\u2082) \u2022 x = (p * a\u2081) \u2022 x\u2081 + (q * a\u2082) \u2022 x\u2082\ny : E\nhyt : y \u2208 t\nhy : (p * b\u2081 + q * b\u2082) \u2022 y = (p * b\u2081) \u2022 y\u2081 + (q * b\u2082) \u2022 y\u2082\n\u22a2 (p * a\u2081) \u2022 x\u2081 + (p * b\u2081) \u2022 y\u2081 + ((q * a\u2082) \u2022 x\u2082 + (q * b\u2082) \u2022 y\u2082) = p \u2022 (a\u2081 \u2022 x\u2081 + b\u2081 \u2022 y\u2081) + q \u2022 (a\u2082 \u2022 x\u2082 + b\u2082 \u2022 y\u2082)\n[PROOFSTEP]\nsimp only [smul_add, smul_smul]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx y : E\nhs : Set.Nonempty s\nht : Set.Nonempty t\n\u22a2 \u2191(convexHull \ud835\udd5c) (s \u222a t) = convexJoin \ud835\udd5c (\u2191(convexHull \ud835\udd5c) s) (\u2191(convexHull \ud835\udd5c) t)\n[PROOFSTEP]\nrw [\u2190 convexHull_convexHull_union_left, \u2190 convexHull_convexHull_union_right]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx y : E\nhs : Set.Nonempty s\nht : Set.Nonempty t\n\u22a2 \u2191(convexHull \ud835\udd5c) (\u2191(convexHull \ud835\udd5c) s \u222a \u2191(convexHull \ud835\udd5c) t) = convexJoin \ud835\udd5c (\u2191(convexHull \ud835\udd5c) s) (\u2191(convexHull \ud835\udd5c) t)\n[PROOFSTEP]\nexact (convex_convexHull \ud835\udd5c s).convexHull_union (convex_convexHull \ud835\udd5c t) hs.convexHull ht.convexHull\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx y : E\nhs : Set.Nonempty s\n\u22a2 \u2191(convexHull \ud835\udd5c) (insert x s) = convexJoin \ud835\udd5c {x} (\u2191(convexHull \ud835\udd5c) s)\n[PROOFSTEP]\nrw [insert_eq, convexHull_union (singleton_nonempty _) hs, convexHull_singleton]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx y a b c d : E\n\u22a2 convexJoin \ud835\udd5c (segment \ud835\udd5c a b) (segment \ud835\udd5c c d) = \u2191(convexHull \ud835\udd5c) {a, b, c, d}\n[PROOFSTEP]\nsimp_rw [\u2190 convexHull_pair, convexHull_insert (insert_nonempty _ _), convexHull_insert (singleton_nonempty _),\n convexJoin_assoc, convexHull_singleton]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx y a b c : E\n\u22a2 convexJoin \ud835\udd5c (segment \ud835\udd5c a b) {c} = \u2191(convexHull \ud835\udd5c) {a, b, c}\n[PROOFSTEP]\nrw [\u2190 pair_eq_singleton, \u2190 convexJoin_segments, segment_same, pair_eq_singleton]\n[GOAL]\n\u03b9 : Sort u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns t u : Set E\nx y a b c : E\n\u22a2 convexJoin \ud835\udd5c {a} (segment \ud835\udd5c b c) = \u2191(convexHull \ud835\udd5c) {a, b, c}\n[PROOFSTEP]\nrw [\u2190 segment_same \ud835\udd5c, convexJoin_segments, insert_idem]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.Join", "llama_tokens": 15486, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.4011471626771152}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\nH : Type u_2\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nE : Type u_3\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nI : ModelWithCorners \ud835\udd5c E H\nF : Type u_4\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\nJ : ModelWithCorners \ud835\udd5c F F\nG : Type u_5\ninst\u271d\u00b9\u00b3 : TopologicalSpace G\ninst\u271d\u00b9\u00b2 : ChartedSpace H G\ninst\u271d\u00b9\u00b9 : Group G\ninst\u271d\u00b9\u2070 : LieGroup I G\nE' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_7\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_8\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H' M\nE'' : Type u_9\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM' : Type u_11\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H'' M'\nn : \u2115\u221e\nf g : M \u2192 G\ns : Set M\nx\u2080 : M\nhf : ContMDiffWithinAt I' I n f s x\u2080\nhg : ContMDiffWithinAt I' I n g s x\u2080\n\u22a2 ContMDiffWithinAt I' I n (fun x => f x / g x) s x\u2080\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\nH : Type u_2\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nE : Type u_3\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nI : ModelWithCorners \ud835\udd5c E H\nF : Type u_4\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\nJ : ModelWithCorners \ud835\udd5c F F\nG : Type u_5\ninst\u271d\u00b9\u00b3 : TopologicalSpace G\ninst\u271d\u00b9\u00b2 : ChartedSpace H G\ninst\u271d\u00b9\u00b9 : Group G\ninst\u271d\u00b9\u2070 : LieGroup I G\nE' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_7\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_8\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H' M\nE'' : Type u_9\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM' : Type u_11\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H'' M'\nn : \u2115\u221e\nf g : M \u2192 G\ns : Set M\nx\u2080 : M\nhf : ContMDiffWithinAt I' I n f s x\u2080\nhg : ContMDiffWithinAt I' I n g s x\u2080\n\u22a2 ContMDiffWithinAt I' I n (fun x => f x * (g x)\u207b\u00b9) s x\u2080\n[PROOFSTEP]\nexact hf.mul hg.inv\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\nH : Type u_2\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nE : Type u_3\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nI : ModelWithCorners \ud835\udd5c E H\nF : Type u_4\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\nJ : ModelWithCorners \ud835\udd5c F F\nG : Type u_5\ninst\u271d\u00b9\u00b3 : TopologicalSpace G\ninst\u271d\u00b9\u00b2 : ChartedSpace H G\ninst\u271d\u00b9\u00b9 : Group G\ninst\u271d\u00b9\u2070 : LieGroup I G\nE' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_7\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_8\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H' M\nE'' : Type u_9\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM' : Type u_11\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H'' M'\nn : \u2115\u221e\nf g : M \u2192 G\nx\u2080 : M\nhf : ContMDiffAt I' I n f x\u2080\nhg : ContMDiffAt I' I n g x\u2080\n\u22a2 ContMDiffAt I' I n (fun x => f x / g x) x\u2080\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\nH : Type u_2\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nE : Type u_3\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nI : ModelWithCorners \ud835\udd5c E H\nF : Type u_4\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\nJ : ModelWithCorners \ud835\udd5c F F\nG : Type u_5\ninst\u271d\u00b9\u00b3 : TopologicalSpace G\ninst\u271d\u00b9\u00b2 : ChartedSpace H G\ninst\u271d\u00b9\u00b9 : Group G\ninst\u271d\u00b9\u2070 : LieGroup I G\nE' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_7\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_8\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H' M\nE'' : Type u_9\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM' : Type u_11\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H'' M'\nn : \u2115\u221e\nf g : M \u2192 G\nx\u2080 : M\nhf : ContMDiffAt I' I n f x\u2080\nhg : ContMDiffAt I' I n g x\u2080\n\u22a2 ContMDiffAt I' I n (fun x => f x * (g x)\u207b\u00b9) x\u2080\n[PROOFSTEP]\nexact hf.mul hg.inv\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\nH : Type u_2\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nE : Type u_3\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nI : ModelWithCorners \ud835\udd5c E H\nF : Type u_4\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\nJ : ModelWithCorners \ud835\udd5c F F\nG : Type u_5\ninst\u271d\u00b9\u00b3 : TopologicalSpace G\ninst\u271d\u00b9\u00b2 : ChartedSpace H G\ninst\u271d\u00b9\u00b9 : Group G\ninst\u271d\u00b9\u2070 : LieGroup I G\nE' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_7\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_8\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H' M\nE'' : Type u_9\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM' : Type u_11\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H'' M'\nn : \u2115\u221e\nf g : M \u2192 G\ns : Set M\nhf : ContMDiffOn I' I n f s\nhg : ContMDiffOn I' I n g s\n\u22a2 ContMDiffOn I' I n (fun x => f x / g x) s\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\nH : Type u_2\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nE : Type u_3\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nI : ModelWithCorners \ud835\udd5c E H\nF : Type u_4\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\nJ : ModelWithCorners \ud835\udd5c F F\nG : Type u_5\ninst\u271d\u00b9\u00b3 : TopologicalSpace G\ninst\u271d\u00b9\u00b2 : ChartedSpace H G\ninst\u271d\u00b9\u00b9 : Group G\ninst\u271d\u00b9\u2070 : LieGroup I G\nE' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_7\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_8\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H' M\nE'' : Type u_9\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM' : Type u_11\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H'' M'\nn : \u2115\u221e\nf g : M \u2192 G\ns : Set M\nhf : ContMDiffOn I' I n f s\nhg : ContMDiffOn I' I n g s\n\u22a2 ContMDiffOn I' I n (fun x => f x * (g x)\u207b\u00b9) s\n[PROOFSTEP]\nexact hf.mul hg.inv\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\nH : Type u_2\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nE : Type u_3\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nI : ModelWithCorners \ud835\udd5c E H\nF : Type u_4\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\nJ : ModelWithCorners \ud835\udd5c F F\nG : Type u_5\ninst\u271d\u00b9\u00b3 : TopologicalSpace G\ninst\u271d\u00b9\u00b2 : ChartedSpace H G\ninst\u271d\u00b9\u00b9 : Group G\ninst\u271d\u00b9\u2070 : LieGroup I G\nE' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_7\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_8\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H' M\nE'' : Type u_9\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM' : Type u_11\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H'' M'\nn : \u2115\u221e\nf g : M \u2192 G\nhf : ContMDiff I' I n f\nhg : ContMDiff I' I n g\n\u22a2 ContMDiff I' I n fun x => f x / g x\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\nH : Type u_2\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nE : Type u_3\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nI : ModelWithCorners \ud835\udd5c E H\nF : Type u_4\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\nJ : ModelWithCorners \ud835\udd5c F F\nG : Type u_5\ninst\u271d\u00b9\u00b3 : TopologicalSpace G\ninst\u271d\u00b9\u00b2 : ChartedSpace H G\ninst\u271d\u00b9\u00b9 : Group G\ninst\u271d\u00b9\u2070 : LieGroup I G\nE' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_7\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_8\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H' M\nE'' : Type u_9\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM' : Type u_11\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H'' M'\nn : \u2115\u221e\nf g : M \u2192 G\nhf : ContMDiff I' I n f\nhg : ContMDiff I' I n g\n\u22a2 ContMDiff I' I n fun x => f x * (g x)\u207b\u00b9\n[PROOFSTEP]\nexact hf.mul hg.inv\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d : CompleteSpace \ud835\udd5c\n\u22a2 \u2200 \u2983x : \ud835\udd5c\u2984, x \u2260 0 \u2192 SmoothAt \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) (fun y => y\u207b\u00b9) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d : CompleteSpace \ud835\udd5c\nx : \ud835\udd5c\nhx : x \u2260 0\n\u22a2 SmoothAt \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) (fun y => y\u207b\u00b9) x\n[PROOFSTEP]\nchange ContMDiffAt \ud835\udcd8(\ud835\udd5c) \ud835\udcd8(\ud835\udd5c) \u22a4 Inv.inv x\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d : CompleteSpace \ud835\udd5c\nx : \ud835\udd5c\nhx : x \u2260 0\n\u22a2 ContMDiffAt \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \u22a4 Inv.inv x\n[PROOFSTEP]\nrw [contMDiffAt_iff_contDiffAt]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d : CompleteSpace \ud835\udd5c\nx : \ud835\udd5c\nhx : x \u2260 0\n\u22a2 ContDiffAt \ud835\udd5c \u22a4 Inv.inv x\n[PROOFSTEP]\nexact contDiffAt_inv \ud835\udd5c hx\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nH : Type u_2\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nE : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nI : ModelWithCorners \ud835\udd5c E H\nG : Type u_4\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : ChartedSpace H G\ninst\u271d\u2077 : GroupWithZero G\ninst\u271d\u2076 : SmoothInv\u2080 I G\ninst\u271d\u2075 : SmoothMul I G\nE' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_7\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ChartedSpace H' M\nf g : M \u2192 G\nn : \u2115\u221e\ns : Set M\na : M\nhf : ContMDiffWithinAt I' I n f s a\nhg : ContMDiffWithinAt I' I n g s a\nh\u2080 : g a \u2260 0\n\u22a2 ContMDiffWithinAt I' I n (f / g) s a\n[PROOFSTEP]\nsimpa [div_eq_mul_inv] using hf.mul (hg.inv\u2080 h\u2080)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nH : Type u_2\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nE : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nI : ModelWithCorners \ud835\udd5c E H\nG : Type u_4\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : ChartedSpace H G\ninst\u271d\u2077 : GroupWithZero G\ninst\u271d\u2076 : SmoothInv\u2080 I G\ninst\u271d\u2075 : SmoothMul I G\nE' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_7\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ChartedSpace H' M\nf g : M \u2192 G\nn : \u2115\u221e\ns : Set M\nhf : ContMDiffOn I' I n f s\nhg : ContMDiffOn I' I n g s\nh\u2080 : \u2200 (x : M), x \u2208 s \u2192 g x \u2260 0\n\u22a2 ContMDiffOn I' I n (f / g) s\n[PROOFSTEP]\nsimpa [div_eq_mul_inv] using hf.mul (hg.inv\u2080 h\u2080)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nH : Type u_2\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nE : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nI : ModelWithCorners \ud835\udd5c E H\nG : Type u_4\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : ChartedSpace H G\ninst\u271d\u2077 : GroupWithZero G\ninst\u271d\u2076 : SmoothInv\u2080 I G\ninst\u271d\u2075 : SmoothMul I G\nE' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_7\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ChartedSpace H' M\nf g : M \u2192 G\nn : \u2115\u221e\na : M\nhf : ContMDiffAt I' I n f a\nhg : ContMDiffAt I' I n g a\nh\u2080 : g a \u2260 0\n\u22a2 ContMDiffAt I' I n (f / g) a\n[PROOFSTEP]\nsimpa [div_eq_mul_inv] using hf.mul (hg.inv\u2080 h\u2080)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nH : Type u_2\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nE : Type u_3\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nI : ModelWithCorners \ud835\udd5c E H\nG : Type u_4\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : ChartedSpace H G\ninst\u271d\u2077 : GroupWithZero G\ninst\u271d\u2076 : SmoothInv\u2080 I G\ninst\u271d\u2075 : SmoothMul I G\nE' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM : Type u_7\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ChartedSpace H' M\nf g : M \u2192 G\nn : \u2115\u221e\nhf : ContMDiff I' I n f\nhg : ContMDiff I' I n g\nh\u2080 : \u2200 (x : M), g x \u2260 0\n\u22a2 ContMDiff I' I n (f / g)\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using hf.mul (hg.inv\u2080 h\u2080)\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Manifold.Algebra.LieGroup", "llama_tokens": 5879, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.766293653760418, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.4010936916389687}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\n\u22a2 OpenEmbedding \u2191(\u03b9 D.toGlueData i).base\n[PROOFSTEP]\nrw [\u2190 show _ = (\ud835\udda3.\u03b9 i).base from \ud835\udda3.\u03b9_gluedIso_inv (PresheafedSpace.forget _) _]\n -- Porting note : added this erewrite\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\n\u22a2 OpenEmbedding \u2191(\u03b9 (mapGlueData D.toGlueData (forget C)) i \u226b (gluedIso D.toGlueData (forget C)).inv)\n[PROOFSTEP]\nerw [coe_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\n\u22a2 OpenEmbedding (\u2191(gluedIso D.toGlueData (forget C)).inv \u2218 \u2191(\u03b9 (mapGlueData D.toGlueData (forget C)) i))\n[PROOFSTEP]\nrefine\n OpenEmbedding.comp (TopCat.homeoOfIso (\ud835\udda3.gluedIso (PresheafedSpace.forget _)).symm).openEmbedding\n (D.toTopGlueData.\u03b9_openEmbedding i)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nS : Set \u2191\u2191(V D.toGlueData (i, j))\n\u22a2 \u2191pullback.snd.base '' (\u2191pullback.fst.base \u207b\u00b9' S) = \u2191(f D.toGlueData i k).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base '' S)\n[PROOFSTEP]\nhave eq\u2081 : _ = (\u03c0\u2081 i, j, k).base := PreservesPullback.iso_hom_fst (forget C) _ _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nS : Set \u2191\u2191(V D.toGlueData (i, j))\neq\u2081 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.fst = pullback.fst.base\n\u22a2 \u2191pullback.snd.base '' (\u2191pullback.fst.base \u207b\u00b9' S) = \u2191(f D.toGlueData i k).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base '' S)\n[PROOFSTEP]\nhave eq\u2082 : _ = (\u03c0\u2082 i, j, k).base := PreservesPullback.iso_hom_snd (forget C) _ _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nS : Set \u2191\u2191(V D.toGlueData (i, j))\neq\u2081 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.fst = pullback.fst.base\neq\u2082 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.snd = pullback.snd.base\n\u22a2 \u2191pullback.snd.base '' (\u2191pullback.fst.base \u207b\u00b9' S) = \u2191(f D.toGlueData i k).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base '' S)\n[PROOFSTEP]\nrw [\u2190 eq\u2081, \u2190 eq\u2082]\n -- Porting note : `rw` to `erw` on `coe_comp`\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nS : Set \u2191\u2191(V D.toGlueData (i, j))\neq\u2081 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.fst = pullback.fst.base\neq\u2082 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.snd = pullback.snd.base\n\u22a2 \u2191((PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.snd) ''\n (\u2191((PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.fst) \u207b\u00b9' S) =\n \u2191(f D.toGlueData i k).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base '' S)\n[PROOFSTEP]\nerw [coe_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nS : Set \u2191\u2191(V D.toGlueData (i, j))\neq\u2081 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.fst = pullback.fst.base\neq\u2082 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.snd = pullback.snd.base\n\u22a2 \u2191pullback.snd \u2218 \u2191(PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom ''\n (\u2191((PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.fst) \u207b\u00b9' S) =\n \u2191(f D.toGlueData i k).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base '' S)\n[PROOFSTEP]\nrw [Set.image_comp]\n -- Porting note : `rw` to `erw` on `coe_comp`\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nS : Set \u2191\u2191(V D.toGlueData (i, j))\neq\u2081 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.fst = pullback.fst.base\neq\u2082 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.snd = pullback.snd.base\n\u22a2 \u2191pullback.snd ''\n (\u2191(PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom ''\n (\u2191((PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.fst) \u207b\u00b9' S)) =\n \u2191(f D.toGlueData i k).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base '' S)\n[PROOFSTEP]\nerw [coe_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nS : Set \u2191\u2191(V D.toGlueData (i, j))\neq\u2081 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.fst = pullback.fst.base\neq\u2082 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.snd = pullback.snd.base\n\u22a2 \u2191pullback.snd ''\n (\u2191(PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom ''\n (\u2191pullback.fst \u2218 \u2191(PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u207b\u00b9' S)) =\n \u2191(f D.toGlueData i k).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base '' S)\n[PROOFSTEP]\nrw [Set.preimage_comp, Set.image_preimage_eq, TopCat.pullback_snd_image_fst_preimage]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nS : Set \u2191\u2191(V D.toGlueData (i, j))\neq\u2081 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.fst = pullback.fst.base\neq\u2082 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.snd = pullback.snd.base\n\u22a2 \u2191((forget C).map (f D.toGlueData i k)) \u207b\u00b9' (\u2191((forget C).map (f D.toGlueData i j)) '' S) =\n \u2191(f D.toGlueData i k).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base '' S)\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nS : Set \u2191\u2191(V D.toGlueData (i, j))\neq\u2081 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.fst = pullback.fst.base\neq\u2082 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.snd = pullback.snd.base\n\u22a2 Function.Surjective \u2191(PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nS : Set \u2191\u2191(V D.toGlueData (i, j))\neq\u2081 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.fst = pullback.fst.base\neq\u2082 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.snd = pullback.snd.base\n\u22a2 Function.Surjective \u2191(PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom\n[PROOFSTEP]\nrw [\u2190 TopCat.epi_iff_surjective]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nS : Set \u2191\u2191(V D.toGlueData (i, j))\neq\u2081 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.fst = pullback.fst.base\neq\u2082 :\n (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom \u226b pullback.snd = pullback.snd.base\n\u22a2 Epi (PreservesPullback.iso (forget C) (f D.toGlueData i j) (f D.toGlueData i k)).hom\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(V D.toGlueData (i, j))\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData i k).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U))\n[PROOFSTEP]\ndelta IsOpenImmersion.openFunctor\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(V D.toGlueData (i, j))\n\u22a2 op\n ((IsOpenMap.functor (_ : IsOpenMap \u2191pullback.snd.base)).obj\n {\n unop :=\n { carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData i k).base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(f D.toGlueData i j).base)).obj U))\n[PROOFSTEP]\ndsimp only [Functor.op, IsOpenMap.functor, Opens.map, unop_op]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(V D.toGlueData (i, j))\n\u22a2 op\n {\n carrier :=\n \u2191pullback.snd.base ''\n \u2191{ carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191U, is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191U)) },\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.snd.base ''\n \u2191{ carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191U, is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191U)) })) } =\n op\n {\n carrier :=\n \u2191(f D.toGlueData i k).base \u207b\u00b9'\n \u2191{ carrier := \u2191(f D.toGlueData i j).base '' \u2191U,\n is_open' := (_ : IsOpen (\u2191(f D.toGlueData i j).base '' \u2191U)) },\n is_open' :=\n (_ :\n IsOpen\n (\u2191(f D.toGlueData i k).base \u207b\u00b9'\n \u2191{ carrier := \u2191(f D.toGlueData i j).base '' \u2191U,\n is_open' := (_ : IsOpen (\u2191(f D.toGlueData i j).base '' \u2191U)) })) }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_x.e_carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(V D.toGlueData (i, j))\n\u22a2 \u2191pullback.snd.base ''\n \u2191{ carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191U, is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191U)) } =\n \u2191(f D.toGlueData i k).base \u207b\u00b9'\n \u2191{ carrier := \u2191(f D.toGlueData i j).base '' \u2191U, is_open' := (_ : IsOpen (\u2191(f D.toGlueData i j).base '' \u2191U)) }\n[PROOFSTEP]\napply pullback_base\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(V D.toGlueData (i, j))\n\u22a2 IsOpenImmersion.invApp (_ : IsOpenImmersion (f D.toGlueData i j)) U \u226b\n NatTrans.app (f D.toGlueData i k).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)) =\n NatTrans.app pullback.fst.c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n unop :=\n { carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop)) } }.unop \u226b\n (V D.toGlueData (i, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData i k).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U))))\n[PROOFSTEP]\nhave := PresheafedSpace.congr_app (@pullback.condition _ _ _ _ _ (D.f i j) (D.f i k) _)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(V D.toGlueData (i, j))\nthis :\n \u2200 (U : (Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i))\u1d52\u1d56),\n NatTrans.app (pullback.fst \u226b f D.toGlueData i j).c U =\n NatTrans.app (pullback.snd \u226b f D.toGlueData i k).c U \u226b\n (pullback (f D.toGlueData i j) (f D.toGlueData i k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (pullback.snd \u226b f D.toGlueData i k).base).op.obj U =\n (Opens.map (pullback.fst \u226b f D.toGlueData i j).base).op.obj U))\n\u22a2 IsOpenImmersion.invApp (_ : IsOpenImmersion (f D.toGlueData i j)) U \u226b\n NatTrans.app (f D.toGlueData i k).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)) =\n NatTrans.app pullback.fst.c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n unop :=\n { carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop)) } }.unop \u226b\n (V D.toGlueData (i, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData i k).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U))))\n[PROOFSTEP]\ndsimp only [comp_c_app] at this \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(V D.toGlueData (i, j))\nthis :\n \u2200 (U : (Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i))\u1d52\u1d56),\n NatTrans.app (f D.toGlueData i j).c U \u226b\n NatTrans.app pullback.fst.c (op ((Opens.map (f D.toGlueData i j).base).obj U.unop)) =\n (NatTrans.app (f D.toGlueData i k).c U \u226b\n NatTrans.app pullback.snd.c (op ((Opens.map (f D.toGlueData i k).base).obj U.unop))) \u226b\n (pullback (f D.toGlueData i j) (f D.toGlueData i k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (pullback.snd \u226b f D.toGlueData i k).base).op.obj U =\n (Opens.map (pullback.fst \u226b f D.toGlueData i j).base).op.obj U))\n\u22a2 IsOpenImmersion.invApp (_ : IsOpenImmersion (f D.toGlueData i j)) U \u226b\n NatTrans.app (f D.toGlueData i k).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)) =\n NatTrans.app pullback.fst.c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n unop :=\n { carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop)) } }.unop \u226b\n (V D.toGlueData (i, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData i k).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U))))\n[PROOFSTEP]\nrw [\u2190 cancel_epi (inv ((D.f_open i j).invApp U)), IsIso.inv_hom_id_assoc, IsOpenImmersion.inv_invApp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(V D.toGlueData (i, j))\nthis :\n \u2200 (U : (Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i))\u1d52\u1d56),\n NatTrans.app (f D.toGlueData i j).c U \u226b\n NatTrans.app pullback.fst.c (op ((Opens.map (f D.toGlueData i j).base).obj U.unop)) =\n (NatTrans.app (f D.toGlueData i k).c U \u226b\n NatTrans.app pullback.snd.c (op ((Opens.map (f D.toGlueData i k).base).obj U.unop))) \u226b\n (pullback (f D.toGlueData i j) (f D.toGlueData i k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (pullback.snd \u226b f D.toGlueData i k).base).op.obj U =\n (Opens.map (pullback.fst \u226b f D.toGlueData i j).base).op.obj U))\n\u22a2 NatTrans.app (f D.toGlueData i k).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)) =\n (NatTrans.app (f D.toGlueData i j).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)) \u226b\n (V D.toGlueData (i, j)).presheaf.map\n (eqToHom\n (_ :\n op\n ((Opens.map (f D.toGlueData i j).base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)).unop) =\n op U))) \u226b\n NatTrans.app pullback.fst.c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n unop :=\n { carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop)) } }.unop \u226b\n (V D.toGlueData (i, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData i k).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U))))\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(V D.toGlueData (i, j))\nthis :\n \u2200 (U : (Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i))\u1d52\u1d56),\n NatTrans.app (f D.toGlueData i j).c U \u226b\n NatTrans.app pullback.fst.c (op ((Opens.map (f D.toGlueData i j).base).obj U.unop)) =\n (NatTrans.app (f D.toGlueData i k).c U \u226b\n NatTrans.app pullback.snd.c (op ((Opens.map (f D.toGlueData i k).base).obj U.unop))) \u226b\n (pullback (f D.toGlueData i j) (f D.toGlueData i k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (pullback.snd \u226b f D.toGlueData i k).base).op.obj U =\n (Opens.map (pullback.fst \u226b f D.toGlueData i j).base).op.obj U))\n\u22a2 NatTrans.app (f D.toGlueData i k).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)) =\n NatTrans.app (f D.toGlueData i j).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)) \u226b\n (V D.toGlueData (i, j)).presheaf.map\n (eqToHom\n (_ :\n op\n ((Opens.map (f D.toGlueData i j).base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)).unop) =\n op U)) \u226b\n NatTrans.app pullback.fst.c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n (V D.toGlueData (i, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData i k).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U))))\n[PROOFSTEP]\nerw [(\u03c0\u2081 i, j, k).c.naturality_assoc, reassoc_of% this, \u2190 Functor.map_comp_assoc, IsOpenImmersion.inv_naturality_assoc,\n IsOpenImmersion.app_invApp_assoc, \u2190 (D.V (i, k)).presheaf.map_comp, \u2190 (D.V (i, k)).presheaf.map_comp]\n -- Porting note : need to provide an explicit argument, otherwise Lean does not know which\n -- category we are talking about\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(V D.toGlueData (i, j))\nthis :\n \u2200 (U : (Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i))\u1d52\u1d56),\n NatTrans.app (f D.toGlueData i j).c U \u226b\n NatTrans.app pullback.fst.c (op ((Opens.map (f D.toGlueData i j).base).obj U.unop)) =\n (NatTrans.app (f D.toGlueData i k).c U \u226b\n NatTrans.app pullback.snd.c (op ((Opens.map (f D.toGlueData i k).base).obj U.unop))) \u226b\n (pullback (f D.toGlueData i j) (f D.toGlueData i k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (pullback.snd \u226b f D.toGlueData i k).base).op.obj U =\n (Opens.map (pullback.fst \u226b f D.toGlueData i j).base).op.obj U))\n\u22a2 NatTrans.app (f D.toGlueData i k).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)) =\n NatTrans.app (f D.toGlueData i k).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)) \u226b\n (V D.toGlueData (i, k)).presheaf.map\n ((homOfLE\n (_ :\n \u2191pullback.snd.base ''\n (\u2191pullback.snd.base \u207b\u00b9'\n ((Opens.map (f D.toGlueData i k).base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj\n U)).unop).carrier) \u2286\n ((Opens.map (f D.toGlueData i k).base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj\n U)).unop).carrier)).op \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).op.map\n (eqToHom\n (_ :\n (Opens.map (pullback.snd \u226b f D.toGlueData i k).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)) =\n (Opens.map (pullback.fst \u226b f D.toGlueData i j).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U))) \u226b\n (Opens.map pullback.fst.base).op.map\n (eqToHom\n (_ :\n op\n ((Opens.map (f D.toGlueData i j).base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj\n U)).unop) =\n op U))) \u226b\n eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData i k).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U))))\n[PROOFSTEP]\nconvert (Category.comp_id ((f D.toGlueData i k).c.app _)).symm\n[GOAL]\ncase h.e'_3.h.h.e'_7\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(V D.toGlueData (i, j))\nthis :\n \u2200 (U : (Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i))\u1d52\u1d56),\n NatTrans.app (f D.toGlueData i j).c U \u226b\n NatTrans.app pullback.fst.c (op ((Opens.map (f D.toGlueData i j).base).obj U.unop)) =\n (NatTrans.app (f D.toGlueData i k).c U \u226b\n NatTrans.app pullback.snd.c (op ((Opens.map (f D.toGlueData i k).base).obj U.unop))) \u226b\n (pullback (f D.toGlueData i j) (f D.toGlueData i k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (pullback.snd \u226b f D.toGlueData i k).base).op.obj U =\n (Opens.map (pullback.fst \u226b f D.toGlueData i j).base).op.obj U))\n\u22a2 (V D.toGlueData (i, k)).presheaf.map\n ((homOfLE\n (_ :\n \u2191pullback.snd.base ''\n (\u2191pullback.snd.base \u207b\u00b9'\n ((Opens.map (f D.toGlueData i k).base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj\n U)).unop).carrier) \u2286\n ((Opens.map (f D.toGlueData i k).base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj\n U)).unop).carrier)).op \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).op.map\n (eqToHom\n (_ :\n (Opens.map (pullback.snd \u226b f D.toGlueData i k).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)) =\n (Opens.map (pullback.fst \u226b f D.toGlueData i j).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U))) \u226b\n (Opens.map pullback.fst.base).op.map\n (eqToHom\n (_ :\n op\n ((Opens.map (f D.toGlueData i j).base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)).unop) =\n op U))) \u226b\n eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191pullback.fst.base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData i k).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)))) =\n \ud835\udfd9\n (((f D.toGlueData i k).base _* (V D.toGlueData (i, k)).presheaf).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)))\n[PROOFSTEP]\nerw [(D.V (i, k)).presheaf.map_id]\n[GOAL]\ncase h.e'_3.h.h.e'_7\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(V D.toGlueData (i, j))\nthis :\n \u2200 (U : (Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i))\u1d52\u1d56),\n NatTrans.app (f D.toGlueData i j).c U \u226b\n NatTrans.app pullback.fst.c (op ((Opens.map (f D.toGlueData i j).base).obj U.unop)) =\n (NatTrans.app (f D.toGlueData i k).c U \u226b\n NatTrans.app pullback.snd.c (op ((Opens.map (f D.toGlueData i k).base).obj U.unop))) \u226b\n (pullback (f D.toGlueData i j) (f D.toGlueData i k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (pullback.snd \u226b f D.toGlueData i k).base).op.obj U =\n (Opens.map (pullback.fst \u226b f D.toGlueData i j).base).op.obj U))\n\u22a2 \ud835\udfd9\n ((V D.toGlueData (i, k)).presheaf.obj\n (op\n ((Opens.map (f D.toGlueData i k).base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)).unop))) =\n \ud835\udfd9\n (((f D.toGlueData i k).base _* (V D.toGlueData (i, k)).presheaf).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i j))).obj U)))\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 \u2203 eq,\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd) U \u226b\n NatTrans.app (t D.toGlueData k i).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n (V D.toGlueData (k, i)).presheaf.map (eqToHom eq) =\n NatTrans.app (t' D.toGlueData k i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop\n[PROOFSTEP]\nfconstructor\n -- Porting note: I don't know what the magic was in Lean3 proof, it just skipped the proof of `eq`\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop)\n[PROOFSTEP]\ndelta IsOpenImmersion.openFunctor\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 (Opens.map (t D.toGlueData k i).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191pullback.snd.base)).obj U)) =\n op\n ((IsOpenMap.functor (_ : IsOpenMap \u2191pullback.fst.base)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop)\n[PROOFSTEP]\ndsimp only [Functor.op, Opens.map, IsOpenMap.functor, unop_op, Opens.coe_mk]\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 op\n { carrier := \u2191(t D.toGlueData k i).base \u207b\u00b9' (\u2191pullback.snd.base '' \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData k i).base \u207b\u00b9'\n \u2191{ carrier := \u2191pullback.snd.base '' \u2191U, is_open' := (_ : IsOpen (\u2191pullback.snd.base '' \u2191U)) })) } =\n op\n { carrier := \u2191pullback.fst.base '' (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.fst.base ''\n \u2191{ carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191U,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191U)) })) }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase w.e_x.e_carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 \u2191(t D.toGlueData k i).base \u207b\u00b9' (\u2191pullback.snd.base '' \u2191U) =\n \u2191pullback.fst.base '' (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191U)\n[PROOFSTEP]\nhave := (\ud835\udda3.t_fac k i j).symm\n[GOAL]\ncase w.e_x.e_carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : pullback.fst \u226b t D.toGlueData k i = t' D.toGlueData k i j \u226b pullback.snd\n\u22a2 \u2191(t D.toGlueData k i).base \u207b\u00b9' (\u2191pullback.snd.base '' \u2191U) =\n \u2191pullback.fst.base '' (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191U)\n[PROOFSTEP]\nrw [\u2190 IsIso.inv_comp_eq] at this \n[GOAL]\ncase w.e_x.e_carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : inv (t' D.toGlueData k i j) \u226b pullback.fst \u226b t D.toGlueData k i = pullback.snd\n\u22a2 \u2191(t D.toGlueData k i).base \u207b\u00b9' (\u2191pullback.snd.base '' \u2191U) =\n \u2191pullback.fst.base '' (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191U)\n[PROOFSTEP]\nreplace this := (congr_arg ((PresheafedSpace.Hom.base \u00b7)) this).symm\n[GOAL]\ncase w.e_x.e_carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : pullback.snd.base = (inv (t' D.toGlueData k i j) \u226b pullback.fst \u226b t D.toGlueData k i).base\n\u22a2 \u2191(t D.toGlueData k i).base \u207b\u00b9' (\u2191pullback.snd.base '' \u2191U) =\n \u2191pullback.fst.base '' (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191U)\n[PROOFSTEP]\nreplace this := congr_arg (ContinuousMap.toFun \u00b7) this\n[GOAL]\ncase w.e_x.e_carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis :\n (fun x => x.toFun) pullback.snd.base =\n (fun x => x.toFun) (inv (t' D.toGlueData k i j) \u226b pullback.fst \u226b t D.toGlueData k i).base\n\u22a2 \u2191(t D.toGlueData k i).base \u207b\u00b9' (\u2191pullback.snd.base '' \u2191U) =\n \u2191pullback.fst.base '' (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191U)\n[PROOFSTEP]\ndsimp at this \n[GOAL]\ncase w.e_x.e_carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : \u2191pullback.snd.base = \u2191((inv (t' D.toGlueData k i j)).base \u226b pullback.fst.base \u226b (t D.toGlueData k i).base)\n\u22a2 \u2191(t D.toGlueData k i).base \u207b\u00b9' (\u2191pullback.snd.base '' \u2191U) =\n \u2191pullback.fst.base '' (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191U)\n[PROOFSTEP]\nrw [coe_comp, coe_comp] at this \n[GOAL]\ncase w.e_x.e_carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : \u2191pullback.snd.base = (\u2191(t D.toGlueData k i).base \u2218 \u2191pullback.fst.base) \u2218 \u2191(inv (t' D.toGlueData k i j)).base\n\u22a2 \u2191(t D.toGlueData k i).base \u207b\u00b9' (\u2191pullback.snd.base '' \u2191U) =\n \u2191pullback.fst.base '' (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191U)\n[PROOFSTEP]\nrw [this, Set.image_comp, Set.image_comp, Set.preimage_image_eq]\n[GOAL]\ncase w.e_x.e_carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : \u2191pullback.snd.base = (\u2191(t D.toGlueData k i).base \u2218 \u2191pullback.fst.base) \u2218 \u2191(inv (t' D.toGlueData k i j)).base\n\u22a2 \u2191pullback.fst.base '' (\u2191(inv (t' D.toGlueData k i j)).base '' \u2191U) =\n \u2191pullback.fst.base '' (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191U)\ncase w.e_x.e_carrier.h\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : \u2191pullback.snd.base = (\u2191(t D.toGlueData k i).base \u2218 \u2191pullback.fst.base) \u2218 \u2191(inv (t' D.toGlueData k i j)).base\n\u22a2 Function.Injective \u2191(t D.toGlueData k i).base\n[PROOFSTEP]\nswap\n[GOAL]\ncase w.e_x.e_carrier.h\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : \u2191pullback.snd.base = (\u2191(t D.toGlueData k i).base \u2218 \u2191pullback.fst.base) \u2218 \u2191(inv (t' D.toGlueData k i j)).base\n\u22a2 Function.Injective \u2191(t D.toGlueData k i).base\n[PROOFSTEP]\nrefine Function.HasLeftInverse.injective \u27e8(D.t i k).base, fun x => ?_\u27e9\n[GOAL]\ncase w.e_x.e_carrier.h\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : \u2191pullback.snd.base = (\u2191(t D.toGlueData k i).base \u2218 \u2191pullback.fst.base) \u2218 \u2191(inv (t' D.toGlueData k i j)).base\nx : \u2191\u2191(V D.toGlueData (k, i))\n\u22a2 \u2191(t D.toGlueData i k).base (\u2191(t D.toGlueData k i).base x) = x\n[PROOFSTEP]\nrw [\u2190 comp_apply, \u2190 comp_base, D.t_inv, id_base, id_apply]\n[GOAL]\ncase w.e_x.e_carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : \u2191pullback.snd.base = (\u2191(t D.toGlueData k i).base \u2218 \u2191pullback.fst.base) \u2218 \u2191(inv (t' D.toGlueData k i j)).base\n\u22a2 \u2191pullback.fst.base '' (\u2191(inv (t' D.toGlueData k i j)).base '' \u2191U) =\n \u2191pullback.fst.base '' (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191U)\n[PROOFSTEP]\nrefine congr_arg (_ '' \u00b7) ?_\n[GOAL]\ncase w.e_x.e_carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : \u2191pullback.snd.base = (\u2191(t D.toGlueData k i).base \u2218 \u2191pullback.fst.base) \u2218 \u2191(inv (t' D.toGlueData k i j)).base\n\u22a2 \u2191(inv (t' D.toGlueData k i j)).base '' \u2191U = \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191U\n[PROOFSTEP]\nrefine congr_fun ?_ _\n[GOAL]\ncase w.e_x.e_carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : \u2191pullback.snd.base = (\u2191(t D.toGlueData k i).base \u2218 \u2191pullback.fst.base) \u2218 \u2191(inv (t' D.toGlueData k i j)).base\n\u22a2 Set.image \u2191(inv (t' D.toGlueData k i j)).base = Set.preimage \u2191(t' D.toGlueData k i j).base\n[PROOFSTEP]\nrefine Set.image_eq_preimage_of_inverse ?_ ?_\n[GOAL]\ncase w.e_x.e_carrier.refine_1\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : \u2191pullback.snd.base = (\u2191(t D.toGlueData k i).base \u2218 \u2191pullback.fst.base) \u2218 \u2191(inv (t' D.toGlueData k i j)).base\n\u22a2 Function.LeftInverse \u2191(t' D.toGlueData k i j).base \u2191(inv (t' D.toGlueData k i j)).base\n[PROOFSTEP]\nintro x\n[GOAL]\ncase w.e_x.e_carrier.refine_1\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : \u2191pullback.snd.base = (\u2191(t D.toGlueData k i).base \u2218 \u2191pullback.fst.base) \u2218 \u2191(inv (t' D.toGlueData k i j)).base\nx : (CategoryTheory.forget TopCat).obj \u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 \u2191(t' D.toGlueData k i j).base (\u2191(inv (t' D.toGlueData k i j)).base x) = x\n[PROOFSTEP]\nrw [\u2190 comp_apply, \u2190 comp_base, IsIso.inv_hom_id, id_base, id_apply]\n[GOAL]\ncase w.e_x.e_carrier.refine_2\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : \u2191pullback.snd.base = (\u2191(t D.toGlueData k i).base \u2218 \u2191pullback.fst.base) \u2218 \u2191(inv (t' D.toGlueData k i j)).base\n\u22a2 Function.RightInverse \u2191(t' D.toGlueData k i j).base \u2191(inv (t' D.toGlueData k i j)).base\n[PROOFSTEP]\nintro x\n[GOAL]\ncase w.e_x.e_carrier.refine_2\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nthis : \u2191pullback.snd.base = (\u2191(t D.toGlueData k i).base \u2218 \u2191pullback.fst.base) \u2218 \u2191(inv (t' D.toGlueData k i j)).base\nx : (CategoryTheory.forget TopCat).obj \u2191(pullback (f D.toGlueData k i) (f D.toGlueData k j))\n\u22a2 \u2191(inv (t' D.toGlueData k i j)).base (\u2191(t' D.toGlueData k i j).base x) = x\n[PROOFSTEP]\nrw [\u2190 comp_apply, \u2190 comp_base, IsIso.hom_inv_id, id_base, id_apply]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd) U \u226b\n NatTrans.app (t D.toGlueData k i).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop))) =\n NatTrans.app (t' D.toGlueData k i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop\n[PROOFSTEP]\nrw [\u2190 IsIso.eq_inv_comp, IsOpenImmersion.inv_invApp, Category.assoc, (D.t' k i j).c.naturality_assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 NatTrans.app (t D.toGlueData k i).c (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop))) =\n NatTrans.app pullback.snd.c (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n NatTrans.app (t' D.toGlueData k i j).c\n ((Opens.map pullback.snd.base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U))) \u226b\n ((t' D.toGlueData k i j).base _* (pullback (f D.toGlueData k i) (f D.toGlueData k j)).presheaf).map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)).unop) =\n op U)) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop\n[PROOFSTEP]\nsimp_rw [\u2190 Category.assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 NatTrans.app (t D.toGlueData k i).c (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop))) =\n ((NatTrans.app pullback.snd.c (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n NatTrans.app (t' D.toGlueData k i j).c\n ((Opens.map pullback.snd.base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)))) \u226b\n ((t' D.toGlueData k i j).base _* (pullback (f D.toGlueData k i) (f D.toGlueData k j)).presheaf).map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)).unop) =\n op U))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) }\n[PROOFSTEP]\nerw [\u2190 comp_c_app]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 NatTrans.app (t D.toGlueData k i).c (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop))) =\n (NatTrans.app (t' D.toGlueData k i j \u226b pullback.snd).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n ((t' D.toGlueData k i j).base _* (pullback (f D.toGlueData k i) (f D.toGlueData k j)).presheaf).map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)).unop) =\n op U))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) }\n[PROOFSTEP]\nrw [congr_app (D.t_fac k i j), comp_c_app]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 NatTrans.app (t D.toGlueData k i).c (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop))) =\n (((NatTrans.app (t D.toGlueData k i).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n NatTrans.app pullback.fst.c\n (op\n ((Opens.map (t D.toGlueData k i).base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)).unop))) \u226b\n (pullback (f D.toGlueData k i) (f D.toGlueData k j)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (pullback.fst \u226b t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n (Opens.map (t' D.toGlueData k i j \u226b pullback.snd).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U))))) \u226b\n ((t' D.toGlueData k i j).base _* (pullback (f D.toGlueData k i) (f D.toGlueData k j)).presheaf).map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)).unop) =\n op U))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) }\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 NatTrans.app (t D.toGlueData k i).c (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop))) =\n NatTrans.app (t D.toGlueData k i).c (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n NatTrans.app pullback.fst.c\n (op\n ((Opens.map (t D.toGlueData k i).base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)).unop)) \u226b\n (pullback (f D.toGlueData k i) (f D.toGlueData k j)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (pullback.fst \u226b t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n (Opens.map (t' D.toGlueData k i j \u226b pullback.snd).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)))) \u226b\n ((t' D.toGlueData k i j).base _* (pullback (f D.toGlueData k i) (f D.toGlueData k j)).presheaf).map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)).unop) =\n op U)) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) }\n[PROOFSTEP]\nerw [IsOpenImmersion.inv_naturality, IsOpenImmersion.inv_naturality_assoc, IsOpenImmersion.app_inv_app'_assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 NatTrans.app (t D.toGlueData k i).c (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop))) =\n NatTrans.app (t D.toGlueData k i).c (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n ((Opens.map pullback.fst.base).obj\n ((Opens.map (t D.toGlueData k i).base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)).unop)) =\n (Opens.map (t D.toGlueData k i).base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)).unop)).op \u226b\n (V D.toGlueData (k, i)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map (pullback.fst \u226b t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n (Opens.map (t' D.toGlueData k i j \u226b pullback.snd).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U))))) \u226b\n (V D.toGlueData (k, i)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n ((Opens.map (t' D.toGlueData k i j).base).op.map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)).unop) =\n op U))))\ncase h.hU\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 \u2191((Opens.map (t D.toGlueData k i).base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)).unop) \u2286\n Set.range \u2191pullback.fst.base\n[PROOFSTEP]\nsimp_rw [\u2190 (\ud835\udda3.V (k, i)).presheaf.map_comp, eqToHom_map (Functor.op _), eqToHom_op, eqToHom_trans]\n[GOAL]\ncase h.hU\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 \u2191((Opens.map (t D.toGlueData k i).base).obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)).unop) \u2286\n Set.range \u2191pullback.fst.base\n[PROOFSTEP]\nrintro x \u27e8y, -, eq\u27e9\n[GOAL]\ncase h.hU.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nx : \u2191\u2191(V D.toGlueData (k, i))\ny : (CategoryTheory.forget TopCat).obj \u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\neq : \u2191pullback.snd.base y = \u2191(t D.toGlueData k i).base x\n\u22a2 x \u2208 Set.range \u2191pullback.fst.base\n[PROOFSTEP]\nreplace eq := ConcreteCategory.congr_arg (\ud835\udda3.t i k).base eq\n[GOAL]\ncase h.hU.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nx : \u2191\u2191(V D.toGlueData (k, i))\ny : (CategoryTheory.forget TopCat).obj \u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\neq : \u2191(t D.toGlueData i k).base (\u2191pullback.snd.base y) = \u2191(t D.toGlueData i k).base (\u2191(t D.toGlueData k i).base x)\n\u22a2 x \u2208 Set.range \u2191pullback.fst.base\n[PROOFSTEP]\nchange ((\u03c0\u2082 i, j, k) \u226b D.t i k).base y = (D.t k i \u226b D.t i k).base x at eq \n[GOAL]\ncase h.hU.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nx : \u2191\u2191(V D.toGlueData (k, i))\ny : (CategoryTheory.forget TopCat).obj \u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\neq : \u2191(pullback.snd \u226b t D.toGlueData i k).base y = \u2191(t D.toGlueData k i \u226b t D.toGlueData i k).base x\n\u22a2 x \u2208 Set.range \u2191pullback.fst.base\n[PROOFSTEP]\nrw [\ud835\udda3.t_inv, id_base, TopCat.id_app] at eq \n[GOAL]\ncase h.hU.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\nx : \u2191\u2191(V D.toGlueData (k, i))\ny : (CategoryTheory.forget TopCat).obj \u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\neq : \u2191(pullback.snd \u226b t D.toGlueData i k).base y = x\n\u22a2 x \u2208 Set.range \u2191pullback.fst.base\n[PROOFSTEP]\nsubst eq\n[GOAL]\ncase h.hU.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\ny : (CategoryTheory.forget TopCat).obj \u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 \u2191(pullback.snd \u226b t D.toGlueData i k).base y \u2208 Set.range \u2191pullback.fst.base\n[PROOFSTEP]\nuse(inv (D.t' k i j)).base y\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\ny : (CategoryTheory.forget TopCat).obj \u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 \u2191pullback.fst.base (\u2191(inv (t' D.toGlueData k i j)).base y) = \u2191(pullback.snd \u226b t D.toGlueData i k).base y\n[PROOFSTEP]\nchange (inv (D.t' k i j) \u226b \u03c0\u2081 k, i, j).base y = _\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\ny : (CategoryTheory.forget TopCat).obj \u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 \u2191(inv (t' D.toGlueData k i j) \u226b pullback.fst).base y = \u2191(pullback.snd \u226b t D.toGlueData i k).base y\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase h.e_a.e_self\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\ny : (CategoryTheory.forget TopCat).obj \u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 inv (t' D.toGlueData k i j) \u226b pullback.fst = pullback.snd \u226b t D.toGlueData i k\n[PROOFSTEP]\nrw [IsIso.inv_comp_eq, \ud835\udda3.t_fac_assoc, \ud835\udda3.t_inv, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\n\u22a2 IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd) U \u226b\n NatTrans.app (t D.toGlueData k i).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n NatTrans.app (t' D.toGlueData k i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U))))\n[PROOFSTEP]\nhave e := (D.snd_invApp_t_app' i j k U).choose_spec\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\ne :\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd) U \u226b\n NatTrans.app (t D.toGlueData k i).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop))) =\n NatTrans.app (t' D.toGlueData k i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop\n\u22a2 IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd) U \u226b\n NatTrans.app (t D.toGlueData k i).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n NatTrans.app (t' D.toGlueData k i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U))))\n[PROOFSTEP]\nreplace e := reassoc_of% e\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\ne :\n \u2200 {Z : C}\n (h :\n (V D.toGlueData (k, i)).presheaf.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop)) \u27f6\n Z),\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd) U \u226b\n NatTrans.app (t D.toGlueData k i).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop))) \u226b\n h =\n NatTrans.app (t' D.toGlueData k i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n h\n\u22a2 IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd) U \u226b\n NatTrans.app (t D.toGlueData k i).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n NatTrans.app (t' D.toGlueData k i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U))))\n[PROOFSTEP]\nrw [\u2190 e]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens \u2191\u2191(pullback (f D.toGlueData i j) (f D.toGlueData i k))\ne :\n \u2200 {Z : C}\n (h :\n (V D.toGlueData (k, i)).presheaf.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop)) \u27f6\n Z),\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd) U \u226b\n NatTrans.app (t D.toGlueData k i).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop))) \u226b\n h =\n NatTrans.app (t' D.toGlueData k i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n h\n\u22a2 IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd) U \u226b\n NatTrans.app (t D.toGlueData k i).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd) U \u226b\n NatTrans.app (t D.toGlueData k i).c\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U)) =\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop))) \u226b\n (V D.toGlueData (k, i)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n { carrier := \u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(t' D.toGlueData k i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (t D.toGlueData k i).base).op.obj\n (op ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj U))))\n[PROOFSTEP]\nsimp [eqToHom_map]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 (Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n ((Opens.map (t D.toGlueData j i).base).obj ((Opens.map (f D.toGlueData i j).base).obj U))\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)) =\n \u2191((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n ((Opens.map (t D.toGlueData j i).base).obj ((Opens.map (f D.toGlueData i j).base).obj U)))\n[PROOFSTEP]\ndsimp only [Opens.map_coe, IsOpenMap.functor_obj_coe]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191(\u03b9 D.toGlueData j).base \u207b\u00b9' (\u2191(\u03b9 D.toGlueData i).base '' \u2191U) =\n \u2191(f D.toGlueData j i).base '' (\u2191(t D.toGlueData j i).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U))\n[PROOFSTEP]\nrw [\u2190 show _ = (\ud835\udda3.\u03b9 i).base from \ud835\udda3.\u03b9_gluedIso_inv (PresheafedSpace.forget _) i, \u2190\n show _ = (\ud835\udda3.\u03b9 j).base from \ud835\udda3.\u03b9_gluedIso_inv (PresheafedSpace.forget _) j]\n -- Porting note : change `rw` to `erw` on `coe_comp`\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191(\u03b9 (mapGlueData D.toGlueData (forget C)) j \u226b (gluedIso D.toGlueData (forget C)).inv) \u207b\u00b9'\n (\u2191(\u03b9 (mapGlueData D.toGlueData (forget C)) i \u226b (gluedIso D.toGlueData (forget C)).inv) '' \u2191U) =\n \u2191(f D.toGlueData j i).base '' (\u2191(t D.toGlueData j i).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U))\n[PROOFSTEP]\nerw [coe_comp, coe_comp, coe_comp]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 (\u2191(preservesColimitIso (forget C) (MultispanIndex.multispan (diagram D.toGlueData))).inv \u2218\n \u2191(HasColimit.isoOfNatIso (diagramIso D.toGlueData (forget C))).inv) \u2218\n \u2191(\u03b9 (mapGlueData D.toGlueData (forget C)) j) \u207b\u00b9'\n (\u2191(gluedIso D.toGlueData (forget C)).inv \u2218 \u2191(\u03b9 (mapGlueData D.toGlueData (forget C)) i) '' \u2191U) =\n \u2191(f D.toGlueData j i).base '' (\u2191(t D.toGlueData j i).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U))\n[PROOFSTEP]\nrw [Set.image_comp, Set.preimage_comp]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191(\u03b9 (mapGlueData D.toGlueData (forget C)) j) \u207b\u00b9'\n (\u2191(preservesColimitIso (forget C) (MultispanIndex.multispan (diagram D.toGlueData))).inv \u2218\n \u2191(HasColimit.isoOfNatIso (diagramIso D.toGlueData (forget C))).inv \u207b\u00b9'\n (\u2191(gluedIso D.toGlueData (forget C)).inv '' (\u2191(\u03b9 (mapGlueData D.toGlueData (forget C)) i) '' \u2191U))) =\n \u2191(f D.toGlueData j i).base '' (\u2191(t D.toGlueData j i).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U))\n[PROOFSTEP]\nerw [Set.preimage_image_eq]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191(\u03b9 (mapGlueData D.toGlueData (forget C)) j) \u207b\u00b9' (\u2191(\u03b9 (mapGlueData D.toGlueData (forget C)) i) '' \u2191U) =\n \u2191(f D.toGlueData j i).base '' (\u2191(t D.toGlueData j i).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U))\n[PROOFSTEP]\nrefine' Eq.trans (D.toTopGlueData.preimage_image_eq_image' _ _ _) _\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191(t (toTopGlueData D).toGlueData i j \u226b f (toTopGlueData D).toGlueData j i) ''\n (\u2191(f (toTopGlueData D).toGlueData i j) \u207b\u00b9' \u2191U) =\n \u2191(f D.toGlueData j i).base '' (\u2191(t D.toGlueData j i).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191((t D.toGlueData i j).base \u226b (f D.toGlueData j i).base) '' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U) =\n \u2191(f D.toGlueData j i).base '' (\u2191(t D.toGlueData j i).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U))\n[PROOFSTEP]\nrw [coe_comp, Set.image_comp]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191(f D.toGlueData j i).base '' (\u2191(t D.toGlueData i j).base '' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U)) =\n \u2191(f D.toGlueData j i).base '' (\u2191(t D.toGlueData j i).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U))\n[PROOFSTEP]\nrefine congr_arg (_ '' \u00b7) ?_\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191(t D.toGlueData i j).base '' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U) =\n \u2191(t D.toGlueData j i).base \u207b\u00b9' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U)\n[PROOFSTEP]\nrw [Set.eq_preimage_iff_image_eq, \u2190 Set.image_comp]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191(t D.toGlueData j i).base \u2218 \u2191(t D.toGlueData i j).base '' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U) =\n \u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U\ncase h.hf\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Function.Bijective \u2191(t D.toGlueData j i).base\n[PROOFSTEP]\nswap\n[GOAL]\ncase h.hf\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Function.Bijective \u2191(t D.toGlueData j i).base\n[PROOFSTEP]\napply CategoryTheory.ConcreteCategory.bijective_of_isIso\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191(t D.toGlueData j i).base \u2218 \u2191(t D.toGlueData i j).base '' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U) =\n \u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U\n[PROOFSTEP]\nchange (D.t i j \u226b D.t j i).base '' _ = _\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191(t D.toGlueData i j \u226b t D.toGlueData j i).base '' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U) =\n \u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U\n[PROOFSTEP]\nrw [\ud835\udda3.t_inv]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191(\ud835\udfd9 (V D.toGlueData (i, j))).base '' (\u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U) = \u2191(f D.toGlueData i j).base \u207b\u00b9' \u2191U\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Function.Injective\n (\u2191(preservesColimitIso (forget C) (MultispanIndex.multispan (diagram D.toGlueData))).inv \u2218\n \u2191(HasColimit.isoOfNatIso (diagramIso D.toGlueData (forget C))).inv)\n[PROOFSTEP]\nrw [\u2190 coe_comp, \u2190 TopCat.mono_iff_injective]\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Mono\n ((HasColimit.isoOfNatIso (diagramIso D.toGlueData (forget C))).inv \u226b\n (preservesColimitIso (forget C) (MultispanIndex.multispan (diagram D.toGlueData))).inv)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2203 eq,\n opensImagePreimageMap D i j U \u226b\n NatTrans.app (f D.toGlueData j k).c\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop \u226b\n (V D.toGlueData (j, k)).presheaf.map (eqToHom eq)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 opensImagePreimageMap D i j U \u226b\n NatTrans.app (f D.toGlueData j k).c\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop \u226b\n (V D.toGlueData (j, k)).presheaf.map (eqToHom ?w)\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\n[PROOFSTEP]\ndelta opensImagePreimageMap\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 (NatTrans.app (f D.toGlueData i j).c (op U) \u226b\n NatTrans.app (t D.toGlueData j i).c ((Opens.map (f D.toGlueData i j).base).op.obj (op U)) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion (f D.toGlueData j i))\n {\n unop :=\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) } }.unop \u226b\n (CategoryTheory.GlueData.U D.toGlueData j).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n ((Opens.map (t D.toGlueData j i).base).obj\n ((Opens.map (f D.toGlueData i j).base).obj U)))).op) \u226b\n NatTrans.app (f D.toGlueData j k).c\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop \u226b\n (V D.toGlueData (j, k)).presheaf.map (eqToHom ?w)\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 NatTrans.app (f D.toGlueData i j).c (op U) \u226b\n NatTrans.app (t D.toGlueData j i).c ((Opens.map (f D.toGlueData i j).base).op.obj (op U)) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion (f D.toGlueData j i))\n { carrier := \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) } \u226b\n (CategoryTheory.GlueData.U D.toGlueData j).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n ((Opens.map (t D.toGlueData j i).base).obj ((Opens.map (f D.toGlueData i j).base).obj U)))).op \u226b\n NatTrans.app (f D.toGlueData j k).c\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map (eqToHom ?w)\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\n[PROOFSTEP]\nrw [(D.f j k).c.naturality, f_invApp_f_app_assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 NatTrans.app (f D.toGlueData i j).c (op U) \u226b\n NatTrans.app (t D.toGlueData j i).c ((Opens.map (f D.toGlueData i j).base).op.obj (op U)) \u226b\n NatTrans.app pullback.fst.c\n (op\n { carrier := \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n carrier :=\n \u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj\n (op U)).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj\n (op U)).unop)) }).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) })))) \u226b\n ((f D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n ((Opens.map (t D.toGlueData j i).base).obj\n ((Opens.map (f D.toGlueData i j).base).obj U)))).op =\n NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map (eqToHom ?w)\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\n[PROOFSTEP]\nerw [\u2190 (D.V (j, k)).presheaf.map_comp]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 NatTrans.app (f D.toGlueData i j).c (op U) \u226b\n NatTrans.app (t D.toGlueData j i).c ((Opens.map (f D.toGlueData i j).base).op.obj (op U)) \u226b\n NatTrans.app pullback.fst.c\n (op\n { carrier := \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n carrier :=\n \u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj\n (op U)).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj\n (op U)).unop)) }).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }))) \u226b\n (Opens.map (f D.toGlueData j k).base).op.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n ((Opens.map (t D.toGlueData j i).base).obj\n ((Opens.map (f D.toGlueData i j).base).obj U)))).op) =\n NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map (eqToHom ?w)\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\n[PROOFSTEP]\nsimp_rw [\u2190 Category.assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 (((NatTrans.app (f D.toGlueData i j).c (op U) \u226b\n NatTrans.app (t D.toGlueData j i).c ((Opens.map (f D.toGlueData i j).base).op.obj (op U))) \u226b\n NatTrans.app pullback.fst.c\n (op\n { carrier := \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) })) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n carrier :=\n \u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }).unop)) }) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj\n (op U)).unop)) }).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }))) \u226b\n (Opens.map (f D.toGlueData j k).base).op.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n ((Opens.map (t D.toGlueData j i).base).obj ((Opens.map (f D.toGlueData i j).base).obj U)))).op) =\n (NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) }) \u226b\n (V D.toGlueData (j, k)).presheaf.map (eqToHom ?w)\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\n[PROOFSTEP]\nerw [\u2190 comp_c_app, \u2190 comp_c_app]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 (NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n carrier :=\n \u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }).unop)) }) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj\n (op U)).unop)) }).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }))) \u226b\n (Opens.map (f D.toGlueData j k).base).op.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n ((Opens.map (t D.toGlueData j i).base).obj ((Opens.map (f D.toGlueData i j).base).obj U)))).op) =\n (NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) }) \u226b\n (V D.toGlueData (j, k)).presheaf.map (eqToHom ?w)\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n carrier :=\n \u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.fst.base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj\n (op U)).unop)) }).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n {\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }))) \u226b\n (Opens.map (f D.toGlueData j k).base).op.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n ((Opens.map (t D.toGlueData j i).base).obj\n ((Opens.map (f D.toGlueData i j).base).obj U)))).op) =\n NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map (eqToHom ?w)\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\n[PROOFSTEP]\ndsimp only [Functor.op, unop_op, Quiver.Hom.unop_op]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n carrier :=\n \u2191pullback.fst.base \u207b\u00b9'\n \u2191{ carrier := \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).obj U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) },\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.fst.base \u207b\u00b9'\n \u2191{ carrier := \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).obj U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) })) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n carrier :=\n \u2191pullback.fst.base \u207b\u00b9'\n \u2191{ carrier := \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).obj U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) },\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.fst.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).obj U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) })) }) =\n op\n ((Opens.map (f D.toGlueData j k).base).obj\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n { carrier := \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).obj U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) }))) \u226b\n ((Opens.map (f D.toGlueData j k).base).map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n ((Opens.map (t D.toGlueData j i).base).obj\n ((Opens.map (f D.toGlueData i j).base).obj U))))).op) =\n NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191U,\n is_open' := (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191U)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map (eqToHom ?w)\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\n[PROOFSTEP]\nrw [eqToHom_map (Opens.map _), eqToHom_op, eqToHom_trans]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n carrier :=\n \u2191pullback.fst.base \u207b\u00b9'\n \u2191{ carrier := \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).obj U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) },\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.fst.base \u207b\u00b9'\n \u2191{ carrier := \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).obj U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) })) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n carrier :=\n \u2191pullback.fst.base \u207b\u00b9'\n \u2191{ carrier := \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).obj U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) },\n is_open' :=\n (_ :\n IsOpen\n (\u2191pullback.fst.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(t D.toGlueData j i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i j).base).obj U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData j i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) })) }) =\n op\n ((Opens.map (f D.toGlueData j k).base).obj\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) =\n NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191U,\n is_open' := (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191U)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map (eqToHom ?w)\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\n[PROOFSTEP]\ncongr\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j k : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nX' : C\nf' :\n ((f D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u27f6\n X'\n\u22a2 opensImagePreimageMap D i j U \u226b\n NatTrans.app (f D.toGlueData j k).c\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n f' =\n NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n f'\n[PROOFSTEP]\nsimpa only [Category.assoc] using congr_arg (\u00b7 \u226b f') (opensImagePreimageMap_app D i j k U)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj : WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom\n\u22a2 (CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U) \u27f6 (diagramOverOpen D U).obj (op j)\n[PROOFSTEP]\nrcases j with (\u27e8j, k\u27e9 | j)\n[GOAL]\ncase left.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 (CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U) \u27f6\n (diagramOverOpen D U).obj (op (WalkingMultispan.left (j, k)))\n[PROOFSTEP]\nrefine' D.opensImagePreimageMap i j U \u226b (D.f j k).c.app _ \u226b (D.V (j, k)).presheaf.map (eqToHom _)\n[GOAL]\ncase left.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))\n[PROOFSTEP]\nrw [Functor.op_obj]\n[GOAL]\ncase left.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 op\n ((Opens.map (f D.toGlueData j k).base).obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))).unop) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase left.mk.e_x\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 (Opens.map (f D.toGlueData j k).base).obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))).unop =\n (Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)\n[PROOFSTEP]\next1\n[GOAL]\ncase left.mk.e_x.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 \u2191((Opens.map (f D.toGlueData j k).base).obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))).unop) =\n \u2191((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))\n[PROOFSTEP]\ndsimp only [Functor.op_obj, Opens.map_coe, unop_op, IsOpenMap.functor_obj_coe]\n[GOAL]\ncase left.mk.e_x.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 \u2191(f D.toGlueData j k).base \u207b\u00b9' (\u2191(\u03b9 D.toGlueData j).base \u207b\u00b9' (\u2191(\u03b9 D.toGlueData i).base '' \u2191U)) =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.left (j, k))).base \u207b\u00b9'\n (\u2191(\u03b9 D.toGlueData i).base '' \u2191U)\n[PROOFSTEP]\nrw [Set.preimage_preimage]\n[GOAL]\ncase left.mk.e_x.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 (fun x => \u2191(\u03b9 D.toGlueData j).base (\u2191(f D.toGlueData j k).base x)) \u207b\u00b9' (\u2191(\u03b9 D.toGlueData i).base '' \u2191U) =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.left (j, k))).base \u207b\u00b9'\n (\u2191(\u03b9 D.toGlueData i).base '' \u2191U)\n[PROOFSTEP]\nchange\n (D.f j k \u226b \ud835\udda3.\u03b9 j).base \u207b\u00b9' _ =\n _\n -- Porting note : used to be `congr 3`\n[GOAL]\ncase left.mk.e_x.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 \u2191(f D.toGlueData j k \u226b \u03b9 D.toGlueData j).base \u207b\u00b9' (\u2191(\u03b9 D.toGlueData i).base '' \u2191U) =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.left (j, k))).base \u207b\u00b9'\n (\u2191(\u03b9 D.toGlueData i).base '' \u2191U)\n[PROOFSTEP]\nrefine congr_arg (\u00b7 \u207b\u00b9' _) ?_\n[GOAL]\ncase left.mk.e_x.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 \u2191(f D.toGlueData j k \u226b \u03b9 D.toGlueData j).base =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.left (j, k))).base\n[PROOFSTEP]\nconvert congr_arg (ContinuousMap.toFun (\u03b1 := D.V \u27e8j, k\u27e9) (\u03b2 := D.glued) \u00b7) ?_\n[GOAL]\ncase left.mk.e_x.h.convert_3\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 (f D.toGlueData j k \u226b \u03b9 D.toGlueData j).base =\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.left (j, k))).base\n[PROOFSTEP]\nrefine congr_arg (PresheafedSpace.Hom.base (C := C) \u00b7) ?_\n[GOAL]\ncase left.mk.e_x.h.convert_3\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 f D.toGlueData j k \u226b \u03b9 D.toGlueData j =\n colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.left (j, k))\n[PROOFSTEP]\nexact colimit.w \ud835\udda3.diagram.multispan (WalkingMultispan.Hom.fst (j, k))\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj : (diagram D.toGlueData).R\n\u22a2 (CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U) \u27f6\n (diagramOverOpen D U).obj (op (WalkingMultispan.right j))\n[PROOFSTEP]\nexact D.opensImagePreimageMap i j U\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nX Y : (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56\nf' : X \u27f6 Y\n\u22a2 ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56).obj\n ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map\n f' \u226b\n (fun j => \u03b9InvApp\u03c0App D U j.unop) Y =\n (fun j => \u03b9InvApp\u03c0App D U j.unop) X \u226b (diagramOverOpen D U).map f'\n[PROOFSTEP]\ninduction X using Opposite.rec' with\n| h X => ?_\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nX Y : (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56\nf' : X \u27f6 Y\n\u22a2 ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56).obj\n ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map\n f' \u226b\n (fun j => \u03b9InvApp\u03c0App D U j.unop) Y =\n (fun j => \u03b9InvApp\u03c0App D U j.unop) X \u226b (diagramOverOpen D U).map f'\n[PROOFSTEP]\ninduction X using Opposite.rec' with\n| h X => ?_\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nY : (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56\nX : WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom\nf' : op X \u27f6 Y\n\u22a2 ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56).obj\n ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map\n f' \u226b\n (fun j => \u03b9InvApp\u03c0App D U j.unop) Y =\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op X) \u226b (diagramOverOpen D U).map f'\n[PROOFSTEP]\ninduction Y using Opposite.rec' with\n| h Y => ?_\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nY : (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56\nX : WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom\nf' : op X \u27f6 Y\n\u22a2 ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56).obj\n ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map\n f' \u226b\n (fun j => \u03b9InvApp\u03c0App D U j.unop) Y =\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op X) \u226b (diagramOverOpen D U).map f'\n[PROOFSTEP]\ninduction Y using Opposite.rec' with\n| h Y => ?_\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nX Y : WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom\nf' : op X \u27f6 op Y\n\u22a2 ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56).obj\n ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map\n f' \u226b\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op Y) =\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op X) \u226b (diagramOverOpen D U).map f'\n[PROOFSTEP]\nlet f : Y \u27f6 X := f'.unop\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nX Y : WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom\nf' : op X \u27f6 op Y\nf : Y \u27f6 X := f'.unop\n\u22a2 ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56).obj\n ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map\n f' \u226b\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op Y) =\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op X) \u226b (diagramOverOpen D U).map f'\n[PROOFSTEP]\nhave : f' = f.op := rfl\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nX Y : WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom\nf' : op X \u27f6 op Y\nf : Y \u27f6 X := f'.unop\nthis : f' = f.op\n\u22a2 ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56).obj\n ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map\n f' \u226b\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op Y) =\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op X) \u226b (diagramOverOpen D U).map f'\n[PROOFSTEP]\nclear_value f\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nX Y : WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom\nf' : op X \u27f6 op Y\nf : Y \u27f6 X\nthis : f' = f.op\n\u22a2 ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56).obj\n ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map\n f' \u226b\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op Y) =\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op X) \u226b (diagramOverOpen D U).map f'\n[PROOFSTEP]\nsubst this\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nX Y : WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom\nf : Y \u27f6 X\n\u22a2 ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56).obj\n ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map\n f.op \u226b\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op Y) =\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op X) \u226b (diagramOverOpen D U).map f.op\n[PROOFSTEP]\nrcases f with (_ | \u27e8j, k\u27e9 | \u27e8j, k\u27e9)\n[GOAL]\ncase h.h.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nX : WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom\n\u22a2 ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56).obj\n ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map\n (WalkingMultispan.Hom.id X).op \u226b\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op X) =\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op X) \u226b (diagramOverOpen D U).map (WalkingMultispan.Hom.id X).op\n[PROOFSTEP]\nerw [Category.id_comp, CategoryTheory.Functor.map_id]\n[GOAL]\ncase h.h.id\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nX : WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom\n\u22a2 (fun j => \u03b9InvApp\u03c0App D U j.unop) (op X) =\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op X) \u226b \ud835\udfd9 ((diagramOverOpen D U).obj (op X))\n[PROOFSTEP]\nrw [Category.comp_id]\n[GOAL]\ncase h.h.fst.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56).obj\n ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map\n (WalkingMultispan.Hom.fst (j, k)).op \u226b\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op (WalkingMultispan.left (j, k))) =\n (fun j => \u03b9InvApp\u03c0App D U j.unop)\n (op (WalkingMultispan.right (MultispanIndex.fstFrom (diagram D.toGlueData) (j, k)))) \u226b\n (diagramOverOpen D U).map (WalkingMultispan.Hom.fst (j, k)).op\n[PROOFSTEP]\nerw [Category.id_comp]\n[GOAL]\ncase h.h.fst.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 (fun j => \u03b9InvApp\u03c0App D U j.unop) (op (WalkingMultispan.left (j, k))) =\n (fun j => \u03b9InvApp\u03c0App D U j.unop)\n (op (WalkingMultispan.right (MultispanIndex.fstFrom (diagram D.toGlueData) (j, k)))) \u226b\n (diagramOverOpen D U).map (WalkingMultispan.Hom.fst (j, k)).op\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.h.snd.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56).obj\n ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map\n (WalkingMultispan.Hom.snd (j, k)).op \u226b\n (fun j => \u03b9InvApp\u03c0App D U j.unop) (op (WalkingMultispan.left (j, k))) =\n (fun j => \u03b9InvApp\u03c0App D U j.unop)\n (op (WalkingMultispan.right (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))) \u226b\n (diagramOverOpen D U).map (WalkingMultispan.Hom.snd (j, k)).op\n[PROOFSTEP]\nerw [Category.id_comp]\n -- It remains to show that the blue is equal to red + green in the original diagram.\n -- The proof strategy is illustrated in ![this diagram](https://i.imgur.com/mBzV1Rx.png)\n -- where we prove red = pink = light-blue = green = blue.\n[GOAL]\ncase h.h.snd.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 (fun j => \u03b9InvApp\u03c0App D U j.unop) (op (WalkingMultispan.left (j, k))) =\n (fun j => \u03b9InvApp\u03c0App D U j.unop)\n (op (WalkingMultispan.right (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))) \u226b\n (diagramOverOpen D U).map (WalkingMultispan.Hom.snd (j, k)).op\n[PROOFSTEP]\nchange\n D.opensImagePreimageMap i j U \u226b (D.f j k).c.app _ \u226b (D.V (j, k)).presheaf.map (eqToHom _) =\n D.opensImagePreimageMap _ _ _ \u226b ((D.f k j).c.app _ \u226b (D.t j k).c.app _) \u226b (D.V (j, k)).presheaf.map (eqToHom _)\n[GOAL]\ncase h.h.snd.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 opensImagePreimageMap D i j U \u226b\n NatTrans.app (f D.toGlueData j k).c\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n opensImagePreimageMap D i k U \u226b\n (NatTrans.app (f D.toGlueData k j).c\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n NatTrans.app (t D.toGlueData j k).c\n ((Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [opensImagePreimageMap_app_assoc]\n[GOAL]\ncase h.h.snd.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } }.unop \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n opensImagePreimageMap D i k U \u226b\n (NatTrans.app (f D.toGlueData k j).c\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n NatTrans.app (t D.toGlueData j k).c\n ((Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase h.h.snd.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n opensImagePreimageMap D i k U \u226b\n NatTrans.app (f D.toGlueData k j).c\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n NatTrans.app (t D.toGlueData j k).c\n ((Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [opensImagePreimageMap_app_assoc, (D.t j k).c.naturality_assoc]\n[GOAL]\ncase h.h.snd.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n NatTrans.app (pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop)) } }.unop \u226b\n NatTrans.app (t D.toGlueData j k).c\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nrw [snd_invApp_t_app_assoc]\n[GOAL]\ncase h.h.snd.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n NatTrans.app (pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).c (op U) \u226b\n NatTrans.app (t' D.toGlueData j k i).c\n (op\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop)) } }.unop) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [\u2190 PresheafedSpace.comp_c_app_assoc]\n -- light-blue = green is relatively easy since the part that differs does not involve\n -- partial inverses.\n[GOAL]\ncase h.h.snd.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n NatTrans.app (t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nhave : D.t' j k i \u226b (\u03c0\u2081 k, i, j) \u226b D.t k i \u226b \ud835\udda3.f i k = (pullbackSymmetry _ _).hom \u226b (\u03c0\u2081 j, i, k) \u226b D.t j i \u226b D.f i j :=\n by\n rw [\u2190 \ud835\udda3.t_fac_assoc, \ud835\udda3.t'_comp_eq_pullbackSymmetry_assoc, pullbackSymmetry_hom_comp_snd_assoc, pullback.condition,\n \ud835\udda3.t_fac_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n[PROOFSTEP]\nrw [\u2190 \ud835\udda3.t_fac_assoc, \ud835\udda3.t'_comp_eq_pullbackSymmetry_assoc, pullbackSymmetry_hom_comp_snd_assoc, pullback.condition,\n \ud835\udda3.t_fac_assoc]\n[GOAL]\ncase h.h.snd.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n NatTrans.app (t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nrw [congr_app this]\n[GOAL]\ncase h.h.snd.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n (NatTrans.app\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c\n (op U) \u226b\n (pullback (f D.toGlueData j k) (f D.toGlueData j i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map (t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U)))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [PresheafedSpace.comp_c_app_assoc (pullbackSymmetry _ _).hom]\n[GOAL]\ncase h.h.snd.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n (NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n NatTrans.app (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.c\n (op ((Opens.map (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).obj (op U).unop)) \u226b\n (pullback (f D.toGlueData j k) (f D.toGlueData j i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U)))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase h.h.snd.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n NatTrans.app (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).c (op U) \u226b\n NatTrans.app (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.c\n (op ((Opens.map (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).obj (op U).unop)) \u226b\n (pullback (f D.toGlueData j k) (f D.toGlueData j i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) }).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n NatTrans.app (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.c\n (op ((Opens.map (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).obj (op U).unop)) \u226b\n (pullback (f D.toGlueData j k) (f D.toGlueData j i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map (t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) }).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nrw [\u2190 IsIso.eq_inv_comp]\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n inv\n (IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) }) \u226b\n NatTrans.app (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.c\n (op ((Opens.map (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).obj (op U).unop)) \u226b\n (pullback (f D.toGlueData j k) (f D.toGlueData j i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) }).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [IsOpenImmersion.inv_invApp]\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n (NatTrans.app pullback.snd.c\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) \u226b\n (pullback (f D.toGlueData j i) (f D.toGlueData j k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })).unop) =\n op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) }))) \u226b\n NatTrans.app (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.c\n (op ((Opens.map (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).obj (op U).unop)) \u226b\n (pullback (f D.toGlueData j k) (f D.toGlueData j i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) }).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n NatTrans.app pullback.snd.c\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) \u226b\n (pullback (f D.toGlueData j i) (f D.toGlueData j k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })).unop) =\n op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) \u226b\n NatTrans.app (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.c\n (op ((Opens.map (pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).obj (op U).unop)) \u226b\n (pullback (f D.toGlueData j k) (f D.toGlueData j i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b\n pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) }).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b\n f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [NatTrans.naturality_assoc, \u2190 PresheafedSpace.comp_c_app_assoc, congr_app (pullbackSymmetry_hom_comp_snd _ _)]\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n (NatTrans.app pullback.fst.c\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) \u226b\n (pullback (f D.toGlueData j k) (f D.toGlueData j i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map pullback.fst.base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) =\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b pullback.snd).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }))))) \u226b\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.base _*\n (pullback (f D.toGlueData j k) (f D.toGlueData j i)).presheaf).map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })).unop) =\n op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) \u226b\n (pullback (f D.toGlueData j k) (f D.toGlueData j i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) }).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n NatTrans.app pullback.fst.c\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ : IsOpen (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) \u226b\n (pullback (f D.toGlueData j k) (f D.toGlueData j i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map pullback.fst.base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) =\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b pullback.snd).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })))) \u226b\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.base _*\n (pullback (f D.toGlueData j k) (f D.toGlueData j i)).presheaf).map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })).unop) =\n op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) \u226b\n (pullback (f D.toGlueData j k) (f D.toGlueData j i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b\n pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.fst)\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) }).unop)) } \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b\n f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [IsOpenImmersion.inv_naturality_assoc, IsOpenImmersion.inv_naturality_assoc, IsOpenImmersion.inv_naturality_assoc,\n IsOpenImmersion.app_invApp_assoc]\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n (V D.toGlueData (j, k)).presheaf.map\n (homOfLE\n (_ :\n \u2191pullback.fst.base ''\n (\u2191pullback.fst.base \u207b\u00b9'\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier) \u2286\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier)).op \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map pullback.fst.base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) =\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b pullback.snd).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n ((Opens.map (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.base).op.map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })).unop) =\n op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b\n pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U)))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nrepeat'\n erw [\u2190 (D.V (j, k)).presheaf.map_comp]\n -- Porting note : was just `congr`\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n (V D.toGlueData (j, k)).presheaf.map\n (homOfLE\n (_ :\n \u2191pullback.fst.base ''\n (\u2191pullback.fst.base \u207b\u00b9'\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier) \u2286\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier)).op \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map pullback.fst.base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) =\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b pullback.snd).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n ((Opens.map (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.base).op.map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })).unop) =\n op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b\n pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U)))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [\u2190 (D.V (j, k)).presheaf.map_comp]\n -- Porting note : was just `congr`\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) \u226b\n eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n (V D.toGlueData (j, k)).presheaf.map\n (homOfLE\n (_ :\n \u2191pullback.fst.base ''\n (\u2191pullback.fst.base \u207b\u00b9'\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier) \u2286\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier)).op \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map pullback.fst.base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) =\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b pullback.snd).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n ((Opens.map (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.base).op.map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })).unop) =\n op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b\n pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U)))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n ((t D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [\u2190 (D.V (j, k)).presheaf.map_comp]\n -- Porting note : was just `congr`\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) \u226b\n eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n (V D.toGlueData (j, k)).presheaf.map\n (homOfLE\n (_ :\n \u2191pullback.fst.base ''\n (\u2191pullback.fst.base \u207b\u00b9'\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier) \u2286\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier)).op \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map pullback.fst.base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) =\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b pullback.snd).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n ((Opens.map (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.base).op.map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })).unop) =\n op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b\n pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U)))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop)))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((Opens.map (t D.toGlueData j k).base).op.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [\u2190 (D.V (j, k)).presheaf.map_comp]\n -- Porting note : was just `congr`\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) \u226b\n eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n (V D.toGlueData (j, k)).presheaf.map\n (homOfLE\n (_ :\n \u2191pullback.fst.base ''\n (\u2191pullback.fst.base \u207b\u00b9'\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier) \u2286\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier)).op \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map pullback.fst.base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) =\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b pullback.snd).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n ((Opens.map (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.base).op.map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })).unop) =\n op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b\n pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U)))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop))) \u226b\n (Opens.map (t D.toGlueData j k).base).op.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [\u2190 (D.V (j, k)).presheaf.map_comp]\n -- Porting note : was just `congr`\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) \u226b\n eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n (V D.toGlueData (j, k)).presheaf.map\n (homOfLE\n (_ :\n \u2191pullback.fst.base ''\n (\u2191pullback.fst.base \u207b\u00b9'\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier) \u2286\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier)).op \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map pullback.fst.base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) =\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b pullback.snd).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n ((Opens.map (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.base).op.map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })).unop) =\n op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b\n pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U))) \u226b\n eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop))) \u226b\n (Opens.map (t D.toGlueData j k).base).op.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [\u2190 (D.V (j, k)).presheaf.map_comp]\n -- Porting note : was just `congr`\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) \u226b\n eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n (V D.toGlueData (j, k)).presheaf.map\n (homOfLE\n (_ :\n \u2191pullback.fst.base ''\n (\u2191pullback.fst.base \u207b\u00b9'\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier) \u2286\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier)).op \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map pullback.fst.base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) =\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b pullback.snd).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }))))) \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n ((Opens.map (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.base).op.map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })).unop) =\n op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }))) \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b\n pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U))) \u226b\n eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop))) \u226b\n (Opens.map (t D.toGlueData j k).base).op.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [\u2190 (D.V (j, k)).presheaf.map_comp]\n -- Porting note : was just `congr`\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) \u226b\n eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n (V D.toGlueData (j, k)).presheaf.map\n (homOfLE\n (_ :\n \u2191pullback.fst.base ''\n (\u2191pullback.fst.base \u207b\u00b9'\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier) \u2286\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier)).op \u226b\n (V D.toGlueData (j, k)).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map pullback.fst.base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) =\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b pullback.snd).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })))) \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n ((Opens.map (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.base).op.map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })).unop) =\n op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }))) \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b\n pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U))) \u226b\n eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop))) \u226b\n (Opens.map (t D.toGlueData j k).base).op.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [\u2190 (D.V (j, k)).presheaf.map_comp]\n -- Porting note : was just `congr`\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) \u226b\n eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n (V D.toGlueData (j, k)).presheaf.map\n ((homOfLE\n (_ :\n \u2191pullback.fst.base ''\n (\u2191pullback.fst.base \u207b\u00b9'\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier) \u2286\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier)).op \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map pullback.fst.base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) =\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b pullback.snd).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })))) \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n ((Opens.map (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.base).op.map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })).unop) =\n op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }))) \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b\n pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U))) \u226b\n eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop))) \u226b\n (Opens.map (t D.toGlueData j k).base).op.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [\u2190 (D.V (j, k)).presheaf.map_comp]\n -- Porting note : was just `congr`\n[GOAL]\ncase h.h.snd.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\nthis :\n t' D.toGlueData j k i \u226b pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k =\n (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j\n\u22a2 (V D.toGlueData (j, k)).presheaf.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) \u226b\n eqToHom\n (_ :\n (Opens.map (f D.toGlueData j k).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData j).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n (V D.toGlueData (j, k)).presheaf.map\n ((homOfLE\n (_ :\n \u2191pullback.fst.base ''\n (\u2191pullback.fst.base \u207b\u00b9'\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier) \u2286\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }).carrier)).op \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map pullback.fst.base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop)) })) =\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b pullback.snd).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })))) \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n ((Opens.map (pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom.base).op.map\n (eqToHom\n (_ :\n op\n ((Opens.map pullback.snd.base).obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) })).unop) =\n op\n { carrier := \u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base \u207b\u00b9'\n \u2191(op U).unop)) }))) \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).op.map\n (eqToHom\n (_ :\n (Opens.map\n ((pullbackSymmetry (f D.toGlueData j k) (f D.toGlueData j i)).hom \u226b\n pullback.fst \u226b t D.toGlueData j i \u226b f D.toGlueData i j).base).op.obj\n (op U) =\n (Opens.map\n (t' D.toGlueData j k i \u226b\n pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base).op.obj\n (op U))) \u226b\n eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.fst)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t' D.toGlueData j k i).base \u207b\u00b9'\n \u2191(op\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b\n t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop).unop)) } }.unop) =\n (Opens.map (t D.toGlueData j k).base).op.obj\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop))) \u226b\n (Opens.map (t D.toGlueData j k).base).op.map\n (eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj\n {\n unop :=\n {\n carrier :=\n \u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9' \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(pullback.fst \u226b t D.toGlueData k i \u226b f D.toGlueData i k).base \u207b\u00b9'\n \u2191(op U).unop)) } }.unop) =\n (Opens.map (f D.toGlueData k j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData k).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))) \u226b\n eqToHom\n (_ :\n (Opens.map\n ((MultispanIndex.multispan (diagram D.toGlueData)).map\n (WalkingMultispan.Hom.snd (j, k)).op.unop).base).op.obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (j, k)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (j, k))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nexact congr_arg ((D.V (j, k)).presheaf.map \u00b7) rfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2203 eq, \u03b9InvApp D U \u226b diagramOverOpen\u03c0 D U i = (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map (eqToHom eq)\n[PROOFSTEP]\nfconstructor\n -- Porting note: I don't know what the magic was in Lean3 proof, it just skipped the proof of `eq`\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 op U =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (op (WalkingMultispan.right i)).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase w.e_x\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 U =\n (Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (op (WalkingMultispan.right i)).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)\n[PROOFSTEP]\next1\n[GOAL]\ncase w.e_x.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191U =\n \u2191((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (op (WalkingMultispan.right i)).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))\n[PROOFSTEP]\nchange _ = _ \u207b\u00b9' (_ '' _)\n[GOAL]\ncase w.e_x.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191U =\n (fun x =>\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (op (WalkingMultispan.right i)).unop).base x) \u207b\u00b9'\n ((fun a => \u2191(\u03b9 D.toGlueData i).base a) '' \u2191U)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase w.e_x.h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nx : \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 x \u2208 \u2191U \u2194\n x \u2208\n (fun x =>\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (op (WalkingMultispan.right i)).unop).base\n x) \u207b\u00b9'\n ((fun a => \u2191(\u03b9 D.toGlueData i).base a) '' \u2191U)\n[PROOFSTEP]\nsimp only [SetLike.mem_coe, diagram_l, diagram_r, unop_op, Set.mem_preimage, Set.mem_image]\n[GOAL]\ncase w.e_x.h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nx : \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 x \u2208 U \u2194\n \u2203 x_1,\n x_1 \u2208 U \u2227\n \u2191(\u03b9 D.toGlueData i).base x_1 =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base x\n[PROOFSTEP]\nrefine \u27e8fun h => \u27e8_, h, rfl\u27e9, ?_\u27e9\n[GOAL]\ncase w.e_x.h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nx : \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 (\u2203 x_1,\n x_1 \u2208 U \u2227\n \u2191(\u03b9 D.toGlueData i).base x_1 =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base x) \u2192\n x \u2208 U\n[PROOFSTEP]\nrintro \u27e8y, h1, h2\u27e9\n[GOAL]\ncase w.e_x.h.h.intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nx : \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\ny : (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\nh1 : y \u2208 U\nh2 :\n \u2191(\u03b9 D.toGlueData i).base y =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base x\n\u22a2 x \u2208 U\n[PROOFSTEP]\nconvert h1 using 1\n[GOAL]\ncase h.e'_4.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nx : \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\ny : (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\nh1 : y \u2208 U\nh2 :\n \u2191(\u03b9 D.toGlueData i).base y =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base x\ne_1\u271d :\n \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i) =\n (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 x = y\n[PROOFSTEP]\ndelta \u03b9 Multicoequalizer.\u03c0 at h2 \n[GOAL]\ncase h.e'_4.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nx : \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\ny : (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\nh1 : y \u2208 U\nh2 :\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base y =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base x\ne_1\u271d :\n \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i) =\n (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 x = y\n[PROOFSTEP]\napply_fun (D.\u03b9 _).base\n[GOAL]\ncase h.e'_4.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nx : \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\ny : (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\nh1 : y \u2208 U\nh2 :\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base y =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base x\ne_1\u271d :\n \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i) =\n (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191(\u03b9 D.toGlueData i).base x = \u2191(\u03b9 D.toGlueData i).base y\n[PROOFSTEP]\nexact h2.symm\n[GOAL]\ncase h.e'_4.h.inj\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nx : \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\ny : (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\nh1 : y \u2208 U\nh2 :\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base y =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base x\ne_1\u271d :\n \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i) =\n (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Function.Injective \u2191(\u03b9 D.toGlueData i).base\n[PROOFSTEP]\nhave := D.\u03b9_gluedIso_inv (PresheafedSpace.forget _) i\n[GOAL]\ncase h.e'_4.h.inj\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nx : \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\ny : (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\nh1 : y \u2208 U\nh2 :\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base y =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base x\ne_1\u271d :\n \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i) =\n (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\nthis :\n \u03b9 (mapGlueData D.toGlueData (forget C)) i \u226b (gluedIso D.toGlueData (forget C)).inv = (forget C).map (\u03b9 D.toGlueData i)\n\u22a2 Function.Injective \u2191(\u03b9 D.toGlueData i).base\n[PROOFSTEP]\ndsimp at this \n[GOAL]\ncase h.e'_4.h.inj\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nx : \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\ny : (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\nh1 : y \u2208 U\nh2 :\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base y =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base x\ne_1\u271d :\n \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i) =\n (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\nthis : \u03b9 (mapGlueData D.toGlueData (forget C)) i \u226b (gluedIso D.toGlueData (forget C)).inv = (\u03b9 D.toGlueData i).base\n\u22a2 Function.Injective \u2191(\u03b9 D.toGlueData i).base\n[PROOFSTEP]\nrw [\u2190 this, coe_comp]\n[GOAL]\ncase h.e'_4.h.inj\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nx : \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\ny : (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\nh1 : y \u2208 U\nh2 :\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base y =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base x\ne_1\u271d :\n \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i) =\n (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\nthis : \u03b9 (mapGlueData D.toGlueData (forget C)) i \u226b (gluedIso D.toGlueData (forget C)).inv = (\u03b9 D.toGlueData i).base\n\u22a2 Function.Injective (\u2191(gluedIso D.toGlueData (forget C)).inv \u2218 \u2191(\u03b9 (mapGlueData D.toGlueData (forget C)) i))\n[PROOFSTEP]\nrefine Function.Injective.comp ?_ (TopCat.GlueData.\u03b9_injective D.toTopGlueData i)\n[GOAL]\ncase h.e'_4.h.inj\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nx : \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\ny : (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\nh1 : y \u2208 U\nh2 :\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base y =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base x\ne_1\u271d :\n \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i) =\n (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\nthis : \u03b9 (mapGlueData D.toGlueData (forget C)) i \u226b (gluedIso D.toGlueData (forget C)).inv = (\u03b9 D.toGlueData i).base\n\u22a2 Function.Injective \u2191(gluedIso D.toGlueData (forget C)).inv\n[PROOFSTEP]\nrw [\u2190 TopCat.mono_iff_injective]\n[GOAL]\ncase h.e'_4.h.inj\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nx : \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\ny : (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\nh1 : y \u2208 U\nh2 :\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base y =\n \u2191(colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base x\ne_1\u271d :\n \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i) =\n (CategoryTheory.forget TopCat).obj \u2191(CategoryTheory.GlueData.U D.toGlueData i)\nthis : \u03b9 (mapGlueData D.toGlueData (forget C)) i \u226b (gluedIso D.toGlueData (forget C)).inv = (\u03b9 D.toGlueData i).base\n\u22a2 Mono (gluedIso D.toGlueData (forget C)).inv\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u03b9InvApp D U \u226b diagramOverOpen\u03c0 D U i =\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n op U =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right i)).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\ndelta \u03b9InvApp\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 limit.lift (diagramOverOpen D U)\n { pt := (CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U),\n \u03c0 := NatTrans.mk fun j => \u03b9InvApp\u03c0App D U j.unop } \u226b\n diagramOverOpen\u03c0 D U i =\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n op U =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right i)).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nrw [limit.lift_\u03c0]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 NatTrans.app\n { pt := (CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U),\n \u03c0 := NatTrans.mk fun j => \u03b9InvApp\u03c0App D U j.unop }.\u03c0\n (op (WalkingMultispan.right i)) =\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n op U =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right i)).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nchange D.opensImagePreimageMap i i U = _\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 opensImagePreimageMap D i i U =\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n op U =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right i)).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\ndsimp [opensImagePreimageMap]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 NatTrans.app (f D.toGlueData i i).c (op U) \u226b\n NatTrans.app (t D.toGlueData i i).c (op ((Opens.map (f D.toGlueData i i).base).obj U)) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion (f D.toGlueData i i))\n { carrier := \u2191(t D.toGlueData i i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i i).base).obj U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData i i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i i).base).op.obj (op U)).unop)) } \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj ((Opens.map (f D.toGlueData i i).base).obj U)))).op =\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n op U =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nrw [congr_app (D.t_id _), id_c_app, \u2190 Functor.map_comp]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 NatTrans.app (f D.toGlueData i i).c (op U) \u226b\n (V D.toGlueData (i, i)).presheaf.map\n (\ud835\udfd9 (op ((Opens.map (f D.toGlueData i i).base).obj U)) \u226b\n eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 (V D.toGlueData (i, i))).base).op.obj (op ((Opens.map (f D.toGlueData i i).base).obj U)) =\n (Opens.map (t D.toGlueData i i).base).op.obj (op ((Opens.map (f D.toGlueData i i).base).obj U)))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion (f D.toGlueData i i))\n { carrier := \u2191(t D.toGlueData i i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i i).base).obj U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData i i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i i).base).op.obj (op U)).unop)) } \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj ((Opens.map (f D.toGlueData i i).base).obj U)))).op =\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n op U =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nerw [IsOpenImmersion.inv_naturality_assoc, IsOpenImmersion.app_inv_app'_assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (f D.toGlueData i i).base).obj U) =\n U)).op \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).op.map\n (\ud835\udfd9 (op ((Opens.map (f D.toGlueData i i).base).obj U)) \u226b\n eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 (V D.toGlueData (i, i))).base).op.obj\n (op ((Opens.map (f D.toGlueData i i).base).obj U)) =\n (Opens.map (t D.toGlueData i i).base).op.obj (op ((Opens.map (f D.toGlueData i i).base).obj U))))) \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj ((Opens.map (f D.toGlueData i i).base).obj U)))).op =\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n op U =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nsimp only [eqToHom_op, eqToHom_trans, eqToHom_map (Functor.op _), \u2190 Functor.map_comp]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n op U =\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (f D.toGlueData i i).base).obj U))) \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).op.map\n (\ud835\udfd9 (op ((Opens.map (f D.toGlueData i i).base).obj U)) \u226b\n eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 (V D.toGlueData (i, i))).base).op.obj\n (op ((Opens.map (f D.toGlueData i i).base).obj U)) =\n (Opens.map (t D.toGlueData i i).base).op.obj (op ((Opens.map (f D.toGlueData i i).base).obj U)))) \u226b\n eqToHom\n (_ :\n op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj ((Opens.map (f D.toGlueData i i).base).obj U))) =\n op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n op U =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.right i)).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.hU\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191U \u2286 Set.range \u2191(f D.toGlueData i i).base\n[PROOFSTEP]\nrw [Set.range_iff_surjective.mpr _]\n[GOAL]\ncase h.hU\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u2191U \u2286 Set.univ\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Function.Surjective \u2191(f D.toGlueData i i).base\n[PROOFSTEP]\nrw [\u2190 TopCat.epi_iff_surjective]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Epi (f D.toGlueData i i).base\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b \u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U \u226b diagramOverOpen\u03c0 D U j = diagramOverOpen\u03c0 D U j\n[PROOFSTEP]\nrw [\u2190\n @cancel_mono (f :=\n (componentwiseDiagram \ud835\udda3.diagram.multispan _).map (Quiver.Hom.op (WalkingMultispan.Hom.snd (i, j))) \u226b \ud835\udfd9 _) _ _\n (by\n rw [Category.comp_id]\n apply (config := { allowSynthFailures := true }) mono_comp\n change Mono ((_ \u226b D.f j i).c.app _)\n rw [comp_c_app]\n apply (config := { allowSynthFailures := true }) mono_comp\n erw [D.\u03b9_image_preimage_eq i j U]\n \u00b7 infer_instance\n \u00b7 have : IsIso (D.t i j).c := by apply c_isIso_of_iso\n infer_instance)]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Mono\n ((componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).map\n (WalkingMultispan.Hom.snd (i, j)).op \u226b\n \ud835\udfd9\n ((componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).obj\n (op (WalkingMultispan.left (i, j)))))\n[PROOFSTEP]\nrw [Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Mono\n ((componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).map\n (WalkingMultispan.Hom.snd (i, j)).op)\n[PROOFSTEP]\napply (config := { allowSynthFailures := true }) mono_comp\n[GOAL]\ncase inst\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Mono\n (NatTrans.app ((MultispanIndex.multispan (diagram D.toGlueData)).map (WalkingMultispan.Hom.snd (i, j)).op.unop).c\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right (MultispanIndex.sndFrom (diagram D.toGlueData) (i, j)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nchange Mono ((_ \u226b D.f j i).c.app _)\n[GOAL]\ncase inst\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Mono\n (NatTrans.app (t D.toGlueData i j \u226b f D.toGlueData j i).c\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right (MultispanIndex.sndFrom (diagram D.toGlueData) (i, j)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\n[PROOFSTEP]\nrw [comp_c_app]\n[GOAL]\ncase inst\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Mono\n (NatTrans.app (f D.toGlueData j i).c\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right (MultispanIndex.sndFrom (diagram D.toGlueData) (i, j)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n NatTrans.app (t D.toGlueData i j).c\n (op\n ((Opens.map (f D.toGlueData j i).base).obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (i, j)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))).unop)))\n[PROOFSTEP]\napply (config := { allowSynthFailures := true }) mono_comp\n[GOAL]\ncase inst\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Mono\n (NatTrans.app (f D.toGlueData j i).c\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right (MultispanIndex.sndFrom (diagram D.toGlueData) (i, j)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))))\ncase inst\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Mono\n (NatTrans.app (t D.toGlueData i j).c\n (op\n ((Opens.map (f D.toGlueData j i).base).obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (i, j)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))).unop)))\n[PROOFSTEP]\nerw [D.\u03b9_image_preimage_eq i j U]\n[GOAL]\ncase inst\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Mono\n (NatTrans.app (f D.toGlueData j i).c\n (op\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj\n ((Opens.map (t D.toGlueData j i).base).obj ((Opens.map (f D.toGlueData i j).base).obj U)))))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase inst\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 Mono\n (NatTrans.app (t D.toGlueData i j).c\n (op\n ((Opens.map (f D.toGlueData j i).base).obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (i, j)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))).unop)))\n[PROOFSTEP]\nhave : IsIso (D.t i j).c := by apply c_isIso_of_iso\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 IsIso (t D.toGlueData i j).c\n[PROOFSTEP]\napply c_isIso_of_iso\n[GOAL]\ncase inst\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nthis : IsIso (t D.toGlueData i j).c\n\u22a2 Mono\n (NatTrans.app (t D.toGlueData i j).c\n (op\n ((Opens.map (f D.toGlueData j i).base).obj\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op\n (WalkingMultispan.right\n (MultispanIndex.sndFrom (diagram D.toGlueData) (i, j)))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))).unop)))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 (diagramOverOpen\u03c0 D U i \u226b \u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U \u226b diagramOverOpen\u03c0 D U j) \u226b\n (componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).map\n (WalkingMultispan.Hom.snd (i, j)).op \u226b\n \ud835\udfd9\n ((componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).obj\n (op (WalkingMultispan.left (i, j)))) =\n diagramOverOpen\u03c0 D U j \u226b\n (componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).map\n (WalkingMultispan.Hom.snd (i, j)).op \u226b\n \ud835\udfd9\n ((componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).obj\n (op (WalkingMultispan.left (i, j))))\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n \u03b9InvApp\u03c0EqMap D U \u226b\n \u03b9InvApp D U \u226b\n diagramOverOpen\u03c0 D U j \u226b\n (componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).map\n (WalkingMultispan.Hom.snd (i, j)).op \u226b\n \ud835\udfd9\n ((componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).obj\n (op (WalkingMultispan.left (i, j)))) =\n diagramOverOpen\u03c0 D U j \u226b\n (componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).map\n (WalkingMultispan.Hom.snd (i, j)).op \u226b\n \ud835\udfd9\n ((componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).obj\n (op (WalkingMultispan.left (i, j))))\n[PROOFSTEP]\nrw [limit.w_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n \u03b9InvApp\u03c0EqMap D U \u226b\n \u03b9InvApp D U \u226b\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j))) \u226b\n \ud835\udfd9\n ((componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).obj\n (op (WalkingMultispan.left (i, j)))) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j))) \u226b\n \ud835\udfd9\n ((componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).obj\n (op (WalkingMultispan.left (i, j))))\n[PROOFSTEP]\nerw [limit.lift_\u03c0_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n \u03b9InvApp\u03c0EqMap D U \u226b\n NatTrans.app\n { pt := (CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U),\n \u03c0 := NatTrans.mk fun j => \u03b9InvApp\u03c0App D U j.unop }.\u03c0\n (op (WalkingMultispan.left (i, j))) \u226b\n \ud835\udfd9\n ((componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).obj\n (op (WalkingMultispan.left (i, j)))) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j))) \u226b\n \ud835\udfd9\n ((componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).obj\n (op (WalkingMultispan.left (i, j))))\n[PROOFSTEP]\nrw [Category.comp_id, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n \u03b9InvApp\u03c0EqMap D U \u226b\n NatTrans.app\n { pt := (CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U),\n \u03c0 := NatTrans.mk fun j => \u03b9InvApp\u03c0App D U j.unop }.\u03c0\n (op (WalkingMultispan.left (i, j))) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j)))\n[PROOFSTEP]\nchange _ \u226b _ \u226b (_ \u226b _) \u226b _ = _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n \u03b9InvApp\u03c0EqMap D U \u226b\n (NatTrans.app (f D.toGlueData i i).c (op U) \u226b\n NatTrans.app (t D.toGlueData i i).c ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion (f D.toGlueData i i))\n {\n unop :=\n {\n carrier :=\n \u2191(t D.toGlueData i i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i i).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData i i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i i).base).op.obj (op U)).unop)) } }.unop \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj\n ((Opens.map (f D.toGlueData i i).base).obj U)))).op) \u226b\n NatTrans.app (f D.toGlueData i j).c\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n (V D.toGlueData (i, j)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData i j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (i, j))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j)))\n[PROOFSTEP]\nrw [congr_app (D.t_id _), id_c_app]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n \u03b9InvApp\u03c0EqMap D U \u226b\n (NatTrans.app (f D.toGlueData i i).c (op U) \u226b\n ((V D.toGlueData (i, i)).presheaf.map (\ud835\udfd9 ((Opens.map (f D.toGlueData i i).base).op.obj (op U))) \u226b\n (V D.toGlueData (i, i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 (V D.toGlueData (i, i))).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) =\n (Opens.map (t D.toGlueData i i).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U))))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion (f D.toGlueData i i))\n {\n unop :=\n {\n carrier :=\n \u2191(t D.toGlueData i i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i i).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData i i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i i).base).op.obj (op U)).unop)) } }.unop \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj\n ((Opens.map (f D.toGlueData i i).base).obj U)))).op) \u226b\n NatTrans.app (f D.toGlueData i j).c\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n (V D.toGlueData (i, j)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData i j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (i, j))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j)))\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n \u03b9InvApp\u03c0EqMap D U \u226b\n NatTrans.app (f D.toGlueData i i).c (op U) \u226b\n (V D.toGlueData (i, i)).presheaf.map (\ud835\udfd9 ((Opens.map (f D.toGlueData i i).base).op.obj (op U))) \u226b\n (V D.toGlueData (i, i)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 (V D.toGlueData (i, i))).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) =\n (Opens.map (t D.toGlueData i i).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U)))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion (f D.toGlueData i i))\n {\n carrier :=\n \u2191(t D.toGlueData i i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i i).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData i i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i i).base).op.obj (op U)).unop)) } \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj\n ((Opens.map (f D.toGlueData i i).base).obj U)))).op \u226b\n NatTrans.app (f D.toGlueData i j).c\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n (V D.toGlueData (i, j)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData i j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (i, j))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j)))\n[PROOFSTEP]\nrw [\u2190 Functor.map_comp_assoc]\n -- Porting note : change `rw` to `erw`\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n \u03b9InvApp\u03c0EqMap D U \u226b\n NatTrans.app (f D.toGlueData i i).c (op U) \u226b\n (V D.toGlueData (i, i)).presheaf.map\n (\ud835\udfd9 ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) \u226b\n eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 (V D.toGlueData (i, i))).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) =\n (Opens.map (t D.toGlueData i i).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U)))) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion (f D.toGlueData i i))\n { carrier := \u2191(t D.toGlueData i i).base \u207b\u00b9' \u2191((Opens.map (f D.toGlueData i i).base).op.obj (op U)).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(t D.toGlueData i i).base \u207b\u00b9'\n \u2191((Opens.map (f D.toGlueData i i).base).op.obj (op U)).unop)) } \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj\n ((Opens.map (f D.toGlueData i i).base).obj U)))).op \u226b\n NatTrans.app (f D.toGlueData i j).c\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n (V D.toGlueData (i, j)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData i j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (i, j))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j)))\n[PROOFSTEP]\nerw [IsOpenImmersion.inv_naturality_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n \u03b9InvApp\u03c0EqMap D U \u226b\n NatTrans.app (f D.toGlueData i i).c (op U) \u226b\n IsOpenImmersion.invApp (_ : IsOpenImmersion (f D.toGlueData i i))\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U)).unop \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).op.map\n (\ud835\udfd9 ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) \u226b\n eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 (V D.toGlueData (i, i))).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) =\n (Opens.map (t D.toGlueData i i).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U))))) \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj\n ((Opens.map (f D.toGlueData i i).base).obj U)))).op \u226b\n NatTrans.app (f D.toGlueData i j).c\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n (V D.toGlueData (i, j)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData i j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (i, j))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j)))\n[PROOFSTEP]\nerw [IsOpenImmersion.app_invApp_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n \u03b9InvApp\u03c0EqMap D U \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (homOfLE (_ : \u2191(f D.toGlueData i i).base '' (\u2191(f D.toGlueData i i).base \u207b\u00b9' U.carrier) \u2286 U.carrier)).op \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).op.map\n (\ud835\udfd9 ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) \u226b\n eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 (V D.toGlueData (i, i))).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) =\n (Opens.map (t D.toGlueData i i).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U))))) \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj\n ((Opens.map (f D.toGlueData i i).base).obj U)))).op \u226b\n NatTrans.app (f D.toGlueData i j).c\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n (V D.toGlueData (i, j)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData i j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (i, j))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j)))\n[PROOFSTEP]\niterate 3 rw [\u2190 Functor.map_comp_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n \u03b9InvApp\u03c0EqMap D U \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (homOfLE (_ : \u2191(f D.toGlueData i i).base '' (\u2191(f D.toGlueData i i).base \u207b\u00b9' U.carrier) \u2286 U.carrier)).op \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).op.map\n (\ud835\udfd9 ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) \u226b\n eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 (V D.toGlueData (i, i))).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) =\n (Opens.map (t D.toGlueData i i).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U))))) \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj\n ((Opens.map (f D.toGlueData i i).base).obj U)))).op \u226b\n NatTrans.app (f D.toGlueData i j).c\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n (V D.toGlueData (i, j)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData i j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (i, j))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j)))\n[PROOFSTEP]\nrw [\u2190 Functor.map_comp_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n ((eqToIso\n (_ :\n op U =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right i)).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))).inv \u226b\n (homOfLE (_ : \u2191(f D.toGlueData i i).base '' (\u2191(f D.toGlueData i i).base \u207b\u00b9' U.carrier) \u2286 U.carrier)).op) \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).op.map\n (\ud835\udfd9 ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) \u226b\n eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 (V D.toGlueData (i, i))).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) =\n (Opens.map (t D.toGlueData i i).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U))))) \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj ((Opens.map (f D.toGlueData i i).base).obj U)))).op \u226b\n NatTrans.app (f D.toGlueData i j).c\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n (V D.toGlueData (i, j)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData i j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (i, j))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j)))\n[PROOFSTEP]\nrw [\u2190 Functor.map_comp_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (((eqToIso\n (_ :\n op U =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right i)).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))).inv \u226b\n (homOfLE (_ : \u2191(f D.toGlueData i i).base '' (\u2191(f D.toGlueData i i).base \u207b\u00b9' U.carrier) \u2286 U.carrier)).op) \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).op.map\n (\ud835\udfd9 ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) \u226b\n eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 (V D.toGlueData (i, i))).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) =\n (Opens.map (t D.toGlueData i i).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U))))) \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj ((Opens.map (f D.toGlueData i i).base).obj U)))).op \u226b\n NatTrans.app (f D.toGlueData i j).c\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n (V D.toGlueData (i, j)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData i j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (i, j))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j)))\n[PROOFSTEP]\nrw [\u2190 Functor.map_comp_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n ((((eqToIso\n (_ :\n op U =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right i)).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))).inv \u226b\n (homOfLE\n (_ : \u2191(f D.toGlueData i i).base '' (\u2191(f D.toGlueData i i).base \u207b\u00b9' U.carrier) \u2286 U.carrier)).op) \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).op.map\n (\ud835\udfd9 ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) \u226b\n eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 (V D.toGlueData (i, i))).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) =\n (Opens.map (t D.toGlueData i i).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U))))) \u226b\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj ((Opens.map (f D.toGlueData i i).base).obj U)))).op) \u226b\n NatTrans.app (f D.toGlueData i j).c\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n (V D.toGlueData (i, j)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData i j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (i, j))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j)))\n[PROOFSTEP]\nrw [NatTrans.naturality_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n NatTrans.app (f D.toGlueData i j).c\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right i)).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n ((f D.toGlueData i j).base _* (V D.toGlueData (i, j)).presheaf).map\n ((((eqToIso\n (_ :\n op U =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right i)).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))).inv \u226b\n (homOfLE\n (_ : \u2191(f D.toGlueData i i).base '' (\u2191(f D.toGlueData i i).base \u207b\u00b9' U.carrier) \u2286 U.carrier)).op) \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).op.map\n (\ud835\udfd9 ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) \u226b\n eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 (V D.toGlueData (i, i))).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) =\n (Opens.map (t D.toGlueData i i).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U))))) \u226b\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj\n ((Opens.map (f D.toGlueData i i).base).obj U)))).op) \u226b\n (V D.toGlueData (i, j)).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (f D.toGlueData i j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (i, j))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j)))\n[PROOFSTEP]\nerw [\u2190 (D.V (i, j)).presheaf.map_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n NatTrans.app (f D.toGlueData i j).c\n (op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right i)).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) \u226b\n (V D.toGlueData (i, j)).presheaf.map\n ((Opens.map (f D.toGlueData i j).base).op.map\n ((((eqToIso\n (_ :\n op U =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right i)).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))).inv \u226b\n (homOfLE\n (_ :\n \u2191(f D.toGlueData i i).base '' (\u2191(f D.toGlueData i i).base \u207b\u00b9' U.carrier) \u2286 U.carrier)).op) \u226b\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).op.map\n (\ud835\udfd9 ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) \u226b\n eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 (V D.toGlueData (i, i))).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U)) =\n (Opens.map (t D.toGlueData i i).base).op.obj\n ((Opens.map (f D.toGlueData i i).base).op.obj (op U))))) \u226b\n (eqToHom\n (_ :\n (Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U) =\n (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData i i))).obj\n ((Opens.map (t D.toGlueData i i).base).obj\n ((Opens.map (f D.toGlueData i i).base).obj U)))).op) \u226b\n eqToHom\n (_ :\n (Opens.map (f D.toGlueData i j).base).op.obj\n (op\n ((Opens.map (\u03b9 D.toGlueData i).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))) =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.left (i, j))).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (i, j)))\n[PROOFSTEP]\nconvert limit.w (componentwiseDiagram \ud835\udda3.diagram.multispan _) (Quiver.Hom.op (WalkingMultispan.Hom.fst (i, j)))\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b \u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U = \ud835\udfd9 (limit (diagramOverOpen D U))\n[PROOFSTEP]\next j\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj : (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56\n\u22a2 (diagramOverOpen\u03c0 D U i \u226b \u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U) \u226b limit.\u03c0 (diagramOverOpen D U) j =\n \ud835\udfd9 (limit (diagramOverOpen D U)) \u226b limit.\u03c0 (diagramOverOpen D U) j\n[PROOFSTEP]\ninduction j using Opposite.rec' with\n| h j => ?_\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj : (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)\u1d52\u1d56\n\u22a2 (diagramOverOpen\u03c0 D U i \u226b \u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U) \u226b limit.\u03c0 (diagramOverOpen D U) j =\n \ud835\udfd9 (limit (diagramOverOpen D U)) \u226b limit.\u03c0 (diagramOverOpen D U) j\n[PROOFSTEP]\ninduction j using Opposite.rec' with\n| h j => ?_\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj : WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom\n\u22a2 (diagramOverOpen\u03c0 D U i \u226b \u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U) \u226b limit.\u03c0 (diagramOverOpen D U) (op j) =\n \ud835\udfd9 (limit (diagramOverOpen D U)) \u226b limit.\u03c0 (diagramOverOpen D U) (op j)\n[PROOFSTEP]\nrcases j with (\u27e8j, k\u27e9 | \u27e8j\u27e9)\n[GOAL]\ncase w.h.left.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 (diagramOverOpen\u03c0 D U i \u226b \u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U) \u226b\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (j, k))) =\n \ud835\udfd9 (limit (diagramOverOpen D U)) \u226b limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.left (j, k)))\n[PROOFSTEP]\nrw [\u2190 limit.w (componentwiseDiagram \ud835\udda3.diagram.multispan _) (Quiver.Hom.op (WalkingMultispan.Hom.fst (j, k))), \u2190\n Category.assoc, Category.id_comp]\n[GOAL]\ncase w.h.left.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 ((diagramOverOpen\u03c0 D U i \u226b \u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U) \u226b\n limit.\u03c0\n (componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))\n (op (WalkingMultispan.right (MultispanIndex.fstFrom (diagram D.toGlueData) (j, k))))) \u226b\n (componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).map\n (WalkingMultispan.Hom.fst (j, k)).op =\n limit.\u03c0\n (componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))\n (op (WalkingMultispan.right (MultispanIndex.fstFrom (diagram D.toGlueData) (j, k)))) \u226b\n (componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).map\n (WalkingMultispan.Hom.fst (j, k)).op\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase w.h.left.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 (diagramOverOpen\u03c0 D U i \u226b \u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U) \u226b\n limit.\u03c0\n (componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))\n (op (WalkingMultispan.right (MultispanIndex.fstFrom (diagram D.toGlueData) (j, k)))) =\n limit.\u03c0\n (componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))\n (op (WalkingMultispan.right (MultispanIndex.fstFrom (diagram D.toGlueData) (j, k))))\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase w.h.left.mk.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj k : D.J\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n \u03b9InvApp\u03c0EqMap D U \u226b\n \u03b9InvApp D U \u226b\n limit.\u03c0\n (componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))\n (op (WalkingMultispan.right (MultispanIndex.fstFrom (diagram D.toGlueData) (j, k)))) =\n limit.\u03c0\n (componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))\n (op (WalkingMultispan.right (MultispanIndex.fstFrom (diagram D.toGlueData) (j, k))))\n[PROOFSTEP]\napply \u03c0_\u03b9InvApp_\u03c0\n[GOAL]\ncase w.h.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj : (diagram D.toGlueData).R\n\u22a2 (diagramOverOpen\u03c0 D U i \u226b \u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U) \u226b\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.right j)) =\n \ud835\udfd9 (limit (diagramOverOpen D U)) \u226b limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.right j))\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase w.h.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj : (diagram D.toGlueData).R\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n \u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U \u226b limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.right j)) =\n \ud835\udfd9 (limit (diagramOverOpen D U)) \u226b limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.right j))\n[PROOFSTEP]\nrw [Category.id_comp]\n[GOAL]\ncase w.h.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\nj : (diagram D.toGlueData).R\n\u22a2 diagramOverOpen\u03c0 D U i \u226b\n \u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U \u226b limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.right j)) =\n limit.\u03c0 (diagramOverOpen D U) (op (WalkingMultispan.right j))\n[PROOFSTEP]\napply \u03c0_\u03b9InvApp_\u03c0\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 IsIso (diagramOverOpen\u03c0 D U i)\n[PROOFSTEP]\nuse D.\u03b9InvApp\u03c0EqMap U \u226b D.\u03b9InvApp U\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b \u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U = \ud835\udfd9 (limit (diagramOverOpen D U)) \u2227\n (\u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U) \u226b diagramOverOpen\u03c0 D U i =\n \ud835\udfd9 ((diagramOverOpen D U).obj (op (WalkingMultispan.right i)))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 diagramOverOpen\u03c0 D U i \u226b \u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U = \ud835\udfd9 (limit (diagramOverOpen D U))\n[PROOFSTEP]\napply \u03c0_\u03b9InvApp_eq_id\n[GOAL]\ncase h.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 (\u03b9InvApp\u03c0EqMap D U \u226b \u03b9InvApp D U) \u226b diagramOverOpen\u03c0 D U i =\n \ud835\udfd9 ((diagramOverOpen D U).obj (op (WalkingMultispan.right i)))\n[PROOFSTEP]\nrw [Category.assoc, (D.\u03b9InvApp_\u03c0 _).choose_spec]\n[GOAL]\ncase h.right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 \u03b9InvApp\u03c0EqMap D U \u226b\n (CategoryTheory.GlueData.U D.toGlueData i).presheaf.map\n (eqToHom\n (_ :\n op U =\n op\n ((Opens.map\n (colimit.\u03b9 (MultispanIndex.multispan (diagram D.toGlueData))\n (op (WalkingMultispan.right i)).unop).base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))) =\n \ud835\udfd9 ((diagramOverOpen D U).obj (op (WalkingMultispan.right i)))\n[PROOFSTEP]\nexact Iso.inv_hom_id ((D.U i).presheaf.mapIso (eqToIso _))\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 IsIso (NatTrans.app (\u03b9 D.toGlueData i).c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)))\n[PROOFSTEP]\nerw [\u2190 colimitPresheafObjIsoComponentwiseLimit_hom_\u03c0]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\nU : Opens \u2191\u2191(CategoryTheory.GlueData.U D.toGlueData i)\n\u22a2 IsIso\n ((colimitPresheafObjIsoComponentwiseLimit (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U)).hom \u226b\n limit.\u03c0\n (componentwiseDiagram (MultispanIndex.multispan (diagram D.toGlueData))\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(\u03b9 D.toGlueData i).base)).obj U))\n (op (WalkingMultispan.right i)))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\n\u22a2 { l //\n l \u226b PullbackCone.fst (vPullbackCone D.toGlueData i j) = PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (vPullbackCone D.toGlueData i j) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (vPullbackCone D.toGlueData i j).pt},\n m \u226b PullbackCone.fst (vPullbackCone D.toGlueData i j) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (vPullbackCone D.toGlueData i j) = PullbackCone.snd s \u2192 m = l }\n[PROOFSTEP]\nrefine' \u27e8_, _, _, _\u27e9\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\n\u22a2 s.pt \u27f6 (vPullbackCone D.toGlueData i j).pt\n[PROOFSTEP]\nrefine' PresheafedSpace.IsOpenImmersion.lift (D.f i j) s.fst _\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\n\u22a2 Set.range \u2191(PullbackCone.fst s).base \u2286 Set.range \u2191(f D.toGlueData i j).base\n[PROOFSTEP]\nerw [\u2190 D.toTopGlueData.preimage_range j i]\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\n\u22a2 Set.range \u2191(PullbackCone.fst s).base \u2286\n \u2191(\u03b9 (toTopGlueData D).toGlueData i) \u207b\u00b9' Set.range \u2191(\u03b9 (toTopGlueData D).toGlueData j)\n[PROOFSTEP]\nhave : s.fst.base \u226b D.toTopGlueData.\u03b9 i = s.snd.base \u226b D.toTopGlueData.\u03b9 j :=\n by\n rw [\u2190 \ud835\udda3.\u03b9_gluedIso_hom (PresheafedSpace.forget _) _, \u2190 \ud835\udda3.\u03b9_gluedIso_hom (PresheafedSpace.forget _) _]\n have := congr_arg PresheafedSpace.Hom.base s.condition\n rw [comp_base, comp_base] at this \n replace this := reassoc_of% this\n exact this _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\n\u22a2 (PullbackCone.fst s).base \u226b \u03b9 (toTopGlueData D).toGlueData i =\n (PullbackCone.snd s).base \u226b \u03b9 (toTopGlueData D).toGlueData j\n[PROOFSTEP]\nrw [\u2190 \ud835\udda3.\u03b9_gluedIso_hom (PresheafedSpace.forget _) _, \u2190 \ud835\udda3.\u03b9_gluedIso_hom (PresheafedSpace.forget _) _]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\n\u22a2 (PullbackCone.fst s).base \u226b (forget C).map (\u03b9 D.toGlueData i) \u226b (gluedIso D.toGlueData (forget C)).hom =\n (PullbackCone.snd s).base \u226b (forget C).map (\u03b9 D.toGlueData j) \u226b (gluedIso D.toGlueData (forget C)).hom\n[PROOFSTEP]\nhave := congr_arg PresheafedSpace.Hom.base s.condition\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\nthis : (PullbackCone.fst s \u226b \u03b9 D.toGlueData i).base = (PullbackCone.snd s \u226b \u03b9 D.toGlueData j).base\n\u22a2 (PullbackCone.fst s).base \u226b (forget C).map (\u03b9 D.toGlueData i) \u226b (gluedIso D.toGlueData (forget C)).hom =\n (PullbackCone.snd s).base \u226b (forget C).map (\u03b9 D.toGlueData j) \u226b (gluedIso D.toGlueData (forget C)).hom\n[PROOFSTEP]\nrw [comp_base, comp_base] at this \n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\nthis : (PullbackCone.fst s).base \u226b (\u03b9 D.toGlueData i).base = (PullbackCone.snd s).base \u226b (\u03b9 D.toGlueData j).base\n\u22a2 (PullbackCone.fst s).base \u226b (forget C).map (\u03b9 D.toGlueData i) \u226b (gluedIso D.toGlueData (forget C)).hom =\n (PullbackCone.snd s).base \u226b (forget C).map (\u03b9 D.toGlueData j) \u226b (gluedIso D.toGlueData (forget C)).hom\n[PROOFSTEP]\nreplace this := reassoc_of% this\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\nthis :\n \u2200 {Z : TopCat} (h : \u2191(glued D.toGlueData) \u27f6 Z),\n (PullbackCone.fst s).base \u226b (\u03b9 D.toGlueData i).base \u226b h = (PullbackCone.snd s).base \u226b (\u03b9 D.toGlueData j).base \u226b h\n\u22a2 (PullbackCone.fst s).base \u226b (forget C).map (\u03b9 D.toGlueData i) \u226b (gluedIso D.toGlueData (forget C)).hom =\n (PullbackCone.snd s).base \u226b (forget C).map (\u03b9 D.toGlueData j) \u226b (gluedIso D.toGlueData (forget C)).hom\n[PROOFSTEP]\nexact this _\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\nthis :\n (PullbackCone.fst s).base \u226b \u03b9 (toTopGlueData D).toGlueData i =\n (PullbackCone.snd s).base \u226b \u03b9 (toTopGlueData D).toGlueData j\n\u22a2 Set.range \u2191(PullbackCone.fst s).base \u2286\n \u2191(\u03b9 (toTopGlueData D).toGlueData i) \u207b\u00b9' Set.range \u2191(\u03b9 (toTopGlueData D).toGlueData j)\n[PROOFSTEP]\nrw [\u2190 Set.image_subset_iff, \u2190 Set.image_univ, \u2190 Set.image_comp, Set.image_univ]\n -- Porting note : change `rw` to `erw`\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\nthis :\n (PullbackCone.fst s).base \u226b \u03b9 (toTopGlueData D).toGlueData i =\n (PullbackCone.snd s).base \u226b \u03b9 (toTopGlueData D).toGlueData j\n\u22a2 Set.range (\u2191(\u03b9 (toTopGlueData D).toGlueData i) \u2218 \u2191(PullbackCone.fst s).base) \u2286\n Set.range \u2191(\u03b9 (toTopGlueData D).toGlueData j)\n[PROOFSTEP]\nerw [\u2190 coe_comp]\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\nthis :\n (PullbackCone.fst s).base \u226b \u03b9 (toTopGlueData D).toGlueData i =\n (PullbackCone.snd s).base \u226b \u03b9 (toTopGlueData D).toGlueData j\n\u22a2 Set.range \u2191((PullbackCone.fst s).base \u226b \u03b9 (toTopGlueData D).toGlueData i) \u2286\n Set.range \u2191(\u03b9 (toTopGlueData D).toGlueData j)\n[PROOFSTEP]\nrw [this, coe_comp, \u2190 Set.image_univ, Set.image_comp]\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\nthis :\n (PullbackCone.fst s).base \u226b \u03b9 (toTopGlueData D).toGlueData i =\n (PullbackCone.snd s).base \u226b \u03b9 (toTopGlueData D).toGlueData j\n\u22a2 \u2191(\u03b9 (toTopGlueData D).toGlueData j) '' (\u2191(PullbackCone.snd s).base '' Set.univ) \u2286\n Set.range \u2191(\u03b9 (toTopGlueData D).toGlueData j)\n[PROOFSTEP]\nexact Set.image_subset_range _ _\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\n\u22a2 IsOpenImmersion.lift (f D.toGlueData i j) (PullbackCone.fst s)\n (_ : Set.range \u2191(PullbackCone.fst s).base \u2286 Set.range \u2191(f D.toGlueData i j).base) \u226b\n PullbackCone.fst (vPullbackCone D.toGlueData i j) =\n PullbackCone.fst s\n[PROOFSTEP]\napply IsOpenImmersion.lift_fac\n[GOAL]\ncase refine'_3\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\n\u22a2 IsOpenImmersion.lift (f D.toGlueData i j) (PullbackCone.fst s)\n (_ : Set.range \u2191(PullbackCone.fst s).base \u2286 Set.range \u2191(f D.toGlueData i j).base) \u226b\n PullbackCone.snd (vPullbackCone D.toGlueData i j) =\n PullbackCone.snd s\n[PROOFSTEP]\nrw [\u2190 cancel_mono (\ud835\udda3.\u03b9 j), Category.assoc, \u2190 (\ud835\udda3.vPullbackCone i j).condition]\n[GOAL]\ncase refine'_3\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\n\u22a2 IsOpenImmersion.lift (f D.toGlueData i j) (PullbackCone.fst s)\n (_ : Set.range \u2191(PullbackCone.fst s).base \u2286 Set.range \u2191(f D.toGlueData i j).base) \u226b\n PullbackCone.fst (vPullbackCone D.toGlueData i j) \u226b \u03b9 D.toGlueData i =\n PullbackCone.snd s \u226b \u03b9 D.toGlueData j\n[PROOFSTEP]\nconv_rhs => rw [\u2190 s.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\n| PullbackCone.snd s \u226b \u03b9 D.toGlueData j\n[PROOFSTEP]\nrw [\u2190 s.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\n| PullbackCone.snd s \u226b \u03b9 D.toGlueData j\n[PROOFSTEP]\nrw [\u2190 s.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\n| PullbackCone.snd s \u226b \u03b9 D.toGlueData j\n[PROOFSTEP]\nrw [\u2190 s.condition]\n[GOAL]\ncase refine'_3\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\n\u22a2 IsOpenImmersion.lift (f D.toGlueData i j) (PullbackCone.fst s)\n (_ : Set.range \u2191(PullbackCone.fst s).base \u2286 Set.range \u2191(f D.toGlueData i j).base) \u226b\n PullbackCone.fst (vPullbackCone D.toGlueData i j) \u226b \u03b9 D.toGlueData i =\n PullbackCone.fst s \u226b \u03b9 D.toGlueData i\n[PROOFSTEP]\nerw [IsOpenImmersion.lift_fac_assoc]\n[GOAL]\ncase refine'_4\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\n\u22a2 \u2200 {m : s.pt \u27f6 (vPullbackCone D.toGlueData i j).pt},\n m \u226b PullbackCone.fst (vPullbackCone D.toGlueData i j) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (vPullbackCone D.toGlueData i j) = PullbackCone.snd s \u2192\n m =\n IsOpenImmersion.lift (f D.toGlueData i j) (PullbackCone.fst s)\n (_ : Set.range \u2191(PullbackCone.fst s).base \u2286 Set.range \u2191(f D.toGlueData i j).base)\n[PROOFSTEP]\nintro m e\u2081 _\n[GOAL]\ncase refine'_4\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\nm : s.pt \u27f6 (vPullbackCone D.toGlueData i j).pt\ne\u2081 : m \u226b PullbackCone.fst (vPullbackCone D.toGlueData i j) = PullbackCone.fst s\na\u271d : m \u226b PullbackCone.snd (vPullbackCone D.toGlueData i j) = PullbackCone.snd s\n\u22a2 m =\n IsOpenImmersion.lift (f D.toGlueData i j) (PullbackCone.fst s)\n (_ : Set.range \u2191(PullbackCone.fst s).base \u2286 Set.range \u2191(f D.toGlueData i j).base)\n[PROOFSTEP]\nrw [\u2190 cancel_mono (D.f i j)]\n[GOAL]\ncase refine'_4\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\nm : s.pt \u27f6 (vPullbackCone D.toGlueData i j).pt\ne\u2081 : m \u226b PullbackCone.fst (vPullbackCone D.toGlueData i j) = PullbackCone.fst s\na\u271d : m \u226b PullbackCone.snd (vPullbackCone D.toGlueData i j) = PullbackCone.snd s\n\u22a2 m \u226b f D.toGlueData i j =\n IsOpenImmersion.lift (f D.toGlueData i j) (PullbackCone.fst s)\n (_ : Set.range \u2191(PullbackCone.fst s).base \u2286 Set.range \u2191(f D.toGlueData i j).base) \u226b\n f D.toGlueData i j\n[PROOFSTEP]\nerw [e\u2081]\n[GOAL]\ncase refine'_4\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : GlueData C\ninst\u271d : HasLimits C\ni j : D.J\ns : PullbackCone (\u03b9 D.toGlueData i) (\u03b9 D.toGlueData j)\nm : s.pt \u27f6 (vPullbackCone D.toGlueData i j).pt\ne\u2081 : m \u226b PullbackCone.fst (vPullbackCone D.toGlueData i j) = PullbackCone.fst s\na\u271d : m \u226b PullbackCone.snd (vPullbackCone D.toGlueData i j) = PullbackCone.snd s\n\u22a2 PullbackCone.fst s =\n IsOpenImmersion.lift (f D.toGlueData i j) (PullbackCone.fst s)\n (_ : Set.range \u2191(PullbackCone.fst s).base \u2286 Set.range \u2191(f D.toGlueData i j).base) \u226b\n f D.toGlueData i j\n[PROOFSTEP]\nrw [IsOpenImmersion.lift_fac]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasProducts C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\n\u22a2 IsOpenImmersion (\u03b9 D.toGlueData i)\n[PROOFSTEP]\nrw [\u2190 D.\u03b9_isoPresheafedSpace_inv]\n -- Porting note : was `inferInstance`\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasProducts C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\n\u22a2 IsOpenImmersion (\u03b9 (toPresheafedSpaceGlueData D).toGlueData i \u226b (isoPresheafedSpace D).inv)\n[PROOFSTEP]\nrefine PresheafedSpace.IsOpenImmersion.comp (hf := ?_) (hg := inferInstance)\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasProducts C\nD : GlueData C\ninst\u271d : HasLimits C\ni : D.J\n\u22a2 PresheafedSpace.IsOpenImmersion (\u03b9 (toPresheafedSpaceGlueData D).toGlueData i)\n[PROOFSTEP]\napply PresheafedSpace.GlueData.\u03b9IsOpenImmersion\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData\ni : D.J\n\u22a2 IsOpenImmersion (\u03b9 D.toGlueData i)\n[PROOFSTEP]\ndelta IsOpenImmersion\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData\ni : D.J\n\u22a2 SheafedSpace.IsOpenImmersion (\u03b9 D.toGlueData i).val\n[PROOFSTEP]\nrw [\u2190 D.\u03b9_isoSheafedSpace_inv]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData\ni : D.J\n\u22a2 SheafedSpace.IsOpenImmersion (\u03b9 (toSheafedSpaceGlueData D).toGlueData i \u226b (isoSheafedSpace D).inv)\n[PROOFSTEP]\napply (config := { allowSynthFailures := true }) PresheafedSpace.IsOpenImmersion.comp\n[GOAL]\ncase hf\nC : Type u\ninst\u271d : Category.{v, u} C\nD : GlueData\ni : D.J\n\u22a2 PresheafedSpace.IsOpenImmersion (\u03b9 (toSheafedSpaceGlueData D).toGlueData i)\n[PROOFSTEP]\nexact (D.toSheafedSpaceGlueData).\u03b9IsOpenImmersion i\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.PresheafedSpace.Gluing", "llama_tokens": 183876, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723317123102955, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.40100091850671254}} {"text": "[GOAL]\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\nC : (\u2a01 (i : \u03b9), \u03b2 i) \u2192 Prop\nx : \u2a01 (i : \u03b9), \u03b2 i\nH_zero : C 0\nH_basic : \u2200 (i : \u03b9) (x : \u03b2 i), C (\u2191(of \u03b2 i) x)\nH_plus : \u2200 (x y : \u2a01 (i : \u03b9), \u03b2 i), C x \u2192 C y \u2192 C (x + y)\n\u22a2 C x\n[PROOFSTEP]\napply DFinsupp.induction x H_zero\n[GOAL]\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\nC : (\u2a01 (i : \u03b9), \u03b2 i) \u2192 Prop\nx : \u2a01 (i : \u03b9), \u03b2 i\nH_zero : C 0\nH_basic : \u2200 (i : \u03b9) (x : \u03b2 i), C (\u2191(of \u03b2 i) x)\nH_plus : \u2200 (x y : \u2a01 (i : \u03b9), \u03b2 i), C x \u2192 C y \u2192 C (x + y)\n\u22a2 \u2200 (i : \u03b9) (b : (fun i => \u03b2 i) i) (f : \u03a0\u2080 (i : \u03b9), (fun i => \u03b2 i) i),\n \u2191f i = 0 \u2192 b \u2260 0 \u2192 C f \u2192 C (DFinsupp.single i b + f)\n[PROOFSTEP]\nintro i b f h1 h2 ih\n[GOAL]\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\nC : (\u2a01 (i : \u03b9), \u03b2 i) \u2192 Prop\nx : \u2a01 (i : \u03b9), \u03b2 i\nH_zero : C 0\nH_basic : \u2200 (i : \u03b9) (x : \u03b2 i), C (\u2191(of \u03b2 i) x)\nH_plus : \u2200 (x y : \u2a01 (i : \u03b9), \u03b2 i), C x \u2192 C y \u2192 C (x + y)\ni : \u03b9\nb : \u03b2 i\nf : \u03a0\u2080 (i : \u03b9), (fun i => \u03b2 i) i\nh1 : \u2191f i = 0\nh2 : b \u2260 0\nih : C f\n\u22a2 C (DFinsupp.single i b + f)\n[PROOFSTEP]\nsolve_by_elim\n[GOAL]\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\n\u03c8 : (\u2a01 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nf : \u2a01 (i : \u03b9), \u03b2 i\n\u22a2 \u2191\u03c8 f = \u2191(toAddMonoid fun i => AddMonoidHom.comp \u03c8 (of \u03b2 i)) f\n[PROOFSTEP]\ncongr\n -- Porting note: ext applies addHom_ext' here, which isn't what we want.\n[GOAL]\ncase e_a\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\n\u03c8 : (\u2a01 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nf : \u2a01 (i : \u03b9), \u03b2 i\n\u22a2 \u03c8 = toAddMonoid fun i => AddMonoidHom.comp \u03c8 (of \u03b2 i)\n[PROOFSTEP]\napply DFinsupp.addHom_ext'\n[GOAL]\ncase e_a.H\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\n\u03c8 : (\u2a01 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nf : \u2a01 (i : \u03b9), \u03b2 i\n\u22a2 \u2200 (x : \u03b9),\n AddMonoidHom.comp \u03c8 (DFinsupp.singleAddHom (fun i => (fun i => \u03b2 i) i) x) =\n AddMonoidHom.comp (toAddMonoid fun i => AddMonoidHom.comp \u03c8 (of \u03b2 i))\n (DFinsupp.singleAddHom (fun i => (fun i => \u03b2 i) i) x)\n[PROOFSTEP]\nsimp [toAddMonoid, of]\n[GOAL]\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d : AddCommMonoid \u03b3\ni : \u03b9\nf : \u03b3 \u2192+ \u03b2 i\n\u22a2 \u2191fromAddMonoid (\u2191(of (fun i => \u03b3 \u2192+ \u03b2 i) i) f) = AddMonoidHom.comp (of \u03b2 i) f\n[PROOFSTEP]\nrw [fromAddMonoid, toAddMonoid_of]\n[GOAL]\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d : AddCommMonoid \u03b3\ni : \u03b9\nf : \u03b3 \u2192+ \u03b2 i\n\u22a2 \u2191(\u2191AddMonoidHom.compHom (of \u03b2 i)) f = AddMonoidHom.comp (of \u03b2 i) f\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d : AddCommMonoid \u03b3\ni : \u03b9\nf : \u03b3 \u2192+ \u03b2 i\nx : \u03b3\n\u22a2 \u2191(\u2191fromAddMonoid (\u2191(of (fun i => \u03b3 \u2192+ \u03b2 i) i) f)) x = \u2191(of \u03b2 i) (\u2191f x)\n[PROOFSTEP]\nrw [fromAddMonoid_of, AddMonoidHom.coe_comp, Function.comp]\n[GOAL]\n\u03b9\u271d : Type v\ndec_\u03b9 : DecidableEq \u03b9\u271d\n\u03b2 : \u03b9\u271d \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9\u271d) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d\u00b2 : AddCommMonoid \u03b3\nM : Type v\n\u03b9 : optParam (Type u_1) PUnit\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Unique \u03b9\nsrc\u271d : (\u2a01 (i : \u03b9), M) \u2192+ M := toAddMonoid fun x => AddMonoidHom.id M\nx : \u2a01 (x : \u03b9), M\n\u22a2 \u2191(of (fun x => M) default) (\u2191(toAddMonoid fun x => AddMonoidHom.id M) 0) = 0\n[PROOFSTEP]\nrw [AddMonoidHom.map_zero, AddMonoidHom.map_zero]\n[GOAL]\n\u03b9\u271d : Type v\ndec_\u03b9 : DecidableEq \u03b9\u271d\n\u03b2 : \u03b9\u271d \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9\u271d) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d\u00b2 : AddCommMonoid \u03b3\nM : Type v\n\u03b9 : optParam (Type u_1) PUnit\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Unique \u03b9\nsrc\u271d : (\u2a01 (i : \u03b9), M) \u2192+ M := toAddMonoid fun x => AddMonoidHom.id M\nx\u271d : \u2a01 (x : \u03b9), M\np : \u03b9\nx : M\n\u22a2 \u2191(of (fun x => M) default) (\u2191(toAddMonoid fun x => AddMonoidHom.id M) (\u2191(of (fun i => M) p) x)) =\n \u2191(of (fun i => M) p) x\n[PROOFSTEP]\nrw [Unique.default_eq p, toAddMonoid_of]\n[GOAL]\n\u03b9\u271d : Type v\ndec_\u03b9 : DecidableEq \u03b9\u271d\n\u03b2 : \u03b9\u271d \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9\u271d) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d\u00b2 : AddCommMonoid \u03b3\nM : Type v\n\u03b9 : optParam (Type u_1) PUnit\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Unique \u03b9\nsrc\u271d : (\u2a01 (i : \u03b9), M) \u2192+ M := toAddMonoid fun x => AddMonoidHom.id M\nx\u271d : \u2a01 (x : \u03b9), M\np : \u03b9\nx : M\n\u22a2 \u2191(of (fun x => M) p) (\u2191(AddMonoidHom.id M) x) = \u2191(of (fun i => M) p) x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9\u271d : Type v\ndec_\u03b9 : DecidableEq \u03b9\u271d\n\u03b2 : \u03b9\u271d \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9\u271d) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d\u00b2 : AddCommMonoid \u03b3\nM : Type v\n\u03b9 : optParam (Type u_1) PUnit\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Unique \u03b9\nsrc\u271d : (\u2a01 (i : \u03b9), M) \u2192+ M := toAddMonoid fun x => AddMonoidHom.id M\nx\u271d x y : \u2a01 (i : \u03b9), M\nihx : \u2191(of (fun x => M) default) (\u2191(toAddMonoid fun x => AddMonoidHom.id M) x) = x\nihy : \u2191(of (fun x => M) default) (\u2191(toAddMonoid fun x => AddMonoidHom.id M) y) = y\n\u22a2 \u2191(of (fun x => M) default) (\u2191(toAddMonoid fun x => AddMonoidHom.id M) (x + y)) = x + y\n[PROOFSTEP]\nrw [AddMonoidHom.map_add, AddMonoidHom.map_add, ihx, ihy]\n[GOAL]\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d : AddCommMonoid \u03b3\n\u03ba : Type u_1\nh : \u03b9 \u2243 \u03ba\nf : \u2a01 (i : \u03b9), \u03b2 i\nk : \u03ba\n\u22a2 \u2191(\u2191(equivCongrLeft h) f) k = \u2191f (\u2191h.symm k)\n[PROOFSTEP]\nexact DFinsupp.comapDomain'_apply _ h.right_inv _ _\n[GOAL]\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d\u2074 : AddCommMonoid \u03b3\nM : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SetLike S M\ninst\u271d : AddSubmonoidClass S M\nA : \u03b9 \u2192 S\ni j : \u03b9\nx : { x // x \u2208 A i }\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) i) x) j) = \u2191(if i = j then x else 0)\n[PROOFSTEP]\nobtain rfl | h := Decidable.eq_or_ne i j\n[GOAL]\ncase inl\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d\u2074 : AddCommMonoid \u03b3\nM : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SetLike S M\ninst\u271d : AddSubmonoidClass S M\nA : \u03b9 \u2192 S\ni : \u03b9\nx : { x // x \u2208 A i }\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) i) x) i) = \u2191(if i = i then x else 0)\n[PROOFSTEP]\nrw [DirectSum.of_eq_same, if_pos rfl]\n[GOAL]\ncase inr\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d\u2074 : AddCommMonoid \u03b3\nM : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : SetLike S M\ninst\u271d : AddSubmonoidClass S M\nA : \u03b9 \u2192 S\ni j : \u03b9\nx : { x // x \u2208 A i }\nh : i \u2260 j\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) i) x) j) = \u2191(if i = j then x else 0)\n[PROOFSTEP]\nrw [DirectSum.of_eq_of_ne _ _ _ _ h, if_neg h, ZeroMemClass.coe_zero, ZeroMemClass.coe_zero]\n[GOAL]\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d\u00b2 : AddCommMonoid \u03b3\nM : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\nA : \u03b9 \u2192 AddSubmonoid M\nh : IsInternal A\n\u22a2 iSup A = \u22a4\n[PROOFSTEP]\nrw [AddSubmonoid.iSup_eq_mrange_dfinsupp_sumAddHom, AddMonoidHom.mrange_top_iff_surjective]\n[GOAL]\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d\u00b2 : AddCommMonoid \u03b3\nM : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\nA : \u03b9 \u2192 AddSubmonoid M\nh : IsInternal A\n\u22a2 Function.Surjective \u2191(DFinsupp.sumAddHom fun i => AddSubmonoid.subtype (A i))\n[PROOFSTEP]\nexact Function.Bijective.surjective h\n", "meta": {"mathlib_filename": "Mathlib.Algebra.DirectSum.Basic", "llama_tokens": 4121, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718435083355187, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.4009891597906099}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\n\u22a2 { fst := k, snd := c } \u2208 sigmaLift (fun x => Icc) { fst := i, snd := a } { fst := j, snd := b } \u2194\n { fst := i, snd := a } \u2264 { fst := k, snd := c } \u2227 { fst := k, snd := c } \u2264 { fst := j, snd := b }\n[PROOFSTEP]\nsimp_rw [mem_sigmaLift, le_def, mem_Icc, exists_and_left, \u2190 exists_and_right, \u2190 exists_prop]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\n\u22a2 (\u2203 h _h x, c \u2264 (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b) \u2194\n \u2203 h _h h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c \u2264 b\n[PROOFSTEP]\nexact bex_congr fun _ _ => by constructor <;> rintro \u27e8\u27e8\u27e9, ht\u27e9 <;> exact \u27e8rfl, ht\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\nx\u271d\u00b9 : i = k\nx\u271d : (_ : i = k) \u25b8 a \u2264 c\n\u22a2 (\u2203 x, c \u2264 (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b) \u2194\n \u2203 h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c \u2264 b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\nx\u271d\u00b9 : i = k\nx\u271d : (_ : i = k) \u25b8 a \u2264 c\n\u22a2 (\u2203 x, c \u2264 (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b) \u2192\n \u2203 h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c \u2264 b\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9, ht\u27e9\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\nx\u271d\u00b9 : i = k\nx\u271d : (_ : i = k) \u25b8 a \u2264 c\n\u22a2 (\u2203 h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c \u2264 b) \u2192\n \u2203 x, c \u2264 (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9, ht\u27e9\n[GOAL]\ncase mp.intro.refl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb c : \u03b1 j\nx\u271d\u00b9 : i = j\nx\u271d : (_ : i = j) \u25b8 a \u2264 c\nht : c \u2264 (_ : { fst := j, snd := b }.fst = { fst := j, snd := c }.fst) \u25b8 b\n\u22a2 \u2203 h, (_ : { fst := j, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c \u2264 b\n[PROOFSTEP]\nexact \u27e8rfl, ht\u27e9\n[GOAL]\ncase mpr.intro.refl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb c : \u03b1 j\nx\u271d\u00b9 : i = j\nx\u271d : (_ : i = j) \u25b8 a \u2264 c\nht : (_ : { fst := j, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c \u2264 b\n\u22a2 \u2203 x, c \u2264 (_ : { fst := j, snd := b }.fst = { fst := j, snd := c }.fst) \u25b8 b\n[PROOFSTEP]\nexact \u27e8rfl, ht\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\n\u22a2 { fst := k, snd := c } \u2208 sigmaLift (fun x => Ico) { fst := i, snd := a } { fst := j, snd := b } \u2194\n { fst := i, snd := a } \u2264 { fst := k, snd := c } \u2227 { fst := k, snd := c } < { fst := j, snd := b }\n[PROOFSTEP]\nsimp_rw [mem_sigmaLift, le_def, lt_def, mem_Ico, exists_and_left, \u2190 exists_and_right, \u2190 exists_prop]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\n\u22a2 (\u2203 h _h x, c < (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b) \u2194\n \u2203 h _h h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c < b\n[PROOFSTEP]\nexact bex_congr fun _ _ => by constructor <;> rintro \u27e8\u27e8\u27e9, ht\u27e9 <;> exact \u27e8rfl, ht\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\nx\u271d\u00b9 : i = k\nx\u271d : (_ : i = k) \u25b8 a \u2264 c\n\u22a2 (\u2203 x, c < (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b) \u2194\n \u2203 h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c < b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\nx\u271d\u00b9 : i = k\nx\u271d : (_ : i = k) \u25b8 a \u2264 c\n\u22a2 (\u2203 x, c < (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b) \u2192\n \u2203 h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c < b\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9, ht\u27e9\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\nx\u271d\u00b9 : i = k\nx\u271d : (_ : i = k) \u25b8 a \u2264 c\n\u22a2 (\u2203 h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c < b) \u2192\n \u2203 x, c < (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9, ht\u27e9\n[GOAL]\ncase mp.intro.refl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb c : \u03b1 j\nx\u271d\u00b9 : i = j\nx\u271d : (_ : i = j) \u25b8 a \u2264 c\nht : c < (_ : { fst := j, snd := b }.fst = { fst := j, snd := c }.fst) \u25b8 b\n\u22a2 \u2203 h, (_ : { fst := j, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c < b\n[PROOFSTEP]\nexact \u27e8rfl, ht\u27e9\n[GOAL]\ncase mpr.intro.refl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb c : \u03b1 j\nx\u271d\u00b9 : i = j\nx\u271d : (_ : i = j) \u25b8 a \u2264 c\nht : (_ : { fst := j, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c < b\n\u22a2 \u2203 x, c < (_ : { fst := j, snd := b }.fst = { fst := j, snd := c }.fst) \u25b8 b\n[PROOFSTEP]\nexact \u27e8rfl, ht\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\n\u22a2 { fst := k, snd := c } \u2208 sigmaLift (fun x => Ioc) { fst := i, snd := a } { fst := j, snd := b } \u2194\n { fst := i, snd := a } < { fst := k, snd := c } \u2227 { fst := k, snd := c } \u2264 { fst := j, snd := b }\n[PROOFSTEP]\nsimp_rw [mem_sigmaLift, le_def, lt_def, mem_Ioc, exists_and_left, \u2190 exists_and_right, \u2190 exists_prop]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\n\u22a2 (\u2203 h _h x, c \u2264 (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b) \u2194\n \u2203 h _h h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c \u2264 b\n[PROOFSTEP]\nexact bex_congr fun _ _ => by constructor <;> rintro \u27e8\u27e8\u27e9, ht\u27e9 <;> exact \u27e8rfl, ht\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\nx\u271d\u00b9 : i = k\nx\u271d : (_ : i = k) \u25b8 a < c\n\u22a2 (\u2203 x, c \u2264 (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b) \u2194\n \u2203 h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c \u2264 b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\nx\u271d\u00b9 : i = k\nx\u271d : (_ : i = k) \u25b8 a < c\n\u22a2 (\u2203 x, c \u2264 (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b) \u2192\n \u2203 h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c \u2264 b\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9, ht\u27e9\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\nx\u271d\u00b9 : i = k\nx\u271d : (_ : i = k) \u25b8 a < c\n\u22a2 (\u2203 h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c \u2264 b) \u2192\n \u2203 x, c \u2264 (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9, ht\u27e9\n[GOAL]\ncase mp.intro.refl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb c : \u03b1 j\nx\u271d\u00b9 : i = j\nx\u271d : (_ : i = j) \u25b8 a < c\nht : c \u2264 (_ : { fst := j, snd := b }.fst = { fst := j, snd := c }.fst) \u25b8 b\n\u22a2 \u2203 h, (_ : { fst := j, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c \u2264 b\n[PROOFSTEP]\nexact \u27e8rfl, ht\u27e9\n[GOAL]\ncase mpr.intro.refl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb c : \u03b1 j\nx\u271d\u00b9 : i = j\nx\u271d : (_ : i = j) \u25b8 a < c\nht : (_ : { fst := j, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c \u2264 b\n\u22a2 \u2203 x, c \u2264 (_ : { fst := j, snd := b }.fst = { fst := j, snd := c }.fst) \u25b8 b\n[PROOFSTEP]\nexact \u27e8rfl, ht\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\n\u22a2 { fst := k, snd := c } \u2208 sigmaLift (fun x => Ioo) { fst := i, snd := a } { fst := j, snd := b } \u2194\n { fst := i, snd := a } < { fst := k, snd := c } \u2227 { fst := k, snd := c } < { fst := j, snd := b }\n[PROOFSTEP]\nsimp_rw [mem_sigmaLift, lt_def, mem_Ioo, exists_and_left, \u2190 exists_and_right, \u2190 exists_prop]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\n\u22a2 (\u2203 h _h x, c < (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b) \u2194\n \u2203 h _h h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c < b\n[PROOFSTEP]\nexact bex_congr fun _ _ => by constructor <;> rintro \u27e8\u27e8\u27e9, ht\u27e9 <;> exact \u27e8rfl, ht\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\nx\u271d\u00b9 : i = k\nx\u271d : (_ : i = k) \u25b8 a < c\n\u22a2 (\u2203 x, c < (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b) \u2194\n \u2203 h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c < b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\nx\u271d\u00b9 : i = k\nx\u271d : (_ : i = k) \u25b8 a < c\n\u22a2 (\u2203 x, c < (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b) \u2192\n \u2203 h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c < b\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9, ht\u27e9\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nk : \u03b9\nc : \u03b1 k\nx\u271d\u00b9 : i = k\nx\u271d : (_ : i = k) \u25b8 a < c\n\u22a2 (\u2203 h, (_ : { fst := k, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c < b) \u2192\n \u2203 x, c < (_ : { fst := j, snd := b }.fst = { fst := k, snd := c }.fst) \u25b8 b\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9, ht\u27e9\n[GOAL]\ncase mp.intro.refl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb c : \u03b1 j\nx\u271d\u00b9 : i = j\nx\u271d : (_ : i = j) \u25b8 a < c\nht : c < (_ : { fst := j, snd := b }.fst = { fst := j, snd := c }.fst) \u25b8 b\n\u22a2 \u2203 h, (_ : { fst := j, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c < b\n[PROOFSTEP]\nexact \u27e8rfl, ht\u27e9\n[GOAL]\ncase mpr.intro.refl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i)\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb c : \u03b1 j\nx\u271d\u00b9 : i = j\nx\u271d : (_ : i = j) \u25b8 a < c\nht : (_ : { fst := j, snd := c }.fst = { fst := j, snd := b }.fst) \u25b8 c < b\n\u22a2 \u2203 x, c < (_ : { fst := j, snd := b }.fst = { fst := j, snd := c }.fst) \u25b8 b\n[PROOFSTEP]\nexact \u27e8rfl, ht\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Data.Sigma.Interval", "llama_tokens": 7513, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390747, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.4009891543374835}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : StrictConvexSpace \ud835\udd5c E\nx : E\nr : \u211d\n\u22a2 StrictConvex \ud835\udd5c (closedBall x r)\n[PROOFSTEP]\ncases' le_or_lt r 0 with hr hr\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : StrictConvexSpace \ud835\udd5c E\nx : E\nr : \u211d\nhr : r \u2264 0\n\u22a2 StrictConvex \ud835\udd5c (closedBall x r)\n[PROOFSTEP]\nexact (subsingleton_closedBall x hr).strictConvex\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : StrictConvexSpace \ud835\udd5c E\nx : E\nr : \u211d\nhr : 0 < r\n\u22a2 StrictConvex \ud835\udd5c (closedBall x r)\n[PROOFSTEP]\nrw [\u2190 vadd_closedBall_zero]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : StrictConvexSpace \ud835\udd5c E\nx : E\nr : \u211d\nhr : 0 < r\n\u22a2 StrictConvex \ud835\udd5c (x +\u1d65 closedBall 0 r)\n[PROOFSTEP]\nexact (StrictConvexSpace.strictConvex_closedBall r hr).vadd _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : LinearMap.CompatibleSMul E E \ud835\udd5c \u211d\nh : StrictConvex \ud835\udd5c (closedBall 0 1)\nr : \u211d\nhr : 0 < r\n\u22a2 StrictConvex \ud835\udd5c (closedBall 0 r)\n[PROOFSTEP]\nsimpa only [smul_closedUnitBall_of_nonneg hr.le] using h.smul r\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 (x y : E), \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 x \u2260 y \u2192 \u2203 a b, a + b = 1 \u2227 \u2016a \u2022 x + b \u2022 y\u2016 < 1\n\u22a2 StrictConvexSpace \u211d E\n[PROOFSTEP]\nrefine'\n StrictConvexSpace.of_strictConvex_closed_unit_ball \u211d ((convex_closedBall _ _).strictConvex' fun x hx y hy hne => _)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 (x y : E), \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 x \u2260 y \u2192 \u2203 a b, a + b = 1 \u2227 \u2016a \u2022 x + b \u2022 y\u2016 < 1\nx : E\nhx : x \u2208 closedBall 0 1 \\ interior (closedBall 0 1)\ny : E\nhy : y \u2208 closedBall 0 1 \\ interior (closedBall 0 1)\nhne : x \u2260 y\n\u22a2 (fun x y => \u2203 c, \u2191(AffineMap.lineMap x y) c \u2208 interior (closedBall 0 1)) x y\n[PROOFSTEP]\nrw [interior_closedBall (0 : E) one_ne_zero, closedBall_diff_ball, mem_sphere_zero_iff_norm] at hx hy \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 (x y : E), \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 x \u2260 y \u2192 \u2203 a b, a + b = 1 \u2227 \u2016a \u2022 x + b \u2022 y\u2016 < 1\nx : E\nhx : \u2016x\u2016 = 1\ny : E\nhy : \u2016y\u2016 = 1\nhne : x \u2260 y\n\u22a2 \u2203 c, \u2191(AffineMap.lineMap x y) c \u2208 interior (closedBall 0 1)\n[PROOFSTEP]\nrcases h x y hx hy hne with \u27e8a, b, hab, hlt\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 (x y : E), \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 x \u2260 y \u2192 \u2203 a b, a + b = 1 \u2227 \u2016a \u2022 x + b \u2022 y\u2016 < 1\nx : E\nhx : \u2016x\u2016 = 1\ny : E\nhy : \u2016y\u2016 = 1\nhne : x \u2260 y\na b : \u211d\nhab : a + b = 1\nhlt : \u2016a \u2022 x + b \u2022 y\u2016 < 1\n\u22a2 \u2203 c, \u2191(AffineMap.lineMap x y) c \u2208 interior (closedBall 0 1)\n[PROOFSTEP]\nuse b\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 (x y : E), \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 x \u2260 y \u2192 \u2203 a b, a + b = 1 \u2227 \u2016a \u2022 x + b \u2022 y\u2016 < 1\nx : E\nhx : \u2016x\u2016 = 1\ny : E\nhy : \u2016y\u2016 = 1\nhne : x \u2260 y\na b : \u211d\nhab : a + b = 1\nhlt : \u2016a \u2022 x + b \u2022 y\u2016 < 1\n\u22a2 \u2191(AffineMap.lineMap x y) b \u2208 interior (closedBall 0 1)\n[PROOFSTEP]\nrwa [AffineMap.lineMap_apply_module, interior_closedBall (0 : E) one_ne_zero, mem_ball_zero_iff,\n sub_eq_iff_eq_add.2 hab.symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 (x y : E), \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 x \u2260 y \u2192 \u2203 a b, 0 \u2264 a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 \u2016a \u2022 x + b \u2022 y\u2016 \u2260 1\n\u22a2 StrictConvexSpace \u211d E\n[PROOFSTEP]\nrefine' StrictConvexSpace.of_strictConvex_closed_unit_ball \u211d ((convex_closedBall _ _).strictConvex _)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 (x y : E), \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 x \u2260 y \u2192 \u2203 a b, 0 \u2264 a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 \u2016a \u2022 x + b \u2022 y\u2016 \u2260 1\n\u22a2 Set.Pairwise (closedBall 0 1 \\ interior (closedBall 0 1)) fun x y =>\n Set.Nonempty ([x-[\u211d]y] \\ frontier (closedBall 0 1))\n[PROOFSTEP]\nsimp only [interior_closedBall _ one_ne_zero, closedBall_diff_ball, Set.Pairwise, frontier_closedBall _ one_ne_zero,\n mem_sphere_zero_iff_norm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 (x y : E), \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 x \u2260 y \u2192 \u2203 a b, 0 \u2264 a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 \u2016a \u2022 x + b \u2022 y\u2016 \u2260 1\n\u22a2 \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 x \u2260 y \u2192 Set.Nonempty ([x-[\u211d]y] \\ sphere 0 1)\n[PROOFSTEP]\nintro x hx y hy hne\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 (x y : E), \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 x \u2260 y \u2192 \u2203 a b, 0 \u2264 a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 \u2016a \u2022 x + b \u2022 y\u2016 \u2260 1\nx : E\nhx : \u2016x\u2016 = 1\ny : E\nhy : \u2016y\u2016 = 1\nhne : x \u2260 y\n\u22a2 Set.Nonempty ([x-[\u211d]y] \\ sphere 0 1)\n[PROOFSTEP]\nrcases h x y hx hy hne with \u27e8a, b, ha, hb, hab, hne'\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 (x y : E), \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 x \u2260 y \u2192 \u2203 a b, 0 \u2264 a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 \u2016a \u2022 x + b \u2022 y\u2016 \u2260 1\nx : E\nhx : \u2016x\u2016 = 1\ny : E\nhy : \u2016y\u2016 = 1\nhne : x \u2260 y\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhne' : \u2016a \u2022 x + b \u2022 y\u2016 \u2260 1\n\u22a2 Set.Nonempty ([x-[\u211d]y] \\ sphere 0 1)\n[PROOFSTEP]\nexact \u27e8_, \u27e8a, b, ha, hb, hab, rfl\u27e9, mt mem_sphere_zero_iff_norm.1 hne'\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 \u2983x y : E\u2984, \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 x \u2260 y \u2192 \u2016x + y\u2016 \u2260 2\n\u22a2 StrictConvexSpace \u211d E\n[PROOFSTEP]\nrefine'\n StrictConvexSpace.of_norm_combo_ne_one fun x y hx hy hne =>\n \u27e81 / 2, 1 / 2, one_half_pos.le, one_half_pos.le, add_halves _, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 \u2983x y : E\u2984, \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 x \u2260 y \u2192 \u2016x + y\u2016 \u2260 2\nx y : E\nhx : \u2016x\u2016 = 1\nhy : \u2016y\u2016 = 1\nhne : x \u2260 y\n\u22a2 \u2016(1 / 2) \u2022 x + (1 / 2) \u2022 y\u2016 \u2260 1\n[PROOFSTEP]\nrw [\u2190 smul_add, norm_smul, Real.norm_of_nonneg one_half_pos.le, one_div, \u2190 div_eq_inv_mul, Ne.def,\n div_eq_one_iff_eq (two_ne_zero' \u211d)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 \u2983x y : E\u2984, \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 x \u2260 y \u2192 \u2016x + y\u2016 \u2260 2\nx y : E\nhx : \u2016x\u2016 = 1\nhy : \u2016y\u2016 = 1\nhne : x \u2260 y\n\u22a2 \u00ac\u2016x + y\u2016 = 2\n[PROOFSTEP]\nexact h hx hy hne\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 (x y : E), \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 \u2016x + y\u2016 = 2 \u2192 SameRay \u211d x y\n\u22a2 StrictConvexSpace \u211d E\n[PROOFSTEP]\nrefine' StrictConvexSpace.of_pairwise_sphere_norm_ne_two fun x hx y hy => mt fun h\u2082 => _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 (x y : E), \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 \u2016x + y\u2016 = 2 \u2192 SameRay \u211d x y\nx : E\nhx : x \u2208 sphere 0 1\ny : E\nhy : y \u2208 sphere 0 1\nh\u2082 : \u2016x + y\u2016 = 2\n\u22a2 x = y\n[PROOFSTEP]\nrw [mem_sphere_zero_iff_norm] at hx hy \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nh : \u2200 (x y : E), \u2016x\u2016 = 1 \u2192 \u2016y\u2016 = 1 \u2192 \u2016x + y\u2016 = 2 \u2192 SameRay \u211d x y\nx : E\nhx : \u2016x\u2016 = 1\ny : E\nhy : \u2016y\u2016 = 1\nh\u2082 : \u2016x + y\u2016 = 2\n\u22a2 x = y\n[PROOFSTEP]\nexact (sameRay_iff_of_norm_eq (hx.trans hy.symm)).1 (h x y hx hy h\u2082)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nhx : x \u2208 closedBall z r\nhy : y \u2208 closedBall z r\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2208 ball z r\n[PROOFSTEP]\nrcases eq_or_ne r 0 with (rfl | hr)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b : \u211d\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx : x \u2208 closedBall z 0\nhy : y \u2208 closedBall z 0\n\u22a2 a \u2022 x + b \u2022 y \u2208 ball z 0\n[PROOFSTEP]\nrw [closedBall_zero, mem_singleton_iff] at hx hy \n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b : \u211d\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx : x = z\nhy : y = z\n\u22a2 a \u2022 x + b \u2022 y \u2208 ball z 0\n[PROOFSTEP]\nexact (hne (hx.trans hy.symm)).elim\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nhx : x \u2208 closedBall z r\nhy : y \u2208 closedBall z r\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhr : r \u2260 0\n\u22a2 a \u2022 x + b \u2022 y \u2208 ball z r\n[PROOFSTEP]\nsimp only [\u2190 interior_closedBall _ hr] at hx hy \u22a2\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nhx : x \u2208 closedBall z r\nhy : y \u2208 closedBall z r\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhr : r \u2260 0\n\u22a2 a \u2022 x + b \u2022 y \u2208 interior (closedBall z r)\n[PROOFSTEP]\nexact strictConvex_closedBall \u211d z r hx hy hne ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nhx : \u2016x\u2016 \u2264 r\nhy : \u2016y\u2016 \u2264 r\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 \u2016a \u2022 x + b \u2022 y\u2016 < r\n[PROOFSTEP]\nsimp only [\u2190 mem_ball_zero_iff, \u2190 mem_closedBall_zero_iff] at hx hy \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx : x \u2208 closedBall 0 r\nhy : y \u2208 closedBall 0 r\n\u22a2 a \u2022 x + b \u2022 y \u2208 ball 0 r\n[PROOFSTEP]\nexact combo_mem_ball_of_ne hx hy hne ha hb hab\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nh : \u00acSameRay \u211d x y\n\u22a2 \u2016x + y\u2016 < \u2016x\u2016 + \u2016y\u2016\n[PROOFSTEP]\nsimp only [sameRay_iff_inv_norm_smul_eq, not_or, \u2190 Ne.def] at h \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nh : x \u2260 0 \u2227 y \u2260 0 \u2227 \u2016x\u2016\u207b\u00b9 \u2022 x \u2260 \u2016y\u2016\u207b\u00b9 \u2022 y\n\u22a2 \u2016x + y\u2016 < \u2016x\u2016 + \u2016y\u2016\n[PROOFSTEP]\nrcases h with \u27e8hx, hy, hne\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nhx : x \u2260 0\nhy : y \u2260 0\nhne : \u2016x\u2016\u207b\u00b9 \u2022 x \u2260 \u2016y\u2016\u207b\u00b9 \u2022 y\n\u22a2 \u2016x + y\u2016 < \u2016x\u2016 + \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 norm_pos_iff] at hx hy \n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nhx : 0 < \u2016x\u2016\nhy : 0 < \u2016y\u2016\nhne : \u2016x\u2016\u207b\u00b9 \u2022 x \u2260 \u2016y\u2016\u207b\u00b9 \u2022 y\n\u22a2 \u2016x + y\u2016 < \u2016x\u2016 + \u2016y\u2016\n[PROOFSTEP]\nhave hxy : 0 < \u2016x\u2016 + \u2016y\u2016 := add_pos hx hy\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nhx : 0 < \u2016x\u2016\nhy : 0 < \u2016y\u2016\nhne : \u2016x\u2016\u207b\u00b9 \u2022 x \u2260 \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy : 0 < \u2016x\u2016 + \u2016y\u2016\n\u22a2 \u2016x + y\u2016 < \u2016x\u2016 + \u2016y\u2016\n[PROOFSTEP]\nhave :=\n combo_mem_ball_of_ne (inv_norm_smul_mem_closed_unit_ball x) (inv_norm_smul_mem_closed_unit_ball y) hne\n (div_pos hx hxy) (div_pos hy hxy) (by rw [\u2190 add_div, div_self hxy.ne'])\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nhx : 0 < \u2016x\u2016\nhy : 0 < \u2016y\u2016\nhne : \u2016x\u2016\u207b\u00b9 \u2022 x \u2260 \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy : 0 < \u2016x\u2016 + \u2016y\u2016\n\u22a2 \u2016x\u2016 / (\u2016x\u2016 + \u2016y\u2016) + \u2016y\u2016 / (\u2016x\u2016 + \u2016y\u2016) = 1\n[PROOFSTEP]\nrw [\u2190 add_div, div_self hxy.ne']\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nhx : 0 < \u2016x\u2016\nhy : 0 < \u2016y\u2016\nhne : \u2016x\u2016\u207b\u00b9 \u2022 x \u2260 \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy : 0 < \u2016x\u2016 + \u2016y\u2016\nthis : (\u2016x\u2016 / (\u2016x\u2016 + \u2016y\u2016)) \u2022 \u2016x\u2016\u207b\u00b9 \u2022 x + (\u2016y\u2016 / (\u2016x\u2016 + \u2016y\u2016)) \u2022 \u2016y\u2016\u207b\u00b9 \u2022 y \u2208 ball 0 1\n\u22a2 \u2016x + y\u2016 < \u2016x\u2016 + \u2016y\u2016\n[PROOFSTEP]\nrwa [mem_ball_zero_iff, div_eq_inv_mul, div_eq_inv_mul, mul_smul, mul_smul, smul_inv_smul\u2080 hx.ne',\n smul_inv_smul\u2080 hy.ne', \u2190 smul_add, norm_smul, Real.norm_of_nonneg (inv_pos.2 hxy).le, \u2190 div_eq_inv_mul,\n div_lt_one hxy] at this \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nh : \u00acSameRay \u211d x y\n\u22a2 \u2016x\u2016 - \u2016y\u2016 < \u2016x - y\u2016\n[PROOFSTEP]\nnth_rw 1 [\u2190 sub_add_cancel x y] at h \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nh : \u00acSameRay \u211d (x - y + y) y\n\u22a2 \u2016x - y + y\u2016 - \u2016y\u2016 < \u2016x - y\u2016\n[PROOFSTEP]\nexact sub_lt_iff_lt_add.2 (norm_add_lt_of_not_sameRay fun H' => h <| H'.add_left SameRay.rfl)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nh : \u00acSameRay \u211d x y\n\u22a2 |\u2016x\u2016 - \u2016y\u2016| < \u2016x - y\u2016\n[PROOFSTEP]\nrefine' abs_sub_lt_iff.2 \u27e8lt_norm_sub_of_not_sameRay h, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nh : \u00acSameRay \u211d x y\n\u22a2 \u2016y\u2016 - \u2016x\u2016 < \u2016x - y\u2016\n[PROOFSTEP]\nrw [norm_sub_rev]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nh : \u00acSameRay \u211d x y\n\u22a2 \u2016y\u2016 - \u2016x\u2016 < \u2016y - x\u2016\n[PROOFSTEP]\nexact lt_norm_sub_of_not_sameRay (mt SameRay.symm h)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\n\u22a2 dist x y + dist y z = dist x z \u2194 y \u2208 [x-[\u211d]z]\n[PROOFSTEP]\nsimp only [mem_segment_iff_sameRay, sameRay_iff_norm_add, dist_eq_norm', sub_add_sub_cancel', eq_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nh : \u2016x\u2016 = \u2016y\u2016\n\u22a2 \u2016(1 / 2) \u2022 (x + y)\u2016 < \u2016x\u2016 \u2194 x \u2260 y\n[PROOFSTEP]\nrw [norm_smul, Real.norm_of_nonneg (one_div_nonneg.2 zero_le_two), \u2190 inv_eq_one_div, \u2190 div_eq_inv_mul,\n div_lt_iff (zero_lt_two' \u211d), mul_two, \u2190 not_sameRay_iff_of_norm_eq h, not_sameRay_iff_norm_add_lt, h]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z\u271d : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nx y z : PE\nhxy : dist x y = r * dist x z\nhyz : dist y z = (1 - r) * dist x z\n\u22a2 y = \u2191(AffineMap.lineMap x z) r\n[PROOFSTEP]\nhave : y -\u1d65 x \u2208 [(0 : E)-[\u211d]z -\u1d65 x] := by\n rw [\u2190 dist_add_dist_eq_iff, dist_zero_left, dist_vsub_cancel_right, \u2190 dist_eq_norm_vsub', \u2190 dist_eq_norm_vsub', hxy,\n hyz, \u2190 add_mul, add_sub_cancel'_right, one_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z\u271d : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nx y z : PE\nhxy : dist x y = r * dist x z\nhyz : dist y z = (1 - r) * dist x z\n\u22a2 y -\u1d65 x \u2208 [0-[\u211d]z -\u1d65 x]\n[PROOFSTEP]\nrw [\u2190 dist_add_dist_eq_iff, dist_zero_left, dist_vsub_cancel_right, \u2190 dist_eq_norm_vsub', \u2190 dist_eq_norm_vsub', hxy,\n hyz, \u2190 add_mul, add_sub_cancel'_right, one_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z\u271d : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nx y z : PE\nhxy : dist x y = r * dist x z\nhyz : dist y z = (1 - r) * dist x z\nthis : y -\u1d65 x \u2208 [0-[\u211d]z -\u1d65 x]\n\u22a2 y = \u2191(AffineMap.lineMap x z) r\n[PROOFSTEP]\nrcases eq_or_ne x z with (rfl | hne)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nx y : PE\nhxy : dist x y = r * dist x x\nhyz : dist y x = (1 - r) * dist x x\nthis : y -\u1d65 x \u2208 [0-[\u211d]x -\u1d65 x]\n\u22a2 y = \u2191(AffineMap.lineMap x x) r\n[PROOFSTEP]\nobtain rfl : y = x := by simpa\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nx y : PE\nhxy : dist x y = r * dist x x\nhyz : dist y x = (1 - r) * dist x x\nthis : y -\u1d65 x \u2208 [0-[\u211d]x -\u1d65 x]\n\u22a2 y = x\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx y\u271d z : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\ny : PE\nhxy : dist y y = r * dist y y\nhyz : dist y y = (1 - r) * dist y y\nthis : y -\u1d65 y \u2208 [0-[\u211d]y -\u1d65 y]\n\u22a2 y = \u2191(AffineMap.lineMap y y) r\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z\u271d : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nx y z : PE\nhxy : dist x y = r * dist x z\nhyz : dist y z = (1 - r) * dist x z\nthis : y -\u1d65 x \u2208 [0-[\u211d]z -\u1d65 x]\nhne : x \u2260 z\n\u22a2 y = \u2191(AffineMap.lineMap x z) r\n[PROOFSTEP]\nrw [\u2190 dist_ne_zero] at hne \n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z\u271d : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nx y z : PE\nhxy : dist x y = r * dist x z\nhyz : dist y z = (1 - r) * dist x z\nthis : y -\u1d65 x \u2208 [0-[\u211d]z -\u1d65 x]\nhne\u271d : x \u2260 z\nhne : dist x z \u2260 0\n\u22a2 y = \u2191(AffineMap.lineMap x z) r\n[PROOFSTEP]\nrcases this with \u27e8a, b, _, hb, _, H\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z\u271d : E\na\u271d b\u271d r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nx y z : PE\nhxy : dist x y = r * dist x z\nhyz : dist y z = (1 - r) * dist x z\nhne\u271d : x \u2260 z\nhne : dist x z \u2260 0\na b : \u211d\nw\u271d\u00b9 : 0 \u2264 a\nhb : 0 \u2264 b\nw\u271d : a + b = 1\nH : a \u2022 0 + b \u2022 (z -\u1d65 x) = y -\u1d65 x\n\u22a2 y = \u2191(AffineMap.lineMap x z) r\n[PROOFSTEP]\nrw [smul_zero, zero_add] at H \n[GOAL]\ncase inr.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z\u271d : E\na\u271d b\u271d r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nx y z : PE\nhxy : dist x y = r * dist x z\nhyz : dist y z = (1 - r) * dist x z\nhne\u271d : x \u2260 z\nhne : dist x z \u2260 0\na b : \u211d\nw\u271d\u00b9 : 0 \u2264 a\nhb : 0 \u2264 b\nw\u271d : a + b = 1\nH : b \u2022 (z -\u1d65 x) = y -\u1d65 x\n\u22a2 y = \u2191(AffineMap.lineMap x z) r\n[PROOFSTEP]\nhave H' := congr_arg norm H\n[GOAL]\ncase inr.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z\u271d : E\na\u271d b\u271d r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nx y z : PE\nhxy : dist x y = r * dist x z\nhyz : dist y z = (1 - r) * dist x z\nhne\u271d : x \u2260 z\nhne : dist x z \u2260 0\na b : \u211d\nw\u271d\u00b9 : 0 \u2264 a\nhb : 0 \u2264 b\nw\u271d : a + b = 1\nH : b \u2022 (z -\u1d65 x) = y -\u1d65 x\nH' : \u2016b \u2022 (z -\u1d65 x)\u2016 = \u2016y -\u1d65 x\u2016\n\u22a2 y = \u2191(AffineMap.lineMap x z) r\n[PROOFSTEP]\nrw [norm_smul, Real.norm_of_nonneg hb, \u2190 dist_eq_norm_vsub', \u2190 dist_eq_norm_vsub', hxy, mul_left_inj' hne] at H' \n[GOAL]\ncase inr.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z\u271d : E\na\u271d b\u271d r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nx y z : PE\nhxy : dist x y = r * dist x z\nhyz : dist y z = (1 - r) * dist x z\nhne\u271d : x \u2260 z\nhne : dist x z \u2260 0\na b : \u211d\nw\u271d\u00b9 : 0 \u2264 a\nhb : 0 \u2264 b\nw\u271d : a + b = 1\nH : b \u2022 (z -\u1d65 x) = y -\u1d65 x\nH' : b = r\n\u22a2 y = \u2191(AffineMap.lineMap x z) r\n[PROOFSTEP]\nrw [AffineMap.lineMap_apply, \u2190 H', H, vsub_vadd]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z\u271d : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nx y z : PE\nhx : dist x y = dist x z / 2\nhy : dist y z = dist x z / 2\n\u22a2 y = midpoint \u211d x z\n[PROOFSTEP]\napply eq_lineMap_of_dist_eq_mul_of_dist_eq_mul\n[GOAL]\ncase hxy\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z\u271d : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nx y z : PE\nhx : dist x y = dist x z / 2\nhy : dist y z = dist x z / 2\n\u22a2 dist x y = \u215f2 * dist x z\n[PROOFSTEP]\nrwa [invOf_eq_inv, \u2190 div_eq_inv_mul]\n[GOAL]\ncase hyz\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z\u271d : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nx y z : PE\nhx : dist x y = dist x z / 2\nhy : dist y z = dist x z / 2\n\u22a2 dist y z = (1 - \u215f2) * dist x z\n[PROOFSTEP]\nrwa [invOf_eq_inv, \u2190 one_div, sub_half, one_div, \u2190 div_eq_inv_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nf : PF \u2192 PE\nhi : Isometry f\nx y : PF\n\u22a2 f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)\n[PROOFSTEP]\napply eq_midpoint_of_dist_eq_half\n[GOAL]\ncase hx\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nf : PF \u2192 PE\nhi : Isometry f\nx y : PF\n\u22a2 dist (f x) (f (midpoint \u211d x y)) = dist (f x) (f y) / 2\n[PROOFSTEP]\nrw [hi.dist_eq, hi.dist_eq]\n[GOAL]\ncase hx\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nf : PF \u2192 PE\nhi : Isometry f\nx y : PF\n\u22a2 dist x (midpoint \u211d x y) = dist x y / 2\n[PROOFSTEP]\nsimp only [dist_left_midpoint, Real.norm_of_nonneg zero_le_two, div_eq_inv_mul]\n[GOAL]\ncase hy\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nf : PF \u2192 PE\nhi : Isometry f\nx y : PF\n\u22a2 dist (f (midpoint \u211d x y)) (f y) = dist (f x) (f y) / 2\n[PROOFSTEP]\nrw [hi.dist_eq, hi.dist_eq]\n[GOAL]\ncase hy\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y\u271d z : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nf : PF \u2192 PE\nhi : Isometry f\nx y : PF\n\u22a2 dist (midpoint \u211d x y) y = dist x y / 2\n[PROOFSTEP]\nsimp only [dist_midpoint_right, Real.norm_of_nonneg zero_le_two, div_eq_inv_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : StrictConvexSpace \u211d E\nx\u271d y z : E\na b r : \u211d\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\nPF : Type u\nPE : Type u_4\ninst\u271d\u00b3 : MetricSpace PF\ninst\u271d\u00b2 : MetricSpace PE\ninst\u271d\u00b9 : NormedAddTorsor F PF\ninst\u271d : NormedAddTorsor E PE\nf : PF \u2192 PE\nhi : Isometry f\nsrc\u271d : PF \u2192\u1d43[\u211d] PE :=\n AffineMap.ofMapMidpoint f (_ : \u2200 (x y : PF), f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)) (_ : Continuous f)\nx : F\n\u22a2 \u2016\u2191{ toFun := src\u271d.toFun, linear := src\u271d.linear,\n map_vadd' :=\n (_ :\n \u2200 (p : PF) (v : F), AffineMap.toFun src\u271d (v +\u1d65 p) = \u2191src\u271d.linear v +\u1d65 AffineMap.toFun src\u271d p) }.linear\n x\u2016 =\n \u2016x\u2016\n[PROOFSTEP]\nsimp [AffineMap.ofMapMidpoint, \u2190 dist_eq_norm_vsub E, hi.dist_eq]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.StrictConvexSpace", "llama_tokens": 15662, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390746, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.4009891543374834}} {"text": "[GOAL]\nf : \u2115 \u2192 \u211d\nhf : Tendsto f atTop (\ud835\udcdd 0)\nr\u271d : \u211d\nhr : 0 < r\u271d\nhg : Tendsto (fun x => -f x) atTop (\ud835\udcdd (-0))\n\u22a2 -ofSeq f < \u2191r\u271d\n[PROOFSTEP]\nrw [neg_zero] at hg \n[GOAL]\nf : \u2115 \u2192 \u211d\nhf : Tendsto f atTop (\ud835\udcdd 0)\nr\u271d : \u211d\nhr : 0 < r\u271d\nhg : Tendsto (fun x => -f x) atTop (\ud835\udcdd 0)\n\u22a2 -ofSeq f < \u2191r\u271d\n[PROOFSTEP]\nexact lt_of_tendsto_zero_of_pos hg hr\n[GOAL]\nf : \u2115 \u2192 \u211d\nhf : Tendsto f atTop (\ud835\udcdd 0)\nr : \u211d\nhr : r < 0\n\u22a2 \u2191r < ofSeq f\n[PROOFSTEP]\nrw [\u2190 neg_neg r, coe_neg]\n[GOAL]\nf : \u2115 \u2192 \u211d\nhf : Tendsto f atTop (\ud835\udcdd 0)\nr : \u211d\nhr : r < 0\n\u22a2 -\u2191(-r) < ofSeq f\n[PROOFSTEP]\nexact neg_lt_of_tendsto_zero_of_pos hf (neg_pos.mpr hr)\n[GOAL]\nx : \u211d*\nr : \u211d\n\u22a2 IsSt x r \u2194 Germ.Tendsto x (\ud835\udcdd r)\n[PROOFSTEP]\nrcases ofSeq_surjective x with \u27e8f, rfl\u27e9\n[GOAL]\ncase intro\nr : \u211d\nf : \u2115 \u2192 \u211d\n\u22a2 IsSt (ofSeq f) r \u2194 Germ.Tendsto (ofSeq f) (\ud835\udcdd r)\n[PROOFSTEP]\nexact isSt_ofSeq_iff_tendsto\n[GOAL]\nx y : \u211d*\nr s : \u211d\nhxr : IsSt x r\nhys : IsSt y s\nhrs : r < s\n\u22a2 x < y\n[PROOFSTEP]\nrcases ofSeq_surjective x with \u27e8f, rfl\u27e9\n[GOAL]\ncase intro\ny : \u211d*\nr s : \u211d\nhys : IsSt y s\nhrs : r < s\nf : \u2115 \u2192 \u211d\nhxr : IsSt (ofSeq f) r\n\u22a2 ofSeq f < y\n[PROOFSTEP]\nrcases ofSeq_surjective y with \u27e8g, rfl\u27e9\n[GOAL]\ncase intro.intro\nr s : \u211d\nhrs : r < s\nf : \u2115 \u2192 \u211d\nhxr : IsSt (ofSeq f) r\ng : \u2115 \u2192 \u211d\nhys : IsSt (ofSeq g) s\n\u22a2 ofSeq f < ofSeq g\n[PROOFSTEP]\nrw [isSt_ofSeq_iff_tendsto] at hxr hys \n[GOAL]\ncase intro.intro\nr s : \u211d\nhrs : r < s\nf : \u2115 \u2192 \u211d\nhxr : Tendsto f (\u2191(hyperfilter \u2115)) (\ud835\udcdd r)\ng : \u2115 \u2192 \u211d\nhys : Tendsto g (\u2191(hyperfilter \u2115)) (\ud835\udcdd s)\n\u22a2 ofSeq f < ofSeq g\n[PROOFSTEP]\nexact ofSeq_lt_ofSeq.2 <| hxr.eventually_lt hys hrs\n[GOAL]\nx : \u211d*\nr s : \u211d\nhr : IsSt x r\nhs : IsSt x s\n\u22a2 r = s\n[PROOFSTEP]\nrcases ofSeq_surjective x with \u27e8f, rfl\u27e9\n[GOAL]\ncase intro\nr s : \u211d\nf : \u2115 \u2192 \u211d\nhr : IsSt (ofSeq f) r\nhs : IsSt (ofSeq f) s\n\u22a2 r = s\n[PROOFSTEP]\nrw [isSt_ofSeq_iff_tendsto] at hr hs \n[GOAL]\ncase intro\nr s : \u211d\nf : \u2115 \u2192 \u211d\nhr : Tendsto f (\u2191(hyperfilter \u2115)) (\ud835\udcdd r)\nhs : Tendsto f (\u2191(hyperfilter \u2115)) (\ud835\udcdd s)\n\u22a2 r = s\n[PROOFSTEP]\nexact tendsto_nhds_unique hr hs\n[GOAL]\nx : \u211d*\nr : \u211d\nhxr : IsSt x r\n\u22a2 st x = r\n[PROOFSTEP]\nhave h : \u2203 r, IsSt x r := \u27e8r, hxr\u27e9\n[GOAL]\nx : \u211d*\nr : \u211d\nhxr : IsSt x r\nh : \u2203 r, IsSt x r\n\u22a2 st x = r\n[PROOFSTEP]\nrw [st, dif_pos h]\n[GOAL]\nx : \u211d*\nr : \u211d\nhxr : IsSt x r\nh : \u2203 r, IsSt x r\n\u22a2 choose h = r\n[PROOFSTEP]\nexact (Classical.choose_spec h).unique hxr\n[GOAL]\nx : \u211d*\n\u22a2 st x = sSup {y | \u2191y < x}\n[PROOFSTEP]\nrcases _root_.em (Infinite x) with (hx | hx)\n[GOAL]\ncase inl\nx : \u211d*\nhx : Infinite x\n\u22a2 st x = sSup {y | \u2191y < x}\n[PROOFSTEP]\nrw [hx.st_eq]\n[GOAL]\ncase inl\nx : \u211d*\nhx : Infinite x\n\u22a2 0 = sSup {y | \u2191y < x}\n[PROOFSTEP]\ncases hx with\n| inl hx =>\n convert Real.sSup_univ.symm\n exact Set.eq_univ_of_forall hx\n| inr hx =>\n convert Real.sSup_empty.symm\n exact Set.eq_empty_of_forall_not_mem fun y hy \u21a6 hy.out.not_lt (hx _)\n[GOAL]\ncase inl\nx : \u211d*\nhx : Infinite x\n\u22a2 0 = sSup {y | \u2191y < x}\n[PROOFSTEP]\ncases hx with\n| inl hx =>\n convert Real.sSup_univ.symm\n exact Set.eq_univ_of_forall hx\n| inr hx =>\n convert Real.sSup_empty.symm\n exact Set.eq_empty_of_forall_not_mem fun y hy \u21a6 hy.out.not_lt (hx _)\n[GOAL]\ncase inl.inl\nx : \u211d*\nhx : InfinitePos x\n\u22a2 0 = sSup {y | \u2191y < x}\n[PROOFSTEP]\n\n| inl hx =>\n convert Real.sSup_univ.symm\n exact Set.eq_univ_of_forall hx\n[GOAL]\ncase inl.inl\nx : \u211d*\nhx : InfinitePos x\n\u22a2 0 = sSup {y | \u2191y < x}\n[PROOFSTEP]\nconvert Real.sSup_univ.symm\n[GOAL]\ncase h.e'_3.h.e'_3\nx : \u211d*\nhx : InfinitePos x\n\u22a2 {y | \u2191y < x} = Set.univ\n[PROOFSTEP]\nexact Set.eq_univ_of_forall hx\n[GOAL]\ncase inl.inr\nx : \u211d*\nhx : InfiniteNeg x\n\u22a2 0 = sSup {y | \u2191y < x}\n[PROOFSTEP]\n\n| inr hx =>\n convert Real.sSup_empty.symm\n exact Set.eq_empty_of_forall_not_mem fun y hy \u21a6 hy.out.not_lt (hx _)\n[GOAL]\ncase inl.inr\nx : \u211d*\nhx : InfiniteNeg x\n\u22a2 0 = sSup {y | \u2191y < x}\n[PROOFSTEP]\nconvert Real.sSup_empty.symm\n[GOAL]\ncase h.e'_3.h.e'_3\nx : \u211d*\nhx : InfiniteNeg x\n\u22a2 {y | \u2191y < x} = \u2205\n[PROOFSTEP]\nexact Set.eq_empty_of_forall_not_mem fun y hy \u21a6 hy.out.not_lt (hx _)\n[GOAL]\ncase inr\nx : \u211d*\nhx : \u00acInfinite x\n\u22a2 st x = sSup {y | \u2191y < x}\n[PROOFSTEP]\nexact (isSt_sSup hx).st_eq\n[GOAL]\nx : \u211d*\nr : \u211d\nhxr : IsSt x r\n\u22a2 IsSt x (st x)\n[PROOFSTEP]\nrwa [hxr.st_eq]\n[GOAL]\nr s : \u211d\n\u22a2 IsSt (\u2191r) s \u2194 IsSt (\u2191s) r\n[PROOFSTEP]\nrw [isSt_real_iff_eq, isSt_real_iff_eq, eq_comm]\n[GOAL]\nr s t : \u211d\n\u22a2 IsSt (\u2191r) s \u2192 IsSt (\u2191s) t \u2192 IsSt (\u2191r) t\n[PROOFSTEP]\nrw [isSt_real_iff_eq, isSt_real_iff_eq, isSt_real_iff_eq]\n[GOAL]\nr s t : \u211d\n\u22a2 r = s \u2192 s = t \u2192 r = t\n[PROOFSTEP]\nexact Eq.trans\n[GOAL]\nx : \u211d*\nr : \u211d\n\u22a2 IsSt x r \u2194 \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 |x - \u2191r| < \u2191\u03b4\n[PROOFSTEP]\nsimp only [abs_sub_lt_iff, sub_lt_iff_lt_add, IsSt, and_comm, add_comm]\n[GOAL]\nx : \u211d*\nr : \u211d\nhxr : IsSt x r\nf : \u211d \u2192 \u211d\nhf : ContinuousAt f r\n\u22a2 IsSt (Germ.map f x) (f r)\n[PROOFSTEP]\nrcases ofSeq_surjective x with \u27e8g, rfl\u27e9\n[GOAL]\ncase intro\nr : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousAt f r\ng : \u2115 \u2192 \u211d\nhxr : IsSt (ofSeq g) r\n\u22a2 IsSt (Germ.map f (ofSeq g)) (f r)\n[PROOFSTEP]\nexact isSt_ofSeq_iff_tendsto.2 <| hf.tendsto.comp (isSt_ofSeq_iff_tendsto.1 hxr)\n[GOAL]\nx y : \u211d*\nr s : \u211d\nhxr : IsSt x r\nhys : IsSt y s\nf : \u211d \u2192 \u211d \u2192 \u211d\nhf : ContinuousAt (Function.uncurry f) (r, s)\n\u22a2 IsSt (Germ.map\u2082 f x y) (f r s)\n[PROOFSTEP]\nrcases ofSeq_surjective x with \u27e8x, rfl\u27e9\n[GOAL]\ncase intro\ny : \u211d*\nr s : \u211d\nhys : IsSt y s\nf : \u211d \u2192 \u211d \u2192 \u211d\nhf : ContinuousAt (Function.uncurry f) (r, s)\nx : \u2115 \u2192 \u211d\nhxr : IsSt (ofSeq x) r\n\u22a2 IsSt (Germ.map\u2082 f (ofSeq x) y) (f r s)\n[PROOFSTEP]\nrcases ofSeq_surjective y with \u27e8y, rfl\u27e9\n[GOAL]\ncase intro.intro\nr s : \u211d\nf : \u211d \u2192 \u211d \u2192 \u211d\nhf : ContinuousAt (Function.uncurry f) (r, s)\nx : \u2115 \u2192 \u211d\nhxr : IsSt (ofSeq x) r\ny : \u2115 \u2192 \u211d\nhys : IsSt (ofSeq y) s\n\u22a2 IsSt (Germ.map\u2082 f (ofSeq x) (ofSeq y)) (f r s)\n[PROOFSTEP]\nrw [isSt_ofSeq_iff_tendsto] at hxr hys \n[GOAL]\ncase intro.intro\nr s : \u211d\nf : \u211d \u2192 \u211d \u2192 \u211d\nhf : ContinuousAt (Function.uncurry f) (r, s)\nx : \u2115 \u2192 \u211d\nhxr : Tendsto x (\u2191(hyperfilter \u2115)) (\ud835\udcdd r)\ny : \u2115 \u2192 \u211d\nhys : Tendsto y (\u2191(hyperfilter \u2115)) (\ud835\udcdd s)\n\u22a2 IsSt (Germ.map\u2082 f (ofSeq x) (ofSeq y)) (f r s)\n[PROOFSTEP]\nexact isSt_ofSeq_iff_tendsto.2 <| hf.tendsto.comp (hxr.prod_mk_nhds hys)\n[GOAL]\nx : \u211d*\n\u22a2 Infinite |x| \u2194 Infinite x\n[PROOFSTEP]\ncases le_total 0 x\n[GOAL]\ncase inl\nx : \u211d*\nh\u271d : 0 \u2264 x\n\u22a2 Infinite |x| \u2194 Infinite x\n[PROOFSTEP]\nsimp [*, abs_of_nonneg, abs_of_nonpos, infinite_neg]\n[GOAL]\ncase inr\nx : \u211d*\nh\u271d : x \u2264 0\n\u22a2 Infinite |x| \u2194 Infinite x\n[PROOFSTEP]\nsimp [*, abs_of_nonneg, abs_of_nonpos, infinite_neg]\n[GOAL]\nx y : \u211d*\n\u22a2 InfinitePos x \u2192 \u00acInfiniteNeg y \u2192 InfinitePos (x + y)\n[PROOFSTEP]\nintro hip hnin r\n[GOAL]\nx y : \u211d*\nhip : InfinitePos x\nhnin : \u00acInfiniteNeg y\nr : \u211d\n\u22a2 \u2191r < x + y\n[PROOFSTEP]\ncases' not_forall.mp hnin with r\u2082 hr\u2082\n[GOAL]\ncase intro\nx y : \u211d*\nhip : InfinitePos x\nhnin : \u00acInfiniteNeg y\nr r\u2082 : \u211d\nhr\u2082 : \u00acy < \u2191r\u2082\n\u22a2 \u2191r < x + y\n[PROOFSTEP]\nconvert add_lt_add_of_lt_of_le (hip (r + -r\u2082)) (not_lt.mp hr\u2082) using 1\n[GOAL]\ncase h.e'_3\nx y : \u211d*\nhip : InfinitePos x\nhnin : \u00acInfiniteNeg y\nr r\u2082 : \u211d\nhr\u2082 : \u00acy < \u2191r\u2082\n\u22a2 \u2191r = \u2191(r + -r\u2082) + \u2191r\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nx y : \u211d*\n\u22a2 InfiniteNeg x \u2192 \u00acInfinitePos y \u2192 InfiniteNeg (x + y)\n[PROOFSTEP]\nrw [\u2190 infinitePos_neg, \u2190 infinitePos_neg, \u2190 @infiniteNeg_neg y, neg_add]\n[GOAL]\nx y : \u211d*\n\u22a2 InfinitePos (-x) \u2192 \u00acInfiniteNeg (-y) \u2192 InfinitePos (-x + -y)\n[PROOFSTEP]\nexact infinitePos_add_not_infiniteNeg\n[GOAL]\nf : \u2115 \u2192 \u211d\nhf : Tendsto f atTop atTop\nr : \u211d\nhf' : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 f a\ni : \u2115\nhi : \u2200 (a : \u2115), i \u2264 a \u2192 r + 1 \u2264 f a\nhi' : \u2200 (a : \u2115), f a < r + 1 \u2192 a < i\n\u22a2 {a | r < f a}\u1d9c \u2286 {a | a \u2264 i}\n[PROOFSTEP]\nsimp only [Set.compl_setOf, not_lt]\n[GOAL]\nf : \u2115 \u2192 \u211d\nhf : Tendsto f atTop atTop\nr : \u211d\nhf' : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 b \u2264 f a\ni : \u2115\nhi : \u2200 (a : \u2115), i \u2264 a \u2192 r + 1 \u2264 f a\nhi' : \u2200 (a : \u2115), f a < r + 1 \u2192 a < i\n\u22a2 {a | f a \u2264 r} \u2286 {a | a \u2264 i}\n[PROOFSTEP]\nexact fun a har => le_of_lt (hi' a (lt_of_le_of_lt har (lt_add_one _)))\n[GOAL]\nf : \u2115 \u2192 \u211d\nhf : Tendsto f atTop atBot\nr : \u211d\nhf' : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 f a \u2264 b\ni : \u2115\nhi : \u2200 (a : \u2115), i \u2264 a \u2192 f a \u2264 r - 1\nhi' : \u2200 (a : \u2115), r - 1 < f a \u2192 a < i\n\u22a2 {a | f a < r}\u1d9c \u2286 {a | a \u2264 i}\n[PROOFSTEP]\nsimp only [Set.compl_setOf, not_lt]\n[GOAL]\nf : \u2115 \u2192 \u211d\nhf : Tendsto f atTop atBot\nr : \u211d\nhf' : \u2200 (b : \u211d), \u2203 i, \u2200 (a : \u2115), i \u2264 a \u2192 f a \u2264 b\ni : \u2115\nhi : \u2200 (a : \u2115), i \u2264 a \u2192 f a \u2264 r - 1\nhi' : \u2200 (a : \u2115), r - 1 < f a \u2192 a < i\n\u22a2 {a | r \u2264 f a} \u2286 {a | a \u2264 i}\n[PROOFSTEP]\nexact fun a har => le_of_lt (hi' a (lt_of_lt_of_le (sub_one_lt _) har))\n[GOAL]\nr : \u211d\n\u22a2 \u00acInfinite \u2191r\n[PROOFSTEP]\nrw [not_infinite_iff_exist_lt_gt]\n[GOAL]\nr : \u211d\n\u22a2 \u2203 r_1 s, \u2191r_1 < \u2191r \u2227 \u2191r < \u2191s\n[PROOFSTEP]\nexact \u27e8r - 1, r + 1, coe_lt_coe.2 <| sub_one_lt r, coe_lt_coe.2 <| lt_add_one r\u27e9\n[GOAL]\nx : \u211d*\nh : Infinite x\n\u22a2 st (-x) = -st x\n[PROOFSTEP]\nrw [h.st_eq, (infinite_neg.2 h).st_eq, neg_zero]\n[GOAL]\nx : \u211d*\n\u22a2 Infinitesimal x \u2194 \u2200 (r : \u211d), 0 < r \u2192 -\u2191r < x \u2227 x < \u2191r\n[PROOFSTEP]\nsimp [Infinitesimal, IsSt]\n[GOAL]\nx y : \u211d*\nhx : Infinitesimal x\nhy : Infinitesimal y\n\u22a2 Infinitesimal (x + y)\n[PROOFSTEP]\nsimpa only [add_zero] using hx.add hy\n[GOAL]\nx : \u211d*\nhx : Infinitesimal x\n\u22a2 Infinitesimal (-x)\n[PROOFSTEP]\nsimpa only [neg_zero] using hx.neg\n[GOAL]\nx y : \u211d*\nhx : Infinitesimal x\nhy : Infinitesimal y\n\u22a2 Infinitesimal (x * y)\n[PROOFSTEP]\nsimpa only [mul_zero] using hx.mul hy\n[GOAL]\nx : \u211d*\nr : \u211d\nhxr : IsSt x r\n\u22a2 Infinitesimal (x - \u2191r)\n[PROOFSTEP]\nsimpa only [sub_self] using hxr.sub (isSt_refl_real r)\n[GOAL]\nx : \u211d*\nhip : InfinitePos x\nr : \u211d\nhr : 0 < r\n\u22a2 (\u2191r)\u207b\u00b9 < x\n[PROOFSTEP]\nconvert hip r\u207b\u00b9\n[GOAL]\nx : \u211d*\n\u22a2 InfiniteNeg x \u2194 Infinitesimal x\u207b\u00b9 \u2227 x\u207b\u00b9 < 0\n[PROOFSTEP]\nrw [\u2190 infinitePos_neg, infinitePos_iff_infinitesimal_inv_pos, inv_neg, neg_pos, infinitesimal_neg]\n[GOAL]\nx : \u211d*\nh0 : x \u2260 0\nhi : Infinitesimal x\u207b\u00b9\n\u22a2 Infinite x\n[PROOFSTEP]\ncases' lt_or_gt_of_ne h0 with hn hp\n[GOAL]\ncase inl\nx : \u211d*\nh0 : x \u2260 0\nhi : Infinitesimal x\u207b\u00b9\nhn : x < 0\n\u22a2 Infinite x\n[PROOFSTEP]\nexact Or.inr (infiniteNeg_iff_infinitesimal_inv_neg.mpr \u27e8hi, inv_lt_zero.mpr hn\u27e9)\n[GOAL]\ncase inr\nx : \u211d*\nh0 : x \u2260 0\nhi : Infinitesimal x\u207b\u00b9\nhp : x > 0\n\u22a2 Infinite x\n[PROOFSTEP]\nexact Or.inl (infinitePos_iff_infinitesimal_inv_pos.mpr \u27e8hi, inv_pos.mpr hp\u27e9)\n[GOAL]\nx : \u211d*\n\u22a2 Infinitesimal x\u207b\u00b9\u207b\u00b9 \u2227 0 < x\u207b\u00b9\u207b\u00b9 \u2194 Infinitesimal x \u2227 0 < x\n[PROOFSTEP]\nrw [inv_inv]\n[GOAL]\nx : \u211d*\n\u22a2 Infinitesimal x\u207b\u00b9\u207b\u00b9 \u2227 x\u207b\u00b9\u207b\u00b9 < 0 \u2194 Infinitesimal x \u2227 x < 0\n[PROOFSTEP]\nrw [inv_inv]\n[GOAL]\nx : \u211d*\nh : x \u2260 0\n\u22a2 Infinitesimal x \u2194 Infinitesimal x\u207b\u00b9\u207b\u00b9\n[PROOFSTEP]\nrw [inv_inv]\n[GOAL]\nx : \u211d*\nr : \u211d\nhi : \u00acInfinitesimal x\nhr : IsSt x r\n\u22a2 r \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nx : \u211d*\nhi : \u00acInfinitesimal x\nhr : IsSt x 0\n\u22a2 False\n[PROOFSTEP]\nexact hi hr\n[GOAL]\nx : \u211d*\n\u22a2 st x\u207b\u00b9 = (st x)\u207b\u00b9\n[PROOFSTEP]\nby_cases h0 : x = 0\n[GOAL]\ncase pos\nx : \u211d*\nh0 : x = 0\n\u22a2 st x\u207b\u00b9 = (st x)\u207b\u00b9\ncase neg x : \u211d* h0 : \u00acx = 0 \u22a2 st x\u207b\u00b9 = (st x)\u207b\u00b9\n[PROOFSTEP]\nrw [h0, inv_zero, \u2190 coe_zero, st_id_real, inv_zero]\n[GOAL]\ncase neg\nx : \u211d*\nh0 : \u00acx = 0\n\u22a2 st x\u207b\u00b9 = (st x)\u207b\u00b9\n[PROOFSTEP]\nby_cases h1 : Infinitesimal x\n[GOAL]\ncase pos\nx : \u211d*\nh0 : \u00acx = 0\nh1 : Infinitesimal x\n\u22a2 st x\u207b\u00b9 = (st x)\u207b\u00b9\ncase neg x : \u211d* h0 : \u00acx = 0 h1 : \u00acInfinitesimal x \u22a2 st x\u207b\u00b9 = (st x)\u207b\u00b9\n[PROOFSTEP]\nrw [((infinitesimal_iff_infinite_inv h0).mp h1).st_eq, h1.st_eq, inv_zero]\n[GOAL]\ncase neg\nx : \u211d*\nh0 : \u00acx = 0\nh1 : \u00acInfinitesimal x\n\u22a2 st x\u207b\u00b9 = (st x)\u207b\u00b9\n[PROOFSTEP]\nby_cases h2 : Infinite x\n[GOAL]\ncase pos\nx : \u211d*\nh0 : \u00acx = 0\nh1 : \u00acInfinitesimal x\nh2 : Infinite x\n\u22a2 st x\u207b\u00b9 = (st x)\u207b\u00b9\ncase neg x : \u211d* h0 : \u00acx = 0 h1 : \u00acInfinitesimal x h2 : \u00acInfinite x \u22a2 st x\u207b\u00b9 = (st x)\u207b\u00b9\n[PROOFSTEP]\nrw [(infinitesimal_inv_of_infinite h2).st_eq, h2.st_eq, inv_zero]\n[GOAL]\ncase neg\nx : \u211d*\nh0 : \u00acx = 0\nh1 : \u00acInfinitesimal x\nh2 : \u00acInfinite x\n\u22a2 st x\u207b\u00b9 = (st x)\u207b\u00b9\n[PROOFSTEP]\nexact ((isSt_st' h2).inv h1).st_eq\n[GOAL]\nx y : \u211d*\nhx : InfinitePos x\nhy\u2081 : \u00acInfinitesimal y\nhy\u2082 : 0 < y\nr : \u211d\n\u22a2 \u2191r < x * y\n[PROOFSTEP]\nhave hy\u2081' := not_forall.mp (mt infinitesimal_def.2 hy\u2081)\n[GOAL]\nx y : \u211d*\nhx : InfinitePos x\nhy\u2081 : \u00acInfinitesimal y\nhy\u2082 : 0 < y\nr : \u211d\nhy\u2081' : \u2203 x, \u00ac(0 < x \u2192 -\u2191x < y \u2227 y < \u2191x)\n\u22a2 \u2191r < x * y\n[PROOFSTEP]\nlet \u27e8r\u2081, hy\u2081''\u27e9 := hy\u2081'\n[GOAL]\nx y : \u211d*\nhx : InfinitePos x\nhy\u2081 : \u00acInfinitesimal y\nhy\u2082 : 0 < y\nr : \u211d\nhy\u2081' : \u2203 x, \u00ac(0 < x \u2192 -\u2191x < y \u2227 y < \u2191x)\nr\u2081 : \u211d\nhy\u2081'' : \u00ac(0 < r\u2081 \u2192 -\u2191r\u2081 < y \u2227 y < \u2191r\u2081)\n\u22a2 \u2191r < x * y\n[PROOFSTEP]\nhave hyr : 0 < r\u2081 \u2227 \u2191r\u2081 \u2264 y := by rwa [not_imp, \u2190 abs_lt, not_lt, abs_of_pos hy\u2082] at hy\u2081'' \n[GOAL]\nx y : \u211d*\nhx : InfinitePos x\nhy\u2081 : \u00acInfinitesimal y\nhy\u2082 : 0 < y\nr : \u211d\nhy\u2081' : \u2203 x, \u00ac(0 < x \u2192 -\u2191x < y \u2227 y < \u2191x)\nr\u2081 : \u211d\nhy\u2081'' : \u00ac(0 < r\u2081 \u2192 -\u2191r\u2081 < y \u2227 y < \u2191r\u2081)\n\u22a2 0 < r\u2081 \u2227 \u2191r\u2081 \u2264 y\n[PROOFSTEP]\nrwa [not_imp, \u2190 abs_lt, not_lt, abs_of_pos hy\u2082] at hy\u2081'' \n[GOAL]\nx y : \u211d*\nhx : InfinitePos x\nhy\u2081 : \u00acInfinitesimal y\nhy\u2082 : 0 < y\nr : \u211d\nhy\u2081' : \u2203 x, \u00ac(0 < x \u2192 -\u2191x < y \u2227 y < \u2191x)\nr\u2081 : \u211d\nhy\u2081'' : \u00ac(0 < r\u2081 \u2192 -\u2191r\u2081 < y \u2227 y < \u2191r\u2081)\nhyr : 0 < r\u2081 \u2227 \u2191r\u2081 \u2264 y\n\u22a2 \u2191r < x * y\n[PROOFSTEP]\nrw [\u2190 div_mul_cancel r (ne_of_gt hyr.1), coe_mul]\n[GOAL]\nx y : \u211d*\nhx : InfinitePos x\nhy\u2081 : \u00acInfinitesimal y\nhy\u2082 : 0 < y\nr : \u211d\nhy\u2081' : \u2203 x, \u00ac(0 < x \u2192 -\u2191x < y \u2227 y < \u2191x)\nr\u2081 : \u211d\nhy\u2081'' : \u00ac(0 < r\u2081 \u2192 -\u2191r\u2081 < y \u2227 y < \u2191r\u2081)\nhyr : 0 < r\u2081 \u2227 \u2191r\u2081 \u2264 y\n\u22a2 \u2191(r / r\u2081) * \u2191r\u2081 < x * y\n[PROOFSTEP]\nexact mul_lt_mul (hx (r / r\u2081)) hyr.2 (coe_lt_coe.2 hyr.1) (le_of_lt (hx 0))\n[GOAL]\nx y : \u211d*\n\u22a2 InfiniteNeg x \u2192 \u00acInfinitesimal y \u2192 y < 0 \u2192 InfinitePos (x * y)\n[PROOFSTEP]\nrw [\u2190 infinitePos_neg, \u2190 neg_pos, \u2190 neg_mul_neg, \u2190 infinitesimal_neg]\n[GOAL]\nx y : \u211d*\n\u22a2 InfinitePos (-x) \u2192 \u00acInfinitesimal (-y) \u2192 0 < -y \u2192 InfinitePos (-x * -y)\n[PROOFSTEP]\nexact infinitePos_mul_of_infinitePos_not_infinitesimal_pos\n[GOAL]\nx y : \u211d*\n\u22a2 InfinitePos x \u2192 \u00acInfinitesimal y \u2192 y < 0 \u2192 InfiniteNeg (x * y)\n[PROOFSTEP]\nrw [\u2190 infinitePos_neg, \u2190 neg_pos, neg_mul_eq_mul_neg, \u2190 infinitesimal_neg]\n[GOAL]\nx y : \u211d*\n\u22a2 InfinitePos x \u2192 \u00acInfinitesimal (-y) \u2192 0 < -y \u2192 InfinitePos (x * -y)\n[PROOFSTEP]\nexact infinitePos_mul_of_infinitePos_not_infinitesimal_pos\n[GOAL]\nx y : \u211d*\n\u22a2 InfiniteNeg x \u2192 \u00acInfinitesimal y \u2192 0 < y \u2192 InfiniteNeg (x * y)\n[PROOFSTEP]\nrw [\u2190 infinitePos_neg, \u2190 infinitePos_neg, neg_mul_eq_neg_mul]\n[GOAL]\nx y : \u211d*\n\u22a2 InfinitePos (-x) \u2192 \u00acInfinitesimal y \u2192 0 < y \u2192 InfinitePos (-x * y)\n[PROOFSTEP]\nexact infinitePos_mul_of_infinitePos_not_infinitesimal_pos\n[GOAL]\nx y : \u211d*\nhx : \u00acInfinitesimal x\nhp : 0 < x\nhy : InfiniteNeg y\n\u22a2 InfiniteNeg (x * y)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nx y : \u211d*\nhx : \u00acInfinitesimal x\nhp : 0 < x\nhy : InfiniteNeg y\n\u22a2 InfiniteNeg (y * x)\n[PROOFSTEP]\nexact infiniteNeg_mul_of_infiniteNeg_not_infinitesimal_pos hy hx hp\n[GOAL]\nx y : \u211d*\nhx : \u00acInfinitesimal x\nhy : Infinite y\n\u22a2 Infinite (x * y)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nx y : \u211d*\nhx : \u00acInfinitesimal x\nhy : Infinite y\n\u22a2 Infinite (y * x)\n[PROOFSTEP]\nexact infinite_mul_of_infinite_not_infinitesimal hy hx\n", "meta": {"mathlib_filename": "Mathlib.Data.Real.Hyperreal", "llama_tokens": 7974, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.6477982315512488, "lm_q1q2_score": 0.40084487069778074}} {"text": "[GOAL]\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y\u2081 Y\u2082 : C\nf : Y\u2081 \u27f6 Y\u2082\n\u22a2 \ud835\udfd9 X \u2297 f = X \u25c1 f\n[PROOFSTEP]\nsimp [tensorHom_def]\n[GOAL]\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX\u2081 X\u2082 : C\nf : X\u2081 \u27f6 X\u2082\nY : C\n\u22a2 f \u2297 \ud835\udfd9 Y = f \u25b7 Y\n[PROOFSTEP]\nsimp [tensorHom_def]\n[GOAL]\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nW X Y Z : C\nf : W \u27f6 X\ng : Y \u27f6 Z\n\u22a2 W \u25c1 g \u226b f \u25b7 Z = f \u25b7 Y \u226b X \u25c1 g\n[PROOFSTEP]\nsimp [\u2190 id_tensorHom, \u2190 tensorHom_id, \u2190 tensor_comp]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\nX Y X' Y' : C\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nf : X \u2245 Y\ng : X' \u2245 Y'\n\u22a2 (f.hom \u2297 g.hom) \u226b (f.inv \u2297 g.inv) = \ud835\udfd9 (X \u2297 X')\n[PROOFSTEP]\nrw [\u2190 tensor_comp, Iso.hom_inv_id, Iso.hom_inv_id, \u2190 tensor_id]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\nX Y X' Y' : C\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nf : X \u2245 Y\ng : X' \u2245 Y'\n\u22a2 (f.inv \u2297 g.inv) \u226b (f.hom \u2297 g.hom) = \ud835\udfd9 (Y \u2297 Y')\n[PROOFSTEP]\nrw [\u2190 tensor_comp, Iso.inv_hom_id, Iso.inv_hom_id, \u2190 tensor_id]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u2074 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : MonoidalCategory C\nW X Y Z : C\nf : W \u27f6 X\ninst\u271d\u00b9 : IsIso f\ng : Y \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 inv (f \u2297 g) = inv f \u2297 inv g\n[PROOFSTEP]\naesop_cat_nonterminal\n[GOAL]\ncase hom_inv_id\nC_1 : Type u\n\ud835\udc9e : Category.{v, u} C_1\ninst : MonoidalCategory C_1\nC : Type u\ninst_1 : Category.{v, u} C\ninst_2 : MonoidalCategory C\nW X Y Z : C\nf : W \u27f6 X\ninst_3 : IsIso f\ng : Y \u27f6 Z\ninst_4 : IsIso g\n\u22a2 (f \u2297 g) \u226b (inv f \u2297 inv g) = \ud835\udfd9 (W \u2297 Y)\n[PROOFSTEP]\nsimp [\u2190 tensor_comp]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b3 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : MonoidalCategory C\nU V W\u271d X\u271d Y\u271d Z\u271d : C\nP : Prop\ninst\u271d : Decidable P\nW X Y Z : C\nf : W \u27f6 X\ng : P \u2192 (Y \u27f6 Z)\ng' : \u00acP \u2192 (Y \u27f6 Z)\n\u22a2 (f \u2297 if h : P then g h else g' h) = if h : P then f \u2297 g h else f \u2297 g' h\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b3 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : MonoidalCategory C\nU V W\u271d X\u271d Y\u271d Z\u271d : C\nP : Prop\ninst\u271d : Decidable P\nW X Y Z : C\nf : W \u27f6 X\ng : P \u2192 (Y \u27f6 Z)\ng' : \u00acP \u2192 (Y \u27f6 Z)\nh\u271d : P\n\u22a2 f \u2297 g h\u271d = f \u2297 g h\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b3 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : MonoidalCategory C\nU V W\u271d X\u271d Y\u271d Z\u271d : C\nP : Prop\ninst\u271d : Decidable P\nW X Y Z : C\nf : W \u27f6 X\ng : P \u2192 (Y \u27f6 Z)\ng' : \u00acP \u2192 (Y \u27f6 Z)\nh\u271d : \u00acP\n\u22a2 f \u2297 g' h\u271d = f \u2297 g' h\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b3 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : MonoidalCategory C\nU V W\u271d X\u271d Y\u271d Z\u271d : C\nP : Prop\ninst\u271d : Decidable P\nW X Y Z : C\nf : W \u27f6 X\ng : P \u2192 (Y \u27f6 Z)\ng' : \u00acP \u2192 (Y \u27f6 Z)\n\u22a2 (if h : P then g h else g' h) \u2297 f = if h : P then g h \u2297 f else g' h \u2297 f\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b3 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : MonoidalCategory C\nU V W\u271d X\u271d Y\u271d Z\u271d : C\nP : Prop\ninst\u271d : Decidable P\nW X Y Z : C\nf : W \u27f6 X\ng : P \u2192 (Y \u27f6 Z)\ng' : \u00acP \u2192 (Y \u27f6 Z)\nh\u271d : P\n\u22a2 g h\u271d \u2297 f = g h\u271d \u2297 f\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b3 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : MonoidalCategory C\nU V W\u271d X\u271d Y\u271d Z\u271d : C\nP : Prop\ninst\u271d : Decidable P\nW X Y Z : C\nf : W \u27f6 X\ng : P \u2192 (Y \u27f6 Z)\ng' : \u00acP \u2192 (Y \u27f6 Z)\nh\u271d : \u00acP\n\u22a2 g' h\u271d \u2297 f = g' h\u271d \u2297 f\n[PROOFSTEP]\nrfl\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : X \u27f6 Y\n\u22a2 f \u226b g \u2297 \ud835\udfd9 Z = (f \u2297 \ud835\udfd9 Z) \u226b (g \u2297 \ud835\udfd9 Z)\n[PROOFSTEP]\nrw [\u2190 tensor_comp]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : X \u27f6 Y\n\u22a2 f \u226b g \u2297 \ud835\udfd9 Z = f \u226b g \u2297 \ud835\udfd9 Z \u226b \ud835\udfd9 Z\n[PROOFSTEP]\nsimp\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : X \u27f6 Y\n\u22a2 \ud835\udfd9 Z \u2297 f \u226b g = (\ud835\udfd9 Z \u2297 f) \u226b (\ud835\udfd9 Z \u2297 g)\n[PROOFSTEP]\nrw [\u2190 tensor_comp]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : X \u27f6 Y\n\u22a2 \ud835\udfd9 Z \u2297 f \u226b g = \ud835\udfd9 Z \u226b \ud835\udfd9 Z \u2297 f \u226b g\n[PROOFSTEP]\nsimp\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : Y \u27f6 Z\n\u22a2 (\ud835\udfd9 Y \u2297 f) \u226b (g \u2297 \ud835\udfd9 X) = g \u2297 f\n[PROOFSTEP]\nrw [\u2190 tensor_comp]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : Y \u27f6 Z\n\u22a2 \ud835\udfd9 Y \u226b g \u2297 f \u226b \ud835\udfd9 X = g \u2297 f\n[PROOFSTEP]\nsimp\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : Y \u27f6 Z\n\u22a2 (g \u2297 \ud835\udfd9 W) \u226b (\ud835\udfd9 Z \u2297 f) = g \u2297 f\n[PROOFSTEP]\nrw [\u2190 tensor_comp]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : Y \u27f6 Z\n\u22a2 g \u226b \ud835\udfd9 Z \u2297 \ud835\udfd9 W \u226b f = g \u2297 f\n[PROOFSTEP]\nsimp\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z X Y : C\nf : X \u27f6 Y\n\u22a2 f \u2297 \ud835\udfd9 (\ud835\udfd9_ C) = (\u03c1_ X).hom \u226b f \u226b (\u03c1_ Y).inv\n[PROOFSTEP]\nrw [\u2190 rightUnitor_naturality_assoc, Iso.hom_inv_id, Category.comp_id]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z X Y : C\nf : X \u27f6 Y\n\u22a2 \ud835\udfd9 (\ud835\udfd9_ C) \u2297 f = (\u03bb_ X).hom \u226b f \u226b (\u03bb_ Y).inv\n[PROOFSTEP]\nrw [\u2190 leftUnitor_naturality_assoc, Iso.hom_inv_id, Category.comp_id]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y Z X X' : C\nf : X \u27f6 X'\n\u22a2 f \u226b (\u03bb_ X').inv = (\u03bb_ X).inv \u226b (\ud835\udfd9 tensorUnit' \u2297 f)\n[PROOFSTEP]\nsimp\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y Z X X' : C\nf : X \u27f6 X'\n\u22a2 f \u226b (\u03c1_ X').inv = (\u03c1_ X).inv \u226b (f \u2297 \ud835\udfd9 tensorUnit')\n[PROOFSTEP]\nsimp\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z X Y : C\nf g : X \u27f6 Y\n\u22a2 \ud835\udfd9 (\ud835\udfd9_ C) \u2297 f = \ud835\udfd9 (\ud835\udfd9_ C) \u2297 g \u2194 f = g\n[PROOFSTEP]\nsimp\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z X Y : C\nf g : X \u27f6 Y\n\u22a2 f \u2297 \ud835\udfd9 (\ud835\udfd9_ C) = g \u2297 \ud835\udfd9 (\ud835\udfd9_ C) \u2194 f = g\n[PROOFSTEP]\nsimp\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W\u271d X\u271d Y\u271d Z\u271d W X Y Z : C\n\u22a2 inv ((\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv) \u226b (\u03b1_ W (X \u2297 Y) Z).inv \u226b ((\u03b1_ W X Y).inv \u2297 \ud835\udfd9 Z)) =\n inv ((\u03b1_ W X (Y \u2297 Z)).inv \u226b (\u03b1_ (W \u2297 X) Y Z).inv)\n[PROOFSTEP]\nsimp [pentagon]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z X Y : C\n\u22a2 (\u03c1_ (X \u2297 Y)).hom = (\u03b1_ X Y (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X \u2297 (\u03c1_ Y).hom)\n[PROOFSTEP]\nrw [\u2190 tensor_right_iff, comp_tensor_id, \u2190 cancel_mono (\u03b1_ X Y (\ud835\udfd9_ C)).hom, assoc, associator_naturality, \u2190\n triangle_assoc, \u2190 triangle, id_tensor_comp, pentagon_assoc, \u2190 associator_naturality, tensor_id]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z X Y : C\n\u22a2 inv (\u03c1_ (X \u2297 Y)).inv = inv ((\ud835\udfd9 X \u2297 (\u03c1_ Y).inv) \u226b (\u03b1_ X Y (\ud835\udfd9_ C)).inv)\n[PROOFSTEP]\nsimp\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z X Y : C\n\u22a2 (\u03b1_ X (\ud835\udfd9_ C) Y).inv \u226b ((\u03c1_ X).hom \u2297 \ud835\udfd9 Y) = \ud835\udfd9 X \u2297 (\u03bb_ Y).hom\n[PROOFSTEP]\nrw [\u2190 triangle, Iso.inv_hom_id_assoc]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z X Y : C\n\u22a2 (\ud835\udfd9 X \u2297 (\u03bb_ Y).inv) \u226b (\u03b1_ X (\ud835\udfd9_ C) Y).inv = (\u03c1_ X).inv \u2297 \ud835\udfd9 Y\n[PROOFSTEP]\napply (cancel_mono ((\u03c1_ X).hom \u2297 \ud835\udfd9 Y)).1\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z X Y : C\n\u22a2 ((\ud835\udfd9 X \u2297 (\u03bb_ Y).inv) \u226b (\u03b1_ X (\ud835\udfd9_ C) Y).inv) \u226b ((\u03c1_ X).hom \u2297 \ud835\udfd9 Y) = ((\u03c1_ X).inv \u2297 \ud835\udfd9 Y) \u226b ((\u03c1_ X).hom \u2297 \ud835\udfd9 Y)\n[PROOFSTEP]\nsimp only [triangle_assoc_comp_right, assoc]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z X Y : C\n\u22a2 (\ud835\udfd9 X \u2297 (\u03bb_ Y).inv) \u226b (\ud835\udfd9 X \u2297 (\u03bb_ Y).hom) = ((\u03c1_ X).inv \u2297 \ud835\udfd9 Y) \u226b ((\u03c1_ X).hom \u2297 \ud835\udfd9 Y)\n[PROOFSTEP]\nrw [\u2190 id_tensor_comp, Iso.inv_hom_id, \u2190 comp_tensor_id, Iso.inv_hom_id]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z\u271d X Y Z X' Y' Z' : C\nf : X \u27f6 X'\ng : Y \u27f6 Y'\nh : Z \u27f6 Z'\n\u22a2 (f \u2297 g \u2297 h) \u226b (\u03b1_ X' Y' Z').inv = (\u03b1_ X Y Z).inv \u226b ((f \u2297 g) \u2297 h)\n[PROOFSTEP]\nrw [comp_inv_eq, assoc, associator_naturality]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z\u271d X Y Z X' Y' Z' : C\nf : X \u27f6 X'\ng : Y \u27f6 Y'\nh : Z \u27f6 Z'\n\u22a2 f \u2297 g \u2297 h = (\u03b1_ X Y Z).inv \u226b (\u03b1_ X Y Z).hom \u226b (f \u2297 g \u2297 h)\n[PROOFSTEP]\nsimp\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z\u271d X X' Y Y' Z Z' : C\nf : X \u27f6 X'\ng : Y \u27f6 Y'\nh : Z \u27f6 Z'\n\u22a2 (f \u2297 g) \u2297 h = (\u03b1_ X Y Z).hom \u226b (f \u2297 g \u2297 h) \u226b (\u03b1_ X' Y' Z').inv\n[PROOFSTEP]\nrw [associator_inv_naturality, hom_inv_id_assoc]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z\u271d X X' Y Y' Z Z' : C\nf : X \u27f6 X'\ng : Y \u27f6 Y'\nh : Z \u27f6 Z'\n\u22a2 f \u2297 g \u2297 h = (\u03b1_ X Y Z).inv \u226b ((f \u2297 g) \u2297 h) \u226b (\u03b1_ X' Y' Z').hom\n[PROOFSTEP]\nrw [associator_naturality, inv_hom_id_assoc]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z\u271d X Y Z Z' : C\nh : Z \u27f6 Z'\n\u22a2 (\ud835\udfd9 (X \u2297 Y) \u2297 h) \u226b (\u03b1_ X Y Z').hom = (\u03b1_ X Y Z).hom \u226b (\ud835\udfd9 X \u2297 \ud835\udfd9 Y \u2297 h)\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_naturality]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z\u271d X Y Z X' : C\nf : X \u27f6 X'\n\u22a2 (f \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b (\u03b1_ X' Y Z).inv = (\u03b1_ X Y Z).inv \u226b ((f \u2297 \ud835\udfd9 Y) \u2297 \ud835\udfd9 Z)\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_inv_naturality]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V\u271d W\u271d X\u271d Y\u271d Z\u271d V W X Y Z : C\nf : V \u2245 W\ng : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 (f.hom \u2297 g) \u226b (f.inv \u2297 h) = (\ud835\udfd9 V \u2297 g) \u226b (\ud835\udfd9 V \u2297 h)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, f.hom_inv_id, id_tensor_comp]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V\u271d W\u271d X\u271d Y\u271d Z\u271d V W X Y Z : C\nf : V \u2245 W\ng : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 (f.inv \u2297 g) \u226b (f.hom \u2297 h) = (\ud835\udfd9 W \u2297 g) \u226b (\ud835\udfd9 W \u2297 h)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, f.inv_hom_id, id_tensor_comp]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V\u271d W\u271d X\u271d Y\u271d Z\u271d V W X Y Z : C\nf : V \u2245 W\ng : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 (g \u2297 f.hom) \u226b (h \u2297 f.inv) = (g \u2297 \ud835\udfd9 V) \u226b (h \u2297 \ud835\udfd9 V)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, f.hom_inv_id, comp_tensor_id]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V\u271d W\u271d X\u271d Y\u271d Z\u271d V W X Y Z : C\nf : V \u2245 W\ng : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 (g \u2297 f.inv) \u226b (h \u2297 f.hom) = (g \u2297 \ud835\udfd9 W) \u226b (h \u2297 \ud835\udfd9 W)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, f.inv_hom_id, comp_tensor_id]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b3 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : MonoidalCategory C\nU V\u271d W\u271d X\u271d Y\u271d Z\u271d V W X Y Z : C\nf : V \u27f6 W\ninst\u271d : IsIso f\ng : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 (f \u2297 g) \u226b (inv f \u2297 h) = (\ud835\udfd9 V \u2297 g) \u226b (\ud835\udfd9 V \u2297 h)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, IsIso.hom_inv_id, id_tensor_comp]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b3 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : MonoidalCategory C\nU V\u271d W\u271d X\u271d Y\u271d Z\u271d V W X Y Z : C\nf : V \u27f6 W\ninst\u271d : IsIso f\ng : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 (inv f \u2297 g) \u226b (f \u2297 h) = (\ud835\udfd9 W \u2297 g) \u226b (\ud835\udfd9 W \u2297 h)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, IsIso.inv_hom_id, id_tensor_comp]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b3 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : MonoidalCategory C\nU V\u271d W\u271d X\u271d Y\u271d Z\u271d V W X Y Z : C\nf : V \u27f6 W\ninst\u271d : IsIso f\ng : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 (g \u2297 f) \u226b (h \u2297 inv f) = (g \u2297 \ud835\udfd9 V) \u226b (h \u2297 \ud835\udfd9 V)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, IsIso.hom_inv_id, comp_tensor_id]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b3 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : MonoidalCategory C\nU V\u271d W\u271d X\u271d Y\u271d Z\u271d V W X Y Z : C\nf : V \u27f6 W\ninst\u271d : IsIso f\ng : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 (g \u2297 inv f) \u226b (h \u2297 f) = (g \u2297 \ud835\udfd9 W) \u226b (h \u2297 \ud835\udfd9 W)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, IsIso.inv_hom_id, comp_tensor_id]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\ntensorObj : C \u2192 C \u2192 C\ntensorHom : {X\u2081 Y\u2081 X\u2082 Y\u2082 : C} \u2192 (X\u2081 \u27f6 Y\u2081) \u2192 (X\u2082 \u27f6 Y\u2082) \u2192 (tensorObj X\u2081 X\u2082 \u27f6 tensorObj Y\u2081 Y\u2082)\nwhiskerLeft :\n optParam ((X : C) \u2192 {Y\u2081 Y\u2082 : C} \u2192 (Y\u2081 \u27f6 Y\u2082) \u2192 (tensorObj X Y\u2081 \u27f6 tensorObj X Y\u2082)) fun X x x_1 f => tensorHom (\ud835\udfd9 X) f\nwhiskerRight :\n optParam ({X\u2081 X\u2082 : C} \u2192 (X\u2081 \u27f6 X\u2082) \u2192 (Y : C) \u2192 tensorObj X\u2081 Y \u27f6 tensorObj X\u2082 Y) fun {X\u2081 X\u2082} f Y => tensorHom f (\ud835\udfd9 Y)\ntensor_id : autoParam (\u2200 (X\u2081 X\u2082 : C), tensorHom (\ud835\udfd9 X\u2081) (\ud835\udfd9 X\u2082) = \ud835\udfd9 (tensorObj X\u2081 X\u2082)) _auto\u271d\nid_tensorHom : autoParam (\u2200 (X : C) {Y\u2081 Y\u2082 : C} (f : Y\u2081 \u27f6 Y\u2082), tensorHom (\ud835\udfd9 X) f = whiskerLeft X f) _auto\u271d\ntensorHom_id : autoParam (\u2200 {X\u2081 X\u2082 : C} (f : X\u2081 \u27f6 X\u2082) (Y : C), tensorHom f (\ud835\udfd9 Y) = whiskerRight f Y) _auto\u271d\ntensor_comp :\n autoParam\n (\u2200 {X\u2081 Y\u2081 Z\u2081 X\u2082 Y\u2082 Z\u2082 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (g\u2081 : Y\u2081 \u27f6 Z\u2081) (g\u2082 : Y\u2082 \u27f6 Z\u2082),\n tensorHom (f\u2081 \u226b g\u2081) (f\u2082 \u226b g\u2082) = tensorHom f\u2081 f\u2082 \u226b tensorHom g\u2081 g\u2082)\n _auto\u271d\ntensorUnit' : C\nassociator : (X Y Z : C) \u2192 tensorObj (tensorObj X Y) Z \u2245 tensorObj X (tensorObj Y Z)\nassociator_naturality :\n autoParam\n (\u2200 {X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083),\n tensorHom (tensorHom f\u2081 f\u2082) f\u2083 \u226b (associator Y\u2081 Y\u2082 Y\u2083).hom =\n (associator X\u2081 X\u2082 X\u2083).hom \u226b tensorHom f\u2081 (tensorHom f\u2082 f\u2083))\n _auto\u271d\nleftUnitor : (X : C) \u2192 tensorObj tensorUnit' X \u2245 X\nleftUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom (\ud835\udfd9 tensorUnit') f \u226b (leftUnitor Y).hom = (leftUnitor X).hom \u226b f) _auto\u271d\nrightUnitor : (X : C) \u2192 tensorObj X tensorUnit' \u2245 X\nrightUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom f (\ud835\udfd9 tensorUnit') \u226b (rightUnitor Y).hom = (rightUnitor X).hom \u226b f)\n _auto\u271d\npentagon :\n autoParam\n (\u2200 (W X Y Z : C),\n tensorHom (associator W X Y).hom (\ud835\udfd9 Z) \u226b\n (associator W (tensorObj X Y) Z).hom \u226b tensorHom (\ud835\udfd9 W) (associator X Y Z).hom =\n (associator (tensorObj W X) Y Z).hom \u226b (associator W X (tensorObj Y Z)).hom)\n _auto\u271d\ntriangle :\n autoParam\n (\u2200 (X Y : C),\n (associator X tensorUnit' Y).hom \u226b tensorHom (\ud835\udfd9 X) (leftUnitor Y).hom = tensorHom (rightUnitor X).hom (\ud835\udfd9 Y))\n _auto\u271d\n\u22a2 \u2200 {X\u2081 Y\u2081 X\u2082 Y\u2082 : C} (f : X\u2081 \u27f6 Y\u2081) (g : X\u2082 \u27f6 Y\u2082),\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} => tensorHom) f g =\n (fun {X\u2081 X\u2082} f X => whiskerRight f X) f X\u2082 \u226b (fun X x x_1 f => whiskerLeft X f) Y\u2081 X\u2082 Y\u2082 g\n[PROOFSTEP]\nintros\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\ntensorObj : C \u2192 C \u2192 C\ntensorHom : {X\u2081 Y\u2081 X\u2082 Y\u2082 : C} \u2192 (X\u2081 \u27f6 Y\u2081) \u2192 (X\u2082 \u27f6 Y\u2082) \u2192 (tensorObj X\u2081 X\u2082 \u27f6 tensorObj Y\u2081 Y\u2082)\nwhiskerLeft :\n optParam ((X : C) \u2192 {Y\u2081 Y\u2082 : C} \u2192 (Y\u2081 \u27f6 Y\u2082) \u2192 (tensorObj X Y\u2081 \u27f6 tensorObj X Y\u2082)) fun X x x_1 f => tensorHom (\ud835\udfd9 X) f\nwhiskerRight :\n optParam ({X\u2081 X\u2082 : C} \u2192 (X\u2081 \u27f6 X\u2082) \u2192 (Y : C) \u2192 tensorObj X\u2081 Y \u27f6 tensorObj X\u2082 Y) fun {X\u2081 X\u2082} f Y => tensorHom f (\ud835\udfd9 Y)\ntensor_id : autoParam (\u2200 (X\u2081 X\u2082 : C), tensorHom (\ud835\udfd9 X\u2081) (\ud835\udfd9 X\u2082) = \ud835\udfd9 (tensorObj X\u2081 X\u2082)) _auto\u271d\nid_tensorHom : autoParam (\u2200 (X : C) {Y\u2081 Y\u2082 : C} (f : Y\u2081 \u27f6 Y\u2082), tensorHom (\ud835\udfd9 X) f = whiskerLeft X f) _auto\u271d\ntensorHom_id : autoParam (\u2200 {X\u2081 X\u2082 : C} (f : X\u2081 \u27f6 X\u2082) (Y : C), tensorHom f (\ud835\udfd9 Y) = whiskerRight f Y) _auto\u271d\ntensor_comp :\n autoParam\n (\u2200 {X\u2081 Y\u2081 Z\u2081 X\u2082 Y\u2082 Z\u2082 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (g\u2081 : Y\u2081 \u27f6 Z\u2081) (g\u2082 : Y\u2082 \u27f6 Z\u2082),\n tensorHom (f\u2081 \u226b g\u2081) (f\u2082 \u226b g\u2082) = tensorHom f\u2081 f\u2082 \u226b tensorHom g\u2081 g\u2082)\n _auto\u271d\ntensorUnit' : C\nassociator : (X Y Z : C) \u2192 tensorObj (tensorObj X Y) Z \u2245 tensorObj X (tensorObj Y Z)\nassociator_naturality :\n autoParam\n (\u2200 {X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083),\n tensorHom (tensorHom f\u2081 f\u2082) f\u2083 \u226b (associator Y\u2081 Y\u2082 Y\u2083).hom =\n (associator X\u2081 X\u2082 X\u2083).hom \u226b tensorHom f\u2081 (tensorHom f\u2082 f\u2083))\n _auto\u271d\nleftUnitor : (X : C) \u2192 tensorObj tensorUnit' X \u2245 X\nleftUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom (\ud835\udfd9 tensorUnit') f \u226b (leftUnitor Y).hom = (leftUnitor X).hom \u226b f) _auto\u271d\nrightUnitor : (X : C) \u2192 tensorObj X tensorUnit' \u2245 X\nrightUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom f (\ud835\udfd9 tensorUnit') \u226b (rightUnitor Y).hom = (rightUnitor X).hom \u226b f)\n _auto\u271d\npentagon :\n autoParam\n (\u2200 (W X Y Z : C),\n tensorHom (associator W X Y).hom (\ud835\udfd9 Z) \u226b\n (associator W (tensorObj X Y) Z).hom \u226b tensorHom (\ud835\udfd9 W) (associator X Y Z).hom =\n (associator (tensorObj W X) Y Z).hom \u226b (associator W X (tensorObj Y Z)).hom)\n _auto\u271d\ntriangle :\n autoParam\n (\u2200 (X Y : C),\n (associator X tensorUnit' Y).hom \u226b tensorHom (\ud835\udfd9 X) (leftUnitor Y).hom = tensorHom (rightUnitor X).hom (\ud835\udfd9 Y))\n _auto\u271d\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : C\nf\u271d : X\u2081\u271d \u27f6 Y\u2081\u271d\ng\u271d : X\u2082\u271d \u27f6 Y\u2082\u271d\n\u22a2 (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} => tensorHom) f\u271d g\u271d =\n (fun {X\u2081 X\u2082} f X => whiskerRight f X) f\u271d X\u2082\u271d \u226b (fun X x x_1 f => whiskerLeft X f) Y\u2081\u271d X\u2082\u271d Y\u2082\u271d g\u271d\n[PROOFSTEP]\nsimp [\u2190 id_tensorHom, \u2190 tensorHom_id, \u2190 tensor_comp]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\ntensorObj : C \u2192 C \u2192 C\ntensorHom : {X\u2081 Y\u2081 X\u2082 Y\u2082 : C} \u2192 (X\u2081 \u27f6 Y\u2081) \u2192 (X\u2082 \u27f6 Y\u2082) \u2192 (tensorObj X\u2081 X\u2082 \u27f6 tensorObj Y\u2081 Y\u2082)\nwhiskerLeft :\n optParam ((X : C) \u2192 {Y\u2081 Y\u2082 : C} \u2192 (Y\u2081 \u27f6 Y\u2082) \u2192 (tensorObj X Y\u2081 \u27f6 tensorObj X Y\u2082)) fun X x x_1 f => tensorHom (\ud835\udfd9 X) f\nwhiskerRight :\n optParam ({X\u2081 X\u2082 : C} \u2192 (X\u2081 \u27f6 X\u2082) \u2192 (Y : C) \u2192 tensorObj X\u2081 Y \u27f6 tensorObj X\u2082 Y) fun {X\u2081 X\u2082} f Y => tensorHom f (\ud835\udfd9 Y)\ntensor_id : autoParam (\u2200 (X\u2081 X\u2082 : C), tensorHom (\ud835\udfd9 X\u2081) (\ud835\udfd9 X\u2082) = \ud835\udfd9 (tensorObj X\u2081 X\u2082)) _auto\u271d\nid_tensorHom : autoParam (\u2200 (X : C) {Y\u2081 Y\u2082 : C} (f : Y\u2081 \u27f6 Y\u2082), tensorHom (\ud835\udfd9 X) f = whiskerLeft X f) _auto\u271d\ntensorHom_id : autoParam (\u2200 {X\u2081 X\u2082 : C} (f : X\u2081 \u27f6 X\u2082) (Y : C), tensorHom f (\ud835\udfd9 Y) = whiskerRight f Y) _auto\u271d\ntensor_comp :\n autoParam\n (\u2200 {X\u2081 Y\u2081 Z\u2081 X\u2082 Y\u2082 Z\u2082 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (g\u2081 : Y\u2081 \u27f6 Z\u2081) (g\u2082 : Y\u2082 \u27f6 Z\u2082),\n tensorHom (f\u2081 \u226b g\u2081) (f\u2082 \u226b g\u2082) = tensorHom f\u2081 f\u2082 \u226b tensorHom g\u2081 g\u2082)\n _auto\u271d\ntensorUnit' : C\nassociator : (X Y Z : C) \u2192 tensorObj (tensorObj X Y) Z \u2245 tensorObj X (tensorObj Y Z)\nassociator_naturality :\n autoParam\n (\u2200 {X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083),\n tensorHom (tensorHom f\u2081 f\u2082) f\u2083 \u226b (associator Y\u2081 Y\u2082 Y\u2083).hom =\n (associator X\u2081 X\u2082 X\u2083).hom \u226b tensorHom f\u2081 (tensorHom f\u2082 f\u2083))\n _auto\u271d\nleftUnitor : (X : C) \u2192 tensorObj tensorUnit' X \u2245 X\nleftUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom (\ud835\udfd9 tensorUnit') f \u226b (leftUnitor Y).hom = (leftUnitor X).hom \u226b f) _auto\u271d\nrightUnitor : (X : C) \u2192 tensorObj X tensorUnit' \u2245 X\nrightUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom f (\ud835\udfd9 tensorUnit') \u226b (rightUnitor Y).hom = (rightUnitor X).hom \u226b f)\n _auto\u271d\npentagon :\n autoParam\n (\u2200 (W X Y Z : C),\n tensorHom (associator W X Y).hom (\ud835\udfd9 Z) \u226b\n (associator W (tensorObj X Y) Z).hom \u226b tensorHom (\ud835\udfd9 W) (associator X Y Z).hom =\n (associator (tensorObj W X) Y Z).hom \u226b (associator W X (tensorObj Y Z)).hom)\n _auto\u271d\ntriangle :\n autoParam\n (\u2200 (X Y : C),\n (associator X tensorUnit' Y).hom \u226b tensorHom (\ud835\udfd9 X) (leftUnitor Y).hom = tensorHom (rightUnitor X).hom (\ud835\udfd9 Y))\n _auto\u271d\n\u22a2 \u2200 (X Y : C), (fun X x x_1 f => whiskerLeft X f) X Y Y (\ud835\udfd9 Y) = \ud835\udfd9 (tensorObj X Y)\n[PROOFSTEP]\nintros\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\ntensorObj : C \u2192 C \u2192 C\ntensorHom : {X\u2081 Y\u2081 X\u2082 Y\u2082 : C} \u2192 (X\u2081 \u27f6 Y\u2081) \u2192 (X\u2082 \u27f6 Y\u2082) \u2192 (tensorObj X\u2081 X\u2082 \u27f6 tensorObj Y\u2081 Y\u2082)\nwhiskerLeft :\n optParam ((X : C) \u2192 {Y\u2081 Y\u2082 : C} \u2192 (Y\u2081 \u27f6 Y\u2082) \u2192 (tensorObj X Y\u2081 \u27f6 tensorObj X Y\u2082)) fun X x x_1 f => tensorHom (\ud835\udfd9 X) f\nwhiskerRight :\n optParam ({X\u2081 X\u2082 : C} \u2192 (X\u2081 \u27f6 X\u2082) \u2192 (Y : C) \u2192 tensorObj X\u2081 Y \u27f6 tensorObj X\u2082 Y) fun {X\u2081 X\u2082} f Y => tensorHom f (\ud835\udfd9 Y)\ntensor_id : autoParam (\u2200 (X\u2081 X\u2082 : C), tensorHom (\ud835\udfd9 X\u2081) (\ud835\udfd9 X\u2082) = \ud835\udfd9 (tensorObj X\u2081 X\u2082)) _auto\u271d\nid_tensorHom : autoParam (\u2200 (X : C) {Y\u2081 Y\u2082 : C} (f : Y\u2081 \u27f6 Y\u2082), tensorHom (\ud835\udfd9 X) f = whiskerLeft X f) _auto\u271d\ntensorHom_id : autoParam (\u2200 {X\u2081 X\u2082 : C} (f : X\u2081 \u27f6 X\u2082) (Y : C), tensorHom f (\ud835\udfd9 Y) = whiskerRight f Y) _auto\u271d\ntensor_comp :\n autoParam\n (\u2200 {X\u2081 Y\u2081 Z\u2081 X\u2082 Y\u2082 Z\u2082 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (g\u2081 : Y\u2081 \u27f6 Z\u2081) (g\u2082 : Y\u2082 \u27f6 Z\u2082),\n tensorHom (f\u2081 \u226b g\u2081) (f\u2082 \u226b g\u2082) = tensorHom f\u2081 f\u2082 \u226b tensorHom g\u2081 g\u2082)\n _auto\u271d\ntensorUnit' : C\nassociator : (X Y Z : C) \u2192 tensorObj (tensorObj X Y) Z \u2245 tensorObj X (tensorObj Y Z)\nassociator_naturality :\n autoParam\n (\u2200 {X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083),\n tensorHom (tensorHom f\u2081 f\u2082) f\u2083 \u226b (associator Y\u2081 Y\u2082 Y\u2083).hom =\n (associator X\u2081 X\u2082 X\u2083).hom \u226b tensorHom f\u2081 (tensorHom f\u2082 f\u2083))\n _auto\u271d\nleftUnitor : (X : C) \u2192 tensorObj tensorUnit' X \u2245 X\nleftUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom (\ud835\udfd9 tensorUnit') f \u226b (leftUnitor Y).hom = (leftUnitor X).hom \u226b f) _auto\u271d\nrightUnitor : (X : C) \u2192 tensorObj X tensorUnit' \u2245 X\nrightUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom f (\ud835\udfd9 tensorUnit') \u226b (rightUnitor Y).hom = (rightUnitor X).hom \u226b f)\n _auto\u271d\npentagon :\n autoParam\n (\u2200 (W X Y Z : C),\n tensorHom (associator W X Y).hom (\ud835\udfd9 Z) \u226b\n (associator W (tensorObj X Y) Z).hom \u226b tensorHom (\ud835\udfd9 W) (associator X Y Z).hom =\n (associator (tensorObj W X) Y Z).hom \u226b (associator W X (tensorObj Y Z)).hom)\n _auto\u271d\ntriangle :\n autoParam\n (\u2200 (X Y : C),\n (associator X tensorUnit' Y).hom \u226b tensorHom (\ud835\udfd9 X) (leftUnitor Y).hom = tensorHom (rightUnitor X).hom (\ud835\udfd9 Y))\n _auto\u271d\nX\u271d Y\u271d : C\n\u22a2 (fun X x x_1 f => whiskerLeft X f) X\u271d Y\u271d Y\u271d (\ud835\udfd9 Y\u271d) = \ud835\udfd9 (tensorObj X\u271d Y\u271d)\n[PROOFSTEP]\nsimp [\u2190 id_tensorHom, \u2190 tensor_id]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\ntensorObj : C \u2192 C \u2192 C\ntensorHom : {X\u2081 Y\u2081 X\u2082 Y\u2082 : C} \u2192 (X\u2081 \u27f6 Y\u2081) \u2192 (X\u2082 \u27f6 Y\u2082) \u2192 (tensorObj X\u2081 X\u2082 \u27f6 tensorObj Y\u2081 Y\u2082)\nwhiskerLeft :\n optParam ((X : C) \u2192 {Y\u2081 Y\u2082 : C} \u2192 (Y\u2081 \u27f6 Y\u2082) \u2192 (tensorObj X Y\u2081 \u27f6 tensorObj X Y\u2082)) fun X x x_1 f => tensorHom (\ud835\udfd9 X) f\nwhiskerRight :\n optParam ({X\u2081 X\u2082 : C} \u2192 (X\u2081 \u27f6 X\u2082) \u2192 (Y : C) \u2192 tensorObj X\u2081 Y \u27f6 tensorObj X\u2082 Y) fun {X\u2081 X\u2082} f Y => tensorHom f (\ud835\udfd9 Y)\ntensor_id : autoParam (\u2200 (X\u2081 X\u2082 : C), tensorHom (\ud835\udfd9 X\u2081) (\ud835\udfd9 X\u2082) = \ud835\udfd9 (tensorObj X\u2081 X\u2082)) _auto\u271d\nid_tensorHom : autoParam (\u2200 (X : C) {Y\u2081 Y\u2082 : C} (f : Y\u2081 \u27f6 Y\u2082), tensorHom (\ud835\udfd9 X) f = whiskerLeft X f) _auto\u271d\ntensorHom_id : autoParam (\u2200 {X\u2081 X\u2082 : C} (f : X\u2081 \u27f6 X\u2082) (Y : C), tensorHom f (\ud835\udfd9 Y) = whiskerRight f Y) _auto\u271d\ntensor_comp :\n autoParam\n (\u2200 {X\u2081 Y\u2081 Z\u2081 X\u2082 Y\u2082 Z\u2082 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (g\u2081 : Y\u2081 \u27f6 Z\u2081) (g\u2082 : Y\u2082 \u27f6 Z\u2082),\n tensorHom (f\u2081 \u226b g\u2081) (f\u2082 \u226b g\u2082) = tensorHom f\u2081 f\u2082 \u226b tensorHom g\u2081 g\u2082)\n _auto\u271d\ntensorUnit' : C\nassociator : (X Y Z : C) \u2192 tensorObj (tensorObj X Y) Z \u2245 tensorObj X (tensorObj Y Z)\nassociator_naturality :\n autoParam\n (\u2200 {X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083),\n tensorHom (tensorHom f\u2081 f\u2082) f\u2083 \u226b (associator Y\u2081 Y\u2082 Y\u2083).hom =\n (associator X\u2081 X\u2082 X\u2083).hom \u226b tensorHom f\u2081 (tensorHom f\u2082 f\u2083))\n _auto\u271d\nleftUnitor : (X : C) \u2192 tensorObj tensorUnit' X \u2245 X\nleftUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom (\ud835\udfd9 tensorUnit') f \u226b (leftUnitor Y).hom = (leftUnitor X).hom \u226b f) _auto\u271d\nrightUnitor : (X : C) \u2192 tensorObj X tensorUnit' \u2245 X\nrightUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom f (\ud835\udfd9 tensorUnit') \u226b (rightUnitor Y).hom = (rightUnitor X).hom \u226b f)\n _auto\u271d\npentagon :\n autoParam\n (\u2200 (W X Y Z : C),\n tensorHom (associator W X Y).hom (\ud835\udfd9 Z) \u226b\n (associator W (tensorObj X Y) Z).hom \u226b tensorHom (\ud835\udfd9 W) (associator X Y Z).hom =\n (associator (tensorObj W X) Y Z).hom \u226b (associator W X (tensorObj Y Z)).hom)\n _auto\u271d\ntriangle :\n autoParam\n (\u2200 (X Y : C),\n (associator X tensorUnit' Y).hom \u226b tensorHom (\ud835\udfd9 X) (leftUnitor Y).hom = tensorHom (rightUnitor X).hom (\ud835\udfd9 Y))\n _auto\u271d\n\u22a2 \u2200 (X Y : C), (fun {X\u2081 X\u2082} f X => whiskerRight f X) (\ud835\udfd9 X) Y = \ud835\udfd9 (tensorObj X Y)\n[PROOFSTEP]\nintros\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\ntensorObj : C \u2192 C \u2192 C\ntensorHom : {X\u2081 Y\u2081 X\u2082 Y\u2082 : C} \u2192 (X\u2081 \u27f6 Y\u2081) \u2192 (X\u2082 \u27f6 Y\u2082) \u2192 (tensorObj X\u2081 X\u2082 \u27f6 tensorObj Y\u2081 Y\u2082)\nwhiskerLeft :\n optParam ((X : C) \u2192 {Y\u2081 Y\u2082 : C} \u2192 (Y\u2081 \u27f6 Y\u2082) \u2192 (tensorObj X Y\u2081 \u27f6 tensorObj X Y\u2082)) fun X x x_1 f => tensorHom (\ud835\udfd9 X) f\nwhiskerRight :\n optParam ({X\u2081 X\u2082 : C} \u2192 (X\u2081 \u27f6 X\u2082) \u2192 (Y : C) \u2192 tensorObj X\u2081 Y \u27f6 tensorObj X\u2082 Y) fun {X\u2081 X\u2082} f Y => tensorHom f (\ud835\udfd9 Y)\ntensor_id : autoParam (\u2200 (X\u2081 X\u2082 : C), tensorHom (\ud835\udfd9 X\u2081) (\ud835\udfd9 X\u2082) = \ud835\udfd9 (tensorObj X\u2081 X\u2082)) _auto\u271d\nid_tensorHom : autoParam (\u2200 (X : C) {Y\u2081 Y\u2082 : C} (f : Y\u2081 \u27f6 Y\u2082), tensorHom (\ud835\udfd9 X) f = whiskerLeft X f) _auto\u271d\ntensorHom_id : autoParam (\u2200 {X\u2081 X\u2082 : C} (f : X\u2081 \u27f6 X\u2082) (Y : C), tensorHom f (\ud835\udfd9 Y) = whiskerRight f Y) _auto\u271d\ntensor_comp :\n autoParam\n (\u2200 {X\u2081 Y\u2081 Z\u2081 X\u2082 Y\u2082 Z\u2082 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (g\u2081 : Y\u2081 \u27f6 Z\u2081) (g\u2082 : Y\u2082 \u27f6 Z\u2082),\n tensorHom (f\u2081 \u226b g\u2081) (f\u2082 \u226b g\u2082) = tensorHom f\u2081 f\u2082 \u226b tensorHom g\u2081 g\u2082)\n _auto\u271d\ntensorUnit' : C\nassociator : (X Y Z : C) \u2192 tensorObj (tensorObj X Y) Z \u2245 tensorObj X (tensorObj Y Z)\nassociator_naturality :\n autoParam\n (\u2200 {X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083),\n tensorHom (tensorHom f\u2081 f\u2082) f\u2083 \u226b (associator Y\u2081 Y\u2082 Y\u2083).hom =\n (associator X\u2081 X\u2082 X\u2083).hom \u226b tensorHom f\u2081 (tensorHom f\u2082 f\u2083))\n _auto\u271d\nleftUnitor : (X : C) \u2192 tensorObj tensorUnit' X \u2245 X\nleftUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom (\ud835\udfd9 tensorUnit') f \u226b (leftUnitor Y).hom = (leftUnitor X).hom \u226b f) _auto\u271d\nrightUnitor : (X : C) \u2192 tensorObj X tensorUnit' \u2245 X\nrightUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom f (\ud835\udfd9 tensorUnit') \u226b (rightUnitor Y).hom = (rightUnitor X).hom \u226b f)\n _auto\u271d\npentagon :\n autoParam\n (\u2200 (W X Y Z : C),\n tensorHom (associator W X Y).hom (\ud835\udfd9 Z) \u226b\n (associator W (tensorObj X Y) Z).hom \u226b tensorHom (\ud835\udfd9 W) (associator X Y Z).hom =\n (associator (tensorObj W X) Y Z).hom \u226b (associator W X (tensorObj Y Z)).hom)\n _auto\u271d\ntriangle :\n autoParam\n (\u2200 (X Y : C),\n (associator X tensorUnit' Y).hom \u226b tensorHom (\ud835\udfd9 X) (leftUnitor Y).hom = tensorHom (rightUnitor X).hom (\ud835\udfd9 Y))\n _auto\u271d\nX\u271d Y\u271d : C\n\u22a2 (fun {X\u2081 X\u2082} f X => whiskerRight f X) (\ud835\udfd9 X\u271d) Y\u271d = \ud835\udfd9 (tensorObj X\u271d Y\u271d)\n[PROOFSTEP]\nsimp [\u2190 tensorHom_id, tensor_id]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\ntensorObj : C \u2192 C \u2192 C\ntensorHom : {X\u2081 Y\u2081 X\u2082 Y\u2082 : C} \u2192 (X\u2081 \u27f6 Y\u2081) \u2192 (X\u2082 \u27f6 Y\u2082) \u2192 (tensorObj X\u2081 X\u2082 \u27f6 tensorObj Y\u2081 Y\u2082)\nwhiskerLeft :\n optParam ((X : C) \u2192 {Y\u2081 Y\u2082 : C} \u2192 (Y\u2081 \u27f6 Y\u2082) \u2192 (tensorObj X Y\u2081 \u27f6 tensorObj X Y\u2082)) fun X x x_1 f => tensorHom (\ud835\udfd9 X) f\nwhiskerRight :\n optParam ({X\u2081 X\u2082 : C} \u2192 (X\u2081 \u27f6 X\u2082) \u2192 (Y : C) \u2192 tensorObj X\u2081 Y \u27f6 tensorObj X\u2082 Y) fun {X\u2081 X\u2082} f Y => tensorHom f (\ud835\udfd9 Y)\ntensor_id : autoParam (\u2200 (X\u2081 X\u2082 : C), tensorHom (\ud835\udfd9 X\u2081) (\ud835\udfd9 X\u2082) = \ud835\udfd9 (tensorObj X\u2081 X\u2082)) _auto\u271d\nid_tensorHom : autoParam (\u2200 (X : C) {Y\u2081 Y\u2082 : C} (f : Y\u2081 \u27f6 Y\u2082), tensorHom (\ud835\udfd9 X) f = whiskerLeft X f) _auto\u271d\ntensorHom_id : autoParam (\u2200 {X\u2081 X\u2082 : C} (f : X\u2081 \u27f6 X\u2082) (Y : C), tensorHom f (\ud835\udfd9 Y) = whiskerRight f Y) _auto\u271d\ntensor_comp :\n autoParam\n (\u2200 {X\u2081 Y\u2081 Z\u2081 X\u2082 Y\u2082 Z\u2082 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (g\u2081 : Y\u2081 \u27f6 Z\u2081) (g\u2082 : Y\u2082 \u27f6 Z\u2082),\n tensorHom (f\u2081 \u226b g\u2081) (f\u2082 \u226b g\u2082) = tensorHom f\u2081 f\u2082 \u226b tensorHom g\u2081 g\u2082)\n _auto\u271d\ntensorUnit' : C\nassociator : (X Y Z : C) \u2192 tensorObj (tensorObj X Y) Z \u2245 tensorObj X (tensorObj Y Z)\nassociator_naturality :\n autoParam\n (\u2200 {X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083),\n tensorHom (tensorHom f\u2081 f\u2082) f\u2083 \u226b (associator Y\u2081 Y\u2082 Y\u2083).hom =\n (associator X\u2081 X\u2082 X\u2083).hom \u226b tensorHom f\u2081 (tensorHom f\u2082 f\u2083))\n _auto\u271d\nleftUnitor : (X : C) \u2192 tensorObj tensorUnit' X \u2245 X\nleftUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom (\ud835\udfd9 tensorUnit') f \u226b (leftUnitor Y).hom = (leftUnitor X).hom \u226b f) _auto\u271d\nrightUnitor : (X : C) \u2192 tensorObj X tensorUnit' \u2245 X\nrightUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom f (\ud835\udfd9 tensorUnit') \u226b (rightUnitor Y).hom = (rightUnitor X).hom \u226b f)\n _auto\u271d\npentagon :\n autoParam\n (\u2200 (W X Y Z : C),\n tensorHom (associator W X Y).hom (\ud835\udfd9 Z) \u226b\n (associator W (tensorObj X Y) Z).hom \u226b tensorHom (\ud835\udfd9 W) (associator X Y Z).hom =\n (associator (tensorObj W X) Y Z).hom \u226b (associator W X (tensorObj Y Z)).hom)\n _auto\u271d\ntriangle :\n autoParam\n (\u2200 (X Y : C),\n (associator X tensorUnit' Y).hom \u226b tensorHom (\ud835\udfd9 X) (leftUnitor Y).hom = tensorHom (rightUnitor X).hom (\ud835\udfd9 Y))\n _auto\u271d\n\u22a2 \u2200 (W X Y Z : C),\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} => tensorHom) (associator W X Y).hom (\ud835\udfd9 Z) \u226b\n (associator W (tensorObj X Y) Z).hom \u226b (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} => tensorHom) (\ud835\udfd9 W) (associator X Y Z).hom =\n (associator (tensorObj W X) Y Z).hom \u226b (associator W X (tensorObj Y Z)).hom\n[PROOFSTEP]\nintros\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\ntensorObj : C \u2192 C \u2192 C\ntensorHom : {X\u2081 Y\u2081 X\u2082 Y\u2082 : C} \u2192 (X\u2081 \u27f6 Y\u2081) \u2192 (X\u2082 \u27f6 Y\u2082) \u2192 (tensorObj X\u2081 X\u2082 \u27f6 tensorObj Y\u2081 Y\u2082)\nwhiskerLeft :\n optParam ((X : C) \u2192 {Y\u2081 Y\u2082 : C} \u2192 (Y\u2081 \u27f6 Y\u2082) \u2192 (tensorObj X Y\u2081 \u27f6 tensorObj X Y\u2082)) fun X x x_1 f => tensorHom (\ud835\udfd9 X) f\nwhiskerRight :\n optParam ({X\u2081 X\u2082 : C} \u2192 (X\u2081 \u27f6 X\u2082) \u2192 (Y : C) \u2192 tensorObj X\u2081 Y \u27f6 tensorObj X\u2082 Y) fun {X\u2081 X\u2082} f Y => tensorHom f (\ud835\udfd9 Y)\ntensor_id : autoParam (\u2200 (X\u2081 X\u2082 : C), tensorHom (\ud835\udfd9 X\u2081) (\ud835\udfd9 X\u2082) = \ud835\udfd9 (tensorObj X\u2081 X\u2082)) _auto\u271d\nid_tensorHom : autoParam (\u2200 (X : C) {Y\u2081 Y\u2082 : C} (f : Y\u2081 \u27f6 Y\u2082), tensorHom (\ud835\udfd9 X) f = whiskerLeft X f) _auto\u271d\ntensorHom_id : autoParam (\u2200 {X\u2081 X\u2082 : C} (f : X\u2081 \u27f6 X\u2082) (Y : C), tensorHom f (\ud835\udfd9 Y) = whiskerRight f Y) _auto\u271d\ntensor_comp :\n autoParam\n (\u2200 {X\u2081 Y\u2081 Z\u2081 X\u2082 Y\u2082 Z\u2082 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (g\u2081 : Y\u2081 \u27f6 Z\u2081) (g\u2082 : Y\u2082 \u27f6 Z\u2082),\n tensorHom (f\u2081 \u226b g\u2081) (f\u2082 \u226b g\u2082) = tensorHom f\u2081 f\u2082 \u226b tensorHom g\u2081 g\u2082)\n _auto\u271d\ntensorUnit' : C\nassociator : (X Y Z : C) \u2192 tensorObj (tensorObj X Y) Z \u2245 tensorObj X (tensorObj Y Z)\nassociator_naturality :\n autoParam\n (\u2200 {X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083),\n tensorHom (tensorHom f\u2081 f\u2082) f\u2083 \u226b (associator Y\u2081 Y\u2082 Y\u2083).hom =\n (associator X\u2081 X\u2082 X\u2083).hom \u226b tensorHom f\u2081 (tensorHom f\u2082 f\u2083))\n _auto\u271d\nleftUnitor : (X : C) \u2192 tensorObj tensorUnit' X \u2245 X\nleftUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom (\ud835\udfd9 tensorUnit') f \u226b (leftUnitor Y).hom = (leftUnitor X).hom \u226b f) _auto\u271d\nrightUnitor : (X : C) \u2192 tensorObj X tensorUnit' \u2245 X\nrightUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom f (\ud835\udfd9 tensorUnit') \u226b (rightUnitor Y).hom = (rightUnitor X).hom \u226b f)\n _auto\u271d\npentagon :\n autoParam\n (\u2200 (W X Y Z : C),\n tensorHom (associator W X Y).hom (\ud835\udfd9 Z) \u226b\n (associator W (tensorObj X Y) Z).hom \u226b tensorHom (\ud835\udfd9 W) (associator X Y Z).hom =\n (associator (tensorObj W X) Y Z).hom \u226b (associator W X (tensorObj Y Z)).hom)\n _auto\u271d\ntriangle :\n autoParam\n (\u2200 (X Y : C),\n (associator X tensorUnit' Y).hom \u226b tensorHom (\ud835\udfd9 X) (leftUnitor Y).hom = tensorHom (rightUnitor X).hom (\ud835\udfd9 Y))\n _auto\u271d\nW\u271d X\u271d Y\u271d Z\u271d : C\n\u22a2 (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} => tensorHom) (associator W\u271d X\u271d Y\u271d).hom (\ud835\udfd9 Z\u271d) \u226b\n (associator W\u271d (tensorObj X\u271d Y\u271d) Z\u271d).hom \u226b (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} => tensorHom) (\ud835\udfd9 W\u271d) (associator X\u271d Y\u271d Z\u271d).hom =\n (associator (tensorObj W\u271d X\u271d) Y\u271d Z\u271d).hom \u226b (associator W\u271d X\u271d (tensorObj Y\u271d Z\u271d)).hom\n[PROOFSTEP]\nsimp [\u2190 id_tensorHom, \u2190 tensorHom_id, pentagon]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\ntensorObj : C \u2192 C \u2192 C\ntensorHom : {X\u2081 Y\u2081 X\u2082 Y\u2082 : C} \u2192 (X\u2081 \u27f6 Y\u2081) \u2192 (X\u2082 \u27f6 Y\u2082) \u2192 (tensorObj X\u2081 X\u2082 \u27f6 tensorObj Y\u2081 Y\u2082)\nwhiskerLeft :\n optParam ((X : C) \u2192 {Y\u2081 Y\u2082 : C} \u2192 (Y\u2081 \u27f6 Y\u2082) \u2192 (tensorObj X Y\u2081 \u27f6 tensorObj X Y\u2082)) fun X x x_1 f => tensorHom (\ud835\udfd9 X) f\nwhiskerRight :\n optParam ({X\u2081 X\u2082 : C} \u2192 (X\u2081 \u27f6 X\u2082) \u2192 (Y : C) \u2192 tensorObj X\u2081 Y \u27f6 tensorObj X\u2082 Y) fun {X\u2081 X\u2082} f Y => tensorHom f (\ud835\udfd9 Y)\ntensor_id : autoParam (\u2200 (X\u2081 X\u2082 : C), tensorHom (\ud835\udfd9 X\u2081) (\ud835\udfd9 X\u2082) = \ud835\udfd9 (tensorObj X\u2081 X\u2082)) _auto\u271d\nid_tensorHom : autoParam (\u2200 (X : C) {Y\u2081 Y\u2082 : C} (f : Y\u2081 \u27f6 Y\u2082), tensorHom (\ud835\udfd9 X) f = whiskerLeft X f) _auto\u271d\ntensorHom_id : autoParam (\u2200 {X\u2081 X\u2082 : C} (f : X\u2081 \u27f6 X\u2082) (Y : C), tensorHom f (\ud835\udfd9 Y) = whiskerRight f Y) _auto\u271d\ntensor_comp :\n autoParam\n (\u2200 {X\u2081 Y\u2081 Z\u2081 X\u2082 Y\u2082 Z\u2082 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (g\u2081 : Y\u2081 \u27f6 Z\u2081) (g\u2082 : Y\u2082 \u27f6 Z\u2082),\n tensorHom (f\u2081 \u226b g\u2081) (f\u2082 \u226b g\u2082) = tensorHom f\u2081 f\u2082 \u226b tensorHom g\u2081 g\u2082)\n _auto\u271d\ntensorUnit' : C\nassociator : (X Y Z : C) \u2192 tensorObj (tensorObj X Y) Z \u2245 tensorObj X (tensorObj Y Z)\nassociator_naturality :\n autoParam\n (\u2200 {X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083),\n tensorHom (tensorHom f\u2081 f\u2082) f\u2083 \u226b (associator Y\u2081 Y\u2082 Y\u2083).hom =\n (associator X\u2081 X\u2082 X\u2083).hom \u226b tensorHom f\u2081 (tensorHom f\u2082 f\u2083))\n _auto\u271d\nleftUnitor : (X : C) \u2192 tensorObj tensorUnit' X \u2245 X\nleftUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom (\ud835\udfd9 tensorUnit') f \u226b (leftUnitor Y).hom = (leftUnitor X).hom \u226b f) _auto\u271d\nrightUnitor : (X : C) \u2192 tensorObj X tensorUnit' \u2245 X\nrightUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom f (\ud835\udfd9 tensorUnit') \u226b (rightUnitor Y).hom = (rightUnitor X).hom \u226b f)\n _auto\u271d\npentagon :\n autoParam\n (\u2200 (W X Y Z : C),\n tensorHom (associator W X Y).hom (\ud835\udfd9 Z) \u226b\n (associator W (tensorObj X Y) Z).hom \u226b tensorHom (\ud835\udfd9 W) (associator X Y Z).hom =\n (associator (tensorObj W X) Y Z).hom \u226b (associator W X (tensorObj Y Z)).hom)\n _auto\u271d\ntriangle :\n autoParam\n (\u2200 (X Y : C),\n (associator X tensorUnit' Y).hom \u226b tensorHom (\ud835\udfd9 X) (leftUnitor Y).hom = tensorHom (rightUnitor X).hom (\ud835\udfd9 Y))\n _auto\u271d\n\u22a2 \u2200 (X Y : C),\n (associator X tensorUnit' Y).hom \u226b (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} => tensorHom) (\ud835\udfd9 X) (leftUnitor Y).hom =\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} => tensorHom) (rightUnitor X).hom (\ud835\udfd9 Y)\n[PROOFSTEP]\nintros\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\ntensorObj : C \u2192 C \u2192 C\ntensorHom : {X\u2081 Y\u2081 X\u2082 Y\u2082 : C} \u2192 (X\u2081 \u27f6 Y\u2081) \u2192 (X\u2082 \u27f6 Y\u2082) \u2192 (tensorObj X\u2081 X\u2082 \u27f6 tensorObj Y\u2081 Y\u2082)\nwhiskerLeft :\n optParam ((X : C) \u2192 {Y\u2081 Y\u2082 : C} \u2192 (Y\u2081 \u27f6 Y\u2082) \u2192 (tensorObj X Y\u2081 \u27f6 tensorObj X Y\u2082)) fun X x x_1 f => tensorHom (\ud835\udfd9 X) f\nwhiskerRight :\n optParam ({X\u2081 X\u2082 : C} \u2192 (X\u2081 \u27f6 X\u2082) \u2192 (Y : C) \u2192 tensorObj X\u2081 Y \u27f6 tensorObj X\u2082 Y) fun {X\u2081 X\u2082} f Y => tensorHom f (\ud835\udfd9 Y)\ntensor_id : autoParam (\u2200 (X\u2081 X\u2082 : C), tensorHom (\ud835\udfd9 X\u2081) (\ud835\udfd9 X\u2082) = \ud835\udfd9 (tensorObj X\u2081 X\u2082)) _auto\u271d\nid_tensorHom : autoParam (\u2200 (X : C) {Y\u2081 Y\u2082 : C} (f : Y\u2081 \u27f6 Y\u2082), tensorHom (\ud835\udfd9 X) f = whiskerLeft X f) _auto\u271d\ntensorHom_id : autoParam (\u2200 {X\u2081 X\u2082 : C} (f : X\u2081 \u27f6 X\u2082) (Y : C), tensorHom f (\ud835\udfd9 Y) = whiskerRight f Y) _auto\u271d\ntensor_comp :\n autoParam\n (\u2200 {X\u2081 Y\u2081 Z\u2081 X\u2082 Y\u2082 Z\u2082 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (g\u2081 : Y\u2081 \u27f6 Z\u2081) (g\u2082 : Y\u2082 \u27f6 Z\u2082),\n tensorHom (f\u2081 \u226b g\u2081) (f\u2082 \u226b g\u2082) = tensorHom f\u2081 f\u2082 \u226b tensorHom g\u2081 g\u2082)\n _auto\u271d\ntensorUnit' : C\nassociator : (X Y Z : C) \u2192 tensorObj (tensorObj X Y) Z \u2245 tensorObj X (tensorObj Y Z)\nassociator_naturality :\n autoParam\n (\u2200 {X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083),\n tensorHom (tensorHom f\u2081 f\u2082) f\u2083 \u226b (associator Y\u2081 Y\u2082 Y\u2083).hom =\n (associator X\u2081 X\u2082 X\u2083).hom \u226b tensorHom f\u2081 (tensorHom f\u2082 f\u2083))\n _auto\u271d\nleftUnitor : (X : C) \u2192 tensorObj tensorUnit' X \u2245 X\nleftUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom (\ud835\udfd9 tensorUnit') f \u226b (leftUnitor Y).hom = (leftUnitor X).hom \u226b f) _auto\u271d\nrightUnitor : (X : C) \u2192 tensorObj X tensorUnit' \u2245 X\nrightUnitor_naturality :\n autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), tensorHom f (\ud835\udfd9 tensorUnit') \u226b (rightUnitor Y).hom = (rightUnitor X).hom \u226b f)\n _auto\u271d\npentagon :\n autoParam\n (\u2200 (W X Y Z : C),\n tensorHom (associator W X Y).hom (\ud835\udfd9 Z) \u226b\n (associator W (tensorObj X Y) Z).hom \u226b tensorHom (\ud835\udfd9 W) (associator X Y Z).hom =\n (associator (tensorObj W X) Y Z).hom \u226b (associator W X (tensorObj Y Z)).hom)\n _auto\u271d\ntriangle :\n autoParam\n (\u2200 (X Y : C),\n (associator X tensorUnit' Y).hom \u226b tensorHom (\ud835\udfd9 X) (leftUnitor Y).hom = tensorHom (rightUnitor X).hom (\ud835\udfd9 Y))\n _auto\u271d\nX\u271d Y\u271d : C\n\u22a2 (associator X\u271d tensorUnit' Y\u271d).hom \u226b (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} => tensorHom) (\ud835\udfd9 X\u271d) (leftUnitor Y\u271d).hom =\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} => tensorHom) (rightUnitor X\u271d).hom (\ud835\udfd9 Y\u271d)\n[PROOFSTEP]\nsimp [\u2190 id_tensorHom, \u2190 tensorHom_id, triangle]\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\n\u22a2 (X : C \u00d7 C \u00d7 C) \u2192 (leftAssocTensor C).obj X \u2245 (rightAssocTensor C).obj X\n[PROOFSTEP]\nintros\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX\u271d : C \u00d7 C \u00d7 C\n\u22a2 (leftAssocTensor C).obj X\u271d \u2245 (rightAssocTensor C).obj X\u271d\n[PROOFSTEP]\napply MonoidalCategory.associator\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\n\u22a2 \u2200 {X Y : C \u00d7 C \u00d7 C} (f : X \u27f6 Y),\n (leftAssocTensor C).map f \u226b (\u03b1_ Y.fst Y.snd.fst Y.snd.snd).hom =\n (\u03b1_ X.fst X.snd.fst X.snd.snd).hom \u226b (rightAssocTensor C).map f\n[PROOFSTEP]\nintros\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX\u271d Y\u271d : C \u00d7 C \u00d7 C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (leftAssocTensor C).map f\u271d \u226b (\u03b1_ Y\u271d.fst Y\u271d.snd.fst Y\u271d.snd.snd).hom =\n (\u03b1_ X\u271d.fst X\u271d.snd.fst X\u271d.snd.snd).hom \u226b (rightAssocTensor C).map f\u271d\n[PROOFSTEP]\napply MonoidalCategory.associator_naturality\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\n\u22a2 (X : C) \u2192 (tensorUnitLeft C).obj X \u2245 (\ud835\udfed C).obj X\n[PROOFSTEP]\nintros\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX\u271d : C\n\u22a2 (tensorUnitLeft C).obj X\u271d \u2245 (\ud835\udfed C).obj X\u271d\n[PROOFSTEP]\napply MonoidalCategory.leftUnitor\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\n\u22a2 \u2200 {X Y : C} (f : X \u27f6 Y), (tensorUnitLeft C).map f \u226b (\u03bb_ Y).hom = (\u03bb_ X).hom \u226b (\ud835\udfed C).map f\n[PROOFSTEP]\nintros\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (tensorUnitLeft C).map f\u271d \u226b (\u03bb_ Y\u271d).hom = (\u03bb_ X\u271d).hom \u226b (\ud835\udfed C).map f\u271d\n[PROOFSTEP]\napply MonoidalCategory.leftUnitor_naturality\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\n\u22a2 (X : C) \u2192 (tensorUnitRight C).obj X \u2245 (\ud835\udfed C).obj X\n[PROOFSTEP]\nintros\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX\u271d : C\n\u22a2 (tensorUnitRight C).obj X\u271d \u2245 (\ud835\udfed C).obj X\u271d\n[PROOFSTEP]\napply MonoidalCategory.rightUnitor\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\n\u22a2 \u2200 {X Y : C} (f : X \u27f6 Y), (tensorUnitRight C).map f \u226b (\u03c1_ Y).hom = (\u03c1_ X).hom \u226b (\ud835\udfed C).map f\n[PROOFSTEP]\nintros\n[GOAL]\nC\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (tensorUnitRight C).map f\u271d \u226b (\u03c1_ Y\u271d).hom = (\u03c1_ X\u271d).hom \u226b (\ud835\udfed C).map f\u271d\n[PROOFSTEP]\napply MonoidalCategory.rightUnitor_naturality\n[GOAL]\nC\u271d\u00b9 : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\u00b9\ninst\u271d\u2074 : MonoidalCategory C\u271d\u00b9\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y Z Z' : C\nf : Z \u27f6 Z'\n\u22a2 (tensorLeft (X \u2297 Y)).map f \u226b (\u03b1_ X Y Z').hom = (\u03b1_ X Y Z).hom \u226b (tensorLeft Y \u22d9 tensorLeft X).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC\u271d\u00b9 : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\u00b9\ninst\u271d\u2074 : MonoidalCategory C\u271d\u00b9\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y Z Z' : C\nf : Z \u27f6 Z'\n\u22a2 (\ud835\udfd9 (X \u2297 Y) \u2297 f) \u226b (\u03b1_ X Y Z').hom = (\u03b1_ X Y Z).hom \u226b (\ud835\udfd9 X \u2297 \ud835\udfd9 Y \u2297 f)\n[PROOFSTEP]\nrw [\u2190 tensor_id]\n[GOAL]\nC\u271d\u00b9 : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\u00b9\ninst\u271d\u2074 : MonoidalCategory C\u271d\u00b9\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y Z Z' : C\nf : Z \u27f6 Z'\n\u22a2 ((\ud835\udfd9 X \u2297 \ud835\udfd9 Y) \u2297 f) \u226b (\u03b1_ X Y Z').hom = (\u03b1_ X Y Z).hom \u226b (\ud835\udfd9 X \u2297 \ud835\udfd9 Y \u2297 f)\n[PROOFSTEP]\napply associator_naturality\n[GOAL]\nC\u271d\u00b9 : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\u00b9\ninst\u271d\u2074 : MonoidalCategory C\u271d\u00b9\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y Z : C\n\u22a2 NatTrans.app (tensorLeftTensor X Y).inv Z = (\u03b1_ X Y Z).inv\n[PROOFSTEP]\nsimp [tensorLeftTensor]\n[GOAL]\nC\u271d\u00b2 : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\u00b2\ninst\u271d\u2076 : MonoidalCategory C\u271d\u00b2\nC\u271d\u00b9 : Type u\ninst\u271d\u2075 : Category.{v, u} C\u271d\u00b9\ninst\u271d\u2074 : MonoidalCategory C\u271d\u00b9\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y : C\nf g : X \u27f6 Y\nh : (tensoringLeft C).map f = (tensoringLeft C).map g\n\u22a2 f = g\n[PROOFSTEP]\ninjections h\n[GOAL]\nC\u271d\u00b2 : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\u00b2\ninst\u271d\u2076 : MonoidalCategory C\u271d\u00b2\nC\u271d\u00b9 : Type u\ninst\u271d\u2075 : Category.{v, u} C\u271d\u00b9\ninst\u271d\u2074 : MonoidalCategory C\u271d\u00b9\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y : C\nf g : X \u27f6 Y\nh : (fun Z => f \u2297 \ud835\udfd9 Z) = fun Z => g \u2297 \ud835\udfd9 Z\n\u22a2 f = g\n[PROOFSTEP]\nreplace h := congr_fun h (\ud835\udfd9_ C)\n[GOAL]\nC\u271d\u00b2 : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\u00b2\ninst\u271d\u2076 : MonoidalCategory C\u271d\u00b2\nC\u271d\u00b9 : Type u\ninst\u271d\u2075 : Category.{v, u} C\u271d\u00b9\ninst\u271d\u2074 : MonoidalCategory C\u271d\u00b9\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y : C\nf g : X \u27f6 Y\nh : f \u2297 \ud835\udfd9 (\ud835\udfd9_ C) = g \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\n\u22a2 f = g\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nC\u271d\u00b2 : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\u00b2\ninst\u271d\u2076 : MonoidalCategory C\u271d\u00b2\nC\u271d\u00b9 : Type u\ninst\u271d\u2075 : Category.{v, u} C\u271d\u00b9\ninst\u271d\u2074 : MonoidalCategory C\u271d\u00b9\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y : C\nf g : X \u27f6 Y\nh : (tensoringRight C).map f = (tensoringRight C).map g\n\u22a2 f = g\n[PROOFSTEP]\ninjections h\n[GOAL]\nC\u271d\u00b2 : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\u00b2\ninst\u271d\u2076 : MonoidalCategory C\u271d\u00b2\nC\u271d\u00b9 : Type u\ninst\u271d\u2075 : Category.{v, u} C\u271d\u00b9\ninst\u271d\u2074 : MonoidalCategory C\u271d\u00b9\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y : C\nf g : X \u27f6 Y\nh : (fun Z => \ud835\udfd9 Z \u2297 f) = fun Z => \ud835\udfd9 Z \u2297 g\n\u22a2 f = g\n[PROOFSTEP]\nreplace h := congr_fun h (\ud835\udfd9_ C)\n[GOAL]\nC\u271d\u00b2 : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\u00b2\ninst\u271d\u2076 : MonoidalCategory C\u271d\u00b2\nC\u271d\u00b9 : Type u\ninst\u271d\u2075 : Category.{v, u} C\u271d\u00b9\ninst\u271d\u2074 : MonoidalCategory C\u271d\u00b9\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y : C\nf g : X \u27f6 Y\nh : \ud835\udfd9 (\ud835\udfd9_ C) \u2297 f = \ud835\udfd9 (\ud835\udfd9_ C) \u2297 g\n\u22a2 f = g\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nC\u271d\u00b3 : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\u00b3\ninst\u271d\u2078 : MonoidalCategory C\u271d\u00b3\nC\u271d\u00b2 : Type u\ninst\u271d\u2077 : Category.{v, u} C\u271d\u00b2\ninst\u271d\u2076 : MonoidalCategory C\u271d\u00b2\nC\u271d\u00b9 : Type u\ninst\u271d\u2075 : Category.{v, u} C\u271d\u00b9\ninst\u271d\u2074 : MonoidalCategory C\u271d\u00b9\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y Z Z' : C\nf : Z \u27f6 Z'\n\u22a2 (tensorRight (X \u2297 Y)).map f \u226b ((fun Z => (\u03b1_ Z X Y).symm) Z').hom =\n ((fun Z => (\u03b1_ Z X Y).symm) Z).hom \u226b (tensorRight X \u22d9 tensorRight Y).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC\u271d\u00b3 : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\u00b3\ninst\u271d\u2078 : MonoidalCategory C\u271d\u00b3\nC\u271d\u00b2 : Type u\ninst\u271d\u2077 : Category.{v, u} C\u271d\u00b2\ninst\u271d\u2076 : MonoidalCategory C\u271d\u00b2\nC\u271d\u00b9 : Type u\ninst\u271d\u2075 : Category.{v, u} C\u271d\u00b9\ninst\u271d\u2074 : MonoidalCategory C\u271d\u00b9\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y Z Z' : C\nf : Z \u27f6 Z'\n\u22a2 (f \u2297 \ud835\udfd9 (X \u2297 Y)) \u226b (\u03b1_ Z' X Y).inv = (\u03b1_ Z X Y).inv \u226b ((f \u2297 \ud835\udfd9 X) \u2297 \ud835\udfd9 Y)\n[PROOFSTEP]\nrw [\u2190 tensor_id]\n[GOAL]\nC\u271d\u00b3 : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\u00b3\ninst\u271d\u2078 : MonoidalCategory C\u271d\u00b3\nC\u271d\u00b2 : Type u\ninst\u271d\u2077 : Category.{v, u} C\u271d\u00b2\ninst\u271d\u2076 : MonoidalCategory C\u271d\u00b2\nC\u271d\u00b9 : Type u\ninst\u271d\u2075 : Category.{v, u} C\u271d\u00b9\ninst\u271d\u2074 : MonoidalCategory C\u271d\u00b9\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y Z Z' : C\nf : Z \u27f6 Z'\n\u22a2 (f \u2297 \ud835\udfd9 X \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ Z' X Y).inv = (\u03b1_ Z X Y).inv \u226b ((f \u2297 \ud835\udfd9 X) \u2297 \ud835\udfd9 Y)\n[PROOFSTEP]\napply associator_inv_naturality\n[GOAL]\nC\u271d\u00b3 : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\u00b3\ninst\u271d\u2078 : MonoidalCategory C\u271d\u00b3\nC\u271d\u00b2 : Type u\ninst\u271d\u2077 : Category.{v, u} C\u271d\u00b2\ninst\u271d\u2076 : MonoidalCategory C\u271d\u00b2\nC\u271d\u00b9 : Type u\ninst\u271d\u2075 : Category.{v, u} C\u271d\u00b9\ninst\u271d\u2074 : MonoidalCategory C\u271d\u00b9\nC\u271d : Type u\ninst\u271d\u00b3 : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y Z : C\n\u22a2 NatTrans.app (tensorRightTensor X Y).inv Z = (\u03b1_ Z X Y).hom\n[PROOFSTEP]\nsimp [tensorRightTensor]\n[GOAL]\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\n\u22a2 \u2200 {X\u2081 Y\u2081 X\u2082 Y\u2082 : C\u2081 \u00d7 C\u2082} (f : X\u2081 \u27f6 Y\u2081) (g : X\u2082 \u27f6 Y\u2082),\n (fun {X\u2081 Y\u2081 X\u2082 Y\u2082} f g => (f.fst \u2297 g.fst, f.snd \u2297 g.snd)) f g =\n (fun {X\u2081 X\u2082} f X => (f.fst \u25b7 X.fst, f.snd \u25b7 X.snd)) f X\u2082 \u226b\n (fun X x x_1 f => (X.fst \u25c1 f.fst, X.snd \u25c1 f.snd)) Y\u2081 X\u2082 Y\u2082 g\n[PROOFSTEP]\nsimp [tensorHom_def]\n[GOAL]\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nX : C\u2081 \u00d7 C\u2082\n\u22a2 (\u03bb_ X).hom.fst = (\u03bb_ X.fst).hom\n[PROOFSTEP]\ncases X\n[GOAL]\ncase mk\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nfst\u271d : C\u2081\nsnd\u271d : C\u2082\n\u22a2 (\u03bb_ (fst\u271d, snd\u271d)).hom.fst = (\u03bb_ (fst\u271d, snd\u271d).fst).hom\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nX : C\u2081 \u00d7 C\u2082\n\u22a2 (\u03bb_ X).hom.snd = (\u03bb_ X.snd).hom\n[PROOFSTEP]\ncases X\n[GOAL]\ncase mk\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nfst\u271d : C\u2081\nsnd\u271d : C\u2082\n\u22a2 (\u03bb_ (fst\u271d, snd\u271d)).hom.snd = (\u03bb_ (fst\u271d, snd\u271d).snd).hom\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nX : C\u2081 \u00d7 C\u2082\n\u22a2 (\u03bb_ X).inv.fst = (\u03bb_ X.fst).inv\n[PROOFSTEP]\ncases X\n[GOAL]\ncase mk\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nfst\u271d : C\u2081\nsnd\u271d : C\u2082\n\u22a2 (\u03bb_ (fst\u271d, snd\u271d)).inv.fst = (\u03bb_ (fst\u271d, snd\u271d).fst).inv\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nX : C\u2081 \u00d7 C\u2082\n\u22a2 (\u03bb_ X).inv.snd = (\u03bb_ X.snd).inv\n[PROOFSTEP]\ncases X\n[GOAL]\ncase mk\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nfst\u271d : C\u2081\nsnd\u271d : C\u2082\n\u22a2 (\u03bb_ (fst\u271d, snd\u271d)).inv.snd = (\u03bb_ (fst\u271d, snd\u271d).snd).inv\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nX : C\u2081 \u00d7 C\u2082\n\u22a2 (\u03c1_ X).hom.fst = (\u03c1_ X.fst).hom\n[PROOFSTEP]\ncases X\n[GOAL]\ncase mk\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nfst\u271d : C\u2081\nsnd\u271d : C\u2082\n\u22a2 (\u03c1_ (fst\u271d, snd\u271d)).hom.fst = (\u03c1_ (fst\u271d, snd\u271d).fst).hom\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nX : C\u2081 \u00d7 C\u2082\n\u22a2 (\u03c1_ X).hom.snd = (\u03c1_ X.snd).hom\n[PROOFSTEP]\ncases X\n[GOAL]\ncase mk\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nfst\u271d : C\u2081\nsnd\u271d : C\u2082\n\u22a2 (\u03c1_ (fst\u271d, snd\u271d)).hom.snd = (\u03c1_ (fst\u271d, snd\u271d).snd).hom\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nX : C\u2081 \u00d7 C\u2082\n\u22a2 (\u03c1_ X).inv.fst = (\u03c1_ X.fst).inv\n[PROOFSTEP]\ncases X\n[GOAL]\ncase mk\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nfst\u271d : C\u2081\nsnd\u271d : C\u2082\n\u22a2 (\u03c1_ (fst\u271d, snd\u271d)).inv.fst = (\u03c1_ (fst\u271d, snd\u271d).fst).inv\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nX : C\u2081 \u00d7 C\u2082\n\u22a2 (\u03c1_ X).inv.snd = (\u03c1_ X.snd).inv\n[PROOFSTEP]\ncases X\n[GOAL]\ncase mk\nC : Type u\n\ud835\udc9e : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d : MonoidalCategory C\u2082\nfst\u271d : C\u2081\nsnd\u271d : C\u2082\n\u22a2 (\u03c1_ (fst\u271d, snd\u271d)).inv.snd = (\u03c1_ (fst\u271d, snd\u271d).snd).inv\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monoidal.Category", "llama_tokens": 28392, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.40084486228289995}} {"text": "[GOAL]\nA : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid A\ninst\u271d\u00b9 : Pow A \u03b1\ninst\u271d : Zero \u03b1\nH : \u2200 {n : \u03b1}, n \u2260 0 \u2192 Function.Surjective fun a => a ^ n\nx\u271d : A\n\u22a2 (fun a n => if x : n = 0 then 1 else Exists.choose (_ : \u2203 a_1, (fun a => a ^ n) a_1 = a)) x\u271d 0 = 1\n[PROOFSTEP]\nclassical exact dif_pos rfl\n[GOAL]\nA : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid A\ninst\u271d\u00b9 : Pow A \u03b1\ninst\u271d : Zero \u03b1\nH : \u2200 {n : \u03b1}, n \u2260 0 \u2192 Function.Surjective fun a => a ^ n\nx\u271d : A\n\u22a2 (fun a n => if x : n = 0 then 1 else Exists.choose (_ : \u2203 a_1, (fun a => a ^ n) a_1 = a)) x\u271d 0 = 1\n[PROOFSTEP]\nexact dif_pos rfl\n[GOAL]\nA : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid A\ninst\u271d\u00b9 : Pow A \u03b1\ninst\u271d : Zero \u03b1\nH : \u2200 {n : \u03b1}, n \u2260 0 \u2192 Function.Surjective fun a => a ^ n\nn\u271d : \u03b1\na : A\nhn : n\u271d \u2260 0\n\u22a2 (fun a n => if x : n = 0 then 1 else Exists.choose (_ : \u2203 a_1, (fun a => a ^ n) a_1 = a)) a n\u271d ^ n\u271d = a\n[PROOFSTEP]\ndsimp only\n[GOAL]\nA : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid A\ninst\u271d\u00b9 : Pow A \u03b1\ninst\u271d : Zero \u03b1\nH : \u2200 {n : \u03b1}, n \u2260 0 \u2192 Function.Surjective fun a => a ^ n\nn\u271d : \u03b1\na : A\nhn : n\u271d \u2260 0\n\u22a2 (if x : n\u271d = 0 then 1 else Exists.choose (_ : \u2203 a_1, (fun a => a ^ n\u271d) a_1 = a)) ^ n\u271d = a\n[PROOFSTEP]\nrw [dif_neg hn]\n[GOAL]\nA : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Monoid A\ninst\u271d\u00b9 : Pow A \u03b1\ninst\u271d : Zero \u03b1\nH : \u2200 {n : \u03b1}, n \u2260 0 \u2192 Function.Surjective fun a => a ^ n\nn\u271d : \u03b1\na : A\nhn : n\u271d \u2260 0\n\u22a2 Exists.choose (_ : \u2203 a_1, (fun a => a ^ n\u271d) a_1 = a) ^ n\u271d = a\n[PROOFSTEP]\nexact (H hn a).choose_spec\n[GOAL]\nA : Type u_1\ninst\u271d : AddCommGroup A\nH : \u2200 {n : \u2124}, n \u2260 0 \u2192 n \u2022 \u22a4 = \u22a4\na : A\nn : \u2124\nhn : \u00acn = 0\n\u22a2 a \u2208 n \u2022 \u22a4\n[PROOFSTEP]\nrw [H hn]\n[GOAL]\nA : Type u_1\ninst\u271d : AddCommGroup A\nH : \u2200 {n : \u2124}, n \u2260 0 \u2192 n \u2022 \u22a4 = \u22a4\na : A\nn : \u2124\nhn : \u00acn = 0\n\u22a2 a \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\nA : Type u_1\ninst\u271d : AddCommGroup A\nH : \u2200 {n : \u2124}, n \u2260 0 \u2192 n \u2022 \u22a4 = \u22a4\nn\u271d : \u2124\na : A\nhn : n\u271d \u2260 0\n\u22a2 n\u271d \u2022 (fun a n => if hn : n = 0 then 0 else Exists.choose (_ : a \u2208 n \u2022 \u22a4)) a n\u271d = a\n[PROOFSTEP]\nsimp_rw [dif_neg hn]\n[GOAL]\nA : Type u_1\ninst\u271d : AddCommGroup A\nH : \u2200 {n : \u2124}, n \u2260 0 \u2192 n \u2022 \u22a4 = \u22a4\nn\u271d : \u2124\na : A\nhn : n\u271d \u2260 0\n\u22a2 n\u271d \u2022 Exists.choose (_ : a \u2208 n\u271d \u2022 \u22a4) = a\n[PROOFSTEP]\ngeneralize_proofs h1\n[GOAL]\nA : Type u_1\ninst\u271d : AddCommGroup A\nH : \u2200 {n : \u2124}, n \u2260 0 \u2192 n \u2022 \u22a4 = \u22a4\nn\u271d : \u2124\na : A\nhn : n\u271d \u2260 0\nh1 : a \u2208 n\u271d \u2022 \u22a4\n\u22a2 n\u271d \u2022 Exists.choose h1 = a\n[PROOFSTEP]\nexact h1.choose_spec.2\n[GOAL]\n\ud835\udd5c : Type ?u.71834\ninst\u271d\u00b9 : DivisionRing \ud835\udd5c\ninst\u271d : CharZero \ud835\udd5c\nq : \ud835\udd5c\n\u22a2 (fun q n => q / \u2191n) q 0 = 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type ?u.71834\ninst\u271d\u00b9 : DivisionRing \ud835\udd5c\ninst\u271d : CharZero \ud835\udd5c\nn : \u2124\nq : \ud835\udd5c\nhn : n \u2260 0\n\u22a2 n \u2022 (fun q n => q / \u2191n) q n = q\n[PROOFSTEP]\nrw [zsmul_eq_mul, (Int.cast_commute n _).eq, div_mul_cancel q (Int.cast_ne_zero.mpr hn)]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : Group A\ninst\u271d : RootableBy A \u2115\nn : \u2124\na : A\nhn : n \u2260 0\n\u22a2 (fun a z =>\n match z with\n | ofNat n => RootableBy.root a n\n | -[n+1] => (RootableBy.root a (n + 1))\u207b\u00b9)\n a n ^\n n =\n a\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase ofNat\nA : Type u_1\ninst\u271d\u00b9 : Group A\ninst\u271d : RootableBy A \u2115\na : A\na\u271d : \u2115\nhn : ofNat a\u271d \u2260 0\n\u22a2 (fun a z =>\n match z with\n | ofNat n => RootableBy.root a n\n | -[n+1] => (RootableBy.root a (n + 1))\u207b\u00b9)\n a (ofNat a\u271d) ^\n ofNat a\u271d =\n a\n[PROOFSTEP]\nchange RootableBy.root a _ ^ _ = a\n[GOAL]\ncase ofNat\nA : Type u_1\ninst\u271d\u00b9 : Group A\ninst\u271d : RootableBy A \u2115\na : A\na\u271d : \u2115\nhn : ofNat a\u271d \u2260 0\n\u22a2 RootableBy.root a a\u271d ^ ofNat a\u271d = a\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase ofNat\nA : Type u_1\ninst\u271d\u00b9 : Group A\ninst\u271d : RootableBy A \u2115\na : A\na\u271d : \u2115\nhn : ofNat a\u271d \u2260 0\n\u22a2 RootableBy.root a a\u271d ^ a\u271d = a\n[PROOFSTEP]\nrw [RootableBy.root_cancel]\n[GOAL]\ncase ofNat.a\nA : Type u_1\ninst\u271d\u00b9 : Group A\ninst\u271d : RootableBy A \u2115\na : A\na\u271d : \u2115\nhn : ofNat a\u271d \u2260 0\n\u22a2 a\u271d \u2260 0\n[PROOFSTEP]\nrw [Int.ofNat_eq_coe] at hn \n[GOAL]\ncase ofNat.a\nA : Type u_1\ninst\u271d\u00b9 : Group A\ninst\u271d : RootableBy A \u2115\na : A\na\u271d : \u2115\nhn : \u2191a\u271d \u2260 0\n\u22a2 a\u271d \u2260 0\n[PROOFSTEP]\nexact_mod_cast hn\n[GOAL]\ncase negSucc\nA : Type u_1\ninst\u271d\u00b9 : Group A\ninst\u271d : RootableBy A \u2115\na : A\na\u271d : \u2115\nhn : -[a\u271d+1] \u2260 0\n\u22a2 (fun a z =>\n match z with\n | ofNat n => RootableBy.root a n\n | -[n+1] => (RootableBy.root a (n + 1))\u207b\u00b9)\n a -[a\u271d+1] ^\n -[a\u271d+1] =\n a\n[PROOFSTEP]\nchange (RootableBy.root a _)\u207b\u00b9 ^ _ = a\n[GOAL]\ncase negSucc\nA : Type u_1\ninst\u271d\u00b9 : Group A\ninst\u271d : RootableBy A \u2115\na : A\na\u271d : \u2115\nhn : -[a\u271d+1] \u2260 0\n\u22a2 (RootableBy.root a (a\u271d + 1))\u207b\u00b9 ^ -[a\u271d+1] = a\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase negSucc\nA : Type u_1\ninst\u271d\u00b9 : Group A\ninst\u271d : RootableBy A \u2115\na : A\na\u271d : \u2115\nhn : -[a\u271d+1] \u2260 0\n\u22a2 RootableBy.root a (a\u271d + 1) ^ (a\u271d + 1) = a\n[PROOFSTEP]\nrw [RootableBy.root_cancel]\n[GOAL]\ncase negSucc.a\nA : Type u_1\ninst\u271d\u00b9 : Group A\ninst\u271d : RootableBy A \u2115\na : A\na\u271d : \u2115\nhn : -[a\u271d+1] \u2260 0\n\u22a2 a\u271d + 1 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : Group A\ninst\u271d : RootableBy A \u2124\nn : \u2115\na : A\nhn : n \u2260 0\n\u22a2 (fun a n => RootableBy.root a \u2191n) a n ^ n = a\n[PROOFSTEP]\nsimpa only [zpow_coe_nat] using RootableBy.root_cancel a (show (n : \u2124) \u2260 0 by exact_mod_cast hn)\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : Group A\ninst\u271d : RootableBy A \u2124\nn : \u2115\na : A\nhn : n \u2260 0\n\u22a2 \u2191n \u2260 0\n[PROOFSTEP]\nexact_mod_cast hn\n[GOAL]\nA : Type u_1\nB : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Zero \u03b1\ninst\u271d\u2074 : Monoid A\ninst\u271d\u00b3 : Monoid B\ninst\u271d\u00b2 : Pow A \u03b1\ninst\u271d\u00b9 : Pow B \u03b1\ninst\u271d : RootableBy A \u03b1\nf : A \u2192 B\nhf : Surjective f\nhpow : \u2200 (a : A) (n : \u03b1), f (a ^ n) = f a ^ n\nn : \u03b1\nhn : n \u2260 0\nx : B\ny : A\nhy : f y = x\n\u22a2 f (RootableBy.root y n) ^ n = x\n[PROOFSTEP]\nrw [\u2190 hpow (RootableBy.root y n) n, RootableBy.root_cancel _ hn, hy]\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Divisible", "llama_tokens": 3070, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754607093178, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.40082865059546036}} {"text": "[GOAL]\n\u22a2 factors 0 = []\n[PROOFSTEP]\nrw [factors]\n[GOAL]\n\u22a2 factors 1 = []\n[PROOFSTEP]\nrw [factors]\n[GOAL]\nn : \u2115\n\u22a2 \u2200 {p : \u2115}, p \u2208 factors n \u2192 Prime p\n[PROOFSTEP]\nmatch n with\n| 0 => simp\n| 1 => simp\n| k + 2 =>\n intro p h\n let m := minFac (k + 2)\n have : (k + 2) / m < (k + 2) := factors_lemma\n have h\u2081 : p = m \u2228 p \u2208 factors ((k + 2) / m) := List.mem_cons.1 (by rwa [factors] at h )\n exact Or.casesOn h\u2081 (fun h\u2082 => h\u2082.symm \u25b8 minFac_prime (by simp)) prime_of_mem_factors\n[GOAL]\nn : \u2115\n\u22a2 \u2200 {p : \u2115}, p \u2208 factors 0 \u2192 Prime p\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 \u2200 {p : \u2115}, p \u2208 factors 1 \u2192 Prime p\n[PROOFSTEP]\nsimp\n[GOAL]\nn k : \u2115\n\u22a2 \u2200 {p : \u2115}, p \u2208 factors (k + 2) \u2192 Prime p\n[PROOFSTEP]\nintro p h\n[GOAL]\nn k p : \u2115\nh : p \u2208 factors (k + 2)\n\u22a2 Prime p\n[PROOFSTEP]\nlet m := minFac (k + 2)\n[GOAL]\nn k p : \u2115\nh : p \u2208 factors (k + 2)\nm : \u2115 := minFac (k + 2)\n\u22a2 Prime p\n[PROOFSTEP]\nhave : (k + 2) / m < (k + 2) := factors_lemma\n[GOAL]\nn k p : \u2115\nh : p \u2208 factors (k + 2)\nm : \u2115 := minFac (k + 2)\nthis : (k + 2) / m < k + 2\n\u22a2 Prime p\n[PROOFSTEP]\nhave h\u2081 : p = m \u2228 p \u2208 factors ((k + 2) / m) := List.mem_cons.1 (by rwa [factors] at h )\n[GOAL]\nn k p : \u2115\nh : p \u2208 factors (k + 2)\nm : \u2115 := minFac (k + 2)\nthis : (k + 2) / m < k + 2\n\u22a2 p \u2208 m :: factors ((k + 2) / m)\n[PROOFSTEP]\nrwa [factors] at h \n[GOAL]\nn k p : \u2115\nh : p \u2208 factors (k + 2)\nm : \u2115 := minFac (k + 2)\nthis : (k + 2) / m < k + 2\nh\u2081 : p = m \u2228 p \u2208 factors ((k + 2) / m)\n\u22a2 Prime p\n[PROOFSTEP]\nexact Or.casesOn h\u2081 (fun h\u2082 => h\u2082.symm \u25b8 minFac_prime (by simp)) prime_of_mem_factors\n[GOAL]\nn k p : \u2115\nh : p \u2208 factors (k + 2)\nm : \u2115 := minFac (k + 2)\nthis : (k + 2) / m < k + 2\nh\u2081 : p = m \u2228 p \u2208 factors ((k + 2) / m)\nh\u2082 : p = m\n\u22a2 k + 2 \u2260 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 0 \u2260 0 \u2192 List.prod (factors 0) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 1 \u2260 0 \u2192 List.prod (factors 1) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nk : \u2115\nx\u271d : k + 2 \u2260 0\nm : \u2115 := minFac (k + 2)\nthis : (k + 2) / m < k + 2\n\u22a2 List.prod (factors (k + 2)) = k + 2\n[PROOFSTEP]\nhave h\u2081 : (k + 2) / m \u2260 0 := fun h =>\n by\n have : (k + 2) = 0 * m := (Nat.div_eq_iff_eq_mul_left (minFac_pos _) (minFac_dvd _)).1 h\n rw [zero_mul] at this ; exact (show k + 2 \u2260 0 by simp) this\n[GOAL]\nk : \u2115\nx\u271d : k + 2 \u2260 0\nm : \u2115 := minFac (k + 2)\nthis : (k + 2) / m < k + 2\nh : (k + 2) / m = 0\n\u22a2 False\n[PROOFSTEP]\nhave : (k + 2) = 0 * m := (Nat.div_eq_iff_eq_mul_left (minFac_pos _) (minFac_dvd _)).1 h\n[GOAL]\nk : \u2115\nx\u271d : k + 2 \u2260 0\nm : \u2115 := minFac (k + 2)\nthis\u271d : (k + 2) / m < k + 2\nh : (k + 2) / m = 0\nthis : k + 2 = 0 * m\n\u22a2 False\n[PROOFSTEP]\nrw [zero_mul] at this \n[GOAL]\nk : \u2115\nx\u271d : k + 2 \u2260 0\nm : \u2115 := minFac (k + 2)\nthis\u271d : (k + 2) / m < k + 2\nh : (k + 2) / m = 0\nthis : k + 2 = 0\n\u22a2 False\n[PROOFSTEP]\nexact (show k + 2 \u2260 0 by simp) this\n[GOAL]\nk : \u2115\nx\u271d : k + 2 \u2260 0\nm : \u2115 := minFac (k + 2)\nthis\u271d : (k + 2) / m < k + 2\nh : (k + 2) / m = 0\nthis : k + 2 = 0\n\u22a2 k + 2 \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\nk : \u2115\nx\u271d : k + 2 \u2260 0\nm : \u2115 := minFac (k + 2)\nthis : (k + 2) / m < k + 2\nh\u2081 : (k + 2) / m \u2260 0\n\u22a2 List.prod (factors (k + 2)) = k + 2\n[PROOFSTEP]\nrw [factors, List.prod_cons, prod_factors h\u2081, Nat.mul_div_cancel' (minFac_dvd _)]\n[GOAL]\np : \u2115\nhp : Prime p\n\u22a2 factors p = [p]\n[PROOFSTEP]\nhave : p = p - 2 + 2 := (tsub_eq_iff_eq_add_of_le hp.two_le).mp rfl\n[GOAL]\np : \u2115\nhp : Prime p\nthis : p = p - 2 + 2\n\u22a2 factors p = [p]\n[PROOFSTEP]\nrw [this, Nat.factors]\n[GOAL]\np : \u2115\nhp : Prime p\nthis : p = p - 2 + 2\n\u22a2 (let m := minFac (p - 2 + 2);\n let_fun this := (_ : (p - 2 + 2) / minFac (p - 2 + 2) < p - 2 + 2);\n m :: factors ((p - 2 + 2) / m)) =\n [p - 2 + 2]\n[PROOFSTEP]\nsimp only [Eq.symm this]\n[GOAL]\np : \u2115\nhp : Prime p\nthis : p = p - 2 + 2\n\u22a2 minFac p :: factors (p / minFac p) = [p]\n[PROOFSTEP]\nhave : Nat.minFac p = p := (Nat.prime_def_minFac.mp hp).2\n[GOAL]\np : \u2115\nhp : Prime p\nthis\u271d : p = p - 2 + 2\nthis : minFac p = p\n\u22a2 minFac p :: factors (p / minFac p) = [p]\n[PROOFSTEP]\nsimp only [this, Nat.factors, Nat.div_self (Nat.Prime.pos hp)]\n[GOAL]\nn : \u2115\n\u22a2 \u2200 {a : \u2115}, (\u2200 (p : \u2115), Prime p \u2192 p \u2223 n \u2192 a \u2264 p) \u2192 List.Chain (fun x x_1 => x \u2264 x_1) a (factors n)\n[PROOFSTEP]\nmatch n with\n| 0 => simp\n| 1 => simp\n| k + 2 =>\n intro a h\n let m := minFac (k + 2)\n have : (k + 2) / m < (k + 2) := factors_lemma\n rw [factors]\n refine' List.Chain.cons ((le_minFac.2 h).resolve_left (by simp)) (factors_chain _)\n exact fun p pp d => minFac_le_of_dvd pp.two_le (d.trans <| div_dvd_of_dvd <| minFac_dvd _)\n[GOAL]\nn : \u2115\n\u22a2 \u2200 {a : \u2115}, (\u2200 (p : \u2115), Prime p \u2192 p \u2223 0 \u2192 a \u2264 p) \u2192 List.Chain (fun x x_1 => x \u2264 x_1) a (factors 0)\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 \u2200 {a : \u2115}, (\u2200 (p : \u2115), Prime p \u2192 p \u2223 1 \u2192 a \u2264 p) \u2192 List.Chain (fun x x_1 => x \u2264 x_1) a (factors 1)\n[PROOFSTEP]\nsimp\n[GOAL]\nn k : \u2115\n\u22a2 \u2200 {a : \u2115}, (\u2200 (p : \u2115), Prime p \u2192 p \u2223 k + 2 \u2192 a \u2264 p) \u2192 List.Chain (fun x x_1 => x \u2264 x_1) a (factors (k + 2))\n[PROOFSTEP]\nintro a h\n[GOAL]\nn k a : \u2115\nh : \u2200 (p : \u2115), Prime p \u2192 p \u2223 k + 2 \u2192 a \u2264 p\n\u22a2 List.Chain (fun x x_1 => x \u2264 x_1) a (factors (k + 2))\n[PROOFSTEP]\nlet m := minFac (k + 2)\n[GOAL]\nn k a : \u2115\nh : \u2200 (p : \u2115), Prime p \u2192 p \u2223 k + 2 \u2192 a \u2264 p\nm : \u2115 := minFac (k + 2)\n\u22a2 List.Chain (fun x x_1 => x \u2264 x_1) a (factors (k + 2))\n[PROOFSTEP]\nhave : (k + 2) / m < (k + 2) := factors_lemma\n[GOAL]\nn k a : \u2115\nh : \u2200 (p : \u2115), Prime p \u2192 p \u2223 k + 2 \u2192 a \u2264 p\nm : \u2115 := minFac (k + 2)\nthis : (k + 2) / m < k + 2\n\u22a2 List.Chain (fun x x_1 => x \u2264 x_1) a (factors (k + 2))\n[PROOFSTEP]\nrw [factors]\n[GOAL]\nn k a : \u2115\nh : \u2200 (p : \u2115), Prime p \u2192 p \u2223 k + 2 \u2192 a \u2264 p\nm : \u2115 := minFac (k + 2)\nthis : (k + 2) / m < k + 2\n\u22a2 List.Chain (fun x x_1 => x \u2264 x_1) a\n (let m := minFac (k + 2);\n let_fun this := (_ : (k + 2) / minFac (k + 2) < k + 2);\n m :: factors ((k + 2) / m))\n[PROOFSTEP]\nrefine' List.Chain.cons ((le_minFac.2 h).resolve_left (by simp)) (factors_chain _)\n[GOAL]\nn k a : \u2115\nh : \u2200 (p : \u2115), Prime p \u2192 p \u2223 k + 2 \u2192 a \u2264 p\nm : \u2115 := minFac (k + 2)\nthis : (k + 2) / m < k + 2\n\u22a2 \u00ack + 2 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nn k a : \u2115\nh : \u2200 (p : \u2115), Prime p \u2192 p \u2223 k + 2 \u2192 a \u2264 p\nm : \u2115 := minFac (k + 2)\nthis : (k + 2) / m < k + 2\n\u22a2 \u2200 (p : \u2115), Prime p \u2192 p \u2223 (k + 2) / minFac (k + 2) \u2192 minFac (k + 2) \u2264 p\n[PROOFSTEP]\nexact fun p pp d => minFac_le_of_dvd pp.two_le (d.trans <| div_dvd_of_dvd <| minFac_dvd _)\n[GOAL]\nn : \u2115\n\u22a2 factors (n + 2) = minFac (n + 2) :: factors ((n + 2) / minFac (n + 2))\n[PROOFSTEP]\nrw [factors]\n[GOAL]\nn : \u2115\n\u22a2 factors n = [] \u2194 n = 0 \u2228 n = 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nn : \u2115\n\u22a2 factors n = [] \u2192 n = 0 \u2228 n = 1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nn : \u2115\n\u22a2 n = 0 \u2228 n = 1 \u2192 factors n = []\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nn : \u2115\nh : factors n = []\n\u22a2 n = 0 \u2228 n = 1\n[PROOFSTEP]\nrcases n with (_ | _ | n)\n[GOAL]\ncase mp.zero\nh : factors zero = []\n\u22a2 zero = 0 \u2228 zero = 1\n[PROOFSTEP]\nexact Or.inl rfl\n[GOAL]\ncase mp.succ.zero\nh : factors (succ zero) = []\n\u22a2 succ zero = 0 \u2228 succ zero = 1\n[PROOFSTEP]\nexact Or.inr rfl\n[GOAL]\ncase mp.succ.succ\nn : \u2115\nh : factors (succ (succ n)) = []\n\u22a2 succ (succ n) = 0 \u2228 succ (succ n) = 1\n[PROOFSTEP]\nrw [factors] at h \n[GOAL]\ncase mp.succ.succ\nn : \u2115\nh :\n (let m := minFac (n + 2);\n let_fun this := (_ : (n + 2) / minFac (n + 2) < n + 2);\n m :: factors ((n + 2) / m)) =\n []\n\u22a2 succ (succ n) = 0 \u2228 succ (succ n) = 1\n[PROOFSTEP]\ninjection h\n[GOAL]\ncase mpr\nn : \u2115\nh : n = 0 \u2228 n = 1\n\u22a2 factors n = []\n[PROOFSTEP]\nrcases h with (rfl | rfl)\n[GOAL]\ncase mpr.inl\n\u22a2 factors 0 = []\n[PROOFSTEP]\nexact factors_zero\n[GOAL]\ncase mpr.inr\n\u22a2 factors 1 = []\n[PROOFSTEP]\nexact factors_one\n[GOAL]\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\nh : factors a ~ factors b\n\u22a2 a = b\n[PROOFSTEP]\nsimpa [prod_factors ha, prod_factors hb] using List.Perm.prod_eq h\n[GOAL]\nn p : \u2115\nh : p \u2208 factors n\n\u22a2 p \u2223 n\n[PROOFSTEP]\nrcases n.eq_zero_or_pos with (rfl | hn)\n[GOAL]\ncase inl\np : \u2115\nh : p \u2208 factors 0\n\u22a2 p \u2223 0\n[PROOFSTEP]\nexact dvd_zero p\n[GOAL]\ncase inr\nn p : \u2115\nh : p \u2208 factors n\nhn : n > 0\n\u22a2 p \u2223 n\n[PROOFSTEP]\nrwa [\u2190 mem_factors_iff_dvd hn.ne' (prime_of_mem_factors h)]\n[GOAL]\nn p : \u2115\nh : p \u2208 factors n\n\u22a2 p \u2264 n\n[PROOFSTEP]\nrcases n.eq_zero_or_pos with (rfl | hn)\n[GOAL]\ncase inl\np : \u2115\nh : p \u2208 factors 0\n\u22a2 p \u2264 0\n[PROOFSTEP]\nrw [factors_zero] at h \n[GOAL]\ncase inl\np : \u2115\nh : p \u2208 []\n\u22a2 p \u2264 0\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inr\nn p : \u2115\nh : p \u2208 factors n\nhn : n > 0\n\u22a2 p \u2264 n\n[PROOFSTEP]\nexact le_of_dvd hn (dvd_of_mem_factors h)\n[GOAL]\nn : \u2115\nl : List \u2115\nh\u2081 : prod l = n\nh\u2082 : \u2200 (p : \u2115), p \u2208 l \u2192 Prime p\n\u22a2 l ~ factors n\n[PROOFSTEP]\nrefine' perm_of_prod_eq_prod _ _ _\n[GOAL]\ncase refine'_1\nn : \u2115\nl : List \u2115\nh\u2081 : prod l = n\nh\u2082 : \u2200 (p : \u2115), p \u2208 l \u2192 Prime p\n\u22a2 prod l = prod (factors n)\n[PROOFSTEP]\nrw [h\u2081]\n[GOAL]\ncase refine'_1\nn : \u2115\nl : List \u2115\nh\u2081 : prod l = n\nh\u2082 : \u2200 (p : \u2115), p \u2208 l \u2192 Prime p\n\u22a2 n = prod (factors n)\n[PROOFSTEP]\nrefine' (prod_factors _).symm\n[GOAL]\ncase refine'_1\nn : \u2115\nl : List \u2115\nh\u2081 : prod l = n\nh\u2082 : \u2200 (p : \u2115), p \u2208 l \u2192 Prime p\n\u22a2 n \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_1\nl : List \u2115\nh\u2082 : \u2200 (p : \u2115), p \u2208 l \u2192 Prime p\nh\u2081 : prod l = 0\n\u22a2 False\n[PROOFSTEP]\nrw [prod_eq_zero_iff] at h\u2081 \n[GOAL]\ncase refine'_1\nl : List \u2115\nh\u2082 : \u2200 (p : \u2115), p \u2208 l \u2192 Prime p\nh\u2081 : 0 \u2208 l\n\u22a2 False\n[PROOFSTEP]\nexact Prime.ne_zero (h\u2082 0 h\u2081) rfl\n[GOAL]\ncase refine'_2\nn : \u2115\nl : List \u2115\nh\u2081 : prod l = n\nh\u2082 : \u2200 (p : \u2115), p \u2208 l \u2192 Prime p\n\u22a2 \u2200 (p : \u2115), p \u2208 l \u2192 _root_.Prime p\n[PROOFSTEP]\nsimp_rw [\u2190 prime_iff]\n[GOAL]\ncase refine'_2\nn : \u2115\nl : List \u2115\nh\u2081 : prod l = n\nh\u2082 : \u2200 (p : \u2115), p \u2208 l \u2192 Prime p\n\u22a2 \u2200 (p : \u2115), p \u2208 l \u2192 Prime p\n[PROOFSTEP]\nexact h\u2082\n[GOAL]\ncase refine'_3\nn : \u2115\nl : List \u2115\nh\u2081 : prod l = n\nh\u2082 : \u2200 (p : \u2115), p \u2208 l \u2192 Prime p\n\u22a2 \u2200 (p : \u2115), p \u2208 factors n \u2192 _root_.Prime p\n[PROOFSTEP]\nsimp_rw [\u2190 prime_iff]\n[GOAL]\ncase refine'_3\nn : \u2115\nl : List \u2115\nh\u2081 : prod l = n\nh\u2082 : \u2200 (p : \u2115), p \u2208 l \u2192 Prime p\n\u22a2 \u2200 (p : \u2115), p \u2208 factors n \u2192 Prime p\n[PROOFSTEP]\nexact fun p => prime_of_mem_factors\n[GOAL]\np : \u2115\nhp : Prime p\nn : \u2115\n\u22a2 factors (p ^ n) = replicate n p\n[PROOFSTEP]\nsymm\n[GOAL]\np : \u2115\nhp : Prime p\nn : \u2115\n\u22a2 replicate n p = factors (p ^ n)\n[PROOFSTEP]\nrw [\u2190 List.replicate_perm]\n[GOAL]\np : \u2115\nhp : Prime p\nn : \u2115\n\u22a2 replicate n p ~ factors (p ^ n)\n[PROOFSTEP]\napply Nat.factors_unique (List.prod_replicate n p)\n[GOAL]\np : \u2115\nhp : Prime p\nn : \u2115\n\u22a2 \u2200 (p_1 : \u2115), p_1 \u2208 replicate n p \u2192 Prime p_1\n[PROOFSTEP]\nintro q hq\n[GOAL]\np : \u2115\nhp : Prime p\nn q : \u2115\nhq : q \u2208 replicate n p\n\u22a2 Prime q\n[PROOFSTEP]\nrwa [eq_of_mem_replicate hq]\n[GOAL]\nn p : \u2115\nhpos : n \u2260 0\nh : \u2200 {d : \u2115}, Prime d \u2192 d \u2223 n \u2192 d = p\n\u22a2 n = p ^ length (factors n)\n[PROOFSTEP]\nset k := n.factors.length\n[GOAL]\nn p : \u2115\nhpos : n \u2260 0\nh : \u2200 {d : \u2115}, Prime d \u2192 d \u2223 n \u2192 d = p\nk : \u2115 := length (factors n)\n\u22a2 n = p ^ k\n[PROOFSTEP]\nrw [\u2190 prod_factors hpos, \u2190 prod_replicate k p,\n eq_replicate_of_mem fun d hd => h (prime_of_mem_factors hd) (dvd_of_mem_factors hd)]\n[GOAL]\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 factors (a * b) ~ factors a ++ factors b\n[PROOFSTEP]\nrefine' (factors_unique _ _).symm\n[GOAL]\ncase refine'_1\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 prod (factors a ++ factors b) = a * b\n[PROOFSTEP]\nrw [List.prod_append, prod_factors ha, prod_factors hb]\n[GOAL]\ncase refine'_2\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 \u2200 (p : \u2115), p \u2208 factors a ++ factors b \u2192 Prime p\n[PROOFSTEP]\nintro p hp\n[GOAL]\ncase refine'_2\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\np : \u2115\nhp : p \u2208 factors a ++ factors b\n\u22a2 Prime p\n[PROOFSTEP]\nrw [List.mem_append] at hp \n[GOAL]\ncase refine'_2\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\np : \u2115\nhp : p \u2208 factors a \u2228 p \u2208 factors b\n\u22a2 Prime p\n[PROOFSTEP]\ncases' hp with hp' hp'\n[GOAL]\ncase refine'_2.inl\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\np : \u2115\nhp' : p \u2208 factors a\n\u22a2 Prime p\n[PROOFSTEP]\nexact prime_of_mem_factors hp'\n[GOAL]\ncase refine'_2.inr\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\np : \u2115\nhp' : p \u2208 factors b\n\u22a2 Prime p\n[PROOFSTEP]\nexact prime_of_mem_factors hp'\n[GOAL]\na b : \u2115\nhab : coprime a b\n\u22a2 factors (a * b) ~ factors a ++ factors b\n[PROOFSTEP]\nrcases a.eq_zero_or_pos with (rfl | ha)\n[GOAL]\ncase inl\nb : \u2115\nhab : coprime 0 b\n\u22a2 factors (0 * b) ~ factors 0 ++ factors b\n[PROOFSTEP]\nsimp [(coprime_zero_left _).mp hab]\n[GOAL]\ncase inr\na b : \u2115\nhab : coprime a b\nha : a > 0\n\u22a2 factors (a * b) ~ factors a ++ factors b\n[PROOFSTEP]\nrcases b.eq_zero_or_pos with (rfl | hb)\n[GOAL]\ncase inr.inl\na : \u2115\nha : a > 0\nhab : coprime a 0\n\u22a2 factors (a * 0) ~ factors a ++ factors 0\n[PROOFSTEP]\nsimp [(coprime_zero_right _).mp hab]\n[GOAL]\ncase inr.inr\na b : \u2115\nhab : coprime a b\nha : a > 0\nhb : b > 0\n\u22a2 factors (a * b) ~ factors a ++ factors b\n[PROOFSTEP]\nexact perm_factors_mul ha.ne' hb.ne'\n[GOAL]\nn k : \u2115\nh : k \u2260 0\n\u22a2 factors n <+ factors (n * k)\n[PROOFSTEP]\ncases' n with hn\n[GOAL]\ncase zero\nk : \u2115\nh : k \u2260 0\n\u22a2 factors zero <+ factors (zero * k)\n[PROOFSTEP]\nsimp [zero_mul]\n[GOAL]\ncase succ\nk : \u2115\nh : k \u2260 0\nhn : \u2115\n\u22a2 factors (succ hn) <+ factors (succ hn * k)\n[PROOFSTEP]\napply sublist_of_subperm_of_sorted _ (factors_sorted _) (factors_sorted _)\n[GOAL]\nk : \u2115\nh : k \u2260 0\nhn : \u2115\n\u22a2 factors (succ hn) <+~ factors (succ hn * k)\n[PROOFSTEP]\nsimp [(perm_factors_mul (Nat.succ_ne_zero _) h).subperm_left]\n[GOAL]\nk : \u2115\nh : k \u2260 0\nhn : \u2115\n\u22a2 factors (succ hn) <+~ factors (succ hn) ++ factors k\n[PROOFSTEP]\nexact (sublist_append_left _ _).subperm\n[GOAL]\nn k : \u2115\nh : n \u2223 k\nh' : k \u2260 0\n\u22a2 factors n <+ factors k\n[PROOFSTEP]\nobtain \u27e8a, rfl\u27e9 := h\n[GOAL]\ncase intro\nn a : \u2115\nh' : n * a \u2260 0\n\u22a2 factors n <+ factors (n * a)\n[PROOFSTEP]\nexact factors_sublist_right (right_ne_zero_of_mul h')\n[GOAL]\na b : \u2115\nha : a \u2260 0\nh : factors a <+~ factors b\n\u22a2 a \u2223 b\n[PROOFSTEP]\nrcases b.eq_zero_or_pos with (rfl | hb)\n[GOAL]\ncase inl\na : \u2115\nha : a \u2260 0\nh : factors a <+~ factors 0\n\u22a2 a \u2223 0\n[PROOFSTEP]\nexact dvd_zero _\n[GOAL]\ncase inr\na b : \u2115\nha : a \u2260 0\nh : factors a <+~ factors b\nhb : b > 0\n\u22a2 a \u2223 b\n[PROOFSTEP]\nrcases a with (_ | _ | a)\n[GOAL]\ncase inr.zero\nb : \u2115\nhb : b > 0\nha : zero \u2260 0\nh : factors zero <+~ factors b\n\u22a2 zero \u2223 b\n[PROOFSTEP]\nexact (ha rfl).elim\n[GOAL]\ncase inr.succ.zero\nb : \u2115\nhb : b > 0\nha : succ zero \u2260 0\nh : factors (succ zero) <+~ factors b\n\u22a2 succ zero \u2223 b\n[PROOFSTEP]\nexact\n one_dvd\n _\n --Porting note: previous proof\n --use (b.factors.diff a.succ.succ.factors).prod\n[GOAL]\ncase inr.succ.succ\nb : \u2115\nhb : b > 0\na : \u2115\nha : succ (succ a) \u2260 0\nh : factors (succ (succ a)) <+~ factors b\n\u22a2 succ (succ a) \u2223 b\n[PROOFSTEP]\nuse(@List.diff _ instBEq b.factors a.succ.succ.factors).prod\n[GOAL]\ncase h\nb : \u2115\nhb : b > 0\na : \u2115\nha : succ (succ a) \u2260 0\nh : factors (succ (succ a)) <+~ factors b\n\u22a2 b = succ (succ a) * prod (List.diff (factors b) (factors (succ (succ a))))\n[PROOFSTEP]\nnth_rw 1 [\u2190 Nat.prod_factors ha]\n[GOAL]\ncase h\nb : \u2115\nhb : b > 0\na : \u2115\nha : succ (succ a) \u2260 0\nh : factors (succ (succ a)) <+~ factors b\n\u22a2 b = prod (factors (succ (succ a))) * prod (List.diff (factors b) (factors (succ (succ a))))\n[PROOFSTEP]\nrw [\u2190 List.prod_append, List.Perm.prod_eq <| List.subperm_append_diff_self_of_count_le <| List.subperm_ext_iff.mp h,\n Nat.prod_factors hb.ne']\n[GOAL]\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\np : \u2115\n\u22a2 p \u2208 factors (a * b) \u2194 p \u2208 factors a \u2228 p \u2208 factors b\n[PROOFSTEP]\nrw [mem_factors (mul_ne_zero ha hb), mem_factors ha, mem_factors hb, \u2190 and_or_left]\n[GOAL]\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\np : \u2115\n\u22a2 Prime p \u2227 p \u2223 a * b \u2194 Prime p \u2227 (p \u2223 a \u2228 p \u2223 b)\n[PROOFSTEP]\nsimpa only [and_congr_right_iff] using Prime.dvd_mul\n[GOAL]\na b : \u2115\nhab : coprime a b\n\u22a2 List.Disjoint (factors a) (factors b)\n[PROOFSTEP]\nintro q hqa hqb\n[GOAL]\na b : \u2115\nhab : coprime a b\nq : \u2115\nhqa : q \u2208 factors a\nhqb : q \u2208 factors b\n\u22a2 False\n[PROOFSTEP]\napply not_prime_one\n[GOAL]\na b : \u2115\nhab : coprime a b\nq : \u2115\nhqa : q \u2208 factors a\nhqb : q \u2208 factors b\n\u22a2 Prime 1\n[PROOFSTEP]\nrw [\u2190 eq_one_of_dvd_coprimes hab (dvd_of_mem_factors hqa) (dvd_of_mem_factors hqb)]\n[GOAL]\na b : \u2115\nhab : coprime a b\nq : \u2115\nhqa : q \u2208 factors a\nhqb : q \u2208 factors b\n\u22a2 Prime q\n[PROOFSTEP]\nexact prime_of_mem_factors hqa\n[GOAL]\na b : \u2115\nhab : coprime a b\np : \u2115\n\u22a2 p \u2208 factors (a * b) \u2194 p \u2208 factors a \u222a factors b\n[PROOFSTEP]\nrcases a.eq_zero_or_pos with (rfl | ha)\n[GOAL]\ncase inl\nb p : \u2115\nhab : coprime 0 b\n\u22a2 p \u2208 factors (0 * b) \u2194 p \u2208 factors 0 \u222a factors b\n[PROOFSTEP]\nsimp [(coprime_zero_left _).mp hab]\n[GOAL]\ncase inr\na b : \u2115\nhab : coprime a b\np : \u2115\nha : a > 0\n\u22a2 p \u2208 factors (a * b) \u2194 p \u2208 factors a \u222a factors b\n[PROOFSTEP]\nrcases b.eq_zero_or_pos with (rfl | hb)\n[GOAL]\ncase inr.inl\na p : \u2115\nha : a > 0\nhab : coprime a 0\n\u22a2 p \u2208 factors (a * 0) \u2194 p \u2208 factors a \u222a factors 0\n[PROOFSTEP]\nsimp [(coprime_zero_right _).mp hab]\n[GOAL]\ncase inr.inr\na b : \u2115\nhab : coprime a b\np : \u2115\nha : a > 0\nhb : b > 0\n\u22a2 p \u2208 factors (a * b) \u2194 p \u2208 factors a \u222a factors b\n[PROOFSTEP]\nrw [mem_factors_mul ha.ne' hb.ne', List.mem_union_iff]\n[GOAL]\np a b : \u2115\nhpa : p \u2208 factors a\nhb : b \u2260 0\n\u22a2 p \u2208 factors (a * b)\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha)\n[GOAL]\ncase inl\np b : \u2115\nhb : b \u2260 0\nhpa : p \u2208 factors 0\n\u22a2 p \u2208 factors (0 * b)\n[PROOFSTEP]\nsimp at hpa \n[GOAL]\ncase inr\np a b : \u2115\nhpa : p \u2208 factors a\nhb : b \u2260 0\nha : a \u2260 0\n\u22a2 p \u2208 factors (a * b)\n[PROOFSTEP]\napply (mem_factors_mul ha hb).2 (Or.inl hpa)\n[GOAL]\np a b : \u2115\nhpb : p \u2208 factors b\nha : a \u2260 0\n\u22a2 p \u2208 factors (a * b)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\np a b : \u2115\nhpb : p \u2208 factors b\nha : a \u2260 0\n\u22a2 p \u2208 factors (b * a)\n[PROOFSTEP]\nexact mem_factors_mul_left hpb ha\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Factors", "llama_tokens": 8916, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124811, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.40055508679400764}} {"text": "[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\n\u22a2 natDegree 0 \u2264 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : Multiset S[X]\n\u22a2 \u2200 (a : List S[X]),\n natDegree (Multiset.sum (Quotient.mk (List.isSetoid S[X]) a)) \u2264\n foldr max (_ : \u2200 (a b c : \u2115), max a (max b c) = max b (max a c)) 0\n (Multiset.map natDegree (Quotient.mk (List.isSetoid S[X]) a))\n[PROOFSTEP]\nsimpa using natDegree_list_sum_le\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nf : \u03b9 \u2192 S[X]\n\u22a2 natDegree (\u2211 i in s, f i) \u2264 Finset.fold max 0 (natDegree \u2218 f) s\n[PROOFSTEP]\nsimpa using natDegree_multiset_sum_le (s.val.map f)\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\n\u22a2 degree (List.sum l) \u2264 List.maximum (List.map natDegree l)\n[PROOFSTEP]\nby_cases h : l.sum = 0\n[GOAL]\ncase pos\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : List.sum l = 0\n\u22a2 degree (List.sum l) \u2264 List.maximum (List.map natDegree l)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\n\u22a2 degree (List.sum l) \u2264 List.maximum (List.map natDegree l)\n[PROOFSTEP]\nrw [degree_eq_natDegree h]\n[GOAL]\ncase neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\n\u22a2 \u2191(natDegree (List.sum l)) \u2264 List.maximum (List.map natDegree l)\n[PROOFSTEP]\nsuffices (l.map natDegree).maximum = ((l.map natDegree).foldr max 0 : \u2115)\n by\n rw [this]\n simpa [this, Nat.cast_withBot] using natDegree_list_sum_le l\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\nthis : List.maximum (List.map natDegree l) = \u2191(List.foldr max 0 (List.map natDegree l))\n\u22a2 \u2191(natDegree (List.sum l)) \u2264 List.maximum (List.map natDegree l)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\nthis : List.maximum (List.map natDegree l) = \u2191(List.foldr max 0 (List.map natDegree l))\n\u22a2 \u2191(natDegree (List.sum l)) \u2264 \u2191(List.foldr max 0 (List.map natDegree l))\n[PROOFSTEP]\nsimpa [this, Nat.cast_withBot] using natDegree_list_sum_le l\n[GOAL]\ncase neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\n\u22a2 List.maximum (List.map natDegree l) = \u2191(List.foldr max 0 (List.map natDegree l))\n[PROOFSTEP]\nrw [\u2190 List.foldr_max_of_ne_nil]\n[GOAL]\ncase neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\n\u22a2 \u2191(List.foldr max \u22a5 (List.map natDegree l)) = \u2191(List.foldr max 0 (List.map natDegree l))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\n\u22a2 List.map natDegree l \u2260 []\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : List.map natDegree l = []\n\u22a2 List.sum l = 0\n[PROOFSTEP]\nrw [List.map_eq_nil] at h \n[GOAL]\ncase neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : l = []\n\u22a2 List.sum l = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\n\u22a2 natDegree (List.prod l) \u2264 List.sum (List.map natDegree l)\n[PROOFSTEP]\ninduction' l with hd tl IH\n[GOAL]\ncase nil\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\n\u22a2 natDegree (List.prod []) \u2264 List.sum (List.map natDegree [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nhd : S[X]\ntl : List S[X]\nIH : natDegree (List.prod tl) \u2264 List.sum (List.map natDegree tl)\n\u22a2 natDegree (List.prod (hd :: tl)) \u2264 List.sum (List.map natDegree (hd :: tl))\n[PROOFSTEP]\nsimpa using natDegree_mul_le.trans (add_le_add_left IH _)\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\n\u22a2 degree (List.prod l) \u2264 List.sum (List.map degree l)\n[PROOFSTEP]\ninduction' l with hd tl IH\n[GOAL]\ncase nil\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\n\u22a2 degree (List.prod []) \u2264 List.sum (List.map degree [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nhd : S[X]\ntl : List S[X]\nIH : degree (List.prod tl) \u2264 List.sum (List.map degree tl)\n\u22a2 degree (List.prod (hd :: tl)) \u2264 List.sum (List.map degree (hd :: tl))\n[PROOFSTEP]\nsimpa using (degree_mul_le _ _).trans (add_le_add_left IH _)\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nn : \u2115\nhl : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 n\n\u22a2 coeff (List.prod l) (List.length l * n) = List.prod (List.map (fun p => coeff p n) l)\n[PROOFSTEP]\ninduction' l with hd tl IH\n[GOAL]\ncase nil\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nn : \u2115\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 n\nhl : \u2200 (p : S[X]), p \u2208 [] \u2192 natDegree p \u2264 n\n\u22a2 coeff (List.prod []) (List.length [] * n) = List.prod (List.map (fun p => coeff p n) [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nn : \u2115\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 n\nhd : S[X]\ntl : List S[X]\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n) \u2192\n coeff (List.prod tl) (List.length tl * n) = List.prod (List.map (fun p => coeff p n) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 n\n\u22a2 coeff (List.prod (hd :: tl)) (List.length (hd :: tl) * n) = List.prod (List.map (fun p => coeff p n) (hd :: tl))\n[PROOFSTEP]\nhave hl' : \u2200 p \u2208 tl, natDegree p \u2264 n := fun p hp => hl p (List.mem_cons_of_mem _ hp)\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nn : \u2115\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 n\nhd : S[X]\ntl : List S[X]\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n) \u2192\n coeff (List.prod tl) (List.length tl * n) = List.prod (List.map (fun p => coeff p n) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 n\nhl' : \u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n\n\u22a2 coeff (List.prod (hd :: tl)) (List.length (hd :: tl) * n) = List.prod (List.map (fun p => coeff p n) (hd :: tl))\n[PROOFSTEP]\nsimp only [List.prod_cons, List.map, List.length]\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nn : \u2115\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 n\nhd : S[X]\ntl : List S[X]\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n) \u2192\n coeff (List.prod tl) (List.length tl * n) = List.prod (List.map (fun p => coeff p n) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 n\nhl' : \u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n\n\u22a2 coeff (hd * List.prod tl) ((List.length tl + 1) * n) = coeff hd n * List.prod (List.map (fun p => coeff p n) tl)\n[PROOFSTEP]\nrw [add_mul, one_mul, add_comm, \u2190 IH hl', mul_comm tl.length]\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nn : \u2115\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 n\nhd : S[X]\ntl : List S[X]\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n) \u2192\n coeff (List.prod tl) (List.length tl * n) = List.prod (List.map (fun p => coeff p n) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 n\nhl' : \u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n\n\u22a2 coeff (hd * List.prod tl) (n + n * List.length tl) = coeff hd n * coeff (List.prod tl) (n * List.length tl)\n[PROOFSTEP]\nhave h : natDegree tl.prod \u2264 n * tl.length :=\n by\n refine' (natDegree_list_prod_le _).trans _\n rw [\u2190 tl.length_map natDegree, mul_comm]\n refine' List.sum_le_card_nsmul _ _ _\n simpa using hl'\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nn : \u2115\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 n\nhd : S[X]\ntl : List S[X]\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n) \u2192\n coeff (List.prod tl) (List.length tl * n) = List.prod (List.map (fun p => coeff p n) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 n\nhl' : \u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n\n\u22a2 natDegree (List.prod tl) \u2264 n * List.length tl\n[PROOFSTEP]\nrefine' (natDegree_list_prod_le _).trans _\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nn : \u2115\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 n\nhd : S[X]\ntl : List S[X]\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n) \u2192\n coeff (List.prod tl) (List.length tl * n) = List.prod (List.map (fun p => coeff p n) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 n\nhl' : \u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n\n\u22a2 List.sum (List.map natDegree tl) \u2264 n * List.length tl\n[PROOFSTEP]\nrw [\u2190 tl.length_map natDegree, mul_comm]\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nn : \u2115\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 n\nhd : S[X]\ntl : List S[X]\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n) \u2192\n coeff (List.prod tl) (List.length tl * n) = List.prod (List.map (fun p => coeff p n) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 n\nhl' : \u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n\n\u22a2 List.sum (List.map natDegree tl) \u2264 List.length (List.map natDegree tl) * n\n[PROOFSTEP]\nrefine' List.sum_le_card_nsmul _ _ _\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nn : \u2115\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 n\nhd : S[X]\ntl : List S[X]\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n) \u2192\n coeff (List.prod tl) (List.length tl * n) = List.prod (List.map (fun p => coeff p n) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 n\nhl' : \u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n\n\u22a2 \u2200 (x : \u2115), x \u2208 List.map natDegree tl \u2192 x \u2264 n\n[PROOFSTEP]\nsimpa using hl'\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nn : \u2115\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 n\nhd : S[X]\ntl : List S[X]\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n) \u2192\n coeff (List.prod tl) (List.length tl * n) = List.prod (List.map (fun p => coeff p n) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 n\nhl' : \u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n\nh : natDegree (List.prod tl) \u2264 n * List.length tl\n\u22a2 coeff (hd * List.prod tl) (n + n * List.length tl) = coeff hd n * coeff (List.prod tl) (n * List.length tl)\n[PROOFSTEP]\nhave hdn : natDegree hd \u2264 n := hl _ (List.mem_cons_self _ _)\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nn : \u2115\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 n\nhd : S[X]\ntl : List S[X]\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n) \u2192\n coeff (List.prod tl) (List.length tl * n) = List.prod (List.map (fun p => coeff p n) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 n\nhl' : \u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n\nh : natDegree (List.prod tl) \u2264 n * List.length tl\nhdn : natDegree hd \u2264 n\n\u22a2 coeff (hd * List.prod tl) (n + n * List.length tl) = coeff hd n * coeff (List.prod tl) (n * List.length tl)\n[PROOFSTEP]\nrcases hdn.eq_or_lt with (rfl | hdn')\n[GOAL]\ncase cons.inl\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nhd : S[X]\ntl : List S[X]\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 natDegree hd\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 natDegree hd) \u2192\n coeff (List.prod tl) (List.length tl * natDegree hd) = List.prod (List.map (fun p => coeff p (natDegree hd)) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 natDegree hd\nhl' : \u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 natDegree hd\nh : natDegree (List.prod tl) \u2264 natDegree hd * List.length tl\nhdn : natDegree hd \u2264 natDegree hd\n\u22a2 coeff (hd * List.prod tl) (natDegree hd + natDegree hd * List.length tl) =\n coeff hd (natDegree hd) * coeff (List.prod tl) (natDegree hd * List.length tl)\n[PROOFSTEP]\ncases' h.eq_or_lt with h' h'\n[GOAL]\ncase cons.inl.inl\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nhd : S[X]\ntl : List S[X]\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 natDegree hd\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 natDegree hd) \u2192\n coeff (List.prod tl) (List.length tl * natDegree hd) = List.prod (List.map (fun p => coeff p (natDegree hd)) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 natDegree hd\nhl' : \u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 natDegree hd\nh : natDegree (List.prod tl) \u2264 natDegree hd * List.length tl\nhdn : natDegree hd \u2264 natDegree hd\nh' : natDegree (List.prod tl) = natDegree hd * List.length tl\n\u22a2 coeff (hd * List.prod tl) (natDegree hd + natDegree hd * List.length tl) =\n coeff hd (natDegree hd) * coeff (List.prod tl) (natDegree hd * List.length tl)\n[PROOFSTEP]\nrw [\u2190 h', coeff_mul_degree_add_degree, leadingCoeff, leadingCoeff]\n[GOAL]\ncase cons.inl.inr\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nhd : S[X]\ntl : List S[X]\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 natDegree hd\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 natDegree hd) \u2192\n coeff (List.prod tl) (List.length tl * natDegree hd) = List.prod (List.map (fun p => coeff p (natDegree hd)) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 natDegree hd\nhl' : \u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 natDegree hd\nh : natDegree (List.prod tl) \u2264 natDegree hd * List.length tl\nhdn : natDegree hd \u2264 natDegree hd\nh' : natDegree (List.prod tl) < natDegree hd * List.length tl\n\u22a2 coeff (hd * List.prod tl) (natDegree hd + natDegree hd * List.length tl) =\n coeff hd (natDegree hd) * coeff (List.prod tl) (natDegree hd * List.length tl)\n[PROOFSTEP]\nrw [coeff_eq_zero_of_natDegree_lt, coeff_eq_zero_of_natDegree_lt h', mul_zero]\n[GOAL]\ncase cons.inl.inr\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nhd : S[X]\ntl : List S[X]\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 natDegree hd\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 natDegree hd) \u2192\n coeff (List.prod tl) (List.length tl * natDegree hd) = List.prod (List.map (fun p => coeff p (natDegree hd)) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 natDegree hd\nhl' : \u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 natDegree hd\nh : natDegree (List.prod tl) \u2264 natDegree hd * List.length tl\nhdn : natDegree hd \u2264 natDegree hd\nh' : natDegree (List.prod tl) < natDegree hd * List.length tl\n\u22a2 natDegree (hd * List.prod tl) < natDegree hd + natDegree hd * List.length tl\n[PROOFSTEP]\nexact natDegree_mul_le.trans_lt (add_lt_add_left h' _)\n[GOAL]\ncase cons.inr\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nn : \u2115\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 n\nhd : S[X]\ntl : List S[X]\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n) \u2192\n coeff (List.prod tl) (List.length tl * n) = List.prod (List.map (fun p => coeff p n) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 n\nhl' : \u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n\nh : natDegree (List.prod tl) \u2264 n * List.length tl\nhdn : natDegree hd \u2264 n\nhdn' : natDegree hd < n\n\u22a2 coeff (hd * List.prod tl) (n + n * List.length tl) = coeff hd n * coeff (List.prod tl) (n * List.length tl)\n[PROOFSTEP]\nrw [coeff_eq_zero_of_natDegree_lt hdn', coeff_eq_zero_of_natDegree_lt, zero_mul]\n[GOAL]\ncase cons.inr\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nn : \u2115\nhl\u271d : \u2200 (p : S[X]), p \u2208 l \u2192 natDegree p \u2264 n\nhd : S[X]\ntl : List S[X]\nIH :\n (\u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n) \u2192\n coeff (List.prod tl) (List.length tl * n) = List.prod (List.map (fun p => coeff p n) tl)\nhl : \u2200 (p : S[X]), p \u2208 hd :: tl \u2192 natDegree p \u2264 n\nhl' : \u2200 (p : S[X]), p \u2208 tl \u2192 natDegree p \u2264 n\nh : natDegree (List.prod tl) \u2264 n * List.length tl\nhdn : natDegree hd \u2264 n\nhdn' : natDegree hd < n\n\u22a2 natDegree (hd * List.prod tl) < n + n * List.length tl\n[PROOFSTEP]\nexact natDegree_mul_le.trans_lt (add_lt_add_of_lt_of_le hdn' h)\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 \u2200 (a : List R[X]),\n natDegree (prod (Quotient.mk (List.isSetoid R[X]) a)) \u2264\n Multiset.sum (Multiset.map natDegree (Quotient.mk (List.isSetoid R[X]) a))\n[PROOFSTEP]\nsimpa using natDegree_list_prod_le\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 natDegree (\u220f i in s, f i) \u2264 \u2211 i in s, natDegree (f i)\n[PROOFSTEP]\nsimpa using natDegree_multiset_prod_le (s.1.map f)\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 \u2200 (a : List R[X]),\n degree (prod (Quotient.mk (List.isSetoid R[X]) a)) \u2264\n Multiset.sum (Multiset.map degree (Quotient.mk (List.isSetoid R[X]) a))\n[PROOFSTEP]\nsimpa using degree_list_prod_le\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 degree (\u220f i in s, f i) \u2264 \u2211 i in s, degree (f i)\n[PROOFSTEP]\nsimpa only [Multiset.map_map] using degree_multiset_prod_le (s.1.map f)\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : prod (Multiset.map leadingCoeff t) \u2260 0\n\u22a2 leadingCoeff (prod t) = prod (Multiset.map leadingCoeff t)\n[PROOFSTEP]\ninduction' t using Multiset.induction_on with a t ih\n[GOAL]\ncase empty\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh\u271d : prod (Multiset.map leadingCoeff t) \u2260 0\nh : prod (Multiset.map leadingCoeff 0) \u2260 0\n\u22a2 leadingCoeff (prod 0) = prod (Multiset.map leadingCoeff 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\nh\u271d : prod (Multiset.map leadingCoeff t\u271d) \u2260 0\na : R[X]\nt : Multiset R[X]\nih : prod (Multiset.map leadingCoeff t) \u2260 0 \u2192 leadingCoeff (prod t) = prod (Multiset.map leadingCoeff t)\nh : prod (Multiset.map leadingCoeff (a ::\u2098 t)) \u2260 0\n\u22a2 leadingCoeff (prod (a ::\u2098 t)) = prod (Multiset.map leadingCoeff (a ::\u2098 t))\n[PROOFSTEP]\nsimp only [Multiset.map_cons, Multiset.prod_cons] at h \u22a2\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\nh\u271d : prod (Multiset.map leadingCoeff t\u271d) \u2260 0\na : R[X]\nt : Multiset R[X]\nih : prod (Multiset.map leadingCoeff t) \u2260 0 \u2192 leadingCoeff (prod t) = prod (Multiset.map leadingCoeff t)\nh : leadingCoeff a * prod (Multiset.map leadingCoeff t) \u2260 0\n\u22a2 leadingCoeff (a * prod t) = leadingCoeff a * prod (Multiset.map leadingCoeff t)\n[PROOFSTEP]\nrw [Polynomial.leadingCoeff_mul']\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\nh\u271d : prod (Multiset.map leadingCoeff t\u271d) \u2260 0\na : R[X]\nt : Multiset R[X]\nih : prod (Multiset.map leadingCoeff t) \u2260 0 \u2192 leadingCoeff (prod t) = prod (Multiset.map leadingCoeff t)\nh : leadingCoeff a * prod (Multiset.map leadingCoeff t) \u2260 0\n\u22a2 leadingCoeff a * leadingCoeff (prod t) = leadingCoeff a * prod (Multiset.map leadingCoeff t)\n[PROOFSTEP]\nrw [ih]\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\nh\u271d : prod (Multiset.map leadingCoeff t\u271d) \u2260 0\na : R[X]\nt : Multiset R[X]\nih : prod (Multiset.map leadingCoeff t) \u2260 0 \u2192 leadingCoeff (prod t) = prod (Multiset.map leadingCoeff t)\nh : leadingCoeff a * prod (Multiset.map leadingCoeff t) \u2260 0\n\u22a2 prod (Multiset.map leadingCoeff t) \u2260 0\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\nh\u271d : prod (Multiset.map leadingCoeff t\u271d) \u2260 0\na : R[X]\nt : Multiset R[X]\nih : prod (Multiset.map leadingCoeff t) \u2260 0 \u2192 leadingCoeff (prod t) = prod (Multiset.map leadingCoeff t)\nh : leadingCoeff a * prod (Multiset.map leadingCoeff t) \u2260 0\n\u22a2 \u00acprod (Multiset.map leadingCoeff t) = 0\n[PROOFSTEP]\napply right_ne_zero_of_mul h\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\nh\u271d : prod (Multiset.map leadingCoeff t\u271d) \u2260 0\na : R[X]\nt : Multiset R[X]\nih : prod (Multiset.map leadingCoeff t) \u2260 0 \u2192 leadingCoeff (prod t) = prod (Multiset.map leadingCoeff t)\nh : leadingCoeff a * prod (Multiset.map leadingCoeff t) \u2260 0\n\u22a2 leadingCoeff a * leadingCoeff (prod t) \u2260 0\n[PROOFSTEP]\nrw [ih]\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\nh\u271d : prod (Multiset.map leadingCoeff t\u271d) \u2260 0\na : R[X]\nt : Multiset R[X]\nih : prod (Multiset.map leadingCoeff t) \u2260 0 \u2192 leadingCoeff (prod t) = prod (Multiset.map leadingCoeff t)\nh : leadingCoeff a * prod (Multiset.map leadingCoeff t) \u2260 0\n\u22a2 leadingCoeff a * prod (Multiset.map leadingCoeff t) \u2260 0\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\nh\u271d : prod (Multiset.map leadingCoeff t\u271d) \u2260 0\na : R[X]\nt : Multiset R[X]\nih : prod (Multiset.map leadingCoeff t) \u2260 0 \u2192 leadingCoeff (prod t) = prod (Multiset.map leadingCoeff t)\nh : leadingCoeff a * prod (Multiset.map leadingCoeff t) \u2260 0\n\u22a2 prod (Multiset.map leadingCoeff t) \u2260 0\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase cons\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\nh\u271d : prod (Multiset.map leadingCoeff t\u271d) \u2260 0\na : R[X]\nt : Multiset R[X]\nih : prod (Multiset.map leadingCoeff t) \u2260 0 \u2192 leadingCoeff (prod t) = prod (Multiset.map leadingCoeff t)\nh : leadingCoeff a * prod (Multiset.map leadingCoeff t) \u2260 0\n\u22a2 prod (Multiset.map leadingCoeff t) \u2260 0\n[PROOFSTEP]\napply right_ne_zero_of_mul h\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u220f i in s, leadingCoeff (f i) \u2260 0\n\u22a2 leadingCoeff (\u220f i in s, f i) = \u220f i in s, leadingCoeff (f i)\n[PROOFSTEP]\nsimpa using leadingCoeff_multiset_prod' (s.1.map f) (by simpa using h)\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u220f i in s, leadingCoeff (f i) \u2260 0\n\u22a2 prod (Multiset.map leadingCoeff (Multiset.map f s.val)) \u2260 0\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n\u22a2 natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\n[PROOFSTEP]\nrevert h\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\n[PROOFSTEP]\nrefine' Multiset.induction_on t _ fun a t ih ht => _\n[GOAL]\ncase refine'_1\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) 0) \u2260 0 \u2192\n natDegree (prod 0) = Multiset.sum (Multiset.map (fun f => natDegree f) 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih :\n prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\nht : prod (Multiset.map (fun f => leadingCoeff f) (a ::\u2098 t)) \u2260 0\n\u22a2 natDegree (prod (a ::\u2098 t)) = Multiset.sum (Multiset.map (fun f => natDegree f) (a ::\u2098 t))\n[PROOFSTEP]\nrw [Multiset.map_cons, Multiset.prod_cons] at ht \u22a2\n[GOAL]\ncase refine'_2\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih :\n prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\nht : leadingCoeff a * prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n\u22a2 natDegree (a * prod t) = Multiset.sum (natDegree a ::\u2098 Multiset.map (fun f => natDegree f) t)\n[PROOFSTEP]\nrw [Multiset.sum_cons, Polynomial.natDegree_mul', ih]\n[GOAL]\ncase refine'_2\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih :\n prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\nht : leadingCoeff a * prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n[PROOFSTEP]\napply right_ne_zero_of_mul ht\n[GOAL]\ncase refine'_2\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih :\n prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\nht : leadingCoeff a * prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n\u22a2 leadingCoeff a * leadingCoeff (prod t) \u2260 0\n[PROOFSTEP]\nrwa [Polynomial.leadingCoeff_multiset_prod']\n[GOAL]\ncase refine'_2.h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih :\n prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\nht : leadingCoeff a * prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n\u22a2 prod (Multiset.map leadingCoeff t) \u2260 0\n[PROOFSTEP]\napply right_ne_zero_of_mul ht\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u220f i in s, leadingCoeff (f i) \u2260 0\n\u22a2 natDegree (\u220f i in s, f i) = \u2211 i in s, natDegree (f i)\n[PROOFSTEP]\nsimpa using natDegree_multiset_prod' (s.1.map f) (by simpa using h)\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u220f i in s, leadingCoeff (f i) \u2260 0\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) (Multiset.map f s.val)) \u2260 0\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (f : R[X]), f \u2208 t \u2192 Monic f\n\u22a2 natDegree (prod t) = Multiset.sum (Multiset.map natDegree t)\n[PROOFSTEP]\nnontriviality R\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (f : R[X]), f \u2208 t \u2192 Monic f\n\u271d : Nontrivial R\n\u22a2 natDegree (prod t) = Multiset.sum (Multiset.map natDegree t)\n[PROOFSTEP]\napply natDegree_multiset_prod'\n[GOAL]\ncase h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (f : R[X]), f \u2208 t \u2192 Monic f\n\u271d : Nontrivial R\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n[PROOFSTEP]\nsuffices (t.map fun f => leadingCoeff f).prod = 1 by\n rw [this]\n simp\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (f : R[X]), f \u2208 t \u2192 Monic f\n\u271d : Nontrivial R\nthis : prod (Multiset.map (fun f => leadingCoeff f) t) = 1\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n[PROOFSTEP]\nrw [this]\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (f : R[X]), f \u2208 t \u2192 Monic f\n\u271d : Nontrivial R\nthis : prod (Multiset.map (fun f => leadingCoeff f) t) = 1\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (f : R[X]), f \u2208 t \u2192 Monic f\n\u271d : Nontrivial R\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) t) = 1\n[PROOFSTEP]\nconvert prod_replicate (Multiset.card t) (1 : R)\n[GOAL]\ncase h.e'_2.h.e'_3\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (f : R[X]), f \u2208 t \u2192 Monic f\n\u271d : Nontrivial R\n\u22a2 Multiset.map (fun f => leadingCoeff f) t = replicate (\u2191Multiset.card t) 1\n[PROOFSTEP]\nsimp only [eq_replicate, Multiset.card_map, eq_self_iff_true, true_and_iff]\n[GOAL]\ncase h.e'_2.h.e'_3\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (f : R[X]), f \u2208 t \u2192 Monic f\n\u271d : Nontrivial R\n\u22a2 \u2200 (b : R), b \u2208 Multiset.map (fun f => leadingCoeff f) t \u2192 b = 1\n[PROOFSTEP]\nrintro i hi\n[GOAL]\ncase h.e'_2.h.e'_3\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (f : R[X]), f \u2208 t \u2192 Monic f\n\u271d : Nontrivial R\ni : R\nhi : i \u2208 Multiset.map (fun f => leadingCoeff f) t\n\u22a2 i = 1\n[PROOFSTEP]\nobtain \u27e8i, hi, rfl\u27e9 := Multiset.mem_map.mp hi\n[GOAL]\ncase h.e'_2.h.e'_3.intro.intro\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (f : R[X]), f \u2208 t \u2192 Monic f\n\u271d : Nontrivial R\ni : R[X]\nhi\u271d : i \u2208 t\nhi : leadingCoeff i \u2208 Multiset.map (fun f => leadingCoeff f) t\n\u22a2 leadingCoeff i = 1\n[PROOFSTEP]\napply h\n[GOAL]\ncase h.e'_2.h.e'_3.intro.intro.a\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (f : R[X]), f \u2208 t \u2192 Monic f\n\u271d : Nontrivial R\ni : R[X]\nhi\u271d : i \u2208 t\nhi : leadingCoeff i \u2208 Multiset.map (fun f => leadingCoeff f) t\n\u22a2 i \u2208 t\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h.e'_3\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (f : R[X]), f \u2208 t \u2192 Monic f\n\u271d : Nontrivial R\n\u22a2 1 = 1 ^ \u2191Multiset.card t\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 Monic (f i)\n\u22a2 natDegree (\u220f i in s, f i) = \u2211 i in s, natDegree (f i)\n[PROOFSTEP]\nsimpa using natDegree_multiset_prod_of_monic (s.1.map f) (by simpa using h)\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 Monic (f i)\n\u22a2 \u2200 (f_1 : R[X]), f_1 \u2208 Multiset.map f s.val \u2192 Monic f_1\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nn : \u2115\nhl : \u2200 (p : R[X]), p \u2208 t \u2192 natDegree p \u2264 n\n\u22a2 coeff (prod t) (\u2191Multiset.card t * n) = prod (Multiset.map (fun p => coeff p n) t)\n[PROOFSTEP]\ninduction t using Quotient.inductionOn\n[GOAL]\ncase h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nn : \u2115\na\u271d : List R[X]\nhl : \u2200 (p : R[X]), p \u2208 Quotient.mk (List.isSetoid R[X]) a\u271d \u2192 natDegree p \u2264 n\n\u22a2 coeff (prod (Quotient.mk (List.isSetoid R[X]) a\u271d)) (\u2191Multiset.card (Quotient.mk (List.isSetoid R[X]) a\u271d) * n) =\n prod (Multiset.map (fun p => coeff p n) (Quotient.mk (List.isSetoid R[X]) a\u271d))\n[PROOFSTEP]\nsimpa using coeff_list_prod_of_natDegree_le _ _ hl\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf\u271d : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nf : \u03b9 \u2192 R[X]\nn : \u2115\nh : \u2200 (p : \u03b9), p \u2208 s \u2192 natDegree (f p) \u2264 n\n\u22a2 coeff (\u220f i in s, f i) (Finset.card s * n) = \u220f i in s, coeff (f i) n\n[PROOFSTEP]\ncases' s with l hl\n[GOAL]\ncase mk\nR : Type u\n\u03b9 : Type w\ninst\u271d : CommSemiring R\nf\u271d : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nf : \u03b9 \u2192 R[X]\nn : \u2115\nl : Multiset \u03b9\nhl : Nodup l\nh : \u2200 (p : \u03b9), p \u2208 { val := l, nodup := hl } \u2192 natDegree (f p) \u2264 n\n\u22a2 coeff (\u220f i in { val := l, nodup := hl }, f i) (Finset.card { val := l, nodup := hl } * n) =\n \u220f i in { val := l, nodup := hl }, coeff (f i) n\n[PROOFSTEP]\nconvert coeff_multiset_prod_of_natDegree_le (l.map f) n ?_\n[GOAL]\ncase h.e'_2.h.e'_4.h.e'_5\nR : Type u\n\u03b9 : Type w\ninst\u271d : CommSemiring R\nf\u271d : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nf : \u03b9 \u2192 R[X]\nn : \u2115\nl : Multiset \u03b9\nhl : Nodup l\nh : \u2200 (p : \u03b9), p \u2208 { val := l, nodup := hl } \u2192 natDegree (f p) \u2264 n\n\u22a2 Finset.card { val := l, nodup := hl } = \u2191Multiset.card (Multiset.map f l)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nR : Type u\n\u03b9 : Type w\ninst\u271d : CommSemiring R\nf\u271d : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nf : \u03b9 \u2192 R[X]\nn : \u2115\nl : Multiset \u03b9\nhl : Nodup l\nh : \u2200 (p : \u03b9), p \u2208 { val := l, nodup := hl } \u2192 natDegree (f p) \u2264 n\n\u22a2 \u220f i in { val := l, nodup := hl }, coeff (f i) n = prod (Multiset.map (fun p => coeff p n) (Multiset.map f l))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk\nR : Type u\n\u03b9 : Type w\ninst\u271d : CommSemiring R\nf\u271d : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nf : \u03b9 \u2192 R[X]\nn : \u2115\nl : Multiset \u03b9\nhl : Nodup l\nh : \u2200 (p : \u03b9), p \u2208 { val := l, nodup := hl } \u2192 natDegree (f p) \u2264 n\n\u22a2 \u2200 (p : R[X]), p \u2208 Multiset.map f l \u2192 natDegree p \u2264 n\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 coeff (prod t) 0 = prod (Multiset.map (fun f => coeff f 0) t)\n[PROOFSTEP]\nrefine' Multiset.induction_on t _ fun a t ht => _\n[GOAL]\ncase refine'_1\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 coeff (prod 0) 0 = prod (Multiset.map (fun f => coeff f 0) 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nht : coeff (prod t) 0 = prod (Multiset.map (fun f => coeff f 0) t)\n\u22a2 coeff (prod (a ::\u2098 t)) 0 = prod (Multiset.map (fun f => coeff f 0) (a ::\u2098 t))\n[PROOFSTEP]\nrw [Multiset.prod_cons, Multiset.map_cons, Multiset.prod_cons, Polynomial.mul_coeff_zero, ht]\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 coeff (\u220f i in s, f i) 0 = \u220f i in s, coeff (f i) 0\n[PROOFSTEP]\nsimpa using coeff_zero_multiset_prod (s.1.map f)\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\n\u22a2 nextCoeff (prod (Multiset.map (fun x => X - \u2191C x) t)) = -Multiset.sum t\n[PROOFSTEP]\nrw [nextCoeff_multiset_prod]\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\n\u22a2 Multiset.sum (Multiset.map (fun i => nextCoeff (X - \u2191C i)) t) = -Multiset.sum t\n[PROOFSTEP]\nsimp only [nextCoeff_X_sub_C]\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\n\u22a2 Multiset.sum (Multiset.map (fun x => -x) t) = -Multiset.sum t\n[PROOFSTEP]\nexact t.sum_hom (-AddMonoidHom.id R)\n[GOAL]\ncase h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\n\u22a2 \u2200 (i : R), i \u2208 t \u2192 Monic (X - \u2191C i)\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\ni\u271d : R\na\u271d : i\u271d \u2208 t\n\u22a2 Monic (X - \u2191C i\u271d)\n[PROOFSTEP]\napply monic_X_sub_C\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns\u271d : Finset \u03b9\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 R\n\u22a2 nextCoeff (\u220f i in s, (X - \u2191C (f i))) = -\u2211 i in s, f i\n[PROOFSTEP]\nsimpa using multiset_prod_X_sub_C_nextCoeff (s.1.map f)\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u22a2 coeff (prod (Multiset.map (fun x => X - \u2191C x) t)) (\u2191Multiset.card t - 1) = -Multiset.sum t\n[PROOFSTEP]\nnontriviality R\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\n\u22a2 coeff (prod (Multiset.map (fun x => X - \u2191C x) t)) (\u2191Multiset.card t - 1) = -Multiset.sum t\n[PROOFSTEP]\nconvert multiset_prod_X_sub_C_nextCoeff (by assumption)\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\n\u22a2 Multiset ?m.927417\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h.e'_2\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\n\u22a2 coeff (prod (Multiset.map (fun x => X - \u2191C x) t)) (\u2191Multiset.card t - 1) =\n nextCoeff (prod (Multiset.map (fun x => X - \u2191C x) t))\n[PROOFSTEP]\nrw [nextCoeff]\n[GOAL]\ncase h.e'_2\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\n\u22a2 coeff (prod (Multiset.map (fun x => X - \u2191C x) t)) (\u2191Multiset.card t - 1) =\n if natDegree (prod (Multiset.map (fun x => X - \u2191C x) t)) = 0 then 0\n else coeff (prod (Multiset.map (fun x => X - \u2191C x) t)) (natDegree (prod (Multiset.map (fun x => X - \u2191C x) t)) - 1)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\nh : natDegree (prod (Multiset.map (fun x => X - \u2191C x) t)) = 0\n\u22a2 coeff (prod (Multiset.map (fun x => X - \u2191C x) t)) (\u2191Multiset.card t - 1) = 0\n[PROOFSTEP]\nrw [natDegree_multiset_prod_of_monic] at h \n[GOAL]\ncase pos\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\nh : Multiset.sum (Multiset.map natDegree (Multiset.map (fun x => X - \u2191C x) t)) = 0\n\u22a2 coeff (prod (Multiset.map (fun x => X - \u2191C x) t)) (\u2191Multiset.card t - 1) = 0\n[PROOFSTEP]\nsimp only [Multiset.mem_map] at *\n[GOAL]\ncase pos.h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\nh : natDegree (prod (Multiset.map (fun x => X - \u2191C x) t)) = 0\n\u22a2 \u2200 (f : R[X]), f \u2208 Multiset.map (fun x => X - \u2191C x) t \u2192 Monic f\n[PROOFSTEP]\nsimp only [Multiset.mem_map] at *\n[GOAL]\ncase pos\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\nh : Multiset.sum (Multiset.map (fun x => natDegree x) (Multiset.map (fun x => X - \u2191C x) t)) = 0\n\u22a2 coeff (prod (Multiset.map (fun x => X - \u2191C x) t)) (\u2191Multiset.card t - 1) = 0\ncase pos.h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\nh : natDegree (prod (Multiset.map (fun x => X - \u2191C x) t)) = 0\n\u22a2 \u2200 (f : R[X]), (\u2203 a, a \u2208 t \u2227 X - \u2191C a = f) \u2192 Monic f\n[PROOFSTEP]\nswap\n[GOAL]\ncase pos.h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\nh : natDegree (prod (Multiset.map (fun x => X - \u2191C x) t)) = 0\n\u22a2 \u2200 (f : R[X]), (\u2203 a, a \u2208 t \u2227 X - \u2191C a = f) \u2192 Monic f\n[PROOFSTEP]\nrintro _ \u27e8_, _, rfl\u27e9\n[GOAL]\ncase pos.h.intro.intro\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\nh : natDegree (prod (Multiset.map (fun x => X - \u2191C x) t)) = 0\nw\u271d : R\nleft\u271d : w\u271d \u2208 t\n\u22a2 Monic (X - \u2191C w\u271d)\n[PROOFSTEP]\napply monic_X_sub_C\n[GOAL]\ncase pos\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\nh : Multiset.sum (Multiset.map (fun x => natDegree x) (Multiset.map (fun x => X - \u2191C x) t)) = 0\n\u22a2 coeff (prod (Multiset.map (fun x => X - \u2191C x) t)) (\u2191Multiset.card t - 1) = 0\n[PROOFSTEP]\nsimp_rw [Multiset.sum_eq_zero_iff, Multiset.mem_map] at h \n[GOAL]\ncase pos\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\nh : \u2200 (x : \u2115), (\u2203 a, (\u2203 a_1, a_1 \u2208 t \u2227 X - \u2191C a_1 = a) \u2227 natDegree a = x) \u2192 x = 0\n\u22a2 coeff (prod (Multiset.map (fun x => X - \u2191C x) t)) (\u2191Multiset.card t - 1) = 0\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase pos\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\nh : coeff (prod (Multiset.map (fun x => X - \u2191C x) t)) (\u2191Multiset.card t - 1) \u2260 0\n\u22a2 \u2203 x, (\u2203 a, (\u2203 a_1, a_1 \u2208 t \u2227 X - \u2191C a_1 = a) \u2227 natDegree a = x) \u2227 x \u2260 0\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := card_pos_iff_exists_mem.mp ht\n[GOAL]\ncase pos.intro\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\nh : coeff (prod (Multiset.map (fun x => X - \u2191C x) t)) (\u2191Multiset.card t - 1) \u2260 0\nx : R\nhx : x \u2208 t\n\u22a2 \u2203 x, (\u2203 a, (\u2203 a_1, a_1 \u2208 t \u2227 X - \u2191C a_1 = a) \u2227 natDegree a = x) \u2227 x \u2260 0\n[PROOFSTEP]\nexact \u27e8_, \u27e8_, \u27e8x, hx, rfl\u27e9, natDegree_X_sub_C _\u27e9, one_ne_zero\u27e9\n[GOAL]\ncase neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\nh : \u00acnatDegree (prod (Multiset.map (fun x => X - \u2191C x) t)) = 0\n\u22a2 coeff (prod (Multiset.map (fun x => X - \u2191C x) t)) (\u2191Multiset.card t - 1) =\n coeff (prod (Multiset.map (fun x => X - \u2191C x) t)) (natDegree (prod (Multiset.map (fun x => X - \u2191C x) t)) - 1)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase neg.e_a.e_a\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\nh : \u00acnatDegree (prod (Multiset.map (fun x => X - \u2191C x) t)) = 0\n\u22a2 \u2191Multiset.card t = natDegree (prod (Multiset.map (fun x => X - \u2191C x) t))\n[PROOFSTEP]\nrw [natDegree_multiset_prod_of_monic]\n[GOAL]\ncase neg.e_a.e_a\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\nh : \u00acnatDegree (prod (Multiset.map (fun x => X - \u2191C x) t)) = 0\n\u22a2 \u2191Multiset.card t = Multiset.sum (Multiset.map natDegree (Multiset.map (fun x => X - \u2191C x) t))\n[PROOFSTEP]\nsimp [natDegree_X_sub_C, monic_X_sub_C]\n[GOAL]\ncase neg.e_a.e_a.h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommRing R\nt : Multiset R\nht : 0 < \u2191Multiset.card t\n\u271d : Nontrivial R\nh : \u00acnatDegree (prod (Multiset.map (fun x => X - \u2191C x) t)) = 0\n\u22a2 \u2200 (f : R[X]), f \u2208 Multiset.map (fun x => X - \u2191C x) t \u2192 Monic f\n[PROOFSTEP]\nsimp [natDegree_X_sub_C, monic_X_sub_C]\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns\u271d : Finset \u03b9\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 R\nhs : 0 < Finset.card s\n\u22a2 coeff (\u220f i in s, (X - \u2191C (f i))) (Finset.card s - 1) = -\u2211 i in s, f i\n[PROOFSTEP]\nsimpa using multiset_prod_X_sub_C_coeff_card_pred (s.1.map f) (by simpa using hs)\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns\u271d : Finset \u03b9\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 R\nhs : 0 < Finset.card s\n\u22a2 0 < \u2191Multiset.card (Multiset.map f s.val)\n[PROOFSTEP]\nsimpa using hs\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 0\n\u22a2 natDegree (\u220f i in s, f i) = \u2211 i in s, natDegree (f i)\n[PROOFSTEP]\nnontriviality R\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 0\n\u271d : Nontrivial R\n\u22a2 natDegree (\u220f i in s, f i) = \u2211 i in s, natDegree (f i)\n[PROOFSTEP]\napply natDegree_prod'\n[GOAL]\ncase h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 0\n\u271d : Nontrivial R\n\u22a2 \u220f i in s, leadingCoeff (f i) \u2260 0\n[PROOFSTEP]\nrw [prod_ne_zero_iff]\n[GOAL]\ncase h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 0\n\u271d : Nontrivial R\n\u22a2 \u2200 (a : \u03b9), a \u2208 s \u2192 leadingCoeff (f a) \u2260 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 0\n\u271d : Nontrivial R\nx : \u03b9\nhx : x \u2208 s\n\u22a2 leadingCoeff (f x) \u2260 0\n[PROOFSTEP]\nsimp [h x hx]\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u00ac0 \u2208 t\n\u22a2 natDegree (prod t) = Multiset.sum (Multiset.map natDegree t)\n[PROOFSTEP]\nnontriviality R\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u00ac0 \u2208 t\n\u271d : Nontrivial R\n\u22a2 natDegree (prod t) = Multiset.sum (Multiset.map natDegree t)\n[PROOFSTEP]\nrw [natDegree_multiset_prod']\n[GOAL]\ncase h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u00ac0 \u2208 t\n\u271d : Nontrivial R\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n[PROOFSTEP]\nsimp_rw [Ne.def, Multiset.prod_eq_zero_iff, Multiset.mem_map, leadingCoeff_eq_zero]\n[GOAL]\ncase h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u00ac0 \u2208 t\n\u271d : Nontrivial R\n\u22a2 \u00ac\u2203 a, a \u2208 t \u2227 a = 0\n[PROOFSTEP]\nrintro \u27e8_, h, rfl\u27e9\n[GOAL]\ncase h.intro.intro\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh\u271d : \u00ac0 \u2208 t\n\u271d : Nontrivial R\nh : 0 \u2208 t\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 leadingCoeff (prod t) = prod (Multiset.map (fun f => leadingCoeff f) t)\n[PROOFSTEP]\nrw [\u2190 leadingCoeffHom_apply, MonoidHom.map_multiset_prod]\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 prod (Multiset.map (\u2191leadingCoeffHom) t) = prod (Multiset.map (fun f => leadingCoeff f) t)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 leadingCoeff (\u220f i in s, f i) = \u220f i in s, leadingCoeff (f i)\n[PROOFSTEP]\nsimpa using leadingCoeff_multiset_prod (s.1.map f)\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Polynomial.BigOperators", "llama_tokens": 22204, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.4005031084552774}} {"text": "[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nx : R\nhx : x \u2208 jacobson I\ny : R\nhxy : I \u2294 span {y * x + 1} = \u22a4\np : R\nhpi : p \u2208 I\nq : R\nhq : q \u2208 span {y * x + 1}\nhpq : p + q = 1\nr : R\nhr : r * (y * x + 1) = q\n\u22a2 r * y * x + r - 1 \u2208 I\n[PROOFSTEP]\nrw [mul_assoc, \u2190 mul_add_one r (y * x), hr, \u2190 hpq, \u2190 neg_sub, add_sub_cancel]\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nx : R\nhx : x \u2208 jacobson I\ny : R\nhxy : I \u2294 span {y * x + 1} = \u22a4\np : R\nhpi : p \u2208 I\nq : R\nhq : q \u2208 span {y * x + 1}\nhpq : p + q = 1\nr : R\nhr : r * (y * x + 1) = q\n\u22a2 -p \u2208 I\n[PROOFSTEP]\nexact I.neg_mem hpi\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nx : R\nhx : \u2200 (y : R), \u2203 z, z * y * x + z - 1 \u2208 I\nM : Ideal R\nx\u271d : M \u2208 {J | I \u2264 J \u2227 IsMaximal J}\nhim : I \u2264 M\nhm : IsMaximal M\nhxm : \u00acx \u2208 M\ny i : R\nhi : i \u2208 M\ndf : y * x + i = 1\nz : R\nhz : z * -y * x + z - 1 \u2208 I\n\u22a2 z * -y * x + z \u2208 M\n[PROOFSTEP]\nrw [mul_assoc, \u2190 mul_add_one z, neg_mul, \u2190 sub_eq_iff_eq_add.mpr df.symm, neg_sub, sub_add_cancel]\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nx : R\nhx : \u2200 (y : R), \u2203 z, z * y * x + z - 1 \u2208 I\nM : Ideal R\nx\u271d : M \u2208 {J | I \u2264 J \u2227 IsMaximal J}\nhim : I \u2264 M\nhm : IsMaximal M\nhxm : \u00acx \u2208 M\ny i : R\nhi : i \u2208 M\ndf : y * x + i = 1\nz : R\nhz : z * -y * x + z - 1 \u2208 I\n\u22a2 z * i \u2208 M\n[PROOFSTEP]\nexact M.mul_mem_left _ hi\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I : Ideal R\nr : R\nh : r - 1 \u2208 jacobson I\n\u22a2 \u2203 s, s * r - 1 \u2208 I\n[PROOFSTEP]\ncases' mem_jacobson_iff.1 h 1 with s hs\n[GOAL]\ncase intro\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I : Ideal R\nr : R\nh : r - 1 \u2208 jacobson I\ns : R\nhs : s * 1 * (r - 1) + s - 1 \u2208 I\n\u22a2 \u2203 s, s * r - 1 \u2208 I\n[PROOFSTEP]\nuse s\n[GOAL]\ncase h\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I : Ideal R\nr : R\nh : r - 1 \u2208 jacobson I\ns : R\nhs : s * 1 * (r - 1) + s - 1 \u2208 I\n\u22a2 s * r - 1 \u2208 I\n[PROOFSTEP]\nsimpa [mul_sub] using hs\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\n\u22a2 jacobson I = I \u2194 \u2203 M, (\u2200 (J : Ideal R), J \u2208 M \u2192 IsMaximal J \u2228 J = \u22a4) \u2227 I = sInf M\n[PROOFSTEP]\nuse fun hI => \u27e8{J : Ideal R | I \u2264 J \u2227 J.IsMaximal}, \u27e8fun _ hJ => Or.inl hJ.right, hI.symm\u27e9\u27e9\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\n\u22a2 (\u2203 M, (\u2200 (J : Ideal R), J \u2208 M \u2192 IsMaximal J \u2228 J = \u22a4) \u2227 I = sInf M) \u2192 jacobson I = I\n[PROOFSTEP]\nrintro \u27e8M, hM, hInf\u27e9\n[GOAL]\ncase mpr.intro.intro\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nM : Set (Ideal R)\nhM : \u2200 (J : Ideal R), J \u2208 M \u2192 IsMaximal J \u2228 J = \u22a4\nhInf : I = sInf M\n\u22a2 jacobson I = I\n[PROOFSTEP]\nrefine le_antisymm (fun x hx => ?_) le_jacobson\n[GOAL]\ncase mpr.intro.intro\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nM : Set (Ideal R)\nhM : \u2200 (J : Ideal R), J \u2208 M \u2192 IsMaximal J \u2228 J = \u22a4\nhInf : I = sInf M\nx : R\nhx : x \u2208 jacobson I\n\u22a2 x \u2208 I\n[PROOFSTEP]\nrw [hInf, mem_sInf]\n[GOAL]\ncase mpr.intro.intro\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nM : Set (Ideal R)\nhM : \u2200 (J : Ideal R), J \u2208 M \u2192 IsMaximal J \u2228 J = \u22a4\nhInf : I = sInf M\nx : R\nhx : x \u2208 jacobson I\n\u22a2 \u2200 \u2983I : Ideal R\u2984, I \u2208 M \u2192 x \u2208 I\n[PROOFSTEP]\nintro I hI\n[GOAL]\ncase mpr.intro.intro\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d : Ideal R\nM : Set (Ideal R)\nhM : \u2200 (J : Ideal R), J \u2208 M \u2192 IsMaximal J \u2228 J = \u22a4\nhInf : I\u271d = sInf M\nx : R\nhx : x \u2208 jacobson I\u271d\nI : Ideal R\nhI : I \u2208 M\n\u22a2 x \u2208 I\n[PROOFSTEP]\ncases' hM I hI with is_max is_top\n[GOAL]\ncase mpr.intro.intro.inl\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d : Ideal R\nM : Set (Ideal R)\nhM : \u2200 (J : Ideal R), J \u2208 M \u2192 IsMaximal J \u2228 J = \u22a4\nhInf : I\u271d = sInf M\nx : R\nhx : x \u2208 jacobson I\u271d\nI : Ideal R\nhI : I \u2208 M\nis_max : IsMaximal I\n\u22a2 x \u2208 I\n[PROOFSTEP]\nexact (mem_sInf.1 hx) \u27e8le_sInf_iff.1 (le_of_eq hInf) I hI, is_max\u27e9\n[GOAL]\ncase mpr.intro.intro.inr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d : Ideal R\nM : Set (Ideal R)\nhM : \u2200 (J : Ideal R), J \u2208 M \u2192 IsMaximal J \u2228 J = \u22a4\nhInf : I\u271d = sInf M\nx : R\nhx : x \u2208 jacobson I\u271d\nI : Ideal R\nhI : I \u2208 M\nis_top : I = \u22a4\n\u22a2 x \u2208 I\n[PROOFSTEP]\nexact is_top.symm \u25b8 Submodule.mem_top\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\n\u22a2 jacobson I = I \u2194 \u2200 (x : R), \u00acx \u2208 I \u2192 \u2203 M, (I \u2264 M \u2227 IsMaximal M) \u2227 \u00acx \u2208 M\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\n\u22a2 jacobson I = I \u2192 \u2200 (x : R), \u00acx \u2208 I \u2192 \u2203 M, (I \u2264 M \u2227 IsMaximal M) \u2227 \u00acx \u2208 M\n[PROOFSTEP]\nintro h x hx\n[GOAL]\ncase mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : jacobson I = I\nx : R\nhx : \u00acx \u2208 I\n\u22a2 \u2203 M, (I \u2264 M \u2227 IsMaximal M) \u2227 \u00acx \u2208 M\n[PROOFSTEP]\nerw [\u2190 h, mem_sInf] at hx \n[GOAL]\ncase mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : jacobson I = I\nx : R\nhx : \u00ac\u2200 \u2983I_1 : Ideal R\u2984, I_1 \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 x \u2208 I_1\n\u22a2 \u2203 M, (I \u2264 M \u2227 IsMaximal M) \u2227 \u00acx \u2208 M\n[PROOFSTEP]\npush_neg at hx \n[GOAL]\ncase mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : jacobson I = I\nx : R\nhx : Exists fun \u2983I_1\u2984 => I_1 \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2227 \u00acx \u2208 I_1\n\u22a2 \u2203 M, (I \u2264 M \u2227 IsMaximal M) \u2227 \u00acx \u2208 M\n[PROOFSTEP]\nexact hx\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\n\u22a2 (\u2200 (x : R), \u00acx \u2208 I \u2192 \u2203 M, (I \u2264 M \u2227 IsMaximal M) \u2227 \u00acx \u2208 M) \u2192 jacobson I = I\n[PROOFSTEP]\nrefine fun h => le_antisymm (fun x hx => ?_) le_jacobson\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : \u2200 (x : R), \u00acx \u2208 I \u2192 \u2203 M, (I \u2264 M \u2227 IsMaximal M) \u2227 \u00acx \u2208 M\nx : R\nhx : x \u2208 jacobson I\n\u22a2 x \u2208 I\n[PROOFSTEP]\ncontrapose hx\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : \u2200 (x : R), \u00acx \u2208 I \u2192 \u2203 M, (I \u2264 M \u2227 IsMaximal M) \u2227 \u00acx \u2208 M\nx : R\nhx : \u00acx \u2208 I\n\u22a2 \u00acx \u2208 jacobson I\n[PROOFSTEP]\nerw [mem_sInf]\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : \u2200 (x : R), \u00acx \u2208 I \u2192 \u2203 M, (I \u2264 M \u2227 IsMaximal M) \u2227 \u00acx \u2208 M\nx : R\nhx : \u00acx \u2208 I\n\u22a2 \u00ac\u2200 \u2983I_1 : Ideal R\u2984, I_1 \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 x \u2208 I_1\n[PROOFSTEP]\npush_neg\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : \u2200 (x : R), \u00acx \u2208 I \u2192 \u2203 M, (I \u2264 M \u2227 IsMaximal M) \u2227 \u00acx \u2208 M\nx : R\nhx : \u00acx \u2208 I\n\u22a2 Exists fun \u2983I_1\u2984 => I_1 \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2227 \u00acx \u2208 I_1\n[PROOFSTEP]\nexact h x hx\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\n\u22a2 RingHom.ker f \u2264 I \u2192 map f (jacobson I) = jacobson (map f I)\n[PROOFSTEP]\nintro h\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : RingHom.ker f \u2264 I\n\u22a2 map f (jacobson I) = jacobson (map f I)\n[PROOFSTEP]\nunfold Ideal.jacobson\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : RingHom.ker f \u2264 I\n\u22a2 map f (sInf {J | I \u2264 J \u2227 IsMaximal J}) = sInf {J | map f I \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\nhave : \u2200 J \u2208 {J : Ideal R | I \u2264 J \u2227 J.IsMaximal}, RingHom.ker f \u2264 J := fun J hJ => le_trans h hJ.left\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : RingHom.ker f \u2264 I\nthis : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J\n\u22a2 map f (sInf {J | I \u2264 J \u2227 IsMaximal J}) = sInf {J | map f I \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\nrefine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)\n[GOAL]\ncase refine_1\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : RingHom.ker f \u2264 I\nthis : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J\n\u22a2 sInf (map f '' {J | I \u2264 J \u2227 IsMaximal J}) \u2264 sInf {J | map f I \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\nrefine' sInf_le_sInf fun J hJ => \u27e8comap f J, \u27e8\u27e8le_comap_of_map_le hJ.1, _\u27e9, map_comap_of_surjective f hf J\u27e9\u27e9\n[GOAL]\ncase refine_1\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : RingHom.ker f \u2264 I\nthis : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J\nJ : Ideal S\nhJ : J \u2208 {J | map f I \u2264 J \u2227 IsMaximal J}\n\u22a2 IsMaximal (comap f J)\n[PROOFSTEP]\nhaveI : J.IsMaximal := hJ.right\n[GOAL]\ncase refine_1\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : RingHom.ker f \u2264 I\nthis\u271d : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J\nJ : Ideal S\nhJ : J \u2208 {J | map f I \u2264 J \u2227 IsMaximal J}\nthis : IsMaximal J\n\u22a2 IsMaximal (comap f J)\n[PROOFSTEP]\nexact comap_isMaximal_of_surjective f hf\n[GOAL]\ncase refine_2\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : RingHom.ker f \u2264 I\nthis : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J\n\u22a2 sInf {J | map f I \u2264 J \u2227 IsMaximal J} \u2264 sInf (map f '' {J | I \u2264 J \u2227 IsMaximal J})\n[PROOFSTEP]\nrefine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _\n[GOAL]\ncase refine_2\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : RingHom.ker f \u2264 I\nthis : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J\nj : Ideal S\nhj : j \u2208 map f '' {J | I \u2264 J \u2227 IsMaximal J}\nJ : Ideal R\nhJ : J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2227 map f J = j\n\u22a2 j \u2208 insert \u22a4 {J | map f I \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\nrw [\u2190 hJ.2]\n[GOAL]\ncase refine_2\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : RingHom.ker f \u2264 I\nthis : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J\nj : Ideal S\nhj : j \u2208 map f '' {J | I \u2264 J \u2227 IsMaximal J}\nJ : Ideal R\nhJ : J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2227 map f J = j\n\u22a2 map f J \u2208 insert \u22a4 {J | map f I \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\ncases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax\n[GOAL]\ncase refine_2.inl\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : RingHom.ker f \u2264 I\nthis : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J\nj : Ideal S\nhj : j \u2208 map f '' {J | I \u2264 J \u2227 IsMaximal J}\nJ : Ideal R\nhJ : J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2227 map f J = j\nhtop : map f J = \u22a4\n\u22a2 map f J \u2208 insert \u22a4 {J | map f I \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\nexact htop.symm \u25b8 Set.mem_insert \u22a4 _\n[GOAL]\ncase refine_2.inr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nh : RingHom.ker f \u2264 I\nthis : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J\nj : Ideal S\nhj : j \u2208 map f '' {J | I \u2264 J \u2227 IsMaximal J}\nJ : Ideal R\nhJ : J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2227 map f J = j\nhmax : IsMaximal (map f J)\n\u22a2 map f J \u2208 insert \u22a4 {J | map f I \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\nexact Set.mem_insert_of_mem \u22a4 \u27e8map_mono hJ.1.1, hmax\u27e9\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nK : Ideal S\n\u22a2 comap f (jacobson K) = jacobson (comap f K)\n[PROOFSTEP]\nunfold Ideal.jacobson\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nK : Ideal S\n\u22a2 comap f (sInf {J | K \u2264 J \u2227 IsMaximal J}) = sInf {J | comap f K \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nK : Ideal S\n\u22a2 comap f (sInf {J | K \u2264 J \u2227 IsMaximal J}) \u2264 sInf {J | comap f K \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\nrefine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_\n[GOAL]\ncase refine'_1\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nK : Ideal S\n\u22a2 comap f (sInf (insert \u22a4 {J | K \u2264 J \u2227 IsMaximal J})) \u2264 sInf {J | comap f K \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\nrw [comap_sInf', sInf_eq_iInf]\n[GOAL]\ncase refine'_1\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nK : Ideal S\n\u22a2 \u2a05 (I : Ideal R) (_ : I \u2208 comap f '' insert \u22a4 {J | K \u2264 J \u2227 IsMaximal J}), I \u2264\n \u2a05 (a : Ideal R) (_ : a \u2208 {J | comap f K \u2264 J \u2227 IsMaximal J}), a\n[PROOFSTEP]\nrefine' iInf_le_iInf_of_subset fun J hJ => _\n[GOAL]\ncase refine'_1\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nK : Ideal S\nJ : Ideal R\nhJ : J \u2208 {J | comap f K \u2264 J \u2227 IsMaximal J}\n\u22a2 J \u2208 comap f '' insert \u22a4 {J | K \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\nhave : comap f (map f J) = J :=\n Trans.trans (comap_map_of_surjective f hf J)\n (le_antisymm (sup_le_iff.2 \u27e8le_of_eq rfl, le_trans (comap_mono bot_le) hJ.left\u27e9) le_sup_left)\n[GOAL]\ncase refine'_1\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nK : Ideal S\nJ : Ideal R\nhJ : J \u2208 {J | comap f K \u2264 J \u2227 IsMaximal J}\nthis : comap f (map f J) = J\n\u22a2 J \u2208 comap f '' insert \u22a4 {J | K \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\ncases' map_eq_top_or_isMaximal_of_surjective _ hf hJ.right with htop hmax\n[GOAL]\ncase refine'_1.inl\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nK : Ideal S\nJ : Ideal R\nhJ : J \u2208 {J | comap f K \u2264 J \u2227 IsMaximal J}\nthis : comap f (map f J) = J\nhtop : map f J = \u22a4\n\u22a2 J \u2208 comap f '' insert \u22a4 {J | K \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\nexact \u27e8\u22a4, \u27e8Set.mem_insert \u22a4 _, htop \u25b8 this\u27e9\u27e9\n[GOAL]\ncase refine'_1.inr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nK : Ideal S\nJ : Ideal R\nhJ : J \u2208 {J | comap f K \u2264 J \u2227 IsMaximal J}\nthis : comap f (map f J) = J\nhmax : IsMaximal (map f J)\n\u22a2 J \u2208 comap f '' insert \u22a4 {J | K \u2264 J \u2227 IsMaximal J}\n[PROOFSTEP]\nexact \u27e8map f J, \u27e8Set.mem_insert_of_mem _ \u27e8le_map_of_comap_le_of_surjective f hf hJ.1, hmax\u27e9, this\u27e9\u27e9\n[GOAL]\ncase refine'_2\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nK : Ideal S\n\u22a2 sInf {J | comap f K \u2264 J \u2227 IsMaximal J} \u2264 comap f (sInf {J | K \u2264 J \u2227 IsMaximal J})\n[PROOFSTEP]\nrw [comap_sInf]\n[GOAL]\ncase refine'_2\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nK : Ideal S\n\u22a2 sInf {J | comap f K \u2264 J \u2227 IsMaximal J} \u2264 \u2a05 (I : Ideal S) (_ : I \u2208 {J | K \u2264 J \u2227 IsMaximal J}), comap f I\n[PROOFSTEP]\nrefine' le_iInf_iff.2 fun J => le_iInf_iff.2 fun hJ => _\n[GOAL]\ncase refine'_2\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nK J : Ideal S\nhJ : J \u2208 {J | K \u2264 J \u2227 IsMaximal J}\n\u22a2 sInf {J | comap f K \u2264 J \u2227 IsMaximal J} \u2264 comap f J\n[PROOFSTEP]\nhaveI : J.IsMaximal := hJ.right\n[GOAL]\ncase refine'_2\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nK J : Ideal S\nhJ : J \u2208 {J | K \u2264 J \u2227 IsMaximal J}\nthis : IsMaximal J\n\u22a2 sInf {J | comap f K \u2264 J \u2227 IsMaximal J} \u2264 comap f J\n[PROOFSTEP]\nrefine' sInf_le \u27e8comap_mono hJ.left, comap_isMaximal_of_surjective _ hf\u27e9\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I J : Ideal R\n\u22a2 I \u2264 J \u2192 jacobson I \u2264 jacobson J\n[PROOFSTEP]\nintro h x hx\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I J : Ideal R\nh : I \u2264 J\nx : R\nhx : x \u2208 jacobson I\n\u22a2 x \u2208 jacobson J\n[PROOFSTEP]\nerw [mem_sInf] at hx \u22a2\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI\u271d I J : Ideal R\nh : I \u2264 J\nx : R\nhx : \u2200 \u2983I_1 : Ideal R\u2984, I_1 \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 x \u2208 I_1\n\u22a2 \u2200 \u2983I : Ideal R\u2984, I \u2208 {J_1 | J \u2264 J_1 \u2227 IsMaximal J_1} \u2192 x \u2208 I\n[PROOFSTEP]\nexact fun K \u27e8hK, hK_max\u27e9 => hx \u27e8Trans.trans h hK, hK_max\u27e9\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nr : R\nh : r - 1 \u2208 jacobson \u22a5\n\u22a2 IsUnit r\n[PROOFSTEP]\ncases' exists_mul_sub_mem_of_sub_one_mem_jacobson r h with s hs\n[GOAL]\ncase intro\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nr : R\nh : r - 1 \u2208 jacobson \u22a5\ns : R\nhs : s * r - 1 \u2208 \u22a5\n\u22a2 IsUnit r\n[PROOFSTEP]\nrw [mem_bot, sub_eq_zero, mul_comm] at hs \n[GOAL]\ncase intro\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nr : R\nh : r - 1 \u2208 jacobson \u22a5\ns : R\nhs : r * s = 1\n\u22a2 IsUnit r\n[PROOFSTEP]\nexact isUnit_of_mul_eq_one _ _ hs\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nx : R\nhx : x \u2208 jacobson \u22a5\ny z : R\nhz : z * y * x + z - 1 \u2208 \u22a5\n\u22a2 (x * y + 1) * z = 1\n[PROOFSTEP]\nrwa [add_mul, one_mul, \u2190 sub_eq_zero, mul_right_comm, mul_comm _ z, mul_right_comm]\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nx : R\nh : \u2200 (y : R), IsUnit (x * y + 1)\ny b : R\nhb : (x * y + 1) * b = 1\n\u22a2 b * y * x + b - (x * y + 1) * b = 0\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\n\u22a2 jacobson I = I \u2194 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nhave hf : Function.Surjective (Ideal.Quotient.mk I) := Submodule.Quotient.mk_surjective I\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\n\u22a2 jacobson I = I \u2194 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\n\u22a2 jacobson I = I \u2192 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\nh : jacobson I = I\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nreplace h := congr_arg (Ideal.map (Ideal.Quotient.mk I)) h\n[GOAL]\ncase mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\nh : map (Quotient.mk I) (jacobson I) = map (Quotient.mk I) I\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nrw [map_jacobson_of_surjective hf (le_of_eq mk_ker)] at h \n[GOAL]\ncase mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\nh : jacobson (map (Quotient.mk I) I) = map (Quotient.mk I) I\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\n\u22a2 jacobson \u22a5 = \u22a5 \u2192 jacobson I = I\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\nh : jacobson \u22a5 = \u22a5\n\u22a2 jacobson I = I\n[PROOFSTEP]\nreplace h := congr_arg (comap (Ideal.Quotient.mk I)) h\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\nh : comap (Quotient.mk I) (jacobson \u22a5) = comap (Quotient.mk I) \u22a5\n\u22a2 jacobson I = I\n[PROOFSTEP]\nrw [comap_jacobson_of_surjective hf, \u2190 RingHom.ker_eq_comap_bot (Ideal.Quotient.mk I)] at h \n[GOAL]\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\nh : jacobson (RingHom.ker (Quotient.mk I)) = RingHom.ker (Quotient.mk I)\n\u22a2 jacobson I = I\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\n\u22a2 radical I = jacobson I \u2194 radical \u22a5 = jacobson \u22a5\n[PROOFSTEP]\nhave hf : Function.Surjective (Ideal.Quotient.mk I) := Submodule.Quotient.mk_surjective I\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\n\u22a2 radical I = jacobson I \u2194 radical \u22a5 = jacobson \u22a5\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\n\u22a2 radical I = jacobson I \u2192 radical \u22a5 = jacobson \u22a5\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\nh : radical I = jacobson I\n\u22a2 radical \u22a5 = jacobson \u22a5\n[PROOFSTEP]\nhave := congr_arg (map (Ideal.Quotient.mk I)) h\n[GOAL]\ncase mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\nh : radical I = jacobson I\nthis : map (Quotient.mk I) (radical I) = map (Quotient.mk I) (jacobson I)\n\u22a2 radical \u22a5 = jacobson \u22a5\n[PROOFSTEP]\nrw [map_radical_of_surjective hf (le_of_eq mk_ker), map_jacobson_of_surjective hf (le_of_eq mk_ker)] at this \n[GOAL]\ncase mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\nh : radical I = jacobson I\nthis : radical (map (Quotient.mk I) I) = jacobson (map (Quotient.mk I) I)\n\u22a2 radical \u22a5 = jacobson \u22a5\n[PROOFSTEP]\nsimpa using this\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\n\u22a2 radical \u22a5 = jacobson \u22a5 \u2192 radical I = jacobson I\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\nh : radical \u22a5 = jacobson \u22a5\n\u22a2 radical I = jacobson I\n[PROOFSTEP]\nhave := congr_arg (comap (Ideal.Quotient.mk I)) h\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\nh : radical \u22a5 = jacobson \u22a5\nthis : comap (Quotient.mk I) (radical \u22a5) = comap (Quotient.mk I) (jacobson \u22a5)\n\u22a2 radical I = jacobson I\n[PROOFSTEP]\nrw [comap_radical, comap_jacobson_of_surjective hf, \u2190 RingHom.ker_eq_comap_bot (Ideal.Quotient.mk I)] at this \n[GOAL]\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI : Ideal R\nhf : Function.Surjective \u2191(Quotient.mk I)\nh : radical \u22a5 = jacobson \u22a5\nthis : radical (RingHom.ker (Quotient.mk I)) = jacobson (RingHom.ker (Quotient.mk I))\n\u22a2 radical I = jacobson I\n[PROOFSTEP]\nsimpa using this\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\n\u22a2 jacobson \u22a5 \u2264 sInf (map C '' {J | IsMaximal J})\n[PROOFSTEP]\nrefine' le_sInf fun J => exists_imp.2 fun j hj => _\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\n\u22a2 jacobson \u22a5 \u2264 J\n[PROOFSTEP]\nhaveI : j.IsMaximal := hj.1\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\nthis : IsMaximal j\n\u22a2 jacobson \u22a5 \u2264 J\n[PROOFSTEP]\nrefine' Trans.trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson \u2264 J)\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\nthis : IsMaximal j\n\u22a2 jacobson J = J\n[PROOFSTEP]\nsuffices t : (\u22a5 : Ideal (Polynomial (R \u29f8 j))).jacobson = \u22a5\n by\n rw [\u2190 hj.2, jacobson_eq_iff_jacobson_quotient_eq_bot]\n replace t := congr_arg (map (polynomialQuotientEquivQuotientPolynomial j).toRingHom) t\n rwa [map_jacobson_of_bijective _, map_bot] at t \n exact RingEquiv.bijective (polynomialQuotientEquivQuotientPolynomial j)\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\nthis : IsMaximal j\nt : jacobson \u22a5 = \u22a5\n\u22a2 jacobson J = J\n[PROOFSTEP]\nrw [\u2190 hj.2, jacobson_eq_iff_jacobson_quotient_eq_bot]\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\nthis : IsMaximal j\nt : jacobson \u22a5 = \u22a5\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nreplace t := congr_arg (map (polynomialQuotientEquivQuotientPolynomial j).toRingHom) t\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\nthis : IsMaximal j\nt :\n map (RingEquiv.toRingHom (polynomialQuotientEquivQuotientPolynomial j)) (jacobson \u22a5) =\n map (RingEquiv.toRingHom (polynomialQuotientEquivQuotientPolynomial j)) \u22a5\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nrwa [map_jacobson_of_bijective _, map_bot] at t \n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\nthis : IsMaximal j\nt :\n map (RingEquiv.toRingHom (polynomialQuotientEquivQuotientPolynomial j)) (jacobson \u22a5) =\n map (RingEquiv.toRingHom (polynomialQuotientEquivQuotientPolynomial j)) \u22a5\n\u22a2 Function.Bijective \u2191(RingEquiv.toRingHom (polynomialQuotientEquivQuotientPolynomial j))\n[PROOFSTEP]\nexact RingEquiv.bijective (polynomialQuotientEquivQuotientPolynomial j)\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\nthis : IsMaximal j\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nrefine' eq_bot_iff.2 fun f hf => _\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\nthis : IsMaximal j\nf : (R \u29f8 j)[X]\nhf : f \u2208 jacobson \u22a5\n\u22a2 f \u2208 \u22a5\n[PROOFSTEP]\nhave r1 : (X : (R \u29f8 j)[X]) \u2260 0 := fun hX =>\n by\n replace hX := congr_arg (fun f => coeff f 1) hX\n simp only [coeff_X_one, coeff_zero] at hX \n exact zero_ne_one hX.symm\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\nthis : IsMaximal j\nf : (R \u29f8 j)[X]\nhf : f \u2208 jacobson \u22a5\nhX : X = 0\n\u22a2 False\n[PROOFSTEP]\nreplace hX := congr_arg (fun f => coeff f 1) hX\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\nthis : IsMaximal j\nf : (R \u29f8 j)[X]\nhf : f \u2208 jacobson \u22a5\nhX : (fun f => coeff f 1) X = (fun f => coeff f 1) 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [coeff_X_one, coeff_zero] at hX \n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\nthis : IsMaximal j\nf : (R \u29f8 j)[X]\nhf : f \u2208 jacobson \u22a5\nhX : 1 = 0\n\u22a2 False\n[PROOFSTEP]\nexact zero_ne_one hX.symm\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\nthis : IsMaximal j\nf : (R \u29f8 j)[X]\nhf : f \u2208 jacobson \u22a5\nr1 : X \u2260 0\n\u22a2 f \u2208 \u22a5\n[PROOFSTEP]\nhave r2 := eq_C_of_degree_eq_zero (degree_eq_zero_of_isUnit ((mem_jacobson_bot.1 hf) X))\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\nthis : IsMaximal j\nf : (R \u29f8 j)[X]\nhf : f \u2208 jacobson \u22a5\nr1 : X \u2260 0\nr2 : f * X + 1 = \u2191C (coeff (f * X + 1) 0)\n\u22a2 f \u2208 \u22a5\n[PROOFSTEP]\nsimp only [coeff_add, mul_coeff_zero, coeff_X_zero, mul_zero, coeff_one_zero, zero_add] at r2 \n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\nthis : IsMaximal j\nf : (R \u29f8 j)[X]\nhf : f \u2208 jacobson \u22a5\nr1 : X \u2260 0\nr2 : f * X + 1 = \u2191C 1\n\u22a2 f \u2208 \u22a5\n[PROOFSTEP]\nerw [add_left_eq_self] at r2 \n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j \u2208 {J | IsMaximal J} \u2227 map C j = J\nthis : IsMaximal j\nf : (R \u29f8 j)[X]\nhf : f \u2208 jacobson \u22a5\nr1 : X \u2260 0\nr2 : f * X = 0\n\u22a2 f \u2208 \u22a5\n[PROOFSTEP]\nsimpa using (mul_eq_zero.mp r2).resolve_right r1\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nh : jacobson \u22a5 = \u22a5\n\u22a2 jacobson \u22a5 = \u22a5\n[PROOFSTEP]\nrefine' eq_bot_iff.2 (le_trans jacobson_bot_polynomial_le_sInf_map_maximal _)\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nh : jacobson \u22a5 = \u22a5\n\u22a2 sInf (map C '' {J | IsMaximal J}) \u2264 \u22a5\n[PROOFSTEP]\nrefine' fun f hf =>\n (Submodule.mem_bot R[X]).2 <| Polynomial.ext fun n => Trans.trans (?_ : coeff f n = 0) (coeff_zero n).symm\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nh : jacobson \u22a5 = \u22a5\nf : R[X]\nhf : f \u2208 sInf (map C '' {J | IsMaximal J})\nn : \u2115\n\u22a2 coeff f n = 0\n[PROOFSTEP]\nsuffices f.coeff n \u2208 Ideal.jacobson \u22a5 by rwa [h, Submodule.mem_bot] at this \n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nh : jacobson \u22a5 = \u22a5\nf : R[X]\nhf : f \u2208 sInf (map C '' {J | IsMaximal J})\nn : \u2115\nthis : coeff f n \u2208 jacobson \u22a5\n\u22a2 coeff f n = 0\n[PROOFSTEP]\nrwa [h, Submodule.mem_bot] at this \n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nh : jacobson \u22a5 = \u22a5\nf : R[X]\nhf : f \u2208 sInf (map C '' {J | IsMaximal J})\nn : \u2115\n\u22a2 coeff f n \u2208 jacobson \u22a5\n[PROOFSTEP]\nexact mem_sInf.2 fun j hj => (mem_map_C_iff.1 ((mem_sInf.1 hf) \u27e8j, \u27e8hj.2, rfl\u27e9\u27e9)) n\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nI : Ideal R\nhi : IsLocal I\nx : R\nh : I \u2294 span {x} = \u22a4\np : R\nhpi : p \u2208 I\nq : R\nhq : q \u2208 span {x}\nhpq : p + q = 1\nr : R\nhr : q = x * r\n\u22a2 r * x - 1 \u2208 I\n[PROOFSTEP]\nrw [\u2190 hpq, mul_comm, \u2190 hr, \u2190 neg_sub, add_sub_cancel]\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nI : Ideal R\nhi : IsLocal I\nx : R\nh : I \u2294 span {x} = \u22a4\np : R\nhpi : p \u2208 I\nq : R\nhq : q \u2208 span {x}\nhpq : p + q = 1\nr : R\nhr : q = x * r\n\u22a2 -p \u2208 I\n[PROOFSTEP]\nexact I.neg_mem hpi\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nI : Ideal R\nhi : IsMaximal (radical I)\nthis : radical I = jacobson I\nx y : R\nhxy : x * y \u2208 I\nx\u271d : \u2203 y_1, y_1 * y - 1 \u2208 I\nz : R\nhz : z * y - 1 \u2208 I\n\u22a2 x \u2208 I\n[PROOFSTEP]\nrw [\u2190 mul_one x, \u2190 sub_sub_cancel (z * y) 1, mul_sub, mul_left_comm]\n[GOAL]\nR : Type u\nS : Type v\ninst\u271d : CommRing R\nI : Ideal R\nhi : IsMaximal (radical I)\nthis : radical I = jacobson I\nx y : R\nhxy : x * y \u2208 I\nx\u271d : \u2203 y_1, y_1 * y - 1 \u2208 I\nz : R\nhz : z * y - 1 \u2208 I\n\u22a2 z * (x * y) - x * (z * y - 1) \u2208 I\n[PROOFSTEP]\nexact I.sub_mem (I.mul_mem_left _ hxy) (I.mul_mem_left _ hz)\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.JacobsonIdeal", "llama_tokens": 15013, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952052, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.4003386159047977}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 \u2016inclusionInDoubleDual \ud835\udd5c E\u2016 \u2264 1\n[PROOFSTEP]\nrw [inclusionInDoubleDual_norm_eq]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 \u2016ContinuousLinearMap.id \ud835\udd5c (Dual \ud835\udd5c E)\u2016 \u2264 1\n[PROOFSTEP]\nexact ContinuousLinearMap.norm_id_le\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nx : E\n\u22a2 \u2016\u2191(inclusionInDoubleDual \ud835\udd5c E) x\u2016 \u2264 \u2016x\u2016\n[PROOFSTEP]\nsimpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le \ud835\udd5c E) x\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 LinearMap.SeparatingLeft (dualPairing \ud835\udd5c E)\n[PROOFSTEP]\nrw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 Function.Injective \u2191(dualPairing \ud835\udd5c E)\n[PROOFSTEP]\nexact ContinuousLinearMap.coe_injective\n[GOAL]\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016\u2191f x\u2016 \u2264 M * \u2016f\u2016\n\u22a2 \u2016x\u2016 \u2264 M\n[PROOFSTEP]\nclassical\nby_cases h : x = 0\n\u00b7 simp only [h, hMp, norm_zero]\n\u00b7 obtain \u27e8f, hf\u2081, hfx\u27e9 : \u2203 f : E \u2192L[\ud835\udd5c] \ud835\udd5c, \u2016f\u2016 = 1 \u2227 f x = \u2016x\u2016 := exists_dual_vector \ud835\udd5c x h\n calc\n \u2016x\u2016 = \u2016(\u2016x\u2016 : \ud835\udd5c)\u2016 := IsROrC.norm_coe_norm.symm\n _ = \u2016f x\u2016 := by rw [hfx]\n _ \u2264 M * \u2016f\u2016 := (hM f)\n _ = M := by rw [hf\u2081, mul_one]\n[GOAL]\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016\u2191f x\u2016 \u2264 M * \u2016f\u2016\n\u22a2 \u2016x\u2016 \u2264 M\n[PROOFSTEP]\nby_cases h : x = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016\u2191f x\u2016 \u2264 M * \u2016f\u2016\nh : x = 0\n\u22a2 \u2016x\u2016 \u2264 M\n[PROOFSTEP]\nsimp only [h, hMp, norm_zero]\n[GOAL]\ncase neg\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016\u2191f x\u2016 \u2264 M * \u2016f\u2016\nh : \u00acx = 0\n\u22a2 \u2016x\u2016 \u2264 M\n[PROOFSTEP]\nobtain \u27e8f, hf\u2081, hfx\u27e9 : \u2203 f : E \u2192L[\ud835\udd5c] \ud835\udd5c, \u2016f\u2016 = 1 \u2227 f x = \u2016x\u2016 := exists_dual_vector \ud835\udd5c x h\n[GOAL]\ncase neg.intro.intro\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016\u2191f x\u2016 \u2264 M * \u2016f\u2016\nh : \u00acx = 0\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nhf\u2081 : \u2016f\u2016 = 1\nhfx : \u2191f x = \u2191\u2016x\u2016\n\u22a2 \u2016x\u2016 \u2264 M\n[PROOFSTEP]\ncalc\n \u2016x\u2016 = \u2016(\u2016x\u2016 : \ud835\udd5c)\u2016 := IsROrC.norm_coe_norm.symm\n _ = \u2016f x\u2016 := by rw [hfx]\n _ \u2264 M * \u2016f\u2016 := (hM f)\n _ = M := by rw [hf\u2081, mul_one]\n[GOAL]\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016\u2191f x\u2016 \u2264 M * \u2016f\u2016\nh : \u00acx = 0\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nhf\u2081 : \u2016f\u2016 = 1\nhfx : \u2191f x = \u2191\u2016x\u2016\n\u22a2 \u2016\u2191\u2016x\u2016\u2016 = \u2016\u2191f x\u2016\n[PROOFSTEP]\nrw [hfx]\n[GOAL]\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016\u2191f x\u2016 \u2264 M * \u2016f\u2016\nh : \u00acx = 0\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nhf\u2081 : \u2016f\u2016 = 1\nhfx : \u2191f x = \u2191\u2016x\u2016\n\u22a2 M * \u2016f\u2016 = M\n[PROOFSTEP]\nrw [hf\u2081, mul_one]\n[GOAL]\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nh : \u2200 (f : Dual \ud835\udd5c E), \u2191f x = 0\nf : Dual \ud835\udd5c E\n\u22a2 \u2016\u2191f x\u2016 \u2264 0 * \u2016f\u2016\n[PROOFSTEP]\nsimp [h f]\n[GOAL]\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nhx : x = 0\n\u22a2 \u2200 (g : Dual \ud835\udd5c E), \u2191g x = 0\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx y : E\n\u22a2 x = y \u2194 \u2200 (g : Dual \ud835\udd5c E), \u2191g x = \u2191g y\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, eq_zero_iff_forall_dual_eq_zero \ud835\udd5c (x - y)]\n[GOAL]\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx y : E\n\u22a2 (\u2200 (g : Dual \ud835\udd5c E), \u2191g (x - y) = 0) \u2194 \u2200 (g : Dual \ud835\udd5c E), \u2191g x = \u2191g y\n[PROOFSTEP]\nsimp [sub_eq_zero]\n[GOAL]\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nsrc\u271d : E \u2192L[\ud835\udd5c] Dual \ud835\udd5c (Dual \ud835\udd5c E) := inclusionInDoubleDual \ud835\udd5c E\n\u22a2 \u2200 (x : E), \u2016\u2191\u2191src\u271d x\u2016 = \u2016x\u2016\n[PROOFSTEP]\nintro x\n[GOAL]\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nsrc\u271d : E \u2192L[\ud835\udd5c] Dual \ud835\udd5c (Dual \ud835\udd5c E) := inclusionInDoubleDual \ud835\udd5c E\nx : E\n\u22a2 \u2016\u2191\u2191src\u271d x\u2016 = \u2016x\u2016\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nsrc\u271d : E \u2192L[\ud835\udd5c] Dual \ud835\udd5c (Dual \ud835\udd5c E) := inclusionInDoubleDual \ud835\udd5c E\nx : E\n\u22a2 \u2016\u2191\u2191src\u271d x\u2016 \u2264 \u2016x\u2016\n[PROOFSTEP]\nexact double_dual_bound \ud835\udd5c E x\n[GOAL]\ncase a\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nsrc\u271d : E \u2192L[\ud835\udd5c] Dual \ud835\udd5c (Dual \ud835\udd5c E) := inclusionInDoubleDual \ud835\udd5c E\nx : E\n\u22a2 \u2016x\u2016 \u2264 \u2016\u2191\u2191src\u271d x\u2016\n[PROOFSTEP]\nrw [ContinuousLinearMap.norm_def]\n[GOAL]\ncase a\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nsrc\u271d : E \u2192L[\ud835\udd5c] Dual \ud835\udd5c (Dual \ud835\udd5c E) := inclusionInDoubleDual \ud835\udd5c E\nx : E\n\u22a2 \u2016x\u2016 \u2264 sInf {c | 0 \u2264 c \u2227 \u2200 (x_1 : Dual \ud835\udd5c E), \u2016\u2191(\u2191\u2191src\u271d x) x_1\u2016 \u2264 c * \u2016x_1\u2016}\n[PROOFSTEP]\nrefine' le_csInf ContinuousLinearMap.bounds_nonempty _\n[GOAL]\ncase a\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nsrc\u271d : E \u2192L[\ud835\udd5c] Dual \ud835\udd5c (Dual \ud835\udd5c E) := inclusionInDoubleDual \ud835\udd5c E\nx : E\n\u22a2 \u2200 (b : \u211d), b \u2208 {c | 0 \u2264 c \u2227 \u2200 (x_1 : Dual \ud835\udd5c E), \u2016\u2191(\u2191\u2191src\u271d x) x_1\u2016 \u2264 c * \u2016x_1\u2016} \u2192 \u2016x\u2016 \u2264 b\n[PROOFSTEP]\nrintro c \u27e8hc1, hc2\u27e9\n[GOAL]\ncase a.intro\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nsrc\u271d : E \u2192L[\ud835\udd5c] Dual \ud835\udd5c (Dual \ud835\udd5c E) := inclusionInDoubleDual \ud835\udd5c E\nx : E\nc : \u211d\nhc1 : 0 \u2264 c\nhc2 : \u2200 (x_1 : Dual \ud835\udd5c E), \u2016\u2191(\u2191\u2191src\u271d x) x_1\u2016 \u2264 c * \u2016x_1\u2016\n\u22a2 \u2016x\u2016 \u2264 c\n[PROOFSTEP]\nexact norm_le_dual_bound \ud835\udd5c x hc1 hc2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 IsClosed (polar \ud835\udd5c s)\n[PROOFSTEP]\ndsimp only [NormedSpace.polar]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 IsClosed (LinearMap.polar (LinearMap.flip (dualPairing \ud835\udd5c E)) s)\n[PROOFSTEP]\nsimp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 IsClosed (\u22c2 (x : E) (_ : x \u2208 s), {y | \u2016\u2191(\u2191(dualPairing \ud835\udd5c E) y) x\u2016 \u2264 1})\n[PROOFSTEP]\nrefine' isClosed_biInter fun z _ => _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nz : E\nx\u271d : z \u2208 s\n\u22a2 IsClosed {y | \u2016\u2191(\u2191(dualPairing \ud835\udd5c E) y) z\u2016 \u2264 1}\n[PROOFSTEP]\nexact isClosed_Iic.preimage (ContinuousLinearMap.apply \ud835\udd5c \ud835\udd5c z).continuous.norm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 IsClosed\n ((LinearMap.polar (LinearMap.flip (LinearMap.flip (dualPairing \ud835\udd5c E))) \u2218 \u2191OrderDual.ofDual)\n (LinearMap.polar (LinearMap.flip (dualPairing \ud835\udd5c E)) s))\n[PROOFSTEP]\nsimpa [LinearMap.flip_flip] using (isClosed_polar _ _).preimage (inclusionInDoubleDual \ud835\udd5c E).continuous\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 (z : E), z \u2208 s \u2192 \u2016\u2191x' z\u2016 \u2264 \u2016c\u2016\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s\n[PROOFSTEP]\nby_cases c_zero : c = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 (z : E), z \u2208 s \u2192 \u2016\u2191x' z\u2016 \u2264 \u2016c\u2016\nc_zero : c = 0\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s\n[PROOFSTEP]\nsimp only [c_zero, inv_zero, zero_smul]\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 (z : E), z \u2208 s \u2192 \u2016\u2191x' z\u2016 \u2264 \u2016c\u2016\nc_zero : c = 0\n\u22a2 0 \u2208 polar \ud835\udd5c s\n[PROOFSTEP]\nexact (dualPairing \ud835\udd5c E).flip.zero_mem_polar _\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 (z : E), z \u2208 s \u2192 \u2016\u2191x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s\n[PROOFSTEP]\nhave eq : \u2200 z, \u2016c\u207b\u00b9 \u2022 x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016 := fun z => norm_smul c\u207b\u00b9 _\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 (z : E), z \u2208 s \u2192 \u2016\u2191x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 \u2191x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016\u2191x' z\u2016\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s\n[PROOFSTEP]\nhave le : \u2200 z, z \u2208 s \u2192 \u2016c\u207b\u00b9 \u2022 x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016 := by\n intro z hzs\n rw [eq z]\n apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 (z : E), z \u2208 s \u2192 \u2016\u2191x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 \u2191x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016\u2191x' z\u2016\n\u22a2 \u2200 (z : E), z \u2208 s \u2192 \u2016c\u207b\u00b9 \u2022 \u2191x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016\n[PROOFSTEP]\nintro z hzs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 (z : E), z \u2208 s \u2192 \u2016\u2191x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 \u2191x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016\u2191x' z\u2016\nz : E\nhzs : z \u2208 s\n\u22a2 \u2016c\u207b\u00b9 \u2022 \u2191x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016\n[PROOFSTEP]\nrw [eq z]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 (z : E), z \u2208 s \u2192 \u2016\u2191x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 \u2191x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016\u2191x' z\u2016\nz : E\nhzs : z \u2208 s\n\u22a2 \u2016c\u207b\u00b9\u2016 * \u2016\u2191x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016\n[PROOFSTEP]\napply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 (z : E), z \u2208 s \u2192 \u2016\u2191x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 \u2191x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016\u2191x' z\u2016\nle : \u2200 (z : E), z \u2208 s \u2192 \u2016c\u207b\u00b9 \u2022 \u2191x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s\n[PROOFSTEP]\nhave cancel : \u2016c\u207b\u00b9\u2016 * \u2016c\u2016 = 1 := by simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 (z : E), z \u2208 s \u2192 \u2016\u2191x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 \u2191x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016\u2191x' z\u2016\nle : \u2200 (z : E), z \u2208 s \u2192 \u2016c\u207b\u00b9 \u2022 \u2191x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016\n\u22a2 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016 = 1\n[PROOFSTEP]\nsimp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 (z : E), z \u2208 s \u2192 \u2016\u2191x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 \u2191x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016\u2191x' z\u2016\nle : \u2200 (z : E), z \u2208 s \u2192 \u2016c\u207b\u00b9 \u2022 \u2191x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016\ncancel : \u2016c\u207b\u00b9\u2016 * \u2016c\u2016 = 1\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s\n[PROOFSTEP]\nrwa [cancel] at le \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\n\u22a2 polar \ud835\udd5c (ball 0 r) \u2286 closedBall 0 (\u2016c\u2016 / r)\n[PROOFSTEP]\nintro x' hx'\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : x' \u2208 polar \ud835\udd5c (ball 0 r)\n\u22a2 x' \u2208 closedBall 0 (\u2016c\u2016 / r)\n[PROOFSTEP]\nrw [mem_polar_iff] at hx' \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : \u2200 (z : E), z \u2208 ball 0 r \u2192 \u2016\u2191x' z\u2016 \u2264 1\n\u22a2 x' \u2208 closedBall 0 (\u2016c\u2016 / r)\n[PROOFSTEP]\nsimp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at *\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : \u2200 (z : E), \u2016z\u2016 < r \u2192 \u2016\u2191x' z\u2016 \u2264 1\n\u22a2 \u2016x'\u2016 \u2264 \u2016c\u2016 / r\n[PROOFSTEP]\nhave hcr : 0 < \u2016c\u2016 / r := div_pos (zero_lt_one.trans hc) hr\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : \u2200 (z : E), \u2016z\u2016 < r \u2192 \u2016\u2191x' z\u2016 \u2264 1\nhcr : 0 < \u2016c\u2016 / r\n\u22a2 \u2016x'\u2016 \u2264 \u2016c\u2016 / r\n[PROOFSTEP]\nrefine' ContinuousLinearMap.op_norm_le_of_shell hr hcr.le hc fun x h\u2081 h\u2082 => _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : \u2200 (z : E), \u2016z\u2016 < r \u2192 \u2016\u2191x' z\u2016 \u2264 1\nhcr : 0 < \u2016c\u2016 / r\nx : E\nh\u2081 : r / \u2016c\u2016 \u2264 \u2016x\u2016\nh\u2082 : \u2016x\u2016 < r\n\u22a2 \u2016\u2191x' x\u2016 \u2264 \u2016c\u2016 / r * \u2016x\u2016\n[PROOFSTEP]\ncalc\n \u2016x' x\u2016 \u2264 1 := hx' _ h\u2082\n _ \u2264 \u2016c\u2016 / r * \u2016x\u2016 := (inv_pos_le_iff_one_le_mul' hcr).1 (by rwa [inv_div])\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : \u2200 (z : E), \u2016z\u2016 < r \u2192 \u2016\u2191x' z\u2016 \u2264 1\nhcr : 0 < \u2016c\u2016 / r\nx : E\nh\u2081 : r / \u2016c\u2016 \u2264 \u2016x\u2016\nh\u2082 : \u2016x\u2016 < r\n\u22a2 (\u2016c\u2016 / r)\u207b\u00b9 \u2264 \u2016x\u2016\n[PROOFSTEP]\nrwa [inv_div]\n[GOAL]\n\ud835\udd5c\u271d : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\u271d\nE\u271d : Type u_2\ninst\u271d\u2074 : SeminormedAddCommGroup E\u271d\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u271d E\u271d\n\ud835\udd5c : Type u_3\nE : Type u_4\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nhr : 0 < r\n\u22a2 polar \ud835\udd5c (closedBall 0 r) = closedBall 0 r\u207b\u00b9\n[PROOFSTEP]\nrefine' Subset.antisymm _ (closedBall_inv_subset_polar_closedBall \ud835\udd5c)\n[GOAL]\n\ud835\udd5c\u271d : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\u271d\nE\u271d : Type u_2\ninst\u271d\u2074 : SeminormedAddCommGroup E\u271d\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u271d E\u271d\n\ud835\udd5c : Type u_3\nE : Type u_4\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nhr : 0 < r\n\u22a2 polar \ud835\udd5c (closedBall 0 r) \u2286 closedBall 0 r\u207b\u00b9\n[PROOFSTEP]\nintro x' h\n[GOAL]\n\ud835\udd5c\u271d : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\u271d\nE\u271d : Type u_2\ninst\u271d\u2074 : SeminormedAddCommGroup E\u271d\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u271d E\u271d\n\ud835\udd5c : Type u_3\nE : Type u_4\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nh : x' \u2208 polar \ud835\udd5c (closedBall 0 r)\n\u22a2 x' \u2208 closedBall 0 r\u207b\u00b9\n[PROOFSTEP]\nsimp only [mem_closedBall_zero_iff]\n[GOAL]\n\ud835\udd5c\u271d : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\u271d\nE\u271d : Type u_2\ninst\u271d\u2074 : SeminormedAddCommGroup E\u271d\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u271d E\u271d\n\ud835\udd5c : Type u_3\nE : Type u_4\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nh : x' \u2208 polar \ud835\udd5c (closedBall 0 r)\n\u22a2 \u2016x'\u2016 \u2264 r\u207b\u00b9\n[PROOFSTEP]\nrefine' ContinuousLinearMap.op_norm_le_of_ball hr (inv_nonneg.mpr hr.le) fun z _ => _\n[GOAL]\n\ud835\udd5c\u271d : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\u271d\nE\u271d : Type u_2\ninst\u271d\u2074 : SeminormedAddCommGroup E\u271d\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u271d E\u271d\n\ud835\udd5c : Type u_3\nE : Type u_4\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nh : x' \u2208 polar \ud835\udd5c (closedBall 0 r)\nz : E\nx\u271d : z \u2208 ball 0 r\n\u22a2 \u2016\u2191x' z\u2016 \u2264 r\u207b\u00b9 * \u2016z\u2016\n[PROOFSTEP]\nsimpa only [one_div] using LinearMap.bound_of_ball_bound' hr 1 x'.toLinearMap h z\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\ns_nhd : s \u2208 \ud835\udcdd 0\n\u22a2 Metric.Bounded (polar \ud835\udd5c s)\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 : \u2203 a : \ud835\udd5c, 1 < \u2016a\u2016 := NormedField.exists_one_lt_norm \ud835\udd5c\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\ns_nhd : s \u2208 \ud835\udcdd 0\na : \ud835\udd5c\nha : 1 < \u2016a\u2016\n\u22a2 Metric.Bounded (polar \ud835\udd5c s)\n[PROOFSTEP]\nobtain \u27e8r, r_pos, r_ball\u27e9 : \u2203 r : \u211d, 0 < r \u2227 ball 0 r \u2286 s := Metric.mem_nhds_iff.1 s_nhd\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\ns_nhd : s \u2208 \ud835\udcdd 0\na : \ud835\udd5c\nha : 1 < \u2016a\u2016\nr : \u211d\nr_pos : 0 < r\nr_ball : ball 0 r \u2286 s\n\u22a2 Metric.Bounded (polar \ud835\udd5c s)\n[PROOFSTEP]\nexact\n bounded_closedBall.mono\n (((dualPairing \ud835\udd5c E).flip.polar_antitone r_ball).trans <| polar_ball_subset_closedBall_div ha r_pos)\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.Dual", "llama_tokens": 9575, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300449389326, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.40033860880655764}} {"text": "[GOAL]\n\u22a2 CharZero \u2124\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 val a < n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\ninst\u271d : NeZero Nat.zero\na : ZMod Nat.zero\n\u22a2 val a < Nat.zero\n[PROOFSTEP]\ncases NeZero.ne 0 rfl\n[GOAL]\ncase succ\nn\u271d : \u2115\ninst\u271d : NeZero (Nat.succ n\u271d)\na : ZMod (Nat.succ n\u271d)\n\u22a2 val a < Nat.succ n\u271d\n[PROOFSTEP]\nexact Fin.is_lt a\n[GOAL]\nn a : \u2115\n\u22a2 val \u2191a = a % n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\na : \u2115\n\u22a2 val \u2191a = a % Nat.zero\n[PROOFSTEP]\nrw [Nat.mod_zero]\n[GOAL]\ncase zero\na : \u2115\n\u22a2 val \u2191a = a\n[PROOFSTEP]\nexact Int.natAbs_ofNat a\n[GOAL]\ncase succ\na n\u271d : \u2115\n\u22a2 val \u2191a = a % Nat.succ n\u271d\n[PROOFSTEP]\nrw [\u2190 Fin.ofNat_eq_val]\n[GOAL]\ncase succ\na n\u271d : \u2115\n\u22a2 val (Fin.ofNat'' a) = a % Nat.succ n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u22a2 \u2200 (x : \u2115), \u2191x = 0 \u2194 n \u2223 x\n[PROOFSTEP]\nintro k\n[GOAL]\nn k : \u2115\n\u22a2 \u2191k = 0 \u2194 n \u2223 k\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nk : \u2115\n\u22a2 \u2191k = 0 \u2194 Nat.zero \u2223 k\n[PROOFSTEP]\nsimp [zero_dvd_iff, Int.coe_nat_eq_zero, Nat.zero_eq]\n[GOAL]\ncase succ\nk n : \u2115\n\u22a2 \u2191k = 0 \u2194 Nat.succ n \u2223 k\n[PROOFSTEP]\nrw [Fin.eq_iff_veq]\n[GOAL]\ncase succ\nk n : \u2115\n\u22a2 \u2191\u2191k = \u21910 \u2194 Nat.succ n \u2223 k\n[PROOFSTEP]\nshow (k : ZMod (n + 1)).val = (0 : ZMod (n + 1)).val \u2194 _\n[GOAL]\ncase succ\nk n : \u2115\n\u22a2 val \u2191k = val 0 \u2194 Nat.succ n \u2223 k\n[PROOFSTEP]\nrw [val_nat_cast, val_zero, Nat.dvd_iff_mod_eq_zero]\n[GOAL]\na n : \u2115\nn0 : n \u2260 0\n\u22a2 addOrderOf \u2191a = n / Nat.gcd n a\n[PROOFSTEP]\ncases' a with a\n[GOAL]\ncase zero\nn : \u2115\nn0 : n \u2260 0\n\u22a2 addOrderOf \u2191Nat.zero = n / Nat.gcd n Nat.zero\ncase succ n : \u2115 n0 : n \u2260 0 a : \u2115 \u22a2 addOrderOf \u2191(Nat.succ a) = n / Nat.gcd n (Nat.succ a)\n[PROOFSTEP]\nsimp [Nat.pos_of_ne_zero n0]\n[GOAL]\ncase succ\nn : \u2115\nn0 : n \u2260 0\na : \u2115\n\u22a2 addOrderOf \u2191(Nat.succ a) = n / Nat.gcd n (Nat.succ a)\n[PROOFSTEP]\nrw [\u2190 Nat.smul_one_eq_coe, addOrderOf_nsmul' _ a.succ_ne_zero, ZMod.addOrderOf_one]\n[GOAL]\na n : \u2115\na0 : a \u2260 0\n\u22a2 addOrderOf \u2191a = n / Nat.gcd n a\n[PROOFSTEP]\nrw [\u2190 Nat.smul_one_eq_coe, addOrderOf_nsmul' _ a0, ZMod.addOrderOf_one]\n[GOAL]\nn : \u2115\n\u22a2 ringChar (ZMod n) = n\n[PROOFSTEP]\nrw [ringChar.eq_iff]\n[GOAL]\nn : \u2115\n\u22a2 CharP (ZMod n) n\n[PROOFSTEP]\nexact ZMod.charP n\n[GOAL]\nn : \u2115\n\u22a2 \u2191n + 1 = 0\n[PROOFSTEP]\nrw [\u2190 Nat.cast_add_one, nat_cast_self (n + 1)]\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d : AddGroupWithOne R\n\u22a2 \u21910 = 0\n[PROOFSTEP]\ndelta ZMod.cast\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d : AddGroupWithOne R\n\u22a2 (match (motive := (x : \u2115) \u2192 ZMod x \u2192 R) n with\n | 0 => Int.cast\n | Nat.succ n => fun i => \u2191(val i))\n 0 =\n 0\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d : AddGroupWithOne R\n\u22a2 (match (motive := (x : \u2115) \u2192 ZMod x \u2192 R) Nat.zero with\n | 0 => Int.cast\n | Nat.succ n => fun i => \u2191(val i))\n 0 =\n 0\n[PROOFSTEP]\nexact Int.cast_zero\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : AddGroupWithOne R\nn\u271d : \u2115\n\u22a2 (match (motive := (x : \u2115) \u2192 ZMod x \u2192 R) Nat.succ n\u271d with\n | 0 => Int.cast\n | Nat.succ n => fun i => \u2191(val i))\n 0 =\n 0\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 \u2191a = \u2191(val a)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\ninst\u271d : NeZero Nat.zero\na : ZMod Nat.zero\n\u22a2 \u2191a = \u2191(val a)\n[PROOFSTEP]\ncases NeZero.ne 0 rfl\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\nn\u271d : \u2115\ninst\u271d : NeZero (Nat.succ n\u271d)\na : ZMod (Nat.succ n\u271d)\n\u22a2 \u2191a = \u2191(val a)\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\nS : Type u_2\ninst\u271d : AddGroupWithOne S\na : ZMod n\n\u22a2 (\u2191a).fst = \u2191a\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\nS : Type u_2\ninst\u271d : AddGroupWithOne S\na : ZMod Nat.zero\n\u22a2 (\u2191a).fst = \u2191a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\nS : Type u_2\ninst\u271d : AddGroupWithOne S\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\n\u22a2 (\u2191a).fst = \u2191a\n[PROOFSTEP]\nsimp [ZMod.cast]\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\nS : Type u_2\ninst\u271d : AddGroupWithOne S\na : ZMod n\n\u22a2 (\u2191a).snd = \u2191a\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\nS : Type u_2\ninst\u271d : AddGroupWithOne S\na : ZMod Nat.zero\n\u22a2 (\u2191a).snd = \u2191a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\nS : Type u_2\ninst\u271d : AddGroupWithOne S\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\n\u22a2 (\u2191a).snd = \u2191a\n[PROOFSTEP]\nsimp [ZMod.cast]\n[GOAL]\nn\u271d : \u2115\nR : Type u_1\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 \u2191(val a) = a\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nn : \u2115\nR : Type u_1\ninst\u271d : NeZero Nat.zero\na : ZMod Nat.zero\n\u22a2 \u2191(val a) = a\n[PROOFSTEP]\ncases NeZero.ne 0 rfl\n[GOAL]\ncase succ\nn : \u2115\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : NeZero (Nat.succ n\u271d)\na : ZMod (Nat.succ n\u271d)\n\u22a2 \u2191(val a) = a\n[PROOFSTEP]\napply Fin.cast_val_eq_self\n[GOAL]\nn : \u2115\nR : Type u_1\na : ZMod n\n\u22a2 \u2191\u2191a = a\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nR : Type u_1\na : ZMod Nat.zero\n\u22a2 \u2191\u2191a = a\n[PROOFSTEP]\nsimp [ZMod.cast, ZMod]\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\n\u22a2 \u2191\u2191a = a\n[PROOFSTEP]\ndsimp [ZMod.cast, ZMod]\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\n\u22a2 \u2191\u2191(val a) = a\n[PROOFSTEP]\nerw [Int.cast_ofNat, Fin.cast_val_eq_self]\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : NeZero n\n\u22a2 Nat.cast \u2218 val = cast\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : NeZero Nat.zero\n\u22a2 Nat.cast \u2218 val = cast\n[PROOFSTEP]\ncases NeZero.ne 0 rfl\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nn\u271d : \u2115\ninst\u271d : NeZero (Nat.succ n\u271d)\n\u22a2 Nat.cast \u2218 val = cast\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d : Ring R\n\u22a2 Int.cast \u2218 cast = cast\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d : Ring R\n\u22a2 Int.cast \u2218 cast = cast\n[PROOFSTEP]\nexact congr_arg ((\u00b7 \u2218 \u00b7) Int.cast) ZMod.cast_id'\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\n\u22a2 Int.cast \u2218 cast = cast\n[PROOFSTEP]\next\n[GOAL]\ncase succ.h\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nx\u271d : ZMod (Nat.succ n\u271d)\n\u22a2 (Int.cast \u2218 cast) x\u271d = \u2191x\u271d\n[PROOFSTEP]\nsimp [ZMod, ZMod.cast]\n[GOAL]\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\na b : ZMod n\n\u22a2 \u2191(a + b) = if \u2191n \u2264 \u2191a + \u2191b then \u2191a + \u2191b - \u2191n else \u2191a + \u2191b\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nn : \u2115\nR : Type u_1\ninst\u271d : Ring R\na b : ZMod Nat.zero\n\u22a2 \u2191(a + b) = if \u2191Nat.zero \u2264 \u2191a + \u2191b then \u2191a + \u2191b - \u2191Nat.zero else \u2191a + \u2191b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase zero\nn : \u2115\nR : Type u_1\ninst\u271d : Ring R\na b : ZMod Nat.zero\n\u22a2 \u2191(a + b) = \u2191a + \u2191b\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\na b : ZMod (Nat.succ n)\n\u22a2 \u2191(a + b) = if \u2191(Nat.succ n) \u2264 \u2191a + \u2191b then \u2191a + \u2191b - \u2191(Nat.succ n) else \u2191a + \u2191b\n[PROOFSTEP]\nchange Fin (n + 1) at a b \n[GOAL]\ncase succ\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\na b : Fin (n + 1)\n\u22a2 \u2191(a + b) = if \u2191(Nat.succ n) \u2264 \u2191a + \u2191b then \u2191a + \u2191b - \u2191(Nat.succ n) else \u2191a + \u2191b\n[PROOFSTEP]\nchange ((((a + b) : Fin (n + 1)) : \u2115) : \u2124) = if ((n + 1 : \u2115) : \u2124) \u2264 (a : \u2115) + b then _ else _\n[GOAL]\ncase succ\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\na b : Fin (n + 1)\n\u22a2 \u2191\u2191(a + b) = if \u2191(n + 1) \u2264 \u2191\u2191a + \u2191\u2191b then \u2191a + \u2191b - \u2191(Nat.succ n) else \u2191a + \u2191b\n[PROOFSTEP]\nsimp only [Fin.val_add_eq_ite, Int.ofNat_succ, Int.ofNat_le]\n[GOAL]\ncase succ\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\na b : Fin (n + 1)\n\u22a2 \u2191(if n + 1 \u2264 \u2191a + \u2191b then \u2191a + \u2191b - (n + 1) else \u2191a + \u2191b) = if \u2191n + 1 \u2264 \u2191\u2191a + \u2191\u2191b then \u2191a + \u2191b - (\u2191n + 1) else \u2191a + \u2191b\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase succ\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\na b : Fin (n + 1)\n\u22a2 \u2191(if n + 1 \u2264 \u2191a + \u2191b then \u2191a + \u2191b - (n + 1) else \u2191a + \u2191b) = if n + 1 \u2264 \u2191a + \u2191b then \u2191a + \u2191b - \u2191(n + 1) else \u2191a + \u2191b\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\na b : Fin (n + 1)\nh : n + 1 \u2264 \u2191a + \u2191b\n\u22a2 \u2191(\u2191a + \u2191b - (n + 1)) = \u2191a + \u2191b - \u2191(n + 1)\n[PROOFSTEP]\nrw [Nat.cast_sub h]\n[GOAL]\ncase pos\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\na b : Fin (n + 1)\nh : n + 1 \u2264 \u2191a + \u2191b\n\u22a2 \u2191(\u2191a + \u2191b) - \u2191(n + 1) = \u2191a + \u2191b - \u2191(n + 1)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase neg\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\na b : Fin (n + 1)\nh : \u00acn + 1 \u2264 \u2191a + \u2191b\n\u22a2 \u2191(\u2191a + \u2191b) = \u2191a + \u2191b\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nh : m \u2223 n\n\u22a2 \u21911 = 1\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nh : m \u2223 Nat.zero\n\u22a2 \u21911 = 1\n[PROOFSTEP]\nexact Int.cast_one\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nn : \u2115\nh : m \u2223 Nat.succ n\n\u22a2 \u21911 = 1\n[PROOFSTEP]\nshow ((1 % (n + 1) : \u2115) : R) = 1\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nn : \u2115\nh : m \u2223 Nat.succ n\n\u22a2 \u2191(1 % (n + 1)) = 1\n[PROOFSTEP]\ncases n\n[GOAL]\ncase succ.zero\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nh : m \u2223 Nat.succ Nat.zero\n\u22a2 \u2191(1 % (Nat.zero + 1)) = 1\n[PROOFSTEP]\nrw [Nat.dvd_one] at h \n[GOAL]\ncase succ.zero\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nh : m = 1\n\u22a2 \u2191(1 % (Nat.zero + 1)) = 1\n[PROOFSTEP]\nsubst m\n[GOAL]\ncase succ.zero\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R 1\n\u22a2 \u2191(1 % (Nat.zero + 1)) = 1\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\ncase succ.succ\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nn\u271d : \u2115\nh : m \u2223 Nat.succ (Nat.succ n\u271d)\n\u22a2 \u2191(1 % (Nat.succ n\u271d + 1)) = 1\n[PROOFSTEP]\nrw [Nat.mod_eq_of_lt]\n[GOAL]\ncase succ.succ\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nn\u271d : \u2115\nh : m \u2223 Nat.succ (Nat.succ n\u271d)\n\u22a2 \u21911 = 1\n[PROOFSTEP]\nexact Nat.cast_one\n[GOAL]\ncase succ.succ\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nn\u271d : \u2115\nh : m \u2223 Nat.succ (Nat.succ n\u271d)\n\u22a2 1 < Nat.succ n\u271d + 1\n[PROOFSTEP]\nexact Nat.lt_of_sub_eq_succ rfl\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nh : m \u2223 n\na b : ZMod n\n\u22a2 \u2191(a + b) = \u2191a + \u2191b\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nh : m \u2223 Nat.zero\na b : ZMod Nat.zero\n\u22a2 \u2191(a + b) = \u2191a + \u2191b\n[PROOFSTEP]\napply Int.cast_add\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nn\u271d : \u2115\nh : m \u2223 Nat.succ n\u271d\na b : ZMod (Nat.succ n\u271d)\n\u22a2 \u2191(a + b) = \u2191a + \u2191b\n[PROOFSTEP]\nsymm\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nn\u271d : \u2115\nh : m \u2223 Nat.succ n\u271d\na b : ZMod (Nat.succ n\u271d)\n\u22a2 \u2191a + \u2191b = \u2191(a + b)\n[PROOFSTEP]\ndsimp [ZMod, ZMod.cast]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nn\u271d : \u2115\nh : m \u2223 Nat.succ n\u271d\na b : ZMod (Nat.succ n\u271d)\n\u22a2 \u2191(val a) + \u2191(val b) = \u2191(val (a + b))\n[PROOFSTEP]\nerw [\u2190 Nat.cast_add, \u2190 sub_eq_zero, \u2190 Nat.cast_sub (Nat.mod_le _ _), @CharP.cast_eq_zero_iff R _ m]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nn\u271d : \u2115\nh : m \u2223 Nat.succ n\u271d\na b : ZMod (Nat.succ n\u271d)\n\u22a2 m \u2223 val a + val b - (val a + val b) % Nat.succ n\u271d\n[PROOFSTEP]\nexact h.trans (Nat.dvd_sub_mod _)\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nh : m \u2223 n\na b : ZMod n\n\u22a2 \u2191(a * b) = \u2191a * \u2191b\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nh : m \u2223 Nat.zero\na b : ZMod Nat.zero\n\u22a2 \u2191(a * b) = \u2191a * \u2191b\n[PROOFSTEP]\napply Int.cast_mul\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nn\u271d : \u2115\nh : m \u2223 Nat.succ n\u271d\na b : ZMod (Nat.succ n\u271d)\n\u22a2 \u2191(a * b) = \u2191a * \u2191b\n[PROOFSTEP]\nsymm\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nn\u271d : \u2115\nh : m \u2223 Nat.succ n\u271d\na b : ZMod (Nat.succ n\u271d)\n\u22a2 \u2191a * \u2191b = \u2191(a * b)\n[PROOFSTEP]\ndsimp [ZMod, ZMod.cast]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nn\u271d : \u2115\nh : m \u2223 Nat.succ n\u271d\na b : ZMod (Nat.succ n\u271d)\n\u22a2 \u2191(val a) * \u2191(val b) = \u2191(val (a * b))\n[PROOFSTEP]\nerw [\u2190 Nat.cast_mul, \u2190 sub_eq_zero, \u2190 Nat.cast_sub (Nat.mod_le _ _), @CharP.cast_eq_zero_iff R _ m]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : Ring R\nm : \u2115\ninst\u271d : CharP R m\nn\u271d : \u2115\nh : m \u2223 Nat.succ n\u271d\na b : ZMod (Nat.succ n\u271d)\n\u22a2 m \u2223 val a * val b - val a * val b % Nat.succ n\u271d\n[PROOFSTEP]\nexact h.trans (Nat.dvd_sub_mod _)\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\n\u22a2 Injective \u2191(castHom (_ : n \u2223 n) R)\n[PROOFSTEP]\nrw [injective_iff_map_eq_zero]\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\n\u22a2 \u2200 (a : ZMod n), \u2191(castHom (_ : n \u2223 n) R) a = 0 \u2192 a = 0\n[PROOFSTEP]\nintro x\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\nx : ZMod n\n\u22a2 \u2191(castHom (_ : n \u2223 n) R) x = 0 \u2192 x = 0\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := ZMod.int_cast_surjective x\n[GOAL]\ncase intro\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\nk : \u2124\n\u22a2 \u2191(castHom (_ : n \u2223 n) R) \u2191k = 0 \u2192 \u2191k = 0\n[PROOFSTEP]\nrw [map_intCast, CharP.int_cast_eq_zero_iff R n, CharP.int_cast_eq_zero_iff (ZMod n) n]\n[GOAL]\ncase intro\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\nk : \u2124\n\u22a2 \u2191n \u2223 k \u2192 \u2191n \u2223 k\n[PROOFSTEP]\nexact id\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CharP R n\ninst\u271d : Fintype R\nh : Fintype.card R = n\n\u22a2 Bijective \u2191(castHom (_ : n \u2223 n) R)\n[PROOFSTEP]\nhaveI : NeZero n :=\n \u27e8by\n intro hn\n rw [hn] at h \n exact (Fintype.card_eq_zero_iff.mp h).elim' 0\u27e9\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CharP R n\ninst\u271d : Fintype R\nh : Fintype.card R = n\n\u22a2 n \u2260 0\n[PROOFSTEP]\nintro hn\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CharP R n\ninst\u271d : Fintype R\nh : Fintype.card R = n\nhn : n = 0\n\u22a2 False\n[PROOFSTEP]\nrw [hn] at h \n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CharP R n\ninst\u271d : Fintype R\nh : Fintype.card R = 0\nhn : n = 0\n\u22a2 False\n[PROOFSTEP]\nexact (Fintype.card_eq_zero_iff.mp h).elim' 0\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CharP R n\ninst\u271d : Fintype R\nh : Fintype.card R = n\nthis : NeZero n\n\u22a2 Bijective \u2191(castHom (_ : n \u2223 n) R)\n[PROOFSTEP]\nrw [Fintype.bijective_iff_injective_and_card, ZMod.card, h, eq_self_iff_true, and_true_iff]\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CharP R n\ninst\u271d : Fintype R\nh : Fintype.card R = n\nthis : NeZero n\n\u22a2 Injective \u2191(castHom (_ : n \u2223 n) R)\n[PROOFSTEP]\napply ZMod.castHom_injective\n[GOAL]\nn\u271d : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\u271d\nm n : \u2115\nh : m = n\n\u22a2 ZMod m \u2243+* ZMod n\n[PROOFSTEP]\ncases' m with m\n[GOAL]\ncase zero\nn\u271d : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\u271d\nn : \u2115\nh : Nat.zero = n\n\u22a2 ZMod Nat.zero \u2243+* ZMod n\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase succ\nn\u271d : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\u271d\nn m : \u2115\nh : Nat.succ m = n\n\u22a2 ZMod (Nat.succ m) \u2243+* ZMod n\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero.zero\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\nh : Nat.zero = Nat.zero\n\u22a2 ZMod Nat.zero \u2243+* ZMod Nat.zero\n[PROOFSTEP]\nexact RingEquiv.refl _\n[GOAL]\ncase zero.succ\nn\u271d : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\u271d\nn : \u2115\nh : Nat.zero = Nat.succ n\n\u22a2 ZMod Nat.zero \u2243+* ZMod (Nat.succ n)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase zero.succ.h\nn\u271d : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\u271d\nn : \u2115\nh : Nat.zero = Nat.succ n\n\u22a2 False\n[PROOFSTEP]\nexact n.succ_ne_zero h.symm\n[GOAL]\ncase succ.zero\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\nm : \u2115\nh : Nat.succ m = Nat.zero\n\u22a2 ZMod (Nat.succ m) \u2243+* ZMod Nat.zero\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase succ.zero.h\nn : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\nm : \u2115\nh : Nat.succ m = Nat.zero\n\u22a2 False\n[PROOFSTEP]\nexact m.succ_ne_zero h\n[GOAL]\ncase succ.succ\nn\u271d : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\u271d\nm n : \u2115\nh : Nat.succ m = Nat.succ n\n\u22a2 ZMod (Nat.succ m) \u2243+* ZMod (Nat.succ n)\n[PROOFSTEP]\nexact\n {\n Fin.castIso\n h with\n map_mul' := fun a b => by\n dsimp [ZMod]\n ext\n rw [Fin.coe_castIso, Fin.coe_mul, Fin.coe_mul, Fin.coe_castIso, Fin.coe_castIso, \u2190 h]\n map_add' := fun a b => by\n dsimp [ZMod]\n ext\n rw [Fin.coe_castIso, Fin.val_add, Fin.val_add, Fin.coe_castIso, Fin.coe_castIso, \u2190 h] }\n[GOAL]\nn\u271d : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\u271d\nm n : \u2115\nh : Nat.succ m = Nat.succ n\nsrc\u271d : Fin (Nat.succ m) \u2243o Fin (Nat.succ n) := Fin.castIso h\na b : ZMod (Nat.succ m)\n\u22a2 Equiv.toFun src\u271d.toEquiv (a * b) = Equiv.toFun src\u271d.toEquiv a * Equiv.toFun src\u271d.toEquiv b\n[PROOFSTEP]\ndsimp [ZMod]\n[GOAL]\nn\u271d : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\u271d\nm n : \u2115\nh : Nat.succ m = Nat.succ n\nsrc\u271d : Fin (Nat.succ m) \u2243o Fin (Nat.succ n) := Fin.castIso h\na b : ZMod (Nat.succ m)\n\u22a2 \u2191(Fin.castIso h) (a * b) = \u2191(Fin.castIso h) a * \u2191(Fin.castIso h) b\n[PROOFSTEP]\next\n[GOAL]\ncase h\nn\u271d : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\u271d\nm n : \u2115\nh : Nat.succ m = Nat.succ n\nsrc\u271d : Fin (Nat.succ m) \u2243o Fin (Nat.succ n) := Fin.castIso h\na b : ZMod (Nat.succ m)\n\u22a2 \u2191(\u2191(Fin.castIso h) (a * b)) = \u2191(\u2191(Fin.castIso h) a * \u2191(Fin.castIso h) b)\n[PROOFSTEP]\nrw [Fin.coe_castIso, Fin.coe_mul, Fin.coe_mul, Fin.coe_castIso, Fin.coe_castIso, \u2190 h]\n[GOAL]\nn\u271d : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\u271d\nm n : \u2115\nh : Nat.succ m = Nat.succ n\nsrc\u271d : Fin (Nat.succ m) \u2243o Fin (Nat.succ n) := Fin.castIso h\na b : ZMod (Nat.succ m)\n\u22a2 Equiv.toFun src\u271d.toEquiv (a + b) = Equiv.toFun src\u271d.toEquiv a + Equiv.toFun src\u271d.toEquiv b\n[PROOFSTEP]\ndsimp [ZMod]\n[GOAL]\nn\u271d : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\u271d\nm n : \u2115\nh : Nat.succ m = Nat.succ n\nsrc\u271d : Fin (Nat.succ m) \u2243o Fin (Nat.succ n) := Fin.castIso h\na b : ZMod (Nat.succ m)\n\u22a2 \u2191(Fin.castIso h) (a + b) = \u2191(Fin.castIso h) a + \u2191(Fin.castIso h) b\n[PROOFSTEP]\next\n[GOAL]\ncase h\nn\u271d : \u2115\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R n\u271d\nm n : \u2115\nh : Nat.succ m = Nat.succ n\nsrc\u271d : Fin (Nat.succ m) \u2243o Fin (Nat.succ n) := Fin.castIso h\na b : ZMod (Nat.succ m)\n\u22a2 \u2191(\u2191(Fin.castIso h) (a + b)) = \u2191(\u2191(Fin.castIso h) a + \u2191(Fin.castIso h) b)\n[PROOFSTEP]\nrw [Fin.coe_castIso, Fin.val_add, Fin.val_add, Fin.coe_castIso, Fin.coe_castIso, \u2190 h]\n[GOAL]\na b c : \u2115\n\u22a2 \u2191a = \u2191b \u2194 a \u2261 b [MOD c]\n[PROOFSTEP]\nsimpa [Int.coe_nat_modEq_iff] using ZMod.int_cast_eq_int_cast_iff a b c\n[GOAL]\na : \u2124\nb : \u2115\n\u22a2 \u2191a = 0 \u2194 \u2191b \u2223 a\n[PROOFSTEP]\nrw [\u2190 Int.cast_zero, ZMod.int_cast_eq_int_cast_iff, Int.modEq_zero_iff_dvd]\n[GOAL]\na b : \u2124\nc : \u2115\n\u22a2 \u2191a = \u2191b \u2194 \u2191c \u2223 b - a\n[PROOFSTEP]\nrw [ZMod.int_cast_eq_int_cast_iff, Int.modEq_iff_dvd]\n[GOAL]\na b : \u2115\n\u22a2 \u2191a = 0 \u2194 b \u2223 a\n[PROOFSTEP]\nrw [\u2190 Nat.cast_zero, ZMod.nat_cast_eq_nat_cast_iff, Nat.modEq_zero_iff_dvd]\n[GOAL]\nn : \u2115\na : \u2124\ninst\u271d : NeZero n\n\u22a2 \u2191(val \u2191a) = a % \u2191n\n[PROOFSTEP]\nhave hle : (0 : \u2124) \u2264 \u2191(a : ZMod n).val := Int.coe_nat_nonneg _\n[GOAL]\nn : \u2115\na : \u2124\ninst\u271d : NeZero n\nhle : 0 \u2264 \u2191(val \u2191a)\n\u22a2 \u2191(val \u2191a) = a % \u2191n\n[PROOFSTEP]\nhave hlt : \u2191(a : ZMod n).val < (n : \u2124) := Int.ofNat_lt.mpr (ZMod.val_lt a)\n[GOAL]\nn : \u2115\na : \u2124\ninst\u271d : NeZero n\nhle : 0 \u2264 \u2191(val \u2191a)\nhlt : \u2191(val \u2191a) < \u2191n\n\u22a2 \u2191(val \u2191a) = a % \u2191n\n[PROOFSTEP]\nrefine' (Int.emod_eq_of_lt hle hlt).symm.trans _\n[GOAL]\nn : \u2115\na : \u2124\ninst\u271d : NeZero n\nhle : 0 \u2264 \u2191(val \u2191a)\nhlt : \u2191(val \u2191a) < \u2191n\n\u22a2 \u2191(val \u2191a) % \u2191n = a % \u2191n\n[PROOFSTEP]\nrw [\u2190 ZMod.int_cast_eq_int_cast_iff', Int.cast_ofNat, ZMod.nat_cast_val, ZMod.cast_id]\n[GOAL]\nn : \u2115\na : \u2124\n\u22a2 \u2191\u2191a = a % \u2191n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\na : \u2124\n\u22a2 \u2191\u2191a = a % \u2191Nat.zero\n[PROOFSTEP]\nrw [Int.ofNat_zero, Int.emod_zero, Int.cast_id]\n[GOAL]\ncase zero\na : \u2124\n\u22a2 \u2191a = a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\na : \u2124\nn\u271d : \u2115\n\u22a2 \u2191\u2191a = a % \u2191(Nat.succ n\u271d)\n[PROOFSTEP]\nrw [\u2190 val_int_cast, val]\n[GOAL]\ncase succ\na : \u2124\nn\u271d : \u2115\n\u22a2 \u2191\u2191a =\n \u2191((match (motive := (x : \u2115) \u2192 ZMod x \u2192 \u2115) Nat.succ n\u271d with\n | 0 => Int.natAbs\n | Nat.succ n => Fin.val)\n \u2191a)\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u22a2 val (-1) = n\n[PROOFSTEP]\ndsimp [val, Fin.coe_neg]\n[GOAL]\nn : \u2115\n\u22a2 \u2191(-1) = n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u22a2 \u2191(-1) = Nat.zero\n[PROOFSTEP]\nsimp [Nat.mod_one]\n[GOAL]\ncase succ\nn\u271d : \u2115\n\u22a2 \u2191(-1) = Nat.succ n\u271d\n[PROOFSTEP]\ndsimp [ZMod, ZMod.cast]\n[GOAL]\ncase succ\nn\u271d : \u2115\n\u22a2 \u2191(-1) = Nat.succ n\u271d\n[PROOFSTEP]\nrw [Fin.coe_neg_one]\n[GOAL]\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\n\u22a2 \u2191(-1) = \u2191n - 1\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d : Ring R\n\u22a2 \u2191(-1) = \u2191Nat.zero - 1\n[PROOFSTEP]\ndsimp [ZMod, ZMod.cast]\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d : Ring R\n\u22a2 \u2191(-1) = \u21910 - 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\n\u22a2 \u2191(-1) = \u2191(Nat.succ n) - 1\n[PROOFSTEP]\nrw [\u2190 nat_cast_val, val_neg_one, Nat.cast_succ, add_sub_cancel]\n[GOAL]\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\nk : ZMod n\n\u22a2 \u2191(k - 1) = (if k = 0 then \u2191n else \u2191k) - 1\n[PROOFSTEP]\nsplit_ifs with hk\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\nk : ZMod n\nhk : k = 0\n\u22a2 \u2191(k - 1) = \u2191n - 1\n[PROOFSTEP]\nrw [hk, zero_sub, ZMod.cast_neg_one]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\nk : ZMod n\nhk : \u00ack = 0\n\u22a2 \u2191(k - 1) = \u2191k - 1\n[PROOFSTEP]\ncases n\n[GOAL]\ncase neg.zero\nR : Type u_1\ninst\u271d : Ring R\nk : ZMod Nat.zero\nhk : \u00ack = 0\n\u22a2 \u2191(k - 1) = \u2191k - 1\n[PROOFSTEP]\ndsimp [ZMod, ZMod.cast]\n[GOAL]\ncase neg.zero\nR : Type u_1\ninst\u271d : Ring R\nk : ZMod Nat.zero\nhk : \u00ack = 0\n\u22a2 \u2191(k - 1) = \u2191k - 1\n[PROOFSTEP]\nrw [Int.cast_sub, Int.cast_one]\n[GOAL]\ncase neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (Nat.succ n\u271d)\nhk : \u00ack = 0\n\u22a2 \u2191(k - 1) = \u2191k - 1\n[PROOFSTEP]\ndsimp [ZMod, ZMod.cast, ZMod.val]\n[GOAL]\ncase neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (Nat.succ n\u271d)\nhk : \u00ack = 0\n\u22a2 \u2191\u2191(k - 1) = \u2191\u2191k - 1\n[PROOFSTEP]\nrw [Fin.coe_sub_one, if_neg]\n[GOAL]\ncase neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (Nat.succ n\u271d)\nhk : \u00ack = 0\n\u22a2 \u2191(\u2191k - 1) = \u2191\u2191k - 1\n[PROOFSTEP]\nrw [Nat.cast_sub, Nat.cast_one]\n[GOAL]\ncase neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (Nat.succ n\u271d)\nhk : \u00ack = 0\n\u22a2 1 \u2264 \u2191k\n[PROOFSTEP]\nrwa [Fin.ext_iff, Fin.val_zero, \u2190 Ne, \u2190 Nat.one_le_iff_ne_zero] at hk \n[GOAL]\ncase neg.succ.hnc\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (Nat.succ n\u271d)\nhk : \u00ack = 0\n\u22a2 \u00ack = 0\n[PROOFSTEP]\nexact hk\n[GOAL]\np n : \u2115\nz : ZMod p\ninst\u271d : NeZero p\n\u22a2 \u2191n = z \u2194 \u2203 k, n = val z + p * k\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\np n : \u2115\nz : ZMod p\ninst\u271d : NeZero p\n\u22a2 \u2191n = z \u2192 \u2203 k, n = val z + p * k\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mp\np n : \u2115\ninst\u271d : NeZero p\n\u22a2 \u2203 k, n = val \u2191n + p * k\n[PROOFSTEP]\nrefine' \u27e8n / p, _\u27e9\n[GOAL]\ncase mp\np n : \u2115\ninst\u271d : NeZero p\n\u22a2 n = val \u2191n + p * (n / p)\n[PROOFSTEP]\nrw [val_nat_cast, Nat.mod_add_div]\n[GOAL]\ncase mpr\np n : \u2115\nz : ZMod p\ninst\u271d : NeZero p\n\u22a2 (\u2203 k, n = val z + p * k) \u2192 \u2191n = z\n[PROOFSTEP]\nrintro \u27e8k, rfl\u27e9\n[GOAL]\ncase mpr.intro\np : \u2115\nz : ZMod p\ninst\u271d : NeZero p\nk : \u2115\n\u22a2 \u2191(val z + p * k) = z\n[PROOFSTEP]\nrw [Nat.cast_add, nat_cast_zmod_val, Nat.cast_mul, nat_cast_self, zero_mul, add_zero]\n[GOAL]\np : \u2115\nn : \u2124\nz : ZMod p\ninst\u271d : NeZero p\n\u22a2 \u2191n = z \u2194 \u2203 k, n = \u2191(val z) + \u2191p * k\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\np : \u2115\nn : \u2124\nz : ZMod p\ninst\u271d : NeZero p\n\u22a2 \u2191n = z \u2192 \u2203 k, n = \u2191(val z) + \u2191p * k\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mp\np : \u2115\nn : \u2124\ninst\u271d : NeZero p\n\u22a2 \u2203 k, n = \u2191(val \u2191n) + \u2191p * k\n[PROOFSTEP]\nrefine' \u27e8n / p, _\u27e9\n[GOAL]\ncase mp\np : \u2115\nn : \u2124\ninst\u271d : NeZero p\n\u22a2 n = \u2191(val \u2191n) + \u2191p * (n / \u2191p)\n[PROOFSTEP]\nrw [val_int_cast, Int.emod_add_ediv]\n[GOAL]\ncase mpr\np : \u2115\nn : \u2124\nz : ZMod p\ninst\u271d : NeZero p\n\u22a2 (\u2203 k, n = \u2191(val z) + \u2191p * k) \u2192 \u2191n = z\n[PROOFSTEP]\nrintro \u27e8k, rfl\u27e9\n[GOAL]\ncase mpr.intro\np : \u2115\nz : ZMod p\ninst\u271d : NeZero p\nk : \u2124\n\u22a2 \u2191(\u2191(val z) + \u2191p * k) = z\n[PROOFSTEP]\nrw [Int.cast_add, Int.cast_mul, Int.cast_ofNat, Int.cast_ofNat, nat_cast_val, ZMod.nat_cast_self, zero_mul, add_zero,\n cast_id]\n[GOAL]\na : \u2124\nb : \u2115\n\u22a2 \u2191(a % \u2191b) = \u2191a\n[PROOFSTEP]\nrw [ZMod.int_cast_eq_int_cast_iff]\n[GOAL]\na : \u2124\nb : \u2115\n\u22a2 a % \u2191b \u2261 a [ZMOD \u2191b]\n[PROOFSTEP]\napply Int.mod_modEq\n[GOAL]\nn : \u2115\n\u22a2 AddMonoidHom.ker (Int.castAddHom (ZMod n)) = AddSubgroup.zmultiples \u2191n\n[PROOFSTEP]\next\n[GOAL]\ncase h\nn : \u2115\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod n)) \u2194 x\u271d \u2208 AddSubgroup.zmultiples \u2191n\n[PROOFSTEP]\nrw [Int.mem_zmultiples_iff, AddMonoidHom.mem_ker, Int.coe_castAddHom, int_cast_zmod_eq_zero_iff_dvd]\n[GOAL]\nn : \u2115\n\u22a2 RingHom.ker (Int.castRingHom (ZMod n)) = Ideal.span {\u2191n}\n[PROOFSTEP]\next\n[GOAL]\ncase h\nn : \u2115\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 RingHom.ker (Int.castRingHom (ZMod n)) \u2194 x\u271d \u2208 Ideal.span {\u2191n}\n[PROOFSTEP]\nrw [Ideal.mem_span_singleton, RingHom.mem_ker, Int.coe_castRingHom, int_cast_zmod_eq_zero_iff_dvd]\n[GOAL]\np n : \u2115\n_h : 0 \u2264 \u2191n\n\u22a2 \u2191(Int.toNat \u2191n) = \u2191\u2191n\n[PROOFSTEP]\nsimp only [Int.cast_ofNat, Int.toNat_coe_nat]\n[GOAL]\np n : \u2115\nh : 0 \u2264 Int.negSucc n\n\u22a2 \u2191(Int.toNat (Int.negSucc n)) = \u2191(Int.negSucc n)\n[PROOFSTEP]\nsimp at h \n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\n\u22a2 Injective val\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\ninst\u271d : NeZero Nat.zero\n\u22a2 Injective val\n[PROOFSTEP]\ncases NeZero.ne 0 rfl\n[GOAL]\ncase succ\nn\u271d : \u2115\ninst\u271d : NeZero (Nat.succ n\u271d)\n\u22a2 Injective val\n[PROOFSTEP]\nintro a b h\n[GOAL]\ncase succ\nn\u271d : \u2115\ninst\u271d : NeZero (Nat.succ n\u271d)\na b : ZMod (Nat.succ n\u271d)\nh : val a = val b\n\u22a2 a = b\n[PROOFSTEP]\ndsimp [ZMod]\n[GOAL]\ncase succ\nn\u271d : \u2115\ninst\u271d : NeZero (Nat.succ n\u271d)\na b : ZMod (Nat.succ n\u271d)\nh : val a = val b\n\u22a2 a = b\n[PROOFSTEP]\next\n[GOAL]\ncase succ.h\nn\u271d : \u2115\ninst\u271d : NeZero (Nat.succ n\u271d)\na b : ZMod (Nat.succ n\u271d)\nh : val a = val b\n\u22a2 \u2191a = \u2191b\n[PROOFSTEP]\nexact h\n[GOAL]\nn : \u2115\n\u22a2 val 1 = 1 % n\n[PROOFSTEP]\nrw [\u2190 Nat.cast_one, val_nat_cast]\n[GOAL]\nn : \u2115\ninst\u271d : Fact (1 < n)\n\u22a2 val 1 = 1\n[PROOFSTEP]\nrw [val_one_eq_one_mod]\n[GOAL]\nn : \u2115\ninst\u271d : Fact (1 < n)\n\u22a2 1 % n = 1\n[PROOFSTEP]\nexact Nat.mod_eq_of_lt Fact.out\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\na b : ZMod n\n\u22a2 val (a + b) = (val a + val b) % n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\ninst\u271d : NeZero Nat.zero\na b : ZMod Nat.zero\n\u22a2 val (a + b) = (val a + val b) % Nat.zero\n[PROOFSTEP]\ncases NeZero.ne 0 rfl\n[GOAL]\ncase succ\nn\u271d : \u2115\ninst\u271d : NeZero (Nat.succ n\u271d)\na b : ZMod (Nat.succ n\u271d)\n\u22a2 val (a + b) = (val a + val b) % Nat.succ n\u271d\n[PROOFSTEP]\napply Fin.val_add\n[GOAL]\nn : \u2115\na b : ZMod n\n\u22a2 val (a * b) = val a * val b % n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\na b : ZMod Nat.zero\n\u22a2 val (a * b) = val a * val b % Nat.zero\n[PROOFSTEP]\nrw [Nat.mod_zero]\n[GOAL]\ncase zero\na b : ZMod Nat.zero\n\u22a2 val (a * b) = val a * val b\n[PROOFSTEP]\napply Int.natAbs_mul\n[GOAL]\ncase succ\nn\u271d : \u2115\na b : ZMod (Nat.succ n\u271d)\n\u22a2 val (a * b) = val a * val b % Nat.succ n\u271d\n[PROOFSTEP]\napply Fin.val_mul\n[GOAL]\nn : \u2115\ninst\u271d : Fact (1 < n)\nh : 0 = 1\n\u22a2 0 = val 0\n[PROOFSTEP]\nrw [val_zero]\n[GOAL]\n\u22a2 Nontrivial (ZMod 0)\n[PROOFSTEP]\ndelta ZMod\n[GOAL]\n\u22a2 Nontrivial\n (match 0 with\n | 0 => \u2124\n | Nat.succ n => Fin (n + 1))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nn : \u2115\n\u22a2 \u2191(Nat.gcdA (val 0) (n + 1)) = 0\n[PROOFSTEP]\nrw [val_zero]\n[GOAL]\nn : \u2115\n\u22a2 \u2191(Nat.gcdA 0 (n + 1)) = 0\n[PROOFSTEP]\nunfold Nat.gcdA Nat.xgcd Nat.xgcdAux\n[GOAL]\nn : \u2115\n\u22a2 \u2191(n + 1, 0, 1).snd.fst = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\na : ZMod n\n\u22a2 a * a\u207b\u00b9 = \u2191(Nat.gcd (val a) n)\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\na : ZMod Nat.zero\n\u22a2 a * a\u207b\u00b9 = \u2191(Nat.gcd (val a) Nat.zero)\n[PROOFSTEP]\ndsimp [ZMod] at a \u22a2\n[GOAL]\ncase zero\na : \u2124\n\u22a2 a * a\u207b\u00b9 = \u2191(Nat.gcd (val a) 0)\n[PROOFSTEP]\ncalc\n _ = a * Int.sign a := rfl\n _ = a.natAbs := by rw [Int.mul_sign]\n _ = a.natAbs.gcd 0 := by rw [Nat.gcd_zero_right]\n[GOAL]\na : \u2124\n\u22a2 a * Int.sign a = \u2191(Int.natAbs a)\n[PROOFSTEP]\nrw [Int.mul_sign]\n[GOAL]\na : \u2124\n\u22a2 \u2191(Int.natAbs a) = \u2191(Nat.gcd (Int.natAbs a) 0)\n[PROOFSTEP]\nrw [Nat.gcd_zero_right]\n[GOAL]\ncase succ\nn : \u2115\na : ZMod (Nat.succ n)\n\u22a2 a * a\u207b\u00b9 = \u2191(Nat.gcd (val a) (Nat.succ n))\n[PROOFSTEP]\ncalc\n a * a\u207b\u00b9 = a * a\u207b\u00b9 + n.succ * Nat.gcdB (val a) n.succ := by rw [nat_cast_self, zero_mul, add_zero]\n _ = \u2191(\u2191a.val * Nat.gcdA (val a) n.succ + n.succ * Nat.gcdB (val a) n.succ) :=\n by\n push_cast\n rw [nat_cast_zmod_val]\n rfl\n _ = Nat.gcd a.val n.succ := by rw [\u2190 Nat.gcd_eq_gcd_ab a.val n.succ]; rfl\n[GOAL]\nn : \u2115\na : ZMod (Nat.succ n)\n\u22a2 a * a\u207b\u00b9 = a * a\u207b\u00b9 + \u2191(Nat.succ n) * \u2191(Nat.gcdB (val a) (Nat.succ n))\n[PROOFSTEP]\nrw [nat_cast_self, zero_mul, add_zero]\n[GOAL]\nn : \u2115\na : ZMod (Nat.succ n)\n\u22a2 a * a\u207b\u00b9 + \u2191(Nat.succ n) * \u2191(Nat.gcdB (val a) (Nat.succ n)) =\n \u2191(\u2191(val a) * Nat.gcdA (val a) (Nat.succ n) + \u2191(Nat.succ n) * Nat.gcdB (val a) (Nat.succ n))\n[PROOFSTEP]\npush_cast\n[GOAL]\nn : \u2115\na : ZMod (Nat.succ n)\n\u22a2 a * a\u207b\u00b9 + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n)) =\n \u2191(val a) * \u2191(Nat.gcdA (val a) (Nat.succ n)) + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n))\n[PROOFSTEP]\nrw [nat_cast_zmod_val]\n[GOAL]\nn : \u2115\na : ZMod (Nat.succ n)\n\u22a2 a * a\u207b\u00b9 + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n)) =\n a * \u2191(Nat.gcdA (val a) (Nat.succ n)) + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n))\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\na : ZMod (Nat.succ n)\n\u22a2 \u2191(\u2191(val a) * Nat.gcdA (val a) (Nat.succ n) + \u2191(Nat.succ n) * Nat.gcdB (val a) (Nat.succ n)) =\n \u2191(Nat.gcd (val a) (Nat.succ n))\n[PROOFSTEP]\nrw [\u2190 Nat.gcd_eq_gcd_ab a.val n.succ]\n[GOAL]\nn : \u2115\na : ZMod (Nat.succ n)\n\u22a2 \u2191\u2191(Nat.gcd (val a) (Nat.succ n)) = \u2191(Nat.gcd (val a) (Nat.succ n))\n[PROOFSTEP]\nrfl\n[GOAL]\na n : \u2115\n\u22a2 \u2191(a % n) = \u2191a\n[PROOFSTEP]\nconv =>\n rhs\n rw [\u2190 Nat.mod_add_div a n]\n[GOAL]\na n : \u2115\n| \u2191(a % n) = \u2191a\n[PROOFSTEP]\n rhs\n rw [\u2190 Nat.mod_add_div a n]\n[GOAL]\na n : \u2115\n| \u2191(a % n) = \u2191a\n[PROOFSTEP]\n rhs\n rw [\u2190 Nat.mod_add_div a n]\n[GOAL]\na n : \u2115\n| \u2191(a % n) = \u2191a\n[PROOFSTEP]\nrhs\n[GOAL]\na n : \u2115\n| \u2191a\n[PROOFSTEP]\nrw [\u2190 Nat.mod_add_div a n]\n[GOAL]\na n : \u2115\n\u22a2 \u2191(a % n) = \u2191(a % n + n * (a / n))\n[PROOFSTEP]\nsimp\n[GOAL]\nn a b : \u2115\n\u22a2 \u2191a = \u2191b \u2194 a \u2261 b [MOD n]\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\na b : \u2115\n\u22a2 \u2191a = \u2191b \u2194 a \u2261 b [MOD Nat.zero]\n[PROOFSTEP]\nsimp [Nat.ModEq, Int.coe_nat_inj', Nat.mod_zero]\n[GOAL]\ncase succ\na b n\u271d : \u2115\n\u22a2 \u2191a = \u2191b \u2194 a \u2261 b [MOD Nat.succ n\u271d]\n[PROOFSTEP]\nrw [Fin.ext_iff, Nat.ModEq, \u2190 val_nat_cast, \u2190 val_nat_cast]\n[GOAL]\ncase succ\na b n\u271d : \u2115\n\u22a2 \u2191\u2191a = \u2191\u2191b \u2194 val \u2191a = val \u2191b\n[PROOFSTEP]\nexact Iff.rfl\n[GOAL]\nn x : \u2115\nh : Nat.coprime x n\n\u22a2 \u2191x * (\u2191x)\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [Nat.coprime, Nat.gcd_comm, Nat.gcd_rec] at h \n[GOAL]\nn x : \u2115\nh : Nat.gcd (x % n) n = 1\n\u22a2 \u2191x * (\u2191x)\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [mul_inv_eq_gcd, val_nat_cast, h, Nat.cast_one]\n[GOAL]\nn x : \u2115\nh : Nat.coprime x n\n\u22a2 (\u2191x)\u207b\u00b9 * \u2191x = 1\n[PROOFSTEP]\nrw [mul_comm, coe_mul_inv_eq_one x h]\n[GOAL]\nn : \u2115\nu : (ZMod n)\u02e3\n\u22a2 Nat.coprime (val \u2191u) n\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nu : (ZMod Nat.zero)\u02e3\n\u22a2 Nat.coprime (val \u2191u) Nat.zero\n[PROOFSTEP]\nrcases Int.units_eq_one_or u with (rfl | rfl)\n[GOAL]\ncase zero.inl\n\u22a2 Nat.coprime (val \u21911) Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase zero.inr\n\u22a2 Nat.coprime (val \u2191(-1)) Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\n\u22a2 Nat.coprime (val \u2191u) (Nat.succ n)\n[PROOFSTEP]\napply Nat.coprime_of_mul_modEq_one ((u\u207b\u00b9 : Units (ZMod (n + 1))) : ZMod (n + 1)).val\n[GOAL]\ncase succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\n\u22a2 val \u2191u * val \u2191u\u207b\u00b9 \u2261 1 [MOD Nat.succ n]\n[PROOFSTEP]\nhave := Units.ext_iff.1 (mul_right_inv u)\n[GOAL]\ncase succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\nthis : \u2191(u * u\u207b\u00b9) = \u21911\n\u22a2 val \u2191u * val \u2191u\u207b\u00b9 \u2261 1 [MOD Nat.succ n]\n[PROOFSTEP]\nrw [Units.val_one] at this \n[GOAL]\ncase succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\nthis : \u2191(u * u\u207b\u00b9) = 1\n\u22a2 val \u2191u * val \u2191u\u207b\u00b9 \u2261 1 [MOD Nat.succ n]\n[PROOFSTEP]\nrw [\u2190 eq_iff_modEq_nat, Nat.cast_one, \u2190 this]\n[GOAL]\ncase succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\nthis : \u2191(u * u\u207b\u00b9) = 1\n\u22a2 \u2191(val \u2191u * val \u2191u\u207b\u00b9) = \u2191(u * u\u207b\u00b9)\n[PROOFSTEP]\nclear this\n[GOAL]\ncase succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\n\u22a2 \u2191(val \u2191u * val \u2191u\u207b\u00b9) = \u2191(u * u\u207b\u00b9)\n[PROOFSTEP]\nrw [\u2190 nat_cast_zmod_val ((u * u\u207b\u00b9 : Units (ZMod (n + 1))) : ZMod (n + 1))]\n[GOAL]\ncase succ\nn : \u2115\nu : (ZMod (Nat.succ n))\u02e3\n\u22a2 \u2191(val \u2191u * val \u2191u\u207b\u00b9) = \u2191(val \u2191(u * u\u207b\u00b9))\n[PROOFSTEP]\nrw [Units.val_mul, val_mul, nat_cast_mod]\n[GOAL]\nn : \u2115\nu : (ZMod n)\u02e3\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9\n[PROOFSTEP]\nhave := congr_arg ((\u2191) : \u2115 \u2192 ZMod n) (val_coe_unit_coprime u)\n[GOAL]\nn : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191(Nat.gcd (val \u2191u) n) = \u21911\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 mul_inv_eq_gcd, Nat.cast_one] at this \n[GOAL]\nn : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9\n[PROOFSTEP]\nlet u' : (ZMod n)\u02e3 := \u27e8u, (u : ZMod n)\u207b\u00b9, this, by rwa [mul_comm]\u27e9\n[GOAL]\nn : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\n\u22a2 (\u2191u)\u207b\u00b9 * \u2191u = 1\n[PROOFSTEP]\nrwa [mul_comm]\n[GOAL]\nn : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9\n[PROOFSTEP]\nhave h : u = u' := by\n apply Units.ext\n rfl\n[GOAL]\nn : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\n\u22a2 u = u'\n[PROOFSTEP]\napply Units.ext\n[GOAL]\ncase a\nn : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\n\u22a2 \u2191u = \u2191u'\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\nh : u = u'\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9\n[PROOFSTEP]\nrw [h]\n[GOAL]\nn : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\nh : u = u'\n\u22a2 (\u2191u')\u207b\u00b9 = \u2191u'\u207b\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\na : ZMod n\nh : IsUnit a\n\u22a2 a * a\u207b\u00b9 = 1\n[PROOFSTEP]\nrcases h with \u27e8u, rfl\u27e9\n[GOAL]\ncase intro\nn : \u2115\nu : (ZMod n)\u02e3\n\u22a2 \u2191u * (\u2191u)\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [inv_coe_unit, u.mul_inv]\n[GOAL]\nn : \u2115\na : ZMod n\nh : IsUnit a\n\u22a2 a\u207b\u00b9 * a = 1\n[PROOFSTEP]\nrw [mul_comm, mul_inv_of_unit a h]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\nx\u271d : { x // Nat.coprime (val x) n }\nval\u271d : ZMod n\nproperty\u271d : Nat.coprime (val val\u271d) n\n\u22a2 (fun x => { val := \u2191x, property := (_ : Nat.coprime (val \u2191x) n) })\n ((fun x => unitOfCoprime (val \u2191x) (_ : Nat.coprime (val \u2191x) n)) { val := val\u271d, property := property\u271d }) =\n { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nsimp\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\n\u22a2 Nat.lcm m n \u2223 m * n\n[PROOFSTEP]\nsimp [Nat.lcm_dvd_iff]\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : m * n = 0\n\u22a2 LeftInverse inv_fun to_fun \u2227 Function.RightInverse inv_fun to_fun\n[PROOFSTEP]\nrcases h.eq_of_mul_eq_zero hmn0 with (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9)\n[GOAL]\ncase inl.intro\nh : Nat.coprime 0 1\nto_fun : ZMod (0 * 1) \u2192 ZMod 0 \u00d7 ZMod 1 := \u2191(castHom (_ : Nat.lcm 0 1 \u2223 0 * 1) (ZMod 0 \u00d7 ZMod 1))\ninv_fun : ZMod 0 \u00d7 ZMod 1 \u2192 ZMod (0 * 1) :=\n fun x =>\n if 0 * 1 = 0 then if 0 = 1 then \u2191(\u2191(RingHom.snd (ZMod 0) (ZMod 1)) x) else \u2191(\u2191(RingHom.fst (ZMod 0) (ZMod 1)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : 0 * 1 = 0\n\u22a2 LeftInverse inv_fun to_fun \u2227 Function.RightInverse inv_fun to_fun\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase inl.intro.left\nh : Nat.coprime 0 1\nto_fun : ZMod (0 * 1) \u2192 ZMod 0 \u00d7 ZMod 1 := \u2191(castHom (_ : Nat.lcm 0 1 \u2223 0 * 1) (ZMod 0 \u00d7 ZMod 1))\ninv_fun : ZMod 0 \u00d7 ZMod 1 \u2192 ZMod (0 * 1) :=\n fun x =>\n if 0 * 1 = 0 then if 0 = 1 then \u2191(\u2191(RingHom.snd (ZMod 0) (ZMod 1)) x) else \u2191(\u2191(RingHom.fst (ZMod 0) (ZMod 1)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : 0 * 1 = 0\n\u22a2 LeftInverse inv_fun to_fun\n[PROOFSTEP]\nintro x\n[GOAL]\ncase inl.intro.left\nh : Nat.coprime 0 1\nto_fun : ZMod (0 * 1) \u2192 ZMod 0 \u00d7 ZMod 1 := \u2191(castHom (_ : Nat.lcm 0 1 \u2223 0 * 1) (ZMod 0 \u00d7 ZMod 1))\ninv_fun : ZMod 0 \u00d7 ZMod 1 \u2192 ZMod (0 * 1) :=\n fun x =>\n if 0 * 1 = 0 then if 0 = 1 then \u2191(\u2191(RingHom.snd (ZMod 0) (ZMod 1)) x) else \u2191(\u2191(RingHom.fst (ZMod 0) (ZMod 1)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : 0 * 1 = 0\nx : ZMod (0 * 1)\n\u22a2 inv_fun (to_fun x) = x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inl.intro.right\nh : Nat.coprime 0 1\nto_fun : ZMod (0 * 1) \u2192 ZMod 0 \u00d7 ZMod 1 := \u2191(castHom (_ : Nat.lcm 0 1 \u2223 0 * 1) (ZMod 0 \u00d7 ZMod 1))\ninv_fun : ZMod 0 \u00d7 ZMod 1 \u2192 ZMod (0 * 1) :=\n fun x =>\n if 0 * 1 = 0 then if 0 = 1 then \u2191(\u2191(RingHom.snd (ZMod 0) (ZMod 1)) x) else \u2191(\u2191(RingHom.fst (ZMod 0) (ZMod 1)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : 0 * 1 = 0\n\u22a2 Function.RightInverse inv_fun to_fun\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9\n[GOAL]\ncase inl.intro.right.mk\nh : Nat.coprime 0 1\nto_fun : ZMod (0 * 1) \u2192 ZMod 0 \u00d7 ZMod 1 := \u2191(castHom (_ : Nat.lcm 0 1 \u2223 0 * 1) (ZMod 0 \u00d7 ZMod 1))\ninv_fun : ZMod 0 \u00d7 ZMod 1 \u2192 ZMod (0 * 1) :=\n fun x =>\n if 0 * 1 = 0 then if 0 = 1 then \u2191(\u2191(RingHom.snd (ZMod 0) (ZMod 1)) x) else \u2191(\u2191(RingHom.fst (ZMod 0) (ZMod 1)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : 0 * 1 = 0\nx : ZMod 0\ny : ZMod 1\n\u22a2 to_fun (inv_fun (x, y)) = (x, y)\n[PROOFSTEP]\nfin_cases y\n[GOAL]\ncase inl.intro.right.mk.head\nh : Nat.coprime 0 1\nto_fun : ZMod (0 * 1) \u2192 ZMod 0 \u00d7 ZMod 1 := \u2191(castHom (_ : Nat.lcm 0 1 \u2223 0 * 1) (ZMod 0 \u00d7 ZMod 1))\ninv_fun : ZMod 0 \u00d7 ZMod 1 \u2192 ZMod (0 * 1) :=\n fun x =>\n if 0 * 1 = 0 then if 0 = 1 then \u2191(\u2191(RingHom.snd (ZMod 0) (ZMod 1)) x) else \u2191(\u2191(RingHom.fst (ZMod 0) (ZMod 1)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : 0 * 1 = 0\nx : ZMod 0\n\u22a2 to_fun (inv_fun (x, { val := 0, isLt := (_ : 0 < 0 + 1) })) = (x, { val := 0, isLt := (_ : 0 < 0 + 1) })\n[PROOFSTEP]\nsimp [castHom, Prod.ext_iff]\n[GOAL]\ncase inr.intro\nh : Nat.coprime 1 0\nto_fun : ZMod (1 * 0) \u2192 ZMod 1 \u00d7 ZMod 0 := \u2191(castHom (_ : Nat.lcm 1 0 \u2223 1 * 0) (ZMod 1 \u00d7 ZMod 0))\ninv_fun : ZMod 1 \u00d7 ZMod 0 \u2192 ZMod (1 * 0) :=\n fun x =>\n if 1 * 0 = 0 then if 1 = 1 then \u2191(\u2191(RingHom.snd (ZMod 1) (ZMod 0)) x) else \u2191(\u2191(RingHom.fst (ZMod 1) (ZMod 0)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : 1 * 0 = 0\n\u22a2 LeftInverse inv_fun to_fun \u2227 Function.RightInverse inv_fun to_fun\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase inr.intro.left\nh : Nat.coprime 1 0\nto_fun : ZMod (1 * 0) \u2192 ZMod 1 \u00d7 ZMod 0 := \u2191(castHom (_ : Nat.lcm 1 0 \u2223 1 * 0) (ZMod 1 \u00d7 ZMod 0))\ninv_fun : ZMod 1 \u00d7 ZMod 0 \u2192 ZMod (1 * 0) :=\n fun x =>\n if 1 * 0 = 0 then if 1 = 1 then \u2191(\u2191(RingHom.snd (ZMod 1) (ZMod 0)) x) else \u2191(\u2191(RingHom.fst (ZMod 1) (ZMod 0)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : 1 * 0 = 0\n\u22a2 LeftInverse inv_fun to_fun\n[PROOFSTEP]\nintro x\n[GOAL]\ncase inr.intro.left\nh : Nat.coprime 1 0\nto_fun : ZMod (1 * 0) \u2192 ZMod 1 \u00d7 ZMod 0 := \u2191(castHom (_ : Nat.lcm 1 0 \u2223 1 * 0) (ZMod 1 \u00d7 ZMod 0))\ninv_fun : ZMod 1 \u00d7 ZMod 0 \u2192 ZMod (1 * 0) :=\n fun x =>\n if 1 * 0 = 0 then if 1 = 1 then \u2191(\u2191(RingHom.snd (ZMod 1) (ZMod 0)) x) else \u2191(\u2191(RingHom.fst (ZMod 1) (ZMod 0)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : 1 * 0 = 0\nx : ZMod (1 * 0)\n\u22a2 inv_fun (to_fun x) = x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr.intro.right\nh : Nat.coprime 1 0\nto_fun : ZMod (1 * 0) \u2192 ZMod 1 \u00d7 ZMod 0 := \u2191(castHom (_ : Nat.lcm 1 0 \u2223 1 * 0) (ZMod 1 \u00d7 ZMod 0))\ninv_fun : ZMod 1 \u00d7 ZMod 0 \u2192 ZMod (1 * 0) :=\n fun x =>\n if 1 * 0 = 0 then if 1 = 1 then \u2191(\u2191(RingHom.snd (ZMod 1) (ZMod 0)) x) else \u2191(\u2191(RingHom.fst (ZMod 1) (ZMod 0)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : 1 * 0 = 0\n\u22a2 Function.RightInverse inv_fun to_fun\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9\n[GOAL]\ncase inr.intro.right.mk\nh : Nat.coprime 1 0\nto_fun : ZMod (1 * 0) \u2192 ZMod 1 \u00d7 ZMod 0 := \u2191(castHom (_ : Nat.lcm 1 0 \u2223 1 * 0) (ZMod 1 \u00d7 ZMod 0))\ninv_fun : ZMod 1 \u00d7 ZMod 0 \u2192 ZMod (1 * 0) :=\n fun x =>\n if 1 * 0 = 0 then if 1 = 1 then \u2191(\u2191(RingHom.snd (ZMod 1) (ZMod 0)) x) else \u2191(\u2191(RingHom.fst (ZMod 1) (ZMod 0)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : 1 * 0 = 0\nx : ZMod 1\ny : ZMod 0\n\u22a2 to_fun (inv_fun (x, y)) = (x, y)\n[PROOFSTEP]\nfin_cases x\n[GOAL]\ncase inr.intro.right.mk.head\nh : Nat.coprime 1 0\nto_fun : ZMod (1 * 0) \u2192 ZMod 1 \u00d7 ZMod 0 := \u2191(castHom (_ : Nat.lcm 1 0 \u2223 1 * 0) (ZMod 1 \u00d7 ZMod 0))\ninv_fun : ZMod 1 \u00d7 ZMod 0 \u2192 ZMod (1 * 0) :=\n fun x =>\n if 1 * 0 = 0 then if 1 = 1 then \u2191(\u2191(RingHom.snd (ZMod 1) (ZMod 0)) x) else \u2191(\u2191(RingHom.fst (ZMod 1) (ZMod 0)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : 1 * 0 = 0\ny : ZMod 0\n\u22a2 to_fun (inv_fun ({ val := 0, isLt := (_ : 0 < 0 + 1) }, y)) = ({ val := 0, isLt := (_ : 0 < 0 + 1) }, y)\n[PROOFSTEP]\nsimp [castHom, Prod.ext_iff]\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\n\u22a2 LeftInverse inv_fun to_fun \u2227 Function.RightInverse inv_fun to_fun\n[PROOFSTEP]\nhaveI : NeZero (m * n) := \u27e8hmn0\u27e9\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\nthis : NeZero (m * n)\n\u22a2 LeftInverse inv_fun to_fun \u2227 Function.RightInverse inv_fun to_fun\n[PROOFSTEP]\nhaveI : NeZero m := \u27e8left_ne_zero_of_mul hmn0\u27e9\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\nthis\u271d : NeZero (m * n)\nthis : NeZero m\n\u22a2 LeftInverse inv_fun to_fun \u2227 Function.RightInverse inv_fun to_fun\n[PROOFSTEP]\nhaveI : NeZero n := \u27e8right_ne_zero_of_mul hmn0\u27e9\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\nthis\u271d\u00b9 : NeZero (m * n)\nthis\u271d : NeZero m\nthis : NeZero n\n\u22a2 LeftInverse inv_fun to_fun \u2227 Function.RightInverse inv_fun to_fun\n[PROOFSTEP]\nhave left_inv : Function.LeftInverse inv_fun to_fun := by\n intro x\n dsimp only [dvd_mul_left, dvd_mul_right, ZMod.castHom_apply]\n conv_rhs => rw [\u2190 ZMod.nat_cast_zmod_val x]\n rw [if_neg hmn0, ZMod.eq_iff_modEq_nat, \u2190 Nat.modEq_and_modEq_iff_modEq_mul h, Prod.fst_zmod_cast, Prod.snd_zmod_cast]\n refine'\n \u27e8(Nat.chineseRemainder h (x : ZMod m).val (x : ZMod n).val).2.left.trans _,\n (Nat.chineseRemainder h (x : ZMod m).val (x : ZMod n).val).2.right.trans _\u27e9\n \u00b7 rw [\u2190 ZMod.eq_iff_modEq_nat, ZMod.nat_cast_zmod_val, ZMod.nat_cast_val]\n \u00b7 rw [\u2190 ZMod.eq_iff_modEq_nat, ZMod.nat_cast_zmod_val, ZMod.nat_cast_val]\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\nthis\u271d\u00b9 : NeZero (m * n)\nthis\u271d : NeZero m\nthis : NeZero n\n\u22a2 LeftInverse inv_fun to_fun\n[PROOFSTEP]\nintro x\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\nthis\u271d\u00b9 : NeZero (m * n)\nthis\u271d : NeZero m\nthis : NeZero n\nx : ZMod (m * n)\n\u22a2 inv_fun (to_fun x) = x\n[PROOFSTEP]\ndsimp only [dvd_mul_left, dvd_mul_right, ZMod.castHom_apply]\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\nthis\u271d\u00b9 : NeZero (m * n)\nthis\u271d : NeZero m\nthis : NeZero n\nx : ZMod (m * n)\n\u22a2 (if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) \u2191x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) \u2191x)\n else \u2191\u2191(Nat.chineseRemainder h (val (\u2191x).fst) (val (\u2191x).snd))) =\n x\n[PROOFSTEP]\nconv_rhs => rw [\u2190 ZMod.nat_cast_zmod_val x]\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\nthis\u271d\u00b9 : NeZero (m * n)\nthis\u271d : NeZero m\nthis : NeZero n\nx : ZMod (m * n)\n| x\n[PROOFSTEP]\nrw [\u2190 ZMod.nat_cast_zmod_val x]\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\nthis\u271d\u00b9 : NeZero (m * n)\nthis\u271d : NeZero m\nthis : NeZero n\nx : ZMod (m * n)\n| x\n[PROOFSTEP]\nrw [\u2190 ZMod.nat_cast_zmod_val x]\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\nthis\u271d\u00b9 : NeZero (m * n)\nthis\u271d : NeZero m\nthis : NeZero n\nx : ZMod (m * n)\n| x\n[PROOFSTEP]\nrw [\u2190 ZMod.nat_cast_zmod_val x]\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\nthis\u271d\u00b9 : NeZero (m * n)\nthis\u271d : NeZero m\nthis : NeZero n\nx : ZMod (m * n)\n\u22a2 (if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) \u2191x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) \u2191x)\n else \u2191\u2191(Nat.chineseRemainder h (val (\u2191x).fst) (val (\u2191x).snd))) =\n \u2191(val x)\n[PROOFSTEP]\nrw [if_neg hmn0, ZMod.eq_iff_modEq_nat, \u2190 Nat.modEq_and_modEq_iff_modEq_mul h, Prod.fst_zmod_cast, Prod.snd_zmod_cast]\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\nthis\u271d\u00b9 : NeZero (m * n)\nthis\u271d : NeZero m\nthis : NeZero n\nx : ZMod (m * n)\n\u22a2 \u2191(Nat.chineseRemainder h (val \u2191x) (val \u2191x)) \u2261 val x [MOD m] \u2227\n \u2191(Nat.chineseRemainder h (val \u2191x) (val \u2191x)) \u2261 val x [MOD n]\n[PROOFSTEP]\nrefine'\n \u27e8(Nat.chineseRemainder h (x : ZMod m).val (x : ZMod n).val).2.left.trans _,\n (Nat.chineseRemainder h (x : ZMod m).val (x : ZMod n).val).2.right.trans _\u27e9\n[GOAL]\ncase refine'_1\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\nthis\u271d\u00b9 : NeZero (m * n)\nthis\u271d : NeZero m\nthis : NeZero n\nx : ZMod (m * n)\n\u22a2 val \u2191x \u2261 val x [MOD m]\n[PROOFSTEP]\nrw [\u2190 ZMod.eq_iff_modEq_nat, ZMod.nat_cast_zmod_val, ZMod.nat_cast_val]\n[GOAL]\ncase refine'_2\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\nthis\u271d\u00b9 : NeZero (m * n)\nthis\u271d : NeZero m\nthis : NeZero n\nx : ZMod (m * n)\n\u22a2 val \u2191x \u2261 val x [MOD n]\n[PROOFSTEP]\nrw [\u2190 ZMod.eq_iff_modEq_nat, ZMod.nat_cast_zmod_val, ZMod.nat_cast_val]\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\nthis\u271d\u00b9 : NeZero (m * n)\nthis\u271d : NeZero m\nthis : NeZero n\nleft_inv : LeftInverse inv_fun to_fun\n\u22a2 LeftInverse inv_fun to_fun \u2227 Function.RightInverse inv_fun to_fun\n[PROOFSTEP]\nexact \u27e8left_inv, left_inv.rightInverse_of_card_le (by simp)\u27e9\n[GOAL]\nm n : \u2115\nh : Nat.coprime m n\nto_fun : ZMod (m * n) \u2192 ZMod m \u00d7 ZMod n := \u2191(castHom (_ : Nat.lcm m n \u2223 m * n) (ZMod m \u00d7 ZMod n))\ninv_fun : ZMod m \u00d7 ZMod n \u2192 ZMod (m * n) :=\n fun x =>\n if m * n = 0 then if m = 1 then \u2191(\u2191(RingHom.snd (ZMod m) (ZMod n)) x) else \u2191(\u2191(RingHom.fst (ZMod m) (ZMod n)) x)\n else \u2191\u2191(Nat.chineseRemainder h (val x.fst) (val x.snd))\nhmn0 : \u00acm * n = 0\nthis\u271d\u00b9 : NeZero (m * n)\nthis\u271d : NeZero m\nthis : NeZero n\nleft_inv : LeftInverse inv_fun to_fun\n\u22a2 Fintype.card (ZMod m \u00d7 ZMod n) \u2264 Fintype.card (ZMod (m * n))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2200 (a b : (ZMod 2)\u02e3), a = b\n[PROOFSTEP]\ndecide\n[GOAL]\nn : \u2115\nhn : Odd n\na : ZMod n\n\u22a2 a + a = 0 \u2194 a = 0\n[PROOFSTEP]\nrw [Nat.odd_iff, \u2190 Nat.two_dvd_ne_zero, \u2190 Nat.prime_two.coprime_iff_not_dvd] at hn \n[GOAL]\nn : \u2115\nhn : Nat.coprime 2 n\na : ZMod n\n\u22a2 a + a = 0 \u2194 a = 0\n[PROOFSTEP]\nrw [\u2190 mul_two, \u2190 @Nat.cast_two (ZMod n), \u2190 ZMod.coe_unitOfCoprime 2 hn, Units.mul_left_eq_zero]\n[GOAL]\nn : \u2115\nhn : Odd n\na : ZMod n\nha : a \u2260 0\n\u22a2 a \u2260 -a\n[PROOFSTEP]\nrwa [Ne, eq_neg_iff_add_eq_zero, add_self_eq_zero_iff_eq_zero hn]\n[GOAL]\na : ZMod 2\n\u22a2 -a = a\n[PROOFSTEP]\nfin_cases a\n[GOAL]\ncase head\n\u22a2 -{ val := 0, isLt := (_ : 0 < 1 + 1) } = { val := 0, isLt := (_ : 0 < 1 + 1) }\n[PROOFSTEP]\napply Fin.ext\n[GOAL]\ncase tail.head\n\u22a2 -{ val := 1, isLt := (_ : (fun a => a < 1 + 1) 1) } = { val := 1, isLt := (_ : (fun a => a < 1 + 1) 1) }\n[PROOFSTEP]\napply Fin.ext\n[GOAL]\ncase head.h\n\u22a2 \u2191(-{ val := 0, isLt := (_ : 0 < 1 + 1) }) = \u2191{ val := 0, isLt := (_ : 0 < 1 + 1) }\n[PROOFSTEP]\nsimp [Fin.coe_neg, Int.natMod]\n[GOAL]\ncase tail.head.h\n\u22a2 \u2191(-{ val := 1, isLt := (_ : (fun a => a < 1 + 1) 1) }) = \u2191{ val := 1, isLt := (_ : (fun a => a < 1 + 1) 1) }\n[PROOFSTEP]\nsimp [Fin.coe_neg, Int.natMod]\n[GOAL]\na : \u2124\n\u22a2 \u2191(Int.natAbs a) = \u2191a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase ofNat\na\u271d : \u2115\n\u22a2 \u2191(Int.natAbs (Int.ofNat a\u271d)) = \u2191(Int.ofNat a\u271d)\n[PROOFSTEP]\nsimp only [Int.natAbs_ofNat, Int.cast_ofNat, Int.ofNat_eq_coe]\n[GOAL]\ncase negSucc\na\u271d : \u2115\n\u22a2 \u2191(Int.natAbs (Int.negSucc a\u271d)) = \u2191(Int.negSucc a\u271d)\n[PROOFSTEP]\nsimp only [neg_eq_self_mod_two, Nat.cast_succ, Int.natAbs, Int.cast_negSucc]\n[GOAL]\nn : \u2115\na : ZMod (n + 1)\n\u22a2 val a = 0 \u2194 a = 0\n[PROOFSTEP]\nrw [Fin.ext_iff]\n[GOAL]\nn : \u2115\na : ZMod (n + 1)\n\u22a2 val a = 0 \u2194 \u2191a = \u21910\n[PROOFSTEP]\nexact Iff.rfl\n[GOAL]\nn : \u2115\na : ZMod n\n\u22a2 -a = a \u2194 a = 0 \u2228 2 * val a = n\n[PROOFSTEP]\nrw [neg_eq_iff_add_eq_zero, \u2190 two_mul]\n[GOAL]\nn : \u2115\na : ZMod n\n\u22a2 2 * a = 0 \u2194 a = 0 \u2228 2 * val a = n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\na : ZMod Nat.zero\n\u22a2 2 * a = 0 \u2194 a = 0 \u2228 2 * val a = Nat.zero\n[PROOFSTEP]\nerw [@mul_eq_zero \u2124, @mul_eq_zero \u2115, val_eq_zero]\n[GOAL]\ncase zero\na : ZMod Nat.zero\n\u22a2 2 = 0 \u2228 a = 0 \u2194 a = 0 \u2228 2 = 0 \u2228 a = 0\n[PROOFSTEP]\nexact \u27e8fun h => h.elim (by simp) Or.inl, fun h => Or.inr (h.elim id fun h => h.elim (by simp) id)\u27e9\n[GOAL]\na : ZMod Nat.zero\nh : 2 = 0 \u2228 a = 0\n\u22a2 2 = 0 \u2192 a = 0 \u2228 2 = 0 \u2228 a = 0\n[PROOFSTEP]\nsimp\n[GOAL]\na : ZMod Nat.zero\nh\u271d : a = 0 \u2228 2 = 0 \u2228 a = 0\nh : 2 = 0 \u2228 a = 0\n\u22a2 2 = 0 \u2192 a = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\n\u22a2 2 * a = 0 \u2194 a = 0 \u2228 2 * val a = Nat.succ n\u271d\n[PROOFSTEP]\nconv_lhs => rw [\u2190 a.nat_cast_zmod_val, \u2190 Nat.cast_two, \u2190 Nat.cast_mul, nat_cast_zmod_eq_zero_iff_dvd]\n[GOAL]\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\n| 2 * a = 0\n[PROOFSTEP]\nrw [\u2190 a.nat_cast_zmod_val, \u2190 Nat.cast_two, \u2190 Nat.cast_mul, nat_cast_zmod_eq_zero_iff_dvd]\n[GOAL]\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\n| 2 * a = 0\n[PROOFSTEP]\nrw [\u2190 a.nat_cast_zmod_val, \u2190 Nat.cast_two, \u2190 Nat.cast_mul, nat_cast_zmod_eq_zero_iff_dvd]\n[GOAL]\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\n| 2 * a = 0\n[PROOFSTEP]\nrw [\u2190 a.nat_cast_zmod_val, \u2190 Nat.cast_two, \u2190 Nat.cast_mul, nat_cast_zmod_eq_zero_iff_dvd]\n[GOAL]\ncase succ\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\n\u22a2 Nat.succ n\u271d \u2223 2 * val a \u2194 a = 0 \u2228 2 * val a = Nat.succ n\u271d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase succ.mp\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\n\u22a2 Nat.succ n\u271d \u2223 2 * val a \u2192 a = 0 \u2228 2 * val a = Nat.succ n\u271d\n[PROOFSTEP]\nrintro \u27e8m, he\u27e9\n[GOAL]\ncase succ.mp.intro\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\nm : \u2115\nhe : 2 * val a = Nat.succ n\u271d * m\n\u22a2 a = 0 \u2228 2 * val a = Nat.succ n\u271d\n[PROOFSTEP]\ncases' m with m\n[GOAL]\ncase succ.mp.intro.zero\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\nhe : 2 * val a = Nat.succ n\u271d * Nat.zero\n\u22a2 a = 0 \u2228 2 * val a = Nat.succ n\u271d\n[PROOFSTEP]\nerw [mul_zero, mul_eq_zero] at he \n[GOAL]\ncase succ.mp.intro.zero\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\nhe : 2 = 0 \u2228 val a = 0\n\u22a2 a = 0 \u2228 2 * val a = Nat.succ n\u271d\n[PROOFSTEP]\nrcases he with (\u27e8\u27e8\u27e9\u27e9 | he)\n[GOAL]\ncase succ.mp.intro.zero.inr\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\nhe : val a = 0\n\u22a2 a = 0 \u2228 2 * val a = Nat.succ n\u271d\n[PROOFSTEP]\nexact Or.inl (a.val_eq_zero.1 he)\n[GOAL]\ncase succ.mp.intro.succ\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\nm : \u2115\nhe : 2 * val a = Nat.succ n\u271d * Nat.succ m\n\u22a2 a = 0 \u2228 2 * val a = Nat.succ n\u271d\n[PROOFSTEP]\ncases m\n[GOAL]\ncase succ.mp.intro.succ.zero\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\nhe : 2 * val a = Nat.succ n\u271d * Nat.succ Nat.zero\n\u22a2 a = 0 \u2228 2 * val a = Nat.succ n\u271d\n[PROOFSTEP]\nright\n[GOAL]\ncase succ.mp.intro.succ.zero.h\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\nhe : 2 * val a = Nat.succ n\u271d * Nat.succ Nat.zero\n\u22a2 2 * val a = Nat.succ n\u271d\n[PROOFSTEP]\nrwa [show Nat.succ Nat.zero = 1 from rfl, mul_one] at he \n[GOAL]\ncase succ.mp.intro.succ.succ\nn\u271d\u00b9 : \u2115\na : ZMod (Nat.succ n\u271d\u00b9)\nn\u271d : \u2115\nhe : 2 * val a = Nat.succ n\u271d\u00b9 * Nat.succ (Nat.succ n\u271d)\n\u22a2 a = 0 \u2228 2 * val a = Nat.succ n\u271d\u00b9\n[PROOFSTEP]\nrefine' (a.val_lt.not_le <| Nat.le_of_mul_le_mul_left _ zero_lt_two).elim\n[GOAL]\ncase succ.mp.intro.succ.succ\nn\u271d\u00b9 : \u2115\na : ZMod (Nat.succ n\u271d\u00b9)\nn\u271d : \u2115\nhe : 2 * val a = Nat.succ n\u271d\u00b9 * Nat.succ (Nat.succ n\u271d)\n\u22a2 2 * Nat.succ n\u271d\u00b9 \u2264 2 * val a\n[PROOFSTEP]\nrw [he, mul_comm]\n[GOAL]\ncase succ.mp.intro.succ.succ\nn\u271d\u00b9 : \u2115\na : ZMod (Nat.succ n\u271d\u00b9)\nn\u271d : \u2115\nhe : 2 * val a = Nat.succ n\u271d\u00b9 * Nat.succ (Nat.succ n\u271d)\n\u22a2 Nat.succ n\u271d\u00b9 * 2 \u2264 Nat.succ n\u271d\u00b9 * Nat.succ (Nat.succ n\u271d)\n[PROOFSTEP]\napply Nat.mul_le_mul_left\n[GOAL]\ncase succ.mp.intro.succ.succ.h\nn\u271d\u00b9 : \u2115\na : ZMod (Nat.succ n\u271d\u00b9)\nn\u271d : \u2115\nhe : 2 * val a = Nat.succ n\u271d\u00b9 * Nat.succ (Nat.succ n\u271d)\n\u22a2 2 \u2264 Nat.succ (Nat.succ n\u271d)\n[PROOFSTEP]\nerw [Nat.succ_le_succ_iff, Nat.succ_le_succ_iff]\n[GOAL]\ncase succ.mp.intro.succ.succ.h\nn\u271d\u00b9 : \u2115\na : ZMod (Nat.succ n\u271d\u00b9)\nn\u271d : \u2115\nhe : 2 * val a = Nat.succ n\u271d\u00b9 * Nat.succ (Nat.succ n\u271d)\n\u22a2 0 \u2264 n\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.mpr\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\n\u22a2 a = 0 \u2228 2 * val a = Nat.succ n\u271d \u2192 Nat.succ n\u271d \u2223 2 * val a\n[PROOFSTEP]\nrintro (rfl | h)\n[GOAL]\ncase succ.mpr.inl\nn\u271d : \u2115\n\u22a2 Nat.succ n\u271d \u2223 2 * val 0\n[PROOFSTEP]\nrw [val_zero, mul_zero]\n[GOAL]\ncase succ.mpr.inl\nn\u271d : \u2115\n\u22a2 Nat.succ n\u271d \u2223 0\n[PROOFSTEP]\napply dvd_zero\n[GOAL]\ncase succ.mpr.inr\nn\u271d : \u2115\na : ZMod (Nat.succ n\u271d)\nh : 2 * val a = Nat.succ n\u271d\n\u22a2 Nat.succ n\u271d \u2223 2 * val a\n[PROOFSTEP]\nrw [h]\n[GOAL]\nn a : \u2115\nh : a < n\n\u22a2 val \u2191a = a\n[PROOFSTEP]\nrw [val_nat_cast, Nat.mod_eq_of_lt h]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 val (-a) = val (-a) % n\n[PROOFSTEP]\nrw [Nat.mod_eq_of_lt (-a).val_lt]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 val (-a) + ?m.561595 a \u2261 n - val a + ?m.561595 a [MOD n]\n[PROOFSTEP]\nrw [Nat.ModEq, \u2190 val_add, add_left_neg, tsub_add_cancel_of_le a.val_le, Nat.mod_self, val_zero]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 val (-a) = if a = 0 then 0 else n - val a\n[PROOFSTEP]\nrw [neg_val']\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 (n - val a) % n = if a = 0 then 0 else n - val a\n[PROOFSTEP]\nby_cases h : a = 0\n[GOAL]\ncase pos\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : a = 0\n\u22a2 (n - val a) % n = if a = 0 then 0 else n - val a\n[PROOFSTEP]\nrw [if_pos h, h, val_zero, tsub_zero, Nat.mod_self]\n[GOAL]\ncase neg\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : \u00aca = 0\n\u22a2 (n - val a) % n = if a = 0 then 0 else n - val a\n[PROOFSTEP]\nrw [if_neg h]\n[GOAL]\ncase neg\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : \u00aca = 0\n\u22a2 (n - val a) % n = n - val a\n[PROOFSTEP]\napply Nat.mod_eq_of_lt\n[GOAL]\ncase neg.h\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : \u00aca = 0\n\u22a2 n - val a < n\n[PROOFSTEP]\napply Nat.sub_lt (NeZero.pos n)\n[GOAL]\ncase neg.h\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : \u00aca = 0\n\u22a2 0 < val a\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase neg.h\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh : val a \u2264 0\n\u22a2 a = 0\n[PROOFSTEP]\nrwa [le_zero_iff, val_eq_zero] at h \n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\n\u22a2 valMinAbs x = if val x \u2264 n / 2 then \u2191(val x) else \u2191(val x) - \u2191n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\ninst\u271d : NeZero Nat.zero\nx : ZMod Nat.zero\n\u22a2 valMinAbs x = if val x \u2264 Nat.zero / 2 then \u2191(val x) else \u2191(val x) - \u2191Nat.zero\n[PROOFSTEP]\ncases NeZero.ne 0 rfl\n[GOAL]\ncase succ\nn\u271d : \u2115\ninst\u271d : NeZero (Nat.succ n\u271d)\nx : ZMod (Nat.succ n\u271d)\n\u22a2 valMinAbs x = if val x \u2264 Nat.succ n\u271d / 2 then \u2191(val x) else \u2191(val x) - \u2191(Nat.succ n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nk n : \u2115\nh\u271d : k = n + 1\nx : ZMod (namedPattern k (n + 1) h\u271d)\n\u22a2 \u2191(valMinAbs x) = x\n[PROOFSTEP]\nrw [valMinAbs_def_pos]\n[GOAL]\nk n : \u2115\nh\u271d : k = n + 1\nx : ZMod (namedPattern k (n + 1) h\u271d)\n\u22a2 \u2191(if val x \u2264 namedPattern k (n + 1) h\u271d / 2 then \u2191(val x) else \u2191(val x) - \u2191(namedPattern k (n + 1) h\u271d)) = x\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nk n : \u2115\nh\u271d\u00b9 : k = n + 1\nx : ZMod (namedPattern k (n + 1) h\u271d\u00b9)\nh\u271d : val x \u2264 namedPattern k (n + 1) h\u271d\u00b9 / 2\n\u22a2 \u2191\u2191(val x) = x\n[PROOFSTEP]\nrw [Int.cast_ofNat, nat_cast_zmod_val]\n[GOAL]\ncase neg\nk n : \u2115\nh\u271d\u00b9 : k = n + 1\nx : ZMod (namedPattern k (n + 1) h\u271d\u00b9)\nh\u271d : \u00acval x \u2264 namedPattern k (n + 1) h\u271d\u00b9 / 2\n\u22a2 \u2191(\u2191(val x) - \u2191(namedPattern k (n + 1) h\u271d\u00b9)) = x\n[PROOFSTEP]\nrw [Int.cast_sub, Int.cast_ofNat, nat_cast_zmod_val, Int.cast_ofNat, nat_cast_self, sub_zero]\n[GOAL]\nn m : \u2115\n\u22a2 m \u2264 n / 2 \u2194 \u2191m * 2 \u2264 \u2191n\n[PROOFSTEP]\nrw [Nat.le_div_iff_mul_le zero_lt_two, \u2190 Int.ofNat_le, Int.ofNat_mul, Nat.cast_two]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\n\u22a2 0 \u2264 valMinAbs x \u2194 val x \u2264 n / 2\n[PROOFSTEP]\nrw [valMinAbs_def_pos]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\n\u22a2 (0 \u2264 if val x \u2264 n / 2 then \u2191(val x) else \u2191(val x) - \u2191n) \u2194 val x \u2264 n / 2\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : val x \u2264 n / 2\n\u22a2 0 \u2264 \u2191(val x) \u2194 val x \u2264 n / 2\n[PROOFSTEP]\nexact iff_of_true (Nat.cast_nonneg _) h\n[GOAL]\ncase neg\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u00acval x \u2264 n / 2\n\u22a2 0 \u2264 \u2191(val x) - \u2191n \u2194 val x \u2264 n / 2\n[PROOFSTEP]\nexact iff_of_false (sub_lt_zero.2 <| Int.ofNat_lt.2 x.val_lt).not_le h\n[GOAL]\nn : \u2115\na : ZMod n\n\u22a2 valMinAbs a * 2 = \u2191n \u2194 2 * val a = n\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\na : ZMod Nat.zero\n\u22a2 valMinAbs a * 2 = \u2191Nat.zero \u2194 2 * val a = Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nn : \u2115\na : ZMod (Nat.succ n)\n\u22a2 valMinAbs a * 2 = \u2191(Nat.succ n) \u2194 2 * val a = Nat.succ n\n[PROOFSTEP]\nby_cases h : a.val \u2264 n.succ / 2\n[GOAL]\ncase pos\nn : \u2115\na : ZMod (Nat.succ n)\nh : val a \u2264 Nat.succ n / 2\n\u22a2 valMinAbs a * 2 = \u2191(Nat.succ n) \u2194 2 * val a = Nat.succ n\n[PROOFSTEP]\ndsimp [valMinAbs]\n[GOAL]\ncase pos\nn : \u2115\na : ZMod (Nat.succ n)\nh : val a \u2264 Nat.succ n / 2\n\u22a2 (if val a \u2264 Nat.succ n / 2 then \u2191(val a) else \u2191(val a) - \u2191(Nat.succ n)) * 2 = \u2191(Nat.succ n) \u2194 2 * val a = Nat.succ n\n[PROOFSTEP]\nrw [if_pos h, \u2190 Int.coe_nat_inj', Nat.cast_mul, Nat.cast_two, mul_comm]\n[GOAL]\ncase neg\nn : \u2115\na : ZMod (Nat.succ n)\nh : \u00acval a \u2264 Nat.succ n / 2\n\u22a2 valMinAbs a * 2 = \u2191(Nat.succ n) \u2194 2 * val a = Nat.succ n\n[PROOFSTEP]\napply iff_of_false _ (mt _ h)\n[GOAL]\nn : \u2115\na : ZMod (Nat.succ n)\nh : \u00acval a \u2264 Nat.succ n / 2\n\u22a2 \u00acvalMinAbs a * 2 = \u2191(Nat.succ n)\n[PROOFSTEP]\nintro he\n[GOAL]\nn : \u2115\na : ZMod (Nat.succ n)\nh : \u00acval a \u2264 Nat.succ n / 2\nhe : valMinAbs a * 2 = \u2191(Nat.succ n)\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 a.valMinAbs_nonneg_iff, \u2190 mul_nonneg_iff_left_nonneg_of_pos, he] at h \n[GOAL]\nn : \u2115\na : ZMod (Nat.succ n)\nh\u271d : \u00ac0 \u2264 valMinAbs a\nh : \u00ac0 \u2264 \u2191(Nat.succ n)\nhe : valMinAbs a * 2 = \u2191(Nat.succ n)\n\u22a2 False\nn : \u2115 a : ZMod (Nat.succ n) h : \u00ac0 \u2264 valMinAbs a he : valMinAbs a * 2 = \u2191(Nat.succ n) \u22a2 0 < 2\n[PROOFSTEP]\nexacts [h (Nat.cast_nonneg _), zero_lt_two]\n[GOAL]\nn : \u2115\na : ZMod (Nat.succ n)\nh : \u00acval a \u2264 Nat.succ n / 2\n\u22a2 2 * val a = Nat.succ n \u2192 val a \u2264 Nat.succ n / 2\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nn : \u2115\na : ZMod (Nat.succ n)\nh : \u00acval a \u2264 Nat.succ n / 2\n\u22a2 val a * 2 = Nat.succ n \u2192 val a \u2264 Nat.succ n / 2\n[PROOFSTEP]\nexact fun h => (Nat.le_div_iff_mul_le zero_lt_two).2 h.le\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\n\u22a2 valMinAbs x * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n[PROOFSTEP]\nsimp_rw [valMinAbs_def_pos, Nat.le_div_two_iff_mul_two_le]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\n\u22a2 (if \u2191(val x) * 2 \u2264 \u2191n then \u2191(val x) else \u2191(val x) - \u2191n) * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u2191(val x) * 2 \u2264 \u2191n\n\u22a2 \u2191(val x) * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n[PROOFSTEP]\nrefine' \u27e8(neg_lt_zero.2 <| by exact_mod_cast NeZero.pos n).trans_le (mul_nonneg _ _), h\u27e9\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u2191(val x) * 2 \u2264 \u2191n\n\u22a2 0 < \u2191n\n[PROOFSTEP]\nexact_mod_cast NeZero.pos n\n[GOAL]\ncase pos.refine'_1\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u2191(val x) * 2 \u2264 \u2191n\n\u22a2 0 \u2264 \u2191(val x)\ncase pos.refine'_2 n : \u2115 inst\u271d : NeZero n x : ZMod n h : \u2191(val x) * 2 \u2264 \u2191n \u22a2 0 \u2264 2\n[PROOFSTEP]\nexacts [Nat.cast_nonneg _, zero_le_two]\n[GOAL]\ncase neg\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u00ac\u2191(val x) * 2 \u2264 \u2191n\n\u22a2 (\u2191(val x) - \u2191n) * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n[PROOFSTEP]\nrefine' \u27e8_, le_trans (mul_nonpos_of_nonpos_of_nonneg _ zero_le_two) <| Nat.cast_nonneg _\u27e9\n[GOAL]\ncase neg.refine'_1\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u00ac\u2191(val x) * 2 \u2264 \u2191n\n\u22a2 -\u2191n < (\u2191(val x) - \u2191n) * 2\n[PROOFSTEP]\nlinarith only [h]\n[GOAL]\ncase neg.refine'_2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u00ac\u2191(val x) * 2 \u2264 \u2191n\n\u22a2 \u2191(val x) - \u2191n \u2264 0\n[PROOFSTEP]\nrw [sub_nonpos, Int.ofNat_le]\n[GOAL]\ncase neg.refine'_2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : \u00ac\u2191(val x) * 2 \u2264 \u2191n\n\u22a2 val x \u2264 n\n[PROOFSTEP]\nexact x.val_lt.le\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\n\u22a2 valMinAbs x = y \u2192 x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\n\u22a2 x = \u2191(valMinAbs x) \u2227 valMinAbs x * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n[PROOFSTEP]\nexact \u27e8x.coe_valMinAbs.symm, x.valMinAbs_mem_Ioc\u27e9\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 valMinAbs x = y\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 valMinAbs x - y = 0\n[PROOFSTEP]\napply @Int.eq_zero_of_abs_lt_dvd n\n[GOAL]\ncase h1\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 \u2191n \u2223 valMinAbs x - y\n[PROOFSTEP]\nrw [\u2190 int_cast_zmod_eq_zero_iff_dvd, Int.cast_sub, coe_valMinAbs, h.1, sub_self]\n[GOAL]\ncase h2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 |valMinAbs x - y| < \u2191n\n[PROOFSTEP]\nrw [\u2190 mul_lt_mul_right (@zero_lt_two \u2124 _ _ _ _ _)]\n[GOAL]\ncase h2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 |valMinAbs x - y| * 2 < \u2191n * 2\n[PROOFSTEP]\nnth_rw 1 [\u2190 abs_eq_self.2 (@zero_le_two \u2124 _ _ _ _)]\n[GOAL]\ncase h2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 |valMinAbs x - y| * |2| < \u2191n * 2\n[PROOFSTEP]\nrw [\u2190 abs_mul, sub_mul, abs_lt]\n[GOAL]\ncase h2\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 -(\u2191n * 2) < valMinAbs x * 2 - y * 2 \u2227 valMinAbs x * 2 - y * 2 < \u2191n * 2\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h2.left\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 -(\u2191n * 2) < valMinAbs x * 2 - y * 2\n[PROOFSTEP]\nlinarith only [x.valMinAbs_mem_Ioc.1, x.valMinAbs_mem_Ioc.2, h.2.1, h.2.2]\n[GOAL]\ncase h2.right\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\ny : \u2124\nh : x = \u2191y \u2227 y * 2 \u2208 Set.Ioc (-\u2191n) \u2191n\n\u22a2 valMinAbs x * 2 - y * 2 < \u2191n * 2\n[PROOFSTEP]\nlinarith only [x.valMinAbs_mem_Ioc.1, x.valMinAbs_mem_Ioc.2, h.2.1, h.2.2]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\n\u22a2 Int.natAbs (valMinAbs x) \u2264 n / 2\n[PROOFSTEP]\nrw [Nat.le_div_two_iff_mul_two_le]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\n\u22a2 \u2191(Int.natAbs (valMinAbs x)) * 2 \u2264 \u2191n\n[PROOFSTEP]\ncases' x.valMinAbs.natAbs_eq with h h\n[GOAL]\ncase inl\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : valMinAbs x = \u2191(Int.natAbs (valMinAbs x))\n\u22a2 \u2191(Int.natAbs (valMinAbs x)) * 2 \u2264 \u2191n\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase inl\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : valMinAbs x = \u2191(Int.natAbs (valMinAbs x))\n\u22a2 valMinAbs x * 2 \u2264 \u2191n\n[PROOFSTEP]\nexact x.valMinAbs_mem_Ioc.2\n[GOAL]\ncase inr\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : valMinAbs x = -\u2191(Int.natAbs (valMinAbs x))\n\u22a2 \u2191(Int.natAbs (valMinAbs x)) * 2 \u2264 \u2191n\n[PROOFSTEP]\nrw [\u2190 neg_le_neg_iff, \u2190 neg_mul, \u2190 h]\n[GOAL]\ncase inr\nn : \u2115\ninst\u271d : NeZero n\nx : ZMod n\nh : valMinAbs x = -\u2191(Int.natAbs (valMinAbs x))\n\u22a2 -\u2191n \u2264 valMinAbs x * 2\n[PROOFSTEP]\nexact x.valMinAbs_mem_Ioc.1.le\n[GOAL]\n\u22a2 valMinAbs 0 = 0\n[PROOFSTEP]\nsimp only [valMinAbs_def_zero]\n[GOAL]\nn : \u2115\n\u22a2 valMinAbs 0 = 0\n[PROOFSTEP]\nsimp only [valMinAbs_def_pos, if_true, Int.ofNat_zero, zero_le, val_zero]\n[GOAL]\nn : \u2115\nx : ZMod n\n\u22a2 valMinAbs x = 0 \u2194 x = 0\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nx : ZMod Nat.zero\n\u22a2 valMinAbs x = 0 \u2194 x = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nn : \u2115\nx : ZMod (Nat.succ n)\n\u22a2 valMinAbs x = 0 \u2194 x = 0\n[PROOFSTEP]\nrw [\u2190 valMinAbs_zero n.succ]\n[GOAL]\ncase succ\nn : \u2115\nx : ZMod (Nat.succ n)\n\u22a2 valMinAbs x = valMinAbs 0 \u2194 x = 0\n[PROOFSTEP]\napply injective_valMinAbs.eq_iff\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 \u2191(Int.natAbs (valMinAbs a)) = if val a \u2264 n / 2 then a else -a\n[PROOFSTEP]\nhave : (a.val : \u2124) - n \u2264 0 := by\n erw [sub_nonpos, Int.ofNat_le]\n exact a.val_le\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 \u2191(val a) - \u2191n \u2264 0\n[PROOFSTEP]\nerw [sub_nonpos, Int.ofNat_le]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 val a \u2264 n\n[PROOFSTEP]\nexact a.val_le\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nthis : \u2191(val a) - \u2191n \u2264 0\n\u22a2 \u2191(Int.natAbs (valMinAbs a)) = if val a \u2264 n / 2 then a else -a\n[PROOFSTEP]\nrw [valMinAbs_def_pos]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nthis : \u2191(val a) - \u2191n \u2264 0\n\u22a2 \u2191(Int.natAbs (if val a \u2264 n / 2 then \u2191(val a) else \u2191(val a) - \u2191n)) = if val a \u2264 n / 2 then a else -a\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nthis : \u2191(val a) - \u2191n \u2264 0\nh\u271d : val a \u2264 n / 2\n\u22a2 \u2191(Int.natAbs \u2191(val a)) = a\n[PROOFSTEP]\nrw [Int.natAbs_ofNat, nat_cast_zmod_val]\n[GOAL]\ncase neg\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nthis : \u2191(val a) - \u2191n \u2264 0\nh\u271d : \u00acval a \u2264 n / 2\n\u22a2 \u2191(Int.natAbs (\u2191(val a) - \u2191n)) = -a\n[PROOFSTEP]\nrw [\u2190 Int.cast_ofNat, Int.ofNat_natAbs_of_nonpos this, Int.cast_neg, Int.cast_sub, Int.cast_ofNat, Int.cast_ofNat,\n nat_cast_self, sub_zero, nat_cast_zmod_val]\n[GOAL]\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\n\u22a2 valMinAbs (-a) = -valMinAbs a\n[PROOFSTEP]\ncases' eq_zero_or_neZero n with h h\n[GOAL]\ncase inl\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : n = 0\n\u22a2 valMinAbs (-a) = -valMinAbs a\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase inl\na : ZMod 0\nha : 2 * val a \u2260 0\n\u22a2 valMinAbs (-a) = -valMinAbs a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : NeZero n\n\u22a2 valMinAbs (-a) = -valMinAbs a\n[PROOFSTEP]\nrefine' (valMinAbs_spec _ _).2 \u27e8_, _, _\u27e9\n[GOAL]\ncase inr.refine'_1\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : NeZero n\n\u22a2 -a = \u2191(-valMinAbs a)\n[PROOFSTEP]\nrw [Int.cast_neg, coe_valMinAbs]\n[GOAL]\ncase inr.refine'_2\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : NeZero n\n\u22a2 -\u2191n < -valMinAbs a * 2\n[PROOFSTEP]\nrw [neg_mul, neg_lt_neg_iff]\n[GOAL]\ncase inr.refine'_2\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : NeZero n\n\u22a2 valMinAbs a * 2 < \u2191n\n[PROOFSTEP]\nexact a.valMinAbs_mem_Ioc.2.lt_of_ne (mt a.valMinAbs_mul_two_eq_iff.1 ha)\n[GOAL]\ncase inr.refine'_3\nn : \u2115\na : ZMod n\nha : 2 * val a \u2260 n\nh : NeZero n\n\u22a2 -valMinAbs a * 2 \u2264 \u2191n\n[PROOFSTEP]\nlinarith only [a.valMinAbs_mem_Ioc.1]\n[GOAL]\nn : \u2115\na : ZMod n\n\u22a2 Int.natAbs (valMinAbs (-a)) = Int.natAbs (valMinAbs a)\n[PROOFSTEP]\nby_cases h2a : 2 * a.val = n\n[GOAL]\ncase pos\nn : \u2115\na : ZMod n\nh2a : 2 * val a = n\n\u22a2 Int.natAbs (valMinAbs (-a)) = Int.natAbs (valMinAbs a)\n[PROOFSTEP]\nrw [a.neg_eq_self_iff.2 (Or.inr h2a)]\n[GOAL]\ncase neg\nn : \u2115\na : ZMod n\nh2a : \u00ac2 * val a = n\n\u22a2 Int.natAbs (valMinAbs (-a)) = Int.natAbs (valMinAbs a)\n[PROOFSTEP]\nrw [valMinAbs_neg_of_ne_half h2a, Int.natAbs_neg]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 \u2191(val a) = valMinAbs a + \u2191(if val a \u2264 n / 2 then 0 else n)\n[PROOFSTEP]\nrw [valMinAbs_def_pos]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 \u2191(val a) = (if val a \u2264 n / 2 then \u2191(val a) else \u2191(val a) - \u2191n) + \u2191(if val a \u2264 n / 2 then 0 else n)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh\u271d : val a \u2264 n / 2\n\u22a2 \u2191(val a) = \u2191(val a) + \u21910\n[PROOFSTEP]\nsimp [add_zero, sub_add_cancel]\n[GOAL]\ncase neg\nn : \u2115\ninst\u271d : NeZero n\na : ZMod n\nh\u271d : \u00acval a \u2264 n / 2\n\u22a2 \u2191(val a) = \u2191(val a) - \u2191n + \u2191n\n[PROOFSTEP]\nsimp [add_zero, sub_add_cancel]\n[GOAL]\np q : \u2115\nhp : Fact (Nat.Prime p)\nhq : Fact (Nat.Prime q)\nhpq : p \u2260 q\n\u22a2 \u2191q \u2260 0\n[PROOFSTEP]\nrwa [\u2190 Nat.cast_zero, Ne.def, eq_iff_modEq_nat, Nat.modEq_zero_iff_dvd, \u2190 hp.1.coprime_iff_not_dvd,\n Nat.coprime_primes hp.1 hq.1]\n[GOAL]\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\n\u22a2 Int.natAbs (valMinAbs a) = min (val a) (n - val a)\n[PROOFSTEP]\nrw [valMinAbs_def_pos]\n[GOAL]\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\n\u22a2 Int.natAbs (if val a \u2264 n / 2 then \u2191(val a) else \u2191(val a) - \u2191n) = min (val a) (n - val a)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : val a \u2264 n / 2\n\u22a2 Int.natAbs \u2191(val a) = min (val a) (n - val a)\n[PROOFSTEP]\nrw [Int.natAbs_ofNat]\n[GOAL]\ncase pos\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : val a \u2264 n / 2\n\u22a2 val a = min (val a) (n - val a)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase pos\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : val a \u2264 n / 2\n\u22a2 min (val a) (n - val a) = val a\n[PROOFSTEP]\napply min_eq_left (le_trans h (le_trans (Nat.half_le_of_sub_le_half _) (Nat.sub_le_sub_left n h)))\n[GOAL]\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : val a \u2264 n / 2\n\u22a2 n - (n - n / 2) \u2264 n / 2\n[PROOFSTEP]\nrw [Nat.sub_sub_self (Nat.div_le_self _ _)]\n[GOAL]\ncase neg\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : \u00acval a \u2264 n / 2\n\u22a2 Int.natAbs (\u2191(val a) - \u2191n) = min (val a) (n - val a)\n[PROOFSTEP]\nrw [\u2190 Int.natAbs_neg, neg_sub, \u2190 Nat.cast_sub a.val_le]\n[GOAL]\ncase neg\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : \u00acval a \u2264 n / 2\n\u22a2 Int.natAbs \u2191(n - val a) = min (val a) (n - val a)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase neg\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : \u00acval a \u2264 n / 2\n\u22a2 min (val a) (n - val a) = Int.natAbs \u2191(n - val a)\n[PROOFSTEP]\napply\n min_eq_right\n (le_trans (le_trans (Nat.sub_le_sub_left n (lt_of_not_ge h)) (Nat.le_half_of_half_lt_sub _)) (le_of_not_ge h))\n[GOAL]\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : \u00acval a \u2264 n / 2\n\u22a2 n / 2 < n - (n - Nat.succ (n / 2))\n[PROOFSTEP]\nrw [Nat.sub_sub_self (Nat.div_lt_self (lt_of_le_of_ne' (Nat.zero_le _) hpos.1) one_lt_two)]\n[GOAL]\nn\u271d a\u271d n : \u2115\nhpos : NeZero n\na : ZMod n\nh : \u00acval a \u2264 n / 2\n\u22a2 n / 2 < Nat.succ (n / 2)\n[PROOFSTEP]\napply Nat.lt_succ_self\n[GOAL]\nn a : \u2115\nha : a \u2264 n / 2\n\u22a2 valMinAbs \u2191a = \u2191a\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\na : \u2115\nha : a \u2264 Nat.zero / 2\n\u22a2 valMinAbs \u2191a = \u2191a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\na n\u271d : \u2115\nha : a \u2264 Nat.succ n\u271d / 2\n\u22a2 valMinAbs \u2191a = \u2191a\n[PROOFSTEP]\nsimp [valMinAbs_def_pos, val_nat_cast, Nat.mod_eq_of_lt (ha.trans_lt <| Nat.div_lt_self' _ 0), ha]\n[GOAL]\nn a : \u2115\nha : n / 2 < a\nha' : a < n\n\u22a2 valMinAbs \u2191a = \u2191a - \u2191n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\na : \u2115\nha : Nat.zero / 2 < a\nha' : a < Nat.zero\n\u22a2 valMinAbs \u2191a = \u2191a - \u2191Nat.zero\n[PROOFSTEP]\ncases not_lt_bot ha'\n[GOAL]\ncase succ\na n\u271d : \u2115\nha : Nat.succ n\u271d / 2 < a\nha' : a < Nat.succ n\u271d\n\u22a2 valMinAbs \u2191a = \u2191a - \u2191(Nat.succ n\u271d)\n[PROOFSTEP]\nsimp [valMinAbs_def_pos, val_nat_cast, Nat.mod_eq_of_lt ha', ha.not_le]\n[GOAL]\nn a : \u2115\ninst\u271d : NeZero n\n\u22a2 valMinAbs \u2191a = \u2191a \u2194 a \u2264 n / 2\n[PROOFSTEP]\nrefine' \u27e8fun ha => _, valMinAbs_natCast_of_le_half\u27e9\n[GOAL]\nn a : \u2115\ninst\u271d : NeZero n\nha : valMinAbs \u2191a = \u2191a\n\u22a2 a \u2264 n / 2\n[PROOFSTEP]\nrw [\u2190 Int.natAbs_ofNat a, \u2190 ha]\n[GOAL]\nn a : \u2115\ninst\u271d : NeZero n\nha : valMinAbs \u2191a = \u2191a\n\u22a2 Int.natAbs (valMinAbs \u2191a) \u2264 n / 2\n[PROOFSTEP]\nexact natAbs_valMinAbs_le a\n[GOAL]\nn\u271d a n : \u2115\nx y : \u2124\nhe : \u2191x = \u2191y\nhl : Int.natAbs x \u2264 n / 2\n\u22a2 Int.natAbs x \u2264 Int.natAbs y\n[PROOFSTEP]\nrw [int_cast_eq_int_cast_iff_dvd_sub] at he \n[GOAL]\nn\u271d a n : \u2115\nx y : \u2124\nhe : \u2191n \u2223 y - x\nhl : Int.natAbs x \u2264 n / 2\n\u22a2 Int.natAbs x \u2264 Int.natAbs y\n[PROOFSTEP]\nobtain \u27e8m, he\u27e9 := he\n[GOAL]\ncase intro\nn\u271d a n : \u2115\nx y : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhe : y - x = \u2191n * m\n\u22a2 Int.natAbs x \u2264 Int.natAbs y\n[PROOFSTEP]\nrw [sub_eq_iff_eq_add] at he \n[GOAL]\ncase intro\nn\u271d a n : \u2115\nx y : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhe : y = \u2191n * m + x\n\u22a2 Int.natAbs x \u2264 Int.natAbs y\n[PROOFSTEP]\nsubst he\n[GOAL]\ncase intro\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\n\u22a2 Int.natAbs x \u2264 Int.natAbs (\u2191n * m + x)\n[PROOFSTEP]\nobtain rfl | hm := eq_or_ne m 0\n[GOAL]\ncase intro.inl\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\n\u22a2 Int.natAbs x \u2264 Int.natAbs (\u2191n * 0 + x)\n[PROOFSTEP]\nrw [mul_zero, zero_add]\n[GOAL]\ncase intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 Int.natAbs x \u2264 Int.natAbs (\u2191n * m + x)\n[PROOFSTEP]\napply hl.trans\n[GOAL]\ncase intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 \u2264 Int.natAbs (\u2191n * m + x)\n[PROOFSTEP]\nrw [\u2190 add_le_add_iff_right x.natAbs]\n[GOAL]\ncase intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 + Int.natAbs x \u2264 Int.natAbs (\u2191n * m + x) + Int.natAbs x\n[PROOFSTEP]\nrefine' le_trans (le_trans ((add_le_add_iff_left _).2 hl) _) (Int.natAbs_sub_le _ _)\n[GOAL]\ncase intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 + n / 2 \u2264 Int.natAbs (\u2191n * m + x - x)\n[PROOFSTEP]\nrw [add_sub_cancel, Int.natAbs_mul, Int.natAbs_ofNat]\n[GOAL]\ncase intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 + n / 2 \u2264 n * Int.natAbs m\n[PROOFSTEP]\nrefine' le_trans _ (Nat.le_mul_of_pos_right <| Int.natAbs_pos.2 hm)\n[GOAL]\ncase intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 + n / 2 \u2264 n\n[PROOFSTEP]\nrw [\u2190 mul_two]\n[GOAL]\ncase intro.inr\nn\u271d a n : \u2115\nx : \u2124\nhl : Int.natAbs x \u2264 n / 2\nm : \u2124\nhm : m \u2260 0\n\u22a2 n / 2 * 2 \u2264 n\n[PROOFSTEP]\napply Nat.div_mul_le_self\n[GOAL]\nn\u271d a\u271d n : \u2115\na b : ZMod n\n\u22a2 Int.natAbs (valMinAbs (a + b)) \u2264 Int.natAbs (valMinAbs a + valMinAbs b)\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nn a\u271d : \u2115\na b : ZMod Nat.zero\n\u22a2 Int.natAbs (valMinAbs (a + b)) \u2264 Int.natAbs (valMinAbs a + valMinAbs b)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nn\u271d a\u271d n : \u2115\na b : ZMod (Nat.succ n)\n\u22a2 Int.natAbs (valMinAbs (a + b)) \u2264 Int.natAbs (valMinAbs a + valMinAbs b)\n[PROOFSTEP]\napply natAbs_min_of_le_div_two n.succ\n[GOAL]\ncase succ.he\nn\u271d a\u271d n : \u2115\na b : ZMod (Nat.succ n)\n\u22a2 \u2191(valMinAbs (a + b)) = \u2191(valMinAbs a + valMinAbs b)\n[PROOFSTEP]\nsimp_rw [Int.cast_add, coe_valMinAbs]\n[GOAL]\ncase succ.hl\nn\u271d a\u271d n : \u2115\na b : ZMod (Nat.succ n)\n\u22a2 Int.natAbs (valMinAbs (a + b)) \u2264 Nat.succ n / 2\n[PROOFSTEP]\napply natAbs_valMinAbs_le\n[GOAL]\nn a\u271d p : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nh : a \u2260 0\n\u22a2 a * a\u207b\u00b9 = 1\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := nat_cast_zmod_surjective a\n[GOAL]\ncase intro\nn a p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nk : \u2115\nh : \u2191k \u2260 0\n\u22a2 \u2191k * (\u2191k)\u207b\u00b9 = 1\n[PROOFSTEP]\napply coe_mul_inv_eq_one\n[GOAL]\ncase intro.h\nn a p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nk : \u2115\nh : \u2191k \u2260 0\n\u22a2 Nat.coprime k p\n[PROOFSTEP]\napply Nat.coprime.symm\n[GOAL]\ncase intro.h.a\nn a p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nk : \u2115\nh : \u2191k \u2260 0\n\u22a2 Nat.coprime p k\n[PROOFSTEP]\nrwa [Nat.Prime.coprime_iff_not_dvd Fact.out, \u2190 CharP.cast_eq_zero_iff (ZMod p)]\n[GOAL]\nn a p\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 IsDomain (ZMod p)\n[PROOFSTEP]\ncases p\n[GOAL]\ncase zero\nn a p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : Fact (Nat.Prime Nat.zero)\n\u22a2 IsDomain (ZMod Nat.zero)\n[PROOFSTEP]\nexact (Nat.not_prime_zero hp.out).elim\n[GOAL]\ncase succ\nn a p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d : \u2115\nhp : Fact (Nat.Prime (Nat.succ n\u271d))\n\u22a2 IsDomain (ZMod (Nat.succ n\u271d))\n[PROOFSTEP]\nexact @Field.isDomain (ZMod _) (inferInstanceAs (Field (ZMod _)))\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d : Semiring R\nf g : ZMod n \u2192+* R\n\u22a2 f = g\n[PROOFSTEP]\next a\n[GOAL]\ncase a\nn : \u2115\nR : Type u_1\ninst\u271d : Semiring R\nf g : ZMod n \u2192+* R\na : ZMod n\n\u22a2 \u2191f a = \u2191g a\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := ZMod.int_cast_surjective a\n[GOAL]\ncase a.intro\nn : \u2115\nR : Type u_1\ninst\u271d : Semiring R\nf g : ZMod n \u2192+* R\nk : \u2124\n\u22a2 \u2191f \u2191k = \u2191g \u2191k\n[PROOFSTEP]\nlet \u03c6 : \u2124 \u2192+* R := f.comp (Int.castRingHom (ZMod n))\n[GOAL]\ncase a.intro\nn : \u2115\nR : Type u_1\ninst\u271d : Semiring R\nf g : ZMod n \u2192+* R\nk : \u2124\n\u03c6 : \u2124 \u2192+* R := comp f (Int.castRingHom (ZMod n))\n\u22a2 \u2191f \u2191k = \u2191g \u2191k\n[PROOFSTEP]\nlet \u03c8 : \u2124 \u2192+* R := g.comp (Int.castRingHom (ZMod n))\n[GOAL]\ncase a.intro\nn : \u2115\nR : Type u_1\ninst\u271d : Semiring R\nf g : ZMod n \u2192+* R\nk : \u2124\n\u03c6 : \u2124 \u2192+* R := comp f (Int.castRingHom (ZMod n))\n\u03c8 : \u2124 \u2192+* R := comp g (Int.castRingHom (ZMod n))\n\u22a2 \u2191f \u2191k = \u2191g \u2191k\n[PROOFSTEP]\nshow \u03c6 k = \u03c8 k\n[GOAL]\ncase a.intro\nn : \u2115\nR : Type u_1\ninst\u271d : Semiring R\nf g : ZMod n \u2192+* R\nk : \u2124\n\u03c6 : \u2124 \u2192+* R := comp f (Int.castRingHom (ZMod n))\n\u03c8 : \u2124 \u2192+* R := comp g (Int.castRingHom (ZMod n))\n\u22a2 \u2191\u03c6 k = \u2191\u03c8 k\n[PROOFSTEP]\nrw [\u03c6.ext_int \u03c8]\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d : Semiring R\nf g : ZMod n \u2243+* R\n\u22a2 f = g\n[PROOFSTEP]\nrw [RingEquiv.coe_ringHom_inj_iff]\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d : Semiring R\nf g : ZMod n \u2243+* R\n\u22a2 \u2191f = \u2191g\n[PROOFSTEP]\napply RingHom.ext_zmod _ _\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d : Ring R\nf : R \u2192+* ZMod n\nk : ZMod n\n\u22a2 \u2191f \u2191k = k\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d : Ring R\nf : R \u2192+* ZMod Nat.zero\nk : ZMod Nat.zero\n\u22a2 \u2191f \u2191k = k\n[PROOFSTEP]\ndsimp [ZMod, ZMod.cast] at f k \u22a2\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d : Ring R\nf : R \u2192+* \u2124\nk : \u2124\n\u22a2 \u2191f \u2191k = k\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nf : R \u2192+* ZMod (Nat.succ n\u271d)\nk : ZMod (Nat.succ n\u271d)\n\u22a2 \u2191f \u2191k = k\n[PROOFSTEP]\ndsimp [ZMod, ZMod.cast] at f k \u22a2\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nf : R \u2192+* Fin (n\u271d + 1)\nk : Fin (n\u271d + 1)\n\u22a2 \u2191f \u2191(val k) = k\n[PROOFSTEP]\nerw [map_natCast, Fin.cast_val_eq_self]\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nf g : R \u2192+* ZMod n\nh : RingHom.ker f = RingHom.ker g\n\u22a2 f = g\n[PROOFSTEP]\nhave := f.liftOfRightInverse_comp _ (ZMod.ringHom_rightInverse f) \u27e8g, le_of_eq h\u27e9\n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nf g : R \u2192+* ZMod n\nh : RingHom.ker f = RingHom.ker g\nthis :\n RingHom.comp\n (\u2191(RingHom.liftOfRightInverse f cast (_ : Function.RightInverse cast \u2191f))\n { val := g, property := (_ : RingHom.ker f \u2264 RingHom.ker g) })\n f =\n \u2191{ val := g, property := (_ : RingHom.ker f \u2264 RingHom.ker g) }\n\u22a2 f = g\n[PROOFSTEP]\nrw [Subtype.coe_mk] at this \n[GOAL]\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nf g : R \u2192+* ZMod n\nh : RingHom.ker f = RingHom.ker g\nthis :\n RingHom.comp\n (\u2191(RingHom.liftOfRightInverse f cast (_ : Function.RightInverse cast \u2191f))\n { val := g, property := (_ : RingHom.ker f \u2264 RingHom.ker g) })\n f =\n g\n\u22a2 f = g\n[PROOFSTEP]\nrw [\u2190 this, RingHom.ext_zmod (f.liftOfRightInverse _ _ \u27e8g, _\u27e9) _, RingHom.id_comp]\n[GOAL]\nn : \u2115\nR : Type u_1\nA : Type u_2\ninst\u271d : AddGroup A\n\u22a2 \u2200 (x : \u2124 \u2192+ A), \u2191x \u2191n = 0 \u2194 AddMonoidHom.ker (Int.castAddHom (ZMod n)) \u2264 AddMonoidHom.ker x\n[PROOFSTEP]\nintro f\n[GOAL]\nn : \u2115\nR : Type u_1\nA : Type u_2\ninst\u271d : AddGroup A\nf : \u2124 \u2192+ A\n\u22a2 \u2191f \u2191n = 0 \u2194 AddMonoidHom.ker (Int.castAddHom (ZMod n)) \u2264 AddMonoidHom.ker f\n[PROOFSTEP]\nrw [ker_int_castAddHom]\n[GOAL]\nn : \u2115\nR : Type u_1\nA : Type u_2\ninst\u271d : AddGroup A\nf : \u2124 \u2192+ A\n\u22a2 \u2191f \u2191n = 0 \u2194 AddSubgroup.zmultiples \u2191n \u2264 AddMonoidHom.ker f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nn : \u2115\nR : Type u_1\nA : Type u_2\ninst\u271d : AddGroup A\nf : \u2124 \u2192+ A\n\u22a2 \u2191f \u2191n = 0 \u2192 AddSubgroup.zmultiples \u2191n \u2264 AddMonoidHom.ker f\n[PROOFSTEP]\nrintro hf _ \u27e8x, rfl\u27e9\n[GOAL]\ncase mp.intro\nn : \u2115\nR : Type u_1\nA : Type u_2\ninst\u271d : AddGroup A\nf : \u2124 \u2192+ A\nhf : \u2191f \u2191n = 0\nx : \u2124\n\u22a2 (fun x => x \u2022 \u2191n) x \u2208 AddMonoidHom.ker f\n[PROOFSTEP]\nsimp only [f.map_zsmul, zsmul_zero, f.mem_ker, hf]\n[GOAL]\ncase mpr\nn : \u2115\nR : Type u_1\nA : Type u_2\ninst\u271d : AddGroup A\nf : \u2124 \u2192+ A\n\u22a2 AddSubgroup.zmultiples \u2191n \u2264 AddMonoidHom.ker f \u2192 \u2191f \u2191n = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nn : \u2115\nR : Type u_1\nA : Type u_2\ninst\u271d : AddGroup A\nf : \u2124 \u2192+ A\nh : AddSubgroup.zmultiples \u2191n \u2264 AddMonoidHom.ker f\n\u22a2 \u2191f \u2191n = 0\n[PROOFSTEP]\nrefine' h (AddSubgroup.mem_zmultiples _)\n", "meta": {"mathlib_filename": "Mathlib.Data.ZMod.Basic", "llama_tokens": 45392, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.4001852911778587}} {"text": "[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\n\u03c3 \u03c4 : Gal p\nh : \u2200 (x : SplittingField p), x \u2208 rootSet p (SplittingField p) \u2192 \u2191\u03c3 x = \u2191\u03c4 x\n\u22a2 \u03c3 = \u03c4\n[PROOFSTEP]\nrefine'\n AlgEquiv.ext fun x => (AlgHom.mem_equalizer \u03c3.toAlgHom \u03c4.toAlgHom x).mp ((SetLike.ext_iff.mp _ x).mpr Algebra.mem_top)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\n\u03c3 \u03c4 : Gal p\nh : \u2200 (x : SplittingField p), x \u2208 rootSet p (SplittingField p) \u2192 \u2191\u03c3 x = \u2191\u03c4 x\nx : SplittingField p\n\u22a2 AlgHom.equalizer \u2191\u03c3 \u2191\u03c4 = \u22a4\n[PROOFSTEP]\nrwa [eq_top_iff, \u2190 SplittingField.adjoin_rootSet, Algebra.adjoin_le_iff]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nh : Splits (RingHom.id F) p\nf : Gal p\nx : SplittingField p\n\u22a2 \u2191f x = \u2191default x\n[PROOFSTEP]\nobtain \u27e8y, rfl\u27e9 :=\n Algebra.mem_bot.mp ((SetLike.ext_iff.mp ((IsSplittingField.splits_iff _ p).mp h) x).mp Algebra.mem_top)\n[GOAL]\ncase intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nh : Splits (RingHom.id F) p\nf : Gal p\ny : F\n\u22a2 \u2191f (\u2191(algebraMap F (SplittingField p)) y) = \u2191default (\u2191(algebraMap F (SplittingField p)) y)\n[PROOFSTEP]\nrw [AlgEquiv.commutes, AlgEquiv.commutes]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nh : Fact (Splits (algebraMap F E) p)\n\u22a2 Function.Bijective (mapRoots p E)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nh : Fact (Splits (algebraMap F E) p)\n\u22a2 Function.Injective (mapRoots p E)\n[PROOFSTEP]\nexact fun _ _ h => Subtype.ext (RingHom.injective _ (Subtype.ext_iff.mp h))\n[GOAL]\ncase right\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nh : Fact (Splits (algebraMap F E) p)\n\u22a2 Function.Surjective (mapRoots p E)\n[PROOFSTEP]\nintro y\n[GOAL]\ncase right\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nh : Fact (Splits (algebraMap F E) p)\ny : \u2191(rootSet p E)\n\u22a2 \u2203 a, mapRoots p E a = y\n[PROOFSTEP]\nhave key :=\n roots_map (IsScalarTower.toAlgHom F p.SplittingField E : p.SplittingField \u2192+* E)\n ((splits_id_iff_splits _).mpr (IsSplittingField.splits p.SplittingField p))\n[GOAL]\ncase right\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nh : Fact (Splits (algebraMap F E) p)\ny : \u2191(rootSet p E)\nkey :\n roots (map (\u2191(IsScalarTower.toAlgHom F (SplittingField p) E)) (map (algebraMap F (SplittingField p)) p)) =\n Multiset.map (\u2191\u2191(IsScalarTower.toAlgHom F (SplittingField p) E)) (roots (map (algebraMap F (SplittingField p)) p))\n\u22a2 \u2203 a, mapRoots p E a = y\n[PROOFSTEP]\nrw [map_map, AlgHom.comp_algebraMap] at key \n[GOAL]\ncase right\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nh : Fact (Splits (algebraMap F E) p)\ny : \u2191(rootSet p E)\nkey :\n roots (map (algebraMap F E) p) =\n Multiset.map (\u2191\u2191(IsScalarTower.toAlgHom F (SplittingField p) E)) (roots (map (algebraMap F (SplittingField p)) p))\n\u22a2 \u2203 a, mapRoots p E a = y\n[PROOFSTEP]\nhave hy := Subtype.mem y\n[GOAL]\ncase right\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nh : Fact (Splits (algebraMap F E) p)\ny : \u2191(rootSet p E)\nkey :\n roots (map (algebraMap F E) p) =\n Multiset.map (\u2191\u2191(IsScalarTower.toAlgHom F (SplittingField p) E)) (roots (map (algebraMap F (SplittingField p)) p))\nhy : \u2191y \u2208 rootSet p E\n\u22a2 \u2203 a, mapRoots p E a = y\n[PROOFSTEP]\nsimp only [rootSet, Finset.mem_coe, (Multiset.mem_toFinset), key, Multiset.mem_map] at hy \n[GOAL]\ncase right\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nh : Fact (Splits (algebraMap F E) p)\ny : \u2191(rootSet p E)\nkey :\n roots (map (algebraMap F E) p) =\n Multiset.map (\u2191\u2191(IsScalarTower.toAlgHom F (SplittingField p) E)) (roots (map (algebraMap F (SplittingField p)) p))\nhy : \u2203 a, a \u2208 roots (map (algebraMap F (SplittingField p)) p) \u2227 \u2191\u2191(IsScalarTower.toAlgHom F (SplittingField p) E) a = \u2191y\n\u22a2 \u2203 a, mapRoots p E a = y\n[PROOFSTEP]\nrcases hy with \u27e8x, hx1, hx2\u27e9\n[GOAL]\ncase right.intro.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nh : Fact (Splits (algebraMap F E) p)\ny : \u2191(rootSet p E)\nkey :\n roots (map (algebraMap F E) p) =\n Multiset.map (\u2191\u2191(IsScalarTower.toAlgHom F (SplittingField p) E)) (roots (map (algebraMap F (SplittingField p)) p))\nx : SplittingField p\nhx1 : x \u2208 roots (map (algebraMap F (SplittingField p)) p)\nhx2 : \u2191\u2191(IsScalarTower.toAlgHom F (SplittingField p) E) x = \u2191y\n\u22a2 \u2203 a, mapRoots p E a = y\n[PROOFSTEP]\nexact \u27e8\u27e8x, (@Multiset.mem_toFinset _ (Classical.decEq _) _ _).mpr hx1\u27e9, Subtype.ext hx2\u27e9\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nx\u271d : \u2191(rootSet p (SplittingField p))\n\u22a2 1 \u2022 x\u271d = x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nx\u271d : \u2191(rootSet p (SplittingField p))\n\u22a2 \u2191(1 \u2022 x\u271d) = \u2191x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nx\u271d\u00b2 x\u271d\u00b9 : Gal p\nx\u271d : \u2191(rootSet p (SplittingField p))\n\u22a2 (x\u271d\u00b2 * x\u271d\u00b9) \u2022 x\u271d = x\u271d\u00b2 \u2022 x\u271d\u00b9 \u2022 x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nx\u271d\u00b2 x\u271d\u00b9 : Gal p\nx\u271d : \u2191(rootSet p (SplittingField p))\n\u22a2 \u2191((x\u271d\u00b2 * x\u271d\u00b9) \u2022 x\u271d) = \u2191(x\u271d\u00b2 \u2022 x\u271d\u00b9 \u2022 x\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Fact (Splits (algebraMap F E) p)\nx\u271d : \u2191(rootSet p E)\n\u22a2 1 \u2022 x\u271d = x\u271d\n[PROOFSTEP]\nsimp only [smul_def, Equiv.apply_symm_apply, one_smul]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Fact (Splits (algebraMap F E) p)\nx\u271d\u00b2 x\u271d\u00b9 : Gal p\nx\u271d : \u2191(rootSet p E)\n\u22a2 (x\u271d\u00b2 * x\u271d\u00b9) \u2022 x\u271d = x\u271d\u00b2 \u2022 x\u271d\u00b9 \u2022 x\u271d\n[PROOFSTEP]\nsimp only [smul_def, Equiv.apply_symm_apply, Equiv.symm_apply_apply, mul_smul]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Fact (Splits (algebraMap F E) p)\n\u03d5 : E \u2243\u2090[F] E\nx : \u2191(rootSet p E)\n\u22a2 \u2191(\u2191(restrict p E) \u03d5 \u2022 x) = \u2191\u03d5 \u2191x\n[PROOFSTEP]\nlet \u03c8 := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F p.SplittingField E)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Fact (Splits (algebraMap F E) p)\n\u03d5 : E \u2243\u2090[F] E\nx : \u2191(rootSet p E)\n\u03c8 : SplittingField p \u2243\u2090[F] { x // x \u2208 AlgHom.range (IsScalarTower.toAlgHom F (SplittingField p) E) } :=\n AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F (SplittingField p) E)\n\u22a2 \u2191(\u2191(restrict p E) \u03d5 \u2022 x) = \u2191\u03d5 \u2191x\n[PROOFSTEP]\nchange \u2191(\u03c8 (\u03c8.symm _)) = \u03d5 x\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Fact (Splits (algebraMap F E) p)\n\u03d5 : E \u2243\u2090[F] E\nx : \u2191(rootSet p E)\n\u03c8 : SplittingField p \u2243\u2090[F] { x // x \u2208 AlgHom.range (IsScalarTower.toAlgHom F (SplittingField p) E) } :=\n AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F (SplittingField p) E)\n\u22a2 \u2191(\u2191\u03c8\n (\u2191(AlgEquiv.symm \u03c8)\n (\u2191\u2191(AlgHom.restrictNormalAux (\u2191\u03d5) (SplittingField p))\n (\u2191\u2191\u2191(AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F (SplittingField p) E))\n \u2191(\u2191(rootsEquivRoots p E).symm x))))) =\n \u2191\u03d5 \u2191x\n[PROOFSTEP]\nrw [AlgEquiv.apply_symm_apply \u03c8]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Fact (Splits (algebraMap F E) p)\n\u03d5 : E \u2243\u2090[F] E\nx : \u2191(rootSet p E)\n\u03c8 : SplittingField p \u2243\u2090[F] { x // x \u2208 AlgHom.range (IsScalarTower.toAlgHom F (SplittingField p) E) } :=\n AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F (SplittingField p) E)\n\u22a2 \u2191(\u2191\u2191(AlgHom.restrictNormalAux (\u2191\u03d5) (SplittingField p))\n (\u2191\u2191\u2191(AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F (SplittingField p) E))\n \u2191(\u2191(rootsEquivRoots p E).symm x))) =\n \u2191\u03d5 \u2191x\n[PROOFSTEP]\nchange \u03d5 (rootsEquivRoots p E ((rootsEquivRoots p E).symm x)) = \u03d5 x\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Fact (Splits (algebraMap F E) p)\n\u03d5 : E \u2243\u2090[F] E\nx : \u2191(rootSet p E)\n\u03c8 : SplittingField p \u2243\u2090[F] { x // x \u2208 AlgHom.range (IsScalarTower.toAlgHom F (SplittingField p) E) } :=\n AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F (SplittingField p) E)\n\u22a2 \u2191\u03d5 \u2191(\u2191(rootsEquivRoots p E) (\u2191(rootsEquivRoots p E).symm x)) = \u2191\u03d5 \u2191x\n[PROOFSTEP]\nrw [Equiv.apply_symm_apply (rootsEquivRoots p E)]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Fact (Splits (algebraMap F E) p)\n\u22a2 Function.Injective \u2191(galActionHom p E)\n[PROOFSTEP]\nrw [injective_iff_map_eq_one]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Fact (Splits (algebraMap F E) p)\n\u22a2 \u2200 (a : Gal p), \u2191(galActionHom p E) a = 1 \u2192 a = 1\n[PROOFSTEP]\nintro \u03d5 h\u03d5\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Fact (Splits (algebraMap F E) p)\n\u03d5 : Gal p\nh\u03d5 : \u2191(galActionHom p E) \u03d5 = 1\n\u22a2 \u03d5 = 1\n[PROOFSTEP]\next (x hx)\n[GOAL]\ncase h\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Fact (Splits (algebraMap F E) p)\n\u03d5 : Gal p\nh\u03d5 : \u2191(galActionHom p E) \u03d5 = 1\nx : SplittingField p\nhx : x \u2208 rootSet p (SplittingField p)\n\u22a2 \u2191\u03d5 x = \u21911 x\n[PROOFSTEP]\nhave key := Equiv.Perm.ext_iff.mp h\u03d5 (rootsEquivRoots p E \u27e8x, hx\u27e9)\n[GOAL]\ncase h\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Fact (Splits (algebraMap F E) p)\n\u03d5 : Gal p\nh\u03d5 : \u2191(galActionHom p E) \u03d5 = 1\nx : SplittingField p\nhx : x \u2208 rootSet p (SplittingField p)\nkey :\n \u2191(\u2191(galActionHom p E) \u03d5) (\u2191(rootsEquivRoots p E) { val := x, property := hx }) =\n \u21911 (\u2191(rootsEquivRoots p E) { val := x, property := hx })\n\u22a2 \u2191\u03d5 x = \u21911 x\n[PROOFSTEP]\nchange\n rootsEquivRoots p E (\u03d5 \u2022 (rootsEquivRoots p E).symm (rootsEquivRoots p E \u27e8x, hx\u27e9)) = rootsEquivRoots p E \u27e8x, hx\u27e9 at\n key \n[GOAL]\ncase h\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Fact (Splits (algebraMap F E) p)\n\u03d5 : Gal p\nh\u03d5 : \u2191(galActionHom p E) \u03d5 = 1\nx : SplittingField p\nhx : x \u2208 rootSet p (SplittingField p)\nkey :\n \u2191(rootsEquivRoots p E) (\u03d5 \u2022 \u2191(rootsEquivRoots p E).symm (\u2191(rootsEquivRoots p E) { val := x, property := hx })) =\n \u2191(rootsEquivRoots p E) { val := x, property := hx }\n\u22a2 \u2191\u03d5 x = \u21911 x\n[PROOFSTEP]\nrw [Equiv.symm_apply_apply] at key \n[GOAL]\ncase h\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Fact (Splits (algebraMap F E) p)\n\u03d5 : Gal p\nh\u03d5 : \u2191(galActionHom p E) \u03d5 = 1\nx : SplittingField p\nhx : x \u2208 rootSet p (SplittingField p)\nkey : \u2191(rootsEquivRoots p E) (\u03d5 \u2022 { val := x, property := hx }) = \u2191(rootsEquivRoots p E) { val := x, property := hx }\n\u22a2 \u2191\u03d5 x = \u21911 x\n[PROOFSTEP]\nexact Subtype.ext_iff.mp (Equiv.injective (rootsEquivRoots p E) key)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Decidable (q = 0)\nhpq : p \u2223 q\n\u22a2 restrictDvd hpq = if hq : q = 0 then 1 else restrict p (SplittingField q)\n[PROOFSTEP]\nunfold restrictDvd\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : Decidable (q = 0)\nhpq : p \u2223 q\n\u22a2 (if hq : q = 0 then 1 else restrict p (SplittingField q)) = if hq : q = 0 then 1 else restrict p (SplittingField q)\n[PROOFSTEP]\nconvert rfl\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : p \u2223 q\nhq : q \u2260 0\n\u22a2 Function.Surjective \u2191(restrictDvd hpq)\n[PROOFSTEP]\nclassical\n -- Porting note: was `simp only [restrictDvd_def, dif_neg hq, restrict_surjective]`\nhaveI := Fact.mk <| splits_of_splits_of_dvd (algebraMap F q.SplittingField) hq (SplittingField.splits q) hpq\nsimp only [restrictDvd_def, dif_neg hq]\nexact restrict_surjective _ _\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : p \u2223 q\nhq : q \u2260 0\n\u22a2 Function.Surjective \u2191(restrictDvd hpq)\n[PROOFSTEP]\nhaveI := Fact.mk <| splits_of_splits_of_dvd (algebraMap F q.SplittingField) hq (SplittingField.splits q) hpq\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : p \u2223 q\nhq : q \u2260 0\nthis : Fact (Splits (algebraMap F (SplittingField q)) p)\n\u22a2 Function.Surjective \u2191(restrictDvd hpq)\n[PROOFSTEP]\nsimp only [restrictDvd_def, dif_neg hq]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : p \u2223 q\nhq : q \u2260 0\nthis : Fact (Splits (algebraMap F (SplittingField q)) p)\n\u22a2 Function.Surjective \u2191(restrict p (SplittingField q))\n[PROOFSTEP]\nexact restrict_surjective _ _\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\n\u22a2 Function.Injective \u2191(restrictProd p q)\n[PROOFSTEP]\nby_cases hpq : p * q = 0\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : p * q = 0\n\u22a2 Function.Injective \u2191(restrictProd p q)\n[PROOFSTEP]\nhave : Unique (p * q).Gal := by rw [hpq]; infer_instance\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : p * q = 0\n\u22a2 Unique (Gal (p * q))\n[PROOFSTEP]\nrw [hpq]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : p * q = 0\n\u22a2 Unique (Gal 0)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : p * q = 0\nthis : Unique (Gal (p * q))\n\u22a2 Function.Injective \u2191(restrictProd p q)\n[PROOFSTEP]\nexact fun f g _ => Eq.trans (Unique.eq_default f) (Unique.eq_default g).symm\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\n\u22a2 Function.Injective \u2191(restrictProd p q)\n[PROOFSTEP]\nintro f g hfg\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg : \u2191(restrictProd p q) f = \u2191(restrictProd p q) g\n\u22a2 f = g\n[PROOFSTEP]\nclassical\nsimp only [restrictProd, restrictDvd_def] at hfg \nsimp only [dif_neg hpq, MonoidHom.prod_apply, Prod.mk.inj_iff] at hfg \next (x hx)\nrw [rootSet_def, Polynomial.map_mul, Polynomial.roots_mul] at hx \ncases' Multiset.mem_add.mp (Multiset.mem_toFinset.mp hx) with h h\n\u00b7 haveI : Fact (p.Splits (algebraMap F (p * q).SplittingField)) :=\n \u27e8splits_of_splits_of_dvd _ hpq (SplittingField.splits (p * q)) (dvd_mul_right p q)\u27e9\n have key :\n x =\n algebraMap p.SplittingField (p * q).SplittingField\n ((rootsEquivRoots p _).invFun \u27e8x, (@Multiset.mem_toFinset _ (Classical.decEq _) _ _).mpr h\u27e9) :=\n Subtype.ext_iff.mp (Equiv.apply_symm_apply (rootsEquivRoots p _) \u27e8x, _\u27e9).symm\n rw [key, \u2190 AlgEquiv.restrictNormal_commutes, \u2190 AlgEquiv.restrictNormal_commutes]\n exact congr_arg _ (AlgEquiv.ext_iff.mp hfg.1 _)\n\u00b7 haveI : Fact (q.Splits (algebraMap F (p * q).SplittingField)) :=\n \u27e8splits_of_splits_of_dvd _ hpq (SplittingField.splits (p * q)) (dvd_mul_left q p)\u27e9\n have key :\n x =\n algebraMap q.SplittingField (p * q).SplittingField\n ((rootsEquivRoots q _).invFun \u27e8x, (@Multiset.mem_toFinset _ (Classical.decEq _) _ _).mpr h\u27e9) :=\n Subtype.ext_iff.mp (Equiv.apply_symm_apply (rootsEquivRoots q _) \u27e8x, _\u27e9).symm\n rw [key, \u2190 AlgEquiv.restrictNormal_commutes, \u2190 AlgEquiv.restrictNormal_commutes]\n exact congr_arg _ (AlgEquiv.ext_iff.mp hfg.2 _)\n\u00b7 rwa [Ne.def, mul_eq_zero, map_eq_zero, map_eq_zero, \u2190 mul_eq_zero]\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg : \u2191(restrictProd p q) f = \u2191(restrictProd p q) g\n\u22a2 f = g\n[PROOFSTEP]\nsimp only [restrictProd, restrictDvd_def] at hfg \n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg :\n \u2191(MonoidHom.prod (if hq : p * q = 0 then 1 else restrict p (SplittingField (p * q)))\n (if hq : p * q = 0 then 1 else restrict q (SplittingField (p * q))))\n f =\n \u2191(MonoidHom.prod (if hq : p * q = 0 then 1 else restrict p (SplittingField (p * q)))\n (if hq : p * q = 0 then 1 else restrict q (SplittingField (p * q))))\n g\n\u22a2 f = g\n[PROOFSTEP]\nsimp only [dif_neg hpq, MonoidHom.prod_apply, Prod.mk.inj_iff] at hfg \n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg :\n \u2191(restrict p (SplittingField (p * q))) f = \u2191(restrict p (SplittingField (p * q))) g \u2227\n \u2191(restrict q (SplittingField (p * q))) f = \u2191(restrict q (SplittingField (p * q))) g\n\u22a2 f = g\n[PROOFSTEP]\next (x hx)\n[GOAL]\ncase neg.h\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg :\n \u2191(restrict p (SplittingField (p * q))) f = \u2191(restrict p (SplittingField (p * q))) g \u2227\n \u2191(restrict q (SplittingField (p * q))) f = \u2191(restrict q (SplittingField (p * q))) g\nx : SplittingField (p * q)\nhx : x \u2208 rootSet (p * q) (SplittingField (p * q))\n\u22a2 \u2191f x = \u2191g x\n[PROOFSTEP]\nrw [rootSet_def, Polynomial.map_mul, Polynomial.roots_mul] at hx \n[GOAL]\ncase neg.h\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg :\n \u2191(restrict p (SplittingField (p * q))) f = \u2191(restrict p (SplittingField (p * q))) g \u2227\n \u2191(restrict q (SplittingField (p * q))) f = \u2191(restrict q (SplittingField (p * q))) g\nx : SplittingField (p * q)\nhx :\n x \u2208\n \u2191(Multiset.toFinset\n (roots (map (algebraMap F (SplittingField (p * q))) p) + roots (map (algebraMap F (SplittingField (p * q))) q)))\n\u22a2 \u2191f x = \u2191g x\ncase neg.h\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg :\n \u2191(restrict p (SplittingField (p * q))) f = \u2191(restrict p (SplittingField (p * q))) g \u2227\n \u2191(restrict q (SplittingField (p * q))) f = \u2191(restrict q (SplittingField (p * q))) g\nx : SplittingField (p * q)\nhx :\n x \u2208\n \u2191(Multiset.toFinset\n (roots (map (algebraMap F (SplittingField (p * q))) p * map (algebraMap F (SplittingField (p * q))) q)))\n\u22a2 map (algebraMap F (SplittingField (p * q))) p * map (algebraMap F (SplittingField (p * q))) q \u2260 0\n[PROOFSTEP]\ncases' Multiset.mem_add.mp (Multiset.mem_toFinset.mp hx) with h h\n[GOAL]\ncase neg.h.inl\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg :\n \u2191(restrict p (SplittingField (p * q))) f = \u2191(restrict p (SplittingField (p * q))) g \u2227\n \u2191(restrict q (SplittingField (p * q))) f = \u2191(restrict q (SplittingField (p * q))) g\nx : SplittingField (p * q)\nhx :\n x \u2208\n \u2191(Multiset.toFinset\n (roots (map (algebraMap F (SplittingField (p * q))) p) + roots (map (algebraMap F (SplittingField (p * q))) q)))\nh : x \u2208 roots (map (algebraMap F (SplittingField (p * q))) p)\n\u22a2 \u2191f x = \u2191g x\n[PROOFSTEP]\nhaveI : Fact (p.Splits (algebraMap F (p * q).SplittingField)) :=\n \u27e8splits_of_splits_of_dvd _ hpq (SplittingField.splits (p * q)) (dvd_mul_right p q)\u27e9\n[GOAL]\ncase neg.h.inl\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg :\n \u2191(restrict p (SplittingField (p * q))) f = \u2191(restrict p (SplittingField (p * q))) g \u2227\n \u2191(restrict q (SplittingField (p * q))) f = \u2191(restrict q (SplittingField (p * q))) g\nx : SplittingField (p * q)\nhx :\n x \u2208\n \u2191(Multiset.toFinset\n (roots (map (algebraMap F (SplittingField (p * q))) p) + roots (map (algebraMap F (SplittingField (p * q))) q)))\nh : x \u2208 roots (map (algebraMap F (SplittingField (p * q))) p)\nthis : Fact (Splits (algebraMap F (SplittingField (p * q))) p)\n\u22a2 \u2191f x = \u2191g x\n[PROOFSTEP]\nhave key :\n x =\n algebraMap p.SplittingField (p * q).SplittingField\n ((rootsEquivRoots p _).invFun \u27e8x, (@Multiset.mem_toFinset _ (Classical.decEq _) _ _).mpr h\u27e9) :=\n Subtype.ext_iff.mp (Equiv.apply_symm_apply (rootsEquivRoots p _) \u27e8x, _\u27e9).symm\n[GOAL]\ncase neg.h.inl\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg :\n \u2191(restrict p (SplittingField (p * q))) f = \u2191(restrict p (SplittingField (p * q))) g \u2227\n \u2191(restrict q (SplittingField (p * q))) f = \u2191(restrict q (SplittingField (p * q))) g\nx : SplittingField (p * q)\nhx :\n x \u2208\n \u2191(Multiset.toFinset\n (roots (map (algebraMap F (SplittingField (p * q))) p) + roots (map (algebraMap F (SplittingField (p * q))) q)))\nh : x \u2208 roots (map (algebraMap F (SplittingField (p * q))) p)\nthis : Fact (Splits (algebraMap F (SplittingField (p * q))) p)\nkey :\n x =\n \u2191(algebraMap (SplittingField p) (SplittingField (p * q)))\n \u2191(Equiv.invFun (rootsEquivRoots p (SplittingField (p * q)))\n { val := x, property := (_ : x \u2208 Multiset.toFinset (roots (map (algebraMap F (SplittingField (p * q))) p))) })\n\u22a2 \u2191f x = \u2191g x\n[PROOFSTEP]\nrw [key, \u2190 AlgEquiv.restrictNormal_commutes, \u2190 AlgEquiv.restrictNormal_commutes]\n[GOAL]\ncase neg.h.inl\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg :\n \u2191(restrict p (SplittingField (p * q))) f = \u2191(restrict p (SplittingField (p * q))) g \u2227\n \u2191(restrict q (SplittingField (p * q))) f = \u2191(restrict q (SplittingField (p * q))) g\nx : SplittingField (p * q)\nhx :\n x \u2208\n \u2191(Multiset.toFinset\n (roots (map (algebraMap F (SplittingField (p * q))) p) + roots (map (algebraMap F (SplittingField (p * q))) q)))\nh : x \u2208 roots (map (algebraMap F (SplittingField (p * q))) p)\nthis : Fact (Splits (algebraMap F (SplittingField (p * q))) p)\nkey :\n x =\n \u2191(algebraMap (SplittingField p) (SplittingField (p * q)))\n \u2191(Equiv.invFun (rootsEquivRoots p (SplittingField (p * q)))\n { val := x, property := (_ : x \u2208 Multiset.toFinset (roots (map (algebraMap F (SplittingField (p * q))) p))) })\n\u22a2 \u2191(algebraMap (SplittingField p) (SplittingField (p * q)))\n (\u2191(AlgEquiv.restrictNormal f (SplittingField p))\n \u2191(Equiv.invFun (rootsEquivRoots p (SplittingField (p * q)))\n { val := x,\n property := (_ : x \u2208 Multiset.toFinset (roots (map (algebraMap F (SplittingField (p * q))) p))) })) =\n \u2191(algebraMap (SplittingField p) (SplittingField (p * q)))\n (\u2191(AlgEquiv.restrictNormal g (SplittingField p))\n \u2191(Equiv.invFun (rootsEquivRoots p (SplittingField (p * q)))\n { val := x,\n property := (_ : x \u2208 Multiset.toFinset (roots (map (algebraMap F (SplittingField (p * q))) p))) }))\n[PROOFSTEP]\nexact congr_arg _ (AlgEquiv.ext_iff.mp hfg.1 _)\n[GOAL]\ncase neg.h.inr\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg :\n \u2191(restrict p (SplittingField (p * q))) f = \u2191(restrict p (SplittingField (p * q))) g \u2227\n \u2191(restrict q (SplittingField (p * q))) f = \u2191(restrict q (SplittingField (p * q))) g\nx : SplittingField (p * q)\nhx :\n x \u2208\n \u2191(Multiset.toFinset\n (roots (map (algebraMap F (SplittingField (p * q))) p) + roots (map (algebraMap F (SplittingField (p * q))) q)))\nh : x \u2208 roots (map (algebraMap F (SplittingField (p * q))) q)\n\u22a2 \u2191f x = \u2191g x\n[PROOFSTEP]\nhaveI : Fact (q.Splits (algebraMap F (p * q).SplittingField)) :=\n \u27e8splits_of_splits_of_dvd _ hpq (SplittingField.splits (p * q)) (dvd_mul_left q p)\u27e9\n[GOAL]\ncase neg.h.inr\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg :\n \u2191(restrict p (SplittingField (p * q))) f = \u2191(restrict p (SplittingField (p * q))) g \u2227\n \u2191(restrict q (SplittingField (p * q))) f = \u2191(restrict q (SplittingField (p * q))) g\nx : SplittingField (p * q)\nhx :\n x \u2208\n \u2191(Multiset.toFinset\n (roots (map (algebraMap F (SplittingField (p * q))) p) + roots (map (algebraMap F (SplittingField (p * q))) q)))\nh : x \u2208 roots (map (algebraMap F (SplittingField (p * q))) q)\nthis : Fact (Splits (algebraMap F (SplittingField (p * q))) q)\n\u22a2 \u2191f x = \u2191g x\n[PROOFSTEP]\nhave key :\n x =\n algebraMap q.SplittingField (p * q).SplittingField\n ((rootsEquivRoots q _).invFun \u27e8x, (@Multiset.mem_toFinset _ (Classical.decEq _) _ _).mpr h\u27e9) :=\n Subtype.ext_iff.mp (Equiv.apply_symm_apply (rootsEquivRoots q _) \u27e8x, _\u27e9).symm\n[GOAL]\ncase neg.h.inr\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg :\n \u2191(restrict p (SplittingField (p * q))) f = \u2191(restrict p (SplittingField (p * q))) g \u2227\n \u2191(restrict q (SplittingField (p * q))) f = \u2191(restrict q (SplittingField (p * q))) g\nx : SplittingField (p * q)\nhx :\n x \u2208\n \u2191(Multiset.toFinset\n (roots (map (algebraMap F (SplittingField (p * q))) p) + roots (map (algebraMap F (SplittingField (p * q))) q)))\nh : x \u2208 roots (map (algebraMap F (SplittingField (p * q))) q)\nthis : Fact (Splits (algebraMap F (SplittingField (p * q))) q)\nkey :\n x =\n \u2191(algebraMap (SplittingField q) (SplittingField (p * q)))\n \u2191(Equiv.invFun (rootsEquivRoots q (SplittingField (p * q)))\n { val := x, property := (_ : x \u2208 Multiset.toFinset (roots (map (algebraMap F (SplittingField (p * q))) q))) })\n\u22a2 \u2191f x = \u2191g x\n[PROOFSTEP]\nrw [key, \u2190 AlgEquiv.restrictNormal_commutes, \u2190 AlgEquiv.restrictNormal_commutes]\n[GOAL]\ncase neg.h.inr\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg :\n \u2191(restrict p (SplittingField (p * q))) f = \u2191(restrict p (SplittingField (p * q))) g \u2227\n \u2191(restrict q (SplittingField (p * q))) f = \u2191(restrict q (SplittingField (p * q))) g\nx : SplittingField (p * q)\nhx :\n x \u2208\n \u2191(Multiset.toFinset\n (roots (map (algebraMap F (SplittingField (p * q))) p) + roots (map (algebraMap F (SplittingField (p * q))) q)))\nh : x \u2208 roots (map (algebraMap F (SplittingField (p * q))) q)\nthis : Fact (Splits (algebraMap F (SplittingField (p * q))) q)\nkey :\n x =\n \u2191(algebraMap (SplittingField q) (SplittingField (p * q)))\n \u2191(Equiv.invFun (rootsEquivRoots q (SplittingField (p * q)))\n { val := x, property := (_ : x \u2208 Multiset.toFinset (roots (map (algebraMap F (SplittingField (p * q))) q))) })\n\u22a2 \u2191(algebraMap (SplittingField q) (SplittingField (p * q)))\n (\u2191(AlgEquiv.restrictNormal f (SplittingField q))\n \u2191(Equiv.invFun (rootsEquivRoots q (SplittingField (p * q)))\n { val := x,\n property := (_ : x \u2208 Multiset.toFinset (roots (map (algebraMap F (SplittingField (p * q))) q))) })) =\n \u2191(algebraMap (SplittingField q) (SplittingField (p * q)))\n (\u2191(AlgEquiv.restrictNormal g (SplittingField q))\n \u2191(Equiv.invFun (rootsEquivRoots q (SplittingField (p * q)))\n { val := x,\n property := (_ : x \u2208 Multiset.toFinset (roots (map (algebraMap F (SplittingField (p * q))) q))) }))\n[PROOFSTEP]\nexact congr_arg _ (AlgEquiv.ext_iff.mp hfg.2 _)\n[GOAL]\ncase neg.h\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhpq : \u00acp * q = 0\nf g : Gal (p * q)\nhfg :\n \u2191(restrict p (SplittingField (p * q))) f = \u2191(restrict p (SplittingField (p * q))) g \u2227\n \u2191(restrict q (SplittingField (p * q))) f = \u2191(restrict q (SplittingField (p * q))) g\nx : SplittingField (p * q)\nhx :\n x \u2208\n \u2191(Multiset.toFinset\n (roots (map (algebraMap F (SplittingField (p * q))) p * map (algebraMap F (SplittingField (p * q))) q)))\n\u22a2 map (algebraMap F (SplittingField (p * q))) p * map (algebraMap F (SplittingField (p * q))) q \u2260 0\n[PROOFSTEP]\nrwa [Ne.def, mul_eq_zero, map_eq_zero, map_eq_zero, \u2190 mul_eq_zero]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np\u2081 q\u2081 p\u2082 q\u2082 : F[X]\nhq\u2081 : q\u2081 \u2260 0\nhq\u2082 : q\u2082 \u2260 0\nh\u2081 : Splits (algebraMap F (SplittingField q\u2081)) p\u2081\nh\u2082 : Splits (algebraMap F (SplittingField q\u2082)) p\u2082\n\u22a2 Splits (algebraMap F (SplittingField (q\u2081 * q\u2082))) (p\u2081 * p\u2082)\n[PROOFSTEP]\napply splits_mul\n[GOAL]\ncase hf\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np\u2081 q\u2081 p\u2082 q\u2082 : F[X]\nhq\u2081 : q\u2081 \u2260 0\nhq\u2082 : q\u2082 \u2260 0\nh\u2081 : Splits (algebraMap F (SplittingField q\u2081)) p\u2081\nh\u2082 : Splits (algebraMap F (SplittingField q\u2082)) p\u2082\n\u22a2 Splits (algebraMap F (SplittingField (q\u2081 * q\u2082))) p\u2081\n[PROOFSTEP]\nrw [\u2190\n (SplittingField.lift q\u2081\n (splits_of_splits_of_dvd (algebraMap F (q\u2081 * q\u2082).SplittingField) (mul_ne_zero hq\u2081 hq\u2082) (SplittingField.splits _)\n (dvd_mul_right q\u2081 q\u2082))).comp_algebraMap]\n[GOAL]\ncase hf\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np\u2081 q\u2081 p\u2082 q\u2082 : F[X]\nhq\u2081 : q\u2081 \u2260 0\nhq\u2082 : q\u2082 \u2260 0\nh\u2081 : Splits (algebraMap F (SplittingField q\u2081)) p\u2081\nh\u2082 : Splits (algebraMap F (SplittingField q\u2082)) p\u2082\n\u22a2 Splits\n (RingHom.comp (\u2191(SplittingField.lift q\u2081 (_ : Splits (algebraMap F (SplittingField (q\u2081 * q\u2082))) q\u2081)))\n (algebraMap F (SplittingField q\u2081)))\n p\u2081\n[PROOFSTEP]\nexact splits_comp_of_splits _ _ h\u2081\n[GOAL]\ncase hg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np\u2081 q\u2081 p\u2082 q\u2082 : F[X]\nhq\u2081 : q\u2081 \u2260 0\nhq\u2082 : q\u2082 \u2260 0\nh\u2081 : Splits (algebraMap F (SplittingField q\u2081)) p\u2081\nh\u2082 : Splits (algebraMap F (SplittingField q\u2082)) p\u2082\n\u22a2 Splits (algebraMap F (SplittingField (q\u2081 * q\u2082))) p\u2082\n[PROOFSTEP]\nrw [\u2190\n (SplittingField.lift q\u2082\n (splits_of_splits_of_dvd (algebraMap F (q\u2081 * q\u2082).SplittingField) (mul_ne_zero hq\u2081 hq\u2082) (SplittingField.splits _)\n (dvd_mul_left q\u2082 q\u2081))).comp_algebraMap]\n[GOAL]\ncase hg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np\u2081 q\u2081 p\u2082 q\u2082 : F[X]\nhq\u2081 : q\u2081 \u2260 0\nhq\u2082 : q\u2082 \u2260 0\nh\u2081 : Splits (algebraMap F (SplittingField q\u2081)) p\u2081\nh\u2082 : Splits (algebraMap F (SplittingField q\u2082)) p\u2082\n\u22a2 Splits\n (RingHom.comp (\u2191(SplittingField.lift q\u2082 (_ : Splits (algebraMap F (SplittingField (q\u2081 * q\u2082))) q\u2082)))\n (algebraMap F (SplittingField q\u2082)))\n p\u2082\n[PROOFSTEP]\nexact splits_comp_of_splits _ _ h\u2082\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\n\u22a2 Splits (algebraMap F (SplittingField (comp p q))) p\n[PROOFSTEP]\nlet P : F[X] \u2192 Prop := fun r => r.Splits (algebraMap F (r.comp q).SplittingField)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\n\u22a2 Splits (algebraMap F (SplittingField (comp p q))) p\n[PROOFSTEP]\nhave key1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r := by\n intro r hr\n by_cases hr' : natDegree r = 0\n \u00b7 exact splits_of_natDegree_le_one _ (le_trans (le_of_eq hr') zero_le_one)\n obtain \u27e8x, hx\u27e9 :=\n exists_root_of_splits _ (SplittingField.splits (r.comp q)) fun h =>\n hr' ((mul_eq_zero.mp (natDegree_comp.symm.trans (natDegree_eq_of_degree_eq_some h))).resolve_right hq)\n rw [\u2190 aeval_def, aeval_comp] at hx \n have h_normal : Normal F (r.comp q).SplittingField := SplittingField.instNormal (r.comp q)\n have qx_int := Normal.isIntegral h_normal (aeval x q)\n exact\n splits_of_splits_of_dvd _ (minpoly.ne_zero qx_int) (Normal.splits h_normal _)\n ((minpoly.irreducible qx_int).dvd_symm hr (minpoly.dvd F _ hx))\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\n\u22a2 \u2200 {r : F[X]}, Irreducible r \u2192 P r\n[PROOFSTEP]\nintro r hr\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nr : F[X]\nhr : Irreducible r\n\u22a2 P r\n[PROOFSTEP]\nby_cases hr' : natDegree r = 0\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nr : F[X]\nhr : Irreducible r\nhr' : natDegree r = 0\n\u22a2 P r\n[PROOFSTEP]\nexact splits_of_natDegree_le_one _ (le_trans (le_of_eq hr') zero_le_one)\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nr : F[X]\nhr : Irreducible r\nhr' : \u00acnatDegree r = 0\n\u22a2 P r\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 :=\n exists_root_of_splits _ (SplittingField.splits (r.comp q)) fun h =>\n hr' ((mul_eq_zero.mp (natDegree_comp.symm.trans (natDegree_eq_of_degree_eq_some h))).resolve_right hq)\n[GOAL]\ncase neg.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nr : F[X]\nhr : Irreducible r\nhr' : \u00acnatDegree r = 0\nx : SplittingField (comp r q)\nhx : eval\u2082 (algebraMap F (SplittingField (comp r q))) x (comp r q) = 0\n\u22a2 P r\n[PROOFSTEP]\nrw [\u2190 aeval_def, aeval_comp] at hx \n[GOAL]\ncase neg.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nr : F[X]\nhr : Irreducible r\nhr' : \u00acnatDegree r = 0\nx : SplittingField (comp r q)\nhx : \u2191(aeval (\u2191(aeval x) q)) r = 0\n\u22a2 P r\n[PROOFSTEP]\nhave h_normal : Normal F (r.comp q).SplittingField := SplittingField.instNormal (r.comp q)\n[GOAL]\ncase neg.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nr : F[X]\nhr : Irreducible r\nhr' : \u00acnatDegree r = 0\nx : SplittingField (comp r q)\nhx : \u2191(aeval (\u2191(aeval x) q)) r = 0\nh_normal : Normal F (SplittingField (comp r q))\n\u22a2 P r\n[PROOFSTEP]\nhave qx_int := Normal.isIntegral h_normal (aeval x q)\n[GOAL]\ncase neg.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nr : F[X]\nhr : Irreducible r\nhr' : \u00acnatDegree r = 0\nx : SplittingField (comp r q)\nhx : \u2191(aeval (\u2191(aeval x) q)) r = 0\nh_normal : Normal F (SplittingField (comp r q))\nqx_int : IsIntegral F (\u2191(aeval x) q)\n\u22a2 P r\n[PROOFSTEP]\nexact\n splits_of_splits_of_dvd _ (minpoly.ne_zero qx_int) (Normal.splits h_normal _)\n ((minpoly.irreducible qx_int).dvd_symm hr (minpoly.dvd F _ hx))\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\n\u22a2 Splits (algebraMap F (SplittingField (comp p q))) p\n[PROOFSTEP]\nhave key2 : \u2200 {p\u2081 p\u2082 : F[X]}, P p\u2081 \u2192 P p\u2082 \u2192 P (p\u2081 * p\u2082) :=\n by\n intro p\u2081 p\u2082 hp\u2081 hp\u2082\n by_cases h\u2081 : p\u2081.comp q = 0\n \u00b7 cases' comp_eq_zero_iff.mp h\u2081 with h h\n \u00b7 rw [h, zero_mul]\n exact splits_zero _\n \u00b7 exact False.elim (hq (by rw [h.2, natDegree_C]))\n by_cases h\u2082 : p\u2082.comp q = 0\n \u00b7 cases' comp_eq_zero_iff.mp h\u2082 with h h\n \u00b7 rw [h, mul_zero]\n exact splits_zero _\n \u00b7 exact False.elim (hq (by rw [h.2, natDegree_C]))\n have key := mul_splits_in_splittingField_of_mul h\u2081 h\u2082 hp\u2081 hp\u2082\n rwa [\u2190 mul_comp] at key \n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\n\u22a2 \u2200 {p\u2081 p\u2082 : F[X]}, P p\u2081 \u2192 P p\u2082 \u2192 P (p\u2081 * p\u2082)\n[PROOFSTEP]\nintro p\u2081 p\u2082 hp\u2081 hp\u2082\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\np\u2081 p\u2082 : F[X]\nhp\u2081 : P p\u2081\nhp\u2082 : P p\u2082\n\u22a2 P (p\u2081 * p\u2082)\n[PROOFSTEP]\nby_cases h\u2081 : p\u2081.comp q = 0\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\np\u2081 p\u2082 : F[X]\nhp\u2081 : P p\u2081\nhp\u2082 : P p\u2082\nh\u2081 : comp p\u2081 q = 0\n\u22a2 P (p\u2081 * p\u2082)\n[PROOFSTEP]\ncases' comp_eq_zero_iff.mp h\u2081 with h h\n[GOAL]\ncase pos.inl\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\np\u2081 p\u2082 : F[X]\nhp\u2081 : P p\u2081\nhp\u2082 : P p\u2082\nh\u2081 : comp p\u2081 q = 0\nh : p\u2081 = 0\n\u22a2 P (p\u2081 * p\u2082)\n[PROOFSTEP]\nrw [h, zero_mul]\n[GOAL]\ncase pos.inl\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\np\u2081 p\u2082 : F[X]\nhp\u2081 : P p\u2081\nhp\u2082 : P p\u2082\nh\u2081 : comp p\u2081 q = 0\nh : p\u2081 = 0\n\u22a2 P 0\n[PROOFSTEP]\nexact splits_zero _\n[GOAL]\ncase pos.inr\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\np\u2081 p\u2082 : F[X]\nhp\u2081 : P p\u2081\nhp\u2082 : P p\u2082\nh\u2081 : comp p\u2081 q = 0\nh : eval (coeff q 0) p\u2081 = 0 \u2227 q = \u2191C (coeff q 0)\n\u22a2 P (p\u2081 * p\u2082)\n[PROOFSTEP]\nexact False.elim (hq (by rw [h.2, natDegree_C]))\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\np\u2081 p\u2082 : F[X]\nhp\u2081 : P p\u2081\nhp\u2082 : P p\u2082\nh\u2081 : comp p\u2081 q = 0\nh : eval (coeff q 0) p\u2081 = 0 \u2227 q = \u2191C (coeff q 0)\n\u22a2 natDegree q = 0\n[PROOFSTEP]\nrw [h.2, natDegree_C]\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\np\u2081 p\u2082 : F[X]\nhp\u2081 : P p\u2081\nhp\u2082 : P p\u2082\nh\u2081 : \u00accomp p\u2081 q = 0\n\u22a2 P (p\u2081 * p\u2082)\n[PROOFSTEP]\nby_cases h\u2082 : p\u2082.comp q = 0\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\np\u2081 p\u2082 : F[X]\nhp\u2081 : P p\u2081\nhp\u2082 : P p\u2082\nh\u2081 : \u00accomp p\u2081 q = 0\nh\u2082 : comp p\u2082 q = 0\n\u22a2 P (p\u2081 * p\u2082)\n[PROOFSTEP]\ncases' comp_eq_zero_iff.mp h\u2082 with h h\n[GOAL]\ncase pos.inl\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\np\u2081 p\u2082 : F[X]\nhp\u2081 : P p\u2081\nhp\u2082 : P p\u2082\nh\u2081 : \u00accomp p\u2081 q = 0\nh\u2082 : comp p\u2082 q = 0\nh : p\u2082 = 0\n\u22a2 P (p\u2081 * p\u2082)\n[PROOFSTEP]\nrw [h, mul_zero]\n[GOAL]\ncase pos.inl\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\np\u2081 p\u2082 : F[X]\nhp\u2081 : P p\u2081\nhp\u2082 : P p\u2082\nh\u2081 : \u00accomp p\u2081 q = 0\nh\u2082 : comp p\u2082 q = 0\nh : p\u2082 = 0\n\u22a2 P 0\n[PROOFSTEP]\nexact splits_zero _\n[GOAL]\ncase pos.inr\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\np\u2081 p\u2082 : F[X]\nhp\u2081 : P p\u2081\nhp\u2082 : P p\u2082\nh\u2081 : \u00accomp p\u2081 q = 0\nh\u2082 : comp p\u2082 q = 0\nh : eval (coeff q 0) p\u2082 = 0 \u2227 q = \u2191C (coeff q 0)\n\u22a2 P (p\u2081 * p\u2082)\n[PROOFSTEP]\nexact False.elim (hq (by rw [h.2, natDegree_C]))\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\np\u2081 p\u2082 : F[X]\nhp\u2081 : P p\u2081\nhp\u2082 : P p\u2082\nh\u2081 : \u00accomp p\u2081 q = 0\nh\u2082 : comp p\u2082 q = 0\nh : eval (coeff q 0) p\u2082 = 0 \u2227 q = \u2191C (coeff q 0)\n\u22a2 natDegree q = 0\n[PROOFSTEP]\nrw [h.2, natDegree_C]\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\np\u2081 p\u2082 : F[X]\nhp\u2081 : P p\u2081\nhp\u2082 : P p\u2082\nh\u2081 : \u00accomp p\u2081 q = 0\nh\u2082 : \u00accomp p\u2082 q = 0\n\u22a2 P (p\u2081 * p\u2082)\n[PROOFSTEP]\nhave key := mul_splits_in_splittingField_of_mul h\u2081 h\u2082 hp\u2081 hp\u2082\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\np\u2081 p\u2082 : F[X]\nhp\u2081 : P p\u2081\nhp\u2082 : P p\u2082\nh\u2081 : \u00accomp p\u2081 q = 0\nh\u2082 : \u00accomp p\u2082 q = 0\nkey : Splits (algebraMap F (SplittingField (comp p\u2081 q * comp p\u2082 q))) (p\u2081 * p\u2082)\n\u22a2 P (p\u2081 * p\u2082)\n[PROOFSTEP]\nrwa [\u2190 mul_comp] at key \n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\nkey2 : \u2200 {p\u2081 p\u2082 : F[X]}, P p\u2081 \u2192 P p\u2082 \u2192 P (p\u2081 * p\u2082)\n\u22a2 Splits (algebraMap F (SplittingField (comp p q))) p\n[PROOFSTEP]\ninduction p using WfDvdMonoid.induction_on_irreducible with\n| h0 => exact splits_zero _\n| hu u hu => exact splits_of_isUnit (algebraMap F (SplittingField (comp u q))) hu\n| hi p\u2081 p\u2082 _ hp\u2082 hp\u2081 => apply key2 (key1 hp\u2082) hp\u2081\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\nkey2 : \u2200 {p\u2081 p\u2082 : F[X]}, P p\u2081 \u2192 P p\u2082 \u2192 P (p\u2081 * p\u2082)\n\u22a2 Splits (algebraMap F (SplittingField (comp p q))) p\n[PROOFSTEP]\ninduction p using WfDvdMonoid.induction_on_irreducible with\n| h0 => exact splits_zero _\n| hu u hu => exact splits_of_isUnit (algebraMap F (SplittingField (comp u q))) hu\n| hi p\u2081 p\u2082 _ hp\u2082 hp\u2081 => apply key2 (key1 hp\u2082) hp\u2081\n[GOAL]\ncase h0\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\nkey2 : \u2200 {p\u2081 p\u2082 : F[X]}, P p\u2081 \u2192 P p\u2082 \u2192 P (p\u2081 * p\u2082)\n\u22a2 Splits (algebraMap F (SplittingField (comp 0 q))) 0\n[PROOFSTEP]\n\n| h0 => exact splits_zero _\n[GOAL]\ncase h0\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\nkey2 : \u2200 {p\u2081 p\u2082 : F[X]}, P p\u2081 \u2192 P p\u2082 \u2192 P (p\u2081 * p\u2082)\n\u22a2 Splits (algebraMap F (SplittingField (comp 0 q))) 0\n[PROOFSTEP]\nexact splits_zero _\n[GOAL]\ncase hu\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\nkey2 : \u2200 {p\u2081 p\u2082 : F[X]}, P p\u2081 \u2192 P p\u2082 \u2192 P (p\u2081 * p\u2082)\nu : F[X]\nhu : IsUnit u\n\u22a2 Splits (algebraMap F (SplittingField (comp u q))) u\n[PROOFSTEP]\n\n| hu u hu => exact splits_of_isUnit (algebraMap F (SplittingField (comp u q))) hu\n[GOAL]\ncase hu\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\nkey2 : \u2200 {p\u2081 p\u2082 : F[X]}, P p\u2081 \u2192 P p\u2082 \u2192 P (p\u2081 * p\u2082)\nu : F[X]\nhu : IsUnit u\n\u22a2 Splits (algebraMap F (SplittingField (comp u q))) u\n[PROOFSTEP]\nexact splits_of_isUnit (algebraMap F (SplittingField (comp u q))) hu\n[GOAL]\ncase hi\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\nkey2 : \u2200 {p\u2081 p\u2082 : F[X]}, P p\u2081 \u2192 P p\u2082 \u2192 P (p\u2081 * p\u2082)\np\u2081 p\u2082 : F[X]\na\u271d : p\u2081 \u2260 0\nhp\u2082 : Irreducible p\u2082\nhp\u2081 : Splits (algebraMap F (SplittingField (comp p\u2081 q))) p\u2081\n\u22a2 Splits (algebraMap F (SplittingField (comp (p\u2082 * p\u2081) q))) (p\u2082 * p\u2081)\n[PROOFSTEP]\n\n| hi p\u2081 p\u2082 _ hp\u2082 hp\u2081 => apply key2 (key1 hp\u2082) hp\u2081\n[GOAL]\ncase hi\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nP : F[X] \u2192 Prop := fun r => Splits (algebraMap F (SplittingField (comp r q))) r\nkey1 : \u2200 {r : F[X]}, Irreducible r \u2192 P r\nkey2 : \u2200 {p\u2081 p\u2082 : F[X]}, P p\u2081 \u2192 P p\u2082 \u2192 P (p\u2081 * p\u2082)\np\u2081 p\u2082 : F[X]\na\u271d : p\u2081 \u2260 0\nhp\u2082 : Irreducible p\u2082\nhp\u2081 : Splits (algebraMap F (SplittingField (comp p\u2081 q))) p\u2081\n\u22a2 Splits (algebraMap F (SplittingField (comp (p\u2082 * p\u2081) q))) (p\u2082 * p\u2081)\n[PROOFSTEP]\napply key2 (key1 hp\u2082) hp\u2081\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\n\u22a2 Function.Surjective \u2191(restrictComp p q hq)\n[PROOFSTEP]\nhaveI : Fact (Splits (algebraMap F (SplittingField (comp p q))) p) := \u27e8splits_in_splittingField_of_comp p q hq\u27e9\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nthis : Fact (Splits (algebraMap F (SplittingField (comp p q))) p)\n\u22a2 Function.Surjective \u2191(restrictComp p q hq)\n[PROOFSTEP]\nrw [restrictComp]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nhq : natDegree q \u2260 0\nthis : Fact (Splits (algebraMap F (SplittingField (comp p q))) p)\n\u22a2 Function.Surjective\n \u2191(let h := (_ : Fact (Splits (algebraMap F (SplittingField (comp p q))) p));\n restrict p (SplittingField (comp p q)))\n[PROOFSTEP]\nexact restrict_surjective _ _\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\n\u22a2 natDegree p \u2223 Fintype.card (Gal p)\n[PROOFSTEP]\nrw [Gal.card_of_separable p_irr.separable]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\n\u22a2 natDegree p \u2223 finrank F (SplittingField p)\n[PROOFSTEP]\nhave hp : p.degree \u2260 0 := fun h => Nat.Prime.ne_zero p_deg (natDegree_eq_zero_iff_degree_le_zero.mpr (le_of_eq h))\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\nhp : degree p \u2260 0\n\u22a2 natDegree p \u2223 finrank F (SplittingField p)\n[PROOFSTEP]\nlet \u03b1 : p.SplittingField := rootOfSplits (algebraMap F p.SplittingField) (SplittingField.splits p) hp\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\nhp : degree p \u2260 0\n\u03b1 : SplittingField p :=\n rootOfSplits (algebraMap F (SplittingField p)) (_ : Splits (algebraMap F (SplittingField p)) p) hp\n\u22a2 natDegree p \u2223 finrank F (SplittingField p)\n[PROOFSTEP]\nhave h\u03b1 : IsIntegral F \u03b1 := Algebra.isIntegral_of_finite _ _ \u03b1\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\nhp : degree p \u2260 0\n\u03b1 : SplittingField p :=\n rootOfSplits (algebraMap F (SplittingField p)) (_ : Splits (algebraMap F (SplittingField p)) p) hp\nh\u03b1 : IsIntegral F \u03b1\n\u22a2 natDegree p \u2223 finrank F (SplittingField p)\n[PROOFSTEP]\nuse FiniteDimensional.finrank F\u27ee\u03b1\u27ef p.SplittingField\n[GOAL]\ncase h\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\nhp : degree p \u2260 0\n\u03b1 : SplittingField p :=\n rootOfSplits (algebraMap F (SplittingField p)) (_ : Splits (algebraMap F (SplittingField p)) p) hp\nh\u03b1 : IsIntegral F \u03b1\n\u22a2 finrank F (SplittingField p) = natDegree p * finrank { x // x \u2208 F\u27ee\u03b1\u27ef } (SplittingField p)\n[PROOFSTEP]\nsuffices (minpoly F \u03b1).natDegree = p.natDegree by\n rw [\u2190 FiniteDimensional.finrank_mul_finrank F F\u27ee\u03b1\u27ef p.SplittingField, IntermediateField.adjoin.finrank h\u03b1, this]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\nhp : degree p \u2260 0\n\u03b1 : SplittingField p :=\n rootOfSplits (algebraMap F (SplittingField p)) (_ : Splits (algebraMap F (SplittingField p)) p) hp\nh\u03b1 : IsIntegral F \u03b1\nthis : natDegree (minpoly F \u03b1) = natDegree p\n\u22a2 finrank F (SplittingField p) = natDegree p * finrank { x // x \u2208 F\u27ee\u03b1\u27ef } (SplittingField p)\n[PROOFSTEP]\nrw [\u2190 FiniteDimensional.finrank_mul_finrank F F\u27ee\u03b1\u27ef p.SplittingField, IntermediateField.adjoin.finrank h\u03b1, this]\n[GOAL]\ncase h\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\nhp : degree p \u2260 0\n\u03b1 : SplittingField p :=\n rootOfSplits (algebraMap F (SplittingField p)) (_ : Splits (algebraMap F (SplittingField p)) p) hp\nh\u03b1 : IsIntegral F \u03b1\n\u22a2 natDegree (minpoly F \u03b1) = natDegree p\n[PROOFSTEP]\nsuffices minpoly F \u03b1 \u2223 p by\n have key := (minpoly.irreducible h\u03b1).dvd_symm p_irr this\n apply le_antisymm\n \u00b7 exact natDegree_le_of_dvd this p_irr.ne_zero\n \u00b7 exact natDegree_le_of_dvd key (minpoly.ne_zero h\u03b1)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\nhp : degree p \u2260 0\n\u03b1 : SplittingField p :=\n rootOfSplits (algebraMap F (SplittingField p)) (_ : Splits (algebraMap F (SplittingField p)) p) hp\nh\u03b1 : IsIntegral F \u03b1\nthis : minpoly F \u03b1 \u2223 p\n\u22a2 natDegree (minpoly F \u03b1) = natDegree p\n[PROOFSTEP]\nhave key := (minpoly.irreducible h\u03b1).dvd_symm p_irr this\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\nhp : degree p \u2260 0\n\u03b1 : SplittingField p :=\n rootOfSplits (algebraMap F (SplittingField p)) (_ : Splits (algebraMap F (SplittingField p)) p) hp\nh\u03b1 : IsIntegral F \u03b1\nthis : minpoly F \u03b1 \u2223 p\nkey : p \u2223 minpoly F \u03b1\n\u22a2 natDegree (minpoly F \u03b1) = natDegree p\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\nhp : degree p \u2260 0\n\u03b1 : SplittingField p :=\n rootOfSplits (algebraMap F (SplittingField p)) (_ : Splits (algebraMap F (SplittingField p)) p) hp\nh\u03b1 : IsIntegral F \u03b1\nthis : minpoly F \u03b1 \u2223 p\nkey : p \u2223 minpoly F \u03b1\n\u22a2 natDegree (minpoly F \u03b1) \u2264 natDegree p\n[PROOFSTEP]\nexact natDegree_le_of_dvd this p_irr.ne_zero\n[GOAL]\ncase a\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\nhp : degree p \u2260 0\n\u03b1 : SplittingField p :=\n rootOfSplits (algebraMap F (SplittingField p)) (_ : Splits (algebraMap F (SplittingField p)) p) hp\nh\u03b1 : IsIntegral F \u03b1\nthis : minpoly F \u03b1 \u2223 p\nkey : p \u2223 minpoly F \u03b1\n\u22a2 natDegree p \u2264 natDegree (minpoly F \u03b1)\n[PROOFSTEP]\nexact natDegree_le_of_dvd key (minpoly.ne_zero h\u03b1)\n[GOAL]\ncase h\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\nhp : degree p \u2260 0\n\u03b1 : SplittingField p :=\n rootOfSplits (algebraMap F (SplittingField p)) (_ : Splits (algebraMap F (SplittingField p)) p) hp\nh\u03b1 : IsIntegral F \u03b1\n\u22a2 minpoly F \u03b1 \u2223 p\n[PROOFSTEP]\napply minpoly.dvd F \u03b1\n[GOAL]\ncase h\nF : Type u_1\ninst\u271d\u00b3 : Field F\np q : F[X]\nE : Type u_2\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra F E\ninst\u271d : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\nhp : degree p \u2260 0\n\u03b1 : SplittingField p :=\n rootOfSplits (algebraMap F (SplittingField p)) (_ : Splits (algebraMap F (SplittingField p)) p) hp\nh\u03b1 : IsIntegral F \u03b1\n\u22a2 \u2191(aeval \u03b1) p = 0\n[PROOFSTEP]\nrw [aeval_def, map_rootOfSplits _ (SplittingField.splits p) hp]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n Finset.card (Set.toFinset (rootSet p \u211d)) +\n Finset.card\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n[PROOFSTEP]\nby_cases hp : p = 0\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : p = 0\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n Finset.card (Set.toFinset (rootSet p \u211d)) +\n Finset.card\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n[PROOFSTEP]\nhaveI : IsEmpty (p.rootSet \u2102) := by rw [hp, rootSet_zero]; infer_instance\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : p = 0\n\u22a2 IsEmpty \u2191(rootSet p \u2102)\n[PROOFSTEP]\nrw [hp, rootSet_zero]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : p = 0\n\u22a2 IsEmpty \u2191\u2205\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : p = 0\nthis : IsEmpty \u2191(rootSet p \u2102)\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n Finset.card (Set.toFinset (rootSet p \u211d)) +\n Finset.card\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n[PROOFSTEP]\nsimp_rw [(galActionHom p \u2102 _).support.eq_empty_of_isEmpty, hp, rootSet_zero, Set.toFinset_empty, Finset.card_empty]\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n Finset.card (Set.toFinset (rootSet p \u211d)) +\n Finset.card\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n[PROOFSTEP]\nhave inj : Function.Injective (IsScalarTower.toAlgHom \u211a \u211d \u2102) := (algebraMap \u211d \u2102).injective\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n Finset.card (Set.toFinset (rootSet p \u211d)) +\n Finset.card\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n[PROOFSTEP]\nrw [\u2190 Finset.card_image_of_injective _ Subtype.coe_injective, \u2190 Finset.card_image_of_injective _ inj]\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) +\n Finset.card\n (Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))))\n[PROOFSTEP]\nlet a : Finset \u2102 := ?_\n[GOAL]\ncase neg.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) +\n Finset.card\n (Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))))\ncase neg.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\n\u22a2 Finset \u2102\n[PROOFSTEP]\nlet b : Finset \u2102 := ?_\n[GOAL]\ncase neg.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\nb : Finset \u2102 := ?neg.refine_2.refine_1\u271d\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) +\n Finset.card\n (Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))))\ncase neg.refine_2.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\n\u22a2 Finset \u2102\ncase neg.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\n\u22a2 Finset \u2102\n[PROOFSTEP]\nlet c : Finset \u2102 :=\n ?_\n -- Porting note: was\n -- change a.card = b.card + c.card\n[GOAL]\ncase neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\nb : Finset \u2102 := ?neg.refine_2.refine_1\u271d\nc : Finset \u2102 := ?neg.refine_2.refine_2.refine_1\u271d\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) +\n Finset.card\n (Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))))\ncase neg.refine_2.refine_2.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\nb : Finset \u2102 := ?neg.refine_2.refine_1\u271d\n\u22a2 Finset \u2102\ncase neg.refine_2.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\n\u22a2 Finset \u2102\ncase neg.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\n\u22a2 Finset \u2102\n[PROOFSTEP]\nsuffices a.card = b.card + c.card by exact this\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\nb : Finset \u2102 := ?neg.refine_2.refine_1\u271d\nc : Finset \u2102 := ?neg.refine_2.refine_2.refine_1\u271d\nthis : Finset.card a = Finset.card b + Finset.card c\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) +\n Finset.card\n (Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))))\n[PROOFSTEP]\nexact this\n[GOAL]\ncase neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n\u22a2 Finset.card a = Finset.card b + Finset.card c\n[PROOFSTEP]\nhave ha : \u2200 z : \u2102, z \u2208 a \u2194 aeval z p = 0 := by intro z; rw [Set.mem_toFinset, mem_rootSet_of_ne hp]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n\u22a2 \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\n[PROOFSTEP]\nintro z\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nz : \u2102\n\u22a2 z \u2208 a \u2194 \u2191(aeval z) p = 0\n[PROOFSTEP]\nrw [Set.mem_toFinset, mem_rootSet_of_ne hp]\n[GOAL]\ncase neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\n\u22a2 Finset.card a = Finset.card b + Finset.card c\n[PROOFSTEP]\nhave hb : \u2200 z : \u2102, z \u2208 b \u2194 aeval z p = 0 \u2227 z.im = 0 := by\n intro z\n simp_rw [Finset.mem_image, Set.mem_toFinset, mem_rootSet_of_ne hp]\n constructor\n \u00b7 rintro \u27e8w, hw, rfl\u27e9\n exact \u27e8by rw [aeval_algHom_apply, hw, AlgHom.map_zero], rfl\u27e9\n \u00b7 rintro \u27e8hz1, hz2\u27e9\n have key : IsScalarTower.toAlgHom \u211a \u211d \u2102 z.re = z := by ext; rfl; rw [hz2]; rfl\n exact \u27e8z.re, inj (by rwa [\u2190 aeval_algHom_apply, key, AlgHom.map_zero]), key\u27e9\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\n\u22a2 \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\n[PROOFSTEP]\nintro z\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\n\u22a2 z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\n[PROOFSTEP]\nsimp_rw [Finset.mem_image, Set.mem_toFinset, mem_rootSet_of_ne hp]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\n\u22a2 (\u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z) \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\n\u22a2 (\u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z) \u2192 \u2191(aeval z) p = 0 \u2227 z.im = 0\n[PROOFSTEP]\nrintro \u27e8w, hw, rfl\u27e9\n[GOAL]\ncase mp.intro.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nw : \u211d\nhw : \u2191(aeval w) p = 0\n\u22a2 \u2191(aeval (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) w)) p = 0 \u2227 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) w).im = 0\n[PROOFSTEP]\nexact \u27e8by rw [aeval_algHom_apply, hw, AlgHom.map_zero], rfl\u27e9\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nw : \u211d\nhw : \u2191(aeval w) p = 0\n\u22a2 \u2191(aeval (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) w)) p = 0\n[PROOFSTEP]\nrw [aeval_algHom_apply, hw, AlgHom.map_zero]\n[GOAL]\ncase mpr\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\n\u22a2 \u2191(aeval z) p = 0 \u2227 z.im = 0 \u2192 \u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z\n[PROOFSTEP]\nrintro \u27e8hz1, hz2\u27e9\n[GOAL]\ncase mpr.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 \u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z\n[PROOFSTEP]\nhave key : IsScalarTower.toAlgHom \u211a \u211d \u2102 z.re = z := by ext; rfl; rw [hz2]; rfl\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re = z\n[PROOFSTEP]\next\n[GOAL]\ncase a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).re = z.re\ncase a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).im = z.im\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).im = z.im\n[PROOFSTEP]\nrw [hz2]\n[GOAL]\ncase a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).im = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mpr.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\nkey : \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re = z\n\u22a2 \u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z\n[PROOFSTEP]\nexact \u27e8z.re, inj (by rwa [\u2190 aeval_algHom_apply, key, AlgHom.map_zero]), key\u27e9\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\nkey : \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re = z\n\u22a2 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) (\u2191(aeval z.re) p) = \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) 0\n[PROOFSTEP]\nrwa [\u2190 aeval_algHom_apply, key, AlgHom.map_zero]\n[GOAL]\ncase neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\n\u22a2 Finset.card a = Finset.card b + Finset.card c\n[PROOFSTEP]\nhave hc0 : \u2200 w : p.rootSet \u2102, galActionHom p \u2102 (restrict p \u2102 (Complex.conjAe.restrictScalars \u211a)) w = w \u2194 w.val.im = 0 :=\n by\n intro w\n rw [Subtype.ext_iff, galActionHom_restrict]\n exact Complex.conj_eq_iff_im\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\n\u22a2 \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\n[PROOFSTEP]\nintro w\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nw : \u2191(rootSet p \u2102)\n\u22a2 \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\n[PROOFSTEP]\nrw [Subtype.ext_iff, galActionHom_restrict]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nw : \u2191(rootSet p \u2102)\n\u22a2 \u2191(AlgEquiv.restrictScalars \u211a Complex.conjAe) \u2191w = \u2191w \u2194 (\u2191w).im = 0\n[PROOFSTEP]\nexact Complex.conj_eq_iff_im\n[GOAL]\ncase neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\n\u22a2 Finset.card a = Finset.card b + Finset.card c\n[PROOFSTEP]\nhave hc : \u2200 z : \u2102, z \u2208 c \u2194 aeval z p = 0 \u2227 z.im \u2260 0 := by\n intro z\n simp_rw [Finset.mem_image]\n constructor\n \u00b7 rintro \u27e8w, hw, rfl\u27e9\n exact \u27e8(mem_rootSet.mp w.2).2, mt (hc0 w).mpr (Equiv.Perm.mem_support.mp hw)\u27e9\n \u00b7 rintro \u27e8hz1, hz2\u27e9\n exact \u27e8\u27e8z, mem_rootSet.mpr \u27e8hp, hz1\u27e9\u27e9, Equiv.Perm.mem_support.mpr (mt (hc0 _).mp hz2), rfl\u27e9\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\n\u22a2 \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n[PROOFSTEP]\nintro z\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nz : \u2102\n\u22a2 z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n[PROOFSTEP]\nsimp_rw [Finset.mem_image]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nz : \u2102\n\u22a2 (\u2203 a,\n a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227\n \u2191a = z) \u2194\n \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nz : \u2102\n\u22a2 (\u2203 a,\n a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227\n \u2191a = z) \u2192\n \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n[PROOFSTEP]\nrintro \u27e8w, hw, rfl\u27e9\n[GOAL]\ncase mp.intro.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nw : { x // x \u2208 rootSet p \u2102 }\nhw : w \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))\n\u22a2 \u2191(aeval \u2191w) p = 0 \u2227 (\u2191w).im \u2260 0\n[PROOFSTEP]\nexact \u27e8(mem_rootSet.mp w.2).2, mt (hc0 w).mpr (Equiv.Perm.mem_support.mp hw)\u27e9\n[GOAL]\ncase mpr\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nz : \u2102\n\u22a2 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 \u2192\n \u2203 a,\n a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227\n \u2191a = z\n[PROOFSTEP]\nrintro \u27e8hz1, hz2\u27e9\n[GOAL]\ncase mpr.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im \u2260 0\n\u22a2 \u2203 a,\n a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227 \u2191a = z\n[PROOFSTEP]\nexact \u27e8\u27e8z, mem_rootSet.mpr \u27e8hp, hz1\u27e9\u27e9, Equiv.Perm.mem_support.mpr (mt (hc0 _).mp hz2), rfl\u27e9\n[GOAL]\ncase neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 Finset.card a = Finset.card b + Finset.card c\n[PROOFSTEP]\nrw [\u2190 Finset.card_disjoint_union]\n[GOAL]\ncase neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 Finset.card a = Finset.card (b \u222a c)\n[PROOFSTEP]\napply congr_arg Finset.card\n[GOAL]\ncase neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 a = b \u222a c\n[PROOFSTEP]\nsimp_rw [Finset.ext_iff, Finset.mem_union, ha, hb, hc]\n[GOAL]\ncase neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 \u2200 (a : \u2102), \u2191(aeval a) p = 0 \u2194 \u2191(aeval a) p = 0 \u2227 a.im = 0 \u2228 \u2191(aeval a) p = 0 \u2227 a.im \u2260 0\n[PROOFSTEP]\ntauto\n[GOAL]\ncase neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 Disjoint b c\n[PROOFSTEP]\nrw [Finset.disjoint_left]\n[GOAL]\ncase neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 \u2200 \u2983a : \u2102\u2984, a \u2208 b \u2192 \u00aca \u2208 c\n[PROOFSTEP]\nintro z\n[GOAL]\ncase neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\nz : \u2102\n\u22a2 z \u2208 b \u2192 \u00acz \u2208 c\n[PROOFSTEP]\nrw [hb, hc]\n[GOAL]\ncase neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n Finset.image (fun a => \u2191a)\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n \u2200 (w : \u2191(rootSet p \u2102)),\n \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\nz : \u2102\n\u22a2 \u2191(aeval z) p = 0 \u2227 z.im = 0 \u2192 \u00ac(\u2191(aeval z) p = 0 \u2227 z.im \u2260 0)\n[PROOFSTEP]\ntauto\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\n\u22a2 Function.Bijective \u2191(galActionHom p \u2102)\n[PROOFSTEP]\nclassical\nhave h1 : Fintype.card (p.rootSet \u2102) = p.natDegree :=\n by\n simp_rw [rootSet_def, Finset.coe_sort_coe, Fintype.card_coe]\n rw [Multiset.toFinset_card_of_nodup, \u2190 natDegree_eq_card_roots]\n \u00b7 exact IsAlgClosed.splits_codomain p\n \u00b7 exact nodup_roots ((separable_map (algebraMap \u211a \u2102)).mpr p_irr.separable)\nhave h2 : Fintype.card p.Gal = Fintype.card (galActionHom p \u2102).range :=\n Fintype.card_congr (MonoidHom.ofInjective (galActionHom_injective p \u2102)).toEquiv\nlet conj := restrict p \u2102 (Complex.conjAe.restrictScalars \u211a)\nrefine'\n \u27e8galActionHom_injective p \u2102, fun x =>\n (congr_arg (Membership.mem x) (show (galActionHom p \u2102).range = \u22a4 from _)).mpr (Subgroup.mem_top x)\u27e9\napply Equiv.Perm.subgroup_eq_top_of_swap_mem\n\u00b7 rwa [h1]\n\u00b7 rw [h1]\n convert prime_degree_dvd_card p_irr p_deg using 1\n convert h2.symm\n\u00b7 exact \u27e8conj, rfl\u27e9\n\u00b7 rw [\u2190 Equiv.Perm.card_support_eq_two]\n apply Nat.add_left_cancel\n rw [\u2190 p_roots, \u2190 Set.toFinset_card (rootSet p \u211d), \u2190 Set.toFinset_card (rootSet p \u2102)]\n exact (card_complex_roots_eq_card_real_add_card_not_gal_inv p).symm\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\n\u22a2 Function.Bijective \u2191(galActionHom p \u2102)\n[PROOFSTEP]\nhave h1 : Fintype.card (p.rootSet \u2102) = p.natDegree :=\n by\n simp_rw [rootSet_def, Finset.coe_sort_coe, Fintype.card_coe]\n rw [Multiset.toFinset_card_of_nodup, \u2190 natDegree_eq_card_roots]\n \u00b7 exact IsAlgClosed.splits_codomain p\n \u00b7 exact nodup_roots ((separable_map (algebraMap \u211a \u2102)).mpr p_irr.separable)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\n\u22a2 Fintype.card \u2191(rootSet p \u2102) = natDegree p\n[PROOFSTEP]\nsimp_rw [rootSet_def, Finset.coe_sort_coe, Fintype.card_coe]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\n\u22a2 Finset.card (Multiset.toFinset (roots (map (algebraMap \u211a \u2102) p))) = natDegree p\n[PROOFSTEP]\nrw [Multiset.toFinset_card_of_nodup, \u2190 natDegree_eq_card_roots]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\n\u22a2 Splits (algebraMap \u211a \u2102) p\n[PROOFSTEP]\nexact IsAlgClosed.splits_codomain p\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\n\u22a2 Multiset.Nodup (roots (map (algebraMap \u211a \u2102) p))\n[PROOFSTEP]\nexact nodup_roots ((separable_map (algebraMap \u211a \u2102)).mpr p_irr.separable)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\nh1 : Fintype.card \u2191(rootSet p \u2102) = natDegree p\n\u22a2 Function.Bijective \u2191(galActionHom p \u2102)\n[PROOFSTEP]\nhave h2 : Fintype.card p.Gal = Fintype.card (galActionHom p \u2102).range :=\n Fintype.card_congr (MonoidHom.ofInjective (galActionHom_injective p \u2102)).toEquiv\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\nh1 : Fintype.card \u2191(rootSet p \u2102) = natDegree p\nh2 : Fintype.card (Gal p) = Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) }\n\u22a2 Function.Bijective \u2191(galActionHom p \u2102)\n[PROOFSTEP]\nlet conj := restrict p \u2102 (Complex.conjAe.restrictScalars \u211a)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\nh1 : Fintype.card \u2191(rootSet p \u2102) = natDegree p\nh2 : Fintype.card (Gal p) = Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) }\nconj : (fun x => Gal p) (AlgEquiv.restrictScalars \u211a Complex.conjAe) :=\n \u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)\n\u22a2 Function.Bijective \u2191(galActionHom p \u2102)\n[PROOFSTEP]\nrefine'\n \u27e8galActionHom_injective p \u2102, fun x =>\n (congr_arg (Membership.mem x) (show (galActionHom p \u2102).range = \u22a4 from _)).mpr (Subgroup.mem_top x)\u27e9\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\nh1 : Fintype.card \u2191(rootSet p \u2102) = natDegree p\nh2 : Fintype.card (Gal p) = Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) }\nconj : (fun x => Gal p) (AlgEquiv.restrictScalars \u211a Complex.conjAe) :=\n \u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)\nx : Equiv.Perm \u2191(rootSet p \u2102)\n\u22a2 MonoidHom.range (galActionHom p \u2102) = \u22a4\n[PROOFSTEP]\napply Equiv.Perm.subgroup_eq_top_of_swap_mem\n[GOAL]\ncase h0\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\nh1 : Fintype.card \u2191(rootSet p \u2102) = natDegree p\nh2 : Fintype.card (Gal p) = Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) }\nconj : (fun x => Gal p) (AlgEquiv.restrictScalars \u211a Complex.conjAe) :=\n \u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)\nx : Equiv.Perm \u2191(rootSet p \u2102)\n\u22a2 Nat.Prime (Fintype.card \u2191(rootSet p \u2102))\n[PROOFSTEP]\nrwa [h1]\n[GOAL]\ncase h1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\nh1 : Fintype.card \u2191(rootSet p \u2102) = natDegree p\nh2 : Fintype.card (Gal p) = Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) }\nconj : (fun x => Gal p) (AlgEquiv.restrictScalars \u211a Complex.conjAe) :=\n \u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)\nx : Equiv.Perm \u2191(rootSet p \u2102)\n\u22a2 Fintype.card \u2191(rootSet p \u2102) \u2223 Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) }\n[PROOFSTEP]\nrw [h1]\n[GOAL]\ncase h1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\nh1 : Fintype.card \u2191(rootSet p \u2102) = natDegree p\nh2 : Fintype.card (Gal p) = Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) }\nconj : (fun x => Gal p) (AlgEquiv.restrictScalars \u211a Complex.conjAe) :=\n \u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)\nx : Equiv.Perm \u2191(rootSet p \u2102)\n\u22a2 natDegree p \u2223 Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) }\n[PROOFSTEP]\nconvert prime_degree_dvd_card p_irr p_deg using 1\n[GOAL]\ncase h.e'_4\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\nh1 : Fintype.card \u2191(rootSet p \u2102) = natDegree p\nh2 : Fintype.card (Gal p) = Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) }\nconj : (fun x => Gal p) (AlgEquiv.restrictScalars \u211a Complex.conjAe) :=\n \u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)\nx : Equiv.Perm \u2191(rootSet p \u2102)\n\u22a2 Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) } = Fintype.card (Gal p)\n[PROOFSTEP]\nconvert h2.symm\n[GOAL]\ncase h2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\nh1 : Fintype.card \u2191(rootSet p \u2102) = natDegree p\nh2 : Fintype.card (Gal p) = Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) }\nconj : (fun x => Gal p) (AlgEquiv.restrictScalars \u211a Complex.conjAe) :=\n \u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)\nx : Equiv.Perm \u2191(rootSet p \u2102)\n\u22a2 ?\u03c4 \u2208 MonoidHom.range (galActionHom p \u2102)\n[PROOFSTEP]\nexact \u27e8conj, rfl\u27e9\n[GOAL]\ncase h3\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\nh1 : Fintype.card \u2191(rootSet p \u2102) = natDegree p\nh2 : Fintype.card (Gal p) = Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) }\nconj : (fun x => Gal p) (AlgEquiv.restrictScalars \u211a Complex.conjAe) :=\n \u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)\nx : Equiv.Perm \u2191(rootSet p \u2102)\n\u22a2 Equiv.Perm.IsSwap (\u2191(galActionHom p \u2102) conj)\n[PROOFSTEP]\nrw [\u2190 Equiv.Perm.card_support_eq_two]\n[GOAL]\ncase h3\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\nh1 : Fintype.card \u2191(rootSet p \u2102) = natDegree p\nh2 : Fintype.card (Gal p) = Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) }\nconj : (fun x => Gal p) (AlgEquiv.restrictScalars \u211a Complex.conjAe) :=\n \u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)\nx : Equiv.Perm \u2191(rootSet p \u2102)\n\u22a2 Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) conj)) = 2\n[PROOFSTEP]\napply Nat.add_left_cancel\n[GOAL]\ncase h3.a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\nh1 : Fintype.card \u2191(rootSet p \u2102) = natDegree p\nh2 : Fintype.card (Gal p) = Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) }\nconj : (fun x => Gal p) (AlgEquiv.restrictScalars \u211a Complex.conjAe) :=\n \u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)\nx : Equiv.Perm \u2191(rootSet p \u2102)\n\u22a2 ?h3.n + Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) conj)) = ?h3.n + 2\ncase h3.n\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\nh1 : Fintype.card \u2191(rootSet p \u2102) = natDegree p\nh2 : Fintype.card (Gal p) = Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) }\nconj : (fun x => Gal p) (AlgEquiv.restrictScalars \u211a Complex.conjAe) :=\n \u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)\nx : Equiv.Perm \u2191(rootSet p \u2102)\n\u22a2 \u2115\n[PROOFSTEP]\nrw [\u2190 p_roots, \u2190 Set.toFinset_card (rootSet p \u211d), \u2190 Set.toFinset_card (rootSet p \u2102)]\n[GOAL]\ncase h3.a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots : Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\nh1 : Fintype.card \u2191(rootSet p \u2102) = natDegree p\nh2 : Fintype.card (Gal p) = Fintype.card { x // x \u2208 MonoidHom.range (galActionHom p \u2102) }\nconj : (fun x => Gal p) (AlgEquiv.restrictScalars \u211a Complex.conjAe) :=\n \u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)\nx : Equiv.Perm \u2191(rootSet p \u2102)\n\u22a2 Finset.card (Set.toFinset (rootSet p \u211d)) + Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) conj)) =\n Finset.card (Set.toFinset (rootSet p \u2102))\n[PROOFSTEP]\nexact (card_complex_roots_eq_card_real_add_card_not_gal_inv p).symm\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots1 : Fintype.card \u2191(rootSet p \u211d) + 1 \u2264 Fintype.card \u2191(rootSet p \u2102)\np_roots2 : Fintype.card \u2191(rootSet p \u2102) \u2264 Fintype.card \u2191(rootSet p \u211d) + 3\n\u22a2 Function.Bijective \u2191(galActionHom p \u2102)\n[PROOFSTEP]\napply galActionHom_bijective_of_prime_degree p_irr p_deg\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots1 : Fintype.card \u2191(rootSet p \u211d) + 1 \u2264 Fintype.card \u2191(rootSet p \u2102)\np_roots2 : Fintype.card \u2191(rootSet p \u2102) \u2264 Fintype.card \u2191(rootSet p \u211d) + 3\n\u22a2 Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\n[PROOFSTEP]\nlet n := (galActionHom p \u2102 (restrict p \u2102 (Complex.conjAe.restrictScalars \u211a))).support.card\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots1 : Fintype.card \u2191(rootSet p \u211d) + 1 \u2264 Fintype.card \u2191(rootSet p \u2102)\np_roots2 : Fintype.card \u2191(rootSet p \u2102) \u2264 Fintype.card \u2191(rootSet p \u211d) + 3\nn : \u2115 :=\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n\u22a2 Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\n[PROOFSTEP]\nhave hn : 2 \u2223 n :=\n Equiv.Perm.two_dvd_card_support\n (by\n rw [\u2190 MonoidHom.map_pow, \u2190 MonoidHom.map_pow,\n show AlgEquiv.restrictScalars \u211a Complex.conjAe ^ 2 = 1 from AlgEquiv.ext Complex.conj_conj, MonoidHom.map_one,\n MonoidHom.map_one])\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots1 : Fintype.card \u2191(rootSet p \u211d) + 1 \u2264 Fintype.card \u2191(rootSet p \u2102)\np_roots2 : Fintype.card \u2191(rootSet p \u2102) \u2264 Fintype.card \u2191(rootSet p \u211d) + 3\nn : \u2115 :=\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n\u22a2 \u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)) ^ 2 = 1\n[PROOFSTEP]\nrw [\u2190 MonoidHom.map_pow, \u2190 MonoidHom.map_pow,\n show AlgEquiv.restrictScalars \u211a Complex.conjAe ^ 2 = 1 from AlgEquiv.ext Complex.conj_conj, MonoidHom.map_one,\n MonoidHom.map_one]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots1 : Fintype.card \u2191(rootSet p \u211d) + 1 \u2264 Fintype.card \u2191(rootSet p \u2102)\np_roots2 : Fintype.card \u2191(rootSet p \u2102) \u2264 Fintype.card \u2191(rootSet p \u211d) + 3\nn : \u2115 :=\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nhn : 2 \u2223 n\n\u22a2 Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\n[PROOFSTEP]\nhave key := card_complex_roots_eq_card_real_add_card_not_gal_inv p\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots1 : Fintype.card \u2191(rootSet p \u211d) + 1 \u2264 Fintype.card \u2191(rootSet p \u2102)\np_roots2 : Fintype.card \u2191(rootSet p \u2102) \u2264 Fintype.card \u2191(rootSet p \u211d) + 3\nn : \u2115 :=\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nhn : 2 \u2223 n\nkey :\n Finset.card (Set.toFinset (rootSet p \u2102)) =\n Finset.card (Set.toFinset (rootSet p \u211d)) +\n Finset.card\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n\u22a2 Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\n[PROOFSTEP]\nsimp_rw [Set.toFinset_card] at key \n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots1 : Fintype.card \u2191(rootSet p \u211d) + 1 \u2264 Fintype.card \u2191(rootSet p \u2102)\np_roots2 : Fintype.card \u2191(rootSet p \u2102) \u2264 Fintype.card \u2191(rootSet p \u211d) + 3\nn : \u2115 :=\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nhn : 2 \u2223 n\nkey :\n Fintype.card \u2191(rootSet p \u2102) =\n Fintype.card \u2191(rootSet p \u211d) +\n Finset.card\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n\u22a2 Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\n[PROOFSTEP]\nrw [key, add_le_add_iff_left] at p_roots1 p_roots2 \n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots1 :\n 1 \u2264\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\np_roots2 :\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) \u2264\n 3\nn : \u2115 :=\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nhn : 2 \u2223 n\nkey :\n Fintype.card \u2191(rootSet p \u2102) =\n Fintype.card \u2191(rootSet p \u211d) +\n Finset.card\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n\u22a2 Fintype.card \u2191(rootSet p \u2102) = Fintype.card \u2191(rootSet p \u211d) + 2\n[PROOFSTEP]\nrw [key, add_right_inj]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots1 :\n 1 \u2264\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\np_roots2 :\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) \u2264\n 3\nn : \u2115 :=\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nhn : 2 \u2223 n\nkey :\n Fintype.card \u2191(rootSet p \u2102) =\n Fintype.card \u2191(rootSet p \u211d) +\n Finset.card\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n\u22a2 Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) =\n 2\n[PROOFSTEP]\nsuffices \u2200 m : \u2115, 2 \u2223 m \u2192 1 \u2264 m \u2192 m \u2264 3 \u2192 m = 2 by exact this n hn p_roots1 p_roots2\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots1 :\n 1 \u2264\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\np_roots2 :\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) \u2264\n 3\nn : \u2115 :=\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nhn : 2 \u2223 n\nkey :\n Fintype.card \u2191(rootSet p \u2102) =\n Fintype.card \u2191(rootSet p \u211d) +\n Finset.card\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nthis : \u2200 (m : \u2115), 2 \u2223 m \u2192 1 \u2264 m \u2192 m \u2264 3 \u2192 m = 2\n\u22a2 Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) =\n 2\n[PROOFSTEP]\nexact this n hn p_roots1 p_roots2\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots1 :\n 1 \u2264\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\np_roots2 :\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) \u2264\n 3\nn : \u2115 :=\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nhn : 2 \u2223 n\nkey :\n Fintype.card \u2191(rootSet p \u2102) =\n Fintype.card \u2191(rootSet p \u211d) +\n Finset.card\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n\u22a2 \u2200 (m : \u2115), 2 \u2223 m \u2192 1 \u2264 m \u2192 m \u2264 3 \u2192 m = 2\n[PROOFSTEP]\nrintro m \u27e8k, rfl\u27e9 h2 h3\n[GOAL]\ncase intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\np_irr : Irreducible p\np_deg : Nat.Prime (natDegree p)\np_roots1 :\n 1 \u2264\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\np_roots2 :\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))) \u2264\n 3\nn : \u2115 :=\n Finset.card (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nhn : 2 \u2223 n\nkey :\n Fintype.card \u2191(rootSet p \u2102) =\n Fintype.card \u2191(rootSet p \u211d) +\n Finset.card\n (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nk : \u2115\nh2 : 1 \u2264 2 * k\nh3 : 2 * k \u2264 3\n\u22a2 2 * k = 2\n[PROOFSTEP]\nexact\n le_antisymm (Nat.lt_succ_iff.mp (lt_of_le_of_ne h3 (show 2 * k \u2260 2 * 1 + 1 from Nat.two_mul_ne_two_mul_add_one)))\n (Nat.succ_le_iff.mpr (lt_of_le_of_ne h2 (show 2 * 0 + 1 \u2260 2 * k from Nat.two_mul_ne_two_mul_add_one.symm)))\n", "meta": {"mathlib_filename": "Mathlib.FieldTheory.PolynomialGaloisGroup", "llama_tokens": 55471, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.4001852911778587}}